LMFDB - Embedded newform 3762.2.a.bg.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3762,2,Mod(1,3762)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3762, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3762.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3762 = 2 \cdot 3^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3762.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:	$30.0397212404$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.621.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 418)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.14510$ of defining polynomial
Character	$\chi$	$=$	3762.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.60147 q^{5} +2.89167 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.60147 q^{5} +2.89167 q^{7} +1.00000 q^{8} +1.60147 q^{10} +1.00000 q^{11} +4.89167 q^{13} +2.89167 q^{14} +1.00000 q^{16} +5.74657 q^{17} -1.00000 q^{19} +1.60147 q^{20} +1.00000 q^{22} +7.74657 q^{23} -2.43531 q^{25} +4.89167 q^{26} +2.89167 q^{28} -5.34803 q^{29} -7.43531 q^{31} +1.00000 q^{32} +5.74657 q^{34} +4.63091 q^{35} -7.49314 q^{37} -1.00000 q^{38} +1.60147 q^{40} +2.05783 q^{41} -10.6887 q^{43} +1.00000 q^{44} +7.74657 q^{46} +7.08727 q^{47} +1.36176 q^{49} -2.43531 q^{50} +4.89167 q^{52} -8.32698 q^{53} +1.60147 q^{55} +2.89167 q^{56} -5.34803 q^{58} +2.54364 q^{59} +13.4931 q^{61} -7.43531 q^{62} +1.00000 q^{64} +7.83384 q^{65} -10.3848 q^{67} +5.74657 q^{68} +4.63091 q^{70} -14.9789 q^{71} -3.74657 q^{73} -7.49314 q^{74} -1.00000 q^{76} +2.89167 q^{77} -8.69607 q^{79} +1.60147 q^{80} +2.05783 q^{82} +6.10833 q^{83} +9.20293 q^{85} -10.6887 q^{86} +1.00000 q^{88} -3.20293 q^{89} +14.1451 q^{91} +7.74657 q^{92} +7.08727 q^{94} -1.60147 q^{95} -6.98627 q^{97} +1.36176 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 6 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^5 - 6 * q^7 + 3 * q^8	$ 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 6 q^{7} + 3 q^{8} + 3 q^{10} + 3 q^{11} - 6 q^{14} + 3 q^{16} + 9 q^{17} - 3 q^{19} + 3 q^{20} + 3 q^{22} + 15 q^{23} + 12 q^{25} - 6 q^{28} - 6 q^{29} - 3 q^{31} + 3 q^{32} + 9 q^{34} - 6 q^{37} - 3 q^{38} + 3 q^{40} + 9 q^{41} - 21 q^{43} + 3 q^{44} + 15 q^{46} + 12 q^{47} + 27 q^{49} + 12 q^{50} + 9 q^{53} + 3 q^{55} - 6 q^{56} - 6 q^{58} + 3 q^{59} + 24 q^{61} - 3 q^{62} + 3 q^{64} + 6 q^{65} + 9 q^{68} - 21 q^{71} - 3 q^{73} - 6 q^{74} - 3 q^{76} - 6 q^{77} - 6 q^{79} + 3 q^{80} + 9 q^{82} + 33 q^{83} + 24 q^{85} - 21 q^{86} + 3 q^{88} - 6 q^{89} + 36 q^{91} + 15 q^{92} + 12 q^{94} - 3 q^{95} + 12 q^{97} + 27 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^5 - 6 * q^7 + 3 * q^8 + 3 * q^10 + 3 * q^11 - 6 * q^14 + 3 * q^16 + 9 * q^17 - 3 * q^19 + 3 * q^20 + 3 * q^22 + 15 * q^23 + 12 * q^25 - 6 * q^28 - 6 * q^29 - 3 * q^31 + 3 * q^32 + 9 * q^34 - 6 * q^37 - 3 * q^38 + 3 * q^40 + 9 * q^41 - 21 * q^43 + 3 * q^44 + 15 * q^46 + 12 * q^47 + 27 * q^49 + 12 * q^50 + 9 * q^53 + 3 * q^55 - 6 * q^56 - 6 * q^58 + 3 * q^59 + 24 * q^61 - 3 * q^62 + 3 * q^64 + 6 * q^65 + 9 * q^68 - 21 * q^71 - 3 * q^73 - 6 * q^74 - 3 * q^76 - 6 * q^77 - 6 * q^79 + 3 * q^80 + 9 * q^82 + 33 * q^83 + 24 * q^85 - 21 * q^86 + 3 * q^88 - 6 * q^89 + 36 * q^91 + 15 * q^92 + 12 * q^94 - 3 * q^95 + 12 * q^97 + 27 * 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
$5$	1.60147	0.716197	0.358099	−	0.933684i	$-0.383425\pi$
$5$	1.60147	0.716197	0.358099	+	0.933684i	$0.383425\pi$
$6$	0	0
$7$	2.89167	1.09295	0.546474	−	0.837476i	$-0.315970\pi$
$7$	2.89167	1.09295	0.546474	+	0.837476i	$0.315970\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.60147	0.506428
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	4.89167	1.35671	0.678353	−	0.734736i	$-0.262694\pi$
$13$	4.89167	1.35671	0.678353	+	0.734736i	$0.262694\pi$
$14$	2.89167	0.772832
$15$	0	0
$16$	1.00000	0.250000
$17$	5.74657	1.39375	0.696874	−	0.717194i	$-0.254574\pi$
$17$	5.74657	1.39375	0.696874	+	0.717194i	$0.254574\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	1.60147	0.358099
$21$	0	0
$22$	1.00000	0.213201
$23$	7.74657	1.61527	0.807636	−	0.589682i	$-0.200747\pi$
$23$	7.74657	1.61527	0.807636	+	0.589682i	$0.200747\pi$
$24$	0	0
$25$	−2.43531	−0.487062
$26$	4.89167	0.959336
$27$	0	0
$28$	2.89167	0.546474
$29$	−5.34803	−0.993105	−0.496552	−	0.868007i	$-0.665401\pi$
$29$	−5.34803	−0.993105	−0.496552	+	0.868007i	$0.665401\pi$
$30$	0	0
$31$	−7.43531	−1.33542	−0.667710	−	0.744421i	$-0.732726\pi$
$31$	−7.43531	−1.33542	−0.667710	+	0.744421i	$0.732726\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	5.74657	0.985528
$35$	4.63091	0.782767
$36$	0	0
$37$	−7.49314	−1.23186	−0.615932	−	0.787799i	$-0.711220\pi$
$37$	−7.49314	−1.23186	−0.615932	+	0.787799i	$0.711220\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	1.60147	0.253214
$41$	2.05783	0.321379	0.160689	−	0.987005i	$-0.448628\pi$
$41$	2.05783	0.321379	0.160689	+	0.987005i	$0.448628\pi$
$42$	0	0
$43$	−10.6887	−1.63002	−0.815009	−	0.579449i	$-0.803268\pi$
$43$	−10.6887	−1.63002	−0.815009	+	0.579449i	$0.803268\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	7.74657	1.14217
$47$	7.08727	1.03379	0.516893	−	0.856050i	$-0.327089\pi$
$47$	7.08727	1.03379	0.516893	+	0.856050i	$0.327089\pi$
