LMFDB - Embedded newform 2925.2.c.p.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:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
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			$-2.56155i$ of defining polynomial
Character	$\chi$	$=$	2925.2224
Dual form			2925.2.c.p.2224.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.56155i q^{2} -4.56155 q^{4} +0.438447i q^{7} +6.56155i q^{8} +O(q^{10})$	$q-2.56155i q^{2} -4.56155 q^{4} +0.438447i q^{7} +6.56155i q^{8} -1.56155 q^{11} -1.00000i q^{13} +1.12311 q^{14} +7.68466 q^{16} -1.56155i q^{17} +5.12311 q^{19} +4.00000i q^{22} +2.43845i q^{23} -2.56155 q^{26} -2.00000i q^{28} +7.12311 q^{29} +6.00000 q^{31} -6.56155i q^{32} -4.00000 q^{34} +10.6847i q^{37} -13.1231i q^{38} +3.56155 q^{41} +3.12311i q^{43} +7.12311 q^{44} +6.24621 q^{46} -11.1231i q^{47} +6.80776 q^{49} +4.56155i q^{52} -4.68466i q^{53} -2.87689 q^{56} -18.2462i q^{58} -12.0000 q^{59} -6.68466 q^{61} -15.3693i q^{62} -1.43845 q^{64} +11.3693i q^{67} +7.12311i q^{68} +10.4384 q^{71} -6.00000i q^{73} +27.3693 q^{74} -23.3693 q^{76} -0.684658i q^{77} -4.68466 q^{79} -9.12311i q^{82} -16.4924i q^{83} +8.00000 q^{86} -10.2462i q^{88} -10.6847 q^{89} +0.438447 q^{91} -11.1231i q^{92} -28.4924 q^{94} -16.9309i q^{97} -17.4384i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 10 q^{4}+O(q^{10}) $ 4 * q - 10 * q^4	$ 4 q - 10 q^{4} + 2 q^{11} - 12 q^{14} + 6 q^{16} + 4 q^{19} - 2 q^{26} + 12 q^{29} + 24 q^{31} - 16 q^{34} + 6 q^{41} + 12 q^{44} - 8 q^{46} - 14 q^{49} - 28 q^{56} - 48 q^{59} - 2 q^{61} - 14 q^{64} + 50 q^{71} + 60 q^{74} - 44 q^{76} + 6 q^{79} + 32 q^{86} - 18 q^{89} + 10 q^{91} - 48 q^{94}+O(q^{100}) $ 4 * q - 10 * q^4 + 2 * q^11 - 12 * q^14 + 6 * q^16 + 4 * q^19 - 2 * q^26 + 12 * q^29 + 24 * q^31 - 16 * q^34 + 6 * q^41 + 12 * q^44 - 8 * q^46 - 14 * q^49 - 28 * q^56 - 48 * q^59 - 2 * q^61 - 14 * q^64 + 50 * q^71 + 60 * q^74 - 44 * q^76 + 6 * q^79 + 32 * q^86 - 18 * q^89 + 10 * q^91 - 48 * 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 $

$2$	− 2.56155i	− 1.81129i	−0.424035	−	0.905646i	$-0.639387\pi$
$2$	− 2.56155i	− 1.81129i	0.424035	−	0.905646i	$-0.360613\pi$
$3$	0	0
$3$	0	0
$4$	−4.56155	−2.28078
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.438447i	0.165717i	0.996561	+	0.0828587i	$0.0264050\pi$
$7$	0.438447i	0.165717i	−0.996561	+	0.0828587i	$0.973595\pi$
$8$	6.56155i	2.31986i
$9$	0	0
$10$	0	0
$11$	−1.56155	−0.470826	−0.235413	−	0.971895i	$-0.575644\pi$
$11$	−1.56155	−0.470826	−0.235413	+	0.971895i	$0.575644\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	1.12311	0.300163
$15$	0	0
$16$	7.68466	1.92116
$17$	− 1.56155i	− 0.378732i	−0.981907	−	0.189366i	$-0.939357\pi$
$17$	− 1.56155i	− 0.378732i	0.981907	−	0.189366i	$-0.0606433\pi$
$18$	0	0
$19$	5.12311	1.17532	0.587661	−	0.809108i	$-0.300049\pi$
$19$	5.12311	1.17532	0.587661	+	0.809108i	$0.300049\pi$
$20$	0	0
$21$	0	0
$22$	4.00000i	0.852803i
$23$	2.43845i	0.508451i	0.967145	+	0.254226i	$0.0818206\pi$
$23$	2.43845i	0.508451i	−0.967145	+	0.254226i	$0.918179\pi$
$24$	0	0
$25$	0	0
$26$	−2.56155	−0.502362
$27$	0	0
$28$	− 2.00000i	− 0.377964i
$29$	7.12311	1.32273	0.661364	−	0.750065i	$-0.269978\pi$
$29$	7.12311	1.32273	0.661364	+	0.750065i	$0.269978\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	− 6.56155i	− 1.15993i
$33$	0	0
$34$	−4.00000	−0.685994
$35$	0	0
$36$	0	0
$37$	10.6847i	1.75655i	0.478159	+	0.878274i	$0.341304\pi$
$37$	10.6847i	1.75655i	−0.478159	+	0.878274i	$0.658696\pi$
$38$	− 13.1231i	− 2.12885i
$39$	0	0
$40$	0	0
$41$	3.56155	0.556221	0.278111	−	0.960549i	$-0.410292\pi$
$41$	3.56155	0.556221	0.278111	+	0.960549i	$0.410292\pi$
$42$	0	0
$43$	3.12311i	0.476269i	0.971232	+	0.238135i	$0.0765359\pi$
$43$	3.12311i	0.476269i	−0.971232	+	0.238135i	$0.923464\pi$
$44$	7.12311	1.07385
$45$	0	0
$46$	6.24621	0.920954
$47$	− 11.1231i	− 1.62247i	−0.584719	−	0.811236i	$-0.698795\pi$
$47$	− 11.1231i	− 1.62247i	0.584719	−	0.811236i	$-0.301205\pi$
$48$	0	0
$49$	6.80776	0.972538
$50$	0	0
$51$	0	0
$52$	4.56155i	0.632574i
$53$	− 4.68466i	− 0.643487i	−0.946827	−	0.321744i	$-0.895731\pi$
$53$	− 4.68466i	− 0.643487i	0.946827	−	0.321744i	$-0.104269\pi$
$54$	0	0
$55$	0	0
$56$	−2.87689	−0.384441
$57$	0	0
$58$	− 18.2462i	− 2.39584i
$59$	−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
$61$	−6.68466	−0.855883	−0.427941	−	0.903806i	$-0.640761\pi$
$61$	−6.68466	−0.855883	−0.427941	+	0.903806i	$0.640761\pi$
$62$	− 15.3693i	− 1.95191i
$63$	0	0
$64$	−1.43845	−0.179806
$65$	0	0
$66$	0	0
$67$	11.3693i	1.38898i	0.719501	+	0.694492i	$0.244371\pi$
$67$	11.3693i	1.38898i	−0.719501	+	0.694492i	$0.755629\pi$
$68$	7.12311i	0.863803i
$69$	0	0
$70$	0	0
$71$	10.4384	1.23882	0.619408	−	0.785069i	$-0.287373\pi$
$71$	10.4384	1.23882	0.619408	+	0.785069i	$0.287373\pi$
$72$	0	0
