LMFDB - Embedded newform 8032.2.a.j.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8032,2,Mod(1,8032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8032, 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("8032.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8032 = 2^{5} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8032.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:	$64.1358429035$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	8032.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.26361 q^{3} -0.762700 q^{5} -1.25553 q^{7} +2.12395 q^{9} +O(q^{10})$	$q-2.26361 q^{3} -0.762700 q^{5} -1.25553 q^{7} +2.12395 q^{9} +1.90757 q^{11} -2.67531 q^{13} +1.72646 q^{15} +7.34812 q^{17} +6.08177 q^{19} +2.84203 q^{21} +0.363781 q^{23} -4.41829 q^{25} +1.98304 q^{27} +8.19457 q^{29} +5.80736 q^{31} -4.31800 q^{33} +0.957592 q^{35} -6.69072 q^{37} +6.05586 q^{39} -5.03158 q^{41} +8.74544 q^{43} -1.61994 q^{45} -7.61386 q^{47} -5.42365 q^{49} -16.6333 q^{51} -3.07763 q^{53} -1.45490 q^{55} -13.7668 q^{57} +5.91226 q^{59} -3.36756 q^{61} -2.66668 q^{63} +2.04046 q^{65} +13.7336 q^{67} -0.823460 q^{69} -5.45275 q^{71} +7.16674 q^{73} +10.0013 q^{75} -2.39501 q^{77} -9.28766 q^{79} -10.8607 q^{81} +1.97053 q^{83} -5.60441 q^{85} -18.5493 q^{87} -8.78253 q^{89} +3.35893 q^{91} -13.1456 q^{93} -4.63857 q^{95} +16.7232 q^{97} +4.05157 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + 9 q^{3} + 5 q^{7} + 35 q^{9}+O(q^{10}) $ 30 * q + 9 * q^3 + 5 * q^7 + 35 * q^9	$ 30 q + 9 q^{3} + 5 q^{7} + 35 q^{9} + 31 q^{11} + 5 q^{13} + 8 q^{15} - q^{17} + 29 q^{19} + 6 q^{21} + 27 q^{23} + 34 q^{25} + 36 q^{27} + q^{29} - 9 q^{31} - 8 q^{33} + 51 q^{35} + 5 q^{37} - 8 q^{39} - 3 q^{41} + 43 q^{43} + 42 q^{47} + 41 q^{49} + 54 q^{51} - 11 q^{53} - 10 q^{55} + 2 q^{57} + 49 q^{59} + 21 q^{61} + 15 q^{63} - 14 q^{65} + 68 q^{67} - 10 q^{69} + 44 q^{71} + 25 q^{73} + 51 q^{75} - 20 q^{77} - 49 q^{79} + 6 q^{81} + 88 q^{83} - 5 q^{85} + 16 q^{87} - 16 q^{89} + 46 q^{91} - 4 q^{93} + 28 q^{95} - 4 q^{97} + 71 q^{99}+O(q^{100}) $ 30 * q + 9 * q^3 + 5 * q^7 + 35 * q^9 + 31 * q^11 + 5 * q^13 + 8 * q^15 - q^17 + 29 * q^19 + 6 * q^21 + 27 * q^23 + 34 * q^25 + 36 * q^27 + q^29 - 9 * q^31 - 8 * q^33 + 51 * q^35 + 5 * q^37 - 8 * q^39 - 3 * q^41 + 43 * q^43 + 42 * q^47 + 41 * q^49 + 54 * q^51 - 11 * q^53 - 10 * q^55 + 2 * q^57 + 49 * q^59 + 21 * q^61 + 15 * q^63 - 14 * q^65 + 68 * q^67 - 10 * q^69 + 44 * q^71 + 25 * q^73 + 51 * q^75 - 20 * q^77 - 49 * q^79 + 6 * q^81 + 88 * q^83 - 5 * q^85 + 16 * q^87 - 16 * q^89 + 46 * q^91 - 4 * q^93 + 28 * q^95 - 4 * q^97 + 71 * q^99

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	0
$2$	0	0
$3$	−2.26361	−1.30690	−0.653449	−	0.756971i	$-0.726679\pi$
$3$	−2.26361	−1.30690	−0.653449	+	0.756971i	$0.726679\pi$
$4$	0	0
$5$	−0.762700	−0.341090	−0.170545	−	0.985350i	$-0.554553\pi$
$5$	−0.762700	−0.341090	−0.170545	+	0.985350i	$0.554553\pi$
$6$	0	0
$7$	−1.25553	−0.474545	−0.237273	−	0.971443i	$-0.576254\pi$
$7$	−1.25553	−0.474545	−0.237273	+	0.971443i	$0.576254\pi$
$8$	0	0
$9$	2.12395	0.707983
$10$	0	0
$11$	1.90757	0.575153	0.287577	−	0.957758i	$-0.407150\pi$
$11$	1.90757	0.575153	0.287577	+	0.957758i	$0.407150\pi$
$12$	0	0
$13$	−2.67531	−0.741996	−0.370998	−	0.928634i	$-0.620984\pi$
$13$	−2.67531	−0.741996	−0.370998	+	0.928634i	$0.620984\pi$
$14$	0	0
$15$	1.72646	0.445770
$16$	0	0
$17$	7.34812	1.78218	0.891090	−	0.453826i	$-0.149941\pi$
$17$	7.34812	1.78218	0.891090	+	0.453826i	$0.149941\pi$
$18$	0	0
$19$	6.08177	1.39525	0.697627	−	0.716461i	$-0.254239\pi$
$19$	6.08177	1.39525	0.697627	+	0.716461i	$0.254239\pi$
$20$	0	0
$21$	2.84203	0.620183
$22$	0	0
$23$	0.363781	0.0758536	0.0379268	−	0.999281i	$-0.487925\pi$
$23$	0.363781	0.0758536	0.0379268	+	0.999281i	$0.487925\pi$
$24$	0	0
$25$	−4.41829	−0.883658
$26$	0	0
$27$	1.98304	0.381637
$28$	0	0
$29$	8.19457	1.52169	0.760847	−	0.648931i	$-0.224784\pi$
$29$	8.19457	1.52169	0.760847	+	0.648931i	$0.224784\pi$
$30$	0	0
$31$	5.80736	1.04303	0.521516	−	0.853241i	$-0.325367\pi$
$31$	5.80736	1.04303	0.521516	+	0.853241i	$0.325367\pi$
$32$	0	0
$33$	−4.31800	−0.751667
$34$	0	0
$35$	0.957592	0.161863
$36$	0	0
$37$	−6.69072	−1.09995	−0.549973	−	0.835182i	$-0.685362\pi$
$37$	−6.69072	−1.09995	−0.549973	+	0.835182i	$0.685362\pi$
$38$	0	0
$39$	6.05586	0.969714
$40$	0	0
$41$	−5.03158	−0.785801	−0.392900	−	0.919581i	$-0.628528\pi$
$41$	−5.03158	−0.785801	−0.392900	+	0.919581i	$0.628528\pi$
$42$	0	0
$43$	8.74544	1.33367	0.666834	−	0.745207i	$-0.267649\pi$
$43$	8.74544	1.33367	0.666834	+	0.745207i	$0.267649\pi$
$44$	0	0
$45$	−1.61994	−0.241486
$46$	0	0
$47$	−7.61386	−1.11060	−0.555298	−	0.831651i	$-0.687396\pi$
$47$	−7.61386	−1.11060	−0.555298	+	0.831651i	$0.687396\pi$
$48$	0	0
