LMFDB - Embedded newform 2925.2.c.g.2224.1

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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 585)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000 q^{4} +2.00000i q^{7} -3.00000i q^{8} -4.00000 q^{11} +1.00000i q^{13} +2.00000 q^{14} -1.00000 q^{16} -4.00000i q^{17} -6.00000 q^{19} +4.00000i q^{22} +1.00000 q^{26} +2.00000i q^{28} +4.00000 q^{29} -10.0000 q^{31} -5.00000i q^{32} -4.00000 q^{34} -2.00000i q^{37} +6.00000i q^{38} -6.00000 q^{41} +8.00000i q^{43} -4.00000 q^{44} -8.00000i q^{47} +3.00000 q^{49} +1.00000i q^{52} +4.00000i q^{53} +6.00000 q^{56} -4.00000i q^{58} -12.0000 q^{59} +2.00000 q^{61} +10.0000i q^{62} -7.00000 q^{64} -10.0000i q^{67} -4.00000i q^{68} +6.00000i q^{73} -2.00000 q^{74} -6.00000 q^{76} -8.00000i q^{77} -12.0000 q^{79} +6.00000i q^{82} +4.00000i q^{83} +8.00000 q^{86} +12.0000i q^{88} -14.0000 q^{89} -2.00000 q^{91} -8.00000 q^{94} -14.0000i q^{97} -3.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} - 8 q^{11} + 4 q^{14} - 2 q^{16} - 12 q^{19} + 2 q^{26} + 8 q^{29} - 20 q^{31} - 8 q^{34} - 12 q^{41} - 8 q^{44} + 6 q^{49} + 12 q^{56} - 24 q^{59} + 4 q^{61} - 14 q^{64} - 4 q^{74} - 12 q^{76}+ \cdots - 16 q^{94}+O(q^{100}) $ 2 * q + 2 * q^4 - 8 * q^11 + 4 * q^14 - 2 * q^16 - 12 * q^19 + 2 * q^26 + 8 * q^29 - 20 * q^31 - 8 * q^34 - 12 * q^41 - 8 * q^44 + 6 * q^49 + 12 * q^56 - 24 * q^59 + 4 * q^61 - 14 * q^64 - 4 * q^74 - 12 * q^76 - 24 * q^79 + 16 * q^86 - 28 * q^89 - 4 * q^91 - 16 * q^94

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 $

$2$	− 1.00000i	− 0.707107i	−0.935414	−	0.353553i	$-0.884973\pi$
$2$	− 1.00000i	− 0.707107i	0.935414	−	0.353553i	$-0.115027\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	− 3.00000i	− 1.06066i
$9$	0	0
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	2.00000	0.534522
$15$	0	0
$16$	−1.00000	−0.250000
$17$	− 4.00000i	− 0.970143i	−0.874475	−	0.485071i	$-0.838794\pi$
$17$	− 4.00000i	− 0.970143i	0.874475	−	0.485071i	$-0.161206\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	0	0
$22$	4.00000i	0.852803i
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	0	0
$26$	1.00000	0.196116
$27$	0	0
$28$	2.00000i	0.377964i
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	− 5.00000i	− 0.883883i
$33$	0	0
$34$	−4.00000	−0.685994
$35$	0	0
$36$	0	0
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	6.00000i	0.973329i
$39$	0	0
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	8.00000i	1.21999i	0.792406	+	0.609994i	$0.208828\pi$
$43$	8.00000i	1.21999i	−0.792406	+	0.609994i	$0.791172\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	0	0
$47$	− 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
$52$	1.00000i	0.138675i
$53$	4.00000i	0.549442i	0.961524	+	0.274721i	$0.0885855\pi$
$53$	4.00000i	0.549442i	−0.961524	+	0.274721i	$0.911414\pi$
$54$	0	0
$55$	0	0
$56$	6.00000	0.801784
$57$	0	0
$58$	− 4.00000i	− 0.525226i
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	10.0000i	1.27000i
$63$	0	0
$64$	−7.00000	−0.875000
$65$	0	0
$66$	0	0
$67$	− 10.0000i	− 1.22169i	−0.791748	−	0.610847i	$-0.790829\pi$
$67$	− 10.0000i	− 1.22169i	0.791748	−	0.610847i	$-0.209171\pi$
$68$	− 4.00000i	− 0.485071i
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	6.00000i	0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$	6.00000i	0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	−6.00000	−0.688247
$77$	− 8.00000i	− 0.911685i
$78$	0	0
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	0	0
$81$	0	0
$82$	6.00000i	0.662589i
$83$	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
$86$	8.00000	0.862662
$87$	0	0
$88$	12.0000i	1.27920i
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	0	0
$93$	0	0
$94$	−8.00000	−0.825137
$95$	0	0
$96$	0	0
$97$	− 14.0000i	− 1.42148i	−0.703452	−	0.710742i	$-0.748359\pi$
$97$	− 14.0000i	− 1.42148i	0.703452	−	0.710742i	$-0.251641\pi$
$98$	− 3.00000i	− 0.303046i
$99$	0	0
$100$	0	0
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	0	0
$103$	4.00000i	0.394132i	0.980390	+	0.197066i	$0.0631413\pi$
$103$	4.00000i	0.394132i	−0.980390	+	0.197066i	$0.936859\pi$
$104$	3.00000	0.294174
$105$	0	0
$106$	4.00000	0.388514
$107$	4.00000i	0.386695i	0.981130	+	0.193347i	$0.0619344\pi$