$73$	− 6.00000i	− 0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	− 6.00000i	− 0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$	27.3693	3.18162
$75$	0	0
$76$	−23.3693	−2.68064
$77$	− 0.684658i	− 0.0780241i
$78$	0	0
$79$	−4.68466	−0.527065	−0.263533	−	0.964650i	$-0.584888\pi$
$79$	−4.68466	−0.527065	−0.263533	+	0.964650i	$0.584888\pi$
$80$	0	0
$81$	0	0
$82$	− 9.12311i	− 1.00748i
$83$	− 16.4924i	− 1.81028i	−0.425115	−	0.905139i	$-0.639766\pi$
$83$	− 16.4924i	− 1.81028i	0.425115	−	0.905139i	$-0.360234\pi$
$84$	0	0
$85$	0	0
$86$	8.00000	0.862662
$87$	0	0
$88$	− 10.2462i	− 1.09225i
$89$	−10.6847	−1.13257	−0.566286	−	0.824209i	$-0.691620\pi$
$89$	−10.6847	−1.13257	−0.566286	+	0.824209i	$0.691620\pi$
$90$	0	0
$91$	0.438447	0.0459618
$92$	− 11.1231i	− 1.15966i
$93$	0	0
$94$	−28.4924	−2.93877
$95$	0	0
$96$	0	0
$97$	− 16.9309i	− 1.71907i	−0.511077	−	0.859535i	$-0.670753\pi$
$97$	− 16.9309i	− 1.71907i	0.511077	−	0.859535i	$-0.329247\pi$
$98$	− 17.4384i	− 1.76155i
$99$	0	0
$100$	0	0
$101$	10.2462	1.01954	0.509768	−	0.860312i	$-0.329731\pi$
$101$	10.2462	1.01954	0.509768	+	0.860312i	$0.329731\pi$
$102$	0	0
$103$	− 15.1231i	− 1.49012i	−0.666995	−	0.745062i	$-0.732420\pi$
$103$	− 15.1231i	− 1.49012i	0.666995	−	0.745062i	$-0.267580\pi$
$104$	6.56155	0.643413
$105$	0	0
$106$	−12.0000	−1.16554
$107$	− 10.9309i	− 1.05673i	−0.849018	−	0.528364i	$-0.822806\pi$
$107$	− 10.9309i	− 1.05673i	0.849018	−	0.528364i	$-0.177194\pi$
$108$	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
$112$	3.36932i	0.318371i
$113$	− 4.87689i	− 0.458780i	−0.973335	−	0.229390i	$-0.926327\pi$
$113$	− 4.87689i	− 0.458780i	0.973335	−	0.229390i	$-0.0736731\pi$
$114$	0	0
$115$	0	0
$116$	−32.4924	−3.01685
$117$	0	0
$118$	30.7386i	2.82972i
$119$	0.684658	0.0627625
$120$	0	0
$121$	−8.56155	−0.778323
$122$	17.1231i	1.55025i
$123$	0	0
$124$	−27.3693	−2.45784
$125$	0	0
$126$	0	0
$127$	− 1.75379i	− 0.155624i	−0.996968	−	0.0778118i	$-0.975207\pi$
$127$	− 1.75379i	− 0.155624i	0.996968	−	0.0778118i	$-0.0247933\pi$
$128$	− 9.43845i	− 0.834249i
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	2.24621i	0.194771i
$134$	29.1231	2.51585
$135$	0	0
$136$	10.2462	0.878605
$137$	− 1.12311i	− 0.0959534i	−0.998848	−	0.0479767i	$-0.984723\pi$
$137$	− 1.12311i	− 0.0959534i	0.998848	−	0.0479767i	$-0.0152773\pi$
$138$	0	0
$139$	−3.31534	−0.281204	−0.140602	−	0.990066i	$-0.544904\pi$
$139$	−3.31534	−0.281204	−0.140602	+	0.990066i	$0.544904\pi$
$140$	0	0
$141$	0	0
$142$	− 26.7386i	− 2.24386i
$143$	1.56155i	0.130584i
$144$	0	0
$145$	0	0
$146$	−15.3693	−1.27197
$147$	0	0
$148$	− 48.7386i	− 4.00629i
$149$	17.8078	1.45887	0.729434	−	0.684051i	$-0.239783\pi$
$149$	17.8078	1.45887	0.729434	+	0.684051i	$0.239783\pi$
$150$	0	0
$151$	11.3693	0.925222	0.462611	−	0.886561i	$-0.346913\pi$
$151$	11.3693	0.925222	0.462611	+	0.886561i	$0.346913\pi$
$152$	33.6155i	2.72658i
$153$	0	0
$154$	−1.75379	−0.141324
$155$	0	0
$156$	0	0
$157$	− 3.36932i	− 0.268901i	−0.990920	−	0.134450i	$-0.957073\pi$
$157$	− 3.36932i	− 0.268901i	0.990920	−	0.134450i	$-0.0429269\pi$
$158$	12.0000i	0.954669i
$159$	0	0
$160$	0	0
$161$	−1.06913	−0.0842593
$162$	0	0
$163$	16.0540i	1.25744i	0.777630	+	0.628722i	$0.216422\pi$
$163$	16.0540i	1.25744i	−0.777630	+	0.628722i	$0.783578\pi$
$164$	−16.2462	−1.26862
$165$	0	0
$166$	−42.2462	−3.27894
$167$	− 4.87689i	− 0.377385i	−0.982036	−	0.188693i	$-0.939575\pi$
$167$	− 4.87689i	− 0.377385i	0.982036	−	0.188693i	$-0.0604250\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	− 14.2462i	− 1.08626i
$173$	12.8769i	0.979012i	0.872000	+	0.489506i	$0.162823\pi$
$173$	12.8769i	0.979012i	−0.872000	+	0.489506i	$0.837177\pi$
$174$	0	0
$175$	0	0
$176$	−12.0000	−0.904534
$177$	0	0
$178$	27.3693i	2.05142i
$179$	4.87689	0.364516	0.182258	−	0.983251i	$-0.441659\pi$
$179$	4.87689	0.364516	0.182258	+	0.983251i	$0.441659\pi$
$180$	0	0
$181$	13.3153	0.989722	0.494861	−	0.868972i	$-0.335219\pi$
$181$	13.3153	0.989722	0.494861	+	0.868972i	$0.335219\pi$
$182$	− 1.12311i	− 0.0832501i
$183$	0	0
$184$	−16.0000	−1.17954
$185$	0	0
$186$	0	0
$187$	2.43845i	0.178317i
$188$	50.7386i	3.70050i
$189$	0	0
$190$	0	0
$191$	19.6155	1.41933	0.709665	−	0.704539i	$-0.248846\pi$
$191$	19.6155	1.41933	0.709665	+	0.704539i	$0.248846\pi$
$192$	0	0
$193$	19.5616i	1.40807i	0.710165	+	0.704036i	$0.248621\pi$
$193$	19.5616i	1.40807i	−0.710165	+	0.704036i	$0.751379\pi$
$194$	−43.3693	−3.11374
$195$	0	0
$196$	−31.0540	−2.21814
$197$	3.36932i	0.240054i	0.992771	+	0.120027i	$0.0382981\pi$
$197$	3.36932i	0.240054i	−0.992771	+	0.120027i	$0.961702\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	0	0
$202$	− 26.2462i	− 1.84668i
$203$	3.12311i	0.219199i
$204$	0	0
$205$	0	0
$206$	−38.7386	−2.69905
$207$	0	0
$208$	− 7.68466i	− 0.532835i
$209$	−8.00000	−0.553372
$210$	0	0
$211$	6.24621	0.430007	0.215003	−	0.976613i	$-0.431024\pi$
