LMFDB - Embedded newform 2394.2.a.ba.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2394,2,Mod(1,2394)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2394.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2394, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2394.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-3,0,3,-5,0,-3,-3,0,5,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$19.1161862439$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.469.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 4 $ x^3 - x^2 - 5*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 266)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.39138$ of defining polynomial
Character	$\chi$	$=$	2394.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{4} +0.391382 q^{5} -1.00000 q^{7} -1.00000 q^{8} -0.391382 q^{10} -3.06406 q^{11} +2.78276 q^{13} +1.00000 q^{14} +1.00000 q^{16} +6.22018 q^{17} -1.00000 q^{19} +0.391382 q^{20} +3.06406 q^{22} +4.78276 q^{23} -4.84682 q^{25} -2.78276 q^{26} -1.00000 q^{28} -7.82880 q^{29} +1.34535 q^{31} -1.00000 q^{32} -6.22018 q^{34} -0.391382 q^{35} -3.82880 q^{37} +1.00000 q^{38} -0.391382 q^{40} +4.39138 q^{41} +5.82880 q^{43} -3.06406 q^{44} -4.78276 q^{46} +5.71871 q^{47} +1.00000 q^{49} +4.84682 q^{50} +2.78276 q^{52} -7.15613 q^{53} -1.19922 q^{55} +1.00000 q^{56} +7.82880 q^{58} +7.17415 q^{59} +8.50147 q^{61} -1.34535 q^{62} +1.00000 q^{64} +1.08913 q^{65} +12.2202 q^{67} +6.22018 q^{68} +0.391382 q^{70} -0.935945 q^{71} -4.12811 q^{73} +3.82880 q^{74} -1.00000 q^{76} +3.06406 q^{77} +2.95396 q^{79} +0.391382 q^{80} -4.39138 q^{82} +13.5655 q^{83} +2.43447 q^{85} -5.82880 q^{86} +3.06406 q^{88} -1.06406 q^{89} -2.78276 q^{91} +4.78276 q^{92} -5.71871 q^{94} -0.391382 q^{95} +4.50147 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} - 5 q^{5} - 3 q^{7} - 3 q^{8} + 5 q^{10} - 3 q^{11} - 4 q^{13} + 3 q^{14} + 3 q^{16} - 6 q^{17} - 3 q^{19} - 5 q^{20} + 3 q^{22} + 2 q^{23} + 4 q^{25} + 4 q^{26} - 3 q^{28} - 5 q^{29}+ \cdots - 3 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 - 5 * q^5 - 3 * q^7 - 3 * q^8 + 5 * q^10 - 3 * q^11 - 4 * q^13 + 3 * q^14 + 3 * q^16 - 6 * q^17 - 3 * q^19 - 5 * q^20 + 3 * q^22 + 2 * q^23 + 4 * q^25 + 4 * q^26 - 3 * q^28 - 5 * q^29 + 4 * q^31 - 3 * q^32 + 6 * q^34 + 5 * q^35 + 7 * q^37 + 3 * q^38 + 5 * q^40 + 7 * q^41 - q^43 - 3 * q^44 - 2 * q^46 + 11 * q^47 + 3 * q^49 - 4 * q^50 - 4 * q^52 - 3 * q^53 - 16 * q^55 + 3 * q^56 + 5 * q^58 + 3 * q^59 + 7 * q^61 - 4 * q^62 + 3 * q^64 + 28 * q^65 + 12 * q^67 - 6 * q^68 - 5 * q^70 - 9 * q^71 - 7 * q^74 - 3 * q^76 + 3 * q^77 + 15 * q^79 - 5 * q^80 - 7 * q^82 + 16 * q^83 + 32 * q^85 + q^86 + 3 * q^88 + 3 * q^89 + 4 * q^91 + 2 * q^92 - 11 * q^94 + 5 * q^95 - 5 * q^97 - 3 * q^98

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
$5$	0.391382	0.175032	0.0875158	−	0.996163i	$-0.472107\pi$
$5$	0.391382	0.175032	0.0875158	+	0.996163i	$0.472107\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−0.391382	−0.123766
$11$	−3.06406	−0.923847	−0.461924	−	0.886920i	$-0.652841\pi$
$11$	−3.06406	−0.923847	−0.461924	+	0.886920i	$0.652841\pi$
$12$	0	0
$13$	2.78276	0.771800	0.385900	−	0.922541i	$-0.373891\pi$
$13$	2.78276	0.771800	0.385900	+	0.922541i	$0.373891\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	6.22018	1.50862	0.754308	−	0.656521i	$-0.227972\pi$
$17$	6.22018	1.50862	0.754308	+	0.656521i	$0.227972\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0.391382	0.0875158
$21$	0	0
$22$	3.06406	0.653259
$23$	4.78276	0.997275	0.498638	−	0.866811i	$-0.333834\pi$
$23$	4.78276	0.997275	0.498638	+	0.866811i	$0.333834\pi$
$24$	0	0
$25$	−4.84682	−0.969364
$26$	−2.78276	−0.545745
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−7.82880	−1.45377	−0.726886	−	0.686758i	$-0.759033\pi$
$29$	−7.82880	−1.45377	−0.726886	+	0.686758i	$0.759033\pi$
$30$	0	0
$31$	1.34535	0.241631	0.120816	−	0.992675i	$-0.461449\pi$
$31$	1.34535	0.241631	0.120816	+	0.992675i	$0.461449\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−6.22018	−1.06675
$35$	−0.391382	−0.0661557
$36$	0	0
$37$	−3.82880	−0.629451	−0.314726	−	0.949183i	$-0.601912\pi$
$37$	−3.82880	−0.629451	−0.314726	+	0.949183i	$0.601912\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	−0.391382	−0.0618830
$41$	4.39138	0.685819	0.342909	−	0.939368i	$-0.388588\pi$
$41$	4.39138	0.685819	0.342909	+	0.939368i	$0.388588\pi$
$42$	0	0
$43$	5.82880	0.888884	0.444442	−	0.895808i	$-0.353402\pi$
$43$	5.82880	0.888884	0.444442	+	0.895808i	$0.353402\pi$
$44$	−3.06406	−0.461924
$45$	0	0
$46$	−4.78276	−0.705180
$47$	5.71871	0.834160	0.417080	−	0.908870i	$-0.363054\pi$
$47$	5.71871	0.834160	0.417080	+	0.908870i	$0.363054\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	4.84682	0.685444
$51$	0	0
$52$	2.78276	0.385900
$53$	−7.15613	−0.982970	−0.491485	−	0.870886i	$-0.663546\pi$
$53$	−7.15613	−0.982970	−0.491485	+	0.870886i	$0.663546\pi$
$54$	0	0
$55$	−1.19922	−0.161702
$56$	1.00000	0.133631
$57$	0	0
$58$	7.82880	1.02797
$59$	7.17415	0.933994	0.466997	−	0.884259i	$-0.345336\pi$
$59$	7.17415	0.933994	0.466997	+	0.884259i	$0.345336\pi$
