LMFDB - Embedded newform 2842.2.a.bb.1.4

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:	6.6.401917952.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 14x^{4} + 49x^{2} - 18 $ x^6 - 14x^4 + 49x^2 - 18
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.644301$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +0.644301 q^{3} +1.00000 q^{4} +2.05851 q^{5} +0.644301 q^{6} +1.00000 q^{8} -2.58488 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +0.644301 q^{3} +1.00000 q^{4} +2.05851 q^{5} +0.644301 q^{6} +1.00000 q^{8} -2.58488 q^{9} +2.05851 q^{10} +4.67370 q^{11} +0.644301 q^{12} -3.01127 q^{13} +1.32630 q^{15} +1.00000 q^{16} +5.19539 q^{17} -2.58488 q^{18} +2.70281 q^{19} +2.05851 q^{20} +4.67370 q^{22} +5.34740 q^{23} +0.644301 q^{24} -0.762519 q^{25} -3.01127 q^{26} -3.59834 q^{27} -1.00000 q^{29} +1.32630 q^{30} -3.59834 q^{31} +1.00000 q^{32} +3.01127 q^{33} +5.19539 q^{34} -2.58488 q^{36} -2.00000 q^{37} +2.70281 q^{38} -1.94016 q^{39} +2.05851 q^{40} +0.461461 q^{41} +11.1099 q^{43} +4.67370 q^{44} -5.32101 q^{45} +5.34740 q^{46} -3.59834 q^{47} +0.644301 q^{48} -0.762519 q^{50} +3.34740 q^{51} -3.01127 q^{52} -0.673698 q^{53} -3.59834 q^{54} +9.62087 q^{55} +1.74143 q^{57} -1.00000 q^{58} -10.6010 q^{59} +1.32630 q^{60} +11.9306 q^{61} -3.59834 q^{62} +1.00000 q^{64} -6.19874 q^{65} +3.01127 q^{66} +7.16975 q^{67} +5.19539 q^{68} +3.44533 q^{69} +8.00000 q^{71} -2.58488 q^{72} -10.6010 q^{73} -2.00000 q^{74} -0.491292 q^{75} +2.70281 q^{76} -1.94016 q^{78} +6.41512 q^{79} +2.05851 q^{80} +5.43622 q^{81} +0.461461 q^{82} -0.461461 q^{83} +10.6948 q^{85} +11.1099 q^{86} -0.644301 q^{87} +4.67370 q^{88} -0.827140 q^{89} -5.32101 q^{90} +5.34740 q^{92} -2.31841 q^{93} -3.59834 q^{94} +5.56378 q^{95} +0.644301 q^{96} +0.712685 q^{97} -12.0809 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{2} + 6 q^{4} + 6 q^{8} + 10 q^{9}+O(q^{10}) $ 6 * q + 6 * q^2 + 6 * q^4 + 6 * q^8 + 10 * q^9	$ 6 q + 6 q^{2} + 6 q^{4} + 6 q^{8} + 10 q^{9} + 8 q^{11} + 28 q^{15} + 6 q^{16} + 10 q^{18} + 8 q^{22} - 8 q^{23} + 10 q^{25} - 6 q^{29} + 28 q^{30} + 6 q^{32} + 10 q^{36} - 12 q^{37} - 8 q^{39} + 12 q^{43} + 8 q^{44} - 8 q^{46} + 10 q^{50} - 20 q^{51} + 16 q^{53} + 56 q^{57} - 6 q^{58} + 28 q^{60} + 6 q^{64} + 12 q^{65} - 8 q^{67} + 48 q^{71} + 10 q^{72} - 12 q^{74} - 8 q^{78} + 64 q^{79} - 2 q^{81} - 16 q^{85} + 12 q^{86} + 8 q^{88} - 8 q^{92} + 28 q^{93} + 68 q^{95} - 16 q^{99}+O(q^{100}) $ 6 * q + 6 * q^2 + 6 * q^4 + 6 * q^8 + 10 * q^9 + 8 * q^11 + 28 * q^15 + 6 * q^16 + 10 * q^18 + 8 * q^22 - 8 * q^23 + 10 * q^25 - 6 * q^29 + 28 * q^30 + 6 * q^32 + 10 * q^36 - 12 * q^37 - 8 * q^39 + 12 * q^43 + 8 * q^44 - 8 * q^46 + 10 * q^50 - 20 * q^51 + 16 * q^53 + 56 * q^57 - 6 * q^58 + 28 * q^60 + 6 * q^64 + 12 * q^65 - 8 * q^67 + 48 * q^71 + 10 * q^72 - 12 * q^74 - 8 * q^78 + 64 * q^79 - 2 * q^81 - 16 * q^85 + 12 * q^86 + 8 * q^88 - 8 * q^92 + 28 * q^93 + 68 * q^95 - 16 * 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
$3$	0.644301	0.371987	0.185994	−	0.982551i	$-0.440450\pi$
$3$	0.644301	0.371987	0.185994	+	0.982551i	$0.440450\pi$
$4$	1.00000	0.500000
$5$	2.05851	0.920596	0.460298	−	0.887765i	$-0.347743\pi$
$5$	2.05851	0.920596	0.460298	+	0.887765i	$0.347743\pi$
$6$	0.644301	0.263035
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	−2.58488	−0.861626
$10$	2.05851	0.650959
$11$	4.67370	1.40917	0.704586	−	0.709618i	$-0.251133\pi$
$11$	4.67370	1.40917	0.704586	+	0.709618i	$0.251133\pi$
$12$	0.644301	0.185994
$13$	−3.01127	−0.835175	−0.417588	−	0.908637i	$-0.637124\pi$
$13$	−3.01127	−0.835175	−0.417588	+	0.908637i	$0.637124\pi$
$14$	0	0
$15$	1.32630	0.342450
$16$	1.00000	0.250000
$17$	5.19539	1.26007	0.630034	−	0.776568i	$-0.283041\pi$
$17$	5.19539	1.26007	0.630034	+	0.776568i	$0.283041\pi$
$18$	−2.58488	−0.609261
$19$	2.70281	0.620068	0.310034	−	0.950725i	$-0.399660\pi$
$19$	2.70281	0.620068	0.310034	+	0.950725i	$0.399660\pi$
$20$	2.05851	0.460298
$21$	0	0
$22$	4.67370	0.996436
$23$	5.34740	1.11501	0.557505	−	0.830174i	$-0.311759\pi$
$23$	5.34740	1.11501	0.557505	+	0.830174i	$0.311759\pi$
$24$	0.644301	0.131517
$25$	−0.762519	−0.152504
$26$	−3.01127	−0.590558
$27$	−3.59834	−0.692501
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	1.32630	0.242149
$31$	−3.59834	−0.646281	−0.323140	−	0.946351i	$-0.604739\pi$
$31$	−3.59834	−0.646281	−0.323140	+	0.946351i	$0.604739\pi$
$32$	1.00000	0.176777
$33$	3.01127	0.524194
$34$	5.19539	0.891003
$35$	0	0
$36$	−2.58488	−0.430813
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	2.70281	0.438454
$39$	−1.94016	−0.310674
$40$	2.05851	0.325480
$41$	0.461461	0.0720681	0.0360341	−	0.999351i	$-0.488528\pi$
$41$	0.461461	0.0720681	0.0360341	+	0.999351i	$0.488528\pi$
$42$	0	0
$43$	11.1099	1.69425	0.847123	−	0.531397i	$-0.178333\pi$
$43$	11.1099	1.69425	0.847123	+	0.531397i	$0.178333\pi$
$44$	4.67370	0.704586
$45$	−5.32101	−0.793209
$46$	5.34740	0.788430
$47$	−3.59834	−0.524872	−0.262436	−	0.964949i	$-0.584526\pi$
$47$	−3.59834	−0.524872	−0.262436	+	0.964949i	$0.584526\pi$
$48$	0.644301	0.0929968
$49$	0	0
$50$	−0.762519	−0.107836
$51$	3.34740	0.468729
$52$	−3.01127	−0.417588
$53$	−0.673698	−0.0925395	−0.0462698	−	0.998929i	$-0.514733\pi$
$53$	−0.673698	−0.0925395	−0.0462698	+	0.998929i	$0.514733\pi$
$54$	−3.59834	−0.489672
$55$	9.62087	1.29728
$56$	0	0
$57$	1.74143	0.230657
$58$	−1.00000	−0.131306
$59$	−10.6010	−1.38014	−0.690068	−	0.723745i	$-0.742419\pi$
$59$	−10.6010	−1.38014	−0.690068	+	0.723745i	$0.742419\pi$
