LMFDB - Embedded newform 7360.2.a.cm.1.1

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:	$5$
Coefficient field:	5.5.2255384.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 8x^{3} + 5x^{2} + 13x - 2 $ x^5 - x^4 - 8x^3 + 5x^2 + 13*x - 2
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			1.1
Root			$2.64870$ 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)

$f(q)$	$=$	$q-2.64870 q^{3} +1.00000 q^{5} -0.777466 q^{7} +4.01559 q^{9} +O(q^{10})$	$q-2.64870 q^{3} +1.00000 q^{5} -0.777466 q^{7} +4.01559 q^{9} +0.100554 q^{11} +5.88682 q^{13} -2.64870 q^{15} -4.46066 q^{17} +0.266337 q^{19} +2.05927 q^{21} -1.00000 q^{23} +1.00000 q^{25} -2.68998 q^{27} -1.56800 q^{29} -2.89945 q^{31} -0.266337 q^{33} -0.777466 q^{35} +1.38236 q^{37} -15.5924 q^{39} -0.958715 q^{41} -9.63619 q^{43} +4.01559 q^{45} +0.121980 q^{47} -6.39555 q^{49} +11.8149 q^{51} +9.63858 q^{53} +0.100554 q^{55} -0.705447 q^{57} -1.48887 q^{59} +5.39794 q^{61} -3.12198 q^{63} +5.88682 q^{65} +9.85860 q^{67} +2.64870 q^{69} -13.9437 q^{71} -2.46054 q^{73} -2.64870 q^{75} -0.0781774 q^{77} -5.57384 q^{79} -4.92182 q^{81} -9.43244 q^{83} -4.46066 q^{85} +4.15315 q^{87} -5.80612 q^{89} -4.57680 q^{91} +7.67975 q^{93} +0.266337 q^{95} +10.6644 q^{97} +0.403784 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{3} + 5 q^{5} - 4 q^{7} + 2 q^{9}+O(q^{10}) $ 5 * q - q^3 + 5 * q^5 - 4 * q^7 + 2 * q^9	$ 5 q - q^{3} + 5 q^{5} - 4 q^{7} + 2 q^{9} - 3 q^{11} - q^{13} - q^{15} - 4 q^{17} - q^{19} + 4 q^{21} - 5 q^{23} + 5 q^{25} - 4 q^{27} + 5 q^{29} - 18 q^{31} + q^{33} - 4 q^{35} - 3 q^{37} - 20 q^{39} - 2 q^{41} + 12 q^{43} + 2 q^{45} + 4 q^{47} + 5 q^{49} + q^{51} - 3 q^{53} - 3 q^{55} + 10 q^{57} - 5 q^{59} - q^{61} - 19 q^{63} - q^{65} + 3 q^{67} + q^{69} - 24 q^{71} - 8 q^{73} - q^{75} - 6 q^{77} - 40 q^{79} - 19 q^{81} - 13 q^{83} - 4 q^{85} - 12 q^{87} + 2 q^{89} + 17 q^{91} + 4 q^{93} - q^{95} + 9 q^{97} - q^{99}+O(q^{100}) $ 5 * q - q^3 + 5 * q^5 - 4 * q^7 + 2 * q^9 - 3 * q^11 - q^13 - q^15 - 4 * q^17 - q^19 + 4 * q^21 - 5 * q^23 + 5 * q^25 - 4 * q^27 + 5 * q^29 - 18 * q^31 + q^33 - 4 * q^35 - 3 * q^37 - 20 * q^39 - 2 * q^41 + 12 * q^43 + 2 * q^45 + 4 * q^47 + 5 * q^49 + q^51 - 3 * q^53 - 3 * q^55 + 10 * q^57 - 5 * q^59 - q^61 - 19 * q^63 - q^65 + 3 * q^67 + q^69 - 24 * q^71 - 8 * q^73 - q^75 - 6 * q^77 - 40 * q^79 - 19 * q^81 - 13 * q^83 - 4 * q^85 - 12 * q^87 + 2 * q^89 + 17 * q^91 + 4 * q^93 - q^95 + 9 * 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
$3$	−2.64870	−1.52922	−0.764612	−	0.644490i	$-0.777070\pi$
$3$	−2.64870	−1.52922	−0.764612	+	0.644490i	$0.777070\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.777466	−0.293854	−0.146927	−	0.989147i	$-0.546938\pi$
$7$	−0.777466	−0.293854	−0.146927	+	0.989147i	$0.546938\pi$
$8$	0	0
$9$	4.01559	1.33853
$10$	0	0
$11$	0.100554	0.0303182	0.0151591	−	0.999885i	$-0.495175\pi$
$11$	0.100554	0.0303182	0.0151591	+	0.999885i	$0.495175\pi$
$12$	0	0
$13$	5.88682	1.63271	0.816355	−	0.577551i	$-0.195992\pi$
$13$	5.88682	1.63271	0.816355	+	0.577551i	$0.195992\pi$
$14$	0	0
$15$	−2.64870	−0.683890
$16$	0	0
$17$	−4.46066	−1.08187	−0.540934	−	0.841065i	$-0.681929\pi$
$17$	−4.46066	−1.08187	−0.540934	+	0.841065i	$0.681929\pi$
$18$	0	0
$19$	0.266337	0.0611020	0.0305510	−	0.999533i	$-0.490274\pi$
$19$	0.266337	0.0611020	0.0305510	+	0.999533i	$0.490274\pi$
$20$	0	0
$21$	2.05927	0.449369
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−2.68998	−0.517687
$28$	0	0
$29$	−1.56800	−0.291170	−0.145585	−	0.989346i	$-0.546506\pi$
$29$	−1.56800	−0.291170	−0.145585	+	0.989346i	$0.546506\pi$
$30$	0	0
$31$	−2.89945	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$31$	−2.89945	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$32$	0	0
$33$	−0.266337	−0.0463634
$34$	0	0
$35$	−0.777466	−0.131416
$36$	0	0
$37$	1.38236	0.227258	0.113629	−	0.993523i	$-0.463752\pi$
$37$	1.38236	0.227258	0.113629	+	0.993523i	$0.463752\pi$
$38$	0	0
$39$	−15.5924	−2.49678
$40$	0	0
$41$	−0.958715	−0.149726	−0.0748631	−	0.997194i	$-0.523852\pi$
$41$	−0.958715	−0.149726	−0.0748631	+	0.997194i	$0.523852\pi$
$42$	0	0
$43$	−9.63619	−1.46950	−0.734752	−	0.678335i	$-0.762702\pi$
$43$	−9.63619	−1.46950	−0.734752	+	0.678335i	$0.762702\pi$
$44$	0	0
$45$	4.01559	0.598608
$46$	0	0
$47$	0.121980	0.0177927	0.00889633	−	0.999960i	$-0.497168\pi$
$47$	0.121980	0.0177927	0.00889633	+	0.999960i	$0.497168\pi$
$48$	0	0
$49$	−6.39555	−0.913650
$50$	0	0
$51$	11.8149	1.65442
$52$	0	0
$53$	9.63858	1.32396	0.661981	−	0.749521i	$-0.269716\pi$
$53$	9.63858	1.32396	0.661981	+	0.749521i	$0.269716\pi$
$54$	0	0
$55$	0.100554	0.0135587
$56$	0	0
$57$	−0.705447	−0.0934387
$58$	0	0
$59$	−1.48887	−0.193835	−0.0969173	−	0.995292i	$-0.530898\pi$
$59$	−1.48887	−0.193835	−0.0969173	+	0.995292i	$0.530898\pi$
