LMFDB - Embedded newform 8325.2.a.cr.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8325,2,Mod(1,8325)] 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("8325.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8325 = 3^{2} \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8325.a (trivial)

Newform invariants

comment:select newform

sage:traces = [9,5,0,11,0,0,-8,15,0,0,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$66.4754596827$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 6x^{7} + 30x^{6} + 15x^{5} - 70x^{4} - 22x^{3} + 44x^{2} + 4x - 4 $ x^9 - 4x^8 - 6x^7 + 30x^6 + 15x^5 - 70x^4 - 22x^3 + 44x^2 + 4x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 185)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.356037$ of defining polynomial
Character	$\chi$	$=$	8325.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.35604 q^{2} -0.161165 q^{4} -0.748140 q^{7} -2.93062 q^{8} -5.40521 q^{11} -5.16556 q^{13} -1.01451 q^{14} -3.65170 q^{16} +3.44151 q^{17} +5.38802 q^{19} -7.32966 q^{22} -3.48474 q^{23} -7.00469 q^{26} +0.120574 q^{28} -6.47076 q^{29} -2.23046 q^{31} +0.909404 q^{32} +4.66682 q^{34} +1.00000 q^{37} +7.30635 q^{38} +6.44821 q^{41} -2.71018 q^{43} +0.871130 q^{44} -4.72543 q^{46} +10.3708 q^{47} -6.44029 q^{49} +0.832507 q^{52} -4.64473 q^{53} +2.19251 q^{56} -8.77459 q^{58} +12.7207 q^{59} -10.3592 q^{61} -3.02459 q^{62} +8.53658 q^{64} +6.09691 q^{67} -0.554651 q^{68} +2.80433 q^{71} +1.51542 q^{73} +1.35604 q^{74} -0.868359 q^{76} +4.04385 q^{77} -13.9090 q^{79} +8.74401 q^{82} +1.14982 q^{83} -3.67511 q^{86} +15.8406 q^{88} +0.373310 q^{89} +3.86456 q^{91} +0.561617 q^{92} +14.0632 q^{94} -8.96381 q^{97} -8.73326 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 5 q^{2} + 11 q^{4} - 8 q^{7} + 15 q^{8} - 6 q^{13} + 4 q^{14} + 11 q^{16} + 18 q^{17} - 4 q^{19} - 6 q^{22} + 16 q^{23} + 6 q^{26} + 20 q^{28} + 2 q^{29} - 6 q^{31} + 35 q^{32} + 6 q^{34} + 9 q^{37}+ \cdots + 21 q^{98}+O(q^{100}) $ 9 * q + 5 * q^2 + 11 * q^4 - 8 * q^7 + 15 * q^8 - 6 * q^13 + 4 * q^14 + 11 * q^16 + 18 * q^17 - 4 * q^19 - 6 * q^22 + 16 * q^23 + 6 * q^26 + 20 * q^28 + 2 * q^29 - 6 * q^31 + 35 * q^32 + 6 * q^34 + 9 * q^37 + 4 * q^38 - 2 * q^41 - 14 * q^43 - 4 * q^44 + 16 * q^46 + 34 * q^47 - 7 * q^49 - 16 * q^52 - 4 * q^53 + 24 * q^56 + 22 * q^58 - 10 * q^59 + 8 * q^61 + 26 * q^62 + 15 * q^64 - 16 * q^67 + 36 * q^68 - 8 * q^71 - 8 * q^73 + 5 * q^74 - 48 * q^76 + 14 * q^77 - 26 * q^79 + 4 * q^82 + 70 * q^83 + 16 * q^86 + 36 * q^88 + 8 * q^89 - 18 * q^91 + 16 * q^92 - 26 * q^94 + 2 * q^97 + 21 * q^98

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 $

$2$	1.35604	0.958863	0.479431	−	0.877579i	$-0.340843\pi$
$2$	1.35604	0.958863	0.479431	+	0.877579i	$0.340843\pi$
$3$	0	0
$3$	0	0
$4$	−0.161165	−0.0805825
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.748140	−0.282770	−0.141385	−	0.989955i	$-0.545156\pi$
$7$	−0.748140	−0.282770	−0.141385	+	0.989955i	$0.545156\pi$
$8$	−2.93062	−1.03613
$9$	0	0
$10$	0	0
$11$	−5.40521	−1.62973	−0.814866	−	0.579650i	$-0.803189\pi$
$11$	−5.40521	−1.62973	−0.814866	+	0.579650i	$0.803189\pi$
$12$	0	0
$13$	−5.16556	−1.43267	−0.716334	−	0.697757i	$-0.754181\pi$
$13$	−5.16556	−1.43267	−0.716334	+	0.697757i	$0.754181\pi$
$14$	−1.01451	−0.271138
$15$	0	0
$16$	−3.65170	−0.912924
$17$	3.44151	0.834690	0.417345	−	0.908748i	$-0.362961\pi$
$17$	3.44151	0.834690	0.417345	+	0.908748i	$0.362961\pi$
$18$	0	0
$19$	5.38802	1.23610	0.618048	−	0.786140i	$-0.287924\pi$
$19$	5.38802	1.23610	0.618048	+	0.786140i	$0.287924\pi$
$20$	0	0
$21$	0	0
$22$	−7.32966	−1.56269
$23$	−3.48474	−0.726618	−0.363309	−	0.931669i	$-0.618353\pi$
$23$	−3.48474	−0.726618	−0.363309	+	0.931669i	$0.618353\pi$
$24$	0	0
$25$	0	0
$26$	−7.00469	−1.37373
$27$	0	0
$28$	0.120574	0.0227863
$29$	−6.47076	−1.20159	−0.600795	−	0.799403i	$-0.705149\pi$
$29$	−6.47076	−1.20159	−0.600795	+	0.799403i	$0.705149\pi$
$30$	0	0
$31$	−2.23046	−0.400603	−0.200301	−	0.979734i	$-0.564192\pi$
$31$	−2.23046	−0.400603	−0.200301	+	0.979734i	$0.564192\pi$
$32$	0.909404	0.160761
$33$	0	0
$34$	4.66682	0.800353
$35$	0	0
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	7.30635	1.18525
$39$	0	0
$40$	0	0
$41$	6.44821	1.00704	0.503521	−	0.863983i	$-0.332038\pi$
$41$	6.44821	1.00704	0.503521	+	0.863983i	$0.332038\pi$
$42$	0	0
$43$	−2.71018	−0.413299	−0.206649	−	0.978415i	$-0.566256\pi$
$43$	−2.71018	−0.413299	−0.206649	+	0.978415i	$0.566256\pi$
$44$	0.871130	0.131328
$45$	0	0
$46$	−4.72543	−0.696726
$47$	10.3708	1.51274	0.756370	−	0.654144i	$-0.226971\pi$
$47$	10.3708	1.51274	0.756370	+	0.654144i	$0.226971\pi$
$48$	0	0
$49$	−6.44029	−0.920041
$50$	0	0
$51$	0	0
$52$	0.832507	0.115448
$53$	−4.64473	−0.638002	−0.319001	−	0.947754i	$-0.603347\pi$
$53$	−4.64473	−0.638002	−0.319001	+	0.947754i	$0.603347\pi$
$54$	0	0
