LMFDB - Embedded newform 1682.2.b.j.1681.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1682,2,Mod(1681,1682)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1682, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1682.1681"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1682 = 2 \cdot 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1682.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,-12,12,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

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

Self dual:	no
Analytic conductor:	$13.4308376200$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\Q(\zeta_{28})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 58)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1681.4
Root			$0.781831 + 0.623490i$ of defining polynomial
Character	$\chi$	$=$	1682.1681
Dual form			1682.2.b.j.1681.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} +1.44504i q^{3} -1.00000 q^{4} +1.38619 q^{5} +1.44504 q^{6} -1.98639 q^{7} +1.00000i q^{8} +0.911854 q^{9} -1.38619i q^{10} -3.45467i q^{11} -1.44504i q^{12} -6.11352 q^{13} +1.98639i q^{14} +2.00311i q^{15} +1.00000 q^{16} +4.25956i q^{17} -0.911854i q^{18} +2.16857i q^{19} -1.38619 q^{20} -2.87041i q^{21} -3.45467 q^{22} -0.571217 q^{23} -1.44504 q^{24} -3.07847 q^{25} +6.11352i q^{26} +5.65279i q^{27} +1.98639 q^{28} +2.00311 q^{30} +6.03525i q^{31} -1.00000i q^{32} +4.99214 q^{33} +4.25956 q^{34} -2.75352 q^{35} -0.911854 q^{36} +2.71423i q^{37} +2.16857 q^{38} -8.83429i q^{39} +1.38619i q^{40} -9.14522i q^{41} -2.87041 q^{42} +11.0585i q^{43} +3.45467i q^{44} +1.26401 q^{45} +0.571217i q^{46} -4.98582i q^{47} +1.44504i q^{48} -3.05426 q^{49} +3.07847i q^{50} -6.15524 q^{51} +6.11352 q^{52} -10.4268 q^{53} +5.65279 q^{54} -4.78884i q^{55} -1.98639i q^{56} -3.13368 q^{57} +9.01438 q^{59} -2.00311i q^{60} +9.59621i q^{61} +6.03525 q^{62} -1.81130 q^{63} -1.00000 q^{64} -8.47452 q^{65} -4.99214i q^{66} -12.5522 q^{67} -4.25956i q^{68} -0.825432i q^{69} +2.75352i q^{70} -4.27886 q^{71} +0.911854i q^{72} -6.98180i q^{73} +2.71423 q^{74} -4.44852i q^{75} -2.16857i q^{76} +6.86232i q^{77} -8.83429 q^{78} +11.6286i q^{79} +1.38619 q^{80} -5.43296 q^{81} -9.14522 q^{82} -10.8350 q^{83} +2.87041i q^{84} +5.90457i q^{85} +11.0585 q^{86} +3.45467 q^{88} -2.85162i q^{89} -1.26401i q^{90} +12.1438 q^{91} +0.571217 q^{92} -8.72118 q^{93} -4.98582 q^{94} +3.00606i q^{95} +1.44504 q^{96} +2.42504i q^{97} +3.05426i q^{98} -3.15015i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{4} + 12 q^{5} + 16 q^{6} + 4 q^{7} - 4 q^{9} + 16 q^{13} + 12 q^{16} - 12 q^{20} + 4 q^{22} + 12 q^{23} - 16 q^{24} + 8 q^{25} - 4 q^{28} + 16 q^{30} - 24 q^{33} + 24 q^{34} + 32 q^{35} + 4 q^{36}+ \cdots + 16 q^{96}+O(q^{100}) $ 12 * q - 12 * q^4 + 12 * q^5 + 16 * q^6 + 4 * q^7 - 4 * q^9 + 16 * q^13 + 12 * q^16 - 12 * q^20 + 4 * q^22 + 12 * q^23 - 16 * q^24 + 8 * q^25 - 4 * q^28 + 16 * q^30 - 24 * q^33 + 24 * q^34 + 32 * q^35 + 4 * q^36 + 8 * q^38 + 24 * q^42 - 4 * q^45 - 8 * q^49 - 32 * q^51 - 16 * q^52 + 16 * q^53 - 4 * q^54 - 20 * q^57 - 48 * q^59 + 52 * q^62 - 48 * q^63 - 12 * q^64 - 12 * q^65 - 16 * q^67 - 40 * q^71 - 40 * q^74 + 12 * q^78 + 12 * q^80 + 12 * q^81 - 48 * q^82 - 56 * q^83 + 52 * q^86 - 4 * q^88 + 24 * q^91 - 12 * q^92 - 60 * q^93 - 36 * q^94 + 16 * q^96

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1682\mathbb{Z}\right)^\times$.

$n$	$843$
$\chi(n)$	$-1$

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
$3$	1.44504i	0.834295i	0.908839	+	0.417148i	$0.136970\pi$
$3$	1.44504i	0.834295i	−0.908839	+	0.417148i	$0.863030\pi$
$4$	−1.00000	−0.500000
$5$	1.38619	0.619924	0.309962	−	0.950749i	$-0.399684\pi$
$5$	1.38619	0.619924	0.309962	+	0.950749i	$0.399684\pi$
$6$	1.44504	0.589936
$7$	−1.98639	−0.750784	−0.375392	−	0.926866i	$-0.622492\pi$
$7$	−1.98639	−0.750784	−0.375392	+	0.926866i	$0.622492\pi$
$8$	1.00000i	0.353553i
$9$	0.911854	0.303951
$10$	− 1.38619i	− 0.438353i
$11$	− 3.45467i	− 1.04162i	−0.853672	−	0.520811i	$-0.825630\pi$
$11$	− 3.45467i	− 1.04162i	0.853672	−	0.520811i	$-0.174370\pi$
$12$	− 1.44504i	− 0.417148i
$13$	−6.11352	−1.69559	−0.847793	−	0.530328i	$-0.822069\pi$
$13$	−6.11352	−1.69559	−0.847793	+	0.530328i	$0.822069\pi$
$14$	1.98639i	0.530885i
$15$	2.00311i	0.517200i
$16$	1.00000	0.250000
$17$	4.25956i	1.03309i	0.856259	+	0.516547i	$0.172783\pi$
$17$	4.25956i	1.03309i	−0.856259	+	0.516547i	$0.827217\pi$
$18$	− 0.911854i	− 0.214926i
$19$	2.16857i	0.497504i	0.968567	+	0.248752i	$0.0800205\pi$
$19$	2.16857i	0.497504i	−0.968567	+	0.248752i	$0.919980\pi$
$20$	−1.38619	−0.309962
$21$	− 2.87041i	− 0.626376i
$22$	−3.45467	−0.736538
$23$	−0.571217	−0.119107	−0.0595535	−	0.998225i	$-0.518968\pi$
$23$	−0.571217	−0.119107	−0.0595535	+	0.998225i	$0.518968\pi$
$24$	−1.44504	−0.294968
$25$	−3.07847	−0.615694
$26$	6.11352i	1.19896i
$27$	5.65279i	1.08788i
$28$	1.98639	0.375392
$29$	0	0
$29$	0	0
$30$	2.00311	0.365716
$31$	6.03525i	1.08396i	0.840391	+	0.541981i	$0.182326\pi$
$31$	6.03525i	1.08396i	−0.840391	+	0.541981i	$0.817674\pi$
$32$	− 1.00000i	− 0.176777i
$33$	4.99214	0.869020
$34$	4.25956	0.730508
$35$	−2.75352	−0.465429
$36$	−0.911854	−0.151976
$37$	2.71423i	0.446216i	0.974794	+	0.223108i	$0.0716203\pi$
$37$	2.71423i	0.446216i	−0.974794	+	0.223108i	$0.928380\pi$
$38$	2.16857	0.351789
$39$	− 8.83429i	− 1.41462i
$40$	1.38619i	0.219176i
$41$	− 9.14522i	− 1.42824i	−0.700021	−	0.714122i	$-0.746826\pi$
$41$	− 9.14522i	− 1.42824i	0.700021	−	0.714122i	$-0.253174\pi$
$42$	−2.87041	−0.442915
$43$	11.0585i	1.68641i	0.537590	+	0.843206i	$0.319335\pi$
$43$	11.0585i	1.68641i	−0.537590	+	0.843206i	$0.680665\pi$
$44$	3.45467i	0.520811i
$45$	1.26401	0.188427
$46$	0.571217i	0.0842213i
$47$	− 4.98582i	− 0.727256i	−0.931544	−	0.363628i	$-0.881538\pi$
$47$	− 4.98582i	− 0.727256i	0.931544	−	0.363628i	$-0.118462\pi$
$48$	1.44504i	0.208574i
$49$	−3.05426	−0.436323
$50$	3.07847i	0.435361i
$51$	−6.15524	−0.861906
$52$	6.11352	0.847793
$53$	−10.4268	−1.43223	−0.716113	−	0.697984i	$-0.754081\pi$