$48$	0	0
$49$	1.36176	0.194537
$50$	−2.43531	−0.344405
$51$	0	0
$52$	4.89167	0.678353
$53$	−8.32698	−1.14380	−0.571899	−	0.820324i	$-0.693793\pi$
$53$	−8.32698	−1.14380	−0.571899	+	0.820324i	$0.693793\pi$
$54$	0	0
$55$	1.60147	0.215942
$56$	2.89167	0.386416
$57$	0	0
$58$	−5.34803	−0.702231
$59$	2.54364	0.331153	0.165577	−	0.986197i	$-0.447051\pi$
$59$	2.54364	0.331153	0.165577	+	0.986197i	$0.447051\pi$
$60$	0	0
$61$	13.4931	1.72762	0.863810	−	0.503818i	$-0.168072\pi$
$61$	13.4931	1.72762	0.863810	+	0.503818i	$0.168072\pi$
$62$	−7.43531	−0.944285
$63$	0	0
$64$	1.00000	0.125000
$65$	7.83384	0.971669
$66$	0	0
$67$	−10.3848	−1.26871	−0.634353	−	0.773043i	$-0.718733\pi$
$67$	−10.3848	−1.26871	−0.634353	+	0.773043i	$0.718733\pi$
$68$	5.74657	0.696874
$69$	0	0
$70$	4.63091	0.553500
$71$	−14.9789	−1.77767	−0.888837	−	0.458224i	$-0.848486\pi$
$71$	−14.9789	−1.77767	−0.888837	+	0.458224i	$0.848486\pi$
$72$	0	0
$73$	−3.74657	−0.438503	−0.219251	−	0.975668i	$-0.570361\pi$
$73$	−3.74657	−0.438503	−0.219251	+	0.975668i	$0.570361\pi$
$74$	−7.49314	−0.871059
$75$	0	0
$76$	−1.00000	−0.114708
$77$	2.89167	0.329536
$78$	0	0
$79$	−8.69607	−0.978384	−0.489192	−	0.872176i	$-0.662708\pi$
$79$	−8.69607	−0.978384	−0.489192	+	0.872176i	$0.662708\pi$
$80$	1.60147	0.179049
$81$	0	0
$82$	2.05783	0.227249
$83$	6.10833	0.670476	0.335238	−	0.942133i	$-0.391183\pi$
$83$	6.10833	0.670476	0.335238	+	0.942133i	$0.391183\pi$
$84$	0	0
$85$	9.20293	0.998198
$86$	−10.6887	−1.15260
$87$	0	0
$88$	1.00000	0.106600
$89$	−3.20293	−0.339510	−0.169755	−	0.985486i	$-0.554298\pi$
$89$	−3.20293	−0.339510	−0.169755	+	0.985486i	$0.554298\pi$
$90$	0	0
$91$	14.1451	1.48281
$92$	7.74657	0.807636
$93$	0	0
$94$	7.08727	0.730997
$95$	−1.60147	−0.164307
$96$	0	0
$97$	−6.98627	−0.709349	−0.354674	−	0.934990i	$-0.615408\pi$
$97$	−6.98627	−0.709349	−0.354674	+	0.934990i	$0.615408\pi$
$98$	1.36176	0.137559
$99$	0	0
$100$	−2.43531	−0.243531
$101$	1.20293	0.119696	0.0598481	−	0.998207i	$-0.480938\pi$
$101$	1.20293	0.119696	0.0598481	+	0.998207i	$0.480938\pi$
$102$	0	0
$103$	16.8779	1.66303	0.831517	−	0.555500i	$-0.187473\pi$
$103$	16.8779	1.66303	0.831517	+	0.555500i	$0.187473\pi$
$104$	4.89167	0.479668
$105$	0	0
$106$	−8.32698	−0.808788
$107$	8.03677	0.776944	0.388472	−	0.921460i	$-0.373003\pi$
$107$	8.03677	0.776944	0.388472	+	0.921460i	$0.373003\pi$
$108$	0	0
$109$	11.5299	1.10437	0.552183	−	0.833723i	$-0.313795\pi$
$109$	11.5299	1.10437	0.552183	+	0.833723i	$0.313795\pi$
$110$	1.60147	0.152694
$111$	0	0
$112$	2.89167	0.273237
$113$	8.00000	0.752577	0.376288	−	0.926503i	$-0.377200\pi$
$113$	8.00000	0.752577	0.376288	+	0.926503i	$0.377200\pi$
$114$	0	0
$115$	12.4059	1.15685
$116$	−5.34803	−0.496552
$117$	0	0
$118$	2.54364	0.234161
$119$	16.6172	1.52329
$120$	0	0
$121$	1.00000	0.0909091
$122$	13.4931	1.22161
$123$	0	0
$124$	−7.43531	−0.667710
$125$	−11.9074	−1.06503
$126$	0	0
$127$	−21.2765	−1.88798	−0.943991	−	0.329971i	$-0.892961\pi$
$127$	−21.2765	−1.88798	−0.943991	+	0.329971i	$0.892961\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	7.83384	0.687073
$131$	3.43531	0.300144	0.150072	−	0.988675i	$-0.452049\pi$
$131$	3.43531	0.300144	0.150072	+	0.988675i	$0.452049\pi$
$132$	0	0
$133$	−2.89167	−0.250740
$134$	−10.3848	−0.897111
$135$	0	0
$136$	5.74657	0.492764
$137$	−13.5877	−1.16088	−0.580439	−	0.814303i	$-0.697119\pi$
$137$	−13.5877	−1.16088	−0.580439	+	0.814303i	$0.697119\pi$
$138$	0	0
$139$	3.19560	0.271048	0.135524	−	0.990774i	$-0.456728\pi$
$139$	3.19560	0.271048	0.135524	+	0.990774i	$0.456728\pi$
$140$	4.63091	0.391383
$141$	0	0
$142$	−14.9789	−1.25701
$143$	4.89167	0.409062
$144$	0	0
$145$	−8.56469	−0.711259
$146$	−3.74657	−0.310068
$147$	0	0
$148$	−7.49314	−0.615932
$149$	12.0735	0.989104	0.494552	−	0.869148i	$-0.335332\pi$
$149$	12.0735	0.989104	0.494552	+	0.869148i	$0.335332\pi$
$150$	0	0
$151$	−4.91273	−0.399792	−0.199896	−	0.979817i	$-0.564060\pi$
$151$	−4.91273	−0.399792	−0.199896	+	0.979817i	$0.564060\pi$
$152$	−1.00000	−0.0811107
$153$	0	0
$154$	2.89167	0.233017
$155$	−11.9074	−0.956425
$156$	0	0
$157$	8.00733	0.639054	0.319527	−	0.947577i	$-0.396476\pi$
$157$	8.00733	0.639054	0.319527	+	0.947577i	$0.396476\pi$
$158$	−8.69607	−0.691822
$159$	0	0
$160$	1.60147	0.126607
$161$	22.4005	1.76541
$162$	0	0
$163$	1.88434	0.147593	0.0737966	−	0.997273i	$-0.476488\pi$
$163$	1.88434	0.147593	0.0737966	+	0.997273i	$0.476488\pi$
$164$	2.05783	0.160689
$165$	0	0
$166$	6.10833	0.474098
$167$	4.98627	0.385849	0.192925	−	0.981214i	$-0.438203\pi$
$167$	4.98627	0.385849	0.192925	+	0.981214i	$0.438203\pi$
$168$	0	0
$169$	10.9284	0.840650
$170$	9.20293	0.705833
$171$	0	0
$172$	−10.6887	−0.815009
$173$	−6.51419	−0.495265	−0.247632	−	0.968854i	$-0.579653\pi$
$173$	−6.51419	−0.495265	−0.247632	+	0.968854i	$0.579653\pi$
$174$	0	0
$175$	−7.04211	−0.532333
$176$	1.00000	0.0753778
$177$	0	0
$178$	−3.20293	−0.240070
$179$	5.55096	0.414899	0.207449	−	0.978246i	$-0.433484\pi$
$179$	5.55096	0.414899	0.207449	+	0.978246i	$0.433484\pi$
$180$	0	0
$181$	1.88434	0.140062	0.0700311	−	0.997545i	$-0.477690\pi$
$181$	1.88434	0.140062	0.0700311	+	0.997545i	$0.477690\pi$
$182$	14.1451	1.04850
$183$	0	0
$184$	7.74657	0.571085
$185$	−12.0000	−0.882258
$186$	0	0