$211$	6.24621	0.430007	0.215003	+	0.976613i	$0.431024\pi$
$212$	21.3693i	1.46765i
$213$	0	0
$214$	−28.0000	−1.91404
$215$	0	0
$216$	0	0
$217$	2.63068i	0.178582i
$218$	− 5.12311i	− 0.346980i
$219$	0	0
$220$	0	0
$221$	−1.56155	−0.105041
$222$	0	0
$223$	− 15.3693i	− 1.02921i	−0.857429	−	0.514603i	$-0.827939\pi$
$223$	− 15.3693i	− 1.02921i	0.857429	−	0.514603i	$-0.172061\pi$
$224$	2.87689	0.192221
$225$	0	0
$226$	−12.4924	−0.830984
$227$	− 5.75379i	− 0.381892i	−0.981601	−	0.190946i	$-0.938844\pi$
$227$	− 5.75379i	− 0.381892i	0.981601	−	0.190946i	$-0.0611556\pi$
$228$	0	0
$229$	−17.1231	−1.13153	−0.565763	−	0.824568i	$-0.691418\pi$
$229$	−17.1231	−1.13153	−0.565763	+	0.824568i	$0.691418\pi$
$230$	0	0
$231$	0	0
$232$	46.7386i	3.06854i
$233$	− 27.8078i	− 1.82175i	−0.412685	−	0.910874i	$-0.635409\pi$
$233$	− 27.8078i	− 1.82175i	0.412685	−	0.910874i	$-0.364591\pi$
$234$	0	0
$235$	0	0
$236$	54.7386	3.56318
$237$	0	0
$238$	− 1.75379i	− 0.113681i
$239$	−22.9309	−1.48327	−0.741637	−	0.670801i	$-0.765950\pi$
$239$	−22.9309	−1.48327	−0.741637	+	0.670801i	$0.765950\pi$
$240$	0	0
$241$	24.7386	1.59356	0.796778	−	0.604272i	$-0.206536\pi$
$241$	24.7386	1.59356	0.796778	+	0.604272i	$0.206536\pi$
$242$	21.9309i	1.40977i
$243$	0	0
$244$	30.4924	1.95208
$245$	0	0
$246$	0	0
$247$	− 5.12311i	− 0.325975i
$248$	39.3693i	2.49995i
$249$	0	0
$250$	0	0
$251$	26.2462	1.65665	0.828323	−	0.560251i	$-0.189295\pi$
$251$	26.2462	1.65665	0.828323	+	0.560251i	$0.189295\pi$
$252$	0	0
$253$	− 3.80776i	− 0.239392i
$254$	−4.49242	−0.281880
$255$	0	0
$256$	−27.0540	−1.69087
$257$	12.8769i	0.803239i	0.915807	+	0.401619i	$0.131552\pi$
$257$	12.8769i	0.803239i	−0.915807	+	0.401619i	$0.868448\pi$
$258$	0	0
$259$	−4.68466	−0.291091
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 2.24621i	− 0.138507i	−0.997599	−	0.0692537i	$-0.977938\pi$
$263$	− 2.24621i	− 0.138507i	0.997599	−	0.0692537i	$-0.0220618\pi$
$264$	0	0
$265$	0	0
$266$	5.75379	0.352787
$267$	0	0
$268$	− 51.8617i	− 3.16796i
$269$	0.876894	0.0534652	0.0267326	−	0.999643i	$-0.491490\pi$
$269$	0.876894	0.0534652	0.0267326	+	0.999643i	$0.491490\pi$
$270$	0	0
$271$	−19.3693	−1.17660	−0.588301	−	0.808642i	$-0.700203\pi$
$271$	−19.3693	−1.17660	−0.588301	+	0.808642i	$0.700203\pi$
$272$	− 12.0000i	− 0.727607i
$273$	0	0
$274$	−2.87689	−0.173800
$275$	0	0
$276$	0	0
$277$	12.2462i	0.735804i	0.929865	+	0.367902i	$0.119924\pi$
$277$	12.2462i	0.735804i	−0.929865	+	0.367902i	$0.880076\pi$
$278$	8.49242i	0.509342i
$279$	0	0
$280$	0	0
$281$	4.24621	0.253308	0.126654	−	0.991947i	$-0.459576\pi$
$281$	4.24621	0.253308	0.126654	+	0.991947i	$0.459576\pi$
$282$	0	0
$283$	− 4.00000i	− 0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$	− 4.00000i	− 0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$	−47.6155	−2.82546
$285$	0	0
$286$	4.00000	0.236525
$287$	1.56155i	0.0921755i
$288$	0	0
$289$	14.5616	0.856562
$290$	0	0
$291$	0	0
$292$	27.3693i	1.60167i
$293$	− 20.2462i	− 1.18280i	−0.806380	−	0.591398i	$-0.798576\pi$
$293$	− 20.2462i	− 1.18280i	0.806380	−	0.591398i	$-0.201424\pi$
$294$	0	0
$295$	0	0
$296$	−70.1080	−4.07494
$297$	0	0
$298$	− 45.6155i	− 2.64244i
$299$	2.43845	0.141019
$300$	0	0
$301$	−1.36932	−0.0789261
$302$	− 29.1231i	− 1.67585i
$303$	0	0
$304$	39.3693	2.25799
$305$	0	0
$306$	0	0
$307$	− 7.56155i	− 0.431561i	−0.976442	−	0.215780i	$-0.930770\pi$
$307$	− 7.56155i	− 0.431561i	0.976442	−	0.215780i	$-0.0692295\pi$
$308$	3.12311i	0.177955i
$309$	0	0
$310$	0	0
$311$	2.63068	0.149172	0.0745862	−	0.997215i	$-0.476236\pi$
$311$	2.63068	0.149172	0.0745862	+	0.997215i	$0.476236\pi$
$312$	0	0
$313$	29.1231i	1.64614i	0.567943	+	0.823068i	$0.307739\pi$
$313$	29.1231i	1.64614i	−0.567943	+	0.823068i	$0.692261\pi$
$314$	−8.63068	−0.487058
$315$	0	0
$316$	21.3693	1.20212
$317$	− 1.50758i	− 0.0846740i	−0.999103	−	0.0423370i	$-0.986520\pi$
$317$	− 1.50758i	− 0.0846740i	0.999103	−	0.0423370i	$-0.0134803\pi$
$318$	0	0
$319$	−11.1231	−0.622774
$320$	0	0
$321$	0	0
$322$	2.73863i	0.152618i
$323$	− 8.00000i	− 0.445132i
$324$	0	0
$325$	0	0
$326$	41.1231	2.27760
$327$	0	0
$328$	23.3693i	1.29035i
$329$	4.87689	0.268872
$330$	0	0
$331$	29.1231	1.60075	0.800375	−	0.599499i	$-0.204634\pi$
$331$	29.1231	1.60075	0.800375	+	0.599499i	$0.204634\pi$
$332$	75.2311i	4.12884i
$333$	0	0
$334$	−12.4924	−0.683555
$335$	0	0
$336$	0	0
$337$	30.4924i	1.66103i	0.556998	+	0.830514i	$0.311953\pi$
$337$	30.4924i	1.66103i	−0.556998	+	0.830514i	$0.688047\pi$
$338$	2.56155i	0.139330i
$339$	0	0
$340$	0	0
$341$	−9.36932	−0.507377
$342$	0	0
$343$	6.05398i	0.326884i
$344$	−20.4924	−1.10488
$345$	0	0
$346$	32.9848	1.77328
$347$	26.0540i	1.39865i	0.714804	+	0.699325i	$0.246516\pi$
$347$	26.0540i	1.39865i	−0.714804	+	0.699325i	$0.753484\pi$
$348$	0	0
$349$	23.3693	1.25093	0.625465	−	0.780252i	$-0.284909\pi$
$349$	23.3693	1.25093	0.625465	+	0.780252i	$0.284909\pi$
