LMFDB - Embedded newform 726.3.d.e.241.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [726,3,Mod(241,726)] 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("726.241"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(726, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 726 = 2 \cdot 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	726.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,-32,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$19.7820671926$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.6879707136000000000000.7
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4x^{14} + 15x^{12} - 56x^{10} + 209x^{8} - 56x^{6} + 15x^{4} - 4x^{2} + 1 $ x^16 - 4x^14 + 15x^12 - 56x^10 + 209x^8 - 56x^6 + 15x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12}\cdot 11^{2} $
Twist minimal:	no (minimal twist has level 66)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			241.10
Root			$0.492303 + 0.159959i$ of defining polynomial
Character	$\chi$	$=$	726.241
Dual form			726.3.d.e.241.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.41421i q^{2} -1.73205 q^{3} -2.00000 q^{4} +0.353386 q^{5} -2.44949i q^{6} +0.790744i q^{7} -2.82843i q^{8} +3.00000 q^{9} +0.499764i q^{10} +3.46410 q^{12} +0.209228i q^{13} -1.11828 q^{14} -0.612083 q^{15} +4.00000 q^{16} +14.9696i q^{17} +4.24264i q^{18} +2.16953i q^{19} -0.706772 q^{20} -1.36961i q^{21} -24.1342 q^{23} +4.89898i q^{24} -24.8751 q^{25} -0.295893 q^{26} -5.19615 q^{27} -1.58149i q^{28} -17.8240i q^{29} -0.865616i q^{30} +33.8376 q^{31} +5.65685i q^{32} -21.1702 q^{34} +0.279438i q^{35} -6.00000 q^{36} -60.0856 q^{37} -3.06818 q^{38} -0.362394i q^{39} -0.999527i q^{40} -53.5564i q^{41} +1.93692 q^{42} -15.5727i q^{43} +1.06016 q^{45} -34.1309i q^{46} -61.1556 q^{47} -6.92820 q^{48} +48.3747 q^{49} -35.1787i q^{50} -25.9281i q^{51} -0.418456i q^{52} +58.9673 q^{53} -7.34847i q^{54} +2.23656 q^{56} -3.75774i q^{57} +25.2069 q^{58} -51.4857 q^{59} +1.22417 q^{60} -40.7522i q^{61} +47.8535i q^{62} +2.37223i q^{63} -8.00000 q^{64} +0.0739383i q^{65} -72.9302 q^{67} -29.9392i q^{68} +41.8016 q^{69} -0.395185 q^{70} -95.0520 q^{71} -8.48528i q^{72} -113.523i q^{73} -84.9738i q^{74} +43.0850 q^{75} -4.33906i q^{76} +0.512502 q^{78} -147.127i q^{79} +1.41354 q^{80} +9.00000 q^{81} +75.7401 q^{82} +19.8752i q^{83} +2.73922i q^{84} +5.29005i q^{85} +22.0232 q^{86} +30.8721i q^{87} -74.9710 q^{89} +1.49929i q^{90} -0.165446 q^{91} +48.2684 q^{92} -58.6084 q^{93} -86.4870i q^{94} +0.766683i q^{95} -9.79796i q^{96} +101.506 q^{97} +68.4122i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 32 q^{4} + 8 q^{5} + 48 q^{9} + 48 q^{14} - 48 q^{15} + 64 q^{16} - 16 q^{20} - 8 q^{23} - 8 q^{25} + 128 q^{26} - 8 q^{31} - 64 q^{34} - 96 q^{36} + 96 q^{37} - 208 q^{38} - 144 q^{42} + 24 q^{45}+ \cdots + 184 q^{97}+O(q^{100}) $ 16 * q - 32 * q^4 + 8 * q^5 + 48 * q^9 + 48 * q^14 - 48 * q^15 + 64 * q^16 - 16 * q^20 - 8 * q^23 - 8 * q^25 + 128 * q^26 - 8 * q^31 - 64 * q^34 - 96 * q^36 + 96 * q^37 - 208 * q^38 - 144 * q^42 + 24 * q^45 - 128 * q^47 - 64 * q^49 + 16 * q^53 - 96 * q^56 + 64 * q^58 - 152 * q^59 + 96 * q^60 - 128 * q^64 + 248 * q^67 + 168 * q^69 - 32 * q^70 - 72 * q^71 - 48 * q^75 + 32 * q^80 + 144 * q^81 + 176 * q^82 + 464 * q^86 - 360 * q^89 - 32 * q^91 + 16 * q^92 + 336 * q^93 + 184 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/726\mathbb{Z}\right)^\times$.

$n$	$485$	$607$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$	1.41421i	0.707107i
$2$	1.41421i	0.707107i
$3$	−1.73205	−0.577350
$3$	−1.73205	−0.577350
$4$	−2.00000	−0.500000
$5$	0.353386	0.0706772	0.0353386	−	0.999375i	$-0.488749\pi$
$5$	0.353386	0.0706772	0.0353386	+	0.999375i	$0.488749\pi$
$6$	− 2.44949i	− 0.408248i
$7$	0.790744i	0.112963i	0.998404	+	0.0564817i	$0.0179883\pi$
$7$	0.790744i	0.112963i	−0.998404	+	0.0564817i	$0.982012\pi$
$8$	− 2.82843i	− 0.353553i
$9$	3.00000	0.333333
$10$	0.499764i	0.0499764i
$11$	0	0
$11$	0	0
$12$	3.46410	0.288675
$13$	0.209228i	0.0160945i	0.999968	+	0.00804724i	$0.00256154\pi$
$13$	0.209228i	0.0160945i	−0.999968	+	0.00804724i	$0.997438\pi$
$14$	−1.11828	−0.0798772
$15$	−0.612083	−0.0408055
$16$	4.00000	0.250000
$17$	14.9696i	0.880565i	0.897859	+	0.440282i	$0.145122\pi$
$17$	14.9696i	0.880565i	−0.897859	+	0.440282i	$0.854878\pi$
$18$	4.24264i	0.235702i
$19$	2.16953i	0.114186i	0.998369	+	0.0570930i	$0.0181831\pi$
$19$	2.16953i	0.114186i	−0.998369	+	0.0570930i	$0.981817\pi$
$20$	−0.706772	−0.0353386
$21$	− 1.36961i	− 0.0652194i
$22$	0	0
$23$	−24.1342	−1.04931	−0.524656	−	0.851314i	$-0.675806\pi$
$23$	−24.1342	−1.04931	−0.524656	+	0.851314i	$0.675806\pi$
$24$	4.89898i	0.204124i
$25$	−24.8751	−0.995005
$26$	−0.295893	−0.0113805
$27$	−5.19615	−0.192450
$28$	− 1.58149i	− 0.0564817i
$29$	− 17.8240i	− 0.614621i	−0.951609	−	0.307310i	$-0.900571\pi$
$29$	− 17.8240i	− 0.614621i	0.951609	−	0.307310i	$-0.0994290\pi$
$30$	− 0.865616i	− 0.0288539i
$31$	33.8376	1.09153	0.545767	−	0.837937i	$-0.316238\pi$
$31$	33.8376	1.09153	0.545767	+	0.837937i	$0.316238\pi$
$32$	5.65685i	0.176777i
$33$	0	0
$34$	−21.1702	−0.622653
$35$	0.279438i	0.00798394i
$36$	−6.00000	−0.166667
$37$	−60.0856	−1.62393	−0.811967	−	0.583703i	$-0.801603\pi$
$37$	−60.0856	−1.62393	−0.811967	+	0.583703i	$0.801603\pi$
$38$	−3.06818	−0.0807416
$39$	− 0.362394i	− 0.00929215i
$40$	− 0.999527i	− 0.0249882i
$41$	− 53.5564i	− 1.30625i	−0.757249	−	0.653126i	$-0.773457\pi$
$41$	− 53.5564i	− 1.30625i	0.757249	−	0.653126i	$-0.226543\pi$
$42$	1.93692	0.0461171
$43$	− 15.5727i	− 0.362157i	−0.983469	−	0.181078i	$-0.942041\pi$
$43$	− 15.5727i	− 0.362157i	0.983469	−	0.181078i	$-0.0579588\pi$
$44$	0	0
$45$	1.06016	0.0235591
$46$	− 34.1309i	− 0.741976i
$47$	−61.1556	−1.30118	−0.650591	−	0.759428i	$-0.725479\pi$
$47$	−61.1556	−1.30118	−0.650591	+	0.759428i	$0.725479\pi$
$48$	−6.92820	−0.144338
$49$	48.3747	0.987239
$50$	− 35.1787i	− 0.703575i
$51$	− 25.9281i	− 0.508394i
$52$	− 0.418456i	− 0.00804724i
