LMFDB - Embedded newform 690.3.c.a.91.31

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			91.31
Character	$\chi$	$=$	690.91
Dual form			690.3.c.a.91.26

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} +1.73205 q^{3} +2.00000 q^{4} +2.23607i q^{5} +2.44949 q^{6} +6.64524i q^{7} +2.82843 q^{8} +3.00000 q^{9} +O(q^{10})$	\(q+1.41421 q^{2} +1.73205 q^{3} +2.00000 q^{4} +2.23607i q^{5} +2.44949 q^{6} +6.64524i q^{7} +2.82843 q^{8} +3.00000 q^{9} +3.16228i q^{10} -9.54001i q^{11} +3.46410 q^{12} +10.3241 q^{13} +9.39779i q^{14} +3.87298i q^{15} +4.00000 q^{16} +15.5981i q^{17} +4.24264 q^{18} -0.664113i q^{19} +4.47214i q^{20} +11.5099i q^{21} -13.4916i q^{22} +(16.5468 + 15.9751i) q^{23} +4.89898 q^{24} -5.00000 q^{25} +14.6005 q^{26} +5.19615 q^{27} +13.2905i q^{28} +48.3347 q^{29} +5.47723i q^{30} -42.5781 q^{31} +5.65685 q^{32} -16.5238i q^{33} +22.0590i q^{34} -14.8592 q^{35} +6.00000 q^{36} +37.9575i q^{37} -0.939198i q^{38} +17.8819 q^{39} +6.32456i q^{40} -32.8401 q^{41} +16.2775i q^{42} +56.3079i q^{43} -19.0800i q^{44} +6.70820i q^{45} +(23.4007 + 22.5922i) q^{46} +26.7091 q^{47} +6.92820 q^{48} +4.84077 q^{49} -7.07107 q^{50} +27.0167i q^{51} +20.6482 q^{52} +13.8360i q^{53} +7.34847 q^{54} +21.3321 q^{55} +18.7956i q^{56} -1.15028i q^{57} +68.3556 q^{58} -92.5057 q^{59} +7.74597i q^{60} -114.986i q^{61} -60.2145 q^{62} +19.9357i q^{63} +8.00000 q^{64} +23.0854i q^{65} -23.3682i q^{66} -119.439i q^{67} +31.1962i q^{68} +(28.6599 + 27.6697i) q^{69} -21.0141 q^{70} +104.336 q^{71} +8.48528 q^{72} +4.46718 q^{73} +53.6800i q^{74} -8.66025 q^{75} -1.32823i q^{76} +63.3957 q^{77} +25.2888 q^{78} -29.4634i q^{79} +8.94427i q^{80} +9.00000 q^{81} -46.4429 q^{82} -136.169i q^{83} +23.0198i q^{84} -34.8784 q^{85} +79.6313i q^{86} +83.7182 q^{87} -26.9832i q^{88} -19.1748i q^{89} +9.48683i q^{90} +68.6063i q^{91} +(33.0935 + 31.9502i) q^{92} -73.7474 q^{93} +37.7723 q^{94} +1.48500 q^{95} +9.79796 q^{96} +3.15840i q^{97} +6.84588 q^{98} -28.6200i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$			6.64524i			0.949320i	0.880169	+	0.474660i	$0.157429\pi$
$7$			6.64524i			0.949320i	−0.880169	+	0.474660i	$0.842571\pi$
$8$	2.82843			0.353553
$9$	3.00000			0.333333
$10$			3.16228i			0.316228i
$11$		−	9.54001i		−	0.867274i	−0.901088	−	0.433637i	$-0.857230\pi$
$11$		−	9.54001i		−	0.867274i	0.901088	−	0.433637i	$-0.142770\pi$
$12$	3.46410			0.288675
$13$	10.3241			0.794163			0.397082	−	0.917783i	$-0.370023\pi$
$13$	10.3241			0.794163			0.397082	+	0.917783i	$0.370023\pi$
$14$			9.39779i			0.671271i
$15$			3.87298i			0.258199i
$16$	4.00000			0.250000
$17$			15.5981i			0.917534i	0.888557	+	0.458767i	$0.151709\pi$
$17$			15.5981i			0.917534i	−0.888557	+	0.458767i	$0.848291\pi$
$18$	4.24264			0.235702
$19$		−	0.664113i		−	0.0349533i	−0.999847	−	0.0174767i	$-0.994437\pi$
$19$		−	0.664113i		−	0.0349533i	0.999847	−	0.0174767i	$-0.00556328\pi$
$20$			4.47214i			0.223607i
$21$			11.5099i			0.548090i
$22$		−	13.4916i		−	0.613255i
$23$	16.5468	+	15.9751i	0.719425	+	0.694570i
$23$	16.5468	+	15.9751i	0.719425	+	0.694570i
$24$	4.89898			0.204124
$25$	−5.00000			−0.200000
$26$	14.6005			0.561558
$27$	5.19615			0.192450
$28$			13.2905i			0.474660i
$29$	48.3347			1.66672			0.833358	−	0.552734i	$-0.186416\pi$
$29$	48.3347			1.66672			0.833358	+	0.552734i	$0.186416\pi$
$30$			5.47723i			0.182574i
$31$	−42.5781			−1.37349			−0.686743	−	0.726900i	$-0.740960\pi$
$31$	−42.5781			−1.37349			−0.686743	+	0.726900i	$0.740960\pi$
$32$	5.65685			0.176777
$33$		−	16.5238i		−	0.500721i
$34$			22.0590i			0.648795i
$35$	−14.8592			−0.424549
$36$	6.00000			0.166667
$37$			37.9575i			1.02588i	0.858425	+	0.512939i	$0.171443\pi$
$37$			37.9575i			1.02588i	−0.858425	+	0.512939i	$0.828557\pi$
$38$		−	0.939198i		−	0.0247157i
$39$	17.8819			0.458510
$40$			6.32456i			0.158114i
$41$	−32.8401			−0.800977			−0.400489	−	0.916302i	$-0.631160\pi$
$41$	−32.8401			−0.800977			−0.400489	+	0.916302i	$0.631160\pi$
$42$			16.2775i			0.387558i
$43$			56.3079i			1.30948i	0.755852	+	0.654742i	$0.227223\pi$
$43$			56.3079i			1.30948i	−0.755852	+	0.654742i	$0.772777\pi$
$44$		−	19.0800i		−	0.433637i
$45$			6.70820i			0.149071i
$46$	23.4007	+	22.5922i	0.508710	+	0.491135i
$47$	26.7091			0.568278			0.284139	−	0.958783i	$-0.408292\pi$
$47$	26.7091			0.568278			0.284139	+	0.958783i	$0.408292\pi$
$48$	6.92820			0.144338
$49$	4.84077			0.0987912
$50$	−7.07107			−0.141421
$51$			27.0167i			0.529739i
$52$	20.6482			0.397082
$53$			13.8360i			0.261056i	0.991445	+	0.130528i	$0.0416672\pi$
$53$			13.8360i			0.261056i	−0.991445	+	0.130528i	$0.958333\pi$
$54$	7.34847			0.136083
$55$	21.3321			0.387857
$56$			18.7956i			0.335635i
$57$		−	1.15028i		−	0.0201803i
$58$	68.3556			1.17855
$59$	−92.5057			−1.56789			−0.783947	−	0.620828i	$-0.786797\pi$
$59$	−92.5057			−1.56789			−0.783947	+	0.620828i	$0.786797\pi$
$60$			7.74597i			0.129099i
$61$		−	114.986i		−	1.88502i	−0.334185	−	0.942508i	$-0.608461\pi$
$61$		−	114.986i		−	1.88502i	0.334185	−	0.942508i	$-0.391539\pi$
$62$	−60.2145			−0.971201
$63$			19.9357i			0.316440i
$64$	8.00000			0.125000
$65$			23.0854i			0.355161i
$66$		−	23.3682i		−	0.354063i
$67$		−	119.439i		−	1.78268i	−0.453337	−	0.891339i	$-0.649767\pi$
$67$		−	119.439i		−	1.78268i	0.453337	−	0.891339i	$-0.350233\pi$
$68$			31.1962i			0.458767i
$69$	28.6599	+	27.6697i	0.415360	+	0.401010i
$70$	−21.0141			−0.300201
$71$	104.336			1.46952			0.734761	−	0.678326i	$-0.237294\pi$
$71$	104.336			1.46952			0.734761	+	0.678326i	$0.237294\pi$
$72$	8.48528			0.117851
$73$	4.46718			0.0611943			0.0305972	−	0.999532i	$-0.490259\pi$
$73$	4.46718			0.0611943			0.0305972	+	0.999532i	$0.490259\pi$
$74$			53.6800i			0.725405i
$75$	−8.66025			−0.115470
$76$		−	1.32823i		−	0.0174767i
$77$	63.3957			0.823320
$78$	25.2888			0.324216
$79$		−	29.4634i		−	0.372954i	−0.982459	−	0.186477i	$-0.940293\pi$
$79$		−	29.4634i		−	0.372954i	0.982459	−	0.186477i	$-0.0597070\pi$
$80$			8.94427i			0.111803i
$81$	9.00000			0.111111
$82$	−46.4429			−0.566376
$83$		−	136.169i		−	1.64060i	−0.571937	−	0.820298i	$-0.693808\pi$
$83$		−	136.169i		−	1.64060i	0.571937	−	0.820298i	$-0.306192\pi$
$84$			23.0198i			0.274045i
$85$	−34.8784			−0.410334
$86$			79.6313i			0.925946i
$87$	83.7182			0.962278
$88$		−	26.9832i		−	0.306627i
