LMFDB - Embedded newform 531.5.c.d.235.38

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			235.38
Character	$\chi$	$=$	531.235
Dual form			531.5.c.d.235.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.65331i q^{2} -42.5731 q^{4} +38.1687 q^{5} -35.1454 q^{7} -203.372i q^{8} +O(q^{10})$	\(q+7.65331i q^{2} -42.5731 q^{4} +38.1687 q^{5} -35.1454 q^{7} -203.372i q^{8} +292.117i q^{10} +147.824i q^{11} -17.0918i q^{13} -268.979i q^{14} +875.302 q^{16} +354.287 q^{17} -647.468 q^{19} -1624.96 q^{20} -1131.34 q^{22} +862.278i q^{23} +831.852 q^{25} +130.809 q^{26} +1496.25 q^{28} -440.875 q^{29} +347.019i q^{31} +3445.00i q^{32} +2711.47i q^{34} -1341.45 q^{35} +972.830i q^{37} -4955.27i q^{38} -7762.47i q^{40} -2783.36 q^{41} -2837.80i q^{43} -6293.33i q^{44} -6599.28 q^{46} -1287.07i q^{47} -1165.80 q^{49} +6366.42i q^{50} +727.650i q^{52} +1103.72 q^{53} +5642.25i q^{55} +7147.61i q^{56} -3374.15i q^{58} +(-1692.03 - 3042.10i) q^{59} -5980.77i q^{61} -2655.84 q^{62} -12360.8 q^{64} -652.371i q^{65} +3352.44i q^{67} -15083.1 q^{68} -10266.6i q^{70} +1396.21 q^{71} -1649.77i q^{73} -7445.37 q^{74} +27564.7 q^{76} -5195.33i q^{77} -8357.83 q^{79} +33409.2 q^{80} -21301.9i q^{82} +5025.77i q^{83} +13522.7 q^{85} +21718.6 q^{86} +30063.3 q^{88} -5292.92i q^{89} +600.697i q^{91} -36709.9i q^{92} +9850.36 q^{94} -24713.0 q^{95} +14051.7i q^{97} -8922.24i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$			7.65331i			1.91333i	0.291195	+	0.956664i	$0.405947\pi$
$2$			7.65331i			1.91333i	−0.291195	+	0.956664i	$0.594053\pi$
$3$		0			0
$3$		0			0
$4$	−42.5731			−2.66082
$5$	38.1687			1.52675			0.763375	−	0.645956i	$-0.223541\pi$
$5$	38.1687			1.52675			0.763375	+	0.645956i	$0.223541\pi$
$6$		0			0
$7$	−35.1454			−0.717253			−0.358626	−	0.933481i	$-0.616755\pi$
$7$	−35.1454			−0.717253			−0.358626	+	0.933481i	$0.616755\pi$
$8$		−	203.372i		−	3.17769i
$9$		0			0
$10$			292.117i			2.92117i
$11$			147.824i			1.22169i	0.791752	+	0.610843i	$0.209169\pi$
$11$			147.824i			1.22169i	−0.791752	+	0.610843i	$0.790831\pi$
$12$		0			0
$13$		−	17.0918i		−	0.101135i	−0.998721	−	0.0505674i	$-0.983897\pi$
$13$		−	17.0918i		−	0.101135i	0.998721	−	0.0505674i	$-0.0161030\pi$
$14$		−	268.979i		−	1.37234i
$15$		0			0
$16$	875.302			3.41915
$17$	354.287			1.22591			0.612953	−	0.790119i	$-0.289981\pi$
$17$	354.287			1.22591			0.612953	+	0.790119i	$0.289981\pi$
$18$		0			0
$19$	−647.468			−1.79354			−0.896770	−	0.442498i	$-0.854092\pi$
$19$	−647.468			−1.79354			−0.896770	+	0.442498i	$0.854092\pi$
$20$	−1624.96			−4.06241
$21$		0			0
$22$	−1131.34			−2.33748
$23$			862.278i			1.63001i	0.579451	+	0.815007i	$0.303267\pi$
$23$			862.278i			1.63001i	−0.579451	+	0.815007i	$0.696733\pi$
$24$		0			0
$25$	831.852			1.33096
$26$	130.809			0.193504
$27$		0			0
$28$	1496.25			1.90848
$29$	−440.875			−0.524227			−0.262113	−	0.965037i	$-0.584419\pi$
$29$	−440.875			−0.524227			−0.262113	+	0.965037i	$0.584419\pi$
$30$		0			0
$31$			347.019i			0.361101i	0.983566	+	0.180551i	$0.0577880\pi$
$31$			347.019i			0.361101i	−0.983566	+	0.180551i	$0.942212\pi$
$32$			3445.00i			3.36426i
$33$		0			0
$34$			2711.47i			2.34556i
$35$	−1341.45			−1.09507
$36$		0			0
$37$			972.830i			0.710613i	0.934750	+	0.355307i	$0.115624\pi$
$37$			972.830i			0.710613i	−0.934750	+	0.355307i	$0.884376\pi$
$38$		−	4955.27i		−	3.43163i
$39$		0			0
$40$		−	7762.47i		−	4.85154i
$41$	−2783.36			−1.65578			−0.827889	−	0.560892i	$-0.810458\pi$
$41$	−2783.36			−1.65578			−0.827889	+	0.560892i	$0.810458\pi$
$42$		0			0
$43$		−	2837.80i		−	1.53478i	−0.641183	−	0.767388i	$-0.721556\pi$
$43$		−	2837.80i		−	1.53478i	0.641183	−	0.767388i	$-0.278444\pi$
$44$		−	6293.33i		−	3.25069i
$45$		0			0
$46$	−6599.28			−3.11875
$47$		−	1287.07i		−	0.582649i	−0.956624	−	0.291324i	$-0.905904\pi$
$47$		−	1287.07i		−	0.582649i	0.956624	−	0.291324i	$-0.0940959\pi$
$48$		0			0
$49$	−1165.80			−0.485548
$50$			6366.42i			2.54657i
$51$		0			0
$52$			727.650i			0.269101i
$53$	1103.72			0.392923			0.196462	−	0.980511i	$-0.437055\pi$
$53$	1103.72			0.392923			0.196462	+	0.980511i	$0.437055\pi$
$54$		0			0
$55$			5642.25i			1.86521i
$56$			7147.61i			2.27921i
$57$		0			0
$58$		−	3374.15i		−	1.00302i
$59$	−1692.03	−	3042.10i	−0.486076	−	0.873917i
$59$	−1692.03	−	3042.10i	−0.486076	−	0.873917i
$60$		0			0
$61$		−	5980.77i		−	1.60730i	−0.595100	−	0.803651i	$-0.702888\pi$
$61$		−	5980.77i		−	1.60730i	0.595100	−	0.803651i	$-0.297112\pi$
$62$	−2655.84			−0.690905
$63$		0			0
$64$	−12360.8			−3.01777
$65$		−	652.371i		−	0.154407i
$66$		0			0
$67$			3352.44i			0.746812i	0.927668	+	0.373406i	$0.121810\pi$
$67$			3352.44i			0.746812i	−0.927668	+	0.373406i	$0.878190\pi$
$68$	−15083.1			−3.26192
$69$		0			0
$70$		−	10266.6i		−	2.09522i
$71$	1396.21			0.276971			0.138486	−	0.990364i	$-0.455777\pi$
$71$	1396.21			0.276971			0.138486	+	0.990364i	$0.455777\pi$
$72$		0			0
$73$		−	1649.77i		−	0.309584i	−0.987947	−	0.154792i	$-0.950529\pi$
$73$		−	1649.77i		−	0.309584i	0.987947	−	0.154792i	$-0.0494707\pi$
$74$	−7445.37			−1.35964
$75$		0			0
$76$	27564.7			4.77229
$77$		−	5195.33i		−	0.876258i
$78$		0			0
$79$	−8357.83			−1.33918			−0.669591	−	0.742730i	$-0.733530\pi$
$79$	−8357.83			−1.33918			−0.669591	+	0.742730i	$0.733530\pi$
$80$	33409.2			5.22018
$81$		0			0
$82$		−	21301.9i		−	3.16805i
$83$			5025.77i			0.729536i	0.931098	+	0.364768i	$0.118852\pi$
$83$			5025.77i			0.729536i	−0.931098	+	0.364768i	$0.881148\pi$
$84$		0			0
