LMFDB - Embedded newform 531.5.c.d.235.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,5,Mod(235,531)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(531, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("531.235");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	531.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:	$54.8894503975$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	no (minimal twist has level 177)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			235.6
Character	$\chi$	$=$	531.235
Dual form			531.5.c.d.235.35

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.10324i q^{2} -21.2495 q^{4} +12.8499 q^{5} +4.61608 q^{7} +32.0389i q^{8} +O(q^{10})$	\(q-6.10324i q^{2} -21.2495 q^{4} +12.8499 q^{5} +4.61608 q^{7} +32.0389i q^{8} -78.4262i q^{10} +159.523i q^{11} -148.997i q^{13} -28.1731i q^{14} -144.451 q^{16} -154.411 q^{17} +313.342 q^{19} -273.055 q^{20} +973.607 q^{22} -420.147i q^{23} -459.879 q^{25} -909.363 q^{26} -98.0895 q^{28} -1422.43 q^{29} +1100.16i q^{31} +1394.24i q^{32} +942.405i q^{34} +59.3164 q^{35} +186.690i q^{37} -1912.40i q^{38} +411.698i q^{40} -435.578 q^{41} -2365.57i q^{43} -3389.78i q^{44} -2564.25 q^{46} +2367.77i q^{47} -2379.69 q^{49} +2806.75i q^{50} +3166.11i q^{52} -3945.26 q^{53} +2049.86i q^{55} +147.894i q^{56} +8681.45i q^{58} +(-2040.07 + 2820.55i) q^{59} -1734.71i q^{61} +6714.53 q^{62} +6198.17 q^{64} -1914.60i q^{65} +2714.23i q^{67} +3281.15 q^{68} -362.022i q^{70} -5431.15 q^{71} +2072.73i q^{73} +1139.41 q^{74} -6658.35 q^{76} +736.372i q^{77} +10773.3 q^{79} -1856.18 q^{80} +2658.44i q^{82} +10956.5i q^{83} -1984.17 q^{85} -14437.6 q^{86} -5110.95 q^{88} +1731.25i q^{89} -687.782i q^{91} +8927.90i q^{92} +14451.1 q^{94} +4026.42 q^{95} -3594.71i q^{97} +14523.8i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 320 q^{4} + 80 q^{7}+O(q^{10}) $ 40 * q - 320 * q^4 + 80 * q^7	$ 40 q - 320 q^{4} + 80 q^{7} + 3944 q^{16} + 528 q^{17} + 444 q^{19} - 444 q^{20} + 1304 q^{22} + 4880 q^{25} + 1452 q^{26} - 1160 q^{28} + 996 q^{29} - 10320 q^{35} + 5196 q^{41} - 10476 q^{46} + 5104 q^{49} + 2184 q^{53} + 11736 q^{59} - 15240 q^{62} - 81012 q^{64} - 29568 q^{68} + 5964 q^{71} - 14376 q^{74} + 3480 q^{76} + 19020 q^{79} - 33096 q^{80} + 20220 q^{85} + 65880 q^{86} - 14932 q^{88} - 17864 q^{94} - 11004 q^{95}+O(q^{100}) $ 40 * q - 320 * q^4 + 80 * q^7 + 3944 * q^16 + 528 * q^17 + 444 * q^19 - 444 * q^20 + 1304 * q^22 + 4880 * q^25 + 1452 * q^26 - 1160 * q^28 + 996 * q^29 - 10320 * q^35 + 5196 * q^41 - 10476 * q^46 + 5104 * q^49 + 2184 * q^53 + 11736 * q^59 - 15240 * q^62 - 81012 * q^64 - 29568 * q^68 + 5964 * q^71 - 14376 * q^74 + 3480 * q^76 + 19020 * q^79 - 33096 * q^80 + 20220 * q^85 + 65880 * q^86 - 14932 * q^88 - 17864 * q^94 - 11004 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/531\mathbb{Z}\right)^\times$.

$n$	$119$	$415$
$\chi(n)$	$1$	$-1$

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 $

$2$		−	6.10324i		−	1.52581i	−0.646511	−	0.762905i	$-0.723773\pi$
$2$		−	6.10324i		−	1.52581i	0.646511	−	0.762905i	$-0.276227\pi$
$3$		0			0
$3$		0			0
$4$	−21.2495			−1.32809
$5$	12.8499			0.513997			0.256999	−	0.966412i	$-0.417266\pi$
$5$	12.8499			0.513997			0.256999	+	0.966412i	$0.417266\pi$
$6$		0			0
$7$	4.61608			0.0942058			0.0471029	−	0.998890i	$-0.485001\pi$
$7$	4.61608			0.0942058			0.0471029	+	0.998890i	$0.485001\pi$
$8$			32.0389i			0.500608i
$9$		0			0
$10$		−	78.4262i		−	0.784262i
$11$			159.523i			1.31837i	0.751980	+	0.659186i	$0.229099\pi$
$11$			159.523i			1.31837i	−0.751980	+	0.659186i	$0.770901\pi$
$12$		0			0
$13$		−	148.997i		−	0.881638i	−0.897596	−	0.440819i	$-0.854688\pi$
$13$		−	148.997i		−	0.881638i	0.897596	−	0.440819i	$-0.145312\pi$
$14$		−	28.1731i		−	0.143740i
$15$		0			0
$16$	−144.451			−0.564261
$17$	−154.411			−0.534293			−0.267147	−	0.963656i	$-0.586081\pi$
$17$	−154.411			−0.534293			−0.267147	+	0.963656i	$0.586081\pi$
$18$		0			0
$19$	313.342			0.867982			0.433991	−	0.900917i	$-0.357105\pi$
$19$	313.342			0.867982			0.433991	+	0.900917i	$0.357105\pi$
$20$	−273.055			−0.682636
$21$		0			0
$22$	973.607			2.01158
$23$		−	420.147i		−	0.794228i	−0.917769	−	0.397114i	$-0.870012\pi$
$23$		−	420.147i		−	0.794228i	0.917769	−	0.397114i	$-0.129988\pi$
$24$		0			0
$25$	−459.879			−0.735807
$26$	−909.363			−1.34521
$27$		0			0
$28$	−98.0895			−0.125114
$29$	−1422.43			−1.69136			−0.845680	−	0.533690i	$-0.820805\pi$
$29$	−1422.43			−1.69136			−0.845680	+	0.533690i	$0.820805\pi$
$30$		0			0
$31$			1100.16i			1.14481i	0.819972	+	0.572403i	$0.193989\pi$
$31$			1100.16i			1.14481i	−0.819972	+	0.572403i	$0.806011\pi$
$32$			1394.24i			1.36156i
$33$		0			0
$34$			942.405i			0.815230i
$35$	59.3164			0.0484215
$36$		0			0
$37$			186.690i			0.136370i	0.997673	+	0.0681848i	$0.0217207\pi$
$37$			186.690i			0.136370i	−0.997673	+	0.0681848i	$0.978279\pi$
$38$		−	1912.40i		−	1.32437i
$39$		0			0
$40$			411.698i			0.257311i
$41$	−435.578			−0.259118			−0.129559	−	0.991572i	$-0.541356\pi$
$41$	−435.578			−0.259118			−0.129559	+	0.991572i	$0.541356\pi$
$42$		0			0
$43$		−	2365.57i		−	1.27938i	−0.768634	−	0.639689i	$-0.779063\pi$
$43$		−	2365.57i		−	1.27938i	0.768634	−	0.639689i	$-0.220937\pi$
$44$		−	3389.78i		−	1.75092i
$45$		0			0
$46$	−2564.25			−1.21184
$47$			2367.77i			1.07187i	0.844258	+	0.535937i	$0.180042\pi$
$47$			2367.77i			1.07187i	−0.844258	+	0.535937i	$0.819958\pi$
$48$		0			0
$49$	−2379.69			−0.991125
$50$			2806.75i			1.12270i
$51$		0			0
$52$			3166.11i			1.17090i
$53$	−3945.26			−1.40451			−0.702254	−	0.711927i	$-0.747823\pi$
$53$	−3945.26			−1.40451			−0.702254	+	0.711927i	$0.747823\pi$
$54$		0			0
$55$			2049.86i			0.677640i
$56$			147.894i			0.0471602i
$57$		0			0
$58$			8681.45i			2.58069i
