LMFDB - Embedded newform 108.4.e.a.37.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [108,4,Mod(37,108)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(108, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

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

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

chi := DirichletCharacter("108.37");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 108 = 2^{2} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	108.e (of order $3$, degree $2$, 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:	$6.37220628062$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.6831243.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 13x^{4} + 49x^{2} + 48 $ x^6 + 13x^4 + 49x^2 + 48
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{5} $
Twist minimal:	no (minimal twist has level 36)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			37.3
Root			$-2.13353i$ of defining polynomial
Character	$\chi$	$=$	108.37
Dual form			108.4.e.a.73.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(6.37096 - 11.0348i) q^{5} +(7.02674 + 12.1707i) q^{7} +O(q^{10})$	$q+(6.37096 - 11.0348i) q^{5} +(7.02674 + 12.1707i) q^{7} +(-21.2745 - 36.8486i) q^{11} +(36.2316 - 62.7550i) q^{13} +59.6114 q^{17} +105.570 q^{19} +(-0.112590 + 0.195011i) q^{23} +(-18.6781 - 32.3515i) q^{25} +(-112.855 - 195.470i) q^{29} +(-100.597 + 174.239i) q^{31} +179.068 q^{35} -152.926 q^{37} +(-244.824 + 424.047i) q^{41} +(3.79372 + 6.57091i) q^{43} +(186.696 + 323.366i) q^{47} +(72.7498 - 126.006i) q^{49} +43.6780 q^{53} -542.157 q^{55} +(-335.949 + 581.881i) q^{59} +(-37.0177 - 64.1165i) q^{61} +(-461.660 - 799.619i) q^{65} +(-210.436 + 364.485i) q^{67} +730.840 q^{71} +473.927 q^{73} +(298.982 - 517.851i) q^{77} +(264.811 + 458.666i) q^{79} +(13.0767 + 22.6495i) q^{83} +(379.781 - 657.801i) q^{85} -415.949 q^{89} +1018.36 q^{91} +(672.583 - 1164.95i) q^{95} +(-463.743 - 803.226i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{5} - 6 q^{7}+O(q^{10}) $ 6 * q - 6 * q^5 - 6 * q^7	$ 6 q - 6 q^{5} - 6 q^{7} - 51 q^{11} + 12 q^{13} + 222 q^{17} + 30 q^{19} - 210 q^{23} - 3 q^{25} - 456 q^{29} + 48 q^{31} + 1104 q^{35} - 96 q^{37} - 897 q^{41} + 129 q^{43} - 522 q^{47} - 225 q^{49} + 2208 q^{53} - 216 q^{55} - 453 q^{59} - 402 q^{61} - 1110 q^{65} - 213 q^{67} - 120 q^{71} + 750 q^{73} - 1128 q^{77} + 552 q^{79} + 612 q^{83} + 1188 q^{85} + 924 q^{89} - 264 q^{91} + 2184 q^{95} + 93 q^{97}+O(q^{100}) $ 6 * q - 6 * q^5 - 6 * q^7 - 51 * q^11 + 12 * q^13 + 222 * q^17 + 30 * q^19 - 210 * q^23 - 3 * q^25 - 456 * q^29 + 48 * q^31 + 1104 * q^35 - 96 * q^37 - 897 * q^41 + 129 * q^43 - 522 * q^47 - 225 * q^49 + 2208 * q^53 - 216 * q^55 - 453 * q^59 - 402 * q^61 - 1110 * q^65 - 213 * q^67 - 120 * q^71 + 750 * q^73 - 1128 * q^77 + 552 * q^79 + 612 * q^83 + 1188 * q^85 + 924 * q^89 - 264 * q^91 + 2184 * q^95 + 93 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$.

$n$	$29$	$55$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	6.37096	−	11.0348i	0.569836	−	0.986984i	−0.426746	−	0.904371i	$-0.640340\pi$
$5$	6.37096	−	11.0348i	0.569836	−	0.986984i	0.996582	−	0.0826127i	$-0.0263264\pi$
$6$		0			0
$7$	7.02674	+	12.1707i	0.379408	+	0.657155i	0.990976	−	0.134037i	$-0.0427942\pi$
$7$	7.02674	+	12.1707i	0.379408	+	0.657155i	−0.611568	+	0.791192i	$0.709461\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−21.2745	−	36.8486i	−0.583138	−	1.01002i	−0.995105	−	0.0988256i	$-0.968491\pi$
$11$	−21.2745	−	36.8486i	−0.583138	−	1.01002i	0.411967	−	0.911199i	$-0.364842\pi$
$12$		0			0
$13$	36.2316	−	62.7550i	0.772988	−	1.33885i	−0.162930	−	0.986638i	$-0.552095\pi$
$13$	36.2316	−	62.7550i	0.772988	−	1.33885i	0.935918	−	0.352217i	$-0.114572\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	59.6114			0.850464			0.425232	−	0.905084i	$-0.360193\pi$
$17$	59.6114			0.850464			0.425232	+	0.905084i	$0.360193\pi$
$18$		0			0
$19$	105.570			1.27471			0.637354	−	0.770571i	$-0.280029\pi$
$19$	105.570			1.27471			0.637354	+	0.770571i	$0.280029\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.112590	+	0.195011i	−0.00102072	+	0.00176794i	−0.866535	−	0.499116i	$-0.833658\pi$
$23$	−0.112590	+	0.195011i	−0.00102072	+	0.00176794i	0.865515	+	0.500884i	$0.166992\pi$
$24$		0			0
$25$	−18.6781	−	32.3515i	−0.149425	−	0.258812i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−112.855	−	195.470i	−0.722642	−	1.25165i	−0.959937	−	0.280214i	$-0.909594\pi$
$29$	−112.855	−	195.470i	−0.722642	−	1.25165i	0.237296	−	0.971437i	$-0.423739\pi$
$30$		0			0
$31$	−100.597	+	174.239i	−0.582831	+	1.00949i	0.412312	+	0.911043i	$0.364722\pi$
$31$	−100.597	+	174.239i	−0.582831	+	1.00949i	−0.995142	+	0.0984492i	$0.968612\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	179.068			0.864802
$36$		0			0
$37$	−152.926			−0.679485			−0.339743	−	0.940518i	$-0.610340\pi$
