LMFDB - Embedded newform 690.3.c.a.91.22

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			91.22
Character	$\chi$	$=$	690.91
Dual form			690.3.c.a.91.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} -1.73205 q^{3} +2.00000 q^{4} +2.23607i q^{5} -2.44949 q^{6} -1.37049i q^{7} +2.82843 q^{8} +3.00000 q^{9} +O(q^{10})$	\(q+1.41421 q^{2} -1.73205 q^{3} +2.00000 q^{4} +2.23607i q^{5} -2.44949 q^{6} -1.37049i q^{7} +2.82843 q^{8} +3.00000 q^{9} +3.16228i q^{10} -12.6432i q^{11} -3.46410 q^{12} -17.1953 q^{13} -1.93817i q^{14} -3.87298i q^{15} +4.00000 q^{16} -17.1987i q^{17} +4.24264 q^{18} -5.73865i q^{19} +4.47214i q^{20} +2.37376i q^{21} -17.8802i q^{22} +(22.4818 - 4.85478i) q^{23} -4.89898 q^{24} -5.00000 q^{25} -24.3178 q^{26} -5.19615 q^{27} -2.74098i q^{28} +19.0719 q^{29} -5.47723i q^{30} +27.6139 q^{31} +5.65685 q^{32} +21.8987i q^{33} -24.3226i q^{34} +3.06451 q^{35} +6.00000 q^{36} -28.5685i q^{37} -8.11568i q^{38} +29.7831 q^{39} +6.32456i q^{40} -40.8110 q^{41} +3.35700i q^{42} -33.6014i q^{43} -25.2864i q^{44} +6.70820i q^{45} +(31.7941 - 6.86570i) q^{46} -36.0593 q^{47} -6.92820 q^{48} +47.1218 q^{49} -7.07107 q^{50} +29.7890i q^{51} -34.3906 q^{52} -72.1605i q^{53} -7.34847 q^{54} +28.2711 q^{55} -3.87633i q^{56} +9.93963i q^{57} +26.9718 q^{58} +33.0861 q^{59} -7.74597i q^{60} -102.589i q^{61} +39.0519 q^{62} -4.11147i q^{63} +8.00000 q^{64} -38.4498i q^{65} +30.9694i q^{66} -83.7272i q^{67} -34.3974i q^{68} +(-38.9396 + 8.40873i) q^{69} +4.33387 q^{70} -53.3503 q^{71} +8.48528 q^{72} -6.09068 q^{73} -40.4019i q^{74} +8.66025 q^{75} -11.4773i q^{76} -17.3274 q^{77} +42.1197 q^{78} +16.0407i q^{79} +8.94427i q^{80} +9.00000 q^{81} -57.7155 q^{82} +119.789i q^{83} +4.74752i q^{84} +38.4574 q^{85} -47.5196i q^{86} -33.0335 q^{87} -35.7604i q^{88} +71.6620i q^{89} +9.48683i q^{90} +23.5660i q^{91} +(44.9636 - 9.70956i) q^{92} -47.8286 q^{93} -50.9955 q^{94} +12.8320 q^{95} -9.79796 q^{96} +47.4489i q^{97} +66.6402 q^{98} -37.9297i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$		−	1.37049i		−	0.195784i	−0.995197	−	0.0978922i	$-0.968790\pi$
$7$		−	1.37049i		−	0.195784i	0.995197	−	0.0978922i	$-0.0312100\pi$
$8$	2.82843			0.353553
$9$	3.00000			0.333333
$10$			3.16228i			0.316228i
$11$		−	12.6432i		−	1.14938i	−0.818370	−	0.574692i	$-0.805122\pi$
$11$		−	12.6432i		−	1.14938i	0.818370	−	0.574692i	$-0.194878\pi$
$12$	−3.46410			−0.288675
$13$	−17.1953			−1.32271			−0.661357	−	0.750071i	$-0.730019\pi$
$13$	−17.1953			−1.32271			−0.661357	+	0.750071i	$0.730019\pi$
$14$		−	1.93817i		−	0.138440i
$15$		−	3.87298i		−	0.258199i
$16$	4.00000			0.250000
$17$		−	17.1987i		−	1.01169i	−0.862625	−	0.505844i	$-0.831181\pi$
$17$		−	17.1987i		−	1.01169i	0.862625	−	0.505844i	$-0.168819\pi$
$18$	4.24264			0.235702
$19$		−	5.73865i		−	0.302034i	−0.988531	−	0.151017i	$-0.951745\pi$
$19$		−	5.73865i		−	0.302034i	0.988531	−	0.151017i	$-0.0482548\pi$
$20$			4.47214i			0.223607i
$21$			2.37376i			0.113036i
$22$		−	17.8802i		−	0.812737i
$23$	22.4818	−	4.85478i	0.977469	−	0.211077i
$23$	22.4818	−	4.85478i	0.977469	−	0.211077i
$24$	−4.89898			−0.204124
$25$	−5.00000			−0.200000
$26$	−24.3178			−0.935301
$27$	−5.19615			−0.192450
$28$		−	2.74098i		−	0.0978922i
$29$	19.0719			0.657653			0.328826	−	0.944390i	$-0.393347\pi$
$29$	19.0719			0.657653			0.328826	+	0.944390i	$0.393347\pi$
$30$		−	5.47723i		−	0.182574i
$31$	27.6139			0.890770			0.445385	−	0.895339i	$-0.353067\pi$
$31$	27.6139			0.890770			0.445385	+	0.895339i	$0.353067\pi$
$32$	5.65685			0.176777
$33$			21.8987i			0.663597i
$34$		−	24.3226i		−	0.715371i
$35$	3.06451			0.0875574
$36$	6.00000			0.166667
$37$		−	28.5685i		−	0.772120i	−0.922474	−	0.386060i	$-0.873836\pi$
$37$		−	28.5685i		−	0.772120i	0.922474	−	0.386060i	$-0.126164\pi$
$38$		−	8.11568i		−	0.213570i
$39$	29.7831			0.763670
$40$			6.32456i			0.158114i
$41$	−40.8110			−0.995391			−0.497695	−	0.867352i	$-0.665820\pi$
$41$	−40.8110			−0.995391			−0.497695	+	0.867352i	$0.665820\pi$
$42$			3.35700i			0.0799286i
$43$		−	33.6014i		−	0.781428i	−0.920512	−	0.390714i	$-0.872228\pi$
$43$		−	33.6014i		−	0.781428i	0.920512	−	0.390714i	$-0.127772\pi$
$44$		−	25.2864i		−	0.574692i
$45$			6.70820i			0.149071i
$46$	31.7941	−	6.86570i	0.691175	−	0.149254i
$47$	−36.0593			−0.767218			−0.383609	−	0.923496i	$-0.625319\pi$
$47$	−36.0593			−0.767218			−0.383609	+	0.923496i	$0.625319\pi$
$48$	−6.92820			−0.144338
$49$	47.1218			0.961668
$50$	−7.07107			−0.141421
$51$			29.7890i			0.584098i
$52$	−34.3906			−0.661357
$53$		−	72.1605i		−	1.36152i	−0.732507	−	0.680760i	$-0.761650\pi$
$53$		−	72.1605i		−	1.36152i	0.732507	−	0.680760i	$-0.238350\pi$
$54$	−7.34847			−0.136083
$55$	28.2711			0.514020
$56$		−	3.87633i		−	0.0692202i
$57$			9.93963i			0.174380i
$58$	26.9718			0.465031
$59$	33.0861			0.560781			0.280391	−	0.959886i	$-0.409536\pi$
$59$	33.0861			0.560781			0.280391	+	0.959886i	$0.409536\pi$
$60$		−	7.74597i		−	0.129099i
$61$		−	102.589i		−	1.68179i	−0.541196	−	0.840896i	$-0.682028\pi$
$61$		−	102.589i		−	1.68179i	0.541196	−	0.840896i	$-0.317972\pi$
$62$	39.0519			0.629869
$63$		−	4.11147i		−	0.0652615i
$64$	8.00000			0.125000
$65$		−	38.4498i		−	0.591536i
$66$			30.9694i			0.469234i
$67$		−	83.7272i		−	1.24966i	−0.780761	−	0.624830i	$-0.785168\pi$
$67$		−	83.7272i		−	1.24966i	0.780761	−	0.624830i	$-0.214832\pi$
$68$		−	34.3974i		−	0.505844i
$69$	−38.9396	+	8.40873i	−0.564342	+	0.121866i
$70$	4.33387			0.0619125
$71$	−53.3503			−0.751413			−0.375706	−	0.926739i	$-0.622600\pi$
$71$	−53.3503			−0.751413			−0.375706	+	0.926739i	$0.622600\pi$
$72$	8.48528			0.117851
$73$	−6.09068			−0.0834340			−0.0417170	−	0.999129i	$-0.513283\pi$
$73$	−6.09068			−0.0834340			−0.0417170	+	0.999129i	$0.513283\pi$
$74$		−	40.4019i		−	0.545971i
$75$	8.66025			0.115470
$76$		−	11.4773i		−	0.151017i
$77$	−17.3274			−0.225031
$78$	42.1197			0.539996
$79$			16.0407i			0.203047i	0.994833	+	0.101524i	$0.0323718\pi$
$79$			16.0407i			0.203047i	−0.994833	+	0.101524i	$0.967628\pi$
$80$			8.94427i			0.111803i
$81$	9.00000			0.111111
$82$	−57.7155			−0.703848
$83$			119.789i			1.44325i	0.692287	+	0.721623i	$0.256603\pi$
$83$			119.789i			1.44325i	−0.692287	+	0.721623i	$0.743397\pi$
$84$			4.74752i			0.0565181i
$85$	38.4574			0.452440
$86$		−	47.5196i		−	0.552553i
