LMFDB - Embedded newform 8325.2.a.cf.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8325,2,Mod(1,8325)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$66.4754596827$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.457904.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5x^{3} + 8x^{2} + 5x - 6 $ x^5 - 2x^4 - 5x^3 + 8x^2 + 5x - 6
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1665)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.51432$ 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)

$f(q)$	$=$	$q+1.51432 q^{2} +0.293161 q^{4} +1.43000 q^{7} -2.58470 q^{8} -1.00554 q^{11} -3.60102 q^{13} +2.16548 q^{14} -4.50038 q^{16} +6.24140 q^{17} -0.485681 q^{19} -1.52271 q^{22} +0.828979 q^{23} -5.45309 q^{26} +0.419221 q^{28} -8.26057 q^{29} +8.91565 q^{31} -1.64561 q^{32} +9.45147 q^{34} -1.00000 q^{37} -0.735476 q^{38} +0.301089 q^{41} -2.65670 q^{43} -0.294787 q^{44} +1.25534 q^{46} +6.02864 q^{47} -4.95510 q^{49} -1.05568 q^{52} -9.83973 q^{53} -3.69612 q^{56} -12.5091 q^{58} -0.849222 q^{59} -7.73517 q^{61} +13.5011 q^{62} +6.50877 q^{64} +0.424456 q^{67} +1.82974 q^{68} -13.1515 q^{71} +2.76126 q^{73} -1.51432 q^{74} -0.142383 q^{76} -1.43793 q^{77} +1.35514 q^{79} +0.455945 q^{82} +3.64439 q^{83} -4.02309 q^{86} +2.59903 q^{88} -7.97497 q^{89} -5.14946 q^{91} +0.243025 q^{92} +9.12928 q^{94} -10.2603 q^{97} -7.50360 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 2 q^{2} + 4 q^{4} + 6 q^{8} - 12 q^{11} + 3 q^{13} - 10 q^{14} - 2 q^{16} + 4 q^{17} - 8 q^{19} - 10 q^{22} + 18 q^{23} - 2 q^{26} + 4 q^{28} - 13 q^{29} - 16 q^{31} + 14 q^{32} - 2 q^{34} - 5 q^{37}+ \cdots - 6 q^{98}+O(q^{100}) $ 5 * q + 2 * q^2 + 4 * q^4 + 6 * q^8 - 12 * q^11 + 3 * q^13 - 10 * q^14 - 2 * q^16 + 4 * q^17 - 8 * q^19 - 10 * q^22 + 18 * q^23 - 2 * q^26 + 4 * q^28 - 13 * q^29 - 16 * q^31 + 14 * q^32 - 2 * q^34 - 5 * q^37 + 10 * q^38 - 6 * q^41 - 5 * q^43 - 28 * q^44 - 6 * q^46 + 19 * q^47 - 3 * q^49 - 18 * q^52 + 5 * q^53 + 10 * q^56 - 38 * q^58 + q^59 - 2 * q^61 + 6 * q^62 + 20 * q^64 - 12 * q^67 - 18 * q^68 - 30 * q^71 - 12 * q^73 - 2 * q^74 + 2 * q^76 + 10 * q^77 - 12 * q^79 - 22 * q^82 + 19 * q^83 - 2 * q^86 - 20 * q^88 - 11 * q^89 - 30 * q^91 - 2 * q^92 + 34 * q^94 - 6 * 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 $

$2$	1.51432	1.07079	0.535393	−	0.844603i	$-0.320164\pi$
$2$	1.51432	1.07079	0.535393	+	0.844603i	$0.320164\pi$
$3$	0	0
$3$	0	0
$4$	0.293161	0.146581
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.43000	0.540489	0.270245	−	0.962792i	$-0.412895\pi$
$7$	1.43000	0.540489	0.270245	+	0.962792i	$0.412895\pi$
$8$	−2.58470	−0.913829
$9$	0	0
$10$	0	0
$11$	−1.00554	−0.303183	−0.151591	−	0.988443i	$-0.548440\pi$
$11$	−1.00554	−0.303183	−0.151591	+	0.988443i	$0.548440\pi$
$12$	0	0
$13$	−3.60102	−0.998744	−0.499372	−	0.866388i	$-0.666436\pi$
$13$	−3.60102	−0.998744	−0.499372	+	0.866388i	$0.666436\pi$
$14$	2.16548	0.578748
$15$	0	0
$16$	−4.50038	−1.12509
$17$	6.24140	1.51376	0.756881	−	0.653553i	$-0.226722\pi$
$17$	6.24140	1.51376	0.756881	+	0.653553i	$0.226722\pi$
$18$	0	0
$19$	−0.485681	−0.111423	−0.0557115	−	0.998447i	$-0.517743\pi$
$19$	−0.485681	−0.111423	−0.0557115	+	0.998447i	$0.517743\pi$
$20$	0	0
$21$	0	0
$22$	−1.52271	−0.324644
$23$	0.828979	0.172854	0.0864271	−	0.996258i	$-0.472455\pi$
$23$	0.828979	0.172854	0.0864271	+	0.996258i	$0.472455\pi$
$24$	0	0
$25$	0	0
$26$	−5.45309	−1.06944
$27$	0	0
$28$	0.419221	0.0792253
$29$	−8.26057	−1.53395	−0.766975	−	0.641677i	$-0.778239\pi$
$29$	−8.26057	−1.53395	−0.766975	+	0.641677i	$0.778239\pi$
$30$	0	0
$31$	8.91565	1.60130	0.800649	−	0.599134i	$-0.204488\pi$
$31$	8.91565	1.60130	0.800649	+	0.599134i	$0.204488\pi$
$32$	−1.64561	−0.290906
$33$	0	0
$34$	9.45147	1.62091
$35$	0	0
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	−0.735476	−0.119310
$39$	0	0
$40$	0	0
$41$	0.301089	0.0470223	0.0235111	−	0.999724i	$-0.492515\pi$
$41$	0.301089	0.0470223	0.0235111	+	0.999724i	$0.492515\pi$
$42$	0	0
$43$	−2.65670	−0.405143	−0.202572	−	0.979267i	$-0.564930\pi$
$43$	−2.65670	−0.405143	−0.202572	+	0.979267i	$0.564930\pi$
$44$	−0.294787	−0.0444408
$45$	0	0
$46$	1.25534	0.185090
$47$	6.02864	0.879367	0.439684	−	0.898153i	$-0.355091\pi$
$47$	6.02864	0.879367	0.439684	+	0.898153i	$0.355091\pi$
$48$	0	0
$49$	−4.95510	−0.707871
$50$	0	0
$51$	0	0
$52$	−1.05568	−0.146397
$53$	−9.83973	−1.35159	−0.675795	−	0.737089i	$-0.736200\pi$
$53$	−9.83973	−1.35159	−0.675795	+	0.737089i	$0.736200\pi$
$54$	0	0
$55$	0	0
$56$	−3.69612	−0.493915
$57$	0	0
$58$	−12.5091	−1.64253
$59$	−0.849222	−0.110559	−0.0552796	−	0.998471i	$-0.517605\pi$
$59$	−0.849222	−0.110559	−0.0552796	+	0.998471i	$0.517605\pi$
$60$	0	0
$61$	−7.73517	−0.990387	−0.495193	−	0.868783i	$-0.664903\pi$
