LMFDB - Embedded newform 325.2.b.c.274.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [325,2,Mod(274,325)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("325.274"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	325.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$2.59513806569$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	325.274
Dual form			325.2.b.c.274.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +2.00000 q^{4} -4.00000i q^{7} +2.00000 q^{9} -6.00000 q^{11} -2.00000i q^{12} -1.00000i q^{13} +4.00000 q^{16} +6.00000i q^{17} +4.00000 q^{19} -4.00000 q^{21} -3.00000i q^{23} -5.00000i q^{27} -8.00000i q^{28} +3.00000 q^{29} -4.00000 q^{31} +6.00000i q^{33} +4.00000 q^{36} +2.00000i q^{37} -1.00000 q^{39} +6.00000 q^{41} +7.00000i q^{43} -12.0000 q^{44} -4.00000i q^{48} -9.00000 q^{49} +6.00000 q^{51} -2.00000i q^{52} +9.00000i q^{53} -4.00000i q^{57} +6.00000 q^{59} -1.00000 q^{61} -8.00000i q^{63} +8.00000 q^{64} +14.0000i q^{67} +12.0000i q^{68} -3.00000 q^{69} -6.00000 q^{71} +4.00000i q^{73} +8.00000 q^{76} +24.0000i q^{77} -11.0000 q^{79} +1.00000 q^{81} +6.00000i q^{83} -8.00000 q^{84} -3.00000i q^{87} -4.00000 q^{91} -6.00000i q^{92} +4.00000i q^{93} -10.0000i q^{97} -12.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{4} + 4 q^{9} - 12 q^{11} + 8 q^{16} + 8 q^{19} - 8 q^{21} + 6 q^{29} - 8 q^{31} + 8 q^{36} - 2 q^{39} + 12 q^{41} - 24 q^{44} - 18 q^{49} + 12 q^{51} + 12 q^{59} - 2 q^{61} + 16 q^{64} - 6 q^{69}+ \cdots - 24 q^{99}+O(q^{100}) $ 2 * q + 4 * q^4 + 4 * q^9 - 12 * q^11 + 8 * q^16 + 8 * q^19 - 8 * q^21 + 6 * q^29 - 8 * q^31 + 8 * q^36 - 2 * q^39 + 12 * q^41 - 24 * q^44 - 18 * q^49 + 12 * q^51 + 12 * q^59 - 2 * q^61 + 16 * q^64 - 6 * q^69 - 12 * q^71 + 16 * q^76 - 22 * q^79 + 2 * q^81 - 16 * q^84 - 8 * q^91 - 24 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/325\mathbb{Z}\right)^\times$.

$n$	$27$	$301$
$\chi(n)$	$-1$	$1$

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	1.00000			$0$
$2$	0	0	−1.00000			$\pi$
$3$	− 1.00000i	− 0.577350i	−0.957427	−	0.288675i	$-0.906785\pi$
$3$	− 1.00000i	− 0.577350i	0.957427	−	0.288675i	$-0.0932147\pi$
$4$	2.00000	1.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 4.00000i	− 1.51186i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	− 4.00000i	− 1.51186i	0.654654	−	0.755929i	$-0.272814\pi$
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	−6.00000	−1.80907	−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000	−1.80907	−0.904534	+	0.426401i	$0.859781\pi$
$12$	− 2.00000i	− 0.577350i
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	0	0
$15$	0	0
$16$	4.00000	1.00000
$17$	6.00000i	1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$	6.00000i	1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	−4.00000	−0.872872
$22$	0	0
$23$	− 3.00000i	− 0.625543i	−0.949828	−	0.312772i	$-0.898743\pi$
$23$	− 3.00000i	− 0.625543i	0.949828	−	0.312772i	$-0.101257\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 5.00000i	− 0.962250i
$28$	− 8.00000i	− 1.51186i
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	6.00000i	1.04447i
$34$	0	0
$35$	0	0
$36$	4.00000	0.666667
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	7.00000i	1.06749i	0.845645	+	0.533745i	$0.179216\pi$
$43$	7.00000i	1.06749i	−0.845645	+	0.533745i	$0.820784\pi$
$44$	−12.0000	−1.80907
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	− 4.00000i	− 0.577350i
$49$	−9.00000	−1.28571
$50$	0	0
$51$	6.00000	0.840168
$52$	− 2.00000i	− 0.277350i
$53$	9.00000i	1.23625i	0.786082	+	0.618123i	$0.212106\pi$
$53$	9.00000i	1.23625i	−0.786082	+	0.618123i	$0.787894\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 4.00000i	− 0.529813i
$58$	0	0
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	0	0
$63$	− 8.00000i	− 1.00791i
$64$	8.00000	1.00000
$65$	0	0
$66$	0	0
$67$	14.0000i	1.71037i	0.518321	+	0.855186i	$0.326557\pi$
$67$	14.0000i	1.71037i	−0.518321	+	0.855186i	$0.673443\pi$
$68$	12.0000i	1.45521i
$69$	−3.00000	−0.361158
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	4.00000i	0.468165i	0.972217	+	0.234082i	$0.0752085\pi$
$73$	4.00000i	0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$	0	0
$75$	0	0
$76$	8.00000	0.917663
$77$	24.0000i	2.73505i
$78$	0	0
$79$	−11.0000	−1.23760	−0.618798	−	0.785550i	$-0.712380\pi$