$187$	5.74657	0.420231
$188$	7.08727	0.516893
$189$	0	0
$190$	−1.60147	−0.116183
$191$	−2.44264	−0.176743	−0.0883715	−	0.996088i	$-0.528166\pi$
$191$	−2.44264	−0.176743	−0.0883715	+	0.996088i	$0.528166\pi$
$192$	0	0
$193$	−9.67501	−0.696423	−0.348211	−	0.937416i	$-0.613211\pi$
$193$	−9.67501	−0.696423	−0.348211	+	0.937416i	$0.613211\pi$
$194$	−6.98627	−0.501585
$195$	0	0
$196$	1.36176	0.0972686
$197$	17.4931	1.24633	0.623167	−	0.782089i	$-0.285846\pi$
$197$	17.4931	1.24633	0.623167	+	0.782089i	$0.285846\pi$
$198$	0	0
$199$	−25.2397	−1.78920	−0.894598	−	0.446873i	$-0.852538\pi$
$199$	−25.2397	−1.78920	−0.894598	+	0.446873i	$0.852538\pi$
$200$	−2.43531	−0.172202
$201$	0	0
$202$	1.20293	0.0846379
$203$	−15.4648	−1.08541
$204$	0	0
$205$	3.29554	0.230171
$206$	16.8779	1.17594
$207$	0	0
$208$	4.89167	0.339176
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	−14.8338	−1.02120	−0.510602	−	0.859817i	$-0.670577\pi$
$211$	−14.8338	−1.02120	−0.510602	+	0.859817i	$0.670577\pi$
$212$	−8.32698	−0.571899
$213$	0	0
$214$	8.03677	0.549383
$215$	−17.1176	−1.16741
$216$	0	0
$217$	−21.5005	−1.45955
$218$	11.5299	0.780904
$219$	0	0
$220$	1.60147	0.107971
$221$	28.1103	1.89090
$222$	0	0
$223$	11.4931	0.769637	0.384819	−	0.922992i	$-0.374264\pi$
$223$	11.4931	0.769637	0.384819	+	0.922992i	$0.374264\pi$
$224$	2.89167	0.193208
$225$	0	0
$226$	8.00000	0.532152
$227$	−8.73284	−0.579619	−0.289810	−	0.957084i	$-0.593592\pi$
$227$	−8.73284	−0.579619	−0.289810	+	0.957084i	$0.593592\pi$
$228$	0	0
$229$	7.84117	0.518159	0.259080	−	0.965856i	$-0.416581\pi$
$229$	7.84117	0.518159	0.259080	+	0.965856i	$0.416581\pi$
$230$	12.4059	0.818018
$231$	0	0
$232$	−5.34803	−0.351116
$233$	12.5069	0.819352	0.409676	−	0.912231i	$-0.365642\pi$
$233$	12.5069	0.819352	0.409676	+	0.912231i	$0.365642\pi$
$234$	0	0
$235$	11.3500	0.740394
$236$	2.54364	0.165577
$237$	0	0
$238$	16.6172	1.07713
$239$	−24.6245	−1.59283	−0.796414	−	0.604752i	$-0.793272\pi$
$239$	−24.6245	−1.59283	−0.796414	+	0.604752i	$0.793272\pi$
$240$	0	0
$241$	16.0809	1.03586	0.517930	−	0.855423i	$-0.326703\pi$
$241$	16.0809	1.03586	0.517930	+	0.855423i	$0.326703\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	13.4931	0.863810
$245$	2.18081	0.139327
$246$	0	0
$247$	−4.89167	−0.311250
$248$	−7.43531	−0.472143
$249$	0	0
$250$	−11.9074	−0.753089
$251$	−13.0873	−0.826061	−0.413031	−	0.910717i	$-0.635530\pi$
$251$	−13.0873	−0.826061	−0.413031	+	0.910717i	$0.635530\pi$
$252$	0	0
$253$	7.74657	0.487023
$254$	−21.2765	−1.33500
$255$	0	0
$256$	1.00000	0.0625000
$257$	−0.723522	−0.0451320	−0.0225660	−	0.999745i	$-0.507184\pi$
$257$	−0.723522	−0.0451320	−0.0225660	+	0.999745i	$0.507184\pi$
$258$	0	0
$259$	−21.6677	−1.34636
$260$	7.83384	0.485834
$261$	0	0
$262$	3.43531	0.212234
$263$	−13.7760	−0.849465	−0.424733	−	0.905319i	$-0.639632\pi$
$263$	−13.7760	−0.849465	−0.424733	+	0.905319i	$0.639632\pi$
$264$	0	0
$265$	−13.3354	−0.819185
$266$	−2.89167	−0.177300
$267$	0	0
$268$	−10.3848	−0.634353
$269$	19.9579	1.21685	0.608427	−	0.793610i	$-0.291801\pi$
$269$	19.9579	1.21685	0.608427	+	0.793610i	$0.291801\pi$
$270$	0	0
$271$	28.7486	1.74635	0.873175	−	0.487406i	$-0.162057\pi$
$271$	28.7486	1.74635	0.873175	+	0.487406i	$0.162057\pi$
$272$	5.74657	0.348437
$273$	0	0
$274$	−13.5877	−0.820865
$275$	−2.43531	−0.146855
$276$	0	0
$277$	3.08727	0.185496	0.0927482	−	0.995690i	$-0.470435\pi$
$277$	3.08727	0.185496	0.0927482	+	0.995690i	$0.470435\pi$
$278$	3.19560	0.191660
$279$	0	0
$280$	4.63091	0.276750
$281$	−9.09460	−0.542538	−0.271269	−	0.962504i	$-0.587443\pi$
$281$	−9.09460	−0.542538	−0.271269	+	0.962504i	$0.587443\pi$
$282$	0	0
$283$	13.1451	0.781395	0.390698	−	0.920519i	$-0.372234\pi$
$283$	13.1451	0.781395	0.390698	+	0.920519i	$0.372234\pi$
$284$	−14.9789	−0.888837
$285$	0	0
$286$	4.89167	0.289251
$287$	5.95056	0.351251
$288$	0	0
$289$	16.0230	0.942532
$290$	−8.56469	−0.502936
$291$	0	0
$292$	−3.74657	−0.219251
$293$	−3.10833	−0.181591	−0.0907953	−	0.995870i	$-0.528941\pi$
$293$	−3.10833	−0.181591	−0.0907953	+	0.995870i	$0.528941\pi$
$294$	0	0
$295$	4.07355	0.237171
$296$	−7.49314	−0.435530
$297$	0	0
$298$	12.0735	0.699402
$299$	37.8937	2.19145
$300$	0	0
$301$	−30.9083	−1.78153
$302$	−4.91273	−0.282696
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	21.6088	1.23732
$306$	0	0
$307$	2.18920	0.124944	0.0624722	−	0.998047i	$-0.480102\pi$
$307$	2.18920	0.124944	0.0624722	+	0.998047i	$0.480102\pi$
$308$	2.89167	0.164768
$309$	0	0
$310$	−11.9074	−0.676294
$311$	−3.92112	−0.222346	−0.111173	−	0.993801i	$-0.535461\pi$
$311$	−3.92112	−0.222346	−0.111173	+	0.993801i	$0.535461\pi$
$312$	0	0
$313$	−21.7118	−1.22722	−0.613611	−	0.789608i	$-0.710284\pi$
$313$	−21.7118	−1.22722	−0.613611	+	0.789608i	$0.710284\pi$
$314$	8.00733	0.451880
$315$	0	0
$316$	−8.69607	−0.489192
$317$	−6.90739	−0.387958	−0.193979	−	0.981006i	$-0.562139\pi$
$317$	−6.90739	−0.387958	−0.193979	+	0.981006i	$0.562139\pi$
$318$	0	0
$319$	−5.34803	−0.299432
$320$	1.60147	0.0895246
$321$	0	0
$322$	22.4005	1.24833
$323$	−5.74657	−0.319748
$324$	0	0
$325$	−11.9127	−0.660799
$326$	1.88434	0.104364
$327$	0	0
$328$	2.05783	0.113625
$329$	20.4941	1.12987
$330$	0	0
$331$	32.1030	1.76454	0.882270	−	0.470744i	$-0.156014\pi$