$87$	−33.0335			−0.379696
$88$		−	35.7604i		−	0.406369i
$89$			71.6620i			0.805191i	0.915378	+	0.402596i	$0.131892\pi$
$89$			71.6620i			0.805191i	−0.915378	+	0.402596i	$0.868108\pi$
$90$			9.48683i			0.105409i
$91$			23.5660i			0.258967i
$92$	44.9636	−	9.70956i	0.488735	−	0.105539i
$93$	−47.8286			−0.514286
$94$	−50.9955			−0.542505
$95$	12.8320			0.135074
$96$	−9.79796			−0.102062
$97$			47.4489i			0.489164i	0.969629	+	0.244582i	$0.0786507\pi$
$97$			47.4489i			0.489164i	−0.969629	+	0.244582i	$0.921349\pi$
$98$	66.6402			0.680002
$99$		−	37.9297i		−	0.383128i
$100$	−10.0000			−0.100000
$101$	−25.2315			−0.249817			−0.124908	−	0.992168i	$-0.539864\pi$
$101$	−25.2315			−0.249817			−0.124908	+	0.992168i	$0.539864\pi$
$102$			42.1280i			0.413020i
$103$			102.361i			0.993800i	0.867808	+	0.496900i	$0.165529\pi$
$103$			102.361i			0.993800i	−0.867808	+	0.496900i	$0.834471\pi$
$104$	−48.6356			−0.467650
$105$	−5.30789			−0.0505513
$106$		−	102.050i		−	0.962740i
$107$			99.6919i			0.931700i	0.884864	+	0.465850i	$0.154251\pi$
$107$			99.6919i			0.931700i	−0.884864	+	0.465850i	$0.845749\pi$
$108$	−10.3923			−0.0962250
$109$		−	70.6544i		−	0.648205i	−0.946022	−	0.324103i	$-0.894938\pi$
$109$		−	70.6544i		−	0.648205i	0.946022	−	0.324103i	$-0.105062\pi$
$110$	39.9814			0.363467
$111$			49.4820i			0.445784i
$112$		−	5.48196i		−	0.0489461i
$113$		−	117.729i		−	1.04185i	−0.853603	−	0.520924i	$-0.825587\pi$
$113$		−	117.729i		−	1.04185i	0.853603	−	0.520924i	$-0.174413\pi$
$114$			14.0568i			0.123305i
$115$	10.8556	+	50.2708i	0.0943967	+	0.437138i
$116$	38.1438			0.328826
$117$	−51.5859			−0.440905
$118$	46.7908			0.396532
$119$	−23.5706			−0.198073
$120$		−	10.9545i		−	0.0912871i
$121$	−38.8511			−0.321084
$122$		−	145.083i		−	1.18921i
$123$	70.6868			0.574689
$124$	55.2277			0.445385
$125$		−	11.1803i		−	0.0894427i
$126$		−	5.81450i		−	0.0461468i
$127$	−210.520			−1.65764			−0.828819	−	0.559516i	$-0.810987\pi$
$127$	−210.520			−1.65764			−0.828819	+	0.559516i	$0.810987\pi$
$128$	11.3137			0.0883883
$129$			58.1994i			0.451158i
$130$		−	54.3763i		−	0.418279i
$131$	179.091			1.36711			0.683553	−	0.729901i	$-0.260434\pi$
$131$	179.091			1.36711			0.683553	+	0.729901i	$0.260434\pi$
$132$			43.7974i			0.331799i
$133$	−7.86477			−0.0591336
$134$		−	118.408i		−	0.883643i
$135$		−	11.6190i		−	0.0860663i
$136$		−	48.6452i		−	0.357685i
$137$			68.6261i			0.500921i	0.968127	+	0.250460i	$0.0805820\pi$
$137$			68.6261i			0.500921i	−0.968127	+	0.250460i	$0.919418\pi$
$138$	−55.0689	+	11.8917i	−0.399050	+	0.0861720i
$139$	35.6149			0.256222			0.128111	−	0.991760i	$-0.459109\pi$
$139$	35.6149			0.256222			0.128111	+	0.991760i	$0.459109\pi$
$140$	6.12902			0.0437787
$141$	62.4565			0.442954
$142$	−75.4487			−0.531329
$143$			217.404i			1.52031i
$144$	12.0000			0.0833333
$145$			42.6461i			0.294111i
$146$	−8.61352			−0.0589967
$147$	−81.6173			−0.555220
$148$		−	57.1369i		−	0.386060i
$149$			169.122i			1.13505i	0.823356	+	0.567525i	$0.192099\pi$
$149$			169.122i			1.13505i	−0.823356	+	0.567525i	$0.807901\pi$
$150$	12.2474			0.0816497
$151$	51.9046			0.343739			0.171870	−	0.985120i	$-0.445019\pi$
$151$	51.9046			0.343739			0.171870	+	0.985120i	$0.445019\pi$
$152$		−	16.2314i		−	0.106785i
$153$		−	51.5960i		−	0.337229i
$154$	−24.5047			−0.159121
$155$			61.7465i			0.398364i
$156$	59.5662			0.381835
$157$		−	103.070i		−	0.656500i	−0.944591	−	0.328250i	$-0.893541\pi$
$157$		−	103.070i		−	0.656500i	0.944591	−	0.328250i	$-0.106459\pi$
$158$			22.6850i			0.143576i
$159$			124.986i			0.786074i
$160$			12.6491i			0.0790569i
$161$	−6.65343	−	30.8111i	−0.0413257	−	0.191373i
$162$	12.7279			0.0785674
$163$	56.7509			0.348165			0.174083	−	0.984731i	$-0.444304\pi$
$163$	56.7509			0.348165			0.174083	+	0.984731i	$0.444304\pi$
$164$	−81.6220			−0.497695
$165$	−48.9670			−0.296770
$166$			169.408i			1.02053i
$167$	−309.988			−1.85622			−0.928108	−	0.372310i	$-0.878566\pi$
$167$	−309.988			−1.85622			−0.928108	+	0.372310i	$0.878566\pi$
$168$			6.71401i			0.0399643i
$169$	126.678			0.749575
$170$	54.3870			0.319924
$171$		−	17.2159i		−	0.100678i
$172$		−	67.2028i		−	0.390714i
$173$	301.774			1.74436			0.872179	−	0.489186i	$-0.162706\pi$
$173$	301.774			1.74436			0.872179	+	0.489186i	$0.162706\pi$
$174$	−46.7165			−0.268486
$175$			6.85245i			0.0391569i
$176$		−	50.5729i		−	0.287346i
$177$	−57.3068			−0.323767
$178$			101.345i			0.569356i
$179$	−66.8168			−0.373278			−0.186639	−	0.982429i	$-0.559760\pi$
$179$	−66.8168			−0.373278			−0.186639	+	0.982429i	$0.559760\pi$
$180$			13.4164i			0.0745356i
$181$			140.236i			0.774785i	0.921915	+	0.387392i	$0.126624\pi$
$181$			140.236i			0.774785i	−0.921915	+	0.387392i	$0.873376\pi$
$182$			33.3273i			0.183117i
$183$			177.690i			0.970983i
$184$	63.5881	−	13.7314i	0.345588	−	0.0746271i
$185$	63.8810			0.345303
$186$	−67.6399			−0.363655
$187$	−217.447			−1.16282
$188$	−72.1185			−0.383609
$189$			7.12128i			0.0376787i
$190$	18.1472			0.0955116
$191$			345.209i			1.80738i	0.428190	+	0.903689i	$0.359151\pi$
$191$			345.209i			1.80738i	−0.428190	+	0.903689i	$0.640849\pi$
$192$	−13.8564			−0.0721688
$193$	186.257			0.965062			0.482531	−	0.875879i	$-0.339718\pi$
$193$	186.257			0.965062			0.482531	+	0.875879i	$0.339718\pi$
$194$			67.1028i			0.345891i
$195$			66.5971i			0.341524i
$196$	94.2435			0.480834
$197$	−279.423			−1.41839			−0.709196	−	0.705012i	$-0.750942\pi$
$197$	−279.423			−1.41839			−0.709196	+	0.705012i	$0.750942\pi$
$198$		−	53.6407i		−	0.270912i
$199$			45.7328i			0.229813i	0.993376	+	0.114907i	$0.0366569\pi$
$199$			45.7328i			0.229813i	−0.993376	+	0.114907i	$0.963343\pi$
$200$	−14.1421			−0.0707107
$201$			145.020i			0.721491i
$202$	−35.6827			−0.176647
$203$		−	26.1379i		−	0.128758i
$204$			59.5780i			0.292049i
$205$		−	91.2562i		−	0.445152i
$206$			144.761i			0.702723i
$207$	67.4454	−	14.5643i	0.325823	−	0.0703591i
$208$	−68.7812			−0.330679
$209$	−72.5550			−0.347153
$210$	−7.50649			−0.0357452
$211$	307.575			1.45770			0.728851	−	0.684673i	$-0.240055\pi$
$211$	307.575			1.45770			0.728851	+	0.684673i	$0.240055\pi$
$212$		−	144.321i		−	0.680760i
$213$	92.4055			0.433828
$214$			140.986i			0.658811i
$215$	75.1351			0.349465
$216$	−14.6969			−0.0680414
$217$		−	37.8445i		−	0.174399i
$218$		−	99.9204i		−	0.458350i
