LMFDB - Embedded newform 6223.2.a.k.1.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6223,2,Mod(1,6223)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6223.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6223, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6223 = 7^{2} \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6223.a (trivial)

Newform invariants

comment:select newform

sage:traces = [16,-2,4,12,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$49.6909051778$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} - 20 x^{14} + 38 x^{13} + 155 x^{12} - 275 x^{11} - 593 x^{10} + 957 x^{9} + 1177 x^{8} + \cdots + 1 $ x^16 - 2x^15 - 20x^14 + 38x^13 + 155x^12 - 275x^11 - 593x^10 + 957x^9 + 1177x^8 - 1655x^7 - 1150x^6 + 1279x^5 + 474x^4 - 280x^3 - 83x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 889)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-0.191617$ of defining polynomial
Character	$\chi$	$=$	6223.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.191617 q^{2} +2.32908 q^{3} -1.96328 q^{4} -3.92161 q^{5} +0.446291 q^{6} -0.759431 q^{8} +2.42462 q^{9} -0.751445 q^{10} -5.26691 q^{11} -4.57265 q^{12} -6.07655 q^{13} -9.13374 q^{15} +3.78105 q^{16} +4.12322 q^{17} +0.464598 q^{18} -0.0374083 q^{19} +7.69922 q^{20} -1.00923 q^{22} -3.57588 q^{23} -1.76878 q^{24} +10.3790 q^{25} -1.16437 q^{26} -1.34010 q^{27} -2.19905 q^{29} -1.75018 q^{30} -4.14885 q^{31} +2.24337 q^{32} -12.2671 q^{33} +0.790078 q^{34} -4.76022 q^{36} -9.31219 q^{37} -0.00716804 q^{38} -14.1528 q^{39} +2.97819 q^{40} +9.48327 q^{41} -11.4342 q^{43} +10.3404 q^{44} -9.50842 q^{45} -0.685199 q^{46} +3.87164 q^{47} +8.80637 q^{48} +1.98879 q^{50} +9.60333 q^{51} +11.9300 q^{52} -9.97552 q^{53} -0.256785 q^{54} +20.6547 q^{55} -0.0871269 q^{57} -0.421374 q^{58} +9.91129 q^{59} +17.9321 q^{60} +0.382013 q^{61} -0.794988 q^{62} -7.13223 q^{64} +23.8298 q^{65} -2.35057 q^{66} +4.40596 q^{67} -8.09505 q^{68} -8.32853 q^{69} -10.3123 q^{71} -1.84133 q^{72} +3.37816 q^{73} -1.78437 q^{74} +24.1735 q^{75} +0.0734430 q^{76} -2.71191 q^{78} +9.73339 q^{79} -14.8278 q^{80} -10.3951 q^{81} +1.81715 q^{82} -16.5257 q^{83} -16.1697 q^{85} -2.19098 q^{86} -5.12177 q^{87} +3.99985 q^{88} +2.53028 q^{89} -1.82197 q^{90} +7.02047 q^{92} -9.66301 q^{93} +0.741870 q^{94} +0.146700 q^{95} +5.22500 q^{96} +5.85331 q^{97} -12.7703 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 2 q^{2} + 4 q^{3} + 12 q^{4} + 9 q^{5} + 12 q^{6} - 6 q^{8} + 14 q^{9} + 2 q^{10} - 22 q^{11} + 10 q^{12} + 4 q^{13} - 14 q^{15} + 12 q^{16} + 18 q^{17} - 5 q^{18} + 15 q^{19} + 40 q^{20} - 11 q^{22}+ \cdots - 73 q^{99}+O(q^{100}) $ 16 * q - 2 * q^2 + 4 * q^3 + 12 * q^4 + 9 * q^5 + 12 * q^6 - 6 * q^8 + 14 * q^9 + 2 * q^10 - 22 * q^11 + 10 * q^12 + 4 * q^13 - 14 * q^15 + 12 * q^16 + 18 * q^17 - 5 * q^18 + 15 * q^19 + 40 * q^20 - 11 * q^22 - 5 * q^23 + 26 * q^24 + 15 * q^25 + 24 * q^26 + 10 * q^27 - 12 * q^29 + 3 * q^30 + 32 * q^31 - 9 * q^32 + 10 * q^33 + 14 * q^34 + 10 * q^36 - 2 * q^37 - 3 * q^38 - 41 * q^39 + 14 * q^40 + 45 * q^41 - 3 * q^43 - 54 * q^44 + 22 * q^45 + 49 * q^47 - 10 * q^48 - 6 * q^50 - 12 * q^51 - 38 * q^52 + 16 * q^53 + 67 * q^54 - 7 * q^55 + 8 * q^57 + 16 * q^58 + 35 * q^59 + 56 * q^60 + 11 * q^61 - 17 * q^62 - 2 * q^64 + 14 * q^65 - 86 * q^66 + 17 * q^67 + 71 * q^68 + 17 * q^69 - 81 * q^71 + 13 * q^72 + 15 * q^73 + 13 * q^74 + 48 * q^75 - 14 * q^76 - 10 * q^78 - 34 * q^79 + 33 * q^80 + 40 * q^81 + 14 * q^82 + 39 * q^83 - 17 * q^85 + 36 * q^86 - 7 * q^87 + 61 * q^88 + 32 * q^89 - 79 * q^90 + 37 * q^92 + 27 * q^93 - 13 * q^94 - 33 * q^95 + 55 * q^96 + 4 * q^97 - 73 * q^99

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 $

$2$	0.191617	0.135493	0.0677467	−	0.997703i	$-0.478419\pi$
$2$	0.191617	0.135493	0.0677467	+	0.997703i	$0.478419\pi$
$3$	2.32908	1.34470	0.672348	−	0.740235i	$-0.265286\pi$
$3$	2.32908	1.34470	0.672348	+	0.740235i	$0.265286\pi$
$4$	−1.96328	−0.981642
$5$	−3.92161	−1.75380	−0.876898	−	0.480677i	$-0.840391\pi$
$5$	−3.92161	−1.75380	−0.876898	+	0.480677i	$0.840391\pi$
$6$	0.446291	0.182197
$7$	0	0
$7$	0	0
$8$	−0.759431	−0.268499
$9$	2.42462	0.808208
$10$	−0.751445	−0.237628
$11$	−5.26691	−1.58803	−0.794016	−	0.607897i	$-0.792014\pi$
$11$	−5.26691	−1.58803	−0.794016	+	0.607897i	$0.792014\pi$
$12$	−4.57265	−1.32001
$13$	−6.07655	−1.68533	−0.842665	−	0.538438i	$-0.819015\pi$
$13$	−6.07655	−1.68533	−0.842665	+	0.538438i	$0.819015\pi$
$14$	0	0
$15$	−9.13374	−2.35832
$16$	3.78105	0.945262
$17$	4.12322	1.00003	0.500014	−	0.866017i	$-0.333328\pi$
$17$	4.12322	1.00003	0.500014	+	0.866017i	$0.333328\pi$
$18$	0.464598	0.109507
$19$	−0.0374083	−0.00858204	−0.00429102	−	0.999991i	$-0.501366\pi$
$19$	−0.0374083	−0.00858204	−0.00429102	+	0.999991i	$0.501366\pi$
$20$	7.69922	1.72160
$21$	0	0
$22$	−1.00923	−0.215168
$23$	−3.57588	−0.745623	−0.372812	−	0.927907i	$-0.621606\pi$
$23$	−3.57588	−0.745623	−0.372812	+	0.927907i	$0.621606\pi$
$24$	−1.76878	−0.361050
$25$	10.3790	2.07580
$26$	−1.16437	−0.228351
$27$	−1.34010	−0.257902
$28$	0	0
$29$	−2.19905	−0.408353	−0.204177	−	0.978934i	$-0.565452\pi$
$29$	−2.19905	−0.408353	−0.204177	+	0.978934i	$0.565452\pi$
$30$	−1.75018	−0.319537
$31$	−4.14885	−0.745155	−0.372577	−	0.928001i	$-0.621526\pi$
$31$	−4.14885	−0.745155	−0.372577	+	0.928001i	$0.621526\pi$
$32$	2.24337	0.396576
$33$	−12.2671	−2.13542
$34$	0.790078	0.135497
$35$	0	0
$36$	−4.76022	−0.793371
$37$	−9.31219	−1.53091	−0.765457	−	0.643487i	$-0.777487\pi$
$37$	−9.31219	−1.53091	−0.765457	+	0.643487i	$0.777487\pi$
$38$	−0.00716804	−0.00116281
$39$	−14.1528	−2.26626
$40$	2.97819	0.470893