$331$	32.1030	1.76454	0.882270	+	0.470744i	$0.156014\pi$
$332$	6.10833	0.335238
$333$	0	0
$334$	4.98627	0.272837
$335$	−16.6309	−0.908644
$336$	0	0
$337$	−1.55096	−0.0844864	−0.0422432	−	0.999107i	$-0.513450\pi$
$337$	−1.55096	−0.0844864	−0.0422432	+	0.999107i	$0.513450\pi$
$338$	10.9284	0.594429
$339$	0	0
$340$	9.20293	0.499099
$341$	−7.43531	−0.402645
$342$	0	0
$343$	−16.3039	−0.880330
$344$	−10.6887	−0.576298
$345$	0	0
$346$	−6.51419	−0.350205
$347$	0.506864	0.0272099	0.0136049	−	0.999907i	$-0.495669\pi$
$347$	0.506864	0.0272099	0.0136049	+	0.999907i	$0.495669\pi$
$348$	0	0
$349$	10.8117	0.578738	0.289369	−	0.957218i	$-0.406554\pi$
$349$	10.8117	0.578738	0.289369	+	0.957218i	$0.406554\pi$
$350$	−7.04211	−0.376417
$351$	0	0
$352$	1.00000	0.0533002
$353$	−27.0966	−1.44221	−0.721103	−	0.692828i	$-0.756365\pi$
$353$	−27.0966	−1.44221	−0.721103	+	0.692828i	$0.756365\pi$
$354$	0	0
$355$	−23.9883	−1.27316
$356$	−3.20293	−0.169755
$357$	0	0
$358$	5.55096	0.293378
$359$	20.7486	1.09507	0.547534	−	0.836784i	$-0.315567\pi$
$359$	20.7486	1.09507	0.547534	+	0.836784i	$0.315567\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	1.88434	0.0990389
$363$	0	0
$364$	14.1451	0.741405
$365$	−6.00000	−0.314054
$366$	0	0
$367$	25.6402	1.33841	0.669205	−	0.743078i	$-0.266635\pi$
$367$	25.6402	1.33841	0.669205	+	0.743078i	$0.266635\pi$
$368$	7.74657	0.403818
$369$	0	0
$370$	−12.0000	−0.623850
$371$	−24.0789	−1.25011
$372$	0	0
$373$	25.2049	1.30506	0.652531	−	0.757762i	$-0.273707\pi$
$373$	25.2049	1.30506	0.652531	+	0.757762i	$0.273707\pi$
$374$	5.74657	0.297148
$375$	0	0
$376$	7.08727	0.365498
$377$	−26.1608	−1.34735
$378$	0	0
$379$	3.47208	0.178349	0.0891744	−	0.996016i	$-0.471577\pi$
$379$	3.47208	0.178349	0.0891744	+	0.996016i	$0.471577\pi$
$380$	−1.60147	−0.0821534
$381$	0	0
$382$	−2.44264	−0.124976
$383$	−23.1176	−1.18126	−0.590628	−	0.806944i	$-0.701120\pi$
$383$	−23.1176	−1.18126	−0.590628	+	0.806944i	$0.701120\pi$
$384$	0	0
$385$	4.63091	0.236013
$386$	−9.67501	−0.492445
$387$	0	0
$388$	−6.98627	−0.354674
$389$	26.7623	1.35690	0.678451	−	0.734646i	$-0.262652\pi$
$389$	26.7623	1.35690	0.678451	+	0.734646i	$0.262652\pi$
$390$	0	0
$391$	44.5162	2.25128
$392$	1.36176	0.0687793
$393$	0	0
$394$	17.4931	0.881291
$395$	−13.9265	−0.700716
$396$	0	0
$397$	−31.1030	−1.56101	−0.780507	−	0.625147i	$-0.785039\pi$
$397$	−31.1030	−1.56101	−0.780507	+	0.625147i	$0.785039\pi$
$398$	−25.2397	−1.26515
$399$	0	0
$400$	−2.43531	−0.121765
$401$	−29.3500	−1.46567	−0.732835	−	0.680406i	$-0.761803\pi$
$401$	−29.3500	−1.46567	−0.732835	+	0.680406i	$0.761803\pi$
$402$	0	0
$403$	−36.3711	−1.81177
$404$	1.20293	0.0598481
$405$	0	0
$406$	−15.4648	−0.767503
$407$	−7.49314	−0.371421
$408$	0	0
$409$	8.58774	0.424636	0.212318	−	0.977201i	$-0.431899\pi$
$409$	8.58774	0.424636	0.212318	+	0.977201i	$0.431899\pi$
$410$	3.29554	0.162755
$411$	0	0
$412$	16.8779	0.831517
$413$	7.35536	0.361934
$414$	0	0
$415$	9.78228	0.480193
$416$	4.89167	0.239834
$417$	0	0
$418$	−1.00000	−0.0489116
$419$	−7.59414	−0.370998	−0.185499	−	0.982644i	$-0.559390\pi$
$419$	−7.59414	−0.370998	−0.185499	+	0.982644i	$0.559390\pi$
$420$	0	0
$421$	8.76030	0.426951	0.213475	−	0.976948i	$-0.431522\pi$
$421$	8.76030	0.426951	0.213475	+	0.976948i	$0.431522\pi$
$422$	−14.8338	−0.722100
$423$	0	0
$424$	−8.32698	−0.404394
$425$	−13.9947	−0.678841
$426$	0	0
$427$	39.0177	1.88820
$428$	8.03677	0.388472
$429$	0	0
$430$	−17.1176	−0.825486
$431$	33.6677	1.62172	0.810858	−	0.585243i	$-0.199001\pi$
$431$	33.6677	1.62172	0.810858	+	0.585243i	$0.199001\pi$
$432$	0	0
$433$	30.5383	1.46758	0.733789	−	0.679378i	$-0.237750\pi$
$433$	30.5383	1.46758	0.733789	+	0.679378i	$0.237750\pi$
$434$	−21.5005	−1.03206
$435$	0	0
$436$	11.5299	0.552183
$437$	−7.74657	−0.370569
$438$	0	0
$439$	−12.5069	−0.596920	−0.298460	−	0.954422i	$-0.596473\pi$
$439$	−12.5069	−0.596920	−0.298460	+	0.954422i	$0.596473\pi$
$440$	1.60147	0.0763469
$441$	0	0
$442$	28.1103	1.33707
$443$	17.6677	0.839417	0.419709	−	0.907659i	$-0.362132\pi$
$443$	17.6677	0.839417	0.419709	+	0.907659i	$0.362132\pi$
$444$	0	0
$445$	−5.12938	−0.243156
$446$	11.4931	0.544216
$447$	0	0
$448$	2.89167	0.136619
$449$	17.7098	0.835777	0.417888	−	0.908498i	$-0.362770\pi$
$449$	17.7098	0.835777	0.417888	+	0.908498i	$0.362770\pi$
$450$	0	0
$451$	2.05783	0.0968994
$452$	8.00000	0.376288
$453$	0	0
$454$	−8.73284	−0.409853
$455$	22.6529	1.06198
$456$	0	0
$457$	−24.8064	−1.16039	−0.580197	−	0.814476i	$-0.697024\pi$
$457$	−24.8064	−1.16039	−0.580197	+	0.814476i	$0.697024\pi$
$458$	7.84117	0.366394
$459$	0	0
$460$	12.4059	0.578426
$461$	2.98627	0.139085	0.0695423	−	0.997579i	$-0.477846\pi$
$461$	2.98627	0.139085	0.0695423	+	0.997579i	$0.477846\pi$
$462$	0	0
$463$	−39.5667	−1.83882	−0.919410	−	0.393301i	$-0.871333\pi$
$463$	−39.5667	−1.83882	−0.919410	+	0.393301i	$0.871333\pi$
$464$	−5.34803	−0.248276
$465$	0	0
$466$	12.5069	0.579369
$467$	−4.46475	−0.206604	−0.103302	−	0.994650i	$-0.532941\pi$
$467$	−4.46475	−0.206604	−0.103302	+	0.994650i	$0.532941\pi$
$468$	0	0
$469$	−30.0294	−1.38663
$470$	11.3500	0.523538
$471$	0	0
$472$	2.54364	0.117080
$473$	−10.6887	−0.491469
$474$	0	0
$475$	2.43531	0.111740
$476$	16.6172	0.761647
$477$	0	0
$478$	−24.6245	−1.12630