$219$	10.5494			0.0481706
$220$	56.5422			0.257010
$221$			295.736i			1.33817i
$222$			69.9781i			0.315217i
$223$	−356.509			−1.59870			−0.799348	−	0.600869i	$-0.794821\pi$
$223$	−356.509			−1.59870			−0.799348	+	0.600869i	$0.794821\pi$
$224$		−	7.75267i		−	0.0346101i
$225$	−15.0000			−0.0666667
$226$		−	166.494i		−	0.736698i
$227$			167.195i			0.736542i	0.929719	+	0.368271i	$0.120050\pi$
$227$			167.195i			0.736542i	−0.929719	+	0.368271i	$0.879950\pi$
$228$			19.8793i			0.0871898i
$229$		−	281.250i		−	1.22817i	−0.789242	−	0.614083i	$-0.789526\pi$
$229$		−	281.250i		−	1.22817i	0.789242	−	0.614083i	$-0.210474\pi$
$230$	15.3522	+	71.0937i	0.0667485	+	0.309103i
$231$	30.0120			0.129922
$232$	53.9435			0.232515
$233$	60.2115			0.258419			0.129209	−	0.991617i	$-0.458756\pi$
$233$	60.2115			0.258419			0.129209	+	0.991617i	$0.458756\pi$
$234$	−72.9535			−0.311767
$235$		−	80.6309i		−	0.343110i
$236$	66.1722			0.280391
$237$		−	27.7834i		−	0.117229i
$238$	−33.3339			−0.140058
$239$	44.8445			0.187634			0.0938170	−	0.995589i	$-0.470093\pi$
$239$	44.8445			0.187634			0.0938170	+	0.995589i	$0.470093\pi$
$240$		−	15.4919i		−	0.0645497i
$241$		−	0.284271i		−	0.00117955i	−1.00000		0.000589774i	$-0.999812\pi$
$241$		−	0.284271i		−	0.00117955i	1.00000		0.000589774i	$-0.000187731\pi$
$242$	−54.9438			−0.227040
$243$	−15.5885			−0.0641500
$244$		−	205.179i		−	0.840896i
$245$			105.367i			0.430071i
$246$	99.9662			0.406367
$247$			98.6778i			0.399505i
$248$	78.1038			0.314935
$249$		−	207.481i		−	0.833258i
$250$		−	15.8114i		−	0.0632456i
$251$		−	219.635i		−	0.875041i	−0.899208	−	0.437521i	$-0.855857\pi$
$251$		−	219.635i		−	0.875041i	0.899208	−	0.437521i	$-0.144143\pi$
$252$		−	8.22294i		−	0.0326307i
$253$	−61.3801	−	284.242i	−0.242609	−	1.12349i
$254$	−297.720			−1.17213
$255$	−66.6102			−0.261216
$256$	16.0000			0.0625000
$257$	−83.7888			−0.326026			−0.163013	−	0.986624i	$-0.552121\pi$
$257$	−83.7888			−0.326026			−0.163013	+	0.986624i	$0.552121\pi$
$258$			82.3063i			0.319017i
$259$	−39.1528			−0.151169
$260$		−	76.8997i		−	0.295768i
$261$	57.2158			0.219218
$262$	253.273			0.966690
$263$			52.1022i			0.198107i	0.995082	+	0.0990536i	$0.0315815\pi$
$263$			52.1022i			0.198107i	−0.995082	+	0.0990536i	$0.968418\pi$
$264$			61.9389i			0.234617i
$265$	161.356			0.608890
$266$	−11.1225			−0.0418138
$267$		−	124.122i		−	0.464877i
$268$		−	167.454i		−	0.624830i
$269$	−317.806			−1.18143			−0.590717	−	0.806879i	$-0.701155\pi$
$269$	−317.806			−1.18143			−0.590717	+	0.806879i	$0.701155\pi$
$270$		−	16.4317i		−	0.0608581i
$271$	−249.647			−0.921207			−0.460604	−	0.887606i	$-0.652367\pi$
$271$	−249.647			−0.921207			−0.460604	+	0.887606i	$0.652367\pi$
$272$		−	68.7947i		−	0.252922i
$273$		−	40.8175i		−	0.149515i
$274$			97.0520i			0.354204i
$275$			63.2161i			0.229877i
$276$	−77.8792	+	16.8175i	−0.282171	+	0.0609328i
$277$	−35.6857			−0.128829			−0.0644146	−	0.997923i	$-0.520518\pi$
$277$	−35.6857			−0.128829			−0.0644146	+	0.997923i	$0.520518\pi$
$278$	50.3670			0.181176
$279$	82.8416			0.296923
$280$	8.66774			0.0309562
$281$		−	237.163i		−	0.843996i	−0.906597	−	0.421998i	$-0.861329\pi$
$281$		−	237.163i		−	0.843996i	0.906597	−	0.421998i	$-0.138671\pi$
$282$	88.3268			0.313215
$283$		−	284.190i		−	1.00420i	−0.864809	−	0.502102i	$-0.832560\pi$
$283$		−	284.190i		−	1.00420i	0.864809	−	0.502102i	$-0.167440\pi$
$284$	−106.701			−0.375706
$285$	−22.2257			−0.0779849
$286$			307.456i			1.07502i
$287$			55.9311i			0.194882i
$288$	16.9706			0.0589256
$289$	−6.79461			−0.0235108
$290$			60.3107i			0.207968i
$291$		−	82.1839i		−	0.282419i
$292$	−12.1814			−0.0417170
$293$			551.696i			1.88292i	0.337124	+	0.941460i	$0.390546\pi$
$293$			551.696i			1.88292i	−0.337124	+	0.941460i	$0.609454\pi$
$294$	−115.424			−0.392600
$295$			73.9828i			0.250789i
$296$		−	80.8038i		−	0.272986i
$297$			65.6961i			0.221199i
$298$			239.175i			0.802601i
$299$	−386.581	+	83.4794i	−1.29291	+	0.279195i
$300$	17.3205			0.0577350
$301$	−46.0504			−0.152991
$302$	73.4042			0.243060
$303$	43.7022			0.144232
$304$		−	22.9546i		−	0.0755085i
$305$	229.397			0.752120
$306$		−	72.9678i		−	0.238457i
$307$	96.3675			0.313901			0.156950	−	0.987606i	$-0.449834\pi$
$307$	96.3675			0.313901			0.156950	+	0.987606i	$0.449834\pi$
$308$	−34.6548			−0.112516
$309$		−	177.295i		−	0.573771i
$310$			87.3227i			0.281686i
$311$	423.308			1.36112			0.680560	−	0.732693i	$-0.261737\pi$
$311$	423.308			1.36112			0.680560	+	0.732693i	$0.261737\pi$
$312$	84.2394			0.269998
$313$			58.3745i			0.186500i	0.995643	+	0.0932500i	$0.0297256\pi$
$313$			58.3745i			0.186500i	−0.995643	+	0.0932500i	$0.970274\pi$
$314$		−	145.764i		−	0.464215i
$315$	9.19353			0.0291858
$316$			32.0815i			0.101524i
$317$	60.9213			0.192181			0.0960904	−	0.995373i	$-0.469366\pi$
$317$	60.9213			0.192181			0.0960904	+	0.995373i	$0.469366\pi$
$318$			176.757i			0.555838i
$319$		−	241.131i		−	0.755895i
$320$			17.8885i			0.0559017i
$321$		−	172.671i		−	0.537917i
$322$	−9.40937	−	43.5735i	−0.0292217	−	0.135321i
$323$	−98.6972			−0.305564
$324$	18.0000			0.0555556
$325$	85.9765			0.264543
$326$	80.2579			0.246190
$327$			122.377i			0.374242i
$328$	−115.431			−0.351924
$329$			49.4189i			0.150209i
$330$	−69.2498			−0.209848
$331$	245.050			0.740332			0.370166	−	0.928966i	$-0.379301\pi$
$331$	245.050			0.740332			0.370166	+	0.928966i	$0.379301\pi$
$332$			239.579i			0.721623i
$333$		−	85.7054i		−	0.257373i
$334$	−438.390			−1.31254
$335$	187.220			0.558865
$336$			9.49504i			0.0282590i
$337$		−	570.626i		−	1.69325i	−0.532188	−	0.846626i	$-0.678630\pi$
$337$		−	570.626i		−	1.69325i	0.532188	−	0.846626i	$-0.321370\pi$
$338$	179.150			0.530029
$339$			203.912i			0.601512i
$340$	76.9148			0.226220
$341$		−	349.128i		−	1.02384i
$342$		−	24.3470i		−	0.0711901i
$343$		−	131.734i		−	0.384064i
$344$		−	95.0392i		−	0.276277i
$345$	−18.8025	−	87.0716i	−0.0545000	−	0.252381i
$346$	426.773			1.23345
$347$	72.5341			0.209032			0.104516	−	0.994523i	$-0.466671\pi$
$347$	72.5341			0.209032			0.104516	+	0.994523i	$0.466671\pi$
$348$	−66.0671			−0.189848
$349$	436.542			1.25084			0.625418	−	0.780290i	$-0.284928\pi$
$349$	436.542			1.25084			0.625418	+	0.780290i	$0.284928\pi$
$350$			9.69083i			0.0276881i
$351$	89.3494			0.254557
$352$		−	71.5209i		−	0.203184i
