LMFDB - Embedded newform 538.2.b.b.537.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [538,2,Mod(537,538)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(538, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("538.537");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 538 = 2 \cdot 269 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	538.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$4.29595162874$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} + 40 x^{16} + 656 x^{14} + 5672 x^{12} + 27720 x^{10} + 76589 x^{8} + 114310 x^{6} + 83081 x^{4} + \cdots + 784 $ x^18 + 40x^16 + 656x^14 + 5672x^12 + 27720x^10 + 76589x^8 + 114310x^6 + 83081x^4 + 23804x^2 + 784
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			537.12
Root			$0.919863i$ of defining polynomial
Character	$\chi$	$=$	538.537
Dual form			538.2.b.b.537.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} -0.919863i q^{3} -1.00000 q^{4} -3.72117 q^{5} +0.919863 q^{6} +1.30781i q^{7} -1.00000i q^{8} +2.15385 q^{9} -3.72117i q^{10} +1.03413 q^{11} +0.919863i q^{12} +1.66196 q^{13} -1.30781 q^{14} +3.42297i q^{15} +1.00000 q^{16} -6.82187i q^{17} +2.15385i q^{18} -6.54429i q^{19} +3.72117 q^{20} +1.20300 q^{21} +1.03413i q^{22} +3.49794 q^{23} -0.919863 q^{24} +8.84710 q^{25} +1.66196i q^{26} -4.74084i q^{27} -1.30781i q^{28} +0.382780i q^{29} -3.42297 q^{30} -2.21107i q^{31} +1.00000i q^{32} -0.951260i q^{33} +6.82187 q^{34} -4.86656i q^{35} -2.15385 q^{36} +6.26054 q^{37} +6.54429 q^{38} -1.52878i q^{39} +3.72117i q^{40} -2.06152 q^{41} +1.20300i q^{42} -0.324584 q^{43} -1.03413 q^{44} -8.01485 q^{45} +3.49794i q^{46} -1.93107 q^{47} -0.919863i q^{48} +5.28965 q^{49} +8.84710i q^{50} -6.27518 q^{51} -1.66196 q^{52} -7.81065 q^{53} +4.74084 q^{54} -3.84818 q^{55} +1.30781 q^{56} -6.01985 q^{57} -0.382780 q^{58} -0.0368237i q^{59} -3.42297i q^{60} -13.6182 q^{61} +2.21107 q^{62} +2.81682i q^{63} -1.00000 q^{64} -6.18444 q^{65} +0.951260 q^{66} +9.44008 q^{67} +6.82187i q^{68} -3.21763i q^{69} +4.86656 q^{70} +5.82099i q^{71} -2.15385i q^{72} +3.62943 q^{73} +6.26054i q^{74} -8.13812i q^{75} +6.54429i q^{76} +1.35244i q^{77} +1.52878 q^{78} +7.49443 q^{79} -3.72117 q^{80} +2.10064 q^{81} -2.06152i q^{82} +4.15931i q^{83} -1.20300 q^{84} +25.3853i q^{85} -0.324584i q^{86} +0.352105 q^{87} -1.03413i q^{88} -9.17684 q^{89} -8.01485i q^{90} +2.17352i q^{91} -3.49794 q^{92} -2.03388 q^{93} -1.93107i q^{94} +24.3524i q^{95} +0.919863 q^{96} +13.3281 q^{97} +5.28965i q^{98} +2.22737 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 18 q^{4} - 4 q^{5} - 4 q^{6} - 26 q^{9} - 2 q^{11} + 10 q^{13} - 24 q^{14} + 18 q^{16} + 4 q^{20} - 6 q^{21} - 20 q^{23} + 4 q^{24} + 30 q^{25} + 26 q^{36} - 10 q^{37} + 14 q^{38} - 12 q^{41} - 22 q^{43}+ \cdots - 8 q^{99}+O(q^{100}) $ 18 * q - 18 * q^4 - 4 * q^5 - 4 * q^6 - 26 * q^9 - 2 * q^11 + 10 * q^13 - 24 * q^14 + 18 * q^16 + 4 * q^20 - 6 * q^21 - 20 * q^23 + 4 * q^24 + 30 * q^25 + 26 * q^36 - 10 * q^37 + 14 * q^38 - 12 * q^41 - 22 * q^43 + 2 * q^44 + 38 * q^45 - 18 * q^47 - 14 * q^49 - 6 * q^51 - 10 * q^52 - 14 * q^53 + 4 * q^54 - 6 * q^55 + 24 * q^56 - 12 * q^57 - 4 * q^58 + 6 * q^61 + 20 * q^62 - 18 * q^64 + 14 * q^65 - 50 * q^66 - 14 * q^67 - 18 * q^70 + 8 * q^73 - 10 * q^78 - 10 * q^79 - 4 * q^80 + 18 * q^81 + 6 * q^84 + 96 * q^87 - 18 * q^89 + 20 * q^92 + 70 * q^93 - 4 * q^96 + 72 * q^97 - 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/538\mathbb{Z}\right)^\times$.

$n$	$271$
$\chi(n)$	$-1$

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 $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		−	0.919863i		−	0.531083i	−0.964099	−	0.265542i	$-0.914449\pi$
$3$		−	0.919863i		−	0.531083i	0.964099	−	0.265542i	$-0.0855507\pi$
$4$	−1.00000			−0.500000
$5$	−3.72117			−1.66416			−0.832079	−	0.554657i	$-0.812849\pi$
$5$	−3.72117			−1.66416			−0.832079	+	0.554657i	$0.812849\pi$
$6$	0.919863			0.375532
$7$			1.30781i			0.494304i	0.968977	+	0.247152i	$0.0794947\pi$
$7$			1.30781i			0.494304i	−0.968977	+	0.247152i	$0.920505\pi$
$8$		−	1.00000i		−	0.353553i
$9$	2.15385			0.717951
$10$		−	3.72117i		−	1.17674i
$11$	1.03413			0.311803			0.155901	−	0.987773i	$-0.450172\pi$
$11$	1.03413			0.311803			0.155901	+	0.987773i	$0.450172\pi$
$12$			0.919863i			0.265542i
$13$	1.66196			0.460945			0.230472	−	0.973079i	$-0.425973\pi$
$13$	1.66196			0.460945			0.230472	+	0.973079i	$0.425973\pi$
$14$	−1.30781			−0.349526
$15$			3.42297i			0.883806i
$16$	1.00000			0.250000
$17$		−	6.82187i		−	1.65455i	−0.561801	−	0.827273i	$-0.689891\pi$
$17$		−	6.82187i		−	1.65455i	0.561801	−	0.827273i	$-0.310109\pi$
$18$			2.15385i			0.507668i
$19$		−	6.54429i		−	1.50136i	−0.660664	−	0.750682i	$-0.729725\pi$
$19$		−	6.54429i		−	1.50136i	0.660664	−	0.750682i	$-0.270275\pi$
$20$	3.72117			0.832079
$21$	1.20300			0.262516
$22$			1.03413i			0.220478i
$23$	3.49794			0.729372			0.364686	−	0.931131i	$-0.381176\pi$
$23$	3.49794			0.729372			0.364686	+	0.931131i	$0.381176\pi$
$24$	−0.919863			−0.187766
$25$	8.84710			1.76942
$26$			1.66196i			0.325937i
$27$		−	4.74084i		−	0.912375i
$28$		−	1.30781i		−	0.247152i
$29$			0.382780i			0.0710805i	0.999368	+	0.0355402i	$0.0113152\pi$
$29$			0.382780i			0.0710805i	−0.999368	+	0.0355402i	$0.988685\pi$
$30$	−3.42297			−0.624945
$31$		−	2.21107i		−	0.397119i	−0.980089	−	0.198560i	$-0.936374\pi$
$31$		−	2.21107i		−	0.397119i	0.980089	−	0.198560i	$-0.0636264\pi$
$32$			1.00000i			0.176777i
$33$		−	0.951260i		−	0.165593i
$34$	6.82187			1.16994
$35$		−	4.86656i		−	0.822600i
$36$	−2.15385			−0.358975
$37$	6.26054			1.02923			0.514613	−	0.857422i	$-0.327935\pi$
$37$	6.26054			1.02923			0.514613	+	0.857422i	$0.327935\pi$
$38$	6.54429			1.06162
$39$		−	1.52878i		−	0.244800i
$40$			3.72117i			0.588369i
$41$	−2.06152			−0.321956			−0.160978	−	0.986958i	$-0.551465\pi$
$41$	−2.06152			−0.321956			−0.160978	+	0.986958i	$0.551465\pi$
$42$			1.20300i			0.185627i
$43$	−0.324584			−0.0494986			−0.0247493	−	0.999694i	$-0.507879\pi$
$43$	−0.324584			−0.0494986			−0.0247493	+	0.999694i	$0.507879\pi$
$44$	−1.03413			−0.155901
$45$	−8.01485			−1.19478
$46$			3.49794i			0.515744i
$47$	−1.93107			−0.281676			−0.140838	−	0.990033i	$-0.544980\pi$
$47$	−1.93107			−0.281676			−0.140838	+	0.990033i	$0.544980\pi$
$48$		−	0.919863i		−	0.132771i
$49$	5.28965			0.755664
$50$			8.84710i			1.25117i
$51$	−6.27518			−0.878701
$52$	−1.66196			−0.230472
$53$	−7.81065			−1.07288			−0.536438	−	0.843940i	$-0.680230\pi$