$60$	1.32630	0.171225
$61$	11.9306	1.52756	0.763779	−	0.645478i	$-0.223342\pi$
$61$	11.9306	1.52756	0.763779	+	0.645478i	$0.223342\pi$
$62$	−3.59834	−0.456990
$63$	0	0
$64$	1.00000	0.125000
$65$	−6.19874	−0.768859
$66$	3.01127	0.370661
$67$	7.16975	0.875925	0.437962	−	0.898993i	$-0.355700\pi$
$67$	7.16975	0.875925	0.437962	+	0.898993i	$0.355700\pi$
$68$	5.19539	0.630034
$69$	3.44533	0.414769
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	−2.58488	−0.304631
$73$	−10.6010	−1.24076	−0.620378	−	0.784303i	$-0.713021\pi$
$73$	−10.6010	−1.24076	−0.620378	+	0.784303i	$0.713021\pi$
$74$	−2.00000	−0.232495
$75$	−0.491292	−0.0567295
$76$	2.70281	0.310034
$77$	0	0
$78$	−1.94016	−0.219680
$79$	6.41512	0.721758	0.360879	−	0.932613i	$-0.382477\pi$
$79$	6.41512	0.721758	0.360879	+	0.932613i	$0.382477\pi$
$80$	2.05851	0.230149
$81$	5.43622	0.604024
$82$	0.461461	0.0509598
$83$	−0.461461	−0.0506519	−0.0253260	−	0.999679i	$-0.508062\pi$
$83$	−0.461461	−0.0506519	−0.0253260	+	0.999679i	$0.508062\pi$
$84$	0	0
$85$	10.6948	1.16001
$86$	11.1099	1.19801
$87$	−0.644301	−0.0690763
$88$	4.67370	0.498218
$89$	−0.827140	−0.0876767	−0.0438384	−	0.999039i	$-0.513959\pi$
$89$	−0.827140	−0.0876767	−0.0438384	+	0.999039i	$0.513959\pi$
$90$	−5.32101	−0.560883
$91$	0	0
$92$	5.34740	0.557505
$93$	−2.31841	−0.240408
$94$	−3.59834	−0.371140
$95$	5.56378	0.570832
$96$	0.644301	0.0657587
$97$	0.712685	0.0723622	0.0361811	−	0.999345i	$-0.488481\pi$
$97$	0.712685	0.0723622	0.0361811	+	0.999345i	$0.488481\pi$
$98$	0	0
$99$	−12.0809	−1.21418
$100$	−0.762519	−0.0762519
$101$	−16.0476	−1.59680	−0.798400	−	0.602127i	$-0.794320\pi$
$101$	−16.0476	−1.59680	−0.798400	+	0.602127i	$0.794320\pi$
$102$	3.34740	0.331442
$103$	8.48528	0.836080	0.418040	−	0.908429i	$-0.362717\pi$
$103$	8.48528	0.836080	0.418040	+	0.908429i	$0.362717\pi$
$104$	−3.01127	−0.295279
$105$	0	0
$106$	−0.673698	−0.0654353
$107$	−9.34740	−0.903647	−0.451823	−	0.892107i	$-0.649226\pi$
$107$	−9.34740	−0.903647	−0.451823	+	0.892107i	$0.649226\pi$
$108$	−3.59834	−0.346250
$109$	10.0211	0.959847	0.479923	−	0.877310i	$-0.340664\pi$
$109$	10.0211	0.959847	0.479923	+	0.877310i	$0.340664\pi$
$110$	9.62087	0.917314
$111$	−1.28860	−0.122309
$112$	0	0
$113$	11.6447	1.09544	0.547721	−	0.836661i	$-0.315495\pi$
$113$	11.6447	1.09544	0.547721	+	0.836661i	$0.315495\pi$
$114$	1.74143	0.163099
$115$	11.0077	1.02647
$116$	−1.00000	−0.0928477
$117$	7.78375	0.719608
$118$	−10.6010	−0.975903
$119$	0	0
$120$	1.32630	0.121074
$121$	10.8435	0.985768
$122$	11.9306	1.08015
$123$	0.297320	0.0268084
$124$	−3.59834	−0.323140
$125$	−11.8622	−1.06099
$126$	0	0
$127$	−7.16975	−0.636213	−0.318106	−	0.948055i	$-0.603047\pi$
$127$	−7.16975	−0.636213	−0.318106	+	0.948055i	$0.603047\pi$
$128$	1.00000	0.0883883
$129$	7.15813	0.630238
$130$	−6.19874	−0.543665
$131$	−20.4047	−1.78277	−0.891385	−	0.453247i	$-0.850266\pi$
$131$	−20.4047	−1.78277	−0.891385	+	0.453247i	$0.850266\pi$
$132$	3.01127	0.262097
$133$	0	0
$134$	7.16975	0.619372
$135$	−7.40723	−0.637513
$136$	5.19539	0.445501
$137$	−19.6869	−1.68197	−0.840983	−	0.541062i	$-0.818022\pi$
$137$	−19.6869	−1.68197	−0.840983	+	0.541062i	$0.818022\pi$
$138$	3.44533	0.293286
$139$	−2.00129	−0.169747	−0.0848735	−	0.996392i	$-0.527049\pi$
$139$	−2.00129	−0.169747	−0.0848735	+	0.996392i	$0.527049\pi$
$140$	0	0
$141$	−2.31841	−0.195246
$142$	8.00000	0.671345
$143$	−14.0738	−1.17691
$144$	−2.58488	−0.215406
$145$	−2.05851	−0.170950
$146$	−10.6010	−0.877347
$147$	0	0
$148$	−2.00000	−0.164399
$149$	−0.415123	−0.0340082	−0.0170041	−	0.999855i	$-0.505413\pi$
$149$	−0.415123	−0.0340082	−0.0170041	+	0.999855i	$0.505413\pi$
$150$	−0.491292	−0.0401138
$151$	3.52504	0.286864	0.143432	−	0.989660i	$-0.454186\pi$
$151$	3.52504	0.286864	0.143432	+	0.989660i	$0.454186\pi$
$152$	2.70281	0.219227
$153$	−13.4295	−1.08571
$154$	0	0
$155$	−7.40723	−0.594963
$156$	−1.94016	−0.155337
$157$	−21.4533	−1.71216	−0.856079	−	0.516845i	$-0.827106\pi$
$157$	−21.4533	−1.71216	−0.856079	+	0.516845i	$0.827106\pi$
$158$	6.41512	0.510360
$159$	−0.434064	−0.0344235
$160$	2.05851	0.162740
$161$	0	0
$162$	5.43622	0.427110
$163$	17.6658	1.38369	0.691846	−	0.722045i	$-0.256797\pi$
$163$	17.6658	1.38369	0.691846	+	0.722045i	$0.256797\pi$
$164$	0.461461	0.0360341
$165$	6.19874	0.482571
$166$	−0.461461	−0.0358163
$167$	0.251224	0.0194403	0.00972016	−	0.999953i	$-0.496906\pi$
$167$	0.251224	0.0194403	0.00972016	+	0.999953i	$0.496906\pi$
$168$	0	0
$169$	−3.93227	−0.302482
$170$	10.6948	0.820253
$171$	−6.98644	−0.534267
$172$	11.1099	0.847123
$173$	12.2665	0.932602	0.466301	−	0.884626i	$-0.345586\pi$
$173$	12.2665	0.932602	0.466301	+	0.884626i	$0.345586\pi$
$174$	−0.644301	−0.0488443
$175$	0	0
$176$	4.67370	0.352293
$177$	−6.83025	−0.513393
$178$	−0.827140	−0.0619968
$179$	10.3395	0.772811	0.386405	−	0.922329i	$-0.373717\pi$
$179$	10.3395	0.772811	0.386405	+	0.922329i	$0.373717\pi$
$180$	−5.32101	−0.396604
$181$	3.71280	0.275970	0.137985	−	0.990434i	$-0.455937\pi$
$181$	3.71280	0.275970	0.137985	+	0.990434i	$0.455937\pi$
$182$	0	0
$183$	7.68690	0.568232
$184$	5.34740	0.394215
$185$	−4.11703	−0.302690
$186$	−2.31841	−0.169994
$187$	24.2817	1.77565
$188$	−3.59834	−0.262436
$189$	0	0
$190$	5.56378	0.403639
$191$	11.1698	0.808215	0.404107	−	0.914712i	$-0.367582\pi$
$191$	11.1698	0.808215	0.404107	+	0.914712i	$0.367582\pi$
$192$	0.644301	0.0464984
$193$	−26.3395	−1.89596	−0.947980	−	0.318331i	$-0.896878\pi$
$193$	−26.3395	−1.89596	−0.947980	+	0.318331i	$0.896878\pi$