$353$	−33.3967			−0.0946081			−0.0473040	−	0.998881i	$-0.515063\pi$
$353$	−33.3967			−0.0946081			−0.0473040	+	0.998881i	$0.515063\pi$
$354$	−81.0441			−0.228938
$355$		−	119.295i		−	0.336042i
$356$			143.324i			0.402596i
$357$	40.8255			0.114357
$358$	−94.4932			−0.263948
$359$			348.492i			0.970730i	0.874312	+	0.485365i	$0.161313\pi$
$359$			348.492i			0.970730i	−0.874312	+	0.485365i	$0.838687\pi$
$360$			18.9737i			0.0527046i
$361$	328.068			0.908775
$362$			198.324i			0.547855i
$363$	67.2921			0.185378
$364$			47.1320i			0.129483i
$365$		−	13.6192i		−	0.0373128i
$366$			251.292i			0.686589i
$367$		−	42.8645i		−	0.116797i	−0.998293	−	0.0583985i	$-0.981401\pi$
$367$		−	42.8645i		−	0.116797i	0.998293	−	0.0583985i	$-0.0185994\pi$
$368$	89.9272	−	19.4191i	0.244367	−	0.0527694i
$369$	−122.433			−0.331797
$370$	90.3414			0.244166
$371$	−98.8954			−0.266564
$372$	−95.6572			−0.257143
$373$		−	548.173i		−	1.46963i	−0.678266	−	0.734817i	$-0.737268\pi$
$373$		−	548.173i		−	1.46963i	0.678266	−	0.734817i	$-0.262732\pi$
$374$	−307.516			−0.822236
$375$			19.3649i			0.0516398i
$376$	−101.991			−0.271253
$377$	−327.947			−0.869887
$378$			10.0710i			0.0266429i
$379$			575.949i			1.51966i	0.650125	+	0.759828i	$0.274717\pi$
$379$			575.949i			1.51966i	−0.650125	+	0.759828i	$0.725283\pi$
$380$	25.6640			0.0675369
$381$	364.632			0.957038
$382$			488.199i			1.27801i
$383$			485.343i			1.26721i	0.773655	+	0.633607i	$0.218426\pi$
$383$			485.343i			1.26721i	−0.773655	+	0.633607i	$0.781574\pi$
$384$	−19.5959			−0.0510310
$385$		−	38.7453i		−	0.100637i
$386$	263.407			0.682402
$387$		−	100.804i		−	0.260476i
$388$			94.8977i			0.244582i
$389$		−	34.6469i		−	0.0890666i	−0.999008	−	0.0445333i	$-0.985820\pi$
$389$		−	34.6469i		−	0.0890666i	0.999008	−	0.0445333i	$-0.0141801\pi$
$390$			94.1825i			0.241494i
$391$	−83.4958	−	386.657i	−0.213544	−	0.988893i
$392$	133.280			0.340001
$393$	−310.194			−0.789299
$394$	−395.164			−1.00295
$395$	−35.8682			−0.0908055
$396$		−	75.8593i		−	0.191564i
$397$	−576.948			−1.45327			−0.726635	−	0.687024i	$-0.758917\pi$
$397$	−576.948			−1.45327			−0.726635	+	0.687024i	$0.758917\pi$
$398$			64.6760i			0.162502i
$399$	13.6222			0.0341408
$400$	−20.0000			−0.0500000
$401$			627.475i			1.56477i	0.622792	+	0.782387i	$0.285998\pi$
$401$			627.475i			1.56477i	−0.622792	+	0.782387i	$0.714002\pi$
$402$			205.089i			0.510171i
$403$	−474.828			−1.17823
$404$	−50.4630			−0.124908
$405$			20.1246i			0.0496904i
$406$		−	36.9646i		−	0.0910457i
$407$	−361.197			−0.887463
$408$			84.2560i			0.206510i
$409$	22.7131			0.0555333			0.0277666	−	0.999614i	$-0.491160\pi$
$409$	22.7131			0.0555333			0.0277666	+	0.999614i	$0.491160\pi$
$410$		−	129.056i		−	0.314770i
$411$		−	118.864i		−	0.289207i
$412$			204.723i			0.496900i
$413$		−	45.3442i		−	0.109792i
$414$	95.3822	−	20.5971i	0.230392	−	0.0497514i
$415$	−267.857			−0.645439
$416$	−97.2713			−0.233825
$417$	−61.6868			−0.147930
$418$	−102.608			−0.245474
$419$		−	212.213i		−	0.506474i	−0.967404	−	0.253237i	$-0.918505\pi$
$419$		−	212.213i		−	0.506474i	0.967404	−	0.253237i	$-0.0814952\pi$
$420$	−10.6158			−0.0252757
$421$		−	592.278i		−	1.40684i	−0.710776	−	0.703419i	$-0.751656\pi$
$421$		−	592.278i		−	1.40684i	0.710776	−	0.703419i	$-0.248344\pi$
$422$	434.977			1.03075
$423$	−108.178			−0.255739
$424$		−	204.101i		−	0.481370i
$425$			85.9934i			0.202337i
$426$	130.681			0.306763
$427$	−140.598			−0.329269
$428$			199.384i			0.465850i
$429$		−	376.555i		−	0.877750i
$430$	106.257			0.247109
$431$			333.504i			0.773791i	0.922124	+	0.386895i	$0.126453\pi$
$431$			333.504i			0.773791i	−0.922124	+	0.386895i	$0.873547\pi$
$432$	−20.7846			−0.0481125
$433$		−	61.7200i		−	0.142540i	−0.997457	−	0.0712702i	$-0.977295\pi$
$433$		−	61.7200i		−	0.142540i	0.997457	−	0.0712702i	$-0.0227053\pi$
$434$		−	53.5203i		−	0.123319i
$435$		−	73.8652i		−	0.169805i
$436$		−	141.309i		−	0.324103i
$437$	−27.8599	−	129.015i	−0.0637526	−	0.295229i
$438$	14.9191			0.0340618
$439$	858.490			1.95556			0.977779	−	0.209640i	$-0.0672293\pi$
$439$	858.490			1.95556			0.977779	+	0.209640i	$0.0672293\pi$
$440$	79.9628			0.181734
$441$	141.365			0.320556
$442$			418.234i			0.946232i
$443$	707.452			1.59696			0.798479	−	0.602023i	$-0.205638\pi$
$443$	707.452			1.59696			0.798479	+	0.602023i	$0.205638\pi$
$444$			98.9640i			0.222892i
$445$	−160.241			−0.360092
$446$	−504.180			−1.13045
$447$		−	292.929i		−	0.655321i
$448$		−	10.9639i		−	0.0244730i
$449$	−27.9221			−0.0621873			−0.0310936	−	0.999516i	$-0.509899\pi$
$449$	−27.9221			−0.0621873			−0.0310936	+	0.999516i	$0.509899\pi$
$450$	−21.2132			−0.0471405
$451$			515.983i			1.14409i
$452$		−	235.458i		−	0.520924i
$453$	−89.9014			−0.198458
$454$			236.449i			0.520813i
$455$	−52.6952			−0.115814
$456$			28.1135i			0.0616525i
$457$		−	391.345i		−	0.856335i	−0.903699	−	0.428167i	$-0.859159\pi$
$457$		−	391.345i		−	0.856335i	0.903699	−	0.428167i	$-0.140841\pi$
$458$		−	397.748i		−	0.868444i
$459$			89.3670i			0.194699i
$460$	21.7112	+	100.542i	0.0471983	+	0.218569i
$461$	358.584			0.777841			0.388920	−	0.921271i	$-0.372848\pi$
$461$	358.584			0.777841			0.388920	+	0.921271i	$0.372848\pi$
$462$	42.4433			0.0918687
$463$	−49.7419			−0.107434			−0.0537169	−	0.998556i	$-0.517107\pi$
$463$	−49.7419			−0.107434			−0.0537169	+	0.998556i	$0.517107\pi$
$464$	76.2877			0.164413
$465$		−	106.948i		−	0.229996i
$466$	85.1519			0.182729
$467$			7.85966i			0.0168301i	0.999965	+	0.00841505i	$0.00267863\pi$
$467$			7.85966i			0.0168301i	−0.999965	+	0.00841505i	$0.997321\pi$
$468$	−103.172			−0.220452
$469$	−114.747			−0.244664
$470$		−	114.029i		−	0.242616i
$471$			178.523i			0.379030i
$472$	93.5816			0.198266
$473$	−424.830			−0.898161
$474$		−	39.2916i		−	0.0828937i
$475$			28.6932i			0.0604068i
$476$	−47.1413			−0.0990363
$477$		−	216.482i		−	0.453840i
$478$	63.4197			0.132677
$479$		−	567.601i		−	1.18497i	−0.805581	−	0.592486i	$-0.798147\pi$
$479$		−	567.601i		−	1.18497i	0.805581	−	0.592486i	$-0.201853\pi$
$480$		−	21.9089i		−	0.0456435i
$481$			491.243i			1.02129i
$482$		−	0.402020i		−	0.000834067i
$483$	11.5241	+	53.3664i	0.0238594	+	0.110489i
$484$	−77.7022			−0.160542
$485$	−106.099			−0.218761
$486$	−22.0454			−0.0453609
$487$	374.905			0.769825			0.384913	−	0.922953i	$-0.374232\pi$
