LMFDB - Embedded newform 690.3.c.a.91.30

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			91.30
Character	$\chi$	$=$	690.91
Dual form			690.3.c.a.91.27

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} +1.73205 q^{3} +2.00000 q^{4} +2.23607i q^{5} +2.44949 q^{6} -3.80775i q^{7} +2.82843 q^{8} +3.00000 q^{9} +O(q^{10})$	\(q+1.41421 q^{2} +1.73205 q^{3} +2.00000 q^{4} +2.23607i q^{5} +2.44949 q^{6} -3.80775i q^{7} +2.82843 q^{8} +3.00000 q^{9} +3.16228i q^{10} +16.1827i q^{11} +3.46410 q^{12} +12.9618 q^{13} -5.38497i q^{14} +3.87298i q^{15} +4.00000 q^{16} +10.8189i q^{17} +4.24264 q^{18} -17.1074i q^{19} +4.47214i q^{20} -6.59521i q^{21} +22.8858i q^{22} +(11.6478 + 19.8325i) q^{23} +4.89898 q^{24} -5.00000 q^{25} +18.3308 q^{26} +5.19615 q^{27} -7.61550i q^{28} -10.1177 q^{29} +5.47723i q^{30} +29.2531 q^{31} +5.65685 q^{32} +28.0292i q^{33} +15.3003i q^{34} +8.51438 q^{35} +6.00000 q^{36} -3.06540i q^{37} -24.1935i q^{38} +22.4505 q^{39} +6.32456i q^{40} +55.3042 q^{41} -9.32704i q^{42} -34.1448i q^{43} +32.3653i q^{44} +6.70820i q^{45} +(16.4724 + 28.0475i) q^{46} -49.2975 q^{47} +6.92820 q^{48} +34.5011 q^{49} -7.07107 q^{50} +18.7389i q^{51} +25.9237 q^{52} -28.6856i q^{53} +7.34847 q^{54} -36.1856 q^{55} -10.7699i q^{56} -29.6309i q^{57} -14.3086 q^{58} +24.2529 q^{59} +7.74597i q^{60} +52.9773i q^{61} +41.3702 q^{62} -11.4232i q^{63} +8.00000 q^{64} +28.9835i q^{65} +39.6393i q^{66} +7.48607i q^{67} +21.6379i q^{68} +(20.1745 + 34.3510i) q^{69} +12.0412 q^{70} -121.879 q^{71} +8.48528 q^{72} -132.268 q^{73} -4.33513i q^{74} -8.66025 q^{75} -34.2148i q^{76} +61.6195 q^{77} +31.7499 q^{78} -18.6463i q^{79} +8.94427i q^{80} +9.00000 q^{81} +78.2120 q^{82} +46.3088i q^{83} -13.1904i q^{84} -24.1919 q^{85} -48.2881i q^{86} -17.5244 q^{87} +45.7715i q^{88} +17.2346i q^{89} +9.48683i q^{90} -49.3554i q^{91} +(23.2955 + 39.6651i) q^{92} +50.6679 q^{93} -69.7172 q^{94} +38.2533 q^{95} +9.79796 q^{96} -31.4987i q^{97} +48.7919 q^{98} +48.5480i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 64 q^{4} + 96 q^{9}+O(q^{10}) $ 32 * q + 64 * q^4 + 96 * q^9	$ 32 q + 64 q^{4} + 96 q^{9} - 48 q^{13} + 128 q^{16} - 80 q^{23} - 160 q^{25} + 120 q^{29} + 248 q^{31} - 120 q^{35} + 192 q^{36} - 48 q^{39} + 72 q^{41} + 160 q^{46} + 400 q^{47} - 344 q^{49} - 96 q^{52} - 256 q^{58} + 120 q^{59} + 160 q^{62} + 256 q^{64} + 192 q^{69} + 104 q^{71} + 16 q^{73} + 240 q^{77} + 192 q^{78} + 288 q^{81} + 64 q^{82} - 120 q^{85} + 144 q^{87} - 160 q^{92} - 192 q^{93} + 96 q^{94} - 160 q^{95} + 64 q^{98}+O(q^{100}) $ 32 * q + 64 * q^4 + 96 * q^9 - 48 * q^13 + 128 * q^16 - 80 * q^23 - 160 * q^25 + 120 * q^29 + 248 * q^31 - 120 * q^35 + 192 * q^36 - 48 * q^39 + 72 * q^41 + 160 * q^46 + 400 * q^47 - 344 * q^49 - 96 * q^52 - 256 * q^58 + 120 * q^59 + 160 * q^62 + 256 * q^64 + 192 * q^69 + 104 * q^71 + 16 * q^73 + 240 * q^77 + 192 * q^78 + 288 * q^81 + 64 * q^82 - 120 * q^85 + 144 * q^87 - 160 * q^92 - 192 * q^93 + 96 * q^94 - 160 * q^95 + 64 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/690\mathbb{Z}\right)^\times$.

$n$	$277$	$461$	$511$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	1.41421			0.707107
$2$	1.41421			0.707107
$3$	1.73205			0.577350
$3$	1.73205			0.577350
$4$	2.00000			0.500000
$5$			2.23607i			0.447214i
$5$			2.23607i			0.447214i
$6$	2.44949			0.408248
$7$		−	3.80775i		−	0.543964i	−0.962302	−	0.271982i	$-0.912321\pi$
$7$		−	3.80775i		−	0.543964i	0.962302	−	0.271982i	$-0.0876791\pi$
$8$	2.82843			0.353553
$9$	3.00000			0.333333
$10$			3.16228i			0.316228i
$11$			16.1827i			1.47115i	0.677442	+	0.735576i	$0.263088\pi$
$11$			16.1827i			1.47115i	−0.677442	+	0.735576i	$0.736912\pi$
$12$	3.46410			0.288675
$13$	12.9618			0.997064			0.498532	−	0.866871i	$-0.333873\pi$
$13$	12.9618			0.997064			0.498532	+	0.866871i	$0.333873\pi$
$14$		−	5.38497i		−	0.384641i
$15$			3.87298i			0.258199i
$16$	4.00000			0.250000
$17$			10.8189i			0.636408i	0.948022	+	0.318204i	$0.103080\pi$
$17$			10.8189i			0.636408i	−0.948022	+	0.318204i	$0.896920\pi$
$18$	4.24264			0.235702
$19$		−	17.1074i		−	0.900390i	−0.892930	−	0.450195i	$-0.851354\pi$
$19$		−	17.1074i		−	0.900390i	0.892930	−	0.450195i	$-0.148646\pi$
$20$			4.47214i			0.223607i
$21$		−	6.59521i		−	0.314058i
$22$			22.8858i			1.04026i
$23$	11.6478	+	19.8325i	0.506424	+	0.862285i
$23$	11.6478	+	19.8325i	0.506424	+	0.862285i
$24$	4.89898			0.204124
$25$	−5.00000			−0.200000
$26$	18.3308			0.705031
$27$	5.19615			0.192450
$28$		−	7.61550i		−	0.271982i
$29$	−10.1177			−0.348887			−0.174443	−	0.984667i	$-0.555813\pi$
$29$	−10.1177			−0.348887			−0.174443	+	0.984667i	$0.555813\pi$
$30$			5.47723i			0.182574i
$31$	29.2531			0.943650			0.471825	−	0.881692i	$-0.343595\pi$
$31$	29.2531			0.943650			0.471825	+	0.881692i	$0.343595\pi$
$32$	5.65685			0.176777
$33$			28.0292i			0.849370i
$34$			15.3003i			0.450008i
$35$	8.51438			0.243268
$36$	6.00000			0.166667
$37$		−	3.06540i		−	0.0828486i	−0.999142	−	0.0414243i	$-0.986810\pi$
$37$		−	3.06540i		−	0.0828486i	0.999142	−	0.0414243i	$-0.0131895\pi$
$38$		−	24.1935i		−	0.636672i
$39$	22.4505			0.575655
$40$			6.32456i			0.158114i
$41$	55.3042			1.34888			0.674441	−	0.738328i	$-0.264384\pi$
$41$	55.3042			1.34888			0.674441	+	0.738328i	$0.264384\pi$
$42$		−	9.32704i		−	0.222072i
$43$		−	34.1448i		−	0.794065i	−0.917804	−	0.397033i	$-0.870040\pi$
$43$		−	34.1448i		−	0.794065i	0.917804	−	0.397033i	$-0.129960\pi$
$44$			32.3653i			0.735576i
$45$			6.70820i			0.149071i
$46$	16.4724	+	28.0475i	0.358096	+	0.609727i
$47$	−49.2975			−1.04888			−0.524442	−	0.851446i	$-0.675726\pi$
$47$	−49.2975			−1.04888			−0.524442	+	0.851446i	$0.675726\pi$
$48$	6.92820			0.144338
$49$	34.5011			0.704103
$50$	−7.07107			−0.141421
$51$			18.7389i			0.367430i
$52$	25.9237			0.498532
$53$		−	28.6856i		−	0.541238i	−0.962687	−	0.270619i	$-0.912772\pi$
$53$		−	28.6856i		−	0.541238i	0.962687	−	0.270619i	$-0.0872283\pi$
$54$	7.34847			0.136083
$55$	−36.1856			−0.657919
$56$		−	10.7699i		−	0.192320i
$57$		−	29.6309i		−	0.519840i
$58$	−14.3086			−0.246700
$59$	24.2529			0.411066			0.205533	−	0.978650i	$-0.434107\pi$
$59$	24.2529			0.411066			0.205533	+	0.978650i	$0.434107\pi$
$60$			7.74597i			0.129099i
$61$			52.9773i			0.868481i	0.900797	+	0.434240i	$0.142983\pi$
$61$			52.9773i			0.868481i	−0.900797	+	0.434240i	$0.857017\pi$
$62$	41.3702			0.667261
$63$		−	11.4232i		−	0.181321i
$64$	8.00000			0.125000
$65$			28.9835i			0.445900i
