LMFDB - Embedded newform 690.3.c.a.91.18

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			91.18
Character	$\chi$	$=$	690.91
Dual form			690.3.c.a.91.23

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} -1.73205 q^{3} +2.00000 q^{4} -2.23607i q^{5} -2.44949 q^{6} -7.96402i q^{7} +2.82843 q^{8} +3.00000 q^{9} +O(q^{10})$	\(q+1.41421 q^{2} -1.73205 q^{3} +2.00000 q^{4} -2.23607i q^{5} -2.44949 q^{6} -7.96402i q^{7} +2.82843 q^{8} +3.00000 q^{9} -3.16228i q^{10} -13.6667i q^{11} -3.46410 q^{12} -14.0318 q^{13} -11.2628i q^{14} +3.87298i q^{15} +4.00000 q^{16} +17.2668i q^{17} +4.24264 q^{18} +26.4661i q^{19} -4.47214i q^{20} +13.7941i q^{21} -19.3277i q^{22} +(-11.7666 - 19.7623i) q^{23} -4.89898 q^{24} -5.00000 q^{25} -19.8440 q^{26} -5.19615 q^{27} -15.9280i q^{28} -54.4984 q^{29} +5.47723i q^{30} -22.1628 q^{31} +5.65685 q^{32} +23.6715i q^{33} +24.4189i q^{34} -17.8081 q^{35} +6.00000 q^{36} -32.2493i q^{37} +37.4287i q^{38} +24.3038 q^{39} -6.32456i q^{40} +9.59193 q^{41} +19.5078i q^{42} -65.3706i q^{43} -27.3335i q^{44} -6.70820i q^{45} +(-16.6405 - 27.9481i) q^{46} +82.5547 q^{47} -6.92820 q^{48} -14.4257 q^{49} -7.07107 q^{50} -29.9070i q^{51} -28.0636 q^{52} -21.1000i q^{53} -7.34847 q^{54} -30.5598 q^{55} -22.5257i q^{56} -45.8406i q^{57} -77.0724 q^{58} -105.537 q^{59} +7.74597i q^{60} -33.4803i q^{61} -31.3429 q^{62} -23.8921i q^{63} +8.00000 q^{64} +31.3760i q^{65} +33.4765i q^{66} +34.9712i q^{67} +34.5336i q^{68} +(20.3804 + 34.2292i) q^{69} -25.1845 q^{70} +65.1127 q^{71} +8.48528 q^{72} -126.274 q^{73} -45.6075i q^{74} +8.66025 q^{75} +52.9321i q^{76} -108.842 q^{77} +34.3707 q^{78} -45.0370i q^{79} -8.94427i q^{80} +9.00000 q^{81} +13.5650 q^{82} -10.3487i q^{83} +27.5882i q^{84} +38.6097 q^{85} -92.4480i q^{86} +94.3941 q^{87} -38.6554i q^{88} +20.8576i q^{89} -9.48683i q^{90} +111.750i q^{91} +(-23.5332 - 39.5245i) q^{92} +38.3871 q^{93} +116.750 q^{94} +59.1799 q^{95} -9.79796 q^{96} -146.441i q^{97} -20.4010 q^{98} -41.0002i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$		−	7.96402i		−	1.13772i	−0.822435	−	0.568859i	$-0.807385\pi$
$7$		−	7.96402i		−	1.13772i	0.822435	−	0.568859i	$-0.192615\pi$
$8$	2.82843			0.353553
$9$	3.00000			0.333333
$10$		−	3.16228i		−	0.316228i
$11$		−	13.6667i		−	1.24243i	−0.783640	−	0.621216i	$-0.786639\pi$
$11$		−	13.6667i		−	1.24243i	0.783640	−	0.621216i	$-0.213361\pi$
$12$	−3.46410			−0.288675
$13$	−14.0318			−1.07937			−0.539684	−	0.841867i	$-0.681456\pi$
$13$	−14.0318			−1.07937			−0.539684	+	0.841867i	$0.681456\pi$
$14$		−	11.2628i		−	0.804488i
$15$			3.87298i			0.258199i
$16$	4.00000			0.250000
$17$			17.2668i			1.01569i	0.861447	+	0.507847i	$0.169558\pi$
$17$			17.2668i			1.01569i	−0.861447	+	0.507847i	$0.830442\pi$
$18$	4.24264			0.235702
$19$			26.4661i			1.39295i	0.717581	+	0.696475i	$0.245250\pi$
$19$			26.4661i			1.39295i	−0.717581	+	0.696475i	$0.754750\pi$
$20$		−	4.47214i		−	0.223607i
$21$			13.7941i			0.656862i
$22$		−	19.3277i		−	0.878532i
$23$	−11.7666	−	19.7623i	−0.511592	−	0.859229i
$23$	−11.7666	−	19.7623i	−0.511592	−	0.859229i
$24$	−4.89898			−0.204124
$25$	−5.00000			−0.200000
$26$	−19.8440			−0.763229
$27$	−5.19615			−0.192450
$28$		−	15.9280i		−	0.568859i
$29$	−54.4984			−1.87926			−0.939628	−	0.342197i	$-0.888829\pi$
$29$	−54.4984			−1.87926			−0.939628	+	0.342197i	$0.888829\pi$
$30$			5.47723i			0.182574i
$31$	−22.1628			−0.714929			−0.357465	−	0.933927i	$-0.616359\pi$
$31$	−22.1628			−0.714929			−0.357465	+	0.933927i	$0.616359\pi$
$32$	5.65685			0.176777
$33$			23.6715i			0.717318i
$34$			24.4189i			0.718204i
$35$	−17.8081			−0.508803
$36$	6.00000			0.166667
$37$		−	32.2493i		−	0.871604i	−0.900043	−	0.435802i	$-0.856465\pi$
$37$		−	32.2493i		−	0.871604i	0.900043	−	0.435802i	$-0.143535\pi$
$38$			37.4287i			0.984965i
$39$	24.3038			0.623174
$40$		−	6.32456i		−	0.158114i
$41$	9.59193			0.233949			0.116975	−	0.993135i	$-0.462680\pi$
$41$	9.59193			0.233949			0.116975	+	0.993135i	$0.462680\pi$
$42$			19.5078i			0.464471i
$43$		−	65.3706i		−	1.52025i	−0.649778	−	0.760124i	$-0.725138\pi$
$43$		−	65.3706i		−	1.52025i	0.649778	−	0.760124i	$-0.274862\pi$
$44$		−	27.3335i		−	0.621216i
$45$		−	6.70820i		−	0.149071i
$46$	−16.6405	−	27.9481i	−0.361750	−	0.607566i
$47$	82.5547			1.75648			0.878242	−	0.478217i	$-0.158717\pi$
$47$	82.5547			1.75648			0.878242	+	0.478217i	$0.158717\pi$
$48$	−6.92820			−0.144338
$49$	−14.4257			−0.294402
$50$	−7.07107			−0.141421
$51$		−	29.9070i		−	0.586411i
$52$	−28.0636			−0.539684
$53$		−	21.1000i		−	0.398113i	−0.979988	−	0.199056i	$-0.936212\pi$
$53$		−	21.1000i		−	0.398113i	0.979988	−	0.199056i	$-0.0637877\pi$
$54$	−7.34847			−0.136083
$55$	−30.5598			−0.555632
$56$		−	22.5257i		−	0.402244i
$57$		−	45.8406i		−	0.804221i
$58$	−77.0724			−1.32883
$59$	−105.537			−1.78877			−0.894385	−	0.447299i	$-0.852386\pi$
$59$	−105.537			−1.78877			−0.894385	+	0.447299i	$0.852386\pi$
$60$			7.74597i			0.129099i
$61$		−	33.4803i		−	0.548858i	−0.961607	−	0.274429i	$-0.911511\pi$
$61$		−	33.4803i		−	0.548858i	0.961607	−	0.274429i	$-0.0884888\pi$
$62$	−31.3429			−0.505531
$63$		−	23.8921i		−	0.379239i
$64$	8.00000			0.125000
$65$			31.3760i			0.482708i
$66$			33.4765i			0.507220i
$67$			34.9712i			0.521958i	0.965344	+	0.260979i	$0.0840453\pi$
$67$			34.9712i			0.521958i	−0.965344	+	0.260979i	$0.915955\pi$
$68$			34.5336i			0.507847i
$69$	20.3804	+	34.2292i	0.295368	+	0.496076i
$70$	−25.1845			−0.359778
$71$	65.1127			0.917080			0.458540	−	0.888674i	$-0.348373\pi$
$71$	65.1127			0.917080			0.458540	+	0.888674i	$0.348373\pi$
$72$	8.48528			0.117851
$73$	−126.274			−1.72978			−0.864891	−	0.501959i	$-0.832613\pi$
$73$	−126.274			−1.72978			−0.864891	+	0.501959i	$0.832613\pi$
$74$		−	45.6075i		−	0.616317i
$75$	8.66025			0.115470
$76$			52.9321i			0.696475i
$77$	−108.842			−1.41354
$78$	34.3707			0.440651
$79$		−	45.0370i		−	0.570089i	−0.958514	−	0.285045i	$-0.907992\pi$
$79$		−	45.0370i		−	0.570089i	0.958514	−	0.285045i	$-0.0920084\pi$
$80$		−	8.94427i		−	0.111803i
$81$	9.00000			0.111111
$82$	13.5650			0.165427
$83$		−	10.3487i		−	0.124684i	−0.998055	−	0.0623418i	$-0.980143\pi$
$83$		−	10.3487i		−	0.124684i	0.998055	−	0.0623418i	$-0.0198569\pi$
$84$			27.5882i			0.328431i
$85$	38.6097			0.454232
$86$		−	92.4480i		−	1.07498i
