LMFDB - Embedded newform 5194.2.a.bi.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5194,2,Mod(1,5194)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5194, 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("5194.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5194 = 2 \cdot 7^{2} \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5194.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:	$41.4742988099$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 1 $ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.93185$ of defining polynomial
Character	$\chi$	$=$	5194.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.93185 q^{3} +1.00000 q^{4} +3.86370 q^{5} -1.93185 q^{6} -1.00000 q^{8} +0.732051 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.93185 q^{3} +1.00000 q^{4} +3.86370 q^{5} -1.93185 q^{6} -1.00000 q^{8} +0.732051 q^{9} -3.86370 q^{10} +3.46410 q^{11} +1.93185 q^{12} +3.34607 q^{13} +7.46410 q^{15} +1.00000 q^{16} -0.896575 q^{17} -0.732051 q^{18} +0.517638 q^{19} +3.86370 q^{20} -3.46410 q^{22} +3.00000 q^{23} -1.93185 q^{24} +9.92820 q^{25} -3.34607 q^{26} -4.38134 q^{27} -1.73205 q^{29} -7.46410 q^{30} +4.52004 q^{31} -1.00000 q^{32} +6.69213 q^{33} +0.896575 q^{34} +0.732051 q^{36} -3.00000 q^{37} -0.517638 q^{38} +6.46410 q^{39} -3.86370 q^{40} +1.03528 q^{41} -2.00000 q^{43} +3.46410 q^{44} +2.82843 q^{45} -3.00000 q^{46} +11.3137 q^{47} +1.93185 q^{48} -9.92820 q^{50} -1.73205 q^{51} +3.34607 q^{52} +1.00000 q^{53} +4.38134 q^{54} +13.3843 q^{55} +1.00000 q^{57} +1.73205 q^{58} -7.45001 q^{59} +7.46410 q^{60} -4.52004 q^{62} +1.00000 q^{64} +12.9282 q^{65} -6.69213 q^{66} -7.46410 q^{67} -0.896575 q^{68} +5.79555 q^{69} -13.1962 q^{71} -0.732051 q^{72} -7.72741 q^{73} +3.00000 q^{74} +19.1798 q^{75} +0.517638 q^{76} -6.46410 q^{78} -11.0000 q^{79} +3.86370 q^{80} -10.6603 q^{81} -1.03528 q^{82} +7.58871 q^{83} -3.46410 q^{85} +2.00000 q^{86} -3.34607 q^{87} -3.46410 q^{88} -1.41421 q^{89} -2.82843 q^{90} +3.00000 q^{92} +8.73205 q^{93} -11.3137 q^{94} +2.00000 q^{95} -1.93185 q^{96} +13.1440 q^{97} +2.53590 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 - 4 * q^9	$ 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} - 4 q^{9} + 16 q^{15} + 4 q^{16} + 4 q^{18} + 12 q^{23} + 12 q^{25} - 16 q^{30} - 4 q^{32} - 4 q^{36} - 12 q^{37} + 12 q^{39} - 8 q^{43} - 12 q^{46} - 12 q^{50} + 4 q^{53} + 4 q^{57} + 16 q^{60} + 4 q^{64} + 24 q^{65} - 16 q^{67} - 32 q^{71} + 4 q^{72} + 12 q^{74} - 12 q^{78} - 44 q^{79} - 8 q^{81} + 8 q^{86} + 12 q^{92} + 28 q^{93} + 8 q^{95} + 24 q^{99}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 - 4 * q^9 + 16 * q^15 + 4 * q^16 + 4 * q^18 + 12 * q^23 + 12 * q^25 - 16 * q^30 - 4 * q^32 - 4 * q^36 - 12 * q^37 + 12 * q^39 - 8 * q^43 - 12 * q^46 - 12 * q^50 + 4 * q^53 + 4 * q^57 + 16 * q^60 + 4 * q^64 + 24 * q^65 - 16 * q^67 - 32 * q^71 + 4 * q^72 + 12 * q^74 - 12 * q^78 - 44 * q^79 - 8 * q^81 + 8 * q^86 + 12 * q^92 + 28 * q^93 + 8 * q^95 + 24 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.93185	1.11536	0.557678	−	0.830058i	$-0.311693\pi$
$3$	1.93185	1.11536	0.557678	+	0.830058i	$0.311693\pi$
$4$	1.00000	0.500000
$5$	3.86370	1.72790	0.863950	−	0.503577i	$-0.167983\pi$
$5$	3.86370	1.72790	0.863950	+	0.503577i	$0.167983\pi$
$6$	−1.93185	−0.788675
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	0.732051	0.244017
$10$	−3.86370	−1.22181
$11$	3.46410	1.04447	0.522233	−	0.852803i	$-0.325099\pi$
$11$	3.46410	1.04447	0.522233	+	0.852803i	$0.325099\pi$
$12$	1.93185	0.557678
$13$	3.34607	0.928032	0.464016	−	0.885827i	$-0.346408\pi$
$13$	3.34607	0.928032	0.464016	+	0.885827i	$0.346408\pi$
$14$	0	0
$15$	7.46410	1.92722
$16$	1.00000	0.250000
$17$	−0.896575	−0.217451	−0.108726	−	0.994072i	$-0.534677\pi$
$17$	−0.896575	−0.217451	−0.108726	+	0.994072i	$0.534677\pi$
$18$	−0.732051	−0.172546
$19$	0.517638	0.118754	0.0593772	−	0.998236i	$-0.481089\pi$
$19$	0.517638	0.118754	0.0593772	+	0.998236i	$0.481089\pi$
$20$	3.86370	0.863950
$21$	0	0
$22$	−3.46410	−0.738549
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	−1.93185	−0.394338
$25$	9.92820	1.98564
$26$	−3.34607	−0.656217
$27$	−4.38134	−0.843190
$28$	0	0
$29$	−1.73205	−0.321634	−0.160817	−	0.986984i	$-0.551413\pi$
$29$	−1.73205	−0.321634	−0.160817	+	0.986984i	$0.551413\pi$
$30$	−7.46410	−1.36275
$31$	4.52004	0.811824	0.405912	−	0.913912i	$-0.366954\pi$
$31$	4.52004	0.811824	0.405912	+	0.913912i	$0.366954\pi$
$32$	−1.00000	−0.176777
$33$	6.69213	1.16495
$34$	0.896575	0.153761
$35$	0	0
$36$	0.732051	0.122008
$37$	−3.00000	−0.493197	−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000	−0.493197	−0.246598	+	0.969118i	$0.579313\pi$
$38$	−0.517638	−0.0839720
$39$	6.46410	1.03508
$40$	−3.86370	−0.610905
$41$	1.03528	0.161683	0.0808415	−	0.996727i	$-0.474239\pi$
$41$	1.03528	0.161683	0.0808415	+	0.996727i	$0.474239\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	3.46410	0.522233
$45$	2.82843	0.421637
$46$	−3.00000	−0.442326
$47$	11.3137	1.65027	0.825137	−	0.564933i	$-0.191098\pi$
$47$	11.3137	1.65027	0.825137	+	0.564933i	$0.191098\pi$
$48$	1.93185	0.278839
$49$	0	0
$50$	−9.92820	−1.40406
$51$	−1.73205	−0.242536
$52$	3.34607	0.464016
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	4.38134	0.596225
$55$	13.3843	1.80473
$56$	0	0
$57$	1.00000	0.132453
$58$	1.73205	0.227429
$59$	−7.45001	−0.969908	−0.484954	−	0.874540i	$-0.661164\pi$
