LMFDB - Embedded newform 324.4.i.a.253.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,4,Mod(37,324)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 14]))

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

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

chi := DirichletCharacter("324.37");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	324.i (of order $9$, degree $6$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$19.1166188419$
Analytic rank:	$0$
Dimension:	$54$
Relative dimension:	$9$ over $\Q(\zeta_{9})$
Twist minimal:	no (minimal twist has level 108)
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			253.7
Character	$\chi$	$=$	324.253
Dual form			324.4.i.a.73.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(5.19451 + 4.35871i) q^{5} +(-17.8009 - 6.47901i) q^{7} +O(q^{10})$	$q+(5.19451 + 4.35871i) q^{5} +(-17.8009 - 6.47901i) q^{7} +(0.134044 - 0.112476i) q^{11} +(-1.24242 + 7.04614i) q^{13} +(26.6029 - 46.0776i) q^{17} +(-65.4831 - 113.420i) q^{19} +(-129.122 + 46.9965i) q^{23} +(-13.7215 - 77.8183i) q^{25} +(9.32414 + 52.8798i) q^{29} +(139.135 - 50.6411i) q^{31} +(-64.2270 - 111.244i) q^{35} +(58.6266 - 101.544i) q^{37} +(27.9173 - 158.327i) q^{41} +(51.8941 - 43.5443i) q^{43} +(-597.576 - 217.500i) q^{47} +(12.1427 + 10.1889i) q^{49} -36.5908 q^{53} +1.18654 q^{55} +(-574.476 - 482.043i) q^{59} +(-64.9040 - 23.6231i) q^{61} +(-37.1659 + 31.1859i) q^{65} +(109.760 - 622.479i) q^{67} +(66.9800 - 116.013i) q^{71} +(-435.761 - 754.761i) q^{73} +(-3.11484 + 1.13371i) q^{77} +(55.7069 + 315.930i) q^{79} +(197.919 + 1122.46i) q^{83} +(339.028 - 123.396i) q^{85} +(259.505 + 449.475i) q^{89} +(67.7684 - 117.378i) q^{91} +(154.213 - 874.583i) q^{95} +(-1153.80 + 968.155i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 54 q - 12 q^{5}+O(q^{10}) $ 54 * q - 12 * q^5	$ 54 q - 12 q^{5} + 87 q^{11} - 204 q^{17} - 96 q^{23} - 216 q^{25} - 318 q^{29} - 54 q^{31} - 6 q^{35} - 867 q^{41} - 513 q^{43} + 1548 q^{47} + 594 q^{49} + 1068 q^{53} + 1218 q^{59} - 54 q^{61} - 96 q^{65} - 2997 q^{67} + 120 q^{71} - 216 q^{73} - 3480 q^{77} + 2808 q^{79} - 4464 q^{83} + 2160 q^{85} - 4029 q^{89} + 270 q^{91} + 1650 q^{95} - 3483 q^{97}+O(q^{100}) $ 54 * q - 12 * q^5 + 87 * q^11 - 204 * q^17 - 96 * q^23 - 216 * q^25 - 318 * q^29 - 54 * q^31 - 6 * q^35 - 867 * q^41 - 513 * q^43 + 1548 * q^47 + 594 * q^49 + 1068 * q^53 + 1218 * q^59 - 54 * q^61 - 96 * q^65 - 2997 * q^67 + 120 * q^71 - 216 * q^73 - 3480 * q^77 + 2808 * q^79 - 4464 * q^83 + 2160 * q^85 - 4029 * q^89 + 270 * q^91 + 1650 * q^95 - 3483 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$163$	$245$
$\chi(n)$	$1$	$e\left(\frac{4}{9}\right)$

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
$5$	5.19451	+	4.35871i	0.464611	+	0.389855i	0.844824	−	0.535044i	$-0.179705\pi$
$5$	5.19451	+	4.35871i	0.464611	+	0.389855i	−0.380213	+	0.924899i	$0.624149\pi$
$6$		0			0
$7$	−17.8009	−	6.47901i	−0.961160	−	0.349834i	−0.186672	−	0.982422i	$-0.559770\pi$
$7$	−17.8009	−	6.47901i	−0.961160	−	0.349834i	−0.774488	+	0.632588i	$0.781992\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.134044	−	0.112476i	0.00367416	−	0.00308299i	−0.640949	−	0.767584i	$-0.721459\pi$
$11$	0.134044	−	0.112476i	0.00367416	−	0.00308299i	0.644623	+	0.764501i	$0.277014\pi$
$12$		0			0
$13$	−1.24242	+	7.04614i	−0.0265067	+	0.150327i	−0.995189	−	0.0979783i	$-0.968762\pi$
$13$	−1.24242	+	7.04614i	−0.0265067	+	0.150327i	0.968682	+	0.248305i	$0.0798735\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	26.6029	−	46.0776i	0.379538	−	0.657380i	−0.611457	−	0.791278i	$-0.709416\pi$
$17$	26.6029	−	46.0776i	0.379538	−	0.657380i	0.990995	+	0.133898i	$0.0427495\pi$
$18$		0			0
$19$	−65.4831	−	113.420i	−0.790677	−	1.36949i	−0.925548	−	0.378629i	$-0.876396\pi$
$19$	−65.4831	−	113.420i	−0.790677	−	1.36949i	0.134872	−	0.990863i	$-0.456938\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−129.122	+	46.9965i	−1.17060	+	0.426063i	−0.852872	−	0.522119i	$-0.825142\pi$
$23$	−129.122	+	46.9965i	−1.17060	+	0.426063i	−0.317726	+	0.948182i	$0.602919\pi$
$24$		0			0
$25$	−13.7215	−	77.8183i	−0.109772	−	0.622546i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	9.32414	+	52.8798i	0.0597052	+	0.338605i	0.999998	−	0.00176047i	$-0.000560374\pi$
$29$	9.32414	+	52.8798i	0.0597052	+	0.338605i	−0.940293	+	0.340365i	$0.889449\pi$
$30$		0			0
$31$	139.135	−	50.6411i	0.806111	−	0.293401i	0.0940949	−	0.995563i	$-0.470004\pi$
$31$	139.135	−	50.6411i	0.806111	−	0.293401i	0.712017	+	0.702163i	$0.247782\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−64.2270	−	111.244i	−0.310181	−	0.537249i
$36$		0			0
$37$	58.6266	−	101.544i	0.260491	−	0.451183i	−0.705882	−	0.708330i	$-0.749449\pi$
$37$	58.6266	−	101.544i	0.260491	−	0.451183i	0.966372	+	0.257147i	$0.0827824\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	27.9173	−	158.327i	0.106340	−	0.603085i	−0.884336	−	0.466850i	$-0.845389\pi$
$41$	27.9173	−	158.327i	0.106340	−	0.603085i	0.990677	−	0.136235i	$-0.0435002\pi$
$42$		0			0
$43$	51.8941	−	43.5443i	0.184041	−	0.154429i	−0.546113	−	0.837712i	$-0.683893\pi$
$43$	51.8941	−	43.5443i	0.184041	−	0.154429i	0.730154	+	0.683283i	$0.239448\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−597.576	−	217.500i	−1.85458	−	0.675013i	−0.982674	−	0.185345i	$-0.940660\pi$
$47$	−597.576	−	217.500i	−1.85458	−	0.675013i	−0.871909	−	0.489668i	$-0.837118\pi$
$48$		0			0
$49$	12.1427	+	10.1889i	0.0354013	+	0.0297052i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−36.5908			−0.0948328			−0.0474164	−	0.998875i	$-0.515099\pi$
$53$	−36.5908			−0.0948328			−0.0474164	+	0.998875i	$0.515099\pi$
$54$		0			0
$55$	1.18654			0.00290897
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−574.476	−	482.043i	−1.26763	−	1.06367i	−0.994825	−	0.101603i	$-0.967603\pi$
$59$	−574.476	−	482.043i	−1.26763	−	1.06367i	−0.272809	−	0.962068i	$-0.587953\pi$
$60$		0			0
$61$	−64.9040	−	23.6231i	−0.136231	−	0.0495841i	0.273005	−	0.962013i	$-0.411982\pi$
$61$	−64.9040	−	23.6231i	−0.136231	−	0.0495841i	−0.409236	+	0.912429i	$0.634205\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−37.1659	+	31.1859i	−0.0709209	+	0.0595097i
$66$		0			0
