LMFDB - Embedded newform 378.4.g.e.109.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,4,Mod(109,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

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

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

chi := DirichletCharacter("378.109");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	378.g (of order $3$, degree $2$, 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:	$22.3027219822$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + 66x^{6} + 59x^{5} + 3770x^{4} + 721x^{3} + 29779x^{2} + 1374x + 209764 $ x^8 - x^7 + 66x^6 + 59x^5 + 3770x^4 + 721x^3 + 29779x^2 + 1374x + 209764
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			109.2
Root			$-3.50329 + 6.06788i$ of defining polynomial
Character	$\chi$	$=$	378.109
Dual form			378.4.g.e.163.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 + 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(-2.21575 - 3.83779i) q^{5} +(-11.5959 - 14.4407i) q^{7} -8.00000 q^{8} +O(q^{10})$	\(q+(1.00000 + 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(-2.21575 - 3.83779i) q^{5} +(-11.5959 - 14.4407i) q^{7} -8.00000 q^{8} +(4.43150 - 7.67558i) q^{10} +(-19.8193 + 34.3281i) q^{11} +37.4863 q^{13} +(13.4162 - 34.5255i) q^{14} +(-8.00000 - 13.8564i) q^{16} +(57.0112 - 98.7463i) q^{17} +(67.5634 + 117.023i) q^{19} +17.7260 q^{20} -79.2774 q^{22} +(96.1989 + 166.621i) q^{23} +(52.6809 - 91.2460i) q^{25} +(37.4863 + 64.9282i) q^{26} +(73.2160 - 11.2880i) q^{28} +129.302 q^{29} +(122.291 - 211.813i) q^{31} +(16.0000 - 27.7128i) q^{32} +228.045 q^{34} +(-29.7269 + 76.4998i) q^{35} +(-136.629 - 236.649i) q^{37} +(-135.127 + 234.047i) q^{38} +(17.7260 + 30.7023i) q^{40} +365.448 q^{41} +57.2537 q^{43} +(-79.2774 - 137.312i) q^{44} +(-192.398 + 333.243i) q^{46} +(172.495 + 298.771i) q^{47} +(-74.0693 + 334.907i) q^{49} +210.724 q^{50} +(-74.9727 + 129.856i) q^{52} +(-300.911 + 521.193i) q^{53} +175.659 q^{55} +(92.7674 + 115.526i) q^{56} +(129.302 + 223.957i) q^{58} +(15.7631 - 27.3024i) q^{59} +(-431.259 - 746.963i) q^{61} +489.162 q^{62} +64.0000 q^{64} +(-83.0604 - 143.865i) q^{65} +(341.295 - 591.141i) q^{67} +(228.045 + 394.985i) q^{68} +(-162.228 + 25.0114i) q^{70} +664.860 q^{71} +(-471.122 + 816.007i) q^{73} +(273.259 - 473.298i) q^{74} -540.507 q^{76} +(725.546 - 111.860i) q^{77} +(1.66271 + 2.87990i) q^{79} +(-35.4520 + 61.4047i) q^{80} +(365.448 + 632.975i) q^{82} -54.0900 q^{83} -505.290 q^{85} +(57.2537 + 99.1664i) q^{86} +(158.555 - 274.625i) q^{88} +(142.700 + 247.163i) q^{89} +(-434.689 - 541.330i) q^{91} -769.591 q^{92} +(-344.991 + 597.542i) q^{94} +(299.407 - 518.589i) q^{95} +240.309 q^{97} +(-654.145 + 206.615i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} - 16 q^{4} - 4 q^{5} + 25 q^{7} - 64 q^{8}+O(q^{10}) $ 8 * q + 8 * q^2 - 16 * q^4 - 4 * q^5 + 25 * q^7 - 64 * q^8	$ 8 q + 8 q^{2} - 16 q^{4} - 4 q^{5} + 25 q^{7} - 64 q^{8} + 8 q^{10} - 56 