$87$	94.3941			1.08499
$88$		−	38.6554i		−	0.439266i
$89$			20.8576i			0.234356i	0.993111	+	0.117178i	$0.0373847\pi$
$89$			20.8576i			0.234356i	−0.993111	+	0.117178i	$0.962615\pi$
$90$		−	9.48683i		−	0.105409i
$91$			111.750i			1.22802i
$92$	−23.5332	−	39.5245i	−0.255796	−	0.429614i
$93$	38.3871			0.412764
$94$	116.750			1.24202
$95$	59.1799			0.622947
$96$	−9.79796			−0.102062
$97$		−	146.441i		−	1.50970i	−0.655895	−	0.754852i	$-0.727709\pi$
$97$		−	146.441i		−	1.50970i	0.655895	−	0.754852i	$-0.272291\pi$
$98$	−20.4010			−0.208173
$99$		−	41.0002i		−	0.414144i
$100$	−10.0000			−0.100000
$101$	110.767			1.09671			0.548354	−	0.836246i	$-0.315255\pi$
$101$	110.767			1.09671			0.548354	+	0.836246i	$0.315255\pi$
$102$		−	42.2948i		−	0.414655i
$103$			75.0106i			0.728259i	0.931348	+	0.364129i	$0.118633\pi$
$103$			75.0106i			0.728259i	−0.931348	+	0.364129i	$0.881367\pi$
$104$	−39.6879			−0.381615
$105$	30.8445			0.293757
$106$		−	29.8399i		−	0.281508i
$107$		−	1.13721i		−	0.0106281i	−0.999986	−	0.00531405i	$-0.998308\pi$
$107$		−	1.13721i		−	0.0106281i	0.999986	−	0.00531405i	$-0.00169152\pi$
$108$	−10.3923			−0.0962250
$109$			74.1091i			0.679900i	0.940443	+	0.339950i	$0.110410\pi$
$109$			74.1091i			0.679900i	−0.940443	+	0.339950i	$0.889590\pi$
$110$	−43.2180			−0.392891
$111$			55.8575i			0.503221i
$112$		−	31.8561i		−	0.284429i
$113$			121.550i			1.07566i	0.843053	+	0.537830i	$0.180756\pi$
$113$			121.550i			1.07566i	−0.843053	+	0.537830i	$0.819244\pi$
$114$		−	64.8284i		−	0.568670i
$115$	−44.1897	+	26.3110i	−0.384259	+	0.228791i
$116$	−108.997			−0.939628
$117$	−42.0954			−0.359790
$118$	−149.252			−1.26485
$119$	137.513			1.15557
$120$			10.9545i			0.0912871i
$121$	−65.7799			−0.543635
$122$		−	47.3484i		−	0.388101i
$123$	−16.6137			−0.135071
$124$	−44.3256			−0.357465
$125$			11.1803i			0.0894427i
$126$		−	33.7885i		−	0.268163i
$127$	−46.6213			−0.367097			−0.183549	−	0.983011i	$-0.558758\pi$
$127$	−46.6213			−0.367097			−0.183549	+	0.983011i	$0.558758\pi$
$128$	11.3137			0.0883883
$129$			113.225i			0.877715i
$130$			44.3724i			0.341326i
$131$	172.666			1.31806			0.659030	−	0.752116i	$-0.270967\pi$
$131$	172.666			1.31806			0.659030	+	0.752116i	$0.270967\pi$
$132$			47.3430i			0.358659i
$133$	210.776			1.58478
$134$			49.4567i			0.369080i
$135$			11.6190i			0.0860663i
$136$			48.8379i			0.359102i
$137$		−	269.189i		−	1.96488i	−0.186572	−	0.982441i	$-0.559738\pi$
$137$		−	269.189i		−	1.96488i	0.186572	−	0.982441i	$-0.440262\pi$
$138$	28.8222	+	48.4074i	0.208857	+	0.350779i
$139$	−112.626			−0.810260			−0.405130	−	0.914259i	$-0.632774\pi$
$139$	−112.626			−0.810260			−0.405130	+	0.914259i	$0.632774\pi$
$140$	−35.6162			−0.254401
$141$	−142.989			−1.01411
$142$	92.0832			0.648474
$143$			191.769i			1.34104i
$144$	12.0000			0.0833333
$145$			121.862i			0.840429i
$146$	−178.579			−1.22314
$147$	24.9860			0.169973
$148$		−	64.4987i		−	0.435802i
$149$		−	53.3780i		−	0.358241i	−0.983827	−	0.179121i	$-0.942675\pi$
$149$		−	53.3780i		−	0.358241i	0.983827	−	0.179121i	$-0.0573253\pi$
$150$	12.2474			0.0816497
$151$	39.7896			0.263507			0.131753	−	0.991283i	$-0.457939\pi$
$151$	39.7896			0.263507			0.131753	+	0.991283i	$0.457939\pi$
$152$			74.8573i			0.492483i
$153$			51.8004i			0.338565i
$154$	−153.926			−0.999521
$155$			49.5575i			0.319726i
$156$	48.6076			0.311587
$157$		−	263.251i		−	1.67676i	−0.545086	−	0.838380i	$-0.683503\pi$
$157$		−	263.251i		−	1.67676i	0.545086	−	0.838380i	$-0.316497\pi$
$158$		−	63.6920i		−	0.403114i
$159$			36.5462i			0.229850i
$160$		−	12.6491i		−	0.0790569i
$161$	−157.387	+	93.7096i	−0.977559	+	0.582047i
$162$	12.7279			0.0785674
$163$	239.678			1.47042			0.735209	−	0.677840i	$-0.237084\pi$
$163$	239.678			1.47042			0.735209	+	0.677840i	$0.237084\pi$
$164$	19.1839			0.116975
$165$	52.9311			0.320794
$166$		−	14.6353i		−	0.0881646i
$167$	316.884			1.89751			0.948753	−	0.316017i	$-0.102346\pi$
$167$	316.884			1.89751			0.948753	+	0.316017i	$0.102346\pi$
$168$			39.0156i			0.232236i
$169$	27.8913			0.165037
$170$	54.6024			0.321191
$171$			79.3982i			0.464317i
$172$		−	130.741i		−	0.760124i
$173$	115.891			0.669890			0.334945	−	0.942238i	$-0.391282\pi$
$173$	115.891			0.669890			0.334945	+	0.942238i	$0.391282\pi$
$174$	133.493			0.767203
$175$			39.8201i			0.227544i
$176$		−	54.6670i		−	0.310608i
$177$	182.796			1.03275
$178$			29.4972i			0.165714i
$179$	−62.1264			−0.347075			−0.173537	−	0.984827i	$-0.555520\pi$
$179$	−62.1264			−0.347075			−0.173537	+	0.984827i	$0.555520\pi$
$180$		−	13.4164i		−	0.0745356i
$181$		−	34.2525i		−	0.189240i	−0.995513	−	0.0946202i	$-0.969836\pi$
$181$		−	34.2525i		−	0.189240i	0.995513	−	0.0946202i	$-0.0301637\pi$
$182$			158.038i			0.868339i
$183$			57.9897i			0.316883i
$184$	−33.2810	−	55.8961i	−0.180875	−	0.303783i
$185$	−72.1117			−0.389793
$186$	54.2876			0.291869
$187$	235.981			1.26193
$188$	165.109			0.878242
$189$			41.3823i			0.218954i
$190$	83.6931			0.440490
$191$			365.807i			1.91522i	0.288065	+	0.957611i	$0.406988\pi$
$191$			365.807i			1.91522i	−0.288065	+	0.957611i	$0.593012\pi$
$192$	−13.8564			−0.0721688
$193$	−223.842			−1.15980			−0.579901	−	0.814687i	$-0.696909\pi$
$193$	−223.842			−1.15980			−0.579901	+	0.814687i	$0.696909\pi$
$194$		−	207.099i		−	1.06752i
$195$		−	54.3449i		−	0.278692i
$196$	−28.8513			−0.147201
$197$	89.6193			0.454920			0.227460	−	0.973787i	$-0.426958\pi$
$197$	89.6193			0.454920			0.227460	+	0.973787i	$0.426958\pi$
$198$		−	57.9831i		−	0.292844i
$199$		−	5.90005i		−	0.0296485i	−0.999890	−	0.0148243i	$-0.995281\pi$
$199$		−	5.90005i		−	0.0296485i	0.999890	−	0.0148243i	$-0.00471888\pi$
$200$	−14.1421			−0.0707107
$201$		−	60.5719i		−	0.301353i
$202$	156.649			0.775489
$203$			434.027i			2.13806i
$204$		−	59.8139i		−	0.293205i
$205$		−	21.4482i		−	0.104625i
$206$			106.081i			0.514957i
$207$	−35.2998	−	59.2868i	−0.170531	−	0.286410i
$208$	−56.1272			−0.269842
$209$	361.705			1.73065
$210$	43.6208			0.207718
$211$	−231.698			−1.09810			−0.549048	−	0.835791i	$-0.685009\pi$
$211$	−231.698			−1.09810			−0.549048	+	0.835791i	$0.685009\pi$
$212$		−	42.1999i		−	0.199056i
$213$	−112.778			−0.529476
$214$		−	1.60825i		−	0.00751521i
$215$	−146.173			−0.679875
$216$	−14.6969			−0.0680414
$217$			176.505i			0.813387i