$350$	0	0
$351$	0	0
$352$	10.2462i	0.546125i
$353$	− 22.4924i	− 1.19715i	−0.801066	−	0.598575i	$-0.795734\pi$
$353$	− 22.4924i	− 1.19715i	0.801066	−	0.598575i	$-0.204266\pi$
$354$	0	0
$355$	0	0
$356$	48.7386	2.58314
$357$	0	0
$358$	− 12.4924i	− 0.660245i
$359$	14.2462	0.751886	0.375943	−	0.926643i	$-0.377319\pi$
$359$	14.2462	0.751886	0.375943	+	0.926643i	$0.377319\pi$
$360$	0	0
$361$	7.24621	0.381380
$362$	− 34.1080i	− 1.79267i
$363$	0	0
$364$	−2.00000	−0.104828
$365$	0	0
$366$	0	0
$367$	1.75379i	0.0915470i	0.998952	+	0.0457735i	$0.0145753\pi$
$367$	1.75379i	0.0915470i	−0.998952	+	0.0457735i	$0.985425\pi$
$368$	18.7386i	0.976819i
$369$	0	0
$370$	0	0
$371$	2.05398	0.106637
$372$	0	0
$373$	12.2462i	0.634085i	0.948411	+	0.317042i	$0.102690\pi$
$373$	12.2462i	0.634085i	−0.948411	+	0.317042i	$0.897310\pi$
$374$	6.24621	0.322984
$375$	0	0
$376$	72.9848	3.76391
$377$	− 7.12311i	− 0.366859i
$378$	0	0
$379$	30.4924	1.56629	0.783145	−	0.621839i	$-0.213614\pi$
$379$	30.4924	1.56629	0.783145	+	0.621839i	$0.213614\pi$
$380$	0	0
$381$	0	0
$382$	− 50.2462i	− 2.57082i
$383$	3.50758i	0.179229i	0.995977	+	0.0896144i	$0.0285635\pi$
$383$	3.50758i	0.179229i	−0.995977	+	0.0896144i	$0.971437\pi$
$384$	0	0
$385$	0	0
$386$	50.1080	2.55043
$387$	0	0
$388$	77.2311i	3.92081i
$389$	−37.8617	−1.91967	−0.959833	−	0.280571i	$-0.909476\pi$
$389$	−37.8617	−1.91967	−0.959833	+	0.280571i	$0.909476\pi$
$390$	0	0
$391$	3.80776	0.192567
$392$	44.6695i	2.25615i
$393$	0	0
$394$	8.63068	0.434808
$395$	0	0
$396$	0	0
$397$	4.43845i	0.222759i	0.993778	+	0.111380i	$0.0355270\pi$
$397$	4.43845i	0.222759i	−0.993778	+	0.111380i	$0.964473\pi$
$398$	− 20.4924i	− 1.02719i
$399$	0	0
$400$	0	0
$401$	3.75379	0.187455	0.0937276	−	0.995598i	$-0.470122\pi$
$401$	3.75379	0.187455	0.0937276	+	0.995598i	$0.470122\pi$
$402$	0	0
$403$	− 6.00000i	− 0.298881i
$404$	−46.7386	−2.32533
$405$	0	0
$406$	8.00000	0.397033
$407$	− 16.6847i	− 0.827028i
$408$	0	0
$409$	6.87689	0.340041	0.170020	−	0.985441i	$-0.445617\pi$
$409$	6.87689	0.340041	0.170020	+	0.985441i	$0.445617\pi$
$410$	0	0
$411$	0	0
$412$	68.9848i	3.39864i
$413$	− 5.26137i	− 0.258895i
$414$	0	0
$415$	0	0
$416$	−6.56155	−0.321707
$417$	0	0
$418$	20.4924i	1.00232i
$419$	−7.61553	−0.372043	−0.186021	−	0.982546i	$-0.559559\pi$
$419$	−7.61553	−0.372043	−0.186021	+	0.982546i	$0.559559\pi$
$420$	0	0
$421$	39.3693	1.91874	0.959372	−	0.282146i	$-0.0910462\pi$
$421$	39.3693	1.91874	0.959372	+	0.282146i	$0.0910462\pi$
$422$	− 16.0000i	− 0.778868i
$423$	0	0
$424$	30.7386	1.49280
$425$	0	0
$426$	0	0
$427$	− 2.93087i	− 0.141835i
$428$	49.8617i	2.41016i
$429$	0	0
$430$	0	0
$431$	−3.50758	−0.168954	−0.0844770	−	0.996425i	$-0.526922\pi$
$431$	−3.50758	−0.168954	−0.0844770	+	0.996425i	$0.526922\pi$
$432$	0	0
$433$	− 9.61553i	− 0.462093i	−0.972943	−	0.231046i	$-0.925785\pi$
$433$	− 9.61553i	− 0.462093i	0.972943	−	0.231046i	$-0.0742149\pi$
$434$	6.73863	0.323465
$435$	0	0
$436$	−9.12311	−0.436918
$437$	12.4924i	0.597594i
$438$	0	0
$439$	−22.0540	−1.05258	−0.526289	−	0.850306i	$-0.676417\pi$
$439$	−22.0540	−1.05258	−0.526289	+	0.850306i	$0.676417\pi$
$440$	0	0
$441$	0	0
$442$	4.00000i	0.190261i
$443$	− 7.80776i	− 0.370958i	−0.982648	−	0.185479i	$-0.940616\pi$
$443$	− 7.80776i	− 0.370958i	0.982648	−	0.185479i	$-0.0593837\pi$
$444$	0	0
$445$	0	0
$446$	−39.3693	−1.86419
$447$	0	0
$448$	− 0.630683i	− 0.0297970i
$449$	35.1771	1.66011	0.830055	−	0.557682i	$-0.188309\pi$
$449$	35.1771	1.66011	0.830055	+	0.557682i	$0.188309\pi$
$450$	0	0
$451$	−5.56155	−0.261883
$452$	22.2462i	1.04637i
$453$	0	0
$454$	−14.7386	−0.691718
$455$	0	0
$456$	0	0
$457$	13.3153i	0.622865i	0.950268	+	0.311433i	$0.100809\pi$
$457$	13.3153i	0.622865i	−0.950268	+	0.311433i	$0.899191\pi$
$458$	43.8617i	2.04952i
$459$	0	0
$460$	0	0
$461$	8.05398	0.375111	0.187556	−	0.982254i	$-0.439944\pi$
$461$	8.05398	0.375111	0.187556	+	0.982254i	$0.439944\pi$
$462$	0	0
$463$	− 28.9309i	− 1.34453i	−0.740310	−	0.672266i	$-0.765321\pi$
$463$	− 28.9309i	− 1.34453i	0.740310	−	0.672266i	$-0.234679\pi$
$464$	54.7386	2.54118
$465$	0	0
$466$	−71.2311	−3.29971
$467$	− 6.93087i	− 0.320722i	−0.987058	−	0.160361i	$-0.948734\pi$
$467$	− 6.93087i	− 0.320722i	0.987058	−	0.160361i	$-0.0512659\pi$
$468$	0	0
$469$	−4.98485	−0.230179
$470$	0	0
$471$	0	0
$472$	− 78.7386i	− 3.62424i
$473$	− 4.87689i	− 0.224240i
$474$	0	0
$475$	0	0
$476$	−3.12311	−0.143147
$477$	0	0
$478$	58.7386i	2.68664i
$479$	−26.0540	−1.19044	−0.595218	−	0.803564i	$-0.702934\pi$
$479$	−26.0540	−1.19044	−0.595218	+	0.803564i	$0.702934\pi$
$480$	0	0
$481$	10.6847	0.487178
$482$	− 63.3693i	− 2.88639i
$483$	0	0
$484$	39.0540	1.77518
$485$	0	0
$486$	0	0
$487$	32.0540i	1.45250i	0.687428	+	0.726252i	$0.258740\pi$
$487$	32.0540i	1.45250i	−0.687428	+	0.726252i	$0.741260\pi$
$488$	− 43.8617i	− 1.98553i
$489$	0	0
$490$	0	0
