LMFDB - Embedded newform 7360.2.a.cu.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7360,2,Mod(1,7360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7360, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7360.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7360 = 2^{6} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7360.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:	$58.7698958877$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.255601784.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 10x^{4} + 5x^{3} + 25x^{2} - 4 $ x^6 - x^5 - 10x^4 + 5x^3 + 25*x^2 - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3680)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.15791$ of defining polynomial
Character	$\chi$	$=$	7360.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.15791 q^{3} -1.00000 q^{5} -0.465759 q^{7} -1.65924 q^{9} +O(q^{10})$	$q+1.15791 q^{3} -1.00000 q^{5} -0.465759 q^{7} -1.65924 q^{9} -2.23109 q^{11} +4.62874 q^{13} -1.15791 q^{15} -5.79023 q^{17} +8.52771 q^{19} -0.539307 q^{21} +1.00000 q^{23} +1.00000 q^{25} -5.39499 q^{27} -0.849812 q^{29} +0.420427 q^{31} -2.58340 q^{33} +0.465759 q^{35} -8.02132 q^{37} +5.35967 q^{39} +9.83556 q^{41} -10.7547 q^{43} +1.65924 q^{45} +9.83315 q^{47} -6.78307 q^{49} -6.70457 q^{51} -0.117646 q^{53} +2.23109 q^{55} +9.87434 q^{57} -5.74613 q^{59} +11.1069 q^{61} +0.772806 q^{63} -4.62874 q^{65} -13.0240 q^{67} +1.15791 q^{69} -5.33274 q^{71} -10.3431 q^{73} +1.15791 q^{75} +1.03915 q^{77} +2.81567 q^{79} -1.26920 q^{81} +4.53129 q^{83} +5.79023 q^{85} -0.984007 q^{87} -2.72798 q^{89} -2.15587 q^{91} +0.486818 q^{93} -8.52771 q^{95} -9.33207 q^{97} +3.70191 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 5 q^{3} - 6 q^{5} + 4 q^{7} + 7 q^{9}+O(q^{10}) $ 6 * q - 5 * q^3 - 6 * q^5 + 4 * q^7 + 7 * q^9	$ 6 q - 5 q^{3} - 6 q^{5} + 4 q^{7} + 7 q^{9} - 7 q^{11} - q^{13} + 5 q^{15} - 5 q^{19} + 6 q^{23} + 6 q^{25} - 20 q^{27} + 3 q^{29} + 12 q^{31} - 3 q^{33} - 4 q^{35} - 7 q^{37} + 8 q^{39} + 8 q^{41} - 28 q^{43} - 7 q^{45} + 8 q^{47} + 6 q^{49} - 7 q^{51} + 5 q^{53} + 7 q^{55} + 18 q^{57} - 9 q^{59} + 3 q^{61} - 5 q^{63} + q^{65} - 27 q^{67} - 5 q^{69} - 2 q^{71} - 2 q^{73} - 5 q^{75} + 14 q^{77} + 4 q^{79} + 14 q^{81} - 23 q^{83} + 12 q^{87} - 4 q^{89} + q^{91} + 4 q^{93} + 5 q^{95} - q^{97} - q^{99}+O(q^{100}) $ 6 * q - 5 * q^3 - 6 * q^5 + 4 * q^7 + 7 * q^9 - 7 * q^11 - q^13 + 5 * q^15 - 5 * q^19 + 6 * q^23 + 6 * q^25 - 20 * q^27 + 3 * q^29 + 12 * q^31 - 3 * q^33 - 4 * q^35 - 7 * q^37 + 8 * q^39 + 8 * q^41 - 28 * q^43 - 7 * q^45 + 8 * q^47 + 6 * q^49 - 7 * q^51 + 5 * q^53 + 7 * q^55 + 18 * q^57 - 9 * q^59 + 3 * q^61 - 5 * q^63 + q^65 - 27 * q^67 - 5 * q^69 - 2 * q^71 - 2 * q^73 - 5 * q^75 + 14 * q^77 + 4 * q^79 + 14 * q^81 - 23 * q^83 + 12 * q^87 - 4 * q^89 + q^91 + 4 * q^93 + 5 * q^95 - q^97 - 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
$3$	1.15791	0.668521	0.334260	−	0.942481i	$-0.391514\pi$
$3$	1.15791	0.668521	0.334260	+	0.942481i	$0.391514\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−0.465759	−0.176040	−0.0880201	−	0.996119i	$-0.528054\pi$
$7$	−0.465759	−0.176040	−0.0880201	+	0.996119i	$0.528054\pi$
$8$	0	0
$9$	−1.65924	−0.553080
$10$	0	0
$11$	−2.23109	−0.672699	−0.336349	−	0.941737i	$-0.609192\pi$
$11$	−2.23109	−0.672699	−0.336349	+	0.941737i	$0.609192\pi$
$12$	0	0
$13$	4.62874	1.28378	0.641890	−	0.766797i	$-0.278150\pi$
$13$	4.62874	1.28378	0.641890	+	0.766797i	$0.278150\pi$
$14$	0	0
$15$	−1.15791	−0.298972
$16$	0	0
$17$	−5.79023	−1.40434	−0.702168	−	0.712011i	$-0.747785\pi$
$17$	−5.79023	−1.40434	−0.702168	+	0.712011i	$0.747785\pi$
$18$	0	0
$19$	8.52771	1.95639	0.978196	−	0.207686i	$-0.0665931\pi$
$19$	8.52771	1.95639	0.978196	+	0.207686i	$0.0665931\pi$
$20$	0	0
$21$	−0.539307	−0.117687
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.39499	−1.03827
$28$	0	0
$29$	−0.849812	−0.157806	−0.0789030	−	0.996882i	$-0.525142\pi$
$29$	−0.849812	−0.157806	−0.0789030	+	0.996882i	$0.525142\pi$
$30$	0	0
$31$	0.420427	0.0755110	0.0377555	−	0.999287i	$-0.487979\pi$
$31$	0.420427	0.0755110	0.0377555	+	0.999287i	$0.487979\pi$
$32$	0	0
$33$	−2.58340	−0.449713
$34$	0	0
$35$	0.465759	0.0787276
$36$	0	0
$37$	−8.02132	−1.31870	−0.659348	−	0.751838i	$-0.729168\pi$
$37$	−8.02132	−1.31870	−0.659348	+	0.751838i	$0.729168\pi$
$38$	0	0
$39$	5.35967	0.858234
$40$	0	0
$41$	9.83556	1.53606	0.768028	−	0.640416i	$-0.221238\pi$
$41$	9.83556	1.53606	0.768028	+	0.640416i	$0.221238\pi$
$42$	0	0
$43$	−10.7547	−1.64007	−0.820035	−	0.572313i	$-0.806046\pi$
$43$	−10.7547	−1.64007	−0.820035	+	0.572313i	$0.806046\pi$
$44$	0	0
$45$	1.65924	0.247345
$46$	0	0
$47$	9.83315	1.43431	0.717156	−	0.696912i	$-0.245443\pi$
$47$	9.83315	1.43431	0.717156	+	0.696912i	$0.245443\pi$
$48$	0	0
$49$	−6.78307	−0.969010
