LMFDB - Embedded newform 2842.2.a.bc.1.1

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:	$0$
Dimension:	$6$
Coefficient field:	6.6.373409792.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 13x^{4} + 48x^{2} - 50 $ x^6 - 13x^4 + 48x^2 - 50
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.93372$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -3.06767 q^{3} +1.00000 q^{4} -1.76375 q^{5} -3.06767 q^{6} +1.00000 q^{8} +6.41061 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -3.06767 q^{3} +1.00000 q^{4} -1.76375 q^{5} -3.06767 q^{6} +1.00000 q^{8} +6.41061 q^{9} -1.76375 q^{10} -5.93203 q^{11} -3.06767 q^{12} -1.76375 q^{13} +5.41061 q^{15} +1.00000 q^{16} -1.30392 q^{17} +6.41061 q^{18} -3.86745 q^{19} -1.76375 q^{20} -5.93203 q^{22} +4.52142 q^{23} -3.06767 q^{24} -1.88918 q^{25} -1.76375 q^{26} -10.4626 q^{27} +1.00000 q^{29} +5.41061 q^{30} -0.504144 q^{31} +1.00000 q^{32} +18.1975 q^{33} -1.30392 q^{34} +6.41061 q^{36} -5.34264 q^{37} -3.86745 q^{38} +5.41061 q^{39} -1.76375 q^{40} +9.03882 q^{41} -5.41061 q^{43} -5.93203 q^{44} -11.3067 q^{45} +4.52142 q^{46} +4.03164 q^{47} -3.06767 q^{48} -1.88918 q^{50} +4.00000 q^{51} -1.76375 q^{52} -3.41061 q^{53} -10.4626 q^{54} +10.4626 q^{55} +11.8641 q^{57} +1.00000 q^{58} -11.3067 q^{59} +5.41061 q^{60} -1.25961 q^{61} -0.504144 q^{62} +1.00000 q^{64} +3.11082 q^{65} +18.1975 q^{66} +10.8212 q^{67} -1.30392 q^{68} -13.8702 q^{69} -6.29979 q^{71} +6.41061 q^{72} +1.30392 q^{73} -5.34264 q^{74} +5.79540 q^{75} -3.86745 q^{76} +5.41061 q^{78} -1.41061 q^{79} -1.76375 q^{80} +12.8641 q^{81} +9.03882 q^{82} -16.4338 q^{83} +2.29979 q^{85} -5.41061 q^{86} -3.06767 q^{87} -5.93203 q^{88} -0.295633 q^{89} -11.3067 q^{90} +4.52142 q^{92} +1.54655 q^{93} +4.03164 q^{94} +6.82121 q^{95} -3.06767 q^{96} +18.0218 q^{97} -38.0279 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{2} + 6 q^{4} + 6 q^{8} + 12 q^{9}+O(q^{10}) $ 6 * q + 6 * q^2 + 6 * q^4 + 6 * q^8 + 12 * q^9	$ 6 q + 6 q^{2} + 6 q^{4} + 6 q^{8} + 12 q^{9} - 2 q^{11} + 6 q^{15} + 6 q^{16} + 12 q^{18} - 2 q^{22} + 20 q^{23} + 8 q^{25} + 6 q^{29} + 6 q^{30} + 6 q^{32} + 12 q^{36} + 28 q^{37} + 6 q^{39} - 6 q^{43} - 2 q^{44} + 20 q^{46} + 8 q^{50} + 24 q^{51} + 6 q^{53} + 4 q^{57} + 6 q^{58} + 6 q^{60} + 6 q^{64} + 38 q^{65} + 12 q^{67} + 8 q^{71} + 12 q^{72} + 28 q^{74} + 6 q^{78} + 18 q^{79} + 10 q^{81} - 32 q^{85} - 6 q^{86} - 2 q^{88} + 20 q^{92} + 50 q^{93} - 12 q^{95} - 48 q^{99}+O(q^{100}) $ 6 * q + 6 * q^2 + 6 * q^4 + 6 * q^8 + 12 * q^9 - 2 * q^11 + 6 * q^15 + 6 * q^16 + 12 * q^18 - 2 * q^22 + 20 * q^23 + 8 * q^25 + 6 * q^29 + 6 * q^30 + 6 * q^32 + 12 * q^36 + 28 * q^37 + 6 * q^39 - 6 * q^43 - 2 * q^44 + 20 * q^46 + 8 * q^50 + 24 * q^51 + 6 * q^53 + 4 * q^57 + 6 * q^58 + 6 * q^60 + 6 * q^64 + 38 * q^65 + 12 * q^67 + 8 * q^71 + 12 * q^72 + 28 * q^74 + 6 * q^78 + 18 * q^79 + 10 * q^81 - 32 * q^85 - 6 * q^86 - 2 * q^88 + 20 * q^92 + 50 * q^93 - 12 * q^95 - 48 * 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
$3$	−3.06767	−1.77112	−0.885560	−	0.464524i	$-0.846225\pi$
$3$	−3.06767	−1.77112	−0.885560	+	0.464524i	$0.846225\pi$
$4$	1.00000	0.500000
$5$	−1.76375	−0.788773	−0.394387	−	0.918945i	$-0.629043\pi$
$5$	−1.76375	−0.788773	−0.394387	+	0.918945i	$0.629043\pi$
$6$	−3.06767	−1.25237
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	6.41061	2.13687
$10$	−1.76375	−0.557747
$11$	−5.93203	−1.78857	−0.894287	−	0.447494i	$-0.852317\pi$
$11$	−5.93203	−1.78857	−0.894287	+	0.447494i	$0.852317\pi$
$12$	−3.06767	−0.885560
$13$	−1.76375	−0.489176	−0.244588	−	0.969627i	$-0.578653\pi$
$13$	−1.76375	−0.489176	−0.244588	+	0.969627i	$0.578653\pi$
$14$	0	0
$15$	5.41061	1.39701
$16$	1.00000	0.250000
$17$	−1.30392	−0.316247	−0.158124	−	0.987419i	$-0.550544\pi$
$17$	−1.30392	−0.316247	−0.158124	+	0.987419i	$0.550544\pi$
$18$	6.41061	1.51099
$19$	−3.86745	−0.887253	−0.443627	−	0.896212i	$-0.646308\pi$
$19$	−3.86745	−0.887253	−0.443627	+	0.896212i	$0.646308\pi$
$20$	−1.76375	−0.394387
$21$	0	0
$22$	−5.93203	−1.26471
$23$	4.52142	0.942782	0.471391	−	0.881924i	$-0.343752\pi$
$23$	4.52142	0.942782	0.471391	+	0.881924i	$0.343752\pi$
$24$	−3.06767	−0.626186
$25$	−1.88918	−0.377837
$26$	−1.76375	−0.345900
$27$	−10.4626	−2.01353
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	5.41061	0.987837
$31$	−0.504144	−0.0905469	−0.0452735	−	0.998975i	$-0.514416\pi$
$31$	−0.504144	−0.0905469	−0.0452735	+	0.998975i	$0.514416\pi$
$32$	1.00000	0.176777
$33$	18.1975	3.16778
$34$	−1.30392	−0.223621
$35$	0	0
$36$	6.41061	1.06843
$37$	−5.34264	−0.878324	−0.439162	−	0.898408i	$-0.644725\pi$
$37$	−5.34264	−0.878324	−0.439162	+	0.898408i	$0.644725\pi$
$38$	−3.86745	−0.627383
$39$	5.41061	0.866390
$40$	−1.76375	−0.278873
$41$	9.03882	1.41163	0.705813	−	0.708398i	$-0.250582\pi$
$41$	9.03882	1.41163	0.705813	+	0.708398i	$0.250582\pi$
$42$	0	0
$43$	−5.41061	−0.825110	−0.412555	−	0.910933i	$-0.635364\pi$
$43$	−5.41061	−0.825110	−0.412555	+	0.910933i	$0.635364\pi$
$44$	−5.93203	−0.894287
$45$	−11.3067	−1.68550
$46$	4.52142	0.666647
$47$	4.03164	0.588076	0.294038	−	0.955794i	$-0.405001\pi$
$47$	4.03164	0.588076	0.294038	+	0.955794i	$0.405001\pi$
$48$	−3.06767	−0.442780
$49$	0	0
$50$	−1.88918	−0.267171
$51$	4.00000	0.560112
$52$	−1.76375	−0.244588
$53$	−3.41061	−0.468483	−0.234241	−	0.972178i	$-0.575261\pi$
$53$	−3.41061	−0.468483	−0.234241	+	0.972178i	$0.575261\pi$
$54$	−10.4626	−1.42378
$55$	10.4626	1.41078
$56$	0	0
$57$	11.8641	1.57143