$60$	0	0
$61$	8.50147	1.08850	0.544251	−	0.838922i	$-0.316814\pi$
$61$	8.50147	1.08850	0.544251	+	0.838922i	$0.316814\pi$
$62$	−1.34535	−0.170859
$63$	0	0
$64$	1.00000	0.125000
$65$	1.08913	0.135089
$66$	0	0
$67$	12.2202	1.49293	0.746467	−	0.665423i	$-0.231749\pi$
$67$	12.2202	1.49293	0.746467	+	0.665423i	$0.231749\pi$
$68$	6.22018	0.754308
$69$	0	0
$70$	0.391382	0.0467791
$71$	−0.935945	−0.111076	−0.0555381	−	0.998457i	$-0.517687\pi$
$71$	−0.935945	−0.111076	−0.0555381	+	0.998457i	$0.517687\pi$
$72$	0	0
$73$	−4.12811	−0.483159	−0.241579	−	0.970381i	$-0.577665\pi$
$73$	−4.12811	−0.483159	−0.241579	+	0.970381i	$0.577665\pi$
$74$	3.82880	0.445089
$75$	0	0
$76$	−1.00000	−0.114708
$77$	3.06406	0.349181
$78$	0	0
$79$	2.95396	0.332347	0.166173	−	0.986097i	$-0.446859\pi$
$79$	2.95396	0.332347	0.166173	+	0.986097i	$0.446859\pi$
$80$	0.391382	0.0437579
$81$	0	0
$82$	−4.39138	−0.484947
$83$	13.5655	1.48901	0.744505	−	0.667617i	$-0.232685\pi$
$83$	13.5655	1.48901	0.744505	+	0.667617i	$0.232685\pi$
$84$	0	0
$85$	2.43447	0.264055
$86$	−5.82880	−0.628536
$87$	0	0
$88$	3.06406	0.326629
$89$	−1.06406	−0.112790	−0.0563948	−	0.998409i	$-0.517961\pi$
$89$	−1.06406	−0.112790	−0.0563948	+	0.998409i	$0.517961\pi$
$90$	0	0
$91$	−2.78276	−0.291713
$92$	4.78276	0.498638
$93$	0	0
$94$	−5.71871	−0.589840
$95$	−0.391382	−0.0401550
$96$	0	0
$97$	4.50147	0.457055	0.228528	−	0.973537i	$-0.426609\pi$
$97$	4.50147	0.457055	0.228528	+	0.973537i	$0.426609\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	−4.84682	−0.484682
$101$	−17.1311	−1.70460	−0.852302	−	0.523050i	$-0.824794\pi$
$101$	−17.1311	−1.70460	−0.852302	+	0.523050i	$0.824794\pi$
$102$	0	0
$103$	−2.09207	−0.206138	−0.103069	−	0.994674i	$-0.532866\pi$
$103$	−2.09207	−0.206138	−0.103069	+	0.994674i	$0.532866\pi$
$104$	−2.78276	−0.272873
$105$	0	0
$106$	7.15613	0.695065
$107$	14.0921	1.36233	0.681166	−	0.732129i	$-0.261473\pi$
$107$	14.0921	1.36233	0.681166	+	0.732129i	$0.261473\pi$
$108$	0	0
$109$	15.1561	1.45169	0.725847	−	0.687856i	$-0.241448\pi$
$109$	15.1561	1.45169	0.725847	+	0.687856i	$0.241448\pi$
$110$	1.19922	0.114341
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	−19.7857	−1.86128	−0.930642	−	0.365932i	$-0.880750\pi$
$113$	−19.7857	−1.86128	−0.930642	+	0.365932i	$0.880750\pi$
$114$	0	0
$115$	1.87189	0.174555
$116$	−7.82880	−0.726886
$117$	0	0
$118$	−7.17415	−0.660434
$119$	−6.22018	−0.570203
$120$	0	0
$121$	−1.61157	−0.146506
$122$	−8.50147	−0.769687
$123$	0	0
$124$	1.34535	0.120816
$125$	−3.85387	−0.344701
$126$	0	0
$127$	2.50147	0.221970	0.110985	−	0.993822i	$-0.464599\pi$
$127$	2.50147	0.221970	0.110985	+	0.993822i	$0.464599\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−1.08913	−0.0955226
$131$	15.4374	1.34877	0.674387	−	0.738378i	$-0.264408\pi$
$131$	15.4374	1.34877	0.674387	+	0.738378i	$0.264408\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	−12.2202	−1.05566
$135$	0	0
$136$	−6.22018	−0.533376
$137$	−13.9209	−1.18934	−0.594670	−	0.803970i	$-0.702717\pi$
$137$	−13.9209	−1.18934	−0.594670	+	0.803970i	$0.702717\pi$
$138$	0	0
$139$	10.8748	0.922392	0.461196	−	0.887298i	$-0.347421\pi$
$139$	10.8748	0.922392	0.461196	+	0.887298i	$0.347421\pi$
$140$	−0.391382	−0.0330778
$141$	0	0
$142$	0.935945	0.0785428
$143$	−8.52654	−0.713025
$144$	0	0
$145$	−3.06406	−0.254456
$146$	4.12811	0.341645
$147$	0	0
$148$	−3.82880	−0.314726
$149$	1.77982	0.145808	0.0729041	−	0.997339i	$-0.476773\pi$
$149$	1.77982	0.145808	0.0729041	+	0.997339i	$0.476773\pi$
$150$	0	0
$151$	5.30931	0.432065	0.216033	−	0.976386i	$-0.430688\pi$
$151$	5.30931	0.432065	0.216033	+	0.976386i	$0.430688\pi$
$152$	1.00000	0.0811107
$153$	0	0
$154$	−3.06406	−0.246909
$155$	0.526544	0.0422931
$156$	0	0
$157$	15.3763	1.22716	0.613582	−	0.789631i	$-0.289728\pi$
$157$	15.3763	1.22716	0.613582	+	0.789631i	$0.289728\pi$
$158$	−2.95396	−0.235005
$159$	0	0
$160$	−0.391382	−0.0309415
$161$	−4.78276	−0.376935
$162$	0	0
$163$	17.9389	1.40508	0.702541	−	0.711643i	$-0.252049\pi$
$163$	17.9389	1.40508	0.702541	+	0.711643i	$0.252049\pi$
$164$	4.39138	0.342909
$165$	0	0
$166$	−13.5655	−1.05289
$167$	13.5295	1.04694	0.523472	−	0.852043i	$-0.324637\pi$
$167$	13.5295	1.04694	0.523472	+	0.852043i	$0.324637\pi$
$168$	0	0
$169$	−5.25622	−0.404325
$170$	−2.43447	−0.186715
$171$	0	0
$172$	5.82880	0.444442
$173$	8.12811	0.617969	0.308984	−	0.951067i	$-0.400011\pi$
$173$	8.12811	0.617969	0.308984	+	0.951067i	$0.400011\pi$
$174$	0	0
$175$	4.84682	0.366385
$176$	−3.06406	−0.230962
$177$	0	0
$178$	1.06406	0.0797543
$179$	11.4735	0.857566	0.428783	−	0.903407i	$-0.358942\pi$
$179$	11.4735	0.857566	0.428783	+	0.903407i	$0.358942\pi$
$180$	0	0
$181$	16.3123	1.21248	0.606240	−	0.795282i	$-0.292677\pi$
$181$	16.3123	1.21248	0.606240	+	0.795282i	$0.292677\pi$
$182$	2.78276	0.206272
$183$	0	0
$184$	−4.78276	−0.352590
$185$	−1.49853	−0.110174
$186$	0	0
$187$	−19.0590	−1.39373
$188$	5.71871	0.417080
$189$	0	0
$190$	0.391382	0.0283939
$191$	15.4735	1.11962	0.559810	−	0.828621i	$-0.310874\pi$
$191$	15.4735	1.11962	0.559810	+	0.828621i	$0.310874\pi$
$192$	0	0
$193$	7.00295	0.504083	0.252042	−	0.967716i	$-0.418898\pi$
