LMFDB - Embedded newform 2925.2.c.b.2224.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2925,2,Mod(2224,2925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2925, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2925.2224");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2925 = 3^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2925.c (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:	$23.3562425912$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 325)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2224.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	2925.2224
Dual form			2925.2.c.b.2224.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{2} -2.00000 q^{4} -2.00000i q^{7} +O(q^{10})$	$q+2.00000i q^{2} -2.00000 q^{4} -2.00000i q^{7} -2.00000 q^{11} -1.00000i q^{13} +4.00000 q^{14} -4.00000 q^{16} +2.00000i q^{17} -4.00000i q^{22} +9.00000i q^{23} +2.00000 q^{26} +4.00000i q^{28} +5.00000 q^{29} +2.00000 q^{31} -8.00000i q^{32} -4.00000 q^{34} +8.00000i q^{37} -12.0000 q^{41} +1.00000i q^{43} +4.00000 q^{44} -18.0000 q^{46} -8.00000i q^{47} +3.00000 q^{49} +2.00000i q^{52} -11.0000i q^{53} +10.0000i q^{58} -13.0000 q^{61} +4.00000i q^{62} +8.00000 q^{64} -2.00000i q^{67} -4.00000i q^{68} -12.0000 q^{71} +6.00000i q^{73} -16.0000 q^{74} +4.00000i q^{77} -15.0000 q^{79} -24.0000i q^{82} +4.00000i q^{83} -2.00000 q^{86} -10.0000 q^{89} -2.00000 q^{91} -18.0000i q^{92} +16.0000 q^{94} +8.00000i q^{97} +6.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4}+O(q^{10}) $ 2 * q - 4 * q^4	$ 2 q - 4 q^{4} - 4 q^{11} + 8 q^{14} - 8 q^{16} + 4 q^{26} + 10 q^{29} + 4 q^{31} - 8 q^{34} - 24 q^{41} + 8 q^{44} - 36 q^{46} + 6 q^{49} - 26 q^{61} + 16 q^{64} - 24 q^{71} - 32 q^{74} - 30 q^{79} - 4 q^{86} - 20 q^{89} - 4 q^{91} + 32 q^{94}+O(q^{100}) $ 2 * q - 4 * q^4 - 4 * q^11 + 8 * q^14 - 8 * q^16 + 4 * q^26 + 10 * q^29 + 4 * q^31 - 8 * q^34 - 24 * q^41 + 8 * q^44 - 36 * q^46 + 6 * q^49 - 26 * q^61 + 16 * q^64 - 24 * q^71 - 32 * q^74 - 30 * q^79 - 4 * q^86 - 20 * q^89 - 4 * q^91 + 32 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2925\mathbb{Z}\right)^\times$.

$n$	$326$	$352$	$2251$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	2.00000i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$2$	2.00000i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$3$	0	0
$3$	0	0
$4$	−2.00000	−1.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	4.00000	1.06904
$15$	0	0
$16$	−4.00000	−1.00000
$17$	2.00000i	0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$	2.00000i	0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	− 4.00000i	− 0.852803i
$23$	9.00000i	1.87663i	0.345782	+	0.938315i	$0.387614\pi$
$23$	9.00000i	1.87663i	−0.345782	+	0.938315i	$0.612386\pi$
$24$	0	0
$25$	0	0
$26$	2.00000	0.392232
$27$	0	0
$28$	4.00000i	0.755929i
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	− 8.00000i	− 1.41421i
$33$	0	0
$34$	−4.00000	−0.685994
$35$	0	0
$36$	0	0
$37$	8.00000i	1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$	8.00000i	1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−12.0000	−1.87409	−0.937043	−	0.349215i	$-0.886448\pi$
$41$	−12.0000	−1.87409	−0.937043	+	0.349215i	$0.886448\pi$
$42$	0	0
$43$	1.00000i	0.152499i	0.997089	+	0.0762493i	$0.0242945\pi$
$43$	1.00000i	0.152499i	−0.997089	+	0.0762493i	$0.975706\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	−18.0000	−2.65396
$47$	− 8.00000i	− 1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$	− 8.00000i	− 1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	0	0
$52$	2.00000i	0.277350i
$53$	− 11.0000i	− 1.51097i	−0.655168	−	0.755483i	$-0.727402\pi$
$53$	− 11.0000i	− 1.51097i	0.655168	−	0.755483i	$-0.272598\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	10.0000i	1.31306i
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−13.0000	−1.66448	−0.832240	−	0.554416i	$-0.812942\pi$
$61$	−13.0000	−1.66448	−0.832240	+	0.554416i	$0.812942\pi$
$62$	4.00000i	0.508001i
$63$	0	0
$64$	8.00000	1.00000
$65$	0	0
$66$	0	0
$67$	− 2.00000i	− 0.244339i	−0.992509	−	0.122169i	$-0.961015\pi$
$67$	− 2.00000i	− 0.244339i	0.992509	−	0.122169i	$-0.0389851\pi$
$68$	− 4.00000i	− 0.485071i
$69$	0	0
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	6.00000i	0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$	6.00000i	0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$	−16.0000	−1.85996
$75$	0	0
$76$	0	0
$77$	4.00000i	0.455842i
