LMFDB - Embedded newform 690.3.c.a.91.29

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			91.29
Character	$\chi$	$=$	690.91
Dual form			690.3.c.a.91.28

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} +1.73205 q^{3} +2.00000 q^{4} +2.23607i q^{5} +2.44949 q^{6} -11.4203i q^{7} +2.82843 q^{8} +3.00000 q^{9} +O(q^{10})$	\(q+1.41421 q^{2} +1.73205 q^{3} +2.00000 q^{4} +2.23607i q^{5} +2.44949 q^{6} -11.4203i q^{7} +2.82843 q^{8} +3.00000 q^{9} +3.16228i q^{10} -13.8723i q^{11} +3.46410 q^{12} -2.98777 q^{13} -16.1508i q^{14} +3.87298i q^{15} +4.00000 q^{16} -15.3412i q^{17} +4.24264 q^{18} +2.52942i q^{19} +4.47214i q^{20} -19.7806i q^{21} -19.6184i q^{22} +(-20.9247 + 9.54768i) q^{23} +4.89898 q^{24} -5.00000 q^{25} -4.22535 q^{26} +5.19615 q^{27} -22.8407i q^{28} +11.6231 q^{29} +5.47723i q^{30} +2.71207 q^{31} +5.65685 q^{32} -24.0275i q^{33} -21.6958i q^{34} +25.5367 q^{35} +6.00000 q^{36} -31.7707i q^{37} +3.57714i q^{38} -5.17497 q^{39} +6.32456i q^{40} +36.3226 q^{41} -27.9740i q^{42} +19.9718i q^{43} -27.7446i q^{44} +6.70820i q^{45} +(-29.5920 + 13.5025i) q^{46} +52.2318 q^{47} +6.92820 q^{48} -81.4243 q^{49} -7.07107 q^{50} -26.5718i q^{51} -5.97554 q^{52} +59.4952i q^{53} +7.34847 q^{54} +31.0194 q^{55} -32.3016i q^{56} +4.38108i q^{57} +16.4376 q^{58} +81.6065 q^{59} +7.74597i q^{60} -42.2136i q^{61} +3.83545 q^{62} -34.2610i q^{63} +8.00000 q^{64} -6.68086i q^{65} -33.9801i q^{66} +58.5139i q^{67} -30.6825i q^{68} +(-36.2426 + 16.5371i) q^{69} +36.1143 q^{70} +100.588 q^{71} +8.48528 q^{72} -16.7595 q^{73} -44.9306i q^{74} -8.66025 q^{75} +5.05884i q^{76} -158.427 q^{77} -7.31852 q^{78} +60.6495i q^{79} +8.94427i q^{80} +9.00000 q^{81} +51.3679 q^{82} +44.1530i q^{83} -39.5612i q^{84} +34.3041 q^{85} +28.2444i q^{86} +20.1319 q^{87} -39.2368i q^{88} -106.245i q^{89} +9.48683i q^{90} +34.1214i q^{91} +(-41.8493 + 19.0954i) q^{92} +4.69744 q^{93} +73.8669 q^{94} -5.65595 q^{95} +9.79796 q^{96} +37.9918i q^{97} -115.151 q^{98} -41.6169i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$		−	11.4203i		−	1.63148i	−0.578420	−	0.815739i	$-0.696331\pi$
$7$		−	11.4203i		−	1.63148i	0.578420	−	0.815739i	$-0.303669\pi$
$8$	2.82843			0.353553
$9$	3.00000			0.333333
$10$			3.16228i			0.316228i
$11$		−	13.8723i		−	1.26112i	−0.776141	−	0.630559i	$-0.782825\pi$
$11$		−	13.8723i		−	1.26112i	0.776141	−	0.630559i	$-0.217175\pi$
$12$	3.46410			0.288675
$13$	−2.98777			−0.229829			−0.114914	−	0.993375i	$-0.536659\pi$
$13$	−2.98777			−0.229829			−0.114914	+	0.993375i	$0.536659\pi$
$14$		−	16.1508i		−	1.15363i
$15$			3.87298i			0.258199i
$16$	4.00000			0.250000
$17$		−	15.3412i		−	0.902426i	−0.892416	−	0.451213i	$-0.850991\pi$
$17$		−	15.3412i		−	0.902426i	0.892416	−	0.451213i	$-0.149009\pi$
$18$	4.24264			0.235702
$19$			2.52942i			0.133127i	0.997782	+	0.0665636i	$0.0212035\pi$
$19$			2.52942i			0.133127i	−0.997782	+	0.0665636i	$0.978796\pi$
$20$			4.47214i			0.223607i
$21$		−	19.7806i		−	0.941934i
$22$		−	19.6184i		−	0.891746i
$23$	−20.9247	+	9.54768i	−0.909768	+	0.415117i
$23$	−20.9247	+	9.54768i	−0.909768	+	0.415117i
$24$	4.89898			0.204124
$25$	−5.00000			−0.200000
$26$	−4.22535			−0.162513
$27$	5.19615			0.192450
$28$		−	22.8407i		−	0.815739i
$29$	11.6231			0.400798			0.200399	−	0.979714i	$-0.435776\pi$
$29$	11.6231			0.400798			0.200399	+	0.979714i	$0.435776\pi$
$30$			5.47723i			0.182574i
$31$	2.71207			0.0874861			0.0437431	−	0.999043i	$-0.486072\pi$
$31$	2.71207			0.0874861			0.0437431	+	0.999043i	$0.486072\pi$
$32$	5.65685			0.176777
$33$		−	24.0275i		−	0.728107i
$34$		−	21.6958i		−	0.638112i
$35$	25.5367			0.729619
$36$	6.00000			0.166667
$37$		−	31.7707i		−	0.858668i	−0.903146	−	0.429334i	$-0.858748\pi$
$37$		−	31.7707i		−	0.858668i	0.903146	−	0.429334i	$-0.141252\pi$
$38$			3.57714i			0.0941352i
$39$	−5.17497			−0.132692
$40$			6.32456i			0.158114i
$41$	36.3226			0.885917			0.442959	−	0.896542i	$-0.353929\pi$
$41$	36.3226			0.885917			0.442959	+	0.896542i	$0.353929\pi$
$42$		−	27.9740i		−	0.666048i
$43$			19.9718i			0.464461i	0.972661	+	0.232231i	$0.0746024\pi$
$43$			19.9718i			0.464461i	−0.972661	+	0.232231i	$0.925398\pi$
$44$		−	27.7446i		−	0.630559i
$45$			6.70820i			0.149071i
$46$	−29.5920	+	13.5025i	−0.643303	+	0.293532i
$47$	52.2318			1.11131			0.555657	−	0.831411i	$-0.312467\pi$
$47$	52.2318			1.11131			0.555657	+	0.831411i	$0.312467\pi$
$48$	6.92820			0.144338
$49$	−81.4243			−1.66172
$50$	−7.07107			−0.141421
$51$		−	26.5718i		−	0.521016i
$52$	−5.97554			−0.114914
$53$			59.4952i			1.12255i	0.827629	+	0.561275i	$0.189689\pi$
$53$			59.4952i			1.12255i	−0.827629	+	0.561275i	$0.810311\pi$
$54$	7.34847			0.136083
$55$	31.0194			0.563990
$56$		−	32.3016i		−	0.576815i
$57$			4.38108i			0.0768611i
$58$	16.4376			0.283407
$59$	81.6065			1.38316			0.691581	−	0.722299i	$-0.256915\pi$
$59$	81.6065			1.38316			0.691581	+	0.722299i	$0.256915\pi$
$60$			7.74597i			0.129099i
$61$		−	42.2136i		−	0.692026i	−0.938230	−	0.346013i	$-0.887535\pi$
$61$		−	42.2136i		−	0.692026i	0.938230	−	0.346013i	$-0.112465\pi$
$62$	3.83545			0.0618620
$63$		−	34.2610i		−	0.543826i
$64$	8.00000			0.125000
$65$		−	6.68086i		−	0.102782i
$66$		−	33.9801i		−	0.514850i
$67$			58.5139i			0.873342i	0.899621	+	0.436671i	$0.143843\pi$
$67$			58.5139i			0.873342i	−0.899621	+	0.436671i	$0.856157\pi$
$68$		−	30.6825i		−	0.451213i
$69$	−36.2426	+	16.5371i	−0.525255	+	0.239668i
$70$	36.1143			0.515919
$71$	100.588			1.41673			0.708367	−	0.705844i	$-0.249432\pi$
$71$	100.588			1.41673			0.708367	+	0.705844i	$0.249432\pi$
$72$	8.48528			0.117851
$73$	−16.7595			−0.229582			−0.114791	−	0.993390i	$-0.536620\pi$
$73$	−16.7595			−0.229582			−0.114791	+	0.993390i	$0.536620\pi$
$74$		−	44.9306i		−	0.607170i
$75$	−8.66025			−0.115470
$76$			5.05884i			0.0665636i
$77$	−158.427			−2.05749
$78$	−7.31852			−0.0938271
$79$			60.6495i			0.767715i	0.923392	+	0.383858i	$0.125405\pi$
$79$			60.6495i			0.767715i	−0.923392	+	0.383858i	$0.874595\pi$
$80$			8.94427i			0.111803i
$81$	9.00000			0.111111
$82$	51.3679			0.626438
$83$			44.1530i			0.531964i	0.963978	+	0.265982i	$0.0856962\pi$
$83$			44.1530i			0.531964i	−0.963978	+	0.265982i	$0.914304\pi$
$84$		−	39.5612i		−	0.470967i
$85$	34.3041			0.403577
$86$			28.2444i			0.328424i
$87$	20.1319			0.231401
$88$		−	39.2368i		−	0.445873i
$89$		−	106.245i		−	1.19376i	−0.802330	−	0.596881i	$-0.796406\pi$
