LMFDB - Embedded newform 324.5.g.f.53.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [324,5,Mod(53,324)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("324.53"); S:= CuspForms(chi, 5); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(324, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 5])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	324.g (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,-52] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$33.4918680392$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
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} + 6242x^{12} + 11954017x^{8} + 7856017920x^{4} + 1485512441856 $ x^16 + 6242x^12 + 11954017x^8 + 7856017920*x^4 + 1485512441856
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{44} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			53.2
Root			$-4.65176 - 4.65176i$ of defining polynomial
Character	$\chi$	$=$	324.53
Dual form			324.5.g.f.269.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-22.8264 - 13.1788i) q^{5} +(-10.2177 - 17.6977i) q^{7} +(46.8767 - 27.0643i) q^{11} +(119.150 - 206.375i) q^{13} +195.038i q^{17} +59.1338 q^{19} +(-233.412 - 134.761i) q^{23} +(34.8634 + 60.3852i) q^{25} +(-1033.77 + 596.849i) q^{29} +(-186.923 + 323.761i) q^{31} +538.632i q^{35} -1793.37 q^{37} +(313.370 + 180.924i) q^{41} +(1620.96 + 2807.58i) q^{43} +(-3048.20 + 1759.88i) q^{47} +(991.695 - 1717.67i) q^{49} +1522.31i q^{53} -1426.70 q^{55} +(-5255.44 - 3034.23i) q^{59} +(-449.890 - 779.233i) q^{61} +(-5439.56 + 3140.53i) q^{65} +(3477.06 - 6022.45i) q^{67} +9017.89i q^{71} -3063.58 q^{73} +(-957.948 - 553.072i) q^{77} +(2789.76 + 4832.01i) q^{79} +(-3679.19 + 2124.18i) q^{83} +(2570.37 - 4452.01i) q^{85} +6532.00i q^{89} -4869.80 q^{91} +(-1349.81 - 779.315i) q^{95} +(2283.77 + 3955.60i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 52 q^{7} - 220 q^{13} - 616 q^{19} + 328 q^{25} + 1472 q^{31} + 2456 q^{37} - 388 q^{43} - 3432 q^{49} - 72 q^{55} - 1804 q^{61} + 2180 q^{67} - 9088 q^{73} + 5012 q^{79} + 5364 q^{85} - 3512 q^{91}+ \cdots + 2528 q^{97}+O(q^{100}) $ 16 * q - 52 * q^7 - 220 * q^13 - 616 * q^19 + 328 * q^25 + 1472 * q^31 + 2456 * q^37 - 388 * q^43 - 3432 * q^49 - 72 * q^55 - 1804 * q^61 + 2180 * q^67 - 9088 * q^73 + 5012 * q^79 + 5364 * q^85 - 3512 * q^91 + 2528 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$.

$n$	$163$	$245$
$\chi(n)$	$1$	$e\left(\frac{5}{6}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−22.8264	−	13.1788i	−0.913056	−	0.527153i	−0.0316432	−	0.999499i	$-0.510074\pi$
$5$	−22.8264	−	13.1788i	−0.913056	−	0.527153i	−0.881413	+	0.472346i	$0.843407\pi$
$6$		0			0
$7$	−10.2177	−	17.6977i	−0.208525	−	0.361177i	0.742725	−	0.669597i	$-0.233533\pi$
$7$	−10.2177	−	17.6977i	−0.208525	−	0.361177i	−0.951250	+	0.308420i	$0.900200\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	46.8767	−	27.0643i	0.387411	−	0.223672i	−0.293627	−	0.955920i	$-0.594862\pi$
$11$	46.8767	−	27.0643i	0.387411	−	0.223672i	0.681038	+	0.732248i	$0.261529\pi$
$12$		0			0
$13$	119.150	−	206.375i	0.705032	−	1.22115i	−0.261648	−	0.965163i	$-0.584266\pi$
$13$	119.150	−	206.375i	0.705032	−	1.22115i	0.966680	−	0.255988i	$-0.0824009\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			195.038i			0.674870i	0.941349	+	0.337435i	$0.109559\pi$
$17$			195.038i			0.674870i	−0.941349	+	0.337435i	$0.890441\pi$
$18$		0			0
$19$	59.1338			0.163806			0.0819028	−	0.996640i	$-0.473900\pi$
$19$	59.1338			0.163806			0.0819028	+	0.996640i	$0.473900\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−233.412	−	134.761i	−0.441233	−	0.254746i	0.262887	−	0.964827i	$-0.415325\pi$
$23$	−233.412	−	134.761i	−0.441233	−	0.254746i	−0.704121	+	0.710080i	$0.748659\pi$
$24$		0			0
$25$	34.8634	+	60.3852i	0.0557814	+	0.0966163i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1033.77	+	596.849i	−1.22922	+	0.709690i	−0.966867	−	0.255282i	$-0.917832\pi$
$29$	−1033.77	+	596.849i	−1.22922	+	0.709690i	−0.262353	+	0.964972i	$0.584498\pi$
$30$		0			0
$31$	−186.923	+	323.761i	−0.194509	+	0.336900i	−0.946740	−	0.322000i	$-0.895645\pi$
$31$	−186.923	+	323.761i	−0.194509	+	0.336900i	0.752230	+	0.658900i	$0.228978\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			538.632i			0.439699i
$36$		0			0
$37$	−1793.37			−1.30999			−0.654994	−	0.755634i	$-0.727329\pi$
$37$	−1793.37			−1.30999			−0.654994	+	0.755634i	$0.727329\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	313.370	+	180.924i	0.186419	+	0.107629i	0.590305	−	0.807180i	$-0.299007\pi$
$41$	313.370	+	180.924i	0.186419	+	0.107629i	−0.403886	+	0.914809i	$0.632341\pi$
$42$		0			0
$43$	1620.96	+	2807.58i	0.876666	+	1.51843i	0.854977	+	0.518666i	$0.173571\pi$
$43$	1620.96	+	2807.58i	0.876666	+	1.51843i	0.0216893	+	0.999765i	$0.493096\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3048.20	+	1759.88i	−1.37990	+	0.796686i	−0.992147	−	0.125077i	$-0.960082\pi$
$47$	−3048.20	+	1759.88i	−1.37990	+	0.796686i	−0.387753	+	0.921763i	$0.626749\pi$
$48$		0			0
$49$	991.695	−	1717.67i	0.413034	−	0.715396i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			1522.31i			0.541940i	0.962588	+	0.270970i	$0.0873443\pi$
$53$			1522.31i			0.541940i	−0.962588	+	0.270970i	$0.912656\pi$
$54$		0			0
$55$	−1426.70			−0.471637
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−5255.44	−	3034.23i	−1.50975	−	0.871655i	−0.999935	−	0.0113702i	$-0.996381\pi$
$59$	−5255.44	−	3034.23i	−1.50975	−	0.871655i	−0.509815	−	0.860284i	$-0.670286\pi$
$60$		0			0
$61$	−449.890	−	779.233i	−0.120906	−	0.209415i	0.799219	−	0.601039i	$-0.205247\pi$
$61$	−449.890	−	779.233i	−0.120906	−	0.209415i	−0.920125	+	0.391625i	$0.871913\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−5439.56	+	3140.53i	−1.28747	+	0.743320i
$66$		0			0
$67$	3477.06	−	6022.45i	0.774574	−	1.34160i	−0.160460	−	0.987042i	$-0.551298\pi$