$218$			104.806i			0.480762i
$219$	218.713			0.998690
$220$	−61.1195			−0.277816
$221$		−	242.284i		−	1.09631i
$222$			78.9944i			0.355831i
$223$	130.000			0.582962			0.291481	−	0.956577i	$-0.405852\pi$
$223$	130.000			0.582962			0.291481	+	0.956577i	$0.405852\pi$
$224$		−	45.0513i		−	0.201122i
$225$	−15.0000			−0.0666667
$226$			171.897i			0.760606i
$227$		−	167.115i		−	0.736190i	−0.929788	−	0.368095i	$-0.880010\pi$
$227$		−	167.115i		−	0.736190i	0.929788	−	0.368095i	$-0.119990\pi$
$228$		−	91.6812i		−	0.402110i
$229$			19.2126i			0.0838976i	0.999120	+	0.0419488i	$0.0133566\pi$
$229$			19.2126i			0.0838976i	−0.999120	+	0.0419488i	$0.986643\pi$
$230$	−62.4937	+	37.2093i	−0.271712	+	0.161780i
$231$	188.520			0.816105
$232$	−154.145			−0.664417
$233$	101.698			0.436470			0.218235	−	0.975896i	$-0.429970\pi$
$233$	101.698			0.436470			0.218235	+	0.975896i	$0.429970\pi$
$234$	−59.5319			−0.254410
$235$		−	184.598i		−	0.785523i
$236$	−211.075			−0.894385
$237$			78.0064i			0.329141i
$238$	194.473			0.817113
$239$	−65.7260			−0.275004			−0.137502	−	0.990501i	$-0.543907\pi$
$239$	−65.7260			−0.275004			−0.137502	+	0.990501i	$0.543907\pi$
$240$			15.4919i			0.0645497i
$241$		−	191.368i		−	0.794059i	−0.917806	−	0.397029i	$-0.870041\pi$
$241$		−	191.368i		−	0.794059i	0.917806	−	0.397029i	$-0.129959\pi$
$242$	−93.0268			−0.384408
$243$	−15.5885			−0.0641500
$244$		−	66.9607i		−	0.274429i
$245$			32.2568i			0.131660i
$246$	−23.4953			−0.0955094
$247$		−	371.366i		−	1.50351i
$248$	−62.6859			−0.252766
$249$			17.9245i			0.0719861i
$250$			15.8114i			0.0632456i
$251$			269.641i			1.07427i	0.843498	+	0.537133i	$0.180493\pi$
$251$			269.641i			1.07427i	−0.843498	+	0.537133i	$0.819507\pi$
$252$		−	47.7841i		−	0.189620i
$253$	−270.086	+	160.811i	−1.06753	+	0.635618i
$254$	−65.9325			−0.259577
$255$	−66.8740			−0.262251
$256$	16.0000			0.0625000
$257$	130.359			0.507234			0.253617	−	0.967305i	$-0.418380\pi$
$257$	130.359			0.507234			0.253617	+	0.967305i	$0.418380\pi$
$258$			160.125i			0.620638i
$259$	−256.834			−0.991639
$260$			62.7521i			0.241354i
$261$	−163.495			−0.626419
$262$	244.187			0.932010
$263$		−	179.446i		−	0.682306i	−0.940008	−	0.341153i	$-0.889183\pi$
$263$		−	179.446i		−	0.682306i	0.940008	−	0.341153i	$-0.110817\pi$
$264$			66.9531i			0.253610i
$265$	−47.1810			−0.178041
$266$	298.083			1.12061
$267$		−	36.1265i		−	0.135305i
$268$			69.9424i			0.260979i
$269$	483.252			1.79648			0.898239	−	0.439508i	$-0.144847\pi$
$269$	483.252			1.79648			0.898239	+	0.439508i	$0.144847\pi$
$270$			16.4317i			0.0608581i
$271$	25.8751			0.0954799			0.0477400	−	0.998860i	$-0.484798\pi$
$271$	25.8751			0.0954799			0.0477400	+	0.998860i	$0.484798\pi$
$272$			69.0672i			0.253923i
$273$		−	193.556i		−	0.708996i
$274$		−	380.691i		−	1.38938i
$275$			68.3337i			0.248486i
$276$	40.7608	+	68.4585i	0.147684	+	0.248038i
$277$	−322.130			−1.16293			−0.581463	−	0.813573i	$-0.697519\pi$
$277$	−322.130			−1.16293			−0.581463	+	0.813573i	$0.697519\pi$
$278$	−159.277			−0.572940
$279$	−66.4884			−0.238310
$280$	−50.3689			−0.179889
$281$			152.134i			0.541401i	0.962664	+	0.270701i	$0.0872553\pi$
$281$			152.134i			0.541401i	−0.962664	+	0.270701i	$0.912745\pi$
$282$	−202.217			−0.717081
$283$		−	256.411i		−	0.906047i	−0.891499	−	0.453024i	$-0.850345\pi$
$283$		−	256.411i		−	0.906047i	0.891499	−	0.453024i	$-0.149655\pi$
$284$	130.225			0.458540
$285$	−102.503			−0.359658
$286$			271.202i			0.948260i
$287$		−	76.3903i		−	0.266168i
$288$	16.9706			0.0589256
$289$	−9.14209			−0.0316335
$290$			172.339i			0.594273i
$291$			253.644i			0.871628i
$292$	−252.548			−0.864891
$293$		−	371.098i		−	1.26655i	−0.773929	−	0.633273i	$-0.781711\pi$
$293$		−	371.098i		−	1.26655i	0.773929	−	0.633273i	$-0.218289\pi$
$294$	35.3355			0.120189
$295$			235.989i			0.799962i
$296$		−	91.2149i		−	0.308158i
$297$			71.0145i			0.239106i
$298$		−	75.4878i		−	0.253315i
$299$	165.107	+	277.300i	0.552196	+	0.927425i
$300$	17.3205			0.0577350
$301$	−520.613			−1.72961
$302$	56.2709			0.186328
$303$	−191.855			−0.633184
$304$			105.864i			0.348238i
$305$	−74.8643			−0.245457
$306$			73.2568i			0.239401i
$307$	526.653			1.71548			0.857742	−	0.514081i	$-0.171867\pi$
$307$	526.653			1.71548			0.857742	+	0.514081i	$0.171867\pi$
$308$	−217.685			−0.706768
$309$		−	129.922i		−	0.420460i
$310$			70.0849i			0.226080i
$311$	537.154			1.72718			0.863592	−	0.504192i	$-0.168210\pi$
$311$	537.154			1.72718			0.863592	+	0.504192i	$0.168210\pi$
$312$	68.7415			0.220325
$313$		−	296.354i		−	0.946819i	−0.880842	−	0.473409i	$-0.843023\pi$
$313$		−	296.354i		−	0.946819i	0.880842	−	0.473409i	$-0.156977\pi$
$314$		−	372.294i		−	1.18565i
$315$	−53.4243			−0.169601
$316$		−	90.0741i		−	0.285045i
$317$	−144.472			−0.455748			−0.227874	−	0.973691i	$-0.573177\pi$
$317$	−144.472			−0.455748			−0.227874	+	0.973691i	$0.573177\pi$
$318$			51.6842i			0.162529i
$319$			744.816i			2.33485i
$320$		−	17.8885i		−	0.0559017i
$321$			1.96970i			0.00613614i
$322$	−222.579	+	132.525i	−0.691239	+	0.411570i
$323$	−456.984			−1.41481
$324$	18.0000			0.0555556
$325$	70.1590			0.215874
$326$	338.956			1.03974
$327$		−	128.361i		−	0.392541i
$328$	27.1301			0.0827136
$329$		−	657.468i		−	1.99838i
$330$	74.8558			0.226836
$331$	−481.196			−1.45376			−0.726882	−	0.686763i	$-0.759031\pi$
$331$	−481.196			−1.45376			−0.726882	+	0.686763i	$0.759031\pi$
$332$		−	20.6975i		−	0.0623418i
$333$		−	96.7480i		−	0.290535i
$334$	448.141			1.34174
$335$	78.1980			0.233427
$336$			55.1764i			0.164215i
$337$			227.981i			0.676501i	0.941056	+	0.338251i	$0.109835\pi$
$337$			227.981i			0.676501i	−0.941056	+	0.338251i	$0.890165\pi$
$338$	39.4443			0.116699
$339$		−	210.530i		−	0.621033i
$340$	77.2194			0.227116
$341$			302.893i			0.888250i
$342$			112.286i			0.328322i
$343$		−	275.351i		−	0.802772i
$344$		−	184.896i		−	0.537489i
$345$	76.5389	−	45.5719i	0.221852	−	0.132092i
$346$	163.895			0.473684
$347$	284.411			0.819628			0.409814	−	0.912169i	$-0.365594\pi$
$347$	284.411			0.819628			0.409814	+	0.912169i	$0.365594\pi$
$348$	188.788			0.542495
$349$	−426.741			−1.22275			−0.611377	−	0.791340i	$-0.709384\pi$
$349$	−426.741			−1.22275			−0.611377	+	0.791340i	$0.709384\pi$
$350$			56.3142i			0.160898i
$351$	72.9114			0.207725
$352$		−	77.3108i		−	0.219633i