$66$			39.6393i			0.600595i
$67$			7.48607i			0.111732i	0.998438	+	0.0558662i	$0.0177920\pi$
$67$			7.48607i			0.111732i	−0.998438	+	0.0558662i	$0.982208\pi$
$68$			21.6379i			0.318204i
$69$	20.1745	+	34.3510i	0.292384	+	0.497840i
$70$	12.0412			0.172017
$71$	−121.879			−1.71661			−0.858305	−	0.513140i	$-0.828482\pi$
$71$	−121.879			−1.71661			−0.858305	+	0.513140i	$0.828482\pi$
$72$	8.48528			0.117851
$73$	−132.268			−1.81189			−0.905947	−	0.423391i	$-0.860840\pi$
$73$	−132.268			−1.81189			−0.905947	+	0.423391i	$0.860840\pi$
$74$		−	4.33513i		−	0.0585828i
$75$	−8.66025			−0.115470
$76$		−	34.2148i		−	0.450195i
$77$	61.6195			0.800254
$78$	31.7499			0.407050
$79$		−	18.6463i		−	0.236030i	−0.993012	−	0.118015i	$-0.962347\pi$
$79$		−	18.6463i		−	0.236030i	0.993012	−	0.118015i	$-0.0376530\pi$
$80$			8.94427i			0.111803i
$81$	9.00000			0.111111
$82$	78.2120			0.953804
$83$			46.3088i			0.557938i	0.960300	+	0.278969i	$0.0899926\pi$
$83$			46.3088i			0.557938i	−0.960300	+	0.278969i	$0.910007\pi$
$84$		−	13.1904i		−	0.157029i
$85$	−24.1919			−0.284610
$86$		−	48.2881i		−	0.561489i
$87$	−17.5244			−0.201430
$88$			45.7715i			0.520131i
$89$			17.2346i			0.193647i	0.995302	+	0.0968234i	$0.0308682\pi$
$89$			17.2346i			0.193647i	−0.995302	+	0.0968234i	$0.969132\pi$
$90$			9.48683i			0.105409i
$91$		−	49.3554i		−	0.542367i
$92$	23.2955	+	39.6651i	0.253212	+	0.431142i
$93$	50.6679			0.544817
$94$	−69.7172			−0.741672
$95$	38.2533			0.402667
$96$	9.79796			0.102062
$97$		−	31.4987i		−	0.324728i	−0.986731	−	0.162364i	$-0.948088\pi$
$97$		−	31.4987i		−	0.324728i	0.986731	−	0.162364i	$-0.0519120\pi$
$98$	48.7919			0.497876
$99$			48.5480i			0.490384i
$100$	−10.0000			−0.100000
$101$	11.8823			0.117646			0.0588232	−	0.998268i	$-0.481265\pi$
$101$	11.8823			0.117646			0.0588232	+	0.998268i	$0.481265\pi$
$102$			26.5009i			0.259812i
$103$			61.0381i			0.592603i	0.955094	+	0.296301i	$0.0957533\pi$
$103$			61.0381i			0.592603i	−0.955094	+	0.296301i	$0.904247\pi$
$104$	36.6616			0.352515
$105$	14.7473			0.140451
$106$		−	40.5676i		−	0.382713i
$107$		−	144.705i		−	1.35238i	−0.736728	−	0.676190i	$-0.763630\pi$
$107$		−	144.705i		−	1.35238i	0.736728	−	0.676190i	$-0.236370\pi$
$108$	10.3923			0.0962250
$109$		−	162.558i		−	1.49135i	−0.666308	−	0.745677i	$-0.732126\pi$
$109$		−	162.558i		−	1.49135i	0.666308	−	0.745677i	$-0.267874\pi$
$110$	−51.1741			−0.465219
$111$		−	5.30942i		−	0.0478326i
$112$		−	15.2310i		−	0.135991i
$113$		−	141.903i		−	1.25578i	−0.778304	−	0.627888i	$-0.783920\pi$
$113$		−	141.903i		−	1.25578i	0.778304	−	0.627888i	$-0.216080\pi$
$114$		−	41.9044i		−	0.367583i
$115$	−44.3469	+	26.0452i	−0.385625	+	0.226480i
$116$	−20.2354			−0.174443
$117$	38.8855			0.332355
$118$	34.2988			0.290668
$119$	41.1958			0.346183
$120$			10.9545i			0.0912871i
$121$	−140.879			−1.16429
$122$			74.9213i			0.614109i
$123$	95.7897			0.778778
$124$	58.5063			0.471825
$125$		−	11.1803i		−	0.0894427i
$126$		−	16.1549i		−	0.128214i
$127$	36.2207			0.285202			0.142601	−	0.989780i	$-0.454453\pi$
$127$	36.2207			0.285202			0.142601	+	0.989780i	$0.454453\pi$
$128$	11.3137			0.0883883
$129$		−	59.1405i		−	0.458454i
$130$			40.9889i			0.315299i
$131$	−0.512699			−0.00391373			−0.00195687	−	0.999998i	$-0.500623\pi$
$131$	−0.512699			−0.00391373			−0.00195687	+	0.999998i	$0.500623\pi$
$132$			56.0584i			0.424685i
$133$	−65.1407			−0.489780
$134$			10.5869i			0.0790067i
$135$			11.6190i			0.0860663i
$136$			30.6006i			0.225004i
$137$		−	64.7952i		−	0.472958i	−0.971637	−	0.236479i	$-0.924007\pi$
$137$		−	64.7952i		−	0.472958i	0.971637	−	0.236479i	$-0.0759934\pi$
$138$	28.5311	+	48.5796i	0.206747	+	0.352026i
$139$	−170.051			−1.22339			−0.611693	−	0.791095i	$-0.709511\pi$
$139$	−170.051			−1.22339			−0.611693	+	0.791095i	$0.709511\pi$
$140$	17.0288			0.121634
$141$	−85.3858			−0.605573
$142$	−172.363			−1.21383
$143$			209.757i			1.46683i
$144$	12.0000			0.0833333
$145$		−	22.6239i		−	0.156027i
$146$	−187.056			−1.28120
$147$	59.7576			0.406514
$148$		−	6.13079i		−	0.0414243i
$149$			54.6548i			0.366810i	0.983037	+	0.183405i	$0.0587120\pi$
$149$			54.6548i			0.366810i	−0.983037	+	0.183405i	$0.941288\pi$
$150$	−12.2474			−0.0816497
$151$	131.579			0.871382			0.435691	−	0.900096i	$-0.356504\pi$
$151$	131.579			0.871382			0.435691	+	0.900096i	$0.356504\pi$
$152$		−	48.3871i		−	0.318336i
$153$			32.4568i			0.212136i
$154$	87.1432			0.565865
$155$			65.4120i			0.422013i
$156$	44.9011			0.287828
$157$		−	62.7071i		−	0.399409i	−0.979856	−	0.199704i	$-0.936002\pi$
$157$		−	62.7071i		−	0.399409i	0.979856	−	0.199704i	$-0.0639982\pi$
$158$		−	26.3699i		−	0.166898i
$159$		−	49.6849i		−	0.312484i
$160$			12.6491i			0.0790569i
$161$	75.5173	−	44.3517i	0.469052	−	0.275477i
$162$	12.7279			0.0785674
$163$	65.2642			0.400394			0.200197	−	0.979756i	$-0.435842\pi$
$163$	65.2642			0.400394			0.200197	+	0.979756i	$0.435842\pi$
$164$	110.608			0.674441
$165$	−62.6752			−0.379850
$166$			65.4906i			0.394521i
$167$	−89.9666			−0.538722			−0.269361	−	0.963039i	$-0.586813\pi$
$167$	−89.9666			−0.538722			−0.269361	+	0.963039i	$0.586813\pi$
$168$		−	18.6541i		−	0.111036i
$169$	−0.991006			−0.00586394
$170$	−34.2125			−0.201250
$171$		−	51.3222i		−	0.300130i
$172$		−	68.2896i		−	0.397033i
$173$	−177.669			−1.02699			−0.513494	−	0.858093i	$-0.671649\pi$
$173$	−177.669			−1.02699			−0.513494	+	0.858093i	$0.671649\pi$
$174$	−24.7832			−0.142432
$175$			19.0387i			0.108793i
$176$			64.7307i			0.367788i
$177$	42.0073			0.237329
$178$			24.3734i			0.136929i
$179$	−193.456			−1.08076			−0.540379	−	0.841421i	$-0.681719\pi$
$179$	−193.456			−1.08076			−0.540379	+	0.841421i	$0.681719\pi$
$180$			13.4164i			0.0745356i
$181$			186.354i			1.02958i	0.857316	+	0.514790i	$0.172130\pi$
$181$			186.354i			1.02958i	−0.857316	+	0.514790i	$0.827870\pi$
$182$		−	69.7991i		−	0.383511i
$183$			91.7594i			0.501418i
$184$	32.9448	+	56.0949i	0.179048	+	0.304864i
$185$	6.85443			0.0370510
$186$	71.6553			0.385243
$187$	−175.079			−0.936253
$188$	−98.5950			−0.524442
$189$		−	19.7856i		−	0.104686i
$190$	54.0984			0.284728
$191$		−	18.7678i		−	0.0982605i	−0.998792	−	0.0491303i	$-0.984355\pi$
$191$		−	18.7678i		−	0.0982605i	0.998792	−	0.0491303i	$-0.0156449\pi$
$192$	13.8564			0.0721688
$193$	−161.771			−0.838193			−0.419097	−	0.907942i	$-0.637653\pi$
$193$	−161.771			−0.838193			−0.419097	+	0.907942i	$0.637653\pi$
$194$		−	44.5458i		−	0.229618i
$195$			50.2009i			0.257441i
