LMFDB - Embedded newform 405.4.b.f.244.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,4,Mod(244,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("405.244");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	405.b (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:	$23.8957735523$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 91 x^{14} + 3268 x^{12} + 59128 x^{10} + 571975 x^{8} + 2881141 x^{6} + 6555196 x^{4} + \cdots + 614656 $ x^16 + 91x^14 + 3268x^12 + 59128x^10 + 571975x^8 + 2881141x^6 + 6555196x^4 + 4069504*x^2 + 614656
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{12}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			244.5
Root			$-2.67506i$ of defining polynomial
Character	$\chi$	$=$	405.244
Dual form			405.4.b.f.244.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.67506i q^{2} +0.844033 q^{4} +(10.9788 - 2.11350i) q^{5} -15.4153i q^{7} -23.6584i q^{8} +O(q^{10})$	\(q-2.67506i q^{2} +0.844033 q^{4} +(10.9788 - 2.11350i) q^{5} -15.4153i q^{7} -23.6584i q^{8} +(-5.65374 - 29.3689i) q^{10} -44.5598 q^{11} +28.0233i q^{13} -41.2370 q^{14} -56.5353 q^{16} -92.6615i q^{17} -49.5811 q^{19} +(9.26644 - 1.78386i) q^{20} +119.200i q^{22} -0.922809i q^{23} +(116.066 - 46.4071i) q^{25} +74.9642 q^{26} -13.0110i q^{28} +189.849 q^{29} -299.180 q^{31} -38.0312i q^{32} -247.875 q^{34} +(-32.5802 - 169.241i) q^{35} -57.8330i q^{37} +132.633i q^{38} +(-50.0019 - 259.739i) q^{40} +287.880 q^{41} -0.591953i q^{43} -37.6099 q^{44} -2.46857 q^{46} -598.302i q^{47} +105.368 q^{49} +(-124.142 - 310.485i) q^{50} +23.6526i q^{52} +146.339i q^{53} +(-489.211 + 94.1769i) q^{55} -364.701 q^{56} -507.859i q^{58} -193.066 q^{59} -566.153 q^{61} +800.326i q^{62} -554.019 q^{64} +(59.2272 + 307.661i) q^{65} +355.163i q^{67} -78.2093i q^{68} +(-452.731 + 87.1542i) q^{70} +320.703 q^{71} +636.782i q^{73} -154.707 q^{74} -41.8481 q^{76} +686.903i q^{77} +287.765 q^{79} +(-620.688 + 119.487i) q^{80} -770.097i q^{82} +285.058i q^{83} +(-195.840 - 1017.31i) q^{85} -1.58351 q^{86} +1054.21i q^{88} -331.615 q^{89} +431.988 q^{91} -0.778881i q^{92} -1600.50 q^{94} +(-544.339 + 104.789i) q^{95} -1821.63i q^{97} -281.866i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 54 q^{4} + 3 q^{5}+O(q^{10}) $ 16 * q - 54 * q^4 + 3 * q^5	$ 16 q - 54 q^{4} + 3 q^{5} - 10 q^{10} + 90 q^{11} - 102 q^{14} + 146 q^{16} - 4 q^{19} - 6 q^{20} - 71 q^{25} + 468 q^{26} - 516 q^{29} + 38 q^{31} - 212 q^{34} + 267 q^{35} - 44 q^{40} + 576 q^{41} - 1644 q^{44} - 290 q^{46} + 4 q^{49} + 558 q^{50} + 15 q^{55} + 2430 q^{56} - 2202 q^{59} + 20 q^{61} + 322 q^{64} + 339 q^{65} - 636 q^{70} + 2952 q^{71} - 4080 q^{74} - 396 q^{76} + 218 q^{79} - 1266 q^{80} + 704 q^{85} + 6108 q^{86} - 4074 q^{89} - 942 q^{91} + 1078 q^{94} - 1692 q^{95}+O(q^{100}) $ 16 * q - 54 * q^4 + 3 * q^5 - 10 * q^10 + 90 * q^11 - 102 * q^14 + 146 * q^16 - 4 * q^19 - 6 * q^20 - 71 * q^25 + 468 * q^26 - 516 * q^29 + 38 * q^31 - 212 * q^34 + 267 * q^35 - 44 * q^40 + 576 * q^41 - 1644 * q^44 - 290 * q^46 + 4 * q^49 + 558 * q^50 + 15 * q^55 + 2430 * q^56 - 2202 * q^59 + 20 * q^61 + 322 * q^64 + 339 * q^65 - 636 * q^70 + 2952 * q^71 - 4080 * q^74 - 396 * q^76 + 218 * q^79 - 1266 * q^80 + 704 * q^85 + 6108 * q^86 - 4074 * q^89 - 942 * q^91 + 1078 * q^94 - 1692 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$.

$n$	$82$	$326$
$\chi(n)$	$-1$	$1$

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 $

$2$		−	2.67506i		−	0.945778i	−0.881122	−	0.472889i	$-0.843211\pi$
$2$		−	2.67506i		−	0.945778i	0.881122	−	0.472889i	$-0.156789\pi$
$3$		0			0
$3$		0			0
$4$	0.844033			0.105504
$5$	10.9788	−	2.11350i	0.981970	−	0.189037i
$5$	10.9788	−	2.11350i	0.981970	−	0.189037i
$6$		0			0
$7$		−	15.4153i		−	0.832349i	−0.909285	−	0.416175i	$-0.863371\pi$
$7$		−	15.4153i		−	0.832349i	0.909285	−	0.416175i	$-0.136629\pi$
$8$		−	23.6584i		−	1.04556i
$9$		0			0
$10$	−5.65374	−	29.3689i	−0.178787	−	0.928726i
$11$	−44.5598			−1.22139			−0.610694	−	0.791866i	$-0.709110\pi$
$11$	−44.5598			−1.22139			−0.610694	+	0.791866i	$0.709110\pi$
$12$		0			0
$13$			28.0233i			0.597867i	0.954274	+	0.298933i	$0.0966309\pi$
$13$			28.0233i			0.597867i	−0.954274	+	0.298933i	$0.903369\pi$
$14$	−41.2370			−0.787217
$15$		0			0
$16$	−56.5353			−0.883365
$17$		−	92.6615i		−	1.32198i	−0.750394	−	0.660991i	$-0.770136\pi$
$17$		−	92.6615i		−	1.32198i	0.750394	−	0.660991i	$-0.229864\pi$
$18$		0			0
$19$	−49.5811			−0.598667			−0.299334	−	0.954149i	$-0.596764\pi$
$19$	−49.5811			−0.598667			−0.299334	+	0.954149i	$0.596764\pi$
$20$	9.26644	−	1.78386i	0.103602	−	0.0199442i
$21$		0			0
$22$			119.200i			1.15516i
$23$		−	0.922809i		−	0.00836604i	−0.999991	−	0.00418302i	$-0.998668\pi$
$23$		−	0.922809i		−	0.00836604i	0.999991	−	0.00418302i	$-0.00133150\pi$
$24$		0			0
$25$	116.066	−	46.4071i	0.928530	−	0.371257i
$26$	74.9642			0.565449
$27$		0			0
$28$		−	13.0110i		−	0.0878163i
$29$	189.849			1.21566			0.607830	−	0.794067i	$-0.292040\pi$
$29$	189.849			1.21566			0.607830	+	0.794067i	$0.292040\pi$
$30$		0			0
$31$	−299.180			−1.73337			−0.866683	−	0.498859i	$-0.833752\pi$
$31$	−299.180			−1.73337			−0.866683	+	0.498859i	$0.833752\pi$
$32$		−	38.0312i		−	0.210095i
$33$		0			0
$34$	−247.875			−1.25030
$35$	−32.5802	−	169.241i	−0.157345	−	0.817342i
$36$		0			0
$37$		−	57.8330i		−	0.256965i	−0.991712	−	0.128482i	$-0.958989\pi$
$37$		−	57.8330i		−	0.256965i	0.991712	−	0.128482i	$-0.0410105\pi$
$38$			132.633i			0.566206i
$39$		0			0
$40$	−50.0019	−	259.739i	−0.197650	−	1.02671i
$41$	287.880			1.09657			0.548284	−	0.836292i	$-0.315281\pi$
$41$	287.880			1.09657			0.548284	+	0.836292i	$0.315281\pi$
$42$		0			0
$43$		−	0.591953i		−	0.00209935i	−0.999999	−	0.00104967i	$-0.999666\pi$
$43$		−	0.591953i		−	0.00209935i	0.999999	−	0.00104967i	$-0.000334122\pi$
$44$	−37.6099			−0.128862
$45$		0			0
$46$	−2.46857			−0.00791242
$47$		−	598.302i		−	1.85684i	−0.371536	−	0.928418i	$-0.621169\pi$
$47$		−	598.302i		−	1.85684i	0.371536	−	0.928418i	$-0.378831\pi$
$48$		0			0
$49$	105.368			0.307195
$50$	−124.142	−	310.485i	−0.351127	−	0.878183i
$51$		0			0
$52$			23.6526i			0.0630774i
$53$			146.339i			0.379267i	0.981855	+	0.189634i	$0.0607301\pi$
$53$			146.339i			0.379267i	−0.981855	+	0.189634i	$0.939270\pi$
$54$		0			0
$55$	−489.211	+	94.1769i	−1.19937	+	0.230888i
