LMFDB - Embedded newform 75.3.f.b.7.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,3,Mod(7,75)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("75.7");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	75.f (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$2.04360198270$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			7.2
Root			$1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	75.7
Dual form			75.3.f.b.43.2

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.44949 + 2.44949i) q^{2} +(1.22474 - 1.22474i) q^{3} +8.00000i q^{4} +6.00000 q^{6} +(-6.12372 - 6.12372i) q^{7} +(-9.79796 + 9.79796i) q^{8} -3.00000i q^{9} +O(q^{10})$	\(q+(2.44949 + 2.44949i) q^{2} +(1.22474 - 1.22474i) q^{3} +8.00000i q^{4} +6.00000 q^{6} +(-6.12372 - 6.12372i) q^{7} +(-9.79796 + 9.79796i) q^{8} -3.00000i q^{9} +6.00000 q^{11} +(9.79796 + 9.79796i) q^{12} +(-3.67423 + 3.67423i) q^{13} -30.0000i q^{14} -16.0000 q^{16} +(-17.1464 - 17.1464i) q^{17} +(7.34847 - 7.34847i) q^{18} +23.0000i q^{19} -15.0000 q^{21} +(14.6969 + 14.6969i) q^{22} +(12.2474 - 12.2474i) q^{23} +24.0000i q^{24} -18.0000 q^{26} +(-3.67423 - 3.67423i) q^{27} +(48.9898 - 48.9898i) q^{28} +6.00000i q^{29} +25.0000 q^{31} +(7.34847 - 7.34847i) q^{33} -84.0000i q^{34} +24.0000 q^{36} +(24.4949 + 24.4949i) q^{37} +(-56.3383 + 56.3383i) q^{38} +9.00000i q^{39} -60.0000 q^{41} +(-36.7423 - 36.7423i) q^{42} +(-60.0125 + 60.0125i) q^{43} +48.0000i q^{44} +60.0000 q^{46} +(-7.34847 - 7.34847i) q^{47} +(-19.5959 + 19.5959i) q^{48} +26.0000i q^{49} -42.0000 q^{51} +(-29.3939 - 29.3939i) q^{52} +(24.4949 - 24.4949i) q^{53} -18.0000i q^{54} +120.000 q^{56} +(28.1691 + 28.1691i) q^{57} +(-14.6969 + 14.6969i) q^{58} -18.0000i q^{59} -37.0000 q^{61} +(61.2372 + 61.2372i) q^{62} +(-18.3712 + 18.3712i) q^{63} +64.0000i q^{64} +36.0000 q^{66} +(25.7196 + 25.7196i) q^{67} +(137.171 - 137.171i) q^{68} -30.0000i q^{69} +132.000 q^{71} +(29.3939 + 29.3939i) q^{72} +(24.4949 - 24.4949i) q^{73} +120.000i q^{74} -184.000 q^{76} +(-36.7423 - 36.7423i) q^{77} +(-22.0454 + 22.0454i) q^{78} +10.0000i q^{79} -9.00000 q^{81} +(-146.969 - 146.969i) q^{82} +(-2.44949 + 2.44949i) q^{83} -120.000i q^{84} -294.000 q^{86} +(7.34847 + 7.34847i) q^{87} +(-58.7878 + 58.7878i) q^{88} -132.000i q^{89} +45.0000 q^{91} +(97.9796 + 97.9796i) q^{92} +(30.6186 - 30.6186i) q^{93} -36.0000i q^{94} +(-23.2702 - 23.2702i) q^{97} +(-63.6867 + 63.6867i) q^{98} -18.0000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 24 q^{6}+O(q^{10}) $ 4 * q + 24 * q^6	$ 4 q + 24 q^{6} + 24 q^{11} - 64 q^{16} - 60 q^{21} - 72 q^{26} + 100 q^{31} + 96 q^{36} - 240 q^{41} + 240 q^{46} - 168 q^{51} + 480 q^{56} - 148 q^{61} + 144 q^{66} + 528 q^{71} - 736 q^{76} - 36 q^{81} - 1176 q^{86} + 180 q^{91}+O(q^{100}) $ 4 * q + 24 * q^6 + 24 * q^11 - 64 * q^16 - 60 * q^21 - 72 * q^26 + 100 * q^31 + 96 * q^36 - 240 * q^41 + 240 * q^46 - 168 * q^51 + 480 * q^56 - 148 * q^61 + 144 * q^66 + 528 * q^71 - 736 * q^76 - 36 * q^81 - 1176 * q^86 + 180 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$.

$n$	$26$	$52$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	2.44949	+	2.44949i	1.22474	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$2$	2.44949	+	2.44949i	1.22474	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$3$	1.22474	−	1.22474i	0.408248	−	0.408248i
$3$	1.22474	−	1.22474i	0.408248	−	0.408248i
$4$			8.00000i			2.00000i
$5$		0			0
$5$		0			0
$6$	6.00000			1.00000
$7$	−6.12372	−	6.12372i	−0.874818	−	0.874818i	0.118175	−	0.992993i	$-0.462296\pi$
$7$	−6.12372	−	6.12372i	−0.874818	−	0.874818i	−0.992993	+	0.118175i	$0.962296\pi$
$8$	−9.79796	+	9.79796i	−1.22474	+	1.22474i
$9$		−	3.00000i		−	0.333333i
$10$		0			0
$11$	6.00000			0.545455			0.272727	−	0.962091i	$-0.412074\pi$
$11$	6.00000			0.545455			0.272727	+	0.962091i	$0.412074\pi$
$12$	9.79796	+	9.79796i	0.816497	+	0.816497i
$13$	−3.67423	+	3.67423i	−0.282633	+	0.282633i	−0.834158	−	0.551525i	$-0.814046\pi$
$13$	−3.67423	+	3.67423i	−0.282633	+	0.282633i	0.551525	+	0.834158i	$0.314046\pi$
$14$		−	30.0000i		−	2.14286i
$15$		0			0
$16$	−16.0000			−1.00000
$17$	−17.1464	−	17.1464i	−1.00861	−	1.00861i	−0.999963	−	0.00865084i	$-0.997246\pi$
$17$	−17.1464	−	17.1464i	−1.00861	−	1.00861i	−0.00865084	−	0.999963i	$-0.502754\pi$
$18$	7.34847	−	7.34847i	0.408248	−	0.408248i
$19$			23.0000i			1.21053i	0.796025	+	0.605263i	$0.206932\pi$
$19$			23.0000i			1.21053i	−0.796025	+	0.605263i	$0.793068\pi$
$20$		0			0
$21$	−15.0000			−0.714286
$22$	14.6969	+	14.6969i	0.668043	+	0.668043i
$23$	12.2474	−	12.2474i	0.532498	−	0.532498i	−0.388817	−	0.921315i	$-0.627116\pi$
$23$	12.2474	−	12.2474i	0.532498	−	0.532498i	0.921315	+	0.388817i	$0.127116\pi$
$24$			24.0000i			1.00000i
$25$		0			0
$26$	−18.0000			−0.692308
$27$	−3.67423	−	3.67423i	−0.136083	−	0.136083i
$28$	48.9898	−	48.9898i	1.74964	−	1.74964i
$29$			6.00000i			0.206897i	0.994635	+	0.103448i	$0.0329876\pi$
$29$			6.00000i			0.206897i	−0.994635	+	0.103448i	$0.967012\pi$
$30$		0			0
$31$	25.0000			0.806452			0.403226	−	0.915101i	$-0.367889\pi$
$31$	25.0000			0.806452			0.403226	+	0.915101i	$0.367889\pi$
$32$		0			0
$33$	7.34847	−	7.34847i	0.222681	−	0.222681i
$34$		−	84.0000i		−	2.47059i
$35$		0			0
$36$	24.0000			0.666667
$37$	24.4949	+	24.4949i	0.662024	+	0.662024i	0.955857	−	0.293833i	$-0.0949308\pi$
$37$	24.4949	+	24.4949i	0.662024	+	0.662024i	−0.293833	+	0.955857i	$0.594931\pi$
$38$	−56.3383	+	56.3383i	−1.48259	+	1.48259i
$39$			9.00000i			0.230769i
$40$		0			0
$41$	−60.0000			−1.46341			−0.731707	−	0.681619i	$-0.761276\pi$
$41$	−60.0000			−1.46341			−0.731707	+	0.681619i	$0.761276\pi$
$42$	−36.7423	−	36.7423i	−0.874818	−	0.874818i
$43$	−60.0125	+	60.0125i	−1.39564	+	1.39564i	−0.583594	+	0.812046i	$0.698354\pi$
$43$	−60.0125	+	60.0125i	−1.39564	+	1.39564i	−0.812046	+	0.583594i	$0.801646\pi$
$44$			48.0000i			1.09091i
$45$		0			0
$46$	60.0000			1.30435
$47$	−7.34847	−	7.34847i	−0.156350	−	0.156350i	0.624597	−	0.780947i	$-0.285263\pi$
$47$	−7.34847	−	7.34847i	−0.156350	−	0.156350i	−0.780947	+	0.624597i	$0.785263\pi$
$48$	−19.5959	+	19.5959i	−0.408248	+	0.408248i
$49$			26.0000i			0.530612i
$50$		0			0
$51$	−42.0000			−0.823529
$52$	−29.3939	−	29.3939i	−0.565267	−	0.565267i
$53$	24.4949	−	24.4949i	0.462168	−	0.462168i	−0.437198	−	0.899365i	$-0.644029\pi$
$53$	24.4949	−	24.4949i	0.462168	−	0.462168i	0.899365	+	0.437198i	$0.144029\pi$
$54$		−	18.0000i		−	0.333333i
$55$		0			0
$56$	120.000			2.14286
$57$	28.1691	+	28.1691i	0.494195	+	0.494195i
$58$	−14.6969	+	14.6969i	−0.253395	+	0.253395i
$59$		−	18.0000i		−	0.305085i	−0.988297	−	0.152542i	$-0.951254\pi$
$59$		−	18.0000i		−	0.305085i	0.988297	−	0.152542i	$-0.0487461\pi$
$60$		0			0
$61$	−37.0000			−0.606557			−0.303279	−	0.952902i	$-0.598081\pi$
$61$	−37.0000			−0.606557			−0.303279	+	0.952902i	$0.598081\pi$
$62$	61.2372	+	61.2372i	0.987697	+	0.987697i
$63$	−18.3712	+	18.3712i	−0.291606	+	0.291606i
$64$			64.0000i			1.00000i
$65$		0			0
$66$	36.0000			0.545455
$67$	25.7196	+	25.7196i	0.383875	+	0.383875i	0.872496	−	0.488621i	$-0.162500\pi$
$67$	25.7196	+	25.7196i	0.383875	+	0.383875i	−0.488621	+	0.872496i	$0.662500\pi$
$68$	137.171	−	137.171i	2.01723	−	2.01723i