$61$	−7.73517	−0.990387	−0.495193	+	0.868783i	$0.664903\pi$
$62$	13.5011	1.71465
$63$	0	0
$64$	6.50877	0.813597
$65$	0	0
$66$	0	0
$67$	0.424456	0.0518556	0.0259278	−	0.999664i	$-0.491746\pi$
$67$	0.424456	0.0518556	0.0259278	+	0.999664i	$0.491746\pi$
$68$	1.82974	0.221888
$69$	0	0
$70$	0	0
$71$	−13.1515	−1.56080	−0.780400	−	0.625280i	$-0.784985\pi$
$71$	−13.1515	−1.56080	−0.780400	+	0.625280i	$0.784985\pi$
$72$	0	0
$73$	2.76126	0.323181	0.161591	−	0.986858i	$-0.448338\pi$
$73$	2.76126	0.323181	0.161591	+	0.986858i	$0.448338\pi$
$74$	−1.51432	−0.176036
$75$	0	0
$76$	−0.142383	−0.0163325
$77$	−1.43793	−0.163867
$78$	0	0
$79$	1.35514	0.152466	0.0762329	−	0.997090i	$-0.475711\pi$
$79$	1.35514	0.152466	0.0762329	+	0.997090i	$0.475711\pi$
$80$	0	0
$81$	0	0
$82$	0.455945	0.0503508
$83$	3.64439	0.400024	0.200012	−	0.979793i	$-0.435902\pi$
$83$	3.64439	0.400024	0.200012	+	0.979793i	$0.435902\pi$
$84$	0	0
$85$	0	0
$86$	−4.02309	−0.433821
$87$	0	0
$88$	2.59903	0.277057
$89$	−7.97497	−0.845345	−0.422673	−	0.906282i	$-0.638908\pi$
$89$	−7.97497	−0.845345	−0.422673	+	0.906282i	$0.638908\pi$
$90$	0	0
$91$	−5.14946	−0.539810
$92$	0.243025	0.0253371
$93$	0	0
$94$	9.12928	0.941613
$95$	0	0
$96$	0	0
$97$	−10.2603	−1.04177	−0.520886	−	0.853626i	$-0.674398\pi$
$97$	−10.2603	−1.04177	−0.520886	+	0.853626i	$0.674398\pi$
$98$	−7.50360	−0.757978
$99$	0	0
$100$	0	0
$101$	9.04018	0.899531	0.449766	−	0.893147i	$-0.351508\pi$
$101$	9.04018	0.899531	0.449766	+	0.893147i	$0.351508\pi$
$102$	0	0
$103$	0.773667	0.0762317	0.0381158	−	0.999273i	$-0.487864\pi$
$103$	0.773667	0.0762317	0.0381158	+	0.999273i	$0.487864\pi$
$104$	9.30755	0.912680
$105$	0	0
$106$	−14.9005	−1.44726
$107$	8.63769	0.835037	0.417518	−	0.908669i	$-0.362900\pi$
$107$	8.63769	0.835037	0.417518	+	0.908669i	$0.362900\pi$
$108$	0	0
$109$	−17.5007	−1.67626	−0.838132	−	0.545467i	$-0.816352\pi$
$109$	−17.5007	−1.67626	−0.838132	+	0.545467i	$0.816352\pi$
$110$	0	0
$111$	0	0
$112$	−6.43554	−0.608102
$113$	−1.40458	−0.132132	−0.0660660	−	0.997815i	$-0.521045\pi$
$113$	−1.40458	−0.132132	−0.0660660	+	0.997815i	$0.521045\pi$
$114$	0	0
$115$	0	0
$116$	−2.42168	−0.224848
$117$	0	0
$118$	−1.28599	−0.118385
$119$	8.92520	0.818172
$120$	0	0
$121$	−9.98888	−0.908080
$122$	−11.7135	−1.06049
$123$	0	0
$124$	2.61373	0.234719
$125$	0	0
$126$	0	0
$127$	−4.65718	−0.413258	−0.206629	−	0.978419i	$-0.566249\pi$
$127$	−4.65718	−0.413258	−0.206629	+	0.978419i	$0.566249\pi$
$128$	13.1476	1.16209
$129$	0	0
$130$	0	0
$131$	−20.1789	−1.76304	−0.881520	−	0.472146i	$-0.843479\pi$
$131$	−20.1789	−1.76304	−0.881520	+	0.472146i	$0.843479\pi$
$132$	0	0
$133$	−0.694524	−0.0602229
$134$	0.642762	0.0555262
$135$	0	0
$136$	−16.1321	−1.38332
$137$	−8.39419	−0.717164	−0.358582	−	0.933498i	$-0.616740\pi$
$137$	−8.39419	−0.717164	−0.358582	+	0.933498i	$0.616740\pi$
$138$	0	0
$139$	−14.4736	−1.22764	−0.613819	−	0.789447i	$-0.710368\pi$
$139$	−14.4736	−1.22764	−0.613819	+	0.789447i	$0.710368\pi$
$140$	0	0
$141$	0	0
$142$	−19.9156	−1.67128
$143$	3.62099	0.302802
$144$	0	0
$145$	0	0
$146$	4.18143	0.346058
$147$	0	0
$148$	−0.293161	−0.0240977
$149$	−2.37980	−0.194961	−0.0974806	−	0.995237i	$-0.531078\pi$
$149$	−2.37980	−0.194961	−0.0974806	+	0.995237i	$0.531078\pi$
$150$	0	0
$151$	−10.0175	−0.815217	−0.407608	−	0.913157i	$-0.633637\pi$
$151$	−10.0175	−0.815217	−0.407608	+	0.913157i	$0.633637\pi$
$152$	1.25534	0.101821
$153$	0	0
$154$	−2.17748	−0.175467
$155$	0	0
$156$	0	0
$157$	−10.8952	−0.869536	−0.434768	−	0.900543i	$-0.643170\pi$
$157$	−10.8952	−0.869536	−0.434768	+	0.900543i	$0.643170\pi$
$158$	2.05212	0.163258
$159$	0	0
$160$	0	0
$161$	1.18544	0.0934258
$162$	0	0
$163$	22.9713	1.79925	0.899626	−	0.436662i	$-0.143839\pi$
$163$	22.9713	1.79925	0.899626	+	0.436662i	$0.143839\pi$
$164$	0.0882678	0.00689256
$165$	0	0
$166$	5.51876	0.428339
$167$	16.3091	1.26204	0.631020	−	0.775767i	$-0.282637\pi$
$167$	16.3091	1.26204	0.631020	+	0.775767i	$0.282637\pi$
$168$	0	0
$169$	−0.0326469	−0.00251130
$170$	0	0
$171$	0	0
$172$	−0.778843	−0.0593862
$173$	−3.35883	−0.255367	−0.127684	−	0.991815i	$-0.540754\pi$
$173$	−3.35883	−0.255367	−0.127684	+	0.991815i	$0.540754\pi$
$174$	0	0
$175$	0	0
$176$	4.52533	0.341110
$177$	0	0
$178$	−12.0767	−0.905183
$179$	−9.01947	−0.674147	−0.337073	−	0.941478i	$-0.609437\pi$
$179$	−9.01947	−0.674147	−0.337073	+	0.941478i	$0.609437\pi$
$180$	0	0
$181$	−10.3388	−0.768479	−0.384239	−	0.923234i	$-0.625536\pi$
$181$	−10.3388	−0.768479	−0.384239	+	0.923234i	$0.625536\pi$
$182$	−7.79793	−0.578021
$183$	0	0
$184$	−2.14266	−0.157959
$185$	0	0
$186$	0	0
$187$	−6.27600	−0.458947
$188$	1.76736	0.128898
$189$	0	0
$190$	0	0
$191$	−11.7668	−0.851416	−0.425708	−	0.904861i	$-0.639975\pi$
$191$	−11.7668	−0.851416	−0.425708	+	0.904861i	$0.639975\pi$
$192$	0	0
