LMFDB - Embedded newform 690.3.c.a.91.20

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			91.20
Character	$\chi$	$=$	690.91
Dual form			690.3.c.a.91.21

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} -1.73205 q^{3} +2.00000 q^{4} -2.23607i q^{5} -2.44949 q^{6} +3.95881i q^{7} +2.82843 q^{8} +3.00000 q^{9} +O(q^{10})$	\(q+1.41421 q^{2} -1.73205 q^{3} +2.00000 q^{4} -2.23607i q^{5} -2.44949 q^{6} +3.95881i q^{7} +2.82843 q^{8} +3.00000 q^{9} -3.16228i q^{10} -5.67111i q^{11} -3.46410 q^{12} +11.1376 q^{13} +5.59860i q^{14} +3.87298i q^{15} +4.00000 q^{16} +2.12953i q^{17} +4.24264 q^{18} -20.9865i q^{19} -4.47214i q^{20} -6.85686i q^{21} -8.02016i q^{22} +(-10.6801 + 20.3700i) q^{23} -4.89898 q^{24} -5.00000 q^{25} +15.7510 q^{26} -5.19615 q^{27} +7.91762i q^{28} +36.0592 q^{29} +5.47723i q^{30} +39.7941 q^{31} +5.65685 q^{32} +9.82265i q^{33} +3.01161i q^{34} +8.85217 q^{35} +6.00000 q^{36} -50.2613i q^{37} -29.6794i q^{38} -19.2910 q^{39} -6.32456i q^{40} +41.6563 q^{41} -9.69706i q^{42} -41.3234i q^{43} -11.3422i q^{44} -6.70820i q^{45} +(-15.1039 + 28.8075i) q^{46} +60.1764 q^{47} -6.92820 q^{48} +33.3278 q^{49} -7.07107 q^{50} -3.68845i q^{51} +22.2753 q^{52} -3.29435i q^{53} -7.34847 q^{54} -12.6810 q^{55} +11.1972i q^{56} +36.3497i q^{57} +50.9954 q^{58} -83.2918 q^{59} +7.74597i q^{60} -37.8857i q^{61} +56.2773 q^{62} +11.8764i q^{63} +8.00000 q^{64} -24.9045i q^{65} +13.8913i q^{66} +61.1024i q^{67} +4.25906i q^{68} +(18.4984 - 35.2819i) q^{69} +12.5189 q^{70} -0.572102 q^{71} +8.48528 q^{72} +40.8795 q^{73} -71.0803i q^{74} +8.66025 q^{75} -41.9730i q^{76} +22.4509 q^{77} -27.2815 q^{78} -86.8569i q^{79} -8.94427i q^{80} +9.00000 q^{81} +58.9109 q^{82} -84.6886i q^{83} -13.7137i q^{84} +4.76177 q^{85} -58.4401i q^{86} -62.4563 q^{87} -16.0403i q^{88} +94.7533i q^{89} -9.48683i q^{90} +44.0918i q^{91} +(-21.3601 + 40.7400i) q^{92} -68.9254 q^{93} +85.1023 q^{94} -46.9273 q^{95} -9.79796 q^{96} +171.701i q^{97} +47.1327 q^{98} -17.0133i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$			3.95881i			0.565544i	0.959187	+	0.282772i	$0.0912540\pi$
$7$			3.95881i			0.565544i	−0.959187	+	0.282772i	$0.908746\pi$
$8$	2.82843			0.353553
$9$	3.00000			0.333333
$10$		−	3.16228i		−	0.316228i
$11$		−	5.67111i		−	0.515556i	−0.966204	−	0.257778i	$-0.917010\pi$
$11$		−	5.67111i		−	0.515556i	0.966204	−	0.257778i	$-0.0829903\pi$
$12$	−3.46410			−0.288675
$13$	11.1376			0.856742			0.428371	−	0.903603i	$-0.359088\pi$
$13$	11.1376			0.856742			0.428371	+	0.903603i	$0.359088\pi$
$14$			5.59860i			0.399900i
$15$			3.87298i			0.258199i
$16$	4.00000			0.250000
$17$			2.12953i			0.125266i	0.998037	+	0.0626332i	$0.0199498\pi$
$17$			2.12953i			0.125266i	−0.998037	+	0.0626332i	$0.980050\pi$
$18$	4.24264			0.235702
$19$		−	20.9865i		−	1.10455i	−0.833661	−	0.552277i	$-0.813759\pi$
$19$		−	20.9865i		−	1.10455i	0.833661	−	0.552277i	$-0.186241\pi$
$20$		−	4.47214i		−	0.223607i
$21$		−	6.85686i		−	0.326517i
$22$		−	8.02016i		−	0.364553i
$23$	−10.6801	+	20.3700i	−0.464350	+	0.885652i
$23$	−10.6801	+	20.3700i	−0.464350	+	0.885652i
$24$	−4.89898			−0.204124
$25$	−5.00000			−0.200000
$26$	15.7510			0.605808
$27$	−5.19615			−0.192450
$28$			7.91762i			0.282772i
$29$	36.0592			1.24342			0.621710	−	0.783248i	$-0.286438\pi$
$29$	36.0592			1.24342			0.621710	+	0.783248i	$0.286438\pi$
$30$			5.47723i			0.182574i
$31$	39.7941			1.28368			0.641840	−	0.766838i	$-0.278171\pi$
$31$	39.7941			1.28368			0.641840	+	0.766838i	$0.278171\pi$
$32$	5.65685			0.176777
$33$			9.82265i			0.297656i
$34$			3.01161i			0.0885767i
$35$	8.85217			0.252919
$36$	6.00000			0.166667
$37$		−	50.2613i		−	1.35841i	−0.733946	−	0.679207i	$-0.762324\pi$
$37$		−	50.2613i		−	1.35841i	0.733946	−	0.679207i	$-0.237676\pi$
$38$		−	29.6794i		−	0.781037i
$39$	−19.2910			−0.494640
$40$		−	6.32456i		−	0.158114i
$41$	41.6563			1.01601			0.508003	−	0.861355i	$-0.330384\pi$
$41$	41.6563			1.01601			0.508003	+	0.861355i	$0.330384\pi$
$42$		−	9.69706i		−	0.230882i
$43$		−	41.3234i		−	0.961009i	−0.876992	−	0.480505i	$-0.840453\pi$
$43$		−	41.3234i		−	0.961009i	0.876992	−	0.480505i	$-0.159547\pi$
$44$		−	11.3422i		−	0.257778i
$45$		−	6.70820i		−	0.149071i
$46$	−15.1039	+	28.8075i	−0.328345	+	0.626250i
$47$	60.1764			1.28035			0.640175	−	0.768229i	$-0.278862\pi$
$47$	60.1764			1.28035			0.640175	+	0.768229i	$0.278862\pi$
$48$	−6.92820			−0.144338
$49$	33.3278			0.680160
$50$	−7.07107			−0.141421
$51$		−	3.68845i		−	0.0723226i
$52$	22.2753			0.428371
$53$		−	3.29435i		−	0.0621575i	−0.999517	−	0.0310787i	$-0.990106\pi$
$53$		−	3.29435i		−	0.0621575i	0.999517	−	0.0310787i	$-0.00989427\pi$
$54$	−7.34847			−0.136083
$55$	−12.6810			−0.230563
$56$			11.1972i			0.199950i
$57$			36.3497i			0.637714i
$58$	50.9954			0.879231
$59$	−83.2918			−1.41173			−0.705863	−	0.708349i	$-0.749440\pi$
$59$	−83.2918			−1.41173			−0.705863	+	0.708349i	$0.749440\pi$
$60$			7.74597i			0.129099i
$61$		−	37.8857i		−	0.621077i	−0.950561	−	0.310538i	$-0.899491\pi$
$61$		−	37.8857i		−	0.621077i	0.950561	−	0.310538i	$-0.100509\pi$
$62$	56.2773			0.907699
$63$			11.8764i			0.188515i
$64$	8.00000			0.125000
$65$		−	24.9045i		−	0.383147i
$66$			13.8913i			0.210475i
$67$			61.1024i			0.911975i	0.889986	+	0.455988i	$0.150714\pi$
$67$			61.1024i			0.911975i	−0.889986	+	0.455988i	$0.849286\pi$
$68$			4.25906i			0.0626332i
$69$	18.4984	−	35.2819i	0.268093	−	0.511331i
$70$	12.5189			0.178841
$71$	−0.572102			−0.00805777			−0.00402888	−	0.999992i	$-0.501282\pi$
$71$	−0.572102			−0.00805777			−0.00402888	+	0.999992i	$0.501282\pi$
$72$	8.48528			0.117851
$73$	40.8795			0.559994			0.279997	−	0.960001i	$-0.409667\pi$
$73$	40.8795			0.559994			0.279997	+	0.960001i	$0.409667\pi$
$74$		−	71.0803i		−	0.960544i
$75$	8.66025			0.115470
$76$		−	41.9730i		−	0.552277i
$77$	22.4509			0.291570
$78$	−27.2815			−0.349763
$79$		−	86.8569i		−	1.09945i	−0.835344	−	0.549727i	$-0.814732\pi$
$79$		−	86.8569i		−	1.09945i	0.835344	−	0.549727i	$-0.185268\pi$
$80$		−	8.94427i		−	0.111803i
$81$	9.00000			0.111111
$82$	58.9109			0.718425
$83$		−	84.6886i		−	1.02034i	−0.860072	−	0.510172i	$-0.829582\pi$
$83$		−	84.6886i		−	1.02034i	0.860072	−	0.510172i	$-0.170418\pi$
$84$		−	13.7137i		−	0.163259i
$85$	4.76177			0.0560208
$86$		−	58.4401i		−	0.679536i