$85$	13522.7			1.87165
$86$	21718.6			2.93653
$87$		0			0
$88$	30063.3			3.88214
$89$		−	5292.92i		−	0.668213i	−0.942535	−	0.334107i	$-0.891565\pi$
$89$		−	5292.92i		−	0.668213i	0.942535	−	0.334107i	$-0.108435\pi$
$90$		0			0
$91$			600.697i			0.0725392i
$92$		−	36709.9i		−	4.33718i
$93$		0			0
$94$	9850.36			1.11480
$95$	−24713.0			−2.73828
$96$		0			0
$97$			14051.7i			1.49343i	0.665142	+	0.746716i	$0.268371\pi$
$97$			14051.7i			1.49343i	−0.665142	+	0.746716i	$0.731629\pi$
$98$		−	8922.24i		−	0.929013i
$99$		0			0
$100$	−35414.5			−3.54145
$101$		−	623.626i		−	0.0611338i	−0.999533	−	0.0305669i	$-0.990269\pi$
$101$		−	623.626i		−	0.0611338i	0.999533	−	0.0305669i	$-0.00973126\pi$
$102$		0			0
$103$			8032.20i			0.757112i	0.925578	+	0.378556i	$0.123579\pi$
$103$			8032.20i			0.757112i	−0.925578	+	0.378556i	$0.876421\pi$
$104$	−3475.99			−0.321375
$105$		0			0
$106$			8447.13i			0.751791i
$107$	−11098.7			−0.969402			−0.484701	−	0.874680i	$-0.661072\pi$
$107$	−11098.7			−0.969402			−0.484701	+	0.874680i	$0.661072\pi$
$108$		0			0
$109$		−	7809.17i		−	0.657283i	−0.944455	−	0.328641i	$-0.893409\pi$
$109$		−	7809.17i		−	0.657283i	0.944455	−	0.328641i	$-0.106591\pi$
$110$	−43181.9			−3.56875
$111$		0			0
$112$	−30762.8			−2.45239
$113$		−	3130.68i		−	0.245178i	−0.992457	−	0.122589i	$-0.960880\pi$
$113$		−	3130.68i		−	0.245178i	0.992457	−	0.122589i	$-0.0391198\pi$
$114$		0			0
$115$			32912.0i			2.48862i
$116$	18769.4			1.39487
$117$		0			0
$118$	23282.2	−	12949.6i	1.67209	−	0.930022i
$119$	−12451.6			−0.879285
$120$		0			0
$121$	−7210.93			−0.492516
$122$	45772.7			3.07530
$123$		0			0
$124$		−	14773.7i		−	0.960827i
$125$	7895.26			0.505297
$126$		0			0
$127$	−18711.5			−1.16011			−0.580056	−	0.814576i	$-0.696969\pi$
$127$	−18711.5			−1.16011			−0.580056	+	0.814576i	$0.696969\pi$
$128$		−	39481.1i		−	2.40973i
$129$		0			0
$130$	4992.80			0.295432
$131$			12408.9i			0.723089i	0.932355	+	0.361545i	$0.117751\pi$
$131$			12408.9i			0.723089i	−0.932355	+	0.361545i	$0.882249\pi$
$132$		0			0
$133$	22755.5			1.28642
$134$	−25657.2			−1.42889
$135$		0			0
$136$		−	72052.2i		−	3.89556i
$137$	24025.7			1.28007			0.640037	−	0.768344i	$-0.278919\pi$
$137$	24025.7			1.28007			0.640037	+	0.768344i	$0.278919\pi$
$138$		0			0
$139$	−3014.36			−0.156014			−0.0780072	−	0.996953i	$-0.524856\pi$
$139$	−3014.36			−0.156014			−0.0780072	+	0.996953i	$0.524856\pi$
$140$	57109.9			2.91377
$141$		0			0
$142$			10685.6i			0.529937i
$143$	2526.57			0.123555
$144$		0			0
$145$	−16827.6			−0.800362
$146$	12626.2			0.592335
$147$		0			0
$148$		−	41416.4i		−	1.89082i
$149$			10398.6i			0.468383i	0.972190	+	0.234192i	$0.0752443\pi$
$149$			10398.6i			0.468383i	−0.972190	+	0.234192i	$0.924756\pi$
$150$		0			0
$151$		−	35440.0i		−	1.55432i	−0.629305	−	0.777159i	$-0.716660\pi$
$151$		−	35440.0i		−	1.55432i	0.629305	−	0.777159i	$-0.283340\pi$
$152$			131677.i			5.69932i
$153$		0			0
$154$	39761.5			1.67657
$155$			13245.3i			0.551311i
$156$		0			0
$157$			23302.5i			0.945373i	0.881231	+	0.472686i	$0.156716\pi$
$157$			23302.5i			0.945373i	−0.881231	+	0.472686i	$0.843284\pi$
$158$		−	63965.0i		−	2.56229i
$159$		0			0
$160$			131491.i			5.13637i
$161$		−	30305.1i		−	1.16913i
$162$		0			0
$163$	−8996.62			−0.338613			−0.169307	−	0.985563i	$-0.554153\pi$
$163$	−8996.62			−0.338613			−0.169307	+	0.985563i	$0.554153\pi$
$164$	118497.			4.40573
$165$		0			0
$166$	−38463.8			−1.39584
$167$	441.902			0.0158450			0.00792252	−	0.999969i	$-0.497478\pi$
$167$	441.902			0.0158450			0.00792252	+	0.999969i	$0.497478\pi$
$168$		0			0
$169$	28268.9			0.989772
$170$			103493.i			3.58108i
$171$		0			0
$172$			120814.i			4.08376i
$173$			2947.42i			0.0984803i	0.998787	+	0.0492402i	$0.0156800\pi$
$173$			2947.42i			0.0984803i	−0.998787	+	0.0492402i	$0.984320\pi$
$174$		0			0
$175$	−29235.8			−0.954637
$176$			129391.i			4.17713i
$177$		0			0
$178$	40508.3			1.27851
$179$		−	17326.2i		−	0.540750i	−0.962755	−	0.270375i	$-0.912852\pi$
$179$		−	17326.2i		−	0.540750i	0.962755	−	0.270375i	$-0.0871477\pi$
$180$		0			0
$181$	−9276.44			−0.283155			−0.141578	−	0.989927i	$-0.545217\pi$
$181$	−9276.44			−0.283155			−0.141578	+	0.989927i	$0.545217\pi$
$182$	−4597.32			−0.138791
$183$		0			0
$184$	175364.			5.17969
$185$			37131.7i			1.08493i
$186$		0			0
$187$			52372.1i			1.49767i
$188$			54794.7i			1.55032i
$189$		0			0
$190$		−	189136.i		−	5.23923i
$191$		−	14709.0i		−	0.403195i	−0.979468	−	0.201598i	$-0.935387\pi$
$191$		−	14709.0i		−	0.403195i	0.979468	−	0.201598i	$-0.0646134\pi$
$192$		0			0
$193$	−34249.4			−0.919472			−0.459736	−	0.888056i	$-0.652056\pi$
$193$	−34249.4			−0.919472			−0.459736	+	0.888056i	$0.652056\pi$
$194$	−107542.			−2.85743
$195$		0			0
$196$	49631.8			1.29196
$197$	−7911.51			−0.203858			−0.101929	−	0.994792i	$-0.532501\pi$
$197$	−7911.51			−0.203858			−0.101929	+	0.994792i	$0.532501\pi$
$198$		0			0
$199$	−57454.5			−1.45084			−0.725418	−	0.688309i	$-0.758353\pi$
$199$	−57454.5			−1.45084			−0.725418	+	0.688309i	$0.758353\pi$
$200$		−	169176.i		−	4.22939i
$201$		0			0
$202$	4772.80			0.116969
$203$	15494.7			0.376003
$204$		0			0
$205$	−106237.			−2.52796
$206$	−61472.9			−1.44860
$207$		0			0
$208$		−	14960.5i		−	0.345795i
$209$		−	95711.3i		−	2.19114i
$210$		0			0
$211$			49004.2i			1.10070i	0.834935	+	0.550349i	$0.185505\pi$
$211$			49004.2i			1.10070i	−0.834935	+	0.550349i	$0.814495\pi$
$212$	−46988.9			−1.04550
$213$		0			0
$214$		−	84941.6i		−	1.85478i
