LMFDB - Embedded newform 8028.2.a.j.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8028,2,Mod(1,8028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8028, 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("8028.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8028 = 2^{2} \cdot 3^{2} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8028.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:	$64.1039027427$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 11x^{5} + 24x^{4} + 38x^{3} - 46x^{2} - 36x + 9 $ x^7 - 3x^6 - 11x^5 + 24x^4 + 38x^3 - 46x^2 - 36x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 892)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.49109$ of defining polynomial
Character	$\chi$	$=$	8028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.18873 q^{5} +2.92270 q^{7} +O(q^{10})$	$q+1.18873 q^{5} +2.92270 q^{7} +5.93560 q^{11} -2.54915 q^{13} +4.43759 q^{17} -0.744803 q^{19} -1.44452 q^{23} -3.58693 q^{25} +1.97621 q^{29} +6.39491 q^{31} +3.47429 q^{35} +11.9944 q^{37} +3.41379 q^{41} -8.18938 q^{43} +0.374907 q^{47} +1.54218 q^{49} +2.62196 q^{53} +7.05580 q^{55} +4.22750 q^{59} -13.5536 q^{61} -3.03024 q^{65} +1.75883 q^{67} +2.48376 q^{71} +5.45709 q^{73} +17.3480 q^{77} -11.3839 q^{79} -5.96481 q^{83} +5.27507 q^{85} +0.688587 q^{89} -7.45040 q^{91} -0.885366 q^{95} +17.1583 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{5} - q^{7}+O(q^{10}) $ 7 * q + 7 * q^5 - q^7	$ 7 q + 7 q^{5} - q^{7} + 12 q^{11} - 9 q^{13} - 8 q^{17} - 12 q^{19} + 12 q^{23} + 12 q^{25} + 24 q^{29} + q^{31} + 15 q^{35} - 13 q^{37} - 5 q^{41} - 13 q^{43} + 21 q^{47} + 4 q^{49} + 35 q^{53} + q^{55} + 23 q^{59} - 17 q^{61} - 18 q^{67} + 4 q^{71} + 23 q^{73} - 3 q^{77} + 4 q^{79} + 44 q^{83} + 20 q^{85} + 2 q^{91} + 12 q^{95} + 32 q^{97}+O(q^{100}) $ 7 * q + 7 * q^5 - q^7 + 12 * q^11 - 9 * q^13 - 8 * q^17 - 12 * q^19 + 12 * q^23 + 12 * q^25 + 24 * q^29 + q^31 + 15 * q^35 - 13 * q^37 - 5 * q^41 - 13 * q^43 + 21 * q^47 + 4 * q^49 + 35 * q^53 + q^55 + 23 * q^59 - 17 * q^61 - 18 * q^67 + 4 * q^71 + 23 * q^73 - 3 * q^77 + 4 * q^79 + 44 * q^83 + 20 * q^85 + 2 * q^91 + 12 * q^95 + 32 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.18873	0.531614	0.265807	−	0.964026i	$-0.414362\pi$
$5$	1.18873	0.531614	0.265807	+	0.964026i	$0.414362\pi$
$6$	0	0
$7$	2.92270	1.10468	0.552339	−	0.833620i	$-0.313736\pi$
$7$	2.92270	1.10468	0.552339	+	0.833620i	$0.313736\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.93560	1.78965	0.894826	−	0.446415i	$-0.147299\pi$
$11$	5.93560	1.78965	0.894826	+	0.446415i	$0.147299\pi$
$12$	0	0
$13$	−2.54915	−0.707006	−0.353503	−	0.935433i	$-0.615010\pi$
$13$	−2.54915	−0.707006	−0.353503	+	0.935433i	$0.615010\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.43759	1.07627	0.538136	−	0.842858i	$-0.319129\pi$
$17$	4.43759	1.07627	0.538136	+	0.842858i	$0.319129\pi$
$18$	0	0
$19$	−0.744803	−0.170870	−0.0854348	−	0.996344i	$-0.527228\pi$
$19$	−0.744803	−0.170870	−0.0854348	+	0.996344i	$0.527228\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.44452	−0.301202	−0.150601	−	0.988595i	$-0.548121\pi$
$23$	−1.44452	−0.301202	−0.150601	+	0.988595i	$0.548121\pi$
$24$	0	0
$25$	−3.58693	−0.717386
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.97621	0.366972	0.183486	−	0.983022i	$-0.441262\pi$
$29$	1.97621	0.366972	0.183486	+	0.983022i	$0.441262\pi$
$30$	0	0
$31$	6.39491	1.14856	0.574280	−	0.818659i	$-0.305282\pi$
$31$	6.39491	1.14856	0.574280	+	0.818659i	$0.305282\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.47429	0.587262
$36$	0	0
$37$	11.9944	1.97186	0.985932	−	0.167144i	$-0.0534546\pi$
$37$	11.9944	1.97186	0.985932	+	0.167144i	$0.0534546\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.41379	0.533144	0.266572	−	0.963815i	$-0.414109\pi$
$41$	3.41379	0.533144	0.266572	+	0.963815i	$0.414109\pi$
$42$	0	0
$43$	−8.18938	−1.24887	−0.624434	−	0.781077i	$-0.714671\pi$
$43$	−8.18938	−1.24887	−0.624434	+	0.781077i	$0.714671\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.374907	0.0546857	0.0273429	−	0.999626i	$-0.491295\pi$
$47$	0.374907	0.0546857	0.0273429	+	0.999626i	$0.491295\pi$
$48$	0	0
$49$	1.54218	0.220311
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2.62196	0.360153	0.180077	−	0.983653i	$-0.442365\pi$
$53$	2.62196	0.360153	0.180077	+	0.983653i	$0.442365\pi$
$54$	0	0
$55$	7.05580	0.951404
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.22750	0.550374	0.275187	−	0.961391i	$-0.411260\pi$
$59$	4.22750	0.550374	0.275187	+	0.961391i	$0.411260\pi$
$60$	0	0