$41$	9.48327	1.48104	0.740519	−	0.672035i	$-0.234580\pi$
$41$	9.48327	1.48104	0.740519	+	0.672035i	$0.234580\pi$
$42$	0	0
$43$	−11.4342	−1.74370	−0.871850	−	0.489774i	$-0.837079\pi$
$43$	−11.4342	−1.74370	−0.871850	+	0.489774i	$0.837079\pi$
$44$	10.3404	1.55888
$45$	−9.50842	−1.41743
$46$	−0.685199	−0.101027
$47$	3.87164	0.564737	0.282368	−	0.959306i	$-0.408880\pi$
$47$	3.87164	0.564737	0.282368	+	0.959306i	$0.408880\pi$
$48$	8.80637	1.27109
$49$	0	0
$50$	1.98879	0.281257
$51$	9.60333	1.34473
$52$	11.9300	1.65439
$53$	−9.97552	−1.37024	−0.685122	−	0.728429i	$-0.740251\pi$
$53$	−9.97552	−1.37024	−0.685122	+	0.728429i	$0.740251\pi$
$54$	−0.256785	−0.0349440
$55$	20.6547	2.78508
$56$	0	0
$57$	−0.0871269	−0.0115402
$58$	−0.421374	−0.0553292
$59$	9.91129	1.29034	0.645170	−	0.764039i	$-0.276786\pi$
$59$	9.91129	1.29034	0.645170	+	0.764039i	$0.276786\pi$
$60$	17.9321	2.31503
$61$	0.382013	0.0489118	0.0244559	−	0.999701i	$-0.492215\pi$
$61$	0.382013	0.0489118	0.0244559	+	0.999701i	$0.492215\pi$
$62$	−0.794988	−0.100964
$63$	0	0
$64$	−7.13223	−0.891528
$65$	23.8298	2.95573
$66$	−2.35057	−0.289335
$67$	4.40596	0.538273	0.269137	−	0.963102i	$-0.413262\pi$
$67$	4.40596	0.538273	0.269137	+	0.963102i	$0.413262\pi$
$68$	−8.09505	−0.981669
$69$	−8.32853	−1.00264
$70$	0	0
$71$	−10.3123	−1.22385	−0.611925	−	0.790916i	$-0.709604\pi$
$71$	−10.3123	−1.22385	−0.611925	+	0.790916i	$0.709604\pi$
$72$	−1.84133	−0.217003
$73$	3.37816	0.395383	0.197692	−	0.980264i	$-0.436656\pi$
$73$	3.37816	0.395383	0.197692	+	0.980264i	$0.436656\pi$
$74$	−1.78437	−0.207429
$75$	24.1735	2.79132
$76$	0.0734430	0.00842449
$77$	0	0
$78$	−2.71191	−0.307063
$79$	9.73339	1.09509	0.547546	−	0.836776i	$-0.315562\pi$
$79$	9.73339	1.09509	0.547546	+	0.836776i	$0.315562\pi$
$80$	−14.8278	−1.65780
$81$	−10.3951	−1.15501
$82$	1.81715	0.200671
$83$	−16.5257	−1.81393	−0.906964	−	0.421209i	$-0.861606\pi$
$83$	−16.5257	−1.81393	−0.906964	+	0.421209i	$0.861606\pi$
$84$	0	0
$85$	−16.1697	−1.75385
$86$	−2.19098	−0.236260
$87$	−5.12177	−0.549111
$88$	3.99985	0.426386
$89$	2.53028	0.268209	0.134105	−	0.990967i	$-0.457184\pi$
$89$	2.53028	0.268209	0.134105	+	0.990967i	$0.457184\pi$
$90$	−1.82197	−0.192053
$91$	0	0
$92$	7.02047	0.731935
$93$	−9.66301	−1.00201
$94$	0.741870	0.0765181
$95$	0.146700	0.0150512
$96$	5.22500	0.533274
$97$	5.85331	0.594314	0.297157	−	0.954829i	$-0.403962\pi$
$97$	5.85331	0.594314	0.297157	+	0.954829i	$0.403962\pi$
$98$	0	0
$99$	−12.7703	−1.28346
$100$	−20.3769	−2.03769
$101$	1.78412	0.177527	0.0887633	−	0.996053i	$-0.471709\pi$
$101$	1.78412	0.177527	0.0887633	+	0.996053i	$0.471709\pi$
$102$	1.84016	0.182203
$103$	8.17058	0.805071	0.402536	−	0.915404i	$-0.368129\pi$
$103$	8.17058	0.805071	0.402536	+	0.915404i	$0.368129\pi$
$104$	4.61471	0.452510
$105$	0	0
$106$	−1.91148	−0.185659
$107$	3.22757	0.312021	0.156010	−	0.987755i	$-0.450137\pi$
$107$	3.22757	0.312021	0.156010	+	0.987755i	$0.450137\pi$
$108$	2.63099	0.253167
$109$	16.9092	1.61961	0.809805	−	0.586700i	$-0.199573\pi$
$109$	16.9092	1.61961	0.809805	+	0.586700i	$0.199573\pi$
$110$	3.95779	0.377360
$111$	−21.6889	−2.05861
$112$	0	0
$113$	5.58588	0.525476	0.262738	−	0.964867i	$-0.415375\pi$
$113$	5.58588	0.525476	0.262738	+	0.964867i	$0.415375\pi$
$114$	−0.0166950	−0.00156363
$115$	14.0232	1.30767
$116$	4.31736	0.400857
$117$	−14.7333	−1.36210
$118$	1.89917	0.174833
$119$	0	0
$120$	6.93644	0.633208
$121$	16.7403	1.52185
$122$	0.0732001	0.00662722
$123$	22.0873	1.99155
$124$	8.14536	0.731475
$125$	−21.0943	−1.88673
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−5.85340	−0.517372
$129$	−26.6312	−2.34475
$130$	4.56619	0.400481
$131$	15.7005	1.37176	0.685879	−	0.727716i	$-0.259418\pi$
$131$	15.7005	1.37176	0.685879	+	0.727716i	$0.259418\pi$
$132$	24.0837	2.09622
$133$	0	0
$134$	0.844254	0.0729325
$135$	5.25533	0.452307
$136$	−3.13130	−0.268507
$137$	7.97414	0.681277	0.340639	−	0.940194i	$-0.389357\pi$
$137$	7.97414	0.681277	0.340639	+	0.940194i	$0.389357\pi$
$138$	−1.59588	−0.135851
$139$	3.40873	0.289125	0.144562	−	0.989496i	$-0.453823\pi$
$139$	3.40873	0.289125	0.144562	+	0.989496i	$0.453823\pi$
$140$	0	0
$141$	9.01737	0.759399
$142$	−1.97601	−0.165823
$143$	32.0046	2.67636
$144$	9.16762	0.763968
$145$	8.62381	0.716169
$146$	0.647310	0.0535718
$147$	0	0
$148$	18.2825	1.50281
$149$	−2.34920	−0.192454	−0.0962271	−	0.995359i	$-0.530678\pi$
$149$	−2.34920	−0.192454	−0.0962271	+	0.995359i	$0.530678\pi$
$150$	4.63205	0.378205
$151$	−3.42902	−0.279050	−0.139525	−	0.990219i	$-0.544558\pi$
$151$	−3.42902	−0.279050	−0.139525	+	0.990219i	$0.544558\pi$
$152$	0.0284090	0.00230427
$153$	9.99727	0.808231
$154$	0	0
$155$	16.2701	1.30685
$156$	27.7859	2.22465
$157$	19.8947	1.58777	0.793886	−	0.608067i	$-0.208055\pi$
$157$	19.8947	1.58777	0.793886	+	0.608067i	$0.208055\pi$
$158$	1.86508	0.148378
$159$	−23.2338	−1.84256
$160$	−8.79762	−0.695513
$161$	0	0
$162$	−1.99187	−0.156496
$163$	6.68481	0.523595	0.261797	−	0.965123i	$-0.415685\pi$
$163$	6.68481	0.523595	0.261797	+	0.965123i	$0.415685\pi$
$164$	−18.6183	−1.45385
$165$	48.1066	3.74509
$166$	−3.16659	−0.245775
$167$	−16.0276	−1.24025	−0.620126	−	0.784502i	$-0.712918\pi$
$167$	−16.0276	−1.24025	−0.620126	+	0.784502i	$0.712918\pi$
$168$	0	0
$169$	23.9244	1.84034
$170$	−3.09837	−0.237634
$171$	−0.0907010	−0.00693608
$172$	22.4486	1.71169
$173$	14.8583	1.12965	0.564827	−	0.825209i	$-0.308943\pi$
$173$	14.8583	1.12965	0.564827	+	0.825209i	$0.308943\pi$
$174$	−0.981416	−0.0744009
$175$	0	0
$176$	−19.9144	−1.50111
$177$	23.0842	1.73512
$178$	0.484844	0.0363406
$179$	−10.4016	−0.777456	−0.388728	−	0.921353i	$-0.627085\pi$
