LMFDB - Embedded newform 690.3.c.a.91.6

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			91.6
Character	$\chi$	$=$	690.91
Dual form			690.3.c.a.91.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} -1.73205 q^{3} +2.00000 q^{4} +2.23607i q^{5} +2.44949 q^{6} -1.52229i q^{7} -2.82843 q^{8} +3.00000 q^{9} +O(q^{10})$	\(q-1.41421 q^{2} -1.73205 q^{3} +2.00000 q^{4} +2.23607i q^{5} +2.44949 q^{6} -1.52229i q^{7} -2.82843 q^{8} +3.00000 q^{9} -3.16228i q^{10} +2.20036i q^{11} -3.46410 q^{12} -1.82994 q^{13} +2.15284i q^{14} -3.87298i q^{15} +4.00000 q^{16} -23.2496i q^{17} -4.24264 q^{18} +2.85690i q^{19} +4.47214i q^{20} +2.63668i q^{21} -3.11178i q^{22} +(-14.5452 + 17.8168i) q^{23} +4.89898 q^{24} -5.00000 q^{25} +2.58793 q^{26} -5.19615 q^{27} -3.04458i q^{28} -7.23088 q^{29} +5.47723i q^{30} +49.9104 q^{31} -5.65685 q^{32} -3.81114i q^{33} +32.8800i q^{34} +3.40394 q^{35} +6.00000 q^{36} +9.68006i q^{37} -4.04027i q^{38} +3.16955 q^{39} -6.32456i q^{40} -62.1103 q^{41} -3.72883i q^{42} -76.6308i q^{43} +4.40073i q^{44} +6.70820i q^{45} +(20.5700 - 25.1967i) q^{46} +17.1464 q^{47} -6.92820 q^{48} +46.6826 q^{49} +7.07107 q^{50} +40.2696i q^{51} -3.65988 q^{52} +33.9334i q^{53} +7.34847 q^{54} -4.92016 q^{55} +4.30569i q^{56} -4.94830i q^{57} +10.2260 q^{58} +35.2164 q^{59} -7.74597i q^{60} +8.54346i q^{61} -70.5840 q^{62} -4.56687i q^{63} +8.00000 q^{64} -4.09187i q^{65} +5.38977i q^{66} -78.9586i q^{67} -46.4993i q^{68} +(25.1930 - 30.8596i) q^{69} -4.81390 q^{70} +111.407 q^{71} -8.48528 q^{72} +134.217 q^{73} -13.6897i q^{74} +8.66025 q^{75} +5.71380i q^{76} +3.34959 q^{77} -4.48242 q^{78} -72.4039i q^{79} +8.94427i q^{80} +9.00000 q^{81} +87.8373 q^{82} -119.919i q^{83} +5.27337i q^{84} +51.9878 q^{85} +108.372i q^{86} +12.5243 q^{87} -6.22357i q^{88} +73.9324i q^{89} -9.48683i q^{90} +2.78570i q^{91} +(-29.0904 + 35.6335i) q^{92} -86.4474 q^{93} -24.2486 q^{94} -6.38822 q^{95} +9.79796 q^{96} -178.439i q^{97} -66.0192 q^{98} +6.60109i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$		−	1.52229i		−	0.217470i	−0.994071	−	0.108735i	$-0.965320\pi$
$7$		−	1.52229i		−	0.217470i	0.994071	−	0.108735i	$-0.0346800\pi$
$8$	−2.82843			−0.353553
$9$	3.00000			0.333333
$10$		−	3.16228i		−	0.316228i
$11$			2.20036i			0.200033i	0.994986	+	0.100016i	$0.0318896\pi$
$11$			2.20036i			0.200033i	−0.994986	+	0.100016i	$0.968110\pi$
$12$	−3.46410			−0.288675
$13$	−1.82994			−0.140765			−0.0703824	−	0.997520i	$-0.522422\pi$
$13$	−1.82994			−0.140765			−0.0703824	+	0.997520i	$0.522422\pi$
$14$			2.15284i			0.153774i
$15$		−	3.87298i		−	0.258199i
$16$	4.00000			0.250000
$17$		−	23.2496i		−	1.36763i	−0.729657	−	0.683813i	$-0.760320\pi$
$17$		−	23.2496i		−	1.36763i	0.729657	−	0.683813i	$-0.239680\pi$
$18$	−4.24264			−0.235702
$19$			2.85690i			0.150363i	0.997170	+	0.0751816i	$0.0239536\pi$
$19$			2.85690i			0.150363i	−0.997170	+	0.0751816i	$0.976046\pi$
$20$			4.47214i			0.223607i
$21$			2.63668i			0.125556i
$22$		−	3.11178i		−	0.141445i
$23$	−14.5452	+	17.8168i	−0.632400	+	0.774642i
$23$	−14.5452	+	17.8168i	−0.632400	+	0.774642i
$24$	4.89898			0.204124
$25$	−5.00000			−0.200000
$26$	2.58793			0.0995357
$27$	−5.19615			−0.192450
$28$		−	3.04458i		−	0.108735i
$29$	−7.23088			−0.249341			−0.124670	−	0.992198i	$-0.539787\pi$
$29$	−7.23088			−0.249341			−0.124670	+	0.992198i	$0.539787\pi$
$30$			5.47723i			0.182574i
$31$	49.9104			1.61001			0.805007	−	0.593266i	$-0.202162\pi$
$31$	49.9104			1.61001			0.805007	+	0.593266i	$0.202162\pi$
$32$	−5.65685			−0.176777
$33$		−	3.81114i		−	0.115489i
$34$			32.8800i			0.967058i
$35$	3.40394			0.0972555
$36$	6.00000			0.166667
$37$			9.68006i			0.261623i	0.991407	+	0.130812i	$0.0417583\pi$
$37$			9.68006i			0.261623i	−0.991407	+	0.130812i	$0.958242\pi$
$38$		−	4.04027i		−	0.106323i
$39$	3.16955			0.0812706
$40$		−	6.32456i		−	0.158114i
$41$	−62.1103			−1.51489			−0.757443	−	0.652901i	$-0.773552\pi$
$41$	−62.1103			−1.51489			−0.757443	+	0.652901i	$0.773552\pi$
$42$		−	3.72883i		−	0.0887817i
$43$		−	76.6308i		−	1.78211i	−0.453893	−	0.891056i	$-0.649965\pi$
$43$		−	76.6308i		−	1.78211i	0.453893	−	0.891056i	$-0.350035\pi$
$44$			4.40073i			0.100016i
$45$			6.70820i			0.149071i
$46$	20.5700	−	25.1967i	0.447174	−	0.547755i
$47$	17.1464			0.364817			0.182408	−	0.983223i	$-0.441611\pi$
$47$	17.1464			0.364817			0.182408	+	0.983223i	$0.441611\pi$
$48$	−6.92820			−0.144338
$49$	46.6826			0.952707
$50$	7.07107			0.141421
$51$			40.2696i			0.789599i
$52$	−3.65988			−0.0703824
$53$			33.9334i			0.640253i	0.947375	+	0.320127i	$0.103725\pi$
$53$			33.9334i			0.640253i	−0.947375	+	0.320127i	$0.896275\pi$
$54$	7.34847			0.136083
$55$	−4.92016			−0.0894575
$56$			4.30569i			0.0768872i
$57$		−	4.94830i		−	0.0868122i
$58$	10.2260			0.176311
$59$	35.2164			0.596889			0.298444	−	0.954427i	$-0.403532\pi$
$59$	35.2164			0.596889			0.298444	+	0.954427i	$0.403532\pi$
$60$		−	7.74597i		−	0.129099i
$61$			8.54346i			0.140057i	0.997545	+	0.0700284i	$0.0223090\pi$
$61$			8.54346i			0.140057i	−0.997545	+	0.0700284i	$0.977691\pi$
$62$	−70.5840			−1.13845
$63$		−	4.56687i		−	0.0724900i
$64$	8.00000			0.125000
$65$		−	4.09187i		−	0.0629519i
$66$			5.38977i			0.0816631i
$67$		−	78.9586i		−	1.17849i	−0.807956	−	0.589243i	$-0.799426\pi$
$67$		−	78.9586i		−	1.17849i	0.807956	−	0.589243i	$-0.200574\pi$
$68$		−	46.4993i		−	0.683813i
$69$	25.1930	−	30.8596i	0.365116	−	0.447240i
$70$	−4.81390			−0.0687700
$71$	111.407			1.56911			0.784554	−	0.620061i	$-0.212892\pi$
$71$	111.407			1.56911			0.784554	+	0.620061i	$0.212892\pi$
$72$	−8.48528			−0.117851
$73$	134.217			1.83858			0.919291	−	0.393578i	$-0.128763\pi$
$73$	134.217			1.83858			0.919291	+	0.393578i	$0.128763\pi$
$74$		−	13.6897i		−	0.184996i
$75$	8.66025			0.115470
$76$			5.71380i			0.0751816i
$77$	3.34959			0.0435012
$78$	−4.48242			−0.0574670
$79$		−	72.4039i		−	0.916505i	−0.888822	−	0.458252i	$-0.848476\pi$
$79$		−	72.4039i		−	0.916505i	0.888822	−	0.458252i	$-0.151524\pi$
$80$			8.94427i			0.111803i
$81$	9.00000			0.111111
$82$	87.8373			1.07119
$83$		−	119.919i		−	1.44481i	−0.691472	−	0.722403i	$-0.743037\pi$
$83$		−	119.919i		−	1.44481i	0.691472	−	0.722403i	$-0.256963\pi$
$84$			5.27337i			0.0627782i
$85$	51.9878			0.611621
$86$			108.372i			1.26014i
$87$	12.5243			0.143957
$88$		−	6.22357i		−	0.0707223i