$87$	−62.4563			−0.717889
$88$		−	16.0403i		−	0.182276i
$89$			94.7533i			1.06464i	0.846542	+	0.532322i	$0.178680\pi$
$89$			94.7533i			1.06464i	−0.846542	+	0.532322i	$0.821320\pi$
$90$		−	9.48683i		−	0.105409i
$91$			44.0918i			0.484525i
$92$	−21.3601	+	40.7400i	−0.232175	+	0.442826i
$93$	−68.9254			−0.741133
$94$	85.1023			0.905344
$95$	−46.9273			−0.493971
$96$	−9.79796			−0.102062
$97$			171.701i			1.77011i	0.465485	+	0.885056i	$0.345880\pi$
$97$			171.701i			1.77011i	−0.465485	+	0.885056i	$0.654120\pi$
$98$	47.1327			0.480946
$99$		−	17.0133i		−	0.171852i
$100$	−10.0000			−0.100000
$101$	−48.8836			−0.483997			−0.241998	−	0.970277i	$-0.577803\pi$
$101$	−48.8836			−0.483997			−0.241998	+	0.970277i	$0.577803\pi$
$102$		−	5.21626i		−	0.0511398i
$103$		−	3.88018i		−	0.0376716i	−0.999823	−	0.0188358i	$-0.994004\pi$
$103$		−	3.88018i		−	0.0376716i	0.999823	−	0.0188358i	$-0.00599598\pi$
$104$	31.5020			0.302904
$105$	−15.3324			−0.146023
$106$		−	4.65891i		−	0.0439520i
$107$			121.583i			1.13629i	0.822927	+	0.568147i	$0.192339\pi$
$107$			121.583i			1.13629i	−0.822927	+	0.568147i	$0.807661\pi$
$108$	−10.3923			−0.0962250
$109$		−	140.035i		−	1.28473i	−0.766401	−	0.642363i	$-0.777954\pi$
$109$		−	140.035i		−	1.28473i	0.766401	−	0.642363i	$-0.222046\pi$
$110$	−17.9336			−0.163033
$111$			87.0552i			0.784281i
$112$			15.8352i			0.141386i
$113$			3.73201i			0.0330267i	0.999864	+	0.0165133i	$0.00525660\pi$
$113$			3.73201i			0.0330267i	−0.999864	+	0.0165133i	$0.994743\pi$
$114$			51.4063i			0.450932i
$115$	45.5487	+	23.8813i	0.396075	+	0.207664i
$116$	72.1184			0.621710
$117$	33.4129			0.285581
$118$	−117.792			−0.998240
$119$	−8.43040			−0.0708437
$120$			10.9545i			0.0912871i
$121$	88.8385			0.734202
$122$		−	53.5784i		−	0.439167i
$123$	−72.1508			−0.586592
$124$	79.5882			0.641840
$125$			11.1803i			0.0894427i
$126$			16.7958i			0.133300i
$127$	189.032			1.48844			0.744221	−	0.667933i	$-0.232821\pi$
$127$	189.032			1.48844			0.744221	+	0.667933i	$0.232821\pi$
$128$	11.3137			0.0883883
$129$			71.5742i			0.554839i
$130$		−	35.2203i		−	0.270926i
$131$	−102.753			−0.784374			−0.392187	−	0.919885i	$-0.628281\pi$
$131$	−102.753			−0.784374			−0.392187	+	0.919885i	$0.628281\pi$
$132$			19.6453i			0.148828i
$133$	83.0816			0.624674
$134$			86.4118i			0.644864i
$135$			11.6190i			0.0860663i
$136$			6.02322i			0.0442884i
$137$			53.0102i			0.386936i	0.981107	+	0.193468i	$0.0619736\pi$
$137$			53.0102i			0.386936i	−0.981107	+	0.193468i	$0.938026\pi$
$138$	26.1607	−	49.8961i	0.189570	−	0.361566i
$139$	−230.317			−1.65696			−0.828478	−	0.560022i	$-0.810793\pi$
$139$	−230.317			−1.65696			−0.828478	+	0.560022i	$0.810793\pi$
$140$	17.7043			0.126460
$141$	−104.229			−0.739210
$142$	−0.809074			−0.00569770
$143$		−	63.1628i		−	0.441698i
$144$	12.0000			0.0833333
$145$		−	80.6308i		−	0.556074i
$146$	57.8124			0.395975
$147$	−57.7255			−0.392690
$148$		−	100.523i		−	0.679207i
$149$			34.0099i			0.228254i	0.993466	+	0.114127i	$0.0364071\pi$
$149$			34.0099i			0.228254i	−0.993466	+	0.114127i	$0.963593\pi$
$150$	12.2474			0.0816497
$151$	−273.426			−1.81077			−0.905383	−	0.424596i	$-0.860416\pi$
$151$	−273.426			−1.81077			−0.905383	+	0.424596i	$0.860416\pi$
$152$		−	59.3588i		−	0.390519i
$153$			6.38859i			0.0417555i
$154$	31.7503			0.206171
$155$		−	88.9823i		−	0.574079i
$156$	−38.5819			−0.247320
$157$			302.701i			1.92803i	0.265841	+	0.964017i	$0.414350\pi$
$157$			302.701i			1.92803i	−0.265841	+	0.964017i	$0.585650\pi$
$158$		−	122.834i		−	0.777431i
$159$			5.70598i			0.0358866i
$160$		−	12.6491i		−	0.0790569i
$161$	−80.6409	−	42.2803i	−0.500875	−	0.262611i
$162$	12.7279			0.0785674
$163$	−101.769			−0.624352			−0.312176	−	0.950024i	$-0.601058\pi$
$163$	−101.769			−0.624352			−0.312176	+	0.950024i	$0.601058\pi$
$164$	83.3126			0.508003
$165$	21.9641			0.133116
$166$		−	119.768i		−	0.721493i
$167$	−124.606			−0.746143			−0.373071	−	0.927803i	$-0.621695\pi$
$167$	−124.606			−0.746143			−0.373071	+	0.927803i	$0.621695\pi$
$168$		−	19.3941i		−	0.115441i
$169$	−44.9529			−0.265994
$170$	6.73416			0.0396127
$171$		−	62.9595i		−	0.368184i
$172$		−	82.6468i		−	0.480505i
$173$	−278.331			−1.60885			−0.804425	−	0.594054i	$-0.797526\pi$
$173$	−278.331			−1.60885			−0.804425	+	0.594054i	$0.797526\pi$
$174$	−88.3266			−0.507624
$175$		−	19.7940i		−	0.113109i
$176$		−	22.6844i		−	0.128889i
$177$	144.266			0.815060
$178$			134.001i			0.752817i
$179$	−22.7044			−0.126840			−0.0634202	−	0.997987i	$-0.520201\pi$
$179$	−22.7044			−0.126840			−0.0634202	+	0.997987i	$0.520201\pi$
$180$		−	13.4164i		−	0.0745356i
$181$		−	151.385i		−	0.836379i	−0.908360	−	0.418189i	$-0.862665\pi$
$181$		−	151.385i		−	0.836379i	0.908360	−	0.418189i	$-0.137335\pi$
$182$			62.3552i			0.342611i
$183$			65.6199i			0.358579i
$184$	−30.2078	+	57.6150i	−0.164173	+	0.313125i
$185$	−112.388			−0.607502
$186$	−97.4752			−0.524060
$187$	12.0768			0.0645818
$188$	120.353			0.640175
$189$		−	20.5706i		−	0.108839i
$190$	−66.3652			−0.349290
$191$			68.1039i			0.356565i	0.983979	+	0.178282i	$0.0570541\pi$
$191$			68.1039i			0.356565i	−0.983979	+	0.178282i	$0.942946\pi$
$192$	−13.8564			−0.0721688
$193$	243.269			1.26046			0.630230	−	0.776408i	$-0.282961\pi$
$193$	243.269			1.26046			0.630230	+	0.776408i	$0.282961\pi$
$194$			242.822i			1.25166i
$195$			43.1359i			0.221210i
$196$	66.6556			0.340080
$197$	−131.358			−0.666790			−0.333395	−	0.942787i	$-0.608194\pi$
$197$	−131.358			−0.666790			−0.333395	+	0.942787i	$0.608194\pi$
$198$		−	24.0605i		−	0.121518i
$199$			313.700i			1.57638i	0.615432	+	0.788190i	$0.288982\pi$
$199$			313.700i			1.57638i	−0.615432	+	0.788190i	$0.711018\pi$
$200$	−14.1421			−0.0707107
$201$		−	105.832i		−	0.526529i
$202$	−69.1319			−0.342237
$203$			142.751i			0.703209i
$204$		−	7.37690i		−	0.0361613i
$205$		−	93.1463i		−	0.454372i
$206$		−	5.48740i		−	0.0266379i
$207$	−32.0402	+	61.1100i	−0.154783	+	0.295217i
$208$	44.5506			0.214185
$209$	−119.017			−0.569459
$210$	−21.6833			−0.103254
$211$	75.8630			0.359540			0.179770	−	0.983709i	$-0.442465\pi$
$211$	75.8630			0.359540			0.179770	+	0.983709i	$0.442465\pi$
$212$		−	6.58869i		−	0.0310787i
$213$	0.990909			0.00465216
$214$			171.945i			0.803481i
$215$	−92.4019			−0.429777
$216$	−14.6969			−0.0680414
$217$			157.537i			0.725978i