$79$	−11.0000	−1.23760	−0.618798	+	0.785550i	$0.712380\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	6.00000i	0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$	6.00000i	0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$	−8.00000	−0.872872
$85$	0	0
$86$	0	0
$87$	− 3.00000i	− 0.321634i
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	− 6.00000i	− 0.625543i
$93$	4.00000i	0.414781i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 10.0000i	− 1.01535i	−0.861550	−	0.507673i	$-0.830506\pi$
$97$	− 10.0000i	− 1.01535i	0.861550	−	0.507673i	$-0.169494\pi$
$98$	0	0
$99$	−12.0000	−1.20605
$100$	0	0
$101$	−15.0000	−1.49256	−0.746278	−	0.665635i	$-0.768161\pi$
$101$	−15.0000	−1.49256	−0.746278	+	0.665635i	$0.768161\pi$
$102$	0	0
$103$	− 11.0000i	− 1.08386i	−0.840423	−	0.541931i	$-0.817693\pi$
$103$	− 11.0000i	− 1.08386i	0.840423	−	0.541931i	$-0.182307\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.00000i	0.290021i	0.989430	+	0.145010i	$0.0463216\pi$
$107$	3.00000i	0.290021i	−0.989430	+	0.145010i	$0.953678\pi$
$108$	− 10.0000i	− 0.962250i
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	− 16.0000i	− 1.51186i
$113$	3.00000i	0.282216i	0.989994	+	0.141108i	$0.0450665\pi$
$113$	3.00000i	0.282216i	−0.989994	+	0.141108i	$0.954933\pi$
$114$	0	0
$115$	0	0
$116$	6.00000	0.557086
$117$	− 2.00000i	− 0.184900i
$118$	0	0
$119$	24.0000	2.20008
$120$	0	0
$121$	25.0000	2.27273
$122$	0	0
$123$	− 6.00000i	− 0.541002i
$124$	−8.00000	−0.718421
$125$	0	0
$126$	0	0
$127$	11.0000i	0.976092i	0.872818	+	0.488046i	$0.162290\pi$
$127$	11.0000i	0.976092i	−0.872818	+	0.488046i	$0.837710\pi$
$128$	0	0
$129$	7.00000	0.616316
$130$	0	0
$131$	−15.0000	−1.31056	−0.655278	−	0.755388i	$-0.727449\pi$
$131$	−15.0000	−1.31056	−0.655278	+	0.755388i	$0.727449\pi$
$132$	12.0000i	1.04447i
$133$	− 16.0000i	− 1.38738i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 18.0000i	− 1.53784i	−0.639343	−	0.768922i	$-0.720793\pi$
$137$	− 18.0000i	− 1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$	0	0
$139$	1.00000	0.0848189	0.0424094	−	0.999100i	$-0.486497\pi$
$139$	1.00000	0.0848189	0.0424094	+	0.999100i	$0.486497\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	6.00000i	0.501745i
$144$	8.00000	0.666667
$145$	0	0
$146$	0	0
$147$	9.00000i	0.742307i
$148$	4.00000i	0.328798i
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	0	0
$153$	12.0000i	0.970143i
$154$	0	0
$155$	0	0
$156$	−2.00000	−0.160128
$157$	− 10.0000i	− 0.798087i	−0.916932	−	0.399043i	$-0.869342\pi$
$157$	− 10.0000i	− 0.798087i	0.916932	−	0.399043i	$-0.130658\pi$
$158$	0	0
$159$	9.00000	0.713746
$160$	0	0
$161$	−12.0000	−0.945732
$162$	0	0
$163$	− 2.00000i	− 0.156652i	−0.996928	−	0.0783260i	$-0.975042\pi$
$163$	− 2.00000i	− 0.156652i	0.996928	−	0.0783260i	$-0.0249575\pi$
$164$	12.0000	0.937043
$165$	0	0
$166$	0	0
$167$	− 18.0000i	− 1.39288i	−0.717614	−	0.696441i	$-0.754766\pi$
$167$	− 18.0000i	− 1.39288i	0.717614	−	0.696441i	$-0.245234\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	8.00000	0.611775
$172$	14.0000i	1.06749i
$173$	− 18.0000i	− 1.36851i	−0.729241	−	0.684257i	$-0.760127\pi$
$173$	− 18.0000i	− 1.36851i	0.729241	−	0.684257i	$-0.239873\pi$
$174$	0	0
$175$	0	0
$176$	−24.0000	−1.80907
$177$	− 6.00000i	− 0.450988i
$178$	0	0
$179$	21.0000	1.56961	0.784807	−	0.619740i	$-0.212762\pi$
$179$	21.0000	1.56961	0.784807	+	0.619740i	$0.212762\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	1.00000i	0.0739221i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 36.0000i	− 2.63258i
$188$	0	0
$189$	−20.0000	−1.45479
$190$	0	0
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	− 8.00000i	− 0.577350i
$193$	16.0000i	1.15171i	0.817554	+	0.575853i	$0.195330\pi$
$193$	16.0000i	1.15171i	−0.817554	+	0.575853i	$0.804670\pi$
$194$	0	0
$195$	0	0
$196$	−18.0000	−1.28571
$197$	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
$199$	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
$201$	14.0000	0.987484
$202$	0	0
$203$	− 12.0000i	− 0.842235i
$204$	12.0000	0.840168
$205$	0	0
$206$	0	0
$207$	− 6.00000i	− 0.417029i
$208$	− 4.00000i	− 0.277350i
$209$	−24.0000	−1.66011
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	18.0000i	1.23625i
$213$	6.00000i	0.411113i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	16.0000i	1.08615i
$218$	0	0
$219$	4.00000	0.270295
$220$	0	0
$221$	6.00000	0.403604
$222$	0	0
$223$	− 14.0000i	− 0.937509i	−0.883328	−	0.468755i	$-0.844703\pi$