$194$	0.712685	0.0511678
$195$	−3.99385	−0.286006
$196$	0	0
$197$	8.65260	0.616473	0.308236	−	0.951310i	$-0.400261\pi$
$197$	8.65260	0.616473	0.308236	+	0.951310i	$0.400261\pi$
$198$	−12.0809	−0.858554
$199$	−11.0625	−0.784199	−0.392099	−	0.919923i	$-0.628251\pi$
$199$	−11.0625	−0.784199	−0.392099	+	0.919923i	$0.628251\pi$
$200$	−0.762519	−0.0539182
$201$	4.61948	0.325833
$202$	−16.0476	−1.12911
$203$	0	0
$204$	3.34740	0.234365
$205$	0.949924	0.0663456
$206$	8.48528	0.591198
$207$	−13.8224	−0.960720
$208$	−3.01127	−0.208794
$209$	12.6321	0.873783
$210$	0	0
$211$	−14.3184	−0.985720	−0.492860	−	0.870109i	$-0.664049\pi$
$211$	−14.3184	−0.985720	−0.492860	+	0.870109i	$0.664049\pi$
$212$	−0.673698	−0.0462698
$213$	5.15441	0.353174
$214$	−9.34740	−0.638975
$215$	22.8699	1.55972
$216$	−3.59834	−0.244836
$217$	0	0
$218$	10.0211	0.678714
$219$	−6.83025	−0.461545
$220$	9.62087	0.648639
$221$	−15.6447	−1.05238
$222$	−1.28860	−0.0864853
$223$	12.0451	0.806597	0.403299	−	0.915068i	$-0.367864\pi$
$223$	12.0451	0.806597	0.403299	+	0.915068i	$0.367864\pi$
$224$	0	0
$225$	1.97102	0.131401
$226$	11.6447	0.774595
$227$	−8.64072	−0.573505	−0.286752	−	0.958005i	$-0.592576\pi$
$227$	−8.64072	−0.573505	−0.286752	+	0.958005i	$0.592576\pi$
$228$	1.74143	0.115329
$229$	28.6500	1.89324	0.946621	−	0.322349i	$-0.104472\pi$
$229$	28.6500	1.89324	0.946621	+	0.322349i	$0.104472\pi$
$230$	11.0077	0.725826
$231$	0	0
$232$	−1.00000	−0.0656532
$233$	−15.9244	−1.04324	−0.521621	−	0.853178i	$-0.674672\pi$
$233$	−15.9244	−1.04324	−0.521621	+	0.853178i	$0.674672\pi$
$234$	7.78375	0.508840
$235$	−7.40723	−0.483195
$236$	−10.6010	−0.690068
$237$	4.13327	0.268485
$238$	0	0
$239$	−18.3395	−1.18628	−0.593142	−	0.805098i	$-0.702113\pi$
$239$	−18.3395	−1.18628	−0.593142	+	0.805098i	$0.702113\pi$
$240$	1.32630	0.0856124
$241$	2.80103	0.180430	0.0902151	−	0.995922i	$-0.471245\pi$
$241$	2.80103	0.180430	0.0902151	+	0.995922i	$0.471245\pi$
$242$	10.8435	0.697043
$243$	14.2976	0.917190
$244$	11.9306	0.763779
$245$	0	0
$246$	0.297320	0.0189564
$247$	−8.13890	−0.517866
$248$	−3.59834	−0.228495
$249$	−0.297320	−0.0188419
$250$	−11.8622	−0.750233
$251$	−11.7366	−0.740809	−0.370404	−	0.928871i	$-0.620781\pi$
$251$	−11.7366	−0.740809	−0.370404	+	0.928871i	$0.620781\pi$
$252$	0	0
$253$	24.9921	1.57124
$254$	−7.16975	−0.449870
$255$	6.89066	0.431510
$256$	1.00000	0.0625000
$257$	−9.74649	−0.607969	−0.303985	−	0.952677i	$-0.598317\pi$
$257$	−9.74649	−0.607969	−0.303985	+	0.952677i	$0.598317\pi$
$258$	7.15813	0.445645
$259$	0	0
$260$	−6.19874	−0.384429
$261$	2.58488	0.160000
$262$	−20.4047	−1.26061
$263$	−13.2296	−0.815772	−0.407886	−	0.913033i	$-0.633734\pi$
$263$	−13.2296	−0.815772	−0.407886	+	0.913033i	$0.633734\pi$
$264$	3.01127	0.185331
$265$	−1.38682	−0.0851915
$266$	0	0
$267$	−0.532927	−0.0326146
$268$	7.16975	0.437962
$269$	21.9557	1.33866	0.669332	−	0.742964i	$-0.266580\pi$
$269$	21.9557	1.33866	0.669332	+	0.742964i	$0.266580\pi$
$270$	−7.40723	−0.450790
$271$	−6.45660	−0.392210	−0.196105	−	0.980583i	$-0.562829\pi$
$271$	−6.45660	−0.392210	−0.196105	+	0.980583i	$0.562829\pi$
$272$	5.19539	0.315017
$273$	0	0
$274$	−19.6869	−1.18933
$275$	−3.56378	−0.214904
$276$	3.44533	0.207385
$277$	−29.3316	−1.76237	−0.881183	−	0.472775i	$-0.843252\pi$
$277$	−29.3316	−1.76237	−0.881183	+	0.472775i	$0.843252\pi$
$278$	−2.00129	−0.120029
$279$	9.30126	0.556852
$280$	0	0
$281$	21.4882	1.28188	0.640938	−	0.767592i	$-0.278545\pi$
$281$	21.4882	1.28188	0.640938	+	0.767592i	$0.278545\pi$
$282$	−2.31841	−0.138059
$283$	−1.32959	−0.0790359	−0.0395179	−	0.999219i	$-0.512582\pi$
$283$	−1.32959	−0.0790359	−0.0395179	+	0.999219i	$0.512582\pi$
$284$	8.00000	0.474713
$285$	3.58475	0.212342
$286$	−14.0738	−0.832198
$287$	0	0
$288$	−2.58488	−0.152315
$289$	9.99211	0.587771
$290$	−2.05851	−0.120880
$291$	0.459184	0.0269178
$292$	−10.6010	−0.620378
$293$	8.85096	0.517079	0.258539	−	0.966001i	$-0.416759\pi$
$293$	8.85096	0.517079	0.258539	+	0.966001i	$0.416759\pi$
$294$	0	0
$295$	−21.8224	−1.27055
$296$	−2.00000	−0.116248
$297$	−16.8176	−0.975853
$298$	−0.415123	−0.0240475
$299$	−16.1024	−0.931228
$300$	−0.491292	−0.0283647
$301$	0	0
$302$	3.52504	0.202843
$303$	−10.3395	−0.593989
$304$	2.70281	0.155017
$305$	24.5593	1.40626
$306$	−13.4295	−0.767711
$307$	−30.5281	−1.74233	−0.871164	−	0.490992i	$-0.836634\pi$
$307$	−30.5281	−1.74233	−0.871164	+	0.490992i	$0.836634\pi$
$308$	0	0
$309$	5.46707	0.311011
$310$	−7.40723	−0.420703
$311$	19.4631	1.10365	0.551827	−	0.833959i	$-0.313931\pi$
$311$	19.4631	1.10365	0.551827	+	0.833959i	$0.313931\pi$
$312$	−1.94016	−0.109840
$313$	−10.8137	−0.611226	−0.305613	−	0.952156i	$-0.598861\pi$
$313$	−10.8137	−0.611226	−0.305613	+	0.952156i	$0.598861\pi$
$314$	−21.4533	−1.21068
$315$	0	0
$316$	6.41512	0.360879
$317$	19.4671	1.09338	0.546690	−	0.837335i	$-0.315888\pi$
$317$	19.4671	1.09338	0.546690	+	0.837335i	$0.315888\pi$
$318$	−0.434064	−0.0243411
$319$	−4.67370	−0.261677
$320$	2.05851	0.115074
$321$	−6.02253	−0.336145
$322$	0	0
$323$	14.0422	0.781328
$324$	5.43622	0.302012
$325$	2.29615	0.127367
$326$	17.6658	0.978419
$327$	6.45660	0.357051
$328$	0.461461	0.0254799
$329$	0	0
$330$	6.19874	0.341229
$331$	−30.6350	−1.68385	−0.841925	−	0.539595i	$-0.818577\pi$
$331$	−30.6350	−1.68385	−0.841925	+	0.539595i	$0.818577\pi$
$332$	−0.461461	−0.0253260
$333$	5.16975	0.283301
$334$	0.251224	0.0137464
$335$	14.7590	0.806372
$336$	0	0
$337$	10.0000	0.544735	0.272367	−	0.962193i	$-0.412193\pi$