q^{11} - 18 q^{13} + 22 q^{14} - 64 q^{16} + 118 q^{17} + 37 q^{19} + 32 q^{20} - 224 q^{22} - 200 q^{23} - 104 q^{25} - 18 q^{26} - 56 q^{28} + 524 q^{29} + 276 q^{31} + 128 q^{32} + 472 q^{34} - 290 q^{35} - 185 q^{37} - 74 q^{38} + 32 q^{40} - 60 q^{41} - 1556 q^{43} - 224 q^{44} + 400 q^{46} - 30 q^{47} - 1159 q^{49} - 416 q^{50} + 36 q^{52} - 480 q^{53} + 1456 q^{55} - 200 q^{56} + 524 q^{58} + 296 q^{59} + 474 q^{61} + 1104 q^{62} + 512 q^{64} - 1542 q^{65} + 1319 q^{67} + 472 q^{68} - 32 q^{70} + 1852 q^{71} - 1423 q^{73} + 370 q^{74} - 296 q^{76} - 1228 q^{77} + 765 q^{79} - 64 q^{80} - 60 q^{82} + 1660 q^{83} - 584 q^{85} - 1556 q^{86} + 448 q^{88} + 864 q^{89} - 738 q^{91} + 1600 q^{92} + 60 q^{94} - 1766 q^{95} + 1088 q^{97} - 2704 q^{98}+O(q^{100}) $ 8 * q + 8 * q^2 - 16 * q^4 - 4 * q^5 + 25 * q^7 - 64 * q^8 + 8 * q^10 - 56 * q^11 - 18 * q^13 + 22 * q^14 - 64 * q^16 + 118 * q^17 + 37 * q^19 + 32 * q^20 - 224 * q^22 - 200 * q^23 - 104 * q^25 - 18 * q^26 - 56 * q^28 + 524 * q^29 + 276 * q^31 + 128 * q^32 + 472 * q^34 - 290 * q^35 - 185 * q^37 - 74 * q^38 + 32 * q^40 - 60 * q^41 - 1556 * q^43 - 224 * q^44 + 400 * q^46 - 30 * q^47 - 1159 * q^49 - 416 * q^50 + 36 * q^52 - 480 * q^53 + 1456 * q^55 - 200 * q^56 + 524 * q^58 + 296 * q^59 + 474 * q^61 + 1104 * q^62 + 512 * q^64 - 1542 * q^65 + 1319 * q^67 + 472 * q^68 - 32 * q^70 + 1852 * q^71 - 1423 * q^73 + 370 * q^74 - 296 * q^76 - 1228 * q^77 + 765 * q^79 - 64 * q^80 - 60 * q^82 + 1660 * q^83 - 584 * q^85 - 1556 * q^86 + 448 * q^88 + 864 * q^89 - 738 * q^91 + 1600 * q^92 + 60 * q^94 - 1766 * q^95 + 1088 * q^97 - 2704 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$

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 $

$2$	1.00000	+	1.73205i	0.353553	+	0.612372i
$2$	1.00000	+	1.73205i	0.353553	+	0.612372i
$3$		0			0
$3$		0			0
$4$	−2.00000	+	3.46410i	−0.250000	+	0.433013i
$5$	−2.21575	−	3.83779i	−0.198183	−	0.343263i	0.749756	−	0.661714i	$-0.230171\pi$
$5$	−2.21575	−	3.83779i	−0.198183	−	0.343263i	−0.947939	+	0.318451i	$0.896837\pi$
$6$		0			0
$7$	−11.5959	−	14.4407i	−0.626121	−	0.779726i
$7$	−11.5959	−	14.4407i	−0.626121	−	0.779726i
$8$	−8.00000			−0.353553
$9$		0			0
$10$	4.43150	−	7.67558i	0.140136	−	0.242723i
$11$	−19.8193	+	34.3281i	−0.543251	+	0.940938i	0.455464	+	0.890254i	$0.349473\pi$
$11$	−19.8193	+	34.3281i	−0.543251	+	0.940938i	−0.998715	+	0.0506835i	$0.983860\pi$
$12$		0			0
$13$	37.4863			0.799757			0.399878	−	0.916568i	$-0.369052\pi$
$13$	37.4863			0.799757			0.399878	+	0.916568i	$0.369052\pi$
$14$	13.4162	−	34.5255i	0.256116	−	0.659094i
$15$		0			0
$16$	−8.00000	−	13.8564i	−0.125000	−	0.216506i
$17$	57.0112	−	98.7463i	0.813367	−	1.40879i	−0.0971273	−	0.995272i	$-0.530965\pi$
$17$	57.0112	−	98.7463i	0.813367	−	1.40879i	0.910494	−	0.413521i	$-0.135701\pi$
$18$		0			0
$19$	67.5634	+	117.023i	0.815796	+	1.41300i	0.908755	+	0.417329i	$0.137034\pi$
$19$	67.5634	+	117.023i	0.815796	+	1.41300i	−0.0929597	+	0.995670i	$0.529633\pi$