$223$	− 14.0000i	− 0.937509i	0.883328	−	0.468755i	$-0.155297\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	12.0000i	0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$	12.0000i	0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$	− 8.00000i	− 0.529813i
$229$	−20.0000	−1.32164	−0.660819	−	0.750546i	$-0.729791\pi$
$229$	−20.0000	−1.32164	−0.660819	+	0.750546i	$0.729791\pi$
$230$	0	0
$231$	24.0000	1.57908
$232$	0	0
$233$	3.00000i	0.196537i	0.995160	+	0.0982683i	$0.0313303\pi$
$233$	3.00000i	0.196537i	−0.995160	+	0.0982683i	$0.968670\pi$
$234$	0	0
$235$	0	0
$236$	12.0000	0.781133
$237$	11.0000i	0.714527i
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	− 16.0000i	− 1.02640i
$244$	−2.00000	−0.128037
$245$	0	0
$246$	0	0
$247$	− 4.00000i	− 0.254514i
$248$	0	0
$249$	6.00000	0.380235
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	− 16.0000i	− 1.00791i
$253$	18.0000i	1.13165i
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	− 27.0000i	− 1.68421i	−0.539311	−	0.842107i	$-0.681315\pi$
$257$	− 27.0000i	− 1.68421i	0.539311	−	0.842107i	$-0.318685\pi$
$258$	0	0
$259$	8.00000	0.497096
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	− 24.0000i	− 1.47990i	−0.672660	−	0.739952i	$-0.734848\pi$
$263$	− 24.0000i	− 1.47990i	0.672660	−	0.739952i	$-0.265152\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	28.0000i	1.71037i
$269$	−21.0000	−1.28039	−0.640196	−	0.768211i	$-0.721147\pi$
$269$	−21.0000	−1.28039	−0.640196	+	0.768211i	$0.721147\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	24.0000i	1.45521i
$273$	4.00000i	0.242091i
$274$	0	0
$275$	0	0
$276$	−6.00000	−0.361158
$277$	5.00000i	0.300421i	0.988654	+	0.150210i	$0.0479951\pi$
$277$	5.00000i	0.300421i	−0.988654	+	0.150210i	$0.952005\pi$
$278$	0	0
$279$	−8.00000	−0.478947
$280$	0	0
$281$	−24.0000	−1.43172	−0.715860	−	0.698244i	$-0.753965\pi$
$281$	−24.0000	−1.43172	−0.715860	+	0.698244i	$0.753965\pi$
$282$	0	0
$283$	4.00000i	0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$	4.00000i	0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	0	0
$287$	− 24.0000i	− 1.41668i
$288$	0	0
$289$	−19.0000	−1.11765
$290$	0	0
$291$	−10.0000	−0.586210
$292$	8.00000i	0.468165i
$293$	6.00000i	0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$	6.00000i	0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	30.0000i	1.74078i
$298$	0	0
$299$	−3.00000	−0.173494
$300$	0	0
$301$	28.0000	1.61389
$302$	0	0
$303$	15.0000i	0.861727i
$304$	16.0000	0.917663
$305$	0	0
$306$	0	0
$307$	2.00000i	0.114146i	0.998370	+	0.0570730i	$0.0181768\pi$
$307$	2.00000i	0.114146i	−0.998370	+	0.0570730i	$0.981823\pi$
$308$	48.0000i	2.73505i
$309$	−11.0000	−0.625768
$310$	0	0
$311$	27.0000	1.53103	0.765515	−	0.643418i	$-0.222484\pi$
$311$	27.0000	1.53103	0.765515	+	0.643418i	$0.222484\pi$
$312$	0	0
$313$	− 17.0000i	− 0.960897i	−0.877023	−	0.480448i	$-0.840474\pi$
$313$	− 17.0000i	− 0.960897i	0.877023	−	0.480448i	$-0.159526\pi$
$314$	0	0
$315$	0	0
$316$	−22.0000	−1.23760
$317$	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
$319$	−18.0000	−1.00781
$320$	0	0
$321$	3.00000	0.167444
$322$	0	0
$323$	24.0000i	1.33540i
$324$	2.00000	0.111111
$325$	0	0
$326$	0	0
$327$	2.00000i	0.110600i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−16.0000	−0.879440	−0.439720	−	0.898135i	$-0.644922\pi$
$331$	−16.0000	−0.879440	−0.439720	+	0.898135i	$0.644922\pi$
$332$	12.0000i	0.658586i
$333$	4.00000i	0.219199i
$334$	0	0
$335$	0	0
$336$	−16.0000	−0.872872
$337$	23.0000i	1.25289i	0.779466	+	0.626445i	$0.215491\pi$
$337$	23.0000i	1.25289i	−0.779466	+	0.626445i	$0.784509\pi$
$338$	0	0
$339$	3.00000	0.162938
$340$	0	0
$341$	24.0000	1.29967
$342$	0	0
$343$	8.00000i	0.431959i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 3.00000i	− 0.161048i	−0.996753	−	0.0805242i	$-0.974341\pi$
$347$	− 3.00000i	− 0.161048i	0.996753	−	0.0805242i	$-0.0256594\pi$
$348$	− 6.00000i	− 0.321634i
$349$	16.0000	0.856460	0.428230	−	0.903670i	$-0.359137\pi$
$349$	16.0000	0.856460	0.428230	+	0.903670i	$0.359137\pi$
$350$	0	0
$351$	−5.00000	−0.266880
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 24.0000i	− 1.27021i
$358$	0	0
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	− 25.0000i	− 1.31216i
$364$	−8.00000	−0.419314
$365$	0	0
$366$	0	0
$367$	− 19.0000i	− 0.991792i	−0.868382	−	0.495896i	$-0.834840\pi$
$367$	− 19.0000i	− 0.991792i	0.868382	−	0.495896i	$-0.165160\pi$
$368$	− 12.0000i	− 0.625543i
$369$	12.0000	0.624695
$370$	0	0
$371$	36.0000	1.86903