$107$	4.00000i	0.386695i	−0.981130	+	0.193347i	$0.938066\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	0	0
$112$	− 2.00000i	− 0.188982i
$113$	− 16.0000i	− 1.50515i	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	− 16.0000i	− 1.50515i	0.658505	−	0.752577i	$-0.271189\pi$
$114$	0	0
$115$	0	0
$116$	4.00000	0.371391
$117$	0	0
$118$	12.0000i	1.10469i
$119$	8.00000	0.733359
$120$	0	0
$121$	5.00000	0.454545
$122$	− 2.00000i	− 0.181071i
$123$	0	0
$124$	−10.0000	−0.898027
$125$	0	0
$126$	0	0
$127$	− 8.00000i	− 0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$	− 8.00000i	− 0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$	− 3.00000i	− 0.265165i
$129$	0	0
$130$	0	0
$131$	−16.0000	−1.39793	−0.698963	−	0.715158i	$-0.746355\pi$
$131$	−16.0000	−1.39793	−0.698963	+	0.715158i	$0.746355\pi$
$132$	0	0
$133$	− 12.0000i	− 1.04053i
$134$	−10.0000	−0.863868
$135$	0	0
$136$	−12.0000	−1.02899
$137$	10.0000i	0.854358i	0.904167	+	0.427179i	$0.140493\pi$
$137$	10.0000i	0.854358i	−0.904167	+	0.427179i	$0.859507\pi$
$138$	0	0
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 4.00000i	− 0.334497i
$144$	0	0
$145$	0	0
$146$	6.00000	0.496564
$147$	0	0
$148$	− 2.00000i	− 0.164399i
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	10.0000	0.813788	0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000	0.813788	0.406894	+	0.913475i	$0.366612\pi$
$152$	18.0000i	1.45999i
$153$	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	$0.744931\pi$
$158$	12.0000i	0.954669i
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 6.00000i	− 0.469956i	−0.972001	−	0.234978i	$-0.924498\pi$
$163$	− 6.00000i	− 0.469956i	0.972001	−	0.234978i	$-0.0755019\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	4.00000	0.310460
$167$	− 24.0000i	− 1.85718i	−0.371113	−	0.928588i	$-0.621024\pi$
$167$	− 24.0000i	− 1.85718i	0.371113	−	0.928588i	$-0.378976\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	8.00000i	0.609994i
$173$	− 24.0000i	− 1.82469i	−0.409426	−	0.912343i	$-0.634271\pi$
$173$	− 24.0000i	− 1.82469i	0.409426	−	0.912343i	$-0.365729\pi$
$174$	0	0
$175$	0	0
$176$	4.00000	0.301511
$177$	0	0
$178$	14.0000i	1.04934i
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	2.00000i	0.148250i
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	16.0000i	1.17004i
$188$	− 8.00000i	− 0.583460i
$189$	0	0
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0	0
$193$	26.0000i	1.87152i	0.352636	+	0.935760i	$0.385285\pi$
$193$	26.0000i	1.87152i	−0.352636	+	0.935760i	$0.614715\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	3.00000	0.214286
$197$	18.0000i	1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$	18.0000i	1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	0	0
$202$	− 12.0000i	− 0.844317i
$203$	8.00000i	0.561490i
$204$	0	0
$205$	0	0
$206$	4.00000	0.278693
$207$	0	0
$208$	− 1.00000i	− 0.0693375i
$209$	24.0000	1.66011
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	4.00000i	0.274721i
$213$	0	0
$214$	4.00000	0.273434
$215$	0	0
$216$	0	0
$217$	− 20.0000i	− 1.35769i
$218$	− 2.00000i	− 0.135457i
$219$	0	0
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	14.0000i	0.937509i	0.883328	+	0.468755i	$0.155297\pi$
$223$	14.0000i	0.937509i	−0.883328	+	0.468755i	$0.844703\pi$
$224$	10.0000	0.668153
$225$	0	0
$226$	−16.0000	−1.06430
$227$	− 20.0000i	− 1.32745i	−0.747978	−	0.663723i	$-0.768975\pi$
$227$	− 20.0000i	− 1.32745i	0.747978	−	0.663723i	$-0.231025\pi$
$228$	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
$232$	− 12.0000i	− 0.787839i
$233$	8.00000i	0.524097i	0.965055	+	0.262049i	$0.0843981\pi$
$233$	8.00000i	0.524097i	−0.965055	+	0.262049i	$0.915602\pi$
$234$	0	0
$235$	0	0
$236$	−12.0000	−0.781133
$237$	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	$0.249340\pi$
$242$	− 5.00000i	− 0.321412i
$243$	0	0
$244$	2.00000	0.128037
$245$	0	0
$246$	0	0
$247$	− 6.00000i	− 0.381771i
$248$	30.0000i	1.90500i
$249$	0	0
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	0	0
$254$	−8.00000	−0.501965
$255$	0	0
$256$	−17.0000	−1.06250
$257$	8.00000i	0.499026i	0.968371	+	0.249513i	$0.0802706\pi$
$257$	8.00000i	0.499026i	−0.968371	+	0.249513i	$0.919729\pi$
$258$	0	0
$259$	4.00000	0.248548
$260$	0	0
$261$	0	0
$262$	16.0000i	0.988483i