$69$		−	30.0000i		−	0.434783i
$70$		0			0
$71$	132.000			1.85915			0.929577	−	0.368627i	$-0.120172\pi$
$71$	132.000			1.85915			0.929577	+	0.368627i	$0.120172\pi$
$72$	29.3939	+	29.3939i	0.408248	+	0.408248i
$73$	24.4949	−	24.4949i	0.335547	−	0.335547i	−0.519142	−	0.854688i	$-0.673748\pi$
$73$	24.4949	−	24.4949i	0.335547	−	0.335547i	0.854688	+	0.519142i	$0.173748\pi$
$74$			120.000i			1.62162i
$75$		0			0
$76$	−184.000			−2.42105
$77$	−36.7423	−	36.7423i	−0.477173	−	0.477173i
$78$	−22.0454	+	22.0454i	−0.282633	+	0.282633i
$79$			10.0000i			0.126582i	0.997995	+	0.0632911i	$0.0201597\pi$
$79$			10.0000i			0.126582i	−0.997995	+	0.0632911i	$0.979840\pi$
$80$		0			0
$81$	−9.00000			−0.111111
$82$	−146.969	−	146.969i	−1.79231	−	1.79231i
$83$	−2.44949	+	2.44949i	−0.0295119	+	0.0295119i	−0.721709	−	0.692197i	$-0.756643\pi$
$83$	−2.44949	+	2.44949i	−0.0295119	+	0.0295119i	0.692197	+	0.721709i	$0.256643\pi$
$84$		−	120.000i		−	1.42857i
$85$		0			0
$86$	−294.000			−3.41860
$87$	7.34847	+	7.34847i	0.0844652	+	0.0844652i
$88$	−58.7878	+	58.7878i	−0.668043	+	0.668043i
$89$		−	132.000i		−	1.48315i	−0.670872	−	0.741573i	$-0.734080\pi$
$89$		−	132.000i		−	1.48315i	0.670872	−	0.741573i	$-0.265920\pi$
$90$		0			0
$91$	45.0000			0.494505
$92$	97.9796	+	97.9796i	1.06500	+	1.06500i
$93$	30.6186	−	30.6186i	0.329232	−	0.329232i
$94$		−	36.0000i		−	0.382979i
$95$		0			0
$96$		0			0
$97$	−23.2702	−	23.2702i	−0.239898	−	0.239898i	0.576910	−	0.816808i	$-0.304259\pi$
$97$	−23.2702	−	23.2702i	−0.239898	−	0.239898i	−0.816808	+	0.576910i	$0.804259\pi$
$98$	−63.6867	+	63.6867i	−0.649865	+	0.649865i
$99$		−	18.0000i		−	0.181818i
$100$		0			0
$101$	96.0000			0.950495			0.475248	−	0.879852i	$-0.342359\pi$
$101$	96.0000			0.950495			0.475248	+	0.879852i	$0.342359\pi$
$102$	−102.879	−	102.879i	−1.00861	−	1.00861i
$103$	19.5959	−	19.5959i	0.190252	−	0.190252i	−0.605553	−	0.795805i	$-0.707048\pi$
$103$	19.5959	−	19.5959i	0.190252	−	0.190252i	0.795805	+	0.605553i	$0.207048\pi$
$104$		−	72.0000i		−	0.692308i
$105$		0			0
$106$	120.000			1.13208
$107$	29.3939	+	29.3939i	0.274709	+	0.274709i	0.830993	−	0.556283i	$-0.187773\pi$
$107$	29.3939	+	29.3939i	0.274709	+	0.274709i	−0.556283	+	0.830993i	$0.687773\pi$
$108$	29.3939	−	29.3939i	0.272166	−	0.272166i
$109$		−	167.000i		−	1.53211i	−0.642775	−	0.766055i	$-0.722217\pi$
$109$		−	167.000i		−	1.53211i	0.642775	−	0.766055i	$-0.277783\pi$
$110$		0			0
$111$	60.0000			0.540541
$112$	97.9796	+	97.9796i	0.874818	+	0.874818i
$113$	58.7878	−	58.7878i	0.520246	−	0.520246i	−0.397400	−	0.917646i	$-0.630087\pi$
$113$	58.7878	−	58.7878i	0.520246	−	0.520246i	0.917646	+	0.397400i	$0.130087\pi$
$114$			138.000i			1.21053i
$115$		0			0
$116$	−48.0000			−0.413793
$117$	11.0227	+	11.0227i	0.0942111	+	0.0942111i
$118$	44.0908	−	44.0908i	0.373651	−	0.373651i
$119$			210.000i			1.76471i
$120$		0			0
$121$	−85.0000			−0.702479
$122$	−90.6311	−	90.6311i	−0.742878	−	0.742878i
$123$	−73.4847	+	73.4847i	−0.597437	+	0.597437i
$124$			200.000i			1.61290i
$125$		0			0
$126$	−90.0000			−0.714286
$127$	44.0908	+	44.0908i	0.347172	+	0.347172i	0.859055	−	0.511883i	$-0.171052\pi$
$127$	44.0908	+	44.0908i	0.347172	+	0.347172i	−0.511883	+	0.859055i	$0.671052\pi$
$128$	−156.767	+	156.767i	−1.22474	+	1.22474i
$129$			147.000i			1.13953i
$130$		0			0
$131$	108.000			0.824427			0.412214	−	0.911087i	$-0.364756\pi$
$131$	108.000			0.824427			0.412214	+	0.911087i	$0.364756\pi$
$132$	58.7878	+	58.7878i	0.445362	+	0.445362i
$133$	140.846	−	140.846i	1.05899	−	1.05899i
$134$			126.000i			0.940299i
$135$		0			0
$136$	336.000			2.47059
$137$	−120.025	−	120.025i	−0.876095	−	0.876095i	0.117033	−	0.993128i	$-0.462662\pi$
$137$	−120.025	−	120.025i	−0.876095	−	0.876095i	−0.993128	+	0.117033i	$0.962662\pi$
$138$	73.4847	−	73.4847i	0.532498	−	0.532498i
$139$		−	58.0000i		−	0.417266i	−0.977994	−	0.208633i	$-0.933099\pi$
$139$		−	58.0000i		−	0.417266i	0.977994	−	0.208633i	$-0.0669014\pi$
$140$		0			0
$141$	−18.0000			−0.127660
$142$	323.333	+	323.333i	2.27699	+	2.27699i
$143$	−22.0454	+	22.0454i	−0.154164	+	0.154164i
$144$			48.0000i			0.333333i
$145$		0			0
$146$	120.000			0.821918
$147$	31.8434	+	31.8434i	0.216622	+	0.216622i
$148$	−195.959	+	195.959i	−1.32405	+	1.32405i
$149$		−	186.000i		−	1.24832i	−0.781296	−	0.624161i	$-0.785441\pi$
$149$		−	186.000i		−	1.24832i	0.781296	−	0.624161i	$-0.214559\pi$
$150$		0			0
$151$	−83.0000			−0.549669			−0.274834	−	0.961492i	$-0.588623\pi$
$151$	−83.0000			−0.549669			−0.274834	+	0.961492i	$0.588623\pi$
$152$	−225.353	−	225.353i	−1.48259	−	1.48259i
$153$	−51.4393	+	51.4393i	−0.336204	+	0.336204i
$154$		−	180.000i		−	1.16883i
$155$		0			0
$156$	−72.0000			−0.461538
$157$	45.3156	+	45.3156i	0.288634	+	0.288634i	0.836540	−	0.547906i	$-0.184575\pi$
$157$	45.3156	+	45.3156i	0.288634	+	0.288634i	−0.547906	+	0.836540i	$0.684575\pi$
$158$	−24.4949	+	24.4949i	−0.155031	+	0.155031i
$159$		−	60.0000i		−	0.377358i
$160$		0			0
$161$	−150.000			−0.931677
$162$	−22.0454	−	22.0454i	−0.136083	−	0.136083i
$163$	−99.2043	+	99.2043i	−0.608616	+	0.608616i	−0.942584	−	0.333969i	$-0.891612\pi$
$163$	−99.2043	+	99.2043i	−0.608616	+	0.608616i	0.333969	+	0.942584i	$0.391612\pi$
$164$		−	480.000i		−	2.92683i
$165$		0			0
$166$	−12.0000			−0.0722892
$167$	97.9796	+	97.9796i	0.586704	+	0.586704i	0.936737	−	0.350033i	$-0.113830\pi$
$167$	97.9796	+	97.9796i	0.586704	+	0.586704i	−0.350033	+	0.936737i	$0.613830\pi$
$168$	146.969	−	146.969i	0.874818	−	0.874818i
$169$			142.000i			0.840237i
$170$		0			0
$171$	69.0000			0.403509
$172$	−480.100	−	480.100i	−2.79128	−	2.79128i
$173$	−173.914	+	173.914i	−1.00528	+	1.00528i	−0.00529594	+	0.999986i	$0.501686\pi$
$173$	−173.914	+	173.914i	−1.00528	+	1.00528i	−0.999986	+	0.00529594i	$0.998314\pi$
$174$			36.0000i			0.206897i
$175$		0			0
$176$	−96.0000			−0.545455
$177$	−22.0454	−	22.0454i	−0.124550	−	0.124550i
$178$	323.333	−	323.333i	1.81648	−	1.81648i
$179$			150.000i			0.837989i	0.907989	+	0.418994i	$0.137617\pi$
$179$			150.000i			0.837989i	−0.907989	+	0.418994i	$0.862383\pi$
$180$		0			0
$181$	−215.000			−1.18785			−0.593923	−	0.804522i	$-0.702421\pi$
$181$	−215.000			−1.18785			−0.593923	+	0.804522i	$0.702421\pi$
$182$	110.227	+	110.227i	0.605643	+	0.605643i
$183$	−45.3156	+	45.3156i	−0.247626	+	0.247626i
$184$			240.000i			1.30435i
$185$		0			0
$186$	150.000			0.806452
$187$	−102.879	−	102.879i	−0.550153	−	0.550153i
$188$	58.7878	−	58.7878i	0.312701	−	0.312701i
$189$			45.0000i			0.238095i
$190$		0			0
$191$	−234.000			−1.22513			−0.612565	−	0.790420i	$-0.709862\pi$
$191$	−234.000			−1.22513			−0.612565	+	0.790420i	$0.709862\pi$
$192$	78.3837	+	78.3837i	0.408248	+	0.408248i
$193$	64.9115	−	64.9115i	0.336329	−	0.336329i	−0.518655	−	0.854984i	$-0.673567\pi$
$193$	64.9115	−	64.9115i	0.336329	−	0.336329i	0.854984	+	0.518655i	$0.173567\pi$
$194$		−	114.000i		−	0.587629i
$195$		0			0
$196$	−208.000			−1.06122
$197$	193.510	+	193.510i	0.982283	+	0.982283i	0.999846	−	0.0175631i	$-0.00559079\pi$
$197$	193.510	+	193.510i	0.982283	+	0.982283i	−0.0175631	+	0.999846i	$0.505591\pi$
$198$	44.0908	−	44.0908i	0.222681	−	0.222681i
$199$		−	11.0000i		−	0.0552764i	−0.999618	−	0.0276382i	$-0.991201\pi$