$218$		−	198.040i		−	0.908438i
$219$	−70.8054			−0.323313
$220$	−25.3620			−0.115282
$221$			23.7179i			0.107321i
$222$			123.115i			0.554571i
$223$	−165.794			−0.743472			−0.371736	−	0.928338i	$-0.621237\pi$
$223$	−165.794			−0.743472			−0.371736	+	0.928338i	$0.621237\pi$
$224$			22.3944i			0.0999750i
$225$	−15.0000			−0.0666667
$226$			5.27786i			0.0233534i
$227$			405.393i			1.78587i	0.450183	+	0.892937i	$0.351359\pi$
$227$			405.393i			1.78587i	−0.450183	+	0.892937i	$0.648641\pi$
$228$			72.6994i			0.318857i
$229$		−	382.310i		−	1.66947i	−0.550649	−	0.834737i	$-0.685620\pi$
$229$		−	382.310i		−	1.66947i	0.550649	−	0.834737i	$-0.314380\pi$
$230$	64.4156	+	33.7733i	0.280068	+	0.146840i
$231$	−38.8860			−0.168338
$232$	101.991			0.439615
$233$	−125.133			−0.537050			−0.268525	−	0.963273i	$-0.586536\pi$
$233$	−125.133			−0.537050			−0.268525	+	0.963273i	$0.586536\pi$
$234$	47.2530			0.201936
$235$		−	134.559i		−	0.572590i
$236$	−166.584			−0.705863
$237$			150.441i			0.634770i
$238$	−11.9224			−0.0500941
$239$	−129.207			−0.540616			−0.270308	−	0.962774i	$-0.587126\pi$
$239$	−129.207			−0.540616			−0.270308	+	0.962774i	$0.587126\pi$
$240$			15.4919i			0.0645497i
$241$		−	313.373i		−	1.30030i	−0.759805	−	0.650151i	$-0.774705\pi$
$241$		−	313.373i		−	1.30030i	0.759805	−	0.650151i	$-0.225295\pi$
$242$	125.637			0.519160
$243$	−15.5885			−0.0641500
$244$		−	75.7713i		−	0.310538i
$245$		−	74.5233i		−	0.304177i
$246$	−102.037			−0.414783
$247$		−	233.740i		−	0.946317i
$248$	112.555			0.453850
$249$			146.685i			0.589096i
$250$			15.8114i			0.0632456i
$251$			152.125i			0.606075i	0.952979	+	0.303037i	$0.0980007\pi$
$251$			152.125i			0.606075i	−0.952979	+	0.303037i	$0.901999\pi$
$252$			23.7529i			0.0942574i
$253$	115.520	+	60.5678i	0.456603	+	0.239398i
$254$	267.332			1.05249
$255$	−8.24763			−0.0323436
$256$	16.0000			0.0625000
$257$	19.5904			0.0762272			0.0381136	−	0.999273i	$-0.487865\pi$
$257$	19.5904			0.0762272			0.0381136	+	0.999273i	$0.487865\pi$
$258$			101.221i			0.392330i
$259$	198.975			0.768244
$260$		−	49.8090i		−	0.191573i
$261$	108.178			0.414473
$262$	−145.315			−0.554636
$263$		−	127.348i		−	0.484212i	−0.970250	−	0.242106i	$-0.922162\pi$
$263$		−	127.348i		−	0.484212i	0.970250	−	0.242106i	$-0.0778381\pi$
$264$			27.7827i			0.105237i
$265$	−7.36638			−0.0277977
$266$	117.495			0.441711
$267$		−	164.118i		−	0.614672i
$268$			122.205i			0.455988i
$269$	142.959			0.531448			0.265724	−	0.964049i	$-0.414389\pi$
$269$	142.959			0.531448			0.265724	+	0.964049i	$0.414389\pi$
$270$			16.4317i			0.0608581i
$271$	−63.7008			−0.235058			−0.117529	−	0.993069i	$-0.537497\pi$
$271$	−63.7008			−0.235058			−0.117529	+	0.993069i	$0.537497\pi$
$272$			8.51812i			0.0313166i
$273$		−	76.3692i		−	0.279741i
$274$			74.9678i			0.273605i
$275$			28.3556i			0.103111i
$276$	36.9968	−	70.5637i	0.134046	−	0.255666i
$277$	448.470			1.61903			0.809513	−	0.587101i	$-0.199731\pi$
$277$	448.470			1.61903			0.809513	+	0.587101i	$0.199731\pi$
$278$	−325.717			−1.17164
$279$	119.382			0.427893
$280$	25.0377			0.0894204
$281$			154.698i			0.550525i	0.961369	+	0.275263i	$0.0887648\pi$
$281$			154.698i			0.550525i	−0.961369	+	0.275263i	$0.911235\pi$
$282$	−147.402			−0.522701
$283$		−	19.1545i		−	0.0676837i	−0.999427	−	0.0338418i	$-0.989226\pi$
$283$		−	19.1545i		−	0.0676837i	0.999427	−	0.0338418i	$-0.0107742\pi$
$284$	−1.14420			−0.00402888
$285$	81.2804			0.285194
$286$		−	89.3257i		−	0.312328i
$287$			164.909i			0.574597i
$288$	16.9706			0.0589256
$289$	284.465			0.984308
$290$		−	114.029i		−	0.393204i
$291$		−	297.395i		−	1.02197i
$292$	81.7591			0.279997
$293$		−	269.403i		−	0.919463i	−0.888058	−	0.459732i	$-0.847946\pi$
$293$		−	269.403i		−	0.919463i	0.888058	−	0.459732i	$-0.152054\pi$
$294$	−81.6362			−0.277674
$295$			186.246i			0.631343i
$296$		−	142.161i		−	0.480272i
$297$			29.4680i			0.0992187i
$298$			48.0972i			0.161400i
$299$	−118.951	+	226.874i	−0.397828	+	0.758775i
$300$	17.3205			0.0577350
$301$	163.592			0.543493
$302$	−386.682			−1.28040
$303$	84.6690			0.279436
$304$		−	83.9461i		−	0.276138i
$305$	−84.7149			−0.277754
$306$			9.03483i			0.0295256i
$307$	331.722			1.08053			0.540263	−	0.841496i	$-0.318325\pi$
$307$	331.722			1.08053			0.540263	+	0.841496i	$0.318325\pi$
$308$	44.9017			0.145785
$309$			6.72067i			0.0217497i
$310$		−	125.840i		−	0.405935i
$311$	−397.584			−1.27840			−0.639202	−	0.769039i	$-0.720735\pi$
$311$	−397.584			−1.27840			−0.639202	+	0.769039i	$0.720735\pi$
$312$	−54.5631			−0.174882
$313$		−	124.944i		−	0.399181i	−0.979879	−	0.199590i	$-0.936039\pi$
$313$		−	124.944i		−	0.399181i	0.979879	−	0.199590i	$-0.0639611\pi$
$314$			428.084i			1.36333i
$315$	26.5565			0.0843064
$316$		−	173.714i		−	0.549727i
$317$	272.823			0.860642			0.430321	−	0.902676i	$-0.358400\pi$
$317$	272.823			0.860642			0.430321	+	0.902676i	$0.358400\pi$
$318$			8.06947i			0.0253757i
$319$		−	204.496i		−	0.641052i
$320$		−	17.8885i		−	0.0559017i
$321$		−	210.589i		−	0.656039i
$322$	−114.043	−	59.7934i	−0.354172	−	0.185694i
$323$	44.6914			0.138363
$324$	18.0000			0.0555556
$325$	−55.6882			−0.171348
$326$	−143.924			−0.441483
$327$			242.548i			0.741737i
$328$	117.822			0.359213
$329$			238.227i			0.724094i
$330$	31.0620			0.0941271
$331$	158.093			0.477621			0.238811	−	0.971066i	$-0.423243\pi$
$331$	158.093			0.477621			0.238811	+	0.971066i	$0.423243\pi$
$332$		−	169.377i		−	0.510172i
$333$		−	150.784i		−	0.452805i
$334$	−176.219			−0.527603
$335$	136.629			0.407848
$336$		−	27.4274i		−	0.0816293i
$337$		−	187.915i		−	0.557610i	−0.960348	−	0.278805i	$-0.910062\pi$
$337$		−	187.915i		−	0.557610i	0.960348	−	0.278805i	$-0.0899383\pi$
$338$	−63.5730			−0.188086
$339$		−	6.46404i		−	0.0190680i
$340$	9.52354			0.0280104
$341$		−	225.677i		−	0.661809i
$342$		−	89.0382i		−	0.260346i
$343$			325.920i			0.950205i
$344$		−	116.880i		−	0.339768i
$345$	−78.8926	−	41.3637i	−0.228674	−	0.119895i
$346$	−393.620			−1.13763
$347$	−313.436			−0.903274			−0.451637	−	0.892202i	$-0.649160\pi$
$347$	−313.436			−0.903274			−0.451637	+	0.892202i	$0.649160\pi$
$348$	−124.913			−0.358944
$349$	363.470			1.04146			0.520730	−	0.853721i	$-0.325660\pi$
$349$	363.470			1.04146			0.520730	+	0.853721i	$0.325660\pi$
$350$		−	27.9930i		−	0.0799800i
$351$	−57.8729			−0.164880
