LMFDB - Embedded newform 2070.3.c.b.91.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2070,3,Mod(91,2070)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2070, 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("2070.91");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2070.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:	$56.4034147226$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 690)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			91.19
Character	$\chi$	$=$	2070.91
Dual form			2070.3.c.b.91.30

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} +2.00000 q^{4} -2.23607i q^{5} -1.52229i q^{7} +2.82843 q^{8} +O(q^{10})$	\(q+1.41421 q^{2} +2.00000 q^{4} -2.23607i q^{5} -1.52229i q^{7} +2.82843 q^{8} -3.16228i q^{10} -2.20036i q^{11} -1.82994 q^{13} -2.15284i q^{14} +4.00000 q^{16} +23.2496i q^{17} +2.85690i q^{19} -4.47214i q^{20} -3.11178i q^{22} +(14.5452 - 17.8168i) q^{23} -5.00000 q^{25} -2.58793 q^{26} -3.04458i q^{28} +7.23088 q^{29} +49.9104 q^{31} +5.65685 q^{32} +32.8800i q^{34} -3.40394 q^{35} +9.68006i q^{37} +4.04027i q^{38} -6.32456i q^{40} +62.1103 q^{41} -76.6308i q^{43} -4.40073i q^{44} +(20.5700 - 25.1967i) q^{46} -17.1464 q^{47} +46.6826 q^{49} -7.07107 q^{50} -3.65988 q^{52} -33.9334i q^{53} -4.92016 q^{55} -4.30569i q^{56} +10.2260 q^{58} -35.2164 q^{59} +8.54346i q^{61} +70.5840 q^{62} +8.00000 q^{64} +4.09187i q^{65} -78.9586i q^{67} +46.4993i q^{68} -4.81390 q^{70} -111.407 q^{71} +134.217 q^{73} +13.6897i q^{74} +5.71380i q^{76} -3.34959 q^{77} -72.4039i q^{79} -8.94427i q^{80} +87.8373 q^{82} +119.919i q^{83} +51.9878 q^{85} -108.372i q^{86} -6.22357i q^{88} -73.9324i q^{89} +2.78570i q^{91} +(29.0904 - 35.6335i) q^{92} -24.2486 q^{94} +6.38822 q^{95} -178.439i q^{97} +66.0192 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 64 q^{4}+O(q^{10}) $ 32 * q + 64 * q^4	$ 32 q + 64 q^{4} - 48 q^{13} + 128 q^{16} + 80 q^{23} - 160 q^{25} - 120 q^{29} + 248 q^{31} + 120 q^{35} - 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} - 104 q^{71} + 16 q^{73} - 240 q^{77} + 64 q^{82} - 120 q^{85} + 160 q^{92} + 96 q^{94} + 160 q^{95} - 64 q^{98}+O(q^{100}) $ 32 * q + 64 * q^4 - 48 * q^13 + 128 * q^16 + 80 * q^23 - 160 * q^25 - 120 * q^29 + 248 * q^31 + 120 * q^35 - 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 - 104 * q^71 + 16 * q^73 - 240 * q^77 + 64 * q^82 - 120 * q^85 + 160 * q^92 + 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 $\left(\mathbb{Z}/2070\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$1891$
$\chi(n)$	$1$	$1$	$-1$

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
$3$		0			0
$3$		0			0
$4$	2.00000			0.500000
$5$		−	2.23607i		−	0.447214i
$5$		−	2.23607i		−	0.447214i
$6$		0			0
$7$		−	1.52229i		−	0.217470i	−0.994071	−	0.108735i	$-0.965320\pi$
$7$		−	1.52229i		−	0.217470i	0.994071	−	0.108735i	$-0.0346800\pi$
$8$	2.82843			0.353553
$9$		0			0
$10$		−	3.16228i		−	0.316228i
$11$		−	2.20036i		−	0.200033i	−0.994986	−	0.100016i	$-0.968110\pi$
$11$		−	2.20036i		−	0.200033i	0.994986	−	0.100016i	$-0.0318896\pi$
$12$		0			0
$13$	−1.82994			−0.140765			−0.0703824	−	0.997520i	$-0.522422\pi$
$13$	−1.82994			−0.140765			−0.0703824	+	0.997520i	$0.522422\pi$
$14$		−	2.15284i		−	0.153774i
$15$		0			0
$16$	4.00000			0.250000
$17$			23.2496i			1.36763i	0.729657	+	0.683813i	$0.239680\pi$
$17$			23.2496i			1.36763i	−0.729657	+	0.683813i	$0.760320\pi$
$18$		0			0
$19$			2.85690i			0.150363i	0.997170	+	0.0751816i	$0.0239536\pi$
$19$			2.85690i			0.150363i	−0.997170	+	0.0751816i	$0.976046\pi$
$20$		−	4.47214i		−	0.223607i
$21$		0			0
$22$		−	3.11178i		−	0.141445i
$23$	14.5452	−	17.8168i	0.632400	−	0.774642i
$23$	14.5452	−	17.8168i	0.632400	−	0.774642i
$24$		0			0
$25$	−5.00000			−0.200000
$26$	−2.58793			−0.0995357
$27$		0			0
$28$		−	3.04458i		−	0.108735i
$29$	7.23088			0.249341			0.124670	−	0.992198i	$-0.460213\pi$
$29$	7.23088			0.249341			0.124670	+	0.992198i	$0.460213\pi$
$30$		0			0
$31$	49.9104			1.61001			0.805007	−	0.593266i	$-0.202162\pi$
$31$	49.9104			1.61001			0.805007	+	0.593266i	$0.202162\pi$
$32$	5.65685			0.176777
$33$		0			0
$34$			32.8800i			0.967058i
$35$	−3.40394			−0.0972555
$36$		0			0
$37$			9.68006i			0.261623i	0.991407	+	0.130812i	$0.0417583\pi$
$37$			9.68006i			0.261623i	−0.991407	+	0.130812i	$0.958242\pi$
$38$			4.04027i			0.106323i
$39$		0			0
$40$		−	6.32456i		−	0.158114i
$41$	62.1103			1.51489			0.757443	−	0.652901i	$-0.226448\pi$
$41$	62.1103			1.51489			0.757443	+	0.652901i	$0.226448\pi$
$42$		0			0
$43$		−	76.6308i		−	1.78211i	−0.453893	−	0.891056i	$-0.649965\pi$
$43$		−	76.6308i		−	1.78211i	0.453893	−	0.891056i	$-0.350035\pi$
$44$		−	4.40073i		−	0.100016i
$45$		0			0
$46$	20.5700	−	25.1967i	0.447174	−	0.547755i
$47$	−17.1464			−0.364817			−0.182408	−	0.983223i	$-0.558389\pi$
$47$	−17.1464			−0.364817			−0.182408	+	0.983223i	$0.558389\pi$
$48$		0			0
$49$	46.6826			0.952707
$50$	−7.07107			−0.141421
$51$		0			0
$52$	−3.65988			−0.0703824
$53$		−	33.9334i		−	0.640253i	−0.947375	−	0.320127i	$-0.896275\pi$
$53$		−	33.9334i		−	0.640253i	0.947375	−	0.320127i	$-0.103725\pi$
$54$		0			0
$55$	−4.92016			−0.0894575
$56$		−	4.30569i		−	0.0768872i
$57$		0			0
$58$	10.2260			0.176311
$59$	−35.2164			−0.596889			−0.298444	−	0.954427i	$-0.596468\pi$
$59$	−35.2164			−0.596889			−0.298444	+	0.954427i	$0.596468\pi$
$60$		0			0
$61$			8.54346i			0.140057i	0.997545	+	0.0700284i	$0.0223090\pi$
$61$			8.54346i			0.140057i	−0.997545	+	0.0700284i	$0.977691\pi$
$62$	70.5840			1.13845
$63$		0			0
$64$	8.00000			0.125000
$65$			4.09187i			0.0629519i
$66$		0			0
$67$		−	78.9586i		−	1.17849i	−0.807956	−	0.589243i	$-0.799426\pi$
$67$		−	78.9586i		−	1.17849i	0.807956	−	0.589243i	$-0.200574\pi$