$60$	0	0
$61$	5.39794	0.691136	0.345568	−	0.938394i	$-0.387686\pi$
$61$	5.39794	0.691136	0.345568	+	0.938394i	$0.387686\pi$
$62$	0	0
$63$	−3.12198	−0.393333
$64$	0	0
$65$	5.88682	0.730170
$66$	0	0
$67$	9.85860	1.20442	0.602210	−	0.798338i	$-0.294287\pi$
$67$	9.85860	1.20442	0.602210	+	0.798338i	$0.294287\pi$
$68$	0	0
$69$	2.64870	0.318865
$70$	0	0
$71$	−13.9437	−1.65481	−0.827406	−	0.561605i	$-0.810184\pi$
$71$	−13.9437	−1.65481	−0.827406	+	0.561605i	$0.810184\pi$
$72$	0	0
$73$	−2.46054	−0.287984	−0.143992	−	0.989579i	$-0.545994\pi$
$73$	−2.46054	−0.287984	−0.143992	+	0.989579i	$0.545994\pi$
$74$	0	0
$75$	−2.64870	−0.305845
$76$	0	0
$77$	−0.0781774	−0.00890914
$78$	0	0
$79$	−5.57384	−0.627106	−0.313553	−	0.949571i	$-0.601519\pi$
$79$	−5.57384	−0.627106	−0.313553	+	0.949571i	$0.601519\pi$
$80$	0	0
$81$	−4.92182	−0.546869
$82$	0	0
$83$	−9.43244	−1.03534	−0.517672	−	0.855579i	$-0.673201\pi$
$83$	−9.43244	−1.03534	−0.517672	+	0.855579i	$0.673201\pi$
$84$	0	0
$85$	−4.46066	−0.483826
$86$	0	0
$87$	4.15315	0.445265
$88$	0	0
$89$	−5.80612	−0.615448	−0.307724	−	0.951476i	$-0.599567\pi$
$89$	−5.80612	−0.615448	−0.307724	+	0.951476i	$0.599567\pi$
$90$	0	0
$91$	−4.57680	−0.479779
$92$	0	0
$93$	7.67975	0.796353
$94$	0	0
$95$	0.266337	0.0273256
$96$	0	0
$97$	10.6644	1.08281	0.541403	−	0.840763i	$-0.317893\pi$
$97$	10.6644	1.08281	0.541403	+	0.840763i	$0.317893\pi$
$98$	0	0
$99$	0.403784	0.0405818
$100$	0	0
$101$	12.7434	1.26802	0.634008	−	0.773326i	$-0.281408\pi$
$101$	12.7434	1.26802	0.634008	+	0.773326i	$0.281408\pi$
$102$	0	0
$103$	2.35678	0.232220	0.116110	−	0.993236i	$-0.462957\pi$
$103$	2.35678	0.232220	0.116110	+	0.993236i	$0.462957\pi$
$104$	0	0
$105$	2.05927	0.200964
$106$	0	0
$107$	−1.09364	−0.105727	−0.0528633	−	0.998602i	$-0.516835\pi$
$107$	−1.09364	−0.105727	−0.0528633	+	0.998602i	$0.516835\pi$
$108$	0	0
$109$	−4.78709	−0.458520	−0.229260	−	0.973365i	$-0.573631\pi$
$109$	−4.78709	−0.458520	−0.229260	+	0.973365i	$0.573631\pi$
$110$	0	0
$111$	−3.66144	−0.347529
$112$	0	0
$113$	16.7371	1.57449	0.787245	−	0.616640i	$-0.211507\pi$
$113$	16.7371	1.57449	0.787245	+	0.616640i	$0.211507\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	23.6390	2.18543
$118$	0	0
$119$	3.46801	0.317912
$120$	0	0
$121$	−10.9899	−0.999081
$122$	0	0
$123$	2.53934	0.228965
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	15.6828	1.39163	0.695813	−	0.718223i	$-0.255044\pi$
$127$	15.6828	1.39163	0.695813	+	0.718223i	$0.255044\pi$
$128$	0	0
$129$	25.5233	2.24720
$130$	0	0
$131$	−8.86299	−0.774363	−0.387182	−	0.922003i	$-0.626551\pi$
$131$	−8.86299	−0.774363	−0.387182	+	0.922003i	$0.626551\pi$
$132$	0	0
$133$	−0.207068	−0.0179551
$134$	0	0
$135$	−2.68998	−0.231517
$136$	0	0
$137$	2.12865	0.181863	0.0909314	−	0.995857i	$-0.471016\pi$
$137$	2.12865	0.181863	0.0909314	+	0.995857i	$0.471016\pi$
$138$	0	0
$139$	19.1166	1.62145	0.810724	−	0.585429i	$-0.199074\pi$
$139$	19.1166	1.62145	0.810724	+	0.585429i	$0.199074\pi$
$140$	0	0
$141$	−0.323089	−0.0272090
$142$	0	0
$143$	0.591944	0.0495008
$144$	0	0
$145$	−1.56800	−0.130215
$146$	0	0
$147$	16.9399	1.39718
$148$	0	0
$149$	−9.90228	−0.811227	−0.405613	−	0.914045i	$-0.632942\pi$
$149$	−9.90228	−0.811227	−0.405613	+	0.914045i	$0.632942\pi$
$150$	0	0
$151$	6.44124	0.524180	0.262090	−	0.965043i	$-0.415588\pi$
$151$	6.44124	0.524180	0.262090	+	0.965043i	$0.415588\pi$
$152$	0	0
$153$	−17.9122	−1.44811
$154$	0	0
$155$	−2.89945	−0.232889
$156$	0	0
$157$	−16.2659	−1.29816	−0.649078	−	0.760722i	$-0.724845\pi$
$157$	−16.2659	−1.29816	−0.649078	+	0.760722i	$0.724845\pi$
$158$	0	0
$159$	−25.5297	−2.02463
$160$	0	0
$161$	0.777466	0.0612729
$162$	0	0
$163$	13.5054	1.05783	0.528913	−	0.848676i	$-0.322600\pi$
$163$	13.5054	1.05783	0.528913	+	0.848676i	$0.322600\pi$
$164$	0	0
$165$	−0.266337	−0.0207343
$166$	0	0
$167$	−5.09628	−0.394362	−0.197181	−	0.980367i	$-0.563179\pi$
$167$	−5.09628	−0.394362	−0.197181	+	0.980367i	$0.563179\pi$
$168$	0	0
$169$	21.6546	1.66574
$170$	0	0
$171$	1.06950	0.0817868
$172$	0	0
$173$	−4.97203	−0.378016	−0.189008	−	0.981976i	$-0.560527\pi$
$173$	−4.97203	−0.378016	−0.189008	+	0.981976i	$0.560527\pi$
$174$	0	0
$175$	−0.777466	−0.0587709
$176$	0	0
$177$	3.94357	0.296417
$178$	0	0
$179$	1.65800	0.123925	0.0619625	−	0.998078i	$-0.480264\pi$
$179$	1.65800	0.123925	0.0619625	+	0.998078i	$0.480264\pi$
$180$	0	0
$181$	−8.41101	−0.625186	−0.312593	−	0.949887i	$-0.601198\pi$
$181$	−8.41101	−0.625186	−0.312593	+	0.949887i	$0.601198\pi$
$182$	0	0
$183$	−14.2975	−1.05690
$184$	0	0
$185$	1.38236	0.101633
$186$	0	0
$187$	−0.448538	−0.0328003
$188$	0	0
$189$	2.09137	0.152125
$190$	0	0
$191$	−19.0399	−1.37768	−0.688838	−	0.724916i	$-0.741879\pi$
$191$	−19.0399	−1.37768	−0.688838	+	0.724916i	$0.741879\pi$
$192$	0	0