$487$	374.905			0.769825			0.384913	+	0.922953i	$0.374232\pi$
$488$		−	290.166i		−	0.594603i
$489$	−98.2955			−0.201013
$490$			149.012i			0.304106i
$491$	−412.426			−0.839971			−0.419985	−	0.907531i	$-0.637965\pi$
$491$	−412.426			−0.839971			−0.419985	+	0.907531i	$0.637965\pi$
$492$	141.374			0.287345
$493$		−	328.012i		−	0.665339i
$494$			139.551i			0.282493i
$495$	84.8133			0.171340
$496$	110.455			0.222692
$497$			73.1161i			0.147115i
$498$		−	293.423i		−	0.589202i
$499$	327.529			0.656370			0.328185	−	0.944613i	$-0.393563\pi$
$499$	327.529			0.656370			0.328185	+	0.944613i	$0.393563\pi$
$500$		−	22.3607i		−	0.0447214i
$501$	536.915			1.07169
$502$		−	310.611i		−	0.618748i
$503$		−	636.770i		−	1.26595i	−0.774174	−	0.632973i	$-0.781835\pi$
$503$		−	636.770i		−	1.26595i	0.774174	−	0.632973i	$-0.218165\pi$
$504$		−	11.6290i		−	0.0230734i
$505$		−	56.4193i		−	0.111721i
$506$	−86.8045	−	401.979i	−0.171550	−	0.794426i
$507$	−219.413			−0.432767
$508$	−421.040			−0.828819
$509$	125.717			0.246988			0.123494	−	0.992345i	$-0.460590\pi$
$509$	125.717			0.246988			0.123494	+	0.992345i	$0.460590\pi$
$510$	−94.2011			−0.184708
$511$			8.34722i			0.0163351i
$512$	22.6274			0.0441942
$513$			29.8189i			0.0581265i
$514$	−118.495			−0.230535
$515$	−228.887			−0.444441
$516$			116.399i			0.225579i
$517$			455.905i			0.881828i
$518$	−55.3704			−0.106893
$519$	−522.688			−1.00711
$520$		−	108.753i		−	0.209140i
$521$		−	668.477i		−	1.28306i	−0.767096	−	0.641532i	$-0.778299\pi$
$521$		−	668.477i		−	1.28306i	0.767096	−	0.641532i	$-0.221701\pi$
$522$	80.9153			0.155010
$523$			421.745i			0.806396i	0.915113	+	0.403198i	$0.132101\pi$
$523$			421.745i			0.806396i	−0.915113	+	0.403198i	$0.867899\pi$
$524$	358.182			0.683553
$525$		−	11.8688i		−	0.0226072i
$526$			73.6837i			0.140083i
$527$		−	474.922i		−	0.901180i
$528$			87.5948i			0.165899i
$529$	481.862	−	218.288i	0.910893	−	0.412643i
$530$	228.192			0.430550
$531$	99.2583			0.186927
$532$	−15.7295			−0.0295668
$533$	701.757			1.31662
$534$		−	175.535i		−	0.328718i
$535$	−222.918			−0.416669
$536$		−	236.816i		−	0.441821i
$537$	115.730			0.215512
$538$	−449.445			−0.835400
$539$		−	595.771i		−	1.10533i
$540$		−	23.2379i		−	0.0430331i
$541$	769.598			1.42255			0.711273	−	0.702916i	$-0.248119\pi$
$541$	769.598			1.42255			0.711273	+	0.702916i	$0.248119\pi$
$542$	−353.054			−0.651392
$543$		−	242.896i		−	0.447322i
$544$		−	97.2904i		−	0.178843i
$545$	157.988			0.289886
$546$		−	57.7246i		−	0.105723i
$547$	406.464			0.743079			0.371539	−	0.928417i	$-0.378830\pi$
$547$	406.464			0.743079			0.371539	+	0.928417i	$0.378830\pi$
$548$			137.252i			0.250460i
$549$		−	307.768i		−	0.560597i
$550$			89.4011i			0.162547i
$551$		−	109.447i		−	0.198634i
$552$	−110.138	+	23.7835i	−0.199525	+	0.0430860i
$553$	21.9837			0.0397535
$554$	−50.4672			−0.0910960
$555$	−110.645			−0.199361
$556$	71.2297			0.128111
$557$		−	753.052i		−	1.35198i	−0.736912	−	0.675989i	$-0.763717\pi$
$557$		−	753.052i		−	1.35198i	0.736912	−	0.675989i	$-0.236283\pi$
$558$	117.156			0.209956
$559$			577.786i			1.03361i
$560$	12.2580			0.0218894
$561$	376.629			0.671353
$562$		−	335.399i		−	0.596796i
$563$		−	79.5409i		−	0.141280i	−0.997502	−	0.0706402i	$-0.977496\pi$
$563$		−	79.5409i		−	0.141280i	0.997502	−	0.0706402i	$-0.0225042\pi$
$564$	124.913			0.221477
$565$	263.250			0.465929
$566$		−	401.905i		−	0.710079i
$567$		−	12.3344i		−	0.0217538i
$568$	−150.897			−0.265665
$569$		−	210.524i		−	0.369989i	−0.982740	−	0.184994i	$-0.940773\pi$
$569$		−	210.524i		−	0.369989i	0.982740	−	0.184994i	$-0.0592267\pi$
$570$	−31.4319			−0.0551436
$571$		−	243.033i		−	0.425626i	−0.977093	−	0.212813i	$-0.931737\pi$
$571$		−	243.033i		−	0.425626i	0.977093	−	0.212813i	$-0.0682626\pi$
$572$			434.808i			0.760154i
$573$		−	597.920i		−	1.04349i
$574$			79.0986i			0.137802i
$575$	−112.409	+	24.2739i	−0.195494	+	0.0422155i
$576$	24.0000			0.0416667
$577$	−343.119			−0.594661			−0.297330	−	0.954775i	$-0.596096\pi$
$577$	−343.119			−0.594661			−0.297330	+	0.954775i	$0.596096\pi$
$578$	−9.60903			−0.0166246
$579$	−322.606			−0.557179
$580$			85.2922i			0.147056i
$581$	164.170			0.282565
$582$		−	116.226i		−	0.199700i
$583$	−912.342			−1.56491
$584$	−17.2270			−0.0294984
$585$		−	115.350i		−	0.197179i
$586$			780.216i			1.33143i
$587$	−664.216			−1.13154			−0.565772	−	0.824562i	$-0.691422\pi$
$587$	−664.216			−1.13154			−0.565772	+	0.824562i	$0.691422\pi$
$588$	−163.235			−0.277610
$589$		−	158.466i		−	0.269043i
$590$			104.627i			0.177335i
$591$	483.975			0.818908
$592$		−	114.274i		−	0.193030i
$593$	614.904			1.03694			0.518469	−	0.855097i	$-0.326502\pi$
$593$	614.904			1.03694			0.518469	+	0.855097i	$0.326502\pi$
$594$			92.9083i			0.156411i
$595$		−	52.7055i		−	0.0885807i
$596$			338.245i			0.567525i
$597$		−	79.2116i		−	0.132683i
$598$	−546.708	+	118.058i	−0.914228	+	0.197421i
$599$	606.007			1.01170			0.505849	−	0.862622i	$-0.331179\pi$
$599$	606.007			1.01170			0.505849	+	0.862622i	$0.331179\pi$
$600$	24.4949			0.0408248
$601$	−281.807			−0.468896			−0.234448	−	0.972129i	$-0.575328\pi$
$601$	−281.807			−0.468896			−0.234448	+	0.972129i	$0.575328\pi$
$602$	−65.1252			−0.108181
$603$		−	251.182i		−	0.416553i
$604$	103.809			0.171870
$605$		−	86.8737i		−	0.143593i
$606$	61.8042			0.101987
$607$	2.23403			0.00368044			0.00184022	−	0.999998i	$-0.499414\pi$
$607$	2.23403			0.00368044			0.00184022	+	0.999998i	$0.499414\pi$
$608$		−	32.4627i		−	0.0533926i
$609$			45.2722i			0.0743385i
$610$	324.416			0.531829
$611$	620.049			1.01481
$612$		−	103.192i		−	0.168615i
$613$			24.5222i			0.0400035i	0.999800	+	0.0200018i	$0.00636719\pi$
$613$			24.5222i			0.0400035i	−0.999800	+	0.0200018i	$0.993633\pi$
$614$	136.284			0.221961
$615$			158.060i			0.257009i
$616$	−49.0093			−0.0795606
$617$		−	1084.85i		−	1.75827i	−0.476570	−	0.879137i	$-0.658120\pi$
$617$		−	1084.85i		−	1.75827i	0.476570	−	0.879137i	$-0.341880\pi$
$618$		−	250.733i		−	0.405717i
$619$			817.729i			1.32105i	0.750804	+	0.660525i	$0.229666\pi$
$619$			817.729i			1.32105i	−0.750804	+	0.660525i	$0.770334\pi$
$620$			123.493i			0.199182i
$621$	−116.819	+	25.2262i	−0.188114	+	0.0406219i
$622$	598.648			0.962457
$623$	98.2121			0.157644
$624$	119.132			0.190917
$625$	25.0000			0.0400000
$626$			82.5540i			0.131875i
$627$	125.669			0.200429