$215$		−	108315.i		−	2.34322i
$216$		0			0
$217$		−	12196.1i		−	0.259001i
$218$	59766.0			1.25760
$219$		0			0
$220$		−	240208.i		−	4.96298i
$221$		−	6055.39i		−	0.123982i
$222$		0			0
$223$	94322.8			1.89674			0.948368	−	0.317171i	$-0.102733\pi$
$223$	94322.8			1.89674			0.948368	+	0.317171i	$0.102733\pi$
$224$		−	121076.i		−	2.41302i
$225$		0			0
$226$	23960.1			0.469107
$227$			73496.8i			1.42632i	0.701002	+	0.713159i	$0.252736\pi$
$227$			73496.8i			1.42632i	−0.701002	+	0.713159i	$0.747264\pi$
$228$		0			0
$229$		−	71268.4i		−	1.35902i	−0.733666	−	0.679510i	$-0.762192\pi$
$229$		−	71268.4i		−	1.35902i	0.733666	−	0.679510i	$-0.237808\pi$
$230$	−251886.			−4.76155
$231$		0			0
$232$			89661.7i			1.66583i
$233$			21675.4i			0.399259i	0.979871	+	0.199630i	$0.0639739\pi$
$233$			21675.4i			0.399259i	−0.979871	+	0.199630i	$0.936026\pi$
$234$		0			0
$235$		−	49125.9i		−	0.889559i
$236$	72035.0	+	129512.i	1.29336	+	2.32534i
$237$		0			0
$238$		−	95295.6i		−	1.68236i
$239$	556.432			0.00974129			0.00487065	−	0.999988i	$-0.498450\pi$
$239$	556.432			0.00974129			0.00487065	+	0.999988i	$0.498450\pi$
$240$		0			0
$241$	34167.1			0.588266			0.294133	−	0.955764i	$-0.404969\pi$
$241$	34167.1			0.588266			0.294133	+	0.955764i	$0.404969\pi$
$242$		−	55187.5i		−	0.942345i
$243$		0			0
$244$			254620.i			4.27675i
$245$	−44497.2			−0.741310
$246$		0			0
$247$			11066.4i			0.181389i
$248$	70574.0			1.14747
$249$		0			0
$250$			60424.9i			0.966798i
$251$	−57018.6			−0.905043			−0.452522	−	0.891753i	$-0.649475\pi$
$251$	−57018.6			−0.905043			−0.452522	+	0.891753i	$0.649475\pi$
$252$		0			0
$253$	−127465.			−1.99137
$254$		−	143205.i		−	2.21967i
$255$		0			0
$256$	104388.			1.59283
$257$	3529.50			0.0534375			0.0267188	−	0.999643i	$-0.491494\pi$
$257$	3529.50			0.0534375			0.0267188	+	0.999643i	$0.491494\pi$
$258$		0			0
$259$		−	34190.5i		−	0.509689i
$260$			27773.5i			0.410850i
$261$		0			0
$262$	−94969.4			−1.38351
$263$	−39797.7			−0.575369			−0.287684	−	0.957725i	$-0.592885\pi$
$263$	−39797.7			−0.575369			−0.287684	+	0.957725i	$0.592885\pi$
$264$		0			0
$265$	42127.7			0.599896
$266$			174155.i			2.46134i
$267$		0			0
$268$		−	142724.i		−	1.98713i
$269$		−	99994.5i		−	1.38188i	−0.722910	−	0.690942i	$-0.757196\pi$
$269$		−	99994.5i		−	1.38188i	0.722910	−	0.690942i	$-0.242804\pi$
$270$		0			0
$271$	−70921.8			−0.965698			−0.482849	−	0.875704i	$-0.660398\pi$
$271$	−70921.8			−0.965698			−0.482849	+	0.875704i	$0.660398\pi$
$272$	310108.			4.19156
$273$		0			0
$274$			183876.i			2.44920i
$275$			122968.i			1.62602i
$276$		0			0
$277$	−27566.1			−0.359265			−0.179633	−	0.983734i	$-0.557491\pi$
$277$	−27566.1			−0.359265			−0.179633	+	0.983734i	$0.557491\pi$
$278$		−	23069.8i		−	0.298507i
$279$		0			0
$280$			272815.i			3.47978i
$281$	−116989.			−1.48160			−0.740802	−	0.671723i	$-0.765554\pi$
$281$	−116989.			−1.48160			−0.740802	+	0.671723i	$0.765554\pi$
$282$		0			0
$283$		−	124680.i		−	1.55676i	−0.627791	−	0.778382i	$-0.716041\pi$
$283$		−	124680.i		−	1.55676i	0.627791	−	0.778382i	$-0.283959\pi$
$284$	−59441.1			−0.736971
$285$		0			0
$286$			19336.6i			0.236401i
$287$	97822.4			1.18761
$288$		0			0
$289$	41998.2			0.502846
$290$		−	128787.i		−	1.53136i
$291$		0			0
$292$			70236.0i			0.823747i
$293$	117066.			1.36362			0.681812	−	0.731528i	$-0.261192\pi$
$293$	117066.			1.36362			0.681812	+	0.731528i	$0.261192\pi$
$294$		0			0
$295$	−64582.6	−	116113.i	−0.742116	−	1.33425i
$296$	197847.			2.25811
$297$		0			0
$298$	−79583.5			−0.896170
$299$	14737.8			0.164851
$300$		0			0
$301$			99735.6i			1.10082i
$302$	271233.			2.97392
$303$		0			0
$304$	−566730.			−6.13238
$305$		−	228278.i		−	2.45395i
$306$		0			0
$307$	−74586.4			−0.791376			−0.395688	−	0.918385i	$-0.629494\pi$
$307$	−74586.4			−0.791376			−0.395688	+	0.918385i	$0.629494\pi$
$308$			221182.i			2.33157i
$309$		0			0
$310$	−101370.			−1.05484
$311$	−164539.			−1.70117			−0.850587	−	0.525834i	$-0.823753\pi$
$311$	−164539.			−1.70117			−0.850587	+	0.525834i	$0.823753\pi$
$312$		0			0
$313$			135789.i			1.38604i	0.720919	+	0.693019i	$0.243720\pi$
$313$			135789.i			1.38604i	−0.720919	+	0.693019i	$0.756280\pi$
$314$	−178341.			−1.80881
$315$		0			0
$316$	355819.			3.56332
$317$	92280.2			0.918311			0.459156	−	0.888356i	$-0.348152\pi$
$317$	92280.2			0.918311			0.459156	+	0.888356i	$0.348152\pi$
$318$		0			0
$319$		−	65171.8i		−	0.640440i
$320$	−471796.			−4.60738
$321$		0			0
$322$	231934.			2.23693
$323$	−229389.			−2.19871
$324$		0			0
$325$		−	14217.8i		−	0.134607i
$326$		−	68853.9i		−	0.647878i
$327$		0			0
$328$			566059.i			5.26156i
$329$			45234.6i			0.417907i
$330$		0			0
$331$	10830.5			0.0988536			0.0494268	−	0.998778i	$-0.484261\pi$
$331$	10830.5			0.0988536			0.0494268	+	0.998778i	$0.484261\pi$
$332$		−	213963.i		−	1.94116i
$333$		0			0
$334$			3382.02i			0.0303167i
$335$			127958.i			1.14019i
$336$		0			0
$337$		−	124414.i		−	1.09549i	−0.836645	−	0.547746i	$-0.815486\pi$
$337$		−	124414.i		−	1.09549i	0.836645	−	0.547746i	$-0.184514\pi$
$338$			216350.i			1.89376i
$339$		0			0
$340$	−575703.			−4.98013
$341$	−51297.7			−0.441153
$342$		0			0
$343$	125357.			1.06551
$344$	−577130.			−4.87705
$345$		0			0
$346$	−22557.5			−0.188425
$347$			226048.i			1.87733i	0.344829	+	0.938665i	$0.387937\pi$
$347$			226048.i			1.87733i	−0.344829	+	0.938665i	$0.612063\pi$
$348$		0			0
$349$			223773.i			1.83720i	0.395185	+	0.918602i	$0.370681\pi$