$263$	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
$266$	−12.0000	−0.735767
$267$	0	0
$268$	− 10.0000i	− 0.610847i
$269$	−12.0000	−0.731653	−0.365826	−	0.930683i	$-0.619214\pi$
$269$	−12.0000	−0.731653	−0.365826	+	0.930683i	$0.619214\pi$
$270$	0	0
$271$	14.0000	0.850439	0.425220	−	0.905090i	$-0.360197\pi$
$271$	14.0000	0.850439	0.425220	+	0.905090i	$0.360197\pi$
$272$	4.00000i	0.242536i
$273$	0	0
$274$	10.0000	0.604122
$275$	0	0
$276$	0	0
$277$	10.0000i	0.600842i	0.953807	+	0.300421i	$0.0971271\pi$
$277$	10.0000i	0.600842i	−0.953807	+	0.300421i	$0.902873\pi$
$278$	12.0000i	0.719712i
$279$	0	0
$280$	0	0
$281$	−30.0000	−1.78965	−0.894825	−	0.446417i	$-0.852700\pi$
$281$	−30.0000	−1.78965	−0.894825	+	0.446417i	$0.852700\pi$
$282$	0	0
$283$	4.00000i	0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$	4.00000i	0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	0	0
$285$	0	0
$286$	−4.00000	−0.236525
$287$	− 12.0000i	− 0.708338i
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0	0
$292$	6.00000i	0.351123i
$293$	2.00000i	0.116841i	0.998292	+	0.0584206i	$0.0186065\pi$
$293$	2.00000i	0.116841i	−0.998292	+	0.0584206i	$0.981394\pi$
$294$	0	0
$295$	0	0
$296$	−6.00000	−0.348743
$297$	0	0
$298$	− 18.0000i	− 1.04271i
$299$	0	0
$300$	0	0
$301$	−16.0000	−0.922225
$302$	− 10.0000i	− 0.575435i
$303$	0	0
$304$	6.00000	0.344124
$305$	0	0
$306$	0	0
$307$	− 22.0000i	− 1.25561i	−0.778372	−	0.627803i	$-0.783954\pi$
$307$	− 22.0000i	− 1.25561i	0.778372	−	0.627803i	$-0.216046\pi$
$308$	− 8.00000i	− 0.455842i
$309$	0	0
$310$	0	0
$311$	−4.00000	−0.226819	−0.113410	−	0.993548i	$-0.536177\pi$
$311$	−4.00000	−0.226819	−0.113410	+	0.993548i	$0.536177\pi$
$312$	0	0
$313$	− 2.00000i	− 0.113047i	−0.998401	−	0.0565233i	$-0.981998\pi$
$313$	− 2.00000i	− 0.113047i	0.998401	−	0.0565233i	$-0.0180015\pi$
$314$	18.0000	1.01580
$315$	0	0
$316$	−12.0000	−0.675053
$317$	18.0000i	1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$	18.0000i	1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$	0	0
$319$	−16.0000	−0.895828
$320$	0	0
$321$	0	0
$322$	0	0
$323$	24.0000i	1.33540i
$324$	0	0
$325$	0	0
$326$	−6.00000	−0.332309
$327$	0	0
$328$	18.0000i	0.993884i
$329$	16.0000	0.882109
$330$	0	0
$331$	18.0000	0.989369	0.494685	−	0.869072i	$-0.335284\pi$
$331$	18.0000	0.989369	0.494685	+	0.869072i	$0.335284\pi$
$332$	4.00000i	0.219529i
$333$	0	0
$334$	−24.0000	−1.31322
$335$	0	0
$336$	0	0
$337$	14.0000i	0.762629i	0.924445	+	0.381314i	$0.124528\pi$
$337$	14.0000i	0.762629i	−0.924445	+	0.381314i	$0.875472\pi$
$338$	1.00000i	0.0543928i
$339$	0	0
$340$	0	0
$341$	40.0000	2.16612
$342$	0	0
$343$	20.0000i	1.07990i
$344$	24.0000	1.29399
$345$	0	0
$346$	−24.0000	−1.29025
$347$	32.0000i	1.71785i	0.512101	+	0.858925i	$0.328867\pi$
$347$	32.0000i	1.71785i	−0.512101	+	0.858925i	$0.671133\pi$
$348$	0	0
$349$	6.00000	0.321173	0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000	0.321173	0.160586	+	0.987022i	$0.448662\pi$
$350$	0	0
$351$	0	0
$352$	20.0000i	1.06600i
$353$	6.00000i	0.319348i	0.987170	+	0.159674i	$0.0510443\pi$
$353$	6.00000i	0.319348i	−0.987170	+	0.159674i	$0.948956\pi$
$354$	0	0
$355$	0	0
$356$	−14.0000	−0.741999
$357$	0	0
$358$	0	0
$359$	16.0000	0.844448	0.422224	−	0.906492i	$-0.361250\pi$
$359$	16.0000	0.844448	0.422224	+	0.906492i	$0.361250\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	10.0000i	0.525588i
$363$	0	0
$364$	−2.00000	−0.104828
$365$	0	0
$366$	0	0
$367$	− 8.00000i	− 0.417597i	−0.977959	−	0.208798i	$-0.933045\pi$
$367$	− 8.00000i	− 0.417597i	0.977959	−	0.208798i	$-0.0669552\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−8.00000	−0.415339
$372$	0	0
$373$	− 22.0000i	− 1.13912i	−0.821951	−	0.569558i	$-0.807114\pi$
$373$	− 22.0000i	− 1.13912i	0.821951	−	0.569558i	$-0.192886\pi$
$374$	16.0000	0.827340
$375$	0	0
$376$	−24.0000	−1.23771
$377$	4.00000i	0.206010i
$378$	0	0
$379$	2.00000	0.102733	0.0513665	−	0.998680i	$-0.483642\pi$
$379$	2.00000	0.102733	0.0513665	+	0.998680i	$0.483642\pi$
$380$	0	0
$381$	0	0
$382$	12.0000i	0.613973i
$383$	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
$386$	26.0000	1.32337
$387$	0	0
$388$	− 14.0000i	− 0.710742i
$389$	−8.00000	−0.405616	−0.202808	−	0.979219i	$-0.565007\pi$
$389$	−8.00000	−0.405616	−0.202808	+	0.979219i	$0.565007\pi$
