LMFDB - Embedded newform 405.4.b.f.244.6

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			244.6
Root			$-2.46385i$ of defining polynomial
Character	$\chi$	$=$	405.244
Dual form			405.4.b.f.244.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.46385i q^{2} +1.92944 q^{4} +(0.0773702 + 11.1801i) q^{5} +19.2401i q^{7} -24.4647i q^{8} +O(q^{10})$	\(q-2.46385i q^{2} +1.92944 q^{4} +(0.0773702 + 11.1801i) q^{5} +19.2401i q^{7} -24.4647i q^{8} +(27.5460 - 0.190629i) q^{10} -39.8548 q^{11} +1.00910i q^{13} +47.4048 q^{14} -44.8417 q^{16} +52.6170i q^{17} +49.5371 q^{19} +(0.149282 + 21.5713i) q^{20} +98.1964i q^{22} -27.4019i q^{23} +(-124.988 + 1.73001i) q^{25} +2.48626 q^{26} +37.1228i q^{28} -254.953 q^{29} -168.656 q^{31} -85.2341i q^{32} +129.640 q^{34} +(-215.106 + 1.48861i) q^{35} +419.938i q^{37} -122.052i q^{38} +(273.517 - 1.89284i) q^{40} -398.570 q^{41} +358.828i q^{43} -76.8977 q^{44} -67.5141 q^{46} +141.302i q^{47} -27.1829 q^{49} +(4.26248 + 307.952i) q^{50} +1.94699i q^{52} -290.878i q^{53} +(-3.08358 - 445.580i) q^{55} +470.703 q^{56} +628.166i q^{58} +28.7319 q^{59} +732.727 q^{61} +415.544i q^{62} -568.737 q^{64} +(-11.2818 + 0.0780740i) q^{65} +176.546i q^{67} +101.522i q^{68} +(3.66772 + 529.989i) q^{70} +802.814 q^{71} +512.820i q^{73} +1034.66 q^{74} +95.5791 q^{76} -766.813i q^{77} -612.348 q^{79} +(-3.46941 - 501.333i) q^{80} +982.017i q^{82} -80.8162i q^{83} +(-588.262 + 4.07099i) q^{85} +884.098 q^{86} +975.035i q^{88} +24.0097 q^{89} -19.4151 q^{91} -52.8704i q^{92} +348.147 q^{94} +(3.83270 + 553.828i) q^{95} -1367.92i q^{97} +66.9747i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	2.46385i		−	0.871102i	−0.900164	−	0.435551i	$-0.856553\pi$
$2$		−	2.46385i		−	0.871102i	0.900164	−	0.435551i	$-0.143447\pi$
$3$		0			0
$3$		0			0
$4$	1.92944			0.241181
$5$	0.0773702	+	11.1801i	0.00692020	+	0.999976i
$5$	0.0773702	+	11.1801i	0.00692020	+	0.999976i
$6$		0			0
$7$			19.2401i			1.03887i	0.854510	+	0.519435i	$0.173858\pi$
$7$			19.2401i			1.03887i	−0.854510	+	0.519435i	$0.826142\pi$
$8$		−	24.4647i		−	1.08120i
$9$		0			0
$10$	27.5460	−	0.190629i	0.871082	−	0.00602821i
$11$	−39.8548			−1.09243			−0.546213	−	0.837646i	$-0.683931\pi$
$11$	−39.8548			−1.09243			−0.546213	+	0.837646i	$0.683931\pi$
$12$		0			0
$13$			1.00910i			0.0215287i	0.999942	+	0.0107643i	$0.00342646\pi$
$13$			1.00910i			0.0215287i	−0.999942	+	0.0107643i	$0.996574\pi$
$14$	47.4048			0.904962
$15$		0			0
$16$	−44.8417			−0.700651
$17$			52.6170i			0.750676i	0.926888	+	0.375338i	$0.122473\pi$
$17$			52.6170i			0.750676i	−0.926888	+	0.375338i	$0.877527\pi$
$18$		0			0
$19$	49.5371			0.598136			0.299068	−	0.954232i	$-0.403324\pi$
$19$	49.5371			0.598136			0.299068	+	0.954232i	$0.403324\pi$
$20$	0.149282	+	21.5713i	0.00166902	+	0.241175i
$21$		0			0
$22$			98.1964i			0.951615i
$23$		−	27.4019i		−	0.248421i	−0.992256	−	0.124210i	$-0.960360\pi$
$23$		−	27.4019i		−	0.248421i	0.992256	−	0.124210i	$-0.0396398\pi$
$24$		0			0
$25$	−124.988	+	1.73001i	−0.999904	+	0.0138401i
$26$	2.48626			0.0187537
$27$		0			0
$28$			37.1228i			0.250555i
$29$	−254.953			−1.63254			−0.816269	−	0.577672i	$-0.803961\pi$
$29$	−254.953			−1.63254			−0.816269	+	0.577672i	$0.803961\pi$
$30$		0			0
$31$	−168.656			−0.977147			−0.488574	−	0.872523i	$-0.662483\pi$
$31$	−168.656			−0.977147			−0.488574	+	0.872523i	$0.662483\pi$
$32$		−	85.2341i		−	0.470856i
$33$		0			0
$34$	129.640			0.653916
$35$	−215.106	+	1.48861i	−1.03884	+	0.00718919i
$36$		0			0
$37$			419.938i			1.86588i	0.360038	+	0.932938i	$0.382764\pi$
$37$			419.938i			1.86588i	−0.360038	+	0.932938i	$0.617236\pi$
$38$		−	122.052i		−	0.521038i
$39$		0			0
$40$	273.517	−	1.89284i	1.08117	−	0.00748209i
$41$	−398.570			−1.51820			−0.759100	−	0.650974i	$-0.774361\pi$
$41$	−398.570			−1.51820			−0.759100	+	0.650974i	$0.774361\pi$
$42$		0			0
$43$			358.828i			1.27258i	0.771452	+	0.636288i	$0.219531\pi$
$43$			358.828i			1.27258i	−0.771452	+	0.636288i	$0.780469\pi$
$44$	−76.8977			−0.263472
$45$		0			0
$46$	−67.5141			−0.216400
$47$			141.302i			0.438532i	0.975665	+	0.219266i	$0.0703663\pi$
$47$			141.302i			0.438532i	−0.975665	+	0.219266i	$0.929634\pi$
$48$		0			0
$49$	−27.1829			−0.0792505
$50$	4.26248	+	307.952i	0.0120561	+	0.871019i
$51$		0			0
$52$			1.94699i			0.00519230i
$53$		−	290.878i		−	0.753872i	−0.926239	−	0.376936i	$-0.876978\pi$
$53$		−	290.878i		−	0.753872i	0.926239	−	0.376936i	$-0.123022\pi$
$54$		0			0
$55$	−3.08358	−	445.580i	−0.00755981	−	1.09240i
$56$	470.703			1.12322
$57$		0			0
$58$			628.166i			1.42211i
$59$	28.7319			0.0633995			0.0316998	−	0.999497i	$-0.489908\pi$
$59$	28.7319			0.0633995			0.0316998	+	0.999497i	$0.489908\pi$
$60$		0			0
$61$	732.727			1.53797			0.768985	−	0.639267i	$-0.220762\pi$
$61$	732.727			1.53797			0.768985	+	0.639267i	$0.220762\pi$
$62$			415.544i			0.851195i
$63$		0			0
$64$	−568.737			−1.11082
$65$	−11.2818	+	0.0780740i	−0.0215282	+	0.000148983i
$66$		0			0
$67$			176.546i			0.321918i	0.986961	+	0.160959i	$0.0514587\pi$
$67$			176.546i			0.321918i	−0.986961	+	0.160959i	$0.948541\pi$
$68$			101.522i			0.181049i
$69$		0			0
$70$	3.66772	+	529.989i	0.00626252	+	0.904940i
$71$	802.814			1.34192			0.670961	−	0.741493i	$-0.265882\pi$
$71$	802.814			1.34192			0.670961	+	0.741493i	$0.265882\pi$
$72$		0			0
$73$			512.820i			0.822205i	0.911589	+	0.411103i	$0.134856\pi$
$73$			512.820i			0.822205i	−0.911589	+	0.411103i	$0.865144\pi$
$74$	1034.66			1.62537
$75$		0			0
$76$	95.5791			0.144259
$77$		−	766.813i		−	1.13489i
$78$		0			0
$79$	−612.348			−0.872082			−0.436041	−	0.899927i	$-0.643620\pi$
$79$	−612.348			−0.872082			−0.436041	+	0.899927i	$0.643620\pi$
$80$	−3.46941	−	501.333i	−0.00484865	−	0.700635i
$81$		0			0
$82$			982.017i			1.32251i
$83$		−	80.8162i		−	0.106876i	−0.998571	−	0.0534382i	$-0.982982\pi$
$83$		−	80.8162i		−	0.106876i	0.998571	−	0.0534382i	$-0.0170180\pi$
$84$		0			0
$85$	−588.262	+	4.07099i	−0.750658	+	0.00519483i
