LMFDB - Embedded newform 75.3.f.b.7.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [75,3,Mod(7,75)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [4,0,0,0,0,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

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			7.1
Root			$-1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	75.7
Dual form			75.3.f.b.43.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

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

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	$-0.916667\pi$
$2$	−2.44949	−	2.44949i	−1.22474	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$3$	−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	0.992993	−	0.118175i	$-0.0377044\pi$
$7$	6.12372	+	6.12372i	0.874818	+	0.874818i	−0.118175	+	0.992993i	$0.537704\pi$
$8$	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	−0.551525	−	0.834158i	$-0.685954\pi$
$13$	3.67423	−	3.67423i	0.282633	−	0.282633i	0.834158	+	0.551525i	$0.185954\pi$
$14$		−	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	$0.00275368\pi$
$17$	17.1464	+	17.1464i	1.00861	+	1.00861i	0.00865084	+	0.999963i	$0.497246\pi$
$18$	−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	−0.921315	−	0.388817i	$-0.872884\pi$
$23$	−12.2474	+	12.2474i	−0.532498	+	0.532498i	0.388817	+	0.921315i	$0.372884\pi$
$24$			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	0.293833	−	0.955857i	$-0.405069\pi$
$37$	−24.4949	−	24.4949i	−0.662024	−	0.662024i	−0.955857	+	0.293833i	$0.905069\pi$
$38$	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	$-0.301646\pi$
$43$	60.0125	−	60.0125i	1.39564	−	1.39564i	0.812046	−	0.583594i	$-0.198354\pi$
$44$			48.0000i			1.09091i
$45$		0			0
$46$	60.0000			1.30435
$47$	7.34847	+	7.34847i	0.156350	+	0.156350i	0.780947	−	0.624597i	$-0.214737\pi$
$47$	7.34847	+	7.34847i	0.156350	+	0.156350i	−0.624597	+	0.780947i	$0.714737\pi$
$48$	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	−0.899365	−	0.437198i	$-0.855971\pi$
$53$	−24.4949	+	24.4949i	−0.462168	+	0.462168i	0.437198	+	0.899365i	$0.355971\pi$
$54$		−	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	0.488621	−	0.872496i	$-0.337500\pi$
$67$	−25.7196	−	25.7196i	−0.383875	−	0.383875i	−0.872496	+	0.488621i	$0.837500\pi$
$68$	−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	−0.854688	−	0.519142i	$-0.826252\pi$
$73$	−24.4949	+	24.4949i	−0.335547	+	0.335547i	0.519142	+	0.854688i	$0.326252\pi$
$74$			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	−0.692197	−	0.721709i	$-0.743357\pi$
$83$	2.44949	−	2.44949i	0.0295119	−	0.0295119i	0.721709	+	0.692197i	$0.243357\pi$
$84$		−	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	0.816808	−	0.576910i	$-0.195741\pi$
$97$	23.2702	+	23.2702i	0.239898	+	0.239898i	−0.576910	+	0.816808i	$0.695741\pi$
$98$	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	−0.795805	−	0.605553i	$-0.792952\pi$
$103$	−19.5959	+	19.5959i	−0.190252	+	0.190252i	0.605553	+	0.795805i	$0.292952\pi$
$104$		−	72.0000i		−	0.692308i
$105$		0			0
$106$	120.000			1.13208
$107$	−29.3939	−	29.3939i	−0.274709	−	0.274709i	0.556283	−	0.830993i	$-0.312227\pi$
$107$	−29.3939	−	29.3939i	−0.274709	−	0.274709i	−0.830993	+	0.556283i	$0.812227\pi$
$108$	−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	−0.917646	−	0.397400i	$-0.869913\pi$
$113$	−58.7878	+	58.7878i	−0.520246	+	0.520246i	0.397400	+	0.917646i	$0.369913\pi$
$114$			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	0.511883	−	0.859055i	$-0.328948\pi$
