LMFDB - Embedded newform 690.3.c.a.91.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [690,3,Mod(91,690)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(690, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("690.91");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 690 = 2 \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	690.c (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:	$18.8011382409$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			91.14
Character	$\chi$	$=$	690.91
Dual form			690.3.c.a.91.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +1.73205 q^{3} +2.00000 q^{4} +2.23607i q^{5} -2.44949 q^{6} +0.411309i q^{7} -2.82843 q^{8} +3.00000 q^{9} +O(q^{10})$	\(q-1.41421 q^{2} +1.73205 q^{3} +2.00000 q^{4} +2.23607i q^{5} -2.44949 q^{6} +0.411309i q^{7} -2.82843 q^{8} +3.00000 q^{9} -3.16228i q^{10} -6.98697i q^{11} +3.46410 q^{12} -7.65791 q^{13} -0.581679i q^{14} +3.87298i q^{15} +4.00000 q^{16} -22.0292i q^{17} -4.24264 q^{18} -8.31107i q^{19} +4.47214i q^{20} +0.712409i q^{21} +9.88107i q^{22} +(17.3790 - 15.0656i) q^{23} -4.89898 q^{24} -5.00000 q^{25} +10.8299 q^{26} +5.19615 q^{27} +0.822619i q^{28} +32.6032 q^{29} -5.47723i q^{30} -17.7255 q^{31} -5.65685 q^{32} -12.1018i q^{33} +31.1540i q^{34} -0.919716 q^{35} +6.00000 q^{36} -22.9773i q^{37} +11.7536i q^{38} -13.2639 q^{39} -6.32456i q^{40} +28.8797 q^{41} -1.00750i q^{42} -7.58017i q^{43} -13.9739i q^{44} +6.70820i q^{45} +(-24.5776 + 21.3060i) q^{46} +49.4935 q^{47} +6.92820 q^{48} +48.8308 q^{49} +7.07107 q^{50} -38.1557i q^{51} -15.3158 q^{52} -17.5382i q^{53} -7.34847 q^{54} +15.6233 q^{55} -1.16336i q^{56} -14.3952i q^{57} -46.1078 q^{58} -33.8653 q^{59} +7.74597i q^{60} +39.2567i q^{61} +25.0676 q^{62} +1.23393i q^{63} +8.00000 q^{64} -17.1236i q^{65} +17.1145i q^{66} +70.8924i q^{67} -44.0585i q^{68} +(30.1012 - 26.0944i) q^{69} +1.30067 q^{70} +26.8165 q^{71} -8.48528 q^{72} -62.0148 q^{73} +32.4948i q^{74} -8.66025 q^{75} -16.6221i q^{76} +2.87381 q^{77} +18.7580 q^{78} -35.2805i q^{79} +8.94427i q^{80} +9.00000 q^{81} -40.8420 q^{82} -143.692i q^{83} +1.42482i q^{84} +49.2589 q^{85} +10.7200i q^{86} +56.4704 q^{87} +19.7621i q^{88} -73.4091i q^{89} -9.48683i q^{90} -3.14977i q^{91} +(34.7579 - 30.1312i) q^{92} -30.7014 q^{93} -69.9943 q^{94} +18.5841 q^{95} -9.79796 q^{96} +95.9147i q^{97} -69.0572 q^{98} -20.9609i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 64 q^{4} + 96 q^{9}+O(q^{10}) $ 32 * q + 64 * q^4 + 96 * q^9	$ 32 q + 64 q^{4} + 96 q^{9} - 48 q^{13} + 128 q^{16} - 80 q^{23} - 160 q^{25} + 120 q^{29} + 248 q^{31} - 120 q^{35} + 192 q^{36} - 48 q^{39} + 72 q^{41} + 160 q^{46} + 400 q^{47} - 344 q^{49} - 96 q^{52} - 256 q^{58} + 120 q^{59} + 160 q^{62} + 256 q^{64} + 192 q^{69} + 104 q^{71} + 16 q^{73} + 240 q^{77} + 192 q^{78} + 288 q^{81} + 64 q^{82} - 120 q^{85} + 144 q^{87} - 160 q^{92} - 192 q^{93} + 96 q^{94} - 160 q^{95} + 64 q^{98}+O(q^{100}) $ 32 * q + 64 * q^4 + 96 * q^9 - 48 * q^13 + 128 * q^16 - 80 * q^23 - 160 * q^25 + 120 * q^29 + 248 * q^31 - 120 * q^35 + 192 * q^36 - 48 * q^39 + 72 * q^41 + 160 * q^46 + 400 * q^47 - 344 * q^49 - 96 * q^52 - 256 * q^58 + 120 * q^59 + 160 * q^62 + 256 * q^64 + 192 * q^69 + 104 * q^71 + 16 * q^73 + 240 * q^77 + 192 * q^78 + 288 * q^81 + 64 * q^82 - 120 * q^85 + 144 * q^87 - 160 * q^92 - 192 * q^93 + 96 * q^94 - 160 * q^95 + 64 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/690\mathbb{Z}\right)^\times$.

$n$	$277$	$461$	$511$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	−1.41421			−0.707107
$2$	−1.41421			−0.707107
$3$	1.73205			0.577350
$3$	1.73205			0.577350
$4$	2.00000			0.500000
$5$			2.23607i			0.447214i
$5$			2.23607i			0.447214i
$6$	−2.44949			−0.408248
$7$			0.411309i			0.0587585i	0.999568	+	0.0293792i	$0.00935305\pi$
$7$			0.411309i			0.0587585i	−0.999568	+	0.0293792i	$0.990647\pi$
$8$	−2.82843			−0.353553
$9$	3.00000			0.333333
$10$		−	3.16228i		−	0.316228i
$11$		−	6.98697i		−	0.635179i	−0.948228	−	0.317589i	$-0.897127\pi$
$11$		−	6.98697i		−	0.635179i	0.948228	−	0.317589i	$-0.102873\pi$
$12$	3.46410			0.288675
$13$	−7.65791			−0.589070			−0.294535	−	0.955641i	$-0.595165\pi$
$13$	−7.65791			−0.589070			−0.294535	+	0.955641i	$0.595165\pi$
$14$		−	0.581679i		−	0.0415485i
$15$			3.87298i			0.258199i
$16$	4.00000			0.250000
$17$		−	22.0292i		−	1.29584i	−0.761710	−	0.647918i	$-0.775640\pi$
$17$		−	22.0292i		−	1.29584i	0.761710	−	0.647918i	$-0.224360\pi$
$18$	−4.24264			−0.235702
$19$		−	8.31107i		−	0.437425i	−0.975789	−	0.218712i	$-0.929814\pi$
$19$		−	8.31107i		−	0.437425i	0.975789	−	0.218712i	$-0.0701857\pi$
$20$			4.47214i			0.223607i
$21$			0.712409i			0.0339242i
$22$			9.88107i			0.449139i
$23$	17.3790	−	15.0656i	0.755607	−	0.655026i
$23$	17.3790	−	15.0656i	0.755607	−	0.655026i
$24$	−4.89898			−0.204124
$25$	−5.00000			−0.200000
$26$	10.8299			0.416535
$27$	5.19615			0.192450
$28$			0.822619i			0.0293792i
$29$	32.6032			1.12425			0.562124	−	0.827053i	$-0.309984\pi$
$29$	32.6032			1.12425			0.562124	+	0.827053i	$0.309984\pi$
$30$		−	5.47723i		−	0.182574i
$31$	−17.7255			−0.571789			−0.285894	−	0.958261i	$-0.592291\pi$
$31$	−17.7255			−0.571789			−0.285894	+	0.958261i	$0.592291\pi$
$32$	−5.65685			−0.176777
$33$		−	12.1018i		−	0.366721i
$34$			31.1540i			0.916295i
$35$	−0.919716			−0.0262776
$36$	6.00000			0.166667
$37$		−	22.9773i		−	0.621008i	−0.950572	−	0.310504i	$-0.899502\pi$
$37$		−	22.9773i		−	0.621008i	0.950572	−	0.310504i	$-0.100498\pi$
$38$			11.7536i			0.309306i
$39$	−13.2639			−0.340100
$40$		−	6.32456i		−	0.158114i
$41$	28.8797			0.704382			0.352191	−	0.935928i	$-0.385437\pi$
$41$	28.8797			0.704382			0.352191	+	0.935928i	$0.385437\pi$
$42$		−	1.00750i		−	0.0239881i
$43$		−	7.58017i		−	0.176283i	−0.996108	−	0.0881415i	$-0.971907\pi$
$43$		−	7.58017i		−	0.176283i	0.996108	−	0.0881415i	$-0.0280928\pi$
$44$		−	13.9739i		−	0.317589i
$45$			6.70820i			0.149071i
$46$	−24.5776	+	21.3060i	−0.534295	+	0.463173i
$47$	49.4935			1.05305			0.526526	−	0.850159i	$-0.323494\pi$
$47$	49.4935			1.05305			0.526526	+	0.850159i	$0.323494\pi$
$48$	6.92820			0.144338
$49$	48.8308			0.996547
$50$	7.07107			0.141421
$51$		−	38.1557i		−	0.748152i
$52$	−15.3158			−0.294535
$53$		−	17.5382i		−	0.330910i	−0.986217	−	0.165455i	$-0.947091\pi$
$53$		−	17.5382i		−	0.330910i	0.986217	−	0.165455i	$-0.0529093\pi$
$54$	−7.34847			−0.136083
$55$	15.6233			0.284061
$56$		−	1.16336i		−	0.0207743i
$57$		−	14.3952i		−	0.252547i
$58$	−46.1078			−0.794963
$59$	−33.8653			−0.573989			−0.286994	−	0.957932i	$-0.592656\pi$
$59$	−33.8653			−0.573989			−0.286994	+	0.957932i	$0.592656\pi$