$49$	−5.42365	−0.774807
$50$	0	0
$51$	−16.6333	−2.32913
$52$	0	0
$53$	−3.07763	−0.422746	−0.211373	−	0.977406i	$-0.567793\pi$
$53$	−3.07763	−0.422746	−0.211373	+	0.977406i	$0.567793\pi$
$54$	0	0
$55$	−1.45490	−0.196179
$56$	0	0
$57$	−13.7668	−1.82345
$58$	0	0
$59$	5.91226	0.769711	0.384855	−	0.922977i	$-0.374251\pi$
$59$	5.91226	0.769711	0.384855	+	0.922977i	$0.374251\pi$
$60$	0	0
$61$	−3.36756	−0.431172	−0.215586	−	0.976485i	$-0.569166\pi$
$61$	−3.36756	−0.431172	−0.215586	+	0.976485i	$0.569166\pi$
$62$	0	0
$63$	−2.66668	−0.335970
$64$	0	0
$65$	2.04046	0.253087
$66$	0	0
$67$	13.7336	1.67782	0.838911	−	0.544268i	$-0.183193\pi$
$67$	13.7336	1.67782	0.838911	+	0.544268i	$0.183193\pi$
$68$	0	0
$69$	−0.823460	−0.0991329
$70$	0	0
$71$	−5.45275	−0.647122	−0.323561	−	0.946207i	$-0.604880\pi$
$71$	−5.45275	−0.647122	−0.323561	+	0.946207i	$0.604880\pi$
$72$	0	0
$73$	7.16674	0.838803	0.419402	−	0.907801i	$-0.362240\pi$
$73$	7.16674	0.838803	0.419402	+	0.907801i	$0.362240\pi$
$74$	0	0
$75$	10.0013	1.15485
$76$	0	0
$77$	−2.39501	−0.272936
$78$	0	0
$79$	−9.28766	−1.04494	−0.522472	−	0.852656i	$-0.674990\pi$
$79$	−9.28766	−1.04494	−0.522472	+	0.852656i	$0.674990\pi$
$80$	0	0
$81$	−10.8607	−1.20674
$82$	0	0
$83$	1.97053	0.216294	0.108147	−	0.994135i	$-0.465508\pi$
$83$	1.97053	0.216294	0.108147	+	0.994135i	$0.465508\pi$
$84$	0	0
$85$	−5.60441	−0.607884
$86$	0	0
$87$	−18.5493	−1.98870
$88$	0	0
$89$	−8.78253	−0.930947	−0.465473	−	0.885062i	$-0.654116\pi$
$89$	−8.78253	−0.930947	−0.465473	+	0.885062i	$0.654116\pi$
$90$	0	0
$91$	3.35893	0.352111
$92$	0	0
$93$	−13.1456	−1.36314
$94$	0	0
$95$	−4.63857	−0.475907
$96$	0	0
$97$	16.7232	1.69799	0.848994	−	0.528402i	$-0.177209\pi$
$97$	16.7232	1.69799	0.848994	+	0.528402i	$0.177209\pi$
$98$	0	0
$99$	4.05157	0.407198
$100$	0	0
$101$	−7.99893	−0.795924	−0.397962	−	0.917402i	$-0.630282\pi$
$101$	−7.99893	−0.795924	−0.397962	+	0.917402i	$0.630282\pi$
$102$	0	0
$103$	−10.8130	−1.06544	−0.532719	−	0.846292i	$-0.678830\pi$
$103$	−10.8130	−1.06544	−0.532719	+	0.846292i	$0.678830\pi$
$104$	0	0
$105$	−2.16762	−0.211538
$106$	0	0
$107$	−6.70348	−0.648050	−0.324025	−	0.946049i	$-0.605036\pi$
$107$	−6.70348	−0.648050	−0.324025	+	0.946049i	$0.605036\pi$
$108$	0	0
$109$	−7.29009	−0.698264	−0.349132	−	0.937074i	$-0.613523\pi$
$109$	−7.29009	−0.698264	−0.349132	+	0.937074i	$0.613523\pi$
$110$	0	0
$111$	15.1452	1.43752
$112$	0	0
$113$	5.73622	0.539618	0.269809	−	0.962914i	$-0.413039\pi$
$113$	5.73622	0.539618	0.269809	+	0.962914i	$0.413039\pi$
$114$	0	0
$115$	−0.277456	−0.0258729
$116$	0	0
$117$	−5.68221	−0.525321
$118$	0	0
$119$	−9.22578	−0.845726
$120$	0	0
$121$	−7.36119	−0.669199
$122$	0	0
$123$	11.3896	1.02696
$124$	0	0
$125$	7.18333	0.642497
$126$	0	0
$127$	5.82884	0.517226	0.258613	−	0.965981i	$-0.416735\pi$
$127$	5.82884	0.517226	0.258613	+	0.965981i	$0.416735\pi$
$128$	0	0
$129$	−19.7963	−1.74297
$130$	0	0
$131$	−6.41950	−0.560874	−0.280437	−	0.959872i	$-0.590479\pi$
$131$	−6.41950	−0.560874	−0.280437	+	0.959872i	$0.590479\pi$
$132$	0	0
$133$	−7.63584	−0.662111
$134$	0	0
$135$	−1.51247	−0.130173
$136$	0	0
$137$	−3.65074	−0.311904	−0.155952	−	0.987765i	$-0.549844\pi$
$137$	−3.65074	−0.311904	−0.155952	+	0.987765i	$0.549844\pi$
$138$	0	0
$139$	5.62398	0.477020	0.238510	−	0.971140i	$-0.423341\pi$
$139$	5.62398	0.477020	0.238510	+	0.971140i	$0.423341\pi$
$140$	0	0
$141$	17.2348	1.45144
$142$	0	0
$143$	−5.10333	−0.426762
$144$	0	0
$145$	−6.25000	−0.519034
$146$	0	0
$147$	12.2770	1.01259
$148$	0	0
$149$	8.61887	0.706085	0.353043	−	0.935607i	$-0.385147\pi$
$149$	8.61887	0.706085	0.353043	+	0.935607i	$0.385147\pi$
$150$	0	0
$151$	10.2806	0.836622	0.418311	−	0.908304i	$-0.362622\pi$
$151$	10.2806	0.836622	0.418311	+	0.908304i	$0.362622\pi$
$152$	0	0
$153$	15.6070	1.26175
$154$	0	0
$155$	−4.42927	−0.355768
$156$	0	0
$157$	16.8574	1.34536	0.672682	−	0.739931i	$-0.265142\pi$
$157$	16.8574	1.34536	0.672682	+	0.739931i	$0.265142\pi$
$158$	0	0
$159$	6.96658	0.552485
$160$	0	0
$161$	−0.456738	−0.0359960
$162$	0	0
$163$	9.45262	0.740386	0.370193	−	0.928955i	$-0.379291\pi$
$163$	9.45262	0.740386	0.370193	+	0.928955i	$0.379291\pi$
$164$	0	0
$165$	3.29334	0.256386
$166$	0	0
$167$	19.3432	1.49682	0.748410	−	0.663236i	$-0.230818\pi$
$167$	19.3432	1.49682	0.748410	+	0.663236i	$0.230818\pi$
$168$	0	0
$169$	−5.84274	−0.449441
$170$	0	0
$171$	12.9174	0.987815
$172$	0	0
$173$	−9.89693	−0.752450	−0.376225	−	0.926528i	$-0.622778\pi$
$173$	−9.89693	−0.752450	−0.376225	+	0.926528i	$0.622778\pi$
$174$	0	0
$175$	5.54729	0.419336
$176$	0	0
$177$	−13.3831	−1.00593
$178$	0	0
$179$	−9.18617	−0.686606	−0.343303	−	0.939225i	$-0.611546\pi$
$179$	−9.18617	−0.686606	−0.343303	+	0.939225i	$0.611546\pi$
$180$	0	0
$181$	3.17729	0.236166	0.118083	−	0.993004i	$-0.462325\pi$