$352$		−	32.0807i		−	0.0911382i
$353$	−524.725			−1.48647			−0.743236	−	0.669029i	$-0.766710\pi$
$353$	−524.725			−1.48647			−0.743236	+	0.669029i	$0.766710\pi$
$354$	204.022			0.576334
$355$			1.27926i			0.00360354i
$356$			189.507i			0.532322i
$357$	14.6019			0.0409016
$358$	−32.1089			−0.0896897
$359$			311.701i			0.868249i	0.900853	+	0.434124i	$0.142942\pi$
$359$			311.701i			0.868249i	−0.900853	+	0.434124i	$0.857058\pi$
$360$		−	18.9737i		−	0.0527046i
$361$	−79.4338			−0.220038
$362$		−	214.090i		−	0.591409i
$363$	−153.873			−0.423892
$364$			88.1836i			0.242263i
$365$		−	91.4094i		−	0.250437i
$366$			92.8006i			0.253553i
$367$			468.490i			1.27654i	0.769813	+	0.638269i	$0.220349\pi$
$367$			468.490i			1.27654i	−0.769813	+	0.638269i	$0.779651\pi$
$368$	−42.7202	+	81.4799i	−0.116088	+	0.221413i
$369$	124.969			0.338669
$370$	−158.940			−0.429568
$371$	13.0417			0.0351528
$372$	−137.851			−0.370567
$373$			7.45302i			0.0199813i	0.999950	+	0.00999064i	$0.00318017\pi$
$373$			7.45302i			0.0199813i	−0.999950	+	0.00999064i	$0.996820\pi$
$374$	17.0792			0.0456662
$375$		−	19.3649i		−	0.0516398i
$376$	170.205			0.452672
$377$	401.614			1.06529
$378$		−	29.0912i		−	0.0769608i
$379$		−	444.034i		−	1.17159i	−0.810458	−	0.585796i	$-0.800782\pi$
$379$		−	444.034i		−	1.17159i	0.810458	−	0.585796i	$-0.199218\pi$
$380$	−93.8545			−0.246986
$381$	−327.413			−0.859353
$382$			96.3134i			0.252129i
$383$			315.599i			0.824017i	0.911180	+	0.412009i	$0.135173\pi$
$383$			315.599i			0.824017i	−0.911180	+	0.412009i	$0.864827\pi$
$384$	−19.5959			−0.0510310
$385$		−	50.2016i		−	0.130394i
$386$	344.034			0.891280
$387$		−	123.970i		−	0.320336i
$388$			343.402i			0.885056i
$389$			126.333i			0.324763i	0.986728	+	0.162381i	$0.0519175\pi$
$389$			126.333i			0.324763i	−0.986728	+	0.162381i	$0.948083\pi$
$390$			61.0034i			0.156419i
$391$	−43.3785	−	22.7435i	−0.110942	−	0.0581675i
$392$	94.2653			0.240473
$393$	177.974			0.452859
$394$	−185.768			−0.471492
$395$	−194.218			−0.491691
$396$		−	34.0267i		−	0.0859259i
$397$	−165.211			−0.416148			−0.208074	−	0.978113i	$-0.566719\pi$
$397$	−165.211			−0.416148			−0.208074	+	0.978113i	$0.566719\pi$
$398$			443.638i			1.11467i
$399$	−143.902			−0.360656
$400$	−20.0000			−0.0500000
$401$		−	298.998i		−	0.745630i	−0.927906	−	0.372815i	$-0.878393\pi$
$401$		−	298.998i		−	0.745630i	0.927906	−	0.372815i	$-0.121607\pi$
$402$		−	149.670i		−	0.372312i
$403$	443.212			1.09978
$404$	−97.7673			−0.241998
$405$		−	20.1246i		−	0.0496904i
$406$			201.881i			0.497244i
$407$	−285.038			−0.700338
$408$		−	10.4325i		−	0.0255699i
$409$	−55.5554			−0.135832			−0.0679162	−	0.997691i	$-0.521635\pi$
$409$	−55.5554			−0.135832			−0.0679162	+	0.997691i	$0.521635\pi$
$410$		−	131.729i		−	0.321290i
$411$		−	91.8164i		−	0.223398i
$412$		−	7.76036i		−	0.0188358i
$413$		−	329.736i		−	0.798393i
$414$	−45.3117	+	86.4225i	−0.109448	+	0.208750i
$415$	−189.369			−0.456312
$416$	63.0040			0.151452
$417$	398.920			0.956644
$418$	−168.315			−0.402668
$419$		−	422.645i		−	1.00870i	−0.863500	−	0.504349i	$-0.831732\pi$
$419$		−	422.645i		−	1.00870i	0.863500	−	0.504349i	$-0.168268\pi$
$420$	−30.6648			−0.0730114
$421$			106.651i			0.253328i	0.991946	+	0.126664i	$0.0404269\pi$
$421$			106.651i			0.253328i	−0.991946	+	0.126664i	$0.959573\pi$
$422$	107.287			0.254233
$423$	180.529			0.426783
$424$		−	9.31782i		−	0.0219760i
$425$		−	10.6476i		−	0.0250533i
$426$	1.40136			0.00328957
$427$	149.982			0.351246
$428$			243.167i			0.568147i
$429$			109.401i			0.255014i
$430$	−130.676			−0.303898
$431$		−	305.614i		−	0.709080i	−0.935041	−	0.354540i	$-0.884637\pi$
$431$		−	305.614i		−	0.709080i	0.935041	−	0.354540i	$-0.115363\pi$
$432$	−20.7846			−0.0481125
$433$			285.890i			0.660254i	0.943936	+	0.330127i	$0.107092\pi$
$433$			285.890i			0.660254i	−0.943936	+	0.330127i	$0.892908\pi$
$434$			222.791i			0.513344i
$435$			139.657i			0.321050i
$436$		−	280.070i		−	0.642363i
$437$	427.495	+	224.137i	0.978249	+	0.512900i
$438$	−100.134			−0.228616
$439$	−461.780			−1.05189			−0.525945	−	0.850519i	$-0.676288\pi$
$439$	−461.780			−1.05189			−0.525945	+	0.850519i	$0.676288\pi$
$440$	−35.8673			−0.0815165
$441$	99.9835			0.226720
$442$			33.5422i			0.0758874i
$443$	−65.0709			−0.146887			−0.0734434	−	0.997299i	$-0.523399\pi$
$443$	−65.0709			−0.146887			−0.0734434	+	0.997299i	$0.523399\pi$
$444$			174.110i			0.392141i
$445$	211.875			0.476123
$446$	−234.469			−0.525714
$447$		−	58.9068i		−	0.131783i
$448$			31.6705i			0.0706930i
$449$	547.308			1.21895			0.609475	−	0.792806i	$-0.291380\pi$
$449$	547.308			1.21895			0.609475	+	0.792806i	$0.291380\pi$
$450$	−21.2132			−0.0471405
$451$		−	236.237i		−	0.523808i
$452$			7.46403i			0.0165133i
$453$	473.587			1.04545
$454$			573.313i			1.26280i
$455$	98.5923			0.216686
$456$			102.813i			0.225466i
$457$		−	334.369i		−	0.731660i	−0.930682	−	0.365830i	$-0.880785\pi$
$457$		−	334.369i		−	0.731660i	0.930682	−	0.365830i	$-0.119215\pi$
$458$		−	540.667i		−	1.18050i
$459$		−	11.0654i		−	0.0241075i
$460$	91.0973	+	47.7627i	0.198038	+	0.103832i
$461$	−384.546			−0.834156			−0.417078	−	0.908871i	$-0.636946\pi$
$461$	−384.546			−0.834156			−0.417078	+	0.908871i	$0.636946\pi$
$462$	−54.9931			−0.119033
$463$	741.501			1.60151			0.800757	−	0.598990i	$-0.204431\pi$
$463$	741.501			1.60151			0.800757	+	0.598990i	$0.204431\pi$
$464$	144.237			0.310855
$465$			154.122i			0.331445i
$466$	−176.964			−0.379752
$467$			183.855i			0.393695i	0.980434	+	0.196847i	$0.0630704\pi$
$467$			183.855i			0.393695i	−0.980434	+	0.196847i	$0.936930\pi$
$468$	66.8259			0.142790
$469$	−241.893			−0.515763
$470$		−	190.295i		−	0.404882i
$471$		−	524.294i		−	1.11315i
$472$	−235.585			−0.499120
$473$	−234.350			−0.495454
$474$			212.755i			0.448850i
$475$			104.933i			0.220911i
$476$	−16.8608			−0.0354218
$477$		−	9.88304i		−	0.0207192i
$478$	−182.727			−0.382273
$479$			643.670i			1.34378i	0.740652	+	0.671889i	$0.234517\pi$
$479$			643.670i			1.34378i	−0.740652	+	0.671889i	$0.765483\pi$
$480$			21.9089i			0.0456435i
$481$		−	559.793i		−	1.16381i
$482$		−	443.176i		−	0.919453i
$483$	139.674	+	73.2317i	0.289180	+	0.151618i
$484$	177.677			0.367101
$485$	383.935			0.791618
$486$	−22.0454			−0.0453609
$487$	−404.712			−0.831031			−0.415516	−	0.909586i	$-0.636399\pi$