$68$			46.4993i			0.683813i
$69$		0			0
$70$	−4.81390			−0.0687700
$71$	−111.407			−1.56911			−0.784554	−	0.620061i	$-0.787108\pi$
$71$	−111.407			−1.56911			−0.784554	+	0.620061i	$0.787108\pi$
$72$		0			0
$73$	134.217			1.83858			0.919291	−	0.393578i	$-0.128763\pi$
$73$	134.217			1.83858			0.919291	+	0.393578i	$0.128763\pi$
$74$			13.6897i			0.184996i
$75$		0			0
$76$			5.71380i			0.0751816i
$77$	−3.34959			−0.0435012
$78$		0			0
$79$		−	72.4039i		−	0.916505i	−0.888822	−	0.458252i	$-0.848476\pi$
$79$		−	72.4039i		−	0.916505i	0.888822	−	0.458252i	$-0.151524\pi$
$80$		−	8.94427i		−	0.111803i
$81$		0			0
$82$	87.8373			1.07119
$83$			119.919i			1.44481i	0.691472	+	0.722403i	$0.256963\pi$
$83$			119.919i			1.44481i	−0.691472	+	0.722403i	$0.743037\pi$
$84$		0			0
$85$	51.9878			0.611621
$86$		−	108.372i		−	1.26014i
$87$		0			0
$88$		−	6.22357i		−	0.0707223i
$89$		−	73.9324i		−	0.830701i	−0.909661	−	0.415351i	$-0.863659\pi$
$89$		−	73.9324i		−	0.830701i	0.909661	−	0.415351i	$-0.136341\pi$
$90$		0			0
$91$			2.78570i			0.0306121i
$92$	29.0904	−	35.6335i	0.316200	−	0.387321i
$93$		0			0
$94$	−24.2486			−0.257964
$95$	6.38822			0.0672445
$96$		0			0
$97$		−	178.439i		−	1.83958i	−0.392408	−	0.919791i	$-0.628358\pi$
$97$		−	178.439i		−	1.83958i	0.392408	−	0.919791i	$-0.371642\pi$
$98$	66.0192			0.673665
$99$		0			0
$100$	−10.0000			−0.100000
$101$	−147.269			−1.45811			−0.729053	−	0.684457i	$-0.760039\pi$
$101$	−147.269			−1.45811			−0.729053	+	0.684457i	$0.760039\pi$
$102$		0			0
$103$		−	58.9552i		−	0.572380i	−0.958173	−	0.286190i	$-0.907611\pi$
$103$		−	58.9552i		−	0.572380i	0.958173	−	0.286190i	$-0.0923889\pi$
$104$	−5.17586			−0.0497679
$105$		0			0
$106$		−	47.9891i		−	0.452727i
$107$			111.196i			1.03922i	0.854404	+	0.519609i	$0.173922\pi$
$107$			111.196i			1.03922i	−0.854404	+	0.519609i	$0.826078\pi$
$108$		0			0
$109$		−	52.7876i		−	0.484290i	−0.970240	−	0.242145i	$-0.922149\pi$
$109$		−	52.7876i		−	0.484290i	0.970240	−	0.242145i	$-0.0778509\pi$
$110$	−6.95816			−0.0632560
$111$		0			0
$112$		−	6.08916i		−	0.0543675i
$113$		−	25.4121i		−	0.224886i	−0.993658	−	0.112443i	$-0.964132\pi$
$113$		−	25.4121i		−	0.224886i	0.993658	−	0.112443i	$-0.0358676\pi$
$114$		0			0
$115$	−39.8395	−	32.5240i	−0.346431	−	0.282818i
$116$	14.4618			0.124670
$117$		0			0
$118$	−49.8035			−0.422064
$119$	35.3927			0.297418
$120$		0			0
$121$	116.158			0.959987
$122$			12.0823i			0.0990350i
$123$		0			0
$124$	99.8208			0.805007
$125$			11.1803i			0.0894427i
$126$		0			0
$127$	100.861			0.794177			0.397089	−	0.917780i	$-0.370021\pi$
$127$	100.861			0.794177			0.397089	+	0.917780i	$0.370021\pi$
$128$	11.3137			0.0883883
$129$		0			0
$130$			5.78678i			0.0445137i
$131$	28.6539			0.218732			0.109366	−	0.994002i	$-0.465118\pi$
$131$	28.6539			0.218732			0.109366	+	0.994002i	$0.465118\pi$
$132$		0			0
$133$	4.34903			0.0326995
$134$		−	111.664i		−	0.833316i
$135$		0			0
$136$			65.7599i			0.483529i
$137$		−	159.048i		−	1.16094i	−0.814282	−	0.580469i	$-0.802869\pi$
$137$		−	159.048i		−	1.16094i	0.814282	−	0.580469i	$-0.197131\pi$
$138$		0			0
$139$	119.112			0.856918			0.428459	−	0.903561i	$-0.359057\pi$
$139$	119.112			0.856918			0.428459	+	0.903561i	$0.359057\pi$
$140$	−6.80789			−0.0486278
$141$		0			0
$142$	−157.553			−1.10953
$143$			4.02654i			0.0281576i
$144$		0			0
$145$		−	16.1687i		−	0.111509i
$146$	189.811			1.30007
$147$		0			0
$148$			19.3601i			0.130812i
$149$			245.833i			1.64989i	0.565214	+	0.824945i	$0.308794\pi$
$149$			245.833i			1.64989i	−0.565214	+	0.824945i	$0.691206\pi$
$150$		0			0
$151$	209.449			1.38708			0.693541	−	0.720417i	$-0.256050\pi$
$151$	209.449			1.38708			0.693541	+	0.720417i	$0.256050\pi$
$152$			8.08053i			0.0531614i
$153$		0			0
$154$	−4.73704			−0.0307600
$155$		−	111.603i		−	0.720020i
$156$		0			0
$157$			211.566i			1.34755i	0.738935	+	0.673777i	$0.235329\pi$
$157$			211.566i			1.34755i	−0.738935	+	0.673777i	$0.764671\pi$
$158$		−	102.395i		−	0.648067i
$159$		0			0
$160$		−	12.6491i		−	0.0790569i
$161$	−27.1223	−	22.1420i	−0.168461	−	0.137528i
$162$		0			0
$163$	−14.1494			−0.0868061			−0.0434030	−	0.999058i	$-0.513820\pi$
$163$	−14.1494			−0.0868061			−0.0434030	+	0.999058i	$0.513820\pi$
$164$	124.221			0.757443
$165$		0			0
$166$			169.591i			1.02163i
$167$	−52.0666			−0.311776			−0.155888	−	0.987775i	$-0.549824\pi$
$167$	−52.0666			−0.311776			−0.155888	+	0.987775i	$0.549824\pi$
$168$		0			0
$169$	−165.651			−0.980185
$170$	73.5218			0.432481
$171$		0			0
$172$		−	153.262i		−	0.891056i
$173$	219.256			1.26738			0.633688	−	0.773589i	$-0.281540\pi$
$173$	219.256			1.26738			0.633688	+	0.773589i	$0.281540\pi$
$174$		0			0
$175$			7.61145i			0.0434940i
$176$		−	8.80145i		−	0.0500082i
$177$		0			0
$178$		−	104.556i		−	0.587395i
$179$	84.3515			0.471237			0.235619	−	0.971846i	$-0.424288\pi$
$179$	84.3515			0.471237			0.235619	+	0.971846i	$0.424288\pi$
$180$		0			0
$181$			272.635i			1.50627i	0.657867	+	0.753134i	$0.271459\pi$
$181$			272.635i			1.50627i	−0.657867	+	0.753134i	$0.728541\pi$
$182$			3.93958i			0.0216460i
$183$		0			0
$184$	41.1400	−	50.3934i	0.223587	−	0.273877i
$185$	21.6453			0.117001
$186$		0			0
$187$	51.1577			0.273570
$188$	−34.2928			−0.182408
$189$		0			0
$190$	9.03431			0.0475490
$191$		−	235.645i		−	1.23375i	−0.787063	−	0.616873i	$-0.788399\pi$
$191$		−	235.645i		−	1.23375i	0.787063	−	0.616873i	$-0.211601\pi$
$192$		0			0
$193$	−340.126			−1.76231			−0.881155	−	0.472828i	$-0.843233\pi$
$193$	−340.126			−1.76231			−0.881155	+	0.472828i	$0.843233\pi$
$194$		−	252.352i		−	1.30078i
$195$		0			0
$196$	93.3653			0.476353
$197$	221.673			1.12524			0.562621	−	0.826715i	$-0.309793\pi$
$197$	221.673			1.12524			0.562621	+	0.826715i	$0.309793\pi$
$198$		0			0