$53$	−7.81065			−1.07288			−0.536438	+	0.843940i	$0.680230\pi$
$54$	4.74084			0.645146
$55$	−3.84818			−0.518889
$56$	1.30781			0.174763
$57$	−6.01985			−0.797349
$58$	−0.382780			−0.0502615
$59$		−	0.0368237i		−	0.00479404i	−0.999997	−	0.00239702i	$-0.999237\pi$
$59$		−	0.0368237i		−	0.00479404i	0.999997	−	0.00239702i	$-0.000762996\pi$
$60$		−	3.42297i		−	0.441903i
$61$	−13.6182			−1.74363			−0.871814	−	0.489837i	$-0.837056\pi$
$61$	−13.6182			−1.74363			−0.871814	+	0.489837i	$0.837056\pi$
$62$	2.21107			0.280806
$63$			2.81682i			0.354886i
$64$	−1.00000			−0.125000
$65$	−6.18444			−0.767085
$66$	0.951260			0.117092
$67$	9.44008			1.15329			0.576645	−	0.816995i	$-0.304362\pi$
$67$	9.44008			1.15329			0.576645	+	0.816995i	$0.304362\pi$
$68$			6.82187i			0.827273i
$69$		−	3.21763i		−	0.387357i
$70$	4.86656			0.581666
$71$			5.82099i			0.690825i	0.938451	+	0.345412i	$0.112261\pi$
$71$			5.82099i			0.690825i	−0.938451	+	0.345412i	$0.887739\pi$
$72$		−	2.15385i		−	0.253834i
$73$	3.62943			0.424793			0.212396	−	0.977184i	$-0.431873\pi$
$73$	3.62943			0.424793			0.212396	+	0.977184i	$0.431873\pi$
$74$			6.26054i			0.727773i
$75$		−	8.13812i		−	0.939709i
$76$			6.54429i			0.750682i
$77$			1.35244i			0.154125i
$78$	1.52878			0.173100
$79$	7.49443			0.843190			0.421595	−	0.906784i	$-0.361470\pi$
$79$	7.49443			0.843190			0.421595	+	0.906784i	$0.361470\pi$
$80$	−3.72117			−0.416039
$81$	2.10064			0.233404
$82$		−	2.06152i		−	0.227657i
$83$			4.15931i			0.456543i	0.973597	+	0.228272i	$0.0733075\pi$
$83$			4.15931i			0.456543i	−0.973597	+	0.228272i	$0.926693\pi$
$84$	−1.20300			−0.131258
$85$			25.3853i			2.75342i
$86$		−	0.324584i		−	0.0350008i
$87$	0.352105			0.0377496
$88$		−	1.03413i		−	0.110239i
$89$	−9.17684			−0.972743			−0.486371	−	0.873752i	$-0.661680\pi$
$89$	−9.17684			−0.972743			−0.486371	+	0.873752i	$0.661680\pi$
$90$		−	8.01485i		−	0.844839i
$91$			2.17352i			0.227847i
$92$	−3.49794			−0.364686
$93$	−2.03388			−0.210903
$94$		−	1.93107i		−	0.199175i
$95$			24.3524i			2.49851i
$96$	0.919863			0.0938831
$97$	13.3281			1.35326			0.676632	−	0.736321i	$-0.263439\pi$
$97$	13.3281			1.35326			0.676632	+	0.736321i	$0.263439\pi$
$98$			5.28965i			0.534335i
$99$	2.22737			0.223859
$100$	−8.84710			−0.884710
$101$		−	18.0226i		−	1.79332i	−0.442723	−	0.896659i	$-0.645987\pi$
$101$		−	18.0226i		−	1.79332i	0.442723	−	0.896659i	$-0.354013\pi$
$102$		−	6.27518i		−	0.621336i
$103$	7.79555			0.768118			0.384059	−	0.923309i	$-0.374526\pi$
$103$	7.79555			0.768118			0.384059	+	0.923309i	$0.374526\pi$
$104$		−	1.66196i		−	0.162969i
$105$	−4.47657			−0.436869
$106$		−	7.81065i		−	0.758638i
$107$		−	11.8777i		−	1.14826i	−0.818765	−	0.574128i	$-0.805341\pi$
$107$		−	11.8777i		−	1.14826i	0.818765	−	0.574128i	$-0.194659\pi$
$108$			4.74084i			0.456187i
$109$		−	1.71325i		−	0.164099i	−0.996628	−	0.0820496i	$-0.973853\pi$
$109$		−	1.71325i		−	0.164099i	0.996628	−	0.0820496i	$-0.0261466\pi$
$110$		−	3.84818i		−	0.366910i
$111$		−	5.75884i		−	0.546605i
$112$			1.30781i			0.123576i
$113$			0.614627i			0.0578192i	0.999582	+	0.0289096i	$0.00920350\pi$
$113$			0.614627i			0.0578192i	−0.999582	+	0.0289096i	$0.990796\pi$
$114$		−	6.01985i		−	0.563811i
$115$	−13.0164			−1.21379
$116$		−	0.382780i		−	0.0355402i
$117$	3.57962			0.330936
$118$	0.0368237			0.00338990
$119$	8.92167			0.817848
$120$	3.42297			0.312473
$121$	−9.93057			−0.902779
$122$		−	13.6182i		−	1.23293i
$123$			1.89632i			0.170985i
$124$			2.21107i			0.198560i
$125$	−14.3157			−1.28044
$126$	−2.81682			−0.250942
$127$	−13.3370			−1.18347			−0.591734	−	0.806133i	$-0.701556\pi$
$127$	−13.3370			−1.18347			−0.591734	+	0.806133i	$0.701556\pi$
$128$		−	1.00000i		−	0.0883883i
$129$			0.298573i			0.0262879i
$130$		−	6.18444i		−	0.542411i
$131$	−13.0568			−1.14078			−0.570388	−	0.821376i	$-0.693207\pi$
$131$	−13.0568			−1.14078			−0.570388	+	0.821376i	$0.693207\pi$
$132$			0.951260i			0.0827966i
$133$	8.55866			0.742130
$134$			9.44008i			0.815499i
$135$			17.6415i			1.51834i
$136$	−6.82187			−0.584970
$137$		−	19.9430i		−	1.70384i	−0.523669	−	0.851922i	$-0.675437\pi$
$137$		−	19.9430i		−	1.70384i	0.523669	−	0.851922i	$-0.324563\pi$
$138$	3.21763			0.273903
$139$			13.5191i			1.14667i	0.819320	+	0.573336i	$0.194351\pi$
$139$			13.5191i			1.14667i	−0.819320	+	0.573336i	$0.805649\pi$
$140$			4.86656i			0.411300i
$141$			1.77632i			0.149593i
$142$	−5.82099			−0.488487
$143$	1.71869			0.143724
$144$	2.15385			0.179488
$145$		−	1.42439i		−	0.118289i
$146$			3.62943i			0.300374i
$147$		−	4.86575i		−	0.401320i
$148$	−6.26054			−0.514613
$149$	−8.03999			−0.658662			−0.329331	−	0.944215i	$-0.606823\pi$
$149$	−8.03999			−0.658662			−0.329331	+	0.944215i	$0.606823\pi$
$150$	8.13812			0.664475
$151$	16.5679			1.34828			0.674139	−	0.738604i	$-0.264515\pi$
$151$	16.5679			1.34828			0.674139	+	0.738604i	$0.264515\pi$
$152$	−6.54429			−0.530812
$153$		−	14.6933i		−	1.18788i
$154$	−1.35244			−0.108983
$155$			8.22776i			0.660869i
$156$			1.52878i			0.122400i
$157$			0.364326i			0.0290764i	0.999894	+	0.0145382i	$0.00462782\pi$
$157$			0.364326i			0.0290764i	−0.999894	+	0.0145382i	$0.995372\pi$
$158$			7.49443i			0.596225i
$159$			7.18473i			0.569786i
$160$		−	3.72117i		−	0.294184i
$161$			4.57463i			0.360531i
$162$			2.10064i			0.165042i
$163$			4.60521i			0.360708i	0.983602	+	0.180354i	$0.0577243\pi$
$163$			4.60521i			0.360708i	−0.983602	+	0.180354i	$0.942276\pi$
$164$	2.06152			0.160978
$165$			3.53980i			0.275573i
$166$	−4.15931			−0.322825
$167$			5.91606i			0.457799i	0.973450	+	0.228899i	$0.0735127\pi$
$167$			5.91606i			0.457799i	−0.973450	+	0.228899i	$0.926487\pi$
$168$		−	1.20300i		−	0.0928136i
$169$	−10.2379			−0.787530
$170$	−25.3853			−1.94696
$171$		−	14.0954i		−	1.07791i
$172$	0.324584			0.0247493
$173$	7.27887			0.553403			0.276701	−	0.960956i	$-0.410759\pi$
$173$	7.27887			0.553403			0.276701	+	0.960956i	$0.410759\pi$
$174$			0.352105i			0.0266930i
$175$			11.5703i			0.874631i
$176$	1.03413			0.0779507
$177$	−0.0338728			−0.00254603
$178$		−	9.17684i		−	0.687833i
$179$		−	9.12777i		−	0.682241i	−0.940020	−	0.341121i	$-0.889194\pi$
$179$		−	9.12777i		−	0.682241i	0.940020	−	0.341121i	$-0.110806\pi$
$180$	8.01485			0.597392
$181$		−	23.3017i		−	1.73200i	−0.500044	−	0.866000i	$-0.666683\pi$
$181$		−	23.3017i		−	1.73200i	0.500044	−	0.866000i	$-0.333317\pi$
$182$	−2.17352			−0.161112
$183$			12.5268i			0.926011i
$184$		−	3.49794i		−	0.257872i
$185$	−23.2965			−1.71280
$186$		−	2.03388i		−	0.149131i