$353$	159.245			0.451118			0.225559	−	0.974230i	$-0.427579\pi$
$353$	159.245			0.451118			0.225559	+	0.974230i	$0.427579\pi$
$354$	258.513			0.730262
$355$		−	145.596i		−	0.410131i
$356$			41.7153i			0.117178i
$357$	−238.180			−0.667170
$358$	−87.8600			−0.245419
$359$		−	660.255i		−	1.83915i	−0.392914	−	0.919575i	$-0.628533\pi$
$359$		−	660.255i		−	1.83915i	0.392914	−	0.919575i	$-0.371467\pi$
$360$		−	18.9737i		−	0.0527046i
$361$	−339.453			−0.940312
$362$		−	48.4404i		−	0.133813i
$363$	113.934			0.313868
$364$			223.499i			0.614009i
$365$			282.358i			0.773582i
$366$			82.0098i			0.224070i
$367$		−	153.906i		−	0.419363i	−0.977770	−	0.209681i	$-0.932757\pi$
$367$		−	153.906i		−	0.419363i	0.977770	−	0.209681i	$-0.0672427\pi$
$368$	−47.0665	−	79.0490i	−0.127898	−	0.214807i
$369$	28.7758			0.0779831
$370$	−101.981			−0.275625
$371$	−168.041			−0.452940
$372$	76.7742			0.206382
$373$		−	114.077i		−	0.305836i	−0.988239	−	0.152918i	$-0.951133\pi$
$373$		−	114.077i		−	0.305836i	0.988239	−	0.152918i	$-0.0488671\pi$
$374$	333.727			0.892319
$375$		−	19.3649i		−	0.0516398i
$376$	233.500			0.621011
$377$	764.711			2.02841
$378$			58.5234i			0.154824i
$379$		−	568.948i		−	1.50118i	−0.660767	−	0.750591i	$-0.729769\pi$
$379$		−	568.948i		−	1.50118i	0.660767	−	0.750591i	$-0.270231\pi$
$380$	118.360			0.311473
$381$	80.7505			0.211944
$382$			517.330i			1.35427i
$383$			598.817i			1.56349i	0.623597	+	0.781746i	$0.285671\pi$
$383$			598.817i			1.56349i	−0.623597	+	0.781746i	$0.714329\pi$
$384$	−19.5959			−0.0510310
$385$			243.379i			0.632153i
$386$	−316.560			−0.820103
$387$		−	196.112i		−	0.506749i
$388$		−	292.883i		−	0.754852i
$389$			296.143i			0.761293i	0.924721	+	0.380647i	$0.124299\pi$
$389$			296.143i			0.761293i	−0.924721	+	0.380647i	$0.875701\pi$
$390$		−	76.8553i		−	0.197065i
$391$	341.231	−	203.172i	0.872713	−	0.519621i
$392$	−40.8020			−0.104087
$393$	−299.066			−0.760983
$394$	126.741			0.321677
$395$	−100.706			−0.254952
$396$		−	82.0005i		−	0.207072i
$397$	−706.135			−1.77868			−0.889338	−	0.457249i	$-0.848835\pi$
$397$	−706.135			−1.77868			−0.889338	+	0.457249i	$0.848835\pi$
$398$		−	8.34394i		−	0.0209647i
$399$	−365.075			−0.914976
$400$	−20.0000			−0.0500000
$401$			501.908i			1.25164i	0.779967	+	0.625821i	$0.215236\pi$
$401$			501.908i			1.25164i	−0.779967	+	0.625821i	$0.784764\pi$
$402$		−	85.6616i		−	0.213088i
$403$	310.984			0.771672
$404$	221.535			0.548354
$405$		−	20.1246i		−	0.0496904i
$406$			613.807i			1.51184i
$407$	−440.743			−1.08291
$408$		−	84.5897i		−	0.207328i
$409$	−351.306			−0.858939			−0.429470	−	0.903081i	$-0.641299\pi$
$409$	−351.306			−0.858939			−0.429470	+	0.903081i	$0.641299\pi$
$410$		−	30.3323i		−	0.0739813i
$411$			466.249i			1.13443i
$412$			150.021i			0.364129i
$413$			840.502i			2.03511i
$414$	−49.9215	−	83.8442i	−0.120583	−	0.202522i
$415$	−23.1405			−0.0557602
$416$	−79.3758			−0.190807
$417$	195.074			0.467804
$418$	511.528			1.22375
$419$			497.695i			1.18782i	0.804533	+	0.593908i	$0.202416\pi$
$419$			497.695i			1.18782i	−0.804533	+	0.593908i	$0.797584\pi$
$420$	61.6891			0.146879
$421$		−	323.572i		−	0.768581i	−0.923212	−	0.384290i	$-0.874446\pi$
$421$		−	323.572i		−	0.768581i	0.923212	−	0.384290i	$-0.125554\pi$
$422$	−327.671			−0.776470
$423$	247.664			0.585495
$424$		−	59.6797i		−	0.140754i
$425$		−	86.3340i		−	0.203139i
$426$	−159.493			−0.374396
$427$	−266.638			−0.624446
$428$		−	2.27442i		−	0.00531405i
$429$		−	332.154i		−	0.774251i
$430$	−206.720			−0.480744
$431$		−	539.731i		−	1.25228i	−0.779712	−	0.626138i	$-0.784635\pi$
$431$		−	539.731i		−	1.25228i	0.779712	−	0.626138i	$-0.215365\pi$
$432$	−20.7846			−0.0481125
$433$			421.843i			0.974234i	0.873337	+	0.487117i	$0.161951\pi$
$433$			421.843i			0.974234i	−0.873337	+	0.487117i	$0.838049\pi$
$434$			249.616i			0.575152i
$435$		−	211.072i		−	0.485222i
$436$			148.218i			0.339950i
$437$	523.029	−	311.416i	1.19686	−	0.712623i
$438$	309.307			0.706181
$439$	324.408			0.738970			0.369485	−	0.929237i	$-0.379534\pi$
$439$	324.408			0.738970			0.369485	+	0.929237i	$0.379534\pi$
$440$	−86.4361			−0.196446
$441$	−43.2770			−0.0981338
$442$		−	342.641i		−	0.775207i
$443$	221.797			0.500671			0.250335	−	0.968159i	$-0.419459\pi$
$443$	221.797			0.500671			0.250335	+	0.968159i	$0.419459\pi$
$444$			111.715i			0.251610i
$445$	46.6391			0.104807
$446$	183.848			0.412216
$447$			92.4533i			0.206831i
$448$		−	63.7122i		−	0.142215i
$449$	−523.920			−1.16686			−0.583430	−	0.812164i	$-0.698290\pi$
$449$	−523.920			−1.16686			−0.583430	+	0.812164i	$0.698290\pi$
$450$	−21.2132			−0.0471405
$451$		−	131.090i		−	0.290666i
$452$			243.099i			0.537830i
$453$	−68.9175			−0.152136
$454$		−	236.337i		−	0.520565i
$455$	249.880			0.549186
$456$		−	129.657i		−	0.284335i
$457$		−	379.414i		−	0.830227i	−0.909770	−	0.415114i	$-0.863742\pi$
$457$		−	379.414i		−	0.830227i	0.909770	−	0.415114i	$-0.136258\pi$
$458$			27.1707i			0.0593246i
$459$		−	89.7209i		−	0.195470i
$460$	−88.3795	+	52.6219i	−0.192129	+	0.114395i
$461$	−449.296			−0.974611			−0.487306	−	0.873231i	$-0.662020\pi$
$461$	−449.296			−0.974611			−0.487306	+	0.873231i	$0.662020\pi$
$462$	266.608			0.577074
$463$	581.451			1.25583			0.627917	−	0.778280i	$-0.283908\pi$
$463$	581.451			1.25583			0.627917	+	0.778280i	$0.283908\pi$
$464$	−217.994			−0.469814
$465$		−	85.8362i		−	0.184594i
$466$	143.822			0.308631
$467$			622.103i			1.33213i	0.745895	+	0.666064i	$0.232022\pi$
$467$			622.103i			1.33213i	−0.745895	+	0.666064i	$0.767978\pi$
$468$	−84.1908			−0.179895
$469$	278.511			0.593841
$470$		−	261.061i		−	0.555449i
$471$			455.965i			0.968078i
$472$	−298.505			−0.632425
$473$	−893.404			−1.88880
$474$			110.318i			0.232738i
$475$		−	132.330i		−	0.278590i
$476$	275.026			0.577786
$477$		−	63.2999i		−	0.132704i
$478$	−92.9506			−0.194457
$479$			269.556i			0.562747i	0.959598	+	0.281373i	$0.0907900\pi$
$479$			269.556i			0.562747i	−0.959598	+	0.281373i	$0.909210\pi$
$480$			21.9089i			0.0456435i
$481$			452.516i			0.940782i
$482$		−	270.635i		−	0.561484i
$483$	272.602	−	162.310i	0.564394	−	0.336045i
$484$	−131.560			−0.271818
$485$	−327.453			−0.675160
$486$	−22.0454			−0.0453609
$487$	321.930			0.661048			0.330524	−	0.943798i	$-0.392775\pi$
$487$	321.930			0.661048			0.330524	+	0.943798i	$0.392775\pi$