$61$	−13.5536	−1.73536	−0.867678	−	0.497126i	$-0.834389\pi$
$61$	−13.5536	−1.73536	−0.867678	+	0.497126i	$0.834389\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.03024	−0.375855
$66$	0	0
$67$	1.75883	0.214875	0.107438	−	0.994212i	$-0.465735\pi$
$67$	1.75883	0.214875	0.107438	+	0.994212i	$0.465735\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.48376	0.294768	0.147384	−	0.989079i	$-0.452915\pi$
$71$	2.48376	0.294768	0.147384	+	0.989079i	$0.452915\pi$
$72$	0	0
$73$	5.45709	0.638704	0.319352	−	0.947636i	$-0.396535\pi$
$73$	5.45709	0.638704	0.319352	+	0.947636i	$0.396535\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	17.3480	1.97699
$78$	0	0
$79$	−11.3839	−1.28079	−0.640393	−	0.768047i	$-0.721229\pi$
$79$	−11.3839	−1.28079	−0.640393	+	0.768047i	$0.721229\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−5.96481	−0.654723	−0.327362	−	0.944899i	$-0.606160\pi$
$83$	−5.96481	−0.654723	−0.327362	+	0.944899i	$0.606160\pi$
$84$	0	0
$85$	5.27507	0.572162
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0.688587	0.0729901	0.0364950	−	0.999334i	$-0.488381\pi$
$89$	0.688587	0.0729901	0.0364950	+	0.999334i	$0.488381\pi$
$90$	0	0
$91$	−7.45040	−0.781014
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−0.885366	−0.0908367
$96$	0	0
$97$	17.1583	1.74216	0.871078	−	0.491144i	$-0.163421\pi$
$97$	17.1583	1.74216	0.871078	+	0.491144i	$0.163421\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	11.8084	1.17498	0.587490	−	0.809231i	$-0.300116\pi$
$101$	11.8084	1.17498	0.587490	+	0.809231i	$0.300116\pi$
$102$	0	0
$103$	4.68008	0.461142	0.230571	−	0.973055i	$-0.425941\pi$
$103$	4.68008	0.461142	0.230571	+	0.973055i	$0.425941\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.96783	0.286911	0.143455	−	0.989657i	$-0.454179\pi$
$107$	2.96783	0.286911	0.143455	+	0.989657i	$0.454179\pi$
$108$	0	0
$109$	4.93407	0.472598	0.236299	−	0.971680i	$-0.424066\pi$
$109$	4.93407	0.472598	0.236299	+	0.971680i	$0.424066\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−12.6737	−1.19224	−0.596121	−	0.802895i	$-0.703292\pi$
$113$	−12.6737	−1.19224	−0.596121	+	0.802895i	$0.703292\pi$
$114$	0	0
$115$	−1.71713	−0.160123
$116$	0	0
$117$	0	0
$118$	0	0
$119$	12.9697	1.18893
$120$	0	0
$121$	24.2314	2.20285
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−10.2075	−0.912987
$126$	0	0
$127$	−11.8065	−1.04766	−0.523828	−	0.851824i	$-0.675497\pi$
$127$	−11.8065	−1.04766	−0.523828	+	0.851824i	$0.675497\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	18.4709	1.61381	0.806904	−	0.590683i	$-0.201142\pi$
$131$	18.4709	1.61381	0.806904	+	0.590683i	$0.201142\pi$
$132$	0	0
$133$	−2.17684	−0.188756
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.41775	−0.462870	−0.231435	−	0.972850i	$-0.574342\pi$
$137$	−5.41775	−0.462870	−0.231435	+	0.972850i	$0.574342\pi$
$138$	0	0
$139$	1.56547	0.132782	0.0663909	−	0.997794i	$-0.478852\pi$
$139$	1.56547	0.132782	0.0663909	+	0.997794i	$0.478852\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−15.1307	−1.26530
$144$	0	0
$145$	2.34916	0.195088
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−13.9872	−1.14588	−0.572939	−	0.819598i	$-0.694197\pi$
$149$	−13.9872	−1.14588	−0.572939	+	0.819598i	$0.694197\pi$
$150$	0	0
$151$	2.39980	0.195293	0.0976465	−	0.995221i	$-0.468869\pi$
$151$	2.39980	0.195293	0.0976465	+	0.995221i	$0.468869\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	7.60179	0.610590
$156$	0	0
$157$	6.80615	0.543190	0.271595	−	0.962412i	$-0.412449\pi$
$157$	6.80615	0.543190	0.271595	+	0.962412i	$0.412449\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−4.22189	−0.332731
$162$	0	0
$163$	−10.4304	−0.816971	−0.408486	−	0.912765i	$-0.633943\pi$
$163$	−10.4304	−0.816971	−0.408486	+	0.912765i	$0.633943\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0.476861	0.0369006	0.0184503	−	0.999830i	$-0.494127\pi$
$167$	0.476861	0.0369006	0.0184503	+	0.999830i	$0.494127\pi$
$168$	0	0
$169$	−6.50185	−0.500142
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1.12076	−0.0852099	−0.0426050	−	0.999092i	$-0.513566\pi$
$173$	−1.12076	−0.0852099	−0.0426050	+	0.999092i	$0.513566\pi$
$174$	0	0
$175$	−10.4835	−0.792480
$176$	0	0
$177$	0	0
$178$	0	0
$179$	8.69918	0.650207	0.325104	−	0.945678i	$-0.394601\pi$
$179$	8.69918	0.650207	0.325104	+	0.945678i	$0.394601\pi$
$180$	0	0
$181$	−19.9119	−1.48004	−0.740019	−	0.672586i	$-0.765183\pi$
$181$	−19.9119	−1.48004	−0.740019	+	0.672586i	$0.765183\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	14.2580	1.04827
$186$	0	0
$187$	26.3398	1.92615
