LMFDB - Embedded newform 1620.4.i.m.1081.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,4,Mod(541,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1620.541");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1620.i (of order $3$, degree $2$, not 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:	$95.5830942093$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{41})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 11x^{2} + 10x + 100 $ x^4 - x^3 + 11x^2 + 10x + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 540)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1081.2
Root			$1.85078 + 3.20565i$ of defining polynomial
Character	$\chi$	$=$	1620.1081
Dual form			1620.4.i.m.541.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.50000 + 4.33013i) q^{5} +(1.55234 + 2.68874i) q^{7} +O(q^{10})$	$q+(-2.50000 + 4.33013i) q^{5} +(1.55234 + 2.68874i) q^{7} +(20.2617 + 35.0943i) q^{11} +(16.7094 - 28.9415i) q^{13} +70.5234 q^{17} +146.361 q^{19} +(-50.2617 + 87.0558i) q^{23} +(-12.5000 - 21.6506i) q^{25} +(-50.2617 - 87.0558i) q^{29} +(12.9477 - 22.4260i) q^{31} -15.5234 q^{35} -218.570 q^{37} +(46.5703 - 80.6621i) q^{41} +(-8.63360 - 14.9538i) q^{43} +(-206.309 - 357.337i) q^{47} +(166.680 - 288.699i) q^{49} +471.047 q^{53} -202.617 q^{55} +(-321.047 + 556.069i) q^{59} +(-245.170 - 424.646i) q^{61} +(83.5469 + 144.707i) q^{65} +(-218.798 + 378.969i) q^{67} +1113.14 q^{71} +222.570 q^{73} +(-62.9063 + 108.957i) q^{77} +(201.641 + 349.252i) q^{79} +(330.234 + 571.983i) q^{83} +(-176.309 + 305.375i) q^{85} +173.719 q^{89} +103.755 q^{91} +(-365.902 + 633.761i) q^{95} +(295.018 + 510.986i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 10 q^{5} - 13 q^{7}+O(q^{10}) $ 4 * q - 10 * q^5 - 13 * q^7	$ 4 q - 10 q^{5} - 13 q^{7} - 15 q^{11} - 10 q^{13} + 90 q^{17} + 86 q^{19} - 105 q^{23} - 50 q^{25} - 105 q^{29} + 71 q^{31} + 130 q^{35} - 298 q^{37} - 390 q^{41} - 169 q^{43} - 345 q^{47} + 417 q^{49} + 1500 q^{53} + 150 q^{55} - 900 q^{59} - q^{61} - 50 q^{65} + 335 q^{67} + 3300 q^{71} + 314 q^{73} - 1020 q^{77} - 346 q^{79} - 600 q^{83} - 225 q^{85} + 3000 q^{89} + 1606 q^{91} - 215 q^{95} + 623 q^{97}+O(q^{100}) $ 4 * q - 10 * q^5 - 13 * q^7 - 15 * q^11 - 10 * q^13 + 90 * q^17 + 86 * q^19 - 105 * q^23 - 50 * q^25 - 105 * q^29 + 71 * q^31 + 130 * q^35 - 298 * q^37 - 390 * q^41 - 169 * q^43 - 345 * q^47 + 417 * q^49 + 1500 * q^53 + 150 * q^55 - 900 * q^59 - q^61 - 50 * q^65 + 335 * q^67 + 3300 * q^71 + 314 * q^73 - 1020 * q^77 - 346 * q^79 - 600 * q^83 - 225 * q^85 + 3000 * q^89 + 1606 * q^91 - 215 * q^95 + 623 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−2.50000	+	4.33013i	−0.223607	+	0.387298i
$5$	−2.50000	+	4.33013i	−0.223607	+	0.387298i
$6$		0			0
$7$	1.55234	+	2.68874i	0.0838187	+	0.145178i	0.904887	−	0.425651i	$-0.139955\pi$
$7$	1.55234	+	2.68874i	0.0838187	+	0.145178i	−0.821069	+	0.570830i	$0.806622\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	20.2617	+	35.0943i	0.555376	+	0.961940i	0.997874	+	0.0651702i	$0.0207590\pi$
$11$	20.2617	+	35.0943i	0.555376	+	0.961940i	−0.442498	+	0.896769i	$0.645908\pi$
$12$		0			0
$13$	16.7094	−	28.9415i	0.356488	−	0.617456i	−0.630883	−	0.775878i	$-0.717307\pi$
$13$	16.7094	−	28.9415i	0.356488	−	0.617456i	0.987371	+	0.158422i	$0.0506407\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	70.5234			1.00614			0.503072	−	0.864245i	$-0.332203\pi$
$17$	70.5234			1.00614			0.503072	+	0.864245i	$0.332203\pi$
$18$		0			0
$19$	146.361			1.76724			0.883618	−	0.468208i	$-0.155100\pi$
$19$	146.361			1.76724			0.883618	+	0.468208i	$0.155100\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−50.2617	+	87.0558i	−0.455665	+	0.789235i	−0.998726	−	0.0504583i	$-0.983932\pi$
$23$	−50.2617	+	87.0558i	−0.455665	+	0.789235i	0.543061	+	0.839693i	$0.317265\pi$
$24$		0			0
$25$	−12.5000	−	21.6506i	−0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−50.2617	−	87.0558i	−0.321840	−	0.557444i	0.659028	−	0.752119i	$-0.270968\pi$
$29$	−50.2617	−	87.0558i	−0.321840	−	0.557444i	−0.980868	+	0.194675i	$0.937635\pi$
$30$		0			0
$31$	12.9477	−	22.4260i	0.0750151	−	0.129930i	−0.826078	−	0.563556i	$-0.809433\pi$
$31$	12.9477	−	22.4260i	0.0750151	−	0.129930i	0.901093	+	0.433626i	$0.142766\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−15.5234			−0.0749697
$36$		0			0
$37$	−218.570			−0.971155			−0.485578	−	0.874194i	$-0.661391\pi$
$37$	−218.570			−0.971155			−0.485578	+	0.874194i	$0.661391\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	46.5703	−	80.6621i	0.177392	−	0.307251i	−0.763595	−	0.645696i	$-0.776567\pi$
$41$	46.5703	−	80.6621i	0.177392	−	0.307251i	0.940986	+	0.338444i	$0.109901\pi$
$42$		0			0
$43$	−8.63360	−	14.9538i	−0.0306189	−	0.0530334i	0.850310	−	0.526282i	$-0.176414\pi$
$43$	−8.63360	−	14.9538i	−0.0306189	−	0.0530334i	−0.880929	+	0.473249i	$0.843081\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−206.309	−	357.337i	−0.640281	−	1.10900i	−0.985370	−	0.170429i	$-0.945485\pi$
$47$	−206.309	−	357.337i	−0.640281	−	1.10900i	0.345089	−	0.938570i	$-0.387849\pi$
$48$		0			0
$49$	166.680	−	288.699i	0.485949	−	0.841688i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	471.047			1.22082			0.610408	−	0.792087i	$-0.291005\pi$
$53$	471.047			1.22082			0.610408	+	0.792087i	$0.291005\pi$
$54$		0			0
$55$	−202.617			−0.496743
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−321.047	+	556.069i	−0.708419	+	1.22702i	0.257024	+	0.966405i	$0.417258\pi$
$59$	−321.047	+	556.069i	−0.708419	+	1.22702i	−0.965443	+	0.260613i	$0.916075\pi$
$60$		0			0
$61$	−245.170	−	424.646i	−0.514602	−	0.891317i	−0.999856	−	0.0169441i	$-0.994606\pi$
$61$	−245.170	−	424.646i	−0.514602	−	0.891317i	0.485254	−	0.874373i	$-0.338727\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	83.5469	+	144.707i	0.159426	+	0.276135i
$66$		0			0
$67$	−218.798	+	378.969i	−0.398961	+	0.691021i	−0.993598	−	0.112974i	$-0.963962\pi$