$58$	1.00000	0.131306
$59$	−11.3067	−1.47201	−0.736004	−	0.676977i	$-0.763290\pi$
$59$	−11.3067	−1.47201	−0.736004	+	0.676977i	$0.763290\pi$
$60$	5.41061	0.698506
$61$	−1.25961	−0.161276	−0.0806381	−	0.996743i	$-0.525696\pi$
$61$	−1.25961	−0.161276	−0.0806381	+	0.996743i	$0.525696\pi$
$62$	−0.504144	−0.0640263
$63$	0	0
$64$	1.00000	0.125000
$65$	3.11082	0.385849
$66$	18.1975	2.23996
$67$	10.8212	1.32202	0.661011	−	0.750376i	$-0.270128\pi$
$67$	10.8212	1.32202	0.661011	+	0.750376i	$0.270128\pi$
$68$	−1.30392	−0.158124
$69$	−13.8702	−1.66978
$70$	0	0
$71$	−6.29979	−0.747648	−0.373824	−	0.927500i	$-0.621954\pi$
$71$	−6.29979	−0.747648	−0.373824	+	0.927500i	$0.621954\pi$
$72$	6.41061	0.755497
$73$	1.30392	0.152612	0.0763062	−	0.997084i	$-0.475687\pi$
$73$	1.30392	0.152612	0.0763062	+	0.997084i	$0.475687\pi$
$74$	−5.34264	−0.621069
$75$	5.79540	0.669195
$76$	−3.86745	−0.443627
$77$	0	0
$78$	5.41061	0.612631
$79$	−1.41061	−0.158706	−0.0793528	−	0.996847i	$-0.525285\pi$
$79$	−1.41061	−0.158706	−0.0793528	+	0.996847i	$0.525285\pi$
$80$	−1.76375	−0.197193
$81$	12.8641	1.42934
$82$	9.03882	0.998170
$83$	−16.4338	−1.80384	−0.901920	−	0.431903i	$-0.857842\pi$
$83$	−16.4338	−1.80384	−0.901920	+	0.431903i	$0.857842\pi$
$84$	0	0
$85$	2.29979	0.249447
$86$	−5.41061	−0.583441
$87$	−3.06767	−0.328889
$88$	−5.93203	−0.632356
$89$	−0.295633	−0.0313370	−0.0156685	−	0.999877i	$-0.504988\pi$
$89$	−0.295633	−0.0313370	−0.0156685	+	0.999877i	$0.504988\pi$
$90$	−11.3067	−1.19183
$91$	0	0
$92$	4.52142	0.471391
$93$	1.54655	0.160370
$94$	4.03164	0.415832
$95$	6.82121	0.699842
$96$	−3.06767	−0.313093
$97$	18.0218	1.82983	0.914917	−	0.403643i	$-0.132256\pi$
$97$	18.0218	1.82983	0.914917	+	0.403643i	$0.132256\pi$
$98$	0	0
$99$	−38.0279	−3.82195
$100$	−1.88918	−0.188918
$101$	19.6656	1.95680	0.978402	−	0.206712i	$-0.0662763\pi$
$101$	19.6656	1.95680	0.978402	+	0.206712i	$0.0662763\pi$
$102$	4.00000	0.396059
$103$	17.3977	1.71425	0.857125	−	0.515108i	$-0.172248\pi$
$103$	17.3977	1.71425	0.857125	+	0.515108i	$0.172248\pi$
$104$	−1.76375	−0.172950
$105$	0	0
$106$	−3.41061	−0.331267
$107$	14.8212	1.43282	0.716410	−	0.697679i	$-0.245784\pi$
$107$	14.8212	1.43282	0.716410	+	0.697679i	$0.245784\pi$
$108$	−10.4626	−1.00677
$109$	18.2318	1.74629	0.873146	−	0.487459i	$-0.162076\pi$
$109$	18.2318	1.74629	0.873146	+	0.487459i	$0.162076\pi$
$110$	10.4626	0.997572
$111$	16.3895	1.55562
$112$	0	0
$113$	7.77837	0.731727	0.365864	−	0.930668i	$-0.380774\pi$
$113$	7.77837	0.731727	0.365864	+	0.930668i	$0.380774\pi$
$114$	11.8641	1.11117
$115$	−7.97466	−0.743641
$116$	1.00000	0.0928477
$117$	−11.3067	−1.04531
$118$	−11.3067	−1.04087
$119$	0	0
$120$	5.41061	0.493919
$121$	24.1890	2.19900
$122$	−1.25961	−0.114039
$123$	−27.7281	−2.50016
$124$	−0.504144	−0.0452735
$125$	12.1508	1.08680
$126$	0	0
$127$	−13.0428	−1.15737	−0.578683	−	0.815553i	$-0.696433\pi$
$127$	−13.0428	−1.15737	−0.578683	+	0.815553i	$0.696433\pi$
$128$	1.00000	0.0883883
$129$	16.5980	1.46137
$130$	3.11082	0.272837
$131$	18.6573	1.63010	0.815050	−	0.579391i	$-0.196709\pi$
$131$	18.6573	1.63010	0.815050	+	0.579391i	$0.196709\pi$
$132$	18.1975	1.58389
$133$	0	0
$134$	10.8212	0.934810
$135$	18.4535	1.58822
$136$	−1.30392	−0.111810
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	−13.8702	−1.18071
$139$	12.2264	1.03703	0.518514	−	0.855069i	$-0.326486\pi$
$139$	12.2264	1.03703	0.518514	+	0.855069i	$0.326486\pi$
$140$	0	0
$141$	−12.3678	−1.04155
$142$	−6.29979	−0.528667
$143$	10.4626	0.874928
$144$	6.41061	0.534217
$145$	−1.76375	−0.146472
$146$	1.30392	0.107913
$147$	0	0
$148$	−5.34264	−0.439162
$149$	23.2747	1.90673	0.953367	−	0.301812i	$-0.0975915\pi$
$149$	23.2747	1.90673	0.953367	+	0.301812i	$0.0975915\pi$
$150$	5.79540	0.473192
$151$	−17.1210	−1.39329	−0.696644	−	0.717417i	$-0.745324\pi$
$151$	−17.1210	−1.39329	−0.696644	+	0.717417i	$0.745324\pi$
$152$	−3.86745	−0.313691
$153$	−8.35892	−0.675779
$154$	0	0
$155$	0.889184	0.0714210
$156$	5.41061	0.433195
$157$	−4.78711	−0.382053	−0.191026	−	0.981585i	$-0.561182\pi$
$157$	−4.78711	−0.382053	−0.191026	+	0.981585i	$0.561182\pi$
$158$	−1.41061	−0.112222
$159$	10.4626	0.829740
$160$	−1.76375	−0.139437
$161$	0	0
$162$	12.8641	1.01070
$163$	−6.97488	−0.546314	−0.273157	−	0.961969i	$-0.588068\pi$
$163$	−6.97488	−0.546314	−0.273157	+	0.961969i	$0.588068\pi$
$164$	9.03882	0.705813
$165$	−32.0959	−2.49866
$166$	−16.4338	−1.27551
$167$	4.53579	0.350990	0.175495	−	0.984480i	$-0.443847\pi$
$167$	4.53579	0.350990	0.175495	+	0.984480i	$0.443847\pi$
$168$	0	0
$169$	−9.88918	−0.760706
$170$	2.29979	0.176386
$171$	−24.7927	−1.89594
$172$	−5.41061	−0.412555
$173$	−18.3617	−1.39602	−0.698008	−	0.716090i	$-0.745930\pi$
$173$	−18.3617	−1.39602	−0.698008	+	0.716090i	$0.745930\pi$
$174$	−3.06767	−0.232560
$175$	0	0
$176$	−5.93203	−0.447144
$177$	34.6853	2.60710
$178$	−0.295633	−0.0221586
$179$	10.0857	0.753840	0.376920	−	0.926246i	$-0.376983\pi$
$179$	10.0857	0.753840	0.376920	+	0.926246i	$0.376983\pi$
$180$	−11.3067	−0.842752
$181$	1.85238	0.137686	0.0688431	−	0.997627i	$-0.478069\pi$
$181$	1.85238	0.137686	0.0688431	+	0.997627i	$0.478069\pi$
$182$	0	0
$183$	3.86406	0.285639
$184$	4.52142	0.333324
$185$	9.42308	0.692798
$186$	1.54655	0.113398
$187$	7.73490	0.565632
$188$	4.03164	0.294038
$189$	0	0
$190$	6.82121	0.494863
$191$	−22.6853	−1.64145	−0.820724	−	0.571324i	$-0.806430\pi$
$191$	−22.6853	−1.64145	−0.820724	+	0.571324i	$0.806430\pi$
$192$	−3.06767	−0.221390
