LMFDB - Embedded newform 4024.2.a.e.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4024,2,Mod(1,4024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4024, 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("4024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4024 = 2^{3} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4024.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:	$32.1318017734$
Analytic rank:	$1$
Dimension:	$29$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	4024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.46401 q^{3} -4.08257 q^{5} -4.62698 q^{7} +3.07134 q^{9} +O(q^{10})$	$q-2.46401 q^{3} -4.08257 q^{5} -4.62698 q^{7} +3.07134 q^{9} -4.22967 q^{11} +4.60311 q^{13} +10.0595 q^{15} +0.221030 q^{17} -0.311525 q^{19} +11.4009 q^{21} -7.93188 q^{23} +11.6674 q^{25} -0.175772 q^{27} -4.36735 q^{29} +3.02767 q^{31} +10.4220 q^{33} +18.8900 q^{35} +7.63378 q^{37} -11.3421 q^{39} -8.18730 q^{41} -4.18112 q^{43} -12.5389 q^{45} +2.06218 q^{47} +14.4090 q^{49} -0.544621 q^{51} +10.6360 q^{53} +17.2679 q^{55} +0.767600 q^{57} +9.22303 q^{59} +8.58075 q^{61} -14.2110 q^{63} -18.7925 q^{65} +13.9989 q^{67} +19.5442 q^{69} +6.79903 q^{71} -0.575248 q^{73} -28.7485 q^{75} +19.5706 q^{77} +3.32197 q^{79} -8.78090 q^{81} -10.0986 q^{83} -0.902372 q^{85} +10.7612 q^{87} -14.3155 q^{89} -21.2985 q^{91} -7.46020 q^{93} +1.27182 q^{95} -15.8228 q^{97} -12.9907 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9}+O(q^{10}) $ 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9	$ 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9} - 27 q^{11} + 16 q^{13} - 14 q^{15} - 15 q^{17} - 14 q^{19} + q^{21} - 25 q^{23} + 21 q^{25} - 25 q^{27} - 13 q^{29} - 27 q^{31} - 9 q^{33} - 29 q^{35} + 35 q^{37} - 38 q^{39} - 30 q^{41} - 38 q^{43} + q^{45} - 35 q^{47} + 14 q^{49} - 21 q^{51} + 2 q^{53} - 25 q^{55} - 25 q^{57} - 40 q^{59} + 10 q^{61} - 56 q^{63} - 50 q^{65} - 31 q^{67} + 11 q^{69} - 65 q^{71} - 23 q^{73} - 32 q^{75} + 13 q^{77} - 44 q^{79} - 7 q^{81} - 41 q^{83} + 26 q^{85} - 25 q^{87} - 48 q^{89} - 44 q^{91} + 25 q^{93} - 75 q^{95} - 18 q^{97} - 80 q^{99}+O(q^{100}) $ 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9 - 27 * q^11 + 16 * q^13 - 14 * q^15 - 15 * q^17 - 14 * q^19 + q^21 - 25 * q^23 + 21 * q^25 - 25 * q^27 - 13 * q^29 - 27 * q^31 - 9 * q^33 - 29 * q^35 + 35 * q^37 - 38 * q^39 - 30 * q^41 - 38 * q^43 + q^45 - 35 * q^47 + 14 * q^49 - 21 * q^51 + 2 * q^53 - 25 * q^55 - 25 * q^57 - 40 * q^59 + 10 * q^61 - 56 * q^63 - 50 * q^65 - 31 * q^67 + 11 * q^69 - 65 * q^71 - 23 * q^73 - 32 * q^75 + 13 * q^77 - 44 * q^79 - 7 * q^81 - 41 * q^83 + 26 * q^85 - 25 * q^87 - 48 * q^89 - 44 * q^91 + 25 * q^93 - 75 * q^95 - 18 * q^97 - 80 * 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
$3$	−2.46401	−1.42260	−0.711298	−	0.702891i	$-0.751892\pi$
$3$	−2.46401	−1.42260	−0.711298	+	0.702891i	$0.751892\pi$
$4$	0	0
$5$	−4.08257	−1.82578	−0.912891	−	0.408204i	$-0.866155\pi$
$5$	−4.08257	−1.82578	−0.912891	+	0.408204i	$0.866155\pi$
$6$	0	0
$7$	−4.62698	−1.74884	−0.874418	−	0.485174i	$-0.838756\pi$
$7$	−4.62698	−1.74884	−0.874418	+	0.485174i	$0.838756\pi$
$8$	0	0
$9$	3.07134	1.02378
$10$	0	0
$11$	−4.22967	−1.27529	−0.637647	−	0.770328i	$-0.720092\pi$
$11$	−4.22967	−1.27529	−0.637647	+	0.770328i	$0.720092\pi$
$12$	0	0
$13$	4.60311	1.27667	0.638337	−	0.769757i	$-0.279623\pi$
$13$	4.60311	1.27667	0.638337	+	0.769757i	$0.279623\pi$
$14$	0	0
$15$	10.0595	2.59735
$16$	0	0
$17$	0.221030	0.0536077	0.0268039	−	0.999641i	$-0.491467\pi$
$17$	0.221030	0.0536077	0.0268039	+	0.999641i	$0.491467\pi$
$18$	0	0
$19$	−0.311525	−0.0714687	−0.0357344	−	0.999361i	$-0.511377\pi$
$19$	−0.311525	−0.0714687	−0.0357344	+	0.999361i	$0.511377\pi$
$20$	0	0
$21$	11.4009	2.48789
$22$	0	0
$23$	−7.93188	−1.65391	−0.826955	−	0.562268i	$-0.809929\pi$
$23$	−7.93188	−1.65391	−0.826955	+	0.562268i	$0.809929\pi$
$24$	0	0
$25$	11.6674	2.33348
$26$	0	0
$27$	−0.175772	−0.0338273
$28$	0	0
$29$	−4.36735	−0.810996	−0.405498	−	0.914096i	$-0.632902\pi$
$29$	−4.36735	−0.810996	−0.405498	+	0.914096i	$0.632902\pi$
$30$	0	0
$31$	3.02767	0.543785	0.271893	−	0.962328i	$-0.412350\pi$
$31$	3.02767	0.543785	0.271893	+	0.962328i	$0.412350\pi$
$32$	0	0
$33$	10.4220	1.81423
$34$	0	0
$35$	18.8900	3.19299
$36$	0	0
$37$	7.63378	1.25499	0.627493	−	0.778623i	$-0.284081\pi$
$37$	7.63378	1.25499	0.627493	+	0.778623i	$0.284081\pi$
$38$	0	0
$39$	−11.3421	−1.81619
$40$	0	0
$41$	−8.18730	−1.27864	−0.639320	−	0.768940i	$-0.720784\pi$
$41$	−8.18730	−1.27864	−0.639320	+	0.768940i	$0.720784\pi$
$42$	0	0
$43$	−4.18112	−0.637615	−0.318807	−	0.947820i	$-0.603282\pi$
$43$	−4.18112	−0.637615	−0.318807	+	0.947820i	$0.603282\pi$
$44$	0	0
$45$	−12.5389	−1.86920
$46$	0	0
$47$	2.06218	0.300799	0.150400	−	0.988625i	$-0.451944\pi$