$67$	3477.06	−	6022.45i	0.774574	−	1.34160i	0.935034	−	0.354559i	$-0.115369\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			9017.89i			1.78891i	0.447160	+	0.894454i	$0.352436\pi$
$71$			9017.89i			1.78891i	−0.447160	+	0.894454i	$0.647564\pi$
$72$		0			0
$73$	−3063.58			−0.574888			−0.287444	−	0.957797i	$-0.592806\pi$
$73$	−3063.58			−0.574888			−0.287444	+	0.957797i	$0.592806\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−957.948	−	553.072i	−0.161570	−	0.0932824i
$78$		0			0
$79$	2789.76	+	4832.01i	0.447006	+	0.774237i	0.998190	−	0.0601472i	$-0.0191570\pi$
$79$	2789.76	+	4832.01i	0.447006	+	0.774237i	−0.551184	+	0.834384i	$0.685824\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−3679.19	+	2124.18i	−0.534067	+	0.308344i	−0.742671	−	0.669656i	$-0.766441\pi$
$83$	−3679.19	+	2124.18i	−0.534067	+	0.308344i	0.208604	+	0.978000i	$0.433108\pi$
$84$		0			0
$85$	2570.37	−	4452.01i	0.355760	−	0.616195i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			6532.00i			0.824644i	0.911038	+	0.412322i	$0.135282\pi$
$89$			6532.00i			0.824644i	−0.911038	+	0.412322i	$0.864718\pi$
$90$		0			0
$91$	−4869.80			−0.588069
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1349.81	−	779.315i	−0.149564	−	0.0863507i
$96$		0			0
$97$	2283.77	+	3955.60i	0.242722	+	0.420406i	0.961489	−	0.274845i	$-0.0886265\pi$
$97$	2283.77	+	3955.60i	0.242722	+	0.420406i	−0.718767	+	0.695251i	$0.755293\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−16074.1	+	9280.36i	−1.57573	+	0.909750i	−0.580288	+	0.814411i	$0.697060\pi$
$101$	−16074.1	+	9280.36i	−1.57573	+	0.909750i	−0.995445	+	0.0953389i	$0.969607\pi$
$102$		0			0
$103$	−2535.31	+	4391.29i	−0.238978	+	0.413921i	−0.960421	−	0.278552i	$-0.910146\pi$
$103$	−2535.31	+	4391.29i	−0.238978	+	0.413921i	0.721444	+	0.692473i	$0.243479\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	16248.3i		−	1.41919i	−0.704609	−	0.709595i	$-0.748878\pi$
$107$		−	16248.3i		−	1.41919i	0.704609	−	0.709595i	$-0.251122\pi$
$108$		0			0
$109$	−2690.46			−0.226451			−0.113225	−	0.993569i	$-0.536118\pi$
$109$	−2690.46			−0.226451			−0.113225	+	0.993569i	$0.536118\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−4414.56	−	2548.74i	−0.345724	−	0.199604i	0.317076	−	0.948400i	$-0.397299\pi$
$113$	−4414.56	−	2548.74i	−0.345724	−	0.199604i	−0.662801	+	0.748796i	$0.730632\pi$
$114$		0			0
$115$	3551.98	+	6152.21i	0.268581	+	0.465195i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	3451.71	−	1992.84i	0.243747	−	0.140728i
$120$		0			0
$121$	−5855.55	+	10142.1i	−0.399942	+	0.692720i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			14635.7i			0.936685i
$126$		0			0
$127$	−11705.0			−0.725714			−0.362857	−	0.931845i	$-0.618199\pi$
$127$	−11705.0			−0.725714			−0.362857	+	0.931845i	$0.618199\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	10498.3	+	6061.17i	0.611751	+	0.353194i	0.773650	−	0.633613i	$-0.218429\pi$
$131$	10498.3	+	6061.17i	0.611751	+	0.353194i	−0.161900	+	0.986807i	$0.551762\pi$
$132$		0			0
$133$	−604.214	−	1046.53i	−0.0341576	−	0.0591628i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	20748.6	−	11979.2i	1.10547	−	0.638246i	0.167820	−	0.985818i	$-0.446327\pi$
$137$	20748.6	−	11979.2i	1.10547	−	0.638246i	0.937653	+	0.347572i	$0.112994\pi$
$138$		0			0
$139$	9773.69	−	16928.5i	0.505858	−	0.876172i	−0.494119	−	0.869394i	$-0.664509\pi$
$139$	9773.69	−	16928.5i	0.505858	−	0.876172i	0.999977	−	0.00677780i	$-0.00215746\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	12898.9i		−	0.630783i
$144$		0			0
$145$	31463.1			1.49646
$146$		0			0
$147$		0			0
$148$		0			0
$149$	8026.22	+	4633.94i	0.361525	+	0.208727i	0.669750	−	0.742587i	$-0.266401\pi$
$149$	8026.22	+	4633.94i	0.361525	+	0.208727i	−0.308224	+	0.951314i	$0.599735\pi$
$150$		0			0
$151$	−15891.7	−	27525.2i	−0.696973	−	1.20719i	−0.969511	−	0.245047i	$-0.921197\pi$
$151$	−15891.7	−	27525.2i	−0.696973	−	1.20719i	0.272539	−	0.962145i	$-0.412137\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	8533.58	−	4926.86i	0.355196	−	0.205072i
$156$		0			0
$157$	−8551.76	+	14812.1i	−0.346941	+	0.600920i	−0.985704	−	0.168483i	$-0.946113\pi$
$157$	−8551.76	+	14812.1i	−0.346941	+	0.600920i	0.638763	+	0.769403i	$0.279446\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			5507.80i			0.212484i
$162$		0			0
$163$	14463.6			0.544379			0.272189	−	0.962244i	$-0.412252\pi$
$163$	14463.6			0.544379			0.272189	+	0.962244i	$0.412252\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−21922.8	−	12657.1i	−0.786072	−	0.453839i	0.0525059	−	0.998621i	$-0.483279\pi$
$167$	−21922.8	−	12657.1i	−0.786072	−	0.453839i	−0.838578	+	0.544782i	$0.816612\pi$
$168$		0			0
$169$	−14113.2	−	24444.7i	−0.494141	−	0.855878i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	35424.7	−	20452.4i	1.18362	−	0.683365i	0.226773	−	0.973948i	$-0.427182\pi$
$173$	35424.7	−	20452.4i	1.18362	−	0.683365i	0.956850	+	0.290583i	$0.0938491\pi$
$174$		0			0
$175$	712.450	−	1234.00i	0.0232637	−	0.0402939i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	45433.1i		−	1.41797i	−0.705225	−	0.708984i	$-0.749154\pi$
$179$		−	45433.1i		−	1.41797i	0.705225	−	0.708984i	$-0.250846\pi$
$180$		0			0
$181$	41107.8			1.25478			0.627389	−	0.778706i	$-0.284124\pi$
$181$	41107.8			1.25478			0.627389	+	0.778706i	$0.284124\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	40936.3	+	23634.6i	1.19609	+	0.690564i
$186$		0			0
$187$	5278.55	+	9142.71i	0.150949	+	0.261452i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	52547.2	−	30338.1i	1.44040	−	0.831615i	0.442523	−	0.896757i	$-0.354083\pi$
$191$	52547.2	−	30338.1i	1.44040	−	0.831615i	0.997876	+	0.0651421i	$0.0207501\pi$
$192$		0			0
$193$	−22365.7	+	38738.4i	−0.600436	+	1.03999i	0.392319	+	0.919829i	$0.371673\pi$