$78$	0	0
$79$	−15.0000	−1.68763	−0.843816	−	0.536633i	$-0.819696\pi$
$79$	−15.0000	−1.68763	−0.843816	+	0.536633i	$0.819696\pi$
$80$	0	0
$81$	0	0
$82$	− 24.0000i	− 2.65036i
$83$	4.00000i	0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$	4.00000i	0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$	0	0
$85$	0	0
$86$	−2.00000	−0.215666
$87$	0	0
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	− 18.0000i	− 1.87663i
$93$	0	0
$94$	16.0000	1.65027
$95$	0	0
$96$	0	0
$97$	8.00000i	0.812277i	0.913812	+	0.406138i	$0.133125\pi$
$97$	8.00000i	0.812277i	−0.913812	+	0.406138i	$0.866875\pi$
$98$	6.00000i	0.606092i
$99$	0	0
$100$	0	0
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	0	0
$103$	11.0000i	1.08386i	0.840423	+	0.541931i	$0.182307\pi$
$103$	11.0000i	1.08386i	−0.840423	+	0.541931i	$0.817693\pi$
$104$	0	0
$105$	0	0
$106$	22.0000	2.13683
$107$	7.00000i	0.676716i	0.941018	+	0.338358i	$0.109871\pi$
$107$	7.00000i	0.676716i	−0.941018	+	0.338358i	$0.890129\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	0	0
$112$	8.00000i	0.755929i
$113$	− 1.00000i	− 0.0940721i	−0.998893	−	0.0470360i	$-0.985022\pi$
$113$	− 1.00000i	− 0.0940721i	0.998893	−	0.0470360i	$-0.0149776\pi$
$114$	0	0
$115$	0	0
$116$	−10.0000	−0.928477
$117$	0	0
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	−7.00000	−0.636364
$122$	− 26.0000i	− 2.35393i
$123$	0	0
$124$	−4.00000	−0.359211
$125$	0	0
$126$	0	0
$127$	− 7.00000i	− 0.621150i	−0.950549	−	0.310575i	$-0.899478\pi$
$127$	− 7.00000i	− 0.621150i	0.950549	−	0.310575i	$-0.100522\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−17.0000	−1.48530	−0.742648	−	0.669681i	$-0.766431\pi$
$131$	−17.0000	−1.48530	−0.742648	+	0.669681i	$0.766431\pi$
$132$	0	0
$133$	0	0
$134$	4.00000	0.345547
$135$	0	0
$136$	0	0
$137$	− 8.00000i	− 0.683486i	−0.939793	−	0.341743i	$-0.888983\pi$
$137$	− 8.00000i	− 0.683486i	0.939793	−	0.341743i	$-0.111017\pi$
$138$	0	0
$139$	−15.0000	−1.27228	−0.636142	−	0.771572i	$-0.719471\pi$
$139$	−15.0000	−1.27228	−0.636142	+	0.771572i	$0.719471\pi$
$140$	0	0
$141$	0	0
$142$	− 24.0000i	− 2.01404i
$143$	2.00000i	0.167248i
$144$	0	0
$145$	0	0
$146$	−12.0000	−0.993127
$147$	0	0
$148$	− 16.0000i	− 1.31519i
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	0	0
$154$	−8.00000	−0.644658
$155$	0	0
$156$	0	0
$157$	18.0000i	1.43656i	0.695756	+	0.718278i	$0.255069\pi$
$157$	18.0000i	1.43656i	−0.695756	+	0.718278i	$0.744931\pi$
$158$	− 30.0000i	− 2.38667i
$159$	0	0
$160$	0	0
$161$	18.0000	1.41860
$162$	0	0
$163$	− 24.0000i	− 1.87983i	−0.341415	−	0.939913i	$-0.610906\pi$
$163$	− 24.0000i	− 1.87983i	0.341415	−	0.939913i	$-0.389094\pi$
$164$	24.0000	1.87409
$165$	0	0
$166$	−8.00000	−0.620920
$167$	12.0000i	0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$	12.0000i	0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	− 2.00000i	− 0.152499i
$173$	− 6.00000i	− 0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$	− 6.00000i	− 0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$	0	0
$175$	0	0
$176$	8.00000	0.603023
$177$	0	0
$178$	− 20.0000i	− 1.49906i
$179$	15.0000	1.12115	0.560576	−	0.828103i	$-0.310580\pi$
$179$	15.0000	1.12115	0.560576	+	0.828103i	$0.310580\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	− 4.00000i	− 0.296500i
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 4.00000i	− 0.292509i
$188$	16.0000i	1.16692i
$189$	0	0
$190$	0	0
$191$	3.00000	0.217072	0.108536	−	0.994092i	$-0.465384\pi$
$191$	3.00000	0.217072	0.108536	+	0.994092i	$0.465384\pi$
$192$	0	0
$193$	16.0000i	1.15171i	0.817554	+	0.575853i	$0.195330\pi$
$193$	16.0000i	1.15171i	−0.817554	+	0.575853i	$0.804670\pi$
$194$	−16.0000	−1.14873
$195$	0	0
$196$	−6.00000	−0.428571
$197$	12.0000i	0.854965i	0.904024	+	0.427482i	$0.140599\pi$
$197$	12.0000i	0.854965i	−0.904024	+	0.427482i	$0.859401\pi$
$198$	0	0
$199$	25.0000	1.77220	0.886102	−	0.463491i	$-0.153403\pi$
$199$	25.0000	1.77220	0.886102	+	0.463491i	$0.153403\pi$
$200$	0	0
$201$	0	0
$202$	6.00000i	0.422159i
$203$	− 10.0000i	− 0.701862i
$204$	0	0
$205$	0	0
$206$	−22.0000	−1.53281
$207$	0	0
$208$	4.00000i	0.277350i
$209$	0	0
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	22.0000i	1.51097i
$213$	0	0
$214$	−14.0000	−0.957020
$215$	0	0
$216$	0	0
$217$	− 4.00000i	− 0.271538i
$218$	− 20.0000i	− 1.35457i
$219$	0	0
$220$	0	0
$221$	2.00000	0.134535
$222$	0	0
$223$	− 14.0000i	− 0.937509i	−0.883328	−	0.468755i	$-0.844703\pi$