$199$		−	198.065i		−	0.995303i	−0.867377	−	0.497651i	$-0.834196\pi$
$199$		−	198.065i		−	0.995303i	0.867377	−	0.497651i	$-0.165804\pi$
$200$	−14.1421			−0.0707107
$201$		0			0
$202$	−208.269			−1.03104
$203$		−	11.0075i		−	0.0542241i
$204$		0			0
$205$		−	138.883i		−	0.677478i
$206$		−	83.3752i		−	0.404734i
$207$		0			0
$208$	−7.31977			−0.0351912
$209$	6.28622			0.0300776
$210$		0			0
$211$	−105.845			−0.501636			−0.250818	−	0.968034i	$-0.580700\pi$
$211$	−105.845			−0.501636			−0.250818	+	0.968034i	$0.580700\pi$
$212$		−	67.8668i		−	0.320127i
$213$		0			0
$214$			157.255i			0.734838i
$215$	−171.352			−0.796985
$216$		0			0
$217$		−	75.9781i		−	0.350130i
$218$		−	74.6529i		−	0.342445i
$219$		0			0
$220$	−9.84032			−0.0447287
$221$		−	42.5455i		−	0.192514i
$222$		0			0
$223$	230.988			1.03582			0.517910	−	0.855435i	$-0.326710\pi$
$223$	230.988			1.03582			0.517910	+	0.855435i	$0.326710\pi$
$224$		−	8.61137i		−	0.0384436i
$225$		0			0
$226$		−	35.9382i		−	0.159019i
$227$			146.750i			0.646475i	0.946318	+	0.323237i	$0.104771\pi$
$227$			146.750i			0.646475i	−0.946318	+	0.323237i	$0.895229\pi$
$228$		0			0
$229$		−	203.857i		−	0.890203i	−0.895480	−	0.445102i	$-0.853168\pi$
$229$		−	203.857i		−	0.890203i	0.895480	−	0.445102i	$-0.146832\pi$
$230$	−56.3416	−	45.9959i	−0.244963	−	0.199982i
$231$		0			0
$232$	20.4520			0.0881553
$233$	−296.931			−1.27438			−0.637191	−	0.770706i	$-0.719904\pi$
$233$	−296.931			−1.27438			−0.637191	+	0.770706i	$0.719904\pi$
$234$		0			0
$235$			38.3405i			0.163151i
$236$	−70.4328			−0.298444
$237$		0			0
$238$	50.0528			0.210306
$239$	266.458			1.11488			0.557442	−	0.830216i	$-0.311783\pi$
$239$	266.458			1.11488			0.557442	+	0.830216i	$0.311783\pi$
$240$		0			0
$241$			327.615i			1.35940i	0.733490	+	0.679700i	$0.237890\pi$
$241$			327.615i			1.35940i	−0.733490	+	0.679700i	$0.762110\pi$
$242$	164.273			0.678813
$243$		0			0
$244$			17.0869i			0.0700284i
$245$		−	104.386i		−	0.426063i
$246$		0			0
$247$		−	5.22796i		−	0.0211658i
$248$	141.168			0.569226
$249$		0			0
$250$			15.8114i			0.0632456i
$251$		−	5.89337i		−	0.0234796i	−0.999931	−	0.0117398i	$-0.996263\pi$
$251$		−	5.89337i		−	0.0234796i	0.999931	−	0.0117398i	$-0.00373698\pi$
$252$		0			0
$253$	−39.2034	−	32.0047i	−0.154954	−	0.126501i
$254$	142.638			0.561568
$255$		0			0
$256$	16.0000			0.0625000
$257$	45.2730			0.176160			0.0880798	−	0.996113i	$-0.471927\pi$
$257$	45.2730			0.176160			0.0880798	+	0.996113i	$0.471927\pi$
$258$		0			0
$259$	14.7359			0.0568952
$260$			8.18375i			0.0314760i
$261$		0			0
$262$	40.5228			0.154667
$263$			211.508i			0.804215i	0.915593	+	0.402107i	$0.131722\pi$
$263$			211.508i			0.804215i	−0.915593	+	0.402107i	$0.868278\pi$
$264$		0			0
$265$	−75.8774			−0.286330
$266$	6.15046			0.0231220
$267$		0			0
$268$		−	157.917i		−	0.589243i
$269$	133.758			0.497242			0.248621	−	0.968601i	$-0.420023\pi$
$269$	133.758			0.497242			0.248621	+	0.968601i	$0.420023\pi$
$270$		0			0
$271$	186.815			0.689353			0.344676	−	0.938722i	$-0.387989\pi$
$271$	186.815			0.689353			0.344676	+	0.938722i	$0.387989\pi$
$272$			92.9986i			0.341907i
$273$		0			0
$274$		−	224.929i		−	0.820907i
$275$			11.0018i			0.0400066i
$276$		0			0
$277$	−128.144			−0.462615			−0.231308	−	0.972881i	$-0.574300\pi$
$277$	−128.144			−0.462615			−0.231308	+	0.972881i	$0.574300\pi$
$278$	168.449			0.605933
$279$		0			0
$280$	−9.62781			−0.0343850
$281$			183.268i			0.652200i	0.945335	+	0.326100i	$0.105735\pi$
$281$			183.268i			0.652200i	−0.945335	+	0.326100i	$0.894265\pi$
$282$		0			0
$283$			436.731i			1.54322i	0.636097	+	0.771609i	$0.280548\pi$
$283$			436.731i			1.54322i	−0.636097	+	0.771609i	$0.719452\pi$
$284$	−222.813			−0.784554
$285$		0			0
$286$			5.69438i			0.0199104i
$287$		−	94.5499i		−	0.329442i
$288$		0			0
$289$	−251.546			−0.870401
$290$		−	22.8661i		−	0.0788485i
$291$		0			0
$292$	268.433			0.919291
$293$			34.4017i			0.117412i	0.998275	+	0.0587060i	$0.0186974\pi$
$293$			34.4017i			0.117412i	−0.998275	+	0.0587060i	$0.981303\pi$
$294$		0			0
$295$			78.7463i			0.266937i
$296$			27.3793i			0.0924978i
$297$		0			0
$298$			347.661i			1.16665i
$299$	−26.6169	+	32.6037i	−0.0890196	+	0.109042i
$300$		0			0
$301$	−116.654			−0.387556
$302$	296.206			0.980815
$303$		0			0
$304$			11.4276i			0.0375908i
$305$	19.1038			0.0626353
$306$		0			0
$307$	−263.872			−0.859518			−0.429759	−	0.902944i	$-0.641401\pi$
$307$	−263.872			−0.859518			−0.429759	+	0.902944i	$0.641401\pi$
$308$	−6.69918			−0.0217506
$309$		0			0
$310$		−	157.831i		−	0.509131i
$311$	−12.9685			−0.0416992			−0.0208496	−	0.999783i	$-0.506637\pi$
$311$	−12.9685			−0.0416992			−0.0208496	+	0.999783i	$0.506637\pi$
$312$		0			0
$313$		−	71.7752i		−	0.229314i	−0.993405	−	0.114657i	$-0.963423\pi$
$313$		−	71.7752i		−	0.229314i	0.993405	−	0.114657i	$-0.0365769\pi$
$314$			299.199i			0.952864i
$315$		0			0
$316$		−	144.808i		−	0.458252i
$317$	−193.888			−0.611633			−0.305816	−	0.952090i	$-0.598929\pi$
$317$	−193.888			−0.611633			−0.305816	+	0.952090i	$0.598929\pi$
$318$		0			0
$319$		−	15.9106i		−	0.0498764i
$320$		−	17.8885i		−	0.0559017i
$321$		0			0
$322$	−38.3567	−	31.3135i	−0.119120	−	0.0972469i
$323$	−66.4219			−0.205641
$324$		0			0
$325$	9.14971			0.0281530
$326$	−20.0103			−0.0613812
$327$		0			0
$328$	175.675			0.535593
$329$			26.1018i			0.0793367i
$330$		0			0
$331$	393.325			1.18829			0.594147	−	0.804356i	$-0.297490\pi$
$331$	393.325			1.18829			0.594147	+	0.804356i	$0.297490\pi$
$332$			239.838i			0.722403i
$333$		0			0
$334$	−73.6333			−0.220459
$335$	−176.557			−0.527035
$336$		0			0
$337$		−	39.3542i		−	0.116778i	−0.998294	−	0.0583891i	$-0.981404\pi$
$337$		−	39.3542i		−	0.116778i	0.998294	−	0.0583891i	$-0.0185964\pi$
$338$	−234.266			−0.693096