$628$		−	206.141i		−	0.328250i
$629$	−491.340			−0.781144
$630$	13.0016			0.0206375
$631$			393.978i			0.624371i	0.950021	+	0.312185i	$0.101061\pi$
$631$			393.978i			0.624371i	−0.950021	+	0.312185i	$0.898939\pi$
$632$			45.3701i			0.0717881i
$633$	−532.736			−0.841604
$634$	86.1558			0.135892
$635$		−	470.737i		−	0.741319i
$636$			249.971i			0.393037i
$637$	−810.272			−1.27201
$638$		−	341.010i		−	0.534499i
$639$	−160.051			−0.250471
$640$			25.2982i			0.0395285i
$641$			803.224i			1.25308i	0.779389	+	0.626540i	$0.215529\pi$
$641$			803.224i			1.25308i	−0.779389	+	0.626540i	$0.784471\pi$
$642$		−	244.194i		−	0.380365i
$643$			590.715i			0.918686i	0.888259	+	0.459343i	$0.151915\pi$
$643$			590.715i			0.918686i	−0.888259	+	0.459343i	$0.848085\pi$
$644$	−13.3069	−	61.6222i	−0.0206628	−	0.0956866i
$645$	−130.138			−0.201764
$646$	−139.579			−0.216066
$647$	328.095			0.507102			0.253551	−	0.967322i	$-0.418401\pi$
$647$	328.095			0.507102			0.253551	+	0.967322i	$0.418401\pi$
$648$	25.4558			0.0392837
$649$		−	418.315i		−	0.644553i
$650$	121.589			0.187060
$651$			65.5487i			0.100689i
$652$	113.502			0.174083
$653$	291.306			0.446104			0.223052	−	0.974806i	$-0.428398\pi$
$653$	291.306			0.446104			0.223052	+	0.974806i	$0.428398\pi$
$654$			173.067i			0.264629i
$655$			400.459i			0.611388i
$656$	−163.244			−0.248848
$657$	−18.2720			−0.0278113
$658$			69.8888i			0.106214i
$659$		−	500.076i		−	0.758841i	−0.925224	−	0.379420i	$-0.876123\pi$
$659$		−	500.076i		−	0.758841i	0.925224	−	0.379420i	$-0.123877\pi$
$660$	−97.9340			−0.148385
$661$		−	803.159i		−	1.21507i	−0.794294	−	0.607533i	$-0.792159\pi$
$661$		−	803.159i		−	1.21507i	0.794294	−	0.607533i	$-0.207841\pi$
$662$	346.553			0.523494
$663$		−	512.230i		−	0.772595i
$664$			338.815i			0.510264i
$665$		−	17.5862i		−	0.0264453i
$666$		−	121.206i		−	0.181990i
$667$	428.771	−	92.5900i	0.642835	−	0.138816i
$668$	−619.976			−0.928108
$669$	617.492			0.923007
$670$	264.769			0.395177
$671$	−1297.06			−1.93303
$672$			13.4280i			0.0199822i
$673$	−908.196			−1.34947			−0.674737	−	0.738058i	$-0.735743\pi$
$673$	−908.196			−1.34947			−0.674737	+	0.738058i	$0.735743\pi$
$674$		−	806.987i		−	1.19731i
$675$	25.9808			0.0384900
$676$	253.356			0.374787
$677$			968.574i			1.43069i	0.698774	+	0.715343i	$0.253729\pi$
$677$			968.574i			1.43069i	−0.698774	+	0.715343i	$0.746271\pi$
$678$			288.376i			0.425333i
$679$	65.0282			0.0957706
$680$	108.774			0.159962
$681$		−	289.590i		−	0.425242i
$682$		−	493.742i		−	0.723962i
$683$	455.566			0.667008			0.333504	−	0.942749i	$-0.391769\pi$
$683$	455.566			0.667008			0.333504	+	0.942749i	$0.391769\pi$
$684$		−	34.4319i		−	0.0503390i
$685$	−153.453			−0.224019
$686$		−	186.300i		−	0.271574i
$687$			487.139i			0.709082i
$688$		−	134.406i		−	0.195357i
$689$			1240.82i			1.80090i
$690$	−26.5907	−	123.138i	−0.0385373	−	0.178461i
$691$	47.1105			0.0681772			0.0340886	−	0.999419i	$-0.489147\pi$
$691$	47.1105			0.0681772			0.0340886	+	0.999419i	$0.489147\pi$
$692$	603.548			0.872179
$693$	−51.9823			−0.0750105
$694$	102.579			0.147808
$695$			79.6373i			0.114586i
$696$	−93.4330			−0.134243
$697$			701.896i			1.00702i
$698$	617.364			0.884475
$699$	−104.289			−0.149198
$700$			13.7049i			0.0195784i
$701$		−	364.053i		−	0.519334i	−0.965698	−	0.259667i	$-0.916387\pi$
$701$		−	364.053i		−	0.519334i	0.965698	−	0.259667i	$-0.0836127\pi$
$702$	126.359			0.179999
$703$	−163.944			−0.233207
$704$		−	101.146i		−	0.143673i
$705$			139.657i			0.198095i
$706$	−47.2300			−0.0668980
$707$			34.5795i			0.0489102i
$708$	−114.614			−0.161884
$709$		−	853.680i		−	1.20406i	−0.798472	−	0.602031i	$-0.794358\pi$
$709$		−	853.680i		−	1.20406i	0.798472	−	0.602031i	$-0.205642\pi$
$710$		−	168.709i		−	0.237618i
$711$			48.1222i			0.0676824i
$712$			202.691i			0.284678i
$713$	620.809	−	134.059i	0.870700	−	0.188021i
$714$	57.7360			0.0808628
$715$	−486.130			−0.679902
$716$	−133.634			−0.186639
$717$	−77.6730			−0.108330
$718$			492.842i			0.686410i
$719$	825.637			1.14831			0.574157	−	0.818745i	$-0.305330\pi$
$719$	825.637			1.14831			0.574157	+	0.818745i	$0.305330\pi$
$720$			26.8328i			0.0372678i
$721$	140.285			0.194571
$722$	463.958			0.642601
$723$			0.492372i			0.000681013i
$724$			280.472i			0.387392i
$725$	−95.3596			−0.131531
$726$	95.1654			0.131082
$727$			1157.96i			1.59279i	0.604778	+	0.796394i	$0.293262\pi$
$727$			1157.96i			1.59279i	−0.604778	+	0.796394i	$0.706738\pi$
$728$			66.6547i			0.0915586i
$729$	27.0000			0.0370370
$730$		−	19.2604i		−	0.0263841i
$731$	−577.900			−0.790561
$732$			355.380i			0.485492i
$733$			937.923i			1.27957i	0.768555	+	0.639784i	$0.220976\pi$
$733$			937.923i			1.27957i	−0.768555	+	0.639784i	$0.779024\pi$
$734$		−	60.6195i		−	0.0825879i
$735$		−	182.502i		−	0.248302i
$736$	127.176	−	27.4628i	0.172794	−	0.0373136i
$737$	−1058.58			−1.43634
$738$	−173.146			−0.234616
$739$	276.870			0.374655			0.187327	−	0.982298i	$-0.440017\pi$
$739$	276.870			0.374655			0.187327	+	0.982298i	$0.440017\pi$
$740$	127.762			0.172651
$741$		−	170.915i		−	0.230654i
$742$	−139.859			−0.188489
$743$		−	953.464i		−	1.28326i	−0.767013	−	0.641631i	$-0.778258\pi$
$743$		−	953.464i		−	1.28326i	0.767013	−	0.641631i	$-0.221742\pi$
$744$	−135.280			−0.181828
$745$	−378.169			−0.507610
$746$		−	775.234i		−	1.03919i
$747$			359.368i			0.481082i
$748$	−434.894			−0.581408
$749$	136.627			0.182412
$750$			27.3861i			0.0365148i
$751$		−	101.682i		−	0.135396i	−0.997706	−	0.0676980i	$-0.978435\pi$
$751$		−	101.682i		−	0.135396i	0.997706	−	0.0676980i	$-0.0215654\pi$
$752$	−144.237			−0.191805
$753$			380.420i			0.505205i
$754$	−463.788			−0.615103
$755$			116.062i			0.153725i
$756$			14.2426i			0.0188394i
$757$			979.734i			1.29423i	0.762391	+	0.647116i	$0.224025\pi$
$757$			979.734i			1.29423i	−0.762391	+	0.647116i	$0.775975\pi$
$758$			814.515i			1.07456i
$759$	106.313	+	492.322i	0.140070	+	0.648646i
$760$	36.2944			0.0477558
$761$	−113.842			−0.149595			−0.0747974	−	0.997199i	$-0.523831\pi$
$761$	−113.842			−0.149595			−0.0747974	+	0.997199i	$0.523831\pi$
$762$	515.667			0.676728
$763$	−96.8312			−0.126908
$764$			690.418i			0.903689i
$765$	115.372			0.150813
$766$			686.378i			0.896055i
$767$	−568.925			−0.741754
$768$	−27.7128			−0.0360844
$769$			683.546i			0.888876i	0.895810	+	0.444438i	$0.146597\pi$