$37$	−152.926			−0.679485			−0.339743	+	0.940518i	$0.610340\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−244.824	+	424.047i	−0.932563	+	1.61525i	−0.153639	+	0.988127i	$0.549099\pi$
$41$	−244.824	+	424.047i	−0.932563	+	1.61525i	−0.778923	+	0.627119i	$0.784234\pi$
$42$		0			0
$43$	3.79372	+	6.57091i	0.0134543	+	0.0233036i	0.872674	−	0.488303i	$-0.162384\pi$
$43$	3.79372	+	6.57091i	0.0134543	+	0.0233036i	−0.859220	+	0.511607i	$0.829051\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	186.696	+	323.366i	0.579412	+	1.00357i	0.995547	+	0.0942681i	$0.0300511\pi$
$47$	186.696	+	323.366i	0.579412	+	1.00357i	−0.416135	+	0.909303i	$0.636616\pi$
$48$		0			0
$49$	72.7498	−	126.006i	0.212098	−	0.367365i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	43.6780			0.113201			0.0566003	−	0.998397i	$-0.481974\pi$
$53$	43.6780			0.113201			0.0566003	+	0.998397i	$0.481974\pi$
$54$		0			0
$55$	−542.157			−1.32917
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−335.949	+	581.881i	−0.741303	+	1.28397i	0.210599	+	0.977572i	$0.432458\pi$
$59$	−335.949	+	581.881i	−0.741303	+	1.28397i	−0.951902	+	0.306402i	$0.900875\pi$
$60$		0			0
$61$	−37.0177	−	64.1165i	−0.0776988	−	0.134578i	0.824558	−	0.565778i	$-0.191424\pi$
$61$	−37.0177	−	64.1165i	−0.0776988	−	0.134578i	−0.902257	+	0.431199i	$0.858091\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−461.660	−	799.619i	−0.880952	−	1.52585i
$66$		0			0
$67$	−210.436	+	364.485i	−0.383713	+	0.664611i	−0.991590	−	0.129421i	$-0.958688\pi$
$67$	−210.436	+	364.485i	−0.383713	+	0.664611i	0.607876	+	0.794032i	$0.292022\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	730.840			1.22162			0.610808	−	0.791778i	$-0.290845\pi$
$71$	730.840			1.22162			0.610808	+	0.791778i	$0.290845\pi$
$72$		0			0
$73$	473.927			0.759849			0.379925	−	0.925017i	$-0.375950\pi$
$73$	473.927			0.759849			0.379925	+	0.925017i	$0.375950\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	298.982	−	517.851i	0.442495	−	0.766424i
$78$		0			0
$79$	264.811	+	458.666i	0.377134	+	0.653215i	0.990644	−	0.136472i	$-0.0435764\pi$
$79$	264.811	+	458.666i	0.377134	+	0.653215i	−0.613510	+	0.789687i	$0.710243\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	13.0767	+	22.6495i	0.0172934	+	0.0299531i	0.874543	−	0.484949i	$-0.161162\pi$
$83$	13.0767	+	22.6495i	0.0172934	+	0.0299531i	−0.857249	+	0.514902i	$0.827828\pi$
$84$		0			0
$85$	379.781	−	657.801i	0.484624	−	0.839394i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−415.949			−0.495399			−0.247700	−	0.968837i	$-0.579675\pi$
$89$	−415.949			−0.495399			−0.247700	+	0.968837i	$0.579675\pi$
$90$		0			0
$91$	1018.36			1.17311
$92$		0			0
$93$		0			0
$94$		0			0
$95$	672.583	−	1164.95i	0.726374	−	1.25812i
$96$		0			0
$97$	−463.743	−	803.226i	−0.485422	−	0.840776i	0.514438	−	0.857528i	$-0.328001\pi$
$97$	−463.743	−	803.226i	−0.485422	−	0.840776i	−0.999860	+	0.0167522i	$0.994667\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−45.0590	−	78.0445i	−0.0443915	−	0.0768883i	0.842976	−	0.537951i	$-0.180802\pi$
$101$	−45.0590	−	78.0445i	−0.0443915	−	0.0768883i	−0.887367	+	0.461063i	$0.847468\pi$
$102$		0			0
$103$	162.826	−	282.023i	0.155765	−	0.269792i	−0.777573	−	0.628793i	$-0.783549\pi$
$103$	162.826	−	282.023i	0.155765	−	0.269792i	0.933337	+	0.359001i	$0.116883\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1073.49			−0.969888			−0.484944	−	0.874545i	$-0.661160\pi$
$107$	−1073.49			−0.969888			−0.484944	+	0.874545i	$0.661160\pi$
$108$		0			0
$109$	601.488			0.528552			0.264276	−	0.964447i	$-0.414867\pi$
$109$	601.488			0.528552			0.264276	+	0.964447i	$0.414867\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−210.019	+	363.764i	−0.174840	+	0.302832i	−0.940106	−	0.340882i	$-0.889274\pi$
$113$	−210.019	+	363.764i	−0.174840	+	0.302832i	0.765266	+	0.643715i	$0.222608\pi$
$114$		0			0
$115$	1.43461	+	2.48482i	0.00116329	+	0.00201487i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	418.874	+	725.511i	0.322673	+	0.558886i
$120$		0			0
$121$	−239.713	+	415.194i	−0.180100	+	0.311942i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1116.75			0.799080
$126$		0			0
$127$	−980.264			−0.684916			−0.342458	−	0.939533i	$-0.611259\pi$
$127$	−980.264			−0.684916			−0.342458	+	0.939533i	$0.611259\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	677.695	−	1173.80i	0.451988	−	0.782866i	−0.546521	−	0.837445i	$-0.684048\pi$
$131$	677.695	−	1173.80i	0.451988	−	0.782866i	0.998509	+	0.0545787i	$0.0173816\pi$
$132$		0			0
$133$	741.815	+	1284.86i	0.483635	+	0.837681i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	453.261	+	785.071i	0.282662	+	0.489585i	0.972040	−	0.234817i	$-0.0754492\pi$
$137$	453.261	+	785.071i	0.282662	+	0.489585i	−0.689378	+	0.724402i	$0.742116\pi$
$138$		0			0
$139$	409.209	−	708.771i	0.249703	−	0.432498i	−0.713741	−	0.700410i	$-0.753000\pi$