$53$	−10.4268	−1.43223	−0.716113	+	0.697984i	$0.754081\pi$
$54$	5.65279	0.769248
$55$	− 4.78884i	− 0.645727i
$56$	− 1.98639i	− 0.265442i
$57$	−3.13368	−0.415066
$58$	0	0
$59$	9.01438	1.17357	0.586786	−	0.809742i	$-0.300393\pi$
$59$	9.01438	1.17357	0.586786	+	0.809742i	$0.300393\pi$
$60$	− 2.00311i	− 0.258600i
$61$	9.59621i	1.22867i	0.789046	+	0.614335i	$0.210576\pi$
$61$	9.59621i	1.22867i	−0.789046	+	0.614335i	$0.789424\pi$
$62$	6.03525	0.766477
$63$	−1.81130	−0.228202
$64$	−1.00000	−0.125000
$65$	−8.47452	−1.05113
$66$	− 4.99214i	− 0.614490i
$67$	−12.5522	−1.53349	−0.766747	−	0.641950i	$-0.778126\pi$
$67$	−12.5522	−1.53349	−0.766747	+	0.641950i	$0.778126\pi$
$68$	− 4.25956i	− 0.516547i
$69$	− 0.825432i	− 0.0993703i
$70$	2.75352i	0.329108i
$71$	−4.27886	−0.507808	−0.253904	−	0.967229i	$-0.581715\pi$
$71$	−4.27886	−0.507808	−0.253904	+	0.967229i	$0.581715\pi$
$72$	0.911854i	0.107463i
$73$	− 6.98180i	− 0.817158i	−0.912723	−	0.408579i	$-0.866024\pi$
$73$	− 6.98180i	− 0.817158i	0.912723	−	0.408579i	$-0.133976\pi$
$74$	2.71423	0.315523
$75$	− 4.44852i	− 0.513671i
$76$	− 2.16857i	− 0.248752i
$77$	6.86232i	0.782034i
$78$	−8.83429	−1.00029
$79$	11.6286i	1.30832i	0.756354	+	0.654162i	$0.226979\pi$
$79$	11.6286i	1.30832i	−0.756354	+	0.654162i	$0.773021\pi$
$80$	1.38619	0.154981
$81$	−5.43296	−0.603662
$82$	−9.14522	−1.00992
$83$	−10.8350	−1.18930	−0.594650	−	0.803985i	$-0.702709\pi$
$83$	−10.8350	−1.18930	−0.594650	+	0.803985i	$0.702709\pi$
$84$	2.87041i	0.313188i
$85$	5.90457i	0.640440i
$86$	11.0585	1.19247
$87$	0	0
$88$	3.45467	0.368269
$89$	− 2.85162i	− 0.302271i	−0.988513	−	0.151136i	$-0.951707\pi$
$89$	− 2.85162i	− 0.302271i	0.988513	−	0.151136i	$-0.0482930\pi$
$90$	− 1.26401i	− 0.133238i
$91$	12.1438	1.27302
$92$	0.571217	0.0595535
$93$	−8.72118	−0.904344
$94$	−4.98582	−0.514248
$95$	3.00606i	0.308415i
$96$	1.44504	0.147484
$97$	2.42504i	0.246226i	0.992393	+	0.123113i	$0.0392877\pi$
$97$	2.42504i	0.246226i	−0.992393	+	0.123113i	$0.960712\pi$
$98$	3.05426i	0.308527i
$99$	− 3.15015i	− 0.316602i
$100$	3.07847	0.307847
$101$	− 7.23806i	− 0.720214i	−0.932911	−	0.360107i	$-0.882740\pi$
$101$	− 7.23806i	− 0.720214i	0.932911	−	0.360107i	$-0.117260\pi$
$102$	6.15524i	0.609460i
$103$	−7.94165	−0.782514	−0.391257	−	0.920281i	$-0.627960\pi$
$103$	−7.94165	−0.782514	−0.391257	+	0.920281i	$0.627960\pi$
$104$	− 6.11352i	− 0.599480i
$105$	− 3.97895i	− 0.388306i
$106$	10.4268i	1.01274i
$107$	13.1191	1.26827	0.634135	−	0.773222i	$-0.281356\pi$
$107$	13.1191	1.26827	0.634135	+	0.773222i	$0.281356\pi$
$108$	− 5.65279i	− 0.543940i
$109$	−5.23723	−0.501636	−0.250818	−	0.968034i	$-0.580700\pi$
$109$	−5.23723	−0.501636	−0.250818	+	0.968034i	$0.580700\pi$
$110$	−4.78884	−0.456598
$111$	−3.92217	−0.372276
$112$	−1.98639	−0.187696
$113$	0.138924i	0.0130689i	0.999979	+	0.00653443i	$0.00207999\pi$
$113$	0.138924i	0.0130689i	−0.999979	+	0.00653443i	$0.997920\pi$
$114$	3.13368i	0.293496i
$115$	−0.791817	−0.0738373
$116$	0	0
$117$	−5.57464	−0.515375
$118$	− 9.01438i	− 0.829841i
$119$	− 8.46114i	− 0.775631i
$120$	−2.00311	−0.182858
$121$	−0.934743	−0.0849767
$122$	9.59621	0.868800
$123$	13.2152	1.19158
$124$	− 6.03525i	− 0.541981i
$125$	−11.1983	−1.00161
$126$	1.81130i	0.161363i
$127$	6.49143i	0.576021i	0.957627	+	0.288011i	$0.0929939\pi$
$127$	6.49143i	0.576021i	−0.957627	+	0.288011i	$0.907006\pi$
$128$	1.00000i	0.0883883i
$129$	−15.9801	−1.40697
$130$	8.47452i	0.743264i
$131$	2.25044i	0.196622i	0.995156	+	0.0983111i	$0.0313440\pi$
$131$	2.25044i	0.196622i	−0.995156	+	0.0983111i	$0.968656\pi$
$132$	−4.99214	−0.434510
$133$	− 4.30762i	− 0.373518i
$134$	12.5522i	1.08434i
$135$	7.83586i	0.674404i
$136$	−4.25956	−0.365254
$137$	7.56230i	0.646091i	0.946383	+	0.323045i	$0.104707\pi$
$137$	7.56230i	0.646091i	−0.946383	+	0.323045i	$0.895293\pi$
$138$	−0.825432	−0.0702654
$139$	6.23282	0.528661	0.264331	−	0.964432i	$-0.414849\pi$
$139$	6.23282	0.528661	0.264331	+	0.964432i	$0.414849\pi$
$140$	2.75352	0.232715
$141$	7.20471	0.606746
$142$	4.27886i	0.359074i
$143$	21.1202i	1.76616i
$144$	0.911854	0.0759878
$145$	0	0
$146$	−6.98180	−0.577818
$147$	− 4.41353i	− 0.364022i
$148$	− 2.71423i	− 0.223108i
$149$	0.725140	0.0594058	0.0297029	−	0.999559i	$-0.490544\pi$
$149$	0.725140	0.0594058	0.0297029	+	0.999559i	$0.490544\pi$
$150$	−4.44852	−0.363220
$151$	−9.36422	−0.762049	−0.381025	−	0.924565i	$-0.624429\pi$
$151$	−9.36422	−0.762049	−0.381025	+	0.924565i	$0.624429\pi$
$152$	−2.16857	−0.175894
$153$	3.88410i	0.314011i
$154$	6.86232	0.552981
$155$	8.36601i	0.671974i
$156$	8.83429i	0.707309i
$157$	− 11.8225i	− 0.943535i	−0.881723	−	0.471767i	$-0.843616\pi$
$157$	− 11.8225i	− 0.943535i	0.881723	−	0.471767i	$-0.156384\pi$
$158$	11.6286	0.925125
$159$	− 15.0671i	− 1.19490i
$160$	− 1.38619i	− 0.109588i
$161$	1.13466	0.0894236
$162$	5.43296i	0.426854i
$163$	13.5610i	1.06218i	0.847315	+	0.531091i	$0.178218\pi$
$163$	13.5610i	1.06218i	−0.847315	+	0.531091i	$0.821782\pi$
$164$	9.14522i	0.714122i
$165$	6.92007	0.538727
$166$	10.8350i	0.840962i
$167$	4.64283	0.359273	0.179637	−	0.983733i	$-0.442508\pi$
$167$	4.64283	0.359273	0.179637	+	0.983733i	$0.442508\pi$
$168$	2.87041	0.221457
$169$	24.3751	1.87501
$170$	5.90457	0.452860
$171$	1.97742i	0.151217i
$172$	− 11.0585i	− 0.843206i
$173$	9.49522	0.721908	0.360954	−	0.932584i	$-0.382451\pi$
$173$	9.49522	0.721908	0.360954	+	0.932584i	$0.382451\pi$
$174$	0	0
$175$	6.11504	0.462253
$176$	− 3.45467i	− 0.260406i
$177$	13.0262i	0.979106i
$178$	−2.85162	−0.213738
$179$	7.53207	0.562974	0.281487	−	0.959565i	$-0.409172\pi$
$179$	7.53207	0.562974	0.281487	+	0.959565i	$0.409172\pi$
$180$	−1.26401	−0.0942134
$181$	−18.4798	−1.37359	−0.686796	−	0.726850i	$-0.740984\pi$
$181$	−18.4798	−1.37359	−0.686796	+	0.726850i	$0.740984\pi$
$182$	− 12.1438i	− 0.900160i
$183$	−13.8669	−1.02507
$184$	− 0.571217i	− 0.0421107i
$185$	3.76244i	0.276620i
$186$	8.72118i	0.639468i
$187$	14.7154	1.07609
$188$	4.98582i	0.363628i
$189$	− 11.2286i	− 0.816764i
$190$	3.00606	0.218082