$193$	−12.6209	−0.908472	−0.454236	−	0.890881i	$-0.650088\pi$
$193$	−12.6209	−0.908472	−0.454236	+	0.890881i	$0.650088\pi$
$194$	−15.5373	−1.11551
$195$	0	0
$196$	−1.45264	−0.103760
$197$	1.08156	0.0770578	0.0385289	−	0.999257i	$-0.487733\pi$
$197$	1.08156	0.0770578	0.0385289	+	0.999257i	$0.487733\pi$
$198$	0	0
$199$	−4.67462	−0.331375	−0.165688	−	0.986178i	$-0.552984\pi$
$199$	−4.67462	−0.331375	−0.165688	+	0.986178i	$0.552984\pi$
$200$	0	0
$201$	0	0
$202$	13.6897	0.963205
$203$	−11.8126	−0.829084
$204$	0	0
$205$	0	0
$206$	1.17158	0.0816278
$207$	0	0
$208$	16.2060	1.12368
$209$	0.488374	0.0337815
$210$	0	0
$211$	−0.377789	−0.0260080	−0.0130040	−	0.999915i	$-0.504139\pi$
$211$	−0.377789	−0.0260080	−0.0130040	+	0.999915i	$0.504139\pi$
$212$	−2.88463	−0.198117
$213$	0	0
$214$	13.0802	0.894145
$215$	0	0
$216$	0	0
$217$	12.7494	0.865485
$218$	−26.5017	−1.79492
$219$	0	0
$220$	0	0
$221$	−22.4754	−1.51186
$222$	0	0
$223$	22.4985	1.50661	0.753306	−	0.657671i	$-0.228458\pi$
$223$	22.4985	1.50661	0.753306	+	0.657671i	$0.228458\pi$
$224$	−2.35323	−0.157232
$225$	0	0
$226$	−2.12699	−0.141485
$227$	23.3995	1.55308	0.776538	−	0.630070i	$-0.216974\pi$
$227$	23.3995	1.55308	0.776538	+	0.630070i	$0.216974\pi$
$228$	0	0
$229$	27.5246	1.81887	0.909437	−	0.415841i	$-0.136513\pi$
$229$	27.5246	1.81887	0.909437	+	0.415841i	$0.136513\pi$
$230$	0	0
$231$	0	0
$232$	21.3511	1.40177
$233$	−8.18885	−0.536469	−0.268235	−	0.963354i	$-0.586440\pi$
$233$	−8.18885	−0.536469	−0.268235	+	0.963354i	$0.586440\pi$
$234$	0	0
$235$	0	0
$236$	−0.248959	−0.0162059
$237$	0	0
$238$	13.5156	0.876087
$239$	−7.84205	−0.507260	−0.253630	−	0.967301i	$-0.581625\pi$
$239$	−7.84205	−0.507260	−0.253630	+	0.967301i	$0.581625\pi$
$240$	0	0
$241$	2.81826	0.181540	0.0907700	−	0.995872i	$-0.471067\pi$
$241$	2.81826	0.181540	0.0907700	+	0.995872i	$0.471067\pi$
$242$	−15.1264	−0.972359
$243$	0	0
$244$	−2.26765	−0.145172
$245$	0	0
$246$	0	0
$247$	1.74895	0.111283
$248$	−23.0443	−1.46331
$249$	0	0
$250$	0	0
$251$	−25.9733	−1.63942	−0.819710	−	0.572779i	$-0.805865\pi$
$251$	−25.9733	−1.63942	−0.819710	+	0.572779i	$0.805865\pi$
$252$	0	0
$253$	−0.833575	−0.0524064
$254$	−7.05246	−0.442511
$255$	0	0
$256$	6.89209	0.430756
$257$	5.90010	0.368038	0.184019	−	0.982923i	$-0.441089\pi$
$257$	5.90010	0.368038	0.184019	+	0.982923i	$0.441089\pi$
$258$	0	0
$259$	−1.43000	−0.0888559
$260$	0	0
$261$	0	0
$262$	−30.5573	−1.88784
$263$	−22.6113	−1.39427	−0.697135	−	0.716940i	$-0.745542\pi$
$263$	−22.6113	−1.39427	−0.697135	+	0.716940i	$0.745542\pi$
$264$	0	0
$265$	0	0
$266$	−1.05173	−0.0644858
$267$	0	0
$268$	0.124434	0.00760103
$269$	29.5891	1.80408	0.902039	−	0.431655i	$-0.142070\pi$
$269$	29.5891	1.80408	0.902039	+	0.431655i	$0.142070\pi$
$270$	0	0
$271$	−7.91061	−0.480535	−0.240268	−	0.970707i	$-0.577235\pi$
$271$	−7.91061	−0.480535	−0.240268	+	0.970707i	$0.577235\pi$
$272$	−28.0887	−1.70313
$273$	0	0
$274$	−12.7115	−0.767929
$275$	0	0
$276$	0	0
$277$	−20.0183	−1.20278	−0.601392	−	0.798954i	$-0.705387\pi$
$277$	−20.0183	−1.20278	−0.601392	+	0.798954i	$0.705387\pi$
$278$	−21.9177	−1.31454
$279$	0	0
$280$	0	0
$281$	1.95583	0.116675	0.0583375	−	0.998297i	$-0.481420\pi$
$281$	1.95583	0.116675	0.0583375	+	0.998297i	$0.481420\pi$
$282$	0	0
$283$	−26.0247	−1.54701	−0.773503	−	0.633793i	$-0.781497\pi$
$283$	−26.0247	−1.54701	−0.773503	+	0.633793i	$0.781497\pi$
$284$	−3.85552	−0.228783
$285$	0	0
$286$	5.48333	0.324236
$287$	0.430558	0.0254150
$288$	0	0
$289$	21.9551	1.29147
$290$	0	0
$291$	0	0
$292$	0.809496	0.0473722
$293$	13.5422	0.791143	0.395571	−	0.918435i	$-0.370547\pi$
$293$	13.5422	0.791143	0.395571	+	0.918435i	$0.370547\pi$
$294$	0	0
$295$	0	0
$296$	2.58470	0.150233
$297$	0	0
$298$	−3.60378	−0.208761
$299$	−2.98517	−0.172637
$300$	0	0
$301$	−3.79908	−0.218976
$302$	−15.1698	−0.872922
$303$	0	0
$304$	2.18575	0.125361
$305$	0	0
$306$	0	0
$307$	24.6104	1.40459	0.702295	−	0.711886i	$-0.252159\pi$
$307$	24.6104	1.40459	0.702295	+	0.711886i	$0.252159\pi$
$308$	−0.421545	−0.0240198
$309$	0	0
$310$	0	0
$311$	28.1674	1.59723	0.798614	−	0.601843i	$-0.205567\pi$
$311$	28.1674	1.59723	0.798614	+	0.601843i	$0.205567\pi$
$312$	0	0
$313$	16.2843	0.920441	0.460221	−	0.887805i	$-0.347770\pi$
$313$	16.2843	0.920441	0.460221	+	0.887805i	$0.347770\pi$
$314$	−16.4989	−0.931086
$315$	0	0
$316$	0.397276	0.0223485
$317$	−10.7976	−0.606451	−0.303225	−	0.952919i	$-0.598064\pi$
$317$	−10.7976	−0.606451	−0.303225	+	0.952919i	$0.598064\pi$
$318$	0	0
$319$	8.30637	0.465068
$320$	0	0
$321$	0	0
$322$	1.79514	0.100039
$323$	−3.03133	−0.168668
$324$	0	0
$325$	0	0
$326$	34.7859	1.92661
$327$	0	0
$328$	−0.778225	−0.0429703
$329$	8.62095	0.475289
$330$	0	0
$331$	−11.1507	−0.612896	−0.306448	−	0.951887i	$-0.599140\pi$