$67$	−218.798	+	378.969i	−0.398961	+	0.691021i	0.594637	+	0.803994i	$0.297296\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	1113.14			1.86064			0.930320	−	0.366748i	$-0.119529\pi$
$71$	1113.14			1.86064			0.930320	+	0.366748i	$0.119529\pi$
$72$		0			0
$73$	222.570			0.356848			0.178424	−	0.983954i	$-0.442900\pi$
$73$	222.570			0.356848			0.178424	+	0.983954i	$0.442900\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−62.9063	+	108.957i	−0.0931018	+	0.161257i
$78$		0			0
$79$	201.641	+	349.252i	0.287169	+	0.497391i	0.973133	−	0.230244i	$-0.0739526\pi$
$79$	201.641	+	349.252i	0.287169	+	0.497391i	−0.685964	+	0.727636i	$0.740619\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	330.234	+	571.983i	0.436722	+	0.756425i	0.997434	−	0.0715859i	$-0.0228060\pi$
$83$	330.234	+	571.983i	0.436722	+	0.756425i	−0.560712	+	0.828011i	$0.689473\pi$
$84$		0			0
$85$	−176.309	+	305.375i	−0.224981	+	0.389678i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	173.719			0.206901			0.103450	−	0.994635i	$-0.467012\pi$
$89$	173.719			0.206901			0.103450	+	0.994635i	$0.467012\pi$
$90$		0			0
$91$	103.755			0.119521
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−365.902	+	633.761i	−0.395166	+	0.684448i
$96$		0			0
$97$	295.018	+	510.986i	0.308810	+	0.534874i	0.978102	−	0.208125i	$-0.0667361\pi$
$97$	295.018	+	510.986i	0.308810	+	0.534874i	−0.669293	+	0.742999i	$0.733403\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−482.590	−	835.870i	−0.475440	−	0.823487i	0.524164	−	0.851617i	$-0.324378\pi$
$101$	−482.590	−	835.870i	−0.475440	−	0.823487i	−0.999604	+	0.0281306i	$0.991045\pi$
$102$		0			0
$103$	320.996	−	555.981i	0.307075	−	0.531869i	−0.670647	−	0.741777i	$-0.733983\pi$
$103$	320.996	−	555.981i	0.307075	−	0.531869i	0.977721	+	0.209908i	$0.0673166\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1145.81			1.03523			0.517616	−	0.855613i	$-0.326819\pi$
$107$	1145.81			1.03523			0.517616	+	0.855613i	$0.326819\pi$
$108$		0			0
$109$	165.978			0.145852			0.0729258	−	0.997337i	$-0.476766\pi$
$109$	165.978			0.145852			0.0729258	+	0.997337i	$0.476766\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−868.113	+	1503.62i	−0.722701	+	1.25175i	0.237212	+	0.971458i	$0.423766\pi$
$113$	−868.113	+	1503.62i	−0.722701	+	1.25175i	−0.959913	+	0.280297i	$0.909567\pi$
$114$		0			0
$115$	−251.309	−	435.279i	−0.203780	−	0.352956i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	109.477	+	189.619i	0.0843336	+	0.146070i
$120$		0			0
$121$	−155.574	+	269.463i	−0.116885	+	0.202451i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	125.000			0.0894427
$126$		0			0
$127$	29.4218			0.0205572			0.0102786	−	0.999947i	$-0.496728\pi$
$127$	29.4218			0.0205572			0.0102786	+	0.999947i	$0.496728\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	29.4492	−	51.0075i	0.0196411	−	0.0340194i	−0.856038	−	0.516913i	$-0.827081\pi$
$131$	29.4492	−	51.0075i	0.0196411	−	0.0340194i	0.875679	+	0.482894i	$0.160414\pi$
$132$		0			0
$133$	227.202	+	393.526i	0.148127	+	0.256564i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1188.14	+	2057.92i	0.740947	+	1.28336i	0.952065	+	0.305897i	$0.0989562\pi$
$137$	1188.14	+	2057.92i	0.740947	+	1.28336i	−0.211118	+	0.977461i	$0.567710\pi$
$138$		0			0
$139$	−512.137	+	887.047i	−0.312510	+	0.541283i	−0.978905	−	0.204316i	$-0.934503\pi$
$139$	−512.137	+	887.047i	−0.312510	+	0.541283i	0.666395	+	0.745599i	$0.267836\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1354.24			0.791940
$144$		0			0
$145$	502.617			0.287863
$146$		0			0
$147$		0			0
$148$		0			0
$149$	590.496	−	1022.77i	0.324667	−	0.562339i	−0.656778	−	0.754084i	$-0.728081\pi$
$149$	590.496	−	1022.77i	0.324667	−	0.562339i	0.981445	+	0.191745i	$0.0614146\pi$
$150$		0			0
$151$	−648.756	−	1123.68i	−0.349636	−	0.605587i	0.636549	−	0.771236i	$-0.280361\pi$
$151$	−648.756	−	1123.68i	−0.349636	−	0.605587i	−0.986185	+	0.165649i	$0.947028\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	64.7383	+	112.130i	0.0335478	+	0.0581064i
$156$		0			0
$157$	1493.99	−	2587.66i	0.759447	−	1.31540i	−0.183687	−	0.982985i	$-0.558803\pi$
$157$	1493.99	−	2587.66i	0.759447	−	1.31540i	0.943133	−	0.332415i	$-0.107864\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−312.094			−0.152773
$162$		0			0
$163$	−1273.44			−0.611924			−0.305962	−	0.952044i	$-0.598978\pi$
$163$	−1273.44			−0.611924			−0.305962	+	0.952044i	$0.598978\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	72.0937	−	124.870i	0.0334059	−	0.0578607i	−0.848839	−	0.528651i	$-0.822698\pi$
$167$	72.0937	−	124.870i	0.0334059	−	0.0578607i	0.882245	+	0.470791i	$0.156031\pi$
$168$		0			0
$169$	540.094	+	935.470i	0.245832	+	0.425794i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1699.48	+	2943.58i	0.746872	+	1.29362i	0.949315	+	0.314326i	$0.101778\pi$
$173$	1699.48	+	2943.58i	0.746872	+	1.29362i	−0.202443	+	0.979294i	$0.564888\pi$
$174$		0			0
$175$	38.8086	−	67.2184i	0.0167637	−	0.0290356i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	3994.08			1.66777			0.833887	−	0.551936i	$-0.186111\pi$
$179$	3994.08			1.66777			0.833887	+	0.551936i	$0.186111\pi$
$180$		0			0
$181$	2244.20			0.921601			0.460800	−	0.887504i	$-0.347562\pi$
$181$	2244.20			0.921601			0.460800	+	0.887504i	$0.347562\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	546.426	−	946.437i	0.217157	−	0.376127i
$186$		0			0
$187$	1428.93	+	2474.97i	0.558788	+	0.967850i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1846.80	+	3198.76i	0.699634	+	1.21180i	0.968593	+	0.248650i	$0.0799869\pi$
$191$	1846.80	+	3198.76i	0.699634	+	1.21180i	−0.268959	+	0.963152i	$0.586680\pi$
$192$		0			0
$193$	−2061.05	+	3569.84i	−0.768691	+	1.33141i	0.169582	+	0.985516i	$0.445758\pi$
$193$	−2061.05	+	3569.84i	−0.768691	+	1.33141i	−0.938273	+	0.345896i	$0.887575\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1423.03			0.514654			0.257327	−	0.966324i	$-0.417158\pi$
