LMFDB - Embedded newform 690.3.c.a.91.1

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			91.1
Character	$\chi$	$=$	690.91
Dual form			690.3.c.a.91.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} -1.73205 q^{3} +2.00000 q^{4} -2.23607i q^{5} +2.44949 q^{6} -12.6313i q^{7} -2.82843 q^{8} +3.00000 q^{9} +O(q^{10})$	\(q-1.41421 q^{2} -1.73205 q^{3} +2.00000 q^{4} -2.23607i q^{5} +2.44949 q^{6} -12.6313i q^{7} -2.82843 q^{8} +3.00000 q^{9} +3.16228i q^{10} -6.64640i q^{11} -3.46410 q^{12} -21.6127 q^{13} +17.8634i q^{14} +3.87298i q^{15} +4.00000 q^{16} -26.9012i q^{17} -4.24264 q^{18} -5.80470i q^{19} -4.47214i q^{20} +21.8781i q^{21} +9.39943i q^{22} +(-11.1853 + 20.0970i) q^{23} +4.89898 q^{24} -5.00000 q^{25} +30.5649 q^{26} -5.19615 q^{27} -25.2627i q^{28} +50.7850 q^{29} -5.47723i q^{30} +0.790554 q^{31} -5.65685 q^{32} +11.5119i q^{33} +38.0440i q^{34} -28.2445 q^{35} +6.00000 q^{36} -11.7874i q^{37} +8.20908i q^{38} +37.4342 q^{39} +6.32456i q^{40} -0.0875420 q^{41} -30.9403i q^{42} +78.4553i q^{43} -13.2928i q^{44} -6.70820i q^{45} +(15.8184 - 28.4214i) q^{46} +65.2980 q^{47} -6.92820 q^{48} -110.551 q^{49} +7.07107 q^{50} +46.5942i q^{51} -43.2253 q^{52} -35.8320i q^{53} +7.34847 q^{54} -14.8618 q^{55} +35.7268i q^{56} +10.0540i q^{57} -71.8208 q^{58} -28.0844 q^{59} +7.74597i q^{60} +18.6748i q^{61} -1.11801 q^{62} -37.8940i q^{63} +8.00000 q^{64} +48.3274i q^{65} -16.2803i q^{66} -23.9607i q^{67} -53.8024i q^{68} +(19.3736 - 34.8090i) q^{69} +39.9438 q^{70} -101.896 q^{71} -8.48528 q^{72} -111.078 q^{73} +16.6699i q^{74} +8.66025 q^{75} -11.6094i q^{76} -83.9530 q^{77} -52.9400 q^{78} +84.0979i q^{79} -8.94427i q^{80} +9.00000 q^{81} +0.123803 q^{82} -137.812i q^{83} +43.7563i q^{84} -60.1529 q^{85} -110.953i q^{86} -87.9622 q^{87} +18.7989i q^{88} +46.7519i q^{89} +9.48683i q^{90} +272.997i q^{91} +(-22.3707 + 40.1940i) q^{92} -1.36928 q^{93} -92.3454 q^{94} -12.9797 q^{95} +9.79796 q^{96} +20.6627i q^{97} +156.343 q^{98} -19.9392i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$		−	12.6313i		−	1.80448i	−0.431237	−	0.902239i	$-0.641923\pi$
$7$		−	12.6313i		−	1.80448i	0.431237	−	0.902239i	$-0.358077\pi$
$8$	−2.82843			−0.353553
$9$	3.00000			0.333333
$10$			3.16228i			0.316228i
$11$		−	6.64640i		−	0.604218i	−0.953273	−	0.302109i	$-0.902309\pi$
$11$		−	6.64640i		−	0.604218i	0.953273	−	0.302109i	$-0.0976907\pi$
$12$	−3.46410			−0.288675
$13$	−21.6127			−1.66251			−0.831256	−	0.555890i	$-0.812378\pi$
$13$	−21.6127			−1.66251			−0.831256	+	0.555890i	$0.812378\pi$
$14$			17.8634i			1.27596i
$15$			3.87298i			0.258199i
$16$	4.00000			0.250000
$17$		−	26.9012i		−	1.58242i	−0.611543	−	0.791211i	$-0.709451\pi$
$17$		−	26.9012i		−	1.58242i	0.611543	−	0.791211i	$-0.290549\pi$
$18$	−4.24264			−0.235702
$19$		−	5.80470i		−	0.305510i	−0.988264	−	0.152755i	$-0.951185\pi$
$19$		−	5.80470i		−	0.305510i	0.988264	−	0.152755i	$-0.0488146\pi$
$20$		−	4.47214i		−	0.223607i
$21$			21.8781i			1.04182i
$22$			9.39943i			0.427247i
$23$	−11.1853	+	20.0970i	−0.486319	+	0.873782i
$23$	−11.1853	+	20.0970i	−0.486319	+	0.873782i
$24$	4.89898			0.204124
$25$	−5.00000			−0.200000
$26$	30.5649			1.17557
$27$	−5.19615			−0.192450
$28$		−	25.2627i		−	0.902239i
$29$	50.7850			1.75121			0.875604	−	0.483031i	$-0.160464\pi$
$29$	50.7850			1.75121			0.875604	+	0.483031i	$0.160464\pi$
$30$		−	5.47723i		−	0.182574i
$31$	0.790554			0.0255017			0.0127509	−	0.999919i	$-0.495941\pi$
$31$	0.790554			0.0255017			0.0127509	+	0.999919i	$0.495941\pi$
$32$	−5.65685			−0.176777
$33$			11.5119i			0.348846i
$34$			38.0440i			1.11894i
$35$	−28.2445			−0.806987
$36$	6.00000			0.166667
$37$		−	11.7874i		−	0.318578i	−0.987232	−	0.159289i	$-0.949080\pi$
$37$		−	11.7874i		−	0.318578i	0.987232	−	0.159289i	$-0.0509202\pi$
$38$			8.20908i			0.216028i
$39$	37.4342			0.959852
$40$			6.32456i			0.158114i
$41$	−0.0875420			−0.00213517			−0.00106759	−	0.999999i	$-0.500340\pi$
$41$	−0.0875420			−0.00213517			−0.00106759	+	0.999999i	$0.500340\pi$
$42$		−	30.9403i		−	0.736675i
$43$			78.4553i			1.82454i	0.409588	+	0.912271i	$0.365673\pi$
$43$			78.4553i			1.82454i	−0.409588	+	0.912271i	$0.634327\pi$
$44$		−	13.2928i		−	0.302109i
$45$		−	6.70820i		−	0.149071i
$46$	15.8184	−	28.4214i	0.343879	−	0.617857i
$47$	65.2980			1.38932			0.694660	−	0.719338i	$-0.255555\pi$
$47$	65.2980			1.38932			0.694660	+	0.719338i	$0.255555\pi$
$48$	−6.92820			−0.144338
$49$	−110.551			−2.25614
$50$	7.07107			0.141421
$51$			46.5942i			0.913612i
$52$	−43.2253			−0.831256
$53$		−	35.8320i		−	0.676076i	−0.941132	−	0.338038i	$-0.890237\pi$
$53$		−	35.8320i		−	0.676076i	0.941132	−	0.338038i	$-0.109763\pi$
$54$	7.34847			0.136083
$55$	−14.8618			−0.270215
$56$			35.7268i			0.637979i
$57$			10.0540i			0.176387i
$58$	−71.8208			−1.23829
$59$	−28.0844			−0.476006			−0.238003	−	0.971264i	$-0.576493\pi$
$59$	−28.0844			−0.476006			−0.238003	+	0.971264i	$0.576493\pi$
$60$			7.74597i			0.129099i
$61$			18.6748i			0.306145i	0.988215	+	0.153072i	$0.0489168\pi$
$61$			18.6748i			0.306145i	−0.988215	+	0.153072i	$0.951083\pi$
$62$	−1.11801			−0.0180324
$63$		−	37.8940i		−	0.601493i
$64$	8.00000			0.125000
$65$			48.3274i			0.743498i
$66$		−	16.2803i		−	0.246671i
$67$		−	23.9607i		−	0.357622i	−0.983883	−	0.178811i	$-0.942775\pi$
$67$		−	23.9607i		−	0.357622i	0.983883	−	0.178811i	$-0.0572250\pi$
$68$		−	53.8024i		−	0.791211i
$69$	19.3736	−	34.8090i	0.280776	−	0.504478i
$70$	39.9438			0.570626
$71$	−101.896			−1.43515			−0.717576	−	0.696481i	$-0.754748\pi$
$71$	−101.896			−1.43515			−0.717576	+	0.696481i	$0.754748\pi$
$72$	−8.48528			−0.117851
$73$	−111.078			−1.52162			−0.760811	−	0.648973i	$-0.775199\pi$
$73$	−111.078			−1.52162			−0.760811	+	0.648973i	$0.775199\pi$
$74$			16.6699i			0.225269i
$75$	8.66025			0.115470
$76$		−	11.6094i		−	0.152755i
$77$	−83.9530			−1.09030
$78$	−52.9400			−0.678718
$79$			84.0979i			1.06453i	0.846578	+	0.532265i	$0.178659\pi$
$79$			84.0979i			1.06453i	−0.846578	+	0.532265i	$0.821341\pi$
$80$		−	8.94427i		−	0.111803i
$81$	9.00000			0.111111
$82$	0.123803			0.00150979
$83$		−	137.812i		−	1.66039i	−0.557476	−	0.830193i	$-0.688230\pi$
$83$		−	137.812i		−	1.66039i	0.557476	−	0.830193i	$-0.311770\pi$
$84$			43.7563i			0.520908i
$85$	−60.1529			−0.707681
$86$		−	110.953i		−	1.29015i