$199$		−	11.0000i		−	0.0552764i	0.999618	−	0.0276382i	$-0.00879863\pi$
$200$		0			0
$201$	63.0000			0.313433
$202$	235.151	+	235.151i	1.16411	+	1.16411i
$203$	36.7423	−	36.7423i	0.180997	−	0.180997i
$204$		−	336.000i		−	1.64706i
$205$		0			0
$206$	96.0000			0.466019
$207$	−36.7423	−	36.7423i	−0.177499	−	0.177499i
$208$	58.7878	−	58.7878i	0.282633	−	0.282633i
$209$			138.000i			0.660287i
$210$		0			0
$211$	−85.0000			−0.402844			−0.201422	−	0.979505i	$-0.564556\pi$
$211$	−85.0000			−0.402844			−0.201422	+	0.979505i	$0.564556\pi$
$212$	195.959	+	195.959i	0.924336	+	0.924336i
$213$	161.666	−	161.666i	0.758997	−	0.758997i
$214$			144.000i			0.672897i
$215$		0			0
$216$	72.0000			0.333333
$217$	−153.093	−	153.093i	−0.705498	−	0.705498i
$218$	409.065	−	409.065i	1.87644	−	1.87644i
$219$		−	60.0000i		−	0.273973i
$220$		0			0
$221$	126.000			0.570136
$222$	146.969	+	146.969i	0.662024	+	0.662024i
$223$	−101.654	+	101.654i	−0.455847	+	0.455847i	−0.897289	−	0.441443i	$-0.854467\pi$
$223$	−101.654	+	101.654i	−0.455847	+	0.455847i	0.441443	+	0.897289i	$0.354467\pi$
$224$		0			0
$225$		0			0
$226$	288.000			1.27434
$227$	−195.959	−	195.959i	−0.863256	−	0.863256i	0.128459	−	0.991715i	$-0.458997\pi$
$227$	−195.959	−	195.959i	−0.863256	−	0.863256i	−0.991715	+	0.128459i	$0.958997\pi$
$228$	−225.353	+	225.353i	−0.988391	+	0.988391i
$229$			227.000i			0.991266i	0.868532	+	0.495633i	$0.165064\pi$
$229$			227.000i			0.991266i	−0.868532	+	0.495633i	$0.834936\pi$
$230$		0			0
$231$	−90.0000			−0.389610
$232$	−58.7878	−	58.7878i	−0.253395	−	0.253395i
$233$	−102.879	+	102.879i	−0.441539	+	0.441539i	−0.892529	−	0.450990i	$-0.851071\pi$
$233$	−102.879	+	102.879i	−0.441539	+	0.441539i	0.450990	+	0.892529i	$0.351071\pi$
$234$			54.0000i			0.230769i
$235$		0			0
$236$	144.000			0.610169
$237$	12.2474	+	12.2474i	0.0516770	+	0.0516770i
$238$	−514.393	+	514.393i	−2.16131	+	2.16131i
$239$			228.000i			0.953975i	0.878910	+	0.476987i	$0.158271\pi$
$239$			228.000i			0.953975i	−0.878910	+	0.476987i	$0.841729\pi$
$240$		0			0
$241$	191.000			0.792531			0.396266	−	0.918136i	$-0.370306\pi$
$241$	191.000			0.792531			0.396266	+	0.918136i	$0.370306\pi$
$242$	−208.207	−	208.207i	−0.860358	−	0.860358i
$243$	−11.0227	+	11.0227i	−0.0453609	+	0.0453609i
$244$		−	296.000i		−	1.21311i
$245$		0			0
$246$	−360.000			−1.46341
$247$	−84.5074	−	84.5074i	−0.342135	−	0.342135i
$248$	−244.949	+	244.949i	−0.987697	+	0.987697i
$249$			6.00000i			0.0240964i
$250$		0			0
$251$	192.000			0.764940			0.382470	−	0.923968i	$-0.375073\pi$
$251$	192.000			0.764940			0.382470	+	0.923968i	$0.375073\pi$
$252$	−146.969	−	146.969i	−0.583212	−	0.583212i
$253$	73.4847	−	73.4847i	0.290453	−	0.290453i
$254$			216.000i			0.850394i
$255$		0			0
$256$	−512.000			−2.00000
$257$	−142.070	−	142.070i	−0.552803	−	0.552803i	0.374446	−	0.927249i	$-0.377833\pi$
$257$	−142.070	−	142.070i	−0.552803	−	0.552803i	−0.927249	+	0.374446i	$0.877833\pi$
$258$	−360.075	+	360.075i	−1.39564	+	1.39564i
$259$		−	300.000i		−	1.15830i
$260$		0			0
$261$	18.0000			0.0689655
$262$	264.545	+	264.545i	1.00971	+	1.00971i
$263$	249.848	−	249.848i	0.949992	−	0.949992i	−0.0488156	−	0.998808i	$-0.515545\pi$
$263$	249.848	−	249.848i	0.949992	−	0.949992i	0.998808	+	0.0488156i	$0.0155447\pi$
$264$			144.000i			0.545455i
$265$		0			0
$266$	690.000			2.59398
$267$	−161.666	−	161.666i	−0.605492	−	0.605492i
$268$	−205.757	+	205.757i	−0.767751	+	0.767751i
$269$		−	462.000i		−	1.71747i	−0.512418	−	0.858736i	$-0.671250\pi$
$269$		−	462.000i		−	1.71747i	0.512418	−	0.858736i	$-0.328750\pi$
$270$		0			0
$271$	446.000			1.64576			0.822878	−	0.568218i	$-0.192367\pi$
$271$	446.000			1.64576			0.822878	+	0.568218i	$0.192367\pi$
$272$	274.343	+	274.343i	1.00861	+	1.00861i
$273$	55.1135	−	55.1135i	0.201881	−	0.201881i
$274$		−	588.000i		−	2.14599i
$275$		0			0
$276$	240.000			0.869565
$277$	175.139	+	175.139i	0.632269	+	0.632269i	0.948637	−	0.316368i	$-0.102463\pi$
$277$	175.139	+	175.139i	0.632269	+	0.632269i	−0.316368	+	0.948637i	$0.602463\pi$
$278$	142.070	−	142.070i	0.511045	−	0.511045i
$279$		−	75.0000i		−	0.268817i
$280$		0			0
$281$	−546.000			−1.94306			−0.971530	−	0.236916i	$-0.923864\pi$
$281$	−546.000			−1.94306			−0.971530	+	0.236916i	$0.923864\pi$
$282$	−44.0908	−	44.0908i	−0.156350	−	0.156350i
$283$	192.285	−	192.285i	0.679452	−	0.679452i	−0.280424	−	0.959876i	$-0.590475\pi$
$283$	192.285	−	192.285i	0.679452	−	0.679452i	0.959876	+	0.280424i	$0.0904751\pi$
$284$			1056.00i			3.71831i
$285$		0			0
$286$	−108.000			−0.377622
$287$	367.423	+	367.423i	1.28022	+	1.28022i
$288$		0			0
$289$			299.000i			1.03460i
$290$		0			0
$291$	−57.0000			−0.195876
$292$	195.959	+	195.959i	0.671093	+	0.671093i
$293$	−276.792	+	276.792i	−0.944684	+	0.944684i	−0.998548	−	0.0538645i	$-0.982846\pi$
$293$	−276.792	+	276.792i	−0.944684	+	0.944684i	0.0538645	+	0.998548i	$0.482846\pi$
$294$			156.000i			0.530612i
$295$		0			0
$296$	−480.000			−1.62162
$297$	−22.0454	−	22.0454i	−0.0742270	−	0.0742270i
$298$	455.605	−	455.605i	1.52888	−	1.52888i
$299$			90.0000i			0.301003i
$300$		0			0
$301$	735.000			2.44186
$302$	−203.308	−	203.308i	−0.673204	−	0.673204i
$303$	117.576	−	117.576i	0.388038	−	0.388038i
$304$		−	368.000i		−	1.21053i
$305$		0			0
$306$	−252.000			−0.823529
$307$	−86.9569	−	86.9569i	−0.283247	−	0.283247i	0.551155	−	0.834403i	$-0.314187\pi$
$307$	−86.9569	−	86.9569i	−0.283247	−	0.283247i	−0.834403	+	0.551155i	$0.814187\pi$
$308$	293.939	−	293.939i	0.954347	−	0.954347i
$309$		−	48.0000i		−	0.155340i
$310$		0			0
$311$	−294.000			−0.945338			−0.472669	−	0.881240i	$-0.656709\pi$
$311$	−294.000			−0.945338			−0.472669	+	0.881240i	$0.656709\pi$
$312$	−88.1816	−	88.1816i	−0.282633	−	0.282633i
$313$	−393.143	+	393.143i	−1.25605	+	1.25605i	−0.303085	+	0.952964i	$0.598016\pi$
$313$	−393.143	+	393.143i	−1.25605	+	1.25605i	−0.952964	+	0.303085i	$0.901984\pi$
$314$			222.000i			0.707006i
$315$		0			0
$316$	−80.0000			−0.253165
$317$	−93.0806	−	93.0806i	−0.293630	−	0.293630i	0.544883	−	0.838512i	$-0.316574\pi$
$317$	−93.0806	−	93.0806i	−0.293630	−	0.293630i	−0.838512	+	0.544883i	$0.816574\pi$
$318$	146.969	−	146.969i	0.462168	−	0.462168i
$319$			36.0000i			0.112853i
$320$		0			0
$321$	72.0000			0.224299
$322$	−367.423	−	367.423i	−1.14107	−	1.14107i
$323$	394.368	−	394.368i	1.22095	−	1.22095i
$324$		−	72.0000i		−	0.222222i
$325$		0			0
$326$	−486.000			−1.49080
$327$	−204.532	−	204.532i	−0.625481	−	0.625481i
$328$	587.878	−	587.878i	1.79231	−	1.79231i
$329$			90.0000i			0.273556i
$330$		0			0
$331$	−178.000			−0.537764			−0.268882	−	0.963173i	$-0.586654\pi$
$331$	−178.000			−0.537764			−0.268882	+	0.963173i	$0.586654\pi$
$332$	−19.5959	−	19.5959i	−0.0590238	−	0.0590238i
$333$	73.4847	−	73.4847i	0.220675	−	0.220675i
$334$			480.000i			1.43713i
$335$		0			0
$336$	240.000			0.714286
$337$	150.644	+	150.644i	0.447014	+	0.447014i	0.894361	−	0.447347i	$-0.147631\pi$
$337$	150.644	+	150.644i	0.447014	+	0.447014i	−0.447347	+	0.894361i	$0.647631\pi$
$338$	−347.828	+	347.828i	−1.02908	+	1.02908i
$339$		−	144.000i		−	0.424779i
$340$		0			0
$341$	150.000			0.439883
$342$	169.015	+	169.015i	0.494195	+	0.494195i
$343$	−140.846	+	140.846i	−0.410629	+	0.410629i
$344$		−	1176.00i		−	3.41860i
$345$		0			0