$56$	−364.701			−0.870272
$57$		0			0
$58$		−	507.859i		−	1.14975i
$59$	−193.066			−0.426018			−0.213009	−	0.977050i	$-0.568326\pi$
$59$	−193.066			−0.426018			−0.213009	+	0.977050i	$0.568326\pi$
$60$		0			0
$61$	−566.153			−1.18833			−0.594167	−	0.804341i	$-0.702518\pi$
$61$	−566.153			−1.18833			−0.594167	+	0.804341i	$0.702518\pi$
$62$			800.326i			1.63938i
$63$		0			0
$64$	−554.019			−1.08207
$65$	59.2272	+	307.661i	0.113019	+	0.587087i
$66$		0			0
$67$			355.163i			0.647613i	0.946123	+	0.323806i	$0.104963\pi$
$67$			355.163i			0.647613i	−0.946123	+	0.323806i	$0.895037\pi$
$68$		−	78.2093i		−	0.139475i
$69$		0			0
$70$	−452.731	+	87.1542i	−0.773024	+	0.148813i
$71$	320.703			0.536062			0.268031	−	0.963410i	$-0.413627\pi$
$71$	320.703			0.536062			0.268031	+	0.963410i	$0.413627\pi$
$72$		0			0
$73$			636.782i			1.02095i	0.859891	+	0.510477i	$0.170531\pi$
$73$			636.782i			1.02095i	−0.859891	+	0.510477i	$0.829469\pi$
$74$	−154.707			−0.243031
$75$		0			0
$76$	−41.8481			−0.0631619
$77$			686.903i			1.01662i
$78$		0			0
$79$	287.765			0.409824			0.204912	−	0.978780i	$-0.434309\pi$
$79$	287.765			0.409824			0.204912	+	0.978780i	$0.434309\pi$
$80$	−620.688	+	119.487i	−0.867438	+	0.166989i
$81$		0			0
$82$		−	770.097i		−	1.03711i
$83$			285.058i			0.376978i	0.982075	+	0.188489i	$0.0603590\pi$
$83$			285.058i			0.376978i	−0.982075	+	0.188489i	$0.939641\pi$
$84$		0			0
$85$	−195.840	−	1017.31i	−0.249904	−	1.29815i
$86$	−1.58351			−0.00198552
$87$		0			0
$88$			1054.21i			1.27704i
$89$	−331.615			−0.394957			−0.197478	−	0.980307i	$-0.563275\pi$
$89$	−331.615			−0.394957			−0.197478	+	0.980307i	$0.563275\pi$
$90$		0			0
$91$	431.988			0.497634
$92$		−	0.778881i		−	0.000882652i
$93$		0			0
$94$	−1600.50			−1.75616
$95$	−544.339	+	104.789i	−0.587873	+	0.113170i
$96$		0			0
$97$		−	1821.63i		−	1.90679i	−0.301730	−	0.953393i	$-0.597564\pi$
$97$		−	1821.63i		−	1.90679i	0.301730	−	0.953393i	$-0.402436\pi$
$98$		−	281.866i		−	0.290538i
$99$		0			0
$100$	97.9638	−	39.1692i	0.0979638	−	0.0391692i
$101$	1147.37			1.13037			0.565184	−	0.824965i	$-0.308805\pi$
$101$	1147.37			1.13037			0.565184	+	0.824965i	$0.308805\pi$
$102$		0			0
$103$			1582.54i			1.51391i	0.653467	+	0.756955i	$0.273314\pi$
$103$			1582.54i			1.51391i	−0.653467	+	0.756955i	$0.726686\pi$
$104$	662.986			0.625107
$105$		0			0
$106$	391.466			0.358703
$107$		−	385.387i		−	0.348194i	−0.984728	−	0.174097i	$-0.944299\pi$
$107$		−	385.387i		−	0.348194i	0.984728	−	0.174097i	$-0.0557007\pi$
$108$		0			0
$109$	203.502			0.178825			0.0894126	−	0.995995i	$-0.471501\pi$
$109$	203.502			0.178825			0.0894126	+	0.995995i	$0.471501\pi$
$110$	251.929	+	1308.67i	0.218368	+	1.13433i
$111$		0			0
$112$			871.510i			0.735268i
$113$		−	66.0150i		−	0.0549573i	−0.999622	−	0.0274786i	$-0.991252\pi$
$113$		−	66.0150i		−	0.0549573i	0.999622	−	0.0274786i	$-0.00874782\pi$
$114$		0			0
$115$	−1.95035	−	10.1313i	−0.00158149	−	0.00821520i
$116$	160.239			0.128257
$117$		0			0
$118$			516.464i			0.402918i
$119$	−1428.41			−1.10035
$120$		0			0
$121$	654.573			0.491790
$122$			1514.49i			1.12390i
$123$		0			0
$124$	−252.518			−0.182877
$125$	1176.18	−	754.799i	0.841607	−	0.540090i
$126$		0			0
$127$		−	1821.52i		−	1.27270i	−0.771399	−	0.636352i	$-0.780443\pi$
$127$		−	1821.52i		−	1.27270i	0.771399	−	0.636352i	$-0.219557\pi$
$128$			1177.79i			0.813301i
$129$		0			0
$130$	823.013	−	158.437i	0.555254	−	0.106891i
$131$	−238.258			−0.158906			−0.0794530	−	0.996839i	$-0.525317\pi$
$131$	−238.258			−0.158906			−0.0794530	+	0.996839i	$0.525317\pi$
$132$		0			0
$133$			764.308i			0.498300i
$134$	950.083			0.612498
$135$		0			0
$136$	−2192.22			−1.38221
$137$		−	2047.48i		−	1.27685i	−0.769685	−	0.638424i	$-0.779587\pi$
$137$		−	2047.48i		−	1.27685i	0.769685	−	0.638424i	$-0.220413\pi$
$138$		0			0
$139$	2179.37			1.32987			0.664934	−	0.746902i	$-0.268460\pi$
$139$	2179.37			1.32987			0.664934	+	0.746902i	$0.268460\pi$
$140$	−27.4988	−	142.845i	−0.0166005	−	0.0862330i
$141$		0			0
$142$		−	857.901i		−	0.506996i
$143$		−	1248.71i		−	0.730228i
$144$		0			0
$145$	2084.31	−	401.246i	1.19374	−	0.229805i
$146$	1703.43			0.965596
$147$		0			0
$148$		−	48.8130i		−	0.0271108i
$149$	155.948			0.0857435			0.0428717	−	0.999081i	$-0.486349\pi$
$149$	155.948			0.0857435			0.0428717	+	0.999081i	$0.486349\pi$
$150$		0			0
$151$	1614.14			0.869914			0.434957	−	0.900451i	$-0.356764\pi$
$151$	1614.14			0.869914			0.434957	+	0.900451i	$0.356764\pi$
$152$			1173.01i			0.625943i
$153$		0			0
$154$	1837.51			0.961498
$155$	−3284.63	+	632.316i	−1.70211	+	0.327670i
$156$		0			0
$157$			988.508i			0.502494i	0.967923	+	0.251247i	$0.0808406\pi$
$157$			988.508i			0.502494i	−0.967923	+	0.251247i	$0.919159\pi$
$158$		−	769.791i		−	0.387603i
$159$		0			0
$160$	−80.3788	−	417.535i	−0.0397156	−	0.206307i
$161$	−14.2254			−0.00696347
$162$		0			0
$163$			2974.11i			1.42914i	0.699564	+	0.714570i	$0.253378\pi$
$163$			2974.11i			1.42914i	−0.699564	+	0.714570i	$0.746622\pi$
$164$	242.980			0.115692
$165$		0			0
$166$	762.548			0.356537
$167$		−	2438.96i		−	1.13013i	−0.825045	−	0.565067i	$-0.808850\pi$
$167$		−	2438.96i		−	1.13013i	0.825045	−	0.565067i	$-0.191150\pi$
$168$		0			0
$169$	1411.69			0.642555
$170$	−2721.36	+	523.884i	−1.22776	+	0.236353i
$171$		0			0
$172$		−	0.499628i		−	0.000221490i
$173$		−	2653.59i		−	1.16618i	−0.812409	−	0.583089i	$-0.801844\pi$
$173$		−	2653.59i		−	1.16618i	0.812409	−	0.583089i	$-0.198156\pi$
$174$		0			0
$175$	−715.381	−	1789.20i	−0.309016	−	0.772861i
$176$	2519.20			1.07893
$177$		0			0
$178$			887.092i			0.373541i
$179$	−2102.30			−0.877841			−0.438921	−	0.898526i	$-0.644639\pi$
$179$	−2102.30			−0.877841			−0.438921	+	0.898526i	$0.644639\pi$
$180$		0			0
$181$	1597.36			0.655973			0.327987	−	0.944682i	$-0.393630\pi$
$181$	1597.36			0.655973			0.327987	+	0.944682i	$0.393630\pi$
$182$		−	1155.60i		−	0.470651i
$183$		0			0
$184$	−21.8321			−0.00874721
$185$	−122.230	−	634.935i	−0.0485758	−	0.252331i
$186$		0			0
$187$			4128.97i			1.61465i
$188$		−	504.987i		−	0.195904i
$189$		0			0
$190$	280.318	+	1456.14i	0.107034	+	0.555997i
$191$	3407.43			1.29085			0.645426	−	0.763823i	$-0.276680\pi$
$191$	3407.43			1.29085			0.645426	+	0.763823i	$0.276680\pi$
$192$		0			0
$193$		−	819.130i		−	0.305504i	−0.988265	−	0.152752i	$-0.951186\pi$
