LMFDB - Embedded newform 2842.2.a.bb.1.3

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			1.3
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	$-0.559550\pi$
$3$	−0.644301	−0.371987	−0.185994	+	0.982551i	$0.559550\pi$
$4$	1.00000	0.500000
$5$	−2.05851	−0.920596	−0.460298	−	0.887765i	$-0.652257\pi$
$5$	−2.05851	−0.920596	−0.460298	+	0.887765i	$0.652257\pi$
$6$	−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	$-0.362876\pi$
$13$	3.01127	0.835175	0.417588	+	0.908637i	$0.362876\pi$
$14$	0	0
$15$	1.32630	0.342450
$16$	1.00000	0.250000
$17$	−5.19539	−1.26007	−0.630034	−	0.776568i	$-0.716959\pi$
$17$	−5.19539	−1.26007	−0.630034	+	0.776568i	$0.716959\pi$
$18$	−2.58488	−0.609261
$19$	−2.70281	−0.620068	−0.310034	−	0.950725i	$-0.600340\pi$
$19$	−2.70281	−0.620068	−0.310034	+	0.950725i	$0.600340\pi$
$20$	−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	$-0.395261\pi$
$31$	3.59834	0.646281	0.323140	+	0.946351i	$0.395261\pi$
$32$	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	$-0.511472\pi$
$41$	−0.461461	−0.0720681	−0.0360341	+	0.999351i	$0.511472\pi$
$42$	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	$-0.415474\pi$
$47$	3.59834	0.524872	0.262436	+	0.964949i	$0.415474\pi$
$48$	−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	$-0.257581\pi$
$59$	10.6010	1.38014	0.690068	+	0.723745i	$0.257581\pi$
$60$	1.32630	0.171225
$61$	−11.9306	−1.52756	−0.763779	−	0.645478i	$-0.776658\pi$
$61$	−11.9306	−1.52756	−0.763779	+	0.645478i	$0.776658\pi$
$62$	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	$-0.286979\pi$
$73$	10.6010	1.24076	0.620378	+	0.784303i	$0.286979\pi$
$74$	−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	$-0.491938\pi$
$83$	0.461461	0.0506519	0.0253260	+	0.999679i	$0.491938\pi$
$84$	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	$-0.486041\pi$
$89$	0.827140	0.0876767	0.0438384	+	0.999039i	$0.486041\pi$
$90$	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	$-0.511519\pi$
$97$	−0.712685	−0.0723622	−0.0361811	+	0.999345i	$0.511519\pi$
$98$	0	0
$99$	−12.0809	−1.21418
$100$	−0.762519	−0.0762519
$101$	16.0476	1.59680	0.798400	−	0.602127i	$-0.205680\pi$
$101$	16.0476	1.59680	0.798400	+	0.602127i	$0.205680\pi$
$102$	3.34740	0.331442
$103$	−8.48528	−0.836080	−0.418040	−	0.908429i	$-0.637283\pi$
$103$	−8.48528	−0.836080	−0.418040	+	0.908429i	$0.637283\pi$
$104$	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	$-0.149734\pi$
$131$	20.4047	1.78277	0.891385	+	0.453247i	$0.149734\pi$
$132$	−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	$-0.472951\pi$
$139$	2.00129	0.169747	0.0848735	+	0.996392i	$0.472951\pi$
$140$	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	$-0.172894\pi$
$157$	21.4533	1.71216	0.856079	+	0.516845i	$0.172894\pi$
$158$	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	$-0.503094\pi$
$167$	−0.251224	−0.0194403	−0.00972016	+	0.999953i	$0.503094\pi$
$168$	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	$-0.654414\pi$
$173$	−12.2665	−0.932602	−0.466301	+	0.884626i	$0.654414\pi$
$174$	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	$-0.544063\pi$
$181$	−3.71280	−0.275970	−0.137985	+	0.990434i	$0.544063\pi$
$182$	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	$-0.371749\pi$
$199$	11.0625	0.784199	0.392099	+	0.919923i	$0.371749\pi$
$200$	−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	$-0.632136\pi$
$223$	−12.0451	−0.806597	−0.403299	+	0.915068i	$0.632136\pi$
$224$	0	0
$225$	1.97102	0.131401
$226$	11.6447	0.774595
$227$	8.64072	0.573505	0.286752	−	0.958005i	$-0.407424\pi$
$227$	8.64072	0.573505	0.286752	+	0.958005i	$0.407424\pi$
$228$	1.74143	0.115329
$229$	−28.6500	−1.89324	−0.946621	−	0.322349i	$-0.895528\pi$
$229$	−28.6500	−1.89324	−0.946621	+	0.322349i	$0.895528\pi$
$230$	−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	$-0.528755\pi$
$241$	−2.80103	−0.180430	−0.0902151	+	0.995922i	$0.528755\pi$
$242$	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	$-0.379219\pi$
$251$	11.7366	0.740809	0.370404	+	0.928871i	$0.379219\pi$
$252$	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	$-0.401683\pi$
$257$	9.74649	0.607969	0.303985	+	0.952677i	$0.401683\pi$