$479$	20.8863	0.954321	0.477161	−	0.878816i	$-0.341666\pi$
$479$	20.8863	0.954321	0.477161	+	0.878816i	$0.341666\pi$
$480$	0	0
$481$	−36.6540	−1.67128
$482$	16.0809	0.732464
$483$	0	0
$484$	1.00000	0.0454545
$485$	−11.1883	−0.508033
$486$	0	0
$487$	15.2838	0.692575	0.346288	−	0.938128i	$-0.387442\pi$
$487$	15.2838	0.692575	0.346288	+	0.938128i	$0.387442\pi$
$488$	13.4931	0.610806
$489$	0	0
$490$	2.18081	0.0985191
$491$	−3.31965	−0.149814	−0.0749069	−	0.997191i	$-0.523866\pi$
$491$	−3.31965	−0.149814	−0.0749069	+	0.997191i	$0.523866\pi$
$492$	0	0
$493$	−30.7328	−1.38414
$494$	−4.89167	−0.220087
$495$	0	0
$496$	−7.43531	−0.333855
$497$	−43.3142	−1.94291
$498$	0	0
$499$	−2.07355	−0.0928247	−0.0464124	−	0.998922i	$-0.514779\pi$
$499$	−2.07355	−0.0928247	−0.0464124	+	0.998922i	$0.514779\pi$
$500$	−11.9074	−0.532515
$501$	0	0
$502$	−13.0873	−0.584114
$503$	1.11672	0.0497921	0.0248960	−	0.999690i	$-0.492075\pi$
$503$	1.11672	0.0497921	0.0248960	+	0.999690i	$0.492075\pi$
$504$	0	0
$505$	1.92645	0.0857260
$506$	7.74657	0.344377
$507$	0	0
$508$	−21.2765	−0.943991
$509$	38.2481	1.69532	0.847659	−	0.530542i	$-0.178012\pi$
$509$	38.2481	1.69532	0.847659	+	0.530542i	$0.178012\pi$
$510$	0	0
$511$	−10.8338	−0.479261
$512$	1.00000	0.0441942
$513$	0	0
$514$	−0.723522	−0.0319132
$515$	27.0294	1.19106
$516$	0	0
$517$	7.08727	0.311698
$518$	−21.6677	−0.952023
$519$	0	0
$520$	7.83384	0.343537
$521$	−10.8853	−0.476892	−0.238446	−	0.971156i	$-0.576638\pi$
$521$	−10.8853	−0.476892	−0.238446	+	0.971156i	$0.576638\pi$
$522$	0	0
$523$	−15.7191	−0.687349	−0.343674	−	0.939089i	$-0.611672\pi$
$523$	−15.7191	−0.687349	−0.343674	+	0.939089i	$0.611672\pi$
$524$	3.43531	0.150072
$525$	0	0
$526$	−13.7760	−0.600663
$527$	−42.7275	−1.86124
$528$	0	0
$529$	37.0093	1.60910
$530$	−13.3354	−0.579251
$531$	0	0
$532$	−2.89167	−0.125370
$533$	10.0662	0.436016
$534$	0	0
$535$	12.8706	0.556445
$536$	−10.3848	−0.448555
$537$	0	0
$538$	19.9579	0.860446
$539$	1.36176	0.0586552
$540$	0	0
$541$	−26.4373	−1.13663	−0.568314	−	0.822812i	$-0.692404\pi$
$541$	−26.4373	−1.13663	−0.568314	+	0.822812i	$0.692404\pi$
$542$	28.7486	1.23486
$543$	0	0
$544$	5.74657	0.246382
$545$	18.4648	0.790943
$546$	0	0
$547$	−24.9588	−1.06716	−0.533581	−	0.845749i	$-0.679154\pi$
$547$	−24.9588	−1.06716	−0.533581	+	0.845749i	$0.679154\pi$
$548$	−13.5877	−0.580439
$549$	0	0
$550$	−2.43531	−0.103842
$551$	5.34803	0.227834
$552$	0	0
$553$	−25.1462	−1.06932
$554$	3.08727	0.131166
$555$	0	0
$556$	3.19560	0.135524
$557$	−8.30486	−0.351888	−0.175944	−	0.984400i	$-0.556298\pi$
$557$	−8.30486	−0.351888	−0.175944	+	0.984400i	$0.556298\pi$
$558$	0	0
$559$	−52.2858	−2.21145
$560$	4.63091	0.195692
$561$	0	0
$562$	−9.09460	−0.383633
$563$	−41.6402	−1.75493	−0.877463	−	0.479644i	$-0.840766\pi$
$563$	−41.6402	−1.75493	−0.877463	+	0.479644i	$0.840766\pi$
$564$	0	0
$565$	12.8117	0.538993
$566$	13.1451	0.552530
$567$	0	0
$568$	−14.9789	−0.628503
$569$	14.9138	0.625219	0.312609	−	0.949882i	$-0.398797\pi$
$569$	14.9138	0.625219	0.312609	+	0.949882i	$0.398797\pi$
$570$	0	0
$571$	−23.0020	−0.962603	−0.481302	−	0.876555i	$-0.659836\pi$
$571$	−23.0020	−0.962603	−0.481302	+	0.876555i	$0.659836\pi$
$572$	4.89167	0.204531
$573$	0	0
$574$	5.95056	0.248372
$575$	−18.8653	−0.786737
$576$	0	0
$577$	−33.6613	−1.40134	−0.700669	−	0.713487i	$-0.747115\pi$
$577$	−33.6613	−1.40134	−0.700669	+	0.713487i	$0.747115\pi$
$578$	16.0230	0.666471
$579$	0	0
$580$	−8.56469	−0.355629
$581$	17.6633	0.732796
$582$	0	0
$583$	−8.32698	−0.344868
$584$	−3.74657	−0.155034
$585$	0	0
$586$	−3.10833	−0.128404
$587$	−4.00000	−0.165098	−0.0825488	−	0.996587i	$-0.526306\pi$
$587$	−4.00000	−0.165098	−0.0825488	+	0.996587i	$0.526306\pi$
$588$	0	0
$589$	7.43531	0.306367
$590$	4.07355	0.167705
$591$	0	0
$592$	−7.49314	−0.307966
$593$	30.9588	1.27133	0.635663	−	0.771967i	$-0.280727\pi$
$593$	30.9588	1.27133	0.635663	+	0.771967i	$0.280727\pi$
$594$	0	0
$595$	26.6118	1.09098
$596$	12.0735	0.494552
$597$	0	0
$598$	37.8937	1.54959
$599$	−3.93378	−0.160730	−0.0803650	−	0.996766i	$-0.525609\pi$
$599$	−3.93378	−0.160730	−0.0803650	+	0.996766i	$0.525609\pi$
$600$	0	0
$601$	23.1451	0.944108	0.472054	−	0.881570i	$-0.343513\pi$
$601$	23.1451	0.944108	0.472054	+	0.881570i	$0.343513\pi$
$602$	−30.9083	−1.25973
$603$	0	0
$604$	−4.91273	−0.199896
$605$	1.60147	0.0651088
$606$	0	0
$607$	−1.78334	−0.0723836	−0.0361918	−	0.999345i	$-0.511523\pi$
$607$	−1.78334	−0.0723836	−0.0361918	+	0.999345i	$0.511523\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	21.6088	0.874914
$611$	34.6686	1.40254
$612$	0	0
$613$	−33.0598	−1.33527	−0.667637	−	0.744487i	$-0.732694\pi$
$613$	−33.0598	−1.33527	−0.667637	+	0.744487i	$0.732694\pi$
$614$	2.18920	0.0883491
$615$	0	0
$616$	2.89167	0.116509
$617$	21.8412	0.879292	0.439646	−	0.898171i	$-0.355104\pi$
$617$	21.8412	0.879292	0.439646	+	0.898171i	$0.355104\pi$
$618$	0	0
$619$	−46.2206	−1.85776	−0.928882	−	0.370375i	$-0.879229\pi$
$619$	−46.2206	−1.85776	−0.928882	+	0.370375i	$0.879229\pi$
$620$	−11.9074	−0.478212
$621$	0	0
$622$	−3.92112	−0.157222
$623$	−9.26182	−0.371067
$624$	0	0
$625$	−6.89273	−0.275709
$626$	−21.7118	−0.867778
$627$	0	0
$628$	8.00733	0.319527
$629$	−43.0598	−1.71691
$630$	0	0