$89$			73.9324i			0.830701i	0.909661	+	0.415351i	$0.136341\pi$
$89$			73.9324i			0.830701i	−0.909661	+	0.415351i	$0.863659\pi$
$90$		−	9.48683i		−	0.105409i
$91$			2.78570i			0.0306121i
$92$	−29.0904	+	35.6335i	−0.316200	+	0.387321i
$93$	−86.4474			−0.929542
$94$	−24.2486			−0.257964
$95$	−6.38822			−0.0672445
$96$	9.79796			0.102062
$97$		−	178.439i		−	1.83958i	−0.392408	−	0.919791i	$-0.628358\pi$
$97$		−	178.439i		−	1.83958i	0.392408	−	0.919791i	$-0.371642\pi$
$98$	−66.0192			−0.673665
$99$			6.60109i			0.0666777i
$100$	−10.0000			−0.100000
$101$	147.269			1.45811			0.729053	−	0.684457i	$-0.239961\pi$
$101$	147.269			1.45811			0.729053	+	0.684457i	$0.239961\pi$
$102$		−	56.9498i		−	0.558331i
$103$		−	58.9552i		−	0.572380i	−0.958173	−	0.286190i	$-0.907611\pi$
$103$		−	58.9552i		−	0.572380i	0.958173	−	0.286190i	$-0.0923889\pi$
$104$	5.17586			0.0497679
$105$	−5.89580			−0.0561505
$106$		−	47.9891i		−	0.452727i
$107$		−	111.196i		−	1.03922i	−0.854404	−	0.519609i	$-0.826078\pi$
$107$		−	111.196i		−	1.03922i	0.854404	−	0.519609i	$-0.173922\pi$
$108$	−10.3923			−0.0962250
$109$		−	52.7876i		−	0.484290i	−0.970240	−	0.242145i	$-0.922149\pi$
$109$		−	52.7876i		−	0.484290i	0.970240	−	0.242145i	$-0.0778509\pi$
$110$	6.95816			0.0632560
$111$		−	16.7664i		−	0.151048i
$112$		−	6.08916i		−	0.0543675i
$113$			25.4121i			0.224886i	0.993658	+	0.112443i	$0.0358676\pi$
$113$			25.4121i			0.224886i	−0.993658	+	0.112443i	$0.964132\pi$
$114$			6.99795i			0.0613855i
$115$	−39.8395	−	32.5240i	−0.346431	−	0.282818i
$116$	−14.4618			−0.124670
$117$	−5.48983			−0.0469216
$118$	−49.8035			−0.422064
$119$	−35.3927			−0.297418
$120$			10.9545i			0.0912871i
$121$	116.158			0.959987
$122$		−	12.0823i		−	0.0990350i
$123$	107.578			0.874620
$124$	99.8208			0.805007
$125$		−	11.1803i		−	0.0894427i
$126$			6.45853i			0.0512582i
$127$	100.861			0.794177			0.397089	−	0.917780i	$-0.370021\pi$
$127$	100.861			0.794177			0.397089	+	0.917780i	$0.370021\pi$
$128$	−11.3137			−0.0883883
$129$			132.728i			1.02890i
$130$			5.78678i			0.0445137i
$131$	−28.6539			−0.218732			−0.109366	−	0.994002i	$-0.534882\pi$
$131$	−28.6539			−0.218732			−0.109366	+	0.994002i	$0.534882\pi$
$132$		−	7.62228i		−	0.0577445i
$133$	4.34903			0.0326995
$134$			111.664i			0.833316i
$135$		−	11.6190i		−	0.0860663i
$136$			65.7599i			0.483529i
$137$			159.048i			1.16094i	0.814282	+	0.580469i	$0.197131\pi$
$137$			159.048i			1.16094i	−0.814282	+	0.580469i	$0.802869\pi$
$138$	−35.6283	+	43.6420i	−0.258176	+	0.316246i
$139$	119.112			0.856918			0.428459	−	0.903561i	$-0.359057\pi$
$139$	119.112			0.856918			0.428459	+	0.903561i	$0.359057\pi$
$140$	6.80789			0.0486278
$141$	−29.6984			−0.210627
$142$	−157.553			−1.10953
$143$		−	4.02654i		−	0.0281576i
$144$	12.0000			0.0833333
$145$		−	16.1687i		−	0.111509i
$146$	−189.811			−1.30007
$147$	−80.8567			−0.550046
$148$			19.3601i			0.130812i
$149$		−	245.833i		−	1.64989i	−0.565214	−	0.824945i	$-0.691206\pi$
$149$		−	245.833i		−	1.64989i	0.565214	−	0.824945i	$-0.308794\pi$
$150$	−12.2474			−0.0816497
$151$	209.449			1.38708			0.693541	−	0.720417i	$-0.256050\pi$
$151$	209.449			1.38708			0.693541	+	0.720417i	$0.256050\pi$
$152$		−	8.08053i		−	0.0531614i
$153$		−	69.7489i		−	0.455875i
$154$	−4.73704			−0.0307600
$155$			111.603i			0.720020i
$156$	6.33911			0.0406353
$157$			211.566i			1.34755i	0.738935	+	0.673777i	$0.235329\pi$
$157$			211.566i			1.34755i	−0.738935	+	0.673777i	$0.764671\pi$
$158$			102.395i			0.648067i
$159$		−	58.7744i		−	0.369650i
$160$		−	12.6491i		−	0.0790569i
$161$	27.1223	+	22.1420i	0.168461	+	0.137528i
$162$	−12.7279			−0.0785674
$163$	−14.1494			−0.0868061			−0.0434030	−	0.999058i	$-0.513820\pi$
$163$	−14.1494			−0.0868061			−0.0434030	+	0.999058i	$0.513820\pi$
$164$	−124.221			−0.757443
$165$	8.52197			0.0516483
$166$			169.591i			1.02163i
$167$	52.0666			0.311776			0.155888	−	0.987775i	$-0.450176\pi$
$167$	52.0666			0.311776			0.155888	+	0.987775i	$0.450176\pi$
$168$		−	7.45767i		−	0.0443909i
$169$	−165.651			−0.980185
$170$	−73.5218			−0.432481
$171$			8.57070i			0.0501211i
$172$		−	153.262i		−	0.891056i
$173$	−219.256			−1.26738			−0.633688	−	0.773589i	$-0.718460\pi$
$173$	−219.256			−1.26738			−0.633688	+	0.773589i	$0.718460\pi$
$174$	−17.7120			−0.101793
$175$			7.61145i			0.0434940i
$176$			8.80145i			0.0500082i
$177$	−60.9966			−0.344614
$178$		−	104.556i		−	0.587395i
$179$	−84.3515			−0.471237			−0.235619	−	0.971846i	$-0.575712\pi$
$179$	−84.3515			−0.471237			−0.235619	+	0.971846i	$0.575712\pi$
$180$			13.4164i			0.0745356i
$181$			272.635i			1.50627i	0.657867	+	0.753134i	$0.271459\pi$
$181$			272.635i			1.50627i	−0.657867	+	0.753134i	$0.728541\pi$
$182$		−	3.93958i		−	0.0216460i
$183$		−	14.7977i		−	0.0808618i
$184$	41.1400	−	50.3934i	0.223587	−	0.273877i
$185$	−21.6453			−0.117001
$186$	122.255			0.657285
$187$	51.1577			0.273570
$188$	34.2928			0.182408
$189$			7.91005i			0.0418521i
$190$	9.03431			0.0475490
$191$			235.645i			1.23375i	0.787063	+	0.616873i	$0.211601\pi$
$191$			235.645i			1.23375i	−0.787063	+	0.616873i	$0.788399\pi$
$192$	−13.8564			−0.0721688
$193$	−340.126			−1.76231			−0.881155	−	0.472828i	$-0.843233\pi$
$193$	−340.126			−1.76231			−0.881155	+	0.472828i	$0.843233\pi$
$194$			252.352i			1.30078i
$195$			7.08734i			0.0363453i
$196$	93.3653			0.476353
$197$	−221.673			−1.12524			−0.562621	−	0.826715i	$-0.690207\pi$
$197$	−221.673			−1.12524			−0.562621	+	0.826715i	$0.690207\pi$
$198$		−	9.33535i		−	0.0471482i
$199$		−	198.065i		−	0.995303i	−0.867377	−	0.497651i	$-0.834196\pi$
$199$		−	198.065i		−	0.995303i	0.867377	−	0.497651i	$-0.165804\pi$
$200$	14.1421			0.0707107
$201$			136.760i			0.680400i
$202$	−208.269			−1.03104
$203$			11.0075i			0.0542241i
$204$			80.5391i			0.394800i
$205$		−	138.883i		−	0.677478i
$206$			83.3752i			0.404734i
$207$	−43.6356	+	53.4503i	−0.210800	+	0.258214i
$208$	−7.31977			−0.0351912
$209$	−6.28622			−0.0300776
$210$	8.33792			0.0397044
$211$	−105.845			−0.501636			−0.250818	−	0.968034i	$-0.580700\pi$
$211$	−105.845			−0.501636			−0.250818	+	0.968034i	$0.580700\pi$
$212$			67.8668i			0.320127i
$213$	−192.962			−0.905925
$214$			157.255i			0.734838i
$215$	171.352			0.796985
$216$	14.6969			0.0680414
$217$		−	75.9781i		−	0.350130i
$218$			74.6529i			0.342445i
$219$	−232.470			−1.06151