$55$	0	0
$56$	2.19251	0.292987
$57$	0	0
$58$	−8.77459	−1.15216
$59$	12.7207	1.65609	0.828046	−	0.560660i	$-0.189452\pi$
$59$	12.7207	1.65609	0.828046	+	0.560660i	$0.189452\pi$
$60$	0	0
$61$	−10.3592	−1.32635	−0.663177	−	0.748462i	$-0.730792\pi$
$61$	−10.3592	−1.32635	−0.663177	+	0.748462i	$0.730792\pi$
$62$	−3.02459	−0.384123
$63$	0	0
$64$	8.53658	1.06707
$65$	0	0
$66$	0	0
$67$	6.09691	0.744857	0.372428	−	0.928061i	$-0.378525\pi$
$67$	6.09691	0.744857	0.372428	+	0.928061i	$0.378525\pi$
$68$	−0.554651	−0.0672613
$69$	0	0
$70$	0	0
$71$	2.80433	0.332813	0.166407	−	0.986057i	$-0.446784\pi$
$71$	2.80433	0.332813	0.166407	+	0.986057i	$0.446784\pi$
$72$	0	0
$73$	1.51542	0.177367	0.0886833	−	0.996060i	$-0.471734\pi$
$73$	1.51542	0.177367	0.0886833	+	0.996060i	$0.471734\pi$
$74$	1.35604	0.157636
$75$	0	0
$76$	−0.868359	−0.0996076
$77$	4.04385	0.460840
$78$	0	0
$79$	−13.9090	−1.56489	−0.782444	−	0.622721i	$-0.786027\pi$
$79$	−13.9090	−1.56489	−0.782444	+	0.622721i	$0.786027\pi$
$80$	0	0
$81$	0	0
$82$	8.74401	0.965614
$83$	1.14982	0.126209	0.0631047	−	0.998007i	$-0.479900\pi$
$83$	1.14982	0.126209	0.0631047	+	0.998007i	$0.479900\pi$
$84$	0	0
$85$	0	0
$86$	−3.67511	−0.396297
$87$	0	0
$88$	15.8406	1.68861
$89$	0.373310	0.0395707	0.0197854	−	0.999804i	$-0.493702\pi$
$89$	0.373310	0.0395707	0.0197854	+	0.999804i	$0.493702\pi$
$90$	0	0
$91$	3.86456	0.405116
$92$	0.561617	0.0585526
$93$	0	0
$94$	14.0632	1.45051
$95$	0	0
$96$	0	0
$97$	−8.96381	−0.910137	−0.455069	−	0.890456i	$-0.650385\pi$
$97$	−8.96381	−0.910137	−0.455069	+	0.890456i	$0.650385\pi$
$98$	−8.73326	−0.882193
$99$	0	0
$100$	0	0
$101$	13.9235	1.38544	0.692722	−	0.721205i	$-0.256411\pi$
$101$	13.9235	1.38544	0.692722	+	0.721205i	$0.256411\pi$
$102$	0	0
$103$	−2.98700	−0.294318	−0.147159	−	0.989113i	$-0.547013\pi$
$103$	−2.98700	−0.294318	−0.147159	+	0.989113i	$0.547013\pi$
$104$	15.1383	1.48443
$105$	0	0
$106$	−6.29842	−0.611756
$107$	6.58927	0.637009	0.318504	−	0.947921i	$-0.396819\pi$
$107$	6.58927	0.637009	0.318504	+	0.947921i	$0.396819\pi$
$108$	0	0
$109$	17.9577	1.72004	0.860019	−	0.510263i	$-0.170452\pi$
$109$	17.9577	1.72004	0.860019	+	0.510263i	$0.170452\pi$
$110$	0	0
$111$	0	0
$112$	2.73198	0.258148
$113$	19.7970	1.86235	0.931173	−	0.364576i	$-0.118786\pi$
$113$	19.7970	1.86235	0.931173	+	0.364576i	$0.118786\pi$
$114$	0	0
$115$	0	0
$116$	1.04286	0.0968271
$117$	0	0
$118$	17.2497	1.58797
$119$	−2.57473	−0.236026
$120$	0	0
$121$	18.2163	1.65602
$122$	−14.0474	−1.27179
$123$	0	0
$124$	0.359472	0.0322815
$125$	0	0
$126$	0	0
$127$	0.141983	0.0125989	0.00629947	−	0.999980i	$-0.497995\pi$
$127$	0.141983	0.0125989	0.00629947	+	0.999980i	$0.497995\pi$
$128$	9.75710	0.862414
$129$	0	0
$130$	0	0
$131$	0.435890	0.0380839	0.0190420	−	0.999819i	$-0.493938\pi$
$131$	0.435890	0.0380839	0.0190420	+	0.999819i	$0.493938\pi$
$132$	0	0
$133$	−4.03099	−0.349531
$134$	8.26764	0.714215
$135$	0	0
$136$	−10.0858	−0.864847
$137$	3.78307	0.323209	0.161605	−	0.986856i	$-0.448333\pi$
$137$	3.78307	0.323209	0.161605	+	0.986856i	$0.448333\pi$
$138$	0	0
$139$	−12.7579	−1.08211	−0.541055	−	0.840987i	$-0.681975\pi$
$139$	−12.7579	−1.08211	−0.541055	+	0.840987i	$0.681975\pi$
$140$	0	0
$141$	0	0
$142$	3.80278	0.319122
$143$	27.9209	2.33486
$144$	0	0
$145$	0	0
$146$	2.05497	0.170070
$147$	0	0
$148$	−0.161165	−0.0132477
$149$	3.72013	0.304765	0.152382	−	0.988322i	$-0.451305\pi$
$149$	3.72013	0.304765	0.152382	+	0.988322i	$0.451305\pi$
$150$	0	0
$151$	−7.59391	−0.617984	−0.308992	−	0.951065i	$-0.599992\pi$
$151$	−7.59391	−0.617984	−0.308992	+	0.951065i	$0.599992\pi$
$152$	−15.7902	−1.28076
$153$	0	0
$154$	5.48361	0.441882
$155$	0	0
$156$	0	0
$157$	16.0657	1.28218	0.641092	−	0.767464i	$-0.278481\pi$
$157$	16.0657	1.28218	0.641092	+	0.767464i	$0.278481\pi$
$158$	−18.8611	−1.50051
$159$	0	0
$160$	0	0
$161$	2.60707	0.205466
$162$	0	0
$163$	6.99182	0.547641	0.273821	−	0.961781i	$-0.411713\pi$
$163$	6.99182	0.547641	0.273821	+	0.961781i	$0.411713\pi$
$164$	−1.03923	−0.0811499
$165$	0	0
$166$	1.55920	0.121017
$167$	−16.3483	−1.26507	−0.632534	−	0.774532i	$-0.717985\pi$
$167$	−16.3483	−1.26507	−0.632534	+	0.774532i	$0.717985\pi$
$168$	0	0
$169$	13.6830	1.05254
$170$	0	0
$171$	0	0
$172$	0.436786	0.0333046
$173$	6.21965	0.472871	0.236436	−	0.971647i	$-0.424021\pi$
$173$	6.21965	0.472871	0.236436	+	0.971647i	$0.424021\pi$
$174$	0	0
$175$	0	0
$176$	19.7382	1.48782
$177$	0	0
$178$	0.506222	0.0379429
$179$	10.9497	0.818418	0.409209	−	0.912441i	$-0.365805\pi$
$179$	10.9497	0.818418	0.409209	+	0.912441i	$0.365805\pi$
$180$	0	0
$181$	8.75050	0.650419	0.325210	−	0.945642i	$-0.394565\pi$
$181$	8.75050	0.650419	0.325210	+	0.945642i	$0.394565\pi$
$182$	5.24049	0.388451
$183$	0	0
$184$	10.2124	0.752870
$185$	0	0
$186$	0	0
$187$	−18.6021	−1.36032
$188$	−1.67141	−0.121900
$189$	0	0
$190$	0	0
