LMFDB - Embedded newform 8325.2.a.cr.1.5

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			1.5
Root			$0.296889$ 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)

$f(q)$	$=$	$q+0.703111 q^{2} -1.50564 q^{4} -0.501390 q^{7} -2.46485 q^{8} +1.80401 q^{11} -1.63218 q^{13} -0.352533 q^{14} +1.27821 q^{16} +1.92844 q^{17} -0.869173 q^{19} +1.26842 q^{22} +4.86522 q^{23} -1.14761 q^{26} +0.754910 q^{28} +3.15000 q^{29} +2.39134 q^{31} +5.82842 q^{32} +1.35591 q^{34} +1.00000 q^{37} -0.611125 q^{38} -8.24040 q^{41} -11.6931 q^{43} -2.71619 q^{44} +3.42079 q^{46} -1.43311 q^{47} -6.74861 q^{49} +2.45747 q^{52} +10.7692 q^{53} +1.23585 q^{56} +2.21480 q^{58} -8.71701 q^{59} -7.84555 q^{61} +1.68138 q^{62} +1.54161 q^{64} -11.8599 q^{67} -2.90353 q^{68} +6.06069 q^{71} +5.56086 q^{73} +0.703111 q^{74} +1.30866 q^{76} -0.904514 q^{77} +7.44924 q^{79} -5.79391 q^{82} -4.70795 q^{83} -8.22154 q^{86} -4.44662 q^{88} -5.16332 q^{89} +0.818361 q^{91} -7.32524 q^{92} -1.00764 q^{94} +6.48639 q^{97} -4.74502 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.703111	0.497174	0.248587	−	0.968610i	$-0.420034\pi$
$2$	0.703111	0.497174	0.248587	+	0.968610i	$0.420034\pi$
$3$	0	0
$3$	0	0
$4$	−1.50564	−0.752818
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.501390	−0.189508	−0.0947538	−	0.995501i	$-0.530206\pi$
$7$	−0.501390	−0.189508	−0.0947538	+	0.995501i	$0.530206\pi$
$8$	−2.46485	−0.871456
$9$	0	0
$10$	0	0
$11$	1.80401	0.543931	0.271965	−	0.962307i	$-0.412326\pi$
$11$	1.80401	0.543931	0.271965	+	0.962307i	$0.412326\pi$
$12$	0	0
$13$	−1.63218	−0.452686	−0.226343	−	0.974048i	$-0.572677\pi$
$13$	−1.63218	−0.452686	−0.226343	+	0.974048i	$0.572677\pi$
$14$	−0.352533	−0.0942183
$15$	0	0
$16$	1.27821	0.319552
$17$	1.92844	0.467717	0.233858	−	0.972271i	$-0.424865\pi$
$17$	1.92844	0.467717	0.233858	+	0.972271i	$0.424865\pi$
$18$	0	0
$19$	−0.869173	−0.199402	−0.0997010	−	0.995017i	$-0.531789\pi$
$19$	−0.869173	−0.199402	−0.0997010	+	0.995017i	$0.531789\pi$
$20$	0	0
$21$	0	0
$22$	1.26842	0.270428
$23$	4.86522	1.01447	0.507234	−	0.861808i	$-0.330668\pi$
$23$	4.86522	1.01447	0.507234	+	0.861808i	$0.330668\pi$
$24$	0	0
$25$	0	0
$26$	−1.14761	−0.225064
$27$	0	0
$28$	0.754910	0.142665
$29$	3.15000	0.584940	0.292470	−	0.956275i	$-0.405523\pi$
$29$	3.15000	0.584940	0.292470	+	0.956275i	$0.405523\pi$
$30$	0	0
$31$	2.39134	0.429498	0.214749	−	0.976669i	$-0.431107\pi$
$31$	2.39134	0.429498	0.214749	+	0.976669i	$0.431107\pi$
$32$	5.82842	1.03033
$33$	0	0
$34$	1.35591	0.232537
$35$	0	0
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	−0.611125	−0.0991375
$39$	0	0
$40$	0	0
$41$	−8.24040	−1.28693	−0.643467	−	0.765474i	$-0.722505\pi$
$41$	−8.24040	−1.28693	−0.643467	+	0.765474i	$0.722505\pi$
$42$	0	0
$43$	−11.6931	−1.78318	−0.891590	−	0.452844i	$-0.850410\pi$
$43$	−11.6931	−1.78318	−0.891590	+	0.452844i	$0.850410\pi$
$44$	−2.71619	−0.409481
$45$	0	0
$46$	3.42079	0.504368
$47$	−1.43311	−0.209041	−0.104521	−	0.994523i	$-0.533331\pi$
$47$	−1.43311	−0.209041	−0.104521	+	0.994523i	$0.533331\pi$
$48$	0	0
$49$	−6.74861	−0.964087
$50$	0	0
$51$	0	0
$52$	2.45747	0.340790
$53$	10.7692	1.47927	0.739633	−	0.673010i	$-0.234999\pi$
$53$	10.7692	1.47927	0.739633	+	0.673010i	$0.234999\pi$
$54$	0	0
$55$	0	0
$56$	1.23585	0.165148
$57$	0	0
$58$	2.21480	0.290817
$59$	−8.71701	−1.13486	−0.567429	−	0.823422i	$-0.692062\pi$
$59$	−8.71701	−1.13486	−0.567429	+	0.823422i	$0.692062\pi$
$60$	0	0
$61$	−7.84555	−1.00452	−0.502260	−	0.864717i	$-0.667498\pi$
$61$	−7.84555	−1.00452	−0.502260	+	0.864717i	$0.667498\pi$
$62$	1.68138	0.213535
$63$	0	0
$64$	1.54161	0.192701
$65$	0	0
$66$	0	0
$67$	−11.8599	−1.44891	−0.724456	−	0.689321i	$-0.757909\pi$
$67$	−11.8599	−1.44891	−0.724456	+	0.689321i	$0.757909\pi$
$68$	−2.90353	−0.352105
$69$	0	0
$70$	0	0
$71$	6.06069	0.719272	0.359636	−	0.933093i	$-0.382901\pi$
$71$	6.06069	0.719272	0.359636	+	0.933093i	$0.382901\pi$
$72$	0	0
$73$	5.56086	0.650849	0.325424	−	0.945568i	$-0.394493\pi$
$73$	5.56086	0.650849	0.325424	+	0.945568i	$0.394493\pi$
$74$	0.703111	0.0817350
$75$	0	0
$76$	1.30866	0.150113
$77$	−0.904514	−0.103079
$78$	0	0
$79$	7.44924	0.838105	0.419052	−	0.907962i	$-0.362362\pi$
$79$	7.44924	0.838105	0.419052	+	0.907962i	$0.362362\pi$
$80$	0	0
$81$	0	0
$82$	−5.79391	−0.639830
$83$	−4.70795	−0.516765	−0.258382	−	0.966043i	$-0.583189\pi$
$83$	−4.70795	−0.516765	−0.258382	+	0.966043i	$0.583189\pi$
$84$	0	0
$85$	0	0
$86$	−8.22154	−0.886551
$87$	0	0
$88$	−4.44662	−0.474012
$89$	−5.16332	−0.547311	−0.273656	−	0.961828i	$-0.588233\pi$