$86$	884.098			1.10854
$87$		0			0
$88$			975.035i			1.18113i
$89$	24.0097			0.0285957			0.0142979	−	0.999898i	$-0.495449\pi$
$89$	24.0097			0.0285957			0.0142979	+	0.999898i	$0.495449\pi$
$90$		0			0
$91$	−19.4151			−0.0223655
$92$		−	52.8704i		−	0.0599143i
$93$		0			0
$94$	348.147			0.382006
$95$	3.83270	+	553.828i	0.00413922	+	0.598122i
$96$		0			0
$97$		−	1367.92i		−	1.43187i	−0.698168	−	0.715934i	$-0.746001\pi$
$97$		−	1367.92i		−	1.43187i	0.698168	−	0.715934i	$-0.253999\pi$
$98$			66.9747i			0.0690353i
$99$		0			0
$100$	−241.157	+	3.33796i	−0.241157	+	0.00333796i
$101$	−568.413			−0.559992			−0.279996	−	0.960001i	$-0.590333\pi$
$101$	−568.413			−0.559992			−0.279996	+	0.960001i	$0.590333\pi$
$102$		0			0
$103$			475.159i			0.454551i	0.973830	+	0.227276i	$0.0729818\pi$
$103$			475.159i			0.454551i	−0.973830	+	0.227276i	$0.927018\pi$
$104$	24.6872			0.0232767
$105$		0			0
$106$	−716.680			−0.656700
$107$			1452.60i			1.31242i	0.754580	+	0.656208i	$0.227841\pi$
$107$			1452.60i			1.31242i	−0.754580	+	0.656208i	$0.772159\pi$
$108$		0			0
$109$	1962.92			1.72490			0.862448	−	0.506146i	$-0.168930\pi$
$109$	1962.92			1.72490			0.862448	+	0.506146i	$0.168930\pi$
$110$	−1097.84	+	7.59747i	−0.951592	+	0.00658537i
$111$		0			0
$112$		−	862.760i		−	0.727886i
$113$			978.286i			0.814419i	0.913335	+	0.407210i	$0.133498\pi$
$113$			978.286i			0.814419i	−0.913335	+	0.407210i	$0.866502\pi$
$114$		0			0
$115$	306.355	−	2.12009i	0.248415	−	0.00171912i
$116$	−491.918			−0.393736
$117$		0			0
$118$		−	70.7910i		−	0.0552275i
$119$	−1012.36			−0.779855
$120$		0			0
$121$	257.409			0.193395
$122$		−	1805.33i		−	1.33973i
$123$		0			0
$124$	−325.413			−0.235669
$125$	−29.0120	−	1397.24i	−0.0207593	−	0.999785i
$126$		0			0
$127$		−	865.941i		−	0.605038i	−0.953143	−	0.302519i	$-0.902172\pi$
$127$		−	865.941i		−	0.605038i	0.953143	−	0.302519i	$-0.0978276\pi$
$128$			719.411i			0.496778i
$129$		0			0
$130$	0.192363	+	27.7966i	0.000129779	+	0.0187532i
$131$	1468.96			0.979720			0.489860	−	0.871801i	$-0.337048\pi$
$131$	1468.96			0.979720			0.489860	+	0.871801i	$0.337048\pi$
$132$		0			0
$133$			953.101i			0.621386i
$134$	434.983			0.280424
$135$		0			0
$136$	1287.26			0.811628
$137$		−	71.5353i		−	0.0446108i	−0.999751	−	0.0223054i	$-0.992899\pi$
$137$		−	71.5353i		−	0.0446108i	0.999751	−	0.0223054i	$-0.00710061\pi$
$138$		0			0
$139$	−229.065			−0.139777			−0.0698886	−	0.997555i	$-0.522264\pi$
$139$	−229.065			−0.139777			−0.0698886	+	0.997555i	$0.522264\pi$
$140$	−415.035	+	2.87220i	−0.250549	+	0.00173389i
$141$		0			0
$142$		−	1978.01i		−	1.16895i
$143$		−	40.2174i		−	0.0235185i
$144$		0			0
$145$	−19.7258	−	2850.39i	−0.0112975	−	1.63250i
$146$	1263.51			0.716225
$147$		0			0
$148$			810.247i			0.450013i
$149$	−835.144			−0.459179			−0.229589	−	0.973288i	$-0.573738\pi$
$149$	−835.144			−0.459179			−0.229589	+	0.973288i	$0.573738\pi$
$150$		0			0
$151$	−1866.58			−1.00596			−0.502982	−	0.864297i	$-0.667764\pi$
$151$	−1866.58			−1.00596			−0.502982	+	0.864297i	$0.667764\pi$
$152$		−	1211.91i		−	0.646702i
$153$		0			0
$154$	−1889.31			−0.988604
$155$	−13.0490	−	1885.59i	−0.00676206	−	0.977124i
$156$		0			0
$157$		−	2299.91i		−	1.16913i	−0.811348	−	0.584563i	$-0.801266\pi$
$157$		−	2299.91i		−	1.16913i	0.811348	−	0.584563i	$-0.198734\pi$
$158$			1508.73i			0.759673i
$159$		0			0
$160$	952.923	−	6.59458i	0.470845	−	0.00325842i
$161$	527.215			0.258077
$162$		0			0
$163$		−	44.9023i		−	0.0215768i	−0.999942	−	0.0107884i	$-0.996566\pi$
$163$		−	44.9023i		−	0.0215768i	0.999942	−	0.0107884i	$-0.00343412\pi$
$164$	−769.019			−0.366160
$165$		0			0
$166$	−199.119			−0.0931002
$167$			1176.56i			0.545180i	0.962130	+	0.272590i	$0.0878803\pi$
$167$			1176.56i			0.545180i	−0.962130	+	0.272590i	$0.912120\pi$
$168$		0			0
$169$	2195.98			0.999537
$170$	10.0303	+	1449.39i	0.00452523	+	0.653900i
$171$		0			0
$172$			692.339i			0.306921i
$173$			1068.09i			0.469396i	0.972068	+	0.234698i	$0.0754100\pi$
$173$			1068.09i			0.469396i	−0.972068	+	0.234698i	$0.924590\pi$
$174$		0			0
$175$	−33.2856	−	2404.79i	−0.0143780	−	1.03877i
$176$	1787.16			0.765410
$177$		0			0
$178$		−	59.1562i		−	0.0249098i
$179$	−2779.66			−1.16068			−0.580339	−	0.814375i	$-0.697080\pi$
$179$	−2779.66			−1.16068			−0.580339	+	0.814375i	$0.697080\pi$
$180$		0			0
$181$	−3568.66			−1.46550			−0.732752	−	0.680496i	$-0.761764\pi$
$181$	−3568.66			−1.46550			−0.732752	+	0.680496i	$0.761764\pi$
$182$			47.8360i			0.0194826i
$183$		0			0
$184$	−670.377			−0.268592
$185$	−4694.94	+	32.4907i	−1.86583	+	0.0129122i
$186$		0			0
$187$		−	2097.04i		−	0.820059i
$188$			272.634i			0.105765i
$189$		0			0
$190$	1364.55	−	9.44319i	0.521026	−	0.00360569i
$191$	3338.62			1.26478			0.632392	−	0.774648i	$-0.282073\pi$
$191$	3338.62			1.26478			0.632392	+	0.774648i	$0.282073\pi$
$192$		0			0
$193$		−	1940.87i		−	0.723868i	−0.932204	−	0.361934i	$-0.882117\pi$
$193$		−	1940.87i		−	0.723868i	0.932204	−	0.361934i	$-0.117883\pi$
$194$	−3370.35			−1.24730
$195$		0			0
$196$	−52.4480			−0.0191137
$197$			1035.75i			0.374588i	0.982304	+	0.187294i	$0.0599717\pi$
$197$			1035.75i			0.374588i	−0.982304	+	0.187294i	$0.940028\pi$
$198$		0			0
$199$	−1565.53			−0.557675			−0.278838	−	0.960338i	$-0.589949\pi$
$199$	−1565.53			−0.557675			−0.278838	+	0.960338i	$0.589949\pi$
$200$	42.3241	+	3057.79i	0.0149638	+	1.08109i
$201$		0			0
$202$			1400.48i			0.487810i
$203$		−	4905.33i		−	1.69599i
$204$		0			0
$205$	−30.8375	−	4456.04i	−0.0105063	−	1.51816i
$206$	1170.72			0.395961
$207$		0			0
$208$		−	45.2496i		−	0.0150841i
$209$	−1974.29			−0.653420
$210$		0			0
$211$	3385.27			1.10451			0.552255	−	0.833675i	$-0.313767\pi$
$211$	3385.27			1.10451			0.552255	+	0.833675i	$0.313767\pi$
$212$		−	561.234i		−	0.181819i
$213$		0			0
$214$	3579.00			1.14325
$215$	−4011.72	+	27.7626i	−1.27255	+	0.00880648i
$216$		0			0
$217$		−	3244.97i		−	1.01513i
$218$		−	4836.34i		−	1.50256i
$219$		0			0
$220$	−5.94959	−	859.722i	−0.00182328	−	0.263466i
