LMFDB - Embedded newform 405.4.b.e.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.e.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	$-0.826142\pi$
$7$		−	19.2401i		−	1.03887i	0.854510	−	0.519435i	$-0.173858\pi$
$8$		−	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	$-0.316069\pi$
$11$	39.8548			1.09243			0.546213	+	0.837646i	$0.316069\pi$
$12$		0			0
$13$		−	1.00910i		−	0.0215287i	−0.999942	−	0.0107643i	$-0.996574\pi$
$13$		−	1.00910i		−	0.0215287i	0.999942	−	0.0107643i	$-0.00342646\pi$
$14$	−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	$-0.196039\pi$
$29$	254.953			1.63254			0.816269	+	0.577672i	$0.196039\pi$
$30$		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	$-0.617236\pi$
$37$		−	419.938i		−	1.86588i	0.360038	−	0.932938i	$-0.382764\pi$
$38$		−	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	$-0.225639\pi$
$41$	398.570			1.51820			0.759100	+	0.650974i	$0.225639\pi$
$42$		0			0
$43$		−	358.828i		−	1.27258i	−0.771452	−	0.636288i	$-0.780469\pi$
$43$		−	358.828i		−	1.27258i	0.771452	−	0.636288i	$-0.219531\pi$
$44$	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	$-0.510092\pi$
$59$	−28.7319			−0.0633995			−0.0316998	+	0.999497i	$0.510092\pi$
$60$		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	$-0.948541\pi$
$67$		−	176.546i		−	0.321918i	0.986961	−	0.160959i	$-0.0514587\pi$
$68$			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	$-0.734118\pi$
$71$	−802.814			−1.34192			−0.670961	+	0.741493i	$0.734118\pi$
$72$		0			0
$73$		−	512.820i		−	0.822205i	−0.911589	−	0.411103i	$-0.865144\pi$
$73$		−	512.820i		−	0.822205i	0.911589	−	0.411103i	$-0.134856\pi$
$74$	−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	$-0.504551\pi$
$89$	−24.0097			−0.0285957			−0.0142979	+	0.999898i	$0.504551\pi$
$90$		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	$0.253999\pi$
$97$			1367.92i			1.43187i	−0.698168	+	0.715934i	$0.746001\pi$
$98$			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	$-0.409667\pi$
$101$	568.413			0.559992			0.279996	+	0.960001i	$0.409667\pi$
$102$		0			0
$103$		−	475.159i		−	0.454551i	−0.973830	−	0.227276i	$-0.927018\pi$
$103$		−	475.159i		−	0.454551i	0.973830	−	0.227276i	$-0.0729818\pi$
$104$	−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	$0.0978276\pi$
$127$			865.941i			0.605038i	−0.953143	+	0.302519i	$0.902172\pi$
$128$			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	$-0.662952\pi$
$131$	−1468.96			−0.979720			−0.489860	+	0.871801i	$0.662952\pi$
$132$		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	$-0.426262\pi$
$149$	835.144			0.459179			0.229589	+	0.973288i	$0.426262\pi$
$150$		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	$0.198734\pi$
$157$			2299.91i			1.16913i	−0.811348	+	0.584563i	$0.801266\pi$
$158$			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	$0.00343412\pi$
$163$			44.9023i			0.0215768i	−0.999942	+	0.0107884i	$0.996566\pi$
$164$	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	$-0.302920\pi$
$179$	2779.66			1.16068			0.580339	+	0.814375i	$0.302920\pi$
$180$		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	$-0.717927\pi$
$191$	−3338.62			−1.26478			−0.632392	+	0.774648i	$0.717927\pi$
$192$		0			0
$193$			1940.87i			0.723868i	0.932204	+	0.361934i	$0.117883\pi$
$193$			1940.87i			0.723868i	−0.932204	+	0.361934i	$0.882117\pi$
$194$	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	$0.115579\pi$
$223$			2365.54i			0.710352i	−0.934799	+	0.355176i	$0.884421\pi$