$59$	−2040.07	+	2820.55i	−0.586057	+	0.810270i
$59$	−2040.07	+	2820.55i	−0.586057	+	0.810270i
$60$		0			0
$61$		−	1734.71i		−	0.466194i	−0.972453	−	0.233097i	$-0.925114\pi$
$61$		−	1734.71i		−	0.466194i	0.972453	−	0.233097i	$-0.0748860\pi$
$62$	6714.53			1.74676
$63$		0			0
$64$	6198.17			1.51322
$65$		−	1914.60i		−	0.453160i
$66$		0			0
$67$			2714.23i			0.604640i	0.953206	+	0.302320i	$0.0977611\pi$
$67$			2714.23i			0.604640i	−0.953206	+	0.302320i	$0.902239\pi$
$68$	3281.15			0.709592
$69$		0			0
$70$		−	362.022i		−	0.0738820i
$71$	−5431.15			−1.07740			−0.538698	−	0.842499i	$-0.681084\pi$
$71$	−5431.15			−1.07740			−0.538698	+	0.842499i	$0.681084\pi$
$72$		0			0
$73$			2072.73i			0.388953i	0.980907	+	0.194476i	$0.0623008\pi$
$73$			2072.73i			0.388953i	−0.980907	+	0.194476i	$0.937699\pi$
$74$	1139.41			0.208074
$75$		0			0
$76$	−6658.35			−1.15276
$77$			736.372i			0.124198i
$78$		0			0
$79$	10773.3			1.72622			0.863108	−	0.505019i	$-0.168515\pi$
$79$	10773.3			1.72622			0.863108	+	0.505019i	$0.168515\pi$
$80$	−1856.18			−0.290029
$81$		0			0
$82$			2658.44i			0.395365i
$83$			10956.5i			1.59044i	0.606321	+	0.795220i	$0.292645\pi$
$83$			10956.5i			1.59044i	−0.606321	+	0.795220i	$0.707355\pi$
$84$		0			0
$85$	−1984.17			−0.274625
$86$	−14437.6			−1.95209
$87$		0			0
$88$	−5110.95			−0.659988
$89$			1731.25i			0.218565i	0.994011	+	0.109283i	$0.0348553\pi$
$89$			1731.25i			0.218565i	−0.994011	+	0.109283i	$0.965145\pi$
$90$		0			0
$91$		−	687.782i		−	0.0830555i
$92$			8927.90i			1.05481i
$93$		0			0
$94$	14451.1			1.63547
$95$	4026.42			0.446140
$96$		0			0
$97$		−	3594.71i		−	0.382050i	−0.981585	−	0.191025i	$-0.938819\pi$
$97$		−	3594.71i		−	0.382050i	0.981585	−	0.191025i	$-0.0611812\pi$
$98$			14523.8i			1.51227i
$99$		0			0
$100$	9772.20			0.977220
$101$			9157.70i			0.897726i	0.893601	+	0.448863i	$0.148171\pi$
$101$			9157.70i			0.897726i	−0.893601	+	0.448863i	$0.851829\pi$
$102$		0			0
$103$		−	2682.66i		−	0.252866i	−0.991975	−	0.126433i	$-0.959647\pi$
$103$		−	2682.66i		−	0.252866i	0.991975	−	0.126433i	$-0.0403529\pi$
$104$	4773.70			0.441355
$105$		0			0
$106$			24078.9i			2.14301i
$107$	3938.64			0.344016			0.172008	−	0.985096i	$-0.444974\pi$
$107$	3938.64			0.344016			0.172008	+	0.985096i	$0.444974\pi$
$108$		0			0
$109$		−	11260.3i		−	0.947754i	−0.880591	−	0.473877i	$-0.842854\pi$
$109$		−	11260.3i		−	0.947754i	0.880591	−	0.473877i	$-0.157146\pi$
$110$	12510.8			1.03395
$111$		0			0
$112$	−666.797			−0.0531567
$113$			9397.40i			0.735954i	0.929835	+	0.367977i	$0.119950\pi$
$113$			9397.40i			0.735954i	−0.929835	+	0.367977i	$0.880050\pi$
$114$		0			0
$115$		−	5398.85i		−	0.408231i
$116$	30226.0			2.24629
$117$		0			0
$118$	17214.5	+	12451.0i	1.23632	+	0.894212i
$119$	−712.773			−0.0503335
$120$		0			0
$121$	−10806.6			−0.738105
$122$	−10587.3			−0.711323
$123$		0			0
$124$		−	23377.8i		−	1.52041i
$125$	−13940.6			−0.892200
$126$		0			0
$127$	−9401.70			−0.582906			−0.291453	−	0.956585i	$-0.594139\pi$
$127$	−9401.70			−0.582906			−0.291453	+	0.956585i	$0.594139\pi$
$128$		−	15521.0i		−	0.947328i
$129$		0			0
$130$	−11685.3			−0.691435
$131$		−	28050.5i		−	1.63455i	−0.576249	−	0.817274i	$-0.695484\pi$
$131$		−	28050.5i		−	1.63455i	0.576249	−	0.817274i	$-0.304516\pi$
$132$		0			0
$133$	1446.41			0.0817690
$134$	16565.6			0.922566
$135$		0			0
$136$		−	4947.16i		−	0.267472i
$137$	10546.1			0.561888			0.280944	−	0.959724i	$-0.409352\pi$
$137$	10546.1			0.561888			0.280944	+	0.959724i	$0.409352\pi$
$138$		0			0
$139$	13525.1			0.700020			0.350010	−	0.936746i	$-0.386178\pi$
$139$	13525.1			0.700020			0.350010	+	0.936746i	$0.386178\pi$
$140$	−1260.44			−0.0643083
$141$		0			0
$142$			33147.6i			1.64390i
$143$	23768.4			1.16233
$144$		0			0
$145$	−18278.2			−0.869355
$146$	12650.4			0.593468
$147$		0			0
$148$		−	3967.07i		−	0.181112i
$149$		−	4534.58i		−	0.204251i	−0.994772	−	0.102126i	$-0.967436\pi$
$149$		−	4534.58i		−	0.204251i	0.994772	−	0.102126i	$-0.0325644\pi$
$150$		0			0
$151$		−	28293.6i		−	1.24089i	−0.784249	−	0.620446i	$-0.786952\pi$
$151$		−	28293.6i		−	1.24089i	0.784249	−	0.620446i	$-0.213048\pi$
$152$			10039.1i			0.434519i
$153$		0			0
$154$	4494.25			0.189503
$155$			14137.0i			0.588427i
$156$		0			0
$157$			8264.64i			0.335293i	0.985847	+	0.167647i	$0.0536167\pi$
$157$			8264.64i			0.335293i	−0.985847	+	0.167647i	$0.946383\pi$
$158$		−	65752.1i		−	2.63388i
$159$		0			0
$160$			17915.9i			0.699839i
$161$		−	1939.43i		−	0.0748209i
$162$		0			0
$163$	−22072.4			−0.830759			−0.415379	−	0.909648i	$-0.636351\pi$
$163$	−22072.4			−0.830759			−0.415379	+	0.909648i	$0.636351\pi$
$164$	9255.81			0.344133
$165$		0			0
$166$	66870.4			2.42671
$167$	13873.2			0.497445			0.248723	−	0.968575i	$-0.419989\pi$
$167$	13873.2			0.497445			0.248723	+	0.968575i	$0.419989\pi$
$168$		0			0
$169$	6360.93			0.222714
$170$			12109.8i			0.419026i
$171$		0			0
$172$			50267.2i			1.69913i
$173$			26589.0i			0.888402i	0.895927	+	0.444201i	$0.146512\pi$
$173$			26589.0i			0.888402i	−0.895927	+	0.444201i	$0.853488\pi$
$174$		0			0
$175$	−2122.84			−0.0693173
$176$		−	23043.2i		−	0.743906i
$177$		0			0
$178$	10566.3			0.333489
$179$		−	23765.9i		−	0.741735i	−0.928686	−	0.370867i	$-0.879060\pi$
$179$		−	23765.9i		−	0.741735i	0.928686	−	0.370867i	$-0.120940\pi$
$180$		0			0
$181$	−25062.8			−0.765019			−0.382510	−	0.923951i	$-0.624940\pi$
$181$	−25062.8			−0.765019			−0.382510	+	0.923951i	$0.624940\pi$
$182$	−4197.70			−0.126727
$183$		0			0
$184$	13461.0			0.397597
$185$			2398.95i			0.0700936i
$186$		0			0
$187$		−	24632.1i		−	0.704397i
$188$		−	50313.9i		−	1.42355i
$189$		0			0
$190$		−	24574.2i		−	0.680725i
$191$		−	8426.68i		−	0.230988i	−0.993308	−	0.115494i	$-0.963155\pi$
$191$		−	8426.68i		−	0.230988i	0.993308	−	0.115494i	$-0.0368451\pi$