$349$			223773.i			1.83720i	−0.395185	+	0.918602i	$0.629319\pi$
$350$		−	223750.i		−	1.82653i
$351$		0			0
$352$	−509253.			−4.11006
$353$		−	132923.i		−	1.06672i	−0.845889	−	0.533358i	$-0.820930\pi$
$353$		−	132923.i		−	1.06672i	0.845889	−	0.533358i	$-0.179070\pi$
$354$		0			0
$355$	53291.6			0.422866
$356$			225336.i			1.77800i
$357$		0			0
$358$	132603.			1.03463
$359$	−52427.0			−0.406786			−0.203393	−	0.979097i	$-0.565197\pi$
$359$	−52427.0			−0.406786			−0.203393	+	0.979097i	$0.565197\pi$
$360$		0			0
$361$	288893.			2.21678
$362$		−	70995.5i		−	0.541768i
$363$		0			0
$364$		−	25573.6i		−	0.193014i
$365$		−	62969.7i		−	0.472657i
$366$		0			0
$367$			30282.9i			0.224835i	0.993661	+	0.112418i	$0.0358595\pi$
$367$			30282.9i			0.224835i	−0.993661	+	0.112418i	$0.964141\pi$
$368$			754753.i			5.57326i
$369$		0			0
$370$	−284180.			−2.07582
$371$	−38790.7			−0.281826
$372$		0			0
$373$	47523.7			0.341580			0.170790	−	0.985307i	$-0.445368\pi$
$373$	47523.7			0.341580			0.170790	+	0.985307i	$0.445368\pi$
$374$	−400820.			−2.86554
$375$		0			0
$376$	−261755.			−1.85148
$377$			7535.33i			0.0530175i
$378$		0			0
$379$	185993.			1.29485			0.647425	−	0.762130i	$-0.275846\pi$
$379$	185993.			1.29485			0.647425	+	0.762130i	$0.275846\pi$
$380$	1.05211e6			7.28609
$381$		0			0
$382$	112572.			0.771445
$383$	141142.			0.962188			0.481094	−	0.876669i	$-0.340240\pi$
$383$	141142.			0.962188			0.481094	+	0.876669i	$0.340240\pi$
$384$		0			0
$385$		−	198299.i		−	1.33783i
$386$		−	262121.i		−	1.75925i
$387$		0			0
$388$		−	598225.i		−	3.97376i
$389$	−58679.8			−0.387784			−0.193892	−	0.981023i	$-0.562111\pi$
$389$	−58679.8			−0.387784			−0.193892	+	0.981023i	$0.562111\pi$
$390$		0			0
$391$			305494.i			1.99825i
$392$			237092.i			1.54292i
$393$		0			0
$394$		−	60549.3i		−	0.390046i
$395$	−319008.			−2.04459
$396$		0			0
$397$			277665.i			1.76173i	0.473365	+	0.880866i	$0.343039\pi$
$397$			277665.i			1.76173i	−0.473365	+	0.880866i	$0.656961\pi$
$398$		−	439717.i		−	2.77592i
$399$		0			0
$400$	728121.			4.55076
$401$			59794.3i			0.371853i	0.982564	+	0.185927i	$0.0595286\pi$
$401$			59794.3i			0.371853i	−0.982564	+	0.185927i	$0.940471\pi$
$402$		0			0
$403$	5931.16			0.0365199
$404$			26549.7i			0.162666i
$405$		0			0
$406$			118586.i			0.719417i
$407$	−143808.			−0.868146
$408$		0			0
$409$		−	92288.3i		−	0.551696i	−0.961201	−	0.275848i	$-0.911041\pi$
$409$		−	92288.3i		−	0.551696i	0.961201	−	0.275848i	$-0.0889587\pi$
$410$		−	813068.i		−	4.83681i
$411$		0			0
$412$		−	341956.i		−	2.01454i
$413$	59467.0	+	106916.i	0.348639	+	0.626819i
$414$		0			0
$415$			191827.i			1.11382i
$416$	58881.1			0.340243
$417$		0			0
$418$	732508.			4.19237
$419$			100145.i			0.570429i	0.958464	+	0.285214i	$0.0920648\pi$
$419$			100145.i			0.570429i	−0.958464	+	0.285214i	$0.907935\pi$
$420$		0			0
$421$			102049.i			0.575762i	0.957666	+	0.287881i	$0.0929507\pi$
$421$			102049.i			0.575762i	−0.957666	+	0.287881i	$0.907049\pi$
$422$	−375044.			−2.10600
$423$		0			0
$424$		−	224467.i		−	1.24859i
$425$	294714.			1.63164
$426$		0			0
$427$			210197.i			1.15284i
$428$	472506.			2.57941
$429$		0			0
$430$	828970.			4.48334
$431$			259464.i			1.39676i	0.715725	+	0.698382i	$0.246096\pi$
$431$			259464.i			1.39676i	−0.715725	+	0.698382i	$0.753904\pi$
$432$		0			0
$433$	−332067.			−1.77113			−0.885565	−	0.464516i	$-0.846228\pi$
$433$	−332067.			−1.77113			−0.885565	+	0.464516i	$0.846228\pi$
$434$	93340.5			0.495554
$435$		0			0
$436$			332461.i			1.74891i
$437$		−	558297.i		−	2.92350i
$438$		0			0
$439$	70862.7			0.367696			0.183848	−	0.982955i	$-0.441145\pi$
$439$	70862.7			0.367696			0.183848	+	0.982955i	$0.441145\pi$
$440$	1.14748e6			5.92706
$441$		0			0
$442$	46343.8			0.237218
$443$			85593.5i			0.436147i	0.975932	+	0.218074i	$0.0699773\pi$
$443$			85593.5i			0.436147i	−0.975932	+	0.218074i	$0.930023\pi$
$444$		0			0
$445$		−	202024.i		−	1.02019i
$446$			721882.i			3.62908i
$447$		0			0
$448$	434425.			2.16451
$449$	−70020.7			−0.347323			−0.173662	−	0.984805i	$-0.555560\pi$
$449$	−70020.7			−0.347323			−0.173662	+	0.984805i	$0.555560\pi$
$450$		0			0
$451$		−	411448.i		−	2.02284i
$452$			133283.i			0.652376i
$453$		0			0
$454$	−562494.			−2.72901
$455$			22927.8i			0.110749i
$456$		0			0
$457$		−	97704.0i		−	0.467821i	−0.972258	−	0.233911i	$-0.924848\pi$
$457$		−	97704.0i		−	0.467821i	0.972258	−	0.233911i	$-0.0751523\pi$
$458$	545439.			2.60025
$459$		0			0
$460$		−	1.40117e6i		−	6.62178i
$461$	−149231.			−0.702194			−0.351097	−	0.936339i	$-0.614191\pi$
$461$	−149231.			−0.702194			−0.351097	+	0.936339i	$0.614191\pi$
$462$		0			0
$463$			350096.i			1.63315i	0.577242	+	0.816573i	$0.304129\pi$
$463$			350096.i			1.63315i	−0.577242	+	0.816573i	$0.695871\pi$
$464$	−385898.			−1.79241
$465$		0			0
$466$	−165888.			−0.763914
$467$			370985.i			1.70107i	0.525917	+	0.850536i	$0.323722\pi$
$467$			370985.i			1.70107i	−0.525917	+	0.850536i	$0.676278\pi$
$468$		0			0
$469$		−	117823.i		−	0.535653i
$470$	375976.			1.70202
$471$		0			0
$472$	−618680.	+	344112.i	−2.77704	+	1.54460i
$473$	419495.			1.87501
$474$		0			0
$475$	−538597.			−2.38713
$476$	530102.			2.33962
$477$		0			0
$478$			4258.55i			0.0186383i
$479$	64617.2			0.281629			0.140814	−	0.990036i	$-0.455028\pi$
$479$	64617.2			0.281629			0.140814	+	0.990036i	$0.455028\pi$
$480$		0			0
$481$	16627.4			0.0718677
$482$			261491.i			1.12555i
$483$		0			0
$484$	306992.			1.31050
$485$			536336.i			2.28010i
$486$		0			0