$50$	0	0
$51$	−6.70457	−0.938828
$52$	0	0
$53$	−0.117646	−0.0161600	−0.00807998	−	0.999967i	$-0.502572\pi$
$53$	−0.117646	−0.0161600	−0.00807998	+	0.999967i	$0.502572\pi$
$54$	0	0
$55$	2.23109	0.300840
$56$	0	0
$57$	9.87434	1.30789
$58$	0	0
$59$	−5.74613	−0.748082	−0.374041	−	0.927412i	$-0.622028\pi$
$59$	−5.74613	−0.748082	−0.374041	+	0.927412i	$0.622028\pi$
$60$	0	0
$61$	11.1069	1.42209	0.711047	−	0.703144i	$-0.248221\pi$
$61$	11.1069	1.42209	0.711047	+	0.703144i	$0.248221\pi$
$62$	0	0
$63$	0.772806	0.0973644
$64$	0	0
$65$	−4.62874	−0.574124
$66$	0	0
$67$	−13.0240	−1.59113	−0.795566	−	0.605867i	$-0.792826\pi$
$67$	−13.0240	−1.59113	−0.795566	+	0.605867i	$0.792826\pi$
$68$	0	0
$69$	1.15791	0.139396
$70$	0	0
$71$	−5.33274	−0.632880	−0.316440	−	0.948612i	$-0.602488\pi$
$71$	−5.33274	−0.632880	−0.316440	+	0.948612i	$0.602488\pi$
$72$	0	0
$73$	−10.3431	−1.21057	−0.605286	−	0.796008i	$-0.706941\pi$
$73$	−10.3431	−1.21057	−0.605286	+	0.796008i	$0.706941\pi$
$74$	0	0
$75$	1.15791	0.133704
$76$	0	0
$77$	1.03915	0.118422
$78$	0	0
$79$	2.81567	0.316787	0.158394	−	0.987376i	$-0.449369\pi$
$79$	2.81567	0.316787	0.158394	+	0.987376i	$0.449369\pi$
$80$	0	0
$81$	−1.26920	−0.141022
$82$	0	0
$83$	4.53129	0.497374	0.248687	−	0.968584i	$-0.420001\pi$
$83$	4.53129	0.497374	0.248687	+	0.968584i	$0.420001\pi$
$84$	0	0
$85$	5.79023	0.628038
$86$	0	0
$87$	−0.984007	−0.105497
$88$	0	0
$89$	−2.72798	−0.289165	−0.144582	−	0.989493i	$-0.546184\pi$
$89$	−2.72798	−0.289165	−0.144582	+	0.989493i	$0.546184\pi$
$90$	0	0
$91$	−2.15587	−0.225997
$92$	0	0
$93$	0.486818	0.0504806
$94$	0	0
$95$	−8.52771	−0.874925
$96$	0	0
$97$	−9.33207	−0.947529	−0.473764	−	0.880652i	$-0.657105\pi$
$97$	−9.33207	−0.947529	−0.473764	+	0.880652i	$0.657105\pi$
$98$	0	0
$99$	3.70191	0.372056
$100$	0	0
$101$	−10.8298	−1.07761	−0.538804	−	0.842431i	$-0.681124\pi$
$101$	−10.8298	−1.07761	−0.538804	+	0.842431i	$0.681124\pi$
$102$	0	0
$103$	18.4020	1.81321	0.906604	−	0.421983i	$-0.138666\pi$
$103$	18.4020	1.81321	0.906604	+	0.421983i	$0.138666\pi$
$104$	0	0
$105$	0.539307	0.0526310
$106$	0	0
$107$	−13.5591	−1.31081	−0.655406	−	0.755277i	$-0.727502\pi$
$107$	−13.5591	−1.31081	−0.655406	+	0.755277i	$0.727502\pi$
$108$	0	0
$109$	8.90936	0.853362	0.426681	−	0.904402i	$-0.359683\pi$
$109$	8.90936	0.853362	0.426681	+	0.904402i	$0.359683\pi$
$110$	0	0
$111$	−9.28798	−0.881576
$112$	0	0
$113$	14.7407	1.38668	0.693342	−	0.720609i	$-0.256138\pi$
$113$	14.7407	1.38668	0.693342	+	0.720609i	$0.256138\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	−7.68019	−0.710034
$118$	0	0
$119$	2.69685	0.247220
$120$	0	0
$121$	−6.02224	−0.547476
$122$	0	0
$123$	11.3887	1.02689
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−5.21214	−0.462503	−0.231251	−	0.972894i	$-0.574282\pi$
$127$	−5.21214	−0.462503	−0.231251	+	0.972894i	$0.574282\pi$
$128$	0	0
$129$	−12.4529	−1.09642
$130$	0	0
$131$	−9.39432	−0.820785	−0.410393	−	0.911909i	$-0.634608\pi$
$131$	−9.39432	−0.820785	−0.410393	+	0.911909i	$0.634608\pi$
$132$	0	0
$133$	−3.97186	−0.344404
$134$	0	0
$135$	5.39499	0.464327
$136$	0	0
$137$	−8.10617	−0.692557	−0.346278	−	0.938132i	$-0.612555\pi$
$137$	−8.10617	−0.692557	−0.346278	+	0.938132i	$0.612555\pi$
$138$	0	0
$139$	10.9301	0.927080	0.463540	−	0.886076i	$-0.346579\pi$
$139$	10.9301	0.927080	0.463540	+	0.886076i	$0.346579\pi$
$140$	0	0
$141$	11.3859	0.958867
$142$	0	0
$143$	−10.3271	−0.863598
$144$	0	0
$145$	0.849812	0.0705730
$146$	0	0
$147$	−7.85419	−0.647803
$148$	0	0
$149$	−10.3994	−0.851951	−0.425975	−	0.904735i	$-0.640069\pi$
$149$	−10.3994	−0.851951	−0.425975	+	0.904735i	$0.640069\pi$
$150$	0	0
$151$	11.7585	0.956897	0.478449	−	0.878116i	$-0.341199\pi$
$151$	11.7585	0.956897	0.478449	+	0.878116i	$0.341199\pi$
$152$	0	0
$153$	9.60738	0.776711
$154$	0	0
$155$	−0.420427	−0.0337695
$156$	0	0
$157$	−20.6898	−1.65122	−0.825612	−	0.564238i	$-0.809170\pi$
$157$	−20.6898	−1.65122	−0.825612	+	0.564238i	$0.809170\pi$
$158$	0	0
$159$	−0.136224	−0.0108033
$160$	0	0
$161$	−0.465759	−0.0367069
$162$	0	0
$163$	−2.33329	−0.182757	−0.0913785	−	0.995816i	$-0.529127\pi$
$163$	−2.33329	−0.182757	−0.0913785	+	0.995816i	$0.529127\pi$
$164$	0	0
$165$	2.58340	0.201118
$166$	0	0
$167$	−11.7369	−0.908232	−0.454116	−	0.890943i	$-0.650045\pi$
$167$	−11.7369	−0.908232	−0.454116	+	0.890943i	$0.650045\pi$
$168$	0	0
$169$	8.42520	0.648092
$170$	0	0
$171$	−14.1495	−1.08204
$172$	0	0
$173$	−3.51375	−0.267146	−0.133573	−	0.991039i	$-0.542645\pi$
$173$	−3.51375	−0.267146	−0.133573	+	0.991039i	$0.542645\pi$
$174$	0	0
$175$	−0.465759	−0.0352080
$176$	0	0
$177$	−6.65351	−0.500109
$178$	0	0
$179$	−17.5028	−1.30822	−0.654109	−	0.756400i	$-0.726956\pi$
$179$	−17.5028	−1.30822	−0.654109	+	0.756400i	$0.726956\pi$
$180$	0	0
$181$	−3.75553	−0.279146	−0.139573	−	0.990212i	$-0.544573\pi$