$89$	−5.16332	−0.547311	−0.273656	+	0.961828i	$0.588233\pi$
$90$	0	0
$91$	0.818361	0.0857875
$92$	−7.32524	−0.763709
$93$	0	0
$94$	−1.00764	−0.103930
$95$	0	0
$96$	0	0
$97$	6.48639	0.658593	0.329297	−	0.944227i	$-0.393188\pi$
$97$	6.48639	0.658593	0.329297	+	0.944227i	$0.393188\pi$
$98$	−4.74502	−0.479319
$99$	0	0
$100$	0	0
$101$	−2.50611	−0.249367	−0.124684	−	0.992197i	$-0.539792\pi$
$101$	−2.50611	−0.249367	−0.124684	+	0.992197i	$0.539792\pi$
$102$	0	0
$103$	6.40079	0.630688	0.315344	−	0.948977i	$-0.397880\pi$
$103$	6.40079	0.630688	0.315344	+	0.948977i	$0.397880\pi$
$104$	4.02309	0.394496
$105$	0	0
$106$	7.57195	0.735453
$107$	11.6711	1.12829	0.564146	−	0.825675i	$-0.309206\pi$
$107$	11.6711	1.12829	0.564146	+	0.825675i	$0.309206\pi$
$108$	0	0
$109$	7.93573	0.760105	0.380053	−	0.924965i	$-0.375906\pi$
$109$	7.93573	0.760105	0.380053	+	0.924965i	$0.375906\pi$
$110$	0	0
$111$	0	0
$112$	−0.640881	−0.0605575
$113$	16.2261	1.52643	0.763213	−	0.646146i	$-0.223620\pi$
$113$	16.2261	1.52643	0.763213	+	0.646146i	$0.223620\pi$
$114$	0	0
$115$	0	0
$116$	−4.74275	−0.440353
$117$	0	0
$118$	−6.12902	−0.564222
$119$	−0.966903	−0.0886358
$120$	0	0
$121$	−7.74553	−0.704139
$122$	−5.51629	−0.499421
$123$	0	0
$124$	−3.60049	−0.323333
$125$	0	0
$126$	0	0
$127$	9.56408	0.848675	0.424337	−	0.905504i	$-0.360507\pi$
$127$	9.56408	0.848675	0.424337	+	0.905504i	$0.360507\pi$
$128$	−10.5729	−0.934523
$129$	0	0
$130$	0	0
$131$	20.2974	1.77340	0.886698	−	0.462350i	$-0.152994\pi$
$131$	20.2974	1.77340	0.886698	+	0.462350i	$0.152994\pi$
$132$	0	0
$133$	0.435795	0.0377882
$134$	−8.33879	−0.720362
$135$	0	0
$136$	−4.75333	−0.407594
$137$	−11.0879	−0.947300	−0.473650	−	0.880713i	$-0.657064\pi$
$137$	−11.0879	−0.947300	−0.473650	+	0.880713i	$0.657064\pi$
$138$	0	0
$139$	14.1716	1.20202	0.601009	−	0.799242i	$-0.294765\pi$
$139$	14.1716	1.20202	0.601009	+	0.799242i	$0.294765\pi$
$140$	0	0
$141$	0	0
$142$	4.26134	0.357603
$143$	−2.94448	−0.246230
$144$	0	0
$145$	0	0
$146$	3.90990	0.323585
$147$	0	0
$148$	−1.50564	−0.123762
$149$	17.5785	1.44008	0.720042	−	0.693930i	$-0.244122\pi$
$149$	17.5785	1.44008	0.720042	+	0.693930i	$0.244122\pi$
$150$	0	0
$151$	10.4870	0.853422	0.426711	−	0.904388i	$-0.359672\pi$
$151$	10.4870	0.853422	0.426711	+	0.904388i	$0.359672\pi$
$152$	2.14238	0.173770
$153$	0	0
$154$	−0.635974	−0.0512482
$155$	0	0
$156$	0	0
$157$	14.1135	1.12638	0.563189	−	0.826328i	$-0.309574\pi$
$157$	14.1135	1.12638	0.563189	+	0.826328i	$0.309574\pi$
$158$	5.23764	0.416684
$159$	0	0
$160$	0	0
$161$	−2.43937	−0.192249
$162$	0	0
$163$	−0.261155	−0.0204553	−0.0102276	−	0.999948i	$-0.503256\pi$
$163$	−0.261155	−0.0204553	−0.0102276	+	0.999948i	$0.503256\pi$
$164$	12.4070	0.968826
$165$	0	0
$166$	−3.31021	−0.256922
$167$	11.7929	0.912560	0.456280	−	0.889836i	$-0.349182\pi$
$167$	11.7929	0.912560	0.456280	+	0.889836i	$0.349182\pi$
$168$	0	0
$169$	−10.3360	−0.795075
$170$	0	0
$171$	0	0
$172$	17.6055	1.34241
$173$	13.5257	1.02834	0.514170	−	0.857688i	$-0.328100\pi$
$173$	13.5257	1.02834	0.514170	+	0.857688i	$0.328100\pi$
$174$	0	0
$175$	0	0
$176$	2.30591	0.173814
$177$	0	0
$178$	−3.63039	−0.272109
$179$	−0.443280	−0.0331323	−0.0165661	−	0.999863i	$-0.505273\pi$
$179$	−0.443280	−0.0331323	−0.0165661	+	0.999863i	$0.505273\pi$
$180$	0	0
$181$	11.4380	0.850182	0.425091	−	0.905151i	$-0.360242\pi$
$181$	11.4380	0.850182	0.425091	+	0.905151i	$0.360242\pi$
$182$	0.575398	0.0426513
$183$	0	0
$184$	−11.9920	−0.884064
$185$	0	0
$186$	0	0
$187$	3.47894	0.254405
$188$	2.15775	0.157370
$189$	0	0
$190$	0	0
$191$	−10.1652	−0.735531	−0.367765	−	0.929919i	$-0.619877\pi$
$191$	−10.1652	−0.735531	−0.367765	+	0.929919i	$0.619877\pi$
$192$	0	0
$193$	8.94479	0.643860	0.321930	−	0.946763i	$-0.395668\pi$
$193$	8.94479	0.643860	0.321930	+	0.946763i	$0.395668\pi$
$194$	4.56065	0.327436
$195$	0	0
$196$	10.1609	0.725782
$197$	−8.65157	−0.616399	−0.308199	−	0.951322i	$-0.599726\pi$
$197$	−8.65157	−0.616399	−0.308199	+	0.951322i	$0.599726\pi$
$198$	0	0
$199$	10.2145	0.724084	0.362042	−	0.932162i	$-0.382080\pi$
$199$	10.2145	0.724084	0.362042	+	0.932162i	$0.382080\pi$
$200$	0	0
$201$	0	0
$202$	−1.76207	−0.123979
$203$	−1.57938	−0.110851
$204$	0	0
$205$	0	0
$206$	4.50046	0.313562
$207$	0	0
$208$	−2.08627	−0.144657
$209$	−1.56800	−0.108461
$210$	0	0
$211$	13.6414	0.939113	0.469557	−	0.882902i	$-0.344414\pi$
$211$	13.6414	0.939113	0.469557	+	0.882902i	$0.344414\pi$
$212$	−16.2145	−1.11362
$213$	0	0
$214$	8.20611	0.560958
$215$	0	0
$216$	0	0
$217$	−1.19899	−0.0813930
$218$	5.57970	0.377905
$219$	0	0
$220$	0	0