$331$	−11.1507	−0.612896	−0.306448	+	0.951887i	$0.599140\pi$
$332$	1.06839	0.0586357
$333$	0	0
$334$	24.6972	1.35137
$335$	0	0
$336$	0	0
$337$	−35.3432	−1.92527	−0.962634	−	0.270807i	$-0.912709\pi$
$337$	−35.3432	−1.92527	−0.962634	+	0.270807i	$0.912709\pi$
$338$	−0.0494378	−0.00268906
$339$	0	0
$340$	0	0
$341$	−8.96508	−0.485486
$342$	0	0
$343$	−17.0958	−0.923086
$344$	6.86677	0.370231
$345$	0	0
$346$	−5.08634	−0.273443
$347$	−7.89341	−0.423740	−0.211870	−	0.977298i	$-0.567955\pi$
$347$	−7.89341	−0.423740	−0.211870	+	0.977298i	$0.567955\pi$
$348$	0	0
$349$	−33.8866	−1.81391	−0.906955	−	0.421227i	$-0.861600\pi$
$349$	−33.8866	−1.81391	−0.906955	+	0.421227i	$0.861600\pi$
$350$	0	0
$351$	0	0
$352$	1.65474	0.0881978
$353$	−13.0296	−0.693498	−0.346749	−	0.937958i	$-0.612714\pi$
$353$	−13.0296	−0.693498	−0.346749	+	0.937958i	$0.612714\pi$
$354$	0	0
$355$	0	0
$356$	−2.33795	−0.123911
$357$	0	0
$358$	−13.6583	−0.721866
$359$	1.77204	0.0935248	0.0467624	−	0.998906i	$-0.485110\pi$
$359$	1.77204	0.0935248	0.0467624	+	0.998906i	$0.485110\pi$
$360$	0	0
$361$	−18.7641	−0.987585
$362$	−15.6563	−0.822875
$363$	0	0
$364$	−1.50962	−0.0791258
$365$	0	0
$366$	0	0
$367$	6.37821	0.332940	0.166470	−	0.986047i	$-0.446763\pi$
$367$	6.37821	0.332940	0.166470	+	0.986047i	$0.446763\pi$
$368$	−3.73072	−0.194477
$369$	0	0
$370$	0	0
$371$	−14.0708	−0.730520
$372$	0	0
$373$	−1.64642	−0.0852484	−0.0426242	−	0.999091i	$-0.513572\pi$
$373$	−1.64642	−0.0852484	−0.0426242	+	0.999091i	$0.513572\pi$
$374$	−9.50387	−0.491433
$375$	0	0
$376$	−15.5822	−0.803591
$377$	29.7465	1.53202
$378$	0	0
$379$	1.86269	0.0956801	0.0478401	−	0.998855i	$-0.484766\pi$
$379$	1.86269	0.0956801	0.0478401	+	0.998855i	$0.484766\pi$
$380$	0	0
$381$	0	0
$382$	−17.8187	−0.911684
$383$	17.5264	0.895555	0.447777	−	0.894145i	$-0.352216\pi$
$383$	17.5264	0.895555	0.447777	+	0.894145i	$0.352216\pi$
$384$	0	0
$385$	0	0
$386$	−19.1121	−0.972778
$387$	0	0
$388$	−3.00791	−0.152704
$389$	1.60428	0.0813402	0.0406701	−	0.999173i	$-0.487051\pi$
$389$	1.60428	0.0813402	0.0406701	+	0.999173i	$0.487051\pi$
$390$	0	0
$391$	5.17399	0.261660
$392$	12.8074	0.646873
$393$	0	0
$394$	1.63782	0.0825123
$395$	0	0
$396$	0	0
$397$	−15.9428	−0.800147	−0.400073	−	0.916483i	$-0.631015\pi$
$397$	−15.9428	−0.800147	−0.400073	+	0.916483i	$0.631015\pi$
$398$	−7.07886	−0.354831
$399$	0	0
$400$	0	0
$401$	13.6963	0.683962	0.341981	−	0.939707i	$-0.388902\pi$
$401$	13.6963	0.683962	0.341981	+	0.939707i	$0.388902\pi$
$402$	0	0
$403$	−32.1054	−1.59929
$404$	2.65023	0.131854
$405$	0	0
$406$	−17.8881	−0.887771
$407$	1.00554	0.0498430
$408$	0	0
$409$	0.355950	0.0176006	0.00880030	−	0.999961i	$-0.497199\pi$
$409$	0.355950	0.0176006	0.00880030	+	0.999961i	$0.497199\pi$
$410$	0	0
$411$	0	0
$412$	0.226809	0.0111741
$413$	−1.21439	−0.0597561
$414$	0	0
$415$	0	0
$416$	5.92589	0.290541
$417$	0	0
$418$	0.739554	0.0361728
$419$	31.8129	1.55416	0.777082	−	0.629400i	$-0.216699\pi$
$419$	31.8129	1.55416	0.777082	+	0.629400i	$0.216699\pi$
$420$	0	0
$421$	25.1473	1.22560	0.612801	−	0.790237i	$-0.290043\pi$
$421$	25.1473	1.22560	0.612801	+	0.790237i	$0.290043\pi$
$422$	−0.572092	−0.0278490
$423$	0	0
$424$	25.4327	1.23512
$425$	0	0
$426$	0	0
$427$	−11.0613	−0.535293
$428$	2.53224	0.122400
$429$	0	0
$430$	0	0
$431$	1.18312	0.0569887	0.0284944	−	0.999594i	$-0.490929\pi$
$431$	1.18312	0.0569887	0.0284944	+	0.999594i	$0.490929\pi$
$432$	0	0
$433$	18.5064	0.889359	0.444679	−	0.895690i	$-0.353318\pi$
$433$	18.5064	0.889359	0.444679	+	0.895690i	$0.353318\pi$
$434$	19.3066	0.926748
$435$	0	0
$436$	−5.13054	−0.245708
$437$	−0.402620	−0.0192599
$438$	0	0
$439$	22.3714	1.06773	0.533865	−	0.845570i	$-0.320739\pi$
$439$	22.3714	1.06773	0.533865	+	0.845570i	$0.320739\pi$
$440$	0	0
$441$	0	0
$442$	−34.0349	−1.61888
$443$	−5.58381	−0.265295	−0.132647	−	0.991163i	$-0.542348\pi$
$443$	−5.58381	−0.265295	−0.132647	+	0.991163i	$0.542348\pi$
$444$	0	0
$445$	0	0
$446$	34.0699	1.61326
$447$	0	0
$448$	9.30755	0.439740
$449$	0.244500	0.0115387	0.00576934	−	0.999983i	$-0.498164\pi$
$449$	0.244500	0.0115387	0.00576934	+	0.999983i	$0.498164\pi$
$450$	0	0
$451$	−0.302759	−0.0142564
$452$	−0.411770	−0.0193680
$453$	0	0
$454$	35.4342	1.66301
$455$	0	0
$456$	0	0
$457$	17.7132	0.828591	0.414295	−	0.910142i	$-0.364028\pi$
$457$	17.7132	0.828591	0.414295	+	0.910142i	$0.364028\pi$
$458$	41.6810	1.94762
$459$	0	0
$460$	0	0
$461$	33.8452	1.57633	0.788165	−	0.615464i	$-0.211031\pi$
$461$	33.8452	1.57633	0.788165	+	0.615464i	$0.211031\pi$
$462$	0	0
$463$	−17.6019	−0.818029	−0.409015	−	0.912528i	$-0.634127\pi$
$463$	−17.6019	−0.818029	−0.409015	+	0.912528i	$0.634127\pi$
$464$	37.1757	1.72584
$465$	0	0
$466$	−12.4005	−0.574443
$467$	−1.23902	−0.0573348	−0.0286674	−	0.999589i	$-0.509126\pi$