$20$	17.7260			0.198183
$21$		0			0
$22$	−79.2774			−0.768272
$23$	96.1989	+	166.621i	0.872124	+	1.51056i	0.859795	+	0.510639i	$0.170591\pi$
$23$	96.1989	+	166.621i	0.872124	+	1.51056i	0.0123284	+	0.999924i	$0.496076\pi$
$24$		0			0
$25$	52.6809	−	91.2460i	0.421447	−	0.729968i
$26$	37.4863	+	64.9282i	0.282757	+	0.489749i
$27$		0			0
$28$	73.2160	−	11.2880i	0.494161	−	0.0761867i
$29$	129.302			0.827957			0.413978	−	0.910287i	$-0.364139\pi$
$29$	129.302			0.827957			0.413978	+	0.910287i	$0.364139\pi$
$30$		0			0
$31$	122.291	−	211.813i	0.708517	−	1.22719i	−0.256890	−	0.966441i	$-0.582698\pi$
$31$	122.291	−	211.813i	0.708517	−	1.22719i	0.965407	−	0.260747i	$-0.0839687\pi$
$32$	16.0000	−	27.7128i	0.0883883	−	0.153093i
$33$		0			0
$34$	228.045			1.15027
$35$	−29.7269	+	76.4998i	−0.143564	+	0.369452i
$36$		0			0
$37$	−136.629	−	236.649i	−0.607073	−	1.05148i	−0.991720	−	0.128418i	$-0.959010\pi$
$37$	−136.629	−	236.649i	−0.607073	−	1.05148i	0.384647	−	0.923064i	$-0.374323\pi$
$38$	−135.127	+	234.047i	−0.576855	+	0.999141i
$39$		0			0
$40$	17.7260	+	30.7023i	0.0700682	+	0.121362i
$41$	365.448			1.39203			0.696017	−	0.718025i	$-0.254954\pi$
$41$	365.448			1.39203			0.696017	+	0.718025i	$0.254954\pi$
$42$		0			0
$43$	57.2537			0.203049			0.101525	−	0.994833i	$-0.467628\pi$
$43$	57.2537			0.203049			0.101525	+	0.994833i	$0.467628\pi$
$44$	−79.2774	−	137.312i	−0.271625	−	0.470469i
$45$		0			0
$46$	−192.398	+	333.243i	−0.616685	+	1.06813i
$47$	172.495	+	298.771i	0.535341	+	0.927238i	0.999147	+	0.0413012i	$0.0131503\pi$
$47$	172.495	+	298.771i	0.535341	+	0.927238i	−0.463805	+	0.885937i	$0.653516\pi$
$48$		0			0
$49$	−74.0693	+	334.907i	−0.215946	+	0.976405i
$50$	210.724			0.596016
$51$		0			0
$52$	−74.9727	+	129.856i	−0.199939	+	0.346305i
$53$	−300.911	+	521.193i	−0.779874	+	1.35078i	0.152140	+	0.988359i	$0.451384\pi$
$53$	−300.911	+	521.193i	−0.779874	+	1.35078i	−0.932014	+	0.362422i	$0.881950\pi$
$54$		0			0
$55$	175.659			0.430652
$56$	92.7674	+	115.526i	0.221367	+	0.275675i
$57$		0			0
$58$	129.302	+	223.957i	0.292727	+	0.507018i
$59$	15.7631	−	27.3024i	0.0347827	−	0.0602454i	−0.848110	−	0.529820i	$-0.822259\pi$
$59$	15.7631	−	27.3024i	0.0347827	−	0.0602454i	0.882893	+	0.469575i	$0.155593\pi$
$60$		0			0
$61$	−431.259	−	746.963i	−0.905199	−	1.56785i	−0.820650	−	0.571431i	$-0.806389\pi$
$61$	−431.259	−	746.963i	−0.905199	−	1.56785i	−0.0845483	−	0.996419i	$-0.526945\pi$
$62$	489.162			1.00199
$63$		0			0
$64$	64.0000			0.125000
$65$	−83.0604	−	143.865i	−0.158498	−	0.274527i
$66$		0			0
$67$	341.295	−	591.141i	0.622326	−	1.07790i	−0.366725	−	0.930329i	$-0.619521\pi$
$67$	341.295	−	591.141i	0.622326	−	1.07790i	0.989051	−	0.147571i	$-0.0471456\pi$
$68$	228.045	+	394.985i	0.406684	+	0.704397i
$69$		0			0
$70$	−162.228	+	25.0114i	−0.277000	+	0.0427061i
$71$	664.860			1.11133			0.555665	−	0.831406i	$-0.312464\pi$
$71$	664.860			1.11133			0.555665	+	0.831406i	$0.312464\pi$
$72$		0			0