$139$	409.209	−	708.771i	0.249703	−	0.432498i	0.963443	+	0.267912i	$0.0863338\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−3083.25			−1.80303
$144$		0			0
$145$	−2875.97			−1.64715
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−579.748	+	1004.15i	−0.318757	+	0.552103i	−0.980229	−	0.197867i	$-0.936599\pi$
$149$	−579.748	+	1004.15i	−0.318757	+	0.552103i	0.661472	+	0.749970i	$0.269932\pi$
$150$		0			0
$151$	318.987	+	552.501i	0.171912	+	0.297761i	0.939088	−	0.343676i	$-0.111672\pi$
$151$	318.987	+	552.501i	0.171912	+	0.297761i	−0.767176	+	0.641437i	$0.778339\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1281.80	+	2220.14i	0.664235	+	1.15049i
$156$		0			0
$157$	−1645.86	+	2850.71i	−0.836649	+	1.44912i	0.0560311	+	0.998429i	$0.482155\pi$
$157$	−1645.86	+	2850.71i	−0.836649	+	1.44912i	−0.892680	+	0.450690i	$0.851178\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−3.16456			−0.00154908
$162$		0			0
$163$	−1197.14			−0.575258			−0.287629	−	0.957742i	$-0.592867\pi$
$163$	−1197.14			−0.575258			−0.287629	+	0.957742i	$0.592867\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	419.419	−	726.455i	0.194345	−	0.336615i	−0.752341	−	0.658774i	$-0.771075\pi$
$167$	419.419	−	726.455i	0.194345	−	0.336615i	0.946686	+	0.322159i	$0.104409\pi$
$168$		0			0
$169$	−1526.96	−	2644.77i	−0.695021	−	1.20381i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1153.59	+	1998.08i	0.506970	+	0.878098i	0.999967	+	0.00806725i	$0.00256791\pi$
$173$	1153.59	+	1998.08i	0.506970	+	0.878098i	−0.492997	+	0.870031i	$0.664099\pi$
$174$		0			0
$175$	262.493	−	454.651i	0.113386	−	0.196391i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−3114.47			−1.30048			−0.650241	−	0.759728i	$-0.725332\pi$
$179$	−3114.47			−1.30048			−0.650241	+	0.759728i	$0.725332\pi$
$180$		0			0
$181$	3902.75			1.60270			0.801350	−	0.598195i	$-0.204115\pi$
$181$	3902.75			1.60270			0.801350	+	0.598195i	$0.204115\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−974.288	+	1687.52i	−0.387195	+	0.670641i
$186$		0			0
$187$	−1268.20	−	2196.60i	−0.495938	−	0.858989i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	52.8697	+	91.5730i	0.0200289	+	0.0346911i	0.875866	−	0.482554i	$-0.160291\pi$
$191$	52.8697	+	91.5730i	0.0200289	+	0.0346911i	−0.855837	+	0.517245i	$0.826958\pi$
$192$		0			0
$193$	792.685	−	1372.97i	0.295641	−	0.512065i	−0.679493	−	0.733682i	$-0.737800\pi$
$193$	792.685	−	1372.97i	0.295641	−	0.512065i	0.975134	+	0.221617i	$0.0711334\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	3905.37			1.41242			0.706208	−	0.708005i	$-0.250405\pi$
$197$	3905.37			1.41242			0.706208	+	0.708005i	$0.250405\pi$
$198$		0			0
$199$	1538.77			0.548143			0.274072	−	0.961709i	$-0.411629\pi$
$199$	1538.77			0.548143			0.274072	+	0.961709i	$0.411629\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1586.00	−	2747.04i	0.548353	−	0.949775i
$204$		0			0
$205$	3119.52	+	5403.17i	1.06281	+	1.84085i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−2245.96	−	3890.11i	−0.743331	−	1.28749i
$210$		0			0
$211$	−470.733	+	815.334i	−0.153586	+	0.266019i	−0.932543	−	0.361058i	$-0.882416\pi$
$211$	−470.733	+	815.334i	−0.153586	+	0.266019i	0.778957	+	0.627077i	$0.215749\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	96.6784			0.0306670
$216$		0			0
$217$	−2827.48			−0.884523
$218$		0			0
$219$		0			0
$220$		0			0
$221$	2159.82	−	3740.91i	0.657398	−	1.13865i
$222$		0			0
$223$	−1660.78	−	2876.56i	−0.498719	−	0.863807i	0.501280	−	0.865285i	$-0.332863\pi$
$223$	−1660.78	−	2876.56i	−0.498719	−	0.863807i	−0.999999	+	0.00147850i	$0.999529\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−2105.92	−	3647.56i	−0.615749	−	1.06651i	−0.990253	−	0.139283i	$-0.955520\pi$
$227$	−2105.92	−	3647.56i	−0.615749	−	1.06651i	0.374504	−	0.927225i	$-0.377813\pi$
$228$		0			0
$229$	177.072	−	306.698i	0.0510973	−	0.0885031i	−0.839345	−	0.543598i	$-0.817061\pi$
$229$	177.072	−	306.698i	0.0510973	−	0.0885031i	0.890443	+	0.455095i	$0.150395\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−868.789			−0.244276			−0.122138	−	0.992513i	$-0.538975\pi$
$233$	−868.789			−0.244276			−0.122138	+	0.992513i	$0.538975\pi$
$234$		0			0
$235$	4757.72			1.32068
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−602.177	+	1043.00i	−0.162977	+	0.282285i	−0.935935	−	0.352172i	$-0.885443\pi$
$239$	−602.177	+	1043.00i	−0.162977	+	0.282285i	0.772958	+	0.634457i	$0.218776\pi$
$240$		0			0
$241$	2543.70	+	4405.82i	0.679893	+	1.17761i	0.975013	+	0.222149i	$0.0713072\pi$
$241$	2543.70	+	4405.82i	0.679893	+	1.17761i	−0.295120	+	0.955460i	$0.595360\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−926.971	−	1605.56i	−0.241722	−	0.418676i
$246$		0			0
$247$	3824.98	−	6625.06i	0.985335	−	1.70665i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	5463.91			1.37402			0.687010	−	0.726648i	$-0.258923\pi$