$193$	−22365.7	+	38738.4i	−0.600436	+	1.03999i	−0.992755	+	0.120157i	$0.961660\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	464.194i		−	0.0119610i	−0.999982	−	0.00598050i	$-0.998096\pi$
$197$		−	464.194i		−	0.0119610i	0.999982	−	0.00598050i	$-0.00190366\pi$
$198$		0			0
$199$	−48133.7			−1.21547			−0.607733	−	0.794142i	$-0.707921\pi$
$199$	−48133.7			−1.21547			−0.607733	+	0.794142i	$0.707921\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	21125.7	+	12196.9i	0.512647	+	0.295977i
$204$		0			0
$205$	−4768.75	−	8259.71i	−0.113474	−	0.196543i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2772.00	−	1600.41i	0.0634601	−	0.0366387i
$210$		0			0
$211$	−16097.4	+	27881.5i	−0.361569	+	0.626256i	−0.988219	−	0.153045i	$-0.951092\pi$
$211$	−16097.4	+	27881.5i	−0.361569	+	0.626256i	0.626650	+	0.779300i	$0.284425\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	85449.2i		−	1.84855i
$216$		0			0
$217$	7639.74			0.162240
$218$		0			0
$219$		0			0
$220$		0			0
$221$	40250.8	+	23238.8i	0.824119	+	0.475805i
$222$		0			0
$223$	−1146.07	−	1985.06i	−0.0230464	−	0.0399175i	0.854272	−	0.519826i	$-0.174003\pi$
$223$	−1146.07	−	1985.06i	−0.0230464	−	0.0399175i	−0.877319	+	0.479908i	$0.840670\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	23205.1	−	13397.5i	0.450331	−	0.259998i	−0.257639	−	0.966241i	$-0.582945\pi$
$227$	23205.1	−	13397.5i	0.450331	−	0.259998i	0.707970	+	0.706243i	$0.249611\pi$
$228$		0			0
$229$	−24030.0	+	41621.3i	−0.458230	+	0.793678i	−0.998868	−	0.0475778i	$-0.984850\pi$
$229$	−24030.0	+	41621.3i	−0.458230	+	0.793678i	0.540637	+	0.841256i	$0.318183\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			27107.9i			0.499326i	0.968333	+	0.249663i	$0.0803199\pi$
$233$			27107.9i			0.499326i	−0.968333	+	0.249663i	$0.919680\pi$
$234$		0			0
$235$	92772.6			1.67990
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−53063.5	−	30636.2i	−0.928967	−	0.536339i	−0.0424821	−	0.999097i	$-0.513527\pi$
$239$	−53063.5	−	30636.2i	−0.928967	−	0.536339i	−0.886485	+	0.462758i	$0.846860\pi$
$240$		0			0
$241$	−18925.0	−	32779.0i	−0.325837	−	0.564367i	0.655844	−	0.754896i	$-0.272313\pi$
$241$	−18925.0	−	32779.0i	−0.325837	−	0.564367i	−0.981681	+	0.190529i	$0.938980\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−45273.7	+	26138.8i	−0.754247	+	0.435465i
$246$		0			0
$247$	7045.82	−	12203.7i	0.115488	−	0.200032i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	25694.6i		−	0.407844i	−0.978987	−	0.203922i	$-0.934631\pi$
$251$		−	25694.6i		−	0.407844i	0.978987	−	0.203922i	$-0.0653688\pi$
$252$		0			0
$253$	−14588.8			−0.227918
$254$		0			0
$255$		0			0
$256$		0			0
$257$	22272.0	+	12858.7i	0.337204	+	0.194685i	0.659035	−	0.752113i	$-0.270965\pi$
$257$	22272.0	+	12858.7i	0.337204	+	0.194685i	−0.321831	+	0.946797i	$0.604298\pi$
$258$		0			0
$259$	18324.2	+	31738.5i	0.273166	+	0.473137i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	86343.9	−	49850.7i	1.24830	−	0.720708i	0.277532	−	0.960716i	$-0.410483\pi$
$263$	86343.9	−	49850.7i	1.24830	−	0.720708i	0.970771	+	0.240008i	$0.0771501\pi$
$264$		0			0
$265$	20062.2	−	34748.8i	0.285685	−	0.494821i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			126040.i			1.74182i	0.491445	+	0.870909i	$0.336469\pi$
$269$			126040.i			1.74182i	−0.491445	+	0.870909i	$0.663531\pi$
$270$		0			0
$271$	−73237.4			−0.997228			−0.498614	−	0.866824i	$-0.666157\pi$
$271$	−73237.4			−0.997228			−0.498614	+	0.866824i	$0.666157\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3268.56	+	1887.10i	0.0432206	+	0.0249535i
$276$		0			0
$277$	−52217.7	−	90443.8i	−0.680547	−	1.17874i	−0.974814	−	0.223019i	$-0.928409\pi$
$277$	−52217.7	−	90443.8i	−0.680547	−	1.17874i	0.294267	−	0.955723i	$-0.404925\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−47540.0	+	27447.2i	−0.602069	+	0.347605i	−0.769855	−	0.638218i	$-0.779672\pi$
$281$	−47540.0	+	27447.2i	−0.602069	+	0.347605i	0.167786	+	0.985823i	$0.446338\pi$
$282$		0			0
$283$	57600.3	−	99766.7i	0.719204	−	1.24570i	−0.242112	−	0.970248i	$-0.577840\pi$
$283$	57600.3	−	99766.7i	0.719204	−	1.24570i	0.961316	−	0.275449i	$-0.0888266\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	7394.56i		−	0.0897735i
$288$		0			0
$289$	45481.4			0.544550
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−31631.0	−	18262.2i	−0.368449	−	0.212724i	0.304332	−	0.952566i	$-0.401567\pi$
$293$	−31631.0	−	18262.2i	−0.368449	−	0.212724i	−0.672781	+	0.739842i	$0.734900\pi$
$294$		0			0
$295$	79975.2	+	138521.i	0.918991	+	1.59174i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−55622.4	+	32113.6i	−0.622167	+	0.359209i
$300$		0			0
$301$	33125.0	−	57374.2i	0.365614	−	0.633263i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			23716.1i			0.254944i
$306$		0			0
$307$	−95335.4			−1.01153			−0.505763	−	0.862672i	$-0.668789\pi$
$307$	−95335.4			−1.01153			−0.505763	+	0.862672i	$0.668789\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−157452.	−	90904.7i	−1.62789	−	0.939865i	−0.984720	−	0.174146i	$-0.944283\pi$
$311$	−157452.	−	90904.7i	−1.62789	−	0.939865i	−0.643175	−	0.765719i	$-0.722383\pi$
$312$		0			0
$313$	−30195.7	−	52300.5i	−0.308217	−	0.533848i	0.669755	−	0.742582i	$-0.266399\pi$
$313$	−30195.7	−	52300.5i	−0.308217	−	0.533848i	−0.977972	+	0.208734i	$0.933066\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−82260.6	+	47493.2i	−0.818603	+	0.472621i	−0.849935	−	0.526888i	$-0.823359\pi$
$317$	−82260.6	+	47493.2i	−0.818603	+	0.472621i	0.0313312	+	0.999509i	$0.490025\pi$
$318$		0			0
$319$	−32306.6	+	55956.7i	−0.317475	+	0.549883i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			11533.3i			0.110548i
$324$		0			0
$325$	16616.0			0.157311
$326$		0			0
$327$		0			0
$328$		0			0
$329$	62291.5	+	35964.0i	0.575489	+	0.332258i
$330$		0			0
$331$	69064.7	+	119624.i	0.630377	+	1.09184i	0.987475	+	0.157778i	$0.0504330\pi$