$59$	−7.45001	−0.969908	−0.484954	+	0.874540i	$0.661164\pi$
$60$	7.46410	0.963611
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	−4.52004	−0.574046
$63$	0	0
$64$	1.00000	0.125000
$65$	12.9282	1.60355
$66$	−6.69213	−0.823744
$67$	−7.46410	−0.911885	−0.455943	−	0.890009i	$-0.650698\pi$
$67$	−7.46410	−0.911885	−0.455943	+	0.890009i	$0.650698\pi$
$68$	−0.896575	−0.108726
$69$	5.79555	0.697703
$70$	0	0
$71$	−13.1962	−1.56610	−0.783048	−	0.621962i	$-0.786336\pi$
$71$	−13.1962	−1.56610	−0.783048	+	0.621962i	$0.786336\pi$
$72$	−0.732051	−0.0862730
$73$	−7.72741	−0.904425	−0.452212	−	0.891910i	$-0.649365\pi$
$73$	−7.72741	−0.904425	−0.452212	+	0.891910i	$0.649365\pi$
$74$	3.00000	0.348743
$75$	19.1798	2.21469
$76$	0.517638	0.0593772
$77$	0	0
$78$	−6.46410	−0.731915
$79$	−11.0000	−1.23760	−0.618798	−	0.785550i	$-0.712380\pi$
$79$	−11.0000	−1.23760	−0.618798	+	0.785550i	$0.712380\pi$
$80$	3.86370	0.431975
$81$	−10.6603	−1.18447
$82$	−1.03528	−0.114327
$83$	7.58871	0.832969	0.416484	−	0.909143i	$-0.363262\pi$
$83$	7.58871	0.832969	0.416484	+	0.909143i	$0.363262\pi$
$84$	0	0
$85$	−3.46410	−0.375735
$86$	2.00000	0.215666
$87$	−3.34607	−0.358736
$88$	−3.46410	−0.369274
$89$	−1.41421	−0.149906	−0.0749532	−	0.997187i	$-0.523881\pi$
$89$	−1.41421	−0.149906	−0.0749532	+	0.997187i	$0.523881\pi$
$90$	−2.82843	−0.298142
$91$	0	0
$92$	3.00000	0.312772
$93$	8.73205	0.905471
$94$	−11.3137	−1.16692
$95$	2.00000	0.205196
$96$	−1.93185	−0.197169
$97$	13.1440	1.33457	0.667287	−	0.744801i	$-0.267456\pi$
$97$	13.1440	1.33457	0.667287	+	0.744801i	$0.267456\pi$
$98$	0	0
$99$	2.53590	0.254867
$100$	9.92820	0.992820
$101$	−9.52056	−0.947331	−0.473665	−	0.880705i	$-0.657069\pi$
$101$	−9.52056	−0.947331	−0.473665	+	0.880705i	$0.657069\pi$
$102$	1.73205	0.171499
$103$	8.24504	0.812408	0.406204	−	0.913782i	$-0.366852\pi$
$103$	8.24504	0.812408	0.406204	+	0.913782i	$0.366852\pi$
$104$	−3.34607	−0.328109
$105$	0	0
$106$	−1.00000	−0.0971286
$107$	−1.46410	−0.141540	−0.0707700	−	0.997493i	$-0.522546\pi$
$107$	−1.46410	−0.141540	−0.0707700	+	0.997493i	$0.522546\pi$
$108$	−4.38134	−0.421595
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	−13.3843	−1.27614
$111$	−5.79555	−0.550090
$112$	0	0
$113$	0.0717968	0.00675407	0.00337704	−	0.999994i	$-0.498925\pi$
$113$	0.0717968	0.00675407	0.00337704	+	0.999994i	$0.498925\pi$
$114$	−1.00000	−0.0936586
$115$	11.5911	1.08088
$116$	−1.73205	−0.160817
$117$	2.44949	0.226455
$118$	7.45001	0.685829
$119$	0	0
$120$	−7.46410	−0.681376
$121$	1.00000	0.0909091
$122$	0	0
$123$	2.00000	0.180334
$124$	4.52004	0.405912
$125$	19.0411	1.70309
$126$	0	0
$127$	6.07180	0.538785	0.269392	−	0.963030i	$-0.413177\pi$
$127$	6.07180	0.538785	0.269392	+	0.963030i	$0.413177\pi$
$128$	−1.00000	−0.0883883
$129$	−3.86370	−0.340180
$130$	−12.9282	−1.13388
$131$	−6.69213	−0.584694	−0.292347	−	0.956312i	$-0.594436\pi$
$131$	−6.69213	−0.584694	−0.292347	+	0.956312i	$0.594436\pi$
$132$	6.69213	0.582475
$133$	0	0
$134$	7.46410	0.644800
$135$	−16.9282	−1.45695
$136$	0.896575	0.0768807
$137$	−10.3923	−0.887875	−0.443937	−	0.896058i	$-0.646419\pi$
$137$	−10.3923	−0.887875	−0.443937	+	0.896058i	$0.646419\pi$
$138$	−5.79555	−0.493350
$139$	2.44949	0.207763	0.103882	−	0.994590i	$-0.466874\pi$
$139$	2.44949	0.207763	0.103882	+	0.994590i	$0.466874\pi$
$140$	0	0
$141$	21.8564	1.84064
$142$	13.1962	1.10740
$143$	11.5911	0.969297
$144$	0.732051	0.0610042
$145$	−6.69213	−0.555751
$146$	7.72741	0.639525
$147$	0	0
$148$	−3.00000	−0.246598
$149$	19.0000	1.55654	0.778270	−	0.627929i	$-0.216097\pi$
$149$	19.0000	1.55654	0.778270	+	0.627929i	$0.216097\pi$
$150$	−19.1798	−1.56603
$151$	−0.660254	−0.0537307	−0.0268654	−	0.999639i	$-0.508553\pi$
$151$	−0.660254	−0.0537307	−0.0268654	+	0.999639i	$0.508553\pi$
$152$	−0.517638	−0.0419860
$153$	−0.656339	−0.0530618
$154$	0	0
$155$	17.4641	1.40275
$156$	6.46410	0.517542
$157$	6.21166	0.495744	0.247872	−	0.968793i	$-0.420269\pi$
$157$	6.21166	0.495744	0.247872	+	0.968793i	$0.420269\pi$
$158$	11.0000	0.875113
$159$	1.93185	0.153206
$160$	−3.86370	−0.305453
$161$	0	0
$162$	10.6603	0.837549
$163$	−4.53590	−0.355279	−0.177639	−	0.984096i	$-0.556846\pi$
$163$	−4.53590	−0.355279	−0.177639	+	0.984096i	$0.556846\pi$
$164$	1.03528	0.0808415
$165$	25.8564	2.01292
$166$	−7.58871	−0.588998
$167$	15.9725	1.23599	0.617993	−	0.786184i	$-0.287946\pi$
$167$	15.9725	1.23599	0.617993	+	0.786184i	$0.287946\pi$
$168$	0	0
$169$	−1.80385	−0.138758
$170$	3.46410	0.265684
$171$	0.378937	0.0289781
$172$	−2.00000	−0.152499
$173$	−14.4195	−1.09630	−0.548149	−	0.836381i	$-0.684667\pi$
$173$	−14.4195	−1.09630	−0.548149	+	0.836381i	$0.684667\pi$
$174$	3.34607	0.253665
$175$	0	0
$176$	3.46410	0.261116
$177$	−14.3923	−1.08179
$178$	1.41421	0.106000
$179$	10.2679	0.767463	0.383731	−	0.923445i	$-0.374639\pi$
$179$	10.2679	0.767463	0.383731	+	0.923445i	$0.374639\pi$
$180$	2.82843	0.210819
$181$	19.0411	1.41531	0.707657	−	0.706556i	$-0.249752\pi$
$181$	19.0411	1.41531	0.707657	+	0.706556i	$0.249752\pi$
$182$	0	0
$183$	0	0
$184$	−3.00000	−0.221163
$185$	−11.5911	−0.852195
$186$	−8.73205	−0.640265
$187$	−3.10583	−0.227121
$188$	11.3137	0.825137
$189$	0	0
$190$	−2.00000	−0.145095
$191$	−20.1244	−1.45615	−0.728074	−	0.685499i	$-0.759584\pi$
$191$	−20.1244	−1.45615	−0.728074	+	0.685499i	$0.759584\pi$
$192$	1.93185	0.139419
$193$	−23.8564	−1.71722	−0.858611	−	0.512628i	$-0.828672\pi$