$339$		0			0
$340$	103.976			0.305811
$341$		−	109.821i		−	0.322056i
$342$		0			0
$343$		−	145.657i		−	0.424655i
$344$		−	216.745i		−	0.630072i
$345$		0			0
$346$	310.075			0.896170
$347$	−309.497			−0.891922			−0.445961	−	0.895052i	$-0.647138\pi$
$347$	−309.497			−0.891922			−0.445961	+	0.895052i	$0.647138\pi$
$348$		0			0
$349$	−328.191			−0.940375			−0.470188	−	0.882567i	$-0.655814\pi$
$349$	−328.191			−0.940375			−0.470188	+	0.882567i	$0.655814\pi$
$350$			10.7642i			0.0307549i
$351$		0			0
$352$		−	12.4471i		−	0.0353612i
$353$	111.495			0.315851			0.157925	−	0.987451i	$-0.449519\pi$
$353$	111.495			0.315851			0.157925	+	0.987451i	$0.449519\pi$
$354$		0			0
$355$			249.113i			0.701726i
$356$		−	147.865i		−	0.415351i
$357$		0			0
$358$	119.291			0.333215
$359$			516.911i			1.43986i	0.694045	+	0.719932i	$0.255827\pi$
$359$			516.911i			1.43986i	−0.694045	+	0.719932i	$0.744173\pi$
$360$		0			0
$361$	352.838			0.977391
$362$			385.564i			1.06509i
$363$		0			0
$364$			5.57140i			0.0153061i
$365$		−	300.117i		−	0.822239i
$366$		0			0
$367$		−	32.4661i		−	0.0884635i	−0.999021	−	0.0442318i	$-0.985916\pi$
$367$		−	32.4661i		−	0.0884635i	0.999021	−	0.0442318i	$-0.0140840\pi$
$368$	58.1808	−	71.2671i	0.158100	−	0.193661i
$369$		0			0
$370$	30.6110			0.0827325
$371$	−51.6565			−0.139236
$372$		0			0
$373$			153.444i			0.411377i	0.978617	+	0.205689i	$0.0659434\pi$
$373$			153.444i			0.411377i	−0.978617	+	0.205689i	$0.934057\pi$
$374$	72.3478			0.193443
$375$		0			0
$376$	−48.4973			−0.128982
$377$	−13.2321			−0.0350984
$378$		0			0
$379$		−	597.537i		−	1.57662i	−0.615281	−	0.788308i	$-0.710958\pi$
$379$		−	597.537i		−	1.57662i	0.615281	−	0.788308i	$-0.289042\pi$
$380$	12.7764			0.0336222
$381$		0			0
$382$		−	333.253i		−	0.872390i
$383$		−	309.730i		−	0.808694i	−0.914606	−	0.404347i	$-0.867499\pi$
$383$		−	309.730i		−	0.808694i	0.914606	−	0.404347i	$-0.132501\pi$
$384$		0			0
$385$			7.48991i			0.0194543i
$386$	−481.010			−1.24614
$387$		0			0
$388$		−	356.879i		−	0.919791i
$389$			295.042i			0.758461i	0.925302	+	0.379231i	$0.123811\pi$
$389$			295.042i			0.758461i	−0.925302	+	0.379231i	$0.876189\pi$
$390$		0			0
$391$	414.234	+	338.171i	1.05942	+	0.864886i
$392$	132.038			0.336833
$393$		0			0
$394$	313.492			0.795666
$395$	−161.900			−0.409873
$396$		0			0
$397$	45.4461			0.114474			0.0572368	−	0.998361i	$-0.481771\pi$
$397$	45.4461			0.114474			0.0572368	+	0.998361i	$0.481771\pi$
$398$		−	280.107i		−	0.703785i
$399$		0			0
$400$	−20.0000			−0.0500000
$401$		−	51.4990i		−	0.128427i	−0.997936	−	0.0642133i	$-0.979546\pi$
$401$		−	51.4990i		−	0.128427i	0.997936	−	0.0642133i	$-0.0204538\pi$
$402$		0			0
$403$	−91.3332			−0.226633
$404$	−294.537			−0.729053
$405$		0			0
$406$		−	15.5669i		−	0.0383422i
$407$	21.2996			0.0523333
$408$		0			0
$409$	−507.527			−1.24090			−0.620449	−	0.784247i	$-0.713050\pi$
$409$	−507.527			−1.24090			−0.620449	+	0.784247i	$0.713050\pi$
$410$		−	196.410i		−	0.479049i
$411$		0			0
$412$		−	117.910i		−	0.286190i
$413$			53.6096i			0.129805i
$414$		0			0
$415$	268.147			0.646137
$416$	−10.3517			−0.0248839
$417$		0			0
$418$	8.89005			0.0212681
$419$			471.677i			1.12572i	0.826552	+	0.562861i	$0.190299\pi$
$419$			471.677i			1.12572i	−0.826552	+	0.562861i	$0.809701\pi$
$420$		0			0
$421$			760.247i			1.80581i	0.429838	+	0.902906i	$0.358570\pi$
$421$			760.247i			1.80581i	−0.429838	+	0.902906i	$0.641430\pi$
$422$	−149.688			−0.354711
$423$		0			0
$424$		−	95.9782i		−	0.226364i
$425$		−	116.248i		−	0.273525i
$426$		0			0
$427$	13.0056			0.0304581
$428$			222.393i			0.519609i
$429$		0			0
$430$	−242.328			−0.563553
$431$			601.309i			1.39515i	0.716512	+	0.697575i	$0.245738\pi$
$431$			601.309i			1.39515i	−0.716512	+	0.697575i	$0.754262\pi$
$432$		0			0
$433$		−	466.438i		−	1.07722i	−0.842554	−	0.538612i	$-0.818949\pi$
$433$		−	466.438i		−	1.07722i	0.842554	−	0.538612i	$-0.181051\pi$
$434$		−	107.449i		−	0.247579i
$435$		0			0
$436$		−	105.575i		−	0.242145i
$437$	50.9007	+	41.5542i	0.116478	+	0.0950896i
$438$		0			0
$439$	−294.320			−0.670432			−0.335216	−	0.942141i	$-0.608809\pi$
$439$	−294.320			−0.670432			−0.335216	+	0.942141i	$0.608809\pi$
$440$	−13.9163			−0.0316280
$441$		0			0
$442$		−	60.1684i		−	0.136128i
$443$	−642.170			−1.44959			−0.724797	−	0.688962i	$-0.758066\pi$
$443$	−642.170			−1.44959			−0.724797	+	0.688962i	$0.758066\pi$
$444$		0			0
$445$	−165.318			−0.371501
$446$	326.666			0.732435
$447$		0			0
$448$		−	12.1783i		−	0.0271837i
$449$	−663.924			−1.47867			−0.739337	−	0.673336i	$-0.764861\pi$
$449$	−663.924			−1.47867			−0.739337	+	0.673336i	$0.764861\pi$
$450$		0			0
$451$		−	136.665i		−	0.303027i
$452$		−	50.8243i		−	0.112443i
$453$		0			0
$454$			207.536i			0.457127i
$455$	6.22902			0.0136902
$456$		0			0
$457$			22.1190i			0.0484005i	0.999707	+	0.0242002i	$0.00770393\pi$
$457$			22.1190i			0.0484005i	−0.999707	+	0.0242002i	$0.992296\pi$
$458$		−	288.297i		−	0.629469i
$459$		0			0
$460$	−79.6790	−	65.0481i	−0.173215	−	0.141409i
$461$	332.992			0.722325			0.361162	−	0.932503i	$-0.382380\pi$
$461$	332.992			0.722325			0.361162	+	0.932503i	$0.382380\pi$
$462$		0			0
$463$	−58.4021			−0.126139			−0.0630693	−	0.998009i	$-0.520089\pi$
$463$	−58.4021			−0.126139			−0.0630693	+	0.998009i	$0.520089\pi$
$464$	28.9235			0.0623352
$465$		0			0
$466$	−419.924			−0.901124
$467$		−	609.288i		−	1.30469i	−0.757924	−	0.652343i	$-0.773786\pi$
$467$		−	609.288i		−	1.30469i	0.757924	−	0.652343i	$-0.226214\pi$
$468$		0			0
$469$	−120.198			−0.256285
$470$			54.2216i			0.115365i
$471$		0			0
$472$	−99.6071			−0.211032
$473$	−168.616			−0.356481
$474$		0			0
$475$		−	14.2845i		−	0.0300726i
$476$	70.7854			0.148709
$477$		0			0
$478$	376.828			0.788343