$89$		−	106.245i		−	1.19376i	0.802330	−	0.596881i	$-0.203594\pi$
$90$			9.48683i			0.105409i
$91$			34.1214i			0.374960i
$92$	−41.8493	+	19.0954i	−0.454884	+	0.207558i
$93$	4.69744			0.0505101
$94$	73.8669			0.785818
$95$	−5.65595			−0.0595363
$96$	9.79796			0.102062
$97$			37.9918i			0.391668i	0.980637	+	0.195834i	$0.0627414\pi$
$97$			37.9918i			0.391668i	−0.980637	+	0.195834i	$0.937259\pi$
$98$	−115.151			−1.17501
$99$		−	41.6169i		−	0.420373i
$100$	−10.0000			−0.100000
$101$	−147.465			−1.46005			−0.730024	−	0.683422i	$-0.760491\pi$
$101$	−147.465			−1.46005			−0.730024	+	0.683422i	$0.760491\pi$
$102$		−	37.5782i		−	0.368414i
$103$			74.8003i			0.726217i	0.931747	+	0.363108i	$0.118285\pi$
$103$			74.8003i			0.726217i	−0.931747	+	0.363108i	$0.881715\pi$
$104$	−8.45069			−0.0812567
$105$	44.2308			0.421246
$106$			84.1389i			0.793763i
$107$			36.8895i			0.344762i	0.985030	+	0.172381i	$0.0551460\pi$
$107$			36.8895i			0.344762i	−0.985030	+	0.172381i	$0.944854\pi$
$108$	10.3923			0.0962250
$109$			89.1884i			0.818242i	0.912480	+	0.409121i	$0.134165\pi$
$109$			89.1884i			0.818242i	−0.912480	+	0.409121i	$0.865835\pi$
$110$	43.8681			0.398801
$111$		−	55.0285i		−	0.495752i
$112$		−	45.6814i		−	0.407869i
$113$		−	39.7138i		−	0.351450i	−0.984439	−	0.175725i	$-0.943773\pi$
$113$		−	39.7138i		−	0.351450i	0.984439	−	0.175725i	$-0.0562269\pi$
$114$			6.19578i			0.0543490i
$115$	−21.3493	−	46.7890i	−0.185646	−	0.406861i
$116$	23.2463			0.200399
$117$	−8.96331			−0.0766095
$118$	115.409			0.978043
$119$	−175.202			−1.47229
$120$			10.9545i			0.0912871i
$121$	−71.4409			−0.590421
$122$		−	59.6990i		−	0.489336i
$123$	62.9126			0.511485
$124$	5.42414			0.0437431
$125$		−	11.1803i		−	0.0894427i
$126$		−	48.4524i		−	0.384543i
$127$	−130.656			−1.02878			−0.514392	−	0.857555i	$-0.671982\pi$
$127$	−130.656			−1.02878			−0.514392	+	0.857555i	$0.671982\pi$
$128$	11.3137			0.0883883
$129$			34.5922i			0.268157i
$130$		−	9.44816i		−	0.0726782i
$131$	2.98786			0.0228081			0.0114041	−	0.999935i	$-0.496370\pi$
$131$	2.98786			0.0228081			0.0114041	+	0.999935i	$0.496370\pi$
$132$		−	48.0551i		−	0.364054i
$133$	28.8868			0.217194
$134$			82.7512i			0.617546i
$135$			11.6190i			0.0860663i
$136$		−	43.3916i		−	0.319056i
$137$		−	120.177i		−	0.877207i	−0.898681	−	0.438603i	$-0.855473\pi$
$137$		−	120.177i		−	0.877207i	0.898681	−	0.438603i	$-0.144527\pi$
$138$	−51.2548	+	23.3869i	−0.371411	+	0.169471i
$139$	−127.272			−0.915628			−0.457814	−	0.889048i	$-0.651367\pi$
$139$	−127.272			−0.915628			−0.457814	+	0.889048i	$0.651367\pi$
$140$	51.0733			0.364810
$141$	90.4681			0.641618
$142$	142.253			1.00178
$143$			41.4473i			0.289841i
$144$	12.0000			0.0833333
$145$			25.9901i			0.179242i
$146$	−23.7015			−0.162339
$147$	−141.031			−0.959395
$148$		−	63.5414i		−	0.429334i
$149$		−	131.641i		−	0.883494i	−0.897140	−	0.441747i	$-0.854359\pi$
$149$		−	131.641i		−	0.883494i	0.897140	−	0.441747i	$-0.145641\pi$
$150$	−12.2474			−0.0816497
$151$	149.135			0.987650			0.493825	−	0.869561i	$-0.335598\pi$
$151$	149.135			0.987650			0.493825	+	0.869561i	$0.335598\pi$
$152$			7.15427i			0.0470676i
$153$		−	46.0237i		−	0.300809i
$154$	−224.049			−1.45486
$155$			6.06437i			0.0391250i
$156$	−10.3499			−0.0663458
$157$			228.188i			1.45343i	0.686941	+	0.726713i	$0.258953\pi$
$157$			228.188i			1.45343i	−0.686941	+	0.726713i	$0.741047\pi$
$158$			85.7713i			0.542857i
$159$			103.049i			0.648105i
$160$			12.6491i			0.0790569i
$161$	109.038	+	238.967i	0.677253	+	1.48427i
$162$	12.7279			0.0785674
$163$	260.536			1.59838			0.799191	−	0.601077i	$-0.205261\pi$
$163$	260.536			1.59838			0.799191	+	0.601077i	$0.205261\pi$
$164$	72.6452			0.442959
$165$	53.7272			0.325620
$166$			62.4418i			0.376155i
$167$	−177.110			−1.06054			−0.530271	−	0.847828i	$-0.677910\pi$
$167$	−177.110			−1.06054			−0.530271	+	0.847828i	$0.677910\pi$
$168$		−	55.9480i		−	0.333024i
$169$	−160.073			−0.947179
$170$	48.5133			0.285372
$171$			7.58825i			0.0443758i
$172$			39.9437i			0.232231i
$173$	7.60898			0.0439825			0.0219913	−	0.999758i	$-0.492999\pi$
$173$	7.60898			0.0439825			0.0219913	+	0.999758i	$0.492999\pi$
$174$	28.4708			0.163625
$175$			57.1017i			0.326296i
$176$		−	55.4892i		−	0.315280i
$177$	141.347			0.798568
$178$		−	150.253i		−	0.844118i
$179$	319.384			1.78427			0.892133	−	0.451772i	$-0.149208\pi$
$179$	319.384			1.78427			0.892133	+	0.451772i	$0.149208\pi$
$180$			13.4164i			0.0745356i
$181$			115.161i			0.636248i	0.948049	+	0.318124i	$0.103053\pi$
$181$			115.161i			0.636248i	−0.948049	+	0.318124i	$0.896947\pi$
$182$			48.2549i			0.265137i
$183$		−	73.1161i		−	0.399541i
$184$	−59.1839	+	27.0049i	−0.321652	+	0.146766i
$185$	71.0414			0.384008
$186$	6.64319			0.0357161
$187$	−212.818			−1.13807
$188$	104.464			0.555657
$189$		−	59.3419i		−	0.313978i
$190$	−7.99872			−0.0420985
$191$			104.681i			0.548068i	0.961720	+	0.274034i	$0.0883581\pi$
$191$			104.681i			0.548068i	−0.961720	+	0.274034i	$0.911642\pi$
$192$	13.8564			0.0721688
$193$	58.0026			0.300532			0.150266	−	0.988646i	$-0.451987\pi$
$193$	58.0026			0.300532			0.150266	+	0.988646i	$0.451987\pi$
$194$			53.7285i			0.276951i
$195$		−	11.5716i		−	0.0593415i
$196$	−162.849			−0.830860
$197$	96.0748			0.487689			0.243845	−	0.969814i	$-0.421591\pi$
$197$	96.0748			0.487689			0.243845	+	0.969814i	$0.421591\pi$
$198$		−	58.8552i		−	0.297249i
$199$			198.494i			0.997459i	0.866758	+	0.498729i	$0.166200\pi$
$199$			198.494i			0.997459i	−0.866758	+	0.498729i	$0.833800\pi$
$200$	−14.1421			−0.0707107
$201$			101.349i			0.504224i
$202$	−208.547			−1.03241
$203$		−	132.740i		−	0.653893i
$204$		−	53.1436i		−	0.260508i
$205$			81.2198i			0.396194i
$206$			105.784i			0.513513i
$207$	−62.7740	+	28.6430i	−0.303256	+	0.138372i
$208$	−11.9511			−0.0574571
$209$	35.0889			0.167889
$210$	62.5518			0.297866
$211$	−154.074			−0.730207			−0.365103	−	0.930967i	$-0.618966\pi$
$211$	−154.074			−0.730207			−0.365103	+	0.930967i	$0.618966\pi$
$212$			118.990i			0.561275i
$213$	174.224			0.817952
$214$			52.1697i			0.243784i
$215$	−44.6584			−0.207713
$216$	14.6969			0.0680414
$217$		−	30.9728i		−	0.142732i
$218$			126.131i			0.578584i
$219$	−29.0282			−0.132549
$220$	62.0388			0.281995
$221$			45.8361i			0.207403i
$222$		−	77.8220i		−	0.350550i
$223$	−88.2708			−0.395833			−0.197916	−	0.980219i	$-0.563417\pi$
$223$	−88.2708			−0.395833			−0.197916	+	0.980219i	$0.563417\pi$
