LMFDB - Embedded newform 7497.2.a.ce.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7497,2,Mod(1,7497)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7497, 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("7497.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7497 = 3^{2} \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7497.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:	$59.8638463954$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.17314349056.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 12x^{6} + 40x^{4} - 32x^{2} + 4 $ x^8 - 12x^6 + 40x^4 - 32*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.391024$ of defining polynomial
Character	$\chi$	$=$	7497.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.391024 q^{2} -1.84710 q^{4} +1.78339 q^{5} -1.50431 q^{8} +O(q^{10})$	$q+0.391024 q^{2} -1.84710 q^{4} +1.78339 q^{5} -1.50431 q^{8} +0.697349 q^{10} -2.39447 q^{11} +0.824980 q^{13} +3.10598 q^{16} +1.00000 q^{17} +2.47917 q^{19} -3.29411 q^{20} -0.936294 q^{22} +6.42088 q^{23} -1.81950 q^{25} +0.322587 q^{26} -0.325841 q^{29} -9.36282 q^{31} +4.22313 q^{32} +0.391024 q^{34} +5.73648 q^{37} +0.969413 q^{38} -2.68277 q^{40} +3.20281 q^{41} +5.52750 q^{43} +4.42283 q^{44} +2.51071 q^{46} -9.68339 q^{47} -0.711468 q^{50} -1.52382 q^{52} -2.50798 q^{53} -4.27029 q^{55} -0.127411 q^{58} +2.90318 q^{59} +1.02319 q^{61} -3.66108 q^{62} -4.56062 q^{64} +1.47127 q^{65} +1.50773 q^{67} -1.84710 q^{68} -9.16070 q^{71} +0.0705787 q^{73} +2.24310 q^{74} -4.57927 q^{76} -0.730309 q^{79} +5.53919 q^{80} +1.25237 q^{82} +5.53919 q^{83} +1.78339 q^{85} +2.16138 q^{86} +3.60202 q^{88} +0.313430 q^{89} -11.8600 q^{92} -3.78643 q^{94} +4.42134 q^{95} +7.79755 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{4} + 12 q^{5}+O(q^{10}) $ 8 * q + 8 * q^4 + 12 * q^5	$ 8 q + 8 q^{4} + 12 q^{5} + 16 q^{16} + 8 q^{17} - 8 q^{20} - 28 q^{22} + 8 q^{25} - 4 q^{26} + 12 q^{37} + 60 q^{38} + 28 q^{41} - 24 q^{43} + 4 q^{46} + 20 q^{47} + 40 q^{58} + 12 q^{59} + 48 q^{62} + 48 q^{67} + 8 q^{68} + 60 q^{79} + 40 q^{80} + 40 q^{83} + 12 q^{85} - 8 q^{88} + 16 q^{89}+O(q^{100}) $ 8 * q + 8 * q^4 + 12 * q^5 + 16 * q^16 + 8 * q^17 - 8 * q^20 - 28 * q^22 + 8 * q^25 - 4 * q^26 + 12 * q^37 + 60 * q^38 + 28 * q^41 - 24 * q^43 + 4 * q^46 + 20 * q^47 + 40 * q^58 + 12 * q^59 + 48 * q^62 + 48 * q^67 + 8 * q^68 + 60 * q^79 + 40 * q^80 + 40 * q^83 + 12 * q^85 - 8 * q^88 + 16 * q^89

Display 100 coefficients 10 coefficients

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 $

$2$	0.391024	0.276495	0.138248	−	0.990398i	$-0.455853\pi$
$2$	0.391024	0.276495	0.138248	+	0.990398i	$0.455853\pi$
$3$	0	0
$3$	0	0
$4$	−1.84710	−0.923550
$5$	1.78339	0.797558	0.398779	−	0.917047i	$-0.369434\pi$
$5$	1.78339	0.797558	0.398779	+	0.917047i	$0.369434\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.50431	−0.531853
$9$	0	0
$10$	0.697349	0.220521
$11$	−2.39447	−0.721960	−0.360980	−	0.932574i	$-0.617558\pi$
$11$	−2.39447	−0.721960	−0.360980	+	0.932574i	$0.617558\pi$
$12$	0	0
$13$	0.824980	0.228808	0.114404	−	0.993434i	$-0.463504\pi$
$13$	0.824980	0.228808	0.114404	+	0.993434i	$0.463504\pi$
$14$	0	0
$15$	0	0
$16$	3.10598	0.776495
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	2.47917	0.568760	0.284380	−	0.958712i	$-0.408212\pi$
$19$	2.47917	0.568760	0.284380	+	0.958712i	$0.408212\pi$
$20$	−3.29411	−0.736585
$21$	0	0
$22$	−0.936294	−0.199619
$23$	6.42088	1.33885	0.669423	−	0.742882i	$-0.266541\pi$
$23$	6.42088	1.33885	0.669423	+	0.742882i	$0.266541\pi$
$24$	0	0
$25$	−1.81950	−0.363901
$26$	0.322587	0.0632645
$27$	0	0
$28$	0	0
$29$	−0.325841	−0.0605071	−0.0302535	−	0.999542i	$-0.509631\pi$
$29$	−0.325841	−0.0605071	−0.0302535	+	0.999542i	$0.509631\pi$
$30$	0	0
$31$	−9.36282	−1.68161	−0.840806	−	0.541337i	$-0.817918\pi$
$31$	−9.36282	−1.68161	−0.840806	+	0.541337i	$0.817918\pi$
$32$	4.22313	0.746550
$33$	0	0
$34$	0.391024	0.0670600
$35$	0	0
$36$	0	0
$37$	5.73648	0.943071	0.471536	−	0.881847i	$-0.343700\pi$
$37$	5.73648	0.943071	0.471536	+	0.881847i	$0.343700\pi$
$38$	0.969413	0.157260
$39$	0	0
$40$	−2.68277	−0.424184
$41$	3.20281	0.500194	0.250097	−	0.968221i	$-0.419537\pi$
$41$	3.20281	0.500194	0.250097	+	0.968221i	$0.419537\pi$
$42$	0	0
$43$	5.52750	0.842936	0.421468	−	0.906843i	$-0.361515\pi$
$43$	5.52750	0.842936	0.421468	+	0.906843i	$0.361515\pi$
$44$	4.42283	0.666766
$45$	0	0
$46$	2.51071	0.370185
$47$	−9.68339	−1.41247	−0.706234	−	0.707979i	$-0.749607\pi$
$47$	−9.68339	−1.41247	−0.706234	+	0.707979i	$0.749607\pi$
$48$	0	0
$49$	0	0
$50$	−0.711468	−0.100617
$51$	0	0
$52$	−1.52382	−0.211316
$53$	−2.50798	−0.344498	−0.172249	−	0.985053i	$-0.555103\pi$
$53$	−2.50798	−0.344498	−0.172249	+	0.985053i	$0.555103\pi$
$54$	0	0
$55$	−4.27029	−0.575805
$56$	0	0
$57$	0	0
$58$	−0.127411	−0.0167299
$59$	2.90318	0.377961	0.188981	−	0.981981i	$-0.439482\pi$