$196$	69.0021			0.352052
$197$	62.3846			0.316673			0.158337	−	0.987385i	$-0.449387\pi$
$197$	62.3846			0.316673			0.158337	+	0.987385i	$0.449387\pi$
$198$			68.6573i			0.346754i
$199$		−	280.675i		−	1.41043i	−0.708994	−	0.705214i	$-0.750851\pi$
$199$		−	280.675i		−	1.41043i	0.708994	−	0.705214i	$-0.249149\pi$
$200$	−14.1421			−0.0707107
$201$			12.9663i			0.0645087i
$202$	16.8041			0.0831885
$203$			38.5257i			0.189782i
$204$			37.4779i			0.183715i
$205$			123.664i			0.603239i
$206$			86.3209i			0.419033i
$207$	34.9433	+	59.4976i	0.168808	+	0.287428i
$208$	51.8473			0.249266
$209$	276.844			1.32461
$210$	20.8559			0.0993138
$211$	−252.159			−1.19507			−0.597533	−	0.801844i	$-0.703852\pi$
$211$	−252.159			−1.19507			−0.597533	+	0.801844i	$0.703852\pi$
$212$		−	57.3712i		−	0.270619i
$213$	−211.101			−0.991085
$214$		−	204.643i		−	0.956276i
$215$	76.3501			0.355117
$216$	14.6969			0.0680414
$217$		−	111.389i		−	0.513312i
$218$		−	229.891i		−	1.05455i
$219$	−229.095			−1.04610
$220$	−72.3711			−0.328960
$221$			140.233i			0.634539i
$222$		−	7.50866i		−	0.0338228i
$223$	−225.671			−1.01198			−0.505989	−	0.862540i	$-0.668872\pi$
$223$	−225.671			−1.01198			−0.505989	+	0.862540i	$0.668872\pi$
$224$		−	21.5399i		−	0.0961602i
$225$	−15.0000			−0.0666667
$226$		−	200.681i		−	0.887967i
$227$		−	397.011i		−	1.74895i	−0.485074	−	0.874473i	$-0.661207\pi$
$227$		−	397.011i		−	1.74895i	0.485074	−	0.874473i	$-0.338793\pi$
$228$		−	59.2618i		−	0.259920i
$229$		−	269.388i		−	1.17637i	−0.808728	−	0.588183i	$-0.799844\pi$
$229$		−	269.388i		−	1.17637i	0.808728	−	0.588183i	$-0.200156\pi$
$230$	−62.7160	+	36.8334i	−0.272678	+	0.160145i
$231$	106.728			0.462027
$232$	−28.6172			−0.123350
$233$	350.279			1.50334			0.751672	−	0.659537i	$-0.229248\pi$
$233$	350.279			1.50334			0.751672	+	0.659537i	$0.229248\pi$
$234$	54.9924			0.235010
$235$		−	110.233i		−	0.469075i
$236$	48.5058			0.205533
$237$		−	32.2964i		−	0.136272i
$238$	58.2596			0.244788
$239$	34.7609			0.145443			0.0727215	−	0.997352i	$-0.476832\pi$
$239$	34.7609			0.145443			0.0727215	+	0.997352i	$0.476832\pi$
$240$			15.4919i			0.0645497i
$241$		−	343.644i		−	1.42591i	−0.701211	−	0.712954i	$-0.747357\pi$
$241$		−	343.644i		−	1.42591i	0.701211	−	0.712954i	$-0.252643\pi$
$242$	−199.233			−0.823276
$243$	15.5885			0.0641500
$244$			105.955i			0.434240i
$245$			77.1467i			0.314884i
$246$	135.467			0.550679
$247$		−	221.743i		−	0.897746i
$248$	82.7404			0.333631
$249$			80.2092i			0.322125i
$250$		−	15.8114i		−	0.0632456i
$251$			442.427i			1.76266i	0.472505	+	0.881328i	$0.343350\pi$
$251$			442.427i			1.76266i	−0.472505	+	0.881328i	$0.656650\pi$
$252$		−	22.8465i		−	0.0906607i
$253$	−320.944	+	188.492i	−1.26855	+	0.745027i
$254$	51.2238			0.201669
$255$	−41.9016			−0.164320
$256$	16.0000			0.0625000
$257$	377.957			1.47065			0.735325	−	0.677715i	$-0.237029\pi$
$257$	377.957			1.47065			0.735325	+	0.677715i	$0.237029\pi$
$258$		−	83.6374i		−	0.324176i
$259$	−11.6723			−0.0450666
$260$			57.9671i			0.222950i
$261$	−30.3531			−0.116296
$262$	−0.725066			−0.00276743
$263$		−	233.084i		−	0.886251i	−0.896459	−	0.443126i	$-0.853870\pi$
$263$		−	233.084i		−	0.886251i	0.896459	−	0.443126i	$-0.146130\pi$
$264$			79.2786i			0.300298i
$265$	64.1430			0.242049
$266$	−92.1229			−0.346327
$267$			29.8512i			0.111802i
$268$			14.9721i			0.0558662i
$269$	−278.353			−1.03477			−0.517385	−	0.855753i	$-0.673094\pi$
$269$	−278.353			−1.03477			−0.517385	+	0.855753i	$0.673094\pi$
$270$			16.4317i			0.0608581i
$271$	−508.787			−1.87744			−0.938722	−	0.344676i	$-0.887989\pi$
$271$	−508.787			−1.87744			−0.938722	+	0.344676i	$0.887989\pi$
$272$			43.2757i			0.159102i
$273$		−	85.4860i		−	0.313136i
$274$		−	91.6343i		−	0.334432i
$275$		−	80.9134i		−	0.294230i
$276$	40.3490	+	68.7019i	0.146192	+	0.248920i
$277$	−19.7229			−0.0712019			−0.0356009	−	0.999366i	$-0.511335\pi$
$277$	−19.7229			−0.0712019			−0.0356009	+	0.999366i	$0.511335\pi$
$278$	−240.488			−0.865065
$279$	87.7594			0.314550
$280$	24.0823			0.0860083
$281$			343.453i			1.22225i	0.791533	+	0.611126i	$0.209283\pi$
$281$			343.453i			1.22225i	−0.791533	+	0.611126i	$0.790717\pi$
$282$	−120.754			−0.428205
$283$			149.531i			0.528378i	0.964471	+	0.264189i	$0.0851043\pi$
$283$			149.531i			0.528378i	−0.964471	+	0.264189i	$0.914896\pi$
$284$	−243.759			−0.858305
$285$	66.2567			0.232480
$286$			296.641i			1.03721i
$287$		−	210.584i		−	0.733744i
$288$	16.9706			0.0589256
$289$	171.951			0.594985
$290$		−	31.9950i		−	0.110328i
$291$		−	54.5573i		−	0.187482i
$292$	−264.537			−0.905947
$293$			80.2344i			0.273838i	0.990582	+	0.136919i	$0.0437199\pi$
$293$			80.2344i			0.273838i	−0.990582	+	0.136919i	$0.956280\pi$
$294$	84.5100			0.287449
$295$			54.2312i			0.183834i
$296$		−	8.67025i		−	0.0292914i
$297$			84.0876i			0.283123i
$298$			77.2935i			0.259374i
$299$	150.976	+	257.066i	0.504937	+	0.859753i
$300$	−17.3205			−0.0577350
$301$	−130.015			−0.431943
$302$	186.080			0.616160
$303$	20.5807			0.0679232
$304$		−	68.4296i		−	0.225097i
$305$	−118.461			−0.388396
$306$			45.9009i			0.150003i
$307$	153.780			0.500911			0.250456	−	0.968128i	$-0.419420\pi$
$307$	153.780			0.500911			0.250456	+	0.968128i	$0.419420\pi$
$308$	123.239			0.400127
$309$			105.721i			0.342139i
$310$			92.5066i			0.298408i
$311$	−124.567			−0.400536			−0.200268	−	0.979741i	$-0.564181\pi$
$311$	−124.567			−0.400536			−0.200268	+	0.979741i	$0.564181\pi$
$312$	63.4997			0.203525
$313$			222.424i			0.710619i	0.934749	+	0.355310i	$0.115625\pi$
$313$			222.424i			0.710619i	−0.934749	+	0.355310i	$0.884375\pi$
$314$		−	88.6813i		−	0.282425i
$315$	25.5432			0.0810894
$316$		−	37.2927i		−	0.118015i
$317$	303.301			0.956785			0.478393	−	0.878146i	$-0.341220\pi$
$317$	303.301			0.956785			0.478393	+	0.878146i	$0.341220\pi$
$318$		−	70.2651i		−	0.220959i
$319$		−	163.732i		−	0.513265i
$320$			17.8885i			0.0559017i
$321$		−	250.636i		−	0.780796i
$322$	106.798	−	62.7228i	0.331670	−	0.194791i
$323$	185.084			0.573015
$324$	18.0000			0.0555556
$325$	−64.8091			−0.199413
$326$	92.2975			0.283121
$327$		−	281.558i		−	0.861034i
$328$	156.424			0.476902
$329$			187.713i			0.570555i
$330$	−88.6361			−0.268594
$331$	−281.157			−0.849416			−0.424708	−	0.905330i	$-0.639623\pi$
$331$	−281.157			−0.849416			−0.424708	+	0.905330i	$0.639623\pi$
$332$			92.6176i			0.278969i
$333$		−	9.19619i		−	0.0276162i
$334$	−127.232			−0.380934
$335$	−16.7394			−0.0499683