$193$		−	819.130i		−	0.305504i	0.988265	−	0.152752i	$-0.0488135\pi$
$194$	−4872.97			−1.80340
$195$		0			0
$196$	88.9340			0.0324103
$197$		−	359.979i		−	0.130190i	−0.997879	−	0.0650950i	$-0.979265\pi$
$197$		−	359.979i		−	0.130190i	0.997879	−	0.0650950i	$-0.0207350\pi$
$198$		0			0
$199$	1338.34			0.476745			0.238372	−	0.971174i	$-0.423386\pi$
$199$	1338.34			0.476745			0.238372	+	0.971174i	$0.423386\pi$
$200$	−1097.92	−	2745.94i	−0.388172	−	0.970835i
$201$		0			0
$202$		−	3069.28i		−	1.06908i
$203$		−	2926.59i		−	1.01185i
$204$		0			0
$205$	3160.56	−	608.433i	1.07680	−	0.207292i
$206$	4233.41			1.43182
$207$		0			0
$208$		−	1584.31i		−	0.528135i
$209$	2209.32			0.731205
$210$		0			0
$211$	1068.23			0.348530			0.174265	−	0.984699i	$-0.444245\pi$
$211$	1068.23			0.348530			0.174265	+	0.984699i	$0.444245\pi$
$212$			123.515i			0.0400143i
$213$		0			0
$214$	−1030.94			−0.329315
$215$	−1.25109	−	6.49891i	−0.000396855	−	0.00206150i
$216$		0			0
$217$			4611.96i			1.44277i
$218$		−	544.381i		−	0.169129i
$219$		0			0
$220$	−412.910	+	79.4885i	−0.126538	+	0.0243596i
$221$	2596.68			0.790370
$222$		0			0
$223$			1518.83i			0.456091i	0.973650	+	0.228046i	$0.0732335\pi$
$223$			1518.83i			0.456091i	−0.973650	+	0.228046i	$0.926766\pi$
$224$	−586.263			−0.174872
$225$		0			0
$226$	−176.594			−0.0519774
$227$			4235.12i			1.23830i	0.785271	+	0.619152i	$0.212524\pi$
$227$			4235.12i			1.23830i	−0.785271	+	0.619152i	$0.787476\pi$
$228$		0			0
$229$	5536.52			1.59766			0.798829	−	0.601558i	$-0.205453\pi$
$229$	5536.52			1.59766			0.798829	+	0.601558i	$0.205453\pi$
$230$	−27.1019	+	5.21732i	−0.00776976	+	0.00149574i
$231$		0			0
$232$		−	4491.53i		−	1.27105i
$233$			6337.30i			1.78185i	0.454154	+	0.890923i	$0.349941\pi$
$233$			6337.30i			1.78185i	−0.454154	+	0.890923i	$0.650059\pi$
$234$		0			0
$235$	−1264.51	−	6568.61i	−0.351011	−	1.82336i
$236$	−162.954			−0.0449466
$237$		0			0
$238$			3821.08i			1.04069i
$239$	2542.87			0.688220			0.344110	−	0.938929i	$-0.388181\pi$
$239$	2542.87			0.688220			0.344110	+	0.938929i	$0.388181\pi$
$240$		0			0
$241$	2051.42			0.548314			0.274157	−	0.961685i	$-0.411601\pi$
$241$	2051.42			0.548314			0.274157	+	0.961685i	$0.411601\pi$
$242$		−	1751.02i		−	0.465124i
$243$		0			0
$244$	−477.852			−0.125374
$245$	1156.81	−	222.695i	0.301656	−	0.0580712i
$246$		0			0
$247$		−	1389.43i		−	0.357923i
$248$			7078.11i			1.81234i
$249$		0			0
$250$	−2019.13	−	3146.36i	−0.510805	−	0.795974i
$251$	2770.89			0.696801			0.348401	−	0.937346i	$-0.386725\pi$
$251$	2770.89			0.696801			0.348401	+	0.937346i	$0.386725\pi$
$252$		0			0
$253$			41.1201i			0.0102182i
$254$	−4872.67			−1.20370
$255$		0			0
$256$	−1281.50			−0.312865
$257$			426.755i			0.103581i	0.998658	+	0.0517904i	$0.0164928\pi$
$257$			426.755i			0.103581i	−0.998658	+	0.0517904i	$0.983507\pi$
$258$		0			0
$259$	−891.514			−0.213884
$260$	49.9897	+	259.676i	0.0119240	+	0.0619402i
$261$		0			0
$262$			637.355i			0.150290i
$263$			3762.39i			0.882125i	0.897476	+	0.441063i	$0.145398\pi$
$263$			3762.39i			0.882125i	−0.897476	+	0.441063i	$0.854602\pi$
$264$		0			0
$265$	309.287	+	1606.62i	0.0716956	+	0.372429i
$266$	2044.57			0.471281
$267$		0			0
$268$			299.769i			0.0683258i
$269$	−1980.25			−0.448839			−0.224420	−	0.974493i	$-0.572049\pi$
$269$	−1980.25			−0.448839			−0.224420	+	0.974493i	$0.572049\pi$
$270$		0			0
$271$	5659.70			1.26864			0.634321	−	0.773070i	$-0.281280\pi$
$271$	5659.70			1.26864			0.634321	+	0.773070i	$0.281280\pi$
$272$			5238.65i			1.16779i
$273$		0			0
$274$	−5477.14			−1.20761
$275$	−5171.89	+	2067.89i	−1.13410	+	0.453449i
$276$		0			0
$277$			6169.57i			1.33824i	0.743153	+	0.669121i	$0.233329\pi$
$277$			6169.57i			1.33824i	−0.743153	+	0.669121i	$0.766671\pi$
$278$		−	5829.95i		−	1.25776i
$279$		0			0
$280$	−4003.97	+	770.795i	−0.854581	+	0.164514i
$281$	−3343.45			−0.709800			−0.354900	−	0.934904i	$-0.615485\pi$
$281$	−3343.45			−0.709800			−0.354900	+	0.934904i	$0.615485\pi$
$282$		0			0
$283$			5103.32i			1.07195i	0.844235	+	0.535973i	$0.180055\pi$
$283$			5103.32i			1.07195i	−0.844235	+	0.535973i	$0.819945\pi$
$284$	270.684			0.0565568
$285$		0			0
$286$	−3340.39			−0.690633
$287$		−	4437.76i		−	0.912727i
$288$		0			0
$289$	−3673.14			−0.747638
$290$	−1073.36	−	5575.67i	−0.217344	−	1.12902i
$291$		0			0
$292$			537.465i			0.107715i
$293$			4130.28i			0.823527i	0.911291	+	0.411764i	$0.135087\pi$
$293$			4130.28i			0.823527i	−0.911291	+	0.411764i	$0.864913\pi$
$294$		0			0
$295$	−2119.62	+	408.044i	−0.418337	+	0.0805331i
$296$	−1368.23			−0.268672
$297$		0			0
$298$		−	417.172i		−	0.0810943i
$299$	25.8602			0.00500178
$300$		0			0
$301$	−9.12515			−0.00174739
$302$		−	4317.93i		−	0.822746i
$303$		0			0
$304$	2803.08			0.528841
$305$	−6215.65	+	1196.56i	−1.16691	+	0.224639i
$306$		0			0
$307$		−	3382.52i		−	0.628830i	−0.949286	−	0.314415i	$-0.898192\pi$
$307$		−	3382.52i		−	0.628830i	0.949286	−	0.314415i	$-0.101808\pi$
$308$			579.769i			0.107258i
$309$		0			0
$310$	1691.49	+	8786.59i	0.309903	+	1.60982i
$311$	6021.22			1.09785			0.548926	−	0.835871i	$-0.315037\pi$
$311$	6021.22			1.09785			0.548926	+	0.835871i	$0.315037\pi$
$312$		0			0
$313$		−	10192.9i		−	1.84070i	−0.391096	−	0.920350i	$-0.627904\pi$
$313$		−	10192.9i		−	1.84070i	0.391096	−	0.920350i	$-0.372096\pi$
$314$	2644.32			0.475247
$315$		0			0
$316$	242.884			0.0432382
$317$			5400.59i			0.956869i	0.878123	+	0.478435i	$0.158796\pi$
$317$			5400.59i			0.956869i	−0.878123	+	0.478435i	$0.841204\pi$
$318$		0			0
$319$	−8459.65			−1.48479
$320$	−6082.44	+	1170.92i	−1.06256	+	0.204551i
$321$		0			0
$322$			38.0538i			0.00658589i
$323$			4594.25i			0.791428i
$324$		0			0
$325$	1300.48	+	3252.56i	0.221962	+	0.555137i
$326$	7955.92			1.35165
$327$		0			0
$328$		−	6810.76i		−	1.14653i
$329$	−9223.02			−1.54554
$330$		0			0
$331$	−2302.90			−0.382413			−0.191206	−	0.981550i	$-0.561240\pi$
$331$	−2302.90			−0.382413			−0.191206	+	0.981550i	$0.561240\pi$
$332$			240.598i			0.0397727i
$333$		0			0
$334$	−6524.37			−1.06886
$335$	750.636	+	3899.25i	0.122423	+	0.635936i
$336$		0			0
$337$		−	9566.33i		−	1.54632i	−0.634209	−	0.773162i	$-0.718674\pi$
$337$		−	9566.33i		−	1.54632i	0.634209	−	0.773162i	$-0.281326\pi$
$338$		−	3776.37i		−	0.607714i
$339$		0			0