$67$	109.760	−	622.479i	0.200139	−	1.13504i	−0.704769	−	0.709437i	$-0.748949\pi$
$67$	109.760	−	622.479i	0.200139	−	1.13504i	0.904908	−	0.425607i	$-0.139939\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	66.9800	−	116.013i	0.111959	−	0.193918i	−0.804601	−	0.593816i	$-0.797621\pi$
$71$	66.9800	−	116.013i	0.111959	−	0.193918i	0.916560	+	0.399897i	$0.130954\pi$
$72$		0			0
$73$	−435.761	−	754.761i	−0.698658	−	1.21011i	−0.968932	−	0.247327i	$-0.920448\pi$
$73$	−435.761	−	754.761i	−0.698658	−	1.21011i	0.270274	−	0.962783i	$-0.412886\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−3.11484	+	1.13371i	−0.00460999	+	0.00167790i
$78$		0			0
$79$	55.7069	+	315.930i	0.0793357	+	0.449935i	0.998436	+	0.0559108i	$0.0178062\pi$
$79$	55.7069	+	315.930i	0.0793357	+	0.449935i	−0.919100	+	0.394024i	$0.871083\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	197.919	+	1122.46i	0.261741	+	1.48440i	0.778159	+	0.628068i	$0.216154\pi$
$83$	197.919	+	1122.46i	0.261741	+	1.48440i	−0.516418	+	0.856337i	$0.672735\pi$
$84$		0			0
$85$	339.028	−	123.396i	0.432620	−	0.157461i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	259.505	+	449.475i	0.309073	+	0.535329i	0.978160	−	0.207855i	$-0.0666481\pi$
$89$	259.505	+	449.475i	0.309073	+	0.535329i	−0.669087	+	0.743184i	$0.733315\pi$
$90$		0			0
$91$	67.7684	−	117.378i	0.0780665	−	0.135215i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	154.213	−	874.583i	0.166546	−	0.944530i
$96$		0			0
$97$	−1153.80	+	968.155i	−1.20774	+	1.01341i	−0.208366	+	0.978051i	$0.566814\pi$
$97$	−1153.80	+	968.155i	−1.20774	+	1.01341i	−0.999375	+	0.0353635i	$0.988741\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	956.621	+	348.182i	0.942449	+	0.343023i	0.767132	−	0.641489i	$-0.221683\pi$
$101$	956.621	+	348.182i	0.942449	+	0.343023i	0.175317	+	0.984512i	$0.443905\pi$
$102$		0			0
$103$	749.339	+	628.770i	0.716840	+	0.601500i	0.926509	−	0.376272i	$-0.122794\pi$
$103$	749.339	+	628.770i	0.716840	+	0.601500i	−0.209669	+	0.977772i	$0.567239\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1665.48			1.50475			0.752374	−	0.658736i	$-0.228909\pi$
$107$	1665.48			1.50475			0.752374	+	0.658736i	$0.228909\pi$
$108$		0			0
$109$	531.528			0.467075			0.233538	−	0.972348i	$-0.424970\pi$
$109$	531.528			0.467075			0.233538	+	0.972348i	$0.424970\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	277.167	+	232.571i	0.230740	+	0.193614i	0.750826	−	0.660500i	$-0.229656\pi$
$113$	277.167	+	232.571i	0.230740	+	0.193614i	−0.520086	+	0.854114i	$0.674100\pi$
$114$		0			0
$115$	−875.568	−	318.681i	−0.709975	−	0.258410i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−772.094	+	647.864i	−0.594771	+	0.499072i
$120$		0			0
$121$	−231.120	+	1310.75i	−0.173644	+	0.984785i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	691.720	−	1198.09i	0.494955	−	0.857287i
$126$		0			0
$127$	−947.516	−	1641.15i	−0.662035	−	1.14668i	−0.980080	−	0.198603i	$-0.936360\pi$
$127$	−947.516	−	1641.15i	−0.662035	−	1.14668i	0.318045	−	0.948076i	$-0.396974\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−2543.01	+	925.581i	−1.69606	+	0.617316i	−0.995367	−	0.0961447i	$-0.969349\pi$
$131$	−2543.01	+	925.581i	−1.69606	+	0.617316i	−0.700695	+	0.713461i	$0.747127\pi$
$132$		0			0
$133$	430.811	+	2443.25i	0.280873	+	1.59291i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	255.797	+	1450.70i	0.159520	+	0.904681i	0.954537	+	0.298094i	$0.0963508\pi$
$137$	255.797	+	1450.70i	0.159520	+	0.904681i	−0.795017	+	0.606587i	$0.792538\pi$
$138$		0			0
$139$	2088.85	−	760.279i	1.27463	−	0.463928i	0.385979	−	0.922507i	$-0.373864\pi$
$139$	2088.85	−	760.279i	1.27463	−	0.463928i	0.888654	+	0.458579i	$0.151641\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	0.625984	+	1.08424i	0.000366066	+	0.000634045i
$144$		0			0
$145$	−182.053	+	315.326i	−0.104267	+	0.180596i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	387.086	−	2195.27i	0.212828	−	1.20701i	−0.671809	−	0.740724i	$-0.734482\pi$
$149$	387.086	−	2195.27i	0.212828	−	1.20701i	0.884637	−	0.466281i	$-0.154406\pi$
$150$		0			0
$151$	−1652.75	+	1386.82i	−0.890720	+	0.747403i	−0.968354	−	0.249579i	$-0.919708\pi$
$151$	−1652.75	+	1386.82i	−0.890720	+	0.747403i	0.0776345	+	0.996982i	$0.475263\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	943.470	+	343.395i	0.488912	+	0.177949i
$156$		0			0
$157$	2419.53	+	2030.22i	1.22993	+	1.03204i	0.998243	+	0.0592452i	$0.0188694\pi$
$157$	2419.53	+	2030.22i	1.22993	+	1.03204i	0.231688	+	0.972790i	$0.425575\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	2602.98			1.27418
$162$		0			0
$163$	−1919.61			−0.922428			−0.461214	−	0.887289i	$-0.652586\pi$
$163$	−1919.61			−0.922428			−0.461214	+	0.887289i	$0.652586\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	531.834	+	446.262i	0.246435	+	0.206783i	0.757635	−	0.652678i	$-0.226355\pi$
$167$	531.834	+	446.262i	0.246435	+	0.206783i	−0.511201	+	0.859461i	$0.670799\pi$
$168$		0			0
$169$	2016.40	+	733.910i	0.917797	+	0.334051i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−2035.08	+	1707.63i	−0.894358	+	0.750456i	−0.969080	−	0.246749i	$-0.920638\pi$
$173$	−2035.08	+	1707.63i	−0.894358	+	0.750456i	0.0747211	+	0.997204i	$0.476193\pi$
$174$		0			0
$175$	−259.931	+	1474.14i	−0.112280	+	0.636769i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	554.255	−	959.998i	0.231436	−	0.400858i	−0.726795	−	0.686854i	$-0.758991\pi$
$179$	554.255	−	959.998i	0.231436	−	0.400858i	0.958231	+	0.285996i	$0.0923244\pi$
$180$		0			0
$181$	1019.98	+	1766.65i	0.418864	+	0.725493i	0.995825	−	0.0912779i	$-0.0290952\pi$
$181$	1019.98	+	1766.65i	0.418864	+	0.725493i	−0.576962	+	0.816771i	$0.695762\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	747.138	−	271.936i	0.296922	−	0.108071i
$186$		0			0
$187$	−1.61667	−	9.16862i	−0.000632208	−	0.00358543i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−187.456	−	1063.12i	−0.0710149	−	0.402745i	−0.999507	−	0.0313936i	$-0.990005\pi$
$191$	−187.456	−	1063.12i	−0.0710149	−	0.402745i	0.928492	−	0.371352i	$-0.121106\pi$
$192$		0			0
$193$	−231.941	+	84.4197i	−0.0865051	+	0.0314853i	−0.384910	−	0.922954i	$-0.625768\pi$