$181$	3.17729	0.236166	0.118083	+	0.993004i	$0.462325\pi$
$182$	0	0
$183$	7.62286	0.563498
$184$	0	0
$185$	5.10301	0.375181
$186$	0	0
$187$	14.0170	1.02503
$188$	0	0
$189$	−2.48977	−0.181104
$190$	0	0
$191$	−6.34768	−0.459302	−0.229651	−	0.973273i	$-0.573758\pi$
$191$	−6.34768	−0.459302	−0.229651	+	0.973273i	$0.573758\pi$
$192$	0	0
$193$	−6.34101	−0.456436	−0.228218	−	0.973610i	$-0.573290\pi$
$193$	−6.34101	−0.456436	−0.228218	+	0.973610i	$0.573290\pi$
$194$	0	0
$195$	−4.61881	−0.330760
$196$	0	0
$197$	17.8865	1.27436	0.637181	−	0.770714i	$-0.280100\pi$
$197$	17.8865	1.27436	0.637181	+	0.770714i	$0.280100\pi$
$198$	0	0
$199$	15.2600	1.08175	0.540875	−	0.841103i	$-0.318093\pi$
$199$	15.2600	1.08175	0.540875	+	0.841103i	$0.318093\pi$
$200$	0	0
$201$	−31.0875	−2.19274
$202$	0	0
$203$	−10.2885	−0.722113
$204$	0	0
$205$	3.83759	0.268029
$206$	0	0
$207$	0.772652	0.0537030
$208$	0	0
$209$	11.6014	0.802484
$210$	0	0
$211$	4.47859	0.308319	0.154159	−	0.988046i	$-0.450733\pi$
$211$	4.47859	0.308319	0.154159	+	0.988046i	$0.450733\pi$
$212$	0	0
$213$	12.3429	0.845723
$214$	0	0
$215$	−6.67015	−0.454900
$216$	0	0
$217$	−7.29131	−0.494966
$218$	0	0
$219$	−16.2227	−1.09623
$220$	0	0
$221$	−19.6585	−1.32237
$222$	0	0
$223$	3.25514	0.217981	0.108990	−	0.994043i	$-0.465238\pi$
$223$	3.25514	0.217981	0.108990	+	0.994043i	$0.465238\pi$
$224$	0	0
$225$	−9.38421	−0.625614
$226$	0	0
$227$	18.0601	1.19869	0.599347	−	0.800489i	$-0.295427\pi$
$227$	18.0601	1.19869	0.599347	+	0.800489i	$0.295427\pi$
$228$	0	0
$229$	3.83650	0.253523	0.126762	−	0.991933i	$-0.459542\pi$
$229$	3.83650	0.253523	0.126762	+	0.991933i	$0.459542\pi$
$230$	0	0
$231$	5.42137	0.356700
$232$	0	0
$233$	3.16626	0.207428	0.103714	−	0.994607i	$-0.466927\pi$
$233$	3.16626	0.207428	0.103714	+	0.994607i	$0.466927\pi$
$234$	0	0
$235$	5.80709	0.378813
$236$	0	0
$237$	21.0237	1.36564
$238$	0	0
$239$	4.69685	0.303814	0.151907	−	0.988395i	$-0.451459\pi$
$239$	4.69685	0.303814	0.151907	+	0.988395i	$0.451459\pi$
$240$	0	0
$241$	−14.1080	−0.908776	−0.454388	−	0.890804i	$-0.650142\pi$
$241$	−14.1080	−0.908776	−0.454388	+	0.890804i	$0.650142\pi$
$242$	0	0
$243$	18.6353	1.19545
$244$	0	0
$245$	4.13662	0.264279
$246$	0	0
$247$	−16.2706	−1.03527
$248$	0	0
$249$	−4.46052	−0.282674
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	0.693937	0.0436274
$254$	0	0
$255$	12.6862	0.794442
$256$	0	0
$257$	−2.04897	−0.127811	−0.0639055	−	0.997956i	$-0.520356\pi$
$257$	−2.04897	−0.127811	−0.0639055	+	0.997956i	$0.520356\pi$
$258$	0	0
$259$	8.40039	0.521975
$260$	0	0
$261$	17.4048	1.07733
$262$	0	0
$263$	−0.574065	−0.0353984	−0.0176992	−	0.999843i	$-0.505634\pi$
$263$	−0.574065	−0.0353984	−0.0176992	+	0.999843i	$0.505634\pi$
$264$	0	0
$265$	2.34731	0.144194
$266$	0	0
$267$	19.8803	1.21665
$268$	0	0
$269$	−23.4361	−1.42892	−0.714462	−	0.699675i	$-0.753328\pi$
$269$	−23.4361	−1.42892	−0.714462	+	0.699675i	$0.753328\pi$
$270$	0	0
$271$	−2.62765	−0.159619	−0.0798093	−	0.996810i	$-0.525431\pi$
$271$	−2.62765	−0.159619	−0.0798093	+	0.996810i	$0.525431\pi$
$272$	0	0
$273$	−7.60331	−0.460173
$274$	0	0
$275$	−8.42818	−0.508239
$276$	0	0
$277$	−6.14634	−0.369298	−0.184649	−	0.982805i	$-0.559115\pi$
$277$	−6.14634	−0.369298	−0.184649	+	0.982805i	$0.559115\pi$
$278$	0	0
$279$	12.3345	0.738449
$280$	0	0
$281$	−20.4781	−1.22162	−0.610810	−	0.791777i	$-0.709156\pi$
$281$	−20.4781	−1.22162	−0.610810	+	0.791777i	$0.709156\pi$
$282$	0	0
$283$	11.5726	0.687918	0.343959	−	0.938985i	$-0.388232\pi$
$283$	11.5726	0.687918	0.343959	+	0.938985i	$0.388232\pi$
$284$	0	0
$285$	10.4999	0.621962
$286$	0	0
$287$	6.31729	0.372898
$288$	0	0
$289$	36.9948	2.17617
$290$	0	0
$291$	−37.8550	−2.21910
$292$	0	0
$293$	−25.5100	−1.49031	−0.745154	−	0.666892i	$-0.767624\pi$
$293$	−25.5100	−1.49031	−0.745154	+	0.666892i	$0.767624\pi$
$294$	0	0
$295$	−4.50928	−0.262540
$296$	0	0
$297$	3.78279	0.219500
$298$	0	0
$299$	−0.973226	−0.0562831
$300$	0	0
$301$	−10.9802	−0.632886
$302$	0	0
$303$	18.1065	1.04019
$304$	0	0
$305$	2.56844	0.147068
$306$	0	0
$307$	−24.0300	−1.37146	−0.685732	−	0.727854i	$-0.740518\pi$
$307$	−24.0300	−1.37146	−0.685732	+	0.727854i	$0.740518\pi$
$308$	0	0
$309$	24.4765	1.39242
$310$	0	0
$311$	−27.6653	−1.56875	−0.784377	−	0.620284i	$-0.787017\pi$
$311$	−27.6653	−1.56875	−0.784377	+	0.620284i	$0.787017\pi$
$312$	0	0
$313$	−8.59890	−0.486039	−0.243019	−	0.970021i	$-0.578138\pi$
$313$	−8.59890	−0.486039	−0.243019	+	0.970021i	$0.578138\pi$
$314$	0	0
$315$	2.03388	0.114596
$316$	0	0
$317$	31.4055	1.76391	0.881955	−	0.471333i	$-0.156227\pi$
$317$	31.4055	1.76391	0.881955	+	0.471333i	$0.156227\pi$
$318$	0	0
$319$	15.6317	0.875207
$320$	0	0
$321$	15.1741	0.846935
$322$	0	0
$323$	44.6896	2.48659
$324$	0	0
$325$	11.8203	0.655671
$326$	0	0
$327$	16.5019	0.912560