$193$	−0.919425	−0.0661817	−0.0330908	−	0.999452i	$-0.510535\pi$
$193$	−0.919425	−0.0661817	−0.0330908	+	0.999452i	$0.510535\pi$
$194$	0	0
$195$	−15.5924	−1.11659
$196$	0	0
$197$	−5.72103	−0.407607	−0.203803	−	0.979012i	$-0.565330\pi$
$197$	−5.72103	−0.407607	−0.203803	+	0.979012i	$0.565330\pi$
$198$	0	0
$199$	−7.09760	−0.503135	−0.251568	−	0.967840i	$-0.580946\pi$
$199$	−7.09760	−0.503135	−0.251568	+	0.967840i	$0.580946\pi$
$200$	0	0
$201$	−26.1124	−1.84183
$202$	0	0
$203$	1.21907	0.0855617
$204$	0	0
$205$	−0.958715	−0.0669596
$206$	0	0
$207$	−4.01559	−0.279103
$208$	0	0
$209$	0.0267813	0.00185250
$210$	0	0
$211$	27.6395	1.90278	0.951390	−	0.307990i	$-0.0996564\pi$
$211$	27.6395	1.90278	0.951390	+	0.307990i	$0.0996564\pi$
$212$	0	0
$213$	36.9326	2.53058
$214$	0	0
$215$	−9.63619	−0.657182
$216$	0	0
$217$	2.25422	0.153026
$218$	0	0
$219$	6.51721	0.440392
$220$	0	0
$221$	−26.2591	−1.76638
$222$	0	0
$223$	−13.4985	−0.903926	−0.451963	−	0.892037i	$-0.649276\pi$
$223$	−13.4985	−0.903926	−0.451963	+	0.892037i	$0.649276\pi$
$224$	0	0
$225$	4.01559	0.267706
$226$	0	0
$227$	−9.49127	−0.629958	−0.314979	−	0.949099i	$-0.601997\pi$
$227$	−9.49127	−0.629958	−0.314979	+	0.949099i	$0.601997\pi$
$228$	0	0
$229$	−24.6947	−1.63187	−0.815934	−	0.578145i	$-0.803777\pi$
$229$	−24.6947	−1.63187	−0.815934	+	0.578145i	$0.803777\pi$
$230$	0	0
$231$	0.207068	0.0136241
$232$	0	0
$233$	−4.17206	−0.273321	−0.136660	−	0.990618i	$-0.543637\pi$
$233$	−4.17206	−0.273321	−0.136660	+	0.990618i	$0.543637\pi$
$234$	0	0
$235$	0.121980	0.00795712
$236$	0	0
$237$	14.7634	0.958986
$238$	0	0
$239$	6.02533	0.389746	0.194873	−	0.980828i	$-0.437570\pi$
$239$	6.02533	0.389746	0.194873	+	0.980828i	$0.437570\pi$
$240$	0	0
$241$	6.38864	0.411528	0.205764	−	0.978602i	$-0.434032\pi$
$241$	6.38864	0.411528	0.205764	+	0.978602i	$0.434032\pi$
$242$	0	0
$243$	21.1063	1.35397
$244$	0	0
$245$	−6.39555	−0.408597
$246$	0	0
$247$	1.56788	0.0997618
$248$	0	0
$249$	24.9837	1.58328
$250$	0	0
$251$	1.14236	0.0721053	0.0360526	−	0.999350i	$-0.488522\pi$
$251$	1.14236	0.0721053	0.0360526	+	0.999350i	$0.488522\pi$
$252$	0	0
$253$	−0.100554	−0.00632179
$254$	0	0
$255$	11.8149	0.739879
$256$	0	0
$257$	1.32644	0.0827409	0.0413704	−	0.999144i	$-0.486828\pi$
$257$	1.32644	0.0827409	0.0413704	+	0.999144i	$0.486828\pi$
$258$	0	0
$259$	−1.07474	−0.0667808
$260$	0	0
$261$	−6.29644	−0.389740
$262$	0	0
$263$	−14.3408	−0.884292	−0.442146	−	0.896943i	$-0.645783\pi$
$263$	−14.3408	−0.884292	−0.442146	+	0.896943i	$0.645783\pi$
$264$	0	0
$265$	9.63858	0.592094
$266$	0	0
$267$	15.3786	0.941158
$268$	0	0
$269$	−8.38060	−0.510974	−0.255487	−	0.966812i	$-0.582236\pi$
$269$	−8.38060	−0.510974	−0.255487	+	0.966812i	$0.582236\pi$
$270$	0	0
$271$	−11.8070	−0.717221	−0.358611	−	0.933487i	$-0.616749\pi$
$271$	−11.8070	−0.717221	−0.358611	+	0.933487i	$0.616749\pi$
$272$	0	0
$273$	12.1225	0.733690
$274$	0	0
$275$	0.100554	0.00606365
$276$	0	0
$277$	−4.66164	−0.280091	−0.140046	−	0.990145i	$-0.544725\pi$
$277$	−4.66164	−0.280091	−0.140046	+	0.990145i	$0.544725\pi$
$278$	0	0
$279$	−11.6430	−0.697047
$280$	0	0
$281$	16.6965	0.996030	0.498015	−	0.867168i	$-0.334062\pi$
$281$	16.6965	0.996030	0.498015	+	0.867168i	$0.334062\pi$
$282$	0	0
$283$	2.64159	0.157026	0.0785130	−	0.996913i	$-0.474983\pi$
$283$	2.64159	0.157026	0.0785130	+	0.996913i	$0.474983\pi$
$284$	0	0
$285$	−0.705447	−0.0417871
$286$	0	0
$287$	0.745368	0.0439977
$288$	0	0
$289$	2.89745	0.170438
$290$	0	0
$291$	−28.2468	−1.65585
$292$	0	0
$293$	23.4072	1.36746	0.683731	−	0.729734i	$-0.260356\pi$
$293$	23.4072	1.36746	0.683731	+	0.729734i	$0.260356\pi$
$294$	0	0
$295$	−1.48887	−0.0866855
$296$	0	0
$297$	−0.270489	−0.0156953
$298$	0	0
$299$	−5.88682	−0.340443
$300$	0	0
$301$	7.49180	0.431820
$302$	0	0
$303$	−33.7534	−1.93908
$304$	0	0
$305$	5.39794	0.309085
$306$	0	0
$307$	−6.20203	−0.353969	−0.176984	−	0.984214i	$-0.556634\pi$
$307$	−6.20203	−0.353969	−0.176984	+	0.984214i	$0.556634\pi$
$308$	0	0
$309$	−6.24239	−0.355117
$310$	0	0
$311$	10.4656	0.593448	0.296724	−	0.954963i	$-0.404106\pi$
$311$	10.4656	0.593448	0.296724	+	0.954963i	$0.404106\pi$
$312$	0	0
$313$	5.04537	0.285181	0.142591	−	0.989782i	$-0.454457\pi$
$313$	5.04537	0.285181	0.142591	+	0.989782i	$0.454457\pi$
$314$	0	0
$315$	−3.12198	−0.175904
$316$	0	0
$317$	−30.1008	−1.69063	−0.845314	−	0.534270i	$-0.820586\pi$
$317$	−30.1008	−1.69063	−0.845314	+	0.534270i	$0.820586\pi$
$318$	0	0
$319$	−0.157669	−0.00882777
$320$	0	0
$321$	2.89673	0.161680
$322$	0	0
$323$	−1.18804	−0.0661043
$324$	0	0
$325$	5.88682	0.326542
$326$	0	0
$327$	12.6795	0.701181
$328$	0	0
$329$	−0.0948355	−0.00522845
$330$	0	0
$331$	−21.6804	−1.19166	−0.595832	−	0.803109i	$-0.703178\pi$
$331$	−21.6804	−1.19166	−0.595832	+	0.803109i	$0.703178\pi$
$332$	0	0