$47$	2.06218	0.300799	0.150400	+	0.988625i	$0.451944\pi$
$48$	0	0
$49$	14.4090	2.05843
$50$	0	0
$51$	−0.544621	−0.0762621
$52$	0	0
$53$	10.6360	1.46097	0.730487	−	0.682927i	$-0.239293\pi$
$53$	10.6360	1.46097	0.730487	+	0.682927i	$0.239293\pi$
$54$	0	0
$55$	17.2679	2.32841
$56$	0	0
$57$	0.767600	0.101671
$58$	0	0
$59$	9.22303	1.20074	0.600368	−	0.799724i	$-0.295021\pi$
$59$	9.22303	1.20074	0.600368	+	0.799724i	$0.295021\pi$
$60$	0	0
$61$	8.58075	1.09865	0.549326	−	0.835608i	$-0.314884\pi$
$61$	8.58075	1.09865	0.549326	+	0.835608i	$0.314884\pi$
$62$	0	0
$63$	−14.2110	−1.79042
$64$	0	0
$65$	−18.7925	−2.33093
$66$	0	0
$67$	13.9989	1.71024	0.855121	−	0.518428i	$-0.173483\pi$
$67$	13.9989	1.71024	0.855121	+	0.518428i	$0.173483\pi$
$68$	0	0
$69$	19.5442	2.35285
$70$	0	0
$71$	6.79903	0.806897	0.403448	−	0.915002i	$-0.367811\pi$
$71$	6.79903	0.806897	0.403448	+	0.915002i	$0.367811\pi$
$72$	0	0
$73$	−0.575248	−0.0673277	−0.0336638	−	0.999433i	$-0.510718\pi$
$73$	−0.575248	−0.0673277	−0.0336638	+	0.999433i	$0.510718\pi$
$74$	0	0
$75$	−28.7485	−3.31959
$76$	0	0
$77$	19.5706	2.23028
$78$	0	0
$79$	3.32197	0.373751	0.186875	−	0.982384i	$-0.440164\pi$
$79$	3.32197	0.373751	0.186875	+	0.982384i	$0.440164\pi$
$80$	0	0
$81$	−8.78090	−0.975656
$82$	0	0
$83$	−10.0986	−1.10847	−0.554233	−	0.832362i	$-0.686988\pi$
$83$	−10.0986	−1.10847	−0.554233	+	0.832362i	$0.686988\pi$
$84$	0	0
$85$	−0.902372	−0.0978760
$86$	0	0
$87$	10.7612	1.15372
$88$	0	0
$89$	−14.3155	−1.51744	−0.758719	−	0.651418i	$-0.774174\pi$
$89$	−14.3155	−1.51744	−0.758719	+	0.651418i	$0.774174\pi$
$90$	0	0
$91$	−21.2985	−2.23269
$92$	0	0
$93$	−7.46020	−0.773587
$94$	0	0
$95$	1.27182	0.130486
$96$	0	0
$97$	−15.8228	−1.60657	−0.803283	−	0.595598i	$-0.796915\pi$
$97$	−15.8228	−1.60657	−0.803283	+	0.595598i	$0.796915\pi$
$98$	0	0
$99$	−12.9907	−1.30562
$100$	0	0
$101$	−11.8284	−1.17697	−0.588486	−	0.808507i	$-0.700276\pi$
$101$	−11.8284	−1.17697	−0.588486	+	0.808507i	$0.700276\pi$
$102$	0	0
$103$	−4.41802	−0.435320	−0.217660	−	0.976025i	$-0.569842\pi$
$103$	−4.41802	−0.435320	−0.217660	+	0.976025i	$0.569842\pi$
$104$	0	0
$105$	−46.5451	−4.54234
$106$	0	0
$107$	6.79070	0.656481	0.328241	−	0.944594i	$-0.393544\pi$
$107$	6.79070	0.656481	0.328241	+	0.944594i	$0.393544\pi$
$108$	0	0
$109$	7.65591	0.733303	0.366651	−	0.930358i	$-0.380504\pi$
$109$	7.65591	0.733303	0.366651	+	0.930358i	$0.380504\pi$
$110$	0	0
$111$	−18.8097	−1.78534
$112$	0	0
$113$	16.3866	1.54152	0.770761	−	0.637125i	$-0.219876\pi$
$113$	16.3866	1.54152	0.770761	+	0.637125i	$0.219876\pi$
$114$	0	0
$115$	32.3824	3.01968
$116$	0	0
$117$	14.1377	1.30703
$118$	0	0
$119$	−1.02270	−0.0937511
$120$	0	0
$121$	6.89014	0.626376
$122$	0	0
$123$	20.1736	1.81899
$124$	0	0
$125$	−27.2201	−2.43464
$126$	0	0
$127$	−18.2872	−1.62273	−0.811363	−	0.584543i	$-0.801274\pi$
$127$	−18.2872	−1.62273	−0.811363	+	0.584543i	$0.801274\pi$
$128$	0	0
$129$	10.3023	0.907068
$130$	0	0
$131$	−3.34826	−0.292539	−0.146270	−	0.989245i	$-0.546727\pi$
$131$	−3.34826	−0.292539	−0.146270	+	0.989245i	$0.546727\pi$
$132$	0	0
$133$	1.44142	0.124987
$134$	0	0
$135$	0.717602	0.0617613
$136$	0	0
$137$	−15.2507	−1.30295	−0.651476	−	0.758669i	$-0.725850\pi$
$137$	−15.2507	−1.30295	−0.651476	+	0.758669i	$0.725850\pi$
$138$	0	0
$139$	−1.11961	−0.0949637	−0.0474818	−	0.998872i	$-0.515120\pi$
$139$	−1.11961	−0.0949637	−0.0474818	+	0.998872i	$0.515120\pi$
$140$	0	0
$141$	−5.08122	−0.427916
$142$	0	0
$143$	−19.4697	−1.62814
$144$	0	0
$145$	17.8300	1.48070
$146$	0	0
$147$	−35.5039	−2.92831
$148$	0	0
$149$	0.546372	0.0447606	0.0223803	−	0.999750i	$-0.492876\pi$
$149$	0.546372	0.0447606	0.0223803	+	0.999750i	$0.492876\pi$
$150$	0	0
$151$	1.51037	0.122913	0.0614563	−	0.998110i	$-0.480426\pi$
$151$	1.51037	0.122913	0.0614563	+	0.998110i	$0.480426\pi$
$152$	0	0
$153$	0.678858	0.0548824
$154$	0	0
$155$	−12.3607	−0.992833
$156$	0	0
$157$	12.0421	0.961061	0.480531	−	0.876978i	$-0.340444\pi$
$157$	12.0421	0.961061	0.480531	+	0.876978i	$0.340444\pi$
$158$	0	0
$159$	−26.2073	−2.07837
$160$	0	0
$161$	36.7007	2.89242
$162$	0	0
$163$	−11.3557	−0.889448	−0.444724	−	0.895668i	$-0.646698\pi$
$163$	−11.3557	−0.889448	−0.444724	+	0.895668i	$0.646698\pi$
$164$	0	0
$165$	−42.5483	−3.31238
$166$	0	0
$167$	9.83129	0.760768	0.380384	−	0.924829i	$-0.375792\pi$
$167$	9.83129	0.760768	0.380384	+	0.924829i	$0.375792\pi$
$168$	0	0
$169$	8.18866	0.629897
$170$	0	0
$171$	−0.956798	−0.0731682
$172$	0	0
$173$	6.64081	0.504891	0.252446	−	0.967611i	$-0.418765\pi$
$173$	6.64081	0.504891	0.252446	+	0.967611i	$0.418765\pi$
$174$	0	0
$175$	−53.9848	−4.08087
$176$	0	0
$177$	−22.7256	−1.70816
$178$	0	0
$179$	−11.6448	−0.870374	−0.435187	−	0.900340i	$-0.643318\pi$
$179$	−11.6448	−0.870374	−0.435187	+	0.900340i	$0.643318\pi$
$180$	0	0