$181$	−3.75553	−0.279146	−0.139573	+	0.990212i	$0.544573\pi$
$182$	0	0
$183$	12.8608	0.950700
$184$	0	0
$185$	8.02132	0.589739
$186$	0	0
$187$	12.9185	0.944695
$188$	0	0
$189$	2.51276	0.182777
$190$	0	0
$191$	−1.43816	−0.104061	−0.0520307	−	0.998645i	$-0.516569\pi$
$191$	−1.43816	−0.104061	−0.0520307	+	0.998645i	$0.516569\pi$
$192$	0	0
$193$	24.1344	1.73723	0.868617	−	0.495484i	$-0.165009\pi$
$193$	24.1344	1.73723	0.868617	+	0.495484i	$0.165009\pi$
$194$	0	0
$195$	−5.35967	−0.383814
$196$	0	0
$197$	−6.42860	−0.458019	−0.229009	−	0.973424i	$-0.573549\pi$
$197$	−6.42860	−0.458019	−0.229009	+	0.973424i	$0.573549\pi$
$198$	0	0
$199$	−26.1963	−1.85701	−0.928503	−	0.371324i	$-0.878904\pi$
$199$	−26.1963	−1.85701	−0.928503	+	0.371324i	$0.878904\pi$
$200$	0	0
$201$	−15.0806	−1.06370
$202$	0	0
$203$	0.395807	0.0277802
$204$	0	0
$205$	−9.83556	−0.686945
$206$	0	0
$207$	−1.65924	−0.115325
$208$	0	0
$209$	−19.0261	−1.31606
$210$	0	0
$211$	−10.8223	−0.745036	−0.372518	−	0.928025i	$-0.621506\pi$
$211$	−10.8223	−0.745036	−0.372518	+	0.928025i	$0.621506\pi$
$212$	0	0
$213$	−6.17485	−0.423093
$214$	0	0
$215$	10.7547	0.733462
$216$	0	0
$217$	−0.195818	−0.0132930
$218$	0	0
$219$	−11.9764	−0.809292
$220$	0	0
$221$	−26.8014	−1.80286
$222$	0	0
$223$	−23.7476	−1.59026	−0.795128	−	0.606442i	$-0.792596\pi$
$223$	−23.7476	−1.59026	−0.795128	+	0.606442i	$0.792596\pi$
$224$	0	0
$225$	−1.65924	−0.110616
$226$	0	0
$227$	−25.2198	−1.67390	−0.836949	−	0.547281i	$-0.815663\pi$
$227$	−25.2198	−1.67390	−0.836949	+	0.547281i	$0.815663\pi$
$228$	0	0
$229$	2.98229	0.197075	0.0985375	−	0.995133i	$-0.468584\pi$
$229$	2.98229	0.197075	0.0985375	+	0.995133i	$0.468584\pi$
$230$	0	0
$231$	1.20324	0.0791676
$232$	0	0
$233$	12.5812	0.824219	0.412109	−	0.911134i	$-0.364792\pi$
$233$	12.5812	0.824219	0.412109	+	0.911134i	$0.364792\pi$
$234$	0	0
$235$	−9.83315	−0.641444
$236$	0	0
$237$	3.26029	0.211779
$238$	0	0
$239$	0.448297	0.0289979	0.0144990	−	0.999895i	$-0.495385\pi$
$239$	0.448297	0.0289979	0.0144990	+	0.999895i	$0.495385\pi$
$240$	0	0
$241$	3.29364	0.212162	0.106081	−	0.994357i	$-0.466170\pi$
$241$	3.29364	0.212162	0.106081	+	0.994357i	$0.466170\pi$
$242$	0	0
$243$	14.7153	0.943990
$244$	0	0
$245$	6.78307	0.433354
$246$	0	0
$247$	39.4725	2.51158
$248$	0	0
$249$	5.24683	0.332505
$250$	0	0
$251$	−12.3994	−0.782642	−0.391321	−	0.920254i	$-0.627982\pi$
$251$	−12.3994	−0.782642	−0.391321	+	0.920254i	$0.627982\pi$
$252$	0	0
$253$	−2.23109	−0.140267
$254$	0	0
$255$	6.70457	0.419857
$256$	0	0
$257$	−4.38450	−0.273498	−0.136749	−	0.990606i	$-0.543665\pi$
$257$	−4.38450	−0.273498	−0.136749	+	0.990606i	$0.543665\pi$
$258$	0	0
$259$	3.73600	0.232144
$260$	0	0
$261$	1.41004	0.0872794
$262$	0	0
$263$	−20.7891	−1.28191	−0.640954	−	0.767579i	$-0.721461\pi$
$263$	−20.7891	−1.28191	−0.640954	+	0.767579i	$0.721461\pi$
$264$	0	0
$265$	0.117646	0.00722695
$266$	0	0
$267$	−3.15876	−0.193313
$268$	0	0
$269$	−25.9124	−1.57991	−0.789953	−	0.613167i	$-0.789895\pi$
$269$	−25.9124	−1.57991	−0.789953	+	0.613167i	$0.789895\pi$
$270$	0	0
$271$	10.9864	0.667374	0.333687	−	0.942684i	$-0.391707\pi$
$271$	10.9864	0.667374	0.333687	+	0.942684i	$0.391707\pi$
$272$	0	0
$273$	−2.49631	−0.151084
$274$	0	0
$275$	−2.23109	−0.134540
$276$	0	0
$277$	30.1433	1.81114	0.905568	−	0.424201i	$-0.139445\pi$
$277$	30.1433	1.81114	0.905568	+	0.424201i	$0.139445\pi$
$278$	0	0
$279$	−0.697590	−0.0417636
$280$	0	0
$281$	−30.9734	−1.84772	−0.923859	−	0.382732i	$-0.874983\pi$
$281$	−30.9734	−1.84772	−0.923859	+	0.382732i	$0.874983\pi$
$282$	0	0
$283$	−17.3146	−1.02925	−0.514623	−	0.857416i	$-0.672068\pi$
$283$	−17.3146	−1.02925	−0.514623	+	0.857416i	$0.672068\pi$
$284$	0	0
$285$	−9.87434	−0.584905
$286$	0	0
$287$	−4.58100	−0.270408
$288$	0	0
$289$	16.5267	0.972161
$290$	0	0
$291$	−10.8057	−0.633442
$292$	0	0
$293$	12.1473	0.709653	0.354826	−	0.934932i	$-0.384540\pi$
$293$	12.1473	0.709653	0.354826	+	0.934932i	$0.384540\pi$
$294$	0	0
$295$	5.74613	0.334553
$296$	0	0
$297$	12.0367	0.698440
$298$	0	0
$299$	4.62874	0.267687
$300$	0	0
$301$	5.00907	0.288718
$302$	0	0
$303$	−12.5400	−0.720403
$304$	0	0
$305$	−11.1069	−0.635980
$306$	0	0
$307$	−5.18321	−0.295822	−0.147911	−	0.989001i	$-0.547255\pi$
$307$	−5.18321	−0.295822	−0.147911	+	0.989001i	$0.547255\pi$
$308$	0	0
$309$	21.3079	1.21217
$310$	0	0
$311$	−33.8310	−1.91838	−0.959189	−	0.282766i	$-0.908748\pi$
$311$	−33.8310	−1.91838	−0.959189	+	0.282766i	$0.908748\pi$
$312$	0	0
$313$	−4.21244	−0.238101	−0.119050	−	0.992888i	$-0.537985\pi$
$313$	−4.21244	−0.238101	−0.119050	+	0.992888i	$0.537985\pi$
$314$	0	0
$315$	−0.772806	−0.0435427
$316$	0	0
$317$	21.2598	1.19407	0.597034	−	0.802216i	$-0.296346\pi$
$317$	21.2598	1.19407	0.597034	+	0.802216i	$0.296346\pi$
$318$	0	0
$319$	1.89601	0.106156
$320$	0	0
$321$	−15.7003	−0.876304
$322$	0	0
$323$	−49.3774	−2.74743