$193$	−231.941	+	84.4197i	−0.0865051	+	0.0314853i	0.298405	+	0.954439i	$0.403545\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	2154.60	+	3731.87i	0.779232	+	1.34967i	0.932385	+	0.361466i	$0.117724\pi$
$197$	2154.60	+	3731.87i	0.779232	+	1.34967i	−0.153154	+	0.988202i	$0.548943\pi$
$198$		0			0
$199$	1279.40	−	2215.99i	0.455751	−	0.789383i	−0.542980	−	0.839745i	$-0.682704\pi$
$199$	1279.40	−	2215.99i	0.455751	−	0.789383i	0.998731	+	0.0503621i	$0.0160376\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	176.631	−	1001.72i	0.0610692	−	0.346340i
$204$		0			0
$205$	835.117	−	700.746i	0.284522	−	0.238743i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−21.5347	−	7.83798i	−0.00712720	−	0.00259409i
$210$		0			0
$211$	−1331.28	−	1117.08i	−0.434357	−	0.364468i	0.399236	−	0.916848i	$-0.369275\pi$
$211$	−1331.28	−	1117.08i	−0.434357	−	0.364468i	−0.833593	+	0.552380i	$0.813720\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	459.361			0.145712
$216$		0			0
$217$	−2804.85			−0.877444
$218$		0			0
$219$		0			0
$220$		0			0
$221$	291.617	+	244.696i	0.0887615	+	0.0744797i
$222$		0			0
$223$	−3665.27	−	1334.05i	−1.10065	−	0.400603i	−0.273093	−	0.961988i	$-0.588047\pi$
$223$	−3665.27	−	1334.05i	−1.10065	−	0.400603i	−0.827555	+	0.561385i	$0.810269\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−2227.65	+	1869.22i	−0.651342	+	0.546541i	−0.907478	−	0.420100i	$-0.861995\pi$
$227$	−2227.65	+	1869.22i	−0.651342	+	0.546541i	0.256136	+	0.966641i	$0.417551\pi$
$228$		0			0
$229$	862.951	−	4894.04i	0.249019	−	1.41226i	−0.561950	−	0.827171i	$-0.689949\pi$
$229$	862.951	−	4894.04i	0.249019	−	1.41226i	0.810970	−	0.585088i	$-0.198940\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	254.515	−	440.832i	0.0715614	−	0.123948i	−0.828024	−	0.560692i	$-0.810535\pi$
$233$	254.515	−	440.832i	0.0715614	−	0.123948i	0.899586	+	0.436744i	$0.143868\pi$
$234$		0			0
$235$	−2156.09	−	3734.46i	−0.598502	−	1.03664i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−4164.81	+	1515.87i	−1.12719	+	0.410265i	−0.837272	−	0.546786i	$-0.815851\pi$
$239$	−4164.81	+	1515.87i	−1.12719	+	0.410265i	−0.289921	+	0.957051i	$0.593629\pi$
$240$		0			0
$241$	90.6192	+	513.927i	0.0242211	+	0.137365i	0.994520	−	0.104543i	$-0.0333380\pi$
$241$	90.6192	+	513.927i	0.0242211	+	0.137365i	−0.970299	+	0.241908i	$0.922227\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	18.6647	+	105.853i	0.00486711	+	0.0276027i
$246$		0			0
$247$	880.532	−	320.487i	0.226830	−	0.0825592i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−3252.48	−	5633.46i	−0.817907	−	1.41666i	−0.907221	−	0.420653i	$-0.861801\pi$
$251$	−3252.48	−	5633.46i	−0.817907	−	1.41666i	0.0893143	−	0.996003i	$-0.471532\pi$
$252$		0			0
$253$	−12.0220	+	20.8227i	−0.00298742	+	0.00517437i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	993.934	−	5636.88i	0.241245	−	1.36817i	−0.587810	−	0.808999i	$-0.700010\pi$
$257$	993.934	−	5636.88i	0.241245	−	1.36817i	0.829054	−	0.559168i	$-0.188879\pi$
$258$		0			0
$259$	−1701.51	+	1427.74i	−0.408212	+	0.342531i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	2466.09	+	897.584i	0.578197	+	0.210446i	0.614530	−	0.788893i	$-0.289346\pi$
$263$	2466.09	+	897.584i	0.578197	+	0.210446i	−0.0363333	+	0.999340i	$0.511568\pi$
$264$		0			0
$265$	−190.071	−	159.489i	−0.0440603	−	0.0369710i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−2872.70			−0.651122			−0.325561	−	0.945521i	$-0.605553\pi$
$269$	−2872.70			−0.651122			−0.325561	+	0.945521i	$0.605553\pi$
$270$		0			0
$271$	3701.45			0.829695			0.414847	−	0.909891i	$-0.363835\pi$
$271$	3701.45			0.829695			0.414847	+	0.909891i	$0.363835\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−10.5920	−	8.88773i	−0.00232262	−	0.00194891i
$276$		0			0
$277$	1213.14	+	441.548i	0.263143	+	0.0957763i	0.470223	−	0.882548i	$-0.344174\pi$
$277$	1213.14	+	441.548i	0.263143	+	0.0957763i	−0.207079	+	0.978324i	$0.566396\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	3031.24	−	2543.52i	0.643519	−	0.539977i	−0.261578	−	0.965182i	$-0.584243\pi$
$281$	3031.24	−	2543.52i	0.643519	−	0.539977i	0.905097	+	0.425206i	$0.139798\pi$
$282$		0			0
$283$	−174.948	+	992.178i	−0.0367476	+	0.208406i	−0.997653	−	0.0684704i	$-0.978188\pi$
$283$	−174.948	+	992.178i	−0.0367476	+	0.208406i	0.960906	+	0.276876i	$0.0892992\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1522.75	+	2637.49i	−0.313189	+	0.542460i
$288$		0			0
$289$	1041.07	+	1803.19i	0.211901	+	0.367024i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−7284.28	+	2651.26i	−1.45240	+	0.528629i	−0.943259	−	0.332057i	$-0.892257\pi$
$293$	−7284.28	+	2651.26i	−1.45240	+	0.528629i	−0.509137	+	0.860686i	$0.670035\pi$
$294$		0			0
$295$	−883.036	−	5007.95i	−0.174279	−	0.988387i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−170.720	−	968.201i	−0.0330200	−	0.187266i
$300$		0			0
$301$	−1205.89	+	438.907i	−0.230918	+	0.0840471i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−234.178	−	405.608i	−0.0439639	−	0.0761477i
$306$		0			0
$307$	4498.36	−	7791.39i	0.836271	−	1.44846i	−0.0567205	−	0.998390i	$-0.518064\pi$
$307$	4498.36	−	7791.39i	0.836271	−	1.44846i	0.892991	−	0.450074i	$-0.148602\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	933.478	−	5294.02i	0.170202	−	0.965261i	−0.773336	−	0.633996i	$-0.781413\pi$
$311$	933.478	−	5294.02i	0.170202	−	0.965261i	0.943538	−	0.331265i	$-0.107475\pi$
$312$		0			0
$313$	2795.71	−	2345.88i	0.504865	−	0.423632i	−0.354453	−	0.935074i	$-0.615333\pi$
$313$	2795.71	−	2345.88i	0.504865	−	0.423632i	0.859318	+	0.511442i	$0.170888\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−9666.93	−	3518.47i	−1.71277	−	0.623398i	−0.715597	−	0.698513i	$-0.753845\pi$
$317$	−9666.93	−	3518.47i	−1.71277	−	0.623398i	−0.997175	+	0.0751153i	$0.976068\pi$
$318$		0			0
$319$	7.19757	+	6.03948i	0.00126328	+	0.00106002i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−6968.17			−1.20037
$324$		0			0
$325$	565.367			0.0964951
$326$		0			0
$327$		0			0
$328$		0			0
$329$	9228.23	+	7743.40i	1.54641	+	1.29759i
$330$		0			0
$331$	807.479	+	293.898i	0.134088	+	0.0488039i	0.408192	−	0.912896i	$-0.366159\pi$
$331$	807.479	+	293.898i	0.134088	+	0.0488039i	−0.274105	+	0.961700i	$0.588381\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	3283.35	−	2755.06i	0.535489	−	0.449329i