$221$	−3.14758	−0.211729
$222$	0	0
$223$	6.66288	0.446180	0.223090	−	0.974798i	$-0.428386\pi$
$223$	6.66288	0.446180	0.223090	+	0.974798i	$0.428386\pi$
$224$	−2.92231	−0.195255
$225$	0	0
$226$	11.4088	0.758900
$227$	5.71859	0.379556	0.189778	−	0.981827i	$-0.439223\pi$
$227$	5.71859	0.379556	0.189778	+	0.981827i	$0.439223\pi$
$228$	0	0
$229$	29.9549	1.97948	0.989738	−	0.142895i	$-0.0456412\pi$
$229$	29.9549	1.97948	0.989738	+	0.142895i	$0.0456412\pi$
$230$	0	0
$231$	0	0
$232$	−7.76427	−0.509750
$233$	−25.4277	−1.66583	−0.832913	−	0.553404i	$-0.813329\pi$
$233$	−25.4277	−1.66583	−0.832913	+	0.553404i	$0.813329\pi$
$234$	0	0
$235$	0	0
$236$	13.1246	0.854341
$237$	0	0
$238$	−0.679840	−0.0440675
$239$	10.9704	0.709615	0.354808	−	0.934939i	$-0.384546\pi$
$239$	10.9704	0.709615	0.354808	+	0.934939i	$0.384546\pi$
$240$	0	0
$241$	23.8516	1.53642	0.768208	−	0.640200i	$-0.221149\pi$
$241$	23.8516	1.53642	0.768208	+	0.640200i	$0.221149\pi$
$242$	−5.44597	−0.350080
$243$	0	0
$244$	11.8125	0.756220
$245$	0	0
$246$	0	0
$247$	1.41865	0.0902666
$248$	−5.89430	−0.374288
$249$	0	0
$250$	0	0
$251$	−18.9886	−1.19855	−0.599275	−	0.800543i	$-0.704545\pi$
$251$	−18.9886	−1.19855	−0.599275	+	0.800543i	$0.704545\pi$
$252$	0	0
$253$	8.77692	0.551800
$254$	6.72461	0.421939
$255$	0	0
$256$	−10.5172	−0.657322
$257$	5.12928	0.319956	0.159978	−	0.987121i	$-0.448858\pi$
$257$	5.12928	0.319956	0.159978	+	0.987121i	$0.448858\pi$
$258$	0	0
$259$	−0.501390	−0.0311549
$260$	0	0
$261$	0	0
$262$	14.2713	0.881687
$263$	27.1591	1.67470	0.837351	−	0.546666i	$-0.184103\pi$
$263$	27.1591	1.67470	0.837351	+	0.546666i	$0.184103\pi$
$264$	0	0
$265$	0	0
$266$	0.306412	0.0187873
$267$	0	0
$268$	17.8566	1.09077
$269$	−19.0743	−1.16298	−0.581492	−	0.813552i	$-0.697531\pi$
$269$	−19.0743	−1.16298	−0.581492	+	0.813552i	$0.697531\pi$
$270$	0	0
$271$	−18.0630	−1.09725	−0.548626	−	0.836068i	$-0.684849\pi$
$271$	−18.0630	−1.09725	−0.548626	+	0.836068i	$0.684849\pi$
$272$	2.46495	0.149460
$273$	0	0
$274$	−7.79600	−0.470973
$275$	0	0
$276$	0	0
$277$	−22.9367	−1.37813	−0.689067	−	0.724698i	$-0.741979\pi$
$277$	−22.9367	−1.37813	−0.689067	+	0.724698i	$0.741979\pi$
$278$	9.96420	0.597613
$279$	0	0
$280$	0	0
$281$	−14.1793	−0.845869	−0.422934	−	0.906160i	$-0.639000\pi$
$281$	−14.1793	−0.845869	−0.422934	+	0.906160i	$0.639000\pi$
$282$	0	0
$283$	8.55472	0.508525	0.254263	−	0.967135i	$-0.418167\pi$
$283$	8.55472	0.508525	0.254263	+	0.967135i	$0.418167\pi$
$284$	−9.12519	−0.541480
$285$	0	0
$286$	−2.07030	−0.122419
$287$	4.13165	0.243884
$288$	0	0
$289$	−13.2811	−0.781241
$290$	0	0
$291$	0	0
$292$	−8.37262	−0.489971
$293$	26.2220	1.53191	0.765953	−	0.642896i	$-0.222267\pi$
$293$	26.2220	1.53191	0.765953	+	0.642896i	$0.222267\pi$
$294$	0	0
$295$	0	0
$296$	−2.46485	−0.143266
$297$	0	0
$298$	12.3596	0.715973
$299$	−7.94093	−0.459236
$300$	0	0
$301$	5.86280	0.337926
$302$	7.37354	0.424300
$303$	0	0
$304$	−1.11098	−0.0637193
$305$	0	0
$306$	0	0
$307$	−20.8170	−1.18809	−0.594046	−	0.804431i	$-0.702470\pi$
$307$	−20.8170	−1.18809	−0.594046	+	0.804431i	$0.702470\pi$
$308$	1.36187	0.0775997
$309$	0	0
$310$	0	0
$311$	−15.9789	−0.906078	−0.453039	−	0.891491i	$-0.649660\pi$
$311$	−15.9789	−0.906078	−0.453039	+	0.891491i	$0.649660\pi$
$312$	0	0
$313$	4.27429	0.241597	0.120799	−	0.992677i	$-0.461454\pi$
$313$	4.27429	0.241597	0.120799	+	0.992677i	$0.461454\pi$
$314$	9.92333	0.560006
$315$	0	0
$316$	−11.2158	−0.630940
$317$	−6.19424	−0.347903	−0.173952	−	0.984754i	$-0.555654\pi$
$317$	−6.19424	−0.347903	−0.173952	+	0.984754i	$0.555654\pi$
$318$	0	0
$319$	5.68264	0.318167
$320$	0	0
$321$	0	0
$322$	−1.71515	−0.0955815
$323$	−1.67615	−0.0932636
$324$	0	0
$325$	0	0
$326$	−0.183621	−0.0101698
$327$	0	0
$328$	20.3113	1.12151
$329$	0.718549	0.0396149
$330$	0	0
$331$	−32.0400	−1.76108	−0.880540	−	0.473972i	$-0.842820\pi$
$331$	−32.0400	−1.76108	−0.880540	+	0.473972i	$0.842820\pi$
$332$	7.08846	0.389030
$333$	0	0
$334$	8.29169	0.453701
$335$	0	0
$336$	0	0
$337$	−12.5183	−0.681917	−0.340959	−	0.940078i	$-0.610752\pi$
$337$	−12.5183	−0.681917	−0.340959	+	0.940078i	$0.610752\pi$
$338$	−7.26733	−0.395291
$339$	0	0
$340$	0	0
$341$	4.31401	0.233617
$342$	0	0
$343$	6.89341	0.372209
$344$	28.8217	1.55396
$345$	0	0
$346$	9.51007	0.511264
$347$	3.30472	0.177407	0.0887033	−	0.996058i	$-0.471728\pi$
$347$	3.30472	0.177407	0.0887033	+	0.996058i	$0.471728\pi$
$348$	0	0
$349$	−4.45130	−0.238273	−0.119136	−	0.992878i	$-0.538013\pi$
$349$	−4.45130	−0.238273	−0.119136	+	0.992878i	$0.538013\pi$
$350$	0	0
$351$	0	0
$352$	10.5146	0.560428