$197$	1423.03			0.514654			0.257327	+	0.966324i	$0.417158\pi$
$198$		0			0
$199$	−3962.59			−1.41156			−0.705781	−	0.708430i	$-0.749404\pi$
$199$	−3962.59			−1.41156			−0.705781	+	0.708430i	$0.749404\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	156.047	−	270.281i	0.0539524	−	0.0934483i
$204$		0			0
$205$	232.851	+	403.311i	0.0793320	+	0.137407i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2965.52	+	5136.44i	0.981481	+	1.69998i
$210$		0			0
$211$	−1792.90	+	3105.40i	−0.584969	+	1.01320i	0.409910	+	0.912126i	$0.365560\pi$
$211$	−1792.90	+	3105.40i	−0.584969	+	1.01320i	−0.994879	+	0.101070i	$0.967773\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	86.3360			0.0273863
$216$		0			0
$217$	80.3968			0.0251507
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1178.40	−	2041.05i	0.358678	−	0.621249i
$222$		0			0
$223$	−1391.28	−	2409.77i	−0.417791	−	0.723634i	0.577926	−	0.816089i	$-0.303862\pi$
$223$	−1391.28	−	2409.77i	−0.417791	−	0.723634i	−0.995717	+	0.0924545i	$0.970529\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	600.867	+	1040.73i	0.175687	+	0.304299i	0.940399	−	0.340074i	$-0.110452\pi$
$227$	600.867	+	1040.73i	0.175687	+	0.304299i	−0.764712	+	0.644373i	$0.777119\pi$
$228$		0			0
$229$	440.427	−	762.841i	0.127093	−	0.220131i	−0.795456	−	0.606011i	$-0.792769\pi$
$229$	440.427	−	762.841i	0.127093	−	0.220131i	0.922549	+	0.385880i	$0.126102\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−3309.42			−0.930504			−0.465252	−	0.885178i	$-0.654036\pi$
$233$	−3309.42			−0.930504			−0.465252	+	0.885178i	$0.654036\pi$
$234$		0			0
$235$	2063.09			0.572685
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−60.2343	+	104.329i	−0.0163022	+	0.0282363i	−0.874061	−	0.485815i	$-0.838523\pi$
$239$	−60.2343	+	104.329i	−0.0163022	+	0.0282363i	0.857759	+	0.514052i	$0.171856\pi$
$240$		0			0
$241$	−1277.17	−	2212.13i	−0.341369	−	0.591268i	0.643318	−	0.765599i	$-0.277557\pi$
$241$	−1277.17	−	2212.13i	−0.341369	−	0.591268i	−0.984687	+	0.174331i	$0.944224\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	833.402	+	1443.50i	0.217323	+	0.376414i
$246$		0			0
$247$	2445.60	−	4235.90i	0.629999	−	1.09119i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1380.63			0.347190			0.173595	−	0.984817i	$-0.444462\pi$
$251$	1380.63			0.347190			0.173595	+	0.984817i	$0.444462\pi$
$252$		0			0
$253$	−4073.55			−1.01226
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−170.965	+	296.119i	−0.0414960	+	0.0718733i	−0.886027	−	0.463633i	$-0.846546\pi$
$257$	−170.965	+	296.119i	−0.0414960	+	0.0718733i	0.844531	+	0.535506i	$0.179879\pi$
$258$		0			0
$259$	−339.296	−	587.678i	−0.0814009	−	0.140991i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	1061.63	+	1838.79i	0.248907	+	0.431120i	0.963223	−	0.268704i	$-0.0865952\pi$
$263$	1061.63	+	1838.79i	0.248907	+	0.431120i	−0.714316	+	0.699824i	$0.753262\pi$
$264$		0			0
$265$	−1177.62	+	2039.69i	−0.272983	+	0.472820i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1939.95			−0.439705			−0.219852	−	0.975533i	$-0.570558\pi$
$269$	−1939.95			−0.439705			−0.219852	+	0.975533i	$0.570558\pi$
$270$		0			0
$271$	8278.84			1.85573			0.927866	−	0.372913i	$-0.121641\pi$
$271$	8278.84			1.85573			0.927866	+	0.372913i	$0.121641\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	506.543	−	877.358i	0.111075	−	0.192388i
$276$		0			0
$277$	−99.1935	−	171.808i	−0.0215161	−	0.0372670i	0.855067	−	0.518518i	$-0.173516\pi$
$277$	−99.1935	−	171.808i	−0.0215161	−	0.0372670i	−0.876583	+	0.481251i	$0.840183\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	4438.90	+	7688.40i	0.942358	+	1.63221i	0.760957	+	0.648802i	$0.224730\pi$
$281$	4438.90	+	7688.40i	0.942358	+	1.63221i	0.181400	+	0.983409i	$0.441937\pi$
$282$		0			0
$283$	3795.25	−	6573.56i	0.797187	−	1.38077i	−0.124254	−	0.992250i	$-0.539654\pi$
$283$	3795.25	−	6573.56i	0.797187	−	1.38077i	0.921441	−	0.388518i	$-0.127013\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	289.172			0.0594749
$288$		0			0
$289$	60.5544			0.0123253
$290$		0			0
$291$		0			0
$292$		0			0
$293$	2526.98	−	4376.86i	0.503850	−	0.872693i	−0.496140	−	0.868242i	$-0.665250\pi$
$293$	2526.98	−	4376.86i	0.503850	−	0.872693i	0.999990	−	0.00445108i	$-0.00141683\pi$
$294$		0			0
$295$	−1605.23	−	2780.35i	−0.316815	−	0.548739i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1679.68	+	2909.30i	0.324878	+	0.562706i
$300$		0			0
$301$	26.8046	−	46.4270i	0.00513286	−	0.00889038i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2451.70			0.460274
$306$		0			0
$307$	−7370.65			−1.37024			−0.685122	−	0.728428i	$-0.740251\pi$
$307$	−7370.65			−1.37024			−0.685122	+	0.728428i	$0.740251\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−4445.02	+	7698.99i	−0.810462	+	1.40376i	0.102079	+	0.994776i	$0.467451\pi$
$311$	−4445.02	+	7698.99i	−0.810462	+	1.40376i	−0.912541	+	0.408985i	$0.865883\pi$
$312$		0			0
$313$	2228.91	+	3860.59i	0.402510	+	0.697167i	0.994028	−	0.109124i	$-0.0348047\pi$
$313$	2228.91	+	3860.59i	0.402510	+	0.697167i	−0.591518	+	0.806291i	$0.701471\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	5062.45	+	8768.43i	0.896958	+	1.55358i	0.831363	+	0.555730i	$0.187561\pi$
$317$	5062.45	+	8768.43i	0.896958	+	1.55358i	0.0655954	+	0.997846i	$0.479105\pi$
$318$		0			0
$319$	2036.78	−	3527.80i	0.357485	−	0.619182i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	10321.9			1.77809
$324$		0			0
$325$	−835.469			−0.142595
$326$		0			0
$327$		0			0
$328$		0			0
$329$	640.523	−	1109.42i	0.107335	−	0.185910i
$330$		0			0
$331$	−763.012	−	1321.58i	−0.126704	−	0.219457i	0.795694	−	0.605699i	$-0.207106\pi$
$331$	−763.012	−	1321.58i	−0.126704	−	0.219457i	−0.922398	+	0.386242i	$0.873773\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1093.99	−	1894.84i	−0.178421	−	0.309034i
$336$		0			0
$337$	1001.58	−	1734.79i	0.161898	−	0.280415i	−0.773652	−	0.633611i	$-0.781572\pi$