$73$	−471.122	+	816.007i	−0.755351	+	1.30831i	0.189848	+	0.981813i	$0.439200\pi$
$73$	−471.122	+	816.007i	−0.755351	+	1.30831i	−0.945199	+	0.326494i	$0.894133\pi$
$74$	273.259	−	473.298i	0.429266	−	0.743510i
$75$		0			0
$76$	−540.507			−0.815796
$77$	725.546	−	111.860i	1.07381	−	0.165554i
$78$		0			0
$79$	1.66271	+	2.87990i	0.00236797	+	0.00410145i	0.867207	−	0.497948i	$-0.165913\pi$
$79$	1.66271	+	2.87990i	0.00236797	+	0.00410145i	−0.864839	+	0.502049i	$0.832580\pi$
$80$	−35.4520	+	61.4047i	−0.0495457	+	0.0858156i
$81$		0			0
$82$	365.448	+	632.975i	0.492159	+	0.852444i
$83$	−54.0900			−0.0715319			−0.0357660	−	0.999360i	$-0.511387\pi$
$83$	−54.0900			−0.0715319			−0.0357660	+	0.999360i	$0.511387\pi$
$84$		0			0
$85$	−505.290			−0.644781
$86$	57.2537	+	99.1664i	0.0717887	+	0.124342i
$87$		0			0
$88$	158.555	−	274.625i	0.192068	−	0.332672i
$89$	142.700	+	247.163i	0.169957	+	0.294374i	0.938404	−	0.345539i	$-0.112304\pi$
$89$	142.700	+	247.163i	0.169957	+	0.294374i	−0.768448	+	0.639913i	$0.778971\pi$
$90$		0			0
$91$	−434.689	−	541.330i	−0.500744	−	0.623591i
$92$	−769.591			−0.872124
$93$		0			0
$94$	−344.991	+	597.542i	−0.378543	+	0.655656i
$95$	299.407	−	518.589i	0.323353	−	0.560064i
$96$		0			0
$97$	240.309			0.251543			0.125771	−	0.992059i	$-0.459859\pi$
$97$	240.309			0.251543			0.125771	+	0.992059i	$0.459859\pi$
$98$	−654.145	+	206.615i	−0.674272	+	0.212972i
$99$		0			0
$100$	210.724	+	364.984i	0.210724	+	0.364984i
$101$	−222.821	+	385.937i	−0.219520	+	0.380220i	−0.954661	−	0.297694i	$-0.903782\pi$
$101$	−222.821	+	385.937i	−0.219520	+	0.380220i	0.735141	+	0.677914i	$0.237116\pi$
$102$		0			0
$103$	423.094	+	732.821i	0.404745	+	0.701039i	0.994292	−	0.106695i	$-0.0340270\pi$
$103$	423.094	+	732.821i	0.404745	+	0.701039i	−0.589547	+	0.807734i	$0.700694\pi$
$104$	−299.891			−0.282757
$105$		0			0
$106$	−1203.64			−1.10291
$107$	−281.156	−	486.976i	−0.254022	−	0.439979i	0.710607	−	0.703589i	$-0.248420\pi$
$107$	−281.156	−	486.976i	−0.254022	−	0.439979i	−0.964629	+	0.263610i	$0.915087\pi$
$108$		0			0
$109$	557.817	−	966.168i	0.490176	−	0.849010i	−0.509760	−	0.860317i	$-0.670266\pi$
$109$	557.817	−	966.168i	0.490176	−	0.849010i	0.999936	+	0.0113064i	$0.00359902\pi$
$110$	175.659	+	304.250i	0.152258	+	0.263719i
$111$		0			0
$112$	−107.329	+	276.204i	−0.0905506	+	0.233025i
$113$	1518.31			1.26399			0.631995	−	0.774972i	$-0.282236\pi$
$113$	1518.31			1.26399			0.631995	+	0.774972i	$0.282236\pi$
$114$		0			0
$115$	426.305	−	738.382i	0.345680	−	0.598735i
$116$	−258.604	+	447.915i	−0.206989	+	0.358516i
$117$		0			0
$118$	63.0523			0.0491901
$119$	−2087.06	+	321.771i	−1.60774	+	0.247871i
$120$		0			0
$121$	−120.113	−	208.041i	−0.0902423	−	0.156304i
$122$	862.519	−	1493.93i	0.640072	−	1.10864i
$123$		0			0
$124$	489.162	+	847.253i	0.354258	+	0.613594i
$125$	−1020.85			−0.730460
$126$		0			0
$127$	−803.079			−0.561116			−0.280558	−	0.959837i	$-0.590519\pi$
$127$	−803.079			−0.561116			−0.280558	+	0.959837i	$0.590519\pi$
