LMFDB - Embedded newform 2842.2.a.ba.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2842,2,Mod(1,2842)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2842, 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("2842.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2842 = 2 \cdot 7^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2842.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:	$22.6934842544$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.52756992.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 13x^{4} + 18x^{3} + 22x^{2} - 20x - 7 $ x^6 - 2x^5 - 13x^4 + 18x^3 + 22x^2 - 20*x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.17509$ of defining polynomial
Character	$\chi$	$=$	2842.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} +4.16178 q^{5} -1.41421 q^{6} -1.00000 q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} +4.16178 q^{5} -1.41421 q^{6} -1.00000 q^{8} -1.00000 q^{9} -4.16178 q^{10} -6.36389 q^{11} +1.41421 q^{12} -4.16178 q^{13} +5.88564 q^{15} +1.00000 q^{16} -6.25233 q^{17} +1.00000 q^{18} -0.737871 q^{19} +4.16178 q^{20} +6.36389 q^{22} -5.52175 q^{23} -1.41421 q^{24} +12.3204 q^{25} +4.16178 q^{26} -5.65685 q^{27} -1.00000 q^{29} -5.88564 q^{30} +5.51446 q^{31} -1.00000 q^{32} -8.99989 q^{33} +6.25233 q^{34} -1.00000 q^{36} +8.24953 q^{37} +0.737871 q^{38} -5.88564 q^{39} -4.16178 q^{40} -8.40442 q^{41} -2.95649 q^{43} -6.36389 q^{44} -4.16178 q^{45} +5.52175 q^{46} -11.0096 q^{47} +1.41421 q^{48} -12.3204 q^{50} -8.84213 q^{51} -4.16178 q^{52} +5.04351 q^{53} +5.65685 q^{54} -26.4851 q^{55} -1.04351 q^{57} +1.00000 q^{58} +3.42391 q^{59} +5.88564 q^{60} +3.50477 q^{61} -5.51446 q^{62} +1.00000 q^{64} -17.3204 q^{65} +8.99989 q^{66} -4.00000 q^{67} -6.25233 q^{68} -7.80894 q^{69} -4.56526 q^{71} +1.00000 q^{72} -1.27182 q^{73} -8.24953 q^{74} +17.4236 q^{75} -0.737871 q^{76} +5.88564 q^{78} -6.84213 q^{79} +4.16178 q^{80} -5.00000 q^{81} +8.40442 q^{82} +12.4238 q^{83} -26.0208 q^{85} +2.95649 q^{86} -1.41421 q^{87} +6.36389 q^{88} +11.0711 q^{89} +4.16178 q^{90} -5.52175 q^{92} +7.79863 q^{93} +11.0096 q^{94} -3.07085 q^{95} -1.41421 q^{96} -11.9092 q^{97} +6.36389 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{2} + 6 q^{4} - 6 q^{8} - 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^2 + 6 * q^4 - 6 * q^8 - 6 * q^9	$ 6 q - 6 q^{2} + 6 q^{4} - 6 q^{8} - 6 q^{9} - 4 q^{11} + 6 q^{16} + 6 q^{18} + 4 q^{22} - 32 q^{23} + 42 q^{25} - 6 q^{29} - 6 q^{32} - 6 q^{36} - 20 q^{37} - 20 q^{43} - 4 q^{44} + 32 q^{46} - 42 q^{50} - 20 q^{51} + 28 q^{53} - 4 q^{57} + 6 q^{58} + 6 q^{64} - 72 q^{65} - 24 q^{67} - 24 q^{71} + 6 q^{72} + 20 q^{74} - 8 q^{79} - 30 q^{81} - 16 q^{85} + 20 q^{86} + 4 q^{88} - 32 q^{92} + 16 q^{93} - 56 q^{95} + 4 q^{99}+O(q^{100}) $ 6 * q - 6 * q^2 + 6 * q^4 - 6 * q^8 - 6 * q^9 - 4 * q^11 + 6 * q^16 + 6 * q^18 + 4 * q^22 - 32 * q^23 + 42 * q^25 - 6 * q^29 - 6 * q^32 - 6 * q^36 - 20 * q^37 - 20 * q^43 - 4 * q^44 + 32 * q^46 - 42 * q^50 - 20 * q^51 + 28 * q^53 - 4 * q^57 + 6 * q^58 + 6 * q^64 - 72 * q^65 - 24 * q^67 - 24 * q^71 + 6 * q^72 + 20 * q^74 - 8 * q^79 - 30 * q^81 - 16 * q^85 + 20 * q^86 + 4 * q^88 - 32 * q^92 + 16 * q^93 - 56 * q^95 + 4 * 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
$3$	1.41421	0.816497	0.408248	−	0.912871i	$-0.366140\pi$
$3$	1.41421	0.816497	0.408248	+	0.912871i	$0.366140\pi$
$4$	1.00000	0.500000
$5$	4.16178	1.86120	0.930601	−	0.366034i	$-0.119285\pi$
$5$	4.16178	1.86120	0.930601	+	0.366034i	$0.119285\pi$
$6$	−1.41421	−0.577350
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	−1.00000	−0.333333
$10$	−4.16178	−1.31607
$11$	−6.36389	−1.91878	−0.959392	−	0.282076i	$-0.908977\pi$
$11$	−6.36389	−1.91878	−0.959392	+	0.282076i	$0.908977\pi$
$12$	1.41421	0.408248
$13$	−4.16178	−1.15427	−0.577134	−	0.816649i	$-0.695829\pi$
$13$	−4.16178	−1.15427	−0.577134	+	0.816649i	$0.695829\pi$
$14$	0	0
$15$	5.88564	1.51967
$16$	1.00000	0.250000
$17$	−6.25233	−1.51641	−0.758207	−	0.652014i	$-0.773924\pi$
$17$	−6.25233	−1.51641	−0.758207	+	0.652014i	$0.773924\pi$
$18$	1.00000	0.235702
$19$	−0.737871	−0.169279	−0.0846396	−	0.996412i	$-0.526974\pi$
$19$	−0.737871	−0.169279	−0.0846396	+	0.996412i	$0.526974\pi$
$20$	4.16178	0.930601
$21$	0	0
$22$	6.36389	1.35679
$23$	−5.52175	−1.15137	−0.575683	−	0.817673i	$-0.695264\pi$
$23$	−5.52175	−1.15137	−0.575683	+	0.817673i	$0.695264\pi$
$24$	−1.41421	−0.288675
$25$	12.3204	2.46408
$26$	4.16178	0.816191
$27$	−5.65685	−1.08866
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	−5.88564	−1.07457
$31$	5.51446	0.990427	0.495213	−	0.868771i	$-0.335090\pi$
$31$	5.51446	0.990427	0.495213	+	0.868771i	$0.335090\pi$
$32$	−1.00000	−0.176777
$33$	−8.99989	−1.56668
$34$	6.25233	1.07227
$35$	0	0
$36$	−1.00000	−0.166667
$37$	8.24953	1.35621	0.678107	−	0.734963i	$-0.262801\pi$
$37$	8.24953	1.35621	0.678107	+	0.734963i	$0.262801\pi$
$38$	0.737871	0.119698
$39$	−5.88564	−0.942457
$40$	−4.16178	−0.658035
$41$	−8.40442	−1.31255	−0.656275	−	0.754522i	$-0.727869\pi$
$41$	−8.40442	−1.31255	−0.656275	+	0.754522i	$0.727869\pi$
$42$	0	0
$43$	−2.95649	−0.450861	−0.225430	−	0.974259i	$-0.572379\pi$
$43$	−2.95649	−0.450861	−0.225430	+	0.974259i	$0.572379\pi$
$44$	−6.36389	−0.959392
$45$	−4.16178	−0.620401
$46$	5.52175	0.814138
$47$	−11.0096	−1.60591	−0.802957	−	0.596037i	$-0.796741\pi$
$47$	−11.0096	−1.60591	−0.802957	+	0.596037i	$0.796741\pi$
$48$	1.41421	0.204124
$49$	0	0
$50$	−12.3204	−1.74236
$51$	−8.84213	−1.23815
$52$	−4.16178	−0.577134