$487$	−404.712			−0.831031			−0.415516	+	0.909586i	$0.636399\pi$
$488$		−	107.157i		−	0.219584i
$489$	176.270			0.360470
$490$		−	105.392i		−	0.215085i
$491$	173.587			0.353537			0.176769	−	0.984252i	$-0.443436\pi$
$491$	173.587			0.353537			0.176769	+	0.984252i	$0.443436\pi$
$492$	−144.302			−0.293296
$493$			76.7891i			0.155759i
$494$		−	330.559i		−	0.669147i
$495$	−38.0430			−0.0768545
$496$	159.176			0.320920
$497$		−	2.26484i		−	0.00455703i
$498$			207.444i			0.416554i
$499$	6.44061			0.0129070			0.00645352	−	0.999979i	$-0.497946\pi$
$499$	6.44061			0.0129070			0.00645352	+	0.999979i	$0.497946\pi$
$500$			22.3607i			0.0447214i
$501$	215.824			0.430786
$502$			215.137i			0.428559i
$503$		−	617.629i		−	1.22789i	−0.789348	−	0.613945i	$-0.789581\pi$
$503$		−	617.629i		−	1.22789i	0.789348	−	0.613945i	$-0.210419\pi$
$504$			33.5916i			0.0666500i
$505$			109.307i			0.216450i
$506$	163.371	+	85.6558i	0.322867	+	0.169280i
$507$	77.8608			0.153572
$508$	378.064			0.744221
$509$	200.449			0.393809			0.196904	−	0.980423i	$-0.436911\pi$
$509$	200.449			0.393809			0.196904	+	0.980423i	$0.436911\pi$
$510$	−11.6639			−0.0228704
$511$			161.834i			0.316701i
$512$	22.6274			0.0441942
$513$			109.049i			0.212571i
$514$	27.7050			0.0539008
$515$	−8.67634			−0.0168473
$516$			143.148i			0.277420i
$517$		−	341.267i		−	0.660091i
$518$	281.393			0.543230
$519$	482.083			0.928870
$520$		−	70.4406i		−	0.135463i
$521$			201.610i			0.386968i	0.981103	+	0.193484i	$0.0619787\pi$
$521$			201.610i			0.386968i	−0.981103	+	0.193484i	$0.938021\pi$
$522$	152.986			0.293077
$523$			678.779i			1.29786i	0.760850	+	0.648928i	$0.224782\pi$
$523$			678.779i			1.29786i	−0.760850	+	0.648928i	$0.775218\pi$
$524$	−205.506			−0.392187
$525$			34.2843i			0.0653034i
$526$		−	180.097i		−	0.342389i
$527$			84.7427i			0.160802i
$528$			39.2906i			0.0744140i
$529$	−300.873	−	435.105i	−0.568757	−	0.822505i
$530$	−10.4176			−0.0196559
$531$	−249.875			−0.470575
$532$	166.163			0.312337
$533$	463.953			0.870455
$534$		−	232.097i		−	0.434639i
$535$	271.869			0.508166
$536$			172.824i			0.322432i
$537$	39.3252			0.0732313
$538$	202.175			0.375790
$539$		−	189.006i		−	0.350660i
$540$			23.2379i			0.0430331i
$541$	465.332			0.860133			0.430067	−	0.902797i	$-0.358490\pi$
$541$	465.332			0.860133			0.430067	+	0.902797i	$0.358490\pi$
$542$	−90.0865			−0.166211
$543$			262.206i			0.482884i
$544$			12.0464i			0.0221442i
$545$	−313.128			−0.574547
$546$		−	108.002i		−	0.197807i
$547$	−309.522			−0.565853			−0.282926	−	0.959142i	$-0.591305\pi$
$547$	−309.522			−0.565853			−0.282926	+	0.959142i	$0.591305\pi$
$548$			106.020i			0.193468i
$549$		−	113.657i		−	0.207026i
$550$			40.1008i			0.0729106i
$551$		−	756.756i		−	1.37342i
$552$	52.3214	−	99.7921i	0.0947851	−	0.180783i
$553$	343.850			0.621790
$554$	634.233			1.14482
$555$	194.661			0.350741
$556$	−460.634			−0.828478
$557$			826.642i			1.48410i	0.670346	+	0.742049i	$0.266146\pi$
$557$			826.642i			1.48410i	−0.670346	+	0.742049i	$0.733854\pi$
$558$	168.832			0.302566
$559$		−	460.245i		−	0.823337i
$560$	35.4087			0.0632298
$561$	−20.9176			−0.0372863
$562$			218.775i			0.389280i
$563$		−	1018.64i		−	1.80931i	−0.426140	−	0.904657i	$-0.640127\pi$
$563$		−	1018.64i		−	1.80931i	0.426140	−	0.904657i	$-0.359873\pi$
$564$	−208.457			−0.369605
$565$	8.34503			0.0147700
$566$		−	27.0885i		−	0.0478596i
$567$			35.6293i			0.0628383i
$568$	−1.61815			−0.00284885
$569$		−	766.176i		−	1.34653i	−0.739401	−	0.673266i	$-0.764891\pi$
$569$		−	766.176i		−	1.34653i	0.739401	−	0.673266i	$-0.235109\pi$
$570$	114.948			0.201663
$571$		−	119.265i		−	0.208870i	−0.994532	−	0.104435i	$-0.966697\pi$
$571$		−	119.265i		−	0.208870i	0.994532	−	0.104435i	$-0.0333035\pi$
$572$		−	126.326i		−	0.220849i
$573$		−	117.959i		−	0.205863i
$574$			233.217i			0.406301i
$575$	53.4003	−	101.850i	0.0928701	−	0.177130i
$576$	24.0000			0.0416667
$577$	−73.2216			−0.126901			−0.0634503	−	0.997985i	$-0.520210\pi$
$577$	−73.2216			−0.126901			−0.0634503	+	0.997985i	$0.520210\pi$
$578$	402.294			0.696011
$579$	−421.354			−0.727727
$580$		−	161.262i		−	0.278037i
$581$	335.266			0.577050
$582$		−	420.579i		−	0.722645i
$583$	−18.6826			−0.0320456
$584$	115.625			0.197988
$585$		−	74.7136i		−	0.127716i
$586$		−	380.993i		−	0.650159i
$587$	829.222			1.41264			0.706322	−	0.707891i	$-0.250353\pi$
$587$	829.222			1.41264			0.706322	+	0.707891i	$0.250353\pi$
$588$	−115.451			−0.196345
$589$		−	835.139i		−	1.41789i
$590$			263.392i			0.446427i
$591$	227.518			0.384972
$592$		−	201.045i		−	0.339604i
$593$	82.6719			0.139413			0.0697065	−	0.997568i	$-0.477794\pi$
$593$	82.6719			0.139413			0.0697065	+	0.997568i	$0.477794\pi$
$594$			41.6740i			0.0701582i
$595$			18.8509i			0.0316823i
$596$			68.0198i			0.114127i
$597$		−	543.344i		−	0.910123i
$598$	−168.222	+	320.848i	−0.281307	+	0.536535i
$599$	390.672			0.652208			0.326104	−	0.945334i	$-0.394264\pi$
$599$	390.672			0.652208			0.326104	+	0.945334i	$0.394264\pi$
$600$	24.4949			0.0408248
$601$	−726.400			−1.20865			−0.604326	−	0.796737i	$-0.706558\pi$
$601$	−726.400			−1.20865			−0.604326	+	0.796737i	$0.706558\pi$
$602$	231.353			0.384308
$603$			183.307i			0.303992i
$604$	−546.851			−0.905383
$605$		−	198.649i		−	0.328345i
$606$	119.740			0.197591
$607$	−289.612			−0.477120			−0.238560	−	0.971128i	$-0.576675\pi$
$607$	−289.612			−0.477120			−0.238560	+	0.971128i	$0.576675\pi$
$608$		−	118.718i		−	0.195259i
$609$		−	247.253i		−	0.405998i
$610$	−119.805			−0.196402
$611$	670.224			1.09693
$612$			12.7772i			0.0208777i
$613$			457.354i			0.746091i	0.927813	+	0.373045i	$0.121686\pi$
$613$			457.354i			0.746091i	−0.927813	+	0.373045i	$0.878314\pi$
$614$	469.125			0.764048
$615$			161.334i			0.262332i
$616$	63.5006			0.103085
$617$			934.894i			1.51523i	0.652705	+	0.757613i	$0.273634\pi$
$617$			934.894i			1.51523i	−0.652705	+	0.757613i	$0.726366\pi$
$618$			9.50446i			0.0153794i
$619$		−	79.2136i		−	0.127970i	−0.997951	−	0.0639851i	$-0.979619\pi$
$619$		−	79.2136i		−	0.127970i	0.997951	−	0.0639851i	$-0.0203810\pi$
$620$		−	177.965i		−	0.287040i
$621$	55.4952	−	105.846i	0.0893643	−	0.170444i
$622$	−562.268			−0.903968
$623$	−375.110			−0.602103
$624$	−77.1638			−0.123660
$625$	25.0000			0.0400000
$626$		−	176.697i		−	0.282263i
$627$	206.143			0.328777