$128$	64.0000	+	110.851i	0.0441942	+	0.0765466i
$129$		0			0
$130$	166.121	−	287.730i	0.112075	−	0.194120i
$131$	−445.107	−	770.948i	−0.296864	−	0.514183i	0.678553	−	0.734552i	$-0.262607\pi$
$131$	−445.107	−	770.948i	−0.296864	−	0.514183i	−0.975417	+	0.220368i	$0.929274\pi$
$132$		0			0
$133$	906.442	−	2332.66i	0.590966	−	1.52081i
$134$	1365.18			0.880102
$135$		0			0
$136$	−456.089	+	789.970i	−0.287569	+	0.498084i
$137$	−1022.28	+	1770.64i	−0.637514	+	1.10421i	0.348462	+	0.937323i	$0.386704\pi$
$137$	−1022.28	+	1770.64i	−0.637514	+	1.10421i	−0.985976	+	0.166884i	$0.946629\pi$
$138$		0			0
$139$	587.471			0.358479			0.179240	−	0.983805i	$-0.442636\pi$
$139$	587.471			0.358479			0.179240	+	0.983805i	$0.442636\pi$
$140$	−205.549	−	255.976i	−0.124086	−	0.154528i
$141$		0			0
$142$	664.860	+	1151.57i	0.392914	+	0.680547i
$143$	−742.955	+	1286.84i	−0.434468	+	0.752521i
$144$		0			0
$145$	−286.501	−	496.233i	−0.164087	−	0.284207i
$146$	−1884.49			−1.06823
$147$		0			0
$148$	1093.03			0.607073
$149$	−1356.66	−	2349.80i	−0.745918	−	1.29197i	−0.949765	−	0.312964i	$-0.898678\pi$
$149$	−1356.66	−	2349.80i	−0.745918	−	1.29197i	0.203847	−	0.979003i	$-0.434655\pi$
$150$		0			0
$151$	83.2159	−	144.134i	0.0448478	−	0.0776786i	−0.842730	−	0.538336i	$-0.819053\pi$
$151$	83.2159	−	144.134i	0.0448478	−	0.0776786i	0.887578	+	0.460658i	$0.152386\pi$
$152$	−540.507	−	936.186i	−0.288427	−	0.499571i
$153$		0			0
$154$	919.294	+	1144.82i	0.481031	+	0.599042i
$155$	−1083.86			−0.561663
$156$		0			0
$157$	267.786	−	463.819i	0.136125	−	0.235776i	−0.789901	−	0.613234i	$-0.789868\pi$
$157$	267.786	−	463.819i	0.136125	−	0.235776i	0.926027	+	0.377458i	$0.123202\pi$
$158$	−3.32542	+	5.75980i	−0.00167441	+	0.00290016i
$159$		0			0
$160$	−141.808			−0.0700682
$161$	1290.62	−	3321.31i	0.631770	−	1.62581i
$162$		0			0
$163$	−601.378	−	1041.62i	−0.288979	−	0.500526i	0.684587	−	0.728931i	$-0.259982\pi$
$163$	−601.378	−	1041.62i	−0.288979	−	0.500526i	−0.973566	+	0.228404i	$0.926649\pi$
$164$	−730.896	+	1265.95i	−0.348009	+	0.602769i
$165$		0			0
$166$	−54.0900	−	93.6866i	−0.0252903	−	0.0438042i
$167$	3467.14			1.60656			0.803279	−	0.595604i	$-0.203087\pi$
$167$	3467.14			1.60656			0.803279	+	0.595604i	$0.203087\pi$
$168$		0			0
$169$	−791.774			−0.360389
$170$	−505.290	−	875.188i	−0.227965	−	0.394846i
$171$		0			0
$172$	−114.507	+	198.333i	−0.0507623	+	0.0879229i
$173$	−208.096	−	360.433i	−0.0914524	−	0.158400i	0.816670	−	0.577105i	$-0.195818\pi$
$173$	−208.096	−	360.433i	−0.0914524	−	0.158400i	−0.908122	+	0.418705i	$0.862484\pi$
$174$		0			0
$175$	−1928.54	+	297.331i	−0.833052	+	0.128435i
$176$	634.219			0.271625
$177$		0			0
$178$	−285.400	+	494.327i	−0.120178	+	0.208154i
$179$	−187.817	+	325.309i	−0.0784251	+	0.135836i	−0.902571	−	0.430542i	$-0.858322\pi$
$179$	−187.817	+	325.309i	−0.0784251	+	0.135836i	0.824145	+	0.566378i	$0.191656\pi$
$180$		0			0
$181$	2201.29			0.903981			0.451990	−	0.892023i	$-0.350714\pi$