$127$	−44.0908	−	44.0908i	−0.347172	−	0.347172i	−0.859055	+	0.511883i	$0.828948\pi$
$128$	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	0.993128	−	0.117033i	$-0.0373384\pi$
$137$	120.025	+	120.025i	0.876095	+	0.876095i	−0.117033	+	0.993128i	$0.537338\pi$
$138$	−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	0.547906	−	0.836540i	$-0.315425\pi$
$157$	−45.3156	−	45.3156i	−0.288634	−	0.288634i	−0.836540	+	0.547906i	$0.815425\pi$
$158$	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	−0.333969	−	0.942584i	$-0.608388\pi$
$163$	99.2043	−	99.2043i	0.608616	−	0.608616i	0.942584	+	0.333969i	$0.108388\pi$
$164$		−	480.000i		−	2.92683i
$165$		0			0
$166$	−12.0000			−0.0722892
$167$	−97.9796	−	97.9796i	−0.586704	−	0.586704i	0.350033	−	0.936737i	$-0.386170\pi$
$167$	−97.9796	−	97.9796i	−0.586704	−	0.586704i	−0.936737	+	0.350033i	$0.886170\pi$
$168$	−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	$-0.498314\pi$
$173$	173.914	−	173.914i	1.00528	−	1.00528i	0.999986	−	0.00529594i	$-0.00168576\pi$
$174$			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	−0.854984	−	0.518655i	$-0.826433\pi$
$193$	−64.9115	+	64.9115i	−0.336329	+	0.336329i	0.518655	+	0.854984i	$0.326433\pi$
$194$		−	114.000i		−	0.587629i
$195$		0			0
$196$	−208.000			−1.06122
$197$	−193.510	−	193.510i	−0.982283	−	0.982283i	0.0175631	−	0.999846i	$-0.494409\pi$
$197$	−193.510	−	193.510i	−0.982283	−	0.982283i	−0.999846	+	0.0175631i	$0.994409\pi$
$198$	−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	−0.441443	−	0.897289i	$-0.645533\pi$
$223$	101.654	−	101.654i	0.455847	−	0.455847i	0.897289	+	0.441443i	$0.145533\pi$
$224$		0			0
$225$		0			0
$226$	288.000			1.27434
$227$	195.959	+	195.959i	0.863256	+	0.863256i	0.991715	−	0.128459i	$-0.0410029\pi$
$227$	195.959	+	195.959i	0.863256	+	0.863256i	−0.128459	+	0.991715i	$0.541003\pi$
$228$	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	−0.450990	−	0.892529i	$-0.648929\pi$
$233$	102.879	−	102.879i	0.441539	−	0.441539i	0.892529	+	0.450990i	$0.148929\pi$
$234$			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	0.927249	−	0.374446i	$-0.122167\pi$
$257$	142.070	+	142.070i	0.552803	+	0.552803i	−0.374446	+	0.927249i	$0.622167\pi$
$258$	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	−0.998808	−	0.0488156i	$-0.984455\pi$
$263$	−249.848	+	249.848i	−0.949992	+	0.949992i	0.0488156	+	0.998808i	$0.484455\pi$
$264$			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	0.316368	−	0.948637i	$-0.397537\pi$
$277$	−175.139	−	175.139i	−0.632269	−	0.632269i	−0.948637	+	0.316368i	$0.897537\pi$
$278$	−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	−0.959876	−	0.280424i	$-0.909525\pi$
$283$	−192.285	+	192.285i	−0.679452	+	0.679452i	0.280424	+	0.959876i	$0.409525\pi$
$284$			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	−0.0538645	−	0.998548i	$-0.517154\pi$
$293$	276.792	−	276.792i	0.944684	−	0.944684i	0.998548	+	0.0538645i	$0.0171539\pi$
$294$			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	0.834403	−	0.551155i	$-0.185813\pi$
$307$	86.9569	+	86.9569i	0.283247	+	0.283247i	−0.551155	+	0.834403i	$0.685813\pi$
$308$	−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	$-0.401984\pi$
$313$	393.143	−	393.143i	1.25605	−	1.25605i	0.952964	−	0.303085i	$-0.0980165\pi$
$314$			222.000i			0.707006i
$315$		0			0
$316$	−80.0000			−0.253165
$317$	93.0806	+	93.0806i	0.293630	+	0.293630i	0.838512	−	0.544883i	$-0.183426\pi$