$324$	0	0
$325$	4.62874	0.256756
$326$	0	0
$327$	10.3163	0.570490
$328$	0	0
$329$	−4.57987	−0.252497
$330$	0	0
$331$	9.46343	0.520157	0.260079	−	0.965587i	$-0.416252\pi$
$331$	9.46343	0.520157	0.260079	+	0.965587i	$0.416252\pi$
$332$	0	0
$333$	13.3093	0.729345
$334$	0	0
$335$	13.0240	0.711576
$336$	0	0
$337$	22.4773	1.22442	0.612209	−	0.790696i	$-0.290281\pi$
$337$	22.4773	1.22442	0.612209	+	0.790696i	$0.290281\pi$
$338$	0	0
$339$	17.0684	0.927027
$340$	0	0
$341$	−0.938011	−0.0507961
$342$	0	0
$343$	6.41958	0.346625
$344$	0	0
$345$	−1.15791	−0.0623399
$346$	0	0
$347$	−6.40869	−0.344037	−0.172018	−	0.985094i	$-0.555029\pi$
$347$	−6.40869	−0.344037	−0.172018	+	0.985094i	$0.555029\pi$
$348$	0	0
$349$	31.4659	1.68433	0.842166	−	0.539218i	$-0.181280\pi$
$349$	31.4659	1.68433	0.842166	+	0.539218i	$0.181280\pi$
$350$	0	0
$351$	−24.9720	−1.33291
$352$	0	0
$353$	27.7243	1.47561	0.737807	−	0.675012i	$-0.235861\pi$
$353$	27.7243	1.47561	0.737807	+	0.675012i	$0.235861\pi$
$354$	0	0
$355$	5.33274	0.283033
$356$	0	0
$357$	3.12271	0.165271
$358$	0	0
$359$	−8.59766	−0.453767	−0.226884	−	0.973922i	$-0.572854\pi$
$359$	−8.59766	−0.453767	−0.226884	+	0.973922i	$0.572854\pi$
$360$	0	0
$361$	53.7219	2.82747
$362$	0	0
$363$	−6.97322	−0.365999
$364$	0	0
$365$	10.3431	0.541384
$366$	0	0
$367$	−13.1007	−0.683850	−0.341925	−	0.939727i	$-0.611079\pi$
$367$	−13.1007	−0.683850	−0.341925	+	0.939727i	$0.611079\pi$
$368$	0	0
$369$	−16.3196	−0.849562
$370$	0	0
$371$	0.0547948	0.00284480
$372$	0	0
$373$	−37.5034	−1.94185	−0.970927	−	0.239378i	$-0.923057\pi$
$373$	−37.5034	−1.94185	−0.970927	+	0.239378i	$0.923057\pi$
$374$	0	0
$375$	−1.15791	−0.0597943
$376$	0	0
$377$	−3.93356	−0.202588
$378$	0	0
$379$	3.18372	0.163537	0.0817684	−	0.996651i	$-0.473943\pi$
$379$	3.18372	0.163537	0.0817684	+	0.996651i	$0.473943\pi$
$380$	0	0
$381$	−6.03520	−0.309193
$382$	0	0
$383$	−34.3673	−1.75609	−0.878044	−	0.478581i	$-0.841151\pi$
$383$	−34.3673	−1.75609	−0.878044	+	0.478581i	$0.841151\pi$
$384$	0	0
$385$	−1.03915	−0.0529600
$386$	0	0
$387$	17.8446	0.907090
$388$	0	0
$389$	−11.5186	−0.584014	−0.292007	−	0.956416i	$-0.594323\pi$
$389$	−11.5186	−0.584014	−0.292007	+	0.956416i	$0.594323\pi$
$390$	0	0
$391$	−5.79023	−0.292824
$392$	0	0
$393$	−10.8778	−0.548712
$394$	0	0
$395$	−2.81567	−0.141671
$396$	0	0
$397$	8.53073	0.428145	0.214073	−	0.976818i	$-0.431327\pi$
$397$	8.53073	0.428145	0.214073	+	0.976818i	$0.431327\pi$
$398$	0	0
$399$	−4.59906	−0.230241
$400$	0	0
$401$	30.5701	1.52660	0.763298	−	0.646047i	$-0.223579\pi$
$401$	30.5701	1.52660	0.763298	+	0.646047i	$0.223579\pi$
$402$	0	0
$403$	1.94605	0.0969395
$404$	0	0
$405$	1.26920	0.0630670
$406$	0	0
$407$	17.8963	0.887086
$408$	0	0
$409$	0.225779	0.0111640	0.00558201	−	0.999984i	$-0.498223\pi$
$409$	0.225779	0.0111640	0.00558201	+	0.999984i	$0.498223\pi$
$410$	0	0
$411$	−9.38623	−0.462988
$412$	0	0
$413$	2.67631	0.131693
$414$	0	0
$415$	−4.53129	−0.222432
$416$	0	0
$417$	12.6561	0.619772
$418$	0	0
$419$	−21.9026	−1.07001	−0.535006	−	0.844848i	$-0.679691\pi$
$419$	−21.9026	−1.07001	−0.535006	+	0.844848i	$0.679691\pi$
$420$	0	0
$421$	−21.4368	−1.04476	−0.522382	−	0.852711i	$-0.674957\pi$
$421$	−21.4368	−1.04476	−0.522382	+	0.852711i	$0.674957\pi$
$422$	0	0
$423$	−16.3156	−0.793290
$424$	0	0
$425$	−5.79023	−0.280867
$426$	0	0
$427$	−5.17314	−0.250346
$428$	0	0
$429$	−11.9579	−0.577333
$430$	0	0
$431$	−0.0808822	−0.00389596	−0.00194798	−	0.999998i	$-0.500620\pi$
$431$	−0.0808822	−0.00389596	−0.00194798	+	0.999998i	$0.500620\pi$
$432$	0	0
$433$	−4.84185	−0.232684	−0.116342	−	0.993209i	$-0.537117\pi$
$433$	−4.84185	−0.232684	−0.116342	+	0.993209i	$0.537117\pi$
$434$	0	0
$435$	0.984007	0.0471795
$436$	0	0
$437$	8.52771	0.407936
$438$	0	0
$439$	26.6511	1.27199	0.635994	−	0.771694i	$-0.280590\pi$
$439$	26.6511	1.27199	0.635994	+	0.771694i	$0.280590\pi$
$440$	0	0
$441$	11.2547	0.535940
$442$	0	0
$443$	−23.7170	−1.12683	−0.563415	−	0.826174i	$-0.690513\pi$
$443$	−23.7170	−1.12683	−0.563415	+	0.826174i	$0.690513\pi$
$444$	0	0
$445$	2.72798	0.129318
$446$	0	0
$447$	−12.0416	−0.569547
$448$	0	0
$449$	20.4809	0.966552	0.483276	−	0.875468i	$-0.339447\pi$
$449$	20.4809	0.966552	0.483276	+	0.875468i	$0.339447\pi$
$450$	0	0
$451$	−21.9440	−1.03330
$452$	0	0
$453$	13.6154	0.639705
$454$	0	0
$455$	2.15587	0.101069
$456$	0	0
$457$	13.9529	0.652690	0.326345	−	0.945251i	$-0.394183\pi$
$457$	13.9529	0.652690	0.326345	+	0.945251i	$0.394183\pi$
$458$	0	0
$459$	31.2382	1.45807
$460$	0	0
$461$	−16.4742	−0.767279	−0.383640	−	0.923483i	$-0.625330\pi$
$461$	−16.4742	−0.767279	−0.383640	+	0.923483i	$0.625330\pi$
$462$	0	0
$463$	16.5416	0.768755	0.384377	−	0.923176i	$-0.374416\pi$
$463$	16.5416	0.768755	0.384377	+	0.923176i	$0.374416\pi$
$464$	0	0
$465$	−0.486818	−0.0225756
$466$	0	0
$467$	−25.0041	−1.15705	−0.578526	−	0.815664i	$-0.696372\pi$