$337$	1001.58	−	1734.79i	0.161898	−	0.280415i	0.935549	+	0.353196i	$0.114905\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1049.37			0.166646
$342$		0			0
$343$	2099.89			0.330564
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1325.41	−	2295.68i	0.205049	−	0.355155i	−0.745099	−	0.666953i	$-0.767598\pi$
$347$	1325.41	−	2295.68i	0.205049	−	0.355155i	0.950148	+	0.311798i	$0.100931\pi$
$348$		0			0
$349$	2381.29	+	4124.52i	0.365237	+	0.632609i	0.988814	−	0.149153i	$-0.0476546\pi$
$349$	2381.29	+	4124.52i	0.365237	+	0.632609i	−0.623577	+	0.781762i	$0.714321\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1187.19	−	2056.28i	−0.179002	−	0.310041i	0.762537	−	0.646945i	$-0.223954\pi$
$353$	−1187.19	−	2056.28i	−0.179002	−	0.310041i	−0.941539	+	0.336904i	$0.890620\pi$
$354$		0			0
$355$	−2782.85	+	4820.04i	−0.416052	+	0.720623i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	3973.36			0.584139			0.292070	−	0.956397i	$-0.405656\pi$
$359$	3973.36			0.584139			0.292070	+	0.956397i	$0.405656\pi$
$360$		0			0
$361$	14562.5			2.12313
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−556.426	+	963.758i	−0.0797936	+	0.138207i
$366$		0			0
$367$	1717.81	+	2975.33i	0.244329	+	0.423190i	0.961943	−	0.273251i	$-0.0880990\pi$
$367$	1717.81	+	2975.33i	0.244329	+	0.423190i	−0.717614	+	0.696441i	$0.754766\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	731.226	+	1266.52i	0.102327	+	0.177236i
$372$		0			0
$373$	−4607.95	+	7981.20i	−0.639653	+	1.10791i	0.345856	+	0.938288i	$0.387589\pi$
$373$	−4607.95	+	7981.20i	−0.639653	+	1.10791i	−0.985509	+	0.169624i	$0.945745\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−3359.37			−0.458929
$378$		0			0
$379$	3064.74			0.415370			0.207685	−	0.978196i	$-0.433407\pi$
$379$	3064.74			0.415370			0.207685	+	0.978196i	$0.433407\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	864.574	−	1497.49i	0.115346	−	0.199786i	−0.802572	−	0.596556i	$-0.796536\pi$
$383$	864.574	−	1497.49i	0.115346	−	0.199786i	0.917918	+	0.396770i	$0.129869\pi$
$384$		0			0
$385$	−314.531	−	544.784i	−0.0416364	−	0.0721163i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−2975.10	−	5153.02i	−0.387772	−	0.671641i	0.604377	−	0.796698i	$-0.293422\pi$
$389$	−2975.10	−	5153.02i	−0.387772	−	0.671641i	−0.992150	+	0.125057i	$0.960089\pi$
$390$		0			0
$391$	−3544.63	+	6139.48i	−0.458464	+	0.794084i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−2016.41			−0.256852
$396$		0			0
$397$	−13663.9			−1.72739			−0.863694	−	0.504016i	$-0.831855\pi$
$397$	−13663.9			−1.72739			−0.863694	+	0.504016i	$0.831855\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−925.992	+	1603.87i	−0.115316	+	0.199734i	−0.917906	−	0.396798i	$-0.870121\pi$
$401$	−925.992	+	1603.87i	−0.115316	+	0.199734i	0.802590	+	0.596531i	$0.203455\pi$
$402$		0			0
$403$	−432.694	−	749.449i	−0.0534840	−	0.0926370i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−4428.61	−	7670.58i	−0.539356	−	0.934192i
$408$		0			0
$409$	6154.37	−	10659.7i	0.744044	−	1.28872i	−0.206597	−	0.978426i	$-0.566239\pi$
$409$	6154.37	−	10659.7i	0.744044	−	1.28872i	0.950640	−	0.310295i	$-0.100428\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−1993.50			−0.237515
$414$		0			0
$415$	−3302.34			−0.390616
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−2643.69	+	4579.01i	−0.308241	+	0.533888i	−0.977978	−	0.208710i	$-0.933073\pi$
$419$	−2643.69	+	4579.01i	−0.308241	+	0.533888i	0.669737	+	0.742598i	$0.266407\pi$
$420$		0			0
$421$	−7564.55	−	13102.2i	−0.875709	−	1.51677i	−0.856005	−	0.516967i	$-0.827061\pi$
$421$	−7564.55	−	13102.2i	−0.875709	−	1.51677i	−0.0197041	−	0.999806i	$-0.506272\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−881.543	−	1526.88i	−0.100614	−	0.174269i
$426$		0			0
$427$	761.174	−	1318.39i	0.0862665	−	0.149418i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	15509.7			1.73335			0.866676	−	0.498872i	$-0.166252\pi$
$431$	15509.7			1.73335			0.866676	+	0.498872i	$0.166252\pi$
$432$		0			0
$433$	−11171.2			−1.23984			−0.619922	−	0.784663i	$-0.712836\pi$
$433$	−11171.2			−1.23984			−0.619922	+	0.784663i	$0.712836\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−7356.35	+	12741.6i	−0.805268	+	1.39476i
$438$		0			0
$439$	1667.77	+	2888.66i	0.181317	+	0.314051i	0.942329	−	0.334687i	$-0.108631\pi$
$439$	1667.77	+	2888.66i	0.181317	+	0.314051i	−0.761012	+	0.648738i	$0.775297\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−2271.05	−	3933.57i	−0.243568	−	0.421872i	0.718160	−	0.695878i	$-0.244985\pi$
$443$	−2271.05	−	3933.57i	−0.243568	−	0.421872i	−0.961728	+	0.274006i	$0.911651\pi$
$444$		0			0
$445$	−434.297	+	752.225i	−0.0462644	+	0.0801323i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−7617.80			−0.800682			−0.400341	−	0.916366i	$-0.631108\pi$
$449$	−7617.80			−0.800682			−0.400341	+	0.916366i	$0.631108\pi$
$450$		0			0
$451$	3774.38			0.394076
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−259.387	+	449.271i	−0.0267258	+	0.0462905i
$456$		0			0
$457$	7384.84	+	12790.9i	0.755904	+	1.30926i	0.944924	+	0.327291i	$0.106136\pi$
$457$	7384.84	+	12790.9i	0.755904	+	1.30926i	−0.189020	+	0.981973i	$0.560531\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−4492.15	−	7780.63i	−0.453840	−	0.786074i	0.544781	−	0.838579i	$-0.316613\pi$
$461$	−4492.15	−	7780.63i	−0.453840	−	0.786074i	−0.998621	+	0.0525047i	$0.983280\pi$
$462$		0			0
$463$	−4982.11	+	8629.27i	−0.500083	+	0.866169i	0.499917	+	0.866073i	$0.333364\pi$
$463$	−4982.11	+	8629.27i	−0.500083	+	0.866169i	−1.00000		9.58172e-5i	$0.999970\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	15489.1			1.53480			0.767398	−	0.641171i	$-0.221551\pi$
$467$	15489.1			1.53480			0.767398	+	0.641171i	$0.221551\pi$
$468$		0			0
$469$	−1358.60			−0.133761
$470$		0			0
$471$		0			0
$472$		0			0
$473$	349.863	−	605.981i	0.0340100	−	0.0589070i