$187$		−	7.05471i		−	0.515892i
$188$	1.93107			0.140838
$189$	6.20009			0.450990
$190$	−24.3524			−1.76671
$191$	4.89860			0.354451			0.177225	−	0.984170i	$-0.443288\pi$
$191$	4.89860			0.354451			0.177225	+	0.984170i	$0.443288\pi$
$192$			0.919863i			0.0663854i
$193$			13.3856i			0.963518i	0.876304	+	0.481759i	$0.160002\pi$
$193$			13.3856i			0.963518i	−0.876304	+	0.481759i	$0.839998\pi$
$194$			13.3281i			0.956903i
$195$			5.68883i			0.407386i
$196$	−5.28965			−0.377832
$197$		−	6.79324i		−	0.483998i	−0.970276	−	0.241999i	$-0.922197\pi$
$197$		−	6.79324i		−	0.483998i	0.970276	−	0.241999i	$-0.0778031\pi$
$198$			2.22737i			0.158292i
$199$	12.1456			0.860981			0.430490	−	0.902595i	$-0.358341\pi$
$199$	12.1456			0.860981			0.430490	+	0.902595i	$0.358341\pi$
$200$		−	8.84710i		−	0.625585i
$201$		−	8.68358i		−	0.612493i
$202$	18.0226			1.26807
$203$	−0.500602			−0.0351354
$204$	6.27518			0.439351
$205$	7.67128			0.535785
$206$			7.79555i			0.543142i
$207$	7.53406			0.523653
$208$	1.66196			0.115236
$209$		−	6.76767i		−	0.468129i
$210$		−	4.47657i		−	0.308913i
$211$	−18.0139			−1.24013			−0.620063	−	0.784552i	$-0.712893\pi$
$211$	−18.0139			−1.24013			−0.620063	+	0.784552i	$0.712893\pi$
$212$	7.81065			0.536438
$213$	5.35452			0.366885
$214$	11.8777			0.811940
$215$	1.20783			0.0823735
$216$	−4.74084			−0.322573
$217$	2.89165			0.196298
$218$	1.71325			0.116036
$219$		−	3.33858i		−	0.225600i
$220$	3.84818			0.259444
$221$		−	11.3377i		−	0.762654i
$222$	5.75884			0.386508
$223$			22.6418i			1.51620i	0.652136	+	0.758102i	$0.273873\pi$
$223$			22.6418i			1.51620i	−0.652136	+	0.758102i	$0.726127\pi$
$224$	−1.30781			−0.0873814
$225$	19.0553			1.27036
$226$	−0.614627			−0.0408844
$227$			17.2433i			1.14448i	0.820088	+	0.572238i	$0.193925\pi$
$227$			17.2433i			1.14448i	−0.820088	+	0.572238i	$0.806075\pi$
$228$	6.01985			0.398675
$229$		−	11.2567i		−	0.743861i	−0.928261	−	0.371931i	$-0.878696\pi$
$229$		−	11.2567i		−	0.743861i	0.928261	−	0.371931i	$-0.121304\pi$
$230$		−	13.0164i		−	0.858279i
$231$	1.24406			0.0818533
$232$	0.382780			0.0251307
$233$	−13.0664			−0.856008			−0.428004	−	0.903777i	$-0.640783\pi$
$233$	−13.0664			−0.856008			−0.428004	+	0.903777i	$0.640783\pi$
$234$			3.57962i			0.234007i
$235$	7.18585			0.468753
$236$			0.0368237i			0.00239702i
$237$		−	6.89385i		−	0.447804i
$238$			8.92167i			0.578306i
$239$	28.8237			1.86445			0.932225	−	0.361880i	$-0.117865\pi$
$239$	28.8237			1.86445			0.932225	+	0.361880i	$0.117865\pi$
$240$			3.42297i			0.220951i
$241$			13.7305i			0.884457i	0.896902	+	0.442229i	$0.145812\pi$
$241$			13.7305i			0.884457i	−0.896902	+	0.442229i	$0.854188\pi$
$242$		−	9.93057i		−	0.638361i
$243$		−	16.1548i		−	1.03633i
$244$	13.6182			0.871814
$245$	−19.6837			−1.25754
$246$	−1.89632			−0.120905
$247$		−	10.8764i		−	0.692046i
$248$	−2.21107			−0.140403
$249$	3.82599			0.242463
$250$		−	14.3157i		−	0.905405i
$251$			25.6111i			1.61656i	0.588800	+	0.808279i	$0.299600\pi$
$251$			25.6111i			1.61656i	−0.588800	+	0.808279i	$0.700400\pi$
$252$		−	2.81682i		−	0.177443i
$253$	3.61734			0.227420
$254$		−	13.3370i		−	0.836838i
$255$	23.3510			1.46230
$256$	1.00000			0.0625000
$257$		−	8.89335i		−	0.554752i	−0.960761	−	0.277376i	$-0.910535\pi$
$257$		−	8.89335i		−	0.554752i	0.960761	−	0.277376i	$-0.0894648\pi$
$258$	−0.298573			−0.0185883
$259$			8.18757i			0.508751i
$260$	6.18444			0.383542
$261$			0.824452i			0.0510323i
$262$		−	13.0568i		−	0.806650i
$263$	18.1107			1.11676			0.558378	−	0.829587i	$-0.311424\pi$
$263$	18.1107			1.11676			0.558378	+	0.829587i	$0.311424\pi$
$264$	−0.951260			−0.0585460
$265$	29.0648			1.78543
$266$			8.55866i			0.524765i
$267$			8.44143i			0.516607i
$268$	−9.44008			−0.576645
$269$	−8.02478	+	14.3039i	−0.489280	+	0.872127i
$269$	−8.02478	+	14.3039i	−0.489280	+	0.872127i
$270$	−17.6415			−1.07363
$271$			23.3751i			1.41994i	0.704233	+	0.709969i	$0.251291\pi$
$271$			23.3751i			1.41994i	−0.704233	+	0.709969i	$0.748709\pi$
$272$		−	6.82187i		−	0.413636i
$273$	1.99934			0.121006
$274$	19.9430			1.20480
$275$	9.14908			0.551710
$276$			3.21763i			0.193679i
$277$			9.44473i			0.567479i	0.958901	+	0.283739i	$0.0915751\pi$
$277$			9.44473i			0.567479i	−0.958901	+	0.283739i	$0.908425\pi$
$278$	−13.5191			−0.810819
$279$		−	4.76231i		−	0.285112i
$280$	−4.86656			−0.290833
$281$		−	6.68455i		−	0.398767i	−0.979922	−	0.199383i	$-0.936106\pi$
$281$		−	6.68455i		−	0.398767i	0.979922	−	0.199383i	$-0.0638939\pi$
$282$	−1.77632			−0.105779
$283$	30.4554			1.81039			0.905193	−	0.425000i	$-0.139726\pi$
$283$	30.4554			1.81039			0.905193	+	0.425000i	$0.139726\pi$
$284$		−	5.82099i		−	0.345412i
$285$	22.4009			1.32691
$286$			1.71869i			0.101628i
$287$		−	2.69607i		−	0.159144i
$288$			2.15385i			0.126917i
$289$	−29.5378			−1.73752
$290$	1.42439			0.0836431
$291$		−	12.2600i		−	0.718696i
$292$	−3.62943			−0.212396
$293$	−5.74177			−0.335438			−0.167719	−	0.985835i	$-0.553640\pi$
$293$	−5.74177			−0.335438			−0.167719	+	0.985835i	$0.553640\pi$
$294$	4.86575			0.283776
$295$			0.137027i			0.00797803i
$296$		−	6.26054i		−	0.363887i
$297$		−	4.90266i		−	0.284481i
$298$		−	8.03999i		−	0.465744i
$299$	5.81345			0.336200
$300$			8.13812i			0.469855i
$301$		−	0.424493i		−	0.0244674i
$302$			16.5679i			0.953377i
$303$	−16.5783			−0.952401
$304$		−	6.54429i		−	0.375341i
$305$	50.6755			2.90167
$306$	14.6933			0.839959
$307$	9.25181			0.528029			0.264014	−	0.964519i	$-0.414953\pi$
$307$	9.25181			0.528029			0.264014	+	0.964519i	$0.414953\pi$
$308$		−	1.35244i		−	0.0770627i
$309$		−	7.17084i		−	0.407935i
$310$	−8.22776			−0.467305
$311$			23.2300i			1.31725i	0.752471	+	0.658626i	$0.228862\pi$
$311$			23.2300i			1.31725i	−0.752471	+	0.658626i	$0.771138\pi$
$312$	−1.52878			−0.0865499
$313$	5.67790			0.320934			0.160467	−	0.987041i	$-0.448700\pi$
$313$	5.67790			0.320934			0.160467	+	0.987041i	$0.448700\pi$
$314$	−0.364326			−0.0205601
$315$		−	10.4819i		−	0.590586i
$316$	−7.49443			−0.421595
$317$			16.6550i			0.935441i	0.883877	+	0.467720i	$0.154925\pi$
$317$			16.6550i			0.935441i	−0.883877	+	0.467720i	$0.845075\pi$
$318$	−7.18473			−0.402900
$319$			0.395845i			0.0221631i
$320$	3.72117			0.208020
$321$	−10.9258			−0.609820
$322$	−4.57463			−0.254934
$323$	−44.6443			−2.48408
$324$	−2.10064			−0.116702
$325$	14.7035			0.815605
$326$	−4.60521			−0.255059
$327$	−1.57595			−0.0871503
$328$			2.06152i			0.113829i
$329$		−	2.52547i		−	0.139234i
$330$	−3.53980			−0.194860
$331$	3.47181			0.190828			0.0954140	−	0.995438i	$-0.469583\pi$