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−19.1753	−1.38748	−0.693738	−	0.720228i	$-0.744037\pi$
$191$	−19.1753	−1.38748	−0.693738	+	0.720228i	$0.744037\pi$
$192$	0	0
$193$	−5.48599	−0.394891	−0.197445	−	0.980314i	$-0.563264\pi$
$193$	−5.48599	−0.394891	−0.197445	+	0.980314i	$0.563264\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−6.23781	−0.444425	−0.222213	−	0.974998i	$-0.571328\pi$
$197$	−6.23781	−0.444425	−0.222213	+	0.974998i	$0.571328\pi$
$198$	0	0
$199$	−13.8258	−0.980082	−0.490041	−	0.871699i	$-0.663018\pi$
$199$	−13.8258	−0.980082	−0.490041	+	0.871699i	$0.663018\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	5.77586	0.405386
$204$	0	0
$205$	4.05806	0.283427
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4.42086	−0.305797
$210$	0	0
$211$	−10.1278	−0.697228	−0.348614	−	0.937266i	$-0.613348\pi$
$211$	−10.1278	−0.697228	−0.348614	+	0.937266i	$0.613348\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−9.73492	−0.663916
$216$	0	0
$217$	18.6904	1.26879
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−11.3121	−0.760932
$222$	0	0
$223$	1.00000	0.0669650
$223$	1.00000	0.0669650
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.27870	0.615849	0.307925	−	0.951411i	$-0.400366\pi$
$227$	9.27870	0.615849	0.307925	+	0.951411i	$0.400366\pi$
$228$	0	0
$229$	16.5331	1.09254	0.546268	−	0.837610i	$-0.316048\pi$
$229$	16.5331	1.09254	0.546268	+	0.837610i	$0.316048\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6.20355	−0.406408	−0.203204	−	0.979136i	$-0.565135\pi$
$233$	−6.20355	−0.406408	−0.203204	+	0.979136i	$0.565135\pi$
$234$	0	0
$235$	0.445661	0.0290717
$236$	0	0
$237$	0	0
$238$	0	0
$239$	5.04947	0.326623	0.163311	−	0.986575i	$-0.447782\pi$
$239$	5.04947	0.326623	0.163311	+	0.986575i	$0.447782\pi$
$240$	0	0
$241$	−3.13371	−0.201860	−0.100930	−	0.994894i	$-0.532182\pi$
$241$	−3.13371	−0.201860	−0.100930	+	0.994894i	$0.532182\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.83323	0.117121
$246$	0	0
$247$	1.89861	0.120806
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−29.1416	−1.83940	−0.919699	−	0.392623i	$-0.871568\pi$
$251$	−29.1416	−1.83940	−0.919699	+	0.392623i	$0.871568\pi$
$252$	0	0
$253$	−8.57407	−0.539047
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−28.5471	−1.78072	−0.890359	−	0.455258i	$-0.849547\pi$
$257$	−28.5471	−1.78072	−0.890359	+	0.455258i	$0.849547\pi$
$258$	0	0
$259$	35.0560	2.17827
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1.24189	0.0765783	0.0382891	−	0.999267i	$-0.487809\pi$
$263$	1.24189	0.0765783	0.0382891	+	0.999267i	$0.487809\pi$
$264$	0	0
$265$	3.11679	0.191463
$266$	0	0
$267$	0	0
$268$	0	0
$269$	4.31732	0.263231	0.131616	−	0.991301i	$-0.457984\pi$
$269$	4.31732	0.263231	0.131616	+	0.991301i	$0.457984\pi$
$270$	0	0
$271$	21.6133	1.31291	0.656457	−	0.754364i	$-0.272054\pi$
$271$	21.6133	1.31291	0.656457	+	0.754364i	$0.272054\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−21.2906	−1.28387
$276$	0	0
$277$	26.6627	1.60201	0.801004	−	0.598660i	$-0.204300\pi$
$277$	26.6627	1.60201	0.801004	+	0.598660i	$0.204300\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	24.9737	1.48981	0.744903	−	0.667172i	$-0.232496\pi$
$281$	24.9737	1.48981	0.744903	+	0.667172i	$0.232496\pi$
$282$	0	0
$283$	−29.1700	−1.73398	−0.866989	−	0.498327i	$-0.833948\pi$
$283$	−29.1700	−1.73398	−0.866989	+	0.498327i	$0.833948\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	9.97749	0.588952
$288$	0	0
$289$	2.69216	0.158362
$290$	0	0
$291$	0	0
$292$	0	0
$293$	7.51573	0.439073	0.219537	−	0.975604i	$-0.429545\pi$
$293$	7.51573	0.439073	0.219537	+	0.975604i	$0.429545\pi$
$294$	0	0
$295$	5.02534	0.292587
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3.68228	0.212952
$300$	0	0
$301$	−23.9351	−1.37960
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−16.1115	−0.922540
$306$	0	0
$307$	−13.0902	−0.747095	−0.373548	−	0.927611i	$-0.621859\pi$
$307$	−13.0902	−0.747095	−0.373548	+	0.927611i	$0.621859\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	19.4825	1.10475	0.552376	−	0.833595i	$-0.313721\pi$
$311$	19.4825	1.10475	0.552376	+	0.833595i	$0.313721\pi$
$312$	0	0
$313$	8.89999	0.503057	0.251529	−	0.967850i	$-0.419067\pi$
$313$	8.89999	0.503057	0.251529	+	0.967850i	$0.419067\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	30.3478	1.70450	0.852252	−	0.523132i	$-0.175237\pi$
$317$	30.3478	1.70450	0.852252	+	0.523132i	$0.175237\pi$
$318$	0	0
$319$	11.7300	0.656752
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3.30513	−0.183902
$324$	0	0
$325$	9.14362	0.507197