$491$	−27.6155	−1.24627	−0.623136	−	0.782114i	$-0.714142\pi$
$491$	−27.6155	−1.24627	−0.623136	+	0.782114i	$0.714142\pi$
$492$	0	0
$493$	− 11.1231i	− 0.500959i
$494$	−13.1231	−0.590436
$495$	0	0
$496$	46.1080	2.07031
$497$	4.57671i	0.205293i
$498$	0	0
$499$	−34.9848	−1.56614	−0.783068	−	0.621936i	$-0.786347\pi$
$499$	−34.9848	−1.56614	−0.783068	+	0.621936i	$0.786347\pi$
$500$	0	0
$501$	0	0
$502$	− 67.2311i	− 3.00067i
$503$	13.7538i	0.613251i	0.951830	+	0.306626i	$0.0991999\pi$
$503$	13.7538i	0.613251i	−0.951830	+	0.306626i	$0.900800\pi$
$504$	0	0
$505$	0	0
$506$	−9.75379	−0.433609
$507$	0	0
$508$	8.00000i	0.354943i
$509$	23.5616	1.04435	0.522174	−	0.852839i	$-0.325121\pi$
$509$	23.5616	1.04435	0.522174	+	0.852839i	$0.325121\pi$
$510$	0	0
$511$	2.63068	0.116375
$512$	50.4233i	2.22842i
$513$	0	0
$514$	32.9848	1.45490
$515$	0	0
$516$	0	0
$517$	17.3693i	0.763902i
$518$	12.0000i	0.527250i
$519$	0	0
$520$	0	0
$521$	−25.8617	−1.13302	−0.566512	−	0.824054i	$-0.691707\pi$
$521$	−25.8617	−1.13302	−0.566512	+	0.824054i	$0.691707\pi$
$522$	0	0
$523$	− 30.7386i	− 1.34411i	−0.740503	−	0.672053i	$-0.765413\pi$
$523$	− 30.7386i	− 1.34411i	0.740503	−	0.672053i	$-0.234587\pi$
$524$	0	0
$525$	0	0
$526$	−5.75379	−0.250877
$527$	− 9.36932i	− 0.408134i
$528$	0	0
$529$	17.0540	0.741477
$530$	0	0
$531$	0	0
$532$	− 10.2462i	− 0.444230i
$533$	− 3.56155i	− 0.154268i
$534$	0	0
$535$	0	0
$536$	−74.6004	−3.22225
$537$	0	0
$538$	− 2.24621i	− 0.0968410i
$539$	−10.6307	−0.457896
$540$	0	0
$541$	18.8769	0.811581	0.405791	−	0.913966i	$-0.366996\pi$
$541$	18.8769	0.811581	0.405791	+	0.913966i	$0.366996\pi$
$542$	49.6155i	2.13117i
$543$	0	0
$544$	−10.2462	−0.439303
$545$	0	0
$546$	0	0
$547$	5.36932i	0.229575i	0.993390	+	0.114788i	$0.0366188\pi$
$547$	5.36932i	0.229575i	−0.993390	+	0.114788i	$0.963381\pi$
$548$	5.12311i	0.218848i
$549$	0	0
$550$	0	0
$551$	36.4924	1.55463
$552$	0	0
$553$	− 2.05398i	− 0.0873439i
$554$	31.3693	1.33275
$555$	0	0
$556$	15.1231	0.641363
$557$	− 6.49242i	− 0.275093i	−0.990495	−	0.137546i	$-0.956078\pi$
$557$	− 6.49242i	− 0.275093i	0.990495	−	0.137546i	$-0.0439216\pi$
$558$	0	0
$559$	3.12311	0.132093
$560$	0	0
$561$	0	0
$562$	− 10.8769i	− 0.458814i
$563$	− 19.3153i	− 0.814045i	−0.913418	−	0.407022i	$-0.866567\pi$
$563$	− 19.3153i	− 0.814045i	0.913418	−	0.407022i	$-0.133433\pi$
$564$	0	0
$565$	0	0
$566$	−10.2462	−0.430680
$567$	0	0
$568$	68.4924i	2.87388i
$569$	32.8769	1.37827	0.689136	−	0.724632i	$-0.257990\pi$
$569$	32.8769	1.37827	0.689136	+	0.724632i	$0.257990\pi$
$570$	0	0
$571$	−22.0540	−0.922930	−0.461465	−	0.887158i	$-0.652676\pi$
$571$	−22.0540	−0.922930	−0.461465	+	0.887158i	$0.652676\pi$
$572$	− 7.12311i	− 0.297832i
$573$	0	0
$574$	4.00000	0.166957
$575$	0	0
$576$	0	0
$577$	− 24.4384i	− 1.01739i	−0.860948	−	0.508693i	$-0.830129\pi$
$577$	− 24.4384i	− 1.01739i	0.860948	−	0.508693i	$-0.169871\pi$
$578$	− 37.3002i	− 1.55148i
$579$	0	0
$580$	0	0
$581$	7.23106	0.299995
$582$	0	0
$583$	7.31534i	0.302970i
$584$	39.3693	1.62911
$585$	0	0
$586$	−51.8617	−2.14239
$587$	32.4924i	1.34111i	0.741862	+	0.670553i	$0.233943\pi$
$587$	32.4924i	1.34111i	−0.741862	+	0.670553i	$0.766057\pi$
$588$	0	0
$589$	30.7386	1.26656
$590$	0	0
$591$	0	0
$592$	82.1080i	3.37462i
$593$	24.2462i	0.995673i	0.867271	+	0.497836i	$0.165872\pi$
$593$	24.2462i	0.995673i	−0.867271	+	0.497836i	$0.834128\pi$
$594$	0	0
$595$	0	0
$596$	−81.2311	−3.32735
$597$	0	0
$598$	− 6.24621i	− 0.255427i
$599$	9.36932	0.382820	0.191410	−	0.981510i	$-0.438694\pi$
$599$	9.36932	0.382820	0.191410	+	0.981510i	$0.438694\pi$
$600$	0	0
$601$	−28.5464	−1.16443	−0.582216	−	0.813034i	$-0.697814\pi$
$601$	−28.5464	−1.16443	−0.582216	+	0.813034i	$0.697814\pi$
$602$	3.50758i	0.142958i
$603$	0	0
$604$	−51.8617	−2.11022
$605$	0	0
$606$	0	0
$607$	24.0000i	0.974130i	0.873366	+	0.487065i	$0.161933\pi$
$607$	24.0000i	0.974130i	−0.873366	+	0.487065i	$0.838067\pi$
$608$	− 33.6155i	− 1.36329i
$609$	0	0
$610$	0	0
$611$	−11.1231	−0.449993
$612$	0	0
$613$	− 32.5464i	− 1.31454i	−0.753657	−	0.657268i	$-0.771712\pi$
$613$	− 32.5464i	− 1.31454i	0.753657	−	0.657268i	$-0.228288\pi$
$614$	−19.3693	−0.781682
$615$	0	0
$616$	4.49242	0.181005
$617$	− 0.738634i	− 0.0297363i	−0.999889	−	0.0148681i	$-0.995267\pi$
$617$	− 0.738634i	− 0.0297363i	0.999889	−	0.0148681i	$-0.00473285\pi$
$618$	0	0
$619$	8.63068	0.346896	0.173448	−	0.984843i	$-0.444509\pi$
$619$	8.63068	0.346896	0.173448	+	0.984843i	$0.444509\pi$
$620$	0	0
$621$	0	0
$622$	− 6.73863i	− 0.270195i
$623$	− 4.68466i	− 0.187687i
$624$	0	0
$625$	0	0
$626$	74.6004	2.98163
$627$	0	0
$628$	15.3693i	0.613303i
$629$	16.6847	0.665261
$630$	0	0
$631$	−32.7386	−1.30330	−0.651652	−	0.758518i	$-0.725924\pi$
$631$	−32.7386	−1.30330	−0.651652	+	0.758518i	$0.725924\pi$
$632$	− 30.7386i	− 1.22272i
$633$	0	0
$634$	−3.86174	−0.153369
$635$	0	0
$636$	0	0