$317$	93.0806	+	93.0806i	0.293630	+	0.293630i	−0.544883	+	0.838512i	$0.683426\pi$
$318$	−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	0.447347	−	0.894361i	$-0.352369\pi$
$337$	−150.644	−	150.644i	−0.447014	−	0.447014i	−0.894361	+	0.447347i	$0.852369\pi$
$338$	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	$-0.973381\pi$
$347$	−374.772	−	374.772i	−1.08003	−	1.08003i	−0.0835290	−	0.996505i	$-0.526619\pi$
$348$	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	−0.274593	−	0.961561i	$-0.588543\pi$
$353$	242.499	−	242.499i	0.686967	−	0.686967i	0.961561	+	0.274593i	$0.0885432\pi$
$354$		−	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	0.874848	−	0.484398i	$-0.160961\pi$
$367$	143.295	+	143.295i	0.390450	+	0.390450i	−0.484398	+	0.874848i	$0.660961\pi$
$368$	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	−0.657891	−	0.753113i	$-0.728551\pi$
$373$	35.5176	−	35.5176i	0.0952215	−	0.0952215i	0.753113	+	0.657891i	$0.228551\pi$
$374$		−	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	−0.681062	−	0.732226i	$-0.738481\pi$
$383$	19.5959	−	19.5959i	0.0511643	−	0.0511643i	0.732226	+	0.681062i	$0.238481\pi$
$384$			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	0.0747848	−	0.997200i	$-0.476173\pi$
$397$	−366.199	−	366.199i	−0.922415	−	0.922415i	−0.997200	+	0.0747848i	$0.976173\pi$
$398$	−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	−0.985736	−	0.168297i	$-0.946173\pi$
$433$	−353.951	+	353.951i	−0.817439	+	0.817439i	0.168297	+	0.985736i	$0.446173\pi$
$434$		−	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	−0.822751	−	0.568402i	$-0.807562\pi$
$443$	−112.677	+	112.677i	−0.254349	+	0.254349i	0.568402	+	0.822751i	$0.307562\pi$
$444$			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	0.999401	−	0.0346127i	$-0.0110198\pi$
$457$	440.908	+	440.908i	0.964788	+	0.964788i	−0.0346127	+	0.999401i	$0.511020\pi$
$458$	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	−0.998392	−	0.0566875i	$-0.981946\pi$
$463$	−436.009	+	436.009i	−0.941704	+	0.941704i	0.0566875	+	0.998392i	$0.481946\pi$
$464$		−	96.0000i		−	0.206897i
$465$		0			0
$466$	−504.000			−1.08155
$467$	276.792	+	276.792i	0.592703	+	0.592703i	0.938361	−	0.345658i	$-0.112344\pi$
$467$	276.792	+	276.792i	0.592703	+	0.592703i	−0.345658	+	0.938361i	$0.612344\pi$
$468$	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	$0.00109552\pi$
$487$	488.673	+	488.673i	1.00344	+	1.00344i	0.00344166	+	0.999994i	$0.498904\pi$
$488$	−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	−0.830189	−	0.557482i	$-0.811767\pi$
$503$	−137.171	+	137.171i	−0.272707	+	0.272707i	0.557482	+	0.830189i	$0.311767\pi$
$504$		−	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	−0.418697	−	0.908126i	$-0.637513\pi$
$523$	255.972	−	255.972i	0.489430	−	0.489430i	0.908126	+	0.418697i	$0.137513\pi$
$524$			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	0.0994353	−	0.995044i	$-0.468296\pi$
$547$	−489.898	−	489.898i	−0.895609	−	0.895609i	−0.995044	+	0.0994353i	$0.968296\pi$
$548$	−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	0.554879	−	0.831931i	$-0.312764\pi$
$557$	−154.318	−	154.318i	−0.277052	−	0.277052i	−0.831931	+	0.554879i	$0.812764\pi$
$558$	−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	$0.527776\pi$
$563$	−609.923	+	609.923i	−1.08334	+	1.08334i	−0.996195	+	0.0871492i	$0.972224\pi$
$564$		−	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	0.886029	−	0.463630i	$-0.153453\pi$
$577$	243.724	+	243.724i	0.422399	+	0.422399i	−0.463630	+	0.886029i	$0.653453\pi$
