LMFDB - Embedded newform 7500.2.d.g.1249.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7500,2,Mod(1249,7500)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7500, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("7500.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7500 = 2^{2} \cdot 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7500.d (of order $2$, degree $1$, 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:	$59.8878015160$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 300)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.3
Character	$\chi$	$=$	7500.1249
Dual form			7500.2.d.g.1249.22

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -1.57893i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} -1.57893i q^{7} -1.00000 q^{9} +3.88059 q^{11} -0.343277i q^{13} +6.07054i q^{17} -3.69189 q^{19} -1.57893 q^{21} +1.39041i q^{23} +1.00000i q^{27} +3.32990 q^{29} -5.25496 q^{31} -3.88059i q^{33} +8.56667i q^{37} -0.343277 q^{39} -1.27815 q^{41} +1.42438i q^{43} +0.375462i q^{47} +4.50698 q^{49} +6.07054 q^{51} -11.2992i q^{53} +3.69189i q^{57} -11.5818 q^{59} -10.9673 q^{61} +1.57893i q^{63} +10.4591i q^{67} +1.39041 q^{69} -10.1261 q^{71} +13.1900i q^{73} -6.12718i q^{77} -13.7995 q^{79} +1.00000 q^{81} -4.62458i q^{83} -3.32990i q^{87} +7.26904 q^{89} -0.542010 q^{91} +5.25496i q^{93} +6.05557i q^{97} -3.88059 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 24 q^{9}+O(q^{10}) $ 24 * q - 24 * q^9	$ 24 q - 24 q^{9} + 4 q^{11} - 20 q^{19} + 16 q^{21} - 16 q^{29} - 4 q^{31} + 20 q^{41} - 56 q^{49} + 16 q^{51} + 4 q^{59} + 68 q^{61} - 36 q^{69} - 12 q^{79} + 24 q^{81} - 20 q^{89} + 40 q^{91} - 4 q^{99}+O(q^{100}) $ 24 * q - 24 * q^9 + 4 * q^11 - 20 * q^19 + 16 * q^21 - 16 * q^29 - 4 * q^31 + 20 * q^41 - 56 * q^49 + 16 * q^51 + 4 * q^59 + 68 * q^61 - 36 * q^69 - 12 * q^79 + 24 * q^81 - 20 * q^89 + 40 * q^91 - 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/7500\mathbb{Z}\right)^\times$.

$n$	$2501$	$3751$	$6877$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.57893i	− 0.596780i	−0.954444	−	0.298390i	$-0.903550\pi$
$7$	− 1.57893i	− 0.596780i	0.954444	−	0.298390i	$-0.0964496\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	3.88059	1.17004	0.585021	−	0.811018i	$-0.301086\pi$
$11$	3.88059	1.17004	0.585021	+	0.811018i	$0.301086\pi$
$12$	0	0
$13$	− 0.343277i	− 0.0952078i	−0.998866	−	0.0476039i	$-0.984841\pi$
$13$	− 0.343277i	− 0.0952078i	0.998866	−	0.0476039i	$-0.0151585\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.07054i	1.47232i	0.676807	+	0.736161i	$0.263363\pi$
$17$	6.07054i	1.47232i	−0.676807	+	0.736161i	$0.736637\pi$
$18$	0	0
$19$	−3.69189	−0.846977	−0.423489	−	0.905901i	$-0.639195\pi$
$19$	−3.69189	−0.846977	−0.423489	+	0.905901i	$0.639195\pi$
$20$	0	0
$21$	−1.57893	−0.344551
$22$	0	0
$23$	1.39041i	0.289921i	0.989437	+	0.144961i	$0.0463056\pi$
$23$	1.39041i	0.289921i	−0.989437	+	0.144961i	$0.953694\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	3.32990	0.618347	0.309174	−	0.951006i	$-0.399948\pi$
$29$	3.32990	0.618347	0.309174	+	0.951006i	$0.399948\pi$
$30$	0	0
$31$	−5.25496	−0.943818	−0.471909	−	0.881647i	$-0.656435\pi$
$31$	−5.25496	−0.943818	−0.471909	+	0.881647i	$0.656435\pi$
$32$	0	0
$33$	− 3.88059i	− 0.675524i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.56667i	1.40835i	0.710025	+	0.704176i	$0.248683\pi$
$37$	8.56667i	1.40835i	−0.710025	+	0.704176i	$0.751317\pi$
$38$	0	0
$39$	−0.343277	−0.0549682
$40$	0	0
$41$	−1.27815	−0.199613	−0.0998067	−	0.995007i	$-0.531822\pi$
$41$	−1.27815	−0.199613	−0.0998067	+	0.995007i	$0.531822\pi$
$42$	0	0
$43$	1.42438i	0.217216i	0.994085	+	0.108608i	$0.0346394\pi$
$43$	1.42438i	0.217216i	−0.994085	+	0.108608i	$0.965361\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.375462i	0.0547667i	0.999625	+	0.0273834i	$0.00871748\pi$
$47$	0.375462i	0.0547667i	−0.999625	+	0.0273834i	$0.991283\pi$
$48$	0	0
$49$	4.50698	0.643854
$50$	0	0
$51$	6.07054	0.850045
$52$	0	0
$53$	− 11.2992i	− 1.55207i	−0.630692	−	0.776033i	$-0.717229\pi$
$53$	− 11.2992i	− 1.55207i	0.630692	−	0.776033i	$-0.282771\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.69189i	0.489002i
$58$	0	0
$59$	−11.5818	−1.50783	−0.753914	−	0.656973i	$-0.771837\pi$
$59$	−11.5818	−1.50783	−0.753914	+	0.656973i	$0.771837\pi$
$60$	0	0
$61$	−10.9673	−1.40422	−0.702111	−	0.712068i	$-0.747759\pi$
$61$	−10.9673	−1.40422	−0.702111	+	0.712068i	$0.747759\pi$
$62$	0	0
$63$	1.57893i	0.198927i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	10.4591i	1.27778i	0.769296	+	0.638892i	$0.220607\pi$
$67$	10.4591i	1.27778i	−0.769296	+	0.638892i	$0.779393\pi$
$68$	0	0
$69$	1.39041	0.167386
$70$	0	0
$71$	−10.1261	−1.20175	−0.600874	−	0.799344i	$-0.705181\pi$
$71$	−10.1261	−1.20175	−0.600874	+	0.799344i	$0.705181\pi$