$89$		−	19.1748i		−	0.215447i	−0.994181	−	0.107724i	$-0.965644\pi$
$89$		−	19.1748i		−	0.215447i	0.994181	−	0.107724i	$-0.0343562\pi$
$90$			9.48683i			0.105409i
$91$			68.6063i			0.753915i
$92$	33.0935	+	31.9502i	0.359712	+	0.347285i
$93$	−73.7474			−0.792983
$94$	37.7723			0.401833
$95$	1.48500			0.0156316
$96$	9.79796			0.102062
$97$			3.15840i			0.0325609i	0.999867	+	0.0162804i	$0.00518245\pi$
$97$			3.15840i			0.0325609i	−0.999867	+	0.0162804i	$0.994818\pi$
$98$	6.84588			0.0698559
$99$		−	28.6200i		−	0.289091i
$100$	−10.0000			−0.100000
$101$	−134.694			−1.33361			−0.666803	−	0.745234i	$-0.732338\pi$
$101$	−134.694			−1.33361			−0.666803	+	0.745234i	$0.732338\pi$
$102$			38.2073i			0.374582i
$103$		−	39.6863i		−	0.385303i	−0.981267	−	0.192652i	$-0.938291\pi$
$103$		−	39.6863i		−	0.385303i	0.981267	−	0.192652i	$-0.0617088\pi$
$104$	29.2010			0.280779
$105$	−25.7369			−0.245113
$106$			19.5670i			0.184594i
$107$			37.7240i			0.352561i	0.984340	+	0.176281i	$0.0564066\pi$
$107$			37.7240i			0.352561i	−0.984340	+	0.176281i	$0.943593\pi$
$108$	10.3923			0.0962250
$109$		−	180.885i		−	1.65949i	−0.558142	−	0.829746i	$-0.688485\pi$
$109$		−	180.885i		−	1.65949i	0.558142	−	0.829746i	$-0.311515\pi$
$110$	30.1682			0.274256
$111$			65.7443i			0.592291i
$112$			26.5810i			0.237330i
$113$			168.729i			1.49317i	0.665288	+	0.746587i	$0.268309\pi$
$113$			168.729i			1.49317i	−0.665288	+	0.746587i	$0.731691\pi$
$114$		−	1.62674i		−	0.0142696i
$115$	−35.7214	+	36.9997i	−0.310621	+	0.321737i
$116$	96.6695			0.833358
$117$	30.9724			0.264721
$118$	−130.823			−1.10867
$119$	−103.653			−0.871034
$120$			10.9545i			0.0912871i
$121$	29.9882			0.247837
$122$		−	162.615i		−	1.33291i
$123$	−56.8807			−0.462444
$124$	−85.1561			−0.686743
$125$		−	11.1803i		−	0.0894427i
$126$			28.1934i			0.223757i
$127$	−206.225			−1.62382			−0.811911	−	0.583781i	$-0.801573\pi$
$127$	−206.225			−1.62382			−0.811911	+	0.583781i	$0.801573\pi$
$128$	11.3137			0.0883883
$129$			97.5281i			0.756031i
$130$			32.6477i			0.251136i
$131$	26.2454			0.200347			0.100173	−	0.994970i	$-0.468060\pi$
$131$	26.2454			0.200347			0.100173	+	0.994970i	$0.468060\pi$
$132$		−	33.0476i		−	0.250360i
$133$	4.41319			0.0331819
$134$		−	168.913i		−	1.26054i
$135$			11.6190i			0.0860663i
$136$			44.1180i			0.324397i
$137$		−	68.4377i		−	0.499545i	−0.968305	−	0.249773i	$-0.919644\pi$
$137$		−	68.4377i		−	0.499545i	0.968305	−	0.249773i	$-0.0803559\pi$
$138$	40.5311	+	39.1309i	0.293704	+	0.283557i
$139$	256.704			1.84679			0.923396	−	0.383849i	$-0.125402\pi$
$139$	256.704			1.84679			0.923396	+	0.383849i	$0.125402\pi$
$140$	−29.7184			−0.212274
$141$	46.2615			0.328096
$142$	147.553			1.03911
$143$		−	98.4922i		−	0.688757i
$144$	12.0000			0.0833333
$145$			108.080i			0.745378i
$146$	6.31755			0.0432709
$147$	8.38446			0.0570371
$148$			75.9150i			0.512939i
$149$		−	14.9048i		−	0.100032i	−0.998748	−	0.0500161i	$-0.984073\pi$
$149$		−	14.9048i		−	0.100032i	0.998748	−	0.0500161i	$-0.0159273\pi$
$150$	−12.2474			−0.0816497
$151$	−179.587			−1.18931			−0.594657	−	0.803979i	$-0.702712\pi$
$151$	−179.587			−1.18931			−0.594657	+	0.803979i	$0.702712\pi$
$152$		−	1.87840i		−	0.0123579i
$153$			46.7942i			0.305845i
$154$	89.6550			0.582175
$155$		−	95.2075i		−	0.614242i
$156$	35.7638			0.229255
$157$		−	164.415i		−	1.04723i	−0.851956	−	0.523614i	$-0.824583\pi$
$157$		−	164.415i		−	1.04723i	0.851956	−	0.523614i	$-0.175417\pi$
$158$		−	41.6675i		−	0.263719i
$159$			23.9646i			0.150721i
$160$			12.6491i			0.0790569i
$161$	−106.158	+	109.957i	−0.659370	+	0.682965i
$162$	12.7279			0.0785674
$163$	−283.644			−1.74015			−0.870074	−	0.492921i	$-0.835929\pi$
$163$	−283.644			−1.74015			−0.870074	+	0.492921i	$0.835929\pi$
$164$	−65.6801			−0.400489
$165$	36.9483			0.223929
$166$		−	192.573i		−	1.16008i
$167$	213.320			1.27737			0.638684	−	0.769469i	$-0.279479\pi$
$167$	213.320			1.27737			0.638684	+	0.769469i	$0.279479\pi$
$168$			32.5549i			0.193779i
$169$	−62.4126			−0.369305
$170$	−49.3255			−0.290150
$171$		−	1.99234i		−	0.0116511i
$172$			112.616i			0.654742i
$173$	158.006			0.913327			0.456664	−	0.889639i	$-0.349044\pi$
$173$	158.006			0.913327			0.456664	+	0.889639i	$0.349044\pi$
$174$	118.395			0.680434
$175$		−	33.2262i		−	0.189864i
$176$		−	38.1600i		−	0.216818i
$177$	−160.225			−0.905224
$178$		−	27.1173i		−	0.152344i
$179$	88.4216			0.493975			0.246988	−	0.969019i	$-0.420559\pi$
$179$	88.4216			0.493975			0.246988	+	0.969019i	$0.420559\pi$
$180$			13.4164i			0.0745356i
$181$		−	142.050i		−	0.784805i	−0.919794	−	0.392403i	$-0.871644\pi$
$181$		−	142.050i		−	0.784805i	0.919794	−	0.392403i	$-0.128356\pi$
$182$			97.0239i			0.533098i
$183$		−	199.161i		−	1.08831i
$184$	46.8013	+	45.1845i	0.254355	+	0.245568i
$185$	−84.8755			−0.458787
$186$	−104.295			−0.560723
$187$	148.806			0.795753
$188$	53.4181			0.284139
$189$			34.5297i			0.182697i
$190$	2.10011			0.0110532
$191$			58.6375i			0.307002i	0.988148	+	0.153501i	$0.0490549\pi$
$191$			58.6375i			0.307002i	−0.988148	+	0.153501i	$0.950945\pi$
$192$	13.8564			0.0721688
$193$	−265.804			−1.37722			−0.688612	−	0.725130i	$-0.741780\pi$
$193$	−265.804			−1.37722			−0.688612	+	0.725130i	$0.741780\pi$
$194$			4.46666i			0.0230240i
$195$			39.9851i			0.205052i
$196$	9.68154			0.0493956
$197$	−27.0148			−0.137131			−0.0685655	−	0.997647i	$-0.521842\pi$
$197$	−27.0148			−0.137131			−0.0685655	+	0.997647i	$0.521842\pi$
$198$		−	40.4748i		−	0.204418i
$199$		−	4.41948i		−	0.0222085i	−0.999938	−	0.0111042i	$-0.996465\pi$
$199$		−	4.41948i		−	0.0222085i	0.999938	−	0.0111042i	$-0.00353466\pi$
$200$	−14.1421			−0.0707107
$201$		−	206.875i		−	1.02923i
$202$	−190.486			−0.943002
$203$			321.196i			1.58225i
$204$			54.0333i			0.264869i
$205$		−	73.4326i		−	0.358208i
$206$		−	56.1248i		−	0.272451i
$207$	49.6403	+	47.9253i	0.239808	+	0.231523i
$208$	41.2965			0.198541
$209$	−6.33564			−0.0303141
$210$	−36.3975			−0.173321
$211$	−65.1177			−0.308615			−0.154307	−	0.988023i	$-0.549315\pi$
$211$	−65.1177			−0.308615			−0.154307	+	0.988023i	$0.549315\pi$
$212$			27.6719i			0.130528i
$213$	180.715			0.848429
$214$			53.3498i			0.249298i
$215$	−125.908			−0.585619
$216$	14.6969			0.0680414
$217$		−	282.942i		−	1.30388i
$218$		−	255.809i		−	1.17344i
$219$	7.73739			0.0353305