$467$	−1.23902	−0.0573348	−0.0286674	+	0.999589i	$0.509126\pi$
$468$	0	0
$469$	0.606973	0.0280274
$470$	0	0
$471$	0	0
$472$	2.19498	0.101032
$473$	2.67143	0.122833
$474$	0	0
$475$	0	0
$476$	2.61653	0.119928
$477$	0	0
$478$	−11.8754	−0.543167
$479$	7.20703	0.329298	0.164649	−	0.986352i	$-0.447351\pi$
$479$	7.20703	0.329298	0.164649	+	0.986352i	$0.447351\pi$
$480$	0	0
$481$	3.60102	0.164192
$482$	4.26774	0.194390
$483$	0	0
$484$	−2.92836	−0.133107
$485$	0	0
$486$	0	0
$487$	−34.8356	−1.57855	−0.789276	−	0.614038i	$-0.789544\pi$
$487$	−34.8356	−1.57855	−0.789276	+	0.614038i	$0.789544\pi$
$488$	19.9931	0.905044
$489$	0	0
$490$	0	0
$491$	−1.08796	−0.0490989	−0.0245495	−	0.999699i	$-0.507815\pi$
$491$	−1.08796	−0.0490989	−0.0245495	+	0.999699i	$0.507815\pi$
$492$	0	0
$493$	−51.5575	−2.32204
$494$	2.64847	0.119160
$495$	0	0
$496$	−40.1238	−1.80161
$497$	−18.8067	−0.843596
$498$	0	0
$499$	18.4025	0.823809	0.411905	−	0.911227i	$-0.364864\pi$
$499$	18.4025	0.823809	0.411905	+	0.911227i	$0.364864\pi$
$500$	0	0
$501$	0	0
$502$	−39.3318	−1.75547
$503$	−7.66267	−0.341661	−0.170831	−	0.985300i	$-0.554645\pi$
$503$	−7.66267	−0.341661	−0.170831	+	0.985300i	$0.554645\pi$
$504$	0	0
$505$	0	0
$506$	−1.26230	−0.0561160
$507$	0	0
$508$	−1.36531	−0.0605757
$509$	−5.94536	−0.263523	−0.131762	−	0.991281i	$-0.542063\pi$
$509$	−5.94536	−0.263523	−0.131762	+	0.991281i	$0.542063\pi$
$510$	0	0
$511$	3.94861	0.174676
$512$	−15.8584	−0.700847
$513$	0	0
$514$	8.93463	0.394089
$515$	0	0
$516$	0	0
$517$	−6.06206	−0.266609
$518$	−2.16548	−0.0951456
$519$	0	0
$520$	0	0
$521$	37.7815	1.65524	0.827620	−	0.561290i	$-0.189695\pi$
$521$	37.7815	1.65524	0.827620	+	0.561290i	$0.189695\pi$
$522$	0	0
$523$	33.5417	1.46667	0.733337	−	0.679865i	$-0.237962\pi$
$523$	33.5417	1.46667	0.733337	+	0.679865i	$0.237962\pi$
$524$	−5.91568	−0.258428
$525$	0	0
$526$	−34.2407	−1.49296
$527$	55.6461	2.42398
$528$	0	0
$529$	−22.3128	−0.970121
$530$	0	0
$531$	0	0
$532$	−0.203608	−0.00882752
$533$	−1.08423	−0.0469632
$534$	0	0
$535$	0	0
$536$	−1.09709	−0.0473871
$537$	0	0
$538$	44.8073	1.93178
$539$	4.98257	0.214614
$540$	0	0
$541$	−9.97375	−0.428805	−0.214402	−	0.976745i	$-0.568780\pi$
$541$	−9.97375	−0.428805	−0.214402	+	0.976745i	$0.568780\pi$
$542$	−11.9792	−0.514550
$543$	0	0
$544$	−10.2709	−0.440363
$545$	0	0
$546$	0	0
$547$	5.30442	0.226801	0.113400	−	0.993549i	$-0.463826\pi$
$547$	5.30442	0.226801	0.113400	+	0.993549i	$0.463826\pi$
$548$	−2.46085	−0.105122
$549$	0	0
$550$	0	0
$551$	4.01201	0.170917
$552$	0	0
$553$	1.93786	0.0824061
$554$	−30.3141	−1.28792
$555$	0	0
$556$	−4.24312	−0.179948
$557$	15.6542	0.663289	0.331645	−	0.943404i	$-0.392397\pi$
$557$	15.6542	0.663289	0.331645	+	0.943404i	$0.392397\pi$
$558$	0	0
$559$	9.56684	0.404634
$560$	0	0
$561$	0	0
$562$	2.96175	0.124934
$563$	−11.4875	−0.484140	−0.242070	−	0.970259i	$-0.577826\pi$
$563$	−11.4875	−0.484140	−0.242070	+	0.970259i	$0.577826\pi$
$564$	0	0
$565$	0	0
$566$	−39.4096	−1.65651
$567$	0	0
$568$	33.9927	1.42630
$569$	14.6116	0.612550	0.306275	−	0.951943i	$-0.400917\pi$
$569$	14.6116	0.612550	0.306275	+	0.951943i	$0.400917\pi$
$570$	0	0
$571$	−27.6908	−1.15882	−0.579411	−	0.815035i	$-0.696718\pi$
$571$	−27.6908	−1.15882	−0.579411	+	0.815035i	$0.696718\pi$
$572$	1.06153	0.0443849
$573$	0	0
$574$	0.652002	0.0272140
$575$	0	0
$576$	0	0
$577$	14.2052	0.591371	0.295686	−	0.955285i	$-0.404452\pi$
$577$	14.2052	0.591371	0.295686	+	0.955285i	$0.404452\pi$
$578$	33.2470	1.38289
$579$	0	0
$580$	0	0
$581$	5.21148	0.216208
$582$	0	0
$583$	9.89428	0.409779
$584$	−7.13703	−0.295332
$585$	0	0
$586$	20.5072	0.847144
$587$	−28.6303	−1.18170	−0.590849	−	0.806782i	$-0.701207\pi$
$587$	−28.6303	−1.18170	−0.590849	+	0.806782i	$0.701207\pi$
$588$	0	0
$589$	−4.33016	−0.178421
$590$	0	0
$591$	0	0
$592$	4.50038	0.184964
$593$	−13.8042	−0.566872	−0.283436	−	0.958991i	$-0.591474\pi$
$593$	−13.8042	−0.566872	−0.283436	+	0.958991i	$0.591474\pi$
$594$	0	0
$595$	0	0
$596$	−0.697667	−0.0285775
$597$	0	0
$598$	−4.52050	−0.184857
$599$	−24.2711	−0.991689	−0.495845	−	0.868411i	$-0.665141\pi$
$599$	−24.2711	−0.991689	−0.495845	+	0.868411i	$0.665141\pi$
$600$	0	0
$601$	−26.5945	−1.08481	−0.542406	−	0.840117i	$-0.682487\pi$
$601$	−26.5945	−1.08481	−0.542406	+	0.840117i	$0.682487\pi$
$602$	−5.75303	−0.234476
$603$	0	0
$604$	−2.93676	−0.119495
$605$	0	0
$606$	0	0
$607$	−35.5776	−1.44405	−0.722026	−	0.691866i	$-0.756789\pi$
$607$	−35.5776	−1.44405	−0.722026	+	0.691866i	$0.756789\pi$
$608$	0.799244	0.0324136
$609$	0	0
$610$	0	0
$611$	−21.7093	−0.878262
$612$	0	0
$613$	−4.87831	−0.197033	−0.0985165	−	0.995135i	$-0.531410\pi$
$613$	−4.87831	−0.197033	−0.0985165	+	0.995135i	$0.531410\pi$
$614$	37.2680	1.50402
$615$	0	0
$616$	3.71661	0.149746
$617$	13.4467	0.541344	0.270672	−	0.962672i	$-0.412754\pi$