$87$	−87.9622			−1.01106
$88$			18.7989i			0.213623i
$89$			46.7519i			0.525302i	0.964891	+	0.262651i	$0.0845967\pi$
$89$			46.7519i			0.525302i	−0.964891	+	0.262651i	$0.915403\pi$
$90$			9.48683i			0.105409i
$91$			272.997i			2.99997i
$92$	−22.3707	+	40.1940i	−0.243159	+	0.436891i
$93$	−1.36928			−0.0147234
$94$	−92.3454			−0.982398
$95$	−12.9797			−0.136628
$96$	9.79796			0.102062
$97$			20.6627i			0.213018i	0.994312	+	0.106509i	$0.0339673\pi$
$97$			20.6627i			0.213018i	−0.994312	+	0.106509i	$0.966033\pi$
$98$	156.343			1.59533
$99$		−	19.9392i		−	0.201406i
$100$	−10.0000			−0.100000
$101$	109.820			1.08733			0.543665	−	0.839303i	$-0.317036\pi$
$101$	109.820			1.08733			0.543665	+	0.839303i	$0.317036\pi$
$102$		−	65.8942i		−	0.646021i
$103$		−	9.00887i		−	0.0874647i	−0.999043	−	0.0437324i	$-0.986075\pi$
$103$		−	9.00887i		−	0.0874647i	0.999043	−	0.0437324i	$-0.0139249\pi$
$104$	61.1298			0.587787
$105$	48.9210			0.465914
$106$			50.6742i			0.478058i
$107$			168.462i			1.57441i	0.616689	+	0.787207i	$0.288474\pi$
$107$			168.462i			1.57441i	−0.616689	+	0.787207i	$0.711526\pi$
$108$	−10.3923			−0.0962250
$109$			32.8697i			0.301557i	0.988568	+	0.150778i	$0.0481780\pi$
$109$			32.8697i			0.301557i	−0.988568	+	0.150778i	$0.951822\pi$
$110$	21.0178			0.191071
$111$			20.4164i			0.183931i
$112$		−	50.5254i		−	0.451119i
$113$			40.3701i			0.357257i	0.983917	+	0.178629i	$0.0571661\pi$
$113$			40.3701i			0.357257i	−0.983917	+	0.178629i	$0.942834\pi$
$114$		−	14.2185i		−	0.124724i
$115$	44.9382	+	25.0112i	0.390767	+	0.217488i
$116$	101.570			0.875604
$117$	−64.8380			−0.554171
$118$	39.7173			0.336587
$119$	−339.798			−2.85545
$120$		−	10.9545i		−	0.0912871i
$121$	76.8254			0.634920
$122$		−	26.4102i		−	0.216477i
$123$	0.151627			0.00123274
$124$	1.58111			0.0127509
$125$			11.1803i			0.0894427i
$126$			53.5903i			0.425319i
$127$	−222.002			−1.74805			−0.874023	−	0.485884i	$-0.838498\pi$
$127$	−222.002			−1.74805			−0.874023	+	0.485884i	$0.838498\pi$
$128$	−11.3137			−0.0883883
$129$		−	135.889i		−	1.05340i
$130$		−	68.3452i		−	0.525733i
$131$	−104.077			−0.794482			−0.397241	−	0.917714i	$-0.630032\pi$
$131$	−104.077			−0.794482			−0.397241	+	0.917714i	$0.630032\pi$
$132$			23.0238i			0.174423i
$133$	−73.3211			−0.551287
$134$			33.8855i			0.252877i
$135$			11.6190i			0.0860663i
$136$			76.0880i			0.559471i
$137$		−	153.744i		−	1.12222i	−0.827741	−	0.561110i	$-0.810375\pi$
$137$		−	153.744i		−	1.12222i	0.827741	−	0.561110i	$-0.189625\pi$
$138$	−27.3983	+	49.2273i	−0.198539	+	0.356720i
$139$	35.8886			0.258191			0.129096	−	0.991632i	$-0.458793\pi$
$139$	35.8886			0.258191			0.129096	+	0.991632i	$0.458793\pi$
$140$	−56.4891			−0.403493
$141$	−113.100			−0.802124
$142$	144.102			1.01481
$143$			143.646i			1.00452i
$144$	12.0000			0.0833333
$145$		−	113.559i		−	0.783164i
$146$	157.089			1.07595
$147$	191.480			1.30258
$148$		−	23.5748i		−	0.159289i
$149$		−	5.22632i		−	0.0350760i	−0.999846	−	0.0175380i	$-0.994417\pi$
$149$		−	5.22632i		−	0.0350760i	0.999846	−	0.0175380i	$-0.00558280\pi$
$150$	−12.2474			−0.0816497
$151$	−277.637			−1.83865			−0.919326	−	0.393496i	$-0.871266\pi$
$151$	−277.637			−1.83865			−0.919326	+	0.393496i	$0.871266\pi$
$152$			16.4182i			0.108014i
$153$		−	80.7035i		−	0.527474i
$154$	118.727			0.770957
$155$		−	1.76773i		−	0.0114047i
$156$	74.8684			0.479926
$157$		−	188.330i		−	1.19955i	−0.800168	−	0.599776i	$-0.795257\pi$
$157$		−	188.330i		−	1.19955i	0.800168	−	0.599776i	$-0.204743\pi$
$158$		−	118.932i		−	0.752737i
$159$			62.0629i			0.390333i
$160$			12.6491i			0.0790569i
$161$	253.852	+	141.286i	1.57672	+	0.877551i
$162$	−12.7279			−0.0785674
$163$	−119.568			−0.733543			−0.366772	−	0.930311i	$-0.619537\pi$
$163$	−119.568			−0.733543			−0.366772	+	0.930311i	$0.619537\pi$
$164$	−0.175084			−0.00106759
$165$	25.7414			0.156008
$166$			194.896i			1.17407i
$167$	106.259			0.636281			0.318140	−	0.948044i	$-0.396942\pi$
$167$	106.259			0.636281			0.318140	+	0.948044i	$0.396942\pi$
$168$		−	61.8807i		−	0.368337i
$169$	298.107			1.76395
$170$	85.0690			0.500406
$171$		−	17.4141i		−	0.101837i
$172$			156.911i			0.912271i
$173$	343.403			1.98499			0.992494	−	0.122293i	$-0.0390249\pi$
$173$	343.403			1.98499			0.992494	+	0.122293i	$0.0390249\pi$
$174$	124.397			0.714927
$175$			63.1567i			0.360896i
$176$		−	26.5856i		−	0.151055i
$177$	48.6435			0.274822
$178$		−	66.1171i		−	0.371444i
$179$	172.782			0.965260			0.482630	−	0.875824i	$-0.339682\pi$
$179$	172.782			0.965260			0.482630	+	0.875824i	$0.339682\pi$
$180$		−	13.4164i		−	0.0745356i
$181$			20.5336i			0.113445i	0.998390	+	0.0567226i	$0.0180651\pi$
$181$			20.5336i			0.113445i	−0.998390	+	0.0567226i	$0.981935\pi$
$182$		−	386.076i		−	2.12130i
$183$		−	32.3457i		−	0.176753i
$184$	31.6369	−	56.8428i	0.171940	−	0.308928i
$185$	−26.3574			−0.142473
$186$	1.93645			0.0104110
$187$	−178.796			−0.956129
$188$	130.596			0.694660
$189$			65.6344i			0.347272i
$190$	18.3561			0.0966109
$191$		−	126.392i		−	0.661741i	−0.943676	−	0.330870i	$-0.892658\pi$
$191$		−	126.392i		−	0.661741i	0.943676	−	0.330870i	$-0.107342\pi$
$192$	−13.8564			−0.0721688
$193$	−119.288			−0.618073			−0.309037	−	0.951050i	$-0.600007\pi$
$193$	−119.288			−0.618073			−0.309037	+	0.951050i	$0.600007\pi$
$194$		−	29.2215i		−	0.150626i
$195$		−	83.7055i		−	0.429259i
$196$	−221.102			−1.12807
$197$	32.0842			0.162864			0.0814319	−	0.996679i	$-0.474051\pi$
$197$	32.0842			0.162864			0.0814319	+	0.996679i	$0.474051\pi$
$198$			28.1983i			0.142416i
$199$		−	308.976i		−	1.55264i	−0.630336	−	0.776322i	$-0.717083\pi$
$199$		−	308.976i		−	1.55264i	0.630336	−	0.776322i	$-0.282917\pi$
$200$	14.1421			0.0707107
$201$			41.5011i			0.206473i
$202$	−155.309			−0.768858
$203$		−	641.483i		−	3.16001i
$204$			93.1884i			0.456806i
$205$			0.195750i			0.000954878i
$206$			12.7405i			0.0618469i
$207$	−33.5560	+	60.2909i	−0.162106	+	0.291261i
$208$	−86.4506			−0.415628
$209$	−38.5803			−0.184595
$210$	−69.1847			−0.329451
$211$	284.915			1.35031			0.675153	−	0.737678i	$-0.264078\pi$
$211$	284.915			1.35031			0.675153	+	0.737678i	$0.264078\pi$
$212$		−	71.6641i		−	0.338038i
$213$	176.489			0.828585
$214$		−	238.242i		−	1.11328i
$215$	175.431			0.815960
$216$	14.6969			0.0680414
$217$		−	9.98576i		−	0.0460173i
$218$		−	46.4848i		−	0.213233i