$336$		0			0
$337$	475.226	−	2695.14i	0.0768166	−	0.435648i	−0.922008	−	0.387171i	$-0.873452\pi$
$337$	475.226	−	2695.14i	0.0768166	−	0.435648i	0.998824	−	0.0484770i	$-0.0154368\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	12.9543	−	22.4376i	0.00205723	−	0.00356323i
$342$		0			0
$343$	3098.65	+	5367.02i	0.487788	+	0.844874i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−9139.24	+	3326.41i	−1.41389	+	0.514614i	−0.932269	−	0.361765i	$-0.882174\pi$
$347$	−9139.24	+	3326.41i	−1.41389	+	0.514614i	−0.481622	+	0.876379i	$0.659952\pi$
$348$		0			0
$349$	−244.747	−	1388.03i	−0.0375387	−	0.212892i	0.960269	−	0.279077i	$-0.0900285\pi$
$349$	−244.747	−	1388.03i	−0.0375387	−	0.212892i	−0.997807	+	0.0661847i	$0.978917\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−145.344	−	824.287i	−0.0219147	−	0.124284i	0.971888	−	0.235444i	$-0.0756545\pi$
$353$	−145.344	−	824.287i	−0.0219147	−	0.124284i	−0.993803	+	0.111160i	$0.964543\pi$
$354$		0			0
$355$	853.594	−	310.683i	0.127617	−	0.0464488i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−2517.07	−	4359.69i	−0.370044	−	0.640935i	0.619528	−	0.784975i	$-0.287324\pi$
$359$	−2517.07	−	4359.69i	−0.370044	−	0.640935i	−0.989572	+	0.144040i	$0.953991\pi$
$360$		0			0
$361$	−5146.58	+	8914.13i	−0.750339	+	1.29963i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1026.22	−	5819.97i	0.147163	−	0.834606i
$366$		0			0
$367$	4798.61	−	4026.51i	0.682521	−	0.572703i	−0.234221	−	0.972183i	$-0.575254\pi$
$367$	4798.61	−	4026.51i	0.682521	−	0.572703i	0.916742	+	0.399480i	$0.130809\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	651.351	+	237.073i	0.0911495	+	0.0331757i
$372$		0			0
$373$	−9425.34	−	7908.80i	−1.30838	−	1.09786i	−0.988632	−	0.150359i	$-0.951957\pi$
$373$	−9425.34	−	7908.80i	−1.30838	−	1.09786i	−0.319749	−	0.947502i	$-0.603598\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−384.183			−0.0524840
$378$		0			0
$379$	−4996.58			−0.677195			−0.338598	−	0.940931i	$-0.609953\pi$
$379$	−4996.58			−0.677195			−0.338598	+	0.940931i	$0.609953\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	10520.9	+	8828.08i	1.40364	+	1.17779i	0.959458	+	0.281851i	$0.0909484\pi$
$383$	10520.9	+	8828.08i	1.40364	+	1.17779i	0.444177	+	0.895939i	$0.353496\pi$
$384$		0			0
$385$	−21.1216	−	7.68763i	−0.00279599	−	0.00101766i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	6156.52	−	5165.93i	0.802437	−	0.673324i	−0.146353	−	0.989232i	$-0.546754\pi$
$389$	6156.52	−	5165.93i	0.802437	−	0.673324i	0.948790	+	0.315908i	$0.102309\pi$
$390$		0			0
$391$	−1269.53	+	7199.87i	−0.164202	+	0.931235i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−1087.68	+	1883.91i	−0.138549	+	0.239974i
$396$		0			0
$397$	3689.69	+	6390.72i	0.466448	+	0.807912i	0.999266	−	0.0383180i	$-0.0122000\pi$
$397$	3689.69	+	6390.72i	0.466448	+	0.807912i	−0.532817	+	0.846230i	$0.678867\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	7940.72	−	2890.19i	0.988880	−	0.359923i	0.203593	−	0.979056i	$-0.434738\pi$
$401$	7940.72	−	2890.19i	0.988880	−	0.359923i	0.785286	+	0.619133i	$0.212516\pi$
$402$		0			0
$403$	183.959	+	1043.29i	0.0227386	+	0.128957i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−3.56277	−	20.2055i	−0.000433907	−	0.00246081i
$408$		0			0
$409$	−12312.8	+	4481.48i	−1.48857	+	0.541796i	−0.953073	−	0.302739i	$-0.902099\pi$
$409$	−12312.8	+	4481.48i	−1.48857	+	0.541796i	−0.535500	+	0.844535i	$0.679877\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	7103.06	+	12302.9i	0.846292	+	1.46582i
$414$		0			0
$415$	−3864.37	+	6693.28i	−0.457095	+	0.791711i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−978.770	+	5550.88i	−0.114119	+	0.647204i	0.873063	+	0.487607i	$0.162130\pi$
$419$	−978.770	+	5550.88i	−0.114119	+	0.647204i	−0.987182	+	0.159596i	$0.948981\pi$
$420$		0			0
$421$	8502.48	−	7134.42i	0.984288	−	0.825916i	−0.000442623	−	1.00000i	$-0.500141\pi$
$421$	8502.48	−	7134.42i	0.984288	−	0.825916i	0.984731	+	0.174084i	$0.0556964\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3950.71	−	1437.94i	−0.450912	−	0.164119i
$426$		0			0
$427$	1002.30	+	841.027i	0.113594	+	0.0953165i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	3944.54			0.440839			0.220420	−	0.975405i	$-0.429257\pi$
$431$	3944.54			0.440839			0.220420	+	0.975405i	$0.429257\pi$
$432$		0			0
$433$	1909.68			0.211948			0.105974	−	0.994369i	$-0.466204\pi$
$433$	1909.68			0.211948			0.105974	+	0.994369i	$0.466204\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	13785.7	+	11567.5i	1.50906	+	1.26625i
$438$		0			0
$439$	7600.00	+	2766.17i	0.826260	+	0.300734i	0.720323	−	0.693639i	$-0.243994\pi$
$439$	7600.00	+	2766.17i	0.826260	+	0.300734i	0.105937	+	0.994373i	$0.466216\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	4879.88	−	4094.71i	0.523364	−	0.439155i	−0.342439	−	0.939540i	$-0.611253\pi$
$443$	4879.88	−	4094.71i	0.523364	−	0.439155i	0.865803	+	0.500386i	$0.166808\pi$
$444$		0			0
$445$	−611.133	+	3465.91i	−0.0651022	+	0.369213i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−6820.97	+	11814.3i	−0.716929	+	1.24176i	0.245281	+	0.969452i	$0.421120\pi$
$449$	−6820.97	+	11814.3i	−0.716929	+	1.24176i	−0.962211	+	0.272306i	$0.912214\pi$
$450$		0			0
$451$	−14.0659	−	24.3628i	−0.00146859	−	0.00254368i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	863.641	−	314.340i	0.0889849	−	0.0323878i
$456$		0			0
$457$	−2344.18	−	13294.5i	−0.239947	−	1.36081i	−0.831940	−	0.554865i	$-0.812770\pi$
$457$	−2344.18	−	13294.5i	−0.239947	−	1.36081i	0.591993	−	0.805943i	$-0.298341\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−2081.24	−	11803.3i	−0.210267	−	1.19248i	−0.888933	−	0.458037i	$-0.848553\pi$
$461$	−2081.24	−	11803.3i	−0.210267	−	1.19248i	0.678666	−	0.734447i	$-0.262558\pi$
$462$		0			0
$463$	−6171.60	+	2246.28i	−0.619479	+	0.225472i	−0.632646	−	0.774442i	$-0.718031\pi$
$463$	−6171.60	+	2246.28i	−0.619479	+	0.225472i	0.0131668	+	0.999913i	$0.495809\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−2152.08	−	3727.52i	−0.213247	−	0.369355i	0.739482	−	0.673177i	$-0.235071\pi$
$467$	−2152.08	−	3727.52i	−0.213247	−	0.369355i	−0.952729	+	0.303822i	$0.901737\pi$