$617$	13.4467	0.541344	0.270672	+	0.962672i	$0.412754\pi$
$618$	0	0
$619$	−24.1560	−0.970911	−0.485456	−	0.874261i	$-0.661346\pi$
$619$	−24.1560	−0.970911	−0.485456	+	0.874261i	$0.661346\pi$
$620$	0	0
$621$	0	0
$622$	42.6545	1.71029
$623$	−11.4042	−0.456900
$624$	0	0
$625$	0	0
$626$	24.6596	0.985595
$627$	0	0
$628$	−3.19407	−0.127457
$629$	−6.24140	−0.248861
$630$	0	0
$631$	34.5765	1.37647	0.688235	−	0.725487i	$-0.258386\pi$
$631$	34.5765	1.37647	0.688235	+	0.725487i	$0.258386\pi$
$632$	−3.50264	−0.139328
$633$	0	0
$634$	−16.3509	−0.649379
$635$	0	0
$636$	0	0
$637$	17.8434	0.706982
$638$	12.5785	0.497987
$639$	0	0
$640$	0	0
$641$	−31.6135	−1.24866	−0.624329	−	0.781161i	$-0.714628\pi$
$641$	−31.6135	−1.24866	−0.624329	+	0.781161i	$0.714628\pi$
$642$	0	0
$643$	−30.0703	−1.18586	−0.592929	−	0.805255i	$-0.702028\pi$
$643$	−30.0703	−1.18586	−0.592929	+	0.805255i	$0.702028\pi$
$644$	0.347526	0.0136944
$645$	0	0
$646$	−4.59040	−0.180607
$647$	16.6914	0.656205	0.328103	−	0.944642i	$-0.393591\pi$
$647$	16.6914	0.656205	0.328103	+	0.944642i	$0.393591\pi$
$648$	0	0
$649$	0.853930	0.0335197
$650$	0	0
$651$	0	0
$652$	6.73430	0.263736
$653$	32.8997	1.28746	0.643732	−	0.765251i	$-0.277385\pi$
$653$	32.8997	1.28746	0.643732	+	0.765251i	$0.277385\pi$
$654$	0	0
$655$	0	0
$656$	−1.35502	−0.0529045
$657$	0	0
$658$	13.0549	0.508932
$659$	−28.2476	−1.10037	−0.550184	−	0.835043i	$-0.685443\pi$
$659$	−28.2476	−1.10037	−0.550184	+	0.835043i	$0.685443\pi$
$660$	0	0
$661$	15.9506	0.620407	0.310203	−	0.950670i	$-0.399603\pi$
$661$	15.9506	0.620407	0.310203	+	0.950670i	$0.399603\pi$
$662$	−16.8857	−0.656279
$663$	0	0
$664$	−9.41964	−0.365553
$665$	0	0
$666$	0	0
$667$	−6.84785	−0.265150
$668$	4.78121	0.184991
$669$	0	0
$670$	0	0
$671$	7.77805	0.300268
$672$	0	0
$673$	−42.4140	−1.63494	−0.817471	−	0.575970i	$-0.804624\pi$
$673$	−42.4140	−1.63494	−0.817471	+	0.575970i	$0.804624\pi$
$674$	−53.5209	−2.06155
$675$	0	0
$676$	−0.00957081	−0.000368108 0
$677$	27.5448	1.05863	0.529316	−	0.848425i	$-0.322449\pi$
$677$	27.5448	1.05863	0.529316	+	0.848425i	$0.322449\pi$
$678$	0	0
$679$	−14.6722	−0.563067
$680$	0	0
$681$	0	0
$682$	−13.5760	−0.519851
$683$	7.46557	0.285662	0.142831	−	0.989747i	$-0.454379\pi$
$683$	7.46557	0.285662	0.142831	+	0.989747i	$0.454379\pi$
$684$	0	0
$685$	0	0
$686$	−25.8885	−0.988427
$687$	0	0
$688$	11.9562	0.455825
$689$	35.4331	1.34989
$690$	0	0
$691$	−1.14169	−0.0434320	−0.0217160	−	0.999764i	$-0.506913\pi$
$691$	−1.14169	−0.0434320	−0.0217160	+	0.999764i	$0.506913\pi$
$692$	−0.984680	−0.0374319
$693$	0	0
$694$	−11.9531	−0.453735
$695$	0	0
$696$	0	0
$697$	1.87922	0.0711805
$698$	−51.3152	−1.94231
$699$	0	0
$700$	0	0
$701$	−4.34341	−0.164048	−0.0820242	−	0.996630i	$-0.526138\pi$
$701$	−4.34341	−0.164048	−0.0820242	+	0.996630i	$0.526138\pi$
$702$	0	0
$703$	0.485681	0.0183178
$704$	−6.54486	−0.246669
$705$	0	0
$706$	−19.7310	−0.742587
$707$	12.9275	0.486187
$708$	0	0
$709$	−41.6107	−1.56272	−0.781362	−	0.624079i	$-0.785474\pi$
$709$	−41.6107	−1.56272	−0.781362	+	0.624079i	$0.785474\pi$
$710$	0	0
$711$	0	0
$712$	20.6129	0.772501
$713$	7.39089	0.276791
$714$	0	0
$715$	0	0
$716$	−2.64416	−0.0988169
$717$	0	0
$718$	2.68344	0.100145
$719$	26.3710	0.983471	0.491736	−	0.870745i	$-0.336363\pi$
$719$	26.3710	0.983471	0.491736	+	0.870745i	$0.336363\pi$
$720$	0	0
$721$	1.10634	0.0412024
$722$	−28.4149	−1.05749
$723$	0	0
$724$	−3.03094	−0.112644
$725$	0	0
$726$	0	0
$727$	−24.0658	−0.892552	−0.446276	−	0.894895i	$-0.647250\pi$
$727$	−24.0658	−0.892552	−0.446276	+	0.894895i	$0.647250\pi$
$728$	13.3098	0.493294
$729$	0	0
$730$	0	0
$731$	−16.5815	−0.613290
$732$	0	0
$733$	−19.8486	−0.733126	−0.366563	−	0.930393i	$-0.619466\pi$
$733$	−19.8486	−0.733126	−0.366563	+	0.930393i	$0.619466\pi$
$734$	9.65864	0.356507
$735$	0	0
$736$	−1.36418	−0.0502843
$737$	−0.426810	−0.0157217
$738$	0	0
$739$	−5.32613	−0.195925	−0.0979624	−	0.995190i	$-0.531232\pi$
$739$	−5.32613	−0.195925	−0.0979624	+	0.995190i	$0.531232\pi$
$740$	0	0
$741$	0	0
$742$	−21.3077	−0.782230
$743$	41.3416	1.51668	0.758339	−	0.651861i	$-0.226011\pi$
$743$	41.3416	1.51668	0.758339	+	0.651861i	$0.226011\pi$
$744$	0	0
$745$	0	0
$746$	−2.49321	−0.0912827
$747$	0	0
$748$	−1.83988	−0.0672728
$749$	12.3519	0.451328
$750$	0	0
$751$	−27.7021	−1.01087	−0.505433	−	0.862866i	$-0.668667\pi$
$751$	−27.7021	−1.01087	−0.505433	+	0.862866i	$0.668667\pi$
$752$	−27.1312	−0.989371
$753$	0	0
$754$	45.0457	1.64047
$755$	0	0
$756$	0	0
$757$	2.43292	0.0884261	0.0442131	−	0.999022i	$-0.485922\pi$
$757$	2.43292	0.0884261	0.0442131	+	0.999022i	$0.485922\pi$
$758$	2.82071	0.102453
$759$	0	0
$760$	0	0
$761$	41.1185	1.49054	0.745272	−	0.666761i	$-0.232320\pi$
$761$	41.1185	1.49054	0.745272	+	0.666761i	$0.232320\pi$
$762$	0	0