$53$	5.04351	0.692779	0.346389	−	0.938091i	$-0.387408\pi$
$53$	5.04351	0.692779	0.346389	+	0.938091i	$0.387408\pi$
$54$	5.65685	0.769800
$55$	−26.4851	−3.57125
$56$	0	0
$57$	−1.04351	−0.138216
$58$	1.00000	0.131306
$59$	3.42391	0.445754	0.222877	−	0.974847i	$-0.428455\pi$
$59$	3.42391	0.445754	0.222877	+	0.974847i	$0.428455\pi$
$60$	5.88564	0.759833
$61$	3.50477	0.448740	0.224370	−	0.974504i	$-0.427968\pi$
$61$	3.50477	0.448740	0.224370	+	0.974504i	$0.427968\pi$
$62$	−5.51446	−0.700337
$63$	0	0
$64$	1.00000	0.125000
$65$	−17.3204	−2.14833
$66$	8.99989	1.10781
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	−6.25233	−0.758207
$69$	−7.80894	−0.940086
$70$	0	0
$71$	−4.56526	−0.541797	−0.270898	−	0.962608i	$-0.587321\pi$
$71$	−4.56526	−0.541797	−0.270898	+	0.962608i	$0.587321\pi$
$72$	1.00000	0.117851
$73$	−1.27182	−0.148855	−0.0744277	−	0.997226i	$-0.523713\pi$
$73$	−1.27182	−0.148855	−0.0744277	+	0.997226i	$0.523713\pi$
$74$	−8.24953	−0.958988
$75$	17.4236	2.01191
$76$	−0.737871	−0.0846396
$77$	0	0
$78$	5.88564	0.666418
$79$	−6.84213	−0.769800	−0.384900	−	0.922958i	$-0.625764\pi$
$79$	−6.84213	−0.769800	−0.384900	+	0.922958i	$0.625764\pi$
$80$	4.16178	0.465301
$81$	−5.00000	−0.555556
$82$	8.40442	0.928113
$83$	12.4238	1.36369	0.681845	−	0.731497i	$-0.261178\pi$
$83$	12.4238	1.36369	0.681845	+	0.731497i	$0.261178\pi$
$84$	0	0
$85$	−26.0208	−2.82235
$86$	2.95649	0.318807
$87$	−1.41421	−0.151620
$88$	6.36389	0.678393
$89$	11.0711	1.17354	0.586768	−	0.809755i	$-0.300400\pi$
$89$	11.0711	1.17354	0.586768	+	0.809755i	$0.300400\pi$
$90$	4.16178	0.438690
$91$	0	0
$92$	−5.52175	−0.575683
$93$	7.79863	0.808680
$94$	11.0096	1.13555
$95$	−3.07085	−0.315063
$96$	−1.41421	−0.144338
$97$	−11.9092	−1.20919	−0.604597	−	0.796531i	$-0.706666\pi$
$97$	−11.9092	−1.20919	−0.604597	+	0.796531i	$0.706666\pi$
$98$	0	0
$99$	6.36389	0.639595
$100$	12.3204	1.23204
$101$	−9.16162	−0.911616	−0.455808	−	0.890078i	$-0.650650\pi$
$101$	−9.16162	−0.911616	−0.455808	+	0.890078i	$0.650650\pi$
$102$	8.84213	0.875502
$103$	−6.84781	−0.674735	−0.337367	−	0.941373i	$-0.609536\pi$
$103$	−6.84781	−0.674735	−0.337367	+	0.941373i	$0.609536\pi$
$104$	4.16178	0.408096
$105$	0	0
$106$	−5.04351	−0.489869
$107$	8.72777	0.843746	0.421873	−	0.906655i	$-0.361373\pi$
$107$	8.72777	0.843746	0.421873	+	0.906655i	$0.361373\pi$
$108$	−5.65685	−0.544331
$109$	5.04351	0.483080	0.241540	−	0.970391i	$-0.422347\pi$
$109$	5.04351	0.483080	0.241540	+	0.970391i	$0.422347\pi$
$110$	26.4851	2.52525
$111$	11.6666	1.10734
$112$	0	0
$113$	−13.8856	−1.30625	−0.653126	−	0.757250i	$-0.726543\pi$
$113$	−13.8856	−1.30625	−0.653126	+	0.757250i	$0.726543\pi$
$114$	1.04351	0.0977334
$115$	−22.9803	−2.14292
$116$	−1.00000	−0.0928477
$117$	4.16178	0.384756
$118$	−3.42391	−0.315196
$119$	0	0
$120$	−5.88564	−0.537283
$121$	29.4991	2.68173
$122$	−3.50477	−0.317307
$123$	−11.8856	−1.07169
$124$	5.51446	0.495213
$125$	30.4658	2.72494
$126$	0	0
$127$	−3.79863	−0.337074	−0.168537	−	0.985695i	$-0.553904\pi$
$127$	−3.79863	−0.337074	−0.168537	+	0.985695i	$0.553904\pi$
$128$	−1.00000	−0.0883883
$129$	−4.18111	−0.368126
$130$	17.3204	1.51910
$131$	12.0516	1.05295	0.526476	−	0.850190i	$-0.323513\pi$
$131$	12.0516	1.05295	0.526476	+	0.850190i	$0.323513\pi$
$132$	−8.99989	−0.783340
$133$	0	0
$134$	4.00000	0.345547
$135$	−23.5426	−2.02622
$136$	6.25233	0.536133
$137$	11.5699	0.988484	0.494242	−	0.869324i	$-0.335446\pi$
$137$	11.5699	0.988484	0.494242	+	0.869324i	$0.335446\pi$
$138$	7.80894	0.664741
$139$	−8.24269	−0.699136	−0.349568	−	0.936911i	$-0.613672\pi$
$139$	−8.24269	−0.699136	−0.349568	+	0.936911i	$0.613672\pi$
$140$	0	0
$141$	−15.5699	−1.31122
$142$	4.56526	0.383108
$143$	26.4851	2.21479
$144$	−1.00000	−0.0833333
$145$	−4.16178	−0.345617
$146$	1.27182	0.105257
$147$	0	0
$148$	8.24953	0.678107
$149$	−13.7986	−1.13043	−0.565214	−	0.824944i	$-0.691206\pi$
$149$	−13.7986	−1.13043	−0.565214	+	0.824944i	$0.691206\pi$
$150$	−17.4236	−1.42263
$151$	15.2060	1.23745	0.618724	−	0.785608i	$-0.287650\pi$
$151$	15.2060	1.23745	0.618724	+	0.785608i	$0.287650\pi$
$152$	0.737871	0.0598492
$153$	6.25233	0.505471
$154$	0	0
$155$	22.9500	1.84338
$156$	−5.88564	−0.471228
$157$	23.4949	1.87510	0.937549	−	0.347853i	$-0.113089\pi$
$157$	23.4949	1.87510	0.937549	+	0.347853i	$0.113089\pi$
$158$	6.84213	0.544331
$159$	7.13260	0.565652
$160$	−4.16178	−0.329017
$161$	0	0
$162$	5.00000	0.392837
$163$	−6.59261	−0.516373	−0.258186	−	0.966095i	$-0.583125\pi$
$163$	−6.59261	−0.516373	−0.258186	+	0.966095i	$0.583125\pi$
$164$	−8.40442	−0.656275
$165$	−37.4555	−2.91591
$166$	−12.4238	−0.964274
$167$	−10.6374	−0.823144	−0.411572	−	0.911377i	$-0.635020\pi$
$167$	−10.6374	−0.823144	−0.411572	+	0.911377i	$0.635020\pi$
$168$	0	0
$169$	4.32038	0.332337
$170$	26.0208	1.99570
$171$	0.737871	0.0564264
$172$	−2.95649	−0.225430
$173$	−8.46594	−0.643654	−0.321827	−	0.946799i	$-0.604297\pi$
$173$	−8.46594	−0.643654	−0.321827	+	0.946799i	$0.604297\pi$
$174$	1.41421	0.107211
$175$	0	0
$176$	−6.36389	−0.479696
$177$	4.84213	0.363957
$178$	−11.0711	−0.829815
$179$	5.91299	0.441957	0.220979	−	0.975279i	$-0.429075\pi$
$179$	5.91299	0.441957	0.220979	+	0.975279i	$0.429075\pi$
$180$	−4.16178	−0.310200
$181$	5.79925	0.431055	0.215527	−	0.976498i	$-0.430853\pi$
$181$	5.79925	0.431055	0.215527	+	0.976498i	$0.430853\pi$
$182$	0	0
$183$	4.95649	0.366395
$184$	5.52175	0.407069
$185$	34.3327	2.52419
$186$	−7.79863	−0.571823
$187$	39.7891	2.90967
$188$	−11.0096	−0.802957
$189$	0	0
$190$	3.07085	0.222783