$337$	10.0000	0.544735	0.272367	+	0.962193i	$0.412193\pi$
$338$	−3.93227	−0.213887
$339$	7.50270	0.407491
$340$	10.6948	0.580006
$341$	−16.8176	−0.910722
$342$	−6.98644	−0.377784
$343$	0	0
$344$	11.1099	0.599006
$345$	7.09226	0.381835
$346$	12.2665	0.659449
$347$	6.00000	0.322097	0.161048	−	0.986947i	$-0.448512\pi$
$347$	6.00000	0.322097	0.161048	+	0.986947i	$0.448512\pi$
$348$	−0.644301	−0.0345381
$349$	−10.1781	−0.544822	−0.272411	−	0.962181i	$-0.587821\pi$
$349$	−10.1781	−0.544822	−0.272411	+	0.962181i	$0.587821\pi$
$350$	0	0
$351$	10.8356	0.578359
$352$	4.67370	0.249109
$353$	20.5415	1.09331	0.546657	−	0.837357i	$-0.315900\pi$
$353$	20.5415	1.09331	0.546657	+	0.837357i	$0.315900\pi$
$354$	−6.83025	−0.363024
$355$	16.4681	0.874037
$356$	−0.827140	−0.0438384
$357$	0	0
$358$	10.3395	0.546460
$359$	−16.2797	−0.859208	−0.429604	−	0.903017i	$-0.641347\pi$
$359$	−16.2797	−0.859208	−0.429604	+	0.903017i	$0.641347\pi$
$360$	−5.32101	−0.280442
$361$	−11.6948	−0.615515
$362$	3.71280	0.195140
$363$	6.98644	0.366693
$364$	0	0
$365$	−21.8224	−1.14223
$366$	7.68690	0.401801
$367$	−5.57223	−0.290868	−0.145434	−	0.989368i	$-0.546458\pi$
$367$	−5.57223	−0.290868	−0.145434	+	0.989368i	$0.546458\pi$
$368$	5.34740	0.278752
$369$	−1.19282	−0.0620957
$370$	−4.11703	−0.214034
$371$	0	0
$372$	−2.31841	−0.120204
$373$	33.8047	1.75034	0.875171	−	0.483814i	$-0.160749\pi$
$373$	33.8047	1.75034	0.875171	+	0.483814i	$0.160749\pi$
$374$	24.2817	1.25558
$375$	−7.64284	−0.394675
$376$	−3.59834	−0.185570
$377$	3.01127	0.155088
$378$	0	0
$379$	−22.1619	−1.13838	−0.569189	−	0.822207i	$-0.692743\pi$
$379$	−22.1619	−1.13838	−0.569189	+	0.822207i	$0.692743\pi$
$380$	5.56378	0.285416
$381$	−4.61948	−0.236663
$382$	11.1698	0.571494
$383$	−34.5580	−1.76583	−0.882916	−	0.469530i	$-0.844423\pi$
$383$	−34.5580	−1.76583	−0.882916	+	0.469530i	$0.844423\pi$
$384$	0.644301	0.0328793
$385$	0	0
$386$	−26.3395	−1.34065
$387$	−28.7178	−1.45981
$388$	0.712685	0.0361811
$389$	−27.5093	−1.39477	−0.697387	−	0.716694i	$-0.745654\pi$
$389$	−27.5093	−1.39477	−0.697387	+	0.716694i	$0.745654\pi$
$390$	−3.99385	−0.202236
$391$	27.7818	1.40499
$392$	0	0
$393$	−13.1468	−0.663168
$394$	8.65260	0.435912
$395$	13.2056	0.664447
$396$	−12.0809	−0.607090
$397$	26.8725	1.34869	0.674346	−	0.738416i	$-0.264426\pi$
$397$	26.8725	1.34869	0.674346	+	0.738416i	$0.264426\pi$
$398$	−11.0625	−0.554512
$399$	0	0
$400$	−0.762519	−0.0381259
$401$	9.50394	0.474604	0.237302	−	0.971436i	$-0.423737\pi$
$401$	9.50394	0.474604	0.237302	+	0.971436i	$0.423737\pi$
$402$	4.61948	0.230399
$403$	10.8356	0.539758
$404$	−16.0476	−0.798400
$405$	11.1905	0.556062
$406$	0	0
$407$	−9.34740	−0.463333
$408$	3.34740	0.165721
$409$	7.04124	0.348167	0.174083	−	0.984731i	$-0.444304\pi$
$409$	7.04124	0.348167	0.174083	+	0.984731i	$0.444304\pi$
$410$	0.949924	0.0469134
$411$	−12.6843	−0.625670
$412$	8.48528	0.418040
$413$	0	0
$414$	−13.8224	−0.679332
$415$	−0.949924	−0.0466299
$416$	−3.01127	−0.147640
$417$	−1.28943	−0.0631437
$418$	12.6321	0.617858
$419$	−17.9122	−0.875066	−0.437533	−	0.899202i	$-0.644148\pi$
$419$	−17.9122	−0.875066	−0.437533	+	0.899202i	$0.644148\pi$
$420$	0	0
$421$	16.8567	0.821543	0.410772	−	0.911738i	$-0.365259\pi$
$421$	16.8567	0.821543	0.410772	+	0.911738i	$0.365259\pi$
$422$	−14.3184	−0.697009
$423$	9.30126	0.452243
$424$	−0.673698	−0.0327177
$425$	−3.96159	−0.192165
$426$	5.15441	0.249732
$427$	0	0
$428$	−9.34740	−0.451823
$429$	−9.06773	−0.437794
$430$	22.8699	1.10289
$431$	25.8224	1.24382	0.621910	−	0.783089i	$-0.286357\pi$
$431$	25.8224	1.24382	0.621910	+	0.783089i	$0.286357\pi$
$432$	−3.59834	−0.173125
$433$	−14.7181	−0.707304	−0.353652	−	0.935377i	$-0.615060\pi$
$433$	−14.7181	−0.707304	−0.353652	+	0.935377i	$0.615060\pi$
$434$	0	0
$435$	−1.32630	−0.0635913
$436$	10.0211	0.479923
$437$	14.4530	0.691382
$438$	−6.83025	−0.326362
$439$	5.52009	0.263459	0.131730	−	0.991286i	$-0.457947\pi$
$439$	5.52009	0.263459	0.131730	+	0.991286i	$0.457947\pi$
$440$	9.62087	0.458657
$441$	0	0
$442$	−15.6447	−0.744143
$443$	18.5751	0.882530	0.441265	−	0.897377i	$-0.354530\pi$
$443$	18.5751	0.882530	0.441265	+	0.897377i	$0.354530\pi$
$444$	−1.28860	−0.0611543
$445$	−1.70268	−0.0807148
$446$	12.0451	0.570350
$447$	−0.267464	−0.0126506
$448$	0	0
$449$	−3.94203	−0.186036	−0.0930181	−	0.995664i	$-0.529651\pi$
$449$	−3.94203	−0.186036	−0.0930181	+	0.995664i	$0.529651\pi$
$450$	1.97102	0.0929147
$451$	2.15673	0.101556
$452$	11.6447	0.547721
$453$	2.27118	0.106710
$454$	−8.64072	−0.405529
$455$	0	0
$456$	1.74143	0.0815497
$457$	−12.6368	−0.591126	−0.295563	−	0.955323i	$-0.595507\pi$
$457$	−12.6368	−0.591126	−0.295563	+	0.955323i	$0.595507\pi$
$458$	28.6500	1.33872
$459$	−18.6948	−0.872598
$460$	11.0077	0.513236
$461$	−8.06481	−0.375616	−0.187808	−	0.982206i	$-0.560138\pi$
$461$	−8.06481	−0.375616	−0.187808	+	0.982206i	$0.560138\pi$
$462$	0	0
$463$	8.35528	0.388303	0.194152	−	0.980972i	$-0.437805\pi$
$463$	8.35528	0.388303	0.194152	+	0.980972i	$0.437805\pi$
$464$	−1.00000	−0.0464238
$465$	−4.77249	−0.221319
$466$	−15.9244	−0.737683
$467$	−16.8585	−0.780120	−0.390060	−	0.920789i	$-0.627546\pi$
$467$	−16.8585	−0.780120	−0.390060	+	0.920789i	$0.627546\pi$
$468$	7.78375	0.359804
$469$	0	0
$470$	−7.40723	−0.341670
$471$	−13.8224	−0.636901
$472$	−10.6010	−0.487952
$473$	51.9244	2.38749
$474$	4.13327	0.189847
$475$	−2.06095	−0.0945628
$476$	0	0
$477$	1.74143	0.0797344
$478$	−18.3395	−0.838829
$479$	−26.7905	−1.22409	−0.612045	−	0.790823i	$-0.709653\pi$