$637$	− 6.80776i	− 0.269733i
$638$	28.4924i	1.12803i
$639$	0	0
$640$	0	0
$641$	−32.9848	−1.30282	−0.651412	−	0.758725i	$-0.725823\pi$
$641$	−32.9848	−1.30282	−0.651412	+	0.758725i	$0.725823\pi$
$642$	0	0
$643$	− 10.6847i	− 0.421362i	−0.977555	−	0.210681i	$-0.932432\pi$
$643$	− 10.6847i	− 0.421362i	0.977555	−	0.210681i	$-0.0675681\pi$
$644$	4.87689	0.192177
$645$	0	0
$646$	−20.4924	−0.806264
$647$	− 31.8078i	− 1.25049i	−0.780428	−	0.625246i	$-0.784999\pi$
$647$	− 31.8078i	− 1.25049i	0.780428	−	0.625246i	$-0.215001\pi$
$648$	0	0
$649$	18.7386	0.735556
$650$	0	0
$651$	0	0
$652$	− 73.2311i	− 2.86795i
$653$	− 33.3693i	− 1.30584i	−0.757426	−	0.652921i	$-0.773543\pi$
$653$	− 33.3693i	− 1.30584i	0.757426	−	0.652921i	$-0.226457\pi$
$654$	0	0
$655$	0	0
$656$	27.3693	1.06859
$657$	0	0
$658$	− 12.4924i	− 0.487005i
$659$	−0.876894	−0.0341590	−0.0170795	−	0.999854i	$-0.505437\pi$
$659$	−0.876894	−0.0341590	−0.0170795	+	0.999854i	$0.505437\pi$
$660$	0	0
$661$	6.49242	0.252526	0.126263	−	0.991997i	$-0.459702\pi$
$661$	6.49242	0.252526	0.126263	+	0.991997i	$0.459702\pi$
$662$	− 74.6004i	− 2.89943i
$663$	0	0
$664$	108.216	4.19959
$665$	0	0
$666$	0	0
$667$	17.3693i	0.672543i
$668$	22.2462i	0.860732i
$669$	0	0
$670$	0	0
$671$	10.4384	0.402972
$672$	0	0
$673$	16.7386i	0.645227i	0.946531	+	0.322613i	$0.104561\pi$
$673$	16.7386i	0.645227i	−0.946531	+	0.322613i	$0.895439\pi$
$674$	78.1080	3.00861
$675$	0	0
$676$	4.56155	0.175444
$677$	14.4384i	0.554915i	0.960738	+	0.277457i	$0.0894917\pi$
$677$	14.4384i	0.554915i	−0.960738	+	0.277457i	$0.910508\pi$
$678$	0	0
$679$	7.42329	0.284880
$680$	0	0
$681$	0	0
$682$	24.0000i	0.919007i
$683$	32.4924i	1.24329i	0.783300	+	0.621644i	$0.213535\pi$
$683$	32.4924i	1.24329i	−0.783300	+	0.621644i	$0.786465\pi$
$684$	0	0
$685$	0	0
$686$	15.5076	0.592082
$687$	0	0
$688$	24.0000i	0.914991i
$689$	−4.68466	−0.178471
$690$	0	0
$691$	−21.6155	−0.822293	−0.411147	−	0.911569i	$-0.634872\pi$
$691$	−21.6155	−0.822293	−0.411147	+	0.911569i	$0.634872\pi$
$692$	− 58.7386i	− 2.23291i
$693$	0	0
$694$	66.7386	2.53336
$695$	0	0
$696$	0	0
$697$	− 5.56155i	− 0.210659i
$698$	− 59.8617i	− 2.26580i
$699$	0	0
$700$	0	0
$701$	−48.9848	−1.85013	−0.925066	−	0.379806i	$-0.875991\pi$
$701$	−48.9848	−1.85013	−0.925066	+	0.379806i	$0.875991\pi$
$702$	0	0
$703$	54.7386i	2.06451i
$704$	2.24621	0.0846573
$705$	0	0
$706$	−57.6155	−2.16839
$707$	4.49242i	0.168955i
$708$	0	0
$709$	−9.12311	−0.342625	−0.171313	−	0.985217i	$-0.554801\pi$
$709$	−9.12311	−0.342625	−0.171313	+	0.985217i	$0.554801\pi$
$710$	0	0
$711$	0	0
$712$	− 70.1080i	− 2.62741i
$713$	14.6307i	0.547923i
$714$	0	0
$715$	0	0
$716$	−22.2462	−0.831380
$717$	0	0
$718$	− 36.4924i	− 1.36189i
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	0	0
$721$	6.63068	0.246940
$722$	− 18.5616i	− 0.690789i
$723$	0	0
$724$	−60.7386	−2.25733
$725$	0	0
$726$	0	0
$727$	− 12.8769i	− 0.477578i	−0.971072	−	0.238789i	$-0.923250\pi$
$727$	− 12.8769i	− 0.477578i	0.971072	−	0.238789i	$-0.0767504\pi$
$728$	2.87689i	0.106625i
$729$	0	0
$730$	0	0
$731$	4.87689	0.180378
$732$	0	0
$733$	− 16.4384i	− 0.607168i	−0.952805	−	0.303584i	$-0.901817\pi$
$733$	− 16.4384i	− 0.607168i	0.952805	−	0.303584i	$-0.0981833\pi$
$734$	4.49242	0.165818
$735$	0	0
$736$	16.0000	0.589768
$737$	− 17.7538i	− 0.653969i
$738$	0	0
$739$	−48.7386	−1.79288	−0.896440	−	0.443166i	$-0.853855\pi$
$739$	−48.7386	−1.79288	−0.896440	+	0.443166i	$0.853855\pi$
$740$	0	0
$741$	0	0
$742$	− 5.26137i	− 0.193151i
$743$	18.7386i	0.687454i	0.939070	+	0.343727i	$0.111689\pi$
$743$	18.7386i	0.687454i	−0.939070	+	0.343727i	$0.888311\pi$
$744$	0	0
$745$	0	0
$746$	31.3693	1.14851
$747$	0	0
$748$	− 11.1231i	− 0.406701i
$749$	4.79261	0.175118
$750$	0	0
$751$	14.0540	0.512837	0.256418	−	0.966566i	$-0.417458\pi$
$751$	14.0540	0.512837	0.256418	+	0.966566i	$0.417458\pi$
$752$	− 85.4773i	− 3.11704i
$753$	0	0
$754$	−18.2462	−0.664488
$755$	0	0
$756$	0	0
$757$	14.4924i	0.526736i	0.964695	+	0.263368i	$0.0848334\pi$
$757$	14.4924i	0.526736i	−0.964695	+	0.263368i	$0.915167\pi$
$758$	− 78.1080i	− 2.83701i
$759$	0	0
$760$	0	0
$761$	45.2311	1.63962	0.819812	−	0.572632i	$-0.194078\pi$
$761$	45.2311	1.63962	0.819812	+	0.572632i	$0.194078\pi$
$762$	0	0
$763$	0.876894i	0.0317457i
$764$	−89.4773	−3.23717
$765$	0	0
$766$	8.98485	0.324636
$767$	12.0000i	0.433295i
$768$	0	0
$769$	30.0000	1.08183	0.540914	−	0.841078i	$-0.318079\pi$
$769$	30.0000	1.08183	0.540914	+	0.841078i	$0.318079\pi$
$770$	0	0
$771$	0	0
$772$	− 89.2311i	− 3.21150i
$773$	− 9.12311i	− 0.328135i	−0.986449	−	0.164068i	$-0.947538\pi$
$773$	− 9.12311i	− 0.328135i	0.986449	−	0.164068i	$-0.0524615\pi$
$774$	0	0
$775$	0	0
$776$	111.093	3.98800
$777$	0	0
$778$	96.9848i	3.47708i
$779$	18.2462	0.653738
$780$	0	0
$781$	−16.3002	−0.583267
$782$	− 9.75379i	− 0.348795i
$783$	0	0
$784$	52.3153	1.86841
$785$	0	0