$193$	−5.86406	−0.422104	−0.211052	−	0.977475i	$-0.567689\pi$
$193$	−5.86406	−0.422104	−0.211052	+	0.977475i	$0.567689\pi$
$194$	18.0218	1.29389
$195$	−9.54296	−0.683386
$196$	0	0
$197$	24.6853	1.75875	0.879376	−	0.476127i	$-0.157960\pi$
$197$	24.6853	1.75875	0.879376	+	0.476127i	$0.157960\pi$
$198$	−38.0279	−2.70253
$199$	12.1821	0.863563	0.431782	−	0.901978i	$-0.357885\pi$
$199$	12.1821	0.863563	0.431782	+	0.901978i	$0.357885\pi$
$200$	−1.88918	−0.133585
$201$	−33.1959	−2.34146
$202$	19.6656	1.38367
$203$	0	0
$204$	4.00000	0.280056
$205$	−15.9422	−1.11345
$206$	17.3977	1.21216
$207$	28.9851	2.01460
$208$	−1.76375	−0.122294
$209$	22.9418	1.58692
$210$	0	0
$211$	−15.1890	−1.04565	−0.522826	−	0.852439i	$-0.675122\pi$
$211$	−15.1890	−1.04565	−0.522826	+	0.852439i	$0.675122\pi$
$212$	−3.41061	−0.234241
$213$	19.3257	1.32417
$214$	14.8212	1.01316
$215$	9.54296	0.650825
$216$	−10.4626	−0.711891
$217$	0	0
$218$	18.2318	1.23481
$219$	−4.00000	−0.270295
$220$	10.4626	0.705390
$221$	2.29979	0.154701
$222$	16.3895	1.09999
$223$	4.44716	0.297804	0.148902	−	0.988852i	$-0.452426\pi$
$223$	4.44716	0.297804	0.148902	+	0.988852i	$0.452426\pi$
$224$	0	0
$225$	−12.1108	−0.807388
$226$	7.77837	0.517409
$227$	14.1543	0.939455	0.469728	−	0.882811i	$-0.344352\pi$
$227$	14.1543	0.939455	0.469728	+	0.882811i	$0.344352\pi$
$228$	11.8641	0.785717
$229$	−15.4582	−1.02151	−0.510755	−	0.859727i	$-0.670634\pi$
$229$	−15.4582	−1.02151	−0.510755	+	0.859727i	$0.670634\pi$
$230$	−7.97466	−0.525834
$231$	0	0
$232$	1.00000	0.0656532
$233$	8.45345	0.553804	0.276902	−	0.960898i	$-0.410692\pi$
$233$	8.45345	0.553804	0.276902	+	0.960898i	$0.410692\pi$
$234$	−11.3067	−0.739143
$235$	−7.11082	−0.463858
$236$	−11.3067	−0.736004
$237$	4.32728	0.281087
$238$	0	0
$239$	−1.04285	−0.0674561	−0.0337280	−	0.999431i	$-0.510738\pi$
$239$	−1.04285	−0.0674561	−0.0337280	+	0.999431i	$0.510738\pi$
$240$	5.41061	0.349253
$241$	−3.40762	−0.219504	−0.109752	−	0.993959i	$-0.535006\pi$
$241$	−3.40762	−0.219504	−0.109752	+	0.993959i	$0.535006\pi$
$242$	24.1890	1.55493
$243$	−8.07484	−0.518001
$244$	−1.25961	−0.0806381
$245$	0	0
$246$	−27.7281	−1.76788
$247$	6.82121	0.434023
$248$	−0.504144	−0.0320132
$249$	50.4134	3.19482
$250$	12.1508	0.768484
$251$	13.4104	0.846457	0.423229	−	0.906023i	$-0.360897\pi$
$251$	13.4104	0.846457	0.423229	+	0.906023i	$0.360897\pi$
$252$	0	0
$253$	−26.8212	−1.68624
$254$	−13.0428	−0.818381
$255$	−7.05500	−0.441801
$256$	1.00000	0.0625000
$257$	−12.3906	−0.772902	−0.386451	−	0.922310i	$-0.626299\pi$
$257$	−12.3906	−0.772902	−0.386451	+	0.922310i	$0.626299\pi$
$258$	16.5980	1.03334
$259$	0	0
$260$	3.11082	0.192925
$261$	6.41061	0.396807
$262$	18.6573	1.15265
$263$	9.41061	0.580283	0.290141	−	0.956984i	$-0.406298\pi$
$263$	9.41061	0.580283	0.290141	+	0.956984i	$0.406298\pi$
$264$	18.1975	1.11998
$265$	6.01546	0.369527
$266$	0	0
$267$	0.906905	0.0555017
$268$	10.8212	0.661011
$269$	−28.0804	−1.71209	−0.856047	−	0.516898i	$-0.827087\pi$
$269$	−28.0804	−1.71209	−0.856047	+	0.516898i	$0.827087\pi$
$270$	18.4535	1.12304
$271$	−30.8525	−1.87415	−0.937077	−	0.349123i	$-0.886479\pi$
$271$	−30.8525	−1.87415	−0.937077	+	0.349123i	$0.886479\pi$
$272$	−1.30392	−0.0790618
$273$	0	0
$274$	−2.00000	−0.120824
$275$	11.2067	0.675789
$276$	−13.8702	−0.834890
$277$	8.95715	0.538183	0.269092	−	0.963115i	$-0.413277\pi$
$277$	8.95715	0.538183	0.269092	+	0.963115i	$0.413277\pi$
$278$	12.2264	0.733289
$279$	−3.23187	−0.193487
$280$	0	0
$281$	11.9320	0.711805	0.355903	−	0.934523i	$-0.384173\pi$
$281$	11.9320	0.711805	0.355903	+	0.934523i	$0.384173\pi$
$282$	−12.3678	−0.736489
$283$	−4.49147	−0.266990	−0.133495	−	0.991049i	$-0.542620\pi$
$283$	−4.49147	−0.266990	−0.133495	+	0.991049i	$0.542620\pi$
$284$	−6.29979	−0.373824
$285$	−20.9252	−1.23950
$286$	10.4626	0.618668
$287$	0	0
$288$	6.41061	0.377749
$289$	−15.2998	−0.899988
$290$	−1.76375	−0.103571
$291$	−55.2849	−3.24086
$292$	1.30392	0.0763062
$293$	13.4417	0.785270	0.392635	−	0.919694i	$-0.371564\pi$
$293$	13.4417	0.785270	0.392635	+	0.919694i	$0.371564\pi$
$294$	0	0
$295$	19.9422	1.16108
$296$	−5.34264	−0.310534
$297$	62.0646	3.60135
$298$	23.2747	1.34827
$299$	−7.97466	−0.461187
$300$	5.79540	0.334597
$301$	0	0
$302$	−17.1210	−0.985203
$303$	−60.3277	−3.46574
$304$	−3.86745	−0.221813
$305$	2.22163	0.127210
$306$	−8.35892	−0.477848
$307$	8.28335	0.472756	0.236378	−	0.971661i	$-0.424040\pi$
$307$	8.28335	0.472756	0.236378	+	0.971661i	$0.424040\pi$
$308$	0	0
$309$	−53.3705	−3.03614
$310$	0.889184	0.0505023
$311$	−24.9256	−1.41340	−0.706701	−	0.707512i	$-0.749818\pi$
$311$	−24.9256	−1.41340	−0.706701	+	0.707512i	$0.749818\pi$
$312$	5.41061	0.306315
$313$	0.208511	0.0117857	0.00589287	−	0.999983i	$-0.498124\pi$
$313$	0.208511	0.0117857	0.00589287	+	0.999983i	$0.498124\pi$
$314$	−4.78711	−0.270152
$315$	0	0
$316$	−1.41061	−0.0793528
$317$	−0.0856911	−0.00481289	−0.00240645	−	0.999997i	$-0.500766\pi$
$317$	−0.0856911	−0.00481289	−0.00240645	+	0.999997i	$0.500766\pi$
$318$	10.4626	0.586715
$319$	−5.93203	−0.332130
$320$	−1.76375	−0.0985967
$321$	−45.4666	−2.53770
$322$	0	0
$323$	5.04285	0.280591
$324$	12.8641	0.714670
$325$	3.33205	0.184829
$326$	−6.97488	−0.386303
$327$	−55.9292	−3.09289
$328$	9.03882	0.499085
$329$	0	0
$330$	−32.0959	−1.76682
$331$	1.27467	0.0700620	0.0350310	−	0.999386i	$-0.488847\pi$
$331$	1.27467	0.0700620	0.0350310	+	0.999386i	$0.488847\pi$
$332$	−16.4338	−0.901920
$333$	−34.2495	−1.87686
$334$	4.53579	0.248187