$60$			7.74597i			0.129099i
$61$			39.2567i			0.643552i	0.946816	+	0.321776i	$0.104280\pi$
$61$			39.2567i			0.643552i	−0.946816	+	0.321776i	$0.895720\pi$
$62$	25.0676			0.404316
$63$			1.23393i			0.0195862i
$64$	8.00000			0.125000
$65$		−	17.1236i		−	0.263440i
$66$			17.1145i			0.259311i
$67$			70.8924i			1.05810i	0.848592	+	0.529048i	$0.177451\pi$
$67$			70.8924i			1.05810i	−0.848592	+	0.529048i	$0.822549\pi$
$68$		−	44.0585i		−	0.647918i
$69$	30.1012	−	26.0944i	0.436250	−	0.378179i
$70$	1.30067			0.0185811
$71$	26.8165			0.377698			0.188849	−	0.982006i	$-0.439524\pi$
$71$	26.8165			0.377698			0.188849	+	0.982006i	$0.439524\pi$
$72$	−8.48528			−0.117851
$73$	−62.0148			−0.849518			−0.424759	−	0.905306i	$-0.639641\pi$
$73$	−62.0148			−0.849518			−0.424759	+	0.905306i	$0.639641\pi$
$74$			32.4948i			0.439119i
$75$	−8.66025			−0.115470
$76$		−	16.6221i		−	0.218712i
$77$	2.87381			0.0373222
$78$	18.7580			0.240487
$79$		−	35.2805i		−	0.446589i	−0.974751	−	0.223294i	$-0.928319\pi$
$79$		−	35.2805i		−	0.446589i	0.974751	−	0.223294i	$-0.0716811\pi$
$80$			8.94427i			0.111803i
$81$	9.00000			0.111111
$82$	−40.8420			−0.498074
$83$		−	143.692i		−	1.73123i	−0.500712	−	0.865614i	$-0.666928\pi$
$83$		−	143.692i		−	1.73123i	0.500712	−	0.865614i	$-0.333072\pi$
$84$			1.42482i			0.0169621i
$85$	49.2589			0.579516
$86$			10.7200i			0.124651i
$87$	56.4704			0.649084
$88$			19.7621i			0.224570i
$89$		−	73.4091i		−	0.824821i	−0.910998	−	0.412411i	$-0.864687\pi$
$89$		−	73.4091i		−	0.824821i	0.910998	−	0.412411i	$-0.135313\pi$
$90$		−	9.48683i		−	0.105409i
$91$		−	3.14977i		−	0.0346129i
$92$	34.7579	−	30.1312i	0.377803	−	0.327513i
$93$	−30.7014			−0.330123
$94$	−69.9943			−0.744620
$95$	18.5841			0.195622
$96$	−9.79796			−0.102062
$97$			95.9147i			0.988811i	0.869231	+	0.494405i	$0.164614\pi$
$97$			95.9147i			0.988811i	−0.869231	+	0.494405i	$0.835386\pi$
$98$	−69.0572			−0.704665
$99$		−	20.9609i		−	0.211726i
$100$	−10.0000			−0.100000
$101$	144.998			1.43563			0.717814	−	0.696235i	$-0.245143\pi$
$101$	144.998			1.43563			0.717814	+	0.696235i	$0.245143\pi$
$102$			53.9604i			0.529023i
$103$		−	123.567i		−	1.19968i	−0.800121	−	0.599838i	$-0.795232\pi$
$103$		−	123.567i		−	1.19968i	0.800121	−	0.599838i	$-0.204768\pi$
$104$	21.6598			0.208268
$105$	−1.59299			−0.0151714
$106$			24.8028i			0.233989i
$107$		−	92.0932i		−	0.860684i	−0.902666	−	0.430342i	$-0.858393\pi$
$107$		−	92.0932i		−	0.860684i	0.902666	−	0.430342i	$-0.141607\pi$
$108$	10.3923			0.0962250
$109$		−	45.0093i		−	0.412929i	−0.978454	−	0.206465i	$-0.933804\pi$
$109$		−	45.0093i		−	0.412929i	0.978454	−	0.206465i	$-0.0661958\pi$
$110$	−22.0947			−0.200861
$111$		−	39.7978i		−	0.358539i
$112$			1.64524i			0.0146896i
$113$		−	54.3464i		−	0.480942i	−0.970656	−	0.240471i	$-0.922698\pi$
$113$		−	54.3464i		−	0.480942i	0.970656	−	0.240471i	$-0.0773018\pi$
$114$			20.3579i			0.178578i
$115$	33.6877	+	38.8605i	0.292936	+	0.337918i
$116$	65.2063			0.562124
$117$	−22.9737			−0.196357
$118$	47.8928			0.405871
$119$	9.06083			0.0761414
$120$		−	10.9545i		−	0.0912871i
$121$	72.1823			0.596548
$122$		−	55.5173i		−	0.455060i
$123$	50.0211			0.406675
$124$	−35.4509			−0.285894
$125$		−	11.1803i		−	0.0894427i
$126$		−	1.74504i		−	0.0138495i
$127$	114.199			0.899203			0.449602	−	0.893229i	$-0.351566\pi$
$127$	114.199			0.899203			0.449602	+	0.893229i	$0.351566\pi$
$128$	−11.3137			−0.0883883
$129$		−	13.1292i		−	0.101777i
$130$			24.2164i			0.186280i
$131$	87.1429			0.665213			0.332606	−	0.943066i	$-0.392072\pi$
$131$	87.1429			0.665213			0.332606	+	0.943066i	$0.392072\pi$
$132$		−	24.2036i		−	0.183360i
$133$	3.41842			0.0257024
$134$		−	100.257i		−	0.748186i
$135$			11.6190i			0.0860663i
$136$			62.3081i			0.458148i
$137$		−	3.00953i		−	0.0219674i	−0.999940	−	0.0109837i	$-0.996504\pi$
$137$		−	3.00953i		−	0.0219674i	0.999940	−	0.0109837i	$-0.00349628\pi$
$138$	−42.5696	+	36.9030i	−0.308475	+	0.267413i
$139$	−124.206			−0.893567			−0.446783	−	0.894642i	$-0.647431\pi$
$139$	−124.206			−0.893567			−0.446783	+	0.894642i	$0.647431\pi$
$140$	−1.83943			−0.0131388
$141$	85.7252			0.607980
$142$	−37.9243			−0.267073
$143$			53.5056i			0.374165i
$144$	12.0000			0.0833333
$145$			72.9029i			0.502779i
$146$	87.7022			0.600700
$147$	84.5775			0.575357
$148$		−	45.9546i		−	0.310504i
$149$			11.8453i			0.0794988i	0.999210	+	0.0397494i	$0.0126560\pi$
$149$			11.8453i			0.0794988i	−0.999210	+	0.0397494i	$0.987344\pi$
$150$	12.2474			0.0816497
$151$	135.282			0.895908			0.447954	−	0.894057i	$-0.352153\pi$
$151$	135.282			0.895908			0.447954	+	0.894057i	$0.352153\pi$
$152$			23.5073i			0.154653i
$153$		−	66.0877i		−	0.431946i
$154$	−4.06418			−0.0263908
$155$		−	39.6353i		−	0.255712i
$156$	−26.5278			−0.170050
$157$		−	172.890i		−	1.10121i	−0.834765	−	0.550606i	$-0.814396\pi$
$157$		−	172.890i		−	1.10121i	0.834765	−	0.550606i	$-0.185604\pi$
$158$			49.8942i			0.315786i
$159$		−	30.3771i		−	0.191051i
$160$		−	12.6491i		−	0.0790569i
$161$	6.19662	+	7.14813i	0.0384883	+	0.0443983i
$162$	−12.7279			−0.0785674
$163$	−247.308			−1.51723			−0.758613	−	0.651541i	$-0.774123\pi$
$163$	−247.308			−1.51723			−0.758613	+	0.651541i	$0.774123\pi$
$164$	57.7594			0.352191
$165$	27.0604			0.164003
$166$			203.211i			1.22416i
$167$	−218.390			−1.30773			−0.653863	−	0.756613i	$-0.726853\pi$
$167$	−218.390			−1.30773			−0.653863	+	0.756613i	$0.726853\pi$
$168$		−	2.01500i		−	0.0119940i
$169$	−110.356			−0.652997
$170$	−69.6625			−0.409780
$171$		−	24.9332i		−	0.145808i
$172$		−	15.1603i		−	0.0881415i
$173$	−172.122			−0.994927			−0.497463	−	0.867485i	$-0.665735\pi$
$173$	−172.122			−0.994927			−0.497463	+	0.867485i	$0.665735\pi$
$174$	−79.8611			−0.458972
$175$		−	2.05655i		−	0.0117517i
$176$		−	27.9479i		−	0.158795i
$177$	−58.6565			−0.331392
$178$			103.816i			0.583237i
$179$	84.0652			0.469638			0.234819	−	0.972039i	$-0.424550\pi$
$179$	84.0652			0.469638			0.234819	+	0.972039i	$0.424550\pi$
$180$			13.4164i			0.0745356i
$181$			277.980i			1.53580i	0.640569	+	0.767901i	$0.278699\pi$
$181$			277.980i			1.53580i	−0.640569	+	0.767901i	$0.721301\pi$
$182$			4.45445i			0.0244750i
$183$			67.9946i			0.371555i
$184$	−49.1551	+	42.6119i	−0.267147	+	0.231587i
$185$	51.3788			0.277723
$186$	43.4183			0.233432
$187$	−153.918			−0.823088
$188$	98.9869			0.526526
$189$			2.13723i			0.0113081i
$190$	−26.2819			−0.138326
$191$			66.0363i			0.345740i	0.984945	+	0.172870i	$0.0553040\pi$
$191$			66.0363i			0.345740i	−0.984945	+	0.172870i	$0.944696\pi$
$192$	13.8564			0.0721688
$193$	−91.4052			−0.473602			−0.236801	−	0.971558i	$-0.576099\pi$
$193$	−91.4052			−0.473602			−0.236801	+	0.971558i	$0.576099\pi$