$220$	42.6642			0.193928
$221$			161.036i			0.728672i
$222$			92.9765i			0.418813i
$223$	−269.495			−1.20850			−0.604248	−	0.796796i	$-0.706526\pi$
$223$	−269.495			−1.20850			−0.604248	+	0.796796i	$0.706526\pi$
$224$			37.5912i			0.167818i
$225$	−15.0000			−0.0666667
$226$			238.618i			1.05583i
$227$		−	312.590i		−	1.37705i	−0.725214	−	0.688523i	$-0.758259\pi$
$227$		−	312.590i		−	1.37705i	0.725214	−	0.688523i	$-0.241741\pi$
$228$		−	2.30055i		−	0.0100902i
$229$		−	213.240i		−	0.931179i	−0.885001	−	0.465589i	$-0.845842\pi$
$229$		−	213.240i		−	0.931179i	0.885001	−	0.465589i	$-0.154158\pi$
$230$	−50.5178	+	52.3255i	−0.219642	+	0.227502i
$231$	109.805			0.475344
$232$	136.711			0.589273
$233$	−241.296			−1.03560			−0.517802	−	0.855500i	$-0.673250\pi$
$233$	−241.296			−1.03560			−0.517802	+	0.855500i	$0.673250\pi$
$234$	43.8015			0.187186
$235$			59.7233i			0.254142i
$236$	−185.011			−0.783947
$237$		−	51.0321i		−	0.215325i
$238$	−146.588			−0.615914
$239$	49.0519			0.205238			0.102619	−	0.994721i	$-0.467278\pi$
$239$	49.0519			0.205238			0.102619	+	0.994721i	$0.467278\pi$
$240$			15.4919i			0.0645497i
$241$			125.719i			0.521654i	0.965386	+	0.260827i	$0.0839953\pi$
$241$			125.719i			0.521654i	−0.965386	+	0.260827i	$0.916005\pi$
$242$	42.4098			0.175247
$243$	15.5885			0.0641500
$244$		−	229.972i		−	0.942508i
$245$			10.8243i			0.0441808i
$246$	−80.4414			−0.326998
$247$		−	6.85638i		−	0.0277586i
$248$	−120.429			−0.485601
$249$		−	235.852i		−	0.947198i
$250$		−	15.8114i		−	0.0632456i
$251$		−	167.693i		−	0.668099i	−0.942556	−	0.334049i	$-0.891585\pi$
$251$		−	167.693i		−	0.668099i	0.942556	−	0.334049i	$-0.108415\pi$
$252$			39.8714i			0.158220i
$253$	152.403	−	157.856i	0.602382	−	0.623938i
$254$	−291.647			−1.14822
$255$	−60.4111			−0.236906
$256$	16.0000			0.0625000
$257$	447.839			1.74256			0.871282	−	0.490783i	$-0.163289\pi$
$257$	447.839			1.74256			0.871282	+	0.490783i	$0.163289\pi$
$258$			137.926i			0.534595i
$259$	−252.237			−0.973887
$260$			46.1709i			0.177580i
$261$	145.004			0.555572
$262$	37.1166			0.141666
$263$			206.991i			0.787039i	0.919316	+	0.393519i	$0.128743\pi$
$263$			206.991i			0.787039i	−0.919316	+	0.393519i	$0.871257\pi$
$264$		−	46.7363i		−	0.177031i
$265$	−30.9382			−0.116748
$266$	6.24119			0.0234631
$267$		−	33.2118i		−	0.124389i
$268$		−	238.879i		−	0.891339i
$269$	60.4178			0.224602			0.112301	−	0.993674i	$-0.464178\pi$
$269$	60.4178			0.224602			0.112301	+	0.993674i	$0.464178\pi$
$270$			16.4317i			0.0608581i
$271$	−374.057			−1.38028			−0.690141	−	0.723675i	$-0.742452\pi$
$271$	−374.057			−1.38028			−0.690141	+	0.723675i	$0.742452\pi$
$272$			62.3923i			0.229384i
$273$			118.830i			0.435273i
$274$		−	96.7855i		−	0.353232i
$275$			47.7000i			0.173455i
$276$	57.3197	+	55.3394i	0.207680	+	0.200505i
$277$	168.436			0.608074			0.304037	−	0.952660i	$-0.401665\pi$
$277$	168.436			0.608074			0.304037	+	0.952660i	$0.401665\pi$
$278$	363.034			1.30588
$279$	−127.734			−0.457829
$280$	−42.0282			−0.150101
$281$			451.713i			1.60752i	0.594953	+	0.803760i	$0.297171\pi$
$281$			451.713i			1.60752i	−0.594953	+	0.803760i	$0.702829\pi$
$282$	65.4236			0.231999
$283$		−	272.170i		−	0.961733i	−0.876794	−	0.480866i	$-0.840322\pi$
$283$		−	272.170i		−	0.961733i	0.876794	−	0.480866i	$-0.159678\pi$
$284$	208.672			0.734761
$285$	2.57210			0.00902491
$286$		−	139.289i		−	0.487024i
$287$		−	218.230i		−	0.760384i
$288$	16.9706			0.0589256
$289$	45.6998			0.158131
$290$			152.848i			0.527062i
$291$			5.47052i			0.0187990i
$292$	8.93437			0.0305972
$293$			97.2149i			0.331792i	0.986143	+	0.165896i	$0.0530515\pi$
$293$			97.2149i			0.331792i	−0.986143	+	0.165896i	$0.946948\pi$
$294$	11.8574			0.0403313
$295$		−	206.849i		−	0.701183i
$296$			107.360i			0.362703i
$297$		−	49.5713i		−	0.166907i
$298$		−	21.0786i		−	0.0707334i
$299$	170.831	+	164.929i	0.571341	+	0.551602i
$300$	−17.3205			−0.0577350
$301$	−374.179			−1.24312
$302$	−253.974			−0.840973
$303$	−233.297			−0.769958
$304$		−	2.65645i		−	0.00873833i
$305$	257.116			0.843004
$306$			66.1771i			0.216265i
$307$	64.9944			0.211708			0.105854	−	0.994382i	$-0.466242\pi$
$307$	64.9944			0.211708			0.105854	+	0.994382i	$0.466242\pi$
$308$	126.791			0.411660
$309$		−	68.7386i		−	0.222455i
$310$		−	134.644i		−	0.434334i
$311$	−277.605			−0.892621			−0.446310	−	0.894878i	$-0.647262\pi$
$311$	−277.605			−0.892621			−0.446310	+	0.894878i	$0.647262\pi$
$312$	50.5777			0.162108
$313$			424.841i			1.35732i	0.734453	+	0.678660i	$0.237439\pi$
$313$			424.841i			1.35732i	−0.734453	+	0.678660i	$0.762561\pi$
$314$		−	232.518i		−	0.740502i
$315$	−44.5776			−0.141516
$316$		−	58.9268i		−	0.186477i
$317$	222.185			0.700900			0.350450	−	0.936581i	$-0.386029\pi$
$317$	222.185			0.700900			0.350450	+	0.936581i	$0.386029\pi$
$318$			33.8911i			0.106576i
$319$		−	461.114i		−	1.44550i
$320$			17.8885i			0.0559017i
$321$			65.3399i			0.203551i
$322$	−150.131	+	155.503i	−0.466245	+	0.482929i
$323$	10.3589			0.0320709
$324$	18.0000			0.0555556
$325$	−51.6206			−0.158833
$326$	−401.134			−1.23047
$327$		−	313.301i		−	0.958108i
$328$	−92.8857			−0.283188
$329$			177.488i			0.539478i
$330$	52.2528			0.158342
$331$	−228.815			−0.691284			−0.345642	−	0.938366i	$-0.612339\pi$
$331$	−228.815			−0.691284			−0.345642	+	0.938366i	$0.612339\pi$
$332$		−	272.339i		−	0.820298i
$333$			113.872i			0.341959i
$334$	301.680			0.903235
$335$	267.075			0.797238
$336$			46.0396i			0.137023i
$337$		−	309.895i		−	0.919571i	−0.888030	−	0.459786i	$-0.847926\pi$
$337$		−	309.895i		−	0.919571i	0.888030	−	0.459786i	$-0.152074\pi$
$338$	−88.2647			−0.261138
$339$			292.247i			0.862084i
$340$	−69.7567			−0.205167
$341$			406.195i			1.19119i
$342$		−	2.81759i		−	0.00823858i
$343$			357.785i			1.04310i
$344$			159.263i			0.462973i
$345$	−61.8714	+	64.0854i	−0.179337	+	0.185755i
$346$	223.454			0.645820
$347$	−347.850			−1.00245			−0.501224	−	0.865318i	$-0.667117\pi$
$347$	−347.850			−1.00245			−0.501224	+	0.865318i	$0.667117\pi$
$348$	167.436			0.481139
$349$	50.4420			0.144533			0.0722665	−	0.997385i	$-0.476977\pi$
$349$	50.4420			0.144533			0.0722665	+	0.997385i	$0.476977\pi$
$350$		−	46.9890i		−	0.134254i
$351$	53.6457			0.152837
$352$		−	53.9664i		−	0.153314i
$353$	521.736			1.47801			0.739003	−	0.673702i	$-0.235297\pi$