$631$	0.317659	0.0126458	0.00632291	−	0.999980i	$-0.497987\pi$
$631$	0.317659	0.0126458	0.00632291	+	0.999980i	$0.497987\pi$
$632$	−8.69607	−0.345911
$633$	0	0
$634$	−6.90739	−0.274327
$635$	−34.0735	−1.35217
$636$	0	0
$637$	6.66129	0.263930
$638$	−5.34803	−0.211731
$639$	0	0
$640$	1.60147	0.0633035
$641$	41.5246	1.64012	0.820061	−	0.572276i	$-0.193939\pi$
$641$	41.5246	1.64012	0.820061	+	0.572276i	$0.193939\pi$
$642$	0	0
$643$	7.26182	0.286378	0.143189	−	0.989695i	$-0.454264\pi$
$643$	7.26182	0.286378	0.143189	+	0.989695i	$0.454264\pi$
$644$	22.4005	0.882704
$645$	0	0
$646$	−5.74657	−0.226096
$647$	−29.0819	−1.14333	−0.571664	−	0.820487i	$-0.693702\pi$
$647$	−29.0819	−1.14333	−0.571664	+	0.820487i	$0.693702\pi$
$648$	0	0
$649$	2.54364	0.0998465
$650$	−11.9127	−0.467256
$651$	0	0
$652$	1.88434	0.0737966
$653$	−12.6529	−0.495146	−0.247573	−	0.968869i	$-0.579633\pi$
$653$	−12.6529	−0.495146	−0.247573	+	0.968869i	$0.579633\pi$
$654$	0	0
$655$	5.50153	0.214962
$656$	2.05783	0.0803447
$657$	0	0
$658$	20.4941	0.798942
$659$	−49.4878	−1.92777	−0.963886	−	0.266317i	$-0.914193\pi$
$659$	−49.4878	−1.92777	−0.963886	+	0.266317i	$0.914193\pi$
$660$	0	0
$661$	4.94950	0.192513	0.0962566	−	0.995357i	$-0.469313\pi$
$661$	4.94950	0.192513	0.0962566	+	0.995357i	$0.469313\pi$
$662$	32.1030	1.24772
$663$	0	0
$664$	6.10833	0.237049
$665$	−4.63091	−0.179579
$666$	0	0
$667$	−41.4289	−1.60413
$668$	4.98627	0.192925
$669$	0	0
$670$	−16.6309	−0.642508
$671$	13.4931	0.520897
$672$	0	0
$673$	21.9001	0.844185	0.422093	−	0.906553i	$-0.361296\pi$
$673$	21.9001	0.844185	0.422093	+	0.906553i	$0.361296\pi$
$674$	−1.55096	−0.0597409
$675$	0	0
$676$	10.9284	0.420325
$677$	−26.4353	−1.01599	−0.507996	−	0.861360i	$-0.669613\pi$
$677$	−26.4353	−1.01599	−0.507996	+	0.861360i	$0.669613\pi$
$678$	0	0
$679$	−20.2020	−0.775282
$680$	9.20293	0.352916
$681$	0	0
$682$	−7.43531	−0.284713
$683$	−9.82545	−0.375960	−0.187980	−	0.982173i	$-0.560194\pi$
$683$	−9.82545	−0.375960	−0.187980	+	0.982173i	$0.560194\pi$
$684$	0	0
$685$	−21.7603	−0.831418
$686$	−16.3039	−0.622487
$687$	0	0
$688$	−10.6887	−0.407504
$689$	−40.7328	−1.55180
$690$	0	0
$691$	−19.8255	−0.754196	−0.377098	−	0.926173i	$-0.623078\pi$
$691$	−19.8255	−0.754196	−0.377098	+	0.926173i	$0.623078\pi$
$692$	−6.51419	−0.247632
$693$	0	0
$694$	0.506864	0.0192403
$695$	5.11765	0.194123
$696$	0	0
$697$	11.8255	0.447921
$698$	10.8117	0.409230
$699$	0	0
$700$	−7.04211	−0.266167
$701$	10.7550	0.406209	0.203105	−	0.979157i	$-0.434897\pi$
$701$	10.7550	0.406209	0.203105	+	0.979157i	$0.434897\pi$
$702$	0	0
$703$	7.49314	0.282609
$704$	1.00000	0.0376889
$705$	0	0
$706$	−27.0966	−1.01979
$707$	3.47848	0.130822
$708$	0	0
$709$	−1.82651	−0.0685962	−0.0342981	−	0.999412i	$-0.510920\pi$
$709$	−1.82651	−0.0685962	−0.0342981	+	0.999412i	$0.510920\pi$
$710$	−23.9883	−0.900264
$711$	0	0
$712$	−3.20293	−0.120035
$713$	−57.5981	−2.15707
$714$	0	0
$715$	7.83384	0.292969
$716$	5.55096	0.207449
$717$	0	0
$718$	20.7486	0.774329
$719$	4.65929	0.173762	0.0868812	−	0.996219i	$-0.472310\pi$
$719$	4.65929	0.173762	0.0868812	+	0.996219i	$0.472310\pi$
$720$	0	0
$721$	48.8055	1.81761
$722$	1.00000	0.0372161
$723$	0	0
$724$	1.88434	0.0700311
$725$	13.0241	0.483703
$726$	0	0
$727$	19.9358	0.739377	0.369688	−	0.929156i	$-0.379464\pi$
$727$	19.9358	0.739377	0.369688	+	0.929156i	$0.379464\pi$
$728$	14.1451	0.524252
$729$	0	0
$730$	−6.00000	−0.222070
$731$	−61.4236	−2.27183
$732$	0	0
$733$	13.7687	0.508558	0.254279	−	0.967131i	$-0.418162\pi$
$733$	13.7687	0.508558	0.254279	+	0.967131i	$0.418162\pi$
$734$	25.6402	0.946398
$735$	0	0
$736$	7.74657	0.285542
$737$	−10.3848	−0.382529
$738$	0	0
$739$	22.4867	0.827188	0.413594	−	0.910461i	$-0.364273\pi$
$739$	22.4867	0.827188	0.413594	+	0.910461i	$0.364273\pi$
$740$	−12.0000	−0.441129
$741$	0	0
$742$	−24.0789	−0.883964
$743$	50.6540	1.85831	0.929157	−	0.369686i	$-0.120535\pi$
$743$	50.6540	1.85831	0.929157	+	0.369686i	$0.120535\pi$
$744$	0	0
$745$	19.3354	0.708393
$746$	25.2049	0.922818
$747$	0	0
$748$	5.74657	0.210115
$749$	23.2397	0.849160
$750$	0	0
$751$	−38.9127	−1.41995	−0.709973	−	0.704229i	$-0.751293\pi$
$751$	−38.9127	−1.41995	−0.709973	+	0.704229i	$0.751293\pi$
$752$	7.08727	0.258446
$753$	0	0
$754$	−26.1608	−0.952721
$755$	−7.86756	−0.286330
$756$	0	0
$757$	−20.5877	−0.748274	−0.374137	−	0.927373i	$-0.622061\pi$
$757$	−20.5877	−0.748274	−0.374137	+	0.927373i	$0.622061\pi$
$758$	3.47208	0.126112
$759$	0	0
$760$	−1.60147	−0.0580913
$761$	−10.6025	−0.384341	−0.192171	−	0.981362i	$-0.561553\pi$
$761$	−10.6025	−0.384341	−0.192171	+	0.981362i	$0.561553\pi$
$762$	0	0
$763$	33.3407	1.20701
$764$	−2.44264	−0.0883715
$765$	0	0
$766$	−23.1176	−0.835275
$767$	12.4426	0.449278
$768$	0	0
$769$	−8.77495	−0.316433	−0.158216	−	0.987404i	$-0.550574\pi$
$769$	−8.77495	−0.316433	−0.158216	+	0.987404i	$0.550574\pi$
$770$	4.63091	0.166886
$771$	0	0
$772$	−9.67501	−0.348211
$773$	46.8653	1.68563	0.842813	−	0.538206i	$-0.180898\pi$
$773$	46.8653	1.68563	0.842813	+	0.538206i	$0.180898\pi$
$774$	0	0
$775$	18.1073	0.650432
$776$	−6.98627	−0.250793
$777$	0	0
$778$	26.7623	0.959474
$779$	−2.05783	−0.0737294
$780$	0	0
$781$	−14.9789	−0.535989
$782$	44.5162	1.59190
$783$	0	0
$784$	1.36176	0.0486343
$785$	12.8235	0.457689
