LMFDB - Embedded newform 1200.3.bg.g.193.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,3,Mod(193,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 0, 3]))

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

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

chi := DirichletCharacter("1200.193");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1200.bg (of order $4$, degree $2$, not 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:	$32.6976317232$
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:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 75)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			193.1
Root			$1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1200.193
Dual form			1200.3.bg.g.1057.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.22474 - 1.22474i) q^{3} +(6.12372 - 6.12372i) q^{7} +3.00000i q^{9} +O(q^{10})$	$q+(-1.22474 - 1.22474i) q^{3} +(6.12372 - 6.12372i) q^{7} +3.00000i q^{9} -6.00000 q^{11} +(-3.67423 - 3.67423i) q^{13} +(-17.1464 + 17.1464i) q^{17} +23.0000i q^{19} -15.0000 q^{21} +(-12.2474 - 12.2474i) q^{23} +(3.67423 - 3.67423i) q^{27} -6.00000i q^{29} -25.0000 q^{31} +(7.34847 + 7.34847i) q^{33} +(24.4949 - 24.4949i) q^{37} +9.00000i q^{39} -60.0000 q^{41} +(60.0125 + 60.0125i) q^{43} +(7.34847 - 7.34847i) q^{47} -26.0000i q^{49} +42.0000 q^{51} +(24.4949 + 24.4949i) q^{53} +(28.1691 - 28.1691i) q^{57} -18.0000i q^{59} -37.0000 q^{61} +(18.3712 + 18.3712i) q^{63} +(-25.7196 + 25.7196i) q^{67} +30.0000i q^{69} -132.000 q^{71} +(24.4949 + 24.4949i) q^{73} +(-36.7423 + 36.7423i) q^{77} +10.0000i q^{79} -9.00000 q^{81} +(2.44949 + 2.44949i) q^{83} +(-7.34847 + 7.34847i) q^{87} +132.000i q^{89} -45.0000 q^{91} +(30.6186 + 30.6186i) q^{93} +(-23.2702 + 23.2702i) q^{97} -18.0000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 24 q^{11} - 60 q^{21} - 100 q^{31} - 240 q^{41} + 168 q^{51} - 148 q^{61} - 528 q^{71} - 36 q^{81} - 180 q^{91}+O(q^{100}) $ 4 * q - 24 * q^11 - 60 * q^21 - 100 * q^31 - 240 * q^41 + 168 * q^51 - 148 * q^61 - 528 * q^71 - 36 * q^81 - 180 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$.

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	−1.22474	−	1.22474i	−0.408248	−	0.408248i
$3$	−1.22474	−	1.22474i	−0.408248	−	0.408248i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	6.12372	−	6.12372i	0.874818	−	0.874818i	−0.118175	−	0.992993i	$-0.537704\pi$
$7$	6.12372	−	6.12372i	0.874818	−	0.874818i	0.992993	+	0.118175i	$0.0377044\pi$
$8$		0			0
$9$			3.00000i			0.333333i
$10$		0			0
$11$	−6.00000			−0.545455			−0.272727	−	0.962091i	$-0.587926\pi$
$11$	−6.00000			−0.545455			−0.272727	+	0.962091i	$0.587926\pi$
$12$		0			0
$13$	−3.67423	−	3.67423i	−0.282633	−	0.282633i	0.551525	−	0.834158i	$-0.314046\pi$
$13$	−3.67423	−	3.67423i	−0.282633	−	0.282633i	−0.834158	+	0.551525i	$0.814046\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−17.1464	+	17.1464i	−1.00861	+	1.00861i	−0.00865084	+	0.999963i	$0.502754\pi$
$17$	−17.1464	+	17.1464i	−1.00861	+	1.00861i	−0.999963	+	0.00865084i	$0.997246\pi$
$18$		0			0
$19$			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
$22$		0			0
$23$	−12.2474	−	12.2474i	−0.532498	−	0.532498i	0.388817	−	0.921315i	$-0.372884\pi$
$23$	−12.2474	−	12.2474i	−0.532498	−	0.532498i	−0.921315	+	0.388817i	$0.872884\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	3.67423	−	3.67423i	0.136083	−	0.136083i
$28$		0			0
$29$		−	6.00000i		−	0.206897i	−0.994635	−	0.103448i	$-0.967012\pi$
$29$		−	6.00000i		−	0.206897i	0.994635	−	0.103448i	$-0.0329876\pi$
$30$		0			0
$31$	−25.0000			−0.806452			−0.403226	−	0.915101i	$-0.632111\pi$
$31$	−25.0000			−0.806452			−0.403226	+	0.915101i	$0.632111\pi$
$32$		0			0
$33$	7.34847	+	7.34847i	0.222681	+	0.222681i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	24.4949	−	24.4949i	0.662024	−	0.662024i	−0.293833	−	0.955857i	$-0.594931\pi$
$37$	24.4949	−	24.4949i	0.662024	−	0.662024i	0.955857	+	0.293833i	$0.0949308\pi$
$38$		0			0
$39$			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	$0.761276\pi$
$42$		0			0
$43$	60.0125	+	60.0125i	1.39564	+	1.39564i	0.812046	+	0.583594i	$0.198354\pi$
$43$	60.0125	+	60.0125i	1.39564	+	1.39564i	0.583594	+	0.812046i	$0.301646\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	7.34847	−	7.34847i	0.156350	−	0.156350i	−0.624597	−	0.780947i	$-0.714737\pi$
$47$	7.34847	−	7.34847i	0.156350	−	0.156350i	0.780947	+	0.624597i	$0.214737\pi$
$48$		0			0
$49$		−	26.0000i		−	0.530612i
$50$		0			0
$51$	42.0000			0.823529
$52$		0			0
$53$	24.4949	+	24.4949i	0.462168	+	0.462168i	0.899365	−	0.437198i	$-0.144029\pi$
$53$	24.4949	+	24.4949i	0.462168	+	0.462168i	−0.437198	+	0.899365i	$0.644029\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	28.1691	−	28.1691i	0.494195	−	0.494195i
$58$		0			0
$59$		−	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	$0.598081\pi$
$62$		0			0
$63$	18.3712	+	18.3712i	0.291606	+	0.291606i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−25.7196	+	25.7196i	−0.383875	+	0.383875i	−0.872496	−	0.488621i	$-0.837500\pi$
$67$	−25.7196	+	25.7196i	−0.383875	+	0.383875i	0.488621	+	0.872496i	$0.337500\pi$
$68$		0			0
$69$			30.0000i			0.434783i
$70$		0			0
$71$	−132.000			−1.85915			−0.929577	−	0.368627i	$-0.879828\pi$