$72$	0	0
$73$	13.1900i	1.54377i	0.635760	+	0.771887i	$0.280687\pi$
$73$	13.1900i	1.54377i	−0.635760	+	0.771887i	$0.719313\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 6.12718i	− 0.698257i
$78$	0	0
$79$	−13.7995	−1.55256	−0.776282	−	0.630386i	$-0.782897\pi$
$79$	−13.7995	−1.55256	−0.776282	+	0.630386i	$0.782897\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 4.62458i	− 0.507613i	−0.967255	−	0.253807i	$-0.918317\pi$
$83$	− 4.62458i	− 0.507613i	0.967255	−	0.253807i	$-0.0816827\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 3.32990i	− 0.357003i
$88$	0	0
$89$	7.26904	0.770516	0.385258	−	0.922809i	$-0.374112\pi$
$89$	7.26904	0.770516	0.385258	+	0.922809i	$0.374112\pi$
$90$	0	0
$91$	−0.542010	−0.0568181
$92$	0	0
$93$	5.25496i	0.544914i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	6.05557i	0.614850i	0.951572	+	0.307425i	$0.0994674\pi$
$97$	6.05557i	0.614850i	−0.951572	+	0.307425i	$0.900533\pi$
$98$	0	0
$99$	−3.88059	−0.390014
$100$	0	0
$101$	3.23036	0.321432	0.160716	−	0.987001i	$-0.448620\pi$
$101$	3.23036	0.321432	0.160716	+	0.987001i	$0.448620\pi$
$102$	0	0
$103$	7.69725i	0.758433i	0.925308	+	0.379216i	$0.123806\pi$
$103$	7.69725i	0.758433i	−0.925308	+	0.379216i	$0.876194\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	18.0376i	1.74376i	0.489722	+	0.871878i	$0.337098\pi$
$107$	18.0376i	1.74376i	−0.489722	+	0.871878i	$0.662902\pi$
$108$	0	0
$109$	−16.7259	−1.60205	−0.801027	−	0.598629i	$-0.795712\pi$
$109$	−16.7259	−1.60205	−0.801027	+	0.598629i	$0.795712\pi$
$110$	0	0
$111$	8.56667	0.813113
$112$	0	0
$113$	− 2.26898i	− 0.213448i	−0.994289	−	0.106724i	$-0.965964\pi$
$113$	− 2.26898i	− 0.213448i	0.994289	−	0.106724i	$-0.0340361\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0.343277i	0.0317359i
$118$	0	0
$119$	9.58496	0.878652
$120$	0	0
$121$	4.05896	0.368996
$122$	0	0
$123$	1.27815i	0.115247i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 1.50725i	− 0.133747i	−0.997761	−	0.0668735i	$-0.978698\pi$
$127$	− 1.50725i	− 0.133747i	0.997761	−	0.0668735i	$-0.0213024\pi$
$128$	0	0
$129$	1.42438	0.125410
$130$	0	0
$131$	−15.9388	−1.39258	−0.696289	−	0.717761i	$-0.745167\pi$
$131$	−15.9388	−1.39258	−0.696289	+	0.717761i	$0.745167\pi$
$132$	0	0
$133$	5.82924i	0.505459i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 5.44654i	− 0.465329i	−0.972557	−	0.232665i	$-0.925256\pi$
$137$	− 5.44654i	− 0.465329i	0.972557	−	0.232665i	$-0.0747444\pi$
$138$	0	0
$139$	9.88325	0.838286	0.419143	−	0.907920i	$-0.362331\pi$
$139$	9.88325	0.838286	0.419143	+	0.907920i	$0.362331\pi$
$140$	0	0
$141$	0.375462	0.0316196
$142$	0	0
$143$	− 1.33211i	− 0.111397i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 4.50698i	− 0.371729i
$148$	0	0
$149$	−0.0649364	−0.00531979	−0.00265990	−	0.999996i	$-0.500847\pi$
$149$	−0.0649364	−0.00531979	−0.00265990	+	0.999996i	$0.500847\pi$
$150$	0	0
$151$	−12.1221	−0.986481	−0.493240	−	0.869893i	$-0.664188\pi$
$151$	−12.1221	−0.986481	−0.493240	+	0.869893i	$0.664188\pi$
$152$	0	0
$153$	− 6.07054i	− 0.490774i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 23.4721i	− 1.87328i	−0.350300	−	0.936638i	$-0.613920\pi$
$157$	− 23.4721i	− 1.87328i	0.350300	−	0.936638i	$-0.386080\pi$
$158$	0	0
$159$	−11.2992	−0.896086
$160$	0	0
$161$	2.19537	0.173019
$162$	0	0
$163$	− 5.99585i	− 0.469632i	−0.972040	−	0.234816i	$-0.924551\pi$
$163$	− 5.99585i	− 0.469632i	0.972040	−	0.234816i	$-0.0754487\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 6.39850i	− 0.495130i	−0.968871	−	0.247565i	$-0.920370\pi$
$167$	− 6.39850i	− 0.495130i	0.968871	−	0.247565i	$-0.0796304\pi$
$168$	0	0
$169$	12.8822	0.990935
$170$	0	0
$171$	3.69189	0.282326
$172$	0	0
$173$	5.22138i	0.396974i	0.980104	+	0.198487i	$0.0636028\pi$
$173$	5.22138i	0.396974i	−0.980104	+	0.198487i	$0.936397\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	11.5818i	0.870545i
$178$	0	0
$179$	−23.3040	−1.74183	−0.870913	−	0.491437i	$-0.836472\pi$
$179$	−23.3040	−1.74183	−0.870913	+	0.491437i	$0.836472\pi$
$180$	0	0
$181$	25.1739	1.87116	0.935582	−	0.353108i	$-0.114875\pi$
$181$	25.1739	1.87116	0.935582	+	0.353108i	$0.114875\pi$
$182$	0	0
$183$	10.9673i	0.810728i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	23.5573i	1.72268i
$188$	0	0
$189$	1.57893	0.114850
$190$	0	0
$191$	8.03152	0.581140	0.290570	−	0.956854i	$-0.406155\pi$
$191$	8.03152	0.581140	0.290570	+	0.956854i	$0.406155\pi$
$192$	0	0
$193$	− 20.2575i	− 1.45817i	−0.684424	−	0.729084i	$-0.739946\pi$
$193$	− 20.2575i	− 1.45817i	0.684424	−	0.729084i	$-0.260054\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	20.0158i	1.42607i	0.701130	+	0.713034i	$0.252679\pi$
$197$	20.0158i	1.42607i	−0.701130	+	0.713034i	$0.747321\pi$
$198$	0	0
$199$	22.4180	1.58917	0.794585	−	0.607153i	$-0.207689\pi$