$335$	−19.0859	−1.04278
$336$	0	0
$337$	−6.00000	−0.326841	−0.163420	−	0.986557i	$-0.552253\pi$
$337$	−6.00000	−0.326841	−0.163420	+	0.986557i	$0.552253\pi$
$338$	−9.88918	−0.537901
$339$	−23.8615	−1.29598
$340$	2.29979	0.124724
$341$	2.99060	0.161950
$342$	−24.7927	−1.34064
$343$	0	0
$344$	−5.41061	−0.291720
$345$	24.4636	1.31708
$346$	−18.3617	−0.987132
$347$	8.90690	0.478148	0.239074	−	0.971001i	$-0.423156\pi$
$347$	8.90690	0.478148	0.239074	+	0.971001i	$0.423156\pi$
$348$	−3.06767	−0.164444
$349$	10.5956	0.567167	0.283584	−	0.958947i	$-0.408477\pi$
$349$	10.5956	0.567167	0.283584	+	0.958947i	$0.408477\pi$
$350$	0	0
$351$	18.4535	0.984972
$352$	−5.93203	−0.316178
$353$	31.5964	1.68171	0.840853	−	0.541264i	$-0.182054\pi$
$353$	31.5964	1.68171	0.840853	+	0.541264i	$0.182054\pi$
$354$	34.6853	1.84350
$355$	11.1113	0.589724
$356$	−0.295633	−0.0156685
$357$	0	0
$358$	10.0857	0.533045
$359$	−35.0530	−1.85003	−0.925014	−	0.379933i	$-0.875947\pi$
$359$	−35.0530	−1.85003	−0.925014	+	0.379933i	$0.875947\pi$
$360$	−11.3067	−0.595916
$361$	−4.04285	−0.212781
$362$	1.85238	0.0973589
$363$	−74.2038	−3.89469
$364$	0	0
$365$	−2.29979	−0.120377
$366$	3.86406	0.201978
$367$	−17.6934	−0.923587	−0.461793	−	0.886988i	$-0.652794\pi$
$367$	−17.6934	−0.923587	−0.461793	+	0.886988i	$0.652794\pi$
$368$	4.52142	0.235695
$369$	57.9443	3.01646
$370$	9.42308	0.489882
$371$	0	0
$372$	1.54655	0.0801848
$373$	−21.0530	−1.09008	−0.545042	−	0.838408i	$-0.683486\pi$
$373$	−21.0530	−1.09008	−0.545042	+	0.838408i	$0.683486\pi$
$374$	7.73490	0.399962
$375$	−37.2747	−1.92486
$376$	4.03164	0.207916
$377$	−1.76375	−0.0908378
$378$	0	0
$379$	2.68527	0.137933	0.0689666	−	0.997619i	$-0.478030\pi$
$379$	2.68527	0.137933	0.0689666	+	0.997619i	$0.478030\pi$
$380$	6.82121	0.349921
$381$	40.0112	2.04983
$382$	−22.6853	−1.16068
$383$	20.2453	1.03449	0.517244	−	0.855838i	$-0.326958\pi$
$383$	20.2453	1.03449	0.517244	+	0.855838i	$0.326958\pi$
$384$	−3.06767	−0.156546
$385$	0	0
$386$	−5.86406	−0.298473
$387$	−34.6853	−1.76315
$388$	18.0218	0.914917
$389$	4.43573	0.224901	0.112450	−	0.993657i	$-0.464130\pi$
$389$	4.43573	0.224901	0.112450	+	0.993657i	$0.464130\pi$
$390$	−9.54296	−0.483227
$391$	−5.89558	−0.298152
$392$	0	0
$393$	−57.2346	−2.88710
$394$	24.6853	1.24363
$395$	2.48796	0.125183
$396$	−38.0279	−1.91097
$397$	0.755463	0.0379156	0.0189578	−	0.999820i	$-0.493965\pi$
$397$	0.755463	0.0379156	0.0189578	+	0.999820i	$0.493965\pi$
$398$	12.1821	0.610631
$399$	0	0
$400$	−1.88918	−0.0944592
$401$	−11.4106	−0.569819	−0.284909	−	0.958554i	$-0.591963\pi$
$401$	−11.4106	−0.569819	−0.284909	+	0.958554i	$0.591963\pi$
$402$	−33.1959	−1.65566
$403$	0.889184	0.0442934
$404$	19.6656	0.978402
$405$	−22.6890	−1.12743
$406$	0	0
$407$	31.6927	1.57095
$408$	4.00000	0.198030
$409$	−36.0994	−1.78500	−0.892500	−	0.451047i	$-0.851051\pi$
$409$	−36.0994	−1.78500	−0.892500	+	0.451047i	$0.851051\pi$
$410$	−15.9422	−0.787330
$411$	6.13534	0.302634
$412$	17.3977	0.857125
$413$	0	0
$414$	28.9851	1.42454
$415$	28.9851	1.42282
$416$	−1.76375	−0.0864750
$417$	−37.5065	−1.83670
$418$	22.9418	1.12212
$419$	21.5608	1.05332	0.526658	−	0.850078i	$-0.323445\pi$
$419$	21.5608	1.05332	0.526658	+	0.850078i	$0.323445\pi$
$420$	0	0
$421$	26.9851	1.31517	0.657586	−	0.753380i	$-0.271578\pi$
$421$	26.9851	1.31517	0.657586	+	0.753380i	$0.271578\pi$
$422$	−15.1890	−0.739388
$423$	25.8453	1.25664
$424$	−3.41061	−0.165634
$425$	2.46335	0.119490
$426$	19.3257	0.936333
$427$	0	0
$428$	14.8212	0.716410
$429$	−32.0959	−1.54960
$430$	9.54296	0.460202
$431$	5.64243	0.271786	0.135893	−	0.990724i	$-0.456610\pi$
$431$	5.64243	0.271786	0.135893	+	0.990724i	$0.456610\pi$
$432$	−10.4626	−0.503383
$433$	−8.68732	−0.417486	−0.208743	−	0.977971i	$-0.566937\pi$
$433$	−8.68732	−0.417486	−0.208743	+	0.977971i	$0.566937\pi$
$434$	0	0
$435$	5.41061	0.259419
$436$	18.2318	0.873146
$437$	−17.4864	−0.836486
$438$	−4.00000	−0.191127
$439$	−29.3400	−1.40032	−0.700162	−	0.713984i	$-0.746889\pi$
$439$	−29.3400	−1.40032	−0.700162	+	0.713984i	$0.746889\pi$
$440$	10.4626	0.498786
$441$	0	0
$442$	2.29979	0.109390
$443$	−15.3426	−0.728951	−0.364475	−	0.931213i	$-0.618752\pi$
$443$	−15.3426	−0.728951	−0.364475	+	0.931213i	$0.618752\pi$
$444$	16.3895	0.777809
$445$	0.521423	0.0247178
$446$	4.44716	0.210579
$447$	−71.3990	−3.37706
$448$	0	0
$449$	35.5065	1.67565	0.837827	−	0.545935i	$-0.183826\pi$
$449$	35.5065	1.67565	0.837827	+	0.545935i	$0.183826\pi$
$450$	−12.1108	−0.570909
$451$	−53.6185	−2.52480
$452$	7.77837	0.365864
$453$	52.5216	2.46768
$454$	14.1543	0.664295
$455$	0	0
$456$	11.8641	0.555585
$457$	19.4283	0.908819	0.454409	−	0.890793i	$-0.349850\pi$
$457$	19.4283	0.908819	0.454409	+	0.890793i	$0.349850\pi$
$458$	−15.4582	−0.722316
$459$	13.6424	0.636774
$460$	−7.97466	−0.371821
$461$	6.71506	0.312751	0.156376	−	0.987698i	$-0.450019\pi$
$461$	6.71506	0.312751	0.156376	+	0.987698i	$0.450019\pi$
$462$	0	0
$463$	23.3426	1.08482	0.542412	−	0.840113i	$-0.317511\pi$
$463$	23.3426	1.08482	0.542412	+	0.840113i	$0.317511\pi$
$464$	1.00000	0.0464238
$465$	−2.72772	−0.126495
$466$	8.45345	0.391599
$467$	23.4016	1.08290	0.541450	−	0.840733i	$-0.317876\pi$
$467$	23.4016	1.08290	0.541450	+	0.840733i	$0.317876\pi$
$468$	−11.3067	−0.522653
$469$	0	0
$470$	−7.11082	−0.327997
$471$	14.6853	0.676662
$472$	−11.3067	−0.520434
$473$	32.0959	1.47577
$474$	4.32728	0.198758
$475$	7.30632	0.335237
$476$	0	0
$477$	−21.8641	−1.00109
$478$	−1.04285	−0.0476986