$224$		−	64.6032i		−	0.288407i
$225$	−15.0000			−0.0666667
$226$		−	56.1638i		−	0.248512i
$227$		−	220.495i		−	0.971345i	−0.874141	−	0.485673i	$-0.838575\pi$
$227$		−	220.495i		−	0.971345i	0.874141	−	0.485673i	$-0.161425\pi$
$228$			8.76216i			0.0384305i
$229$		−	138.113i		−	0.603115i	−0.953448	−	0.301557i	$-0.902494\pi$
$229$		−	138.113i		−	0.603115i	0.953448	−	0.301557i	$-0.0975065\pi$
$230$	−30.1924	−	66.1696i	−0.131271	−	0.287694i
$231$	−274.403			−1.18789
$232$	32.8752			0.141703
$233$	98.1877			0.421406			0.210703	−	0.977550i	$-0.432425\pi$
$233$	98.1877			0.421406			0.210703	+	0.977550i	$0.432425\pi$
$234$	−12.6760			−0.0541711
$235$			116.794i			0.496995i
$236$	163.213			0.691581
$237$			105.048i			0.443240i
$238$	−247.773			−1.04106
$239$	−421.219			−1.76242			−0.881211	−	0.472722i	$-0.843271\pi$
$239$	−421.219			−1.76242			−0.881211	+	0.472722i	$0.843271\pi$
$240$			15.4919i			0.0645497i
$241$			181.752i			0.754160i	0.926181	+	0.377080i	$0.123072\pi$
$241$			181.752i			0.754160i	−0.926181	+	0.377080i	$0.876928\pi$
$242$	−101.033			−0.417491
$243$	15.5885			0.0641500
$244$		−	84.4272i		−	0.346013i
$245$		−	182.070i		−	0.743144i
$246$	88.9718			0.361674
$247$		−	7.55732i		−	0.0305964i
$248$	7.67089			0.0309310
$249$			76.4752i			0.307130i
$250$		−	15.8114i		−	0.0632456i
$251$			208.037i			0.828835i	0.910087	+	0.414417i	$0.136015\pi$
$251$			208.037i			0.828835i	−0.910087	+	0.414417i	$0.863985\pi$
$252$		−	68.5221i		−	0.271913i
$253$	132.448	+	290.273i	0.523511	+	1.14733i
$254$	−184.775			−0.727460
$255$	59.4164			0.233005
$256$	16.0000			0.0625000
$257$	−463.102			−1.80196			−0.900978	−	0.433866i	$-0.857149\pi$
$257$	−463.102			−1.80196			−0.900978	+	0.433866i	$0.857149\pi$
$258$			48.9208i			0.189615i
$259$	−362.832			−1.40090
$260$		−	13.3617i		−	0.0513912i
$261$	34.8694			0.133599
$262$	4.22548			0.0161278
$263$			34.0014i			0.129283i	0.997909	+	0.0646415i	$0.0205904\pi$
$263$			34.0014i			0.129283i	−0.997909	+	0.0646415i	$0.979410\pi$
$264$		−	67.9602i		−	0.257425i
$265$	−133.035			−0.502020
$266$	40.8521			0.153579
$267$		−	184.022i		−	0.689219i
$268$			117.028i			0.436671i
$269$	499.913			1.85841			0.929207	−	0.369560i	$-0.120491\pi$
$269$	499.913			1.85841			0.929207	+	0.369560i	$0.120491\pi$
$270$			16.4317i			0.0608581i
$271$	246.666			0.910207			0.455104	−	0.890439i	$-0.349602\pi$
$271$	246.666			0.910207			0.455104	+	0.890439i	$0.349602\pi$
$272$		−	61.3650i		−	0.225607i
$273$			59.1000i			0.216483i
$274$		−	169.956i		−	0.620279i
$275$			69.3615i			0.252224i
$276$	−72.4852	+	33.0741i	−0.262627	+	0.119834i
$277$	−350.841			−1.26657			−0.633287	−	0.773917i	$-0.718295\pi$
$277$	−350.841			−1.26657			−0.633287	+	0.773917i	$0.718295\pi$
$278$	−179.990			−0.647447
$279$	8.13621			0.0291620
$280$	72.2286			0.257959
$281$		−	232.377i		−	0.826965i	−0.910512	−	0.413482i	$-0.864312\pi$
$281$		−	232.377i		−	0.826965i	0.910512	−	0.413482i	$-0.135688\pi$
$282$	127.941			0.453692
$283$			462.787i			1.63529i	0.575723	+	0.817645i	$0.304721\pi$
$283$			462.787i			1.63529i	−0.575723	+	0.817645i	$0.695279\pi$
$284$	201.176			0.708367
$285$	−9.79639			−0.0343733
$286$			58.6153i			0.204949i
$287$		−	414.817i		−	1.44535i
$288$	16.9706			0.0589256
$289$	53.6463			0.185627
$290$			36.7556i			0.126743i
$291$			65.8037i			0.226130i
$292$	−33.5189			−0.114791
$293$		−	163.785i		−	0.558995i	−0.960146	−	0.279497i	$-0.909832\pi$
$293$		−	163.785i		−	0.558995i	0.960146	−	0.279497i	$-0.0901678\pi$
$294$	−199.448			−0.678394
$295$			182.478i			0.618568i
$296$		−	89.8611i		−	0.303585i
$297$		−	72.0826i		−	0.242702i
$298$		−	186.168i		−	0.624725i
$299$	62.5181	−	28.5263i	0.209091	−	0.0954056i
$300$	−17.3205			−0.0577350
$301$	228.085			0.757758
$302$	210.909			0.698374
$303$	−255.416			−0.842959
$304$			10.1177i			0.0332818i
$305$	94.3924			0.309483
$306$		−	65.0874i		−	0.212704i
$307$	−359.655			−1.17152			−0.585758	−	0.810486i	$-0.699203\pi$
$307$	−359.655			−1.17152			−0.585758	+	0.810486i	$0.699203\pi$
$308$	−316.853			−1.02874
$309$			129.558i			0.419281i
$310$			8.57632i			0.0276655i
$311$	314.588			1.01154			0.505768	−	0.862669i	$-0.331209\pi$
$311$	314.588			1.01154			0.505768	+	0.862669i	$0.331209\pi$
$312$	−14.6370			−0.0469136
$313$			451.729i			1.44322i	0.692298	+	0.721612i	$0.256599\pi$
$313$			451.729i			1.44322i	−0.692298	+	0.721612i	$0.743401\pi$
$314$			322.706i			1.02773i
$315$	76.6100			0.243206
$316$			121.299i			0.383858i
$317$	179.052			0.564834			0.282417	−	0.959292i	$-0.408864\pi$
$317$	179.052			0.564834			0.282417	+	0.959292i	$0.408864\pi$
$318$			145.733i			0.458279i
$319$		−	161.240i		−	0.505454i
$320$			17.8885i			0.0559017i
$321$			63.8946i			0.199048i
$322$	154.203	+	337.950i	0.478891	+	1.04954i
$323$	38.8044			0.120138
$324$	18.0000			0.0555556
$325$	14.9389			0.0459657
$326$	368.454			1.13023
$327$			154.479i			0.472412i
$328$	102.736			0.313219
$329$		−	596.505i		−	1.81309i
$330$	75.9818			0.230248
$331$	528.798			1.59758			0.798788	−	0.601612i	$-0.205475\pi$
$331$	528.798			1.59758			0.798788	+	0.601612i	$0.205475\pi$
$332$			88.3060i			0.265982i
$333$		−	95.3121i		−	0.286223i
$334$	−250.472			−0.749916
$335$	−130.841			−0.390570
$336$		−	79.1225i		−	0.235484i
$337$		−	493.496i		−	1.46438i	−0.681100	−	0.732191i	$-0.738498\pi$
$337$		−	493.496i		−	1.46438i	0.681100	−	0.732191i	$-0.261502\pi$
$338$	−226.378			−0.669757
$339$		−	68.7863i		−	0.202910i
$340$	68.6081			0.201789
$341$		−	37.6227i		−	0.110330i
$342$			10.7314i			0.0313784i
$343$			370.297i			1.07958i
$344$			56.4889i			0.164212i
$345$	−36.9780	−	81.0409i	−0.107183	−	0.234901i
$346$	10.7607			0.0311003
$347$	133.171			0.383778			0.191889	−	0.981417i	$-0.438539\pi$
$347$	133.171			0.383778			0.191889	+	0.981417i	$0.438539\pi$
$348$	40.2637			0.115700
$349$	62.9714			0.180434			0.0902169	−	0.995922i	$-0.471244\pi$
$349$	62.9714			0.180434			0.0902169	+	0.995922i	$0.471244\pi$
$350$			80.7540i			0.230726i
$351$	−15.5249			−0.0442305
$352$		−	78.4736i		−	0.222936i
$353$	−103.337			−0.292739			−0.146370	−	0.989230i	$-0.546759\pi$
$353$	−103.337			−0.292739			−0.146370	+	0.989230i	$0.546759\pi$
$354$	199.894			0.564673
$355$			224.922i			0.633583i
$356$		−	212.490i		−	0.596881i
$357$	−303.459			−0.850026
$358$	451.677			1.26167
$359$			608.721i			1.69560i	0.530314	+	0.847801i	$0.322074\pi$