$221$	−53.0956			−0.0161611
$222$		0			0
$223$		−	2365.54i		−	0.710352i	−0.934799	−	0.355176i	$-0.884421\pi$
$223$		−	2365.54i		−	0.710352i	0.934799	−	0.355176i	$-0.115579\pi$
$224$	1639.92			0.489158
$225$		0			0
$226$	2410.35			0.709443
$227$			976.230i			0.285439i	0.989763	+	0.142720i	$0.0455847\pi$
$227$			976.230i			0.285439i	−0.989763	+	0.142720i	$0.954415\pi$
$228$		0			0
$229$	−1595.28			−0.460345			−0.230173	−	0.973150i	$-0.573929\pi$
$229$	−1595.28			−0.460345			−0.230173	+	0.973150i	$0.573929\pi$
$230$	−5.22358	−	754.812i	−0.00149753	−	0.216395i
$231$		0			0
$232$			6237.34i			1.76509i
$233$		−	943.571i		−	0.265302i	−0.991163	−	0.132651i	$-0.957651\pi$
$233$		−	943.571i		−	0.265302i	0.991163	−	0.132651i	$-0.0423490\pi$
$234$		0			0
$235$	−1579.77	+	10.9326i	−0.438522	+	0.00303473i
$236$	55.4366			0.0152907
$237$		0			0
$238$			2494.30i			0.679334i
$239$	2731.50			0.739271			0.369635	−	0.929177i	$-0.379483\pi$
$239$	2731.50			0.739271			0.369635	+	0.929177i	$0.379483\pi$
$240$		0			0
$241$	3417.23			0.913375			0.456687	−	0.889627i	$-0.349036\pi$
$241$	3417.23			0.913375			0.456687	+	0.889627i	$0.349036\pi$
$242$		−	634.217i		−	0.168467i
$243$		0			0
$244$	1413.76			0.370928
$245$	−2.10315	−	303.907i	−0.000548430	−	0.0792486i
$246$		0			0
$247$			49.9877i			0.0128771i
$248$			4126.12i			1.05649i
$249$		0			0
$250$	−3442.59	+	71.4812i	−0.870915	+	0.0180835i
$251$	3164.50			0.795782			0.397891	−	0.917433i	$-0.369742\pi$
$251$	3164.50			0.795782			0.397891	+	0.917433i	$0.369742\pi$
$252$		0			0
$253$			1092.10i			0.271382i
$254$	−2133.55			−0.527050
$255$		0			0
$256$	−2777.38			−0.678071
$257$			5549.43i			1.34694i	0.739214	+	0.673471i	$0.235197\pi$
$257$			5549.43i			1.34694i	−0.739214	+	0.673471i	$0.764803\pi$
$258$		0			0
$259$	−8079.67			−1.93840
$260$	−21.7675	+	0.150639i	−0.00519218	+	3.59318e-5i
$261$		0			0
$262$		−	3619.29i		−	0.853436i
$263$			4137.64i			0.970106i	0.874485	+	0.485053i	$0.161200\pi$
$263$			4137.64i			0.970106i	−0.874485	+	0.485053i	$0.838800\pi$
$264$		0			0
$265$	3252.04	−	22.5053i	0.753854	−	0.00521695i
$266$	2348.30			0.541291
$267$		0			0
$268$			340.636i			0.0776404i
$269$	−1026.74			−0.232719			−0.116359	−	0.993207i	$-0.537122\pi$
$269$	−1026.74			−0.232719			−0.116359	+	0.993207i	$0.537122\pi$
$270$		0			0
$271$	3400.66			0.762270			0.381135	−	0.924519i	$-0.375533\pi$
$271$	3400.66			0.762270			0.381135	+	0.924519i	$0.375533\pi$
$272$		−	2359.44i		−	0.525962i
$273$		0			0
$274$	−176.252			−0.0388605
$275$	4981.38	−	68.9493i	1.09232	−	0.0151193i
$276$		0			0
$277$			3568.81i			0.774112i	0.922056	+	0.387056i	$0.126508\pi$
$277$			3568.81i			0.774112i	−0.922056	+	0.387056i	$0.873492\pi$
$278$			564.381i			0.121760i
$279$		0			0
$280$	36.4184	+	5262.50i	0.00777292	+	1.12319i
$281$	6018.68			1.27774			0.638869	−	0.769316i	$-0.279403\pi$
$281$	6018.68			1.27774			0.638869	+	0.769316i	$0.279403\pi$
$282$		0			0
$283$		−	4578.17i		−	0.961638i	−0.876820	−	0.480819i	$-0.840339\pi$
$283$		−	4578.17i		−	0.961638i	0.876820	−	0.480819i	$-0.159661\pi$
$284$	1548.98			0.323646
$285$		0			0
$286$	−99.0895			−0.0204870
$287$		−	7668.55i		−	1.57721i
$288$		0			0
$289$	2144.45			0.436485
$290$	−7022.94	+	48.6013i	−1.42207	+	0.00984127i
$291$		0			0
$292$			989.457i			0.198300i
$293$		−	6479.73i		−	1.29198i	−0.763346	−	0.645990i	$-0.776445\pi$
$293$		−	6479.73i		−	1.29198i	0.763346	−	0.645990i	$-0.223555\pi$
$294$		0			0
$295$	2.22299	+	321.224i	0.000438737	+	0.0633980i
$296$	10273.6			2.01738
$297$		0			0
$298$			2057.67i			0.399992i
$299$	27.6511			0.00534818
$300$		0			0
$301$	−6903.90			−1.32204
$302$			4598.98i			0.876297i
$303$		0			0
$304$	−2221.33			−0.419085
$305$	56.6913	+	8191.95i	0.0106431	+	1.53793i
$306$		0			0
$307$		−	708.452i		−	0.131705i	−0.997829	−	0.0658526i	$-0.979023\pi$
$307$		−	708.452i		−	0.131705i	0.997829	−	0.0658526i	$-0.0209767\pi$
$308$		−	1479.52i		−	0.273713i
$309$		0			0
$310$	−4645.81	+	32.1507i	−0.851175	+	0.00589044i
$311$	6234.64			1.13676			0.568382	−	0.822765i	$-0.307570\pi$
$311$	6234.64			1.13676			0.568382	+	0.822765i	$0.307570\pi$
$312$		0			0
$313$		−	2631.41i		−	0.475196i	−0.971364	−	0.237598i	$-0.923640\pi$
$313$		−	2631.41i		−	0.475196i	0.971364	−	0.237598i	$-0.0763600\pi$
$314$	−5666.63			−1.01843
$315$		0			0
$316$	−1181.49			−0.210329
$317$		−	502.534i		−	0.0890383i	−0.999009	−	0.0445191i	$-0.985824\pi$
$317$		−	502.534i		−	0.0890383i	0.999009	−	0.0445191i	$-0.0141756\pi$
$318$		0			0
$319$	10161.1			1.78343
$320$	−44.0033	−	6358.53i	−0.00768707	−	1.11079i
$321$		0			0
$322$		−	1298.98i		−	0.224812i
$323$			2606.49i			0.449007i
$324$		0			0
$325$	−1.74575	−	126.125i	−0.000297959	−	0.0215266i
$326$	−110.632			−0.0187956
$327$		0			0
$328$			9750.89i			1.64147i
$329$	−2718.67			−0.455578
$330$		0			0
$331$	3087.14			0.512642			0.256321	−	0.966592i	$-0.417490\pi$
$331$	3087.14			0.512642			0.256321	+	0.966592i	$0.417490\pi$
$332$		−	155.930i		−	0.0257765i
$333$		0			0
$334$	2898.87			0.474908
$335$	−1973.80	+	13.6594i	−0.321911	+	0.00222774i
$336$		0			0
$337$			5985.80i			0.967559i	0.875190	+	0.483780i	$0.160736\pi$
$337$			5985.80i			0.967559i	−0.875190	+	0.483780i	$0.839264\pi$
$338$		−	5410.57i		−	0.870699i
$339$		0			0
$340$	−1135.02	+	7.85475i	−0.181044	+	0.00125289i
$341$	6721.77			1.06746
$342$		0			0
$343$			6076.36i			0.956539i
$344$	8778.60			1.37590
$345$		0			0
$346$	2631.61			0.408892
$347$			1006.65i			0.155734i	0.996964	+	0.0778671i	$0.0248110\pi$
$347$			1006.65i			0.155734i	−0.996964	+	0.0778671i	$0.975189\pi$
$348$		0			0
$349$	1949.36			0.298987			0.149494	−	0.988763i	$-0.452236\pi$
$349$	1949.36			0.298987			0.149494	+	0.988763i	$0.452236\pi$
$350$	−5925.03	+	82.0108i	−0.904875	+	0.0125247i
$351$		0			0
$352$			3396.99i			0.514376i
$353$		−	2591.73i		−	0.390776i	−0.980726	−	0.195388i	$-0.937403\pi$
$353$		−	2591.73i		−	0.390776i	0.980726	−	0.195388i	$-0.0625966\pi$
$354$		0			0