$353$	31.9145	1.69864	0.849319	−	0.527879i	$-0.177013\pi$
$353$	31.9145	1.69864	0.849319	+	0.527879i	$0.177013\pi$
$354$	0	0
$355$	0	0
$356$	7.77408	0.412026
$357$	0	0
$358$	−0.311675	−0.0164725
$359$	−31.8986	−1.68354	−0.841771	−	0.539835i	$-0.818487\pi$
$359$	−31.8986	−1.68354	−0.841771	+	0.539835i	$0.818487\pi$
$360$	0	0
$361$	−18.2445	−0.960239
$362$	8.04220	0.422689
$363$	0	0
$364$	−1.23215	−0.0645823
$365$	0	0
$366$	0	0
$367$	22.7089	1.18540	0.592698	−	0.805425i	$-0.298063\pi$
$367$	22.7089	1.18540	0.592698	+	0.805425i	$0.298063\pi$
$368$	6.21876	0.324175
$369$	0	0
$370$	0	0
$371$	−5.39958	−0.280332
$372$	0	0
$373$	28.7105	1.48657	0.743287	−	0.668973i	$-0.233266\pi$
$373$	28.7105	1.48657	0.743287	+	0.668973i	$0.233266\pi$
$374$	2.44608	0.126484
$375$	0	0
$376$	3.53241	0.182170
$377$	−5.14138	−0.264794
$378$	0	0
$379$	30.3515	1.55905	0.779526	−	0.626370i	$-0.215460\pi$
$379$	30.3515	1.55905	0.779526	+	0.626370i	$0.215460\pi$
$380$	0	0
$381$	0	0
$382$	−7.14729	−0.365687
$383$	35.4366	1.81073	0.905363	−	0.424639i	$-0.139599\pi$
$383$	35.4366	1.81073	0.905363	+	0.424639i	$0.139599\pi$
$384$	0	0
$385$	0	0
$386$	6.28918	0.320111
$387$	0	0
$388$	−9.76614	−0.495801
$389$	17.2794	0.876102	0.438051	−	0.898950i	$-0.355669\pi$
$389$	17.2794	0.876102	0.438051	+	0.898950i	$0.355669\pi$
$390$	0	0
$391$	9.38230	0.474483
$392$	16.6343	0.840159
$393$	0	0
$394$	−6.08301	−0.306458
$395$	0	0
$396$	0	0
$397$	4.68683	0.235225	0.117613	−	0.993060i	$-0.462476\pi$
$397$	4.68683	0.235225	0.117613	+	0.993060i	$0.462476\pi$
$398$	7.18190	0.359996
$399$	0	0
$400$	0	0
$401$	22.3427	1.11574	0.557871	−	0.829928i	$-0.311618\pi$
$401$	22.3427	1.11574	0.557871	+	0.829928i	$0.311618\pi$
$402$	0	0
$403$	−3.90311	−0.194428
$404$	3.77329	0.187728
$405$	0	0
$406$	−1.11048	−0.0551121
$407$	1.80401	0.0894216
$408$	0	0
$409$	−5.17691	−0.255982	−0.127991	−	0.991775i	$-0.540853\pi$
$409$	−5.17691	−0.255982	−0.127991	+	0.991775i	$0.540853\pi$
$410$	0	0
$411$	0	0
$412$	−9.63725	−0.474793
$413$	4.37062	0.215064
$414$	0	0
$415$	0	0
$416$	−9.51306	−0.466416
$417$	0	0
$418$	−1.10248	−0.0539239
$419$	15.8651	0.775063	0.387531	−	0.921857i	$-0.373328\pi$
$419$	15.8651	0.775063	0.387531	+	0.921857i	$0.373328\pi$
$420$	0	0
$421$	6.46992	0.315325	0.157662	−	0.987493i	$-0.449604\pi$
$421$	6.46992	0.315325	0.157662	+	0.987493i	$0.449604\pi$
$422$	9.59142	0.466903
$423$	0	0
$424$	−26.5445	−1.28912
$425$	0	0
$426$	0	0
$427$	3.93368	0.190364
$428$	−17.5725	−0.849398
$429$	0	0
$430$	0	0
$431$	24.9727	1.20289	0.601445	−	0.798914i	$-0.294592\pi$
$431$	24.9727	1.20289	0.601445	+	0.798914i	$0.294592\pi$
$432$	0	0
$433$	38.7308	1.86128	0.930641	−	0.365934i	$-0.119250\pi$
$433$	38.7308	1.86128	0.930641	+	0.365934i	$0.119250\pi$
$434$	−0.843026	−0.0404665
$435$	0	0
$436$	−11.9483	−0.572220
$437$	−4.22872	−0.202287
$438$	0	0
$439$	−32.8425	−1.56749	−0.783743	−	0.621085i	$-0.786692\pi$
$439$	−32.8425	−1.56749	−0.783743	+	0.621085i	$0.786692\pi$
$440$	0	0
$441$	0	0
$442$	−2.21310	−0.105266
$443$	24.4262	1.16052	0.580262	−	0.814430i	$-0.302950\pi$
$443$	24.4262	1.16052	0.580262	+	0.814430i	$0.302950\pi$
$444$	0	0
$445$	0	0
$446$	4.68475	0.221829
$447$	0	0
$448$	−0.772947	−0.0365183
$449$	23.8188	1.12408	0.562040	−	0.827110i	$-0.310017\pi$
$449$	23.8188	1.12408	0.562040	+	0.827110i	$0.310017\pi$
$450$	0	0
$451$	−14.8658	−0.700003
$452$	−24.4307	−1.14912
$453$	0	0
$454$	4.02080	0.188706
$455$	0	0
$456$	0	0
$457$	−35.0439	−1.63928	−0.819642	−	0.572876i	$-0.805828\pi$
$457$	−35.0439	−1.63928	−0.819642	+	0.572876i	$0.805828\pi$
$458$	21.0616	0.984145
$459$	0	0
$460$	0	0
$461$	−7.94572	−0.370069	−0.185035	−	0.982732i	$-0.559240\pi$
$461$	−7.94572	−0.370069	−0.185035	+	0.982732i	$0.559240\pi$
$462$	0	0
$463$	−23.9463	−1.11288	−0.556440	−	0.830888i	$-0.687833\pi$
$463$	−23.9463	−1.11288	−0.556440	+	0.830888i	$0.687833\pi$
$464$	4.02635	0.186919
$465$	0	0
$466$	−17.8785	−0.828206
$467$	42.2147	1.95346	0.976732	−	0.214464i	$-0.0688005\pi$
$467$	42.2147	1.95346	0.976732	+	0.214464i	$0.0688005\pi$
$468$	0	0
$469$	5.94641	0.274580
$470$	0	0
$471$	0	0
$472$	21.4861	0.988979
$473$	−21.0945	−0.969926
$474$	0	0
$475$	0	0
$476$	1.45580	0.0667266
$477$	0	0
$478$	7.71340	0.352803
$479$	−34.3187	−1.56806	−0.784031	−	0.620721i	$-0.786840\pi$
$479$	−34.3187	−1.56806	−0.784031	+	0.620721i	$0.786840\pi$
$480$	0	0
$481$	−1.63218	−0.0744212
$482$	16.7703	0.763867
$483$	0	0
$484$	11.6619	0.530089
$485$	0	0
$486$	0	0
$487$	3.03995	0.137753	0.0688766	−	0.997625i	$-0.478059\pi$
$487$	3.03995	0.137753	0.0688766	+	0.997625i	$0.478059\pi$