$340$	−165.295	−	858.641i	−0.0263659	−	0.136960i
$341$	13331.4			2.11711
$342$		0			0
$343$		−	6911.73i		−	1.08804i
$344$	−14.0046			−0.00219500
$345$		0			0
$346$	−7098.52			−1.10294
$347$		−	81.5218i		−	0.0126119i	−0.999980	−	0.00630593i	$-0.997993\pi$
$347$		−	81.5218i		−	0.0126119i	0.999980	−	0.00630593i	$-0.00200725\pi$
$348$		0			0
$349$	−93.5719			−0.0143518			−0.00717591	−	0.999974i	$-0.502284\pi$
$349$	−93.5719			−0.0143518			−0.00717591	+	0.999974i	$0.502284\pi$
$350$	−4786.22	+	1913.69i	−0.730955	+	0.292260i
$351$		0			0
$352$			1694.66i			0.256607i
$353$		−	6675.41i		−	1.00651i	−0.864139	−	0.503253i	$-0.832136\pi$
$353$		−	6675.41i		−	1.00651i	0.864139	−	0.503253i	$-0.167864\pi$
$354$		0			0
$355$	3520.92	−	677.805i	0.526397	−	0.101336i
$356$	−279.894			−0.0416696
$357$		0			0
$358$			5623.80i			0.830243i
$359$	−8334.50			−1.22529			−0.612644	−	0.790359i	$-0.709894\pi$
$359$	−8334.50			−1.22529			−0.612644	+	0.790359i	$0.709894\pi$
$360$		0			0
$361$	−4400.72			−0.641598
$362$		−	4273.05i		−	0.620405i
$363$		0			0
$364$	364.613			0.0525025
$365$	1345.84	+	6991.07i	0.192998	+	1.00255i
$366$		0			0
$367$		−	125.761i		−	0.0178874i	−0.999960	−	0.00894369i	$-0.997153\pi$
$367$		−	125.761i		−	0.0178874i	0.999960	−	0.00894369i	$-0.00284690\pi$
$368$			52.1713i			0.00739026i
$369$		0			0
$370$	−1698.49	+	326.973i	−0.238649	+	0.0459419i
$371$	2255.86			0.315683
$372$		0			0
$373$		−	1435.76i		−	0.199305i	−0.995022	−	0.0996525i	$-0.968227\pi$
$373$		−	1435.76i		−	0.199305i	0.995022	−	0.0996525i	$-0.0317731\pi$
$374$	11045.3			1.52710
$375$		0			0
$376$	−14154.8			−1.94144
$377$			5320.21i			0.726803i
$378$		0			0
$379$	2125.04			0.288010			0.144005	−	0.989577i	$-0.454002\pi$
$379$	2125.04			0.288010			0.144005	+	0.989577i	$0.454002\pi$
$380$	−459.440	+	88.4458i	−0.0620231	+	0.0119399i
$381$		0			0
$382$		−	9115.09i		−	1.22086i
$383$		−	7060.76i		−	0.942005i	−0.882132	−	0.471002i	$-0.843892\pi$
$383$		−	7060.76i		−	0.942005i	0.882132	−	0.471002i	$-0.156108\pi$
$384$		0			0
$385$	1451.77	+	7541.34i	0.192179	+	0.998292i
$386$	−2191.22			−0.288939
$387$		0			0
$388$		−	1537.51i		−	0.201174i
$389$	−9819.12			−1.27982			−0.639909	−	0.768451i	$-0.721028\pi$
$389$	−9819.12			−1.27982			−0.639909	+	0.768451i	$0.721028\pi$
$390$		0			0
$391$	−85.5088			−0.0110598
$392$		−	2492.83i		−	0.321191i
$393$		0			0
$394$	−962.967			−0.123131
$395$	3159.31	−	608.191i	0.402435	−	0.0774720i
$396$		0			0
$397$			10995.5i			1.39005i	0.718987	+	0.695023i	$0.244606\pi$
$397$			10995.5i			1.39005i	−0.718987	+	0.695023i	$0.755394\pi$
$398$		−	3580.14i		−	0.450895i
$399$		0			0
$400$	−6561.85	+	2623.64i	−0.820231	+	0.327956i
$401$	6531.81			0.813424			0.406712	−	0.913556i	$-0.366675\pi$
$401$	6531.81			0.813424			0.406712	+	0.913556i	$0.366675\pi$
$402$		0			0
$403$		−	8384.02i		−	1.03632i
$404$	968.415			0.119259
$405$		0			0
$406$	−7828.82			−0.956989
$407$			2577.02i			0.313854i
$408$		0			0
$409$	5005.34			0.605130			0.302565	−	0.953129i	$-0.402157\pi$
$409$	5005.34			0.605130			0.302565	+	0.953129i	$0.402157\pi$
$410$	−1627.60	−	8454.71i	−0.196052	−	1.01841i
$411$		0			0
$412$			1335.72i			0.159724i
$413$			2976.17i			0.354596i
$414$		0			0
$415$	602.469	+	3129.58i	0.0712627	+	0.370181i
$416$	1065.76			0.125609
$417$		0			0
$418$		−	5910.07i		−	0.691558i
$419$	−73.4944			−0.00856907			−0.00428453	−	0.999991i	$-0.501364\pi$
$419$	−73.4944			−0.00856907			−0.00428453	+	0.999991i	$0.501364\pi$
$420$		0			0
$421$	5030.81			0.582391			0.291195	−	0.956664i	$-0.405947\pi$
$421$	5030.81			0.582391			0.291195	+	0.956664i	$0.405947\pi$
$422$		−	2857.58i		−	0.329632i
$423$		0			0
$424$	3462.13			0.396547
$425$	−4300.15	−	10754.9i	−0.490796	−	1.22750i
$426$		0			0
$427$			8727.43i			0.989110i
$428$		−	325.280i		−	0.0367360i
$429$		0			0
$430$	−17.3850	+	3.34675i	−0.00194972	+	0.000375336i
$431$	1951.29			0.218075			0.109037	−	0.994038i	$-0.465223\pi$
$431$	1951.29			0.218075			0.109037	+	0.994038i	$0.465223\pi$
$432$		0			0
$433$			3805.50i			0.422357i	0.977448	+	0.211178i	$0.0677301\pi$
$433$			3805.50i			0.422357i	−0.977448	+	0.211178i	$0.932270\pi$
$434$	12337.3			1.36454
$435$		0			0
$436$	171.762			0.0188668
$437$			45.7538i			0.00500847i
$438$		0			0
$439$	−11904.4			−1.29422			−0.647111	−	0.762395i	$-0.724023\pi$
$439$	−11904.4			−1.29422			−0.647111	+	0.762395i	$0.724023\pi$
$440$	2228.07	+	11573.9i	0.241407	+	1.25401i
$441$		0			0
$442$		−	6946.29i		−	0.747514i
$443$		−	9068.45i		−	0.972585i	−0.873796	−	0.486292i	$-0.838349\pi$
$443$		−	9068.45i		−	0.972585i	0.873796	−	0.486292i	$-0.161651\pi$
$444$		0			0
$445$	−3640.72	+	700.868i	−0.387836	+	0.0746614i
$446$	4062.97			0.431361
$447$		0			0
$448$			8540.37i			0.900658i
$449$	7332.48			0.770693			0.385346	−	0.922772i	$-0.374082\pi$
$449$	7332.48			0.770693			0.385346	+	0.922772i	$0.374082\pi$
$450$		0			0
$451$	−12827.9			−1.33934
$452$		−	55.7189i		−	0.00579822i
$453$		0			0
$454$	11329.2			1.17116
$455$	4742.70	−	913.006i	0.488662	−	0.0940712i
$456$		0			0
$457$			1170.43i			0.119804i	0.998204	+	0.0599019i	$0.0190788\pi$
$457$			1170.43i			0.119804i	−0.998204	+	0.0599019i	$0.980921\pi$
$458$		−	14810.5i		−	1.51103i
$459$		0			0
$460$	−1.64616	−	8.55115i	−0.000166854	−	0.000866738i
$461$	14747.8			1.48996			0.744981	−	0.667086i	$-0.232459\pi$
$461$	14747.8			1.48996			0.744981	+	0.667086i	$0.232459\pi$
$462$		0			0
$463$		−	2085.51i		−	0.209334i	−0.994507	−	0.104667i	$-0.966622\pi$
$463$		−	2085.51i		−	0.209334i	0.994507	−	0.104667i	$-0.0333777\pi$
$464$	−10733.2			−1.07387
$465$		0			0
$466$	16952.7			1.68523
$467$		−	9943.76i		−	0.985315i	−0.870223	−	0.492658i	$-0.836025\pi$
$467$		−	9943.76i		−	0.985315i	0.870223	−	0.492658i	$-0.163975\pi$
$468$		0			0
$469$	5474.95			0.539040
$470$	−17571.5	+	3382.64i	−1.72449	+	0.331978i
$471$		0			0
$472$			4567.62i			0.445428i
$473$			26.3773i			0.00256412i
$474$		0			0
$475$	−5754.69	+	2300.92i	−0.555880	+	0.222260i
$476$	−1205.62			−0.116092
$477$		0			0
$478$		−	6802.34i		−	0.650903i
$479$	−2664.20			−0.254134			−0.127067	−	0.991894i	$-0.540556\pi$
$479$	−2664.20			−0.254134			−0.127067	+	0.991894i	$0.540556\pi$
$480$		0			0
$481$	1620.67			0.153631
$482$		−	5487.68i		−	0.518583i
$483$		0			0
$484$	552.481			0.0518859
$485$	−3850.01	−	19999.2i	−0.360453	−	1.87241i