$474$		0			0
$475$	−1829.51	−	3168.81i	−0.176724	−	0.306094i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	1675.19	+	2901.52i	0.159795	+	0.276772i	0.934795	−	0.355189i	$-0.115583\pi$
$479$	1675.19	+	2901.52i	0.159795	+	0.276772i	−0.775000	+	0.631961i	$0.782250\pi$
$480$		0			0
$481$	−3652.17	+	6325.75i	−0.346205	+	0.599645i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−2950.18			−0.276208
$486$		0			0
$487$	5435.64			0.505775			0.252887	−	0.967496i	$-0.418620\pi$
$487$	5435.64			0.505775			0.252887	+	0.967496i	$0.418620\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	3967.29	−	6871.55i	0.364646	−	0.631586i	−0.624073	−	0.781366i	$-0.714523\pi$
$491$	3967.29	−	6871.55i	0.364646	−	0.631586i	0.988719	+	0.149780i	$0.0478567\pi$
$492$		0			0
$493$	−3544.63	−	6139.48i	−0.323818	−	0.560868i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1727.98	+	2992.94i	0.155956	+	0.270124i
$498$		0			0
$499$	−3833.06	+	6639.05i	−0.343870	+	0.595601i	−0.985148	−	0.171708i	$-0.945071\pi$
$499$	−3833.06	+	6639.05i	−0.343870	+	0.595601i	0.641278	+	0.767309i	$0.278405\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	10069.6			0.892608			0.446304	−	0.894881i	$-0.352740\pi$
$503$	10069.6			0.892608			0.446304	+	0.894881i	$0.352740\pi$
$504$		0			0
$505$	4825.90			0.425247
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−8845.75	+	15321.3i	−0.770296	+	1.33419i	0.167104	+	0.985939i	$0.446558\pi$
$509$	−8845.75	+	15321.3i	−0.770296	+	1.33419i	−0.937401	+	0.348253i	$0.886775\pi$
$510$		0			0
$511$	345.505	+	598.433i	0.0299105	+	0.0518065i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1604.98	+	2779.91i	0.137328	+	0.237859i
$516$		0			0
$517$	8360.33	−	14480.5i	0.711193	−	1.23182i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−20972.5			−1.76357			−0.881786	−	0.471649i	$-0.843659\pi$
$521$	−20972.5			−1.76357			−0.881786	+	0.471649i	$0.843659\pi$
$522$		0			0
$523$	−18424.0			−1.54039			−0.770196	−	0.637807i	$-0.779842\pi$
$523$	−18424.0			−1.54039			−0.770196	+	0.637807i	$0.779842\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	913.113	−	1581.56i	0.0754760	−	0.130728i
$528$		0			0
$529$	1031.02	+	1785.78i	0.0847390	+	0.146772i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1556.32	−	2695.63i	−0.126476	−	0.219063i
$534$		0			0
$535$	−2864.53	+	4961.51i	−0.231485	+	0.400944i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	13508.9			1.07954
$540$		0			0
$541$	13367.9			1.06235			0.531174	−	0.847263i	$-0.321751\pi$
$541$	13367.9			1.06235			0.531174	+	0.847263i	$0.321751\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−414.945	+	718.706i	−0.0326134	+	0.0564881i
$546$		0			0
$547$	7804.60	+	13518.0i	0.610056	+	1.05665i	0.991230	+	0.132145i	$0.0421864\pi$
$547$	7804.60	+	13518.0i	0.610056	+	1.05665i	−0.381174	+	0.924503i	$0.624480\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−7356.35	−	12741.6i	−0.568768	−	0.985135i
$552$		0			0
$553$	−626.031	+	1084.32i	−0.0481402	+	0.0833813i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	1379.34			0.104927			0.0524637	−	0.998623i	$-0.483293\pi$
$557$	1379.34			0.104927			0.0524637	+	0.998623i	$0.483293\pi$
$558$		0			0
$559$	−577.048			−0.0436611
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−5237.88	+	9072.28i	−0.392097	+	0.679131i	−0.992726	−	0.120396i	$-0.961583\pi$
$563$	−5237.88	+	9072.28i	−0.392097	+	0.679131i	0.600629	+	0.799528i	$0.294917\pi$
$564$		0			0
$565$	−4340.57	−	7518.08i	−0.323202	−	0.559802i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−10557.5	−	18286.1i	−0.777842	−	1.34726i	−0.933183	−	0.359401i	$-0.882981\pi$
$569$	−10557.5	−	18286.1i	−0.777842	−	1.34726i	0.155341	−	0.987861i	$-0.450352\pi$
$570$		0			0
$571$	8776.02	−	15200.5i	0.643196	−	1.11405i	−0.341519	−	0.939875i	$-0.610941\pi$
$571$	8776.02	−	15200.5i	0.643196	−	1.11405i	0.984715	−	0.174173i	$-0.0557253\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	2513.09			0.182266
$576$		0			0
$577$	22028.2			1.58933			0.794667	−	0.607046i	$-0.207646\pi$
$577$	22028.2			1.58933			0.794667	+	0.607046i	$0.207646\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−1025.27	+	1775.83i	−0.0732109	+	0.126805i
$582$		0			0
$583$	9544.22	+	16531.1i	0.678012	+	1.17435i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−11171.2	−	19349.1i	−0.785494	−	1.36052i	−0.928704	−	0.370823i	$-0.879076\pi$
$587$	−11171.2	−	19349.1i	−0.785494	−	1.36052i	0.143210	−	0.989692i	$-0.454258\pi$
$588$		0			0
$589$	1895.03	−	3282.29i	0.132569	−	0.229617i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−13126.8			−0.909027			−0.454514	−	0.890740i	$-0.650187\pi$
$593$	−13126.8			−0.909027			−0.454514	+	0.890740i	$0.650187\pi$
$594$		0			0
$595$	−1094.77			−0.0754303
$596$		0			0
$597$		0			0
$598$		0			0
$599$	11874.1	−	20566.5i	0.809954	−	1.40288i	−0.102942	−	0.994687i	$-0.532825\pi$
$599$	11874.1	−	20566.5i	0.809954	−	1.40288i	0.912895	−	0.408194i	$-0.133841\pi$
$600$		0			0
$601$	−2131.75	−	3692.30i	−0.144685	−	0.250602i	0.784570	−	0.620040i	$-0.212884\pi$
$601$	−2131.75	−	3692.30i	−0.144685	−	0.250602i	−0.929255	+	0.369438i	$0.879550\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−777.871	−	1347.31i	−0.0522727	−	0.0905389i
$606$		0			0
$607$	9277.26	−	16068.7i	0.620350	−	1.07448i	−0.369071	−	0.929401i	$-0.620324\pi$
$607$	9277.26	−	16068.7i	0.620350	−	1.07448i	0.989420	−	0.145076i	$-0.0463427\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−13789.1			−0.913010
$612$		0			0
$613$	−3110.17			−0.204924			−0.102462	−	0.994737i	$-0.532672\pi$
$613$	−3110.17			−0.204924			−0.102462	+	0.994737i	$0.532672\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	12074.4	−	20913.5i	0.787841	−	1.36458i	−0.139446	−	0.990230i	$-0.544532\pi$
$617$	12074.4	−	20913.5i	0.787841	−	1.36458i	0.927287	−	0.374351i	$-0.122135\pi$
$618$		0			0
$619$	−2029.62	−	3515.40i	−0.131789	−	0.228265i	0.792577	−	0.609771i	$-0.208739\pi$