$53$	58.9673	1.11259	0.556295	−	0.830985i	$-0.312222\pi$
$53$	58.9673	1.11259	0.556295	+	0.830985i	$0.312222\pi$
$54$	− 7.34847i	− 0.136083i
$55$	0	0
$56$	2.23656	0.0399386
$57$	− 3.75774i	− 0.0659253i
$58$	25.2069	0.434602
$59$	−51.4857	−0.872639	−0.436320	−	0.899792i	$-0.643718\pi$
$59$	−51.4857	−0.872639	−0.436320	+	0.899792i	$0.643718\pi$
$60$	1.22417	0.0204028
$61$	− 40.7522i	− 0.668069i	−0.942561	−	0.334034i	$-0.891590\pi$
$61$	− 40.7522i	− 0.668069i	0.942561	−	0.334034i	$-0.108410\pi$
$62$	47.8535i	0.771831i
$63$	2.37223i	0.0376545i
$64$	−8.00000	−0.125000
$65$	0.0739383i	0.00113751i
$66$	0	0
$67$	−72.9302	−1.08851	−0.544255	−	0.838920i	$-0.683188\pi$
$67$	−72.9302	−1.08851	−0.544255	+	0.838920i	$0.683188\pi$
$68$	− 29.9392i	− 0.440282i
$69$	41.8016	0.605821
$70$	−0.395185	−0.00564550
$71$	−95.0520	−1.33876	−0.669380	−	0.742920i	$-0.733440\pi$
$71$	−95.0520	−1.33876	−0.669380	+	0.742920i	$0.733440\pi$
$72$	− 8.48528i	− 0.117851i
$73$	− 113.523i	− 1.55511i	−0.628814	−	0.777555i	$-0.716459\pi$
$73$	− 113.523i	− 1.55511i	0.628814	−	0.777555i	$-0.283541\pi$
$74$	− 84.9738i	− 1.14829i
$75$	43.0850	0.574466
$76$	− 4.33906i	− 0.0570930i
$77$	0	0
$78$	0.512502	0.00657054
$79$	− 147.127i	− 1.86236i	−0.364556	−	0.931182i	$-0.618779\pi$
$79$	− 147.127i	− 1.86236i	0.364556	−	0.931182i	$-0.381221\pi$
$80$	1.41354	0.0176693
$81$	9.00000	0.111111
$82$	75.7401	0.923660
$83$	19.8752i	0.239461i	0.992806	+	0.119730i	$0.0382030\pi$
$83$	19.8752i	0.239461i	−0.992806	+	0.119730i	$0.961797\pi$
$84$	2.73922i	0.0326097i
$85$	5.29005i	0.0622359i
$86$	22.0232	0.256084
$87$	30.8721i	0.354851i
$88$	0	0
$89$	−74.9710	−0.842371	−0.421185	−	0.906975i	$-0.638386\pi$
$89$	−74.9710	−0.842371	−0.421185	+	0.906975i	$0.638386\pi$
$90$	1.49929i	0.0166588i
$91$	−0.165446	−0.00181809
$92$	48.2684	0.524656
$93$	−58.6084	−0.630197
$94$	− 86.4870i	− 0.920075i
$95$	0.766683i	0.00807034i
$96$	− 9.79796i	− 0.102062i
$97$	101.506	1.04645	0.523227	−	0.852194i	$-0.324728\pi$
$97$	101.506	1.04645	0.523227	+	0.852194i	$0.324728\pi$
$98$	68.4122i	0.698084i
$99$	0	0
$100$	49.7502	0.497502
$101$	26.0855i	0.258272i	0.991627	+	0.129136i	$0.0412204\pi$
$101$	26.0855i	0.258272i	−0.991627	+	0.129136i	$0.958780\pi$
$102$	36.6679	0.359489
$103$	−31.5362	−0.306177	−0.153088	−	0.988212i	$-0.548922\pi$
$103$	−31.5362	−0.306177	−0.153088	+	0.988212i	$0.548922\pi$
$104$	0.591787	0.00569026
$105$	− 0.484001i	− 0.00460953i
$106$	83.3923i	0.786720i
$107$	− 80.5378i	− 0.752690i	−0.926480	−	0.376345i	$-0.877181\pi$
$107$	− 80.5378i	− 0.752690i	0.926480	−	0.376345i	$-0.122819\pi$
$108$	10.3923	0.0962250
$109$	32.1811i	0.295239i	0.989044	+	0.147620i	$0.0471612\pi$
$109$	32.1811i	0.295239i	−0.989044	+	0.147620i	$0.952839\pi$
$110$	0	0
$111$	104.071	0.937579
$112$	3.16297i	0.0282408i
$113$	−29.2039	−0.258441	−0.129221	−	0.991616i	$-0.541248\pi$
$113$	−29.2039	−0.258441	−0.129221	+	0.991616i	$0.541248\pi$
$114$	5.31425	0.0466162
$115$	−8.52869	−0.0741625
$116$	35.6480i	0.307310i
$117$	0.627684i	0.00536482i
$118$	− 72.8118i	− 0.617049i
$119$	−11.8371	−0.0994716
$120$	1.73123i	0.0144269i
$121$	0	0
$122$	57.6323	0.472396
$123$	92.7623i	0.754165i
$124$	−67.6751	−0.545767
$125$	−17.6252	−0.141001
$126$	−3.35484	−0.0266257
$127$	− 149.052i	− 1.17364i	−0.809719	−	0.586818i	$-0.800380\pi$
$127$	− 149.052i	− 1.17364i	0.809719	−	0.586818i	$-0.199620\pi$
$128$	− 11.3137i	− 0.0883883i
$129$	26.9728i	0.209091i
$130$	−0.104565	−0.000804343 0
$131$	165.965i	1.26691i	0.773780	+	0.633454i	$0.218364\pi$
$131$	165.965i	1.26691i	−0.773780	+	0.633454i	$0.781636\pi$
$132$	0	0
$133$	−1.71554	−0.0128988
$134$	− 103.139i	− 0.769693i
$135$	−1.83625	−0.0136018
$136$	42.3404	0.311327
$137$	−146.868	−1.07203	−0.536014	−	0.844209i	$-0.680070\pi$
$137$	−146.868	−1.07203	−0.536014	+	0.844209i	$0.680070\pi$
$138$	59.1164i	0.428380i
$139$	− 213.774i	− 1.53795i	−0.639281	−	0.768973i	$-0.720768\pi$
$139$	− 213.774i	− 1.53795i	0.639281	−	0.768973i	$-0.279232\pi$
$140$	− 0.558876i	− 0.00399197i
$141$	105.925	0.751238
$142$	− 134.424i	− 0.946646i
$143$	0	0
$144$	12.0000	0.0833333
$145$	− 6.29876i	− 0.0434397i
$146$	160.546	1.09963
$147$	−83.7875	−0.569983
$148$	120.171	0.811967
$149$	231.242i	1.55196i	0.630757	+	0.775981i	$0.282745\pi$
$149$	231.242i	1.55196i	−0.630757	+	0.775981i	$0.717255\pi$
$150$	60.9313i	0.406209i
$151$	266.405i	1.76427i	0.470992	+	0.882137i	$0.343896\pi$
$151$	266.405i	1.76427i	−0.470992	+	0.882137i	$0.656104\pi$
$152$	6.13636	0.0403708
$153$	44.9088i	0.293522i
$154$	0	0
$155$	11.9577	0.0771466
$156$	0.724788i	0.00464607i
$157$	−104.278	−0.664193	−0.332097	−	0.943245i	$-0.607756\pi$
$157$	−104.278	−0.664193	−0.332097	+	0.943245i	$0.607756\pi$
$158$	208.069	1.31689
$159$	−102.134	−0.642354
$160$	1.99905i	0.0124941i
$161$	− 19.0839i	− 0.118534i
$162$	12.7279i	0.0785674i
$163$	−279.668	−1.71575	−0.857877	−	0.513855i	$-0.828217\pi$
$163$	−279.668	−1.71575	−0.857877	+	0.513855i	$0.828217\pi$
$164$	107.113i	0.653126i
$165$	0	0
$166$	−28.1078	−0.169324
$167$	256.946i	1.53860i	0.638889	+	0.769299i	$0.279394\pi$
$167$	256.946i	1.53860i	−0.638889	+	0.769299i	$0.720606\pi$
$168$	−3.87384	−0.0230586
$169$	168.956	0.999741
$170$	−7.48126	−0.0440074
$171$	6.50860i	0.0380620i
$172$	31.1455i	0.181078i
$173$	110.189i	0.636929i	0.947935	+	0.318464i	$0.103167\pi$
$173$	110.189i	0.636929i	−0.947935	+	0.318464i	$0.896833\pi$
$174$	−43.6597	−0.250918
$175$	− 19.6698i	− 0.112399i
$176$	0	0
$177$	89.1759	0.503819
$178$	− 106.025i	− 0.595646i
$179$	27.1267	0.151546	0.0757729	−	0.997125i	$-0.475858\pi$
$179$	27.1267	0.151546	0.0757729	+	0.997125i	$0.475858\pi$
$180$	−2.12032	−0.0117795
$181$	259.571	1.43410	0.717048	−	0.697024i	$-0.245493\pi$
$181$	259.571	1.43410	0.717048	+	0.697024i	$0.245493\pi$
$182$	− 0.233976i	− 0.00128558i
$183$	70.5849i	0.385710i
$184$	68.2618i	0.370988i
$185$	−21.2334	−0.114775
$186$	− 82.8847i	− 0.445617i
$187$	0	0
$188$	122.311	0.650591
$189$	− 4.10882i	− 0.0217398i