$191$	11.8574i	0.857973i	0.903311	+	0.428986i	$0.141129\pi$
$191$	11.8574i	0.857973i	−0.903311	+	0.428986i	$0.858871\pi$
$192$	− 1.44504i	− 0.104287i
$193$	− 16.9564i	− 1.22055i	−0.792189	−	0.610276i	$-0.791059\pi$
$193$	− 16.9564i	− 1.22055i	0.792189	−	0.610276i	$-0.208941\pi$
$194$	2.42504	0.174108
$195$	− 12.2460i	− 0.876957i
$196$	3.05426	0.218161
$197$	1.43781	0.102439	0.0512197	−	0.998687i	$-0.483689\pi$
$197$	1.43781	0.102439	0.0512197	+	0.998687i	$0.483689\pi$
$198$	−3.15015	−0.223872
$199$	4.53854	0.321729	0.160864	−	0.986977i	$-0.448572\pi$
$199$	4.53854	0.321729	0.160864	+	0.986977i	$0.448572\pi$
$200$	− 3.07847i	− 0.217681i
$201$	− 18.1384i	− 1.27939i
$202$	−7.23806	−0.509268
$203$	0	0
$204$	6.15524	0.430953
$205$	− 12.6770i	− 0.885403i
$206$	7.94165i	0.553321i
$207$	−0.520866	−0.0362027
$208$	−6.11352	−0.423896
$209$	7.49170	0.518211
$210$	−3.97895	−0.274574
$211$	10.4852i	0.721831i	0.932598	+	0.360916i	$0.117536\pi$
$211$	10.4852i	0.721831i	−0.932598	+	0.360916i	$0.882464\pi$
$212$	10.4268	0.716113
$213$	− 6.18314i	− 0.423662i
$214$	− 13.1191i	− 0.896803i
$215$	15.3293i	1.04545i
$216$	−5.65279	−0.384624
$217$	− 11.9883i	− 0.813822i
$218$	5.23723i	0.354710i
$219$	10.0890	0.681751
$220$	4.78884i	0.322863i
$221$	− 26.0409i	− 1.75170i
$222$	3.92217i	0.263239i
$223$	−21.7825	−1.45867	−0.729333	−	0.684159i	$-0.760169\pi$
$223$	−21.7825	−1.45867	−0.729333	+	0.684159i	$0.760169\pi$
$224$	1.98639i	0.132721i
$225$	−2.80711	−0.187141
$226$	0.138924	0.00924109
$227$	19.1320	1.26983	0.634916	−	0.772581i	$-0.281035\pi$
$227$	19.1320	1.26983	0.634916	+	0.772581i	$0.281035\pi$
$228$	3.13368	0.207533
$229$	− 25.5961i	− 1.69144i	−0.533631	−	0.845718i	$-0.679173\pi$
$229$	− 25.5961i	− 1.69144i	0.533631	−	0.845718i	$-0.320827\pi$
$230$	0.791817i	0.0522108i
$231$	−9.91633	−0.652447
$232$	0	0
$233$	−27.0319	−1.77092	−0.885461	−	0.464714i	$-0.846157\pi$
$233$	−27.0319	−1.77092	−0.885461	+	0.464714i	$0.846157\pi$
$234$	5.57464i	0.364425i
$235$	− 6.91130i	− 0.450844i
$236$	−9.01438	−0.586786
$237$	−16.8039	−1.09153
$238$	−8.46114	−0.548454
$239$	26.0650	1.68600	0.843002	−	0.537910i	$-0.180786\pi$
$239$	26.0650	1.68600	0.843002	+	0.537910i	$0.180786\pi$
$240$	2.00311i	0.129300i
$241$	18.9186	1.21866	0.609328	−	0.792918i	$-0.291439\pi$
$241$	18.9186	1.21866	0.609328	+	0.792918i	$0.291439\pi$
$242$	0.934743i	0.0600876i
$243$	9.10752i	0.584248i
$244$	− 9.59621i	− 0.614335i
$245$	−4.23379	−0.270487
$246$	− 13.2152i	− 0.842572i
$247$	− 13.2576i	− 0.843561i
$248$	−6.03525	−0.383238
$249$	− 15.6571i	− 0.992227i
$250$	11.1983i	0.708244i
$251$	− 23.2682i	− 1.46868i	−0.678783	−	0.734339i	$-0.737492\pi$
$251$	− 23.2682i	− 1.46868i	0.678783	−	0.734339i	$-0.262508\pi$
$252$	1.81130	0.114101
$253$	1.97337i	0.124064i
$254$	6.49143	0.407309
$255$	−8.53235	−0.534316
$256$	1.00000	0.0625000
$257$	−11.4731	−0.715669	−0.357835	−	0.933785i	$-0.616485\pi$
$257$	−11.4731	−0.715669	−0.357835	+	0.933785i	$0.616485\pi$
$258$	15.9801i	0.994875i
$259$	− 5.39151i	− 0.335012i
$260$	8.47452	0.525567
$261$	0	0
$262$	2.25044	0.139033
$263$	21.6210i	1.33321i	0.745412	+	0.666605i	$0.232253\pi$
$263$	21.6210i	1.33321i	−0.745412	+	0.666605i	$0.767747\pi$
$264$	4.99214i	0.307245i
$265$	−14.4535	−0.887872
$266$	−4.30762	−0.264117
$267$	4.12071	0.252184
$268$	12.5522	0.766747
$269$	4.55084i	0.277469i	0.990330	+	0.138735i	$0.0443035\pi$
$269$	4.55084i	0.277469i	−0.990330	+	0.138735i	$0.955696\pi$
$270$	7.83586	0.476875
$271$	8.03071i	0.487831i	0.969797	+	0.243915i	$0.0784319\pi$
$271$	8.03071i	0.487831i	−0.969797	+	0.243915i	$0.921568\pi$
$272$	4.25956i	0.258274i
$273$	17.5483i	1.06207i
$274$	7.56230	0.456855
$275$	10.6351i	0.641320i
$276$	0.825432i	0.0496852i
$277$	28.4727	1.71076	0.855380	−	0.518001i	$-0.173323\pi$
$277$	28.4727	1.71076	0.855380	+	0.518001i	$0.173323\pi$
$278$	− 6.23282i	− 0.373820i
$279$	5.50326i	0.329472i
$280$	− 2.75352i	− 0.164554i
$281$	19.9725	1.19146	0.595729	−	0.803185i	$-0.296863\pi$
$281$	19.9725	1.19146	0.595729	+	0.803185i	$0.296863\pi$
$282$	− 7.20471i	− 0.429034i
$283$	−4.28248	−0.254567	−0.127284	−	0.991866i	$-0.540626\pi$
$283$	−4.28248	−0.254567	−0.127284	+	0.991866i	$0.540626\pi$
$284$	4.27886	0.253904
$285$	−4.34388	−0.257309
$286$	21.1202	1.24886
$287$	18.1660i	1.07230i
$288$	− 0.911854i	− 0.0537315i
$289$	−1.14384	−0.0672846
$290$	0	0
$291$	−3.50429	−0.205425
$292$	6.98180i	0.408579i
$293$	− 3.04294i	− 0.177771i	−0.996042	−	0.0888853i	$-0.971670\pi$
$293$	− 3.04294i	− 0.177771i	0.996042	−	0.0888853i	$-0.0283305\pi$
$294$	−4.41353	−0.257403
$295$	12.4957	0.727526
$296$	−2.71423	−0.157761
$297$	19.5285	1.13316
$298$	− 0.725140i	− 0.0420062i
$299$	3.49215	0.201956
$300$	4.44852i	0.256835i
$301$	− 21.9666i	− 1.26613i
$302$	9.36422i	0.538850i
$303$	10.4593	0.600871
$304$	2.16857i	0.124376i
$305$	13.3022i	0.761682i
$306$	3.88410	0.222039
$307$	23.6143i	1.34774i	0.738850	+	0.673870i	$0.235369\pi$
$307$	23.6143i	1.34774i	−0.738850	+	0.673870i	$0.764631\pi$
$308$	− 6.86232i	− 0.391017i
$309$	− 11.4760i	− 0.652848i
$310$	8.36601	0.475158
$311$	− 13.8346i	− 0.784490i	−0.919861	−	0.392245i	$-0.871699\pi$
$311$	− 13.8346i	− 0.784490i	0.919861	−	0.392245i	$-0.128301\pi$
$312$	8.83429	0.500143
$313$	−8.59812	−0.485994	−0.242997	−	0.970027i	$-0.578131\pi$
$313$	−8.59812	−0.485994	−0.242997	+	0.970027i	$0.578131\pi$
$314$	−11.8225	−0.667180
$315$	−2.51081	−0.141468
$316$	− 11.6286i	− 0.654162i
$317$	− 12.6827i	− 0.712333i	−0.934423	−	0.356166i	$-0.884084\pi$
$317$	− 12.6827i	− 0.712333i	0.934423	−	0.356166i	$-0.115916\pi$
$318$	−15.0671	−0.844922
$319$	0	0
$320$	−1.38619	−0.0774905
$321$	18.9576i	1.05811i
$322$	− 1.13466i	− 0.0632320i
$323$	−9.23715	−0.513969
$324$	5.43296	0.301831
$325$	18.8203	1.04396
$326$	13.5610	0.751075
$327$	− 7.56802i	− 0.418512i
$328$	9.14522	0.504960
$329$	9.90377i	0.546012i
$330$	− 6.92007i	− 0.380937i
$331$	5.21791i	0.286802i	0.989665	+	0.143401i	$0.0458039\pi$
$331$	5.21791i	0.286802i	−0.989665	+	0.143401i	$0.954196\pi$
$332$	10.8350	0.594650
$333$	2.47498i	0.135628i