$763$	−25.0260	−0.906003
$764$	−3.44957	−0.124801
$765$	0	0
$766$	26.5405	0.958947
$767$	3.05806	0.110420
$768$	0	0
$769$	21.6760	0.781655	0.390827	−	0.920464i	$-0.372189\pi$
$769$	21.6760	0.781655	0.390827	+	0.920464i	$0.372189\pi$
$770$	0	0
$771$	0	0
$772$	−3.69996	−0.133164
$773$	16.1814	0.582003	0.291002	−	0.956723i	$-0.406012\pi$
$773$	16.1814	0.582003	0.291002	+	0.956723i	$0.406012\pi$
$774$	0	0
$775$	0	0
$776$	26.5197	0.952001
$777$	0	0
$778$	2.42939	0.0870979
$779$	−0.146233	−0.00523936
$780$	0	0
$781$	13.2244	0.473208
$782$	7.83507	0.280182
$783$	0	0
$784$	22.2998	0.796422
$785$	0	0
$786$	0	0
$787$	44.4560	1.58468	0.792342	−	0.610077i	$-0.208862\pi$
$787$	44.4560	1.58468	0.792342	+	0.610077i	$0.208862\pi$
$788$	0.317071	0.0112952
$789$	0	0
$790$	0	0
$791$	−2.00855	−0.0714160
$792$	0	0
$793$	27.8545	0.989142
$794$	−24.1425	−0.856785
$795$	0	0
$796$	−1.37042	−0.0485732
$797$	19.5738	0.693338	0.346669	−	0.937988i	$-0.387313\pi$
$797$	19.5738	0.693338	0.346669	+	0.937988i	$0.387313\pi$
$798$	0	0
$799$	37.6271	1.33115
$800$	0	0
$801$	0	0
$802$	20.7406	0.732377
$803$	−2.77657	−0.0979831
$804$	0	0
$805$	0	0
$806$	−48.6179	−1.71249
$807$	0	0
$808$	−23.3661	−0.822017
$809$	25.6304	0.901116	0.450558	−	0.892747i	$-0.351225\pi$
$809$	25.6304	0.901116	0.450558	+	0.892747i	$0.351225\pi$
$810$	0	0
$811$	−43.7556	−1.53647	−0.768234	−	0.640170i	$-0.778864\pi$
$811$	−43.7556	−1.53647	−0.768234	+	0.640170i	$0.778864\pi$
$812$	−3.46301	−0.121528
$813$	0	0
$814$	1.52271	0.0533711
$815$	0	0
$816$	0	0
$817$	1.29031	0.0451422
$818$	0.539022	0.0188465
$819$	0	0
$820$	0	0
$821$	20.2095	0.705317	0.352659	−	0.935752i	$-0.385278\pi$
$821$	20.2095	0.705317	0.352659	+	0.935752i	$0.385278\pi$
$822$	0	0
$823$	19.9936	0.696934	0.348467	−	0.937321i	$-0.386702\pi$
$823$	19.9936	0.696934	0.348467	+	0.937321i	$0.386702\pi$
$824$	−1.99970	−0.0696627
$825$	0	0
$826$	−1.83897	−0.0639859
$827$	−50.8207	−1.76721	−0.883604	−	0.468234i	$-0.844890\pi$
$827$	−50.8207	−1.76721	−0.883604	+	0.468234i	$0.844890\pi$
$828$	0	0
$829$	−56.8590	−1.97479	−0.987397	−	0.158260i	$-0.949411\pi$
$829$	−56.8590	−1.97479	−0.987397	+	0.158260i	$0.949411\pi$
$830$	0	0
$831$	0	0
$832$	−23.4382	−0.812575
$833$	−30.9268	−1.07155
$834$	0	0
$835$	0	0
$836$	0.143172	0.00495172
$837$	0	0
$838$	48.1749	1.66418
$839$	1.68762	0.0582633	0.0291316	−	0.999576i	$-0.490726\pi$
$839$	1.68762	0.0582633	0.0291316	+	0.999576i	$0.490726\pi$
$840$	0	0
$841$	39.2371	1.35300
$842$	38.0810	1.31236
$843$	0	0
$844$	−0.110753	−0.00381228
$845$	0	0
$846$	0	0
$847$	−14.2841	−0.490808
$848$	44.2825	1.52067
$849$	0	0
$850$	0	0
$851$	−0.828979	−0.0284170
$852$	0	0
$853$	28.0236	0.959510	0.479755	−	0.877403i	$-0.340726\pi$
$853$	28.0236	0.959510	0.479755	+	0.877403i	$0.340726\pi$
$854$	−16.7503	−0.573184
$855$	0	0
$856$	−22.3258	−0.763080
$857$	−47.8661	−1.63508	−0.817538	−	0.575875i	$-0.804662\pi$
$857$	−47.8661	−1.63508	−0.817538	+	0.575875i	$0.804662\pi$
$858$	0	0
$859$	−21.3156	−0.727280	−0.363640	−	0.931540i	$-0.618466\pi$
$859$	−21.3156	−0.727280	−0.363640	+	0.931540i	$0.618466\pi$
$860$	0	0
$861$	0	0
$862$	1.79162	0.0610227
$863$	30.1233	1.02541	0.512705	−	0.858565i	$-0.328644\pi$
$863$	30.1233	1.02541	0.512705	+	0.858565i	$0.328644\pi$
$864$	0	0
$865$	0	0
$866$	28.0245	0.952312
$867$	0	0
$868$	3.73763	0.126863
$869$	−1.36266	−0.0462250
$870$	0	0
$871$	−1.52848	−0.0517905
$872$	45.2341	1.53182
$873$	0	0
$874$	−0.609695	−0.0206232
$875$	0	0
$876$	0	0
$877$	31.9006	1.07721	0.538603	−	0.842560i	$-0.318952\pi$
$877$	31.9006	1.07721	0.538603	+	0.842560i	$0.318952\pi$
$878$	33.8775	1.14331
$879$	0	0
$880$	0	0
$881$	15.7688	0.531265	0.265632	−	0.964074i	$-0.414419\pi$
$881$	15.7688	0.531265	0.265632	+	0.964074i	$0.414419\pi$
$882$	0	0
$883$	−32.3052	−1.08716	−0.543578	−	0.839359i	$-0.682931\pi$
$883$	−32.3052	−1.08716	−0.543578	+	0.839359i	$0.682931\pi$
$884$	−6.58893	−0.221610
$885$	0	0
$886$	−8.45567	−0.284074
$887$	56.2557	1.88888	0.944440	−	0.328684i	$-0.106605\pi$
$887$	56.2557	1.88888	0.944440	+	0.328684i	$0.106605\pi$
$888$	0	0
$889$	−6.65977	−0.223362
$890$	0	0
$891$	0	0
$892$	6.59570	0.220840
$893$	−2.92800	−0.0979816
$894$	0	0
$895$	0	0
$896$	18.8011	0.628099
$897$	0	0
$898$	0.370251	0.0123554
$899$	−73.6484	−2.45631
$900$	0	0
$901$	−61.4137	−2.04599
$902$	−0.458473	−0.0152655
$903$	0	0
$904$	3.63042	0.120746
$905$	0	0
$906$	0	0
$907$	−49.2033	−1.63377	−0.816884	−	0.576802i	$-0.804301\pi$
$907$	−49.2033	−1.63377	−0.816884	+	0.576802i	$0.804301\pi$
$908$	6.85982	0.227651
$909$	0	0
$910$	0	0
$911$	−35.1411	−1.16428	−0.582139	−	0.813090i	$-0.697784\pi$
$911$	−35.1411	−1.16428	−0.582139	+	0.813090i	$0.697784\pi$
$912$	0	0
$913$	−3.66459	−0.121280
$914$	26.8235	0.887243
$915$	0	0
$916$	8.06914	0.266612
$917$	−28.8559	−0.952905