$190$	−1.08425	−0.00570660
$191$	−124.782	−0.653310	−0.326655	−	0.945144i	$-0.605922\pi$
$191$	−124.782	−0.653310	−0.326655	+	0.945144i	$0.605922\pi$
$192$	13.8564	0.0721688
$193$	− 41.9892i	− 0.217560i	−0.994066	−	0.108780i	$-0.965306\pi$
$193$	− 41.9892i	− 0.217560i	0.994066	−	0.108780i	$-0.0346945\pi$
$194$	143.551i	0.739954i
$195$	− 0.128065i	0 0.000656743i
$196$	−96.7494	−0.493620
$197$	− 274.338i	− 1.39258i	−0.717760	−	0.696290i	$-0.754833\pi$
$197$	− 274.338i	− 1.39258i	0.717760	−	0.696290i	$-0.245167\pi$
$198$	0	0
$199$	−150.979	−0.758690	−0.379345	−	0.925255i	$-0.623851\pi$
$199$	−150.979	−0.758690	−0.379345	+	0.925255i	$0.623851\pi$
$200$	70.3575i	0.351787i
$201$	126.319	0.628452
$202$	−36.8904	−0.182626
$203$	14.0942	0.0694296
$204$	51.8562i	0.254197i
$205$	− 18.9261i	− 0.0923223i
$206$	− 44.5989i	− 0.216500i
$207$	−72.4025	−0.349771
$208$	0.836913i	0.00402362i
$209$	0	0
$210$	0.684480	0.00325943
$211$	− 18.6837i	− 0.0885486i	−0.999019	−	0.0442743i	$-0.985902\pi$
$211$	− 18.6837i	− 0.0885486i	0.999019	−	0.0442743i	$-0.0140976\pi$
$212$	−117.935	−0.556295
$213$	164.635	0.772933
$214$	113.898	0.532232
$215$	− 5.50319i	− 0.0255963i
$216$	14.6969i	0.0680414i
$217$	26.7568i	0.123303i
$218$	−45.5109	−0.208766
$219$	196.628i	0.897844i
$220$	0	0
$221$	−3.13206	−0.0141722
$222$	147.179i	0.662968i
$223$	−94.2808	−0.422784	−0.211392	−	0.977401i	$-0.567800\pi$
$223$	−94.2808	−0.422784	−0.211392	+	0.977401i	$0.567800\pi$
$224$	−4.47312	−0.0199693
$225$	−74.6254	−0.331668
$226$	− 41.3005i	− 0.182746i
$227$	289.791i	1.27661i	0.769782	+	0.638306i	$0.220365\pi$
$227$	289.791i	1.27661i	−0.769782	+	0.638306i	$0.779635\pi$
$228$	7.51548i	0.0329626i
$229$	−228.228	−0.996631	−0.498315	−	0.866996i	$-0.666048\pi$
$229$	−228.228	−0.996631	−0.498315	+	0.866996i	$0.666048\pi$
$230$	− 12.0614i	− 0.0524408i
$231$	0	0
$232$	−50.4139	−0.217301
$233$	298.827i	1.28252i	0.767324	+	0.641260i	$0.221588\pi$
$233$	298.827i	1.28252i	−0.767324	+	0.641260i	$0.778412\pi$
$234$	−0.887680	−0.00379350
$235$	−21.6115	−0.0919640
$236$	102.971	0.436320
$237$	254.831i	1.07524i
$238$	− 16.7402i	− 0.0703370i
$239$	245.929i	1.02899i	0.857492	+	0.514496i	$0.172021\pi$
$239$	245.929i	1.02899i	−0.857492	+	0.514496i	$0.827979\pi$
$240$	−2.44833	−0.0102014
$241$	− 373.418i	− 1.54945i	−0.632297	−	0.774726i	$-0.717888\pi$
$241$	− 373.418i	− 1.54945i	0.632297	−	0.774726i	$-0.282112\pi$
$242$	0	0
$243$	−15.5885	−0.0641500
$244$	81.5044i	0.334034i
$245$	17.0950	0.0697753
$246$	−131.186	−0.533275
$247$	−0.453927	−0.00183776
$248$	− 95.7071i	− 0.385916i
$249$	− 34.4249i	− 0.138253i
$250$	− 24.9258i	− 0.0997031i
$251$	−359.422	−1.43196	−0.715980	−	0.698121i	$-0.754020\pi$
$251$	−359.422	−1.43196	−0.715980	+	0.698121i	$0.754020\pi$
$252$	− 4.74446i	− 0.0188272i
$253$	0	0
$254$	210.791	0.829886
$255$	− 9.16264i	− 0.0359319i
$256$	16.0000	0.0625000
$257$	−35.7842	−0.139238	−0.0696191	−	0.997574i	$-0.522178\pi$
$257$	−35.7842	−0.139238	−0.0696191	+	0.997574i	$0.522178\pi$
$258$	−38.1453	−0.147850
$259$	− 47.5123i	− 0.183445i
$260$	− 0.147877i	0 0.000568756i
$261$	− 53.4720i	− 0.204874i
$262$	−234.710	−0.895840
$263$	− 364.668i	− 1.38657i	−0.720663	−	0.693286i	$-0.756162\pi$
$263$	− 364.668i	− 1.38657i	0.720663	−	0.693286i	$-0.243838\pi$
$264$	0	0
$265$	20.8382	0.0786348
$266$	− 2.42615i	− 0.00912085i
$267$	129.854	0.486343
$268$	145.860	0.544255
$269$	−13.3769	−0.0497283	−0.0248641	−	0.999691i	$-0.507915\pi$
$269$	−13.3769	−0.0497283	−0.0248641	+	0.999691i	$0.507915\pi$
$270$	− 2.59685i	− 0.00961795i
$271$	70.8425i	0.261411i	0.991421	+	0.130706i	$0.0417243\pi$
$271$	70.8425i	0.261411i	−0.991421	+	0.130706i	$0.958276\pi$
$272$	59.8784i	0.220141i
$273$	0.286561	0.00104967
$274$	− 207.702i	− 0.758038i
$275$	0	0
$276$	−83.6033	−0.302910
$277$	− 79.3467i	− 0.286450i	−0.989690	−	0.143225i	$-0.954253\pi$
$277$	− 79.3467i	− 0.286450i	0.989690	−	0.143225i	$-0.0457473\pi$
$278$	302.323	1.08749
$279$	101.513	0.363845
$280$	0.790370	0.00282275
$281$	343.092i	1.22097i	0.792029	+	0.610484i	$0.209025\pi$
$281$	343.092i	1.22097i	−0.792029	+	0.610484i	$0.790975\pi$
$282$	149.800i	0.531205i
$283$	119.679i	0.422895i	0.977389	+	0.211447i	$0.0678177\pi$
$283$	119.679i	0.422895i	−0.977389	+	0.211447i	$0.932182\pi$
$284$	190.104	0.669380
$285$	− 1.32793i	− 0.00465942i
$286$	0	0
$287$	42.3493	0.147559
$288$	16.9706i	0.0589256i
$289$	64.9111	0.224606
$290$	8.90779	0.0307165
$291$	−175.813	−0.604170
$292$	227.046i	0.777555i
$293$	11.5161i	0.0393042i	0.999807	+	0.0196521i	$0.00625585\pi$
$293$	11.5161i	0.0393042i	−0.999807	+	0.0196521i	$0.993744\pi$
$294$	− 118.493i	− 0.403039i
$295$	−18.1943	−0.0616757
$296$	169.948i	0.574147i
$297$	0	0
$298$	−327.026	−1.09740
$299$	− 5.04955i	− 0.0168881i
$300$	−86.1699	−0.287233
$301$	12.3141	0.0409105
$302$	−376.754	−1.24753
$303$	− 45.1814i	− 0.149113i
$304$	8.67813i	0.0285465i
$305$	− 14.4013i	− 0.0472172i
$306$	−63.5106	−0.207551
$307$	196.858i	0.641231i	0.947210	+	0.320615i	$0.103890\pi$
$307$	196.858i	0.641231i	−0.947210	+	0.320615i	$0.896110\pi$
$308$	0	0
$309$	54.6223	0.176771
$310$	16.9108i	0.0545509i
$311$	270.416	0.869504	0.434752	−	0.900550i	$-0.356836\pi$
$311$	270.416	0.869504	0.434752	+	0.900550i	$0.356836\pi$
$312$	−1.02500	−0.00328527
$313$	344.610	1.10099	0.550495	−	0.834839i	$-0.314439\pi$
$313$	344.610	1.10099	0.550495	+	0.834839i	$0.314439\pi$
$314$	− 147.472i	− 0.469656i
$315$	0.838314i	0.00266131i
$316$	294.253i	0.931182i
$317$	65.5714	0.206850	0.103425	−	0.994637i	$-0.467020\pi$
$317$	65.5714	0.206850	0.103425	+	0.994637i	$0.467020\pi$
$318$	− 144.440i	− 0.454213i
$319$	0	0
$320$	−2.82709	−0.00883465
$321$	139.496i	0.434566i
$322$	26.9888	0.0838161
$323$	−32.4770	−0.100548
$324$	−18.0000	−0.0555556
$325$	− 5.20458i	− 0.0160141i
$326$	− 395.510i	− 1.21322i
$327$	− 55.7393i	− 0.170457i
$328$	−151.480	−0.461830
$329$	− 48.3584i	− 0.146986i
$330$	0	0
$331$	105.818	0.319693	0.159847	−	0.987142i	$-0.448900\pi$