$628$			605.403i			0.964017i
$629$	107.033			0.170164
$630$	37.5566			0.0596136
$631$			405.115i			0.642021i	0.947076	+	0.321011i	$0.104023\pi$
$631$			405.115i			0.642021i	−0.947076	+	0.321011i	$0.895977\pi$
$632$		−	245.668i		−	0.388716i
$633$	−131.399			−0.207581
$634$	385.831			0.608566
$635$		−	422.689i		−	0.665652i
$636$			11.4120i			0.0179433i
$637$	371.193			0.582721
$638$		−	289.200i		−	0.453292i
$639$	−1.71630			−0.00268592
$640$		−	25.2982i		−	0.0395285i
$641$			984.727i			1.53623i	0.640309	+	0.768117i	$0.278806\pi$
$641$			984.727i			1.53623i	−0.640309	+	0.768117i	$0.721194\pi$
$642$		−	297.817i		−	0.463890i
$643$			330.197i			0.513526i	0.966474	+	0.256763i	$0.0826560\pi$
$643$			330.197i			0.513526i	−0.966474	+	0.256763i	$0.917344\pi$
$644$	−161.282	−	84.5606i	−0.250438	−	0.131305i
$645$	160.045			0.248132
$646$	63.2032			0.0978377
$647$	948.869			1.46657			0.733284	−	0.679923i	$-0.237987\pi$
$647$	948.869			1.46657			0.733284	+	0.679923i	$0.237987\pi$
$648$	25.4558			0.0392837
$649$			472.357i			0.727823i
$650$	−78.7550			−0.121162
$651$		−	272.863i		−	0.419144i
$652$	−203.539			−0.312176
$653$	−378.498			−0.579630			−0.289815	−	0.957083i	$-0.593594\pi$
$653$	−378.498			−0.579630			−0.289815	+	0.957083i	$0.593594\pi$
$654$			343.014i			0.524487i
$655$			229.763i			0.350783i
$656$	166.625			0.254002
$657$	122.639			0.186665
$658$			336.904i			0.512012i
$659$		−	470.040i		−	0.713262i	−0.934245	−	0.356631i	$-0.883925\pi$
$659$		−	470.040i		−	0.713262i	0.934245	−	0.356631i	$-0.116075\pi$
$660$	43.9282			0.0665579
$661$		−	244.609i		−	0.370058i	−0.982733	−	0.185029i	$-0.940762\pi$
$661$		−	244.609i		−	0.370058i	0.982733	−	0.185029i	$-0.0592380\pi$
$662$	223.577			0.337729
$663$		−	41.0807i		−	0.0619618i
$664$		−	239.536i		−	0.360746i
$665$		−	185.776i		−	0.279363i
$666$		−	213.241i		−	0.320181i
$667$	−385.114	+	734.525i	−0.577383	+	1.10124i
$668$	−249.212			−0.373071
$669$	287.164			0.429244
$670$	193.223			0.288392
$671$	−214.854			−0.320199
$672$		−	38.7883i		−	0.0577206i
$673$	297.873			0.442605			0.221302	−	0.975205i	$-0.428969\pi$
$673$	297.873			0.442605			0.221302	+	0.975205i	$0.428969\pi$
$674$		−	265.751i		−	0.394290i
$675$	25.9808			0.0384900
$676$	−89.9059			−0.132997
$677$			306.899i			0.453322i	0.973974	+	0.226661i	$0.0727809\pi$
$677$			306.899i			0.453322i	−0.973974	+	0.226661i	$0.927219\pi$
$678$		−	9.14153i		−	0.0134831i
$679$	−679.731			−1.00108
$680$	13.4683			0.0198064
$681$		−	702.162i		−	1.03107i
$682$		−	319.155i		−	0.467969i
$683$	270.473			0.396007			0.198003	−	0.980201i	$-0.436554\pi$
$683$	270.473			0.396007			0.198003	+	0.980201i	$0.436554\pi$
$684$		−	125.919i		−	0.184092i
$685$	118.534			0.173043
$686$			460.921i			0.671896i
$687$			662.180i			0.963871i
$688$		−	165.294i		−	0.240252i
$689$		−	36.6913i		−	0.0532529i
$690$	−111.571	−	58.4971i	−0.161697	−	0.0847784i
$691$	−994.734			−1.43956			−0.719779	−	0.694204i	$-0.755757\pi$
$691$	−994.734			−1.43956			−0.719779	+	0.694204i	$0.755757\pi$
$692$	−556.662			−0.804425
$693$	67.3526			0.0971898
$694$	−443.266			−0.638711
$695$			515.004i			0.741013i
$696$	−176.653			−0.253812
$697$			88.7082i			0.127272i
$698$	514.024			0.736424
$699$	216.736			0.310066
$700$		−	39.5881i		−	0.0565544i
$701$		−	615.121i		−	0.877490i	−0.898612	−	0.438745i	$-0.855423\pi$
$701$		−	615.121i		−	0.877490i	0.898612	−	0.438745i	$-0.144577\pi$
$702$	−81.8446			−0.116588
$703$	−1054.81			−1.50044
$704$		−	45.3689i		−	0.0644444i
$705$			233.062i			0.330585i
$706$	−742.073			−1.05109
$707$		−	193.521i		−	0.273721i
$708$	288.531			0.407530
$709$			768.270i			1.08360i	0.840509	+	0.541798i	$0.182256\pi$
$709$			768.270i			1.08360i	−0.840509	+	0.541798i	$0.817744\pi$
$710$			1.80914i			0.00254809i
$711$		−	260.571i		−	0.366485i
$712$			268.003i			0.376408i
$713$	−425.003	+	810.605i	−0.596078	+	1.13689i
$714$	20.6502			0.0289218
$715$	−141.236			−0.197533
$716$	−45.4088			−0.0634202
$717$	223.793			0.312125
$718$			440.812i			0.613944i
$719$	−613.243			−0.852910			−0.426455	−	0.904509i	$-0.640238\pi$
$719$	−613.243			−0.852910			−0.426455	+	0.904509i	$0.640238\pi$
$720$		−	26.8328i		−	0.0372678i
$721$	15.3609			0.0213050
$722$	−112.336			−0.155590
$723$			542.778i			0.750730i
$724$		−	302.769i		−	0.418189i
$725$	−180.296			−0.248684
$726$	−217.609			−0.299737
$727$		−	1148.29i		−	1.57949i	−0.613436	−	0.789744i	$-0.710213\pi$
$727$		−	1148.29i		−	1.57949i	0.613436	−	0.789744i	$-0.289787\pi$
$728$			124.710i			0.171306i
$729$	27.0000			0.0370370
$730$		−	129.272i		−	0.177086i
$731$	87.9994			0.120382
$732$			131.240i			0.179289i
$733$			570.538i			0.778361i	0.921162	+	0.389180i	$0.127242\pi$
$733$			570.538i			0.778361i	−0.921162	+	0.389180i	$0.872758\pi$
$734$			662.544i			0.902649i
$735$			129.078i			0.175616i
$736$	−60.4155	+	115.230i	−0.0820863	+	0.156563i
$737$	346.518			0.470174
$738$	176.733			0.239475
$739$	−1062.30			−1.43749			−0.718744	−	0.695275i	$-0.755283\pi$
$739$	−1062.30			−1.43749			−0.718744	+	0.695275i	$0.755283\pi$
$740$	−224.776			−0.303751
$741$			404.850i			0.546356i
$742$	18.4437			0.0248568
$743$		−	166.131i		−	0.223595i	−0.993731	−	0.111797i	$-0.964339\pi$
$743$		−	166.131i		−	0.223595i	0.993731	−	0.111797i	$-0.0356608\pi$
$744$	−194.950			−0.262030
$745$	76.0484			0.102078
$746$			10.5402i			0.0141289i
$747$		−	254.066i		−	0.340115i
$748$	24.1536			0.0322909
$749$	−481.325			−0.642624
$750$		−	27.3861i		−	0.0365148i
$751$			1210.26i			1.61153i	0.592236	+	0.805765i	$0.298245\pi$
$751$			1210.26i			1.61153i	−0.592236	+	0.805765i	$0.701755\pi$
$752$	240.706			0.320087
$753$		−	263.488i		−	0.349917i
$754$	567.968			0.753274
$755$			611.398i			0.809799i
$756$		−	41.1412i		−	0.0544195i
$757$			500.893i			0.661681i	0.943687	+	0.330841i	$0.107332\pi$
$757$			500.893i			0.661681i	−0.943687	+	0.330841i	$0.892668\pi$
$758$		−	627.958i		−	0.828441i
$759$	−200.087	−	104.907i	−0.263620	−	0.138217i
$760$	−132.730			−0.174645
$761$	788.375			1.03597			0.517987	−	0.855389i	$-0.326682\pi$
$761$	788.375			1.03597			0.517987	+	0.855389i	$0.326682\pi$
$762$	−463.032			−0.607654
$763$	554.372			0.726569
$764$			136.208i			0.178282i
$765$	14.2853			0.0186736
$766$			446.324i			0.582668i
$767$	−927.674			−1.20948
$768$	−27.7128			−0.0360844