$333$	5.55098	0.304192
$334$	0	0
$335$	9.85860	0.538633
$336$	0	0
$337$	−1.73366	−0.0944386	−0.0472193	−	0.998885i	$-0.515036\pi$
$337$	−1.73366	−0.0944386	−0.0472193	+	0.998885i	$0.515036\pi$
$338$	0	0
$339$	−44.3314	−2.40775
$340$	0	0
$341$	−0.291551	−0.0157884
$342$	0	0
$343$	10.4146	0.562334
$344$	0	0
$345$	2.64870	0.142601
$346$	0	0
$347$	−23.8536	−1.28053	−0.640265	−	0.768154i	$-0.721175\pi$
$347$	−23.8536	−1.28053	−0.640265	+	0.768154i	$0.721175\pi$
$348$	0	0
$349$	−27.0066	−1.44563	−0.722814	−	0.691043i	$-0.757152\pi$
$349$	−27.0066	−1.44563	−0.722814	+	0.691043i	$0.757152\pi$
$350$	0	0
$351$	−15.8354	−0.845232
$352$	0	0
$353$	−7.27689	−0.387310	−0.193655	−	0.981070i	$-0.562034\pi$
$353$	−7.27689	−0.387310	−0.193655	+	0.981070i	$0.562034\pi$
$354$	0	0
$355$	−13.9437	−0.740054
$356$	0	0
$357$	−9.18569	−0.486158
$358$	0	0
$359$	−28.2824	−1.49269	−0.746343	−	0.665561i	$-0.768192\pi$
$359$	−28.2824	−1.49269	−0.746343	+	0.665561i	$0.768192\pi$
$360$	0	0
$361$	−18.9291	−0.996267
$362$	0	0
$363$	29.1089	1.52782
$364$	0	0
$365$	−2.46054	−0.128790
$366$	0	0
$367$	1.44602	0.0754816	0.0377408	−	0.999288i	$-0.487984\pi$
$367$	1.44602	0.0754816	0.0377408	+	0.999288i	$0.487984\pi$
$368$	0	0
$369$	−3.84980	−0.200413
$370$	0	0
$371$	−7.49367	−0.389052
$372$	0	0
$373$	7.87292	0.407644	0.203822	−	0.979008i	$-0.434664\pi$
$373$	7.87292	0.407644	0.203822	+	0.979008i	$0.434664\pi$
$374$	0	0
$375$	−2.64870	−0.136778
$376$	0	0
$377$	−9.23053	−0.475396
$378$	0	0
$379$	24.5704	1.26210	0.631050	−	0.775743i	$-0.282624\pi$
$379$	24.5704	1.26210	0.631050	+	0.775743i	$0.282624\pi$
$380$	0	0
$381$	−41.5390	−2.12811
$382$	0	0
$383$	27.4573	1.40300	0.701500	−	0.712669i	$-0.252514\pi$
$383$	27.4573	1.40300	0.701500	+	0.712669i	$0.252514\pi$
$384$	0	0
$385$	−0.0781774	−0.00398429
$386$	0	0
$387$	−38.6949	−1.96697
$388$	0	0
$389$	0.0592571	0.00300446	0.00150223	−	0.999999i	$-0.499522\pi$
$389$	0.0592571	0.00300446	0.00150223	+	0.999999i	$0.499522\pi$
$390$	0	0
$391$	4.46066	0.225585
$392$	0	0
$393$	23.4754	1.18418
$394$	0	0
$395$	−5.57384	−0.280450
$396$	0	0
$397$	−15.9445	−0.800233	−0.400117	−	0.916464i	$-0.631030\pi$
$397$	−15.9445	−0.800233	−0.400117	+	0.916464i	$0.631030\pi$
$398$	0	0
$399$	0.548460	0.0274574
$400$	0	0
$401$	−18.0321	−0.900478	−0.450239	−	0.892908i	$-0.648661\pi$
$401$	−18.0321	−0.900478	−0.450239	+	0.892908i	$0.648661\pi$
$402$	0	0
$403$	−17.0685	−0.850243
$404$	0	0
$405$	−4.92182	−0.244567
$406$	0	0
$407$	0.139002	0.00689007
$408$	0	0
$409$	−32.2269	−1.59352	−0.796760	−	0.604296i	$-0.793454\pi$
$409$	−32.2269	−1.59352	−0.796760	+	0.604296i	$0.793454\pi$
$410$	0	0
$411$	−5.63814	−0.278109
$412$	0	0
$413$	1.15755	0.0569591
$414$	0	0
$415$	−9.43244	−0.463020
$416$	0	0
$417$	−50.6340	−2.47956
$418$	0	0
$419$	−38.9648	−1.90355	−0.951776	−	0.306793i	$-0.900744\pi$
$419$	−38.9648	−1.90355	−0.951776	+	0.306793i	$0.900744\pi$
$420$	0	0
$421$	27.6077	1.34552	0.672759	−	0.739862i	$-0.265109\pi$
$421$	27.6077	1.34552	0.672759	+	0.739862i	$0.265109\pi$
$422$	0	0
$423$	0.489823	0.0238160
$424$	0	0
$425$	−4.46066	−0.216374
$426$	0	0
$427$	−4.19672	−0.203093
$428$	0	0
$429$	−1.56788	−0.0756979
$430$	0	0
$431$	−26.0677	−1.25564	−0.627818	−	0.778360i	$-0.716052\pi$
$431$	−26.0677	−1.25564	−0.627818	+	0.778360i	$0.716052\pi$
$432$	0	0
$433$	16.6390	0.799619	0.399810	−	0.916598i	$-0.369076\pi$
$433$	16.6390	0.799619	0.399810	+	0.916598i	$0.369076\pi$
$434$	0	0
$435$	4.15315	0.199128
$436$	0	0
$437$	−0.266337	−0.0127406
$438$	0	0
$439$	14.1055	0.673220	0.336610	−	0.941644i	$-0.390720\pi$
$439$	14.1055	0.673220	0.336610	+	0.941644i	$0.390720\pi$
$440$	0	0
$441$	−25.6819	−1.22295
$442$	0	0
$443$	11.8184	0.561507	0.280754	−	0.959780i	$-0.409416\pi$
$443$	11.8184	0.561507	0.280754	+	0.959780i	$0.409416\pi$
$444$	0	0
$445$	−5.80612	−0.275237
$446$	0	0
$447$	26.2281	1.24055
$448$	0	0
$449$	14.4760	0.683165	0.341582	−	0.939852i	$-0.389037\pi$
$449$	14.4760	0.683165	0.341582	+	0.939852i	$0.389037\pi$
$450$	0	0
$451$	−0.0964028	−0.00453943
$452$	0	0
$453$	−17.0609	−0.801590
$454$	0	0
$455$	−4.57680	−0.214564
$456$	0	0
$457$	−6.62991	−0.310134	−0.155067	−	0.987904i	$-0.549559\pi$
$457$	−6.62991	−0.310134	−0.155067	+	0.987904i	$0.549559\pi$
$458$	0	0
$459$	11.9991	0.560069
$460$	0	0
$461$	−5.33201	−0.248336	−0.124168	−	0.992261i	$-0.539626\pi$
$461$	−5.33201	−0.248336	−0.124168	+	0.992261i	$0.539626\pi$
$462$	0	0
$463$	−37.1024	−1.72430	−0.862148	−	0.506656i	$-0.830881\pi$
$463$	−37.1024	−1.72430	−0.862148	+	0.506656i	$0.830881\pi$
$464$	0	0
$465$	7.67975	0.356140
$466$	0	0
$467$	−27.0886	−1.25351	−0.626754	−	0.779217i	$-0.715617\pi$
$467$	−27.0886	−1.25351	−0.626754	+	0.779217i	$0.715617\pi$
$468$	0	0
$469$	−7.66472	−0.353924
$470$	0	0
$471$	43.0833	1.98517
$472$	0	0
$473$	−0.968959	−0.0445528