$258$	−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	$-0.733420\pi$
$269$	−21.9557	−1.33866	−0.669332	+	0.742964i	$0.733420\pi$
$270$	−7.40723	−0.450790
$271$	6.45660	0.392210	0.196105	−	0.980583i	$-0.437171\pi$
$271$	6.45660	0.392210	0.196105	+	0.980583i	$0.437171\pi$
$272$	−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	$-0.487418\pi$
$283$	1.32959	0.0790359	0.0395179	+	0.999219i	$0.487418\pi$
$284$	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	$-0.583241\pi$
$293$	−8.85096	−0.517079	−0.258539	+	0.966001i	$0.583241\pi$
$294$	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	$-0.163366\pi$
$307$	30.5281	1.74233	0.871164	+	0.490992i	$0.163366\pi$
$308$	0	0
$309$	5.46707	0.311011
$310$	−7.40723	−0.420703
$311$	−19.4631	−1.10365	−0.551827	−	0.833959i	$-0.686069\pi$
$311$	−19.4631	−1.10365	−0.551827	+	0.833959i	$0.686069\pi$
$312$	−1.94016	−0.109840
$313$	10.8137	0.611226	0.305613	−	0.952156i	$-0.401139\pi$
$313$	10.8137	0.611226	0.305613	+	0.952156i	$0.401139\pi$
$314$	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	$-0.412179\pi$
$349$	10.1781	0.544822	0.272411	+	0.962181i	$0.412179\pi$
$350$	0	0
$351$	10.8356	0.578359
$352$	4.67370	0.249109
$353$	−20.5415	−1.09331	−0.546657	−	0.837357i	$-0.684100\pi$
$353$	−20.5415	−1.09331	−0.546657	+	0.837357i	$0.684100\pi$
$354$	−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	$-0.453542\pi$
$367$	5.57223	0.290868	0.145434	+	0.989368i	$0.453542\pi$
$368$	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	$-0.155577\pi$
$383$	34.5580	1.76583	0.882916	+	0.469530i	$0.155577\pi$
$384$	−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	$-0.735574\pi$
$397$	−26.8725	−1.34869	−0.674346	+	0.738416i	$0.735574\pi$
$398$	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	$-0.555696\pi$
$409$	−7.04124	−0.348167	−0.174083	+	0.984731i	$0.555696\pi$
$410$	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	$-0.355852\pi$
$419$	17.9122	0.875066	0.437533	+	0.899202i	$0.355852\pi$
$420$	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	$-0.384940\pi$
$433$	14.7181	0.707304	0.353652	+	0.935377i	$0.384940\pi$
$434$	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	$-0.542053\pi$
$439$	−5.52009	−0.263459	−0.131730	+	0.991286i	$0.542053\pi$
$440$	−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	$-0.439862\pi$
$461$	8.06481	0.375616	0.187808	+	0.982206i	$0.439862\pi$
$462$	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	$-0.372454\pi$
$467$	16.8585	0.780120	0.390060	+	0.920789i	$0.372454\pi$
$468$	−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	$-0.290347\pi$
$479$	26.7905	1.22409	0.612045	+	0.790823i	$0.290347\pi$
$480$	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	$-0.472436\pi$
$503$	3.87939	0.172974	0.0864868	+	0.996253i	$0.472436\pi$
$504$	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	$-0.420043\pi$
$509$	11.2155	0.497118	0.248559	+	0.968617i	$0.420043\pi$
$510$	−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	$-0.500989\pi$
$521$	−0.141852	−0.00621466	−0.00310733	+	0.999995i	$0.500989\pi$
$522$	2.58488	0.113137
$523$	7.46658	0.326491	0.163245	−	0.986586i	$-0.447804\pi$
$523$	7.46658	0.326491	0.163245	+	0.986586i	$0.447804\pi$
$524$	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	$-0.491940\pi$
$563$	1.20154	0.0506390	0.0253195	+	0.999679i	$0.491940\pi$
$564$	−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	$-0.319698\pi$
$577$	25.7805	1.07326	0.536629	+	0.843818i	$0.319698\pi$
$578$	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	$-0.874121\pi$
$587$	−44.7163	−1.84564	−0.922819	+	0.385234i	$0.874121\pi$
$588$	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	$-0.512278\pi$
$593$	−1.87811	−0.0771247	−0.0385623	+	0.999256i	$0.512278\pi$
$594$	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	$-0.557146\pi$
$601$	−8.75518	−0.357131	−0.178566	+	0.983928i	$0.557146\pi$
$602$	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	$-0.367091\pi$
$607$	19.9818	0.811037	0.405519	+	0.914087i	$0.367091\pi$
$608$	−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	$-0.913871\pi$
$619$	−47.9489	−1.92723	−0.963615	+	0.267294i	$0.913871\pi$