$193$	−23.8564	−1.71722	−0.858611	+	0.512628i	$0.828672\pi$
$194$	−13.1440	−0.943686
$195$	24.9754	1.78852
$196$	0	0
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	−2.53590	−0.180218
$199$	13.6617	0.968450	0.484225	−	0.874944i	$-0.339102\pi$
$199$	13.6617	0.968450	0.484225	+	0.874944i	$0.339102\pi$
$200$	−9.92820	−0.702030
$201$	−14.4195	−1.01708
$202$	9.52056	0.669864
$203$	0	0
$204$	−1.73205	−0.121268
$205$	4.00000	0.279372
$206$	−8.24504	−0.574459
$207$	2.19615	0.152643
$208$	3.34607	0.232008
$209$	1.79315	0.124035
$210$	0	0
$211$	22.7846	1.56856	0.784279	−	0.620409i	$-0.213033\pi$
$211$	22.7846	1.56856	0.784279	+	0.620409i	$0.213033\pi$
$212$	1.00000	0.0686803
$213$	−25.4930	−1.74675
$214$	1.46410	0.100084
$215$	−7.72741	−0.527005
$216$	4.38134	0.298113
$217$	0	0
$218$	−6.00000	−0.406371
$219$	−14.9282	−1.00875
$220$	13.3843	0.902367
$221$	−3.00000	−0.201802
$222$	5.79555	0.388972
$223$	−12.9038	−0.864102	−0.432051	−	0.901849i	$-0.642210\pi$
$223$	−12.9038	−0.864102	−0.432051	+	0.901849i	$0.642210\pi$
$224$	0	0
$225$	7.26795	0.484530
$226$	−0.0717968	−0.00477585
$227$	−19.7990	−1.31411	−0.657053	−	0.753845i	$-0.728197\pi$
$227$	−19.7990	−1.31411	−0.657053	+	0.753845i	$0.728197\pi$
$228$	1.00000	0.0662266
$229$	26.1493	1.72800	0.863999	−	0.503494i	$-0.167952\pi$
$229$	26.1493	1.72800	0.863999	+	0.503494i	$0.167952\pi$
$230$	−11.5911	−0.764295
$231$	0	0
$232$	1.73205	0.113715
$233$	11.8564	0.776739	0.388370	−	0.921504i	$-0.373038\pi$
$233$	11.8564	0.776739	0.388370	+	0.921504i	$0.373038\pi$
$234$	−2.44949	−0.160128
$235$	43.7128	2.85151
$236$	−7.45001	−0.484954
$237$	−21.2504	−1.38036
$238$	0	0
$239$	20.5359	1.32836	0.664178	−	0.747574i	$-0.268782\pi$
$239$	20.5359	1.32836	0.664178	+	0.747574i	$0.268782\pi$
$240$	7.46410	0.481806
$241$	−25.4930	−1.64215	−0.821075	−	0.570821i	$-0.806625\pi$
$241$	−25.4930	−1.64215	−0.821075	+	0.570821i	$0.806625\pi$
$242$	−1.00000	−0.0642824
$243$	−7.45001	−0.477918
$244$	0	0
$245$	0	0
$246$	−2.00000	−0.127515
$247$	1.73205	0.110208
$248$	−4.52004	−0.287023
$249$	14.6603	0.929056
$250$	−19.0411	−1.20427
$251$	−6.03579	−0.380976	−0.190488	−	0.981690i	$-0.561007\pi$
$251$	−6.03579	−0.380976	−0.190488	+	0.981690i	$0.561007\pi$
$252$	0	0
$253$	10.3923	0.653359
$254$	−6.07180	−0.380978
$255$	−6.69213	−0.419077
$256$	1.00000	0.0625000
$257$	−9.52056	−0.593876	−0.296938	−	0.954897i	$-0.595965\pi$
$257$	−9.52056	−0.593876	−0.296938	+	0.954897i	$0.595965\pi$
$258$	3.86370	0.240544
$259$	0	0
$260$	12.9282	0.801773
$261$	−1.26795	−0.0784841
$262$	6.69213	0.413441
$263$	−27.7128	−1.70885	−0.854423	−	0.519579i	$-0.826089\pi$
$263$	−27.7128	−1.70885	−0.854423	+	0.519579i	$0.826089\pi$
$264$	−6.69213	−0.411872
$265$	3.86370	0.237345
$266$	0	0
$267$	−2.73205	−0.167199
$268$	−7.46410	−0.455943
$269$	4.86181	0.296430	0.148215	−	0.988955i	$-0.452647\pi$
$269$	4.86181	0.296430	0.148215	+	0.988955i	$0.452647\pi$
$270$	16.9282	1.03022
$271$	−24.9754	−1.51715	−0.758573	−	0.651588i	$-0.774103\pi$
$271$	−24.9754	−1.51715	−0.758573	+	0.651588i	$0.774103\pi$
$272$	−0.896575	−0.0543629
$273$	0	0
$274$	10.3923	0.627822
$275$	34.3923	2.07393
$276$	5.79555	0.348851
$277$	−17.8564	−1.07289	−0.536444	−	0.843936i	$-0.680233\pi$
$277$	−17.8564	−1.07289	−0.536444	+	0.843936i	$0.680233\pi$
$278$	−2.44949	−0.146911
$279$	3.30890	0.198099
$280$	0	0
$281$	13.9282	0.830887	0.415443	−	0.909619i	$-0.363626\pi$
$281$	13.9282	0.830887	0.415443	+	0.909619i	$0.363626\pi$
$282$	−21.8564	−1.30153
$283$	−2.03339	−0.120872	−0.0604362	−	0.998172i	$-0.519249\pi$
$283$	−2.03339	−0.120872	−0.0604362	+	0.998172i	$0.519249\pi$
$284$	−13.1962	−0.783048
$285$	3.86370	0.228866
$286$	−11.5911	−0.685397
$287$	0	0
$288$	−0.732051	−0.0431365
$289$	−16.1962	−0.952715
$290$	6.69213	0.392975
$291$	25.3923	1.48852
$292$	−7.72741	−0.452212
$293$	−10.4543	−0.610747	−0.305373	−	0.952233i	$-0.598781\pi$
$293$	−10.4543	−0.610747	−0.305373	+	0.952233i	$0.598781\pi$
$294$	0	0
$295$	−28.7846	−1.67590
$296$	3.00000	0.174371
$297$	−15.1774	−0.880683
$298$	−19.0000	−1.10064
$299$	10.0382	0.580524
$300$	19.1798	1.10735
$301$	0	0
$302$	0.660254	0.0379934
$303$	−18.3923	−1.05661
$304$	0.517638	0.0296886
$305$	0	0
$306$	0.656339	0.0375204
$307$	−31.1870	−1.77994	−0.889969	−	0.456021i	$-0.849274\pi$
$307$	−31.1870	−1.77994	−0.889969	+	0.456021i	$0.849274\pi$
$308$	0	0
$309$	15.9282	0.906124
$310$	−17.4641	−0.991894
$311$	−22.4243	−1.27157	−0.635784	−	0.771867i	$-0.719323\pi$
$311$	−22.4243	−1.27157	−0.635784	+	0.771867i	$0.719323\pi$
$312$	−6.46410	−0.365958
$313$	30.9096	1.74712	0.873558	−	0.486721i	$-0.161807\pi$
$313$	30.9096	1.74712	0.873558	+	0.486721i	$0.161807\pi$
$314$	−6.21166	−0.350544
$315$	0	0
$316$	−11.0000	−0.618798
$317$	16.1244	0.905634	0.452817	−	0.891603i	$-0.350419\pi$
$317$	16.1244	0.905634	0.452817	+	0.891603i	$0.350419\pi$
$318$	−1.93185	−0.108333
$319$	−6.00000	−0.335936
$320$	3.86370	0.215988
$321$	−2.82843	−0.157867
$322$	0	0
$323$	−0.464102	−0.0258233
$324$	−10.6603	−0.592236
$325$	33.2204	1.84274
$326$	4.53590	0.251220
$327$	11.5911	0.640990
$328$	−1.03528	−0.0571636
$329$	0	0
$330$	−25.8564	−1.42335
$331$	16.5359	0.908895	0.454448	−	0.890773i	$-0.349837\pi$
$331$	16.5359	0.908895	0.454448	+	0.890773i	$0.349837\pi$
$332$	7.58871	0.416484
$333$	−2.19615	−0.120348
$334$	−15.9725	−0.873974
$335$	−28.8391	−1.57565
$336$	0	0
$337$	20.2487	1.10302	0.551509	−	0.834169i	$-0.314052\pi$