$219$	192.394			0.878509
$220$	−29.7236			−0.135107
$221$			581.406i			2.63080i
$222$		−	28.8731i		−	0.130059i
$223$	−58.1222			−0.260638			−0.130319	−	0.991472i	$-0.541600\pi$
$223$	−58.1222			−0.260638			−0.130319	+	0.991472i	$0.541600\pi$
$224$			71.4537i			0.318990i
$225$	−15.0000			−0.0666667
$226$		−	57.0919i		−	0.252619i
$227$		−	61.9517i		−	0.272915i	−0.990646	−	0.136458i	$-0.956428\pi$
$227$		−	61.9517i		−	0.272915i	0.990646	−	0.136458i	$-0.0435717\pi$
$228$			20.1081i			0.0881933i
$229$		−	274.171i		−	1.19725i	−0.801029	−	0.598626i	$-0.795714\pi$
$229$		−	274.171i		−	1.19725i	0.801029	−	0.598626i	$-0.204286\pi$
$230$	−63.5522	−	35.3711i	−0.276314	−	0.153787i
$231$	145.411			0.629484
$232$	−143.642			−0.619145
$233$	−12.4541			−0.0534511			−0.0267255	−	0.999643i	$-0.508508\pi$
$233$	−12.4541			−0.0534511			−0.0267255	+	0.999643i	$0.508508\pi$
$234$	91.6947			0.391858
$235$		−	146.011i		−	0.621323i
$236$	−56.1687			−0.238003
$237$		−	145.662i		−	0.614607i
$238$	480.547			2.01911
$239$	−348.698			−1.45899			−0.729494	−	0.683987i	$-0.760244\pi$
$239$	−348.698			−1.45899			−0.729494	+	0.683987i	$0.760244\pi$
$240$			15.4919i			0.0645497i
$241$			333.130i			1.38228i	0.722719	+	0.691142i	$0.242892\pi$
$241$			333.130i			1.38228i	−0.722719	+	0.691142i	$0.757108\pi$
$242$	−108.647			−0.448956
$243$	−15.5885			−0.0641500
$244$			37.3497i			0.153072i
$245$			247.199i			1.00898i
$246$	−0.214433			−0.000871680
$247$			125.455i			0.507915i
$248$	−2.23602			−0.00901622
$249$			238.697i			0.958624i
$250$		−	15.8114i		−	0.0632456i
$251$			218.644i			0.871093i	0.900166	+	0.435546i	$0.143445\pi$
$251$			218.644i			0.871093i	−0.900166	+	0.435546i	$0.856555\pi$
$252$		−	75.7881i		−	0.300746i
$253$	133.573	+	74.3422i	0.527955	+	0.293843i
$254$	313.958			1.23606
$255$	104.188			0.408580
$256$	16.0000			0.0625000
$257$	436.402			1.69806			0.849032	−	0.528342i	$-0.177186\pi$
$257$	436.402			1.69806			0.849032	+	0.528342i	$0.177186\pi$
$258$			192.175i			0.744866i
$259$	−148.891			−0.574867
$260$			96.6548i			0.371749i
$261$	152.355			0.583736
$262$	147.187			0.561784
$263$			431.615i			1.64112i	0.571558	+	0.820562i	$0.306339\pi$
$263$			431.615i			1.64112i	−0.571558	+	0.820562i	$0.693661\pi$
$264$		−	32.5606i		−	0.123336i
$265$	−80.1229			−0.302351
$266$	103.692			0.389819
$267$		−	80.9766i		−	0.303283i
$268$		−	47.9213i		−	0.178811i
$269$	−312.123			−1.16031			−0.580155	−	0.814506i	$-0.697008\pi$
$269$	−312.123			−1.16031			−0.580155	+	0.814506i	$0.697008\pi$
$270$		−	16.4317i		−	0.0608581i
$271$	−70.7048			−0.260903			−0.130452	−	0.991455i	$-0.541643\pi$
$271$	−70.7048			−0.260903			−0.130452	+	0.991455i	$0.541643\pi$
$272$		−	107.605i		−	0.395606i
$273$		−	472.845i		−	1.73203i
$274$			217.427i			0.793529i
$275$			33.2320i			0.120844i
$276$	38.7471	−	69.6180i	0.140388	−	0.252239i
$277$	344.720			1.24448			0.622239	−	0.782827i	$-0.286223\pi$
$277$	344.720			1.24448			0.622239	+	0.782827i	$0.286223\pi$
$278$	−50.7541			−0.182569
$279$	2.37166			0.00850058
$280$	79.8876			0.285313
$281$		−	299.210i		−	1.06481i	−0.846491	−	0.532403i	$-0.821289\pi$
$281$		−	299.210i		−	1.06481i	0.846491	−	0.532403i	$-0.178711\pi$
$282$	159.947			0.567188
$283$			49.7476i			0.175787i	0.996130	+	0.0878933i	$0.0280135\pi$
$283$			49.7476i			0.175787i	−0.996130	+	0.0878933i	$0.971987\pi$
$284$	−203.791			−0.717576
$285$	22.4815			0.0788824
$286$		−	203.147i		−	0.710303i
$287$			1.10577i			0.00385287i
$288$	−16.9706			−0.0589256
$289$	−434.674			−1.50406
$290$			160.596i			0.553780i
$291$		−	35.7889i		−	0.122986i
$292$	−222.157			−0.760811
$293$			37.1215i			0.126695i	0.997992	+	0.0633473i	$0.0201776\pi$
$293$			37.1215i			0.126695i	−0.997992	+	0.0633473i	$0.979822\pi$
$294$	−270.793			−0.921065
$295$			62.7985i			0.212876i
$296$			33.3398i			0.112634i
$297$			34.5357i			0.116282i
$298$			7.39113i			0.0248025i
$299$	241.745	−	434.349i	0.808511	−	1.45267i
$300$	17.3205			0.0577350
$301$	990.996			3.29234
$302$	392.637			1.30012
$303$	−190.214			−0.627770
$304$		−	23.2188i		−	0.0763776i
$305$	41.7582			0.136912
$306$			114.132i			0.372981i
$307$	−0.623436			−0.00203074			−0.00101537	−	0.999999i	$-0.500323\pi$
$307$	−0.623436			−0.00203074			−0.00101537	+	0.999999i	$0.500323\pi$
$308$	−167.906			−0.545149
$309$			15.6038i			0.0504978i
$310$			2.49995i			0.00806436i
$311$	118.290			0.380352			0.190176	−	0.981750i	$-0.439094\pi$
$311$	118.290			0.380352			0.190176	+	0.981750i	$0.439094\pi$
$312$	−105.880			−0.339359
$313$		−	318.068i		−	1.01619i	−0.861300	−	0.508096i	$-0.830349\pi$
$313$		−	318.068i		−	1.01619i	0.861300	−	0.508096i	$-0.169651\pi$
$314$			266.338i			0.848211i
$315$	−84.7336			−0.268996
$316$			168.196i			0.532265i
$317$	−291.589			−0.919839			−0.459919	−	0.887961i	$-0.652122\pi$
$317$	−291.589			−0.919839			−0.459919	+	0.887961i	$0.652122\pi$
$318$		−	87.7702i		−	0.276007i
$319$		−	337.537i		−	1.05811i
$320$		−	17.8885i		−	0.0559017i
$321$		−	291.785i		−	0.908988i
$322$	−359.001	−	199.808i	−1.11491	−	0.620522i
$323$	−156.153			−0.483447
$324$	18.0000			0.0555556
$325$	108.063			0.332502
$326$	169.094			0.518693
$327$		−	56.9320i		−	0.174104i
$328$	0.247606			0.000754897
$329$		−	824.802i		−	2.50700i
$330$	−36.4038			−0.110315
$331$	64.2594			0.194137			0.0970686	−	0.995278i	$-0.469053\pi$
$331$	64.2594			0.194137			0.0970686	+	0.995278i	$0.469053\pi$
$332$		−	275.624i		−	0.830193i
$333$		−	35.3622i		−	0.106193i
$334$	−150.273			−0.449919
$335$	−53.5777			−0.159933
$336$			87.5125i			0.260454i
$337$			353.359i			1.04854i	0.851552	+	0.524271i	$0.175662\pi$
$337$			353.359i			1.04854i	−0.851552	+	0.524271i	$0.824338\pi$
$338$	−421.587			−1.24730
$339$		−	69.9230i		−	0.206263i
$340$	−120.306			−0.353840
$341$		−	5.25434i		−	0.0154086i
$342$			24.6272i			0.0720095i
$343$			777.470i			2.26668i
$344$		−	221.905i		−	0.645073i
$345$	−77.8352	−	43.3206i	−0.225609	−	0.125567i
$346$	−485.645			−1.40360
$347$	−424.699			−1.22392			−0.611958	−	0.790890i	$-0.709618\pi$
$347$	−424.699			−1.22392			−0.611958	+	0.790890i	$0.709618\pi$
$348$	−175.924			−0.505530
$349$	−111.111			−0.318370			−0.159185	−	0.987249i	$-0.550887\pi$
$349$	−111.111			−0.318370			−0.159185	+	0.987249i	$0.550887\pi$
$350$		−	89.3171i		−	0.255192i
$351$	112.303			0.319951
$352$			37.5977i			0.106812i