$328$	0	0
$329$	9.55943	0.527028
$330$	0	0
$331$	−6.56509	−0.360850	−0.180425	−	0.983589i	$-0.557747\pi$
$331$	−6.56509	−0.360850	−0.180425	+	0.983589i	$0.557747\pi$
$332$	0	0
$333$	−14.2107	−0.778743
$334$	0	0
$335$	−10.4746	−0.572288
$336$	0	0
$337$	33.7019	1.83586	0.917930	−	0.396743i	$-0.129860\pi$
$337$	33.7019	1.83586	0.917930	+	0.396743i	$0.129860\pi$
$338$	0	0
$339$	−12.9846	−0.705225
$340$	0	0
$341$	11.0779	0.599904
$342$	0	0
$343$	15.5983	0.842226
$344$	0	0
$345$	0.628053	0.0338132
$346$	0	0
$347$	2.22753	0.119580	0.0597902	−	0.998211i	$-0.480957\pi$
$347$	2.22753	0.119580	0.0597902	+	0.998211i	$0.480957\pi$
$348$	0	0
$349$	15.5343	0.831533	0.415767	−	0.909471i	$-0.363513\pi$
$349$	15.5343	0.831533	0.415767	+	0.909471i	$0.363513\pi$
$350$	0	0
$351$	−5.30525	−0.283173
$352$	0	0
$353$	18.6261	0.991369	0.495685	−	0.868503i	$-0.334917\pi$
$353$	18.6261	0.991369	0.495685	+	0.868503i	$0.334917\pi$
$354$	0	0
$355$	4.15881	0.220727
$356$	0	0
$357$	20.8836	1.10528
$358$	0	0
$359$	34.6264	1.82751	0.913755	−	0.406266i	$-0.133169\pi$
$359$	34.6264	1.82751	0.913755	+	0.406266i	$0.133169\pi$
$360$	0	0
$361$	17.9879	0.946732
$362$	0	0
$363$	16.6629	0.874575
$364$	0	0
$365$	−5.46607	−0.286107
$366$	0	0
$367$	6.97460	0.364071	0.182036	−	0.983292i	$-0.441731\pi$
$367$	6.97460	0.364071	0.182036	+	0.983292i	$0.441731\pi$
$368$	0	0
$369$	−10.6868	−0.556333
$370$	0	0
$371$	3.86406	0.200612
$372$	0	0
$373$	−12.3624	−0.640098	−0.320049	−	0.947401i	$-0.603699\pi$
$373$	−12.3624	−0.640098	−0.320049	+	0.947401i	$0.603699\pi$
$374$	0	0
$375$	−16.2603	−0.839677
$376$	0	0
$377$	−21.9230	−1.12909
$378$	0	0
$379$	−30.9789	−1.59128	−0.795639	−	0.605770i	$-0.792865\pi$
$379$	−30.9789	−1.59128	−0.795639	+	0.605770i	$0.792865\pi$
$380$	0	0
$381$	−13.1943	−0.675962
$382$	0	0
$383$	20.2972	1.03714	0.518569	−	0.855036i	$-0.326465\pi$
$383$	20.2972	1.03714	0.518569	+	0.855036i	$0.326465\pi$
$384$	0	0
$385$	1.82667	0.0930958
$386$	0	0
$387$	18.5749	0.944213
$388$	0	0
$389$	22.3172	1.13153	0.565764	−	0.824567i	$-0.308581\pi$
$389$	22.3172	1.13153	0.565764	+	0.824567i	$0.308581\pi$
$390$	0	0
$391$	2.67311	0.135185
$392$	0	0
$393$	14.5313	0.733005
$394$	0	0
$395$	7.08370	0.356420
$396$	0	0
$397$	−3.73173	−0.187290	−0.0936452	−	0.995606i	$-0.529852\pi$
$397$	−3.73173	−0.187290	−0.0936452	+	0.995606i	$0.529852\pi$
$398$	0	0
$399$	17.2846	0.865312
$400$	0	0
$401$	−12.5028	−0.624361	−0.312180	−	0.950023i	$-0.601059\pi$
$401$	−12.5028	−0.624361	−0.312180	+	0.950023i	$0.601059\pi$
$402$	0	0
$403$	−15.5365	−0.773927
$404$	0	0
$405$	8.28345	0.411608
$406$	0	0
$407$	−12.7630	−0.632638
$408$	0	0
$409$	−9.13411	−0.451653	−0.225826	−	0.974168i	$-0.572508\pi$
$409$	−9.13411	−0.451653	−0.225826	+	0.974168i	$0.572508\pi$
$410$	0	0
$411$	8.26386	0.407626
$412$	0	0
$413$	−7.42302	−0.365263
$414$	0	0
$415$	−1.50292	−0.0737757
$416$	0	0
$417$	−12.7305	−0.623416
$418$	0	0
$419$	40.2980	1.96868	0.984342	−	0.176269i	$-0.0564030\pi$
$419$	40.2980	1.96868	0.984342	+	0.176269i	$0.0564030\pi$
$420$	0	0
$421$	20.6338	1.00563	0.502814	−	0.864394i	$-0.332298\pi$
$421$	20.6338	1.00563	0.502814	+	0.864394i	$0.332298\pi$
$422$	0	0
$423$	−16.1714	−0.786283
$424$	0	0
$425$	−32.4661	−1.57484
$426$	0	0
$427$	4.22807	0.204611
$428$	0	0
$429$	11.5520	0.557734
$430$	0	0
$431$	−12.1253	−0.584055	−0.292027	−	0.956410i	$-0.594330\pi$
$431$	−12.1253	−0.584055	−0.292027	+	0.956410i	$0.594330\pi$
$432$	0	0
$433$	10.6103	0.509897	0.254948	−	0.966955i	$-0.417942\pi$
$433$	10.6103	0.509897	0.254948	+	0.966955i	$0.417942\pi$
$434$	0	0
$435$	14.1476	0.678325
$436$	0	0
$437$	2.21243	0.105835
$438$	0	0
$439$	−14.8261	−0.707610	−0.353805	−	0.935319i	$-0.615112\pi$
$439$	−14.8261	−0.707610	−0.353805	+	0.935319i	$0.615112\pi$
$440$	0	0
$441$	−11.5195	−0.548550
$442$	0	0
$443$	−1.42021	−0.0674762	−0.0337381	−	0.999431i	$-0.510741\pi$
$443$	−1.42021	−0.0674762	−0.0337381	+	0.999431i	$0.510741\pi$
$444$	0	0
$445$	6.69844	0.317536
$446$	0	0
$447$	−19.5098	−0.922782
$448$	0	0
$449$	−36.8547	−1.73928	−0.869641	−	0.493684i	$-0.835650\pi$
$449$	−36.8547	−1.73928	−0.869641	+	0.493684i	$0.835650\pi$
$450$	0	0
$451$	−9.59807	−0.451956
$452$	0	0
$453$	−23.2713	−1.09338
$454$	0	0
$455$	−2.56185	−0.120102
$456$	0	0
$457$	−12.6424	−0.591384	−0.295692	−	0.955283i	$-0.595550\pi$
$457$	−12.6424	−0.591384	−0.295692	+	0.955283i	$0.595550\pi$
$458$	0	0
$459$	14.5716	0.680146
$460$	0	0
$461$	−14.6727	−0.683377	−0.341689	−	0.939813i	$-0.610999\pi$
$461$	−14.6727	−0.683377	−0.341689	+	0.939813i	$0.610999\pi$
$462$	0	0
$463$	16.0463	0.745733	0.372866	−	0.927885i	$-0.378375\pi$
$463$	16.0463	0.745733	0.372866	+	0.927885i	$0.378375\pi$
$464$	0	0
$465$	10.0262	0.464952
$466$	0	0
$467$	34.5843	1.60037	0.800186	−	0.599753i	$-0.204734\pi$