$224$	−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	$-0.620517\pi$
$239$	−2731.50			−0.739271			−0.369635	+	0.929177i	$0.620517\pi$
$240$		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	$-0.630258\pi$
$251$	−3164.50			−0.795782			−0.397891	+	0.917433i	$0.630258\pi$
$252$		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	$-0.462878\pi$
$269$	1026.74			0.232719			0.116359	+	0.993207i	$0.462878\pi$
$270$		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	$-0.873492\pi$
$277$		−	3568.81i		−	0.774112i	0.922056	−	0.387056i	$-0.126508\pi$
$278$			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	$-0.720597\pi$
$281$	−6018.68			−1.27774			−0.638869	+	0.769316i	$0.720597\pi$
$282$		0			0
$283$			4578.17i			0.961638i	0.876820	+	0.480819i	$0.159661\pi$
$283$			4578.17i			0.961638i	−0.876820	+	0.480819i	$0.840339\pi$
$284$	−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	$0.0209767\pi$
$307$			708.452i			0.131705i	−0.997829	+	0.0658526i	$0.979023\pi$
$308$		−	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	$-0.692430\pi$
$311$	−6234.64			−1.13676			−0.568382	+	0.822765i	$0.692430\pi$
$312$		0			0
$313$			2631.41i			0.475196i	0.971364	+	0.237598i	$0.0763600\pi$
$313$			2631.41i			0.475196i	−0.971364	+	0.237598i	$0.923640\pi$
$314$	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	$-0.839264\pi$
$337$		−	5985.80i		−	0.967559i	0.875190	−	0.483780i	$-0.160736\pi$
$338$		−	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	$-0.485817\pi$
$359$	605.954			0.0890837			0.0445418	+	0.999008i	$0.485817\pi$
$360$		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	$-0.755795\pi$
$367$		−	9760.26i		−	1.38823i	0.719863	−	0.694117i	$-0.244205\pi$
$368$			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	$0.182719\pi$
$373$			7823.62i			1.08604i	−0.839721	+	0.543018i	$0.817281\pi$
$374$	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	$-0.395590\pi$
$389$	4943.46			0.644327			0.322163	+	0.946684i	$0.395590\pi$
$390$		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	$0.147403\pi$
$397$			7067.06i			0.893415i	−0.894680	+	0.446707i	$0.852597\pi$
$398$			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	$-0.518196\pi$
$401$	−917.584			−0.114269			−0.0571346	+	0.998366i	$0.518196\pi$
$402$		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	$-0.318599\pi$
$419$	9254.92			1.07907			0.539537	+	0.841962i	$0.318599\pi$
$420$		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	$-0.351341\pi$
$431$	8057.20			0.900468			0.450234	+	0.892911i	$0.351341\pi$
$432$		0			0
$433$			4306.26i			0.477935i	0.971028	+	0.238967i	$0.0768089\pi$
$433$			4306.26i			0.477935i	−0.971028	+	0.238967i	$0.923191\pi$
$434$	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	$-0.592440\pi$
$449$	−5448.64			−0.572688			−0.286344	+	0.958127i	$0.592440\pi$
$450$		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	$0.000859033\pi$
$457$			52.7307i			0.00539746i	−0.999996	+	0.00269873i	$0.999141\pi$
$458$			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	$-0.551143\pi$
$461$	−3167.02			−0.319963			−0.159982	+	0.987120i	$0.551143\pi$
$462$		0			0
$463$			13092.4i			1.31416i	0.753820	+	0.657081i	$0.228209\pi$
$463$			13092.4i			1.31416i	−0.753820	+	0.657081i	$0.771791\pi$
$464$	−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	$-0.366158\pi$
$479$	8558.59			0.816393			0.408196	+	0.912894i	$0.366158\pi$