$353$	521.736			1.47801			0.739003	+	0.673702i	$0.235297\pi$
$354$	−226.592			−0.640090
$355$			233.302i			0.657190i
$356$		−	38.3497i		−	0.107724i
$357$	−179.532			−0.502892
$358$	125.047			0.349293
$359$			0.869077i			0.00242083i	0.999999	+	0.00121041i	$0.000385287\pi$
$359$			0.869077i			0.00242083i	−0.999999	+	0.00121041i	$0.999615\pi$
$360$			18.9737i			0.0527046i
$361$	360.559			0.998778
$362$		−	200.889i		−	0.554941i
$363$	51.9411			0.143089
$364$			137.213i			0.376958i
$365$			9.98893i			0.0273669i
$366$		−	281.657i		−	0.769554i
$367$		−	589.026i		−	1.60498i	−0.596668	−	0.802488i	$-0.703509\pi$
$367$		−	589.026i		−	1.60498i	0.596668	−	0.802488i	$-0.296491\pi$
$368$	66.1871	+	63.9005i	0.179856	+	0.173643i
$369$	−98.5202			−0.266992
$370$	−120.032			−0.324411
$371$	−91.9433			−0.247826
$372$	−147.495			−0.396491
$373$			335.776i			0.900205i	0.892977	+	0.450102i	$0.148613\pi$
$373$			335.776i			0.900205i	−0.892977	+	0.450102i	$0.851387\pi$
$374$	210.443			0.562682
$375$		−	19.3649i		−	0.0516398i
$376$	75.5447			0.200917
$377$	499.014			1.32364
$378$			48.8324i			0.129186i
$379$		−	530.492i		−	1.39971i	−0.714283	−	0.699857i	$-0.753247\pi$
$379$		−	530.492i		−	1.39971i	0.714283	−	0.699857i	$-0.246753\pi$
$380$	2.97000			0.00781580
$381$	−357.193			−0.937514
$382$			82.9259i			0.217083i
$383$		−	617.105i		−	1.61124i	−0.592433	−	0.805620i	$-0.701833\pi$
$383$		−	617.105i		−	1.61124i	0.592433	−	0.805620i	$-0.298167\pi$
$384$	19.5959			0.0510310
$385$			141.757i			0.368200i
$386$	−375.904			−0.973844
$387$			168.924i			0.436495i
$388$			6.31681i			0.0162804i
$389$			543.176i			1.39634i	0.715933	+	0.698169i	$0.246002\pi$
$389$			543.176i			1.39634i	−0.715933	+	0.698169i	$0.753998\pi$
$390$			56.5475i			0.144994i
$391$	−249.181	+	258.098i	−0.637292	+	0.660097i
$392$	13.6918			0.0349280
$393$	45.4584			0.115670
$394$	−38.2047			−0.0969662
$395$	65.8821			0.166790
$396$		−	57.2401i		−	0.144546i
$397$	−6.47467			−0.0163090			−0.00815450	−	0.999967i	$-0.502596\pi$
$397$	−6.47467			−0.0163090			−0.00815450	+	0.999967i	$0.502596\pi$
$398$		−	6.25010i		−	0.0157038i
$399$	7.64387			0.0191576
$400$	−20.0000			−0.0500000
$401$			132.267i			0.329843i	0.986307	+	0.164921i	$0.0527371\pi$
$401$			132.267i			0.329843i	−0.986307	+	0.164921i	$0.947263\pi$
$402$		−	292.566i		−	0.727775i
$403$	−439.581			−1.09077
$404$	−269.388			−0.666803
$405$			20.1246i			0.0496904i
$406$			454.240i			1.11882i
$407$	362.115			0.889717
$408$			76.4147i			0.187291i
$409$	−290.718			−0.710802			−0.355401	−	0.934714i	$-0.615656\pi$
$409$	−290.718			−0.710802			−0.355401	+	0.934714i	$0.615656\pi$
$410$		−	103.849i		−	0.253291i
$411$		−	118.538i		−	0.288413i
$412$		−	79.3725i		−	0.192652i
$413$		−	614.723i		−	1.48843i
$414$	70.2020	+	67.7767i	0.169570	+	0.163712i
$415$	304.484			0.733696
$416$	58.4020			0.140390
$417$	444.625			1.06625
$418$	−8.95995			−0.0214353
$419$			715.612i			1.70790i	0.520352	+	0.853952i	$0.325801\pi$
$419$			715.612i			1.70790i	−0.520352	+	0.853952i	$0.674199\pi$
$420$	−51.4738			−0.122557
$421$			236.715i			0.562268i	0.959669	+	0.281134i	$0.0907105\pi$
$421$			236.715i			0.562268i	−0.959669	+	0.281134i	$0.909289\pi$
$422$	−92.0903			−0.218223
$423$	80.1272			0.189426
$424$			39.1340i			0.0922972i
$425$		−	77.9904i		−	0.183507i
$426$	255.570			0.599930
$427$	764.109			1.78948
$428$			75.4481i			0.176281i
$429$		−	170.593i		−	0.397654i
$430$	−178.061			−0.414096
$431$			719.166i			1.66860i	0.551312	+	0.834299i	$0.314127\pi$
$431$			719.166i			1.66860i	−0.551312	+	0.834299i	$0.685873\pi$
$432$	20.7846			0.0481125
$433$			344.163i			0.794834i	0.917638	+	0.397417i	$0.130093\pi$
$433$			344.163i			0.794834i	−0.917638	+	0.397417i	$0.869907\pi$
$434$		−	400.140i		−	0.921981i
$435$			187.200i			0.430344i
$436$		−	361.769i		−	0.829746i
$437$	10.6093	−	10.9889i	0.0242775	−	0.0251463i
$438$	10.9423			0.0249825
$439$	481.890			1.09770			0.548850	−	0.835921i	$-0.315066\pi$
$439$	481.890			1.09770			0.548850	+	0.835921i	$0.315066\pi$
$440$	60.3363			0.137128
$441$	14.5223			0.0329304
$442$			227.740i			0.515249i
$443$	499.957			1.12857			0.564286	−	0.825580i	$-0.309152\pi$
$443$	499.957			1.12857			0.564286	+	0.825580i	$0.309152\pi$
$444$			131.489i			0.296145i
$445$	42.8762			0.0963510
$446$	−381.123			−0.854536
$447$		−	25.8159i		−	0.0577536i
$448$			53.1619i			0.118665i
$449$	−312.769			−0.696590			−0.348295	−	0.937385i	$-0.613239\pi$
$449$	−312.769			−0.696590			−0.348295	+	0.937385i	$0.613239\pi$
$450$	−21.2132			−0.0471405
$451$			313.295i			0.694666i
$452$			337.457i			0.746587i
$453$	−311.053			−0.686651
$454$		−	442.068i		−	0.973719i
$455$	−153.408			−0.337161
$456$		−	3.25348i		−	0.00713482i
$457$			91.3525i			0.199896i	0.994993	+	0.0999481i	$0.0318677\pi$
$457$			91.3525i			0.199896i	−0.994993	+	0.0999481i	$0.968132\pi$
$458$		−	301.567i		−	0.658443i
$459$			81.0500i			0.176580i
$460$	−71.4429	+	73.9994i	−0.155311	+	0.160868i
$461$	−32.8781			−0.0713191			−0.0356596	−	0.999364i	$-0.511353\pi$
$461$	−32.8781			−0.0713191			−0.0356596	+	0.999364i	$0.511353\pi$
$462$	155.287			0.336119
$463$	439.384			0.948995			0.474497	−	0.880257i	$-0.342630\pi$
$463$	439.384			0.948995			0.474497	+	0.880257i	$0.342630\pi$
$464$	193.339			0.416679
$465$		−	164.904i		−	0.354633i
$466$	−341.244			−0.732283
$467$			175.531i			0.375870i	0.982181	+	0.187935i	$0.0601794\pi$
$467$			175.531i			0.375870i	−0.982181	+	0.187935i	$0.939821\pi$
$468$	61.9447			0.132361
$469$	793.704			1.69233
$470$			84.4615i			0.179705i
$471$		−	284.775i		−	0.604617i
$472$	−261.646			−0.554334
$473$	537.177			1.13568
$474$		−	72.1703i		−	0.152258i
$475$			3.32056i			0.00699066i
$476$	−207.306			−0.435517
$477$			41.5079i			0.0870187i
$478$	69.3698			0.145125
$479$		−	50.8178i		−	0.106091i	−0.998592	−	0.0530457i	$-0.983107\pi$
$479$		−	50.8178i		−	0.106091i	0.998592	−	0.0530457i	$-0.0168929\pi$
$480$			21.9089i			0.0456435i
$481$			391.878i			0.814714i
$482$			177.793i			0.368865i
$483$	−183.872	+	190.452i	−0.380687	+	0.394310i
$484$	59.9765			0.123918
$485$	−7.06241			−0.0145617
$486$	22.0454			0.0453609
$487$	14.3618			0.0294903			0.0147451	−	0.999891i	$-0.495306\pi$
$487$	14.3618			0.0294903			0.0147451	+	0.999891i	$0.495306\pi$
$488$		−	325.229i		−	0.666454i