$353$	−306.094			−0.867121			−0.433561	−	0.901124i	$-0.642743\pi$
$353$	−306.094			−0.867121			−0.433561	+	0.901124i	$0.642743\pi$
$354$	−68.7924			−0.194329
$355$			227.846i			0.641819i
$356$			93.5037i			0.262651i
$357$	588.548			1.64859
$358$	−244.350			−0.682542
$359$			77.5601i			0.216045i	0.994148	+	0.108022i	$0.0344518\pi$
$359$			77.5601i			0.216045i	−0.994148	+	0.108022i	$0.965548\pi$
$360$			18.9737i			0.0527046i
$361$	327.305			0.906663
$362$		−	29.0389i		−	0.0802178i
$363$	−133.065			−0.366571
$364$			545.994i			1.49998i
$365$			248.379i			0.680490i
$366$			45.7438i			0.124983i
$367$		−	85.7495i		−	0.233650i	−0.993153	−	0.116825i	$-0.962728\pi$
$367$		−	85.7495i		−	0.233650i	0.993153	−	0.116825i	$-0.0372717\pi$
$368$	−44.7413	+	80.3879i	−0.121580	+	0.218445i
$369$	−0.262626			−0.000711724
$370$	37.2750			0.100743
$371$	−452.607			−1.21996
$372$	−2.73856			−0.00736172
$373$			144.680i			0.387882i	0.981013	+	0.193941i	$0.0621271\pi$
$373$			144.680i			0.387882i	−0.981013	+	0.193941i	$0.937873\pi$
$374$	252.856			0.676085
$375$		−	19.3649i		−	0.0516398i
$376$	−184.691			−0.491199
$377$	−1097.60			−2.91140
$378$		−	92.8210i		−	0.245558i
$379$		−	303.886i		−	0.801810i	−0.916120	−	0.400905i	$-0.868696\pi$
$379$		−	303.886i		−	0.801810i	0.916120	−	0.400905i	$-0.131304\pi$
$380$	−25.9594			−0.0683142
$381$	384.518			1.00923
$382$			178.746i			0.467921i
$383$			215.968i			0.563884i	0.959431	+	0.281942i	$0.0909785\pi$
$383$			215.968i			0.563884i	−0.959431	+	0.281942i	$0.909021\pi$
$384$	19.5959			0.0510310
$385$			187.725i			0.487596i
$386$	168.699			0.437044
$387$			235.366i			0.608180i
$388$			41.3254i			0.106509i
$389$			41.3140i			0.106206i	0.998589	+	0.0531028i	$0.0169111\pi$
$389$			41.3140i			0.106206i	−0.998589	+	0.0531028i	$0.983089\pi$
$390$			118.377i			0.303532i
$391$	540.632	+	300.899i	1.38269	+	0.769562i
$392$	312.685			0.797666
$393$	180.267			0.458694
$394$	−45.3739			−0.115162
$395$	188.049			0.476072
$396$		−	39.8784i		−	0.100703i
$397$	400.330			1.00839			0.504194	−	0.863590i	$-0.331790\pi$
$397$	400.330			1.00839			0.504194	+	0.863590i	$0.331790\pi$
$398$			436.958i			1.09789i
$399$	126.996			0.318286
$400$	−20.0000			−0.0500000
$401$		−	442.777i		−	1.10418i	−0.833784	−	0.552091i	$-0.813830\pi$
$401$		−	442.777i		−	1.10418i	0.833784	−	0.552091i	$-0.186170\pi$
$402$		−	58.6914i		−	0.145998i
$403$	−17.0860			−0.0423969
$404$	219.641			0.543665
$405$		−	20.1246i		−	0.0496904i
$406$			907.194i			2.23447i
$407$	−78.3438			−0.192491
$408$		−	131.788i		−	0.323011i
$409$	−781.825			−1.91155			−0.955776	−	0.294096i	$-0.904981\pi$
$409$	−781.825			−1.91155			−0.955776	+	0.294096i	$0.904981\pi$
$410$		−	0.276832i		−	0.000675200i
$411$			266.293i			0.647914i
$412$		−	18.0177i		−	0.0437324i
$413$			354.743i			0.858943i
$414$	47.4553	−	85.2642i	0.114626	−	0.205952i
$415$	−308.157			−0.742547
$416$	122.260			0.293893
$417$	−62.1608			−0.149067
$418$	54.5609			0.130528
$419$		−	417.722i		−	0.996950i	−0.866904	−	0.498475i	$-0.833894\pi$
$419$		−	417.722i		−	0.996950i	0.866904	−	0.498475i	$-0.166106\pi$
$420$	97.8420			0.232957
$421$		−	140.248i		−	0.333131i	−0.986030	−	0.166566i	$-0.946732\pi$
$421$		−	140.248i		−	0.333131i	0.986030	−	0.166566i	$-0.0532678\pi$
$422$	−402.930			−0.954811
$423$	195.894			0.463107
$424$			101.348i			0.239029i
$425$			134.506i			0.316484i
$426$	−249.593			−0.585898
$427$	235.888			0.552431
$428$			336.925i			0.787207i
$429$		−	248.803i		−	0.579960i
$430$	−248.097			−0.576971
$431$			3.48703i			0.00809055i	0.999992	+	0.00404528i	$0.00128765\pi$
$431$			3.48703i			0.00809055i	−0.999992	+	0.00404528i	$0.998712\pi$
$432$	−20.7846			−0.0481125
$433$		−	372.607i		−	0.860523i	−0.902704	−	0.430262i	$-0.858421\pi$
$433$		−	372.607i		−	0.860523i	0.902704	−	0.430262i	$-0.141579\pi$
$434$			14.1220i			0.0325391i
$435$			196.689i			0.452160i
$436$			65.7394i			0.150778i
$437$	116.657	+	64.9275i	0.266949	+	0.148575i
$438$	−272.086			−0.621200
$439$	103.728			0.236283			0.118142	−	0.992997i	$-0.462306\pi$
$439$	103.728			0.236283			0.118142	+	0.992997i	$0.462306\pi$
$440$	42.0355			0.0955353
$441$	−331.653			−0.752047
$442$		−	822.232i		−	1.86025i
$443$	−693.747			−1.56602			−0.783010	−	0.622009i	$-0.786317\pi$
$443$	−693.747			−1.56602			−0.783010	+	0.622009i	$0.786317\pi$
$444$			40.8327i			0.0919656i
$445$	104.540			0.234922
$446$	82.1972			0.184299
$447$			9.05225i			0.0202511i
$448$		−	101.051i		−	0.225560i
$449$	274.366			0.611061			0.305531	−	0.952182i	$-0.401166\pi$
$449$	274.366			0.611061			0.305531	+	0.952182i	$0.401166\pi$
$450$	21.2132			0.0471405
$451$			0.581839i			0.00129011i
$452$			80.7401i			0.178629i
$453$	480.881			1.06155
$454$			87.6130i			0.192980i
$455$	610.440			1.34163
$456$		−	28.4371i		−	0.0623621i
$457$		−	588.108i		−	1.28689i	−0.765493	−	0.643444i	$-0.777505\pi$
$457$		−	588.108i		−	1.28689i	0.765493	−	0.643444i	$-0.222495\pi$
$458$			387.736i			0.846585i
$459$			139.783i			0.304537i
$460$	89.8764	+	50.0223i	0.195383	+	0.108744i
$461$	−219.244			−0.475584			−0.237792	−	0.971316i	$-0.576424\pi$
$461$	−219.244			−0.475584			−0.237792	+	0.971316i	$0.576424\pi$
$462$	−205.642			−0.445112
$463$	−105.290			−0.227409			−0.113705	−	0.993515i	$-0.536272\pi$
$463$	−105.290			−0.227409			−0.113705	+	0.993515i	$0.536272\pi$
$464$	203.140			0.437802
$465$			3.06180i			0.00658452i
$466$	17.6128			0.0377956
$467$		−	297.818i		−	0.637727i	−0.947801	−	0.318863i	$-0.896699\pi$
$467$		−	297.818i		−	0.637727i	0.947801	−	0.318863i	$-0.103301\pi$
$468$	−129.676			−0.277085
$469$	−302.655			−0.645321
$470$			206.491i			0.439342i
$471$			326.196i			0.692561i
$472$	79.4346			0.168294
$473$	521.445			1.10242
$474$			205.997i			0.434593i
$475$			29.0235i			0.0611021i
$476$	−679.596			−1.42772
$477$		−	107.496i		−	0.225359i
$478$	493.134			1.03166
$479$			573.980i			1.19829i	0.800641	+	0.599144i	$0.204492\pi$
$479$			573.980i			1.19829i	−0.800641	+	0.599144i	$0.795508\pi$
$480$		−	21.9089i		−	0.0456435i
$481$			254.757i			0.529640i
$482$		−	471.117i		−	0.977422i
$483$	−439.684	−	244.714i	−0.910319	−	0.506654i
$484$	153.651			0.317460
$485$	46.2033			0.0952644
$486$	22.0454			0.0453609
$487$	−934.765			−1.91944			−0.959718	−	0.280965i	$-0.909346\pi$
$487$	−934.765			−1.91944			−0.959718	+	0.280965i	$0.909346\pi$
