LMFDB - Embedded newform 3726.2.a.y.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3726,2,Mod(1,3726)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3726, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3726.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3726 = 2 \cdot 3^{4} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3726.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:	$29.7522597931$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 34x^{6} - 4x^{5} + 370x^{4} + 104x^{3} - 1466x^{2} - 564x + 1401 $ x^8 - 34x^6 - 4x^5 + 370x^4 + 104x^3 - 1466x^2 - 564x + 1401
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.72222$ of defining polynomial
Character	$\chi$	$=$	3726.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +2.72222 q^{5} +4.49173 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +2.72222 q^{5} +4.49173 q^{7} -1.00000 q^{8} -2.72222 q^{10} -4.42858 q^{11} +4.77816 q^{13} -4.49173 q^{14} +1.00000 q^{16} +2.45659 q^{17} +2.15285 q^{19} +2.72222 q^{20} +4.42858 q^{22} -1.00000 q^{23} +2.41047 q^{25} -4.77816 q^{26} +4.49173 q^{28} -4.60224 q^{29} +4.01849 q^{31} -1.00000 q^{32} -2.45659 q^{34} +12.2275 q^{35} -11.6201 q^{37} -2.15285 q^{38} -2.72222 q^{40} +8.26989 q^{41} -2.37264 q^{43} -4.42858 q^{44} +1.00000 q^{46} +13.5459 q^{47} +13.1756 q^{49} -2.41047 q^{50} +4.77816 q^{52} +4.52801 q^{53} -12.0556 q^{55} -4.49173 q^{56} +4.60224 q^{58} -8.35277 q^{59} +8.33917 q^{61} -4.01849 q^{62} +1.00000 q^{64} +13.0072 q^{65} -3.12197 q^{67} +2.45659 q^{68} -12.2275 q^{70} -10.0507 q^{71} +15.2502 q^{73} +11.6201 q^{74} +2.15285 q^{76} -19.8920 q^{77} +5.32771 q^{79} +2.72222 q^{80} -8.26989 q^{82} +9.30715 q^{83} +6.68736 q^{85} +2.37264 q^{86} +4.42858 q^{88} -11.7827 q^{89} +21.4622 q^{91} -1.00000 q^{92} -13.5459 q^{94} +5.86052 q^{95} +13.9997 q^{97} -13.1756 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + 8 q^{4} + 6 q^{7} - 8 q^{8}+O(q^{10}) $ 8 * q - 8 * q^2 + 8 * q^4 + 6 * q^7 - 8 * q^8	$ 8 q - 8 q^{2} + 8 q^{4} + 6 q^{7} - 8 q^{8} + 6 q^{13} - 6 q^{14} + 8 q^{16} + 10 q^{17} + 12 q^{19} - 8 q^{23} + 28 q^{25} - 6 q^{26} + 6 q^{28} - 6 q^{29} + 16 q^{31} - 8 q^{32} - 10 q^{34} - 18 q^{35} + 4 q^{37} - 12 q^{38} + 4 q^{41} + 6 q^{43} + 8 q^{46} - 14 q^{47} + 28 q^{49} - 28 q^{50} + 6 q^{52} - 20 q^{53} + 16 q^{55} - 6 q^{56} + 6 q^{58} - 6 q^{59} + 12 q^{61} - 16 q^{62} + 8 q^{64} + 10 q^{65} + 10 q^{67} + 10 q^{68} + 18 q^{70} + 18 q^{71} + 44 q^{73} - 4 q^{74} + 12 q^{76} - 28 q^{77} + 22 q^{79} - 4 q^{82} - 4 q^{83} + 22 q^{85} - 6 q^{86} + 18 q^{89} + 14 q^{91} - 8 q^{92} + 14 q^{94} + 20 q^{95} + 36 q^{97} - 28 q^{98}+O(q^{100}) $ 8 * q - 8 * q^2 + 8 * q^4 + 6 * q^7 - 8 * q^8 + 6 * q^13 - 6 * q^14 + 8 * q^16 + 10 * q^17 + 12 * q^19 - 8 * q^23 + 28 * q^25 - 6 * q^26 + 6 * q^28 - 6 * q^29 + 16 * q^31 - 8 * q^32 - 10 * q^34 - 18 * q^35 + 4 * q^37 - 12 * q^38 + 4 * q^41 + 6 * q^43 + 8 * q^46 - 14 * q^47 + 28 * q^49 - 28 * q^50 + 6 * q^52 - 20 * q^53 + 16 * q^55 - 6 * q^56 + 6 * q^58 - 6 * q^59 + 12 * q^61 - 16 * q^62 + 8 * q^64 + 10 * q^65 + 10 * q^67 + 10 * q^68 + 18 * q^70 + 18 * q^71 + 44 * q^73 - 4 * q^74 + 12 * q^76 - 28 * q^77 + 22 * q^79 - 4 * q^82 - 4 * q^83 + 22 * q^85 - 6 * q^86 + 18 * q^89 + 14 * q^91 - 8 * q^92 + 14 * q^94 + 20 * q^95 + 36 * q^97 - 28 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	2.72222	1.21741	0.608707	−	0.793395i	$-0.291689\pi$
$5$	2.72222	1.21741	0.608707	+	0.793395i	$0.291689\pi$
$6$	0	0
$7$	4.49173	1.69771	0.848857	−	0.528623i	$-0.177291\pi$
$7$	4.49173	1.69771	0.848857	+	0.528623i	$0.177291\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−2.72222	−0.860841
$11$	−4.42858	−1.33527	−0.667634	−	0.744490i	$-0.732693\pi$
$11$	−4.42858	−1.33527	−0.667634	+	0.744490i	$0.732693\pi$
$12$	0	0
$13$	4.77816	1.32522	0.662612	−	0.748963i	$-0.269448\pi$
$13$	4.77816	1.32522	0.662612	+	0.748963i	$0.269448\pi$
$14$	−4.49173	−1.20046
$15$	0	0
$16$	1.00000	0.250000
$17$	2.45659	0.595810	0.297905	−	0.954596i	$-0.403712\pi$
$17$	2.45659	0.595810	0.297905	+	0.954596i	$0.403712\pi$
$18$	0	0
$19$	2.15285	0.493897	0.246949	−	0.969029i	$-0.420572\pi$
$19$	2.15285	0.493897	0.246949	+	0.969029i	$0.420572\pi$
$20$	2.72222	0.608707
$21$	0	0
$22$	4.42858	0.944177
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	2.41047	0.482095
$26$	−4.77816	−0.937075
$27$	0	0
$28$	4.49173	0.848857
$29$	−4.60224	−0.854614	−0.427307	−	0.904107i	$-0.640538\pi$
$29$	−4.60224	−0.854614	−0.427307	+	0.904107i	$0.640538\pi$
$30$	0	0
$31$	4.01849	0.721742	0.360871	−	0.932616i	$-0.382480\pi$
$31$	4.01849	0.721742	0.360871	+	0.932616i	$0.382480\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−2.45659	−0.421301
$35$	12.2275	2.06682
$36$	0	0
$37$	−11.6201	−1.91034	−0.955170	−	0.296058i	$-0.904328\pi$
$37$	−11.6201	−1.91034	−0.955170	+	0.296058i	$0.904328\pi$
$38$	−2.15285	−0.349238
$39$	0	0
$40$	−2.72222	−0.430421
$41$	8.26989	1.29154	0.645770	−	0.763532i	$-0.276536\pi$
$41$	8.26989	1.29154	0.645770	+	0.763532i	$0.276536\pi$
$42$	0	0
$43$	−2.37264	−0.361824	−0.180912	−	0.983499i	$-0.557905\pi$
$43$	−2.37264	−0.361824	−0.180912	+	0.983499i	$0.557905\pi$
$44$	−4.42858	−0.667634
$45$	0	0
$46$	1.00000	0.147442