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.09574	0.0604101
$330$	0	0
$331$	18.6493	1.02506	0.512528	−	0.858670i	$-0.328709\pi$
$331$	18.6493	1.02506	0.512528	+	0.858670i	$0.328709\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2.09077	0.114231
$336$	0	0
$337$	2.52591	0.137595	0.0687975	−	0.997631i	$-0.478084\pi$
$337$	2.52591	0.137595	0.0687975	+	0.997631i	$0.478084\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	37.9576	2.05552
$342$	0	0
$343$	−15.9516	−0.861304
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8.18250	0.439260	0.219630	−	0.975583i	$-0.429515\pi$
$347$	8.18250	0.439260	0.219630	+	0.975583i	$0.429515\pi$
$348$	0	0
$349$	−14.2371	−0.762095	−0.381048	−	0.924555i	$-0.624437\pi$
$349$	−14.2371	−0.762095	−0.381048	+	0.924555i	$0.624437\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	20.9719	1.11622	0.558111	−	0.829766i	$-0.311526\pi$
$353$	20.9719	1.11622	0.558111	+	0.829766i	$0.311526\pi$
$354$	0	0
$355$	2.95251	0.156703
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−0.643422	−0.0339585	−0.0169793	−	0.999856i	$-0.505405\pi$
$359$	−0.643422	−0.0339585	−0.0169793	+	0.999856i	$0.505405\pi$
$360$	0	0
$361$	−18.4453	−0.970804
$362$	0	0
$363$	0	0
$364$	0	0
$365$	6.48698	0.339544
$366$	0	0
$367$	11.7036	0.610925	0.305463	−	0.952204i	$-0.401189\pi$
$367$	11.7036	0.610925	0.305463	+	0.952204i	$0.401189\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	7.66319	0.397853
$372$	0	0
$373$	26.6751	1.38119	0.690593	−	0.723244i	$-0.257350\pi$
$373$	26.6751	1.38119	0.690593	+	0.723244i	$0.257350\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−5.03764	−0.259452
$378$	0	0
$379$	−14.3668	−0.737972	−0.368986	−	0.929435i	$-0.620295\pi$
$379$	−14.3668	−0.737972	−0.368986	+	0.929435i	$0.620295\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	29.5866	1.51181	0.755903	−	0.654684i	$-0.227198\pi$
$383$	29.5866	1.51181	0.755903	+	0.654684i	$0.227198\pi$
$384$	0	0
$385$	20.6220	1.05099
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−30.6072	−1.55185	−0.775924	−	0.630827i	$-0.782716\pi$
$389$	−30.6072	−1.55185	−0.775924	+	0.630827i	$0.782716\pi$
$390$	0	0
$391$	−6.41016	−0.324176
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−13.5323	−0.680884
$396$	0	0
$397$	−17.9822	−0.902499	−0.451250	−	0.892398i	$-0.649022\pi$
$397$	−17.9822	−0.902499	−0.451250	+	0.892398i	$0.649022\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	15.8720	0.792610	0.396305	−	0.918119i	$-0.370292\pi$
$401$	15.8720	0.792610	0.396305	+	0.918119i	$0.370292\pi$
$402$	0	0
$403$	−16.3016	−0.812039
$404$	0	0
$405$	0	0
$406$	0	0
$407$	71.1939	3.52895
$408$	0	0
$409$	19.0180	0.940382	0.470191	−	0.882565i	$-0.344185\pi$
$409$	19.0180	0.940382	0.470191	+	0.882565i	$0.344185\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	12.3557	0.607986
$414$	0	0
$415$	−7.09052	−0.348060
$416$	0	0
$417$	0	0
$418$	0	0
$419$	10.0164	0.489332	0.244666	−	0.969607i	$-0.421322\pi$
$419$	10.0164	0.489332	0.244666	+	0.969607i	$0.421322\pi$
$420$	0	0
$421$	−22.1541	−1.07972	−0.539862	−	0.841754i	$-0.681523\pi$
$421$	−22.1541	−1.07972	−0.539862	+	0.841754i	$0.681523\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−15.9173	−0.772103
$426$	0	0
$427$	−39.6130	−1.91701
$428$	0	0
$429$	0	0
$430$	0	0
$431$	20.5826	0.991429	0.495715	−	0.868485i	$-0.334906\pi$
$431$	20.5826	0.991429	0.495715	+	0.868485i	$0.334906\pi$
$432$	0	0
$433$	−33.1603	−1.59358	−0.796792	−	0.604254i	$-0.793471\pi$
$433$	−33.1603	−1.59358	−0.796792	+	0.604254i	$0.793471\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.07588	0.0514663
$438$	0	0
$439$	15.7630	0.752326	0.376163	−	0.926554i	$-0.377243\pi$
$439$	15.7630	0.752326	0.376163	+	0.926554i	$0.377243\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	18.3984	0.874137	0.437068	−	0.899428i	$-0.356017\pi$
$443$	18.3984	0.874137	0.437068	+	0.899428i	$0.356017\pi$
$444$	0	0
$445$	0.818541	0.0388025
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−21.4449	−1.01205	−0.506023	−	0.862520i	$-0.668885\pi$
$449$	−21.4449	−1.01205	−0.506023	+	0.862520i	$0.668885\pi$
$450$	0	0
$451$	20.2629	0.954143
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−8.85647	−0.415198
$456$	0	0
$457$	−5.46684	−0.255728	−0.127864	−	0.991792i	$-0.540812\pi$
$457$	−5.46684	−0.255728	−0.127864	+	0.991792i	$0.540812\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	6.65790	0.310089	0.155045	−	0.987907i	$-0.450448\pi$
$461$	6.65790	0.310089	0.155045	+	0.987907i	$0.450448\pi$
$462$	0	0
$463$	27.5414	1.27996	0.639978	−	0.768393i	$-0.278943\pi$