$468$		0			0
$469$	−5986.88	+	10369.6i	−0.589442	+	1.02094i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	2.05839	−	11.6737i	0.000200095	−	0.00113479i
$474$		0			0
$475$	−7927.63	+	6652.07i	−0.765778	+	0.642564i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	14016.5	+	5101.57i	1.33701	+	0.486632i	0.908870	−	0.417080i	$-0.136947\pi$
$479$	14016.5	+	5101.57i	1.33701	+	0.486632i	0.428141	+	0.903712i	$0.359169\pi$
$480$		0			0
$481$	642.656	+	539.252i	0.0609201	+	0.0511181i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−10213.3			−0.956214
$486$		0			0
$487$	21294.9			1.98144			0.990722	−	0.135901i	$-0.0433931\pi$
$487$	21294.9			1.98144			0.990722	+	0.135901i	$0.0433931\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	8618.26	+	7231.58i	0.792132	+	0.664677i	0.946272	−	0.323372i	$-0.104816\pi$
$491$	8618.26	+	7231.58i	0.792132	+	0.664677i	−0.154140	+	0.988049i	$0.549261\pi$
$492$		0			0
$493$	2684.63	+	977.124i	0.245252	+	0.0892646i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1943.96	+	1631.17i	−0.175449	+	0.147220i
$498$		0			0
$499$	137.225	−	778.243i	0.0123107	−	0.0698175i	−0.978033	−	0.208448i	$-0.933159\pi$
$499$	137.225	−	778.243i	0.0123107	−	0.0698175i	0.990344	+	0.138630i	$0.0442700\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	5970.64	−	10341.5i	0.529260	−	0.916705i	−0.470158	−	0.882583i	$-0.655803\pi$
$503$	5970.64	−	10341.5i	0.529260	−	0.916705i	0.999418	−	0.0341229i	$-0.0108638\pi$
$504$		0			0
$505$	3451.55	+	5978.26i	0.304143	+	0.526791i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−10169.7	+	3701.49i	−0.885592	+	0.322329i	−0.744464	−	0.667662i	$-0.767295\pi$
$509$	−10169.7	+	3701.49i	−0.885592	+	0.322329i	−0.141128	+	0.989991i	$0.545073\pi$
$510$		0			0
$511$	2866.86	+	16258.8i	0.248185	+	1.40752i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1151.82	+	6532.30i	0.0985539	+	0.558927i
$516$		0			0
$517$	−104.565	+	38.0585i	−0.00889509	+	0.00323755i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−6195.35	−	10730.7i	−0.520966	−	0.902339i	−0.999703	−	0.0243806i	$-0.992239\pi$
$521$	−6195.35	−	10730.7i	−0.520966	−	0.902339i	0.478737	−	0.877958i	$-0.341095\pi$
$522$		0			0
$523$	−4693.00	+	8128.52i	−0.392372	+	0.679609i	−0.992762	−	0.120099i	$-0.961679\pi$
$523$	−4693.00	+	8128.52i	−0.392372	+	0.679609i	0.600390	+	0.799708i	$0.295012\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1367.98	−	7758.23i	0.113075	−	0.641278i
$528$		0			0
$529$	5143.32	−	4315.76i	0.422727	−	0.354710i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1080.91	+	393.418i	0.0878411	+	0.0319715i
$534$		0			0
$535$	8651.35	+	7259.34i	0.699122	+	0.586633i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	2.77366			0.000221651
$540$		0			0
$541$	1596.07			0.126840			0.0634202	−	0.997987i	$-0.479799\pi$
$541$	1596.07			0.126840			0.0634202	+	0.997987i	$0.479799\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	2761.03	+	2316.78i	0.217008	+	0.182091i
$546$		0			0
$547$	−9720.40	−	3537.94i	−0.759807	−	0.276547i	−0.0670804	−	0.997748i	$-0.521368\pi$
$547$	−9720.40	−	3537.94i	−0.759807	−	0.276547i	−0.692726	+	0.721201i	$0.743591\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	5387.06	−	4520.28i	0.416509	−	0.349493i
$552$		0			0
$553$	1055.28	−	5984.77i	0.0811481	−	0.460214i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−4716.05	+	8168.44i	−0.358753	+	0.621379i	−0.987753	−	0.156027i	$-0.950131\pi$
$557$	−4716.05	+	8168.44i	−0.358753	+	0.621379i	0.629000	+	0.777406i	$0.283465\pi$
$558$		0			0
$559$	242.345	+	419.753i	0.0183365	+	0.0317597i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−20619.8	+	7504.98i	−1.54355	+	0.561807i	−0.966894	−	0.255179i	$-0.917866\pi$
$563$	−20619.8	+	7504.98i	−1.54355	+	0.561807i	−0.576658	+	0.816986i	$0.695644\pi$
$564$		0			0
$565$	426.038	+	2416.18i	0.0317231	+	0.179911i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1838.39	−	10426.1i	−0.135447	−	0.768160i	−0.974547	−	0.224182i	$-0.928029\pi$
$569$	−1838.39	−	10426.1i	−0.135447	−	0.768160i	0.839100	−	0.543977i	$-0.183082\pi$
$570$		0			0
$571$	−2257.85	+	821.791i	−0.165478	+	0.0602292i	−0.423431	−	0.905928i	$-0.639174\pi$
$571$	−2257.85	+	821.791i	−0.165478	+	0.0602292i	0.257953	+	0.966158i	$0.416952\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	5428.93	+	9403.18i	0.393743	+	0.681982i
$576$		0			0
$577$	7640.34	−	13233.5i	0.551251	−	0.954794i	−0.446934	−	0.894567i	$-0.647484\pi$
$577$	7640.34	−	13233.5i	0.551251	−	0.954794i	0.998185	−	0.0602273i	$-0.0191825\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	3749.26	−	21263.1i	0.267720	−	1.51832i
$582$		0			0
$583$	−4.90478	+	4.11560i	−0.000348431	+	0.000292368i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	12260.7	+	4462.53i	0.862101	+	0.313779i	0.734964	−	0.678106i	$-0.237199\pi$
$587$	12260.7	+	4462.53i	0.862101	+	0.313779i	0.127137	+	0.991885i	$0.459421\pi$
$588$		0			0
$589$	−14854.7	−	12464.6i	−1.03918	−	0.871978i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−16271.5			−1.12680			−0.563400	−	0.826185i	$-0.690507\pi$
$593$	−16271.5			−1.12680			−0.563400	+	0.826185i	$0.690507\pi$
$594$		0			0
$595$	−6834.50			−0.470903
$596$		0			0
$597$		0			0
$598$		0			0
$599$	12021.9	+	10087.6i	0.820038	+	0.688094i	0.952981	−	0.303030i	$-0.0979982\pi$
$599$	12021.9	+	10087.6i	0.820038	+	0.688094i	−0.132943	+	0.991124i	$0.542443\pi$
$600$		0			0
$601$	9308.20	+	3387.91i	0.631763	+	0.229943i	0.637998	−	0.770038i	$-0.279763\pi$
$601$	9308.20	+	3387.91i	0.631763	+	0.229943i	−0.00623571	+	0.999981i	$0.501985\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−6913.73	+	5801.31i	−0.464600	+	0.389846i
$606$		0			0
$607$	−2791.61	+	15832.0i	−0.186669	+	1.05865i	0.737123	+	0.675758i	$0.236184\pi$
$607$	−2791.61	+	15832.0i	−0.186669	+	1.05865i	−0.923792	+	0.382894i	$0.874927\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	2274.98	−	3940.38i	0.150631	−	0.260901i
$612$		0			0
$613$	−2802.75	−	4854.51i	−0.184669	−	0.319856i	0.758796	−	0.651328i	$-0.225788\pi$
$613$	−2802.75	−	4854.51i	−0.184669	−	0.319856i	−0.943465	+	0.331472i	$0.892455\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−3393.93	+	1235.29i	−0.221450	+	0.0806012i	−0.450362	−	0.892846i	$-0.648705\pi$