$331$	3.47181			0.190828			0.0954140	+	0.995438i	$0.469583\pi$
$332$		−	4.15931i		−	0.228272i
$333$	13.4843			0.738934
$334$	−5.91606			−0.323713
$335$	−35.1282			−1.91926
$336$	1.20300			0.0656291
$337$			5.29794i			0.288597i	0.989534	+	0.144299i	$0.0460926\pi$
$337$			5.29794i			0.288597i	−0.989534	+	0.144299i	$0.953907\pi$
$338$		−	10.2379i		−	0.556868i
$339$	0.565373			0.0307068
$340$		−	25.3853i		−	1.37671i
$341$		−	2.28654i		−	0.123823i
$342$	14.0954			0.762194
$343$			16.0725i			0.867831i
$344$			0.324584i			0.0175004i
$345$			11.9733i			0.644623i
$346$			7.27887i			0.391315i
$347$	−11.7899			−0.632915			−0.316457	−	0.948607i	$-0.602493\pi$
$347$	−11.7899			−0.632915			−0.316457	+	0.948607i	$0.602493\pi$
$348$	−0.352105			−0.0188748
$349$	−30.0098			−1.60639			−0.803195	−	0.595716i	$-0.796868\pi$
$349$	−30.0098			−1.60639			−0.803195	+	0.595716i	$0.796868\pi$
$350$	−11.5703			−0.618458
$351$		−	7.87908i		−	0.420554i
$352$			1.03413i			0.0551195i
$353$	21.4599			1.14220			0.571098	−	0.820882i	$-0.306518\pi$
$353$	21.4599			1.14220			0.571098	+	0.820882i	$0.306518\pi$
$354$		−	0.0338728i		−	0.00180032i
$355$		−	21.6609i		−	1.14964i
$356$	9.17684			0.486371
$357$		−	8.20671i		−	0.434345i
$358$	9.12777			0.482417
$359$		−	1.69504i		−	0.0894607i	−0.998999	−	0.0447303i	$-0.985757\pi$
$359$		−	1.69504i		−	0.0894607i	0.998999	−	0.0447303i	$-0.0142429\pi$
$360$			8.01485i			0.422420i
$361$	−23.8278			−1.25409
$362$	23.3017			1.22471
$363$			9.13476i			0.479451i
$364$		−	2.17352i		−	0.113923i
$365$	−13.5057			−0.706922
$366$	−12.5268			−0.654789
$367$			13.5954i			0.709674i	0.934928	+	0.354837i	$0.115464\pi$
$367$			13.5954i			0.709674i	−0.934928	+	0.354837i	$0.884536\pi$
$368$	3.49794			0.182343
$369$	−4.44022			−0.231148
$370$		−	23.2965i		−	1.21113i
$371$		−	10.2148i		−	0.530327i
$372$	2.03388			0.105452
$373$		−	30.6750i		−	1.58829i	−0.607726	−	0.794147i	$-0.707918\pi$
$373$		−	30.6750i		−	1.58829i	0.607726	−	0.794147i	$-0.292082\pi$
$374$	7.05471			0.364791
$375$			13.1685i			0.680018i
$376$			1.93107i			0.0995875i
$377$			0.636165i			0.0327642i
$378$			6.20009i			0.318898i
$379$		−	35.4530i		−	1.82110i	−0.413401	−	0.910549i	$-0.635659\pi$
$379$		−	35.4530i		−	1.82110i	0.413401	−	0.910549i	$-0.364341\pi$
$380$		−	24.3524i		−	1.24925i
$381$			12.2682i			0.628520i
$382$			4.89860i			0.250634i
$383$		−	6.04327i		−	0.308797i	−0.988009	−	0.154398i	$-0.950656\pi$
$383$		−	6.04327i		−	0.308797i	0.988009	−	0.154398i	$-0.0493439\pi$
$384$	−0.919863			−0.0469416
$385$		−	5.03267i		−	0.256489i
$386$	−13.3856			−0.681310
$387$	−0.699106			−0.0355376
$388$	−13.3281			−0.676632
$389$	14.7955			0.750160			0.375080	−	0.926992i	$-0.377615\pi$
$389$	14.7955			0.750160			0.375080	+	0.926992i	$0.377615\pi$
$390$	−5.68883			−0.288065
$391$		−	23.8625i		−	1.20678i
$392$		−	5.28965i		−	0.267167i
$393$			12.0104i			0.605846i
$394$	6.79324			0.342239
$395$	−27.8881			−1.40320
$396$	−2.22737			−0.111930
$397$			28.2174i			1.41619i	0.706116	+	0.708096i	$0.250446\pi$
$397$			28.2174i			1.41619i	−0.706116	+	0.708096i	$0.749554\pi$
$398$			12.1456i			0.608805i
$399$		−	7.87280i		−	0.394133i
$400$	8.84710			0.442355
$401$			35.7100i			1.78327i	0.452754	+	0.891636i	$0.350442\pi$
$401$			35.7100i			1.78327i	−0.452754	+	0.891636i	$0.649558\pi$
$402$	8.68358			0.433098
$403$		−	3.67471i		−	0.183050i
$404$			18.0226i			0.896659i
$405$	−7.81682			−0.388421
$406$		−	0.500602i		−	0.0248445i
$407$	6.47423			0.320916
$408$			6.27518i			0.310668i
$409$		−	22.8670i		−	1.13070i	−0.824851	−	0.565351i	$-0.808741\pi$
$409$		−	22.8670i		−	1.13070i	0.824851	−	0.565351i	$-0.191259\pi$
$410$			7.67128i			0.378857i
$411$	−18.3448			−0.904883
$412$	−7.79555			−0.384059
$413$	0.0481582			0.00236971
$414$			7.53406i			0.370279i
$415$		−	15.4775i		−	0.759760i
$416$			1.66196i			0.0814843i
$417$	12.4357			0.608978
$418$	6.76767			0.331018
$419$	32.9463			1.60953			0.804766	−	0.593592i	$-0.202291\pi$
$419$	32.9463			1.60953			0.804766	+	0.593592i	$0.202291\pi$
$420$	4.47657			0.218434
$421$	−9.14584			−0.445741			−0.222870	−	0.974848i	$-0.571543\pi$
$421$	−9.14584			−0.445741			−0.222870	+	0.974848i	$0.571543\pi$
$422$		−	18.0139i		−	0.876901i
$423$	−4.15925			−0.202230
$424$			7.81065i			0.379319i
$425$		−	60.3537i		−	2.92759i
$426$			5.35452i			0.259427i
$427$		−	17.8099i		−	0.861882i
$428$			11.8777i			0.574128i
$429$		−	1.58096i		−	0.0763293i
$430$			1.20783i			0.0582468i
$431$			34.7872i			1.67564i	0.545948	+	0.837819i	$0.316170\pi$
$431$			34.7872i			1.67564i	−0.545948	+	0.837819i	$0.683830\pi$
$432$		−	4.74084i		−	0.228094i
$433$	−14.6378			−0.703450			−0.351725	−	0.936103i	$-0.614405\pi$
$433$	−14.6378			−0.703450			−0.351725	+	0.936103i	$0.614405\pi$
$434$			2.89165i			0.138803i
$435$	−1.31024			−0.0628214
$436$			1.71325i			0.0820496i
$437$		−	22.8916i		−	1.09505i
$438$	3.33858			0.159523
$439$	−29.5184			−1.40884			−0.704419	−	0.709784i	$-0.748792\pi$
$439$	−29.5184			−1.40884			−0.704419	+	0.709784i	$0.748792\pi$
$440$			3.84818i			0.183455i
$441$	11.3931			0.542529
$442$	11.3377			0.539278
$443$			29.4519i			1.39930i	0.714484	+	0.699652i	$0.246662\pi$
$443$			29.4519i			1.39930i	−0.714484	+	0.699652i	$0.753338\pi$
$444$			5.75884i			0.273302i
$445$	34.1486			1.61880
$446$	−22.6418			−1.07212
$447$			7.39569i			0.349804i
$448$		−	1.30781i		−	0.0617880i
$449$	15.5814			0.735330			0.367665	−	0.929958i	$-0.380157\pi$
$449$	15.5814			0.735330			0.367665	+	0.929958i	$0.380157\pi$
$450$			19.0553i			0.898278i
$451$	−2.13189			−0.100387
$452$		−	0.614627i		−	0.0289096i
$453$		−	15.2402i		−	0.716048i
$454$	−17.2433			−0.809267
$455$		−	8.08804i		−	0.379173i
$456$			6.01985i			0.281906i
$457$	13.0025			0.608232			0.304116	−	0.952635i	$-0.401639\pi$
$457$	13.0025			0.608232			0.304116	+	0.952635i	$0.401639\pi$
$458$	11.2567			0.525989
$459$	−32.3414			−1.50957
$460$	13.0164			0.606895
$461$			9.56016i			0.445261i	0.974903	+	0.222631i	$0.0714644\pi$
$461$			9.56016i			0.445261i	−0.974903	+	0.222631i	$0.928536\pi$
$462$			1.24406i			0.0578791i
$463$		−	34.5791i		−	1.60703i	−0.595287	−	0.803513i	$-0.702962\pi$
$463$		−	34.5791i		−	1.60703i	0.595287	−	0.803513i	$-0.297038\pi$
$464$			0.382780i			0.0177701i
$465$	7.56841			0.350977
$466$		−	13.0664i		−	0.605289i
$467$			26.8002i			1.24017i	0.784536	+	0.620083i	$0.212901\pi$
$467$			26.8002i			1.24017i	−0.784536	+	0.620083i	$0.787099\pi$
$468$	−3.57962			−0.165468
$469$			12.3458i			0.570076i
$470$			7.18585i			0.331459i