$480$		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	$-0.637096\pi$
$487$		−	19531.3i		−	1.81735i	0.417506	−	0.908674i	$-0.362904\pi$
$488$		−	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	$-0.853262\pi$
$491$	−19488.2			−1.79122			−0.895612	+	0.444835i	$0.853262\pi$
$492$		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	$-0.579103\pi$
$509$	−5648.99			−0.491920			−0.245960	+	0.969280i	$0.579103\pi$
$510$		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	$-0.655450\pi$
$521$	−11159.0			−0.938359			−0.469180	+	0.883103i	$0.655450\pi$
$522$		0			0
$523$			9476.61i			0.792320i	0.918182	+	0.396160i	$0.129657\pi$
$523$			9476.61i			0.792320i	−0.918182	+	0.396160i	$0.870343\pi$
$524$	−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	$0.0871522\pi$
$547$			6918.30i			0.540777i	−0.962751	+	0.270389i	$0.912848\pi$
$548$		−	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	$-0.597380\pi$
$569$	−8175.67			−0.602359			−0.301179	+	0.953568i	$0.597380\pi$
$570$		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	$-0.794573\pi$
$577$		−	16673.4i		−	1.20299i	0.798878	−	0.601493i	$-0.205427\pi$
$578$		−	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	$-0.220226\pi$
$599$	22578.5			1.54012			0.770061	+	0.637970i	$0.220226\pi$
$600$		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	$-0.849859\pi$
$607$		−	13590.6i		−	0.908771i	0.890805	−	0.454385i	$-0.150141\pi$
$608$		−	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	$0.0575436\pi$
$613$			5457.57i			0.359591i	−0.983704	+	0.179796i	$0.942456\pi$
$614$	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	$-0.578659\pi$
$641$	−7939.36			−0.489214			−0.244607	+	0.969622i	$0.578659\pi$
$642$		0			0
$643$		−	7635.95i		−	0.468324i	−0.972198	−	0.234162i	$-0.924765\pi$
$643$		−	7635.95i		−	0.468324i	0.972198	−	0.234162i	$-0.0752347\pi$
$644$	−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	$-0.748510\pi$
$659$	−23812.3			−1.40758			−0.703789	+	0.710409i	$0.748510\pi$
$660$		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	$0.0983884\pi$
$673$			10622.1i			0.608396i	−0.952609	+	0.304198i	$0.901612\pi$
$674$	−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	$-0.495055\pi$
$701$	576.680			0.0310712			0.0155356	+	0.999879i	$0.495055\pi$
$702$		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	$-0.374606\pi$
$719$	14799.9			0.767655			0.383827	+	0.923405i	$0.374606\pi$
$720$		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	$-0.947717\pi$
$727$		−	6410.40i		−	0.327027i	0.986541	−	0.163513i	$-0.0522827\pi$
$728$			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	$-0.851347\pi$
$733$		−	17869.2i		−	0.900431i	0.892920	−	0.450215i	$-0.148653\pi$
$734$	−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	$0.0725344\pi$
$757$			9410.28i			0.451813i	−0.974149	+	0.225906i	$0.927466\pi$
$758$			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	$-0.189445\pi$
$761$	34767.1			1.65612			0.828060	+	0.560640i	$0.189445\pi$
$762$		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	$-0.722183\pi$
$787$		−	33829.1i		−	1.53225i	0.642693	−	0.766124i	$-0.277817\pi$
$788$			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	$-0.267857\pi$
$809$	30665.7			1.33269			0.666347	+	0.745642i	$0.267857\pi$
$810$		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	$-0.238611\pi$
$821$	34437.0			1.46390			0.731948	+	0.681361i	$0.238611\pi$
$822$		0			0
$823$		−	42801.8i		−	1.81285i	−0.422365	−	0.906426i	$-0.638800\pi$