$179$	−10.4016	−0.777456	−0.388728	+	0.921353i	$0.627085\pi$
$180$	18.6677	1.39141
$181$	6.10810	0.454011	0.227006	−	0.973893i	$-0.427106\pi$
$181$	6.10810	0.454011	0.227006	+	0.973893i	$0.427106\pi$
$182$	0	0
$183$	0.889740	0.0657715
$184$	2.71564	0.200199
$185$	36.5187	2.68491
$186$	−1.85159	−0.135765
$187$	−21.7166	−1.58808
$188$	−7.60113	−0.554369
$189$	0	0
$190$	0.0281102	0.00203933
$191$	8.67944	0.628022	0.314011	−	0.949419i	$-0.398327\pi$
$191$	8.67944	0.628022	0.314011	+	0.949419i	$0.398327\pi$
$192$	−16.6115	−1.19883
$193$	−4.28438	−0.308396	−0.154198	−	0.988040i	$-0.549279\pi$
$193$	−4.28438	−0.308396	−0.154198	+	0.988040i	$0.549279\pi$
$194$	1.12159	0.0805255
$195$	55.5016	3.97455
$196$	0	0
$197$	20.8740	1.48721	0.743606	−	0.668618i	$-0.233114\pi$
$197$	20.8740	1.48721	0.743606	+	0.668618i	$0.233114\pi$
$198$	−2.44700	−0.173900
$199$	−23.3191	−1.65305	−0.826525	−	0.562900i	$-0.809686\pi$
$199$	−23.3191	−1.65305	−0.826525	+	0.562900i	$0.809686\pi$
$200$	−7.88213	−0.557350
$201$	10.2618	0.723814
$202$	0.341867	0.0240537
$203$	0	0
$204$	−18.8540	−1.32005
$205$	−37.1897	−2.59744
$206$	1.56562	0.109082
$207$	−8.67018	−0.602619
$208$	−22.9757	−1.59308
$209$	0.197026	0.0136286
$210$	0	0
$211$	−17.2768	−1.18938	−0.594691	−	0.803954i	$-0.702726\pi$
$211$	−17.2768	−1.18938	−0.594691	+	0.803954i	$0.702726\pi$
$212$	19.5848	1.34509
$213$	−24.0183	−1.64571
$214$	0.618456	0.0422768
$215$	44.8404	3.05809
$216$	1.01771	0.0692465
$217$	0	0
$218$	3.24009	0.219446
$219$	7.86800	0.531670
$220$	−40.5511	−2.73395
$221$	−25.0550	−1.68538
$222$	−4.15594	−0.278929
$223$	−26.8248	−1.79632	−0.898162	−	0.439665i	$-0.855097\pi$
$223$	−26.8248	−1.79632	−0.898162	+	0.439665i	$0.855097\pi$
$224$	0	0
$225$	25.1652	1.67768
$226$	1.07035	0.0711985
$227$	−18.2819	−1.21341	−0.606707	−	0.794925i	$-0.707510\pi$
$227$	−18.2819	−1.21341	−0.606707	+	0.794925i	$0.707510\pi$
$228$	0.171055	0.0113284
$229$	−4.27326	−0.282385	−0.141192	−	0.989982i	$-0.545094\pi$
$229$	−4.27326	−0.282385	−0.141192	+	0.989982i	$0.545094\pi$
$230$	2.68708	0.177181
$231$	0	0
$232$	1.67003	0.109643
$233$	−0.982041	−0.0643357	−0.0321678	−	0.999482i	$-0.510241\pi$
$233$	−0.982041	−0.0643357	−0.0321678	+	0.999482i	$0.510241\pi$
$234$	−2.82315	−0.184555
$235$	−15.1830	−0.990433
$236$	−19.4587	−1.26665
$237$	22.6699	1.47257
$238$	0	0
$239$	−8.88724	−0.574868	−0.287434	−	0.957800i	$-0.592802\pi$
$239$	−8.88724	−0.574868	−0.287434	+	0.957800i	$0.592802\pi$
$240$	−34.5351	−2.22923
$241$	27.4668	1.76929	0.884646	−	0.466264i	$-0.154400\pi$
$241$	27.4668	1.76929	0.884646	+	0.466264i	$0.154400\pi$
$242$	3.20772	0.206200
$243$	−20.1907	−1.29523
$244$	−0.750000	−0.0480138
$245$	0	0
$246$	4.23230	0.269841
$247$	0.227313	0.0144636
$248$	3.15076	0.200074
$249$	−38.4896	−2.43918
$250$	−4.04202	−0.255640
$251$	7.65870	0.483413	0.241706	−	0.970349i	$-0.422293\pi$
$251$	7.65870	0.483413	0.241706	+	0.970349i	$0.422293\pi$
$252$	0	0
$253$	18.8339	1.18407
$254$	−0.191617	−0.0120231
$255$	−37.6605	−2.35839
$256$	13.1428	0.821428
$257$	19.3455	1.20674	0.603369	−	0.797462i	$-0.293824\pi$
$257$	19.3455	1.20674	0.603369	+	0.797462i	$0.293824\pi$
$258$	−5.10298	−0.317697
$259$	0	0
$260$	−46.7847	−2.90146
$261$	−5.33187	−0.330035
$262$	3.00847	0.185864
$263$	12.7413	0.785660	0.392830	−	0.919611i	$-0.371496\pi$
$263$	12.7413	0.785660	0.392830	+	0.919611i	$0.371496\pi$
$264$	9.31598	0.573359
$265$	39.1201	2.40313
$266$	0	0
$267$	5.89323	0.360660
$268$	−8.65014	−0.528392
$269$	−9.02960	−0.550545	−0.275272	−	0.961366i	$-0.588768\pi$
$269$	−9.02960	−0.550545	−0.275272	+	0.961366i	$0.588768\pi$
$270$	1.00701	0.0612846
$271$	−16.6046	−1.00866	−0.504328	−	0.863512i	$-0.668260\pi$
$271$	−16.6046	−1.00866	−0.504328	+	0.863512i	$0.668260\pi$
$272$	15.5901	0.945289
$273$	0	0
$274$	1.52798	0.0923085
$275$	−54.6652	−3.29644
$276$	16.3513	0.984230
$277$	−7.79112	−0.468123	−0.234061	−	0.972222i	$-0.575202\pi$
$277$	−7.79112	−0.468123	−0.234061	+	0.972222i	$0.575202\pi$
$278$	0.653169	0.0391745
$279$	−10.0594	−0.602240
$280$	0	0
$281$	−8.94643	−0.533700	−0.266850	−	0.963738i	$-0.585983\pi$
$281$	−8.94643	−0.533700	−0.266850	+	0.963738i	$0.585983\pi$
$282$	1.72788	0.102894
$283$	11.5107	0.684243	0.342121	−	0.939656i	$-0.388855\pi$
$283$	11.5107	0.684243	0.342121	+	0.939656i	$0.388855\pi$
$284$	20.2460	1.20138
$285$	0.341678	0.0202392
$286$	6.13261	0.362629
$287$	0	0
$288$	5.43933	0.320516
$289$	0.000967701 0	5.69236e−5 0
$290$	1.65246	0.0970361
$291$	13.6328	0.799171
$292$	−6.63228	−0.388125
$293$	4.41505	0.257930	0.128965	−	0.991649i	$-0.458834\pi$
$293$	4.41505	0.257930	0.128965	+	0.991649i	$0.458834\pi$
$294$	0	0
$295$	−38.8682	−2.26299
$296$	7.07196	0.411049
$297$	7.05817	0.409557
$298$	−0.450146	−0.0260763
$299$	21.7290	1.25662
$300$	−47.4595	−2.74007
$301$	0	0
$302$	−0.657057	−0.0378094
$303$	4.15536	0.238719
$304$	−0.141442	−0.00811228
$305$	−1.49811	−0.0857813
$306$	1.91564	0.109510
$307$	−4.82910	−0.275611	−0.137806	−	0.990459i	$-0.544005\pi$
$307$	−4.82910	−0.275611	−0.137806	+	0.990459i	$0.544005\pi$
$308$	0	0
$309$	19.0300	1.08258
$310$	3.11763	0.177069
$311$	15.7618	0.893772	0.446886	−	0.894591i	$-0.352533\pi$
$311$	15.7618	0.893772	0.446886	+	0.894591i	$0.352533\pi$
$312$	10.7480	0.608488
$313$	−13.2573	−0.749349	−0.374675	−	0.927156i	$-0.622246\pi$
$313$	−13.2573	−0.749349	−0.374675	+	0.927156i	$0.622246\pi$
$314$	3.81216	0.215132
$315$	0	0
$316$	−19.1094	−1.07499
$317$	5.19102	0.291557	0.145778	−	0.989317i	$-0.453431\pi$
$317$	5.19102	0.291557	0.145778	+	0.989317i	$0.453431\pi$
$318$	−4.45198	−0.249655
$319$	11.5822	0.648479
$320$	27.9698	1.56356
$321$	7.51727	0.419573