$488$		−	94.6967i		−	0.194051i
$489$	−415.135			−0.848946
$490$			45.6180i			0.0930979i
$491$	−420.200			−0.855805			−0.427903	−	0.903825i	$-0.640747\pi$
$491$	−420.200			−0.855805			−0.427903	+	0.903825i	$0.640747\pi$
$492$	−33.2274			−0.0675354
$493$		−	941.013i		−	1.90875i
$494$		−	525.192i		−	1.06314i
$495$	−91.6793			−0.185211
$496$	−88.6512			−0.178732
$497$		−	518.559i		−	1.04338i
$498$			25.3491i			0.0509018i
$499$	−259.403			−0.519845			−0.259922	−	0.965629i	$-0.583697\pi$
$499$	−259.403			−0.519845			−0.259922	+	0.965629i	$0.583697\pi$
$500$			22.3607i			0.0447214i
$501$	−548.859			−1.09553
$502$			381.330i			0.759621i
$503$		−	687.217i		−	1.36624i	−0.730307	−	0.683119i	$-0.760623\pi$
$503$		−	687.217i		−	1.36624i	0.730307	−	0.683119i	$-0.239377\pi$
$504$		−	67.5770i		−	0.134081i
$505$		−	247.684i		−	0.490463i
$506$	−381.959	+	227.422i	−0.754859	+	0.449450i
$507$	−48.3092			−0.0952843
$508$	−93.2427			−0.183549
$509$	−537.856			−1.05669			−0.528345	−	0.849030i	$-0.677187\pi$
$509$	−537.856			−1.05669			−0.528345	+	0.849030i	$0.677187\pi$
$510$	−94.5741			−0.185439
$511$			1005.65i			1.96800i
$512$	22.6274			0.0441942
$513$		−	137.522i		−	0.268074i
$514$	184.356			0.358669
$515$	167.729			0.325687
$516$			226.451i			0.438858i
$517$		−	1128.25i		−	2.18231i
$518$	−363.219			−0.701195
$519$	−200.729			−0.386761
$520$			88.7449i			0.170663i
$521$		−	295.782i		−	0.567720i	−0.958866	−	0.283860i	$-0.908385\pi$
$521$		−	295.782i		−	0.567720i	0.958866	−	0.283860i	$-0.0916151\pi$
$522$	−231.217			−0.442945
$523$		−	889.701i		−	1.70115i	−0.525855	−	0.850575i	$-0.676254\pi$
$523$		−	889.701i		−	1.70115i	0.525855	−	0.850575i	$-0.323746\pi$
$524$	345.332			0.659030
$525$		−	68.9705i		−	0.131372i
$526$		−	253.776i		−	0.482463i
$527$		−	382.680i		−	0.726149i
$528$			94.6860i			0.179330i
$529$	−252.094	+	465.070i	−0.476547	+	0.879149i
$530$	−66.7240			−0.125894
$531$	−316.612			−0.596256
$532$	421.553			0.792392
$533$	−134.592			−0.252518
$534$		−	51.0906i		−	0.0956752i
$535$	−2.54287			−0.00475303
$536$			98.9135i			0.184540i
$537$	107.606			0.200384
$538$	683.422			1.27030
$539$			197.152i			0.365774i
$540$			23.2379i			0.0430331i
$541$	704.031			1.30135			0.650676	−	0.759356i	$-0.274486\pi$
$541$	704.031			1.30135			0.650676	+	0.759356i	$0.274486\pi$
$542$	36.5929			0.0675145
$543$			59.3271i			0.109258i
$544$			97.6757i			0.179551i
$545$	165.713			0.304061
$546$		−	273.729i		−	0.501336i
$547$	469.176			0.857726			0.428863	−	0.903370i	$-0.358914\pi$
$547$	469.176			0.857726			0.428863	+	0.903370i	$0.358914\pi$
$548$		−	538.378i		−	0.982441i
$549$		−	100.441i		−	0.182953i
$550$			96.6385i			0.175706i
$551$		−	1442.36i		−	2.61771i
$552$	57.6444	+	96.8149i	0.104428	+	0.175389i
$553$	−358.676			−0.648600
$554$	−455.561			−0.822313
$555$	124.901			0.225047
$556$	−225.252			−0.405130
$557$		−	280.095i		−	0.502864i	−0.967875	−	0.251432i	$-0.919099\pi$
$557$		−	280.095i		−	0.502864i	0.967875	−	0.251432i	$-0.0809015\pi$
$558$	−94.0288			−0.168510
$559$			917.268i			1.64091i
$560$	−71.2324			−0.127201
$561$	−408.731			−0.728575
$562$			215.150i			0.382828i
$563$			30.3039i			0.0538257i	0.999638	+	0.0269129i	$0.00856767\pi$
$563$			30.3039i			0.0538257i	−0.999638	+	0.0269129i	$0.991432\pi$
$564$	−285.978			−0.507053
$565$	271.793			0.481050
$566$		−	362.620i		−	0.640672i
$567$		−	71.6762i		−	0.126413i
$568$	184.166			0.324237
$569$			253.630i			0.445746i	0.974847	+	0.222873i	$0.0715436\pi$
$569$			253.630i			0.445746i	−0.974847	+	0.222873i	$0.928456\pi$
$570$	−144.961			−0.254317
$571$			329.218i			0.576564i	0.957546	+	0.288282i	$0.0930841\pi$
$571$			329.218i			0.576564i	−0.957546	+	0.288282i	$0.906916\pi$
$572$			383.538i			0.670521i
$573$		−	633.597i		−	1.10575i
$574$		−	108.032i		−	0.188209i
$575$	58.8331	+	98.8113i	0.102318	+	0.171846i
$576$	24.0000			0.0416667
$577$	172.230			0.298492			0.149246	−	0.988800i	$-0.452315\pi$
$577$	172.230			0.298492			0.149246	+	0.988800i	$0.452315\pi$
$578$	−12.9289			−0.0223683
$579$	387.705			0.669612
$580$			243.724i			0.420214i
$581$	−82.4175			−0.141855
$582$			358.707i			0.616334i
$583$	−288.368			−0.494628
$584$	−357.157			−0.611570
$585$			94.1281i			0.160903i
$586$		−	524.811i		−	0.895583i
$587$	−627.853			−1.06960			−0.534798	−	0.844980i	$-0.679612\pi$
$587$	−627.853			−1.06960			−0.534798	+	0.844980i	$0.679612\pi$
$588$	49.9720			0.0849864
$589$		−	586.562i		−	0.995861i
$590$			333.739i			0.565659i
$591$	−155.225			−0.262648
$592$		−	128.997i		−	0.217901i
$593$	−916.147			−1.54494			−0.772468	−	0.635054i	$-0.780978\pi$
$593$	−916.147			−1.54494			−0.772468	+	0.635054i	$0.780978\pi$
$594$			100.430i			0.169073i
$595$		−	307.489i		−	0.516788i
$596$		−	106.756i		−	0.179121i
$597$			10.2192i			0.0171176i
$598$	233.496	+	392.161i	0.390462	+	0.655788i
$599$	767.429			1.28118			0.640592	−	0.767881i	$-0.278689\pi$
$599$	767.429			1.28118			0.640592	+	0.767881i	$0.278689\pi$
$600$	24.4949			0.0408248
$601$	−486.951			−0.810234			−0.405117	−	0.914265i	$-0.632769\pi$
$601$	−486.951			−0.810234			−0.405117	+	0.914265i	$0.632769\pi$
$602$	−736.258			−1.22302
$603$			104.914i			0.173986i
$604$	79.5791			0.131753
$605$			147.088i			0.243121i
$606$	−271.324			−0.447729
$607$	−856.769			−1.41148			−0.705740	−	0.708471i	$-0.749385\pi$
$607$	−856.769			−1.41148			−0.705740	+	0.708471i	$0.749385\pi$
$608$			149.715i			0.246241i
$609$		−	751.757i		−	1.23441i
$610$	−105.874			−0.173564
$611$	−1158.39			−1.89589
$612$			103.601i			0.169282i
$613$			996.963i			1.62637i	0.582008	+	0.813183i	$0.302267\pi$
$613$			996.963i			1.62637i	−0.582008	+	0.813183i	$0.697733\pi$
$614$	744.800			1.21303
$615$			37.1494i			0.0604055i
$616$	−307.852			−0.499760
$617$			403.163i			0.653425i	0.945124	+	0.326712i	$0.105941\pi$
$617$			403.163i			0.653425i	−0.945124	+	0.326712i	$0.894059\pi$
$618$		−	183.738i		−	0.297310i
$619$			6.17899i			0.00998222i	0.999988	+	0.00499111i	$0.00158873\pi$
$619$			6.17899i			0.00998222i	−0.999988	+	0.00499111i	$0.998411\pi$
$620$			99.1151i			0.159863i
$621$	61.1411	+	102.688i	0.0984559	+	0.165359i
$622$	759.651			1.22130
$623$	166.111			0.266630
$624$	97.2151			0.155793
$625$	25.0000			0.0400000
$626$		−	419.108i		−	0.669502i
$627$	−626.491			−0.999189
$628$		−	526.503i		−	0.838380i