$337$	20.2487	1.10302	0.551509	+	0.834169i	$0.314052\pi$
$338$	1.80385	0.0981164
$339$	0.138701	0.00753319
$340$	−3.46410	−0.187867
$341$	15.6579	0.847922
$342$	−0.378937	−0.0204906
$343$	0	0
$344$	2.00000	0.107833
$345$	22.3923	1.20556
$346$	14.4195	0.775199
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	−3.34607	−0.179368
$349$	−35.5312	−1.90194	−0.950971	−	0.309280i	$-0.899912\pi$
$349$	−35.5312	−1.90194	−0.950971	+	0.309280i	$0.899912\pi$
$350$	0	0
$351$	−14.6603	−0.782506
$352$	−3.46410	−0.184637
$353$	−28.8391	−1.53495	−0.767475	−	0.641079i	$-0.778487\pi$
$353$	−28.8391	−1.53495	−0.767475	+	0.641079i	$0.778487\pi$
$354$	14.3923	0.764942
$355$	−50.9860	−2.70606
$356$	−1.41421	−0.0749532
$357$	0	0
$358$	−10.2679	−0.542678
$359$	9.73205	0.513638	0.256819	−	0.966460i	$-0.417326\pi$
$359$	9.73205	0.513638	0.256819	+	0.966460i	$0.417326\pi$
$360$	−2.82843	−0.149071
$361$	−18.7321	−0.985897
$362$	−19.0411	−1.00078
$363$	1.93185	0.101396
$364$	0	0
$365$	−29.8564	−1.56276
$366$	0	0
$367$	26.7685	1.39731	0.698653	−	0.715461i	$-0.253783\pi$
$367$	26.7685	1.39731	0.698653	+	0.715461i	$0.253783\pi$
$368$	3.00000	0.156386
$369$	0.757875	0.0394534
$370$	11.5911	0.602593
$371$	0	0
$372$	8.73205	0.452736
$373$	−27.1769	−1.40717	−0.703584	−	0.710612i	$-0.748418\pi$
$373$	−27.1769	−1.40717	−0.703584	+	0.710612i	$0.748418\pi$
$374$	3.10583	0.160599
$375$	36.7846	1.89955
$376$	−11.3137	−0.583460
$377$	−5.79555	−0.298486
$378$	0	0
$379$	35.0526	1.80053	0.900265	−	0.435343i	$-0.143373\pi$
$379$	35.0526	1.80053	0.900265	+	0.435343i	$0.143373\pi$
$380$	2.00000	0.102598
$381$	11.7298	0.600936
$382$	20.1244	1.02965
$383$	25.6317	1.30972	0.654860	−	0.755751i	$-0.272728\pi$
$383$	25.6317	1.30972	0.654860	+	0.755751i	$0.272728\pi$
$384$	−1.93185	−0.0985844
$385$	0	0
$386$	23.8564	1.21426
$387$	−1.46410	−0.0744245
$388$	13.1440	0.667287
$389$	−12.7846	−0.648205	−0.324103	−	0.946022i	$-0.605062\pi$
$389$	−12.7846	−0.648205	−0.324103	+	0.946022i	$0.605062\pi$
$390$	−24.9754	−1.26468
$391$	−2.68973	−0.136025
$392$	0	0
$393$	−12.9282	−0.652142
$394$	0	0
$395$	−42.5007	−2.13844
$396$	2.53590	0.127434
$397$	−17.2480	−0.865651	−0.432825	−	0.901478i	$-0.642483\pi$
$397$	−17.2480	−0.865651	−0.432825	+	0.901478i	$0.642483\pi$
$398$	−13.6617	−0.684797
$399$	0	0
$400$	9.92820	0.496410
$401$	−12.5359	−0.626013	−0.313006	−	0.949751i	$-0.601336\pi$
$401$	−12.5359	−0.626013	−0.313006	+	0.949751i	$0.601336\pi$
$402$	14.4195	0.719181
$403$	15.1244	0.753398
$404$	−9.52056	−0.473665
$405$	−41.1881	−2.04665
$406$	0	0
$407$	−10.3923	−0.515127
$408$	1.73205	0.0857493
$409$	2.41233	0.119282	0.0596409	−	0.998220i	$-0.481004\pi$
$409$	2.41233	0.119282	0.0596409	+	0.998220i	$0.481004\pi$
$410$	−4.00000	−0.197546
$411$	−20.0764	−0.990295
$412$	8.24504	0.406204
$413$	0	0
$414$	−2.19615	−0.107935
$415$	29.3205	1.43929
$416$	−3.34607	−0.164054
$417$	4.73205	0.231730
$418$	−1.79315	−0.0877059
$419$	27.7023	1.35334	0.676672	−	0.736285i	$-0.263422\pi$
$419$	27.7023	1.35334	0.676672	+	0.736285i	$0.263422\pi$
$420$	0	0
$421$	2.67949	0.130590	0.0652952	−	0.997866i	$-0.479201\pi$
$421$	2.67949	0.130590	0.0652952	+	0.997866i	$0.479201\pi$
$422$	−22.7846	−1.10914
$423$	8.28221	0.402695
$424$	−1.00000	−0.0485643
$425$	−8.90138	−0.431781
$426$	25.4930	1.23514
$427$	0	0
$428$	−1.46410	−0.0707700
$429$	22.3923	1.08111
$430$	7.72741	0.372649
$431$	35.7128	1.72023	0.860113	−	0.510104i	$-0.170393\pi$
$431$	35.7128	1.72023	0.860113	+	0.510104i	$0.170393\pi$
$432$	−4.38134	−0.210797
$433$	35.9473	1.72752	0.863759	−	0.503906i	$-0.168104\pi$
$433$	35.9473	1.72752	0.863759	+	0.503906i	$0.168104\pi$
$434$	0	0
$435$	−12.9282	−0.619860
$436$	6.00000	0.287348
$437$	1.55291	0.0742860
$438$	14.9282	0.713297
$439$	8.20788	0.391741	0.195870	−	0.980630i	$-0.437247\pi$
$439$	8.20788	0.391741	0.195870	+	0.980630i	$0.437247\pi$
$440$	−13.3843	−0.638070
$441$	0	0
$442$	3.00000	0.142695
$443$	−18.8564	−0.895895	−0.447947	−	0.894060i	$-0.647845\pi$
$443$	−18.8564	−0.895895	−0.447947	+	0.894060i	$0.647845\pi$
$444$	−5.79555	−0.275045
$445$	−5.46410	−0.259023
$446$	12.9038	0.611012
$447$	36.7052	1.73610
$448$	0	0
$449$	11.3397	0.535156	0.267578	−	0.963536i	$-0.413777\pi$
$449$	11.3397	0.535156	0.267578	+	0.963536i	$0.413777\pi$
$450$	−7.26795	−0.342614
$451$	3.58630	0.168872
$452$	0.0717968	0.00337704
$453$	−1.27551	−0.0599288
$454$	19.7990	0.929213
$455$	0	0
$456$	−1.00000	−0.0468293
$457$	−26.9282	−1.25965	−0.629824	−	0.776738i	$-0.716873\pi$
$457$	−26.9282	−1.25965	−0.629824	+	0.776738i	$0.716873\pi$
$458$	−26.1493	−1.22188
$459$	3.92820	0.183353
$460$	11.5911	0.540438
$461$	−17.5897	−0.819236	−0.409618	−	0.912257i	$-0.634338\pi$
$461$	−17.5897	−0.819236	−0.409618	+	0.912257i	$0.634338\pi$
$462$	0	0
$463$	7.87564	0.366012	0.183006	−	0.983112i	$-0.441417\pi$
$463$	7.87564	0.366012	0.183006	+	0.983112i	$0.441417\pi$
$464$	−1.73205	−0.0804084
$465$	33.7381	1.56456
$466$	−11.8564	−0.549237
$467$	−1.03528	−0.0479069	−0.0239534	−	0.999713i	$-0.507625\pi$
$467$	−1.03528	−0.0479069	−0.0239534	+	0.999713i	$0.507625\pi$
$468$	2.44949	0.113228
$469$	0	0
$470$	−43.7128	−2.01632
$471$	12.0000	0.552931
$472$	7.45001	0.342914
$473$	−6.92820	−0.318559
$474$	21.2504	0.976062
$475$	5.13922	0.235803
$476$	0	0
$477$	0.732051	0.0335183
$478$	−20.5359	−0.939290
$479$	−26.6298	−1.21675	−0.608374	−	0.793651i	$-0.708178\pi$