$322$	0	0
$323$	−0.154243	−0.00858229
$324$	20.4085	1.13380
$325$	−63.0684	−3.49841
$326$	1.28092	0.0709436
$327$	39.3830	2.17788
$328$	−7.20189	−0.397658
$329$	0	0
$330$	9.21802	0.507435
$331$	−11.1884	−0.614969	−0.307485	−	0.951553i	$-0.599487\pi$
$331$	−11.1884	−0.614969	−0.307485	+	0.951553i	$0.599487\pi$
$332$	32.4446	1.78063
$333$	−22.5786	−1.23730
$334$	−3.07115	−0.168046
$335$	−17.2784	−0.944022
$336$	0	0
$337$	17.1501	0.934223	0.467111	−	0.884198i	$-0.345295\pi$
$337$	17.1501	0.934223	0.467111	+	0.884198i	$0.345295\pi$
$338$	4.58431	0.249354
$339$	13.0100	0.706605
$340$	31.7456	1.72165
$341$	21.8516	1.18333
$342$	−0.0173798	−0.000939792 0
$343$	0	0
$344$	8.68348	0.468182
$345$	32.6612	1.75842
$346$	2.84709	0.153061
$347$	0.0256912	0.00137917	0.000689587	−	1.00000i	$-0.499780\pi$
$347$	0.0256912	0.00137917	0.000689587		1.00000i	$0.499780\pi$
$348$	10.0555	0.539031
$349$	−23.9479	−1.28190	−0.640950	−	0.767583i	$-0.721459\pi$
$349$	−23.9479	−1.28190	−0.640950	+	0.767583i	$0.721459\pi$
$350$	0	0
$351$	8.14316	0.434650
$352$	−11.8156	−0.629775
$353$	6.33830	0.337354	0.168677	−	0.985671i	$-0.446051\pi$
$353$	6.33830	0.337354	0.168677	+	0.985671i	$0.446051\pi$
$354$	4.42332	0.235097
$355$	40.4409	2.14638
$356$	−4.96766	−0.263285
$357$	0	0
$358$	−1.99313	−0.105340
$359$	−10.9811	−0.579563	−0.289781	−	0.957093i	$-0.593583\pi$
$359$	−10.9811	−0.579563	−0.289781	+	0.957093i	$0.593583\pi$
$360$	7.22099	0.380579
$361$	−18.9986	−0.999926
$362$	1.17041	0.0615155
$363$	38.9896	2.04642
$364$	0	0
$365$	−13.2478	−0.693421
$366$	0.170489	0.00891160
$367$	−21.0669	−1.09968	−0.549840	−	0.835270i	$-0.685311\pi$
$367$	−21.0669	−1.09968	−0.549840	+	0.835270i	$0.685311\pi$
$368$	−13.5206	−0.704809
$369$	22.9934	1.19699
$370$	6.99759	0.363787
$371$	0	0
$372$	18.9712	0.983612
$373$	10.6031	0.549009	0.274505	−	0.961586i	$-0.411486\pi$
$373$	10.6031	0.549009	0.274505	+	0.961586i	$0.411486\pi$
$374$	−4.16127	−0.215174
$375$	−49.1304	−2.53708
$376$	−2.94024	−0.151631
$377$	13.3626	0.688211
$378$	0	0
$379$	−5.57679	−0.286460	−0.143230	−	0.989689i	$-0.545749\pi$
$379$	−5.57679	−0.286460	−0.143230	+	0.989689i	$0.545749\pi$
$380$	−0.288015	−0.0147748
$381$	−2.32908	−0.119322
$382$	1.66312	0.0850928
$383$	−19.7345	−1.00838	−0.504192	−	0.863592i	$-0.668209\pi$
$383$	−19.7345	−1.00838	−0.504192	+	0.863592i	$0.668209\pi$
$384$	−13.6330	−0.695708
$385$	0	0
$386$	−0.820958	−0.0417857
$387$	−27.7236	−1.40927
$388$	−11.4917	−0.583403
$389$	16.1098	0.816799	0.408399	−	0.912803i	$-0.366087\pi$
$389$	16.1098	0.816799	0.408399	+	0.912803i	$0.366087\pi$
$390$	10.6350	0.538525
$391$	−14.7442	−0.745645
$392$	0	0
$393$	36.5677	1.84460
$394$	3.99981	0.201507
$395$	−38.1705	−1.92057
$396$	25.0717	1.25990
$397$	6.80662	0.341614	0.170807	−	0.985304i	$-0.445362\pi$
$397$	6.80662	0.341614	0.170807	+	0.985304i	$0.445362\pi$
$398$	−4.46833	−0.223977
$399$	0	0
$400$	39.2435	1.96217
$401$	−17.1989	−0.858873	−0.429437	−	0.903097i	$-0.641288\pi$
$401$	−17.1989	−0.858873	−0.429437	+	0.903097i	$0.641288\pi$
$402$	1.96634	0.0980720
$403$	25.2107	1.25583
$404$	−3.50273	−0.174267
$405$	40.7654	2.02565
$406$	0	0
$407$	49.0464	2.43114
$408$	−7.29306	−0.361060
$409$	−3.13066	−0.154801	−0.0774005	−	0.997000i	$-0.524662\pi$
$409$	−3.13066	−0.154801	−0.0774005	+	0.997000i	$0.524662\pi$
$410$	−7.12615	−0.351936
$411$	18.5724	0.916111
$412$	−16.0412	−0.790291
$413$	0	0
$414$	−1.66135	−0.0816509
$415$	64.8071	3.18126
$416$	−13.6320	−0.668362
$417$	7.93922	0.388785
$418$	0.0377534	0.00184658
$419$	−18.8437	−0.920576	−0.460288	−	0.887770i	$-0.652254\pi$
$419$	−18.8437	−0.920576	−0.460288	+	0.887770i	$0.652254\pi$
$420$	0	0
$421$	−13.1531	−0.641045	−0.320523	−	0.947241i	$-0.603859\pi$
$421$	−13.1531	−0.641045	−0.320523	+	0.947241i	$0.603859\pi$
$422$	−3.31052	−0.161153
$423$	9.38727	0.456425
$424$	7.57572	0.367909
$425$	42.7949	2.07586
$426$	−4.60230	−0.222982
$427$	0	0
$428$	−6.33663	−0.306293
$429$	74.5414	3.59889
$430$	8.59217	0.414351
$431$	−38.5459	−1.85669	−0.928346	−	0.371716i	$-0.878769\pi$
$431$	−38.5459	−1.85669	−0.928346	+	0.371716i	$0.878769\pi$
$432$	−5.06697	−0.243785
$433$	−27.5890	−1.32584	−0.662922	−	0.748689i	$-0.730684\pi$
$433$	−27.5890	−1.32584	−0.662922	+	0.748689i	$0.730684\pi$
$434$	0	0
$435$	20.0856	0.963029
$436$	−33.1976	−1.58988
$437$	0.133768	0.00639897
$438$	1.50764	0.0720378
$439$	18.5614	0.885887	0.442943	−	0.896550i	$-0.353934\pi$
$439$	18.5614	0.885887	0.442943	+	0.896550i	$0.353934\pi$
$440$	−15.6858	−0.747793
$441$	0	0
$442$	−4.80094	−0.228358
$443$	4.02148	0.191066	0.0955332	−	0.995426i	$-0.469544\pi$
$443$	4.02148	0.191066	0.0955332	+	0.995426i	$0.469544\pi$
$444$	42.5814	2.02082
$445$	−9.92277	−0.470384
$446$	−5.14008	−0.243390
$447$	−5.47149	−0.258793
$448$	0	0
$449$	−25.0013	−1.17989	−0.589943	−	0.807445i	$-0.700850\pi$
$449$	−25.0013	−1.17989	−0.589943	+	0.807445i	$0.700850\pi$
$450$	4.82206	0.227314
$451$	−49.9475	−2.35194
$452$	−10.9667	−0.515829
$453$	−7.98647	−0.375237
$454$	−3.50312	−0.164410
$455$	0	0
$456$	0.0661669	0.00309855
$457$	−10.2474	−0.479354	−0.239677	−	0.970853i	$-0.577042\pi$
$457$	−10.2474	−0.479354	−0.239677	+	0.970853i	$0.577042\pi$
$458$	−0.818826	−0.0382612
$459$	−5.52552	−0.257909
$460$	−27.5315	−1.28366
$461$	−10.5197	−0.489951	−0.244975	−	0.969529i	$-0.578780\pi$
$461$	−10.5197	−0.489951	−0.244975	+	0.969529i	$0.578780\pi$
$462$	0	0
$463$	25.2692	1.17436	0.587179	−	0.809457i	$-0.300238\pi$
$463$	25.2692	1.17436	0.587179	+	0.809457i	$0.300238\pi$
$464$	−8.31471	−0.386001
$465$	37.8945	1.75732
$466$	−0.188175	−0.00871705
$467$	−4.08966	−0.189247	−0.0946234	−	0.995513i	$-0.530165\pi$