$47$	13.5459	1.97587	0.987937	−	0.154853i	$-0.0494905\pi$
$47$	13.5459	1.97587	0.987937	+	0.154853i	$0.0494905\pi$
$48$	0	0
$49$	13.1756	1.88223
$50$	−2.41047	−0.340892
$51$	0	0
$52$	4.77816	0.662612
$53$	4.52801	0.621970	0.310985	−	0.950415i	$-0.399341\pi$
$53$	4.52801	0.621970	0.310985	+	0.950415i	$0.399341\pi$
$54$	0	0
$55$	−12.0556	−1.62557
$56$	−4.49173	−0.600232
$57$	0	0
$58$	4.60224	0.604303
$59$	−8.35277	−1.08744	−0.543719	−	0.839267i	$-0.682984\pi$
$59$	−8.35277	−1.08744	−0.543719	+	0.839267i	$0.682984\pi$
$60$	0	0
$61$	8.33917	1.06772	0.533860	−	0.845573i	$-0.320741\pi$
$61$	8.33917	1.06772	0.533860	+	0.845573i	$0.320741\pi$
$62$	−4.01849	−0.510348
$63$	0	0
$64$	1.00000	0.125000
$65$	13.0072	1.61335
$66$	0	0
$67$	−3.12197	−0.381409	−0.190704	−	0.981648i	$-0.561077\pi$
$67$	−3.12197	−0.381409	−0.190704	+	0.981648i	$0.561077\pi$
$68$	2.45659	0.297905
$69$	0	0
$70$	−12.2275	−1.46146
$71$	−10.0507	−1.19279	−0.596397	−	0.802690i	$-0.703402\pi$
$71$	−10.0507	−1.19279	−0.596397	+	0.802690i	$0.703402\pi$
$72$	0	0
$73$	15.2502	1.78490	0.892452	−	0.451142i	$-0.148983\pi$
$73$	15.2502	1.78490	0.892452	+	0.451142i	$0.148983\pi$
$74$	11.6201	1.35081
$75$	0	0
$76$	2.15285	0.246949
$77$	−19.8920	−2.26690
$78$	0	0
$79$	5.32771	0.599414	0.299707	−	0.954031i	$-0.403111\pi$
$79$	5.32771	0.599414	0.299707	+	0.954031i	$0.403111\pi$
$80$	2.72222	0.304353
$81$	0	0
$82$	−8.26989	−0.913257
$83$	9.30715	1.02159	0.510796	−	0.859702i	$-0.329351\pi$
$83$	9.30715	1.02159	0.510796	+	0.859702i	$0.329351\pi$
$84$	0	0
$85$	6.68736	0.725346
$86$	2.37264	0.255848
$87$	0	0
$88$	4.42858	0.472089
$89$	−11.7827	−1.24896	−0.624482	−	0.781039i	$-0.714690\pi$
$89$	−11.7827	−1.24896	−0.624482	+	0.781039i	$0.714690\pi$
$90$	0	0
$91$	21.4622	2.24985
$92$	−1.00000	−0.104257
$93$	0	0
$94$	−13.5459	−1.39715
$95$	5.86052	0.601277
$96$	0	0
$97$	13.9997	1.42145	0.710727	−	0.703468i	$-0.248366\pi$
$97$	13.9997	1.42145	0.710727	+	0.703468i	$0.248366\pi$
$98$	−13.1756	−1.33094
$99$	0	0
$100$	2.41047	0.241047
$101$	−8.84128	−0.879741	−0.439870	−	0.898061i	$-0.644976\pi$
$101$	−8.84128	−0.879741	−0.439870	+	0.898061i	$0.644976\pi$
$102$	0	0
$103$	−0.355336	−0.0350123	−0.0175061	−	0.999847i	$-0.505573\pi$
$103$	−0.355336	−0.0350123	−0.0175061	+	0.999847i	$0.505573\pi$
$104$	−4.77816	−0.468538
$105$	0	0
$106$	−4.52801	−0.439799
$107$	4.64079	0.448642	0.224321	−	0.974515i	$-0.427984\pi$
$107$	4.64079	0.448642	0.224321	+	0.974515i	$0.427984\pi$
$108$	0	0
$109$	−8.76086	−0.839138	−0.419569	−	0.907723i	$-0.637819\pi$
$109$	−8.76086	−0.839138	−0.419569	+	0.907723i	$0.637819\pi$
$110$	12.0556	1.14945
$111$	0	0
$112$	4.49173	0.424428
$113$	8.10095	0.762073	0.381036	−	0.924560i	$-0.375567\pi$
$113$	8.10095	0.762073	0.381036	+	0.924560i	$0.375567\pi$
$114$	0	0
$115$	−2.72222	−0.253848
$116$	−4.60224	−0.427307
$117$	0	0
$118$	8.35277	0.768935
$119$	11.0343	1.01151
$120$	0	0
$121$	8.61236	0.782941
$122$	−8.33917	−0.754993
$123$	0	0
$124$	4.01849	0.360871
$125$	−7.04926	−0.630505
$126$	0	0
$127$	−3.38341	−0.300229	−0.150114	−	0.988669i	$-0.547964\pi$
$127$	−3.38341	−0.300229	−0.150114	+	0.988669i	$0.547964\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−13.0072	−1.14081
$131$	−14.8990	−1.30173	−0.650865	−	0.759194i	$-0.725594\pi$
$131$	−14.8990	−1.30173	−0.650865	+	0.759194i	$0.725594\pi$
$132$	0	0
$133$	9.67001	0.838496
$134$	3.12197	0.269697
$135$	0	0
$136$	−2.45659	−0.210651
$137$	5.86922	0.501441	0.250721	−	0.968060i	$-0.419332\pi$
$137$	5.86922	0.501441	0.250721	+	0.968060i	$0.419332\pi$
$138$	0	0
$139$	−18.5534	−1.57368	−0.786838	−	0.617160i	$-0.788283\pi$
$139$	−18.5534	−1.57368	−0.786838	+	0.617160i	$0.788283\pi$
$140$	12.2275	1.03341
$141$	0	0
$142$	10.0507	0.843433
$143$	−21.1605	−1.76953
$144$	0	0
$145$	−12.5283	−1.04042
$146$	−15.2502	−1.26212
$147$	0	0
$148$	−11.6201	−0.955170
$149$	−7.13113	−0.584205	−0.292103	−	0.956387i	$-0.594355\pi$
$149$	−7.13113	−0.584205	−0.292103	+	0.956387i	$0.594355\pi$
$150$	0	0
$151$	−3.59610	−0.292646	−0.146323	−	0.989237i	$-0.546744\pi$
$151$	−3.59610	−0.292646	−0.146323	+	0.989237i	$0.546744\pi$
$152$	−2.15285	−0.174619
$153$	0	0
$154$	19.8920	1.60294
$155$	10.9392	0.878658
$156$	0	0
$157$	0.111645	0.00891026	0.00445513	−	0.999990i	$-0.498582\pi$
$157$	0.111645	0.00891026	0.00445513	+	0.999990i	$0.498582\pi$
$158$	−5.32771	−0.423850
$159$	0	0
$160$	−2.72222	−0.215210
$161$	−4.49173	−0.353998
$162$	0	0
$163$	−20.9496	−1.64090	−0.820449	−	0.571719i	$-0.806277\pi$
$163$	−20.9496	−1.64090	−0.820449	+	0.571719i	$0.806277\pi$
$164$	8.26989	0.645770
$165$	0	0
$166$	−9.30715	−0.722375
$167$	10.1130	0.782565	0.391282	−	0.920271i	$-0.372032\pi$
$167$	10.1130	0.782565	0.391282	+	0.920271i	$0.372032\pi$
$168$	0	0
$169$	9.83086	0.756220
$170$	−6.68736	−0.512897
$171$	0	0
$172$	−2.37264	−0.180912
$173$	5.83471	0.443605	0.221802	−	0.975092i	$-0.428806\pi$
$173$	5.83471	0.443605	0.221802	+	0.975092i	$0.428806\pi$
$174$	0	0
$175$	10.8272	0.818458
$176$	−4.42858	−0.333817
$177$	0	0
$178$	11.7827	0.883152
$179$	−10.2464	−0.765848	−0.382924	−	0.923780i	$-0.625083\pi$
$179$	−10.2464	−0.765848	−0.382924	+	0.923780i	$0.625083\pi$
$180$	0	0
$181$	−15.0091	−1.11562	−0.557810	−	0.829969i	$-0.688358\pi$