$220$	−9.84032			−0.0447287
$221$			42.5455i			0.192514i
$222$			23.7112i			0.106807i
$223$	230.988			1.03582			0.517910	−	0.855435i	$-0.326710\pi$
$223$	230.988			1.03582			0.517910	+	0.855435i	$0.326710\pi$
$224$			8.61137i			0.0384436i
$225$	−15.0000			−0.0666667
$226$		−	35.9382i		−	0.159019i
$227$		−	146.750i		−	0.646475i	−0.946318	−	0.323237i	$-0.895229\pi$
$227$		−	146.750i		−	0.646475i	0.946318	−	0.323237i	$-0.104771\pi$
$228$		−	9.89659i		−	0.0434061i
$229$		−	203.857i		−	0.890203i	−0.895480	−	0.445102i	$-0.853168\pi$
$229$		−	203.857i		−	0.890203i	0.895480	−	0.445102i	$-0.146832\pi$
$230$	56.3416	+	45.9959i	0.244963	+	0.199982i
$231$	−5.80166			−0.0251154
$232$	20.4520			0.0881553
$233$	296.931			1.27438			0.637191	−	0.770706i	$-0.280096\pi$
$233$	296.931			1.27438			0.637191	+	0.770706i	$0.280096\pi$
$234$	7.76379			0.0331786
$235$			38.3405i			0.163151i
$236$	70.4328			0.298444
$237$			125.407i			0.529144i
$238$	50.0528			0.210306
$239$	−266.458			−1.11488			−0.557442	−	0.830216i	$-0.688217\pi$
$239$	−266.458			−1.11488			−0.557442	+	0.830216i	$0.688217\pi$
$240$		−	15.4919i		−	0.0645497i
$241$			327.615i			1.35940i	0.733490	+	0.679700i	$0.237890\pi$
$241$			327.615i			1.35940i	−0.733490	+	0.679700i	$0.762110\pi$
$242$	−164.273			−0.678813
$243$	−15.5885			−0.0641500
$244$			17.0869i			0.0700284i
$245$			104.386i			0.426063i
$246$	−152.139			−0.618450
$247$		−	5.22796i		−	0.0211658i
$248$	−141.168			−0.569226
$249$			207.706i			0.834160i
$250$			15.8114i			0.0632456i
$251$			5.89337i			0.0234796i	0.999931	+	0.0117398i	$0.00373698\pi$
$251$			5.89337i			0.0234796i	−0.999931	+	0.0117398i	$0.996263\pi$
$252$		−	9.13374i		−	0.0362450i
$253$	−39.2034	−	32.0047i	−0.154954	−	0.126501i
$254$	−142.638			−0.561568
$255$	−90.0455			−0.353120
$256$	16.0000			0.0625000
$257$	−45.2730			−0.176160			−0.0880798	−	0.996113i	$-0.528073\pi$
$257$	−45.2730			−0.176160			−0.0880798	+	0.996113i	$0.528073\pi$
$258$		−	187.706i		−	0.727544i
$259$	14.7359			0.0568952
$260$		−	8.18375i		−	0.0314760i
$261$	−21.6926			−0.0831136
$262$	40.5228			0.154667
$263$		−	211.508i		−	0.804215i	−0.915593	−	0.402107i	$-0.868278\pi$
$263$		−	211.508i		−	0.804215i	0.915593	−	0.402107i	$-0.131722\pi$
$264$			10.7795i			0.0408316i
$265$	−75.8774			−0.286330
$266$	−6.15046			−0.0231220
$267$		−	128.055i		−	0.479606i
$268$		−	157.917i		−	0.589243i
$269$	−133.758			−0.497242			−0.248621	−	0.968601i	$-0.579977\pi$
$269$	−133.758			−0.497242			−0.248621	+	0.968601i	$0.579977\pi$
$270$			16.4317i			0.0608581i
$271$	186.815			0.689353			0.344676	−	0.938722i	$-0.387989\pi$
$271$	186.815			0.689353			0.344676	+	0.938722i	$0.387989\pi$
$272$		−	92.9986i		−	0.341907i
$273$		−	4.82498i		−	0.0176739i
$274$		−	224.929i		−	0.820907i
$275$		−	11.0018i		−	0.0400066i
$276$	50.3860	−	61.7191i	0.182558	−	0.223620i
$277$	−128.144			−0.462615			−0.231308	−	0.972881i	$-0.574300\pi$
$277$	−128.144			−0.462615			−0.231308	+	0.972881i	$0.574300\pi$
$278$	−168.449			−0.605933
$279$	149.731			0.536671
$280$	−9.62781			−0.0343850
$281$		−	183.268i		−	0.652200i	−0.945335	−	0.326100i	$-0.894265\pi$
$281$		−	183.268i		−	0.652200i	0.945335	−	0.326100i	$-0.105735\pi$
$282$	41.9999			0.148936
$283$			436.731i			1.54322i	0.636097	+	0.771609i	$0.280548\pi$
$283$			436.731i			1.54322i	−0.636097	+	0.771609i	$0.719452\pi$
$284$	222.813			0.784554
$285$	11.0647			0.0388236
$286$			5.69438i			0.0199104i
$287$			94.5499i			0.329442i
$288$	−16.9706			−0.0589256
$289$	−251.546			−0.870401
$290$			22.8661i			0.0788485i
$291$			309.066i			1.06208i
$292$	268.433			0.919291
$293$		−	34.4017i		−	0.117412i	−0.998275	−	0.0587060i	$-0.981303\pi$
$293$		−	34.4017i		−	0.117412i	0.998275	−	0.0587060i	$-0.0186974\pi$
$294$	114.349			0.388941
$295$			78.7463i			0.266937i
$296$		−	27.3793i		−	0.0924978i
$297$		−	11.4334i		−	0.0384964i
$298$			347.661i			1.16665i
$299$	26.6169	−	32.6037i	0.0890196	−	0.109042i
$300$	17.3205			0.0577350
$301$	−116.654			−0.387556
$302$	−296.206			−0.980815
$303$	−255.077			−0.841838
$304$			11.4276i			0.0375908i
$305$	−19.1038			−0.0626353
$306$			98.6399i			0.322353i
$307$	−263.872			−0.859518			−0.429759	−	0.902944i	$-0.641401\pi$
$307$	−263.872			−0.859518			−0.429759	+	0.902944i	$0.641401\pi$
$308$	6.69918			0.0217506
$309$			102.113i			0.330464i
$310$		−	157.831i		−	0.509131i
$311$	12.9685			0.0416992			0.0208496	−	0.999783i	$-0.493363\pi$
$311$	12.9685			0.0416992			0.0208496	+	0.999783i	$0.493363\pi$
$312$	−8.96485			−0.0287335
$313$		−	71.7752i		−	0.229314i	−0.993405	−	0.114657i	$-0.963423\pi$
$313$		−	71.7752i		−	0.229314i	0.993405	−	0.114657i	$-0.0365769\pi$
$314$		−	299.199i		−	0.952864i
$315$	10.2118			0.0324185
$316$		−	144.808i		−	0.458252i
$317$	193.888			0.611633			0.305816	−	0.952090i	$-0.401071\pi$
$317$	193.888			0.611633			0.305816	+	0.952090i	$0.401071\pi$
$318$			83.1196i			0.261382i
$319$		−	15.9106i		−	0.0498764i
$320$			17.8885i			0.0559017i
$321$			192.598i			0.599993i
$322$	−38.3567	−	31.3135i	−0.119120	−	0.0972469i
$323$	66.4219			0.205641
$324$	18.0000			0.0555556
$325$	9.14971			0.0281530
$326$	20.0103			0.0613812
$327$			91.4308i			0.279605i
$328$	175.675			0.535593
$329$		−	26.1018i		−	0.0793367i
$330$	−12.0519			−0.0365209
$331$	393.325			1.18829			0.594147	−	0.804356i	$-0.297490\pi$
$331$	393.325			1.18829			0.594147	+	0.804356i	$0.297490\pi$
$332$		−	239.838i		−	0.722403i
$333$			29.0402i			0.0872078i
$334$	−73.6333			−0.220459
$335$	176.557			0.527035
$336$			10.5467i			0.0313891i
$337$		−	39.3542i		−	0.116778i	−0.998294	−	0.0583891i	$-0.981404\pi$
$337$		−	39.3542i		−	0.116778i	0.998294	−	0.0583891i	$-0.0185964\pi$
$338$	234.266			0.693096
$339$		−	44.0151i		−	0.129838i
$340$	103.976			0.305811
$341$			109.821i			0.322056i
$342$		−	12.1208i		−	0.0354409i
$343$		−	145.657i		−	0.424655i
$344$			216.745i			0.630072i
$345$	69.0041	+	56.3333i	0.200012	+	0.163285i
$346$	310.075			0.896170
$347$	309.497			0.891922			0.445961	−	0.895052i	$-0.352862\pi$
$347$	309.497			0.891922			0.445961	+	0.895052i	$0.352862\pi$
$348$	25.0485			0.0719785
$349$	−328.191			−0.940375			−0.470188	−	0.882567i	$-0.655814\pi$
$349$	−328.191			−0.940375			−0.470188	+	0.882567i	$0.655814\pi$
$350$		−	10.7642i		−	0.0307549i
$351$	9.50866			0.0270902
$352$		−	12.4471i		−	0.0353612i