$487$	177831.			0.749808			0.374904	−	0.927064i	$-0.377676\pi$
$487$	177831.			0.749808			0.374904	+	0.927064i	$0.377676\pi$
$488$	−1.21632e6			−5.10752
$489$		0			0
$490$		−	340550.i		−	1.41837i
$491$	146359.			0.607094			0.303547	−	0.952817i	$-0.401829\pi$
$491$	146359.			0.607094			0.303547	+	0.952817i	$0.401829\pi$
$492$		0			0
$493$	−156196.			−0.642653
$494$	−84694.3			−0.347057
$495$		0			0
$496$			303746.i			1.23466i
$497$	−49070.4			−0.198658
$498$		0			0
$499$	297990.			1.19674			0.598372	−	0.801218i	$-0.295815\pi$
$499$	297990.			1.19674			0.598372	+	0.801218i	$0.295815\pi$
$500$	−336126.			−1.34450
$501$		0			0
$502$		−	436381.i		−	1.73164i
$503$		−	379938.i		−	1.50168i	−0.660486	−	0.750839i	$-0.729650\pi$
$503$		−	379938.i		−	1.50168i	0.660486	−	0.750839i	$-0.270350\pi$
$504$		0			0
$505$		−	23803.0i		−	0.0933359i
$506$		−	975532.i		−	3.81013i
$507$		0			0
$508$	796605.			3.08685
$509$			68834.8i			0.265688i	0.991137	+	0.132844i	$0.0424110\pi$
$509$			68834.8i			0.265688i	−0.991137	+	0.132844i	$0.957589\pi$
$510$		0			0
$511$			57981.9i			0.222050i
$512$			167216.i			0.637880i
$513$		0			0
$514$			27012.3i			0.102243i
$515$			306579.i			1.15592i
$516$		0			0
$517$	190260.			0.711814
$518$	261670.			0.975203
$519$		0			0
$520$	−132674.			−0.490659
$521$	−450862.			−1.66099			−0.830497	−	0.557023i	$-0.811943\pi$
$521$	−450862.			−1.66099			−0.830497	+	0.557023i	$0.811943\pi$
$522$		0			0
$523$	−157180.			−0.574639			−0.287319	−	0.957835i	$-0.592764\pi$
$523$	−157180.			−0.574639			−0.287319	+	0.957835i	$0.592764\pi$
$524$		−	528287.i		−	1.92401i
$525$		0			0
$526$		−	304584.i		−	1.10087i
$527$			122944.i			0.442677i
$528$		0			0
$529$	−463682.			−1.65695
$530$			322416.i			1.14780i
$531$		0			0
$532$	−968773.			−3.42294
$533$			47572.6i			0.167457i
$534$		0			0
$535$	−423623.			−1.48003
$536$	681793.			2.37314
$537$		0			0
$538$	765289.			2.64400
$539$		−	172333.i		−	0.593187i
$540$		0			0
$541$			272034.i			0.929455i	0.885454	+	0.464727i	$0.153848\pi$
$541$			272034.i			0.929455i	−0.885454	+	0.464727i	$0.846152\pi$
$542$		−	542787.i		−	1.84770i
$543$		0			0
$544$			1.22052e6i			4.12426i
$545$		−	298066.i		−	1.00351i
$546$		0			0
$547$	351369.			1.17433			0.587163	−	0.809469i	$-0.300245\pi$
$547$	351369.			1.17433			0.587163	+	0.809469i	$0.300245\pi$
$548$	−1.02285e6			−3.40605
$549$		0			0
$550$	−941109.			−3.11110
$551$	285452.			0.940221
$552$		0			0
$553$	293739.			0.960532
$554$		−	210972.i		−	0.687392i
$555$		0			0
$556$	128331.			0.415127
$557$	−88233.8			−0.284397			−0.142198	−	0.989838i	$-0.545417\pi$
$557$	−88233.8			−0.284397			−0.142198	+	0.989838i	$0.545417\pi$
$558$		0			0
$559$	−48503.0			−0.155219
$560$	−1.17418e6			−3.74419
$561$		0			0
$562$		−	895352.i		−	2.83479i
$563$			203437.i			0.641819i	0.947110	+	0.320910i	$0.103989\pi$
$563$			203437.i			0.641819i	−0.947110	+	0.320910i	$0.896011\pi$
$564$		0			0
$565$		−	119494.i		−	0.374326i
$566$	954212.			2.97860
$567$		0			0
$568$		−	283951.i		−	0.880130i
$569$			20875.8i			0.0644791i	0.999480	+	0.0322395i	$0.0102639\pi$
$569$			20875.8i			0.0644791i	−0.999480	+	0.0322395i	$0.989736\pi$
$570$		0			0
$571$			308480.i			0.946139i	0.881025	+	0.473069i	$0.156854\pi$
$571$			308480.i			0.946139i	−0.881025	+	0.473069i	$0.843146\pi$
$572$	−107564.			−0.328757
$573$		0			0
$574$			748665.i			2.27229i
$575$			717287.i			2.16949i
$576$		0			0
$577$	−332414.			−0.998454			−0.499227	−	0.866471i	$-0.666383\pi$
$577$	−332414.			−0.998454			−0.499227	+	0.866471i	$0.666383\pi$
$578$			321425.i			0.962110i
$579$		0			0
$580$	716405.			2.12962
$581$		−	176633.i		−	0.523262i
$582$		0			0
$583$			163157.i			0.480029i
$584$	−335518.			−0.983763
$585$		0			0
$586$			895940.i			2.60906i
$587$			398675.i			1.15703i	0.815673	+	0.578513i	$0.196367\pi$
$587$			398675.i			1.15703i	−0.815673	+	0.578513i	$0.803633\pi$
$588$		0			0
$589$		−	224683.i		−	0.647650i
$590$	888650.	−	494271.i	2.55286	−	1.41991i
$591$		0			0
$592$			851520.i			2.42969i
$593$	−150216.			−0.427177			−0.213589	−	0.976924i	$-0.568515\pi$
$593$	−150216.			−0.427177			−0.213589	+	0.976924i	$0.568515\pi$
$594$		0			0
$595$	−475260.			−1.34245
$596$		−	442700.i		−	1.24628i
$597$		0			0
$598$			112793.i			0.315414i
$599$	79272.8			0.220938			0.110469	−	0.993880i	$-0.464765\pi$
$599$	79272.8			0.220938			0.110469	+	0.993880i	$0.464765\pi$
$600$		0			0
$601$			510334.i			1.41288i	0.707772	+	0.706441i	$0.249700\pi$
$601$			510334.i			1.41288i	−0.707772	+	0.706441i	$0.750300\pi$
$602$	−763307.			−2.10623
$603$		0			0
$604$			1.50879e6i			4.13576i
$605$	−275232.			−0.751949
$606$		0			0
$607$	−108872.			−0.295487			−0.147743	−	0.989026i	$-0.547201\pi$
$607$	−108872.			−0.295487			−0.147743	+	0.989026i	$0.547201\pi$
$608$		−	2.23052e6i		−	6.03392i
$609$		0			0
$610$	1.74709e6			4.69521
$611$	−21998.3			−0.0589260
$612$		0			0
$613$			71190.4i			0.189453i	0.995503	+	0.0947263i	$0.0301976\pi$
$613$			71190.4i			0.189453i	−0.995503	+	0.0947263i	$0.969802\pi$
$614$		−	570833.i		−	1.51416i
$615$		0			0
$616$	−1.05659e6			−2.78448
$617$	463811.			1.21835			0.609173	−	0.793037i	$-0.291502\pi$
$617$	463811.			1.21835			0.609173	+	0.793037i	$0.291502\pi$
$618$		0			0
$619$	192587.			0.502627			0.251313	−	0.967906i	$-0.419138\pi$
$619$	192587.			0.502627			0.251313	+	0.967906i	$0.419138\pi$
$620$		−	563892.i		−	1.46694i
$621$		0			0
$622$		−	1.25927e6i		−	3.25490i
$623$			186022.i			0.479278i
$624$		0			0