$192$		0			0
$193$	5162.04			0.138582			0.0692910	−	0.997596i	$-0.477926\pi$
$193$	5162.04			0.138582			0.0692910	+	0.997596i	$0.477926\pi$
$194$	−21939.4			−0.582936
$195$		0			0
$196$	50567.3			1.31631
$197$	−13261.3			−0.341707			−0.170853	−	0.985296i	$-0.554652\pi$
$197$	−13261.3			−0.341707			−0.170853	+	0.985296i	$0.554652\pi$
$198$		0			0
$199$	−19128.2			−0.483022			−0.241511	−	0.970398i	$-0.577643\pi$
$199$	−19128.2			−0.483022			−0.241511	+	0.970398i	$0.577643\pi$
$200$		−	14734.0i		−	0.368351i
$201$		0			0
$202$	55891.6			1.36976
$203$	−6566.08			−0.159336
$204$		0			0
$205$	−5597.15			−0.133186
$206$	−16372.9			−0.385825
$207$		0			0
$208$			21522.7i			0.497474i
$209$			49985.2i			1.14432i
$210$		0			0
$211$			88561.9i			1.98922i	0.103705	+	0.994608i	$0.466930\pi$
$211$			88561.9i			1.98922i	−0.103705	+	0.994608i	$0.533070\pi$
$212$	83834.9			1.86532
$213$		0			0
$214$		−	24038.5i		−	0.524903i
$215$		−	30397.4i		−	0.657597i
$216$		0			0
$217$			5078.43i			0.107847i
$218$	−68724.1			−1.44609
$219$		0			0
$220$		−	43558.5i		−	0.899969i
$221$			23006.7i			0.471053i
$222$		0			0
$223$	19289.2			0.387885			0.193943	−	0.981013i	$-0.437872\pi$
$223$	19289.2			0.387885			0.193943	+	0.981013i	$0.437872\pi$
$224$			6435.93i			0.128267i
$225$		0			0
$226$	57354.6			1.12293
$227$			27309.5i			0.529983i	0.964251	+	0.264991i	$0.0853691\pi$
$227$			27309.5i			0.529983i	−0.964251	+	0.264991i	$0.914631\pi$
$228$		0			0
$229$			7085.68i			0.135117i	0.997715	+	0.0675586i	$0.0215210\pi$
$229$			7085.68i			0.135117i	−0.997715	+	0.0675586i	$0.978479\pi$
$230$	−32950.5			−0.622882
$231$		0			0
$232$		−	45573.3i		−	0.846709i
$233$			59816.5i			1.10182i	0.834566	+	0.550908i	$0.185719\pi$
$233$			59816.5i			1.10182i	−0.834566	+	0.550908i	$0.814281\pi$
$234$		0			0
$235$			30425.7i			0.550940i
$236$	43350.4	−	59935.2i	0.778339	−	1.07611i
$237$		0			0
$238$			4350.22i			0.0767994i
$239$	−78985.4			−1.38277			−0.691387	−	0.722485i	$-0.743000\pi$
$239$	−78985.4			−1.38277			−0.691387	+	0.722485i	$0.743000\pi$
$240$		0			0
$241$	−50419.4			−0.868088			−0.434044	−	0.900892i	$-0.642914\pi$
$241$	−50419.4			−0.868088			−0.434044	+	0.900892i	$0.642914\pi$
$242$			65955.2i			1.12621i
$243$		0			0
$244$			36861.7i			0.619149i
$245$	−30578.9			−0.509436
$246$		0			0
$247$		−	46686.9i		−	0.765246i
$248$	−35247.9			−0.573099
$249$		0			0
$250$			85082.9i			1.36133i
$251$	−107267.			−1.70262			−0.851312	−	0.524660i	$-0.824193\pi$
$251$	−107267.			−1.70262			−0.851312	+	0.524660i	$0.824193\pi$
$252$		0			0
$253$	67023.0			1.04709
$254$			57380.8i			0.889404i
$255$		0			0
$256$	4442.15			0.0677818
$257$	−52186.5			−0.790118			−0.395059	−	0.918656i	$-0.629276\pi$
$257$	−52186.5			−0.790118			−0.395059	+	0.918656i	$0.629276\pi$
$258$		0			0
$259$			861.776i			0.0128468i
$260$			40684.3i			0.601838i
$261$		0			0
$262$	−171199.			−2.49401
$263$	109997.			1.59026			0.795132	−	0.606436i	$-0.207401\pi$
$263$	109997.			1.59026			0.795132	+	0.606436i	$0.207401\pi$
$264$		0			0
$265$	−50696.3			−0.721913
$266$		−	8827.79i		−	0.124764i
$267$		0			0
$268$		−	57676.0i		−	0.803019i
$269$			60277.7i			0.833014i	0.909133	+	0.416507i	$0.136746\pi$
$269$			60277.7i			0.833014i	−0.909133	+	0.416507i	$0.863254\pi$
$270$		0			0
$271$	−68787.2			−0.936633			−0.468316	−	0.883561i	$-0.655139\pi$
$271$	−68787.2			−0.936633			−0.468316	+	0.883561i	$0.655139\pi$
$272$	22304.8			0.301481
$273$		0			0
$274$		−	64365.2i		−	0.857334i
$275$		−	73361.3i		−	0.970067i
$276$		0			0
$277$	−136750.			−1.78225			−0.891126	−	0.453755i	$-0.850084\pi$
$277$	−136750.			−1.78225			−0.891126	+	0.453755i	$0.850084\pi$
$278$		−	82546.8i		−	1.06810i
$279$		0			0
$280$			1900.43i			0.0242402i
$281$	128919.			1.63270			0.816349	−	0.577559i	$-0.195995\pi$
$281$	128919.			1.63270			0.816349	+	0.577559i	$0.195995\pi$
$282$		0			0
$283$			35381.9i			0.441782i	0.975298	+	0.220891i	$0.0708965\pi$
$283$			35381.9i			0.441782i	−0.975298	+	0.220891i	$0.929103\pi$
$284$	115409.			1.43088
$285$		0			0
$286$		−	145064.i		−	1.77349i
$287$	−2010.66			−0.0244105
$288$		0			0
$289$	−59678.3			−0.714531
$290$			111556.i			1.32647i
$291$		0			0
$292$		−	44044.5i		−	0.516566i
$293$	27782.7			0.323622			0.161811	−	0.986822i	$-0.448266\pi$
$293$	27782.7			0.323622			0.161811	+	0.986822i	$0.448266\pi$
$294$		0			0
$295$	−26214.7	+	36243.8i	−0.301232	+	0.416476i
$296$	−5981.34			−0.0682677
$297$		0			0
$298$	−27675.6			−0.311648
$299$	−62600.5			−0.700222
$300$		0			0
$301$		−	10919.7i		−	0.120525i
$302$	−172682.			−1.89336
$303$		0			0
$304$	−45262.4			−0.489768
$305$		−	22290.9i		−	0.239622i
$306$		0			0
$307$	−87467.0			−0.928042			−0.464021	−	0.885824i	$-0.653594\pi$
$307$	−87467.0			−0.928042			−0.464021	+	0.885824i	$0.653594\pi$
$308$		−	15647.5i		−	0.164947i
$309$		0			0
$310$	86281.2			0.897827
$311$	−122891.			−1.27057			−0.635284	−	0.772279i	$-0.719117\pi$
$311$	−122891.			−1.27057			−0.635284	+	0.772279i	$0.719117\pi$
$312$		0			0
$313$		−	13347.4i		−	0.136242i	−0.997677	−	0.0681208i	$-0.978300\pi$
$313$		−	13347.4i		−	0.136242i	0.997677	−	0.0681208i	$-0.0217003\pi$
$314$	50441.0			0.511593
$315$		0			0
$316$	−228928.			−2.29258
$317$	1111.83			0.0110642			0.00553210	−	0.999985i	$-0.498239\pi$
$317$	1111.83			0.0110642			0.00553210	+	0.999985i	$0.498239\pi$
$318$		0			0
$319$		−	226911.i		−	2.22984i
$320$	79646.0			0.777793
$321$		0			0
$322$	−11836.8			−0.114162
$323$	−48383.3			−0.463757
$324$		0			0
$325$			68520.6i			0.648716i
$326$			134713.i			1.26758i
$327$		0			0
$328$		−	13955.5i		−	0.129717i
$329$			10929.8i			0.100977i
$330$		0			0
$331$	141657.			1.29295			0.646476	−	0.762935i	$-0.276242\pi$
$331$	141657.			1.29295			0.646476	+	0.762935i	$0.276242\pi$
$332$		−	232821.i		−	2.11225i
$333$		0			0