$353$	−111.495			−0.315851			−0.157925	−	0.987451i	$-0.550481\pi$
$353$	−111.495			−0.315851			−0.157925	+	0.987451i	$0.550481\pi$
$354$	86.2623			0.243679
$355$			249.113i			0.701726i
$356$			147.865i			0.415351i
$357$	61.3020			0.171714
$358$	119.291			0.333215
$359$		−	516.911i		−	1.43986i	−0.694045	−	0.719932i	$-0.744173\pi$
$359$		−	516.911i		−	1.43986i	0.694045	−	0.719932i	$-0.255827\pi$
$360$		−	18.9737i		−	0.0527046i
$361$	352.838			0.977391
$362$		−	385.564i		−	1.06509i
$363$	−201.192			−0.554249
$364$			5.57140i			0.0153061i
$365$			300.117i			0.822239i
$366$			20.9271i			0.0571779i
$367$		−	32.4661i		−	0.0884635i	−0.999021	−	0.0442318i	$-0.985916\pi$
$367$		−	32.4661i		−	0.0884635i	0.999021	−	0.0442318i	$-0.0140840\pi$
$368$	−58.1808	+	71.2671i	−0.158100	+	0.193661i
$369$	−186.331			−0.504962
$370$	30.6110			0.0827325
$371$	51.6565			0.139236
$372$	−172.895			−0.464771
$373$			153.444i			0.411377i	0.978617	+	0.205689i	$0.0659434\pi$
$373$			153.444i			0.411377i	−0.978617	+	0.205689i	$0.934057\pi$
$374$	−72.3478			−0.193443
$375$			19.3649i			0.0516398i
$376$	−48.4973			−0.128982
$377$	13.2321			0.0350984
$378$		−	11.1865i		−	0.0295939i
$379$		−	597.537i		−	1.57662i	−0.615281	−	0.788308i	$-0.710958\pi$
$379$		−	597.537i		−	1.57662i	0.615281	−	0.788308i	$-0.289042\pi$
$380$	−12.7764			−0.0336222
$381$	−174.696			−0.458518
$382$		−	333.253i		−	0.872390i
$383$			309.730i			0.808694i	0.914606	+	0.404347i	$0.132501\pi$
$383$			309.730i			0.808694i	−0.914606	+	0.404347i	$0.867499\pi$
$384$	19.5959			0.0510310
$385$			7.48991i			0.0194543i
$386$	481.010			1.24614
$387$		−	229.892i		−	0.594037i
$388$		−	356.879i		−	0.919791i
$389$		−	295.042i		−	0.758461i	−0.925302	−	0.379231i	$-0.876189\pi$
$389$		−	295.042i		−	0.758461i	0.925302	−	0.379231i	$-0.123811\pi$
$390$		−	10.0230i		−	0.0257000i
$391$	414.234	+	338.171i	1.05942	+	0.864886i
$392$	−132.038			−0.336833
$393$	49.6301			0.126285
$394$	313.492			0.795666
$395$	161.900			0.409873
$396$			13.2022i			0.0333388i
$397$	45.4461			0.114474			0.0572368	−	0.998361i	$-0.481771\pi$
$397$	45.4461			0.114474			0.0572368	+	0.998361i	$0.481771\pi$
$398$			280.107i			0.703785i
$399$	−7.53274			−0.0188791
$400$	−20.0000			−0.0500000
$401$			51.4990i			0.128427i	0.997936	+	0.0642133i	$0.0204538\pi$
$401$			51.4990i			0.128427i	−0.997936	+	0.0642133i	$0.979546\pi$
$402$		−	193.408i		−	0.481115i
$403$	−91.3332			−0.226633
$404$	294.537			0.729053
$405$			20.1246i			0.0496904i
$406$		−	15.5669i		−	0.0383422i
$407$	−21.2996			−0.0523333
$408$		−	113.900i		−	0.279166i
$409$	−507.527			−1.24090			−0.620449	−	0.784247i	$-0.713050\pi$
$409$	−507.527			−1.24090			−0.620449	+	0.784247i	$0.713050\pi$
$410$			196.410i			0.479049i
$411$		−	275.480i		−	0.670268i
$412$		−	117.910i		−	0.286190i
$413$		−	53.6096i		−	0.129805i
$414$	61.7100	−	75.5902i	0.149058	−	0.182585i
$415$	268.147			0.646137
$416$	10.3517			0.0248839
$417$	−206.307			−0.494742
$418$	8.89005			0.0212681
$419$		−	471.677i		−	1.12572i	−0.826552	−	0.562861i	$-0.809701\pi$
$419$		−	471.677i		−	1.12572i	0.826552	−	0.562861i	$-0.190299\pi$
$420$	−11.7916			−0.0280753
$421$			760.247i			1.80581i	0.429838	+	0.902906i	$0.358570\pi$
$421$			760.247i			1.80581i	−0.429838	+	0.902906i	$0.641430\pi$
$422$	149.688			0.354711
$423$	51.4391			0.121606
$424$		−	95.9782i		−	0.226364i
$425$			116.248i			0.273525i
$426$	272.889			0.640585
$427$	13.0056			0.0304581
$428$		−	222.393i		−	0.519609i
$429$			6.97417i			0.0162568i
$430$	−242.328			−0.563553
$431$		−	601.309i		−	1.39515i	−0.716512	−	0.697575i	$-0.754262\pi$
$431$		−	601.309i		−	1.39515i	0.716512	−	0.697575i	$-0.245738\pi$
$432$	−20.7846			−0.0481125
$433$		−	466.438i		−	1.07722i	−0.842554	−	0.538612i	$-0.818949\pi$
$433$		−	466.438i		−	1.07722i	0.842554	−	0.538612i	$-0.181051\pi$
$434$			107.449i			0.247579i
$435$			28.0051i			0.0643795i
$436$		−	105.575i		−	0.242145i
$437$	−50.9007	−	41.5542i	−0.116478	−	0.0950896i
$438$	328.762			0.750598
$439$	−294.320			−0.670432			−0.335216	−	0.942141i	$-0.608809\pi$
$439$	−294.320			−0.670432			−0.335216	+	0.942141i	$0.608809\pi$
$440$	13.9163			0.0316280
$441$	140.048			0.317569
$442$		−	60.1684i		−	0.136128i
$443$	642.170			1.44959			0.724797	−	0.688962i	$-0.241934\pi$
$443$	642.170			1.44959			0.724797	+	0.688962i	$0.241934\pi$
$444$		−	33.5327i		−	0.0755241i
$445$	−165.318			−0.371501
$446$	−326.666			−0.732435
$447$			425.796i			0.952564i
$448$		−	12.1783i		−	0.0271837i
$449$	663.924			1.47867			0.739337	−	0.673336i	$-0.235139\pi$
$449$	663.924			1.47867			0.739337	+	0.673336i	$0.235139\pi$
$450$	21.2132			0.0471405
$451$		−	136.665i		−	0.303027i
$452$			50.8243i			0.112443i
$453$	−362.777			−0.800832
$454$			207.536i			0.457127i
$455$	−6.22902			−0.0136902
$456$			13.9959i			0.0306928i
$457$			22.1190i			0.0484005i	0.999707	+	0.0242002i	$0.00770393\pi$
$457$			22.1190i			0.0484005i	−0.999707	+	0.0242002i	$0.992296\pi$
$458$			288.297i			0.629469i
$459$			120.809i			0.263200i
$460$	−79.6790	−	65.0481i	−0.173215	−	0.141409i
$461$	−332.992			−0.722325			−0.361162	−	0.932503i	$-0.617620\pi$
$461$	−332.992			−0.722325			−0.361162	+	0.932503i	$0.617620\pi$
$462$	8.20479			0.0177593
$463$	−58.4021			−0.126139			−0.0630693	−	0.998009i	$-0.520089\pi$
$463$	−58.4021			−0.126139			−0.0630693	+	0.998009i	$0.520089\pi$
$464$	−28.9235			−0.0623352
$465$		−	193.302i		−	0.415704i
$466$	−419.924			−0.901124
$467$			609.288i			1.30469i	0.757924	+	0.652343i	$0.226214\pi$
$467$			609.288i			1.30469i	−0.757924	+	0.652343i	$0.773786\pi$
$468$	−10.9797			−0.0234608
$469$	−120.198			−0.256285
$470$		−	54.2216i		−	0.115365i
$471$		−	366.443i		−	0.778010i
$472$	−99.6071			−0.211032
$473$	168.616			0.356481
$474$		−	177.353i		−	0.374161i
$475$		−	14.2845i		−	0.0300726i
$476$	−70.7854			−0.148709
$477$			101.800i			0.213418i
$478$	376.828			0.788343
$479$			200.294i			0.418150i	0.977900	+	0.209075i	$0.0670453\pi$
$479$			200.294i			0.418150i	−0.977900	+	0.209075i	$0.932955\pi$
$480$			21.9089i			0.0456435i
$481$		−	17.7140i		−	0.0368273i
$482$		−	463.318i		−	0.961241i
$483$	−46.9772	−	38.3511i	−0.0972613	−	0.0794018i
$484$	232.317			0.479993
$485$	399.003			0.822686
$486$	22.0454			0.0453609
$487$	136.133			0.279534			0.139767	−	0.990184i	$-0.455365\pi$
$487$	136.133			0.279534			0.139767	+	0.990184i	$0.455365\pi$