$223$	− 14.0000i	− 0.937509i	0.883328	−	0.468755i	$-0.155297\pi$
$224$	−16.0000	−1.06904
$225$	0	0
$226$	2.00000	0.133038
$227$	− 8.00000i	− 0.530979i	−0.964114	−	0.265489i	$-0.914466\pi$
$227$	− 8.00000i	− 0.530979i	0.964114	−	0.265489i	$-0.0855335\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 1.00000i	− 0.0655122i	−0.999463	−	0.0327561i	$-0.989572\pi$
$233$	− 1.00000i	− 0.0655122i	0.999463	−	0.0327561i	$-0.0104285\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	8.00000i	0.518563i
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	22.0000	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$242$	− 14.0000i	− 0.899954i
$243$	0	0
$244$	26.0000	1.66448
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	− 18.0000i	− 1.13165i
$254$	14.0000	0.878438
$255$	0	0
$256$	16.0000	1.00000
$257$	17.0000i	1.06043i	0.847863	+	0.530215i	$0.177889\pi$
$257$	17.0000i	1.06043i	−0.847863	+	0.530215i	$0.822111\pi$
$258$	0	0
$259$	16.0000	0.994192
$260$	0	0
$261$	0	0
$262$	− 34.0000i	− 2.10053i
$263$	4.00000i	0.246651i	0.992366	+	0.123325i	$0.0393559\pi$
$263$	4.00000i	0.246651i	−0.992366	+	0.123325i	$0.960644\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	4.00000i	0.244339i
$269$	−15.0000	−0.914566	−0.457283	−	0.889321i	$-0.651177\pi$
$269$	−15.0000	−0.914566	−0.457283	+	0.889321i	$0.651177\pi$
$270$	0	0
$271$	−28.0000	−1.70088	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000	−1.70088	−0.850439	+	0.526073i	$0.823664\pi$
$272$	− 8.00000i	− 0.485071i
$273$	0	0
$274$	16.0000	0.966595
$275$	0	0
$276$	0	0
$277$	23.0000i	1.38194i	0.722885	+	0.690968i	$0.242815\pi$
$277$	23.0000i	1.38194i	−0.722885	+	0.690968i	$0.757185\pi$
$278$	− 30.0000i	− 1.79928i
$279$	0	0
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	− 4.00000i	− 0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$	− 4.00000i	− 0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$	24.0000	1.42414
$285$	0	0
$286$	−4.00000	−0.236525
$287$	24.0000i	1.41668i
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	0	0
$292$	− 12.0000i	− 0.702247i
$293$	14.0000i	0.817889i	0.912559	+	0.408944i	$0.134103\pi$
$293$	14.0000i	0.817889i	−0.912559	+	0.408944i	$0.865897\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	9.00000	0.520483
$300$	0	0
$301$	2.00000	0.115278
$302$	− 16.0000i	− 0.920697i
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 2.00000i	− 0.114146i	−0.998370	−	0.0570730i	$-0.981823\pi$
$307$	− 2.00000i	− 0.114146i	0.998370	−	0.0570730i	$-0.0181768\pi$
$308$	− 8.00000i	− 0.455842i
$309$	0	0
$310$	0	0
$311$	13.0000	0.737162	0.368581	−	0.929596i	$-0.379844\pi$
$311$	13.0000	0.737162	0.368581	+	0.929596i	$0.379844\pi$
$312$	0	0
$313$	− 19.0000i	− 1.07394i	−0.843600	−	0.536972i	$-0.819568\pi$
$313$	− 19.0000i	− 1.07394i	0.843600	−	0.536972i	$-0.180432\pi$
$314$	−36.0000	−2.03160
$315$	0	0
$316$	30.0000	1.68763
$317$	− 18.0000i	− 1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$	− 18.0000i	− 1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$	0	0
$319$	−10.0000	−0.559893
$320$	0	0
$321$	0	0
$322$	36.0000i	2.00620i
$323$	0	0
$324$	0	0
$325$	0	0
$326$	48.0000	2.65847
$327$	0	0
$328$	0	0
$329$	−16.0000	−0.882109
$330$	0	0
$331$	−18.0000	−0.989369	−0.494685	−	0.869072i	$-0.664716\pi$
$331$	−18.0000	−0.989369	−0.494685	+	0.869072i	$0.664716\pi$
$332$	− 8.00000i	− 0.439057i
$333$	0	0
$334$	−24.0000	−1.31322
$335$	0	0
$336$	0	0
$337$	13.0000i	0.708155i	0.935216	+	0.354078i	$0.115205\pi$
$337$	13.0000i	0.708155i	−0.935216	+	0.354078i	$0.884795\pi$
$338$	− 2.00000i	− 0.108786i
$339$	0	0
$340$	0	0
$341$	−4.00000	−0.216612
$342$	0	0
$343$	− 20.0000i	− 1.07990i
$344$	0	0
$345$	0	0
$346$	12.0000	0.645124
$347$	17.0000i	0.912608i	0.889824	+	0.456304i	$0.150827\pi$
$347$	17.0000i	0.912608i	−0.889824	+	0.456304i	$0.849173\pi$
$348$	0	0
$349$	30.0000	1.60586	0.802932	−	0.596071i	$-0.203272\pi$
$349$	30.0000	1.60586	0.802932	+	0.596071i	$0.203272\pi$
$350$	0	0
$351$	0	0
$352$	16.0000i	0.852803i
$353$	24.0000i	1.27739i	0.769460	+	0.638696i	$0.220526\pi$
$353$	24.0000i	1.27739i	−0.769460	+	0.638696i	$0.779474\pi$
$354$	0	0
$355$	0	0
$356$	20.0000	1.06000
$357$	0	0
$358$	30.0000i	1.58555i
$359$	20.0000	1.05556	0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000	1.05556	0.527780	+	0.849381i	$0.323025\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	4.00000i	0.210235i
$363$	0	0
$364$	4.00000	0.209657
$365$	0	0
$366$	0	0