$486$		0			0
$487$		−	3071.50i		−	0.285797i	−0.989737	−	0.142898i	$-0.954358\pi$
$487$		−	3071.50i		−	0.285797i	0.989737	−	0.142898i	$-0.0456422\pi$
$488$			13394.2i			1.24248i
$489$		0			0
$490$	−595.723	−	3094.54i	−0.0549225	−	0.285300i
$491$	−16089.6			−1.47885			−0.739423	−	0.673241i	$-0.764902\pi$
$491$	−16089.6			−1.47885			−0.739423	+	0.673241i	$0.764902\pi$
$492$		0			0
$493$		−	17591.7i		−	1.60708i
$494$	−3716.80			−0.338516
$495$		0			0
$496$	16914.3			1.53119
$497$		−	4943.74i		−	0.446191i
$498$		0			0
$499$	−12060.6			−1.08197			−0.540987	−	0.841031i	$-0.681949\pi$
$499$	−12060.6			−1.08197			−0.540987	+	0.841031i	$0.681949\pi$
$500$	992.737	−	637.075i	0.0887931	−	0.0569817i
$501$		0			0
$502$		−	7412.31i		−	0.659019i
$503$			1324.90i			0.117444i	0.998274	+	0.0587222i	$0.0187026\pi$
$503$			1324.90i			0.117444i	−0.998274	+	0.0587222i	$0.981297\pi$
$504$		0			0
$505$	12596.7	−	2424.95i	1.10999	−	0.213681i
$506$	109.999			0.00966413
$507$		0			0
$508$		−	1537.42i		−	0.134276i
$509$	−12121.1			−1.05551			−0.527757	−	0.849395i	$-0.676967\pi$
$509$	−12121.1			−1.05551			−0.527757	+	0.849395i	$0.676967\pi$
$510$		0			0
$511$	9816.20			0.849791
$512$			12850.4i			1.10920i
$513$		0			0
$514$	1141.60			0.0979644
$515$	3344.70	+	17374.4i	0.286185	+	1.48661i
$516$		0			0
$517$			26660.2i			2.26792i
$518$			2384.86i			0.202287i
$519$		0			0
$520$	7278.76	−	1401.22i	0.613836	−	0.118168i
$521$	−17997.6			−1.51342			−0.756708	−	0.653753i	$-0.773193\pi$
$521$	−17997.6			−1.51342			−0.756708	+	0.653753i	$0.773193\pi$
$522$		0			0
$523$		−	6451.03i		−	0.539357i	−0.962950	−	0.269679i	$-0.913083\pi$
$523$		−	6451.03i		−	0.539357i	0.962950	−	0.269679i	$-0.0869174\pi$
$524$	−201.098			−0.0167653
$525$		0			0
$526$	10064.6			0.834294
$527$			27722.5i			2.29148i
$528$		0			0
$529$	12166.1			0.999930
$530$	4297.81	−	827.361i	0.352235	−	0.0678081i
$531$		0			0
$532$			645.101i			0.0525727i
$533$			8067.35i			0.655602i
$534$		0			0
$535$	−814.515	−	4231.07i	−0.0658216	−	0.341916i
$536$	8402.57			0.677119
$537$		0			0
$538$			5297.29i			0.424502i
$539$	−4695.17			−0.375204
$540$		0			0
$541$	−520.899			−0.0413959			−0.0206980	−	0.999786i	$-0.506589\pi$
$541$	−520.899			−0.0413959			−0.0206980	+	0.999786i	$0.506589\pi$
$542$		−	15140.1i		−	1.19985i
$543$		0			0
$544$	−3524.02			−0.277741
$545$	2234.20	−	430.101i	0.175601	−	0.0338046i
$546$		0			0
$547$			8098.33i			0.633015i	0.948590	+	0.316508i	$0.102510\pi$
$547$			8098.33i			0.633015i	−0.948590	+	0.316508i	$0.897490\pi$
$548$		−	1728.14i		−	0.134713i
$549$		0			0
$550$	5531.74	+	13835.1i	0.428862	+	1.07260i
$551$	−9412.94			−0.727776
$552$		0			0
$553$		−	4436.00i		−	0.341117i
$554$	16504.0			1.26568
$555$		0			0
$556$	1839.46			0.140307
$557$			6687.22i			0.508701i	0.967112	+	0.254351i	$0.0818617\pi$
$557$			6687.22i			0.508701i	−0.967112	+	0.254351i	$0.918138\pi$
$558$		0			0
$559$	16.5885			0.00125513
$560$	1841.93	+	9568.10i	0.138993	+	0.722011i
$561$		0			0
$562$			8943.95i			0.671313i
$563$			2427.54i			0.181720i	0.995864	+	0.0908602i	$0.0289616\pi$
$563$			2427.54i			0.181720i	−0.995864	+	0.0908602i	$0.971038\pi$
$564$		0			0
$565$	−139.523	−	724.763i	−0.0103890	−	0.0539664i
$566$	13651.7			1.01382
$567$		0			0
$568$		−	7587.30i		−	0.560486i
$569$	2553.65			0.188145			0.0940727	−	0.995565i	$-0.470011\pi$
$569$	2553.65			0.188145			0.0940727	+	0.995565i	$0.470011\pi$
$570$		0			0
$571$	9036.15			0.662261			0.331130	−	0.943585i	$-0.392570\pi$
$571$	9036.15			0.662261			0.331130	+	0.943585i	$0.392570\pi$
$572$		−	1053.95i		−	0.0770421i
$573$		0			0
$574$	−11871.3			−0.863237
$575$	−42.8249	−	107.107i	−0.00310595	−	0.00776812i
$576$		0			0
$577$			5427.02i			0.391560i	0.980648	+	0.195780i	$0.0627238\pi$
$577$			5427.02i			0.391560i	−0.980648	+	0.195780i	$0.937276\pi$
$578$			9825.90i			0.707099i
$579$		0			0
$580$	1759.23	−	338.665i	0.125945	−	0.0242454i
$581$	4394.26			0.313777
$582$		0			0
$583$		−	6520.82i		−	0.463233i
$584$	15065.2			1.06747
$585$		0			0
$586$	11048.8			0.778874
$587$		−	5339.38i		−	0.375434i	−0.982223	−	0.187717i	$-0.939891\pi$
$587$		−	5339.38i		−	0.375434i	0.982223	−	0.187717i	$-0.0601088\pi$
$588$		0			0
$589$	14833.7			1.03771
$590$	1091.54	+	5670.13i	0.0761664	+	0.395654i
$591$		0			0
$592$			3269.61i			0.226993i
$593$			4617.67i			0.319772i	0.987135	+	0.159886i	$0.0511127\pi$
$593$			4617.67i			0.319772i	−0.987135	+	0.159886i	$0.948887\pi$
$594$		0			0
$595$	−15682.1	+	3018.93i	−1.08051	+	0.208007i
$596$	131.626			0.00904629
$597$		0			0
$598$		−	69.1776i		−	0.00473057i
$599$	10888.4			0.742716			0.371358	−	0.928490i	$-0.378892\pi$
$599$	10888.4			0.742716			0.371358	+	0.928490i	$0.378892\pi$
$600$		0			0
$601$	−11273.2			−0.765131			−0.382566	−	0.923928i	$-0.624959\pi$
$601$	−11273.2			−0.765131			−0.382566	+	0.923928i	$0.624959\pi$
$602$			24.4104i			0.00165264i
$603$		0			0
$604$	1362.39			0.0917795
$605$	7186.39	−	1383.44i	0.482923	−	0.0929665i
$606$		0			0
$607$			10224.7i			0.683702i	0.939754	+	0.341851i	$0.111054\pi$
$607$			10224.7i			0.683702i	−0.939754	+	0.341851i	$0.888946\pi$
$608$			1885.63i			0.125777i
$609$		0			0
$610$	3200.88	+	16627.3i	0.212459	+	1.10364i
$611$	16766.4			1.11014
$612$		0			0
$613$			7119.74i			0.469109i	0.972103	+	0.234554i	$0.0753631\pi$
$613$			7119.74i			0.469109i	−0.972103	+	0.234554i	$0.924637\pi$
$614$	−9048.46			−0.594733
$615$		0			0
$616$	16251.0			1.06294
$617$		−	5358.95i		−	0.349665i	−0.984598	−	0.174832i	$-0.944062\pi$
$617$		−	5358.95i		−	0.349665i	0.984598	−	0.174832i	$-0.0559384\pi$
$618$		0			0
$619$	21597.2			1.40237			0.701183	−	0.712981i	$-0.252656\pi$
$619$	21597.2			1.40237			0.701183	+	0.712981i	$0.252656\pi$
$620$	−2772.33	+	533.696i	−0.179580	+	0.0345706i
$621$		0			0
$622$		−	16107.1i		−	1.03832i
$623$			5111.96i			0.328742i
$624$		0			0
$625$	11317.8	−	10772.6i	0.724336	−	0.689447i
$626$	−27266.8			−1.74089
$627$		0			0
$628$			834.333i			0.0530152i
$629$	−5358.89			−0.339703
$630$		0			0
$631$	−1384.23			−0.0873305			−0.0436652	−	0.999046i	$-0.513903\pi$
$631$	−1384.23			−0.0873305			−0.0436652	+	0.999046i	$0.513903\pi$
$632$		−	6808.06i		−	0.428497i
$633$		0			0
$634$	14446.9			0.904986
$635$	−3849.77	−	19998.0i	−0.240588	−	1.24976i
$636$		0			0
$637$			2952.76i			0.183662i
$638$			22630.1i			1.40429i