$625$	−218555.			−0.559501
$626$	−1.03923e6			−2.65194
$627$		0			0
$628$		−	992060.i		−	2.51547i
$629$			344661.i			0.871145i
$630$		0			0
$631$	−13953.1			−0.0350438			−0.0175219	−	0.999846i	$-0.505578\pi$
$631$	−13953.1			−0.0350438			−0.0175219	+	0.999846i	$0.505578\pi$
$632$			1.69975e6i			4.25551i
$633$		0			0
$634$			706249.i			1.75703i
$635$	−714192.			−1.77120
$636$		0			0
$637$			19925.6i			0.0491058i
$638$	498780.			1.22537
$639$		0			0
$640$		−	1.50694e6i		−	3.67906i
$641$	416645.			1.01403			0.507014	−	0.861938i	$-0.330749\pi$
$641$	416645.			1.01403			0.507014	+	0.861938i	$0.330749\pi$
$642$		0			0
$643$	79601.2			0.192530			0.0962648	−	0.995356i	$-0.469310\pi$
$643$	79601.2			0.192530			0.0962648	+	0.995356i	$0.469310\pi$
$644$			1.29018e6i			3.11085i
$645$		0			0
$646$		−	1.75559e6i		−	4.20685i
$647$	681530.			1.62808			0.814041	−	0.580807i	$-0.197263\pi$
$647$	681530.			1.62808			0.814041	+	0.580807i	$0.197263\pi$
$648$		0			0
$649$	449696.	−	250123.i	1.06765	−	0.593832i
$650$	108813.			0.257546
$651$		0			0
$652$	383014.			0.900990
$653$	383016.			0.898237			0.449118	−	0.893472i	$-0.351738\pi$
$653$	383016.			0.898237			0.449118	+	0.893472i	$0.351738\pi$
$654$		0			0
$655$			473633.i			1.10398i
$656$	−2.43628e6			−5.66135
$657$		0			0
$658$	−346195.			−0.799592
$659$			527534.i			1.21473i	0.794423	+	0.607365i	$0.207773\pi$
$659$			527534.i			1.21473i	−0.794423	+	0.607365i	$0.792227\pi$
$660$		0			0
$661$	282774.			0.647197			0.323599	−	0.946194i	$-0.395107\pi$
$661$	282774.			0.647197			0.323599	+	0.946194i	$0.395107\pi$
$662$			82889.2i			0.189139i
$663$		0			0
$664$	1.02210e6			2.31824
$665$	868549.			1.96404
$666$		0			0
$667$		−	380156.i		−	0.854497i
$668$	−18813.2			−0.0421608
$669$		0			0
$670$	−979304.			−2.18156
$671$	884102.			1.96362
$672$		0			0
$673$			709243.i			1.56590i	0.622083	+	0.782952i	$0.286287\pi$
$673$			709243.i			1.56590i	−0.622083	+	0.782952i	$0.713713\pi$
$674$	952178.			2.09604
$675$		0			0
$676$	−1.20349e6			−2.63361
$677$	419269.			0.914777			0.457389	−	0.889267i	$-0.348785\pi$
$677$	419269.			0.914777			0.457389	+	0.889267i	$0.348785\pi$
$678$		0			0
$679$		−	493853.i		−	1.07117i
$680$		−	2.75014e6i		−	5.94754i
$681$		0			0
$682$		−	392597.i		−	0.844069i
$683$		−	341293.i		−	0.731620i	−0.930690	−	0.365810i	$-0.880792\pi$
$683$		−	341293.i		−	0.731620i	0.930690	−	0.365810i	$-0.119208\pi$
$684$		0			0
$685$	917030.			1.95435
$686$			959393.i			2.03868i
$687$		0			0
$688$		−	2.48393e6i		−	5.24762i
$689$		−	18864.6i		−	0.0397382i
$690$		0			0
$691$		−	57041.7i		−	0.119464i	−0.998214	−	0.0597319i	$-0.980975\pi$
$691$		−	57041.7i		−	0.119464i	0.998214	−	0.0597319i	$-0.0190246\pi$
$692$		−	125481.i		−	0.262039i
$693$		0			0
$694$	−1.73001e6			−3.59195
$695$	−115054.			−0.238195
$696$		0			0
$697$	−986109.			−2.02983
$698$	−1.71261e6			−3.51517
$699$		0			0
$700$	1.24466e6			2.54012
$701$		−	292521.i		−	0.595279i	−0.954678	−	0.297640i	$-0.903801\pi$
$701$		−	292521.i		−	0.595279i	0.954678	−	0.297640i	$-0.0961994\pi$
$702$		0			0
$703$		−	629876.i		−	1.27451i
$704$		−	1.82722e6i		−	3.68677i
$705$		0			0
$706$	1.01730e6			2.04098
$707$			21917.6i			0.0438484i
$708$		0			0
$709$	265410.			0.527988			0.263994	−	0.964524i	$-0.414960\pi$
$709$	265410.			0.527988			0.263994	+	0.964524i	$0.414960\pi$
$710$			407857.i			0.809080i
$711$		0			0
$712$	−1.07643e6			−2.12338
$713$	−299226.			−0.588601
$714$		0			0
$715$	96436.1			0.188637
$716$			737630.i			1.43884i
$717$		0			0
$718$		−	401240.i		−	0.778315i
$719$		−	843328.i		−	1.63132i	−0.578532	−	0.815659i	$-0.696374\pi$
$719$		−	843328.i		−	1.63132i	0.578532	−	0.815659i	$-0.303626\pi$
$720$		0			0
$721$		−	282295.i		−	0.543041i
$722$			2.21099e6i			4.24143i
$723$		0			0
$724$	394927.			0.753425
$725$	−366742.			−0.697726
$726$		0			0
$727$	361987.			0.684895			0.342448	−	0.939537i	$-0.388744\pi$
$727$	361987.			0.684895			0.342448	+	0.939537i	$0.388744\pi$
$728$	122165.			0.230507
$729$		0			0
$730$	481927.			0.904347
$731$		−	1.00540e6i		−	1.88149i
$732$		0			0
$733$	364992.			0.679322			0.339661	−	0.940548i	$-0.389688\pi$
$733$	364992.			0.679322			0.339661	+	0.940548i	$0.389688\pi$
$734$	−231764.			−0.430184
$735$		0			0
$736$	−2.97054e6			−5.48379
$737$	−495571.			−0.912369
$738$		0			0
$739$		−	397416.i		−	0.727707i	−0.931456	−	0.363854i	$-0.881461\pi$
$739$		−	397416.i		−	0.727707i	0.931456	−	0.363854i	$-0.118539\pi$
$740$		−	1.58081e6i		−	2.88680i
$741$		0			0
$742$		−	296878.i		−	0.539224i
$743$	−208889.			−0.378389			−0.189195	−	0.981940i	$-0.560588\pi$
$743$	−208889.			−0.378389			−0.189195	+	0.981940i	$0.560588\pi$
$744$		0			0
$745$			396900.i			0.715104i
$746$			363714.i			0.653555i
$747$		0			0
$748$		−	2.22965e6i		−	3.98504i
$749$	390068.			0.695306
$750$		0			0
$751$			1.12660e6i			1.99751i	0.0498694	+	0.998756i	$0.484120\pi$
$751$			1.12660e6i			1.99751i	−0.0498694	+	0.998756i	$0.515880\pi$
$752$		−	1.12658e6i		−	1.99216i
$753$		0			0
$754$	−57670.2			−0.101440
$755$		−	1.35270e6i		−	2.37305i
$756$		0			0
$757$	−287250.			−0.501266			−0.250633	−	0.968082i	$-0.580639\pi$
$757$	−287250.			−0.501266			−0.250633	+	0.968082i	$0.580639\pi$
$758$			1.42347e6i			2.47747i
$759$		0			0
$760$			5.02595e6i			8.70143i
$761$	39472.2			0.0681587			0.0340794	−	0.999419i	$-0.489150\pi$
$761$	39472.2			0.0681587			0.0340794	+	0.999419i	$0.489150\pi$
$762$		0			0
$763$			274456.i			0.471438i
$764$			626207.i			1.07283i
$765$		0			0