$331$	69064.7	+	119624.i	0.630377	+	1.09184i	−0.357098	+	0.934067i	$0.616234\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−158738.	+	91647.3i	−1.41446	+	0.816639i
$336$		0			0
$337$	−40400.0	+	69974.8i	−0.355731	+	0.616143i	−0.987243	−	0.159223i	$-0.949101\pi$
$337$	−40400.0	+	69974.8i	−0.355731	+	0.616143i	0.631512	+	0.775366i	$0.282435\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			20235.8i			0.174025i
$342$		0			0
$343$	−89597.2			−0.761563
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−10100.6	−	5831.59i	−0.0838858	−	0.0484315i	0.457470	−	0.889225i	$-0.348756\pi$
$347$	−10100.6	−	5831.59i	−0.0838858	−	0.0484315i	−0.541356	+	0.840793i	$0.682089\pi$
$348$		0			0
$349$	78250.7	+	135534.i	0.642447	+	1.11275i	0.984885	+	0.173210i	$0.0554140\pi$
$349$	78250.7	+	135534.i	0.642447	+	1.11275i	−0.342438	+	0.939540i	$0.611253\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−170384.	+	98371.0i	−1.36735	+	0.789437i	−0.990588	−	0.136875i	$-0.956294\pi$
$353$	−170384.	+	98371.0i	−1.36735	+	0.789437i	−0.376757	+	0.926312i	$0.622961\pi$
$354$		0			0
$355$	118845.	−	205846.i	0.943029	−	1.63337i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			36095.4i			0.280068i	0.990147	+	0.140034i	$0.0447211\pi$
$359$			36095.4i			0.280068i	−0.990147	+	0.140034i	$0.955279\pi$
$360$		0			0
$361$	−126824.			−0.973168
$362$		0			0
$363$		0			0
$364$		0			0
$365$	69930.5	+	40374.4i	0.524905	+	0.303054i
$366$		0			0
$367$	−20774.0	−	35981.5i	−0.154236	−	0.267145i	0.778544	−	0.627590i	$-0.215958\pi$
$367$	−20774.0	−	35981.5i	−0.154236	−	0.267145i	−0.932781	+	0.360444i	$0.882625\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	26941.3	−	15554.6i	0.195736	−	0.113008i
$372$		0			0
$373$	−87236.2	+	151098.i	−0.627017	+	1.08602i	0.361130	+	0.932515i	$0.382391\pi$
$373$	−87236.2	+	151098.i	−0.627017	+	1.08602i	−0.988147	+	0.153510i	$0.950942\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			284460.i			2.00142i
$378$		0			0
$379$	−164788.			−1.14722			−0.573612	−	0.819127i	$-0.694458\pi$
$379$	−164788.			−1.14722			−0.573612	+	0.819127i	$0.694458\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−10981.9	−	6340.40i	−0.0748652	−	0.0432234i	0.462100	−	0.886828i	$-0.347096\pi$
$383$	−10981.9	−	6340.40i	−0.0748652	−	0.0432234i	−0.536965	+	0.843604i	$0.680429\pi$
$384$		0			0
$385$	14577.7	+	25249.3i	0.0983483	+	0.170344i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−55798.5	+	32215.3i	−0.368743	+	0.212894i	−0.672909	−	0.739725i	$-0.734955\pi$
$389$	−55798.5	+	32215.3i	−0.368743	+	0.212894i	0.304166	+	0.952619i	$0.401622\pi$
$390$		0			0
$391$	26283.4	−	45524.2i	0.171921	−	0.297775i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	147063.i		−	0.942563i
$396$		0			0
$397$	143446.			0.910135			0.455068	−	0.890457i	$-0.349615\pi$
$397$	143446.			0.910135			0.455068	+	0.890457i	$0.349615\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−183922.	−	106187.i	−1.14378	−	0.660364i	−0.196419	−	0.980520i	$-0.562931\pi$
$401$	−183922.	−	106187.i	−1.14378	−	0.660364i	−0.947365	+	0.320156i	$0.896265\pi$
$402$		0			0
$403$	44544.0	+	77152.5i	0.274271	+	0.475050i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−84067.4	+	48536.3i	−0.507503	+	0.293007i
$408$		0			0
$409$	106107.	−	183782.i	0.634301	−	1.09864i	−0.352361	−	0.935864i	$-0.614621\pi$
$409$	106107.	−	183782.i	0.634301	−	1.09864i	0.986663	−	0.162778i	$-0.0520454\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			124012.i			0.727048i
$414$		0			0
$415$	111977.			0.650178
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−105265.	−	60774.6i	−0.599591	−	0.346174i	0.169290	−	0.985566i	$-0.445853\pi$
$419$	−105265.	−	60774.6i	−0.599591	−	0.346174i	−0.768880	+	0.639393i	$0.779186\pi$
$420$		0			0
$421$	−31505.8	−	54569.6i	−0.177757	−	0.307884i	0.763355	−	0.645979i	$-0.223551\pi$
$421$	−31505.8	−	54569.6i	−0.177757	−	0.307884i	−0.941112	+	0.338095i	$0.890217\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−11777.4	+	6799.67i	−0.0652035	+	0.0376452i
$426$		0			0
$427$	−9193.73	+	15924.0i	−0.0504239	+	0.0873367i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	128883.i		−	0.693812i	−0.937900	−	0.346906i	$-0.887232\pi$
$431$		−	128883.i		−	0.693812i	0.937900	−	0.346906i	$-0.112768\pi$
$432$		0			0
$433$	6365.28			0.0339502			0.0169751	−	0.999856i	$-0.494596\pi$
$433$	6365.28			0.0339502			0.0169751	+	0.999856i	$0.494596\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−13802.6	−	7968.92i	−0.0722765	−	0.0417289i
$438$		0			0
$439$	−175148.	−	303365.i	−0.908815	−	1.57411i	−0.815713	−	0.578457i	$-0.803655\pi$
$439$	−175148.	−	303365.i	−0.908815	−	1.57411i	−0.0931020	−	0.995657i	$-0.529678\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	161380.	−	93172.9i	0.822324	−	0.474769i	−0.0288935	−	0.999582i	$-0.509198\pi$
$443$	161380.	−	93172.9i	0.822324	−	0.474769i	0.851217	+	0.524814i	$0.175865\pi$
$444$		0			0
$445$	86084.2	−	149102.i	0.434714	−	0.752946i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			46711.0i			0.231700i	0.993267	+	0.115850i	$0.0369593\pi$
$449$			46711.0i			0.231700i	−0.993267	+	0.115850i	$0.963041\pi$
$450$		0			0
$451$	19586.3			0.0962943
$452$		0			0
$453$		0			0
$454$		0			0
$455$	111160.	+	64178.2i	0.536940	+	0.310002i
$456$		0			0
$457$	145112.	+	251341.i	0.694816	+	1.20346i	0.970243	+	0.242134i	$0.0778474\pi$
$457$	145112.	+	251341.i	0.694816	+	1.20346i	−0.275427	+	0.961322i	$0.588819\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	205476.	−	118632.i	0.966850	−	0.558211i	0.0685752	−	0.997646i	$-0.478155\pi$
$461$	205476.	−	118632.i	0.966850	−	0.558211i	0.898274	+	0.439435i	$0.144821\pi$
$462$		0			0
$463$	63082.2	−	109262.i	0.294269	−	0.509689i	−0.680545	−	0.732706i	$-0.738257\pi$
$463$	63082.2	−	109262.i	0.294269	−	0.509689i	0.974815	+	0.223017i	$0.0715905\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	20073.4i		−	0.0920421i	−0.998940	−	0.0460210i	$-0.985346\pi$