$488$		−	24.1646i		−	0.0495175i
$489$	24.5075			0.0501175
$490$		−	147.623i		−	0.301272i
$491$	−217.319			−0.442604			−0.221302	−	0.975205i	$-0.571031\pi$
$491$	−217.319			−0.442604			−0.221302	+	0.975205i	$0.571031\pi$
$492$	215.156			0.437310
$493$			168.115i			0.341005i
$494$			7.39345i			0.0149665i
$495$	−14.7605			−0.0298192
$496$	199.642			0.402503
$497$		−	169.593i		−	0.341234i
$498$		−	293.740i		−	0.589840i
$499$	−620.672			−1.24383			−0.621916	−	0.783084i	$-0.713646\pi$
$499$	−620.672			−1.24383			−0.621916	+	0.783084i	$0.713646\pi$
$500$		−	22.3607i		−	0.0447214i
$501$	−90.1821			−0.180004
$502$		−	8.33449i		−	0.0166026i
$503$		−	165.943i		−	0.329906i	−0.986301	−	0.164953i	$-0.947253\pi$
$503$		−	165.943i		−	0.329906i	0.986301	−	0.164953i	$-0.0527473\pi$
$504$			12.9171i			0.0256291i
$505$			329.303i			0.652085i
$506$	55.4419	+	45.2615i	0.109569	+	0.0894496i
$507$	286.916			0.565910
$508$	201.721			0.397089
$509$	−376.623			−0.739928			−0.369964	−	0.929046i	$-0.620630\pi$
$509$	−376.623			−0.739928			−0.369964	+	0.929046i	$0.620630\pi$
$510$	127.344			0.249693
$511$		−	204.316i		−	0.399836i
$512$	−22.6274			−0.0441942
$513$		−	14.8449i		−	0.0289374i
$514$	64.0257			0.124564
$515$	131.828			0.255976
$516$			265.457i			0.514452i
$517$			37.7283i			0.0729754i
$518$	−20.8397			−0.0402310
$519$	379.762			0.731720
$520$			11.5736i			0.0222569i
$521$		−	675.726i		−	1.29698i	−0.761223	−	0.648490i	$-0.775401\pi$
$521$		−	675.726i		−	1.29698i	0.761223	−	0.648490i	$-0.224599\pi$
$522$	30.6780			0.0587702
$523$		−	53.9295i		−	0.103116i	−0.998670	−	0.0515578i	$-0.983581\pi$
$523$		−	53.9295i		−	0.103116i	0.998670	−	0.0515578i	$-0.0164186\pi$
$524$	−57.3079			−0.109366
$525$		−	13.1834i		−	0.0251113i
$526$			299.118i			0.568666i
$527$		−	1160.40i		−	2.20190i
$528$		−	15.2446i		−	0.0288723i
$529$	−105.875	−	518.297i	−0.200141	−	0.979767i
$530$	107.307			0.202466
$531$	105.649			0.198963
$532$	8.69806			0.0163497
$533$	113.658			0.213243
$534$			181.097i			0.339132i
$535$	248.643			0.464752
$536$			223.329i			0.416658i
$537$	146.101			0.272069
$538$	189.163			0.351604
$539$			102.719i			0.190573i
$540$		−	23.2379i		−	0.0430331i
$541$	32.6654			0.0603797			0.0301899	−	0.999544i	$-0.490389\pi$
$541$	32.6654			0.0603797			0.0301899	+	0.999544i	$0.490389\pi$
$542$	−264.196			−0.487446
$543$		−	472.217i		−	0.869645i
$544$			131.520i			0.241764i
$545$	118.037			0.216581
$546$			6.82355i			0.0124973i
$547$	125.227			0.228934			0.114467	−	0.993427i	$-0.463484\pi$
$547$	125.227			0.228934			0.114467	+	0.993427i	$0.463484\pi$
$548$			318.097i			0.580469i
$549$			25.6304i			0.0466856i
$550$			15.5589i			0.0282889i
$551$		−	20.6579i		−	0.0374917i
$552$	−71.2566	+	87.2840i	−0.129088	+	0.158123i
$553$	−110.220			−0.199312
$554$	181.224			0.327118
$555$	37.4907			0.0675508
$556$	238.223			0.428459
$557$		−	18.9083i		−	0.0339466i	−0.999856	−	0.0169733i	$-0.994597\pi$
$557$		−	18.9083i		−	0.0339466i	0.999856	−	0.0169733i	$-0.00540303\pi$
$558$	−211.752			−0.379484
$559$			140.230i			0.250859i
$560$	13.6158			0.0243139
$561$	−88.6077			−0.157946
$562$			259.181i			0.461175i
$563$			663.769i			1.17899i	0.807774	+	0.589493i	$0.200672\pi$
$563$			663.769i			1.17899i	−0.807774	+	0.589493i	$0.799328\pi$
$564$	−59.3968			−0.105313
$565$	−56.8233			−0.100572
$566$		−	617.630i		−	1.09122i
$567$		−	13.7006i		−	0.0241633i
$568$	−315.105			−0.554763
$569$			491.805i			0.864332i	0.901794	+	0.432166i	$0.142251\pi$
$569$			491.805i			0.864332i	−0.901794	+	0.432166i	$0.857749\pi$
$570$	−15.6479			−0.0274524
$571$			880.062i			1.54126i	0.637280	+	0.770632i	$0.280060\pi$
$571$			880.062i			1.54126i	−0.637280	+	0.770632i	$0.719940\pi$
$572$		−	8.05307i		−	0.0140788i
$573$		−	408.150i		−	0.712303i
$574$		−	133.714i		−	0.232951i
$575$	72.7260	−	89.0839i	0.126480	−	0.154928i
$576$	24.0000			0.0416667
$577$	−4.02697			−0.00697914			−0.00348957	−	0.999994i	$-0.501111\pi$
$577$	−4.02697			−0.00697914			−0.00348957	+	0.999994i	$0.501111\pi$
$578$	355.740			0.615467
$579$	589.115			1.01747
$580$		−	32.3375i		−	0.0557543i
$581$	−182.551			−0.314202
$582$		−	437.086i		−	0.751006i
$583$	−74.6658			−0.128072
$584$	−379.622			−0.650037
$585$		−	12.2756i		−	0.0209840i
$586$			48.6513i			0.0830228i
$587$	−313.115			−0.533415			−0.266708	−	0.963778i	$-0.585936\pi$
$587$	−313.115			−0.533415			−0.266708	+	0.963778i	$0.585936\pi$
$588$	−161.713			−0.275023
$589$			142.589i			0.242087i
$590$		−	111.364i		−	0.188753i
$591$	383.948			0.649659
$592$			38.7202i			0.0654058i
$593$	643.656			1.08542			0.542712	−	0.839919i	$-0.317398\pi$
$593$	643.656			1.08542			0.542712	+	0.839919i	$0.317398\pi$
$594$			16.1693i			0.0272210i
$595$		−	79.1405i		−	0.133009i
$596$		−	491.667i		−	0.824945i
$597$			343.059i			0.574638i
$598$	−37.6419	+	46.1085i	−0.0629464	+	0.0771046i
$599$	482.724			0.805884			0.402942	−	0.915226i	$-0.367988\pi$
$599$	482.724			0.805884			0.402942	+	0.915226i	$0.367988\pi$
$600$	−24.4949			−0.0408248
$601$	−510.476			−0.849377			−0.424689	−	0.905340i	$-0.639616\pi$
$601$	−510.476			−0.849377			−0.424689	+	0.905340i	$0.639616\pi$
$602$	164.974			0.274043
$603$		−	236.876i		−	0.392829i
$604$	418.899			0.693541
$605$			259.738i			0.429319i
$606$	360.733			0.595269
$607$	−719.184			−1.18482			−0.592409	−	0.805638i	$-0.701823\pi$
$607$	−719.184			−1.18482			−0.592409	+	0.805638i	$0.701823\pi$
$608$		−	16.1611i		−	0.0265807i
$609$		−	19.0655i		−	0.0313063i
$610$	27.0168			0.0442898
$611$	−31.3769			−0.0513533
$612$		−	139.498i		−	0.227938i
$613$			672.054i			1.09634i	0.836368	+	0.548168i	$0.184675\pi$
$613$			672.054i			1.09634i	−0.836368	+	0.548168i	$0.815325\pi$
$614$	373.171			0.607771
$615$			240.552i			0.391142i
$616$	−9.47407			−0.0153800
$617$			295.731i			0.479305i	0.970859	+	0.239653i	$0.0770336\pi$
$617$			295.731i			0.479305i	−0.970859	+	0.239653i	$0.922966\pi$
$618$		−	144.410i		−	0.233673i
$619$		−	823.442i		−	1.33028i	−0.746720	−	0.665139i	$-0.768372\pi$
$619$		−	823.442i		−	1.33028i	0.746720	−	0.665139i	$-0.231628\pi$
$620$			223.206i			0.360010i
$621$	75.5790	−	92.5787i	0.121705	−	0.149080i
$622$	−18.3402			−0.0294858
$623$	112.547			0.180653
$624$	12.6782			0.0203176
$625$	25.0000			0.0400000
$626$			101.505i			0.162149i
$627$	10.8880			0.0173653
$628$			423.132i			0.673777i