$367$	23.0000i	1.20059i	0.799779	+	0.600295i	$0.204950\pi$
$367$	23.0000i	1.20059i	−0.799779	+	0.600295i	$0.795050\pi$
$368$	− 36.0000i	− 1.87663i
$369$	0	0
$370$	0	0
$371$	−22.0000	−1.14218
$372$	0	0
$373$	1.00000i	0.0517780i	0.999665	+	0.0258890i	$0.00824165\pi$
$373$	1.00000i	0.0517780i	−0.999665	+	0.0258890i	$0.991758\pi$
$374$	8.00000	0.413670
$375$	0	0
$376$	0	0
$377$	− 5.00000i	− 0.257513i
$378$	0	0
$379$	−10.0000	−0.513665	−0.256833	−	0.966456i	$-0.582679\pi$
$379$	−10.0000	−0.513665	−0.256833	+	0.966456i	$0.582679\pi$
$380$	0	0
$381$	0	0
$382$	6.00000i	0.306987i
$383$	24.0000i	1.22634i	0.789950	+	0.613171i	$0.210106\pi$
$383$	24.0000i	1.22634i	−0.789950	+	0.613171i	$0.789894\pi$
$384$	0	0
$385$	0	0
$386$	−32.0000	−1.62876
$387$	0	0
$388$	− 16.0000i	− 0.812277i
$389$	35.0000	1.77457	0.887285	−	0.461221i	$-0.152589\pi$
$389$	35.0000	1.77457	0.887285	+	0.461221i	$0.152589\pi$
$390$	0	0
$391$	−18.0000	−0.910299
$392$	0	0
$393$	0	0
$394$	−24.0000	−1.20910
$395$	0	0
$396$	0	0
$397$	18.0000i	0.903394i	0.892171	+	0.451697i	$0.149181\pi$
$397$	18.0000i	0.903394i	−0.892171	+	0.451697i	$0.850819\pi$
$398$	50.0000i	2.50627i
$399$	0	0
$400$	0	0
$401$	−22.0000	−1.09863	−0.549314	−	0.835616i	$-0.685111\pi$
$401$	−22.0000	−1.09863	−0.549314	+	0.835616i	$0.685111\pi$
$402$	0	0
$403$	− 2.00000i	− 0.0996271i
$404$	−6.00000	−0.298511
$405$	0	0
$406$	20.0000	0.992583
$407$	− 16.0000i	− 0.793091i
$408$	0	0
$409$	30.0000	1.48340	0.741702	−	0.670729i	$-0.234019\pi$
$409$	30.0000	1.48340	0.741702	+	0.670729i	$0.234019\pi$
$410$	0	0
$411$	0	0
$412$	− 22.0000i	− 1.08386i
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−8.00000	−0.392232
$417$	0	0
$418$	0	0
$419$	35.0000	1.70986	0.854931	−	0.518742i	$-0.173599\pi$
$419$	35.0000	1.70986	0.854931	+	0.518742i	$0.173599\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	24.0000i	1.16830i
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	26.0000i	1.25823i
$428$	− 14.0000i	− 0.676716i
$429$	0	0
$430$	0	0
$431$	−2.00000	−0.0963366	−0.0481683	−	0.998839i	$-0.515338\pi$
$431$	−2.00000	−0.0963366	−0.0481683	+	0.998839i	$0.515338\pi$
$432$	0	0
$433$	1.00000i	0.0480569i	0.999711	+	0.0240285i	$0.00764923\pi$
$433$	1.00000i	0.0480569i	−0.999711	+	0.0240285i	$0.992351\pi$
$434$	8.00000	0.384012
$435$	0	0
$436$	20.0000	0.957826
$437$	0	0
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	0	0
$442$	4.00000i	0.190261i
$443$	4.00000i	0.190046i	0.995475	+	0.0950229i	$0.0302924\pi$
$443$	4.00000i	0.190046i	−0.995475	+	0.0950229i	$0.969708\pi$
$444$	0	0
$445$	0	0
$446$	28.0000	1.32584
$447$	0	0
$448$	− 16.0000i	− 0.755929i
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	24.0000	1.13012
$452$	2.00000i	0.0940721i
$453$	0	0
$454$	16.0000	0.750917
$455$	0	0
$456$	0	0
$457$	− 32.0000i	− 1.49690i	−0.663193	−	0.748448i	$-0.730799\pi$
$457$	− 32.0000i	− 1.49690i	0.663193	−	0.748448i	$-0.269201\pi$
$458$	20.0000i	0.934539i
$459$	0	0
$460$	0	0
$461$	38.0000	1.76984	0.884918	−	0.465746i	$-0.154214\pi$
$461$	38.0000	1.76984	0.884918	+	0.465746i	$0.154214\pi$
$462$	0	0
$463$	6.00000i	0.278844i	0.990233	+	0.139422i	$0.0445244\pi$
$463$	6.00000i	0.278844i	−0.990233	+	0.139422i	$0.955476\pi$
$464$	−20.0000	−0.928477
$465$	0	0
$466$	2.00000	0.0926482
$467$	17.0000i	0.786666i	0.919396	+	0.393333i	$0.128678\pi$
$467$	17.0000i	0.786666i	−0.919396	+	0.393333i	$0.871322\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 2.00000i	− 0.0919601i
$474$	0	0
$475$	0	0
$476$	−8.00000	−0.366679
$477$	0	0
$478$	0	0
$479$	−30.0000	−1.37073	−0.685367	−	0.728197i	$-0.740358\pi$
$479$	−30.0000	−1.37073	−0.685367	+	0.728197i	$0.740358\pi$
$480$	0	0
$481$	8.00000	0.364769
$482$	44.0000i	2.00415i
$483$	0	0
$484$	14.0000	0.636364
$485$	0	0
$486$	0	0
$487$	− 32.0000i	− 1.45006i	−0.688718	−	0.725029i	$-0.741826\pi$
$487$	− 32.0000i	− 1.45006i	0.688718	−	0.725029i	$-0.258174\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−27.0000	−1.21849	−0.609246	−	0.792981i	$-0.708528\pi$
$491$	−27.0000	−1.21849	−0.609246	+	0.792981i	$0.708528\pi$
$492$	0	0
$493$	10.0000i	0.450377i
$494$	0	0
$495$	0	0
$496$	−8.00000	−0.359211
$497$	24.0000i	1.07655i
$498$	0	0
$499$	−10.0000	−0.447661	−0.223831	−	0.974628i	$-0.571856\pi$
$499$	−10.0000	−0.447661	−0.223831	+	0.974628i	$0.571856\pi$
$500$	0	0
$501$	0	0
$502$	− 24.0000i	− 1.07117i
$503$	− 21.0000i	− 0.936344i	−0.883637	−	0.468172i	$-0.844913\pi$
$503$	− 21.0000i	− 0.936344i	0.883637	−	0.468172i	$-0.155087\pi$