$769$		−	115.079i		−	0.149647i	−0.997197	−	0.0748237i	$-0.976161\pi$
$769$		−	115.079i		−	0.149647i	0.997197	−	0.0748237i	$-0.0238394\pi$
$770$		−	70.9958i		−	0.0922024i
$771$	−33.9315			−0.0440098
$772$	486.538			0.630230
$773$		−	1396.31i		−	1.80635i	−0.429272	−	0.903175i	$-0.641230\pi$
$773$		−	1396.31i		−	1.80635i	0.429272	−	0.903175i	$-0.358770\pi$
$774$		−	175.320i		−	0.226512i
$775$	−198.970			−0.256736
$776$			485.643i			0.625829i
$777$	−344.635			−0.443546
$778$			178.661i			0.229642i
$779$		−	874.220i		−	1.12223i
$780$			86.2718i			0.110605i
$781$			3.24445i			0.00415423i
$782$	−61.3464	−	32.1642i	−0.0784481	−	0.0411306i
$783$	−187.369			−0.239296
$784$	133.311			0.170040
$785$	676.861			0.862243
$786$	251.693			0.320220
$787$			608.869i			0.773659i	0.922151	+	0.386829i	$0.126430\pi$
$787$			608.869i			0.773659i	−0.922151	+	0.386829i	$0.873570\pi$
$788$	−262.715			−0.333395
$789$			220.573i			0.279560i
$790$	−274.666			−0.347678
$791$	−14.7743			−0.0186780
$792$		−	48.1210i		−	0.0607588i
$793$		−	421.957i		−	0.532102i
$794$	−233.643			−0.294261
$795$	12.7590			0.0160490
$796$			627.399i			0.788190i
$797$			651.157i			0.817009i	0.912756	+	0.408505i	$0.133950\pi$
$797$			651.157i			0.817009i	−0.912756	+	0.408505i	$0.866050\pi$
$798$	−203.508			−0.255022
$799$			128.147i			0.160385i
$800$	−28.2843			−0.0353553
$801$			284.260i			0.354881i
$802$		−	422.847i		−	0.527240i
$803$		−	231.832i		−	0.288708i
$804$		−	211.665i		−	0.263265i
$805$	−94.5417	+	180.319i	−0.117443	+	0.223998i
$806$	626.797			0.777664
$807$	−247.613			−0.306832
$808$	−138.264			−0.171119
$809$	−521.791			−0.644983			−0.322492	−	0.946572i	$-0.604520\pi$
$809$	−521.791			−0.644983			−0.322492	+	0.946572i	$0.604520\pi$
$810$		−	28.4605i		−	0.0351364i
$811$	−1004.88			−1.23907			−0.619533	−	0.784971i	$-0.712678\pi$
$811$	−1004.88			−1.23907			−0.619533	+	0.784971i	$0.712678\pi$
$812$			285.503i			0.351604i
$813$	110.333			0.135711
$814$	−403.104			−0.495214
$815$			227.563i			0.279219i
$816$		−	14.7538i		−	0.0180806i
$817$	−867.234			−1.06149
$818$	−78.5673			−0.0960480
$819$			132.275i			0.161508i
$820$		−	186.293i		−	0.227186i
$821$	−752.571			−0.916652			−0.458326	−	0.888784i	$-0.651551\pi$
$821$	−752.571			−0.916652			−0.458326	+	0.888784i	$0.651551\pi$
$822$		−	129.848i		−	0.157966i
$823$	−1119.05			−1.35972			−0.679858	−	0.733343i	$-0.737959\pi$
$823$	−1119.05			−1.35972			−0.679858	+	0.733343i	$0.737959\pi$
$824$		−	10.9748i		−	0.0133189i
$825$		−	49.1133i		−	0.0595312i
$826$		−	466.318i		−	0.564549i
$827$		−	305.110i		−	0.368936i	−0.982839	−	0.184468i	$-0.940944\pi$
$827$		−	305.110i		−	0.368936i	0.982839	−	0.184468i	$-0.0590562\pi$
$828$	−64.0804	+	122.220i	−0.0773917	+	0.147609i
$829$	−1139.25			−1.37424			−0.687121	−	0.726543i	$-0.741126\pi$
$829$	−1139.25			−1.37424			−0.687121	+	0.726543i	$0.741126\pi$
$830$	−267.809			−0.322661
$831$	−776.774			−0.934746
$832$	89.1011			0.107093
$833$			70.9726i			0.0852012i
$834$	564.159			0.676449
$835$			278.627i			0.333685i
$836$	−238.034			−0.284729
$837$	−206.776			−0.247044
$838$		−	597.710i		−	0.713258i
$839$			811.707i			0.967470i	0.875215	+	0.483735i	$0.160720\pi$
$839$			811.707i			0.967470i	−0.875215	+	0.483735i	$0.839280\pi$
$840$	−43.3666			−0.0516269
$841$	459.264			0.546093
$842$			150.827i			0.179130i
$843$		−	267.944i		−	0.317846i
$844$	151.726			0.179770
$845$			100.518i			0.118956i
$846$	255.307			0.301781
$847$			351.695i			0.415224i
$848$		−	13.1774i		−	0.0155394i
$849$			33.1765i			0.0390772i
$850$		−	15.0580i		−	0.0177153i
$851$	1023.82	+	536.794i	1.20308	+	0.630781i
$852$	1.98182			0.00232608
$853$	−336.775			−0.394812			−0.197406	−	0.980322i	$-0.563252\pi$
$853$	−336.775			−0.394812			−0.197406	+	0.980322i	$0.563252\pi$
$854$	212.107			0.248369
$855$	−140.782			−0.164657
$856$			343.890i			0.401740i
$857$	−764.331			−0.891869			−0.445934	−	0.895066i	$-0.647129\pi$
$857$	−764.331			−0.891869			−0.445934	+	0.895066i	$0.647129\pi$
$858$			154.717i			0.180322i
$859$	−368.659			−0.429172			−0.214586	−	0.976705i	$-0.568840\pi$
$859$	−368.659			−0.429172			−0.214586	+	0.976705i	$0.568840\pi$
$860$	−184.804			−0.214888
$861$		−	285.631i		−	0.331744i
$862$		−	432.203i		−	0.501395i
$863$	−731.744			−0.847907			−0.423954	−	0.905684i	$-0.639358\pi$
$863$	−731.744			−0.847907			−0.423954	+	0.905684i	$0.639358\pi$
$864$	−29.3939			−0.0340207
$865$			622.367i			0.719500i
$866$			404.310i			0.466870i
$867$	−492.708			−0.568291
$868$			315.074i			0.362989i
$869$	−492.575			−0.566830
$870$			197.504i			0.227016i
$871$			680.536i			0.781327i
$872$		−	396.079i		−	0.454219i
$873$			515.102i			0.590037i
$874$	604.569	+	316.978i	0.691727	+	0.362675i
$875$	−44.2608			−0.0505838
$876$	−141.611			−0.161656
$877$	−53.7425			−0.0612799			−0.0306400	−	0.999530i	$-0.509755\pi$
$877$	−53.7425			−0.0612799			−0.0306400	+	0.999530i	$0.509755\pi$
$878$	−653.055			−0.743799
$879$			466.619i			0.530852i
$880$	−50.7240			−0.0576409
$881$			860.235i			0.976430i	0.872723	+	0.488215i	$0.162352\pi$
$881$			860.235i			0.976430i	−0.872723	+	0.488215i	$0.837648\pi$
$882$	141.398			0.160315
$883$	−445.282			−0.504283			−0.252142	−	0.967690i	$-0.581135\pi$
$883$	−445.282			−0.504283			−0.252142	+	0.967690i	$0.581135\pi$
$884$			47.4359i			0.0536605i
$885$		−	322.588i		−	0.364506i
$886$	−92.0241			−0.103865
$887$	121.682			0.137184			0.0685921	−	0.997645i	$-0.478149\pi$
$887$	121.682			0.137184			0.0685921	+	0.997645i	$0.478149\pi$
$888$			246.229i			0.277285i
$889$			748.343i			0.841780i
$890$	299.636			0.336670
$891$		−	51.0400i		−	0.0572840i
$892$	−331.589			−0.371736
$893$		−	1262.89i		−	1.41421i
$894$		−	83.3069i		−	0.0931844i
$895$			50.7686i			0.0567247i
$896$			44.7888i			0.0499875i
$897$	206.029	−	392.957i	0.229686	−	0.438079i
$898$	774.011			0.861927
$899$	1434.94			1.59615
$900$	−30.0000			−0.0333333
$901$	7.01541			0.00778625
$902$		−	334.090i		−	0.370388i
$903$	−283.349			−0.313786
$904$			10.5557i			0.0116767i
$905$	−338.506			−0.374040
$906$	669.753			0.739242
$907$		−	914.181i		−	1.00792i	−0.863728	−	0.503959i	$-0.831876\pi$
$907$		−	914.181i		−	1.00792i	0.863728	−	0.503959i	$-0.168124\pi$
$908$			810.786i			0.892937i
$909$	−146.651			−0.161332
$910$	139.431			0.153220
$911$		−	936.862i		−	1.02839i	−0.857674	−	0.514195i	$-0.828091\pi$