$390$	0	0
$391$	0	0
$392$	− 9.00000i	− 0.454569i
$393$	0	0
$394$	18.0000	0.906827
$395$	0	0
$396$	0	0
$397$	− 18.0000i	− 0.903394i	−0.892171	−	0.451697i	$-0.850819\pi$
$397$	− 18.0000i	− 0.903394i	0.892171	−	0.451697i	$-0.149181\pi$
$398$	8.00000i	0.401004i
$399$	0	0
$400$	0	0
$401$	22.0000	1.09863	0.549314	−	0.835616i	$-0.314889\pi$
$401$	22.0000	1.09863	0.549314	+	0.835616i	$0.314889\pi$
$402$	0	0
$403$	− 10.0000i	− 0.498135i
$404$	12.0000	0.597022
$405$	0	0
$406$	8.00000	0.397033
$407$	8.00000i	0.396545i
$408$	0	0
$409$	−30.0000	−1.48340	−0.741702	−	0.670729i	$-0.765981\pi$
$409$	−30.0000	−1.48340	−0.741702	+	0.670729i	$0.765981\pi$
$410$	0	0
$411$	0	0
$412$	4.00000i	0.197066i
$413$	− 24.0000i	− 1.18096i
$414$	0	0
$415$	0	0
$416$	5.00000	0.245145
$417$	0	0
$418$	− 24.0000i	− 1.17388i
$419$	16.0000	0.781651	0.390826	−	0.920465i	$-0.372190\pi$
$419$	16.0000	0.781651	0.390826	+	0.920465i	$0.372190\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	0	0
$423$	0	0
$424$	12.0000	0.582772
$425$	0	0
$426$	0	0
$427$	4.00000i	0.193574i
$428$	4.00000i	0.193347i
$429$	0	0
$430$	0	0
$431$	−40.0000	−1.92673	−0.963366	−	0.268190i	$-0.913575\pi$
$431$	−40.0000	−1.92673	−0.963366	+	0.268190i	$0.913575\pi$
$432$	0	0
$433$	2.00000i	0.0961139i	0.998845	+	0.0480569i	$0.0153029\pi$
$433$	2.00000i	0.0961139i	−0.998845	+	0.0480569i	$0.984697\pi$
$434$	−20.0000	−0.960031
$435$	0	0
$436$	2.00000	0.0957826
$437$	0	0
$438$	0	0
$439$	36.0000	1.71819	0.859093	−	0.511819i	$-0.171028\pi$
$439$	36.0000	1.71819	0.859093	+	0.511819i	$0.171028\pi$
$440$	0	0
$441$	0	0
$442$	− 4.00000i	− 0.190261i
$443$	− 20.0000i	− 0.950229i	−0.879924	−	0.475114i	$-0.842407\pi$
$443$	− 20.0000i	− 0.950229i	0.879924	−	0.475114i	$-0.157593\pi$
$444$	0	0
$445$	0	0
$446$	14.0000	0.662919
$447$	0	0
$448$	− 14.0000i	− 0.661438i
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	0	0
$451$	24.0000	1.13012
$452$	− 16.0000i	− 0.752577i
$453$	0	0
$454$	−20.0000	−0.938647
$455$	0	0
$456$	0	0
$457$	− 22.0000i	− 1.02912i	−0.857455	−	0.514558i	$-0.827956\pi$
$457$	− 22.0000i	− 1.02912i	0.857455	−	0.514558i	$-0.172044\pi$
$458$	− 10.0000i	− 0.467269i
$459$	0	0
$460$	0	0
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	0	0
$463$	− 18.0000i	− 0.836531i	−0.908325	−	0.418265i	$-0.862638\pi$
$463$	− 18.0000i	− 0.836531i	0.908325	−	0.418265i	$-0.137362\pi$
$464$	−4.00000	−0.185695
$465$	0	0
$466$	8.00000	0.370593
$467$	8.00000i	0.370196i	0.982720	+	0.185098i	$0.0592602\pi$
$467$	8.00000i	0.370196i	−0.982720	+	0.185098i	$0.940740\pi$
$468$	0	0
$469$	20.0000	0.923514
$470$	0	0
$471$	0	0
$472$	36.0000i	1.65703i
$473$	− 32.0000i	− 1.47136i
$474$	0	0
$475$	0	0
$476$	8.00000	0.366679
$477$	0	0
$478$	0	0
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	− 22.0000i	− 1.00207i
$483$	0	0
$484$	5.00000	0.227273
$485$	0	0
$486$	0	0
$487$	− 22.0000i	− 0.996915i	−0.866914	−	0.498458i	$-0.833900\pi$
$487$	− 22.0000i	− 0.996915i	0.866914	−	0.498458i	$-0.166100\pi$
$488$	− 6.00000i	− 0.271607i
$489$	0	0
$490$	0	0
$491$	36.0000	1.62466	0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000	1.62466	0.812329	+	0.583200i	$0.198200\pi$
$492$	0	0
$493$	− 16.0000i	− 0.720604i
$494$	−6.00000	−0.269953
$495$	0	0
$496$	10.0000	0.449013
$497$	0	0
$498$	0	0
$499$	38.0000	1.70111	0.850557	−	0.525883i	$-0.176265\pi$
$499$	38.0000	1.70111	0.850557	+	0.525883i	$0.176265\pi$
$500$	0	0
$501$	0	0
$502$	− 12.0000i	− 0.535586i
$503$	− 12.0000i	− 0.535054i	−0.963550	−	0.267527i	$-0.913794\pi$
$503$	− 12.0000i	− 0.535054i	0.963550	−	0.267527i	$-0.0862064\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	− 8.00000i	− 0.354943i
$509$	−42.0000	−1.86162	−0.930809	−	0.365507i	$-0.880896\pi$
$509$	−42.0000	−1.86162	−0.930809	+	0.365507i	$0.880896\pi$
$510$	0	0
$511$	−12.0000	−0.530849
$512$	11.0000i	0.486136i
$513$	0	0
$514$	8.00000	0.352865
$515$	0	0
$516$	0	0
$517$	32.0000i	1.40736i
$518$	− 4.00000i	− 0.175750i
$519$	0	0
$520$	0	0
$521$	12.0000	0.525730	0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000	0.525730	0.262865	+	0.964833i	$0.415333\pi$
$522$	0	0
$523$	12.0000i	0.524723i	0.964970	+	0.262362i	$0.0845013\pi$
$523$	12.0000i	0.524723i	−0.964970	+	0.262362i	$0.915499\pi$
$524$	−16.0000	−0.698963
$525$	0	0