$471$	0.335130			0.0154420
$472$	−0.0368237			−0.00169495
$473$	−0.335663			−0.0154338
$474$	6.89385			0.316645
$475$		−	57.8980i		−	2.65654i
$476$	−8.92167			−0.408924
$477$	−16.8230			−0.770272
$478$			28.8237i			1.31836i
$479$		−	14.2506i		−	0.651128i	−0.945520	−	0.325564i	$-0.894446\pi$
$479$		−	14.2506i		−	0.651128i	0.945520	−	0.325564i	$-0.105554\pi$
$480$	−3.42297			−0.156236
$481$	10.4048			0.474417
$482$	−13.7305			−0.625406
$483$	4.20803			0.191472
$484$	9.93057			0.451390
$485$	−49.5962			−2.25205
$486$	16.1548			0.732797
$487$	14.2840			0.647271			0.323636	−	0.946182i	$-0.395095\pi$
$487$	14.2840			0.647271			0.323636	+	0.946182i	$0.395095\pi$
$488$			13.6182i			0.616465i
$489$	4.23616			0.191566
$490$		−	19.6837i		−	0.889217i
$491$	−38.9011			−1.75558			−0.877791	−	0.479043i	$-0.840984\pi$
$491$	−38.9011			−1.75558			−0.877791	+	0.479043i	$0.840984\pi$
$492$		−	1.89632i		−	0.0854927i
$493$	2.61127			0.117606
$494$	10.8764			0.489351
$495$	−8.28842			−0.372537
$496$		−	2.21107i		−	0.0992799i
$497$	−7.61272			−0.341477
$498$			3.82599i			0.171447i
$499$		−	2.48954i		−	0.111447i	−0.998446	−	0.0557235i	$-0.982253\pi$
$499$		−	2.48954i		−	0.111447i	0.998446	−	0.0557235i	$-0.0177465\pi$
$500$	14.3157			0.640218
$501$	5.44197			0.243129
$502$	−25.6111			−1.14308
$503$		−	3.50005i		−	0.156059i	−0.996951	−	0.0780297i	$-0.975137\pi$
$503$		−	3.50005i		−	0.156059i	0.996951	−	0.0780297i	$-0.0248629\pi$
$504$	2.81682			0.125471
$505$			67.0652i			2.98436i
$506$			3.61734i			0.160810i
$507$			9.41745i			0.418244i
$508$	13.3370			0.591734
$509$			31.3478i			1.38947i	0.719268	+	0.694733i	$0.244477\pi$
$509$			31.3478i			1.38947i	−0.719268	+	0.694733i	$0.755523\pi$
$510$			23.3510i			1.03400i
$511$			4.74659i			0.209977i
$512$			1.00000i			0.0441942i
$513$	−31.0254			−1.36981
$514$	8.89335			0.392269
$515$	−29.0086			−1.27827
$516$		−	0.298573i		−	0.0131439i
$517$	−1.99699			−0.0878274
$518$	−8.18757			−0.359741
$519$		−	6.69557i		−	0.293903i
$520$			6.18444i			0.271205i
$521$		−	5.07967i		−	0.222544i	−0.993790	−	0.111272i	$-0.964507\pi$
$521$		−	5.07967i		−	0.222544i	0.993790	−	0.111272i	$-0.0354925\pi$
$522$	−0.824452			−0.0360853
$523$		−	33.5497i		−	1.46702i	−0.679676	−	0.733512i	$-0.737880\pi$
$523$		−	33.5497i		−	1.46702i	0.679676	−	0.733512i	$-0.262120\pi$
$524$	13.0568			0.570388
$525$	10.6431			0.464502
$526$			18.1107i			0.789665i
$527$	−15.0836			−0.657052
$528$		−	0.951260i		−	0.0413983i
$529$	−10.7644			−0.468017
$530$			29.0648i			1.26249i
$531$		−	0.0793128i		−	0.00344188i
$532$	−8.55866			−0.371065
$533$	−3.42617			−0.148404
$534$	−8.44143			−0.365297
$535$			44.1988i			1.91088i
$536$		−	9.44008i		−	0.407750i
$537$	−8.39629			−0.362327
$538$	−14.3039	−	8.02478i	−0.616687	−	0.345973i
$539$	5.47020			0.235618
$540$		−	17.6415i		−	0.759168i
$541$			27.8590i			1.19775i	0.800842	+	0.598876i	$0.204386\pi$
$541$			27.8590i			1.19775i	−0.800842	+	0.598876i	$0.795614\pi$
$542$	−23.3751			−1.00405
$543$	−21.4343			−0.919836
$544$	6.82187			0.292485
$545$			6.37528i			0.273087i
$546$			1.99934i			0.0855639i
$547$	35.4227			1.51457			0.757283	−	0.653087i	$-0.226526\pi$
$547$	35.4227			1.51457			0.757283	+	0.653087i	$0.226526\pi$
$548$			19.9430i			0.851922i
$549$	−29.3315			−1.25184
$550$			9.14908i			0.390118i
$551$	2.50503			0.106718
$552$	−3.21763			−0.136951
$553$			9.80126i			0.416792i
$554$	−9.44473			−0.401268
$555$			21.4296i			0.909637i
$556$		−	13.5191i		−	0.573336i
$557$			10.7912i			0.457238i	0.973516	+	0.228619i	$0.0734210\pi$
$557$			10.7912i			0.457238i	−0.973516	+	0.228619i	$0.926579\pi$
$558$	4.76231			0.201605
$559$	−0.539446			−0.0228161
$560$		−	4.86656i		−	0.205650i
$561$	−6.48937			−0.273981
$562$	6.68455			0.281971
$563$	30.5042			1.28560			0.642799	−	0.766035i	$-0.277773\pi$
$563$	30.5042			1.28560			0.642799	+	0.766035i	$0.277773\pi$
$564$		−	1.77632i		−	0.0747967i
$565$		−	2.28713i		−	0.0962203i
$566$			30.4554i			1.28014i
$567$			2.74722i			0.115372i
$568$	5.82099			0.244243
$569$			14.7621i			0.618859i	0.950922	+	0.309430i	$0.100138\pi$
$569$			14.7621i			0.618859i	−0.950922	+	0.309430i	$0.899862\pi$
$570$			22.4009i			0.938270i
$571$		−	15.5741i		−	0.651755i	−0.945412	−	0.325878i	$-0.894340\pi$
$571$		−	15.5741i		−	0.651755i	0.945412	−	0.325878i	$-0.105660\pi$
$572$	−1.71869			−0.0718619
$573$		−	4.50604i		−	0.188243i
$574$	2.69607			0.112532
$575$	30.9467			1.29057
$576$	−2.15385			−0.0897438
$577$			3.80455i			0.158386i	0.996859	+	0.0791928i	$0.0252343\pi$
$577$			3.80455i			0.158386i	−0.996859	+	0.0791928i	$0.974766\pi$
$578$		−	29.5378i		−	1.22861i
$579$	12.3129			0.511708
$580$			1.42439i			0.0591446i
$581$	−5.43957			−0.225671
$582$	12.2600			0.508195
$583$	−8.07725			−0.334526
$584$		−	3.62943i		−	0.150187i
$585$	−13.3204			−0.550729
$586$		−	5.74177i		−	0.237190i
$587$	34.8005			1.43637			0.718185	−	0.695853i	$-0.244973\pi$
$587$	34.8005			1.43637			0.718185	+	0.695853i	$0.244973\pi$
$588$			4.86575i			0.200660i
$589$	−14.4699			−0.596221
$590$	−0.137027			−0.00564132
$591$	−6.24885			−0.257043
$592$	6.26054			0.257307
$593$	4.97852			0.204443			0.102222	−	0.994762i	$-0.467405\pi$
$593$	4.97852			0.204443			0.102222	+	0.994762i	$0.467405\pi$
$594$	4.90266			0.201158
$595$	−33.1990			−1.36103
$596$	8.03999			0.329331
$597$		−	11.1723i		−	0.457252i
$598$			5.81345i			0.237729i
$599$	−35.6144			−1.45516			−0.727582	−	0.686020i	$-0.759356\pi$
$599$	−35.6144			−1.45516			−0.727582	+	0.686020i	$0.759356\pi$
$600$	−8.13812			−0.332237
$601$		−	45.9134i		−	1.87285i	−0.350872	−	0.936423i	$-0.614115\pi$
$601$		−	45.9134i		−	1.87285i	0.350872	−	0.936423i	$-0.385885\pi$
$602$	0.424493			0.0173010
$603$	20.3325			0.828005
$604$	−16.5679			−0.674139
$605$	36.9533			1.50237
$606$		−	16.5783i		−	0.673449i
$607$		−	22.9870i		−	0.933015i	−0.884517	−	0.466508i	$-0.845512\pi$
$607$		−	22.9870i		−	0.933015i	0.884517	−	0.466508i	$-0.154488\pi$
$608$	6.54429			0.265406
$609$			0.460485i			0.0186598i
$610$			50.6755i			2.05179i
$611$	−3.20937			−0.129837
$612$			14.6933i			0.593941i
$613$			33.0223i			1.33376i	0.745166	+	0.666880i	$0.232370\pi$
$613$			33.0223i			1.33376i	−0.745166	+	0.666880i	$0.767630\pi$
$614$			9.25181i			0.373373i
$615$		−	7.05652i		−	0.284547i
$616$	1.35244			0.0544915
$617$	20.7006			0.833376			0.416688	−	0.909050i	$-0.363191\pi$
$617$	20.7006			0.833376			0.416688	+	0.909050i	$0.363191\pi$
$618$	7.17084			0.288453
$619$	37.5867			1.51074			0.755369	−	0.655300i	$-0.227458\pi$