$786$	0	0
$787$	− 11.3693i	− 0.405272i	−0.979254	−	0.202636i	$-0.935049\pi$
$787$	− 11.3693i	− 0.405272i	0.979254	−	0.202636i	$-0.0649509\pi$
$788$	− 15.3693i	− 0.547509i
$789$	0	0
$790$	0	0
$791$	2.13826	0.0760278
$792$	0	0
$793$	6.68466i	0.237379i
$794$	11.3693	0.403482
$795$	0	0
$796$	−36.4924	−1.29344
$797$	− 4.68466i	− 0.165939i	−0.996552	−	0.0829696i	$-0.973560\pi$
$797$	− 4.68466i	− 0.165939i	0.996552	−	0.0829696i	$-0.0264404\pi$
$798$	0	0
$799$	−17.3693	−0.614482
$800$	0	0
$801$	0	0
$802$	− 9.61553i	− 0.339536i
$803$	9.36932i	0.330636i
$804$	0	0
$805$	0	0
$806$	−15.3693	−0.541361
$807$	0	0
$808$	67.2311i	2.36518i
$809$	−7.50758	−0.263952	−0.131976	−	0.991253i	$-0.542132\pi$
$809$	−7.50758	−0.263952	−0.131976	+	0.991253i	$0.542132\pi$
$810$	0	0
$811$	4.24621	0.149105	0.0745523	−	0.997217i	$-0.476247\pi$
$811$	4.24621	0.149105	0.0745523	+	0.997217i	$0.476247\pi$
$812$	− 14.2462i	− 0.499944i
$813$	0	0
$814$	−42.7386	−1.49799
$815$	0	0
$816$	0	0
$817$	16.0000i	0.559769i
$818$	− 17.6155i	− 0.615912i
$819$	0	0
$820$	0	0
$821$	48.5464	1.69428	0.847140	−	0.531369i	$-0.178322\pi$
$821$	48.5464	1.69428	0.847140	+	0.531369i	$0.178322\pi$
$822$	0	0
$823$	− 29.7538i	− 1.03715i	−0.855032	−	0.518576i	$-0.826462\pi$
$823$	− 29.7538i	− 1.03715i	0.855032	−	0.518576i	$-0.173538\pi$
$824$	99.2311	3.45688
$825$	0	0
$826$	−13.4773	−0.468934
$827$	− 7.12311i	− 0.247695i	−0.992301	−	0.123847i	$-0.960477\pi$
$827$	− 7.12311i	− 0.247695i	0.992301	−	0.123847i	$-0.0395233\pi$
$828$	0	0
$829$	−0.738634	−0.0256538	−0.0128269	−	0.999918i	$-0.504083\pi$
$829$	−0.738634	−0.0256538	−0.0128269	+	0.999918i	$0.504083\pi$
$830$	0	0
$831$	0	0
$832$	1.43845i	0.0498692i
$833$	− 10.6307i	− 0.368331i
$834$	0	0
$835$	0	0
$836$	36.4924	1.26212
$837$	0	0
$838$	19.5076i	0.673878i
$839$	44.7926	1.54641	0.773206	−	0.634155i	$-0.218652\pi$
$839$	44.7926	1.54641	0.773206	+	0.634155i	$0.218652\pi$
$840$	0	0
$841$	21.7386	0.749608
$842$	− 100.847i	− 3.47540i
$843$	0	0
$844$	−28.4924	−0.980750
$845$	0	0
$846$	0	0
$847$	− 3.75379i	− 0.128982i
$848$	− 36.0000i	− 1.23625i
$849$	0	0
$850$	0	0
$851$	−26.0540	−0.893119
$852$	0	0
$853$	41.4233i	1.41831i	0.705054	+	0.709153i	$0.250923\pi$
$853$	41.4233i	1.41831i	−0.705054	+	0.709153i	$0.749077\pi$
$854$	−7.50758	−0.256904
$855$	0	0
$856$	71.7235	2.45146
$857$	− 2.43845i	− 0.0832958i	−0.999132	−	0.0416479i	$-0.986739\pi$
$857$	− 2.43845i	− 0.0832958i	0.999132	−	0.0416479i	$-0.0132608\pi$
$858$	0	0
$859$	−3.80776	−0.129919	−0.0649596	−	0.997888i	$-0.520692\pi$
$859$	−3.80776	−0.129919	−0.0649596	+	0.997888i	$0.520692\pi$
$860$	0	0
$861$	0	0
$862$	8.98485i	0.306025i
$863$	− 9.36932i	− 0.318935i	−0.987203	−	0.159468i	$-0.949022\pi$
$863$	− 9.36932i	− 0.318935i	0.987203	−	0.159468i	$-0.0509778\pi$
$864$	0	0
$865$	0	0
$866$	−24.6307	−0.836985
$867$	0	0
$868$	− 12.0000i	− 0.407307i
$869$	7.31534	0.248156
$870$	0	0
$871$	11.3693	0.385235
$872$	13.1231i	0.444404i
$873$	0	0
$874$	32.0000	1.08242
$875$	0	0
$876$	0	0
$877$	− 46.9848i	− 1.58657i	−0.608853	−	0.793283i	$-0.708370\pi$
$877$	− 46.9848i	− 1.58657i	0.608853	−	0.793283i	$-0.291630\pi$
$878$	56.4924i	1.90653i
$879$	0	0
$880$	0	0
$881$	−21.3693	−0.719951	−0.359975	−	0.932962i	$-0.617215\pi$
$881$	−21.3693	−0.719951	−0.359975	+	0.932962i	$0.617215\pi$
$882$	0	0
$883$	− 56.1080i	− 1.88818i	−0.329684	−	0.944091i	$-0.606942\pi$
$883$	− 56.1080i	− 1.88818i	0.329684	−	0.944091i	$-0.393058\pi$
$884$	7.12311	0.239576
$885$	0	0
$886$	−20.0000	−0.671913
$887$	− 5.56155i	− 0.186739i	−0.995632	−	0.0933693i	$-0.970236\pi$
$887$	− 5.56155i	− 0.186739i	0.995632	−	0.0933693i	$-0.0297637\pi$
$888$	0	0
$889$	0.768944	0.0257895
$890$	0	0
$891$	0	0
$892$	70.1080i	2.34739i
$893$	− 56.9848i	− 1.90693i
$894$	0	0
$895$	0	0
$896$	4.13826	0.138250
$897$	0	0
$898$	− 90.1080i	− 3.00694i
$899$	42.7386	1.42541
$900$	0	0
$901$	−7.31534	−0.243709
$902$	14.2462i	0.474347i
$903$	0	0
$904$	32.0000	1.06430
$905$	0	0
$906$	0	0
$907$	2.63068i	0.0873504i	0.999046	+	0.0436752i	$0.0139067\pi$
$907$	2.63068i	0.0873504i	−0.999046	+	0.0436752i	$0.986093\pi$
$908$	26.2462i	0.871011i
$909$	0	0
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	0	0
$913$	25.7538i	0.852326i
$914$	34.1080	1.12819
$915$	0	0
$916$	78.1080	2.58076
$917$	0	0
$918$	0	0
$919$	−1.94602	−0.0641934	−0.0320967	−	0.999485i	$-0.510218\pi$
$919$	−1.94602	−0.0641934	−0.0320967	+	0.999485i	$0.510218\pi$
$920$	0	0
$921$	0	0
$922$	− 20.6307i	− 0.679435i
$923$	− 10.4384i	− 0.343586i
$924$	0	0
$925$	0	0
$926$	−74.1080	−2.43534
$927$	0	0
$928$	− 46.7386i	− 1.53427i
$929$	−39.6695	−1.30151	−0.650757	−	0.759286i	$-0.725548\pi$
$929$	−39.6695	−1.30151	−0.650757	+	0.759286i	$0.725548\pi$
$930$	0	0
$931$	34.8769	1.14304
$932$	126.847i	4.15500i
$933$	0	0
$934$	−17.7538	−0.580922
$935$	0	0
$936$	0	0
$937$	27.3693i	0.894117i	0.894505	+	0.447058i	$0.147528\pi$