$59$	2.90318	0.377961	0.188981	+	0.981981i	$0.439482\pi$
$60$	0	0
$61$	1.02319	0.131006	0.0655030	−	0.997852i	$-0.479135\pi$
$61$	1.02319	0.131006	0.0655030	+	0.997852i	$0.479135\pi$
$62$	−3.66108	−0.464958
$63$	0	0
$64$	−4.56062	−0.570078
$65$	1.47127	0.182488
$66$	0	0
$67$	1.50773	0.184198	0.0920990	−	0.995750i	$-0.470642\pi$
$67$	1.50773	0.184198	0.0920990	+	0.995750i	$0.470642\pi$
$68$	−1.84710	−0.223994
$69$	0	0
$70$	0	0
$71$	−9.16070	−1.08718	−0.543588	−	0.839352i	$-0.682935\pi$
$71$	−9.16070	−1.08718	−0.543588	+	0.839352i	$0.682935\pi$
$72$	0	0
$73$	0.0705787	0.00826062	0.00413031	−	0.999991i	$-0.498685\pi$
$73$	0.0705787	0.00826062	0.00413031	+	0.999991i	$0.498685\pi$
$74$	2.24310	0.260755
$75$	0	0
$76$	−4.57927	−0.525279
$77$	0	0
$78$	0	0
$79$	−0.730309	−0.0821662	−0.0410831	−	0.999156i	$-0.513081\pi$
$79$	−0.730309	−0.0821662	−0.0410831	+	0.999156i	$0.513081\pi$
$80$	5.53919	0.619300
$81$	0	0
$82$	1.25237	0.138301
$83$	5.53919	0.608005	0.304003	−	0.952671i	$-0.401677\pi$
$83$	5.53919	0.608005	0.304003	+	0.952671i	$0.401677\pi$
$84$	0	0
$85$	1.78339	0.193436
$86$	2.16138	0.233068
$87$	0	0
$88$	3.60202	0.383976
$89$	0.313430	0.0332235	0.0166117	−	0.999862i	$-0.494712\pi$
$89$	0.313430	0.0332235	0.0166117	+	0.999862i	$0.494712\pi$
$90$	0	0
$91$	0	0
$92$	−11.8600	−1.23649
$93$	0	0
$94$	−3.78643	−0.390541
$95$	4.42134	0.453619
$96$	0	0
$97$	7.79755	0.791722	0.395861	−	0.918311i	$-0.370446\pi$
$97$	7.79755	0.791722	0.395861	+	0.918311i	$0.370446\pi$
$98$	0	0
$99$	0	0
$100$	3.36080	0.336080
$101$	10.0244	0.997467	0.498733	−	0.866755i	$-0.333799\pi$
$101$	10.0244	0.997467	0.498733	+	0.866755i	$0.333799\pi$
$102$	0	0
$103$	4.37989	0.431564	0.215782	−	0.976442i	$-0.430770\pi$
$103$	4.37989	0.431564	0.215782	+	0.976442i	$0.430770\pi$
$104$	−1.24102	−0.121692
$105$	0	0
$106$	−0.980680	−0.0952521
$107$	11.5552	1.11708	0.558540	−	0.829477i	$-0.311362\pi$
$107$	11.5552	1.11708	0.558540	+	0.829477i	$0.311362\pi$
$108$	0	0
$109$	1.27732	0.122345	0.0611726	−	0.998127i	$-0.480516\pi$
$109$	1.27732	0.122345	0.0611726	+	0.998127i	$0.480516\pi$
$110$	−1.66978	−0.159208
$111$	0	0
$112$	0	0
$113$	−16.1510	−1.51936	−0.759678	−	0.650299i	$-0.774643\pi$
$113$	−16.1510	−1.51936	−0.759678	+	0.650299i	$0.774643\pi$
$114$	0	0
$115$	11.4510	1.06781
$116$	0.601860	0.0558813
$117$	0	0
$118$	1.13521	0.104505
$119$	0	0
$120$	0	0
$121$	−5.26651	−0.478774
$122$	0.400091	0.0362226
$123$	0	0
$124$	17.2941	1.55305
$125$	−12.1619	−1.08779
$126$	0	0
$127$	22.3923	1.98699	0.993496	−	0.113864i	$-0.0363228\pi$
$127$	22.3923	1.98699	0.993496	+	0.113864i	$0.0363228\pi$
$128$	−10.2296	−0.904174
$129$	0	0
$130$	0.575300	0.0504571
$131$	13.9108	1.21539	0.607696	−	0.794169i	$-0.292094\pi$
$131$	13.9108	1.21539	0.607696	+	0.794169i	$0.292094\pi$
$132$	0	0
$133$	0	0
$134$	0.589556	0.0509299
$135$	0	0
$136$	−1.50431	−0.128993
$137$	−4.04833	−0.345873	−0.172936	−	0.984933i	$-0.555325\pi$
$137$	−4.04833	−0.345873	−0.172936	+	0.984933i	$0.555325\pi$
$138$	0	0
$139$	6.50948	0.552127	0.276063	−	0.961139i	$-0.410970\pi$
$139$	6.50948	0.552127	0.276063	+	0.961139i	$0.410970\pi$
$140$	0	0
$141$	0	0
$142$	−3.58205	−0.300599
$143$	−1.97539	−0.165190
$144$	0	0
$145$	−0.581102	−0.0482579
$146$	0.0275979	0.00228402
$147$	0	0
$148$	−10.5959	−0.870974
$149$	10.4695	0.857698	0.428849	−	0.903376i	$-0.358919\pi$
$149$	10.4695	0.857698	0.428849	+	0.903376i	$0.358919\pi$
$150$	0	0
$151$	1.60754	0.130820	0.0654098	−	0.997858i	$-0.479165\pi$
$151$	1.60754	0.130820	0.0654098	+	0.997858i	$0.479165\pi$
$152$	−3.72943	−0.302497
$153$	0	0
$154$	0	0
$155$	−16.6976	−1.34118
$156$	0	0
$157$	−7.35398	−0.586911	−0.293456	−	0.955973i	$-0.594805\pi$
$157$	−7.35398	−0.586911	−0.293456	+	0.955973i	$0.594805\pi$
$158$	−0.285568	−0.0227186
$159$	0	0
$160$	7.53150	0.595417
$161$	0	0
$162$	0	0
$163$	16.2532	1.27305	0.636523	−	0.771258i	$-0.280372\pi$
$163$	16.2532	1.27305	0.636523	+	0.771258i	$0.280372\pi$
$164$	−5.91591	−0.461955
$165$	0	0
$166$	2.16595	0.168111
$167$	9.59527	0.742504	0.371252	−	0.928532i	$-0.378929\pi$
$167$	9.59527	0.742504	0.371252	+	0.928532i	$0.378929\pi$
$168$	0	0
$169$	−12.3194	−0.947647
$170$	0.697349	0.0534843
$171$	0	0
$172$	−10.2099	−0.778494
$173$	−10.4990	−0.798226	−0.399113	−	0.916902i	$-0.630682\pi$
$173$	−10.4990	−0.798226	−0.399113	+	0.916902i	$0.630682\pi$
$174$	0	0
$175$	0	0
$176$	−7.43718	−0.560599
$177$	0	0
$178$	0.122558	0.00918614
$179$	17.8730	1.33589	0.667946	−	0.744210i	$-0.267174\pi$
$179$	17.8730	1.33589	0.667946	+	0.744210i	$0.267174\pi$
$180$	0	0
$181$	13.7724	1.02370	0.511849	−	0.859076i	$-0.328961\pi$
$181$	13.7724	1.02370	0.511849	+	0.859076i	$0.328961\pi$
$182$	0	0
$183$	0	0
$184$	−9.65897	−0.712069
$185$	10.2304	0.752154
$186$	0	0
$187$	−2.39447	−0.175101
$188$	17.8862	1.30449
$189$	0	0
$190$	1.72885	0.125424
$191$	18.9809	1.37341	0.686705	−	0.726936i	$-0.259057\pi$