$193$	7.00295	0.504083	0.252042	+	0.967716i	$0.418898\pi$
$194$	−4.50147	−0.323187
$195$	0	0
$196$	1.00000	0.0714286
$197$	10.4404	0.743845	0.371923	−	0.928264i	$-0.378699\pi$
$197$	10.4404	0.743845	0.371923	+	0.928264i	$0.378699\pi$
$198$	0	0
$199$	−6.83175	−0.484290	−0.242145	−	0.970240i	$-0.577851\pi$
$199$	−6.83175	−0.484290	−0.242145	+	0.970240i	$0.577851\pi$
$200$	4.84682	0.342722
$201$	0	0
$202$	17.1311	1.20534
$203$	7.82880	0.549474
$204$	0	0
$205$	1.71871	0.120040
$206$	2.09207	0.145762
$207$	0	0
$208$	2.78276	0.192950
$209$	3.06406	0.211945
$210$	0	0
$211$	−9.30931	−0.640879	−0.320440	−	0.947269i	$-0.603831\pi$
$211$	−9.30931	−0.640879	−0.320440	+	0.947269i	$0.603831\pi$
$212$	−7.15613	−0.491485
$213$	0	0
$214$	−14.0921	−0.963314
$215$	2.28129	0.155583
$216$	0	0
$217$	−1.34535	−0.0913280
$218$	−15.1561	−1.02650
$219$	0	0
$220$	−1.19922	−0.0808512
$221$	17.3093	1.16435
$222$	0	0
$223$	−2.31226	−0.154840	−0.0774201	−	0.996999i	$-0.524668\pi$
$223$	−2.31226	−0.154840	−0.0774201	+	0.996999i	$0.524668\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	19.7857	1.31613
$227$	−12.4404	−0.825696	−0.412848	−	0.910800i	$-0.635466\pi$
$227$	−12.4404	−0.825696	−0.412848	+	0.910800i	$0.635466\pi$
$228$	0	0
$229$	15.6086	1.03145	0.515723	−	0.856755i	$-0.327523\pi$
$229$	15.6086	1.03145	0.515723	+	0.856755i	$0.327523\pi$
$230$	−1.87189	−0.123429
$231$	0	0
$232$	7.82880	0.513986
$233$	5.28424	0.346182	0.173091	−	0.984906i	$-0.444625\pi$
$233$	5.28424	0.346182	0.173091	+	0.984906i	$0.444625\pi$
$234$	0	0
$235$	2.23820	0.146004
$236$	7.17415	0.466997
$237$	0	0
$238$	6.22018	0.403195
$239$	−1.87189	−0.121082	−0.0605412	−	0.998166i	$-0.519283\pi$
$239$	−1.87189	−0.121082	−0.0605412	+	0.998166i	$0.519283\pi$
$240$	0	0
$241$	−22.4835	−1.44829	−0.724143	−	0.689649i	$-0.757765\pi$
$241$	−22.4835	−1.44829	−0.724143	+	0.689649i	$0.757765\pi$
$242$	1.61157	0.103595
$243$	0	0
$244$	8.50147	0.544251
$245$	0.391382	0.0250045
$246$	0	0
$247$	−2.78276	−0.177063
$248$	−1.34535	−0.0854295
$249$	0	0
$250$	3.85387	0.243740
$251$	−11.6936	−0.738096	−0.369048	−	0.929410i	$-0.620316\pi$
$251$	−11.6936	−0.738096	−0.369048	+	0.929410i	$0.620316\pi$
$252$	0	0
$253$	−14.6547	−0.921330
$254$	−2.50147	−0.156956
$255$	0	0
$256$	1.00000	0.0625000
$257$	2.82585	0.176272	0.0881359	−	0.996108i	$-0.471909\pi$
$257$	2.82585	0.176272	0.0881359	+	0.996108i	$0.471909\pi$
$258$	0	0
$259$	3.82880	0.237910
$260$	1.08913	0.0675447
$261$	0	0
$262$	−15.4374	−0.953727
$263$	29.4433	1.81555	0.907776	−	0.419455i	$-0.137779\pi$
$263$	29.4433	1.81555	0.907776	+	0.419455i	$0.137779\pi$
$264$	0	0
$265$	−2.80078	−0.172051
$266$	−1.00000	−0.0613139
$267$	0	0
$268$	12.2202	0.746467
$269$	10.4404	0.636560	0.318280	−	0.947997i	$-0.396895\pi$
$269$	10.4404	0.636560	0.318280	+	0.947997i	$0.396895\pi$
$270$	0	0
$271$	−4.29931	−0.261164	−0.130582	−	0.991437i	$-0.541685\pi$
$271$	−4.29931	−0.261164	−0.130582	+	0.991437i	$0.541685\pi$
$272$	6.22018	0.377154
$273$	0	0
$274$	13.9209	0.840991
$275$	14.8509	0.895544
$276$	0	0
$277$	−28.7526	−1.72758	−0.863789	−	0.503854i	$-0.831915\pi$
$277$	−28.7526	−1.72758	−0.863789	+	0.503854i	$0.831915\pi$
$278$	−10.8748	−0.652229
$279$	0	0
$280$	0.391382	0.0233896
$281$	20.6547	1.23215	0.616077	−	0.787686i	$-0.288721\pi$
$281$	20.6547	1.23215	0.616077	+	0.787686i	$0.288721\pi$
$282$	0	0
$283$	−9.30931	−0.553381	−0.276690	−	0.960959i	$-0.589238\pi$
$283$	−9.30931	−0.553381	−0.276690	+	0.960959i	$0.589238\pi$
$284$	−0.935945	−0.0555381
$285$	0	0
$286$	8.52654	0.504185
$287$	−4.39138	−0.259215
$288$	0	0
$289$	21.6907	1.27592
$290$	3.06406	0.179928
$291$	0	0
$292$	−4.12811	−0.241579
$293$	−1.65760	−0.0968382	−0.0484191	−	0.998827i	$-0.515418\pi$
$293$	−1.65760	−0.0968382	−0.0484191	+	0.998827i	$0.515418\pi$
$294$	0	0
$295$	2.80783	0.163478
$296$	3.82880	0.222545
$297$	0	0
$298$	−1.77982	−0.103102
$299$	13.3093	0.769697
$300$	0	0
$301$	−5.82880	−0.335967
$302$	−5.30931	−0.305516
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	3.32733	0.190522
$306$	0	0
$307$	−31.0880	−1.77428	−0.887142	−	0.461496i	$-0.847313\pi$
$307$	−31.0880	−1.77428	−0.887142	+	0.461496i	$0.847313\pi$
$308$	3.06406	0.174591
$309$	0	0
$310$	−0.526544	−0.0299057
$311$	−11.8468	−0.671772	−0.335886	−	0.941903i	$-0.609036\pi$
$311$	−11.8468	−0.671772	−0.335886	+	0.941903i	$0.609036\pi$
$312$	0	0
$313$	34.6606	1.95913	0.979565	−	0.201127i	$-0.0644605\pi$
$313$	34.6606	1.95913	0.979565	+	0.201127i	$0.0644605\pi$
$314$	−15.3763	−0.867736
$315$	0	0
$316$	2.95396	0.166173
$317$	16.4654	0.924791	0.462396	−	0.886674i	$-0.346990\pi$
$317$	16.4654	0.924791	0.462396	+	0.886674i	$0.346990\pi$
$318$	0	0
$319$	23.9879	1.34306
$320$	0.391382	0.0218789
$321$	0	0
$322$	4.78276	0.266533
$323$	−6.22018	−0.346100
$324$	0	0
$325$	−13.4876	−0.748155
$326$	−17.9389	−0.993543
$327$	0	0
$328$	−4.39138	−0.242474
$329$	−5.71871	−0.315283
$330$	0	0
$331$	−3.09502	−0.170118	−0.0850589	−	0.996376i	$-0.527108\pi$
$331$	−3.09502	−0.170118	−0.0850589	+	0.996376i	$0.527108\pi$
$332$	13.5655	0.744505
$333$	0	0
$334$	−13.5295	−0.740301
$335$	4.78276	0.261310