$359$			608.721i			1.69560i	−0.530314	+	0.847801i	$0.677926\pi$
$360$			18.9737i			0.0527046i
$361$	354.602			0.982277
$362$			162.862i			0.449895i
$363$	−123.739			−0.340880
$364$			68.2428i			0.187480i
$365$		−	37.4753i		−	0.102672i
$366$		−	103.402i		−	0.282518i
$367$		−	213.718i		−	0.582338i	−0.956672	−	0.291169i	$-0.905956\pi$
$367$		−	213.718i		−	0.582338i	0.956672	−	0.291169i	$-0.0940441\pi$
$368$	−83.6987	+	38.1907i	−0.227442	+	0.103779i
$369$	108.968			0.295306
$370$	100.468			0.271535
$371$	679.455			1.83142
$372$	9.39489			0.0252551
$373$			542.556i			1.45457i	0.686334	+	0.727287i	$0.259219\pi$
$373$			542.556i			1.45457i	−0.686334	+	0.727287i	$0.740781\pi$
$374$	−300.971			−0.804735
$375$		−	19.3649i		−	0.0516398i
$376$	147.734			0.392909
$377$	−34.7273			−0.0921148
$378$		−	83.9221i		−	0.222016i
$379$		−	348.769i		−	0.920235i	−0.887858	−	0.460117i	$-0.847807\pi$
$379$		−	348.769i		−	0.920235i	0.887858	−	0.460117i	$-0.152193\pi$
$380$	−11.3119			−0.0297682
$381$	−226.302			−0.593969
$382$			148.041i			0.387543i
$383$		−	672.210i		−	1.75512i	−0.479470	−	0.877559i	$-0.659171\pi$
$383$		−	672.210i		−	1.75512i	0.479470	−	0.877559i	$-0.340829\pi$
$384$	19.5959			0.0510310
$385$		−	354.253i		−	0.920136i
$386$	82.0281			0.212508
$387$			59.9155i			0.154820i
$388$			75.9836i			0.195834i
$389$		−	464.622i		−	1.19440i	−0.802092	−	0.597200i	$-0.796280\pi$
$389$		−	464.622i		−	1.19440i	0.802092	−	0.597200i	$-0.203720\pi$
$390$		−	16.3647i		−	0.0419608i
$391$	146.473	+	321.010i	0.374612	+	0.820999i
$392$	−230.303			−0.587507
$393$	5.17513			0.0131683
$394$	135.870			0.344848
$395$	−135.616			−0.343333
$396$		−	83.2338i		−	0.210186i
$397$	−1.92370			−0.00484560			−0.00242280	−	0.999997i	$-0.500771\pi$
$397$	−1.92370			−0.00484560			−0.00242280	+	0.999997i	$0.500771\pi$
$398$			280.713i			0.705310i
$399$	50.0335			0.125397
$400$	−20.0000			−0.0500000
$401$		−	417.281i		−	1.04060i	−0.853983	−	0.520301i	$-0.825820\pi$
$401$		−	417.281i		−	1.04060i	0.853983	−	0.520301i	$-0.174180\pi$
$402$			143.329i			0.356540i
$403$	−8.10305			−0.0201068
$404$	−294.930			−0.730024
$405$			20.1246i			0.0496904i
$406$		−	187.723i		−	0.462372i
$407$	−440.733			−1.08288
$408$		−	75.1564i		−	0.184207i
$409$	−812.820			−1.98733			−0.993667	−	0.112361i	$-0.964159\pi$
$409$	−812.820			−1.98733			−0.993667	+	0.112361i	$0.964159\pi$
$410$			114.862i			0.280152i
$411$		−	208.153i		−	0.506455i
$412$			149.601i			0.363108i
$413$		−	931.974i		−	2.25660i
$414$	−88.7759	+	40.5074i	−0.214434	+	0.0978439i
$415$	−98.7291			−0.237901
$416$	−16.9014			−0.0406283
$417$	−220.442			−0.528638
$418$	49.6232			0.118716
$419$			268.155i			0.639988i	0.947420	+	0.319994i	$0.103681\pi$
$419$			268.155i			0.639988i	−0.947420	+	0.319994i	$0.896319\pi$
$420$	88.4616			0.210623
$421$		−	174.098i		−	0.413533i	−0.978390	−	0.206767i	$-0.933706\pi$
$421$		−	174.098i		−	0.413533i	0.978390	−	0.206767i	$-0.0662941\pi$
$422$	−217.893			−0.516334
$423$	156.695			0.370438
$424$			168.278i			0.396881i
$425$			76.7062i			0.180485i
$426$	246.390			0.578380
$427$	−482.094			−1.12903
$428$			73.7791i			0.172381i
$429$			71.7888i			0.167340i
$430$	−63.1565			−0.146876
$431$			291.962i			0.677407i	0.940893	+	0.338704i	$0.109988\pi$
$431$			291.962i			0.677407i	−0.940893	+	0.338704i	$0.890012\pi$
$432$	20.7846			0.0481125
$433$		−	439.111i		−	1.01411i	−0.861913	−	0.507056i	$-0.830734\pi$
$433$		−	439.111i		−	1.01411i	0.861913	−	0.507056i	$-0.169266\pi$
$434$		−	43.8021i		−	0.100927i
$435$			45.0162i			0.103486i
$436$			178.377i			0.409121i
$437$	−24.1501	−	52.9272i	−0.0552633	−	0.121115i
$438$	−41.0521			−0.0937263
$439$	−62.0644			−0.141377			−0.0706884	−	0.997498i	$-0.522520\pi$
$439$	−62.0644			−0.141377			−0.0706884	+	0.997498i	$0.522520\pi$
$440$	87.7362			0.199400
$441$	−244.273			−0.553907
$442$			64.8221i			0.146656i
$443$	−261.415			−0.590102			−0.295051	−	0.955482i	$-0.595337\pi$
$443$	−261.415			−0.590102			−0.295051	+	0.955482i	$0.595337\pi$
$444$		−	110.057i		−	0.247876i
$445$	237.571			0.533867
$446$	−124.834			−0.279896
$447$		−	228.008i		−	0.510086i
$448$		−	91.3628i		−	0.203935i
$449$	431.325			0.960634			0.480317	−	0.877095i	$-0.340522\pi$
$449$	431.325			0.960634			0.480317	+	0.877095i	$0.340522\pi$
$450$	−21.2132			−0.0471405
$451$		−	503.878i		−	1.11725i
$452$		−	79.4276i		−	0.175725i
$453$	258.310			0.570220
$454$		−	311.828i		−	0.686845i
$455$	−76.2977			−0.167687
$456$			12.3916i			0.0271745i
$457$		−	734.574i		−	1.60738i	−0.595046	−	0.803692i	$-0.702866\pi$
$457$		−	734.574i		−	1.60738i	0.595046	−	0.803692i	$-0.297134\pi$
$458$		−	195.322i		−	0.426466i
$459$		−	79.7154i		−	0.173672i
$460$	−42.6985	−	93.5780i	−0.0928229	−	0.203430i
$461$	626.021			1.35796			0.678982	−	0.734155i	$-0.262422\pi$
$461$	626.021			1.35796			0.678982	+	0.734155i	$0.262422\pi$
$462$	−388.064			−0.839966
$463$	−320.725			−0.692712			−0.346356	−	0.938103i	$-0.612581\pi$
$463$	−320.725			−0.692712			−0.346356	+	0.938103i	$0.612581\pi$
$464$	46.4925			0.100199
$465$			10.5038i			0.0225888i
$466$	138.858			0.297979
$467$		−	148.416i		−	0.317807i	−0.987294	−	0.158904i	$-0.949204\pi$
$467$		−	148.416i		−	0.317807i	0.987294	−	0.158904i	$-0.0507959\pi$
$468$	−17.9266			−0.0383048
$469$	668.249			1.42484
$470$			165.171i			0.351429i
$471$			395.233i			0.839136i
$472$	230.818			0.489021
$473$	277.055			0.585741
$474$			148.560i			0.313418i
$475$		−	12.6471i		−	0.0266255i
$476$	−350.405			−0.736144
$477$			178.486i			0.374183i
$478$	−595.694			−1.24622
$479$			749.170i			1.56403i	0.623260	+	0.782014i	$0.285808\pi$
$479$			749.170i			1.56403i	−0.623260	+	0.782014i	$0.714192\pi$
$480$			21.9089i			0.0456435i
$481$			94.9236i			0.197346i
$482$			257.037i			0.533271i
$483$	188.859	+	413.903i	0.391012	+	0.856942i
$484$	−142.882			−0.295210
$485$	−84.9523			−0.175159
$486$	22.0454			0.0453609
$487$	289.122			0.593679			0.296839	−	0.954927i	$-0.404067\pi$
$487$	289.122			0.593679			0.296839	+	0.954927i	$0.404067\pi$
$488$		−	119.398i		−	0.244668i
$489$	451.262			0.922827
$490$		−	257.486i		−	0.525482i
$491$	768.805			1.56579			0.782897	−	0.622152i	$-0.213741\pi$
$491$	768.805			1.56579			0.782897	+	0.622152i	$0.213741\pi$
$492$	125.825			0.255742
$493$		−	178.313i		−	0.361690i
$494$		−	10.6877i		−	0.0216350i
$495$	93.0583			0.187997
$496$	10.8483			0.0218715
$497$		−	1148.75i		−	2.31137i
$498$			108.152i			0.217173i