$467$	34.5843	1.60037	0.800186	+	0.599753i	$0.204734\pi$
$468$	0	0
$469$	−17.2429	−0.796203
$470$	0	0
$471$	−38.1586	−1.75825
$472$	0	0
$473$	16.6825	0.767063
$474$	0	0
$475$	−26.8710	−1.23293
$476$	0	0
$477$	−6.53673	−0.299297
$478$	0	0
$479$	3.87052	0.176848	0.0884242	−	0.996083i	$-0.471817\pi$
$479$	3.87052	0.176848	0.0884242	+	0.996083i	$0.471817\pi$
$480$	0	0
$481$	17.8997	0.816157
$482$	0	0
$483$	1.03388	0.0470431
$484$	0	0
$485$	−12.7548	−0.579167
$486$	0	0
$487$	−29.8932	−1.35459	−0.677295	−	0.735712i	$-0.736848\pi$
$487$	−29.8932	−1.35459	−0.677295	+	0.735712i	$0.736848\pi$
$488$	0	0
$489$	−21.3971	−0.967610
$490$	0	0
$491$	21.2481	0.958912	0.479456	−	0.877566i	$-0.340834\pi$
$491$	21.2481	0.958912	0.479456	+	0.877566i	$0.340834\pi$
$492$	0	0
$493$	60.2147	2.71193
$494$	0	0
$495$	−3.09013	−0.138891
$496$	0	0
$497$	6.84609	0.307089
$498$	0	0
$499$	−0.246489	−0.0110344	−0.00551718	−	0.999985i	$-0.501756\pi$
$499$	−0.246489	−0.0110344	−0.00551718	+	0.999985i	$0.501756\pi$
$500$	0	0
$501$	−43.7855	−1.95619
$502$	0	0
$503$	−24.0932	−1.07426	−0.537131	−	0.843499i	$-0.680492\pi$
$503$	−24.0932	−1.07426	−0.537131	+	0.843499i	$0.680492\pi$
$504$	0	0
$505$	6.10079	0.271482
$506$	0	0
$507$	13.2257	0.587374
$508$	0	0
$509$	7.32582	0.324711	0.162356	−	0.986732i	$-0.448091\pi$
$509$	7.32582	0.324711	0.162356	+	0.986732i	$0.448091\pi$
$510$	0	0
$511$	−8.99805	−0.398050
$512$	0	0
$513$	12.0604	0.532480
$514$	0	0
$515$	8.24708	0.363410
$516$	0	0
$517$	−14.5240	−0.638763
$518$	0	0
$519$	22.4028	0.983375
$520$	0	0
$521$	21.3445	0.935118	0.467559	−	0.883962i	$-0.345133\pi$
$521$	21.3445	0.935118	0.467559	+	0.883962i	$0.345133\pi$
$522$	0	0
$523$	29.7286	1.29994	0.649971	−	0.759959i	$-0.274781\pi$
$523$	29.7286	1.29994	0.649971	+	0.759959i	$0.274781\pi$
$524$	0	0
$525$	−12.5569	−0.548029
$526$	0	0
$527$	42.6732	1.85887
$528$	0	0
$529$	−22.8677	−0.994246
$530$	0	0
$531$	12.5573	0.544942
$532$	0	0
$533$	13.4610	0.583061
$534$	0	0
$535$	5.11274	0.221043
$536$	0	0
$537$	20.7939	0.897325
$538$	0	0
$539$	−10.3460	−0.445632
$540$	0	0
$541$	30.1785	1.29748	0.648738	−	0.761012i	$-0.275297\pi$
$541$	30.1785	1.29748	0.648738	+	0.761012i	$0.275297\pi$
$542$	0	0
$543$	−7.19215	−0.308645
$544$	0	0
$545$	5.56015	0.238171
$546$	0	0
$547$	39.0005	1.66754	0.833770	−	0.552112i	$-0.186178\pi$
$547$	39.0005	1.66754	0.833770	+	0.552112i	$0.186178\pi$
$548$	0	0
$549$	−7.15252	−0.305262
$550$	0	0
$551$	49.8375	2.12315
$552$	0	0
$553$	11.6609	0.495873
$554$	0	0
$555$	−11.5512	−0.490323
$556$	0	0
$557$	28.5077	1.20791	0.603956	−	0.797018i	$-0.293590\pi$
$557$	28.5077	1.20791	0.603956	+	0.797018i	$0.293590\pi$
$558$	0	0
$559$	−23.3967	−0.989576
$560$	0	0
$561$	−31.7291	−1.33961
$562$	0	0
$563$	24.7188	1.04177	0.520886	−	0.853626i	$-0.325602\pi$
$563$	24.7188	1.04177	0.520886	+	0.853626i	$0.325602\pi$
$564$	0	0
$565$	−4.37501	−0.184058
$566$	0	0
$567$	13.6359	0.572655
$568$	0	0
$569$	23.6743	0.992479	0.496239	−	0.868186i	$-0.334714\pi$
$569$	23.6743	0.992479	0.496239	+	0.868186i	$0.334714\pi$
$570$	0	0
$571$	25.3149	1.05940	0.529699	−	0.848186i	$-0.322305\pi$
$571$	25.3149	1.05940	0.529699	+	0.848186i	$0.322305\pi$
$572$	0	0
$573$	14.3687	0.600261
$574$	0	0
$575$	−1.60729	−0.0670286
$576$	0	0
$577$	6.21711	0.258822	0.129411	−	0.991591i	$-0.458691\pi$
$577$	6.21711	0.258822	0.129411	+	0.991591i	$0.458691\pi$
$578$	0	0
$579$	14.3536	0.596515
$580$	0	0
$581$	−2.47406	−0.102641
$582$	0	0
$583$	−5.87079	−0.243143
$584$	0	0
$585$	4.33382	0.179182
$586$	0	0
$587$	−13.6267	−0.562435	−0.281217	−	0.959644i	$-0.590738\pi$
$587$	−13.6267	−0.562435	−0.281217	+	0.959644i	$0.590738\pi$
$588$	0	0
$589$	35.3190	1.45530
$590$	0	0
$591$	−40.4882	−1.66546
$592$	0	0
$593$	35.2246	1.44650	0.723251	−	0.690586i	$-0.242647\pi$
$593$	35.2246	1.44650	0.723251	+	0.690586i	$0.242647\pi$
$594$	0	0
$595$	7.03650	0.288468
$596$	0	0
$597$	−34.5427	−1.41374
$598$	0	0
$599$	28.8118	1.17722	0.588609	−	0.808418i	$-0.299676\pi$
$599$	28.8118	1.17722	0.588609	+	0.808418i	$0.299676\pi$
$600$	0	0
$601$	11.9106	0.485845	0.242922	−	0.970046i	$-0.421894\pi$
$601$	11.9106	0.485845	0.242922	+	0.970046i	$0.421894\pi$
$602$	0	0
$603$	29.1694	1.18787
$604$	0	0
$605$	5.61438	0.228257
$606$	0	0
$607$	−41.7278	−1.69368	−0.846840	−	0.531847i	$-0.821498\pi$
$607$	−41.7278	−1.69368	−0.846840	+	0.531847i	$0.821498\pi$
$608$	0	0
$609$	23.2892	0.943728
$610$	0	0
$611$	20.3694	0.824058
$612$	0	0
$613$	17.2892	0.698305	0.349153	−	0.937066i	$-0.386470\pi$
$613$	17.2892	0.698305	0.349153	+	0.937066i	$0.386470\pi$
$614$	0	0
$615$	−8.68681	−0.350286
$616$	0	0
$617$	−32.3597	−1.30275	−0.651376	−	0.758755i	$-0.725808\pi$
$617$	−32.3597	−1.30275	−0.651376	+	0.758755i	$0.725808\pi$
$618$	0	0
$619$	13.7525	0.552759	0.276380	−	0.961049i	$-0.410865\pi$