$479$		−	200.294i		−	0.418150i	−0.977900	−	0.209075i	$-0.932955\pi$
$479$		−	200.294i		−	0.418150i	0.977900	−	0.209075i	$-0.0670453\pi$
$480$		0			0
$481$		−	17.7140i		−	0.0368273i
$482$			463.318i			0.961241i
$483$		0			0
$484$	232.317			0.479993
$485$	−399.003			−0.822686
$486$		0			0
$487$	136.133			0.279534			0.139767	−	0.990184i	$-0.455365\pi$
$487$	136.133			0.279534			0.139767	+	0.990184i	$0.455365\pi$
$488$			24.1646i			0.0495175i
$489$		0			0
$490$		−	147.623i		−	0.301272i
$491$	217.319			0.442604			0.221302	−	0.975205i	$-0.428969\pi$
$491$	217.319			0.442604			0.221302	+	0.975205i	$0.428969\pi$
$492$		0			0
$493$			168.115i			0.341005i
$494$		−	7.39345i		−	0.0149665i
$495$		0			0
$496$	199.642			0.402503
$497$			169.593i			0.341234i
$498$		0			0
$499$	−620.672			−1.24383			−0.621916	−	0.783084i	$-0.713646\pi$
$499$	−620.672			−1.24383			−0.621916	+	0.783084i	$0.713646\pi$
$500$			22.3607i			0.0447214i
$501$		0			0
$502$		−	8.33449i		−	0.0166026i
$503$			165.943i			0.329906i	0.986301	+	0.164953i	$0.0527473\pi$
$503$			165.943i			0.329906i	−0.986301	+	0.164953i	$0.947253\pi$
$504$		0			0
$505$			329.303i			0.652085i
$506$	−55.4419	−	45.2615i	−0.109569	−	0.0894496i
$507$		0			0
$508$	201.721			0.397089
$509$	376.623			0.739928			0.369964	−	0.929046i	$-0.379370\pi$
$509$	376.623			0.739928			0.369964	+	0.929046i	$0.379370\pi$
$510$		0			0
$511$		−	204.316i		−	0.399836i
$512$	22.6274			0.0441942
$513$		0			0
$514$	64.0257			0.124564
$515$	−131.828			−0.255976
$516$		0			0
$517$			37.7283i			0.0729754i
$518$	20.8397			0.0402310
$519$		0			0
$520$			11.5736i			0.0222569i
$521$			675.726i			1.29698i	0.761223	+	0.648490i	$0.224599\pi$
$521$			675.726i			1.29698i	−0.761223	+	0.648490i	$0.775401\pi$
$522$		0			0
$523$		−	53.9295i		−	0.103116i	−0.998670	−	0.0515578i	$-0.983581\pi$
$523$		−	53.9295i		−	0.103116i	0.998670	−	0.0515578i	$-0.0164186\pi$
$524$	57.3079			0.109366
$525$		0			0
$526$			299.118i			0.568666i
$527$			1160.40i			2.20190i
$528$		0			0
$529$	−105.875	−	518.297i	−0.200141	−	0.979767i
$530$	−107.307			−0.202466
$531$		0			0
$532$	8.69806			0.0163497
$533$	−113.658			−0.213243
$534$		0			0
$535$	248.643			0.464752
$536$		−	223.329i		−	0.416658i
$537$		0			0
$538$	189.163			0.351604
$539$		−	102.719i		−	0.190573i
$540$		0			0
$541$	32.6654			0.0603797			0.0301899	−	0.999544i	$-0.490389\pi$
$541$	32.6654			0.0603797			0.0301899	+	0.999544i	$0.490389\pi$
$542$	264.196			0.487446
$543$		0			0
$544$			131.520i			0.241764i
$545$	−118.037			−0.216581
$546$		0			0
$547$	125.227			0.228934			0.114467	−	0.993427i	$-0.463484\pi$
$547$	125.227			0.228934			0.114467	+	0.993427i	$0.463484\pi$
$548$		−	318.097i		−	0.580469i
$549$		0			0
$550$			15.5589i			0.0282889i
$551$			20.6579i			0.0374917i
$552$		0			0
$553$	−110.220			−0.199312
$554$	−181.224			−0.327118
$555$		0			0
$556$	238.223			0.428459
$557$			18.9083i			0.0339466i	0.999856	+	0.0169733i	$0.00540303\pi$
$557$			18.9083i			0.0339466i	−0.999856	+	0.0169733i	$0.994597\pi$
$558$		0			0
$559$			140.230i			0.250859i
$560$	−13.6158			−0.0243139
$561$		0			0
$562$			259.181i			0.461175i
$563$		−	663.769i		−	1.17899i	−0.807774	−	0.589493i	$-0.799328\pi$
$563$		−	663.769i		−	1.17899i	0.807774	−	0.589493i	$-0.200672\pi$
$564$		0			0
$565$	−56.8233			−0.100572
$566$			617.630i			1.09122i
$567$		0			0
$568$	−315.105			−0.554763
$569$		−	491.805i		−	0.864332i	−0.901794	−	0.432166i	$-0.857749\pi$
$569$		−	491.805i		−	0.864332i	0.901794	−	0.432166i	$-0.142251\pi$
$570$		0			0
$571$			880.062i			1.54126i	0.637280	+	0.770632i	$0.280060\pi$
$571$			880.062i			1.54126i	−0.637280	+	0.770632i	$0.719940\pi$
$572$			8.05307i			0.0140788i
$573$		0			0
$574$		−	133.714i		−	0.232951i
$575$	−72.7260	+	89.0839i	−0.126480	+	0.154928i
$576$		0			0
$577$	−4.02697			−0.00697914			−0.00348957	−	0.999994i	$-0.501111\pi$
$577$	−4.02697			−0.00697914			−0.00348957	+	0.999994i	$0.501111\pi$
$578$	−355.740			−0.615467
$579$		0			0
$580$		−	32.3375i		−	0.0557543i
$581$	182.551			0.314202
$582$		0			0
$583$	−74.6658			−0.128072
$584$	379.622			0.650037
$585$		0			0
$586$			48.6513i			0.0830228i
$587$	313.115			0.533415			0.266708	−	0.963778i	$-0.414064\pi$
$587$	313.115			0.533415			0.266708	+	0.963778i	$0.414064\pi$
$588$		0			0
$589$			142.589i			0.242087i
$590$			111.364i			0.188753i
$591$		0			0
$592$			38.7202i			0.0654058i
$593$	−643.656			−1.08542			−0.542712	−	0.839919i	$-0.682602\pi$
$593$	−643.656			−1.08542			−0.542712	+	0.839919i	$0.682602\pi$
$594$		0			0
$595$		−	79.1405i		−	0.133009i
$596$			491.667i			0.824945i
$597$		0			0
$598$	−37.6419	+	46.1085i	−0.0629464	+	0.0771046i
$599$	−482.724			−0.805884			−0.402942	−	0.915226i	$-0.632012\pi$
$599$	−482.724			−0.805884			−0.402942	+	0.915226i	$0.632012\pi$
$600$		0			0
$601$	−510.476			−0.849377			−0.424689	−	0.905340i	$-0.639616\pi$
$601$	−510.476			−0.849377			−0.424689	+	0.905340i	$0.639616\pi$
$602$	−164.974			−0.274043
$603$		0			0
$604$	418.899			0.693541
$605$		−	259.738i		−	0.429319i
$606$		0			0
$607$	−719.184			−1.18482			−0.592409	−	0.805638i	$-0.701823\pi$
$607$	−719.184			−1.18482			−0.592409	+	0.805638i	$0.701823\pi$
$608$			16.1611i			0.0265807i
$609$		0			0
$610$	27.0168			0.0442898
$611$	31.3769			0.0513533
$612$		0			0
$613$			672.054i			1.09634i	0.836368	+	0.548168i	$0.184675\pi$
$613$			672.054i			1.09634i	−0.836368	+	0.548168i	$0.815325\pi$
$614$	−373.171			−0.607771
$615$		0			0
$616$	−9.47407			−0.0153800
$617$		−	295.731i		−	0.479305i	−0.970859	−	0.239653i	$-0.922966\pi$
$617$		−	295.731i		−	0.479305i	0.970859	−	0.239653i	$-0.0770336\pi$
$618$		0			0
$619$		−	823.442i		−	1.33028i	−0.746720	−	0.665139i	$-0.768372\pi$
$619$		−	823.442i		−	1.33028i	0.746720	−	0.665139i	$-0.231628\pi$
