LMFDB - Embedded newform 2070.3.c.b.91.9

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			91.9
Character	$\chi$	$=$	2070.91
Dual form			2070.3.c.b.91.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +2.00000 q^{4} +2.23607i q^{5} -11.8194i q^{7} -2.82843 q^{8} +O(q^{10})$	\(q-1.41421 q^{2} +2.00000 q^{4} +2.23607i q^{5} -11.8194i q^{7} -2.82843 q^{8} -3.16228i q^{10} -20.7652i q^{11} +7.75559 q^{13} +16.7152i q^{14} +4.00000 q^{16} +22.9081i q^{17} -21.0098i q^{19} +4.47214i q^{20} +29.3664i q^{22} +(21.9968 + 6.71856i) q^{23} -5.00000 q^{25} -10.9681 q^{26} -23.6389i q^{28} +38.2483 q^{29} +33.3493 q^{31} -5.65685 q^{32} -32.3970i q^{34} +26.4291 q^{35} -49.4326i q^{37} +29.7124i q^{38} -6.32456i q^{40} +0.679308 q^{41} +7.99696i q^{43} -41.5303i q^{44} +(-31.1082 - 9.50148i) q^{46} +85.9436 q^{47} -90.6993 q^{49} +7.07107 q^{50} +15.5112 q^{52} +66.6981i q^{53} +46.4323 q^{55} +33.4304i q^{56} -54.0913 q^{58} -110.632 q^{59} -16.6780i q^{61} -47.1630 q^{62} +8.00000 q^{64} +17.3420i q^{65} -117.857i q^{67} +45.8163i q^{68} -37.3764 q^{70} -28.9967 q^{71} +31.0752 q^{73} +69.9083i q^{74} -42.0196i q^{76} -245.433 q^{77} +105.069i q^{79} +8.94427i q^{80} -0.960687 q^{82} -43.1979i q^{83} -51.2241 q^{85} -11.3094i q^{86} +58.7327i q^{88} +49.1782i q^{89} -91.6668i q^{91} +(43.9937 + 13.4371i) q^{92} -121.543 q^{94} +46.9794 q^{95} -69.6738i q^{97} +128.268 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$		−	11.8194i		−	1.68849i	−0.535955	−	0.844246i	$-0.680049\pi$
$7$		−	11.8194i		−	1.68849i	0.535955	−	0.844246i	$-0.319951\pi$
$8$	−2.82843			−0.353553
$9$		0			0
$10$		−	3.16228i		−	0.316228i
$11$		−	20.7652i		−	1.88774i	−0.330315	−	0.943871i	$-0.607155\pi$
$11$		−	20.7652i		−	1.88774i	0.330315	−	0.943871i	$-0.392845\pi$
$12$		0			0
$13$	7.75559			0.596584			0.298292	−	0.954475i	$-0.403583\pi$
$13$	7.75559			0.596584			0.298292	+	0.954475i	$0.403583\pi$
$14$			16.7152i			1.19394i
$15$		0			0
$16$	4.00000			0.250000
$17$			22.9081i			1.34754i	0.738943	+	0.673768i	$0.235325\pi$
$17$			22.9081i			1.34754i	−0.738943	+	0.673768i	$0.764675\pi$
$18$		0			0
$19$		−	21.0098i		−	1.10578i	−0.833255	−	0.552890i	$-0.813525\pi$
$19$		−	21.0098i		−	1.10578i	0.833255	−	0.552890i	$-0.186475\pi$
$20$			4.47214i			0.223607i
$21$		0			0
$22$			29.3664i			1.33483i
$23$	21.9968	+	6.71856i	0.956384	+	0.292111i
$23$	21.9968	+	6.71856i	0.956384	+	0.292111i
$24$		0			0
$25$	−5.00000			−0.200000
$26$	−10.9681			−0.421848
$27$		0			0
$28$		−	23.6389i		−	0.844246i
$29$	38.2483			1.31891			0.659454	−	0.751745i	$-0.270788\pi$
$29$	38.2483			1.31891			0.659454	+	0.751745i	$0.270788\pi$
$30$		0			0
$31$	33.3493			1.07578			0.537892	−	0.843014i	$-0.319221\pi$
$31$	33.3493			1.07578			0.537892	+	0.843014i	$0.319221\pi$
$32$	−5.65685			−0.176777
$33$		0			0
$34$		−	32.3970i		−	0.952852i
$35$	26.4291			0.755117
$36$		0			0
$37$		−	49.4326i		−	1.33602i	−0.744154	−	0.668009i	$-0.767147\pi$
$37$		−	49.4326i		−	1.33602i	0.744154	−	0.668009i	$-0.232853\pi$
$38$			29.7124i			0.781904i
$39$		0			0
$40$		−	6.32456i		−	0.158114i
$41$	0.679308			0.0165685			0.00828424	−	0.999966i	$-0.497363\pi$
$41$	0.679308			0.0165685			0.00828424	+	0.999966i	$0.497363\pi$
$42$		0			0
$43$			7.99696i			0.185976i	0.995667	+	0.0929879i	$0.0296418\pi$
$43$			7.99696i			0.185976i	−0.995667	+	0.0929879i	$0.970358\pi$
$44$		−	41.5303i		−	0.943871i
$45$		0			0
$46$	−31.1082	−	9.50148i	−0.676266	−	0.206554i
$47$	85.9436			1.82859			0.914294	−	0.405052i	$-0.132747\pi$
$47$	85.9436			1.82859			0.914294	+	0.405052i	$0.132747\pi$
$48$		0			0
$49$	−90.6993			−1.85101
$50$	7.07107			0.141421
$51$		0			0
$52$	15.5112			0.298292
$53$			66.6981i			1.25845i	0.777221	+	0.629227i	$0.216629\pi$
$53$			66.6981i			1.25845i	−0.777221	+	0.629227i	$0.783371\pi$
$54$		0			0
$55$	46.4323			0.844224
$56$			33.4304i			0.596972i
$57$		0			0
$58$	−54.0913			−0.932608
$59$	−110.632			−1.87512			−0.937560	−	0.347823i	$-0.886921\pi$
$59$	−110.632			−1.87512			−0.937560	+	0.347823i	$0.886921\pi$
$60$		0			0
$61$		−	16.6780i		−	0.273409i	−0.990612	−	0.136705i	$-0.956349\pi$
$61$		−	16.6780i		−	0.273409i	0.990612	−	0.136705i	$-0.0436511\pi$
$62$	−47.1630			−0.760694
$63$		0			0
$64$	8.00000			0.125000
$65$			17.3420i			0.266800i
$66$		0			0
$67$		−	117.857i		−	1.75906i	−0.475840	−	0.879532i	$-0.657856\pi$
$67$		−	117.857i		−	1.75906i	0.475840	−	0.879532i	$-0.342144\pi$
$68$			45.8163i			0.673768i
$69$		0			0
$70$	−37.3764			−0.533948
$71$	−28.9967			−0.408405			−0.204202	−	0.978929i	$-0.565460\pi$
$71$	−28.9967			−0.408405			−0.204202	+	0.978929i	$0.565460\pi$
$72$		0			0
$73$	31.0752			0.425688			0.212844	−	0.977086i	$-0.431727\pi$
$73$	31.0752			0.425688			0.212844	+	0.977086i	$0.431727\pi$
$74$			69.9083i			0.944707i
$75$		0			0
$76$		−	42.0196i		−	0.552890i
$77$	−245.433			−3.18744
$78$		0			0
$79$			105.069i			1.32998i	0.746850	+	0.664992i	$0.231565\pi$
$79$			105.069i			1.32998i	−0.746850	+	0.664992i	$0.768435\pi$
$80$			8.94427i			0.111803i
$81$		0			0
$82$	−0.960687			−0.0117157
$83$		−	43.1979i		−	0.520456i	−0.965547	−	0.260228i	$-0.916202\pi$
$83$		−	43.1979i		−	0.520456i	0.965547	−	0.260228i	$-0.0837977\pi$
$84$		0			0
$85$	−51.2241			−0.602637
$86$		−	11.3094i		−	0.131505i
$87$		0			0
$88$			58.7327i			0.667417i
$89$			49.1782i			0.552564i	0.961077	+	0.276282i	$0.0891023\pi$
$89$			49.1782i			0.552564i	−0.961077	+	0.276282i	$0.910898\pi$
$90$		0			0
$91$		−	91.6668i		−	1.00733i