$504$	0	0
$505$	0	0
$506$	36.0000	1.60040
$507$	0	0
$508$	14.0000i	0.621150i
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	12.0000	0.530849
$512$	32.0000i	1.41421i
$513$	0	0
$514$	−34.0000	−1.49968
$515$	0	0
$516$	0	0
$517$	16.0000i	0.703679i
$518$	32.0000i	1.40600i
$519$	0	0
$520$	0	0
$521$	3.00000	0.131432	0.0657162	−	0.997838i	$-0.479067\pi$
$521$	3.00000	0.131432	0.0657162	+	0.997838i	$0.479067\pi$
$522$	0	0
$523$	36.0000i	1.57417i	0.616844	+	0.787085i	$0.288411\pi$
$523$	36.0000i	1.57417i	−0.616844	+	0.787085i	$0.711589\pi$
$524$	34.0000	1.48530
$525$	0	0
$526$	−8.00000	−0.348817
$527$	4.00000i	0.174243i
$528$	0	0
$529$	−58.0000	−2.52174
$530$	0	0
$531$	0	0
$532$	0	0
$533$	12.0000i	0.519778i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	− 30.0000i	− 1.29339i
$539$	−6.00000	−0.258438
$540$	0	0
$541$	−28.0000	−1.20381	−0.601907	−	0.798566i	$-0.705592\pi$
$541$	−28.0000	−1.20381	−0.601907	+	0.798566i	$0.705592\pi$
$542$	− 56.0000i	− 2.40541i
$543$	0	0
$544$	16.0000	0.685994
$545$	0	0
$546$	0	0
$547$	− 12.0000i	− 0.513083i	−0.966533	−	0.256541i	$-0.917417\pi$
$547$	− 12.0000i	− 0.513083i	0.966533	−	0.256541i	$-0.0825830\pi$
$548$	16.0000i	0.683486i
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	30.0000i	1.27573i
$554$	−46.0000	−1.95435
$555$	0	0
$556$	30.0000	1.27228
$557$	22.0000i	0.932170i	0.884740	+	0.466085i	$0.154336\pi$
$557$	22.0000i	0.932170i	−0.884740	+	0.466085i	$0.845664\pi$
$558$	0	0
$559$	1.00000	0.0422955
$560$	0	0
$561$	0	0
$562$	36.0000i	1.51857i
$563$	− 11.0000i	− 0.463595i	−0.972764	−	0.231797i	$-0.925539\pi$
$563$	− 11.0000i	− 0.463595i	0.972764	−	0.231797i	$-0.0744606\pi$
$564$	0	0
$565$	0	0
$566$	8.00000	0.336265
$567$	0	0
$568$	0	0
$569$	25.0000	1.04805	0.524027	−	0.851701i	$-0.324429\pi$
$569$	25.0000	1.04805	0.524027	+	0.851701i	$0.324429\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	− 4.00000i	− 0.167248i
$573$	0	0
$574$	−48.0000	−2.00348
$575$	0	0
$576$	0	0
$577$	− 2.00000i	− 0.0832611i	−0.999133	−	0.0416305i	$-0.986745\pi$
$577$	− 2.00000i	− 0.0832611i	0.999133	−	0.0416305i	$-0.0132552\pi$
$578$	26.0000i	1.08146i
$579$	0	0
$580$	0	0
$581$	8.00000	0.331896
$582$	0	0
$583$	22.0000i	0.911147i
$584$	0	0
$585$	0	0
$586$	−28.0000	−1.15667
$587$	− 38.0000i	− 1.56843i	−0.620491	−	0.784214i	$-0.713066\pi$
$587$	− 38.0000i	− 1.56843i	0.620491	−	0.784214i	$-0.286934\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	− 32.0000i	− 1.31519i
$593$	− 26.0000i	− 1.06769i	−0.845582	−	0.533846i	$-0.820746\pi$
$593$	− 26.0000i	− 1.06769i	0.845582	−	0.533846i	$-0.179254\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	18.0000i	0.736075i
$599$	5.00000	0.204294	0.102147	−	0.994769i	$-0.467429\pi$
$599$	5.00000	0.204294	0.102147	+	0.994769i	$0.467429\pi$
$600$	0	0
$601$	−23.0000	−0.938190	−0.469095	−	0.883148i	$-0.655420\pi$
$601$	−23.0000	−0.938190	−0.469095	+	0.883148i	$0.655420\pi$
$602$	4.00000i	0.163028i
$603$	0	0
$604$	16.0000	0.651031
$605$	0	0
$606$	0	0
$607$	8.00000i	0.324710i	0.986732	+	0.162355i	$0.0519090\pi$
$607$	8.00000i	0.324710i	−0.986732	+	0.162355i	$0.948091\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−8.00000	−0.323645
$612$	0	0
$613$	− 24.0000i	− 0.969351i	−0.874694	−	0.484675i	$-0.838938\pi$
$613$	− 24.0000i	− 0.969351i	0.874694	−	0.484675i	$-0.161062\pi$
$614$	4.00000	0.161427
$615$	0	0
$616$	0	0
$617$	− 48.0000i	− 1.93241i	−0.257780	−	0.966204i	$-0.582991\pi$
$617$	− 48.0000i	− 1.93241i	0.257780	−	0.966204i	$-0.417009\pi$
$618$	0	0
$619$	−40.0000	−1.60774	−0.803868	−	0.594808i	$-0.797228\pi$
$619$	−40.0000	−1.60774	−0.803868	+	0.594808i	$0.797228\pi$
$620$	0	0
$621$	0	0
$622$	26.0000i	1.04251i
$623$	20.0000i	0.801283i
$624$	0	0
$625$	0	0
$626$	38.0000	1.51879
$627$	0	0
$628$	− 36.0000i	− 1.43656i
$629$	−16.0000	−0.637962
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	0	0
$633$	0	0
$634$	36.0000	1.42974
$635$	0	0
$636$	0	0
$637$	− 3.00000i	− 0.118864i
$638$	− 20.0000i	− 0.791808i
$639$	0	0
$640$	0	0
$641$	13.0000	0.513469	0.256735	−	0.966482i	$-0.417353\pi$
$641$	13.0000	0.513469	0.256735	+	0.966482i	$0.417353\pi$
$642$	0	0
$643$	26.0000i	1.02534i	0.858586	+	0.512670i	$0.171344\pi$
$643$	26.0000i	1.02534i	−0.858586	+	0.512670i	$0.828656\pi$
$644$	−36.0000	−1.41860
$645$	0	0
$646$	0	0
$647$	32.0000i	1.25805i	0.777385	+	0.629025i	$0.216546\pi$
$647$	32.0000i	1.25805i	−0.777385	+	0.629025i	$0.783454\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	48.0000i	1.87983i