$181$	21.2616	1.58036	0.790179	−	0.612876i	$-0.209987\pi$
$181$	21.2616	1.58036	0.790179	+	0.612876i	$0.209987\pi$
$182$	0	0
$183$	−21.1430	−1.56294
$184$	0	0
$185$	−31.1654	−2.29133
$186$	0	0
$187$	−0.934886	−0.0683656
$188$	0	0
$189$	0.813294	0.0591585
$190$	0	0
$191$	9.29631	0.672658	0.336329	−	0.941745i	$-0.390815\pi$
$191$	9.29631	0.672658	0.336329	+	0.941745i	$0.390815\pi$
$192$	0	0
$193$	−12.4520	−0.896314	−0.448157	−	0.893955i	$-0.647919\pi$
$193$	−12.4520	−0.896314	−0.448157	+	0.893955i	$0.647919\pi$
$194$	0	0
$195$	46.3050	3.31597
$196$	0	0
$197$	−16.2464	−1.15751	−0.578755	−	0.815501i	$-0.696461\pi$
$197$	−16.2464	−1.15751	−0.578755	+	0.815501i	$0.696461\pi$
$198$	0	0
$199$	−7.20845	−0.510993	−0.255497	−	0.966810i	$-0.582239\pi$
$199$	−7.20845	−0.510993	−0.255497	+	0.966810i	$0.582239\pi$
$200$	0	0
$201$	−34.4935	−2.43298
$202$	0	0
$203$	20.2076	1.41830
$204$	0	0
$205$	33.4252	2.33452
$206$	0	0
$207$	−24.3615	−1.69324
$208$	0	0
$209$	1.31765	0.0911437
$210$	0	0
$211$	0.634159	0.0436573	0.0218286	−	0.999762i	$-0.493051\pi$
$211$	0.634159	0.0436573	0.0218286	+	0.999762i	$0.493051\pi$
$212$	0	0
$213$	−16.7529	−1.14789
$214$	0	0
$215$	17.0697	1.16415
$216$	0	0
$217$	−14.0090	−0.950991
$218$	0	0
$219$	1.41742	0.0957801
$220$	0	0
$221$	1.01743	0.0684396
$222$	0	0
$223$	7.18758	0.481316	0.240658	−	0.970610i	$-0.422637\pi$
$223$	7.18758	0.481316	0.240658	+	0.970610i	$0.422637\pi$
$224$	0	0
$225$	35.8345	2.38896
$226$	0	0
$227$	6.53254	0.433580	0.216790	−	0.976218i	$-0.430441\pi$
$227$	6.53254	0.433580	0.216790	+	0.976218i	$0.430441\pi$
$228$	0	0
$229$	−18.8788	−1.24754	−0.623772	−	0.781607i	$-0.714401\pi$
$229$	−18.8788	−1.24754	−0.623772	+	0.781607i	$0.714401\pi$
$230$	0	0
$231$	−48.2222	−3.17279
$232$	0	0
$233$	5.11477	0.335079	0.167540	−	0.985865i	$-0.446418\pi$
$233$	5.11477	0.335079	0.167540	+	0.985865i	$0.446418\pi$
$234$	0	0
$235$	−8.41899	−0.549194
$236$	0	0
$237$	−8.18536	−0.531696
$238$	0	0
$239$	−0.591503	−0.0382611	−0.0191306	−	0.999817i	$-0.506090\pi$
$239$	−0.591503	−0.0382611	−0.0191306	+	0.999817i	$0.506090\pi$
$240$	0	0
$241$	22.5288	1.45121	0.725605	−	0.688111i	$-0.241560\pi$
$241$	22.5288	1.45121	0.725605	+	0.688111i	$0.241560\pi$
$242$	0	0
$243$	22.1635	1.42179
$244$	0	0
$245$	−58.8257	−3.75824
$246$	0	0
$247$	−1.43399	−0.0912423
$248$	0	0
$249$	24.8830	1.57690
$250$	0	0
$251$	−13.1933	−0.832753	−0.416377	−	0.909192i	$-0.636700\pi$
$251$	−13.1933	−0.832753	−0.416377	+	0.909192i	$0.636700\pi$
$252$	0	0
$253$	33.5492	2.10922
$254$	0	0
$255$	2.22345	0.139238
$256$	0	0
$257$	29.2080	1.82195	0.910974	−	0.412464i	$-0.135332\pi$
$257$	29.2080	1.82195	0.910974	+	0.412464i	$0.135332\pi$
$258$	0	0
$259$	−35.3214	−2.19476
$260$	0	0
$261$	−13.4136	−0.830280
$262$	0	0
$263$	8.25496	0.509022	0.254511	−	0.967070i	$-0.418085\pi$
$263$	8.25496	0.509022	0.254511	+	0.967070i	$0.418085\pi$
$264$	0	0
$265$	−43.4224	−2.66742
$266$	0	0
$267$	35.2734	2.15870
$268$	0	0
$269$	20.6411	1.25851	0.629254	−	0.777200i	$-0.283361\pi$
$269$	20.6411	1.25851	0.629254	+	0.777200i	$0.283361\pi$
$270$	0	0
$271$	12.2371	0.743353	0.371676	−	0.928362i	$-0.378783\pi$
$271$	12.2371	0.743353	0.371676	+	0.928362i	$0.378783\pi$
$272$	0	0
$273$	52.4798	3.17622
$274$	0	0
$275$	−49.3492	−2.97587
$276$	0	0
$277$	−25.8279	−1.55185	−0.775924	−	0.630827i	$-0.782716\pi$
$277$	−25.8279	−1.55185	−0.775924	+	0.630827i	$0.782716\pi$
$278$	0	0
$279$	9.29899	0.556716
$280$	0	0
$281$	1.70315	0.101602	0.0508008	−	0.998709i	$-0.483823\pi$
$281$	1.70315	0.101602	0.0508008	+	0.998709i	$0.483823\pi$
$282$	0	0
$283$	16.4714	0.979121	0.489561	−	0.871969i	$-0.337157\pi$
$283$	16.4714	0.979121	0.489561	+	0.871969i	$0.337157\pi$
$284$	0	0
$285$	−3.13378	−0.185629
$286$	0	0
$287$	37.8825	2.23613
$288$	0	0
$289$	−16.9511	−0.997126
$290$	0	0
$291$	38.9876	2.28549
$292$	0	0
$293$	25.3206	1.47924	0.739622	−	0.673022i	$-0.235004\pi$
$293$	25.3206	1.47924	0.739622	+	0.673022i	$0.235004\pi$
$294$	0	0
$295$	−37.6537	−2.19228
$296$	0	0
$297$	0.743458	0.0431398
$298$	0	0
$299$	−36.5113	−2.11151
$300$	0	0
$301$	19.3460	1.11508
$302$	0	0
$303$	29.1453	1.67436
$304$	0	0
$305$	−35.0315	−2.00590
$306$	0	0
$307$	27.7832	1.58567	0.792837	−	0.609434i	$-0.208603\pi$
$307$	27.7832	1.58567	0.792837	+	0.609434i	$0.208603\pi$
$308$	0	0
$309$	10.8860	0.619285
$310$	0	0
$311$	14.3259	0.812345	0.406172	−	0.913797i	$-0.366863\pi$
$311$	14.3259	0.812345	0.406172	+	0.913797i	$0.366863\pi$
$312$	0	0
$313$	−0.739032	−0.0417726	−0.0208863	−	0.999782i	$-0.506649\pi$
$313$	−0.739032	−0.0417726	−0.0208863	+	0.999782i	$0.506649\pi$
$314$	0	0
$315$	58.0175	3.26892
$316$	0	0
$317$	28.6636	1.60991	0.804954	−	0.593337i	$-0.202190\pi$
$317$	28.6636	1.60991	0.804954	+	0.593337i	$0.202190\pi$
$318$	0	0
$319$	18.4724	1.03426
$320$	0	0
$321$	−16.7323	−0.933908
$322$	0	0