$619$	37.5867			1.51074			0.755369	+	0.655300i	$0.227458\pi$
$620$		−	8.22776i		−	0.330435i
$621$		−	16.5832i		−	0.665460i
$622$	−23.2300			−0.931437
$623$		−	12.0015i		−	0.480831i
$624$		−	1.52878i		−	0.0612000i
$625$	9.03570			0.361428
$626$			5.67790i			0.226934i
$627$	−6.22533			−0.248616
$628$		−	0.364326i		−	0.0145382i
$629$		−	42.7086i		−	1.70290i
$630$	10.4819			0.417607
$631$	−26.4150			−1.05156			−0.525782	−	0.850619i	$-0.676227\pi$
$631$	−26.4150			−1.05156			−0.525782	+	0.850619i	$0.676227\pi$
$632$		−	7.49443i		−	0.298113i
$633$			16.5703i			0.658610i
$634$	−16.6550			−0.661456
$635$	49.6293			1.96948
$636$		−	7.18473i		−	0.284893i
$637$	8.79118			0.348319
$638$	−0.395845			−0.0156717
$639$			12.5376i			0.495978i
$640$			3.72117i			0.147092i
$641$	−26.4787			−1.04585			−0.522924	−	0.852380i	$-0.675159\pi$
$641$	−26.4787			−1.04585			−0.522924	+	0.852380i	$0.675159\pi$
$642$		−	10.9258i		−	0.431208i
$643$	−33.2723			−1.31213			−0.656065	−	0.754704i	$-0.727780\pi$
$643$	−33.2723			−1.31213			−0.656065	+	0.754704i	$0.727780\pi$
$644$		−	4.57463i		−	0.180266i
$645$		−	1.11104i		−	0.0437472i
$646$		−	44.6443i		−	1.75651i
$647$		−	25.4487i		−	1.00049i	−0.865884	−	0.500245i	$-0.833243\pi$
$647$		−	25.4487i		−	1.00049i	0.865884	−	0.500245i	$-0.166757\pi$
$648$		−	2.10064i		−	0.0825208i
$649$		−	0.0380806i		−	0.00149479i
$650$			14.7035i			0.576720i
$651$		−	2.65992i		−	0.104250i
$652$		−	4.60521i		−	0.180354i
$653$	0.496062			0.0194124			0.00970621	−	0.999953i	$-0.496910\pi$
$653$	0.496062			0.0194124			0.00970621	+	0.999953i	$0.496910\pi$
$654$		−	1.57595i		−	0.0616246i
$655$	48.5864			1.89843
$656$	−2.06152			−0.0804890
$657$	7.81726			0.304980
$658$	2.52547			0.0984530
$659$	21.8450			0.850961			0.425480	−	0.904968i	$-0.360105\pi$
$659$	21.8450			0.850961			0.425480	+	0.904968i	$0.360105\pi$
$660$		−	3.53980i		−	0.137787i
$661$		−	22.4503i		−	0.873215i	−0.899652	−	0.436608i	$-0.856180\pi$
$661$		−	22.4503i		−	0.873215i	0.899652	−	0.436608i	$-0.143820\pi$
$662$			3.47181i			0.134936i
$663$	−10.4291			−0.405033
$664$	4.15931			0.161412
$665$	−31.8482			−1.23502
$666$			13.4843i			0.522505i
$667$			1.33894i			0.0518441i
$668$		−	5.91606i		−	0.228899i
$669$	20.8273			0.805231
$670$		−	35.1282i		−	1.35712i
$671$	−14.0830			−0.543668
$672$			1.20300i			0.0464068i
$673$			2.89485i			0.111588i	0.998442	+	0.0557941i	$0.0177691\pi$
$673$			2.89485i			0.111588i	−0.998442	+	0.0557941i	$0.982231\pi$
$674$	−5.29794			−0.204069
$675$		−	41.9427i		−	1.61437i
$676$	10.2379			0.393765
$677$			17.5748i			0.675454i	0.941244	+	0.337727i	$0.109658\pi$
$677$			17.5748i			0.675454i	−0.941244	+	0.337727i	$0.890342\pi$
$678$			0.565373i			0.0217130i
$679$			17.4306i			0.668924i
$680$	25.3853			0.973482
$681$	15.8614			0.607812
$682$	2.28654			0.0875560
$683$			6.54991i			0.250625i	0.992117	+	0.125313i	$0.0399934\pi$
$683$			6.54991i			0.250625i	−0.992117	+	0.125313i	$0.960007\pi$
$684$			14.0954i			0.538953i
$685$			74.2112i			2.83546i
$686$	−16.0725			−0.613649
$687$	−10.3546			−0.395052
$688$	−0.324584			−0.0123747
$689$	−12.9810			−0.494537
$690$	−11.9733			−0.455817
$691$			50.9782i			1.93930i	0.244494	+	0.969651i	$0.421378\pi$
$691$			50.9782i			1.93930i	−0.244494	+	0.969651i	$0.578622\pi$
$692$	−7.27887			−0.276701
$693$			2.91296i			0.110654i
$694$		−	11.7899i		−	0.447538i
$695$		−	50.3067i		−	1.90824i
$696$		−	0.352105i		−	0.0133465i
$697$			14.0634i			0.532691i
$698$		−	30.0098i		−	1.13589i
$699$			12.0193i			0.454611i
$700$		−	11.5703i		−	0.437316i
$701$		−	26.2569i		−	0.991709i	−0.868406	−	0.495855i	$-0.834855\pi$
$701$		−	26.2569i		−	0.991709i	0.868406	−	0.495855i	$-0.165145\pi$
$702$	7.87908			0.297377
$703$		−	40.9708i		−	1.54524i
$704$	−1.03413			−0.0389753
$705$		−	6.61000i		−	0.248947i
$706$			21.4599i			0.807655i
$707$	23.5701			0.886444
$708$	0.0338728			0.00127302
$709$			44.8738i			1.68527i	0.538484	+	0.842636i	$0.318997\pi$
$709$			44.8738i			1.68527i	−0.538484	+	0.842636i	$0.681003\pi$
$710$	21.6609			0.812919
$711$	16.1419			0.605369
$712$			9.17684i			0.343917i
$713$		−	7.73419i		−	0.289648i
$714$	8.20671			0.307129
$715$	−6.39553			−0.239179
$716$			9.12777i			0.341121i
$717$		−	26.5138i		−	0.990178i
$718$	1.69504			0.0632583
$719$		−	24.3323i		−	0.907442i	−0.891144	−	0.453721i	$-0.850096\pi$
$719$		−	24.3323i		−	0.907442i	0.891144	−	0.453721i	$-0.149904\pi$
$720$	−8.01485			−0.298696
$721$			10.1951i			0.379684i
$722$		−	23.8278i		−	0.886779i
$723$	12.6302			0.469720
$724$			23.3017i			0.866000i
$725$			3.38650i			0.125771i
$726$	−9.13476			−0.339023
$727$	11.7219			0.434740			0.217370	−	0.976089i	$-0.430252\pi$
$727$	11.7219			0.434740			0.217370	+	0.976089i	$0.430252\pi$
$728$	2.17352			0.0805560
$729$	−8.55830			−0.316974
$730$		−	13.5057i		−	0.499869i
$731$			2.21427i			0.0818977i
$732$		−	12.5268i		−	0.463006i
$733$		−	31.3743i		−	1.15884i	−0.815031	−	0.579418i	$-0.803280\pi$
$733$		−	31.3743i		−	1.15884i	0.815031	−	0.579418i	$-0.196720\pi$
$734$	−13.5954			−0.501815
$735$			18.1063i			0.667860i
$736$			3.49794i			0.128936i
$737$	9.76230			0.359599
$738$		−	4.44022i		−	0.163447i
$739$			26.8045i			0.986021i	0.870023	+	0.493011i	$0.164104\pi$
$739$			26.8045i			0.986021i	−0.870023	+	0.493011i	$0.835896\pi$
$740$	23.2965			0.856398
$741$	−10.0048			−0.367534
$742$	10.2148			0.374998
$743$	21.9137			0.803934			0.401967	−	0.915654i	$-0.368327\pi$
$743$	21.9137			0.803934			0.401967	+	0.915654i	$0.368327\pi$
$744$			2.03388i			0.0745656i
$745$	29.9182			1.09612
$746$	30.6750			1.12309
$747$			8.95854i			0.327776i
$748$			7.05471i			0.257946i
$749$	15.5337			0.567588
$750$	−13.1685			−0.480846
$751$	−27.9564			−1.02014			−0.510071	−	0.860132i	$-0.670381\pi$
$751$	−27.9564			−1.02014			−0.510071	+	0.860132i	$0.670381\pi$
$752$	−1.93107			−0.0704190
$753$	23.5587			0.858526
$754$	−0.636165			−0.0231678
$755$	−61.6520			−2.24375
$756$	−6.20009			−0.225495
$757$		−	26.0610i		−	0.947204i	−0.880739	−	0.473602i	$-0.842954\pi$
$757$		−	26.0610i		−	0.947204i	0.880739	−	0.473602i	$-0.157046\pi$
$758$	35.4530			1.28771
$759$		−	3.32746i		−	0.120779i
$760$	24.3524			0.883355
$761$		−	16.6466i		−	0.603438i	−0.953397	−	0.301719i	$-0.902440\pi$
$761$		−	16.6466i		−	0.603438i	0.953397	−	0.301719i	$-0.0975604\pi$
$762$	−12.2682			−0.444431
$763$	2.24059			0.0811149
$764$	−4.89860			−0.177225
$765$			54.6762i			1.97682i
$766$	6.04327			0.218352
$767$		−	0.0611995i		−	0.00220979i
$768$		−	0.919863i		−	0.0331927i