$619$	−2029.62	−	3515.40i	−0.131789	−	0.228265i	−0.924366	+	0.381507i	$0.875405\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	269.671	+	467.084i	0.0173421	+	0.0300375i
$624$		0			0
$625$	−312.500	+	541.266i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−15414.3			−0.977122
$630$		0			0
$631$	22017.9			1.38909			0.694547	−	0.719447i	$-0.255605\pi$
$631$	22017.9			1.38909			0.694547	+	0.719447i	$0.255605\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−73.5544	+	127.400i	−0.00459672	+	0.00796176i
$636$		0			0
$637$	−5570.25	−	9647.96i	−0.346470	−	0.600104i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−5434.38	−	9412.63i	−0.334860	−	0.579994i	0.648598	−	0.761131i	$-0.275356\pi$
$641$	−5434.38	−	9412.63i	−0.334860	−	0.579994i	−0.983458	+	0.181137i	$0.942022\pi$
$642$		0			0
$643$	−11644.7	+	20169.2i	−0.714188	+	1.23701i	0.249084	+	0.968482i	$0.419870\pi$
$643$	−11644.7	+	20169.2i	−0.714188	+	1.23701i	−0.963272	+	0.268528i	$0.913463\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−31678.8			−1.92492			−0.962460	−	0.271424i	$-0.912505\pi$
$647$	−31678.8			−1.92492			−0.962460	+	0.271424i	$0.912505\pi$
$648$		0			0
$649$	−26019.8			−1.57376
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−13544.1	+	23459.1i	−0.811672	+	1.40586i	0.100021	+	0.994985i	$0.468109\pi$
$653$	−13544.1	+	23459.1i	−0.811672	+	1.40586i	−0.911693	+	0.410872i	$0.865224\pi$
$654$		0			0
$655$	147.246	+	255.037i	0.00878377	+	0.0152139i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	2340.94	+	4054.62i	0.138376	+	0.239675i	0.926882	−	0.375352i	$-0.122478\pi$
$659$	2340.94	+	4054.62i	0.138376	+	0.239675i	−0.788506	+	0.615027i	$0.789145\pi$
$660$		0			0
$661$	−1850.96	+	3205.95i	−0.108917	+	0.188649i	−0.915332	−	0.402701i	$-0.868071\pi$
$661$	−1850.96	+	3205.95i	−0.108917	+	0.188649i	0.806415	+	0.591350i	$0.201405\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−2272.02			−0.132489
$666$		0			0
$667$	10105.0			0.586605
$668$		0			0
$669$		0			0
$670$		0			0
$671$	9935.11	−	17208.1i	0.571596	−	0.990033i
$672$		0			0
$673$	−14150.1	−	24508.7i	−0.810469	−	1.40377i	−0.912536	−	0.408996i	$-0.865879\pi$
$673$	−14150.1	−	24508.7i	−0.810469	−	1.40377i	0.102067	−	0.994778i	$-0.467454\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−7197.66	−	12466.7i	−0.408609	−	0.707732i	0.586125	−	0.810221i	$-0.300653\pi$
$677$	−7197.66	−	12466.7i	−0.408609	−	0.707732i	−0.994734	+	0.102489i	$0.967319\pi$
$678$		0			0
$679$	−915.938	+	1586.45i	−0.0517680	+	0.0896648i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−31758.3			−1.77921			−0.889604	−	0.456733i	$-0.849019\pi$
$683$	−31758.3			−1.77921			−0.889604	+	0.456733i	$0.849019\pi$
$684$		0			0
$685$	−11881.4			−0.662723
$686$		0			0
$687$		0			0
$688$		0			0
$689$	7870.90	−	13632.8i	0.435207	−	0.753800i
$690$		0			0
$691$	−11613.2	−	20114.6i	−0.639343	−	1.10737i	−0.985577	−	0.169227i	$-0.945873\pi$
$691$	−11613.2	−	20114.6i	−0.639343	−	1.10737i	0.346234	−	0.938148i	$-0.387460\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−2560.68	−	4435.23i	−0.139759	−	0.242069i
$696$		0			0
$697$	3284.30	−	5688.57i	0.178482	−	0.309139i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	14493.9			0.780921			0.390461	−	0.920620i	$-0.372316\pi$
$701$	14493.9			0.780921			0.390461	+	0.920620i	$0.372316\pi$
$702$		0			0
$703$	−31990.2			−1.71626
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1498.29	−	2595.11i	0.0797015	−	0.138047i
$708$		0			0
$709$	8502.48	+	14726.7i	0.450377	+	0.780076i	0.998409	−	0.0563811i	$-0.0179562\pi$
$709$	8502.48	+	14726.7i	0.450377	+	0.780076i	−0.548032	+	0.836457i	$0.684623\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1301.54	+	2254.34i	0.0683635	+	0.118409i
$714$		0			0
$715$	−3385.61	+	5864.04i	−0.177083	+	0.306717i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−5275.78			−0.273649			−0.136824	−	0.990595i	$-0.543690\pi$
$719$	−5275.78			−0.273649			−0.136824	+	0.990595i	$0.543690\pi$
$720$		0			0
$721$	1993.18			0.102954
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1256.54	+	2176.40i	−0.0643680	+	0.111489i
$726$		0			0
$727$	2382.68	+	4126.92i	0.121552	+	0.210535i	0.920380	−	0.391025i	$-0.127879\pi$
$727$	2382.68	+	4126.92i	0.121552	+	0.210535i	−0.798828	+	0.601560i	$0.794546\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−608.871	−	1054.60i	−0.0308070	−	0.0533593i
$732$		0			0
$733$	−11709.3	+	20281.1i	−0.590031	+	1.02196i	0.404197	+	0.914672i	$0.367551\pi$
$733$	−11709.3	+	20281.1i	−0.590031	+	1.02196i	−0.994228	+	0.107292i	$0.965782\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−17732.9			−0.886294
$738$		0			0
$739$	11307.4			0.562855			0.281427	−	0.959583i	$-0.409192\pi$
$739$	11307.4			0.562855			0.281427	+	0.959583i	$0.409192\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	9039.50	−	15656.9i	0.446335	−	0.773076i	−0.551809	−	0.833971i	$-0.686062\pi$
$743$	9039.50	−	15656.9i	0.446335	−	0.773076i	0.998144	+	0.0608950i	$0.0193955\pi$
$744$		0			0
$745$	2952.48	+	5113.85i	0.145195	+	0.251486i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1778.69	+	3080.79i	0.0867718	+	0.150293i
$750$		0			0
$751$	13250.0	−	22949.6i	0.643805	−	1.11510i	−0.340771	−	0.940146i	$-0.610688\pi$
$751$	13250.0	−	22949.6i	0.643805	−	1.11510i	0.984576	−	0.174957i	$-0.0559786\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	6487.56			0.312724
$756$		0			0
$757$	−9631.16			−0.462418			−0.231209	−	0.972904i	$-0.574268\pi$
$757$	−9631.16			−0.462418			−0.231209	+	0.972904i	$0.574268\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−4856.30	+	8411.35i	−0.231328	+	0.400672i	−0.958199	−	0.286102i	$-0.907640\pi$
$761$	−4856.30	+	8411.35i	−0.231328	+	0.400672i	0.726871	+	0.686774i	$0.240974\pi$
$762$		0			0
$763$	257.655	+	446.271i	0.0122251	+	0.0211745i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	10729.0	+	18583.1i	0.505086	+	0.874835i