$346$	−852.000			−2.46243
$347$	374.772	+	374.772i	1.08003	+	1.08003i	0.996505	+	0.0835290i	$0.0266191\pi$
$347$	374.772	+	374.772i	1.08003	+	1.08003i	0.0835290	+	0.996505i	$0.473381\pi$
$348$	−58.7878	+	58.7878i	−0.168930	+	0.168930i
$349$		−	514.000i		−	1.47278i	−0.676558	−	0.736390i	$-0.736529\pi$
$349$		−	514.000i		−	1.47278i	0.676558	−	0.736390i	$-0.263471\pi$
$350$		0			0
$351$	27.0000			0.0769231
$352$		0			0
$353$	−242.499	+	242.499i	−0.686967	+	0.686967i	−0.961561	−	0.274593i	$-0.911457\pi$
$353$	−242.499	+	242.499i	−0.686967	+	0.686967i	0.274593	+	0.961561i	$0.411457\pi$
$354$		−	108.000i		−	0.305085i
$355$		0			0
$356$	1056.00			2.96629
$357$	257.196	+	257.196i	0.720438	+	0.720438i
$358$	−367.423	+	367.423i	−1.02632	+	1.02632i
$359$		−	648.000i		−	1.80501i	−0.430675	−	0.902507i	$-0.641725\pi$
$359$		−	648.000i		−	1.80501i	0.430675	−	0.902507i	$-0.358275\pi$
$360$		0			0
$361$	−168.000			−0.465374
$362$	−526.640	−	526.640i	−1.45481	−	1.45481i
$363$	−104.103	+	104.103i	−0.286786	+	0.286786i
$364$			360.000i			0.989011i
$365$		0			0
$366$	−222.000			−0.606557
$367$	−143.295	−	143.295i	−0.390450	−	0.390450i	0.484398	−	0.874848i	$-0.339039\pi$
$367$	−143.295	−	143.295i	−0.390450	−	0.390450i	−0.874848	+	0.484398i	$0.839039\pi$
$368$	−195.959	+	195.959i	−0.532498	+	0.532498i
$369$			180.000i			0.487805i
$370$		0			0
$371$	−300.000			−0.808625
$372$	244.949	+	244.949i	0.658465	+	0.658465i
$373$	−35.5176	+	35.5176i	−0.0952215	+	0.0952215i	−0.753113	−	0.657891i	$-0.771449\pi$
$373$	−35.5176	+	35.5176i	−0.0952215	+	0.0952215i	0.657891	+	0.753113i	$0.271449\pi$
$374$		−	504.000i		−	1.34759i
$375$		0			0
$376$	144.000			0.382979
$377$	−22.0454	−	22.0454i	−0.0584759	−	0.0584759i
$378$	−110.227	+	110.227i	−0.291606	+	0.291606i
$379$			215.000i			0.567282i	0.958930	+	0.283641i	$0.0915425\pi$
$379$			215.000i			0.567282i	−0.958930	+	0.283641i	$0.908458\pi$
$380$		0			0
$381$	108.000			0.283465
$382$	−573.181	−	573.181i	−1.50047	−	1.50047i
$383$	−19.5959	+	19.5959i	−0.0511643	+	0.0511643i	−0.732226	−	0.681062i	$-0.761519\pi$
$383$	−19.5959	+	19.5959i	−0.0511643	+	0.0511643i	0.681062	+	0.732226i	$0.261519\pi$
$384$			384.000i			1.00000i
$385$		0			0
$386$	318.000			0.823834
$387$	180.037	+	180.037i	0.465213	+	0.465213i
$388$	186.161	−	186.161i	0.479797	−	0.479797i
$389$			192.000i			0.493573i	0.969070	+	0.246787i	$0.0793747\pi$
$389$			192.000i			0.493573i	−0.969070	+	0.246787i	$0.920625\pi$
$390$		0			0
$391$	−420.000			−1.07417
$392$	−254.747	−	254.747i	−0.649865	−	0.649865i
$393$	132.272	−	132.272i	0.336571	−	0.336571i
$394$			948.000i			2.40609i
$395$		0			0
$396$	144.000			0.363636
$397$	366.199	+	366.199i	0.922415	+	0.922415i	0.997200	−	0.0747848i	$-0.0238270\pi$
$397$	366.199	+	366.199i	0.922415	+	0.922415i	−0.0747848	+	0.997200i	$0.523827\pi$
$398$	26.9444	−	26.9444i	0.0676995	−	0.0676995i
$399$		−	345.000i		−	0.864662i
$400$		0			0
$401$	228.000			0.568579			0.284289	−	0.958739i	$-0.408242\pi$
$401$	228.000			0.568579			0.284289	+	0.958739i	$0.408242\pi$
$402$	154.318	+	154.318i	0.383875	+	0.383875i
$403$	−91.8559	+	91.8559i	−0.227930	+	0.227930i
$404$			768.000i			1.90099i
$405$		0			0
$406$	180.000			0.443350
$407$	146.969	+	146.969i	0.361104	+	0.361104i
$408$	411.514	−	411.514i	1.00861	−	1.00861i
$409$			707.000i			1.72861i	0.502971	+	0.864303i	$0.332240\pi$
$409$			707.000i			1.72861i	−0.502971	+	0.864303i	$0.667760\pi$
$410$		0			0
$411$	−294.000			−0.715328
$412$	156.767	+	156.767i	0.380503	+	0.380503i
$413$	−110.227	+	110.227i	−0.266894	+	0.266894i
$414$		−	180.000i		−	0.434783i
$415$		0			0
$416$		0			0
$417$	−71.0352	−	71.0352i	−0.170348	−	0.170348i
$418$	−338.030	+	338.030i	−0.808683	+	0.808683i
$419$			48.0000i			0.114558i	0.998358	+	0.0572792i	$0.0182425\pi$
$419$			48.0000i			0.114558i	−0.998358	+	0.0572792i	$0.981757\pi$
$420$		0			0
$421$	514.000			1.22090			0.610451	−	0.792054i	$-0.290988\pi$
$421$	514.000			1.22090			0.610451	+	0.792054i	$0.290988\pi$
$422$	−208.207	−	208.207i	−0.493381	−	0.493381i
$423$	−22.0454	+	22.0454i	−0.0521168	+	0.0521168i
$424$			480.000i			1.13208i
$425$		0			0
$426$	792.000			1.85915
$427$	226.578	+	226.578i	0.530627	+	0.530627i
$428$	−235.151	+	235.151i	−0.549418	+	0.549418i
$429$			54.0000i			0.125874i
$430$		0			0
$431$	30.0000			0.0696056			0.0348028	−	0.999394i	$-0.488920\pi$
$431$	30.0000			0.0696056			0.0348028	+	0.999394i	$0.488920\pi$
$432$	58.7878	+	58.7878i	0.136083	+	0.136083i
$433$	353.951	−	353.951i	0.817439	−	0.817439i	−0.168297	−	0.985736i	$-0.553827\pi$
$433$	353.951	−	353.951i	0.817439	−	0.817439i	0.985736	+	0.168297i	$0.0538267\pi$
$434$		−	750.000i		−	1.72811i
$435$		0			0
$436$	1336.00			3.06422
$437$	281.691	+	281.691i	0.644603	+	0.644603i
$438$	146.969	−	146.969i	0.335547	−	0.335547i
$439$		−	575.000i		−	1.30979i	−0.755718	−	0.654897i	$-0.772712\pi$
$439$		−	575.000i		−	1.30979i	0.755718	−	0.654897i	$-0.227288\pi$
$440$		0			0
$441$	78.0000			0.176871
$442$	308.636	+	308.636i	0.698271	+	0.698271i
$443$	112.677	−	112.677i	0.254349	−	0.254349i	−0.568402	−	0.822751i	$-0.692438\pi$
$443$	112.677	−	112.677i	0.254349	−	0.254349i	0.822751	+	0.568402i	$0.192438\pi$
$444$			480.000i			1.08108i
$445$		0			0
$446$	−498.000			−1.11659
$447$	−227.803	−	227.803i	−0.509625	−	0.509625i
$448$	391.918	−	391.918i	0.874818	−	0.874818i
$449$			204.000i			0.454343i	0.973855	+	0.227171i	$0.0729478\pi$
$449$			204.000i			0.454343i	−0.973855	+	0.227171i	$0.927052\pi$
$450$		0			0
$451$	−360.000			−0.798226
$452$	470.302	+	470.302i	1.04049	+	1.04049i
$453$	−101.654	+	101.654i	−0.224401	+	0.224401i
$454$		−	960.000i		−	2.11454i
$455$		0			0
$456$	−552.000			−1.21053
$457$	−440.908	−	440.908i	−0.964788	−	0.964788i	0.0346127	−	0.999401i	$-0.488980\pi$
$457$	−440.908	−	440.908i	−0.964788	−	0.964788i	−0.999401	+	0.0346127i	$0.988980\pi$
$458$	−556.034	+	556.034i	−1.21405	+	1.21405i
$459$			126.000i			0.274510i
$460$		0			0
$461$	132.000			0.286334			0.143167	−	0.989699i	$-0.454271\pi$
$461$	132.000			0.286334			0.143167	+	0.989699i	$0.454271\pi$
$462$	−220.454	−	220.454i	−0.477173	−	0.477173i
$463$	436.009	−	436.009i	0.941704	−	0.941704i	−0.0566875	−	0.998392i	$-0.518054\pi$
$463$	436.009	−	436.009i	0.941704	−	0.941704i	0.998392	+	0.0566875i	$0.0180539\pi$
$464$		−	96.0000i		−	0.206897i
$465$		0			0
$466$	−504.000			−1.08155
$467$	−276.792	−	276.792i	−0.592703	−	0.592703i	0.345658	−	0.938361i	$-0.387656\pi$
$467$	−276.792	−	276.792i	−0.592703	−	0.592703i	−0.938361	+	0.345658i	$0.887656\pi$
$468$	−88.1816	+	88.1816i	−0.188422	+	0.188422i
$469$		−	315.000i		−	0.671642i
$470$		0			0
$471$	111.000			0.235669
$472$	176.363	+	176.363i	0.373651	+	0.373651i
$473$	−360.075	+	360.075i	−0.761258	+	0.761258i
$474$			60.0000i			0.126582i
$475$		0			0
$476$	−1680.00			−3.52941
$477$	−73.4847	−	73.4847i	−0.154056	−	0.154056i
$478$	−558.484	+	558.484i	−1.16838	+	1.16838i
$479$			810.000i			1.69102i	0.533957	+	0.845511i	$0.320704\pi$
$479$			810.000i			1.69102i	−0.533957	+	0.845511i	$0.679296\pi$
$480$		0			0
$481$	−180.000			−0.374220
$482$	467.853	+	467.853i	0.970648	+	0.970648i
$483$	−183.712	+	183.712i	−0.380356	+	0.380356i
$484$		−	680.000i		−	1.40496i
$485$		0			0
$486$	−54.0000			−0.111111