$526$	4.00000	0.174408
$527$	40.0000i	1.74243i
$528$	0	0
$529$	23.0000	1.00000
$530$	0	0
$531$	0	0
$532$	− 12.0000i	− 0.520266i
$533$	− 6.00000i	− 0.259889i
$534$	0	0
$535$	0	0
$536$	−30.0000	−1.29580
$537$	0	0
$538$	12.0000i	0.517357i
$539$	−12.0000	−0.516877
$540$	0	0
$541$	14.0000	0.601907	0.300954	−	0.953639i	$-0.402695\pi$
$541$	14.0000	0.601907	0.300954	+	0.953639i	$0.402695\pi$
$542$	− 14.0000i	− 0.601351i
$543$	0	0
$544$	−20.0000	−0.857493
$545$	0	0
$546$	0	0
$547$	− 36.0000i	− 1.53925i	−0.638497	−	0.769624i	$-0.720443\pi$
$547$	− 36.0000i	− 1.53925i	0.638497	−	0.769624i	$-0.279557\pi$
$548$	10.0000i	0.427179i
$549$	0	0
$550$	0	0
$551$	−24.0000	−1.02243
$552$	0	0
$553$	− 24.0000i	− 1.02058i
$554$	10.0000	0.424859
$555$	0	0
$556$	−12.0000	−0.508913
$557$	− 26.0000i	− 1.10166i	−0.834619	−	0.550828i	$-0.814312\pi$
$557$	− 26.0000i	− 1.10166i	0.834619	−	0.550828i	$-0.185688\pi$
$558$	0	0
$559$	−8.00000	−0.338364
$560$	0	0
$561$	0	0
$562$	30.0000i	1.26547i
$563$	4.00000i	0.168580i	0.996441	+	0.0842900i	$0.0268622\pi$
$563$	4.00000i	0.168580i	−0.996441	+	0.0842900i	$0.973138\pi$
$564$	0	0
$565$	0	0
$566$	4.00000	0.168133
$567$	0	0
$568$	0	0
$569$	−28.0000	−1.17382	−0.586911	−	0.809652i	$-0.699656\pi$
$569$	−28.0000	−1.17382	−0.586911	+	0.809652i	$0.699656\pi$
$570$	0	0
$571$	−12.0000	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$572$	− 4.00000i	− 0.167248i
$573$	0	0
$574$	−12.0000	−0.500870
$575$	0	0
$576$	0	0
$577$	38.0000i	1.58196i	0.611842	+	0.790980i	$0.290429\pi$
$577$	38.0000i	1.58196i	−0.611842	+	0.790980i	$0.709571\pi$
$578$	− 1.00000i	− 0.0415945i
$579$	0	0
$580$	0	0
$581$	−8.00000	−0.331896
$582$	0	0
$583$	− 16.0000i	− 0.662652i
$584$	18.0000	0.744845
$585$	0	0
$586$	2.00000	0.0826192
$587$	− 20.0000i	− 0.825488i	−0.910847	−	0.412744i	$-0.864570\pi$
$587$	− 20.0000i	− 0.825488i	0.910847	−	0.412744i	$-0.135430\pi$
$588$	0	0
$589$	60.0000	2.47226
$590$	0	0
$591$	0	0
$592$	2.00000i	0.0821995i
$593$	− 14.0000i	− 0.574911i	−0.957794	−	0.287456i	$-0.907191\pi$
$593$	− 14.0000i	− 0.574911i	0.957794	−	0.287456i	$-0.0928094\pi$
$594$	0	0
$595$	0	0
$596$	18.0000	0.737309
$597$	0	0
$598$	0	0
$599$	−8.00000	−0.326871	−0.163436	−	0.986554i	$-0.552258\pi$
$599$	−8.00000	−0.326871	−0.163436	+	0.986554i	$0.552258\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	16.0000i	0.652111i
$603$	0	0
$604$	10.0000	0.406894
$605$	0	0
$606$	0	0
$607$	− 8.00000i	− 0.324710i	−0.986732	−	0.162355i	$-0.948091\pi$
$607$	− 8.00000i	− 0.324710i	0.986732	−	0.162355i	$-0.0519090\pi$
$608$	30.0000i	1.21666i
$609$	0	0
$610$	0	0
$611$	8.00000	0.323645
$612$	0	0
$613$	− 6.00000i	− 0.242338i	−0.992632	−	0.121169i	$-0.961336\pi$
$613$	− 6.00000i	− 0.242338i	0.992632	−	0.121169i	$-0.0386643\pi$
$614$	−22.0000	−0.887848
$615$	0	0
$616$	−24.0000	−0.966988
$617$	18.0000i	0.724653i	0.932051	+	0.362326i	$0.118017\pi$
$617$	18.0000i	0.724653i	−0.932051	+	0.362326i	$0.881983\pi$
$618$	0	0
$619$	−22.0000	−0.884255	−0.442127	−	0.896952i	$-0.645776\pi$
$619$	−22.0000	−0.884255	−0.442127	+	0.896952i	$0.645776\pi$
$620$	0	0
$621$	0	0
$622$	4.00000i	0.160385i
$623$	− 28.0000i	− 1.12180i
$624$	0	0
$625$	0	0
$626$	−2.00000	−0.0799361
$627$	0	0
$628$	18.0000i	0.718278i
$629$	−8.00000	−0.318981
$630$	0	0
$631$	−14.0000	−0.557331	−0.278666	−	0.960388i	$-0.589892\pi$
$631$	−14.0000	−0.557331	−0.278666	+	0.960388i	$0.589892\pi$
$632$	36.0000i	1.43200i
$633$	0	0
$634$	18.0000	0.714871
$635$	0	0
$636$	0	0
$637$	3.00000i	0.118864i
$638$	16.0000i	0.633446i
$639$	0	0
$640$	0	0
$641$	−40.0000	−1.57991	−0.789953	−	0.613168i	$-0.789895\pi$
$641$	−40.0000	−1.57991	−0.789953	+	0.613168i	$0.789895\pi$
$642$	0	0
$643$	− 14.0000i	− 0.552106i	−0.961142	−	0.276053i	$-0.910973\pi$
$643$	− 14.0000i	− 0.552106i	0.961142	−	0.276053i	$-0.0890266\pi$
$644$	0	0
$645$	0	0
$646$	24.0000	0.944267
$647$	− 28.0000i	− 1.10079i	−0.834903	−	0.550397i	$-0.814476\pi$
$647$	− 28.0000i	− 1.10079i	0.834903	−	0.550397i	$-0.185524\pi$
$648$	0	0
$649$	48.0000	1.88416
$650$	0	0
$651$	0	0
$652$	− 6.00000i	− 0.234978i
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	6.00000	0.234261
$657$	0	0
$658$	− 16.0000i	− 0.623745i
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	26.0000	1.01128	0.505641	−	0.862744i	$-0.331256\pi$