$194$		−	135.644i		−	0.699195i
$195$		−	29.6590i		−	0.152097i
$196$	97.6616			0.498274
$197$	34.0865			0.173028			0.0865140	−	0.996251i	$-0.472427\pi$
$197$	34.0865			0.173028			0.0865140	+	0.996251i	$0.472427\pi$
$198$			29.6432i			0.149713i
$199$			276.277i			1.38833i	0.719817	+	0.694164i	$0.244226\pi$
$199$			276.277i			1.38833i	−0.719817	+	0.694164i	$0.755774\pi$
$200$	14.1421			0.0707107
$201$			122.789i			0.610892i
$202$	−205.059			−1.01514
$203$			13.4100i			0.0660591i
$204$		−	76.3115i		−	0.374076i
$205$			64.5769i			0.315009i
$206$			174.750i			0.848299i
$207$	52.1369	−	45.1968i	0.251869	−	0.218342i
$208$	−30.6316			−0.147267
$209$	−58.0692			−0.277843
$210$	2.25283			0.0107278
$211$	−159.869			−0.757675			−0.378838	−	0.925463i	$-0.623676\pi$
$211$	−159.869			−0.757675			−0.378838	+	0.925463i	$0.623676\pi$
$212$		−	35.0765i		−	0.165455i
$213$	46.4476			0.218064
$214$			130.239i			0.608596i
$215$	16.9498			0.0788362
$216$	−14.6969			−0.0680414
$217$		−	7.29065i		−	0.0335975i
$218$			63.6528i			0.291985i
$219$	−107.413			−0.490470
$220$	31.2467			0.142030
$221$			168.698i			0.763339i
$222$			56.2826i			0.253525i
$223$	−26.6387			−0.119456			−0.0597280	−	0.998215i	$-0.519023\pi$
$223$	−26.6387			−0.119456			−0.0597280	+	0.998215i	$0.519023\pi$
$224$		−	2.32672i		−	0.0103871i
$225$	−15.0000			−0.0666667
$226$			76.8574i			0.340077i
$227$			209.504i			0.922924i	0.887160	+	0.461462i	$0.152675\pi$
$227$			209.504i			0.922924i	−0.887160	+	0.461462i	$0.847325\pi$
$228$		−	28.7904i		−	0.126274i
$229$			80.8748i			0.353165i	0.984286	+	0.176583i	$0.0565043\pi$
$229$			80.8748i			0.353165i	−0.984286	+	0.176583i	$0.943496\pi$
$230$	−47.6416	−	54.9571i	−0.207137	−	0.238944i
$231$	4.97758			0.0215480
$232$	−92.2157			−0.397481
$233$	−174.793			−0.750184			−0.375092	−	0.926988i	$-0.622389\pi$
$233$	−174.793			−0.750184			−0.375092	+	0.926988i	$0.622389\pi$
$234$	32.4898			0.138845
$235$			110.671i			0.470939i
$236$	−67.7307			−0.286994
$237$		−	61.1077i		−	0.257838i
$238$	−12.8139			−0.0538401
$239$	99.3102			0.415524			0.207762	−	0.978179i	$-0.433382\pi$
$239$	99.3102			0.415524			0.207762	+	0.978179i	$0.433382\pi$
$240$			15.4919i			0.0645497i
$241$			43.9106i			0.182202i	0.995842	+	0.0911008i	$0.0290386\pi$
$241$			43.9106i			0.182202i	−0.995842	+	0.0911008i	$0.970961\pi$
$242$	−102.081			−0.421823
$243$	15.5885			0.0641500
$244$			78.5134i			0.321776i
$245$			109.189i			0.445670i
$246$	−70.7405			−0.287563
$247$			63.6454i			0.257674i
$248$	50.1352			0.202158
$249$		−	248.882i		−	0.999525i
$250$			15.8114i			0.0632456i
$251$			419.272i			1.67041i	0.549940	+	0.835204i	$0.314651\pi$
$251$			419.272i			1.67041i	−0.549940	+	0.835204i	$0.685349\pi$
$252$			2.46786i			0.00979308i
$253$	−105.263	−	121.426i	−0.416059	−	0.479945i
$254$	−161.501			−0.635833
$255$	85.3188			0.334584
$256$	16.0000			0.0625000
$257$	−207.444			−0.807176			−0.403588	−	0.914941i	$-0.632237\pi$
$257$	−207.444			−0.807176			−0.403588	+	0.914941i	$0.632237\pi$
$258$			18.5676i			0.0719673i
$259$	9.45077			0.0364895
$260$		−	34.2472i		−	0.131720i
$261$	97.8095			0.374749
$262$	−123.239			−0.470376
$263$		−	342.541i		−	1.30244i	−0.758891	−	0.651218i	$-0.774258\pi$
$263$		−	342.541i		−	1.30244i	0.758891	−	0.651218i	$-0.225742\pi$
$264$			34.2290i			0.129655i
$265$	39.2167			0.147987
$266$	−4.83438			−0.0181744
$267$		−	127.148i		−	0.476211i
$268$			141.785i			0.529048i
$269$	−403.239			−1.49903			−0.749514	−	0.661988i	$-0.769713\pi$
$269$	−403.239			−1.49903			−0.749514	+	0.661988i	$0.769713\pi$
$270$		−	16.4317i		−	0.0608581i
$271$	45.3290			0.167266			0.0836328	−	0.996497i	$-0.473348\pi$
$271$	45.3290			0.167266			0.0836328	+	0.996497i	$0.473348\pi$
$272$		−	88.1169i		−	0.323959i
$273$		−	5.45556i		−	0.0199837i
$274$			4.25612i			0.0155333i
$275$			34.9348i			0.127036i
$276$	60.2025	−	52.1887i	0.218125	−	0.189090i
$277$	−241.667			−0.872446			−0.436223	−	0.899839i	$-0.643684\pi$
$277$	−241.667			−0.872446			−0.436223	+	0.899839i	$0.643684\pi$
$278$	175.654			0.631847
$279$	−53.1764			−0.190596
$280$	2.60135			0.00929053
$281$			231.673i			0.824459i	0.911080	+	0.412229i	$0.135250\pi$
$281$			231.673i			0.824459i	−0.911080	+	0.412229i	$0.864750\pi$
$282$	−121.234			−0.429907
$283$		−	175.141i		−	0.618874i	−0.950920	−	0.309437i	$-0.899859\pi$
$283$		−	175.141i		−	0.618874i	0.950920	−	0.309437i	$-0.100141\pi$
$284$	53.6331			0.188849
$285$	32.1886			0.112943
$286$		−	75.6683i		−	0.264575i
$287$			11.8785i			0.0413884i
$288$	−16.9706			−0.0589256
$289$	−196.287			−0.679194
$290$		−	103.100i		−	0.355518i
$291$			166.129i			0.570890i
$292$	−124.030			−0.424759
$293$			306.727i			1.04685i	0.852072	+	0.523424i	$0.175346\pi$
$293$			306.727i			1.04685i	−0.852072	+	0.523424i	$0.824654\pi$
$294$	−119.611			−0.406839
$295$		−	75.7252i		−	0.256696i
$296$			64.9896i			0.219559i
$297$		−	36.3054i		−	0.122240i
$298$		−	16.7518i		−	0.0562141i
$299$	−133.086	+	115.371i	−0.445105	+	0.385856i
$300$	−17.3205			−0.0577350
$301$	3.11780			0.0103581
$302$	−191.318			−0.633503
$303$	251.145			0.828860
$304$		−	33.2443i		−	0.109356i
$305$	−87.7806			−0.287805
$306$			93.4621i			0.305432i
$307$	106.431			0.346682			0.173341	−	0.984862i	$-0.444544\pi$
$307$	106.431			0.346682			0.173341	+	0.984862i	$0.444544\pi$
$308$	5.74761			0.0186611
$309$		−	214.024i		−	0.692633i
$310$			56.0528i			0.180816i
$311$	604.165			1.94265			0.971327	−	0.237749i	$-0.0764097\pi$
$311$	604.165			1.94265			0.971327	+	0.237749i	$0.0764097\pi$
$312$	37.5159			0.120243
$313$			505.257i			1.61424i	0.590388	+	0.807120i	$0.298975\pi$
$313$			505.257i			1.61424i	−0.590388	+	0.807120i	$0.701025\pi$
$314$			244.504i			0.778675i
$315$	−2.75915			−0.00875920
$316$		−	70.5611i		−	0.223294i
$317$	104.465			0.329544			0.164772	−	0.986332i	$-0.447311\pi$
$317$	104.465			0.329544			0.164772	+	0.986332i	$0.447311\pi$
$318$			42.9597i			0.135093i
$319$		−	227.797i		−	0.714098i
$320$			17.8885i			0.0559017i
$321$		−	159.510i		−	0.496916i
$322$	−8.76334	−	10.1090i	−0.0272154	−	0.0313943i
$323$	−183.086			−0.566831
$324$	18.0000			0.0555556
$325$	38.2895			0.117814
$326$	349.746			1.07284
$327$		−	77.9584i		−	0.238405i
$328$	−81.6841			−0.249037
$329$			20.3571i			0.0618758i
$330$	−38.2692			−0.115967
$331$	−41.3629			−0.124963			−0.0624817	−	0.998046i	$-0.519902\pi$
$331$	−41.3629			−0.124963			−0.0624817	+	0.998046i	$0.519902\pi$
$332$		−	287.384i		−	0.865614i
$333$		−	68.9319i		−	0.207003i
$334$	308.851			0.924702
$335$	−158.520			−0.473195
$336$			2.84964i			0.00848106i
$337$			181.417i			0.538331i	0.963094	+	0.269165i	$0.0867478\pi$