$479$	−26.6298	−1.21675	−0.608374	+	0.793651i	$0.708178\pi$
$480$	−7.46410	−0.340688
$481$	−10.0382	−0.457702
$482$	25.4930	1.16117
$483$	0	0
$484$	1.00000	0.0454545
$485$	50.7846	2.30601
$486$	7.45001	0.337939
$487$	30.3923	1.37721	0.688603	−	0.725138i	$-0.258224\pi$
$487$	30.3923	1.37721	0.688603	+	0.725138i	$0.258224\pi$
$488$	0	0
$489$	−8.76268	−0.396262
$490$	0	0
$491$	−22.5167	−1.01616	−0.508081	−	0.861309i	$-0.669645\pi$
$491$	−22.5167	−1.01616	−0.508081	+	0.861309i	$0.669645\pi$
$492$	2.00000	0.0901670
$493$	1.55291	0.0699397
$494$	−1.73205	−0.0779287
$495$	9.79796	0.440386
$496$	4.52004	0.202956
$497$	0	0
$498$	−14.6603	−0.656942
$499$	−40.7128	−1.82256	−0.911278	−	0.411792i	$-0.864903\pi$
$499$	−40.7128	−1.82256	−0.911278	+	0.411792i	$0.864903\pi$
$500$	19.0411	0.851545
$501$	30.8564	1.37856
$502$	6.03579	0.269391
$503$	10.9348	0.487557	0.243779	−	0.969831i	$-0.421613\pi$
$503$	10.9348	0.487557	0.243779	+	0.969831i	$0.421613\pi$
$504$	0	0
$505$	−36.7846	−1.63689
$506$	−10.3923	−0.461994
$507$	−3.48477	−0.154764
$508$	6.07180	0.269392
$509$	19.7990	0.877575	0.438787	−	0.898591i	$-0.355408\pi$
$509$	19.7990	0.877575	0.438787	+	0.898591i	$0.355408\pi$
$510$	6.69213	0.296333
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	−2.26795	−0.100132
$514$	9.52056	0.419934
$515$	31.8564	1.40376
$516$	−3.86370	−0.170090
$517$	39.1918	1.72365
$518$	0	0
$519$	−27.8564	−1.22276
$520$	−12.9282	−0.566939
$521$	−30.2905	−1.32705	−0.663524	−	0.748155i	$-0.730940\pi$
$521$	−30.2905	−1.32705	−0.663524	+	0.748155i	$0.730940\pi$
$522$	1.26795	0.0554966
$523$	26.0106	1.13737	0.568683	−	0.822557i	$-0.307453\pi$
$523$	26.0106	1.13737	0.568683	+	0.822557i	$0.307453\pi$
$524$	−6.69213	−0.292347
$525$	0	0
$526$	27.7128	1.20834
$527$	−4.05256	−0.176532
$528$	6.69213	0.291238
$529$	−14.0000	−0.608696
$530$	−3.86370	−0.167829
$531$	−5.45378	−0.236674
$532$	0	0
$533$	3.46410	0.150047
$534$	2.73205	0.118227
$535$	−5.65685	−0.244567
$536$	7.46410	0.322400
$537$	19.8362	0.855993
$538$	−4.86181	−0.209608
$539$	0	0
$540$	−16.9282	−0.728474
$541$	36.7846	1.58149	0.790747	−	0.612143i	$-0.209692\pi$
$541$	36.7846	1.58149	0.790747	+	0.612143i	$0.209692\pi$
$542$	24.9754	1.07278
$543$	36.7846	1.57858
$544$	0.896575	0.0384404
$545$	23.1822	0.993017
$546$	0	0
$547$	15.8564	0.677971	0.338985	−	0.940792i	$-0.389916\pi$
$547$	15.8564	0.677971	0.338985	+	0.940792i	$0.389916\pi$
$548$	−10.3923	−0.443937
$549$	0	0
$550$	−34.3923	−1.46649
$551$	−0.896575	−0.0381954
$552$	−5.79555	−0.246675
$553$	0	0
$554$	17.8564	0.758646
$555$	−22.3923	−0.950500
$556$	2.44949	0.103882
$557$	6.53590	0.276935	0.138467	−	0.990367i	$-0.455782\pi$
$557$	6.53590	0.276935	0.138467	+	0.990367i	$0.455782\pi$
$558$	−3.30890	−0.140077
$559$	−6.69213	−0.283047
$560$	0	0
$561$	−6.00000	−0.253320
$562$	−13.9282	−0.587526
$563$	25.7704	1.08609	0.543047	−	0.839703i	$-0.317271\pi$
$563$	25.7704	1.08609	0.543047	+	0.839703i	$0.317271\pi$
$564$	21.8564	0.920321
$565$	0.277401	0.0116704
$566$	2.03339	0.0854697
$567$	0	0
$568$	13.1962	0.553698
$569$	40.7846	1.70978	0.854890	−	0.518809i	$-0.173625\pi$
$569$	40.7846	1.70978	0.854890	+	0.518809i	$0.173625\pi$
$570$	−3.86370	−0.161833
$571$	−10.3923	−0.434904	−0.217452	−	0.976071i	$-0.569775\pi$
$571$	−10.3923	−0.434904	−0.217452	+	0.976071i	$0.569775\pi$
$572$	11.5911	0.484649
$573$	−38.8773	−1.62412
$574$	0	0
$575$	29.7846	1.24210
$576$	0.732051	0.0305021
$577$	−19.4944	−0.811562	−0.405781	−	0.913970i	$-0.633000\pi$
$577$	−19.4944	−0.811562	−0.405781	+	0.913970i	$0.633000\pi$
$578$	16.1962	0.673671
$579$	−46.0870	−1.91531
$580$	−6.69213	−0.277876
$581$	0	0
$582$	−25.3923	−1.05254
$583$	3.46410	0.143468
$584$	7.72741	0.319762
$585$	9.46410	0.391292
$586$	10.4543	0.431863
$587$	−33.7381	−1.39252	−0.696259	−	0.717790i	$-0.745154\pi$
$587$	−33.7381	−1.39252	−0.696259	+	0.717790i	$0.745154\pi$
$588$	0	0
$589$	2.33975	0.0964076
$590$	28.7846	1.18504
$591$	0	0
$592$	−3.00000	−0.123299
$593$	−4.76028	−0.195481	−0.0977406	−	0.995212i	$-0.531162\pi$
$593$	−4.76028	−0.195481	−0.0977406	+	0.995212i	$0.531162\pi$
$594$	15.1774	0.622737
$595$	0	0
$596$	19.0000	0.778270
$597$	26.3923	1.08017
$598$	−10.0382	−0.410492
$599$	38.3923	1.56867	0.784334	−	0.620339i	$-0.213005\pi$
$599$	38.3923	1.56867	0.784334	+	0.620339i	$0.213005\pi$
$600$	−19.1798	−0.783013
$601$	−42.2233	−1.72233	−0.861163	−	0.508329i	$-0.830263\pi$
$601$	−42.2233	−1.72233	−0.861163	+	0.508329i	$0.830263\pi$
$602$	0	0
$603$	−5.46410	−0.222515
$604$	−0.660254	−0.0268654
$605$	3.86370	0.157082
$606$	18.3923	0.747136
$607$	30.0774	1.22080	0.610402	−	0.792091i	$-0.291008\pi$
$607$	30.0774	1.22080	0.610402	+	0.792091i	$0.291008\pi$
$608$	−0.517638	−0.0209930
$609$	0	0
$610$	0	0
$611$	37.8564	1.53151
$612$	−0.656339	−0.0265309
$613$	−34.3923	−1.38909	−0.694546	−	0.719448i	$-0.744395\pi$
$613$	−34.3923	−1.38909	−0.694546	+	0.719448i	$0.744395\pi$
$614$	31.1870	1.25861
$615$	7.72741	0.311599
$616$	0	0
$617$	34.9282	1.40616	0.703078	−	0.711112i	$-0.251808\pi$
$617$	34.9282	1.40616	0.703078	+	0.711112i	$0.251808\pi$
$618$	−15.9282	−0.640726
$619$	25.1784	1.01201	0.506004	−	0.862531i	$-0.331122\pi$
$619$	25.1784	1.01201	0.506004	+	0.862531i	$0.331122\pi$
$620$	17.4641	0.701375
$621$	−13.1440	−0.527452
$622$	22.4243	0.899134
$623$	0	0
$624$	6.46410	0.258771
$625$	23.9282	0.957128
$626$	−30.9096	−1.23540
$627$	3.46410	0.138343
$628$	6.21166	0.247872