$191$	20.5264	1.48524	0.742619	−	0.669714i	$-0.233583\pi$
$191$	20.5264	1.48524	0.742619	+	0.669714i	$0.233583\pi$
$192$	1.41421	0.102062
$193$	−10.8421	−0.780434	−0.390217	−	0.920723i	$-0.627600\pi$
$193$	−10.8421	−0.780434	−0.390217	+	0.920723i	$0.627600\pi$
$194$	11.9092	0.855030
$195$	−24.4947	−1.75410
$196$	0	0
$197$	−4.84213	−0.344988	−0.172494	−	0.985011i	$-0.555182\pi$
$197$	−4.84213	−0.344988	−0.172494	+	0.985011i	$0.555182\pi$
$198$	−6.36389	−0.452262
$199$	−11.7053	−0.829764	−0.414882	−	0.909875i	$-0.636177\pi$
$199$	−11.7053	−0.829764	−0.414882	+	0.909875i	$0.636177\pi$
$200$	−12.3204	−0.871182
$201$	−5.65685	−0.399004
$202$	9.16162	0.644610
$203$	0	0
$204$	−8.84213	−0.619073
$205$	−34.9773	−2.44292
$206$	6.84781	0.477110
$207$	5.52175	0.383788
$208$	−4.16178	−0.288567
$209$	4.69573	0.324810
$210$	0	0
$211$	−15.0435	−1.03564	−0.517819	−	0.855490i	$-0.673256\pi$
$211$	−15.0435	−1.03564	−0.517819	+	0.855490i	$0.673256\pi$
$212$	5.04351	0.346389
$213$	−6.45625	−0.442375
$214$	−8.72777	−0.596618
$215$	−12.3043	−0.839144
$216$	5.65685	0.384900
$217$	0	0
$218$	−5.04351	−0.341589
$219$	−1.79863	−0.121540
$220$	−26.4851	−1.78562
$221$	26.0208	1.75035
$222$	−11.6666	−0.783010
$223$	0.838072	0.0561215	0.0280607	−	0.999606i	$-0.491067\pi$
$223$	0.838072	0.0561215	0.0280607	+	0.999606i	$0.491067\pi$
$224$	0	0
$225$	−12.3204	−0.821359
$226$	13.8856	0.923659
$227$	1.55661	0.103316	0.0516578	−	0.998665i	$-0.483549\pi$
$227$	1.55661	0.103316	0.0516578	+	0.998665i	$0.483549\pi$
$228$	−1.04351	−0.0691079
$229$	13.3041	0.879157	0.439579	−	0.898204i	$-0.355128\pi$
$229$	13.3041	0.879157	0.439579	+	0.898204i	$0.355128\pi$
$230$	22.9803	1.51528
$231$	0	0
$232$	1.00000	0.0656532
$233$	−11.7713	−0.771162	−0.385581	−	0.922674i	$-0.625999\pi$
$233$	−11.7713	−0.771162	−0.385581	+	0.922674i	$0.625999\pi$
$234$	−4.16178	−0.272064
$235$	−45.8194	−2.98893
$236$	3.42391	0.222877
$237$	−9.67624	−0.628539
$238$	0	0
$239$	−23.5426	−1.52284	−0.761421	−	0.648258i	$-0.775498\pi$
$239$	−23.5426	−1.52284	−0.761421	+	0.648258i	$0.775498\pi$
$240$	5.88564	0.379916
$241$	27.0612	1.74317	0.871583	−	0.490248i	$-0.163094\pi$
$241$	27.0612	1.74317	0.871583	+	0.490248i	$0.163094\pi$
$242$	−29.4991	−1.89627
$243$	9.89949	0.635053
$244$	3.50477	0.224370
$245$	0	0
$246$	11.8856	0.757801
$247$	3.07085	0.195394
$248$	−5.51446	−0.350169
$249$	17.5699	1.11345
$250$	−30.4658	−1.92683
$251$	−14.9481	−0.943516	−0.471758	−	0.881728i	$-0.656380\pi$
$251$	−14.9481	−0.943516	−0.471758	+	0.881728i	$0.656380\pi$
$252$	0	0
$253$	35.1398	2.20922
$254$	3.79863	0.238347
$255$	−36.7990	−2.30444
$256$	1.00000	0.0625000
$257$	−4.40437	−0.274737	−0.137369	−	0.990520i	$-0.543864\pi$
$257$	−4.40437	−0.274737	−0.137369	+	0.990520i	$0.543864\pi$
$258$	4.18111	0.260305
$259$	0	0
$260$	−17.3204	−1.07416
$261$	1.00000	0.0618984
$262$	−12.0516	−0.744549
$263$	−24.9291	−1.53720	−0.768599	−	0.639731i	$-0.779046\pi$
$263$	−24.9291	−1.53720	−0.768599	+	0.639731i	$0.779046\pi$
$264$	8.99989	0.553905
$265$	20.9899	1.28940
$266$	0	0
$267$	15.6569	0.958188
$268$	−4.00000	−0.244339
$269$	−2.82843	−0.172452	−0.0862261	−	0.996276i	$-0.527481\pi$
$269$	−2.82843	−0.172452	−0.0862261	+	0.996276i	$0.527481\pi$
$270$	23.5426	1.43275
$271$	−15.4755	−0.940069	−0.470034	−	0.882648i	$-0.655758\pi$
$271$	−15.4755	−0.940069	−0.470034	+	0.882648i	$0.655758\pi$
$272$	−6.25233	−0.379103
$273$	0	0
$274$	−11.5699	−0.698964
$275$	−78.4055	−4.72803
$276$	−7.80894	−0.470043
$277$	−14.9291	−0.897006	−0.448503	−	0.893781i	$-0.648043\pi$
$277$	−14.9291	−0.897006	−0.448503	+	0.893781i	$0.648043\pi$
$278$	8.24269	0.494364
$279$	−5.51446	−0.330142
$280$	0	0
$281$	24.8629	1.48320	0.741599	−	0.670843i	$-0.234068\pi$
$281$	24.8629	1.48320	0.741599	+	0.670843i	$0.234068\pi$
$282$	15.5699	0.927174
$283$	14.7376	0.876060	0.438030	−	0.898960i	$-0.355676\pi$
$283$	14.7376	0.876060	0.438030	+	0.898960i	$0.355676\pi$
$284$	−4.56526	−0.270898
$285$	−4.34284	−0.257248
$286$	−26.4851	−1.56610
$287$	0	0
$288$	1.00000	0.0589256
$289$	22.0917	1.29951
$290$	4.16178	0.244388
$291$	−16.8421	−0.987303
$292$	−1.27182	−0.0744277
$293$	12.9512	0.756616	0.378308	−	0.925680i	$-0.376506\pi$
$293$	12.9512	0.756616	0.378308	+	0.925680i	$0.376506\pi$
$294$	0	0
$295$	14.2495	0.829640
$296$	−8.24953	−0.479494
$297$	35.9996	2.08891
$298$	13.7986	0.799333
$299$	22.9803	1.32899
$300$	17.4236	1.00595
$301$	0	0
$302$	−15.2060	−0.875008
$303$	−12.9565	−0.744331
$304$	−0.737871	−0.0423198
$305$	14.5861	0.835196
$306$	−6.25233	−0.357422
$307$	13.1195	0.748768	0.374384	−	0.927274i	$-0.377854\pi$
$307$	13.1195	0.748768	0.374384	+	0.927274i	$0.377854\pi$
$308$	0	0
$309$	−9.68427	−0.550919
$310$	−22.9500	−1.30347
$311$	−13.5146	−0.766340	−0.383170	−	0.923678i	$-0.625168\pi$
$311$	−13.5146	−0.766340	−0.383170	+	0.923678i	$0.625168\pi$
$312$	5.88564	0.333209
$313$	−11.8898	−0.672054	−0.336027	−	0.941852i	$-0.609083\pi$
$313$	−11.8898	−0.672054	−0.336027	+	0.941852i	$0.609083\pi$
$314$	−23.4949	−1.32589
$315$	0	0
$316$	−6.84213	−0.384900
$317$	−21.7713	−1.22280	−0.611398	−	0.791323i	$-0.709393\pi$
$317$	−21.7713	−1.22280	−0.611398	+	0.791323i	$0.709393\pi$
$318$	−7.13260	−0.399976
$319$	6.36389	0.356309
$320$	4.16178	0.232650
$321$	12.3429	0.688915
$322$	0	0
$323$	4.61341	0.256697
$324$	−5.00000	−0.277778
$325$	−51.2747	−2.84421
$326$	6.59261	0.365131
$327$	7.13260	0.394433
$328$	8.40442	0.464056
$329$	0	0
$330$	37.4555	2.06186
$331$	11.8583	0.651791	0.325895	−	0.945406i	$-0.394334\pi$