$463$	27.5414	1.27996	0.639978	+	0.768393i	$0.278943\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	28.8456	1.33481	0.667407	−	0.744693i	$-0.267404\pi$
$467$	28.8456	1.33481	0.667407	+	0.744693i	$0.267404\pi$
$468$	0	0
$469$	5.14054	0.237368
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−48.6089	−2.23504
$474$	0	0
$475$	2.67156	0.122580
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−19.3168	−0.882606	−0.441303	−	0.897358i	$-0.645484\pi$
$479$	−19.3168	−0.882606	−0.441303	+	0.897358i	$0.645484\pi$
$480$	0	0
$481$	−30.5755	−1.39412
$482$	0	0
$483$	0	0
$484$	0	0
$485$	20.3965	0.926155
$486$	0	0
$487$	−33.9975	−1.54057	−0.770286	−	0.637698i	$-0.779887\pi$
$487$	−33.9975	−1.54057	−0.770286	+	0.637698i	$0.779887\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	15.5717	0.702743	0.351371	−	0.936236i	$-0.385715\pi$
$491$	15.5717	0.702743	0.351371	+	0.936236i	$0.385715\pi$
$492$	0	0
$493$	8.76958	0.394962
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7.25929	0.325624
$498$	0	0
$499$	34.4174	1.54074	0.770368	−	0.637600i	$-0.220073\pi$
$499$	34.4174	1.54074	0.770368	+	0.637600i	$0.220073\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	3.25765	0.145252	0.0726258	−	0.997359i	$-0.476862\pi$
$503$	3.25765	0.145252	0.0726258	+	0.997359i	$0.476862\pi$
$504$	0	0
$505$	14.0369	0.624636
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−34.9729	−1.55015	−0.775074	−	0.631871i	$-0.782287\pi$
$509$	−34.9729	−1.55015	−0.775074	+	0.631871i	$0.782287\pi$
$510$	0	0
$511$	15.9494	0.705562
$512$	0	0
$513$	0	0
$514$	0	0
$515$	5.56333	0.245150
$516$	0	0
$517$	2.22530	0.0978685
$518$	0	0
$519$	0	0
$520$	0	0
$521$	9.11893	0.399508	0.199754	−	0.979846i	$-0.435986\pi$
$521$	9.11893	0.399508	0.199754	+	0.979846i	$0.435986\pi$
$522$	0	0
$523$	−17.7156	−0.774651	−0.387325	−	0.921943i	$-0.626601\pi$
$523$	−17.7156	−0.774651	−0.387325	+	0.921943i	$0.626601\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	28.3779	1.23616
$528$	0	0
$529$	−20.9134	−0.909277
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−8.70226	−0.376937
$534$	0	0
$535$	3.52793	0.152526
$536$	0	0
$537$	0	0
$538$	0	0
$539$	9.15377	0.394281
$540$	0	0
$541$	−17.7396	−0.762683	−0.381342	−	0.924434i	$-0.624538\pi$
$541$	−17.7396	−0.762683	−0.381342	+	0.924434i	$0.624538\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	5.86525	0.251240
$546$	0	0
$547$	0.446985	0.0191117	0.00955586	−	0.999954i	$-0.496958\pi$
$547$	0.446985	0.0191117	0.00955586	+	0.999954i	$0.496958\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1.47188	−0.0627044
$552$	0	0
$553$	−33.2717	−1.41486
$554$	0	0
$555$	0	0
$556$	0	0
$557$	43.7352	1.85312	0.926560	−	0.376146i	$-0.122751\pi$
$557$	43.7352	1.85312	0.926560	+	0.376146i	$0.122751\pi$
$558$	0	0
$559$	20.8759	0.882958
$560$	0	0
$561$	0	0
$562$	0	0
$563$	16.0475	0.676320	0.338160	−	0.941089i	$-0.390196\pi$
$563$	16.0475	0.676320	0.338160	+	0.941089i	$0.390196\pi$
$564$	0	0
$565$	−15.0655	−0.633812
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−44.4695	−1.86426	−0.932130	−	0.362124i	$-0.882052\pi$
$569$	−44.4695	−1.86426	−0.932130	+	0.362124i	$0.882052\pi$
$570$	0	0
$571$	36.8825	1.54348	0.771742	−	0.635936i	$-0.219386\pi$
$571$	36.8825	1.54348	0.771742	+	0.635936i	$0.219386\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	5.18138	0.216078
$576$	0	0
$577$	22.0020	0.915954	0.457977	−	0.888964i	$-0.348574\pi$
$577$	22.0020	0.915954	0.457977	+	0.888964i	$0.348574\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−17.4334	−0.723258
$582$	0	0
$583$	15.5629	0.644549
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−41.4511	−1.71087	−0.855434	−	0.517912i	$-0.826710\pi$
$587$	−41.4511	−1.71087	−0.855434	+	0.517912i	$0.826710\pi$
$588$	0	0
$589$	−4.76295	−0.196254
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−9.05921	−0.372017	−0.186009	−	0.982548i	$-0.559555\pi$
$593$	−9.05921	−0.372017	−0.186009	+	0.982548i	$0.559555\pi$
$594$	0	0
$595$	15.4174	0.632054
$596$	0	0
$597$	0	0
$598$	0	0
$599$	4.97221	0.203159	0.101580	−	0.994827i	$-0.467610\pi$
$599$	4.97221	0.203159	0.101580	+	0.994827i	$0.467610\pi$
$600$	0	0
$601$	−23.9654	−0.977570	−0.488785	−	0.872404i	$-0.662560\pi$
$601$	−23.9654	−0.977570	−0.488785	+	0.872404i	$0.662560\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	28.8045	1.17107
$606$	0	0
$607$	7.46332	0.302927	0.151463	−	0.988463i	$-0.451601\pi$
$607$	7.46332	0.302927	0.151463	+	0.988463i	$0.451601\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−0.955692	−0.0386632
$612$	0	0
$613$	−35.0202	−1.41445	−0.707227	−	0.706986i	$-0.750054\pi$