$617$	−3393.93	+	1235.29i	−0.221450	+	0.0806012i	0.228913	+	0.973447i	$0.426483\pi$
$618$		0			0
$619$	−3759.85	−	21323.2i	−0.244138	−	1.38457i	−0.822487	−	0.568784i	$-0.807414\pi$
$619$	−3759.85	−	21323.2i	−0.244138	−	1.38457i	0.578349	−	0.815789i	$-0.303697\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1707.27	−	9682.42i	−0.109792	−	0.622661i
$624$		0			0
$625$	−466.380	+	169.748i	−0.0298483	+	0.0108639i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−3119.28	−	5402.74i	−0.197732	−	0.342482i
$630$		0			0
$631$	4449.63	−	7706.98i	0.280724	−	0.486228i	−0.690839	−	0.723008i	$-0.742759\pi$
$631$	4449.63	−	7706.98i	0.280724	−	0.486228i	0.971563	+	0.236780i	$0.0760921\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	2231.40	−	12654.9i	0.139449	−	0.790857i
$636$		0			0
$637$	−86.8787	+	72.8999i	−0.00540386	+	0.00453438i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−20334.9	−	7401.30i	−1.25301	−	0.456059i	−0.371593	−	0.928396i	$-0.621188\pi$
$641$	−20334.9	−	7401.30i	−1.25301	−	0.456059i	−0.881418	+	0.472337i	$0.843411\pi$
$642$		0			0
$643$	−6779.27	−	5688.49i	−0.415783	−	0.348883i	0.410773	−	0.911738i	$-0.365259\pi$
$643$	−6779.27	−	5688.49i	−0.415783	−	0.348883i	−0.826556	+	0.562854i	$0.809703\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	6445.91			0.391677			0.195838	−	0.980636i	$-0.437257\pi$
$647$	6445.91			0.391677			0.195838	+	0.980636i	$0.437257\pi$
$648$		0			0
$649$	−131.223			−0.00793678
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−2338.71	−	1962.41i	−0.140154	−	0.117603i	0.570015	−	0.821634i	$-0.306937\pi$
$653$	−2338.71	−	1962.41i	−0.140154	−	0.117603i	−0.710170	+	0.704031i	$0.751382\pi$
$654$		0			0
$655$	−17244.0	−	6276.32i	−1.02867	−	0.374406i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−11991.4	+	10062.0i	−0.708828	+	0.594777i	−0.924270	−	0.381739i	$-0.875325\pi$
$659$	−11991.4	+	10062.0i	−0.708828	+	0.594777i	0.215442	+	0.976517i	$0.430881\pi$
$660$		0			0
$661$	5685.36	−	32243.3i	0.334546	−	1.89730i	−0.0971257	−	0.995272i	$-0.530965\pi$
$661$	5685.36	−	32243.3i	0.334546	−	1.89730i	0.431671	−	0.902031i	$-0.357924\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−8411.56	+	14569.3i	−0.490506	+	0.849581i
$666$		0			0
$667$	−3689.12	−	6389.74i	−0.214158	−	0.370932i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−11.3570	+	4.13362i	−0.000653402	+	0.000237819i
$672$		0			0
$673$	−959.177	−	5439.76i	−0.0549384	−	0.311571i	0.944939	−	0.327247i	$-0.106121\pi$
$673$	−959.177	−	5439.76i	−0.0549384	−	0.311571i	−0.999877	+	0.0156762i	$0.995010\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	621.476	+	3524.57i	0.0352810	+	0.200089i	0.997353	−	0.0727055i	$-0.0231633\pi$
$677$	621.476	+	3524.57i	0.0352810	+	0.200089i	−0.962072	+	0.272794i	$0.912052\pi$
$678$		0			0
$679$	26811.4	−	9758.57i	1.51536	−	0.551546i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−813.215	−	1408.53i	−0.0455590	−	0.0789105i	0.842347	−	0.538936i	$-0.181174\pi$
$683$	−813.215	−	1408.53i	−0.0455590	−	0.0789105i	−0.887906	+	0.460026i	$0.847840\pi$
$684$		0			0
$685$	−4994.42	+	8650.59i	−0.278580	+	0.482514i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	45.4614	−	257.824i	0.00251370	−	0.0142559i
$690$		0			0
$691$	−771.115	+	647.042i	−0.0424524	+	0.0356218i	−0.663767	−	0.747939i	$-0.731043\pi$
$691$	−771.115	+	647.042i	−0.0424524	+	0.0356218i	0.621315	+	0.783561i	$0.286599\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	14164.4	+	5155.41i	0.773073	+	0.281376i
$696$		0			0
$697$	−6552.63	−	5498.31i	−0.356096	−	0.298800i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	30593.6			1.64837			0.824183	−	0.566324i	$-0.191635\pi$
$701$	30593.6			1.64837			0.824183	+	0.566324i	$0.191635\pi$
$702$		0			0
$703$	−15356.2			−0.823855
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−14772.9	−	12395.9i	−0.785843	−	0.659401i
$708$		0			0
$709$	15119.8	+	5503.15i	0.800896	+	0.291502i	0.709858	−	0.704345i	$-0.248759\pi$
$709$	15119.8	+	5503.15i	0.800896	+	0.291502i	0.0910385	+	0.995847i	$0.470981\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−15585.5	+	13077.8i	−0.818626	+	0.686909i
$714$		0			0
$715$	−1.47419	+	8.36055i	−7.71072e−5	+	0.000437296i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	13641.9	−	23628.5i	0.707591	−	1.22558i	−0.258157	−	0.966103i	$-0.583115\pi$
$719$	13641.9	−	23628.5i	0.707591	−	1.22558i	0.965748	−	0.259481i	$-0.0835515\pi$
$720$		0			0
$721$	−9265.12	−	16047.7i	−0.478573	−	0.828913i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	3987.08	−	1451.18i	0.204243	−	0.0743385i
$726$		0			0
$727$	3276.60	+	18582.5i	0.167156	+	0.947989i	0.946814	+	0.321783i	$0.104282\pi$
$727$	3276.60	+	18582.5i	0.167156	+	0.947989i	−0.779657	+	0.626206i	$0.784607\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−625.883	−	3549.56i	−0.0316677	−	0.179597i
$732$		0			0
$733$	−7560.06	+	2751.64i	−0.380951	+	0.138655i	−0.525396	−	0.850858i	$-0.676083\pi$
$733$	−7560.06	+	2751.64i	−0.380951	+	0.138655i	0.144445	+	0.989513i	$0.453860\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−55.3015	−	95.7850i	−0.00276398	−	0.00478736i
$738$		0			0
$739$	13107.9	−	22703.6i	0.652480	−	1.13013i	−0.330040	−	0.943967i	$-0.607062\pi$
$739$	13107.9	−	22703.6i	0.652480	−	1.13013i	0.982519	−	0.186161i	$-0.0596045\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−4176.05	+	23683.6i	−0.206197	+	1.16940i	0.689348	+	0.724431i	$0.257897\pi$
$743$	−4176.05	+	23683.6i	−0.206197	+	1.16940i	−0.895545	+	0.444971i	$0.853214\pi$
$744$		0			0
$745$	11579.3	−	9716.17i	0.569439	−	0.477816i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−29647.1	−	10790.7i	−1.44630	−	0.526412i
$750$		0			0
$751$	29364.6	+	24639.8i	1.42680	+	1.19723i	0.947573	+	0.319538i	$0.103528\pi$
$751$	29364.6	+	24639.8i	1.42680	+	1.19723i	0.479228	+	0.877690i	$0.340917\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−14630.0			−0.705217
$756$		0			0
$757$	−13077.8			−0.627900			−0.313950	−	0.949440i	$-0.601652\pi$
$757$	−13077.8			−0.627900			−0.313950	+	0.949440i	$0.601652\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−23600.6	−	19803.3i	−1.12421	−	0.943322i	−0.125398	−	0.992106i	$-0.540021\pi$