$661$	26.0000	1.01128	0.505641	+	0.862744i	$0.331256\pi$
$662$	− 18.0000i	− 0.699590i
$663$	0	0
$664$	12.0000	0.465690
$665$	0	0
$666$	0	0
$667$	0	0
$668$	− 24.0000i	− 0.928588i
$669$	0	0
$670$	0	0
$671$	−8.00000	−0.308837
$672$	0	0
$673$	18.0000i	0.693849i	0.937893	+	0.346925i	$0.112774\pi$
$673$	18.0000i	0.693849i	−0.937893	+	0.346925i	$0.887226\pi$
$674$	14.0000	0.539260
$675$	0	0
$676$	−1.00000	−0.0384615
$677$	− 36.0000i	− 1.38359i	−0.722093	−	0.691796i	$-0.756820\pi$
$677$	− 36.0000i	− 1.38359i	0.722093	−	0.691796i	$-0.243180\pi$
$678$	0	0
$679$	28.0000	1.07454
$680$	0	0
$681$	0	0
$682$	− 40.0000i	− 1.53168i
$683$	− 4.00000i	− 0.153056i	−0.997067	−	0.0765279i	$-0.975617\pi$
$683$	− 4.00000i	− 0.153056i	0.997067	−	0.0765279i	$-0.0243834\pi$
$684$	0	0
$685$	0	0
$686$	20.0000	0.763604
$687$	0	0
$688$	− 8.00000i	− 0.304997i
$689$	−4.00000	−0.152388
$690$	0	0
$691$	34.0000	1.29342	0.646710	−	0.762736i	$-0.276144\pi$
$691$	34.0000	1.29342	0.646710	+	0.762736i	$0.276144\pi$
$692$	− 24.0000i	− 0.912343i
$693$	0	0
$694$	32.0000	1.21470
$695$	0	0
$696$	0	0
$697$	24.0000i	0.909065i
$698$	− 6.00000i	− 0.227103i
$699$	0	0
$700$	0	0
$701$	40.0000	1.51078	0.755390	−	0.655276i	$-0.227448\pi$
$701$	40.0000	1.51078	0.755390	+	0.655276i	$0.227448\pi$
$702$	0	0
$703$	12.0000i	0.452589i
$704$	28.0000	1.05529
$705$	0	0
$706$	6.00000	0.225813
$707$	24.0000i	0.902613i
$708$	0	0
$709$	50.0000	1.87779	0.938895	−	0.344204i	$-0.111851\pi$
$709$	50.0000	1.87779	0.938895	+	0.344204i	$0.111851\pi$
$710$	0	0
$711$	0	0
$712$	42.0000i	1.57402i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	− 16.0000i	− 0.597115i
$719$	20.0000	0.745874	0.372937	−	0.927857i	$-0.378351\pi$
$719$	20.0000	0.745874	0.372937	+	0.927857i	$0.378351\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	− 17.0000i	− 0.632674i
$723$	0	0
$724$	−10.0000	−0.371647
$725$	0	0
$726$	0	0
$727$	− 24.0000i	− 0.890111i	−0.895503	−	0.445055i	$-0.853184\pi$
$727$	− 24.0000i	− 0.890111i	0.895503	−	0.445055i	$-0.146816\pi$
$728$	6.00000i	0.222375i
$729$	0	0
$730$	0	0
$731$	32.0000	1.18356
$732$	0	0
$733$	− 34.0000i	− 1.25582i	−0.778287	−	0.627909i	$-0.783911\pi$
$733$	− 34.0000i	− 1.25582i	0.778287	−	0.627909i	$-0.216089\pi$
$734$	−8.00000	−0.295285
$735$	0	0
$736$	0	0
$737$	40.0000i	1.47342i
$738$	0	0
$739$	−30.0000	−1.10357	−0.551784	−	0.833987i	$-0.686053\pi$
$739$	−30.0000	−1.10357	−0.551784	+	0.833987i	$0.686053\pi$
$740$	0	0
$741$	0	0
$742$	8.00000i	0.293689i
$743$	− 24.0000i	− 0.880475i	−0.897881	−	0.440237i	$-0.854894\pi$
$743$	− 24.0000i	− 0.880475i	0.897881	−	0.440237i	$-0.145106\pi$
$744$	0	0
$745$	0	0
$746$	−22.0000	−0.805477
$747$	0	0
$748$	16.0000i	0.585018i
$749$	−8.00000	−0.292314
$750$	0	0
$751$	20.0000	0.729810	0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000	0.729810	0.364905	+	0.931045i	$0.381101\pi$
$752$	8.00000i	0.291730i
$753$	0	0
$754$	4.00000	0.145671
$755$	0	0
$756$	0	0
$757$	− 34.0000i	− 1.23575i	−0.786276	−	0.617876i	$-0.787994\pi$
$757$	− 34.0000i	− 1.23575i	0.786276	−	0.617876i	$-0.212006\pi$
$758$	− 2.00000i	− 0.0726433i
$759$	0	0
$760$	0	0
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	0	0
$763$	4.00000i	0.144810i
$764$	−12.0000	−0.434145
$765$	0	0
$766$	24.0000	0.867155
$767$	− 12.0000i	− 0.433295i
$768$	0	0
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	0	0
$771$	0	0
$772$	26.0000i	0.935760i
$773$	50.0000i	1.79838i	0.437564	+	0.899188i	$0.355842\pi$
$773$	50.0000i	1.79838i	−0.437564	+	0.899188i	$0.644158\pi$
$774$	0	0
$775$	0	0
$776$	−42.0000	−1.50771
$777$	0	0
$778$	8.00000i	0.286814i
$779$	36.0000	1.28983
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−3.00000	−0.107143
$785$	0	0
$786$	0	0
$787$	− 22.0000i	− 0.784215i	−0.919919	−	0.392108i	$-0.871746\pi$
$787$	− 22.0000i	− 0.784215i	0.919919	−	0.392108i	$-0.128254\pi$
$788$	18.0000i	0.641223i
$789$	0	0
$790$	0	0
$791$	32.0000	1.13779
$792$	0	0
$793$	2.00000i	0.0710221i
$794$	−18.0000	−0.638796
$795$	0	0
$796$	−8.00000	−0.283552
$797$	− 12.0000i	− 0.425062i	−0.977154	−	0.212531i	$-0.931829\pi$
$797$	− 12.0000i	− 0.425062i	0.977154	−	0.212531i	$-0.0681706\pi$
$798$	0	0
$799$	−32.0000	−1.13208
$800$	0	0
$801$	0	0
$802$	− 22.0000i	− 0.776847i
$803$	− 24.0000i	− 0.846942i
$804$	0	0
$805$	0	0