$613$	−35.0202	−1.41445	−0.707227	+	0.706986i	$0.750054\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	30.2837	1.21918	0.609588	−	0.792719i	$-0.291335\pi$
$617$	30.2837	1.21918	0.609588	+	0.792719i	$0.291335\pi$
$618$	0	0
$619$	−0.663450	−0.0266663	−0.0133331	−	0.999911i	$-0.504244\pi$
$619$	−0.663450	−0.0266663	−0.0133331	+	0.999911i	$0.504244\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2.01253	0.0806305
$624$	0	0
$625$	5.80075	0.232030
$626$	0	0
$627$	0	0
$628$	0	0
$629$	53.2261	2.12226
$630$	0	0
$631$	−25.1106	−0.999639	−0.499819	−	0.866130i	$-0.666600\pi$
$631$	−25.1106	−0.999639	−0.499819	+	0.866130i	$0.666600\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−14.0347	−0.556949
$636$	0	0
$637$	−3.93124	−0.155762
$638$	0	0
$639$	0	0
$640$	0	0
$641$	12.1745	0.480865	0.240433	−	0.970666i	$-0.422711\pi$
$641$	12.1745	0.480865	0.240433	+	0.970666i	$0.422711\pi$
$642$	0	0
$643$	11.8363	0.466779	0.233390	−	0.972383i	$-0.425018\pi$
$643$	11.8363	0.466779	0.233390	+	0.972383i	$0.425018\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	35.3742	1.39070	0.695352	−	0.718669i	$-0.255248\pi$
$647$	35.3742	1.39070	0.695352	+	0.718669i	$0.255248\pi$
$648$	0	0
$649$	25.0928	0.984978
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−3.68542	−0.144222	−0.0721108	−	0.997397i	$-0.522974\pi$
$653$	−3.68542	−0.144222	−0.0721108	+	0.997397i	$0.522974\pi$
$654$	0	0
$655$	21.9568	0.857923
$656$	0	0
$657$	0	0
$658$	0	0
$659$	36.9589	1.43972	0.719858	−	0.694122i	$-0.244207\pi$
$659$	36.9589	1.43972	0.719858	+	0.694122i	$0.244207\pi$
$660$	0	0
$661$	−15.7343	−0.611993	−0.305996	−	0.952033i	$-0.598990\pi$
$661$	−15.7343	−0.611993	−0.305996	+	0.952033i	$0.598990\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2.58766	−0.100345
$666$	0	0
$667$	−2.85466	−0.110533
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−80.4486	−3.10568
$672$	0	0
$673$	−5.32283	−0.205180	−0.102590	−	0.994724i	$-0.532713\pi$
$673$	−5.32283	−0.205180	−0.102590	+	0.994724i	$0.532713\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	16.0833	0.618132	0.309066	−	0.951041i	$-0.399984\pi$
$677$	16.0833	0.618132	0.309066	+	0.951041i	$0.399984\pi$
$678$	0	0
$679$	50.1484	1.92452
$680$	0	0
$681$	0	0
$682$	0	0
$683$	47.6305	1.82253	0.911266	−	0.411818i	$-0.135106\pi$
$683$	47.6305	1.82253	0.911266	+	0.411818i	$0.135106\pi$
$684$	0	0
$685$	−6.44022	−0.246068
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−6.68375	−0.254631
$690$	0	0
$691$	−9.48931	−0.360990	−0.180495	−	0.983576i	$-0.557770\pi$
$691$	−9.48931	−0.360990	−0.180495	+	0.983576i	$0.557770\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.86092	0.0705887
$696$	0	0
$697$	15.1490	0.573809
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−5.71735	−0.215941	−0.107971	−	0.994154i	$-0.534435\pi$
$701$	−5.71735	−0.215941	−0.107971	+	0.994154i	$0.534435\pi$
$702$	0	0
$703$	−8.93346	−0.336932
$704$	0	0
$705$	0	0
$706$	0	0
$707$	34.5124	1.29797
$708$	0	0
$709$	−43.9106	−1.64910	−0.824548	−	0.565792i	$-0.808570\pi$
$709$	−43.9106	−1.64910	−0.824548	+	0.565792i	$0.808570\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−9.23754	−0.345949
$714$	0	0
$715$	−17.9863	−0.672649
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−40.4735	−1.50941	−0.754703	−	0.656067i	$-0.772219\pi$
$719$	−40.4735	−1.50941	−0.754703	+	0.656067i	$0.772219\pi$
$720$	0	0
$721$	13.6785	0.509413
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−7.08851	−0.263261
$726$	0	0
$727$	29.6370	1.09918	0.549588	−	0.835436i	$-0.314785\pi$
$727$	29.6370	1.09918	0.549588	+	0.835436i	$0.314785\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−36.3411	−1.34412
$732$	0	0
$733$	31.2063	1.15263	0.576315	−	0.817228i	$-0.304490\pi$
$733$	31.2063	1.15263	0.576315	+	0.817228i	$0.304490\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	10.4397	0.384552
$738$	0	0
$739$	51.2427	1.88499	0.942496	−	0.334217i	$-0.108472\pi$
$739$	51.2427	1.88499	0.942496	+	0.334217i	$0.108472\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	13.0149	0.477471	0.238736	−	0.971085i	$-0.423267\pi$
$743$	13.0149	0.477471	0.238736	+	0.971085i	$0.423267\pi$
$744$	0	0
$745$	−16.6270	−0.609165
$746$	0	0
$747$	0	0
$748$	0	0
$749$	8.67408	0.316944
$750$	0	0
$751$	9.64478	0.351943	0.175971	−	0.984395i	$-0.443693\pi$
$751$	9.64478	0.351943	0.175971	+	0.984395i	$0.443693\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2.85270	0.103820
$756$	0	0
$757$	−24.9478	−0.906743	−0.453372	−	0.891322i	$-0.649779\pi$