$768$		0			0
$769$	1482.08	−	2567.04i	0.0694995	−	0.120377i	−0.829182	−	0.558979i	$-0.811193\pi$
$769$	1482.08	−	2567.04i	0.0694995	−	0.120377i	0.898681	+	0.438603i	$0.144526\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−30022.1			−1.39692			−0.698461	−	0.715648i	$-0.746131\pi$
$773$	−30022.1			−1.39692			−0.698461	+	0.715648i	$0.746131\pi$
$774$		0			0
$775$	−647.383			−0.0300060
$776$		0			0
$777$		0			0
$778$		0			0
$779$	6816.07	−	11805.8i	0.313493	−	0.542986i
$780$		0			0
$781$	22554.1	+	39064.9i	1.03336	+	1.78982i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	7469.93	+	12938.3i	0.339635	+	0.588265i
$786$		0			0
$787$	45.3685	−	78.5805i	0.00205491	−	0.00355920i	−0.864996	−	0.501779i	$-0.832679\pi$
$787$	45.3685	−	78.5805i	0.00205491	−	0.00355920i	0.867051	+	0.498219i	$0.166013\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−5390.44			−0.242303
$792$		0			0
$793$	−16386.5			−0.733798
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−7210.69	+	12489.3i	−0.320471	+	0.555073i	−0.980585	−	0.196093i	$-0.937175\pi$
$797$	−7210.69	+	12489.3i	−0.320471	+	0.555073i	0.660114	+	0.751165i	$0.270508\pi$
$798$		0			0
$799$	−14549.6	−	25200.6i	−0.644215	−	1.11581i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	4509.66	+	7810.95i	0.198185	+	0.343266i
$804$		0			0
$805$	780.234	−	1351.41i	0.0341611	−	0.0591687i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−28357.7			−1.23239			−0.616195	−	0.787594i	$-0.711327\pi$
$809$	−28357.7			−1.23239			−0.616195	+	0.787594i	$0.711327\pi$
$810$		0			0
$811$	28791.5			1.24662			0.623308	−	0.781977i	$-0.285788\pi$
$811$	28791.5			1.24662			0.623308	+	0.781977i	$0.285788\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	3183.60	−	5514.16i	0.136830	−	0.236997i
$816$		0			0
$817$	−1263.62	−	2188.66i	−0.0541108	−	0.0937226i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−3512.84	−	6084.41i	−0.149329	−	0.258645i	0.781651	−	0.623716i	$-0.214378\pi$
$821$	−3512.84	−	6084.41i	−0.149329	−	0.258645i	−0.930980	+	0.365071i	$0.881045\pi$
$822$		0			0
$823$	2386.74	−	4133.95i	0.101089	−	0.175092i	−0.811044	−	0.584985i	$-0.801101\pi$
$823$	2386.74	−	4133.95i	0.101089	−	0.175092i	0.912134	+	0.409893i	$0.134434\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−13428.7			−0.564645			−0.282323	−	0.959319i	$-0.591105\pi$
$827$	−13428.7			−0.564645			−0.282323	+	0.959319i	$0.591105\pi$
$828$		0			0
$829$	32961.5			1.38094			0.690470	−	0.723361i	$-0.257404\pi$
$829$	32961.5			1.38094			0.690470	+	0.723361i	$0.257404\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	11754.9	−	20360.0i	0.488934	−	0.846859i
$834$		0			0
$835$	360.469	+	624.350i	0.0149396	+	0.0258761i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−16466.0	−	28520.0i	−0.677558	−	1.17356i	−0.975714	−	0.219047i	$-0.929705\pi$
$839$	−16466.0	−	28520.0i	−0.677558	−	1.17356i	0.298157	−	0.954517i	$-0.403628\pi$
$840$		0			0
$841$	7142.02	−	12370.3i	0.292838	−	0.507210i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−5400.94			−0.219879
$846$		0			0
$847$	−966.019			−0.0391887
$848$		0			0
$849$		0			0
$850$		0			0
$851$	10985.7	−	19027.8i	0.442521	−	0.766469i
$852$		0			0
$853$	267.138	+	462.697i	0.0107229	+	0.0185726i	0.871337	−	0.490685i	$-0.163253\pi$
$853$	267.138	+	462.697i	0.0107229	+	0.0185726i	−0.860614	+	0.509258i	$0.829920\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−15515.5	−	26873.7i	−0.618437	−	1.07116i	−0.989771	−	0.142665i	$-0.954433\pi$
$857$	−15515.5	−	26873.7i	−0.618437	−	1.07116i	0.371334	−	0.928499i	$-0.378900\pi$
$858$		0			0
$859$	6495.49	−	11250.5i	0.258001	−	0.446871i	−0.707705	−	0.706508i	$-0.750269\pi$
$859$	6495.49	−	11250.5i	0.258001	−	0.446871i	0.965706	+	0.259637i	$0.0836028\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	16721.3			0.659558			0.329779	−	0.944058i	$-0.393026\pi$
$863$	16721.3			0.659558			0.329779	+	0.944058i	$0.393026\pi$
$864$		0			0
$865$	−16994.8			−0.668022
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−8171.17	+	14152.9i	−0.318973	+	0.552478i
$870$		0			0
$871$	7311.94	+	12664.7i	0.284450	+	0.492681i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	194.043	+	336.092i	0.00749697	+	0.0129851i
$876$		0			0
$877$	−12192.7	+	21118.4i	−0.469462	+	0.813133i	−0.999390	−	0.0349099i	$-0.988886\pi$
$877$	−12192.7	+	21118.4i	−0.469462	+	0.813133i	0.529928	+	0.848043i	$0.322219\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−50589.8			−1.93464			−0.967318	−	0.253565i	$-0.918397\pi$
$881$	−50589.8			−1.93464			−0.967318	+	0.253565i	$0.918397\pi$
$882$		0			0
$883$	−37437.1			−1.42679			−0.713396	−	0.700761i	$-0.752844\pi$
$883$	−37437.1			−1.42679			−0.713396	+	0.700761i	$0.752844\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	19565.6	−	33888.6i	0.740640	−	1.28283i	−0.211565	−	0.977364i	$-0.567856\pi$
$887$	19565.6	−	33888.6i	0.740640	−	1.28283i	0.952204	−	0.305462i	$-0.0988108\pi$
$888$		0			0
$889$	45.6727	+	79.1074i	0.00172307	+	0.00298445i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−30195.5	−	52300.2i	−1.13153	−	1.95986i
$894$		0			0
$895$	−9985.19	+	17294.9i	−0.372925	+	0.645926i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−2603.09			−0.0965715
$900$		0			0
$901$	33219.8			1.22832
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−5610.49	+	9717.65i	−0.206076	+	0.356934i
$906$		0			0
$907$	−7809.06	−	13525.7i	−0.285883	−	0.495163i	0.686940	−	0.726714i	$-0.258953\pi$
$907$	−7809.06	−	13525.7i	−0.285883	−	0.495163i	−0.972823	+	0.231551i	$0.925620\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	4316.53	+	7476.45i	0.156985	+	0.271905i	0.933780	−	0.357848i	$-0.116489\pi$
$911$	4316.53	+	7476.45i	0.156985	+	0.271905i	−0.776795	+	0.629753i	$0.783156\pi$
$912$		0			0
$913$	−13382.2	+	23178.7i	−0.485090	+	0.840200i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	182.861			0.00658517