$331$	105.818	0.319693	0.159847	+	0.987142i	$0.448900\pi$
$332$	− 39.7505i	− 0.119730i
$333$	−180.257	−0.541311
$334$	−363.376	−1.08795
$335$	−25.7725	−0.0769329
$336$	− 5.47843i	− 0.0163049i
$337$	225.621i	0.669498i	0.942307	+	0.334749i	$0.108652\pi$
$337$	225.621i	0.669498i	−0.942307	+	0.334749i	$0.891348\pi$
$338$	238.940i	0.706924i
$339$	50.5826	0.149211
$340$	− 10.5801i	− 0.0311179i
$341$	0	0
$342$	−9.20455	−0.0269139
$343$	76.9984i	0.224485i
$344$	−44.0464	−0.128042
$345$	14.7721	0.0428177
$346$	−155.830	−0.450377
$347$	− 135.569i	− 0.390687i	−0.980735	−	0.195344i	$-0.937418\pi$
$347$	− 135.569i	− 0.390687i	0.980735	−	0.195344i	$-0.0625822\pi$
$348$	− 61.7441i	− 0.177426i
$349$	− 500.154i	− 1.43311i	−0.697533	−	0.716553i	$-0.745719\pi$
$349$	− 500.154i	− 1.43311i	0.697533	−	0.716553i	$-0.254281\pi$
$350$	27.8174	0.0794782
$351$	− 1.08718i	− 0.00309738i
$352$	0	0
$353$	320.592	0.908192	0.454096	−	0.890953i	$-0.349962\pi$
$353$	320.592	0.908192	0.454096	+	0.890953i	$0.349962\pi$
$354$	126.114i	0.356253i
$355$	−33.5900	−0.0946199
$356$	149.942	0.421185
$357$	20.5025	0.0574299
$358$	38.3630i	0.107159i
$359$	113.433i	0.315971i	0.987441	+	0.157985i	$0.0504999\pi$
$359$	113.433i	0.315971i	−0.987441	+	0.157985i	$0.949500\pi$
$360$	− 2.99858i	− 0.00832939i
$361$	356.293	0.986962
$362$	367.089i	1.01406i
$363$	0	0
$364$	0.330892	0.000909043 0
$365$	− 40.1175i	− 0.109911i
$366$	−99.8221	−0.272738
$367$	89.5681	0.244055	0.122027	−	0.992527i	$-0.461060\pi$
$367$	89.5681	0.244055	0.122027	+	0.992527i	$0.461060\pi$
$368$	−96.5367	−0.262328
$369$	− 160.669i	− 0.435418i
$370$	− 30.0286i	− 0.0811583i
$371$	46.6280i	0.125682i
$372$	117.217	0.315099
$373$	275.552i	0.738744i	0.929282	+	0.369372i	$0.120427\pi$
$373$	275.552i	0.738744i	−0.929282	+	0.369372i	$0.879573\pi$
$374$	0	0
$375$	30.5277	0.0814072
$376$	172.974i	0.460037i
$377$	3.72928	0.00989200
$378$	5.81075	0.0153724
$379$	−175.545	−0.463179	−0.231590	−	0.972814i	$-0.574393\pi$
$379$	−175.545	−0.463179	−0.231590	+	0.972814i	$0.574393\pi$
$380$	− 1.53337i	− 0.00403517i
$381$	258.165i	0.677599i
$382$	− 176.469i	− 0.461960i
$383$	−51.1788	−0.133626	−0.0668131	−	0.997766i	$-0.521283\pi$
$383$	−51.1788	−0.133626	−0.0668131	+	0.997766i	$0.521283\pi$
$384$	19.5959i	0.0510310i
$385$	0	0
$386$	59.3817	0.153838
$387$	− 46.7182i	− 0.120719i
$388$	−203.012	−0.523227
$389$	−22.9064	−0.0588854	−0.0294427	−	0.999566i	$-0.509373\pi$
$389$	−22.9064	−0.0588854	−0.0294427	+	0.999566i	$0.509373\pi$
$390$	0.181111	0.000464388 0
$391$	− 361.279i	− 0.923987i
$392$	− 136.824i	− 0.349042i
$393$	− 287.460i	− 0.731450i
$394$	387.973	0.984703
$395$	− 51.9925i	− 0.131627i
$396$	0	0
$397$	131.054	0.330112	0.165056	−	0.986284i	$-0.447220\pi$
$397$	131.054	0.330112	0.165056	+	0.986284i	$0.447220\pi$
$398$	− 213.517i	− 0.536475i
$399$	2.97141	0.00744714
$400$	−99.5005	−0.248751
$401$	775.849	1.93479	0.967393	−	0.253279i	$-0.0815090\pi$
$401$	775.849	1.93479	0.967393	+	0.253279i	$0.0815090\pi$
$402$	178.642i	0.444382i
$403$	7.07977i	0.0175677i
$404$	− 52.1710i	− 0.129136i
$405$	3.18048	0.00785303
$406$	19.9322i	0.0490942i
$407$	0	0
$408$	−73.3358	−0.179745
$409$	− 747.379i	− 1.82733i	−0.406464	−	0.913667i	$-0.633238\pi$
$409$	− 747.379i	− 1.82733i	0.406464	−	0.913667i	$-0.366762\pi$
$410$	26.7655	0.0652817
$411$	254.382	0.618935
$412$	63.0724	0.153088
$413$	− 40.7120i	− 0.0985763i
$414$	− 102.393i	− 0.247325i
$415$	7.02364i	0.0169244i
$416$	−1.18357	−0.00284513
$417$	370.268i	0.887934i
$418$	0	0
$419$	−408.427	−0.974765	−0.487383	−	0.873189i	$-0.662048\pi$
$419$	−408.427	−0.974765	−0.487383	+	0.873189i	$0.662048\pi$
$420$	0.968001i	0.00230476i
$421$	104.511	0.248245	0.124123	−	0.992267i	$-0.460388\pi$
$421$	104.511	0.248245	0.124123	+	0.992267i	$0.460388\pi$
$422$	26.4228	0.0626133
$423$	−183.467	−0.433727
$424$	− 166.785i	− 0.393360i
$425$	− 372.371i	− 0.876166i
$426$	232.829i	0.546546i
$427$	32.2245	0.0754673
$428$	161.076i	0.376345i
$429$	0	0
$430$	7.78269	0.0180993
$431$	− 289.063i	− 0.670680i	−0.942097	−	0.335340i	$-0.891149\pi$
$431$	− 289.063i	− 0.670680i	0.942097	−	0.335340i	$-0.108851\pi$
$432$	−20.7846	−0.0481125
$433$	550.999	1.27252	0.636258	−	0.771476i	$-0.280481\pi$
$433$	550.999	1.27252	0.636258	+	0.771476i	$0.280481\pi$
$434$	−37.8399	−0.0871886
$435$	10.9098i	0.0250799i
$436$	− 64.3622i	− 0.147620i
$437$	− 52.3599i	− 0.119817i
$438$	−278.074	−0.634871
$439$	576.891i	1.31410i	0.753846	+	0.657051i	$0.228196\pi$
$439$	576.891i	1.31410i	−0.753846	+	0.657051i	$0.771804\pi$
$440$	0	0
$441$	145.124	0.329080
$442$	− 4.42940i	− 0.0100213i
$443$	−178.241	−0.402349	−0.201175	−	0.979555i	$-0.564476\pi$
$443$	−178.241	−0.402349	−0.201175	+	0.979555i	$0.564476\pi$
$444$	−208.142	−0.468789
$445$	−26.4937	−0.0595364
$446$	− 133.333i	− 0.298953i
$447$	− 400.523i	− 0.896025i
$448$	− 6.32595i	− 0.0141204i
$449$	622.710	1.38688	0.693441	−	0.720513i	$-0.256094\pi$
$449$	622.710	1.38688	0.693441	+	0.720513i	$0.256094\pi$
$450$	− 105.536i	− 0.234525i
$451$	0	0
$452$	58.4078	0.129221
$453$	− 461.428i	− 1.01860i
$454$	−409.827	−0.902702
$455$	−0.0584663	−0.000128497 0
$456$	−10.6285	−0.0233081
$457$	787.808i	1.72387i	0.507021	+	0.861934i	$0.330747\pi$
$457$	787.808i	1.72387i	−0.507021	+	0.861934i	$0.669253\pi$
$458$	− 322.764i	− 0.704725i
$459$	− 77.7843i	− 0.169465i
$460$	17.0574	0.0370812
$461$	− 377.985i	− 0.819924i	−0.912103	−	0.409962i	$-0.865542\pi$
$461$	− 377.985i	− 0.819924i	0.912103	−	0.409962i	$-0.134458\pi$
$462$	0	0
$463$	−49.3159	−0.106514	−0.0532569	−	0.998581i	$-0.516960\pi$
$463$	−49.3159	−0.106514	−0.0532569	+	0.998581i	$0.516960\pi$
$464$	− 71.2960i	− 0.153655i
$465$	−20.7114	−0.0445406
$466$	−422.605	−0.906878
$467$	−117.187	−0.250936	−0.125468	−	0.992098i	$-0.540043\pi$
$467$	−117.187	−0.250936	−0.125468	+	0.992098i	$0.540043\pi$
$468$	− 1.25537i	− 0.00268241i
$469$	− 57.6691i	− 0.122962i
$470$	− 30.5633i	− 0.0650283i
$471$	180.615	0.383472
$472$	145.624i	0.308525i
$473$	0	0