$181$	−15.0091	−1.11562	−0.557810	+	0.829969i	$0.688358\pi$
$182$	−21.4622	−1.59089
$183$	0	0
$184$	1.00000	0.0737210
$185$	−31.6326	−2.32567
$186$	0	0
$187$	−10.8792	−0.795566
$188$	13.5459	0.987937
$189$	0	0
$190$	−5.86052	−0.425167
$191$	−19.1459	−1.38535	−0.692675	−	0.721250i	$-0.743568\pi$
$191$	−19.1459	−1.38535	−0.692675	+	0.721250i	$0.743568\pi$
$192$	0	0
$193$	9.74849	0.701712	0.350856	−	0.936429i	$-0.385891\pi$
$193$	9.74849	0.701712	0.350856	+	0.936429i	$0.385891\pi$
$194$	−13.9997	−1.00512
$195$	0	0
$196$	13.1756	0.941116
$197$	−1.57339	−0.112099	−0.0560497	−	0.998428i	$-0.517851\pi$
$197$	−1.57339	−0.112099	−0.0560497	+	0.998428i	$0.517851\pi$
$198$	0	0
$199$	19.1919	1.36047	0.680237	−	0.732992i	$-0.261877\pi$
$199$	19.1919	1.36047	0.680237	+	0.732992i	$0.261877\pi$
$200$	−2.41047	−0.170446
$201$	0	0
$202$	8.84128	0.622071
$203$	−20.6720	−1.45089
$204$	0	0
$205$	22.5125	1.57234
$206$	0.355336	0.0247574
$207$	0	0
$208$	4.77816	0.331306
$209$	−9.53407	−0.659485
$210$	0	0
$211$	10.4294	0.717991	0.358995	−	0.933339i	$-0.383119\pi$
$211$	10.4294	0.717991	0.358995	+	0.933339i	$0.383119\pi$
$212$	4.52801	0.310985
$213$	0	0
$214$	−4.64079	−0.317238
$215$	−6.45884	−0.440489
$216$	0	0
$217$	18.0500	1.22531
$218$	8.76086	0.593360
$219$	0	0
$220$	−12.0556	−0.812787
$221$	11.7380	0.789581
$222$	0	0
$223$	−26.0351	−1.74344	−0.871720	−	0.490005i	$-0.836995\pi$
$223$	−26.0351	−1.74344	−0.871720	+	0.490005i	$0.836995\pi$
$224$	−4.49173	−0.300116
$225$	0	0
$226$	−8.10095	−0.538867
$227$	2.50420	0.166209	0.0831046	−	0.996541i	$-0.473516\pi$
$227$	2.50420	0.166209	0.0831046	+	0.996541i	$0.473516\pi$
$228$	0	0
$229$	29.5287	1.95131	0.975657	−	0.219301i	$-0.0703777\pi$
$229$	29.5287	1.95131	0.975657	+	0.219301i	$0.0703777\pi$
$230$	2.72222	0.179498
$231$	0	0
$232$	4.60224	0.302152
$233$	29.4447	1.92898	0.964492	−	0.264112i	$-0.0850790\pi$
$233$	29.4447	1.92898	0.964492	+	0.264112i	$0.0850790\pi$
$234$	0	0
$235$	36.8749	2.40546
$236$	−8.35277	−0.543719
$237$	0	0
$238$	−11.0343	−0.715249
$239$	1.04205	0.0674045	0.0337023	−	0.999432i	$-0.489270\pi$
$239$	1.04205	0.0674045	0.0337023	+	0.999432i	$0.489270\pi$
$240$	0	0
$241$	−9.54030	−0.614545	−0.307272	−	0.951622i	$-0.599416\pi$
$241$	−9.54030	−0.614545	−0.307272	+	0.951622i	$0.599416\pi$
$242$	−8.61236	−0.553623
$243$	0	0
$244$	8.33917	0.533860
$245$	35.8669	2.29145
$246$	0	0
$247$	10.2867	0.654524
$248$	−4.01849	−0.255174
$249$	0	0
$250$	7.04926	0.445834
$251$	3.63153	0.229220	0.114610	−	0.993411i	$-0.463438\pi$
$251$	3.63153	0.229220	0.114610	+	0.993411i	$0.463438\pi$
$252$	0	0
$253$	4.42858	0.278423
$254$	3.38341	0.212294
$255$	0	0
$256$	1.00000	0.0625000
$257$	2.08768	0.130226	0.0651129	−	0.997878i	$-0.479259\pi$
$257$	2.08768	0.130226	0.0651129	+	0.997878i	$0.479259\pi$
$258$	0	0
$259$	−52.1945	−3.24321
$260$	13.0072	0.806673
$261$	0	0
$262$	14.8990	0.920462
$263$	26.4900	1.63344	0.816722	−	0.577032i	$-0.195789\pi$
$263$	26.4900	1.63344	0.816722	+	0.577032i	$0.195789\pi$
$264$	0	0
$265$	12.3262	0.757194
$266$	−9.67001	−0.592906
$267$	0	0
$268$	−3.12197	−0.190704
$269$	−1.06849	−0.0651467	−0.0325734	−	0.999469i	$-0.510370\pi$
$269$	−1.06849	−0.0651467	−0.0325734	+	0.999469i	$0.510370\pi$
$270$	0	0
$271$	−20.8160	−1.26448	−0.632240	−	0.774773i	$-0.717864\pi$
$271$	−20.8160	−1.26448	−0.632240	+	0.774773i	$0.717864\pi$
$272$	2.45659	0.148952
$273$	0	0
$274$	−5.86922	−0.354572
$275$	−10.6750	−0.643726
$276$	0	0
$277$	12.3240	0.740476	0.370238	−	0.928937i	$-0.379276\pi$
$277$	12.3240	0.740476	0.370238	+	0.928937i	$0.379276\pi$
$278$	18.5534	1.11276
$279$	0	0
$280$	−12.2275	−0.730731
$281$	−4.79225	−0.285882	−0.142941	−	0.989731i	$-0.545656\pi$
$281$	−4.79225	−0.285882	−0.142941	+	0.989731i	$0.545656\pi$
$282$	0	0
$283$	0.227481	0.0135223	0.00676116	−	0.999977i	$-0.497848\pi$
$283$	0.227481	0.0135223	0.00676116	+	0.999977i	$0.497848\pi$
$284$	−10.0507	−0.596397
$285$	0	0
$286$	21.1605	1.25125
$287$	37.1461	2.19267
$288$	0	0
$289$	−10.9652	−0.645011
$290$	12.5283	0.735687
$291$	0	0
$292$	15.2502	0.892452
$293$	−4.05115	−0.236671	−0.118335	−	0.992974i	$-0.537756\pi$
$293$	−4.05115	−0.236671	−0.118335	+	0.992974i	$0.537756\pi$
$294$	0	0
$295$	−22.7381	−1.32386
$296$	11.6201	0.675407
$297$	0	0
$298$	7.13113	0.413095
$299$	−4.77816	−0.276328
$300$	0	0
$301$	−10.6572	−0.614273
$302$	3.59610	0.206932
$303$	0	0
$304$	2.15285	0.123474
$305$	22.7010	1.29986
$306$	0	0
$307$	7.58376	0.432828	0.216414	−	0.976302i	$-0.430564\pi$
$307$	7.58376	0.432828	0.216414	+	0.976302i	$0.430564\pi$
$308$	−19.8920	−1.13345
$309$	0	0
$310$	−10.9392	−0.621305
$311$	8.84232	0.501402	0.250701	−	0.968065i	$-0.419339\pi$
$311$	8.84232	0.501402	0.250701	+	0.968065i	$0.419339\pi$
$312$	0	0
$313$	0.387473	0.0219013	0.0109506	−	0.999940i	$-0.496514\pi$
$313$	0.387473	0.0219013	0.0109506	+	0.999940i	$0.496514\pi$
$314$	−0.111645	−0.00630051
$315$	0	0
$316$	5.32771	0.299707
$317$	28.2206	1.58502	0.792512	−	0.609856i	$-0.208773\pi$
$317$	28.2206	1.58502	0.792512	+	0.609856i	$0.208773\pi$
$318$	0	0
$319$	20.3814	1.14114
$320$	2.72222	0.152177
$321$	0	0
$322$	4.49173	0.250314
$323$	5.28866	0.294269
$324$	0	0
$325$	11.5176	0.638883
$326$	20.9496	1.16029