$488$	19.3381	0.875394
$489$	0	0
$490$	0	0
$491$	11.4565	0.517023	0.258511	−	0.966008i	$-0.416768\pi$
$491$	11.4565	0.517023	0.258511	+	0.966008i	$0.416768\pi$
$492$	0	0
$493$	6.07460	0.273586
$494$	0.997468	0.0448782
$495$	0	0
$496$	3.05663	0.137247
$497$	−3.03877	−0.136307
$498$	0	0
$499$	−39.9114	−1.78668	−0.893340	−	0.449381i	$-0.851645\pi$
$499$	−39.9114	−1.78668	−0.893340	+	0.449381i	$0.851645\pi$
$500$	0	0
$501$	0	0
$502$	−13.3511	−0.595889
$503$	−14.8201	−0.660794	−0.330397	−	0.943842i	$-0.607183\pi$
$503$	−14.8201	−0.660794	−0.330397	+	0.943842i	$0.607183\pi$
$504$	0	0
$505$	0	0
$506$	6.17115	0.274341
$507$	0	0
$508$	−14.4000	−0.638897
$509$	22.3612	0.991145	0.495572	−	0.868567i	$-0.334958\pi$
$509$	22.3612	0.991145	0.495572	+	0.868567i	$0.334958\pi$
$510$	0	0
$511$	−2.78816	−0.123341
$512$	13.7511	0.607719
$513$	0	0
$514$	3.60645	0.159074
$515$	0	0
$516$	0	0
$517$	−2.58536	−0.113704
$518$	−0.352533	−0.0154894
$519$	0	0
$520$	0	0
$521$	2.48042	0.108669	0.0543346	−	0.998523i	$-0.482696\pi$
$521$	2.48042	0.108669	0.0543346	+	0.998523i	$0.482696\pi$
$522$	0	0
$523$	0.452507	0.0197867	0.00989336	−	0.999951i	$-0.496851\pi$
$523$	0.452507	0.0197867	0.00989336	+	0.999951i	$0.496851\pi$
$524$	−30.5605	−1.33504
$525$	0	0
$526$	19.0958	0.832618
$527$	4.61157	0.200883
$528$	0	0
$529$	0.670348	0.0291455
$530$	0	0
$531$	0	0
$532$	−0.656148	−0.0284476
$533$	13.4498	0.582577
$534$	0	0
$535$	0	0
$536$	29.2328	1.26266
$537$	0	0
$538$	−13.4114	−0.578205
$539$	−12.1746	−0.524396
$540$	0	0
$541$	−16.4102	−0.705531	−0.352765	−	0.935712i	$-0.614759\pi$
$541$	−16.4102	−0.705531	−0.352765	+	0.935712i	$0.614759\pi$
$542$	−12.7003	−0.545525
$543$	0	0
$544$	11.2398	0.481902
$545$	0	0
$546$	0	0
$547$	−8.18537	−0.349981	−0.174991	−	0.984570i	$-0.555990\pi$
$547$	−8.18537	−0.349981	−0.174991	+	0.984570i	$0.555990\pi$
$548$	16.6943	0.713144
$549$	0	0
$550$	0	0
$551$	−2.73789	−0.116638
$552$	0	0
$553$	−3.73497	−0.158827
$554$	−16.1271	−0.685173
$555$	0	0
$556$	−21.3372	−0.904901
$557$	25.8788	1.09652	0.548259	−	0.836309i	$-0.315291\pi$
$557$	25.8788	1.09652	0.548259	+	0.836309i	$0.315291\pi$
$558$	0	0
$559$	19.0853	0.807221
$560$	0	0
$561$	0	0
$562$	−9.96965	−0.420544
$563$	24.4404	1.03004	0.515021	−	0.857178i	$-0.327784\pi$
$563$	24.4404	1.03004	0.515021	+	0.857178i	$0.327784\pi$
$564$	0	0
$565$	0	0
$566$	6.01491	0.252826
$567$	0	0
$568$	−14.9387	−0.626814
$569$	−5.31751	−0.222922	−0.111461	−	0.993769i	$-0.535553\pi$
$569$	−5.31751	−0.222922	−0.111461	+	0.993769i	$0.535553\pi$
$570$	0	0
$571$	−13.2022	−0.552496	−0.276248	−	0.961086i	$-0.589091\pi$
$571$	−13.2022	−0.552496	−0.276248	+	0.961086i	$0.589091\pi$
$572$	4.43332	0.185366
$573$	0	0
$574$	2.90501	0.121253
$575$	0	0
$576$	0	0
$577$	41.4001	1.72351	0.861755	−	0.507325i	$-0.169365\pi$
$577$	41.4001	1.72351	0.861755	+	0.507325i	$0.169365\pi$
$578$	−9.33808	−0.388413
$579$	0	0
$580$	0	0
$581$	2.36052	0.0979308
$582$	0	0
$583$	19.4278	0.804618
$584$	−13.7067	−0.567186
$585$	0	0
$586$	18.4370	0.761625
$587$	11.4771	0.473712	0.236856	−	0.971545i	$-0.423883\pi$
$587$	11.4771	0.473712	0.236856	+	0.971545i	$0.423883\pi$
$588$	0	0
$589$	−2.07849	−0.0856426
$590$	0	0
$591$	0	0
$592$	1.27821	0.0525340
$593$	−11.1191	−0.456608	−0.228304	−	0.973590i	$-0.573318\pi$
$593$	−11.1191	−0.456608	−0.228304	+	0.973590i	$0.573318\pi$
$594$	0	0
$595$	0	0
$596$	−26.4668	−1.08412
$597$	0	0
$598$	−5.58335	−0.228320
$599$	−7.25813	−0.296559	−0.148280	−	0.988945i	$-0.547374\pi$
$599$	−7.25813	−0.296559	−0.148280	+	0.988945i	$0.547374\pi$
$600$	0	0
$601$	−14.6121	−0.596040	−0.298020	−	0.954560i	$-0.596326\pi$
$601$	−14.6121	−0.596040	−0.298020	+	0.954560i	$0.596326\pi$
$602$	4.12220	0.168008
$603$	0	0
$604$	−15.7896	−0.642471
$605$	0	0
$606$	0	0
$607$	−26.3898	−1.07113	−0.535564	−	0.844495i	$-0.679901\pi$
$607$	−26.3898	−1.07113	−0.535564	+	0.844495i	$0.679901\pi$
$608$	−5.06591	−0.205450
$609$	0	0
$610$	0	0
$611$	2.33911	0.0946301
$612$	0	0
$613$	40.8554	1.65013	0.825067	−	0.565035i	$-0.191137\pi$
$613$	40.8554	1.65013	0.825067	+	0.565035i	$0.191137\pi$
$614$	−14.6367	−0.590689
$615$	0	0
$616$	2.22949	0.0898288
$617$	28.2802	1.13852	0.569260	−	0.822158i	$-0.307230\pi$
$617$	28.2802	1.13852	0.569260	+	0.822158i	$0.307230\pi$
$618$	0	0
$619$	−7.17028	−0.288198	−0.144099	−	0.989563i	$-0.546028\pi$
$619$	−7.17028	−0.288198	−0.144099	+	0.989563i	$0.546028\pi$
$620$	0	0
$621$	0	0
$622$	−11.2349	−0.450479
$623$	2.58884	0.103720
$624$	0	0
$625$	0	0
$626$	3.00530	0.120116
$627$	0	0
$628$	−21.2497	−0.847957
$629$	1.92844	0.0768921
$630$	0	0