$251$	5463.91			1.37402			0.687010	+	0.726648i	$0.258923\pi$
$252$		0			0
$253$	9.58119			0.00238089
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−3701.06	+	6410.43i	−0.898311	+	1.55592i	−0.0686576	+	0.997640i	$0.521872\pi$
$257$	−3701.06	+	6410.43i	−0.898311	+	1.55592i	−0.829653	+	0.558279i	$0.811462\pi$
$258$		0			0
$259$	−1074.58	−	1861.22i	−0.257803	−	0.446527i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−100.015	−	173.232i	−0.0234495	−	0.0406156i	0.854063	−	0.520170i	$-0.174132\pi$
$263$	−100.015	−	173.232i	−0.0234495	−	0.0406156i	−0.877512	+	0.479555i	$0.840798\pi$
$264$		0			0
$265$	278.270	−	481.979i	0.0645057	−	0.111727i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	5493.50			1.24515			0.622574	−	0.782561i	$-0.286087\pi$
$269$	5493.50			1.24515			0.622574	+	0.782561i	$0.286087\pi$
$270$		0			0
$271$	−1861.89			−0.417350			−0.208675	−	0.977985i	$-0.566915\pi$
$271$	−1861.89			−0.417350			−0.208675	+	0.977985i	$0.566915\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−794.738	+	1376.53i	−0.174271	+	0.301846i
$276$		0			0
$277$	−2819.52	−	4883.55i	−0.611582	−	1.05929i	−0.990974	−	0.134055i	$-0.957200\pi$
$277$	−2819.52	−	4883.55i	−0.611582	−	1.05929i	0.379391	−	0.925236i	$-0.376133\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1279.02	+	2215.33i	0.271530	+	0.470303i	0.969254	−	0.246063i	$-0.0791371\pi$
$281$	1279.02	+	2215.33i	0.271530	+	0.470303i	−0.697724	+	0.716367i	$0.745804\pi$
$282$		0			0
$283$	−3733.97	+	6467.43i	−0.784317	+	1.35848i	0.145089	+	0.989419i	$0.453653\pi$
$283$	−3733.97	+	6467.43i	−0.784317	+	1.35848i	−0.929406	+	0.369058i	$0.879680\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−6881.26			−1.41529
$288$		0			0
$289$	−1359.48			−0.276712
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1780.66	+	3084.19i	−0.355041	+	0.614950i	−0.987125	−	0.159950i	$-0.948867\pi$
$293$	−1780.66	+	3084.19i	−0.355041	+	0.614950i	0.632084	+	0.774900i	$0.282200\pi$
$294$		0			0
$295$	4280.64	+	7414.28i	0.844842	+	1.46331i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	8.15863	+	14.1312i	0.00157801	+	0.00273320i
$300$		0			0
$301$	−53.3149	+	92.3442i	−0.0102094	+	0.0176832i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−943.352			−0.177102
$306$		0			0
$307$	−6101.93			−1.13438			−0.567192	−	0.823586i	$-0.691970\pi$
$307$	−6101.93			−1.13438			−0.567192	+	0.823586i	$0.691970\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	185.618	−	321.499i	0.0338438	−	0.0586191i	−0.848607	−	0.529023i	$-0.822558\pi$
$311$	185.618	−	321.499i	0.0338438	−	0.0586191i	0.882451	+	0.470404i	$0.155892\pi$
$312$		0			0
$313$	−4660.55	−	8072.31i	−0.841629	−	1.45774i	−0.888517	−	0.458844i	$-0.848264\pi$
$313$	−4660.55	−	8072.31i	−0.841629	−	1.45774i	0.0468881	−	0.998900i	$-0.485070\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3191.79	−	5528.33i	−0.565516	−	0.979502i	−0.997001	−	0.0773824i	$-0.975344\pi$
$317$	−3191.79	−	5528.33i	−0.565516	−	0.979502i	0.431486	−	0.902120i	$-0.357990\pi$
$318$		0			0
$319$	−4801.87	+	8317.08i	−0.842799	+	1.45977i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	6293.19			1.08409
$324$		0			0
$325$	−2706.96			−0.462015
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−2623.73	+	4544.43i	−0.439668	+	0.761527i
$330$		0			0
$331$	1264.71	+	2190.54i	0.210014	+	0.363755i	0.951719	−	0.306971i	$-0.0993156\pi$
$331$	1264.71	+	2190.54i	0.210014	+	0.363755i	−0.741704	+	0.670727i	$0.765982\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	2681.35	+	4644.24i	0.437307	+	0.757438i
$336$		0			0
$337$	−1799.91	+	3117.53i	−0.290941	+	0.503924i	−0.974032	−	0.226408i	$-0.927302\pi$
$337$	−1799.91	+	3117.53i	−0.290941	+	0.503924i	0.683092	+	0.730333i	$0.260635\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	8560.62			1.35948
$342$		0			0
$343$	6865.12			1.08070
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−5686.60	+	9849.48i	−0.879748	+	1.52377i	−0.0281313	+	0.999604i	$0.508956\pi$
$347$	−5686.60	+	9849.48i	−0.879748	+	1.52377i	−0.851617	+	0.524165i	$0.824378\pi$
$348$		0			0
$349$	894.476	+	1549.28i	0.137193	+	0.237624i	0.926433	−	0.376460i	$-0.122859\pi$
$349$	894.476	+	1549.28i	0.137193	+	0.237624i	−0.789240	+	0.614084i	$0.789525\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−555.184	−	961.608i	−0.0837096	−	0.144989i	0.821131	−	0.570740i	$-0.193343\pi$
$353$	−555.184	−	961.608i	−0.0837096	−	0.144989i	−0.904841	+	0.425750i	$0.860010\pi$
$354$		0			0
$355$	4656.15	−	8064.69i	0.696121	−	1.20572i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−2250.71			−0.330886			−0.165443	−	0.986219i	$-0.552905\pi$
$359$	−2250.71			−0.330886			−0.165443	+	0.986219i	$0.552905\pi$
$360$		0			0
$361$	4286.07			0.624883
$362$		0			0
$363$		0			0
$364$		0			0
$365$	3019.37	−	5229.70i	0.432989	−	0.749959i
$366$		0			0
$367$	−1253.69	−	2171.45i	−0.178316	−	0.308852i	0.762988	−	0.646413i	$-0.223732\pi$