$489$	−491.286			−1.00468
$490$			15.3079i			0.0312405i
$491$	−382.245			−0.778503			−0.389252	−	0.921131i	$-0.627266\pi$
$491$	−382.245			−0.778503			−0.389252	+	0.921131i	$0.627266\pi$
$492$	−113.761			−0.231222
$493$			753.929i			1.52927i
$494$		−	9.69639i		−	0.0196283i
$495$	63.9963			0.129286
$496$	−170.312			−0.343372
$497$			693.338i			1.39505i
$498$		−	333.546i		−	0.669770i
$499$	−465.646			−0.933159			−0.466579	−	0.884479i	$-0.654514\pi$
$499$	−465.646			−0.933159			−0.466579	+	0.884479i	$0.654514\pi$
$500$		−	22.3607i		−	0.0447214i
$501$	369.482			0.737488
$502$		−	237.153i		−	0.472417i
$503$		−	445.223i		−	0.885134i	−0.896735	−	0.442567i	$-0.854068\pi$
$503$		−	445.223i		−	0.885134i	0.896735	−	0.442567i	$-0.145932\pi$
$504$			56.3867i			0.111878i
$505$		−	301.185i		−	0.596407i
$506$	215.530	−	223.243i	0.425949	−	0.441191i
$507$	−108.102			−0.213218
$508$	−412.451			−0.811911
$509$	211.434			0.415392			0.207696	−	0.978193i	$-0.433404\pi$
$509$	211.434			0.415392			0.207696	+	0.978193i	$0.433404\pi$
$510$	−85.4342			−0.167518
$511$			29.6855i			0.0580930i
$512$	22.6274			0.0441942
$513$		−	3.45083i		−	0.00672677i
$514$	633.340			1.23218
$515$	88.7412			0.172313
$516$			195.056i			0.378016i
$517$		−	254.805i		−	0.492853i
$518$	−356.716			−0.688642
$519$	273.674			0.527310
$520$			65.2955i			0.125568i
$521$			685.288i			1.31533i	0.753310	+	0.657666i	$0.228456\pi$
$521$			685.288i			1.31533i	−0.753310	+	0.657666i	$0.771544\pi$
$522$	205.067			0.392849
$523$		−	535.072i		−	1.02308i	−0.859259	−	0.511541i	$-0.829075\pi$
$523$		−	535.072i		−	1.02308i	0.859259	−	0.511541i	$-0.170925\pi$
$524$	52.4908			0.100173
$525$		−	57.5495i		−	0.109618i
$526$			292.730i			0.556520i
$527$		−	664.136i		−	1.26022i
$528$		−	66.0951i		−	0.125180i
$529$	18.5913	+	528.673i	0.0351443	+	0.999382i
$530$	−43.7532			−0.0825532
$531$	−277.517			−0.522631
$532$	8.82638			0.0165909
$533$	−339.045			−0.636107
$534$		−	46.9685i		−	0.0879561i
$535$	−84.3535			−0.157670
$536$		−	337.826i		−	0.630272i
$537$	153.151			0.285197
$538$	85.4437			0.158817
$539$		−	46.1810i		−	0.0856790i
$540$			23.2379i			0.0430331i
$541$	240.218			0.444026			0.222013	−	0.975044i	$-0.428737\pi$
$541$	240.218			0.444026			0.222013	+	0.975044i	$0.428737\pi$
$542$	−528.996			−0.976007
$543$		−	246.037i		−	0.453108i
$544$			88.2361i			0.162199i
$545$	404.470			0.742147
$546$			168.050i			0.307785i
$547$	560.977			1.02555			0.512776	−	0.858522i	$-0.328617\pi$
$547$	560.977			1.02555			0.512776	+	0.858522i	$0.328617\pi$
$548$		−	136.875i		−	0.249773i
$549$		−	344.958i		−	0.628338i
$550$			67.4580i			0.122651i
$551$		−	32.0997i		−	0.0582572i
$552$	81.0623	+	78.2618i	0.146852	+	0.141779i
$553$	195.791			0.354053
$554$	238.205			0.429973
$555$	−147.009			−0.264881
$556$	513.408			0.923396
$557$		−	243.211i		−	0.436645i	−0.975877	−	0.218322i	$-0.929942\pi$
$557$		−	243.211i		−	0.436645i	0.975877	−	0.218322i	$-0.0700584\pi$
$558$	−180.643			−0.323734
$559$			581.329i			1.03994i
$560$	−59.4368			−0.106137
$561$	257.739			0.459428
$562$			638.819i			1.13669i
$563$		−	403.682i		−	0.717020i	−0.933526	−	0.358510i	$-0.883285\pi$
$563$		−	403.682i		−	0.717020i	0.933526	−	0.358510i	$-0.116715\pi$
$564$	92.5229			0.164048
$565$	−377.289			−0.667768
$566$		−	384.907i		−	0.680048i
$567$			59.8072i			0.105480i
$568$	295.107			0.519554
$569$			404.655i			0.711169i	0.934644	+	0.355584i	$0.115718\pi$
$569$			404.655i			0.711169i	−0.934644	+	0.355584i	$0.884282\pi$
$570$	3.63750			0.00638157
$571$			747.309i			1.30877i	0.756161	+	0.654386i	$0.227073\pi$
$571$			747.309i			1.30877i	−0.756161	+	0.654386i	$0.772927\pi$
$572$		−	196.984i		−	0.344378i
$573$			101.563i			0.177248i
$574$		−	308.624i		−	0.537673i
$575$	−82.7339	−	79.8756i	−0.143885	−	0.138914i
$576$	24.0000			0.0416667
$577$	952.285			1.65041			0.825203	−	0.564836i	$-0.191060\pi$
$577$	952.285			1.65041			0.825203	+	0.564836i	$0.191060\pi$
$578$	64.6293			0.111815
$579$	−460.386			−0.795141
$580$			216.160i			0.372689i
$581$	904.879			1.55745
$582$			7.73648i			0.0132929i
$583$	131.995			0.226407
$584$	12.6351			0.0216355
$585$			69.2563i			0.118387i
$586$			137.483i			0.234612i
$587$	295.831			0.503972			0.251986	−	0.967731i	$-0.418916\pi$
$587$	295.831			0.503972			0.251986	+	0.967731i	$0.418916\pi$
$588$	16.7689			0.0285186
$589$			28.2767i			0.0480079i
$590$		−	292.529i		−	0.495812i
$591$	−46.7910			−0.0791726
$592$			151.830i			0.256469i
$593$	380.724			0.642031			0.321015	−	0.947074i	$-0.395976\pi$
$593$	380.724			0.642031			0.321015	+	0.947074i	$0.395976\pi$
$594$		−	70.1045i		−	0.118021i
$595$		−	231.775i		−	0.389538i
$596$		−	29.8096i		−	0.0500161i
$597$		−	7.65477i		−	0.0128221i
$598$	241.591	+	233.245i	0.403999	+	0.390042i
$599$	−432.741			−0.722440			−0.361220	−	0.932481i	$-0.617640\pi$
$599$	−432.741			−0.722440			−0.361220	+	0.932481i	$0.617640\pi$
$600$	−24.4949			−0.0408248
$601$	−103.199			−0.171713			−0.0858563	−	0.996308i	$-0.527363\pi$
$601$	−103.199			−0.171713			−0.0858563	+	0.996308i	$0.527363\pi$
$602$	−529.169			−0.879019
$603$		−	358.318i		−	0.594226i
$604$	−359.173			−0.594657
$605$			67.0557i			0.110836i
$606$	−329.932			−0.544442
$607$	−220.607			−0.363439			−0.181719	−	0.983350i	$-0.558166\pi$
$607$	−220.607			−0.363439			−0.181719	+	0.983350i	$0.558166\pi$
$608$		−	3.75679i		−	0.00617893i
$609$			556.328i			0.913510i
$610$	363.617			0.596094
$611$	275.748			0.451305
$612$			93.5885i			0.152922i
$613$			932.888i			1.52184i	0.648846	+	0.760920i	$0.275252\pi$
$613$			932.888i			1.52184i	−0.648846	+	0.760920i	$0.724748\pi$
$614$	91.9159			0.149700
$615$		−	127.189i		−	0.206811i
$616$	179.310			0.291088
$617$			1218.37i			1.97467i	0.158643	+	0.987336i	$0.449288\pi$
$617$			1218.37i			1.97467i	−0.158643	+	0.987336i	$0.550712\pi$
$618$		−	97.2111i		−	0.157299i
$619$			338.996i			0.547652i	0.961779	+	0.273826i	$0.0882892\pi$
$619$			338.996i			0.547652i	−0.961779	+	0.273826i	$0.911711\pi$
$620$		−	190.415i		−	0.307121i
$621$	85.9796	+	83.0091i	0.138453	+	0.133670i
$622$	−392.593			−0.631178
$623$	127.421			0.204529
$624$	71.5276			0.114628
$625$	25.0000			0.0400000
$626$			600.816i			0.959770i
$627$	−10.9737			−0.0175018
$628$		−	328.830i		−	0.523614i
$629$	−592.064			−0.941278
$630$	−63.0423			−0.100067