$467$	−25.0041	−1.15705	−0.578526	+	0.815664i	$0.696372\pi$
$468$	0	0
$469$	6.06603	0.280103
$470$	0	0
$471$	−23.9569	−1.10388
$472$	0	0
$473$	23.9946	1.10327
$474$	0	0
$475$	8.52771	0.391278
$476$	0	0
$477$	0.195203	0.00893775
$478$	0	0
$479$	25.9709	1.18664	0.593321	−	0.804966i	$-0.297817\pi$
$479$	25.9709	1.18664	0.593321	+	0.804966i	$0.297817\pi$
$480$	0	0
$481$	−37.1286	−1.69292
$482$	0	0
$483$	−0.539307	−0.0245393
$484$	0	0
$485$	9.33207	0.423748
$486$	0	0
$487$	30.9401	1.40203	0.701014	−	0.713147i	$-0.252731\pi$
$487$	30.9401	1.40203	0.701014	+	0.713147i	$0.252731\pi$
$488$	0	0
$489$	−2.70174	−0.122177
$490$	0	0
$491$	34.3814	1.55161	0.775805	−	0.630973i	$-0.217344\pi$
$491$	34.3814	1.55161	0.775805	+	0.630973i	$0.217344\pi$
$492$	0	0
$493$	4.92060	0.221613
$494$	0	0
$495$	−3.70191	−0.166389
$496$	0	0
$497$	2.48377	0.111412
$498$	0	0
$499$	−5.77394	−0.258477	−0.129238	−	0.991614i	$-0.541253\pi$
$499$	−5.77394	−0.258477	−0.129238	+	0.991614i	$0.541253\pi$
$500$	0	0
$501$	−13.5903	−0.607172
$502$	0	0
$503$	−31.8853	−1.42170	−0.710848	−	0.703346i	$-0.751689\pi$
$503$	−31.8853	−1.42170	−0.710848	+	0.703346i	$0.751689\pi$
$504$	0	0
$505$	10.8298	0.481921
$506$	0	0
$507$	9.75564	0.433263
$508$	0	0
$509$	10.1536	0.450048	0.225024	−	0.974353i	$-0.427754\pi$
$509$	10.1536	0.450048	0.225024	+	0.974353i	$0.427754\pi$
$510$	0	0
$511$	4.81740	0.213109
$512$	0	0
$513$	−46.0069	−2.03125
$514$	0	0
$515$	−18.4020	−0.810891
$516$	0	0
$517$	−21.9386	−0.964860
$518$	0	0
$519$	−4.06861	−0.178592
$520$	0	0
$521$	−15.3459	−0.672316	−0.336158	−	0.941806i	$-0.609128\pi$
$521$	−15.3459	−0.672316	−0.336158	+	0.941806i	$0.609128\pi$
$522$	0	0
$523$	6.02976	0.263663	0.131832	−	0.991272i	$-0.457914\pi$
$523$	6.02976	0.263663	0.131832	+	0.991272i	$0.457914\pi$
$524$	0	0
$525$	−0.539307	−0.0235373
$526$	0	0
$527$	−2.43437	−0.106043
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	9.53421	0.413750
$532$	0	0
$533$	45.5262	1.97196
$534$	0	0
$535$	13.5591	0.586213
$536$	0	0
$537$	−20.2667	−0.874571
$538$	0	0
$539$	15.1336	0.651852
$540$	0	0
$541$	6.28062	0.270025	0.135013	−	0.990844i	$-0.456893\pi$
$541$	6.28062	0.270025	0.135013	+	0.990844i	$0.456893\pi$
$542$	0	0
$543$	−4.34857	−0.186615
$544$	0	0
$545$	−8.90936	−0.381635
$546$	0	0
$547$	−43.3358	−1.85290	−0.926452	−	0.376414i	$-0.877157\pi$
$547$	−43.3358	−1.85290	−0.926452	+	0.376414i	$0.877157\pi$
$548$	0	0
$549$	−18.4290	−0.786532
$550$	0	0
$551$	−7.24695	−0.308730
$552$	0	0
$553$	−1.31142	−0.0557673
$554$	0	0
$555$	9.28798	0.394253
$556$	0	0
$557$	34.4176	1.45832	0.729160	−	0.684343i	$-0.239911\pi$
$557$	34.4176	1.45832	0.729160	+	0.684343i	$0.239911\pi$
$558$	0	0
$559$	−49.7805	−2.10549
$560$	0	0
$561$	14.9585	0.631548
$562$	0	0
$563$	35.9862	1.51664	0.758318	−	0.651885i	$-0.226021\pi$
$563$	35.9862	1.51664	0.758318	+	0.651885i	$0.226021\pi$
$564$	0	0
$565$	−14.7407	−0.620144
$566$	0	0
$567$	0.591141	0.0248256
$568$	0	0
$569$	−12.1866	−0.510891	−0.255445	−	0.966824i	$-0.582222\pi$
$569$	−12.1866	−0.510891	−0.255445	+	0.966824i	$0.582222\pi$
$570$	0	0
$571$	34.1258	1.42812	0.714060	−	0.700084i	$-0.246854\pi$
$571$	34.1258	1.42812	0.714060	+	0.700084i	$0.246854\pi$
$572$	0	0
$573$	−1.66526	−0.0695671
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	−10.0326	−0.417664	−0.208832	−	0.977952i	$-0.566966\pi$
$577$	−10.0326	−0.417664	−0.208832	+	0.977952i	$0.566966\pi$
$578$	0	0
$579$	27.9455	1.16138
$580$	0	0
$581$	−2.11049	−0.0875578
$582$	0	0
$583$	0.262479	0.0108708
$584$	0	0
$585$	7.68019	0.317537
$586$	0	0
$587$	−2.45278	−0.101237	−0.0506184	−	0.998718i	$-0.516119\pi$
$587$	−2.45278	−0.101237	−0.0506184	+	0.998718i	$0.516119\pi$
$588$	0	0
$589$	3.58528	0.147729
$590$	0	0
$591$	−7.44375	−0.306195
$592$	0	0
$593$	−7.36886	−0.302603	−0.151301	−	0.988488i	$-0.548346\pi$
$593$	−7.36886	−0.302603	−0.151301	+	0.988488i	$0.548346\pi$
$594$	0	0
$595$	−2.69685	−0.110560
$596$	0	0
$597$	−30.3330	−1.24145
$598$	0	0
$599$	−42.2822	−1.72760	−0.863801	−	0.503833i	$-0.831923\pi$
$599$	−42.2822	−1.72760	−0.863801	+	0.503833i	$0.831923\pi$
$600$	0	0
$601$	−10.1309	−0.413249	−0.206624	−	0.978420i	$-0.566248\pi$
$601$	−10.1309	−0.413249	−0.206624	+	0.978420i	$0.566248\pi$
$602$	0	0
$603$	21.6099	0.880023
$604$	0	0
$605$	6.02224	0.244839
$606$	0	0
$607$	19.8910	0.807349	0.403675	−	0.914903i	$-0.367733\pi$
$607$	19.8910	0.807349	0.403675	+	0.914903i	$0.367733\pi$
$608$	0	0
$609$	0.458310	0.0185717
$610$	0	0
$611$	45.5151	1.84134
$612$	0	0
$613$	10.0556	0.406142	0.203071	−	0.979164i	$-0.434908\pi$
$613$	10.0556	0.406142	0.203071	+	0.979164i	$0.434908\pi$
$614$	0	0
$615$	−11.3887	−0.459237
$616$	0	0
$617$	0.200039	0.00805325	0.00402662	−	0.999992i	$-0.498718\pi$
$617$	0.200039	0.00805325	0.00402662	+	0.999992i	$0.498718\pi$
$618$	0	0