$467$	−4.08966	−0.189247	−0.0946234	+	0.995513i	$0.530165\pi$
$468$	28.9257	1.33709
$469$	0	0
$470$	−2.90932	−0.134197
$471$	46.3364	2.13507
$472$	−7.52694	−0.346455
$473$	60.2229	2.76905
$474$	4.34392	0.199523
$475$	−0.388260	−0.0178146
$476$	0	0
$477$	−24.1869	−1.10744
$478$	−1.70294	−0.0778908
$479$	36.7606	1.67963	0.839817	−	0.542869i	$-0.182662\pi$
$479$	36.7606	1.67963	0.839817	+	0.542869i	$0.182662\pi$
$480$	−20.4904	−0.935254
$481$	56.5859	2.58010
$482$	5.26309	0.239727
$483$	0	0
$484$	−32.8660	−1.49391
$485$	−22.9544	−1.04230
$486$	−3.86887	−0.175495
$487$	−33.7554	−1.52960	−0.764802	−	0.644265i	$-0.777163\pi$
$487$	−33.7554	−1.52960	−0.764802	+	0.644265i	$0.777163\pi$
$488$	−0.290113	−0.0131328
$489$	15.5695	0.704076
$490$	0	0
$491$	2.85086	0.128657	0.0643286	−	0.997929i	$-0.479509\pi$
$491$	2.85086	0.128657	0.0643286	+	0.997929i	$0.479509\pi$
$492$	−43.3637	−1.95499
$493$	−9.06718	−0.408365
$494$	0.0435569	0.00195972
$495$	50.0800	2.25093
$496$	−15.6870	−0.704366
$497$	0	0
$498$	−7.37525	−0.330493
$499$	−14.9560	−0.669524	−0.334762	−	0.942303i	$-0.608656\pi$
$499$	−14.9560	−0.669524	−0.334762	+	0.942303i	$0.608656\pi$
$500$	41.4141	1.85209
$501$	−37.3296	−1.66776
$502$	1.46753	0.0654992
$503$	8.56487	0.381889	0.190944	−	0.981601i	$-0.438845\pi$
$503$	8.56487	0.381889	0.190944	+	0.981601i	$0.438845\pi$
$504$	0	0
$505$	−6.99661	−0.311345
$506$	3.60888	0.160434
$507$	55.7219	2.47470
$508$	1.96328	0.0871066
$509$	−24.3428	−1.07898	−0.539489	−	0.841993i	$-0.681382\pi$
$509$	−24.3428	−1.07898	−0.539489	+	0.841993i	$0.681382\pi$
$510$	−7.21637	−0.319546
$511$	0	0
$512$	14.2252	0.628670
$513$	0.0501307	0.00221333
$514$	3.70692	0.163505
$515$	−32.0418	−1.41193
$516$	52.2846	2.30170
$517$	−20.3916	−0.896820
$518$	0	0
$519$	34.6061	1.51904
$520$	−18.0971	−0.793610
$521$	33.9526	1.48749	0.743745	−	0.668464i	$-0.233048\pi$
$521$	33.9526	1.48749	0.743745	+	0.668464i	$0.233048\pi$
$522$	−1.02167	−0.0447175
$523$	−33.7243	−1.47466	−0.737331	−	0.675532i	$-0.763914\pi$
$523$	−33.7243	−1.47466	−0.737331	+	0.675532i	$0.763914\pi$
$524$	−30.8245	−1.34657
$525$	0	0
$526$	2.44144	0.106452
$527$	−17.1066	−0.745176
$528$	−46.3823	−2.01853
$529$	−10.2131	−0.444046
$530$	7.49605	0.325608
$531$	24.0312	1.04286
$532$	0	0
$533$	−57.6255	−2.49604
$534$	1.12924	0.0488670
$535$	−12.6573	−0.547221
$536$	−3.34602	−0.144526
$537$	−24.2263	−1.04544
$538$	−1.73022	−0.0745951
$539$	0	0
$540$	−10.3177	−0.444004
$541$	−22.8612	−0.982879	−0.491439	−	0.870912i	$-0.663529\pi$
$541$	−22.8612	−0.982879	−0.491439	+	0.870912i	$0.663529\pi$
$542$	−3.18171	−0.136666
$543$	14.2263	0.610508
$544$	9.24992	0.396587
$545$	−66.3113	−2.84046
$546$	0	0
$547$	−30.4179	−1.30058	−0.650288	−	0.759688i	$-0.725352\pi$
$547$	−30.4179	−1.30058	−0.650288	+	0.759688i	$0.725352\pi$
$548$	−15.6555	−0.668770
$549$	0.926239	0.0395309
$550$	−10.4748	−0.446645
$551$	0.0822627	0.00350451
$552$	6.32494	0.269207
$553$	0	0
$554$	−1.49291	−0.0634275
$555$	85.0551	3.61039
$556$	−6.69230	−0.283817
$557$	24.3671	1.03247	0.516233	−	0.856448i	$-0.327334\pi$
$557$	24.3671	1.03247	0.516233	+	0.856448i	$0.327334\pi$
$558$	−1.92755	−0.0815995
$559$	69.4804	2.93871
$560$	0	0
$561$	−50.5798	−2.13548
$562$	−1.71428	−0.0723127
$563$	39.1345	1.64932	0.824661	−	0.565627i	$-0.191366\pi$
$563$	39.1345	1.64932	0.824661	+	0.565627i	$0.191366\pi$
$564$	−17.7036	−0.745458
$565$	−21.9056	−0.921577
$566$	2.20565	0.0927103
$567$	0	0
$568$	7.83151	0.328603
$569$	41.9315	1.75786	0.878930	−	0.476951i	$-0.158258\pi$
$569$	41.9315	1.75786	0.878930	+	0.476951i	$0.158258\pi$
$570$	0.0654711	0.00274228
$571$	1.63409	0.0683845	0.0341923	−	0.999415i	$-0.489114\pi$
$571$	1.63409	0.0683845	0.0341923	+	0.999415i	$0.489114\pi$
$572$	−62.8341	−2.62723
$573$	20.2151	0.844499
$574$	0	0
$575$	−37.1141	−1.54776
$576$	−17.2930	−0.720540
$577$	3.28569	0.136785	0.0683925	−	0.997658i	$-0.478213\pi$
$577$	3.28569	0.136785	0.0683925	+	0.997658i	$0.478213\pi$
$578$	0.000185428 0	7.71277e−6 0
$579$	−9.97867	−0.414700
$580$	−16.9310	−0.703021
$581$	0	0
$582$	2.61228	0.108282
$583$	52.5402	2.17599
$584$	−2.56547	−0.106160
$585$	57.7784	2.38884
$586$	0.845997	0.0349478
$587$	37.3079	1.53986	0.769931	−	0.638127i	$-0.220291\pi$
$587$	37.3079	1.53986	0.769931	+	0.638127i	$0.220291\pi$
$588$	0	0
$589$	0.155201	0.00639495
$590$	−7.44779	−0.306621
$591$	48.6173	1.99985
$592$	−35.2098	−1.44711
$593$	10.6325	0.436626	0.218313	−	0.975879i	$-0.429945\pi$
$593$	10.6325	0.436626	0.218313	+	0.975879i	$0.429945\pi$
$594$	1.35246	0.0554922
$595$	0	0
$596$	4.61215	0.188921
$597$	−54.3122	−2.22285
$598$	4.16364	0.170264
$599$	42.0174	1.71678	0.858391	−	0.512995i	$-0.171464\pi$
$599$	42.0174	1.71678	0.858391	+	0.512995i	$0.171464\pi$
$600$	−18.3581	−0.749467
$601$	16.1101	0.657144	0.328572	−	0.944479i	$-0.393433\pi$
$601$	16.1101	0.657144	0.328572	+	0.944479i	$0.393433\pi$
$602$	0	0
$603$	10.6828	0.435037
$604$	6.73214	0.273927
$605$	−65.6489	−2.66901
$606$	0.796236	0.0323449
$607$	14.1660	0.574979	0.287489	−	0.957784i	$-0.407179\pi$
$607$	14.1660	0.574979	0.287489	+	0.957784i	$0.407179\pi$
$608$	−0.0839207	−0.00340343
$609$	0	0
$610$	−0.287062	−0.0116228
$611$	−23.5262	−0.951768
$612$	−19.6275	−0.793393
$613$	17.1420	0.692361	0.346180	−	0.938168i	$-0.387479\pi$
$613$	17.1420	0.692361	0.346180	+	0.938168i	$0.387479\pi$
$614$	−0.925335	−0.0373435
$615$	−86.6178	−3.49277
$616$	0	0
$617$	30.3241	1.22080	0.610401	−	0.792092i	$-0.291008\pi$
$617$	30.3241	1.22080	0.610401	+	0.792092i	$0.291008\pi$
$618$	3.64645	0.146682
$619$	9.91575	0.398547	0.199274	−	0.979944i	$-0.436142\pi$