$372$	8.00000i	0.414781i
$373$	− 29.0000i	− 1.50156i	−0.660551	−	0.750782i	$-0.729677\pi$
$373$	− 29.0000i	− 1.50156i	0.660551	−	0.750782i	$-0.270323\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 3.00000i	− 0.154508i
$378$	0	0
$379$	−2.00000	−0.102733	−0.0513665	−	0.998680i	$-0.516358\pi$
$379$	−2.00000	−0.102733	−0.0513665	+	0.998680i	$0.516358\pi$
$380$	0	0
$381$	11.0000	0.563547
$382$	0	0
$383$	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
$386$	0	0
$387$	14.0000i	0.711660i
$388$	− 20.0000i	− 1.01535i
$389$	9.00000	0.456318	0.228159	−	0.973624i	$-0.426729\pi$
$389$	9.00000	0.456318	0.228159	+	0.973624i	$0.426729\pi$
$390$	0	0
$391$	18.0000	0.910299
$392$	0	0
$393$	15.0000i	0.756650i
$394$	0	0
$395$	0	0
$396$	−24.0000	−1.20605
$397$	− 4.00000i	− 0.200754i	−0.994949	−	0.100377i	$-0.967995\pi$
$397$	− 4.00000i	− 0.200754i	0.994949	−	0.100377i	$-0.0320049\pi$
$398$	0	0
$399$	−16.0000	−0.801002
$400$	0	0
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	4.00000i	0.199254i
$404$	−30.0000	−1.49256
$405$	0	0
$406$	0	0
$407$	− 12.0000i	− 0.594818i
$408$	0	0
$409$	−32.0000	−1.58230	−0.791149	−	0.611623i	$-0.790517\pi$
$409$	−32.0000	−1.58230	−0.791149	+	0.611623i	$0.790517\pi$
$410$	0	0
$411$	−18.0000	−0.887875
$412$	− 22.0000i	− 1.08386i
$413$	− 24.0000i	− 1.18096i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 1.00000i	− 0.0489702i
$418$	0	0
$419$	−3.00000	−0.146560	−0.0732798	−	0.997311i	$-0.523347\pi$
$419$	−3.00000	−0.146560	−0.0732798	+	0.997311i	$0.523347\pi$
$420$	0	0
$421$	−4.00000	−0.194948	−0.0974740	−	0.995238i	$-0.531076\pi$
$421$	−4.00000	−0.194948	−0.0974740	+	0.995238i	$0.531076\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4.00000i	0.193574i
$428$	6.00000i	0.290021i
$429$	6.00000	0.289683
$430$	0	0
$431$	−12.0000	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$432$	− 20.0000i	− 0.962250i
$433$	7.00000i	0.336399i	0.985753	+	0.168199i	$0.0537952\pi$
$433$	7.00000i	0.336399i	−0.985753	+	0.168199i	$0.946205\pi$
$434$	0	0
$435$	0	0
$436$	−4.00000	−0.191565
$437$	− 12.0000i	− 0.574038i
$438$	0	0
$439$	16.0000	0.763638	0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	0	0
$441$	−18.0000	−0.857143
$442$	0	0
$443$	12.0000i	0.570137i	0.958507	+	0.285069i	$0.0920164\pi$
$443$	12.0000i	0.570137i	−0.958507	+	0.285069i	$0.907984\pi$
$444$	4.00000	0.189832
$445$	0	0
$446$	0	0
$447$	− 6.00000i	− 0.283790i
$448$	− 32.0000i	− 1.51186i
$449$	36.0000	1.69895	0.849473	−	0.527633i	$-0.176920\pi$
$449$	36.0000	1.69895	0.849473	+	0.527633i	$0.176920\pi$
$450$	0	0
$451$	−36.0000	−1.69517
$452$	6.00000i	0.282216i
$453$	− 2.00000i	− 0.0939682i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	8.00000i	0.374224i	0.982339	+	0.187112i	$0.0599128\pi$
$457$	8.00000i	0.374224i	−0.982339	+	0.187112i	$0.940087\pi$
$458$	0	0
$459$	30.0000	1.40028
$460$	0	0
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	0	0
$463$	− 26.0000i	− 1.20832i	−0.796862	−	0.604161i	$-0.793508\pi$
$463$	− 26.0000i	− 1.20832i	0.796862	−	0.604161i	$-0.206492\pi$
$464$	12.0000	0.557086
$465$	0	0
$466$	0	0
$467$	− 15.0000i	− 0.694117i	−0.937843	−	0.347059i	$-0.887180\pi$
$467$	− 15.0000i	− 0.694117i	0.937843	−	0.347059i	$-0.112820\pi$
$468$	− 4.00000i	− 0.184900i
$469$	56.0000	2.58584
$470$	0	0
$471$	−10.0000	−0.460776
$472$	0	0
$473$	− 42.0000i	− 1.93116i
$474$	0	0
$475$	0	0
$476$	48.0000	2.20008
$477$	18.0000i	0.824163i
$478$	0	0
$479$	−18.0000	−0.822441	−0.411220	−	0.911536i	$-0.634897\pi$
$479$	−18.0000	−0.822441	−0.411220	+	0.911536i	$0.634897\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	0	0
$483$	12.0000i	0.546019i
$484$	50.0000	2.27273
$485$	0	0
$486$	0	0
$487$	20.0000i	0.906287i	0.891438	+	0.453143i	$0.149697\pi$
$487$	20.0000i	0.906287i	−0.891438	+	0.453143i	$0.850303\pi$
$488$	0	0
$489$	−2.00000	−0.0904431
$490$	0	0
$491$	−33.0000	−1.48927	−0.744635	−	0.667472i	$-0.767376\pi$
$491$	−33.0000	−1.48927	−0.744635	+	0.667472i	$0.767376\pi$
$492$	− 12.0000i	− 0.541002i
$493$	18.0000i	0.810679i
$494$	0	0
$495$	0	0
$496$	−16.0000	−0.718421
$497$	24.0000i	1.07655i
$498$	0	0
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	0	0
$501$	−18.0000	−0.804181
$502$	0	0
$503$	− 9.00000i	− 0.401290i	−0.979664	−	0.200645i	$-0.935696\pi$
$503$	− 9.00000i	− 0.401290i	0.979664	−	0.200645i	$-0.0643038\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.00000i	0.0444116i
$508$	22.0000i	0.976092i