$479$	−26.7905	−1.22409	−0.612045	+	0.790823i	$0.709653\pi$
$480$	1.32630	0.0605371
$481$	6.02253	0.274604
$482$	2.80103	0.127583
$483$	0	0
$484$	10.8435	0.492884
$485$	1.46707	0.0666163
$486$	14.2976	0.648551
$487$	−32.1619	−1.45739	−0.728697	−	0.684837i	$-0.759874\pi$
$487$	−32.1619	−1.45739	−0.728697	+	0.684837i	$0.759874\pi$
$488$	11.9306	0.540073
$489$	11.3821	0.514716
$490$	0	0
$491$	−43.7080	−1.97251	−0.986257	−	0.165218i	$-0.947167\pi$
$491$	−43.7080	−1.97251	−0.986257	+	0.165218i	$0.947167\pi$
$492$	0.297320	0.0134042
$493$	−5.19539	−0.233989
$494$	−8.13890	−0.366186
$495$	−24.8688	−1.11777
$496$	−3.59834	−0.161570
$497$	0	0
$498$	−0.297320	−0.0133232
$499$	−28.9921	−1.29787	−0.648933	−	0.760846i	$-0.724784\pi$
$499$	−28.9921	−1.29787	−0.648933	+	0.760846i	$0.724784\pi$
$500$	−11.8622	−0.530495
$501$	0.161864	0.00723155
$502$	−11.7366	−0.523831
$503$	−3.87939	−0.172974	−0.0864868	−	0.996253i	$-0.527564\pi$
$503$	−3.87939	−0.172974	−0.0864868	+	0.996253i	$0.527564\pi$
$504$	0	0
$505$	−33.0343	−1.47001
$506$	24.9921	1.11103
$507$	−2.53357	−0.112520
$508$	−7.16975	−0.318106
$509$	−11.2155	−0.497118	−0.248559	−	0.968617i	$-0.579957\pi$
$509$	−11.2155	−0.497118	−0.248559	+	0.968617i	$0.579957\pi$
$510$	6.89066	0.305124
$511$	0	0
$512$	1.00000	0.0441942
$513$	−9.72565	−0.429398
$514$	−9.74649	−0.429899
$515$	17.4671	0.769691
$516$	7.15813	0.315119
$517$	−16.8176	−0.739635
$518$	0	0
$519$	7.90329	0.346916
$520$	−6.19874	−0.271833
$521$	0.141852	0.00621466	0.00310733	−	0.999995i	$-0.499011\pi$
$521$	0.141852	0.00621466	0.00310733	+	0.999995i	$0.499011\pi$
$522$	2.58488	0.113137
$523$	−7.46658	−0.326491	−0.163245	−	0.986586i	$-0.552196\pi$
$523$	−7.46658	−0.326491	−0.163245	+	0.986586i	$0.552196\pi$
$524$	−20.4047	−0.891385
$525$	0	0
$526$	−13.2296	−0.576838
$527$	−18.6948	−0.814358
$528$	3.01127	0.131049
$529$	5.59464	0.243245
$530$	−1.38682	−0.0602395
$531$	27.4023	1.18916
$532$	0	0
$533$	−1.38958	−0.0601895
$534$	−0.532927	−0.0230620
$535$	−19.2417	−0.831893
$536$	7.16975	0.309686
$537$	6.66175	0.287476
$538$	21.9557	0.946578
$539$	0	0
$540$	−7.40723	−0.318757
$541$	−34.3975	−1.47886	−0.739431	−	0.673232i	$-0.764906\pi$
$541$	−34.3975	−1.47886	−0.739431	+	0.673232i	$0.764906\pi$
$542$	−6.45660	−0.277335
$543$	2.39216	0.102657
$544$	5.19539	0.222751
$545$	20.6286	0.883631
$546$	0	0
$547$	−34.1619	−1.46066	−0.730328	−	0.683097i	$-0.760633\pi$
$547$	−34.1619	−1.46066	−0.730328	+	0.683097i	$0.760633\pi$
$548$	−19.6869	−0.840983
$549$	−30.8392	−1.31618
$550$	−3.56378	−0.151960
$551$	−2.70281	−0.115144
$552$	3.44533	0.146643
$553$	0	0
$554$	−29.3316	−1.24618
$555$	−2.65260	−0.112597
$556$	−2.00129	−0.0848735
$557$	2.94992	0.124992	0.0624961	−	0.998045i	$-0.480094\pi$
$557$	2.94992	0.124992	0.0624961	+	0.998045i	$0.480094\pi$
$558$	9.30126	0.393754
$559$	−33.4549	−1.41499
$560$	0	0
$561$	15.6447	0.660520
$562$	21.4882	0.906424
$563$	−1.20154	−0.0506390	−0.0253195	−	0.999679i	$-0.508060\pi$
$563$	−1.20154	−0.0506390	−0.0253195	+	0.999679i	$0.508060\pi$
$564$	−2.31841	−0.0976228
$565$	23.9708	1.00846
$566$	−1.32959	−0.0558868
$567$	0	0
$568$	8.00000	0.335673
$569$	27.4671	1.15148	0.575740	−	0.817633i	$-0.304714\pi$
$569$	27.4671	1.15148	0.575740	+	0.817633i	$0.304714\pi$
$570$	3.58475	0.150149
$571$	8.85666	0.370639	0.185320	−	0.982678i	$-0.440668\pi$
$571$	8.85666	0.370639	0.185320	+	0.982678i	$0.440668\pi$
$572$	−14.0738	−0.588453
$573$	7.19668	0.300646
$574$	0	0
$575$	−4.07749	−0.170043
$576$	−2.58488	−0.107703
$577$	−25.7805	−1.07326	−0.536629	−	0.843818i	$-0.680302\pi$
$577$	−25.7805	−1.07326	−0.536629	+	0.843818i	$0.680302\pi$
$578$	9.99211	0.415617
$579$	−16.9706	−0.705273
$580$	−2.05851	−0.0854752
$581$	0	0
$582$	0.459184	0.0190338
$583$	−3.14866	−0.130404
$584$	−10.6010	−0.438673
$585$	16.0230	0.662468
$586$	8.85096	0.365630
$587$	44.7163	1.84564	0.922819	−	0.385234i	$-0.125879\pi$
$587$	44.7163	1.84564	0.922819	+	0.385234i	$0.125879\pi$
$588$	0	0
$589$	−9.72565	−0.400738
$590$	−21.8224	−0.898412
$591$	5.57488	0.229320
$592$	−2.00000	−0.0821995
$593$	1.87811	0.0771247	0.0385623	−	0.999256i	$-0.487722\pi$
$593$	1.87811	0.0771247	0.0385623	+	0.999256i	$0.487722\pi$
$594$	−16.8176	−0.690033
$595$	0	0
$596$	−0.415123	−0.0170041
$597$	−7.12757	−0.291712
$598$	−16.1024	−0.658478
$599$	−14.7704	−0.603503	−0.301751	−	0.953387i	$-0.597571\pi$
$599$	−14.7704	−0.603503	−0.301751	+	0.953387i	$0.597571\pi$
$600$	−0.491292	−0.0200569
$601$	8.75518	0.357131	0.178566	−	0.983928i	$-0.442854\pi$
$601$	8.75518	0.357131	0.178566	+	0.983928i	$0.442854\pi$
$602$	0	0
$603$	−18.5329	−0.754719
$604$	3.52504	0.143432
$605$	22.3214	0.907494
$606$	−10.3395	−0.420014
$607$	−19.9818	−0.811037	−0.405519	−	0.914087i	$-0.632909\pi$
$607$	−19.9818	−0.811037	−0.405519	+	0.914087i	$0.632909\pi$
$608$	2.70281	0.109614
$609$	0	0
$610$	24.5593	0.994378
$611$	10.8356	0.438360
$612$	−13.4295	−0.542853
$613$	28.2639	1.14157	0.570784	−	0.821100i	$-0.306639\pi$
$613$	28.2639	1.14157	0.570784	+	0.821100i	$0.306639\pi$
$614$	−30.5281	−1.23201
$615$	0.612037	0.0246797
$616$	0	0
$617$	−2.47496	−0.0996382	−0.0498191	−	0.998758i	$-0.515864\pi$
$617$	−2.47496	−0.0996382	−0.0498191	+	0.998758i	$0.515864\pi$
$618$	5.46707	0.219918
$619$	47.9489	1.92723	0.963615	−	0.267294i	$-0.0861295\pi$
$619$	47.9489	1.92723	0.963615	+	0.267294i	$0.0861295\pi$
$620$	−7.40723	−0.297482
$621$	−19.2417	−0.772145
$622$	19.4631	0.780401
$623$	0	0
$624$	−1.94016	−0.0776686
$625$	−20.6060	−0.824239
$626$	−10.8137	−0.432202
$627$	8.13890	0.325036