$488$		−	52.8204i		−	0.108238i
$489$	207.097			0.423511
$490$		−	349.592i		−	0.713454i
$491$	−285.271			−0.581001			−0.290500	−	0.956875i	$-0.593822\pi$
$491$	−285.271			−0.581001			−0.290500	+	0.956875i	$0.593822\pi$
$492$	0.303254			0.000616371
$493$		−	1366.18i		−	2.77115i
$494$		−	177.420i		−	0.359150i
$495$	−44.5854			−0.0900715
$496$	3.16221			0.00637543
$497$			1287.08i			2.58970i
$498$		−	337.569i		−	0.677850i
$499$	225.862			0.452628			0.226314	−	0.974054i	$-0.427332\pi$
$499$	225.862			0.452628			0.226314	+	0.974054i	$0.427332\pi$
$500$			22.3607i			0.0447214i
$501$	−184.046			−0.367357
$502$		−	309.210i		−	0.615956i
$503$			689.412i			1.37060i	0.728261	+	0.685300i	$0.240329\pi$
$503$			689.412i			1.37060i	−0.728261	+	0.685300i	$0.759671\pi$
$504$			107.181i			0.212660i
$505$		−	245.566i		−	0.486268i
$506$	−188.900	−	105.136i	−0.373320	−	0.207778i
$507$	−516.337			−1.01842
$508$	−444.004			−0.874023
$509$	203.484			0.399772			0.199886	−	0.979819i	$-0.435943\pi$
$509$	203.484			0.399772			0.199886	+	0.979819i	$0.435943\pi$
$510$	−147.344			−0.288909
$511$			1403.07i			2.74573i
$512$	−22.6274			−0.0441942
$513$			30.1621i			0.0587955i
$514$	−617.166			−1.20071
$515$	−20.1444			−0.0391154
$516$		−	271.777i		−	0.526700i
$517$		−	433.997i		−	0.839453i
$518$	210.563			0.406493
$519$	−594.791			−1.14603
$520$		−	136.690i		−	0.262866i
$521$		−	427.253i		−	0.820063i	−0.912071	−	0.410031i	$-0.865518\pi$
$521$		−	427.253i		−	0.820063i	0.912071	−	0.410031i	$-0.134482\pi$
$522$	−215.463			−0.412763
$523$			824.411i			1.57631i	0.615476	+	0.788156i	$0.288964\pi$
$523$			824.411i			1.57631i	−0.615476	+	0.788156i	$0.711036\pi$
$524$	−208.154			−0.397241
$525$		−	109.391i		−	0.208363i
$526$		−	610.396i		−	1.16045i
$527$		−	21.2668i		−	0.0403545i
$528$			46.0476i			0.0872114i
$529$	−278.777	−	449.583i	−0.526988	−	0.849873i
$530$	113.311			0.213794
$531$	−84.2531			−0.158669
$532$	−146.642			−0.275643
$533$	1.89202			0.00354975
$534$			114.518i			0.214454i
$535$	376.693			0.704099
$536$			67.7710i			0.126438i
$537$	−299.266			−0.557293
$538$	441.409			0.820462
$539$			734.765i			1.36320i
$540$			23.2379i			0.0430331i
$541$	−628.584			−1.16189			−0.580947	−	0.813942i	$-0.697318\pi$
$541$	−628.584			−1.16189			−0.580947	+	0.813942i	$0.697318\pi$
$542$	99.9917			0.184487
$543$		−	35.5652i		−	0.0654976i
$544$			152.176i			0.279735i
$545$	73.4989			0.134860
$546$			668.703i			1.22473i
$547$	419.610			0.767112			0.383556	−	0.923518i	$-0.374699\pi$
$547$	419.610			0.767112			0.383556	+	0.923518i	$0.374699\pi$
$548$		−	307.488i		−	0.561110i
$549$			56.0245i			0.102048i
$550$		−	46.9972i		−	0.0854494i
$551$		−	294.792i		−	0.535012i
$552$	−54.7967	+	98.4547i	−0.0992694	+	0.178360i
$553$	1062.27			1.92092
$554$	−487.508			−0.879979
$555$	45.6524			0.0822566
$556$	71.7771			0.129096
$557$			573.400i			1.02944i	0.857357	+	0.514722i	$0.172105\pi$
$557$			573.400i			1.02944i	−0.857357	+	0.514722i	$0.827895\pi$
$558$	−3.35404			−0.00601082
$559$		−	1695.63i		−	3.03332i
$560$	−112.978			−0.201747
$561$	309.684			0.552021
$562$			423.147i			0.752931i
$563$			38.6963i			0.0687323i	0.999409	+	0.0343661i	$0.0109412\pi$
$563$			38.6963i			0.0687323i	−0.999409	+	0.0343661i	$0.989059\pi$
$564$	−226.199			−0.401062
$565$	90.2702			0.159770
$566$		−	70.3537i		−	0.124300i
$567$		−	113.682i		−	0.200498i
$568$	288.205			0.507403
$569$			426.562i			0.749670i	0.927092	+	0.374835i	$0.122301\pi$
$569$			426.562i			0.749670i	−0.927092	+	0.374835i	$0.877699\pi$
$570$	−31.7936			−0.0557783
$571$		−	761.099i		−	1.33292i	−0.745539	−	0.666462i	$-0.767808\pi$
$571$		−	761.099i		−	1.33292i	0.745539	−	0.666462i	$-0.232192\pi$
$572$			287.293i			0.502260i
$573$			218.918i			0.382056i
$574$		−	1.56380i		−	0.00272439i
$575$	55.9266	−	100.485i	0.0972637	−	0.174756i
$576$	24.0000			0.0416667
$577$	194.028			0.336270			0.168135	−	0.985764i	$-0.446226\pi$
$577$	194.028			0.336270			0.168135	+	0.985764i	$0.446226\pi$
$578$	614.721			1.06353
$579$	206.613			0.356845
$580$		−	227.117i		−	0.391582i
$581$	−1740.75			−2.99613
$582$			50.6131i			0.0869641i
$583$	−238.154			−0.408498
$584$	314.177			0.537975
$585$			144.982i			0.247833i
$586$		−	52.4978i		−	0.0895867i
$587$	575.615			0.980605			0.490302	−	0.871552i	$-0.336886\pi$
$587$	575.615			0.980605			0.490302	+	0.871552i	$0.336886\pi$
$588$	382.959			0.651291
$589$		−	4.58893i		−	0.00779104i
$590$		−	88.8106i		−	0.150526i
$591$	−55.5714			−0.0940295
$592$		−	47.1496i		−	0.0796446i
$593$	270.936			0.456890			0.228445	−	0.973557i	$-0.426636\pi$
$593$	270.936			0.456890			0.228445	+	0.973557i	$0.426636\pi$
$594$		−	48.8409i		−	0.0822237i
$595$			759.812i			1.27699i
$596$		−	10.4526i		−	0.0175380i
$597$			535.163i			0.896420i
$598$	−341.879	+	614.262i	−0.571703	+	1.02719i
$599$	421.554			0.703762			0.351881	−	0.936045i	$-0.385542\pi$
$599$	421.554			0.703762			0.351881	+	0.936045i	$0.385542\pi$
$600$	−24.4949			−0.0408248
$601$	732.875			1.21943			0.609713	−	0.792622i	$-0.291285\pi$
$601$	732.875			1.21943			0.609713	+	0.792622i	$0.291285\pi$
$602$	−1401.48			−2.32804
$603$		−	71.8820i		−	0.119207i
$604$	−555.273			−0.919326
$605$		−	171.787i		−	0.283945i
$606$	269.004			0.443900
$607$	−128.402			−0.211536			−0.105768	−	0.994391i	$-0.533730\pi$
$607$	−128.402			−0.211536			−0.105768	+	0.994391i	$0.533730\pi$
$608$			32.8363i			0.0540071i
$609$			1111.08i			1.82443i
$610$	−59.0550			−0.0968115
$611$	−1411.26			−2.30976
$612$		−	161.407i		−	0.263737i
$613$		−	863.292i		−	1.40831i	−0.710048	−	0.704153i	$-0.751327\pi$
$613$		−	863.292i		−	1.40831i	0.710048	−	0.704153i	$-0.248673\pi$
$614$	0.881672			0.00143595
$615$		−	0.339049i		−	0.000551299i
$616$	237.455			0.385479
$617$		−	883.691i		−	1.43224i	−0.697978	−	0.716119i	$-0.745917\pi$
$617$		−	883.691i		−	1.43224i	0.697978	−	0.716119i	$-0.254083\pi$
$618$		−	22.0671i		−	0.0357073i
$619$			265.802i			0.429405i	0.976680	+	0.214702i	$0.0688782\pi$
$619$			265.802i			0.429405i	−0.976680	+	0.214702i	$0.931122\pi$
$620$		−	3.53546i		−	0.00570236i
$621$	58.1207	−	104.427i	0.0935921	−	0.168159i
$622$	−167.287			−0.268950
$623$	590.539			0.947895
$624$	149.737			0.239963
$625$	25.0000			0.0400000
$626$			449.816i			0.718556i
$627$	66.8231			0.106576
$628$		−	376.659i		−	0.599776i
$629$	−317.095			−0.504125