$918$		0			0
$919$	49659.5			1.78250			0.891249	−	0.453514i	$-0.149830\pi$
$919$	49659.5			1.78250			0.891249	+	0.453514i	$0.149830\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	18599.9	−	32215.9i	0.663296	−	1.14886i
$924$		0			0
$925$	2732.13	+	4732.19i	0.0971155	+	0.168209i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	710.648	+	1230.88i	0.0250975	+	0.0434702i	0.878301	−	0.478107i	$-0.158677\pi$
$929$	710.648	+	1230.88i	0.0250975	+	0.0434702i	−0.853204	+	0.521578i	$0.825344\pi$
$930$		0			0
$931$	24395.5	−	42254.3i	0.858787	−	1.48746i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−14289.3			−0.499795
$936$		0			0
$937$	14023.3			0.488922			0.244461	−	0.969659i	$-0.421389\pi$
$937$	14023.3			0.488922			0.244461	+	0.969659i	$0.421389\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−6742.00	+	11677.5i	−0.233563	+	0.404543i	−0.958854	−	0.283899i	$-0.908372\pi$
$941$	−6742.00	+	11677.5i	−0.233563	+	0.404543i	0.725291	+	0.688442i	$0.241705\pi$
$942$		0			0
$943$	4681.41	+	8108.43i	0.161662	+	0.280007i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	13539.1	+	23450.4i	0.464584	+	0.804682i	0.999183	−	0.0404235i	$-0.0128707\pi$
$947$	13539.1	+	23450.4i	0.464584	+	0.804682i	−0.534599	+	0.845106i	$0.679537\pi$
$948$		0			0
$949$	3719.01	−	6441.51i	0.127212	−	0.220338i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	31779.6			1.08021			0.540107	−	0.841597i	$-0.318384\pi$
$953$	31779.6			1.08021			0.540107	+	0.841597i	$0.318384\pi$
$954$		0			0
$955$	−18468.0			−0.625772
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−3688.80	+	6389.20i	−0.124210	+	0.215139i
$960$		0			0
$961$	14560.2	+	25219.0i	0.488745	+	0.846532i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−10305.2	−	17849.2i	−0.343769	−	0.595425i
$966$		0			0
$967$	−8567.00	+	14838.5i	−0.284898	+	0.493457i	−0.972584	−	0.232550i	$-0.925293\pi$
$967$	−8567.00	+	14838.5i	−0.284898	+	0.493457i	0.687687	+	0.726008i	$0.258626\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	3605.56			0.119164			0.0595818	−	0.998223i	$-0.481023\pi$
$971$	3605.56			0.119164			0.0595818	+	0.998223i	$0.481023\pi$
$972$		0			0
$973$	−3180.05			−0.104777
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−21533.5	+	37297.1i	−0.705136	+	1.22133i	0.261506	+	0.965202i	$0.415781\pi$
$977$	−21533.5	+	37297.1i	−0.705136	+	1.22133i	−0.966642	+	0.256130i	$0.917552\pi$
$978$		0			0
$979$	3519.84	+	6096.54i	0.114908	+	0.199026i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	6802.30	+	11781.9i	0.220712	+	0.382284i	0.955024	−	0.296528i	$-0.0958286\pi$
$983$	6802.30	+	11781.9i	0.220712	+	0.382284i	−0.734313	+	0.678812i	$0.762495\pi$
$984$		0			0
$985$	−3557.58	+	6161.90i	−0.115080	+	0.199324i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1735.76			0.0558078
$990$		0			0
$991$	−17228.8			−0.552260			−0.276130	−	0.961120i	$-0.589052\pi$
$991$	−17228.8			−0.552260			−0.276130	+	0.961120i	$0.589052\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	9906.48	−	17158.5i	0.315635	−	0.546696i
$996$		0			0
$997$	−24359.0	−	42191.0i	−0.773778	−	1.34022i	−0.935479	−	0.353383i	$-0.885031\pi$
$997$	−24359.0	−	42191.0i	−0.773778	−	1.34022i	0.161701	−	0.986840i	$-0.448302\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.4.i.m.1081.2		4
3.2	odd	2		1620.4.i.p.1081.2		4
9.2	odd	6		1620.4.i.p.541.2		4
9.4	even	3		540.4.a.j.1.1	yes	2
9.5	odd	6		540.4.a.g.1.1	✓	2
9.7	even	3	inner	1620.4.i.m.541.2		4
36.23	even	6		2160.4.a.u.1.2		2
36.31	odd	6		2160.4.a.z.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
540.4.a.g.1.1	✓	2	9.5	odd	6
540.4.a.j.1.1	yes	2	9.4	even	3
1620.4.i.m.541.2		4	9.7	even	3	inner
1620.4.i.m.1081.2		4	1.1	even	1	trivial
1620.4.i.p.541.2		4	9.2	odd	6
1620.4.i.p.1081.2		4	3.2	odd	2
2160.4.a.u.1.2		2	36.23	even	6
2160.4.a.z.1.2		2	36.31	odd	6

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:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1620.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.50000 + 4.33013i) q^{5} +(1.55234 + 2.68874i) q^{7} +O(q^{10})\)	\(q+(-2.50000 + 4.33013i) q^{5} +(1.55234 + 2.68874i) q^{7} +(20.2617 + 35.0943i) q^{11} +(16.7094 - 28.9415i) q^{13} +70.5234 q^{17} +146.361 q^{19} +(-50.2617 + 87.0558i) q^{23} +(-12.5000 - 21.6506i) q^{25} +(-50.2617 - 87.0558i) q^{29} +(12.9477 - 22.4260i) q^{31} -15.5234 q^{35} -218.570 q^{37} +(46.5703 - 80.6621i) q^{41} +(-8.63360 - 14.9538i) q^{43} +(-206.309 - 357.337i) q^{47} +(166.680 - 288.699i) q^{49} +471.047 q^{53} -202.617 q^{55} +(-321.047 + 556.069i) q^{59} +(-245.170 - 424.646i) q^{61} +(83.5469 + 144.707i) q^{65} +(-218.798 + 378.969i) q^{67} +1113.14 q^{71} +222.570 q^{73} +(-62.9063 + 108.957i) q^{77} +(201.641 + 349.252i) q^{79} +(330.234 + 571.983i) q^{83} +(-176.309 + 305.375i) q^{85} +173.719 q^{89} +103.755 q^{91} +(-365.902 + 633.761i) q^{95} +(295.018 + 510.986i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 10 q^{5} - 13 q^{7}+O(q^{10}) \) 4 * q - 10 * q^5 - 13 * q^7	\( 4 q - 10 q^{5} - 13 q^{7} - 15 q^{11} - 10 q^{13} + 90 q^{17} + 86 q^{19} - 105 q^{23} - 50 q^{25} - 105 q^{29} + 71 q^{31} + 130 q^{35} - 298 q^{37} - 390 q^{41} - 169 q^{43} - 345 q^{47} + 417 q^{49} + 1500 q^{53} + 150 q^{55} - 900 q^{59} - q^{61} - 50 q^{65} + 335 q^{67} + 3300 q^{71} + 314 q^{73} - 1020 q^{77} - 346 q^{79} - 600 q^{83} - 225 q^{85} + 3000 q^{89} + 1606 q^{91} - 215 q^{95} + 623 q^{97}+O(q^{100}) \) 4 * q - 10 * q^5 - 13 * q^7 - 15 * q^11 - 10 * q^13 + 90 * q^17 + 86 * q^19 - 105 * q^23 - 50 * q^25 - 105 * q^29 + 71 * q^31 + 130 * q^35 - 298 * q^37 - 390 * q^41 - 169 * q^43 - 345 * q^47 + 417 * q^49 + 1500 * q^53 + 150 * q^55 - 900 * q^59 - q^61 - 50 * q^65 + 335 * q^67 + 3300 * q^71 + 314 * q^73 - 1020 * q^77 - 346 * q^79 - 600 * q^83 - 225 * q^85 + 3000 * q^89 + 1606 * q^91 - 215 * q^95 + 623 * q^97

Introduction

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