$757$	−24.9478	−0.906743	−0.453372	+	0.891322i	$0.649779\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	11.3516	0.411496	0.205748	−	0.978605i	$-0.434037\pi$
$761$	11.3516	0.411496	0.205748	+	0.978605i	$0.434037\pi$
$762$	0	0
$763$	14.4208	0.522068
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−10.7765	−0.389118
$768$	0	0
$769$	39.4849	1.42386	0.711931	−	0.702250i	$-0.247821\pi$
$769$	39.4849	1.42386	0.711931	+	0.702250i	$0.247821\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−15.3442	−0.551894	−0.275947	−	0.961173i	$-0.588991\pi$
$773$	−15.3442	−0.551894	−0.275947	+	0.961173i	$0.588991\pi$
$774$	0	0
$775$	−22.9381	−0.823961
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.54260	−0.0910982
$780$	0	0
$781$	14.7426	0.527533
$782$	0	0
$783$	0	0
$784$	0	0
$785$	8.09065	0.288768
$786$	0	0
$787$	29.9811	1.06871	0.534356	−	0.845260i	$-0.320554\pi$
$787$	29.9811	1.06871	0.534356	+	0.845260i	$0.320554\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−37.0414	−1.31704
$792$	0	0
$793$	34.5500	1.22691
$794$	0	0
$795$	0	0
$796$	0	0
$797$	40.4929	1.43433	0.717167	−	0.696902i	$-0.245439\pi$
$797$	40.4929	1.43433	0.717167	+	0.696902i	$0.245439\pi$
$798$	0	0
$799$	1.66368	0.0588568
$800$	0	0
$801$	0	0
$802$	0	0
$803$	32.3911	1.14306
$804$	0	0
$805$	−5.01866	−0.176885
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−8.53290	−0.300001	−0.150000	−	0.988686i	$-0.547928\pi$
$809$	−8.53290	−0.300001	−0.150000	+	0.988686i	$0.547928\pi$
$810$	0	0
$811$	−48.6599	−1.70868	−0.854340	−	0.519714i	$-0.826039\pi$
$811$	−48.6599	−1.70868	−0.854340	+	0.519714i	$0.826039\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−12.3989	−0.434313
$816$	0	0
$817$	6.09948	0.213394
$818$	0	0
$819$	0	0
$820$	0	0
$821$	29.7200	1.03724	0.518618	−	0.855006i	$-0.326447\pi$
$821$	29.7200	1.03724	0.518618	+	0.855006i	$0.326447\pi$
$822$	0	0
$823$	1.92085	0.0669565	0.0334782	−	0.999439i	$-0.489342\pi$
$823$	1.92085	0.0669565	0.0334782	+	0.999439i	$0.489342\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−50.2262	−1.74653	−0.873267	−	0.487241i	$-0.838003\pi$
$827$	−50.2262	−1.74653	−0.873267	+	0.487241i	$0.838003\pi$
$828$	0	0
$829$	32.4968	1.12866	0.564331	−	0.825549i	$-0.309134\pi$
$829$	32.4968	1.12866	0.564331	+	0.825549i	$0.309134\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	6.84355	0.237115
$834$	0	0
$835$	0.566857	0.0196169
$836$	0	0
$837$	0	0
$838$	0	0
$839$	46.8502	1.61745	0.808725	−	0.588188i	$-0.200158\pi$
$839$	46.8502	1.61745	0.808725	+	0.588188i	$0.200158\pi$
$840$	0	0
$841$	−25.0946	−0.865331
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−7.72891	−0.265882
$846$	0	0
$847$	70.8211	2.43344
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−17.3261	−0.593930
$852$	0	0
$853$	−37.9410	−1.29907	−0.649537	−	0.760330i	$-0.725037\pi$
$853$	−37.9410	−1.29907	−0.649537	+	0.760330i	$0.725037\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−53.8013	−1.83782	−0.918909	−	0.394469i	$-0.870929\pi$
$857$	−53.8013	−1.83782	−0.918909	+	0.394469i	$0.870929\pi$
$858$	0	0
$859$	21.2181	0.723953	0.361977	−	0.932187i	$-0.382102\pi$
$859$	21.2181	0.723953	0.361977	+	0.932187i	$0.382102\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−54.2060	−1.84519	−0.922597	−	0.385764i	$-0.873938\pi$
$863$	−54.2060	−1.84519	−0.922597	+	0.385764i	$0.873938\pi$
$864$	0	0
$865$	−1.33228	−0.0452988
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−67.5702	−2.29216
$870$	0	0
$871$	−4.48352	−0.151918
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−29.8335	−1.00856
$876$	0	0
$877$	34.4117	1.16200	0.581000	−	0.813904i	$-0.302662\pi$
$877$	34.4117	1.16200	0.581000	+	0.813904i	$0.302662\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−33.3764	−1.12448	−0.562239	−	0.826975i	$-0.690060\pi$
$881$	−33.3764	−1.12448	−0.562239	+	0.826975i	$0.690060\pi$
$882$	0	0
$883$	−35.5157	−1.19520	−0.597599	−	0.801795i	$-0.703878\pi$
$883$	−35.5157	−1.19520	−0.597599	+	0.801795i	$0.703878\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−15.9236	−0.534662	−0.267331	−	0.963605i	$-0.586142\pi$
$887$	−15.9236	−0.534662	−0.267331	+	0.963605i	$0.586142\pi$
$888$	0	0
$889$	−34.5068	−1.15732
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−0.279232	−0.00934413
$894$	0	0
$895$	10.3409	0.345659
$896$	0	0
$897$	0	0
$898$	0	0
$899$	12.6377	0.421489
$900$	0	0
$901$	11.6352	0.387623
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−23.6697	−0.786809
$906$	0	0
$907$	−31.4373	−1.04386	−0.521929	−	0.852989i	$-0.674787\pi$
$907$	−31.4373	−1.04386	−0.521929	+	0.852989i	$0.674787\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	46.3587	1.53593	0.767966	−	0.640490i	$-0.221269\pi$