$336$		−	26.3809i		−	0.0785145i
$337$		−	258.510i		−	0.767091i	−0.923522	−	0.383546i	$-0.874703\pi$
$337$		−	258.510i		−	0.767091i	0.923522	−	0.383546i	$-0.125297\pi$
$338$	−1.40149			−0.00414643
$339$		−	245.783i		−	0.725022i
$340$	−48.3837			−0.142305
$341$			473.394i			1.38825i
$342$		−	72.5806i		−	0.212224i
$343$		−	317.951i		−	0.926971i
$344$		−	96.5761i		−	0.280744i
$345$	−76.8111	+	45.1116i	−0.222641	+	0.130758i
$346$	−251.262			−0.726191
$347$	613.312			1.76747			0.883735	−	0.467988i	$-0.155021\pi$
$347$	613.312			1.76747			0.883735	+	0.467988i	$0.155021\pi$
$348$	−35.0488			−0.100715
$349$	−75.5131			−0.216370			−0.108185	−	0.994131i	$-0.534504\pi$
$349$	−75.5131			−0.216370			−0.108185	+	0.994131i	$0.534504\pi$
$350$			26.9248i			0.0769281i
$351$	67.3516			0.191885
$352$			91.5430i			0.260065i
$353$	−31.5279			−0.0893142			−0.0446571	−	0.999002i	$-0.514220\pi$
$353$	−31.5279			−0.0893142			−0.0446571	+	0.999002i	$0.514220\pi$
$354$	59.4073			0.167817
$355$		−	272.530i		−	0.767691i
$356$			34.4691i			0.0968234i
$357$	71.3532			0.199869
$358$	−273.588			−0.764212
$359$		−	192.855i		−	0.537199i	−0.963252	−	0.268600i	$-0.913439\pi$
$359$		−	192.855i		−	0.537199i	0.963252	−	0.268600i	$-0.0865608\pi$
$360$			18.9737i			0.0527046i
$361$	68.3365			0.189298
$362$			263.544i			0.728023i
$363$	−244.009			−0.672202
$364$		−	98.7108i		−	0.271183i
$365$		−	295.761i		−	0.810304i
$366$			129.767i			0.354556i
$367$		−	512.314i		−	1.39595i	−0.716121	−	0.697976i	$-0.754084\pi$
$367$		−	512.314i		−	1.39595i	0.716121	−	0.697976i	$-0.245916\pi$
$368$	46.5910	+	79.3302i	0.126606	+	0.215571i
$369$	165.913			0.449628
$370$	9.69363			0.0261990
$371$	−109.228			−0.294414
$372$	101.336			0.272408
$373$			308.198i			0.826267i	0.910670	+	0.413134i	$0.135566\pi$
$373$			308.198i			0.826267i	−0.910670	+	0.413134i	$0.864434\pi$
$374$	−247.599			−0.662031
$375$		−	19.3649i		−	0.0516398i
$376$	−139.434			−0.370836
$377$	−131.144			−0.347862
$378$		−	27.9811i		−	0.0740241i
$379$			274.002i			0.722959i	0.932380	+	0.361480i	$0.117728\pi$
$379$			274.002i			0.722959i	−0.932380	+	0.361480i	$0.882272\pi$
$380$	76.5067			0.201333
$381$	62.7361			0.164662
$382$		−	26.5416i		−	0.0694807i
$383$			184.563i			0.481889i	0.970539	+	0.240944i	$0.0774571\pi$
$383$			184.563i			0.481889i	−0.970539	+	0.240944i	$0.922543\pi$
$384$	19.5959			0.0510310
$385$			137.785i			0.357884i
$386$	−228.779			−0.592692
$387$		−	102.434i		−	0.264688i
$388$		−	62.9973i		−	0.162364i
$389$			384.497i			0.988425i	0.869341	+	0.494212i	$0.164543\pi$
$389$			384.497i			0.988425i	−0.869341	+	0.494212i	$0.835457\pi$
$390$			70.9949i			0.182038i
$391$	−214.567	+	126.016i	−0.548765	+	0.322292i
$392$	97.5837			0.248938
$393$	−0.888020			−0.00225959
$394$	88.2252			0.223922
$395$	41.6945			0.105556
$396$			97.0960i			0.245192i
$397$	340.530			0.857757			0.428879	−	0.903362i	$-0.358909\pi$
$397$	340.530			0.857757			0.428879	+	0.903362i	$0.358909\pi$
$398$		−	396.935i		−	0.997324i
$399$	−112.827			−0.282775
$400$	−20.0000			−0.0500000
$401$			742.537i			1.85171i	0.377875	+	0.925857i	$0.376655\pi$
$401$			742.537i			1.85171i	−0.377875	+	0.925857i	$0.623345\pi$
$402$			18.3371i			0.0456146i
$403$	379.174			0.940879
$404$	23.7646			0.0588232
$405$			20.1246i			0.0496904i
$406$			54.4836i			0.134196i
$407$	49.6063			0.121883
$408$			53.0017i			0.129906i
$409$	757.373			1.85177			0.925884	−	0.377807i	$-0.123322\pi$
$409$	757.373			1.85177			0.925884	+	0.377807i	$0.123322\pi$
$410$			174.887i			0.426554i
$411$		−	112.229i		−	0.273062i
$412$			122.076i			0.296301i
$413$		−	92.3490i		−	0.223605i
$414$	49.4172	+	84.1424i	0.119365	+	0.203242i
$415$	−103.550			−0.249517
$416$	73.3232			0.176258
$417$	−294.537			−0.706323
$418$	391.516			0.936641
$419$		−	213.737i		−	0.510113i	−0.966926	−	0.255056i	$-0.917906\pi$
$419$		−	213.737i		−	0.510113i	0.966926	−	0.255056i	$-0.0820940\pi$
$420$	29.4947			0.0702255
$421$			675.986i			1.60567i	0.596202	+	0.802834i	$0.296676\pi$
$421$			675.986i			1.60567i	−0.596202	+	0.802834i	$0.703324\pi$
$422$	−356.607			−0.845040
$423$	−147.893			−0.349628
$424$		−	81.1351i		−	0.191356i
$425$		−	54.0947i		−	0.127282i
$426$	−298.542			−0.700803
$427$	201.724			0.472422
$428$		−	289.409i		−	0.676190i
$429$			363.310i			0.846876i
$430$	107.975			0.251105
$431$			8.04660i			0.0186696i	0.999956	+	0.00933480i	$0.00297140\pi$
$431$			8.04660i			0.0186696i	−0.999956	+	0.00933480i	$0.997029\pi$
$432$	20.7846			0.0481125
$433$			369.369i			0.853046i	0.904477	+	0.426523i	$0.140262\pi$
$433$			369.369i			0.853046i	−0.904477	+	0.426523i	$0.859738\pi$
$434$		−	157.527i		−	0.362966i
$435$		−	39.1857i		−	0.0900821i
$436$		−	325.115i		−	0.745677i
$437$	339.283	−	199.263i	0.776392	−	0.455979i
$438$	−323.990			−0.739703
$439$	−54.3077			−0.123708			−0.0618539	−	0.998085i	$-0.519701\pi$
$439$	−54.3077			−0.123708			−0.0618539	+	0.998085i	$0.519701\pi$
$440$	−102.348			−0.232610
$441$	103.503			0.234701
$442$			198.320i			0.448687i
$443$	263.570			0.594966			0.297483	−	0.954727i	$-0.403853\pi$
$443$	263.570			0.594966			0.297483	+	0.954727i	$0.403853\pi$
$444$		−	10.6188i		−	0.0239163i
$445$	−38.5377			−0.0866015
$446$	−319.147			−0.715576
$447$			94.6648i			0.211778i
$448$		−	30.4620i		−	0.0679955i
$449$	578.427			1.28826			0.644128	−	0.764917i	$-0.277220\pi$
$449$	578.427			1.28826			0.644128	+	0.764917i	$0.277220\pi$
$450$	−21.2132			−0.0471405
$451$			894.970i			1.98441i
$452$		−	283.805i		−	0.627888i
$453$	227.901			0.503092
$454$		−	561.458i		−	1.23669i
$455$	110.362			0.242554
$456$		−	83.8088i		−	0.183791i
$457$		−	514.251i		−	1.12528i	−0.826703	−	0.562638i	$-0.809786\pi$
$457$		−	514.251i		−	1.12528i	0.826703	−	0.562638i	$-0.190214\pi$
$458$		−	380.972i		−	0.831816i
$459$			56.2168i			0.122477i
$460$	−88.6938	+	52.0903i	−0.192813	+	0.113240i
$461$	545.697			1.18372			0.591862	−	0.806039i	$-0.298393\pi$
$461$	545.697			1.18372			0.591862	+	0.806039i	$0.298393\pi$
$462$	150.936			0.326702
$463$	−481.885			−1.04079			−0.520395	−	0.853926i	$-0.674215\pi$
$463$	−481.885			−1.04079			−0.520395	+	0.853926i	$0.674215\pi$
$464$	−40.4708			−0.0872217
$465$			113.297i			0.243649i
$466$	495.370			1.06302
$467$		−	6.72759i		−	0.0144060i	−0.999974	−	0.00720299i	$-0.997707\pi$
$467$		−	6.72759i		−	0.0144060i	0.999974	−	0.00720299i	$-0.00229280\pi$
$468$	77.7710			0.166177
$469$	28.5051			0.0607784
$470$		−	155.892i		−	0.331686i
$471$		−	108.612i		−	0.230599i