$191$	−15.7914	−1.14263	−0.571314	−	0.820732i	$-0.693566\pi$
$191$	−15.7914	−1.14263	−0.571314	+	0.820732i	$0.693566\pi$
$192$	0	0
$193$	10.9845	0.790681	0.395341	−	0.918535i	$-0.370627\pi$
$193$	10.9845	0.790681	0.395341	+	0.918535i	$0.370627\pi$
$194$	−12.1553	−0.872697
$195$	0	0
$196$	1.03795	0.0741391
$197$	−9.86682	−0.702982	−0.351491	−	0.936191i	$-0.614325\pi$
$197$	−9.86682	−0.702982	−0.351491	+	0.936191i	$0.614325\pi$
$198$	0	0
$199$	5.58756	0.396091	0.198046	−	0.980193i	$-0.436541\pi$
$199$	5.58756	0.396091	0.198046	+	0.980193i	$0.436541\pi$
$200$	0	0
$201$	0	0
$202$	18.8808	1.32845
$203$	4.84104	0.339774
$204$	0	0
$205$	0	0
$206$	−4.05048	−0.282210
$207$	0	0
$208$	18.8630	1.30792
$209$	−29.1233	−2.01450
$210$	0	0
$211$	−1.95780	−0.134781	−0.0673904	−	0.997727i	$-0.521467\pi$
$211$	−1.95780	−0.134781	−0.0673904	+	0.997727i	$0.521467\pi$
$212$	0.748567	0.0514118
$213$	0	0
$214$	8.93529	0.610804
$215$	0	0
$216$	0	0
$217$	1.66870	0.113279
$218$	24.3513	1.64928
$219$	0	0
$220$	0	0
$221$	−17.7773	−1.19583
$222$	0	0
$223$	−8.97271	−0.600857	−0.300429	−	0.953804i	$-0.597130\pi$
$223$	−8.97271	−0.600857	−0.300429	+	0.953804i	$0.597130\pi$
$224$	−0.680362	−0.0454586
$225$	0	0
$226$	26.8455	1.78573
$227$	7.78659	0.516814	0.258407	−	0.966036i	$-0.416802\pi$
$227$	7.78659	0.516814	0.258407	+	0.966036i	$0.416802\pi$
$228$	0	0
$229$	−9.69814	−0.640871	−0.320435	−	0.947270i	$-0.603829\pi$
$229$	−9.69814	−0.640871	−0.320435	+	0.947270i	$0.603829\pi$
$230$	0	0
$231$	0	0
$232$	18.9633	1.24500
$233$	22.4897	1.47335	0.736676	−	0.676246i	$-0.236394\pi$
$233$	22.4897	1.47335	0.736676	+	0.676246i	$0.236394\pi$
$234$	0	0
$235$	0	0
$236$	−2.05013	−0.133452
$237$	0	0
$238$	−3.49143	−0.226316
$239$	22.3620	1.44648	0.723238	−	0.690598i	$-0.242653\pi$
$239$	22.3620	1.44648	0.723238	+	0.690598i	$0.242653\pi$
$240$	0	0
$241$	−4.78481	−0.308217	−0.154108	−	0.988054i	$-0.549250\pi$
$241$	−4.78481	−0.308217	−0.154108	+	0.988054i	$0.549250\pi$
$242$	24.7019	1.58790
$243$	0	0
$244$	1.66953	0.106881
$245$	0	0
$246$	0	0
$247$	−27.8321	−1.77092
$248$	6.53663	0.415076
$249$	0	0
$250$	0	0
$251$	0.357304	0.0225529	0.0112764	−	0.999936i	$-0.496411\pi$
$251$	0.357304	0.0225529	0.0112764	+	0.999936i	$0.496411\pi$
$252$	0	0
$253$	18.8357	1.18419
$254$	0.192534	0.0120807
$255$	0	0
$256$	−3.84217	−0.240135
$257$	7.93068	0.494702	0.247351	−	0.968926i	$-0.420440\pi$
$257$	7.93068	0.494702	0.247351	+	0.968926i	$0.420440\pi$
$258$	0	0
$259$	−0.748140	−0.0464872
$260$	0	0
$261$	0	0
$262$	0.591083	0.0365172
$263$	−2.56626	−0.158242	−0.0791211	−	0.996865i	$-0.525211\pi$
$263$	−2.56626	−0.158242	−0.0791211	+	0.996865i	$0.525211\pi$
$264$	0	0
$265$	0	0
$266$	−5.46617	−0.335153
$267$	0	0
$268$	−0.982609	−0.0600224
$269$	−0.834873	−0.0509031	−0.0254516	−	0.999676i	$-0.508102\pi$
$269$	−0.834873	−0.0509031	−0.0254516	+	0.999676i	$0.508102\pi$
$270$	0	0
$271$	14.5172	0.881855	0.440928	−	0.897543i	$-0.354650\pi$
$271$	14.5172	0.881855	0.440928	+	0.897543i	$0.354650\pi$
$272$	−12.5674	−0.762008
$273$	0	0
$274$	5.12998	0.309913
$275$	0	0
$276$	0	0
$277$	31.2010	1.87469	0.937344	−	0.348406i	$-0.113277\pi$
$277$	31.2010	1.87469	0.937344	+	0.348406i	$0.113277\pi$
$278$	−17.3002	−1.03759
$279$	0	0
$280$	0	0
$281$	13.0674	0.779533	0.389767	−	0.920914i	$-0.372556\pi$
$281$	13.0674	0.779533	0.389767	+	0.920914i	$0.372556\pi$
$282$	0	0
$283$	3.47900	0.206805	0.103402	−	0.994640i	$-0.467027\pi$
$283$	3.47900	0.206805	0.103402	+	0.994640i	$0.467027\pi$
$284$	−0.451960	−0.0268189
$285$	0	0
$286$	37.8618	2.23881
$287$	−4.82417	−0.284762
$288$	0	0
$289$	−5.15599	−0.303293
$290$	0	0
$291$	0	0
$292$	−0.244233	−0.0142926
$293$	−30.0461	−1.75531	−0.877655	−	0.479293i	$-0.840893\pi$
$293$	−30.0461	−1.75531	−0.877655	+	0.479293i	$0.840893\pi$
$294$	0	0
$295$	0	0
$296$	−2.93062	−0.170339
$297$	0	0
$298$	5.04463	0.292228
$299$	18.0006	1.04100
$300$	0	0
$301$	2.02760	0.116869
$302$	−10.2976	−0.592562
$303$	0	0
$304$	−19.6754	−1.12846
$305$	0	0
$306$	0	0
$307$	−26.3957	−1.50648	−0.753241	−	0.657745i	$-0.771510\pi$
$307$	−26.3957	−1.50648	−0.753241	+	0.657745i	$0.771510\pi$
$308$	−0.651727	−0.0371356
$309$	0	0
$310$	0	0
$311$	14.8744	0.843447	0.421724	−	0.906724i	$-0.361425\pi$
$311$	14.8744	0.843447	0.421724	+	0.906724i	$0.361425\pi$
$312$	0	0
$313$	−0.373591	−0.0211166	−0.0105583	−	0.999944i	$-0.503361\pi$
$313$	−0.373591	−0.0211166	−0.0105583	+	0.999944i	$0.503361\pi$
$314$	21.7857	1.22944
$315$	0	0
$316$	2.24165	0.126103
$317$	−5.02971	−0.282497	−0.141248	−	0.989974i	$-0.545112\pi$
$317$	−5.02971	−0.282497	−0.141248	+	0.989974i	$0.545112\pi$
$318$	0	0
$319$	34.9758	1.95827
$320$	0	0
$321$	0	0
$322$	3.53528	0.197014
$323$	18.5429	1.03176
$324$	0	0
$325$	0	0
$326$	9.48116	0.525113
$327$	0	0
$328$	−18.8972	−1.04343
$329$	−7.75883	−0.427758
$330$	0	0
$331$	2.14829	0.118081	0.0590404	−	0.998256i	$-0.481196\pi$