$806$	−10.0000	−0.352235
$807$	0	0
$808$	− 36.0000i	− 1.26648i
$809$	−28.0000	−0.984428	−0.492214	−	0.870474i	$-0.663812\pi$
$809$	−28.0000	−0.984428	−0.492214	+	0.870474i	$0.663812\pi$
$810$	0	0
$811$	30.0000	1.05344	0.526721	−	0.850038i	$-0.323421\pi$
$811$	30.0000	1.05344	0.526721	+	0.850038i	$0.323421\pi$
$812$	8.00000i	0.280745i
$813$	0	0
$814$	8.00000	0.280400
$815$	0	0
$816$	0	0
$817$	− 48.0000i	− 1.67931i
$818$	30.0000i	1.04893i
$819$	0	0
$820$	0	0
$821$	−10.0000	−0.349002	−0.174501	−	0.984657i	$-0.555831\pi$
$821$	−10.0000	−0.349002	−0.174501	+	0.984657i	$0.555831\pi$
$822$	0	0
$823$	4.00000i	0.139431i	0.997567	+	0.0697156i	$0.0222092\pi$
$823$	4.00000i	0.139431i	−0.997567	+	0.0697156i	$0.977791\pi$
$824$	12.0000	0.418040
$825$	0	0
$826$	−24.0000	−0.835067
$827$	− 20.0000i	− 0.695468i	−0.937593	−	0.347734i	$-0.886951\pi$
$827$	− 20.0000i	− 0.695468i	0.937593	−	0.347734i	$-0.113049\pi$
$828$	0	0
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	0	0
$831$	0	0
$832$	− 7.00000i	− 0.242681i
$833$	− 12.0000i	− 0.415775i
$834$	0	0
$835$	0	0
$836$	24.0000	0.830057
$837$	0	0
$838$	− 16.0000i	− 0.552711i
$839$	40.0000	1.38095	0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000	1.38095	0.690477	+	0.723355i	$0.257401\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	− 22.0000i	− 0.758170i
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	10.0000i	0.343604i
$848$	− 4.00000i	− 0.137361i
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 30.0000i	− 1.02718i	−0.858036	−	0.513590i	$-0.828315\pi$
$853$	− 30.0000i	− 1.02718i	0.858036	−	0.513590i	$-0.171685\pi$
$854$	4.00000	0.136877
$855$	0	0
$856$	12.0000	0.410152
$857$	− 48.0000i	− 1.63965i	−0.572615	−	0.819824i	$-0.694071\pi$
$857$	− 48.0000i	− 1.63965i	0.572615	−	0.819824i	$-0.305929\pi$
$858$	0	0
$859$	−16.0000	−0.545913	−0.272956	−	0.962026i	$-0.588002\pi$
$859$	−16.0000	−0.545913	−0.272956	+	0.962026i	$0.588002\pi$
$860$	0	0
$861$	0	0
$862$	40.0000i	1.36241i
$863$	48.0000i	1.63394i	0.576681	+	0.816970i	$0.304348\pi$
$863$	48.0000i	1.63394i	−0.576681	+	0.816970i	$0.695652\pi$
$864$	0	0
$865$	0	0
$866$	2.00000	0.0679628
$867$	0	0
$868$	− 20.0000i	− 0.678844i
$869$	48.0000	1.62829
$870$	0	0
$871$	10.0000	0.338837
$872$	− 6.00000i	− 0.203186i
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 42.0000i	− 1.41824i	−0.705088	−	0.709120i	$-0.749093\pi$
$877$	− 42.0000i	− 1.41824i	0.705088	−	0.709120i	$-0.250907\pi$
$878$	− 36.0000i	− 1.21494i
$879$	0	0
$880$	0	0
$881$	−28.0000	−0.943344	−0.471672	−	0.881774i	$-0.656349\pi$
$881$	−28.0000	−0.943344	−0.471672	+	0.881774i	$0.656349\pi$
$882$	0	0
$883$	4.00000i	0.134611i	0.997732	+	0.0673054i	$0.0214402\pi$
$883$	4.00000i	0.134611i	−0.997732	+	0.0673054i	$0.978560\pi$
$884$	4.00000	0.134535
$885$	0	0
$886$	−20.0000	−0.671913
$887$	− 56.0000i	− 1.88030i	−0.340766	−	0.940148i	$-0.610687\pi$
$887$	− 56.0000i	− 1.88030i	0.340766	−	0.940148i	$-0.389313\pi$
$888$	0	0
$889$	16.0000	0.536623
$890$	0	0
$891$	0	0
$892$	14.0000i	0.468755i
$893$	48.0000i	1.60626i
$894$	0	0
$895$	0	0
$896$	6.00000	0.200446
$897$	0	0
$898$	30.0000i	1.00111i
$899$	−40.0000	−1.33407
$900$	0	0
$901$	16.0000	0.533037
$902$	− 24.0000i	− 0.799113i
$903$	0	0
$904$	−48.0000	−1.59646
$905$	0	0
$906$	0	0
$907$	44.0000i	1.46100i	0.682915	+	0.730498i	$0.260712\pi$
$907$	44.0000i	1.46100i	−0.682915	+	0.730498i	$0.739288\pi$
$908$	− 20.0000i	− 0.663723i
$909$	0	0
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	0	0
$913$	− 16.0000i	− 0.529523i
$914$	−22.0000	−0.727695
$915$	0	0
$916$	10.0000	0.330409
$917$	− 32.0000i	− 1.05673i
$918$	0	0
$919$	−44.0000	−1.45143	−0.725713	−	0.687998i	$-0.758490\pi$
$919$	−44.0000	−1.45143	−0.725713	+	0.687998i	$0.758490\pi$
$920$	0	0
$921$	0	0
$922$	14.0000i	0.461065i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	−18.0000	−0.591517
$927$	0	0
$928$	− 20.0000i	− 0.656532i
$929$	−10.0000	−0.328089	−0.164045	−	0.986453i	$-0.552454\pi$
$929$	−10.0000	−0.328089	−0.164045	+	0.986453i	$0.552454\pi$
$930$	0	0
$931$	−18.0000	−0.589926
$932$	8.00000i	0.262049i
$933$	0	0
$934$	8.00000	0.261768
$935$	0	0
$936$	0	0
$937$	54.0000i	1.76410i	0.471153	+	0.882052i	$0.343838\pi$
$937$	54.0000i	1.76410i	−0.471153	+	0.882052i	$0.656162\pi$
$938$	− 20.0000i	− 0.653023i