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	0	0
$511$	16.0000	0.707798
$512$	0	0
$513$	− 20.0000i	− 0.883022i
$514$	0	0
$515$	0	0
$516$	14.0000	0.616316
$517$	0	0
$518$	0	0
$519$	−18.0000	−0.790112
$520$	0	0
$521$	−27.0000	−1.18289	−0.591446	−	0.806345i	$-0.701443\pi$
$521$	−27.0000	−1.18289	−0.591446	+	0.806345i	$0.701443\pi$
$522$	0	0
$523$	40.0000i	1.74908i	0.484955	+	0.874539i	$0.338836\pi$
$523$	40.0000i	1.74908i	−0.484955	+	0.874539i	$0.661164\pi$
$524$	−30.0000	−1.31056
$525$	0	0
$526$	0	0
$527$	− 24.0000i	− 1.04546i
$528$	24.0000i	1.04447i
$529$	14.0000	0.608696
$530$	0	0
$531$	12.0000	0.520756
$532$	− 32.0000i	− 1.38738i
$533$	− 6.00000i	− 0.259889i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 21.0000i	− 0.906217i
$538$	0	0
$539$	54.0000	2.32594
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	0	0
$543$	− 2.00000i	− 0.0858282i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	20.0000i	0.855138i	0.903983	+	0.427569i	$0.140630\pi$
$547$	20.0000i	0.855138i	−0.903983	+	0.427569i	$0.859370\pi$
$548$	− 36.0000i	− 1.53784i
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	12.0000	0.511217
$552$	0	0
$553$	44.0000i	1.87107i
$554$	0	0
$555$	0	0
$556$	2.00000	0.0848189
$557$	− 30.0000i	− 1.27114i	−0.772043	−	0.635570i	$-0.780765\pi$
$557$	− 30.0000i	− 1.27114i	0.772043	−	0.635570i	$-0.219235\pi$
$558$	0	0
$559$	7.00000	0.296068
$560$	0	0
$561$	−36.0000	−1.51992
$562$	0	0
$563$	− 3.00000i	− 0.126435i	−0.998000	−	0.0632175i	$-0.979864\pi$
$563$	− 3.00000i	− 0.126435i	0.998000	−	0.0632175i	$-0.0201362\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 4.00000i	− 0.167984i
$568$	0	0
$569$	−33.0000	−1.38343	−0.691716	−	0.722170i	$-0.743145\pi$
$569$	−33.0000	−1.38343	−0.691716	+	0.722170i	$0.743145\pi$
$570$	0	0
$571$	−28.0000	−1.17176	−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000	−1.17176	−0.585882	+	0.810397i	$0.699252\pi$
$572$	12.0000i	0.501745i
$573$	3.00000i	0.125327i
$574$	0	0
$575$	0	0
$576$	16.0000	0.666667
$577$	14.0000i	0.582828i	0.956597	+	0.291414i	$0.0941257\pi$
$577$	14.0000i	0.582828i	−0.956597	+	0.291414i	$0.905874\pi$
$578$	0	0
$579$	16.0000	0.664937
$580$	0	0
$581$	24.0000	0.995688
$582$	0	0
$583$	− 54.0000i	− 2.23645i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 42.0000i	− 1.73353i	−0.498721	−	0.866763i	$-0.666197\pi$
$587$	− 42.0000i	− 1.73353i	0.498721	−	0.866763i	$-0.333803\pi$
$588$	18.0000i	0.742307i
$589$	−16.0000	−0.659269
$590$	0	0
$591$	18.0000	0.740421
$592$	8.00000i	0.328798i
$593$	− 48.0000i	− 1.97112i	−0.169316	−	0.985562i	$-0.554156\pi$
$593$	− 48.0000i	− 1.97112i	0.169316	−	0.985562i	$-0.445844\pi$
$594$	0	0
$595$	0	0
$596$	12.0000	0.491539
$597$	− 25.0000i	− 1.02318i
$598$	0	0
$599$	15.0000	0.612883	0.306442	−	0.951889i	$-0.400862\pi$
$599$	15.0000	0.612883	0.306442	+	0.951889i	$0.400862\pi$
$600$	0	0
$601$	5.00000	0.203954	0.101977	−	0.994787i	$-0.467483\pi$
$601$	5.00000	0.203954	0.101977	+	0.994787i	$0.467483\pi$
$602$	0	0
$603$	28.0000i	1.14025i
$604$	4.00000	0.162758
$605$	0	0
$606$	0	0
$607$	32.0000i	1.29884i	0.760430	+	0.649420i	$0.224988\pi$
$607$	32.0000i	1.29884i	−0.760430	+	0.649420i	$0.775012\pi$
$608$	0	0
$609$	−12.0000	−0.486265
$610$	0	0
$611$	0	0
$612$	24.0000i	0.970143i
$613$	40.0000i	1.61558i	0.589467	+	0.807792i	$0.299338\pi$
$613$	40.0000i	1.61558i	−0.589467	+	0.807792i	$0.700662\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 6.00000i	− 0.241551i	−0.992680	−	0.120775i	$-0.961462\pi$
$617$	− 6.00000i	− 0.241551i	0.992680	−	0.120775i	$-0.0385381\pi$
$618$	0	0
$619$	−14.0000	−0.562708	−0.281354	−	0.959604i	$-0.590783\pi$
$619$	−14.0000	−0.562708	−0.281354	+	0.959604i	$0.590783\pi$
$620$	0	0
$621$	−15.0000	−0.601929
$622$	0	0
$623$	0	0
$624$	−4.00000	−0.160128
$625$	0	0
$626$	0	0
$627$	24.0000i	0.958468i
$628$	− 20.0000i	− 0.798087i
$629$	−12.0000	−0.478471
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	0	0
$633$	4.00000i	0.158986i
$634$	0	0
$635$	0	0
$636$	18.0000	0.713746
$637$	9.00000i	0.356593i
$638$	0	0
$639$	−12.0000	−0.474713
$640$	0	0
$641$	15.0000	0.592464	0.296232	−	0.955116i	$-0.404270\pi$
$641$	15.0000	0.592464	0.296232	+	0.955116i	$0.404270\pi$
$642$	0	0
$643$	4.00000i	0.157745i	0.996885	+	0.0788723i	$0.0251319\pi$
$643$	4.00000i	0.157745i	−0.996885	+	0.0788723i	$0.974868\pi$
$644$	−24.0000	−0.945732
$645$	0	0
$646$	0	0
$647$	36.0000i	1.41531i	0.706560	+	0.707653i	$0.250246\pi$