$628$	−21.4533	−0.856079
$629$	−10.3908	−0.414308
$630$	0	0
$631$	−9.82236	−0.391022	−0.195511	−	0.980702i	$-0.562637\pi$
$631$	−9.82236	−0.391022	−0.195511	+	0.980702i	$0.562637\pi$
$632$	6.41512	0.255180
$633$	−9.22536	−0.366675
$634$	19.4671	0.773136
$635$	−14.7590	−0.585695
$636$	−0.434064	−0.0172118
$637$	0	0
$638$	−4.67370	−0.185033
$639$	−20.6790	−0.818049
$640$	2.05851	0.0813699
$641$	−7.94203	−0.313692	−0.156846	−	0.987623i	$-0.550133\pi$
$641$	−7.94203	−0.313692	−0.156846	+	0.987623i	$0.550133\pi$
$642$	−6.02253	−0.237690
$643$	20.9918	0.827836	0.413918	−	0.910314i	$-0.364160\pi$
$643$	20.9918	0.827836	0.413918	+	0.910314i	$0.364160\pi$
$644$	0	0
$645$	14.7351	0.580194
$646$	14.0422	0.552482
$647$	−31.3416	−1.23217	−0.616083	−	0.787681i	$-0.711281\pi$
$647$	−31.3416	−1.23217	−0.616083	+	0.787681i	$0.711281\pi$
$648$	5.43622	0.213555
$649$	−49.5460	−1.94485
$650$	2.29615	0.0900623
$651$	0	0
$652$	17.6658	0.691846
$653$	30.9921	1.21282	0.606408	−	0.795154i	$-0.292610\pi$
$653$	30.9921	1.21282	0.606408	+	0.795154i	$0.292610\pi$
$654$	6.45660	0.252473
$655$	−42.0034	−1.64121
$656$	0.461461	0.0180170
$657$	27.4023	1.06907
$658$	0	0
$659$	38.6579	1.50590	0.752949	−	0.658078i	$-0.228630\pi$
$659$	38.6579	1.50590	0.752949	+	0.658078i	$0.228630\pi$
$660$	6.19874	0.241285
$661$	−20.0800	−0.781023	−0.390512	−	0.920598i	$-0.627702\pi$
$661$	−20.0800	−0.781023	−0.390512	+	0.920598i	$0.627702\pi$
$662$	−30.6350	−1.19066
$663$	−10.0799	−0.391471
$664$	−0.461461	−0.0179082
$665$	0	0
$666$	5.16975	0.200324
$667$	−5.34740	−0.207052
$668$	0.251224	0.00972016
$669$	7.76065	0.300044
$670$	14.7590	0.570191
$671$	55.7601	2.15259
$672$	0	0
$673$	22.7934	0.878620	0.439310	−	0.898336i	$-0.355223\pi$
$673$	22.7934	0.878620	0.439310	+	0.898336i	$0.355223\pi$
$674$	10.0000	0.385186
$675$	2.74380	0.105609
$676$	−3.93227	−0.151241
$677$	14.6223	0.561979	0.280990	−	0.959711i	$-0.409337\pi$
$677$	14.6223	0.561979	0.280990	+	0.959711i	$0.409337\pi$
$678$	7.50270	0.288139
$679$	0	0
$680$	10.6948	0.410127
$681$	−5.56722	−0.213337
$682$	−16.8176	−0.643977
$683$	2.53293	0.0969198	0.0484599	−	0.998825i	$-0.484569\pi$
$683$	2.53293	0.0969198	0.0484599	+	0.998825i	$0.484569\pi$
$684$	−6.98644	−0.267133
$685$	−40.5258	−1.54841
$686$	0	0
$687$	18.4592	0.704262
$688$	11.1099	0.423562
$689$	2.02868	0.0772867
$690$	7.09226	0.269998
$691$	32.7857	1.24723	0.623613	−	0.781734i	$-0.285664\pi$
$691$	32.7857	1.24723	0.623613	+	0.781734i	$0.285664\pi$
$692$	12.2665	0.466301
$693$	0	0
$694$	6.00000	0.227757
$695$	−4.11968	−0.156268
$696$	−0.644301	−0.0244222
$697$	2.39747	0.0908107
$698$	−10.1781	−0.385247
$699$	−10.2601	−0.388072
$700$	0	0
$701$	−22.8743	−0.863951	−0.431975	−	0.901885i	$-0.642183\pi$
$701$	−22.8743	−0.863951	−0.431975	+	0.901885i	$0.642183\pi$
$702$	10.8356	0.408962
$703$	−5.40563	−0.203877
$704$	4.67370	0.176147
$705$	−4.77249	−0.179742
$706$	20.5415	0.773090
$707$	0	0
$708$	−6.83025	−0.256696
$709$	−5.38427	−0.202210	−0.101105	−	0.994876i	$-0.532238\pi$
$709$	−5.38427	−0.202210	−0.101105	+	0.994876i	$0.532238\pi$
$710$	16.4681	0.618037
$711$	−16.5823	−0.621885
$712$	−0.827140	−0.0309984
$713$	−19.2417	−0.720609
$714$	0	0
$715$	−28.9710	−1.08345
$716$	10.3395	0.386405
$717$	−11.8162	−0.441282
$718$	−16.2797	−0.607552
$719$	−43.6927	−1.62946	−0.814731	−	0.579839i	$-0.803116\pi$
$719$	−43.6927	−1.62946	−0.814731	+	0.579839i	$0.803116\pi$
$720$	−5.32101	−0.198302
$721$	0	0
$722$	−11.6948	−0.435235
$723$	1.80471	0.0671177
$724$	3.71280	0.137985
$725$	0.762519	0.0283192
$726$	6.98644	0.259291
$727$	30.0232	1.11350	0.556749	−	0.830681i	$-0.312049\pi$
$727$	30.0232	1.11350	0.556749	+	0.830681i	$0.312049\pi$
$728$	0	0
$729$	−7.09671	−0.262841
$730$	−21.8224	−0.807682
$731$	57.7204	2.13487
$732$	7.68690	0.284116
$733$	−10.2540	−0.378741	−0.189370	−	0.981906i	$-0.560645\pi$
$733$	−10.2540	−0.378741	−0.189370	+	0.981906i	$0.560645\pi$
$734$	−5.57223	−0.205675
$735$	0	0
$736$	5.34740	0.197108
$737$	33.5093	1.23433
$738$	−1.19282	−0.0439083
$739$	−8.29545	−0.305153	−0.152576	−	0.988292i	$-0.548757\pi$
$739$	−8.29545	−0.305153	−0.152576	+	0.988292i	$0.548757\pi$
$740$	−4.11703	−0.151345
$741$	−5.24390	−0.192639
$742$	0	0
$743$	4.59464	0.168561	0.0842805	−	0.996442i	$-0.473141\pi$
$743$	4.59464	0.168561	0.0842805	+	0.996442i	$0.473141\pi$
$744$	−2.31841	−0.0849971
$745$	−0.854537	−0.0313078
$746$	33.8047	1.23768
$747$	1.19282	0.0436430
$748$	24.2817	0.887827
$749$	0	0
$750$	−7.64284	−0.279077
$751$	−27.0501	−0.987071	−0.493536	−	0.869726i	$-0.664296\pi$
$751$	−27.0501	−0.987071	−0.493536	+	0.869726i	$0.664296\pi$
$752$	−3.59834	−0.131218
$753$	−7.56191	−0.275571
$754$	3.01127	0.109664
$755$	7.25634	0.264085
$756$	0	0
$757$	32.4592	1.17975	0.589875	−	0.807495i	$-0.299177\pi$
$757$	32.4592	1.17975	0.589875	+	0.807495i	$0.299177\pi$
$758$	−22.1619	−0.804955
$759$	16.1024	0.584481
$760$	5.56378	0.201820
$761$	−28.3551	−1.02787	−0.513936	−	0.857829i	$-0.671813\pi$
$761$	−28.3551	−1.02787	−0.513936	+	0.857829i	$0.671813\pi$
$762$	−4.61948	−0.167346
$763$	0	0
$764$	11.1698	0.404107
$765$	−27.6447	−0.999497
$766$	−34.5580	−1.24863
$767$	31.9225	1.15266
$768$	0.644301	0.0232492
$769$	20.7406	0.747925	0.373962	−	0.927444i	$-0.377999\pi$
$769$	20.7406	0.747925	0.373962	+	0.927444i	$0.377999\pi$
$770$	0	0
$771$	−6.27967	−0.226157
$772$	−26.3395	−0.947980
$773$	43.6602	1.57035	0.785174	−	0.619275i	$-0.212573\pi$
$773$	43.6602	1.57035	0.785174	+	0.619275i	$0.212573\pi$
$774$	−28.7178	−1.03224