$474$	−360.385	−0.760307
$475$	− 53.9674i	− 0.113616i
$476$	23.6742	0.0497358
$477$	176.902	0.370863
$478$	−347.797	−0.727608
$479$	− 920.732i	− 1.92220i	−0.276209	−	0.961098i	$-0.589078\pi$
$479$	− 920.732i	− 1.92220i	0.276209	−	0.961098i	$-0.410922\pi$
$480$	− 3.46246i	− 0.00721347i
$481$	− 12.5716i	− 0.0261364i
$482$	528.093	1.09563
$483$	33.0544i	0.0684355i
$484$	0	0
$485$	35.8708	0.0739604
$486$	− 22.0454i	− 0.0453609i
$487$	388.108	0.796936	0.398468	−	0.917182i	$-0.369542\pi$
$487$	388.108	0.796936	0.398468	+	0.917182i	$0.369542\pi$
$488$	−115.265	−0.236198
$489$	484.399	0.990591
$490$	24.1759i	0.0493386i
$491$	364.974i	0.743328i	0.928367	+	0.371664i	$0.121213\pi$
$491$	364.974i	0.743328i	−0.928367	+	0.371664i	$0.878787\pi$
$492$	− 185.525i	− 0.377083i
$493$	266.818	0.541213
$494$	− 0.641950i	− 0.00129949i
$495$	0	0
$496$	135.350	0.272884
$497$	− 75.1617i	− 0.151231i
$498$	48.6842	0.0977595
$499$	−245.125	−0.491232	−0.245616	−	0.969367i	$-0.578990\pi$
$499$	−245.125	−0.491232	−0.245616	+	0.969367i	$0.578990\pi$
$500$	35.2504	0.0705007
$501$	− 445.043i	− 0.888310i
$502$	− 508.299i	− 1.01255i
$503$	− 294.256i	− 0.585002i	−0.956265	−	0.292501i	$-0.905513\pi$
$503$	− 294.256i	− 0.585002i	0.956265	−	0.292501i	$-0.0944875\pi$
$504$	6.70968	0.0133129
$505$	9.21825i	0.0182540i
$506$	0	0
$507$	−292.641	−0.577201
$508$	298.103i	0.586818i
$509$	−855.189	−1.68013	−0.840067	−	0.542482i	$-0.817485\pi$
$509$	−855.189	−1.68013	−0.840067	+	0.542482i	$0.817485\pi$
$510$	12.9579	0.0254077
$511$	89.7677	0.175671
$512$	22.6274i	0.0441942i
$513$	− 11.2732i	− 0.0219751i
$514$	− 50.6065i	− 0.0984563i
$515$	−11.1445	−0.0216397
$516$	− 53.9456i	− 0.104546i
$517$	0	0
$518$	67.1925	0.129715
$519$	− 190.852i	− 0.367731i
$520$	0.209129	0.000402172 0
$521$	−607.086	−1.16523	−0.582617	−	0.812747i	$-0.697971\pi$
$521$	−607.086	−1.16523	−0.582617	+	0.812747i	$0.697971\pi$
$522$	75.6208	0.144867
$523$	− 674.935i	− 1.29051i	−0.763969	−	0.645253i	$-0.776752\pi$
$523$	− 674.935i	− 1.29051i	0.763969	−	0.645253i	$-0.223248\pi$
$524$	− 331.930i	− 0.633454i
$525$	34.0692i	0.0648936i
$526$	515.719	0.980454
$527$	506.535i	0.961166i
$528$	0	0
$529$	53.4586	0.101056
$530$	29.4697i	0.0556032i
$531$	−154.457	−0.290880
$532$	3.43109	0.00644941
$533$	11.2055	0.0210234
$534$	183.641i	0.343896i
$535$	− 28.4609i	− 0.0531980i
$536$	206.278i	0.384846i
$537$	−46.9848	−0.0874950
$538$	− 18.9178i	− 0.0351632i
$539$	0	0
$540$	3.67250	0.00680092
$541$	− 233.688i	− 0.431956i	−0.976398	−	0.215978i	$-0.930706\pi$
$541$	− 233.688i	− 0.431956i	0.976398	−	0.215978i	$-0.0692940\pi$
$542$	−100.186	−0.184846
$543$	−449.591	−0.827976
$544$	−84.6808	−0.155663
$545$	11.3724i	0.0208667i
$546$	0.405258i	0 0.000742230i
$547$	− 953.254i	− 1.74269i	−0.490667	−	0.871347i	$-0.663247\pi$
$547$	− 953.254i	− 1.74269i	0.490667	−	0.871347i	$-0.336753\pi$
$548$	293.735	0.536014
$549$	− 122.257i	− 0.222690i
$550$	0	0
$551$	38.6697	0.0701810
$552$	− 118.233i	− 0.214190i
$553$	116.339	0.210379
$554$	112.213	0.202551
$555$	36.7773	0.0662655
$556$	427.549i	0.768973i
$557$	450.083i	0.808049i	0.914748	+	0.404025i	$0.132389\pi$
$557$	450.083i	0.808049i	−0.914748	+	0.404025i	$0.867611\pi$
$558$	143.561i	0.257277i
$559$	3.25826	0.00582873
$560$	1.11775i	0.00199598i
$561$	0	0
$562$	−485.205	−0.863354
$563$	312.343i	0.554783i	0.960757	+	0.277391i	$0.0894698\pi$
$563$	312.343i	0.554783i	−0.960757	+	0.277391i	$0.910530\pi$
$564$	−211.849	−0.375619
$565$	−10.3202	−0.0182659
$566$	−169.252	−0.299032
$567$	7.11669i	0.0125515i
$568$	268.848i	0.473323i
$569$	− 564.562i	− 0.992200i	−0.868265	−	0.496100i	$-0.834765\pi$
$569$	− 564.562i	− 0.992200i	0.868265	−	0.496100i	$-0.165235\pi$
$570$	1.87798	0.00329470
$571$	224.552i	0.393260i	0.980478	+	0.196630i	$0.0629998\pi$
$571$	224.552i	0.393260i	−0.980478	+	0.196630i	$0.937000\pi$
$572$	0	0
$573$	216.129	0.377189
$574$	59.8910i	0.104340i
$575$	600.341	1.04407
$576$	−24.0000	−0.0416667
$577$	−1055.36	−1.82905	−0.914524	−	0.404532i	$-0.867434\pi$
$577$	−1055.36	−1.82905	−0.914524	+	0.404532i	$0.867434\pi$
$578$	91.7981i	0.158820i
$579$	72.7274i	0.125609i
$580$	12.5975i	0.0217198i
$581$	−15.7162	−0.0270503
$582$	− 248.638i	− 0.427213i
$583$	0	0
$584$	−321.092	−0.549815
$585$	0.221815i	0 0.000379171i
$586$	−16.2862	−0.0277922
$587$	−79.4137	−0.135287	−0.0676437	−	0.997710i	$-0.521548\pi$
$587$	−79.4137	−0.135287	−0.0676437	+	0.997710i	$0.521548\pi$
$588$	167.575	0.284991
$589$	73.4117i	0.124638i
$590$	− 25.7307i	− 0.0436113i
$591$	475.168i	0.804007i
$592$	−240.342	−0.405984
$593$	− 709.519i	− 1.19649i	−0.801313	−	0.598245i	$-0.795865\pi$
$593$	− 709.519i	− 1.19649i	0.801313	−	0.598245i	$-0.204135\pi$
$594$	0	0
$595$	−4.18307	−0.00703037
$596$	− 462.484i	− 0.775981i
$597$	261.504	0.438030
$598$	7.14114	0.0119417
$599$	698.251	1.16569	0.582847	−	0.812582i	$-0.301939\pi$
$599$	698.251	1.16569	0.582847	+	0.812582i	$0.301939\pi$
$600$	− 121.863i	− 0.203104i
$601$	223.491i	0.371865i	0.982562	+	0.185933i	$0.0595306\pi$
$601$	223.491i	0.371865i	−0.982562	+	0.185933i	$0.940469\pi$
$602$	17.4147i	0.0289281i
$603$	−218.791	−0.362837
$604$	− 532.811i	− 0.882137i
$605$	0	0
$606$	63.8961	0.105439
$607$	953.367i	1.57062i	0.619102	+	0.785310i	$0.287497\pi$
$607$	953.367i	1.57062i	−0.619102	+	0.785310i	$0.712503\pi$
$608$	−12.2727	−0.0201854
$609$	−24.4119	−0.0400852
$610$	20.3665	0.0333876
$611$	− 12.7955i	− 0.0209418i
$612$	− 89.8176i	− 0.146761i
$613$	27.8874i	0.0454934i	0.999741	+	0.0227467i	$0.00724112\pi$
$613$	27.8874i	0.0454934i	−0.999741	+	0.0227467i	$0.992759\pi$
$614$	−278.399	−0.453418
$615$	32.7809i	0.0533023i
$616$	0	0
$617$	−997.165	−1.61615	−0.808075	−	0.589079i	$-0.799491\pi$
$617$	−997.165	−1.61615	−0.808075	+	0.589079i	$0.799491\pi$
$618$	77.2476i	0.124996i
$619$	−23.2638	−0.0375829	−0.0187915	−	0.999823i	$-0.505982\pi$
$619$	−23.2638	−0.0375829	−0.0187915	+	0.999823i	$0.505982\pi$
$620$	−23.9154	−0.0385733
$621$	125.405	0.201940
$622$	382.426i	0.614832i