$331$	11.8583	0.651791	0.325895	+	0.945406i	$0.394334\pi$
$332$	12.4238	0.681845
$333$	−8.24953	−0.452071
$334$	10.6374	0.582051
$335$	−16.6471	−0.909528
$336$	0	0
$337$	8.81479	0.480172	0.240086	−	0.970752i	$-0.422824\pi$
$337$	8.81479	0.480172	0.240086	+	0.970752i	$0.422824\pi$
$338$	−4.32038	−0.234998
$339$	−19.6373	−1.06655
$340$	−26.0208	−1.41118
$341$	−35.0934	−1.90041
$342$	−0.737871	−0.0398995
$343$	0	0
$344$	2.95649	0.159403
$345$	−32.4991	−1.74969
$346$	8.46594	0.455132
$347$	−3.15787	−0.169523	−0.0847616	−	0.996401i	$-0.527013\pi$
$347$	−3.15787	−0.169523	−0.0847616	+	0.996401i	$0.527013\pi$
$348$	−1.41421	−0.0758098
$349$	−15.3524	−0.821797	−0.410898	−	0.911681i	$-0.634785\pi$
$349$	−15.3524	−0.821797	−0.410898	+	0.911681i	$0.634785\pi$
$350$	0	0
$351$	23.5426	1.25661
$352$	6.36389	0.339196
$353$	17.5467	0.933917	0.466958	−	0.884279i	$-0.345350\pi$
$353$	17.5467	0.933917	0.466958	+	0.884279i	$0.345350\pi$
$354$	−4.84213	−0.257356
$355$	−18.9996	−1.00839
$356$	11.0711	0.586768
$357$	0	0
$358$	−5.91299	−0.312511
$359$	−1.88564	−0.0995203	−0.0497601	−	0.998761i	$-0.515846\pi$
$359$	−1.88564	−0.0995203	−0.0497601	+	0.998761i	$0.515846\pi$
$360$	4.16178	0.219345
$361$	−18.4555	−0.971345
$362$	−5.79925	−0.304802
$363$	41.7180	2.18963
$364$	0	0
$365$	−5.29303	−0.277050
$366$	−4.95649	−0.259080
$367$	1.49508	0.0780424	0.0390212	−	0.999238i	$-0.487576\pi$
$367$	1.49508	0.0780424	0.0390212	+	0.999238i	$0.487576\pi$
$368$	−5.52175	−0.287841
$369$	8.40442	0.437516
$370$	−34.3327	−1.78487
$371$	0	0
$372$	7.79863	0.404340
$373$	20.6134	1.06732	0.533661	−	0.845698i	$-0.320816\pi$
$373$	20.6134	1.06732	0.533661	+	0.845698i	$0.320816\pi$
$374$	−39.7891	−2.05745
$375$	43.0851	2.22491
$376$	11.0096	0.567776
$377$	4.16178	0.214342
$378$	0	0
$379$	−24.4120	−1.25396	−0.626981	−	0.779034i	$-0.715710\pi$
$379$	−24.4120	−1.25396	−0.626981	+	0.779034i	$0.715710\pi$
$380$	−3.07085	−0.157531
$381$	−5.37207	−0.275219
$382$	−20.5264	−1.05022
$383$	−4.98051	−0.254492	−0.127246	−	0.991871i	$-0.540614\pi$
$383$	−4.98051	−0.254492	−0.127246	+	0.991871i	$0.540614\pi$
$384$	−1.41421	−0.0721688
$385$	0	0
$386$	10.8421	0.551850
$387$	2.95649	0.150287
$388$	−11.9092	−0.604597
$389$	7.52175	0.381368	0.190684	−	0.981651i	$-0.438929\pi$
$389$	7.52175	0.381368	0.190684	+	0.981651i	$0.438929\pi$
$390$	24.4947	1.24034
$391$	34.5238	1.74595
$392$	0	0
$393$	17.0435	0.859731
$394$	4.84213	0.243943
$395$	−28.4754	−1.43275
$396$	6.36389	0.319797
$397$	−8.62767	−0.433011	−0.216505	−	0.976281i	$-0.569466\pi$
$397$	−8.62767	−0.433011	−0.216505	+	0.976281i	$0.569466\pi$
$398$	11.7053	0.586732
$399$	0	0
$400$	12.3204	0.616019
$401$	9.77128	0.487954	0.243977	−	0.969781i	$-0.421548\pi$
$401$	9.77128	0.487954	0.243977	+	0.969781i	$0.421548\pi$
$402$	5.65685	0.282138
$403$	−22.9500	−1.14322
$404$	−9.16162	−0.455808
$405$	−20.8089	−1.03400
$406$	0	0
$407$	−52.4991	−2.60228
$408$	8.84213	0.437751
$409$	14.6989	0.726816	0.363408	−	0.931630i	$-0.381613\pi$
$409$	14.6989	0.726816	0.363408	+	0.931630i	$0.381613\pi$
$410$	34.9773	1.72741
$411$	16.3623	0.807094
$412$	−6.84781	−0.337367
$413$	0	0
$414$	−5.52175	−0.271379
$415$	51.7051	2.53810
$416$	4.16178	0.204048
$417$	−11.6569	−0.570842
$418$	−4.69573	−0.229675
$419$	−3.74736	−0.183071	−0.0915354	−	0.995802i	$-0.529177\pi$
$419$	−3.74736	−0.183071	−0.0915354	+	0.995802i	$0.529177\pi$
$420$	0	0
$421$	−12.9773	−0.632475	−0.316237	−	0.948680i	$-0.602420\pi$
$421$	−12.9773	−0.632475	−0.316237	+	0.948680i	$0.602420\pi$
$422$	15.0435	0.732306
$423$	11.0096	0.535304
$424$	−5.04351	−0.244934
$425$	−77.0311	−3.73656
$426$	6.45625	0.312806
$427$	0	0
$428$	8.72777	0.421873
$429$	37.4555	1.80837
$430$	12.3043	0.593364
$431$	−8.95649	−0.431419	−0.215710	−	0.976458i	$-0.569206\pi$
$431$	−8.95649	−0.431419	−0.215710	+	0.976458i	$0.569206\pi$
$432$	−5.65685	−0.272166
$433$	1.71834	0.0825779	0.0412890	−	0.999147i	$-0.486854\pi$
$433$	1.71834	0.0825779	0.0412890	+	0.999147i	$0.486854\pi$
$434$	0	0
$435$	−5.88564	−0.282195
$436$	5.04351	0.241540
$437$	4.07434	0.194902
$438$	1.79863	0.0859417
$439$	−6.45625	−0.308140	−0.154070	−	0.988060i	$-0.549238\pi$
$439$	−6.45625	−0.308140	−0.154070	+	0.988060i	$0.549238\pi$
$440$	26.4851	1.26263
$441$	0	0
$442$	−26.0208	−1.23768
$443$	28.8629	1.37132	0.685660	−	0.727922i	$-0.259514\pi$
$443$	28.8629	1.37132	0.685660	+	0.727922i	$0.259514\pi$
$444$	11.6666	0.553672
$445$	46.0755	2.18419
$446$	−0.838072	−0.0396839
$447$	−19.5142	−0.922990
$448$	0	0
$449$	9.54256	0.450341	0.225171	−	0.974319i	$-0.427706\pi$
$449$	9.54256	0.450341	0.225171	+	0.974319i	$0.427706\pi$
$450$	12.3204	0.580788
$451$	53.4848	2.51850
$452$	−13.8856	−0.653126
$453$	21.5046	1.01037
$454$	−1.55661	−0.0730552
$455$	0	0
$456$	1.04351	0.0488667
$457$	35.1787	1.64559	0.822794	−	0.568339i	$-0.192414\pi$
$457$	35.1787	1.64559	0.822794	+	0.568339i	$0.192414\pi$
$458$	−13.3041	−0.621658
$459$	35.3685	1.65086
$460$	−22.9803	−1.07146
$461$	−1.47574	−0.0687321	−0.0343661	−	0.999409i	$-0.510941\pi$
$461$	−1.47574	−0.0687321	−0.0343661	+	0.999409i	$0.510941\pi$
$462$	0	0
$463$	−1.29303	−0.0600924	−0.0300462	−	0.999549i	$-0.509565\pi$
$463$	−1.29303	−0.0600924	−0.0300462	+	0.999549i	$0.509565\pi$
$464$	−1.00000	−0.0464238
$465$	32.4561	1.50512
$466$	11.7713	0.545294
$467$	−26.7764	−1.23907	−0.619533	−	0.784971i	$-0.712678\pi$
$467$	−26.7764	−1.23907	−0.619533	+	0.784971i	$0.712678\pi$
$468$	4.16178	0.192378
$469$	0	0
$470$	45.8194	2.11349
$471$	33.2268	1.53101
$472$	−3.42391	−0.157598
$473$	18.8148	0.865105