$323$	−0.0688565	−0.00383128
$324$	0	0
$325$	53.7063	2.97909
$326$	0	0
$327$	−18.8642	−1.04319
$328$	0	0
$329$	−9.54166	−0.526049
$330$	0	0
$331$	−20.3598	−1.11907	−0.559537	−	0.828805i	$-0.689021\pi$
$331$	−20.3598	−1.11907	−0.559537	+	0.828805i	$0.689021\pi$
$332$	0	0
$333$	23.4459	1.28483
$334$	0	0
$335$	−57.1516	−3.12253
$336$	0	0
$337$	−14.7548	−0.803745	−0.401873	−	0.915696i	$-0.631641\pi$
$337$	−14.7548	−0.803745	−0.401873	+	0.915696i	$0.631641\pi$
$338$	0	0
$339$	−40.3767	−2.19296
$340$	0	0
$341$	−12.8061	−0.693487
$342$	0	0
$343$	−34.2813	−1.85101
$344$	0	0
$345$	−79.7906	−4.29578
$346$	0	0
$347$	−14.2577	−0.765392	−0.382696	−	0.923874i	$-0.625004\pi$
$347$	−14.2577	−0.765392	−0.382696	+	0.923874i	$0.625004\pi$
$348$	0	0
$349$	−30.4242	−1.62857	−0.814284	−	0.580467i	$-0.802870\pi$
$349$	−30.4242	−1.62857	−0.814284	+	0.580467i	$0.802870\pi$
$350$	0	0
$351$	−0.809099	−0.0431865
$352$	0	0
$353$	17.0913	0.909677	0.454838	−	0.890574i	$-0.349697\pi$
$353$	17.0913	0.909677	0.454838	+	0.890574i	$0.349697\pi$
$354$	0	0
$355$	−27.7575	−1.47322
$356$	0	0
$357$	2.51995	0.133370
$358$	0	0
$359$	−14.9380	−0.788399	−0.394199	−	0.919025i	$-0.628978\pi$
$359$	−14.9380	−0.788399	−0.394199	+	0.919025i	$0.628978\pi$
$360$	0	0
$361$	−18.9030	−0.994892
$362$	0	0
$363$	−16.9774	−0.891080
$364$	0	0
$365$	2.34849	0.122926
$366$	0	0
$367$	22.6526	1.18246	0.591228	−	0.806504i	$-0.298643\pi$
$367$	22.6526	1.18246	0.591228	+	0.806504i	$0.298643\pi$
$368$	0	0
$369$	−25.1459	−1.30905
$370$	0	0
$371$	−49.2128	−2.55500
$372$	0	0
$373$	19.2046	0.994378	0.497189	−	0.867642i	$-0.334365\pi$
$373$	19.2046	0.994378	0.497189	+	0.867642i	$0.334365\pi$
$374$	0	0
$375$	67.0705	3.46350
$376$	0	0
$377$	−20.1034	−1.03538
$378$	0	0
$379$	−2.52712	−0.129809	−0.0649047	−	0.997891i	$-0.520674\pi$
$379$	−2.52712	−0.129809	−0.0649047	+	0.997891i	$0.520674\pi$
$380$	0	0
$381$	45.0598	2.30848
$382$	0	0
$383$	27.6032	1.41046	0.705228	−	0.708981i	$-0.250845\pi$
$383$	27.6032	1.41046	0.705228	+	0.708981i	$0.250845\pi$
$384$	0	0
$385$	−79.8985	−4.07200
$386$	0	0
$387$	−12.8416	−0.652776
$388$	0	0
$389$	29.3154	1.48635	0.743175	−	0.669097i	$-0.233319\pi$
$389$	29.3154	1.48635	0.743175	+	0.669097i	$0.233319\pi$
$390$	0	0
$391$	−1.75319	−0.0886624
$392$	0	0
$393$	8.25015	0.416165
$394$	0	0
$395$	−13.5622	−0.682387
$396$	0	0
$397$	31.3666	1.57424	0.787122	−	0.616797i	$-0.211570\pi$
$397$	31.3666	1.57424	0.787122	+	0.616797i	$0.211570\pi$
$398$	0	0
$399$	−3.55167	−0.177806
$400$	0	0
$401$	−3.11920	−0.155765	−0.0778827	−	0.996963i	$-0.524816\pi$
$401$	−3.11920	−0.155765	−0.0778827	+	0.996963i	$0.524816\pi$
$402$	0	0
$403$	13.9367	0.694237
$404$	0	0
$405$	35.8487	1.78133
$406$	0	0
$407$	−32.2884	−1.60048
$408$	0	0
$409$	1.59345	0.0787911	0.0393955	−	0.999224i	$-0.487457\pi$
$409$	1.59345	0.0787911	0.0393955	+	0.999224i	$0.487457\pi$
$410$	0	0
$411$	37.5778	1.85357
$412$	0	0
$413$	−42.6748	−2.09989
$414$	0	0
$415$	41.2282	2.02381
$416$	0	0
$417$	2.75872	0.135095
$418$	0	0
$419$	−3.94866	−0.192905	−0.0964524	−	0.995338i	$-0.530750\pi$
$419$	−3.94866	−0.192905	−0.0964524	+	0.995338i	$0.530750\pi$
$420$	0	0
$421$	1.64867	0.0803511	0.0401756	−	0.999193i	$-0.487208\pi$
$421$	1.64867	0.0803511	0.0401756	+	0.999193i	$0.487208\pi$
$422$	0	0
$423$	6.33364	0.307952
$424$	0	0
$425$	2.57885	0.125092
$426$	0	0
$427$	−39.7030	−1.92136
$428$	0	0
$429$	47.9734	2.31618
$430$	0	0
$431$	−2.59853	−0.125167	−0.0625834	−	0.998040i	$-0.519934\pi$
$431$	−2.59853	−0.125167	−0.0625834	+	0.998040i	$0.519934\pi$
$432$	0	0
$433$	25.6220	1.23132	0.615658	−	0.788014i	$-0.288890\pi$
$433$	25.6220	1.23132	0.615658	+	0.788014i	$0.288890\pi$
$434$	0	0
$435$	−43.9333	−2.10644
$436$	0	0
$437$	2.47098	0.118203
$438$	0	0
$439$	−0.475905	−0.0227137	−0.0113568	−	0.999936i	$-0.503615\pi$
$439$	−0.475905	−0.0227137	−0.0113568	+	0.999936i	$0.503615\pi$
$440$	0	0
$441$	44.2548	2.10737
$442$	0	0
$443$	29.1429	1.38462	0.692311	−	0.721599i	$-0.256593\pi$
$443$	29.1429	1.38462	0.692311	+	0.721599i	$0.256593\pi$
$444$	0	0
$445$	58.4439	2.77051
$446$	0	0
$447$	−1.34627	−0.0636762
$448$	0	0
$449$	−5.88762	−0.277854	−0.138927	−	0.990303i	$-0.544365\pi$
$449$	−5.88762	−0.277854	−0.138927	+	0.990303i	$0.544365\pi$
$450$	0	0
$451$	34.6296	1.63064
$452$	0	0
$453$	−3.72157	−0.174855
$454$	0	0
$455$	86.9528	4.07641
$456$	0	0
$457$	−1.23704	−0.0578664	−0.0289332	−	0.999581i	$-0.509211\pi$
$457$	−1.23704	−0.0578664	−0.0289332	+	0.999581i	$0.509211\pi$
$458$	0	0
$459$	−0.0388509	−0.00181341
$460$	0	0
$461$	−28.1353	−1.31039	−0.655197	−	0.755458i	$-0.727414\pi$
$461$	−28.1353	−1.31039	−0.655197	+	0.755458i	$0.727414\pi$
$462$	0	0
$463$	−33.2186	−1.54380	−0.771900	−	0.635744i	$-0.780693\pi$
$463$	−33.2186	−1.54380	−0.771900	+	0.635744i	$0.780693\pi$
$464$	0	0
$465$	30.4568	1.41240
$466$	0	0