$472$	68.5976			0.145334
$473$	552.554			1.16819
$474$		−	45.6740i		−	0.0963587i
$475$			85.5370i			0.180078i
$476$	82.3916			0.173092
$477$		−	86.0568i		−	0.180413i
$478$	49.1593			0.102844
$479$			366.353i			0.764828i	0.923991	+	0.382414i	$0.124907\pi$
$479$			366.353i			0.764828i	−0.923991	+	0.382414i	$0.875093\pi$
$480$			21.9089i			0.0456435i
$481$		−	39.7331i		−	0.0826053i
$482$		−	485.985i		−	1.00827i
$483$	130.800	−	76.8194i	0.270807	−	0.159046i
$484$	−281.758			−0.582144
$485$	70.4331			0.145223
$486$	22.0454			0.0453609
$487$	212.080			0.435483			0.217742	−	0.976006i	$-0.430131\pi$
$487$	212.080			0.435483			0.217742	+	0.976006i	$0.430131\pi$
$488$			149.843i			0.307054i
$489$	113.041			0.231167
$490$			109.102i			0.222657i
$491$	−323.833			−0.659537			−0.329768	−	0.944062i	$-0.606971\pi$
$491$	−323.833			−0.659537			−0.329768	+	0.944062i	$0.606971\pi$
$492$	191.579			0.389389
$493$		−	109.463i		−	0.222034i
$494$		−	313.592i		−	0.634802i
$495$	−108.557			−0.219306
$496$	117.013			0.235912
$497$			464.086i			0.933774i
$498$			113.433i			0.227777i
$499$	113.014			0.226481			0.113241	−	0.993568i	$-0.463877\pi$
$499$	113.014			0.226481			0.113241	+	0.993568i	$0.463877\pi$
$500$		−	22.3607i		−	0.0447214i
$501$	−155.827			−0.311032
$502$			625.686i			1.24639i
$503$		−	841.141i		−	1.67225i	−0.548541	−	0.836124i	$-0.684816\pi$
$503$		−	841.141i		−	1.67225i	0.548541	−	0.836124i	$-0.315184\pi$
$504$		−	32.3098i		−	0.0641068i
$505$			26.5696i			0.0526131i
$506$	−453.883	+	266.568i	−0.897001	+	0.526813i
$507$	−1.71647			−0.00338555
$508$	72.4414			0.142601
$509$	−570.869			−1.12155			−0.560775	−	0.827968i	$-0.689497\pi$
$509$	−570.869			−1.12155			−0.560775	+	0.827968i	$0.689497\pi$
$510$	−59.2577			−0.116192
$511$			503.644i			0.985605i
$512$	22.6274			0.0441942
$513$		−	88.8927i		−	0.173280i
$514$	534.512			1.03991
$515$	−136.485			−0.265020
$516$		−	118.281i		−	0.229227i
$517$		−	797.765i		−	1.54307i
$518$	−16.5071			−0.0318669
$519$	−307.732			−0.592932
$520$			81.9778i			0.157650i
$521$		−	823.426i		−	1.58047i	−0.612802	−	0.790236i	$-0.709958\pi$
$521$		−	823.426i		−	1.58047i	0.612802	−	0.790236i	$-0.290042\pi$
$522$	−42.9258			−0.0822334
$523$			692.210i			1.32354i	0.749709	+	0.661768i	$0.230194\pi$
$523$			692.210i			1.32354i	−0.749709	+	0.661768i	$0.769806\pi$
$524$	−1.02540			−0.00195687
$525$			32.9761i			0.0628116i
$526$		−	329.631i		−	0.626674i
$527$			316.488i			0.600546i
$528$			112.117i			0.212342i
$529$	−257.660	+	462.009i	−0.487069	+	0.873363i
$530$	90.7118			0.171154
$531$	72.7587			0.137022
$532$	−130.281			−0.244890
$533$	716.844			1.34492
$534$			42.2159i			0.0790560i
$535$	323.569			0.604802
$536$			21.1738i			0.0395034i
$537$	−335.075			−0.623976
$538$	−393.650			−0.731692
$539$			558.319i			1.03584i
$540$			23.2379i			0.0430331i
$541$	915.849			1.69288			0.846440	−	0.532483i	$-0.178741\pi$
$541$	915.849			1.69288			0.846440	+	0.532483i	$0.178741\pi$
$542$	−719.534			−1.32755
$543$			322.775i			0.594428i
$544$			61.2011i			0.112502i
$545$	363.490			0.666954
$546$		−	120.896i		−	0.221420i
$547$	203.592			0.372197			0.186099	−	0.982531i	$-0.440416\pi$
$547$	203.592			0.372197			0.186099	+	0.982531i	$0.440416\pi$
$548$		−	129.590i		−	0.236479i
$549$			158.932i			0.289494i
$550$		−	114.429i		−	0.208052i
$551$			173.088i			0.314134i
$552$	57.0621	+	97.1592i	0.103373	+	0.176013i
$553$	−71.0006			−0.128392
$554$	−27.8924			−0.0503473
$555$	11.8722			0.0213914
$556$	−340.102			−0.611693
$557$			624.298i			1.12082i	0.828215	+	0.560411i	$0.189357\pi$
$557$			624.298i			1.12082i	−0.828215	+	0.560411i	$0.810643\pi$
$558$	124.111			0.222420
$559$		−	442.579i		−	0.791734i
$560$	34.0575			0.0608170
$561$	−303.246			−0.540546
$562$			485.716i			0.864263i
$563$		−	961.054i		−	1.70702i	−0.521074	−	0.853511i	$-0.674469\pi$
$563$		−	961.054i		−	1.70702i	0.521074	−	0.853511i	$-0.325531\pi$
$564$	−170.772			−0.302786
$565$	317.304			0.561600
$566$			211.469i			0.373620i
$567$		−	34.2697i		−	0.0604405i
$568$	−344.727			−0.606913
$569$			576.159i			1.01258i	0.862363	+	0.506290i	$0.168984\pi$
$569$			576.159i			1.01258i	−0.862363	+	0.506290i	$0.831016\pi$
$570$	93.7011			0.164388
$571$			346.570i			0.606952i	0.952839	+	0.303476i	$0.0981472\pi$
$571$			346.570i			0.606952i	−0.952839	+	0.303476i	$0.901853\pi$
$572$			419.514i			0.733416i
$573$		−	32.5067i		−	0.0567307i
$574$		−	297.811i		−	0.518835i
$575$	−58.2388	−	99.1627i	−0.101285	−	0.172457i
$576$	24.0000			0.0416667
$577$	675.477			1.17067			0.585335	−	0.810791i	$-0.300963\pi$
$577$	675.477			1.17067			0.585335	+	0.810791i	$0.300963\pi$
$578$	243.175			0.420718
$579$	−280.196			−0.483931
$580$		−	45.2478i		−	0.0780134i
$581$	176.332			0.303498
$582$		−	77.1556i		−	0.132570i
$583$	464.210			0.796243
$584$	−374.111			−0.640601
$585$			86.9506i			0.148633i
$586$			113.469i			0.193632i
$587$	−820.083			−1.39708			−0.698538	−	0.715573i	$-0.746165\pi$
$587$	−820.083			−1.39708			−0.698538	+	0.715573i	$0.746165\pi$
$588$	119.515			0.203257
$589$		−	500.446i		−	0.849653i
$590$			76.6944i			0.129991i
$591$	108.053			0.182831
$592$		−	12.2616i		−	0.0207121i
$593$	768.200			1.29545			0.647723	−	0.761876i	$-0.275721\pi$
$593$	768.200			1.29545			0.647723	+	0.761876i	$0.275721\pi$
$594$			118.918i			0.200198i
$595$			92.1166i			0.154818i
$596$			109.310i			0.183405i
$597$		−	486.144i		−	0.814311i
$598$	213.513	+	363.546i	0.357044	+	0.607937i
$599$	−192.340			−0.321102			−0.160551	−	0.987028i	$-0.551327\pi$
$599$	−192.340			−0.321102			−0.160551	+	0.987028i	$0.551327\pi$
$600$	−24.4949			−0.0408248
$601$	829.949			1.38095			0.690474	−	0.723358i	$-0.257402\pi$
$601$	829.949			1.38095			0.690474	+	0.723358i	$0.257402\pi$
$602$	−183.869			−0.305430
$603$			22.4582i			0.0372441i
$604$	263.157			0.435691
$605$		−	315.015i		−	0.520685i
$606$	29.1055			0.0480289
$607$	503.567			0.829600			0.414800	−	0.909913i	$-0.363851\pi$
$607$	503.567			0.829600			0.414800	+	0.909913i	$0.363851\pi$
$608$		−	96.7741i		−	0.159168i
$609$			66.7285i			0.109571i
$610$	−167.529			−0.274638
$611$	−638.986			−1.04580
$612$			64.9136i			0.106068i
$613$			117.049i			0.190944i	0.995432	+	0.0954721i	$0.0304361\pi$
$613$			117.049i			0.190944i	−0.995432	+	0.0954721i	$0.969564\pi$
$614$	217.477			0.354198
$615$			214.192i			0.348280i
$616$	174.286			0.282932
$617$		−	13.5947i		−	0.0220336i	−0.999939	−	0.0110168i	$-0.996493\pi$
$617$		−	13.5947i		−	0.0220336i	0.999939	−	0.0110168i	$-0.00350682\pi$