$181$	2201.29			0.903981			0.451990	+	0.892023i	$0.350714\pi$
$182$	502.923	−	1294.23i	0.204830	−	0.527115i
$183$		0			0
$184$	−769.591	−	1332.97i	−0.308342	−	0.534065i
$185$	−605.473	+	1048.71i	−0.240623	+	0.416771i
$186$		0			0
$187$	2259.85	+	3914.17i	0.883724	+	1.53066i
$188$	−1379.96			−0.535341
$189$		0			0
$190$	1197.63			0.457290
$191$	−2176.64	−	3770.05i	−0.824587	−	1.42823i	−0.902234	−	0.431246i	$-0.858074\pi$
$191$	−2176.64	−	3770.05i	−0.824587	−	1.42823i	0.0776467	−	0.996981i	$-0.475259\pi$
$192$		0			0
$193$	1113.21	−	1928.14i	0.415184	−	0.719121i	−0.580263	−	0.814429i	$-0.697050\pi$
$193$	1113.21	−	1928.14i	0.415184	−	0.719121i	0.995448	+	0.0953083i	$0.0303837\pi$
$194$	240.309	+	416.227i	0.0889339	+	0.154038i
$195$		0			0
$196$	−1012.01	−	926.398i	−0.368810	−	0.337609i
$197$	−3746.39			−1.35492			−0.677460	−	0.735559i	$-0.736919\pi$
$197$	−3746.39			−1.35492			−0.677460	+	0.735559i	$0.736919\pi$
$198$		0			0
$199$	1432.02	−	2480.33i	0.510116	−	0.883547i	−0.489815	−	0.871826i	$-0.662936\pi$
$199$	1432.02	−	2480.33i	0.510116	−	0.883547i	0.999931	−	0.0117211i	$-0.00373101\pi$
$200$	−421.447	+	729.968i	−0.149004	+	0.258083i
$201$		0			0
$202$	−891.284			−0.310448
$203$	−1499.37	−	1867.21i	−0.518401	−	0.645579i
$204$		0			0
$205$	−809.742	−	1402.51i	−0.275877	−	0.477833i
$206$	−846.189	+	1465.64i	−0.286198	+	0.495709i
$207$		0			0
$208$	−299.891	−	519.426i	−0.0999696	−	0.173152i
$209$	−5356.25			−1.77273
$210$		0			0
$211$	3729.13			1.21670			0.608350	−	0.793669i	$-0.291832\pi$
$211$	3729.13			1.21670			0.608350	+	0.793669i	$0.291832\pi$
$212$	−1203.64	−	2084.77i	−0.389937	−	0.675391i
$213$		0			0
$214$	562.311	−	973.952i	0.179621	−	0.311112i
$215$	−126.860	−	219.728i	−0.0402408	−	0.0696992i
$216$		0			0
$217$	−4476.81	+	690.207i	−1.40049	+	0.215918i
$218$	2231.27			0.693214
$219$		0			0
$220$	−351.318	+	608.500i	−0.107663	+	0.186478i
$221$	2137.14	−	3701.64i	0.650496	−	1.12669i
$222$		0			0
$223$	152.890			0.0459115			0.0229557	−	0.999736i	$-0.492692\pi$
$223$	152.890			0.0459115			0.0229557	+	0.999736i	$0.492692\pi$
$224$	−585.728	+	90.3039i	−0.174712	+	0.0269361i
$225$		0			0
$226$	1518.31	+	2629.80i	0.446888	+	0.774033i
$227$	−1693.04	+	2932.43i	−0.495027	+	0.857411i	−0.999984	−	0.00573340i	$-0.998175\pi$
$227$	−1693.04	+	2932.43i	−0.495027	+	0.857411i	0.504957	+	0.863144i	$0.331508\pi$
$228$		0			0
$229$	386.178	+	668.880i	0.111438	+	0.193017i	0.916350	−	0.400377i	$-0.131121\pi$
$229$	386.178	+	668.880i	0.111438	+	0.193017i	−0.804912	+	0.593394i	$0.797788\pi$
$230$	1705.22			0.488865
$231$		0			0
$232$	−1034.41			−0.292727
$233$	1781.93	+	3086.39i	0.501021	+	0.867794i	0.999999	+	0.00117935i	$0.000375398\pi$
$233$	1781.93	+	3086.39i	0.501021	+	0.867794i	−0.498978	+	0.866614i	$0.666291\pi$
$234$		0			0
$235$	764.413	−	1324.00i	0.212191	−	0.367525i
$236$	63.0523	+	109.210i	0.0173913	+	0.0301227i
$237$		0			0
$238$	−2644.39	−	3293.13i	−0.720211	−	0.896899i