$631$		−	55.2614i		−	0.0875775i	−0.999041	−	0.0437887i	$-0.986057\pi$
$631$		−	55.2614i		−	0.0875775i	0.999041	−	0.0437887i	$-0.0139428\pi$
$632$		−	83.3351i		−	0.131859i
$633$	−112.787			−0.178179
$634$	314.218			0.495611
$635$		−	461.134i		−	0.726195i
$636$			47.9292i			0.0753604i
$637$	49.9767			0.0784563
$638$		−	652.113i		−	1.02212i
$639$	313.008			0.489841
$640$			25.2982i			0.0395285i
$641$		−	1079.52i		−	1.68411i	−0.539390	−	0.842056i	$-0.681345\pi$
$641$		−	1079.52i		−	1.68411i	0.539390	−	0.842056i	$-0.318655\pi$
$642$			92.4046i			0.143932i
$643$		−	256.649i		−	0.399143i	−0.979883	−	0.199572i	$-0.936045\pi$
$643$		−	256.649i		−	0.399143i	0.979883	−	0.199572i	$-0.0639550\pi$
$644$	−212.317	+	219.915i	−0.329685	+	0.341482i
$645$	−218.079			−0.338108
$646$	14.6497			0.0226775
$647$	−961.900			−1.48671			−0.743354	−	0.668899i	$-0.766766\pi$
$647$	−961.900			−1.48671			−0.743354	+	0.668899i	$0.766766\pi$
$648$	25.4558			0.0392837
$649$			882.506i			1.35979i
$650$	−73.0025			−0.112312
$651$		−	490.069i		−	0.752794i
$652$	−567.288			−0.870074
$653$	860.220			1.31734			0.658668	−	0.752434i	$-0.271120\pi$
$653$	860.220			1.31734			0.658668	+	0.752434i	$0.271120\pi$
$654$		−	443.075i		−	0.677484i
$655$			58.6865i			0.0895977i
$656$	−131.360			−0.200244
$657$	13.4016			0.0203981
$658$			251.006i			0.381468i
$659$		−	45.8016i		−	0.0695016i	−0.999396	−	0.0347508i	$-0.988936\pi$
$659$		−	45.8016i		−	0.0695016i	0.999396	−	0.0347508i	$-0.0110638\pi$
$660$	73.8966			0.111965
$661$		−	190.025i		−	0.287481i	−0.989615	−	0.143741i	$-0.954087\pi$
$661$		−	190.025i		−	0.287481i	0.989615	−	0.143741i	$-0.0459131\pi$
$662$	−323.593			−0.488812
$663$			278.923i			0.420699i
$664$		−	385.145i		−	0.580038i
$665$			9.86819i			0.0148394i
$666$			161.040i			0.241802i
$667$	799.784	+	772.153i	1.19908	+	1.15765i
$668$	426.641			0.638684
$669$	−466.778			−0.697726
$670$	377.701			0.563732
$671$	−1096.97			−1.63482
$672$			65.1098i			0.0968896i
$673$	−223.701			−0.332394			−0.166197	−	0.986093i	$-0.553149\pi$
$673$	−223.701			−0.332394			−0.166197	+	0.986093i	$0.553149\pi$
$674$		−	438.258i		−	0.650235i
$675$	−25.9808			−0.0384900
$676$	−124.825			−0.184653
$677$			879.263i			1.29876i	0.760463	+	0.649382i	$0.224972\pi$
$677$			879.263i			1.29876i	−0.760463	+	0.649382i	$0.775028\pi$
$678$			413.299i			0.609586i
$679$	−20.9884			−0.0309107
$680$	−98.6509			−0.145075
$681$		−	541.421i		−	0.795038i
$682$			574.447i			0.842297i
$683$	323.930			0.474275			0.237137	−	0.971476i	$-0.423791\pi$
$683$	323.930			0.474275			0.237137	+	0.971476i	$0.423791\pi$
$684$		−	3.98468i		−	0.00582555i
$685$	153.031			0.223403
$686$			505.984i			0.737586i
$687$		−	369.342i		−	0.537616i
$688$			225.231i			0.327371i
$689$			142.844i			0.207321i
$690$	−87.4993	+	90.6304i	−0.126811	+	0.131348i
$691$	−574.754			−0.831772			−0.415886	−	0.909417i	$-0.636528\pi$
$691$	−574.754			−0.831772			−0.415886	+	0.909417i	$0.636528\pi$
$692$	316.011			0.456664
$693$	190.187			0.274440
$694$	−491.934			−0.708838
$695$			574.008i			0.825911i
$696$	236.791			0.340217
$697$		−	512.242i		−	0.734924i
$698$	71.3357			0.102200
$699$	−417.937			−0.597907
$700$		−	66.4524i		−	0.0949320i
$701$		−	206.109i		−	0.294021i	−0.989135	−	0.147011i	$-0.953035\pi$
$701$		−	206.109i		−	0.294021i	0.989135	−	0.147011i	$-0.0469651\pi$
$702$	75.8665			0.108072
$703$	25.2081			0.0358578
$704$		−	76.3201i		−	0.108409i
$705$			103.444i			0.146729i
$706$	737.847			1.04511
$707$		−	895.076i		−	1.26602i
$708$	−320.449			−0.452612
$709$			759.295i			1.07094i	0.844555	+	0.535469i	$0.179865\pi$
$709$			759.295i			1.07094i	−0.844555	+	0.535469i	$0.820135\pi$
$710$			329.939i			0.464704i
$711$		−	88.3902i		−	0.124318i
$712$		−	54.2346i		−	0.0761722i
$713$	−704.530	−	680.190i	−0.988120	−	0.953983i
$714$	−253.897			−0.355598
$715$	220.235			0.308021
$716$	176.843			0.246988
$717$	84.9604			0.118494
$718$			1.22906i			0.00171178i
$719$	−1039.63			−1.44594			−0.722970	−	0.690880i	$-0.757223\pi$
$719$	−1039.63			−1.44594			−0.722970	+	0.690880i	$0.757223\pi$
$720$			26.8328i			0.0372678i
$721$	263.725			0.365776
$722$	509.907			0.706243
$723$			217.751i			0.301177i
$724$		−	284.100i		−	0.392403i
$725$	−241.674			−0.333343
$726$	73.4559			0.101179
$727$		−	911.165i		−	1.25332i	−0.779292	−	0.626661i	$-0.784421\pi$
$727$		−	911.165i		−	1.25332i	0.779292	−	0.626661i	$-0.215579\pi$
$728$			194.048i			0.266549i
$729$	27.0000			0.0370370
$730$			14.1265i			0.0193513i
$731$	−878.295			−1.20150
$732$		−	398.323i		−	0.544157i
$733$			487.839i			0.665538i	0.943008	+	0.332769i	$0.107983\pi$
$733$			487.839i			0.665538i	−0.943008	+	0.332769i	$0.892017\pi$
$734$		−	833.009i		−	1.13489i
$735$			18.7482i			0.0255078i
$736$	93.6027	+	90.3689i	0.127178	+	0.122784i
$737$	−1139.45			−1.54607
$738$	−139.329			−0.188792
$739$	861.878			1.16628			0.583138	−	0.812373i	$-0.301825\pi$
$739$	861.878			1.16628			0.583138	+	0.812373i	$0.301825\pi$
$740$	−169.751			−0.229393
$741$		−	11.8756i		−	0.0160265i
$742$	−130.028			−0.175239
$743$		−	999.092i		−	1.34467i	−0.740246	−	0.672337i	$-0.765291\pi$
$743$		−	999.092i		−	1.34467i	0.740246	−	0.672337i	$-0.234709\pi$
$744$	−208.589			−0.280362
$745$	33.3281			0.0447358
$746$			474.860i			0.636541i
$747$		−	408.508i		−	0.546865i
$748$	297.612			0.397877
$749$	−250.685			−0.334693
$750$		−	27.3861i		−	0.0365148i
$751$		−	80.7630i		−	0.107541i	−0.998553	−	0.0537703i	$-0.982876\pi$
$751$		−	80.7630i		−	0.107541i	0.998553	−	0.0537703i	$-0.0171239\pi$
$752$	106.836			0.142070
$753$		−	290.452i		−	0.385727i
$754$	705.712			0.935957
$755$		−	401.568i		−	0.531878i
$756$			69.0594i			0.0913484i
$757$		−	75.6838i		−	0.0999786i	−0.998750	−	0.0499893i	$-0.984081\pi$
$757$		−	75.6838i		−	0.0999786i	0.998750	−	0.0499893i	$-0.0159187\pi$
$758$		−	750.229i		−	0.989747i
$759$	263.969	−	273.415i	0.347786	−	0.360231i
$760$	4.20022			0.00552660
$761$	249.328			0.327633			0.163816	−	0.986491i	$-0.447620\pi$
$761$	249.328			0.327633			0.163816	+	0.986491i	$0.447620\pi$
$762$	−505.147			−0.662923
$763$	1202.02			1.57539
$764$			117.275i			0.153501i
$765$	−104.635			−0.136778
$766$		−	872.718i		−	1.13932i
$767$	−955.040			−1.24516
$768$	27.7128			0.0360844
$769$		−	1367.54i		−	1.77834i	−0.457581	−	0.889168i	$-0.651284\pi$