$191$	18.9809	1.37341	0.686705	+	0.726936i	$0.259057\pi$
$192$	0	0
$193$	−16.4445	−1.18370	−0.591850	−	0.806048i	$-0.701602\pi$
$193$	−16.4445	−1.18370	−0.591850	+	0.806048i	$0.701602\pi$
$194$	3.04903	0.218907
$195$	0	0
$196$	0	0
$197$	6.09504	0.434253	0.217127	−	0.976143i	$-0.430332\pi$
$197$	6.09504	0.434253	0.217127	+	0.976143i	$0.430332\pi$
$198$	0	0
$199$	17.6917	1.25413	0.627064	−	0.778968i	$-0.284256\pi$
$199$	17.6917	1.25413	0.627064	+	0.778968i	$0.284256\pi$
$200$	2.73709	0.193542
$201$	0	0
$202$	3.91978	0.275795
$203$	0	0
$204$	0	0
$205$	5.71187	0.398934
$206$	1.71264	0.119325
$207$	0	0
$208$	2.56237	0.177669
$209$	−5.93629	−0.410622
$210$	0	0
$211$	3.70125	0.254804	0.127402	−	0.991851i	$-0.459336\pi$
$211$	3.70125	0.254804	0.127402	+	0.991851i	$0.459336\pi$
$212$	4.63250	0.318161
$213$	0	0
$214$	4.51835	0.308868
$215$	9.85772	0.672291
$216$	0	0
$217$	0	0
$218$	0.499463	0.0338279
$219$	0	0
$220$	7.88765	0.531785
$221$	0.824980	0.0554942
$222$	0	0
$223$	−19.8786	−1.33117	−0.665585	−	0.746322i	$-0.731818\pi$
$223$	−19.8786	−1.33117	−0.665585	+	0.746322i	$0.731818\pi$
$224$	0	0
$225$	0	0
$226$	−6.31541	−0.420095
$227$	15.3302	1.01750	0.508751	−	0.860914i	$-0.330107\pi$
$227$	15.3302	1.01750	0.508751	+	0.860914i	$0.330107\pi$
$228$	0	0
$229$	−10.5914	−0.699903	−0.349951	−	0.936768i	$-0.613802\pi$
$229$	−10.5914	−0.699903	−0.349951	+	0.936768i	$0.613802\pi$
$230$	4.47760	0.295244
$231$	0	0
$232$	0.490164	0.0321809
$233$	−11.0326	−0.722771	−0.361385	−	0.932417i	$-0.617696\pi$
$233$	−11.0326	−0.722771	−0.361385	+	0.932417i	$0.617696\pi$
$234$	0	0
$235$	−17.2693	−1.12653
$236$	−5.36246	−0.349066
$237$	0	0
$238$	0	0
$239$	−4.99978	−0.323409	−0.161704	−	0.986839i	$-0.551699\pi$
$239$	−4.99978	−0.323409	−0.161704	+	0.986839i	$0.551699\pi$
$240$	0	0
$241$	14.6013	0.940554	0.470277	−	0.882519i	$-0.344154\pi$
$241$	14.6013	0.940554	0.470277	+	0.882519i	$0.344154\pi$
$242$	−2.05933	−0.132379
$243$	0	0
$244$	−1.88993	−0.120991
$245$	0	0
$246$	0	0
$247$	2.04526	0.130137
$248$	14.0846	0.894370
$249$	0	0
$250$	−4.75558	−0.300769
$251$	27.8108	1.75540	0.877701	−	0.479208i	$-0.159076\pi$
$251$	27.8108	1.75540	0.877701	+	0.479208i	$0.159076\pi$
$252$	0	0
$253$	−15.3746	−0.966593
$254$	8.75591	0.549394
$255$	0	0
$256$	5.12124	0.320078
$257$	20.7172	1.29230	0.646150	−	0.763210i	$-0.276378\pi$
$257$	20.7172	1.29230	0.646150	+	0.763210i	$0.276378\pi$
$258$	0	0
$259$	0	0
$260$	−2.71758	−0.168537
$261$	0	0
$262$	5.43945	0.336051
$263$	11.8435	0.730300	0.365150	−	0.930949i	$-0.381018\pi$
$263$	11.8435	0.730300	0.365150	+	0.930949i	$0.381018\pi$
$264$	0	0
$265$	−4.47272	−0.274757
$266$	0	0
$267$	0	0
$268$	−2.78492	−0.170116
$269$	30.1792	1.84006	0.920030	−	0.391847i	$-0.128164\pi$
$269$	30.1792	1.84006	0.920030	+	0.391847i	$0.128164\pi$
$270$	0	0
$271$	−1.48089	−0.0899576	−0.0449788	−	0.998988i	$-0.514322\pi$
$271$	−1.48089	−0.0899576	−0.0449788	+	0.998988i	$0.514322\pi$
$272$	3.10598	0.188328
$273$	0	0
$274$	−1.58299	−0.0956322
$275$	4.35675	0.262722
$276$	0	0
$277$	17.6156	1.05842	0.529210	−	0.848491i	$-0.322488\pi$
$277$	17.6156	1.05842	0.529210	+	0.848491i	$0.322488\pi$
$278$	2.54536	0.152661
$279$	0	0
$280$	0	0
$281$	−21.6529	−1.29170	−0.645851	−	0.763463i	$-0.723497\pi$
$281$	−21.6529	−1.29170	−0.645851	+	0.763463i	$0.723497\pi$
$282$	0	0
$283$	−9.07601	−0.539513	−0.269756	−	0.962929i	$-0.586943\pi$
$283$	−9.07601	−0.539513	−0.269756	+	0.962929i	$0.586943\pi$
$284$	16.9207	1.00406
$285$	0	0
$286$	−0.772424	−0.0456744
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	−0.227225	−0.0133431
$291$	0	0
$292$	−0.130366	−0.00762909
$293$	20.2684	1.18409	0.592047	−	0.805903i	$-0.298320\pi$
$293$	20.2684	1.18409	0.592047	+	0.805903i	$0.298320\pi$
$294$	0	0
$295$	5.17751	0.301446
$296$	−8.62942	−0.501575
$297$	0	0
$298$	4.09384	0.237150
$299$	5.29710	0.306339
$300$	0	0
$301$	0	0
$302$	0.628586	0.0361710
$303$	0	0
$304$	7.70025	0.441640
$305$	1.82475	0.104485
$306$	0	0
$307$	−8.63815	−0.493005	−0.246503	−	0.969142i	$-0.579281\pi$
$307$	−8.63815	−0.493005	−0.246503	+	0.969142i	$0.579281\pi$
$308$	0	0
$309$	0	0
$310$	−6.52916	−0.370831
$311$	22.4944	1.27554	0.637770	−	0.770227i	$-0.279857\pi$
$311$	22.4944	1.27554	0.637770	+	0.770227i	$0.279857\pi$
$312$	0	0
$313$	−13.4893	−0.762460	−0.381230	−	0.924480i	$-0.624499\pi$
$313$	−13.4893	−0.762460	−0.381230	+	0.924480i	$0.624499\pi$
$314$	−2.87558	−0.162278
$315$	0	0
$316$	1.34895	0.0758846
$317$	11.6200	0.652646	0.326323	−	0.945258i	$-0.394190\pi$
$317$	11.6200	0.652646	0.326323	+	0.945258i	$0.394190\pi$
$318$	0	0
$319$	0.780216	0.0436837
$320$	−8.13339	−0.454670
$321$	0	0
$322$	0	0
$323$	2.47917	0.137945
$324$	0	0
$325$	−1.50105	−0.0832635
$326$	6.35537	0.351992
$327$	0	0
$328$	−4.81800	−0.266030
$329$	0	0
$330$	0	0
$331$	−12.8624	−0.706979	−0.353490	−	0.935438i	$-0.615005\pi$
$331$	−12.8624	−0.706979	−0.353490	+	0.935438i	$0.615005\pi$