$619$	13.7525	0.552759	0.276380	+	0.961049i	$0.410865\pi$
$620$	0	0
$621$	0.721394	0.0289485
$622$	0	0
$623$	11.0267	0.441776
$624$	0	0
$625$	16.6127	0.664509
$626$	0	0
$627$	−26.2611	−1.04877
$628$	0	0
$629$	−49.1642	−1.96030
$630$	0	0
$631$	15.6507	0.623043	0.311522	−	0.950239i	$-0.399161\pi$
$631$	15.6507	0.623043	0.311522	+	0.950239i	$0.399161\pi$
$632$	0	0
$633$	−10.1378	−0.402941
$634$	0	0
$635$	−4.44566	−0.176421
$636$	0	0
$637$	14.5099	0.574904
$638$	0	0
$639$	−11.5814	−0.458151
$640$	0	0
$641$	26.9432	1.06419	0.532096	−	0.846684i	$-0.321405\pi$
$641$	26.9432	1.06419	0.532096	+	0.846684i	$0.321405\pi$
$642$	0	0
$643$	−30.5083	−1.20313	−0.601565	−	0.798824i	$-0.705456\pi$
$643$	−30.5083	−1.20313	−0.601565	+	0.798824i	$0.705456\pi$
$644$	0	0
$645$	15.0986	0.594508
$646$	0	0
$647$	−41.8407	−1.64493	−0.822463	−	0.568818i	$-0.807401\pi$
$647$	−41.8407	−1.64493	−0.822463	+	0.568818i	$0.807401\pi$
$648$	0	0
$649$	11.2780	0.442702
$650$	0	0
$651$	16.5047	0.646871
$652$	0	0
$653$	49.5256	1.93809	0.969044	−	0.246889i	$-0.0794084\pi$
$653$	49.5256	1.93809	0.969044	+	0.246889i	$0.0794084\pi$
$654$	0	0
$655$	4.89615	0.191309
$656$	0	0
$657$	15.2218	0.593858
$658$	0	0
$659$	−15.0062	−0.584559	−0.292280	−	0.956333i	$-0.594414\pi$
$659$	−15.0062	−0.584559	−0.292280	+	0.956333i	$0.594414\pi$
$660$	0	0
$661$	−22.6900	−0.882540	−0.441270	−	0.897374i	$-0.645472\pi$
$661$	−22.6900	−0.882540	−0.441270	+	0.897374i	$0.645472\pi$
$662$	0	0
$663$	44.4992	1.72820
$664$	0	0
$665$	5.82386	0.225839
$666$	0	0
$667$	2.98103	0.115426
$668$	0	0
$669$	−7.36839	−0.284878
$670$	0	0
$671$	−6.42385	−0.247990
$672$	0	0
$673$	49.0249	1.88977	0.944886	−	0.327401i	$-0.106173\pi$
$673$	49.0249	1.88977	0.944886	+	0.327401i	$0.106173\pi$
$674$	0	0
$675$	−8.76166	−0.337236
$676$	0	0
$677$	17.0891	0.656787	0.328394	−	0.944541i	$-0.393493\pi$
$677$	17.0891	0.656787	0.328394	+	0.944541i	$0.393493\pi$
$678$	0	0
$679$	−20.9965	−0.805773
$680$	0	0
$681$	−40.8812	−1.56657
$682$	0	0
$683$	25.4515	0.973875	0.486938	−	0.873437i	$-0.338114\pi$
$683$	25.4515	0.973875	0.486938	+	0.873437i	$0.338114\pi$
$684$	0	0
$685$	2.78442	0.106387
$686$	0	0
$687$	−8.68435	−0.331329
$688$	0	0
$689$	8.23361	0.313676
$690$	0	0
$691$	−31.4227	−1.19538	−0.597688	−	0.801729i	$-0.703914\pi$
$691$	−31.4227	−1.19538	−0.597688	+	0.801729i	$0.703914\pi$
$692$	0	0
$693$	−5.08687	−0.193234
$694$	0	0
$695$	−4.28941	−0.162707
$696$	0	0
$697$	−36.9726	−1.40044
$698$	0	0
$699$	−7.16719	−0.271088
$700$	0	0
$701$	13.4056	0.506321	0.253160	−	0.967424i	$-0.418530\pi$
$701$	13.4056	0.506321	0.253160	+	0.967424i	$0.418530\pi$
$702$	0	0
$703$	−40.6914	−1.53470
$704$	0	0
$705$	−13.1450	−0.495070
$706$	0	0
$707$	10.0429	0.377702
$708$	0	0
$709$	−22.9568	−0.862160	−0.431080	−	0.902314i	$-0.641867\pi$
$709$	−22.9568	−0.862160	−0.431080	+	0.902314i	$0.641867\pi$
$710$	0	0
$711$	−19.7265	−0.739802
$712$	0	0
$713$	2.11261	0.0791178
$714$	0	0
$715$	3.89231	0.145564
$716$	0	0
$717$	−10.6318	−0.397053
$718$	0	0
$719$	7.87923	0.293846	0.146923	−	0.989148i	$-0.453063\pi$
$719$	7.87923	0.293846	0.146923	+	0.989148i	$0.453063\pi$
$720$	0	0
$721$	13.5761	0.505599
$722$	0	0
$723$	31.9351	1.18768
$724$	0	0
$725$	−36.2060	−1.34466
$726$	0	0
$727$	9.18917	0.340808	0.170404	−	0.985374i	$-0.445493\pi$
$727$	9.18917	0.340808	0.170404	+	0.985374i	$0.445493\pi$
$728$	0	0
$729$	−9.60100	−0.355593
$730$	0	0
$731$	64.2625	2.37684
$732$	0	0
$733$	−4.40073	−0.162545	−0.0812723	−	0.996692i	$-0.525898\pi$
$733$	−4.40073	−0.162545	−0.0812723	+	0.996692i	$0.525898\pi$
$734$	0	0
$735$	−9.36370	−0.345385
$736$	0	0
$737$	26.1977	0.965005
$738$	0	0
$739$	24.8942	0.915748	0.457874	−	0.889017i	$-0.348611\pi$
$739$	24.8942	0.915748	0.457874	+	0.889017i	$0.348611\pi$
$740$	0	0
$741$	36.8303	1.35300
$742$	0	0
$743$	21.8678	0.802251	0.401125	−	0.916023i	$-0.368619\pi$
$743$	21.8678	0.802251	0.401125	+	0.916023i	$0.368619\pi$
$744$	0	0
$745$	−6.57361	−0.240839
$746$	0	0
$747$	4.18531	0.153132
$748$	0	0
$749$	8.41642	0.307529
$750$	0	0
$751$	17.6433	0.643813	0.321906	−	0.946771i	$-0.395676\pi$
$751$	17.6433	0.643813	0.321906	+	0.946771i	$0.395676\pi$
$752$	0	0
$753$	2.26361	0.0824907
$754$	0	0
$755$	−7.84100	−0.285363
$756$	0	0
$757$	−21.5830	−0.784447	−0.392223	−	0.919870i	$-0.628294\pi$
$757$	−21.5830	−0.784447	−0.392223	+	0.919870i	$0.628294\pi$
$758$	0	0
$759$	−1.57081	−0.0570166
$760$	0	0
$761$	−32.2336	−1.16847	−0.584234	−	0.811585i	$-0.698605\pi$
$761$	−32.2336	−1.16847	−0.584234	+	0.811585i	$0.698605\pi$
$762$	0	0
$763$	9.15292	0.331358
$764$	0	0
$765$	−11.9035	−0.430371
$766$	0	0
$767$	−15.8171	−0.571123
$768$	0	0
$769$	−42.7465	−1.54148	−0.770738	−	0.637152i	$-0.780112\pi$
$769$	−42.7465	−1.54148	−0.770738	+	0.637152i	$0.780112\pi$
$770$	0	0
$771$	4.63807	0.167036
$772$	0	0