$766$			1.08021e6i			1.84098i
$767$	−51994.9	+	28919.8i	−0.0883833	+	0.0491591i
$768$		0			0
$769$			696042.i			1.17702i	0.808491	+	0.588509i	$0.200285\pi$
$769$			696042.i			1.17702i	−0.808491	+	0.588509i	$0.799715\pi$
$770$	1.51765e6			2.55970
$771$		0			0
$772$	1.45810e6			2.44655
$773$		−	245846.i		−	0.411438i	−0.978611	−	0.205719i	$-0.934047\pi$
$773$		−	245846.i		−	0.411438i	0.978611	−	0.205719i	$-0.0659533\pi$
$774$		0			0
$775$			288668.i			0.480613i
$776$	2.85773e6			4.74567
$777$		0			0
$778$		−	449095.i		−	0.741957i
$779$	1.80214e6			2.96970
$780$		0			0
$781$			206394.i			0.338372i
$782$	−2.33804e6			−3.82330
$783$		0			0
$784$	−1.02043e6			−1.66016
$785$			889426.i			1.44335i
$786$		0			0
$787$	437284.			0.706016			0.353008	−	0.935620i	$-0.385159\pi$
$787$	437284.			0.706016			0.353008	+	0.935620i	$0.385159\pi$
$788$	336818.			0.542429
$789$		0			0
$790$		−	2.44146e6i		−	3.91198i
$791$			110029.i			0.175855i
$792$		0			0
$793$	−102222.			−0.162554
$794$	−2.12506e6			−3.37077
$795$		0			0
$796$	2.44602e6			3.86041
$797$		−	377368.i		−	0.594085i	−0.954864	−	0.297042i	$-0.904000\pi$
$797$		−	377368.i		−	0.594085i	0.954864	−	0.297042i	$-0.0960003\pi$
$798$		0			0
$799$		−	455993.i		−	0.714273i
$800$			2.86573e6i			4.47770i
$801$		0			0
$802$	−457625.			−0.711477
$803$	243876.			0.378214
$804$		0			0
$805$		−	1.15671e6i		−	1.78497i
$806$			45393.0i			0.0698745i
$807$		0			0
$808$	−126828.			−0.194264
$809$			84575.4i			0.129225i	0.997910	+	0.0646126i	$0.0205812\pi$
$809$			84575.4i			0.129225i	−0.997910	+	0.0646126i	$0.979419\pi$
$810$		0			0
$811$		−	280073.i		−	0.425823i	−0.977071	−	0.212911i	$-0.931705\pi$
$811$		−	280073.i		−	0.425823i	0.977071	−	0.212911i	$-0.0682946\pi$
$812$	−659658.			−1.00048
$813$		0			0
$814$		−	1.10060e6i		−	1.66105i
$815$	−343390.			−0.516978
$816$		0			0
$817$			1.83738e6i			2.75268i
$818$	706311.			1.05558
$819$		0			0
$820$	4.52286e6			6.72644
$821$			475894.i			0.706031i	0.935617	+	0.353016i	$0.114844\pi$
$821$			475894.i			0.706031i	−0.935617	+	0.353016i	$0.885156\pi$
$822$		0			0
$823$			448500.i			0.662160i	0.943603	+	0.331080i	$0.107413\pi$
$823$			448500.i			0.662160i	−0.943603	+	0.331080i	$0.892587\pi$
$824$	1.63353e6			2.40587
$825$		0			0
$826$	−818261.	+	455120.i	−1.19931	+	0.667061i
$827$	−794086.			−1.16107			−0.580533	−	0.814237i	$-0.697156\pi$
$827$	−794086.			−1.16107			−0.580533	+	0.814237i	$0.697156\pi$
$828$		0			0
$829$	351649.			0.511682			0.255841	−	0.966719i	$-0.417648\pi$
$829$	351649.			0.511682			0.255841	+	0.966719i	$0.417648\pi$
$830$	−1.46811e6			−2.13110
$831$		0			0
$832$			211268.i			0.305202i
$833$	−413028.			−0.595237
$834$		0			0
$835$	16866.8			0.0241914
$836$			4.07473e6i			5.83024i
$837$		0			0
$838$	−766441.			−1.09142
$839$		−	249257.i		−	0.354098i	−0.984202	−	0.177049i	$-0.943345\pi$
$839$		−	249257.i		−	0.354098i	0.984202	−	0.177049i	$-0.0566551\pi$
$840$		0			0
$841$	−512911.			−0.725186
$842$	−781009.			−1.10162
$843$		0			0
$844$		−	2.08626e6i		−	2.92876i
$845$	1.07899e6			1.51113
$846$		0			0
$847$	253431.			0.353259
$848$	966090.			1.34346
$849$		0			0
$850$			2.25554e6i			3.12185i
$851$	−838849.			−1.15831
$852$		0			0
$853$	−526565.			−0.723692			−0.361846	−	0.932238i	$-0.617853\pi$
$853$	−526565.			−0.723692			−0.361846	+	0.932238i	$0.617853\pi$
$854$	−1.60870e6			−2.20577
$855$		0			0
$856$			2.25717e6i			3.08046i
$857$		−	972800.i		−	1.32453i	−0.749269	−	0.662265i	$-0.769595\pi$
$857$		−	972800.i		−	1.32453i	0.749269	−	0.662265i	$-0.230405\pi$
$858$		0			0
$859$		−	1.07599e6i		−	1.45822i	−0.684396	−	0.729110i	$-0.739934\pi$
$859$		−	1.07599e6i		−	1.45822i	0.684396	−	0.729110i	$-0.260066\pi$
$860$			4.61132e6i			6.23488i
$861$		0			0
$862$	−1.98576e6			−2.67247
$863$			119816.i			0.160876i	0.996760	+	0.0804381i	$0.0256320\pi$
$863$			119816.i			0.160876i	−0.996760	+	0.0804381i	$0.974368\pi$
$864$		0			0
$865$			112499.i			0.150355i
$866$		−	2.54141e6i		−	3.38875i
$867$		0			0
$868$			519226.i			0.689156i
$869$		−	1.23549e6i		−	1.63606i
$870$		0			0
$871$	57299.1			0.0755286
$872$	−1.58817e6			−2.08864
$873$		0			0
$874$	4.27282e6			5.59360
$875$	−277482.			−0.362426
$876$		0			0
$877$	−1.46716e6			−1.90756			−0.953778	−	0.300512i	$-0.902843\pi$
$877$	−1.46716e6			−1.90756			−0.953778	+	0.300512i	$0.902843\pi$
$878$			542334.i			0.703522i
$879$		0			0
$880$			4.93868e6i			6.37742i
$881$			220168.i			0.283663i	0.989891	+	0.141831i	$0.0452991\pi$
$881$			220168.i			0.283663i	−0.989891	+	0.141831i	$0.954701\pi$
$882$		0			0
$883$	1.11202e6			1.42624			0.713119	−	0.701044i	$-0.247282\pi$
$883$	1.11202e6			1.42624			0.713119	+	0.701044i	$0.247282\pi$
$884$			257797.i			0.329893i
$885$		0			0
$886$	−655073.			−0.834493
$887$			23264.8i			0.0295701i	0.999891	+	0.0147851i	$0.00470640\pi$
$887$			23264.8i			0.0295701i	−0.999891	+	0.0147851i	$0.995294\pi$
$888$		0			0
$889$	657621.			0.832094
$890$	1.54615e6			1.95197
$891$		0			0
$892$	−4.01562e6			−5.04688
$893$			833337.i			1.04500i
$894$		0			0
$895$		−	661318.i		−	0.825590i
$896$			1.38758e6i			1.72839i
$897$		0			0
$898$		−	535890.i		−	0.664543i
$899$		−	152992.i		−	0.189299i
$900$		0			0
$901$	391034.			0.481687
$902$	3.14894e6			3.87036
$903$		0			0
$904$	−636695.			−0.779102
$905$	−354070.			−0.432307
$906$		0			0
$907$	−1.05698e6			−1.28485			−0.642425	−	0.766348i	$-0.722072\pi$
$907$	−1.05698e6			−1.28485			−0.642425	+	0.766348i	$0.722072\pi$
$908$		−	3.12899e6i		−	3.79518i