$479$	−30.2612	−1.38267	−0.691335	−	0.722535i	$-0.742977\pi$
$479$	−30.2612	−1.38267	−0.691335	+	0.722535i	$0.742977\pi$
$480$	5.41061	0.246959
$481$	9.42308	0.429655
$482$	−3.40762	−0.155213
$483$	0	0
$484$	24.1890	1.09950
$485$	−31.7859	−1.44332
$486$	−8.07484	−0.366282
$487$	2.08569	0.0945117	0.0472558	−	0.998883i	$-0.484952\pi$
$487$	2.08569	0.0945117	0.0472558	+	0.998883i	$0.484952\pi$
$488$	−1.25961	−0.0570197
$489$	21.3966	0.967589
$490$	0	0
$491$	−12.0959	−0.545879	−0.272940	−	0.962031i	$-0.587996\pi$
$491$	−12.0959	−0.545879	−0.272940	+	0.962031i	$0.587996\pi$
$492$	−27.7281	−1.25008
$493$	−1.30392	−0.0587256
$494$	6.82121	0.306901
$495$	67.0717	3.01465
$496$	−0.504144	−0.0226367
$497$	0	0
$498$	50.4134	2.25908
$499$	29.5065	1.32089	0.660446	−	0.750874i	$-0.270367\pi$
$499$	29.5065	1.32089	0.660446	+	0.750874i	$0.270367\pi$
$500$	12.1508	0.543400
$501$	−13.9143	−0.621645
$502$	13.4104	0.598536
$503$	0.743910	0.0331693	0.0165846	−	0.999862i	$-0.494721\pi$
$503$	0.743910	0.0331693	0.0165846	+	0.999862i	$0.494721\pi$
$504$	0	0
$505$	−34.6853	−1.54347
$506$	−26.8212	−1.19235
$507$	30.3368	1.34730
$508$	−13.0428	−0.578683
$509$	−26.8964	−1.19216	−0.596081	−	0.802925i	$-0.703276\pi$
$509$	−26.8964	−1.19216	−0.596081	+	0.802925i	$0.703276\pi$
$510$	−7.05500	−0.312401
$511$	0	0
$512$	1.00000	0.0441942
$513$	40.4636	1.78651
$514$	−12.3906	−0.546525
$515$	−30.6853	−1.35215
$516$	16.5980	0.730685
$517$	−23.9158	−1.05182
$518$	0	0
$519$	56.3277	2.47251
$520$	3.11082	0.136418
$521$	2.81635	0.123387	0.0616933	−	0.998095i	$-0.480350\pi$
$521$	2.81635	0.123387	0.0616933	+	0.998095i	$0.480350\pi$
$522$	6.41061	0.280585
$523$	−14.8342	−0.648655	−0.324327	−	0.945945i	$-0.605138\pi$
$523$	−14.8342	−0.648655	−0.324327	+	0.945945i	$0.605138\pi$
$524$	18.6573	0.815050
$525$	0	0
$526$	9.41061	0.410322
$527$	0.657364	0.0286352
$528$	18.1975	0.791945
$529$	−2.55674	−0.111162
$530$	6.01546	0.261295
$531$	−72.4829	−3.14549
$532$	0	0
$533$	−15.9422	−0.690534
$534$	0.906905	0.0392456
$535$	−26.1409	−1.13017
$536$	10.8212	0.467405
$537$	−30.9396	−1.33514
$538$	−28.0804	−1.21063
$539$	0	0
$540$	18.4535	0.794110
$541$	23.1788	0.996534	0.498267	−	0.867024i	$-0.333970\pi$
$541$	23.1788	0.996534	0.498267	+	0.867024i	$0.333970\pi$
$542$	−30.8525	−1.32523
$543$	−5.68249	−0.243859
$544$	−1.30392	−0.0559051
$545$	−32.1564	−1.37743
$546$	0	0
$547$	−18.2216	−0.779101	−0.389550	−	0.921005i	$-0.627370\pi$
$547$	−18.2216	−0.779101	−0.389550	+	0.921005i	$0.627370\pi$
$548$	−2.00000	−0.0854358
$549$	−8.07484	−0.344626
$550$	11.2067	0.477855
$551$	−3.86745	−0.164759
$552$	−13.8702	−0.590357
$553$	0	0
$554$	8.95715	0.380553
$555$	−28.9069	−1.22703
$556$	12.2264	0.518514
$557$	12.9572	0.549012	0.274506	−	0.961585i	$-0.411486\pi$
$557$	12.9572	0.549012	0.274506	+	0.961585i	$0.411486\pi$
$558$	−3.23187	−0.136816
$559$	9.54296	0.403624
$560$	0	0
$561$	−23.7281	−1.00180
$562$	11.9320	0.503322
$563$	35.9352	1.51449	0.757244	−	0.653132i	$-0.226545\pi$
$563$	35.9352	1.51449	0.757244	+	0.653132i	$0.226545\pi$
$564$	−12.3678	−0.520777
$565$	−13.7191	−0.577167
$566$	−4.49147	−0.188791
$567$	0	0
$568$	−6.29979	−0.264333
$569$	17.7281	0.743201	0.371601	−	0.928393i	$-0.378809\pi$
$569$	17.7281	0.743201	0.371601	+	0.928393i	$0.378809\pi$
$570$	−20.9252	−0.876462
$571$	1.50649	0.0630445	0.0315222	−	0.999503i	$-0.489964\pi$
$571$	1.50649	0.0630445	0.0315222	+	0.999503i	$0.489964\pi$
$572$	10.4626	0.437464
$573$	69.5910	2.90720
$574$	0	0
$575$	−8.54180	−0.356218
$576$	6.41061	0.267109
$577$	25.4283	1.05859	0.529296	−	0.848437i	$-0.322456\pi$
$577$	25.4283	1.05859	0.529296	+	0.848437i	$0.322456\pi$
$578$	−15.2998	−0.636387
$579$	17.9890	0.747598
$580$	−1.76375	−0.0732358
$581$	0	0
$582$	−55.2849	−2.29163
$583$	20.2318	0.837916
$584$	1.30392	0.0539566
$585$	19.9422	0.824509
$586$	13.4417	0.555270
$587$	43.2546	1.78531	0.892654	−	0.450743i	$-0.148841\pi$
$587$	43.2546	1.78531	0.892654	+	0.450743i	$0.148841\pi$
$588$	0	0
$589$	1.94975	0.0803381
$590$	19.9422	0.821008
$591$	−75.7263	−3.11496
$592$	−5.34264	−0.219581
$593$	−21.6364	−0.888500	−0.444250	−	0.895903i	$-0.646530\pi$
$593$	−21.6364	−0.888500	−0.444250	+	0.895903i	$0.646530\pi$
$594$	62.0646	2.54654
$595$	0	0
$596$	23.2747	0.953367
$597$	−37.3705	−1.52947
$598$	−7.97466	−0.326108
$599$	32.8314	1.34145	0.670727	−	0.741704i	$-0.265982\pi$
$599$	32.8314	1.34145	0.670727	+	0.741704i	$0.265982\pi$
$600$	5.79540	0.236596
$601$	3.82313	0.155949	0.0779744	−	0.996955i	$-0.475155\pi$
$601$	3.82313	0.155949	0.0779744	+	0.996955i	$0.475155\pi$
$602$	0	0
$603$	69.3705	2.82499
$604$	−17.1210	−0.696644
$605$	−42.6633	−1.73451
$606$	−60.3277	−2.45065
$607$	25.3084	1.02724	0.513618	−	0.858019i	$-0.328305\pi$
$607$	25.3084	1.02724	0.513618	+	0.858019i	$0.328305\pi$
$608$	−3.86745	−0.156846
$609$	0	0
$610$	2.22163	0.0899512
$611$	−7.11082	−0.287673
$612$	−8.35892	−0.337889
$613$	28.4535	1.14922	0.574612	−	0.818426i	$-0.305153\pi$
$613$	28.4535	1.14922	0.574612	+	0.818426i	$0.305153\pi$
$614$	8.28335	0.334289
$615$	48.9055	1.97206
$616$	0	0
$617$	30.9069	1.24427	0.622133	−	0.782912i	$-0.286266\pi$
$617$	30.9069	1.24427	0.622133	+	0.782912i	$0.286266\pi$
$618$	−53.3705	−2.14688
$619$	33.6558	1.35274	0.676370	−	0.736562i	$-0.263552\pi$
$619$	33.6558	1.35274	0.676370	+	0.736562i	$0.263552\pi$
$620$	0.889184	0.0357105
$621$	−47.3059	−1.89832
$622$	−24.9256	−0.999427
$623$	0	0
$624$	5.41061	0.216598
$625$	−11.9851	−0.479403