$239$	5832.92			1.57866			0.789331	−	0.613968i	$-0.210428\pi$
$239$	5832.92			1.57866			0.789331	+	0.613968i	$0.210428\pi$
$240$		0			0
$241$	−1846.61	+	3198.42i	−0.493570	+	0.854889i	−0.999973	−	0.00740841i	$-0.997642\pi$
$241$	−1846.61	+	3198.42i	−0.493570	+	0.854889i	0.506402	+	0.862297i	$0.330975\pi$
$242$	240.225	−	416.082i	0.0638110	−	0.110524i
$243$		0			0
$244$	3450.08			0.905199
$245$	1449.42	−	457.808i	0.377960	−	0.119381i
$246$		0			0
$247$	2532.71	+	4386.78i	0.652438	+	1.13006i
$248$	−978.324	+	1694.51i	−0.250499	+	0.433876i
$249$		0			0
$250$	−1020.85	−	1768.16i	−0.258257	−	0.447313i
$251$	−1571.78			−0.395259			−0.197630	−	0.980277i	$-0.563324\pi$
$251$	−1571.78			−0.395259			−0.197630	+	0.980277i	$0.563324\pi$
$252$		0			0
$253$	−7626.39			−1.89513
$254$	−803.079	−	1390.97i	−0.198384	−	0.343612i
$255$		0			0
$256$	−128.000	+	221.703i	−0.0312500	+	0.0541266i
$257$	1826.14	+	3162.97i	0.443236	+	0.767707i	0.997927	−	0.0643490i	$-0.0204971\pi$
$257$	1826.14	+	3162.97i	0.443236	+	0.767707i	−0.554692	+	0.832056i	$0.687164\pi$
$258$		0			0
$259$	−1833.04	+	4717.19i	−0.439767	+	1.13171i
$260$	664.483			0.158498
$261$		0			0
$262$	890.214	−	1541.90i	0.209914	−	0.363582i
$263$	−355.268	+	615.342i	−0.0832956	+	0.144272i	−0.904664	−	0.426126i	$-0.859878\pi$
$263$	−355.268	+	615.342i	−0.0832956	+	0.144272i	0.821368	+	0.570399i	$0.193211\pi$
$264$		0			0
$265$	2666.98			0.618230
$266$	4946.72	−	762.655i	1.14024	−	0.175795i
$267$		0			0
$268$	1365.18	+	2364.56i	0.311163	+	0.538950i
$269$	4.13321	−	7.15894i	0.000936827	−	0.00162263i	−0.865557	−	0.500811i	$-0.833035\pi$
$269$	4.13321	−	7.15894i	0.000936827	−	0.00162263i	0.866493	+	0.499188i	$0.166368\pi$
$270$		0			0
$271$	1361.85	+	2358.79i	0.305264	+	0.528732i	0.977320	−	0.211768i	$-0.0679221\pi$
$271$	1361.85	+	2358.79i	0.305264	+	0.528732i	−0.672056	+	0.740500i	$0.734589\pi$
$272$	−1824.36			−0.406684
$273$		0			0
$274$	−4089.13			−0.901581
$275$	2088.20	+	3616.87i	0.457903	+	0.793111i
$276$		0			0
$277$	−620.284	+	1074.36i	−0.134546	+	0.233041i	−0.925424	−	0.378933i	$-0.876291\pi$
$277$	−620.284	+	1074.36i	−0.134546	+	0.233041i	0.790878	+	0.611974i	$0.209624\pi$
$278$	587.471	+	1017.53i	0.126742	+	0.219523i
$279$		0			0
$280$	237.815	−	611.998i	0.0507577	−	0.130621i
$281$	7346.62			1.55965			0.779826	−	0.625996i	$-0.215307\pi$
$281$	7346.62			1.55965			0.779826	+	0.625996i	$0.215307\pi$
$282$		0			0
$283$	−2058.87	+	3566.06i	−0.432462	+	0.749047i	−0.997085	−	0.0763025i	$-0.975689\pi$
$283$	−2058.87	+	3566.06i	−0.432462	+	0.749047i	0.564622	+	0.825349i	$0.309022\pi$
$284$	−1329.72	+	2303.14i	−0.277832	+	0.481220i
$285$		0			0
$286$	−2971.82			−0.614431
$287$	−4237.71	−	5277.34i	−0.871582	−	1.08541i
$288$		0			0
$289$	−4044.05	−	7004.50i	−0.823132	−	1.42571i
$290$	573.001	−	992.467i	0.116027	−	0.200964i
$291$		0			0
$292$	−1884.49	−	3264.03i	−0.377676	−	0.654154i
$293$	−2410.10			−0.480545			−0.240273	−	0.970705i	$-0.577237\pi$