$499$	201.586			0.403981			0.201990	−	0.979388i	$-0.435259\pi$
$499$	201.586			0.403981			0.201990	+	0.979388i	$0.435259\pi$
$500$		−	22.3607i		−	0.0447214i
$501$	−306.764			−0.612304
$502$			294.209i			0.586075i
$503$		−	385.180i		−	0.765766i	−0.923797	−	0.382883i	$-0.874931\pi$
$503$		−	385.180i		−	0.765766i	0.923797	−	0.382883i	$-0.125069\pi$
$504$		−	96.9048i		−	0.192272i
$505$		−	329.741i		−	0.652953i
$506$	187.310	+	410.509i	0.370178	+	0.811282i
$507$	−277.255			−0.546854
$508$	−261.311			−0.514392
$509$	563.072			1.10623			0.553116	−	0.833104i	$-0.313439\pi$
$509$	563.072			1.10623			0.553116	+	0.833104i	$0.313439\pi$
$510$	84.0274			0.164760
$511$			191.399i			0.374557i
$512$	22.6274			0.0441942
$513$			13.1432i			0.0256204i
$514$	−654.926			−1.27417
$515$	−167.259			−0.324774
$516$			69.1844i			0.134078i
$517$		−	724.576i		−	1.40150i
$518$	−513.122			−0.990584
$519$	13.1791			0.0253933
$520$		−	18.8963i		−	0.0363391i
$521$			687.431i			1.31945i	0.751509	+	0.659723i	$0.229326\pi$
$521$			687.431i			1.31945i	−0.751509	+	0.659723i	$0.770674\pi$
$522$	49.3128			0.0944689
$523$		−	580.256i		−	1.10948i	−0.832025	−	0.554738i	$-0.812819\pi$
$523$		−	580.256i		−	1.10948i	0.832025	−	0.554738i	$-0.187181\pi$
$524$	5.97573			0.0114041
$525$			98.9031i			0.188387i
$526$			48.0853i			0.0914169i
$527$		−	41.6065i		−	0.0789498i
$528$		−	96.1102i		−	0.182027i
$529$	346.684	−	399.564i	0.655357	−	0.755320i
$530$	−188.140			−0.354982
$531$	244.819			0.461054
$532$	57.7737			0.108597
$533$	−108.524			−0.203609
$534$		−	260.246i		−	0.487352i
$535$	−82.4875			−0.154182
$536$			165.502i			0.308773i
$537$	553.189			1.03015
$538$	706.984			1.31410
$539$			1129.54i			2.09563i
$540$			23.2379i			0.0430331i
$541$	148.687			0.274838			0.137419	−	0.990513i	$-0.456119\pi$
$541$	148.687			0.274838			0.137419	+	0.990513i	$0.456119\pi$
$542$	348.839			0.643614
$543$			199.465i			0.367338i
$544$		−	86.7832i		−	0.159528i
$545$	−199.431			−0.365929
$546$			83.5800i			0.153077i
$547$	−648.445			−1.18546			−0.592729	−	0.805402i	$-0.701949\pi$
$547$	−648.445			−1.18546			−0.592729	+	0.805402i	$0.701949\pi$
$548$		−	240.355i		−	0.438603i
$549$		−	126.641i		−	0.230675i
$550$			98.0920i			0.178349i
$551$			29.3998i			0.0533571i
$552$	−102.510	+	46.7739i	−0.185706	+	0.0847353i
$553$	692.638			1.25251
$554$	−496.164			−0.895603
$555$	123.047			0.221707
$556$	−254.545			−0.457814
$557$			697.203i			1.25171i	0.779939	+	0.625856i	$0.215250\pi$
$557$			697.203i			1.25171i	−0.779939	+	0.625856i	$0.784750\pi$
$558$	11.5063			0.0206207
$559$		−	59.6713i		−	0.106746i
$560$	102.147			0.182405
$561$	−368.612			−0.657063
$562$		−	328.631i		−	0.584752i
$563$		−	890.248i		−	1.58126i	−0.612295	−	0.790629i	$-0.709754\pi$
$563$		−	890.248i		−	1.58126i	0.612295	−	0.790629i	$-0.290246\pi$
$564$	180.936			0.320809
$565$	88.8028			0.157173
$566$			654.480i			1.15632i
$567$		−	102.783i		−	0.181275i
$568$	284.506			0.500891
$569$		−	548.456i		−	0.963894i	−0.876200	−	0.481947i	$-0.839930\pi$
$569$		−	548.456i		−	0.963894i	0.876200	−	0.481947i	$-0.160070\pi$
$570$	−13.8542			−0.0243056
$571$			479.794i			0.840270i	0.907462	+	0.420135i	$0.138017\pi$
$571$			479.794i			0.840270i	−0.907462	+	0.420135i	$0.861983\pi$
$572$			82.8946i			0.144921i
$573$			181.313i			0.316427i
$574$		−	586.639i		−	1.02202i
$575$	104.623	−	47.7384i	0.181954	−	0.0830233i
$576$	24.0000			0.0416667
$577$	12.3741			0.0214456			0.0107228	−	0.999943i	$-0.496587\pi$
$577$	12.3741			0.0214456			0.0107228	+	0.999943i	$0.496587\pi$
$578$	75.8673			0.131258
$579$	100.464			0.173512
$580$			51.9802i			0.0896211i
$581$	504.243			0.867887
$582$			93.0605i			0.159898i
$583$	825.335			1.41567
$584$	−47.4029			−0.0811694
$585$		−	20.0426i		−	0.0342608i
$586$		−	231.628i		−	0.395269i
$587$	613.203			1.04464			0.522320	−	0.852750i	$-0.325067\pi$
$587$	613.203			1.04464			0.522320	+	0.852750i	$0.325067\pi$
$588$	−282.062			−0.479697
$589$			6.85996i			0.0116468i
$590$			258.062i			0.437394i
$591$	166.406			0.281568
$592$		−	127.083i		−	0.214667i
$593$	−208.251			−0.351182			−0.175591	−	0.984463i	$-0.556184\pi$
$593$	−208.251			−0.351182			−0.175591	+	0.984463i	$0.556184\pi$
$594$		−	101.940i		−	0.171617i
$595$		−	391.764i		−	0.658427i
$596$		−	263.281i		−	0.441747i
$597$			343.802i			0.575883i
$598$	88.4140	−	40.3423i	0.147849	−	0.0674620i
$599$	925.513			1.54510			0.772548	−	0.634956i	$-0.218982\pi$
$599$	925.513			1.54510			0.772548	+	0.634956i	$0.218982\pi$
$600$	−24.4949			−0.0408248
$601$	−922.104			−1.53428			−0.767142	−	0.641478i	$-0.778322\pi$
$601$	−922.104			−1.53428			−0.767142	+	0.641478i	$0.778322\pi$
$602$	322.561			0.535816
$603$			175.542i			0.291114i
$604$	298.270			0.493825
$605$		−	159.747i		−	0.264044i
$606$	−361.213			−0.596062
$607$	−73.7934			−0.121571			−0.0607853	−	0.998151i	$-0.519360\pi$
$607$	−73.7934			−0.121571			−0.0607853	+	0.998151i	$0.519360\pi$
$608$			14.3085i			0.0235338i
$609$		−	229.913i		−	0.377525i
$610$	133.491			0.218838
$611$	−156.057			−0.255412
$612$		−	92.0475i		−	0.150404i
$613$		−	808.004i		−	1.31811i	−0.752093	−	0.659057i	$-0.770956\pi$
$613$		−	808.004i		−	1.31811i	0.752093	−	0.659057i	$-0.229044\pi$
$614$	−508.629			−0.828387
$615$			140.677i			0.228743i
$616$	−448.098			−0.727432
$617$			484.018i			0.784470i	0.919865	+	0.392235i	$0.128298\pi$
$617$			484.018i			0.784470i	−0.919865	+	0.392235i	$0.871702\pi$
$618$			183.223i			0.296477i
$619$		−	454.285i		−	0.733902i	−0.930240	−	0.366951i	$-0.880402\pi$
$619$		−	454.285i		−	0.733902i	0.930240	−	0.366951i	$-0.119598\pi$
$620$			12.1287i			0.0195625i
$621$	−108.728	+	49.6112i	−0.175085	+	0.0798892i
$622$	444.895			0.715265
$623$	−1213.35			−1.94760
$624$	−20.6999			−0.0331729
$625$	25.0000			0.0400000
$626$			638.841i			1.02051i
$627$	60.7757			0.0969309
$628$			456.376i			0.726713i
$629$	−487.402			−0.774884
$630$	108.343			0.171973
$631$		−	385.780i		−	0.611380i	−0.952131	−	0.305690i	$-0.901113\pi$
$631$		−	385.780i		−	0.611380i	0.952131	−	0.305690i	$-0.0988870\pi$
$632$			171.543i			0.271428i
$633$	−266.863			−0.421585
$634$	253.218			0.399398
$635$		−	292.155i		−	0.460086i
$636$			206.097i			0.324052i
$637$	243.277			0.381911
$638$		−	228.027i		−	0.357410i
$639$	301.765			0.472245
$640$			25.2982i			0.0395285i
$641$		−	1009.69i		−	1.57518i	−0.616199	−	0.787591i	$-0.711328\pi$