$467$	11.5230	0.533223	0.266611	−	0.963804i	$-0.414096\pi$
$467$	11.5230	0.533223	0.266611	+	0.963804i	$0.414096\pi$
$468$	0	0
$469$	−64.7729	−2.99093
$470$	0	0
$471$	−29.6717	−1.36720
$472$	0	0
$473$	17.6848	0.813147
$474$	0	0
$475$	−3.63468	−0.166771
$476$	0	0
$477$	32.6669	1.49571
$478$	0	0
$479$	−17.9475	−0.820043	−0.410022	−	0.912076i	$-0.634479\pi$
$479$	−17.9475	−0.820043	−0.410022	+	0.912076i	$0.634479\pi$
$480$	0	0
$481$	35.1391	1.60221
$482$	0	0
$483$	−90.4307	−4.11474
$484$	0	0
$485$	64.5979	2.93324
$486$	0	0
$487$	5.34290	0.242110	0.121055	−	0.992646i	$-0.461372\pi$
$487$	5.34290	0.242110	0.121055	+	0.992646i	$0.461372\pi$
$488$	0	0
$489$	27.9806	1.26533
$490$	0	0
$491$	−30.7848	−1.38930	−0.694650	−	0.719348i	$-0.744441\pi$
$491$	−30.7848	−1.38930	−0.694650	+	0.719348i	$0.744441\pi$
$492$	0	0
$493$	−0.965316	−0.0434756
$494$	0	0
$495$	53.0356	2.38378
$496$	0	0
$497$	−31.4590	−1.41113
$498$	0	0
$499$	−3.92768	−0.175827	−0.0879137	−	0.996128i	$-0.528020\pi$
$499$	−3.92768	−0.175827	−0.0879137	+	0.996128i	$0.528020\pi$
$500$	0	0
$501$	−24.2244	−1.08227
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	48.2904	2.14889
$506$	0	0
$507$	−20.1769	−0.896089
$508$	0	0
$509$	20.5572	0.911183	0.455592	−	0.890189i	$-0.349428\pi$
$509$	20.5572	0.911183	0.455592	+	0.890189i	$0.349428\pi$
$510$	0	0
$511$	2.66166	0.117745
$512$	0	0
$513$	0.0547574	0.00241760
$514$	0	0
$515$	18.0369	0.794799
$516$	0	0
$517$	−8.72234	−0.383608
$518$	0	0
$519$	−16.3630	−0.718256
$520$	0	0
$521$	−7.05801	−0.309217	−0.154609	−	0.987976i	$-0.549412\pi$
$521$	−7.05801	−0.309217	−0.154609	+	0.987976i	$0.549412\pi$
$522$	0	0
$523$	−10.5812	−0.462684	−0.231342	−	0.972873i	$-0.574312\pi$
$523$	−10.5812	−0.462684	−0.231342	+	0.972873i	$0.574312\pi$
$524$	0	0
$525$	133.019	5.80542
$526$	0	0
$527$	0.669207	0.0291511
$528$	0	0
$529$	39.9147	1.73542
$530$	0	0
$531$	28.3270	1.22929
$532$	0	0
$533$	−37.6871	−1.63241
$534$	0	0
$535$	−27.7235	−1.19859
$536$	0	0
$537$	28.6929	1.23819
$538$	0	0
$539$	−60.9453	−2.62510
$540$	0	0
$541$	−37.4738	−1.61112	−0.805562	−	0.592511i	$-0.798137\pi$
$541$	−37.4738	−1.61112	−0.805562	+	0.592511i	$0.798137\pi$
$542$	0	0
$543$	−52.3886	−2.24821
$544$	0	0
$545$	−31.2558	−1.33885
$546$	0	0
$547$	−38.1102	−1.62948	−0.814738	−	0.579829i	$-0.803119\pi$
$547$	−38.1102	−1.62948	−0.814738	+	0.579829i	$0.803119\pi$
$548$	0	0
$549$	26.3544	1.12478
$550$	0	0
$551$	1.36054	0.0579608
$552$	0	0
$553$	−15.3707	−0.653628
$554$	0	0
$555$	76.7919	3.25963
$556$	0	0
$557$	22.6236	0.958591	0.479295	−	0.877654i	$-0.340892\pi$
$557$	22.6236	0.958591	0.479295	+	0.877654i	$0.340892\pi$
$558$	0	0
$559$	−19.2462	−0.814026
$560$	0	0
$561$	2.30357	0.0972567
$562$	0	0
$563$	21.2518	0.895659	0.447829	−	0.894119i	$-0.352197\pi$
$563$	21.2518	0.895659	0.447829	+	0.894119i	$0.352197\pi$
$564$	0	0
$565$	−66.8994	−2.81448
$566$	0	0
$567$	40.6291	1.70626
$568$	0	0
$569$	−8.83009	−0.370177	−0.185088	−	0.982722i	$-0.559257\pi$
$569$	−8.83009	−0.370177	−0.185088	+	0.982722i	$0.559257\pi$
$570$	0	0
$571$	−15.3780	−0.643550	−0.321775	−	0.946816i	$-0.604279\pi$
$571$	−15.3780	−0.643550	−0.321775	+	0.946816i	$0.604279\pi$
$572$	0	0
$573$	−22.9062	−0.956920
$574$	0	0
$575$	−92.5442	−3.85936
$576$	0	0
$577$	14.1894	0.590712	0.295356	−	0.955387i	$-0.404562\pi$
$577$	14.1894	0.590712	0.295356	+	0.955387i	$0.404562\pi$
$578$	0	0
$579$	30.6818	1.27509
$580$	0	0
$581$	46.7261	1.93852
$582$	0	0
$583$	−44.9870	−1.86317
$584$	0	0
$585$	−57.7182	−2.38635
$586$	0	0
$587$	−3.31440	−0.136800	−0.0684000	−	0.997658i	$-0.521789\pi$
$587$	−3.31440	−0.136800	−0.0684000	+	0.997658i	$0.521789\pi$
$588$	0	0
$589$	−0.943194	−0.0388637
$590$	0	0
$591$	40.0313	1.64667
$592$	0	0
$593$	−42.1146	−1.72944	−0.864721	−	0.502253i	$-0.832505\pi$
$593$	−42.1146	−1.72944	−0.864721	+	0.502253i	$0.832505\pi$
$594$	0	0
$595$	4.17526	0.171169
$596$	0	0
$597$	17.7617	0.726937
$598$	0	0
$599$	38.1473	1.55865	0.779327	−	0.626617i	$-0.215561\pi$
$599$	38.1473	1.55865	0.779327	+	0.626617i	$0.215561\pi$
$600$	0	0
$601$	−23.5996	−0.962648	−0.481324	−	0.876543i	$-0.659844\pi$
$601$	−23.5996	−0.962648	−0.481324	+	0.876543i	$0.659844\pi$
$602$	0	0
$603$	42.9954	1.75091
$604$	0	0
$605$	−28.1295	−1.14363
$606$	0	0
$607$	−1.53225	−0.0621919	−0.0310960	−	0.999516i	$-0.509900\pi$
$607$	−1.53225	−0.0621919	−0.0310960	+	0.999516i	$0.509900\pi$
$608$	0	0
$609$	−49.7918	−2.01766
$610$	0	0
$611$	9.49244	0.384023
$612$	0	0
$613$	−4.80794	−0.194191	−0.0970954	−	0.995275i	$-0.530955\pi$
$613$	−4.80794	−0.194191	−0.0970954	+	0.995275i	$0.530955\pi$
$614$	0	0
$615$	−82.3600	−3.32108
$616$	0	0
$617$	24.0571	0.968504	0.484252	−	0.874928i	$-0.339092\pi$
$617$	24.0571	0.968504	0.484252	+	0.874928i	$0.339092\pi$
$618$	0	0
$619$	−15.6444	−0.628803	−0.314401	−	0.949290i	$-0.601804\pi$