$337$			181.417i			0.538331i	−0.963094	+	0.269165i	$0.913252\pi$
$338$	156.068			0.461738
$339$		−	94.1307i		−	0.277672i
$340$	98.5177			0.289758
$341$			123.847i			0.363188i
$342$			35.2609i			0.103102i
$343$			40.2387i			0.117314i
$344$			21.4400i			0.0623255i
$345$	58.3488	+	67.3084i	0.169127	+	0.195097i
$346$	243.418			0.703519
$347$	202.642			0.583984			0.291992	−	0.956421i	$-0.405682\pi$
$347$	202.642			0.583984			0.291992	+	0.956421i	$0.405682\pi$
$348$	112.941			0.324542
$349$	319.984			0.916859			0.458430	−	0.888731i	$-0.348412\pi$
$349$	319.984			0.916859			0.458430	+	0.888731i	$0.348412\pi$
$350$			2.90840i			0.00830971i
$351$	−39.7917			−0.113367
$352$			39.5243i			0.112285i
$353$	−308.167			−0.872994			−0.436497	−	0.899706i	$-0.643781\pi$
$353$	−308.167			−0.872994			−0.436497	+	0.899706i	$0.643781\pi$
$354$	82.9528			0.234330
$355$			59.9636i			0.168912i
$356$		−	146.818i		−	0.412411i
$357$	15.6938			0.0439603
$358$	−118.886			−0.332084
$359$		−	301.630i		−	0.840196i	−0.907479	−	0.420098i	$-0.861996\pi$
$359$		−	301.630i		−	0.840196i	0.907479	−	0.420098i	$-0.138004\pi$
$360$		−	18.9737i		−	0.0527046i
$361$	291.926			0.808660
$362$		−	393.123i		−	1.08598i
$363$	125.023			0.344417
$364$		−	6.29954i		−	0.0173064i
$365$		−	138.669i		−	0.379916i
$366$		−	96.1588i		−	0.262729i
$367$		−	470.453i		−	1.28189i	−0.767587	−	0.640945i	$-0.778543\pi$
$367$		−	470.453i		−	1.28189i	0.767587	−	0.640945i	$-0.221457\pi$
$368$	69.5158	−	60.2624i	0.188902	−	0.163756i
$369$	86.6390			0.234794
$370$	−72.6606			−0.196380
$371$	7.21364			0.0194438
$372$	−61.4028			−0.165061
$373$		−	249.106i		−	0.667844i	−0.942601	−	0.333922i	$-0.891628\pi$
$373$		−	249.106i		−	0.667844i	0.942601	−	0.333922i	$-0.108372\pi$
$374$	217.672			0.582011
$375$		−	19.3649i		−	0.0516398i
$376$	−139.989			−0.372310
$377$	−249.672			−0.662260
$378$		−	3.02249i		−	0.00799602i
$379$		−	141.700i		−	0.373879i	−0.982371	−	0.186940i	$-0.940143\pi$
$379$		−	141.700i		−	0.373879i	0.982371	−	0.186940i	$-0.0598569\pi$
$380$	37.1682			0.0978112
$381$	197.798			0.519155
$382$		−	93.3895i		−	0.244475i
$383$			110.480i			0.288460i	0.989544	+	0.144230i	$0.0460705\pi$
$383$			110.480i			0.288460i	−0.989544	+	0.144230i	$0.953929\pi$
$384$	−19.5959			−0.0510310
$385$			6.42603i			0.0166910i
$386$	129.266			0.334887
$387$		−	22.7405i		−	0.0587610i
$388$			191.829i			0.494405i
$389$			705.447i			1.81349i	0.421681	+	0.906744i	$0.361440\pi$
$389$			705.447i			1.81349i	−0.421681	+	0.906744i	$0.638560\pi$
$390$			41.9441i			0.107549i
$391$	−331.883	−	382.845i	−0.848807	−	0.979143i
$392$	−138.114			−0.352333
$393$	150.936			0.384061
$394$	−48.2056			−0.122349
$395$	78.8897			0.199721
$396$		−	41.9218i		−	0.105863i
$397$	10.3766			0.0261376			0.0130688	−	0.999915i	$-0.495840\pi$
$397$	10.3766			0.0261376			0.0130688	+	0.999915i	$0.495840\pi$
$398$		−	390.715i		−	0.981696i
$399$	5.92088			0.0148393
$400$	−20.0000			−0.0500000
$401$		−	50.4144i		−	0.125722i	−0.998022	−	0.0628609i	$-0.979978\pi$
$401$		−	50.4144i		−	0.125722i	0.998022	−	0.0628609i	$-0.0200224\pi$
$402$		−	173.650i		−	0.431966i
$403$	135.740			0.336824
$404$	289.997			0.717814
$405$			20.1246i			0.0496904i
$406$		−	18.9646i		−	0.0467108i
$407$	−160.542			−0.394451
$408$			107.921i			0.264512i
$409$	71.8300			0.175624			0.0878118	−	0.996137i	$-0.472013\pi$
$409$	71.8300			0.175624			0.0878118	+	0.996137i	$0.472013\pi$
$410$		−	91.3256i		−	0.222745i
$411$		−	5.21266i		−	0.0126829i
$412$		−	247.133i		−	0.599838i
$413$		−	13.9291i		−	0.0337267i
$414$	−73.7327	+	63.9179i	−0.178098	+	0.154391i
$415$	321.305			0.774229
$416$	43.3197			0.104134
$417$	−215.131			−0.515901
$418$	82.1222			0.196465
$419$		−	355.682i		−	0.848884i	−0.905455	−	0.424442i	$-0.860470\pi$
$419$		−	355.682i		−	0.848884i	0.905455	−	0.424442i	$-0.139530\pi$
$420$	−3.18599			−0.00758569
$421$		−	277.750i		−	0.659738i	−0.944027	−	0.329869i	$-0.892995\pi$
$421$		−	277.750i		−	0.659738i	0.944027	−	0.329869i	$-0.107005\pi$
$422$	226.090			0.535757
$423$	148.480			0.351017
$424$			49.6056i			0.116994i
$425$			110.146i			0.259167i
$426$	−65.6868			−0.154194
$427$	−16.1466			−0.0378141
$428$		−	184.186i		−	0.430342i
$429$			92.6744i			0.216024i
$430$	−23.9706			−0.0557456
$431$			219.586i			0.509481i	0.967009	+	0.254741i	$0.0819901\pi$
$431$			219.586i			0.509481i	−0.967009	+	0.254741i	$0.918010\pi$
$432$	20.7846			0.0481125
$433$			284.730i			0.657574i	0.944404	+	0.328787i	$0.106640\pi$
$433$			284.730i			0.657574i	−0.944404	+	0.328787i	$0.893360\pi$
$434$			10.3105i			0.0237570i
$435$			126.272i			0.290279i
$436$		−	90.0186i		−	0.206465i
$437$	−125.211	−	144.438i	−0.286524	−	0.330521i
$438$	151.905			0.346814
$439$	−80.6153			−0.183634			−0.0918170	−	0.995776i	$-0.529267\pi$
$439$	−80.6153			−0.183634			−0.0918170	+	0.995776i	$0.529267\pi$
$440$	−44.1895			−0.100431
$441$	146.492			0.332182
$442$		−	238.575i		−	0.539762i
$443$	208.114			0.469784			0.234892	−	0.972021i	$-0.424526\pi$
$443$	208.114			0.469784			0.234892	+	0.972021i	$0.424526\pi$
$444$		−	79.5956i		−	0.179269i
$445$	164.148			0.368871
$446$	37.6728			0.0844681
$447$			20.5167i			0.0458986i
$448$			3.29048i			0.00734481i
$449$	−409.241			−0.911450			−0.455725	−	0.890121i	$-0.650620\pi$
$449$	−409.241			−0.911450			−0.455725	+	0.890121i	$0.650620\pi$
$450$	21.2132			0.0471405
$451$		−	201.781i		−	0.447409i
$452$		−	108.693i		−	0.240471i
$453$	234.316			0.517253
$454$		−	296.283i		−	0.652606i
$455$	7.04310			0.0154793
$456$			40.7158i			0.0892890i
$457$			310.013i			0.678365i	0.940721	+	0.339183i	$0.110151\pi$
$457$			310.013i			0.678365i	−0.940721	+	0.339183i	$0.889849\pi$
$458$		−	114.374i		−	0.249726i
$459$		−	114.467i		−	0.249384i
$460$	67.3754	+	77.7210i	0.146468	+	0.168959i
$461$	463.484			1.00539			0.502694	−	0.864465i	$-0.332342\pi$
$461$	463.484			1.00539			0.502694	+	0.864465i	$0.332342\pi$
$462$	−7.03936			−0.0152367
$463$	604.690			1.30603			0.653013	−	0.757347i	$-0.273505\pi$
$463$	604.690			1.30603			0.653013	+	0.757347i	$0.273505\pi$
$464$	130.413			0.281062
$465$		−	68.6504i		−	0.147635i
$466$	247.194			0.530460
$467$		−	271.585i		−	0.581551i	−0.956791	−	0.290776i	$-0.906087\pi$
$467$		−	271.585i		−	0.581551i	0.956791	−	0.290776i	$-0.0939134\pi$
$468$	−45.9475			−0.0981783
$469$	−29.1587			−0.0621721
$470$		−	156.512i		−	0.333004i
$471$		−	299.455i		−	0.635786i
$472$	95.7856			0.202936
$473$	−52.9624			−0.111971
$474$			86.4193i			0.182319i
$475$			41.5554i			0.0874850i
$476$	18.1217			0.0380707
$477$		−	52.6147i		−	0.110303i
$478$	−140.446			−0.293820
$479$		−	793.694i		−	1.65698i	−0.560003	−	0.828491i	$-0.689200\pi$