$620$		−	223.206i		−	0.360010i
$621$		0			0
$622$	−18.3402			−0.0294858
$623$	−112.547			−0.180653
$624$		0			0
$625$	25.0000			0.0400000
$626$		−	101.505i		−	0.162149i
$627$		0			0
$628$			423.132i			0.673777i
$629$	−225.058			−0.357803
$630$		0			0
$631$			471.460i			0.747163i	0.927597	+	0.373582i	$0.121870\pi$
$631$			471.460i			0.747163i	−0.927597	+	0.373582i	$0.878130\pi$
$632$		−	204.789i		−	0.324033i
$633$		0			0
$634$	−274.199			−0.432490
$635$		−	225.531i		−	0.355167i
$636$		0			0
$637$	−85.4265			−0.134108
$638$		−	22.5009i		−	0.0352679i
$639$		0			0
$640$		−	25.2982i		−	0.0395285i
$641$			1114.03i			1.73796i	0.494850	+	0.868979i	$0.335223\pi$
$641$			1114.03i			1.73796i	−0.494850	+	0.868979i	$0.664777\pi$
$642$		0			0
$643$		−	182.160i		−	0.283298i	−0.989917	−	0.141649i	$-0.954760\pi$
$643$		−	182.160i		−	0.283298i	0.989917	−	0.141649i	$-0.0452404\pi$
$644$	−54.2446	−	44.2840i	−0.0842307	−	0.0687640i
$645$		0			0
$646$	−93.9348			−0.145410
$647$	−231.496			−0.357799			−0.178899	−	0.983867i	$-0.557254\pi$
$647$	−231.496			−0.357799			−0.178899	+	0.983867i	$0.557254\pi$
$648$		0			0
$649$			77.4889i			0.119397i
$650$	12.9396			0.0199071
$651$		0			0
$652$	−28.2988			−0.0434030
$653$	−961.708			−1.47275			−0.736377	−	0.676572i	$-0.763465\pi$
$653$	−961.708			−1.47275			−0.736377	+	0.676572i	$0.763465\pi$
$654$		0			0
$655$		−	64.0722i		−	0.0978201i
$656$	248.441			0.378721
$657$		0			0
$658$			36.9135i			0.0560995i
$659$		−	482.196i		−	0.731709i	−0.930672	−	0.365854i	$-0.880777\pi$
$659$		−	482.196i		−	0.731709i	0.930672	−	0.365854i	$-0.119223\pi$
$660$		0			0
$661$		−	514.781i		−	0.778792i	−0.921070	−	0.389396i	$-0.872684\pi$
$661$		−	514.781i		−	0.778792i	0.921070	−	0.389396i	$-0.127316\pi$
$662$	556.246			0.840251
$663$		0			0
$664$			339.182i			0.510816i
$665$		−	9.72473i		−	0.0146236i
$666$		0			0
$667$	105.175	−	128.831i	0.157683	−	0.193150i
$668$	−104.133			−0.155888
$669$		0			0
$670$	−249.689			−0.372670
$671$	18.7987			0.0280160
$672$		0			0
$673$	−1100.79			−1.63565			−0.817824	−	0.575468i	$-0.804820\pi$
$673$	−1100.79			−1.63565			−0.817824	+	0.575468i	$0.804820\pi$
$674$		−	55.6553i		−	0.0825746i
$675$		0			0
$676$	−331.303			−0.490093
$677$			155.504i			0.229695i	0.993383	+	0.114848i	$0.0366380\pi$
$677$			155.504i			0.229695i	−0.993383	+	0.114848i	$0.963362\pi$
$678$		0			0
$679$	−271.637			−0.400054
$680$	147.044			0.216241
$681$		0			0
$682$		−	155.310i		−	0.227728i
$683$	1077.79			1.57803			0.789014	−	0.614376i	$-0.210592\pi$
$683$	1077.79			1.57803			0.789014	+	0.614376i	$0.210592\pi$
$684$		0			0
$685$	−355.643			−0.519187
$686$		−	205.990i		−	0.300276i
$687$		0			0
$688$		−	306.523i		−	0.445528i
$689$			62.0962i			0.0901251i
$690$		0			0
$691$	−884.748			−1.28039			−0.640194	−	0.768214i	$-0.721146\pi$
$691$	−884.748			−1.28039			−0.640194	+	0.768214i	$0.721146\pi$
$692$	438.512			0.633688
$693$		0			0
$694$	−437.695			−0.630684
$695$		−	266.342i		−	0.383225i
$696$		0			0
$697$			1444.04i			2.07180i
$698$	−464.132			−0.664946
$699$		0			0
$700$			15.2229i			0.0217470i
$701$			666.703i			0.951074i	0.879696	+	0.475537i	$0.157746\pi$
$701$			666.703i			0.951074i	−0.879696	+	0.475537i	$0.842254\pi$
$702$		0			0
$703$	−27.6550			−0.0393385
$704$		−	17.6029i		−	0.0250041i
$705$		0			0
$706$	157.678			0.223340
$707$			224.186i			0.317094i
$708$		0			0
$709$			967.418i			1.36448i	0.731127	+	0.682241i	$0.238995\pi$
$709$			967.418i			1.36448i	−0.731127	+	0.682241i	$0.761005\pi$
$710$			352.299i			0.496195i
$711$		0			0
$712$		−	209.112i		−	0.293697i
$713$	725.957	−	889.242i	1.01817	−	1.24718i
$714$		0			0
$715$	9.00361			0.0125925
$716$	168.703			0.235619
$717$		0			0
$718$			731.023i			1.01814i
$719$	216.266			0.300787			0.150393	−	0.988626i	$-0.451946\pi$
$719$	216.266			0.300787			0.150393	+	0.988626i	$0.451946\pi$
$720$		0			0
$721$	−89.7469			−0.124476
$722$	498.988			0.691120
$723$		0			0
$724$			545.269i			0.753134i
$725$	−36.1544			−0.0498681
$726$		0			0
$727$		−	113.587i		−	0.156240i	−0.996944	−	0.0781200i	$-0.975108\pi$
$727$		−	113.587i		−	0.156240i	0.996944	−	0.0781200i	$-0.0248917\pi$
$728$			7.87916i			0.0108230i
$729$		0			0
$730$		−	424.430i		−	0.581411i
$731$	1781.64			2.43726
$732$		0			0
$733$		−	104.468i		−	0.142522i	−0.997458	−	0.0712608i	$-0.977298\pi$
$733$		−	104.468i		−	0.142522i	0.997458	−	0.0712608i	$-0.0227023\pi$
$734$		−	45.9140i		−	0.0625531i
$735$		0			0
$736$	82.2800	−	100.787i	0.111794	−	0.136939i
$737$	−173.738			−0.235736
$738$		0			0
$739$	828.795			1.12151			0.560755	−	0.827982i	$-0.310511\pi$
$739$	828.795			1.12151			0.560755	+	0.827982i	$0.310511\pi$
$740$	43.2905			0.0585007
$741$		0			0
$742$	−73.0533			−0.0984546
$743$		−	1067.78i		−	1.43712i	−0.695465	−	0.718560i	$-0.744802\pi$
$743$		−	1067.78i		−	1.43712i	0.695465	−	0.718560i	$-0.255198\pi$
$744$		0			0
$745$	549.700			0.737853
$746$			217.002i			0.290888i
$747$		0			0
$748$	102.315			0.136785
$749$	169.273			0.225999
$750$		0			0
$751$		−	116.437i		−	0.155043i	−0.996991	−	0.0775213i	$-0.975299\pi$
$751$		−	116.437i		−	0.155043i	0.996991	−	0.0775213i	$-0.0247006\pi$
$752$	−68.5855			−0.0912042
$753$		0			0
$754$	−18.7130			−0.0248183
$755$		−	468.343i		−	0.620322i
$756$		0			0
$757$			893.537i			1.18037i	0.807269	+	0.590183i	$0.200944\pi$
$757$			893.537i			1.18037i	−0.807269	+	0.590183i	$0.799056\pi$
$758$		−	845.045i		−	1.11484i
$759$		0			0
$760$	18.0686			0.0237745
$761$	80.0115			0.105140			0.0525700	−	0.998617i	$-0.483259\pi$
$761$	80.0115			0.105140			0.0525700	+	0.998617i	$0.483259\pi$
$762$		0			0
$763$	−80.3580			−0.105318
$764$		−	471.291i		−	0.616873i
$765$		0			0
$766$		−	438.024i		−	0.571833i
$767$	64.4440			0.0840209
$768$		0			0