$367$	−1253.69	−	2171.45i	−0.178316	−	0.308852i	−0.941304	+	0.337560i	$0.890398\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	306.914	+	531.591i	0.0429493	+	0.0743903i
$372$		0			0
$373$	2482.71	−	4300.17i	0.344637	−	0.596929i	−0.640651	−	0.767833i	$-0.721335\pi$
$373$	2482.71	−	4300.17i	0.344637	−	0.596929i	0.985288	+	0.170903i	$0.0546686\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−16355.6			−2.23437
$378$		0			0
$379$	−13541.7			−1.83534			−0.917668	−	0.397349i	$-0.869930\pi$
$379$	−13541.7			−1.83534			−0.917668	+	0.397349i	$0.869930\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	4175.04	−	7231.38i	0.557009	−	0.964768i	−0.440735	−	0.897637i	$-0.645282\pi$
$383$	4175.04	−	7231.38i	0.557009	−	0.964768i	0.997744	−	0.0671311i	$-0.0213846\pi$
$384$		0			0
$385$	−3809.60	−	6598.41i	−0.504299	−	0.873471i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1162.31	−	2013.18i	−0.151495	−	0.262396i	0.780283	−	0.625427i	$-0.215075\pi$
$389$	−1162.31	−	2013.18i	−0.151495	−	0.262396i	−0.931777	+	0.363031i	$0.881742\pi$
$390$		0			0
$391$	−6.71164	+	11.6249i	−0.000868087	+	0.00150357i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	6748.39			0.859617
$396$		0			0
$397$	13253.5			1.67550			0.837749	−	0.546056i	$-0.183871\pi$
$397$	13253.5			1.67550			0.837749	+	0.546056i	$0.183871\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	7790.28	−	13493.2i	0.970144	−	1.68034i	0.275037	−	0.961434i	$-0.411310\pi$
$401$	7790.28	−	13493.2i	0.970144	−	1.68034i	0.695108	−	0.718906i	$-0.255357\pi$
$402$		0			0
$403$	7289.58	+	12625.9i	0.901042	+	1.56065i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	3253.44	+	5635.13i	0.396234	+	0.686297i
$408$		0			0
$409$	6119.59	−	10599.4i	0.739840	−	1.28144i	−0.212728	−	0.977112i	$-0.568235\pi$
$409$	6119.59	−	10599.4i	0.739840	−	1.28144i	0.952567	−	0.304328i	$-0.0984320\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−9442.52			−1.12503
$414$		0			0
$415$	333.244			0.0394176
$416$		0			0
$417$		0			0
$418$		0			0
$419$	3196.50	−	5536.50i	0.372695	−	0.645526i	−0.617284	−	0.786740i	$-0.711767\pi$
$419$	3196.50	−	5536.50i	0.372695	−	0.645526i	0.989979	+	0.141214i	$0.0451005\pi$
$420$		0			0
$421$	7916.31	+	13711.5i	0.916431	+	1.58730i	0.804793	+	0.593556i	$0.202276\pi$
$421$	7916.31	+	13711.5i	0.916431	+	1.58730i	0.111638	+	0.993749i	$0.464390\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1113.43	−	1928.52i	−0.127081	−	0.220110i
$426$		0			0
$427$	520.227	−	901.060i	0.0589592	−	0.102120i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	9339.01			1.04372			0.521861	−	0.853030i	$-0.325238\pi$
$431$	9339.01			1.04372			0.521861	+	0.853030i	$0.325238\pi$
$432$		0			0
$433$	−3379.19			−0.375043			−0.187522	−	0.982260i	$-0.560045\pi$
$433$	−3379.19			−0.375043			−0.187522	+	0.982260i	$0.560045\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−11.8861	+	20.5874i	−0.00130112	+	0.00225361i
$438$		0			0
$439$	−7273.52	−	12598.1i	−0.790766	−	1.36965i	−0.925493	−	0.378764i	$-0.876349\pi$
$439$	−7273.52	−	12598.1i	−0.790766	−	1.36965i	0.134728	−	0.990883i	$-0.456984\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−2951.00	−	5111.28i	−0.316492	−	0.548181i	0.663261	−	0.748388i	$-0.269172\pi$
$443$	−2951.00	−	5111.28i	−0.316492	−	0.548181i	−0.979754	+	0.200207i	$0.935839\pi$
$444$		0			0
$445$	−2650.00	+	4589.93i	−0.282296	+	0.488951i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−2448.48			−0.257352			−0.128676	−	0.991687i	$-0.541073\pi$
$449$	−2448.48			−0.257352			−0.128676	+	0.991687i	$0.541073\pi$
$450$		0			0
$451$	20834.1			2.17525
$452$		0			0
$453$		0			0
$454$		0			0
$455$	6487.93	−	11237.4i	0.668481	−	1.15784i
$456$		0			0
$457$	−2236.87	−	3874.38i	−0.228964	−	0.396577i	0.728537	−	0.685006i	$-0.240200\pi$
$457$	−2236.87	−	3874.38i	−0.228964	−	0.396577i	−0.957501	+	0.288429i	$0.906867\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	251.107	+	434.929i	0.0253692	+	0.0439407i	0.878431	−	0.477869i	$-0.158591\pi$
$461$	251.107	+	434.929i	0.0253692	+	0.0439407i	−0.853062	+	0.521809i	$0.825257\pi$
$462$		0			0
$463$	−3543.37	+	6137.30i	−0.355668	+	0.616036i	−0.987232	−	0.159288i	$-0.949080\pi$
$463$	−3543.37	+	6137.30i	−0.355668	+	0.616036i	0.631564	+	0.775324i	$0.282413\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−8057.18			−0.798376			−0.399188	−	0.916869i	$-0.630708\pi$
$467$	−8057.18			−0.798376			−0.399188	+	0.916869i	$0.630708\pi$
$468$		0			0
$469$	−5914.71			−0.582336
$470$		0			0
$471$		0			0
$472$		0			0
$473$	161.419	−	279.586i	0.0156915	−	0.0271784i
$474$		0			0
$475$	−1971.85	−	3415.35i	−0.190473	−	0.329910i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	8580.44	+	14861.8i	0.818477	+	1.41764i	0.906804	+	0.421552i	$0.138515\pi$
$479$	8580.44	+	14861.8i	0.818477	+	1.41764i	−0.0883271	+	0.996092i	$0.528152\pi$