$71$	−132.000			−1.85915			−0.929577	+	0.368627i	$0.879828\pi$
$72$		0			0
$73$	24.4949	+	24.4949i	0.335547	+	0.335547i	0.854688	−	0.519142i	$-0.173748\pi$
$73$	24.4949	+	24.4949i	0.335547	+	0.335547i	−0.519142	+	0.854688i	$0.673748\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−36.7423	+	36.7423i	−0.477173	+	0.477173i
$78$		0			0
$79$			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
$82$		0			0
$83$	2.44949	+	2.44949i	0.0295119	+	0.0295119i	0.721709	−	0.692197i	$-0.243357\pi$
$83$	2.44949	+	2.44949i	0.0295119	+	0.0295119i	−0.692197	+	0.721709i	$0.743357\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−7.34847	+	7.34847i	−0.0844652	+	0.0844652i
$88$		0			0
$89$			132.000i			1.48315i	0.670872	+	0.741573i	$0.265920\pi$
$89$			132.000i			1.48315i	−0.670872	+	0.741573i	$0.734080\pi$
$90$		0			0
$91$	−45.0000			−0.494505
$92$		0			0
$93$	30.6186	+	30.6186i	0.329232	+	0.329232i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−23.2702	+	23.2702i	−0.239898	+	0.239898i	−0.816808	−	0.576910i	$-0.804259\pi$
$97$	−23.2702	+	23.2702i	−0.239898	+	0.239898i	0.576910	+	0.816808i	$0.304259\pi$
$98$		0			0
$99$		−	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	$0.342359\pi$
$102$		0			0
$103$	−19.5959	−	19.5959i	−0.190252	−	0.190252i	0.605553	−	0.795805i	$-0.292952\pi$
$103$	−19.5959	−	19.5959i	−0.190252	−	0.190252i	−0.795805	+	0.605553i	$0.792952\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−29.3939	+	29.3939i	−0.274709	+	0.274709i	−0.830993	−	0.556283i	$-0.812227\pi$
$107$	−29.3939	+	29.3939i	−0.274709	+	0.274709i	0.556283	+	0.830993i	$0.312227\pi$
$108$		0			0
$109$			167.000i			1.53211i	0.642775	+	0.766055i	$0.277783\pi$
$109$			167.000i			1.53211i	−0.642775	+	0.766055i	$0.722217\pi$
$110$		0			0
$111$	−60.0000			−0.540541
$112$		0			0
$113$	58.7878	+	58.7878i	0.520246	+	0.520246i	0.917646	−	0.397400i	$-0.130087\pi$
$113$	58.7878	+	58.7878i	0.520246	+	0.520246i	−0.397400	+	0.917646i	$0.630087\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	11.0227	−	11.0227i	0.0942111	−	0.0942111i
$118$		0			0
$119$			210.000i			1.76471i
$120$		0			0
$121$	−85.0000			−0.702479
$122$		0			0
$123$	73.4847	+	73.4847i	0.597437	+	0.597437i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−44.0908	+	44.0908i	−0.347172	+	0.347172i	−0.859055	−	0.511883i	$-0.828948\pi$
$127$	−44.0908	+	44.0908i	−0.347172	+	0.347172i	0.511883	+	0.859055i	$0.328948\pi$
$128$		0			0
$129$		−	147.000i		−	1.13953i
$130$		0			0
$131$	−108.000			−0.824427			−0.412214	−	0.911087i	$-0.635244\pi$
$131$	−108.000			−0.824427			−0.412214	+	0.911087i	$0.635244\pi$
$132$		0			0
$133$	140.846	+	140.846i	1.05899	+	1.05899i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−120.025	+	120.025i	−0.876095	+	0.876095i	−0.993128	−	0.117033i	$-0.962662\pi$
$137$	−120.025	+	120.025i	−0.876095	+	0.876095i	0.117033	+	0.993128i	$0.462662\pi$
$138$		0			0
$139$		−	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
$142$		0			0
$143$	22.0454	+	22.0454i	0.154164	+	0.154164i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−31.8434	+	31.8434i	−0.216622	+	0.216622i
$148$		0			0
$149$			186.000i			1.24832i	0.781296	+	0.624161i	$0.214559\pi$
$149$			186.000i			1.24832i	−0.781296	+	0.624161i	$0.785441\pi$
$150$		0			0
$151$	83.0000			0.549669			0.274834	−	0.961492i	$-0.411377\pi$
$151$	83.0000			0.549669			0.274834	+	0.961492i	$0.411377\pi$
$152$		0			0
$153$	−51.4393	−	51.4393i	−0.336204	−	0.336204i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	45.3156	−	45.3156i	0.288634	−	0.288634i	−0.547906	−	0.836540i	$-0.684575\pi$
$157$	45.3156	−	45.3156i	0.288634	−	0.288634i	0.836540	+	0.547906i	$0.184575\pi$
$158$		0			0
$159$		−	60.0000i		−	0.377358i
$160$		0			0
$161$	−150.000			−0.931677
$162$		0			0
$163$	99.2043	+	99.2043i	0.608616	+	0.608616i	0.942584	−	0.333969i	$-0.108388\pi$
$163$	99.2043	+	99.2043i	0.608616	+	0.608616i	−0.333969	+	0.942584i	$0.608388\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−97.9796	+	97.9796i	−0.586704	+	0.586704i	−0.936737	−	0.350033i	$-0.886170\pi$
$167$	−97.9796	+	97.9796i	−0.586704	+	0.586704i	0.350033	+	0.936737i	$0.386170\pi$
$168$		0			0
$169$		−	142.000i		−	0.840237i
$170$		0			0
$171$	−69.0000			−0.403509
$172$		0			0
$173$	−173.914	−	173.914i	−1.00528	−	1.00528i	−0.999986	−	0.00529594i	$-0.998314\pi$
$173$	−173.914	−	173.914i	−1.00528	−	1.00528i	−0.00529594	−	0.999986i	$-0.501686\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−22.0454	+	22.0454i	−0.124550	+	0.124550i
$178$		0			0
$179$			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	$0.702421\pi$
$182$		0			0
$183$	45.3156	+	45.3156i	0.247626	+	0.247626i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	102.879	−	102.879i	0.550153	−	0.550153i
$188$		0			0
$189$		−	45.0000i		−	0.238095i
$190$		0			0
$191$	234.000			1.22513			0.612565	−	0.790420i	$-0.290138\pi$
$191$	234.000			1.22513			0.612565	+	0.790420i	$0.290138\pi$
$192$		0			0
$193$	64.9115	+	64.9115i	0.336329	+	0.336329i	0.854984	−	0.518655i	$-0.173567\pi$
$193$	64.9115	+	64.9115i	0.336329	+	0.336329i	−0.518655	+	0.854984i	$0.673567\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	193.510	−	193.510i	0.982283	−	0.982283i	−0.0175631	−	0.999846i	$-0.505591\pi$