$769$		−	1367.54i		−	1.77834i	0.457581	−	0.889168i	$-0.348716\pi$
$770$			200.475i			0.260357i
$771$	775.679			1.00607
$772$	−531.608			−0.688612
$773$		−	965.836i		−	1.24946i	−0.780839	−	0.624732i	$-0.785208\pi$
$773$		−	965.836i		−	1.24946i	0.780839	−	0.624732i	$-0.214792\pi$
$774$			238.894i			0.308649i
$775$	212.890			0.274697
$776$			8.93332i			0.0115120i
$777$	−436.887			−0.562274
$778$			768.166i			0.987360i
$779$			21.8095i			0.0279968i
$780$			79.9703i			0.102526i
$781$		−	995.367i		−	1.27448i
$782$	−352.395	+	365.006i	−0.450634	+	0.466759i
$783$	251.155			0.320759
$784$	19.3631			0.0246978
$785$	367.643			0.468335
$786$	64.2879			0.0817912
$787$		−	1.86804i		−	0.00237362i	−0.999999	−	0.00118681i	$-0.999622\pi$
$787$		−	1.86804i		−	0.00237362i	0.999999	−	0.00118681i	$-0.000377773\pi$
$788$	−54.0296			−0.0685655
$789$			358.519i			0.454397i
$790$	93.1714			0.117939
$791$	−1121.24			−1.41750
$792$		−	80.9497i		−	0.102209i
$793$		−	1187.13i		−	1.49701i
$794$	−9.15657			−0.0115322
$795$	−53.5865			−0.0674044
$796$		−	8.83897i		−	0.0111042i
$797$		−	20.0019i		−	0.0250965i	−0.999921	−	0.0125483i	$-0.996006\pi$
$797$		−	20.0019i		−	0.0250965i	0.999921	−	0.0125483i	$-0.00399434\pi$
$798$	10.8101			0.0135464
$799$			416.610i			0.521415i
$800$	−28.2843			−0.0353553
$801$		−	57.5245i		−	0.0718158i
$802$			187.054i			0.233234i
$803$		−	42.6170i		−	0.0530722i
$804$		−	413.750i		−	0.514615i
$805$	−245.872	−	237.378i	−0.305431	−	0.294879i
$806$	−621.662			−0.771292
$807$	104.647			0.129674
$808$	−380.973			−0.471501
$809$	1217.58			1.50504			0.752521	−	0.658568i	$-0.228837\pi$
$809$	1217.58			1.50504			0.752521	+	0.658568i	$0.228837\pi$
$810$			28.4605i			0.0351364i
$811$	−357.718			−0.441082			−0.220541	−	0.975378i	$-0.570782\pi$
$811$	−357.718			−0.441082			−0.220541	+	0.975378i	$0.570782\pi$
$812$			642.392i			0.791123i
$813$	−647.885			−0.796907
$814$	512.108			0.629125
$815$		−	634.248i		−	0.778218i
$816$			108.067i			0.132435i
$817$	37.3948			0.0457708
$818$	−411.138			−0.502613
$819$			205.819i			0.251305i
$820$		−	146.865i		−	0.179104i
$821$	266.233			0.324279			0.162140	−	0.986768i	$-0.448161\pi$
$821$	266.233			0.324279			0.162140	+	0.986768i	$0.448161\pi$
$822$		−	167.637i		−	0.203939i
$823$	−861.623			−1.04693			−0.523465	−	0.852047i	$-0.675361\pi$
$823$	−861.623			−1.04693			−0.523465	+	0.852047i	$0.675361\pi$
$824$		−	112.250i		−	0.136225i
$825$			82.6189i			0.100144i
$826$		−	869.350i		−	1.05248i
$827$		−	400.784i		−	0.484624i	−0.970198	−	0.242312i	$-0.922094\pi$
$827$		−	400.784i		−	0.484624i	0.970198	−	0.242312i	$-0.0779058\pi$
$828$	99.2806	+	95.8507i	0.119904	+	0.115762i
$829$	−91.0241			−0.109800			−0.0548999	−	0.998492i	$-0.517484\pi$
$829$	−91.0241			−0.109800			−0.0548999	+	0.998492i	$0.517484\pi$
$830$	430.605			0.518802
$831$	291.740			0.351072
$832$	82.5930			0.0992704
$833$			75.5067i			0.0906443i
$834$	628.794			0.753950
$835$			476.999i			0.571256i
$836$	−12.6713			−0.0151570
$837$	−221.242			−0.264328
$838$			1012.03i			1.20767i
$839$		−	1436.60i		−	1.71228i	−0.516747	−	0.856138i	$-0.672857\pi$
$839$		−	1436.60i		−	1.71228i	0.516747	−	0.856138i	$-0.327143\pi$
$840$	−72.7950			−0.0866607
$841$	1495.25			1.77794
$842$			334.765i			0.397583i
$843$			782.390i			0.928103i
$844$	−130.235			−0.154307
$845$		−	139.559i		−	0.165158i
$846$	113.317			0.133944
$847$			199.279i			0.235276i
$848$			55.3439i			0.0652640i
$849$		−	471.413i		−	0.555257i
$850$		−	110.295i		−	0.129759i
$851$	−606.375	+	628.074i	−0.712544	+	0.738042i
$852$	361.431			0.424214
$853$	−298.556			−0.350007			−0.175004	−	0.984568i	$-0.555994\pi$
$853$	−298.556			−0.350007			−0.175004	+	0.984568i	$0.555994\pi$
$854$	1080.61			1.26536
$855$	4.45501			0.00521053
$856$			106.700i			0.124649i
$857$	−747.993			−0.872804			−0.436402	−	0.899752i	$-0.643748\pi$
$857$	−747.993			−0.872804			−0.436402	+	0.899752i	$0.643748\pi$
$858$		−	241.256i		−	0.281184i
$859$	−1105.06			−1.28645			−0.643225	−	0.765677i	$-0.722404\pi$
$859$	−1105.06			−1.28645			−0.643225	+	0.765677i	$0.722404\pi$
$860$	−251.816			−0.292810
$861$		−	377.986i		−	0.439008i
$862$			1017.05i			1.17988i
$863$	−115.567			−0.133913			−0.0669567	−	0.997756i	$-0.521329\pi$
$863$	−115.567			−0.133913			−0.0669567	+	0.997756i	$0.521329\pi$
$864$	29.3939			0.0340207
$865$			353.311i			0.408452i
$866$			486.720i			0.562032i
$867$	79.1544			0.0912969
$868$		−	565.883i		−	0.651939i
$869$	−281.081			−0.323453
$870$			264.740i			0.304299i
$871$		−	1233.11i		−	1.41574i
$872$		−	511.619i		−	0.586719i
$873$			9.47521i			0.0108536i
$874$	15.0038	−	15.5407i	0.0171668	−	0.0177811i
$875$	74.2961			0.0849098
$876$	15.4748			0.0176653
$877$	−1523.47			−1.73714			−0.868568	−	0.495570i	$-0.834959\pi$
$877$	−1523.47			−1.73714			−0.868568	+	0.495570i	$0.834959\pi$
$878$	681.496			0.776191
$879$			168.381i			0.191560i
$880$	85.3284			0.0969641
$881$		−	1164.99i		−	1.32235i	−0.750234	−	0.661173i	$-0.770059\pi$
$881$		−	1164.99i		−	1.32235i	0.750234	−	0.661173i	$-0.229941\pi$
$882$	20.5376			0.0232853
$883$	−1045.05			−1.18352			−0.591760	−	0.806114i	$-0.701567\pi$
$883$	−1045.05			−1.18352			−0.591760	+	0.806114i	$0.701567\pi$
$884$			322.073i			0.364336i
$885$		−	358.273i		−	0.404828i
$886$	707.046			0.798021
$887$	−1455.96			−1.64144			−0.820720	−	0.571330i	$-0.806428\pi$
$887$	−1455.96			−1.64144			−0.820720	+	0.571330i	$0.806428\pi$
$888$			185.953i			0.209406i
$889$		−	1370.42i		−	1.54153i
$890$	60.6361			0.0681305
$891$		−	85.8601i		−	0.0963637i
$892$	−538.989			−0.604248
$893$		−	17.7378i		−	0.0198632i
$894$		−	36.5091i		−	0.0408380i
$895$			197.717i			0.220913i
$896$			75.1823i			0.0839088i
$897$	295.888	+	285.665i	0.329864	+	0.318468i
$898$	−442.322			−0.492563
$899$	−2058.00			−2.28921
$900$	−30.0000			−0.0333333
$901$	−215.815			−0.239528
$902$			443.065i			0.491203i
$903$	−648.098			−0.717716
$904$			477.237i			0.527917i
$905$	317.633			0.350976
$906$	−439.895			−0.485536
$907$			1339.28i			1.47660i	0.674470	+	0.738302i	$0.264372\pi$
$907$			1339.28i			1.47660i	−0.674470	+	0.738302i	$0.735628\pi$
$908$		−	625.179i		−	0.688523i
$909$	−404.083			−0.444535
$910$	−216.952			−0.238409
$911$			772.255i			0.847700i	0.905732	+	0.423850i	$0.139322\pi$
$911$			772.255i			0.847700i	−0.905732	+	0.423850i	$0.860678\pi$