$336$	0	0
$337$	−24.5324	−1.33637	−0.668184	−	0.743996i	$-0.732928\pi$
$337$	−24.5324	−1.33637	−0.668184	+	0.743996i	$0.732928\pi$
$338$	5.25622	0.285901
$339$	0	0
$340$	2.43447	0.132028
$341$	−4.12221	−0.223230
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−5.82880	−0.314268
$345$	0	0
$346$	−8.12811	−0.436970
$347$	−5.56553	−0.298773	−0.149387	−	0.988779i	$-0.547730\pi$
$347$	−5.56553	−0.298773	−0.149387	+	0.988779i	$0.547730\pi$
$348$	0	0
$349$	25.1311	1.34523	0.672617	−	0.739990i	$-0.265170\pi$
$349$	25.1311	1.34523	0.672617	+	0.739990i	$0.265170\pi$
$350$	−4.84682	−0.259073
$351$	0	0
$352$	3.06406	0.163315
$353$	3.52949	0.187856	0.0939280	−	0.995579i	$-0.470058\pi$
$353$	3.52949	0.187856	0.0939280	+	0.995579i	$0.470058\pi$
$354$	0	0
$355$	−0.366312	−0.0194418
$356$	−1.06406	−0.0563948
$357$	0	0
$358$	−11.4735	−0.606391
$359$	−24.9669	−1.31770	−0.658852	−	0.752273i	$-0.728958\pi$
$359$	−24.9669	−1.31770	−0.658852	+	0.752273i	$0.728958\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−16.3123	−0.857353
$363$	0	0
$364$	−2.78276	−0.145857
$365$	−1.61567	−0.0845680
$366$	0	0
$367$	−15.2842	−0.797831	−0.398915	−	0.916988i	$-0.630613\pi$
$367$	−15.2842	−0.797831	−0.398915	+	0.916988i	$0.630613\pi$
$368$	4.78276	0.249319
$369$	0	0
$370$	1.49853	0.0779046
$371$	7.15613	0.371528
$372$	0	0
$373$	−17.0519	−0.882916	−0.441458	−	0.897282i	$-0.645539\pi$
$373$	−17.0519	−0.882916	−0.441458	+	0.897282i	$0.645539\pi$
$374$	19.0590	0.985517
$375$	0	0
$376$	−5.71871	−0.294920
$377$	−21.7857	−1.12202
$378$	0	0
$379$	17.0029	0.873383	0.436691	−	0.899611i	$-0.356150\pi$
$379$	17.0029	0.873383	0.436691	+	0.899611i	$0.356150\pi$
$380$	−0.391382	−0.0200775
$381$	0	0
$382$	−15.4735	−0.791691
$383$	−29.2231	−1.49323	−0.746616	−	0.665255i	$-0.768323\pi$
$383$	−29.2231	−1.49323	−0.746616	+	0.665255i	$0.768323\pi$
$384$	0	0
$385$	1.19922	0.0611178
$386$	−7.00295	−0.356441
$387$	0	0
$388$	4.50147	0.228528
$389$	26.4404	1.34058	0.670290	−	0.742099i	$-0.266170\pi$
$389$	26.4404	1.34058	0.670290	+	0.742099i	$0.266170\pi$
$390$	0	0
$391$	29.7497	1.50451
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	−10.4404	−0.525978
$395$	1.15613	0.0581712
$396$	0	0
$397$	18.5074	0.928858	0.464429	−	0.885610i	$-0.346260\pi$
$397$	18.5074	0.928858	0.464429	+	0.885610i	$0.346260\pi$
$398$	6.83175	0.342445
$399$	0	0
$400$	−4.84682	−0.242341
$401$	32.2261	1.60929	0.804647	−	0.593754i	$-0.202355\pi$
$401$	32.2261	1.60929	0.804647	+	0.593754i	$0.202355\pi$
$402$	0	0
$403$	3.74378	0.186491
$404$	−17.1311	−0.852302
$405$	0	0
$406$	−7.82880	−0.388537
$407$	11.7317	0.581517
$408$	0	0
$409$	19.3763	0.958097	0.479049	−	0.877788i	$-0.340982\pi$
$409$	19.3763	0.958097	0.479049	+	0.877788i	$0.340982\pi$
$410$	−1.71871	−0.0848810
$411$	0	0
$412$	−2.09207	−0.103069
$413$	−7.17415	−0.353017
$414$	0	0
$415$	5.30931	0.260624
$416$	−2.78276	−0.136436
$417$	0	0
$418$	−3.06406	−0.149868
$419$	−27.5354	−1.34519	−0.672596	−	0.740010i	$-0.734821\pi$
$419$	−27.5354	−1.34519	−0.672596	+	0.740010i	$0.734821\pi$
$420$	0	0
$421$	−16.4345	−0.800967	−0.400484	−	0.916304i	$-0.631158\pi$
$421$	−16.4345	−0.800967	−0.400484	+	0.916304i	$0.631158\pi$
$422$	9.30931	0.453170
$423$	0	0
$424$	7.15613	0.347532
$425$	−30.1481	−1.46240
$426$	0	0
$427$	−8.50147	−0.411415
$428$	14.0921	0.681166
$429$	0	0
$430$	−2.28129	−0.110014
$431$	−40.9979	−1.97480	−0.987399	−	0.158249i	$-0.949415\pi$
$431$	−40.9979	−1.97480	−0.987399	+	0.158249i	$0.949415\pi$
$432$	0	0
$433$	−22.4835	−1.08049	−0.540243	−	0.841509i	$-0.681668\pi$
$433$	−22.4835	−1.08049	−0.540243	+	0.841509i	$0.681668\pi$
$434$	1.34535	0.0645786
$435$	0	0
$436$	15.1561	0.725847
$437$	−4.78276	−0.228791
$438$	0	0
$439$	−18.9109	−0.902567	−0.451283	−	0.892381i	$-0.649034\pi$
$439$	−18.9109	−0.902567	−0.451283	+	0.892381i	$0.649034\pi$
$440$	1.19922	0.0571704
$441$	0	0
$442$	−17.3093	−0.823320
$443$	10.0850	0.479154	0.239577	−	0.970877i	$-0.422991\pi$
$443$	10.0850	0.479154	0.239577	+	0.970877i	$0.422991\pi$
$444$	0	0
$445$	−0.416452	−0.0197417
$446$	2.31226	0.109489
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−17.9138	−0.845406	−0.422703	−	0.906268i	$-0.638919\pi$
$449$	−17.9138	−0.845406	−0.422703	+	0.906268i	$0.638919\pi$
$450$	0	0
$451$	−13.4554	−0.633592
$452$	−19.7857	−0.930642
$453$	0	0
$454$	12.4404	0.583855
$455$	−1.08913	−0.0510590
$456$	0	0
$457$	23.8527	1.11578	0.557892	−	0.829914i	$-0.311611\pi$
$457$	23.8527	1.11578	0.557892	+	0.829914i	$0.311611\pi$
$458$	−15.6086	−0.729343
$459$	0	0
$460$	1.87189	0.0872773
$461$	−22.2512	−1.03634	−0.518170	−	0.855278i	$-0.673386\pi$
$461$	−22.2512	−1.03634	−0.518170	+	0.855278i	$0.673386\pi$
$462$	0	0
$463$	27.8778	1.29559	0.647795	−	0.761814i	$-0.275691\pi$
$463$	27.8778	1.29559	0.647795	+	0.761814i	$0.275691\pi$
$464$	−7.82880	−0.363443
$465$	0	0
$466$	−5.28424	−0.244788
$467$	24.0980	1.11512	0.557561	−	0.830136i	$-0.311737\pi$
$467$	24.0980	1.11512	0.557561	+	0.830136i	$0.311737\pi$
$468$	0	0
$469$	−12.2202	−0.564276
$470$	−2.23820	−0.103241
$471$	0	0
$472$	−7.17415	−0.330217
$473$	−17.8598	−0.821193
$474$	0	0
$475$	4.84682	0.222387
$476$	−6.22018	−0.285102