$327$	0	0
$328$	−8.26989	−0.456628
$329$	60.8446	3.35447
$330$	0	0
$331$	25.3255	1.39202	0.696008	−	0.718034i	$-0.254958\pi$
$331$	25.3255	1.39202	0.696008	+	0.718034i	$0.254958\pi$
$332$	9.30715	0.510796
$333$	0	0
$334$	−10.1130	−0.553357
$335$	−8.49867	−0.464332
$336$	0	0
$337$	−21.3152	−1.16111	−0.580557	−	0.814220i	$-0.697165\pi$
$337$	−21.3152	−1.16111	−0.580557	+	0.814220i	$0.697165\pi$
$338$	−9.83086	−0.534728
$339$	0	0
$340$	6.68736	0.362673
$341$	−17.7962	−0.963719
$342$	0	0
$343$	27.7392	1.49778
$344$	2.37264	0.127924
$345$	0	0
$346$	−5.83471	−0.313676
$347$	−15.8108	−0.848771	−0.424385	−	0.905482i	$-0.639510\pi$
$347$	−15.8108	−0.848771	−0.424385	+	0.905482i	$0.639510\pi$
$348$	0	0
$349$	35.1179	1.87982	0.939909	−	0.341424i	$-0.110909\pi$
$349$	35.1179	1.87982	0.939909	+	0.341424i	$0.110909\pi$
$350$	−10.8272	−0.578737
$351$	0	0
$352$	4.42858	0.236044
$353$	−34.8077	−1.85262	−0.926312	−	0.376757i	$-0.877039\pi$
$353$	−34.8077	−1.85262	−0.926312	+	0.376757i	$0.877039\pi$
$354$	0	0
$355$	−27.3601	−1.45212
$356$	−11.7827	−0.624482
$357$	0	0
$358$	10.2464	0.541537
$359$	11.3635	0.599743	0.299871	−	0.953980i	$-0.403056\pi$
$359$	11.3635	0.599743	0.299871	+	0.953980i	$0.403056\pi$
$360$	0	0
$361$	−14.3652	−0.756066
$362$	15.0091	0.788862
$363$	0	0
$364$	21.4622	1.12493
$365$	41.5144	2.17297
$366$	0	0
$367$	−6.64275	−0.346749	−0.173374	−	0.984856i	$-0.555467\pi$
$367$	−6.64275	−0.346749	−0.173374	+	0.984856i	$0.555467\pi$
$368$	−1.00000	−0.0521286
$369$	0	0
$370$	31.6326	1.64450
$371$	20.3386	1.05593
$372$	0	0
$373$	11.2504	0.582526	0.291263	−	0.956643i	$-0.405924\pi$
$373$	11.2504	0.582526	0.291263	+	0.956643i	$0.405924\pi$
$374$	10.8792	0.562550
$375$	0	0
$376$	−13.5459	−0.698577
$377$	−21.9902	−1.13255
$378$	0	0
$379$	−26.4976	−1.36109	−0.680546	−	0.732705i	$-0.738257\pi$
$379$	−26.4976	−1.36109	−0.680546	+	0.732705i	$0.738257\pi$
$380$	5.86052	0.300638
$381$	0	0
$382$	19.1459	0.979590
$383$	−18.1665	−0.928266	−0.464133	−	0.885765i	$-0.653634\pi$
$383$	−18.1665	−0.928266	−0.464133	+	0.885765i	$0.653634\pi$
$384$	0	0
$385$	−54.1504	−2.75976
$386$	−9.74849	−0.496185
$387$	0	0
$388$	13.9997	0.710727
$389$	18.4277	0.934323	0.467162	−	0.884172i	$-0.345277\pi$
$389$	18.4277	0.934323	0.467162	+	0.884172i	$0.345277\pi$
$390$	0	0
$391$	−2.45659	−0.124235
$392$	−13.1756	−0.665469
$393$	0	0
$394$	1.57339	0.0792662
$395$	14.5032	0.729735
$396$	0	0
$397$	18.0620	0.906506	0.453253	−	0.891382i	$-0.350263\pi$
$397$	18.0620	0.906506	0.453253	+	0.891382i	$0.350263\pi$
$398$	−19.1919	−0.962001
$399$	0	0
$400$	2.41047	0.120524
$401$	11.1163	0.555123	0.277562	−	0.960708i	$-0.410474\pi$
$401$	11.1163	0.555123	0.277562	+	0.960708i	$0.410474\pi$
$402$	0	0
$403$	19.2010	0.956470
$404$	−8.84128	−0.439870
$405$	0	0
$406$	20.6720	1.02593
$407$	51.4608	2.55082
$408$	0	0
$409$	34.5128	1.70655	0.853273	−	0.521464i	$-0.174614\pi$
$409$	34.5128	1.70655	0.853273	+	0.521464i	$0.174614\pi$
$410$	−22.5125	−1.11181
$411$	0	0
$412$	−0.355336	−0.0175061
$413$	−37.5184	−1.84616
$414$	0	0
$415$	25.3361	1.24370
$416$	−4.77816	−0.234269
$417$	0	0
$418$	9.53407	0.466326
$419$	−12.4583	−0.608629	−0.304315	−	0.952572i	$-0.598427\pi$
$419$	−12.4583	−0.608629	−0.304315	+	0.952572i	$0.598427\pi$
$420$	0	0
$421$	−27.9788	−1.36360	−0.681802	−	0.731536i	$-0.738804\pi$
$421$	−27.9788	−1.36360	−0.681802	+	0.731536i	$0.738804\pi$
$422$	−10.4294	−0.507696
$423$	0	0
$424$	−4.52801	−0.219900
$425$	5.92153	0.287237
$426$	0	0
$427$	37.4573	1.81268
$428$	4.64079	0.224321
$429$	0	0
$430$	6.45884	0.311473
$431$	23.3561	1.12502	0.562512	−	0.826789i	$-0.309835\pi$
$431$	23.3561	1.12502	0.562512	+	0.826789i	$0.309835\pi$
$432$	0	0
$433$	−12.8182	−0.616002	−0.308001	−	0.951386i	$-0.599660\pi$
$433$	−12.8182	−0.616002	−0.308001	+	0.951386i	$0.599660\pi$
$434$	−18.0500	−0.866425
$435$	0	0
$436$	−8.76086	−0.419569
$437$	−2.15285	−0.102985
$438$	0	0
$439$	−26.3267	−1.25651	−0.628253	−	0.778009i	$-0.716230\pi$
$439$	−26.3267	−1.25651	−0.628253	+	0.778009i	$0.716230\pi$
$440$	12.0556	0.574727
$441$	0	0
$442$	−11.7380	−0.558318
$443$	−4.65228	−0.221037	−0.110518	−	0.993874i	$-0.535251\pi$
$443$	−4.65228	−0.221037	−0.110518	+	0.993874i	$0.535251\pi$
$444$	0	0
$445$	−32.0751	−1.52051
$446$	26.0351	1.23280
$447$	0	0
$448$	4.49173	0.212214
$449$	−25.2362	−1.19097	−0.595485	−	0.803367i	$-0.703040\pi$
$449$	−25.2362	−1.19097	−0.595485	+	0.803367i	$0.703040\pi$
$450$	0	0
$451$	−36.6239	−1.72455
$452$	8.10095	0.381036
$453$	0	0
$454$	−2.50420	−0.117528
$455$	58.4248	2.73900
$456$	0	0
$457$	0.965423	0.0451606	0.0225803	−	0.999745i	$-0.492812\pi$
$457$	0.965423	0.0451606	0.0225803	+	0.999745i	$0.492812\pi$
$458$	−29.5287	−1.37979
$459$	0	0
$460$	−2.72222	−0.126924
$461$	−27.5435	−1.28283	−0.641414	−	0.767195i	$-0.721652\pi$
$461$	−27.5435	−1.28283	−0.641414	+	0.767195i	$0.721652\pi$
$462$	0	0
$463$	25.8842	1.20294	0.601471	−	0.798894i	$-0.294581\pi$
$463$	25.8842	1.20294	0.601471	+	0.798894i	$0.294581\pi$
$464$	−4.60224	−0.213653
$465$	0	0
$466$	−29.4447	−1.36400
$467$	−2.82650	−0.130795	−0.0653974	−	0.997859i	$-0.520832\pi$
$467$	−2.82650	−0.130795	−0.0653974	+	0.997859i	$0.520832\pi$
$468$	0	0
$469$	−14.0230	−0.647523
$470$	−36.8749	−1.70091