$639$		0			0
$640$	2489.25	+	12930.6i	0.153744	+	0.798637i
$641$	1623.14			0.100016			0.0500080	−	0.998749i	$-0.484075\pi$
$641$	1623.14			0.100016			0.0500080	+	0.998749i	$0.484075\pi$
$642$		0			0
$643$		−	28170.8i		−	1.72776i	−0.503699	−	0.863879i	$-0.668028\pi$
$643$		−	28170.8i		−	1.72776i	0.503699	−	0.863879i	$-0.331972\pi$
$644$	−12.0067			−0.000734675
$645$		0			0
$646$	12289.9			0.748515
$647$			16695.3i			1.01447i	0.861808	+	0.507235i	$0.169332\pi$
$647$			16695.3i			1.01447i	−0.861808	+	0.507235i	$0.830668\pi$
$648$		0			0
$649$	8602.97			0.520333
$650$	8700.81	−	3478.87i	0.525037	−	0.209927i
$651$		0			0
$652$			2510.24i			0.150780i
$653$			4489.96i			0.269075i	0.990909	+	0.134537i	$0.0429548\pi$
$653$			4489.96i			0.269075i	−0.990909	+	0.134537i	$0.957045\pi$
$654$		0			0
$655$	−2615.78	+	503.558i	−0.156041	+	0.0300391i
$656$	−16275.4			−0.968669
$657$		0			0
$658$			24672.2i			1.46173i
$659$	−15300.9			−0.904459			−0.452229	−	0.891902i	$-0.649371\pi$
$659$	−15300.9			−0.904459			−0.452229	+	0.891902i	$0.649371\pi$
$660$		0			0
$661$	−20634.9			−1.21423			−0.607113	−	0.794615i	$-0.707673\pi$
$661$	−20634.9			−1.21423			−0.607113	+	0.794615i	$0.707673\pi$
$662$			6160.39i			0.361678i
$663$		0			0
$664$	6744.00			0.394154
$665$	1615.36	+	8391.15i	0.0941971	+	0.489316i
$666$		0			0
$667$		−	175.195i		−	0.0101703i
$668$		−	2058.56i		−	0.119234i
$669$		0			0
$670$	10430.7	−	2008.00i	0.601454	−	0.115785i
$671$	25227.6			1.45142
$672$		0			0
$673$			17553.8i			1.00542i	0.864454	+	0.502712i	$0.167664\pi$
$673$			17553.8i			1.00542i	−0.864454	+	0.502712i	$0.832336\pi$
$674$	−25590.5			−1.46248
$675$		0			0
$676$	1191.52			0.0677922
$677$		−	15182.6i		−	0.861915i	−0.902372	−	0.430958i	$-0.858176\pi$
$677$		−	15182.6i		−	0.861915i	0.902372	−	0.430958i	$-0.141824\pi$
$678$		0			0
$679$	−28081.0			−1.58711
$680$	−24067.8	+	4633.25i	−1.35729	+	0.261289i
$681$		0			0
$682$		−	35662.3i		−	2.00232i
$683$			1735.90i			0.0972510i	0.998817	+	0.0486255i	$0.0154841\pi$
$683$			1735.90i			0.0972510i	−0.998817	+	0.0486255i	$0.984516\pi$
$684$		0			0
$685$	−4327.34	−	22478.8i	−0.241371	−	1.25383i
$686$	−18489.3			−1.02905
$687$		0			0
$688$			33.4663i			0.00185449i
$689$	−4100.90			−0.226751
$690$		0			0
$691$	−27648.0			−1.52211			−0.761055	−	0.648687i	$-0.775318\pi$
$691$	−27648.0			−1.52211			−0.761055	+	0.648687i	$0.775318\pi$
$692$		−	2239.72i		−	0.123037i
$693$		0			0
$694$	−218.076			−0.0119280
$695$	23926.8	−	4606.09i	1.30589	−	0.251394i
$696$		0			0
$697$		−	26675.4i		−	1.44964i
$698$			250.311i			0.0135736i
$699$		0			0
$700$	−603.805	−	1510.14i	−0.0326024	−	0.0815401i
$701$	−19116.9			−1.03001			−0.515005	−	0.857187i	$-0.672210\pi$
$701$	−19116.9			−1.03001			−0.515005	+	0.857187i	$0.672210\pi$
$702$		0			0
$703$			2867.42i			0.153836i
$704$	24686.9			1.32162
$705$		0			0
$706$	−17857.2			−0.951931
$707$		−	17687.0i		−	0.940861i
$708$		0			0
$709$	10045.3			0.532099			0.266050	−	0.963959i	$-0.414281\pi$
$709$	10045.3			0.532099			0.266050	+	0.963959i	$0.414281\pi$
$710$	−1813.17	−	9418.68i	−0.0958410	−	0.497855i
$711$		0			0
$712$			7845.47i			0.412952i
$713$			276.086i			0.0145014i
$714$		0			0
$715$	−2639.15	−	13709.3i	−0.138040	−	0.717062i
$716$	−1774.41			−0.0926159
$717$		0			0
$718$			22295.3i			1.15885i
$719$	22559.7			1.17014			0.585072	−	0.810981i	$-0.301066\pi$
$719$	22559.7			1.17014			0.585072	+	0.810981i	$0.301066\pi$
$720$		0			0
$721$	24395.4			1.26010
$722$			11772.2i			0.606809i
$723$		0			0
$724$	1348.23			0.0692079
$725$	22035.1	−	8810.37i	1.12878	−	0.451323i
$726$		0			0
$727$			3668.07i			0.187127i	0.995613	+	0.0935635i	$0.0298258\pi$
$727$			3668.07i			0.187127i	−0.995613	+	0.0935635i	$0.970174\pi$
$728$		−	10220.1i		−	0.520307i
$729$		0			0
$730$	18701.6	−	3600.20i	0.948187	−	0.182533i
$731$	−54.8513			−0.00277530
$732$		0			0
$733$			31428.4i			1.58368i	0.610731	+	0.791838i	$0.290876\pi$
$733$			31428.4i			1.58368i	−0.610731	+	0.791838i	$0.709124\pi$
$734$	−336.419			−0.0169175
$735$		0			0
$736$	−35.0955			−0.00175766
$737$		−	15826.0i		−	0.790987i
$738$		0			0
$739$	−17467.0			−0.869465			−0.434732	−	0.900560i	$-0.643157\pi$
$739$	−17467.0			−0.869465			−0.434732	+	0.900560i	$0.643157\pi$
$740$	−103.166	−	535.906i	−0.00512495	−	0.0266220i
$741$		0			0
$742$		−	6034.57i		−	0.298566i
$743$		−	24993.4i		−	1.23408i	−0.786933	−	0.617038i	$-0.788332\pi$
$743$		−	24993.4i		−	1.23408i	0.786933	−	0.617038i	$-0.211668\pi$
$744$		0			0
$745$	1712.12	−	329.596i	0.0841975	−	0.0162087i
$746$	−3840.74			−0.188498
$747$		0			0
$748$			3484.99i			0.170353i
$749$	−5940.87			−0.289819
$750$		0			0
$751$	−9348.04			−0.454214			−0.227107	−	0.973870i	$-0.572927\pi$
$751$	−9348.04			−0.454214			−0.227107	+	0.973870i	$0.572927\pi$
$752$			33825.2i			1.64026i
$753$		0			0
$754$	14231.9			0.687395
$755$	17721.3	−	3411.48i	0.854230	−	0.164446i
$756$		0			0
$757$		−	11708.4i		−	0.562153i	−0.959685	−	0.281076i	$-0.909309\pi$
$757$		−	11708.4i		−	0.562153i	0.959685	−	0.281076i	$-0.0906915\pi$
$758$		−	5684.61i		−	0.272394i
$759$		0			0
$760$	2479.15	+	12878.2i	0.118326	+	0.614658i
$761$	−19410.7			−0.924621			−0.462310	−	0.886718i	$-0.652979\pi$
$761$	−19410.7			−0.924621			−0.462310	+	0.886718i	$0.652979\pi$
$762$		0			0
$763$		−	3137.05i		−	0.148845i
$764$	2875.98			0.136190
$765$		0			0
$766$	−18888.0			−0.890927
$767$		−	5410.35i		−	0.254702i
$768$		0			0
$769$	9857.66			0.462258			0.231129	−	0.972923i	$-0.425758\pi$
$769$	9857.66			0.462258			0.231129	+	0.972923i	$0.425758\pi$
$770$	20173.6	−	3883.57i	0.944162	−	0.181759i
$771$		0			0
$772$		−	691.373i		−	0.0322319i
$773$		−	27574.5i		−	1.28303i	−0.767108	−	0.641517i	$-0.778305\pi$
$773$		−	27574.5i		−	1.28303i	0.767108	−	0.641517i	$-0.221695\pi$
$774$		0			0
$775$	−34724.7	+	13884.1i	−1.60948	+	0.643525i
$776$	−43096.7			−1.99366
$777$		0			0
$778$			26266.8i			1.21042i
$779$	−14273.4			−0.656479
$780$		0			0
$781$	−14290.4			−0.654741
$782$			228.741i			0.0104601i
$783$		0			0
$784$	−5957.01			−0.271365
$785$	2089.21	+	10852.6i	0.0949899	+	0.493434i
$786$		0			0
$787$			11777.0i			0.533425i	0.963776	+	0.266712i	$0.0859374\pi$
$787$			11777.0i			0.533425i	−0.963776	+	0.266712i	$0.914063\pi$
$788$		−	303.834i		−	0.0137356i
$789$		0			0
$790$	−1626.95	−	8451.35i	−0.0732713	−	0.380614i