$939$	0	0
$940$	0	0
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	0	0
$943$	0	0
$944$	12.0000	0.390567
$945$	0	0
$946$	−32.0000	−1.04041
$947$	12.0000i	0.389948i	0.980808	+	0.194974i	$0.0624622\pi$
$947$	12.0000i	0.389948i	−0.980808	+	0.194974i	$0.937538\pi$
$948$	0	0
$949$	−6.00000	−0.194768
$950$	0	0
$951$	0	0
$952$	− 24.0000i	− 0.777844i
$953$	48.0000i	1.55487i	0.628962	+	0.777436i	$0.283480\pi$
$953$	48.0000i	1.55487i	−0.628962	+	0.777436i	$0.716520\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	− 24.0000i	− 0.775405i
$959$	−20.0000	−0.645834
$960$	0	0
$961$	69.0000	2.22581
$962$	− 2.00000i	− 0.0644826i
$963$	0	0
$964$	22.0000	0.708572
$965$	0	0
$966$	0	0
$967$	58.0000i	1.86515i	0.360971	+	0.932577i	$0.382445\pi$
$967$	58.0000i	1.86515i	−0.360971	+	0.932577i	$0.617555\pi$
$968$	− 15.0000i	− 0.482118i
$969$	0	0
$970$	0	0
$971$	−52.0000	−1.66876	−0.834380	−	0.551190i	$-0.814174\pi$
$971$	−52.0000	−1.66876	−0.834380	+	0.551190i	$0.814174\pi$
$972$	0	0
$973$	− 24.0000i	− 0.769405i
$974$	−22.0000	−0.704925
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	54.0000i	1.72761i	0.503824	+	0.863807i	$0.331926\pi$
$977$	54.0000i	1.72761i	−0.503824	+	0.863807i	$0.668074\pi$
$978$	0	0
$979$	56.0000	1.78977
$980$	0	0
$981$	0	0
$982$	− 36.0000i	− 1.14881i
$983$	16.0000i	0.510321i	0.966899	+	0.255160i	$0.0821283\pi$
$983$	16.0000i	0.510321i	−0.966899	+	0.255160i	$0.917872\pi$
$984$	0	0
$985$	0	0
$986$	−16.0000	−0.509544
$987$	0	0
$988$	− 6.00000i	− 0.190885i
$989$	0	0
$990$	0	0
$991$	−48.0000	−1.52477	−0.762385	−	0.647124i	$-0.775972\pi$
$991$	−48.0000	−1.52477	−0.762385	+	0.647124i	$0.775972\pi$
$992$	50.0000i	1.58750i
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 58.0000i	− 1.83688i	−0.395562	−	0.918439i	$-0.629450\pi$
$997$	− 58.0000i	− 1.83688i	0.395562	−	0.918439i	$-0.370550\pi$
$998$	− 38.0000i	− 1.20287i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2925.2.c.g.2224.1		2
3.2	odd	2		2925.2.c.k.2224.2		2
5.2	odd	4		2925.2.a.m.1.1		1
5.3	odd	4		585.2.a.d.1.1	✓	1
5.4	even	2	inner	2925.2.c.g.2224.2		2
15.2	even	4		2925.2.a.c.1.1		1
15.8	even	4		585.2.a.i.1.1	yes	1
15.14	odd	2		2925.2.c.k.2224.1		2
20.3	even	4		9360.2.a.i.1.1		1
60.23	odd	4		9360.2.a.be.1.1		1
65.38	odd	4		7605.2.a.q.1.1		1
195.38	even	4		7605.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
585.2.a.d.1.1	✓	1	5.3	odd	4
585.2.a.i.1.1	yes	1	15.8	even	4
2925.2.a.c.1.1		1	15.2	even	4
2925.2.a.m.1.1		1	5.2	odd	4
2925.2.c.g.2224.1		2	1.1	even	1	trivial
2925.2.c.g.2224.2		2	5.4	even	2	inner
2925.2.c.k.2224.1		2	15.14	odd	2
2925.2.c.k.2224.2		2	3.2	odd	2
7605.2.a.c.1.1		1	195.38	even	4
7605.2.a.q.1.1		1	65.38	odd	4
9360.2.a.i.1.1		1	20.3	even	4
9360.2.a.be.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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000 q^{4} +2.00000i q^{7} -3.00000i q^{8} -4.00000 q^{11} +1.00000i q^{13} +2.00000 q^{14} -1.00000 q^{16} -4.00000i q^{17} -6.00000 q^{19} +4.00000i q^{22} +1.00000 q^{26} +2.00000i q^{28} +4.00000 q^{29} -10.0000 q^{31} -5.00000i q^{32} -4.00000 q^{34} -2.00000i q^{37} +6.00000i q^{38} -6.00000 q^{41} +8.00000i q^{43} -4.00000 q^{44} -8.00000i q^{47} +3.00000 q^{49} +1.00000i q^{52} +4.00000i q^{53} +6.00000 q^{56} -4.00000i q^{58} -12.0000 q^{59} +2.00000 q^{61} +10.0000i q^{62} -7.00000 q^{64} -10.0000i q^{67} -4.00000i q^{68} +6.00000i q^{73} -2.00000 q^{74} -6.00000 q^{76} -8.00000i q^{77} -12.0000 q^{79} +6.00000i q^{82} +4.00000i q^{83} +8.00000 q^{86} +12.0000i q^{88} -14.0000 q^{89} -2.00000 q^{91} -8.00000 q^{94} -14.0000i q^{97} -3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} - 8 q^{11} + 4 q^{14} - 2 q^{16} - 12 q^{19} + 2 q^{26} + 8 q^{29} - 20 q^{31} - 8 q^{34} - 12 q^{41} - 8 q^{44} + 6 q^{49} + 12 q^{56} - 24 q^{59} + 4 q^{61} - 14 q^{64} - 4 q^{74} - 12 q^{76}+ \cdots - 16 q^{94}+O(q^{100}) \) 2 * q + 2 * q^4 - 8 * q^11 + 4 * q^14 - 2 * q^16 - 12 * q^19 + 2 * q^26 + 8 * q^29 - 20 * q^31 - 8 * q^34 - 12 * q^41 - 8 * q^44 + 6 * q^49 + 12 * q^56 - 24 * q^59 + 4 * q^61 - 14 * q^64 - 4 * q^74 - 12 * q^76 - 24 * q^79 + 16 * q^86 - 28 * q^89 - 4 * q^91 - 16 * q^94

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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