$630$	119.831			0.190209
$631$		−	610.986i		−	0.968283i	−0.874990	−	0.484141i	$-0.839132\pi$
$631$		−	610.986i		−	0.968283i	0.874990	−	0.484141i	$-0.160868\pi$
$632$		−	237.865i		−	0.376368i
$633$	−493.487			−0.779600
$634$	412.369			0.650424
$635$			496.411i			0.781750i
$636$			124.126i			0.195166i
$637$	2389.30			3.75086
$638$			477.350i			0.748198i
$639$	−305.687			−0.478384
$640$			25.2982i			0.0395285i
$641$		−	722.476i		−	1.12711i	−0.826079	−	0.563554i	$-0.809434\pi$
$641$		−	722.476i		−	1.12711i	0.826079	−	0.563554i	$-0.190566\pi$
$642$			412.647i			0.642752i
$643$		−	912.834i		−	1.41965i	−0.704379	−	0.709824i	$-0.748774\pi$
$643$		−	912.834i		−	1.41965i	0.704379	−	0.709824i	$-0.251226\pi$
$644$	507.704	+	282.571i	0.788360	+	0.438776i
$645$	−303.856			−0.471095
$646$	220.834			0.341848
$647$	378.045			0.584305			0.292153	−	0.956372i	$-0.405628\pi$
$647$	378.045			0.584305			0.292153	+	0.956372i	$0.405628\pi$
$648$	−25.4558			−0.0392837
$649$			186.660i			0.287612i
$650$	−152.825			−0.235115
$651$			17.2958i			0.0265681i
$652$	−239.135			−0.366772
$653$	−165.523			−0.253481			−0.126740	−	0.991936i	$-0.540451\pi$
$653$	−165.523			−0.253481			−0.126740	+	0.991936i	$0.540451\pi$
$654$			80.5140i			0.123110i
$655$			232.724i			0.355303i
$656$	−0.350168			−0.000533793
$657$	−333.235			−0.507208
$658$			1166.45i			1.77271i
$659$		−	396.351i		−	0.601443i	−0.953712	−	0.300721i	$-0.902773\pi$
$659$		−	396.351i		−	0.601443i	0.953712	−	0.300721i	$-0.0972274\pi$
$660$	51.4828			0.0780042
$661$			631.845i			0.955893i	0.878389	+	0.477946i	$0.158619\pi$
$661$			631.845i			0.955893i	−0.878389	+	0.477946i	$0.841381\pi$
$662$	−90.8765			−0.137276
$663$		−	1007.02i		−	1.51889i
$664$			389.791i			0.587035i
$665$			163.951i			0.246543i
$666$			50.0097i			0.0750896i
$667$	−568.047	+	1020.62i	−0.851645	+	1.53017i
$668$	212.518			0.318140
$669$	100.671			0.150479
$670$	75.7703			0.113090
$671$	124.120			0.184978
$672$		−	123.761i		−	0.184169i
$673$	197.566			0.293560			0.146780	−	0.989169i	$-0.453109\pi$
$673$	197.566			0.293560			0.146780	+	0.989169i	$0.453109\pi$
$674$		−	499.724i		−	0.741431i
$675$	25.9808			0.0384900
$676$	596.214			0.881974
$677$		−	493.179i		−	0.728478i	−0.931306	−	0.364239i	$-0.881329\pi$
$677$		−	493.179i		−	0.728478i	0.931306	−	0.364239i	$-0.118671\pi$
$678$			98.8861i			0.145850i
$679$	260.998			0.384386
$680$	170.138			0.250203
$681$			107.304i			0.157568i
$682$			7.43075i			0.0108955i
$683$	13.6542			0.0199914			0.00999572	−	0.999950i	$-0.496818\pi$
$683$	13.6542			0.0199914			0.00999572	+	0.999950i	$0.496818\pi$
$684$		−	34.8282i		−	0.0509184i
$685$	−343.782			−0.501872
$686$		−	1099.51i		−	1.60278i
$687$			474.877i			0.691233i
$688$			313.821i			0.456135i
$689$			774.426i			1.12399i
$690$	110.076	+	61.2646i	0.159530	+	0.0887892i
$691$	−753.863			−1.09097			−0.545487	−	0.838119i	$-0.683655\pi$
$691$	−753.863			−1.09097			−0.545487	+	0.838119i	$0.683655\pi$
$692$	686.806			0.992494
$693$	−251.859			−0.363433
$694$	600.615			0.865440
$695$		−	80.2492i		−	0.115467i
$696$	248.795			0.357464
$697$			2.35498i			0.00337874i
$698$	157.135			0.225122
$699$	21.5711			0.0308600
$700$			126.313i			0.180448i
$701$		−	1093.76i		−	1.56029i	−0.625601	−	0.780143i	$-0.715146\pi$
$701$		−	1093.76i		−	1.56029i	0.625601	−	0.780143i	$-0.284854\pi$
$702$	−158.820			−0.226239
$703$	−68.4223			−0.0973290
$704$		−	53.1712i		−	0.0755273i
$705$			252.898i			0.358721i
$706$	432.882			0.613147
$707$		−	1387.18i		−	1.96206i
$708$	97.2871			0.137411
$709$			968.137i			1.36550i	0.730654	+	0.682748i	$0.239215\pi$
$709$			968.137i			1.36550i	−0.730654	+	0.682748i	$0.760785\pi$
$710$		−	322.223i		−	0.453835i
$711$			252.294i			0.354843i
$712$		−	132.234i		−	0.185722i
$713$	−8.84260	+	15.8877i	−0.0124020	+	0.0222829i
$714$	−832.332			−1.16573
$715$	321.203			0.449235
$716$	345.563			0.482630
$717$	603.963			0.842347
$718$		−	109.687i		−	0.152767i
$719$	−831.863			−1.15697			−0.578486	−	0.815692i	$-0.696356\pi$
$719$	−831.863			−1.15697			−0.578486	+	0.815692i	$0.696356\pi$
$720$		−	26.8328i		−	0.0372678i
$721$	−113.794			−0.157828
$722$	−462.880			−0.641108
$723$		−	576.999i		−	0.798062i
$724$			41.0671i			0.0567226i
$725$	−253.925			−0.350241
$726$	188.183			0.259205
$727$			961.589i			1.32268i	0.750086	+	0.661340i	$0.230012\pi$
$727$			961.589i			1.32268i	−0.750086	+	0.661340i	$0.769988\pi$
$728$		−	772.152i		−	1.06065i
$729$	27.0000			0.0370370
$730$		−	351.261i		−	0.481179i
$731$	2110.54			2.88720
$732$		−	64.6915i		−	0.0883764i
$733$			817.386i			1.11512i	0.830135	+	0.557562i	$0.188263\pi$
$733$			817.386i			1.11512i	−0.830135	+	0.557562i	$0.811737\pi$
$734$			121.268i			0.165215i
$735$		−	428.162i		−	0.582533i
$736$	63.2738	−	113.686i	0.0859698	−	0.154464i
$737$	−159.252			−0.216082
$738$	0.371409			0.000503265
$739$	1419.01			1.92018			0.960088	−	0.279698i	$-0.0902343\pi$
$739$	1419.01			1.92018			0.960088	+	0.279698i	$0.0902343\pi$
$740$	−52.7148			−0.0712363
$741$		−	217.294i		−	0.293245i
$742$	640.083			0.862645
$743$			406.342i			0.546894i	0.961887	+	0.273447i	$0.0881638\pi$
$743$			406.342i			0.546894i	−0.961887	+	0.273447i	$0.911836\pi$
$744$	3.87291			0.00520552
$745$	−11.6864			−0.0156865
$746$		−	204.608i		−	0.274274i
$747$		−	413.436i		−	0.553462i
$748$	−357.592			−0.478064
$749$	2127.91			2.84099
$750$			27.3861i			0.0365148i
$751$		−	1221.72i		−	1.62679i	−0.581713	−	0.813394i	$-0.697617\pi$
$751$		−	1221.72i		−	1.62679i	0.581713	−	0.813394i	$-0.302383\pi$
$752$	261.192			0.347330
$753$		−	378.703i		−	0.502926i
$754$	1552.24			2.05867
$755$			620.814i			0.822271i
$756$			131.269i			0.173636i
$757$			113.386i			0.149784i	0.997192	+	0.0748919i	$0.0238612\pi$
$757$			113.386i			0.149784i	−0.997192	+	0.0748919i	$0.976139\pi$
$758$			429.760i			0.566965i
$759$	−231.354	−	128.764i	−0.304815	−	0.169650i
$760$	36.7121			0.0483054
$761$	−224.399			−0.294874			−0.147437	−	0.989071i	$-0.547102\pi$
$761$	−224.399			−0.294874			−0.147437	+	0.989071i	$0.547102\pi$
$762$	−543.791			−0.713637
$763$	415.188			0.544153
$764$		−	252.785i		−	0.330870i
$765$	−180.459			−0.235894
$766$		−	305.424i		−	0.398726i
$767$	606.978			0.791366
$768$	−27.7128			−0.0360844
$769$		−	1448.92i		−	1.88417i	−0.335379	−	0.942083i	$-0.608865\pi$