$332$	−10.2314	−0.561523
$333$	0	0
$334$	3.75198	0.205299
$335$	2.68887	0.146909
$336$	0	0
$337$	−2.40162	−0.130824	−0.0654122	−	0.997858i	$-0.520836\pi$
$337$	−2.40162	−0.130824	−0.0654122	+	0.997858i	$0.520836\pi$
$338$	−4.81718	−0.262020
$339$	0	0
$340$	−3.29411	−0.178648
$341$	22.4190	1.21406
$342$	0	0
$343$	0	0
$344$	−8.31506	−0.448318
$345$	0	0
$346$	−4.10537	−0.220706
$347$	−25.5379	−1.37094	−0.685472	−	0.728099i	$-0.740404\pi$
$347$	−25.5379	−1.37094	−0.685472	+	0.728099i	$0.740404\pi$
$348$	0	0
$349$	−11.5107	−0.616156	−0.308078	−	0.951361i	$-0.599686\pi$
$349$	−11.5107	−0.616156	−0.308078	+	0.951361i	$0.599686\pi$
$350$	0	0
$351$	0	0
$352$	−10.1121	−0.538979
$353$	11.1151	0.591599	0.295800	−	0.955250i	$-0.404414\pi$
$353$	11.1151	0.591599	0.295800	+	0.955250i	$0.404414\pi$
$354$	0	0
$355$	−16.3372	−0.867086
$356$	−0.578936	−0.0306835
$357$	0	0
$358$	6.98877	0.369368
$359$	−29.7631	−1.57083	−0.785417	−	0.618966i	$-0.787552\pi$
$359$	−29.7631	−1.57083	−0.785417	+	0.618966i	$0.787552\pi$
$360$	0	0
$361$	−12.8537	−0.676512
$362$	5.38535	0.283048
$363$	0	0
$364$	0	0
$365$	0.125870	0.00658832
$366$	0	0
$367$	−1.90853	−0.0996247	−0.0498123	−	0.998759i	$-0.515862\pi$
$367$	−1.90853	−0.0996247	−0.0498123	+	0.998759i	$0.515862\pi$
$368$	19.9431	1.03961
$369$	0	0
$370$	4.00033	0.207967
$371$	0	0
$372$	0	0
$373$	15.3923	0.796982	0.398491	−	0.917172i	$-0.369534\pi$
$373$	15.3923	0.796982	0.398491	+	0.917172i	$0.369534\pi$
$374$	−0.936294	−0.0484146
$375$	0	0
$376$	14.5668	0.751225
$377$	−0.268812	−0.0138445
$378$	0	0
$379$	−3.91721	−0.201213	−0.100607	−	0.994926i	$-0.532078\pi$
$379$	−3.91721	−0.201213	−0.100607	+	0.994926i	$0.532078\pi$
$380$	−8.16665	−0.418940
$381$	0	0
$382$	7.42198	0.379741
$383$	14.9720	0.765032	0.382516	−	0.923949i	$-0.375058\pi$
$383$	14.9720	0.765032	0.382516	+	0.923949i	$0.375058\pi$
$384$	0	0
$385$	0	0
$386$	−6.43018	−0.327287
$387$	0	0
$388$	−14.4029	−0.731195
$389$	20.8477	1.05702	0.528511	−	0.848926i	$-0.322750\pi$
$389$	20.8477	1.05702	0.528511	+	0.848926i	$0.322750\pi$
$390$	0	0
$391$	6.42088	0.324718
$392$	0	0
$393$	0	0
$394$	2.38330	0.120069
$395$	−1.30243	−0.0655323
$396$	0	0
$397$	−5.21243	−0.261604	−0.130802	−	0.991408i	$-0.541755\pi$
$397$	−5.21243	−0.261604	−0.130802	+	0.991408i	$0.541755\pi$
$398$	6.91785	0.346761
$399$	0	0
$400$	−5.65134	−0.282567
$401$	0.841168	0.0420059	0.0210030	−	0.999779i	$-0.493314\pi$
$401$	0.841168	0.0420059	0.0210030	+	0.999779i	$0.493314\pi$
$402$	0	0
$403$	−7.72414	−0.384767
$404$	−18.5161	−0.921211
$405$	0	0
$406$	0	0
$407$	−13.7358	−0.680860
$408$	0	0
$409$	24.2234	1.19777	0.598885	−	0.800835i	$-0.295611\pi$
$409$	24.2234	1.19777	0.598885	+	0.800835i	$0.295611\pi$
$410$	2.23348	0.110303
$411$	0	0
$412$	−8.09011	−0.398571
$413$	0	0
$414$	0	0
$415$	9.87857	0.484920
$416$	3.48400	0.170817
$417$	0	0
$418$	−2.32123	−0.113535
$419$	2.89448	0.141404	0.0707022	−	0.997497i	$-0.477476\pi$
$419$	2.89448	0.141404	0.0707022	+	0.997497i	$0.477476\pi$
$420$	0	0
$421$	0.568900	0.0277265	0.0138632	−	0.999904i	$-0.495587\pi$
$421$	0.568900	0.0277265	0.0138632	+	0.999904i	$0.495587\pi$
$422$	1.44728	0.0704523
$423$	0	0
$424$	3.77278	0.183222
$425$	−1.81950	−0.0882588
$426$	0	0
$427$	0	0
$428$	−21.3436	−1.03168
$429$	0	0
$430$	3.85460	0.185885
$431$	19.4543	0.937080	0.468540	−	0.883442i	$-0.344780\pi$
$431$	19.4543	0.937080	0.468540	+	0.883442i	$0.344780\pi$
$432$	0	0
$433$	−18.5292	−0.890458	−0.445229	−	0.895417i	$-0.646878\pi$
$433$	−18.5292	−0.890458	−0.445229	+	0.895417i	$0.646878\pi$
$434$	0	0
$435$	0	0
$436$	−2.35934	−0.112992
$437$	15.9184	0.761482
$438$	0	0
$439$	10.6119	0.506477	0.253238	−	0.967404i	$-0.418504\pi$
$439$	10.6119	0.506477	0.253238	+	0.967404i	$0.418504\pi$
$440$	6.42382	0.306244
$441$	0	0
$442$	0.322587	0.0153439
$443$	−2.45035	−0.116420	−0.0582099	−	0.998304i	$-0.518539\pi$
$443$	−2.45035	−0.116420	−0.0582099	+	0.998304i	$0.518539\pi$
$444$	0	0
$445$	0.558969	0.0264977
$446$	−7.77301	−0.368063
$447$	0	0
$448$	0	0
$449$	−9.73433	−0.459391	−0.229696	−	0.973263i	$-0.573773\pi$
$449$	−9.73433	−0.459391	−0.229696	+	0.973263i	$0.573773\pi$
$450$	0	0
$451$	−7.66902	−0.361120
$452$	29.8325	1.40320
$453$	0	0
$454$	5.99448	0.281335
$455$	0	0
$456$	0	0
$457$	−31.6570	−1.48085	−0.740427	−	0.672137i	$-0.765376\pi$
$457$	−31.6570	−1.48085	−0.740427	+	0.672137i	$0.765376\pi$
$458$	−4.14151	−0.193520
$459$	0	0
$460$	−21.1511	−0.986174
$461$	39.7798	1.85273	0.926365	−	0.376627i	$-0.122916\pi$
$461$	39.7798	1.85273	0.926365	+	0.376627i	$0.122916\pi$
$462$	0	0
$463$	−6.42006	−0.298365	−0.149183	−	0.988810i	$-0.547664\pi$
$463$	−6.42006	−0.298365	−0.149183	+	0.988810i	$0.547664\pi$
$464$	−1.01205	−0.0469835
$465$	0	0
$466$	−4.31401	−0.199843
$467$	−4.81379	−0.222756	−0.111378	−	0.993778i	$-0.535526\pi$
$467$	−4.81379	−0.222756	−0.111378	+	0.993778i	$0.535526\pi$