$578$	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	0.909062	−	0.416660i	$-0.136800\pi$
$587$	289.040	+	289.040i	0.492402	+	0.492402i	−0.416660	+	0.909062i	$0.636800\pi$
$588$	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	−0.256159	−	0.966635i	$-0.582457\pi$
$593$	421.312	−	421.312i	0.710476	−	0.710476i	0.966635	+	0.256159i	$0.0824570\pi$
$594$		−	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	0.711131	−	0.703060i	$-0.248183\pi$
$607$	4.89898	+	4.89898i	0.00807081	+	0.00807081i	−0.703060	+	0.711131i	$0.748183\pi$
$608$		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	−0.462864	−	0.886430i	$-0.653178\pi$
$613$	259.646	−	259.646i	0.423566	−	0.423566i	0.886430	+	0.462864i	$0.153178\pi$
$614$		−	426.000i		−	0.693811i
$615$		0			0
$616$	720.000			1.16883
$617$	9.79796	+	9.79796i	0.0158800	+	0.0158800i	0.715002	−	0.699122i	$-0.246426\pi$
$617$	9.79796	+	9.79796i	0.0158800	+	0.0158800i	−0.699122	+	0.715002i	$0.746426\pi$
$618$	−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	−0.839892	−	0.542753i	$-0.817382\pi$
$643$	−191.060	+	191.060i	−0.297139	+	0.297139i	0.542753	+	0.839892i	$0.317382\pi$
$644$		−	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	$-0.995310\pi$
$647$	−656.463	−	656.463i	−1.01463	−	1.01463i	−0.0147349	−	0.999891i	$-0.504690\pi$
$648$	−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	$0.552880\pi$
$653$	−751.993	+	751.993i	−1.15160	+	1.15160i	−0.986232	+	0.165365i	$0.947120\pi$
$654$		−	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	−0.610236	−	0.792219i	$-0.708926\pi$
$673$	122.474	−	122.474i	0.181983	−	0.181983i	0.792219	+	0.610236i	$0.208926\pi$
$674$			738.000i			1.09496i
$675$		0			0
$676$	−1136.00			−1.68047
$677$	−242.499	−	242.499i	−0.358197	−	0.358197i	0.504951	−	0.863148i	$-0.331511\pi$
$677$	−242.499	−	242.499i	−0.358197	−	0.358197i	−0.863148	+	0.504951i	$0.831511\pi$
$678$	−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	−0.692616	−	0.721307i	$-0.743542\pi$
$683$	19.5959	−	19.5959i	0.0286909	−	0.0286909i	0.721307	+	0.692616i	$0.243542\pi$
$684$			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	0.675253	−	0.737586i	$-0.264034\pi$
$727$	−45.3156	−	45.3156i	−0.0623323	−	0.0623323i	−0.737586	+	0.675253i	$0.764034\pi$
$728$	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	−0.955772	−	0.294109i	$-0.904977\pi$
$733$	−484.999	+	484.999i	−0.661663	+	0.661663i	0.294109	+	0.955772i	$0.404977\pi$
$734$		−	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	−0.518692	−	0.854961i	$-0.673581\pi$
$743$	249.848	−	249.848i	0.336269	−	0.336269i	0.854961	+	0.518692i	$0.173581\pi$
$744$			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	0.0177810	−	0.999842i	$-0.494340\pi$
$757$	−743.420	−	743.420i	−0.982061	−	0.982061i	−0.999842	+	0.0177810i	$0.994340\pi$
$758$	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	−0.958128	−	0.286340i	$-0.907561\pi$
$773$	−519.292	+	519.292i	−0.671788	+	0.671788i	0.286340	+	0.958128i	$0.407561\pi$
$774$			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	0.781315	−	0.624137i	$-0.214549\pi$
$787$	123.699	+	123.699i	0.157178	+	0.157178i	−0.624137	+	0.781315i	$0.714549\pi$
$788$	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	0.934296	−	0.356499i	$-0.116030\pi$
$797$	460.504	+	460.504i	0.577797	+	0.577797i	−0.356499	+	0.934296i	$0.616030\pi$
$798$	−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	−0.895656	−	0.444747i	$-0.853293\pi$