$773$	46.6095	1.67643	0.838214	−	0.545341i	$-0.183600\pi$
$773$	46.6095	1.67643	0.838214	+	0.545341i	$0.183600\pi$
$774$	0	0
$775$	−25.6586	−0.921684
$776$	0	0
$777$	−19.0152	−0.682168
$778$	0	0
$779$	−30.6009	−1.09639
$780$	0	0
$781$	−10.4015	−0.372194
$782$	0	0
$783$	16.2502	0.580735
$784$	0	0
$785$	−12.8571	−0.458890
$786$	0	0
$787$	−14.0042	−0.499197	−0.249598	−	0.968349i	$-0.580299\pi$
$787$	−14.0042	−0.499197	−0.249598	+	0.968349i	$0.580299\pi$
$788$	0	0
$789$	1.29946	0.0462621
$790$	0	0
$791$	−7.20199	−0.256073
$792$	0	0
$793$	9.00925	0.319928
$794$	0	0
$795$	−5.31341	−0.188447
$796$	0	0
$797$	9.21045	0.326251	0.163126	−	0.986605i	$-0.447842\pi$
$797$	9.21045	0.326251	0.163126	+	0.986605i	$0.447842\pi$
$798$	0	0
$799$	−55.9476	−1.97928
$800$	0	0
$801$	−18.6536	−0.659094
$802$	0	0
$803$	13.6710	0.482440
$804$	0	0
$805$	0.348354	0.0122779
$806$	0	0
$807$	53.0502	1.86746
$808$	0	0
$809$	−9.36994	−0.329429	−0.164715	−	0.986341i	$-0.552670\pi$
$809$	−9.36994	−0.329429	−0.164715	+	0.986341i	$0.552670\pi$
$810$	0	0
$811$	−10.0009	−0.351181	−0.175590	−	0.984463i	$-0.556183\pi$
$811$	−10.0009	−0.351181	−0.175590	+	0.984463i	$0.556183\pi$
$812$	0	0
$813$	5.94799	0.208605
$814$	0	0
$815$	−7.20951	−0.252538
$816$	0	0
$817$	53.1877	1.86080
$818$	0	0
$819$	7.13418	0.249289
$820$	0	0
$821$	−25.9195	−0.904596	−0.452298	−	0.891867i	$-0.649396\pi$
$821$	−25.9195	−0.904596	−0.452298	+	0.891867i	$0.649396\pi$
$822$	0	0
$823$	29.5794	1.03107	0.515537	−	0.856867i	$-0.327592\pi$
$823$	29.5794	1.03107	0.515537	+	0.856867i	$0.327592\pi$
$824$	0	0
$825$	19.0781	0.664216
$826$	0	0
$827$	−19.7456	−0.686622	−0.343311	−	0.939222i	$-0.611548\pi$
$827$	−19.7456	−0.686622	−0.343311	+	0.939222i	$0.611548\pi$
$828$	0	0
$829$	12.6337	0.438788	0.219394	−	0.975636i	$-0.429592\pi$
$829$	12.6337	0.438788	0.219394	+	0.975636i	$0.429592\pi$
$830$	0	0
$831$	13.9129	0.482634
$832$	0	0
$833$	−39.8536	−1.38085
$834$	0	0
$835$	−14.7530	−0.510550
$836$	0	0
$837$	11.5163	0.398060
$838$	0	0
$839$	1.58362	0.0546727	0.0273363	−	0.999626i	$-0.491297\pi$
$839$	1.58362	0.0546727	0.0273363	+	0.999626i	$0.491297\pi$
$840$	0	0
$841$	38.1510	1.31555
$842$	0	0
$843$	46.3545	1.59653
$844$	0	0
$845$	4.45625	0.153300
$846$	0	0
$847$	9.24219	0.317565
$848$	0	0
$849$	−26.1958	−0.899039
$850$	0	0
$851$	−2.43396	−0.0834350
$852$	0	0
$853$	−51.2066	−1.75328	−0.876641	−	0.481146i	$-0.840221\pi$
$853$	−51.2066	−1.75328	−0.876641	+	0.481146i	$0.840221\pi$
$854$	0	0
$855$	−9.85207	−0.336934
$856$	0	0
$857$	−12.8565	−0.439168	−0.219584	−	0.975594i	$-0.570470\pi$
$857$	−12.8565	−0.439168	−0.219584	+	0.975594i	$0.570470\pi$
$858$	0	0
$859$	−8.32764	−0.284135	−0.142068	−	0.989857i	$-0.545375\pi$
$859$	−8.32764	−0.284135	−0.142068	+	0.989857i	$0.545375\pi$
$860$	0	0
$861$	−14.2999	−0.487340
$862$	0	0
$863$	−30.2092	−1.02833	−0.514166	−	0.857691i	$-0.671898\pi$
$863$	−30.2092	−1.02833	−0.514166	+	0.857691i	$0.671898\pi$
$864$	0	0
$865$	7.54839	0.256653
$866$	0	0
$867$	−83.7420	−2.84403
$868$	0	0
$869$	−17.7168	−0.601003
$870$	0	0
$871$	−36.7415	−1.24494
$872$	0	0
$873$	35.5193	1.20215
$874$	0	0
$875$	−9.01888	−0.304894
$876$	0	0
$877$	17.7485	0.599324	0.299662	−	0.954046i	$-0.403126\pi$
$877$	17.7485	0.599324	0.299662	+	0.954046i	$0.403126\pi$
$878$	0	0
$879$	57.7447	1.94768
$880$	0	0
$881$	41.2112	1.38844	0.694220	−	0.719762i	$-0.255749\pi$
$881$	41.2112	1.38844	0.694220	+	0.719762i	$0.255749\pi$
$882$	0	0
$883$	−7.59649	−0.255642	−0.127821	−	0.991797i	$-0.540798\pi$
$883$	−7.59649	−0.255642	−0.127821	+	0.991797i	$0.540798\pi$
$884$	0	0
$885$	10.2073	0.343114
$886$	0	0
$887$	46.3583	1.55656	0.778279	−	0.627919i	$-0.216093\pi$
$887$	46.3583	1.55656	0.778279	+	0.627919i	$0.216093\pi$
$888$	0	0
$889$	−7.31828	−0.245447
$890$	0	0
$891$	−20.7175	−0.694062
$892$	0	0
$893$	−46.3058	−1.54956
$894$	0	0
$895$	7.00629	0.234194
$896$	0	0
$897$	2.20301	0.0735563
$898$	0	0
$899$	47.5888	1.58718
$900$	0	0
$901$	−22.6148	−0.753409
$902$	0	0
$903$	24.8548	0.827117
$904$	0	0
$905$	−2.42332	−0.0805538
$906$	0	0
$907$	−52.5270	−1.74413	−0.872065	−	0.489389i	$-0.837220\pi$
$907$	−52.5270	−1.74413	−0.872065	+	0.489389i	$0.837220\pi$
$908$	0	0
$909$	−16.9893	−0.563500
$910$	0	0
$911$	50.9144	1.68687	0.843435	−	0.537232i	$-0.180530\pi$
$911$	50.9144	1.68687	0.843435	+	0.537232i	$0.180530\pi$
$912$	0	0
$913$	3.75892	0.124402
$914$	0	0
$915$	−5.81395	−0.192203
$916$	0	0
$917$	8.05987	0.266160
$918$	0	0
$919$	−28.5898	−0.943089	−0.471544	−	0.881842i	$-0.656303\pi$
$919$	−28.5898	−0.943089	−0.471544	+	0.881842i	$0.656303\pi$
$920$	0	0
$921$	54.3946	1.79236
$922$	0	0
$923$	14.5878	0.480162
$924$	0	0
$925$	29.5615	0.971977
$926$	0	0
$927$	−22.9663	−0.754311
$928$	0	0
$929$	28.9074	0.948422	0.474211	−	0.880411i	$-0.342733\pi$