$618$			149.512i			0.241929i
$619$			351.787i			0.568315i	0.958778	+	0.284158i	$0.0917139\pi$
$619$			351.787i			0.568315i	−0.958778	+	0.284158i	$0.908286\pi$
$620$			130.824i			0.211007i
$621$	60.5235	+	103.053i	0.0974614	+	0.165947i
$622$	−176.164			−0.283222
$623$	65.6249			0.105337
$624$	89.8022			0.143914
$625$	25.0000			0.0400000
$626$			314.555i			0.502484i
$627$	479.507			0.764764
$628$		−	125.414i		−	0.199704i
$629$	33.1643			0.0527255
$630$	36.1235			0.0573389
$631$			719.196i			1.13977i	0.821724	+	0.569886i	$0.193013\pi$
$631$			719.196i			1.13977i	−0.821724	+	0.569886i	$0.806987\pi$
$632$		−	52.7398i		−	0.0834491i
$633$	−436.752			−0.689972
$634$	428.932			0.676549
$635$			80.9920i			0.127546i
$636$		−	99.3698i		−	0.156242i
$637$	447.197			0.702036
$638$		−	231.551i		−	0.362933i
$639$	−365.638			−0.572203
$640$			25.2982i			0.0395285i
$641$			745.251i			1.16264i	0.813676	+	0.581319i	$0.197463\pi$
$641$			745.251i			1.16264i	−0.813676	+	0.581319i	$0.802537\pi$
$642$		−	354.452i		−	0.552106i
$643$		−	371.944i		−	0.578450i	−0.957261	−	0.289225i	$-0.906602\pi$
$643$		−	371.944i		−	0.578450i	0.957261	−	0.289225i	$-0.0933976\pi$
$644$	151.035	−	88.7034i	0.234526	−	0.137738i
$645$	132.242			0.205027
$646$	261.748			0.405183
$647$	−772.723			−1.19432			−0.597159	−	0.802123i	$-0.703704\pi$
$647$	−772.723			−1.19432			−0.597159	+	0.802123i	$0.703704\pi$
$648$	25.4558			0.0392837
$649$			392.477i			0.604741i
$650$	−91.6540			−0.141006
$651$		−	192.931i		−	0.296361i
$652$	130.528			0.200197
$653$	42.1242			0.0645088			0.0322544	−	0.999480i	$-0.489731\pi$
$653$	42.1242			0.0645088			0.0322544	+	0.999480i	$0.489731\pi$
$654$		−	398.183i		−	0.608843i
$655$		−	1.14643i		−	0.00175027i
$656$	221.217			0.337221
$657$	−396.805			−0.603965
$658$			265.466i			0.403443i
$659$			1033.80i			1.56873i	0.620297	+	0.784367i	$0.287012\pi$
$659$			1033.80i			1.56873i	−0.620297	+	0.784367i	$0.712988\pi$
$660$	−125.350			−0.189925
$661$		−	512.065i		−	0.774683i	−0.921936	−	0.387341i	$-0.873393\pi$
$661$		−	512.065i		−	0.774683i	0.921936	−	0.387341i	$-0.126607\pi$
$662$	−397.615			−0.600628
$663$			242.891i			0.366351i
$664$			130.981i			0.197261i
$665$		−	145.659i		−	0.219036i
$666$		−	13.0054i		−	0.0195276i
$667$	−117.849	−	200.660i	−0.176685	−	0.300840i
$668$	−179.933			−0.269361
$669$	−390.874			−0.584265
$670$	−23.6730			−0.0353329
$671$	−857.315			−1.27767
$672$		−	37.3082i		−	0.0555181i
$673$	611.507			0.908628			0.454314	−	0.890842i	$-0.349884\pi$
$673$	611.507			0.908628			0.454314	+	0.890842i	$0.349884\pi$
$674$		−	365.588i		−	0.542415i
$675$	−25.9808			−0.0384900
$676$	−1.98201			−0.00293197
$677$		−	109.122i		−	0.161184i	−0.996747	−	0.0805920i	$-0.974319\pi$
$677$		−	109.122i		−	0.161184i	0.996747	−	0.0805920i	$-0.0256811\pi$
$678$		−	347.589i		−	0.512668i
$679$	−119.939			−0.176641
$680$	−68.4250			−0.100625
$681$		−	687.643i		−	1.00975i
$682$			669.480i			0.981643i
$683$	−426.220			−0.624042			−0.312021	−	0.950075i	$-0.601006\pi$
$683$	−426.220			−0.624042			−0.312021	+	0.950075i	$0.601006\pi$
$684$		−	102.644i		−	0.150065i
$685$	144.886			0.211513
$686$		−	449.651i		−	0.655467i
$687$		−	466.593i		−	0.679175i
$688$		−	136.579i		−	0.198516i
$689$		−	371.818i		−	0.539649i
$690$	−108.627	+	63.7974i	−0.157431	+	0.0924600i
$691$	152.559			0.220780			0.110390	−	0.993888i	$-0.464790\pi$
$691$	152.559			0.220780			0.110390	+	0.993888i	$0.464790\pi$
$692$	−355.338			−0.513494
$693$	184.859			0.266751
$694$	867.354			1.24979
$695$		−	380.245i		−	0.547115i
$696$	−49.5665			−0.0712162
$697$			598.333i			0.858440i
$698$	−106.792			−0.152997
$699$	606.701			0.867956
$700$			38.0775i			0.0543964i
$701$		−	828.197i		−	1.18145i	−0.806873	−	0.590725i	$-0.798842\pi$
$701$		−	828.197i		−	1.18145i	0.806873	−	0.590725i	$-0.201158\pi$
$702$	95.2496			0.135683
$703$	−52.4410			−0.0745960
$704$			129.461i			0.183894i
$705$		−	190.928i		−	0.270820i
$706$	−44.5872			−0.0631547
$707$		−	45.2447i		−	0.0639954i
$708$	84.0145			0.118665
$709$			319.241i			0.450269i	0.974328	+	0.225134i	$0.0722821\pi$
$709$			319.241i			0.450269i	−0.974328	+	0.225134i	$0.927718\pi$
$710$		−	385.416i		−	0.542840i
$711$		−	55.9390i		−	0.0786765i
$712$			48.7467i			0.0684645i
$713$	340.734	+	580.164i	0.477887	+	0.813695i
$714$	100.909			0.141329
$715$	−469.031			−0.655987
$716$	−386.912			−0.540379
$717$	60.2076			0.0839716
$718$		−	272.737i		−	0.379857i
$719$	−894.454			−1.24402			−0.622012	−	0.783008i	$-0.713685\pi$
$719$	−894.454			−1.24402			−0.622012	+	0.783008i	$0.713685\pi$
$720$			26.8328i			0.0372678i
$721$	232.418			0.322355
$722$	96.6425			0.133854
$723$		−	595.208i		−	0.823248i
$724$			372.708i			0.514790i
$725$	50.5886			0.0697773
$726$	−345.081			−0.475319
$727$		−	1349.88i		−	1.85679i	−0.371601	−	0.928393i	$-0.621191\pi$
$727$		−	1349.88i		−	1.85679i	0.371601	−	0.928393i	$-0.378809\pi$
$728$		−	139.598i		−	0.191756i
$729$	27.0000			0.0370370
$730$		−	418.269i		−	0.572971i
$731$	369.410			0.505349
$732$			183.519i			0.250709i
$733$		−	584.473i		−	0.797371i	−0.917088	−	0.398686i	$-0.869466\pi$
$733$		−	584.473i		−	0.797371i	0.917088	−	0.398686i	$-0.130534\pi$
$734$		−	724.522i		−	0.987087i
$735$			133.622i			0.181799i
$736$	65.8896	+	112.190i	0.0895240	+	0.152432i
$737$	−121.145			−0.164375
$738$	234.636			0.317935
$739$	100.840			0.136454			0.0682272	−	0.997670i	$-0.478266\pi$
$739$	100.840			0.136454			0.0682272	+	0.997670i	$0.478266\pi$
$740$	13.7089			0.0185255
$741$		−	384.071i		−	0.518314i
$742$	−154.471			−0.208182
$743$		−	1214.22i		−	1.63422i	−0.576482	−	0.817110i	$-0.695575\pi$
$743$		−	1214.22i		−	1.63422i	0.576482	−	0.817110i	$-0.304425\pi$
$744$	143.311			0.192622
$745$	−122.212			−0.164043
$746$			435.857i			0.584259i
$747$			138.926i			0.185979i
$748$	−350.159			−0.468126
$749$	−550.999			−0.735646
$750$		−	27.3861i		−	0.0365148i
$751$		−	209.572i		−	0.279057i	−0.990218	−	0.139529i	$-0.955441\pi$
$751$		−	209.572i		−	0.279057i	0.990218	−	0.139529i	$-0.0445587\pi$
$752$	−197.190			−0.262221
$753$			766.305i			1.01767i
$754$	−185.466			−0.245976
$755$			294.219i			0.389694i
$756$		−	39.5713i		−	0.0523430i
$757$			1291.76i			1.70643i	0.521563	+	0.853213i	$0.325349\pi$
$757$			1291.76i			1.70643i	−0.521563	+	0.853213i	$0.674651\pi$
$758$			387.497i			0.511209i
$759$	−555.891	+	326.477i	−0.732399	+	0.430141i
$760$	108.197			0.142364
$761$	−1137.06			−1.49417			−0.747083	−	0.664731i	$-0.768546\pi$