$769$		−	1448.92i		−	1.88417i	0.335379	−	0.942083i	$-0.391135\pi$
$770$		−	265.483i		−	0.344783i
$771$	−755.871			−0.980377
$772$	−238.576			−0.309037
$773$		−	422.972i		−	0.547182i	−0.961846	−	0.273591i	$-0.911789\pi$
$773$		−	422.972i		−	0.547182i	0.961846	−	0.273591i	$-0.0882115\pi$
$774$		−	332.858i		−	0.430049i
$775$	−3.95277			−0.00510035
$776$		−	58.4430i		−	0.0753132i
$777$	257.886			0.331900
$778$		−	58.4268i		−	0.0750987i
$779$			0.508155i			0.000652317i
$780$		−	167.411i		−	0.214629i
$781$			677.240i			0.867145i
$782$	−764.570	−	425.535i	−0.977711	−	0.544162i
$783$	−263.887			−0.337020
$784$	−442.203			−0.564035
$785$	−421.118			−0.536456
$786$	−254.936			−0.324346
$787$		−	588.517i		−	0.747798i	−0.927469	−	0.373899i	$-0.878021\pi$
$787$		−	588.517i		−	0.747798i	0.927469	−	0.373899i	$-0.121979\pi$
$788$	64.1683			0.0814319
$789$		−	747.580i		−	0.947503i
$790$	−265.941			−0.336634
$791$	509.928			0.644663
$792$			56.3966i			0.0712078i
$793$		−	403.613i		−	0.508969i
$794$	−566.152			−0.713038
$795$	138.777			0.174562
$796$		−	617.953i		−	0.776322i
$797$			128.126i			0.160760i	0.996764	+	0.0803799i	$0.0256134\pi$
$797$			128.126i			0.160760i	−0.996764	+	0.0803799i	$0.974387\pi$
$798$	−179.599			−0.225062
$799$		−	1756.59i		−	2.19849i
$800$	28.2843			0.0353553
$801$			140.256i			0.175101i
$802$			626.181i			0.780775i
$803$			738.272i			0.919392i
$804$			83.0022i			0.103237i
$805$	315.925	−	567.630i	0.392453	−	0.705130i
$806$	24.1632			0.0299792
$807$	540.613			0.669905
$808$	−310.619			−0.384429
$809$	−603.939			−0.746525			−0.373263	−	0.927726i	$-0.621761\pi$
$809$	−603.939			−0.746525			−0.373263	+	0.927726i	$0.621761\pi$
$810$			28.4605i			0.0351364i
$811$	−669.432			−0.825440			−0.412720	−	0.910858i	$-0.635421\pi$
$811$	−669.432			−0.825440			−0.412720	+	0.910858i	$0.635421\pi$
$812$		−	1282.97i		−	1.58001i
$813$	122.464			0.150633
$814$	110.795			0.136112
$815$			267.361i			0.328050i
$816$			186.377i			0.228403i
$817$	455.409			0.557416
$818$	1105.67			1.35167
$819$			818.991i			0.999989i
$820$			0.391500i			0.000477439i
$821$	−764.409			−0.931070			−0.465535	−	0.885029i	$-0.654138\pi$
$821$	−764.409			−0.931070			−0.465535	+	0.885029i	$0.654138\pi$
$822$		−	376.595i		−	0.458144i
$823$	−37.3947			−0.0454371			−0.0227185	−	0.999742i	$-0.507232\pi$
$823$	−37.3947			−0.0454371			−0.0227185	+	0.999742i	$0.507232\pi$
$824$			25.4809i			0.0309235i
$825$		−	57.5595i		−	0.0697691i
$826$		−	501.683i		−	0.607364i
$827$		−	390.592i		−	0.472300i	−0.971717	−	0.236150i	$-0.924114\pi$
$827$		−	390.592i		−	0.472300i	0.971717	−	0.236150i	$-0.0758857\pi$
$828$	−67.1120	+	120.582i	−0.0810531	+	0.145630i
$829$	101.148			0.122012			0.0610060	−	0.998137i	$-0.480569\pi$
$829$	101.148			0.122012			0.0610060	+	0.998137i	$0.480569\pi$
$830$	435.800			0.525060
$831$	−597.073			−0.718500
$832$	−172.901			−0.207814
$833$			2973.95i			3.57017i
$834$	87.9086			0.105406
$835$		−	237.602i		−	0.284553i
$836$	−77.1607			−0.0922975
$837$	−4.10784			−0.00490781
$838$			590.748i			0.704950i
$839$			160.357i			0.191128i	0.995423	+	0.0955641i	$0.0304655\pi$
$839$			160.357i			0.191128i	−0.995423	+	0.0955641i	$0.969535\pi$
$840$	−138.369			−0.164726
$841$	1738.12			2.06673
$842$			198.341i			0.235559i
$843$			518.248i			0.614766i
$844$	569.829			0.675153
$845$		−	666.588i		−	0.788861i
$846$	−277.036			−0.327466
$847$		−	970.407i		−	1.14570i
$848$		−	143.328i		−	0.169019i
$849$		−	86.1654i		−	0.101490i
$850$		−	190.220i		−	0.223788i
$851$	236.891	+	131.846i	0.278368	+	0.154931i
$852$	352.977			0.414292
$853$	−948.676			−1.11216			−0.556082	−	0.831127i	$-0.687696\pi$
$853$	−948.676			−1.11216			−0.556082	+	0.831127i	$0.687696\pi$
$854$	−333.596			−0.390628
$855$	−38.9391			−0.0455428
$856$		−	476.483i		−	0.556639i
$857$	0.565369			0.000659707			0.000329853	−	1.00000i	$-0.499895\pi$
$857$	0.565369			0.000659707			0.000329853		1.00000i	$0.499895\pi$
$858$			351.860i			0.410094i
$859$	−581.652			−0.677126			−0.338563	−	0.940944i	$-0.609941\pi$
$859$	−581.652			−0.677126			−0.338563	+	0.940944i	$0.609941\pi$
$860$	350.863			0.407980
$861$		−	1.91526i		−	0.00222445i
$862$		−	4.93140i		−	0.00572088i
$863$	−889.126			−1.03027			−0.515137	−	0.857108i	$-0.672259\pi$
$863$	−889.126			−1.03027			−0.515137	+	0.857108i	$0.672259\pi$
$864$	29.3939			0.0340207
$865$		−	767.872i		−	0.887714i
$866$			526.945i			0.608482i
$867$	752.877			0.868370
$868$		−	19.9715i		−	0.0230087i
$869$	558.948			0.643209
$870$		−	278.161i		−	0.319725i
$871$			517.854i			0.594551i
$872$		−	92.9695i		−	0.106616i
$873$			61.9882i			0.0710059i
$874$	−164.978	−	91.8213i	−0.188762	−	0.105059i
$875$	141.223			0.161397
$876$	384.787			0.439255
$877$	−654.887			−0.746736			−0.373368	−	0.927683i	$-0.621797\pi$
$877$	−654.887			−0.746736			−0.373368	+	0.927683i	$0.621797\pi$
$878$	−146.694			−0.167077
$879$		−	64.2964i		−	0.0731472i
$880$	−59.4472			−0.0675537
$881$		−	1410.23i		−	1.60072i	−0.599522	−	0.800358i	$-0.704643\pi$
$881$		−	1410.23i		−	1.60072i	0.599522	−	0.800358i	$-0.295357\pi$
$882$	469.028			0.531777
$883$	1313.93			1.48803			0.744016	−	0.668162i	$-0.232919\pi$
$883$	1313.93			1.48803			0.744016	+	0.668162i	$0.232919\pi$
$884$			1162.81i			1.31540i
$885$		−	108.770i		−	0.122904i
$886$	981.106			1.10734
$887$	−647.345			−0.729814			−0.364907	−	0.931044i	$-0.618899\pi$
$887$	−647.345			−0.729814			−0.364907	+	0.931044i	$0.618899\pi$
$888$		−	57.7462i		−	0.0650295i
$889$			2804.18i			3.15431i
$890$	−147.842			−0.166115
$891$		−	59.8176i		−	0.0671354i
$892$	−116.244			−0.130319
$893$		−	379.035i		−	0.424452i
$894$		−	12.8018i		−	0.0143197i
$895$		−	386.351i		−	0.431677i
$896$			142.907i			0.159495i
$897$	−418.714	+	752.315i	−0.466794	+	0.838701i
$898$	−388.013			−0.432085
$899$	40.1483			0.0446588
$900$	−30.0000			−0.0333333
$901$	−963.924			−1.06984
$902$		−	0.822845i		−	0.000912245i
$903$	−1716.45			−1.90084
$904$		−	114.184i		−	0.126310i
$905$	45.9145			0.0507342
$906$	−680.068			−0.750627
$907$			176.027i			0.194076i	0.995281	+	0.0970381i	$0.0309369\pi$
$907$			176.027i			0.194076i	−0.995281	+	0.0970381i	$0.969063\pi$
$908$		−	123.903i		−	0.136458i
$909$	329.461			0.362443
$910$	−863.292			−0.948673
$911$			91.5962i			0.100545i	0.998736	+	0.0502724i	$0.0160089\pi$
$911$			91.5962i			0.100545i	−0.998736	+	0.0502724i	$0.983991\pi$