$468$	0	0
$469$	0	0
$470$	−6.75271	−0.311479
$471$	0	0
$472$	−4.36727	−0.201020
$473$	−13.2354	−0.608566
$474$	0	0
$475$	−4.51085	−0.206972
$476$	0	0
$477$	0	0
$478$	−1.95503	−0.0894210
$479$	−0.688097	−0.0314400	−0.0157200	−	0.999876i	$-0.505004\pi$
$479$	−0.688097	−0.0314400	−0.0157200	+	0.999876i	$0.505004\pi$
$480$	0	0
$481$	4.73248	0.215783
$482$	5.70946	0.260059
$483$	0	0
$484$	9.72778	0.442172
$485$	13.9061	0.631444
$486$	0	0
$487$	−33.1222	−1.50091	−0.750454	−	0.660923i	$-0.770165\pi$
$487$	−33.1222	−1.50091	−0.750454	+	0.660923i	$0.770165\pi$
$488$	−1.53919	−0.0696759
$489$	0	0
$490$	0	0
$491$	8.46154	0.381864	0.190932	−	0.981603i	$-0.438849\pi$
$491$	8.46154	0.381864	0.190932	+	0.981603i	$0.438849\pi$
$492$	0	0
$493$	−0.325841	−0.0146751
$494$	0.799747	0.0359823
$495$	0	0
$496$	−29.0807	−1.30576
$497$	0	0
$498$	0	0
$499$	25.4753	1.14043	0.570215	−	0.821495i	$-0.306860\pi$
$499$	25.4753	1.14043	0.570215	+	0.821495i	$0.306860\pi$
$500$	22.4642	1.00463
$501$	0	0
$502$	10.8747	0.485361
$503$	33.0872	1.47529	0.737644	−	0.675190i	$-0.235938\pi$
$503$	33.0872	1.47529	0.737644	+	0.675190i	$0.235938\pi$
$504$	0	0
$505$	17.8775	0.795538
$506$	−6.01183	−0.267259
$507$	0	0
$508$	−41.3608	−1.83509
$509$	−10.6821	−0.473474	−0.236737	−	0.971574i	$-0.576078\pi$
$509$	−10.6821	−0.473474	−0.236737	+	0.971574i	$0.576078\pi$
$510$	0	0
$511$	0	0
$512$	22.4617	0.992674
$513$	0	0
$514$	8.10090	0.357315
$515$	7.81108	0.344197
$516$	0	0
$517$	23.1866	1.01975
$518$	0	0
$519$	0	0
$520$	−2.21324	−0.0970568
$521$	−0.0806199	−0.00353202	−0.00176601	−	0.999998i	$-0.500562\pi$
$521$	−0.0806199	−0.00353202	−0.00176601	+	0.999998i	$0.500562\pi$
$522$	0	0
$523$	24.9210	1.08972	0.544859	−	0.838528i	$-0.316583\pi$
$523$	24.9210	1.08972	0.544859	+	0.838528i	$0.316583\pi$
$524$	−25.6947	−1.12248
$525$	0	0
$526$	4.63108	0.201925
$527$	−9.36282	−0.407851
$528$	0	0
$529$	18.2277	0.792508
$530$	−1.74894	−0.0759691
$531$	0	0
$532$	0	0
$533$	2.64225	0.114449
$534$	0	0
$535$	20.6074	0.890937
$536$	−2.26808	−0.0979662
$537$	0	0
$538$	11.8008	0.508768
$539$	0	0
$540$	0	0
$541$	1.65299	0.0710678	0.0355339	−	0.999368i	$-0.488687\pi$
$541$	1.65299	0.0710678	0.0355339	+	0.999368i	$0.488687\pi$
$542$	−0.579062	−0.0248729
$543$	0	0
$544$	4.22313	0.181065
$545$	2.27797	0.0975775
$546$	0	0
$547$	3.78099	0.161663	0.0808317	−	0.996728i	$-0.474242\pi$
$547$	3.78099	0.161663	0.0808317	+	0.996728i	$0.474242\pi$
$548$	7.47768	0.319431
$549$	0	0
$550$	1.70359	0.0726413
$551$	−0.807814	−0.0344140
$552$	0	0
$553$	0	0
$554$	6.88813	0.292648
$555$	0	0
$556$	−12.0237	−0.509917
$557$	13.9866	0.592631	0.296315	−	0.955090i	$-0.404242\pi$
$557$	13.9866	0.592631	0.296315	+	0.955090i	$0.404242\pi$
$558$	0	0
$559$	4.56008	0.192871
$560$	0	0
$561$	0	0
$562$	−8.46679	−0.357150
$563$	26.9628	1.13635	0.568174	−	0.822909i	$-0.307650\pi$
$563$	26.9628	1.13635	0.568174	+	0.822909i	$0.307650\pi$
$564$	0	0
$565$	−28.8036	−1.21178
$566$	−3.54893	−0.149173
$567$	0	0
$568$	13.7805	0.578217
$569$	21.7595	0.912206	0.456103	−	0.889927i	$-0.349245\pi$
$569$	21.7595	0.912206	0.456103	+	0.889927i	$0.349245\pi$
$570$	0	0
$571$	17.5694	0.735255	0.367627	−	0.929973i	$-0.380170\pi$
$571$	17.5694	0.735255	0.367627	+	0.929973i	$0.380170\pi$
$572$	3.64875	0.152562
$573$	0	0
$574$	0	0
$575$	−11.6828	−0.487207
$576$	0	0
$577$	−34.3313	−1.42923	−0.714615	−	0.699518i	$-0.753398\pi$
$577$	−34.3313	−1.42923	−0.714615	+	0.699518i	$0.753398\pi$
$578$	0.391024	0.0162644
$579$	0	0
$580$	1.07335	0.0445686
$581$	0	0
$582$	0	0
$583$	6.00529	0.248714
$584$	−0.106172	−0.00439343
$585$	0	0
$586$	7.92543	0.327397
$587$	7.67741	0.316881	0.158440	−	0.987369i	$-0.449353\pi$
$587$	7.67741	0.316881	0.158440	+	0.987369i	$0.449353\pi$
$588$	0	0
$589$	−23.2120	−0.956434
$590$	2.02453	0.0833485
$591$	0	0
$592$	17.8174	0.732290
$593$	28.3378	1.16370	0.581848	−	0.813298i	$-0.302330\pi$
$593$	28.3378	1.16370	0.581848	+	0.813298i	$0.302330\pi$
$594$	0	0
$595$	0	0
$596$	−19.3383	−0.792127
$597$	0	0
$598$	2.07129	0.0847014
$599$	−31.0115	−1.26710	−0.633549	−	0.773703i	$-0.718402\pi$
$599$	−31.0115	−1.26710	−0.633549	+	0.773703i	$0.718402\pi$
$600$	0	0
$601$	32.1942	1.31323	0.656614	−	0.754227i	$-0.271988\pi$
$601$	32.1942	1.31323	0.656614	+	0.754227i	$0.271988\pi$
$602$	0	0
$603$	0	0
$604$	−2.96929	−0.120819
$605$	−9.39227	−0.381850
$606$	0	0
$607$	−17.0667	−0.692716	−0.346358	−	0.938102i	$-0.612582\pi$
$607$	−17.0667	−0.692716	−0.346358	+	0.938102i	$0.612582\pi$
$608$	10.4698	0.424608
$609$	0	0
$610$	0.713521	0.0288896
$611$	−7.98861	−0.323184
$612$	0	0
$613$	17.8907	0.722599	0.361300	−	0.932450i	$-0.382333\pi$
$613$	17.8907	0.722599	0.361300	+	0.932450i	$0.382333\pi$
$614$	−3.37772	−0.136314
$615$	0	0
$616$	0	0
$617$	3.24625	0.130689	0.0653446	−	0.997863i	$-0.479185\pi$
$617$	3.24625	0.130689	0.0653446	+	0.997863i	$0.479185\pi$
$618$	0	0