$480$		0			0
$481$	−5540.77	+	9596.90i	−0.525234	+	0.909732i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−11817.9			−1.10644
$486$		0			0
$487$	−9909.84			−0.922090			−0.461045	−	0.887377i	$-0.652525\pi$
$487$	−9909.84			−0.922090			−0.461045	+	0.887377i	$0.652525\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−9744.27	+	16877.6i	−0.895627	+	1.55127i	−0.0626007	+	0.998039i	$0.519939\pi$
$491$	−9744.27	+	16877.6i	−0.895627	+	1.55127i	−0.833026	+	0.553233i	$0.813394\pi$
$492$		0			0
$493$	−6727.43	−	11652.2i	−0.614580	−	1.06448i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	5135.43	+	8894.82i	0.463492	+	0.802791i
$498$		0			0
$499$	462.728	−	801.468i	0.0415121	−	0.0719011i	−0.844523	−	0.535520i	$-0.820116\pi$
$499$	462.728	−	801.468i	0.0415121	−	0.0719011i	0.886035	+	0.463618i	$0.153449\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−8723.45			−0.773279			−0.386640	−	0.922231i	$-0.626364\pi$
$503$	−8723.45			−0.773279			−0.386640	+	0.922231i	$0.626364\pi$
$504$		0			0
$505$	−1148.28			−0.101183
$506$		0			0
$507$		0			0
$508$		0			0
$509$	1524.35	−	2640.25i	0.132742	−	0.229916i	−0.791991	−	0.610533i	$-0.790955\pi$
$509$	1524.35	−	2640.25i	0.132742	−	0.229916i	0.924733	+	0.380617i	$0.124288\pi$
$510$		0			0
$511$	3330.17	+	5768.02i	0.288293	+	0.499339i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2074.72	−	3593.52i	−0.177520	−	0.307474i
$516$		0			0
$517$	7943.73	−	13758.9i	0.675754	−	1.17044i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	15593.0			1.31121			0.655607	−	0.755103i	$-0.272413\pi$
$521$	15593.0			1.31121			0.655607	+	0.755103i	$0.272413\pi$
$522$		0			0
$523$	−9052.67			−0.756875			−0.378438	−	0.925627i	$-0.623539\pi$
$523$	−9052.67			−0.756875			−0.378438	+	0.925627i	$0.623539\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−5996.72	+	10386.6i	−0.495676	+	0.858536i
$528$		0			0
$529$	6083.47	+	10536.9i	0.499998	+	0.866022i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	17740.7	+	30727.9i	1.44172	+	2.49713i
$534$		0			0
$535$	−6839.14	+	11845.7i	−0.552677	+	0.957264i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−6190.87			−0.494730
$540$		0			0
$541$	−11742.8			−0.933200			−0.466600	−	0.884469i	$-0.654521\pi$
$541$	−11742.8			−0.933200			−0.466600	+	0.884469i	$0.654521\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	3832.05	−	6637.31i	0.301187	−	0.521672i
$546$		0			0
$547$	−1973.46	−	3418.12i	−0.154257	−	0.267182i	0.778531	−	0.627606i	$-0.215965\pi$
$547$	−1973.46	−	3418.12i	−0.154257	−	0.267182i	−0.932788	+	0.360424i	$0.882632\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−11914.1	−	20635.8i	−0.921158	−	1.59549i
$552$		0			0
$553$	−3721.52	+	6445.86i	−0.286175	+	0.495670i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−3129.69			−0.238078			−0.119039	−	0.992890i	$-0.537981\pi$
$557$	−3129.69			−0.238078			−0.119039	+	0.992890i	$0.537981\pi$
$558$		0			0
$559$	549.810			0.0416001
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−96.0178	+	166.308i	−0.00718769	+	0.0124494i	−0.869597	−	0.493762i	$-0.835621\pi$
$563$	−96.0178	+	166.308i	−0.00718769	+	0.0124494i	0.862409	+	0.506212i	$0.168955\pi$
$564$		0			0
$565$	2676.05	+	4635.05i	0.199260	+	0.345129i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	1630.51	+	2824.12i	0.120131	+	0.208073i	0.919819	−	0.392343i	$-0.128335\pi$
$569$	1630.51	+	2824.12i	0.120131	+	0.208073i	−0.799688	+	0.600415i	$0.795002\pi$
$570$		0			0
$571$	4707.22	−	8153.15i	0.344993	−	0.597546i	−0.640359	−	0.768075i	$-0.721215\pi$
$571$	4707.22	−	8153.15i	0.344993	−	0.597546i	0.985353	+	0.170530i	$0.0545479\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	8.41187			0.000610086
$576$		0			0
$577$	−9739.86			−0.702731			−0.351366	−	0.936238i	$-0.614283\pi$
$577$	−9739.86			−0.702731			−0.351366	+	0.936238i	$0.614283\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−183.773	+	318.304i	−0.0131225	+	0.0227289i
$582$		0			0
$583$	−929.229	−	1609.47i	−0.0660116	−	0.114335i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−12702.8	−	22002.0i	−0.893190	−	1.54705i	−0.836029	−	0.548685i	$-0.815129\pi$
$587$	−12702.8	−	22002.0i	−0.893190	−	1.54705i	−0.0571605	−	0.998365i	$-0.518205\pi$
$588$		0			0
$589$	−10620.0	+	18394.5i	−0.742939	+	1.28681i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−2751.26			−0.190524			−0.0952620	−	0.995452i	$-0.530369\pi$
$593$	−2751.26			−0.190524			−0.0952620	+	0.995452i	$0.530369\pi$
$594$		0			0
$595$	10674.5			0.735482
$596$		0			0
$597$		0			0
$598$		0			0
$599$	1658.11	−	2871.93i	0.113103	−	0.195900i	−0.803917	−	0.594742i	$-0.797254\pi$
$599$	1658.11	−	2871.93i	0.113103	−	0.195900i	0.917020	+	0.398842i	$0.130588\pi$
$600$		0			0
$601$	3959.84	+	6858.65i	0.268761	+	0.465508i	0.968542	−	0.248850i	$-0.0800525\pi$