$909$		0			0
$910$	−175474.			−0.211899
$911$	−673892.			−0.811995			−0.405997	−	0.913874i	$-0.633076\pi$
$911$	−673892.			−0.811995			−0.405997	+	0.913874i	$0.633076\pi$
$912$		0			0
$913$	−742930.			−0.891264
$914$	747759.			0.895095
$915$		0			0
$916$			3.03412e6i			3.61611i
$917$		−	436117.i		−	0.518638i
$918$		0			0
$919$			1.40839e6i			1.66760i	0.552070	+	0.833798i	$0.313838\pi$
$919$			1.40839e6i			1.66760i	−0.552070	+	0.833798i	$0.686162\pi$
$920$	6.69340e6			7.90808
$921$		0			0
$922$		−	1.14211e6i		−	1.34353i
$923$		−	23863.7i		−	0.0280114i
$924$		0			0
$925$			809250.i			0.945800i
$926$	−2.67939e6			−3.12474
$927$		0			0
$928$		−	1.51881e6i		−	1.76363i
$929$			1.48700e6i			1.72298i	0.507773	+	0.861491i	$0.330469\pi$
$929$			1.48700e6i			1.72298i	−0.507773	+	0.861491i	$0.669531\pi$
$930$		0			0
$931$	754819.			0.870850
$932$		−	922789.i		−	1.06236i
$933$		0			0
$934$	−2.83926e6			−3.25471
$935$			1.99898e6i			2.28657i
$936$		0			0
$937$			715876.i			0.815378i	0.913121	+	0.407689i	$0.133665\pi$
$937$			715876.i			0.815378i	−0.913121	+	0.407689i	$0.866335\pi$
$938$	901734.			1.02488
$939$		0			0
$940$			2.09144e6i			2.36696i
$941$			351558.i			0.397025i	0.980098	+	0.198512i	$0.0636110\pi$
$941$			351558.i			0.397025i	−0.980098	+	0.198512i	$0.936389\pi$
$942$		0			0
$943$		−	2.40003e6i		−	2.69894i
$944$	−1.48104e6	−	2.66276e6i	−1.66197	−	2.98805i
$945$		0			0
$946$			3.21052e6i			3.58751i
$947$	−894274.			−0.997174			−0.498587	−	0.866840i	$-0.666148\pi$
$947$	−894274.			−0.997174			−0.498587	+	0.866840i	$0.666148\pi$
$948$		0			0
$949$	−28197.5			−0.0313097
$950$		−	4.12205e6i		−	4.56737i
$951$		0			0
$952$			2.53230e6i			2.79410i
$953$	868548.			0.956330			0.478165	−	0.878270i	$-0.341302\pi$
$953$	868548.			0.956330			0.478165	+	0.878270i	$0.341302\pi$
$954$		0			0
$955$		−	561423.i		−	0.615578i
$956$	−23689.1			−0.0259198
$957$		0			0
$958$			494535.i			0.538848i
$959$	−844393.			−0.918136
$960$		0			0
$961$	803099.			0.869606
$962$			127254.i			0.137506i
$963$		0			0
$964$	−1.45460e6			−1.56527
$965$	−1.30726e6			−1.40380
$966$		0			0
$967$		−	27414.2i		−	0.0293172i	−0.999893	−	0.0146586i	$-0.995334\pi$
$967$		−	27414.2i		−	0.0293172i	0.999893	−	0.0146586i	$-0.00466615\pi$
$968$			1.46651e6i			1.56507i
$969$		0			0
$970$	−4.10474e6			−4.36257
$971$	202062.			0.214312			0.107156	−	0.994242i	$-0.465826\pi$
$971$	202062.			0.214312			0.107156	+	0.994242i	$0.465826\pi$
$972$		0			0
$973$	105941.			0.111902
$974$			1.36100e6i			1.43463i
$975$		0			0
$976$		−	5.23498e6i		−	5.49561i
$977$		−	231436.i		−	0.242461i	−0.992624	−	0.121230i	$-0.961316\pi$
$977$		−	231436.i		−	0.242461i	0.992624	−	0.121230i	$-0.0386840\pi$
$978$		0			0
$979$	782420.			0.816347
$980$	1.89438e6			1.97249
$981$		0			0
$982$			1.12013e6i			1.16157i
$983$		−	492235.i		−	0.509407i	−0.967019	−	0.254704i	$-0.918022\pi$
$983$		−	492235.i		−	0.509407i	0.967019	−	0.254704i	$-0.0819779\pi$
$984$		0			0
$985$	−301972.			−0.311240
$986$		−	1.19542e6i		−	1.22961i
$987$		0			0
$988$		−	471130.i		−	0.482644i
$989$	2.44697e6			2.50171
$990$		0			0
$991$			861884.i			0.877610i	0.898582	+	0.438805i	$0.144598\pi$
$991$			861884.i			0.877610i	−0.898582	+	0.438805i	$0.855402\pi$
$992$	−1.19548e6			−1.21484
$993$		0			0
$994$		−	375551.i		−	0.380099i
$995$	−2.19297e6			−2.21506
$996$		0			0
$997$	740627.			0.745091			0.372546	−	0.928014i	$-0.378485\pi$
$997$	740627.			0.745091			0.372546	+	0.928014i	$0.378485\pi$
$998$			2.28061e6i			2.28976i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.5.c.d.235.38		40
3.2	odd	2		177.5.c.a.58.3	✓	40
59.58	odd	2	inner	531.5.c.d.235.3		40
177.176	even	2		177.5.c.a.58.38	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.5.c.a.58.3	✓	40	3.2	odd	2
177.5.c.a.58.38	yes	40	177.176	even	2
531.5.c.d.235.3		40	59.58	odd	2	inner
531.5.c.d.235.38		40	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:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	531.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+7.65331i q^{2} -42.5731 q^{4} +38.1687 q^{5} -35.1454 q^{7} -203.372i q^{8} +O(q^{10})\)	\(q+7.65331i q^{2} -42.5731 q^{4} +38.1687 q^{5} -35.1454 q^{7} -203.372i q^{8} +292.117i q^{10} +147.824i q^{11} -17.0918i q^{13} -268.979i q^{14} +875.302 q^{16} +354.287 q^{17} -647.468 q^{19} -1624.96 q^{20} -1131.34 q^{22} +862.278i q^{23} +831.852 q^{25} +130.809 q^{26} +1496.25 q^{28} -440.875 q^{29} +347.019i q^{31} +3445.00i q^{32} +2711.47i q^{34} -1341.45 q^{35} +972.830i q^{37} -4955.27i q^{38} -7762.47i q^{40} -2783.36 q^{41} -2837.80i q^{43} -6293.33i q^{44} -6599.28 q^{46} -1287.07i q^{47} -1165.80 q^{49} +6366.42i q^{50} +727.650i q^{52} +1103.72 q^{53} +5642.25i q^{55} +7147.61i q^{56} -3374.15i q^{58} +(-1692.03 - 3042.10i) q^{59} -5980.77i q^{61} -2655.84 q^{62} -12360.8 q^{64} -652.371i q^{65} +3352.44i q^{67} -15083.1 q^{68} -10266.6i q^{70} +1396.21 q^{71} -1649.77i q^{73} -7445.37 q^{74} +27564.7 q^{76} -5195.33i q^{77} -8357.83 q^{79} +33409.2 q^{80} -21301.9i q^{82} +5025.77i q^{83} +13522.7 q^{85} +21718.6 q^{86} +30063.3 q^{88} -5292.92i q^{89} +600.697i q^{91} -36709.9i q^{92} +9850.36 q^{94} -24713.0 q^{95} +14051.7i q^{97} -8922.24i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 320 q^{4} + 80 q^{7}+O(q^{10}) \) 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

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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