$619$	−35.6980	−1.43482	−0.717412	−	0.696649i	$-0.754674\pi$
$619$	−35.6980	−1.43482	−0.717412	+	0.696649i	$0.754674\pi$
$620$	30.8421	1.23865
$621$	0	0
$622$	8.79583	0.352681
$623$	0	0
$624$	0	0
$625$	−12.5919	−0.503676
$626$	−5.27463	−0.210817
$627$	0	0
$628$	13.5835	0.542042
$629$	5.73648	0.228728
$630$	0	0
$631$	3.70037	0.147309	0.0736547	−	0.997284i	$-0.476534\pi$
$631$	3.70037	0.147309	0.0736547	+	0.997284i	$0.476534\pi$
$632$	1.09861	0.0437003
$633$	0	0
$634$	4.54371	0.180454
$635$	39.9343	1.58474
$636$	0	0
$637$	0	0
$638$	0.305083	0.0120783
$639$	0	0
$640$	−18.2433	−0.721132
$641$	−13.7758	−0.544110	−0.272055	−	0.962282i	$-0.587703\pi$
$641$	−13.7758	−0.544110	−0.272055	+	0.962282i	$0.587703\pi$
$642$	0	0
$643$	19.3825	0.764370	0.382185	−	0.924086i	$-0.375172\pi$
$643$	19.3825	0.764370	0.382185	+	0.924086i	$0.375172\pi$
$644$	0	0
$645$	0	0
$646$	0.969413	0.0381410
$647$	11.4980	0.452032	0.226016	−	0.974124i	$-0.427430\pi$
$647$	11.4980	0.452032	0.226016	+	0.974124i	$0.427430\pi$
$648$	0	0
$649$	−6.95157	−0.272873
$650$	−0.586947	−0.0230220
$651$	0	0
$652$	−30.0212	−1.17572
$653$	23.0275	0.901135	0.450568	−	0.892742i	$-0.351222\pi$
$653$	23.0275	0.901135	0.450568	+	0.892742i	$0.351222\pi$
$654$	0	0
$655$	24.8085	0.969347
$656$	9.94786	0.388399
$657$	0	0
$658$	0	0
$659$	13.8554	0.539730	0.269865	−	0.962898i	$-0.413021\pi$
$659$	13.8554	0.539730	0.269865	+	0.962898i	$0.413021\pi$
$660$	0	0
$661$	−3.02000	−0.117464	−0.0587322	−	0.998274i	$-0.518706\pi$
$661$	−3.02000	−0.117464	−0.0587322	+	0.998274i	$0.518706\pi$
$662$	−5.02949	−0.195477
$663$	0	0
$664$	−8.33264	−0.323369
$665$	0	0
$666$	0	0
$667$	−2.09218	−0.0810096
$668$	−17.7234	−0.685740
$669$	0	0
$670$	1.05141	0.0406196
$671$	−2.45000	−0.0945811
$672$	0	0
$673$	37.6955	1.45305	0.726527	−	0.687138i	$-0.241133\pi$
$673$	37.6955	1.45305	0.726527	+	0.687138i	$0.241133\pi$
$674$	−0.939089	−0.0361724
$675$	0	0
$676$	22.7552	0.875199
$677$	−18.9072	−0.726663	−0.363331	−	0.931660i	$-0.618361\pi$
$677$	−18.9072	−0.726663	−0.363331	+	0.931660i	$0.618361\pi$
$678$	0	0
$679$	0	0
$680$	−2.68277	−0.102880
$681$	0	0
$682$	8.76635	0.335681
$683$	−43.9211	−1.68060	−0.840298	−	0.542125i	$-0.817620\pi$
$683$	−43.9211	−1.68060	−0.840298	+	0.542125i	$0.817620\pi$
$684$	0	0
$685$	−7.21978	−0.275854
$686$	0	0
$687$	0	0
$688$	17.1683	0.654536
$689$	−2.06904	−0.0788240
$690$	0	0
$691$	−3.48739	−0.132667	−0.0663334	−	0.997798i	$-0.521130\pi$
$691$	−3.48739	−0.132667	−0.0663334	+	0.997798i	$0.521130\pi$
$692$	19.3928	0.737202
$693$	0	0
$694$	−9.98591	−0.379060
$695$	11.6090	0.440353
$696$	0	0
$697$	3.20281	0.121315
$698$	−4.50097	−0.170364
$699$	0	0
$700$	0	0
$701$	−40.8246	−1.54192	−0.770961	−	0.636882i	$-0.780224\pi$
$701$	−40.8246	−1.54192	−0.770961	+	0.636882i	$0.780224\pi$
$702$	0	0
$703$	14.2217	0.536381
$704$	10.9203	0.411573
$705$	0	0
$706$	4.34628	0.163574
$707$	0	0
$708$	0	0
$709$	−13.5408	−0.508537	−0.254269	−	0.967134i	$-0.581835\pi$
$709$	−13.5408	−0.508537	−0.254269	+	0.967134i	$0.581835\pi$
$710$	−6.38821	−0.239745
$711$	0	0
$712$	−0.471494	−0.0176700
$713$	−60.1175	−2.25142
$714$	0	0
$715$	−3.52290	−0.131749
$716$	−33.0132	−1.23376
$717$	0	0
$718$	−11.6381	−0.434329
$719$	1.76508	0.0658264	0.0329132	−	0.999458i	$-0.489522\pi$
$719$	1.76508	0.0658264	0.0329132	+	0.999458i	$0.489522\pi$
$720$	0	0
$721$	0	0
$722$	−5.02611	−0.187052
$723$	0	0
$724$	−25.4391	−0.945436
$725$	0.592868	0.0220186
$726$	0	0
$727$	−23.0693	−0.855591	−0.427796	−	0.903876i	$-0.640710\pi$
$727$	−23.0693	−0.855591	−0.427796	+	0.903876i	$0.640710\pi$
$728$	0	0
$729$	0	0
$730$	0.0492180	0.00182164
$731$	5.52750	0.204442
$732$	0	0
$733$	−4.41984	−0.163251	−0.0816253	−	0.996663i	$-0.526011\pi$
$733$	−4.41984	−0.163251	−0.0816253	+	0.996663i	$0.526011\pi$
$734$	−0.746282	−0.0275458
$735$	0	0
$736$	27.1162	0.999516
$737$	−3.61020	−0.132984
$738$	0	0
$739$	43.0383	1.58319	0.791594	−	0.611048i	$-0.209252\pi$
$739$	43.0383	1.58319	0.791594	+	0.611048i	$0.209252\pi$
$740$	−18.8966	−0.694652
$741$	0	0
$742$	0	0
$743$	24.3270	0.892472	0.446236	−	0.894915i	$-0.352764\pi$
$743$	24.3270	0.892472	0.446236	+	0.894915i	$0.352764\pi$
$744$	0	0
$745$	18.6713	0.684064
$746$	6.01874	0.220362
$747$	0	0
$748$	4.42283	0.161715
$749$	0	0
$750$	0	0
$751$	−35.8791	−1.30925	−0.654624	−	0.755954i	$-0.727173\pi$
$751$	−35.8791	−1.30925	−0.654624	+	0.755954i	$0.727173\pi$
$752$	−30.0764	−1.09677
$753$	0	0
$754$	−0.105112	−0.00382795
$755$	2.86688	0.104336
$756$	0	0
$757$	−29.0527	−1.05594	−0.527969	−	0.849263i	$-0.677046\pi$
$757$	−29.0527	−1.05594	−0.527969	+	0.849263i	$0.677046\pi$
$758$	−1.53172	−0.0556346
$759$	0	0
$760$	−6.65105	−0.241259
$761$	42.4544	1.53897	0.769486	−	0.638663i	$-0.220512\pi$
$761$	42.4544	1.53897	0.769486	+	0.638663i	$0.220512\pi$
$762$	0	0
$763$	0	0
$764$	−35.0596	−1.26841
$765$	0	0
$766$	5.85440	0.211528
$767$	2.39506	0.0864807
$768$	0	0