$641$		−	1009.69i		−	1.57518i	0.616199	−	0.787591i	$-0.288672\pi$
$642$			90.3606i			0.140749i
$643$		−	849.482i		−	1.32112i	−0.750772	−	0.660561i	$-0.770318\pi$
$643$		−	849.482i		−	1.32112i	0.750772	−	0.660561i	$-0.229682\pi$
$644$	218.076	+	477.934i	0.338627	+	0.742133i
$645$	−77.3506			−0.119923
$646$	54.8777			0.0849501
$647$	−788.852			−1.21924			−0.609622	−	0.792692i	$-0.708679\pi$
$647$	−788.852			−1.21924			−0.609622	+	0.792692i	$0.708679\pi$
$648$	25.4558			0.0392837
$649$		−	1132.07i		−	1.74433i
$650$	21.1267			0.0325027
$651$		−	53.6464i		−	0.0824062i
$652$	521.073			0.799191
$653$	−912.671			−1.39766			−0.698830	−	0.715288i	$-0.746295\pi$
$653$	−912.671			−1.39766			−0.698830	+	0.715288i	$0.746295\pi$
$654$			218.466i			0.334046i
$655$			6.68107i			0.0102001i
$656$	145.290			0.221479
$657$	−50.2784			−0.0765272
$658$		−	843.586i		−	1.28205i
$659$			845.536i			1.28306i	0.767098	+	0.641530i	$0.221700\pi$
$659$			845.536i			1.28306i	−0.767098	+	0.641530i	$0.778300\pi$
$660$	107.454			0.162810
$661$			594.404i			0.899250i	0.893217	+	0.449625i	$0.148442\pi$
$661$			594.404i			0.899250i	−0.893217	+	0.449625i	$0.851558\pi$
$662$	747.833			1.12966
$663$			79.3905i			0.119744i
$664$			124.884i			0.188078i
$665$			64.5929i			0.0971322i
$666$		−	134.792i		−	0.202390i
$667$	−243.210	+	110.974i	−0.364633	+	0.166378i
$668$	−354.221			−0.530271
$669$	−152.889			−0.228534
$670$	−185.037			−0.276175
$671$	−585.600			−0.872727
$672$		−	111.896i		−	0.166512i
$673$	−49.6946			−0.0738404			−0.0369202	−	0.999318i	$-0.511755\pi$
$673$	−49.6946			−0.0738404			−0.0369202	+	0.999318i	$0.511755\pi$
$674$		−	697.909i		−	1.03547i
$675$	−25.9808			−0.0384900
$676$	−320.146			−0.473589
$677$			256.435i			0.378782i	0.981902	+	0.189391i	$0.0606513\pi$
$677$			256.435i			0.378782i	−0.981902	+	0.189391i	$0.939349\pi$
$678$		−	97.2786i		−	0.143479i
$679$	433.880			0.638998
$680$	97.0265			0.142686
$681$		−	381.909i		−	0.560806i
$682$		−	53.2065i		−	0.0780154i
$683$	−1344.91			−1.96912			−0.984560	−	0.175045i	$-0.943993\pi$
$683$	−1344.91			−1.96912			−0.984560	+	0.175045i	$0.943993\pi$
$684$			15.1765i			0.0221879i
$685$	268.725			0.392299
$686$			523.679i			0.763380i
$687$		−	239.219i		−	0.348208i
$688$			79.8873i			0.116115i
$689$		−	177.758i		−	0.257994i
$690$	−52.2948	−	114.609i	−0.0757896	−	0.166100i
$691$	559.144			0.809180			0.404590	−	0.914498i	$-0.367414\pi$
$691$	559.144			0.809180			0.404590	+	0.914498i	$0.367414\pi$
$692$	15.2180			0.0219913
$693$	−475.280			−0.685829
$694$	188.332			0.271372
$695$		−	284.589i		−	0.409481i
$696$	56.9415			0.0818125
$697$		−	557.234i		−	0.799475i
$698$	89.0550			0.127586
$699$	170.066			0.243299
$700$			114.203i			0.163148i
$701$		−	1187.16i		−	1.69352i	−0.531977	−	0.846758i	$-0.678551\pi$
$701$		−	1187.16i		−	1.69352i	0.531977	−	0.846758i	$-0.321449\pi$
$702$	−21.9555			−0.0312757
$703$	80.3614			0.114312
$704$		−	110.978i		−	0.157640i
$705$			202.293i			0.286940i
$706$	−146.140			−0.206998
$707$			1684.10i			2.38203i
$708$	282.693			0.399284
$709$			1093.87i			1.54284i	0.636325	+	0.771421i	$0.280454\pi$
$709$			1093.87i			1.54284i	−0.636325	+	0.771421i	$0.719546\pi$
$710$			318.088i			0.448011i
$711$			181.948i			0.255905i
$712$		−	300.506i		−	0.422059i
$713$	−56.7492	+	25.8940i	−0.0795921	+	0.0363169i
$714$	−429.156			−0.601059
$715$	−92.6789			−0.129621
$716$	638.767			0.892133
$717$	−729.573			−1.01754
$718$			860.862i			1.19897i
$719$	−1191.45			−1.65709			−0.828545	−	0.559923i	$-0.810831\pi$
$719$	−1191.45			−1.65709			−0.828545	+	0.559923i	$0.810831\pi$
$720$			26.8328i			0.0372678i
$721$	854.245			1.18481
$722$	501.483			0.694575
$723$			314.804i			0.435414i
$724$			230.322i			0.318124i
$725$	−58.1157			−0.0801596
$726$	−174.994			−0.241038
$727$			1123.33i			1.54516i	0.634920	+	0.772578i	$0.281033\pi$
$727$			1123.33i			1.54516i	−0.634920	+	0.772578i	$0.718967\pi$
$728$			96.5098i			0.132568i
$729$	27.0000			0.0370370
$730$		−	52.9981i		−	0.0726001i
$731$	306.393			0.419142
$732$		−	146.232i		−	0.199771i
$733$			81.6381i			0.111375i	0.998448	+	0.0556877i	$0.0177351\pi$
$733$			81.6381i			0.111375i	−0.998448	+	0.0556877i	$0.982265\pi$
$734$		−	302.243i		−	0.411775i
$735$		−	315.355i		−	0.429054i
$736$	−118.368	+	54.0098i	−0.160826	+	0.0733829i
$737$	811.723			1.10139
$738$	154.104			0.208813
$739$	−706.438			−0.955938			−0.477969	−	0.878377i	$-0.658627\pi$
$739$	−706.438			−0.955938			−0.477969	+	0.878377i	$0.658627\pi$
$740$	142.083			0.192004
$741$		−	13.0897i		−	0.0176649i
$742$	960.895			1.29501
$743$			1028.94i			1.38484i	0.721494	+	0.692420i	$0.243456\pi$
$743$			1028.94i			1.38484i	−0.721494	+	0.692420i	$0.756544\pi$
$744$	13.2864			0.0178580
$745$	294.357			0.395111
$746$			767.290i			1.02854i
$747$			132.459i			0.177321i
$748$	−425.637			−0.569033
$749$	421.291			0.562472
$750$		−	27.3861i		−	0.0365148i
$751$			650.518i			0.866202i	0.901345	+	0.433101i	$0.142581\pi$
$751$			650.518i			0.866202i	−0.901345	+	0.433101i	$0.857419\pi$
$752$	208.927			0.277829
$753$			360.331i			0.478528i
$754$	−49.1118			−0.0651350
$755$			333.476i			0.441690i
$756$		−	118.684i		−	0.156989i
$757$			372.692i			0.492328i	0.969228	+	0.246164i	$0.0791702\pi$
$757$			372.692i			0.492328i	−0.969228	+	0.246164i	$0.920830\pi$
$758$		−	493.234i		−	0.650704i
$759$	229.407	+	502.768i	0.302249	+	0.662409i
$760$	−15.9974			−0.0210493
$761$	−127.577			−0.167644			−0.0838219	−	0.996481i	$-0.526713\pi$
$761$	−127.577			−0.167644			−0.0838219	+	0.996481i	$0.526713\pi$
$762$	−320.040			−0.419999
$763$	1018.56			1.33494
$764$			209.362i			0.274034i
$765$	102.912			0.134526
$766$		−	950.648i		−	1.24106i
$767$	−243.822			−0.317890
$768$	27.7128			0.0360844
$769$			1274.35i			1.65715i	0.559878	+	0.828575i	$0.310848\pi$
$769$			1274.35i			1.65715i	−0.559878	+	0.828575i	$0.689152\pi$
$770$		−	500.989i		−	0.650635i
$771$	−802.117			−1.04036
$772$	116.005			0.150266
$773$			1009.21i			1.30557i	0.757543	+	0.652785i	$0.226399\pi$
$773$			1009.21i			1.30557i	−0.757543	+	0.652785i	$0.773601\pi$
$774$			84.7333i			0.109475i
$775$	−13.5604			−0.0174972
$776$			107.457i			0.138476i
$777$	−628.444			−0.808808
$778$		−	657.074i		−	0.844569i
$779$			91.8751i			0.117940i
$780$		−	23.1432i		−	0.0296707i
$781$		−	1395.39i		−	1.78667i
$782$	207.145	+	453.977i	0.264891	+	0.580534i
$783$	60.3956			0.0771336
$784$	−325.697			−0.415430
$785$	−510.243			−0.649992