$331$	2.14829	0.118081	0.0590404	+	0.998256i	$0.481196\pi$
$332$	−0.185311	−0.0101703
$333$	0	0
$334$	−22.1689	−1.21303
$335$	0	0
$336$	0	0
$337$	14.2432	0.775878	0.387939	−	0.921685i	$-0.373187\pi$
$337$	14.2432	0.775878	0.387939	+	0.921685i	$0.373187\pi$
$338$	18.5546	1.00924
$339$	0	0
$340$	0	0
$341$	12.0561	0.652875
$342$	0	0
$343$	10.0552	0.542931
$344$	7.94251	0.428231
$345$	0	0
$346$	8.43408	0.453419
$347$	21.3584	1.14658	0.573289	−	0.819353i	$-0.305667\pi$
$347$	21.3584	1.14658	0.573289	+	0.819353i	$0.305667\pi$
$348$	0	0
$349$	−8.65232	−0.463148	−0.231574	−	0.972817i	$-0.574388\pi$
$349$	−8.65232	−0.463148	−0.231574	+	0.972817i	$0.574388\pi$
$350$	0	0
$351$	0	0
$352$	−4.91552	−0.261998
$353$	−24.5145	−1.30477	−0.652387	−	0.757886i	$-0.726232\pi$
$353$	−24.5145	−1.30477	−0.652387	+	0.757886i	$0.726232\pi$
$354$	0	0
$355$	0	0
$356$	−0.0601644	−0.00318871
$357$	0	0
$358$	14.8482	0.784750
$359$	−20.8964	−1.10287	−0.551434	−	0.834218i	$-0.685919\pi$
$359$	−20.8964	−1.10287	−0.551434	+	0.834218i	$0.685919\pi$
$360$	0	0
$361$	10.0307	0.527933
$362$	11.8660	0.623663
$363$	0	0
$364$	−0.622832	−0.0326453
$365$	0	0
$366$	0	0
$367$	0.961719	0.0502013	0.0251007	−	0.999685i	$-0.492009\pi$
$367$	0.961719	0.0502013	0.0251007	+	0.999685i	$0.492009\pi$
$368$	12.7252	0.663347
$369$	0	0
$370$	0	0
$371$	3.47491	0.180408
$372$	0	0
$373$	−16.9978	−0.880114	−0.440057	−	0.897970i	$-0.645042\pi$
$373$	−16.9978	−0.880114	−0.440057	+	0.897970i	$0.645042\pi$
$374$	−25.2251	−1.30436
$375$	0	0
$376$	−30.3929	−1.56740
$377$	33.4251	1.72148
$378$	0	0
$379$	12.9112	0.663203	0.331601	−	0.943420i	$-0.392411\pi$
$379$	12.9112	0.663203	0.331601	+	0.943420i	$0.392411\pi$
$380$	0	0
$381$	0	0
$382$	−21.4138	−1.09562
$383$	13.8233	0.706337	0.353168	−	0.935560i	$-0.385104\pi$
$383$	13.8233	0.706337	0.353168	+	0.935560i	$0.385104\pi$
$384$	0	0
$385$	0	0
$386$	14.8954	0.758155
$387$	0	0
$388$	1.44465	0.0733411
$389$	10.0408	0.509089	0.254544	−	0.967061i	$-0.418074\pi$
$389$	10.0408	0.509089	0.254544	+	0.967061i	$0.418074\pi$
$390$	0	0
$391$	−11.9928	−0.606500
$392$	18.8740	0.953282
$393$	0	0
$394$	−13.3798	−0.674063
$395$	0	0
$396$	0	0
$397$	−4.87476	−0.244657	−0.122329	−	0.992490i	$-0.539036\pi$
$397$	−4.87476	−0.244657	−0.122329	+	0.992490i	$0.539036\pi$
$398$	7.57693	0.379797
$399$	0	0
$400$	0	0
$401$	5.97844	0.298549	0.149275	−	0.988796i	$-0.452306\pi$
$401$	5.97844	0.298549	0.149275	+	0.988796i	$0.452306\pi$
$402$	0	0
$403$	11.5216	0.573930
$404$	−2.24398	−0.111642
$405$	0	0
$406$	6.56462	0.325797
$407$	−5.40521	−0.267926
$408$	0	0
$409$	−28.3419	−1.40142	−0.700709	−	0.713447i	$-0.747133\pi$
$409$	−28.3419	−1.40142	−0.700709	+	0.713447i	$0.747133\pi$
$410$	0	0
$411$	0	0
$412$	0.481399	0.0237168
$413$	−9.51686	−0.468294
$414$	0	0
$415$	0	0
$416$	−4.69758	−0.230318
$417$	0	0
$418$	−39.4923	−1.93163
$419$	−9.61738	−0.469840	−0.234920	−	0.972015i	$-0.575483\pi$
$419$	−9.61738	−0.469840	−0.234920	+	0.972015i	$0.575483\pi$
$420$	0	0
$421$	−8.65638	−0.421886	−0.210943	−	0.977498i	$-0.567653\pi$
$421$	−8.65638	−0.421886	−0.210943	+	0.977498i	$0.567653\pi$
$422$	−2.65485	−0.129236
$423$	0	0
$424$	13.6119	0.661053
$425$	0	0
$426$	0	0
$427$	7.75011	0.375054
$428$	−1.06196	−0.0513317
$429$	0	0
$430$	0	0
$431$	−34.7519	−1.67394	−0.836969	−	0.547250i	$-0.815675\pi$
$431$	−34.7519	−1.67394	−0.836969	+	0.547250i	$0.815675\pi$
$432$	0	0
$433$	−27.9319	−1.34232	−0.671161	−	0.741312i	$-0.734204\pi$
$433$	−27.9319	−1.34232	−0.671161	+	0.741312i	$0.734204\pi$
$434$	2.26281	0.108619
$435$	0	0
$436$	−2.89415	−0.138605
$437$	−18.7758	−0.898169
$438$	0	0
$439$	−23.0305	−1.09919	−0.549593	−	0.835433i	$-0.685217\pi$
$439$	−23.0305	−1.09919	−0.549593	+	0.835433i	$0.685217\pi$
$440$	0	0
$441$	0	0
$442$	−24.1067	−1.14664
$443$	27.7267	1.31733	0.658667	−	0.752435i	$-0.271121\pi$
$443$	27.7267	1.31733	0.658667	+	0.752435i	$0.271121\pi$
$444$	0	0
$445$	0	0
$446$	−12.1673	−0.576139
$447$	0	0
$448$	−6.38656	−0.301736
$449$	−26.4363	−1.24761	−0.623804	−	0.781581i	$-0.714414\pi$
$449$	−26.4363	−1.24761	−0.623804	+	0.781581i	$0.714414\pi$
$450$	0	0
$451$	−34.8539	−1.64121
$452$	−3.19058	−0.150072
$453$	0	0
$454$	10.5589	0.495554
$455$	0	0
$456$	0	0
$457$	−26.3869	−1.23433	−0.617164	−	0.786835i	$-0.711718\pi$
$457$	−26.3869	−1.23433	−0.617164	+	0.786835i	$0.711718\pi$
$458$	−13.1510	−0.614507
$459$	0	0
$460$	0	0
$461$	12.8328	0.597684	0.298842	−	0.954303i	$-0.403400\pi$
$461$	12.8328	0.597684	0.298842	+	0.954303i	$0.403400\pi$
$462$	0	0
$463$	−8.59902	−0.399630	−0.199815	−	0.979834i	$-0.564034\pi$
$463$	−8.59902	−0.399630	−0.199815	+	0.979834i	$0.564034\pi$
$464$	23.6293	1.09696
$465$	0	0
$466$	30.4969	1.41274
$467$	29.1169	1.34737	0.673686	−	0.739018i	$-0.264710\pi$
$467$	29.1169	1.34737	0.673686	+	0.739018i	$0.264710\pi$
$468$	0	0
$469$	−4.56135	−0.210623