$786$	0	0
$787$	−27.4143	−0.977213	−0.488606	−	0.872504i	$-0.662495\pi$
$787$	−27.4143	−0.977213	−0.488606	+	0.872504i	$0.662495\pi$
$788$	17.4931	0.623167
$789$	0	0
$790$	−13.9265	−0.495481
$791$	23.1334	0.822528
$792$	0	0
$793$	66.0040	2.34387
$794$	−31.1030	−1.10380
$795$	0	0
$796$	−25.2397	−0.894598
$797$	9.16616	0.324682	0.162341	−	0.986735i	$-0.448096\pi$
$797$	9.16616	0.324682	0.162341	+	0.986735i	$0.448096\pi$
$798$	0	0
$799$	40.7275	1.44084
$800$	−2.43531	−0.0861011
$801$	0	0
$802$	−29.3500	−1.03639
$803$	−3.74657	−0.132214
$804$	0	0
$805$	35.8737	1.26438
$806$	−36.3711	−1.28112
$807$	0	0
$808$	1.20293	0.0423190
$809$	−48.0829	−1.69050	−0.845252	−	0.534368i	$-0.820550\pi$
$809$	−48.0829	−1.69050	−0.845252	+	0.534368i	$0.820550\pi$
$810$	0	0
$811$	−2.67395	−0.0938951	−0.0469475	−	0.998897i	$-0.514949\pi$
$811$	−2.67395	−0.0938951	−0.0469475	+	0.998897i	$0.514949\pi$
$812$	−15.4648	−0.542706
$813$	0	0
$814$	−7.49314	−0.262634
$815$	3.01771	0.105706
$816$	0	0
$817$	10.6887	0.373952
$818$	8.58774	0.300263
$819$	0	0
$820$	3.29554	0.115085
$821$	−29.5078	−1.02983	−0.514915	−	0.857242i	$-0.672176\pi$
$821$	−29.5078	−1.02983	−0.514915	+	0.857242i	$0.672176\pi$
$822$	0	0
$823$	−2.39654	−0.0835382	−0.0417691	−	0.999127i	$-0.513299\pi$
$823$	−2.39654	−0.0835382	−0.0417691	+	0.999127i	$0.513299\pi$
$824$	16.8779	0.587971
$825$	0	0
$826$	7.35536	0.255926
$827$	−7.78868	−0.270839	−0.135419	−	0.990788i	$-0.543238\pi$
$827$	−7.78868	−0.270839	−0.135419	+	0.990788i	$0.543238\pi$
$828$	0	0
$829$	13.2818	0.461296	0.230648	−	0.973037i	$-0.425915\pi$
$829$	13.2818	0.461296	0.230648	+	0.973037i	$0.425915\pi$
$830$	9.78228	0.339548
$831$	0	0
$832$	4.89167	0.169588
$833$	7.82545	0.271136
$834$	0	0
$835$	7.98534	0.276344
$836$	−1.00000	−0.0345857
$837$	0	0
$838$	−7.59414	−0.262335
$839$	33.9063	1.17058	0.585288	−	0.810825i	$-0.300981\pi$
$839$	33.9063	1.17058	0.585288	+	0.810825i	$0.300981\pi$
$840$	0	0
$841$	−0.398534	−0.0137426
$842$	8.76030	0.301900
$843$	0	0
$844$	−14.8338	−0.510602
$845$	17.5015	0.602071
$846$	0	0
$847$	2.89167	0.0993590
$848$	−8.32698	−0.285950
$849$	0	0
$850$	−13.9947	−0.480013
$851$	−58.0461	−1.98979
$852$	0	0
$853$	40.0735	1.37209	0.686046	−	0.727558i	$-0.259345\pi$
$853$	40.0735	1.37209	0.686046	+	0.727558i	$0.259345\pi$
$854$	39.0177	1.33516
$855$	0	0
$856$	8.03677	0.274691
$857$	−0.665693	−0.0227396	−0.0113698	−	0.999935i	$-0.503619\pi$
$857$	−0.665693	−0.0227396	−0.0113698	+	0.999935i	$0.503619\pi$
$858$	0	0
$859$	13.8990	0.474228	0.237114	−	0.971482i	$-0.423799\pi$
$859$	13.8990	0.474228	0.237114	+	0.971482i	$0.423799\pi$
$860$	−17.1176	−0.583707
$861$	0	0
$862$	33.6677	1.14673
$863$	40.3941	1.37503	0.687516	−	0.726169i	$-0.258701\pi$
$863$	40.3941	1.37503	0.687516	+	0.726169i	$0.258701\pi$
$864$	0	0
$865$	−10.4323	−0.354707
$866$	30.5383	1.03773
$867$	0	0
$868$	−21.5005	−0.729773
$869$	−8.69607	−0.294994
$870$	0	0
$871$	−50.7991	−1.72126
$872$	11.5299	0.390452
$873$	0	0
$874$	−7.74657	−0.262032
$875$	−34.4323	−1.16402
$876$	0	0
$877$	15.6108	0.527139	0.263569	−	0.964640i	$-0.415100\pi$
$877$	15.6108	0.527139	0.263569	+	0.964640i	$0.415100\pi$
$878$	−12.5069	−0.422086
$879$	0	0
$880$	1.60147	0.0539854
$881$	−19.9843	−0.673288	−0.336644	−	0.941632i	$-0.609292\pi$
$881$	−19.9843	−0.673288	−0.336644	+	0.941632i	$0.609292\pi$
$882$	0	0
$883$	20.2167	0.680345	0.340172	−	0.940363i	$-0.389515\pi$
$883$	20.2167	0.680345	0.340172	+	0.940363i	$0.389515\pi$
$884$	28.1103	0.945452
$885$	0	0
$886$	17.6677	0.593557
$887$	−20.1471	−0.676473	−0.338237	−	0.941061i	$-0.609830\pi$
$887$	−20.1471	−0.676473	−0.338237	+	0.941061i	$0.609830\pi$
$888$	0	0
$889$	−61.5246	−2.06347
$890$	−5.12938	−0.171937
$891$	0	0
$892$	11.4931	0.384819
$893$	−7.08727	−0.237167
$894$	0	0
$895$	8.88968	0.297149
$896$	2.89167	0.0966039
$897$	0	0
$898$	17.7098	0.590984
$899$	39.7643	1.32621
$900$	0	0
$901$	−47.8516	−1.59417
$902$	2.05783	0.0685182
$903$	0	0
$904$	8.00000	0.266076
$905$	3.01771	0.100312
$906$	0	0
$907$	4.57109	0.151781	0.0758903	−	0.997116i	$-0.475820\pi$
$907$	4.57109	0.151781	0.0758903	+	0.997116i	$0.475820\pi$
$908$	−8.73284	−0.289810
$909$	0	0
$910$	22.6529	0.750936
$911$	−51.6402	−1.71092	−0.855459	−	0.517871i	$-0.826725\pi$
$911$	−51.6402	−1.71092	−0.855459	+	0.517871i	$0.826725\pi$
$912$	0	0
$913$	6.10833	0.202156
$914$	−24.8064	−0.820522
$915$	0	0
$916$	7.84117	0.259080
$917$	9.93378	0.328042
$918$	0	0
$919$	−48.8412	−1.61112	−0.805561	−	0.592513i	$-0.798136\pi$
$919$	−48.8412	−1.61112	−0.805561	+	0.592513i	$0.798136\pi$
$920$	12.4059	0.409009
$921$	0	0
$922$	2.98627	0.0983477
$923$	−73.2721	−2.41178
$924$	0	0
$925$	18.2481	0.599994
$926$	−39.5667	−1.30024
$927$	0	0
$928$	−5.34803	−0.175558
$929$	57.8819	1.89904	0.949522	−	0.313700i	$-0.101569\pi$
$929$	57.8819	1.89904	0.949522	+	0.313700i	$0.101569\pi$
$930$	0	0
$931$	−1.36176	−0.0446299
$932$	12.5069	0.409676
$933$	0	0
$934$	−4.46475	−0.146091
$935$	9.20293	0.300968
$936$	0	0
$937$	−57.6496	−1.88333	−0.941664	−	0.336553i	$-0.890739\pi$
$937$	−57.6496	−1.88333	−0.941664	+	0.336553i	$0.890739\pi$
$938$	−30.0294	−0.980496
$939$	0	0
$940$	11.3500	0.370197
$941$	15.0966	0.492135	0.246067	−	0.969253i	$-0.420862\pi$
$941$	15.0966	0.492135	0.246067	+	0.969253i	$0.420862\pi$