$769$			683.546i			0.888876i	−0.895810	+	0.444438i	$0.853403\pi$
$770$		−	54.7941i		−	0.0711612i
$771$	145.126			0.188231
$772$	372.514			0.482531
$773$			1180.64i			1.52735i	0.645600	+	0.763676i	$0.276607\pi$
$773$			1180.64i			1.52735i	−0.645600	+	0.763676i	$0.723393\pi$
$774$		−	142.559i		−	0.184184i
$775$	−138.069			−0.178154
$776$			134.206i			0.172945i
$777$	67.8146			0.0872775
$778$		−	48.9981i		−	0.0629796i
$779$			234.200i			0.300642i
$780$			133.194i			0.170762i
$781$			674.520i			0.863662i
$782$	−118.081	−	546.816i	−0.150999	−	0.699253i
$783$	−99.1006			−0.126565
$784$	188.487			0.240417
$785$	230.473			0.293596
$786$	−438.681			−0.558119
$787$		−	838.063i		−	1.06488i	−0.846467	−	0.532442i	$-0.821275\pi$
$787$		−	838.063i		−	1.06488i	0.846467	−	0.532442i	$-0.178725\pi$
$788$	−558.846			−0.709196
$789$		−	90.2437i		−	0.114377i
$790$	−50.7253			−0.0642092
$791$	−161.346			−0.203978
$792$		−	107.281i		−	0.135456i
$793$			1764.05i			2.22453i
$794$	−815.927			−1.02762
$795$	−279.477			−0.351543
$796$			91.4656i			0.114907i
$797$		−	49.2134i		−	0.0617483i	−0.999523	−	0.0308741i	$-0.990171\pi$
$797$		−	49.2134i		−	0.0617483i	0.999523	−	0.0308741i	$-0.00982911\pi$
$798$	19.2647			0.0241412
$799$			620.172i			0.776185i
$800$	−28.2843			−0.0353553
$801$			214.986i			0.268397i
$802$			887.383i			1.10646i
$803$			77.0058i			0.0958977i
$804$			290.039i			0.360746i
$805$	68.8957	−	14.8775i	0.0855847	−	0.0184814i
$806$	−671.509			−0.833138
$807$	550.456			0.682101
$808$	−71.3654			−0.0883235
$809$	−639.240			−0.790161			−0.395080	−	0.918647i	$-0.629283\pi$
$809$	−639.240			−0.790161			−0.395080	+	0.918647i	$0.629283\pi$
$810$			28.4605i			0.0351364i
$811$	1108.15			1.36640			0.683201	−	0.730231i	$-0.260587\pi$
$811$	1108.15			1.36640			0.683201	+	0.730231i	$0.260587\pi$
$812$		−	52.2758i		−	0.0643790i
$813$	432.402			0.531859
$814$	−510.810			−0.627531
$815$			126.899i			0.155704i
$816$			119.156i			0.146024i
$817$	−192.827			−0.236018
$818$	32.1212			0.0392680
$819$			70.6980i			0.0863223i
$820$		−	182.512i		−	0.222576i
$821$	−503.173			−0.612879			−0.306439	−	0.951890i	$-0.599138\pi$
$821$	−503.173			−0.612879			−0.306439	+	0.951890i	$0.599138\pi$
$822$		−	168.099i		−	0.204500i
$823$	−1222.86			−1.48586			−0.742931	−	0.669368i	$-0.766565\pi$
$823$	−1222.86			−1.48586			−0.742931	+	0.669368i	$0.766565\pi$
$824$			289.522i			0.351362i
$825$		−	109.494i		−	0.132719i
$826$		−	64.1264i		−	0.0776348i
$827$			875.358i			1.05847i	0.848474	+	0.529237i	$0.177522\pi$
$827$			875.358i			1.05847i	−0.848474	+	0.529237i	$0.822478\pi$
$828$	134.891	−	29.1287i	0.162912	−	0.0351796i
$829$	−176.619			−0.213051			−0.106525	−	0.994310i	$-0.533973\pi$
$829$	−176.619			−0.213051			−0.106525	+	0.994310i	$0.533973\pi$
$830$	−378.807			−0.456394
$831$	61.8094			0.0743796
$832$	−137.562			−0.165339
$833$		−	810.432i		−	0.972908i
$834$	−87.2382			−0.104602
$835$		−	693.155i		−	0.830125i
$836$	−145.110			−0.173577
$837$	−143.486			−0.171429
$838$		−	300.114i		−	0.358131i
$839$			1034.37i			1.23287i	0.787407	+	0.616433i	$0.211423\pi$
$839$			1034.37i			1.23287i	−0.787407	+	0.616433i	$0.788577\pi$
$840$	−15.0130			−0.0178726
$841$	−477.262			−0.567493
$842$		−	837.608i		−	0.994784i
$843$			410.778i			0.487282i
$844$	615.150			0.728851
$845$			283.261i			0.335220i
$846$	−152.986			−0.180835
$847$			53.2451i			0.0628632i
$848$		−	288.642i		−	0.340380i
$849$			492.231i			0.579777i
$850$			121.613i			0.143074i
$851$	−138.694	−	642.270i	−0.162977	−	0.754724i
$852$	184.811			0.216914
$853$	612.490			0.718042			0.359021	−	0.933329i	$-0.383111\pi$
$853$	612.490			0.718042			0.359021	+	0.933329i	$0.383111\pi$
$854$	−198.835			−0.232828
$855$	38.4960			0.0450246
$856$			281.971i			0.329406i
$857$	−1497.60			−1.74749			−0.873743	−	0.486388i	$-0.838314\pi$
$857$	−1497.60			−1.74749			−0.873743	+	0.486388i	$0.838314\pi$
$858$		−	532.529i		−	0.620663i
$859$	464.073			0.540248			0.270124	−	0.962826i	$-0.412935\pi$
$859$	464.073			0.540248			0.270124	+	0.962826i	$0.412935\pi$
$860$	150.270			0.174733
$861$		−	96.8755i		−	0.112515i
$862$			471.646i			0.547153i
$863$	1150.00			1.33256			0.666282	−	0.745700i	$-0.267885\pi$
$863$	1150.00			1.33256			0.666282	+	0.745700i	$0.267885\pi$
$864$	−29.3939			−0.0340207
$865$			674.787i			0.780101i
$866$		−	87.2852i		−	0.100791i
$867$	11.7686			0.0135739
$868$		−	75.6891i		−	0.0871994i
$869$	202.807			0.233379
$870$		−	104.461i		−	0.120070i
$871$			1439.71i			1.65294i
$872$		−	199.841i		−	0.229175i
$873$			142.347i			0.163055i
$874$	−39.3998	−	182.455i	−0.0450799	−	0.208759i
$875$	−15.3226			−0.0175115
$876$	21.0987			0.0240853
$877$	−30.2402			−0.0344814			−0.0172407	−	0.999851i	$-0.505488\pi$
$877$	−30.2402			−0.0344814			−0.0172407	+	0.999851i	$0.505488\pi$
$878$	1214.09			1.38279
$879$		−	955.565i		−	1.08710i
$880$	113.084			0.128505
$881$		−	402.743i		−	0.457143i	−0.973527	−	0.228572i	$-0.926594\pi$
$881$		−	402.743i		−	0.457143i	0.973527	−	0.228572i	$-0.0734055\pi$
$882$	199.921			0.226667
$883$	734.792			0.832154			0.416077	−	0.909329i	$-0.363405\pi$
$883$	734.792			0.832154			0.416077	+	0.909329i	$0.363405\pi$
$884$			591.473i			0.669087i
$885$		−	128.142i		−	0.144793i
$886$	1000.49			1.12922
$887$	87.1915			0.0982993			0.0491496	−	0.998791i	$-0.484349\pi$
$887$	87.1915			0.0982993			0.0491496	+	0.998791i	$0.484349\pi$
$888$			139.956i			0.157608i
$889$			288.516i			0.324540i
$890$	−226.615			−0.254624
$891$		−	113.789i		−	0.127709i
$892$	−713.018			−0.799348
$893$			206.931i			0.231726i
$894$		−	414.264i		−	0.463382i
$895$		−	149.407i		−	0.166935i
$896$		−	15.5053i		−	0.0173051i
$897$	669.578	−	144.591i	0.746464	−	0.161193i
$898$	−39.4878			−0.0439730
$899$	526.650			0.585817
$900$	−30.0000			−0.0333333
$901$	−1241.07			−1.37743
$902$			729.710i			0.808991i
$903$	79.7617			0.0883297
$904$		−	332.988i		−	0.368349i
$905$	−313.577			−0.346494
$906$	−127.140			−0.140331
$907$			1514.40i			1.66968i	0.550490	+	0.834842i	$0.314441\pi$
$907$			1514.40i			1.66968i	−0.550490	+	0.834842i	$0.685559\pi$
$908$			334.390i			0.368271i
$909$	−75.6944			−0.0832722
$910$	−74.5222			−0.0818925
$911$			1384.87i			1.52016i	0.649830	+	0.760080i	$0.274840\pi$
$911$			1384.87i			1.52016i	−0.649830	+	0.760080i	$0.725160\pi$