$629$	556.843			0.885282
$630$	−75.5534			−0.119926
$631$			478.508i			0.758333i	0.925328	+	0.379166i	$0.123789\pi$
$631$			478.508i			0.758333i	−0.925328	+	0.379166i	$0.876211\pi$
$632$		−	127.384i		−	0.201557i
$633$	401.313			0.633985
$634$	−204.314			−0.322262
$635$			104.248i			0.164171i
$636$			73.0925i			0.114925i
$637$	202.418			0.317768
$638$			1053.33i			1.65099i
$639$	195.338			0.305693
$640$		−	25.2982i		−	0.0395285i
$641$			431.496i			0.673160i	0.941655	+	0.336580i	$0.109270\pi$
$641$			431.496i			0.673160i	−0.941655	+	0.336580i	$0.890730\pi$
$642$			2.78558i			0.00433891i
$643$			155.797i			0.242297i	0.992634	+	0.121149i	$0.0386578\pi$
$643$			155.797i			0.242297i	−0.992634	+	0.121149i	$0.961342\pi$
$644$	−314.774	+	187.419i	−0.488780	+	0.291024i
$645$	253.179			0.392526
$646$	−646.273			−1.00042
$647$	−797.916			−1.23326			−0.616628	−	0.787255i	$-0.711502\pi$
$647$	−797.916			−1.23326			−0.616628	+	0.787255i	$0.711502\pi$
$648$	25.4558			0.0392837
$649$			1442.35i			2.22242i
$650$	99.2198			0.152646
$651$		−	305.716i		−	0.469609i
$652$	479.356			0.735209
$653$	579.230			0.887028			0.443514	−	0.896267i	$-0.353732\pi$
$653$	579.230			0.887028			0.443514	+	0.896267i	$0.353732\pi$
$654$		−	181.530i		−	0.277568i
$655$		−	386.093i		−	0.589455i
$656$	38.3677			0.0584873
$657$	−378.822			−0.576594
$658$		−	929.800i		−	1.41307i
$659$		−	991.515i		−	1.50458i	−0.658835	−	0.752288i	$-0.728950\pi$
$659$		−	991.515i		−	1.50458i	0.658835	−	0.752288i	$-0.271050\pi$
$660$	105.862			0.160397
$661$			985.923i			1.49156i	0.666191	+	0.745781i	$0.267923\pi$
$661$			985.923i			1.49156i	−0.666191	+	0.745781i	$0.732077\pi$
$662$	−680.513			−1.02797
$663$			419.648i			0.632954i
$664$		−	29.2706i		−	0.0440823i
$665$		−	471.310i		−	0.708737i
$666$		−	136.822i		−	0.205439i
$667$	641.262	+	1077.01i	0.961412	+	1.61471i
$668$	633.767			0.948753
$669$	−225.167			−0.336573
$670$	110.589			0.165058
$671$	−457.567			−0.681918
$672$			78.0312i			0.116118i
$673$	809.473			1.20278			0.601392	−	0.798954i	$-0.294613\pi$
$673$	809.473			1.20278			0.601392	+	0.798954i	$0.294613\pi$
$674$			322.414i			0.478358i
$675$	25.9808			0.0384900
$676$	55.7826			0.0825186
$677$		−	1104.73i		−	1.63181i	−0.578188	−	0.815903i	$-0.696240\pi$
$677$		−	1104.73i		−	1.63181i	0.578188	−	0.815903i	$-0.303760\pi$
$678$		−	297.734i		−	0.439136i
$679$	−1166.26			−1.71762
$680$	109.205			0.160595
$681$			289.452i			0.425040i
$682$			428.356i			0.628088i
$683$	−503.585			−0.737313			−0.368657	−	0.929566i	$-0.620182\pi$
$683$	−503.585			−0.737313			−0.368657	+	0.929566i	$0.620182\pi$
$684$			158.796i			0.232158i
$685$	−601.925			−0.878722
$686$		−	389.405i		−	0.567645i
$687$		−	33.2771i		−	0.0484383i
$688$		−	261.483i		−	0.380062i
$689$			296.071i			0.429710i
$690$	108.242	−	64.4484i	0.156873	−	0.0934035i
$691$	693.803			1.00406			0.502028	−	0.864851i	$-0.332587\pi$
$691$	693.803			1.00406			0.502028	+	0.864851i	$0.332587\pi$
$692$	231.782			0.334945
$693$	−326.527			−0.471179
$694$	402.218			0.579565
$695$			251.840i			0.362359i
$696$	266.987			0.383602
$697$			165.622i			0.237621i
$698$	−603.503			−0.864617
$699$	−176.145			−0.251996
$700$			79.6402i			0.113772i
$701$		−	456.823i		−	0.651673i	−0.945426	−	0.325836i	$-0.894354\pi$
$701$		−	456.823i		−	0.651673i	0.945426	−	0.325836i	$-0.105646\pi$
$702$	103.112			0.146884
$703$	853.513			1.21410
$704$		−	109.334i		−	0.155304i
$705$			319.733i			0.453522i
$706$	225.206			0.318989
$707$		−	882.155i		−	1.24774i
$708$	365.592			0.516373
$709$			54.3734i			0.0766903i	0.999265	+	0.0383452i	$0.0122086\pi$
$709$			54.3734i			0.0766903i	−0.999265	+	0.0383452i	$0.987791\pi$
$710$		−	205.904i		−	0.290006i
$711$		−	135.111i		−	0.190030i
$712$			58.9943i			0.0828572i
$713$	260.781	+	437.987i	0.365752	+	0.614287i
$714$	−336.837			−0.471761
$715$	428.808			0.599732
$716$	−124.253			−0.173537
$717$	113.841			0.158774
$718$		−	933.742i		−	1.30048i
$719$	154.273			0.214566			0.107283	−	0.994229i	$-0.465785\pi$
$719$	154.273			0.214566			0.107283	+	0.994229i	$0.465785\pi$
$720$		−	26.8328i		−	0.0372678i
$721$	597.386			0.828553
$722$	−480.059			−0.664901
$723$			331.459i			0.458450i
$724$		−	68.5051i		−	0.0946202i
$725$	272.492			0.375851
$726$	161.127			0.221938
$727$			772.570i			1.06268i	0.847158	+	0.531341i	$0.178312\pi$
$727$			772.570i			1.06268i	−0.847158	+	0.531341i	$0.821688\pi$
$728$			316.075i			0.434170i
$729$	27.0000			0.0370370
$730$			399.314i			0.547005i
$731$	1128.74			1.54411
$732$			115.979i			0.158442i
$733$		−	590.416i		−	0.805479i	−0.915315	−	0.402739i	$-0.868058\pi$
$733$		−	590.416i		−	0.805479i	0.915315	−	0.402739i	$-0.131942\pi$
$734$		−	217.656i		−	0.296534i
$735$		−	55.8704i		−	0.0760141i
$736$	−66.5620	−	111.792i	−0.0904375	−	0.151892i
$737$	477.942			0.648497
$738$	40.6951			0.0551424
$739$	99.2703			0.134331			0.0671653	−	0.997742i	$-0.478605\pi$
$739$	99.2703			0.134331			0.0671653	+	0.997742i	$0.478605\pi$
$740$	−144.223			−0.194897
$741$			643.226i			0.868051i
$742$	−237.645			−0.320277
$743$		−	5.78824i		−	0.00779036i	−0.999992	−	0.00389518i	$-0.998760\pi$
$743$		−	5.78824i		−	0.00779036i	0.999992	−	0.00389518i	$-0.00123988\pi$
$744$	108.575			0.145934
$745$	−119.357			−0.160210
$746$		−	161.329i		−	0.216259i
$747$		−	31.0462i		−	0.0415612i
$748$	471.962			0.630965
$749$	−9.05675			−0.0120918
$750$		−	27.3861i		−	0.0365148i
$751$		−	4.93292i		−	0.00656847i	−0.999995	−	0.00328423i	$-0.998955\pi$
$751$		−	4.93292i		−	0.00656847i	0.999995	−	0.00328423i	$-0.00104541\pi$
$752$	330.219			0.439121
$753$		−	467.032i		−	0.620228i
$754$	1081.46			1.43430
$755$		−	88.9721i		−	0.117844i
$756$			82.7646i			0.109477i
$757$		−	841.391i		−	1.11148i	−0.831356	−	0.555740i	$-0.812435\pi$
$757$		−	841.391i		−	1.11148i	0.831356	−	0.555740i	$-0.187565\pi$
$758$		−	804.614i		−	1.06150i
$759$	467.802	−	278.533i	0.616340	−	0.366974i
$760$	167.386			0.220245
$761$	122.636			0.161151			0.0805754	−	0.996749i	$-0.474324\pi$
$761$	122.636			0.161151			0.0805754	+	0.996749i	$0.474324\pi$
$762$	114.198			0.149867
$763$	590.207			0.773535
$764$			731.615i			0.957611i
$765$	115.829			0.151411
$766$			846.855i			1.10556i
$767$	1480.88			1.93074
$768$	−27.7128			−0.0360844
$769$		−	495.138i		−	0.643873i	−0.946761	−	0.321936i	$-0.895666\pi$