$769$	−52.9510			−1.90946			−0.954731	−	0.297470i	$-0.903857\pi$
$769$	−52.9510			−1.90946			−0.954731	+	0.297470i	$0.903857\pi$
$770$	5.03267			0.181365
$771$	−8.18066			−0.294619
$772$		−	13.3856i		−	0.481759i
$773$	3.90976			0.140624			0.0703122	−	0.997525i	$-0.477600\pi$
$773$	3.90976			0.140624			0.0703122	+	0.997525i	$0.477600\pi$
$774$		−	0.699106i		−	0.0251288i
$775$		−	19.5615i		−	0.702671i
$776$		−	13.3281i		−	0.478451i
$777$	7.53144			0.270189
$778$			14.7955i			0.530444i
$779$			13.4912i			0.483373i
$780$		−	5.68883i		−	0.203693i
$781$			6.01968i			0.215401i
$782$	23.8625			0.853322
$783$	1.81470			0.0648520
$784$	5.28965			0.188916
$785$		−	1.35572i		−	0.0483877i
$786$	−12.0104			−0.428398
$787$	−46.6425			−1.66262			−0.831312	−	0.555806i	$-0.812410\pi$
$787$	−46.6425			−1.66262			−0.831312	+	0.555806i	$0.812410\pi$
$788$			6.79324i			0.241999i
$789$		−	16.6594i		−	0.593090i
$790$		−	27.8881i		−	0.992213i
$791$	−0.803812			−0.0285803
$792$		−	2.22737i		−	0.0791461i
$793$	−22.6329			−0.803716
$794$	−28.2174			−1.00140
$795$		−	26.7356i		−	0.948214i
$796$	−12.1456			−0.430490
$797$		−	14.5674i		−	0.516002i	−0.966145	−	0.258001i	$-0.916936\pi$
$797$		−	14.5674i		−	0.516002i	0.966145	−	0.258001i	$-0.0830638\pi$
$798$	7.87280			0.278694
$799$			13.1735i			0.466046i
$800$			8.84710i			0.312792i
$801$	−19.7656			−0.698381
$802$	−35.7100			−1.26096
$803$	3.75331			0.132452
$804$			8.68358i			0.306246i
$805$		−	17.0230i		−	0.599981i
$806$	3.67471			0.129436
$807$	13.1577	+	7.38170i	0.463172	+	0.259848i
$808$	−18.0226			−0.634033
$809$		−	36.0726i		−	1.26825i	−0.773233	−	0.634123i	$-0.781361\pi$
$809$		−	36.0726i		−	1.26825i	0.773233	−	0.634123i	$-0.218639\pi$
$810$		−	7.81682i		−	0.274655i
$811$	−45.4145			−1.59472			−0.797359	−	0.603506i	$-0.793770\pi$
$811$	−45.4145			−1.59472			−0.797359	+	0.603506i	$0.793770\pi$
$812$	0.500602			0.0175677
$813$	21.5019			0.754105
$814$			6.47423i			0.226922i
$815$		−	17.1368i		−	0.600274i
$816$	−6.27518			−0.219675
$817$			2.12417i			0.0743154i
$818$	22.8670			0.799527
$819$			4.68144i			0.163583i
$820$	−7.67128			−0.267893
$821$	−21.4500			−0.748609			−0.374305	−	0.927306i	$-0.622119\pi$
$821$	−21.4500			−0.748609			−0.374305	+	0.927306i	$0.622119\pi$
$822$		−	18.3448i		−	0.639849i
$823$	4.40136			0.153422			0.0767110	−	0.997053i	$-0.475558\pi$
$823$	4.40136			0.153422			0.0767110	+	0.997053i	$0.475558\pi$
$824$		−	7.79555i		−	0.271571i
$825$		−	8.41590i		−	0.293004i
$826$			0.0481582i			0.00167564i
$827$	47.0678			1.63671			0.818354	−	0.574715i	$-0.194887\pi$
$827$	47.0678			1.63671			0.818354	+	0.574715i	$0.194887\pi$
$828$	−7.53406			−0.261827
$829$		−	39.8839i		−	1.38522i	−0.721310	−	0.692612i	$-0.756460\pi$
$829$		−	39.8839i		−	1.38522i	0.721310	−	0.692612i	$-0.243540\pi$
$830$	15.4775			0.537232
$831$	8.68786			0.301378
$832$	−1.66196			−0.0576181
$833$		−	36.0853i		−	1.25028i
$834$			12.4357i			0.430612i
$835$		−	22.0147i		−	0.761849i
$836$			6.76767i			0.234065i
$837$	−10.4823			−0.362322
$838$			32.9463i			1.13811i
$839$			1.72103i			0.0594165i	0.999559	+	0.0297082i	$0.00945782\pi$
$839$			1.72103i			0.0594165i	−0.999559	+	0.0297082i	$0.990542\pi$
$840$			4.47657i			0.154456i
$841$	28.8535			0.994948
$842$		−	9.14584i		−	0.315186i
$843$	−6.14887			−0.211778
$844$	18.0139			0.620063
$845$	38.0969			1.31057
$846$		−	4.15925i		−	0.142998i
$847$		−	12.9872i		−	0.446247i
$848$	−7.81065			−0.268219
$849$		−	28.0148i		−	0.961466i
$850$	60.3537			2.07012
$851$	21.8990			0.750689
$852$	−5.35452			−0.183443
$853$		−	18.5544i		−	0.635292i	−0.948209	−	0.317646i	$-0.897108\pi$
$853$		−	18.5544i		−	0.635292i	0.948209	−	0.317646i	$-0.102892\pi$
$854$	17.8099			0.609443
$855$			52.4515i			1.79380i
$856$	−11.8777			−0.405970
$857$		−	12.7124i		−	0.434247i	−0.976144	−	0.217124i	$-0.930332\pi$
$857$		−	12.7124i		−	0.434247i	0.976144	−	0.217124i	$-0.0696675\pi$
$858$	1.58096			0.0539730
$859$	−22.0874			−0.753614			−0.376807	−	0.926292i	$-0.622978\pi$
$859$	−22.0874			−0.753614			−0.376807	+	0.926292i	$0.622978\pi$
$860$	−1.20783			−0.0411867
$861$	−2.48002			−0.0845187
$862$	−34.7872			−1.18486
$863$	11.1084			0.378135			0.189068	−	0.981964i	$-0.439453\pi$
$863$	11.1084			0.378135			0.189068	+	0.981964i	$0.439453\pi$
$864$	4.74084			0.161287
$865$	−27.0859			−0.920949
$866$		−	14.6378i		−	0.497414i
$867$			27.1708i			0.922768i
$868$	−2.89165			−0.0981488
$869$	7.75024			0.262909
$870$		−	1.31024i		−	0.0444214i
$871$	15.6890			0.531603
$872$	−1.71325			−0.0580178
$873$	28.7068			0.971577
$874$	22.8916			0.774319
$875$		−	18.7222i		−	0.632925i
$876$			3.33858i			0.112800i
$877$	19.7221			0.665969			0.332984	−	0.942932i	$-0.391944\pi$
$877$	19.7221			0.665969			0.332984	+	0.942932i	$0.391944\pi$
$878$		−	29.5184i		−	0.996199i
$879$			5.28164i			0.178145i
$880$	−3.84818			−0.129722
$881$			16.8249i			0.566846i	0.958995	+	0.283423i	$0.0914700\pi$
$881$			16.8249i			0.566846i	−0.958995	+	0.283423i	$0.908530\pi$
$882$			11.3931i			0.383626i
$883$			28.6944i			0.965644i	0.875719	+	0.482822i	$0.160388\pi$
$883$			28.6944i			0.965644i	−0.875719	+	0.482822i	$0.839612\pi$
$884$			11.3377i			0.381327i
$885$	0.126046			0.00423700
$886$	−29.4519			−0.989457
$887$	33.3888			1.12109			0.560543	−	0.828125i	$-0.310593\pi$
$887$	33.3888			1.12109			0.560543	+	0.828125i	$0.310593\pi$
$888$	−5.75884			−0.193254
$889$		−	17.4422i		−	0.584993i
$890$			34.1486i			1.14466i
$891$	2.17234			0.0727760
$892$		−	22.6418i		−	0.758102i
$893$			12.6375i			0.422898i
$894$	−7.39569			−0.247349
$895$			33.9660i			1.13536i
$896$	1.30781			0.0436907
$897$		−	5.34757i		−	0.178550i
$898$			15.5814i			0.519957i
$899$	0.846353			0.0282274
$900$	−19.0553			−0.635178
$901$			53.2832i			1.77512i
$902$		−	2.13189i		−	0.0709841i
$903$	−0.390475			−0.0129942
$904$	0.614627			0.0204422
$905$			86.7095i			2.88232i
$906$	15.2402			0.506322
$907$	39.2092			1.30192			0.650961	−	0.759111i	$-0.274366\pi$
$907$	39.2092			1.30192			0.650961	+	0.759111i	$0.274366\pi$
$908$		−	17.2433i		−	0.572238i
$909$		−	38.8180i		−	1.28751i
$910$	8.08804			0.268116
$911$		−	4.85288i		−	0.160783i	−0.996763	−	0.0803916i	$-0.974383\pi$
$911$		−	4.85288i		−	0.160783i	0.996763	−	0.0803916i	$-0.0256171\pi$
$912$	−6.01985			−0.199337
$913$			4.30128i			0.142351i
$914$			13.0025i			0.430085i
$915$		−	46.6145i		−	1.54103i
$916$			11.2567i			0.371931i
$917$		−	17.0757i		−	0.563890i
$918$		−	32.3414i		−	1.06742i
$919$			15.2345i			0.502541i	0.967917	+	0.251271i	$0.0808484\pi$