$334$		−	84671.7i		−	0.759006i
$335$			34877.7i			0.310783i
$336$		0			0
$337$		−	163985.i		−	1.44392i	−0.691935	−	0.721960i	$-0.743242\pi$
$337$		−	163985.i		−	1.44392i	0.691935	−	0.721960i	$-0.256758\pi$
$338$		−	38822.2i		−	0.339819i
$339$		0			0
$340$	42162.6			0.364728
$341$	−175501.			−1.50928
$342$		0			0
$343$	−22068.1			−0.187576
$344$	75790.3			0.640467
$345$		0			0
$346$	162279.			1.35553
$347$		−	163886.i		−	1.36108i	−0.732710	−	0.680541i	$-0.761745\pi$
$347$		−	163886.i		−	1.36108i	0.732710	−	0.680541i	$-0.238255\pi$
$348$		0			0
$349$		−	63926.5i		−	0.524844i	−0.964953	−	0.262422i	$-0.915479\pi$
$349$		−	63926.5i		−	0.524844i	0.964953	−	0.262422i	$-0.0845212\pi$
$350$			12956.2i			0.105765i
$351$		0			0
$352$	−222413.			−1.79505
$353$		−	87690.5i		−	0.703725i	−0.936052	−	0.351863i	$-0.885548\pi$
$353$		−	87690.5i		−	0.703725i	0.936052	−	0.351863i	$-0.114452\pi$
$354$		0			0
$355$	−69789.9			−0.553779
$356$		−	36788.3i		−	0.290275i
$357$		0			0
$358$	−145049.			−1.13175
$359$	−119280.			−0.925504			−0.462752	−	0.886488i	$-0.653138\pi$
$359$	−119280.			−0.925504			−0.462752	+	0.886488i	$0.653138\pi$
$360$		0			0
$361$	−32138.1			−0.246607
$362$			152964.i			1.16727i
$363$		0			0
$364$			14615.0i			0.110305i
$365$			26634.4i			0.199921i
$366$		0			0
$367$			56563.1i			0.419953i	0.977706	+	0.209977i	$0.0673388\pi$
$367$			56563.1i			0.419953i	−0.977706	+	0.209977i	$0.932661\pi$
$368$			60690.5i			0.448152i
$369$		0			0
$370$	14641.4			0.106949
$371$	−18211.7			−0.132313
$372$		0			0
$373$	123782.			0.889691			0.444846	−	0.895607i	$-0.353259\pi$
$373$	123782.			0.889691			0.444846	+	0.895607i	$0.353259\pi$
$374$	−150335.			−1.07478
$375$		0			0
$376$	−75860.8			−0.536589
$377$			211938.i			1.49117i
$378$		0			0
$379$	237594.			1.65408			0.827042	−	0.562140i	$-0.190022\pi$
$379$	237594.			1.65408			0.827042	+	0.562140i	$0.190022\pi$
$380$	−85559.3			−0.592516
$381$		0			0
$382$	−51430.0			−0.352444
$383$	−271890.			−1.85351			−0.926757	−	0.375661i	$-0.877416\pi$
$383$	−271890.			−1.85351			−0.926757	+	0.375661i	$0.877416\pi$
$384$		0			0
$385$			9462.33i			0.0638376i
$386$		−	31505.2i		−	0.211450i
$387$		0			0
$388$			76385.8i			0.507398i
$389$	76616.7			0.506319			0.253160	−	0.967425i	$-0.418530\pi$
$389$	76616.7			0.506319			0.253160	+	0.967425i	$0.418530\pi$
$390$		0			0
$391$			64875.1i			0.424351i
$392$		−	76242.8i		−	0.496166i
$393$		0			0
$394$			80936.8i			0.521379i
$395$	138436.			0.887270
$396$		0			0
$397$			2682.43i			0.0170195i	0.999964	+	0.00850977i	$0.00270878\pi$
$397$			2682.43i			0.0170195i	−0.999964	+	0.00850977i	$0.997291\pi$
$398$			116744.i			0.737000i
$399$		0			0
$400$	66429.9			0.415187
$401$		−	208279.i		−	1.29526i	−0.761955	−	0.647630i	$-0.775760\pi$
$401$		−	208279.i		−	1.29526i	0.761955	−	0.647630i	$-0.224240\pi$
$402$		0			0
$403$	163920.			1.00931
$404$		−	194597.i		−	1.19226i
$405$		0			0
$406$			40074.3i			0.243116i
$407$	−29781.3			−0.179786
$408$		0			0
$409$		−	306294.i		−	1.83102i	−0.402299	−	0.915508i	$-0.631789\pi$
$409$		−	306294.i		−	1.83102i	0.402299	−	0.915508i	$-0.368211\pi$
$410$			34160.7i			0.203217i
$411$		0			0
$412$			57005.1i			0.335830i
$413$	−9417.11	+	13019.9i	−0.0552100	+	0.0763321i
$414$		0			0
$415$			140791.i			0.817482i
$416$	207737.			1.20041
$417$		0			0
$418$	305071.			1.74602
$419$			214239.i			1.22031i	0.792282	+	0.610155i	$0.208893\pi$
$419$			214239.i			1.22031i	−0.792282	+	0.610155i	$0.791107\pi$
$420$		0			0
$421$		−	211231.i		−	1.19177i	−0.803069	−	0.595886i	$-0.796801\pi$
$421$		−	211231.i		−	1.19177i	0.803069	−	0.595886i	$-0.203199\pi$
$422$	540514.			3.03516
$423$		0			0
$424$		−	126402.i		−	0.703108i
$425$	71010.3			0.393137
$426$		0			0
$427$		−	8007.56i		−	0.0439182i
$428$	−83694.2			−0.456886
$429$		0			0
$430$	−185523.			−1.00337
$431$		−	77388.2i		−	0.416601i	−0.978065	−	0.208300i	$-0.933207\pi$
$431$		−	77388.2i		−	0.416601i	0.978065	−	0.208300i	$-0.0667932\pi$
$432$		0			0
$433$	246940.			1.31709			0.658546	−	0.752541i	$-0.271172\pi$
$433$	246940.			1.31709			0.658546	+	0.752541i	$0.271172\pi$
$434$	30994.8			0.164555
$435$		0			0
$436$			239275.i			1.25871i
$437$		−	131649.i		−	0.689375i
$438$		0			0
$439$	−43951.4			−0.228057			−0.114028	−	0.993477i	$-0.536376\pi$
$439$	−43951.4			−0.228057			−0.114028	+	0.993477i	$0.536376\pi$
$440$	−65675.3			−0.339232
$441$		0			0
$442$	140415.			0.718738
$443$		−	204422.i		−	1.04165i	−0.853665	−	0.520823i	$-0.825625\pi$
$443$		−	204422.i		−	1.04165i	0.853665	−	0.520823i	$-0.174375\pi$
$444$		0			0
$445$			22246.5i			0.112342i
$446$		−	117726.i		−	0.591839i
$447$		0			0
$448$	28611.3			0.142554
$449$	−37083.0			−0.183943			−0.0919713	−	0.995762i	$-0.529317\pi$
$449$	−37083.0			−0.183943			−0.0919713	+	0.995762i	$0.529317\pi$
$450$		0			0
$451$		−	69484.7i		−	0.341614i
$452$		−	199690.i		−	0.977416i
$453$		0			0
$454$	166676.			0.808653
$455$		−	8837.95i		−	0.0426903i
$456$		0			0
$457$		−	29076.9i		−	0.139225i	−0.997574	−	0.0696123i	$-0.977824\pi$
$457$		−	29076.9i		−	0.139225i	0.997574	−	0.0696123i	$-0.0221762\pi$
$458$	43245.6			0.206163
$459$		0			0
$460$			114723.i			0.542169i
$461$	254103.			1.19566			0.597831	−	0.801622i	$-0.296029\pi$
$461$	254103.			1.19566			0.597831	+	0.801622i	$0.296029\pi$
$462$		0			0
$463$			386615.i			1.80350i	0.432256	+	0.901751i	$0.357718\pi$
$463$			386615.i			1.80350i	−0.432256	+	0.901751i	$0.642282\pi$
$464$	205472.			0.954369
$465$		0			0
$466$	365074.			1.68116
$467$		−	344576.i		−	1.57998i	−0.613122	−	0.789988i	$-0.710087\pi$
$467$		−	344576.i		−	1.57998i	0.613122	−	0.789988i	$-0.289913\pi$
$468$		0			0
$469$			12529.1i			0.0569606i
$470$	185695.			0.840629
$471$		0			0
$472$	−90367.4	−	65361.5i	−0.405628	−	0.293385i
$473$	377363.			1.68670
$474$		0			0