$929$	28.9074	0.948422	0.474211	+	0.880411i	$0.342733\pi$
$930$	0	0
$931$	−32.9854	−1.08105
$932$	0	0
$933$	62.6235	2.05020
$934$	0	0
$935$	−10.6908	−0.349626
$936$	0	0
$937$	−15.6882	−0.512511	−0.256256	−	0.966609i	$-0.582489\pi$
$937$	−15.6882	−0.512511	−0.256256	+	0.966609i	$0.582489\pi$
$938$	0	0
$939$	19.4646	0.635203
$940$	0	0
$941$	28.1439	0.917466	0.458733	−	0.888574i	$-0.348303\pi$
$941$	28.1439	0.917466	0.458733	+	0.888574i	$0.348303\pi$
$942$	0	0
$943$	−1.83039	−0.0596058
$944$	0	0
$945$	1.89895	0.0617728
$946$	0	0
$947$	−60.1309	−1.95399	−0.976996	−	0.213257i	$-0.931593\pi$
$947$	−60.1309	−1.95399	−0.976996	+	0.213257i	$0.931593\pi$
$948$	0	0
$949$	−19.1732	−0.622389
$950$	0	0
$951$	−71.0900	−2.30525
$952$	0	0
$953$	8.28620	0.268416	0.134208	−	0.990953i	$-0.457151\pi$
$953$	8.28620	0.268416	0.134208	+	0.990953i	$0.457151\pi$
$954$	0	0
$955$	4.84137	0.156663
$956$	0	0
$957$	−35.3841	−1.14381
$958$	0	0
$959$	4.58361	0.148012
$960$	0	0
$961$	2.72543	0.0879172
$962$	0	0
$963$	−14.2378	−0.458808
$964$	0	0
$965$	4.83629	0.155686
$966$	0	0
$967$	26.3828	0.848413	0.424207	−	0.905565i	$-0.360553\pi$
$967$	26.3828	0.848413	0.424207	+	0.905565i	$0.360553\pi$
$968$	0	0
$969$	−101.160	−3.24972
$970$	0	0
$971$	−11.5516	−0.370709	−0.185354	−	0.982672i	$-0.559343\pi$
$971$	−11.5516	−0.370709	−0.185354	+	0.982672i	$0.559343\pi$
$972$	0	0
$973$	−7.06107	−0.226368
$974$	0	0
$975$	−26.7565	−0.856895
$976$	0	0
$977$	35.3342	1.13044	0.565220	−	0.824940i	$-0.308791\pi$
$977$	35.3342	1.13044	0.565220	+	0.824940i	$0.308791\pi$
$978$	0	0
$979$	−16.7533	−0.535437
$980$	0	0
$981$	−15.4838	−0.494359
$982$	0	0
$983$	−46.6561	−1.48810	−0.744049	−	0.668125i	$-0.767097\pi$
$983$	−46.6561	−1.48810	−0.744049	+	0.668125i	$0.767097\pi$
$984$	0	0
$985$	−13.6421	−0.434672
$986$	0	0
$987$	−21.6389	−0.688772
$988$	0	0
$989$	3.18143	0.101163
$990$	0	0
$991$	23.1407	0.735087	0.367544	−	0.930006i	$-0.380199\pi$
$991$	23.1407	0.735087	0.367544	+	0.930006i	$0.380199\pi$
$992$	0	0
$993$	14.8608	0.471594
$994$	0	0
$995$	−11.6388	−0.368974
$996$	0	0
$997$	26.2539	0.831469	0.415735	−	0.909486i	$-0.363525\pi$
$997$	26.2539	0.831469	0.415735	+	0.909486i	$0.363525\pi$
$998$	0	0
$999$	−13.2680	−0.419780

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8032.2.a.j.1.5	yes	30
4.3	odd	2		8032.2.a.g.1.26	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8032.2.a.g.1.26	✓	30	4.3	odd	2
8032.2.a.j.1.5	yes	30	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:	\( N \)	\(=\)	\( 8032 = 2^{5} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8032.a (trivial)

\(f(q)\)	\(=\)	\(q-2.26361 q^{3} -0.762700 q^{5} -1.25553 q^{7} +2.12395 q^{9} +O(q^{10})\)	\(q-2.26361 q^{3} -0.762700 q^{5} -1.25553 q^{7} +2.12395 q^{9} +1.90757 q^{11} -2.67531 q^{13} +1.72646 q^{15} +7.34812 q^{17} +6.08177 q^{19} +2.84203 q^{21} +0.363781 q^{23} -4.41829 q^{25} +1.98304 q^{27} +8.19457 q^{29} +5.80736 q^{31} -4.31800 q^{33} +0.957592 q^{35} -6.69072 q^{37} +6.05586 q^{39} -5.03158 q^{41} +8.74544 q^{43} -1.61994 q^{45} -7.61386 q^{47} -5.42365 q^{49} -16.6333 q^{51} -3.07763 q^{53} -1.45490 q^{55} -13.7668 q^{57} +5.91226 q^{59} -3.36756 q^{61} -2.66668 q^{63} +2.04046 q^{65} +13.7336 q^{67} -0.823460 q^{69} -5.45275 q^{71} +7.16674 q^{73} +10.0013 q^{75} -2.39501 q^{77} -9.28766 q^{79} -10.8607 q^{81} +1.97053 q^{83} -5.60441 q^{85} -18.5493 q^{87} -8.78253 q^{89} +3.35893 q^{91} -13.1456 q^{93} -4.63857 q^{95} +16.7232 q^{97} +4.05157 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 9 q^{3} + 5 q^{7} + 35 q^{9}+O(q^{10}) \) 30 * q + 9 * q^3 + 5 * q^7 + 35 * q^9	\( 30 q + 9 q^{3} + 5 q^{7} + 35 q^{9} + 31 q^{11} + 5 q^{13} + 8 q^{15} - q^{17} + 29 q^{19} + 6 q^{21} + 27 q^{23} + 34 q^{25} + 36 q^{27} + q^{29} - 9 q^{31} - 8 q^{33} + 51 q^{35} + 5 q^{37} - 8 q^{39} - 3 q^{41} + 43 q^{43} + 42 q^{47} + 41 q^{49} + 54 q^{51} - 11 q^{53} - 10 q^{55} + 2 q^{57} + 49 q^{59} + 21 q^{61} + 15 q^{63} - 14 q^{65} + 68 q^{67} - 10 q^{69} + 44 q^{71} + 25 q^{73} + 51 q^{75} - 20 q^{77} - 49 q^{79} + 6 q^{81} + 88 q^{83} - 5 q^{85} + 16 q^{87} - 16 q^{89} + 46 q^{91} - 4 q^{93} + 28 q^{95} - 4 q^{97} + 71 q^{99}+O(q^{100}) \) 30 * q + 9 * q^3 + 5 * q^7 + 35 * q^9 + 31 * q^11 + 5 * q^13 + 8 * q^15 - q^17 + 29 * q^19 + 6 * q^21 + 27 * q^23 + 34 * q^25 + 36 * q^27 + q^29 - 9 * q^31 - 8 * q^33 + 51 * q^35 + 5 * q^37 - 8 * q^39 - 3 * q^41 + 43 * q^43 + 42 * q^47 + 41 * q^49 + 54 * q^51 - 11 * q^53 - 10 * q^55 + 2 * q^57 + 49 * q^59 + 21 * q^61 + 15 * q^63 - 14 * q^65 + 68 * q^67 - 10 * q^69 + 44 * q^71 + 25 * q^73 + 51 * q^75 - 20 * q^77 - 49 * q^79 + 6 * q^81 + 88 * q^83 - 5 * q^85 + 16 * q^87 - 16 * q^89 + 46 * q^91 - 4 * q^93 + 28 * q^95 - 4 * q^97 + 71 * q^99

Introduction

L-functions

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