$629$	2.68973	0.107246
$630$	0	0
$631$	9.19615	0.366093	0.183047	−	0.983104i	$-0.441404\pi$
$631$	9.19615	0.366093	0.183047	+	0.983104i	$0.441404\pi$
$632$	11.0000	0.437557
$633$	44.0165	1.74950
$634$	−16.1244	−0.640380
$635$	23.4596	0.930967
$636$	1.93185	0.0766029
$637$	0	0
$638$	6.00000	0.237542
$639$	−9.66025	−0.382154
$640$	−3.86370	−0.152726
$641$	−0.679492	−0.0268383	−0.0134192	−	0.999910i	$-0.504272\pi$
$641$	−0.679492	−0.0268383	−0.0134192	+	0.999910i	$0.504272\pi$
$642$	2.82843	0.111629
$643$	−19.5959	−0.772788	−0.386394	−	0.922334i	$-0.626279\pi$
$643$	−19.5959	−0.772788	−0.386394	+	0.922334i	$0.626279\pi$
$644$	0	0
$645$	−14.9282	−0.587797
$646$	0.464102	0.0182598
$647$	−4.62158	−0.181693	−0.0908465	−	0.995865i	$-0.528957\pi$
$647$	−4.62158	−0.181693	−0.0908465	+	0.995865i	$0.528957\pi$
$648$	10.6603	0.418774
$649$	−25.8076	−1.01304
$650$	−33.2204	−1.30301
$651$	0	0
$652$	−4.53590	−0.177639
$653$	12.9282	0.505920	0.252960	−	0.967477i	$-0.418596\pi$
$653$	12.9282	0.505920	0.252960	+	0.967477i	$0.418596\pi$
$654$	−11.5911	−0.453248
$655$	−25.8564	−1.01029
$656$	1.03528	0.0404207
$657$	−5.65685	−0.220695
$658$	0	0
$659$	38.1051	1.48436	0.742182	−	0.670198i	$-0.233791\pi$
$659$	38.1051	1.48436	0.742182	+	0.670198i	$0.233791\pi$
$660$	25.8564	1.00646
$661$	−5.31508	−0.206733	−0.103366	−	0.994643i	$-0.532961\pi$
$661$	−5.31508	−0.206733	−0.103366	+	0.994643i	$0.532961\pi$
$662$	−16.5359	−0.642686
$663$	−5.79555	−0.225081
$664$	−7.58871	−0.294499
$665$	0	0
$666$	2.19615	0.0850992
$667$	−5.19615	−0.201196
$668$	15.9725	0.617993
$669$	−24.9282	−0.963780
$670$	28.8391	1.11415
$671$	0	0
$672$	0	0
$673$	−4.14359	−0.159724	−0.0798619	−	0.996806i	$-0.525448\pi$
$673$	−4.14359	−0.159724	−0.0798619	+	0.996806i	$0.525448\pi$
$674$	−20.2487	−0.779951
$675$	−43.4988	−1.67427
$676$	−1.80385	−0.0693788
$677$	3.58630	0.137833	0.0689164	−	0.997622i	$-0.478046\pi$
$677$	3.58630	0.137833	0.0689164	+	0.997622i	$0.478046\pi$
$678$	−0.138701	−0.00532677
$679$	0	0
$680$	3.46410	0.132842
$681$	−38.2487	−1.46569
$682$	−15.6579	−0.599571
$683$	28.2487	1.08091	0.540453	−	0.841374i	$-0.318253\pi$
$683$	28.2487	1.08091	0.540453	+	0.841374i	$0.318253\pi$
$684$	0.378937	0.0144890
$685$	−40.1528	−1.53416
$686$	0	0
$687$	50.5167	1.92733
$688$	−2.00000	−0.0762493
$689$	3.34607	0.127475
$690$	−22.3923	−0.852460
$691$	−20.9730	−0.797849	−0.398925	−	0.916984i	$-0.630616\pi$
$691$	−20.9730	−0.797849	−0.398925	+	0.916984i	$0.630616\pi$
$692$	−14.4195	−0.548149
$693$	0	0
$694$	−12.0000	−0.455514
$695$	9.46410	0.358994
$696$	3.34607	0.126832
$697$	−0.928203	−0.0351582
$698$	35.5312	1.34488
$699$	22.9048	0.866340
$700$	0	0
$701$	−16.3923	−0.619129	−0.309564	−	0.950878i	$-0.600183\pi$
$701$	−16.3923	−0.619129	−0.309564	+	0.950878i	$0.600183\pi$
$702$	14.6603	0.553316
$703$	−1.55291	−0.0585693
$704$	3.46410	0.130558
$705$	84.4467	3.18045
$706$	28.8391	1.08537
$707$	0	0
$708$	−14.3923	−0.540896
$709$	−2.14359	−0.0805043	−0.0402522	−	0.999190i	$-0.512816\pi$
$709$	−2.14359	−0.0805043	−0.0402522	+	0.999190i	$0.512816\pi$
$710$	50.9860	1.91347
$711$	−8.05256	−0.301995
$712$	1.41421	0.0529999
$713$	13.5601	0.507831
$714$	0	0
$715$	44.7846	1.67485
$716$	10.2679	0.383731
$717$	39.6723	1.48159
$718$	−9.73205	−0.363197
$719$	33.9139	1.26478	0.632388	−	0.774652i	$-0.282075\pi$
$719$	33.9139	1.26478	0.632388	+	0.774652i	$0.282075\pi$
$720$	2.82843	0.105409
$721$	0	0
$722$	18.7321	0.697135
$723$	−49.2487	−1.83158
$724$	19.0411	0.707657
$725$	−17.1962	−0.638649
$726$	−1.93185	−0.0716977
$727$	−27.0459	−1.00308	−0.501539	−	0.865135i	$-0.667233\pi$
$727$	−27.0459	−1.00308	−0.501539	+	0.865135i	$0.667233\pi$
$728$	0	0
$729$	17.5885	0.651424
$730$	29.8564	1.10504
$731$	1.79315	0.0663221
$732$	0	0
$733$	3.48477	0.128713	0.0643564	−	0.997927i	$-0.479501\pi$
$733$	3.48477	0.128713	0.0643564	+	0.997927i	$0.479501\pi$
$734$	−26.7685	−0.988044
$735$	0	0
$736$	−3.00000	−0.110581
$737$	−25.8564	−0.952433
$738$	−0.757875	−0.0278978
$739$	−14.7128	−0.541220	−0.270610	−	0.962689i	$-0.587225\pi$
$739$	−14.7128	−0.541220	−0.270610	+	0.962689i	$0.587225\pi$
$740$	−11.5911	−0.426098
$741$	3.34607	0.122921
$742$	0	0
$743$	−16.3923	−0.601375	−0.300688	−	0.953723i	$-0.597216\pi$
$743$	−16.3923	−0.601375	−0.300688	+	0.953723i	$0.597216\pi$
$744$	−8.73205	−0.320133
$745$	73.4104	2.68955
$746$	27.1769	0.995018
$747$	5.55532	0.203258
$748$	−3.10583	−0.113560
$749$	0	0
$750$	−36.7846	−1.34318
$751$	−2.00000	−0.0729810	−0.0364905	−	0.999334i	$-0.511618\pi$
$751$	−2.00000	−0.0729810	−0.0364905	+	0.999334i	$0.511618\pi$
$752$	11.3137	0.412568
$753$	−11.6603	−0.424923
$754$	5.79555	0.211062
$755$	−2.55103	−0.0928413
$756$	0	0
$757$	−33.7846	−1.22792	−0.613961	−	0.789336i	$-0.710425\pi$
$757$	−33.7846	−1.22792	−0.613961	+	0.789336i	$0.710425\pi$
$758$	−35.0526	−1.27317
$759$	20.0764	0.728727
$760$	−2.00000	−0.0725476
$761$	−23.7370	−0.860466	−0.430233	−	0.902718i	$-0.641569\pi$
$761$	−23.7370	−0.860466	−0.430233	+	0.902718i	$0.641569\pi$
$762$	−11.7298	−0.424926
$763$	0	0
$764$	−20.1244	−0.728074
$765$	−2.53590	−0.0916856
$766$	−25.6317	−0.926111
$767$	−24.9282	−0.900105
$768$	1.93185	0.0697097
$769$	−15.9353	−0.574641	−0.287321	−	0.957834i	$-0.592764\pi$
$769$	−15.9353	−0.574641	−0.287321	+	0.957834i	$0.592764\pi$
$770$	0	0
$771$	−18.3923	−0.662383
$772$	−23.8564	−0.858611
$773$	34.0155	1.22345	0.611725	−	0.791070i	$-0.290476\pi$