$823$	−371.098	+	371.098i	−0.450909	+	0.450909i	0.444747	+	0.895656i	$0.353293\pi$
$824$			384.000i			0.466019i
$825$		0			0
$826$	−540.000			−0.653753
$827$	788.736	+	788.736i	0.953731	+	0.953731i	0.998976	−	0.0452447i	$-0.0144068\pi$
$827$	788.736	+	788.736i	0.953731	+	0.953731i	−0.0452447	+	0.998976i	$0.514407\pi$
$828$	−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	−0.141391	−	0.989954i	$-0.545157\pi$
$853$	723.824	−	723.824i	0.848563	−	0.848563i	0.989954	+	0.141391i	$0.0451574\pi$
$854$			1110.00i			1.29977i
$855$		0			0
$856$	−576.000			−0.672897
$857$	−739.746	−	739.746i	−0.863181	−	0.863181i	0.128525	−	0.991706i	$-0.458976\pi$
$857$	−739.746	−	739.746i	−0.863181	−	0.863181i	−0.991706	+	0.128525i	$0.958976\pi$
$858$	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	−0.457686	−	0.889114i	$-0.651322\pi$
$863$	372.322	−	372.322i	0.431428	−	0.431428i	0.889114	+	0.457686i	$0.151322\pi$
$864$		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	0.857174	−	0.515027i	$-0.172218\pi$
$877$	300.062	+	300.062i	0.342147	+	0.342147i	−0.515027	+	0.857174i	$0.672218\pi$
$878$	−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	−0.999170	−	0.0407343i	$-0.987030\pi$
$883$	−846.299	+	846.299i	−0.958436	+	0.958436i	0.0407343	+	0.999170i	$0.487030\pi$
$884$			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	$-0.957601\pi$
$887$	−996.942	−	996.942i	−1.12395	−	1.12395i	−0.132807	−	0.991142i	$-0.542399\pi$
$888$	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	0.676775	−	0.736190i	$-0.263377\pi$
$907$	−53.8888	−	53.8888i	−0.0594143	−	0.0594143i	−0.736190	+	0.676775i	$0.763377\pi$
$908$	−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	0.883250	−	0.468902i	$-0.155350\pi$
$937$	388.244	+	388.244i	0.414348	+	0.414348i	−0.468902	+	0.883250i	$0.655350\pi$
$938$	−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	0.652091	−	0.758141i	$-0.273892\pi$
$947$	−100.429	−	100.429i	−0.106050	−	0.106050i	−0.758141	+	0.652091i	$0.773892\pi$
$948$	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	−0.980920	−	0.194410i	$-0.937721\pi$
$953$	−749.544	+	749.544i	−0.786510	+	0.786510i	0.194410	+	0.980920i	$0.437721\pi$
$954$		−	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	0.709635	−	0.704569i	$-0.248860\pi$
$967$	4.89898	+	4.89898i	0.00506616	+	0.00506616i	−0.704569	+	0.709635i	$0.748860\pi$
$968$	−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	0.535901	−	0.844281i	$-0.319972\pi$
$977$	−301.287	−	301.287i	−0.308380	−	0.308380i	−0.844281	+	0.535901i	$0.819972\pi$
$978$	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	−0.429578	−	0.903030i	$-0.641338\pi$
$983$	465.403	−	465.403i	0.473452	−	0.473452i	0.903030	+	0.429578i	$0.141338\pi$
$984$		−	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	0.833142	−	0.553060i	$-0.186540\pi$
$997$	279.242	+	279.242i	0.280082	+	0.280082i	−0.553060	+	0.833142i	$0.686540\pi$
$998$	−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.7.1	✓	4
3.2	odd	2		225.3.g.b.82.2		4
4.3	odd	2		1200.3.bg.g.1057.2		4
5.2	odd	4	inner	75.3.f.b.43.2	yes	4
5.3	odd	4	inner	75.3.f.b.43.1	yes	4
5.4	even	2	inner	75.3.f.b.7.2	yes	4
15.2	even	4		225.3.g.b.118.1		4
15.8	even	4		225.3.g.b.118.2		4
15.14	odd	2		225.3.g.b.82.1		4
20.3	even	4		1200.3.bg.g.193.2		4
20.7	even	4		1200.3.bg.g.193.1		4
20.19	odd	2		1200.3.bg.g.1057.1		4

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

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

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