$334$	− 4.64283i	− 0.254045i
$335$	−17.3998	−0.950650
$336$	− 2.87041i	− 0.156594i
$337$	3.68768i	0.200881i	0.994943	+	0.100440i	$0.0320252\pi$
$337$	3.68768i	0.200881i	−0.994943	+	0.100440i	$0.967975\pi$
$338$	− 24.3751i	− 1.32583i
$339$	−0.200751	−0.0109033
$340$	− 5.90457i	− 0.320220i
$341$	20.8498	1.12908
$342$	1.97742	0.106927
$343$	19.9717	1.07837
$344$	−11.0585	−0.596237
$345$	− 1.14421i	− 0.0616021i
$346$	− 9.49522i	− 0.510466i
$347$	3.61233	0.193920	0.0969599	−	0.995288i	$-0.469088\pi$
$347$	3.61233	0.193920	0.0969599	+	0.995288i	$0.469088\pi$
$348$	0	0
$349$	14.2824	0.764521	0.382260	−	0.924055i	$-0.375146\pi$
$349$	14.2824	0.764521	0.382260	+	0.924055i	$0.375146\pi$
$350$	− 6.11504i	− 0.326862i
$351$	− 34.5585i	− 1.84459i
$352$	−3.45467	−0.184135
$353$	−1.22175	−0.0650271	−0.0325136	−	0.999471i	$-0.510351\pi$
$353$	−1.22175	−0.0650271	−0.0325136	+	0.999471i	$0.510351\pi$
$354$	13.0262	0.692333
$355$	−5.93133	−0.314802
$356$	2.85162i	0.151136i
$357$	12.2267	0.647106
$358$	− 7.53207i	− 0.398083i
$359$	− 31.1679i	− 1.64498i	−0.568779	−	0.822490i	$-0.692584\pi$
$359$	− 31.1679i	− 1.64498i	0.568779	−	0.822490i	$-0.307416\pi$
$360$	1.26401i	0.0666189i
$361$	14.2973	0.752489
$362$	18.4798i	0.971277i
$363$	− 1.35074i	− 0.0708956i
$364$	−12.1438	−0.636509
$365$	− 9.67812i	− 0.506576i
$366$	13.8669i	0.724836i
$367$	− 4.83098i	− 0.252175i	−0.992019	−	0.126088i	$-0.959758\pi$
$367$	− 4.83098i	− 0.252175i	0.992019	−	0.126088i	$-0.0402420\pi$
$368$	−0.571217	−0.0297767
$369$	− 8.33911i	− 0.434117i
$370$	3.76244	0.195600
$371$	20.7116	1.07529
$372$	8.72118	0.452172
$373$	−0.936592	−0.0484949	−0.0242474	−	0.999706i	$-0.507719\pi$
$373$	−0.936592	−0.0484949	−0.0242474	+	0.999706i	$0.507719\pi$
$374$	− 14.7154i	− 0.760914i
$375$	− 16.1820i	− 0.835637i
$376$	4.98582	0.257124
$377$	0	0
$378$	−11.2286	−0.577539
$379$	1.44585i	0.0742685i	0.999310	+	0.0371343i	$0.0118229\pi$
$379$	1.44585i	0.0742685i	−0.999310	+	0.0371343i	$0.988177\pi$
$380$	− 3.00606i	− 0.154208i
$381$	−9.38039	−0.480572
$382$	11.8574	0.606678
$383$	−7.89594	−0.403464	−0.201732	−	0.979441i	$-0.564657\pi$
$383$	−7.89594	−0.403464	−0.201732	+	0.979441i	$0.564657\pi$
$384$	−1.44504	−0.0737420
$385$	9.51249i	0.484802i
$386$	−16.9564	−0.863061
$387$	10.0838i	0.512587i
$388$	− 2.42504i	− 0.123113i
$389$	8.04652i	0.407975i	0.978974	+	0.203987i	$0.0653901\pi$
$389$	8.04652i	0.407975i	−0.978974	+	0.203987i	$0.934610\pi$
$390$	−12.2460	−0.620102
$391$	− 2.43313i	− 0.123049i
$392$	− 3.05426i	− 0.154263i
$393$	−3.25198	−0.164041
$394$	− 1.43781i	− 0.0724356i
$395$	16.1195i	0.811062i
$396$	3.15015i	0.158301i
$397$	−2.56721	−0.128844	−0.0644222	−	0.997923i	$-0.520520\pi$
$397$	−2.56721	−0.128844	−0.0644222	+	0.997923i	$0.520520\pi$
$398$	− 4.53854i	− 0.227497i
$399$	6.22470	0.311625
$400$	−3.07847	−0.153923
$401$	−7.62689	−0.380869	−0.190434	−	0.981700i	$-0.560990\pi$
$401$	−7.62689	−0.380869	−0.190434	+	0.981700i	$0.560990\pi$
$402$	−18.1384	−0.904663
$403$	− 36.8966i	− 1.83795i
$404$	7.23806i	0.360107i
$405$	−7.53113	−0.374225
$406$	0	0
$407$	9.37676	0.464789
$408$	− 6.15524i	− 0.304730i
$409$	2.17471i	0.107533i	0.998554	+	0.0537664i	$0.0171226\pi$
$409$	2.17471i	0.107533i	−0.998554	+	0.0537664i	$0.982877\pi$
$410$	−12.6770	−0.626074
$411$	−10.9278	−0.539031
$412$	7.94165	0.391257
$413$	−17.9061	−0.881100
$414$	0.520866i	0.0255992i
$415$	−15.0194	−0.737276
$416$	6.11352i	0.299740i
$417$	9.00669i	0.441059i
$418$	− 7.49170i	− 0.366431i
$419$	11.0437	0.539521	0.269761	−	0.962927i	$-0.413055\pi$
$419$	11.0437	0.539521	0.269761	+	0.962927i	$0.413055\pi$
$420$	3.97895i	0.194153i
$421$	− 3.65914i	− 0.178336i	−0.996017	−	0.0891679i	$-0.971579\pi$
$421$	− 3.65914i	− 0.178336i	0.996017	−	0.0891679i	$-0.0284208\pi$
$422$	10.4852	0.510412
$423$	− 4.54634i	− 0.221050i
$424$	− 10.4268i	− 0.506369i
$425$	− 13.1129i	− 0.636070i
$426$	−6.18314	−0.299574
$427$	− 19.0618i	− 0.922466i
$428$	−13.1191	−0.634135
$429$	−30.5196	−1.47350
$430$	15.3293	0.739243
$431$	6.36313	0.306501	0.153251	−	0.988187i	$-0.451026\pi$
$431$	6.36313	0.306501	0.153251	+	0.988187i	$0.451026\pi$
$432$	5.65279i	0.271970i
$433$	− 21.8657i	− 1.05080i	−0.850856	−	0.525399i	$-0.823916\pi$
$433$	− 21.8657i	− 1.05080i	0.850856	−	0.525399i	$-0.176084\pi$
$434$	−11.9883	−0.575459
$435$	0	0
$436$	5.23723	0.250818
$437$	− 1.23872i	− 0.0592562i
$438$	− 10.0890i	− 0.482071i
$439$	−20.6588	−0.985993	−0.492996	−	0.870031i	$-0.664098\pi$
$439$	−20.6588	−0.985993	−0.492996	+	0.870031i	$0.664098\pi$
$440$	4.78884	0.228299
$441$	−2.78504	−0.132621
$442$	−26.0409	−1.23864
$443$	− 2.56232i	− 0.121740i	−0.998146	−	0.0608698i	$-0.980613\pi$
$443$	− 2.56232i	− 0.121740i	0.998146	−	0.0608698i	$-0.0193874\pi$
$444$	3.92217	0.186138
$445$	− 3.95290i	− 0.187385i
$446$	21.7825i	1.03143i
$447$	1.04786i	0.0495619i
$448$	1.98639	0.0938480
$449$	− 40.7652i	− 1.92383i	−0.273351	−	0.961914i	$-0.588132\pi$
$449$	− 40.7652i	− 1.92383i	0.273351	−	0.961914i	$-0.411868\pi$
$450$	2.80711i	0.132329i
$451$	−31.5937	−1.48769
$452$	− 0.138924i	− 0.00653443i
$453$	− 13.5317i	− 0.635774i
$454$	− 19.1320i	− 0.897907i
$455$	16.8337	0.789175
$456$	− 3.13368i	− 0.146748i
$457$	25.2066	1.17911	0.589557	−	0.807727i	$-0.299302\pi$
$457$	25.2066	1.17911	0.589557	+	0.807727i	$0.299302\pi$
$458$	−25.5961	−1.19603
$459$	−24.0784	−1.12388
$460$	0.791817	0.0369186
$461$	41.0649i	1.91258i	0.292418	+	0.956290i	$0.405540\pi$
$461$	41.0649i	1.91258i	−0.292418	+	0.956290i	$0.594460\pi$
$462$	9.91633i	0.461350i
$463$	−33.1170	−1.53908	−0.769539	−	0.638600i	$-0.779514\pi$
$463$	−33.1170	−1.53908	−0.769539	+	0.638600i	$0.779514\pi$
$464$	0	0
$465$	−12.0892	−0.560625
$466$	27.0319i	1.25223i
$467$	− 15.1228i	− 0.699800i	−0.936787	−	0.349900i	$-0.886216\pi$
$467$	− 15.1228i	− 0.699800i	0.936787	−	0.349900i	$-0.113784\pi$
$468$	5.57464	0.257688
$469$	24.9335	1.15132
$470$	−6.91130	−0.318795
$471$	17.0839	0.787186
$472$	9.01438i	0.414921i
$473$	38.2036	1.75660
$474$	16.8039i	0.771828i
$475$	− 6.67588i	− 0.306310i
$476$	8.46114i	0.387816i