$487$	−488.673	−	488.673i	−1.00344	−	1.00344i	−0.999994	−	0.00344166i	$-0.998904\pi$
$487$	−488.673	−	488.673i	−1.00344	−	1.00344i	−0.00344166	−	0.999994i	$-0.501096\pi$
$488$	362.524	−	362.524i	0.742878	−	0.742878i
$489$			243.000i			0.496933i
$490$		0			0
$491$	348.000			0.708758			0.354379	−	0.935102i	$-0.384692\pi$
$491$	348.000			0.708758			0.354379	+	0.935102i	$0.384692\pi$
$492$	−587.878	−	587.878i	−1.19487	−	1.19487i
$493$	102.879	−	102.879i	0.208679	−	0.208679i
$494$		−	414.000i		−	0.838057i
$495$		0			0
$496$	−400.000			−0.806452
$497$	−808.332	−	808.332i	−1.62642	−	1.62642i
$498$	−14.6969	+	14.6969i	−0.0295119	+	0.0295119i
$499$		−	227.000i		−	0.454910i	−0.973789	−	0.227455i	$-0.926960\pi$
$499$		−	227.000i		−	0.454910i	0.973789	−	0.227455i	$-0.0730404\pi$
$500$		0			0
$501$	240.000			0.479042
$502$	470.302	+	470.302i	0.936857	+	0.936857i
$503$	137.171	−	137.171i	0.272707	−	0.272707i	−0.557482	−	0.830189i	$-0.688233\pi$
$503$	137.171	−	137.171i	0.272707	−	0.272707i	0.830189	+	0.557482i	$0.188233\pi$
$504$		−	360.000i		−	0.714286i
$505$		0			0
$506$	360.000			0.711462
$507$	173.914	+	173.914i	0.343025	+	0.343025i
$508$	−352.727	+	352.727i	−0.694344	+	0.694344i
$509$		−	618.000i		−	1.21415i	−0.794646	−	0.607073i	$-0.792344\pi$
$509$		−	618.000i		−	1.21415i	0.794646	−	0.607073i	$-0.207656\pi$
$510$		0			0
$511$	−300.000			−0.587084
$512$	−627.069	−	627.069i	−1.22474	−	1.22474i
$513$	84.5074	−	84.5074i	0.164732	−	0.164732i
$514$		−	696.000i		−	1.35409i
$515$		0			0
$516$	−1176.00			−2.27907
$517$	−44.0908	−	44.0908i	−0.0852820	−	0.0852820i
$518$	734.847	−	734.847i	1.41862	−	1.41862i
$519$			426.000i			0.820809i
$520$		0			0
$521$	690.000			1.32438			0.662188	−	0.749338i	$-0.269628\pi$
$521$	690.000			1.32438			0.662188	+	0.749338i	$0.269628\pi$
$522$	44.0908	+	44.0908i	0.0844652	+	0.0844652i
$523$	−255.972	+	255.972i	−0.489430	+	0.489430i	−0.908126	−	0.418697i	$-0.862487\pi$
$523$	−255.972	+	255.972i	−0.489430	+	0.489430i	0.418697	+	0.908126i	$0.362487\pi$
$524$			864.000i			1.64885i
$525$		0			0
$526$	1224.00			2.32700
$527$	−428.661	−	428.661i	−0.813398	−	0.813398i
$528$	−117.576	+	117.576i	−0.222681	+	0.222681i
$529$			229.000i			0.432892i
$530$		0			0
$531$	−54.0000			−0.101695
$532$	1126.77	+	1126.77i	2.11798	+	2.11798i
$533$	220.454	−	220.454i	0.413610	−	0.413610i
$534$		−	792.000i		−	1.48315i
$535$		0			0
$536$	−504.000			−0.940299
$537$	183.712	+	183.712i	0.342108	+	0.342108i
$538$	1131.66	−	1131.66i	2.10347	−	2.10347i
$539$			156.000i			0.289425i
$540$		0			0
$541$	−325.000			−0.600739			−0.300370	−	0.953823i	$-0.597110\pi$
$541$	−325.000			−0.600739			−0.300370	+	0.953823i	$0.597110\pi$
$542$	1092.47	+	1092.47i	2.01563	+	2.01563i
$543$	−263.320	+	263.320i	−0.484936	+	0.484936i
$544$		0			0
$545$		0			0
$546$	270.000			0.494505
$547$	489.898	+	489.898i	0.895609	+	0.895609i	0.995044	−	0.0994353i	$-0.0317036\pi$
$547$	489.898	+	489.898i	0.895609	+	0.895609i	−0.0994353	+	0.995044i	$0.531704\pi$
$548$	960.200	−	960.200i	1.75219	−	1.75219i
$549$			111.000i			0.202186i
$550$		0			0
$551$	−138.000			−0.250454
$552$	293.939	+	293.939i	0.532498	+	0.532498i
$553$	61.2372	−	61.2372i	0.110736	−	0.110736i
$554$			858.000i			1.54874i
$555$		0			0
$556$	464.000			0.834532
$557$	154.318	+	154.318i	0.277052	+	0.277052i	0.831931	−	0.554879i	$-0.187236\pi$
$557$	154.318	+	154.318i	0.277052	+	0.277052i	−0.554879	+	0.831931i	$0.687236\pi$
$558$	183.712	−	183.712i	0.329232	−	0.329232i
$559$		−	441.000i		−	0.788909i
$560$		0			0
$561$	−252.000			−0.449198
$562$	−1337.42	−	1337.42i	−2.37975	−	2.37975i
$563$	609.923	−	609.923i	1.08334	−	1.08334i	0.0871492	−	0.996195i	$-0.472224\pi$
$563$	609.923	−	609.923i	1.08334	−	1.08334i	0.996195	−	0.0871492i	$-0.0277757\pi$
$564$		−	144.000i		−	0.255319i
$565$		0			0
$566$	942.000			1.66431
$567$	55.1135	+	55.1135i	0.0972020	+	0.0972020i
$568$	−1293.33	+	1293.33i	−2.27699	+	2.27699i
$569$			198.000i			0.347979i	0.984748	+	0.173989i	$0.0556659\pi$
$569$			198.000i			0.347979i	−0.984748	+	0.173989i	$0.944334\pi$
$570$		0			0
$571$	−169.000			−0.295972			−0.147986	−	0.988989i	$-0.547279\pi$
$571$	−169.000			−0.295972			−0.147986	+	0.988989i	$0.547279\pi$
$572$	−176.363	−	176.363i	−0.308327	−	0.308327i
$573$	−286.590	+	286.590i	−0.500158	+	0.500158i
$574$			1800.00i			3.13589i
$575$		0			0
$576$	192.000			0.333333
$577$	−243.724	−	243.724i	−0.422399	−	0.422399i	0.463630	−	0.886029i	$-0.346547\pi$
$577$	−243.724	−	243.724i	−0.422399	−	0.422399i	−0.886029	+	0.463630i	$0.846547\pi$
$578$	−732.397	+	732.397i	−1.26712	+	1.26712i
$579$		−	159.000i		−	0.274611i
$580$		0			0
$581$	30.0000			0.0516351
$582$	−139.621	−	139.621i	−0.239898	−	0.239898i
$583$	146.969	−	146.969i	0.252092	−	0.252092i
$584$			480.000i			0.821918i
$585$		0			0
$586$	−1356.00			−2.31399
$587$	−289.040	−	289.040i	−0.492402	−	0.492402i	0.416660	−	0.909062i	$-0.363200\pi$
$587$	−289.040	−	289.040i	−0.492402	−	0.492402i	−0.909062	+	0.416660i	$0.863200\pi$
$588$	−254.747	+	254.747i	−0.433243	+	0.433243i
$589$			575.000i			0.976231i
$590$		0			0
$591$	474.000			0.802030
$592$	−391.918	−	391.918i	−0.662024	−	0.662024i
$593$	−421.312	+	421.312i	−0.710476	+	0.710476i	−0.966635	−	0.256159i	$-0.917543\pi$
$593$	−421.312	+	421.312i	−0.710476	+	0.710476i	0.256159	+	0.966635i	$0.417543\pi$
$594$		−	108.000i		−	0.181818i
$595$		0			0
$596$	1488.00			2.49664
$597$	−13.4722	−	13.4722i	−0.0225665	−	0.0225665i
$598$	−220.454	+	220.454i	−0.368652	+	0.368652i
$599$			144.000i			0.240401i	0.992750	+	0.120200i	$0.0383537\pi$
$599$			144.000i			0.240401i	−0.992750	+	0.120200i	$0.961646\pi$
$600$		0			0
$601$	−899.000			−1.49584			−0.747920	−	0.663789i	$-0.768947\pi$
$601$	−899.000			−1.49584			−0.747920	+	0.663789i	$0.768947\pi$
$602$	1800.37	+	1800.37i	2.99066	+	2.99066i
$603$	77.1589	−	77.1589i	0.127958	−	0.127958i
$604$		−	664.000i		−	1.09934i
$605$		0			0
$606$	576.000			0.950495
$607$	−4.89898	−	4.89898i	−0.00807081	−	0.00807081i	0.703060	−	0.711131i	$-0.251817\pi$
$607$	−4.89898	−	4.89898i	−0.00807081	−	0.00807081i	−0.711131	+	0.703060i	$0.751817\pi$
$608$		0			0
$609$		−	90.0000i		−	0.147783i
$610$		0			0
$611$	54.0000			0.0883797
$612$	−411.514	−	411.514i	−0.672409	−	0.672409i
$613$	−259.646	+	259.646i	−0.423566	+	0.423566i	−0.886430	−	0.462864i	$-0.846822\pi$
$613$	−259.646	+	259.646i	−0.423566	+	0.423566i	0.462864	+	0.886430i	$0.346822\pi$
$614$		−	426.000i		−	0.693811i
$615$		0			0
$616$	720.000			1.16883
$617$	−9.79796	−	9.79796i	−0.0158800	−	0.0158800i	0.699122	−	0.715002i	$-0.253574\pi$
$617$	−9.79796	−	9.79796i	−0.0158800	−	0.0158800i	−0.715002	+	0.699122i	$0.753574\pi$
$618$	117.576	−	117.576i	0.190252	−	0.190252i
$619$			637.000i			1.02908i	0.857467	+	0.514540i	$0.172037\pi$
$619$			637.000i			1.02908i	−0.857467	+	0.514540i	$0.827963\pi$
$620$		0			0
$621$	−90.0000			−0.144928
$622$	−720.150	−	720.150i	−1.15780	−	1.15780i
$623$	−808.332	+	808.332i	−1.29748	+	1.29748i
$624$		−	144.000i		−	0.230769i
$625$		0			0
$626$	−1926.00			−3.07668
$627$	169.015	+	169.015i	0.269561	+	0.269561i
$628$	−362.524	+	362.524i	−0.577268	+	0.577268i
$629$		−	840.000i		−	1.33545i
$630$		0			0
$631$	287.000			0.454834			0.227417	−	0.973798i	$-0.426972\pi$