$474$	9.67624	0.444444
$475$	−9.09085	−0.417117
$476$	0	0
$477$	−5.04351	−0.230926
$478$	23.5426	1.07681
$479$	4.20045	0.191923	0.0959617	−	0.995385i	$-0.469407\pi$
$479$	4.20045	0.191923	0.0959617	+	0.995385i	$0.469407\pi$
$480$	−5.88564	−0.268641
$481$	−34.3327	−1.56544
$482$	−27.0612	−1.23260
$483$	0	0
$484$	29.4991	1.34087
$485$	−49.5634	−2.25056
$486$	−9.89949	−0.449050
$487$	−14.6408	−0.663436	−0.331718	−	0.943379i	$-0.607628\pi$
$487$	−14.6408	−0.663436	−0.331718	+	0.943379i	$0.607628\pi$
$488$	−3.50477	−0.158653
$489$	−9.32335	−0.421617
$490$	0	0
$491$	−25.6296	−1.15665	−0.578323	−	0.815808i	$-0.696293\pi$
$491$	−25.6296	−1.15665	−0.578323	+	0.815808i	$0.696293\pi$
$492$	−11.8856	−0.535846
$493$	6.25233	0.281591
$494$	−3.07085	−0.138164
$495$	26.4851	1.19042
$496$	5.51446	0.247607
$497$	0	0
$498$	−17.5699	−0.787326
$499$	−21.6843	−0.970721	−0.485361	−	0.874314i	$-0.661312\pi$
$499$	−21.6843	−0.970721	−0.485361	+	0.874314i	$0.661312\pi$
$500$	30.4658	1.36247
$501$	−15.0435	−0.672094
$502$	14.9481	0.667167
$503$	−22.1616	−0.988135	−0.494068	−	0.869423i	$-0.664491\pi$
$503$	−22.1616	−0.988135	−0.494068	+	0.869423i	$0.664491\pi$
$504$	0	0
$505$	−38.1286	−1.69670
$506$	−35.1398	−1.56216
$507$	6.10994	0.271352
$508$	−3.79863	−0.168537
$509$	12.2005	0.540780	0.270390	−	0.962751i	$-0.412847\pi$
$509$	12.2005	0.540780	0.270390	+	0.962751i	$0.412847\pi$
$510$	36.7990	1.62949
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	4.17403	0.184288
$514$	4.40437	0.194268
$515$	−28.4991	−1.25582
$516$	−4.18111	−0.184063
$517$	70.0638	3.08140
$518$	0	0
$519$	−11.9727	−0.525541
$520$	17.3204	0.759549
$521$	−31.0806	−1.36167	−0.680833	−	0.732439i	$-0.738382\pi$
$521$	−31.0806	−1.36167	−0.680833	+	0.732439i	$0.738382\pi$
$522$	−1.00000	−0.0437688
$523$	−11.3946	−0.498250	−0.249125	−	0.968471i	$-0.580143\pi$
$523$	−11.3946	−0.498250	−0.249125	+	0.968471i	$0.580143\pi$
$524$	12.0516	0.526476
$525$	0	0
$526$	24.9291	1.08696
$527$	−34.4782	−1.50190
$528$	−8.99989	−0.391670
$529$	7.48976	0.325642
$530$	−20.9899	−0.911745
$531$	−3.42391	−0.148585
$532$	0	0
$533$	34.9773	1.51504
$534$	−15.6569	−0.677541
$535$	36.3230	1.57038
$536$	4.00000	0.172774
$537$	8.36223	0.360857
$538$	2.82843	0.121942
$539$	0	0
$540$	−23.5426	−1.01311
$541$	−40.1833	−1.72762	−0.863808	−	0.503821i	$-0.831927\pi$
$541$	−40.1833	−1.72762	−0.863808	+	0.503821i	$0.831927\pi$
$542$	15.4755	0.664729
$543$	8.20137	0.351955
$544$	6.25233	0.268067
$545$	20.9899	0.899111
$546$	0	0
$547$	18.5264	0.792132	0.396066	−	0.918222i	$-0.370375\pi$
$547$	18.5264	0.792132	0.396066	+	0.918222i	$0.370375\pi$
$548$	11.5699	0.494242
$549$	−3.50477	−0.149580
$550$	78.4055	3.34322
$551$	0.737871	0.0314344
$552$	7.80894	0.332370
$553$	0	0
$554$	14.9291	0.634279
$555$	48.5537	2.06099
$556$	−8.24269	−0.349568
$557$	−0.412040	−0.0174587	−0.00872934	−	0.999962i	$-0.502779\pi$
$557$	−0.412040	−0.0174587	−0.00872934	+	0.999962i	$0.502779\pi$
$558$	5.51446	0.233446
$559$	12.3043	0.520415
$560$	0	0
$561$	56.2703	2.37574
$562$	−24.8629	−1.04878
$563$	22.5953	0.952279	0.476139	−	0.879370i	$-0.342036\pi$
$563$	22.5953	0.952279	0.476139	+	0.879370i	$0.342036\pi$
$564$	−15.5699	−0.655611
$565$	−57.7889	−2.43120
$566$	−14.7376	−0.619468
$567$	0	0
$568$	4.56526	0.191554
$569$	−7.13052	−0.298927	−0.149463	−	0.988767i	$-0.547755\pi$
$569$	−7.13052	−0.298927	−0.149463	+	0.988767i	$0.547755\pi$
$570$	4.34284	0.181902
$571$	−41.7439	−1.74693	−0.873465	−	0.486888i	$-0.838132\pi$
$571$	−41.7439	−1.74693	−0.873465	+	0.486888i	$0.838132\pi$
$572$	26.4851	1.10740
$573$	29.0287	1.21269
$574$	0	0
$575$	−68.0301	−2.83705
$576$	−1.00000	−0.0416667
$577$	9.63405	0.401071	0.200535	−	0.979686i	$-0.435732\pi$
$577$	9.63405	0.401071	0.200535	+	0.979686i	$0.435732\pi$
$578$	−22.0917	−0.918892
$579$	−15.3331	−0.637222
$580$	−4.16178	−0.172808
$581$	0	0
$582$	16.8421	0.698129
$583$	−32.0963	−1.32929
$584$	1.27182	0.0526283
$585$	17.3204	0.716110
$586$	−12.9512	−0.535008
$587$	−15.1841	−0.626716	−0.313358	−	0.949635i	$-0.601454\pi$
$587$	−15.1841	−0.626716	−0.313358	+	0.949635i	$0.601454\pi$
$588$	0	0
$589$	−4.06896	−0.167659
$590$	−14.2495	−0.586644
$591$	−6.84781	−0.281681
$592$	8.24953	0.339053
$593$	−17.9383	−0.736636	−0.368318	−	0.929700i	$-0.620066\pi$
$593$	−17.9383	−0.736636	−0.368318	+	0.929700i	$0.620066\pi$
$594$	−35.9996	−1.47708
$595$	0	0
$596$	−13.7986	−0.565214
$597$	−16.5537	−0.677500
$598$	−22.9803	−0.939734
$599$	17.6569	0.721442	0.360721	−	0.932674i	$-0.382531\pi$
$599$	17.6569	0.721442	0.360721	+	0.932674i	$0.382531\pi$
$600$	−17.4236	−0.711317
$601$	−28.7180	−1.17143	−0.585716	−	0.810516i	$-0.699187\pi$
$601$	−28.7180	−1.17143	−0.585716	+	0.810516i	$0.699187\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	15.2060	0.618724
$605$	122.768	4.99125
$606$	12.9565	0.526322
$607$	−30.9316	−1.25548	−0.627738	−	0.778425i	$-0.716019\pi$
$607$	−30.9316	−1.25548	−0.627738	+	0.778425i	$0.716019\pi$
$608$	0.737871	0.0299246
$609$	0	0
$610$	−14.5861	−0.590573
$611$	45.8194	1.85366
$612$	6.25233	0.252736
$613$	39.8856	1.61097	0.805483	−	0.592619i	$-0.201906\pi$
$613$	39.8856	1.61097	0.805483	+	0.592619i	$0.201906\pi$
$614$	−13.1195	−0.529459
$615$	−49.4654	−1.99464
$616$	0	0
$617$	−40.4120	−1.62693	−0.813464	−	0.581616i	$-0.802421\pi$
$617$	−40.4120	−1.62693	−0.813464	+	0.581616i	$0.802421\pi$
$618$	9.68427	0.389558
$619$	−28.5756	−1.14855	−0.574276	−	0.818662i	$-0.694716\pi$
$619$	−28.5756	−1.14855	−0.574276	+	0.818662i	$0.694716\pi$
$620$	22.9500	0.921692