$647$	36.0000i	1.41531i	−0.706560	+	0.707653i	$0.749754\pi$
$648$	0	0
$649$	−36.0000	−1.41312
$650$	0	0
$651$	16.0000	0.627089
$652$	− 4.00000i	− 0.156652i
$653$	− 30.0000i	− 1.17399i	−0.809590	−	0.586995i	$-0.800311\pi$
$653$	− 30.0000i	− 1.17399i	0.809590	−	0.586995i	$-0.199689\pi$
$654$	0	0
$655$	0	0
$656$	24.0000	0.937043
$657$	8.00000i	0.312110i
$658$	0	0
$659$	15.0000	0.584317	0.292159	−	0.956370i	$-0.405627\pi$
$659$	15.0000	0.584317	0.292159	+	0.956370i	$0.405627\pi$
$660$	0	0
$661$	−34.0000	−1.32245	−0.661223	−	0.750189i	$-0.729962\pi$
$661$	−34.0000	−1.32245	−0.661223	+	0.750189i	$0.729962\pi$
$662$	0	0
$663$	− 6.00000i	− 0.233021i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 9.00000i	− 0.348481i
$668$	− 36.0000i	− 1.39288i
$669$	−14.0000	−0.541271
$670$	0	0
$671$	6.00000	0.231627
$672$	0	0
$673$	34.0000i	1.31060i	0.755367	+	0.655302i	$0.227459\pi$
$673$	34.0000i	1.31060i	−0.755367	+	0.655302i	$0.772541\pi$
$674$	0	0
$675$	0	0
$676$	−2.00000	−0.0769231
$677$	15.0000i	0.576497i	0.957556	+	0.288248i	$0.0930729\pi$
$677$	15.0000i	0.576497i	−0.957556	+	0.288248i	$0.906927\pi$
$678$	0	0
$679$	−40.0000	−1.53506
$680$	0	0
$681$	12.0000	0.459841
$682$	0	0
$683$	12.0000i	0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$	12.0000i	0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	16.0000	0.611775
$685$	0	0
$686$	0	0
$687$	20.0000i	0.763048i
$688$	28.0000i	1.06749i
$689$	9.00000	0.342873
$690$	0	0
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	− 36.0000i	− 1.36851i
$693$	48.0000i	1.82337i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	36.0000i	1.36360i
$698$	0	0
$699$	3.00000	0.113470
$700$	0	0
$701$	−33.0000	−1.24639	−0.623196	−	0.782065i	$-0.714166\pi$
$701$	−33.0000	−1.24639	−0.623196	+	0.782065i	$0.714166\pi$
$702$	0	0
$703$	8.00000i	0.301726i
$704$	−48.0000	−1.80907
$705$	0	0
$706$	0	0
$707$	60.0000i	2.25653i
$708$	− 12.0000i	− 0.450988i
$709$	40.0000	1.50223	0.751116	−	0.660171i	$-0.229516\pi$
$709$	40.0000	1.50223	0.751116	+	0.660171i	$0.229516\pi$
$710$	0	0
$711$	−22.0000	−0.825064
$712$	0	0
$713$	12.0000i	0.449404i
$714$	0	0
$715$	0	0
$716$	42.0000	1.56961
$717$	12.0000i	0.448148i
$718$	0	0
$719$	−27.0000	−1.00693	−0.503465	−	0.864016i	$-0.667942\pi$
$719$	−27.0000	−1.00693	−0.503465	+	0.864016i	$0.667942\pi$
$720$	0	0
$721$	−44.0000	−1.63865
$722$	0	0
$723$	10.0000i	0.371904i
$724$	4.00000	0.148659
$725$	0	0
$726$	0	0
$727$	− 7.00000i	− 0.259616i	−0.991539	−	0.129808i	$-0.958564\pi$
$727$	− 7.00000i	− 0.259616i	0.991539	−	0.129808i	$-0.0414360\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	−42.0000	−1.55343
$732$	2.00000i	0.0739221i
$733$	40.0000i	1.47743i	0.674016	+	0.738717i	$0.264568\pi$
$733$	40.0000i	1.47743i	−0.674016	+	0.738717i	$0.735432\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 84.0000i	− 3.09418i
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	−4.00000	−0.146944
$742$	0	0
$743$	18.0000i	0.660356i	0.943919	+	0.330178i	$0.107109\pi$
$743$	18.0000i	0.660356i	−0.943919	+	0.330178i	$0.892891\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	12.0000i	0.439057i
$748$	− 72.0000i	− 2.63258i
$749$	12.0000	0.438470
$750$	0	0
$751$	23.0000	0.839282	0.419641	−	0.907690i	$-0.362156\pi$
$751$	23.0000	0.839282	0.419641	+	0.907690i	$0.362156\pi$
$752$	0	0
$753$	12.0000i	0.437304i
$754$	0	0
$755$	0	0
$756$	−40.0000	−1.45479
$757$	− 22.0000i	− 0.799604i	−0.916602	−	0.399802i	$-0.869079\pi$
$757$	− 22.0000i	− 0.799604i	0.916602	−	0.399802i	$-0.130921\pi$
$758$	0	0
$759$	18.0000	0.653359
$760$	0	0
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	0	0
$763$	8.00000i	0.289619i
$764$	−6.00000	−0.217072
$765$	0	0
$766$	0	0
$767$	− 6.00000i	− 0.216647i
$768$	− 16.0000i	− 0.577350i
$769$	4.00000	0.144244	0.0721218	−	0.997396i	$-0.477023\pi$
$769$	4.00000	0.144244	0.0721218	+	0.997396i	$0.477023\pi$
$770$	0	0
$771$	−27.0000	−0.972381
$772$	32.0000i	1.15171i
$773$	48.0000i	1.72644i	0.504828	+	0.863220i	$0.331556\pi$
$773$	48.0000i	1.72644i	−0.504828	+	0.863220i	$0.668444\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 8.00000i	− 0.286998i
$778$	0	0
$779$	24.0000	0.859889
$780$	0	0
$781$	36.0000	1.28818
$782$	0	0
$783$	− 15.0000i	− 0.536056i
$784$	−36.0000	−1.28571
$785$	0	0
$786$	0	0
$787$	32.0000i	1.14068i	0.821410	+	0.570338i	$0.193188\pi$
$787$	32.0000i	1.14068i	−0.821410	+	0.570338i	$0.806812\pi$
$788$	36.0000i	1.28245i