$626$	0.208511	0.00833377
$627$	−70.3779	−2.81062
$628$	−4.78711	−0.191026
$629$	6.96637	0.277768
$630$	0	0
$631$	−18.5139	−0.737026	−0.368513	−	0.929623i	$-0.620133\pi$
$631$	−18.5139	−0.737026	−0.368513	+	0.929623i	$0.620133\pi$
$632$	−1.41061	−0.0561109
$633$	46.5948	1.85198
$634$	−0.0856911	−0.00340323
$635$	23.0043	0.912899
$636$	10.4626	0.414870
$637$	0	0
$638$	−5.93203	−0.234851
$639$	−40.3855	−1.59762
$640$	−1.76375	−0.0697184
$641$	−34.9069	−1.37874	−0.689370	−	0.724409i	$-0.742113\pi$
$641$	−34.9069	−1.37874	−0.689370	+	0.724409i	$0.742113\pi$
$642$	−45.4666	−1.79442
$643$	−15.7539	−0.621272	−0.310636	−	0.950529i	$-0.600542\pi$
$643$	−15.7539	−0.621272	−0.310636	+	0.950529i	$0.600542\pi$
$644$	0	0
$645$	−29.2747	−1.15269
$646$	5.04285	0.198408
$647$	−12.3593	−0.485895	−0.242947	−	0.970039i	$-0.578114\pi$
$647$	−12.3593	−0.485895	−0.242947	+	0.970039i	$0.578114\pi$
$648$	12.8641	0.505348
$649$	67.0717	2.63280
$650$	3.33205	0.130694
$651$	0	0
$652$	−6.97488	−0.273157
$653$	46.1918	1.80762	0.903812	−	0.427931i	$-0.140757\pi$
$653$	46.1918	1.80762	0.903812	+	0.427931i	$0.140757\pi$
$654$	−55.9292	−2.18701
$655$	−32.9069	−1.28578
$656$	9.03882	0.352907
$657$	8.35892	0.326113
$658$	0	0
$659$	−17.7961	−0.693237	−0.346619	−	0.938006i	$-0.612670\pi$
$659$	−17.7961	−0.693237	−0.346619	+	0.938006i	$0.612670\pi$
$660$	−32.0959	−1.24933
$661$	−6.26828	−0.243808	−0.121904	−	0.992542i	$-0.538900\pi$
$661$	−6.26828	−0.243808	−0.121904	+	0.992542i	$0.538900\pi$
$662$	1.27467	0.0495413
$663$	−7.05500	−0.273994
$664$	−16.4338	−0.637754
$665$	0	0
$666$	−34.2495	−1.32714
$667$	4.52142	0.175070
$668$	4.53579	0.175495
$669$	−13.6424	−0.527447
$670$	−19.0859	−0.737353
$671$	7.47202	0.288454
$672$	0	0
$673$	−24.9749	−0.962711	−0.481355	−	0.876526i	$-0.659855\pi$
$673$	−24.9749	−0.962711	−0.481355	+	0.876526i	$0.659855\pi$
$674$	−6.00000	−0.231111
$675$	19.7658	0.760787
$676$	−9.88918	−0.380353
$677$	−10.1800	−0.391251	−0.195625	−	0.980679i	$-0.562674\pi$
$677$	−10.1800	−0.391251	−0.195625	+	0.980679i	$0.562674\pi$
$678$	−23.8615	−0.916394
$679$	0	0
$680$	2.29979	0.0881930
$681$	−43.4208	−1.66389
$682$	2.99060	0.114516
$683$	10.3576	0.396322	0.198161	−	0.980170i	$-0.436503\pi$
$683$	10.3576	0.396322	0.198161	+	0.980170i	$0.436503\pi$
$684$	−24.7927	−0.947972
$685$	3.52750	0.134779
$686$	0	0
$687$	47.4208	1.80922
$688$	−5.41061	−0.206277
$689$	6.01546	0.229171
$690$	24.4636	0.931315
$691$	13.8259	0.525963	0.262981	−	0.964801i	$-0.415294\pi$
$691$	13.8259	0.525963	0.262981	+	0.964801i	$0.415294\pi$
$692$	−18.3617	−0.698008
$693$	0	0
$694$	8.90690	0.338101
$695$	−21.5643	−0.817979
$696$	−3.06767	−0.116280
$697$	−11.7859	−0.446423
$698$	10.5956	0.401048
$699$	−25.9324	−0.980854
$700$	0	0
$701$	6.50370	0.245641	0.122821	−	0.992429i	$-0.460806\pi$
$701$	6.50370	0.245641	0.122821	+	0.992429i	$0.460806\pi$
$702$	18.4535	0.696481
$703$	20.6624	0.779296
$704$	−5.93203	−0.223572
$705$	21.8136	0.821549
$706$	31.5964	1.18915
$707$	0	0
$708$	34.6853	1.30355
$709$	20.7812	0.780453	0.390226	−	0.920719i	$-0.372397\pi$
$709$	20.7812	0.780453	0.390226	+	0.920719i	$0.372397\pi$
$710$	11.1113	0.416998
$711$	−9.04285	−0.339133
$712$	−0.295633	−0.0110793
$713$	−2.27945	−0.0853660
$714$	0	0
$715$	−18.4535	−0.690120
$716$	10.0857	0.376920
$717$	3.19911	0.119473
$718$	−35.0530	−1.30817
$719$	−17.6375	−0.657768	−0.328884	−	0.944370i	$-0.606673\pi$
$719$	−17.6375	−0.657768	−0.328884	+	0.944370i	$0.606673\pi$
$720$	−11.3067	−0.421376
$721$	0	0
$722$	−4.04285	−0.150459
$723$	10.4535	0.388768
$724$	1.85238	0.0688431
$725$	−1.88918	−0.0701625
$726$	−74.2038	−2.75396
$727$	−11.2952	−0.418914	−0.209457	−	0.977818i	$-0.567170\pi$
$727$	−11.2952	−0.418914	−0.209457	+	0.977818i	$0.567170\pi$
$728$	0	0
$729$	−13.8212	−0.511897
$730$	−2.29979	−0.0851191
$731$	7.05500	0.260939
$732$	3.86406	0.142820
$733$	−34.8726	−1.28805	−0.644024	−	0.765006i	$-0.722736\pi$
$733$	−34.8726	−1.28805	−0.644024	+	0.765006i	$0.722736\pi$
$734$	−17.6934	−0.653074
$735$	0	0
$736$	4.52142	0.166662
$737$	−64.1918	−2.36453
$738$	57.9443	2.13296
$739$	−34.3175	−1.26239	−0.631195	−	0.775624i	$-0.717435\pi$
$739$	−34.3175	−1.26239	−0.631195	+	0.775624i	$0.717435\pi$
$740$	9.42308	0.346399
$741$	−20.9252	−0.768708
$742$	0	0
$743$	5.31473	0.194978	0.0974892	−	0.995237i	$-0.468919\pi$
$743$	5.31473	0.194978	0.0974892	+	0.995237i	$0.468919\pi$
$744$	1.54655	0.0566992
$745$	−41.0507	−1.50398
$746$	−21.0530	−0.770806
$747$	−105.350	−3.85457
$748$	7.73490	0.282816
$749$	0	0
$750$	−37.2747	−1.36108
$751$	−13.0428	−0.475940	−0.237970	−	0.971272i	$-0.576482\pi$
$751$	−13.0428	−0.475940	−0.237970	+	0.971272i	$0.576482\pi$
$752$	4.03164	0.147019
$753$	−41.1387	−1.49918
$754$	−1.76375	−0.0642320
$755$	30.1972	1.09899
$756$	0	0
$757$	9.03531	0.328394	0.164197	−	0.986428i	$-0.447497\pi$
$757$	9.03531	0.328394	0.164197	+	0.986428i	$0.447497\pi$
$758$	2.68527	0.0975335
$759$	82.2787	2.98653
$760$	6.82121	0.247431
$761$	−44.3697	−1.60840	−0.804200	−	0.594359i	$-0.797406\pi$
$761$	−44.3697	−1.60840	−0.804200	+	0.594359i	$0.797406\pi$
$762$	40.0112	1.44945
$763$	0	0
$764$	−22.6853	−0.820724
$765$	14.7431	0.533036
$766$	20.2453	0.731494
$767$	19.9422	0.720072
$768$	−3.06767	−0.110695
$769$	−36.0994	−1.30178	−0.650889	−	0.759173i	$-0.725604\pi$
$769$	−36.0994	−1.30178	−0.650889	+	0.759173i	$0.725604\pi$
$770$	0	0
$771$	38.0102	1.36890
$772$	−5.86406	−0.211052
$773$	−4.10721	−0.147726	−0.0738631	−	0.997268i	$-0.523533\pi$