$653$	− 6.00000i	− 0.234798i	−0.993085	−	0.117399i	$-0.962544\pi$
$653$	− 6.00000i	− 0.234798i	0.993085	−	0.117399i	$-0.0374557\pi$
$654$	0	0
$655$	0	0
$656$	48.0000	1.87409
$657$	0	0
$658$	− 32.0000i	− 1.24749i
$659$	−15.0000	−0.584317	−0.292159	−	0.956370i	$-0.594373\pi$
$659$	−15.0000	−0.584317	−0.292159	+	0.956370i	$0.594373\pi$
$660$	0	0
$661$	2.00000	0.0777910	0.0388955	−	0.999243i	$-0.487616\pi$
$661$	2.00000	0.0777910	0.0388955	+	0.999243i	$0.487616\pi$
$662$	− 36.0000i	− 1.39918i
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	45.0000i	1.74241i
$668$	− 24.0000i	− 0.928588i
$669$	0	0
$670$	0	0
$671$	26.0000	1.00372
$672$	0	0
$673$	6.00000i	0.231283i	0.993291	+	0.115642i	$0.0368924\pi$
$673$	6.00000i	0.231283i	−0.993291	+	0.115642i	$0.963108\pi$
$674$	−26.0000	−1.00148
$675$	0	0
$676$	2.00000	0.0769231
$677$	27.0000i	1.03769i	0.854867	+	0.518847i	$0.173639\pi$
$677$	27.0000i	1.03769i	−0.854867	+	0.518847i	$0.826361\pi$
$678$	0	0
$679$	16.0000	0.614024
$680$	0	0
$681$	0	0
$682$	− 8.00000i	− 0.306336i
$683$	− 16.0000i	− 0.612223i	−0.951996	−	0.306111i	$-0.900972\pi$
$683$	− 16.0000i	− 0.612223i	0.951996	−	0.306111i	$-0.0990280\pi$
$684$	0	0
$685$	0	0
$686$	40.0000	1.52721
$687$	0	0
$688$	− 4.00000i	− 0.152499i
$689$	−11.0000	−0.419067
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	12.0000i	0.456172i
$693$	0	0
$694$	−34.0000	−1.29062
$695$	0	0
$696$	0	0
$697$	− 24.0000i	− 0.909065i
$698$	60.0000i	2.27103i
$699$	0	0
$700$	0	0
$701$	−7.00000	−0.264386	−0.132193	−	0.991224i	$-0.542202\pi$
$701$	−7.00000	−0.264386	−0.132193	+	0.991224i	$0.542202\pi$
$702$	0	0
$703$	0	0
$704$	−16.0000	−0.603023
$705$	0	0
$706$	−48.0000	−1.80650
$707$	− 6.00000i	− 0.225653i
$708$	0	0
$709$	20.0000	0.751116	0.375558	−	0.926799i	$-0.377451\pi$
$709$	20.0000	0.751116	0.375558	+	0.926799i	$0.377451\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	18.0000i	0.674105i
$714$	0	0
$715$	0	0
$716$	−30.0000	−1.12115
$717$	0	0
$718$	40.0000i	1.49279i
$719$	−5.00000	−0.186469	−0.0932343	−	0.995644i	$-0.529721\pi$
$719$	−5.00000	−0.186469	−0.0932343	+	0.995644i	$0.529721\pi$
$720$	0	0
$721$	22.0000	0.819323
$722$	− 38.0000i	− 1.41421i
$723$	0	0
$724$	−4.00000	−0.148659
$725$	0	0
$726$	0	0
$727$	3.00000i	0.111264i	0.998451	+	0.0556319i	$0.0177173\pi$
$727$	3.00000i	0.111264i	−0.998451	+	0.0556319i	$0.982283\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−2.00000	−0.0739727
$732$	0	0
$733$	16.0000i	0.590973i	0.955347	+	0.295487i	$0.0954818\pi$
$733$	16.0000i	0.590973i	−0.955347	+	0.295487i	$0.904518\pi$
$734$	−46.0000	−1.69789
$735$	0	0
$736$	72.0000	2.65396
$737$	4.00000i	0.147342i
$738$	0	0
$739$	30.0000	1.10357	0.551784	−	0.833987i	$-0.313947\pi$
$739$	30.0000	1.10357	0.551784	+	0.833987i	$0.313947\pi$
$740$	0	0
$741$	0	0
$742$	− 44.0000i	− 1.61529i
$743$	− 6.00000i	− 0.220119i	−0.993925	−	0.110059i	$-0.964896\pi$
$743$	− 6.00000i	− 0.220119i	0.993925	−	0.110059i	$-0.0351041\pi$
$744$	0	0
$745$	0	0
$746$	−2.00000	−0.0732252
$747$	0	0
$748$	8.00000i	0.292509i
$749$	14.0000	0.511549
$750$	0	0
$751$	−13.0000	−0.474377	−0.237188	−	0.971464i	$-0.576226\pi$
$751$	−13.0000	−0.474377	−0.237188	+	0.971464i	$0.576226\pi$
$752$	32.0000i	1.16692i
$753$	0	0
$754$	10.0000	0.364179
$755$	0	0
$756$	0	0
$757$	− 2.00000i	− 0.0726912i	−0.999339	−	0.0363456i	$-0.988428\pi$
$757$	− 2.00000i	− 0.0726912i	0.999339	−	0.0363456i	$-0.0115717\pi$
$758$	− 20.0000i	− 0.726433i
$759$	0	0
$760$	0	0
$761$	−12.0000	−0.435000	−0.217500	−	0.976060i	$-0.569790\pi$
$761$	−12.0000	−0.435000	−0.217500	+	0.976060i	$0.569790\pi$
$762$	0	0
$763$	20.0000i	0.724049i
$764$	−6.00000	−0.217072
$765$	0	0
$766$	−48.0000	−1.73431
$767$	0	0
$768$	0	0
$769$	−50.0000	−1.80305	−0.901523	−	0.432731i	$-0.857550\pi$
$769$	−50.0000	−1.80305	−0.901523	+	0.432731i	$0.857550\pi$
$770$	0	0
$771$	0	0
$772$	− 32.0000i	− 1.15171i
$773$	14.0000i	0.503545i	0.967786	+	0.251773i	$0.0810135\pi$
$773$	14.0000i	0.503545i	−0.967786	+	0.251773i	$0.918987\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	70.0000i	2.50962i
$779$	0	0
$780$	0	0
$781$	24.0000	0.858788
$782$	− 36.0000i	− 1.28736i
$783$	0	0
$784$	−12.0000	−0.428571
$785$	0	0
$786$	0	0
$787$	− 32.0000i	− 1.14068i	−0.821410	−	0.570338i	$-0.806812\pi$
$787$	− 32.0000i	− 1.14068i	0.821410	−	0.570338i	$-0.193188\pi$
$788$	− 24.0000i	− 0.854965i
$789$	0	0
$790$	0	0
$791$	−2.00000	−0.0711118
$792$	0	0