$92$	43.9937	+	13.4371i	0.478192	+	0.146056i
$93$		0			0
$94$	−121.543			−1.29301
$95$	46.9794			0.494520
$96$		0			0
$97$		−	69.6738i		−	0.718287i	−0.933282	−	0.359143i	$-0.883069\pi$
$97$		−	69.6738i		−	0.718287i	0.933282	−	0.359143i	$-0.116931\pi$
$98$	128.268			1.30886
$99$		0			0
$100$	−10.0000			−0.100000
$101$	17.2732			0.171022			0.0855111	−	0.996337i	$-0.472748\pi$
$101$	17.2732			0.171022			0.0855111	+	0.996337i	$0.472748\pi$
$102$		0			0
$103$			51.7425i			0.502355i	0.967941	+	0.251177i	$0.0808177\pi$
$103$			51.7425i			0.502355i	−0.967941	+	0.251177i	$0.919182\pi$
$104$	−21.9361			−0.210924
$105$		0			0
$106$		−	94.3254i		−	0.889862i
$107$		−	54.2161i		−	0.506693i	−0.967376	−	0.253346i	$-0.918469\pi$
$107$		−	54.2161i		−	0.506693i	0.967376	−	0.253346i	$-0.0815313\pi$
$108$		0			0
$109$			0.0647920i			0.000594422i	1.00000		0.000297211i	$9.46053e-5\pi$
$109$			0.0647920i			0.000594422i	−1.00000		0.000297211i	$0.999905\pi$
$110$	−65.6652			−0.596956
$111$		0			0
$112$		−	47.2778i		−	0.422123i
$113$		−	157.283i		−	1.39188i	−0.718098	−	0.695942i	$-0.754987\pi$
$113$		−	157.283i		−	1.39188i	0.718098	−	0.695942i	$-0.245013\pi$
$114$		0			0
$115$	−15.0232	+	49.1864i	−0.130636	+	0.427708i
$116$	76.4966			0.659454
$117$		0			0
$118$	156.457			1.32591
$119$	270.761			2.27531
$120$		0			0
$121$	−310.192			−2.56357
$122$			23.5862i			0.193329i
$123$		0			0
$124$	66.6986			0.537892
$125$		−	11.1803i		−	0.0894427i
$126$		0			0
$127$	133.370			1.05016			0.525080	−	0.851053i	$-0.324035\pi$
$127$	133.370			1.05016			0.525080	+	0.851053i	$0.324035\pi$
$128$	−11.3137			−0.0883883
$129$		0			0
$130$		−	24.5253i		−	0.188656i
$131$	−62.9089			−0.480221			−0.240110	−	0.970746i	$-0.577184\pi$
$131$	−62.9089			−0.480221			−0.240110	+	0.970746i	$0.577184\pi$
$132$		0			0
$133$	−248.324			−1.86710
$134$			166.675i			1.24385i
$135$		0			0
$136$		−	64.7940i		−	0.476426i
$137$		−	152.155i		−	1.11062i	−0.831643	−	0.555311i	$-0.812599\pi$
$137$		−	152.155i		−	1.11062i	0.831643	−	0.555311i	$-0.187401\pi$
$138$		0			0
$139$	63.6449			0.457877			0.228938	−	0.973441i	$-0.426475\pi$
$139$	63.6449			0.457877			0.228938	+	0.973441i	$0.426475\pi$
$140$	52.8582			0.377558
$141$		0			0
$142$	41.0076			0.288786
$143$		−	161.046i		−	1.12620i
$144$		0			0
$145$			85.5258i			0.589833i
$146$	−43.9470			−0.301007
$147$		0			0
$148$		−	98.8653i		−	0.668009i
$149$			133.418i			0.895421i	0.894178	+	0.447711i	$0.147761\pi$
$149$			133.418i			0.895421i	−0.894178	+	0.447711i	$0.852239\pi$
$150$		0			0
$151$	133.515			0.884204			0.442102	−	0.896965i	$-0.354233\pi$
$151$	133.515			0.884204			0.442102	+	0.896965i	$0.354233\pi$
$152$			59.4247i			0.390952i
$153$		0			0
$154$	347.094			2.25386
$155$			74.5713i			0.481105i
$156$		0			0
$157$			117.137i			0.746094i	0.927812	+	0.373047i	$0.121687\pi$
$157$			117.137i			0.746094i	−0.927812	+	0.373047i	$0.878313\pi$
$158$		−	148.590i		−	0.940441i
$159$		0			0
$160$		−	12.6491i		−	0.0790569i
$161$	79.4097	−	259.990i	0.493228	−	1.61485i
$162$		0			0
$163$	34.3108			0.210496			0.105248	−	0.994446i	$-0.466436\pi$
$163$	34.3108			0.210496			0.105248	+	0.994446i	$0.466436\pi$
$164$	1.35862			0.00828424
$165$		0			0
$166$			61.0910i			0.368018i
$167$	33.8574			0.202739			0.101370	−	0.994849i	$-0.467678\pi$
$167$	33.8574			0.202739			0.101370	+	0.994849i	$0.467678\pi$
$168$		0			0
$169$	−108.851			−0.644088
$170$	72.4419			0.426129
$171$		0			0
$172$			15.9939i			0.0929879i
$173$	−207.140			−1.19734			−0.598670	−	0.800996i	$-0.704304\pi$
$173$	−207.140			−1.19734			−0.598670	+	0.800996i	$0.704304\pi$
$174$		0			0
$175$			59.0972i			0.337698i
$176$		−	83.0606i		−	0.471935i
$177$		0			0
$178$		−	69.5484i		−	0.390722i
$179$	74.8759			0.418301			0.209151	−	0.977883i	$-0.432930\pi$
$179$	74.8759			0.418301			0.209151	+	0.977883i	$0.432930\pi$
$180$		0			0
$181$			28.3027i			0.156369i	0.996939	+	0.0781844i	$0.0249123\pi$
$181$			28.3027i			0.156369i	−0.996939	+	0.0781844i	$0.975088\pi$
$182$			129.636i			0.712288i
$183$		0			0
$184$	−62.2165	−	19.0030i	−0.338133	−	0.103277i
$185$	110.535			0.597485
$186$		0			0
$187$	475.691			2.54380
$188$	171.887			0.914294
$189$		0			0
$190$	−66.4388			−0.349678
$191$		−	33.2592i		−	0.174132i	−0.996203	−	0.0870659i	$-0.972251\pi$
$191$		−	33.2592i		−	0.174132i	0.996203	−	0.0870659i	$-0.0277491\pi$
$192$		0			0
$193$	93.6041			0.484995			0.242498	−	0.970152i	$-0.422033\pi$
$193$	93.6041			0.484995			0.242498	+	0.970152i	$0.422033\pi$
$194$			98.5336i			0.507905i
$195$		0			0
$196$	−181.399			−0.925503
$197$	−372.604			−1.89139			−0.945696	−	0.325053i	$-0.894618\pi$
$197$	−372.604			−1.89139			−0.945696	+	0.325053i	$0.894618\pi$
$198$		0			0
$199$			59.8348i			0.300678i	0.988635	+	0.150339i	$0.0480364\pi$
$199$			59.8348i			0.300678i	−0.988635	+	0.150339i	$0.951964\pi$
$200$	14.1421			0.0707107
$201$		0			0
$202$	−24.4281			−0.120931
$203$		−	452.074i		−	2.22696i
$204$		0			0
$205$			1.51898i			0.00740965i
$206$		−	73.1750i		−	0.355218i
$207$		0			0
$208$	31.0224			0.149146
$209$	−436.272			−2.08743
$210$		0			0
$211$	−152.649			−0.723455			−0.361728	−	0.932284i	$-0.617813\pi$
$211$	−152.649			−0.723455			−0.361728	+	0.932284i	$0.617813\pi$
$212$			133.396i			0.629227i
$213$		0			0
$214$			76.6732i			0.358286i
$215$	−17.8817			−0.0831709
$216$		0			0
$217$		−	394.170i		−	1.81645i
$218$		−	0.0916298i		−	0.000420320i
$219$		0			0
$220$	92.8646			0.422112