$474$	0	0
$475$	0.266337	0.0122204
$476$	0	0
$477$	38.7046	1.77216
$478$	0	0
$479$	13.3081	0.608063	0.304031	−	0.952662i	$-0.401667\pi$
$479$	13.3081	0.608063	0.304031	+	0.952662i	$0.401667\pi$
$480$	0	0
$481$	8.13769	0.371047
$482$	0	0
$483$	−2.05927	−0.0937000
$484$	0	0
$485$	10.6644	0.484246
$486$	0	0
$487$	−36.8415	−1.66945	−0.834723	−	0.550670i	$-0.814372\pi$
$487$	−36.8415	−1.66945	−0.834723	+	0.550670i	$0.814372\pi$
$488$	0	0
$489$	−35.7717	−1.61765
$490$	0	0
$491$	23.7854	1.07342	0.536710	−	0.843767i	$-0.319667\pi$
$491$	23.7854	1.07342	0.536710	+	0.843767i	$0.319667\pi$
$492$	0	0
$493$	6.99431	0.315008
$494$	0	0
$495$	0.403784	0.0181487
$496$	0	0
$497$	10.8407	0.486274
$498$	0	0
$499$	17.5960	0.787703	0.393852	−	0.919174i	$-0.371142\pi$
$499$	17.5960	0.787703	0.393852	+	0.919174i	$0.371142\pi$
$500$	0	0
$501$	13.4985	0.603068
$502$	0	0
$503$	3.75277	0.167328	0.0836640	−	0.996494i	$-0.473338\pi$
$503$	3.75277	0.167328	0.0836640	+	0.996494i	$0.473338\pi$
$504$	0	0
$505$	12.7434	0.567074
$506$	0	0
$507$	−57.3565	−2.54729
$508$	0	0
$509$	−30.5341	−1.35340	−0.676701	−	0.736258i	$-0.736591\pi$
$509$	−30.5341	−1.35340	−0.676701	+	0.736258i	$0.736591\pi$
$510$	0	0
$511$	1.91298	0.0846253
$512$	0	0
$513$	−0.716442	−0.0316317
$514$	0	0
$515$	2.35678	0.103852
$516$	0	0
$517$	0.0122656	0.000539442 0
$518$	0	0
$519$	13.1694	0.578072
$520$	0	0
$521$	−20.9059	−0.915903	−0.457952	−	0.888977i	$-0.651417\pi$
$521$	−20.9059	−0.915903	−0.457952	+	0.888977i	$0.651417\pi$
$522$	0	0
$523$	34.0200	1.48759	0.743796	−	0.668407i	$-0.233023\pi$
$523$	34.0200	1.48759	0.743796	+	0.668407i	$0.233023\pi$
$524$	0	0
$525$	2.05927	0.0898739
$526$	0	0
$527$	12.9334	0.563389
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−5.97869	−0.259453
$532$	0	0
$533$	−5.64378	−0.244459
$534$	0	0
$535$	−1.09364	−0.0472823
$536$	0	0
$537$	−4.39155	−0.189509
$538$	0	0
$539$	−0.643099	−0.0277002
$540$	0	0
$541$	−15.7188	−0.675804	−0.337902	−	0.941181i	$-0.609717\pi$
$541$	−15.7188	−0.675804	−0.337902	+	0.941181i	$0.609717\pi$
$542$	0	0
$543$	22.2782	0.956050
$544$	0	0
$545$	−4.78709	−0.205057
$546$	0	0
$547$	−1.10480	−0.0472378	−0.0236189	−	0.999721i	$-0.507519\pi$
$547$	−1.10480	−0.0472378	−0.0236189	+	0.999721i	$0.507519\pi$
$548$	0	0
$549$	21.6759	0.925106
$550$	0	0
$551$	−0.417617	−0.0177911
$552$	0	0
$553$	4.33347	0.184278
$554$	0	0
$555$	−3.66144	−0.155420
$556$	0	0
$557$	−27.2612	−1.15509	−0.577547	−	0.816358i	$-0.695990\pi$
$557$	−27.2612	−1.15509	−0.577547	+	0.816358i	$0.695990\pi$
$558$	0	0
$559$	−56.7265	−2.39927
$560$	0	0
$561$	1.18804	0.0501591
$562$	0	0
$563$	−18.1231	−0.763799	−0.381899	−	0.924204i	$-0.624730\pi$
$563$	−18.1231	−0.763799	−0.381899	+	0.924204i	$0.624730\pi$
$564$	0	0
$565$	16.7371	0.704133
$566$	0	0
$567$	3.82655	0.160700
$568$	0	0
$569$	−15.6423	−0.655759	−0.327880	−	0.944720i	$-0.606334\pi$
$569$	−15.6423	−0.655759	−0.327880	+	0.944720i	$0.606334\pi$
$570$	0	0
$571$	−14.3880	−0.602117	−0.301059	−	0.953606i	$-0.597340\pi$
$571$	−14.3880	−0.602117	−0.301059	+	0.953606i	$0.597340\pi$
$572$	0	0
$573$	50.4308	2.10678
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	−16.9157	−0.704209	−0.352104	−	0.935961i	$-0.614534\pi$
$577$	−16.9157	−0.704209	−0.352104	+	0.935961i	$0.614534\pi$
$578$	0	0
$579$	2.43528	0.101207
$580$	0	0
$581$	7.33340	0.304241
$582$	0	0
$583$	0.969200	0.0401402
$584$	0	0
$585$	23.6390	0.977353
$586$	0	0
$587$	−23.7904	−0.981934	−0.490967	−	0.871178i	$-0.663356\pi$
$587$	−23.7904	−0.981934	−0.490967	+	0.871178i	$0.663356\pi$
$588$	0	0
$589$	−0.772231	−0.0318192
$590$	0	0
$591$	15.1533	0.623322
$592$	0	0
$593$	−5.04864	−0.207323	−0.103661	−	0.994613i	$-0.533056\pi$
$593$	−5.04864	−0.207323	−0.103661	+	0.994613i	$0.533056\pi$
$594$	0	0
$595$	3.46801	0.142174
$596$	0	0
$597$	18.7994	0.769407
$598$	0	0
$599$	−27.1096	−1.10767	−0.553834	−	0.832627i	$-0.686836\pi$
$599$	−27.1096	−1.10767	−0.553834	+	0.832627i	$0.686836\pi$
$600$	0	0
$601$	−21.4549	−0.875162	−0.437581	−	0.899179i	$-0.644165\pi$
$601$	−21.4549	−0.875162	−0.437581	+	0.899179i	$0.644165\pi$
$602$	0	0
$603$	39.5881	1.61215
$604$	0	0
$605$	−10.9899	−0.446803
$606$	0	0
$607$	−14.6833	−0.595975	−0.297988	−	0.954570i	$-0.596315\pi$
$607$	−14.6833	−0.595975	−0.297988	+	0.954570i	$0.596315\pi$
$608$	0	0
$609$	−3.22893	−0.130843
$610$	0	0
$611$	0.718076	0.0290502
$612$	0	0
$613$	9.45399	0.381843	0.190921	−	0.981605i	$-0.438852\pi$
$613$	9.45399	0.381843	0.190921	+	0.981605i	$0.438852\pi$
$614$	0	0
$615$	2.53934	0.102396
$616$	0	0
$617$	−23.6215	−0.950965	−0.475482	−	0.879725i	$-0.657726\pi$
$617$	−23.6215	−0.950965	−0.475482	+	0.879725i	$0.657726\pi$
$618$	0	0
$619$	48.9843	1.96884	0.984422	−	0.175823i	$-0.0562586\pi$
$619$	48.9843	1.96884	0.984422	+	0.175823i	$0.0562586\pi$