$791$	−1017.64			−0.0457436
$792$		0			0
$793$		−	15865.5i		−	0.710466i
$794$	29413.7			1.31468
$795$		0			0
$796$	1129.60			0.0502985
$797$		−	24659.8i		−	1.09598i	−0.836485	−	0.547990i	$-0.815393\pi$
$797$		−	24659.8i		−	1.09598i	0.836485	−	0.547990i	$-0.184607\pi$
$798$		0			0
$799$	−55439.5			−2.45471
$800$	−1764.92	−	4414.14i	−0.0779991	−	0.195079i
$801$		0			0
$802$		−	17473.0i		−	0.769318i
$803$		−	28374.9i		−	1.24698i
$804$		0			0
$805$	−156.177	+	30.0653i	−0.00683791	+	0.00131635i
$806$	−22427.8			−0.980130
$807$		0			0
$808$		−	27144.8i		−	1.18187i
$809$	19263.0			0.837144			0.418572	−	0.908184i	$-0.362531\pi$
$809$	19263.0			0.837144			0.418572	+	0.908184i	$0.362531\pi$
$810$		0			0
$811$	−13597.4			−0.588741			−0.294370	−	0.955691i	$-0.595110\pi$
$811$	−13597.4			−0.588741			−0.294370	+	0.955691i	$0.595110\pi$
$812$		−	2470.14i		−	0.106755i
$813$		0			0
$814$	6893.71			0.296836
$815$	6285.76	+	32652.0i	0.270160	+	1.40337i
$816$		0			0
$817$			29.3497i			0.00125681i
$818$		−	13389.6i		−	0.572318i
$819$		0			0
$820$	2667.62	−	513.538i	0.113607	−	0.0218701i
$821$	−28673.6			−1.21890			−0.609449	−	0.792825i	$-0.708609\pi$
$821$	−28673.6			−1.21890			−0.609449	+	0.792825i	$0.708609\pi$
$822$		0			0
$823$			8437.62i			0.357372i	0.983906	+	0.178686i	$0.0571846\pi$
$823$			8437.62i			0.357372i	−0.983906	+	0.178686i	$0.942815\pi$
$824$	37440.4			1.58289
$825$		0			0
$826$	7961.46			0.335369
$827$		−	11315.7i		−	0.475799i	−0.971290	−	0.237900i	$-0.923541\pi$
$827$		−	11315.7i		−	0.475799i	0.971290	−	0.237900i	$-0.0764590\pi$
$828$		0			0
$829$	−6773.57			−0.283783			−0.141891	−	0.989882i	$-0.545318\pi$
$829$	−6773.57			−0.283783			−0.141891	+	0.989882i	$0.545318\pi$
$830$	8371.83	−	1611.64i	0.350109	−	0.0673987i
$831$		0			0
$832$		−	15525.4i		−	0.646932i
$833$		−	9763.54i		−	0.406106i
$834$		0			0
$835$	−5154.73	−	26776.7i	−0.213637	−	1.10976i
$836$	1864.74			0.0771452
$837$		0			0
$838$			196.602i			0.00810443i
$839$	8913.89			0.366796			0.183398	−	0.983039i	$-0.441290\pi$
$839$	8913.89			0.366796			0.183398	+	0.983039i	$0.441290\pi$
$840$		0			0
$841$	11653.8			0.477831
$842$		−	13457.7i		−	0.550812i
$843$		0			0
$844$	901.621			0.0367714
$845$	15498.6	−	2983.61i	0.630970	−	0.121467i
$846$		0			0
$847$		−	10090.4i		−	0.409341i
$848$		−	8273.31i		−	0.335031i
$849$		0			0
$850$	−28770.0	+	11503.2i	−1.16094	+	0.464184i
$851$	−53.3688			−0.00214978
$852$		0			0
$853$			31669.2i			1.27120i	0.772018	+	0.635600i	$0.219247\pi$
$853$			31669.2i			1.27120i	−0.772018	+	0.635600i	$0.780753\pi$
$854$	23346.4			0.935478
$855$		0			0
$856$	−9117.63			−0.364059
$857$			44054.8i			1.75599i	0.478670	+	0.877995i	$0.341119\pi$
$857$			44054.8i			1.75599i	−0.478670	+	0.877995i	$0.658881\pi$
$858$		0			0
$859$	−19069.4			−0.757438			−0.378719	−	0.925512i	$-0.623635\pi$
$859$	−19069.4			−0.757438			−0.378719	+	0.925512i	$0.623635\pi$
$860$	−1.05596	−	5.48530i	−4.18698e−5	−	0.000217497i
$861$		0			0
$862$		−	5219.82i		−	0.206250i
$863$		−	6256.14i		−	0.246769i	−0.992359	−	0.123384i	$-0.960625\pi$
$863$		−	6256.14i		−	0.246769i	0.992359	−	0.123384i	$-0.0393748\pi$
$864$		0			0
$865$	−5608.35	−	29133.1i	−0.220451	−	1.14515i
$866$	10179.9			0.399456
$867$		0			0
$868$			3892.65i			0.152218i
$869$	−12822.8			−0.500555
$870$		0			0
$871$	−9952.84			−0.387186
$872$		−	4814.52i		−	0.186973i
$873$		0			0
$874$	122.394			0.00473690
$875$	−11635.5	−	18131.2i	−0.449543	−	0.700511i
$876$		0			0
$877$			6957.46i			0.267887i	0.990989	+	0.133943i	$0.0427640\pi$
$877$			6957.46i			0.267887i	−0.990989	+	0.133943i	$0.957236\pi$
$878$			31844.9i			1.22405i
$879$		0			0
$880$	27657.7	−	5324.33i	1.05948	−	0.203958i
$881$	−10626.4			−0.406371			−0.203186	−	0.979140i	$-0.565129\pi$
$881$	−10626.4			−0.406371			−0.203186	+	0.979140i	$0.565129\pi$
$882$		0			0
$883$			31013.5i			1.18198i	0.806679	+	0.590990i	$0.201263\pi$
$883$			31013.5i			1.18198i	−0.806679	+	0.590990i	$0.798737\pi$
$884$	2191.69			0.0833873
$885$		0			0
$886$	−24258.7			−0.919849
$887$			9397.37i			0.355730i	0.984055	+	0.177865i	$0.0569191\pi$
$887$			9397.37i			0.355730i	−0.984055	+	0.177865i	$0.943081\pi$
$888$		0			0
$889$	−28079.3			−1.05933
$890$	1874.87	+	9739.17i	0.0706131	+	0.366806i
$891$		0			0
$892$			1281.94i			0.0481195i
$893$			29664.5i			1.11163i
$894$		0			0
$895$	−23080.7	+	4443.21i	−0.862014	+	0.165944i
$896$	18155.9			0.676950
$897$		0			0
$898$		−	19614.8i		−	0.728904i
$899$	−56799.2			−2.10718
$900$		0			0
$901$	13560.0			0.501385
$902$			34315.3i			1.26671i
$903$		0			0
$904$	−1561.81			−0.0574612
$905$	17537.1	−	3376.03i	0.644146	−	0.124003i
$906$		0			0
$907$			27560.5i			1.00896i	0.863422	+	0.504482i	$0.168316\pi$
$907$			27560.5i			1.00896i	−0.863422	+	0.504482i	$0.831684\pi$
$908$			3574.58i			0.130646i
$909$		0			0
$910$	−2442.35	−	12687.0i	−0.0889705	−	0.462165i
$911$	35870.1			1.30453			0.652266	−	0.757990i	$-0.273819\pi$
$911$	35870.1			1.30453			0.652266	+	0.757990i	$0.273819\pi$
$912$		0			0
$913$		−	12702.1i		−	0.460436i
$914$	3130.97			0.113308
$915$		0			0
$916$	4673.01			0.168560
$917$			3672.82i			0.132265i
$918$		0			0
$919$	−3081.79			−0.110619			−0.0553095	−	0.998469i	$-0.517615\pi$
$919$	−3081.79			−0.110619			−0.0553095	+	0.998469i	$0.517615\pi$
$920$	−239.690	+	46.1422i	−0.00858950	+	0.00165355i
$921$		0			0
$922$		−	39451.2i		−	1.40917i
$923$			8987.16i			0.320494i
$924$		0			0
$925$	−2683.86	−	6712.46i	−0.0953999	−	0.238599i
$926$	−5578.86			−0.197984
$927$		0			0
$928$		−	7220.20i		−	0.255404i
$929$	−4270.03			−0.150802			−0.0754011	−	0.997153i	$-0.524024\pi$
$929$	−4270.03			−0.150802			−0.0754011	+	0.997153i	$0.524024\pi$
$930$		0			0
$931$	−5224.25			−0.183908
$932$			5348.89i			0.187992i
$933$		0			0
$934$	−26600.2			−0.931890
$935$	8726.57	+	45331.0i	0.305229	+	1.58554i
$936$		0			0
$937$			49225.3i			1.71624i	0.513446	+	0.858122i	$0.328369\pi$
$937$			49225.3i			1.71624i	−0.513446	+	0.858122i	$0.671631\pi$
$938$		−	14645.8i		−	0.509812i
$939$		0			0
$940$	−1067.29	−	5544.13i	−0.0370331	−	0.192372i
$941$	−34569.7			−1.19760			−0.598800	−	0.800899i	$-0.704356\pi$
$941$	−34569.7			−1.19760			−0.598800	+	0.800899i	$0.704356\pi$
$942$		0			0
$943$		−	265.658i		−	0.00917393i
$944$	10915.1			0.376329
$945$		0			0
$946$	70.5610			0.00242509