$479$		−	793.694i		−	1.65698i	0.560003	−	0.828491i	$-0.310800\pi$
$480$		−	21.9089i		−	0.0456435i
$481$			175.958i			0.365817i
$482$		−	62.0990i		−	0.128836i
$483$	10.7329	+	12.3809i	0.0222212	+	0.0256334i
$484$	144.365			0.298274
$485$	−214.472			−0.442210
$486$	−22.0454			−0.0453609
$487$	−434.574			−0.892350			−0.446175	−	0.894946i	$-0.647214\pi$
$487$	−434.574			−0.892350			−0.446175	+	0.894946i	$0.647214\pi$
$488$		−	111.035i		−	0.227530i
$489$	−428.350			−0.875971
$490$		−	154.417i		−	0.315136i
$491$	95.0833			0.193652			0.0968261	−	0.995301i	$-0.469131\pi$
$491$	95.0833			0.193652			0.0968261	+	0.995301i	$0.469131\pi$
$492$	100.042			0.203338
$493$		−	718.223i		−	1.45684i
$494$		−	90.0082i		−	0.182203i
$495$	46.8700			0.0946869
$496$	−70.9018			−0.142947
$497$			11.0299i			0.0221929i
$498$			351.972i			0.706771i
$499$	826.993			1.65730			0.828650	−	0.559767i	$-0.189109\pi$
$499$	826.993			1.65730			0.828650	+	0.559767i	$0.189109\pi$
$500$		−	22.3607i		−	0.0447214i
$501$	−378.263			−0.755016
$502$		−	592.941i		−	1.18116i
$503$			424.858i			0.844648i	0.906445	+	0.422324i	$0.138785\pi$
$503$			424.858i			0.844648i	−0.906445	+	0.422324i	$0.861215\pi$
$504$		−	3.49008i		−	0.00692475i
$505$			324.226i			0.642032i
$506$	148.864	+	171.723i	0.294198	+	0.339373i
$507$	−191.143			−0.377008
$508$	228.398			0.449602
$509$	−786.116			−1.54443			−0.772216	−	0.635360i	$-0.780852\pi$
$509$	−786.116			−1.54443			−0.772216	+	0.635360i	$0.780852\pi$
$510$	−120.659			−0.236586
$511$		−	25.5073i		−	0.0499164i
$512$	−22.6274			−0.0441942
$513$		−	43.1856i		−	0.0841824i
$514$	293.371			0.570760
$515$	276.303			0.536511
$516$		−	26.2585i		−	0.0508885i
$517$		−	345.809i		−	0.668877i
$518$	−13.3654			−0.0258020
$519$	−298.125			−0.574421
$520$			48.4329i			0.0931401i
$521$			792.323i			1.52077i	0.649471	+	0.760387i	$0.274991\pi$
$521$			792.323i			1.52077i	−0.649471	+	0.760387i	$0.725009\pi$
$522$	−138.324			−0.264988
$523$		−	123.543i		−	0.236220i	−0.993001	−	0.118110i	$-0.962316\pi$
$523$		−	123.543i		−	0.236220i	0.993001	−	0.118110i	$-0.0376835\pi$
$524$	174.286			0.332606
$525$		−	3.56204i		−	0.00678485i
$526$			484.425i			0.920961i
$527$			390.478i			0.740945i
$528$		−	48.4071i		−	0.0916802i
$529$	75.0560	−	523.648i	0.141883	−	0.989883i
$530$	−55.4608			−0.104643
$531$	−101.596			−0.191330
$532$	6.83684			0.0128512
$533$	−221.158			−0.414930
$534$			179.815i			0.336732i
$535$	205.927			0.384910
$536$		−	200.514i		−	0.374093i
$537$	145.605			0.271145
$538$	570.266			1.05997
$539$		−	341.179i		−	0.632986i
$540$			23.2379i			0.0430331i
$541$	−746.830			−1.38046			−0.690231	−	0.723589i	$-0.742491\pi$
$541$	−746.830			−1.38046			−0.690231	+	0.723589i	$0.742491\pi$
$542$	−64.1048			−0.118275
$543$			481.476i			0.886696i
$544$			124.616i			0.229074i
$545$	100.644			0.184668
$546$			7.71533i			0.0141306i
$547$	760.793			1.39085			0.695423	−	0.718600i	$-0.255217\pi$
$547$	760.793			1.39085			0.695423	+	0.718600i	$0.255217\pi$
$548$		−	6.01906i		−	0.0109837i
$549$			117.770i			0.214517i
$550$		−	49.4053i		−	0.0898279i
$551$		−	270.967i		−	0.491774i
$552$	−85.1391	+	73.8060i	−0.154238	+	0.133707i
$553$	14.5112			0.0262409
$554$	341.769			0.616912
$555$	88.9906			0.160343
$556$	−248.412			−0.446783
$557$			619.940i			1.11300i	0.830848	+	0.556499i	$0.187856\pi$
$557$			619.940i			1.11300i	−0.830848	+	0.556499i	$0.812144\pi$
$558$	75.2028			0.134772
$559$			58.0483i			0.103843i
$560$	−3.67886			−0.00656940
$561$	−266.593			−0.475210
$562$		−	327.635i		−	0.582980i
$563$		−	674.480i		−	1.19801i	−0.800745	−	0.599006i	$-0.795563\pi$
$563$		−	674.480i		−	1.19801i	0.800745	−	0.599006i	$-0.204437\pi$
$564$	171.450			0.303990
$565$	121.522			0.215084
$566$			247.687i			0.437610i
$567$			3.70178i			0.00652872i
$568$	−75.8486			−0.133536
$569$			456.740i			0.802706i	0.915923	+	0.401353i	$0.131460\pi$
$569$			456.740i			0.802706i	−0.915923	+	0.401353i	$0.868540\pi$
$570$	−45.5216			−0.0798625
$571$		−	484.210i		−	0.848004i	−0.905661	−	0.424002i	$-0.860625\pi$
$571$		−	484.210i		−	0.848004i	0.905661	−	0.424002i	$-0.139375\pi$
$572$			107.011i			0.187082i
$573$			114.378i			0.199613i
$574$		−	16.7987i		−	0.0292660i
$575$	−86.8948	+	75.3280i	−0.151121	+	0.131005i
$576$	24.0000			0.0416667
$577$	72.0514			0.124872			0.0624362	−	0.998049i	$-0.480113\pi$
$577$	72.0514			0.124872			0.0624362	+	0.998049i	$0.480113\pi$
$578$	277.592			0.480262
$579$	−158.318			−0.273434
$580$			145.806i			0.251389i
$581$	59.1018			0.101724
$582$		−	234.942i		−	0.403680i
$583$	−122.539			−0.210187
$584$	175.404			0.300350
$585$		−	51.3708i		−	0.0878134i
$586$		−	433.777i		−	0.740234i
$587$	−119.743			−0.203991			−0.101995	−	0.994785i	$-0.532523\pi$
$587$	−119.743			−0.203991			−0.101995	+	0.994785i	$0.532523\pi$
$588$	169.155			0.287678
$589$			147.318i			0.250115i
$590$			107.092i			0.181511i
$591$	59.0396			0.0998978
$592$		−	91.9091i		−	0.155252i
$593$	−648.197			−1.09308			−0.546540	−	0.837433i	$-0.684055\pi$
$593$	−648.197			−1.09308			−0.546540	+	0.837433i	$0.684055\pi$
$594$			51.3435i			0.0864369i
$595$			20.2606i			0.0340515i
$596$			23.6906i			0.0397494i
$597$			478.526i			0.801551i
$598$	188.213	−	163.159i	0.314737	−	0.272841i
$599$	269.000			0.449082			0.224541	−	0.974465i	$-0.427912\pi$
$599$	269.000			0.449082			0.224541	+	0.974465i	$0.427912\pi$
$600$	24.4949			0.0408248
$601$	−339.169			−0.564340			−0.282170	−	0.959364i	$-0.591054\pi$
$601$	−339.169			−0.564340			−0.282170	+	0.959364i	$0.591054\pi$
$602$	−4.40923			−0.00732430
$603$			212.677i			0.352698i
$604$	270.564			0.447954
$605$			161.404i			0.266784i
$606$	−355.172			−0.586093
$607$	372.696			0.613997			0.306998	−	0.951710i	$-0.400675\pi$
$607$	372.696			0.613997			0.306998	+	0.951710i	$0.400675\pi$
$608$			47.0145i			0.0773265i
$609$			23.2268i			0.0381392i
$610$	124.141			0.203509
$611$	−379.016			−0.620322
$612$		−	132.175i		−	0.215973i
$613$		−	125.657i		−	0.204987i	−0.994734	−	0.102493i	$-0.967318\pi$
$613$		−	125.657i		−	0.204987i	0.994734	−	0.102493i	$-0.0326820\pi$
$614$	−150.517			−0.245141
$615$			111.851i			0.181871i
$616$	−8.12835			−0.0131954
$617$			702.136i			1.13798i	0.822343	+	0.568992i	$0.192667\pi$
$617$			702.136i			1.13798i	−0.822343	+	0.568992i	$0.807333\pi$
$618$			302.675i			0.489766i
$619$			888.543i			1.43545i	0.696327	+	0.717724i	$0.254816\pi$
$619$			888.543i			1.43545i	−0.696327	+	0.717724i	$0.745184\pi$
$620$		−	79.2707i		−	0.127856i
$621$	90.3037	−	78.2831i	0.145417	−	0.126060i
$622$	−854.418			−1.37366
$623$	30.1938			0.0484652
$624$	−53.0556			−0.0850249
$625$	25.0000			0.0400000
$626$		−	714.541i		−	1.14144i