$199$	22.4180	1.58917	0.794585	+	0.607153i	$0.207689\pi$
$200$	0	0
$201$	10.4591	0.737729
$202$	0	0
$203$	− 5.25769i	− 0.369017i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 1.39041i	− 0.0966404i
$208$	0	0
$209$	−14.3267	−0.990998
$210$	0	0
$211$	8.66554	0.596560	0.298280	−	0.954478i	$-0.403587\pi$
$211$	8.66554	0.596560	0.298280	+	0.954478i	$0.403587\pi$
$212$	0	0
$213$	10.1261i	0.693830i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	8.29722i	0.563252i
$218$	0	0
$219$	13.1900	0.891298
$220$	0	0
$221$	2.08387	0.140177
$222$	0	0
$223$	− 12.3840i	− 0.829293i	−0.909983	−	0.414647i	$-0.863905\pi$
$223$	− 12.3840i	− 0.829293i	0.909983	−	0.414647i	$-0.136095\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	18.2228i	1.20949i	0.796419	+	0.604745i	$0.206725\pi$
$227$	18.2228i	1.20949i	−0.796419	+	0.604745i	$0.793275\pi$
$228$	0	0
$229$	−16.4164	−1.08483	−0.542413	−	0.840112i	$-0.682489\pi$
$229$	−16.4164	−1.08483	−0.542413	+	0.840112i	$0.682489\pi$
$230$	0	0
$231$	−6.12718	−0.403139
$232$	0	0
$233$	− 5.13782i	− 0.336590i	−0.985737	−	0.168295i	$-0.946174\pi$
$233$	− 5.13782i	− 0.336590i	0.985737	−	0.168295i	$-0.0538260\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	13.7995i	0.896373i
$238$	0	0
$239$	−24.5593	−1.58861	−0.794304	−	0.607520i	$-0.792165\pi$
$239$	−24.5593	−1.58861	−0.794304	+	0.607520i	$0.792165\pi$
$240$	0	0
$241$	4.83511	0.311457	0.155728	−	0.987800i	$-0.450228\pi$
$241$	4.83511	0.311457	0.155728	+	0.987800i	$0.450228\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.26734i	0.0806388i
$248$	0	0
$249$	−4.62458	−0.293071
$250$	0	0
$251$	−15.1395	−0.955594	−0.477797	−	0.878470i	$-0.658565\pi$
$251$	−15.1395	−0.955594	−0.477797	+	0.878470i	$0.658565\pi$
$252$	0	0
$253$	5.39562i	0.339220i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	22.7976i	1.42207i	0.703155	+	0.711036i	$0.251774\pi$
$257$	22.7976i	1.42207i	−0.703155	+	0.711036i	$0.748226\pi$
$258$	0	0
$259$	13.5262	0.840476
$260$	0	0
$261$	−3.32990	−0.206116
$262$	0	0
$263$	9.44456i	0.582376i	0.956666	+	0.291188i	$0.0940506\pi$
$263$	9.44456i	0.582376i	−0.956666	+	0.291188i	$0.905949\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 7.26904i	− 0.444858i
$268$	0	0
$269$	29.3838	1.79156	0.895780	−	0.444497i	$-0.146618\pi$
$269$	29.3838	1.79156	0.895780	+	0.444497i	$0.146618\pi$
$270$	0	0
$271$	15.4883	0.940848	0.470424	−	0.882441i	$-0.344101\pi$
$271$	15.4883	0.940848	0.470424	+	0.882441i	$0.344101\pi$
$272$	0	0
$273$	0.542010i	0.0328039i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	15.3786i	0.924010i	0.886877	+	0.462005i	$0.152870\pi$
$277$	15.3786i	0.924010i	−0.886877	+	0.462005i	$0.847130\pi$
$278$	0	0
$279$	5.25496	0.314606
$280$	0	0
$281$	3.87277	0.231030	0.115515	−	0.993306i	$-0.463148\pi$
$281$	3.87277	0.231030	0.115515	+	0.993306i	$0.463148\pi$
$282$	0	0
$283$	11.8869i	0.706605i	0.935509	+	0.353303i	$0.114941\pi$
$283$	11.8869i	0.706605i	−0.935509	+	0.353303i	$0.885059\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.01811i	0.119125i
$288$	0	0
$289$	−19.8514	−1.16773
$290$	0	0
$291$	6.05557	0.354984
$292$	0	0
$293$	15.4596i	0.903161i	0.892230	+	0.451581i	$0.149140\pi$
$293$	15.4596i	0.903161i	−0.892230	+	0.451581i	$0.850860\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	3.88059i	0.225175i
$298$	0	0
$299$	0.477297	0.0276028
$300$	0	0
$301$	2.24900	0.129630
$302$	0	0
$303$	− 3.23036i	− 0.185579i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	26.6092i	1.51867i	0.650702	+	0.759334i	$0.274475\pi$
$307$	26.6092i	1.51867i	−0.650702	+	0.759334i	$0.725525\pi$
$308$	0	0
$309$	7.69725	0.437881
$310$	0	0
$311$	−5.48062	−0.310778	−0.155389	−	0.987853i	$-0.549663\pi$
$311$	−5.48062	−0.310778	−0.155389	+	0.987853i	$0.549663\pi$
$312$	0	0
$313$	− 7.45099i	− 0.421155i	−0.977577	−	0.210578i	$-0.932466\pi$
$313$	− 7.45099i	− 0.421155i	0.977577	−	0.210578i	$-0.0675344\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 2.96844i	− 0.166724i	−0.996519	−	0.0833622i	$-0.973434\pi$
$317$	− 2.96844i	− 0.166724i	0.996519	−	0.0833622i	$-0.0265658\pi$
$318$	0	0
$319$	12.9220	0.723492
$320$	0	0
$321$	18.0376	1.00676
$322$	0	0
$323$	− 22.4117i	− 1.24702i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	16.7259i	0.924946i
$328$	0	0
$329$	0.592828	0.0326837
$330$	0	0
$331$	27.4282	1.50759	0.753795	−	0.657109i	$-0.228221\pi$
$331$	27.4282	1.50759	0.753795	+	0.657109i	$0.228221\pi$
$332$	0	0
$333$	− 8.56667i	− 0.469451i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 7.51695i	− 0.409474i	−0.978817	−	0.204737i	$-0.934366\pi$
$337$	− 7.51695i	− 0.409474i	0.978817	−	0.204737i	$-0.0656340\pi$
$338$	0	0
$339$	−2.26898	−0.123234
$340$	0	0
$341$	−20.3923	−1.10431
$342$	0	0
$343$	− 18.1687i	− 0.981019i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	11.9946i	0.643902i	0.946756	+	0.321951i	$0.104339\pi$