$631$	−7.70444	−0.306709	−0.153354	−	0.988171i	$-0.549008\pi$
$631$	−7.70444	−0.306709	−0.153354	+	0.988171i	$0.549008\pi$
$632$	−18.3613	−0.730372
$633$	0	0
$634$	−4.35524	−0.172969
$635$	0	0
$636$	0	0
$637$	11.0150	0.436429
$638$	3.99553	0.158184
$639$	0	0
$640$	0	0
$641$	−24.3375	−0.961274	−0.480637	−	0.876920i	$-0.659595\pi$
$641$	−24.3375	−0.961274	−0.480637	+	0.876920i	$0.659595\pi$
$642$	0	0
$643$	−27.8512	−1.09834	−0.549171	−	0.835710i	$-0.685057\pi$
$643$	−27.8512	−1.09834	−0.549171	+	0.835710i	$0.685057\pi$
$644$	3.67280	0.144729
$645$	0	0
$646$	−1.17852	−0.0463683
$647$	24.6280	0.968228	0.484114	−	0.875005i	$-0.339142\pi$
$647$	24.6280	0.968228	0.484114	+	0.875005i	$0.339142\pi$
$648$	0	0
$649$	−15.7256	−0.617284
$650$	0	0
$651$	0	0
$652$	0.393205	0.0153991
$653$	−27.8960	−1.09165	−0.545827	−	0.837898i	$-0.683784\pi$
$653$	−27.8960	−1.09165	−0.545827	+	0.837898i	$0.683784\pi$
$654$	0	0
$655$	0	0
$656$	−10.5329	−0.411242
$657$	0	0
$658$	0.505219	0.0196955
$659$	18.3903	0.716384	0.358192	−	0.933648i	$-0.383393\pi$
$659$	18.3903	0.716384	0.358192	+	0.933648i	$0.383393\pi$
$660$	0	0
$661$	2.05323	0.0798614	0.0399307	−	0.999202i	$-0.487286\pi$
$661$	2.05323	0.0798614	0.0399307	+	0.999202i	$0.487286\pi$
$662$	−22.5277	−0.875564
$663$	0	0
$664$	11.6044	0.450338
$665$	0	0
$666$	0	0
$667$	15.3254	0.593403
$668$	−17.7558	−0.686991
$669$	0	0
$670$	0	0
$671$	−14.1535	−0.546389
$672$	0	0
$673$	−16.8287	−0.648698	−0.324349	−	0.945938i	$-0.605145\pi$
$673$	−16.8287	−0.648698	−0.324349	+	0.945938i	$0.605145\pi$
$674$	−8.80178	−0.339032
$675$	0	0
$676$	15.5622	0.598546
$677$	−20.8182	−0.800108	−0.400054	−	0.916491i	$-0.631009\pi$
$677$	−20.8182	−0.800108	−0.400054	+	0.916491i	$0.631009\pi$
$678$	0	0
$679$	−3.25221	−0.124808
$680$	0	0
$681$	0	0
$682$	3.03323	0.116148
$683$	6.40527	0.245091	0.122545	−	0.992463i	$-0.460894\pi$
$683$	6.40527	0.245091	0.122545	+	0.992463i	$0.460894\pi$
$684$	0	0
$685$	0	0
$686$	4.84683	0.185053
$687$	0	0
$688$	−14.9462	−0.569819
$689$	−17.5774	−0.669644
$690$	0	0
$691$	−0.406659	−0.0154700	−0.00773502	−	0.999970i	$-0.502462\pi$
$691$	−0.406659	−0.0154700	−0.00773502	+	0.999970i	$0.502462\pi$
$692$	−20.3648	−0.774153
$693$	0	0
$694$	2.32358	0.0882020
$695$	0	0
$696$	0	0
$697$	−15.8911	−0.601920
$698$	−3.12976	−0.118463
$699$	0	0
$700$	0	0
$701$	−2.39744	−0.0905501	−0.0452750	−	0.998975i	$-0.514416\pi$
$701$	−2.39744	−0.0905501	−0.0452750	+	0.998975i	$0.514416\pi$
$702$	0	0
$703$	−0.869173	−0.0327815
$704$	2.78108	0.104816
$705$	0	0
$706$	22.4394	0.844520
$707$	1.25654	0.0472570
$708$	0	0
$709$	22.0154	0.826804	0.413402	−	0.910549i	$-0.364340\pi$
$709$	22.0154	0.826804	0.413402	+	0.910549i	$0.364340\pi$
$710$	0	0
$711$	0	0
$712$	12.7268	0.476958
$713$	11.6344	0.435712
$714$	0	0
$715$	0	0
$716$	0.667418	0.0249426
$717$	0	0
$718$	−22.4282	−0.837014
$719$	9.29180	0.346526	0.173263	−	0.984876i	$-0.444569\pi$
$719$	9.29180	0.346526	0.173263	+	0.984876i	$0.444569\pi$
$720$	0	0
$721$	−3.20929	−0.119520
$722$	−12.8279	−0.477406
$723$	0	0
$724$	−17.2215	−0.640032
$725$	0	0
$726$	0	0
$727$	29.4352	1.09169	0.545846	−	0.837886i	$-0.316208\pi$
$727$	29.4352	1.09169	0.545846	+	0.837886i	$0.316208\pi$
$728$	−2.01714	−0.0747600
$729$	0	0
$730$	0	0
$731$	−22.5495	−0.834023
$732$	0	0
$733$	3.71935	0.137377	0.0686887	−	0.997638i	$-0.478118\pi$
$733$	3.71935	0.137377	0.0686887	+	0.997638i	$0.478118\pi$
$734$	15.9669	0.589349
$735$	0	0
$736$	28.3565	1.04524
$737$	−21.3953	−0.788107
$738$	0	0
$739$	20.2734	0.745768	0.372884	−	0.927878i	$-0.378369\pi$
$739$	20.2734	0.745768	0.372884	+	0.927878i	$0.378369\pi$
$740$	0	0
$741$	0	0
$742$	−3.79650	−0.139374
$743$	1.11337	0.0408457	0.0204229	−	0.999791i	$-0.493499\pi$
$743$	1.11337	0.0408457	0.0204229	+	0.999791i	$0.493499\pi$
$744$	0	0
$745$	0	0
$746$	20.1867	0.739086
$747$	0	0
$748$	−5.23802	−0.191521
$749$	−5.85179	−0.213820
$750$	0	0
$751$	−37.9213	−1.38377	−0.691885	−	0.722008i	$-0.743219\pi$
$751$	−37.9213	−1.38377	−0.691885	+	0.722008i	$0.743219\pi$
$752$	−1.83182	−0.0667995
$753$	0	0
$754$	−3.61496	−0.131649
$755$	0	0
$756$	0	0
$757$	41.8794	1.52213	0.761066	−	0.648674i	$-0.224676\pi$
$757$	41.8794	1.52213	0.761066	+	0.648674i	$0.224676\pi$
$758$	21.3405	0.775121
$759$	0	0
$760$	0	0
$761$	12.6827	0.459749	0.229875	−	0.973220i	$-0.426168\pi$
$761$	12.6827	0.459749	0.229875	+	0.973220i	$0.426168\pi$
$762$	0	0
$763$	−3.97889	−0.144046
$764$	15.3051	0.553721
$765$	0	0
$766$	24.9159	0.900247
$767$	14.2278	0.513735
$768$	0	0
$769$	34.0483	1.22781	0.613907	−	0.789378i	$-0.289597\pi$