$631$	287.000			0.454834			0.227417	+	0.973798i	$0.426972\pi$
$632$	−97.9796	−	97.9796i	−0.155031	−	0.155031i
$633$	−104.103	+	104.103i	−0.164460	+	0.164460i
$634$		−	456.000i		−	0.719243i
$635$		0			0
$636$	480.000			0.754717
$637$	−95.5301	−	95.5301i	−0.149969	−	0.149969i
$638$	−88.1816	+	88.1816i	−0.138216	+	0.138216i
$639$		−	396.000i		−	0.619718i
$640$		0			0
$641$	−60.0000			−0.0936037			−0.0468019	−	0.998904i	$-0.514903\pi$
$641$	−60.0000			−0.0936037			−0.0468019	+	0.998904i	$0.514903\pi$
$642$	176.363	+	176.363i	0.274709	+	0.274709i
$643$	191.060	−	191.060i	0.297139	−	0.297139i	−0.542753	−	0.839892i	$-0.682618\pi$
$643$	191.060	−	191.060i	0.297139	−	0.297139i	0.839892	+	0.542753i	$0.182618\pi$
$644$		−	1200.00i		−	1.86335i
$645$		0			0
$646$	1932.00			2.99071
$647$	656.463	+	656.463i	1.01463	+	1.01463i	0.999891	+	0.0147349i	$0.00469044\pi$
$647$	656.463	+	656.463i	1.01463	+	1.01463i	0.0147349	+	0.999891i	$0.495310\pi$
$648$	88.1816	−	88.1816i	0.136083	−	0.136083i
$649$		−	108.000i		−	0.166410i
$650$		0			0
$651$	−375.000			−0.576037
$652$	−793.635	−	793.635i	−1.21723	−	1.21723i
$653$	751.993	−	751.993i	1.15160	−	1.15160i	0.165365	−	0.986232i	$-0.447120\pi$
$653$	751.993	−	751.993i	1.15160	−	1.15160i	0.986232	−	0.165365i	$-0.0528803\pi$
$654$		−	1002.00i		−	1.53211i
$655$		0			0
$656$	960.000			1.46341
$657$	−73.4847	−	73.4847i	−0.111849	−	0.111849i
$658$	−220.454	+	220.454i	−0.335037	+	0.335037i
$659$			420.000i			0.637329i	0.947868	+	0.318665i	$0.103234\pi$
$659$			420.000i			0.637329i	−0.947868	+	0.318665i	$0.896766\pi$
$660$		0			0
$661$	−158.000			−0.239032			−0.119516	−	0.992832i	$-0.538134\pi$
$661$	−158.000			−0.239032			−0.119516	+	0.992832i	$0.538134\pi$
$662$	−436.009	−	436.009i	−0.658624	−	0.658624i
$663$	154.318	−	154.318i	0.232757	−	0.232757i
$664$		−	48.0000i		−	0.0722892i
$665$		0			0
$666$	360.000			0.540541
$667$	73.4847	+	73.4847i	0.110172	+	0.110172i
$668$	−783.837	+	783.837i	−1.17341	+	1.17341i
$669$			249.000i			0.372197i
$670$		0			0
$671$	−222.000			−0.330849
$672$		0			0
$673$	−122.474	+	122.474i	−0.181983	+	0.181983i	−0.792219	−	0.610236i	$-0.791074\pi$
$673$	−122.474	+	122.474i	−0.181983	+	0.181983i	0.610236	+	0.792219i	$0.291074\pi$
$674$			738.000i			1.09496i
$675$		0			0
$676$	−1136.00			−1.68047
$677$	242.499	+	242.499i	0.358197	+	0.358197i	0.863148	−	0.504951i	$-0.168489\pi$
$677$	242.499	+	242.499i	0.358197	+	0.358197i	−0.504951	+	0.863148i	$0.668489\pi$
$678$	352.727	−	352.727i	0.520246	−	0.520246i
$679$			285.000i			0.419735i
$680$		0			0
$681$	−480.000			−0.704846
$682$	367.423	+	367.423i	0.538744	+	0.538744i
$683$	−19.5959	+	19.5959i	−0.0286909	+	0.0286909i	−0.721307	−	0.692616i	$-0.756458\pi$
$683$	−19.5959	+	19.5959i	−0.0286909	+	0.0286909i	0.692616	+	0.721307i	$0.256458\pi$
$684$			552.000i			0.807018i
$685$		0			0
$686$	−690.000			−1.00583
$687$	278.017	+	278.017i	0.404683	+	0.404683i
$688$	960.200	−	960.200i	1.39564	−	1.39564i
$689$			180.000i			0.261248i
$690$		0			0
$691$	−74.0000			−0.107091			−0.0535456	−	0.998565i	$-0.517052\pi$
$691$	−74.0000			−0.107091			−0.0535456	+	0.998565i	$0.517052\pi$
$692$	−1391.31	−	1391.31i	−2.01056	−	2.01056i
$693$	−110.227	+	110.227i	−0.159058	+	0.159058i
$694$			1836.00i			2.64553i
$695$		0			0
$696$	−144.000			−0.206897
$697$	1028.79	+	1028.79i	1.47602	+	1.47602i
$698$	1259.04	−	1259.04i	1.80378	−	1.80378i
$699$			252.000i			0.360515i
$700$		0			0
$701$	−102.000			−0.145506			−0.0727532	−	0.997350i	$-0.523179\pi$
$701$	−102.000			−0.145506			−0.0727532	+	0.997350i	$0.523179\pi$
$702$	66.1362	+	66.1362i	0.0942111	+	0.0942111i
$703$	−563.383	+	563.383i	−0.801398	+	0.801398i
$704$			384.000i			0.545455i
$705$		0			0
$706$	−1188.00			−1.68272
$707$	−587.878	−	587.878i	−0.831510	−	0.831510i
$708$	176.363	−	176.363i	0.249101	−	0.249101i
$709$		−	817.000i		−	1.15233i	−0.817334	−	0.576164i	$-0.804549\pi$
$709$		−	817.000i		−	1.15233i	0.817334	−	0.576164i	$-0.195451\pi$
$710$		0			0
$711$	30.0000			0.0421941
$712$	1293.33	+	1293.33i	1.81648	+	1.81648i
$713$	306.186	−	306.186i	0.429434	−	0.429434i
$714$			1260.00i			1.76471i
$715$		0			0
$716$	−1200.00			−1.67598
$717$	279.242	+	279.242i	0.389459	+	0.389459i
$718$	1587.27	−	1587.27i	2.21068	−	2.21068i
$719$		−	6.00000i		−	0.00834492i	−0.999991	−	0.00417246i	$-0.998672\pi$
$719$		−	6.00000i		−	0.00834492i	0.999991	−	0.00417246i	$-0.00132814\pi$
$720$		0			0
$721$	−240.000			−0.332871
$722$	−411.514	−	411.514i	−0.569964	−	0.569964i
$723$	233.926	−	233.926i	0.323549	−	0.323549i
$724$		−	1720.00i		−	2.37569i
$725$		0			0
$726$	−510.000			−0.702479
$727$	45.3156	+	45.3156i	0.0623323	+	0.0623323i	0.737586	−	0.675253i	$-0.235966\pi$
$727$	45.3156	+	45.3156i	0.0623323	+	0.0623323i	−0.675253	+	0.737586i	$0.735966\pi$
$728$	−440.908	+	440.908i	−0.605643	+	0.605643i
$729$			27.0000i			0.0370370i
$730$		0			0
$731$	2058.00			2.81532
$732$	−362.524	−	362.524i	−0.495252	−	0.495252i
$733$	484.999	−	484.999i	0.661663	−	0.661663i	−0.294109	−	0.955772i	$-0.595023\pi$
$733$	484.999	−	484.999i	0.661663	−	0.661663i	0.955772	+	0.294109i	$0.0950228\pi$
$734$		−	702.000i		−	0.956403i
$735$		0			0
$736$		0			0
$737$	154.318	+	154.318i	0.209387	+	0.209387i
$738$	−440.908	+	440.908i	−0.597437	+	0.597437i
$739$		−	298.000i		−	0.403248i	−0.979463	−	0.201624i	$-0.935378\pi$
$739$		−	298.000i		−	0.403248i	0.979463	−	0.201624i	$-0.0646218\pi$
$740$		0			0
$741$	−207.000			−0.279352
$742$	−734.847	−	734.847i	−0.990360	−	0.990360i
$743$	−249.848	+	249.848i	−0.336269	+	0.336269i	−0.854961	−	0.518692i	$-0.826419\pi$
$743$	−249.848	+	249.848i	−0.336269	+	0.336269i	0.518692	+	0.854961i	$0.326419\pi$
$744$			600.000i			0.806452i
$745$		0			0
$746$	−174.000			−0.233244
$747$	7.34847	+	7.34847i	0.00983731	+	0.00983731i
$748$	823.029	−	823.029i	1.10031	−	1.10031i
$749$		−	360.000i		−	0.480641i
$750$		0			0
$751$	1186.00			1.57923			0.789614	−	0.613604i	$-0.210281\pi$
$751$	1186.00			1.57923			0.789614	+	0.613604i	$0.210281\pi$
$752$	117.576	+	117.576i	0.156350	+	0.156350i
$753$	235.151	−	235.151i	0.312286	−	0.312286i
$754$		−	108.000i		−	0.143236i
$755$		0			0
$756$	−360.000			−0.476190
$757$	743.420	+	743.420i	0.982061	+	0.982061i	0.999842	−	0.0177810i	$-0.00566015\pi$
$757$	743.420	+	743.420i	0.982061	+	0.982061i	−0.0177810	+	0.999842i	$0.505660\pi$
$758$	−526.640	+	526.640i	−0.694776	+	0.694776i
$759$		−	180.000i		−	0.237154i
$760$		0			0
$761$	−576.000			−0.756899			−0.378449	−	0.925622i	$-0.623543\pi$
$761$	−576.000			−0.756899			−0.378449	+	0.925622i	$0.623543\pi$
$762$	264.545	+	264.545i	0.347172	+	0.347172i
$763$	−1022.66	+	1022.66i	−1.34032	+	1.34032i
$764$		−	1872.00i		−	2.45026i
$765$		0			0
$766$	−96.0000			−0.125326
$767$	66.1362	+	66.1362i	0.0862271	+	0.0862271i
$768$	−627.069	+	627.069i	−0.816497	+	0.816497i
$769$		−	491.000i		−	0.638492i	−0.947672	−	0.319246i	$-0.896570\pi$
$769$		−	491.000i		−	0.638492i	0.947672	−	0.319246i	$-0.103430\pi$
$770$		0			0
$771$	−348.000			−0.451362
$772$	519.292	+	519.292i	0.672658	+	0.672658i
$773$	519.292	−	519.292i	0.671788	−	0.671788i	−0.286340	−	0.958128i	$-0.592439\pi$
$773$	519.292	−	519.292i	0.671788	−	0.671788i	0.958128	+	0.286340i	$0.0924388\pi$
$774$			882.000i			1.13953i