$775$	2.74380	0.0985603
$776$	0.712685	0.0255839
$777$	0	0
$778$	−27.5093	−0.986255
$779$	1.24724	0.0446871
$780$	−3.99385	−0.143003
$781$	37.3896	1.33790
$782$	27.7818	0.993476
$783$	3.59834	0.128594
$784$	0	0
$785$	−44.1619	−1.57620
$786$	−13.1468	−0.468930
$787$	32.3652	1.15369	0.576847	−	0.816852i	$-0.304283\pi$
$787$	32.3652	1.15369	0.576847	+	0.816852i	$0.304283\pi$
$788$	8.65260	0.308236
$789$	−8.52384	−0.303457
$790$	13.2056	0.469835
$791$	0	0
$792$	−12.0809	−0.429277
$793$	−35.9263	−1.27578
$794$	26.8725	0.953669
$795$	−0.893527	−0.0316901
$796$	−11.0625	−0.392099
$797$	44.8120	1.58732	0.793662	−	0.608359i	$-0.208172\pi$
$797$	44.8120	1.58732	0.793662	+	0.608359i	$0.208172\pi$
$798$	0	0
$799$	−18.6948	−0.661374
$800$	−0.762519	−0.0269591
$801$	2.13806	0.0755445
$802$	9.50394	0.335596
$803$	−49.5460	−1.74844
$804$	4.61948	0.162916
$805$	0	0
$806$	10.8356	0.381666
$807$	14.1461	0.497966
$808$	−16.0476	−0.564554
$809$	−17.3474	−0.609902	−0.304951	−	0.952368i	$-0.598640\pi$
$809$	−17.3474	−0.609902	−0.304951	+	0.952368i	$0.598640\pi$
$810$	11.1905	0.393195
$811$	3.65557	0.128364	0.0641822	−	0.997938i	$-0.479556\pi$
$811$	3.65557	0.128364	0.0641822	+	0.997938i	$0.479556\pi$
$812$	0	0
$813$	−4.15999	−0.145897
$814$	−9.34740	−0.327626
$815$	36.3653	1.27382
$816$	3.34740	0.117182
$817$	30.0280	1.05055
$818$	7.04124	0.246191
$819$	0	0
$820$	0.949924	0.0331728
$821$	14.6157	0.510093	0.255046	−	0.966929i	$-0.417909\pi$
$821$	14.6157	0.510093	0.255046	+	0.966929i	$0.417909\pi$
$822$	−12.6843	−0.442415
$823$	−20.6790	−0.720825	−0.360413	−	0.932793i	$-0.617364\pi$
$823$	−20.6790	−0.720825	−0.360413	+	0.932793i	$0.617364\pi$
$824$	8.48528	0.295599
$825$	−2.29615	−0.0799416
$826$	0	0
$827$	−13.6271	−0.473859	−0.236930	−	0.971527i	$-0.576141\pi$
$827$	−13.6271	−0.473859	−0.236930	+	0.971527i	$0.576141\pi$
$828$	−13.8224	−0.480360
$829$	19.1273	0.664318	0.332159	−	0.943223i	$-0.392223\pi$
$829$	19.1273	0.664318	0.332159	+	0.943223i	$0.392223\pi$
$830$	−0.949924	−0.0329723
$831$	−18.8984	−0.655578
$832$	−3.01127	−0.104397
$833$	0	0
$834$	−1.28943	−0.0446493
$835$	0.517149	0.0178967
$836$	12.6321	0.436892
$837$	12.9481	0.447550
$838$	−17.9122	−0.618765
$839$	−47.5721	−1.64237	−0.821185	−	0.570661i	$-0.806687\pi$
$839$	−47.5721	−1.64237	−0.821185	+	0.570661i	$0.806687\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	16.8567	0.580919
$843$	13.8448	0.476842
$844$	−14.3184	−0.492860
$845$	−8.09464	−0.278464
$846$	9.30126	0.319784
$847$	0	0
$848$	−0.673698	−0.0231349
$849$	−0.856655	−0.0294003
$850$	−3.96159	−0.135881
$851$	−10.6948	−0.366613
$852$	5.15441	0.176587
$853$	−49.6008	−1.69830	−0.849149	−	0.528153i	$-0.822885\pi$
$853$	−49.6008	−1.69830	−0.849149	+	0.528153i	$0.822885\pi$
$854$	0	0
$855$	−14.3817	−0.491844
$856$	−9.34740	−0.319487
$857$	−30.0256	−1.02566	−0.512828	−	0.858492i	$-0.671402\pi$
$857$	−30.0256	−1.02566	−0.512828	+	0.858492i	$0.671402\pi$
$858$	−9.06773	−0.309567
$859$	50.6331	1.72758	0.863789	−	0.503854i	$-0.168085\pi$
$859$	50.6331	1.72758	0.863789	+	0.503854i	$0.168085\pi$
$860$	22.8699	0.779858
$861$	0	0
$862$	25.8224	0.879513
$863$	27.8803	0.949057	0.474529	−	0.880240i	$-0.342619\pi$
$863$	27.8803	0.949057	0.474529	+	0.880240i	$0.342619\pi$
$864$	−3.59834	−0.122418
$865$	25.2507	0.858549
$866$	−14.7181	−0.500140
$867$	6.43792	0.218643
$868$	0	0
$869$	29.9823	1.01708
$870$	−1.32630	−0.0449659
$871$	−21.5900	−0.731551
$872$	10.0211	0.339357
$873$	−1.84220	−0.0623491
$874$	14.4530	0.488881
$875$	0	0
$876$	−6.83025	−0.230773
$877$	34.2797	1.15754	0.578771	−	0.815490i	$-0.303532\pi$
$877$	34.2797	1.15754	0.578771	+	0.815490i	$0.303532\pi$
$878$	5.52009	0.186294
$879$	5.70268	0.192347
$880$	9.62087	0.324320
$881$	−2.75496	−0.0928169	−0.0464085	−	0.998923i	$-0.514778\pi$
$881$	−2.75496	−0.0928169	−0.0464085	+	0.998923i	$0.514778\pi$
$882$	0	0
$883$	36.0422	1.21292	0.606458	−	0.795115i	$-0.292590\pi$
$883$	36.0422	1.21292	0.606458	+	0.795115i	$0.292590\pi$
$884$	−15.6447	−0.526189
$885$	−14.0602	−0.472627
$886$	18.5751	0.624043
$887$	−21.3003	−0.715193	−0.357596	−	0.933876i	$-0.616404\pi$
$887$	−21.3003	−0.715193	−0.357596	+	0.933876i	$0.616404\pi$
$888$	−1.28860	−0.0432426
$889$	0	0
$890$	−1.70268	−0.0570740
$891$	25.4072	0.851174
$892$	12.0451	0.403299
$893$	−9.72565	−0.325456
$894$	−0.267464	−0.00894534
$895$	21.2840	0.711446
$896$	0	0
$897$	−10.3748	−0.346405
$898$	−3.94203	−0.131547
$899$	3.59834	0.120011
$900$	1.97102	0.0657006
$901$	−3.50012	−0.116606
$902$	2.15673	0.0718112
$903$	0	0
$904$	11.6447	0.387297
$905$	7.64284	0.254057
$906$	2.27118	0.0754550
$907$	−40.9146	−1.35855	−0.679274	−	0.733885i	$-0.737705\pi$
$907$	−40.9146	−1.35855	−0.679274	+	0.733885i	$0.737705\pi$
$908$	−8.64072	−0.286752
$909$	41.4812	1.37584
$910$	0	0
$911$	47.3297	1.56810	0.784052	−	0.620695i	$-0.213149\pi$
$911$	47.3297	1.56810	0.784052	+	0.620695i	$0.213149\pi$
$912$	1.74143	0.0576644
$913$	−2.15673	−0.0713773
$914$	−12.6368	−0.417989
$915$	15.8236	0.523112
$916$	28.6500	0.946621
$917$	0	0
$918$	−18.6948	−0.617020
$919$	32.2041	1.06231	0.531157	−	0.847274i	$-0.321758\pi$
$919$	32.2041	1.06231	0.531157	+	0.847274i	$0.321758\pi$
$920$	11.0077	0.362913
$921$	−19.6692	−0.648124
$922$	−8.06481	−0.265600
$923$	−24.0901	−0.792936
$924$	0	0
$925$	1.52504	0.0501429
$926$	8.35528	0.274572
$927$	−21.9334	−0.720388
$928$	−1.00000	−0.0328266
$929$	59.0250	1.93655	0.968274	−	0.249892i	$-0.0803951\pi$
$929$	59.0250	1.93655	0.968274	+	0.249892i	$0.0803951\pi$