$619$	23.6432	0.950302	0.475151	−	0.879904i	$-0.342393\pi$
$619$	23.6432	0.950302	0.475151	+	0.879904i	$0.342393\pi$
$620$	0	0
$621$	−5.39499	−0.216493
$622$	0	0
$623$	1.27058	0.0509047
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−22.0305	−0.879815
$628$	0	0
$629$	46.4452	1.85189
$630$	0	0
$631$	40.0524	1.59446	0.797231	−	0.603675i	$-0.206297\pi$
$631$	40.0524	1.59446	0.797231	+	0.603675i	$0.206297\pi$
$632$	0	0
$633$	−12.5312	−0.498072
$634$	0	0
$635$	5.21214	0.206838
$636$	0	0
$637$	−31.3970	−1.24400
$638$	0	0
$639$	8.84830	0.350033
$640$	0	0
$641$	−23.6726	−0.935011	−0.467506	−	0.883990i	$-0.654847\pi$
$641$	−23.6726	−0.935011	−0.467506	+	0.883990i	$0.654847\pi$
$642$	0	0
$643$	−32.7422	−1.29122	−0.645612	−	0.763665i	$-0.723398\pi$
$643$	−32.7422	−1.29122	−0.645612	+	0.763665i	$0.723398\pi$
$644$	0	0
$645$	12.4529	0.490334
$646$	0	0
$647$	10.9153	0.429126	0.214563	−	0.976710i	$-0.431167\pi$
$647$	10.9153	0.429126	0.214563	+	0.976710i	$0.431167\pi$
$648$	0	0
$649$	12.8201	0.503234
$650$	0	0
$651$	−0.226739	−0.00888662
$652$	0	0
$653$	−9.00988	−0.352584	−0.176292	−	0.984338i	$-0.556410\pi$
$653$	−9.00988	−0.352584	−0.176292	+	0.984338i	$0.556410\pi$
$654$	0	0
$655$	9.39432	0.367066
$656$	0	0
$657$	17.1617	0.669543
$658$	0	0
$659$	46.7116	1.81963	0.909813	−	0.415019i	$-0.136225\pi$
$659$	46.7116	1.81963	0.909813	+	0.415019i	$0.136225\pi$
$660$	0	0
$661$	−17.9700	−0.698953	−0.349477	−	0.936945i	$-0.613641\pi$
$661$	−17.9700	−0.698953	−0.349477	+	0.936945i	$0.613641\pi$
$662$	0	0
$663$	−31.0337	−1.20525
$664$	0	0
$665$	3.97186	0.154022
$666$	0	0
$667$	−0.849812	−0.0329048
$668$	0	0
$669$	−27.4976	−1.06312
$670$	0	0
$671$	−24.7805	−0.956641
$672$	0	0
$673$	17.9737	0.692836	0.346418	−	0.938080i	$-0.387398\pi$
$673$	17.9737	0.692836	0.346418	+	0.938080i	$0.387398\pi$
$674$	0	0
$675$	−5.39499	−0.207653
$676$	0	0
$677$	−18.4620	−0.709552	−0.354776	−	0.934951i	$-0.615443\pi$
$677$	−18.4620	−0.709552	−0.354776	+	0.934951i	$0.615443\pi$
$678$	0	0
$679$	4.34649	0.166803
$680$	0	0
$681$	−29.2023	−1.11904
$682$	0	0
$683$	−44.7818	−1.71353	−0.856765	−	0.515708i	$-0.827529\pi$
$683$	−44.7818	−1.71353	−0.856765	+	0.515708i	$0.827529\pi$
$684$	0	0
$685$	8.10617	0.309721
$686$	0	0
$687$	3.45322	0.131749
$688$	0	0
$689$	−0.544554	−0.0207458
$690$	0	0
$691$	−35.5155	−1.35107	−0.675537	−	0.737326i	$-0.736088\pi$
$691$	−35.5155	−1.35107	−0.675537	+	0.737326i	$0.736088\pi$
$692$	0	0
$693$	−1.72420	−0.0654969
$694$	0	0
$695$	−10.9301	−0.414603
$696$	0	0
$697$	−56.9501	−2.15714
$698$	0	0
$699$	14.5679	0.551007
$700$	0	0
$701$	12.7966	0.483321	0.241661	−	0.970361i	$-0.422308\pi$
$701$	12.7966	0.483321	0.241661	+	0.970361i	$0.422308\pi$
$702$	0	0
$703$	−68.4035	−2.57989
$704$	0	0
$705$	−11.3859	−0.428819
$706$	0	0
$707$	5.04408	0.189702
$708$	0	0
$709$	38.2861	1.43786	0.718932	−	0.695080i	$-0.244631\pi$
$709$	38.2861	1.43786	0.718932	+	0.695080i	$0.244631\pi$
$710$	0	0
$711$	−4.67187	−0.175209
$712$	0	0
$713$	0.420427	0.0157451
$714$	0	0
$715$	10.3271	0.386213
$716$	0	0
$717$	0.519088	0.0193857
$718$	0	0
$719$	10.4736	0.390599	0.195299	−	0.980744i	$-0.437432\pi$
$719$	10.4736	0.390599	0.195299	+	0.980744i	$0.437432\pi$
$720$	0	0
$721$	−8.57091	−0.319197
$722$	0	0
$723$	3.81375	0.141835
$724$	0	0
$725$	−0.849812	−0.0315612
$726$	0	0
$727$	38.1380	1.41446	0.707231	−	0.706983i	$-0.249944\pi$
$727$	38.1380	1.41446	0.707231	+	0.706983i	$0.249944\pi$
$728$	0	0
$729$	20.8467	0.772099
$730$	0	0
$731$	62.2719	2.30321
$732$	0	0
$733$	16.8532	0.622486	0.311243	−	0.950330i	$-0.399255\pi$
$733$	16.8532	0.622486	0.311243	+	0.950330i	$0.399255\pi$
$734$	0	0
$735$	7.85419	0.289706
$736$	0	0
$737$	29.0577	1.07035
$738$	0	0
$739$	24.3286	0.894943	0.447471	−	0.894298i	$-0.352325\pi$
$739$	24.3286	0.894943	0.447471	+	0.894298i	$0.352325\pi$
$740$	0	0
$741$	45.7057	1.67904
$742$	0	0
$743$	29.0838	1.06698	0.533491	−	0.845806i	$-0.320880\pi$
$743$	29.0838	1.06698	0.533491	+	0.845806i	$0.320880\pi$
$744$	0	0
$745$	10.3994	0.381004
$746$	0	0
$747$	−7.51850	−0.275088
$748$	0	0
$749$	6.31529	0.230756
$750$	0	0
$751$	24.8580	0.907082	0.453541	−	0.891235i	$-0.350160\pi$
$751$	24.8580	0.907082	0.453541	+	0.891235i	$0.350160\pi$
$752$	0	0
$753$	−14.3574	−0.523213
$754$	0	0
$755$	−11.7585	−0.427937
$756$	0	0
$757$	−2.97900	−0.108274	−0.0541368	−	0.998534i	$-0.517241\pi$
$757$	−2.97900	−0.108274	−0.0541368	+	0.998534i	$0.517241\pi$
$758$	0	0
$759$	−2.58340	−0.0937717
$760$	0	0
$761$	40.2258	1.45819	0.729093	−	0.684415i	$-0.239942\pi$
$761$	40.2258	1.45819	0.729093	+	0.684415i	$0.239942\pi$
$762$	0	0
$763$	−4.14961	−0.150226
$764$	0	0
$765$	−9.60738	−0.347356
$766$	0	0
$767$	−26.5973	−0.960374
$768$	0	0
$769$	22.7556	0.820586	0.410293	−	0.911954i	$-0.365426\pi$
$769$	22.7556	0.820586	0.410293	+	0.911954i	$0.365426\pi$
$770$	0	0
$771$	−5.07686	−0.182839