$470$	0	0
$471$	0	0
$472$	−37.2795	−1.71593
$473$	14.6491	0.673566
$474$	0	0
$475$	0	0
$476$	0.414957	0.0190195
$477$	0	0
$478$	30.3237	1.38697
$479$	−35.3714	−1.61616	−0.808081	−	0.589072i	$-0.799493\pi$
$479$	−35.3714	−1.61616	−0.808081	+	0.589072i	$0.799493\pi$
$480$	0	0
$481$	−5.16556	−0.235529
$482$	−6.48837	−0.295537
$483$	0	0
$484$	−2.93582	−0.133446
$485$	0	0
$486$	0	0
$487$	37.1133	1.68176	0.840882	−	0.541219i	$-0.182037\pi$
$487$	37.1133	1.68176	0.840882	+	0.541219i	$0.182037\pi$
$488$	30.3587	1.37428
$489$	0	0
$490$	0	0
$491$	30.6327	1.38243	0.691217	−	0.722647i	$-0.257075\pi$
$491$	30.6327	1.38243	0.691217	+	0.722647i	$0.257075\pi$
$492$	0	0
$493$	−22.2692	−1.00295
$494$	−37.7414	−1.69806
$495$	0	0
$496$	8.14496	0.365720
$497$	−2.09804	−0.0941098
$498$	0	0
$499$	−26.3356	−1.17894	−0.589472	−	0.807789i	$-0.700664\pi$
$499$	−26.3356	−1.17894	−0.589472	+	0.807789i	$0.700664\pi$
$500$	0	0
$501$	0	0
$502$	0.484518	0.0216251
$503$	20.8358	0.929023	0.464511	−	0.885567i	$-0.346230\pi$
$503$	20.8358	0.929023	0.464511	+	0.885567i	$0.346230\pi$
$504$	0	0
$505$	0	0
$506$	25.5419	1.13548
$507$	0	0
$508$	−0.0228827	−0.00101525
$509$	−3.57224	−0.158337	−0.0791685	−	0.996861i	$-0.525227\pi$
$509$	−3.57224	−0.158337	−0.0791685	+	0.996861i	$0.525227\pi$
$510$	0	0
$511$	−1.13375	−0.0501540
$512$	−24.7243	−1.09267
$513$	0	0
$514$	10.7543	0.474352
$515$	0	0
$516$	0	0
$517$	−56.0565	−2.46536
$518$	−1.01451	−0.0445748
$519$	0	0
$520$	0	0
$521$	−36.9568	−1.61911	−0.809554	−	0.587045i	$-0.800291\pi$
$521$	−36.9568	−1.61911	−0.809554	+	0.587045i	$0.800291\pi$
$522$	0	0
$523$	16.0177	0.700405	0.350203	−	0.936674i	$-0.386113\pi$
$523$	16.0177	0.700405	0.350203	+	0.936674i	$0.386113\pi$
$524$	−0.0702502	−0.00306890
$525$	0	0
$526$	−3.47994	−0.151733
$527$	−7.67616	−0.334379
$528$	0	0
$529$	−10.8566	−0.472027
$530$	0	0
$531$	0	0
$532$	0.649655	0.0281661
$533$	−33.3086	−1.44276
$534$	0	0
$535$	0	0
$536$	−17.8677	−0.771768
$537$	0	0
$538$	−1.13212	−0.0488091
$539$	34.8111	1.49942
$540$	0	0
$541$	−9.82948	−0.422602	−0.211301	−	0.977421i	$-0.567770\pi$
$541$	−9.82948	−0.422602	−0.211301	+	0.977421i	$0.567770\pi$
$542$	19.6858	0.845578
$543$	0	0
$544$	3.12973	0.134186
$545$	0	0
$546$	0	0
$547$	16.7689	0.716987	0.358493	−	0.933532i	$-0.383291\pi$
$547$	16.7689	0.716987	0.358493	+	0.933532i	$0.383291\pi$
$548$	−0.609698	−0.0260450
$549$	0	0
$550$	0	0
$551$	−34.8646	−1.48528
$552$	0	0
$553$	10.4059	0.442504
$554$	42.3097	1.79757
$555$	0	0
$556$	2.05612	0.0871990
$557$	35.3749	1.49888	0.749440	−	0.662072i	$-0.230323\pi$
$557$	35.3749	1.49888	0.749440	+	0.662072i	$0.230323\pi$
$558$	0	0
$559$	13.9996	0.592120
$560$	0	0
$561$	0	0
$562$	17.7198	0.747465
$563$	16.7458	0.705751	0.352876	−	0.935670i	$-0.385204\pi$
$563$	16.7458	0.705751	0.352876	+	0.935670i	$0.385204\pi$
$564$	0	0
$565$	0	0
$566$	4.71765	0.198298
$567$	0	0
$568$	−8.21844	−0.344838
$569$	26.0953	1.09397	0.546987	−	0.837141i	$-0.315775\pi$
$569$	26.0953	1.09397	0.546987	+	0.837141i	$0.315775\pi$
$570$	0	0
$571$	24.5326	1.02666	0.513328	−	0.858193i	$-0.328413\pi$
$571$	24.5326	1.02666	0.513328	+	0.858193i	$0.328413\pi$
$572$	−4.49987	−0.188149
$573$	0	0
$574$	−6.54175	−0.273047
$575$	0	0
$576$	0	0
$577$	21.9665	0.914479	0.457239	−	0.889344i	$-0.348838\pi$
$577$	21.9665	0.914479	0.457239	+	0.889344i	$0.348838\pi$
$578$	−6.99171	−0.290817
$579$	0	0
$580$	0	0
$581$	−0.860228	−0.0356883
$582$	0	0
$583$	25.1057	1.03977
$584$	−4.44112	−0.183775
$585$	0	0
$586$	−40.7436	−1.68310
$587$	14.6771	0.605788	0.302894	−	0.953024i	$-0.402047\pi$
$587$	14.6771	0.605788	0.302894	+	0.953024i	$0.402047\pi$
$588$	0	0
$589$	−12.0178	−0.495183
$590$	0	0
$591$	0	0
$592$	−3.65170	−0.150084
$593$	−6.40723	−0.263113	−0.131557	−	0.991309i	$-0.541998\pi$
$593$	−6.40723	−0.263113	−0.131557	+	0.991309i	$0.541998\pi$
$594$	0	0
$595$	0	0
$596$	−0.599554	−0.0245587
$597$	0	0
$598$	24.4095	0.998178
$599$	−19.0689	−0.779133	−0.389566	−	0.920998i	$-0.627375\pi$
$599$	−19.0689	−0.779133	−0.389566	+	0.920998i	$0.627375\pi$
$600$	0	0
$601$	22.2923	0.909320	0.454660	−	0.890665i	$-0.349761\pi$
$601$	22.2923	0.909320	0.454660	+	0.890665i	$0.349761\pi$
$602$	2.74949	0.112061
$603$	0	0
$604$	1.22387	0.0497987
$605$	0	0
$606$	0	0
$607$	16.2888	0.661143	0.330571	−	0.943781i	$-0.392759\pi$
$607$	16.2888	0.661143	0.330571	+	0.943781i	$0.392759\pi$
$608$	4.89988	0.198717
$609$	0	0
$610$	0	0
$611$	−53.5711	−2.16726
$612$	0	0
$613$	1.73076	0.0699046	0.0349523	−	0.999389i	$-0.488872\pi$
$613$	1.73076	0.0699046	0.0349523	+	0.999389i	$0.488872\pi$
$614$	−35.7935	−1.44451
$615$	0	0
$616$	−11.8510	−0.477490
$617$	−18.1783	−0.731830	−0.365915	−	0.930648i	$-0.619244\pi$
$617$	−18.1783	−0.731830	−0.365915	+	0.930648i	$0.619244\pi$
$618$	0	0
$619$	32.8475	1.32025	0.660127	−	0.751154i	$-0.270503\pi$