$620$	0	0
$621$	2.68998	0.107945
$622$	0	0
$623$	4.51406	0.180852
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−0.0709356	−0.00283290
$628$	0	0
$629$	−6.16622	−0.245863
$630$	0	0
$631$	12.9109	0.513977	0.256988	−	0.966415i	$-0.417270\pi$
$631$	12.9109	0.513977	0.256988	+	0.966415i	$0.417270\pi$
$632$	0	0
$633$	−73.2085	−2.90978
$634$	0	0
$635$	15.6828	0.622354
$636$	0	0
$637$	−37.6494	−1.49172
$638$	0	0
$639$	−55.9921	−2.21501
$640$	0	0
$641$	−26.0683	−1.02964	−0.514818	−	0.857299i	$-0.672141\pi$
$641$	−26.0683	−1.02964	−0.514818	+	0.857299i	$0.672141\pi$
$642$	0	0
$643$	10.6999	0.421964	0.210982	−	0.977490i	$-0.432334\pi$
$643$	10.6999	0.421964	0.210982	+	0.977490i	$0.432334\pi$
$644$	0	0
$645$	25.5233	1.00498
$646$	0	0
$647$	29.5031	1.15989	0.579943	−	0.814657i	$-0.303075\pi$
$647$	29.5031	1.15989	0.579943	+	0.814657i	$0.303075\pi$
$648$	0	0
$649$	−0.149712	−0.00587672
$650$	0	0
$651$	−5.97074	−0.234012
$652$	0	0
$653$	40.9578	1.60280	0.801402	−	0.598126i	$-0.204088\pi$
$653$	40.9578	1.60280	0.801402	+	0.598126i	$0.204088\pi$
$654$	0	0
$655$	−8.86299	−0.346306
$656$	0	0
$657$	−9.88049	−0.385475
$658$	0	0
$659$	−34.9799	−1.36262	−0.681312	−	0.731993i	$-0.738590\pi$
$659$	−34.9799	−1.36262	−0.681312	+	0.731993i	$0.738590\pi$
$660$	0	0
$661$	−38.1432	−1.48360	−0.741800	−	0.670622i	$-0.766027\pi$
$661$	−38.1432	−1.48360	−0.741800	+	0.670622i	$0.766027\pi$
$662$	0	0
$663$	69.5523	2.70119
$664$	0	0
$665$	−0.207068	−0.00802976
$666$	0	0
$667$	1.56800	0.0607132
$668$	0	0
$669$	35.7534	1.38231
$670$	0	0
$671$	0.542786	0.0209540
$672$	0	0
$673$	−10.5266	−0.405772	−0.202886	−	0.979202i	$-0.565032\pi$
$673$	−10.5266	−0.405772	−0.202886	+	0.979202i	$0.565032\pi$
$674$	0	0
$675$	−2.68998	−0.103537
$676$	0	0
$677$	2.38595	0.0916994	0.0458497	−	0.998948i	$-0.485400\pi$
$677$	2.38595	0.0916994	0.0458497	+	0.998948i	$0.485400\pi$
$678$	0	0
$679$	−8.29121	−0.318187
$680$	0	0
$681$	25.1395	0.963347
$682$	0	0
$683$	50.4383	1.92997	0.964984	−	0.262308i	$-0.0844837\pi$
$683$	50.4383	1.92997	0.964984	+	0.262308i	$0.0844837\pi$
$684$	0	0
$685$	2.12865	0.0813315
$686$	0	0
$687$	65.4086	2.49549
$688$	0	0
$689$	56.7406	2.16164
$690$	0	0
$691$	−15.7567	−0.599412	−0.299706	−	0.954032i	$-0.596889\pi$
$691$	−15.7567	−0.599412	−0.299706	+	0.954032i	$0.596889\pi$
$692$	0	0
$693$	−0.313928	−0.0119251
$694$	0	0
$695$	19.1166	0.725133
$696$	0	0
$697$	4.27650	0.161984
$698$	0	0
$699$	11.0505	0.417969
$700$	0	0
$701$	−17.1067	−0.646113	−0.323056	−	0.946380i	$-0.604710\pi$
$701$	−17.1067	−0.646113	−0.323056	+	0.946380i	$0.604710\pi$
$702$	0	0
$703$	0.368174	0.0138859
$704$	0	0
$705$	−0.323089	−0.0121682
$706$	0	0
$707$	−9.90756	−0.372612
$708$	0	0
$709$	−28.2990	−1.06279	−0.531396	−	0.847123i	$-0.678332\pi$
$709$	−28.2990	−1.06279	−0.531396	+	0.847123i	$0.678332\pi$
$710$	0	0
$711$	−22.3822	−0.839399
$712$	0	0
$713$	2.89945	0.108585
$714$	0	0
$715$	0.591944	0.0221375
$716$	0	0
$717$	−15.9593	−0.596010
$718$	0	0
$719$	4.69757	0.175190	0.0875949	−	0.996156i	$-0.472082\pi$
$719$	4.69757	0.175190	0.0875949	+	0.996156i	$0.472082\pi$
$720$	0	0
$721$	−1.83232	−0.0682390
$722$	0	0
$723$	−16.9216	−0.629319
$724$	0	0
$725$	−1.56800	−0.0582341
$726$	0	0
$727$	−23.0100	−0.853392	−0.426696	−	0.904395i	$-0.640323\pi$
$727$	−23.0100	−0.853392	−0.426696	+	0.904395i	$0.640323\pi$
$728$	0	0
$729$	−41.1388	−1.52366
$730$	0	0
$731$	42.9837	1.58981
$732$	0	0
$733$	−41.0228	−1.51521	−0.757606	−	0.652712i	$-0.773631\pi$
$733$	−41.0228	−1.51521	−0.757606	+	0.652712i	$0.773631\pi$
$734$	0	0
$735$	16.9399	0.624836
$736$	0	0
$737$	0.991324	0.0365159
$738$	0	0
$739$	−41.1326	−1.51309	−0.756543	−	0.653944i	$-0.773113\pi$
$739$	−41.1326	−1.51309	−0.756543	+	0.653944i	$0.773113\pi$
$740$	0	0
$741$	−4.15283	−0.152558
$742$	0	0
$743$	29.5526	1.08418	0.542090	−	0.840321i	$-0.317633\pi$
$743$	29.5526	1.08418	0.542090	+	0.840321i	$0.317633\pi$
$744$	0	0
$745$	−9.90228	−0.362792
$746$	0	0
$747$	−37.8768	−1.38584
$748$	0	0
$749$	0.850270	0.0310682
$750$	0	0
$751$	−20.7410	−0.756850	−0.378425	−	0.925632i	$-0.623534\pi$
$751$	−20.7410	−0.756850	−0.378425	+	0.925632i	$0.623534\pi$
$752$	0	0
$753$	−3.02577	−0.110265
$754$	0	0
$755$	6.44124	0.234421
$756$	0	0
$757$	15.9579	0.579999	0.290000	−	0.957027i	$-0.406345\pi$
$757$	15.9579	0.579999	0.290000	+	0.957027i	$0.406345\pi$
$758$	0	0
$759$	0.266337	0.00966743
$760$	0	0
$761$	5.14209	0.186401	0.0932004	−	0.995647i	$-0.470290\pi$
$761$	5.14209	0.186401	0.0932004	+	0.995647i	$0.470290\pi$
$762$	0	0
$763$	3.72180	0.134738
$764$	0	0
$765$	−17.9122	−0.647615
$766$	0	0
$767$	−8.76472	−0.316476
$768$	0	0
$769$	32.8415	1.18430	0.592148	−	0.805829i	$-0.298280\pi$
$769$	32.8415	1.18430	0.592148	+	0.805829i	$0.298280\pi$
$770$	0	0
$771$	−3.51333	−0.126529
$772$	0	0