$786$	7.31874			0.00931138
$787$			618.658i			0.786096i	0.919518	+	0.393048i	$0.128579\pi$
$787$			618.658i			0.786096i	−0.919518	+	0.393048i	$0.871421\pi$
$788$	192.150			0.243845
$789$			58.8922i			0.0746416i
$790$	−191.791			−0.242773
$791$	−453.545			−0.573382
$792$		−	117.710i		−	0.148624i
$793$			126.125i			0.159047i
$794$	−2.72053			−0.00342636
$795$	−230.424			−0.289841
$796$			396.989i			0.498729i
$797$			849.216i			1.06552i	0.846268	+	0.532758i	$0.178844\pi$
$797$			849.216i			1.06552i	−0.846268	+	0.532758i	$0.821156\pi$
$798$	70.7580			0.0886692
$799$		−	801.301i		−	1.00288i
$800$	−28.2843			−0.0353553
$801$		−	318.735i		−	0.397921i
$802$		−	590.125i		−	0.735817i
$803$			232.492i			0.289530i
$804$			202.698i			0.252112i
$805$	−534.346	+	243.816i	−0.663784	+	0.302877i
$806$	−11.4594			−0.0142177
$807$	865.875			1.07296
$808$	−417.093			−0.516205
$809$	194.891			0.240904			0.120452	−	0.992719i	$-0.461566\pi$
$809$	194.891			0.240904			0.120452	+	0.992719i	$0.461566\pi$
$810$			28.4605i			0.0351364i
$811$	−224.475			−0.276788			−0.138394	−	0.990377i	$-0.544194\pi$
$811$	−224.475			−0.276788			−0.138394	+	0.990377i	$0.544194\pi$
$812$		−	265.480i		−	0.326946i
$813$	427.238			0.525508
$814$	−623.291			−0.765713
$815$			582.577i			0.714818i
$816$		−	106.287i		−	0.130254i
$817$	−50.5171			−0.0618324
$818$	−1149.50			−1.40526
$819$			102.364i			0.124987i
$820$			162.440i			0.198097i
$821$	−472.157			−0.575100			−0.287550	−	0.957766i	$-0.592841\pi$
$821$	−472.157			−0.575100			−0.287550	+	0.957766i	$0.592841\pi$
$822$		−	294.373i		−	0.358118i
$823$	1081.17			1.31370			0.656849	−	0.754022i	$-0.271889\pi$
$823$	1081.17			1.31370			0.656849	+	0.754022i	$0.271889\pi$
$824$			211.567i			0.256756i
$825$			120.138i			0.145621i
$826$		−	1318.01i		−	1.59565i
$827$			64.5413i			0.0780427i	0.999238	+	0.0390214i	$0.0124240\pi$
$827$			64.5413i			0.0780427i	−0.999238	+	0.0390214i	$0.987576\pi$
$828$	−125.548	+	57.2861i	−0.151628	+	0.0691861i
$829$	−548.619			−0.661784			−0.330892	−	0.943669i	$-0.607350\pi$
$829$	−548.619			−0.661784			−0.330892	+	0.943669i	$0.607350\pi$
$830$	−139.624			−0.168222
$831$	−607.674			−0.731256
$832$	−23.9022			−0.0287286
$833$			1249.15i			1.49958i
$834$	−311.752			−0.373803
$835$		−	396.031i		−	0.474289i
$836$	70.1777			0.0839447
$837$	14.0923			0.0168367
$838$			379.229i			0.452540i
$839$			927.859i			1.10591i	0.833211	+	0.552955i	$0.186500\pi$
$839$			927.859i			1.10591i	−0.833211	+	0.552955i	$0.813500\pi$
$840$	125.104			0.148933
$841$	−705.903			−0.839361
$842$		−	246.211i		−	0.292412i
$843$		−	402.489i		−	0.477448i
$844$	−308.147			−0.365103
$845$		−	357.935i		−	0.423591i
$846$	221.601			0.261939
$847$			815.880i			0.963259i
$848$			237.981i			0.280638i
$849$			801.570i			0.944135i
$850$			108.479i			0.127622i
$851$	303.337	+	664.791i	0.356447	+	0.781189i
$852$	348.448			0.408976
$853$	−348.485			−0.408540			−0.204270	−	0.978915i	$-0.565482\pi$
$853$	−348.485			−0.408540			−0.204270	+	0.978915i	$0.565482\pi$
$854$	−681.783			−0.798341
$855$	−16.9679			−0.0198454
$856$			104.339i			0.121892i
$857$	1026.47			1.19774			0.598872	−	0.800845i	$-0.295616\pi$
$857$	1026.47			1.19774			0.598872	+	0.800845i	$0.295616\pi$
$858$			101.525i			0.118327i
$859$	1188.51			1.38360			0.691801	−	0.722088i	$-0.256817\pi$
$859$	1188.51			1.38360			0.691801	+	0.722088i	$0.256817\pi$
$860$	−89.3167			−0.103857
$861$		−	718.484i		−	0.834476i
$862$			412.897i			0.478999i
$863$	−1055.95			−1.22359			−0.611793	−	0.791018i	$-0.709551\pi$
$863$	−1055.95			−1.22359			−0.611793	+	0.791018i	$0.709551\pi$
$864$	29.3939			0.0340207
$865$			17.0142i			0.0196696i
$866$		−	620.996i		−	0.717086i
$867$	92.9180			0.107172
$868$		−	61.9456i		−	0.0713659i
$869$	841.348			0.968180
$870$			63.6625i			0.0731753i
$871$		−	174.826i		−	0.200719i
$872$			252.263i			0.289292i
$873$			113.975i			0.130556i
$874$	−34.1534	−	74.8504i	−0.0390771	−	0.0856412i
$875$	−127.683			−0.145924
$876$	−58.0565			−0.0662745
$877$	−921.525			−1.05077			−0.525385	−	0.850865i	$-0.676079\pi$
$877$	−921.525			−1.05077			−0.525385	+	0.850865i	$0.676079\pi$
$878$	−87.7724			−0.0999685
$879$		−	283.685i		−	0.322736i
$880$	124.078			0.140997
$881$			1210.60i			1.37412i	0.726600	+	0.687061i	$0.241099\pi$
$881$			1210.60i			1.37412i	−0.726600	+	0.687061i	$0.758901\pi$
$882$	−345.454			−0.391671
$883$	−386.146			−0.437311			−0.218656	−	0.975802i	$-0.570167\pi$
$883$	−386.146			−0.437311			−0.218656	+	0.975802i	$0.570167\pi$
$884$			91.6722i			0.103702i
$885$			316.061i			0.357131i
$886$	−369.697			−0.417265
$887$	−493.959			−0.556887			−0.278444	−	0.960453i	$-0.589819\pi$
$887$	−493.959			−0.556887			−0.278444	+	0.960453i	$0.589819\pi$
$888$		−	155.644i		−	0.175275i
$889$			1492.13i			1.67844i
$890$	335.976			0.377501
$891$		−	124.851i		−	0.140124i
$892$	−176.542			−0.197916
$893$			132.116i			0.147946i
$894$		−	322.452i		−	0.360685i
$895$			714.164i			0.797948i
$896$		−	129.206i		−	0.144204i
$897$	108.285	−	49.4090i	0.120719	−	0.0550825i
$898$	609.985			0.679271
$899$	31.5228			0.0350643
$900$	−30.0000			−0.0333333
$901$	912.730			1.01302
$902$		−	712.592i		−	0.790013i
$903$	395.055			0.437492
$904$		−	112.328i		−	0.124256i
$905$	−257.508			−0.284539
$906$	365.305			0.403206
$907$		−	332.986i		−	0.367129i	−0.983008	−	0.183565i	$-0.941236\pi$
$907$		−	332.986i		−	0.367129i	0.983008	−	0.183565i	$-0.0587637\pi$
$908$		−	440.991i		−	0.485673i
$909$	−442.394			−0.486682
$910$	−107.901			−0.118573
$911$			310.817i			0.341182i	0.985342	+	0.170591i	$0.0545677\pi$
$911$			310.817i			0.341182i	−0.985342	+	0.170591i	$0.945432\pi$
$912$			17.5243i			0.0192153i
$913$	612.504			0.670870
$914$		−	1038.84i		−	1.13659i
$915$	163.493			0.178680
$916$		−	276.227i		−	0.301557i
$917$		−	34.1225i		−	0.0372110i
$918$		−	112.735i		−	0.122805i
$919$		−	487.215i		−	0.530158i	−0.964227	−	0.265079i	$-0.914602\pi$
$919$		−	487.215i		−	0.530158i	0.964227	−	0.265079i	$-0.0853980\pi$
$920$	−60.3848	−	132.339i	−0.0656357	−	0.143847i
$921$	−622.941			−0.676375
$922$	885.327			0.960225
$923$	−300.534			−0.325606
$924$	−548.806			−0.593946
$925$			158.854i			0.171734i
$926$	−453.574			−0.489821
$927$			224.401i			0.242072i
$928$	65.7504			0.0708517
$929$	1224.58			1.31817			0.659085	−	0.752068i	$-0.270944\pi$
$929$	1224.58			1.31817			0.659085	+	0.752068i	$0.270944\pi$