$197$	193.510	−	193.510i	0.982283	−	0.982283i	0.999846	+	0.0175631i	$0.00559079\pi$
$198$		0			0
$199$		−	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
$202$		0			0
$203$	−36.7423	−	36.7423i	−0.180997	−	0.180997i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	36.7423	−	36.7423i	0.177499	−	0.177499i
$208$		0			0
$209$		−	138.000i		−	0.660287i
$210$		0			0
$211$	85.0000			0.402844			0.201422	−	0.979505i	$-0.435444\pi$
$211$	85.0000			0.402844			0.201422	+	0.979505i	$0.435444\pi$
$212$		0			0
$213$	161.666	+	161.666i	0.758997	+	0.758997i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−153.093	+	153.093i	−0.705498	+	0.705498i
$218$		0			0
$219$		−	60.0000i		−	0.273973i
$220$		0			0
$221$	126.000			0.570136
$222$		0			0
$223$	101.654	+	101.654i	0.455847	+	0.455847i	0.897289	−	0.441443i	$-0.145533\pi$
$223$	101.654	+	101.654i	0.455847	+	0.455847i	−0.441443	+	0.897289i	$0.645533\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	195.959	−	195.959i	0.863256	−	0.863256i	−0.128459	−	0.991715i	$-0.541003\pi$
$227$	195.959	−	195.959i	0.863256	−	0.863256i	0.991715	+	0.128459i	$0.0410029\pi$
$228$		0			0
$229$		−	227.000i		−	0.991266i	−0.868532	−	0.495633i	$-0.834936\pi$
$229$		−	227.000i		−	0.991266i	0.868532	−	0.495633i	$-0.165064\pi$
$230$		0			0
$231$	90.0000			0.389610
$232$		0			0
$233$	−102.879	−	102.879i	−0.441539	−	0.441539i	0.450990	−	0.892529i	$-0.351071\pi$
$233$	−102.879	−	102.879i	−0.441539	−	0.441539i	−0.892529	+	0.450990i	$0.851071\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	12.2474	−	12.2474i	0.0516770	−	0.0516770i
$238$		0			0
$239$			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	$0.370306\pi$
$242$		0			0
$243$	11.0227	+	11.0227i	0.0453609	+	0.0453609i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	84.5074	−	84.5074i	0.342135	−	0.342135i
$248$		0			0
$249$		−	6.00000i		−	0.0240964i
$250$		0			0
$251$	−192.000			−0.764940			−0.382470	−	0.923968i	$-0.624927\pi$
$251$	−192.000			−0.764940			−0.382470	+	0.923968i	$0.624927\pi$
$252$		0			0
$253$	73.4847	+	73.4847i	0.290453	+	0.290453i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−142.070	+	142.070i	−0.552803	+	0.552803i	−0.927249	−	0.374446i	$-0.877833\pi$
$257$	−142.070	+	142.070i	−0.552803	+	0.552803i	0.374446	+	0.927249i	$0.377833\pi$
$258$		0			0
$259$		−	300.000i		−	1.15830i
$260$		0			0
$261$	18.0000			0.0689655
$262$		0			0
$263$	−249.848	−	249.848i	−0.949992	−	0.949992i	0.0488156	−	0.998808i	$-0.484455\pi$
$263$	−249.848	−	249.848i	−0.949992	−	0.949992i	−0.998808	+	0.0488156i	$0.984455\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	161.666	−	161.666i	0.605492	−	0.605492i
$268$		0			0
$269$			462.000i			1.71747i	0.512418	+	0.858736i	$0.328750\pi$
$269$			462.000i			1.71747i	−0.512418	+	0.858736i	$0.671250\pi$
$270$		0			0
$271$	−446.000			−1.64576			−0.822878	−	0.568218i	$-0.807633\pi$
$271$	−446.000			−1.64576			−0.822878	+	0.568218i	$0.807633\pi$
$272$		0			0
$273$	55.1135	+	55.1135i	0.201881	+	0.201881i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	175.139	−	175.139i	0.632269	−	0.632269i	−0.316368	−	0.948637i	$-0.602463\pi$
$277$	175.139	−	175.139i	0.632269	−	0.632269i	0.948637	+	0.316368i	$0.102463\pi$
$278$		0			0
$279$		−	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	$0.923864\pi$
$282$		0			0
$283$	−192.285	−	192.285i	−0.679452	−	0.679452i	0.280424	−	0.959876i	$-0.409525\pi$
$283$	−192.285	−	192.285i	−0.679452	−	0.679452i	−0.959876	+	0.280424i	$0.909525\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−367.423	+	367.423i	−1.28022	+	1.28022i
$288$		0			0
$289$		−	299.000i		−	1.03460i
$290$		0			0
$291$	57.0000			0.195876
$292$		0			0
$293$	−276.792	−	276.792i	−0.944684	−	0.944684i	0.0538645	−	0.998548i	$-0.482846\pi$
$293$	−276.792	−	276.792i	−0.944684	−	0.944684i	−0.998548	+	0.0538645i	$0.982846\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−22.0454	+	22.0454i	−0.0742270	+	0.0742270i
$298$		0			0
$299$			90.0000i			0.301003i
$300$		0			0
$301$	735.000			2.44186
$302$		0			0
$303$	−117.576	−	117.576i	−0.388038	−	0.388038i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	86.9569	−	86.9569i	0.283247	−	0.283247i	−0.551155	−	0.834403i	$-0.685813\pi$
$307$	86.9569	−	86.9569i	0.283247	−	0.283247i	0.834403	+	0.551155i	$0.185813\pi$
$308$		0			0
$309$			48.0000i			0.155340i
$310$		0			0
$311$	294.000			0.945338			0.472669	−	0.881240i	$-0.343291\pi$
$311$	294.000			0.945338			0.472669	+	0.881240i	$0.343291\pi$
$312$		0			0
$313$	−393.143	−	393.143i	−1.25605	−	1.25605i	−0.952964	−	0.303085i	$-0.901984\pi$
$313$	−393.143	−	393.143i	−1.25605	−	1.25605i	−0.303085	−	0.952964i	$-0.598016\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−93.0806	+	93.0806i	−0.293630	+	0.293630i	−0.838512	−	0.544883i	$-0.816574\pi$
$317$	−93.0806	+	93.0806i	−0.293630	+	0.293630i	0.544883	+	0.838512i	$0.316574\pi$
$318$		0			0
$319$			36.0000i			0.112853i
$320$		0			0
$321$	72.0000			0.224299
$322$		0			0
$323$	−394.368	−	394.368i	−1.22095	−	1.22095i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	204.532	−	204.532i	0.625481	−	0.625481i
$328$		0			0
$329$		−	90.0000i		−	0.273556i
$330$		0			0
$331$	178.000			0.537764			0.268882	−	0.963173i	$-0.413346\pi$
$331$	178.000			0.537764			0.268882	+	0.963173i	$0.413346\pi$