$475$	−144099.			−0.638667
$476$	15146.1			0.0668476
$477$		0			0
$478$			482067.i			2.10985i
$479$	−304897.			−1.32887			−0.664435	−	0.747346i	$-0.731328\pi$
$479$	−304897.			−1.32887			−0.664435	+	0.747346i	$0.731328\pi$
$480$		0			0
$481$	27816.2			0.120229
$482$			307722.i			1.32454i
$483$		0			0
$484$	229635.			0.980273
$485$		−	46191.8i		−	0.196373i
$486$		0			0
$487$	−327375.			−1.38034			−0.690171	−	0.723646i	$-0.742465\pi$
$487$	−327375.			−1.38034			−0.690171	+	0.723646i	$0.742465\pi$
$488$	55578.2			0.233381
$489$		0			0
$490$			186630.i			0.777301i
$491$	330527.			1.37102			0.685511	−	0.728063i	$-0.259579\pi$
$491$	330527.			1.37102			0.685511	+	0.728063i	$0.259579\pi$
$492$		0			0
$493$	219639.			0.903683
$494$	−284941.			−1.16762
$495$		0			0
$496$		−	158919.i		−	0.645969i
$497$	−25070.7			−0.101497
$498$		0			0
$499$	−207100.			−0.831724			−0.415862	−	0.909428i	$-0.636520\pi$
$499$	−207100.			−0.831724			−0.415862	+	0.909428i	$0.636520\pi$
$500$	296231.			1.18492
$501$		0			0
$502$			654676.i			2.59788i
$503$			187431.i			0.740807i	0.928871	+	0.370403i	$0.120781\pi$
$503$			187431.i			0.740807i	−0.928871	+	0.370403i	$0.879219\pi$
$504$		0			0
$505$			117676.i			0.461429i
$506$		−	409058.i		−	1.59766i
$507$		0			0
$508$	199781.			0.774154
$509$			27177.2i			0.104899i	0.998624	+	0.0524493i	$0.0167028\pi$
$509$			27177.2i			0.104899i	−0.998624	+	0.0524493i	$0.983297\pi$
$510$		0			0
$511$			9567.90i			0.0366416i
$512$		−	275448.i		−	1.05075i
$513$		0			0
$514$			318507.i			1.20557i
$515$		−	34471.9i		−	0.129972i
$516$		0			0
$517$	−377714.			−1.41313
$518$	5259.63			0.0196018
$519$		0			0
$520$	61341.7			0.226855
$521$	−492814.			−1.81555			−0.907774	−	0.419460i	$-0.862219\pi$
$521$	−492814.			−1.81555			−0.907774	+	0.419460i	$0.862219\pi$
$522$		0			0
$523$	−154182.			−0.563678			−0.281839	−	0.959462i	$-0.590944\pi$
$523$	−154182.			−0.563678			−0.281839	+	0.959462i	$0.590944\pi$
$524$			596059.i			2.17083i
$525$		0			0
$526$		−	671338.i		−	2.42644i
$527$		−	169876.i		−	0.611662i
$528$		0			0
$529$	103318.			0.369202
$530$			309412.i			1.10150i
$531$		0			0
$532$	−30735.5			−0.108597
$533$			64899.8i			0.228449i
$534$		0			0
$535$	50611.3			0.176823
$536$	−86961.0			−0.302688
$537$		0			0
$538$	367889.			1.27102
$539$		−	379616.i		−	1.30667i
$540$		0			0
$541$		−	175670.i		−	0.600210i	−0.953906	−	0.300105i	$-0.902978\pi$
$541$		−	175670.i		−	0.600210i	0.953906	−	0.300105i	$-0.0970217\pi$
$542$			419825.i			1.42912i
$543$		0			0
$544$		−	215286.i		−	0.727474i
$545$		−	144694.i		−	0.487143i
$546$		0			0
$547$	22187.9			0.0741552			0.0370776	−	0.999312i	$-0.488195\pi$
$547$	22187.9			0.0741552			0.0370776	+	0.999312i	$0.488195\pi$
$548$	−224099.			−0.746240
$549$		0			0
$550$	−447742.			−1.48014
$551$	−445708.			−1.46807
$552$		0			0
$553$	49730.5			0.162620
$554$			834620.i			2.71938i
$555$		0			0
$556$	−287401.			−0.929692
$557$	475276.			1.53192			0.765959	−	0.642890i	$-0.222265\pi$
$557$	475276.			1.53192			0.765959	+	0.642890i	$0.222265\pi$
$558$		0			0
$559$	−352463.			−1.12795
$560$	−8568.30			−0.0273224
$561$		0			0
$562$		−	786826.i		−	2.49118i
$563$		−	28507.6i		−	0.0899382i	−0.998988	−	0.0449691i	$-0.985681\pi$
$563$		−	28507.6i		−	0.0899382i	0.998988	−	0.0449691i	$-0.0143189\pi$
$564$		0			0
$565$			120756.i			0.378278i
$566$	215944.			0.674075
$567$		0			0
$568$		−	174008.i		−	0.539353i
$569$		−	112438.i		−	0.347287i	−0.984809	−	0.173644i	$-0.944446\pi$
$569$		−	112438.i		−	0.347287i	0.984809	−	0.173644i	$-0.0555541\pi$
$570$		0			0
$571$		−	428720.i		−	1.31493i	−0.753486	−	0.657464i	$-0.771629\pi$
$571$		−	428720.i		−	1.31493i	0.753486	−	0.657464i	$-0.228371\pi$
$572$	−505067.			−1.54368
$573$		0			0
$574$			12271.6i			0.0372457i
$575$			193217.i			0.584398i
$576$		0			0
$577$	−75213.5			−0.225915			−0.112957	−	0.993600i	$-0.536032\pi$
$577$	−75213.5			−0.225915			−0.112957	+	0.993600i	$0.536032\pi$
$578$			364231.i			1.09024i
$579$		0			0
$580$	388402.			1.15458
$581$			50576.3i			0.149829i
$582$		0			0
$583$		−	629360.i		−	1.85166i
$584$	−66408.0			−0.194713
$585$		0			0
$586$		−	169564.i		−	0.493786i
$587$		−	134395.i		−	0.390039i	−0.980799	−	0.195019i	$-0.937523\pi$
$587$		−	134395.i		−	0.390039i	0.980799	−	0.195019i	$-0.0624770\pi$
$588$		0			0
$589$			344725.i			0.993671i
$590$	221205.	+	159994.i	0.635463	+	0.459622i
$591$		0			0
$592$		−	26967.5i		−	0.0769480i
$593$	351119.			0.998493			0.499247	−	0.866460i	$-0.333610\pi$
$593$	351119.			0.998493			0.499247	+	0.866460i	$0.333610\pi$
$594$		0			0
$595$	−9159.08			−0.0258713
$596$			96357.6i			0.271265i
$597$		0			0
$598$			382066.i			1.06840i
$599$	−31310.5			−0.0872641			−0.0436321	−	0.999048i	$-0.513893\pi$
$599$	−31310.5			−0.0872641			−0.0436321	+	0.999048i	$0.513893\pi$
$600$		0			0
$601$		−	583406.i		−	1.61518i	−0.589742	−	0.807592i	$-0.700771\pi$
$601$		−	583406.i		−	1.61518i	0.589742	−	0.807592i	$-0.299229\pi$
$602$	−66645.3			−0.183898
$603$		0			0
$604$			601224.i			1.64802i
$605$	−138864.			−0.379384
$606$		0			0
$607$	470041.			1.27573			0.637864	−	0.770149i	$-0.279818\pi$
$607$	470041.			1.27573			0.637864	+	0.770149i	$0.279818\pi$
$608$			436873.i			1.18181i
$609$		0			0
$610$	−136046.			−0.365618
$611$	352790.			0.945005
$612$		0			0
$613$			463701.i			1.23400i	0.786961	+	0.617002i	$0.211653\pi$
$613$			463701.i			1.23400i	−0.786961	+	0.617002i	$0.788347\pi$
$614$			533832.i			1.41602i
$615$		0			0
$616$	−23592.6			−0.0621747
$617$	−23997.3			−0.0630365			−0.0315183	−	0.999503i	$-0.510034\pi$
$617$	−23997.3			−0.0630365			−0.0315183	+	0.999503i	$0.510034\pi$
$618$		0			0
$619$	479797.			1.25221			0.626104	−	0.779739i	$-0.284648\pi$
$619$	479797.			1.25221			0.626104	+	0.779739i	$0.284648\pi$
$620$		−	300403.i		−	0.781486i
$621$		0			0