$623$	− 59.2828i	− 0.0951571i
$624$	− 1.44958i	− 0.00232304i
$625$	615.649	0.985039
$626$	487.352i	0.778517i
$627$	0	0
$628$	208.557	0.332097
$629$	− 899.457i	− 1.42998i
$630$	−1.18555	−0.00188183
$631$	−523.766	−0.830058	−0.415029	−	0.909808i	$-0.636229\pi$
$631$	−523.766	−0.830058	−0.415029	+	0.909808i	$0.636229\pi$
$632$	−416.137	−0.658445
$633$	32.3612i	0.0511235i
$634$	92.7320i	0.146265i
$635$	− 52.6728i	− 0.0829493i
$636$	204.269	0.321177
$637$	10.1214i	0.0158891i
$638$	0	0
$639$	−285.156	−0.446253
$640$	− 3.99811i	− 0.00624704i
$641$	−67.4115	−0.105166	−0.0525831	−	0.998617i	$-0.516745\pi$
$641$	−67.4115	−0.105166	−0.0525831	+	0.998617i	$0.516745\pi$
$642$	−197.277	−0.307284
$643$	−806.459	−1.25421	−0.627106	−	0.778934i	$-0.715761\pi$
$643$	−806.459	−1.25421	−0.627106	+	0.778934i	$0.715761\pi$
$644$	38.1679i	0.0592669i
$645$	9.53181i	0.0147780i
$646$	− 45.9295i	− 0.0710982i
$647$	−970.912	−1.50064	−0.750319	−	0.661076i	$-0.770100\pi$
$647$	−970.912	−1.50064	−0.750319	+	0.661076i	$0.770100\pi$
$648$	− 25.4558i	− 0.0392837i
$649$	0	0
$650$	7.36038	0.0113237
$651$	− 46.3442i	− 0.0711892i
$652$	559.336	0.857877
$653$	28.0428	0.0429445	0.0214723	−	0.999769i	$-0.493165\pi$
$653$	28.0428	0.0429445	0.0214723	+	0.999769i	$0.493165\pi$
$654$	78.8272	0.120531
$655$	58.6497i	0.0895416i
$656$	− 214.225i	− 0.326563i
$657$	− 340.569i	− 0.518370i
$658$	68.3891	0.103935
$659$	− 570.449i	− 0.865628i	−0.901483	−	0.432814i	$-0.857521\pi$
$659$	− 570.449i	− 0.865628i	0.901483	−	0.432814i	$-0.142479\pi$
$660$	0	0
$661$	854.772	1.29315	0.646575	−	0.762851i	$-0.276201\pi$
$661$	854.772	1.29315	0.646575	+	0.762851i	$0.276201\pi$
$662$	149.650i	0.226057i
$663$	5.42489	0.00818234
$664$	56.2157	0.0846622
$665$	−0.606249	−0.000911653 0
$666$	− 254.921i	− 0.382765i
$667$	430.168i	0.644929i
$668$	− 513.892i	− 0.769299i
$669$	163.299	0.244094
$670$	− 36.4478i	− 0.0543998i
$671$	0	0
$672$	7.74767	0.0115293
$673$	353.405i	0.525119i	0.964916	+	0.262560i	$0.0845667\pi$
$673$	353.405i	0.525119i	−0.964916	+	0.262560i	$0.915433\pi$
$674$	−319.076	−0.473407
$675$	129.255	0.191489
$676$	−337.912	−0.499870
$677$	1242.34i	1.83506i	0.397665	+	0.917531i	$0.369821\pi$
$677$	1242.34i	1.83506i	−0.397665	+	0.917531i	$0.630179\pi$
$678$	71.5346i	0.105508i
$679$	80.2652i	0.118211i
$680$	14.9625	0.0220037
$681$	− 501.933i	− 0.737053i
$682$	0	0
$683$	−818.617	−1.19856	−0.599281	−	0.800539i	$-0.704547\pi$
$683$	−818.617	−1.19856	−0.599281	+	0.800539i	$0.704547\pi$
$684$	− 13.0172i	− 0.0190310i
$685$	−51.9010	−0.0757679
$686$	−108.892	−0.158735
$687$	395.303	0.575405
$688$	− 62.2910i	− 0.0905392i
$689$	12.3376i	0.0179066i
$690$	20.8909i	0.0302767i
$691$	273.464	0.395751	0.197875	−	0.980227i	$-0.436596\pi$
$691$	273.464	0.395751	0.197875	+	0.980227i	$0.436596\pi$
$692$	− 220.377i	− 0.318464i
$693$	0	0
$694$	191.723	0.276258
$695$	− 75.5450i	− 0.108698i
$696$	87.3194	0.125459
$697$	801.717	1.15024
$698$	707.324	1.01336
$699$	− 517.584i	− 0.740463i
$700$	39.3397i	0.0561995i
$701$	507.111i	0.723411i	0.932292	+	0.361706i	$0.117805\pi$
$701$	507.111i	0.723411i	−0.932292	+	0.361706i	$0.882195\pi$
$702$	1.53751	0.00219018
$703$	− 130.358i	− 0.185430i
$704$	0	0
$705$	37.4323	0.0530954
$706$	453.385i	0.642189i
$707$	−20.6269	−0.0291753
$708$	−178.352	−0.251909
$709$	578.587	0.816060	0.408030	−	0.912968i	$-0.366216\pi$
$709$	578.587	0.816060	0.408030	+	0.912968i	$0.366216\pi$
$710$	− 47.5035i	− 0.0669063i
$711$	− 441.380i	− 0.620788i
$712$	212.050i	0.297823i
$713$	−816.642	−1.14536
$714$	28.9949i	0.0406091i
$715$	0	0
$716$	−54.2534	−0.0757729
$717$	− 425.962i	− 0.594089i
$718$	−160.419	−0.223425
$719$	39.2537	0.0545948	0.0272974	−	0.999627i	$-0.491310\pi$
$719$	39.2537	0.0545948	0.0272974	+	0.999627i	$0.491310\pi$
$720$	4.24063	0.00588977
$721$	− 24.9371i	− 0.0345868i
$722$	503.875i	0.697887i
$723$	646.779i	0.894577i
$724$	−519.143	−0.717048
$725$	443.374i	0.611550i
$726$	0	0
$727$	918.828	1.26386	0.631932	−	0.775024i	$-0.282262\pi$
$727$	918.828	1.26386	0.631932	+	0.775024i	$0.282262\pi$
$728$	0.467951i	0 0.000642790i
$729$	27.0000	0.0370370
$730$	56.7347	0.0777188
$731$	233.118	0.318903
$732$	− 141.170i	− 0.192855i
$733$	738.905i	1.00806i	0.863687	+	0.504028i	$0.168149\pi$
$733$	738.905i	1.00806i	−0.863687	+	0.504028i	$0.831851\pi$
$734$	126.668i	0.172573i
$735$	−29.6093	−0.0402848
$736$	− 136.524i	− 0.185494i
$737$	0	0
$738$	227.220	0.307887
$739$	1137.91i	1.53979i	0.638168	+	0.769897i	$0.279692\pi$
$739$	1137.91i	1.53979i	−0.638168	+	0.769897i	$0.720308\pi$
$740$	42.4668	0.0573876
$741$	0.786225	0.00106103
$742$	−65.9420	−0.0888706
$743$	− 188.316i	− 0.253454i	−0.991938	−	0.126727i	$-0.959553\pi$
$743$	− 188.316i	− 0.253454i	0.991938	−	0.126727i	$-0.0404472\pi$
$744$	165.769i	0.222808i
$745$	81.7178i	0.109688i
$746$	−389.689	−0.522371
$747$	59.6257i	0.0798203i
$748$	0	0
$749$	63.6848	0.0850264
$750$	43.1727i	0.0575636i
$751$	605.397	0.806122	0.403061	−	0.915173i	$-0.367946\pi$
$751$	605.397	0.806122	0.403061	+	0.915173i	$0.367946\pi$
$752$	−244.622	−0.325296
$753$	622.537	0.826742
$754$	5.27400i	0.00699470i
$755$	94.1440i	0.124694i
$756$	8.21765i	0.0108699i
$757$	−388.614	−0.513361	−0.256680	−	0.966496i	$-0.582629\pi$
$757$	−388.614	−0.513361	−0.256680	+	0.966496i	$0.582629\pi$
$758$	− 248.258i	− 0.327517i
$759$	0	0
$760$	2.16851	0.00285330
$761$	699.156i	0.918734i	0.888247	+	0.459367i	$0.151924\pi$
$761$	699.156i	0.918734i	−0.888247	+	0.459367i	$0.848076\pi$
$762$	−365.101	−0.479135
$763$	−25.4470	−0.0333512
$764$	249.564	0.326655
$765$	15.8702i	0.0207453i
$766$	− 72.3778i	− 0.0944880i
$767$	− 10.7723i	− 0.0140447i
$768$	−27.7128	−0.0360844
$769$	− 337.074i	− 0.438328i	−0.975688	−	0.219164i	$-0.929667\pi$
$769$	− 337.074i	− 0.438328i	0.975688	−	0.219164i	$-0.0703330\pi$
$770$	0	0
$771$	61.9801	0.0803892
$772$	83.9783i	0.108780i
$773$	−795.193	−1.02871	−0.514355	−	0.857577i	$-0.671969\pi$
$773$	−795.193	−1.02871	−0.514355	+	0.857577i	$0.671969\pi$