$773$	−38.9919	−1.40244	−0.701221	−	0.712944i	$-0.747361\pi$
$773$	−38.9919	−1.40244	−0.701221	+	0.712944i	$0.747361\pi$
$774$	0	0
$775$	−2.89945	−0.104151
$776$	0	0
$777$	2.84665	0.102123
$778$	0	0
$779$	−0.255342	−0.00914856
$780$	0	0
$781$	−1.40210	−0.0501709
$782$	0	0
$783$	4.21789	0.150735
$784$	0	0
$785$	−16.2659	−0.580553
$786$	0	0
$787$	−27.5696	−0.982750	−0.491375	−	0.870948i	$-0.663506\pi$
$787$	−27.5696	−0.982750	−0.491375	+	0.870948i	$0.663506\pi$
$788$	0	0
$789$	37.9844	1.35228
$790$	0	0
$791$	−13.0125	−0.462671
$792$	0	0
$793$	31.7767	1.12842
$794$	0	0
$795$	−25.5297	−0.905444
$796$	0	0
$797$	11.6615	0.413071	0.206535	−	0.978439i	$-0.433781\pi$
$797$	11.6615	0.413071	0.206535	+	0.978439i	$0.433781\pi$
$798$	0	0
$799$	−0.544112	−0.0192493
$800$	0	0
$801$	−23.3150	−0.823794
$802$	0	0
$803$	−0.247417	−0.00873116
$804$	0	0
$805$	0.777466	0.0274021
$806$	0	0
$807$	22.1977	0.781395
$808$	0	0
$809$	−28.4285	−0.999493	−0.499747	−	0.866172i	$-0.666574\pi$
$809$	−28.4285	−0.999493	−0.499747	+	0.866172i	$0.666574\pi$
$810$	0	0
$811$	−10.5687	−0.371116	−0.185558	−	0.982633i	$-0.559409\pi$
$811$	−10.5687	−0.371116	−0.185558	+	0.982633i	$0.559409\pi$
$812$	0	0
$813$	31.2730	1.09679
$814$	0	0
$815$	13.5054	0.473074
$816$	0	0
$817$	−2.56648	−0.0897897
$818$	0	0
$819$	−18.3785	−0.642198
$820$	0	0
$821$	−17.5430	−0.612255	−0.306128	−	0.951990i	$-0.599033\pi$
$821$	−17.5430	−0.612255	−0.306128	+	0.951990i	$0.599033\pi$
$822$	0	0
$823$	26.6004	0.927230	0.463615	−	0.886037i	$-0.346552\pi$
$823$	26.6004	0.927230	0.463615	+	0.886037i	$0.346552\pi$
$824$	0	0
$825$	−0.266337	−0.00927268
$826$	0	0
$827$	−22.5228	−0.783195	−0.391597	−	0.920137i	$-0.628077\pi$
$827$	−22.5228	−0.783195	−0.391597	+	0.920137i	$0.628077\pi$
$828$	0	0
$829$	39.6447	1.37692	0.688459	−	0.725275i	$-0.258287\pi$
$829$	39.6447	1.37692	0.688459	+	0.725275i	$0.258287\pi$
$830$	0	0
$831$	12.3473	0.428322
$832$	0	0
$833$	28.5283	0.988448
$834$	0	0
$835$	−5.09628	−0.176364
$836$	0	0
$837$	7.79945	0.269588
$838$	0	0
$839$	−7.81804	−0.269909	−0.134954	−	0.990852i	$-0.543089\pi$
$839$	−7.81804	−0.269909	−0.134954	+	0.990852i	$0.543089\pi$
$840$	0	0
$841$	−26.5414	−0.915220
$842$	0	0
$843$	−44.2239	−1.52315
$844$	0	0
$845$	21.6546	0.744941
$846$	0	0
$847$	8.54426	0.293584
$848$	0	0
$849$	−6.99676	−0.240128
$850$	0	0
$851$	−1.38236	−0.0473866
$852$	0	0
$853$	30.5728	1.04679	0.523396	−	0.852090i	$-0.324665\pi$
$853$	30.5728	1.04679	0.523396	+	0.852090i	$0.324665\pi$
$854$	0	0
$855$	1.06950	0.0365762
$856$	0	0
$857$	38.0206	1.29876	0.649379	−	0.760465i	$-0.275029\pi$
$857$	38.0206	1.29876	0.649379	+	0.760465i	$0.275029\pi$
$858$	0	0
$859$	12.2120	0.416668	0.208334	−	0.978058i	$-0.433196\pi$
$859$	12.2120	0.416668	0.208334	+	0.978058i	$0.433196\pi$
$860$	0	0
$861$	−1.97425	−0.0672823
$862$	0	0
$863$	34.7586	1.18320	0.591599	−	0.806232i	$-0.298497\pi$
$863$	34.7586	1.18320	0.591599	+	0.806232i	$0.298497\pi$
$864$	0	0
$865$	−4.97203	−0.169054
$866$	0	0
$867$	−7.67446	−0.260638
$868$	0	0
$869$	−0.560473	−0.0190127
$870$	0	0
$871$	58.0358	1.96647
$872$	0	0
$873$	42.8238	1.44937
$874$	0	0
$875$	−0.777466	−0.0262831
$876$	0	0
$877$	−16.6058	−0.560737	−0.280369	−	0.959892i	$-0.590457\pi$
$877$	−16.6058	−0.560737	−0.280369	+	0.959892i	$0.590457\pi$
$878$	0	0
$879$	−61.9985	−2.09116
$880$	0	0
$881$	−3.07791	−0.103697	−0.0518487	−	0.998655i	$-0.516511\pi$
$881$	−3.07791	−0.103697	−0.0518487	+	0.998655i	$0.516511\pi$
$882$	0	0
$883$	19.3316	0.650558	0.325279	−	0.945618i	$-0.394542\pi$
$883$	19.3316	0.650558	0.325279	+	0.945618i	$0.394542\pi$
$884$	0	0
$885$	3.94357	0.132562
$886$	0	0
$887$	49.5971	1.66531	0.832655	−	0.553793i	$-0.186820\pi$
$887$	49.5971	1.66531	0.832655	+	0.553793i	$0.186820\pi$
$888$	0	0
$889$	−12.1929	−0.408935
$890$	0	0
$891$	−0.494910	−0.0165801
$892$	0	0
$893$	0.0324879	0.00108717
$894$	0	0
$895$	1.65800	0.0554209
$896$	0	0
$897$	15.5924	0.520615
$898$	0	0
$899$	4.54633	0.151629
$900$	0	0
$901$	−42.9944	−1.43235
$902$	0	0
$903$	−19.8435	−0.660350
$904$	0	0
$905$	−8.41101	−0.279592
$906$	0	0
$907$	28.8380	0.957550	0.478775	−	0.877938i	$-0.341081\pi$
$907$	28.8380	0.957550	0.478775	+	0.877938i	$0.341081\pi$
$908$	0	0
$909$	51.1723	1.69728
$910$	0	0
$911$	9.80926	0.324995	0.162498	−	0.986709i	$-0.448045\pi$
$911$	9.80926	0.324995	0.162498	+	0.986709i	$0.448045\pi$
$912$	0	0
$913$	−0.948471	−0.0313898
$914$	0	0
$915$	−14.2975	−0.472661
$916$	0	0
$917$	6.89067	0.227550
$918$	0	0
$919$	−34.1743	−1.12731	−0.563653	−	0.826012i	$-0.690604\pi$
$919$	−34.1743	−1.12731	−0.563653	+	0.826012i	$0.690604\pi$
$920$	0	0
$921$	16.4273	0.541298
$922$	0	0
$923$	−82.0839	−2.70183
$924$	0	0
$925$	1.38236	0.0454516
$926$	0	0
$927$	9.46386	0.310834
$928$	0	0
$929$	38.1063	1.25023	0.625114	−	0.780533i	$-0.285052\pi$