$912$		−	4.60111i		−	0.00504508i
$913$	−1299.06			−1.42284
$914$			129.192i			0.141348i
$915$	445.339			0.486709
$916$		−	426.480i		−	0.465589i
$917$			174.407i			0.190193i
$918$			114.622i			0.124861i
$919$		−	913.892i		−	0.994441i	−0.867624	−	0.497221i	$-0.834354\pi$
$919$		−	913.892i		−	0.994441i	0.867624	−	0.497221i	$-0.165646\pi$
$920$	−101.036	+	104.651i	−0.109821	+	0.113751i
$921$	112.574			0.122230
$922$	−46.4967			−0.0504302
$923$	1077.18			1.16704
$924$	219.609			0.237672
$925$		−	189.787i		−	0.205176i
$926$	621.383			0.671040
$927$		−	119.059i		−	0.128434i
$928$	273.423			0.294636
$929$	536.691			0.577709			0.288854	−	0.957373i	$-0.406726\pi$
$929$	536.691			0.577709			0.288854	+	0.957373i	$0.406726\pi$
$930$		−	233.210i		−	0.250763i
$931$		−	3.21482i		−	0.00345308i
$932$	−482.592			−0.517802
$933$	−480.826			−0.515355
$934$			248.239i			0.265780i
$935$			332.740i			0.355872i
$936$	87.6031			0.0935930
$937$			636.687i			0.679495i	0.940517	+	0.339748i	$0.110342\pi$
$937$			636.687i			0.679495i	−0.940517	+	0.339748i	$0.889658\pi$
$938$	1122.47			1.19666
$939$			735.846i			0.783649i
$940$			119.447i			0.127071i
$941$		−	452.052i		−	0.480395i	−0.970724	−	0.240197i	$-0.922788\pi$
$941$		−	452.052i		−	0.480395i	0.970724	−	0.240197i	$-0.0772122\pi$
$942$		−	402.732i		−	0.427529i
$943$	−543.397	−	524.624i	−0.576243	−	0.556335i
$944$	−370.023			−0.391973
$945$	−77.2107			−0.0817045
$946$	759.684			0.803048
$947$	−138.936			−0.146711			−0.0733557	−	0.997306i	$-0.523371\pi$
$947$	−138.936			−0.146711			−0.0733557	+	0.997306i	$0.523371\pi$
$948$		−	102.064i		−	0.107663i
$949$	46.1197			0.0485983
$950$			4.69599i			0.00494315i
$951$	384.836			0.404665
$952$	−293.175			−0.307957
$953$			902.261i			0.946759i	0.880859	+	0.473380i	$0.156966\pi$
$953$			902.261i			0.946759i	−0.880859	+	0.473380i	$0.843034\pi$
$954$			58.7010i			0.0615315i
$955$	−131.117			−0.137296
$956$	98.1038			0.102619
$957$		−	798.673i		−	0.834559i
$958$		−	71.8672i		−	0.0750180i
$959$	454.785			0.474228
$960$			30.9839i			0.0322749i
$961$	851.892			0.886465
$962$			554.199i			0.576090i
$963$			113.172i			0.117520i
$964$			251.437i			0.260827i
$965$		−	594.356i		−	0.615913i
$966$	−260.034	+	269.339i	−0.269186	+	0.278819i
$967$	−110.851			−0.114634			−0.0573172	−	0.998356i	$-0.518255\pi$
$967$	−110.851			−0.114634			−0.0573172	+	0.998356i	$0.518255\pi$
$968$	84.8195			0.0876235
$969$	17.9421			0.0185161
$970$	−9.98775			−0.0102967
$971$		−	1126.21i		−	1.15985i	−0.814672	−	0.579923i	$-0.803083\pi$
$971$		−	1126.21i		−	1.15985i	0.814672	−	0.579923i	$-0.196917\pi$
$972$	31.1769			0.0320750
$973$			1705.86i			1.75320i
$974$	20.3106			0.0208528
$975$	−89.4095			−0.0917020
$976$		−	459.944i		−	0.471254i
$977$		−	1127.18i		−	1.15371i	−0.816845	−	0.576857i	$-0.804279\pi$
$977$		−	1127.18i		−	1.15371i	0.816845	−	0.576857i	$-0.195721\pi$
$978$	−694.784			−0.710413
$979$	−182.928			−0.186852
$980$			21.6486i			0.0220904i
$981$		−	542.654i		−	0.553164i
$982$	−540.576			−0.550485
$983$			843.799i			0.858392i	0.903211	+	0.429196i	$0.141203\pi$
$983$			843.799i			0.858392i	−0.903211	+	0.429196i	$0.858797\pi$
$984$	−160.883			−0.163499
$985$		−	60.4069i		−	0.0613268i
$986$			1066.22i			1.08136i
$987$			307.419i			0.311468i
$988$		−	13.7128i		−	0.0138793i
$989$	−899.524	+	931.713i	−0.909529	+	0.942076i
$990$	90.5045			0.0914187
$991$	712.967			0.719442			0.359721	−	0.933060i	$-0.382872\pi$
$991$	712.967			0.719442			0.359721	+	0.933060i	$0.382872\pi$
$992$	−240.858			−0.242800
$993$	−396.319			−0.399113
$994$			980.528i			0.986447i
$995$	9.88227			0.00993193
$996$		−	471.705i		−	0.473599i
$997$	997.105			1.00011			0.500053	−	0.865995i	$-0.333314\pi$
$997$	997.105			1.00011			0.500053	+	0.865995i	$0.333314\pi$
$998$	−658.523			−0.659843
$999$			197.233i			0.197430i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	690.3.c.a.91.31	yes	32
3.2	odd	2		2070.3.c.b.91.5		32
23.22	odd	2	inner	690.3.c.a.91.26	✓	32
69.68	even	2		2070.3.c.b.91.12		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.3.c.a.91.26	✓	32	23.22	odd	2	inner
690.3.c.a.91.31	yes	32	1.1	even	1	trivial
2070.3.c.b.91.5		32	3.2	odd	2
2070.3.c.b.91.12		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)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +1.73205 q^{3} +2.00000 q^{4} +2.23607i q^{5} +2.44949 q^{6} +6.64524i q^{7} +2.82843 q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q+1.41421 q^{2} +1.73205 q^{3} +2.00000 q^{4} +2.23607i q^{5} +2.44949 q^{6} +6.64524i q^{7} +2.82843 q^{8} +3.00000 q^{9} +3.16228i q^{10} -9.54001i q^{11} +3.46410 q^{12} +10.3241 q^{13} +9.39779i q^{14} +3.87298i q^{15} +4.00000 q^{16} +15.5981i q^{17} +4.24264 q^{18} -0.664113i q^{19} +4.47214i q^{20} +11.5099i q^{21} -13.4916i q^{22} +(16.5468 + 15.9751i) q^{23} +4.89898 q^{24} -5.00000 q^{25} +14.6005 q^{26} +5.19615 q^{27} +13.2905i q^{28} +48.3347 q^{29} +5.47723i q^{30} -42.5781 q^{31} +5.65685 q^{32} -16.5238i q^{33} +22.0590i q^{34} -14.8592 q^{35} +6.00000 q^{36} +37.9575i q^{37} -0.939198i q^{38} +17.8819 q^{39} +6.32456i q^{40} -32.8401 q^{41} +16.2775i q^{42} +56.3079i q^{43} -19.0800i q^{44} +6.70820i q^{45} +(23.4007 + 22.5922i) q^{46} +26.7091 q^{47} +6.92820 q^{48} +4.84077 q^{49} -7.07107 q^{50} +27.0167i q^{51} +20.6482 q^{52} +13.8360i q^{53} +7.34847 q^{54} +21.3321 q^{55} +18.7956i q^{56} -1.15028i q^{57} +68.3556 q^{58} -92.5057 q^{59} +7.74597i q^{60} -114.986i q^{61} -60.2145 q^{62} +19.9357i q^{63} +8.00000 q^{64} +23.0854i q^{65} -23.3682i q^{66} -119.439i q^{67} +31.1962i q^{68} +(28.6599 + 27.6697i) q^{69} -21.0141 q^{70} +104.336 q^{71} +8.48528 q^{72} +4.46718 q^{73} +53.6800i q^{74} -8.66025 q^{75} -1.32823i q^{76} +63.3957 q^{77} +25.2888 q^{78} -29.4634i q^{79} +8.94427i q^{80} +9.00000 q^{81} -46.4429 q^{82} -136.169i q^{83} +23.0198i q^{84} -34.8784 q^{85} +79.6313i q^{86} +83.7182 q^{87} -26.9832i q^{88} -19.1748i q^{89} +9.48683i q^{90} +68.6063i q^{91} +(33.0935 + 31.9502i) q^{92} -73.7474 q^{93} +37.7723 q^{94} +1.48500 q^{95} +9.79796 q^{96} +3.15840i q^{97} +6.84588 q^{98} -28.6200i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 64 q^{4} + 96 q^{9}+O(q^{10}) \) 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

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more