$769$	34.0483	1.22781	0.613907	+	0.789378i	$0.289597\pi$
$770$	0	0
$771$	0	0
$772$	−13.4676	−0.484709
$773$	−20.6092	−0.741261	−0.370630	−	0.928780i	$-0.620858\pi$
$773$	−20.6092	−0.741261	−0.370630	+	0.928780i	$0.620858\pi$
$774$	0	0
$775$	0	0
$776$	−15.9880	−0.573935
$777$	0	0
$778$	12.1494	0.435576
$779$	7.16233	0.256617
$780$	0	0
$781$	10.9336	0.391234
$782$	6.59680	0.235901
$783$	0	0
$784$	−8.62613	−0.308076
$785$	0	0
$786$	0	0
$787$	49.1315	1.75135	0.875674	−	0.482903i	$-0.160417\pi$
$787$	49.1315	1.75135	0.875674	+	0.482903i	$0.160417\pi$
$788$	13.0261	0.464036
$789$	0	0
$790$	0	0
$791$	−8.13562	−0.289269
$792$	0	0
$793$	12.8054	0.454732
$794$	3.29536	0.116948
$795$	0	0
$796$	−15.3793	−0.545103
$797$	1.87847	0.0665389	0.0332694	−	0.999446i	$-0.489408\pi$
$797$	1.87847	0.0665389	0.0332694	+	0.999446i	$0.489408\pi$
$798$	0	0
$799$	−2.76368	−0.0977720
$800$	0	0
$801$	0	0
$802$	15.7094	0.554718
$803$	10.0319	0.354017
$804$	0	0
$805$	0	0
$806$	−2.74432	−0.0966645
$807$	0	0
$808$	6.17719	0.217313
$809$	15.4989	0.544912	0.272456	−	0.962168i	$-0.412164\pi$
$809$	15.4989	0.544912	0.272456	+	0.962168i	$0.412164\pi$
$810$	0	0
$811$	6.06459	0.212956	0.106478	−	0.994315i	$-0.466043\pi$
$811$	6.06459	0.212956	0.106478	+	0.994315i	$0.466043\pi$
$812$	2.37797	0.0834503
$813$	0	0
$814$	1.26842	0.0444582
$815$	0	0
$816$	0	0
$817$	10.1633	0.355570
$818$	−3.63994	−0.127267
$819$	0	0
$820$	0	0
$821$	3.79024	0.132280	0.0661402	−	0.997810i	$-0.478932\pi$
$821$	3.79024	0.132280	0.0661402	+	0.997810i	$0.478932\pi$
$822$	0	0
$823$	−2.64216	−0.0920999	−0.0460499	−	0.998939i	$-0.514663\pi$
$823$	−2.64216	−0.0920999	−0.0460499	+	0.998939i	$0.514663\pi$
$824$	−15.7770	−0.549617
$825$	0	0
$826$	3.07303	0.106924
$827$	21.8953	0.761376	0.380688	−	0.924704i	$-0.375687\pi$
$827$	21.8953	0.761376	0.380688	+	0.924704i	$0.375687\pi$
$828$	0	0
$829$	7.03307	0.244269	0.122134	−	0.992514i	$-0.461026\pi$
$829$	7.03307	0.244269	0.122134	+	0.992514i	$0.461026\pi$
$830$	0	0
$831$	0	0
$832$	−2.51619	−0.0872332
$833$	−13.0143	−0.450919
$834$	0	0
$835$	0	0
$836$	2.36084	0.0816512
$837$	0	0
$838$	11.1549	0.385341
$839$	49.4466	1.70709	0.853544	−	0.521021i	$-0.174449\pi$
$839$	49.4466	1.70709	0.853544	+	0.521021i	$0.174449\pi$
$840$	0	0
$841$	−19.0775	−0.657845
$842$	4.54907	0.156771
$843$	0	0
$844$	−20.5390	−0.706981
$845$	0	0
$846$	0	0
$847$	3.88353	0.133440
$848$	13.7653	0.472703
$849$	0	0
$850$	0	0
$851$	4.86522	0.166778
$852$	0	0
$853$	−2.94665	−0.100891	−0.0504457	−	0.998727i	$-0.516064\pi$
$853$	−2.94665	−0.100891	−0.0504457	+	0.998727i	$0.516064\pi$
$854$	2.76581	0.0946441
$855$	0	0
$856$	−28.7676	−0.983257
$857$	−27.7795	−0.948929	−0.474465	−	0.880275i	$-0.657358\pi$
$857$	−27.7795	−0.948929	−0.474465	+	0.880275i	$0.657358\pi$
$858$	0	0
$859$	−50.2134	−1.71326	−0.856629	−	0.515933i	$-0.827445\pi$
$859$	−50.2134	−1.71326	−0.856629	+	0.515933i	$0.827445\pi$
$860$	0	0
$861$	0	0
$862$	17.5585	0.598046
$863$	−1.11827	−0.0380663	−0.0190331	−	0.999819i	$-0.506059\pi$
$863$	−1.11827	−0.0380663	−0.0190331	+	0.999819i	$0.506059\pi$
$864$	0	0
$865$	0	0
$866$	27.2320	0.925382
$867$	0	0
$868$	1.80525	0.0612741
$869$	13.4385	0.455871
$870$	0	0
$871$	19.3575	0.655903
$872$	−19.5604	−0.662398
$873$	0	0
$874$	−2.97326	−0.100572
$875$	0	0
$876$	0	0
$877$	−23.0843	−0.779502	−0.389751	−	0.920920i	$-0.627439\pi$
$877$	−23.0843	−0.779502	−0.389751	+	0.920920i	$0.627439\pi$
$878$	−23.0919	−0.779314
$879$	0	0
$880$	0	0
$881$	10.1304	0.341301	0.170650	−	0.985332i	$-0.445413\pi$
$881$	10.1304	0.341301	0.170650	+	0.985332i	$0.445413\pi$
$882$	0	0
$883$	−0.620421	−0.0208788	−0.0104394	−	0.999946i	$-0.503323\pi$
$883$	−0.620421	−0.0208788	−0.0104394	+	0.999946i	$0.503323\pi$
$884$	4.73910	0.159393
$885$	0	0
$886$	17.1743	0.576983
$887$	0.762728	0.0256099	0.0128049	−	0.999918i	$-0.495924\pi$
$887$	0.762728	0.0256099	0.0128049	+	0.999918i	$0.495924\pi$
$888$	0	0
$889$	−4.79533	−0.160830
$890$	0	0
$891$	0	0
$892$	−10.0319	−0.335892
$893$	1.24562	0.0416832
$894$	0	0
$895$	0	0
$896$	5.30116	0.177099
$897$	0	0
$898$	16.7473	0.558864
$899$	7.53272	0.251230
$900$	0	0
$901$	20.7678	0.691877
$902$	−10.4523	−0.348023
$903$	0	0
$904$	−39.9950	−1.33021
$905$	0	0
$906$	0	0
$907$	−44.6062	−1.48112	−0.740562	−	0.671988i	$-0.765441\pi$
$907$	−44.6062	−1.48112	−0.740562	+	0.671988i	$0.765441\pi$
$908$	−8.61011	−0.285737
$909$	0	0
$910$	0	0
$911$	−21.5382	−0.713591	−0.356796	−	0.934182i	$-0.616131\pi$
$911$	−21.5382	−0.713591	−0.356796	+	0.934182i	$0.616131\pi$
$912$	0	0
$913$	−8.49321	−0.281084
$914$	−24.6397	−0.815010