$775$		0			0
$776$	456.000			0.587629
$777$	−367.423	−	367.423i	−0.472874	−	0.472874i
$778$	−470.302	+	470.302i	−0.604501	+	0.604501i
$779$		−	1380.00i		−	1.77150i
$780$		0			0
$781$	792.000			1.01408
$782$	−1028.79	−	1028.79i	−1.31558	−	1.31558i
$783$	22.0454	−	22.0454i	0.0281551	−	0.0281551i
$784$		−	416.000i		−	0.530612i
$785$		0			0
$786$	648.000			0.824427
$787$	−123.699	−	123.699i	−0.157178	−	0.157178i	0.624137	−	0.781315i	$-0.285451\pi$
$787$	−123.699	−	123.699i	−0.157178	−	0.157178i	−0.781315	+	0.624137i	$0.785451\pi$
$788$	−1548.08	+	1548.08i	−1.96457	+	1.96457i
$789$		−	612.000i		−	0.775665i
$790$		0			0
$791$	−720.000			−0.910240
$792$	176.363	+	176.363i	0.222681	+	0.222681i
$793$	135.947	−	135.947i	0.171433	−	0.171433i
$794$			1794.00i			2.25945i
$795$		0			0
$796$	88.0000			0.110553
$797$	−460.504	−	460.504i	−0.577797	−	0.577797i	0.356499	−	0.934296i	$-0.383970\pi$
$797$	−460.504	−	460.504i	−0.577797	−	0.577797i	−0.934296	+	0.356499i	$0.883970\pi$
$798$	845.074	−	845.074i	1.05899	−	1.05899i
$799$			252.000i			0.315394i
$800$		0			0
$801$	−396.000			−0.494382
$802$	558.484	+	558.484i	0.696364	+	0.696364i
$803$	146.969	−	146.969i	0.183025	−	0.183025i
$804$			504.000i			0.626866i
$805$		0			0
$806$	−450.000			−0.558313
$807$	−565.832	−	565.832i	−0.701155	−	0.701155i
$808$	−940.604	+	940.604i	−1.16411	+	1.16411i
$809$			900.000i			1.11248i	0.831020	+	0.556242i	$0.187757\pi$
$809$			900.000i			1.11248i	−0.831020	+	0.556242i	$0.812243\pi$
$810$		0			0
$811$	1477.00			1.82121			0.910604	−	0.413280i	$-0.135617\pi$
$811$	1477.00			1.82121			0.910604	+	0.413280i	$0.135617\pi$
$812$	293.939	+	293.939i	0.361994	+	0.361994i
$813$	546.236	−	546.236i	0.671877	−	0.671877i
$814$			720.000i			0.884521i
$815$		0			0
$816$	672.000			0.823529
$817$	−1380.29	−	1380.29i	−1.68946	−	1.68946i
$818$	−1731.79	+	1731.79i	−2.11710	+	2.11710i
$819$		−	135.000i		−	0.164835i
$820$		0			0
$821$	−786.000			−0.957369			−0.478685	−	0.877987i	$-0.658886\pi$
$821$	−786.000			−0.957369			−0.478685	+	0.877987i	$0.658886\pi$
$822$	−720.150	−	720.150i	−0.876095	−	0.876095i
$823$	371.098	−	371.098i	0.450909	−	0.450909i	−0.444747	−	0.895656i	$-0.646707\pi$
$823$	371.098	−	371.098i	0.450909	−	0.450909i	0.895656	+	0.444747i	$0.146707\pi$
$824$			384.000i			0.466019i
$825$		0			0
$826$	−540.000			−0.653753
$827$	−788.736	−	788.736i	−0.953731	−	0.953731i	0.0452447	−	0.998976i	$-0.485593\pi$
$827$	−788.736	−	788.736i	−0.953731	−	0.953731i	−0.998976	+	0.0452447i	$0.985593\pi$
$828$	293.939	−	293.939i	0.354999	−	0.354999i
$829$			170.000i			0.205066i	0.994730	+	0.102533i	$0.0326948\pi$
$829$			170.000i			0.205066i	−0.994730	+	0.102533i	$0.967305\pi$
$830$		0			0
$831$	429.000			0.516245
$832$	−235.151	−	235.151i	−0.282633	−	0.282633i
$833$	445.807	−	445.807i	0.535183	−	0.535183i
$834$		−	348.000i		−	0.417266i
$835$		0			0
$836$	−1104.00			−1.32057
$837$	−91.8559	−	91.8559i	−0.109744	−	0.109744i
$838$	−117.576	+	117.576i	−0.140305	+	0.140305i
$839$		−	990.000i		−	1.17998i	−0.807412	−	0.589988i	$-0.799132\pi$
$839$		−	990.000i		−	1.17998i	0.807412	−	0.589988i	$-0.200868\pi$
$840$		0			0
$841$	805.000			0.957194
$842$	1259.04	+	1259.04i	1.49529	+	1.49529i
$843$	−668.711	+	668.711i	−0.793251	+	0.793251i
$844$		−	680.000i		−	0.805687i
$845$		0			0
$846$	−108.000			−0.127660
$847$	520.517	+	520.517i	0.614541	+	0.614541i
$848$	−391.918	+	391.918i	−0.462168	+	0.462168i
$849$		−	471.000i		−	0.554770i
$850$		0			0
$851$	600.000			0.705053
$852$	1293.33	+	1293.33i	1.51799	+	1.51799i
$853$	−723.824	+	723.824i	−0.848563	+	0.848563i	−0.989954	−	0.141391i	$-0.954843\pi$
$853$	−723.824	+	723.824i	−0.848563	+	0.848563i	0.141391	+	0.989954i	$0.454843\pi$
$854$			1110.00i			1.29977i
$855$		0			0
$856$	−576.000			−0.672897
$857$	739.746	+	739.746i	0.863181	+	0.863181i	0.991706	−	0.128525i	$-0.0410244\pi$
$857$	739.746	+	739.746i	0.863181	+	0.863181i	−0.128525	+	0.991706i	$0.541024\pi$
$858$	−132.272	+	132.272i	−0.154164	+	0.154164i
$859$			1510.00i			1.75786i	0.476953	+	0.878929i	$0.341741\pi$
$859$			1510.00i			1.75786i	−0.476953	+	0.878929i	$0.658259\pi$
$860$		0			0
$861$	900.000			1.04530
$862$	73.4847	+	73.4847i	0.0852491	+	0.0852491i
$863$	−372.322	+	372.322i	−0.431428	+	0.431428i	−0.889114	−	0.457686i	$-0.848678\pi$
$863$	−372.322	+	372.322i	−0.431428	+	0.431428i	0.457686	+	0.889114i	$0.348678\pi$
$864$		0			0
$865$		0			0
$866$	1734.00			2.00231
$867$	366.199	+	366.199i	0.422375	+	0.422375i
$868$	1224.74	−	1224.74i	1.41100	−	1.41100i
$869$			60.0000i			0.0690449i
$870$		0			0
$871$	−189.000			−0.216992
$872$	1636.26	+	1636.26i	1.87644	+	1.87644i
$873$	−69.8105	+	69.8105i	−0.0799662	+	0.0799662i
$874$			1380.00i			1.57895i
$875$		0			0
$876$	480.000			0.547945
$877$	−300.062	−	300.062i	−0.342147	−	0.342147i	0.515027	−	0.857174i	$-0.327782\pi$
$877$	−300.062	−	300.062i	−0.342147	−	0.342147i	−0.857174	+	0.515027i	$0.827782\pi$
$878$	1408.46	−	1408.46i	1.60416	−	1.60416i
$879$			678.000i			0.771331i
$880$		0			0
$881$	−216.000			−0.245176			−0.122588	−	0.992458i	$-0.539119\pi$
$881$	−216.000			−0.245176			−0.122588	+	0.992458i	$0.539119\pi$
$882$	191.060	+	191.060i	0.216622	+	0.216622i
$883$	846.299	−	846.299i	0.958436	−	0.958436i	−0.0407343	−	0.999170i	$-0.512970\pi$
$883$	846.299	−	846.299i	0.958436	−	0.958436i	0.999170	+	0.0407343i	$0.0129697\pi$
$884$			1008.00i			1.14027i
$885$		0			0
$886$	552.000			0.623025
$887$	996.942	+	996.942i	1.12395	+	1.12395i	0.991142	+	0.132807i	$0.0423989\pi$
$887$	996.942	+	996.942i	1.12395	+	1.12395i	0.132807	+	0.991142i	$0.457601\pi$
$888$	−587.878	+	587.878i	−0.662024	+	0.662024i
$889$		−	540.000i		−	0.607424i
$890$		0			0
$891$	−54.0000			−0.0606061
$892$	−813.231	−	813.231i	−0.911693	−	0.911693i
$893$	169.015	−	169.015i	0.189266	−	0.189266i
$894$		−	1116.00i		−	1.24832i
$895$		0			0
$896$	1920.00			2.14286
$897$	110.227	+	110.227i	0.122884	+	0.122884i
$898$	−499.696	+	499.696i	−0.556454	+	0.556454i
$899$			150.000i			0.166852i
$900$		0			0
$901$	−840.000			−0.932297
$902$	−881.816	−	881.816i	−0.977623	−	0.977623i
$903$	900.187	−	900.187i	0.996885	−	0.996885i
$904$			1152.00i			1.27434i
$905$		0			0
$906$	−498.000			−0.549669
$907$	53.8888	+	53.8888i	0.0594143	+	0.0594143i	0.736190	−	0.676775i	$-0.236623\pi$
$907$	53.8888	+	53.8888i	0.0594143	+	0.0594143i	−0.676775	+	0.736190i	$0.736623\pi$
$908$	1567.67	−	1567.67i	1.72651	−	1.72651i
$909$		−	288.000i		−	0.316832i
$910$		0			0
$911$	−558.000			−0.612514			−0.306257	−	0.951949i	$-0.599077\pi$
$911$	−558.000			−0.612514			−0.306257	+	0.951949i	$0.599077\pi$
$912$	−450.706	−	450.706i	−0.494195	−	0.494195i
$913$	−14.6969	+	14.6969i	−0.0160974	+	0.0160974i
$914$		−	2160.00i		−	2.36324i
$915$		0			0
$916$	−1816.00			−1.98253
$917$	−661.362	−	661.362i	−0.721224	−	0.721224i
$918$	−308.636	+	308.636i	−0.336204	+	0.336204i
$919$			755.000i			0.821545i	0.911738	+	0.410773i	$0.134741\pi$
$919$			755.000i			0.821545i	−0.911738	+	0.410773i	$0.865259\pi$
$920$		0			0
$921$	−213.000			−0.231270
$922$	323.333	+	323.333i	0.350686	+	0.350686i
$923$	−484.999	+	484.999i	−0.525459	+	0.525459i
$924$		−	720.000i		−	0.779221i
$925$		0			0
$926$	2136.00			2.30670
$927$	−58.7878	−	58.7878i	−0.0634172	−	0.0634172i