$947$		−	27342.4i		−	0.938234i	−0.883136	−	0.469117i	$-0.844572\pi$
$947$		−	27342.4i		−	0.938234i	0.883136	−	0.469117i	$-0.155428\pi$
$948$		0			0
$949$	−17844.7			−0.610395
$950$	6155.10	+	15394.2i	0.210208	+	0.525739i
$951$		0			0
$952$			33793.7i			1.15048i
$953$			45941.5i			1.56158i	0.624791	+	0.780792i	$0.285184\pi$
$953$			45941.5i			1.56158i	−0.624791	+	0.780792i	$0.714816\pi$
$954$		0			0
$955$	37409.3	−	7201.59i	1.26758	−	0.244019i
$956$	2146.27			0.0726101
$957$		0			0
$958$			7126.90i			0.240355i
$959$	−31562.6			−1.06278
$960$		0			0
$961$	59717.8			2.00456
$962$		−	4335.40i		−	0.145300i
$963$		0			0
$964$	1731.47			0.0578494
$965$	−1731.23	−	8993.03i	−0.0577515	−	0.299996i
$966$		0			0
$967$		−	47338.8i		−	1.57426i	−0.616785	−	0.787132i	$-0.711565\pi$
$967$		−	47338.8i		−	1.57426i	0.616785	−	0.787132i	$-0.288435\pi$
$968$		−	15486.1i		−	0.514197i
$969$		0			0
$970$	−53499.2	+	10299.0i	−1.77088	+	0.340909i
$971$	−50168.3			−1.65806			−0.829031	−	0.559202i	$-0.811107\pi$
$971$	−50168.3			−1.65806			−0.829031	+	0.559202i	$0.811107\pi$
$972$		0			0
$973$		−	33595.7i		−	1.10691i
$974$	−8216.46			−0.270300
$975$		0			0
$976$	32007.6			1.04973
$977$		−	47642.1i		−	1.56009i	−0.625725	−	0.780044i	$-0.715197\pi$
$977$		−	47642.1i		−	1.56009i	0.625725	−	0.780044i	$-0.284803\pi$
$978$		0			0
$979$	14776.7			0.482396
$980$	976.385	−	187.962i	0.0318260	−	0.00612675i
$981$		0			0
$982$			43040.7i			1.39866i
$983$		−	11158.1i		−	0.362044i	−0.983479	−	0.181022i	$-0.942060\pi$
$983$		−	11158.1i		−	0.362044i	0.983479	−	0.181022i	$-0.0579405\pi$
$984$		0			0
$985$	−760.814	−	3952.12i	−0.0246107	−	0.127843i
$986$	−47059.0			−1.51994
$987$		0			0
$988$		−	1172.72i		−	0.0377624i
$989$	−0.546260			−1.75632e−5
$990$		0			0
$991$	−57670.7			−1.84861			−0.924304	−	0.381658i	$-0.875353\pi$
$991$	−57670.7			−1.84861			−0.924304	+	0.381658i	$0.875353\pi$
$992$			11378.2i			0.364171i
$993$		0			0
$994$	−13224.8			−0.421998
$995$	14693.3	−	2828.57i	0.468149	−	0.0901223i
$996$		0			0
$997$		−	51730.8i		−	1.64326i	−0.570021	−	0.821630i	$-0.693065\pi$
$997$		−	51730.8i		−	1.64326i	0.570021	−	0.821630i	$-0.306935\pi$
$998$			32262.8i			1.02331i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.4.b.f.244.5		16
3.2	odd	2		405.4.b.e.244.12		16
5.2	odd	4		2025.4.a.bl.1.12		16
5.3	odd	4		2025.4.a.bl.1.5		16
5.4	even	2	inner	405.4.b.f.244.12		16
9.2	odd	6		45.4.j.a.4.12	yes	32
9.4	even	3		135.4.j.a.19.12		32
9.5	odd	6		45.4.j.a.34.5	yes	32
9.7	even	3		135.4.j.a.64.5		32
15.2	even	4		2025.4.a.bk.1.5		16
15.8	even	4		2025.4.a.bk.1.12		16
15.14	odd	2		405.4.b.e.244.5		16
45.2	even	12		225.4.e.g.76.12		32
45.4	even	6		135.4.j.a.19.5		32
45.14	odd	6		45.4.j.a.34.12	yes	32
45.23	even	12		225.4.e.g.151.5		32
45.29	odd	6		45.4.j.a.4.5	✓	32
45.32	even	12		225.4.e.g.151.12		32
45.34	even	6		135.4.j.a.64.12		32
45.38	even	12		225.4.e.g.76.5		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.j.a.4.5	✓	32	45.29	odd	6
45.4.j.a.4.12	yes	32	9.2	odd	6
45.4.j.a.34.5	yes	32	9.5	odd	6
45.4.j.a.34.12	yes	32	45.14	odd	6
135.4.j.a.19.5		32	45.4	even	6
135.4.j.a.19.12		32	9.4	even	3
135.4.j.a.64.5		32	9.7	even	3
135.4.j.a.64.12		32	45.34	even	6
225.4.e.g.76.5		32	45.38	even	12
225.4.e.g.76.12		32	45.2	even	12
225.4.e.g.151.5		32	45.23	even	12
225.4.e.g.151.12		32	45.32	even	12
405.4.b.e.244.5		16	15.14	odd	2
405.4.b.e.244.12		16	3.2	odd	2
405.4.b.f.244.5		16	1.1	even	1	trivial
405.4.b.f.244.12		16	5.4	even	2	inner
2025.4.a.bk.1.5		16	15.2	even	4
2025.4.a.bk.1.12		16	15.8	even	4
2025.4.a.bl.1.5		16	5.3	odd	4
2025.4.a.bl.1.12		16	5.2	odd	4

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:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	405.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.67506i q^{2} +0.844033 q^{4} +(10.9788 - 2.11350i) q^{5} -15.4153i q^{7} -23.6584i q^{8} +O(q^{10})\)	\(q-2.67506i q^{2} +0.844033 q^{4} +(10.9788 - 2.11350i) q^{5} -15.4153i q^{7} -23.6584i q^{8} +(-5.65374 - 29.3689i) q^{10} -44.5598 q^{11} +28.0233i q^{13} -41.2370 q^{14} -56.5353 q^{16} -92.6615i q^{17} -49.5811 q^{19} +(9.26644 - 1.78386i) q^{20} +119.200i q^{22} -0.922809i q^{23} +(116.066 - 46.4071i) q^{25} +74.9642 q^{26} -13.0110i q^{28} +189.849 q^{29} -299.180 q^{31} -38.0312i q^{32} -247.875 q^{34} +(-32.5802 - 169.241i) q^{35} -57.8330i q^{37} +132.633i q^{38} +(-50.0019 - 259.739i) q^{40} +287.880 q^{41} -0.591953i q^{43} -37.6099 q^{44} -2.46857 q^{46} -598.302i q^{47} +105.368 q^{49} +(-124.142 - 310.485i) q^{50} +23.6526i q^{52} +146.339i q^{53} +(-489.211 + 94.1769i) q^{55} -364.701 q^{56} -507.859i q^{58} -193.066 q^{59} -566.153 q^{61} +800.326i q^{62} -554.019 q^{64} +(59.2272 + 307.661i) q^{65} +355.163i q^{67} -78.2093i q^{68} +(-452.731 + 87.1542i) q^{70} +320.703 q^{71} +636.782i q^{73} -154.707 q^{74} -41.8481 q^{76} +686.903i q^{77} +287.765 q^{79} +(-620.688 + 119.487i) q^{80} -770.097i q^{82} +285.058i q^{83} +(-195.840 - 1017.31i) q^{85} -1.58351 q^{86} +1054.21i q^{88} -331.615 q^{89} +431.988 q^{91} -0.778881i q^{92} -1600.50 q^{94} +(-544.339 + 104.789i) q^{95} -1821.63i q^{97} -281.866i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 54 q^{4} + 3 q^{5}+O(q^{10}) \) 16 * q - 54 * q^4 + 3 * q^5	\( 16 q - 54 q^{4} + 3 q^{5} - 10 q^{10} + 90 q^{11} - 102 q^{14} + 146 q^{16} - 4 q^{19} - 6 q^{20} - 71 q^{25} + 468 q^{26} - 516 q^{29} + 38 q^{31} - 212 q^{34} + 267 q^{35} - 44 q^{40} + 576 q^{41} - 1644 q^{44} - 290 q^{46} + 4 q^{49} + 558 q^{50} + 15 q^{55} + 2430 q^{56} - 2202 q^{59} + 20 q^{61} + 322 q^{64} + 339 q^{65} - 636 q^{70} + 2952 q^{71} - 4080 q^{74} - 396 q^{76} + 218 q^{79} - 1266 q^{80} + 704 q^{85} + 6108 q^{86} - 4074 q^{89} - 942 q^{91} + 1078 q^{94} - 1692 q^{95}+O(q^{100}) \) 16 * q - 54 * q^4 + 3 * q^5 - 10 * q^10 + 90 * q^11 - 102 * q^14 + 146 * q^16 - 4 * q^19 - 6 * q^20 - 71 * q^25 + 468 * q^26 - 516 * q^29 + 38 * q^31 - 212 * q^34 + 267 * q^35 - 44 * q^40 + 576 * q^41 - 1644 * q^44 - 290 * q^46 + 4 * q^49 + 558 * q^50 + 15 * q^55 + 2430 * q^56 - 2202 * q^59 + 20 * q^61 + 322 * q^64 + 339 * q^65 - 636 * q^70 + 2952 * q^71 - 4080 * q^74 - 396 * q^76 + 218 * q^79 - 1266 * q^80 + 704 * q^85 + 6108 * q^86 - 4074 * q^89 - 942 * q^91 + 1078 * q^94 - 1692 * q^95

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