$773$	−4.10721	−0.147726	−0.0738631	+	0.997268i	$0.523533\pi$
$774$	−34.6853	−1.24674
$775$	0.952421	0.0342120
$776$	18.0218	0.646944
$777$	0	0
$778$	4.43573	0.159029
$779$	−34.9572	−1.25247
$780$	−9.54296	−0.341693
$781$	37.3705	1.33722
$782$	−5.89558	−0.210825
$783$	−10.4626	−0.373904
$784$	0	0
$785$	8.44326	0.301353
$786$	−57.2346	−2.04149
$787$	−35.6708	−1.27153	−0.635764	−	0.771884i	$-0.719315\pi$
$787$	−35.6708	−1.27153	−0.635764	+	0.771884i	$0.719315\pi$
$788$	24.6853	0.879376
$789$	−28.8686	−1.02775
$790$	2.48796	0.0885176
$791$	0	0
$792$	−38.0279	−1.35126
$793$	2.22163	0.0788925
$794$	0.755463	0.0268104
$795$	−18.4535	−0.654477
$796$	12.1821	0.431782
$797$	35.2241	1.24770	0.623850	−	0.781544i	$-0.285568\pi$
$797$	35.2241	1.24770	0.623850	+	0.781544i	$0.285568\pi$
$798$	0	0
$799$	−5.25695	−0.185977
$800$	−1.88918	−0.0667927
$801$	−1.89519	−0.0669631
$802$	−11.4106	−0.402923
$803$	−7.73490	−0.272959
$804$	−33.1959	−1.17073
$805$	0	0
$806$	0.889184	0.0313202
$807$	86.1415	3.03232
$808$	19.6656	0.691835
$809$	22.0000	0.773479	0.386739	−	0.922189i	$-0.373601\pi$
$809$	22.0000	0.773479	0.386739	+	0.922189i	$0.373601\pi$
$810$	−22.6890	−0.797210
$811$	44.5913	1.56581	0.782906	−	0.622141i	$-0.213737\pi$
$811$	44.5913	1.56581	0.782906	+	0.622141i	$0.213737\pi$
$812$	0	0
$813$	94.6452	3.31935
$814$	31.6927	1.11083
$815$	12.3019	0.430918
$816$	4.00000	0.140028
$817$	20.9252	0.732082
$818$	−36.0994	−1.26219
$819$	0	0
$820$	−15.9422	−0.556726
$821$	−28.9171	−1.00921	−0.504607	−	0.863349i	$-0.668363\pi$
$821$	−28.9171	−1.00921	−0.504607	+	0.863349i	$0.668363\pi$
$822$	6.13534	0.213995
$823$	14.6853	0.511896	0.255948	−	0.966690i	$-0.417612\pi$
$823$	14.6853	0.511896	0.255948	+	0.966690i	$0.417612\pi$
$824$	17.3977	0.606079
$825$	−34.3785	−1.19690
$826$	0	0
$827$	0.289602	0.0100705	0.00503523	−	0.999987i	$-0.498397\pi$
$827$	0.289602	0.0100705	0.00503523	+	0.999987i	$0.498397\pi$
$828$	28.9851	1.00730
$829$	0.339947	0.0118068	0.00590342	−	0.999983i	$-0.498121\pi$
$829$	0.339947	0.0118068	0.00590342	+	0.999983i	$0.498121\pi$
$830$	28.9851	1.00609
$831$	−27.4776	−0.953188
$832$	−1.76375	−0.0611470
$833$	0	0
$834$	−37.5065	−1.29874
$835$	−8.00000	−0.276851
$836$	22.9418	0.793459
$837$	5.27467	0.182319
$838$	21.5608	0.744806
$839$	5.71983	0.197470	0.0987352	−	0.995114i	$-0.468520\pi$
$839$	5.71983	0.197470	0.0987352	+	0.995114i	$0.468520\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	26.9851	0.929967
$843$	−36.6035	−1.26069
$844$	−15.1890	−0.522826
$845$	17.4421	0.600025
$846$	25.8453	0.888579
$847$	0	0
$848$	−3.41061	−0.117121
$849$	13.7784	0.472872
$850$	2.46335	0.0844921
$851$	−24.1563	−0.828068
$852$	19.3257	0.662087
$853$	4.45871	0.152663	0.0763317	−	0.997082i	$-0.475679\pi$
$853$	4.45871	0.152663	0.0763317	+	0.997082i	$0.475679\pi$
$854$	0	0
$855$	43.7281	1.49547
$856$	14.8212	0.506579
$857$	−5.42419	−0.185287	−0.0926435	−	0.995699i	$-0.529532\pi$
$857$	−5.42419	−0.185287	−0.0926435	+	0.995699i	$0.529532\pi$
$858$	−32.0959	−1.09574
$859$	20.1139	0.686278	0.343139	−	0.939285i	$-0.388510\pi$
$859$	20.1139	0.686278	0.343139	+	0.939285i	$0.388510\pi$
$860$	9.54296	0.325412
$861$	0	0
$862$	5.64243	0.192182
$863$	31.8063	1.08270	0.541349	−	0.840798i	$-0.317914\pi$
$863$	31.8063	1.08270	0.541349	+	0.840798i	$0.317914\pi$
$864$	−10.4626	−0.355946
$865$	32.3855	1.10114
$866$	−8.68732	−0.295207
$867$	46.9347	1.59399
$868$	0	0
$869$	8.36776	0.283857
$870$	5.41061	0.183437
$871$	−19.0859	−0.646702
$872$	18.2318	0.617407
$873$	115.530	3.91011
$874$	−17.4864	−0.591485
$875$	0	0
$876$	−4.00000	−0.135147
$877$	13.9245	0.470197	0.235098	−	0.971972i	$-0.424459\pi$
$877$	13.9245	0.470197	0.235098	+	0.971972i	$0.424459\pi$
$878$	−29.3400	−0.990178
$879$	−41.2346	−1.39081
$880$	10.4626	0.352695
$881$	13.7257	0.462432	0.231216	−	0.972902i	$-0.425730\pi$
$881$	13.7257	0.462432	0.231216	+	0.972902i	$0.425730\pi$
$882$	0	0
$883$	−58.2775	−1.96119	−0.980596	−	0.196039i	$-0.937192\pi$
$883$	−58.2775	−1.96119	−0.980596	+	0.196039i	$0.937192\pi$
$884$	2.29979	0.0773503
$885$	−61.1762	−2.05641
$886$	−15.3426	−0.515446
$887$	12.4464	0.417910	0.208955	−	0.977925i	$-0.432994\pi$
$887$	12.4464	0.417910	0.208955	+	0.977925i	$0.432994\pi$
$888$	16.3895	0.549994
$889$	0	0
$890$	0.521423	0.0174781
$891$	−76.3100	−2.55648
$892$	4.44716	0.148902
$893$	−15.5922	−0.521772
$894$	−71.3990	−2.38794
$895$	−17.7886	−0.594609
$896$	0	0
$897$	24.4636	0.816817
$898$	35.5065	1.18487
$899$	−0.504144	−0.0168141
$900$	−12.1108	−0.403694
$901$	4.44716	0.148156
$902$	−53.6185	−1.78530
$903$	0	0
$904$	7.77837	0.258705
$905$	−3.26713	−0.108603
$906$	52.5216	1.74491
$907$	−16.6574	−0.553099	−0.276549	−	0.961000i	$-0.589191\pi$
$907$	−16.6574	−0.553099	−0.276549	+	0.961000i	$0.589191\pi$
$908$	14.1543	0.469728
$909$	126.069	4.18143
$910$	0	0
$911$	47.1890	1.56344	0.781720	−	0.623629i	$-0.214342\pi$
$911$	47.1890	1.56344	0.781720	+	0.623629i	$0.214342\pi$
$912$	11.8641	0.392858
$913$	97.4856	3.22630
$914$	19.4283	0.642632
$915$	−6.81524	−0.225305
$916$	−15.4582	−0.510755
$917$	0	0
$918$	13.6424	0.450267
$919$	−11.4786	−0.378643	−0.189322	−	0.981915i	$-0.560629\pi$
$919$	−11.4786	−0.378643	−0.189322	+	0.981915i	$0.560629\pi$
$920$	−7.97466	−0.262917
$921$	−25.4106	−0.837308
$922$	6.71506	0.221149
$923$	11.1113	0.365732
$924$	0	0
$925$	10.0932	0.331863
$926$	23.3426	0.767087
$927$	111.530	3.66313
$928$	1.00000	0.0328266
$929$	21.3654	0.700975	0.350488	−	0.936567i	$-0.386016\pi$