$471$	0	0
$472$	8.35277	0.384468
$473$	10.5074	0.483132
$474$	0	0
$475$	5.18938	0.238105
$476$	11.0343	0.505757
$477$	0	0
$478$	−1.04205	−0.0476622
$479$	3.27455	0.149618	0.0748091	−	0.997198i	$-0.476165\pi$
$479$	3.27455	0.149618	0.0748091	+	0.997198i	$0.476165\pi$
$480$	0	0
$481$	−55.5230	−2.53163
$482$	9.54030	0.434549
$483$	0	0
$484$	8.61236	0.391471
$485$	38.1102	1.73050
$486$	0	0
$487$	−39.8211	−1.80447	−0.902233	−	0.431248i	$-0.858073\pi$
$487$	−39.8211	−1.80447	−0.902233	+	0.431248i	$0.858073\pi$
$488$	−8.33917	−0.377496
$489$	0	0
$490$	−35.8669	−1.62030
$491$	10.6318	0.479808	0.239904	−	0.970797i	$-0.422884\pi$
$491$	10.6318	0.479808	0.239904	+	0.970797i	$0.422884\pi$
$492$	0	0
$493$	−11.3058	−0.509187
$494$	−10.2867	−0.462819
$495$	0	0
$496$	4.01849	0.180435
$497$	−45.1448	−2.02502
$498$	0	0
$499$	27.3158	1.22282	0.611411	−	0.791313i	$-0.290602\pi$
$499$	27.3158	1.22282	0.611411	+	0.791313i	$0.290602\pi$
$500$	−7.04926	−0.315252
$501$	0	0
$502$	−3.63153	−0.162083
$503$	15.7955	0.704286	0.352143	−	0.935946i	$-0.385453\pi$
$503$	15.7955	0.704286	0.352143	+	0.935946i	$0.385453\pi$
$504$	0	0
$505$	−24.0679	−1.07101
$506$	−4.42858	−0.196875
$507$	0	0
$508$	−3.38341	−0.150114
$509$	10.6094	0.470255	0.235127	−	0.971965i	$-0.424449\pi$
$509$	10.6094	0.470255	0.235127	+	0.971965i	$0.424449\pi$
$510$	0	0
$511$	68.4999	3.03026
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−2.08768	−0.0920836
$515$	−0.967302	−0.0426244
$516$	0	0
$517$	−59.9892	−2.63832
$518$	52.1945	2.29330
$519$	0	0
$520$	−13.0072	−0.570404
$521$	−8.00389	−0.350657	−0.175328	−	0.984510i	$-0.556099\pi$
$521$	−8.00389	−0.350657	−0.175328	+	0.984510i	$0.556099\pi$
$522$	0	0
$523$	12.5415	0.548403	0.274201	−	0.961672i	$-0.411586\pi$
$523$	12.5415	0.548403	0.274201	+	0.961672i	$0.411586\pi$
$524$	−14.8990	−0.650865
$525$	0	0
$526$	−26.4900	−1.15502
$527$	9.87176	0.430021
$528$	0	0
$529$	1.00000	0.0434783
$530$	−12.3262	−0.535417
$531$	0	0
$532$	9.67001	0.419248
$533$	39.5149	1.71158
$534$	0	0
$535$	12.6333	0.546183
$536$	3.12197	0.134848
$537$	0	0
$538$	1.06849	0.0460657
$539$	−58.3493	−2.51328
$540$	0	0
$541$	−36.3576	−1.56314	−0.781568	−	0.623820i	$-0.785580\pi$
$541$	−36.3576	−1.56314	−0.781568	+	0.623820i	$0.785580\pi$
$542$	20.8160	0.894122
$543$	0	0
$544$	−2.45659	−0.105325
$545$	−23.8490	−1.02158
$546$	0	0
$547$	−19.6519	−0.840253	−0.420127	−	0.907466i	$-0.638014\pi$
$547$	−19.6519	−0.840253	−0.420127	+	0.907466i	$0.638014\pi$
$548$	5.86922	0.250721
$549$	0	0
$550$	10.6750	0.455183
$551$	−9.90791	−0.422091
$552$	0	0
$553$	23.9306	1.01763
$554$	−12.3240	−0.523596
$555$	0	0
$556$	−18.5534	−0.786838
$557$	1.84402	0.0781337	0.0390669	−	0.999237i	$-0.487561\pi$
$557$	1.84402	0.0781337	0.0390669	+	0.999237i	$0.487561\pi$
$558$	0	0
$559$	−11.3369	−0.479498
$560$	12.2275	0.516705
$561$	0	0
$562$	4.79225	0.202149
$563$	28.6923	1.20924	0.604618	−	0.796516i	$-0.293326\pi$
$563$	28.6923	1.20924	0.604618	+	0.796516i	$0.293326\pi$
$564$	0	0
$565$	22.0525	0.927758
$566$	−0.227481	−0.00956173
$567$	0	0
$568$	10.0507	0.421716
$569$	17.3591	0.727731	0.363866	−	0.931451i	$-0.381457\pi$
$569$	17.3591	0.727731	0.363866	+	0.931451i	$0.381457\pi$
$570$	0	0
$571$	−9.50663	−0.397840	−0.198920	−	0.980016i	$-0.563743\pi$
$571$	−9.50663	−0.397840	−0.198920	+	0.980016i	$0.563743\pi$
$572$	−21.1605	−0.884765
$573$	0	0
$574$	−37.1461	−1.55045
$575$	−2.41047	−0.100524
$576$	0	0
$577$	−7.74494	−0.322426	−0.161213	−	0.986920i	$-0.551541\pi$
$577$	−7.74494	−0.322426	−0.161213	+	0.986920i	$0.551541\pi$
$578$	10.9652	0.456092
$579$	0	0
$580$	−12.5283	−0.520209
$581$	41.8052	1.73437
$582$	0	0
$583$	−20.0527	−0.830497
$584$	−15.2502	−0.631059
$585$	0	0
$586$	4.05115	0.167351
$587$	2.52677	0.104291	0.0521454	−	0.998640i	$-0.483394\pi$
$587$	2.52677	0.104291	0.0521454	+	0.998640i	$0.483394\pi$
$588$	0	0
$589$	8.65119	0.356466
$590$	22.7381	0.936112
$591$	0	0
$592$	−11.6201	−0.477585
$593$	32.0461	1.31597	0.657987	−	0.753029i	$-0.271408\pi$
$593$	32.0461	1.31597	0.657987	+	0.753029i	$0.271408\pi$
$594$	0	0
$595$	30.0378	1.23143
$596$	−7.13113	−0.292103
$597$	0	0
$598$	4.77816	0.195394
$599$	−7.78628	−0.318139	−0.159069	−	0.987267i	$-0.550849\pi$
$599$	−7.78628	−0.318139	−0.159069	+	0.987267i	$0.550849\pi$
$600$	0	0
$601$	8.38899	0.342194	0.171097	−	0.985254i	$-0.445269\pi$
$601$	8.38899	0.342194	0.171097	+	0.985254i	$0.445269\pi$
$602$	10.6572	0.434357
$603$	0	0
$604$	−3.59610	−0.146323
$605$	23.4447	0.953163
$606$	0	0
$607$	24.9714	1.01356	0.506778	−	0.862076i	$-0.330836\pi$
$607$	24.9714	1.01356	0.506778	+	0.862076i	$0.330836\pi$
$608$	−2.15285	−0.0873095
$609$	0	0
$610$	−22.7010	−0.919138
$611$	64.7246	2.61848
$612$	0	0
$613$	32.3584	1.30694	0.653472	−	0.756951i	$-0.273312\pi$
$613$	32.3584	1.30694	0.653472	+	0.756951i	$0.273312\pi$
$614$	−7.58376	−0.306056
$615$	0	0
$616$	19.8920	0.801471
$617$	−15.8795	−0.639285	−0.319643	−	0.947538i	$-0.603563\pi$
$617$	−15.8795	−0.639285	−0.319643	+	0.947538i	$0.603563\pi$
$618$	0	0
$619$	−16.2663	−0.653799	−0.326899	−	0.945059i	$-0.606004\pi$
$619$	−16.2663	−0.653799	−0.326899	+	0.945059i	$0.606004\pi$
$620$	10.9392	0.439329
$621$	0	0
$622$	−8.84232	−0.354545
$623$	−52.9247	−2.12038