$347$	11.9946i	0.643902i	−0.946756	+	0.321951i	$0.895661\pi$
$348$	0	0
$349$	3.50169	0.187441	0.0937207	−	0.995599i	$-0.470124\pi$
$349$	3.50169	0.187441	0.0937207	+	0.995599i	$0.470124\pi$
$350$	0	0
$351$	0.343277	0.0183227
$352$	0	0
$353$	24.9521i	1.32807i	0.747704	+	0.664033i	$0.231156\pi$
$353$	24.9521i	1.32807i	−0.747704	+	0.664033i	$0.768844\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 9.58496i	− 0.507290i
$358$	0	0
$359$	−0.802395	−0.0423488	−0.0211744	−	0.999776i	$-0.506741\pi$
$359$	−0.802395	−0.0423488	−0.0211744	+	0.999776i	$0.506741\pi$
$360$	0	0
$361$	−5.36997	−0.282630
$362$	0	0
$363$	− 4.05896i	− 0.213040i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	15.8049i	0.825007i	0.910956	+	0.412504i	$0.135346\pi$
$367$	15.8049i	0.825007i	−0.910956	+	0.412504i	$0.864654\pi$
$368$	0	0
$369$	1.27815	0.0665378
$370$	0	0
$371$	−17.8407	−0.926242
$372$	0	0
$373$	− 3.60958i	− 0.186897i	−0.995624	−	0.0934484i	$-0.970211\pi$
$373$	− 3.60958i	− 0.186897i	0.995624	−	0.0934484i	$-0.0297890\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 1.14308i	− 0.0588715i
$378$	0	0
$379$	−27.3130	−1.40297	−0.701486	−	0.712683i	$-0.747480\pi$
$379$	−27.3130	−1.40297	−0.701486	+	0.712683i	$0.747480\pi$
$380$	0	0
$381$	−1.50725	−0.0772189
$382$	0	0
$383$	19.2816i	0.985244i	0.870243	+	0.492622i	$0.163961\pi$
$383$	19.2816i	0.985244i	−0.870243	+	0.492622i	$0.836039\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 1.42438i	− 0.0724054i
$388$	0	0
$389$	17.0594	0.864946	0.432473	−	0.901647i	$-0.357641\pi$
$389$	17.0594	0.864946	0.432473	+	0.901647i	$0.357641\pi$
$390$	0	0
$391$	−8.44056	−0.426857
$392$	0	0
$393$	15.9388i	0.804006i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	27.3641i	1.37336i	0.726958	+	0.686682i	$0.240934\pi$
$397$	27.3641i	1.37336i	−0.726958	+	0.686682i	$0.759066\pi$
$398$	0	0
$399$	5.82924	0.291827
$400$	0	0
$401$	−14.7793	−0.738042	−0.369021	−	0.929421i	$-0.620307\pi$
$401$	−14.7793	−0.738042	−0.369021	+	0.929421i	$0.620307\pi$
$402$	0	0
$403$	1.80390i	0.0898589i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	33.2437i	1.64783i
$408$	0	0
$409$	−21.7633	−1.07613	−0.538063	−	0.842905i	$-0.680844\pi$
$409$	−21.7633	−1.07613	−0.538063	+	0.842905i	$0.680844\pi$
$410$	0	0
$411$	−5.44654	−0.268658
$412$	0	0
$413$	18.2869i	0.899842i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 9.88325i	− 0.483985i
$418$	0	0
$419$	4.53705	0.221649	0.110825	−	0.993840i	$-0.464651\pi$
$419$	4.53705	0.221649	0.110825	+	0.993840i	$0.464651\pi$
$420$	0	0
$421$	3.05548	0.148915	0.0744574	−	0.997224i	$-0.476278\pi$
$421$	3.05548	0.148915	0.0744574	+	0.997224i	$0.476278\pi$
$422$	0	0
$423$	− 0.375462i	− 0.0182556i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	17.3167i	0.838011i
$428$	0	0
$429$	−1.33211	−0.0643151
$430$	0	0
$431$	−18.8971	−0.910241	−0.455120	−	0.890430i	$-0.650404\pi$
$431$	−18.8971	−0.910241	−0.455120	+	0.890430i	$0.650404\pi$
$432$	0	0
$433$	3.41899i	0.164306i	0.996620	+	0.0821532i	$0.0261797\pi$
$433$	3.41899i	0.164306i	−0.996620	+	0.0821532i	$0.973820\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 5.13325i	− 0.245557i
$438$	0	0
$439$	5.95625	0.284276	0.142138	−	0.989847i	$-0.454602\pi$
$439$	5.95625	0.284276	0.142138	+	0.989847i	$0.454602\pi$
$440$	0	0
$441$	−4.50698	−0.214618
$442$	0	0
$443$	− 33.0705i	− 1.57122i	−0.618719	−	0.785612i	$-0.712348\pi$
$443$	− 33.0705i	− 1.57122i	0.618719	−	0.785612i	$-0.287652\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0.0649364i	0.00307138i
$448$	0	0
$449$	8.68077	0.409671	0.204835	−	0.978796i	$-0.434334\pi$
$449$	8.68077	0.409671	0.204835	+	0.978796i	$0.434334\pi$
$450$	0	0
$451$	−4.95997	−0.233556
$452$	0	0
$453$	12.1221i	0.569545i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	13.3667i	0.625269i	0.949873	+	0.312635i	$0.101212\pi$
$457$	13.3667i	0.625269i	−0.949873	+	0.312635i	$0.898788\pi$
$458$	0	0
$459$	−6.07054	−0.283348
$460$	0	0
$461$	−40.9583	−1.90762	−0.953809	−	0.300413i	$-0.902876\pi$
$461$	−40.9583	−1.90762	−0.953809	+	0.300413i	$0.902876\pi$
$462$	0	0
$463$	− 23.1280i	− 1.07485i	−0.843311	−	0.537425i	$-0.819397\pi$
$463$	− 23.1280i	− 1.07485i	0.843311	−	0.537425i	$-0.180603\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	30.2890i	1.40161i	0.713354	+	0.700804i	$0.247175\pi$
$467$	30.2890i	1.40161i	−0.713354	+	0.700804i	$0.752825\pi$
$468$	0	0
$469$	16.5142	0.762556
$470$	0	0
$471$	−23.4721	−1.08154
$472$	0	0
$473$	5.52744i	0.254152i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	11.2992i	0.517355i
$478$	0	0
$479$	24.0798	1.10023	0.550116	−	0.835088i	$-0.314583\pi$
$479$	24.0798	1.10023	0.550116	+	0.835088i	$0.314583\pi$
$480$	0	0
$481$	2.94074	0.134086
$482$	0	0
$483$	− 2.19537i	− 0.0998927i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	2.25482i	0.102176i	0.998694	+	0.0510878i	$0.0162688\pi$