$928$		0			0
$929$		−	762.000i		−	0.820237i	−0.912032	−	0.410118i	$-0.865487\pi$
$929$		−	762.000i		−	0.820237i	0.912032	−	0.410118i	$-0.134513\pi$
$930$		0			0
$931$	−598.000			−0.642320
$932$	−823.029	−	823.029i	−0.883078	−	0.883078i
$933$	−360.075	+	360.075i	−0.385932	+	0.385932i
$934$		−	1356.00i		−	1.45182i
$935$		0			0
$936$	−216.000			−0.230769
$937$	−388.244	−	388.244i	−0.414348	−	0.414348i	0.468902	−	0.883250i	$-0.344650\pi$
$937$	−388.244	−	388.244i	−0.414348	−	0.414348i	−0.883250	+	0.468902i	$0.844650\pi$
$938$	771.589	−	771.589i	0.822590	−	0.822590i
$939$			963.000i			1.02556i
$940$		0			0
$941$	−690.000			−0.733262			−0.366631	−	0.930366i	$-0.619489\pi$
$941$	−690.000			−0.733262			−0.366631	+	0.930366i	$0.619489\pi$
$942$	271.893	+	271.893i	0.288634	+	0.288634i
$943$	−734.847	+	734.847i	−0.779265	+	0.779265i
$944$			288.000i			0.305085i
$945$		0			0
$946$	−1764.00			−1.86469
$947$	100.429	+	100.429i	0.106050	+	0.106050i	0.758141	−	0.652091i	$-0.226108\pi$
$947$	100.429	+	100.429i	0.106050	+	0.106050i	−0.652091	+	0.758141i	$0.726108\pi$
$948$	−97.9796	+	97.9796i	−0.103354	+	0.103354i
$949$			180.000i			0.189673i
$950$		0			0
$951$	−228.000			−0.239748
$952$	−2057.57	−	2057.57i	−2.16131	−	2.16131i
$953$	749.544	−	749.544i	0.786510	−	0.786510i	−0.194410	−	0.980920i	$-0.562279\pi$
$953$	749.544	−	749.544i	0.786510	−	0.786510i	0.980920	+	0.194410i	$0.0622794\pi$
$954$		−	360.000i		−	0.377358i
$955$		0			0
$956$	−1824.00			−1.90795
$957$	44.0908	+	44.0908i	0.0460719	+	0.0460719i
$958$	−1984.09	+	1984.09i	−2.07107	+	2.07107i
$959$			1470.00i			1.53285i
$960$		0			0
$961$	−336.000			−0.349636
$962$	−440.908	−	440.908i	−0.458324	−	0.458324i
$963$	88.1816	−	88.1816i	0.0915697	−	0.0915697i
$964$			1528.00i			1.58506i
$965$		0			0
$966$	−900.000			−0.931677
$967$	−4.89898	−	4.89898i	−0.00506616	−	0.00506616i	0.704569	−	0.709635i	$-0.251140\pi$
$967$	−4.89898	−	4.89898i	−0.00506616	−	0.00506616i	−0.709635	+	0.704569i	$0.751140\pi$
$968$	832.827	−	832.827i	0.860358	−	0.860358i
$969$		−	966.000i		−	0.996904i
$970$		0			0
$971$	90.0000			0.0926880			0.0463440	−	0.998926i	$-0.485243\pi$
$971$	90.0000			0.0926880			0.0463440	+	0.998926i	$0.485243\pi$
$972$	−88.1816	−	88.1816i	−0.0907218	−	0.0907218i
$973$	−355.176	+	355.176i	−0.365032	+	0.365032i
$974$		−	2394.00i		−	2.45791i
$975$		0			0
$976$	592.000			0.606557
$977$	301.287	+	301.287i	0.308380	+	0.308380i	0.844281	−	0.535901i	$-0.180028\pi$
$977$	301.287	+	301.287i	0.308380	+	0.308380i	−0.535901	+	0.844281i	$0.680028\pi$
$978$	−595.226	+	595.226i	−0.608616	+	0.608616i
$979$		−	792.000i		−	0.808989i
$980$		0			0
$981$	−501.000			−0.510703
$982$	852.422	+	852.422i	0.868047	+	0.868047i
$983$	−465.403	+	465.403i	−0.473452	+	0.473452i	−0.903030	−	0.429578i	$-0.858662\pi$
$983$	−465.403	+	465.403i	−0.473452	+	0.473452i	0.429578	+	0.903030i	$0.358662\pi$
$984$		−	1440.00i		−	1.46341i
$985$		0			0
$986$	504.000			0.511156
$987$	110.227	+	110.227i	0.111679	+	0.111679i
$988$	676.059	−	676.059i	0.684270	−	0.684270i
$989$			1470.00i			1.48635i
$990$		0			0
$991$	1067.00			1.07669			0.538345	−	0.842724i	$-0.319050\pi$
$991$	1067.00			1.07669			0.538345	+	0.842724i	$0.319050\pi$
$992$		0			0
$993$	−218.005	+	218.005i	−0.219541	+	0.219541i
$994$		−	3960.00i		−	3.98390i
$995$		0			0
$996$	−48.0000			−0.0481928
$997$	−279.242	−	279.242i	−0.280082	−	0.280082i	0.553060	−	0.833142i	$-0.313460\pi$
$997$	−279.242	−	279.242i	−0.280082	−	0.280082i	−0.833142	+	0.553060i	$0.813460\pi$
$998$	556.034	−	556.034i	0.557148	−	0.557148i
$999$		−	180.000i		−	0.180180i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.3.f.b.7.2	yes	4
3.2	odd	2		225.3.g.b.82.1		4
4.3	odd	2		1200.3.bg.g.1057.1		4
5.2	odd	4	inner	75.3.f.b.43.1	yes	4
5.3	odd	4	inner	75.3.f.b.43.2	yes	4
5.4	even	2	inner	75.3.f.b.7.1	✓	4
15.2	even	4		225.3.g.b.118.2		4
15.8	even	4		225.3.g.b.118.1		4
15.14	odd	2		225.3.g.b.82.2		4
20.3	even	4		1200.3.bg.g.193.1		4
20.7	even	4		1200.3.bg.g.193.2		4
20.19	odd	2		1200.3.bg.g.1057.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.3.f.b.7.1	✓	4	5.4	even	2	inner
75.3.f.b.7.2	yes	4	1.1	even	1	trivial
75.3.f.b.43.1	yes	4	5.2	odd	4	inner
75.3.f.b.43.2	yes	4	5.3	odd	4	inner
225.3.g.b.82.1		4	3.2	odd	2
225.3.g.b.82.2		4	15.14	odd	2
225.3.g.b.118.1		4	15.8	even	4
225.3.g.b.118.2		4	15.2	even	4
1200.3.bg.g.193.1		4	20.3	even	4
1200.3.bg.g.193.2		4	20.7	even	4
1200.3.bg.g.1057.1		4	4.3	odd	2
1200.3.bg.g.1057.2		4	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	75.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(2.44949 + 2.44949i) q^{2} +(1.22474 - 1.22474i) q^{3} +8.00000i q^{4} +6.00000 q^{6} +(-6.12372 - 6.12372i) q^{7} +(-9.79796 + 9.79796i) q^{8} -3.00000i q^{9} +O(q^{10})\)	\(q+(2.44949 + 2.44949i) q^{2} +(1.22474 - 1.22474i) q^{3} +8.00000i q^{4} +6.00000 q^{6} +(-6.12372 - 6.12372i) q^{7} +(-9.79796 + 9.79796i) q^{8} -3.00000i q^{9} +6.00000 q^{11} +(9.79796 + 9.79796i) q^{12} +(-3.67423 + 3.67423i) q^{13} -30.0000i q^{14} -16.0000 q^{16} +(-17.1464 - 17.1464i) q^{17} +(7.34847 - 7.34847i) q^{18} +23.0000i q^{19} -15.0000 q^{21} +(14.6969 + 14.6969i) q^{22} +(12.2474 - 12.2474i) q^{23} +24.0000i q^{24} -18.0000 q^{26} +(-3.67423 - 3.67423i) q^{27} +(48.9898 - 48.9898i) q^{28} +6.00000i q^{29} +25.0000 q^{31} +(7.34847 - 7.34847i) q^{33} -84.0000i q^{34} +24.0000 q^{36} +(24.4949 + 24.4949i) q^{37} +(-56.3383 + 56.3383i) q^{38} +9.00000i q^{39} -60.0000 q^{41} +(-36.7423 - 36.7423i) q^{42} +(-60.0125 + 60.0125i) q^{43} +48.0000i q^{44} +60.0000 q^{46} +(-7.34847 - 7.34847i) q^{47} +(-19.5959 + 19.5959i) q^{48} +26.0000i q^{49} -42.0000 q^{51} +(-29.3939 - 29.3939i) q^{52} +(24.4949 - 24.4949i) q^{53} -18.0000i q^{54} +120.000 q^{56} +(28.1691 + 28.1691i) q^{57} +(-14.6969 + 14.6969i) q^{58} -18.0000i q^{59} -37.0000 q^{61} +(61.2372 + 61.2372i) q^{62} +(-18.3712 + 18.3712i) q^{63} +64.0000i q^{64} +36.0000 q^{66} +(25.7196 + 25.7196i) q^{67} +(137.171 - 137.171i) q^{68} -30.0000i q^{69} +132.000 q^{71} +(29.3939 + 29.3939i) q^{72} +(24.4949 - 24.4949i) q^{73} +120.000i q^{74} -184.000 q^{76} +(-36.7423 - 36.7423i) q^{77} +(-22.0454 + 22.0454i) q^{78} +10.0000i q^{79} -9.00000 q^{81} +(-146.969 - 146.969i) q^{82} +(-2.44949 + 2.44949i) q^{83} -120.000i q^{84} -294.000 q^{86} +(7.34847 + 7.34847i) q^{87} +(-58.7878 + 58.7878i) q^{88} -132.000i q^{89} +45.0000 q^{91} +(97.9796 + 97.9796i) q^{92} +(30.6186 - 30.6186i) q^{93} -36.0000i q^{94} +(-23.2702 - 23.2702i) q^{97} +(-63.6867 + 63.6867i) q^{98} -18.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 24 q^{6}+O(q^{10}) \) 4 * q + 24 * q^6	\( 4 q + 24 q^{6} + 24 q^{11} - 64 q^{16} - 60 q^{21} - 72 q^{26} + 100 q^{31} + 96 q^{36} - 240 q^{41} + 240 q^{46} - 168 q^{51} + 480 q^{56} - 148 q^{61} + 144 q^{66} + 528 q^{71} - 736 q^{76} - 36 q^{81} - 1176 q^{86} + 180 q^{91}+O(q^{100}) \) 4 * q + 24 * q^6 + 24 * q^11 - 64 * q^16 - 60 * q^21 - 72 * q^26 + 100 * q^31 + 96 * q^36 - 240 * q^41 + 240 * q^46 - 168 * q^51 + 480 * q^56 - 148 * q^61 + 144 * q^66 + 528 * q^71 - 736 * q^76 - 36 * q^81 - 1176 * q^86 + 180 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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