$621$	31.2358	1.25345
$622$	13.5146	0.541884
$623$	0	0
$624$	−5.88564	−0.235614
$625$	65.1899	2.60759
$626$	11.8898	0.475214
$627$	6.64076	0.265206
$628$	23.4949	0.937549
$629$	−51.5788	−2.05658
$630$	0	0
$631$	−29.6296	−1.17953	−0.589767	−	0.807573i	$-0.700781\pi$
$631$	−29.6296	−1.17953	−0.589767	+	0.807573i	$0.700781\pi$
$632$	6.84213	0.272165
$633$	−21.2747	−0.845595
$634$	21.7713	0.864648
$635$	−15.8090	−0.627362
$636$	7.13260	0.282826
$637$	0	0
$638$	−6.36389	−0.251949
$639$	4.56526	0.180599
$640$	−4.16178	−0.164509
$641$	−18.9018	−0.746576	−0.373288	−	0.927715i	$-0.621770\pi$
$641$	−18.9018	−0.746576	−0.373288	+	0.927715i	$0.621770\pi$
$642$	−12.3429	−0.487137
$643$	48.0024	1.89303	0.946515	−	0.322660i	$-0.104577\pi$
$643$	48.0024	1.89303	0.946515	+	0.322660i	$0.104577\pi$
$644$	0	0
$645$	−17.4009	−0.685158
$646$	−4.61341	−0.181512
$647$	−7.40110	−0.290967	−0.145484	−	0.989361i	$-0.546474\pi$
$647$	−7.40110	−0.290967	−0.145484	+	0.989361i	$0.546474\pi$
$648$	5.00000	0.196419
$649$	−21.7893	−0.855307
$650$	51.2747	2.01116
$651$	0	0
$652$	−6.59261	−0.258186
$653$	−2.95649	−0.115697	−0.0578483	−	0.998325i	$-0.518424\pi$
$653$	−2.95649	−0.115697	−0.0578483	+	0.998325i	$0.518424\pi$
$654$	−7.13260	−0.278907
$655$	50.1560	1.95976
$656$	−8.40442	−0.328137
$657$	1.27182	0.0496185
$658$	0	0
$659$	39.2657	1.52957	0.764787	−	0.644283i	$-0.222844\pi$
$659$	39.2657	1.52957	0.764787	+	0.644283i	$0.222844\pi$
$660$	−37.4555	−1.45796
$661$	−13.8767	−0.539741	−0.269870	−	0.962897i	$-0.586981\pi$
$661$	−13.8767	−0.539741	−0.269870	+	0.962897i	$0.586981\pi$
$662$	−11.8583	−0.460886
$663$	36.7990	1.42915
$664$	−12.4238	−0.482137
$665$	0	0
$666$	8.24953	0.319663
$667$	5.52175	0.213803
$668$	−10.6374	−0.411572
$669$	1.18521	0.0458230
$670$	16.6471	0.643134
$671$	−22.3040	−0.861035
$672$	0	0
$673$	14.4509	0.557041	0.278521	−	0.960430i	$-0.410156\pi$
$673$	14.4509	0.557041	0.278521	+	0.960430i	$0.410156\pi$
$674$	−8.81479	−0.339533
$675$	−69.6946	−2.68255
$676$	4.32038	0.166168
$677$	32.3331	1.24266	0.621331	−	0.783549i	$-0.286592\pi$
$677$	32.3331	1.24266	0.621331	+	0.783549i	$0.286592\pi$
$678$	19.6373	0.754164
$679$	0	0
$680$	26.0208	0.997852
$681$	2.20137	0.0843568
$682$	35.0934	1.34380
$683$	10.5861	0.405065	0.202532	−	0.979276i	$-0.435083\pi$
$683$	10.5861	0.405065	0.202532	+	0.979276i	$0.435083\pi$
$684$	0.737871	0.0282132
$685$	48.1514	1.83977
$686$	0	0
$687$	18.8148	0.717829
$688$	−2.95649	−0.112715
$689$	−20.9899	−0.799653
$690$	32.4991	1.23722
$691$	17.2426	0.655939	0.327969	−	0.944688i	$-0.393636\pi$
$691$	17.2426	0.655939	0.327969	+	0.944688i	$0.393636\pi$
$692$	−8.46594	−0.321827
$693$	0	0
$694$	3.15787	0.119871
$695$	−34.3042	−1.30123
$696$	1.41421	0.0536056
$697$	52.5472	1.99037
$698$	15.3524	0.581098
$699$	−16.6471	−0.629651
$700$	0	0
$701$	16.8421	0.636119	0.318059	−	0.948071i	$-0.396969\pi$
$701$	16.8421	0.636119	0.318059	+	0.948071i	$0.396969\pi$
$702$	−23.5426	−0.888557
$703$	−6.08708	−0.229579
$704$	−6.36389	−0.239848
$705$	−64.7985	−2.44045
$706$	−17.5467	−0.660379
$707$	0	0
$708$	4.84213	0.181979
$709$	24.4120	0.916813	0.458407	−	0.888743i	$-0.348420\pi$
$709$	24.4120	0.916813	0.458407	+	0.888743i	$0.348420\pi$
$710$	18.9996	0.713042
$711$	6.84213	0.256600
$712$	−11.0711	−0.414908
$713$	−30.4495	−1.14034
$714$	0	0
$715$	110.225	4.12218
$716$	5.91299	0.220979
$717$	−33.2942	−1.24339
$718$	1.88564	0.0703715
$719$	17.1323	0.638927	0.319463	−	0.947599i	$-0.396497\pi$
$719$	17.1323	0.638927	0.319463	+	0.947599i	$0.396497\pi$
$720$	−4.16178	−0.155100
$721$	0	0
$722$	18.4555	0.686844
$723$	38.2703	1.42329
$724$	5.79925	0.215527
$725$	−12.3204	−0.457567
$726$	−41.7180	−1.54830
$727$	39.1321	1.45133	0.725665	−	0.688048i	$-0.241532\pi$
$727$	39.1321	1.45133	0.725665	+	0.688048i	$0.241532\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	5.29303	0.195904
$731$	18.4850	0.683692
$732$	4.95649	0.183197
$733$	21.1811	0.782342	0.391171	−	0.920318i	$-0.372070\pi$
$733$	21.1811	0.782342	0.391171	+	0.920318i	$0.372070\pi$
$734$	−1.49508	−0.0551843
$735$	0	0
$736$	5.52175	0.203535
$737$	25.4555	0.937667
$738$	−8.40442	−0.309371
$739$	−37.8676	−1.39298	−0.696491	−	0.717566i	$-0.745256\pi$
$739$	−37.8676	−1.39298	−0.696491	+	0.717566i	$0.745256\pi$
$740$	34.3327	1.26209
$741$	4.34284	0.159538
$742$	0	0
$743$	29.2268	1.07223	0.536114	−	0.844145i	$-0.319892\pi$
$743$	29.2268	1.07223	0.536114	+	0.844145i	$0.319892\pi$
$744$	−7.79863	−0.285912
$745$	−57.4268	−2.10395
$746$	−20.6134	−0.754711
$747$	−12.4238	−0.454563
$748$	39.7891	1.45483
$749$	0	0
$750$	−43.0851	−1.57325
$751$	37.2815	1.36042	0.680211	−	0.733016i	$-0.261888\pi$
$751$	37.2815	1.36042	0.680211	+	0.733016i	$0.261888\pi$
$752$	−11.0096	−0.401478
$753$	−21.1398	−0.770378
$754$	−4.16178	−0.151563
$755$	63.2840	2.30314
$756$	0	0
$757$	−8.47825	−0.308147	−0.154074	−	0.988059i	$-0.549239\pi$
$757$	−8.47825	−0.308147	−0.154074	+	0.988059i	$0.549239\pi$
$758$	24.4120	0.886685
$759$	49.6952	1.80382
$760$	3.07085	0.111392
$761$	10.8057	0.391705	0.195853	−	0.980633i	$-0.437253\pi$
$761$	10.8057	0.391705	0.195853	+	0.980633i	$0.437253\pi$
$762$	5.37207	0.194610
$763$	0	0
$764$	20.5264	0.742619
$765$	26.0208	0.940784
$766$	4.98051	0.179953
$767$	−14.2495	−0.514521
$768$	1.41421	0.0510310
$769$	3.54696	0.127907	0.0639533	−	0.997953i	$-0.479629\pi$
$769$	3.54696	0.127907	0.0639533	+	0.997953i	$0.479629\pi$
$770$	0	0
$771$	−6.22872	−0.224322
$772$	−10.8421	−0.390217
$773$	24.4947	0.881014	0.440507	−	0.897749i	$-0.354799\pi$