$624$	0	0
$625$	−31.2420	−1.24968
$626$	−0.387473	−0.0154865
$627$	0	0
$628$	0.111645	0.00445513
$629$	−28.5459	−1.13820
$630$	0	0
$631$	−38.4888	−1.53221	−0.766107	−	0.642713i	$-0.777809\pi$
$631$	−38.4888	−1.53221	−0.766107	+	0.642713i	$0.777809\pi$
$632$	−5.32771	−0.211925
$633$	0	0
$634$	−28.2206	−1.12078
$635$	−9.21037	−0.365502
$636$	0	0
$637$	62.9553	2.49438
$638$	−20.3814	−0.806907
$639$	0	0
$640$	−2.72222	−0.107605
$641$	2.13866	0.0844718	0.0422359	−	0.999108i	$-0.486552\pi$
$641$	2.13866	0.0844718	0.0422359	+	0.999108i	$0.486552\pi$
$642$	0	0
$643$	−44.7173	−1.76348	−0.881739	−	0.471737i	$-0.843627\pi$
$643$	−44.7173	−1.76348	−0.881739	+	0.471737i	$0.843627\pi$
$644$	−4.49173	−0.176999
$645$	0	0
$646$	−5.28866	−0.208079
$647$	−31.7866	−1.24966	−0.624830	−	0.780761i	$-0.714832\pi$
$647$	−31.7866	−1.24966	−0.624830	+	0.780761i	$0.714832\pi$
$648$	0	0
$649$	36.9910	1.45202
$650$	−11.5176	−0.451759
$651$	0	0
$652$	−20.9496	−0.820449
$653$	−31.2050	−1.22115	−0.610573	−	0.791960i	$-0.709061\pi$
$653$	−31.2050	−1.22115	−0.610573	+	0.791960i	$0.709061\pi$
$654$	0	0
$655$	−40.5583	−1.58474
$656$	8.26989	0.322885
$657$	0	0
$658$	−60.8446	−2.37197
$659$	−28.7819	−1.12118	−0.560592	−	0.828092i	$-0.689427\pi$
$659$	−28.7819	−1.12118	−0.560592	+	0.828092i	$0.689427\pi$
$660$	0	0
$661$	−32.1596	−1.25087	−0.625433	−	0.780278i	$-0.715077\pi$
$661$	−32.1596	−1.25087	−0.625433	+	0.780278i	$0.715077\pi$
$662$	−25.3255	−0.984304
$663$	0	0
$664$	−9.30715	−0.361187
$665$	26.3239	1.02080
$666$	0	0
$667$	4.60224	0.178199
$668$	10.1130	0.391282
$669$	0	0
$670$	8.49867	0.328332
$671$	−36.9307	−1.42569
$672$	0	0
$673$	−10.8417	−0.417919	−0.208959	−	0.977924i	$-0.567008\pi$
$673$	−10.8417	−0.417919	−0.208959	+	0.977924i	$0.567008\pi$
$674$	21.3152	0.821031
$675$	0	0
$676$	9.83086	0.378110
$677$	−24.9514	−0.958960	−0.479480	−	0.877553i	$-0.659175\pi$
$677$	−24.9514	−0.958960	−0.479480	+	0.877553i	$0.659175\pi$
$678$	0	0
$679$	62.8828	2.41322
$680$	−6.68736	−0.256449
$681$	0	0
$682$	17.7962	0.681452
$683$	−40.7823	−1.56049	−0.780245	−	0.625474i	$-0.784906\pi$
$683$	−40.7823	−1.56049	−0.780245	+	0.625474i	$0.784906\pi$
$684$	0	0
$685$	15.9773	0.610461
$686$	−27.7392	−1.05909
$687$	0	0
$688$	−2.37264	−0.0904560
$689$	21.6356	0.824250
$690$	0	0
$691$	−1.81806	−0.0691621	−0.0345811	−	0.999402i	$-0.511010\pi$
$691$	−1.81806	−0.0691621	−0.0345811	+	0.999402i	$0.511010\pi$
$692$	5.83471	0.221802
$693$	0	0
$694$	15.8108	0.600172
$695$	−50.5063	−1.91581
$696$	0	0
$697$	20.3157	0.769512
$698$	−35.1179	−1.32923
$699$	0	0
$700$	10.8272	0.409229
$701$	42.3501	1.59954	0.799771	−	0.600305i	$-0.204954\pi$
$701$	42.3501	1.59954	0.799771	+	0.600305i	$0.204954\pi$
$702$	0	0
$703$	−25.0164	−0.943511
$704$	−4.42858	−0.166909
$705$	0	0
$706$	34.8077	1.31000
$707$	−39.7126	−1.49355
$708$	0	0
$709$	11.8336	0.444420	0.222210	−	0.974999i	$-0.428673\pi$
$709$	11.8336	0.444420	0.222210	+	0.974999i	$0.428673\pi$
$710$	27.3601	1.02681
$711$	0	0
$712$	11.7827	0.441576
$713$	−4.01849	−0.150494
$714$	0	0
$715$	−57.6035	−2.15425
$716$	−10.2464	−0.382924
$717$	0	0
$718$	−11.3635	−0.424082
$719$	28.8878	1.07733	0.538667	−	0.842519i	$-0.318928\pi$
$719$	28.8878	1.07733	0.538667	+	0.842519i	$0.318928\pi$
$720$	0	0
$721$	−1.59607	−0.0594408
$722$	14.3652	0.534619
$723$	0	0
$724$	−15.0091	−0.557810
$725$	−11.0936	−0.412005
$726$	0	0
$727$	−14.8197	−0.549631	−0.274815	−	0.961497i	$-0.588617\pi$
$727$	−14.8197	−0.549631	−0.274815	+	0.961497i	$0.588617\pi$
$728$	−21.4622	−0.795443
$729$	0	0
$730$	−41.5144	−1.53652
$731$	−5.82859	−0.215578
$732$	0	0
$733$	5.03655	0.186029	0.0930146	−	0.995665i	$-0.470350\pi$
$733$	5.03655	0.186029	0.0930146	+	0.995665i	$0.470350\pi$
$734$	6.64275	0.245188
$735$	0	0
$736$	1.00000	0.0368605
$737$	13.8259	0.509283
$738$	0	0
$739$	4.82880	0.177630	0.0888152	−	0.996048i	$-0.471692\pi$
$739$	4.82880	0.177630	0.0888152	+	0.996048i	$0.471692\pi$
$740$	−31.6326	−1.16284
$741$	0	0
$742$	−20.3386	−0.746653
$743$	31.7244	1.16386	0.581928	−	0.813240i	$-0.302298\pi$
$743$	31.7244	1.16386	0.581928	+	0.813240i	$0.302298\pi$
$744$	0	0
$745$	−19.4125	−0.711219
$746$	−11.2504	−0.411908
$747$	0	0
$748$	−10.8792	−0.397783
$749$	20.8452	0.761666
$750$	0	0
$751$	19.0026	0.693413	0.346706	−	0.937974i	$-0.387300\pi$
$751$	19.0026	0.693413	0.346706	+	0.937974i	$0.387300\pi$
$752$	13.5459	0.493969
$753$	0	0
$754$	21.9902	0.800837
$755$	−9.78937	−0.356272
$756$	0	0
$757$	11.1255	0.404364	0.202182	−	0.979348i	$-0.435197\pi$
$757$	11.1255	0.404364	0.202182	+	0.979348i	$0.435197\pi$
$758$	26.4976	0.962437
$759$	0	0
$760$	−5.86052	−0.212583
$761$	35.8580	1.29985	0.649926	−	0.759997i	$-0.274800\pi$
$761$	35.8580	1.29985	0.649926	+	0.759997i	$0.274800\pi$
$762$	0	0
$763$	−39.3514	−1.42462
$764$	−19.1459	−0.692675
$765$	0	0
$766$	18.1665	0.656384
$767$	−39.9109	−1.44110
$768$	0	0
$769$	−47.8655	−1.72608	−0.863038	−	0.505139i	$-0.831441\pi$
$769$	−47.8655	−1.72608	−0.863038	+	0.505139i	$0.831441\pi$
$770$	54.1504	1.95144
$771$	0	0
$772$	9.74849	0.350856
$773$	51.3178	1.84577	0.922886	−	0.385073i	$-0.125824\pi$
$773$	51.3178	1.84577	0.922886	+	0.385073i	$0.125824\pi$
$774$	0	0
$775$	9.68645	0.347948
$776$	−13.9997	−0.502560
$777$	0	0