$477$	−9.50769	−0.435327
$478$	− 26.0650i	− 1.19218i
$479$	24.5410i	1.12131i	0.828050	+	0.560654i	$0.189450\pi$
$479$	24.5410i	1.12131i	−0.828050	+	0.560654i	$0.810550\pi$
$480$	2.00311	0.0914289
$481$	− 16.5935i	− 0.756598i
$482$	− 18.9186i	− 0.861720i
$483$	1.63963i	0.0746057i
$484$	0.934743	0.0424883
$485$	3.36158i	0.152641i
$486$	9.10752	0.413126
$487$	28.4159	1.28765	0.643823	−	0.765174i	$-0.277347\pi$
$487$	28.4159	1.28765	0.643823	+	0.765174i	$0.277347\pi$
$488$	−9.59621	−0.434400
$489$	−19.5962	−0.886173
$490$	4.23379i	0.191263i
$491$	22.4508i	1.01319i	0.862184	+	0.506596i	$0.169096\pi$
$491$	22.4508i	1.01319i	−0.862184	+	0.506596i	$0.830904\pi$
$492$	−13.2152	−0.595789
$493$	0	0
$494$	−13.2576	−0.596488
$495$	− 4.36672i	− 0.196270i
$496$	6.03525i	0.270991i
$497$	8.49948	0.381254
$498$	−15.6571	−0.701610
$499$	19.6240	0.878490	0.439245	−	0.898367i	$-0.355246\pi$
$499$	19.6240	0.878490	0.439245	+	0.898367i	$0.355246\pi$
$500$	11.1983	0.500804
$501$	6.70909i	0.299740i
$502$	−23.2682	−1.03851
$503$	− 6.46066i	− 0.288067i	−0.989573	−	0.144033i	$-0.953993\pi$
$503$	− 6.46066i	− 0.288067i	0.989573	−	0.144033i	$-0.0460072\pi$
$504$	− 1.81130i	− 0.0806816i
$505$	− 10.0334i	− 0.446478i
$506$	1.97337	0.0877268
$507$	35.2231i	1.56431i
$508$	− 6.49143i	− 0.288011i
$509$	−32.4400	−1.43788	−0.718938	−	0.695074i	$-0.755372\pi$
$509$	−32.4400	−1.43788	−0.718938	+	0.695074i	$0.755372\pi$
$510$	8.53235i	0.377819i
$511$	13.8686i	0.613509i
$512$	− 1.00000i	− 0.0441942i
$513$	−12.2585	−0.541225
$514$	11.4731i	0.506055i
$515$	−11.0087	−0.485099
$516$	15.9801	0.703483
$517$	−17.2243	−0.757526
$518$	−5.39151	−0.236889
$519$	13.7210i	0.602285i
$520$	− 8.47452i	− 0.371632i
$521$	−22.9436	−1.00518	−0.502588	−	0.864526i	$-0.667619\pi$
$521$	−22.9436	−1.00518	−0.502588	+	0.864526i	$0.667619\pi$
$522$	0	0
$523$	−36.0578	−1.57670	−0.788349	−	0.615228i	$-0.789064\pi$
$523$	−36.0578	−1.57670	−0.788349	+	0.615228i	$0.789064\pi$
$524$	− 2.25044i	− 0.0983111i
$525$	8.83648i	0.385656i
$526$	21.6210	0.942721
$527$	−25.7075	−1.11984
$528$	4.99214	0.217255
$529$	−22.6737	−0.985814
$530$	14.4535i	0.627820i
$531$	8.21980	0.356709
$532$	4.30762i	0.186759i
$533$	55.9095i	2.42171i
$534$	− 4.12071i	− 0.178321i
$535$	18.1856	0.786232
$536$	− 12.5522i	− 0.542172i
$537$	10.8842i	0.469686i
$538$	4.55084	0.196201
$539$	10.5515i	0.454484i
$540$	− 7.83586i	− 0.337202i
$541$	13.2755i	0.570759i	0.958415	+	0.285379i	$0.0921196\pi$
$541$	13.2755i	0.570759i	−0.958415	+	0.285379i	$0.907880\pi$
$542$	8.03071	0.344948
$543$	− 26.7041i	− 1.14598i
$544$	4.25956	0.182627
$545$	−7.25981	−0.310976
$546$	17.5483	0.751000
$547$	26.2472	1.12225	0.561125	−	0.827731i	$-0.310369\pi$
$547$	26.2472	1.12225	0.561125	+	0.827731i	$0.310369\pi$
$548$	− 7.56230i	− 0.323045i
$549$	8.75035i	0.373456i
$550$	10.6351	0.453482
$551$	0	0
$552$	0.825432	0.0351327
$553$	− 23.0990i	− 0.982270i
$554$	− 28.4727i	− 1.20969i
$555$	−5.43689	−0.230783
$556$	−6.23282	−0.264331
$557$	−6.98126	−0.295805	−0.147903	−	0.989002i	$-0.547252\pi$
$557$	−6.98126	−0.295805	−0.147903	+	0.989002i	$0.547252\pi$
$558$	5.50326	0.232972
$559$	− 67.6066i	− 2.85946i
$560$	−2.75352	−0.116357
$561$	21.2643i	0.897780i
$562$	− 19.9725i	− 0.842488i
$563$	− 23.7025i	− 0.998942i	−0.866331	−	0.499471i	$-0.833528\pi$
$563$	− 23.7025i	− 0.998942i	0.866331	−	0.499471i	$-0.166472\pi$
$564$	−7.20471	−0.303373
$565$	0.192575i	0.00810171i
$566$	4.28248i	0.180006i
$567$	10.7920	0.453220
$568$	− 4.27886i	− 0.179537i
$569$	4.44182i	0.186211i	0.995656	+	0.0931054i	$0.0296794\pi$
$569$	4.44182i	0.186211i	−0.995656	+	0.0931054i	$0.970321\pi$
$570$	4.34388i	0.181945i
$571$	−23.8305	−0.997275	−0.498637	−	0.866811i	$-0.666166\pi$
$571$	−23.8305	−0.997275	−0.498637	+	0.866811i	$0.666166\pi$
$572$	− 21.1202i	− 0.883080i
$573$	−17.1345	−0.715803
$574$	18.1660	0.758233
$575$	1.75847	0.0733334
$576$	−0.911854	−0.0379939
$577$	− 29.4294i	− 1.22516i	−0.790408	−	0.612581i	$-0.790131\pi$
$577$	− 29.4294i	− 1.22516i	0.790408	−	0.612581i	$-0.209869\pi$
$578$	1.14384i	0.0475774i
$579$	24.5028	1.01830
$580$	0	0
$581$	21.5226	0.892907
$582$	3.50429i	0.145257i
$583$	36.0210i	1.49184i
$584$	6.98180	0.288909
$585$	−7.72752	−0.319494
$586$	−3.04294	−0.125703
$587$	17.4288	0.719365	0.359683	−	0.933075i	$-0.382885\pi$
$587$	17.4288	0.719365	0.359683	+	0.933075i	$0.382885\pi$
$588$	4.41353i	0.182011i
$589$	−13.0879	−0.539276
$590$	− 12.4957i	− 0.514439i
$591$	2.07769i	0.0854647i
$592$	2.71423i	0.111554i
$593$	29.9077	1.22816	0.614080	−	0.789244i	$-0.289527\pi$
$593$	29.9077	1.22816	0.614080	+	0.789244i	$0.289527\pi$
$594$	− 19.5285i	− 0.801265i
$595$	− 11.7288i	− 0.480833i
$596$	−0.725140	−0.0297029
$597$	6.55839i	0.268417i
$598$	− 3.49215i	− 0.142804i
$599$	26.7834i	1.09434i	0.837021	+	0.547170i	$0.184295\pi$
$599$	26.7834i	1.09434i	−0.837021	+	0.547170i	$0.815705\pi$
$600$	4.44852	0.181610
$601$	25.7155i	1.04896i	0.851424	+	0.524478i	$0.175740\pi$
$601$	25.7155i	1.04896i	−0.851424	+	0.524478i	$0.824260\pi$
$602$	−21.9666	−0.895290
$603$	−11.4458	−0.466107
$604$	9.36422	0.381025
$605$	−1.29573	−0.0526791
$606$	− 10.4593i	− 0.424880i
$607$	− 16.6333i	− 0.675125i	−0.941303	−	0.337562i	$-0.890398\pi$
$607$	− 16.6333i	− 0.675125i	0.941303	−	0.337562i	$-0.109602\pi$
$608$	2.16857	0.0879472
$609$	0	0
$610$	13.3022	0.538590
$611$	30.4809i	1.23312i
$612$	− 3.88410i	− 0.157005i
$613$	0.801874	0.0323874	0.0161937	−	0.999869i	$-0.494845\pi$
$613$	0.801874	0.0323874	0.0161937	+	0.999869i	$0.494845\pi$
$614$	23.6143	0.952997
$615$	18.3189	0.738688
$616$	−6.86232	−0.276491
$617$	5.83648i	0.234968i	0.993075	+	0.117484i	$0.0374829\pi$
$617$	5.83648i	0.234968i	−0.993075	+	0.117484i	$0.962517\pi$
$618$	−11.4760	−0.461633
$619$	− 18.5627i	− 0.746097i	−0.927812	−	0.373048i	$-0.878313\pi$
$619$	− 18.5627i	− 0.746097i	0.927812	−	0.373048i	$-0.121687\pi$
$620$	− 8.36601i	− 0.335987i
$621$	− 3.22897i	− 0.129574i
$622$	−13.8346	−0.554718
$623$	5.66443i	0.226941i
$624$	− 8.83429i	− 0.353655i
$625$	−0.130680	−0.00522719