$467$		−	20073.4i		−	0.0920421i	0.998940	−	0.0460210i	$-0.0146541\pi$
$468$		0			0
$469$	−142111.			−0.646073
$470$		0			0
$471$		0			0
$472$		0			0
$473$	151970.	+	87740.0i	0.679260	+	0.392171i
$474$		0			0
$475$	2061.61	+	3570.81i	0.00913731	+	0.0158263i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−1235.35	+	713.231i	−0.00538418	+	0.00310856i	−0.502690	−	0.864467i	$-0.667656\pi$
$479$	−1235.35	+	713.231i	−0.00538418	+	0.00310856i	0.497305	+	0.867575i	$0.334323\pi$
$480$		0			0
$481$	−213681.	+	370107.i	−0.923583	+	1.59969i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	120390.i		−	0.511806i
$486$		0			0
$487$	74085.9			0.312376			0.156188	−	0.987727i	$-0.450079\pi$
$487$	74085.9			0.312376			0.156188	+	0.987727i	$0.450079\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	98342.5	+	56778.1i	0.407923	+	0.235514i	0.689897	−	0.723908i	$-0.257656\pi$
$491$	98342.5	+	56778.1i	0.407923	+	0.235514i	−0.281974	+	0.959422i	$0.590989\pi$
$492$		0			0
$493$	−116408.	−	201625.i	−0.478949	−	0.829564i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	159595.	−	92142.5i	0.646112	−	0.373033i
$498$		0			0
$499$	−140398.	+	243176.i	−0.563845	+	0.976608i	0.433311	+	0.901244i	$0.357345\pi$
$499$	−140398.	+	243176.i	−0.563845	+	0.976608i	−0.997156	+	0.0753637i	$0.975988\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	89107.8i		−	0.352192i	−0.984373	−	0.176096i	$-0.943653\pi$
$503$		−	89107.8i		−	0.352192i	0.984373	−	0.176096i	$-0.0563470\pi$
$504$		0			0
$505$	489217.			1.91831
$506$		0			0
$507$		0			0
$508$		0			0
$509$	102156.	+	58980.1i	0.394303	+	0.227651i	0.684023	−	0.729460i	$-0.260229\pi$
$509$	102156.	+	58980.1i	0.394303	+	0.227651i	−0.289720	+	0.957112i	$0.593562\pi$
$510$		0			0
$511$	31302.9	+	54218.1i	0.119879	+	0.207636i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	115744.	−	66825.0i	0.436400	−	0.251956i
$516$		0			0
$517$	−95259.7	+	164995.i	−0.356392	+	0.617289i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	316710.i		−	1.16677i	−0.812194	−	0.583387i	$-0.801727\pi$
$521$		−	316710.i		−	1.16677i	0.812194	−	0.583387i	$-0.198273\pi$
$522$		0			0
$523$	−390205.			−1.42656			−0.713280	−	0.700880i	$-0.752791\pi$
$523$	−390205.			−1.42656			−0.713280	+	0.700880i	$0.752791\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−63145.5	−	36457.1i	−0.227364	−	0.131268i
$528$		0			0
$529$	−103600.	−	179440.i	−0.370209	−	0.641220i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	74676.4	−	43114.4i	0.262863	−	0.151764i
$534$		0			0
$535$	−214134.	+	370891.i	−0.748131	+	1.29580i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	107358.i		−	0.369536i
$540$		0			0
$541$	99617.9			0.340363			0.170182	−	0.985413i	$-0.445565\pi$
$541$	99617.9			0.340363			0.170182	+	0.985413i	$0.445565\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	61413.6	+	35457.1i	0.206762	+	0.119374i
$546$		0			0
$547$	−124024.	−	214816.i	−0.414506	−	0.717945i	0.580871	−	0.813996i	$-0.302712\pi$
$547$	−124024.	−	214816.i	−0.414506	−	0.717945i	−0.995376	+	0.0960507i	$0.969379\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−61131.0	+	35294.0i	−0.201353	+	0.116251i
$552$		0			0
$553$	57010.2	−	98744.5i	0.186424	−	0.322896i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	242800.i		−	0.782598i	−0.920264	−	0.391299i	$-0.872026\pi$
$557$		−	242800.i		−	0.782598i	0.920264	−	0.391299i	$-0.127974\pi$
$558$		0			0
$559$	772551.			2.47231
$560$		0			0
$561$		0			0
$562$		0			0
$563$	490600.	+	283248.i	1.54778	+	0.893613i	0.998311	+	0.0581021i	$0.0185049\pi$
$563$	490600.	+	283248.i	1.54778	+	0.893613i	0.549473	+	0.835511i	$0.314828\pi$
$564$		0			0
$565$	67179.0	+	116357.i	0.210444	+	0.364500i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	487139.	−	281250.i	1.50462	−	0.868695i	0.504638	−	0.863331i	$-0.331626\pi$
$569$	487139.	−	281250.i	1.50462	−	0.868695i	0.999986	−	0.00536413i	$-0.00170746\pi$
$570$		0			0
$571$	77999.3	−	135099.i	0.239231	−	0.414361i	−0.721263	−	0.692662i	$-0.756438\pi$
$571$	77999.3	−	135099.i	0.239231	−	0.414361i	0.960494	+	0.278301i	$0.0897712\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	18792.9i		−	0.0568404i
$576$		0			0
$577$	−374817.			−1.12582			−0.562908	−	0.826519i	$-0.690318\pi$
$577$	−374817.			−1.12582			−0.562908	+	0.826519i	$0.690318\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	75186.1	+	43408.7i	0.222733	+	0.128595i
$582$		0			0
$583$	41200.2	+	71360.8i	0.121217	+	0.209953i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	456812.	−	263740.i	1.32575	−	0.765421i	0.341109	−	0.940024i	$-0.389198\pi$
$587$	456812.	−	263740.i	1.32575	−	0.765421i	0.984639	+	0.174603i	$0.0558642\pi$
$588$		0			0
$589$	−11053.5	+	19145.2i	−0.0318617	+	0.0551861i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	316522.i		−	0.900108i	−0.893001	−	0.450054i	$-0.851405\pi$
$593$		−	316522.i		−	0.900108i	0.893001	−	0.450054i	$-0.148595\pi$
$594$		0			0
$595$	−105053.			−0.296740
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−328816.	−	189842.i	−0.916429	−	0.529100i	−0.0339348	−	0.999424i	$-0.510804\pi$
$599$	−328816.	−	189842.i	−0.916429	−	0.529100i	−0.882494	+	0.470324i	$0.844137\pi$
$600$		0			0
$601$	−220839.	−	382504.i	−0.611402	−	1.05898i	−0.991004	−	0.133829i	$-0.957273\pi$
$601$	−220839.	−	382504.i	−0.611402	−	1.05898i	0.379603	−	0.925150i	$-0.376061\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	267322.	−	154339.i	0.730339	−	0.421662i
$606$		0			0
$607$	41181.6	−	71328.6i	0.111770	−	0.193591i	−0.804714	−	0.593663i	$-0.797681\pi$
$607$	41181.6	−	71328.6i	0.111770	−	0.193591i	0.916484	+	0.400071i	$0.131015\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			838762.i			2.24676i
$612$		0			0
$613$	154087.			0.410057			0.205029	−	0.978756i	$-0.434271\pi$