$619$	32.8475	1.32025	0.660127	+	0.751154i	$0.270503\pi$
$620$	0	0
$621$	0	0
$622$	20.1702	0.808750
$623$	−0.279288	−0.0111894
$624$	0	0
$625$	0	0
$626$	−0.506603	−0.0202479
$627$	0	0
$628$	−2.58923	−0.103322
$629$	3.44151	0.137222
$630$	0	0
$631$	−24.9007	−0.991280	−0.495640	−	0.868528i	$-0.665066\pi$
$631$	−24.9007	−0.991280	−0.495640	+	0.868528i	$0.665066\pi$
$632$	40.7621	1.62143
$633$	0	0
$634$	−6.82048	−0.270876
$635$	0	0
$636$	0	0
$637$	33.2677	1.31811
$638$	47.4285	1.87771
$639$	0	0
$640$	0	0
$641$	31.7199	1.25286	0.626430	−	0.779477i	$-0.284515\pi$
$641$	31.7199	1.25286	0.626430	+	0.779477i	$0.284515\pi$
$642$	0	0
$643$	32.4283	1.27885	0.639424	−	0.768854i	$-0.279173\pi$
$643$	32.4283	1.27885	0.639424	+	0.768854i	$0.279173\pi$
$644$	−0.420168	−0.0165570
$645$	0	0
$646$	25.1449	0.989312
$647$	45.5239	1.78973	0.894864	−	0.446339i	$-0.147272\pi$
$647$	45.5239	1.78973	0.894864	+	0.446339i	$0.147272\pi$
$648$	0	0
$649$	−68.7580	−2.69899
$650$	0	0
$651$	0	0
$652$	−1.12684	−0.0441303
$653$	18.4351	0.721421	0.360710	−	0.932678i	$-0.382534\pi$
$653$	18.4351	0.721421	0.360710	+	0.932678i	$0.382534\pi$
$654$	0	0
$655$	0	0
$656$	−23.5469	−0.919352
$657$	0	0
$658$	−10.5213	−0.410161
$659$	−44.1954	−1.72161	−0.860805	−	0.508935i	$-0.830039\pi$
$659$	−44.1954	−1.72161	−0.860805	+	0.508935i	$0.830039\pi$
$660$	0	0
$661$	11.9408	0.464443	0.232222	−	0.972663i	$-0.425401\pi$
$661$	11.9408	0.464443	0.232222	+	0.972663i	$0.425401\pi$
$662$	2.91316	0.113223
$663$	0	0
$664$	−3.36969	−0.130769
$665$	0	0
$666$	0	0
$667$	22.5489	0.873097
$668$	2.63477	0.101942
$669$	0	0
$670$	0	0
$671$	55.9934	2.16160
$672$	0	0
$673$	36.3828	1.40245	0.701227	−	0.712938i	$-0.252636\pi$
$673$	36.3828	1.40245	0.701227	+	0.712938i	$0.252636\pi$
$674$	19.3143	0.743960
$675$	0	0
$676$	−2.20522	−0.0848161
$677$	−6.30279	−0.242236	−0.121118	−	0.992638i	$-0.538648\pi$
$677$	−6.30279	−0.242236	−0.121118	+	0.992638i	$0.538648\pi$
$678$	0	0
$679$	6.70619	0.257360
$680$	0	0
$681$	0	0
$682$	16.3485	0.626017
$683$	−5.75633	−0.220260	−0.110130	−	0.993917i	$-0.535127\pi$
$683$	−5.75633	−0.220260	−0.110130	+	0.993917i	$0.535127\pi$
$684$	0	0
$685$	0	0
$686$	13.6352	0.520596
$687$	0	0
$688$	9.89676	0.377310
$689$	23.9926	0.914045
$690$	0	0
$691$	46.6320	1.77397	0.886983	−	0.461802i	$-0.152797\pi$
$691$	46.6320	1.77397	0.886983	+	0.461802i	$0.152797\pi$
$692$	−1.00239	−0.0381051
$693$	0	0
$694$	28.9627	1.09941
$695$	0	0
$696$	0	0
$697$	22.1916	0.840567
$698$	−11.7329	−0.444095
$699$	0	0
$700$	0	0
$701$	−4.14142	−0.156419	−0.0782096	−	0.996937i	$-0.524920\pi$
$701$	−4.14142	−0.156419	−0.0782096	+	0.996937i	$0.524920\pi$
$702$	0	0
$703$	5.38802	0.203213
$704$	−46.1420	−1.73904
$705$	0	0
$706$	−33.2425	−1.25110
$707$	−10.4168	−0.391762
$708$	0	0
$709$	−42.7621	−1.60596	−0.802981	−	0.596004i	$-0.796754\pi$
$709$	−42.7621	−1.60596	−0.802981	+	0.596004i	$0.796754\pi$
$710$	0	0
$711$	0	0
$712$	−1.09403	−0.0410004
$713$	7.77257	0.291085
$714$	0	0
$715$	0	0
$716$	−1.76470	−0.0659501
$717$	0	0
$718$	−28.3363	−1.05750
$719$	6.85976	0.255826	0.127913	−	0.991785i	$-0.459172\pi$
$719$	6.85976	0.255826	0.127913	+	0.991785i	$0.459172\pi$
$720$	0	0
$721$	2.23469	0.0832243
$722$	13.6020	0.506215
$723$	0	0
$724$	−1.41027	−0.0524124
$725$	0	0
$726$	0	0
$727$	−7.85944	−0.291490	−0.145745	−	0.989322i	$-0.546558\pi$
$727$	−7.85944	−0.291490	−0.145745	+	0.989322i	$0.546558\pi$
$728$	−11.3256	−0.419753
$729$	0	0
$730$	0	0
$731$	−9.32713	−0.344976
$732$	0	0
$733$	12.8442	0.474411	0.237206	−	0.971459i	$-0.423769\pi$
$733$	12.8442	0.474411	0.237206	+	0.971459i	$0.423769\pi$
$734$	1.30413	0.0481362
$735$	0	0
$736$	−3.16903	−0.116812
$737$	−32.9551	−1.21392
$738$	0	0
$739$	13.2976	0.489160	0.244580	−	0.969629i	$-0.421350\pi$
$739$	13.2976	0.489160	0.244580	+	0.969629i	$0.421350\pi$
$740$	0	0
$741$	0	0
$742$	4.71210	0.172987
$743$	36.2989	1.33168	0.665839	−	0.746096i	$-0.268074\pi$
$743$	36.2989	1.33168	0.665839	+	0.746096i	$0.268074\pi$
$744$	0	0
$745$	0	0
$746$	−23.0497	−0.843908
$747$	0	0
$748$	2.99800	0.109618
$749$	−4.92970	−0.180127
$750$	0	0
$751$	−24.5638	−0.896346	−0.448173	−	0.893947i	$-0.647925\pi$
$751$	−24.5638	−0.896346	−0.448173	+	0.893947i	$0.647925\pi$
$752$	−37.8711	−1.38102
$753$	0	0
$754$	45.3257	1.65066
$755$	0	0
$756$	0	0
$757$	−26.5324	−0.964336	−0.482168	−	0.876079i	$-0.660150\pi$
$757$	−26.5324	−0.964336	−0.482168	+	0.876079i	$0.660150\pi$
$758$	17.5080	0.635920
$759$	0	0
$760$	0	0
$761$	2.08409	0.0755481	0.0377740	−	0.999286i	$-0.487973\pi$
$761$	2.08409	0.0755481	0.0377740	+	0.999286i	$0.487973\pi$
$762$	0	0
$763$	−13.4349	−0.486376
$764$	2.54502	0.0920757
$765$	0	0
$766$	18.7449	0.677280
$767$	−65.7095	−2.37263
$768$	0	0
$769$	37.4780	1.35149	0.675746	−	0.737135i	$-0.263822\pi$
$769$	37.4780	1.35149	0.675746	+	0.737135i	$0.263822\pi$
$770$	0	0
$771$	0	0
$772$	−1.77032	−0.0637150