$626$	8.59812i	0.343650i
$627$	10.8258i	0.432341i
$628$	11.8225i	0.471767i
$629$	−11.5614	−0.460984
$630$	2.51081i	0.100033i
$631$	−25.5862	−1.01857	−0.509286	−	0.860597i	$-0.670090\pi$
$631$	−25.5862	−1.01857	−0.509286	+	0.860597i	$0.670090\pi$
$632$	−11.6286	−0.462563
$633$	−15.1516	−0.602220
$634$	−12.6827	−0.503695
$635$	8.99837i	0.357090i
$636$	15.0671i	0.597450i
$637$	18.6723	0.739823
$638$	0	0
$639$	−3.90170	−0.154349
$640$	1.38619i	0.0547941i
$641$	17.0798i	0.674613i	0.941395	+	0.337307i	$0.109516\pi$
$641$	17.0798i	0.674613i	−0.941395	+	0.337307i	$0.890484\pi$
$642$	18.9576	0.748198
$643$	29.6451	1.16909	0.584545	−	0.811361i	$-0.301273\pi$
$643$	29.6451	1.16909	0.584545	+	0.811361i	$0.301273\pi$
$644$	−1.13466	−0.0447118
$645$	−22.1514	−0.872212
$646$	9.23715i	0.363431i
$647$	5.83795	0.229514	0.114757	−	0.993394i	$-0.463391\pi$
$647$	5.83795	0.229514	0.114757	+	0.993394i	$0.463391\pi$
$648$	− 5.43296i	− 0.213427i
$649$	− 31.1417i	− 1.22242i
$650$	− 18.8203i	− 0.738192i
$651$	17.3237	0.678968
$652$	− 13.5610i	− 0.531091i
$653$	3.36970i	0.131867i	0.997824	+	0.0659333i	$0.0210024\pi$
$653$	3.36970i	0.131867i	−0.997824	+	0.0659333i	$0.978998\pi$
$654$	−7.56802	−0.295933
$655$	3.11955i	0.121891i
$656$	− 9.14522i	− 0.357061i
$657$	− 6.36638i	− 0.248376i
$658$	9.90377	0.386089
$659$	39.6531i	1.54466i	0.635219	+	0.772332i	$0.280910\pi$
$659$	39.6531i	1.54466i	−0.635219	+	0.772332i	$0.719090\pi$
$660$	−6.92007	−0.269363
$661$	19.5912	0.762008	0.381004	−	0.924573i	$-0.375578\pi$
$661$	19.5912	0.762008	0.381004	+	0.924573i	$0.375578\pi$
$662$	5.21791	0.202800
$663$	37.6302	1.46144
$664$	− 10.8350i	− 0.420481i
$665$	− 5.97120i	− 0.231553i
$666$	2.47498	0.0959035
$667$	0	0
$668$	−4.64283	−0.179637
$669$	− 31.4767i	− 1.21696i
$670$	17.3998i	0.672211i
$671$	33.1518	1.27981
$672$	−2.87041	−0.110729
$673$	32.1812	1.24049	0.620247	−	0.784406i	$-0.287032\pi$
$673$	32.1812	1.24049	0.620247	+	0.784406i	$0.287032\pi$
$674$	3.68768	0.142044
$675$	− 17.4019i	− 0.669801i
$676$	−24.3751	−0.937505
$677$	11.9918i	0.460883i	0.973086	+	0.230441i	$0.0740170\pi$
$677$	11.9918i	0.460883i	−0.973086	+	0.230441i	$0.925983\pi$
$678$	0.200751i	0.00770979i
$679$	− 4.81708i	− 0.184862i
$680$	−5.90457	−0.226430
$681$	27.6465i	1.05942i
$682$	− 20.8498i	− 0.798379i
$683$	36.1343	1.38264	0.691321	−	0.722548i	$-0.257029\pi$
$683$	36.1343	1.38264	0.691321	+	0.722548i	$0.257029\pi$
$684$	− 1.97742i	− 0.0756085i
$685$	10.4828i	0.400527i
$686$	− 19.9717i	− 0.762522i
$687$	36.9874	1.41116
$688$	11.0585i	0.421603i
$689$	63.7442	2.42846
$690$	−1.14421	−0.0435593
$691$	5.03899	0.191692	0.0958461	−	0.995396i	$-0.469444\pi$
$691$	5.03899	0.191692	0.0958461	+	0.995396i	$0.469444\pi$
$692$	−9.49522	−0.360954
$693$	6.25743i	0.237700i
$694$	− 3.61233i	− 0.137122i
$695$	8.63989	0.327730
$696$	0	0
$697$	38.9546	1.47551
$698$	− 14.2824i	− 0.540598i
$699$	− 39.0623i	− 1.47747i
$700$	−6.11504	−0.231127
$701$	−10.2286	−0.386330	−0.193165	−	0.981166i	$-0.561875\pi$
$701$	−10.2286	−0.386330	−0.193165	+	0.981166i	$0.561875\pi$
$702$	−34.5585	−1.30433
$703$	−5.88600	−0.221995
$704$	3.45467i	0.130203i
$705$	9.98712	0.376137
$706$	1.22175i	0.0459811i
$707$	14.3776i	0.540725i
$708$	− 13.0262i	− 0.489553i
$709$	−46.2810	−1.73812	−0.869059	−	0.494708i	$-0.835275\pi$
$709$	−46.2810	−1.73812	−0.869059	+	0.494708i	$0.835275\pi$
$710$	5.93133i	0.222599i
$711$	10.6036i	0.397667i
$712$	2.85162	0.106869
$713$	− 3.44743i	− 0.129107i
$714$	− 12.2267i	− 0.457573i
$715$	29.2767i	1.09489i
$716$	−7.53207	−0.281487
$717$	37.6650i	1.40663i
$718$	−31.1679	−1.16318
$719$	0.248398	0.00926368	0.00463184	−	0.999989i	$-0.498526\pi$
$719$	0.248398	0.00926368	0.00463184	+	0.999989i	$0.498526\pi$
$720$	1.26401	0.0471067
$721$	15.7752	0.587499
$722$	− 14.2973i	− 0.532090i
$723$	27.3382i	1.01672i
$724$	18.4798	0.686796
$725$	0	0
$726$	−1.35074	−0.0501308
$727$	− 20.7397i	− 0.769193i	−0.923085	−	0.384597i	$-0.874341\pi$
$727$	− 20.7397i	− 0.769193i	0.923085	−	0.384597i	$-0.125659\pi$
$728$	12.1438i	0.450080i
$729$	−29.4596	−1.09110
$730$	−9.67812	−0.358203
$731$	−47.1045	−1.74222
$732$	13.8669	0.512537
$733$	4.30302i	0.158936i	0.996837	+	0.0794678i	$0.0253221\pi$
$733$	4.30302i	0.158936i	−0.996837	+	0.0794678i	$0.974678\pi$
$734$	−4.83098	−0.178315
$735$	− 6.11801i	− 0.225666i
$736$	0.571217i	0.0210553i
$737$	43.3637i	1.59732i
$738$	−8.33911	−0.306967
$739$	15.1636i	0.557803i	0.960320	+	0.278901i	$0.0899702\pi$
$739$	15.1636i	0.557803i	−0.960320	+	0.278901i	$0.910030\pi$
$740$	− 3.76244i	− 0.138310i
$741$	19.1578	0.703779
$742$	− 20.7116i	− 0.760347i
$743$	− 26.3730i	− 0.967533i	−0.875197	−	0.483767i	$-0.839268\pi$
$743$	− 26.3730i	− 0.967533i	0.875197	−	0.483767i	$-0.160732\pi$
$744$	− 8.72118i	− 0.319734i
$745$	1.00518	0.0368271
$746$	0.936592i	0.0342911i
$747$	−9.87997	−0.361489
$748$	−14.7154	−0.538047
$749$	−26.0596	−0.952198
$750$	−16.1820	−0.590884
$751$	28.2430i	1.03060i	0.857010	+	0.515301i	$0.172320\pi$
$751$	28.2430i	1.03060i	−0.857010	+	0.515301i	$0.827680\pi$
$752$	− 4.98582i	− 0.181814i
$753$	33.6236	1.22531
$754$	0	0
$755$	−12.9806	−0.472413
$756$	11.2286i	0.408382i
$757$	39.2000i	1.42475i	0.701801	+	0.712373i	$0.252380\pi$
$757$	39.2000i	1.42475i	−0.701801	+	0.712373i	$0.747620\pi$
$758$	1.44585	0.0525158
$759$	−2.85160	−0.103506
$760$	−3.00606	−0.109041
$761$	−11.0366	−0.400075	−0.200037	−	0.979788i	$-0.564106\pi$
$761$	−11.0366	−0.400075	−0.200037	+	0.979788i	$0.564106\pi$
$762$	9.38039i	0.339816i
$763$	10.4032	0.376620
$764$	− 11.8574i	− 0.428986i
$765$	5.38411i	0.194663i
$766$	7.89594i	0.285292i
$767$	−55.1096	−1.98989
$768$	1.44504i	0.0521435i
$769$	− 20.5297i	− 0.740321i	−0.928968	−	0.370160i	$-0.879303\pi$
$769$	− 20.5297i	− 0.740321i	0.928968	−	0.370160i	$-0.120697\pi$
$770$	9.51249	0.342806
$771$	− 16.5790i	− 0.597080i
$772$	16.9564i	0.610276i
$773$	43.6105i	1.56856i	0.620406	+	0.784281i	$0.286968\pi$
$773$	43.6105i	1.56856i	−0.620406	+	0.784281i	$0.713032\pi$
$774$	10.0838	0.362454
$775$	− 18.5793i	− 0.667389i
$776$	−2.42504	−0.0870539