$601$	3959.84	+	6858.65i	0.268761	+	0.465508i	−0.699781	+	0.714357i	$0.746719\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	3054.40	+	5290.37i	0.205254	+	0.355511i
$606$		0			0
$607$	4737.24	−	8205.13i	0.316768	−	0.548659i	−0.663043	−	0.748581i	$-0.730736\pi$
$607$	4737.24	−	8205.13i	0.316768	−	0.548659i	0.979812	+	0.199922i	$0.0640688\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	27057.2			1.79151
$612$		0			0
$613$	−3165.90			−0.208596			−0.104298	−	0.994546i	$-0.533260\pi$
$613$	−3165.90			−0.208596			−0.104298	+	0.994546i	$0.533260\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−315.172	+	545.894i	−0.0205646	+	0.0356189i	−0.876125	−	0.482085i	$-0.839880\pi$
$617$	−315.172	+	545.894i	−0.0205646	+	0.0356189i	0.855560	+	0.517704i	$0.173213\pi$
$618$		0			0
$619$	−586.272	−	1015.45i	−0.0380683	−	0.0659362i	0.846364	−	0.532606i	$-0.178787\pi$
$619$	−586.272	−	1015.45i	−0.0380683	−	0.0659362i	−0.884432	+	0.466669i	$0.845454\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−2922.77	−	5062.39i	−0.187959	−	0.325554i
$624$		0			0
$625$	9449.52	−	16367.1i	0.604769	−	1.04749i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−9116.16			−0.577878
$630$		0			0
$631$	17925.9			1.13093			0.565465	−	0.824772i	$-0.308697\pi$
$631$	17925.9			1.13093			0.565465	+	0.824772i	$0.308697\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−6245.22	+	10817.0i	−0.390290	+	0.676001i
$636$		0			0
$637$	−5271.68	−	9130.82i	−0.327899	−	0.567938i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−3348.43	−	5799.65i	−0.206326	−	0.357367i	0.744228	−	0.667925i	$-0.232817\pi$
$641$	−3348.43	−	5799.65i	−0.206326	−	0.357367i	−0.950554	+	0.310558i	$0.899484\pi$
$642$		0			0
$643$	14845.5	−	25713.2i	0.910498	−	1.57703i	0.0971358	−	0.995271i	$-0.469032\pi$
$643$	14845.5	−	25713.2i	0.910498	−	1.57703i	0.813362	−	0.581758i	$-0.197635\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	12607.6			0.766084			0.383042	−	0.923731i	$-0.374876\pi$
$647$	12607.6			0.766084			0.383042	+	0.923731i	$0.374876\pi$
$648$		0			0
$649$	28588.7			1.72913
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−469.730	+	813.596i	−0.0281500	+	0.0487573i	−0.879757	−	0.475423i	$-0.842295\pi$
$653$	−469.730	+	813.596i	−0.0281500	+	0.0487573i	0.851607	+	0.524180i	$0.175628\pi$
$654$		0			0
$655$	−8635.12	−	14956.5i	−0.515118	−	0.892210i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−6941.60	−	12023.2i	−0.410328	−	0.710709i	0.584597	−	0.811324i	$-0.301253\pi$
$659$	−6941.60	−	12023.2i	−0.410328	−	0.710709i	−0.994926	+	0.100614i	$0.967919\pi$
$660$		0			0
$661$	−4072.30	+	7053.42i	−0.239628	+	0.415047i	−0.960608	−	0.277909i	$-0.910359\pi$
$661$	−4072.30	+	7053.42i	−0.239628	+	0.415047i	0.720980	+	0.692956i	$0.243692\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	18904.3			1.10237
$666$		0			0
$667$	50.8252			0.00295047
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−1575.07	+	2728.10i	−0.0906182	+	0.156955i
$672$		0			0
$673$	14461.2	+	25047.6i	0.828290	+	1.43464i	0.899379	+	0.437170i	$0.144019\pi$
$673$	14461.2	+	25047.6i	0.828290	+	1.43464i	−0.0710895	+	0.997470i	$0.522648\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−13590.4	−	23539.3i	−0.771524	−	1.33632i	−0.936727	−	0.350060i	$-0.886161\pi$
$677$	−13590.4	−	23539.3i	−0.771524	−	1.33632i	0.165203	−	0.986260i	$-0.447172\pi$
$678$		0			0
$679$	6517.20	−	11288.1i	0.368346	−	0.637995i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	7985.87			0.447395			0.223697	−	0.974659i	$-0.428187\pi$
$683$	7985.87			0.447395			0.223697	+	0.974659i	$0.428187\pi$
$684$		0			0
$685$	11550.8			0.644283
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1582.52	−	2741.01i	0.0875027	−	0.151559i
$690$		0			0
$691$	−4625.03	−	8010.78i	−0.254623	−	0.441020i	0.710170	−	0.704030i	$-0.248618\pi$
$691$	−4625.03	−	8010.78i	−0.254623	−	0.441020i	−0.964793	+	0.263010i	$0.915285\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−5214.11	−	9031.10i	−0.284579	−	0.492905i
$696$		0			0
$697$	−14594.3	+	25278.0i	−0.793111	+	1.37371i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	10267.4			0.553201			0.276601	−	0.960985i	$-0.410792\pi$
$701$	10267.4			0.553201			0.276601	+	0.960985i	$0.410792\pi$
$702$		0			0
$703$	−16144.5			−0.866146
$704$		0			0
$705$		0			0
$706$		0			0
$707$	633.236	−	1096.80i	0.0336850	−	0.0583441i
$708$		0			0
$709$	−6979.33	−	12088.6i	−0.369696	−	0.640332i	0.619822	−	0.784742i	$-0.287205\pi$
$709$	−6979.33	−	12088.6i	−0.369696	−	0.640332i	−0.989518	+	0.144410i	$0.953871\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−22.6524	−	39.2351i	−0.00118982	−	0.00206082i
$714$		0			0
$715$	−19643.2	+	34023.0i	−1.02743	+	1.77957i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−26373.7			−1.36798			−0.683988	−	0.729493i	$-0.739756\pi$
$719$	−26373.7			−1.36798			−0.683988	+	0.729493i	$0.739756\pi$
$720$		0			0
$721$	4576.55			0.236394