$769$	52.1169	1.87938	0.939692	−	0.342021i	$-0.111111\pi$
$769$	52.1169	1.87938	0.939692	+	0.342021i	$0.111111\pi$
$770$	0	0
$771$	0	0
$772$	30.3746	1.09321
$773$	13.4872	0.485100	0.242550	−	0.970139i	$-0.422016\pi$
$773$	13.4872	0.485100	0.242550	+	0.970139i	$0.422016\pi$
$774$	0	0
$775$	17.0357	0.611939
$776$	−11.7299	−0.421079
$777$	0	0
$778$	8.15196	0.292262
$779$	7.94029	0.284491
$780$	0	0
$781$	21.9350	0.784897
$782$	2.51071	0.0897830
$783$	0	0
$784$	0	0
$785$	−13.1150	−0.468096
$786$	0	0
$787$	−5.06066	−0.180393	−0.0901965	−	0.995924i	$-0.528750\pi$
$787$	−5.06066	−0.180393	−0.0901965	+	0.995924i	$0.528750\pi$
$788$	−11.2581	−0.401055
$789$	0	0
$790$	−0.509280	−0.0181194
$791$	0	0
$792$	0	0
$793$	0.844112	0.0299753
$794$	−2.03818	−0.0723324
$795$	0	0
$796$	−32.6783	−1.15825
$797$	−16.1134	−0.570767	−0.285383	−	0.958413i	$-0.592121\pi$
$797$	−16.1134	−0.570767	−0.285383	+	0.958413i	$0.592121\pi$
$798$	0	0
$799$	−9.68339	−0.342574
$800$	−7.68399	−0.271670
$801$	0	0
$802$	0.328916	0.0116144
$803$	−0.168999	−0.00596383
$804$	0	0
$805$	0	0
$806$	−3.02032	−0.106386
$807$	0	0
$808$	−15.0798	−0.530506
$809$	−10.6143	−0.373180	−0.186590	−	0.982438i	$-0.559744\pi$
$809$	−10.6143	−0.373180	−0.186590	+	0.982438i	$0.559744\pi$
$810$	0	0
$811$	−30.5293	−1.07203	−0.536014	−	0.844209i	$-0.680070\pi$
$811$	−30.5293	−1.07203	−0.536014	+	0.844209i	$0.680070\pi$
$812$	0	0
$813$	0	0
$814$	−5.37103	−0.188255
$815$	28.9858	1.01533
$816$	0	0
$817$	13.7036	0.479429
$818$	9.47191	0.331178
$819$	0	0
$820$	−10.5504	−0.368436
$821$	55.9025	1.95101	0.975505	−	0.219978i	$-0.0705986\pi$
$821$	55.9025	1.95101	0.975505	+	0.219978i	$0.0705986\pi$
$822$	0	0
$823$	12.7906	0.445851	0.222926	−	0.974835i	$-0.428439\pi$
$823$	12.7906	0.445851	0.222926	+	0.974835i	$0.428439\pi$
$824$	−6.58871	−0.229528
$825$	0	0
$826$	0	0
$827$	27.8611	0.968826	0.484413	−	0.874840i	$-0.339033\pi$
$827$	27.8611	0.968826	0.484413	+	0.874840i	$0.339033\pi$
$828$	0	0
$829$	−19.1069	−0.663609	−0.331804	−	0.943348i	$-0.607657\pi$
$829$	−19.1069	−0.663609	−0.331804	+	0.943348i	$0.607657\pi$
$830$	3.86275	0.134078
$831$	0	0
$832$	−3.76242	−0.130439
$833$	0	0
$834$	0	0
$835$	17.1121	0.592190
$836$	10.9649	0.379230
$837$	0	0
$838$	1.13181	0.0390977
$839$	−20.9072	−0.721797	−0.360898	−	0.932605i	$-0.617530\pi$
$839$	−20.9072	−0.721797	−0.360898	+	0.932605i	$0.617530\pi$
$840$	0	0
$841$	−28.8938	−0.996339
$842$	0.222453	0.00766624
$843$	0	0
$844$	−6.83658	−0.235325
$845$	−21.9704	−0.755804
$846$	0	0
$847$	0	0
$848$	−7.78975	−0.267501
$849$	0	0
$850$	−0.711468	−0.0244032
$851$	36.8332	1.26263
$852$	0	0
$853$	−50.1615	−1.71750	−0.858748	−	0.512397i	$-0.828757\pi$
$853$	−50.1615	−1.71750	−0.858748	+	0.512397i	$0.828757\pi$
$854$	0	0
$855$	0	0
$856$	−17.3825	−0.594123
$857$	−20.9552	−0.715816	−0.357908	−	0.933757i	$-0.616510\pi$
$857$	−20.9552	−0.715816	−0.357908	+	0.933757i	$0.616510\pi$
$858$	0	0
$859$	40.3665	1.37729	0.688643	−	0.725100i	$-0.258207\pi$
$859$	40.3665	1.37729	0.688643	+	0.725100i	$0.258207\pi$
$860$	−18.2082	−0.620894
$861$	0	0
$862$	7.60708	0.259098
$863$	−16.3915	−0.557975	−0.278987	−	0.960295i	$-0.589999\pi$
$863$	−16.3915	−0.557975	−0.278987	+	0.960295i	$0.589999\pi$
$864$	0	0
$865$	−18.7239	−0.636632
$866$	−7.24537	−0.246208
$867$	0	0
$868$	0	0
$869$	1.74870	0.0593207
$870$	0	0
$871$	1.24384	0.0421461
$872$	−1.92148	−0.0650697
$873$	0	0
$874$	6.22448	0.210546
$875$	0	0
$876$	0	0
$877$	34.4169	1.16218	0.581088	−	0.813841i	$-0.302627\pi$
$877$	34.4169	1.16218	0.581088	+	0.813841i	$0.302627\pi$
$878$	4.14949	0.140038
$879$	0	0
$880$	−13.2634	−0.447110
$881$	−9.28729	−0.312897	−0.156448	−	0.987686i	$-0.550005\pi$
$881$	−9.28729	−0.312897	−0.156448	+	0.987686i	$0.550005\pi$
$882$	0	0
$883$	19.4528	0.654639	0.327319	−	0.944914i	$-0.393855\pi$
$883$	19.4528	0.654639	0.327319	+	0.944914i	$0.393855\pi$
$884$	−1.52382	−0.0512517
$885$	0	0
$886$	−0.958146	−0.0321895
$887$	−20.2566	−0.680151	−0.340075	−	0.940398i	$-0.610453\pi$
$887$	−20.2566	−0.680151	−0.340075	+	0.940398i	$0.610453\pi$
$888$	0	0
$889$	0	0
$890$	0.218570	0.00732648
$891$	0	0
$892$	36.7178	1.22940
$893$	−24.0068	−0.803355
$894$	0	0
$895$	31.8746	1.06545
$896$	0	0
$897$	0	0
$898$	−3.80635	−0.127020
$899$	3.05079	0.101749
$900$	0	0
$901$	−2.50798	−0.0835530
$902$	−2.99877	−0.0998481
$903$	0	0
$904$	24.2960	0.808074
$905$	24.5617	0.816459
$906$	0	0
$907$	−9.87681	−0.327954	−0.163977	−	0.986464i	$-0.552432\pi$
$907$	−9.87681	−0.327954	−0.163977	+	0.986464i	$0.552432\pi$
$908$	−28.3165	−0.939715
$909$	0	0
$910$	0	0
$911$	0.832714	0.0275890	0.0137945	−	0.999905i	$-0.495609\pi$
$911$	0.832714	0.0275890	0.0137945	+	0.999905i	$0.495609\pi$
$912$	0	0
$913$	−13.2634	−0.438955
$914$	−12.3786	−0.409449
$915$	0	0
$916$	19.5635	0.646395
$917$	0	0
$918$	0	0
$919$	19.5390	0.644532	0.322266	−	0.946649i	$-0.395555\pi$
$919$	19.5390	0.644532	0.322266	+	0.946649i	$0.395555\pi$