$355$	62.1139	+	8975.52i	0.00928637	+	1.34189i
$356$	46.3253			0.00689674
$357$		0			0
$358$			6848.65i			1.01107i
$359$	−605.954			−0.0890837			−0.0445418	−	0.999008i	$-0.514183\pi$
$359$	−605.954			−0.0890837			−0.0445418	+	0.999008i	$0.514183\pi$
$360$		0			0
$361$	−4405.08			−0.642233
$362$			8792.63i			1.27660i
$363$		0			0
$364$	−37.4604			−0.00539412
$365$	−5733.36	+	39.6770i	−0.822186	+	0.00568983i
$366$		0			0
$367$			9760.26i			1.38823i	0.719863	+	0.694117i	$0.244205\pi$
$367$			9760.26i			1.38823i	−0.719863	+	0.694117i	$0.755795\pi$
$368$			1228.75i			0.174056i
$369$		0			0
$370$	80.0522	+	11567.6i	0.0112479	+	1.62533i
$371$	5596.54			0.783175
$372$		0			0
$373$		−	7823.62i		−	1.08604i	−0.839721	−	0.543018i	$-0.817281\pi$
$373$		−	7823.62i		−	1.08604i	0.839721	−	0.543018i	$-0.182719\pi$
$374$	−5166.80			−0.714355
$375$		0			0
$376$	3456.90			0.474139
$377$		−	257.272i		−	0.0351464i
$378$		0			0
$379$	−2731.46			−0.370200			−0.185100	−	0.982720i	$-0.559261\pi$
$379$	−2731.46			−0.370200			−0.185100	+	0.982720i	$0.559261\pi$
$380$	7.39497	+	1068.58i	0.000998301	+	0.144255i
$381$		0			0
$382$		−	8225.85i		−	1.10176i
$383$			12599.9i			1.68100i	0.541812	+	0.840500i	$0.317738\pi$
$383$			12599.9i			1.68100i	−0.541812	+	0.840500i	$0.682262\pi$
$384$		0			0
$385$	8573.02	−	59.3285i	1.13486	−	0.00785366i
$386$	−4782.00			−0.630563
$387$		0			0
$388$		−	2639.33i		−	0.345339i
$389$	−4943.46			−0.644327			−0.322163	−	0.946684i	$-0.604410\pi$
$389$	−4943.46			−0.644327			−0.322163	+	0.946684i	$0.604410\pi$
$390$		0			0
$391$	1441.80			0.186484
$392$			665.021i			0.0856853i
$393$		0			0
$394$	2551.92			0.326305
$395$	−47.3775	−	6846.09i	−0.00603499	−	0.872062i
$396$		0			0
$397$		−	7067.06i		−	0.893415i	−0.894680	−	0.446707i	$-0.852597\pi$
$397$		−	7067.06i		−	0.893415i	0.894680	−	0.446707i	$-0.147403\pi$
$398$			3857.23i			0.485792i
$399$		0			0
$400$	5604.67	−	77.5765i	0.700584	−	0.00969707i
$401$	917.584			0.114269			0.0571346	−	0.998366i	$-0.481804\pi$
$401$	917.584			0.114269			0.0571346	+	0.998366i	$0.481804\pi$
$402$		0			0
$403$		−	170.190i		−	0.0210367i
$404$	−1096.72			−0.135059
$405$		0			0
$406$	−12086.0			−1.47738
$407$		−	16736.6i		−	2.03833i
$408$		0			0
$409$	−14735.5			−1.78148			−0.890739	−	0.454515i	$-0.849813\pi$
$409$	−14735.5			−1.78148			−0.890739	+	0.454515i	$0.849813\pi$
$410$	−10979.0	+	75.9789i	−1.32248	+	0.00915202i
$411$		0			0
$412$			916.792i			0.109629i
$413$			552.805i			0.0658638i
$414$		0			0
$415$	903.531	−	6.25277i	0.106874	−	0.000739606i
$416$	86.0094			0.0101369
$417$		0			0
$418$			4864.36i			0.569196i
$419$	−9254.92			−1.07907			−0.539537	−	0.841962i	$-0.681401\pi$
$419$	−9254.92			−1.07907			−0.539537	+	0.841962i	$0.681401\pi$
$420$		0			0
$421$	−10202.0			−1.18103			−0.590514	−	0.807027i	$-0.701075\pi$
$421$	−10202.0			−1.18103			−0.590514	+	0.807027i	$0.701075\pi$
$422$		−	8340.80i		−	0.962142i
$423$		0			0
$424$	−7116.24			−0.815083
$425$	−91.0279	−	6576.50i	−0.0103894	−	0.750604i
$426$		0			0
$427$			14097.8i			1.59775i
$428$			2802.72i			0.316529i
$429$		0			0
$430$	68.4029	+	9884.28i	0.00767135	+	1.10852i
$431$	−8057.20			−0.900468			−0.450234	−	0.892911i	$-0.648659\pi$
$431$	−8057.20			−0.900468			−0.450234	+	0.892911i	$0.648659\pi$
$432$		0			0
$433$		−	4306.26i		−	0.477935i	−0.971028	−	0.238967i	$-0.923191\pi$
$433$		−	4306.26i		−	0.477935i	0.971028	−	0.238967i	$-0.0768089\pi$
$434$	−7995.12			−0.884281
$435$		0			0
$436$	3787.34			0.416011
$437$		−	1357.41i		−	0.148590i
$438$		0			0
$439$	4721.79			0.513346			0.256673	−	0.966498i	$-0.417374\pi$
$439$	4721.79			0.513346			0.256673	+	0.966498i	$0.417374\pi$
$440$	−10901.0	+	75.4387i	−1.18110	+	0.00817363i
$441$		0			0
$442$			130.820i			0.0140779i
$443$			15354.4i			1.64675i	0.567501	+	0.823373i	$0.307910\pi$
$443$			15354.4i			1.64675i	−0.567501	+	0.823373i	$0.692090\pi$
$444$		0			0
$445$	1.85763	+	268.430i	0.000197888	+	0.0285951i
$446$	−5828.35			−0.618790
$447$		0			0
$448$		−	10942.6i		−	1.15399i
$449$	5448.64			0.572688			0.286344	−	0.958127i	$-0.407560\pi$
$449$	5448.64			0.572688			0.286344	+	0.958127i	$0.407560\pi$
$450$		0			0
$451$	15885.0			1.65852
$452$			1887.55i			0.196422i
$453$		0			0
$454$	2405.28			0.248647
$455$	−1.50215	−	217.063i	−0.000154774	−	0.0223650i
$456$		0			0
$457$		−	52.7307i		−	0.00539746i	−0.999996	−	0.00269873i	$-0.999141\pi$
$457$		−	52.7307i		−	0.00539746i	0.999996	−	0.00269873i	$-0.000859033\pi$
$458$			3930.53i			0.401008i
$459$		0			0
$460$	591.094	−	4.09059i	0.0599129	−	0.000414619i
$461$	3167.02			0.319963			0.159982	−	0.987120i	$-0.448857\pi$
$461$	3167.02			0.319963			0.159982	+	0.987120i	$0.448857\pi$
$462$		0			0
$463$		−	13092.4i		−	1.31416i	−0.753820	−	0.657081i	$-0.771791\pi$
$463$		−	13092.4i		−	1.31416i	0.753820	−	0.657081i	$-0.228209\pi$
$464$	11432.5			1.14384
$465$		0			0
$466$	−2324.82			−0.231105
$467$			13225.4i			1.31049i	0.755416	+	0.655246i	$0.227435\pi$
$467$			13225.4i			1.31049i	−0.755416	+	0.655246i	$0.772565\pi$
$468$		0			0
$469$	−3396.77			−0.334431
$470$	26.9362	+	3892.31i	0.00264356	+	0.381997i
$471$		0			0
$472$		−	702.915i		−	0.0685473i
$473$		−	14301.0i		−	1.39020i
$474$		0			0
$475$	−6191.54	+	85.6996i	−0.598079	+	0.00827825i
$476$	−1953.29			−0.188086
$477$		0			0
$478$		−	6729.99i		−	0.643981i
$479$	−8558.59			−0.816393			−0.408196	−	0.912894i	$-0.633842\pi$
$479$	−8558.59			−0.816393			−0.408196	+	0.912894i	$0.633842\pi$
$480$		0			0
$481$	−423.758			−0.0401698
$482$		−	8419.55i		−	0.795643i
$483$		0			0
$484$	496.656			0.0466432
$485$	15293.4	−	105.836i	1.43183	−	0.00990882i
$486$		0			0
$487$			19531.3i			1.81735i	0.417506	+	0.908674i	$0.362904\pi$
$487$			19531.3i			1.81735i	−0.417506	+	0.908674i	$0.637096\pi$
$488$		−	17925.9i		−	1.66285i
$489$		0			0
$490$	−748.782	+	5.18184i	−0.0690337	+	0.000477738i
$491$	19488.2			1.79122			0.895612	−	0.444835i	$-0.146738\pi$
$491$	19488.2			1.79122			0.895612	+	0.444835i	$0.146738\pi$