$487$	2.25482i	0.102176i	−0.998694	+	0.0510878i	$0.983731\pi$
$488$	0	0
$489$	−5.99585	−0.271142
$490$	0	0
$491$	−28.6098	−1.29114	−0.645572	−	0.763699i	$-0.723381\pi$
$491$	−28.6098	−1.29114	−0.645572	+	0.763699i	$0.723381\pi$
$492$	0	0
$493$	20.2143i	0.910406i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	15.9884i	0.717179i
$498$	0	0
$499$	26.9489	1.20640	0.603199	−	0.797590i	$-0.293892\pi$
$499$	26.9489	1.20640	0.603199	+	0.797590i	$0.293892\pi$
$500$	0	0
$501$	−6.39850	−0.285864
$502$	0	0
$503$	− 0.416998i	− 0.0185930i	−0.999957	−	0.00929651i	$-0.997041\pi$
$503$	− 0.416998i	− 0.0185930i	0.999957	−	0.00929651i	$-0.00295921\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 12.8822i	− 0.572117i
$508$	0	0
$509$	−13.4030	−0.594079	−0.297039	−	0.954865i	$-0.595999\pi$
$509$	−13.4030	−0.594079	−0.297039	+	0.954865i	$0.595999\pi$
$510$	0	0
$511$	20.8261	0.921293
$512$	0	0
$513$	− 3.69189i	− 0.163001i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.45701i	0.0640793i
$518$	0	0
$519$	5.22138	0.229193
$520$	0	0
$521$	21.0744	0.923288	0.461644	−	0.887065i	$-0.347260\pi$
$521$	21.0744	0.923288	0.461644	+	0.887065i	$0.347260\pi$
$522$	0	0
$523$	− 11.8755i	− 0.519281i	−0.965705	−	0.259640i	$-0.916396\pi$
$523$	− 11.8755i	− 0.519281i	0.965705	−	0.259640i	$-0.0836041\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 31.9004i	− 1.38960i
$528$	0	0
$529$	21.0667	0.915946
$530$	0	0
$531$	11.5818	0.502609
$532$	0	0
$533$	0.438759i	0.0190047i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	23.3040i	1.00564i
$538$	0	0
$539$	17.4897	0.753335
$540$	0	0
$541$	36.5654	1.57207	0.786034	−	0.618183i	$-0.212131\pi$
$541$	36.5654	1.57207	0.786034	+	0.618183i	$0.212131\pi$
$542$	0	0
$543$	− 25.1739i	− 1.08032i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 39.5812i	− 1.69237i	−0.532890	−	0.846185i	$-0.678894\pi$
$547$	− 39.5812i	− 1.69237i	0.532890	−	0.846185i	$-0.321106\pi$
$548$	0	0
$549$	10.9673	0.468074
$550$	0	0
$551$	−12.2936	−0.523726
$552$	0	0
$553$	21.7885i	0.926539i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	12.1804i	0.516102i	0.966131	+	0.258051i	$0.0830802\pi$
$557$	12.1804i	0.516102i	−0.966131	+	0.258051i	$0.916920\pi$
$558$	0	0
$559$	0.488957	0.0206807
$560$	0	0
$561$	23.5573	0.994588
$562$	0	0
$563$	− 36.5886i	− 1.54203i	−0.636819	−	0.771014i	$-0.719750\pi$
$563$	− 36.5886i	− 1.54203i	0.636819	−	0.771014i	$-0.280250\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 1.57893i	− 0.0663089i
$568$	0	0
$569$	33.8355	1.41846	0.709230	−	0.704978i	$-0.249043\pi$
$569$	33.8355	1.41846	0.709230	+	0.704978i	$0.249043\pi$
$570$	0	0
$571$	−1.20423	−0.0503953	−0.0251976	−	0.999682i	$-0.508022\pi$
$571$	−1.20423	−0.0503953	−0.0251976	+	0.999682i	$0.508022\pi$
$572$	0	0
$573$	− 8.03152i	− 0.335522i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	16.4040i	0.682909i	0.939898	+	0.341455i	$0.110920\pi$
$577$	16.4040i	0.682909i	−0.939898	+	0.341455i	$0.889080\pi$
$578$	0	0
$579$	−20.2575	−0.841874
$580$	0	0
$581$	−7.30189	−0.302933
$582$	0	0
$583$	− 43.8476i	− 1.81598i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 33.6919i	− 1.39062i	−0.718712	−	0.695308i	$-0.755268\pi$
$587$	− 33.6919i	− 1.39062i	0.718712	−	0.695308i	$-0.244732\pi$
$588$	0	0
$589$	19.4007	0.799392
$590$	0	0
$591$	20.0158	0.823340
$592$	0	0
$593$	32.2208i	1.32315i	0.749878	+	0.661576i	$0.230112\pi$
$593$	32.2208i	1.32315i	−0.749878	+	0.661576i	$0.769888\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 22.4180i	− 0.917507i
$598$	0	0
$599$	14.7284	0.601784	0.300892	−	0.953658i	$-0.402716\pi$
$599$	14.7284	0.601784	0.300892	+	0.953658i	$0.402716\pi$
$600$	0	0
$601$	35.5643	1.45070	0.725348	−	0.688382i	$-0.241679\pi$
$601$	35.5643	1.45070	0.725348	+	0.688382i	$0.241679\pi$
$602$	0	0
$603$	− 10.4591i	− 0.425928i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 16.0986i	− 0.653421i	−0.945124	−	0.326710i	$-0.894060\pi$
$607$	− 16.0986i	− 0.653421i	0.945124	−	0.326710i	$-0.105940\pi$
$608$	0	0
$609$	−5.25769	−0.213052
$610$	0	0
$611$	0.128887	0.00521422
$612$	0	0
$613$	43.2663i	1.74751i	0.486368	+	0.873754i	$0.338321\pi$
$613$	43.2663i	1.74751i	−0.486368	+	0.873754i	$0.661679\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.58485i	0.0638036i	0.999491	+	0.0319018i	$0.0101564\pi$
$617$	1.58485i	0.0638036i	−0.999491	+	0.0319018i	$0.989844\pi$
$618$	0	0
$619$	13.4791	0.541770	0.270885	−	0.962612i	$-0.412684\pi$
$619$	13.4791	0.541770	0.270885	+	0.962612i	$0.412684\pi$
$620$	0	0
$621$	−1.39041	−0.0557954
$622$	0	0
$623$	− 11.4773i	− 0.459829i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	14.3267i	0.572153i
$628$	0	0
$629$	−52.0043	−2.07355
$630$	0	0
$631$	−29.2879	−1.16593	−0.582967	−	0.812496i	$-0.698108\pi$