$918$	0	0
$919$	−3.02770	−0.0998747	−0.0499373	−	0.998752i	$-0.515902\pi$
$919$	−3.02770	−0.0998747	−0.0499373	+	0.998752i	$0.515902\pi$
$920$	0	0
$921$	0	0
$922$	51.2525	1.68791
$923$	47.3590	1.55884
$924$	0	0
$925$	0	0
$926$	−26.6549	−0.875933
$927$	0	0
$928$	13.5937	0.446236
$929$	−26.6586	−0.874640	−0.437320	−	0.899306i	$-0.644072\pi$
$929$	−26.6586	−0.874640	−0.437320	+	0.899306i	$0.644072\pi$
$930$	0	0
$931$	2.40660	0.0788731
$932$	−2.40065	−0.0786361
$933$	0	0
$934$	−1.87626	−0.0613933
$935$	0	0
$936$	0	0
$937$	22.3140	0.728966	0.364483	−	0.931210i	$-0.381246\pi$
$937$	22.3140	0.728966	0.364483	+	0.931210i	$0.381246\pi$
$938$	0.919150	0.0300113
$939$	0	0
$940$	0	0
$941$	30.9818	1.00998	0.504989	−	0.863126i	$-0.331497\pi$
$941$	30.9818	1.00998	0.504989	+	0.863126i	$0.331497\pi$
$942$	0	0
$943$	0.249597	0.00812800
$944$	3.82182	0.124390
$945$	0	0
$946$	4.04540	0.131527
$947$	−35.9505	−1.16823	−0.584117	−	0.811669i	$-0.698559\pi$
$947$	−35.9505	−1.16823	−0.584117	+	0.811669i	$0.698559\pi$
$948$	0	0
$949$	−9.94336	−0.322775
$950$	0	0
$951$	0	0
$952$	−23.0690	−0.747669
$953$	59.5051	1.92756	0.963779	−	0.266703i	$-0.0859341\pi$
$953$	59.5051	1.92756	0.963779	+	0.266703i	$0.0859341\pi$
$954$	0	0
$955$	0	0
$956$	−2.29899	−0.0743546
$957$	0	0
$958$	10.9137	0.352607
$959$	−12.0037	−0.387620
$960$	0	0
$961$	48.4888	1.56416
$962$	5.45309	0.175815
$963$	0	0
$964$	0.826205	0.0266103
$965$	0	0
$966$	0	0
$967$	55.7295	1.79214	0.896070	−	0.443912i	$-0.146410\pi$
$967$	55.7295	1.79214	0.896070	+	0.443912i	$0.146410\pi$
$968$	25.8182	0.829830
$969$	0	0
$970$	0	0
$971$	41.7036	1.33833	0.669165	−	0.743114i	$-0.266652\pi$
$971$	41.7036	1.33833	0.669165	+	0.743114i	$0.266652\pi$
$972$	0	0
$973$	−20.6973	−0.663526
$974$	−52.7522	−1.69029
$975$	0	0
$976$	34.8112	1.11428
$977$	18.0352	0.576997	0.288499	−	0.957480i	$-0.406844\pi$
$977$	18.0352	0.576997	0.288499	+	0.957480i	$0.406844\pi$
$978$	0	0
$979$	8.01919	0.256294
$980$	0	0
$981$	0	0
$982$	−1.64752	−0.0525744
$983$	−15.8266	−0.504789	−0.252395	−	0.967624i	$-0.581218\pi$
$983$	−15.8266	−0.504789	−0.252395	+	0.967624i	$0.581218\pi$
$984$	0	0
$985$	0	0
$986$	−78.0746	−2.48640
$987$	0	0
$988$	0.512724	0.0163119
$989$	−2.20235	−0.0700307
$990$	0	0
$991$	9.99354	0.317455	0.158728	−	0.987322i	$-0.449261\pi$
$991$	9.99354	0.317455	0.158728	+	0.987322i	$0.449261\pi$
$992$	−14.6717	−0.465828
$993$	0	0
$994$	−28.4793	−0.903310
$995$	0	0
$996$	0	0
$997$	20.1129	0.636981	0.318491	−	0.947926i	$-0.396824\pi$
$997$	20.1129	0.636981	0.318491	+	0.947926i	$0.396824\pi$
$998$	27.8673	0.882123
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8325.2.a.cf.1.4		5
3.2	odd	2		8325.2.a.by.1.2		5
5.4	even	2		1665.2.a.o.1.2	✓	5
15.14	odd	2		1665.2.a.r.1.4	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1665.2.a.o.1.2	✓	5	5.4	even	2
1665.2.a.r.1.4	yes	5	15.14	odd	2
8325.2.a.by.1.2		5	3.2	odd	2
8325.2.a.cf.1.4		5	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.51432 q^{2} +0.293161 q^{4} +1.43000 q^{7} -2.58470 q^{8} -1.00554 q^{11} -3.60102 q^{13} +2.16548 q^{14} -4.50038 q^{16} +6.24140 q^{17} -0.485681 q^{19} -1.52271 q^{22} +0.828979 q^{23} -5.45309 q^{26} +0.419221 q^{28} -8.26057 q^{29} +8.91565 q^{31} -1.64561 q^{32} +9.45147 q^{34} -1.00000 q^{37} -0.735476 q^{38} +0.301089 q^{41} -2.65670 q^{43} -0.294787 q^{44} +1.25534 q^{46} +6.02864 q^{47} -4.95510 q^{49} -1.05568 q^{52} -9.83973 q^{53} -3.69612 q^{56} -12.5091 q^{58} -0.849222 q^{59} -7.73517 q^{61} +13.5011 q^{62} +6.50877 q^{64} +0.424456 q^{67} +1.82974 q^{68} -13.1515 q^{71} +2.76126 q^{73} -1.51432 q^{74} -0.142383 q^{76} -1.43793 q^{77} +1.35514 q^{79} +0.455945 q^{82} +3.64439 q^{83} -4.02309 q^{86} +2.59903 q^{88} -7.97497 q^{89} -5.14946 q^{91} +0.243025 q^{92} +9.12928 q^{94} -10.2603 q^{97} -7.50360 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{2} + 4 q^{4} + 6 q^{8} - 12 q^{11} + 3 q^{13} - 10 q^{14} - 2 q^{16} + 4 q^{17} - 8 q^{19} - 10 q^{22} + 18 q^{23} - 2 q^{26} + 4 q^{28} - 13 q^{29} - 16 q^{31} + 14 q^{32} - 2 q^{34} - 5 q^{37}+ \cdots - 6 q^{98}+O(q^{100}) \) 5 * q + 2 * q^2 + 4 * q^4 + 6 * q^8 - 12 * q^11 + 3 * q^13 - 10 * q^14 - 2 * q^16 + 4 * q^17 - 8 * q^19 - 10 * q^22 + 18 * q^23 - 2 * q^26 + 4 * q^28 - 13 * q^29 - 16 * q^31 + 14 * q^32 - 2 * q^34 - 5 * q^37 + 10 * q^38 - 6 * q^41 - 5 * q^43 - 28 * q^44 - 6 * q^46 + 19 * q^47 - 3 * q^49 - 18 * q^52 + 5 * q^53 + 10 * q^56 - 38 * q^58 + q^59 - 2 * q^61 + 6 * q^62 + 20 * q^64 - 12 * q^67 - 18 * q^68 - 30 * q^71 - 12 * q^73 - 2 * q^74 + 2 * q^76 + 10 * q^77 - 12 * q^79 - 22 * q^82 + 19 * q^83 - 2 * q^86 - 20 * q^88 - 11 * q^89 - 30 * q^91 - 2 * q^92 + 34 * q^94 - 6 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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