$777$	7.79096	0.279499
$778$	8.04652	0.288482
$779$	19.8321	0.710557
$780$	12.2460i	0.438478i
$781$	14.7821i	0.528944i
$782$	−2.43313	−0.0870086
$783$	0	0
$784$	−3.05426	−0.109081
$785$	− 16.3882i	− 0.584920i
$786$	3.25198i	0.115994i
$787$	3.73634	0.133186	0.0665931	−	0.997780i	$-0.478787\pi$
$787$	3.73634	0.133186	0.0665931	+	0.997780i	$0.478787\pi$
$788$	−1.43781	−0.0512197
$789$	−31.2433	−1.11229
$790$	16.1195	0.573508
$791$	− 0.275957i	− 0.00981190i
$792$	3.15015	0.111936
$793$	− 58.6666i	− 2.08331i
$794$	2.56721i	0.0911068i
$795$	− 20.8859i	− 0.740747i
$796$	−4.53854	−0.160864
$797$	11.5294i	0.408392i	0.978930	+	0.204196i	$0.0654581\pi$
$797$	11.5294i	0.408392i	−0.978930	+	0.204196i	$0.934542\pi$
$798$	− 6.22470i	− 0.220352i
$799$	21.2374	0.751324
$800$	3.07847i	0.108840i
$801$	− 2.60026i	− 0.0918758i
$802$	7.62689i	0.269315i
$803$	−24.1198	−0.851170
$804$	18.1384i	0.639693i
$805$	1.57286	0.0554359
$806$	−36.8966	−1.29963
$807$	−6.57615	−0.231491
$808$	7.23806	0.254634
$809$	46.3046i	1.62798i	0.580877	+	0.813991i	$0.302710\pi$
$809$	46.3046i	1.62798i	−0.580877	+	0.813991i	$0.697290\pi$
$810$	7.53113i	0.264617i
$811$	31.6055	1.10982	0.554909	−	0.831911i	$-0.312753\pi$
$811$	31.6055	1.10982	0.554909	+	0.831911i	$0.312753\pi$
$812$	0	0
$813$	−11.6047	−0.406995
$814$	− 9.37676i	− 0.328655i
$815$	18.7982i	0.658472i
$816$	−6.15524	−0.215477
$817$	−23.9812	−0.838997
$818$	2.17471	0.0760371
$819$	11.0734	0.386936
$820$	12.6770i	0.442701i
$821$	9.95432	0.347408	0.173704	−	0.984798i	$-0.444426\pi$
$821$	9.95432	0.347408	0.173704	+	0.984798i	$0.444426\pi$
$822$	10.9278i	0.381152i
$823$	19.9537i	0.695541i	0.937580	+	0.347771i	$0.113061\pi$
$823$	19.9537i	0.695541i	−0.937580	+	0.347771i	$0.886939\pi$
$824$	− 7.94165i	− 0.276660i
$825$	−15.3682	−0.535051
$826$	17.9061i	0.623032i
$827$	32.1706i	1.11868i	0.828938	+	0.559341i	$0.188946\pi$
$827$	32.1706i	1.11868i	−0.828938	+	0.559341i	$0.811054\pi$
$828$	0.520866	0.0181014
$829$	− 18.4924i	− 0.642269i	−0.947034	−	0.321134i	$-0.895936\pi$
$829$	− 18.4924i	− 0.642269i	0.947034	−	0.321134i	$-0.104064\pi$
$830$	15.0194i	0.521333i
$831$	41.1443i	1.42728i
$832$	6.11352	0.211948
$833$	− 13.0098i	− 0.450763i
$834$	9.00669	0.311876
$835$	6.43586	0.222722
$836$	−7.49170	−0.259106
$837$	−34.1160	−1.17922
$838$	− 11.0437i	− 0.381499i
$839$	− 11.3409i	− 0.391530i	−0.980651	−	0.195765i	$-0.937281\pi$
$839$	− 11.3409i	− 0.391530i	0.980651	−	0.195765i	$-0.0627190\pi$
$840$	3.97895	0.137287
$841$	0	0
$842$	−3.65914	−0.126102
$843$	28.8611i	0.994028i
$844$	− 10.4852i	− 0.360916i
$845$	33.7886	1.16236
$846$	−4.54634	−0.156306
$847$	1.85676	0.0637991
$848$	−10.4268	−0.358057
$849$	− 6.18837i	− 0.212384i
$850$	−13.1129	−0.449769
$851$	− 1.55041i	− 0.0531475i
$852$	6.18314i	0.211831i
$853$	20.9622i	0.717733i	0.933389	+	0.358867i	$0.116837\pi$
$853$	20.9622i	0.717733i	−0.933389	+	0.358867i	$0.883163\pi$
$854$	−19.0618	−0.652282
$855$	2.74109i	0.0937432i
$856$	13.1191i	0.448401i
$857$	15.5712	0.531903	0.265951	−	0.963986i	$-0.414314\pi$
$857$	15.5712	0.531903	0.265951	+	0.963986i	$0.414314\pi$
$858$	30.5196i	1.04192i
$859$	44.1276i	1.50561i	0.658241	+	0.752807i	$0.271301\pi$
$859$	44.1276i	1.50561i	−0.658241	+	0.752807i	$0.728699\pi$
$860$	− 15.3293i	− 0.522724i
$861$	−26.2506	−0.894617
$862$	− 6.36313i	− 0.216729i
$863$	−40.0443	−1.36312	−0.681562	−	0.731760i	$-0.738699\pi$
$863$	−40.0443	−1.36312	−0.681562	+	0.731760i	$0.738699\pi$
$864$	5.65279	0.192312
$865$	13.1622	0.447529
$866$	−21.8657	−0.743026
$867$	− 1.65289i	− 0.0561352i
$868$	11.9883i	0.406911i
$869$	40.1731	1.36278
$870$	0	0
$871$	76.7380	2.60017
$872$	− 5.23723i	− 0.177355i
$873$	2.21128i	0.0748406i
$874$	−1.23872	−0.0419005
$875$	22.2442	0.751991
$876$	−10.0890	−0.340875
$877$	−11.5880	−0.391298	−0.195649	−	0.980674i	$-0.562681\pi$
$877$	−11.5880	−0.391298	−0.195649	+	0.980674i	$0.562681\pi$
$878$	20.6588i	0.697202i
$879$	4.39718	0.148313
$880$	− 4.78884i	− 0.161432i
$881$	3.35136i	0.112910i	0.998405	+	0.0564550i	$0.0179797\pi$
$881$	3.35136i	0.112910i	−0.998405	+	0.0564550i	$0.982020\pi$
$882$	2.78504i	0.0937772i
$883$	51.3940	1.72955	0.864773	−	0.502163i	$-0.167462\pi$
$883$	51.3940	1.72955	0.864773	+	0.502163i	$0.167462\pi$
$884$	26.0409i	0.875850i
$885$	18.0568i	0.606972i
$886$	−2.56232	−0.0860829
$887$	42.3730i	1.42275i	0.702814	+	0.711374i	$0.251927\pi$
$887$	42.3730i	1.42275i	−0.702814	+	0.711374i	$0.748073\pi$
$888$	− 3.92217i	− 0.131620i
$889$	− 12.8945i	− 0.432468i
$890$	−3.95290	−0.132501
$891$	18.7691i	0.628788i
$892$	21.7825	0.729333
$893$	10.8121	0.361813
$894$	1.04786	0.0350456
$895$	10.4409	0.349001
$896$	− 1.98639i	− 0.0663606i
$897$	5.04630i	0.168491i
$898$	−40.7652	−1.36035
$899$	0	0
$900$	2.80711	0.0935705
$901$	− 44.4134i	− 1.47963i
$902$	31.5937i	1.05196i
$903$	31.7426	1.05633
$904$	−0.138924	−0.00462054
$905$	−25.6166	−0.851523
$906$	−13.5317	−0.449560
$907$	40.1367i	1.33272i	0.745632	+	0.666358i	$0.232148\pi$
$907$	40.1367i	1.33272i	−0.745632	+	0.666358i	$0.767852\pi$
$908$	−19.1320	−0.634916
$909$	− 6.60006i	− 0.218910i
$910$	− 16.8337i	− 0.558031i
$911$	39.7506i	1.31700i	0.752582	+	0.658499i	$0.228808\pi$
$911$	39.7506i	1.31700i	−0.752582	+	0.658499i	$0.771192\pi$
$912$	−3.13368	−0.103766
$913$	37.4315i	1.23880i
$914$	− 25.2066i	− 0.833759i
$915$	−19.2222	−0.635468
$916$	25.5961i	0.845718i
$917$	− 4.47025i	− 0.147621i
$918$	24.0784i	0.794706i
$919$	−29.8687	−0.985277	−0.492639	−	0.870234i	$-0.663968\pi$
$919$	−29.8687	−0.985277	−0.492639	+	0.870234i	$0.663968\pi$
$920$	− 0.791817i	− 0.0261054i
$921$	−34.1237	−1.12441
$922$	41.0649	1.35240
$923$	26.1589	0.861031
$924$	9.91633	0.326223
$925$	− 8.35567i	− 0.274733i
$926$	33.1170i	1.08829i
$927$	−7.24162	−0.237846
$928$	0	0
$929$	−0.307389	−0.0100851	−0.00504256	−	0.999987i	$-0.501605\pi$
$929$	−0.307389	−0.0100851	−0.00504256	+	0.999987i	$0.501605\pi$
$930$	12.0892i	0.396422i
$931$	− 6.62338i	− 0.217073i
$932$	27.0319	0.885461
$933$	19.9916	0.654496
$934$	−15.1228	−0.494833
$935$	20.3983	0.667097