$629$	225.058			0.357803
$630$	−14.4417			−0.0229233
$631$			471.460i			0.747163i	0.927597	+	0.373582i	$0.121870\pi$
$631$			471.460i			0.747163i	−0.927597	+	0.373582i	$0.878130\pi$
$632$			204.789i			0.324033i
$633$	183.329			0.289620
$634$	−274.199			−0.432490
$635$			225.531i			0.355167i
$636$		−	117.549i		−	0.184825i
$637$	−85.4265			−0.134108
$638$			22.5009i			0.0352679i
$639$	334.220			0.523036
$640$		−	25.2982i		−	0.0395285i
$641$		−	1114.03i		−	1.73796i	−0.494850	−	0.868979i	$-0.664777\pi$
$641$		−	1114.03i		−	1.73796i	0.494850	−	0.868979i	$-0.335223\pi$
$642$		−	272.374i		−	0.424259i
$643$		−	182.160i		−	0.283298i	−0.989917	−	0.141649i	$-0.954760\pi$
$643$		−	182.160i		−	0.283298i	0.989917	−	0.141649i	$-0.0452404\pi$
$644$	54.2446	+	44.2840i	0.0842307	+	0.0687640i
$645$	−296.790			−0.460139
$646$	−93.9348			−0.145410
$647$	231.496			0.357799			0.178899	−	0.983867i	$-0.442746\pi$
$647$	231.496			0.357799			0.178899	+	0.983867i	$0.442746\pi$
$648$	−25.4558			−0.0392837
$649$			77.4889i			0.119397i
$650$	−12.9396			−0.0199071
$651$			131.598i			0.202147i
$652$	−28.2988			−0.0434030
$653$	961.708			1.47275			0.736377	−	0.676572i	$-0.236535\pi$
$653$	961.708			1.47275			0.736377	+	0.676572i	$0.236535\pi$
$654$		−	129.303i		−	0.197710i
$655$		−	64.0722i		−	0.0978201i
$656$	−248.441			−0.378721
$657$	402.650			0.612861
$658$			36.9135i			0.0560995i
$659$			482.196i			0.731709i	0.930672	+	0.365854i	$0.119223\pi$
$659$			482.196i			0.731709i	−0.930672	+	0.365854i	$0.880777\pi$
$660$	17.0439			0.0258241
$661$		−	514.781i		−	0.778792i	−0.921070	−	0.389396i	$-0.872684\pi$
$661$		−	514.781i		−	0.778792i	0.921070	−	0.389396i	$-0.127316\pi$
$662$	−556.246			−0.840251
$663$		−	73.6910i		−	0.111148i
$664$			339.182i			0.510816i
$665$			9.72473i			0.0146236i
$666$		−	41.0690i		−	0.0616652i
$667$	105.175	−	128.831i	0.157683	−	0.193150i
$668$	104.133			0.155888
$669$	−400.083			−0.598031
$670$	−249.689			−0.372670
$671$	−18.7987			−0.0280160
$672$		−	14.9153i		−	0.0221954i
$673$	−1100.79			−1.63565			−0.817824	−	0.575468i	$-0.804820\pi$
$673$	−1100.79			−1.63565			−0.817824	+	0.575468i	$0.804820\pi$
$674$			55.6553i			0.0825746i
$675$	25.9808			0.0384900
$676$	−331.303			−0.490093
$677$		−	155.504i		−	0.229695i	−0.993383	−	0.114848i	$-0.963362\pi$
$677$		−	155.504i		−	0.229695i	0.993383	−	0.114848i	$-0.0366380\pi$
$678$			62.2468i			0.0918094i
$679$	−271.637			−0.400054
$680$	−147.044			−0.216241
$681$			254.178i			0.373242i
$682$		−	155.310i		−	0.227728i
$683$	−1077.79			−1.57803			−0.789014	−	0.614376i	$-0.789408\pi$
$683$	−1077.79			−1.57803			−0.789014	+	0.614376i	$0.789408\pi$
$684$			17.1414i			0.0250605i
$685$	−355.643			−0.519187
$686$			205.990i			0.300276i
$687$			353.090i			0.513959i
$688$		−	306.523i		−	0.445528i
$689$		−	62.0962i		−	0.0901251i
$690$	−97.5865	−	79.6673i	−0.141430	−	0.115460i
$691$	−884.748			−1.28039			−0.640194	−	0.768214i	$-0.721146\pi$
$691$	−884.748			−1.28039			−0.640194	+	0.768214i	$0.721146\pi$
$692$	−438.512			−0.633688
$693$	10.0488			0.0145004
$694$	−437.695			−0.630684
$695$			266.342i			0.383225i
$696$	−35.4239			−0.0508965
$697$			1444.04i			2.07180i
$698$	464.132			0.664946
$699$	−514.299			−0.735765
$700$			15.2229i			0.0217470i
$701$		−	666.703i		−	0.951074i	−0.879696	−	0.475537i	$-0.842254\pi$
$701$		−	666.703i		−	0.951074i	0.879696	−	0.475537i	$-0.157746\pi$
$702$	−13.4473			−0.0191557
$703$	−27.6550			−0.0393385
$704$			17.6029i			0.0250041i
$705$		−	66.4076i		−	0.0941952i
$706$	157.678			0.223340
$707$		−	224.186i		−	0.317094i
$708$	−121.993			−0.172307
$709$			967.418i			1.36448i	0.731127	+	0.682241i	$0.238995\pi$
$709$			967.418i			1.36448i	−0.731127	+	0.682241i	$0.761005\pi$
$710$		−	352.299i		−	0.496195i
$711$		−	217.212i		−	0.305502i
$712$		−	209.112i		−	0.293697i
$713$	−725.957	+	889.242i	−1.01817	+	1.24718i
$714$	−86.6941			−0.121420
$715$	9.00361			0.0125925
$716$	−168.703			−0.235619
$717$	461.518			0.643679
$718$			731.023i			1.01814i
$719$	−216.266			−0.300787			−0.150393	−	0.988626i	$-0.548054\pi$
$719$	−216.266			−0.300787			−0.150393	+	0.988626i	$0.548054\pi$
$720$			26.8328i			0.0372678i
$721$	−89.7469			−0.124476
$722$	−498.988			−0.691120
$723$		−	567.447i		−	0.784850i
$724$			545.269i			0.753134i
$725$	36.1544			0.0498681
$726$	284.529			0.391913
$727$		−	113.587i		−	0.156240i	−0.996944	−	0.0781200i	$-0.975108\pi$
$727$		−	113.587i		−	0.156240i	0.996944	−	0.0781200i	$-0.0248917\pi$
$728$		−	7.87916i		−	0.0108230i
$729$	27.0000			0.0370370
$730$		−	424.430i		−	0.581411i
$731$	−1781.64			−2.43726
$732$		−	29.5954i		−	0.0404309i
$733$		−	104.468i		−	0.142522i	−0.997458	−	0.0712608i	$-0.977298\pi$
$733$		−	104.468i		−	0.142522i	0.997458	−	0.0712608i	$-0.0227023\pi$
$734$			45.9140i			0.0625531i
$735$		−	180.801i		−	0.245988i
$736$	82.2800	−	100.787i	0.111794	−	0.136939i
$737$	173.738			0.235736
$738$	263.512			0.357062
$739$	828.795			1.12151			0.560755	−	0.827982i	$-0.310511\pi$
$739$	828.795			1.12151			0.560755	+	0.827982i	$0.310511\pi$
$740$	−43.2905			−0.0585007
$741$			9.05510i			0.0122201i
$742$	−73.0533			−0.0984546
$743$			1067.78i			1.43712i	0.695465	+	0.718560i	$0.255198\pi$
$743$			1067.78i			1.43712i	−0.695465	+	0.718560i	$0.744802\pi$
$744$	244.510			0.328643
$745$	549.700			0.737853
$746$		−	217.002i		−	0.290888i
$747$		−	359.757i		−	0.481602i
$748$	102.315			0.136785
$749$	−169.273			−0.225999
$750$		−	27.3861i		−	0.0365148i
$751$		−	116.437i		−	0.155043i	−0.996991	−	0.0775213i	$-0.975299\pi$
$751$		−	116.437i		−	0.155043i	0.996991	−	0.0775213i	$-0.0247006\pi$
$752$	68.5855			0.0912042
$753$		−	10.2076i		−	0.0135559i
$754$	−18.7130			−0.0248183
$755$			468.343i			0.620322i
$756$			15.8201i			0.0209261i
$757$			893.537i			1.18037i	0.807269	+	0.590183i	$0.200944\pi$
$757$			893.537i			1.18037i	−0.807269	+	0.590183i	$0.799056\pi$
$758$			845.045i			1.11484i
$759$	67.9022	+	55.4338i	0.0894627	+	0.0730353i
$760$	18.0686			0.0237745
$761$	−80.0115			−0.105140			−0.0525700	−	0.998617i	$-0.516741\pi$
$761$	−80.0115			−0.105140			−0.0525700	+	0.998617i	$0.516741\pi$
$762$	247.057			0.324221
$763$	−80.3580			−0.105318
$764$			471.291i			0.616873i
$765$	155.963			0.203874
$766$		−	438.024i		−	0.571833i
$767$	−64.4440			−0.0840209
$768$	−27.7128			−0.0360844