$929$	21.3654	0.700975	0.350488	+	0.936567i	$0.386016\pi$
$930$	−2.72772	−0.0894456
$931$	0	0
$932$	8.45345	0.276902
$933$	76.4636	2.50331
$934$	23.4016	0.765725
$935$	−13.6424	−0.446155
$936$	−11.3067	−0.369571
$937$	11.6139	0.379409	0.189705	−	0.981841i	$-0.439247\pi$
$937$	11.6139	0.379409	0.189705	+	0.981841i	$0.439247\pi$
$938$	0	0
$939$	−0.639643	−0.0208740
$940$	−7.11082	−0.231929
$941$	−31.0807	−1.01320	−0.506601	−	0.862181i	$-0.669098\pi$
$941$	−31.0807	−1.01320	−0.506601	+	0.862181i	$0.669098\pi$
$942$	14.6853	0.478472
$943$	40.8683	1.33086
$944$	−11.3067	−0.368002
$945$	0	0
$946$	32.0959	1.04353
$947$	0.503702	0.0163681	0.00818406	−	0.999967i	$-0.497395\pi$
$947$	0.503702	0.0163681	0.00818406	+	0.999967i	$0.497395\pi$
$948$	4.32728	0.140543
$949$	−2.29979	−0.0746544
$950$	7.30632	0.237048
$951$	0.262872	0.00852421
$952$	0	0
$953$	56.1238	1.81803	0.909014	−	0.416766i	$-0.136836\pi$
$953$	56.1238	1.81803	0.909014	+	0.416766i	$0.136836\pi$
$954$	−21.8641	−0.707875
$955$	40.0112	1.29473
$956$	−1.04285	−0.0337280
$957$	18.1975	0.588242
$958$	−30.2612	−0.977695
$959$	0	0
$960$	5.41061	0.174627
$961$	−30.7458	−0.991801
$962$	9.42308	0.303812
$963$	95.0130	3.06175
$964$	−3.40762	−0.109752
$965$	10.3427	0.332945
$966$	0	0
$967$	−2.28207	−0.0733864	−0.0366932	−	0.999327i	$-0.511682\pi$
$967$	−2.28207	−0.0733864	−0.0366932	+	0.999327i	$0.511682\pi$
$968$	24.1890	0.777463
$969$	−15.4698	−0.496961
$970$	−31.7859	−1.02058
$971$	−18.1547	−0.582612	−0.291306	−	0.956630i	$-0.594090\pi$
$971$	−18.1547	−0.582612	−0.291306	+	0.956630i	$0.594090\pi$
$972$	−8.07484	−0.259001
$973$	0	0
$974$	2.08569	0.0668299
$975$	−10.2216	−0.327354
$976$	−1.25961	−0.0403190
$977$	9.11082	0.291481	0.145740	−	0.989323i	$-0.453444\pi$
$977$	9.11082	0.291481	0.145740	+	0.989323i	$0.453444\pi$
$978$	21.3966	0.684189
$979$	1.75370	0.0560486
$980$	0	0
$981$	116.877	3.73160
$982$	−12.0959	−0.385995
$983$	32.1005	1.02385	0.511924	−	0.859031i	$-0.328933\pi$
$983$	32.1005	1.02385	0.511924	+	0.859031i	$0.328933\pi$
$984$	−27.7281	−0.883940
$985$	−43.5387	−1.38726
$986$	−1.30392	−0.0415253
$987$	0	0
$988$	6.82121	0.217012
$989$	−24.4636	−0.777899
$990$	67.0717	2.13168
$991$	−18.5493	−0.589239	−0.294619	−	0.955615i	$-0.595193\pi$
$991$	−18.5493	−0.589239	−0.294619	+	0.955615i	$0.595193\pi$
$992$	−0.504144	−0.0160066
$993$	−3.91026	−0.124088
$994$	0	0
$995$	−21.4861	−0.681155
$996$	50.4134	1.59741
$997$	14.0329	0.444427	0.222214	−	0.974998i	$-0.428672\pi$
$997$	14.0329	0.444427	0.222214	+	0.974998i	$0.428672\pi$
$998$	29.5065	0.934012
$999$	55.8980	1.76853

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2842.2.a.bc.1.1	✓	6	1.1	even	1	trivial
2842.2.a.bc.1.6	yes	6	7.6	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -3.06767 q^{3} +1.00000 q^{4} -1.76375 q^{5} -3.06767 q^{6} +1.00000 q^{8} +6.41061 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -3.06767 q^{3} +1.00000 q^{4} -1.76375 q^{5} -3.06767 q^{6} +1.00000 q^{8} +6.41061 q^{9} -1.76375 q^{10} -5.93203 q^{11} -3.06767 q^{12} -1.76375 q^{13} +5.41061 q^{15} +1.00000 q^{16} -1.30392 q^{17} +6.41061 q^{18} -3.86745 q^{19} -1.76375 q^{20} -5.93203 q^{22} +4.52142 q^{23} -3.06767 q^{24} -1.88918 q^{25} -1.76375 q^{26} -10.4626 q^{27} +1.00000 q^{29} +5.41061 q^{30} -0.504144 q^{31} +1.00000 q^{32} +18.1975 q^{33} -1.30392 q^{34} +6.41061 q^{36} -5.34264 q^{37} -3.86745 q^{38} +5.41061 q^{39} -1.76375 q^{40} +9.03882 q^{41} -5.41061 q^{43} -5.93203 q^{44} -11.3067 q^{45} +4.52142 q^{46} +4.03164 q^{47} -3.06767 q^{48} -1.88918 q^{50} +4.00000 q^{51} -1.76375 q^{52} -3.41061 q^{53} -10.4626 q^{54} +10.4626 q^{55} +11.8641 q^{57} +1.00000 q^{58} -11.3067 q^{59} +5.41061 q^{60} -1.25961 q^{61} -0.504144 q^{62} +1.00000 q^{64} +3.11082 q^{65} +18.1975 q^{66} +10.8212 q^{67} -1.30392 q^{68} -13.8702 q^{69} -6.29979 q^{71} +6.41061 q^{72} +1.30392 q^{73} -5.34264 q^{74} +5.79540 q^{75} -3.86745 q^{76} +5.41061 q^{78} -1.41061 q^{79} -1.76375 q^{80} +12.8641 q^{81} +9.03882 q^{82} -16.4338 q^{83} +2.29979 q^{85} -5.41061 q^{86} -3.06767 q^{87} -5.93203 q^{88} -0.295633 q^{89} -11.3067 q^{90} +4.52142 q^{92} +1.54655 q^{93} +4.03164 q^{94} +6.82121 q^{95} -3.06767 q^{96} +18.0218 q^{97} -38.0279 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} + 6 q^{4} + 6 q^{8} + 12 q^{9}+O(q^{10}) \) 6 * q + 6 * q^2 + 6 * q^4 + 6 * q^8 + 12 * q^9	\( 6 q + 6 q^{2} + 6 q^{4} + 6 q^{8} + 12 q^{9} - 2 q^{11} + 6 q^{15} + 6 q^{16} + 12 q^{18} - 2 q^{22} + 20 q^{23} + 8 q^{25} + 6 q^{29} + 6 q^{30} + 6 q^{32} + 12 q^{36} + 28 q^{37} + 6 q^{39} - 6 q^{43} - 2 q^{44} + 20 q^{46} + 8 q^{50} + 24 q^{51} + 6 q^{53} + 4 q^{57} + 6 q^{58} + 6 q^{60} + 6 q^{64} + 38 q^{65} + 12 q^{67} + 8 q^{71} + 12 q^{72} + 28 q^{74} + 6 q^{78} + 18 q^{79} + 10 q^{81} - 32 q^{85} - 6 q^{86} - 2 q^{88} + 20 q^{92} + 50 q^{93} - 12 q^{95} - 48 q^{99}+O(q^{100}) \) 6 * q + 6 * q^2 + 6 * q^4 + 6 * q^8 + 12 * q^9 - 2 * q^11 + 6 * q^15 + 6 * q^16 + 12 * q^18 - 2 * q^22 + 20 * q^23 + 8 * q^25 + 6 * q^29 + 6 * q^30 + 6 * q^32 + 12 * q^36 + 28 * q^37 + 6 * q^39 - 6 * q^43 - 2 * q^44 + 20 * q^46 + 8 * q^50 + 24 * q^51 + 6 * q^53 + 4 * q^57 + 6 * q^58 + 6 * q^60 + 6 * q^64 + 38 * q^65 + 12 * q^67 + 8 * q^71 + 12 * q^72 + 28 * q^74 + 6 * q^78 + 18 * q^79 + 10 * q^81 - 32 * q^85 - 6 * q^86 - 2 * q^88 + 20 * q^92 + 50 * q^93 - 12 * q^95 - 48 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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