$613$	154087.			0.410057			0.205029	+	0.978756i	$0.434271\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−293845.	−	169652.i	−0.771877	−	0.445644i	0.0616665	−	0.998097i	$-0.480358\pi$
$617$	−293845.	−	169652.i	−0.771877	−	0.445644i	−0.833544	+	0.552453i	$0.813692\pi$
$618$		0			0
$619$	61345.8	+	106254.i	0.160105	+	0.277309i	0.934906	−	0.354895i	$-0.115484\pi$
$619$	61345.8	+	106254.i	0.160105	+	0.277309i	−0.774801	+	0.632205i	$0.782150\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	115601.	−	66742.3i	0.297842	−	0.171959i
$624$		0			0
$625$	214671.	−	371821.i	0.549558	−	0.951863i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	349775.i		−	0.884071i
$630$		0			0
$631$	517548.			1.29985			0.649923	−	0.760000i	$-0.274801\pi$
$631$	517548.			1.29985			0.649923	+	0.760000i	$0.274801\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	267184.	+	154259.i	0.662617	+	0.382562i
$636$		0			0
$637$	−236322.	−	409322.i	−0.582405	−	1.00876i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−574464.	+	331667.i	−1.39813	+	0.807209i	−0.994196	−	0.107580i	$-0.965690\pi$
$641$	−574464.	+	331667.i	−1.39813	+	0.807209i	−0.403931	+	0.914789i	$0.632356\pi$
$642$		0			0
$643$	−194488.	+	336864.i	−0.470405	+	0.814765i	−0.999427	−	0.0338427i	$-0.989225\pi$
$643$	−194488.	+	336864.i	−0.470405	+	0.814765i	0.529022	+	0.848608i	$0.322559\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	464654.i		−	1.11000i	−0.831852	−	0.554998i	$-0.812719\pi$
$647$		−	464654.i		−	1.11000i	0.831852	−	0.554998i	$-0.187281\pi$
$648$		0			0
$649$	−328477.			−0.779858
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−293777.	−	169612.i	−0.688955	−	0.397768i	0.114265	−	0.993450i	$-0.463549\pi$
$653$	−293777.	−	169612.i	−0.688955	−	0.397768i	−0.803220	+	0.595682i	$0.796882\pi$
$654$		0			0
$655$	−159758.	−	276709.i	−0.372375	−	0.644973i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	186995.	−	107961.i	0.430584	−	0.248598i	−0.269011	−	0.963137i	$-0.586697\pi$
$659$	186995.	−	107961.i	0.430584	−	0.248598i	0.699596	+	0.714539i	$0.253364\pi$
$660$		0			0
$661$	−19526.3	+	33820.5i	−0.0446906	+	0.0774065i	−0.887505	−	0.460797i	$-0.847564\pi$
$661$	−19526.3	+	33820.5i	−0.0446906	+	0.0774065i	0.842815	+	0.538204i	$0.180897\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			31851.4i			0.0720253i
$666$		0			0
$667$	321727.			0.723163
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−42178.8	−	24351.9i	−0.0936804	−	0.0540864i
$672$		0			0
$673$	416432.	+	721282.i	0.919421	+	1.59248i	0.800297	+	0.599604i	$0.204675\pi$
$673$	416432.	+	721282.i	0.919421	+	1.59248i	0.119124	+	0.992879i	$0.461992\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−293572.	+	169494.i	−0.640527	+	0.369809i	−0.784818	−	0.619727i	$-0.787243\pi$
$677$	−293572.	+	169494.i	−0.640527	+	0.369809i	0.144290	+	0.989535i	$0.453910\pi$
$678$		0			0
$679$	46669.9	−	80834.6i	0.101227	−	0.175331i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			928607.i			1.99063i	0.0966825	+	0.995315i	$0.469177\pi$
$683$			928607.i			1.99063i	−0.0966825	+	0.995315i	$0.530823\pi$
$684$		0			0
$685$	−631489.			−1.34581
$686$		0			0
$687$		0			0
$688$		0			0
$689$	314166.	+	181384.i	0.661790	+	0.382085i
$690$		0			0
$691$	−109635.	−	189894.i	−0.229612	−	0.397700i	0.728081	−	0.685491i	$-0.240412\pi$
$691$	−109635.	−	189894.i	−0.229612	−	0.397700i	−0.957693	+	0.287791i	$0.907079\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−446196.	+	257612.i	−0.923754	+	0.533330i
$696$		0			0
$697$	−35287.0	+	61119.0i	−0.0726356	+	0.125809i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	118463.i		−	0.241071i	−0.992709	−	0.120536i	$-0.961539\pi$
$701$		−	118463.i		−	0.241071i	0.992709	−	0.120536i	$-0.0384612\pi$
$702$		0			0
$703$	−106049.			−0.214583
$704$		0			0
$705$		0			0
$706$		0			0
$707$	328481.	+	189649.i	0.657161	+	0.379412i
$708$		0			0
$709$	407445.	+	705716.i	0.810544	+	1.40390i	0.912484	+	0.409113i	$0.134162\pi$
$709$	407445.	+	705716.i	0.810544	+	1.40390i	−0.101940	+	0.994791i	$0.532505\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	87260.4	−	50379.8i	0.171648	−	0.0991009i
$714$		0			0
$715$	−169992.	+	294435.i	−0.332519	+	0.575941i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	789.716i		−	0.00152761i	−1.00000		0.000763806i	$-0.999757\pi$
$719$		−	789.716i		−	0.00152761i	1.00000		0.000763806i	$-0.000243127\pi$
$720$		0			0
$721$	103621.			0.199332
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−72081.7	−	41616.4i	−0.137135	−	0.0791751i
$726$		0			0
$727$	300646.	+	520735.i	0.568836	+	0.985253i	0.996681	+	0.0814008i	$0.0259394\pi$
$727$	300646.	+	520735.i	0.568836	+	0.985253i	−0.427846	+	0.903852i	$0.640727\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−547583.	+	316147.i	−1.02474	+	0.591636i
$732$		0			0
$733$	504618.	−	874024.i	0.939192	−	1.62673i	0.172211	−	0.985060i	$-0.444909\pi$
$733$	504618.	−	874024.i	0.939192	−	1.62673i	0.766982	−	0.641669i	$-0.221758\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	376417.i		−	0.693001i
$738$		0			0
$739$	−753999.			−1.38065			−0.690323	−	0.723502i	$-0.742531\pi$
$739$	−753999.			−1.38065			−0.690323	+	0.723502i	$0.742531\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−171330.	−	98917.5i	−0.310353	−	0.179182i	0.336731	−	0.941601i	$-0.390679\pi$
$743$	−171330.	−	98917.5i	−0.310353	−	0.179182i	−0.647085	+	0.762418i	$0.724012\pi$
$744$		0			0
$745$	−122140.	−	211552.i	−0.220062	−	0.381158i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−287557.	+	166021.i	−0.512578	+	0.295937i
$750$		0			0
$751$	−386043.	+	668647.i	−0.684473	+	1.18554i	0.289129	+	0.957290i	$0.406634\pi$
$751$	−386043.	+	668647.i	−0.684473	+	1.18554i	−0.973602	+	0.228252i	$0.926699\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			837735.i			1.46965i
$756$		0			0
$757$	−459844.			−0.802452			−0.401226	−	0.915979i	$-0.631416\pi$