$761$	−23600.6	−	19803.3i	−1.12421	−	0.943322i	−0.998809	+	0.0487844i	$0.984465\pi$
$762$		0			0
$763$	−9461.70	−	3443.78i	−0.448934	−	0.163399i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	4110.29	−	3448.94i	0.193499	−	0.162365i
$768$		0			0
$769$	−2655.63	+	15060.8i	−0.124531	+	0.706251i	0.857054	+	0.515226i	$0.172292\pi$
$769$	−2655.63	+	15060.8i	−0.124531	+	0.706251i	−0.981585	+	0.191024i	$0.938819\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	2757.85	−	4776.74i	0.128322	−	0.222260i	−0.794705	−	0.606996i	$-0.792374\pi$
$773$	2757.85	−	4776.74i	0.128322	−	0.222260i	0.923027	+	0.384736i	$0.125708\pi$
$774$		0			0
$775$	−5849.95	−	10132.4i	−0.271144	−	0.469635i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−19785.5	+	7201.35i	−0.910001	+	0.331213i
$780$		0			0
$781$	−4.07042	−	23.0845i	−0.000186493	−	0.00105765i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	3719.09	+	21092.0i	0.169096	+	0.958989i
$786$		0			0
$787$	7795.37	−	2837.28i	0.353081	−	0.128511i	−0.159389	−	0.987216i	$-0.550953\pi$
$787$	7795.37	−	2837.28i	0.353081	−	0.128511i	0.512471	+	0.858705i	$0.328730\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−3427.00	−	5935.74i	−0.154046	−	0.266815i
$792$		0			0
$793$	247.090	−	427.973i	0.0110649	−	0.0191649i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	6858.09	−	38894.2i	0.304801	−	1.72861i	−0.319641	−	0.947539i	$-0.603562\pi$
$797$	6858.09	−	38894.2i	0.304801	−	1.72861i	0.624442	−	0.781071i	$-0.285327\pi$
$798$		0			0
$799$	−25919.1	+	21748.7i	−1.14763	+	0.962972i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−143.304	−	52.1584i	−0.00629774	−	0.00229219i
$804$		0			0
$805$	13521.2	+	11345.6i	0.592000	+	0.496747i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	7558.62			0.328488			0.164244	−	0.986420i	$-0.447481\pi$
$809$	7558.62			0.328488			0.164244	+	0.986420i	$0.447481\pi$
$810$		0			0
$811$	12908.2			0.558900			0.279450	−	0.960160i	$-0.409848\pi$
$811$	12908.2			0.558900			0.279450	+	0.960160i	$0.409848\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−9971.44	−	8367.03i	−0.428570	−	0.359613i
$816$		0			0
$817$	−8336.98	−	3034.41i	−0.357006	−	0.129940i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	9881.42	−	8291.49i	0.420053	−	0.352467i	−0.408130	−	0.912924i	$-0.633819\pi$
$821$	9881.42	−	8291.49i	0.420053	−	0.352467i	0.828183	+	0.560457i	$0.189375\pi$
$822$		0			0
$823$	−809.125	+	4588.78i	−0.0342701	+	0.194356i	−0.997136	−	0.0756229i	$-0.975905\pi$
$823$	−809.125	+	4588.78i	−0.0342701	+	0.194356i	0.962866	+	0.269978i	$0.0870166\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	10441.8	−	18085.7i	0.439053	−	0.760463i	−0.558563	−	0.829462i	$-0.688647\pi$
$827$	10441.8	−	18085.7i	0.439053	−	0.760463i	0.997617	+	0.0689990i	$0.0219805\pi$
$828$		0			0
$829$	2621.32	+	4540.26i	0.109822	+	0.190217i	0.915698	−	0.401867i	$-0.131639\pi$
$829$	2621.32	+	4540.26i	0.109822	+	0.190217i	−0.805876	+	0.592084i	$0.798305\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	792.510	−	288.450i	0.0329638	−	0.0119978i
$834$		0			0
$835$	817.491	+	4636.22i	0.0338808	+	0.192147i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	2975.93	+	16877.3i	0.122456	+	0.694482i	0.982786	+	0.184746i	$0.0591461\pi$
$839$	2975.93	+	16877.3i	0.122456	+	0.694482i	−0.860330	+	0.509737i	$0.829743\pi$
$840$		0			0
$841$	20208.8	−	7355.41i	0.828604	−	0.301587i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	7275.31	+	12601.2i	0.296187	+	0.513011i
$846$		0			0
$847$	12606.5	−	21835.1i	0.511411	−	0.885790i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−2797.75	+	15866.8i	−0.112698	+	0.639139i
$852$		0			0
$853$	−21828.5	+	18316.3i	−0.876196	+	0.735215i	−0.965393	−	0.260798i	$-0.916014\pi$
$853$	−21828.5	+	18316.3i	−0.876196	+	0.735215i	0.0891976	+	0.996014i	$0.471570\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−9601.95	−	3494.82i	−0.382726	−	0.139301i	0.143490	−	0.989652i	$-0.454167\pi$
$857$	−9601.95	−	3494.82i	−0.382726	−	0.139301i	−0.526216	+	0.850351i	$0.676390\pi$
$858$		0			0
$859$	18718.8	+	15707.0i	0.743514	+	0.623883i	0.933779	−	0.357850i	$-0.116490\pi$
$859$	18718.8	+	15707.0i	0.743514	+	0.623883i	−0.190265	+	0.981733i	$0.560935\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	14442.6			0.569679			0.284840	−	0.958575i	$-0.408060\pi$
$863$	14442.6			0.569679			0.284840	+	0.958575i	$0.408060\pi$
$864$		0			0
$865$	−18014.3			−0.708097
$866$		0			0
$867$		0			0
$868$		0			0
$869$	43.0018	+	36.0828i	0.00167864	+	0.00140854i
$870$		0			0
$871$	4249.71	+	1546.77i	0.165323	+	0.0601725i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−20075.7	+	16845.5i	−0.775639	+	0.650838i
$876$		0			0
$877$	1130.95	−	6413.93i	0.0435455	−	0.246959i	−0.955263	−	0.295757i	$-0.904428\pi$
$877$	1130.95	−	6413.93i	0.0435455	−	0.246959i	0.998809	+	0.0487984i	$0.0155392\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	4888.97	−	8467.95i	0.186962	−	0.323828i	−0.757274	−	0.653098i	$-0.773469\pi$
$881$	4888.97	−	8467.95i	0.186962	−	0.323828i	0.944236	+	0.329270i	$0.106802\pi$
$882$		0			0
$883$	−6874.70	−	11907.3i	−0.262007	−	0.453809i	0.704768	−	0.709438i	$-0.251051\pi$
$883$	−6874.70	−	11907.3i	−0.262007	−	0.453809i	−0.966775	+	0.255628i	$0.917718\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	2269.36	−	825.981i	0.0859051	−	0.0312669i	−0.298710	−	0.954344i	$-0.596556\pi$
$887$	2269.36	−	825.981i	0.0859051	−	0.0312669i	0.384615	+	0.923077i	$0.374334\pi$
$888$		0			0
$889$	6233.67	+	35352.9i	0.235175	+	1.33374i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	14462.3	+	82019.6i	0.541950	+	3.07355i
$894$		0			0
$895$	7063.43	−	2570.88i	0.263804	−	0.0960168i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	3975.21	+	6885.27i	0.147476	+	0.255436i
$900$		0			0
$901$	−973.423	+	1686.02i	−0.0359927	+	0.0623412i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−2402.05	+	13622.7i	−0.0882284	+	0.500368i
$906$		0			0
$907$	−18721.1	+	15708.9i	−0.685362	+	0.575087i	−0.917568	−	0.397580i	$-0.869850\pi$