$619$	9.91575	0.398547	0.199274	+	0.979944i	$0.436142\pi$
$620$	−31.9429	−1.28286
$621$	4.79203	0.192298
$622$	3.02023	0.121100
$623$	0	0
$624$	−53.5123	−2.14221
$625$	30.8286	1.23314
$626$	−2.54033	−0.101532
$627$	0.458890	0.0183263
$628$	−39.0590	−1.55862
$629$	−38.3962	−1.53096
$630$	0	0
$631$	−4.94297	−0.196777	−0.0983884	−	0.995148i	$-0.531369\pi$
$631$	−4.94297	−0.196777	−0.0983884	+	0.995148i	$0.531369\pi$
$632$	−7.39183	−0.294031
$633$	−40.2390	−1.59936
$634$	0.994685	0.0395040
$635$	3.92161	0.155624
$636$	45.6146	1.80873
$637$	0	0
$638$	2.21934	0.0878645
$639$	−25.0035	−0.989125
$640$	22.9547	0.907365
$641$	8.92412	0.352481	0.176241	−	0.984347i	$-0.443606\pi$
$641$	8.92412	0.352481	0.176241	+	0.984347i	$0.443606\pi$
$642$	1.44043	0.0568494
$643$	−8.44158	−0.332903	−0.166452	−	0.986050i	$-0.553231\pi$
$643$	−8.44158	−0.332903	−0.166452	+	0.986050i	$0.553231\pi$
$644$	0	0
$645$	104.437	4.11220
$646$	−0.0295554	−0.00116284
$647$	−50.2006	−1.97359	−0.986795	−	0.161976i	$-0.948213\pi$
$647$	−50.2006	−1.97359	−0.986795	+	0.161976i	$0.948213\pi$
$648$	7.89433	0.310119
$649$	−52.2019	−2.04910
$650$	−12.0850	−0.474011
$651$	0	0
$652$	−13.1242	−0.513982
$653$	40.5316	1.58612	0.793062	−	0.609141i	$-0.208486\pi$
$653$	40.5316	1.58612	0.793062	+	0.609141i	$0.208486\pi$
$654$	7.54643	0.295089
$655$	−61.5711	−2.40578
$656$	35.8567	1.39997
$657$	8.19076	0.319552
$658$	0	0
$659$	41.9115	1.63264	0.816320	−	0.577600i	$-0.196011\pi$
$659$	41.9115	1.63264	0.816320	+	0.577600i	$0.196011\pi$
$660$	−94.4468	−3.67634
$661$	−27.5407	−1.07121	−0.535605	−	0.844469i	$-0.679916\pi$
$661$	−27.5407	−1.07121	−0.535605	+	0.844469i	$0.679916\pi$
$662$	−2.14388	−0.0833242
$663$	−58.3550	−2.26632
$664$	12.5501	0.487038
$665$	0	0
$666$	−4.32642	−0.167646
$667$	7.86355	0.304478
$668$	31.4667	1.21748
$669$	−62.4772	−2.41551
$670$	−3.31083	−0.127909
$671$	−2.01203	−0.0776735
$672$	0	0
$673$	6.81901	0.262853	0.131427	−	0.991326i	$-0.458044\pi$
$673$	6.81901	0.262853	0.131427	+	0.991326i	$0.458044\pi$
$674$	3.28623	0.126581
$675$	−13.9089	−0.535352
$676$	−46.9704	−1.80655
$677$	6.66448	0.256137	0.128068	−	0.991765i	$-0.459122\pi$
$677$	6.66448	0.256137	0.128068	+	0.991765i	$0.459122\pi$
$678$	2.49293	0.0957403
$679$	0	0
$680$	12.2797	0.470906
$681$	−42.5801	−1.63167
$682$	4.18713	0.160333
$683$	23.2496	0.889623	0.444811	−	0.895624i	$-0.353271\pi$
$683$	23.2496	0.889623	0.444811	+	0.895624i	$0.353271\pi$
$684$	0.178072	0.00680874
$685$	−31.2715	−1.19482
$686$	0	0
$687$	−9.95276	−0.379722
$688$	−43.2332	−1.64825
$689$	60.6167	2.30931
$690$	6.25843	0.238254
$691$	−7.85702	−0.298895	−0.149447	−	0.988770i	$-0.547749\pi$
$691$	−7.85702	−0.298895	−0.149447	+	0.988770i	$0.547749\pi$
$692$	−29.1710	−1.10892
$693$	0	0
$694$	0.00492285	0.000186869 0
$695$	−13.3677	−0.507066
$696$	3.88963	0.147436
$697$	39.1016	1.48108
$698$	−4.58881	−0.173689
$699$	−2.28725	−0.0865119
$700$	0	0
$701$	−0.243911	−0.00921239	−0.00460619	−	0.999989i	$-0.501466\pi$
$701$	−0.243911	−0.00921239	−0.00460619	+	0.999989i	$0.501466\pi$
$702$	1.56036	0.0588922
$703$	0.348353	0.0131384
$704$	37.5648	1.41578
$705$	−35.3626	−1.33183
$706$	1.21452	0.0457092
$707$	0	0
$708$	−45.3208	−1.70326
$709$	27.6428	1.03815	0.519073	−	0.854730i	$-0.326277\pi$
$709$	27.6428	1.03815	0.519073	+	0.854730i	$0.326277\pi$
$710$	7.74915	0.290820
$711$	23.5998	0.885062
$712$	−1.92157	−0.0720140
$713$	14.8358	0.555605
$714$	0	0
$715$	−125.509	−4.69379
$716$	20.4214	0.763183
$717$	−20.6991	−0.773023
$718$	−2.10417	−0.0785269
$719$	−13.2320	−0.493470	−0.246735	−	0.969083i	$-0.579358\pi$
$719$	−13.2320	−0.493470	−0.246735	+	0.969083i	$0.579358\pi$
$720$	−35.9518	−1.33984
$721$	0	0
$722$	−3.64045	−0.135483
$723$	63.9724	2.37916
$724$	−11.9919	−0.445677
$725$	−22.8239	−0.847660
$726$	7.47105	0.277277
$727$	19.0077	0.704958	0.352479	−	0.935820i	$-0.385339\pi$
$727$	19.0077	0.704958	0.352479	+	0.935820i	$0.385339\pi$
$728$	0	0
$729$	−15.8405	−0.586687
$730$	−2.53850	−0.0939540
$731$	−47.1458	−1.74375
$732$	−1.74681	−0.0645640
$733$	49.8815	1.84242	0.921208	−	0.389071i	$-0.127204\pi$
$733$	49.8815	1.84242	0.921208	+	0.389071i	$0.127204\pi$
$734$	−4.03676	−0.148999
$735$	0	0
$736$	−8.02204	−0.295696
$737$	−23.2058	−0.854796
$738$	4.40591	0.162184
$739$	−29.8966	−1.09976	−0.549882	−	0.835242i	$-0.685327\pi$
$739$	−29.8966	−1.09976	−0.549882	+	0.835242i	$0.685327\pi$
$740$	−71.6966	−2.63562
$741$	0.529431	0.0194491
$742$	0	0
$743$	7.33058	0.268933	0.134467	−	0.990918i	$-0.457068\pi$
$743$	7.33058	0.268933	0.134467	+	0.990918i	$0.457068\pi$
$744$	7.33838	0.269038
$745$	9.21265	0.337525
$746$	2.03174	0.0743871
$747$	−40.0685	−1.46603
$748$	42.6359	1.55892
$749$	0	0
$750$	−9.41419	−0.343758
$751$	−37.8737	−1.38203	−0.691015	−	0.722840i	$-0.742836\pi$
$751$	−37.8737	−1.38203	−0.691015	+	0.722840i	$0.742836\pi$
$752$	14.6389	0.533824
$753$	17.8377	0.650043
$754$	2.56050	0.0932480
$755$	13.4473	0.489396
$756$	0	0
$757$	22.7557	0.827070	0.413535	−	0.910488i	$-0.364294\pi$
$757$	22.7557	0.827070	0.413535	+	0.910488i	$0.364294\pi$
$758$	−1.06860	−0.0388135
$759$	43.8656	1.59222
$760$	−0.111409	−0.00404122
$761$	−7.34081	−0.266104	−0.133052	−	0.991109i	$-0.542478\pi$
$761$	−7.34081	−0.266104	−0.133052	+	0.991109i	$0.542478\pi$
$762$	−0.446291	−0.0161674
$763$	0	0
$764$	−17.0402	−0.616493
$765$	−39.2053	−1.41747
$766$	−3.78145	−0.136629
$767$	−60.2264	−2.17465
$768$	30.6108	1.10457
$769$	39.6967	1.43150	0.715749	−	0.698358i	$-0.246085\pi$
$769$	39.6967	1.43150	0.715749	+	0.698358i	$0.246085\pi$
$770$	0	0
$771$	45.0573	1.62270
$772$	8.41145	0.302735
$773$	9.92476	0.356969	0.178484	−	0.983943i	$-0.442881\pi$