$757$	−459844.			−0.802452			−0.401226	+	0.915979i	$0.631416\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−744438.	−	429801.i	−1.28546	−	0.742161i	−0.307620	−	0.951509i	$-0.599533\pi$
$761$	−744438.	−	429801.i	−1.28546	−	0.742161i	−0.977841	+	0.209348i	$0.932866\pi$
$762$		0			0
$763$	27490.4	+	47614.9i	0.0472207	+	0.0817887i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1.25238e6	+	723060.i	−2.12885	+	1.22909i
$768$		0			0
$769$	−2959.64	+	5126.24i	−0.00500479	+	0.00866855i	−0.868517	−	0.495659i	$-0.834926\pi$
$769$	−2959.64	+	5126.24i	−0.00500479	+	0.00866855i	0.863512	+	0.504328i	$0.168260\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	80749.0i		−	0.135138i	−0.997715	−	0.0675691i	$-0.978476\pi$
$773$		−	80749.0i		−	0.135138i	0.997715	−	0.0675691i	$-0.0215243\pi$
$774$		0			0
$775$	−26067.1			−0.0434000
$776$		0			0
$777$		0			0
$778$		0			0
$779$	18530.8	+	10698.8i	0.0305365	+	0.0176302i
$780$		0			0
$781$	244063.	+	422729.i	0.400128	+	0.693042i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	390412.	−	225404.i	0.633554	−	0.365783i
$786$		0			0
$787$	530526.	−	918897.i	0.856558	−	1.48360i	−0.0186336	−	0.999826i	$-0.505932\pi$
$787$	530526.	−	918897.i	0.856558	−	1.48360i	0.875192	−	0.483776i	$-0.160735\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			104170.i			0.166490i
$792$		0			0
$793$	−214419.			−0.340970
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−805224.	−	464896.i	−1.26765	−	0.731879i	−0.293109	−	0.956079i	$-0.594690\pi$
$797$	−805224.	−	464896.i	−1.26765	−	0.731879i	−0.974543	+	0.224200i	$0.928023\pi$
$798$		0			0
$799$	−343242.	−	594513.i	−0.537660	−	0.931254i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−143610.	+	82913.5i	−0.222718	+	0.128586i
$804$		0			0
$805$	72586.4	−	125723.i	0.112012	−	0.194010i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			181072.i			0.276665i	0.990386	+	0.138333i	$0.0441743\pi$
$809$			181072.i			0.276665i	−0.990386	+	0.138333i	$0.955826\pi$
$810$		0			0
$811$	49745.9			0.0756338			0.0378169	−	0.999285i	$-0.487960\pi$
$811$	49745.9			0.0756338			0.0378169	+	0.999285i	$0.487960\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−330152.	−	190613.i	−0.497048	−	0.286971i
$816$		0			0
$817$	95853.4	+	166023.i	0.143603	+	0.248728i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	6020.80	−	3476.11i	0.00893239	−	0.00515712i	−0.495527	−	0.868592i	$-0.665025\pi$
$821$	6020.80	−	3476.11i	0.00893239	−	0.00515712i	0.504460	+	0.863435i	$0.331692\pi$
$822$		0			0
$823$	379126.	−	656665.i	0.559737	−	0.969493i	−0.437781	−	0.899082i	$-0.644236\pi$
$823$	379126.	−	656665.i	0.559737	−	0.969493i	0.997518	−	0.0704111i	$-0.0224311\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			641019.i			0.937260i	0.883394	+	0.468630i	$0.155252\pi$
$827$			641019.i			0.937260i	−0.883394	+	0.468630i	$0.844748\pi$
$828$		0			0
$829$	549710.			0.799879			0.399940	−	0.916541i	$-0.369031\pi$
$829$	549710.			0.799879			0.399940	+	0.916541i	$0.369031\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	335009.	+	193418.i	0.482800	+	0.278745i
$834$		0			0
$835$	333612.	+	577833.i	0.478485	+	0.828761i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−794215.	+	458540.i	−1.12827	+	0.651409i	−0.943500	−	0.331372i	$-0.892488\pi$
$839$	−794215.	+	458540.i	−1.12827	+	0.651409i	−0.184773	+	0.982781i	$0.559155\pi$
$840$		0			0
$841$	358818.	−	621491.i	0.507320	−	0.878704i
$842$		0			0
$843$		0			0
$844$		0			0
$845$			743980.i			1.04195i
$846$		0			0
$847$	239322.			0.333592
$848$		0			0
$849$		0			0
$850$		0			0
$851$	418595.	+	241676.i	0.578010	+	0.333714i
$852$		0			0
$853$	500711.	+	867257.i	0.688160	+	1.19193i	0.972433	+	0.233184i	$0.0749146\pi$
$853$	500711.	+	867257.i	0.688160	+	1.19193i	−0.284273	+	0.958743i	$0.591752\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	221771.	−	128039.i	0.301955	−	0.174334i	−0.341366	−	0.939931i	$-0.610889\pi$
$857$	221771.	−	128039.i	0.301955	−	0.174334i	0.643321	+	0.765597i	$0.277556\pi$
$858$		0			0
$859$	163149.	−	282582.i	0.221105	−	0.382964i	−0.734039	−	0.679107i	$-0.762367\pi$
$859$	163149.	−	282582.i	0.221105	−	0.382964i	0.955144	+	0.296143i	$0.0957005\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			952268.i			1.27861i	0.768954	+	0.639304i	$0.220778\pi$
$863$			952268.i			1.27861i	−0.768954	+	0.639304i	$0.779222\pi$
$864$		0			0
$865$	−1.07816e6			−1.44095
$866$		0			0
$867$		0			0
$868$		0			0
$869$	261550.	+	151006.i	0.346350	+	0.199965i
$870$		0			0
$871$	−828587.	−	1.43516e6i	−1.09220	−	1.89174i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	259018.	−	149544.i	0.338309	−	0.195323i
$876$		0			0
$877$	10759.1	−	18635.3i	0.0139886	−	0.0242290i	−0.858946	−	0.512066i	$-0.828880\pi$
$877$	10759.1	−	18635.3i	0.0139886	−	0.0242290i	0.872935	+	0.487837i	$0.162214\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	784125.i		−	1.01026i	−0.863043	−	0.505130i	$-0.831444\pi$
$881$		−	784125.i		−	1.01026i	0.863043	−	0.505130i	$-0.168556\pi$
$882$		0			0
$883$	−67879.2			−0.0870593			−0.0435296	−	0.999052i	$-0.513860\pi$
$883$	−67879.2			−0.0870593			−0.0435296	+	0.999052i	$0.513860\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−524429.	−	302779.i	−0.666560	−	0.384839i	0.128212	−	0.991747i	$-0.459076\pi$
$887$	−524429.	−	302779.i	−0.666560	−	0.384839i	−0.794772	+	0.606908i	$0.792410\pi$
$888$		0			0
$889$	119599.	+	207152.i	0.151330	+	0.262111i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−180252.	+	104068.i	−0.226035	+	0.130502i
$894$		0			0
$895$	−598755.	+	1.03707e6i	−0.747486	+	1.29468i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	446260.i		−	0.552165i
$900$		0			0
$901$	−296907.			−0.365739
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−938343.	−	541752.i	−1.14568	−	0.661460i
$906$		0			0
$907$	25686.8	+	44490.9i	0.0312246	+	0.0540825i	0.881215	−	0.472715i	$-0.156726\pi$