$619$	−15.6444	−0.628803	−0.314401	+	0.949290i	$0.601804\pi$
$620$	0	0
$621$	1.39420	0.0559474
$622$	0	0
$623$	66.2375	2.65375
$624$	0	0
$625$	52.7909	2.11164
$626$	0	0
$627$	−3.24670	−0.129661
$628$	0	0
$629$	1.68730	0.0672769
$630$	0	0
$631$	−41.4871	−1.65158	−0.825788	−	0.563981i	$-0.809269\pi$
$631$	−41.4871	−1.65158	−0.825788	+	0.563981i	$0.809269\pi$
$632$	0	0
$633$	−1.56257	−0.0621067
$634$	0	0
$635$	74.6588	2.96274
$636$	0	0
$637$	66.3262	2.62794
$638$	0	0
$639$	20.8821	0.826084
$640$	0	0
$641$	23.1767	0.915423	0.457712	−	0.889101i	$-0.348669\pi$
$641$	23.1767	0.915423	0.457712	+	0.889101i	$0.348669\pi$
$642$	0	0
$643$	34.9833	1.37961	0.689804	−	0.723996i	$-0.257697\pi$
$643$	34.9833	1.37961	0.689804	+	0.723996i	$0.257697\pi$
$644$	0	0
$645$	−42.0599	−1.65611
$646$	0	0
$647$	−13.0957	−0.514846	−0.257423	−	0.966299i	$-0.582873\pi$
$647$	−13.0957	−0.514846	−0.257423	+	0.966299i	$0.582873\pi$
$648$	0	0
$649$	−39.0104	−1.53129
$650$	0	0
$651$	34.5182	1.35288
$652$	0	0
$653$	−9.15176	−0.358136	−0.179068	−	0.983837i	$-0.557308\pi$
$653$	−9.15176	−0.358136	−0.179068	+	0.983837i	$0.557308\pi$
$654$	0	0
$655$	13.6695	0.534112
$656$	0	0
$657$	−1.76678	−0.0689286
$658$	0	0
$659$	−4.49886	−0.175251	−0.0876254	−	0.996153i	$-0.527928\pi$
$659$	−4.49886	−0.175251	−0.0876254	+	0.996153i	$0.527928\pi$
$660$	0	0
$661$	−14.6203	−0.568663	−0.284332	−	0.958726i	$-0.591772\pi$
$661$	−14.6203	−0.568663	−0.284332	+	0.958726i	$0.591772\pi$
$662$	0	0
$663$	−2.50695	−0.0973619
$664$	0	0
$665$	−5.88470	−0.228199
$666$	0	0
$667$	34.6412	1.34131
$668$	0	0
$669$	−17.7102	−0.684718
$670$	0	0
$671$	−36.2938	−1.40111
$672$	0	0
$673$	−51.3651	−1.97998	−0.989990	−	0.141139i	$-0.954923\pi$
$673$	−51.3651	−1.97998	−0.989990	+	0.141139i	$0.954923\pi$
$674$	0	0
$675$	−2.05080	−0.0789353
$676$	0	0
$677$	5.38004	0.206772	0.103386	−	0.994641i	$-0.467032\pi$
$677$	5.38004	0.206772	0.103386	+	0.994641i	$0.467032\pi$
$678$	0	0
$679$	73.2120	2.80962
$680$	0	0
$681$	−16.0962	−0.616809
$682$	0	0
$683$	40.4754	1.54875	0.774374	−	0.632728i	$-0.218065\pi$
$683$	40.4754	1.54875	0.774374	+	0.632728i	$0.218065\pi$
$684$	0	0
$685$	62.2619	2.37891
$686$	0	0
$687$	46.5174	1.77475
$688$	0	0
$689$	48.9589	1.86519
$690$	0	0
$691$	35.3185	1.34358	0.671789	−	0.740743i	$-0.265526\pi$
$691$	35.3185	1.34358	0.671789	+	0.740743i	$0.265526\pi$
$692$	0	0
$693$	60.1080	2.28331
$694$	0	0
$695$	4.57087	0.173383
$696$	0	0
$697$	−1.80964	−0.0685450
$698$	0	0
$699$	−12.6028	−0.476683
$700$	0	0
$701$	−0.0478169	−0.00180602	−0.000903010	−	1.00000i	$-0.500287\pi$
$701$	−0.0478169	−0.00180602	−0.000903010		1.00000i	$0.500287\pi$
$702$	0	0
$703$	−2.37811	−0.0896922
$704$	0	0
$705$	20.7444	0.781281
$706$	0	0
$707$	54.7300	2.05833
$708$	0	0
$709$	14.2148	0.533847	0.266923	−	0.963718i	$-0.413993\pi$
$709$	14.2148	0.533847	0.266923	+	0.963718i	$0.413993\pi$
$710$	0	0
$711$	10.2029	0.382638
$712$	0	0
$713$	−24.0151	−0.899372
$714$	0	0
$715$	79.4863	2.97262
$716$	0	0
$717$	1.45747	0.0544301
$718$	0	0
$719$	12.8246	0.478276	0.239138	−	0.970986i	$-0.423135\pi$
$719$	12.8246	0.478276	0.239138	+	0.970986i	$0.423135\pi$
$720$	0	0
$721$	20.4421	0.761304
$722$	0	0
$723$	−55.5112	−2.06449
$724$	0	0
$725$	−50.9555	−1.89244
$726$	0	0
$727$	39.9797	1.48276	0.741382	−	0.671083i	$-0.234171\pi$
$727$	39.9797	1.48276	0.741382	+	0.671083i	$0.234171\pi$
$728$	0	0
$729$	−28.2684	−1.04698
$730$	0	0
$731$	−0.924154	−0.0341811
$732$	0	0
$733$	−44.9608	−1.66067	−0.830333	−	0.557268i	$-0.811850\pi$
$733$	−44.9608	−1.66067	−0.830333	+	0.557268i	$0.811850\pi$
$734$	0	0
$735$	144.947	5.34645
$736$	0	0
$737$	−59.2109	−2.18106
$738$	0	0
$739$	−51.3911	−1.89045	−0.945226	−	0.326416i	$-0.894159\pi$
$739$	−51.3911	−1.89045	−0.945226	+	0.326416i	$0.894159\pi$
$740$	0	0
$741$	3.53335	0.129801
$742$	0	0
$743$	26.8381	0.984593	0.492297	−	0.870427i	$-0.336157\pi$
$743$	26.8381	0.984593	0.492297	+	0.870427i	$0.336157\pi$
$744$	0	0
$745$	−2.23060	−0.0817230
$746$	0	0
$747$	−31.0162	−1.13482
$748$	0	0
$749$	−31.4204	−1.14808
$750$	0	0
$751$	4.32202	0.157713	0.0788563	−	0.996886i	$-0.474873\pi$
$751$	4.32202	0.157713	0.0788563	+	0.996886i	$0.474873\pi$
$752$	0	0
$753$	32.5084	1.18467
$754$	0	0
$755$	−6.16621	−0.224411
$756$	0	0
$757$	28.1905	1.02460	0.512300	−	0.858807i	$-0.328794\pi$
$757$	28.1905	1.02460	0.512300	+	0.858807i	$0.328794\pi$
$758$	0	0
$759$	−82.6656	−3.00057
$760$	0	0
$761$	−43.3918	−1.57295	−0.786475	−	0.617622i	$-0.788096\pi$
$761$	−43.3918	−1.57295	−0.786475	+	0.617622i	$0.788096\pi$
$762$	0	0
$763$	−35.4238	−1.28243
$764$	0	0
$765$	−2.77149	−0.100203
$766$	0	0
$767$	42.4546	1.53295
$768$	0	0
$769$	46.7779	1.68685	0.843427	−	0.537243i	$-0.180534\pi$
$769$	46.7779	1.68685	0.843427	+	0.537243i	$0.180534\pi$
$770$	0	0
$771$	−71.9689	−2.59190
$772$	0	0