$477$	0	0
$478$	1.87189	0.0856182
$479$	8.92382	0.407740	0.203870	−	0.978998i	$-0.434648\pi$
$479$	8.92382	0.407740	0.203870	+	0.978998i	$0.434648\pi$
$480$	0	0
$481$	−10.6547	−0.485810
$482$	22.4835	1.02409
$483$	0	0
$484$	−1.61157	−0.0732530
$485$	1.76180	0.0799991
$486$	0	0
$487$	−41.0578	−1.86051	−0.930254	−	0.366916i	$-0.880414\pi$
$487$	−41.0578	−1.86051	−0.930254	+	0.366916i	$0.880414\pi$
$488$	−8.50147	−0.384844
$489$	0	0
$490$	−0.391382	−0.0176809
$491$	20.8807	0.942334	0.471167	−	0.882044i	$-0.343833\pi$
$491$	20.8807	0.942334	0.471167	+	0.882044i	$0.343833\pi$
$492$	0	0
$493$	−48.6966	−2.19318
$494$	2.78276	0.125203
$495$	0	0
$496$	1.34535	0.0604078
$497$	0.935945	0.0419829
$498$	0	0
$499$	12.8619	0.575777	0.287889	−	0.957664i	$-0.407047\pi$
$499$	12.8619	0.575777	0.287889	+	0.957664i	$0.407047\pi$
$500$	−3.85387	−0.172350
$501$	0	0
$502$	11.6936	0.521913
$503$	22.9179	1.02186	0.510930	−	0.859622i	$-0.329301\pi$
$503$	22.9179	1.02186	0.510930	+	0.859622i	$0.329301\pi$
$504$	0	0
$505$	−6.70479	−0.298359
$506$	14.6547	0.651479
$507$	0	0
$508$	2.50147	0.110985
$509$	17.6936	0.784257	0.392128	−	0.919910i	$-0.371739\pi$
$509$	17.6936	0.784257	0.392128	+	0.919910i	$0.371739\pi$
$510$	0	0
$511$	4.12811	0.182617
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−2.82585	−0.124643
$515$	−0.818801	−0.0360807
$516$	0	0
$517$	−17.5224	−0.770636
$518$	−3.82880	−0.168228
$519$	0	0
$520$	−1.08913	−0.0477613
$521$	13.3152	0.583350	0.291675	−	0.956518i	$-0.405787\pi$
$521$	13.3152	0.583350	0.291675	+	0.956518i	$0.405787\pi$
$522$	0	0
$523$	−44.4404	−1.94324	−0.971621	−	0.236544i	$-0.923985\pi$
$523$	−44.4404	−1.94324	−0.971621	+	0.236544i	$0.923985\pi$
$524$	15.4374	0.674387
$525$	0	0
$526$	−29.4433	−1.28379
$527$	8.36830	0.364529
$528$	0	0
$529$	−0.125161	−0.00544179
$530$	2.80078	0.121658
$531$	0	0
$532$	1.00000	0.0433555
$533$	12.2202	0.529315
$534$	0	0
$535$	5.51539	0.238451
$536$	−12.2202	−0.527832
$537$	0	0
$538$	−10.4404	−0.450116
$539$	−3.06406	−0.131978
$540$	0	0
$541$	18.5626	0.798068	0.399034	−	0.916936i	$-0.369346\pi$
$541$	18.5626	0.798068	0.399034	+	0.916936i	$0.369346\pi$
$542$	4.29931	0.184671
$543$	0	0
$544$	−6.22018	−0.266688
$545$	5.93184	0.254092
$546$	0	0
$547$	−26.2762	−1.12349	−0.561745	−	0.827310i	$-0.689870\pi$
$547$	−26.2762	−1.12349	−0.561745	+	0.827310i	$0.689870\pi$
$548$	−13.9209	−0.594670
$549$	0	0
$550$	−14.8509	−0.633245
$551$	7.82880	0.333518
$552$	0	0
$553$	−2.95396	−0.125615
$554$	28.7526	1.22158
$555$	0	0
$556$	10.8748	0.461196
$557$	−32.6045	−1.38150	−0.690749	−	0.723095i	$-0.742719\pi$
$557$	−32.6045	−1.38150	−0.690749	+	0.723095i	$0.742719\pi$
$558$	0	0
$559$	16.2202	0.686041
$560$	−0.391382	−0.0165389
$561$	0	0
$562$	−20.6547	−0.871264
$563$	−35.4182	−1.49270	−0.746351	−	0.665553i	$-0.768196\pi$
$563$	−35.4182	−1.49270	−0.746351	+	0.665553i	$0.768196\pi$
$564$	0	0
$565$	−7.74378	−0.325783
$566$	9.30931	0.391299
$567$	0	0
$568$	0.935945	0.0392714
$569$	13.6936	0.574067	0.287034	−	0.957921i	$-0.407331\pi$
$569$	13.6936	0.574067	0.287034	+	0.957921i	$0.407331\pi$
$570$	0	0
$571$	−17.9389	−0.750719	−0.375360	−	0.926879i	$-0.622481\pi$
$571$	−17.9389	−0.750719	−0.375360	+	0.926879i	$0.622481\pi$
$572$	−8.52654	−0.356513
$573$	0	0
$574$	4.39138	0.183293
$575$	−23.1812	−0.966723
$576$	0	0
$577$	−45.3873	−1.88950	−0.944749	−	0.327796i	$-0.893694\pi$
$577$	−45.3873	−1.88950	−0.944749	+	0.327796i	$0.893694\pi$
$578$	−21.6907	−0.902214
$579$	0	0
$580$	−3.06406	−0.127228
$581$	−13.5655	−0.562793
$582$	0	0
$583$	21.9268	0.908114
$584$	4.12811	0.170822
$585$	0	0
$586$	1.65760	0.0684750
$587$	34.3483	1.41771	0.708853	−	0.705356i	$-0.249213\pi$
$587$	34.3483	1.41771	0.708853	+	0.705356i	$0.249213\pi$
$588$	0	0
$589$	−1.34535	−0.0554340
$590$	−2.80783	−0.115597
$591$	0	0
$592$	−3.82880	−0.157363
$593$	−6.56258	−0.269493	−0.134746	−	0.990880i	$-0.543022\pi$
$593$	−6.56258	−0.269493	−0.134746	+	0.990880i	$0.543022\pi$
$594$	0	0
$595$	−2.43447	−0.0998036
$596$	1.77982	0.0729041
$597$	0	0
$598$	−13.3093	−0.544258
$599$	8.62958	0.352595	0.176298	−	0.984337i	$-0.443588\pi$
$599$	8.62958	0.352595	0.176298	+	0.984337i	$0.443588\pi$
$600$	0	0
$601$	10.4404	0.425872	0.212936	−	0.977066i	$-0.431697\pi$
$601$	10.4404	0.425872	0.212936	+	0.977066i	$0.431697\pi$
$602$	5.82880	0.237564
$603$	0	0
$604$	5.30931	0.216033
$605$	−0.630739	−0.0256432
$606$	0	0
$607$	−1.43152	−0.0581037	−0.0290518	−	0.999578i	$-0.509249\pi$
$607$	−1.43152	−0.0581037	−0.0290518	+	0.999578i	$0.509249\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	−3.32733	−0.134720
$611$	15.9138	0.643804
$612$	0	0
$613$	−41.0950	−1.65981	−0.829906	−	0.557903i	$-0.811606\pi$
$613$	−41.0950	−1.65981	−0.829906	+	0.557903i	$0.811606\pi$
$614$	31.0880	1.25461
$615$	0	0
$616$	−3.06406	−0.123454
$617$	−27.1440	−1.09278	−0.546388	−	0.837532i	$-0.683998\pi$
$617$	−27.1440	−1.09278	−0.546388	+	0.837532i	$0.683998\pi$
$618$	0	0
$619$	1.83585	0.0737892	0.0368946	−	0.999319i	$-0.488253\pi$
$619$	1.83585	0.0737892	0.0368946	+	0.999319i	$0.488253\pi$
$620$	0.526544	0.0211465
$621$	0	0
$622$	11.8468	0.475014