$622$			750031.i			1.93864i
$623$			7991.61i			0.0205901i
$624$		0			0
$625$	108289.			0.277219
$626$	−81462.6			−0.207879
$627$		0			0
$628$		−	175619.i		−	0.445301i
$629$		−	28826.9i		−	0.0728613i
$630$		0			0
$631$	−359273.			−0.902331			−0.451166	−	0.892440i	$-0.648992\pi$
$631$	−359273.			−0.902331			−0.451166	+	0.892440i	$0.648992\pi$
$632$			345166.i			0.864158i
$633$		0			0
$634$		−	6785.76i		−	0.0168818i
$635$	−120811.			−0.299612
$636$		0			0
$637$			354567.i			0.873814i
$638$	−1.38489e6			−3.40232
$639$		0			0
$640$		−	199444.i		−	0.486924i
$641$	107842.			0.262465			0.131232	−	0.991352i	$-0.458107\pi$
$641$	107842.			0.262465			0.131232	+	0.991352i	$0.458107\pi$
$642$		0			0
$643$	727011.			1.75840			0.879202	−	0.476448i	$-0.158076\pi$
$643$	727011.			1.75840			0.879202	+	0.476448i	$0.158076\pi$
$644$			41212.0i			0.0993691i
$645$		0			0
$646$			295295.i			0.707605i
$647$	413352.			0.987443			0.493721	−	0.869620i	$-0.335636\pi$
$647$	413352.			0.987443			0.493721	+	0.869620i	$0.335636\pi$
$648$		0			0
$649$	−449942.	−	325437.i	−1.06824	−	0.772642i
$650$	418197.			0.989816
$651$		0			0
$652$	469028.			1.10333
$653$	189327.			0.444002			0.222001	−	0.975046i	$-0.428741\pi$
$653$	189327.			0.444002			0.222001	+	0.975046i	$0.428741\pi$
$654$		0			0
$655$		−	360447.i		−	0.840153i
$656$	62919.6			0.146210
$657$		0			0
$658$	66707.3			0.154071
$659$		−	496507.i		−	1.14328i	−0.820503	−	0.571642i	$-0.806307\pi$
$659$		−	496507.i		−	1.14328i	0.820503	−	0.571642i	$-0.193693\pi$
$660$		0			0
$661$	−542654.			−1.24199			−0.620997	−	0.783813i	$-0.713272\pi$
$661$	−542654.			−1.24199			−0.620997	+	0.783813i	$0.713272\pi$
$662$		−	864566.i		−	1.97280i
$663$		0			0
$664$	−351036.			−0.796188
$665$	18586.3			0.0420290
$666$		0			0
$667$			597631.i			1.34333i
$668$	−294799.			−0.660654
$669$		0			0
$670$	212867.			0.474196
$671$	276726.			0.614617
$672$		0			0
$673$			243660.i			0.537964i	0.963145	+	0.268982i	$0.0866873\pi$
$673$			243660.i			0.537964i	−0.963145	+	0.268982i	$0.913313\pi$
$674$	−1.00084e6			−2.20315
$675$		0			0
$676$	−135166.			−0.295785
$677$	−430256.			−0.938750			−0.469375	−	0.882999i	$-0.655521\pi$
$677$	−430256.			−0.938750			−0.469375	+	0.882999i	$0.655521\pi$
$678$		0			0
$679$		−	16593.5i		−	0.0359913i
$680$		−	63570.6i		−	0.137480i
$681$		0			0
$682$			1.07112e6i			2.30287i
$683$		−	163345.i		−	0.350158i	−0.984554	−	0.175079i	$-0.943982\pi$
$683$		−	163345.i		−	0.350158i	0.984554	−	0.175079i	$-0.0560181\pi$
$684$		0			0
$685$	135516.			0.288809
$686$			134687.i			0.286205i
$687$		0			0
$688$			341709.i			0.721903i
$689$			587832.i			1.23827i
$690$		0			0
$691$			529874.i			1.10973i	0.831941	+	0.554864i	$0.187230\pi$
$691$			529874.i			1.10973i	−0.831941	+	0.554864i	$0.812770\pi$
$692$		−	565002.i		−	1.17988i
$693$		0			0
$694$	−1.00024e6			−2.07675
$695$	173796.			0.359808
$696$		0			0
$697$	67257.9			0.138445
$698$	−390159.			−0.800812
$699$		0			0
$700$	45109.3			0.0920598
$701$			182341.i			0.371064i	0.982638	+	0.185532i	$0.0594008\pi$
$701$			182341.i			0.371064i	−0.982638	+	0.185532i	$0.940599\pi$
$702$		0			0
$703$			58497.7i			0.118366i
$704$			988750.i			1.99499i
$705$		0			0
$706$	−535196.			−1.07375
$707$			42272.7i			0.0845710i
$708$		0			0
$709$	−860411.			−1.71164			−0.855822	−	0.517271i	$-0.826948\pi$
$709$	−860411.			−1.71164			−0.855822	+	0.517271i	$0.826948\pi$
$710$			425945.i			0.844960i
$711$		0			0
$712$	−55467.5			−0.109415
$713$	462228.			0.909237
$714$		0			0
$715$	305423.			0.597433
$716$			505014.i			0.985093i
$717$		0			0
$718$			727993.i			1.41214i
$719$			198523.i			0.384019i	0.981393	+	0.192010i	$0.0615005\pi$
$719$			198523.i			0.384019i	−0.981393	+	0.192010i	$0.938499\pi$
$720$		0			0
$721$		−	12383.4i		−	0.0238215i
$722$			196146.i			0.376275i
$723$		0			0
$724$	532572.			1.01602
$725$	654148.			1.24452
$726$		0			0
$727$	−649.946			−0.00122973			−0.000614863	−	1.00000i	$-0.500196\pi$
$727$	−649.946			−0.00122973			−0.000614863		1.00000i	$0.500196\pi$
$728$	22035.8			0.0415783
$729$		0			0
$730$	162556.			0.305041
$731$			365269.i			0.683563i
$732$		0			0
$733$	27948.5			0.0520176			0.0260088	−	0.999662i	$-0.491720\pi$
$733$	27948.5			0.0520176			0.0260088	+	0.999662i	$0.491720\pi$
$734$	345218.			0.640769
$735$		0			0
$736$	585785.			1.08139
$737$	−432982.			−0.797141
$738$		0			0
$739$		−	406137.i		−	0.743675i	−0.928298	−	0.371838i	$-0.878728\pi$
$739$		−	406137.i		−	0.743675i	0.928298	−	0.371838i	$-0.121272\pi$
$740$		−	50976.5i		−	0.0930908i
$741$		0			0
$742$			111150.i			0.201884i
$743$	426473.			0.772528			0.386264	−	0.922388i	$-0.373765\pi$
$743$	426473.			0.772528			0.386264	+	0.922388i	$0.373765\pi$
$744$		0			0
$745$		−	58269.1i		−	0.104985i
$746$		−	755470.i		−	1.35750i
$747$		0			0
$748$			523419.i			0.935506i
$749$	18181.1			0.0324083
$750$		0			0
$751$		−	625542.i		−	1.10911i	−0.832146	−	0.554557i	$-0.812888\pi$
$751$		−	625542.i		−	1.10911i	0.832146	−	0.554557i	$-0.187112\pi$
$752$		−	342026.i		−	0.604816i
$753$		0			0
$754$	1.29351e6			2.27524
$755$		−	363570.i		−	0.637815i
$756$		0			0
$757$	−42832.4			−0.0747448			−0.0373724	−	0.999301i	$-0.511899\pi$
$757$	−42832.4			−0.0747448			−0.0373724	+	0.999301i	$0.511899\pi$
$758$		−	1.45009e6i		−	2.52382i
$759$		0			0
$760$			129002.i			0.223342i
$761$	−297397.			−0.513532			−0.256766	−	0.966474i	$-0.582657\pi$
$761$	−297397.			−0.513532			−0.256766	+	0.966474i	$0.582657\pi$
$762$		0			0
$763$		−	51978.3i		−	0.0892839i
$764$			179063.i			0.306774i
$765$		0			0
$766$			1.65941e6i			2.82811i
$767$	420253.	+	303963.i	0.714365	+	0.516691i
$768$		0			0
$769$		−	211683.i		−	0.357960i	−0.983853	−	0.178980i	$-0.942720\pi$
$769$		−	211683.i		−	0.357960i	0.983853	−	0.178980i	$-0.0572797\pi$
$770$	57750.8			0.0974040