$332$		0			0
$333$	73.4847	+	73.4847i	0.220675	+	0.220675i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	150.644	−	150.644i	0.447014	−	0.447014i	−0.447347	−	0.894361i	$-0.647631\pi$
$337$	150.644	−	150.644i	0.447014	−	0.447014i	0.894361	+	0.447347i	$0.147631\pi$
$338$		0			0
$339$		−	144.000i		−	0.424779i
$340$		0			0
$341$	150.000			0.439883
$342$		0			0
$343$	140.846	+	140.846i	0.410629	+	0.410629i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−374.772	+	374.772i	−1.08003	+	1.08003i	−0.0835290	+	0.996505i	$0.526619\pi$
$347$	−374.772	+	374.772i	−1.08003	+	1.08003i	−0.996505	+	0.0835290i	$0.973381\pi$
$348$		0			0
$349$			514.000i			1.47278i	0.676558	+	0.736390i	$0.263471\pi$
$349$			514.000i			1.47278i	−0.676558	+	0.736390i	$0.736529\pi$
$350$		0			0
$351$	−27.0000			−0.0769231
$352$		0			0
$353$	−242.499	−	242.499i	−0.686967	−	0.686967i	0.274593	−	0.961561i	$-0.411457\pi$
$353$	−242.499	−	242.499i	−0.686967	−	0.686967i	−0.961561	+	0.274593i	$0.911457\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	257.196	−	257.196i	0.720438	−	0.720438i
$358$		0			0
$359$		−	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
$362$		0			0
$363$	104.103	+	104.103i	0.286786	+	0.286786i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	143.295	−	143.295i	0.390450	−	0.390450i	−0.484398	−	0.874848i	$-0.660961\pi$
$367$	143.295	−	143.295i	0.390450	−	0.390450i	0.874848	+	0.484398i	$0.160961\pi$
$368$		0			0
$369$		−	180.000i		−	0.487805i
$370$		0			0
$371$	300.000			0.808625
$372$		0			0
$373$	−35.5176	−	35.5176i	−0.0952215	−	0.0952215i	0.657891	−	0.753113i	$-0.271449\pi$
$373$	−35.5176	−	35.5176i	−0.0952215	−	0.0952215i	−0.753113	+	0.657891i	$0.771449\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−22.0454	+	22.0454i	−0.0584759	+	0.0584759i
$378$		0			0
$379$			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
$382$		0			0
$383$	19.5959	+	19.5959i	0.0511643	+	0.0511643i	0.732226	−	0.681062i	$-0.238481\pi$
$383$	19.5959	+	19.5959i	0.0511643	+	0.0511643i	−0.681062	+	0.732226i	$0.738481\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−180.037	+	180.037i	−0.465213	+	0.465213i
$388$		0			0
$389$		−	192.000i		−	0.493573i	−0.969070	−	0.246787i	$-0.920625\pi$
$389$		−	192.000i		−	0.493573i	0.969070	−	0.246787i	$-0.0793747\pi$
$390$		0			0
$391$	420.000			1.07417
$392$		0			0
$393$	132.272	+	132.272i	0.336571	+	0.336571i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	366.199	−	366.199i	0.922415	−	0.922415i	−0.0747848	−	0.997200i	$-0.523827\pi$
$397$	366.199	−	366.199i	0.922415	−	0.922415i	0.997200	+	0.0747848i	$0.0238270\pi$
$398$		0			0
$399$		−	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	$0.408242\pi$
$402$		0			0
$403$	91.8559	+	91.8559i	0.227930	+	0.227930i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−146.969	+	146.969i	−0.361104	+	0.361104i
$408$		0			0
$409$		−	707.000i		−	1.72861i	−0.502971	−	0.864303i	$-0.667760\pi$
$409$		−	707.000i		−	1.72861i	0.502971	−	0.864303i	$-0.332240\pi$
$410$		0			0
$411$	294.000			0.715328
$412$		0			0
$413$	−110.227	−	110.227i	−0.266894	−	0.266894i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−71.0352	+	71.0352i	−0.170348	+	0.170348i
$418$		0			0
$419$			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	$0.290988\pi$
$422$		0			0
$423$	22.0454	+	22.0454i	0.0521168	+	0.0521168i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−226.578	+	226.578i	−0.530627	+	0.530627i
$428$		0			0
$429$		−	54.0000i		−	0.125874i
$430$		0			0
$431$	−30.0000			−0.0696056			−0.0348028	−	0.999394i	$-0.511080\pi$
$431$	−30.0000			−0.0696056			−0.0348028	+	0.999394i	$0.511080\pi$
$432$		0			0
$433$	353.951	+	353.951i	0.817439	+	0.817439i	0.985736	−	0.168297i	$-0.0538267\pi$
$433$	353.951	+	353.951i	0.817439	+	0.817439i	−0.168297	+	0.985736i	$0.553827\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	281.691	−	281.691i	0.644603	−	0.644603i
$438$		0			0
$439$		−	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
$442$		0			0
$443$	−112.677	−	112.677i	−0.254349	−	0.254349i	0.568402	−	0.822751i	$-0.307562\pi$
$443$	−112.677	−	112.677i	−0.254349	−	0.254349i	−0.822751	+	0.568402i	$0.807562\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	227.803	−	227.803i	0.509625	−	0.509625i
$448$		0			0
$449$		−	204.000i		−	0.454343i	−0.973855	−	0.227171i	$-0.927052\pi$
$449$		−	204.000i		−	0.454343i	0.973855	−	0.227171i	$-0.0729478\pi$
$450$		0			0
$451$	360.000			0.798226
$452$		0			0
$453$	−101.654	−	101.654i	−0.224401	−	0.224401i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−440.908	+	440.908i	−0.964788	+	0.964788i	−0.999401	−	0.0346127i	$-0.988980\pi$
$457$	−440.908	+	440.908i	−0.964788	+	0.964788i	0.0346127	+	0.999401i	$0.488980\pi$
$458$		0			0
$459$			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	$0.454271\pi$
$462$		0			0
$463$	−436.009	−	436.009i	−0.941704	−	0.941704i	0.0566875	−	0.998392i	$-0.481946\pi$
$463$	−436.009	−	436.009i	−0.941704	−	0.941704i	−0.998392	+	0.0566875i	$0.981946\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	276.792	−	276.792i	0.592703	−	0.592703i	−0.345658	−	0.938361i	$-0.612344\pi$
$467$	276.792	−	276.792i	0.592703	−	0.592703i	0.938361	+	0.345658i	$0.112344\pi$
$468$		0			0
$469$			315.000i			0.671642i
$470$		0			0