$773$	9.92476	0.356969	0.178484	+	0.983943i	$0.442881\pi$
$774$	−5.31231	−0.190947
$775$	−43.0609	−1.54679
$776$	−4.44518	−0.159573
$777$	0	0
$778$	3.08690	0.110671
$779$	−0.354753	−0.0127103
$780$	−108.965	−3.90159
$781$	54.3141	1.94351
$782$	−2.82523	−0.101030
$783$	2.94694	0.105315
$784$	0	0
$785$	−78.0193	−2.78463
$786$	7.00698	0.249931
$787$	−17.2831	−0.616077	−0.308038	−	0.951374i	$-0.599673\pi$
$787$	−17.2831	−0.616077	−0.308038	+	0.951374i	$0.599673\pi$
$788$	−40.9816	−1.45991
$789$	29.6754	1.05647
$790$	−7.31410	−0.260224
$791$	0	0
$792$	9.69814	0.344608
$793$	−2.32132	−0.0824325
$794$	1.30426	0.0462865
$795$	91.1139	3.23148
$796$	45.7821	1.62270
$797$	−3.27024	−0.115838	−0.0579190	−	0.998321i	$-0.518447\pi$
$797$	−3.27024	−0.115838	−0.0579190	+	0.998321i	$0.518447\pi$
$798$	0	0
$799$	15.9636	0.564753
$800$	23.2839	0.823212
$801$	6.13498	0.216769
$802$	−3.29560	−0.116372
$803$	−17.7924	−0.627881
$804$	−20.1469	−0.710526
$805$	0	0
$806$	4.83078	0.170157
$807$	−21.0307	−0.740315
$808$	−1.35491	−0.0476657
$809$	−45.7392	−1.60810	−0.804052	−	0.594558i	$-0.797327\pi$
$809$	−45.7392	−1.60810	−0.804052	+	0.594558i	$0.797327\pi$
$810$	7.81132	0.274462
$811$	−20.1033	−0.705923	−0.352962	−	0.935638i	$-0.614825\pi$
$811$	−20.1033	−0.705923	−0.352962	+	0.935638i	$0.614825\pi$
$812$	0	0
$813$	−38.6734	−1.35633
$814$	9.39811	0.329403
$815$	−26.2152	−0.918278
$816$	36.3106	1.27113
$817$	0.427734	0.0149645
$818$	−0.599886	−0.0209745
$819$	0	0
$820$	73.0138	2.54975
$821$	−19.1738	−0.669170	−0.334585	−	0.942366i	$-0.608596\pi$
$821$	−19.1738	−0.669170	−0.334585	+	0.942366i	$0.608596\pi$
$822$	3.55879	0.124127
$823$	8.70002	0.303264	0.151632	−	0.988437i	$-0.451547\pi$
$823$	8.70002	0.303264	0.151632	+	0.988437i	$0.451547\pi$
$824$	−6.20499	−0.216161
$825$	−127.320	−4.43271
$826$	0	0
$827$	11.0112	0.382898	0.191449	−	0.981503i	$-0.438681\pi$
$827$	11.0112	0.382898	0.191449	+	0.981503i	$0.438681\pi$
$828$	17.0220	0.591556
$829$	−40.0764	−1.39191	−0.695955	−	0.718086i	$-0.745019\pi$
$829$	−40.0764	−1.39191	−0.695955	+	0.718086i	$0.745019\pi$
$830$	12.4181	0.431039
$831$	−18.1462	−0.629483
$832$	43.3393	1.50252
$833$	0	0
$834$	1.52128	0.0526778
$835$	62.8539	2.17515
$836$	−0.386818	−0.0133784
$837$	5.55986	0.192177
$838$	−3.61077	−0.124732
$839$	5.37182	0.185456	0.0927279	−	0.995691i	$-0.470441\pi$
$839$	5.37182	0.185456	0.0927279	+	0.995691i	$0.470441\pi$
$840$	0	0
$841$	−24.1642	−0.833247
$842$	−2.52036	−0.0868574
$843$	−20.8370	−0.717664
$844$	33.9192	1.16755
$845$	−93.8221	−3.22758
$846$	1.79876	0.0618425
$847$	0	0
$848$	−37.7179	−1.29524
$849$	26.8095	0.920099
$850$	8.20021	0.281265
$851$	33.2993	1.14149
$852$	47.1547	1.61549
$853$	−37.5087	−1.28427	−0.642137	−	0.766590i	$-0.721952\pi$
$853$	−37.5087	−1.28427	−0.642137	+	0.766590i	$0.721952\pi$
$854$	0	0
$855$	0.355694	0.0121645
$856$	−2.45111	−0.0837774
$857$	−17.2234	−0.588339	−0.294169	−	0.955753i	$-0.595043\pi$
$857$	−17.2234	−0.588339	−0.294169	+	0.955753i	$0.595043\pi$
$858$	14.2834	0.487626
$859$	26.0045	0.887262	0.443631	−	0.896210i	$-0.353690\pi$
$859$	26.0045	0.887262	0.443631	+	0.896210i	$0.353690\pi$
$860$	−88.0345	−3.00195
$861$	0	0
$862$	−7.38604	−0.251569
$863$	32.4210	1.10362	0.551812	−	0.833969i	$-0.313937\pi$
$863$	32.4210	1.10362	0.551812	+	0.833969i	$0.313937\pi$
$864$	−3.00634	−0.102278
$865$	−58.2683	−1.98118
$866$	−5.28651	−0.179643
$867$	0.00225386	7.65450e−5 0
$868$	0	0
$869$	−51.2649	−1.73904
$870$	3.84873	0.130484
$871$	−26.7730	−0.907169
$872$	−12.8414	−0.434864
$873$	14.1921	0.480329
$874$	0.0256321	0.000867018 0
$875$	0	0
$876$	−15.4471	−0.521910
$877$	31.8805	1.07653	0.538264	−	0.842776i	$-0.319080\pi$
$877$	31.8805	1.07653	0.538264	+	0.842776i	$0.319080\pi$
$878$	3.55667	0.120032
$879$	10.2830	0.346838
$880$	78.0965	2.63263
$881$	10.9692	0.369563	0.184782	−	0.982780i	$-0.440842\pi$
$881$	10.9692	0.369563	0.184782	+	0.982780i	$0.440842\pi$
$882$	0	0
$883$	−43.5368	−1.46513	−0.732564	−	0.680698i	$-0.761677\pi$
$883$	−43.5368	−1.46513	−0.732564	+	0.680698i	$0.761677\pi$
$884$	49.1900	1.65444
$885$	−90.5272	−3.04304
$886$	0.770582	0.0258882
$887$	12.9767	0.435715	0.217857	−	0.975981i	$-0.430093\pi$
$887$	12.9767	0.435715	0.217857	+	0.975981i	$0.430093\pi$
$888$	16.4712	0.552736
$889$	0	0
$890$	−1.90137	−0.0637339
$891$	54.7499	1.83419
$892$	52.6647	1.76335
$893$	−0.144831	−0.00484660
$894$	−1.04843	−0.0350647
$895$	40.7912	1.36350
$896$	0	0
$897$	50.6087	1.68977
$898$	−4.79067	−0.159867
$899$	9.12353	0.304287
$900$	−49.4063	−1.64688
$901$	−41.1313	−1.37028
$902$	−9.57077	−0.318672
$903$	0	0
$904$	−4.24209	−0.141090
$905$	−23.9536	−0.796243
$906$	−1.53034	−0.0508421
$907$	−2.23113	−0.0740833	−0.0370417	−	0.999314i	$-0.511793\pi$
$907$	−2.23113	−0.0740833	−0.0370417	+	0.999314i	$0.511793\pi$
$908$	35.8926	1.19114
$909$	4.32582	0.143478
$910$	0	0
$911$	29.4118	0.974457	0.487228	−	0.873275i	$-0.338008\pi$
$911$	29.4118	0.974457	0.487228	+	0.873275i	$0.338008\pi$
$912$	−0.329431	−0.0109086
$913$	87.0391	2.88058
$914$	−1.96358	−0.0649493
$915$	−3.48921	−0.115350
$916$	8.38961	0.277201
$917$	0	0
$918$	−1.05878	−0.0349450
$919$	3.92759	0.129559	0.0647796	−	0.997900i	$-0.479366\pi$
$919$	3.92759	0.129559	0.0647796	+	0.997900i	$0.479366\pi$
$920$	−10.6497	−0.351109
$921$	−11.2474	−0.370613
$922$	−2.01575	−0.0663851
$923$	62.6634	2.06259
$924$	0	0
$925$	−96.6511	−3.17787
$926$	4.84199	0.159118
$927$	19.8106	0.650665
$928$	−4.93329	−0.161943
$929$	−26.5871	−0.872293	−0.436147	−	0.899876i	$-0.643657\pi$
$929$	−26.5871	−0.872293	−0.436147	+	0.899876i	$0.643657\pi$
$930$	7.26121	0.238105
$931$	0	0
$932$	1.92802	0.0631546