$907$	25686.8	+	44490.9i	0.0312246	+	0.0540825i	−0.849991	+	0.526797i	$0.823393\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−321203.	+	185447.i	−0.387029	+	0.223451i	−0.680872	−	0.732402i	$-0.738399\pi$
$911$	−321203.	+	185447.i	−0.387029	+	0.223451i	0.293843	+	0.955854i	$0.405066\pi$
$912$		0			0
$913$	−114979.	+	199149.i	−0.137936	+	0.238912i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	247726.i		−	0.294600i
$918$		0			0
$919$	891724.			1.05584			0.527922	−	0.849293i	$-0.322971\pi$
$919$	891724.			1.05584			0.527922	+	0.849293i	$0.322971\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1.86106e6	+	1.07449e6i	2.18453	+	1.26124i
$924$		0			0
$925$	−62523.0	−	108293.i	−0.0730730	−	0.126566i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−591258.	+	341363.i	−0.685087	+	0.395535i	−0.801769	−	0.597634i	$-0.796108\pi$
$929$	−591258.	+	341363.i	−0.685087	+	0.395535i	0.116682	+	0.993169i	$0.462774\pi$
$930$		0			0
$931$	58642.8	−	101572.i	0.0676574	−	0.117186i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	278260.i		−	0.318294i
$936$		0			0
$937$	1.02178e6			1.16380			0.581899	−	0.813261i	$-0.302310\pi$
$937$	1.02178e6			1.16380			0.581899	+	0.813261i	$0.302310\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1.04723e6	−	604618.i	−1.18267	−	0.682813i	−0.226037	−	0.974119i	$-0.572577\pi$
$941$	−1.04723e6	−	604618.i	−1.18267	−	0.682813i	−0.956630	+	0.291306i	$0.905910\pi$
$942$		0			0
$943$	−48763.0	−	84460.0i	−0.0548362	−	0.0949790i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	473810.	−	273554.i	0.528329	−	0.305031i	−0.212007	−	0.977268i	$-0.568000\pi$
$947$	473810.	−	273554.i	0.528329	−	0.305031i	0.740336	+	0.672237i	$0.234667\pi$
$948$		0			0
$949$	−365027.	+	632245.i	−0.405315	+	0.702026i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	251720.i		−	0.277161i	−0.990351	−	0.138581i	$-0.955746\pi$
$953$		−	251720.i		−	0.277161i	0.990351	−	0.138581i	$-0.0442540\pi$
$954$		0			0
$955$	−1.59929e6			−1.75355
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−424008.	−	244801.i	−0.461039	−	0.266181i
$960$		0			0
$961$	391880.	+	678756.i	0.424332	+	0.734965i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1.02106e6	−	589507.i	1.09646	−	0.633044i
$966$		0			0
$967$	−908669.	+	1.57386e6i	−0.971746	+	1.68311i	−0.281467	+	0.959571i	$0.590821\pi$
$967$	−908669.	+	1.57386e6i	−0.971746	+	1.68311i	−0.690280	+	0.723543i	$0.742512\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	870954.i		−	0.923755i	−0.886944	−	0.461878i	$-0.847176\pi$
$971$		−	870954.i		−	0.923755i	0.886944	−	0.461878i	$-0.152824\pi$
$972$		0			0
$973$	−399460.			−0.421937
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.00837e6	+	582183.i	1.05641	+	0.609917i	0.924436	−	0.381337i	$-0.124536\pi$
$977$	1.00837e6	+	582183.i	1.05641	+	0.609917i	0.131971	+	0.991254i	$0.457869\pi$
$978$		0			0
$979$	176784.	+	306199.i	0.184449	+	0.319476i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	881418.	−	508887.i	0.912168	−	0.526641i	0.0310402	−	0.999518i	$-0.490118\pi$
$983$	881418.	−	508887.i	0.912168	−	0.526641i	0.881128	+	0.472877i	$0.156785\pi$
$984$		0			0
$985$	−6117.54	+	10595.9i	−0.00630528	+	0.0109211i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	873765.i		−	0.893309i
$990$		0			0
$991$	758495.			0.772334			0.386167	−	0.922429i	$-0.373799\pi$
$991$	758495.			0.772334			0.386167	+	0.922429i	$0.373799\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1.09872e6	+	634345.i	1.10979	+	0.640737i
$996$		0			0
$997$	−241997.	−	419151.i	−0.243456	−	0.421678i	0.718241	−	0.695795i	$-0.244948\pi$
$997$	−241997.	−	419151.i	−0.243456	−	0.421678i	−0.961696	+	0.274117i	$0.911614\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.5.g.f.53.2		16
3.2	odd	2	inner	324.5.g.f.53.7		16
9.2	odd	6	inner	324.5.g.f.269.2		16
9.4	even	3		324.5.c.b.161.7	yes	8
9.5	odd	6		324.5.c.b.161.2	✓	8
9.7	even	3	inner	324.5.g.f.269.7		16
36.23	even	6		1296.5.e.f.161.2		8
36.31	odd	6		1296.5.e.f.161.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
324.5.c.b.161.2	✓	8	9.5	odd	6
324.5.c.b.161.7	yes	8	9.4	even	3
324.5.g.f.53.2		16	1.1	even	1	trivial
324.5.g.f.53.7		16	3.2	odd	2	inner
324.5.g.f.269.2		16	9.2	odd	6	inner
324.5.g.f.269.7		16	9.7	even	3	inner
1296.5.e.f.161.2		8	36.23	even	6
1296.5.e.f.161.7		8	36.31	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	324.g (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-22.8264 - 13.1788i) q^{5} +(-10.2177 - 17.6977i) q^{7} +(46.8767 - 27.0643i) q^{11} +(119.150 - 206.375i) q^{13} +195.038i q^{17} +59.1338 q^{19} +(-233.412 - 134.761i) q^{23} +(34.8634 + 60.3852i) q^{25} +(-1033.77 + 596.849i) q^{29} +(-186.923 + 323.761i) q^{31} +538.632i q^{35} -1793.37 q^{37} +(313.370 + 180.924i) q^{41} +(1620.96 + 2807.58i) q^{43} +(-3048.20 + 1759.88i) q^{47} +(991.695 - 1717.67i) q^{49} +1522.31i q^{53} -1426.70 q^{55} +(-5255.44 - 3034.23i) q^{59} +(-449.890 - 779.233i) q^{61} +(-5439.56 + 3140.53i) q^{65} +(3477.06 - 6022.45i) q^{67} +9017.89i q^{71} -3063.58 q^{73} +(-957.948 - 553.072i) q^{77} +(2789.76 + 4832.01i) q^{79} +(-3679.19 + 2124.18i) q^{83} +(2570.37 - 4452.01i) q^{85} +6532.00i q^{89} -4869.80 q^{91} +(-1349.81 - 779.315i) q^{95} +(2283.77 + 3955.60i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 52 q^{7} - 220 q^{13} - 616 q^{19} + 328 q^{25} + 1472 q^{31} + 2456 q^{37} - 388 q^{43} - 3432 q^{49} - 72 q^{55} - 1804 q^{61} + 2180 q^{67} - 9088 q^{73} + 5012 q^{79} + 5364 q^{85} - 3512 q^{91}+ \cdots + 2528 q^{97}+O(q^{100}) \) 16 * q - 52 * q^7 - 220 * q^13 - 616 * q^19 + 328 * q^25 + 1472 * q^31 + 2456 * q^37 - 388 * q^43 - 3432 * q^49 - 72 * q^55 - 1804 * q^61 + 2180 * q^67 - 9088 * q^73 + 5012 * q^79 + 5364 * q^85 - 3512 * q^91 + 2528 * q^97

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