$623$	1.06406	0.0426305
$624$	0	0
$625$	22.7258	0.909030
$626$	−34.6606	−1.38531
$627$	0	0
$628$	15.3763	0.613582
$629$	−23.8159	−0.949600
$630$	0	0
$631$	43.5714	1.73455	0.867276	−	0.497828i	$-0.165869\pi$
$631$	43.5714	1.73455	0.867276	+	0.497828i	$0.165869\pi$
$632$	−2.95396	−0.117502
$633$	0	0
$634$	−16.4654	−0.653926
$635$	0.979033	0.0388517
$636$	0	0
$637$	2.78276	0.110257
$638$	−23.9879	−0.949689
$639$	0	0
$640$	−0.391382	−0.0154707
$641$	−43.6275	−1.72318	−0.861591	−	0.507604i	$-0.830531\pi$
$641$	−43.6275	−1.72318	−0.861591	+	0.507604i	$0.830531\pi$
$642$	0	0
$643$	14.6907	0.579344	0.289672	−	0.957126i	$-0.406454\pi$
$643$	14.6907	0.579344	0.289672	+	0.957126i	$0.406454\pi$
$644$	−4.78276	−0.188467
$645$	0	0
$646$	6.22018	0.244730
$647$	−19.6145	−0.771126	−0.385563	−	0.922681i	$-0.625993\pi$
$647$	−19.6145	−0.771126	−0.385563	+	0.922681i	$0.625993\pi$
$648$	0	0
$649$	−21.9820	−0.862868
$650$	13.4876	0.529026
$651$	0	0
$652$	17.9389	0.702541
$653$	−42.1700	−1.65024	−0.825121	−	0.564957i	$-0.808893\pi$
$653$	−42.1700	−1.65024	−0.825121	+	0.564957i	$0.808893\pi$
$654$	0	0
$655$	6.04193	0.236078
$656$	4.39138	0.171455
$657$	0	0
$658$	5.71871	0.222939
$659$	−34.1900	−1.33186	−0.665928	−	0.746016i	$-0.731964\pi$
$659$	−34.1900	−1.33186	−0.665928	+	0.746016i	$0.731964\pi$
$660$	0	0
$661$	−31.1370	−1.21109	−0.605544	−	0.795812i	$-0.707044\pi$
$661$	−31.1370	−1.21109	−0.605544	+	0.795812i	$0.707044\pi$
$662$	3.09502	0.120291
$663$	0	0
$664$	−13.5655	−0.526445
$665$	0.391382	0.0151772
$666$	0	0
$667$	−37.4433	−1.44981
$668$	13.5295	0.523472
$669$	0	0
$670$	−4.78276	−0.184774
$671$	−26.0490	−1.00561
$672$	0	0
$673$	−48.6045	−1.87357	−0.936783	−	0.349910i	$-0.886212\pi$
$673$	−48.6045	−1.87357	−0.936783	+	0.349910i	$0.886212\pi$
$674$	24.5324	0.944954
$675$	0	0
$676$	−5.25622	−0.202162
$677$	−9.81585	−0.377254	−0.188627	−	0.982049i	$-0.560404\pi$
$677$	−9.81585	−0.377254	−0.188627	+	0.982049i	$0.560404\pi$
$678$	0	0
$679$	−4.50147	−0.172751
$680$	−2.43447	−0.0933577
$681$	0	0
$682$	4.12221	0.157848
$683$	12.6606	0.484443	0.242221	−	0.970221i	$-0.422124\pi$
$683$	12.6606	0.484443	0.242221	+	0.970221i	$0.422124\pi$
$684$	0	0
$685$	−5.44839	−0.208172
$686$	1.00000	0.0381802
$687$	0	0
$688$	5.82880	0.222221
$689$	−19.9138	−0.758656
$690$	0	0
$691$	45.4073	1.72737	0.863687	−	0.504028i	$-0.168149\pi$
$691$	45.4073	1.72737	0.863687	+	0.504028i	$0.168149\pi$
$692$	8.12811	0.308984
$693$	0	0
$694$	5.56553	0.211265
$695$	4.25622	0.161448
$696$	0	0
$697$	27.3152	1.03464
$698$	−25.1311	−0.951225
$699$	0	0
$700$	4.84682	0.183193
$701$	11.1871	0.422531	0.211265	−	0.977429i	$-0.432242\pi$
$701$	11.1871	0.422531	0.211265	+	0.977429i	$0.432242\pi$
$702$	0	0
$703$	3.82880	0.144406
$704$	−3.06406	−0.115481
$705$	0	0
$706$	−3.52949	−0.132834
$707$	17.1311	0.644280
$708$	0	0
$709$	13.4014	0.503300	0.251650	−	0.967818i	$-0.419027\pi$
$709$	13.4014	0.503300	0.251650	+	0.967818i	$0.419027\pi$
$710$	0.366312	0.0137475
$711$	0	0
$712$	1.06406	0.0398771
$713$	6.43447	0.240973
$714$	0	0
$715$	−3.33714	−0.124802
$716$	11.4735	0.428783
$717$	0	0
$718$	24.9669	0.931757
$719$	16.0000	0.596699	0.298350	−	0.954457i	$-0.403564\pi$
$719$	16.0000	0.596699	0.298350	+	0.954457i	$0.403564\pi$
$720$	0	0
$721$	2.09207	0.0779129
$722$	−1.00000	−0.0372161
$723$	0	0
$724$	16.3123	0.606240
$725$	37.9448	1.40923
$726$	0	0
$727$	−42.4153	−1.57310	−0.786548	−	0.617529i	$-0.788134\pi$
$727$	−42.4153	−1.57310	−0.786548	+	0.617529i	$0.788134\pi$
$728$	2.78276	0.103136
$729$	0	0
$730$	1.61567	0.0597986
$731$	36.2562	1.34098
$732$	0	0
$733$	−43.9628	−1.62380	−0.811902	−	0.583794i	$-0.801568\pi$
$733$	−43.9628	−1.62380	−0.811902	+	0.583794i	$0.801568\pi$
$734$	15.2842	0.564152
$735$	0	0
$736$	−4.78276	−0.176295
$737$	−37.4433	−1.37924
$738$	0	0
$739$	1.57258	0.0578483	0.0289242	−	0.999582i	$-0.490792\pi$
$739$	1.57258	0.0578483	0.0289242	+	0.999582i	$0.490792\pi$
$740$	−1.49853	−0.0550869
$741$	0	0
$742$	−7.15613	−0.262710
$743$	−37.0460	−1.35909	−0.679544	−	0.733635i	$-0.737822\pi$
$743$	−37.0460	−1.35909	−0.679544	+	0.733635i	$0.737822\pi$
$744$	0	0
$745$	0.696589	0.0255210
$746$	17.0519	0.624316
$747$	0	0
$748$	−19.0590	−0.696866
$749$	−14.0921	−0.514913
$750$	0	0
$751$	15.7367	0.574241	0.287121	−	0.957894i	$-0.407302\pi$
$751$	15.7367	0.574241	0.287121	+	0.957894i	$0.407302\pi$
$752$	5.71871	0.208540
$753$	0	0
$754$	21.7857	0.793389
$755$	2.07797	0.0756251
$756$	0	0
$757$	33.3512	1.21217	0.606086	−	0.795399i	$-0.292739\pi$
$757$	33.3512	1.21217	0.606086	+	0.795399i	$0.292739\pi$
$758$	−17.0029	−0.617575
$759$	0	0
$760$	0.391382	0.0141969
$761$	−20.0059	−0.725213	−0.362607	−	0.931942i	$-0.618113\pi$
$761$	−20.0059	−0.725213	−0.362607	+	0.931942i	$0.618113\pi$
$762$	0	0
$763$	−15.1561	−0.548689
$764$	15.4735	0.559810
$765$	0	0
$766$	29.2231	1.05587
$767$	19.9640	0.720857
$768$	0	0
$769$	55.0029	1.98346	0.991729	−	0.128353i	$-0.0409691\pi$
$769$	55.0029	1.98346	0.991729	+	0.128353i	$0.0409691\pi$
$770$	−1.19922	−0.0432168
$771$	0	0
$772$	7.00295	0.252042
$773$	−32.7526	−1.17803	−0.589015	−	0.808122i	$-0.700484\pi$