$471$	−111.000			−0.235669
$472$		0			0
$473$	−360.075	−	360.075i	−0.761258	−	0.761258i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−73.4847	+	73.4847i	−0.154056	+	0.154056i
$478$		0			0
$479$			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
$482$		0			0
$483$	183.712	+	183.712i	0.380356	+	0.380356i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	488.673	−	488.673i	1.00344	−	1.00344i	0.00344166	−	0.999994i	$-0.498904\pi$
$487$	488.673	−	488.673i	1.00344	−	1.00344i	0.999994	−	0.00344166i	$-0.00109552\pi$
$488$		0			0
$489$		−	243.000i		−	0.496933i
$490$		0			0
$491$	−348.000			−0.708758			−0.354379	−	0.935102i	$-0.615308\pi$
$491$	−348.000			−0.708758			−0.354379	+	0.935102i	$0.615308\pi$
$492$		0			0
$493$	102.879	+	102.879i	0.208679	+	0.208679i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−808.332	+	808.332i	−1.62642	+	1.62642i
$498$		0			0
$499$		−	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
$502$		0			0
$503$	−137.171	−	137.171i	−0.272707	−	0.272707i	0.557482	−	0.830189i	$-0.311767\pi$
$503$	−137.171	−	137.171i	−0.272707	−	0.272707i	−0.830189	+	0.557482i	$0.811767\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−173.914	+	173.914i	−0.343025	+	0.343025i
$508$		0			0
$509$			618.000i			1.21415i	0.794646	+	0.607073i	$0.207656\pi$
$509$			618.000i			1.21415i	−0.794646	+	0.607073i	$0.792344\pi$
$510$		0			0
$511$	300.000			0.587084
$512$		0			0
$513$	84.5074	+	84.5074i	0.164732	+	0.164732i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−44.0908	+	44.0908i	−0.0852820	+	0.0852820i
$518$		0			0
$519$			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	$0.269628\pi$
$522$		0			0
$523$	255.972	+	255.972i	0.489430	+	0.489430i	0.908126	−	0.418697i	$-0.137513\pi$
$523$	255.972	+	255.972i	0.489430	+	0.489430i	−0.418697	+	0.908126i	$0.637513\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	428.661	−	428.661i	0.813398	−	0.813398i
$528$		0			0
$529$		−	229.000i		−	0.432892i
$530$		0			0
$531$	54.0000			0.101695
$532$		0			0
$533$	220.454	+	220.454i	0.413610	+	0.413610i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	183.712	−	183.712i	0.342108	−	0.342108i
$538$		0			0
$539$			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	$0.597110\pi$
$542$		0			0
$543$	263.320	+	263.320i	0.484936	+	0.484936i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−489.898	+	489.898i	−0.895609	+	0.895609i	−0.995044	−	0.0994353i	$-0.968296\pi$
$547$	−489.898	+	489.898i	−0.895609	+	0.895609i	0.0994353	+	0.995044i	$0.468296\pi$
$548$		0			0
$549$		−	111.000i		−	0.202186i
$550$		0			0
$551$	138.000			0.250454
$552$		0			0
$553$	61.2372	+	61.2372i	0.110736	+	0.110736i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	154.318	−	154.318i	0.277052	−	0.277052i	−0.554879	−	0.831931i	$-0.687236\pi$
$557$	154.318	−	154.318i	0.277052	−	0.277052i	0.831931	+	0.554879i	$0.187236\pi$
$558$		0			0
$559$		−	441.000i		−	0.788909i
$560$		0			0
$561$	−252.000			−0.449198
$562$		0			0
$563$	−609.923	−	609.923i	−1.08334	−	1.08334i	−0.996195	−	0.0871492i	$-0.972224\pi$
$563$	−609.923	−	609.923i	−1.08334	−	1.08334i	−0.0871492	−	0.996195i	$-0.527776\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−55.1135	+	55.1135i	−0.0972020	+	0.0972020i
$568$		0			0
$569$		−	198.000i		−	0.347979i	−0.984748	−	0.173989i	$-0.944334\pi$
$569$		−	198.000i		−	0.347979i	0.984748	−	0.173989i	$-0.0556659\pi$
$570$		0			0
$571$	169.000			0.295972			0.147986	−	0.988989i	$-0.452721\pi$
$571$	169.000			0.295972			0.147986	+	0.988989i	$0.452721\pi$
$572$		0			0
$573$	−286.590	−	286.590i	−0.500158	−	0.500158i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−243.724	+	243.724i	−0.422399	+	0.422399i	−0.886029	−	0.463630i	$-0.846547\pi$
$577$	−243.724	+	243.724i	−0.422399	+	0.422399i	0.463630	+	0.886029i	$0.346547\pi$
$578$		0			0
$579$		−	159.000i		−	0.274611i
$580$		0			0
$581$	30.0000			0.0516351
$582$		0			0
$583$	−146.969	−	146.969i	−0.252092	−	0.252092i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	289.040	−	289.040i	0.492402	−	0.492402i	−0.416660	−	0.909062i	$-0.636800\pi$
$587$	289.040	−	289.040i	0.492402	−	0.492402i	0.909062	+	0.416660i	$0.136800\pi$
$588$		0			0
$589$		−	575.000i		−	0.976231i
$590$		0			0
$591$	−474.000			−0.802030
$592$		0			0
$593$	−421.312	−	421.312i	−0.710476	−	0.710476i	0.256159	−	0.966635i	$-0.417543\pi$
$593$	−421.312	−	421.312i	−0.710476	−	0.710476i	−0.966635	+	0.256159i	$0.917543\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−13.4722	+	13.4722i	−0.0225665	+	0.0225665i
$598$		0			0
$599$			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	$0.768947\pi$
$602$		0			0
$603$	−77.1589	−	77.1589i	−0.127958	−	0.127958i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	4.89898	−	4.89898i	0.00807081	−	0.00807081i	−0.703060	−	0.711131i	$-0.748183\pi$
$607$	4.89898	−	4.89898i	0.00807081	−	0.00807081i	0.711131	+	0.703060i	$0.248183\pi$
$608$		0			0
$609$			90.0000i			0.147783i
$610$		0			0
$611$	−54.0000			−0.0883797
$612$		0			0
$613$	−259.646	−	259.646i	−0.423566	−	0.423566i	0.462864	−	0.886430i	$-0.346822\pi$
$613$	−259.646	−	259.646i	−0.423566	−	0.423566i	−0.886430	+	0.462864i	$0.846822\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−9.79796	+	9.79796i	−0.0158800	+	0.0158800i	−0.715002	−	0.699122i	$-0.753574\pi$