$793$	13.0000i	0.461644i
$794$	−36.0000	−1.27759
$795$	0	0
$796$	−50.0000	−1.77220
$797$	27.0000i	0.956389i	0.878254	+	0.478195i	$0.158709\pi$
$797$	27.0000i	0.956389i	−0.878254	+	0.478195i	$0.841291\pi$
$798$	0	0
$799$	16.0000	0.566039
$800$	0	0
$801$	0	0
$802$	− 44.0000i	− 1.55369i
$803$	− 12.0000i	− 0.423471i
$804$	0	0
$805$	0	0
$806$	4.00000	0.140894
$807$	0	0
$808$	0	0
$809$	10.0000	0.351581	0.175791	−	0.984428i	$-0.443752\pi$
$809$	10.0000	0.351581	0.175791	+	0.984428i	$0.443752\pi$
$810$	0	0
$811$	42.0000	1.47482	0.737410	−	0.675446i	$-0.236049\pi$
$811$	42.0000	1.47482	0.737410	+	0.675446i	$0.236049\pi$
$812$	20.0000i	0.701862i
$813$	0	0
$814$	32.0000	1.12160
$815$	0	0
$816$	0	0
$817$	0	0
$818$	60.0000i	2.09785i
$819$	0	0
$820$	0	0
$821$	−2.00000	−0.0698005	−0.0349002	−	0.999391i	$-0.511111\pi$
$821$	−2.00000	−0.0698005	−0.0349002	+	0.999391i	$0.511111\pi$
$822$	0	0
$823$	− 19.0000i	− 0.662298i	−0.943578	−	0.331149i	$-0.892564\pi$
$823$	− 19.0000i	− 0.662298i	0.943578	−	0.331149i	$-0.107436\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 8.00000i	− 0.278187i	−0.990279	−	0.139094i	$-0.955581\pi$
$827$	− 8.00000i	− 0.278187i	0.990279	−	0.139094i	$-0.0444189\pi$
$828$	0	0
$829$	−25.0000	−0.868286	−0.434143	−	0.900844i	$-0.642949\pi$
$829$	−25.0000	−0.868286	−0.434143	+	0.900844i	$0.642949\pi$
$830$	0	0
$831$	0	0
$832$	− 8.00000i	− 0.277350i
$833$	6.00000i	0.207888i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	70.0000i	2.41811i
$839$	−40.0000	−1.38095	−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000	−1.38095	−0.690477	+	0.723355i	$0.742599\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	44.0000i	1.51634i
$843$	0	0
$844$	−24.0000	−0.826114
$845$	0	0
$846$	0	0
$847$	14.0000i	0.481046i
$848$	44.0000i	1.51097i
$849$	0	0
$850$	0	0
$851$	−72.0000	−2.46813
$852$	0	0
$853$	36.0000i	1.23262i	0.787505	+	0.616308i	$0.211372\pi$
$853$	36.0000i	1.23262i	−0.787505	+	0.616308i	$0.788628\pi$
$854$	−52.0000	−1.77940
$855$	0	0
$856$	0	0
$857$	− 3.00000i	− 0.102478i	−0.998686	−	0.0512390i	$-0.983683\pi$
$857$	− 3.00000i	− 0.102478i	0.998686	−	0.0512390i	$-0.0163170\pi$
$858$	0	0
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	0	0
$861$	0	0
$862$	− 4.00000i	− 0.136241i
$863$	− 6.00000i	− 0.204242i	−0.994772	−	0.102121i	$-0.967437\pi$
$863$	− 6.00000i	− 0.204242i	0.994772	−	0.102121i	$-0.0325630\pi$
$864$	0	0
$865$	0	0
$866$	−2.00000	−0.0679628
$867$	0	0
$868$	8.00000i	0.271538i
$869$	30.0000	1.01768
$870$	0	0
$871$	−2.00000	−0.0677674
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	18.0000i	0.607817i	0.952701	+	0.303908i	$0.0982917\pi$
$877$	18.0000i	0.607817i	−0.952701	+	0.303908i	$0.901708\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	43.0000	1.44871	0.724353	−	0.689429i	$-0.242138\pi$
$881$	43.0000	1.44871	0.724353	+	0.689429i	$0.242138\pi$
$882$	0	0
$883$	41.0000i	1.37976i	0.723924	+	0.689880i	$0.242337\pi$
$883$	41.0000i	1.37976i	−0.723924	+	0.689880i	$0.757663\pi$
$884$	−4.00000	−0.134535
$885$	0	0
$886$	−8.00000	−0.268765
$887$	− 8.00000i	− 0.268614i	−0.990940	−	0.134307i	$-0.957119\pi$
$887$	− 8.00000i	− 0.268614i	0.990940	−	0.134307i	$-0.0428808\pi$
$888$	0	0
$889$	−14.0000	−0.469545
$890$	0	0
$891$	0	0
$892$	28.0000i	0.937509i
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	10.0000	0.333519
$900$	0	0
$901$	22.0000	0.732926
$902$	48.0000i	1.59823i
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 17.0000i	− 0.564476i	−0.959344	−	0.282238i	$-0.908923\pi$
$907$	− 17.0000i	− 0.564476i	0.959344	−	0.282238i	$-0.0910767\pi$
$908$	16.0000i	0.530979i
$909$	0	0
$910$	0	0
$911$	−27.0000	−0.894550	−0.447275	−	0.894397i	$-0.647605\pi$
$911$	−27.0000	−0.894550	−0.447275	+	0.894397i	$0.647605\pi$
$912$	0	0
$913$	− 8.00000i	− 0.264761i
$914$	64.0000	2.11693
$915$	0	0
$916$	−20.0000	−0.660819
$917$	34.0000i	1.12278i
$918$	0	0
$919$	−20.0000	−0.659739	−0.329870	−	0.944027i	$-0.607005\pi$
$919$	−20.0000	−0.659739	−0.329870	+	0.944027i	$0.607005\pi$
$920$	0	0
$921$	0	0
$922$	76.0000i	2.50293i
$923$	12.0000i	0.394985i
$924$	0	0
$925$	0	0
$926$	−12.0000	−0.394344
$927$	0	0
$928$	− 40.0000i	− 1.31306i
$929$	40.0000	1.31236	0.656179	−	0.754606i	$-0.272172\pi$
$929$	40.0000	1.31236	0.656179	+	0.754606i	$0.272172\pi$
$930$	0	0
$931$	0	0
$932$	2.00000i	0.0655122i
$933$	0	0
$934$	−34.0000	−1.11251
$935$	0	0
$936$	0	0
$937$	33.0000i	1.07806i	0.842286	+	0.539032i	$0.181210\pi$