$761$	−1137.06			−1.49417			−0.747083	+	0.664731i	$0.768546\pi$
$762$	88.7223			0.116433
$763$	−618.978			−0.811243
$764$		−	37.5355i		−	0.0491303i
$765$	−72.5756			−0.0948701
$766$			261.012i			0.340747i
$767$	314.362			0.409859
$768$	27.7128			0.0360844
$769$			1211.18i			1.57501i	0.616310	+	0.787504i	$0.288627\pi$
$769$			1211.18i			1.57501i	−0.616310	+	0.787504i	$0.711373\pi$
$770$			194.858i			0.253062i
$771$	654.641			0.849080
$772$	−323.543			−0.419097
$773$		−	1006.14i		−	1.30160i	−0.759247	−	0.650802i	$-0.774433\pi$
$773$		−	1006.14i		−	1.30160i	0.759247	−	0.650802i	$-0.225567\pi$
$774$		−	144.864i		−	0.187163i
$775$	−146.266			−0.188730
$776$		−	89.0917i		−	0.114809i
$777$	−20.2169			−0.0260192
$778$			543.761i			0.698922i
$779$		−	946.112i		−	1.21452i
$780$			100.402i			0.128720i
$781$		−	1972.33i		−	2.52539i
$782$	−303.444	+	178.214i	−0.388035	+	0.227895i
$783$	−52.5732			−0.0671433
$784$	138.004			0.176026
$785$	140.217			0.178621
$786$	−1.25585			−0.00159777
$787$			837.892i			1.06467i	0.846535	+	0.532333i	$0.178685\pi$
$787$			837.892i			1.06467i	−0.846535	+	0.532333i	$0.821315\pi$
$788$	124.769			0.158337
$789$		−	403.714i		−	0.511677i
$790$	58.9649			0.0746391
$791$	−540.330			−0.683097
$792$			137.315i			0.173377i
$793$			686.683i			0.865931i
$794$	481.582			0.606526
$795$	111.099			0.139747
$796$		−	561.351i		−	0.705214i
$797$			866.601i			1.08733i	0.839303	+	0.543665i	$0.182964\pi$
$797$			866.601i			1.08733i	−0.839303	+	0.543665i	$0.817036\pi$
$798$	−159.562			−0.199952
$799$		−	533.346i		−	0.667518i
$800$	−28.2843			−0.0353553
$801$			51.7037i			0.0645490i
$802$			1050.11i			1.30936i
$803$		−	2140.45i		−	2.66557i
$804$			25.9325i			0.0322544i
$805$	99.1735	+	168.862i	0.123197	+	0.209766i
$806$	536.233			0.665302
$807$	−482.121			−0.597424
$808$	33.6082			0.0415943
$809$	−1290.76			−1.59550			−0.797749	−	0.602990i	$-0.793976\pi$
$809$	−1290.76			−1.59550			−0.797749	+	0.602990i	$0.793976\pi$
$810$			28.4605i			0.0351364i
$811$	−526.205			−0.648835			−0.324418	−	0.945914i	$-0.605168\pi$
$811$	−526.205			−0.648835			−0.324418	+	0.945914i	$0.605168\pi$
$812$			77.0514i			0.0948909i
$813$	−881.245			−1.08394
$814$	70.1539			0.0861842
$815$			145.935i			0.179061i
$816$			74.9558i			0.0918576i
$817$	−584.129			−0.714968
$818$	1071.09			1.30940
$819$		−	148.066i		−	0.180789i
$820$			247.328i			0.301619i
$821$	−445.751			−0.542937			−0.271468	−	0.962447i	$-0.587509\pi$
$821$	−445.751			−0.542937			−0.271468	+	0.962447i	$0.587509\pi$
$822$		−	158.715i		−	0.193084i
$823$	−1114.24			−1.35388			−0.676938	−	0.736040i	$-0.736693\pi$
$823$	−1114.24			−1.35388			−0.676938	+	0.736040i	$0.736693\pi$
$824$			172.642i			0.209517i
$825$		−	140.146i		−	0.169874i
$826$		−	130.601i		−	0.158113i
$827$		−	426.354i		−	0.515543i	−0.966206	−	0.257771i	$-0.917012\pi$
$827$		−	426.354i		−	0.515543i	0.966206	−	0.257771i	$-0.0829882\pi$
$828$	69.8865	+	118.995i	0.0844040	+	0.143714i
$829$	413.046			0.498246			0.249123	−	0.968472i	$-0.419858\pi$
$829$	413.046			0.498246			0.249123	+	0.968472i	$0.419858\pi$
$830$	−146.441			−0.176435
$831$	−34.1611			−0.0411084
$832$	103.695			0.124633
$833$			373.265i			0.448097i
$834$	−416.538			−0.499446
$835$		−	201.172i		−	0.240924i
$836$	553.687			0.662305
$837$	152.004			0.181606
$838$		−	302.270i		−	0.360704i
$839$		−	1356.51i		−	1.61682i	−0.588619	−	0.808411i	$-0.700328\pi$
$839$		−	1356.51i		−	1.61682i	0.588619	−	0.808411i	$-0.299672\pi$
$840$	41.7118			0.0496569
$841$	−738.632			−0.878278
$842$			955.989i			1.13538i
$843$			594.878i			0.705668i
$844$	−504.318			−0.597533
$845$		−	2.21596i		−	0.00262243i
$846$	−209.152			−0.247224
$847$			536.431i			0.633331i
$848$		−	114.742i		−	0.135309i
$849$			258.995i			0.305059i
$850$		−	76.5014i		−	0.0900017i
$851$	60.7946	−	35.7050i	0.0714390	−	0.0419565i
$852$	−422.202			−0.495543
$853$	−1215.79			−1.42531			−0.712657	−	0.701512i	$-0.752509\pi$
$853$	−1215.79			−1.42531			−0.712657	+	0.701512i	$0.752509\pi$
$854$	285.281			0.334053
$855$	114.760			0.134222
$856$		−	409.286i		−	0.478138i
$857$	−321.853			−0.375558			−0.187779	−	0.982211i	$-0.560129\pi$
$857$	−321.853			−0.375558			−0.187779	+	0.982211i	$0.560129\pi$
$858$			513.798i			0.598832i
$859$	217.358			0.253036			0.126518	−	0.991964i	$-0.459620\pi$
$859$	217.358			0.253036			0.126518	+	0.991964i	$0.459620\pi$
$860$	152.700			0.177558
$861$		−	364.743i		−	0.423627i
$862$			11.3796i			0.0132014i
$863$	−1162.38			−1.34690			−0.673451	−	0.739232i	$-0.735189\pi$
$863$	−1162.38			−1.34690			−0.673451	+	0.739232i	$0.735189\pi$
$864$	29.3939			0.0340207
$865$		−	397.280i		−	0.459283i
$866$			522.366i			0.603194i
$867$	297.827			0.343515
$868$		−	222.777i		−	0.256656i
$869$	301.748			0.347235
$870$		−	55.4170i		−	0.0636977i
$871$			97.0332i			0.111404i
$872$		−	459.782i		−	0.527273i
$873$		−	94.4960i		−	0.108243i
$874$	479.819	−	281.800i	0.548992	−	0.322426i
$875$	−42.5719			−0.0486536
$876$	−458.191			−0.523049
$877$	−143.677			−0.163828			−0.0819139	−	0.996639i	$-0.526103\pi$
$877$	−143.677			−0.163828			−0.0819139	+	0.996639i	$0.526103\pi$
$878$	−76.8027			−0.0874746
$879$			138.970i			0.158100i
$880$	−144.742			−0.164480
$881$		−	26.5389i		−	0.0301237i	−0.999887	−	0.0150618i	$-0.995205\pi$
$881$		−	26.5389i		−	0.0301237i	0.999887	−	0.0150618i	$-0.00479451\pi$
$882$	146.376			0.165959
$883$	1442.75			1.63391			0.816957	−	0.576698i	$-0.195659\pi$
$883$	1442.75			1.63391			0.816957	+	0.576698i	$0.195659\pi$
$884$			280.466i			0.317270i
$885$			93.9311i			0.106137i
$886$	372.744			0.420704
$887$	−373.153			−0.420691			−0.210345	−	0.977627i	$-0.567459\pi$
$887$	−373.153			−0.420691			−0.210345	+	0.977627i	$0.567459\pi$
$888$		−	15.0173i		−	0.0169114i
$889$		−	137.919i		−	0.155140i
$890$	−54.5005			−0.0612365
$891$			145.644i			0.163461i
$892$	−451.342			−0.505989
$893$			843.353i			0.944404i
$894$			133.876i			0.149750i
$895$		−	432.580i		−	0.483330i
$896$		−	43.0798i		−	0.0480801i
$897$	261.498	+	445.251i	0.291526	+	0.496378i
$898$	818.020			0.910935
$899$	−295.975			−0.329227
$900$	−30.0000			−0.0333333
$901$	310.348			0.344448
$902$			1265.68i			1.40319i
$903$	−225.192			−0.249382
$904$		−	401.361i		−	0.443984i
$905$	−416.700			−0.460442
$906$	322.300			0.355740
$907$		−	488.681i		−	0.538788i	−0.963030	−	0.269394i	$-0.913177\pi$
$907$		−	488.681i		−	0.538788i	0.963030	−	0.269394i	$-0.0868234\pi$
$908$		−	794.022i		−	0.874473i
$909$	35.6468			0.0392155