$617$	−9.79796	+	9.79796i	−0.0158800	+	0.0158800i	0.699122	+	0.715002i	$0.253574\pi$
$618$		0			0
$619$			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
$622$		0			0
$623$	808.332	+	808.332i	1.29748	+	1.29748i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−169.015	+	169.015i	−0.269561	+	0.269561i
$628$		0			0
$629$			840.000i			1.33545i
$630$		0			0
$631$	−287.000			−0.454834			−0.227417	−	0.973798i	$-0.573028\pi$
$631$	−287.000			−0.454834			−0.227417	+	0.973798i	$0.573028\pi$
$632$		0			0
$633$	−104.103	−	104.103i	−0.164460	−	0.164460i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−95.5301	+	95.5301i	−0.149969	+	0.149969i
$638$		0			0
$639$		−	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	$0.514903\pi$
$642$		0			0
$643$	−191.060	−	191.060i	−0.297139	−	0.297139i	0.542753	−	0.839892i	$-0.317382\pi$
$643$	−191.060	−	191.060i	−0.297139	−	0.297139i	−0.839892	+	0.542753i	$0.817382\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−656.463	+	656.463i	−1.01463	+	1.01463i	−0.0147349	+	0.999891i	$0.504690\pi$
$647$	−656.463	+	656.463i	−1.01463	+	1.01463i	−0.999891	+	0.0147349i	$0.995310\pi$
$648$		0			0
$649$			108.000i			0.166410i
$650$		0			0
$651$	375.000			0.576037
$652$		0			0
$653$	751.993	+	751.993i	1.15160	+	1.15160i	0.986232	+	0.165365i	$0.0528803\pi$
$653$	751.993	+	751.993i	1.15160	+	1.15160i	0.165365	+	0.986232i	$0.447120\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−73.4847	+	73.4847i	−0.111849	+	0.111849i
$658$		0			0
$659$			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	$0.538134\pi$
$662$		0			0
$663$	−154.318	−	154.318i	−0.232757	−	0.232757i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−73.4847	+	73.4847i	−0.110172	+	0.110172i
$668$		0			0
$669$		−	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	$-0.291074\pi$
$673$	−122.474	−	122.474i	−0.181983	−	0.181983i	−0.792219	+	0.610236i	$0.791074\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	242.499	−	242.499i	0.358197	−	0.358197i	−0.504951	−	0.863148i	$-0.668489\pi$
$677$	242.499	−	242.499i	0.358197	−	0.358197i	0.863148	+	0.504951i	$0.168489\pi$
$678$		0			0
$679$			285.000i			0.419735i
$680$		0			0
$681$	−480.000			−0.704846
$682$		0			0
$683$	19.5959	+	19.5959i	0.0286909	+	0.0286909i	0.721307	−	0.692616i	$-0.243542\pi$
$683$	19.5959	+	19.5959i	0.0286909	+	0.0286909i	−0.692616	+	0.721307i	$0.743542\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−278.017	+	278.017i	−0.404683	+	0.404683i
$688$		0			0
$689$		−	180.000i		−	0.261248i
$690$		0			0
$691$	74.0000			0.107091			0.0535456	−	0.998565i	$-0.482948\pi$
$691$	74.0000			0.107091			0.0535456	+	0.998565i	$0.482948\pi$
$692$		0			0
$693$	−110.227	−	110.227i	−0.159058	−	0.159058i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	1028.79	−	1028.79i	1.47602	−	1.47602i
$698$		0			0
$699$			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	$0.523179\pi$
$702$		0			0
$703$	563.383	+	563.383i	0.801398	+	0.801398i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	587.878	−	587.878i	0.831510	−	0.831510i
$708$		0			0
$709$			817.000i			1.15233i	0.817334	+	0.576164i	$0.195451\pi$
$709$			817.000i			1.15233i	−0.817334	+	0.576164i	$0.804549\pi$
$710$		0			0
$711$	−30.0000			−0.0421941
$712$		0			0
$713$	306.186	+	306.186i	0.429434	+	0.429434i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	279.242	−	279.242i	0.389459	−	0.389459i
$718$		0			0
$719$		−	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
$722$		0			0
$723$	−233.926	−	233.926i	−0.323549	−	0.323549i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−45.3156	+	45.3156i	−0.0623323	+	0.0623323i	−0.737586	−	0.675253i	$-0.764034\pi$
$727$	−45.3156	+	45.3156i	−0.0623323	+	0.0623323i	0.675253	+	0.737586i	$0.264034\pi$
$728$		0			0
$729$		−	27.0000i		−	0.0370370i
$730$		0			0
$731$	−2058.00			−2.81532
$732$		0			0
$733$	484.999	+	484.999i	0.661663	+	0.661663i	0.955772	−	0.294109i	$-0.0950228\pi$
$733$	484.999	+	484.999i	0.661663	+	0.661663i	−0.294109	+	0.955772i	$0.595023\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	154.318	−	154.318i	0.209387	−	0.209387i
$738$		0			0
$739$		−	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
$742$		0			0
$743$	249.848	+	249.848i	0.336269	+	0.336269i	0.854961	−	0.518692i	$-0.173581\pi$
$743$	249.848	+	249.848i	0.336269	+	0.336269i	−0.518692	+	0.854961i	$0.673581\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−7.34847	+	7.34847i	−0.00983731	+	0.00983731i
$748$		0			0
$749$			360.000i			0.480641i
$750$		0			0
$751$	−1186.00			−1.57923			−0.789614	−	0.613604i	$-0.789719\pi$
$751$	−1186.00			−1.57923			−0.789614	+	0.613604i	$0.789719\pi$
$752$		0			0
$753$	235.151	+	235.151i	0.312286	+	0.312286i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	743.420	−	743.420i	0.982061	−	0.982061i	−0.0177810	−	0.999842i	$-0.505660\pi$
$757$	743.420	−	743.420i	0.982061	−	0.982061i	0.999842	+	0.0177810i	$0.00566015\pi$
$758$		0			0
$759$		−	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	$0.623543\pi$
$762$		0			0
$763$	1022.66	+	1022.66i	1.34032	+	1.34032i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−66.1362	+	66.1362i	−0.0862271	+	0.0862271i
$768$		0			0
$769$			491.000i			0.638492i	0.947672	+	0.319246i	$0.103430\pi$