$631$	−29.2879	−1.16593	−0.582967	+	0.812496i	$0.698108\pi$
$632$	0	0
$633$	− 8.66554i	− 0.344424i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 1.54714i	− 0.0612999i
$638$	0	0
$639$	10.1261	0.400583
$640$	0	0
$641$	−17.2911	−0.682958	−0.341479	−	0.939889i	$-0.610928\pi$
$641$	−17.2911	−0.682958	−0.341479	+	0.939889i	$0.610928\pi$
$642$	0	0
$643$	46.8857i	1.84899i	0.381190	+	0.924497i	$0.375514\pi$
$643$	46.8857i	1.84899i	−0.381190	+	0.924497i	$0.624486\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	14.3970i	0.566005i	0.959119	+	0.283003i	$0.0913305\pi$
$647$	14.3970i	0.566005i	−0.959119	+	0.283003i	$0.908669\pi$
$648$	0	0
$649$	−44.9444	−1.76422
$650$	0	0
$651$	8.29722	0.325194
$652$	0	0
$653$	− 31.9828i	− 1.25158i	−0.779991	−	0.625791i	$-0.784776\pi$
$653$	− 31.9828i	− 1.25158i	0.779991	−	0.625791i	$-0.215224\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 13.1900i	− 0.514591i
$658$	0	0
$659$	−18.6889	−0.728014	−0.364007	−	0.931396i	$-0.618592\pi$
$659$	−18.6889	−0.728014	−0.364007	+	0.931396i	$0.618592\pi$
$660$	0	0
$661$	−0.679140	−0.0264155	−0.0132078	−	0.999913i	$-0.504204\pi$
$661$	−0.679140	−0.0264155	−0.0132078	+	0.999913i	$0.504204\pi$
$662$	0	0
$663$	− 2.08387i	− 0.0809309i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4.62994i	0.179272i
$668$	0	0
$669$	−12.3840	−0.478793
$670$	0	0
$671$	−42.5597	−1.64300
$672$	0	0
$673$	7.25680i	0.279729i	0.990171	+	0.139864i	$0.0446667\pi$
$673$	7.25680i	0.279729i	−0.990171	+	0.139864i	$0.955333\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 33.9536i	− 1.30494i	−0.757814	−	0.652471i	$-0.773733\pi$
$677$	− 33.9536i	− 1.30494i	0.757814	−	0.652471i	$-0.226267\pi$
$678$	0	0
$679$	9.56133	0.366930
$680$	0	0
$681$	18.2228	0.698299
$682$	0	0
$683$	− 28.1579i	− 1.07743i	−0.842488	−	0.538716i	$-0.818910\pi$
$683$	− 28.1579i	− 1.07743i	0.842488	−	0.538716i	$-0.181090\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	16.4164i	0.626325i
$688$	0	0
$689$	−3.87876	−0.147769
$690$	0	0
$691$	22.1805	0.843787	0.421893	−	0.906645i	$-0.361366\pi$
$691$	22.1805	0.843787	0.421893	+	0.906645i	$0.361366\pi$
$692$	0	0
$693$	6.12718i	0.232752i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 7.75905i	− 0.293895i
$698$	0	0
$699$	−5.13782	−0.194330
$700$	0	0
$701$	16.9652	0.640768	0.320384	−	0.947288i	$-0.396188\pi$
$701$	16.9652	0.640768	0.320384	+	0.947288i	$0.396188\pi$
$702$	0	0
$703$	− 31.6272i	− 1.19284i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 5.10051i	− 0.191824i
$708$	0	0
$709$	−14.3275	−0.538081	−0.269041	−	0.963129i	$-0.586707\pi$
$709$	−14.3275	−0.538081	−0.269041	+	0.963129i	$0.586707\pi$
$710$	0	0
$711$	13.7995	0.517521
$712$	0	0
$713$	− 7.30657i	− 0.273633i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	24.5593i	0.917184i
$718$	0	0
$719$	−11.8215	−0.440869	−0.220434	−	0.975402i	$-0.570747\pi$
$719$	−11.8215	−0.440869	−0.220434	+	0.975402i	$0.570747\pi$
$720$	0	0
$721$	12.1534	0.452618
$722$	0	0
$723$	− 4.83511i	− 0.179820i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.23634i	0.0458534i	0.999737	+	0.0229267i	$0.00729843\pi$
$727$	1.23634i	0.0458534i	−0.999737	+	0.0229267i	$0.992702\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−8.64677	−0.319812
$732$	0	0
$733$	16.5080i	0.609739i	0.952394	+	0.304869i	$0.0986128\pi$
$733$	16.5080i	0.609739i	−0.952394	+	0.304869i	$0.901387\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	40.5875i	1.49506i
$738$	0	0
$739$	16.8461	0.619693	0.309847	−	0.950787i	$-0.399722\pi$
$739$	16.8461	0.619693	0.309847	+	0.950787i	$0.399722\pi$
$740$	0	0
$741$	1.26734	0.0465568
$742$	0	0
$743$	11.8940i	0.436347i	0.975910	+	0.218174i	$0.0700099\pi$
$743$	11.8940i	0.436347i	−0.975910	+	0.218174i	$0.929990\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	4.62458i	0.169204i
$748$	0	0
$749$	28.4801	1.04064
$750$	0	0
$751$	0.821377	0.0299725	0.0149862	−	0.999888i	$-0.495230\pi$
$751$	0.821377	0.0299725	0.0149862	+	0.999888i	$0.495230\pi$
$752$	0	0
$753$	15.1395i	0.551713i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 32.5591i	− 1.18338i	−0.806166	−	0.591690i	$-0.798461\pi$
$757$	− 32.5591i	− 1.18338i	0.806166	−	0.591690i	$-0.201539\pi$
$758$	0	0
$759$	5.39562	0.195849
$760$	0	0
$761$	−10.8675	−0.393946	−0.196973	−	0.980409i	$-0.563111\pi$
$761$	−10.8675	−0.393946	−0.196973	+	0.980409i	$0.563111\pi$
$762$	0	0
$763$	26.4091i	0.956073i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.97578i	0.143557i
$768$	0	0
$769$	2.38400	0.0859692	0.0429846	−	0.999076i	$-0.486313\pi$
$769$	2.38400	0.0859692	0.0429846	+	0.999076i	$0.486313\pi$
$770$	0	0
$771$	22.7976	0.821034
$772$	0	0
$773$	37.2341i	1.33922i	0.742714	+	0.669609i	$0.233538\pi$
$773$	37.2341i	1.33922i	−0.742714	+	0.669609i	$0.766462\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 13.5262i	− 0.485249i