$930$			14.8546i			0.0159727i
$931$		−	205.956i		−	0.221220i
$932$	196.375			0.210703
$933$	544.882			0.584011
$934$		−	209.892i		−	0.224724i
$935$		−	475.877i		−	0.508959i
$936$	−25.3521			−0.0270856
$937$		−	412.886i		−	0.440647i	−0.975427	−	0.220323i	$-0.929289\pi$
$937$		−	412.886i		−	0.440647i	0.975427	−	0.220323i	$-0.0707112\pi$
$938$	945.047			1.00751
$939$			782.418i			0.833246i
$940$			233.588i			0.248498i
$941$		−	729.608i		−	0.775354i	−0.921795	−	0.387677i	$-0.873278\pi$
$941$		−	729.608i		−	0.775354i	0.921795	−	0.387677i	$-0.126722\pi$
$942$			558.944i			0.593359i
$943$	−760.038	+	346.797i	−0.805979	+	0.367759i
$944$	326.426			0.345790
$945$	132.692			0.140415
$946$	391.815			0.414181
$947$	1691.14			1.78579			0.892894	−	0.450268i	$-0.148671\pi$
$947$	1691.14			1.78579			0.892894	+	0.450268i	$0.148671\pi$
$948$			210.096i			0.221620i
$949$	50.0734			0.0527644
$950$		−	17.8857i		−	0.0188270i
$951$	310.128			0.326107
$952$	−495.547			−0.520532
$953$			133.087i			0.139651i	0.997559	+	0.0698255i	$0.0222443\pi$
$953$			133.087i			0.139651i	−0.997559	+	0.0698255i	$0.977756\pi$
$954$			252.417i			0.264588i
$955$	−234.074			−0.245104
$956$	−842.438			−0.881211
$957$		−	279.275i		−	0.291824i
$958$			1059.49i			1.10594i
$959$	−1372.47			−1.43114
$960$			30.9839i			0.0322749i
$961$	−953.645			−0.992346
$962$			134.242i			0.139545i
$963$			110.669i			0.114921i
$964$			363.505i			0.377080i
$965$			129.698i			0.134402i
$966$	267.087	+	585.347i	0.276488	+	0.605949i
$967$	310.427			0.321020			0.160510	−	0.987034i	$-0.448686\pi$
$967$	310.427			0.321020			0.160510	+	0.987034i	$0.448686\pi$
$968$	−202.065			−0.208745
$969$	67.2112			0.0693614
$970$	−120.141			−0.123856
$971$			1571.54i			1.61847i	0.587483	+	0.809237i	$0.300119\pi$
$971$			1571.54i			1.61847i	−0.587483	+	0.809237i	$0.699881\pi$
$972$	31.1769			0.0320750
$973$			1453.49i			1.49383i
$974$	408.880			0.419794
$975$	25.8749			0.0265383
$976$		−	168.854i		−	0.173006i
$977$			283.092i			0.289757i	0.989449	+	0.144878i	$0.0462791\pi$
$977$			283.092i			0.289757i	−0.989449	+	0.144878i	$0.953721\pi$
$978$	638.181			0.652537
$979$	−1473.86			−1.50548
$980$		−	364.141i		−	0.371572i
$981$			267.565i			0.272747i
$982$	1087.25			1.10718
$983$			1310.77i			1.33344i	0.745310	+	0.666718i	$0.232302\pi$
$983$			1310.77i			1.33344i	−0.745310	+	0.666718i	$0.767698\pi$
$984$	177.944			0.180837
$985$			214.830i			0.218101i
$986$		−	252.173i		−	0.255754i
$987$		−	1033.18i		−	1.04679i
$988$		−	15.1146i		−	0.0152982i
$989$	−190.685	−	417.904i	−0.192806	−	0.422552i
$990$	131.604			0.132934
$991$	−999.520			−1.00860			−0.504299	−	0.863529i	$-0.668249\pi$
$991$	−999.520			−1.00860			−0.504299	+	0.863529i	$0.668249\pi$
$992$	15.3418			0.0154655
$993$	915.905			0.922361
$994$		−	1624.58i		−	1.63439i
$995$	−443.847			−0.446077
$996$			152.950i			0.153565i
$997$	−806.118			−0.808543			−0.404272	−	0.914639i	$-0.632475\pi$
$997$	−806.118			−0.808543			−0.404272	+	0.914639i	$0.632475\pi$
$998$	285.086			0.285657
$999$		−	165.085i		−	0.165251i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	690.3.c.a.91.29	yes	32
3.2	odd	2		2070.3.c.b.91.1		32
23.22	odd	2	inner	690.3.c.a.91.28	✓	32
69.68	even	2		2070.3.c.b.91.16		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.3.c.a.91.28	✓	32	23.22	odd	2	inner
690.3.c.a.91.29	yes	32	1.1	even	1	trivial
2070.3.c.b.91.1		32	3.2	odd	2
2070.3.c.b.91.16		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)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +1.73205 q^{3} +2.00000 q^{4} +2.23607i q^{5} +2.44949 q^{6} -11.4203i q^{7} +2.82843 q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q+1.41421 q^{2} +1.73205 q^{3} +2.00000 q^{4} +2.23607i q^{5} +2.44949 q^{6} -11.4203i q^{7} +2.82843 q^{8} +3.00000 q^{9} +3.16228i q^{10} -13.8723i q^{11} +3.46410 q^{12} -2.98777 q^{13} -16.1508i q^{14} +3.87298i q^{15} +4.00000 q^{16} -15.3412i q^{17} +4.24264 q^{18} +2.52942i q^{19} +4.47214i q^{20} -19.7806i q^{21} -19.6184i q^{22} +(-20.9247 + 9.54768i) q^{23} +4.89898 q^{24} -5.00000 q^{25} -4.22535 q^{26} +5.19615 q^{27} -22.8407i q^{28} +11.6231 q^{29} +5.47723i q^{30} +2.71207 q^{31} +5.65685 q^{32} -24.0275i q^{33} -21.6958i q^{34} +25.5367 q^{35} +6.00000 q^{36} -31.7707i q^{37} +3.57714i q^{38} -5.17497 q^{39} +6.32456i q^{40} +36.3226 q^{41} -27.9740i q^{42} +19.9718i q^{43} -27.7446i q^{44} +6.70820i q^{45} +(-29.5920 + 13.5025i) q^{46} +52.2318 q^{47} +6.92820 q^{48} -81.4243 q^{49} -7.07107 q^{50} -26.5718i q^{51} -5.97554 q^{52} +59.4952i q^{53} +7.34847 q^{54} +31.0194 q^{55} -32.3016i q^{56} +4.38108i q^{57} +16.4376 q^{58} +81.6065 q^{59} +7.74597i q^{60} -42.2136i q^{61} +3.83545 q^{62} -34.2610i q^{63} +8.00000 q^{64} -6.68086i q^{65} -33.9801i q^{66} +58.5139i q^{67} -30.6825i q^{68} +(-36.2426 + 16.5371i) q^{69} +36.1143 q^{70} +100.588 q^{71} +8.48528 q^{72} -16.7595 q^{73} -44.9306i q^{74} -8.66025 q^{75} +5.05884i q^{76} -158.427 q^{77} -7.31852 q^{78} +60.6495i q^{79} +8.94427i q^{80} +9.00000 q^{81} +51.3679 q^{82} +44.1530i q^{83} -39.5612i q^{84} +34.3041 q^{85} +28.2444i q^{86} +20.1319 q^{87} -39.2368i q^{88} -106.245i q^{89} +9.48683i q^{90} +34.1214i q^{91} +(-41.8493 + 19.0954i) q^{92} +4.69744 q^{93} +73.8669 q^{94} -5.65595 q^{95} +9.79796 q^{96} +37.9918i q^{97} -115.151 q^{98} -41.6169i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 64 q^{4} + 96 q^{9}+O(q^{10}) \) 32 * q + 64 * q^4 + 96 * q^9	\( 32 q + 64 q^{4} + 96 q^{9} - 48 q^{13} + 128 q^{16} - 80 q^{23} - 160 q^{25} + 120 q^{29} + 248 q^{31} - 120 q^{35} + 192 q^{36} - 48 q^{39} + 72 q^{41} + 160 q^{46} + 400 q^{47} - 344 q^{49} - 96 q^{52} - 256 q^{58} + 120 q^{59} + 160 q^{62} + 256 q^{64} + 192 q^{69} + 104 q^{71} + 16 q^{73} + 240 q^{77} + 192 q^{78} + 288 q^{81} + 64 q^{82} - 120 q^{85} + 144 q^{87} - 160 q^{92} - 192 q^{93} + 96 q^{94} - 160 q^{95} + 64 q^{98}+O(q^{100}) \) 32 * q + 64 * q^4 + 96 * q^9 - 48 * q^13 + 128 * q^16 - 80 * q^23 - 160 * q^25 + 120 * q^29 + 248 * q^31 - 120 * q^35 + 192 * q^36 - 48 * q^39 + 72 * q^41 + 160 * q^46 + 400 * q^47 - 344 * q^49 - 96 * q^52 - 256 * q^58 + 120 * q^59 + 160 * q^62 + 256 * q^64 + 192 * q^69 + 104 * q^71 + 16 * q^73 + 240 * q^77 + 192 * q^78 + 288 * q^81 + 64 * q^82 - 120 * q^85 + 144 * q^87 - 160 * q^92 - 192 * q^93 + 96 * q^94 - 160 * q^95 + 64 * q^98

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