$221$			177.666i			0.803919i
$222$		0			0
$223$	205.883			0.923243			0.461622	−	0.887077i	$-0.347268\pi$
$223$	205.883			0.923243			0.461622	+	0.887077i	$0.347268\pi$
$224$			66.8609i			0.298486i
$225$		0			0
$226$			222.432i			0.984210i
$227$		−	308.272i		−	1.35802i	−0.734127	−	0.679012i	$-0.762408\pi$
$227$		−	308.272i		−	1.35802i	0.734127	−	0.679012i	$-0.237592\pi$
$228$		0			0
$229$			26.6431i			0.116346i	0.998307	+	0.0581728i	$0.0185274\pi$
$229$			26.6431i			0.116346i	−0.998307	+	0.0581728i	$0.981473\pi$
$230$	21.2460	−	69.5601i	0.0923737	−	0.302435i
$231$		0			0
$232$	−108.183			−0.466304
$233$	362.883			1.55744			0.778720	−	0.627372i	$-0.215869\pi$
$233$	362.883			1.55744			0.778720	+	0.627372i	$0.215869\pi$
$234$		0			0
$235$			192.176i			0.817769i
$236$	−221.264			−0.937560
$237$		0			0
$238$	−382.914			−1.60888
$239$	−375.990			−1.57318			−0.786591	−	0.617474i	$-0.788156\pi$
$239$	−375.990			−1.57318			−0.786591	+	0.617474i	$0.788156\pi$
$240$		0			0
$241$		−	214.703i		−	0.890882i	−0.895311	−	0.445441i	$-0.853047\pi$
$241$		−	214.703i		−	0.890882i	0.895311	−	0.445441i	$-0.146953\pi$
$242$	438.677			1.81272
$243$		0			0
$244$		−	33.3559i		−	0.136705i
$245$		−	202.810i		−	0.827795i
$246$		0			0
$247$		−	162.943i		−	0.659690i
$248$	−94.3261			−0.380347
$249$		0			0
$250$			15.8114i			0.0632456i
$251$		−	164.214i		−	0.654237i	−0.944983	−	0.327119i	$-0.893922\pi$
$251$		−	164.214i		−	0.654237i	0.944983	−	0.327119i	$-0.106078\pi$
$252$		0			0
$253$	139.512	−	456.768i	0.551431	−	1.80541i
$254$	−188.614			−0.742576
$255$		0			0
$256$	16.0000			0.0625000
$257$	−92.0757			−0.358271			−0.179136	−	0.983824i	$-0.557330\pi$
$257$	−92.0757			−0.358271			−0.179136	+	0.983824i	$0.557330\pi$
$258$		0			0
$259$	−584.266			−2.25586
$260$			34.6841i			0.133400i
$261$		0			0
$262$	88.9667			0.339567
$263$		−	340.726i		−	1.29554i	−0.761837	−	0.647769i	$-0.775702\pi$
$263$		−	340.726i		−	1.29554i	0.761837	−	0.647769i	$-0.224298\pi$
$264$		0			0
$265$	−149.141			−0.562798
$266$	351.184			1.32024
$267$		0			0
$268$		−	235.715i		−	0.879532i
$269$	−253.330			−0.941748			−0.470874	−	0.882200i	$-0.656061\pi$
$269$	−253.330			−0.941748			−0.470874	+	0.882200i	$0.656061\pi$
$270$		0			0
$271$	−21.6513			−0.0798939			−0.0399470	−	0.999202i	$-0.512719\pi$
$271$	−21.6513			−0.0798939			−0.0399470	+	0.999202i	$0.512719\pi$
$272$			91.6325i			0.336884i
$273$		0			0
$274$			215.180i			0.785328i
$275$			103.826i			0.377548i
$276$		0			0
$277$	459.389			1.65844			0.829222	−	0.558919i	$-0.188784\pi$
$277$	459.389			1.65844			0.829222	+	0.558919i	$0.188784\pi$
$278$	−90.0074			−0.323768
$279$		0			0
$280$	−74.7527			−0.266974
$281$			318.671i			1.13406i	0.823697	+	0.567030i	$0.191908\pi$
$281$			318.671i			1.13406i	−0.823697	+	0.567030i	$0.808092\pi$
$282$		0			0
$283$		−	5.36172i		−	0.0189460i	−0.999955	−	0.00947301i	$-0.996985\pi$
$283$		−	5.36172i		−	0.0189460i	0.999955	−	0.00947301i	$-0.00301540\pi$
$284$	−57.9935			−0.204202
$285$		0			0
$286$			227.753i			0.796341i
$287$		−	8.02904i		−	0.0279758i
$288$		0			0
$289$	−235.782			−0.815856
$290$		−	120.952i		−	0.417075i
$291$		0			0
$292$	62.1505			0.212844
$293$			519.450i			1.77287i	0.462854	+	0.886434i	$0.346825\pi$
$293$			519.450i			1.77287i	−0.462854	+	0.886434i	$0.653175\pi$
$294$		0			0
$295$		−	247.381i		−	0.838579i
$296$			139.817i			0.472353i
$297$		0			0
$298$		−	188.681i		−	0.633159i
$299$	170.598	+	52.1064i	0.570563	+	0.174269i
$300$		0			0
$301$	94.5196			0.314019
$302$	−188.818			−0.625227
$303$		0			0
$304$		−	84.0392i		−	0.276445i
$305$	37.2930			0.122272
$306$		0			0
$307$	−465.601			−1.51662			−0.758308	−	0.651897i	$-0.773974\pi$
$307$	−465.601			−1.51662			−0.758308	+	0.651897i	$0.773974\pi$
$308$	−490.865			−1.59372
$309$		0			0
$310$		−	105.460i		−	0.340193i
$311$	−299.817			−0.964043			−0.482021	−	0.876159i	$-0.660097\pi$
$311$	−299.817			−0.964043			−0.482021	+	0.876159i	$0.660097\pi$
$312$		0			0
$313$		−	473.837i		−	1.51386i	−0.653498	−	0.756928i	$-0.726699\pi$
$313$		−	473.837i		−	1.51386i	0.653498	−	0.756928i	$-0.273301\pi$
$314$		−	165.656i		−	0.527568i
$315$		0			0
$316$			210.138i			0.664992i
$317$	−217.165			−0.685065			−0.342532	−	0.939506i	$-0.611285\pi$
$317$	−217.165			−0.685065			−0.342532	+	0.939506i	$0.611285\pi$
$318$		0			0
$319$		−	794.232i		−	2.48976i
$320$			17.8885i			0.0559017i
$321$		0			0
$322$	−112.302	+	367.682i	−0.348765	+	1.14187i
$323$	481.295			1.49008
$324$		0			0
$325$	−38.7779			−0.119317
$326$	−48.5229			−0.148843
$327$		0			0
$328$	−1.92137			−0.00585784
$329$		−	1015.81i		−	3.08756i
$330$		0			0
$331$	−275.838			−0.833348			−0.416674	−	0.909056i	$-0.636804\pi$
$331$	−275.838			−0.833348			−0.416674	+	0.909056i	$0.636804\pi$
$332$		−	86.3957i		−	0.260228i
$333$		0			0
$334$	−47.8816			−0.143358
$335$	263.537			0.786677
$336$		0			0
$337$			201.315i			0.597374i	0.954351	+	0.298687i	$0.0965487\pi$
$337$			201.315i			0.597374i	−0.954351	+	0.298687i	$0.903451\pi$
$338$	153.938			0.455439
$339$		0			0
$340$	−102.448			−0.301318
$341$		−	692.503i		−	2.03080i
$342$		0			0
$343$			492.863i			1.43692i
$344$		−	22.6188i		−	0.0657524i
$345$		0			0
$346$	292.940			0.846647
$347$	−152.784			−0.440300			−0.220150	−	0.975466i	$-0.570655\pi$
$347$	−152.784			−0.440300			−0.220150	+	0.975466i	$0.570655\pi$
$348$		0			0
$349$	599.141			1.71674			0.858368	−	0.513035i	$-0.171479\pi$
$349$	599.141			1.71674			0.858368	+	0.513035i	$0.171479\pi$