$773$	−32.7526	−1.17803	−0.589015	+	0.808122i	$0.700484\pi$
$774$	0	0
$775$	−6.52065	−0.234229
$776$	−4.50147	−0.161594
$777$	0	0
$778$	−26.4404	−0.947933
$779$	−4.39138	−0.157338
$780$	0	0
$781$	2.86779	0.102617
$782$	−29.7497	−1.06385
$783$	0	0
$784$	1.00000	0.0357143
$785$	6.01802	0.214792
$786$	0	0
$787$	16.2993	0.581008	0.290504	−	0.956874i	$-0.406177\pi$
$787$	16.2993	0.581008	0.290504	+	0.956874i	$0.406177\pi$
$788$	10.4404	0.371923
$789$	0	0
$790$	−1.15613	−0.0411332
$791$	19.7857	0.703499
$792$	0	0
$793$	23.6576	0.840106
$794$	−18.5074	−0.656802
$795$	0	0
$796$	−6.83175	−0.242145
$797$	−9.50949	−0.336843	−0.168422	−	0.985715i	$-0.553867\pi$
$797$	−9.50949	−0.336843	−0.168422	+	0.985715i	$0.553867\pi$
$798$	0	0
$799$	35.5714	1.25843
$800$	4.84682	0.171361
$801$	0	0
$802$	−32.2261	−1.13794
$803$	12.6488	0.446365
$804$	0	0
$805$	−1.87189	−0.0659754
$806$	−3.74378	−0.131869
$807$	0	0
$808$	17.1311	0.602669
$809$	20.0129	0.703618	0.351809	−	0.936072i	$-0.385567\pi$
$809$	20.0129	0.703618	0.351809	+	0.936072i	$0.385567\pi$
$810$	0	0
$811$	49.8527	1.75057	0.875283	−	0.483611i	$-0.160675\pi$
$811$	49.8527	1.75057	0.875283	+	0.483611i	$0.160675\pi$
$812$	7.82880	0.274737
$813$	0	0
$814$	−11.7317	−0.411194
$815$	7.02097	0.245934
$816$	0	0
$817$	−5.82880	−0.203924
$818$	−19.3763	−0.677477
$819$	0	0
$820$	1.71871	0.0600199
$821$	2.68479	0.0936999	0.0468500	−	0.998902i	$-0.485082\pi$
$821$	2.68479	0.0936999	0.0468500	+	0.998902i	$0.485082\pi$
$822$	0	0
$823$	−28.4764	−0.992625	−0.496313	−	0.868144i	$-0.665313\pi$
$823$	−28.4764	−0.992625	−0.496313	+	0.868144i	$0.665313\pi$
$824$	2.09207	0.0728809
$825$	0	0
$826$	7.17415	0.249621
$827$	29.0390	1.00978	0.504892	−	0.863182i	$-0.331532\pi$
$827$	29.0390	1.00978	0.504892	+	0.863182i	$0.331532\pi$
$828$	0	0
$829$	−19.7136	−0.684683	−0.342342	−	0.939576i	$-0.611220\pi$
$829$	−19.7136	−0.684683	−0.342342	+	0.939576i	$0.611220\pi$
$830$	−5.30931	−0.184289
$831$	0	0
$832$	2.78276	0.0964750
$833$	6.22018	0.215517
$834$	0	0
$835$	5.29521	0.183248
$836$	3.06406	0.105973
$837$	0	0
$838$	27.5354	0.951194
$839$	18.9109	0.652876	0.326438	−	0.945219i	$-0.394152\pi$
$839$	18.9109	0.652876	0.326438	+	0.945219i	$0.394152\pi$
$840$	0	0
$841$	32.2901	1.11345
$842$	16.4345	0.566369
$843$	0	0
$844$	−9.30931	−0.320440
$845$	−2.05719	−0.0707696
$846$	0	0
$847$	1.61157	0.0553741
$848$	−7.15613	−0.245742
$849$	0	0
$850$	30.1481	1.03407
$851$	−18.3123	−0.627736
$852$	0	0
$853$	−23.7928	−0.814649	−0.407324	−	0.913284i	$-0.633538\pi$
$853$	−23.7928	−0.814649	−0.407324	+	0.913284i	$0.633538\pi$
$854$	8.50147	0.290914
$855$	0	0
$856$	−14.0921	−0.481657
$857$	−10.0000	−0.341593	−0.170797	−	0.985306i	$-0.554634\pi$
$857$	−10.0000	−0.341593	−0.170797	+	0.985306i	$0.554634\pi$
$858$	0	0
$859$	−12.9669	−0.442425	−0.221213	−	0.975226i	$-0.571001\pi$
$859$	−12.9669	−0.442425	−0.221213	+	0.975226i	$0.571001\pi$
$860$	2.28129	0.0777914
$861$	0	0
$862$	40.9979	1.39639
$863$	−23.0519	−0.784697	−0.392348	−	0.919817i	$-0.628337\pi$
$863$	−23.0519	−0.784697	−0.392348	+	0.919817i	$0.628337\pi$
$864$	0	0
$865$	3.18120	0.108164
$866$	22.4835	0.764019
$867$	0	0
$868$	−1.34535	−0.0456640
$869$	−9.05111	−0.307038
$870$	0	0
$871$	34.0059	1.15225
$872$	−15.1561	−0.513251
$873$	0	0
$874$	4.78276	0.161779
$875$	3.85387	0.130285
$876$	0	0
$877$	17.8468	0.602644	0.301322	−	0.953522i	$-0.402572\pi$
$877$	17.8468	0.602644	0.301322	+	0.953522i	$0.402572\pi$
$878$	18.9109	0.638211
$879$	0	0
$880$	−1.19922	−0.0404256
$881$	−6.71069	−0.226089	−0.113044	−	0.993590i	$-0.536060\pi$
$881$	−6.71069	−0.226089	−0.113044	+	0.993590i	$0.536060\pi$
$882$	0	0
$883$	13.3144	0.448064	0.224032	−	0.974582i	$-0.428078\pi$
$883$	13.3144	0.448064	0.224032	+	0.974582i	$0.428078\pi$
$884$	17.3093	0.582175
$885$	0	0
$886$	−10.0850	−0.338813
$887$	−17.2733	−0.579980	−0.289990	−	0.957030i	$-0.593652\pi$
$887$	−17.2733	−0.579980	−0.289990	+	0.957030i	$0.593652\pi$
$888$	0	0
$889$	−2.50147	−0.0838968
$890$	0.416452	0.0139595
$891$	0	0
$892$	−2.31226	−0.0774201
$893$	−5.71871	−0.191369
$894$	0	0
$895$	4.49051	0.150101
$896$	1.00000	0.0334077
$897$	0	0
$898$	17.9138	0.597792
$899$	−10.5324	−0.351277
$900$	0	0
$901$	−44.5124	−1.48292
$902$	13.4554	0.448017
$903$	0	0
$904$	19.7857	0.658063
$905$	6.38433	0.212222
$906$	0	0
$907$	6.99705	0.232333	0.116167	−	0.993230i	$-0.462939\pi$
$907$	6.99705	0.232333	0.116167	+	0.993230i	$0.462939\pi$
$908$	−12.4404	−0.412848
$909$	0	0
$910$	1.08913	0.0361041
$911$	−24.1050	−0.798635	−0.399318	−	0.916813i	$-0.630753\pi$
$911$	−24.1050	−0.798635	−0.399318	+	0.916813i	$0.630753\pi$
$912$	0	0
$913$	−41.5655	−1.37562
$914$	−23.8527	−0.788978
$915$	0	0
$916$	15.6086	0.515723
$917$	−15.4374	−0.509789
$918$	0	0
$919$	−55.3011	−1.82422	−0.912108	−	0.409951i	$-0.865546\pi$
$919$	−55.3011	−1.82422	−0.912108	+	0.409951i	$0.865546\pi$
$920$	−1.87189	−0.0617144
$921$	0	0
$922$	22.2512	0.732803
$923$	−2.60451	−0.0857286
$924$	0	0
$925$	18.5575	0.610167
$926$	−27.8778	−0.916121
$927$	0	0
$928$	7.82880	0.256993
$929$	−4.60451	−0.151069	−0.0755346	−	0.997143i	$-0.524066\pi$