$911$	46.3587	1.53593	0.767966	+	0.640490i	$0.221269\pi$
$912$	0	0
$913$	−35.4048	−1.17173
$914$	0	0
$915$	0	0
$916$	0	0
$917$	53.9848	1.78274
$918$	0	0
$919$	35.4799	1.17037	0.585187	−	0.810899i	$-0.301021\pi$
$919$	35.4799	1.17037	0.585187	+	0.810899i	$0.301021\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−6.33148	−0.208403
$924$	0	0
$925$	−43.0231	−1.41459
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−48.1814	−1.58078	−0.790391	−	0.612603i	$-0.790122\pi$
$929$	−48.1814	−1.58078	−0.790391	+	0.612603i	$0.790122\pi$
$930$	0	0
$931$	−1.14862	−0.0376445
$932$	0	0
$933$	0	0
$934$	0	0
$935$	31.3107	1.02397
$936$	0	0
$937$	1.11761	0.0365107	0.0182554	−	0.999833i	$-0.494189\pi$
$937$	1.11761	0.0365107	0.0182554	+	0.999833i	$0.494189\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−46.2983	−1.50928	−0.754642	−	0.656137i	$-0.772189\pi$
$941$	−46.2983	−1.50928	−0.754642	+	0.656137i	$0.772189\pi$
$942$	0	0
$943$	−4.93127	−0.160584
$944$	0	0
$945$	0	0
$946$	0	0
$947$	7.88531	0.256238	0.128119	−	0.991759i	$-0.459106\pi$
$947$	7.88531	0.256238	0.128119	+	0.991759i	$0.459106\pi$
$948$	0	0
$949$	−13.9109	−0.451568
$950$	0	0
$951$	0	0
$952$	0	0
$953$	48.6176	1.57488	0.787439	−	0.616392i	$-0.211406\pi$
$953$	48.6176	1.57488	0.787439	+	0.616392i	$0.211406\pi$
$954$	0	0
$955$	−22.7941	−0.737601
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−15.8345	−0.511322
$960$	0	0
$961$	9.89485	0.319189
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−6.52134	−0.209929
$966$	0	0
$967$	−33.0544	−1.06296	−0.531479	−	0.847071i	$-0.678364\pi$
$967$	−33.0544	−1.06296	−0.531479	+	0.847071i	$0.678364\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−12.7081	−0.407821	−0.203911	−	0.978989i	$-0.565365\pi$
$971$	−12.7081	−0.407821	−0.203911	+	0.978989i	$0.565365\pi$
$972$	0	0
$973$	4.57541	0.146681
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−27.0499	−0.865402	−0.432701	−	0.901538i	$-0.642439\pi$
$977$	−27.0499	−0.865402	−0.432701	+	0.901538i	$0.642439\pi$
$978$	0	0
$979$	4.08718	0.130627
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−52.4479	−1.67283	−0.836415	−	0.548097i	$-0.815353\pi$
$983$	−52.4479	−1.67283	−0.836415	+	0.548097i	$0.815353\pi$
$984$	0	0
$985$	−7.41504	−0.236263
$986$	0	0
$987$	0	0
$988$	0	0
$989$	11.8297	0.376162
$990$	0	0
$991$	−12.1220	−0.385067	−0.192534	−	0.981290i	$-0.561670\pi$
$991$	−12.1220	−0.385067	−0.192534	+	0.981290i	$0.561670\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−16.4350	−0.521025
$996$	0	0
$997$	41.5830	1.31695	0.658474	−	0.752603i	$-0.271202\pi$
$997$	41.5830	1.31695	0.658474	+	0.752603i	$0.271202\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8028.2.a.j.1.4		7
3.2	odd	2		892.2.a.d.1.5	✓	7
12.11	even	2		3568.2.a.m.1.3		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
892.2.a.d.1.5	✓	7	3.2	odd	2
3568.2.a.m.1.3		7	12.11	even	2
8028.2.a.j.1.4		7	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:	\( N \)	\(=\)	\( 8028 = 2^{2} \cdot 3^{2} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8028.a (trivial)

\(f(q)\)	\(=\)	\(q+1.18873 q^{5} +2.92270 q^{7} +O(q^{10})\)	\(q+1.18873 q^{5} +2.92270 q^{7} +5.93560 q^{11} -2.54915 q^{13} +4.43759 q^{17} -0.744803 q^{19} -1.44452 q^{23} -3.58693 q^{25} +1.97621 q^{29} +6.39491 q^{31} +3.47429 q^{35} +11.9944 q^{37} +3.41379 q^{41} -8.18938 q^{43} +0.374907 q^{47} +1.54218 q^{49} +2.62196 q^{53} +7.05580 q^{55} +4.22750 q^{59} -13.5536 q^{61} -3.03024 q^{65} +1.75883 q^{67} +2.48376 q^{71} +5.45709 q^{73} +17.3480 q^{77} -11.3839 q^{79} -5.96481 q^{83} +5.27507 q^{85} +0.688587 q^{89} -7.45040 q^{91} -0.885366 q^{95} +17.1583 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{5} - q^{7}+O(q^{10}) \) 7 * q + 7 * q^5 - q^7	\( 7 q + 7 q^{5} - q^{7} + 12 q^{11} - 9 q^{13} - 8 q^{17} - 12 q^{19} + 12 q^{23} + 12 q^{25} + 24 q^{29} + q^{31} + 15 q^{35} - 13 q^{37} - 5 q^{41} - 13 q^{43} + 21 q^{47} + 4 q^{49} + 35 q^{53} + q^{55} + 23 q^{59} - 17 q^{61} - 18 q^{67} + 4 q^{71} + 23 q^{73} - 3 q^{77} + 4 q^{79} + 44 q^{83} + 20 q^{85} + 2 q^{91} + 12 q^{95} + 32 q^{97}+O(q^{100}) \) 7 * q + 7 * q^5 - q^7 + 12 * q^11 - 9 * q^13 - 8 * q^17 - 12 * q^19 + 12 * q^23 + 12 * q^25 + 24 * q^29 + q^31 + 15 * q^35 - 13 * q^37 - 5 * q^41 - 13 * q^43 + 21 * q^47 + 4 * q^49 + 35 * q^53 + q^55 + 23 * q^59 - 17 * q^61 - 18 * q^67 + 4 * q^71 + 23 * q^73 - 3 * q^77 + 4 * q^79 + 44 * q^83 + 20 * q^85 + 2 * q^91 + 12 * q^95 + 32 * q^97

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