$627$	−100.579			−0.160413
$628$		−	345.781i		−	0.550606i
$629$	−506.172			−0.804725
$630$	3.90202			0.00619369
$631$		−	189.781i		−	0.300763i	−0.988628	−	0.150381i	$-0.951950\pi$
$631$		−	189.781i		−	0.300763i	0.988628	−	0.150381i	$-0.0480501\pi$
$632$			99.7884i			0.157893i
$633$	−276.902			−0.437444
$634$	−147.736			−0.233023
$635$			255.356i			0.402136i
$636$		−	60.7542i		−	0.0955255i
$637$	−373.942			−0.587036
$638$			322.154i			0.504944i
$639$	80.4496			0.125899
$640$		−	25.2982i		−	0.0395285i
$641$			682.549i			1.06482i	0.846487	+	0.532410i	$0.178713\pi$
$641$			682.549i			1.06482i	−0.846487	+	0.532410i	$0.821287\pi$
$642$			225.581i			0.351373i
$643$			436.832i			0.679365i	0.940540	+	0.339682i	$0.110320\pi$
$643$			436.832i			0.679365i	−0.940540	+	0.339682i	$0.889680\pi$
$644$	12.3932	+	14.2963i	0.0192442	+	0.0221992i
$645$	29.3579			0.0455161
$646$	258.923			0.400810
$647$	853.291			1.31884			0.659421	−	0.751774i	$-0.270801\pi$
$647$	853.291			1.31884			0.659421	+	0.751774i	$0.270801\pi$
$648$	−25.4558			−0.0392837
$649$			236.616i			0.364586i
$650$	−54.1496			−0.0833071
$651$		−	12.6278i		−	0.0193975i
$652$	−494.616			−0.758613
$653$	−502.550			−0.769601			−0.384801	−	0.923000i	$-0.625730\pi$
$653$	−502.550			−0.769601			−0.384801	+	0.923000i	$0.625730\pi$
$654$			110.250i			0.168578i
$655$			194.857i			0.297492i
$656$	115.519			0.176096
$657$	−186.044			−0.283173
$658$		−	28.7893i		−	0.0437528i
$659$		−	325.726i		−	0.494273i	−0.968981	−	0.247136i	$-0.920510\pi$
$659$		−	325.726i		−	0.494273i	0.968981	−	0.247136i	$-0.0794895\pi$
$660$	54.1208			0.0820013
$661$		−	578.734i		−	0.875543i	−0.899086	−	0.437771i	$-0.855768\pi$
$661$		−	578.734i		−	0.875543i	0.899086	−	0.437771i	$-0.144232\pi$
$662$	58.4960			0.0883625
$663$			292.193i			0.440714i
$664$			406.422i			0.612081i
$665$			7.64382i			0.0114945i
$666$			97.4844i			0.146373i
$667$	566.609	−	491.186i	0.849489	−	0.736411i
$668$	−436.781			−0.653863
$669$	−46.1396			−0.0689679
$670$	224.181			0.334599
$671$	274.285			0.408771
$672$		−	4.02999i		−	0.00599701i
$673$	787.850			1.17065			0.585327	−	0.810798i	$-0.300966\pi$
$673$	787.850			1.17065			0.585327	+	0.810798i	$0.300966\pi$
$674$		−	256.563i		−	0.380657i
$675$	−25.9808			−0.0384900
$676$	−220.713			−0.326498
$677$		−	1053.03i		−	1.55543i	−0.628616	−	0.777716i	$-0.716378\pi$
$677$		−	1053.03i		−	1.55543i	0.628616	−	0.777716i	$-0.283622\pi$
$678$			133.121i			0.196344i
$679$	−39.4506			−0.0581010
$680$	−139.325			−0.204890
$681$			362.871i			0.532850i
$682$		−	175.146i		−	0.256813i
$683$	−416.716			−0.610127			−0.305063	−	0.952332i	$-0.598678\pi$
$683$	−416.716			−0.610127			−0.305063	+	0.952332i	$0.598678\pi$
$684$		−	49.8664i		−	0.0729041i
$685$	6.72951			0.00982410
$686$		−	56.9062i		−	0.0829536i
$687$			140.079i			0.203900i
$688$		−	30.3207i		−	0.0440708i
$689$			134.306i			0.194929i
$690$	−82.5176	−	95.1884i	−0.119591	−	0.137954i
$691$	144.765			0.209500			0.104750	−	0.994499i	$-0.466596\pi$
$691$	144.765			0.209500			0.104750	+	0.994499i	$0.466596\pi$
$692$	−344.245			−0.497463
$693$	8.62142			0.0124407
$694$	−286.580			−0.412939
$695$		−	277.733i		−	0.399615i
$696$	−159.722			−0.229486
$697$		−	636.197i		−	0.912765i
$698$	−452.525			−0.648317
$699$	−302.750			−0.433119
$700$		−	4.11309i		−	0.00587585i
$701$			723.630i			1.03228i	0.856503	+	0.516142i	$0.172632\pi$
$701$			723.630i			1.03228i	−0.856503	+	0.516142i	$0.827368\pi$
$702$	56.2739			0.0801623
$703$	−190.966			−0.271644
$704$		−	55.8958i		−	0.0793974i
$705$			191.687i			0.271897i
$706$	435.814			0.617300
$707$			59.6392i			0.0843553i
$708$	−117.313			−0.165696
$709$		−	4.77044i		−	0.00672841i	−0.999994	−	0.00336420i	$-0.998929\pi$
$709$		−	4.77044i		−	0.00672841i	0.999994	−	0.00336420i	$-0.00107086\pi$
$710$		−	84.8013i		−	0.119438i
$711$		−	105.842i		−	0.148863i
$712$			207.632i			0.291618i
$713$	−308.050	+	267.044i	−0.432048	+	0.374536i
$714$	−22.1944			−0.0310846
$715$	−119.642			−0.167332
$716$	168.130			0.234819
$717$	172.010			0.239903
$718$			426.570i			0.594108i
$719$	−163.295			−0.227115			−0.113557	−	0.993531i	$-0.536225\pi$
$719$	−163.295			−0.227115			−0.113557	+	0.993531i	$0.536225\pi$
$720$			26.8328i			0.0372678i
$721$	50.8241			0.0704911
$722$	−412.846			−0.571809
$723$			76.0554i			0.105194i
$724$			555.960i			0.767901i
$725$	−163.016			−0.224849
$726$	−176.810			−0.243540
$727$			1100.34i			1.51354i	0.653683	+	0.756769i	$0.273223\pi$
$727$			1100.34i			1.51354i	−0.653683	+	0.756769i	$0.726777\pi$
$728$			8.90890i			0.0122375i
$729$	27.0000			0.0370370
$730$			196.108i			0.268641i
$731$	−166.985			−0.228434
$732$			135.989i			0.185777i
$733$			362.979i			0.495197i	0.968863	+	0.247599i	$0.0796414\pi$
$733$			362.979i			0.495197i	−0.968863	+	0.247599i	$0.920359\pi$
$734$			665.321i			0.906432i
$735$			189.121i			0.257307i
$736$	−98.3102	+	85.2238i	−0.133574	+	0.115793i
$737$	495.323			0.672080
$738$	−122.526			−0.166025
$739$	−475.365			−0.643254			−0.321627	−	0.946866i	$-0.604230\pi$
$739$	−475.365			−0.643254			−0.321627	+	0.946866i	$0.604230\pi$
$740$	102.758			0.138862
$741$			110.237i			0.148768i
$742$	−10.2016			−0.0137488
$743$		−	691.352i		−	0.930487i	−0.885183	−	0.465244i	$-0.845967\pi$
$743$		−	691.352i		−	0.930487i	0.885183	−	0.465244i	$-0.154033\pi$
$744$	86.8367			0.116716
$745$	−26.4869			−0.0355529
$746$			352.289i			0.472237i
$747$		−	431.076i		−	0.577076i
$748$	−307.835			−0.411544
$749$	37.8788			0.0505725
$750$			27.3861i			0.0365148i
$751$		−	971.631i		−	1.29378i	−0.762582	−	0.646892i	$-0.776069\pi$
$751$		−	971.631i		−	1.29378i	0.762582	−	0.646892i	$-0.223931\pi$
$752$	197.974			0.263263
$753$			726.201i			0.964411i
$754$	353.090			0.468289
$755$			302.500i			0.400662i
$756$			4.27445i			0.00565404i
$757$			983.792i			1.29959i	0.760108	+	0.649797i	$0.225146\pi$
$757$			983.792i			1.29959i	−0.760108	+	0.649797i	$0.774854\pi$
$758$			200.394i			0.264373i
$759$	−182.321	−	210.316i	−0.240212	−	0.277097i
$760$	−52.5638			−0.0691629
$761$	−73.1433			−0.0961147			−0.0480574	−	0.998845i	$-0.515303\pi$
$761$	−73.1433			−0.0961147			−0.0480574	+	0.998845i	$0.515303\pi$
$762$	−279.729			−0.367098
$763$	18.5127			0.0242631
$764$			132.073i			0.172870i
$765$	147.777			0.193172
$766$		−	156.243i		−	0.203972i
$767$	259.338			0.338119
$768$	27.7128			0.0360844
$769$		−	687.166i		−	0.893584i	−0.894638	−	0.446792i	$-0.852566\pi$
$769$		−	687.166i		−	0.893584i	0.894638	−	0.446792i	$-0.147434\pi$
$770$		−	9.08777i		−	0.0118023i
$771$	−359.304			−0.466023
$772$	−182.810			−0.236801
$773$			56.9245i			0.0736411i	0.999322	+	0.0368205i	$0.0117230\pi$