$769$			890.267i			1.15769i	0.815436	+	0.578847i	$0.196497\pi$
$769$			890.267i			1.15769i	−0.815436	+	0.578847i	$0.803503\pi$
$770$		−	10.5923i		−	0.0137563i
$771$	78.4151			0.101706
$772$	−680.251			−0.881155
$773$		−	619.222i		−	0.801063i	−0.916283	−	0.400531i	$-0.868826\pi$
$773$		−	619.222i		−	0.801063i	0.916283	−	0.400531i	$-0.131174\pi$
$774$			325.117i			0.420048i
$775$	−249.552			−0.322003
$776$			504.703i			0.650391i
$777$	−25.5233			−0.0328485
$778$			417.252i			0.536313i
$779$		−	177.443i		−	0.227783i
$780$			14.1747i			0.0181727i
$781$			245.135i			0.313873i
$782$	−585.815	−	478.245i	−0.749124	−	0.611567i
$783$	37.5728			0.0479856
$784$	186.731			0.238177
$785$	−473.076			−0.602644
$786$	−70.1875			−0.0892971
$787$		−	375.043i		−	0.476548i	−0.971198	−	0.238274i	$-0.923418\pi$
$787$		−	375.043i		−	0.476548i	0.971198	−	0.238274i	$-0.0765816\pi$
$788$	−443.345			−0.562621
$789$			366.343i			0.464314i
$790$	−228.961			−0.289824
$791$	38.6846			0.0489060
$792$		−	18.6707i		−	0.0235741i
$793$		−	15.6340i		−	0.0197151i
$794$	−64.2704			−0.0809451
$795$	131.424			0.165313
$796$		−	396.131i		−	0.497651i
$797$			493.978i			0.619797i	0.950770	+	0.309898i	$0.100295\pi$
$797$			493.978i			0.619797i	−0.950770	+	0.309898i	$0.899705\pi$
$798$	10.6529			0.0133495
$799$		−	398.647i		−	0.498933i
$800$	28.2843			0.0353553
$801$			221.797i			0.276900i
$802$		−	72.8306i		−	0.0908113i
$803$			295.325i			0.367777i
$804$			273.521i			0.340200i
$805$	−49.5110	+	60.6473i	−0.0615044	+	0.0753382i
$806$	129.165			0.160254
$807$	231.676			0.287083
$808$	−416.539			−0.515518
$809$	−132.182			−0.163389			−0.0816946	−	0.996657i	$-0.526033\pi$
$809$	−132.182			−0.163389			−0.0816946	+	0.996657i	$0.526033\pi$
$810$		−	28.4605i		−	0.0351364i
$811$	−158.090			−0.194932			−0.0974659	−	0.995239i	$-0.531074\pi$
$811$	−158.090			−0.194932			−0.0974659	+	0.995239i	$0.531074\pi$
$812$			22.0150i			0.0271121i
$813$	−323.572			−0.397998
$814$	30.1222			0.0370052
$815$		−	31.6390i		−	0.0388209i
$816$			161.078i			0.197400i
$817$	218.927			0.267964
$818$	717.752			0.877448
$819$			8.35711i			0.0102040i
$820$		−	277.766i		−	0.338739i
$821$	−218.188			−0.265759			−0.132879	−	0.991132i	$-0.542422\pi$
$821$	−218.188			−0.265759			−0.132879	+	0.991132i	$0.542422\pi$
$822$			389.588i			0.473951i
$823$	147.052			0.178678			0.0893391	−	0.996001i	$-0.471525\pi$
$823$	147.052			0.178678			0.0893391	+	0.996001i	$0.471525\pi$
$824$			166.750i			0.202367i
$825$			19.0557i			0.0230978i
$826$			75.8154i			0.0917862i
$827$		−	942.783i		−	1.14000i	−0.821644	−	0.570002i	$-0.806943\pi$
$827$		−	942.783i		−	1.14000i	0.821644	−	0.570002i	$-0.193057\pi$
$828$	−87.2712	+	106.901i	−0.105400	+	0.129107i
$829$	110.053			0.132754			0.0663771	−	0.997795i	$-0.478856\pi$
$829$	110.053			0.132754			0.0663771	+	0.997795i	$0.478856\pi$
$830$	−379.217			−0.456888
$831$	221.953			0.267091
$832$	−14.6395			−0.0175956
$833$		−	1085.35i		−	1.30295i
$834$	291.763			0.349835
$835$			116.425i			0.139431i
$836$	−12.5724			−0.0150388
$837$	−259.342			−0.309847
$838$			667.052i			0.796005i
$839$			724.729i			0.863800i	0.901921	+	0.431900i	$0.142157\pi$
$839$			724.729i			0.863800i	−0.901921	+	0.431900i	$0.857843\pi$
$840$	16.6758			0.0198522
$841$	−788.714			−0.937829
$842$		−	1075.15i		−	1.27690i
$843$			317.430i			0.376548i
$844$	−211.691			−0.250818
$845$		−	370.408i		−	0.438352i
$846$	−72.7459			−0.0859881
$847$		−	176.827i		−	0.208768i
$848$			135.734i			0.160063i
$849$		−	756.440i		−	0.890977i
$850$		−	164.400i		−	0.193412i
$851$	−172.467	−	140.798i	−0.202664	−	0.165450i
$852$	−385.924			−0.452962
$853$	−1575.08			−1.84652			−0.923261	−	0.384172i	$-0.874487\pi$
$853$	−1575.08			−1.84652			−0.923261	+	0.384172i	$0.874487\pi$
$854$	−18.3927			−0.0215371
$855$	−19.1647			−0.0224148
$856$			314.511i			0.367419i
$857$	190.678			0.222495			0.111247	−	0.993793i	$-0.464515\pi$
$857$	190.678			0.222495			0.111247	+	0.993793i	$0.464515\pi$
$858$		−	9.86296i		−	0.0114953i
$859$	81.9616			0.0954151			0.0477076	−	0.998861i	$-0.484808\pi$
$859$	81.9616			0.0954151			0.0477076	+	0.998861i	$0.484808\pi$
$860$	342.704			0.398492
$861$		−	163.765i		−	0.190204i
$862$			850.380i			0.986520i
$863$	1544.93			1.79019			0.895093	−	0.445878i	$-0.147109\pi$
$863$	1544.93			1.79019			0.895093	+	0.445878i	$0.147109\pi$
$864$	29.3939			0.0340207
$865$		−	490.271i		−	0.566788i
$866$			659.642i			0.761712i
$867$	435.690			0.502526
$868$		−	151.956i		−	0.175065i
$869$	159.315			0.183331
$870$		−	39.6052i		−	0.0455232i
$871$			144.490i			0.165889i
$872$			149.306i			0.171222i
$873$		−	535.318i		−	0.613194i
$874$	71.9845	+	58.7665i	0.0823621	+	0.0672385i
$875$	−17.0197			−0.0194511
$876$	−464.940			−0.530753
$877$	277.527			0.316450			0.158225	−	0.987403i	$-0.449423\pi$
$877$	277.527			0.316450			0.158225	+	0.987403i	$0.449423\pi$
$878$	416.231			0.474067
$879$			59.5855i			0.0677878i
$880$	−19.6806			−0.0223644
$881$			946.407i			1.07424i	0.843505	+	0.537121i	$0.180488\pi$
$881$			946.407i			1.07424i	−0.843505	+	0.537121i	$0.819512\pi$
$882$	−198.058			−0.224555
$883$	1235.93			1.39969			0.699845	−	0.714295i	$-0.253252\pi$
$883$	1235.93			1.39969			0.699845	+	0.714295i	$0.253252\pi$
$884$			85.0910i			0.0962568i
$885$		−	136.393i		−	0.154116i
$886$	−908.166			−1.02502
$887$	−243.464			−0.274480			−0.137240	−	0.990538i	$-0.543823\pi$
$887$	−243.464			−0.274480			−0.137240	+	0.990538i	$0.543823\pi$
$888$			47.4224i			0.0534036i
$889$		−	153.539i		−	0.172710i
$890$	233.795			0.262691
$891$			19.8033i			0.0222259i
$892$	461.976			0.517910
$893$			48.9855i			0.0548550i
$894$		−	602.167i		−	0.673564i
$895$		−	188.616i		−	0.210744i
$896$			17.2227i			0.0192218i
$897$	−46.1018	+	56.4712i	−0.0513955	+	0.0629556i
$898$	−938.931			−1.04558
$899$	−360.896			−0.401442
$900$	−30.0000			−0.0333333
$901$	788.940			0.875627
$902$			193.274i			0.214273i
$903$	202.051			0.223756
$904$		−	71.8764i		−	0.0795093i
$905$	−609.630			−0.673624
$906$	513.044			0.566274
$907$			1671.67i			1.84307i	0.388292	+	0.921536i	$0.373065\pi$
$907$			1671.67i			1.84307i	−0.388292	+	0.921536i	$0.626935\pi$
$908$		−	293.500i		−	0.323237i
$909$	441.806			0.486035
$910$	8.80916			0.00968040
$911$		−	1179.25i		−	1.29445i	−0.762297	−	0.647227i	$-0.775929\pi$
$911$		−	1179.25i		−	1.29445i	0.762297	−	0.647227i	$-0.224071\pi$