$492$		0			0
$493$		−	13414.9i		−	1.22551i
$494$	123.162			0.0112173
$495$		0			0
$496$	7562.83			0.684640
$497$			15446.2i			1.39408i
$498$		0			0
$499$	339.530			0.0304598			0.0152299	−	0.999884i	$-0.495152\pi$
$499$	339.530			0.0304598			0.0152299	+	0.999884i	$0.495152\pi$
$500$	−55.9770	−	2695.90i	−0.00500674	−	0.241129i
$501$		0			0
$502$		−	7796.84i		−	0.693208i
$503$		−	5550.25i		−	0.491995i	−0.969270	−	0.245998i	$-0.920884\pi$
$503$		−	5550.25i		−	0.491995i	0.969270	−	0.245998i	$-0.0791156\pi$
$504$		0			0
$505$	−43.9782	−	6354.90i	−0.00387526	−	0.559979i
$506$	2690.76			0.236401
$507$		0			0
$508$		−	1670.79i		−	0.145923i
$509$	5648.99			0.491920			0.245960	−	0.969280i	$-0.420897\pi$
$509$	5648.99			0.491920			0.245960	+	0.969280i	$0.420897\pi$
$510$		0			0
$511$	−9866.72			−0.854164
$512$			12598.3i			1.08745i
$513$		0			0
$514$	13673.0			1.17332
$515$	−5312.31	+	36.7631i	−0.454540	+	0.00314559i
$516$		0			0
$517$		−	5631.57i		−	0.479064i
$518$			19907.1i			1.68855i
$519$		0			0
$520$	1.91005	+	276.005i	0.000161080	+	0.0232762i
$521$	11159.0			0.938359			0.469180	−	0.883103i	$-0.344550\pi$
$521$	11159.0			0.938359			0.469180	+	0.883103i	$0.344550\pi$
$522$		0			0
$523$		−	9476.61i		−	0.792320i	−0.918182	−	0.396160i	$-0.870343\pi$
$523$		−	9476.61i		−	0.792320i	0.918182	−	0.396160i	$-0.129657\pi$
$524$	2834.27			0.236289
$525$		0			0
$526$	10194.5			0.845062
$527$		−	8874.19i		−	0.733521i
$528$		0			0
$529$	11416.1			0.938287
$530$	−55.4497	−	8012.54i	−0.00454449	−	0.656684i
$531$		0			0
$532$			1838.95i			0.149866i
$533$		−	402.196i		−	0.0326849i
$534$		0			0
$535$	−16240.2	+	112.388i	−1.31238	+	0.00908218i
$536$	4319.14			0.348057
$537$		0			0
$538$			2529.73i			0.202722i
$539$	1083.37			0.0865754
$540$		0			0
$541$	8285.40			0.658442			0.329221	−	0.944253i	$-0.393214\pi$
$541$	8285.40			0.658442			0.329221	+	0.944253i	$0.393214\pi$
$542$		−	8378.71i		−	0.664015i
$543$		0			0
$544$	4484.76			0.353461
$545$	151.872	+	21945.6i	0.0119366	+	1.72485i
$546$		0			0
$547$		−	6918.30i		−	0.540777i	−0.962751	−	0.270389i	$-0.912848\pi$
$547$		−	6918.30i		−	0.540777i	0.962751	−	0.270389i	$-0.0871522\pi$
$548$		−	138.023i		−	0.0107593i
$549$		0			0
$550$	−169.881	−	12273.4i	−0.0131704	−	0.951524i
$551$	−12629.6			−0.976480
$552$		0			0
$553$		−	11781.7i		−	0.905980i
$554$	8793.01			0.674331
$555$		0			0
$556$	−441.968			−0.0337115
$557$		−	15622.7i		−	1.18843i	−0.804305	−	0.594216i	$-0.797462\pi$
$557$		−	15622.7i		−	1.18843i	0.804305	−	0.594216i	$-0.202538\pi$
$558$		0			0
$559$	−362.092			−0.0273969
$560$	9645.72	−	66.7519i	0.727868	−	0.00503712i
$561$		0			0
$562$		−	14829.1i		−	1.11304i
$563$		−	8129.30i		−	0.608542i	−0.952585	−	0.304271i	$-0.901587\pi$
$563$		−	8129.30i		−	0.608542i	0.952585	−	0.304271i	$-0.0984129\pi$
$564$		0			0
$565$	−10937.3	+	75.6902i	−0.814400	+	0.00563595i
$566$	−11279.9			−0.837686
$567$		0			0
$568$		−	19640.6i		−	1.45088i
$569$	8175.67			0.602359			0.301179	−	0.953568i	$-0.402620\pi$
$569$	8175.67			0.602359			0.301179	+	0.953568i	$0.402620\pi$
$570$		0			0
$571$	7878.84			0.577442			0.288721	−	0.957413i	$-0.406770\pi$
$571$	7878.84			0.577442			0.288721	+	0.957413i	$0.406770\pi$
$572$		−	77.5972i		−	0.00567220i
$573$		0			0
$574$	−18894.1			−1.37391
$575$	47.4055	+	3424.90i	0.00343816	+	0.248397i
$576$		0			0
$577$			16673.4i			1.20299i	0.798878	+	0.601493i	$0.205427\pi$
$577$			16673.4i			1.20299i	−0.798878	+	0.601493i	$0.794573\pi$
$578$		−	5283.60i		−	0.380223i
$579$		0			0
$580$	−38.0598	−	5499.68i	−0.00272474	−	0.393727i
$581$	1554.92			0.111031
$582$		0			0
$583$			11592.9i			0.823549i
$584$	12546.0			0.888965
$585$		0			0
$586$	−15965.1			−1.12545
$587$			3012.18i			0.211799i	0.994377	+	0.105900i	$0.0337722\pi$
$587$			3012.18i			0.211799i	−0.994377	+	0.105900i	$0.966228\pi$
$588$		0			0
$589$	−8354.74			−0.584467
$590$	791.449	−	5.47712i	0.0552261	−	0.000382185i
$591$		0			0
$592$		−	18830.7i		−	1.30733i
$593$		−	7922.71i		−	0.548645i	−0.961638	−	0.274323i	$-0.911546\pi$
$593$		−	7922.71i		−	0.548645i	0.961638	−	0.274323i	$-0.0884536\pi$
$594$		0			0
$595$	−78.3264	−	11318.2i	−0.00539675	−	0.779836i
$596$	−1611.36			−0.110745
$597$		0			0
$598$		−	68.1282i		−	0.00465881i
$599$	−22578.5			−1.54012			−0.770061	−	0.637970i	$-0.779774\pi$
$599$	−22578.5			−1.54012			−0.770061	+	0.637970i	$0.779774\pi$
$600$		0			0
$601$	10618.7			0.720709			0.360354	−	0.932815i	$-0.382656\pi$
$601$	10618.7			0.720709			0.360354	+	0.932815i	$0.382656\pi$
$602$			17010.2i			1.15163i
$603$		0			0
$604$	−3601.47			−0.242619
$605$	19.9158	+	2877.85i	0.00133833	+	0.193391i
$606$		0			0
$607$			13590.6i			0.908771i	0.890805	+	0.454385i	$0.150141\pi$
$607$			13590.6i			0.908771i	−0.890805	+	0.454385i	$0.849859\pi$
$608$		−	4222.25i		−	0.281636i
$609$		0			0
$610$	20183.7	−	139.679i	1.33970	−	0.00927119i
$611$	−142.587			−0.00944102
$612$		0			0
$613$		−	5457.57i		−	0.359591i	−0.983704	−	0.179796i	$-0.942456\pi$
$613$		−	5457.57i		−	0.359591i	0.983704	−	0.179796i	$-0.0575436\pi$
$614$	−1745.52			−0.114729
$615$		0			0
$616$	−18759.8			−1.22704
$617$		−	5972.10i		−	0.389672i	−0.980836	−	0.194836i	$-0.937583\pi$
$617$		−	5972.10i		−	0.389672i	0.980836	−	0.194836i	$-0.0624174\pi$
$618$		0			0
$619$	839.633			0.0545197			0.0272598	−	0.999628i	$-0.491322\pi$
$619$	839.633			0.0545197			0.0272598	+	0.999628i	$0.491322\pi$
$620$	−25.1773	−	3638.14i	−0.00163088	−	0.235663i
$621$		0			0
$622$		−	15361.2i		−	0.990238i
$623$			461.949i			0.0297072i
$624$		0			0
$625$	15619.0	−	432.461i	0.999617	−	0.0276775i
$626$	−6483.41			−0.413944
$627$		0			0
$628$		−	4437.55i		−	0.281970i
$629$	−22095.9			−1.40067
$630$		0			0
$631$	−13699.4			−0.864284			−0.432142	−	0.901806i	$-0.642242\pi$
$631$	−13699.4			−0.864284			−0.432142	+	0.901806i	$0.642242\pi$
$632$			14980.9i			0.942892i
$633$		0			0
$634$	−1238.17			−0.0775615
$635$	9681.28	−	66.9980i	0.605024	−	0.00418699i
$636$		0			0
$637$		−	27.4302i		−	0.00170616i