$773$			56.9245i			0.0736411i	−0.999322	+	0.0368205i	$0.988277\pi$
$774$			32.1599i			0.0415503i
$775$	88.6273			0.114358
$776$		−	271.288i		−	0.349597i
$777$	16.3692			0.0210672
$778$		−	997.653i		−	1.28233i
$779$		−	240.021i		−	0.308114i
$780$		−	59.3179i		−	0.0760486i
$781$		−	187.366i		−	0.239906i
$782$	469.354	+	541.424i	0.600197	+	0.692359i
$783$	169.411			0.216361
$784$	195.323			0.249137
$785$	386.595			0.492477
$786$	−213.456			−0.271572
$787$		−	339.738i		−	0.431688i	−0.976428	−	0.215844i	$-0.930750\pi$
$787$		−	339.738i		−	0.431688i	0.976428	−	0.215844i	$-0.0692502\pi$
$788$	68.1730			0.0865140
$789$		−	593.298i		−	0.751961i
$790$	−111.567			−0.141224
$791$	22.3532			0.0282594
$792$			59.2864i			0.0748566i
$793$		−	300.624i		−	0.379097i
$794$	−14.6748			−0.0184821
$795$	67.9253			0.0854406
$796$			552.554i			0.694164i
$797$			1124.29i			1.41065i	0.708885	+	0.705324i	$0.249199\pi$
$797$			1124.29i			1.41065i	−0.708885	+	0.705324i	$0.750801\pi$
$798$	−8.37339			−0.0104930
$799$		−	1090.30i		−	1.36458i
$800$	28.2843			0.0353553
$801$		−	220.227i		−	0.274940i
$802$			71.2967i			0.0888987i
$803$			433.296i			0.539596i
$804$			245.578i			0.305446i
$805$	−15.9837	+	13.8561i	−0.0198555	+	0.0172125i
$806$	−191.965			−0.238170
$807$	−698.430			−0.865465
$808$	−410.117			−0.507571
$809$	−12.1761			−0.0150508			−0.00752538	−	0.999972i	$-0.502395\pi$
$809$	−12.1761			−0.0150508			−0.00752538	+	0.999972i	$0.502395\pi$
$810$		−	28.4605i		−	0.0351364i
$811$	−1363.41			−1.68114			−0.840571	−	0.541702i	$-0.817780\pi$
$811$	−1363.41			−1.68114			−0.840571	+	0.541702i	$0.817780\pi$
$812$			26.8200i			0.0330295i
$813$	78.5121			0.0965708
$814$	227.040			0.278919
$815$		−	552.997i		−	0.678524i
$816$		−	152.623i		−	0.187038i
$817$	−62.9993			−0.0771106
$818$	−101.583			−0.124185
$819$		−	9.44931i		−	0.0115376i
$820$			129.154i			0.157505i
$821$	1189.77			1.44917			0.724586	−	0.689184i	$-0.242031\pi$
$821$	1189.77			1.44917			0.724586	+	0.689184i	$0.242031\pi$
$822$			7.37181i			0.00896814i
$823$	522.408			0.634761			0.317381	−	0.948298i	$-0.397197\pi$
$823$	522.408			0.634761			0.317381	+	0.948298i	$0.397197\pi$
$824$			349.499i			0.424150i
$825$			60.5089i			0.0733442i
$826$			19.6988i			0.0238484i
$827$			729.651i			0.882286i	0.897437	+	0.441143i	$0.145427\pi$
$827$			729.651i			0.882286i	−0.897437	+	0.441143i	$0.854573\pi$
$828$	104.274	−	90.3935i	0.125934	−	0.109171i
$829$	−864.893			−1.04330			−0.521648	−	0.853161i	$-0.674683\pi$
$829$	−864.893			−1.04330			−0.521648	+	0.853161i	$0.674683\pi$
$830$	−454.394			−0.547462
$831$	−418.580			−0.503707
$832$	−61.2633			−0.0736337
$833$		−	1075.71i		−	1.29136i
$834$	304.241			0.364797
$835$		−	488.336i		−	0.584833i
$836$	−116.138			−0.138922
$837$	−92.1042			−0.110041
$838$			503.011i			0.600252i
$839$		−	179.637i		−	0.214109i	−0.994253	−	0.107054i	$-0.965858\pi$
$839$		−	179.637i		−	0.214109i	0.994253	−	0.107054i	$-0.0341419\pi$
$840$	4.50567			0.00536389
$841$	221.967			0.263932
$842$			392.797i			0.466505i
$843$			401.269i			0.476001i
$844$	−319.739			−0.378838
$845$		−	246.764i		−	0.292029i
$846$	−209.983			−0.248207
$847$			29.6892i			0.0350522i
$848$		−	70.1529i		−	0.0827275i
$849$		−	303.354i		−	0.357307i
$850$		−	155.770i		−	0.183259i
$851$	−346.166	−	399.321i	−0.406776	−	0.469238i
$852$	92.8952			0.109032
$853$	875.737			1.02666			0.513328	−	0.858193i	$-0.328412\pi$
$853$	875.737			1.02666			0.513328	+	0.858193i	$0.328412\pi$
$854$	22.8348			0.0267386
$855$	55.7524			0.0652074
$856$			260.479i			0.304298i
$857$	1063.40			1.24084			0.620421	−	0.784269i	$-0.286962\pi$
$857$	1063.40			1.24084			0.620421	+	0.784269i	$0.286962\pi$
$858$		−	131.061i		−	0.152752i
$859$	629.092			0.732354			0.366177	−	0.930545i	$-0.380667\pi$
$859$	629.092			0.732354			0.366177	+	0.930545i	$0.380667\pi$
$860$	33.8996			0.0394181
$861$			20.5741i			0.0238956i
$862$		−	310.542i		−	0.360258i
$863$	432.281			0.500905			0.250453	−	0.968129i	$-0.419421\pi$
$863$	432.281			0.500905			0.250453	+	0.968129i	$0.419421\pi$
$864$	−29.3939			−0.0340207
$865$		−	384.877i		−	0.444945i
$866$		−	402.668i		−	0.464975i
$867$	−339.979			−0.392133
$868$		−	14.5813i		−	0.0167987i
$869$	−246.504			−0.283664
$870$		−	178.575i		−	0.205259i
$871$		−	542.887i		−	0.623292i
$872$			127.306i			0.145993i
$873$			287.744i			0.329604i
$874$	177.075	+	204.266i	0.202603	+	0.233714i
$875$	4.59858			0.00525552
$876$	−214.826			−0.245235
$877$	1070.11			1.22019			0.610095	−	0.792329i	$-0.291131\pi$
$877$	1070.11			1.22019			0.610095	+	0.792329i	$0.291131\pi$
$878$	114.007			0.129849
$879$			531.266i			0.604398i
$880$	62.4933			0.0710152
$881$		−	669.640i		−	0.760091i	−0.924968	−	0.380045i	$-0.875908\pi$
$881$		−	669.640i		−	0.760091i	0.924968	−	0.380045i	$-0.124092\pi$
$882$	−207.172			−0.234888
$883$	−24.5754			−0.0278317			−0.0139158	−	0.999903i	$-0.504430\pi$
$883$	−24.5754			−0.0278317			−0.0139158	+	0.999903i	$0.504430\pi$
$884$			337.396i			0.381669i
$885$		−	131.160i		−	0.148203i
$886$	−294.318			−0.332188
$887$	1367.15			1.54132			0.770659	−	0.637248i	$-0.219927\pi$
$887$	1367.15			1.54132			0.770659	+	0.637248i	$0.219927\pi$
$888$			112.565i			0.126763i
$889$			46.9710i			0.0528358i
$890$	−232.140			−0.260831
$891$		−	62.8827i		−	0.0705754i
$892$	−53.2774			−0.0597280
$893$		−	411.344i		−	0.460631i
$894$		−	29.0150i		−	0.0324552i
$895$			187.975i			0.210028i
$896$		−	4.65343i		−	0.00519357i
$897$	−230.512	+	199.828i	−0.256982	+	0.222774i
$898$	578.754			0.644493
$899$	−577.906			−0.642832
$900$	−30.0000			−0.0333333
$901$	−386.354			−0.428805
$902$			285.362i			0.316366i
$903$	5.40018			0.00598027
$904$			153.715i			0.170039i
$905$	−621.582			−0.686831
$906$	−331.372			−0.365753
$907$			763.246i			0.841506i	0.907175	+	0.420753i	$0.138234\pi$
$907$			763.246i			0.841506i	−0.907175	+	0.420753i	$0.861766\pi$
$908$			419.007i			0.461462i
$909$	434.995			0.478543
$910$	−9.96045			−0.0109455
$911$		−	2.42161i		−	0.00265819i	−0.999999	−	0.00132909i	$-0.999577\pi$
$911$		−	2.42161i		−	0.00265819i	0.999999	−	0.00132909i	$-0.000423064\pi$
$912$		−	57.5808i		−	0.0631368i
$913$	−1003.97			−1.09964
$914$		−	438.425i		−	0.479677i
$915$	−152.040			−0.166164
$916$			161.750i			0.176583i
$917$			35.8427i			0.0390869i
$918$			161.881i			0.176341i
$919$			75.2667i			0.0819007i	0.999161	+	0.0409503i	$0.0130385\pi$
$919$			75.2667i			0.0819007i	−0.999161	+	0.0409503i	$0.986961\pi$
$920$	−95.2832	−	109.914i	−0.103569	−	0.119472i
$921$	184.345			0.200157
$922$	−655.465			−0.710916
$923$	−205.359			−0.222490
$924$	9.95516			0.0107740
$925$			114.886i			0.124202i
$926$	−855.160			−0.923499