$936$	− 5.57464i	− 0.182213i
$937$	−54.6031	−1.78381	−0.891903	−	0.452227i	$-0.850630\pi$
$937$	−54.6031	−1.78381	−0.891903	+	0.452227i	$0.850630\pi$
$938$	− 24.9335i	− 0.814108i
$939$	− 12.4246i	− 0.405463i
$940$	6.91130i	0.225422i
$941$	−27.6969	−0.902894	−0.451447	−	0.892298i	$-0.649092\pi$
$941$	−27.6969	−0.902894	−0.451447	+	0.892298i	$0.649092\pi$
$942$	− 17.0839i	− 0.556625i
$943$	5.22390i	0.170114i
$944$	9.01438	0.293393
$945$	− 15.5651i	− 0.506332i
$946$	− 38.2036i	− 1.24211i
$947$	37.5978i	1.22177i	0.791721	+	0.610883i	$0.209185\pi$
$947$	37.5978i	1.22177i	−0.791721	+	0.610883i	$0.790815\pi$
$948$	16.8039	0.545765
$949$	42.6834i	1.38556i
$950$	−6.67588	−0.216594
$951$	18.3271	0.594296
$952$	8.46114	0.274227
$953$	15.4013	0.498897	0.249449	−	0.968388i	$-0.419751\pi$
$953$	15.4013	0.498897	0.249449	+	0.968388i	$0.419751\pi$
$954$	9.50769i	0.307823i
$955$	16.4367i	0.531878i
$956$	−26.0650	−0.843002
$957$	0	0
$958$	24.5410	0.792885
$959$	− 15.0217i	− 0.485075i
$960$	− 2.00311i	− 0.0646500i
$961$	−5.42418	−0.174974
$962$	−16.5935	−0.534996
$963$	11.9627	0.385493
$964$	−18.9186	−0.609328
$965$	− 23.5049i	− 0.756650i
$966$	1.63963	0.0527542
$967$	− 42.1501i	− 1.35546i	−0.735313	−	0.677728i	$-0.762965\pi$
$967$	− 42.1501i	− 1.35546i	0.735313	−	0.677728i	$-0.237035\pi$
$968$	− 0.934743i	− 0.0300438i
$969$	− 13.3481i	− 0.428802i
$970$	3.36158	0.107934
$971$	− 22.0724i	− 0.708337i	−0.935182	−	0.354169i	$-0.884764\pi$
$971$	− 22.0724i	− 0.708337i	0.935182	−	0.354169i	$-0.115236\pi$
$972$	− 9.10752i	− 0.292124i
$973$	−12.3808	−0.396910
$974$	− 28.4159i	− 0.910503i
$975$	27.1961i	0.870972i
$976$	9.59621i	0.307167i
$977$	−28.2838	−0.904881	−0.452440	−	0.891795i	$-0.649446\pi$
$977$	−28.2838	−0.904881	−0.452440	+	0.891795i	$0.649446\pi$
$978$	19.5962i	0.626619i
$979$	−9.85141	−0.314853
$980$	4.23379	0.135244
$981$	−4.77559	−0.152473
$982$	22.4508	0.716434
$983$	− 4.39607i	− 0.140213i	−0.997540	−	0.0701064i	$-0.977666\pi$
$983$	− 4.39607i	− 0.140213i	0.997540	−	0.0701064i	$-0.0223339\pi$
$984$	13.2152i	0.421286i
$985$	1.99308	0.0635047
$986$	0	0
$987$	−14.3114	−0.455536
$988$	13.2576i	0.421781i
$989$	− 6.31683i	− 0.200863i
$990$	−4.36672	−0.138784
$991$	0.985266	0.0312980	0.0156490	−	0.999878i	$-0.495019\pi$
$991$	0.985266	0.0312980	0.0156490	+	0.999878i	$0.495019\pi$
$992$	6.03525	0.191619
$993$	−7.54010	−0.239278
$994$	− 8.49948i	− 0.269587i
$995$	6.29130	0.199448
$996$	15.6571i	0.496113i
$997$	− 3.95435i	− 0.125235i	−0.998038	−	0.0626177i	$-0.980055\pi$
$997$	− 3.95435i	− 0.125235i	0.998038	−	0.0626177i	$-0.0199449\pi$
$998$	− 19.6240i	− 0.621186i
$999$	−15.3430	−0.485430

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1682.2.b.j.1681.4		12
29.5	even	14		58.2.e.a.33.1	✓	12
29.12	odd	4		1682.2.a.s.1.3		6
29.17	odd	4		1682.2.a.r.1.3		6
29.23	even	7		58.2.e.a.51.1	yes	12
29.28	even	2	inner	1682.2.b.j.1681.10		12
87.5	odd	14		522.2.n.a.91.2		12
87.23	odd	14		522.2.n.a.109.2		12
116.23	odd	14		464.2.y.c.225.1		12
116.63	odd	14		464.2.y.c.33.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
58.2.e.a.33.1	✓	12	29.5	even	14
58.2.e.a.51.1	yes	12	29.23	even	7
464.2.y.c.33.1		12	116.63	odd	14
464.2.y.c.225.1		12	116.23	odd	14
522.2.n.a.91.2		12	87.5	odd	14
522.2.n.a.109.2		12	87.23	odd	14
1682.2.a.r.1.3		6	29.17	odd	4
1682.2.a.s.1.3		6	29.12	odd	4
1682.2.b.j.1681.4		12	1.1	even	1	trivial
1682.2.b.j.1681.10		12	29.28	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1682 = 2 \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1682.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.44504i q^{3} -1.00000 q^{4} +1.38619 q^{5} +1.44504 q^{6} -1.98639 q^{7} +1.00000i q^{8} +0.911854 q^{9} -1.38619i q^{10} -3.45467i q^{11} -1.44504i q^{12} -6.11352 q^{13} +1.98639i q^{14} +2.00311i q^{15} +1.00000 q^{16} +4.25956i q^{17} -0.911854i q^{18} +2.16857i q^{19} -1.38619 q^{20} -2.87041i q^{21} -3.45467 q^{22} -0.571217 q^{23} -1.44504 q^{24} -3.07847 q^{25} +6.11352i q^{26} +5.65279i q^{27} +1.98639 q^{28} +2.00311 q^{30} +6.03525i q^{31} -1.00000i q^{32} +4.99214 q^{33} +4.25956 q^{34} -2.75352 q^{35} -0.911854 q^{36} +2.71423i q^{37} +2.16857 q^{38} -8.83429i q^{39} +1.38619i q^{40} -9.14522i q^{41} -2.87041 q^{42} +11.0585i q^{43} +3.45467i q^{44} +1.26401 q^{45} +0.571217i q^{46} -4.98582i q^{47} +1.44504i q^{48} -3.05426 q^{49} +3.07847i q^{50} -6.15524 q^{51} +6.11352 q^{52} -10.4268 q^{53} +5.65279 q^{54} -4.78884i q^{55} -1.98639i q^{56} -3.13368 q^{57} +9.01438 q^{59} -2.00311i q^{60} +9.59621i q^{61} +6.03525 q^{62} -1.81130 q^{63} -1.00000 q^{64} -8.47452 q^{65} -4.99214i q^{66} -12.5522 q^{67} -4.25956i q^{68} -0.825432i q^{69} +2.75352i q^{70} -4.27886 q^{71} +0.911854i q^{72} -6.98180i q^{73} +2.71423 q^{74} -4.44852i q^{75} -2.16857i q^{76} +6.86232i q^{77} -8.83429 q^{78} +11.6286i q^{79} +1.38619 q^{80} -5.43296 q^{81} -9.14522 q^{82} -10.8350 q^{83} +2.87041i q^{84} +5.90457i q^{85} +11.0585 q^{86} +3.45467 q^{88} -2.85162i q^{89} -1.26401i q^{90} +12.1438 q^{91} +0.571217 q^{92} -8.72118 q^{93} -4.98582 q^{94} +3.00606i q^{95} +1.44504 q^{96} +2.42504i q^{97} +3.05426i q^{98} -3.15015i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{4} + 12 q^{5} + 16 q^{6} + 4 q^{7} - 4 q^{9} + 16 q^{13} + 12 q^{16} - 12 q^{20} + 4 q^{22} + 12 q^{23} - 16 q^{24} + 8 q^{25} - 4 q^{28} + 16 q^{30} - 24 q^{33} + 24 q^{34} + 32 q^{35} + 4 q^{36}+ \cdots + 16 q^{96}+O(q^{100}) \) 12 * q - 12 * q^4 + 12 * q^5 + 16 * q^6 + 4 * q^7 - 4 * q^9 + 16 * q^13 + 12 * q^16 - 12 * q^20 + 4 * q^22 + 12 * q^23 - 16 * q^24 + 8 * q^25 - 4 * q^28 + 16 * q^30 - 24 * q^33 + 24 * q^34 + 32 * q^35 + 4 * q^36 + 8 * q^38 + 24 * q^42 - 4 * q^45 - 8 * q^49 - 32 * q^51 - 16 * q^52 + 16 * q^53 - 4 * q^54 - 20 * q^57 - 48 * q^59 + 52 * q^62 - 48 * q^63 - 12 * q^64 - 12 * q^65 - 16 * q^67 - 40 * q^71 - 40 * q^74 + 12 * q^78 + 12 * q^80 + 12 * q^81 - 48 * q^82 - 56 * q^83 + 52 * q^86 - 4 * q^88 + 24 * q^91 - 12 * q^92 - 60 * q^93 - 36 * q^94 + 16 * q^96

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