$920$	−17.2258	−0.567917
$921$	0	0
$922$	15.5548	0.512271
$923$	−7.55740	−0.248755
$924$	0	0
$925$	−10.4375	−0.343184
$926$	−2.51039	−0.0824967
$927$	0	0
$928$	−1.37607	−0.0451716
$929$	−58.7213	−1.92658	−0.963292	−	0.268457i	$-0.913486\pi$
$929$	−58.7213	−1.92658	−0.963292	+	0.268457i	$0.913486\pi$
$930$	0	0
$931$	0	0
$932$	20.3784	0.667515
$933$	0	0
$934$	−1.88231	−0.0615909
$935$	−4.27029	−0.139653
$936$	0	0
$937$	−36.2474	−1.18415	−0.592075	−	0.805883i	$-0.701691\pi$
$937$	−36.2474	−1.18415	−0.592075	+	0.805883i	$0.701691\pi$
$938$	0	0
$939$	0	0
$940$	31.8982	1.04040
$941$	−1.30139	−0.0424241	−0.0212120	−	0.999775i	$-0.506753\pi$
$941$	−1.30139	−0.0424241	−0.0212120	+	0.999775i	$0.506753\pi$
$942$	0	0
$943$	20.5648	0.669683
$944$	9.01721	0.293485
$945$	0	0
$946$	−5.17537	−0.168266
$947$	−28.8185	−0.936477	−0.468238	−	0.883602i	$-0.655111\pi$
$947$	−28.8185	−0.936477	−0.468238	+	0.883602i	$0.655111\pi$
$948$	0	0
$949$	0.0582261	0.00189010
$950$	−1.76385	−0.0572268
$951$	0	0
$952$	0	0
$953$	42.4348	1.37460	0.687299	−	0.726375i	$-0.258796\pi$
$953$	42.4348	1.37460	0.687299	+	0.726375i	$0.258796\pi$
$954$	0	0
$955$	33.8504	1.09537
$956$	9.23509	0.298684
$957$	0	0
$958$	−0.269062	−0.00869300
$959$	0	0
$960$	0	0
$961$	56.6624	1.82782
$962$	1.85051	0.0596629
$963$	0	0
$964$	−26.9701	−0.868649
$965$	−29.3270	−0.944069
$966$	0	0
$967$	−39.9540	−1.28483	−0.642417	−	0.766355i	$-0.722068\pi$
$967$	−39.9540	−1.28483	−0.642417	+	0.766355i	$0.722068\pi$
$968$	7.92245	0.254637
$969$	0	0
$970$	5.43762	0.174591
$971$	−5.01507	−0.160941	−0.0804706	−	0.996757i	$-0.525642\pi$
$971$	−5.01507	−0.160941	−0.0804706	+	0.996757i	$0.525642\pi$
$972$	0	0
$973$	0	0
$974$	−12.9515	−0.414994
$975$	0	0
$976$	3.17801	0.101726
$977$	−51.5778	−1.65012	−0.825060	−	0.565045i	$-0.808859\pi$
$977$	−51.5778	−1.65012	−0.825060	+	0.565045i	$0.808859\pi$
$978$	0	0
$979$	−0.750498	−0.0239860
$980$	0	0
$981$	0	0
$982$	3.30866	0.105584
$983$	12.4383	0.396720	0.198360	−	0.980129i	$-0.436438\pi$
$983$	12.4383	0.396720	0.198360	+	0.980129i	$0.436438\pi$
$984$	0	0
$985$	10.8699	0.346343
$986$	−0.127411	−0.00405760
$987$	0	0
$988$	−3.77781	−0.120188
$989$	35.4914	1.12856
$990$	0	0
$991$	25.7520	0.818038	0.409019	−	0.912526i	$-0.365871\pi$
$991$	25.7520	0.818038	0.409019	+	0.912526i	$0.365871\pi$
$992$	−39.5404	−1.25541
$993$	0	0
$994$	0	0
$995$	31.5512	1.00024
$996$	0	0
$997$	−53.4501	−1.69278	−0.846391	−	0.532562i	$-0.821229\pi$
$997$	−53.4501	−1.69278	−0.846391	+	0.532562i	$0.821229\pi$
$998$	9.96144	0.315324
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7497.2.a.ce.1.5	yes	8
3.2	odd	2		7497.2.a.cd.1.4	✓	8
7.6	odd	2		7497.2.a.cd.1.5	yes	8
21.20	even	2	inner	7497.2.a.ce.1.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7497.2.a.cd.1.4	✓	8	3.2	odd	2
7497.2.a.cd.1.5	yes	8	7.6	odd	2
7497.2.a.ce.1.4	yes	8	21.20	even	2	inner
7497.2.a.ce.1.5	yes	8	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:	\( N \)	\(=\)	\( 7497 = 3^{2} \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7497.a (trivial)

\(f(q)\)	\(=\)	\(q+0.391024 q^{2} -1.84710 q^{4} +1.78339 q^{5} -1.50431 q^{8} +O(q^{10})\)	\(q+0.391024 q^{2} -1.84710 q^{4} +1.78339 q^{5} -1.50431 q^{8} +0.697349 q^{10} -2.39447 q^{11} +0.824980 q^{13} +3.10598 q^{16} +1.00000 q^{17} +2.47917 q^{19} -3.29411 q^{20} -0.936294 q^{22} +6.42088 q^{23} -1.81950 q^{25} +0.322587 q^{26} -0.325841 q^{29} -9.36282 q^{31} +4.22313 q^{32} +0.391024 q^{34} +5.73648 q^{37} +0.969413 q^{38} -2.68277 q^{40} +3.20281 q^{41} +5.52750 q^{43} +4.42283 q^{44} +2.51071 q^{46} -9.68339 q^{47} -0.711468 q^{50} -1.52382 q^{52} -2.50798 q^{53} -4.27029 q^{55} -0.127411 q^{58} +2.90318 q^{59} +1.02319 q^{61} -3.66108 q^{62} -4.56062 q^{64} +1.47127 q^{65} +1.50773 q^{67} -1.84710 q^{68} -9.16070 q^{71} +0.0705787 q^{73} +2.24310 q^{74} -4.57927 q^{76} -0.730309 q^{79} +5.53919 q^{80} +1.25237 q^{82} +5.53919 q^{83} +1.78339 q^{85} +2.16138 q^{86} +3.60202 q^{88} +0.313430 q^{89} -11.8600 q^{92} -3.78643 q^{94} +4.42134 q^{95} +7.79755 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{4} + 12 q^{5}+O(q^{10}) \) 8 * q + 8 * q^4 + 12 * q^5	\( 8 q + 8 q^{4} + 12 q^{5} + 16 q^{16} + 8 q^{17} - 8 q^{20} - 28 q^{22} + 8 q^{25} - 4 q^{26} + 12 q^{37} + 60 q^{38} + 28 q^{41} - 24 q^{43} + 4 q^{46} + 20 q^{47} + 40 q^{58} + 12 q^{59} + 48 q^{62} + 48 q^{67} + 8 q^{68} + 60 q^{79} + 40 q^{80} + 40 q^{83} + 12 q^{85} - 8 q^{88} + 16 q^{89}+O(q^{100}) \) 8 * q + 8 * q^4 + 12 * q^5 + 16 * q^16 + 8 * q^17 - 8 * q^20 - 28 * q^22 + 8 * q^25 - 4 * q^26 + 12 * q^37 + 60 * q^38 + 28 * q^41 - 24 * q^43 + 4 * q^46 + 20 * q^47 + 40 * q^58 + 12 * q^59 + 48 * q^62 + 48 * q^67 + 8 * q^68 + 60 * q^79 + 40 * q^80 + 40 * q^83 + 12 * q^85 - 8 * q^88 + 16 * q^89

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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