$769$		−	495.138i		−	0.643873i	0.946761	−	0.321936i	$-0.104334\pi$
$770$			344.189i			0.446999i
$771$	−225.789			−0.292852
$772$	−447.683			−0.579901
$773$			314.784i			0.407224i	0.979052	+	0.203612i	$0.0652680\pi$
$773$			314.784i			0.407224i	−0.979052	+	0.203612i	$0.934732\pi$
$774$		−	277.344i		−	0.358326i
$775$	110.814			0.142986
$776$		−	414.199i		−	0.533761i
$777$	444.850			0.572523
$778$			418.810i			0.538316i
$779$			253.861i			0.325880i
$780$		−	108.690i		−	0.139346i
$781$		−	889.878i		−	1.13941i
$782$	482.573	−	287.328i	0.617101	−	0.367427i
$783$	283.182			0.361663
$784$	−57.7027			−0.0736004
$785$	−588.648			−0.749870
$786$	−422.943			−0.538096
$787$			870.883i			1.10659i	0.832987	+	0.553293i	$0.186629\pi$
$787$			870.883i			1.10659i	−0.832987	+	0.553293i	$0.813371\pi$
$788$	179.239			0.227460
$789$			310.810i			0.393929i
$790$	−142.420			−0.180278
$791$	968.024			1.22380
$792$		−	115.966i		−	0.146422i
$793$			469.789i			0.592420i
$794$	−998.625			−1.25771
$795$	81.7198			0.102792
$796$		−	11.8001i		−	0.0148243i
$797$			614.333i			0.770807i	0.922748	+	0.385403i	$0.125938\pi$
$797$			614.333i			0.770807i	−0.922748	+	0.385403i	$0.874062\pi$
$798$	−516.295			−0.646986
$799$			1425.46i			1.78405i
$800$	−28.2843			−0.0353553
$801$			62.5729i			0.0781185i
$802$			709.806i			0.885045i
$803$			1725.76i			2.14914i
$804$		−	121.144i		−	0.150676i
$805$	209.541	+	351.928i	0.260299	+	0.437178i
$806$	439.798			0.545655
$807$	−837.018			−1.03720
$808$	313.298			0.387745
$809$	−380.557			−0.470405			−0.235202	−	0.971946i	$-0.575575\pi$
$809$	−380.557			−0.470405			−0.235202	+	0.971946i	$0.575575\pi$
$810$		−	28.4605i		−	0.0351364i
$811$	1032.12			1.27265			0.636324	−	0.771422i	$-0.280454\pi$
$811$	1032.12			1.27265			0.636324	+	0.771422i	$0.280454\pi$
$812$			868.054i			1.06903i
$813$	−44.8169			−0.0551254
$814$	−623.305			−0.765731
$815$		−	535.937i		−	0.657591i
$816$		−	119.628i		−	0.146603i
$817$	1730.10			2.11763
$818$	−496.822			−0.607362
$819$			335.249i			0.409339i
$820$		−	42.8964i		−	0.0523127i
$821$	1218.00			1.48355			0.741777	−	0.670647i	$-0.233984\pi$
$821$	1218.00			1.48355			0.741777	+	0.670647i	$0.233984\pi$
$822$			659.375i			0.802160i
$823$	361.478			0.439220			0.219610	−	0.975588i	$-0.429522\pi$
$823$	361.478			0.439220			0.219610	+	0.975588i	$0.429522\pi$
$824$			212.162i			0.257478i
$825$		−	118.357i		−	0.143464i
$826$			1188.65i			1.43904i
$827$		−	1564.84i		−	1.89219i	−0.323893	−	0.946094i	$-0.604992\pi$
$827$		−	1564.84i		−	1.89219i	0.323893	−	0.946094i	$-0.395008\pi$
$828$	−70.5997	−	118.574i	−0.0852653	−	0.143205i
$829$	817.899			0.986609			0.493304	−	0.869857i	$-0.335789\pi$
$829$	817.899			0.986609			0.493304	+	0.869857i	$0.335789\pi$
$830$	−32.7256			−0.0394284
$831$	557.946			0.671416
$832$	−112.254			−0.134921
$833$		−	249.085i		−	0.299022i
$834$	275.876			0.330787
$835$		−	708.573i		−	0.848591i
$836$	723.410			0.865323
$837$	115.161			0.137588
$838$			703.847i			0.839913i
$839$		−	281.562i		−	0.335592i	−0.985822	−	0.167796i	$-0.946335\pi$
$839$		−	281.562i		−	0.335592i	0.985822	−	0.167796i	$-0.0536650\pi$
$840$	87.2415			0.103859
$841$	2129.08			2.53160
$842$		−	457.600i		−	0.543469i
$843$		−	263.503i		−	0.312578i
$844$	−463.396			−0.549048
$845$		−	62.3668i		−	0.0738069i
$846$	350.250			0.414007
$847$			523.873i			0.618504i
$848$		−	84.3999i		−	0.0995282i
$849$			444.117i			0.523107i
$850$		−	122.095i		−	0.143641i
$851$	−637.320	+	379.466i	−0.748907	+	0.445905i
$852$	−225.557			−0.264738
$853$	−856.416			−1.00400			−0.502002	−	0.864866i	$-0.667403\pi$
$853$	−856.416			−1.00400			−0.502002	+	0.864866i	$0.667403\pi$
$854$	−377.083			−0.441550
$855$	177.540			0.207649
$856$		−	3.21651i		−	0.00375760i
$857$	−261.702			−0.305369			−0.152685	−	0.988275i	$-0.548792\pi$
$857$	−261.702			−0.305369			−0.152685	+	0.988275i	$0.548792\pi$
$858$		−	469.736i		−	0.547478i
$859$	−816.256			−0.950240			−0.475120	−	0.879921i	$-0.657595\pi$
$859$	−816.256			−0.950240			−0.475120	+	0.879921i	$0.657595\pi$
$860$	−292.346			−0.339938
$861$			132.312i			0.153672i
$862$		−	763.295i		−	0.885493i
$863$	−915.224			−1.06051			−0.530257	−	0.847837i	$-0.677905\pi$
$863$	−915.224			−1.06051			−0.530257	+	0.847837i	$0.677905\pi$
$864$	−29.3939			−0.0340207
$865$		−	259.140i		−	0.299584i
$866$			596.576i			0.688887i
$867$	15.8346			0.0182636
$868$			353.010i			0.406694i
$869$	−615.510			−0.708297
$870$		−	298.500i		−	0.343104i
$871$		−	490.709i		−	0.563385i
$872$			209.612i			0.240381i
$873$		−	439.324i		−	0.503235i
$874$	739.675	−	440.409i	0.846310	−	0.503900i
$875$	89.0405			0.101761
$876$	437.426			0.499345
$877$	−891.151			−1.01614			−0.508068	−	0.861317i	$-0.669640\pi$
$877$	−891.151			−1.01614			−0.508068	+	0.861317i	$0.669640\pi$
$878$	458.782			0.522531
$879$			642.760i			0.731240i
$880$	−122.239			−0.138908
$881$		−	330.926i		−	0.375625i	−0.982205	−	0.187813i	$-0.939860\pi$
$881$		−	330.926i		−	0.375625i	0.982205	−	0.187813i	$-0.0601398\pi$
$882$	−61.2030			−0.0693911
$883$	−369.742			−0.418734			−0.209367	−	0.977837i	$-0.567140\pi$
$883$	−369.742			−0.418734			−0.209367	+	0.977837i	$0.567140\pi$
$884$		−	484.568i		−	0.548154i
$885$		−	408.745i		−	0.461858i
$886$	313.668			0.354028
$887$	876.383			0.988031			0.494015	−	0.869453i	$-0.335529\pi$
$887$	876.383			0.988031			0.494015	+	0.869453i	$0.335529\pi$
$888$			157.989i			0.177915i
$889$			371.293i			0.417653i
$890$	65.9577			0.0741097
$891$		−	123.001i		−	0.138048i
$892$	260.001			0.291481
$893$			2184.90i			2.44670i
$894$			130.749i			0.146251i
$895$			138.919i			0.155217i
$896$		−	90.1026i		−	0.100561i
$897$	−285.973	−	480.298i	−0.318811	−	0.535449i
$898$	−740.935			−0.825094
$899$	1207.84			1.34354
$900$	−30.0000			−0.0333333
$901$	364.329			0.404361
$902$		−	185.390i		−	0.205532i
$903$	901.729			0.998592
$904$			343.794i			0.380303i
$905$	−76.5910			−0.0846309
$906$	−97.4641			−0.107576
$907$			1391.06i			1.53369i	0.641832	+	0.766845i	$0.278175\pi$
$907$			1391.06i			1.53369i	−0.641832	+	0.766845i	$0.721825\pi$
$908$		−	334.230i		−	0.368095i
$909$	332.302			0.365569
$910$	353.383			0.388333
$911$			633.904i			0.695833i	0.937525	+	0.347917i	$0.113111\pi$
$911$			633.904i			0.695833i	−0.937525	+	0.347917i	$0.886889\pi$