$350$		−	83.5761i		−	0.238789i
$351$		0			0
$352$			117.465i			0.333709i
$353$	574.915			1.62865			0.814327	−	0.580407i	$-0.197106\pi$
$353$	574.915			1.62865			0.814327	+	0.580407i	$0.197106\pi$
$354$		0			0
$355$		−	64.8387i		−	0.182644i
$356$			98.3563i			0.276282i
$357$		0			0
$358$	−105.890			−0.295783
$359$		−	165.976i		−	0.462330i	−0.972915	−	0.231165i	$-0.925746\pi$
$359$		−	165.976i		−	0.462330i	0.972915	−	0.231165i	$-0.0742537\pi$
$360$		0			0
$361$	−80.4119			−0.222748
$362$		−	40.0261i		−	0.110569i
$363$		0			0
$364$		−	183.334i		−	0.503664i
$365$			69.4863i			0.190373i
$366$		0			0
$367$			433.965i			1.18247i	0.806501	+	0.591233i	$0.201358\pi$
$367$			433.965i			1.18247i	−0.806501	+	0.591233i	$0.798642\pi$
$368$	87.9874	+	26.8742i	0.239096	+	0.0730278i
$369$		0			0
$370$	−156.320			−0.422486
$371$	788.335			2.12489
$372$		0			0
$373$		−	342.059i		−	0.917048i	−0.888682	−	0.458524i	$-0.848378\pi$
$373$		−	342.059i		−	0.917048i	0.888682	−	0.458524i	$-0.151622\pi$
$374$	−672.728			−1.79874
$375$		0			0
$376$	−243.085			−0.646503
$377$	296.638			0.786839
$378$		0			0
$379$		−	76.7922i		−	0.202618i	−0.994855	−	0.101309i	$-0.967697\pi$
$379$		−	76.7922i		−	0.202618i	0.994855	−	0.101309i	$-0.0323031\pi$
$380$	93.9587			0.247260
$381$		0			0
$382$			47.0356i			0.123130i
$383$		−	78.7390i		−	0.205585i	−0.994703	−	0.102792i	$-0.967222\pi$
$383$		−	78.7390i		−	0.205585i	0.994703	−	0.102792i	$-0.0327777\pi$
$384$		0			0
$385$		−	548.804i		−	1.42547i
$386$	−132.376			−0.342944
$387$		0			0
$388$		−	139.348i		−	0.359143i
$389$			65.8586i			0.169302i	0.996411	+	0.0846511i	$0.0269776\pi$
$389$			65.8586i			0.169302i	−0.996411	+	0.0846511i	$0.973022\pi$
$390$		0			0
$391$	−153.910	+	503.906i	−0.393631	+	1.28876i
$392$	256.536			0.654430
$393$		0			0
$394$	526.942			1.33742
$395$	−234.941			−0.594787
$396$		0			0
$397$	−261.799			−0.659443			−0.329721	−	0.944078i	$-0.606955\pi$
$397$	−261.799			−0.659443			−0.329721	+	0.944078i	$0.606955\pi$
$398$		−	84.6192i		−	0.212611i
$399$		0			0
$400$	−20.0000			−0.0500000
$401$			136.192i			0.339630i	0.985476	+	0.169815i	$0.0543171\pi$
$401$			136.192i			0.339630i	−0.985476	+	0.169815i	$0.945683\pi$
$402$		0			0
$403$	258.644			0.641795
$404$	34.5465			0.0855111
$405$		0			0
$406$			639.329i			1.57470i
$407$	−1026.48			−2.52205
$408$		0			0
$409$	−162.202			−0.396583			−0.198291	−	0.980143i	$-0.563539\pi$
$409$	−162.202			−0.396583			−0.198291	+	0.980143i	$0.563539\pi$
$410$		−	2.14816i		−	0.00523942i
$411$		0			0
$412$			103.485i			0.251177i
$413$			1307.61i			3.16613i
$414$		0			0
$415$	96.5933			0.232755
$416$	−43.8722			−0.105462
$417$		0			0
$418$	616.982			1.47603
$419$		−	746.726i		−	1.78216i	−0.453844	−	0.891081i	$-0.649948\pi$
$419$		−	746.726i		−	1.78216i	0.453844	−	0.891081i	$-0.350052\pi$
$420$		0			0
$421$		−	675.466i		−	1.60443i	−0.597033	−	0.802216i	$-0.703654\pi$
$421$		−	675.466i		−	1.60443i	0.597033	−	0.802216i	$-0.296346\pi$
$422$	215.878			0.511560
$423$		0			0
$424$		−	188.651i		−	0.444931i
$425$		−	114.541i		−	0.269507i
$426$		0			0
$427$	−197.124			−0.461649
$428$		−	108.432i		−	0.253346i
$429$		0			0
$430$	25.2886			0.0588107
$431$			270.979i			0.628721i	0.949304	+	0.314360i	$0.101790\pi$
$431$			270.979i			0.628721i	−0.949304	+	0.314360i	$0.898210\pi$
$432$		0			0
$433$			549.884i			1.26994i	0.772537	+	0.634970i	$0.218988\pi$
$433$			549.884i			1.26994i	−0.772537	+	0.634970i	$0.781012\pi$
$434$			557.441i			1.28443i
$435$		0			0
$436$			0.129584i			0.000297211i
$437$	141.156	−	462.149i	0.323011	−	1.05755i
$438$		0			0
$439$	342.257			0.779628			0.389814	−	0.920894i	$-0.372539\pi$
$439$	342.257			0.779628			0.389814	+	0.920894i	$0.372539\pi$
$440$	−131.330			−0.298478
$441$		0			0
$442$		−	251.258i		−	0.568456i
$443$	−476.091			−1.07470			−0.537349	−	0.843360i	$-0.680574\pi$
$443$	−476.091			−1.07470			−0.537349	+	0.843360i	$0.680574\pi$
$444$		0			0
$445$	−109.966			−0.247114
$446$	−291.163			−0.652832
$447$		0			0
$448$		−	94.5556i		−	0.211062i
$449$	−554.189			−1.23427			−0.617137	−	0.786855i	$-0.711708\pi$
$449$	−554.189			−1.23427			−0.617137	+	0.786855i	$0.711708\pi$
$450$		0			0
$451$		−	14.1059i		−	0.0312770i
$452$		−	314.566i		−	0.695942i
$453$		0			0
$454$			435.962i			0.960269i
$455$	204.973			0.450490
$456$		0			0
$457$			142.050i			0.310832i	0.987849	+	0.155416i	$0.0496718\pi$
$457$			142.050i			0.310832i	−0.987849	+	0.155416i	$0.950328\pi$
$458$		−	37.6791i		−	0.0822688i
$459$		0			0
$460$	−30.0463	+	98.3729i	−0.0653181	+	0.213854i
$461$	618.196			1.34099			0.670495	−	0.741914i	$-0.266082\pi$
$461$	618.196			1.34099			0.670495	+	0.741914i	$0.266082\pi$
$462$		0			0
$463$	115.319			0.249069			0.124534	−	0.992215i	$-0.460256\pi$
$463$	115.319			0.249069			0.124534	+	0.992215i	$0.460256\pi$
$464$	152.993			0.329727
$465$		0			0
$466$	−513.195			−1.10128
$467$			778.924i			1.66793i	0.551817	+	0.833966i	$0.313935\pi$
$467$			778.924i			1.66793i	−0.551817	+	0.833966i	$0.686065\pi$
$468$		0			0
$469$	−1393.01			−2.97017
$470$		−	271.777i		−	0.578250i
$471$		0			0
$472$	312.915			0.662955
$473$	166.058			0.351074
$474$		0			0
$475$			105.049i			0.221156i
$476$	541.523			1.13765
$477$		0			0
$478$	531.731			1.11241
$479$			18.0817i			0.0377488i	0.999822	+	0.0188744i	$0.00600826\pi$
$479$			18.0817i			0.0377488i	−0.999822	+	0.0188744i	$0.993992\pi$
$480$		0			0
$481$		−	383.379i		−	0.797046i
$482$			303.635i			0.629949i
$483$		0			0