$937$	33.0000i	1.07806i	−0.842286	+	0.539032i	$0.818790\pi$
$938$	− 8.00000i	− 0.261209i
$939$	0	0
$940$	0	0
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	0	0
$943$	− 108.000i	− 3.51696i
$944$	0	0
$945$	0	0
$946$	4.00000	0.130051
$947$	− 48.0000i	− 1.55979i	−0.625910	−	0.779895i	$-0.715272\pi$
$947$	− 48.0000i	− 1.55979i	0.625910	−	0.779895i	$-0.284728\pi$
$948$	0	0
$949$	6.00000	0.194768
$950$	0	0
$951$	0	0
$952$	0	0
$953$	54.0000i	1.74923i	0.484817	+	0.874616i	$0.338886\pi$
$953$	54.0000i	1.74923i	−0.484817	+	0.874616i	$0.661114\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	− 60.0000i	− 1.93851i
$959$	−16.0000	−0.516667
$960$	0	0
$961$	−27.0000	−0.870968
$962$	16.0000i	0.515861i
$963$	0	0
$964$	−44.0000	−1.41714
$965$	0	0
$966$	0	0
$967$	− 52.0000i	− 1.67221i	−0.548572	−	0.836104i	$-0.684828\pi$
$967$	− 52.0000i	− 1.67221i	0.548572	−	0.836104i	$-0.315172\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	28.0000	0.898563	0.449281	−	0.893390i	$-0.351680\pi$
$971$	28.0000	0.898563	0.449281	+	0.893390i	$0.351680\pi$
$972$	0	0
$973$	30.0000i	0.961756i
$974$	64.0000	2.05069
$975$	0	0
$976$	52.0000	1.66448
$977$	− 18.0000i	− 0.575871i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$	− 18.0000i	− 0.575871i	0.957650	−	0.287936i	$-0.0929689\pi$
$978$	0	0
$979$	20.0000	0.639203
$980$	0	0
$981$	0	0
$982$	− 54.0000i	− 1.72321i
$983$	4.00000i	0.127580i	0.997963	+	0.0637901i	$0.0203188\pi$
$983$	4.00000i	0.127580i	−0.997963	+	0.0637901i	$0.979681\pi$
$984$	0	0
$985$	0	0
$986$	−20.0000	−0.636930
$987$	0	0
$988$	0	0
$989$	−9.00000	−0.286183
$990$	0	0
$991$	−33.0000	−1.04828	−0.524140	−	0.851632i	$-0.675613\pi$
$991$	−33.0000	−1.04828	−0.524140	+	0.851632i	$0.675613\pi$
$992$	− 16.0000i	− 0.508001i
$993$	0	0
$994$	−48.0000	−1.52247
$995$	0	0
$996$	0	0
$997$	− 47.0000i	− 1.48850i	−0.667898	−	0.744252i	$-0.732806\pi$
$997$	− 47.0000i	− 1.48850i	0.667898	−	0.744252i	$-0.267194\pi$
$998$	− 20.0000i	− 0.633089i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2925.2.c.b.2224.2		2
3.2	odd	2		325.2.b.a.274.1		2
5.2	odd	4		2925.2.a.a.1.1		1
5.3	odd	4		2925.2.a.r.1.1		1
5.4	even	2	inner	2925.2.c.b.2224.1		2
15.2	even	4		325.2.a.e.1.1	yes	1
15.8	even	4		325.2.a.a.1.1	✓	1
15.14	odd	2		325.2.b.a.274.2		2
60.23	odd	4		5200.2.a.z.1.1		1
60.47	odd	4		5200.2.a.k.1.1		1
195.38	even	4		4225.2.a.p.1.1		1
195.77	even	4		4225.2.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.2.a.a.1.1	✓	1	15.8	even	4
325.2.a.e.1.1	yes	1	15.2	even	4
325.2.b.a.274.1		2	3.2	odd	2
325.2.b.a.274.2		2	15.14	odd	2
2925.2.a.a.1.1		1	5.2	odd	4
2925.2.a.r.1.1		1	5.3	odd	4
2925.2.c.b.2224.1		2	5.4	even	2	inner
2925.2.c.b.2224.2		2	1.1	even	1	trivial
4225.2.a.b.1.1		1	195.77	even	4
4225.2.a.p.1.1		1	195.38	even	4
5200.2.a.k.1.1		1	60.47	odd	4
5200.2.a.z.1.1		1	60.23	odd	4

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:	\( N \)	\(=\)	\( 2925 = 3^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2925.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} -2.00000 q^{4} -2.00000i q^{7} +O(q^{10})\)	\(q+2.00000i q^{2} -2.00000 q^{4} -2.00000i q^{7} -2.00000 q^{11} -1.00000i q^{13} +4.00000 q^{14} -4.00000 q^{16} +2.00000i q^{17} -4.00000i q^{22} +9.00000i q^{23} +2.00000 q^{26} +4.00000i q^{28} +5.00000 q^{29} +2.00000 q^{31} -8.00000i q^{32} -4.00000 q^{34} +8.00000i q^{37} -12.0000 q^{41} +1.00000i q^{43} +4.00000 q^{44} -18.0000 q^{46} -8.00000i q^{47} +3.00000 q^{49} +2.00000i q^{52} -11.0000i q^{53} +10.0000i q^{58} -13.0000 q^{61} +4.00000i q^{62} +8.00000 q^{64} -2.00000i q^{67} -4.00000i q^{68} -12.0000 q^{71} +6.00000i q^{73} -16.0000 q^{74} +4.00000i q^{77} -15.0000 q^{79} -24.0000i q^{82} +4.00000i q^{83} -2.00000 q^{86} -10.0000 q^{89} -2.00000 q^{91} -18.0000i q^{92} +16.0000 q^{94} +8.00000i q^{97} +6.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4}+O(q^{10}) \) 2 * q - 4 * q^4	\( 2 q - 4 q^{4} - 4 q^{11} + 8 q^{14} - 8 q^{16} + 4 q^{26} + 10 q^{29} + 4 q^{31} - 8 q^{34} - 24 q^{41} + 8 q^{44} - 36 q^{46} + 6 q^{49} - 26 q^{61} + 16 q^{64} - 24 q^{71} - 32 q^{74} - 30 q^{79} - 4 q^{86} - 20 q^{89} - 4 q^{91} + 32 q^{94}+O(q^{100}) \) 2 * q - 4 * q^4 - 4 * q^11 + 8 * q^14 - 8 * q^16 + 4 * q^26 + 10 * q^29 + 4 * q^31 - 8 * q^34 - 24 * q^41 + 8 * q^44 - 36 * q^46 + 6 * q^49 - 26 * q^61 + 16 * q^64 - 24 * q^71 - 32 * q^74 - 30 * q^79 - 4 * q^86 - 20 * q^89 - 4 * q^91 + 32 * q^94

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