$789$	−24.0000	−0.854423
$790$	0	0
$791$	12.0000	0.426671
$792$	0	0
$793$	1.00000i	0.0355110i
$794$	0	0
$795$	0	0
$796$	50.0000	1.77220
$797$	− 21.0000i	− 0.743858i	−0.928261	−	0.371929i	$-0.878696\pi$
$797$	− 21.0000i	− 0.743858i	0.928261	−	0.371929i	$-0.121304\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 24.0000i	− 0.846942i
$804$	28.0000	0.987484
$805$	0	0
$806$	0	0
$807$	21.0000i	0.739235i
$808$	0	0
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	0	0
$811$	50.0000	1.75574	0.877869	−	0.478901i	$-0.158965\pi$
$811$	50.0000	1.75574	0.877869	+	0.478901i	$0.158965\pi$
$812$	− 24.0000i	− 0.842235i
$813$	− 8.00000i	− 0.280572i
$814$	0	0
$815$	0	0
$816$	24.0000	0.840168
$817$	28.0000i	0.979596i
$818$	0	0
$819$	−8.00000	−0.279543
$820$	0	0
$821$	24.0000	0.837606	0.418803	−	0.908077i	$-0.362450\pi$
$821$	24.0000	0.837606	0.418803	+	0.908077i	$0.362450\pi$
$822$	0	0
$823$	31.0000i	1.08059i	0.841475	+	0.540296i	$0.181688\pi$
$823$	31.0000i	1.08059i	−0.841475	+	0.540296i	$0.818312\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	54.0000i	1.87776i	0.344239	+	0.938882i	$0.388137\pi$
$827$	54.0000i	1.87776i	−0.344239	+	0.938882i	$0.611863\pi$
$828$	− 12.0000i	− 0.417029i
$829$	31.0000	1.07667	0.538337	−	0.842729i	$-0.319053\pi$
$829$	31.0000	1.07667	0.538337	+	0.842729i	$0.319053\pi$
$830$	0	0
$831$	5.00000	0.173448
$832$	− 8.00000i	− 0.277350i
$833$	− 54.0000i	− 1.87099i
$834$	0	0
$835$	0	0
$836$	−48.0000	−1.66011
$837$	20.0000i	0.691301i
$838$	0	0
$839$	6.00000	0.207143	0.103572	−	0.994622i	$-0.466973\pi$
$839$	6.00000	0.207143	0.103572	+	0.994622i	$0.466973\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	24.0000i	0.826604i
$844$	−8.00000	−0.275371
$845$	0	0
$846$	0	0
$847$	− 100.000i	− 3.43604i
$848$	36.0000i	1.23625i
$849$	4.00000	0.137280
$850$	0	0
$851$	6.00000	0.205677
$852$	12.0000i	0.411113i
$853$	− 44.0000i	− 1.50653i	−0.657716	−	0.753266i	$-0.728477\pi$
$853$	− 44.0000i	− 1.50653i	0.657716	−	0.753266i	$-0.271523\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 27.0000i	− 0.922302i	−0.887322	−	0.461151i	$-0.847437\pi$
$857$	− 27.0000i	− 0.922302i	0.887322	−	0.461151i	$-0.152563\pi$
$858$	0	0
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	0	0
$861$	−24.0000	−0.817918
$862$	0	0
$863$	− 36.0000i	− 1.22545i	−0.790295	−	0.612727i	$-0.790072\pi$
$863$	− 36.0000i	− 1.22545i	0.790295	−	0.612727i	$-0.209928\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	19.0000i	0.645274i
$868$	32.0000i	1.08615i
$869$	66.0000	2.23890
$870$	0	0
$871$	14.0000	0.474372
$872$	0	0
$873$	− 20.0000i	− 0.676897i
$874$	0	0
$875$	0	0
$876$	8.00000	0.270295
$877$	− 28.0000i	− 0.945493i	−0.881199	−	0.472746i	$-0.843263\pi$
$877$	− 28.0000i	− 0.945493i	0.881199	−	0.472746i	$-0.156737\pi$
$878$	0	0
$879$	6.00000	0.202375
$880$	0	0
$881$	33.0000	1.11180	0.555899	−	0.831250i	$-0.312374\pi$
$881$	33.0000	1.11180	0.555899	+	0.831250i	$0.312374\pi$
$882$	0	0
$883$	− 29.0000i	− 0.975928i	−0.872864	−	0.487964i	$-0.837740\pi$
$883$	− 29.0000i	− 0.975928i	0.872864	−	0.487964i	$-0.162260\pi$
$884$	12.0000	0.403604
$885$	0	0
$886$	0	0
$887$	− 24.0000i	− 0.805841i	−0.915235	−	0.402921i	$-0.867995\pi$
$887$	− 24.0000i	− 0.805841i	0.915235	−	0.402921i	$-0.132005\pi$
$888$	0	0
$889$	44.0000	1.47571
$890$	0	0
$891$	−6.00000	−0.201008
$892$	− 28.0000i	− 0.937509i
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	3.00000i	0.100167i
$898$	0	0
$899$	−12.0000	−0.400222
$900$	0	0
$901$	−54.0000	−1.79900
$902$	0	0
$903$	− 28.0000i	− 0.931782i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 19.0000i	− 0.630885i	−0.948945	−	0.315442i	$-0.897847\pi$
$907$	− 19.0000i	− 0.630885i	0.948945	−	0.315442i	$-0.102153\pi$
$908$	24.0000i	0.796468i
$909$	−30.0000	−0.995037
$910$	0	0
$911$	−21.0000	−0.695761	−0.347881	−	0.937539i	$-0.613099\pi$
$911$	−21.0000	−0.695761	−0.347881	+	0.937539i	$0.613099\pi$
$912$	− 16.0000i	− 0.529813i
$913$	− 36.0000i	− 1.19143i
$914$	0	0
$915$	0	0
$916$	−40.0000	−1.32164
$917$	60.0000i	1.98137i
$918$	0	0
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	0	0
$921$	2.00000	0.0659022
$922$	0	0
$923$	6.00000i	0.197492i
$924$	48.0000	1.57908
$925$	0	0
$926$	0	0
$927$	− 22.0000i	− 0.722575i
$928$	0	0
$929$	36.0000	1.18112	0.590561	−	0.806993i	$-0.298907\pi$
$929$	36.0000	1.18112	0.590561	+	0.806993i	$0.298907\pi$
$930$	0	0
$931$	−36.0000	−1.17985
$932$	6.00000i	0.196537i