$773$	34.0155	1.22345	0.611725	+	0.791070i	$0.290476\pi$
$774$	1.46410	0.0526260
$775$	44.8759	1.61199
$776$	−13.1440	−0.471843
$777$	0	0
$778$	12.7846	0.458350
$779$	0.535898	0.0192006
$780$	24.9754	0.894262
$781$	−45.7128	−1.63573
$782$	2.68973	0.0961844
$783$	7.58871	0.271198
$784$	0	0
$785$	24.0000	0.856597
$786$	12.9282	0.461134
$787$	−18.4591	−0.657996	−0.328998	−	0.944331i	$-0.606711\pi$
$787$	−18.4591	−0.657996	−0.328998	+	0.944331i	$0.606711\pi$
$788$	0	0
$789$	−53.5370	−1.90597
$790$	42.5007	1.51211
$791$	0	0
$792$	−2.53590	−0.0901092
$793$	0	0
$794$	17.2480	0.612107
$795$	7.46410	0.264724
$796$	13.6617	0.484225
$797$	−32.9058	−1.16558	−0.582792	−	0.812621i	$-0.698040\pi$
$797$	−32.9058	−1.16558	−0.582792	+	0.812621i	$0.698040\pi$
$798$	0	0
$799$	−10.1436	−0.358855
$800$	−9.92820	−0.351015
$801$	−1.03528	−0.0365797
$802$	12.5359	0.442658
$803$	−26.7685	−0.944641
$804$	−14.4195	−0.508538
$805$	0	0
$806$	−15.1244	−0.532733
$807$	9.39230	0.330625
$808$	9.52056	0.334932
$809$	−37.8564	−1.33096	−0.665480	−	0.746416i	$-0.731773\pi$
$809$	−37.8564	−1.33096	−0.665480	+	0.746416i	$0.731773\pi$
$810$	41.1881	1.44720
$811$	−34.6990	−1.21845	−0.609223	−	0.792999i	$-0.708519\pi$
$811$	−34.6990	−1.21845	−0.609223	+	0.792999i	$0.708519\pi$
$812$	0	0
$813$	−48.2487	−1.69216
$814$	10.3923	0.364250
$815$	−17.5254	−0.613887
$816$	−1.73205	−0.0606339
$817$	−1.03528	−0.0362197
$818$	−2.41233	−0.0843450
$819$	0	0
$820$	4.00000	0.139686
$821$	44.2487	1.54429	0.772145	−	0.635446i	$-0.219184\pi$
$821$	44.2487	1.54429	0.772145	+	0.635446i	$0.219184\pi$
$822$	20.0764	0.700245
$823$	−5.32051	−0.185461	−0.0927306	−	0.995691i	$-0.529560\pi$
$823$	−5.32051	−0.185461	−0.0927306	+	0.995691i	$0.529560\pi$
$824$	−8.24504	−0.287230
$825$	66.4408	2.31317
$826$	0	0
$827$	12.6603	0.440240	0.220120	−	0.975473i	$-0.429355\pi$
$827$	12.6603	0.440240	0.220120	+	0.975473i	$0.429355\pi$
$828$	2.19615	0.0763216
$829$	−31.6675	−1.09986	−0.549929	−	0.835211i	$-0.685345\pi$
$829$	−31.6675	−1.09986	−0.549929	+	0.835211i	$0.685345\pi$
$830$	−29.3205	−1.01773
$831$	−34.4959	−1.19665
$832$	3.34607	0.116004
$833$	0	0
$834$	−4.73205	−0.163858
$835$	61.7128	2.13566
$836$	1.79315	0.0620174
$837$	−19.8038	−0.684521
$838$	−27.7023	−0.956959
$839$	16.4901	0.569301	0.284651	−	0.958631i	$-0.408122\pi$
$839$	16.4901	0.569301	0.284651	+	0.958631i	$0.408122\pi$
$840$	0	0
$841$	−26.0000	−0.896552
$842$	−2.67949	−0.0923414
$843$	26.9072	0.926734
$844$	22.7846	0.784279
$845$	−6.96953	−0.239759
$846$	−8.28221	−0.284748
$847$	0	0
$848$	1.00000	0.0343401
$849$	−3.92820	−0.134816
$850$	8.90138	0.305315
$851$	−9.00000	−0.308516
$852$	−25.4930	−0.873376
$853$	1.79315	0.0613963	0.0306982	−	0.999529i	$-0.490227\pi$
$853$	1.79315	0.0613963	0.0306982	+	0.999529i	$0.490227\pi$
$854$	0	0
$855$	1.46410	0.0500712
$856$	1.46410	0.0500420
$857$	−46.6690	−1.59418	−0.797092	−	0.603858i	$-0.793630\pi$
$857$	−46.6690	−1.59418	−0.797092	+	0.603858i	$0.793630\pi$
$858$	−22.3923	−0.764461
$859$	37.8792	1.29242	0.646210	−	0.763160i	$-0.276353\pi$
$859$	37.8792	1.29242	0.646210	+	0.763160i	$0.276353\pi$
$860$	−7.72741	−0.263502
$861$	0	0
$862$	−35.7128	−1.21638
$863$	13.3205	0.453435	0.226718	−	0.973961i	$-0.427201\pi$
$863$	13.3205	0.453435	0.226718	+	0.973961i	$0.427201\pi$
$864$	4.38134	0.149056
$865$	−55.7128	−1.89429
$866$	−35.9473	−1.22154
$867$	−31.2886	−1.06262
$868$	0	0
$869$	−38.1051	−1.29263
$870$	12.9282	0.438307
$871$	−24.9754	−0.846258
$872$	−6.00000	−0.203186
$873$	9.62209	0.325659
$874$	−1.55291	−0.0525281
$875$	0	0
$876$	−14.9282	−0.504377
$877$	−42.9282	−1.44958	−0.724791	−	0.688969i	$-0.758064\pi$
$877$	−42.9282	−1.44958	−0.724791	+	0.688969i	$0.758064\pi$
$878$	−8.20788	−0.277003
$879$	−20.1962	−0.681199
$880$	13.3843	0.451183
$881$	14.4939	0.488311	0.244155	−	0.969736i	$-0.421489\pi$
$881$	14.4939	0.488311	0.244155	+	0.969736i	$0.421489\pi$
$882$	0	0
$883$	12.7846	0.430236	0.215118	−	0.976588i	$-0.430986\pi$
$883$	12.7846	0.430236	0.215118	+	0.976588i	$0.430986\pi$
$884$	−3.00000	−0.100901
$885$	−55.6076	−1.86923
$886$	18.8564	0.633493
$887$	−8.69831	−0.292061	−0.146030	−	0.989280i	$-0.546650\pi$
$887$	−8.69831	−0.292061	−0.146030	+	0.989280i	$0.546650\pi$
$888$	5.79555	0.194486
$889$	0	0
$890$	5.46410	0.183157
$891$	−36.9282	−1.23714
$892$	−12.9038	−0.432051
$893$	5.85641	0.195977
$894$	−36.7052	−1.22760
$895$	39.6723	1.32610
$896$	0	0
$897$	19.3923	0.647490
$898$	−11.3397	−0.378412
$899$	−7.82894	−0.261110
$900$	7.26795	0.242265
$901$	−0.896575	−0.0298693
$902$	−3.58630	−0.119411
$903$	0	0
$904$	−0.0717968	−0.00238793
$905$	73.5692	2.44552
$906$	1.27551	0.0423761
$907$	7.07180	0.234815	0.117408	−	0.993084i	$-0.462542\pi$
$907$	7.07180	0.234815	0.117408	+	0.993084i	$0.462542\pi$
$908$	−19.7990	−0.657053
$909$	−6.96953	−0.231165
$910$	0	0
$911$	1.21539	0.0402677	0.0201338	−	0.999797i	$-0.493591\pi$
$911$	1.21539	0.0402677	0.0201338	+	0.999797i	$0.493591\pi$
$912$	1.00000	0.0331133
$913$	26.2880	0.870007
$914$	26.9282	0.890706
$915$	0	0
$916$	26.1493	0.863999
$917$	0	0
$918$	−3.92820	−0.129650
$919$	5.19615	0.171405	0.0857026	−	0.996321i	$-0.472687\pi$
$919$	5.19615	0.171405	0.0857026	+	0.996321i	$0.472687\pi$
$920$	−11.5911	−0.382148
$921$	−60.2487	−1.98526
$922$	17.5897	0.579287
$923$	−44.1552	−1.45339
$924$	0	0
$925$	−29.7846	−0.979312
$926$	−7.87564	−0.258810
$927$	6.03579	0.198241