$778$	0	0
$779$	4.71878	0.169068
$780$	0	0
$781$	−39.2953	−1.40609
$782$	0	0
$783$	3.32990i	0.119001i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	22.8974i	0.816206i	0.912936	+	0.408103i	$0.133809\pi$
$787$	22.8974i	0.816206i	−0.912936	+	0.408103i	$0.866191\pi$
$788$	0	0
$789$	9.44456	0.336235
$790$	0	0
$791$	−3.58257	−0.127382
$792$	0	0
$793$	3.76483i	0.133693i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 8.87469i	− 0.314358i	−0.987570	−	0.157179i	$-0.949760\pi$
$797$	− 8.87469i	− 0.314358i	0.987570	−	0.157179i	$-0.0502399\pi$
$798$	0	0
$799$	−2.27925	−0.0806342
$800$	0	0
$801$	−7.26904	−0.256839
$802$	0	0
$803$	51.1850i	1.80628i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 29.3838i	− 1.03436i
$808$	0	0
$809$	40.7380	1.43227	0.716136	−	0.697961i	$-0.245909\pi$
$809$	40.7380	1.43227	0.716136	+	0.697961i	$0.245909\pi$
$810$	0	0
$811$	44.7586	1.57169	0.785844	−	0.618424i	$-0.212229\pi$
$811$	44.7586	1.57169	0.785844	+	0.618424i	$0.212229\pi$
$812$	0	0
$813$	− 15.4883i	− 0.543199i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 5.25866i	− 0.183977i
$818$	0	0
$819$	0.542010	0.0189394
$820$	0	0
$821$	−49.9466	−1.74315	−0.871574	−	0.490264i	$-0.836901\pi$
$821$	−49.9466	−1.74315	−0.871574	+	0.490264i	$0.836901\pi$
$822$	0	0
$823$	15.9822i	0.557104i	0.960421	+	0.278552i	$0.0898544\pi$
$823$	15.9822i	0.557104i	−0.960421	+	0.278552i	$0.910146\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 45.1514i	− 1.57007i	−0.619453	−	0.785034i	$-0.712646\pi$
$827$	− 45.1514i	− 1.57007i	0.619453	−	0.785034i	$-0.287354\pi$
$828$	0	0
$829$	15.4474	0.536510	0.268255	−	0.963348i	$-0.413553\pi$
$829$	15.4474	0.536510	0.268255	+	0.963348i	$0.413553\pi$
$830$	0	0
$831$	15.3786	0.533477
$832$	0	0
$833$	27.3598i	0.947960i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 5.25496i	− 0.181638i
$838$	0	0
$839$	23.1609	0.799603	0.399802	−	0.916602i	$-0.369079\pi$
$839$	23.1609	0.799603	0.399802	+	0.916602i	$0.369079\pi$
$840$	0	0
$841$	−17.9118	−0.617647
$842$	0	0
$843$	− 3.87277i	− 0.133385i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 6.40882i	− 0.220210i
$848$	0	0
$849$	11.8869	0.407959
$850$	0	0
$851$	−11.9112	−0.408311
$852$	0	0
$853$	0.817929i	0.0280054i	0.999902	+	0.0140027i	$0.00445734\pi$
$853$	0.817929i	0.0280054i	−0.999902	+	0.0140027i	$0.995543\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	38.5882i	1.31815i	0.752078	+	0.659074i	$0.229052\pi$
$857$	38.5882i	1.31815i	−0.752078	+	0.659074i	$0.770948\pi$
$858$	0	0
$859$	−40.3774	−1.37766	−0.688829	−	0.724924i	$-0.741875\pi$
$859$	−40.3774	−1.37766	−0.688829	+	0.724924i	$0.741875\pi$
$860$	0	0
$861$	2.01811	0.0687770
$862$	0	0
$863$	26.8839i	0.915139i	0.889174	+	0.457569i	$0.151280\pi$
$863$	26.8839i	0.915139i	−0.889174	+	0.457569i	$0.848720\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	19.8514i	0.674190i
$868$	0	0
$869$	−53.5501	−1.81656
$870$	0	0
$871$	3.59037	0.121655
$872$	0	0
$873$	− 6.05557i	− 0.204950i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	22.6946i	0.766344i	0.923677	+	0.383172i	$0.125168\pi$
$877$	22.6946i	0.766344i	−0.923677	+	0.383172i	$0.874832\pi$
$878$	0	0
$879$	15.4596	0.521440
$880$	0	0
$881$	−30.0485	−1.01236	−0.506180	−	0.862428i	$-0.668943\pi$
$881$	−30.0485	−1.01236	−0.506180	+	0.862428i	$0.668943\pi$
$882$	0	0
$883$	− 34.8902i	− 1.17415i	−0.809533	−	0.587074i	$-0.800280\pi$
$883$	− 34.8902i	− 1.17415i	0.809533	−	0.587074i	$-0.199720\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 38.8006i	− 1.30280i	−0.758736	−	0.651398i	$-0.774183\pi$
$887$	− 38.8006i	− 1.30280i	0.758736	−	0.651398i	$-0.225817\pi$
$888$	0	0
$889$	−2.37985	−0.0798175
$890$	0	0
$891$	3.88059	0.130005
$892$	0	0
$893$	− 1.38616i	− 0.0463861i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 0.477297i	− 0.0159365i
$898$	0	0
$899$	−17.4985	−0.583608
$900$	0	0
$901$	68.5923	2.28514
$902$	0	0
$903$	− 2.24900i	− 0.0748421i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 10.0886i	− 0.334988i	−0.985873	−	0.167494i	$-0.946433\pi$
$907$	− 10.0886i	− 0.334988i	0.985873	−	0.167494i	$-0.0535675\pi$
$908$	0	0
$909$	−3.23036	−0.107144
$910$	0	0
$911$	−28.7150	−0.951372	−0.475686	−	0.879615i	$-0.657800\pi$
$911$	−28.7150	−0.951372	−0.475686	+	0.879615i	$0.657800\pi$
$912$	0	0
$913$	− 17.9461i	− 0.593928i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	25.1663i	0.831063i
$918$	0	0
$919$	−1.47549	−0.0486719	−0.0243360	−	0.999704i	$-0.507747\pi$
$919$	−1.47549	−0.0486719	−0.0243360	+	0.999704i	$0.507747\pi$
$920$	0	0
$921$	26.6092	0.876803
$922$	0	0
$923$	3.47606i	0.114416i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 7.69725i	− 0.252811i
$928$	0	0
$929$	40.8836	1.34135	0.670674	−	0.741752i	$-0.266005\pi$
$929$	40.8836	1.34135	0.670674	+	0.741752i	$0.266005\pi$