$769$			890.267i			1.15769i	0.815436	+	0.578847i	$0.196497\pi$
$769$			890.267i			1.15769i	−0.815436	+	0.578847i	$0.803503\pi$
$770$			10.5923i			0.0137563i
$771$		0			0
$772$	−680.251			−0.881155
$773$			619.222i			0.801063i	0.916283	+	0.400531i	$0.131174\pi$
$773$			619.222i			0.801063i	−0.916283	+	0.400531i	$0.868826\pi$
$774$		0			0
$775$	−249.552			−0.322003
$776$		−	504.703i		−	0.650391i
$777$		0			0
$778$			417.252i			0.536313i
$779$			177.443i			0.227783i
$780$		0			0
$781$			245.135i			0.313873i
$782$	585.815	+	478.245i	0.749124	+	0.611567i
$783$		0			0
$784$	186.731			0.238177
$785$	473.076			0.602644
$786$		0			0
$787$		−	375.043i		−	0.476548i	−0.971198	−	0.238274i	$-0.923418\pi$
$787$		−	375.043i		−	0.476548i	0.971198	−	0.238274i	$-0.0765816\pi$
$788$	443.345			0.562621
$789$		0			0
$790$	−228.961			−0.289824
$791$	−38.6846			−0.0489060
$792$		0			0
$793$		−	15.6340i		−	0.0197151i
$794$	64.2704			0.0809451
$795$		0			0
$796$		−	396.131i		−	0.497651i
$797$		−	493.978i		−	0.619797i	−0.950770	−	0.309898i	$-0.899705\pi$
$797$		−	493.978i		−	0.619797i	0.950770	−	0.309898i	$-0.100295\pi$
$798$		0			0
$799$		−	398.647i		−	0.498933i
$800$	−28.2843			−0.0353553
$801$		0			0
$802$		−	72.8306i		−	0.0908113i
$803$		−	295.325i		−	0.367777i
$804$		0			0
$805$	−49.5110	+	60.6473i	−0.0615044	+	0.0753382i
$806$	−129.165			−0.160254
$807$		0			0
$808$	−416.539			−0.515518
$809$	132.182			0.163389			0.0816946	−	0.996657i	$-0.473967\pi$
$809$	132.182			0.163389			0.0816946	+	0.996657i	$0.473967\pi$
$810$		0			0
$811$	−158.090			−0.194932			−0.0974659	−	0.995239i	$-0.531074\pi$
$811$	−158.090			−0.194932			−0.0974659	+	0.995239i	$0.531074\pi$
$812$		−	22.0150i		−	0.0271121i
$813$		0			0
$814$	30.1222			0.0370052
$815$			31.6390i			0.0388209i
$816$		0			0
$817$	218.927			0.267964
$818$	−717.752			−0.877448
$819$		0			0
$820$		−	277.766i		−	0.338739i
$821$	218.188			0.265759			0.132879	−	0.991132i	$-0.457578\pi$
$821$	218.188			0.265759			0.132879	+	0.991132i	$0.457578\pi$
$822$		0			0
$823$	147.052			0.178678			0.0893391	−	0.996001i	$-0.471525\pi$
$823$	147.052			0.178678			0.0893391	+	0.996001i	$0.471525\pi$
$824$		−	166.750i		−	0.202367i
$825$		0			0
$826$			75.8154i			0.0917862i
$827$			942.783i			1.14000i	0.821644	+	0.570002i	$0.193057\pi$
$827$			942.783i			1.14000i	−0.821644	+	0.570002i	$0.806943\pi$
$828$		0			0
$829$	110.053			0.132754			0.0663771	−	0.997795i	$-0.478856\pi$
$829$	110.053			0.132754			0.0663771	+	0.997795i	$0.478856\pi$
$830$	379.217			0.456888
$831$		0			0
$832$	−14.6395			−0.0175956
$833$			1085.35i			1.30295i
$834$		0			0
$835$			116.425i			0.139431i
$836$	12.5724			0.0150388
$837$		0			0
$838$			667.052i			0.796005i
$839$		−	724.729i		−	0.863800i	−0.901921	−	0.431900i	$-0.857843\pi$
$839$		−	724.729i		−	0.863800i	0.901921	−	0.431900i	$-0.142157\pi$
$840$		0			0
$841$	−788.714			−0.937829
$842$			1075.15i			1.27690i
$843$		0			0
$844$	−211.691			−0.250818
$845$			370.408i			0.438352i
$846$		0			0
$847$		−	176.827i		−	0.208768i
$848$		−	135.734i		−	0.160063i
$849$		0			0
$850$		−	164.400i		−	0.193412i
$851$	172.467	+	140.798i	0.202664	+	0.165450i
$852$		0			0
$853$	−1575.08			−1.84652			−0.923261	−	0.384172i	$-0.874487\pi$
$853$	−1575.08			−1.84652			−0.923261	+	0.384172i	$0.874487\pi$
$854$	18.3927			0.0215371
$855$		0			0
$856$			314.511i			0.367419i
$857$	−190.678			−0.222495			−0.111247	−	0.993793i	$-0.535485\pi$
$857$	−190.678			−0.222495			−0.111247	+	0.993793i	$0.535485\pi$
$858$		0			0
$859$	81.9616			0.0954151			0.0477076	−	0.998861i	$-0.484808\pi$
$859$	81.9616			0.0954151			0.0477076	+	0.998861i	$0.484808\pi$
$860$	−342.704			−0.398492
$861$		0			0
$862$			850.380i			0.986520i
$863$	−1544.93			−1.79019			−0.895093	−	0.445878i	$-0.852891\pi$
$863$	−1544.93			−1.79019			−0.895093	+	0.445878i	$0.852891\pi$
$864$		0			0
$865$		−	490.271i		−	0.566788i
$866$		−	659.642i		−	0.761712i
$867$		0			0
$868$		−	151.956i		−	0.175065i
$869$	−159.315			−0.183331
$870$		0			0
$871$			144.490i			0.165889i
$872$		−	149.306i		−	0.171222i
$873$		0			0
$874$	71.9845	+	58.7665i	0.0823621	+	0.0672385i
$875$	17.0197			0.0194511
$876$		0			0
$877$	277.527			0.316450			0.158225	−	0.987403i	$-0.449423\pi$
$877$	277.527			0.316450			0.158225	+	0.987403i	$0.449423\pi$
$878$	−416.231			−0.474067
$879$		0			0
$880$	−19.6806			−0.0223644
$881$		−	946.407i		−	1.07424i	−0.843505	−	0.537121i	$-0.819512\pi$
$881$		−	946.407i		−	1.07424i	0.843505	−	0.537121i	$-0.180488\pi$
$882$		0			0
$883$	1235.93			1.39969			0.699845	−	0.714295i	$-0.253252\pi$
$883$	1235.93			1.39969			0.699845	+	0.714295i	$0.253252\pi$
$884$		−	85.0910i		−	0.0962568i
$885$		0			0
$886$	−908.166			−1.02502
$887$	243.464			0.274480			0.137240	−	0.990538i	$-0.456177\pi$
$887$	243.464			0.274480			0.137240	+	0.990538i	$0.456177\pi$
$888$		0			0
$889$		−	153.539i		−	0.172710i
$890$	−233.795			−0.262691
$891$		0			0
$892$	461.976			0.517910
$893$		−	48.9855i		−	0.0548550i
$894$		0			0
$895$		−	188.616i		−	0.210744i
$896$		−	17.2227i		−	0.0192218i
$897$		0			0
$898$	−938.931			−1.04558
$899$	360.896			0.401442
$900$		0			0
$901$	788.940			0.875627
$902$		−	193.274i		−	0.214273i
$903$		0			0
$904$		−	71.8764i		−	0.0795093i
$905$	609.630			0.673624
$906$		0			0
$907$			1671.67i			1.84307i	0.388292	+	0.921536i	$0.373065\pi$
$907$			1671.67i			1.84307i	−0.388292	+	0.921536i	$0.626935\pi$
$908$			293.500i			0.323237i
$909$		0			0
$910$	8.80916			0.00968040
$911$			1179.25i			1.29445i	0.762297	+	0.647227i	$0.224071\pi$
$911$			1179.25i			1.29445i	−0.762297	+	0.647227i	$0.775929\pi$
$912$		0			0
$913$	263.865			0.289009
$914$			31.2810i			0.0342243i
$915$		0			0
$916$		−	407.713i		−	0.445102i
$917$		−	43.6196i		−	0.0475677i
$918$		0			0