$933$	36.7106	1.20185
$934$	−0.783646	−0.0256417
$935$	85.1641	2.78516
$936$	11.1889	0.365722
$937$	46.5758	1.52156	0.760782	−	0.649007i	$-0.224815\pi$
$937$	46.5758	1.52156	0.760782	+	0.649007i	$0.224815\pi$
$938$	0	0
$939$	−30.8774	−1.00765
$940$	29.8086	0.972250
$941$	3.71923	0.121243	0.0606217	−	0.998161i	$-0.480692\pi$
$941$	3.71923	0.121243	0.0606217	+	0.998161i	$0.480692\pi$
$942$	8.87883	0.289288
$943$	−33.9111	−1.10430
$944$	37.4751	1.21971
$945$	0	0
$946$	11.5397	0.375188
$947$	−1.23688	−0.0401934	−0.0200967	−	0.999798i	$-0.506397\pi$
$947$	−1.23688	−0.0401934	−0.0200967	+	0.999798i	$0.506397\pi$
$948$	−44.5074	−1.44553
$949$	−20.5275	−0.666351
$950$	−0.0743971	−0.00241376
$951$	12.0903	0.392055
$952$	0	0
$953$	−52.2025	−1.69101	−0.845503	−	0.533971i	$-0.820699\pi$
$953$	−52.2025	−1.69101	−0.845503	+	0.533971i	$0.820699\pi$
$954$	−4.63461	−0.150051
$955$	−34.0373	−1.10142
$956$	17.4482	0.564314
$957$	26.9759	0.872007
$958$	7.04394	0.227579
$959$	0	0
$960$	65.1439	2.10251
$961$	−13.7871	−0.444744
$962$	10.8428	0.349586
$963$	7.82564	0.252178
$964$	−53.9251	−1.73681
$965$	16.8017	0.540864
$966$	0	0
$967$	25.3043	0.813731	0.406866	−	0.913488i	$-0.366622\pi$
$967$	25.3043	0.813731	0.406866	+	0.913488i	$0.366622\pi$
$968$	−12.7131	−0.408615
$969$	−0.359244	−0.0115406
$970$	−4.39844	−0.141225
$971$	−4.96055	−0.159192	−0.0795958	−	0.996827i	$-0.525363\pi$
$971$	−4.96055	−0.159192	−0.0795958	+	0.996827i	$0.525363\pi$
$972$	39.6400	1.27145
$973$	0	0
$974$	−6.46810	−0.207251
$975$	−146.892	−4.70430
$976$	1.44441	0.0462344
$977$	−3.61365	−0.115611	−0.0578055	−	0.998328i	$-0.518410\pi$
$977$	−3.61365	−0.115611	−0.0578055	+	0.998328i	$0.518410\pi$
$978$	2.98337	0.0953976
$979$	−13.3268	−0.425925
$980$	0	0
$981$	40.9985	1.30898
$982$	0.546271	0.0174322
$983$	51.8644	1.65422	0.827109	−	0.562042i	$-0.189984\pi$
$983$	51.8644	1.65422	0.827109	+	0.562042i	$0.189984\pi$
$984$	−16.7738	−0.534729
$985$	−81.8597	−2.60827
$986$	−1.73742	−0.0553308
$987$	0	0
$988$	−0.446280	−0.0141981
$989$	40.8874	1.30014
$990$	9.59615	0.304986
$991$	6.60298	0.209751	0.104875	−	0.994485i	$-0.466556\pi$
$991$	6.60298	0.209751	0.104875	+	0.994485i	$0.466556\pi$
$992$	−9.30741	−0.295511
$993$	−26.0587	−0.826947
$994$	0	0
$995$	91.4485	2.89911
$996$	75.5660	2.39440
$997$	0.888839	0.0281498	0.0140749	−	0.999901i	$-0.495520\pi$
$997$	0.888839	0.0281498	0.0140749	+	0.999901i	$0.495520\pi$
$998$	−2.86582	−0.0907160
$999$	12.4792	0.394826

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6223.2.a.k.1.10		16
7.6	odd	2		889.2.a.c.1.10	✓	16
21.20	even	2		8001.2.a.t.1.7		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.c.1.10	✓	16	7.6	odd	2
6223.2.a.k.1.10		16	1.1	even	1	trivial
8001.2.a.t.1.7		16	21.20	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6223 = 7^{2} \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6223.a (trivial)

\(f(q)\)	\(=\)	\(q+0.191617 q^{2} +2.32908 q^{3} -1.96328 q^{4} -3.92161 q^{5} +0.446291 q^{6} -0.759431 q^{8} +2.42462 q^{9} -0.751445 q^{10} -5.26691 q^{11} -4.57265 q^{12} -6.07655 q^{13} -9.13374 q^{15} +3.78105 q^{16} +4.12322 q^{17} +0.464598 q^{18} -0.0374083 q^{19} +7.69922 q^{20} -1.00923 q^{22} -3.57588 q^{23} -1.76878 q^{24} +10.3790 q^{25} -1.16437 q^{26} -1.34010 q^{27} -2.19905 q^{29} -1.75018 q^{30} -4.14885 q^{31} +2.24337 q^{32} -12.2671 q^{33} +0.790078 q^{34} -4.76022 q^{36} -9.31219 q^{37} -0.00716804 q^{38} -14.1528 q^{39} +2.97819 q^{40} +9.48327 q^{41} -11.4342 q^{43} +10.3404 q^{44} -9.50842 q^{45} -0.685199 q^{46} +3.87164 q^{47} +8.80637 q^{48} +1.98879 q^{50} +9.60333 q^{51} +11.9300 q^{52} -9.97552 q^{53} -0.256785 q^{54} +20.6547 q^{55} -0.0871269 q^{57} -0.421374 q^{58} +9.91129 q^{59} +17.9321 q^{60} +0.382013 q^{61} -0.794988 q^{62} -7.13223 q^{64} +23.8298 q^{65} -2.35057 q^{66} +4.40596 q^{67} -8.09505 q^{68} -8.32853 q^{69} -10.3123 q^{71} -1.84133 q^{72} +3.37816 q^{73} -1.78437 q^{74} +24.1735 q^{75} +0.0734430 q^{76} -2.71191 q^{78} +9.73339 q^{79} -14.8278 q^{80} -10.3951 q^{81} +1.81715 q^{82} -16.5257 q^{83} -16.1697 q^{85} -2.19098 q^{86} -5.12177 q^{87} +3.99985 q^{88} +2.53028 q^{89} -1.82197 q^{90} +7.02047 q^{92} -9.66301 q^{93} +0.741870 q^{94} +0.146700 q^{95} +5.22500 q^{96} +5.85331 q^{97} -12.7703 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{2} + 4 q^{3} + 12 q^{4} + 9 q^{5} + 12 q^{6} - 6 q^{8} + 14 q^{9} + 2 q^{10} - 22 q^{11} + 10 q^{12} + 4 q^{13} - 14 q^{15} + 12 q^{16} + 18 q^{17} - 5 q^{18} + 15 q^{19} + 40 q^{20} - 11 q^{22}+ \cdots - 73 q^{99}+O(q^{100}) \) 16 * q - 2 * q^2 + 4 * q^3 + 12 * q^4 + 9 * q^5 + 12 * q^6 - 6 * q^8 + 14 * q^9 + 2 * q^10 - 22 * q^11 + 10 * q^12 + 4 * q^13 - 14 * q^15 + 12 * q^16 + 18 * q^17 - 5 * q^18 + 15 * q^19 + 40 * q^20 - 11 * q^22 - 5 * q^23 + 26 * q^24 + 15 * q^25 + 24 * q^26 + 10 * q^27 - 12 * q^29 + 3 * q^30 + 32 * q^31 - 9 * q^32 + 10 * q^33 + 14 * q^34 + 10 * q^36 - 2 * q^37 - 3 * q^38 - 41 * q^39 + 14 * q^40 + 45 * q^41 - 3 * q^43 - 54 * q^44 + 22 * q^45 + 49 * q^47 - 10 * q^48 - 6 * q^50 - 12 * q^51 - 38 * q^52 + 16 * q^53 + 67 * q^54 - 7 * q^55 + 8 * q^57 + 16 * q^58 + 35 * q^59 + 56 * q^60 + 11 * q^61 - 17 * q^62 - 2 * q^64 + 14 * q^65 - 86 * q^66 + 17 * q^67 + 71 * q^68 + 17 * q^69 - 81 * q^71 + 13 * q^72 + 15 * q^73 + 13 * q^74 + 48 * q^75 - 14 * q^76 - 10 * q^78 - 34 * q^79 + 33 * q^80 + 40 * q^81 + 14 * q^82 + 39 * q^83 - 17 * q^85 + 36 * q^86 - 7 * q^87 + 61 * q^88 + 32 * q^89 - 79 * q^90 + 37 * q^92 + 27 * q^93 - 13 * q^94 - 33 * q^95 + 55 * q^96 + 4 * q^97 - 73 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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