$771$		0			0
$772$	−109691.			−0.184050
$773$			869866.i			1.45577i	0.685698	+	0.727886i	$0.259497\pi$
$773$			869866.i			1.45577i	−0.685698	+	0.727886i	$0.740503\pi$
$774$		0			0
$775$		−	505940.i		−	0.842356i
$776$	115171.			0.191257
$777$		0			0
$778$		−	467610.i		−	0.772546i
$779$	−136485.			−0.224910
$780$		0			0
$781$		−	866394.i		−	1.42041i
$782$	395948.			0.647478
$783$		0			0
$784$	343748.			0.559253
$785$			106200.i			0.172340i
$786$		0			0
$787$	−281012.			−0.453707			−0.226853	−	0.973929i	$-0.572844\pi$
$787$	−281012.			−0.453707			−0.226853	+	0.973929i	$0.572844\pi$
$788$	281796.			0.453818
$789$		0			0
$790$		−	844910.i		−	1.35381i
$791$			43379.2i			0.0693312i
$792$		0			0
$793$	−258466.			−0.411015
$794$	16371.5			0.0259686
$795$		0			0
$796$	406464.			0.641499
$797$		−	515861.i		−	0.812112i	−0.913848	−	0.406056i	$-0.866904\pi$
$797$		−	515861.i		−	0.812112i	0.913848	−	0.406056i	$-0.133096\pi$
$798$		0			0
$799$		−	365609.i		−	0.572695i
$800$		−	641182.i		−	1.00185i
$801$		0			0
$802$	−1.27118e6			−1.97632
$803$	−330648.			−0.512785
$804$		0			0
$805$		−	24921.6i		−	0.0384577i
$806$		−	1.00044e6i		−	1.54001i
$807$		0			0
$808$	−293403.			−0.449409
$809$		−	568738.i		−	0.868991i	−0.900674	−	0.434496i	$-0.856927\pi$
$809$		−	568738.i		−	0.868991i	0.900674	−	0.434496i	$-0.143073\pi$
$810$		0			0
$811$		−	2103.38i		−	0.00319798i	−0.999999	−	0.00159899i	$-0.999491\pi$
$811$		−	2103.38i		−	0.00319798i	0.999999	−	0.00159899i	$-0.000508974\pi$
$812$	139526.			0.211613
$813$		0			0
$814$			181763.i			0.274319i
$815$	−283629.			−0.427008
$816$		0			0
$817$		−	741231.i		−	1.11048i
$818$	−1.86939e6			−2.79378
$819$		0			0
$820$	118937.			0.176884
$821$		−	442834.i		−	0.656984i	−0.944507	−	0.328492i	$-0.893460\pi$
$821$		−	442834.i		−	0.656984i	0.944507	−	0.328492i	$-0.106540\pi$
$822$		0			0
$823$			814438.i			1.20243i	0.799089	+	0.601213i	$0.205316\pi$
$823$			814438.i			1.20243i	−0.799089	+	0.601213i	$0.794684\pi$
$824$	85949.4			0.126587
$825$		0			0
$826$	79463.5	+	57474.9i	0.116468	+	0.0842399i
$827$	−842514.			−1.23187			−0.615936	−	0.787796i	$-0.711222\pi$
$827$	−842514.			−1.23187			−0.615936	+	0.787796i	$0.711222\pi$
$828$		0			0
$829$	589973.			0.858467			0.429233	−	0.903194i	$-0.358784\pi$
$829$	589973.			0.858467			0.429233	+	0.903194i	$0.358784\pi$
$830$	859280.			1.24732
$831$		0			0
$832$		−	923507.i		−	1.33412i
$833$	367450.			0.529552
$834$		0			0
$835$	178270.			0.255685
$836$		−	1.06216e6i		−	1.51977i
$837$		0			0
$838$	1.30755e6			1.86196
$839$		−	999621.i		−	1.42008i	−0.704164	−	0.710038i	$-0.748678\pi$
$839$		−	999621.i		−	1.42008i	0.704164	−	0.710038i	$-0.251322\pi$
$840$		0			0
$841$	1.31604e6			1.86070
$842$	−1.28919e6			−1.81842
$843$		0			0
$844$		−	1.88190e6i		−	2.64187i
$845$	81737.4			0.114474
$846$		0			0
$847$	−49884.2			−0.0695338
$848$	569896.			0.792509
$849$		0			0
$850$		−	433393.i		−	0.599852i
$851$	78437.1			0.108308
$852$		0			0
$853$	11376.6			0.0156356			0.00781781	−	0.999969i	$-0.497511\pi$
$853$	11376.6			0.0156356			0.00781781	+	0.999969i	$0.497511\pi$
$854$	−48872.0			−0.0670108
$855$		0			0
$856$			126190.i			0.172217i
$857$			404528.i			0.550791i	0.961331	+	0.275396i	$0.0888089\pi$
$857$			404528.i			0.550791i	−0.961331	+	0.275396i	$0.911191\pi$
$858$		0			0
$859$		−	633417.i		−	0.858427i	−0.903203	−	0.429213i	$-0.858791\pi$
$859$		−	633417.i		−	0.858427i	0.903203	−	0.429213i	$-0.141209\pi$
$860$			645930.i			0.873350i
$861$		0			0
$862$	−472318.			−0.635653
$863$		−	1.26751e6i		−	1.70188i	−0.525262	−	0.850941i	$-0.676033\pi$
$863$		−	1.26751e6i		−	1.70188i	0.525262	−	0.850941i	$-0.323967\pi$
$864$		0			0
$865$			341666.i			0.456636i
$866$		−	1.50713e6i		−	2.00963i
$867$		0			0
$868$		−	107914.i		−	0.143231i
$869$			1.71859e6i			2.27580i
$870$		0			0
$871$	404412.			0.533074
$872$	360767.			0.474453
$873$		0			0
$874$	−803487.			−1.05186
$875$	−64351.1			−0.0840504
$876$		0			0
$877$	496024.			0.644917			0.322458	−	0.946584i	$-0.395491\pi$
$877$	496024.			0.644917			0.322458	+	0.946584i	$0.395491\pi$
$878$			268246.i			0.347971i
$879$		0			0
$880$		−	296104.i		−	0.382366i
$881$			1.10943e6i			1.42938i	0.699442	+	0.714690i	$0.253432\pi$
$881$			1.10943e6i			1.42938i	−0.699442	+	0.714690i	$0.746568\pi$
$882$		0			0
$883$	−1.09641e6			−1.40622			−0.703109	−	0.711082i	$-0.748205\pi$
$883$	−1.09641e6			−1.40622			−0.703109	+	0.711082i	$0.748205\pi$
$884$		−	488881.i		−	0.625603i
$885$		0			0
$886$	−1.24764e6			−1.58935
$887$			1.14912e6i			1.46055i	0.683153	+	0.730275i	$0.260608\pi$
$887$			1.14912e6i			1.46055i	−0.683153	+	0.730275i	$0.739392\pi$
$888$		0			0
$889$	−43399.0			−0.0549132
$890$	135776.			0.171412
$891$		0			0
$892$	−409885.			−0.515148
$893$			741920.i			0.930367i
$894$		0			0
$895$		−	305390.i		−	0.381250i
$896$		−	71646.4i		−	0.0892438i
$897$		0			0
$898$			226326.i			0.280661i
$899$		−	1.56490e6i		−	1.93628i
$900$		0			0
$901$	609191.			0.750419
$902$	−424082.			−0.521238
$903$		0			0
$904$	−301083.			−0.368425
$905$	−322055.			−0.393218
$906$		0			0
$907$	152142.			0.184941			0.0924706	−	0.995715i	$-0.470524\pi$
$907$	152142.			0.184941			0.0924706	+	0.995715i	$0.470524\pi$
$908$		−	580313.i		−	0.703867i
$909$		0			0
$910$	−53940.1			−0.0651372
$911$	−38792.0			−0.0467418			−0.0233709	−	0.999727i	$-0.507440\pi$
$911$	−38792.0			−0.0467418			−0.0233709	+	0.999727i	$0.507440\pi$
$912$		0			0
$913$	−1.74782e6			−2.09679
$914$	−177463.			−0.212430
$915$		0			0
$916$		−	150567.i		−	0.179448i
$917$		−	129483.i		−	0.153984i
$918$		0			0
$919$			1.11869e6i			1.32459i	0.749245	+	0.662293i	$0.230416\pi$
$919$			1.11869e6i			1.32459i	−0.749245	+	0.662293i	$0.769584\pi$
$920$	172973.			0.204364
$921$		0			0
$922$		−	1.55085e6i		−	1.82435i
$923$			809225.i			0.949874i