$773$	−51.5098	−1.85268	−0.926339	−	0.376692i	$-0.877062\pi$
$773$	−51.5098	−1.85268	−0.926339	+	0.376692i	$0.877062\pi$
$774$	0	0
$775$	0	0
$776$	26.2695	0.943021
$777$	0	0
$778$	13.6157	0.488146
$779$	34.7431	1.24480
$780$	0	0
$781$	−15.1580	−0.542396
$782$	−16.2626	−0.581550
$783$	0	0
$784$	23.5180	0.839927
$785$	0	0
$786$	0	0
$787$	30.5759	1.08991	0.544957	−	0.838464i	$-0.316546\pi$
$787$	30.5759	1.08991	0.544957	+	0.838464i	$0.316546\pi$
$788$	1.59019	0.0566480
$789$	0	0
$790$	0	0
$791$	−14.8109	−0.526617
$792$	0	0
$793$	53.5109	1.90023
$794$	−6.61035	−0.234593
$795$	0	0
$796$	−0.900518	−0.0319180
$797$	14.8601	0.526373	0.263186	−	0.964745i	$-0.415227\pi$
$797$	14.8601	0.526373	0.263186	+	0.964745i	$0.415227\pi$
$798$	0	0
$799$	35.6913	1.26267
$800$	0	0
$801$	0	0
$802$	8.10699	0.286268
$803$	−8.19116	−0.289060
$804$	0	0
$805$	0	0
$806$	15.6237	0.550321
$807$	0	0
$808$	−40.8046	−1.43550
$809$	17.9217	0.630094	0.315047	−	0.949076i	$-0.397980\pi$
$809$	17.9217	0.630094	0.315047	+	0.949076i	$0.397980\pi$
$810$	0	0
$811$	−25.4430	−0.893425	−0.446712	−	0.894678i	$-0.647405\pi$
$811$	−25.4430	−0.893425	−0.446712	+	0.894678i	$0.647405\pi$
$812$	−0.780205	−0.0273798
$813$	0	0
$814$	−7.32966	−0.256904
$815$	0	0
$816$	0	0
$817$	−14.6025	−0.510877
$818$	−38.4327	−1.34377
$819$	0	0
$820$	0	0
$821$	−33.2113	−1.15908	−0.579541	−	0.814943i	$-0.696768\pi$
$821$	−33.2113	−1.15908	−0.579541	+	0.814943i	$0.696768\pi$
$822$	0	0
$823$	50.9984	1.77769	0.888847	−	0.458205i	$-0.151507\pi$
$823$	50.9984	1.77769	0.888847	+	0.458205i	$0.151507\pi$
$824$	8.75375	0.304951
$825$	0	0
$826$	−12.9052	−0.449030
$827$	41.5779	1.44580	0.722902	−	0.690951i	$-0.242808\pi$
$827$	41.5779	1.44580	0.722902	+	0.690951i	$0.242808\pi$
$828$	0	0
$829$	−17.8704	−0.620665	−0.310332	−	0.950628i	$-0.600440\pi$
$829$	−17.8704	−0.620665	−0.310332	+	0.950628i	$0.600440\pi$
$830$	0	0
$831$	0	0
$832$	−44.0962	−1.52876
$833$	−22.1643	−0.767948
$834$	0	0
$835$	0	0
$836$	4.69366	0.162334
$837$	0	0
$838$	−13.0415	−0.450512
$839$	−26.2881	−0.907565	−0.453783	−	0.891112i	$-0.649926\pi$
$839$	−26.2881	−0.907565	−0.453783	+	0.891112i	$0.649926\pi$
$840$	0	0
$841$	12.8708	0.443820
$842$	−11.7384	−0.404531
$843$	0	0
$844$	0.315529	0.0108610
$845$	0	0
$846$	0	0
$847$	−13.6283	−0.468275
$848$	16.9611	0.582448
$849$	0	0
$850$	0	0
$851$	−3.48474	−0.119455
$852$	0	0
$853$	32.1407	1.10047	0.550237	−	0.835008i	$-0.314537\pi$
$853$	32.1407	1.10047	0.550237	+	0.835008i	$0.314537\pi$
$854$	10.5094	0.359625
$855$	0	0
$856$	−19.3106	−0.660024
$857$	−47.5492	−1.62425	−0.812125	−	0.583483i	$-0.801690\pi$
$857$	−47.5492	−1.62425	−0.812125	+	0.583483i	$0.801690\pi$
$858$	0	0
$859$	13.3037	0.453917	0.226959	−	0.973904i	$-0.427122\pi$
$859$	13.3037	0.453917	0.226959	+	0.973904i	$0.427122\pi$
$860$	0	0
$861$	0	0
$862$	−47.1248	−1.60508
$863$	−4.07561	−0.138735	−0.0693676	−	0.997591i	$-0.522098\pi$
$863$	−4.07561	−0.138735	−0.0693676	+	0.997591i	$0.522098\pi$
$864$	0	0
$865$	0	0
$866$	−37.8767	−1.28710
$867$	0	0
$868$	−0.268935	−0.00912826
$869$	75.1812	2.55035
$870$	0	0
$871$	−31.4940	−1.06713
$872$	−52.6272	−1.78218
$873$	0	0
$874$	−25.4607	−0.861221
$875$	0	0
$876$	0	0
$877$	26.7909	0.904665	0.452333	−	0.891849i	$-0.350592\pi$
$877$	26.7909	0.904665	0.452333	+	0.891849i	$0.350592\pi$
$878$	−31.2302	−1.05397
$879$	0	0
$880$	0	0
$881$	−10.8047	−0.364018	−0.182009	−	0.983297i	$-0.558260\pi$
$881$	−10.8047	−0.364018	−0.182009	+	0.983297i	$0.558260\pi$
$882$	0	0
$883$	−47.6789	−1.60452	−0.802261	−	0.596974i	$-0.796370\pi$
$883$	−47.6789	−1.60452	−0.802261	+	0.596974i	$0.796370\pi$
$884$	2.86508	0.0963632
$885$	0	0
$886$	37.5984	1.26314
$887$	44.6016	1.49757	0.748787	−	0.662811i	$-0.230637\pi$
$887$	44.6016	1.49757	0.748787	+	0.662811i	$0.230637\pi$
$888$	0	0
$889$	−0.106223	−0.00356261
$890$	0	0
$891$	0	0
$892$	1.44609	0.0484185
$893$	55.8782	1.86989
$894$	0	0
$895$	0	0
$896$	−7.29968	−0.243865
$897$	0	0
$898$	−35.8487	−1.19629
$899$	14.4328	0.481360
$900$	0	0
$901$	−15.9849	−0.532534
$902$	−47.2632	−1.57369
$903$	0	0
$904$	−58.0175	−1.92963
$905$	0	0
$906$	0	0
$907$	−0.577224	−0.0191664	−0.00958321	−	0.999954i	$-0.503050\pi$
$907$	−0.577224	−0.0191664	−0.00958321	+	0.999954i	$0.503050\pi$
$908$	−1.25493	−0.0416461
$909$	0	0
$910$	0	0
$911$	−14.8648	−0.492492	−0.246246	−	0.969207i	$-0.579197\pi$
$911$	−14.8648	−0.492492	−0.246246	+	0.969207i	$0.579197\pi$
$912$	0	0
$913$	−6.21503	−0.205687
$914$	−35.7816	−1.18355
$915$	0	0
$916$	1.56300	0.0516430
$917$	−0.326107	−0.0107690
$918$	0	0
$919$	−5.47645	−0.180651	−0.0903257	−	0.995912i	$-0.528791\pi$
$919$	−5.47645	−0.180651	−0.0903257	+	0.995912i	$0.528791\pi$
$920$	0	0
$921$	0	0
$922$	17.4018	0.573097
$923$	−14.4860	−0.476811
$924$	0	0
$925$	0	0
$926$	−11.6606	−0.383190
$927$	0	0
$928$	−5.88454	−0.193169