$929$	−4.60451	−0.151069	−0.0755346	+	0.997143i	$0.524066\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	5.28424	0.173091
$933$	0	0
$934$	−24.0980	−0.788510
$935$	−7.45935	−0.243947
$936$	0	0
$937$	45.7195	1.49359	0.746796	−	0.665053i	$-0.231591\pi$
$937$	45.7195	1.49359	0.746796	+	0.665053i	$0.231591\pi$
$938$	12.2202	0.399003
$939$	0	0
$940$	2.23820	0.0730021
$941$	−32.5324	−1.06053	−0.530264	−	0.847833i	$-0.677907\pi$
$941$	−32.5324	−1.06053	−0.530264	+	0.847833i	$0.677907\pi$
$942$	0	0
$943$	21.0029	0.683950
$944$	7.17415	0.233499
$945$	0	0
$946$	17.8598	0.580671
$947$	35.0641	1.13943	0.569714	−	0.821843i	$-0.307054\pi$
$947$	35.0641	1.13943	0.569714	+	0.821843i	$0.307054\pi$
$948$	0	0
$949$	−11.4876	−0.372902
$950$	−4.84682	−0.157252
$951$	0	0
$952$	6.22018	0.201597
$953$	−15.9941	−0.518100	−0.259050	−	0.965864i	$-0.583409\pi$
$953$	−15.9941	−0.518100	−0.259050	+	0.965864i	$0.583409\pi$
$954$	0	0
$955$	6.05604	0.195969
$956$	−1.87189	−0.0605412
$957$	0	0
$958$	−8.92382	−0.288316
$959$	13.9209	0.449529
$960$	0	0
$961$	−29.1900	−0.941614
$962$	10.6547	0.343520
$963$	0	0
$964$	−22.4835	−0.724143
$965$	2.74083	0.0882305
$966$	0	0
$967$	16.0721	0.516843	0.258422	−	0.966032i	$-0.416798\pi$
$967$	16.0721	0.516843	0.258422	+	0.966032i	$0.416798\pi$
$968$	1.61157	0.0517977
$969$	0	0
$970$	−1.76180	−0.0565679
$971$	−34.6715	−1.11266	−0.556331	−	0.830961i	$-0.687791\pi$
$971$	−34.6715	−1.11266	−0.556331	+	0.830961i	$0.687791\pi$
$972$	0	0
$973$	−10.8748	−0.348631
$974$	41.0578	1.31558
$975$	0	0
$976$	8.50147	0.272126
$977$	12.5685	0.402101	0.201051	−	0.979581i	$-0.435564\pi$
$977$	12.5685	0.402101	0.201051	+	0.979581i	$0.435564\pi$
$978$	0	0
$979$	3.26032	0.104200
$980$	0.391382	0.0125023
$981$	0	0
$982$	−20.8807	−0.666331
$983$	−1.56553	−0.0499326	−0.0249663	−	0.999688i	$-0.507948\pi$
$983$	−1.56553	−0.0499326	−0.0249663	+	0.999688i	$0.507948\pi$
$984$	0	0
$985$	4.08618	0.130196
$986$	48.6966	1.55082
$987$	0	0
$988$	−2.78276	−0.0885315
$989$	27.8778	0.886462
$990$	0	0
$991$	−42.6355	−1.35436	−0.677180	−	0.735817i	$-0.736798\pi$
$991$	−42.6355	−1.35436	−0.677180	+	0.735817i	$0.736798\pi$
$992$	−1.34535	−0.0427148
$993$	0	0
$994$	−0.935945	−0.0296864
$995$	−2.67383	−0.0847660
$996$	0	0
$997$	61.2721	1.94051	0.970254	−	0.242090i	$-0.0778330\pi$
$997$	61.2721	1.94051	0.970254	+	0.242090i	$0.0778330\pi$
$998$	−12.8619	−0.407136
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2394.2.a.ba.1.3		3
3.2	odd	2		266.2.a.d.1.3	✓	3
12.11	even	2		2128.2.a.s.1.1		3
15.14	odd	2		6650.2.a.cd.1.1		3
21.20	even	2		1862.2.a.r.1.1		3
24.5	odd	2		8512.2.a.bm.1.1		3
24.11	even	2		8512.2.a.bj.1.3		3
57.56	even	2		5054.2.a.r.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
266.2.a.d.1.3	✓	3	3.2	odd	2
1862.2.a.r.1.1		3	21.20	even	2
2128.2.a.s.1.1		3	12.11	even	2
2394.2.a.ba.1.3		3	1.1	even	1	trivial
5054.2.a.r.1.1		3	57.56	even	2
6650.2.a.cd.1.1		3	15.14	odd	2
8512.2.a.bj.1.3		3	24.11	even	2
8512.2.a.bm.1.1		3	24.5	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2394.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +0.391382 q^{5} -1.00000 q^{7} -1.00000 q^{8} -0.391382 q^{10} -3.06406 q^{11} +2.78276 q^{13} +1.00000 q^{14} +1.00000 q^{16} +6.22018 q^{17} -1.00000 q^{19} +0.391382 q^{20} +3.06406 q^{22} +4.78276 q^{23} -4.84682 q^{25} -2.78276 q^{26} -1.00000 q^{28} -7.82880 q^{29} +1.34535 q^{31} -1.00000 q^{32} -6.22018 q^{34} -0.391382 q^{35} -3.82880 q^{37} +1.00000 q^{38} -0.391382 q^{40} +4.39138 q^{41} +5.82880 q^{43} -3.06406 q^{44} -4.78276 q^{46} +5.71871 q^{47} +1.00000 q^{49} +4.84682 q^{50} +2.78276 q^{52} -7.15613 q^{53} -1.19922 q^{55} +1.00000 q^{56} +7.82880 q^{58} +7.17415 q^{59} +8.50147 q^{61} -1.34535 q^{62} +1.00000 q^{64} +1.08913 q^{65} +12.2202 q^{67} +6.22018 q^{68} +0.391382 q^{70} -0.935945 q^{71} -4.12811 q^{73} +3.82880 q^{74} -1.00000 q^{76} +3.06406 q^{77} +2.95396 q^{79} +0.391382 q^{80} -4.39138 q^{82} +13.5655 q^{83} +2.43447 q^{85} -5.82880 q^{86} +3.06406 q^{88} -1.06406 q^{89} -2.78276 q^{91} +4.78276 q^{92} -5.71871 q^{94} -0.391382 q^{95} +4.50147 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} - 5 q^{5} - 3 q^{7} - 3 q^{8} + 5 q^{10} - 3 q^{11} - 4 q^{13} + 3 q^{14} + 3 q^{16} - 6 q^{17} - 3 q^{19} - 5 q^{20} + 3 q^{22} + 2 q^{23} + 4 q^{25} + 4 q^{26} - 3 q^{28} - 5 q^{29}+ \cdots - 3 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 - 5 * q^5 - 3 * q^7 - 3 * q^8 + 5 * q^10 - 3 * q^11 - 4 * q^13 + 3 * q^14 + 3 * q^16 - 6 * q^17 - 3 * q^19 - 5 * q^20 + 3 * q^22 + 2 * q^23 + 4 * q^25 + 4 * q^26 - 3 * q^28 - 5 * q^29 + 4 * q^31 - 3 * q^32 + 6 * q^34 + 5 * q^35 + 7 * q^37 + 3 * q^38 + 5 * q^40 + 7 * q^41 - q^43 - 3 * q^44 - 2 * q^46 + 11 * q^47 + 3 * q^49 - 4 * q^50 - 4 * q^52 - 3 * q^53 - 16 * q^55 + 3 * q^56 + 5 * q^58 + 3 * q^59 + 7 * q^61 - 4 * q^62 + 3 * q^64 + 28 * q^65 + 12 * q^67 - 6 * q^68 - 5 * q^70 - 9 * q^71 - 7 * q^74 - 3 * q^76 + 3 * q^77 + 15 * q^79 - 5 * q^80 - 7 * q^82 + 16 * q^83 + 32 * q^85 + q^86 + 3 * q^88 + 3 * q^89 + 4 * q^91 + 2 * q^92 - 11 * q^94 + 5 * q^95 - 5 * q^97 - 3 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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