$911$		−	936.862i		−	1.02839i	0.857674	−	0.514195i	$-0.171909\pi$
$912$			145.399i			0.159429i
$913$	−480.278			−0.526044
$914$		−	472.869i		−	0.517362i
$915$	146.731			0.160361
$916$		−	764.619i		−	0.834737i
$917$		−	406.780i		−	0.443598i
$918$		−	15.6488i		−	0.0170466i
$919$		−	1272.45i		−	1.38460i	−0.721610	−	0.692300i	$-0.756597\pi$
$919$		−	1272.45i		−	1.38460i	0.721610	−	0.692300i	$-0.243403\pi$
$920$	128.831	+	67.5466i	0.140034	+	0.0734202i
$921$	−574.559			−0.623842
$922$	−543.830			−0.589838
$923$	−6.37186			−0.00690343
$924$	−77.7720			−0.0841689
$925$			251.307i			0.271683i
$926$	1048.64			1.13244
$927$		−	11.6405i		−	0.0125572i
$928$	203.981			0.219808
$929$	−1343.27			−1.44593			−0.722967	−	0.690882i	$-0.757222\pi$
$929$	−1343.27			−1.44593			−0.722967	+	0.690882i	$0.757222\pi$
$930$			217.961i			0.234367i
$931$		−	699.435i		−	0.751273i
$932$	−250.265			−0.268525
$933$	688.635			0.738087
$934$			260.011i			0.278384i
$935$		−	27.0045i		−	0.0288819i
$936$	94.5060			0.100968
$937$			850.505i			0.907690i	0.891081	+	0.453845i	$0.149948\pi$
$937$			850.505i			0.907690i	−0.891081	+	0.453845i	$0.850052\pi$
$938$	−342.088			−0.364699
$939$			216.409i			0.230467i
$940$		−	269.117i		−	0.286295i
$941$			1778.92i			1.89045i	0.326417	+	0.945226i	$0.394159\pi$
$941$			1778.92i			1.89045i	−0.326417	+	0.945226i	$0.605841\pi$
$942$		−	741.464i		−	0.787116i
$943$	−444.892	+	848.538i	−0.471783	+	0.899828i
$944$	−333.167			−0.352931
$945$	−45.9972			−0.0486743
$946$	−331.420			−0.350339
$947$	121.252			0.128038			0.0640190	−	0.997949i	$-0.479608\pi$
$947$	121.252			0.128038			0.0640190	+	0.997949i	$0.479608\pi$
$948$			300.881i			0.317385i
$949$	455.302			0.479770
$950$			148.397i			0.156207i
$951$	−472.544			−0.496892
$952$	−23.8448			−0.0250470
$953$		−	1183.70i		−	1.24207i	−0.783781	−	0.621037i	$-0.786712\pi$
$953$		−	1183.70i		−	1.24207i	0.783781	−	0.621037i	$-0.213288\pi$
$954$		−	13.9767i		−	0.0146507i
$955$	152.285			0.159461
$956$	−258.414			−0.270308
$957$			354.197i			0.370112i
$958$			910.286i			0.950195i
$959$	−209.857			−0.218829
$960$			30.9839i			0.0322749i
$961$	622.570			0.647835
$962$		−	791.667i		−	0.822938i
$963$			364.750i			0.378764i
$964$		−	626.746i		−	0.650151i
$965$		−	543.966i		−	0.563695i
$966$	197.529	+	103.565i	0.204481	+	0.107210i
$967$	1135.29			1.17403			0.587015	−	0.809576i	$-0.300303\pi$
$967$	1135.29			1.17403			0.587015	+	0.809576i	$0.300303\pi$
$968$	251.273			0.259580
$969$	−77.4078			−0.0798842
$970$	542.966			0.559758
$971$			1446.26i			1.48945i	0.667369	+	0.744727i	$0.267421\pi$
$971$			1446.26i			1.48945i	−0.667369	+	0.744727i	$0.732579\pi$
$972$	−31.1769			−0.0320750
$973$		−	911.780i		−	0.937082i
$974$	−572.349			−0.587628
$975$	96.4548			0.0989280
$976$		−	151.543i		−	0.155269i
$977$		−	77.0844i		−	0.0788991i	−0.999222	−	0.0394496i	$-0.987440\pi$
$977$		−	77.0844i		−	0.0788991i	0.999222	−	0.0394496i	$-0.0125604\pi$
$978$	249.283			0.254890
$979$	537.356			0.548883
$980$		−	149.047i		−	0.152088i
$981$		−	420.105i		−	0.428242i
$982$	245.489			0.249989
$983$			496.933i			0.505527i	0.967528	+	0.252763i	$0.0813394\pi$
$983$			496.933i			0.505527i	−0.967528	+	0.252763i	$0.918661\pi$
$984$	−204.073			−0.207392
$985$			293.725i			0.298198i
$986$			108.596i			0.110138i
$987$		−	412.621i		−	0.418056i
$988$		−	467.481i		−	0.473158i
$989$	841.757	+	441.336i	0.851120	+	0.446245i
$990$	−53.8009			−0.0543443
$991$	290.841			0.293482			0.146741	−	0.989175i	$-0.453122\pi$
$991$	290.841			0.293482			0.146741	+	0.989175i	$0.453122\pi$
$992$	225.109			0.226925
$993$	−273.824			−0.275755
$994$		−	3.20297i		−	0.00322230i
$995$	701.454			0.704978
$996$			293.370i			0.294548i
$997$	−1034.44			−1.03756			−0.518778	−	0.854909i	$-0.673613\pi$
$997$	−1034.44			−1.03756			−0.518778	+	0.854909i	$0.673613\pi$
$998$	9.10840			0.00912665
$999$			261.166i			0.261427i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	690.3.c.a.91.20	✓	32
3.2	odd	2		2070.3.c.b.91.15		32
23.22	odd	2	inner	690.3.c.a.91.21	yes	32
69.68	even	2		2070.3.c.b.91.2		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.3.c.a.91.20	✓	32	1.1	even	1	trivial
690.3.c.a.91.21	yes	32	23.22	odd	2	inner
2070.3.c.b.91.2		32	69.68	even	2
2070.3.c.b.91.15		32	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} -1.73205 q^{3} +2.00000 q^{4} -2.23607i q^{5} -2.44949 q^{6} +3.95881i q^{7} +2.82843 q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q+1.41421 q^{2} -1.73205 q^{3} +2.00000 q^{4} -2.23607i q^{5} -2.44949 q^{6} +3.95881i q^{7} +2.82843 q^{8} +3.00000 q^{9} -3.16228i q^{10} -5.67111i q^{11} -3.46410 q^{12} +11.1376 q^{13} +5.59860i q^{14} +3.87298i q^{15} +4.00000 q^{16} +2.12953i q^{17} +4.24264 q^{18} -20.9865i q^{19} -4.47214i q^{20} -6.85686i q^{21} -8.02016i q^{22} +(-10.6801 + 20.3700i) q^{23} -4.89898 q^{24} -5.00000 q^{25} +15.7510 q^{26} -5.19615 q^{27} +7.91762i q^{28} +36.0592 q^{29} +5.47723i q^{30} +39.7941 q^{31} +5.65685 q^{32} +9.82265i q^{33} +3.01161i q^{34} +8.85217 q^{35} +6.00000 q^{36} -50.2613i q^{37} -29.6794i q^{38} -19.2910 q^{39} -6.32456i q^{40} +41.6563 q^{41} -9.69706i q^{42} -41.3234i q^{43} -11.3422i q^{44} -6.70820i q^{45} +(-15.1039 + 28.8075i) q^{46} +60.1764 q^{47} -6.92820 q^{48} +33.3278 q^{49} -7.07107 q^{50} -3.68845i q^{51} +22.2753 q^{52} -3.29435i q^{53} -7.34847 q^{54} -12.6810 q^{55} +11.1972i q^{56} +36.3497i q^{57} +50.9954 q^{58} -83.2918 q^{59} +7.74597i q^{60} -37.8857i q^{61} +56.2773 q^{62} +11.8764i q^{63} +8.00000 q^{64} -24.9045i q^{65} +13.8913i q^{66} +61.1024i q^{67} +4.25906i q^{68} +(18.4984 - 35.2819i) q^{69} +12.5189 q^{70} -0.572102 q^{71} +8.48528 q^{72} +40.8795 q^{73} -71.0803i q^{74} +8.66025 q^{75} -41.9730i q^{76} +22.4509 q^{77} -27.2815 q^{78} -86.8569i q^{79} -8.94427i q^{80} +9.00000 q^{81} +58.9109 q^{82} -84.6886i q^{83} -13.7137i q^{84} +4.76177 q^{85} -58.4401i q^{86} -62.4563 q^{87} -16.0403i q^{88} +94.7533i q^{89} -9.48683i q^{90} +44.0918i q^{91} +(-21.3601 + 40.7400i) q^{92} -68.9254 q^{93} +85.1023 q^{94} -46.9273 q^{95} -9.79796 q^{96} +171.701i q^{97} +47.1327 q^{98} -17.0133i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 64 q^{4} + 96 q^{9}+O(q^{10}) \) 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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