$929$	−8.07412	−0.264903	−0.132452	−	0.991189i	$-0.542285\pi$
$929$	−8.07412	−0.264903	−0.132452	+	0.991189i	$0.542285\pi$
$930$	0	0
$931$	−34.7004	−1.13726
$932$	−3.62456	−0.118726
$933$	0	0
$934$	39.4836	1.29194
$935$	0	0
$936$	0	0
$937$	4.83794	0.158048	0.0790242	−	0.996873i	$-0.474820\pi$
$937$	4.83794	0.158048	0.0790242	+	0.996873i	$0.474820\pi$
$938$	−6.18535	−0.201959
$939$	0	0
$940$	0	0
$941$	9.67126	0.315274	0.157637	−	0.987497i	$-0.449612\pi$
$941$	9.67126	0.315274	0.157637	+	0.987497i	$0.449612\pi$
$942$	0	0
$943$	−22.4703	−0.731734
$944$	−46.4521	−1.51189
$945$	0	0
$946$	19.8647	0.645857
$947$	22.6385	0.735653	0.367827	−	0.929894i	$-0.380102\pi$
$947$	22.6385	0.735653	0.367827	+	0.929894i	$0.380102\pi$
$948$	0	0
$949$	−7.82800	−0.254107
$950$	0	0
$951$	0	0
$952$	7.54557	0.244553
$953$	31.8539	1.03185	0.515924	−	0.856634i	$-0.327449\pi$
$953$	31.8539	1.03185	0.515924	+	0.856634i	$0.327449\pi$
$954$	0	0
$955$	0	0
$956$	−3.60397	−0.116561
$957$	0	0
$958$	−47.9649	−1.54968
$959$	−2.83027	−0.0913941
$960$	0	0
$961$	−26.0250	−0.839518
$962$	−7.00469	−0.225840
$963$	0	0
$964$	0.771143	0.0248368
$965$	0	0
$966$	0	0
$967$	−21.4942	−0.691206	−0.345603	−	0.938381i	$-0.612326\pi$
$967$	−21.4942	−0.691206	−0.345603	+	0.938381i	$0.612326\pi$
$968$	−53.3849	−1.71586
$969$	0	0
$970$	0	0
$971$	−26.3424	−0.845368	−0.422684	−	0.906277i	$-0.638912\pi$
$971$	−26.3424	−0.845368	−0.422684	+	0.906277i	$0.638912\pi$
$972$	0	0
$973$	9.54469	0.305989
$974$	50.3270	1.61258
$975$	0	0
$976$	37.8285	1.21086
$977$	−8.62429	−0.275916	−0.137958	−	0.990438i	$-0.544054\pi$
$977$	−8.62429	−0.275916	−0.137958	+	0.990438i	$0.544054\pi$
$978$	0	0
$979$	−2.01782	−0.0644897
$980$	0	0
$981$	0	0
$982$	41.5391	1.32557
$983$	−48.7498	−1.55488	−0.777438	−	0.628959i	$-0.783481\pi$
$983$	−48.7498	−1.55488	−0.777438	+	0.628959i	$0.783481\pi$
$984$	0	0
$985$	0	0
$986$	−30.1979	−0.961696
$987$	0	0
$988$	4.48556	0.142705
$989$	9.44427	0.300310
$990$	0	0
$991$	9.04494	0.287322	0.143661	−	0.989627i	$-0.454112\pi$
$991$	9.04494	0.287322	0.143661	+	0.989627i	$0.454112\pi$
$992$	−2.02839	−0.0644014
$993$	0	0
$994$	−2.84501	−0.0902383
$995$	0	0
$996$	0	0
$997$	38.5413	1.22062	0.610308	−	0.792164i	$-0.291046\pi$
$997$	38.5413	1.22062	0.610308	+	0.792164i	$0.291046\pi$
$998$	−35.7121	−1.13045
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8325.2.a.cr.1.6		9
3.2	odd	2		925.2.a.l.1.4		9
5.2	odd	4		1665.2.c.e.334.13		18
5.3	odd	4		1665.2.c.e.334.6		18
5.4	even	2		8325.2.a.cq.1.4		9
15.2	even	4		185.2.b.a.149.6	✓	18
15.8	even	4		185.2.b.a.149.13	yes	18
15.14	odd	2		925.2.a.m.1.6		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
185.2.b.a.149.6	✓	18	15.2	even	4
185.2.b.a.149.13	yes	18	15.8	even	4
925.2.a.l.1.4		9	3.2	odd	2
925.2.a.m.1.6		9	15.14	odd	2
1665.2.c.e.334.6		18	5.3	odd	4
1665.2.c.e.334.13		18	5.2	odd	4
8325.2.a.cq.1.4		9	5.4	even	2
8325.2.a.cr.1.6		9	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8325 = 3^{2} \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8325.a (trivial)

\(f(q)\)	\(=\)	\(q+1.35604 q^{2} -0.161165 q^{4} -0.748140 q^{7} -2.93062 q^{8} -5.40521 q^{11} -5.16556 q^{13} -1.01451 q^{14} -3.65170 q^{16} +3.44151 q^{17} +5.38802 q^{19} -7.32966 q^{22} -3.48474 q^{23} -7.00469 q^{26} +0.120574 q^{28} -6.47076 q^{29} -2.23046 q^{31} +0.909404 q^{32} +4.66682 q^{34} +1.00000 q^{37} +7.30635 q^{38} +6.44821 q^{41} -2.71018 q^{43} +0.871130 q^{44} -4.72543 q^{46} +10.3708 q^{47} -6.44029 q^{49} +0.832507 q^{52} -4.64473 q^{53} +2.19251 q^{56} -8.77459 q^{58} +12.7207 q^{59} -10.3592 q^{61} -3.02459 q^{62} +8.53658 q^{64} +6.09691 q^{67} -0.554651 q^{68} +2.80433 q^{71} +1.51542 q^{73} +1.35604 q^{74} -0.868359 q^{76} +4.04385 q^{77} -13.9090 q^{79} +8.74401 q^{82} +1.14982 q^{83} -3.67511 q^{86} +15.8406 q^{88} +0.373310 q^{89} +3.86456 q^{91} +0.561617 q^{92} +14.0632 q^{94} -8.96381 q^{97} -8.73326 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 5 q^{2} + 11 q^{4} - 8 q^{7} + 15 q^{8} - 6 q^{13} + 4 q^{14} + 11 q^{16} + 18 q^{17} - 4 q^{19} - 6 q^{22} + 16 q^{23} + 6 q^{26} + 20 q^{28} + 2 q^{29} - 6 q^{31} + 35 q^{32} + 6 q^{34} + 9 q^{37}+ \cdots + 21 q^{98}+O(q^{100}) \) 9 * q + 5 * q^2 + 11 * q^4 - 8 * q^7 + 15 * q^8 - 6 * q^13 + 4 * q^14 + 11 * q^16 + 18 * q^17 - 4 * q^19 - 6 * q^22 + 16 * q^23 + 6 * q^26 + 20 * q^28 + 2 * q^29 - 6 * q^31 + 35 * q^32 + 6 * q^34 + 9 * q^37 + 4 * q^38 - 2 * q^41 - 14 * q^43 - 4 * q^44 + 16 * q^46 + 34 * q^47 - 7 * q^49 - 16 * q^52 - 4 * q^53 + 24 * q^56 + 22 * q^58 - 10 * q^59 + 8 * q^61 + 26 * q^62 + 15 * q^64 - 16 * q^67 + 36 * q^68 - 8 * q^71 - 8 * q^73 + 5 * q^74 - 48 * q^76 + 14 * q^77 - 26 * q^79 + 4 * q^82 + 70 * q^83 + 16 * q^86 + 36 * q^88 + 8 * q^89 - 18 * q^91 + 16 * q^92 - 26 * q^94 + 2 * q^97 + 21 * q^98

Introduction

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