$774$	66.0696	0.0853612
$775$	−841.713	−1.08608
$776$	− 287.102i	− 0.369977i
$777$	82.2937i	0.105912i
$778$	− 32.3946i	− 0.0416383i
$779$	116.192	0.149156
$780$	0.256130i	0 0.000328372i
$781$	0	0
$782$	510.926	0.653358
$783$	92.6162i	0.118284i
$784$	193.499	0.246810
$785$	−36.8505	−0.0469433
$786$	406.530	0.517213
$787$	535.000i	0.679796i	0.940462	+	0.339898i	$0.110393\pi$
$787$	535.000i	0.679796i	−0.940462	+	0.339898i	$0.889607\pi$
$788$	548.677i	0.696290i
$789$	631.624i	0.800537i
$790$	73.5286	0.0930741
$791$	− 23.0928i	− 0.0291944i
$792$	0	0
$793$	8.52651	0.0107522
$794$	185.339i	0.233424i
$795$	−36.0929	−0.0453998
$796$	301.959	0.379345
$797$	−711.043	−0.892149	−0.446074	−	0.894996i	$-0.647178\pi$
$797$	−711.043	−0.892149	−0.446074	+	0.894996i	$0.647178\pi$
$798$	4.20221i	0.00526592i
$799$	− 915.474i	− 1.14578i
$800$	− 140.715i	− 0.175894i
$801$	−224.913	−0.280790
$802$	1097.22i	1.36810i
$803$	0	0
$804$	−252.638	−0.314226
$805$	− 6.74400i	− 0.00837764i
$806$	−10.0123	−0.0124222
$807$	23.1695	0.0287106
$808$	73.7809	0.0913130
$809$	107.587i	0.132987i	0.997787	+	0.0664937i	$0.0211812\pi$
$809$	107.587i	0.132987i	−0.997787	+	0.0664937i	$0.978819\pi$
$810$	4.49787i	0.00555293i
$811$	1256.75i	1.54963i	0.632189	+	0.774814i	$0.282156\pi$
$811$	1256.75i	1.54963i	−0.632189	+	0.774814i	$0.717844\pi$
$812$	−28.1884	−0.0347148
$813$	− 122.703i	− 0.150926i
$814$	0	0
$815$	−98.8308	−0.121265
$816$	− 103.712i	− 0.127099i
$817$	33.7856	0.0413532
$818$	1056.95	1.29212
$819$	−0.496337	−0.000606029 0
$820$	37.8522i	0.0461612i
$821$	− 1034.51i	− 1.26006i	−0.776572	−	0.630028i	$-0.783043\pi$
$821$	− 1034.51i	− 1.26006i	0.776572	−	0.630028i	$-0.216957\pi$
$822$	359.751i	0.437653i
$823$	−535.829	−0.651067	−0.325534	−	0.945530i	$-0.605544\pi$
$823$	−535.829	−0.651067	−0.325534	+	0.945530i	$0.605544\pi$
$824$	89.1979i	0.108250i
$825$	0	0
$826$	57.5755	0.0697039
$827$	137.866i	0.166706i	0.996520	+	0.0833532i	$0.0265630\pi$
$827$	137.866i	0.166706i	−0.996520	+	0.0833532i	$0.973437\pi$
$828$	144.805	0.174885
$829$	−9.58047	−0.0115567	−0.00577833	−	0.999983i	$-0.501839\pi$
$829$	−9.58047	−0.0115567	−0.00577833	+	0.999983i	$0.501839\pi$
$830$	−9.93292	−0.0119674
$831$	137.432i	0.165382i
$832$	− 1.67383i	− 0.00201181i
$833$	724.150i	0.869328i
$834$	−523.638	−0.627864
$835$	90.8011i	0.108744i
$836$	0	0
$837$	−175.825	−0.210066
$838$	− 577.602i	− 0.689263i
$839$	−480.651	−0.572886	−0.286443	−	0.958097i	$-0.592473\pi$
$839$	−480.651	−0.572886	−0.286443	+	0.958097i	$0.592473\pi$
$840$	−1.36896	−0.00162971
$841$	523.305	0.622241
$842$	147.801i	0.175536i
$843$	− 594.253i	− 0.704926i
$844$	37.3675i	0.0442743i
$845$	59.7068	0.0706589
$846$	− 259.461i	− 0.306692i
$847$	0	0
$848$	235.869	0.278148
$849$	− 207.291i	− 0.244158i
$850$	526.612	0.619543
$851$	1450.12	1.70401
$852$	−329.270	−0.386467
$853$	1518.02i	1.77962i	0.456328	+	0.889812i	$0.349165\pi$
$853$	1518.02i	1.77962i	−0.456328	+	0.889812i	$0.650835\pi$
$854$	45.5724i	0.0533634i
$855$	2.30005i	0.00269011i
$856$	−227.795	−0.266116
$857$	229.336i	0.267603i	0.991008	+	0.133802i	$0.0427185\pi$
$857$	229.336i	0.267603i	−0.991008	+	0.133802i	$0.957282\pi$
$858$	0	0
$859$	−308.792	−0.359478	−0.179739	−	0.983714i	$-0.557525\pi$
$859$	−308.792	−0.359478	−0.179739	+	0.983714i	$0.557525\pi$
$860$	11.0064i	0.0127981i
$861$	−73.3512	−0.0851930
$862$	408.797	0.474242
$863$	517.131	0.599225	0.299613	−	0.954061i	$-0.403143\pi$
$863$	517.131	0.599225	0.299613	+	0.954061i	$0.403143\pi$
$864$	− 29.3939i	− 0.0340207i
$865$	38.9391i	0.0450164i
$866$	779.231i	0.899805i
$867$	−112.429	−0.129676
$868$	− 53.5137i	− 0.0616517i
$869$	0	0
$870$	−15.4287	−0.0177342
$871$	− 15.2590i	− 0.0175190i
$872$	91.0219	0.104383
$873$	304.518	0.348818
$874$	74.0481	0.0847232
$875$	− 13.9370i	− 0.0159280i
$876$	− 393.256i	− 0.448922i
$877$	− 252.227i	− 0.287602i	−0.989607	−	0.143801i	$-0.954067\pi$
$877$	− 252.227i	− 0.287602i	0.989607	−	0.143801i	$-0.0459325\pi$
$878$	−815.847	−0.929210
$879$	− 19.9465i	− 0.0226923i
$880$	0	0
$881$	830.794	0.943012	0.471506	−	0.881863i	$-0.343711\pi$
$881$	830.794	0.943012	0.471506	+	0.881863i	$0.343711\pi$
$882$	205.237i	0.232695i
$883$	537.950	0.609229	0.304615	−	0.952476i	$-0.401472\pi$
$883$	537.950	0.609229	0.304615	+	0.952476i	$0.401472\pi$
$884$	6.26412	0.00708611
$885$	31.5135	0.0356085
$886$	− 252.071i	− 0.284504i
$887$	1430.06i	1.61224i	0.591753	+	0.806120i	$0.298436\pi$
$887$	1430.06i	1.61224i	−0.591753	+	0.806120i	$0.701564\pi$
$888$	− 294.358i	− 0.331484i
$889$	117.862	0.132578
$890$	− 37.4678i	− 0.0420986i
$891$	0	0
$892$	188.562	0.211392
$893$	− 132.679i	− 0.148577i
$894$	566.425	0.633585
$895$	9.58620	0.0107108
$896$	8.94624	0.00998465
$897$	8.74608i	0.00975036i
$898$	880.645i	0.980674i
$899$	− 603.121i	− 0.670879i
$900$	149.251	0.165834
$901$	882.717i	0.979708i
$902$	0	0
$903$	−21.3286	−0.0236197
$904$	82.6010i	0.0913728i
$905$	91.7289	0.101358
$906$	652.558	0.720262
$907$	−1352.76	−1.49147	−0.745734	−	0.666244i	$-0.767901\pi$
$907$	−1352.76	−1.49147	−0.745734	+	0.666244i	$0.767901\pi$
$908$	− 579.582i	− 0.638306i
$909$	78.2565i	0.0860907i
$910$	− 0.0826838i	0 9.08613e-5i
$911$	−854.208	−0.937660	−0.468830	−	0.883288i	$-0.655324\pi$
$911$	−854.208	−0.937660	−0.468830	+	0.883288i	$0.655324\pi$
$912$	− 15.0310i	− 0.0164813i
$913$	0	0
$914$	−1114.13	−1.21896
$915$	24.9437i	0.0272609i
$916$	456.457	0.498315
$917$	−131.236	−0.143114
$918$	110.004	0.119830
$919$	330.219i	0.359324i	0.983728	+	0.179662i	$0.0575004\pi$
$919$	330.219i	0.359324i	−0.983728	+	0.179662i	$0.942500\pi$
$920$	24.1228i	0.0262204i
$921$	− 340.968i	− 0.370215i
$922$	534.551	0.579774
$923$	− 19.8875i	− 0.0215466i
$924$	0	0
$925$	1494.64	1.61582
$926$	− 69.7432i	− 0.0753166i
$927$	−94.6086	−0.102059
$928$	100.828	0.108651
$929$	−114.412	−0.123156	−0.0615778	−	0.998102i	$-0.519613\pi$
$929$	−114.412	−0.123156	−0.0615778	+	0.998102i	$0.519613\pi$
$930$	− 29.2903i	− 0.0314950i
$931$	104.951i	0.112729i