$912$		−	19.7932i		−	0.0217031i
$913$	263.865			0.289009
$914$		−	31.2810i		−	0.0342243i
$915$	33.0887			0.0361625
$916$		−	407.713i		−	0.445102i
$917$			43.6196i			0.0475677i
$918$		−	170.849i		−	0.186110i
$919$		−	734.211i		−	0.798924i	−0.916750	−	0.399462i	$-0.869197\pi$
$919$		−	734.211i		−	0.798924i	0.916750	−	0.399462i	$-0.130803\pi$
$920$	112.683	+	91.9919i	0.122482	+	0.0999912i
$921$	457.040			0.496243
$922$	470.921			0.510761
$923$	−203.868			−0.220875
$924$	−11.6033			−0.0125577
$925$		−	48.4003i		−	0.0523247i
$926$	82.5931			0.0891934
$927$		−	176.866i		−	0.190793i
$928$	40.9040			0.0440776
$929$	−43.9814			−0.0473427			−0.0236714	−	0.999720i	$-0.507536\pi$
$929$	−43.9814			−0.0473427			−0.0236714	+	0.999720i	$0.507536\pi$
$930$			273.371i			0.293947i
$931$			133.368i			0.143252i
$932$	593.862			0.637191
$933$	−22.4620			−0.0240751
$934$		−	861.664i		−	0.922552i
$935$			114.392i			0.122344i
$936$	15.5276			0.0165893
$937$		−	393.511i		−	0.419969i	−0.977705	−	0.209984i	$-0.932659\pi$
$937$		−	393.511i		−	0.419969i	0.977705	−	0.209984i	$-0.0673413\pi$
$938$	169.985			0.181221
$939$			124.318i			0.132394i
$940$			76.6809i			0.0815755i
$941$		−	330.827i		−	0.351569i	−0.984429	−	0.175785i	$-0.943754\pi$
$941$		−	330.827i		−	0.351569i	0.984429	−	0.175785i	$-0.0562463\pi$
$942$			518.228i			0.550136i
$943$	903.407	−	1106.61i	0.958013	−	1.17349i
$944$	140.866			0.149222
$945$	−17.6874			−0.0187168
$946$	−238.459			−0.252070
$947$	−251.483			−0.265557			−0.132779	−	0.991146i	$-0.542390\pi$
$947$	−251.483			−0.265557			−0.132779	+	0.991146i	$0.542390\pi$
$948$			250.814i			0.264572i
$949$	−245.608			−0.258808
$950$			20.2013i			0.0212646i
$951$	−335.823			−0.353126
$952$	100.106			0.105153
$953$			1263.57i			1.32589i	0.748670	+	0.662943i	$0.230693\pi$
$953$			1263.57i			1.32589i	−0.748670	+	0.662943i	$0.769307\pi$
$954$		−	143.967i		−	0.150909i
$955$	−526.919			−0.551748
$956$	−532.915			−0.557442
$957$			27.5579i			0.0287961i
$958$		−	283.259i		−	0.295677i
$959$	242.118			0.252469
$960$		−	30.9839i		−	0.0322749i
$961$	1530.05			1.59214
$962$			25.0513i			0.0260409i
$963$		−	333.589i		−	0.346406i
$964$			655.231i			0.679700i
$965$		−	760.544i		−	0.788129i
$966$	66.4358	+	54.2366i	0.0687741	+	0.0561455i
$967$	−717.203			−0.741678			−0.370839	−	0.928697i	$-0.620930\pi$
$967$	−717.203			−0.741678			−0.370839	+	0.928697i	$0.620930\pi$
$968$	−328.546			−0.339407
$969$	−115.046			−0.118727
$970$	−564.275			−0.581727
$971$		−	1232.61i		−	1.26943i	−0.772748	−	0.634713i	$-0.781118\pi$
$971$		−	1232.61i		−	1.26943i	0.772748	−	0.634713i	$-0.218882\pi$
$972$	−31.1769			−0.0320750
$973$		−	181.322i		−	0.186354i
$974$	−192.521			−0.197661
$975$	−15.8478			−0.0162541
$976$			34.1738i			0.0350142i
$977$			1365.57i			1.39771i	0.715262	+	0.698857i	$0.246307\pi$
$977$			1365.57i			1.39771i	−0.715262	+	0.698857i	$0.753693\pi$
$978$	−34.6588			−0.0354384
$979$	−162.678			−0.166168
$980$			208.771i			0.213032i
$981$		−	158.363i		−	0.161430i
$982$	307.335			0.312968
$983$			1879.04i			1.91154i	0.294119	+	0.955769i	$0.404974\pi$
$983$			1879.04i			1.91154i	−0.294119	+	0.955769i	$0.595026\pi$
$984$	−304.277			−0.309225
$985$		−	495.675i		−	0.503223i
$986$		−	237.751i		−	0.241127i
$987$			45.2096i			0.0458050i
$988$		−	10.4559i		−	0.0105829i
$989$	1365.31	+	1114.61i	1.38050	+	1.12701i
$990$	20.8745			0.0210853
$991$	1192.84			1.20367			0.601837	−	0.798619i	$-0.294436\pi$
$991$	1192.84			1.20367			0.601837	+	0.798619i	$0.294436\pi$
$992$	−282.336			−0.284613
$993$	−681.260			−0.686062
$994$			239.841i			0.241289i
$995$	442.887			0.445113
$996$			415.411i			0.417080i
$997$	1855.93			1.86152			0.930758	−	0.365637i	$-0.119149\pi$
$997$	1855.93			1.86152			0.930758	+	0.365637i	$0.119149\pi$
$998$	877.763			0.879522
$999$		−	50.2991i		−	0.0503494i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	690.3.c.a.91.6	yes	32
3.2	odd	2		2070.3.c.b.91.19		32
23.22	odd	2	inner	690.3.c.a.91.3	✓	32
69.68	even	2		2070.3.c.b.91.30		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.3.c.a.91.3	✓	32	23.22	odd	2	inner
690.3.c.a.91.6	yes	32	1.1	even	1	trivial
2070.3.c.b.91.19		32	3.2	odd	2
2070.3.c.b.91.30		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)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} -1.73205 q^{3} +2.00000 q^{4} +2.23607i q^{5} +2.44949 q^{6} -1.52229i q^{7} -2.82843 q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q-1.41421 q^{2} -1.73205 q^{3} +2.00000 q^{4} +2.23607i q^{5} +2.44949 q^{6} -1.52229i q^{7} -2.82843 q^{8} +3.00000 q^{9} -3.16228i q^{10} +2.20036i q^{11} -3.46410 q^{12} -1.82994 q^{13} +2.15284i q^{14} -3.87298i q^{15} +4.00000 q^{16} -23.2496i q^{17} -4.24264 q^{18} +2.85690i q^{19} +4.47214i q^{20} +2.63668i q^{21} -3.11178i q^{22} +(-14.5452 + 17.8168i) q^{23} +4.89898 q^{24} -5.00000 q^{25} +2.58793 q^{26} -5.19615 q^{27} -3.04458i q^{28} -7.23088 q^{29} +5.47723i q^{30} +49.9104 q^{31} -5.65685 q^{32} -3.81114i q^{33} +32.8800i q^{34} +3.40394 q^{35} +6.00000 q^{36} +9.68006i q^{37} -4.04027i q^{38} +3.16955 q^{39} -6.32456i q^{40} -62.1103 q^{41} -3.72883i q^{42} -76.6308i q^{43} +4.40073i q^{44} +6.70820i q^{45} +(20.5700 - 25.1967i) q^{46} +17.1464 q^{47} -6.92820 q^{48} +46.6826 q^{49} +7.07107 q^{50} +40.2696i q^{51} -3.65988 q^{52} +33.9334i q^{53} +7.34847 q^{54} -4.92016 q^{55} +4.30569i q^{56} -4.94830i q^{57} +10.2260 q^{58} +35.2164 q^{59} -7.74597i q^{60} +8.54346i q^{61} -70.5840 q^{62} -4.56687i q^{63} +8.00000 q^{64} -4.09187i q^{65} +5.38977i q^{66} -78.9586i q^{67} -46.4993i q^{68} +(25.1930 - 30.8596i) q^{69} -4.81390 q^{70} +111.407 q^{71} -8.48528 q^{72} +134.217 q^{73} -13.6897i q^{74} +8.66025 q^{75} +5.71380i q^{76} +3.34959 q^{77} -4.48242 q^{78} -72.4039i q^{79} +8.94427i q^{80} +9.00000 q^{81} +87.8373 q^{82} -119.919i q^{83} +5.27337i q^{84} +51.9878 q^{85} +108.372i q^{86} +12.5243 q^{87} -6.22357i q^{88} +73.9324i q^{89} -9.48683i q^{90} +2.78570i q^{91} +(-29.0904 + 35.6335i) q^{92} -86.4474 q^{93} -24.2486 q^{94} -6.38822 q^{95} +9.79796 q^{96} -178.439i q^{97} -66.0192 q^{98} +6.60109i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 64 q^{4} + 96 q^{9}+O(q^{10}) \) 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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