$930$	−4.77249	−0.156496
$931$	0	0
$932$	−15.9244	−0.521621
$933$	12.5401	0.410545
$934$	−16.8585	−0.551628
$935$	49.9842	1.63466
$936$	7.78375	0.254420
$937$	−4.16067	−0.135923	−0.0679615	−	0.997688i	$-0.521650\pi$
$937$	−4.16067	−0.135923	−0.0679615	+	0.997688i	$0.521650\pi$
$938$	0	0
$939$	−6.96727	−0.227368
$940$	−7.40723	−0.241597
$941$	−48.0745	−1.56719	−0.783593	−	0.621275i	$-0.786615\pi$
$941$	−48.0745	−1.56719	−0.783593	+	0.621275i	$0.786615\pi$
$942$	−13.8224	−0.450357
$943$	2.46761	0.0803566
$944$	−10.6010	−0.345034
$945$	0	0
$946$	51.9244	1.68821
$947$	14.6350	0.475572	0.237786	−	0.971318i	$-0.423578\pi$
$947$	14.6350	0.475572	0.237786	+	0.971318i	$0.423578\pi$
$948$	4.13327	0.134242
$949$	31.9225	1.03625
$950$	−2.06095	−0.0668660
$951$	12.5426	0.406723
$952$	0	0
$953$	21.8822	0.708834	0.354417	−	0.935088i	$-0.384679\pi$
$953$	21.8822	0.708834	0.354417	+	0.935088i	$0.384679\pi$
$954$	1.74143	0.0563807
$955$	22.9931	0.744039
$956$	−18.3395	−0.593142
$957$	−3.01127	−0.0973404
$958$	−26.7905	−0.865562
$959$	0	0
$960$	1.32630	0.0428062
$961$	−18.0519	−0.582321
$962$	6.02253	0.194174
$963$	24.1619	0.778605
$964$	2.80103	0.0902151
$965$	−54.2202	−1.74541
$966$	0	0
$967$	21.8013	0.701081	0.350541	−	0.936547i	$-0.385998\pi$
$967$	21.8013	0.701081	0.350541	+	0.936547i	$0.385998\pi$
$968$	10.8435	0.348522
$969$	9.04739	0.290644
$970$	1.46707	0.0471049
$971$	6.68307	0.214470	0.107235	−	0.994234i	$-0.465800\pi$
$971$	6.68307	0.214470	0.107235	+	0.994234i	$0.465800\pi$
$972$	14.2976	0.458595
$973$	0	0
$974$	−32.1619	−1.03053
$975$	1.47941	0.0473790
$976$	11.9306	0.381890
$977$	−45.8856	−1.46801	−0.734006	−	0.679143i	$-0.762352\pi$
$977$	−45.8856	−1.46801	−0.734006	+	0.679143i	$0.762352\pi$
$978$	11.3821	0.363959
$979$	−3.86580	−0.123552
$980$	0	0
$981$	−25.9033	−0.827028
$982$	−43.7080	−1.39478
$983$	13.0364	0.415796	0.207898	−	0.978151i	$-0.433338\pi$
$983$	13.0364	0.415796	0.207898	+	0.978151i	$0.433338\pi$
$984$	0.297320	0.00947820
$985$	17.8115	0.567522
$986$	−5.19539	−0.165455
$987$	0	0
$988$	−8.13890	−0.258933
$989$	59.4091	1.88910
$990$	−24.8688	−0.790381
$991$	−25.7027	−0.816473	−0.408236	−	0.912876i	$-0.633856\pi$
$991$	−25.7027	−0.816473	−0.408236	+	0.912876i	$0.633856\pi$
$992$	−3.59834	−0.114247
$993$	−19.7381	−0.626370
$994$	0	0
$995$	−22.7723	−0.721930
$996$	−0.297320	−0.00942093
$997$	13.8361	0.438194	0.219097	−	0.975703i	$-0.429689\pi$
$997$	13.8361	0.438194	0.219097	+	0.975703i	$0.429689\pi$
$998$	−28.9921	−0.917729
$999$	7.19668	0.227693

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2842.2.a.bb.1.4	yes	6
7.6	odd	2	inner	2842.2.a.bb.1.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2842.2.a.bb.1.3	✓	6	7.6	odd	2	inner
2842.2.a.bb.1.4	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.644301 q^{3} +1.00000 q^{4} +2.05851 q^{5} +0.644301 q^{6} +1.00000 q^{8} -2.58488 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +0.644301 q^{3} +1.00000 q^{4} +2.05851 q^{5} +0.644301 q^{6} +1.00000 q^{8} -2.58488 q^{9} +2.05851 q^{10} +4.67370 q^{11} +0.644301 q^{12} -3.01127 q^{13} +1.32630 q^{15} +1.00000 q^{16} +5.19539 q^{17} -2.58488 q^{18} +2.70281 q^{19} +2.05851 q^{20} +4.67370 q^{22} +5.34740 q^{23} +0.644301 q^{24} -0.762519 q^{25} -3.01127 q^{26} -3.59834 q^{27} -1.00000 q^{29} +1.32630 q^{30} -3.59834 q^{31} +1.00000 q^{32} +3.01127 q^{33} +5.19539 q^{34} -2.58488 q^{36} -2.00000 q^{37} +2.70281 q^{38} -1.94016 q^{39} +2.05851 q^{40} +0.461461 q^{41} +11.1099 q^{43} +4.67370 q^{44} -5.32101 q^{45} +5.34740 q^{46} -3.59834 q^{47} +0.644301 q^{48} -0.762519 q^{50} +3.34740 q^{51} -3.01127 q^{52} -0.673698 q^{53} -3.59834 q^{54} +9.62087 q^{55} +1.74143 q^{57} -1.00000 q^{58} -10.6010 q^{59} +1.32630 q^{60} +11.9306 q^{61} -3.59834 q^{62} +1.00000 q^{64} -6.19874 q^{65} +3.01127 q^{66} +7.16975 q^{67} +5.19539 q^{68} +3.44533 q^{69} +8.00000 q^{71} -2.58488 q^{72} -10.6010 q^{73} -2.00000 q^{74} -0.491292 q^{75} +2.70281 q^{76} -1.94016 q^{78} +6.41512 q^{79} +2.05851 q^{80} +5.43622 q^{81} +0.461461 q^{82} -0.461461 q^{83} +10.6948 q^{85} +11.1099 q^{86} -0.644301 q^{87} +4.67370 q^{88} -0.827140 q^{89} -5.32101 q^{90} +5.34740 q^{92} -2.31841 q^{93} -3.59834 q^{94} +5.56378 q^{95} +0.644301 q^{96} +0.712685 q^{97} -12.0809 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} + 6 q^{4} + 6 q^{8} + 10 q^{9}+O(q^{10}) \) 6 * q + 6 * q^2 + 6 * q^4 + 6 * q^8 + 10 * q^9	\( 6 q + 6 q^{2} + 6 q^{4} + 6 q^{8} + 10 q^{9} + 8 q^{11} + 28 q^{15} + 6 q^{16} + 10 q^{18} + 8 q^{22} - 8 q^{23} + 10 q^{25} - 6 q^{29} + 28 q^{30} + 6 q^{32} + 10 q^{36} - 12 q^{37} - 8 q^{39} + 12 q^{43} + 8 q^{44} - 8 q^{46} + 10 q^{50} - 20 q^{51} + 16 q^{53} + 56 q^{57} - 6 q^{58} + 28 q^{60} + 6 q^{64} + 12 q^{65} - 8 q^{67} + 48 q^{71} + 10 q^{72} - 12 q^{74} - 8 q^{78} + 64 q^{79} - 2 q^{81} - 16 q^{85} + 12 q^{86} + 8 q^{88} - 8 q^{92} + 28 q^{93} + 68 q^{95} - 16 q^{99}+O(q^{100}) \) 6 * q + 6 * q^2 + 6 * q^4 + 6 * q^8 + 10 * q^9 + 8 * q^11 + 28 * q^15 + 6 * q^16 + 10 * q^18 + 8 * q^22 - 8 * q^23 + 10 * q^25 - 6 * q^29 + 28 * q^30 + 6 * q^32 + 10 * q^36 - 12 * q^37 - 8 * q^39 + 12 * q^43 + 8 * q^44 - 8 * q^46 + 10 * q^50 - 20 * q^51 + 16 * q^53 + 56 * q^57 - 6 * q^58 + 28 * q^60 + 6 * q^64 + 12 * q^65 - 8 * q^67 + 48 * q^71 + 10 * q^72 - 12 * q^74 - 8 * q^78 + 64 * q^79 - 2 * q^81 - 16 * q^85 + 12 * q^86 + 8 * q^88 - 8 * q^92 + 28 * q^93 + 68 * q^95 - 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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