$773$	−44.6836	−1.60716	−0.803579	−	0.595198i	$-0.797074\pi$
$773$	−44.6836	−1.60716	−0.803579	+	0.595198i	$0.797074\pi$
$774$	0	0
$775$	35.3250	1.26891
$776$	0	0
$777$	87.0321	3.12226
$778$	0	0
$779$	2.55055	0.0913828
$780$	0	0
$781$	−28.7577	−1.02903
$782$	0	0
$783$	0.767657	0.0274338
$784$	0	0
$785$	−49.1626	−1.75469
$786$	0	0
$787$	−17.5561	−0.625808	−0.312904	−	0.949785i	$-0.601302\pi$
$787$	−17.5561	−0.625808	−0.312904	+	0.949785i	$0.601302\pi$
$788$	0	0
$789$	−20.3403	−0.724133
$790$	0	0
$791$	−75.8205	−2.69587
$792$	0	0
$793$	39.4982	1.40262
$794$	0	0
$795$	106.993	3.79466
$796$	0	0
$797$	−10.6070	−0.375719	−0.187860	−	0.982196i	$-0.560155\pi$
$797$	−10.6070	−0.375719	−0.187860	+	0.982196i	$0.560155\pi$
$798$	0	0
$799$	0.455804	0.0161252
$800$	0	0
$801$	−43.9676	−1.55352
$802$	0	0
$803$	2.43311	0.0858626
$804$	0	0
$805$	−149.833	−5.28092
$806$	0	0
$807$	−50.8597	−1.79035
$808$	0	0
$809$	−38.7350	−1.36185	−0.680925	−	0.732353i	$-0.738422\pi$
$809$	−38.7350	−1.36185	−0.680925	+	0.732353i	$0.738422\pi$
$810$	0	0
$811$	19.1900	0.673853	0.336927	−	0.941531i	$-0.390613\pi$
$811$	19.1900	0.673853	0.336927	+	0.941531i	$0.390613\pi$
$812$	0	0
$813$	−30.1524	−1.05749
$814$	0	0
$815$	46.3605	1.62394
$816$	0	0
$817$	1.30252	0.0455695
$818$	0	0
$819$	−65.4150	−2.28578
$820$	0	0
$821$	−46.0724	−1.60794	−0.803969	−	0.594671i	$-0.797282\pi$
$821$	−46.0724	−1.60794	−0.803969	+	0.594671i	$0.797282\pi$
$822$	0	0
$823$	48.5015	1.69066	0.845328	−	0.534247i	$-0.179405\pi$
$823$	48.5015	1.69066	0.845328	+	0.534247i	$0.179405\pi$
$824$	0	0
$825$	121.597	4.23346
$826$	0	0
$827$	−10.9062	−0.379247	−0.189624	−	0.981857i	$-0.560727\pi$
$827$	−10.9062	−0.379247	−0.189624	+	0.981857i	$0.560727\pi$
$828$	0	0
$829$	24.6990	0.857832	0.428916	−	0.903344i	$-0.358896\pi$
$829$	24.6990	0.857832	0.428916	+	0.903344i	$0.358896\pi$
$830$	0	0
$831$	63.6401	2.20765
$832$	0	0
$833$	3.18482	0.110348
$834$	0	0
$835$	−40.1369	−1.38900
$836$	0	0
$837$	−0.532179	−0.0183948
$838$	0	0
$839$	−23.6780	−0.817456	−0.408728	−	0.912656i	$-0.634028\pi$
$839$	−23.6780	−0.817456	−0.408728	+	0.912656i	$0.634028\pi$
$840$	0	0
$841$	−9.92630	−0.342286
$842$	0	0
$843$	−4.19658	−0.144538
$844$	0	0
$845$	−33.4308	−1.15005
$846$	0	0
$847$	−31.8806	−1.09543
$848$	0	0
$849$	−40.5856	−1.39289
$850$	0	0
$851$	−60.5502	−2.07563
$852$	0	0
$853$	28.0285	0.959677	0.479839	−	0.877357i	$-0.340695\pi$
$853$	28.0285	0.959677	0.479839	+	0.877357i	$0.340695\pi$
$854$	0	0
$855$	3.90620	0.133589
$856$	0	0
$857$	−31.4078	−1.07287	−0.536435	−	0.843941i	$-0.680229\pi$
$857$	−31.4078	−1.07287	−0.536435	+	0.843941i	$0.680229\pi$
$858$	0	0
$859$	44.9044	1.53212	0.766060	−	0.642769i	$-0.222215\pi$
$859$	44.9044	1.53212	0.766060	+	0.642769i	$0.222215\pi$
$860$	0	0
$861$	−93.3428	−3.18111
$862$	0	0
$863$	−19.4731	−0.662872	−0.331436	−	0.943478i	$-0.607533\pi$
$863$	−19.4731	−0.662872	−0.331436	+	0.943478i	$0.607533\pi$
$864$	0	0
$865$	−27.1116	−0.921821
$866$	0	0
$867$	41.7678	1.41851
$868$	0	0
$869$	−14.0508	−0.476642
$870$	0	0
$871$	64.4387	2.18342
$872$	0	0
$873$	−48.5972	−1.64477
$874$	0	0
$875$	125.947	4.25778
$876$	0	0
$877$	6.45308	0.217905	0.108953	−	0.994047i	$-0.465250\pi$
$877$	6.45308	0.217905	0.108953	+	0.994047i	$0.465250\pi$
$878$	0	0
$879$	−62.3901	−2.10437
$880$	0	0
$881$	19.5110	0.657344	0.328672	−	0.944444i	$-0.393399\pi$
$881$	19.5110	0.657344	0.328672	+	0.944444i	$0.393399\pi$
$882$	0	0
$883$	42.6313	1.43466	0.717329	−	0.696735i	$-0.245364\pi$
$883$	42.6313	1.43466	0.717329	+	0.696735i	$0.245364\pi$
$884$	0	0
$885$	92.7789	3.11873
$886$	0	0
$887$	28.2986	0.950174	0.475087	−	0.879939i	$-0.342417\pi$
$887$	28.2986	0.950174	0.475087	+	0.879939i	$0.342417\pi$
$888$	0	0
$889$	84.6146	2.83788
$890$	0	0
$891$	37.1404	1.24425
$892$	0	0
$893$	−0.642420	−0.0214978
$894$	0	0
$895$	47.5408	1.58911
$896$	0	0
$897$	89.9642	3.00382
$898$	0	0
$899$	−13.2229	−0.441008
$900$	0	0
$901$	2.35089	0.0783195
$902$	0	0
$903$	−47.6686	−1.58631
$904$	0	0
$905$	−86.8018	−2.88539
$906$	0	0
$907$	−9.19588	−0.305344	−0.152672	−	0.988277i	$-0.548788\pi$
$907$	−9.19588	−0.305344	−0.152672	+	0.988277i	$0.548788\pi$
$908$	0	0
$909$	−36.3291	−1.20496
$910$	0	0
$911$	−56.2164	−1.86253	−0.931267	−	0.364338i	$-0.881295\pi$
$911$	−56.2164	−1.86253	−0.931267	+	0.364338i	$0.881295\pi$
$912$	0	0
$913$	42.7138	1.41362
$914$	0	0
$915$	86.3179	2.85358
$916$	0	0
$917$	15.4924	0.511603
$918$	0	0
$919$	−14.0570	−0.463697	−0.231848	−	0.972752i	$-0.574477\pi$
$919$	−14.0570	−0.463697	−0.231848	+	0.972752i	$0.574477\pi$
$920$	0	0
$921$	−68.4581	−2.25577
$922$	0	0
$923$	31.2967	1.03014
$924$	0	0
$925$	89.0662	2.92848
$926$	0	0
$927$	−13.5692	−0.445672
$928$	0	0
$929$	−21.1813	−0.694935	−0.347468	−	0.937692i	$-0.612958\pi$
$929$	−21.1813	−0.694935	−0.347468	+	0.937692i	$0.612958\pi$