$484$	−620.383			−1.28178
$485$	155.795			0.321228
$486$		0			0
$487$	522.588			1.07308			0.536538	−	0.843876i	$-0.319732\pi$
$487$	522.588			1.07308			0.536538	+	0.843876i	$0.319732\pi$
$488$			47.1724i			0.0966647i
$489$		0			0
$490$			286.816i			0.585340i
$491$	−524.138			−1.06749			−0.533746	−	0.845645i	$-0.679216\pi$
$491$	−524.138			−1.06749			−0.533746	+	0.845645i	$0.679216\pi$
$492$		0			0
$493$			876.197i			1.77728i
$494$			230.437i			0.466471i
$495$		0			0
$496$	133.397			0.268946
$497$			342.725i			0.689588i
$498$		0			0
$499$	−384.131			−0.769801			−0.384900	−	0.922958i	$-0.625764\pi$
$499$	−384.131			−0.769801			−0.384900	+	0.922958i	$0.625764\pi$
$500$		−	22.3607i		−	0.0447214i
$501$		0			0
$502$			232.233i			0.462616i
$503$		−	815.894i		−	1.62206i	−0.585007	−	0.811028i	$-0.698908\pi$
$503$		−	815.894i		−	1.62206i	0.585007	−	0.811028i	$-0.301092\pi$
$504$		0			0
$505$			38.6241i			0.0764835i
$506$	−197.300	+	645.967i	−0.389920	+	1.27661i
$507$		0			0
$508$	266.741			0.525080
$509$	138.733			0.272560			0.136280	−	0.990670i	$-0.456485\pi$
$509$	138.733			0.272560			0.136280	+	0.990670i	$0.456485\pi$
$510$		0			0
$511$		−	367.292i		−	0.718771i
$512$	−22.6274			−0.0441942
$513$		0			0
$514$	130.215			0.253336
$515$	−115.700			−0.224660
$516$		0			0
$517$		−	1784.63i		−	3.45190i
$518$	826.278			1.59513
$519$		0			0
$520$		−	49.0507i		−	0.0943282i
$521$		−	79.0746i		−	0.151775i	−0.997116	−	0.0758873i	$-0.975821\pi$
$521$		−	79.0746i		−	0.151775i	0.997116	−	0.0758873i	$-0.0241789\pi$
$522$		0			0
$523$		−	188.376i		−	0.360184i	−0.983650	−	0.180092i	$-0.942360\pi$
$523$		−	188.376i		−	0.360184i	0.983650	−	0.180092i	$-0.0576396\pi$
$524$	−125.818			−0.240110
$525$		0			0
$526$			481.860i			0.916083i
$527$			763.970i			1.44966i
$528$		0			0
$529$	438.722	+	295.574i	0.829342	+	0.558741i
$530$	210.918			0.397958
$531$		0			0
$532$	−496.649			−0.933550
$533$	5.26843			0.00988449
$534$		0			0
$535$	121.231			0.226600
$536$			333.351i			0.621923i
$537$		0			0
$538$	358.263			0.665917
$539$			1883.39i			3.49422i
$540$		0			0
$541$	−362.416			−0.669900			−0.334950	−	0.942236i	$-0.608719\pi$
$541$	−362.416			−0.669900			−0.334950	+	0.942236i	$0.608719\pi$
$542$	30.6195			0.0564935
$543$		0			0
$544$		−	129.588i		−	0.238213i
$545$	−0.144879			−0.000265834
$546$		0			0
$547$	−593.659			−1.08530			−0.542650	−	0.839959i	$-0.682579\pi$
$547$	−593.659			−1.08530			−0.542650	+	0.839959i	$0.682579\pi$
$548$		−	304.310i		−	0.555311i
$549$		0			0
$550$		−	146.832i		−	0.266967i
$551$		−	803.589i		−	1.45842i
$552$		0			0
$553$	1241.85			2.24567
$554$	−649.674			−1.17270
$555$		0			0
$556$	127.290			0.228938
$557$			183.937i			0.330229i	0.986274	+	0.165114i	$0.0527993\pi$
$557$			183.937i			0.330229i	−0.986274	+	0.165114i	$0.947201\pi$
$558$		0			0
$559$			62.0211i			0.110950i
$560$	105.716			0.188779
$561$		0			0
$562$		−	450.668i		−	0.801901i
$563$			447.708i			0.795218i	0.917555	+	0.397609i	$0.130160\pi$
$563$			447.708i			0.795218i	−0.917555	+	0.397609i	$0.869840\pi$
$564$		0			0
$565$	351.695			0.622469
$566$			7.58262i			0.0133969i
$567$		0			0
$568$	82.0151			0.144393
$569$		−	334.865i		−	0.588515i	−0.955726	−	0.294258i	$-0.904928\pi$
$569$		−	334.865i		−	0.588515i	0.955726	−	0.294258i	$-0.0950724\pi$
$570$		0			0
$571$			644.554i			1.12882i	0.825496	+	0.564408i	$0.190896\pi$
$571$			644.554i			1.12882i	−0.825496	+	0.564408i	$0.809104\pi$
$572$		−	322.092i		−	0.563098i
$573$		0			0
$574$			11.3548i			0.0197819i
$575$	−109.984	−	33.5928i	−0.191277	−	0.0584223i
$576$		0			0
$577$	82.7980			0.143497			0.0717487	−	0.997423i	$-0.477142\pi$
$577$	82.7980			0.143497			0.0717487	+	0.997423i	$0.477142\pi$
$578$	333.447			0.576897
$579$		0			0
$580$			171.052i			0.294917i
$581$	−510.575			−0.878786
$582$		0			0
$583$	1385.00			2.37564
$584$	−87.8940			−0.150503
$585$		0			0
$586$		−	734.614i		−	1.25361i
$587$	−48.0652			−0.0818827			−0.0409414	−	0.999162i	$-0.513036\pi$
$587$	−48.0652			−0.0818827			−0.0409414	+	0.999162i	$0.513036\pi$
$588$		0			0
$589$		−	700.662i		−	1.18958i
$590$			349.849i			0.592965i
$591$		0			0
$592$		−	197.731i		−	0.334004i
$593$	115.147			0.194176			0.0970882	−	0.995276i	$-0.469047\pi$
$593$	115.147			0.194176			0.0970882	+	0.995276i	$0.469047\pi$
$594$		0			0
$595$			605.441i			1.01755i
$596$			266.836i			0.447711i
$597$		0			0
$598$	−241.263	−	73.6896i	−0.403449	−	0.123227i
$599$	813.749			1.35851			0.679256	−	0.733901i	$-0.262303\pi$
$599$	813.749			1.35851			0.679256	+	0.733901i	$0.262303\pi$
$600$		0			0
$601$	−550.200			−0.915474			−0.457737	−	0.889088i	$-0.651340\pi$
$601$	−550.200			−0.915474			−0.457737	+	0.889088i	$0.651340\pi$
$602$	−133.671			−0.222045
$603$		0			0
$604$	267.030			0.442102
$605$		−	693.609i		−	1.14646i
$606$		0			0
$607$	287.538			0.473703			0.236851	−	0.971546i	$-0.423885\pi$
$607$	287.538			0.473703			0.236851	+	0.971546i	$0.423885\pi$
$608$			118.849i			0.195476i
$609$		0			0
$610$	−52.7403			−0.0864596
$611$	666.543			1.09091
$612$		0			0
$613$			573.173i			0.935029i	0.883985	+	0.467514i	$0.154850\pi$
$613$			573.173i			0.935029i	−0.883985	+	0.467514i	$0.845150\pi$
$614$	658.459			1.07241
$615$		0			0
$616$	694.188			1.12693
$617$			1156.17i			1.87385i	0.349525	+	0.936927i	$0.386343\pi$
$617$			1156.17i			1.87385i	−0.349525	+	0.936927i	$0.613657\pi$
$618$		0			0
$619$			14.9600i			0.0241680i	0.999927	+	0.0120840i	$0.00384655\pi$
$619$			14.9600i			0.0241680i	−0.999927	+	0.0120840i	$0.996153\pi$