$942$	0	0
$943$	15.9411	0.519114
$944$	2.54364	0.0827883
$945$	0	0
$946$	−10.6887	−0.347521
$947$	−33.3775	−1.08462	−0.542311	−	0.840178i	$-0.682451\pi$
$947$	−33.3775	−1.08462	−0.542311	+	0.840178i	$0.682451\pi$
$948$	0	0
$949$	−18.3270	−0.594919
$950$	2.43531	0.0790118
$951$	0	0
$952$	16.6172	0.538566
$953$	12.4794	0.404248	0.202124	−	0.979360i	$-0.435216\pi$
$953$	12.4794	0.404248	0.202124	+	0.979360i	$0.435216\pi$
$954$	0	0
$955$	−3.91180	−0.126583
$956$	−24.6245	−0.796414
$957$	0	0
$958$	20.8863	0.674807
$959$	−39.2913	−1.26878
$960$	0	0
$961$	24.2838	0.783349
$962$	−36.6540	−1.18177
$963$	0	0
$964$	16.0809	0.517930
$965$	−15.4942	−0.498776
$966$	0	0
$967$	−12.5804	−0.404559	−0.202279	−	0.979328i	$-0.564835\pi$
$967$	−12.5804	−0.404559	−0.202279	+	0.979328i	$0.564835\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	−11.1883	−0.359234
$971$	1.79067	0.0574653	0.0287327	−	0.999587i	$-0.490853\pi$
$971$	1.79067	0.0574653	0.0287327	+	0.999587i	$0.490853\pi$
$972$	0	0
$973$	9.24063	0.296241
$974$	15.2838	0.489725
$975$	0	0
$976$	13.4931	0.431905
$977$	−6.40586	−0.204942	−0.102471	−	0.994736i	$-0.532675\pi$
$977$	−6.40586	−0.204942	−0.102471	+	0.994736i	$0.532675\pi$
$978$	0	0
$979$	−3.20293	−0.102366
$980$	2.18081	0.0696635
$981$	0	0
$982$	−3.31965	−0.105934
$983$	20.5647	0.655912	0.327956	−	0.944693i	$-0.393640\pi$
$983$	20.5647	0.655912	0.327956	+	0.944693i	$0.393640\pi$
$984$	0	0
$985$	28.0147	0.892621
$986$	−30.7328	−0.978733
$987$	0	0
$988$	−4.89167	−0.155625
$989$	−82.8011	−2.63292
$990$	0	0
$991$	−29.0166	−0.921744	−0.460872	−	0.887467i	$-0.652463\pi$
$991$	−29.0166	−0.921744	−0.460872	+	0.887467i	$0.652463\pi$
$992$	−7.43531	−0.236071
$993$	0	0
$994$	−43.3142	−1.37384
$995$	−40.4205	−1.28142
$996$	0	0
$997$	42.3638	1.34167	0.670837	−	0.741605i	$-0.265935\pi$
$997$	42.3638	1.34167	0.670837	+	0.741605i	$0.265935\pi$
$998$	−2.07355	−0.0656370
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3762.2.a.bg.1.2		3
3.2	odd	2		418.2.a.g.1.1	✓	3
12.11	even	2		3344.2.a.q.1.3		3
33.32	even	2		4598.2.a.bo.1.1		3
57.56	even	2		7942.2.a.bi.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
418.2.a.g.1.1	✓	3	3.2	odd	2
3344.2.a.q.1.3		3	12.11	even	2
3762.2.a.bg.1.2		3	1.1	even	1	trivial
4598.2.a.bo.1.1		3	33.32	even	2
7942.2.a.bi.1.3		3	57.56	even	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:	\( N \)	\(=\)	\( 3762 = 2 \cdot 3^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3762.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.60147 q^{5} +2.89167 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.60147 q^{5} +2.89167 q^{7} +1.00000 q^{8} +1.60147 q^{10} +1.00000 q^{11} +4.89167 q^{13} +2.89167 q^{14} +1.00000 q^{16} +5.74657 q^{17} -1.00000 q^{19} +1.60147 q^{20} +1.00000 q^{22} +7.74657 q^{23} -2.43531 q^{25} +4.89167 q^{26} +2.89167 q^{28} -5.34803 q^{29} -7.43531 q^{31} +1.00000 q^{32} +5.74657 q^{34} +4.63091 q^{35} -7.49314 q^{37} -1.00000 q^{38} +1.60147 q^{40} +2.05783 q^{41} -10.6887 q^{43} +1.00000 q^{44} +7.74657 q^{46} +7.08727 q^{47} +1.36176 q^{49} -2.43531 q^{50} +4.89167 q^{52} -8.32698 q^{53} +1.60147 q^{55} +2.89167 q^{56} -5.34803 q^{58} +2.54364 q^{59} +13.4931 q^{61} -7.43531 q^{62} +1.00000 q^{64} +7.83384 q^{65} -10.3848 q^{67} +5.74657 q^{68} +4.63091 q^{70} -14.9789 q^{71} -3.74657 q^{73} -7.49314 q^{74} -1.00000 q^{76} +2.89167 q^{77} -8.69607 q^{79} +1.60147 q^{80} +2.05783 q^{82} +6.10833 q^{83} +9.20293 q^{85} -10.6887 q^{86} +1.00000 q^{88} -3.20293 q^{89} +14.1451 q^{91} +7.74657 q^{92} +7.08727 q^{94} -1.60147 q^{95} -6.98627 q^{97} +1.36176 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 6 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^5 - 6 * q^7 + 3 * q^8	\( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 6 q^{7} + 3 q^{8} + 3 q^{10} + 3 q^{11} - 6 q^{14} + 3 q^{16} + 9 q^{17} - 3 q^{19} + 3 q^{20} + 3 q^{22} + 15 q^{23} + 12 q^{25} - 6 q^{28} - 6 q^{29} - 3 q^{31} + 3 q^{32} + 9 q^{34} - 6 q^{37} - 3 q^{38} + 3 q^{40} + 9 q^{41} - 21 q^{43} + 3 q^{44} + 15 q^{46} + 12 q^{47} + 27 q^{49} + 12 q^{50} + 9 q^{53} + 3 q^{55} - 6 q^{56} - 6 q^{58} + 3 q^{59} + 24 q^{61} - 3 q^{62} + 3 q^{64} + 6 q^{65} + 9 q^{68} - 21 q^{71} - 3 q^{73} - 6 q^{74} - 3 q^{76} - 6 q^{77} - 6 q^{79} + 3 q^{80} + 9 q^{82} + 33 q^{83} + 24 q^{85} - 21 q^{86} + 3 q^{88} - 6 q^{89} + 36 q^{91} + 15 q^{92} + 12 q^{94} - 3 q^{95} + 12 q^{97} + 27 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^5 - 6 * q^7 + 3 * q^8 + 3 * q^10 + 3 * q^11 - 6 * q^14 + 3 * q^16 + 9 * q^17 - 3 * q^19 + 3 * q^20 + 3 * q^22 + 15 * q^23 + 12 * q^25 - 6 * q^28 - 6 * q^29 - 3 * q^31 + 3 * q^32 + 9 * q^34 - 6 * q^37 - 3 * q^38 + 3 * q^40 + 9 * q^41 - 21 * q^43 + 3 * q^44 + 15 * q^46 + 12 * q^47 + 27 * q^49 + 12 * q^50 + 9 * q^53 + 3 * q^55 - 6 * q^56 - 6 * q^58 + 3 * q^59 + 24 * q^61 - 3 * q^62 + 3 * q^64 + 6 * q^65 + 9 * q^68 - 21 * q^71 - 3 * q^73 - 6 * q^74 - 3 * q^76 - 6 * q^77 - 6 * q^79 + 3 * q^80 + 9 * q^82 + 33 * q^83 + 24 * q^85 - 21 * q^86 + 3 * q^88 - 6 * q^89 + 36 * q^91 + 15 * q^92 + 12 * q^94 - 3 * q^95 + 12 * q^97 + 27 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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