$773$	24.4947	0.881014	0.440507	+	0.897749i	$0.354799\pi$
$774$	−2.95649	−0.106269
$775$	67.9403	2.44049
$776$	11.9092	0.427515
$777$	0	0
$778$	−7.52175	−0.269668
$779$	6.20137	0.222187
$780$	−24.4947	−0.877052
$781$	29.0528	1.03959
$782$	−34.5238	−1.23457
$783$	5.65685	0.202159
$784$	0	0
$785$	97.7806	3.48994
$786$	−17.0435	−0.607922
$787$	−29.1645	−1.03960	−0.519802	−	0.854287i	$-0.673994\pi$
$787$	−29.1645	−1.03960	−0.519802	+	0.854287i	$0.673994\pi$
$788$	−4.84213	−0.172494
$789$	−35.2551	−1.25512
$790$	28.4754	1.01311
$791$	0	0
$792$	−6.36389	−0.226131
$793$	−14.5861	−0.517966
$794$	8.62767	0.306185
$795$	29.6843	1.05279
$796$	−11.7053	−0.414882
$797$	−54.4846	−1.92994	−0.964971	−	0.262357i	$-0.915500\pi$
$797$	−54.4846	−1.92994	−0.964971	+	0.262357i	$0.915500\pi$
$798$	0	0
$799$	68.8356	2.43523
$800$	−12.3204	−0.435591
$801$	−11.0711	−0.391179
$802$	−9.77128	−0.345036
$803$	8.09372	0.285621
$804$	−5.65685	−0.199502
$805$	0	0
$806$	22.9500	0.808378
$807$	−4.00000	−0.140807
$808$	9.16162	0.322305
$809$	−40.7004	−1.43095	−0.715475	−	0.698638i	$-0.753790\pi$
$809$	−40.7004	−1.43095	−0.715475	+	0.698638i	$0.753790\pi$
$810$	20.8089	0.731150
$811$	−7.88980	−0.277048	−0.138524	−	0.990359i	$-0.544236\pi$
$811$	−7.88980	−0.277048	−0.138524	+	0.990359i	$0.544236\pi$
$812$	0	0
$813$	−21.8856	−0.767563
$814$	52.4991	1.84009
$815$	−27.4370	−0.961074
$816$	−8.84213	−0.309537
$817$	2.18151	0.0763214
$818$	−14.6989	−0.513936
$819$	0	0
$820$	−34.9773	−1.22146
$821$	33.7713	1.17863	0.589313	−	0.807905i	$-0.299399\pi$
$821$	33.7713	1.17863	0.589313	+	0.807905i	$0.299399\pi$
$822$	−16.3623	−0.570701
$823$	−36.6731	−1.27834	−0.639172	−	0.769064i	$-0.720723\pi$
$823$	−36.6731	−1.27834	−0.639172	+	0.769064i	$0.720723\pi$
$824$	6.84781	0.238555
$825$	−110.882	−3.86042
$826$	0	0
$827$	−15.1787	−0.527814	−0.263907	−	0.964548i	$-0.585011\pi$
$827$	−15.1787	−0.527814	−0.263907	+	0.964548i	$0.585011\pi$
$828$	5.52175	0.191894
$829$	16.1712	0.561648	0.280824	−	0.959759i	$-0.409392\pi$
$829$	16.1712	0.561648	0.280824	+	0.959759i	$0.409392\pi$
$830$	−51.7051	−1.79471
$831$	−21.1130	−0.732402
$832$	−4.16178	−0.144284
$833$	0	0
$834$	11.6569	0.403646
$835$	−44.2703	−1.53204
$836$	4.69573	0.162405
$837$	−31.1945	−1.07824
$838$	3.74736	0.129451
$839$	−10.1421	−0.350144	−0.175072	−	0.984556i	$-0.556016\pi$
$839$	−10.1421	−0.350144	−0.175072	+	0.984556i	$0.556016\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	12.9773	0.447227
$843$	35.1615	1.21103
$844$	−15.0435	−0.517819
$845$	17.9805	0.618546
$846$	−11.0096	−0.378517
$847$	0	0
$848$	5.04351	0.173195
$849$	20.8421	0.715300
$850$	77.0311	2.64215
$851$	−45.5519	−1.56150
$852$	−6.45625	−0.221188
$853$	6.65666	0.227920	0.113960	−	0.993485i	$-0.463646\pi$
$853$	6.65666	0.227920	0.113960	+	0.993485i	$0.463646\pi$
$854$	0	0
$855$	3.07085	0.105021
$856$	−8.72777	−0.298309
$857$	−25.2620	−0.862934	−0.431467	−	0.902129i	$-0.642004\pi$
$857$	−25.2620	−0.862934	−0.431467	+	0.902129i	$0.642004\pi$
$858$	−37.4555	−1.27871
$859$	48.3359	1.64920	0.824601	−	0.565715i	$-0.191400\pi$
$859$	48.3359	1.64920	0.824601	+	0.565715i	$0.191400\pi$
$860$	−12.3043	−0.419572
$861$	0	0
$862$	8.95649	0.305059
$863$	−21.2930	−0.724823	−0.362412	−	0.932018i	$-0.618047\pi$
$863$	−21.2930	−0.724823	−0.362412	+	0.932018i	$0.618047\pi$
$864$	5.65685	0.192450
$865$	−35.2334	−1.19797
$866$	−1.71834	−0.0583914
$867$	31.2423	1.06104
$868$	0	0
$869$	43.5426	1.47708
$870$	5.88564	0.199542
$871$	16.6471	0.564066
$872$	−5.04351	−0.170795
$873$	11.9092	0.403065
$874$	−4.07434	−0.137817
$875$	0	0
$876$	−1.79863	−0.0607700
$877$	−39.4829	−1.33324	−0.666621	−	0.745397i	$-0.732260\pi$
$877$	−39.4829	−1.33324	−0.666621	+	0.745397i	$0.732260\pi$
$878$	6.45625	0.217888
$879$	18.3157	0.617774
$880$	−26.4851	−0.892812
$881$	11.8705	0.399928	0.199964	−	0.979803i	$-0.435918\pi$
$881$	11.8705	0.399928	0.199964	+	0.979803i	$0.435918\pi$
$882$	0	0
$883$	−52.0963	−1.75318	−0.876590	−	0.481238i	$-0.840187\pi$
$883$	−52.0963	−1.75318	−0.876590	+	0.481238i	$0.840187\pi$
$884$	26.0208	0.875174
$885$	20.1519	0.677398
$886$	−28.8629	−0.969669
$887$	−12.6084	−0.423348	−0.211674	−	0.977340i	$-0.567892\pi$
$887$	−12.6084	−0.423348	−0.211674	+	0.977340i	$0.567892\pi$
$888$	−11.6666	−0.391505
$889$	0	0
$890$	−46.0755	−1.54445
$891$	31.8194	1.06599
$892$	0.838072	0.0280607
$893$	8.12365	0.271848
$894$	19.5142	0.652653
$895$	24.6085	0.822572
$896$	0	0
$897$	32.4991	1.08511
$898$	−9.54256	−0.318439
$899$	−5.51446	−0.183918
$900$	−12.3204	−0.410679
$901$	−31.5337	−1.05054
$902$	−53.4848	−1.78085
$903$	0	0
$904$	13.8856	0.461830
$905$	24.1352	0.802280
$906$	−21.5046	−0.714441
$907$	22.7759	0.756262	0.378131	−	0.925752i	$-0.376567\pi$
$907$	22.7759	0.756262	0.378131	+	0.925752i	$0.376567\pi$
$908$	1.55661	0.0516578
$909$	9.16162	0.303872
$910$	0	0
$911$	−37.9819	−1.25840	−0.629199	−	0.777244i	$-0.716617\pi$
$911$	−37.9819	−1.25840	−0.629199	+	0.777244i	$0.716617\pi$
$912$	−1.04351	−0.0345540
$913$	−79.0637	−2.61663
$914$	−35.1787	−1.16361
$915$	20.6278	0.681935
$916$	13.3041	0.439579
$917$	0	0
$918$	−35.3685	−1.16734
$919$	−15.7051	−0.518063	−0.259031	−	0.965869i	$-0.583403\pi$
$919$	−15.7051	−0.518063	−0.259031	+	0.965869i	$0.583403\pi$
$920$	22.9803	0.757638
$921$	18.5537	0.611367
$922$	1.47574	0.0486010
$923$	18.9996	0.625379
$924$	0	0
$925$	101.637	3.34181
$926$	1.29303	0.0424917
$927$	6.84781	0.224912
$928$	1.00000	0.0328266
$929$	26.1550	0.858119	0.429060	−	0.903276i	$-0.358845\pi$