$620$			149.143i			0.240553i
$621$		0			0
$622$	424.006			0.681681
$623$	581.259			0.933000
$624$		0			0
$625$	25.0000			0.0400000
$626$			670.107i			1.07046i
$627$		0			0
$628$			234.273i			0.373047i
$629$	1132.41			1.80033
$630$		0			0
$631$		−	361.056i		−	0.572196i	−0.958200	−	0.286098i	$-0.907642\pi$
$631$		−	361.056i		−	0.572196i	0.958200	−	0.286098i	$-0.0923583\pi$
$632$		−	297.179i		−	0.470220i
$633$		0			0
$634$	307.118			0.484414
$635$			298.225i			0.469646i
$636$		0			0
$637$	−703.427			−1.10428
$638$			1123.21i			1.76052i
$639$		0			0
$640$		−	25.2982i		−	0.0395285i
$641$			1132.34i			1.76652i	0.468881	+	0.883261i	$0.344657\pi$
$641$			1132.34i			1.76652i	−0.468881	+	0.883261i	$0.655343\pi$
$642$		0			0
$643$		−	753.140i		−	1.17129i	−0.810567	−	0.585645i	$-0.800841\pi$
$643$		−	753.140i		−	1.17129i	0.810567	−	0.585645i	$-0.199159\pi$
$644$	158.819	−	519.981i	0.246614	−	0.807424i
$645$		0			0
$646$	−680.654			−1.05364
$647$	32.8271			0.0507375			0.0253687	−	0.999678i	$-0.491924\pi$
$647$	32.8271			0.0507375			0.0253687	+	0.999678i	$0.491924\pi$
$648$		0			0
$649$			2297.29i			3.53974i
$650$	54.8403			0.0843697
$651$		0			0
$652$	68.6217			0.105248
$653$	1288.05			1.97251			0.986257	−	0.165219i	$-0.0528332\pi$
$653$	1288.05			1.97251			0.986257	+	0.165219i	$0.0528332\pi$
$654$		0			0
$655$		−	140.669i		−	0.214761i
$656$	2.71723			0.00414212
$657$		0			0
$658$			1436.57i			2.18323i
$659$			101.045i			0.153330i	0.997057	+	0.0766652i	$0.0244273\pi$
$659$			101.045i			0.153330i	−0.997057	+	0.0766652i	$0.975573\pi$
$660$		0			0
$661$			597.593i			0.904074i	0.891999	+	0.452037i	$0.149302\pi$
$661$			597.593i			0.904074i	−0.891999	+	0.452037i	$0.850698\pi$
$662$	390.094			0.589266
$663$		0			0
$664$			122.182i			0.184009i
$665$		−	555.270i		−	0.834992i
$666$		0			0
$667$	841.342	+	256.974i	1.26138	+	0.385268i
$668$	67.7149			0.101370
$669$		0			0
$670$	−372.697			−0.556265
$671$	−346.320			−0.516126
$672$		0			0
$673$	178.729			0.265570			0.132785	−	0.991145i	$-0.457608\pi$
$673$	178.729			0.265570			0.132785	+	0.991145i	$0.457608\pi$
$674$		−	284.703i		−	0.422407i
$675$		0			0
$676$	−217.702			−0.322044
$677$		−	1197.98i		−	1.76955i	−0.466023	−	0.884773i	$-0.654314\pi$
$677$		−	1197.98i		−	1.76955i	0.466023	−	0.884773i	$-0.345686\pi$
$678$		0			0
$679$	−823.506			−1.21282
$680$	144.884			0.213064
$681$		0			0
$682$			979.348i			1.43599i
$683$	−264.345			−0.387035			−0.193518	−	0.981097i	$-0.561990\pi$
$683$	−264.345			−0.387035			−0.193518	+	0.981097i	$0.561990\pi$
$684$		0			0
$685$	340.229			0.496685
$686$		−	697.014i		−	1.01605i
$687$		0			0
$688$			31.9878i			0.0464939i
$689$			517.283i			0.750774i
$690$		0			0
$691$	236.685			0.342525			0.171262	−	0.985225i	$-0.445215\pi$
$691$	236.685			0.342525			0.171262	+	0.985225i	$0.445215\pi$
$692$	−414.280			−0.598670
$693$		0			0
$694$	216.069			0.311339
$695$			142.314i			0.204769i
$696$		0			0
$697$			15.5617i			0.0223266i
$698$	−847.313			−1.21392
$699$		0			0
$700$			118.194i			0.168849i
$701$		−	322.720i		−	0.460371i	−0.973147	−	0.230186i	$-0.926067\pi$
$701$		−	322.720i		−	0.460371i	0.973147	−	0.230186i	$-0.0739334\pi$
$702$		0			0
$703$	−1038.57			−1.47734
$704$		−	166.121i		−	0.235968i
$705$		0			0
$706$	−813.052			−1.15163
$707$		−	204.160i		−	0.288770i
$708$		0			0
$709$			100.401i			0.141609i	0.997490	+	0.0708046i	$0.0225567\pi$
$709$			100.401i			0.141609i	−0.997490	+	0.0708046i	$0.977443\pi$
$710$			91.6957i			0.129149i
$711$		0			0
$712$		−	139.097i		−	0.195361i
$713$	733.579	+	224.059i	1.02886	+	0.314249i
$714$		0			0
$715$	360.110			0.503650
$716$	149.752			0.209151
$717$		0			0
$718$			234.726i			0.326917i
$719$	−1250.73			−1.73953			−0.869767	−	0.493462i	$-0.835731\pi$
$719$	−1250.73			−1.73953			−0.869767	+	0.493462i	$0.835731\pi$
$720$		0			0
$721$	611.568			0.848222
$722$	113.720			0.157506
$723$		0			0
$724$			56.6055i			0.0781844i
$725$	−191.242			−0.263781
$726$		0			0
$727$			528.487i			0.726942i	0.931605	+	0.363471i	$0.118408\pi$
$727$			528.487i			0.726942i	−0.931605	+	0.363471i	$0.881592\pi$
$728$			259.273i			0.356144i
$729$		0			0
$730$		−	98.2685i		−	0.134614i
$731$	−183.195			−0.250609
$732$		0			0
$733$			37.5960i			0.0512905i	0.999671	+	0.0256453i	$0.00816403\pi$
$733$			37.5960i			0.0512905i	−0.999671	+	0.0256453i	$0.991836\pi$
$734$		−	613.719i		−	0.836129i
$735$		0			0
$736$	−124.433	−	38.0059i	−0.169066	−	0.0516385i
$737$	−2447.32			−3.32066
$738$		0			0
$739$	−1383.68			−1.87237			−0.936187	−	0.351502i	$-0.885671\pi$
$739$	−1383.68			−1.87237			−0.936187	+	0.351502i	$0.885671\pi$
$740$	221.069			0.298743
$741$		0			0
$742$	−1114.87			−1.50253
$743$			143.332i			0.192909i	0.995337	+	0.0964547i	$0.0307503\pi$
$743$			143.332i			0.192909i	−0.995337	+	0.0964547i	$0.969250\pi$
$744$		0			0
$745$	−298.331			−0.400445
$746$			483.745i			0.648451i
$747$		0			0
$748$	951.382			1.27190
$749$	−640.805			−0.855547
$750$		0			0
$751$			1057.29i			1.40784i	0.710277	+	0.703922i	$0.248569\pi$
$751$			1057.29i			1.40784i	−0.710277	+	0.703922i	$0.751431\pi$
$752$	343.774			0.457147
$753$		0			0
$754$	−419.510			−0.556379
$755$			298.548i			0.395428i
$756$		0			0
$757$		−	85.5898i		−	0.113065i	−0.998401	−	0.0565323i	$-0.981996\pi$
$757$		−	85.5898i		−	0.113065i	0.998401	−	0.0565323i	$-0.0180044\pi$
$758$			108.601i			0.143272i
$759$		0			0
$760$	−132.878			−0.174839