$929$	38.1063	1.25023	0.625114	+	0.780533i	$0.285052\pi$
$930$	0	0
$931$	−1.70337	−0.0558258
$932$	0	0
$933$	−27.7201	−0.907516
$934$	0	0
$935$	−0.448538	−0.0146687
$936$	0	0
$937$	−35.1943	−1.14975	−0.574875	−	0.818242i	$-0.694949\pi$
$937$	−35.1943	−1.14975	−0.574875	+	0.818242i	$0.694949\pi$
$938$	0	0
$939$	−13.3636	−0.436106
$940$	0	0
$941$	43.2131	1.40871	0.704354	−	0.709849i	$-0.251237\pi$
$941$	43.2131	1.40871	0.704354	+	0.709849i	$0.251237\pi$
$942$	0	0
$943$	0.958715	0.0312201
$944$	0	0
$945$	2.09137	0.0680322
$946$	0	0
$947$	7.92332	0.257473	0.128737	−	0.991679i	$-0.458908\pi$
$947$	7.92332	0.257473	0.128737	+	0.991679i	$0.458908\pi$
$948$	0	0
$949$	−14.4847	−0.470194
$950$	0	0
$951$	79.7278	2.58535
$952$	0	0
$953$	44.7235	1.44874	0.724369	−	0.689413i	$-0.242131\pi$
$953$	44.7235	1.44874	0.724369	+	0.689413i	$0.242131\pi$
$954$	0	0
$955$	−19.0399	−0.616115
$956$	0	0
$957$	0.417617	0.0134996
$958$	0	0
$959$	−1.65495	−0.0534412
$960$	0	0
$961$	−22.5932	−0.728813
$962$	0	0
$963$	−4.39162	−0.141518
$964$	0	0
$965$	−0.919425	−0.0295973
$966$	0	0
$967$	−60.3979	−1.94227	−0.971133	−	0.238539i	$-0.923331\pi$
$967$	−60.3979	−1.94227	−0.971133	+	0.238539i	$0.923331\pi$
$968$	0	0
$969$	3.14675	0.101088
$970$	0	0
$971$	−12.0204	−0.385754	−0.192877	−	0.981223i	$-0.561782\pi$
$971$	−12.0204	−0.385754	−0.192877	+	0.981223i	$0.561782\pi$
$972$	0	0
$973$	−14.8625	−0.476469
$974$	0	0
$975$	−15.5924	−0.499356
$976$	0	0
$977$	0.0388875	0.00124412	0.000622060	−	1.00000i	$-0.499802\pi$
$977$	0.0388875	0.00124412	0.000622060		1.00000i	$0.499802\pi$
$978$	0	0
$979$	−0.583830	−0.0186593
$980$	0	0
$981$	−19.2230	−0.613743
$982$	0	0
$983$	10.1370	0.323320	0.161660	−	0.986847i	$-0.448315\pi$
$983$	10.1370	0.323320	0.161660	+	0.986847i	$0.448315\pi$
$984$	0	0
$985$	−5.72103	−0.182287
$986$	0	0
$987$	0.251190	0.00799548
$988$	0	0
$989$	9.63619	0.306413
$990$	0	0
$991$	−35.9013	−1.14044	−0.570221	−	0.821491i	$-0.693143\pi$
$991$	−35.9013	−1.14044	−0.570221	+	0.821491i	$0.693143\pi$
$992$	0	0
$993$	57.4249	1.82232
$994$	0	0
$995$	−7.09760	−0.225009
$996$	0	0
$997$	30.6073	0.969344	0.484672	−	0.874696i	$-0.338939\pi$
$997$	30.6073	0.969344	0.484672	+	0.874696i	$0.338939\pi$
$998$	0	0
$999$	−3.71852	−0.117649

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7360.2.a.cm.1.1		5
4.3	odd	2		7360.2.a.cs.1.5		5
8.3	odd	2		3680.2.a.w.1.1	✓	5
8.5	even	2		3680.2.a.y.1.5	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3680.2.a.w.1.1	✓	5	8.3	odd	2
3680.2.a.y.1.5	yes	5	8.5	even	2
7360.2.a.cm.1.1		5	1.1	even	1	trivial
7360.2.a.cs.1.5		5	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)

\(f(q)\)	\(=\)	\(q-2.64870 q^{3} +1.00000 q^{5} -0.777466 q^{7} +4.01559 q^{9} +O(q^{10})\)	\(q-2.64870 q^{3} +1.00000 q^{5} -0.777466 q^{7} +4.01559 q^{9} +0.100554 q^{11} +5.88682 q^{13} -2.64870 q^{15} -4.46066 q^{17} +0.266337 q^{19} +2.05927 q^{21} -1.00000 q^{23} +1.00000 q^{25} -2.68998 q^{27} -1.56800 q^{29} -2.89945 q^{31} -0.266337 q^{33} -0.777466 q^{35} +1.38236 q^{37} -15.5924 q^{39} -0.958715 q^{41} -9.63619 q^{43} +4.01559 q^{45} +0.121980 q^{47} -6.39555 q^{49} +11.8149 q^{51} +9.63858 q^{53} +0.100554 q^{55} -0.705447 q^{57} -1.48887 q^{59} +5.39794 q^{61} -3.12198 q^{63} +5.88682 q^{65} +9.85860 q^{67} +2.64870 q^{69} -13.9437 q^{71} -2.46054 q^{73} -2.64870 q^{75} -0.0781774 q^{77} -5.57384 q^{79} -4.92182 q^{81} -9.43244 q^{83} -4.46066 q^{85} +4.15315 q^{87} -5.80612 q^{89} -4.57680 q^{91} +7.67975 q^{93} +0.266337 q^{95} +10.6644 q^{97} +0.403784 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{3} + 5 q^{5} - 4 q^{7} + 2 q^{9}+O(q^{10}) \) 5 * q - q^3 + 5 * q^5 - 4 * q^7 + 2 * q^9	\( 5 q - q^{3} + 5 q^{5} - 4 q^{7} + 2 q^{9} - 3 q^{11} - q^{13} - q^{15} - 4 q^{17} - q^{19} + 4 q^{21} - 5 q^{23} + 5 q^{25} - 4 q^{27} + 5 q^{29} - 18 q^{31} + q^{33} - 4 q^{35} - 3 q^{37} - 20 q^{39} - 2 q^{41} + 12 q^{43} + 2 q^{45} + 4 q^{47} + 5 q^{49} + q^{51} - 3 q^{53} - 3 q^{55} + 10 q^{57} - 5 q^{59} - q^{61} - 19 q^{63} - q^{65} + 3 q^{67} + q^{69} - 24 q^{71} - 8 q^{73} - q^{75} - 6 q^{77} - 40 q^{79} - 19 q^{81} - 13 q^{83} - 4 q^{85} - 12 q^{87} + 2 q^{89} + 17 q^{91} + 4 q^{93} - q^{95} + 9 q^{97} - q^{99}+O(q^{100}) \) 5 * q - q^3 + 5 * q^5 - 4 * q^7 + 2 * q^9 - 3 * q^11 - q^13 - q^15 - 4 * q^17 - q^19 + 4 * q^21 - 5 * q^23 + 5 * q^25 - 4 * q^27 + 5 * q^29 - 18 * q^31 + q^33 - 4 * q^35 - 3 * q^37 - 20 * q^39 - 2 * q^41 + 12 * q^43 + 2 * q^45 + 4 * q^47 + 5 * q^49 + q^51 - 3 * q^53 - 3 * q^55 + 10 * q^57 - 5 * q^59 - q^61 - 19 * q^63 - q^65 + 3 * q^67 + q^69 - 24 * q^71 - 8 * q^73 - q^75 - 6 * q^77 - 40 * q^79 - 19 * q^81 - 13 * q^83 - 4 * q^85 - 12 * q^87 + 2 * q^89 + 17 * q^91 + 4 * q^93 - q^95 + 9 * q^97 - q^99

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