$930$	0	0
$931$	−16.6392	−0.545329
$932$	0	0
$933$	5.48062i	0.179428i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	32.4829i	1.06117i	0.847632	+	0.530584i	$0.178028\pi$
$937$	32.4829i	1.06117i	−0.847632	+	0.530584i	$0.821972\pi$
$938$	0	0
$939$	−7.45099	−0.243154
$940$	0	0
$941$	−7.64940	−0.249363	−0.124682	−	0.992197i	$-0.539791\pi$
$941$	−7.64940	−0.249363	−0.124682	+	0.992197i	$0.539791\pi$
$942$	0	0
$943$	− 1.77716i	− 0.0578722i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	52.1075i	1.69326i	0.532178	+	0.846632i	$0.321374\pi$
$947$	52.1075i	1.69326i	−0.532178	+	0.846632i	$0.678626\pi$
$948$	0	0
$949$	4.52782	0.146979
$950$	0	0
$951$	−2.96844	−0.0962584
$952$	0	0
$953$	− 4.20767i	− 0.136300i	−0.997675	−	0.0681500i	$-0.978290\pi$
$953$	− 4.20767i	− 0.136300i	0.997675	−	0.0681500i	$-0.0217096\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 12.9220i	− 0.417708i
$958$	0	0
$959$	−8.59971	−0.277699
$960$	0	0
$961$	−3.38541	−0.109207
$962$	0	0
$963$	− 18.0376i	− 0.581252i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 50.5324i	− 1.62501i	−0.582952	−	0.812506i	$-0.698103\pi$
$967$	− 50.5324i	− 1.62501i	0.582952	−	0.812506i	$-0.301897\pi$
$968$	0	0
$969$	−22.4117	−0.719969
$970$	0	0
$971$	15.8566	0.508863	0.254432	−	0.967091i	$-0.418112\pi$
$971$	15.8566	0.508863	0.254432	+	0.967091i	$0.418112\pi$
$972$	0	0
$973$	− 15.6050i	− 0.500272i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 43.3101i	− 1.38561i	−0.721124	−	0.692806i	$-0.756374\pi$
$977$	− 43.3101i	− 1.38561i	0.721124	−	0.692806i	$-0.243626\pi$
$978$	0	0
$979$	28.2081	0.901536
$980$	0	0
$981$	16.7259	0.534018
$982$	0	0
$983$	− 34.9708i	− 1.11540i	−0.830044	−	0.557698i	$-0.811685\pi$
$983$	− 34.9708i	− 1.11540i	0.830044	−	0.557698i	$-0.188315\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 0.592828i	− 0.0188699i
$988$	0	0
$989$	−1.98048	−0.0629757
$990$	0	0
$991$	−15.4006	−0.489216	−0.244608	−	0.969622i	$-0.578659\pi$
$991$	−15.4006	−0.489216	−0.244608	+	0.969622i	$0.578659\pi$
$992$	0	0
$993$	− 27.4282i	− 0.870408i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	30.7935i	0.975239i	0.873056	+	0.487620i	$0.162135\pi$
$997$	30.7935i	0.975239i	−0.873056	+	0.487620i	$0.837865\pi$
$998$	0	0
$999$	−8.56667	−0.271038

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7500.2.d.g.1249.3		24
5.2	odd	4		7500.2.a.m.1.10		12
5.3	odd	4		7500.2.a.n.1.3		12
5.4	even	2	inner	7500.2.d.g.1249.22		24
25.2	odd	20		1500.2.m.d.601.5		24
25.9	even	10		300.2.o.a.169.1	✓	24
25.11	even	5		300.2.o.a.229.1	yes	24
25.12	odd	20		1500.2.m.d.901.5		24
25.13	odd	20		1500.2.m.c.901.2		24
25.14	even	10		1500.2.o.c.649.6		24
25.16	even	5		1500.2.o.c.349.6		24
25.23	odd	20		1500.2.m.c.601.2		24
75.11	odd	10		900.2.w.c.829.6		24
75.59	odd	10		900.2.w.c.469.6		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.2.o.a.169.1	✓	24	25.9	even	10
300.2.o.a.229.1	yes	24	25.11	even	5
900.2.w.c.469.6		24	75.59	odd	10
900.2.w.c.829.6		24	75.11	odd	10
1500.2.m.c.601.2		24	25.23	odd	20
1500.2.m.c.901.2		24	25.13	odd	20
1500.2.m.d.601.5		24	25.2	odd	20
1500.2.m.d.901.5		24	25.12	odd	20
1500.2.o.c.349.6		24	25.16	even	5
1500.2.o.c.649.6		24	25.14	even	10
7500.2.a.m.1.10		12	5.2	odd	4
7500.2.a.n.1.3		12	5.3	odd	4
7500.2.d.g.1249.3		24	1.1	even	1	trivial
7500.2.d.g.1249.22		24	5.4	even	2	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:	\( N \)	\(=\)	\( 7500 = 2^{2} \cdot 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7500.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -1.57893i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} -1.57893i q^{7} -1.00000 q^{9} +3.88059 q^{11} -0.343277i q^{13} +6.07054i q^{17} -3.69189 q^{19} -1.57893 q^{21} +1.39041i q^{23} +1.00000i q^{27} +3.32990 q^{29} -5.25496 q^{31} -3.88059i q^{33} +8.56667i q^{37} -0.343277 q^{39} -1.27815 q^{41} +1.42438i q^{43} +0.375462i q^{47} +4.50698 q^{49} +6.07054 q^{51} -11.2992i q^{53} +3.69189i q^{57} -11.5818 q^{59} -10.9673 q^{61} +1.57893i q^{63} +10.4591i q^{67} +1.39041 q^{69} -10.1261 q^{71} +13.1900i q^{73} -6.12718i q^{77} -13.7995 q^{79} +1.00000 q^{81} -4.62458i q^{83} -3.32990i q^{87} +7.26904 q^{89} -0.542010 q^{91} +5.25496i q^{93} +6.05557i q^{97} -3.88059 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{9}+O(q^{10}) \) 24 * q - 24 * q^9	\( 24 q - 24 q^{9} + 4 q^{11} - 20 q^{19} + 16 q^{21} - 16 q^{29} - 4 q^{31} + 20 q^{41} - 56 q^{49} + 16 q^{51} + 4 q^{59} + 68 q^{61} - 36 q^{69} - 12 q^{79} + 24 q^{81} - 20 q^{89} + 40 q^{91} - 4 q^{99}+O(q^{100}) \) 24 * q - 24 * q^9 + 4 * q^11 - 20 * q^19 + 16 * q^21 - 16 * q^29 - 4 * q^31 + 20 * q^41 - 56 * q^49 + 16 * q^51 + 4 * q^59 + 68 * q^61 - 36 * q^69 - 12 * q^79 + 24 * q^81 - 20 * q^89 + 40 * q^91 - 4 * q^99

Introduction

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