$912$			39.7585i			0.0435949i
$913$	1514.52			1.65884
$914$		−	553.446i		−	0.605520i
$915$	−397.327			−0.434237
$916$		−	562.500i		−	0.614083i
$917$		−	245.442i		−	0.267658i
$918$			126.384i			0.137673i
$919$			1089.53i			1.18556i	0.805364	+	0.592780i	$0.201970\pi$
$919$			1089.53i			1.18556i	−0.805364	+	0.592780i	$0.798030\pi$
$920$	30.7043	+	142.187i	0.0333743	+	0.154551i
$921$	−166.913			−0.181231
$922$	507.115			0.550016
$923$	917.374			0.993905
$924$	60.0239			0.0649610
$925$			142.842i			0.154424i
$926$	−70.3456			−0.0759672
$927$			307.084i			0.331267i
$928$	107.887			0.116258
$929$	914.187			0.984055			0.492027	−	0.870580i	$-0.336256\pi$
$929$	914.187			0.984055			0.492027	+	0.870580i	$0.336256\pi$
$930$		−	151.247i		−	0.162632i
$931$		−	270.415i		−	0.290457i
$932$	120.423			0.129209
$933$	−733.191			−0.785843
$934$			11.1152i			0.0119007i
$935$		−	486.226i		−	0.520028i
$936$	−145.907			−0.155883
$937$		−	1594.66i		−	1.70188i	−0.525263	−	0.850940i	$-0.676033\pi$
$937$		−	1594.66i		−	1.70188i	0.525263	−	0.850940i	$-0.323967\pi$
$938$	−162.277			−0.173003
$939$		−	101.108i		−	0.107676i
$940$		−	161.262i		−	0.171555i
$941$			77.9212i			0.0828068i	0.999143	+	0.0414034i	$0.0131829\pi$
$941$			77.9212i			0.0828068i	−0.999143	+	0.0414034i	$0.986817\pi$
$942$			252.470i			0.268015i
$943$	−917.505	+	198.129i	−0.972964	+	0.210105i
$944$	132.344			0.140195
$945$	−15.9237			−0.0168504
$946$	−600.801			−0.635096
$947$	102.848			0.108604			0.0543018	−	0.998525i	$-0.482707\pi$
$947$	102.848			0.108604			0.0543018	+	0.998525i	$0.482707\pi$
$948$		−	55.5667i		−	0.0586147i
$949$	104.731			0.110359
$950$			40.5784i			0.0427141i
$951$	−105.519			−0.110956
$952$	−66.6678			−0.0700292
$953$		−	401.943i		−	0.421766i	−0.977511	−	0.210883i	$-0.932366\pi$
$953$		−	401.943i		−	0.421766i	0.977511	−	0.210883i	$-0.0676339\pi$
$954$		−	306.151i		−	0.320913i
$955$	−771.911			−0.808284
$956$	89.6890			0.0938170
$957$			417.650i			0.436416i
$958$		−	802.710i		−	0.837902i
$959$	94.0515			0.0980724
$960$		−	30.9839i		−	0.0322749i
$961$	−198.474			−0.206529
$962$			694.722i			0.722165i
$963$			299.076i			0.310567i
$964$		−	0.568543i		−	0.000589774i
$965$			416.483i			0.431589i
$966$	16.2975	+	75.4715i	0.0168711	+	0.0781278i
$967$	893.378			0.923865			0.461933	−	0.886915i	$-0.347156\pi$
$967$	893.378			0.923865			0.461933	+	0.886915i	$0.347156\pi$
$968$	−109.888			−0.113520
$969$	170.949			0.176418
$970$	−150.047			−0.154687
$971$		−	502.813i		−	0.517830i	−0.965900	−	0.258915i	$-0.916635\pi$
$971$		−	502.813i		−	0.517830i	0.965900	−	0.258915i	$-0.0833649\pi$
$972$	−31.1769			−0.0320750
$973$		−	48.8098i		−	0.0501643i
$974$	530.195			0.544349
$975$	−148.916			−0.152734
$976$		−	410.357i		−	0.420448i
$977$		−	274.182i		−	0.280637i	−0.990106	−	0.140318i	$-0.955187\pi$
$977$		−	274.182i		−	0.280637i	0.990106	−	0.140318i	$-0.0448126\pi$
$978$	−139.011			−0.142138
$979$	906.039			0.925474
$980$			210.735i			0.215036i
$981$		−	211.963i		−	0.216068i
$982$	−583.258			−0.593949
$983$			1541.49i			1.56815i	0.620664	+	0.784076i	$0.286863\pi$
$983$			1541.49i			1.56815i	−0.620664	+	0.784076i	$0.713137\pi$
$984$	199.932			0.203183
$985$		−	624.809i		−	0.634324i
$986$		−	463.879i		−	0.470465i
$987$		−	85.5960i		−	0.0867234i
$988$			197.356i			0.199753i
$989$	−163.128	−	755.420i	−0.164942	−	0.763822i
$990$	119.944			0.121156
$991$	−1777.97			−1.79412			−0.897061	−	0.441907i	$-0.854302\pi$
$991$	−1777.97			−1.79412			−0.897061	+	0.441907i	$0.854302\pi$
$992$	156.208			0.157467
$993$	−424.439			−0.427431
$994$			103.402i			0.104026i
$995$	−102.262			−0.102776
$996$		−	414.963i		−	0.416629i
$997$	1971.04			1.97698			0.988488	−	0.151302i	$-0.0483465\pi$
$997$	1971.04			1.97698			0.988488	+	0.151302i	$0.0483465\pi$
$998$	463.196			0.464124
$999$			148.446i			0.148595i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	690.3.c.a.91.22	yes	32
3.2	odd	2		2070.3.c.b.91.4		32
23.22	odd	2	inner	690.3.c.a.91.19	✓	32
69.68	even	2		2070.3.c.b.91.13		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.3.c.a.91.19	✓	32	23.22	odd	2	inner
690.3.c.a.91.22	yes	32	1.1	even	1	trivial
2070.3.c.b.91.4		32	3.2	odd	2
2070.3.c.b.91.13		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)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} -1.73205 q^{3} +2.00000 q^{4} +2.23607i q^{5} -2.44949 q^{6} -1.37049i q^{7} +2.82843 q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q+1.41421 q^{2} -1.73205 q^{3} +2.00000 q^{4} +2.23607i q^{5} -2.44949 q^{6} -1.37049i q^{7} +2.82843 q^{8} +3.00000 q^{9} +3.16228i q^{10} -12.6432i q^{11} -3.46410 q^{12} -17.1953 q^{13} -1.93817i q^{14} -3.87298i q^{15} +4.00000 q^{16} -17.1987i q^{17} +4.24264 q^{18} -5.73865i q^{19} +4.47214i q^{20} +2.37376i q^{21} -17.8802i q^{22} +(22.4818 - 4.85478i) q^{23} -4.89898 q^{24} -5.00000 q^{25} -24.3178 q^{26} -5.19615 q^{27} -2.74098i q^{28} +19.0719 q^{29} -5.47723i q^{30} +27.6139 q^{31} +5.65685 q^{32} +21.8987i q^{33} -24.3226i q^{34} +3.06451 q^{35} +6.00000 q^{36} -28.5685i q^{37} -8.11568i q^{38} +29.7831 q^{39} +6.32456i q^{40} -40.8110 q^{41} +3.35700i q^{42} -33.6014i q^{43} -25.2864i q^{44} +6.70820i q^{45} +(31.7941 - 6.86570i) q^{46} -36.0593 q^{47} -6.92820 q^{48} +47.1218 q^{49} -7.07107 q^{50} +29.7890i q^{51} -34.3906 q^{52} -72.1605i q^{53} -7.34847 q^{54} +28.2711 q^{55} -3.87633i q^{56} +9.93963i q^{57} +26.9718 q^{58} +33.0861 q^{59} -7.74597i q^{60} -102.589i q^{61} +39.0519 q^{62} -4.11147i q^{63} +8.00000 q^{64} -38.4498i q^{65} +30.9694i q^{66} -83.7272i q^{67} -34.3974i q^{68} +(-38.9396 + 8.40873i) q^{69} +4.33387 q^{70} -53.3503 q^{71} +8.48528 q^{72} -6.09068 q^{73} -40.4019i q^{74} +8.66025 q^{75} -11.4773i q^{76} -17.3274 q^{77} +42.1197 q^{78} +16.0407i q^{79} +8.94427i q^{80} +9.00000 q^{81} -57.7155 q^{82} +119.789i q^{83} +4.74752i q^{84} +38.4574 q^{85} -47.5196i q^{86} -33.0335 q^{87} -35.7604i q^{88} +71.6620i q^{89} +9.48683i q^{90} +23.5660i q^{91} +(44.9636 - 9.70956i) q^{92} -47.8286 q^{93} -50.9955 q^{94} +12.8320 q^{95} -9.79796 q^{96} +47.4489i q^{97} +66.6402 q^{98} -37.9297i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 64 q^{4} + 96 q^{9}+O(q^{10}) \) 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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