$638$		−	25035.5i		−	1.55355i
$639$		0			0
$640$	−8043.07	+	55.6610i	−0.496766	+	0.00343780i
$641$	7939.36			0.489214			0.244607	−	0.969622i	$-0.421341\pi$
$641$	7939.36			0.489214			0.244607	+	0.969622i	$0.421341\pi$
$642$		0			0
$643$			7635.95i			0.468324i	0.972198	+	0.234162i	$0.0752347\pi$
$643$			7635.95i			0.468324i	−0.972198	+	0.234162i	$0.924765\pi$
$644$	1017.23			0.0622432
$645$		0			0
$646$	6422.01			0.391131
$647$			22084.3i			1.34192i	0.741491	+	0.670962i	$0.234119\pi$
$647$			22084.3i			1.34192i	−0.741491	+	0.670962i	$0.765881\pi$
$648$		0			0
$649$	−1145.10			−0.0692593
$650$	−310.753	+	4.30125i	−0.0187519	+	0.000259552i
$651$		0			0
$652$		−	86.6364i		−	0.00520390i
$653$			6637.97i			0.397801i	0.980020	+	0.198900i	$0.0637370\pi$
$653$			6637.97i			0.397801i	−0.980020	+	0.198900i	$0.936263\pi$
$654$		0			0
$655$	113.653	+	16423.0i	0.00677986	+	0.979696i
$656$	17872.6			1.06373
$657$		0			0
$658$			6698.39i			0.396855i
$659$	23812.3			1.40758			0.703789	−	0.710409i	$-0.251490\pi$
$659$	23812.3			1.40758			0.703789	+	0.710409i	$0.251490\pi$
$660$		0			0
$661$	−999.824			−0.0588330			−0.0294165	−	0.999567i	$-0.509365\pi$
$661$	−999.824			−0.0588330			−0.0294165	+	0.999567i	$0.509365\pi$
$662$		−	7606.24i		−	0.446563i
$663$		0			0
$664$	−1977.14			−0.115554
$665$	−10655.7	+	73.7416i	−0.621371	+	0.00430012i
$666$		0			0
$667$			6986.19i			0.405557i
$668$			2270.11i			0.131487i
$669$		0			0
$670$	33.6547	+	4863.14i	0.00194059	+	0.280417i
$671$	−29202.7			−1.68012
$672$		0			0
$673$		−	10622.1i		−	0.608396i	−0.952609	−	0.304198i	$-0.901612\pi$
$673$		−	10622.1i		−	0.608396i	0.952609	−	0.304198i	$-0.0983884\pi$
$674$	14748.1			0.842843
$675$		0			0
$676$	4237.02			0.241069
$677$		−	24907.8i		−	1.41401i	−0.707209	−	0.707005i	$-0.750046\pi$
$677$		−	24907.8i		−	1.41401i	0.707209	−	0.707005i	$-0.249954\pi$
$678$		0			0
$679$	26319.0			1.48753
$680$	99.5954	+	14391.6i	0.00561663	+	0.811608i
$681$		0			0
$682$		−	16561.4i		−	0.929868i
$683$			17115.6i			0.958874i	0.877576	+	0.479437i	$0.159159\pi$
$683$			17115.6i			0.958874i	−0.877576	+	0.479437i	$0.840841\pi$
$684$		0			0
$685$	799.770	−	5.53470i	0.0446097	−	0.000308716i
$686$	14971.2			0.833243
$687$		0			0
$688$		−	16090.5i		−	0.891632i
$689$	293.524			0.0162299
$690$		0			0
$691$	28899.8			1.59103			0.795514	−	0.605935i	$-0.207201\pi$
$691$	28899.8			1.59103			0.795514	+	0.605935i	$0.207201\pi$
$692$			2060.82i			0.113209i
$693$		0			0
$694$	2480.23			0.135660
$695$	−17.7228	−	2560.96i	−0.000967286	−	0.139774i
$696$		0			0
$697$		−	20971.6i		−	1.13968i
$698$		−	4802.92i		−	0.260449i
$699$		0			0
$700$	−64.2228	−	4639.90i	−0.00346770	−	0.250531i
$701$	−576.680			−0.0310712			−0.0155356	−	0.999879i	$-0.504945\pi$
$701$	−576.680			−0.0310712			−0.0155356	+	0.999879i	$0.504945\pi$
$702$		0			0
$703$			20802.5i			1.11605i
$704$	22666.9			1.21348
$705$		0			0
$706$	−6385.63			−0.340406
$707$		−	10936.3i		−	0.581759i
$708$		0			0
$709$	−5717.55			−0.302859			−0.151430	−	0.988468i	$-0.548388\pi$
$709$	−5717.55			−0.302859			−0.151430	+	0.988468i	$0.548388\pi$
$710$	22114.3	−	153.039i	1.16892	−	0.00808938i
$711$		0			0
$712$		−	587.389i		−	0.0309176i
$713$			4621.50i			0.242744i
$714$		0			0
$715$	449.633	−	3.11163i	0.0235179	−	0.000162753i
$716$	−5363.19			−0.279933
$717$		0			0
$718$			1492.98i			0.0776010i
$719$	−14799.9			−0.767655			−0.383827	−	0.923405i	$-0.625394\pi$
$719$	−14799.9			−0.767655			−0.383827	+	0.923405i	$0.625394\pi$
$720$		0			0
$721$	−9142.12			−0.472219
$722$			10853.4i			0.559451i
$723$		0			0
$724$	−6885.52			−0.353451
$725$	31866.1	−	441.071i	1.63238	−	0.0225944i
$726$		0			0
$727$			6410.40i			0.327027i	0.986541	+	0.163513i	$0.0522827\pi$
$727$			6410.40i			0.327027i	−0.986541	+	0.163513i	$0.947717\pi$
$728$			474.985i			0.0241815i
$729$		0			0
$730$	97.7581	+	14126.1i	0.00495642	+	0.716208i
$731$	−18880.5			−0.955292
$732$		0			0
$733$			17869.2i			0.900431i	0.892920	+	0.450215i	$0.148653\pi$
$733$			17869.2i			0.900431i	−0.892920	+	0.450215i	$0.851347\pi$
$734$	24047.8			1.20929
$735$		0			0
$736$	−2335.57			−0.116971
$737$		−	7036.21i		−	0.351672i
$738$		0			0
$739$	−11099.7			−0.552516			−0.276258	−	0.961084i	$-0.589094\pi$
$739$	−11099.7			−0.552516			−0.276258	+	0.961084i	$0.589094\pi$
$740$	−9058.62	+	62.6890i	−0.450002	+	0.00311418i
$741$		0			0
$742$		−	13789.0i		−	0.682225i
$743$			10857.6i			0.536107i	0.963404	+	0.268053i	$0.0863803\pi$
$743$			10857.6i			0.536107i	−0.963404	+	0.268053i	$0.913620\pi$
$744$		0			0
$745$	−64.6153	−	9336.97i	−0.00317761	−	0.459168i
$746$	−19276.2			−0.946049
$747$		0			0
$748$		−	4046.13i		−	0.197782i
$749$	−27948.3			−1.36343
$750$		0			0
$751$	−32233.9			−1.56622			−0.783111	−	0.621882i	$-0.786368\pi$
$751$	−32233.9			−1.56622			−0.783111	+	0.621882i	$0.786368\pi$
$752$		−	6336.22i		−	0.307258i
$753$		0			0
$754$	−633.880			−0.0306161
$755$	−144.418	−	20868.5i	−0.00696147	−	1.00594i
$756$		0			0
$757$		−	9410.28i		−	0.451813i	−0.974149	−	0.225906i	$-0.927466\pi$
$757$		−	9410.28i		−	0.451813i	0.974149	−	0.225906i	$-0.0725344\pi$
$758$			6729.91i			0.322482i
$759$		0			0
$760$	13549.2	−	93.7656i	0.646687	−	0.00447531i
$761$	−34767.1			−1.65612			−0.828060	−	0.560640i	$-0.810555\pi$
$761$	−34767.1			−1.65612			−0.828060	+	0.560640i	$0.810555\pi$
$762$		0			0
$763$			37766.8i			1.79194i
$764$	6441.68			0.305042
$765$		0			0
$766$	31044.2			1.46432
$767$			28.9932i			0.00136491i
$768$		0			0
$769$	29086.5			1.36396			0.681980	−	0.731371i	$-0.261119\pi$
$769$	29086.5			1.36396			0.681980	+	0.731371i	$0.261119\pi$
$770$	−146.176	−	21122.6i	−0.00684134	−	0.988581i
$771$		0			0
$772$		−	3744.79i		−	0.174583i
$773$		−	2942.77i		−	0.136927i	−0.997654	−	0.0684633i	$-0.978190\pi$
$773$		−	2942.77i		−	0.136927i	0.997654	−	0.0684633i	$-0.0218096\pi$
$774$		0			0
$775$	21080.0	−	291.777i	0.977054	−	0.0135238i
$776$	−33465.7			−1.54813
$777$		0			0
$778$			12179.9i			0.561275i