$919$			15.2345i			0.502541i	−0.967917	+	0.251271i	$0.919152\pi$
$920$			13.0164i			0.429139i
$921$		−	8.51040i		−	0.280427i
$922$	−9.56016			−0.314847
$923$			9.67426i			0.318432i
$924$	−1.24406			−0.0409267
$925$	55.3876			1.82113
$926$	34.5791			1.13634
$927$	16.7905			0.551471
$928$	−0.382780			−0.0125654
$929$		−	30.8070i		−	1.01075i	−0.862901	−	0.505373i	$-0.831355\pi$
$929$		−	30.8070i		−	1.01075i	0.862901	−	0.505373i	$-0.168645\pi$
$930$			7.56841i			0.248178i
$931$		−	34.6170i		−	1.13453i
$932$	13.0664			0.428004
$933$	21.3684			0.699570
$934$	−26.8002			−0.876930
$935$			26.2518i			0.858525i
$936$		−	3.57962i		−	0.117003i
$937$			52.6618i			1.72039i	0.509969	+	0.860193i	$0.329657\pi$
$937$			52.6618i			1.72039i	−0.509969	+	0.860193i	$0.670343\pi$
$938$	−12.3458			−0.403104
$939$		−	5.22289i		−	0.170442i
$940$	−7.18585			−0.234377
$941$			46.9242i			1.52968i	0.644217	+	0.764842i	$0.277183\pi$
$941$			46.9242i			1.52968i	−0.644217	+	0.764842i	$0.722817\pi$
$942$			0.335130i			0.0109191i
$943$	−7.21109			−0.234826
$944$		−	0.0368237i		−	0.00119851i
$945$	−23.0716			−0.750519
$946$		−	0.335663i		−	0.0109133i
$947$		−	21.7491i		−	0.706752i	−0.935481	−	0.353376i	$-0.885034\pi$
$947$		−	21.7491i		−	0.706752i	0.935481	−	0.353376i	$-0.114966\pi$
$948$			6.89385i			0.223902i
$949$	6.03197			0.195806
$950$	57.8980			1.87846
$951$	15.3204			0.496797
$952$		−	8.92167i		−	0.289153i
$953$			17.1576i			0.555788i	0.960612	+	0.277894i	$0.0896364\pi$
$953$			17.1576i			0.555788i	−0.960612	+	0.277894i	$0.910364\pi$
$954$		−	16.8230i		−	0.544665i
$955$	−18.2285			−0.589862
$956$	−28.8237			−0.932225
$957$	0.364124			0.0117704
$958$	14.2506			0.460417
$959$	26.0815			0.842217
$960$		−	3.42297i		−	0.110476i
$961$	26.1112			0.842296
$962$			10.4048i			0.335463i
$963$		−	25.5827i		−	0.824391i
$964$		−	13.7305i		−	0.442229i
$965$		−	49.8102i		−	1.60345i
$966$			4.20803i			0.135391i
$967$		−	13.6697i		−	0.439589i	−0.975546	−	0.219794i	$-0.929461\pi$
$967$		−	13.6697i		−	0.439589i	0.975546	−	0.219794i	$-0.0705386\pi$
$968$			9.93057i			0.319181i
$969$			41.0666i			1.31925i
$970$		−	49.5962i		−	1.59244i
$971$	−15.8659			−0.509161			−0.254580	−	0.967052i	$-0.581937\pi$
$971$	−15.8659			−0.509161			−0.254580	+	0.967052i	$0.581937\pi$
$972$			16.1548i			0.518166i
$973$	−17.6803			−0.566804
$974$			14.2840i			0.457690i
$975$		−	13.5252i		−	0.433154i
$976$	−13.6182			−0.435907
$977$	33.1179			1.05953			0.529767	−	0.848143i	$-0.322279\pi$
$977$	33.1179			1.05953			0.529767	+	0.848143i	$0.322279\pi$
$978$			4.23616i			0.135457i
$979$	−9.49007			−0.303304
$980$	19.6837			0.628772
$981$		−	3.69008i		−	0.117815i
$982$		−	38.9011i		−	1.24138i
$983$	−40.9969			−1.30760			−0.653799	−	0.756668i	$-0.726826\pi$
$983$	−40.9969			−1.30760			−0.653799	+	0.756668i	$0.726826\pi$
$984$	1.89632			0.0604524
$985$			25.2788i			0.805449i
$986$			2.61127i			0.0831599i
$987$	−2.32308			−0.0739446
$988$			10.8764i			0.346023i
$989$	−1.13538			−0.0361029
$990$		−	8.28842i		−	0.263423i
$991$			28.5196i			0.905956i	0.891522	+	0.452978i	$0.149638\pi$
$991$			28.5196i			0.905956i	−0.891522	+	0.452978i	$0.850362\pi$
$992$	2.21107			0.0702015
$993$		−	3.19359i		−	0.101345i
$994$		−	7.61272i		−	0.241461i
$995$	−45.1959			−1.43281
$996$	−3.82599			−0.121231
$997$	−0.361615			−0.0114525			−0.00572623	−	0.999984i	$-0.501823\pi$
$997$	−0.361615			−0.0114525			−0.00572623	+	0.999984i	$0.501823\pi$
$998$	2.48954			0.0788050
$999$		−	29.6802i		−	0.939040i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	538.2.b.b.537.12	yes	18
269.268	even	2	inner	538.2.b.b.537.7	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
538.2.b.b.537.7	✓	18	269.268	even	2	inner
538.2.b.b.537.12	yes	18	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 538 = 2 \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	538.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -0.919863i q^{3} -1.00000 q^{4} -3.72117 q^{5} +0.919863 q^{6} +1.30781i q^{7} -1.00000i q^{8} +2.15385 q^{9} -3.72117i q^{10} +1.03413 q^{11} +0.919863i q^{12} +1.66196 q^{13} -1.30781 q^{14} +3.42297i q^{15} +1.00000 q^{16} -6.82187i q^{17} +2.15385i q^{18} -6.54429i q^{19} +3.72117 q^{20} +1.20300 q^{21} +1.03413i q^{22} +3.49794 q^{23} -0.919863 q^{24} +8.84710 q^{25} +1.66196i q^{26} -4.74084i q^{27} -1.30781i q^{28} +0.382780i q^{29} -3.42297 q^{30} -2.21107i q^{31} +1.00000i q^{32} -0.951260i q^{33} +6.82187 q^{34} -4.86656i q^{35} -2.15385 q^{36} +6.26054 q^{37} +6.54429 q^{38} -1.52878i q^{39} +3.72117i q^{40} -2.06152 q^{41} +1.20300i q^{42} -0.324584 q^{43} -1.03413 q^{44} -8.01485 q^{45} +3.49794i q^{46} -1.93107 q^{47} -0.919863i q^{48} +5.28965 q^{49} +8.84710i q^{50} -6.27518 q^{51} -1.66196 q^{52} -7.81065 q^{53} +4.74084 q^{54} -3.84818 q^{55} +1.30781 q^{56} -6.01985 q^{57} -0.382780 q^{58} -0.0368237i q^{59} -3.42297i q^{60} -13.6182 q^{61} +2.21107 q^{62} +2.81682i q^{63} -1.00000 q^{64} -6.18444 q^{65} +0.951260 q^{66} +9.44008 q^{67} +6.82187i q^{68} -3.21763i q^{69} +4.86656 q^{70} +5.82099i q^{71} -2.15385i q^{72} +3.62943 q^{73} +6.26054i q^{74} -8.13812i q^{75} +6.54429i q^{76} +1.35244i q^{77} +1.52878 q^{78} +7.49443 q^{79} -3.72117 q^{80} +2.10064 q^{81} -2.06152i q^{82} +4.15931i q^{83} -1.20300 q^{84} +25.3853i q^{85} -0.324584i q^{86} +0.352105 q^{87} -1.03413i q^{88} -9.17684 q^{89} -8.01485i q^{90} +2.17352i q^{91} -3.49794 q^{92} -2.03388 q^{93} -1.93107i q^{94} +24.3524i q^{95} +0.919863 q^{96} +13.3281 q^{97} +5.28965i q^{98} +2.22737 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 18 q^{4} - 4 q^{5} - 4 q^{6} - 26 q^{9} - 2 q^{11} + 10 q^{13} - 24 q^{14} + 18 q^{16} + 4 q^{20} - 6 q^{21} - 20 q^{23} + 4 q^{24} + 30 q^{25} + 26 q^{36} - 10 q^{37} + 14 q^{38} - 12 q^{41} - 22 q^{43}+ \cdots - 8 q^{99}+O(q^{100}) \) 18 * q - 18 * q^4 - 4 * q^5 - 4 * q^6 - 26 * q^9 - 2 * q^11 + 10 * q^13 - 24 * q^14 + 18 * q^16 + 4 * q^20 - 6 * q^21 - 20 * q^23 + 4 * q^24 + 30 * q^25 + 26 * q^36 - 10 * q^37 + 14 * q^38 - 12 * q^41 - 22 * q^43 + 2 * q^44 + 38 * q^45 - 18 * q^47 - 14 * q^49 - 6 * q^51 - 10 * q^52 - 14 * q^53 + 4 * q^54 - 6 * q^55 + 24 * q^56 - 12 * q^57 - 4 * q^58 + 6 * q^61 + 20 * q^62 - 18 * q^64 + 14 * q^65 - 50 * q^66 - 14 * q^67 - 18 * q^70 + 8 * q^73 - 10 * q^78 - 10 * q^79 - 4 * q^80 + 18 * q^81 + 6 * q^84 + 96 * q^87 - 18 * q^89 + 20 * q^92 + 70 * q^93 - 4 * q^96 + 72 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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