$907$	−18721.1	+	15708.9i	−0.685362	+	0.575087i	0.232206	+	0.972667i	$0.425406\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	47391.2	+	17249.0i	1.72353	+	0.627315i	0.998137	−	0.0610158i	$-0.0194340\pi$
$911$	47391.2	+	17249.0i	1.72353	+	0.627315i	0.725397	+	0.688331i	$0.241656\pi$
$912$		0			0
$913$	152.780	+	128.197i	0.00553808	+	0.00464700i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	51264.9			1.84615
$918$		0			0
$919$	−360.717			−0.0129477			−0.00647386	−	0.999979i	$-0.502061\pi$
$919$	−360.717			−0.0129477			−0.00647386	+	0.999979i	$0.502061\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	734.225	+	616.088i	0.0261834	+	0.0219705i
$924$		0			0
$925$	−8706.44	−	3168.88i	−0.309477	−	0.112640i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	40315.5	−	33828.7i	1.42380	−	1.19471i	0.474530	−	0.880239i	$-0.342618\pi$
$929$	40315.5	−	33828.7i	1.42380	−	1.19471i	0.949268	−	0.314469i	$-0.101826\pi$
$930$		0			0
$931$	360.487	−	2044.42i	0.0126901	−	0.0719691i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	31.5655	−	54.6731i	0.00110407	−	0.00191230i
$936$		0			0
$937$	8458.48	+	14650.5i	0.294905	+	0.510791i	0.974963	−	0.222368i	$-0.0713785\pi$
$937$	8458.48	+	14650.5i	0.294905	+	0.510791i	−0.680057	+	0.733159i	$0.738045\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	12795.9	−	4657.31i	0.443287	−	0.161343i	−0.110727	−	0.993851i	$-0.535318\pi$
$941$	12795.9	−	4657.31i	0.443287	−	0.161343i	0.554013	+	0.832508i	$0.313096\pi$
$942$		0			0
$943$	3836.07	+	21755.5i	0.132471	+	0.751278i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−2895.19	−	16419.4i	−0.0993463	−	0.563421i	−0.993329	−	0.115319i	$-0.963211\pi$
$947$	−2895.19	−	16419.4i	−0.0993463	−	0.563421i	0.893982	−	0.448102i	$-0.147900\pi$
$948$		0			0
$949$	5859.55	−	2132.70i	0.200431	−	0.0729510i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	15893.0	+	27527.6i	0.540216	+	0.935682i	0.998891	+	0.0470781i	$0.0149910\pi$
$953$	15893.0	+	27527.6i	0.540216	+	0.935682i	−0.458675	+	0.888604i	$0.651676\pi$
$954$		0			0
$955$	3660.07	−	6339.43i	0.124018	−	0.214805i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	4845.65	−	27481.1i	0.163164	−	0.925349i
$960$		0			0
$961$	−6027.10	+	5057.33i	−0.202313	+	0.169760i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1572.78	−	572.445i	−0.0524659	−	0.0190960i
$966$		0			0
$967$	5283.91	+	4433.72i	0.175718	+	0.147445i	0.726405	−	0.687267i	$-0.241190\pi$
$967$	5283.91	+	4433.72i	0.175718	+	0.147445i	−0.550687	+	0.834712i	$0.685634\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	5291.80			0.174894			0.0874469	−	0.996169i	$-0.472129\pi$
$971$	5291.80			0.174894			0.0874469	+	0.996169i	$0.472129\pi$
$972$		0			0
$973$	−42109.4			−1.38742
$974$		0			0
$975$		0			0
$976$		0			0
$977$	11643.4	+	9770.01i	0.381276	+	0.319929i	0.813203	−	0.581980i	$-0.197722\pi$
$977$	11643.4	+	9770.01i	0.381276	+	0.319929i	−0.431927	+	0.901909i	$0.642166\pi$
$978$		0			0
$979$	85.3404	+	31.0614i	0.00278600	+	0.00101402i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−27467.9	+	23048.3i	−0.891242	+	0.747841i	−0.968459	−	0.249173i	$-0.919841\pi$
$983$	−27467.9	+	23048.3i	−0.891242	+	0.747841i	0.0772168	+	0.997014i	$0.475397\pi$
$984$		0			0
$985$	−5074.07	+	28776.5i	−0.164135	+	0.930858i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−4654.23	+	8061.36i	−0.149642	+	0.259187i
$990$		0			0
$991$	12217.3	+	21161.0i	0.391621	+	0.678307i	0.992663	−	0.120910i	$-0.0385812\pi$
$991$	12217.3	+	21161.0i	0.391621	+	0.678307i	−0.601043	+	0.799217i	$0.705248\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	16304.7	−	5934.43i	0.519491	−	0.189079i
$996$		0			0
$997$	3628.99	+	20581.0i	0.115277	+	0.653768i	0.986613	+	0.163081i	$0.0521431\pi$
$997$	3628.99	+	20581.0i	0.115277	+	0.653768i	−0.871336	+	0.490687i	$0.836746\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.4.i.a.253.7		54
3.2	odd	2		108.4.i.a.13.6	✓	54
27.2	odd	18		108.4.i.a.25.6	yes	54
27.25	even	9	inner	324.4.i.a.73.7		54

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.i.a.13.6	✓	54	3.2	odd	2
108.4.i.a.25.6	yes	54	27.2	odd	18
324.4.i.a.73.7		54	27.25	even	9	inner
324.4.i.a.253.7		54	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	324.i (of order \(9\), degree \(6\), not minimal)

\(f(q)\)	\(=\)	\(q+(5.19451 + 4.35871i) q^{5} +(-17.8009 - 6.47901i) q^{7} +O(q^{10})\)	\(q+(5.19451 + 4.35871i) q^{5} +(-17.8009 - 6.47901i) q^{7} +(0.134044 - 0.112476i) q^{11} +(-1.24242 + 7.04614i) q^{13} +(26.6029 - 46.0776i) q^{17} +(-65.4831 - 113.420i) q^{19} +(-129.122 + 46.9965i) q^{23} +(-13.7215 - 77.8183i) q^{25} +(9.32414 + 52.8798i) q^{29} +(139.135 - 50.6411i) q^{31} +(-64.2270 - 111.244i) q^{35} +(58.6266 - 101.544i) q^{37} +(27.9173 - 158.327i) q^{41} +(51.8941 - 43.5443i) q^{43} +(-597.576 - 217.500i) q^{47} +(12.1427 + 10.1889i) q^{49} -36.5908 q^{53} +1.18654 q^{55} +(-574.476 - 482.043i) q^{59} +(-64.9040 - 23.6231i) q^{61} +(-37.1659 + 31.1859i) q^{65} +(109.760 - 622.479i) q^{67} +(66.9800 - 116.013i) q^{71} +(-435.761 - 754.761i) q^{73} +(-3.11484 + 1.13371i) q^{77} +(55.7069 + 315.930i) q^{79} +(197.919 + 1122.46i) q^{83} +(339.028 - 123.396i) q^{85} +(259.505 + 449.475i) q^{89} +(67.7684 - 117.378i) q^{91} +(154.213 - 874.583i) q^{95} +(-1153.80 + 968.155i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 12 q^{5}+O(q^{10}) \) 54 * q - 12 * q^5	\( 54 q - 12 q^{5} + 87 q^{11} - 204 q^{17} - 96 q^{23} - 216 q^{25} - 318 q^{29} - 54 q^{31} - 6 q^{35} - 867 q^{41} - 513 q^{43} + 1548 q^{47} + 594 q^{49} + 1068 q^{53} + 1218 q^{59} - 54 q^{61} - 96 q^{65} - 2997 q^{67} + 120 q^{71} - 216 q^{73} - 3480 q^{77} + 2808 q^{79} - 4464 q^{83} + 2160 q^{85} - 4029 q^{89} + 270 q^{91} + 1650 q^{95} - 3483 q^{97}+O(q^{100}) \) 54 * q - 12 * q^5 + 87 * q^11 - 204 * q^17 - 96 * q^23 - 216 * q^25 - 318 * q^29 - 54 * q^31 - 6 * q^35 - 867 * q^41 - 513 * q^43 + 1548 * q^47 + 594 * q^49 + 1068 * q^53 + 1218 * q^59 - 54 * q^61 - 96 * q^65 - 2997 * q^67 + 120 * q^71 - 216 * q^73 - 3480 * q^77 + 2808 * q^79 - 4464 * q^83 + 2160 * q^85 - 4029 * q^89 + 270 * q^91 + 1650 * q^95 - 3483 * q^97

Introduction

L-functions

Modular forms

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Database

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