$779$	−19744.0			−0.908091
$780$		0			0
$781$	−31996.0			−1.46595
$782$		−	3552.39i		−	0.162446i
$783$		0			0
$784$	1218.93			0.0555270
$785$	25713.2	−	177.944i	1.16910	−	0.00809059i
$786$		0			0
$787$			33829.1i			1.53225i	0.642693	+	0.766124i	$0.277817\pi$
$787$			33829.1i			1.53225i	−0.642693	+	0.766124i	$0.722183\pi$
$788$			1998.41i			0.0903434i
$789$		0			0
$790$	−16867.7	+	116.731i	−0.759655	+	0.00525709i
$791$	−18822.4			−0.846076
$792$		0			0
$793$			739.392i			0.0331105i
$794$	−17412.2			−0.778256
$795$		0			0
$796$	−3020.60			−0.134500
$797$			22904.7i			1.01797i	0.860774	+	0.508987i	$0.169980\pi$
$797$			22904.7i			1.01797i	−0.860774	+	0.508987i	$0.830020\pi$
$798$		0			0
$799$	−7434.89			−0.329196
$800$	147.456	+	10653.2i	0.00651669	+	0.470811i
$801$		0			0
$802$		−	2260.79i		−	0.0995402i
$803$		−	20438.3i		−	0.898199i
$804$		0			0
$805$	40.7908	+	5894.31i	0.00178595	+	0.258071i
$806$	−419.324			−0.0183251
$807$		0			0
$808$			13906.0i			0.605461i
$809$	−30665.7			−1.33269			−0.666347	−	0.745642i	$-0.732143\pi$
$809$	−30665.7			−1.33269			−0.666347	+	0.745642i	$0.732143\pi$
$810$		0			0
$811$	−35135.2			−1.52129			−0.760644	−	0.649169i	$-0.775117\pi$
$811$	−35135.2			−1.52129			−0.760644	+	0.649169i	$0.775117\pi$
$812$		−	9464.57i		−	0.409041i
$813$		0			0
$814$	−41236.4			−1.77560
$815$	502.011	−	3.47410i	0.0215763	−	0.000149316i
$816$		0			0
$817$			17775.3i			0.761174i
$818$			36306.1i			1.55185i
$819$		0			0
$820$	−59.4992	−	8597.69i	−0.00253390	−	0.366152i
$821$	−34437.0			−1.46390			−0.731948	−	0.681361i	$-0.761389\pi$
$821$	−34437.0			−1.46390			−0.731948	+	0.681361i	$0.761389\pi$
$822$		0			0
$823$			42801.8i			1.81285i	0.422365	+	0.906426i	$0.361200\pi$
$823$			42801.8i			1.81285i	−0.422365	+	0.906426i	$0.638800\pi$
$824$	11624.6			0.491459
$825$		0			0
$826$	1362.03			0.0573742
$827$		−	23589.1i		−	0.991868i	−0.868360	−	0.495934i	$-0.834826\pi$
$827$		−	23589.1i		−	0.991868i	0.868360	−	0.495934i	$-0.165174\pi$
$828$		0			0
$829$	26766.2			1.12139			0.560693	−	0.828024i	$-0.310535\pi$
$829$	26766.2			1.12139			0.560693	+	0.828024i	$0.310535\pi$
$830$	−15.4059	−	2226.17i	−0.000644272	−	0.0930980i
$831$		0			0
$832$		−	573.911i		−	0.0239144i
$833$		−	1430.28i		−	0.0594915i
$834$		0			0
$835$	−13154.0	+	91.0309i	−0.545167	+	0.00377276i
$836$	−3809.29			−0.157592
$837$		0			0
$838$			22802.7i			0.939985i
$839$	−30334.3			−1.24822			−0.624110	−	0.781336i	$-0.714538\pi$
$839$	−30334.3			−1.24822			−0.624110	+	0.781336i	$0.714538\pi$
$840$		0			0
$841$	40612.1			1.66518
$842$			25136.1i			1.02880i
$843$		0			0
$844$	6531.69			0.266386
$845$	169.904	+	24551.2i	0.00691700	+	0.999513i
$846$		0			0
$847$			4952.58i			0.200912i
$848$			13043.5i			0.528201i
$849$		0			0
$850$	−16203.5	+	224.279i	−0.653853	+	0.00905025i
$851$	11507.1			0.463523
$852$		0			0
$853$			6533.77i			0.262265i	0.991365	+	0.131133i	$0.0418614\pi$
$853$			6533.77i			0.262265i	−0.991365	+	0.131133i	$0.958139\pi$
$854$	34734.8			1.39180
$855$		0			0
$856$	35537.5			1.41898
$857$			37553.0i			1.49683i	0.663228	+	0.748417i	$0.269186\pi$
$857$			37553.0i			1.49683i	−0.663228	+	0.748417i	$0.730814\pi$
$858$		0			0
$859$	−36637.9			−1.45526			−0.727631	−	0.685969i	$-0.759379\pi$
$859$	−36637.9			−1.45526			−0.727631	+	0.685969i	$0.759379\pi$
$860$	−7740.40	+	53.5664i	−0.306913	+	0.00212395i
$861$		0			0
$862$			19851.7i			0.784400i
$863$		−	14614.7i		−	0.576467i	−0.957560	−	0.288233i	$-0.906932\pi$
$863$		−	14614.7i		−	0.576467i	0.957560	−	0.288233i	$-0.0930679\pi$
$864$		0			0
$865$	−11941.3	+	82.6384i	−0.469384	+	0.00324831i
$866$	−10610.0			−0.416330
$867$		0			0
$868$		−	6260.99i		−	0.244829i
$869$	24405.0			0.952686
$870$		0			0
$871$	−178.152			−0.00693047
$872$		−	48022.2i		−	1.86495i
$873$		0			0
$874$	−3344.45			−0.129437
$875$	26883.1	−	558.194i	1.03865	−	0.0215662i
$876$		0			0
$877$		−	1388.56i		−	0.0534646i	−0.999643	−	0.0267323i	$-0.991490\pi$
$877$		−	1388.56i		−	0.0534646i	0.999643	−	0.0267323i	$-0.00851017\pi$
$878$		−	11633.8i		−	0.447177i
$879$		0			0
$880$	138.273	+	19980.6i	0.00529679	+	0.765392i
$881$	−23664.2			−0.904957			−0.452478	−	0.891775i	$-0.649460\pi$
$881$	−23664.2			−0.904957			−0.452478	+	0.891775i	$0.649460\pi$
$882$		0			0
$883$			49607.6i			1.89063i	0.326152	+	0.945317i	$0.394248\pi$
$883$			49607.6i			1.89063i	−0.326152	+	0.945317i	$0.605752\pi$
$884$	−102.445			−0.00389774
$885$		0			0
$886$	37830.9			1.43448
$887$			11477.0i			0.434454i	0.976121	+	0.217227i	$0.0697012\pi$
$887$			11477.0i			0.434454i	−0.976121	+	0.217227i	$0.930299\pi$
$888$		0			0
$889$	16660.8			0.628556
$890$	661.371	−	4.57693i	0.0249092	−	0.000172381i
$891$		0			0
$892$		−	4564.19i		−	0.171323i
$893$			6999.69i			0.262302i
$894$		0			0
$895$	−215.063	−	31076.8i	−0.00803212	−	1.16065i
$896$	−13841.6			−0.516087
$897$		0			0
$898$		−	13424.6i		−	0.498870i
$899$	42999.4			1.59523
$900$		0			0
$901$	15305.1			0.565914
$902$		−	39138.1i		−	1.44474i
$903$		0			0
$904$	23933.4			0.880547
$905$	−276.108	−	39897.8i	−0.0101416	−	1.46547i
$906$		0			0
$907$			11733.8i			0.429563i	0.976662	+	0.214782i	$0.0689040\pi$
$907$			11733.8i			0.429563i	−0.976662	+	0.214782i	$0.931096\pi$
$908$			1883.58i			0.0688424i
$909$		0			0
$910$	−534.810	+	3.70108i	−0.0194822	+	0.000134824i
$911$	35013.4			1.27338			0.636688	−	0.771122i	$-0.280304\pi$
$911$	35013.4			1.27338			0.636688	+	0.771122i	$0.280304\pi$
$912$		0			0
$913$			3220.92i			0.116755i
$914$	−129.920			−0.00470174
$915$		0			0
$916$	−3078.00			−0.111026
$917$			28262.9i			1.01780i
$918$		0			0
$919$	−16026.6			−0.575264			−0.287632	−	0.957741i	$-0.592868\pi$
$919$	−16026.6			−0.575264			−0.287632	+	0.957741i	$0.592868\pi$
$920$	−51.8672	−	7494.86i	−0.00185871	−	0.268585i
$921$		0			0
$922$		−	7803.07i		−	0.278721i
$923$			810.116i			0.0288898i
$924$		0			0
$925$	−726.497	−	52487.2i	−0.0258239	−	1.86570i
$926$	−32257.8			−1.14477
$927$		0			0
$928$			21730.7i			0.768691i