$772$	0	0
$773$	−43.5956	−1.56802	−0.784011	−	0.620746i	$-0.786830\pi$
$773$	−43.5956	−1.56802	−0.784011	+	0.620746i	$0.786830\pi$
$774$	0	0
$775$	0.420427	0.0151022
$776$	0	0
$777$	4.32596	0.155193
$778$	0	0
$779$	83.8748	3.00513
$780$	0	0
$781$	11.8978	0.425738
$782$	0	0
$783$	4.58473	0.163845
$784$	0	0
$785$	20.6898	0.738450
$786$	0	0
$787$	23.0259	0.820783	0.410392	−	0.911909i	$-0.365392\pi$
$787$	23.0259	0.820783	0.410392	+	0.911909i	$0.365392\pi$
$788$	0	0
$789$	−24.0719	−0.856982
$790$	0	0
$791$	−6.86559	−0.244112
$792$	0	0
$793$	51.4110	1.82566
$794$	0	0
$795$	0.136224	0.00483137
$796$	0	0
$797$	28.9225	1.02449	0.512243	−	0.858840i	$-0.328815\pi$
$797$	28.9225	1.02449	0.512243	+	0.858840i	$0.328815\pi$
$798$	0	0
$799$	−56.9362	−2.01426
$800$	0	0
$801$	4.52637	0.159931
$802$	0	0
$803$	23.0764	0.814350
$804$	0	0
$805$	0.465759	0.0164158
$806$	0	0
$807$	−30.0043	−1.05620
$808$	0	0
$809$	22.0026	0.773569	0.386784	−	0.922170i	$-0.373586\pi$
$809$	22.0026	0.773569	0.386784	+	0.922170i	$0.373586\pi$
$810$	0	0
$811$	−32.5940	−1.14453	−0.572265	−	0.820069i	$-0.693935\pi$
$811$	−32.5940	−1.14453	−0.572265	+	0.820069i	$0.693935\pi$
$812$	0	0
$813$	12.7212	0.446154
$814$	0	0
$815$	2.33329	0.0817314
$816$	0	0
$817$	−91.7126	−3.20862
$818$	0	0
$819$	3.57711	0.124994
$820$	0	0
$821$	18.1922	0.634912	0.317456	−	0.948273i	$-0.397171\pi$
$821$	18.1922	0.634912	0.317456	+	0.948273i	$0.397171\pi$
$822$	0	0
$823$	−28.6190	−0.997594	−0.498797	−	0.866719i	$-0.666225\pi$
$823$	−28.6190	−0.997594	−0.498797	+	0.866719i	$0.666225\pi$
$824$	0	0
$825$	−2.58340	−0.0899426
$826$	0	0
$827$	−14.6331	−0.508842	−0.254421	−	0.967094i	$-0.581885\pi$
$827$	−14.6331	−0.508842	−0.254421	+	0.967094i	$0.581885\pi$
$828$	0	0
$829$	53.9955	1.87534	0.937670	−	0.347528i	$-0.112979\pi$
$829$	53.9955	1.87534	0.937670	+	0.347528i	$0.112979\pi$
$830$	0	0
$831$	34.9033	1.21078
$832$	0	0
$833$	39.2755	1.36082
$834$	0	0
$835$	11.7369	0.406174
$836$	0	0
$837$	−2.26820	−0.0784005
$838$	0	0
$839$	−42.9737	−1.48362	−0.741809	−	0.670612i	$-0.766032\pi$
$839$	−42.9737	−1.48362	−0.741809	+	0.670612i	$0.766032\pi$
$840$	0	0
$841$	−28.2778	−0.975097
$842$	0	0
$843$	−35.8645	−1.23524
$844$	0	0
$845$	−8.42520	−0.289836
$846$	0	0
$847$	2.80491	0.0963778
$848$	0	0
$849$	−20.0488	−0.688073
$850$	0	0
$851$	−8.02132	−0.274967
$852$	0	0
$853$	2.69097	0.0921370	0.0460685	−	0.998938i	$-0.485331\pi$
$853$	2.69097	0.0921370	0.0460685	+	0.998938i	$0.485331\pi$
$854$	0	0
$855$	14.1495	0.483904
$856$	0	0
$857$	4.37139	0.149324	0.0746620	−	0.997209i	$-0.476212\pi$
$857$	4.37139	0.149324	0.0746620	+	0.997209i	$0.476212\pi$
$858$	0	0
$859$	−53.4919	−1.82512	−0.912560	−	0.408942i	$-0.865898\pi$
$859$	−53.4919	−1.82512	−0.912560	+	0.408942i	$0.865898\pi$
$860$	0	0
$861$	−5.30439	−0.180773
$862$	0	0
$863$	−51.0473	−1.73767	−0.868835	−	0.495101i	$-0.835131\pi$
$863$	−51.0473	−1.73767	−0.868835	+	0.495101i	$0.835131\pi$
$864$	0	0
$865$	3.51375	0.119471
$866$	0	0
$867$	19.1365	0.649909
$868$	0	0
$869$	−6.28200	−0.213102
$870$	0	0
$871$	−60.2845	−2.04266
$872$	0	0
$873$	15.4842	0.524059
$874$	0	0
$875$	0.465759	0.0157455
$876$	0	0
$877$	−26.0132	−0.878402	−0.439201	−	0.898389i	$-0.644738\pi$
$877$	−26.0132	−0.878402	−0.439201	+	0.898389i	$0.644738\pi$
$878$	0	0
$879$	14.0655	0.474418
$880$	0	0
$881$	11.8046	0.397707	0.198854	−	0.980029i	$-0.436278\pi$
$881$	11.8046	0.397707	0.198854	+	0.980029i	$0.436278\pi$
$882$	0	0
$883$	−4.40753	−0.148325	−0.0741626	−	0.997246i	$-0.523628\pi$
$883$	−4.40753	−0.148325	−0.0741626	+	0.997246i	$0.523628\pi$
$884$	0	0
$885$	6.65351	0.223655
$886$	0	0
$887$	16.0308	0.538261	0.269130	−	0.963104i	$-0.413264\pi$
$887$	16.0308	0.538261	0.269130	+	0.963104i	$0.413264\pi$
$888$	0	0
$889$	2.42760	0.0814191
$890$	0	0
$891$	2.83170	0.0948655
$892$	0	0
$893$	83.8543	2.80608
$894$	0	0
$895$	17.5028	0.585053
$896$	0	0
$897$	5.35967	0.178954
$898$	0	0
$899$	−0.357284	−0.0119161
$900$	0	0
$901$	0.681199	0.0226940
$902$	0	0
$903$	5.80007	0.193014
$904$	0	0
$905$	3.75553	0.124838
$906$	0	0
$907$	−3.53356	−0.117330	−0.0586649	−	0.998278i	$-0.518684\pi$
$907$	−3.53356	−0.117330	−0.0586649	+	0.998278i	$0.518684\pi$
$908$	0	0
$909$	17.9693	0.596003
$910$	0	0
$911$	−10.4696	−0.346872	−0.173436	−	0.984845i	$-0.555487\pi$
$911$	−10.4696	−0.346872	−0.173436	+	0.984845i	$0.555487\pi$
$912$	0	0
$913$	−10.1097	−0.334583
$914$	0	0
$915$	−12.8608	−0.425166
$916$	0	0
$917$	4.37548	0.144491
$918$	0	0
$919$	−31.7163	−1.04622	−0.523111	−	0.852264i	$-0.675229\pi$
$919$	−31.7163	−1.04622	−0.523111	+	0.852264i	$0.675229\pi$
$920$	0	0
$921$	−6.00170	−0.197763
$922$	0	0
$923$	−24.6839	−0.812479
$924$	0	0
$925$	−8.02132	−0.263739
$926$	0	0
$927$	−30.5334	−1.00285
$928$	0	0
$929$	20.4993	0.672561	0.336281	−	0.941762i	$-0.390831\pi$