$927$		−	370.700i		−	0.399892i
$928$	−184.431			−0.198741
$929$	329.571			0.354759			0.177380	−	0.984143i	$-0.443238\pi$
$929$	329.571			0.354759			0.177380	+	0.984143i	$0.443238\pi$
$930$			97.0863i			0.104394i
$931$		−	405.836i		−	0.435915i
$932$	−349.586			−0.375092
$933$	1046.44			1.12159
$934$			384.079i			0.411219i
$935$		−	344.170i		−	0.368096i
$936$	64.9795			0.0694226
$937$			731.100i			0.780256i	0.920761	+	0.390128i	$0.127569\pi$
$937$			731.100i			0.780256i	−0.920761	+	0.390128i	$0.872431\pi$
$938$	41.2366			0.0439623
$939$			875.131i			0.931982i
$940$			221.341i			0.235470i
$941$			1744.46i			1.85384i	0.375260	+	0.926920i	$0.377553\pi$
$941$			1744.46i			1.85384i	−0.375260	+	0.926920i	$0.622447\pi$
$942$			423.493i			0.449568i
$943$	501.899	−	435.089i	0.532236	−	0.461389i
$944$	−135.461			−0.143497
$945$	−4.77898			−0.00505713
$946$	74.9002			0.0791757
$947$	338.428			0.357369			0.178684	−	0.983906i	$-0.442816\pi$
$947$	338.428			0.357369			0.178684	+	0.983906i	$0.442816\pi$
$948$		−	122.215i		−	0.128919i
$949$	474.904			0.500426
$950$		−	58.7681i		−	0.0618612i
$951$	180.939			0.190262
$952$	−25.6279			−0.0269201
$953$		−	1272.04i		−	1.33478i	−0.744711	−	0.667388i	$-0.767412\pi$
$953$		−	1272.04i		−	1.33478i	0.744711	−	0.667388i	$-0.232588\pi$
$954$			74.4084i			0.0779962i
$955$	−147.662			−0.154620
$956$	198.620			0.207762
$957$		−	394.557i		−	0.412285i
$958$			1122.45i			1.17166i
$959$	1.23785			0.00129077
$960$			30.9839i			0.0322749i
$961$	−646.808			−0.673057
$962$		−	248.842i		−	0.258672i
$963$		−	276.280i		−	0.286895i
$964$			87.8212i			0.0911008i
$965$		−	204.388i		−	0.211801i
$966$	−15.1786	−	17.5093i	−0.0157128	−	0.0181255i
$967$	824.863			0.853013			0.426506	−	0.904485i	$-0.359744\pi$
$967$	824.863			0.853013			0.426506	+	0.904485i	$0.359744\pi$
$968$	−204.162			−0.210911
$969$	−317.115			−0.327260
$970$	303.309			0.312689
$971$			149.116i			0.153570i	0.997048	+	0.0767849i	$0.0244655\pi$
$971$			149.116i			0.153570i	−0.997048	+	0.0767849i	$0.975535\pi$
$972$	31.1769			0.0320750
$973$		−	51.0870i		−	0.0525046i
$974$	614.581			0.630987
$975$	66.3194			0.0680199
$976$			157.027i			0.160888i
$977$			1600.76i			1.63845i	0.573474	+	0.819224i	$0.305595\pi$
$977$			1600.76i			1.63845i	−0.573474	+	0.819224i	$0.694405\pi$
$978$	605.778			0.619405
$979$	−512.907			−0.523909
$980$			218.378i			0.222835i
$981$		−	135.028i		−	0.137643i
$982$	−134.468			−0.136933
$983$		−	1542.13i		−	1.56880i	−0.620256	−	0.784400i	$-0.712971\pi$
$983$		−	1542.13i		−	1.56880i	0.620256	−	0.784400i	$-0.287029\pi$
$984$	−141.481			−0.143781
$985$			76.2198i			0.0773805i
$986$			1015.72i			1.03014i
$987$			35.2596i			0.0357240i
$988$			127.291i			0.128837i
$989$	−114.200	−	131.735i	−0.115470	−	0.133201i
$990$	−66.2842			−0.0669537
$991$	418.435			0.422235			0.211117	−	0.977461i	$-0.432290\pi$
$991$	418.435			0.422235			0.211117	+	0.977461i	$0.432290\pi$
$992$	100.270			0.101079
$993$	−71.6427			−0.0721477
$994$		−	15.5986i		−	0.0156928i
$995$	−617.775			−0.620879
$996$		−	497.763i		−	0.499762i
$997$	1274.10			1.27793			0.638965	−	0.769236i	$-0.279363\pi$
$997$	1274.10			1.27793			0.638965	+	0.769236i	$0.279363\pi$
$998$	−1169.54			−1.17189
$999$		−	119.393i		−	0.119513i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	690.3.c.a.91.14	yes	32
3.2	odd	2		2070.3.c.b.91.20		32
23.22	odd	2	inner	690.3.c.a.91.11	✓	32
69.68	even	2		2070.3.c.b.91.29		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.3.c.a.91.11	✓	32	23.22	odd	2	inner
690.3.c.a.91.14	yes	32	1.1	even	1	trivial
2070.3.c.b.91.20		32	3.2	odd	2
2070.3.c.b.91.29		32	69.68	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	690.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +1.73205 q^{3} +2.00000 q^{4} +2.23607i q^{5} -2.44949 q^{6} +0.411309i q^{7} -2.82843 q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q-1.41421 q^{2} +1.73205 q^{3} +2.00000 q^{4} +2.23607i q^{5} -2.44949 q^{6} +0.411309i q^{7} -2.82843 q^{8} +3.00000 q^{9} -3.16228i q^{10} -6.98697i q^{11} +3.46410 q^{12} -7.65791 q^{13} -0.581679i q^{14} +3.87298i q^{15} +4.00000 q^{16} -22.0292i q^{17} -4.24264 q^{18} -8.31107i q^{19} +4.47214i q^{20} +0.712409i q^{21} +9.88107i q^{22} +(17.3790 - 15.0656i) q^{23} -4.89898 q^{24} -5.00000 q^{25} +10.8299 q^{26} +5.19615 q^{27} +0.822619i q^{28} +32.6032 q^{29} -5.47723i q^{30} -17.7255 q^{31} -5.65685 q^{32} -12.1018i q^{33} +31.1540i q^{34} -0.919716 q^{35} +6.00000 q^{36} -22.9773i q^{37} +11.7536i q^{38} -13.2639 q^{39} -6.32456i q^{40} +28.8797 q^{41} -1.00750i q^{42} -7.58017i q^{43} -13.9739i q^{44} +6.70820i q^{45} +(-24.5776 + 21.3060i) q^{46} +49.4935 q^{47} +6.92820 q^{48} +48.8308 q^{49} +7.07107 q^{50} -38.1557i q^{51} -15.3158 q^{52} -17.5382i q^{53} -7.34847 q^{54} +15.6233 q^{55} -1.16336i q^{56} -14.3952i q^{57} -46.1078 q^{58} -33.8653 q^{59} +7.74597i q^{60} +39.2567i q^{61} +25.0676 q^{62} +1.23393i q^{63} +8.00000 q^{64} -17.1236i q^{65} +17.1145i q^{66} +70.8924i q^{67} -44.0585i q^{68} +(30.1012 - 26.0944i) q^{69} +1.30067 q^{70} +26.8165 q^{71} -8.48528 q^{72} -62.0148 q^{73} +32.4948i q^{74} -8.66025 q^{75} -16.6221i q^{76} +2.87381 q^{77} +18.7580 q^{78} -35.2805i q^{79} +8.94427i q^{80} +9.00000 q^{81} -40.8420 q^{82} -143.692i q^{83} +1.42482i q^{84} +49.2589 q^{85} +10.7200i q^{86} +56.4704 q^{87} +19.7621i q^{88} -73.4091i q^{89} -9.48683i q^{90} -3.14977i q^{91} +(34.7579 - 30.1312i) q^{92} -30.7014 q^{93} -69.9943 q^{94} +18.5841 q^{95} -9.79796 q^{96} +95.9147i q^{97} -69.0572 q^{98} -20.9609i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 64 q^{4} + 96 q^{9}+O(q^{10}) \) 32 * q + 64 * q^4 + 96 * q^9	\( 32 q + 64 q^{4} + 96 q^{9} - 48 q^{13} + 128 q^{16} - 80 q^{23} - 160 q^{25} + 120 q^{29} + 248 q^{31} - 120 q^{35} + 192 q^{36} - 48 q^{39} + 72 q^{41} + 160 q^{46} + 400 q^{47} - 344 q^{49} - 96 q^{52} - 256 q^{58} + 120 q^{59} + 160 q^{62} + 256 q^{64} + 192 q^{69} + 104 q^{71} + 16 q^{73} + 240 q^{77} + 192 q^{78} + 288 q^{81} + 64 q^{82} - 120 q^{85} + 144 q^{87} - 160 q^{92} - 192 q^{93} + 96 q^{94} - 160 q^{95} + 64 q^{98}+O(q^{100}) \) 32 * q + 64 * q^4 + 96 * q^9 - 48 * q^13 + 128 * q^16 - 80 * q^23 - 160 * q^25 + 120 * q^29 + 248 * q^31 - 120 * q^35 + 192 * q^36 - 48 * q^39 + 72 * q^41 + 160 * q^46 + 400 * q^47 - 344 * q^49 - 96 * q^52 - 256 * q^58 + 120 * q^59 + 160 * q^62 + 256 * q^64 + 192 * q^69 + 104 * q^71 + 16 * q^73 + 240 * q^77 + 192 * q^78 + 288 * q^81 + 64 * q^82 - 120 * q^85 + 144 * q^87 - 160 * q^92 - 192 * q^93 + 96 * q^94 - 160 * q^95 + 64 * q^98

Introduction

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