$912$			40.2161i			0.0440966i
$913$	−915.954			−1.00324
$914$			831.710i			0.909967i
$915$	−72.3273			−0.0790462
$916$		−	548.341i		−	0.598626i
$917$			1314.63i			1.43362i
$918$		−	197.683i		−	0.215340i
$919$			1462.85i			1.59179i	0.605437	+	0.795893i	$0.292998\pi$
$919$			1462.85i			1.59179i	−0.605437	+	0.795893i	$0.707002\pi$
$920$	−127.104	−	70.7422i	−0.138157	−	0.0768937i
$921$	1.07982			0.00117245
$922$	310.058			0.336288
$923$	2202.24			2.38596
$924$	290.822			0.314742
$925$			58.9370i			0.0637157i
$926$	148.903			0.160802
$927$		−	27.0266i		−	0.0291549i
$928$	−287.283			−0.309573
$929$	−546.761			−0.588548			−0.294274	−	0.955721i	$-0.595078\pi$
$929$	−546.761			−0.588548			−0.294274	+	0.955721i	$0.595078\pi$
$930$		−	4.33004i		−	0.00465596i
$931$			641.714i			0.689274i
$932$	−24.9082			−0.0267255
$933$	−204.883			−0.219596
$934$			421.179i			0.450941i
$935$			399.800i			0.427594i
$936$	183.389			0.195929
$937$			1340.83i			1.43098i	0.698621	+	0.715492i	$0.253798\pi$
$937$			1340.83i			1.43098i	−0.698621	+	0.715492i	$0.746202\pi$
$938$	428.019			0.456311
$939$			550.910i			0.586699i
$940$		−	292.022i		−	0.310661i
$941$		−	1514.97i		−	1.60996i	−0.593301	−	0.804981i	$-0.702175\pi$
$941$		−	1514.97i		−	1.60996i	0.593301	−	0.804981i	$-0.297825\pi$
$942$		−	461.311i		−	0.489715i
$943$	0.979186	−	1.75933i	0.00103837	−	0.00186567i
$944$	−112.337			−0.119002
$945$	146.763			0.155305
$946$	−737.435			−0.779530
$947$	240.031			0.253464			0.126732	−	0.991937i	$-0.459551\pi$
$947$	240.031			0.253464			0.126732	+	0.991937i	$0.459551\pi$
$948$		−	291.324i		−	0.307303i
$949$	2400.70			2.52972
$950$		−	41.0454i		−	0.0432057i
$951$	505.047			0.531069
$952$	961.094			1.00955
$953$		−	904.817i		−	0.949440i	−0.880137	−	0.474720i	$-0.842549\pi$
$953$		−	904.817i		−	0.949440i	0.880137	−	0.474720i	$-0.157451\pi$
$954$			152.022i			0.159353i
$955$	−282.622			−0.295939
$956$	−697.396			−0.729494
$957$			584.632i			0.610901i
$958$		−	811.730i		−	0.847318i
$959$	−1941.99			−2.02502
$960$			30.9839i			0.0322749i
$961$	−960.375			−0.999350
$962$		−	360.281i		−	0.374512i
$963$			505.387i			0.524805i
$964$			666.261i			0.691142i
$965$			266.736i			0.276411i
$966$	621.807	+	346.078i	0.643693	+	0.358259i
$967$	−1649.74			−1.70604			−0.853022	−	0.521875i	$-0.825233\pi$
$967$	−1649.74			−1.70604			−0.853022	+	0.521875i	$0.825233\pi$
$968$	−217.295			−0.224478
$969$	270.465			0.279118
$970$	−65.3413			−0.0673621
$971$		−	610.803i		−	0.629045i	−0.949250	−	0.314523i	$-0.898156\pi$
$971$		−	610.803i		−	0.629045i	0.949250	−	0.314523i	$-0.101844\pi$
$972$	−31.1769			−0.0320750
$973$		−	453.321i		−	0.465900i
$974$	1321.96			1.35725
$975$	−187.171			−0.191970
$976$			74.6993i			0.0765362i
$977$			1230.78i			1.25976i	0.776694	+	0.629878i	$0.216895\pi$
$977$			1230.78i			1.25976i	−0.776694	+	0.629878i	$0.783105\pi$
$978$	−292.879			−0.299468
$979$	310.732			0.317397
$980$			494.398i			0.504488i
$981$			98.6091i			0.100519i
$982$	403.435			0.410829
$983$			74.5801i			0.0758699i	0.999280	+	0.0379349i	$0.0120780\pi$
$983$			74.5801i			0.0758699i	−0.999280	+	0.0379349i	$0.987922\pi$
$984$	−0.428867			−0.000435840
$985$		−	71.7424i		−	0.0728349i
$986$			1932.07i			1.95950i
$987$			1428.60i			1.44742i
$988$			250.910i			0.253957i
$989$	−1576.71	−	877.548i	−1.59425	−	0.887309i
$990$	63.0533			0.0636902
$991$	1179.77			1.19049			0.595244	−	0.803545i	$-0.297056\pi$
$991$	1179.77			1.19049			0.595244	+	0.803545i	$0.297056\pi$
$992$	−4.47205			−0.00450811
$993$	−111.301			−0.112085
$994$		−	1820.21i		−	1.83119i
$995$	−690.892			−0.694364
$996$			477.395i			0.479312i
$997$	−111.498			−0.111833			−0.0559166	−	0.998435i	$-0.517808\pi$
$997$	−111.498			−0.111833			−0.0559166	+	0.998435i	$0.517808\pi$
$998$	−319.416			−0.320057
$999$			61.2491i			0.0613104i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	690.3.c.a.91.1	✓	32
3.2	odd	2		2070.3.c.b.91.25		32
23.22	odd	2	inner	690.3.c.a.91.8	yes	32
69.68	even	2		2070.3.c.b.91.24		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.3.c.a.91.1	✓	32	1.1	even	1	trivial
690.3.c.a.91.8	yes	32	23.22	odd	2	inner
2070.3.c.b.91.24		32	69.68	even	2
2070.3.c.b.91.25		32	3.2	odd	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)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} -1.73205 q^{3} +2.00000 q^{4} -2.23607i q^{5} +2.44949 q^{6} -12.6313i q^{7} -2.82843 q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q-1.41421 q^{2} -1.73205 q^{3} +2.00000 q^{4} -2.23607i q^{5} +2.44949 q^{6} -12.6313i q^{7} -2.82843 q^{8} +3.00000 q^{9} +3.16228i q^{10} -6.64640i q^{11} -3.46410 q^{12} -21.6127 q^{13} +17.8634i q^{14} +3.87298i q^{15} +4.00000 q^{16} -26.9012i q^{17} -4.24264 q^{18} -5.80470i q^{19} -4.47214i q^{20} +21.8781i q^{21} +9.39943i q^{22} +(-11.1853 + 20.0970i) q^{23} +4.89898 q^{24} -5.00000 q^{25} +30.5649 q^{26} -5.19615 q^{27} -25.2627i q^{28} +50.7850 q^{29} -5.47723i q^{30} +0.790554 q^{31} -5.65685 q^{32} +11.5119i q^{33} +38.0440i q^{34} -28.2445 q^{35} +6.00000 q^{36} -11.7874i q^{37} +8.20908i q^{38} +37.4342 q^{39} +6.32456i q^{40} -0.0875420 q^{41} -30.9403i q^{42} +78.4553i q^{43} -13.2928i q^{44} -6.70820i q^{45} +(15.8184 - 28.4214i) q^{46} +65.2980 q^{47} -6.92820 q^{48} -110.551 q^{49} +7.07107 q^{50} +46.5942i q^{51} -43.2253 q^{52} -35.8320i q^{53} +7.34847 q^{54} -14.8618 q^{55} +35.7268i q^{56} +10.0540i q^{57} -71.8208 q^{58} -28.0844 q^{59} +7.74597i q^{60} +18.6748i q^{61} -1.11801 q^{62} -37.8940i q^{63} +8.00000 q^{64} +48.3274i q^{65} -16.2803i q^{66} -23.9607i q^{67} -53.8024i q^{68} +(19.3736 - 34.8090i) q^{69} +39.9438 q^{70} -101.896 q^{71} -8.48528 q^{72} -111.078 q^{73} +16.6699i q^{74} +8.66025 q^{75} -11.6094i q^{76} -83.9530 q^{77} -52.9400 q^{78} +84.0979i q^{79} -8.94427i q^{80} +9.00000 q^{81} +0.123803 q^{82} -137.812i q^{83} +43.7563i q^{84} -60.1529 q^{85} -110.953i q^{86} -87.9622 q^{87} +18.7989i q^{88} +46.7519i q^{89} +9.48683i q^{90} +272.997i q^{91} +(-22.3707 + 40.1940i) q^{92} -1.36928 q^{93} -92.3454 q^{94} -12.9797 q^{95} +9.79796 q^{96} +20.6627i q^{97} +156.343 q^{98} -19.9392i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 64 q^{4} + 96 q^{9}+O(q^{10}) \) 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

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