$930$	0	0
$931$	−4.48876	−0.147113
$932$	0	0
$933$	−35.2990	−1.15564
$934$	0	0
$935$	3.81674	0.124821
$936$	0	0
$937$	−38.1349	−1.24581	−0.622906	−	0.782297i	$-0.714048\pi$
$937$	−38.1349	−1.24581	−0.622906	+	0.782297i	$0.714048\pi$
$938$	0	0
$939$	1.82098	0.0594255
$940$	0	0
$941$	6.19825	0.202057	0.101029	−	0.994884i	$-0.467787\pi$
$941$	6.19825	0.202057	0.101029	+	0.994884i	$0.467787\pi$
$942$	0	0
$943$	64.9406	2.11476
$944$	0	0
$945$	−3.32033	−0.108010
$946$	0	0
$947$	−18.9627	−0.616205	−0.308102	−	0.951353i	$-0.599694\pi$
$947$	−18.9627	−0.616205	−0.308102	+	0.951353i	$0.599694\pi$
$948$	0	0
$949$	−2.64793	−0.0859555
$950$	0	0
$951$	−70.6274	−2.29025
$952$	0	0
$953$	1.96656	0.0637030	0.0318515	−	0.999493i	$-0.489860\pi$
$953$	1.96656	0.0637030	0.0318515	+	0.999493i	$0.489860\pi$
$954$	0	0
$955$	−37.9529	−1.22813
$956$	0	0
$957$	−45.5163	−1.47133
$958$	0	0
$959$	70.5646	2.27865
$960$	0	0
$961$	−21.8332	−0.704298
$962$	0	0
$963$	20.8565	0.672092
$964$	0	0
$965$	50.8362	1.63647
$966$	0	0
$967$	−29.6624	−0.953877	−0.476939	−	0.878937i	$-0.658254\pi$
$967$	−29.6624	−0.953877	−0.476939	+	0.878937i	$0.658254\pi$
$968$	0	0
$969$	0.169663	0.00545036
$970$	0	0
$971$	28.7301	0.921994	0.460997	−	0.887402i	$-0.347492\pi$
$971$	28.7301	0.921994	0.460997	+	0.887402i	$0.347492\pi$
$972$	0	0
$973$	5.18040	0.166076
$974$	0	0
$975$	−132.333	−4.23804
$976$	0	0
$977$	−25.7237	−0.822974	−0.411487	−	0.911416i	$-0.634990\pi$
$977$	−25.7237	−0.822974	−0.411487	+	0.911416i	$0.634990\pi$
$978$	0	0
$979$	60.5498	1.93518
$980$	0	0
$981$	23.5139	0.750740
$982$	0	0
$983$	−7.37069	−0.235088	−0.117544	−	0.993068i	$-0.537502\pi$
$983$	−7.37069	−0.235088	−0.117544	+	0.993068i	$0.537502\pi$
$984$	0	0
$985$	66.3272	2.11336
$986$	0	0
$987$	23.5107	0.748355
$988$	0	0
$989$	33.1641	1.05456
$990$	0	0
$991$	−28.4456	−0.903603	−0.451801	−	0.892119i	$-0.649218\pi$
$991$	−28.4456	−0.903603	−0.451801	+	0.892119i	$0.649218\pi$
$992$	0	0
$993$	50.1667	1.59199
$994$	0	0
$995$	29.4290	0.932962
$996$	0	0
$997$	50.7606	1.60761	0.803803	−	0.594896i	$-0.202807\pi$
$997$	50.7606	1.60761	0.803803	+	0.594896i	$0.202807\pi$
$998$	0	0
$999$	−1.34180	−0.0424528

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4024.2.a.e.1.5	✓	29
4.3	odd	2		8048.2.a.w.1.25		29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.e.1.5	✓	29	1.1	even	1	trivial
8048.2.a.w.1.25		29	4.3	odd	2

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:	\( N \)	\(=\)	\( 4024 = 2^{3} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4024.a (trivial)

\(f(q)\)	\(=\)	\(q-2.46401 q^{3} -4.08257 q^{5} -4.62698 q^{7} +3.07134 q^{9} +O(q^{10})\)	\(q-2.46401 q^{3} -4.08257 q^{5} -4.62698 q^{7} +3.07134 q^{9} -4.22967 q^{11} +4.60311 q^{13} +10.0595 q^{15} +0.221030 q^{17} -0.311525 q^{19} +11.4009 q^{21} -7.93188 q^{23} +11.6674 q^{25} -0.175772 q^{27} -4.36735 q^{29} +3.02767 q^{31} +10.4220 q^{33} +18.8900 q^{35} +7.63378 q^{37} -11.3421 q^{39} -8.18730 q^{41} -4.18112 q^{43} -12.5389 q^{45} +2.06218 q^{47} +14.4090 q^{49} -0.544621 q^{51} +10.6360 q^{53} +17.2679 q^{55} +0.767600 q^{57} +9.22303 q^{59} +8.58075 q^{61} -14.2110 q^{63} -18.7925 q^{65} +13.9989 q^{67} +19.5442 q^{69} +6.79903 q^{71} -0.575248 q^{73} -28.7485 q^{75} +19.5706 q^{77} +3.32197 q^{79} -8.78090 q^{81} -10.0986 q^{83} -0.902372 q^{85} +10.7612 q^{87} -14.3155 q^{89} -21.2985 q^{91} -7.46020 q^{93} +1.27182 q^{95} -15.8228 q^{97} -12.9907 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9}+O(q^{10}) \) 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9	\( 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9} - 27 q^{11} + 16 q^{13} - 14 q^{15} - 15 q^{17} - 14 q^{19} + q^{21} - 25 q^{23} + 21 q^{25} - 25 q^{27} - 13 q^{29} - 27 q^{31} - 9 q^{33} - 29 q^{35} + 35 q^{37} - 38 q^{39} - 30 q^{41} - 38 q^{43} + q^{45} - 35 q^{47} + 14 q^{49} - 21 q^{51} + 2 q^{53} - 25 q^{55} - 25 q^{57} - 40 q^{59} + 10 q^{61} - 56 q^{63} - 50 q^{65} - 31 q^{67} + 11 q^{69} - 65 q^{71} - 23 q^{73} - 32 q^{75} + 13 q^{77} - 44 q^{79} - 7 q^{81} - 41 q^{83} + 26 q^{85} - 25 q^{87} - 48 q^{89} - 44 q^{91} + 25 q^{93} - 75 q^{95} - 18 q^{97} - 80 q^{99}+O(q^{100}) \) 29 * q - 7 * q^3 - 4 * q^5 - 13 * q^7 + 20 * q^9 - 27 * q^11 + 16 * q^13 - 14 * q^15 - 15 * q^17 - 14 * q^19 + q^21 - 25 * q^23 + 21 * q^25 - 25 * q^27 - 13 * q^29 - 27 * q^31 - 9 * q^33 - 29 * q^35 + 35 * q^37 - 38 * q^39 - 30 * q^41 - 38 * q^43 + q^45 - 35 * q^47 + 14 * q^49 - 21 * q^51 + 2 * q^53 - 25 * q^55 - 25 * q^57 - 40 * q^59 + 10 * q^61 - 56 * q^63 - 50 * q^65 - 31 * q^67 + 11 * q^69 - 65 * q^71 - 23 * q^73 - 32 * q^75 + 13 * q^77 - 44 * q^79 - 7 * q^81 - 41 * q^83 + 26 * q^85 - 25 * q^87 - 48 * q^89 - 44 * q^91 + 25 * q^93 - 75 * q^95 - 18 * q^97 - 80 * q^99

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