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−4215.83	+	7302.04i	−0.215962	+	0.374056i
$726$		0			0
$727$	4396.92	+	7615.69i	0.224309	+	0.388515i	0.956112	−	0.293002i	$-0.0946541\pi$
$727$	4396.92	+	7615.69i	0.224309	+	0.388515i	−0.731803	+	0.681516i	$0.761321\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	226.149	+	391.701i	0.0114424	+	0.0198189i
$732$		0			0
$733$	5337.00	−	9243.95i	0.268931	−	0.465802i	−0.699655	−	0.714481i	$-0.746663\pi$
$733$	5337.00	−	9243.95i	0.268931	−	0.465802i	0.968586	+	0.248678i	$0.0799963\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	17907.7			0.895031
$738$		0			0
$739$	−12678.1			−0.631084			−0.315542	−	0.948912i	$-0.602186\pi$
$739$	−12678.1			−0.631084			−0.315542	+	0.948912i	$0.602186\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	11055.6	−	19148.9i	0.545884	−	0.945499i	−0.452666	−	0.891680i	$-0.649527\pi$
$743$	11055.6	−	19148.9i	0.545884	−	0.945499i	0.998551	−	0.0538194i	$-0.0171395\pi$
$744$		0			0
$745$	7387.10	+	12794.8i	0.363278	+	0.629216i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−7543.12	−	13065.1i	−0.367984	−	0.637366i
$750$		0			0
$751$	11075.5	−	19183.4i	0.538151	−	0.932105i	−0.460853	−	0.887477i	$-0.652456\pi$
$751$	11075.5	−	19183.4i	0.538151	−	0.932105i	0.999004	−	0.0446283i	$-0.0142104\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	8129.00			0.391847
$756$		0			0
$757$	−25282.2			−1.21387			−0.606933	−	0.794753i	$-0.707601\pi$
$757$	−25282.2			−1.21387			−0.606933	+	0.794753i	$0.707601\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−5064.47	+	8771.91i	−0.241244	+	0.417847i	−0.961069	−	0.276309i	$-0.910889\pi$
$761$	−5064.47	+	8771.91i	−0.241244	+	0.417847i	0.719825	+	0.694156i	$0.244222\pi$
$762$		0			0
$763$	4226.50	+	7320.52i	0.200537	+	0.347340i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	24344.0	+	42165.0i	1.14604	+	1.98499i
$768$		0			0
$769$	12617.2	−	21853.7i	0.591663	−	1.02479i	−0.402345	−	0.915488i	$-0.631805\pi$
$769$	12617.2	−	21853.7i	0.591663	−	1.02479i	0.994009	−	0.109303i	$-0.0348618\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−30384.6			−1.41379			−0.706895	−	0.707319i	$-0.749904\pi$
$773$	−30384.6			−1.41379			−0.706895	+	0.707319i	$0.749904\pi$
$774$		0			0
$775$	7515.85			0.348358
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−25846.1	+	44766.8i	−1.18875	+	2.05897i
$780$		0			0
$781$	−15548.3	−	26930.4i	−0.712371	−	1.23386i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	20971.4	+	36323.5i	0.953505	+	1.65152i
$786$		0			0
$787$	−15072.3	+	26106.0i	−0.682682	+	1.18244i	0.291478	+	0.956578i	$0.405853\pi$
$787$	−15072.3	+	26106.0i	−0.682682	+	1.18244i	−0.974159	+	0.225862i	$0.927480\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−5903.01			−0.265344
$792$		0			0
$793$	−5364.84			−0.240241
$794$		0			0

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:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	108.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(6.37096 - 11.0348i) q^{5} +(7.02674 + 12.1707i) q^{7} +O(q^{10})\)	\(q+(6.37096 - 11.0348i) q^{5} +(7.02674 + 12.1707i) q^{7} +(-21.2745 - 36.8486i) q^{11} +(36.2316 - 62.7550i) q^{13} +59.6114 q^{17} +105.570 q^{19} +(-0.112590 + 0.195011i) q^{23} +(-18.6781 - 32.3515i) q^{25} +(-112.855 - 195.470i) q^{29} +(-100.597 + 174.239i) q^{31} +179.068 q^{35} -152.926 q^{37} +(-244.824 + 424.047i) q^{41} +(3.79372 + 6.57091i) q^{43} +(186.696 + 323.366i) q^{47} +(72.7498 - 126.006i) q^{49} +43.6780 q^{53} -542.157 q^{55} +(-335.949 + 581.881i) q^{59} +(-37.0177 - 64.1165i) q^{61} +(-461.660 - 799.619i) q^{65} +(-210.436 + 364.485i) q^{67} +730.840 q^{71} +473.927 q^{73} +(298.982 - 517.851i) q^{77} +(264.811 + 458.666i) q^{79} +(13.0767 + 22.6495i) q^{83} +(379.781 - 657.801i) q^{85} -415.949 q^{89} +1018.36 q^{91} +(672.583 - 1164.95i) q^{95} +(-463.743 - 803.226i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{5} - 6 q^{7}+O(q^{10}) \) 6 * q - 6 * q^5 - 6 * q^7	\( 6 q - 6 q^{5} - 6 q^{7} - 51 q^{11} + 12 q^{13} + 222 q^{17} + 30 q^{19} - 210 q^{23} - 3 q^{25} - 456 q^{29} + 48 q^{31} + 1104 q^{35} - 96 q^{37} - 897 q^{41} + 129 q^{43} - 522 q^{47} - 225 q^{49} + 2208 q^{53} - 216 q^{55} - 453 q^{59} - 402 q^{61} - 1110 q^{65} - 213 q^{67} - 120 q^{71} + 750 q^{73} - 1128 q^{77} + 552 q^{79} + 612 q^{83} + 1188 q^{85} + 924 q^{89} - 264 q^{91} + 2184 q^{95} + 93 q^{97}+O(q^{100}) \) 6 * q - 6 * q^5 - 6 * q^7 - 51 * q^11 + 12 * q^13 + 222 * q^17 + 30 * q^19 - 210 * q^23 - 3 * q^25 - 456 * q^29 + 48 * q^31 + 1104 * q^35 - 96 * q^37 - 897 * q^41 + 129 * q^43 - 522 * q^47 - 225 * q^49 + 2208 * q^53 - 216 * q^55 - 453 * q^59 - 402 * q^61 - 1110 * q^65 - 213 * q^67 - 120 * q^71 + 750 * q^73 - 1128 * q^77 + 552 * q^79 + 612 * q^83 + 1188 * q^85 + 924 * q^89 - 264 * q^91 + 2184 * q^95 + 93 * q^97

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