$823$		−	42801.8i		−	1.81285i	0.422365	−	0.906426i	$-0.361200\pi$
$824$	−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	$-0.285462\pi$
$839$	30334.3			1.24822			0.624110	+	0.781336i	$0.285462\pi$
$840$		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	$-0.958139\pi$
$853$		−	6533.77i		−	0.262265i	0.991365	−	0.131133i	$-0.0418614\pi$
$854$	−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	$0.00851017\pi$
$877$			1388.56i			0.0534646i	−0.999643	+	0.0267323i	$0.991490\pi$
$878$		−	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	$-0.350540\pi$
$881$	23664.2			0.904957			0.452478	+	0.891775i	$0.350540\pi$
$882$		0			0
$883$		−	49607.6i		−	1.89063i	−0.326152	−	0.945317i	$-0.605752\pi$
$883$		−	49607.6i		−	1.89063i	0.326152	−	0.945317i	$-0.394248\pi$
$884$	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	$-0.931096\pi$
$907$		−	11733.8i		−	0.429563i	0.976662	−	0.214782i	$-0.0689040\pi$
$908$			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	$-0.719696\pi$
$911$	−35013.4			−1.27338			−0.636688	+	0.771122i	$0.719696\pi$
$912$		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	$-0.651117\pi$
$929$	−25886.8			−0.914229			−0.457114	+	0.889408i	$0.651117\pi$
$930$		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	$0.0352029\pi$
$937$			6331.15i			0.220736i	−0.993891	+	0.110368i	$0.964797\pi$
$938$			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	$-0.665959\pi$
$941$	−28754.7			−0.996149			−0.498075	+	0.867134i	$0.665959\pi$
$942$		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	$-0.733320\pi$
$967$		−	44695.1i		−	1.48635i	0.669099	−	0.743174i	$-0.266680\pi$
$968$		−	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	$-0.667914\pi$
$971$	−30462.4			−1.00678			−0.503391	+	0.864059i	$0.667914\pi$
$972$		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	$-0.726604\pi$
$997$		−	47669.3i		−	1.51425i	0.653273	−	0.757123i	$-0.273396\pi$
$998$		−	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.e.244.6		16
3.2	odd	2		405.4.b.f.244.11		16
5.2	odd	4		2025.4.a.bk.1.11		16
5.3	odd	4		2025.4.a.bk.1.6		16
5.4	even	2	inner	405.4.b.e.244.11		16
9.2	odd	6		135.4.j.a.64.11		32
9.4	even	3		45.4.j.a.34.11	yes	32
9.5	odd	6		135.4.j.a.19.6		32
9.7	even	3		45.4.j.a.4.6	✓	32
15.2	even	4		2025.4.a.bl.1.6		16
15.8	even	4		2025.4.a.bl.1.11		16
15.14	odd	2		405.4.b.f.244.6		16
45.4	even	6		45.4.j.a.34.6	yes	32
45.7	odd	12		225.4.e.g.76.6		32
45.13	odd	12		225.4.e.g.151.11		32
45.14	odd	6		135.4.j.a.19.11		32
45.22	odd	12		225.4.e.g.151.6		32
45.29	odd	6		135.4.j.a.64.6		32
45.34	even	6		45.4.j.a.4.11	yes	32
45.43	odd	12		225.4.e.g.76.11		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.j.a.4.6	✓	32	9.7	even	3
45.4.j.a.4.11	yes	32	45.34	even	6
45.4.j.a.34.6	yes	32	45.4	even	6
45.4.j.a.34.11	yes	32	9.4	even	3
135.4.j.a.19.6		32	9.5	odd	6
135.4.j.a.19.11		32	45.14	odd	6
135.4.j.a.64.6		32	45.29	odd	6
135.4.j.a.64.11		32	9.2	odd	6
225.4.e.g.76.6		32	45.7	odd	12
225.4.e.g.76.11		32	45.43	odd	12
225.4.e.g.151.6		32	45.22	odd	12
225.4.e.g.151.11		32	45.13	odd	12
405.4.b.e.244.6		16	1.1	even	1	trivial
405.4.b.e.244.11		16	5.4	even	2	inner
405.4.b.f.244.6		16	15.14	odd	2
405.4.b.f.244.11		16	3.2	odd	2
2025.4.a.bk.1.6		16	5.3	odd	4
2025.4.a.bk.1.11		16	5.2	odd	4
2025.4.a.bl.1.6		16	15.2	even	4
2025.4.a.bl.1.11		16	15.8	even	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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