$929$	20.4993	0.672561	0.336281	+	0.941762i	$0.390831\pi$
$930$	0	0
$931$	−57.8441	−1.89576
$932$	0	0
$933$	−39.1733	−1.28248
$934$	0	0
$935$	−12.9185	−0.422481
$936$	0	0
$937$	61.0402	1.99410	0.997049	−	0.0767683i	$-0.0244602\pi$
$937$	61.0402	1.99410	0.997049	+	0.0767683i	$0.0244602\pi$
$938$	0	0
$939$	−4.87763	−0.159175
$940$	0	0
$941$	−18.1187	−0.590653	−0.295327	−	0.955396i	$-0.595428\pi$
$941$	−18.1187	−0.590653	−0.295327	+	0.955396i	$0.595428\pi$
$942$	0	0
$943$	9.83556	0.320290
$944$	0	0
$945$	−2.51276	−0.0817402
$946$	0	0
$947$	−22.8617	−0.742906	−0.371453	−	0.928452i	$-0.621140\pi$
$947$	−22.8617	−0.742906	−0.371453	+	0.928452i	$0.621140\pi$
$948$	0	0
$949$	−47.8756	−1.55411
$950$	0	0
$951$	24.6169	0.798259
$952$	0	0
$953$	−20.2214	−0.655034	−0.327517	−	0.944845i	$-0.606212\pi$
$953$	−20.2214	−0.655034	−0.327517	+	0.944845i	$0.606212\pi$
$954$	0	0
$955$	1.43816	0.0465376
$956$	0	0
$957$	2.19541	0.0709675
$958$	0	0
$959$	3.77552	0.121918
$960$	0	0
$961$	−30.8232	−0.994298
$962$	0	0
$963$	22.4979	0.724984
$964$	0	0
$965$	−24.1344	−0.776915
$966$	0	0
$967$	−17.1231	−0.550643	−0.275321	−	0.961352i	$-0.588784\pi$
$967$	−17.1231	−0.550643	−0.275321	+	0.961352i	$0.588784\pi$
$968$	0	0
$969$	−57.1747	−1.83671
$970$	0	0
$971$	−5.36430	−0.172149	−0.0860744	−	0.996289i	$-0.527432\pi$
$971$	−5.36430	−0.172149	−0.0860744	+	0.996289i	$0.527432\pi$
$972$	0	0
$973$	−5.09079	−0.163203
$974$	0	0
$975$	5.35967	0.171647
$976$	0	0
$977$	7.24966	0.231937	0.115969	−	0.993253i	$-0.463003\pi$
$977$	7.24966	0.231937	0.115969	+	0.993253i	$0.463003\pi$
$978$	0	0
$979$	6.08636	0.194521
$980$	0	0
$981$	−14.7828	−0.471978
$982$	0	0
$983$	12.8030	0.408353	0.204176	−	0.978934i	$-0.434548\pi$
$983$	12.8030	0.408353	0.204176	+	0.978934i	$0.434548\pi$
$984$	0	0
$985$	6.42860	0.204832
$986$	0	0
$987$	−5.30309	−0.168799
$988$	0	0
$989$	−10.7547	−0.341978
$990$	0	0
$991$	9.13537	0.290195	0.145097	−	0.989417i	$-0.453650\pi$
$991$	9.13537	0.290195	0.145097	+	0.989417i	$0.453650\pi$
$992$	0	0
$993$	10.9578	0.347736
$994$	0	0
$995$	26.1963	0.830479
$996$	0	0
$997$	11.0596	0.350261	0.175130	−	0.984545i	$-0.443965\pi$
$997$	11.0596	0.350261	0.175130	+	0.984545i	$0.443965\pi$
$998$	0	0
$999$	43.2749	1.36916

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7360.2.a.cu.1.5		6
4.3	odd	2		7360.2.a.cv.1.2		6
8.3	odd	2		3680.2.a.bc.1.5	✓	6
8.5	even	2		3680.2.a.bd.1.2	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3680.2.a.bc.1.5	✓	6	8.3	odd	2
3680.2.a.bd.1.2	yes	6	8.5	even	2
7360.2.a.cu.1.5		6	1.1	even	1	trivial
7360.2.a.cv.1.2		6	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:	\( N \)	\(=\)	\( 7360 = 2^{6} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7360.a (trivial)

\(f(q)\)	\(=\)	\(q+1.15791 q^{3} -1.00000 q^{5} -0.465759 q^{7} -1.65924 q^{9} +O(q^{10})\)	\(q+1.15791 q^{3} -1.00000 q^{5} -0.465759 q^{7} -1.65924 q^{9} -2.23109 q^{11} +4.62874 q^{13} -1.15791 q^{15} -5.79023 q^{17} +8.52771 q^{19} -0.539307 q^{21} +1.00000 q^{23} +1.00000 q^{25} -5.39499 q^{27} -0.849812 q^{29} +0.420427 q^{31} -2.58340 q^{33} +0.465759 q^{35} -8.02132 q^{37} +5.35967 q^{39} +9.83556 q^{41} -10.7547 q^{43} +1.65924 q^{45} +9.83315 q^{47} -6.78307 q^{49} -6.70457 q^{51} -0.117646 q^{53} +2.23109 q^{55} +9.87434 q^{57} -5.74613 q^{59} +11.1069 q^{61} +0.772806 q^{63} -4.62874 q^{65} -13.0240 q^{67} +1.15791 q^{69} -5.33274 q^{71} -10.3431 q^{73} +1.15791 q^{75} +1.03915 q^{77} +2.81567 q^{79} -1.26920 q^{81} +4.53129 q^{83} +5.79023 q^{85} -0.984007 q^{87} -2.72798 q^{89} -2.15587 q^{91} +0.486818 q^{93} -8.52771 q^{95} -9.33207 q^{97} +3.70191 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 5 q^{3} - 6 q^{5} + 4 q^{7} + 7 q^{9}+O(q^{10}) \) 6 * q - 5 * q^3 - 6 * q^5 + 4 * q^7 + 7 * q^9	\( 6 q - 5 q^{3} - 6 q^{5} + 4 q^{7} + 7 q^{9} - 7 q^{11} - q^{13} + 5 q^{15} - 5 q^{19} + 6 q^{23} + 6 q^{25} - 20 q^{27} + 3 q^{29} + 12 q^{31} - 3 q^{33} - 4 q^{35} - 7 q^{37} + 8 q^{39} + 8 q^{41} - 28 q^{43} - 7 q^{45} + 8 q^{47} + 6 q^{49} - 7 q^{51} + 5 q^{53} + 7 q^{55} + 18 q^{57} - 9 q^{59} + 3 q^{61} - 5 q^{63} + q^{65} - 27 q^{67} - 5 q^{69} - 2 q^{71} - 2 q^{73} - 5 q^{75} + 14 q^{77} + 4 q^{79} + 14 q^{81} - 23 q^{83} + 12 q^{87} - 4 q^{89} + q^{91} + 4 q^{93} + 5 q^{95} - q^{97} - q^{99}+O(q^{100}) \) 6 * q - 5 * q^3 - 6 * q^5 + 4 * q^7 + 7 * q^9 - 7 * q^11 - q^13 + 5 * q^15 - 5 * q^19 + 6 * q^23 + 6 * q^25 - 20 * q^27 + 3 * q^29 + 12 * q^31 - 3 * q^33 - 4 * q^35 - 7 * q^37 + 8 * q^39 + 8 * q^41 - 28 * q^43 - 7 * q^45 + 8 * q^47 + 6 * q^49 - 7 * q^51 + 5 * q^53 + 7 * q^55 + 18 * q^57 - 9 * q^59 + 3 * q^61 - 5 * q^63 + q^65 - 27 * q^67 - 5 * q^69 - 2 * q^71 - 2 * q^73 - 5 * q^75 + 14 * q^77 + 4 * q^79 + 14 * q^81 - 23 * q^83 + 12 * q^87 - 4 * q^89 + q^91 + 4 * q^93 + 5 * q^95 - q^97 - q^99

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