$778$	−18.4277	−0.660666
$779$	17.8038	0.637888
$780$	0	0
$781$	44.5102	1.59270
$782$	2.45659	0.0878473
$783$	0	0
$784$	13.1756	0.470558
$785$	0.303923	0.0108475
$786$	0	0
$787$	−38.2354	−1.36295	−0.681473	−	0.731844i	$-0.738660\pi$
$787$	−38.2354	−1.36295	−0.681473	+	0.731844i	$0.738660\pi$
$788$	−1.57339	−0.0560497
$789$	0	0
$790$	−14.5032	−0.516000
$791$	36.3873	1.29378
$792$	0	0
$793$	39.8459	1.41497
$794$	−18.0620	−0.640997
$795$	0	0
$796$	19.1919	0.680237
$797$	16.6243	0.588865	0.294432	−	0.955672i	$-0.404869\pi$
$797$	16.6243	0.588865	0.294432	+	0.955672i	$0.404869\pi$
$798$	0	0
$799$	33.2767	1.17725
$800$	−2.41047	−0.0852231
$801$	0	0
$802$	−11.1163	−0.392531
$803$	−67.5369	−2.38333
$804$	0	0
$805$	−12.2275	−0.430961
$806$	−19.2010	−0.676326
$807$	0	0
$808$	8.84128	0.311035
$809$	−11.8419	−0.416338	−0.208169	−	0.978093i	$-0.566750\pi$
$809$	−11.8419	−0.416338	−0.208169	+	0.978093i	$0.566750\pi$
$810$	0	0
$811$	32.8032	1.15188	0.575938	−	0.817494i	$-0.304637\pi$
$811$	32.8032	1.15188	0.575938	+	0.817494i	$0.304637\pi$
$812$	−20.6720	−0.725445
$813$	0	0
$814$	−51.4608	−1.80370
$815$	−57.0294	−1.99765
$816$	0	0
$817$	−5.10793	−0.178704
$818$	−34.5128	−1.20671
$819$	0	0
$820$	22.5125	0.786169
$821$	−8.14213	−0.284162	−0.142081	−	0.989855i	$-0.545379\pi$
$821$	−8.14213	−0.284162	−0.142081	+	0.989855i	$0.545379\pi$
$822$	0	0
$823$	42.0166	1.46460	0.732302	−	0.680980i	$-0.238446\pi$
$823$	42.0166	1.46460	0.732302	+	0.680980i	$0.238446\pi$
$824$	0.355336	0.0123787
$825$	0	0
$826$	37.5184	1.30543
$827$	12.4258	0.432086	0.216043	−	0.976384i	$-0.430685\pi$
$827$	12.4258	0.432086	0.216043	+	0.976384i	$0.430685\pi$
$828$	0	0
$829$	−10.4231	−0.362010	−0.181005	−	0.983482i	$-0.557935\pi$
$829$	−10.4231	−0.362010	−0.181005	+	0.983482i	$0.557935\pi$
$830$	−25.3361	−0.879428
$831$	0	0
$832$	4.77816	0.165653
$833$	32.3670	1.12145
$834$	0	0
$835$	27.5297	0.952705
$836$	−9.53407	−0.329743
$837$	0	0
$838$	12.4583	0.430366
$839$	−0.187333	−0.00646746	−0.00323373	−	0.999995i	$-0.501029\pi$
$839$	−0.187333	−0.00646746	−0.00323373	+	0.999995i	$0.501029\pi$
$840$	0	0
$841$	−7.81943	−0.269635
$842$	27.9788	0.964214
$843$	0	0
$844$	10.4294	0.358995
$845$	26.7617	0.920632
$846$	0	0
$847$	38.6844	1.32921
$848$	4.52801	0.155492
$849$	0	0
$850$	−5.92153	−0.203107
$851$	11.6201	0.398333
$852$	0	0
$853$	−9.25481	−0.316879	−0.158439	−	0.987369i	$-0.550646\pi$
$853$	−9.25481	−0.316879	−0.158439	+	0.987369i	$0.550646\pi$
$854$	−37.4573	−1.28176
$855$	0	0
$856$	−4.64079	−0.158619
$857$	0.548439	0.0187343	0.00936716	−	0.999956i	$-0.497018\pi$
$857$	0.548439	0.0187343	0.00936716	+	0.999956i	$0.497018\pi$
$858$	0	0
$859$	43.7783	1.49370	0.746848	−	0.664995i	$-0.231566\pi$
$859$	43.7783	1.49370	0.746848	+	0.664995i	$0.231566\pi$
$860$	−6.45884	−0.220245
$861$	0	0
$862$	−23.3561	−0.795512
$863$	−30.6567	−1.04356	−0.521782	−	0.853079i	$-0.674733\pi$
$863$	−30.6567	−1.04356	−0.521782	+	0.853079i	$0.674733\pi$
$864$	0	0
$865$	15.8833	0.540050
$866$	12.8182	0.435579
$867$	0	0
$868$	18.0500	0.612655
$869$	−23.5942	−0.800379
$870$	0	0
$871$	−14.9173	−0.505452
$872$	8.76086	0.296680
$873$	0	0
$874$	2.15285	0.0728212
$875$	−31.6634	−1.07042
$876$	0	0
$877$	32.7168	1.10477	0.552383	−	0.833590i	$-0.313718\pi$
$877$	32.7168	1.10477	0.552383	+	0.833590i	$0.313718\pi$
$878$	26.3267	0.888484
$879$	0	0
$880$	−12.0556	−0.406393
$881$	43.3951	1.46202	0.731010	−	0.682367i	$-0.239049\pi$
$881$	43.3951	1.46202	0.731010	+	0.682367i	$0.239049\pi$
$882$	0	0
$883$	−43.8587	−1.47596	−0.737981	−	0.674822i	$-0.764221\pi$
$883$	−43.8587	−1.47596	−0.737981	+	0.674822i	$0.764221\pi$
$884$	11.7380	0.394791
$885$	0	0
$886$	4.65228	0.156296
$887$	−29.5569	−0.992422	−0.496211	−	0.868202i	$-0.665276\pi$
$887$	−29.5569	−0.992422	−0.496211	+	0.868202i	$0.665276\pi$
$888$	0	0
$889$	−15.1973	−0.509702
$890$	32.0751	1.07516
$891$	0	0
$892$	−26.0351	−0.871720
$893$	29.1623	0.975879
$894$	0	0
$895$	−27.8928	−0.932354
$896$	−4.49173	−0.150058
$897$	0	0
$898$	25.2362	0.842142
$899$	−18.4940	−0.616810
$900$	0	0
$901$	11.1234	0.370576
$902$	36.6239	1.21944
$903$	0	0
$904$	−8.10095	−0.269433
$905$	−40.8581	−1.35817
$906$	0	0
$907$	17.9178	0.594951	0.297476	−	0.954729i	$-0.403855\pi$
$907$	17.9178	0.594951	0.297476	+	0.954729i	$0.403855\pi$
$908$	2.50420	0.0831046
$909$	0	0
$910$	−58.4248	−1.93676
$911$	−39.2928	−1.30183	−0.650915	−	0.759151i	$-0.725615\pi$
$911$	−39.2928	−1.30183	−0.650915	+	0.759151i	$0.725615\pi$
$912$	0	0
$913$	−41.2175	−1.36410
$914$	−0.965423	−0.0319333
$915$	0	0
$916$	29.5287	0.975657
$917$	−66.9222	−2.20996
$918$	0	0
$919$	−4.60706	−0.151973	−0.0759865	−	0.997109i	$-0.524211\pi$
$919$	−4.60706	−0.151973	−0.0759865	+	0.997109i	$0.524211\pi$
$920$	2.72222	0.0897489
$921$	0	0
$922$	27.5435	0.907096
$923$	−48.0237	−1.58072
$924$	0	0
$925$	−28.0100	−0.920964
$926$	−25.8842	−0.850609
$927$	0	0
$928$	4.60224	0.151076
$929$	−0.797273	−0.0261577	−0.0130788	−	0.999914i	$-0.504163\pi$
$929$	−0.797273	−0.0261577	−0.0130788	+	0.999914i	$0.504163\pi$
$930$	0	0
$931$	28.3651	0.929629
$932$	29.4447	0.964492
$933$	0	0
$934$	2.82650	0.0924859
$935$	−29.6156	−0.968532
$936$	0	0
$937$	−30.5446	−0.997849	−0.498925	−	0.866645i	$-0.666272\pi$