$761$	751.679			0.987752			0.493876	−	0.869532i	$-0.335580\pi$
$761$	751.679			0.987752			0.493876	+	0.869532i	$0.335580\pi$
$762$		0			0
$763$	0.765806			0.00100368
$764$		−	66.5184i		−	0.0870659i
$765$		0			0
$766$			111.354i			0.145370i
$767$	−858.017			−1.11867
$768$		0			0
$769$		−	80.5080i		−	0.104692i	−0.998629	−	0.0523459i	$-0.983330\pi$
$769$		−	80.5080i		−	0.104692i	0.998629	−	0.0523459i	$-0.0166698\pi$
$770$			776.126i			1.00796i
$771$		0			0
$772$	187.208			0.242498
$773$			1038.03i			1.34286i	0.741070	+	0.671428i	$0.234319\pi$
$773$			1038.03i			1.34286i	−0.741070	+	0.671428i	$0.765681\pi$
$774$		0			0
$775$	−166.747			−0.215157
$776$			197.067i			0.253953i
$777$		0			0
$778$		−	93.1381i		−	0.119715i
$779$		−	14.2721i		−	0.0183211i
$780$		0			0
$781$			602.122i			0.770962i
$782$	217.661	−	712.631i	0.278339	−	0.911293i
$783$		0			0
$784$	−362.797			−0.462752
$785$	−261.926			−0.333663
$786$		0			0
$787$			1078.55i			1.37045i	0.728330	+	0.685227i	$0.240297\pi$
$787$			1078.55i			1.37045i	−0.728330	+	0.685227i	$0.759703\pi$
$788$	−745.208			−0.945696
$789$		0			0
$790$	332.257			0.420578
$791$	−1859.00			−2.35019
$792$		0			0
$793$		−	129.347i		−	0.163111i
$794$	370.239			0.466297
$795$		0			0
$796$			119.670i			0.150339i
$797$		−	407.609i		−	0.511429i	−0.966752	−	0.255714i	$-0.917689\pi$
$797$		−	407.609i		−	0.511429i	0.966752	−	0.255714i	$-0.0823106\pi$
$798$		0			0
$799$			1968.81i			2.46409i
$800$	28.2843			0.0353553
$801$		0			0
$802$		−	192.604i		−	0.240155i
$803$		−	645.282i		−	0.803589i
$804$		0			0
$805$	581.356	+	177.565i	0.722182	+	0.220578i
$806$	−365.777			−0.453818
$807$		0			0
$808$	−48.8561			−0.0604655
$809$	−788.093			−0.974158			−0.487079	−	0.873358i	$-0.661938\pi$
$809$	−788.093			−0.974158			−0.487079	+	0.873358i	$0.661938\pi$
$810$		0			0
$811$	779.565			0.961239			0.480619	−	0.876929i	$-0.340412\pi$
$811$	779.565			0.961239			0.480619	+	0.876929i	$0.340412\pi$
$812$		−	904.148i		−	1.11348i
$813$		0			0
$814$	1451.66			1.78336
$815$			76.7214i			0.0941366i
$816$		0			0
$817$	168.015			0.205648
$818$	229.389			0.280426
$819$		0			0
$820$			3.03796i			0.00370483i
$821$	−1297.12			−1.57992			−0.789962	−	0.613156i	$-0.789900\pi$
$821$	−1297.12			−1.57992			−0.789962	+	0.613156i	$0.789900\pi$
$822$		0			0
$823$	−562.355			−0.683299			−0.341649	−	0.939828i	$-0.610986\pi$
$823$	−562.355			−0.683299			−0.341649	+	0.939828i	$0.610986\pi$
$824$		−	146.350i		−	0.177609i
$825$		0			0
$826$		−	1849.24i		−	2.23879i
$827$			1203.99i			1.45585i	0.685656	+	0.727926i	$0.259515\pi$
$827$			1203.99i			1.45585i	−0.685656	+	0.727926i	$0.740485\pi$
$828$		0			0
$829$	1175.99			1.41856			0.709282	−	0.704925i	$-0.249019\pi$
$829$	1175.99			1.41856			0.709282	+	0.704925i	$0.249019\pi$
$830$	−136.604			−0.164583
$831$		0			0
$832$	62.0447			0.0745730
$833$		−	2077.75i		−	2.49430i
$834$		0			0
$835$			75.7075i			0.0906677i
$836$	−872.544			−1.04371
$837$		0			0
$838$			1056.03i			1.26018i
$839$			620.713i			0.739825i	0.929067	+	0.369912i	$0.120612\pi$
$839$			620.713i			0.739825i	−0.929067	+	0.369912i	$0.879388\pi$
$840$		0			0
$841$	621.933			0.739516
$842$			955.254i			1.13451i
$843$		0			0
$844$	−305.298			−0.361728
$845$		−	243.398i		−	0.288045i
$846$		0			0
$847$			3666.29i			4.32856i
$848$			266.792i			0.314614i
$849$		0			0
$850$			161.985i			0.190570i
$851$	332.116	−	1087.36i	0.390266	−	1.27775i
$852$		0			0
$853$	−576.042			−0.675314			−0.337657	−	0.941269i	$-0.609634\pi$
$853$	−576.042			−0.675314			−0.337657	+	0.941269i	$0.609634\pi$
$854$	278.776			0.326435
$855$		0			0
$856$			153.346i			0.179143i
$857$	−1079.53			−1.25966			−0.629832	−	0.776731i	$-0.716876\pi$
$857$	−1079.53			−1.25966			−0.629832	+	0.776731i	$0.716876\pi$
$858$		0			0
$859$	−314.625			−0.366269			−0.183134	−	0.983088i	$-0.558624\pi$
$859$	−314.625			−0.366269			−0.183134	+	0.983088i	$0.558624\pi$
$860$	−35.7635			−0.0415854
$861$		0			0
$862$		−	383.222i		−	0.444573i
$863$	1073.21			1.24358			0.621791	−	0.783183i	$-0.286405\pi$
$863$	1073.21			1.24358			0.621791	+	0.783183i	$0.286405\pi$
$864$		0			0
$865$		−	463.179i		−	0.535467i
$866$		−	777.654i		−	0.897983i
$867$		0			0
$868$		−	788.341i		−	0.908227i
$869$	2181.77			2.51067
$870$		0			0
$871$		−	914.053i		−	1.04943i
$872$		−	0.183260i		−	0.000210160i
$873$		0			0
$874$	−199.624	+	653.578i	−0.228403	+	0.747801i
$875$	−132.145			−0.151023
$876$		0			0
$877$	−26.5292			−0.0302500			−0.0151250	−	0.999886i	$-0.504815\pi$
$877$	−26.5292			−0.0302500			−0.0151250	+	0.999886i	$0.504815\pi$
$878$	−484.024			−0.551280
$879$		0			0
$880$	185.729			0.211056
$881$			528.674i			0.600084i	0.953926	+	0.300042i	$0.0970007\pi$
$881$			528.674i			0.600084i	−0.953926	+	0.300042i	$0.902999\pi$
$882$		0			0
$883$	1219.35			1.38092			0.690461	−	0.723370i	$-0.257408\pi$
$883$	1219.35			1.38092			0.690461	+	0.723370i	$0.257408\pi$
$884$			355.332i			0.401959i
$885$		0			0
$886$	673.294			0.759926
$887$	1586.65			1.78878			0.894390	−	0.447289i	$-0.147610\pi$
$887$	1586.65			1.78878			0.894390	+	0.447289i	$0.147610\pi$
$888$		0			0
$889$		−	1576.36i		−	1.77319i
$890$	155.515			0.174736
$891$		0			0
$892$	411.767			0.461622
$893$		−	1805.66i		−	2.02201i
$894$		0			0
$895$			167.428i			0.187070i
$896$			133.722i			0.149243i
$897$		0			0
$898$	783.742			0.872764
$899$	1275.55			1.41886
$900$		0			0
$901$	−1527.93			−1.69581
$902$			19.9488i			0.0221162i
$903$		0			0
$904$			444.863i			0.492105i
$905$	−63.2868			−0.0699302