$912$		−	183.362i		−	0.201055i
$913$	−141.433			−0.154911
$914$		−	536.572i		−	0.587059i
$915$	129.669			0.141715
$916$			38.4251i			0.0419488i
$917$		−	1375.12i		−	1.49958i
$918$		−	126.884i		−	0.138218i
$919$		−	89.6435i		−	0.0975446i	−0.998810	−	0.0487723i	$-0.984469\pi$
$919$		−	89.6435i		−	0.0975446i	0.998810	−	0.0487723i	$-0.0155309\pi$
$920$	−124.987	+	74.4186i	−0.135856	+	0.0808898i
$921$	−912.190			−0.990435
$922$	−635.400			−0.689154
$923$	−913.648			−0.989868
$924$	377.041			0.408053
$925$			161.247i			0.174321i
$926$	822.296			0.888008
$927$			225.032i			0.242753i
$928$	−308.290			−0.332209
$929$	129.658			0.139568			0.0697838	−	0.997562i	$-0.477769\pi$
$929$	129.658			0.139568			0.0697838	+	0.997562i	$0.477769\pi$
$930$		−	121.391i		−	0.130528i
$931$		−	381.791i		−	0.410087i
$932$	203.395			0.218235
$933$	−930.378			−0.997190
$934$			879.787i			0.941956i
$935$		−	527.669i		−	0.564352i
$936$	−119.064			−0.127205
$937$			171.827i			0.183380i	0.995788	+	0.0916900i	$0.0292269\pi$
$937$			171.827i			0.183380i	−0.995788	+	0.0916900i	$0.970773\pi$
$938$	393.875			0.419909
$939$			513.301i			0.546646i
$940$		−	369.196i		−	0.392762i
$941$		−	6.11599i		−	0.00649946i	−0.999995	−	0.00324973i	$-0.998966\pi$
$941$		−	6.11599i		−	0.00649946i	0.999995	−	0.00324973i	$-0.00103442\pi$
$942$			644.831i			0.684534i
$943$	−112.864	−	189.558i	−0.119687	−	0.201016i
$944$	−422.150			−0.447192
$945$	92.5336			0.0979191
$946$	−1263.46			−1.33559
$947$	131.841			0.139219			0.0696097	−	0.997574i	$-0.477825\pi$
$947$	131.841			0.139219			0.0696097	+	0.997574i	$0.477825\pi$
$948$			156.013i			0.164571i
$949$	1771.85			1.86707
$950$		−	187.143i		−	0.196993i
$951$	250.233			0.263126
$952$	388.946			0.408557
$953$			609.278i			0.639326i	0.947531	+	0.319663i	$0.103570\pi$
$953$			609.278i			0.639326i	−0.947531	+	0.319663i	$0.896430\pi$
$954$		−	89.5196i		−	0.0938361i
$955$	817.970			0.856513
$956$	−131.452			−0.137502
$957$		−	1290.06i		−	1.34802i
$958$			381.209i			0.397922i
$959$	−2143.83			−2.23548
$960$			30.9839i			0.0322749i
$961$	−469.810			−0.488876
$962$			639.954i			0.665233i
$963$		−	3.41162i		−	0.00354270i
$964$		−	382.736i		−	0.397029i
$965$			500.525i			0.518679i
$966$	385.518	−	229.541i	0.399087	−	0.237620i
$967$	443.855			0.459002			0.229501	−	0.973308i	$-0.426291\pi$
$967$	443.855			0.459002			0.229501	+	0.973308i	$0.426291\pi$
$968$	−186.054			−0.192204
$969$	791.520			0.816842
$970$	−463.088			−0.477410
$971$		−	1070.02i		−	1.10198i	−0.834511	−	0.550991i	$-0.814250\pi$
$971$		−	1070.02i		−	1.10198i	0.834511	−	0.550991i	$-0.185750\pi$
$972$	−31.1769			−0.0320750
$973$			896.957i			0.921847i
$974$	455.278			0.467431
$975$	−121.519			−0.124635
$976$		−	133.921i		−	0.137215i
$977$			1707.50i			1.74770i	0.486195	+	0.873850i	$0.338384\pi$
$977$			1707.50i			1.74770i	−0.486195	+	0.873850i	$0.661616\pi$
$978$	−587.089			−0.600296
$979$	285.056			0.291171
$980$			64.5136i			0.0658302i
$981$			222.327i			0.226633i
$982$	−594.253			−0.605146
$983$			1086.75i			1.10555i	0.833332	+	0.552773i	$0.186431\pi$
$983$			1086.75i			1.10555i	−0.833332	+	0.552773i	$0.813569\pi$
$984$	−46.9906			−0.0477547
$985$		−	200.395i		−	0.203446i
$986$		−	1330.79i		−	1.34969i
$987$			1138.77i			1.15377i
$988$		−	742.733i		−	0.751754i
$989$	−1291.87	+	769.191i	−1.30624	+	0.777746i
$990$	−129.654			−0.130964
$991$	374.649			0.378051			0.189026	−	0.981972i	$-0.439467\pi$
$991$	374.649			0.378051			0.189026	+	0.981972i	$0.439467\pi$
$992$	−125.372			−0.126383
$993$	833.455			0.839331
$994$		−	733.353i		−	0.737780i
$995$	−13.1929			−0.0132592
$996$			35.8491i			0.0359930i
$997$	−761.430			−0.763721			−0.381860	−	0.924220i	$-0.624716\pi$
$997$	−761.430			−0.763721			−0.381860	+	0.924220i	$0.624716\pi$
$998$	−366.851			−0.367586
$999$			167.572i			0.167740i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	690.3.c.a.91.18	✓	32
3.2	odd	2		2070.3.c.b.91.10		32
23.22	odd	2	inner	690.3.c.a.91.23	yes	32
69.68	even	2		2070.3.c.b.91.7		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.3.c.a.91.18	✓	32	1.1	even	1	trivial
690.3.c.a.91.23	yes	32	23.22	odd	2	inner
2070.3.c.b.91.7		32	69.68	even	2
2070.3.c.b.91.10		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)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} -1.73205 q^{3} +2.00000 q^{4} -2.23607i q^{5} -2.44949 q^{6} -7.96402i q^{7} +2.82843 q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q+1.41421 q^{2} -1.73205 q^{3} +2.00000 q^{4} -2.23607i q^{5} -2.44949 q^{6} -7.96402i q^{7} +2.82843 q^{8} +3.00000 q^{9} -3.16228i q^{10} -13.6667i q^{11} -3.46410 q^{12} -14.0318 q^{13} -11.2628i q^{14} +3.87298i q^{15} +4.00000 q^{16} +17.2668i q^{17} +4.24264 q^{18} +26.4661i q^{19} -4.47214i q^{20} +13.7941i q^{21} -19.3277i q^{22} +(-11.7666 - 19.7623i) q^{23} -4.89898 q^{24} -5.00000 q^{25} -19.8440 q^{26} -5.19615 q^{27} -15.9280i q^{28} -54.4984 q^{29} +5.47723i q^{30} -22.1628 q^{31} +5.65685 q^{32} +23.6715i q^{33} +24.4189i q^{34} -17.8081 q^{35} +6.00000 q^{36} -32.2493i q^{37} +37.4287i q^{38} +24.3038 q^{39} -6.32456i q^{40} +9.59193 q^{41} +19.5078i q^{42} -65.3706i q^{43} -27.3335i q^{44} -6.70820i q^{45} +(-16.6405 - 27.9481i) q^{46} +82.5547 q^{47} -6.92820 q^{48} -14.4257 q^{49} -7.07107 q^{50} -29.9070i q^{51} -28.0636 q^{52} -21.1000i q^{53} -7.34847 q^{54} -30.5598 q^{55} -22.5257i q^{56} -45.8406i q^{57} -77.0724 q^{58} -105.537 q^{59} +7.74597i q^{60} -33.4803i q^{61} -31.3429 q^{62} -23.8921i q^{63} +8.00000 q^{64} +31.3760i q^{65} +33.4765i q^{66} +34.9712i q^{67} +34.5336i q^{68} +(20.3804 + 34.2292i) q^{69} -25.1845 q^{70} +65.1127 q^{71} +8.48528 q^{72} -126.274 q^{73} -45.6075i q^{74} +8.66025 q^{75} +52.9321i q^{76} -108.842 q^{77} +34.3707 q^{78} -45.0370i q^{79} -8.94427i q^{80} +9.00000 q^{81} +13.5650 q^{82} -10.3487i q^{83} +27.5882i q^{84} +38.6097 q^{85} -92.4480i q^{86} +94.3941 q^{87} -38.6554i q^{88} +20.8576i q^{89} -9.48683i q^{90} +111.750i q^{91} +(-23.5332 - 39.5245i) q^{92} +38.3871 q^{93} +116.750 q^{94} +59.1799 q^{95} -9.79796 q^{96} -146.441i q^{97} -20.4010 q^{98} -41.0002i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 64 q^{4} + 96 q^{9}+O(q^{10}) \) 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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