$928$	1.73205	0.0568574
$929$	44.9502	1.47477	0.737385	−	0.675473i	$-0.236061\pi$
$929$	44.9502	1.47477	0.737385	+	0.675473i	$0.236061\pi$
$930$	−33.7381	−1.10631
$931$	0	0
$932$	11.8564	0.388370
$933$	−43.3205	−1.41825
$934$	1.03528	0.0338753
$935$	−12.0000	−0.392442
$936$	−2.44949	−0.0800641
$937$	25.1884	0.822869	0.411435	−	0.911439i	$-0.365028\pi$
$937$	25.1884	0.822869	0.411435	+	0.911439i	$0.365028\pi$
$938$	0	0
$939$	59.7128	1.94865
$940$	43.7128	1.42575
$941$	52.8807	1.72386	0.861931	−	0.507026i	$-0.169255\pi$
$941$	52.8807	1.72386	0.861931	+	0.507026i	$0.169255\pi$
$942$	−12.0000	−0.390981
$943$	3.10583	0.101140
$944$	−7.45001	−0.242477
$945$	0	0
$946$	6.92820	0.225255
$947$	5.60770	0.182226	0.0911128	−	0.995841i	$-0.470958\pi$
$947$	5.60770	0.182226	0.0911128	+	0.995841i	$0.470958\pi$
$948$	−21.2504	−0.690180
$949$	−25.8564	−0.839334
$950$	−5.13922	−0.166738
$951$	31.1499	1.01010
$952$	0	0
$953$	34.6410	1.12213	0.561066	−	0.827771i	$-0.310391\pi$
$953$	34.6410	1.12213	0.561066	+	0.827771i	$0.310391\pi$
$954$	−0.732051	−0.0237010
$955$	−77.7545	−2.51608
$956$	20.5359	0.664178
$957$	−11.5911	−0.374687
$958$	26.6298	0.860370
$959$	0	0
$960$	7.46410	0.240903
$961$	−10.5692	−0.340943
$962$	10.0382	0.323644
$963$	−1.07180	−0.0345382
$964$	−25.4930	−0.821075
$965$	−92.1741	−2.96719
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	−1.00000	−0.0321412
$969$	−0.896575	−0.0288022
$970$	−50.7846	−1.63060
$971$	−14.4195	−0.462745	−0.231372	−	0.972865i	$-0.574322\pi$
$971$	−14.4195	−0.462745	−0.231372	+	0.972865i	$0.574322\pi$
$972$	−7.45001	−0.238959
$973$	0	0
$974$	−30.3923	−0.973832
$975$	64.1769	2.05531
$976$	0	0
$977$	−16.1436	−0.516479	−0.258240	−	0.966081i	$-0.583142\pi$
$977$	−16.1436	−0.516479	−0.258240	+	0.966081i	$0.583142\pi$
$978$	8.76268	0.280200
$979$	−4.89898	−0.156572
$980$	0	0
$981$	4.39230	0.140236
$982$	22.5167	0.718536
$983$	5.45378	0.173949	0.0869743	−	0.996211i	$-0.472280\pi$
$983$	5.45378	0.173949	0.0869743	+	0.996211i	$0.472280\pi$
$984$	−2.00000	−0.0637577
$985$	0	0
$986$	−1.55291	−0.0494549
$987$	0	0
$988$	1.73205	0.0551039
$989$	−6.00000	−0.190789
$990$	−9.79796	−0.311400
$991$	−3.21539	−0.102140	−0.0510701	−	0.998695i	$-0.516263\pi$
$991$	−3.21539	−0.102140	−0.0510701	+	0.998695i	$0.516263\pi$
$992$	−4.52004	−0.143511
$993$	31.9449	1.01374
$994$	0	0
$995$	52.7846	1.67338
$996$	14.6603	0.464528
$997$	−23.1178	−0.732150	−0.366075	−	0.930585i	$-0.619299\pi$
$997$	−23.1178	−0.732150	−0.366075	+	0.930585i	$0.619299\pi$
$998$	40.7128	1.28874
$999$	13.1440	0.415859

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5194.2.a.bi.1.4	yes	4
7.6	odd	2	inner	5194.2.a.bi.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5194.2.a.bi.1.1	✓	4	7.6	odd	2	inner
5194.2.a.bi.1.4	yes	4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 5194 = 2 \cdot 7^{2} \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5194.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.93185 q^{3} +1.00000 q^{4} +3.86370 q^{5} -1.93185 q^{6} -1.00000 q^{8} +0.732051 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.93185 q^{3} +1.00000 q^{4} +3.86370 q^{5} -1.93185 q^{6} -1.00000 q^{8} +0.732051 q^{9} -3.86370 q^{10} +3.46410 q^{11} +1.93185 q^{12} +3.34607 q^{13} +7.46410 q^{15} +1.00000 q^{16} -0.896575 q^{17} -0.732051 q^{18} +0.517638 q^{19} +3.86370 q^{20} -3.46410 q^{22} +3.00000 q^{23} -1.93185 q^{24} +9.92820 q^{25} -3.34607 q^{26} -4.38134 q^{27} -1.73205 q^{29} -7.46410 q^{30} +4.52004 q^{31} -1.00000 q^{32} +6.69213 q^{33} +0.896575 q^{34} +0.732051 q^{36} -3.00000 q^{37} -0.517638 q^{38} +6.46410 q^{39} -3.86370 q^{40} +1.03528 q^{41} -2.00000 q^{43} +3.46410 q^{44} +2.82843 q^{45} -3.00000 q^{46} +11.3137 q^{47} +1.93185 q^{48} -9.92820 q^{50} -1.73205 q^{51} +3.34607 q^{52} +1.00000 q^{53} +4.38134 q^{54} +13.3843 q^{55} +1.00000 q^{57} +1.73205 q^{58} -7.45001 q^{59} +7.46410 q^{60} -4.52004 q^{62} +1.00000 q^{64} +12.9282 q^{65} -6.69213 q^{66} -7.46410 q^{67} -0.896575 q^{68} +5.79555 q^{69} -13.1962 q^{71} -0.732051 q^{72} -7.72741 q^{73} +3.00000 q^{74} +19.1798 q^{75} +0.517638 q^{76} -6.46410 q^{78} -11.0000 q^{79} +3.86370 q^{80} -10.6603 q^{81} -1.03528 q^{82} +7.58871 q^{83} -3.46410 q^{85} +2.00000 q^{86} -3.34607 q^{87} -3.46410 q^{88} -1.41421 q^{89} -2.82843 q^{90} +3.00000 q^{92} +8.73205 q^{93} -11.3137 q^{94} +2.00000 q^{95} -1.93185 q^{96} +13.1440 q^{97} +2.53590 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 - 4 * q^9	\( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} - 4 q^{9} + 16 q^{15} + 4 q^{16} + 4 q^{18} + 12 q^{23} + 12 q^{25} - 16 q^{30} - 4 q^{32} - 4 q^{36} - 12 q^{37} + 12 q^{39} - 8 q^{43} - 12 q^{46} - 12 q^{50} + 4 q^{53} + 4 q^{57} + 16 q^{60} + 4 q^{64} + 24 q^{65} - 16 q^{67} - 32 q^{71} + 4 q^{72} + 12 q^{74} - 12 q^{78} - 44 q^{79} - 8 q^{81} + 8 q^{86} + 12 q^{92} + 28 q^{93} + 8 q^{95} + 24 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 - 4 * q^9 + 16 * q^15 + 4 * q^16 + 4 * q^18 + 12 * q^23 + 12 * q^25 - 16 * q^30 - 4 * q^32 - 4 * q^36 - 12 * q^37 + 12 * q^39 - 8 * q^43 - 12 * q^46 - 12 * q^50 + 4 * q^53 + 4 * q^57 + 16 * q^60 + 4 * q^64 + 24 * q^65 - 16 * q^67 - 32 * q^71 + 4 * q^72 + 12 * q^74 - 12 * q^78 - 44 * q^79 - 8 * q^81 + 8 * q^86 + 12 * q^92 + 28 * q^93 + 8 * q^95 + 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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