$933$	− 27.0000i	− 0.883940i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	23.0000i	0.751377i	0.926746	+	0.375689i	$0.122594\pi$
$937$	23.0000i	0.751377i	−0.926746	+	0.375689i	$0.877406\pi$
$938$	0	0
$939$	−17.0000	−0.554774
$940$	0	0
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	0	0
$943$	− 18.0000i	− 0.586161i
$944$	24.0000	0.781133
$945$	0	0
$946$	0	0
$947$	− 42.0000i	− 1.36482i	−0.730971	−	0.682408i	$-0.760933\pi$
$947$	− 42.0000i	− 1.36482i	0.730971	−	0.682408i	$-0.239067\pi$
$948$	22.0000i	0.714527i
$949$	4.00000	0.129845
$950$	0	0
$951$	18.0000	0.583690
$952$	0	0
$953$	6.00000i	0.194359i	0.995267	+	0.0971795i	$0.0309821\pi$
$953$	6.00000i	0.194359i	−0.995267	+	0.0971795i	$0.969018\pi$
$954$	0	0
$955$	0	0
$956$	−24.0000	−0.776215
$957$	18.0000i	0.581857i
$958$	0	0
$959$	−72.0000	−2.32500
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	6.00000i	0.193347i
$964$	−20.0000	−0.644157
$965$	0	0
$966$	0	0
$967$	14.0000i	0.450210i	0.974335	+	0.225105i	$0.0722725\pi$
$967$	14.0000i	0.450210i	−0.974335	+	0.225105i	$0.927728\pi$
$968$	0	0
$969$	24.0000	0.770991
$970$	0	0
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	− 32.0000i	− 1.02640i
$973$	− 4.00000i	− 0.128234i
$974$	0	0
$975$	0	0
$976$	−4.00000	−0.128037
$977$	− 42.0000i	− 1.34370i	−0.740688	−	0.671850i	$-0.765500\pi$
$977$	− 42.0000i	− 1.34370i	0.740688	−	0.671850i	$-0.234500\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−4.00000	−0.127710
$982$	0	0
$983$	42.0000i	1.33959i	0.742545	+	0.669796i	$0.233618\pi$
$983$	42.0000i	1.33959i	−0.742545	+	0.669796i	$0.766382\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	− 8.00000i	− 0.254514i
$989$	21.0000	0.667761
$990$	0	0
$991$	−49.0000	−1.55654	−0.778268	−	0.627932i	$-0.783902\pi$
$991$	−49.0000	−1.55654	−0.778268	+	0.627932i	$0.783902\pi$
$992$	0	0
$993$	16.0000i	0.507745i
$994$	0	0
$995$	0	0
$996$	12.0000	0.380235
$997$	− 1.00000i	− 0.0316703i	−0.999875	−	0.0158352i	$-0.994959\pi$
$997$	− 1.00000i	− 0.0316703i	0.999875	−	0.0158352i	$-0.00504070\pi$
$998$	0	0
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.2.b.c.274.1		2
3.2	odd	2		2925.2.c.n.2224.1		2
5.2	odd	4		325.2.a.b.1.1	✓	1
5.3	odd	4		325.2.a.c.1.1	yes	1
5.4	even	2	inner	325.2.b.c.274.2		2
15.2	even	4		2925.2.a.l.1.1		1
15.8	even	4		2925.2.a.g.1.1		1
15.14	odd	2		2925.2.c.n.2224.2		2
20.3	even	4		5200.2.a.n.1.1		1
20.7	even	4		5200.2.a.v.1.1		1
65.12	odd	4		4225.2.a.i.1.1		1
65.38	odd	4		4225.2.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.2.a.b.1.1	✓	1	5.2	odd	4
325.2.a.c.1.1	yes	1	5.3	odd	4
325.2.b.c.274.1		2	1.1	even	1	trivial
325.2.b.c.274.2		2	5.4	even	2	inner
2925.2.a.g.1.1		1	15.8	even	4
2925.2.a.l.1.1		1	15.2	even	4
2925.2.c.n.2224.1		2	3.2	odd	2
2925.2.c.n.2224.2		2	15.14	odd	2
4225.2.a.i.1.1		1	65.12	odd	4
4225.2.a.j.1.1		1	65.38	odd	4
5200.2.a.n.1.1		1	20.3	even	4
5200.2.a.v.1.1		1	20.7	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	325.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +2.00000 q^{4} -4.00000i q^{7} +2.00000 q^{9} -6.00000 q^{11} -2.00000i q^{12} -1.00000i q^{13} +4.00000 q^{16} +6.00000i q^{17} +4.00000 q^{19} -4.00000 q^{21} -3.00000i q^{23} -5.00000i q^{27} -8.00000i q^{28} +3.00000 q^{29} -4.00000 q^{31} +6.00000i q^{33} +4.00000 q^{36} +2.00000i q^{37} -1.00000 q^{39} +6.00000 q^{41} +7.00000i q^{43} -12.0000 q^{44} -4.00000i q^{48} -9.00000 q^{49} +6.00000 q^{51} -2.00000i q^{52} +9.00000i q^{53} -4.00000i q^{57} +6.00000 q^{59} -1.00000 q^{61} -8.00000i q^{63} +8.00000 q^{64} +14.0000i q^{67} +12.0000i q^{68} -3.00000 q^{69} -6.00000 q^{71} +4.00000i q^{73} +8.00000 q^{76} +24.0000i q^{77} -11.0000 q^{79} +1.00000 q^{81} +6.00000i q^{83} -8.00000 q^{84} -3.00000i q^{87} -4.00000 q^{91} -6.00000i q^{92} +4.00000i q^{93} -10.0000i q^{97} -12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{4} + 4 q^{9} - 12 q^{11} + 8 q^{16} + 8 q^{19} - 8 q^{21} + 6 q^{29} - 8 q^{31} + 8 q^{36} - 2 q^{39} + 12 q^{41} - 24 q^{44} - 18 q^{49} + 12 q^{51} + 12 q^{59} - 2 q^{61} + 16 q^{64} - 6 q^{69}+ \cdots - 24 q^{99}+O(q^{100}) \) 2 * q + 4 * q^4 + 4 * q^9 - 12 * q^11 + 8 * q^16 + 8 * q^19 - 8 * q^21 + 6 * q^29 - 8 * q^31 + 8 * q^36 - 2 * q^39 + 12 * q^41 - 24 * q^44 - 18 * q^49 + 12 * q^51 + 12 * q^59 - 2 * q^61 + 16 * q^64 - 6 * q^69 - 12 * q^71 + 16 * q^76 - 22 * q^79 + 2 * q^81 - 16 * q^84 - 8 * q^91 - 24 * q^99

Introduction

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