$932$	− 597.654i	− 0.641260i
$933$	−468.374	−0.502008
$934$	− 165.728i	− 0.177439i
$935$	0	0
$936$	1.77536	0.00189675
$937$	1619.74i	1.72864i	0.502938	+	0.864322i	$0.332252\pi$
$937$	1619.74i	1.72864i	−0.502938	+	0.864322i	$0.667748\pi$
$938$	81.5564	0.0869471
$939$	−596.881	−0.635656
$940$	43.2231	0.0459820
$941$	− 759.245i	− 0.806849i	−0.915013	−	0.403424i	$-0.867820\pi$
$941$	− 759.245i	− 0.806849i	0.915013	−	0.403424i	$-0.132180\pi$
$942$	255.429i	0.271156i
$943$	1292.54i	1.37067i
$944$	−205.943	−0.218160
$945$	− 1.45200i	− 0.00153651i
$946$	0	0
$947$	1855.32	1.95915	0.979575	−	0.201079i	$-0.0644447\pi$
$947$	1855.32	1.95915	0.979575	+	0.201079i	$0.0644447\pi$
$948$	− 509.662i	− 0.537618i
$949$	23.7522	0.0250287
$950$	76.3214	0.0803383
$951$	−113.573	−0.119425
$952$	33.4804i	0.0351685i
$953$	− 1624.78i	− 1.70492i	−0.522796	−	0.852458i	$-0.675111\pi$
$953$	− 1624.78i	− 1.70492i	0.522796	−	0.852458i	$-0.324889\pi$
$954$	250.177i	0.262240i
$955$	−44.0963	−0.0461742
$956$	− 491.859i	− 0.514496i
$957$	0	0
$958$	1302.11	1.35920
$959$	− 116.135i	− 0.121100i
$960$	4.89666	0.00510069
$961$	183.980	0.191447
$962$	17.7789	0.0184812
$963$	− 241.613i	− 0.250897i
$964$	746.836i	0.774726i
$965$	− 14.8384i	− 0.0153766i
$966$	−46.7459	−0.0483912
$967$	325.104i	0.336199i	0.985770	+	0.168099i	$0.0537629\pi$
$967$	325.104i	0.336199i	−0.985770	+	0.168099i	$0.946237\pi$
$968$	0	0
$969$	56.2519	0.0580515
$970$	50.7290i	0.0522979i
$971$	280.767	0.289152	0.144576	−	0.989494i	$-0.453818\pi$
$971$	280.767	0.289152	0.144576	+	0.989494i	$0.453818\pi$
$972$	31.1769	0.0320750
$973$	169.041	0.173732
$974$	548.867i	0.563519i
$975$	9.01459i	0.00924573i
$976$	− 163.009i	− 0.167017i
$977$	1609.79	1.64769	0.823843	−	0.566818i	$-0.191826\pi$
$977$	1609.79	1.64769	0.823843	+	0.566818i	$0.191826\pi$
$978$	685.044i	0.700454i
$979$	0	0
$980$	−34.1899	−0.0348877
$981$	96.5433i	0.0984131i
$982$	−516.151	−0.525612
$983$	−374.382	−0.380856	−0.190428	−	0.981701i	$-0.560988\pi$
$983$	−374.382	−0.380856	−0.190428	+	0.981701i	$0.560988\pi$
$984$	262.371	0.266638
$985$	− 96.9474i	− 0.0984238i
$986$	377.338i	0.382696i
$987$	83.7591i	0.0848624i
$988$	0.907854	0.000918881 0
$989$	375.836i	0.380016i
$990$	0	0
$991$	−943.875	−0.952447	−0.476223	−	0.879324i	$-0.657995\pi$
$991$	−943.875	−0.952447	−0.476223	+	0.879324i	$0.657995\pi$
$992$	191.414i	0.192958i
$993$	−183.283	−0.184575
$994$	106.295	0.106936
$995$	−53.3540	−0.0536221
$996$	68.8499i	0.0691264i
$997$	681.654i	0.683705i	0.939754	+	0.341853i	$0.111054\pi$
$997$	681.654i	0.683705i	−0.939754	+	0.341853i	$0.888946\pi$
$998$	− 346.659i	− 0.347353i
$999$	312.214	0.312526

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	726.3.d.e.241.10		16
3.2	odd	2		2178.3.d.m.1693.4		16
11.3	even	5		66.3.f.a.13.4	✓	16
11.7	odd	10		66.3.f.a.61.4	yes	16
11.10	odd	2	inner	726.3.d.e.241.2		16
33.14	odd	10		198.3.j.b.145.2		16
33.29	even	10		198.3.j.b.127.2		16
33.32	even	2		2178.3.d.m.1693.12		16
44.3	odd	10		528.3.bf.c.145.1		16
44.7	even	10		528.3.bf.c.193.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
66.3.f.a.13.4	✓	16	11.3	even	5
66.3.f.a.61.4	yes	16	11.7	odd	10
198.3.j.b.127.2		16	33.29	even	10
198.3.j.b.145.2		16	33.14	odd	10
528.3.bf.c.145.1		16	44.3	odd	10
528.3.bf.c.193.1		16	44.7	even	10
726.3.d.e.241.2		16	11.10	odd	2	inner
726.3.d.e.241.10		16	1.1	even	1	trivial
2178.3.d.m.1693.4		16	3.2	odd	2
2178.3.d.m.1693.12		16	33.32	even	2

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:	\( N \)	\(=\)	\( 726 = 2 \cdot 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	726.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -1.73205 q^{3} -2.00000 q^{4} +0.353386 q^{5} -2.44949i q^{6} +0.790744i q^{7} -2.82843i q^{8} +3.00000 q^{9} +0.499764i q^{10} +3.46410 q^{12} +0.209228i q^{13} -1.11828 q^{14} -0.612083 q^{15} +4.00000 q^{16} +14.9696i q^{17} +4.24264i q^{18} +2.16953i q^{19} -0.706772 q^{20} -1.36961i q^{21} -24.1342 q^{23} +4.89898i q^{24} -24.8751 q^{25} -0.295893 q^{26} -5.19615 q^{27} -1.58149i q^{28} -17.8240i q^{29} -0.865616i q^{30} +33.8376 q^{31} +5.65685i q^{32} -21.1702 q^{34} +0.279438i q^{35} -6.00000 q^{36} -60.0856 q^{37} -3.06818 q^{38} -0.362394i q^{39} -0.999527i q^{40} -53.5564i q^{41} +1.93692 q^{42} -15.5727i q^{43} +1.06016 q^{45} -34.1309i q^{46} -61.1556 q^{47} -6.92820 q^{48} +48.3747 q^{49} -35.1787i q^{50} -25.9281i q^{51} -0.418456i q^{52} +58.9673 q^{53} -7.34847i q^{54} +2.23656 q^{56} -3.75774i q^{57} +25.2069 q^{58} -51.4857 q^{59} +1.22417 q^{60} -40.7522i q^{61} +47.8535i q^{62} +2.37223i q^{63} -8.00000 q^{64} +0.0739383i q^{65} -72.9302 q^{67} -29.9392i q^{68} +41.8016 q^{69} -0.395185 q^{70} -95.0520 q^{71} -8.48528i q^{72} -113.523i q^{73} -84.9738i q^{74} +43.0850 q^{75} -4.33906i q^{76} +0.512502 q^{78} -147.127i q^{79} +1.41354 q^{80} +9.00000 q^{81} +75.7401 q^{82} +19.8752i q^{83} +2.73922i q^{84} +5.29005i q^{85} +22.0232 q^{86} +30.8721i q^{87} -74.9710 q^{89} +1.49929i q^{90} -0.165446 q^{91} +48.2684 q^{92} -58.6084 q^{93} -86.4870i q^{94} +0.766683i q^{95} -9.79796i q^{96} +101.506 q^{97} +68.4122i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{4} + 8 q^{5} + 48 q^{9} + 48 q^{14} - 48 q^{15} + 64 q^{16} - 16 q^{20} - 8 q^{23} - 8 q^{25} + 128 q^{26} - 8 q^{31} - 64 q^{34} - 96 q^{36} + 96 q^{37} - 208 q^{38} - 144 q^{42} + 24 q^{45}+ \cdots + 184 q^{97}+O(q^{100}) \) 16 * q - 32 * q^4 + 8 * q^5 + 48 * q^9 + 48 * q^14 - 48 * q^15 + 64 * q^16 - 16 * q^20 - 8 * q^23 - 8 * q^25 + 128 * q^26 - 8 * q^31 - 64 * q^34 - 96 * q^36 + 96 * q^37 - 208 * q^38 - 144 * q^42 + 24 * q^45 - 128 * q^47 - 64 * q^49 + 16 * q^53 - 96 * q^56 + 64 * q^58 - 152 * q^59 + 96 * q^60 - 128 * q^64 + 248 * q^67 + 168 * q^69 - 32 * q^70 - 72 * q^71 - 48 * q^75 + 32 * q^80 + 144 * q^81 + 176 * q^82 + 464 * q^86 - 360 * q^89 - 32 * q^91 + 16 * q^92 + 336 * q^93 + 184 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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