$906$		0			0
$907$			107.955i			0.119024i	0.998228	+	0.0595122i	$0.0189545\pi$
$907$			107.955i			0.119024i	−0.998228	+	0.0595122i	$0.981045\pi$
$908$		−	616.543i		−	0.679012i
$909$		0			0
$910$	−289.876			−0.318545
$911$			949.537i			1.04230i	0.853464	+	0.521151i	$0.174497\pi$
$911$			949.537i			1.04230i	−0.853464	+	0.521151i	$0.825503\pi$
$912$		0			0
$913$	−897.010			−0.982486
$914$		−	200.889i		−	0.219791i
$915$		0			0
$916$			53.2863i			0.0581728i
$917$			743.549i			0.810849i
$918$		0			0
$919$		−	88.6314i		−	0.0964433i	−0.998837	−	0.0482217i	$-0.984645\pi$
$919$		−	88.6314i		−	0.0964433i	0.998837	−	0.0482217i	$-0.0153554\pi$
$920$	42.4919	−	139.120i	0.0461869	−	0.151218i
$921$		0			0
$922$	−874.261			−0.948223
$923$	−224.887			−0.243648
$924$		0			0
$925$			247.163i			0.267203i
$926$	−163.085			−0.176118
$927$		0			0
$928$	−216.365			−0.233152
$929$	614.940			0.661938			0.330969	−	0.943642i	$-0.392625\pi$
$929$	614.940			0.661938			0.330969	+	0.943642i	$0.392625\pi$
$930$		0			0
$931$			1905.58i			2.04680i
$932$	725.767			0.778720
$933$		0			0
$934$		−	1101.56i		−	1.17941i
$935$			1063.68i			1.13762i
$936$		0			0
$937$		−	79.7852i		−	0.0851496i	−0.999093	−	0.0425748i	$-0.986444\pi$
$937$		−	79.7852i		−	0.0851496i	0.999093	−	0.0425748i	$-0.0135561\pi$
$938$	1970.01			2.10022
$939$		0			0
$940$			384.351i			0.408884i
$941$		−	382.492i		−	0.406474i	−0.979130	−	0.203237i	$-0.934854\pi$
$941$		−	382.492i		−	0.406474i	0.979130	−	0.203237i	$-0.0651462\pi$
$942$		0			0
$943$	14.9426	+	4.56397i	0.0158458	+	0.00483984i
$944$	−442.528			−0.468780
$945$		0			0
$946$	−234.842			−0.248247
$947$	1415.40			1.49462			0.747309	−	0.664476i	$-0.231345\pi$
$947$	1415.40			1.49462			0.747309	+	0.664476i	$0.231345\pi$
$948$		0			0
$949$	241.007			0.253959
$950$		−	148.562i		−	0.156381i
$951$		0			0
$952$	−765.829			−0.804442
$953$		−	1298.22i		−	1.36224i	−0.732170	−	0.681122i	$-0.761492\pi$
$953$		−	1298.22i		−	1.36224i	0.732170	−	0.681122i	$-0.238508\pi$
$954$		0			0
$955$	74.3698			0.0778741
$956$	−751.981			−0.786591
$957$		0			0
$958$		−	25.5713i		−	0.0266924i
$959$	−1798.39			−1.87528
$960$		0			0
$961$	151.176			0.157311
$962$			542.180i			0.563597i
$963$		0			0
$964$		−	429.405i		−	0.445441i
$965$			209.305i			0.216897i
$966$		0			0
$967$	−1707.55			−1.76582			−0.882912	−	0.469538i	$-0.844420\pi$
$967$	−1707.55			−1.76582			−0.882912	+	0.469538i	$0.844420\pi$
$968$	877.354			0.906358
$969$		0			0
$970$	−220.328			−0.227142
$971$		−	195.262i		−	0.201094i	−0.994932	−	0.100547i	$-0.967941\pi$
$971$		−	195.262i		−	0.201094i	0.994932	−	0.100547i	$-0.0320593\pi$
$972$		0			0
$973$		−	752.247i		−	0.773121i
$974$	−739.051			−0.758779
$975$		0			0
$976$		−	66.7118i		−	0.0683523i
$977$		−	216.403i		−	0.221497i	−0.993848	−	0.110748i	$-0.964675\pi$
$977$		−	216.403i		−	0.221497i	0.993848	−	0.110748i	$-0.0353248\pi$
$978$		0			0
$979$	1021.19			1.04310
$980$		−	405.620i		−	0.413898i
$981$		0			0
$982$	741.243			0.754830
$983$			1395.17i			1.41929i	0.704557	+	0.709647i	$0.251146\pi$
$983$			1395.17i			1.41929i	−0.704557	+	0.709647i	$0.748854\pi$
$984$		0			0
$985$		−	833.168i		−	0.845856i
$986$		−	1239.13i		−	1.25672i
$987$		0			0
$988$		−	325.887i		−	0.329845i
$989$	−53.7280	+	175.908i	−0.0543256	+	0.177864i
$990$		0			0
$991$	1736.34			1.75211			0.876054	−	0.482213i	$-0.160167\pi$
$991$	1736.34			1.75211			0.876054	+	0.482213i	$0.160167\pi$
$992$	−188.652			−0.190174
$993$		0			0
$994$		−	484.687i		−	0.487612i
$995$	−133.795			−0.134467
$996$		0			0
$997$	705.759			0.707883			0.353941	−	0.935268i	$-0.384841\pi$
$997$	705.759			0.707883			0.353941	+	0.935268i	$0.384841\pi$
$998$	543.243			0.544331
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2070.3.c.b.91.9		32
3.2	odd	2		690.3.c.a.91.17	✓	32
23.22	odd	2	inner	2070.3.c.b.91.8		32
69.68	even	2		690.3.c.a.91.24	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.3.c.a.91.17	✓	32	3.2	odd	2
690.3.c.a.91.24	yes	32	69.68	even	2
2070.3.c.b.91.8		32	23.22	odd	2	inner
2070.3.c.b.91.9		32	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +2.00000 q^{4} +2.23607i q^{5} -11.8194i q^{7} -2.82843 q^{8} +O(q^{10})\)	\(q-1.41421 q^{2} +2.00000 q^{4} +2.23607i q^{5} -11.8194i q^{7} -2.82843 q^{8} -3.16228i q^{10} -20.7652i q^{11} +7.75559 q^{13} +16.7152i q^{14} +4.00000 q^{16} +22.9081i q^{17} -21.0098i q^{19} +4.47214i q^{20} +29.3664i q^{22} +(21.9968 + 6.71856i) q^{23} -5.00000 q^{25} -10.9681 q^{26} -23.6389i q^{28} +38.2483 q^{29} +33.3493 q^{31} -5.65685 q^{32} -32.3970i q^{34} +26.4291 q^{35} -49.4326i q^{37} +29.7124i q^{38} -6.32456i q^{40} +0.679308 q^{41} +7.99696i q^{43} -41.5303i q^{44} +(-31.1082 - 9.50148i) q^{46} +85.9436 q^{47} -90.6993 q^{49} +7.07107 q^{50} +15.5112 q^{52} +66.6981i q^{53} +46.4323 q^{55} +33.4304i q^{56} -54.0913 q^{58} -110.632 q^{59} -16.6780i q^{61} -47.1630 q^{62} +8.00000 q^{64} +17.3420i q^{65} -117.857i q^{67} +45.8163i q^{68} -37.3764 q^{70} -28.9967 q^{71} +31.0752 q^{73} +69.9083i q^{74} -42.0196i q^{76} -245.433 q^{77} +105.069i q^{79} +8.94427i q^{80} -0.960687 q^{82} -43.1979i q^{83} -51.2241 q^{85} -11.3094i q^{86} +58.7327i q^{88} +49.1782i q^{89} -91.6668i q^{91} +(43.9937 + 13.4371i) q^{92} -121.543 q^{94} +46.9794 q^{95} -69.6738i q^{97} +128.268 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 64 q^{4}+O(q^{10}) \) 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

Related objects

Downloads

Learn more