$924$		0			0
$925$		−	85854.8i		−	0.100342i
$926$	2.35960e6			2.75180
$927$		0			0
$928$		−	1.98322e6i		−	2.30289i
$929$			872704.i			1.01120i	0.862769	+	0.505598i	$0.168728\pi$
$929$			872704.i			1.01120i	−0.862769	+	0.505598i	$0.831272\pi$
$930$		0			0
$931$	−745656.			−0.860279
$932$		−	1.27107e6i		−	1.46332i
$933$		0			0
$934$	−2.10303e6			−2.41074
$935$		−	316520.i		−	0.362058i
$936$		0			0
$937$		−	1.09424e6i		−	1.24633i	−0.782089	−	0.623167i	$-0.785846\pi$
$937$		−	1.09424e6i		−	1.24633i	0.782089	−	0.623167i	$-0.214154\pi$
$938$	76468.1			0.0869110
$939$		0			0
$940$		−	646530.i		−	0.731700i
$941$		−	145369.i		−	0.164169i	−0.996625	−	0.0820845i	$-0.973842\pi$
$941$		−	145369.i		−	0.164169i	0.996625	−	0.0820845i	$-0.0261578\pi$
$942$		0			0
$943$			183007.i			0.205799i
$944$	294689.	−	407431.i	0.330689	−	0.457204i
$945$		0			0
$946$		−	2.30314e6i		−	2.57358i
$947$	1.33630e6			1.49006			0.745032	−	0.667028i	$-0.232434\pi$
$947$	1.33630e6			1.49006			0.745032	+	0.667028i	$0.232434\pi$
$948$		0			0
$949$	308830.			0.342916
$950$			879472.i			0.974484i
$951$		0			0
$952$		−	22836.5i		−	0.0251974i
$953$	699411.			0.770099			0.385050	−	0.922896i	$-0.374184\pi$
$953$	699411.			0.770099			0.385050	+	0.922896i	$0.374184\pi$
$954$		0			0
$955$		−	108282.i		−	0.118727i
$956$	1.67840e6			1.83645
$957$		0			0
$958$			1.86086e6i			2.02760i
$959$	48681.6			0.0529331
$960$		0			0
$961$	−286828.			−0.310581
$962$		−	169769.i		−	0.183446i
$963$		0			0
$964$	1.07139e6			1.15290
$965$	66331.9			0.0712308
$966$		0			0
$967$			776605.i			0.830514i	0.909704	+	0.415257i	$0.136308\pi$
$967$			776605.i			0.830514i	−0.909704	+	0.415257i	$0.863692\pi$
$968$		−	346232.i		−	0.369502i
$969$		0			0
$970$	−281919.			−0.299627
$971$	−1.06596e6			−1.13058			−0.565292	−	0.824891i	$-0.691236\pi$
$971$	−1.06596e6			−1.13058			−0.565292	+	0.824891i	$0.691236\pi$
$972$		0			0
$973$	62433.0			0.0659460
$974$			1.99804e6i			2.10614i
$975$		0			0
$976$			250580.i			0.263055i
$977$		−	637748.i		−	0.668129i	−0.942550	−	0.334064i	$-0.891580\pi$
$977$		−	637748.i		−	0.668129i	0.942550	−	0.334064i	$-0.108420\pi$
$978$		0			0
$979$	−276175.			−0.288150
$980$	649786.			0.676578
$981$		0			0
$982$		−	2.01729e6i		−	2.09192i
$983$		−	317307.i		−	0.328377i	−0.986429	−	0.164189i	$-0.947499\pi$
$983$		−	317307.i		−	0.328377i	0.986429	−	0.164189i	$-0.0525006\pi$
$984$		0			0
$985$	−170407.			−0.175636
$986$		−	1.34051e6i		−	1.37885i
$987$		0			0
$988$			992073.i			1.01632i
$989$	−993886.			−1.01612
$990$		0			0
$991$			784221.i			0.798530i	0.916836	+	0.399265i	$0.130735\pi$
$991$			784221.i			0.798530i	−0.916836	+	0.399265i	$0.869265\pi$
$992$	−1.53389e6			−1.55873
$993$		0			0
$994$			153012.i			0.154865i
$995$	−245796.			−0.248272
$996$		0			0
$997$	353252.			0.355381			0.177691	−	0.984086i	$-0.443137\pi$
$997$	353252.			0.355381			0.177691	+	0.984086i	$0.443137\pi$
$998$			1.26398e6i			1.26905i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.5.c.d.235.6		40
3.2	odd	2		177.5.c.a.58.35	yes	40
59.58	odd	2	inner	531.5.c.d.235.35		40
177.176	even	2		177.5.c.a.58.6	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.5.c.a.58.6	✓	40	177.176	even	2
177.5.c.a.58.35	yes	40	3.2	odd	2
531.5.c.d.235.6		40	1.1	even	1	trivial
531.5.c.d.235.35		40	59.58	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:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	531.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-6.10324i q^{2} -21.2495 q^{4} +12.8499 q^{5} +4.61608 q^{7} +32.0389i q^{8} +O(q^{10})\)	\(q-6.10324i q^{2} -21.2495 q^{4} +12.8499 q^{5} +4.61608 q^{7} +32.0389i q^{8} -78.4262i q^{10} +159.523i q^{11} -148.997i q^{13} -28.1731i q^{14} -144.451 q^{16} -154.411 q^{17} +313.342 q^{19} -273.055 q^{20} +973.607 q^{22} -420.147i q^{23} -459.879 q^{25} -909.363 q^{26} -98.0895 q^{28} -1422.43 q^{29} +1100.16i q^{31} +1394.24i q^{32} +942.405i q^{34} +59.3164 q^{35} +186.690i q^{37} -1912.40i q^{38} +411.698i q^{40} -435.578 q^{41} -2365.57i q^{43} -3389.78i q^{44} -2564.25 q^{46} +2367.77i q^{47} -2379.69 q^{49} +2806.75i q^{50} +3166.11i q^{52} -3945.26 q^{53} +2049.86i q^{55} +147.894i q^{56} +8681.45i q^{58} +(-2040.07 + 2820.55i) q^{59} -1734.71i q^{61} +6714.53 q^{62} +6198.17 q^{64} -1914.60i q^{65} +2714.23i q^{67} +3281.15 q^{68} -362.022i q^{70} -5431.15 q^{71} +2072.73i q^{73} +1139.41 q^{74} -6658.35 q^{76} +736.372i q^{77} +10773.3 q^{79} -1856.18 q^{80} +2658.44i q^{82} +10956.5i q^{83} -1984.17 q^{85} -14437.6 q^{86} -5110.95 q^{88} +1731.25i q^{89} -687.782i q^{91} +8927.90i q^{92} +14451.1 q^{94} +4026.42 q^{95} -3594.71i q^{97} +14523.8i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 320 q^{4} + 80 q^{7}+O(q^{10}) \) 40 * q - 320 * q^4 + 80 * q^7	\( 40 q - 320 q^{4} + 80 q^{7} + 3944 q^{16} + 528 q^{17} + 444 q^{19} - 444 q^{20} + 1304 q^{22} + 4880 q^{25} + 1452 q^{26} - 1160 q^{28} + 996 q^{29} - 10320 q^{35} + 5196 q^{41} - 10476 q^{46} + 5104 q^{49} + 2184 q^{53} + 11736 q^{59} - 15240 q^{62} - 81012 q^{64} - 29568 q^{68} + 5964 q^{71} - 14376 q^{74} + 3480 q^{76} + 19020 q^{79} - 33096 q^{80} + 20220 q^{85} + 65880 q^{86} - 14932 q^{88} - 17864 q^{94} - 11004 q^{95}+O(q^{100}) \) 40 * q - 320 * q^4 + 80 * q^7 + 3944 * q^16 + 528 * q^17 + 444 * q^19 - 444 * q^20 + 1304 * q^22 + 4880 * q^25 + 1452 * q^26 - 1160 * q^28 + 996 * q^29 - 10320 * q^35 + 5196 * q^41 - 10476 * q^46 + 5104 * q^49 + 2184 * q^53 + 11736 * q^59 - 15240 * q^62 - 81012 * q^64 - 29568 * q^68 + 5964 * q^71 - 14376 * q^74 + 3480 * q^76 + 19020 * q^79 - 33096 * q^80 + 20220 * q^85 + 65880 * q^86 - 14932 * q^88 - 17864 * q^94 - 11004 * q^95

Introduction

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Groups

Database

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