$910$	156.075			0.171511
$911$		−	978.988i		−	1.07463i	−0.843382	−	0.537315i	$-0.819439\pi$
$911$		−	978.988i		−	1.07463i	0.843382	−	0.537315i	$-0.180561\pi$
$912$		−	118.524i		−	0.129960i
$913$	−749.400			−0.820811
$914$		−	727.261i		−	0.795690i
$915$	−205.180			−0.224241
$916$		−	538.775i		−	0.588183i
$917$			1.95223i			0.00212893i
$918$			79.5026i			0.0866041i
$919$		−	346.891i		−	0.377466i	−0.982028	−	0.188733i	$-0.939562\pi$
$919$		−	346.891i		−	0.377466i	0.982028	−	0.188733i	$-0.0604381\pi$
$920$	−125.432	+	73.6669i	−0.136339	+	0.0800727i
$921$	266.354			0.289201
$922$	771.732			0.837019
$923$	−1579.78			−1.71157
$924$	213.456			0.231013
$925$			15.3270i			0.0165697i
$926$	−681.489			−0.735949
$927$			183.114i			0.197534i
$928$	−57.2344			−0.0616750
$929$	700.599			0.754143			0.377072	−	0.926184i	$-0.376931\pi$
$929$	700.599			0.754143			0.377072	+	0.926184i	$0.376931\pi$
$930$			160.226i			0.172286i
$931$		−	590.224i		−	0.633967i
$932$	700.558			0.751672
$933$	−215.756			−0.231249
$934$		−	9.51426i		−	0.0101866i
$935$		−	391.489i		−	0.418705i
$936$	109.985			0.117505
$937$		−	1140.57i		−	1.21726i	−0.793455	−	0.608628i	$-0.791720\pi$
$937$		−	1140.57i		−	1.21726i	0.793455	−	0.608628i	$-0.208280\pi$
$938$	40.3123			0.0429768
$939$			385.249i			0.410276i
$940$		−	220.465i		−	0.234537i
$941$		−	561.160i		−	0.596344i	−0.954512	−	0.298172i	$-0.903623\pi$
$941$		−	561.160i		−	0.596344i	0.954512	−	0.298172i	$-0.0963769\pi$
$942$		−	153.601i		−	0.163058i
$943$	644.170	+	1096.82i	0.683107	+	1.16312i
$944$	97.0116			0.102767
$945$	44.2420			0.0468170
$946$	781.430			0.826036
$947$	−974.501			−1.02904			−0.514520	−	0.857478i	$-0.672030\pi$
$947$	−974.501			−1.02904			−0.514520	+	0.857478i	$0.672030\pi$
$948$		−	64.5928i		−	0.0681359i
$949$	−1714.44			−1.80657
$950$			120.968i			0.127334i
$951$	525.333			0.552400
$952$	116.519			0.122394
$953$		−	245.849i		−	0.257974i	−0.991646	−	0.128987i	$-0.958827\pi$
$953$		−	245.849i		−	0.257974i	0.991646	−	0.128987i	$-0.0411725\pi$
$954$		−	121.703i		−	0.127571i
$955$	41.9660			0.0439434
$956$	69.5218			0.0727215
$957$		−	283.591i		−	0.296334i
$958$			518.101i			0.540815i
$959$	−246.724			−0.257272
$960$			30.9839i			0.0322749i
$961$	−105.253			−0.109525
$962$		−	56.1911i		−	0.0584108i
$963$		−	434.114i		−	0.450793i
$964$		−	687.287i		−	0.712954i
$965$		−	361.732i		−	0.374852i
$966$	184.979	−	108.639i	0.191490	−	0.112463i
$967$	−958.537			−0.991248			−0.495624	−	0.868537i	$-0.665061\pi$
$967$	−958.537			−0.991248			−0.495624	+	0.868537i	$0.665061\pi$
$968$	−398.465			−0.411638
$969$	320.575			0.330831
$970$	99.6075			0.102688
$971$			797.683i			0.821506i	0.911747	+	0.410753i	$0.134734\pi$
$971$			797.683i			0.821506i	−0.911747	+	0.410753i	$0.865266\pi$
$972$	31.1769			0.0320750
$973$			647.511i			0.665479i
$974$	299.927			0.307933
$975$	−112.253			−0.115131
$976$			211.909i			0.217120i
$977$			968.304i			0.991099i	0.868580	+	0.495550i	$0.165033\pi$
$977$			968.304i			0.991099i	−0.868580	+	0.495550i	$0.834967\pi$
$978$	159.864			0.163460
$979$	−278.901			−0.284884
$980$			154.293i			0.157442i
$981$		−	487.673i		−	0.497118i
$982$	−457.968			−0.466363
$983$			242.603i			0.246799i	0.992357	+	0.123399i	$0.0393796\pi$
$983$			242.603i			0.246799i	−0.992357	+	0.123399i	$0.960620\pi$
$984$	270.934			0.275340
$985$			139.496i			0.141621i
$986$		−	154.804i		−	0.157002i
$987$			325.128i			0.329410i
$988$		−	443.487i		−	0.448873i
$989$	677.178	−	397.710i	0.684710	−	0.402134i
$990$	−153.522			−0.155073
$991$	1231.78			1.24297			0.621483	−	0.783428i	$-0.286530\pi$
$991$	1231.78			1.24297			0.621483	+	0.783428i	$0.286530\pi$
$992$	165.481			0.166815
$993$	−486.978			−0.490410
$994$			656.316i			0.660278i
$995$	627.609			0.630763
$996$			160.418i			0.161063i
$997$	−689.979			−0.692055			−0.346027	−	0.938224i	$-0.612470\pi$
$997$	−689.979			−0.692055			−0.346027	+	0.938224i	$0.612470\pi$
$998$	159.826			0.160146
$999$		−	15.9283i		−	0.0159442i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	690.3.c.a.91.30	yes	32
3.2	odd	2		2070.3.c.b.91.3		32
23.22	odd	2	inner	690.3.c.a.91.27	✓	32
69.68	even	2		2070.3.c.b.91.14		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.3.c.a.91.27	✓	32	23.22	odd	2	inner
690.3.c.a.91.30	yes	32	1.1	even	1	trivial
2070.3.c.b.91.3		32	3.2	odd	2
2070.3.c.b.91.14		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)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +1.73205 q^{3} +2.00000 q^{4} +2.23607i q^{5} +2.44949 q^{6} -3.80775i q^{7} +2.82843 q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q+1.41421 q^{2} +1.73205 q^{3} +2.00000 q^{4} +2.23607i q^{5} +2.44949 q^{6} -3.80775i q^{7} +2.82843 q^{8} +3.00000 q^{9} +3.16228i q^{10} +16.1827i q^{11} +3.46410 q^{12} +12.9618 q^{13} -5.38497i q^{14} +3.87298i q^{15} +4.00000 q^{16} +10.8189i q^{17} +4.24264 q^{18} -17.1074i q^{19} +4.47214i q^{20} -6.59521i q^{21} +22.8858i q^{22} +(11.6478 + 19.8325i) q^{23} +4.89898 q^{24} -5.00000 q^{25} +18.3308 q^{26} +5.19615 q^{27} -7.61550i q^{28} -10.1177 q^{29} +5.47723i q^{30} +29.2531 q^{31} +5.65685 q^{32} +28.0292i q^{33} +15.3003i q^{34} +8.51438 q^{35} +6.00000 q^{36} -3.06540i q^{37} -24.1935i q^{38} +22.4505 q^{39} +6.32456i q^{40} +55.3042 q^{41} -9.32704i q^{42} -34.1448i q^{43} +32.3653i q^{44} +6.70820i q^{45} +(16.4724 + 28.0475i) q^{46} -49.2975 q^{47} +6.92820 q^{48} +34.5011 q^{49} -7.07107 q^{50} +18.7389i q^{51} +25.9237 q^{52} -28.6856i q^{53} +7.34847 q^{54} -36.1856 q^{55} -10.7699i q^{56} -29.6309i q^{57} -14.3086 q^{58} +24.2529 q^{59} +7.74597i q^{60} +52.9773i q^{61} +41.3702 q^{62} -11.4232i q^{63} +8.00000 q^{64} +28.9835i q^{65} +39.6393i q^{66} +7.48607i q^{67} +21.6379i q^{68} +(20.1745 + 34.3510i) q^{69} +12.0412 q^{70} -121.879 q^{71} +8.48528 q^{72} -132.268 q^{73} -4.33513i q^{74} -8.66025 q^{75} -34.2148i q^{76} +61.6195 q^{77} +31.7499 q^{78} -18.6463i q^{79} +8.94427i q^{80} +9.00000 q^{81} +78.2120 q^{82} +46.3088i q^{83} -13.1904i q^{84} -24.1919 q^{85} -48.2881i q^{86} -17.5244 q^{87} +45.7715i q^{88} +17.2346i q^{89} +9.48683i q^{90} -49.3554i q^{91} +(23.2955 + 39.6651i) q^{92} +50.6679 q^{93} -69.7172 q^{94} +38.2533 q^{95} +9.79796 q^{96} -31.4987i q^{97} +48.7919 q^{98} +48.5480i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 64 q^{4} + 96 q^{9}+O(q^{10}) \) 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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