$937$	27.3693i	0.894117i	−0.894505	+	0.447058i	$0.852472\pi$
$938$	12.7689i	0.416921i
$939$	0	0
$940$	0	0
$941$	−41.8078	−1.36289	−0.681447	−	0.731867i	$-0.738649\pi$
$941$	−41.8078	−1.36289	−0.681447	+	0.731867i	$0.738649\pi$
$942$	0	0
$943$	8.68466i	0.282811i
$944$	−92.2159	−3.00137
$945$	0	0
$946$	−12.4924	−0.406164
$947$	− 60.6004i	− 1.96925i	−0.174688	−	0.984624i	$-0.555892\pi$
$947$	− 60.6004i	− 1.96925i	0.174688	−	0.984624i	$-0.444108\pi$
$948$	0	0
$949$	−6.00000	−0.194768
$950$	0	0
$951$	0	0
$952$	4.49242i	0.145600i
$953$	29.5616i	0.957593i	0.877926	+	0.478796i	$0.158927\pi$
$953$	29.5616i	0.957593i	−0.877926	+	0.478796i	$0.841073\pi$
$954$	0	0
$955$	0	0
$956$	104.600	3.38302
$957$	0	0
$958$	66.7386i	2.15623i
$959$	0.492423	0.0159012
$960$	0	0
$961$	5.00000	0.161290
$962$	− 27.3693i	− 0.882422i
$963$	0	0
$964$	−112.847	−3.63454
$965$	0	0
$966$	0	0
$967$	7.36932i	0.236981i	0.992955	+	0.118491i	$0.0378056\pi$
$967$	7.36932i	0.236981i	−0.992955	+	0.118491i	$0.962194\pi$
$968$	− 56.1771i	− 1.80560i
$969$	0	0
$970$	0	0
$971$	−24.4924	−0.785999	−0.393000	−	0.919539i	$-0.628563\pi$
$971$	−24.4924	−0.785999	−0.393000	+	0.919539i	$0.628563\pi$
$972$	0	0
$973$	− 1.45360i	− 0.0466003i
$974$	82.1080	2.63091
$975$	0	0
$976$	−51.3693	−1.64429
$977$	43.4773i	1.39096i	0.718545	+	0.695481i	$0.244808\pi$
$977$	43.4773i	1.39096i	−0.718545	+	0.695481i	$0.755192\pi$
$978$	0	0
$979$	16.6847	0.533244
$980$	0	0
$981$	0	0
$982$	70.7386i	2.25736i
$983$	42.7386i	1.36315i	0.731748	+	0.681575i	$0.238705\pi$
$983$	42.7386i	1.36315i	−0.731748	+	0.681575i	$0.761295\pi$
$984$	0	0
$985$	0	0
$986$	−28.4924	−0.907384
$987$	0	0
$988$	23.3693i	0.743477i
$989$	−7.61553	−0.242160
$990$	0	0
$991$	10.0540	0.319375	0.159688	−	0.987168i	$-0.448951\pi$
$991$	10.0540	0.319375	0.159688	+	0.987168i	$0.448951\pi$
$992$	− 39.3693i	− 1.24998i
$993$	0	0
$994$	11.7235	0.371846
$995$	0	0
$996$	0	0
$997$	− 28.2462i	− 0.894566i	−0.894392	−	0.447283i	$-0.852392\pi$
$997$	− 28.2462i	− 0.894566i	0.894392	−	0.447283i	$-0.147608\pi$
$998$	89.6155i	2.83673i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2925.2.c.p.2224.1		4
3.2	odd	2		2925.2.c.o.2224.4		4
5.2	odd	4		585.2.a.l.1.2	yes	2
5.3	odd	4		2925.2.a.x.1.1		2
5.4	even	2	inner	2925.2.c.p.2224.4		4
15.2	even	4		585.2.a.j.1.1	✓	2
15.8	even	4		2925.2.a.bc.1.2		2
15.14	odd	2		2925.2.c.o.2224.1		4
20.7	even	4		9360.2.a.cw.1.1		2
60.47	odd	4		9360.2.a.cl.1.1		2
65.12	odd	4		7605.2.a.bd.1.1		2
195.77	even	4		7605.2.a.bi.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
585.2.a.j.1.1	✓	2	15.2	even	4
585.2.a.l.1.2	yes	2	5.2	odd	4
2925.2.a.x.1.1		2	5.3	odd	4
2925.2.a.bc.1.2		2	15.8	even	4
2925.2.c.o.2224.1		4	15.14	odd	2
2925.2.c.o.2224.4		4	3.2	odd	2
2925.2.c.p.2224.1		4	1.1	even	1	trivial
2925.2.c.p.2224.4		4	5.4	even	2	inner
7605.2.a.bd.1.1		2	65.12	odd	4
7605.2.a.bi.1.2		2	195.77	even	4
9360.2.a.cl.1.1		2	60.47	odd	4
9360.2.a.cw.1.1		2	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 $\F_{q}$

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)

\(f(q)\)	\(=\)	\(q-2.56155i q^{2} -4.56155 q^{4} +0.438447i q^{7} +6.56155i q^{8} +O(q^{10})\)	\(q-2.56155i q^{2} -4.56155 q^{4} +0.438447i q^{7} +6.56155i q^{8} -1.56155 q^{11} -1.00000i q^{13} +1.12311 q^{14} +7.68466 q^{16} -1.56155i q^{17} +5.12311 q^{19} +4.00000i q^{22} +2.43845i q^{23} -2.56155 q^{26} -2.00000i q^{28} +7.12311 q^{29} +6.00000 q^{31} -6.56155i q^{32} -4.00000 q^{34} +10.6847i q^{37} -13.1231i q^{38} +3.56155 q^{41} +3.12311i q^{43} +7.12311 q^{44} +6.24621 q^{46} -11.1231i q^{47} +6.80776 q^{49} +4.56155i q^{52} -4.68466i q^{53} -2.87689 q^{56} -18.2462i q^{58} -12.0000 q^{59} -6.68466 q^{61} -15.3693i q^{62} -1.43845 q^{64} +11.3693i q^{67} +7.12311i q^{68} +10.4384 q^{71} -6.00000i q^{73} +27.3693 q^{74} -23.3693 q^{76} -0.684658i q^{77} -4.68466 q^{79} -9.12311i q^{82} -16.4924i q^{83} +8.00000 q^{86} -10.2462i q^{88} -10.6847 q^{89} +0.438447 q^{91} -11.1231i q^{92} -28.4924 q^{94} -16.9309i q^{97} -17.4384i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 10 q^{4}+O(q^{10}) \) 4 * q - 10 * q^4	\( 4 q - 10 q^{4} + 2 q^{11} - 12 q^{14} + 6 q^{16} + 4 q^{19} - 2 q^{26} + 12 q^{29} + 24 q^{31} - 16 q^{34} + 6 q^{41} + 12 q^{44} - 8 q^{46} - 14 q^{49} - 28 q^{56} - 48 q^{59} - 2 q^{61} - 14 q^{64} + 50 q^{71} + 60 q^{74} - 44 q^{76} + 6 q^{79} + 32 q^{86} - 18 q^{89} + 10 q^{91} - 48 q^{94}+O(q^{100}) \) 4 * q - 10 * q^4 + 2 * q^11 - 12 * q^14 + 6 * q^16 + 4 * q^19 - 2 * q^26 + 12 * q^29 + 24 * q^31 - 16 * q^34 + 6 * q^41 + 12 * q^44 - 8 * q^46 - 14 * q^49 - 28 * q^56 - 48 * q^59 - 2 * q^61 - 14 * q^64 + 50 * q^71 + 60 * q^74 - 44 * q^76 + 6 * q^79 + 32 * q^86 - 18 * q^89 + 10 * q^91 - 48 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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