$620$	−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	$-0.635840\pi$
$643$	−20.9918	−0.827836	−0.413918	+	0.910314i	$0.635840\pi$
$644$	0	0
$645$	14.7351	0.580194
$646$	14.0422	0.552482
$647$	31.3416	1.23217	0.616083	−	0.787681i	$-0.288719\pi$
$647$	31.3416	1.23217	0.616083	+	0.787681i	$0.288719\pi$
$648$	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	$-0.372298\pi$
$661$	20.0800	0.781023	0.390512	+	0.920598i	$0.372298\pi$
$662$	−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	$-0.590663\pi$
$677$	−14.6223	−0.561979	−0.280990	+	0.959711i	$0.590663\pi$
$678$	−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	$-0.714336\pi$
$691$	−32.7857	−1.24723	−0.623613	+	0.781734i	$0.714336\pi$
$692$	−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	$-0.196884\pi$
$719$	43.6927	1.62946	0.814731	+	0.579839i	$0.196884\pi$
$720$	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	$-0.687951\pi$
$727$	−30.0232	−1.11350	−0.556749	+	0.830681i	$0.687951\pi$
$728$	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	$-0.439355\pi$
$733$	10.2540	0.378741	0.189370	+	0.981906i	$0.439355\pi$
$734$	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	$-0.328187\pi$
$761$	28.3551	1.02787	0.513936	+	0.857829i	$0.328187\pi$
$762$	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	$-0.622001\pi$
$769$	−20.7406	−0.747925	−0.373962	+	0.927444i	$0.622001\pi$
$770$	0	0
$771$	−6.27967	−0.226157
$772$	−26.3395	−0.947980
$773$	−43.6602	−1.57035	−0.785174	−	0.619275i	$-0.787427\pi$
$773$	−43.6602	−1.57035	−0.785174	+	0.619275i	$0.787427\pi$
$774$	−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	$-0.695717\pi$
$787$	−32.3652	−1.15369	−0.576847	+	0.816852i	$0.695717\pi$
$788$	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	$-0.791828\pi$
$797$	−44.8120	−1.58732	−0.793662	+	0.608359i	$0.791828\pi$
$798$	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	$-0.520444\pi$
$811$	−3.65557	−0.128364	−0.0641822	+	0.997938i	$0.520444\pi$
$812$	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	$-0.607777\pi$
$829$	−19.1273	−0.664318	−0.332159	+	0.943223i	$0.607777\pi$
$830$	−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	$-0.193313\pi$
$839$	47.5721	1.64237	0.821185	+	0.570661i	$0.193313\pi$
$840$	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	$-0.177115\pi$
$853$	49.6008	1.69830	0.849149	+	0.528153i	$0.177115\pi$
$854$	0	0
$855$	−14.3817	−0.491844
$856$	−9.34740	−0.319487
$857$	30.0256	1.02566	0.512828	−	0.858492i	$-0.328598\pi$
$857$	30.0256	1.02566	0.512828	+	0.858492i	$0.328598\pi$
$858$	−9.06773	−0.309567
$859$	−50.6331	−1.72758	−0.863789	−	0.503854i	$-0.831915\pi$
$859$	−50.6331	−1.72758	−0.863789	+	0.503854i	$0.831915\pi$
$860$	−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	$-0.485222\pi$
$881$	2.75496	0.0928169	0.0464085	+	0.998923i	$0.485222\pi$
$882$	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	$-0.383596\pi$
$887$	21.3003	0.715193	0.357596	+	0.933876i	$0.383596\pi$
$888$	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	$-0.919605\pi$
$929$	−59.0250	−1.93655	−0.968274	+	0.249892i	$0.919605\pi$
$930$	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	$-0.478350\pi$
$937$	4.16067	0.135923	0.0679615	+	0.997688i	$0.478350\pi$
$938$	0	0
$939$	−6.96727	−0.227368
$940$	−7.40723	−0.241597
$941$	48.0745	1.56719	0.783593	−	0.621275i	$-0.213385\pi$
$941$	48.0745	1.56719	0.783593	+	0.621275i	$0.213385\pi$
$942$	−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	$-0.534200\pi$
$971$	−6.68307	−0.214470	−0.107235	+	0.994234i	$0.534200\pi$
$972$	−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	$-0.566662\pi$
$983$	−13.0364	−0.415796	−0.207898	+	0.978151i	$0.566662\pi$
$984$	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	$-0.570311\pi$
$997$	−13.8361	−0.438194	−0.219097	+	0.975703i	$0.570311\pi$
$998$	−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.1.3	✓	6
7.6	odd	2	inner	2842.2.a.bb.1.4	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2842.2.a.bb.1.3	✓	6	1.1	even	1	trivial
2842.2.a.bb.1.4	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)

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