$929$	26.1550	0.858119	0.429060	+	0.903276i	$0.358845\pi$
$930$	−32.4561	−1.06428
$931$	0	0
$932$	−11.7713	−0.385581
$933$	−19.1125	−0.625714
$934$	26.7764	0.876151
$935$	165.593	5.41549
$936$	−4.16178	−0.136032
$937$	−10.6602	−0.348254	−0.174127	−	0.984723i	$-0.555710\pi$
$937$	−10.6602	−0.348254	−0.174127	+	0.984723i	$0.555710\pi$
$938$	0	0
$939$	−16.8148	−0.548730
$940$	−45.8194	−1.49447
$941$	18.6274	0.607235	0.303617	−	0.952794i	$-0.401806\pi$
$941$	18.6274	0.607235	0.303617	+	0.952794i	$0.401806\pi$
$942$	−33.2268	−1.08259
$943$	46.4071	1.51122
$944$	3.42391	0.111439
$945$	0	0
$946$	−18.8148	−0.611721
$947$	−3.13052	−0.101728	−0.0508641	−	0.998706i	$-0.516198\pi$
$947$	−3.13052	−0.101728	−0.0508641	+	0.998706i	$0.516198\pi$
$948$	−9.67624	−0.314270
$949$	5.29303	0.171819
$950$	9.09085	0.294946
$951$	−30.7892	−0.998410
$952$	0	0
$953$	−5.90645	−0.191329	−0.0956643	−	0.995414i	$-0.530498\pi$
$953$	−5.90645	−0.191329	−0.0956643	+	0.995414i	$0.530498\pi$
$954$	5.04351	0.163290
$955$	85.4263	2.76433
$956$	−23.5426	−0.761421
$957$	8.99989	0.290925
$958$	−4.20045	−0.135710
$959$	0	0
$960$	5.88564	0.189958
$961$	−0.590714	−0.0190553
$962$	34.3327	1.10693
$963$	−8.72777	−0.281249
$964$	27.0612	0.871583
$965$	−45.1225	−1.45255
$966$	0	0
$967$	12.3250	0.396346	0.198173	−	0.980167i	$-0.436499\pi$
$967$	12.3250	0.396346	0.198173	+	0.980167i	$0.436499\pi$
$968$	−29.4991	−0.948135
$969$	6.52435	0.209592
$970$	49.5634	1.59138
$971$	57.0511	1.83086	0.915428	−	0.402482i	$-0.131853\pi$
$971$	57.0511	1.83086	0.915428	+	0.402482i	$0.131853\pi$
$972$	9.89949	0.317526
$973$	0	0
$974$	14.6408	0.469120
$975$	−72.5133	−2.32228
$976$	3.50477	0.112185
$977$	−37.2657	−1.19223	−0.596117	−	0.802897i	$-0.703291\pi$
$977$	−37.2657	−1.19223	−0.596117	+	0.802897i	$0.703291\pi$
$978$	9.32335	0.298128
$979$	−70.4553	−2.25176
$980$	0	0
$981$	−5.04351	−0.161027
$982$	25.6296	0.817873
$983$	−29.4946	−0.940730	−0.470365	−	0.882472i	$-0.655878\pi$
$983$	−29.4946	−0.940730	−0.470365	+	0.882472i	$0.655878\pi$
$984$	11.8856	0.378900
$985$	−20.1519	−0.642092
$986$	−6.25233	−0.199115
$987$	0	0
$988$	3.07085	0.0976968
$989$	16.3250	0.519106
$990$	−26.4851	−0.841751
$991$	10.5861	0.336278	0.168139	−	0.985763i	$-0.446224\pi$
$991$	10.5861	0.336278	0.168139	+	0.985763i	$0.446224\pi$
$992$	−5.51446	−0.175084
$993$	16.7702	0.532185
$994$	0	0
$995$	−48.7147	−1.54436
$996$	17.5699	0.556724
$997$	−55.0286	−1.74277	−0.871387	−	0.490597i	$-0.836779\pi$
$997$	−55.0286	−1.74277	−0.871387	+	0.490597i	$0.836779\pi$
$998$	21.6843	0.686403
$999$	−46.6664	−1.47646

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2842.2.a.ba.1.6	yes	6
7.6	odd	2	inner	2842.2.a.ba.1.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2842.2.a.ba.1.1	✓	6	7.6	odd	2	inner
2842.2.a.ba.1.6	yes	6	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:	\( N \)	\(=\)	\( 2842 = 2 \cdot 7^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2842.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} +4.16178 q^{5} -1.41421 q^{6} -1.00000 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} +4.16178 q^{5} -1.41421 q^{6} -1.00000 q^{8} -1.00000 q^{9} -4.16178 q^{10} -6.36389 q^{11} +1.41421 q^{12} -4.16178 q^{13} +5.88564 q^{15} +1.00000 q^{16} -6.25233 q^{17} +1.00000 q^{18} -0.737871 q^{19} +4.16178 q^{20} +6.36389 q^{22} -5.52175 q^{23} -1.41421 q^{24} +12.3204 q^{25} +4.16178 q^{26} -5.65685 q^{27} -1.00000 q^{29} -5.88564 q^{30} +5.51446 q^{31} -1.00000 q^{32} -8.99989 q^{33} +6.25233 q^{34} -1.00000 q^{36} +8.24953 q^{37} +0.737871 q^{38} -5.88564 q^{39} -4.16178 q^{40} -8.40442 q^{41} -2.95649 q^{43} -6.36389 q^{44} -4.16178 q^{45} +5.52175 q^{46} -11.0096 q^{47} +1.41421 q^{48} -12.3204 q^{50} -8.84213 q^{51} -4.16178 q^{52} +5.04351 q^{53} +5.65685 q^{54} -26.4851 q^{55} -1.04351 q^{57} +1.00000 q^{58} +3.42391 q^{59} +5.88564 q^{60} +3.50477 q^{61} -5.51446 q^{62} +1.00000 q^{64} -17.3204 q^{65} +8.99989 q^{66} -4.00000 q^{67} -6.25233 q^{68} -7.80894 q^{69} -4.56526 q^{71} +1.00000 q^{72} -1.27182 q^{73} -8.24953 q^{74} +17.4236 q^{75} -0.737871 q^{76} +5.88564 q^{78} -6.84213 q^{79} +4.16178 q^{80} -5.00000 q^{81} +8.40442 q^{82} +12.4238 q^{83} -26.0208 q^{85} +2.95649 q^{86} -1.41421 q^{87} +6.36389 q^{88} +11.0711 q^{89} +4.16178 q^{90} -5.52175 q^{92} +7.79863 q^{93} +11.0096 q^{94} -3.07085 q^{95} -1.41421 q^{96} -11.9092 q^{97} +6.36389 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{2} + 6 q^{4} - 6 q^{8} - 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^2 + 6 * q^4 - 6 * q^8 - 6 * q^9	\( 6 q - 6 q^{2} + 6 q^{4} - 6 q^{8} - 6 q^{9} - 4 q^{11} + 6 q^{16} + 6 q^{18} + 4 q^{22} - 32 q^{23} + 42 q^{25} - 6 q^{29} - 6 q^{32} - 6 q^{36} - 20 q^{37} - 20 q^{43} - 4 q^{44} + 32 q^{46} - 42 q^{50} - 20 q^{51} + 28 q^{53} - 4 q^{57} + 6 q^{58} + 6 q^{64} - 72 q^{65} - 24 q^{67} - 24 q^{71} + 6 q^{72} + 20 q^{74} - 8 q^{79} - 30 q^{81} - 16 q^{85} + 20 q^{86} + 4 q^{88} - 32 q^{92} + 16 q^{93} - 56 q^{95} + 4 q^{99}+O(q^{100}) \) 6 * q - 6 * q^2 + 6 * q^4 - 6 * q^8 - 6 * q^9 - 4 * q^11 + 6 * q^16 + 6 * q^18 + 4 * q^22 - 32 * q^23 + 42 * q^25 - 6 * q^29 - 6 * q^32 - 6 * q^36 - 20 * q^37 - 20 * q^43 - 4 * q^44 + 32 * q^46 - 42 * q^50 - 20 * q^51 + 28 * q^53 - 4 * q^57 + 6 * q^58 + 6 * q^64 - 72 * q^65 - 24 * q^67 - 24 * q^71 + 6 * q^72 + 20 * q^74 - 8 * q^79 - 30 * q^81 - 16 * q^85 + 20 * q^86 + 4 * q^88 - 32 * q^92 + 16 * q^93 - 56 * q^95 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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