$929$	25886.8			0.914229			0.457114	−	0.889408i	$-0.348883\pi$
$929$	25886.8			0.914229			0.457114	+	0.889408i	$0.348883\pi$
$930$		0			0
$931$	−1346.56			−0.0474026
$932$		−	1820.57i		−	0.0639857i
$933$		0			0
$934$	32585.5			1.14157
$935$	23445.1	−	162.249i	0.820039	−	0.00567497i
$936$		0			0
$937$		−	6331.15i		−	0.220736i	−0.993891	−	0.110368i	$-0.964797\pi$
$937$		−	6331.15i		−	0.220736i	0.993891	−	0.110368i	$-0.0352029\pi$
$938$			8369.13i			0.291324i
$939$		0			0
$940$	−3048.07	+	21.0938i	−0.105763	+	0.000731918i
$941$	28754.7			0.996149			0.498075	−	0.867134i	$-0.334041\pi$
$941$	28754.7			0.996149			0.498075	+	0.867134i	$0.334041\pi$
$942$		0			0
$943$			10921.6i			0.377153i
$944$	−1288.39			−0.0444210
$945$		0			0
$946$	−35235.6			−1.21100
$947$		−	29240.7i		−	1.00337i	−0.865049	−	0.501687i	$-0.832713\pi$
$947$		−	29240.7i		−	1.00337i	0.865049	−	0.501687i	$-0.167287\pi$
$948$		0			0
$949$	−517.484			−0.0177010
$950$	211.151	+	15255.0i	0.00721120	+	0.520988i
$951$		0			0
$952$			24767.0i			0.843176i
$953$			28758.3i			0.977517i	0.872419	+	0.488758i	$0.162550\pi$
$953$			28758.3i			0.977517i	−0.872419	+	0.488758i	$0.837450\pi$
$954$		0			0
$955$	258.310	+	37326.0i	0.00875257	+	1.26475i
$956$	5270.27			0.178298
$957$		0			0
$958$			21087.1i			0.711161i
$959$	1376.35			0.0463448
$960$		0			0
$961$	−1346.05			−0.0451831
$962$			1044.08i			0.0349920i
$963$		0			0
$964$	6593.36			0.220288
$965$	21699.0	−	150.165i	0.723851	−	0.00500931i
$966$		0			0
$967$			44695.1i			1.48635i	0.669099	+	0.743174i	$0.266680\pi$
$967$			44695.1i			1.48635i	−0.669099	+	0.743174i	$0.733320\pi$
$968$		−	6297.42i		−	0.209098i
$969$		0			0
$970$	−260.765	−	37680.8i	−0.00863160	−	1.24727i
$971$	30462.4			1.00678			0.503391	−	0.864059i	$-0.332086\pi$
$971$	30462.4			1.00678			0.503391	+	0.864059i	$0.332086\pi$
$972$		0			0
$973$		−	4407.24i		−	0.145210i
$974$	48122.2			1.58310
$975$		0			0
$976$	−32856.7			−1.07758
$977$		−	11808.0i		−	0.386665i	−0.981133	−	0.193332i	$-0.938070\pi$
$977$		−	11808.0i		−	0.386665i	0.981133	−	0.193332i	$-0.0619295\pi$
$978$		0			0
$979$	−956.902			−0.0312387
$980$	−4.05791	−	586.372i	−0.000132271	−	0.0191132i
$981$		0			0
$982$		−	48016.1i		−	1.56034i
$983$			24702.9i			0.801526i	0.916182	+	0.400763i	$0.131255\pi$
$983$			24702.9i			0.801526i	−0.916182	+	0.400763i	$0.868745\pi$
$984$		0			0
$985$	−11579.7	+	80.1359i	−0.374579	+	0.00259223i
$986$	−33052.2			−1.06754
$987$		0			0
$988$			96.4485i			0.00310570i
$989$	9832.55			0.316134
$990$		0			0
$991$	14443.0			0.462965			0.231482	−	0.972839i	$-0.425642\pi$
$991$	14443.0			0.462965			0.231482	+	0.972839i	$0.425642\pi$
$992$			14375.3i			0.460096i
$993$		0			0
$994$	38057.2			1.21439
$995$	−121.125	−	17502.7i	−0.00385923	−	0.557662i
$996$		0			0
$997$			47669.3i			1.51425i	0.653273	+	0.757123i	$0.273396\pi$
$997$			47669.3i			1.51425i	−0.653273	+	0.757123i	$0.726604\pi$
$998$		−	836.551i		−	0.0265336i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.4.b.f.244.6		16
3.2	odd	2		405.4.b.e.244.11		16
5.2	odd	4		2025.4.a.bl.1.11		16
5.3	odd	4		2025.4.a.bl.1.6		16
5.4	even	2	inner	405.4.b.f.244.11		16
9.2	odd	6		45.4.j.a.4.11	yes	32
9.4	even	3		135.4.j.a.19.11		32
9.5	odd	6		45.4.j.a.34.6	yes	32
9.7	even	3		135.4.j.a.64.6		32
15.2	even	4		2025.4.a.bk.1.6		16
15.8	even	4		2025.4.a.bk.1.11		16
15.14	odd	2		405.4.b.e.244.6		16
45.2	even	12		225.4.e.g.76.11		32
45.4	even	6		135.4.j.a.19.6		32
45.14	odd	6		45.4.j.a.34.11	yes	32
45.23	even	12		225.4.e.g.151.6		32
45.29	odd	6		45.4.j.a.4.6	✓	32
45.32	even	12		225.4.e.g.151.11		32
45.34	even	6		135.4.j.a.64.11		32
45.38	even	12		225.4.e.g.76.6		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.j.a.4.6	✓	32	45.29	odd	6
45.4.j.a.4.11	yes	32	9.2	odd	6
45.4.j.a.34.6	yes	32	9.5	odd	6
45.4.j.a.34.11	yes	32	45.14	odd	6
135.4.j.a.19.6		32	45.4	even	6
135.4.j.a.19.11		32	9.4	even	3
135.4.j.a.64.6		32	9.7	even	3
135.4.j.a.64.11		32	45.34	even	6
225.4.e.g.76.6		32	45.38	even	12
225.4.e.g.76.11		32	45.2	even	12
225.4.e.g.151.6		32	45.23	even	12
225.4.e.g.151.11		32	45.32	even	12
405.4.b.e.244.6		16	15.14	odd	2
405.4.b.e.244.11		16	3.2	odd	2
405.4.b.f.244.6		16	1.1	even	1	trivial
405.4.b.f.244.11		16	5.4	even	2	inner
2025.4.a.bk.1.6		16	15.2	even	4
2025.4.a.bk.1.11		16	15.8	even	4
2025.4.a.bl.1.6		16	5.3	odd	4
2025.4.a.bl.1.11		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)

\(f(q)\)	\(=\)	\(q-2.46385i q^{2} +1.92944 q^{4} +(0.0773702 + 11.1801i) q^{5} +19.2401i q^{7} -24.4647i q^{8} +O(q^{10})\)	\(q-2.46385i q^{2} +1.92944 q^{4} +(0.0773702 + 11.1801i) q^{5} +19.2401i q^{7} -24.4647i q^{8} +(27.5460 - 0.190629i) q^{10} -39.8548 q^{11} +1.00910i q^{13} +47.4048 q^{14} -44.8417 q^{16} +52.6170i q^{17} +49.5371 q^{19} +(0.149282 + 21.5713i) q^{20} +98.1964i q^{22} -27.4019i q^{23} +(-124.988 + 1.73001i) q^{25} +2.48626 q^{26} +37.1228i q^{28} -254.953 q^{29} -168.656 q^{31} -85.2341i q^{32} +129.640 q^{34} +(-215.106 + 1.48861i) q^{35} +419.938i q^{37} -122.052i q^{38} +(273.517 - 1.89284i) q^{40} -398.570 q^{41} +358.828i q^{43} -76.8977 q^{44} -67.5141 q^{46} +141.302i q^{47} -27.1829 q^{49} +(4.26248 + 307.952i) q^{50} +1.94699i q^{52} -290.878i q^{53} +(-3.08358 - 445.580i) q^{55} +470.703 q^{56} +628.166i q^{58} +28.7319 q^{59} +732.727 q^{61} +415.544i q^{62} -568.737 q^{64} +(-11.2818 + 0.0780740i) q^{65} +176.546i q^{67} +101.522i q^{68} +(3.66772 + 529.989i) q^{70} +802.814 q^{71} +512.820i q^{73} +1034.66 q^{74} +95.5791 q^{76} -766.813i q^{77} -612.348 q^{79} +(-3.46941 - 501.333i) q^{80} +982.017i q^{82} -80.8162i q^{83} +(-588.262 + 4.07099i) q^{85} +884.098 q^{86} +975.035i q^{88} +24.0097 q^{89} -19.4151 q^{91} -52.8704i q^{92} +348.147 q^{94} +(3.83270 + 553.828i) q^{95} -1367.92i q^{97} +66.9747i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 54 q^{4} + 3 q^{5}+O(q^{10}) \) 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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