$919$		−	734.211i		−	0.798924i	−0.916750	−	0.399462i	$-0.869197\pi$
$919$		−	734.211i		−	0.798924i	0.916750	−	0.399462i	$-0.130803\pi$
$920$	−112.683	−	91.9919i	−0.122482	−	0.0999912i
$921$		0			0
$922$	470.921			0.510761
$923$	203.868			0.220875
$924$		0			0
$925$		−	48.4003i		−	0.0523247i
$926$	−82.5931			−0.0891934
$927$		0			0
$928$	40.9040			0.0440776
$929$	43.9814			0.0473427			0.0236714	−	0.999720i	$-0.492464\pi$
$929$	43.9814			0.0473427			0.0236714	+	0.999720i	$0.492464\pi$
$930$		0			0
$931$			133.368i			0.143252i
$932$	−593.862			−0.637191
$933$		0			0
$934$		−	861.664i		−	0.922552i
$935$		−	114.392i		−	0.122344i
$936$		0			0
$937$		−	393.511i		−	0.419969i	−0.977705	−	0.209984i	$-0.932659\pi$
$937$		−	393.511i		−	0.419969i	0.977705	−	0.209984i	$-0.0673413\pi$
$938$	−169.985			−0.181221
$939$		0			0
$940$			76.6809i			0.0815755i
$941$			330.827i			0.351569i	0.984429	+	0.175785i	$0.0562463\pi$
$941$			330.827i			0.351569i	−0.984429	+	0.175785i	$0.943754\pi$
$942$		0			0
$943$	903.407	−	1106.61i	0.958013	−	1.17349i
$944$	−140.866			−0.149222
$945$		0			0
$946$	−238.459			−0.252070
$947$	251.483			0.265557			0.132779	−	0.991146i	$-0.457610\pi$
$947$	251.483			0.265557			0.132779	+	0.991146i	$0.457610\pi$
$948$		0			0
$949$	−245.608			−0.258808
$950$		−	20.2013i		−	0.0212646i
$951$		0			0
$952$	100.106			0.105153
$953$		−	1263.57i		−	1.32589i	−0.748670	−	0.662943i	$-0.769307\pi$
$953$		−	1263.57i		−	1.32589i	0.748670	−	0.662943i	$-0.230693\pi$
$954$		0			0
$955$	−526.919			−0.551748
$956$	532.915			0.557442
$957$		0			0
$958$		−	283.259i		−	0.295677i
$959$	−242.118			−0.252469
$960$		0			0
$961$	1530.05			1.59214
$962$		−	25.0513i		−	0.0260409i
$963$		0			0
$964$			655.231i			0.679700i
$965$			760.544i			0.788129i
$966$		0			0
$967$	−717.203			−0.741678			−0.370839	−	0.928697i	$-0.620930\pi$
$967$	−717.203			−0.741678			−0.370839	+	0.928697i	$0.620930\pi$
$968$	328.546			0.339407
$969$		0			0
$970$	−564.275			−0.581727
$971$			1232.61i			1.26943i	0.772748	+	0.634713i	$0.218882\pi$
$971$			1232.61i			1.26943i	−0.772748	+	0.634713i	$0.781118\pi$
$972$		0			0
$973$		−	181.322i		−	0.186354i
$974$	192.521			0.197661
$975$		0			0
$976$			34.1738i			0.0350142i
$977$		−	1365.57i		−	1.39771i	−0.715262	−	0.698857i	$-0.753693\pi$
$977$		−	1365.57i		−	1.39771i	0.715262	−	0.698857i	$-0.246307\pi$
$978$		0			0
$979$	−162.678			−0.166168
$980$		−	208.771i		−	0.213032i
$981$		0			0
$982$	307.335			0.312968
$983$		−	1879.04i		−	1.91154i	−0.294119	−	0.955769i	$-0.595026\pi$
$983$		−	1879.04i		−	1.91154i	0.294119	−	0.955769i	$-0.404974\pi$
$984$		0			0
$985$		−	495.675i		−	0.503223i
$986$			237.751i			0.241127i
$987$		0			0
$988$		−	10.4559i		−	0.0105829i
$989$	−1365.31	−	1114.61i	−1.38050	−	1.12701i
$990$		0			0
$991$	1192.84			1.20367			0.601837	−	0.798619i	$-0.294436\pi$
$991$	1192.84			1.20367			0.601837	+	0.798619i	$0.294436\pi$
$992$	282.336			0.284613
$993$		0			0
$994$			239.841i			0.241289i
$995$	−442.887			−0.445113
$996$		0			0
$997$	1855.93			1.86152			0.930758	−	0.365637i	$-0.119149\pi$
$997$	1855.93			1.86152			0.930758	+	0.365637i	$0.119149\pi$
$998$	−877.763			−0.879522
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2070.3.c.b.91.19		32
3.2	odd	2		690.3.c.a.91.6	yes	32
23.22	odd	2	inner	2070.3.c.b.91.30		32
69.68	even	2		690.3.c.a.91.3	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.3.c.a.91.3	✓	32	69.68	even	2
690.3.c.a.91.6	yes	32	3.2	odd	2
2070.3.c.b.91.19		32	1.1	even	1	trivial
2070.3.c.b.91.30		32	23.22	odd	2	inner

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:	\( N \)	\(=\)	\( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2070.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +2.00000 q^{4} -2.23607i q^{5} -1.52229i q^{7} +2.82843 q^{8} +O(q^{10})\)	\(q+1.41421 q^{2} +2.00000 q^{4} -2.23607i q^{5} -1.52229i q^{7} +2.82843 q^{8} -3.16228i q^{10} -2.20036i q^{11} -1.82994 q^{13} -2.15284i q^{14} +4.00000 q^{16} +23.2496i q^{17} +2.85690i q^{19} -4.47214i q^{20} -3.11178i q^{22} +(14.5452 - 17.8168i) q^{23} -5.00000 q^{25} -2.58793 q^{26} -3.04458i q^{28} +7.23088 q^{29} +49.9104 q^{31} +5.65685 q^{32} +32.8800i q^{34} -3.40394 q^{35} +9.68006i q^{37} +4.04027i q^{38} -6.32456i q^{40} +62.1103 q^{41} -76.6308i q^{43} -4.40073i q^{44} +(20.5700 - 25.1967i) q^{46} -17.1464 q^{47} +46.6826 q^{49} -7.07107 q^{50} -3.65988 q^{52} -33.9334i q^{53} -4.92016 q^{55} -4.30569i q^{56} +10.2260 q^{58} -35.2164 q^{59} +8.54346i q^{61} +70.5840 q^{62} +8.00000 q^{64} +4.09187i q^{65} -78.9586i q^{67} +46.4993i q^{68} -4.81390 q^{70} -111.407 q^{71} +134.217 q^{73} +13.6897i q^{74} +5.71380i q^{76} -3.34959 q^{77} -72.4039i q^{79} -8.94427i q^{80} +87.8373 q^{82} +119.919i q^{83} +51.9878 q^{85} -108.372i q^{86} -6.22357i q^{88} -73.9324i q^{89} +2.78570i q^{91} +(29.0904 - 35.6335i) q^{92} -24.2486 q^{94} +6.38822 q^{95} -178.439i q^{97} +66.0192 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 64 q^{4}+O(q^{10}) \) 32 * q + 64 * q^4	\( 32 q + 64 q^{4} - 48 q^{13} + 128 q^{16} + 80 q^{23} - 160 q^{25} - 120 q^{29} + 248 q^{31} + 120 q^{35} - 72 q^{41} + 160 q^{46} - 400 q^{47} - 344 q^{49} - 96 q^{52} - 256 q^{58} - 120 q^{59} - 160 q^{62} + 256 q^{64} - 104 q^{71} + 16 q^{73} - 240 q^{77} + 64 q^{82} - 120 q^{85} + 160 q^{92} + 96 q^{94} + 160 q^{95} - 64 q^{98}+O(q^{100}) \) 32 * q + 64 * q^4 - 48 * q^13 + 128 * q^16 + 80 * q^23 - 160 * q^25 - 120 * q^29 + 248 * q^31 + 120 * q^35 - 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 - 104 * q^71 + 16 * q^73 - 240 * q^77 + 64 * q^82 - 120 * q^85 + 160 * q^92 + 96 * q^94 + 160 * q^95 - 64 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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