$937$	−30.5446	−0.997849	−0.498925	+	0.866645i	$0.666272\pi$
$938$	14.0230	0.457868
$939$	0	0
$940$	36.8749	1.20273
$941$	−5.92532	−0.193160	−0.0965800	−	0.995325i	$-0.530790\pi$
$941$	−5.92532	−0.193160	−0.0965800	+	0.995325i	$0.530790\pi$
$942$	0	0
$943$	−8.26989	−0.269305
$944$	−8.35277	−0.271860
$945$	0	0
$946$	−10.5074	−0.341626
$947$	−28.6182	−0.929965	−0.464983	−	0.885320i	$-0.653939\pi$
$947$	−28.6182	−0.929965	−0.464983	+	0.885320i	$0.653939\pi$
$948$	0	0
$949$	72.8681	2.36540
$950$	−5.18938	−0.168366
$951$	0	0
$952$	−11.0343	−0.357624
$953$	−8.53590	−0.276505	−0.138252	−	0.990397i	$-0.544149\pi$
$953$	−8.53590	−0.276505	−0.138252	+	0.990397i	$0.544149\pi$
$954$	0	0
$955$	−52.1193	−1.68654
$956$	1.04205	0.0337023
$957$	0	0
$958$	−3.27455	−0.105796
$959$	26.3629	0.851303
$960$	0	0
$961$	−14.8518	−0.479089
$962$	55.5230	1.79013
$963$	0	0
$964$	−9.54030	−0.307272
$965$	26.5375	0.854273
$966$	0	0
$967$	−49.3881	−1.58821	−0.794107	−	0.607778i	$-0.792061\pi$
$967$	−49.3881	−1.58821	−0.794107	+	0.607778i	$0.792061\pi$
$968$	−8.61236	−0.276812
$969$	0	0
$970$	−38.1102	−1.22365
$971$	−0.999000	−0.0320594	−0.0160297	−	0.999872i	$-0.505103\pi$
$971$	−0.999000	−0.0320594	−0.0160297	+	0.999872i	$0.505103\pi$
$972$	0	0
$973$	−83.3367	−2.67165
$974$	39.8211	1.27595
$975$	0	0
$976$	8.33917	0.266930
$977$	18.2786	0.584783	0.292392	−	0.956299i	$-0.405549\pi$
$977$	18.2786	0.584783	0.292392	+	0.956299i	$0.405549\pi$
$978$	0	0
$979$	52.1807	1.66770
$980$	35.8669	1.14573
$981$	0	0
$982$	−10.6318	−0.339275
$983$	−10.1224	−0.322854	−0.161427	−	0.986885i	$-0.551610\pi$
$983$	−10.1224	−0.322854	−0.161427	+	0.986885i	$0.551610\pi$
$984$	0	0
$985$	−4.28311	−0.136471
$986$	11.3058	0.360050
$987$	0	0
$988$	10.2867	0.327262
$989$	2.37264	0.0754455
$990$	0	0
$991$	−57.3627	−1.82219	−0.911094	−	0.412200i	$-0.864761\pi$
$991$	−57.3627	−1.82219	−0.911094	+	0.412200i	$0.864761\pi$
$992$	−4.01849	−0.127587
$993$	0	0
$994$	45.1448	1.43191
$995$	52.2444	1.65626
$996$	0	0
$997$	17.4855	0.553771	0.276885	−	0.960903i	$-0.410698\pi$
$997$	17.4855	0.553771	0.276885	+	0.960903i	$0.410698\pi$
$998$	−27.3158	−0.864666
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3726.2.a.y.1.6	✓	8
3.2	odd	2		3726.2.a.z.1.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3726.2.a.y.1.6	✓	8	1.1	even	1	trivial
3726.2.a.z.1.3	yes	8	3.2	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:	\( N \)	\(=\)	\( 3726 = 2 \cdot 3^{4} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3726.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.72222 q^{5} +4.49173 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.72222 q^{5} +4.49173 q^{7} -1.00000 q^{8} -2.72222 q^{10} -4.42858 q^{11} +4.77816 q^{13} -4.49173 q^{14} +1.00000 q^{16} +2.45659 q^{17} +2.15285 q^{19} +2.72222 q^{20} +4.42858 q^{22} -1.00000 q^{23} +2.41047 q^{25} -4.77816 q^{26} +4.49173 q^{28} -4.60224 q^{29} +4.01849 q^{31} -1.00000 q^{32} -2.45659 q^{34} +12.2275 q^{35} -11.6201 q^{37} -2.15285 q^{38} -2.72222 q^{40} +8.26989 q^{41} -2.37264 q^{43} -4.42858 q^{44} +1.00000 q^{46} +13.5459 q^{47} +13.1756 q^{49} -2.41047 q^{50} +4.77816 q^{52} +4.52801 q^{53} -12.0556 q^{55} -4.49173 q^{56} +4.60224 q^{58} -8.35277 q^{59} +8.33917 q^{61} -4.01849 q^{62} +1.00000 q^{64} +13.0072 q^{65} -3.12197 q^{67} +2.45659 q^{68} -12.2275 q^{70} -10.0507 q^{71} +15.2502 q^{73} +11.6201 q^{74} +2.15285 q^{76} -19.8920 q^{77} +5.32771 q^{79} +2.72222 q^{80} -8.26989 q^{82} +9.30715 q^{83} +6.68736 q^{85} +2.37264 q^{86} +4.42858 q^{88} -11.7827 q^{89} +21.4622 q^{91} -1.00000 q^{92} -13.5459 q^{94} +5.86052 q^{95} +13.9997 q^{97} -13.1756 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 8 q^{4} + 6 q^{7} - 8 q^{8}+O(q^{10}) \) 8 * q - 8 * q^2 + 8 * q^4 + 6 * q^7 - 8 * q^8	\( 8 q - 8 q^{2} + 8 q^{4} + 6 q^{7} - 8 q^{8} + 6 q^{13} - 6 q^{14} + 8 q^{16} + 10 q^{17} + 12 q^{19} - 8 q^{23} + 28 q^{25} - 6 q^{26} + 6 q^{28} - 6 q^{29} + 16 q^{31} - 8 q^{32} - 10 q^{34} - 18 q^{35} + 4 q^{37} - 12 q^{38} + 4 q^{41} + 6 q^{43} + 8 q^{46} - 14 q^{47} + 28 q^{49} - 28 q^{50} + 6 q^{52} - 20 q^{53} + 16 q^{55} - 6 q^{56} + 6 q^{58} - 6 q^{59} + 12 q^{61} - 16 q^{62} + 8 q^{64} + 10 q^{65} + 10 q^{67} + 10 q^{68} + 18 q^{70} + 18 q^{71} + 44 q^{73} - 4 q^{74} + 12 q^{76} - 28 q^{77} + 22 q^{79} - 4 q^{82} - 4 q^{83} + 22 q^{85} - 6 q^{86} + 18 q^{89} + 14 q^{91} - 8 q^{92} + 14 q^{94} + 20 q^{95} + 36 q^{97} - 28 q^{98}+O(q^{100}) \) 8 * q - 8 * q^2 + 8 * q^4 + 6 * q^7 - 8 * q^8 + 6 * q^13 - 6 * q^14 + 8 * q^16 + 10 * q^17 + 12 * q^19 - 8 * q^23 + 28 * q^25 - 6 * q^26 + 6 * q^28 - 6 * q^29 + 16 * q^31 - 8 * q^32 - 10 * q^34 - 18 * q^35 + 4 * q^37 - 12 * q^38 + 4 * q^41 + 6 * q^43 + 8 * q^46 - 14 * q^47 + 28 * q^49 - 28 * q^50 + 6 * q^52 - 20 * q^53 + 16 * q^55 - 6 * q^56 + 6 * q^58 - 6 * q^59 + 12 * q^61 - 16 * q^62 + 8 * q^64 + 10 * q^65 + 10 * q^67 + 10 * q^68 + 18 * q^70 + 18 * q^71 + 44 * q^73 - 4 * q^74 + 12 * q^76 - 28 * q^77 + 22 * q^79 - 4 * q^82 - 4 * q^83 + 22 * q^85 - 6 * q^86 + 18 * q^89 + 14 * q^91 - 8 * q^92 + 14 * q^94 + 20 * q^95 + 36 * q^97 - 28 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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