$769$			491.000i			0.638492i	−0.947672	+	0.319246i	$0.896570\pi$
$770$		0			0
$771$	348.000			0.451362
$772$		0			0
$773$	519.292	+	519.292i	0.671788	+	0.671788i	0.958128	−	0.286340i	$-0.0924388\pi$
$773$	519.292	+	519.292i	0.671788	+	0.671788i	−0.286340	+	0.958128i	$0.592439\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−367.423	+	367.423i	−0.472874	+	0.472874i
$778$		0			0
$779$		−	1380.00i		−	1.77150i
$780$		0			0
$781$	792.000			1.01408
$782$		0			0
$783$	−22.0454	−	22.0454i	−0.0281551	−	0.0281551i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	123.699	−	123.699i	0.157178	−	0.157178i	−0.624137	−	0.781315i	$-0.714549\pi$
$787$	123.699	−	123.699i	0.157178	−	0.157178i	0.781315	+	0.624137i	$0.214549\pi$
$788$		0			0
$789$			612.000i			0.775665i
$790$		0			0
$791$	720.000			0.910240
$792$		0			0
$793$	135.947	+	135.947i	0.171433	+	0.171433i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−460.504	+	460.504i	−0.577797	+	0.577797i	−0.934296	−	0.356499i	$-0.883970\pi$
$797$	−460.504	+	460.504i	−0.577797	+	0.577797i	0.356499	+	0.934296i	$0.383970\pi$
$798$		0			0
$799$			252.000i			0.315394i
$800$		0			0
$801$	−396.000			−0.494382
$802$		0			0
$803$	−146.969	−	146.969i	−0.183025	−	0.183025i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	565.832	−	565.832i	0.701155	−	0.701155i
$808$		0			0
$809$		−	900.000i		−	1.11248i	−0.831020	−	0.556242i	$-0.812243\pi$
$809$		−	900.000i		−	1.11248i	0.831020	−	0.556242i	$-0.187757\pi$
$810$		0			0
$811$	−1477.00			−1.82121			−0.910604	−	0.413280i	$-0.864383\pi$
$811$	−1477.00			−1.82121			−0.910604	+	0.413280i	$0.864383\pi$
$812$		0			0
$813$	546.236	+	546.236i	0.671877	+	0.671877i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−1380.29	+	1380.29i	−1.68946	+	1.68946i
$818$		0			0
$819$		−	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	$0.658886\pi$
$822$		0			0
$823$	−371.098	−	371.098i	−0.450909	−	0.450909i	0.444747	−	0.895656i	$-0.353293\pi$
$823$	−371.098	−	371.098i	−0.450909	−	0.450909i	−0.895656	+	0.444747i	$0.853293\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	788.736	−	788.736i	0.953731	−	0.953731i	−0.0452447	−	0.998976i	$-0.514407\pi$
$827$	788.736	−	788.736i	0.953731	−	0.953731i	0.998976	+	0.0452447i	$0.0144068\pi$
$828$		0			0
$829$		−	170.000i		−	0.205066i	−0.994730	−	0.102533i	$-0.967305\pi$
$829$		−	170.000i		−	0.205066i	0.994730	−	0.102533i	$-0.0326948\pi$
$830$		0			0
$831$	−429.000			−0.516245
$832$		0			0
$833$	445.807	+	445.807i	0.535183	+	0.535183i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−91.8559	+	91.8559i	−0.109744	+	0.109744i
$838$		0			0
$839$		−	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
$842$		0			0
$843$	668.711	+	668.711i	0.793251	+	0.793251i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−520.517	+	520.517i	−0.614541	+	0.614541i
$848$		0			0
$849$			471.000i			0.554770i
$850$		0			0
$851$	−600.000			−0.705053
$852$		0			0
$853$	−723.824	−	723.824i	−0.848563	−	0.848563i	0.141391	−	0.989954i	$-0.454843\pi$
$853$	−723.824	−	723.824i	−0.848563	−	0.848563i	−0.989954	+	0.141391i	$0.954843\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	739.746	−	739.746i	0.863181	−	0.863181i	−0.128525	−	0.991706i	$-0.541024\pi$
$857$	739.746	−	739.746i	0.863181	−	0.863181i	0.991706	+	0.128525i	$0.0410244\pi$
$858$		0			0
$859$			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
$862$		0			0
$863$	372.322	+	372.322i	0.431428	+	0.431428i	0.889114	−	0.457686i	$-0.151322\pi$
$863$	372.322	+	372.322i	0.431428	+	0.431428i	−0.457686	+	0.889114i	$0.651322\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−366.199	+	366.199i	−0.422375	+	0.422375i
$868$		0			0
$869$		−	60.0000i		−	0.0690449i
$870$		0			0
$871$	189.000			0.216992
$872$		0			0
$873$	−69.8105	−	69.8105i	−0.0799662	−	0.0799662i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−300.062	+	300.062i	−0.342147	+	0.342147i	−0.857174	−	0.515027i	$-0.827782\pi$
$877$	−300.062	+	300.062i	−0.342147	+	0.342147i	0.515027	+	0.857174i	$0.327782\pi$
$878$		0			0
$879$			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	$0.539119\pi$
$882$		0			0
$883$	−846.299	−	846.299i	−0.958436	−	0.958436i	0.0407343	−	0.999170i	$-0.487030\pi$
$883$	−846.299	−	846.299i	−0.958436	−	0.958436i	−0.999170	+	0.0407343i	$0.987030\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−996.942	+	996.942i	−1.12395	+	1.12395i	−0.132807	+	0.991142i	$0.542399\pi$
$887$	−996.942	+	996.942i	−1.12395	+	1.12395i	−0.991142	+	0.132807i	$0.957601\pi$
$888$		0			0
$889$			540.000i			0.607424i
$890$		0			0
$891$	54.0000			0.0606061
$892$		0			0
$893$	169.015	+	169.015i	0.189266	+	0.189266i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	110.227	−	110.227i	0.122884	−	0.122884i
$898$		0			0
$899$			150.000i			0.166852i
$900$		0			0
$901$	−840.000			−0.932297
$902$		0			0
$903$	−900.187	−	900.187i	−0.996885	−	0.996885i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−53.8888	+	53.8888i	−0.0594143	+	0.0594143i	−0.736190	−	0.676775i	$-0.763377\pi$
$907$	−53.8888	+	53.8888i	−0.0594143	+	0.0594143i	0.676775	+	0.736190i	$0.263377\pi$
$908$		0			0
$909$			288.000i			0.316832i
$910$		0			0
$911$	558.000			0.612514			0.306257	−	0.951949i	$-0.400923\pi$