$915$	0	0
$916$	−45.1012	−1.49018
$917$	−10.1769	−0.336072
$918$	0	0
$919$	−32.1570	−1.06076	−0.530380	−	0.847760i	$-0.677951\pi$
$919$	−32.1570	−1.06076	−0.530380	+	0.847760i	$0.677951\pi$
$920$	0	0
$921$	0	0
$922$	−5.58672	−0.183989
$923$	−9.89217	−0.325605
$924$	0	0
$925$	0	0
$926$	−16.8369	−0.553295
$927$	0	0
$928$	18.3595	0.602681
$929$	−53.7540	−1.76361	−0.881805	−	0.471614i	$-0.843672\pi$
$929$	−53.7540	−1.76361	−0.881805	+	0.471614i	$0.843672\pi$
$930$	0	0
$931$	5.86571	0.192241
$932$	38.2849	1.25406
$933$	0	0
$934$	29.6816	0.971212
$935$	0	0
$936$	0	0
$937$	32.7426	1.06965	0.534827	−	0.844961i	$-0.320377\pi$
$937$	32.7426	1.06965	0.534827	+	0.844961i	$0.320377\pi$
$938$	4.18098	0.136514
$939$	0	0
$940$	0	0
$941$	7.12766	0.232355	0.116178	−	0.993228i	$-0.462936\pi$
$941$	7.12766	0.232355	0.116178	+	0.993228i	$0.462936\pi$
$942$	0	0
$943$	−40.0913	−1.30555
$944$	−11.1422	−0.362646
$945$	0	0
$946$	−14.8318	−0.482222
$947$	−14.6389	−0.475702	−0.237851	−	0.971302i	$-0.576443\pi$
$947$	−14.6389	−0.475702	−0.237851	+	0.971302i	$0.576443\pi$
$948$	0	0
$949$	−9.07634	−0.294630
$950$	0	0
$951$	0	0
$952$	2.38327	0.0772422
$953$	−51.2467	−1.66004	−0.830021	−	0.557731i	$-0.811672\pi$
$953$	−51.2467	−1.66004	−0.830021	+	0.557731i	$0.811672\pi$
$954$	0	0
$955$	0	0
$956$	−16.5174	−0.534211
$957$	0	0
$958$	−24.1299	−0.779601
$959$	5.55934	0.179521
$960$	0	0
$961$	−25.2815	−0.815532
$962$	−1.14761	−0.0370003
$963$	0	0
$964$	−35.9118	−1.15664
$965$	0	0
$966$	0	0
$967$	12.5599	0.403899	0.201949	−	0.979396i	$-0.435272\pi$
$967$	12.5599	0.403899	0.201949	+	0.979396i	$0.435272\pi$
$968$	19.0916	0.613627
$969$	0	0
$970$	0	0
$971$	5.56513	0.178594	0.0892968	−	0.996005i	$-0.471538\pi$
$971$	5.56513	0.178594	0.0892968	+	0.996005i	$0.471538\pi$
$972$	0	0
$973$	−7.10549	−0.227792
$974$	2.13742	0.0684874
$975$	0	0
$976$	−10.0282	−0.320996
$977$	−25.6883	−0.821843	−0.410922	−	0.911671i	$-0.634793\pi$
$977$	−25.6883	−0.821843	−0.410922	+	0.911671i	$0.634793\pi$
$978$	0	0
$979$	−9.31471	−0.297699
$980$	0	0
$981$	0	0
$982$	8.05516	0.257050
$983$	−21.5725	−0.688057	−0.344028	−	0.938959i	$-0.611792\pi$
$983$	−21.5725	−0.688057	−0.344028	+	0.938959i	$0.611792\pi$
$984$	0	0
$985$	0	0
$986$	4.27112	0.136020
$987$	0	0
$988$	−2.13597	−0.0679543
$989$	−56.8894	−1.80898
$990$	0	0
$991$	9.15527	0.290827	0.145413	−	0.989371i	$-0.453549\pi$
$991$	9.15527	0.290827	0.145413	+	0.989371i	$0.453549\pi$
$992$	13.9377	0.442524
$993$	0	0
$994$	−2.13659	−0.0677686
$995$	0	0
$996$	0	0
$997$	−21.1522	−0.669896	−0.334948	−	0.942237i	$-0.608719\pi$
$997$	−21.1522	−0.669896	−0.334948	+	0.942237i	$0.608719\pi$
$998$	−28.0621	−0.888292
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8325.2.a.cr.1.5		9
3.2	odd	2		925.2.a.l.1.5		9
5.2	odd	4		1665.2.c.e.334.11		18
5.3	odd	4		1665.2.c.e.334.8		18
5.4	even	2		8325.2.a.cq.1.5		9
15.2	even	4		185.2.b.a.149.8	✓	18
15.8	even	4		185.2.b.a.149.11	yes	18
15.14	odd	2		925.2.a.m.1.5		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
185.2.b.a.149.8	✓	18	15.2	even	4
185.2.b.a.149.11	yes	18	15.8	even	4
925.2.a.l.1.5		9	3.2	odd	2
925.2.a.m.1.5		9	15.14	odd	2
1665.2.c.e.334.8		18	5.3	odd	4
1665.2.c.e.334.11		18	5.2	odd	4
8325.2.a.cq.1.5		9	5.4	even	2
8325.2.a.cr.1.5		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)

\(f(q)\)	\(=\)	\(q+0.703111 q^{2} -1.50564 q^{4} -0.501390 q^{7} -2.46485 q^{8} +1.80401 q^{11} -1.63218 q^{13} -0.352533 q^{14} +1.27821 q^{16} +1.92844 q^{17} -0.869173 q^{19} +1.26842 q^{22} +4.86522 q^{23} -1.14761 q^{26} +0.754910 q^{28} +3.15000 q^{29} +2.39134 q^{31} +5.82842 q^{32} +1.35591 q^{34} +1.00000 q^{37} -0.611125 q^{38} -8.24040 q^{41} -11.6931 q^{43} -2.71619 q^{44} +3.42079 q^{46} -1.43311 q^{47} -6.74861 q^{49} +2.45747 q^{52} +10.7692 q^{53} +1.23585 q^{56} +2.21480 q^{58} -8.71701 q^{59} -7.84555 q^{61} +1.68138 q^{62} +1.54161 q^{64} -11.8599 q^{67} -2.90353 q^{68} +6.06069 q^{71} +5.56086 q^{73} +0.703111 q^{74} +1.30866 q^{76} -0.904514 q^{77} +7.44924 q^{79} -5.79391 q^{82} -4.70795 q^{83} -8.22154 q^{86} -4.44662 q^{88} -5.16332 q^{89} +0.818361 q^{91} -7.32524 q^{92} -1.00764 q^{94} +6.48639 q^{97} -4.74502 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 5 q^{2} + 11 q^{4} - 8 q^{7} + 15 q^{8} - 6 q^{13} + 4 q^{14} + 11 q^{16} + 18 q^{17} - 4 q^{19} - 6 q^{22} + 16 q^{23} + 6 q^{26} + 20 q^{28} + 2 q^{29} - 6 q^{31} + 35 q^{32} + 6 q^{34} + 9 q^{37}+ \cdots + 21 q^{98}+O(q^{100}) \) 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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