$911$	558.000			0.612514			0.306257	+	0.951949i	$0.400923\pi$
$912$		0			0
$913$	−14.6969	−	14.6969i	−0.0160974	−	0.0160974i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−661.362	+	661.362i	−0.721224	+	0.721224i
$918$		0			0
$919$			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
$922$		0			0
$923$	484.999	+	484.999i	0.525459	+	0.525459i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	58.7878	−	58.7878i	0.0634172	−	0.0634172i
$928$		0			0
$929$			762.000i			0.820237i	0.912032	+	0.410118i	$0.134513\pi$
$929$			762.000i			0.820237i	−0.912032	+	0.410118i	$0.865487\pi$
$930$		0			0
$931$	598.000			0.642320
$932$		0			0
$933$	−360.075	−	360.075i	−0.385932	−	0.385932i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−388.244	+	388.244i	−0.414348	+	0.414348i	−0.883250	−	0.468902i	$-0.844650\pi$
$937$	−388.244	+	388.244i	−0.414348	+	0.414348i	0.468902	+	0.883250i	$0.344650\pi$
$938$		0			0
$939$			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	$0.619489\pi$
$942$		0			0
$943$	734.847	+	734.847i	0.779265	+	0.779265i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−100.429	+	100.429i	−0.106050	+	0.106050i	−0.758141	−	0.652091i	$-0.773892\pi$
$947$	−100.429	+	100.429i	−0.106050	+	0.106050i	0.652091	+	0.758141i	$0.273892\pi$
$948$		0			0
$949$		−	180.000i		−	0.189673i
$950$		0			0
$951$	228.000			0.239748
$952$		0			0
$953$	749.544	+	749.544i	0.786510	+	0.786510i	0.980920	−	0.194410i	$-0.0622794\pi$
$953$	749.544	+	749.544i	0.786510	+	0.786510i	−0.194410	+	0.980920i	$0.562279\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	44.0908	−	44.0908i	0.0460719	−	0.0460719i
$958$		0			0
$959$			1470.00i			1.53285i
$960$		0			0
$961$	−336.000			−0.349636
$962$		0			0
$963$	−88.1816	−	88.1816i	−0.0915697	−	0.0915697i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	4.89898	−	4.89898i	0.00506616	−	0.00506616i	−0.704569	−	0.709635i	$-0.748860\pi$
$967$	4.89898	−	4.89898i	0.00506616	−	0.00506616i	0.709635	+	0.704569i	$0.248860\pi$
$968$		0			0
$969$			966.000i			0.996904i
$970$		0			0
$971$	−90.0000			−0.0926880			−0.0463440	−	0.998926i	$-0.514757\pi$
$971$	−90.0000			−0.0926880			−0.0463440	+	0.998926i	$0.514757\pi$
$972$		0			0
$973$	−355.176	−	355.176i	−0.365032	−	0.365032i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	301.287	−	301.287i	0.308380	−	0.308380i	−0.535901	−	0.844281i	$-0.680028\pi$
$977$	301.287	−	301.287i	0.308380	−	0.308380i	0.844281	+	0.535901i	$0.180028\pi$
$978$		0			0
$979$		−	792.000i		−	0.808989i
$980$		0			0
$981$	−501.000			−0.510703
$982$		0			0
$983$	465.403	+	465.403i	0.473452	+	0.473452i	0.903030	−	0.429578i	$-0.141338\pi$
$983$	465.403	+	465.403i	0.473452	+	0.473452i	−0.429578	+	0.903030i	$0.641338\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−110.227	+	110.227i	−0.111679	+	0.111679i
$988$		0			0
$989$		−	1470.00i		−	1.48635i
$990$		0			0
$991$	−1067.00			−1.07669			−0.538345	−	0.842724i	$-0.680950\pi$
$991$	−1067.00			−1.07669			−0.538345	+	0.842724i	$0.680950\pi$
$992$		0			0
$993$	−218.005	−	218.005i	−0.219541	−	0.219541i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−279.242	+	279.242i	−0.280082	+	0.280082i	−0.833142	−	0.553060i	$-0.813460\pi$
$997$	−279.242	+	279.242i	−0.280082	+	0.280082i	0.553060	+	0.833142i	$0.313460\pi$
$998$		0			0
$999$		−	180.000i		−	0.180180i

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.3.f.b.7.1	✓	4	20.3	even	4
75.3.f.b.7.2	yes	4	20.7	even	4
75.3.f.b.43.1	yes	4	20.19	odd	2
75.3.f.b.43.2	yes	4	4.3	odd	2
225.3.g.b.82.1		4	60.47	odd	4
225.3.g.b.82.2		4	60.23	odd	4
225.3.g.b.118.1		4	12.11	even	2
225.3.g.b.118.2		4	60.59	even	2
1200.3.bg.g.193.1		4	1.1	even	1	trivial
1200.3.bg.g.193.2		4	5.4	even	2	inner
1200.3.bg.g.1057.1		4	5.2	odd	4	inner
1200.3.bg.g.1057.2		4	5.3	odd	4	inner

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:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1200.bg (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.22474 - 1.22474i) q^{3} +(6.12372 - 6.12372i) q^{7} +3.00000i q^{9} +O(q^{10})\)	\(q+(-1.22474 - 1.22474i) q^{3} +(6.12372 - 6.12372i) q^{7} +3.00000i q^{9} -6.00000 q^{11} +(-3.67423 - 3.67423i) q^{13} +(-17.1464 + 17.1464i) q^{17} +23.0000i q^{19} -15.0000 q^{21} +(-12.2474 - 12.2474i) q^{23} +(3.67423 - 3.67423i) q^{27} -6.00000i q^{29} -25.0000 q^{31} +(7.34847 + 7.34847i) q^{33} +(24.4949 - 24.4949i) q^{37} +9.00000i q^{39} -60.0000 q^{41} +(60.0125 + 60.0125i) q^{43} +(7.34847 - 7.34847i) q^{47} -26.0000i q^{49} +42.0000 q^{51} +(24.4949 + 24.4949i) q^{53} +(28.1691 - 28.1691i) q^{57} -18.0000i q^{59} -37.0000 q^{61} +(18.3712 + 18.3712i) q^{63} +(-25.7196 + 25.7196i) q^{67} +30.0000i q^{69} -132.000 q^{71} +(24.4949 + 24.4949i) q^{73} +(-36.7423 + 36.7423i) q^{77} +10.0000i q^{79} -9.00000 q^{81} +(2.44949 + 2.44949i) q^{83} +(-7.34847 + 7.34847i) q^{87} +132.000i q^{89} -45.0000 q^{91} +(30.6186 + 30.6186i) q^{93} +(-23.2702 + 23.2702i) q^{97} -18.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 24 q^{11} - 60 q^{21} - 100 q^{31} - 240 q^{41} + 168 q^{51} - 148 q^{61} - 528 q^{71} - 36 q^{81} - 180 q^{91}+O(q^{100}) \) 4 * q - 24 * q^11 - 60 * q^21 - 100 * q^31 - 240 * q^41 + 168 * q^51 - 148 * q^61 - 528 * q^71 - 36 * q^81 - 180 * q^91

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