$293$	−2410.10			−0.480545			−0.240273	+	0.970705i	$0.577237\pi$
$294$		0			0
$295$	−139.708			−0.0275733
$296$	1093.03	+	1893.19i	0.214633	+	0.371755i
$297$		0			0
$298$	2713.32	−	4699.60i	0.527443	−	0.913559i
$299$	3606.14	+	6246.02i	0.697487	+	1.20808i
$300$		0			0
$301$	−663.910	−	826.786i	−0.127133	−	0.158323i
$302$	332.864			0.0634244
$303$		0			0
$304$	1081.01	−	1872.37i	0.203949	−	0.353250i
$305$	−1911.13	+	3310.17i	−0.358789	+	0.621442i
$306$		0			0
$307$	−5172.84			−0.961661			−0.480830	−	0.876814i	$-0.659665\pi$
$307$	−5172.84			−0.961661			−0.480830	+	0.876814i	$0.659665\pi$
$308$	−1063.60	+	2737.09i	−0.196767	+	0.506364i
$309$		0			0
$310$	−1083.86	−	1877.30i	−0.198578	−	0.343947i
$311$	−1714.23	+	2969.14i	−0.312557	+	0.541365i	−0.978915	−	0.204267i	$-0.934519\pi$
$311$	−1714.23	+	2969.14i	−0.312557	+	0.541365i	0.666358	+	0.745632i	$0.267852\pi$
$312$		0			0
$313$	2292.03	+	3969.91i	0.413908	+	0.716909i	0.995313	−	0.0967050i	$-0.0308303\pi$
$313$	2292.03	+	3969.91i	0.413908	+	0.716909i	−0.581406	+	0.813614i	$0.697497\pi$
$314$	1071.14			0.192510
$315$		0			0
$316$	−13.3017			−0.00236797
$317$	−74.0671	−	128.288i	−0.0131231	−	0.0227299i	0.859389	−	0.511322i	$-0.170844\pi$
$317$	−74.0671	−	128.288i	−0.0131231	−	0.0227299i	−0.872512	+	0.488592i	$0.837511\pi$
$318$		0			0
$319$	−2562.68	+	4438.69i	−0.449788	+	0.779055i
$320$	−141.808	−	245.619i	−0.0247728	−	0.0429078i
$321$		0			0
$322$	7043.29	−	1085.89i	1.21897	−	0.187933i
$323$	15407.5			2.65417
$324$		0			0
$325$	1974.81	−	3420.48i	0.337055	−	0.583797i
$326$	1202.76	−	2083.24i	0.204339	−	0.353926i
$327$		0			0
$328$	−2923.59			−0.492159
$329$	2314.23	−	5955.48i	0.387804	−	0.997983i
$330$		0			0
$331$	3893.30	+	6743.40i	0.646512	+	1.11979i	0.983950	+	0.178444i	$0.0571062\pi$
$331$	3893.30	+	6743.40i	0.646512	+	1.11979i	−0.337438	+	0.941348i	$0.609560\pi$
$332$	108.180	−	187.373i	0.0178830	−	0.0309742i
$333$		0			0
$334$	3467.14	+	6005.26i	0.568004	+	0.983811i
$335$	−3024.90			−0.493337
$336$		0			0
$337$	906.591			0.146544			0.0732718	−	0.997312i	$-0.476656\pi$
$337$	906.591			0.146544			0.0732718	+	0.997312i	$0.476656\pi$
$338$	−791.774	−	1371.39i	−0.127417	−	0.220692i
$339$		0			0
$340$	1010.58	−	1750.38i	0.161195	−	0.279199i
$341$	4847.43	+	8396.00i	0.769804	+	1.33334i
$342$		0			0
$343$	5695.20	−	2813.94i	0.896537	−	0.442969i
$344$	−458.030			−0.0717887
$345$		0			0
$346$	416.192	−	720.866i	0.0646666	−	0.112006i
$347$	−2098.16	+	3634.12i	−0.324597	+	0.562219i	−0.981431	−	0.191817i	$-0.938562\pi$
$347$	−2098.16	+	3634.12i	−0.324597	+	0.562219i	0.656834	+	0.754036i	$0.271895\pi$
$348$		0			0
$349$	−5970.17			−0.915690			−0.457845	−	0.889032i	$-0.651379\pi$
$349$	−5970.17			−0.915690			−0.457845	+	0.889032i	$0.651379\pi$
$350$	−2443.53	−	3043.00i	−0.373178	−	0.464729i
$351$		0			0
$352$	634.219	+	1098.50i	0.0960340	+	0.166336i
$353$	−5523.89	+	9567.65i	−0.832880	+	1.44259i	0.0628644	+	0.998022i	$0.479976\pi$
$353$	−5523.89	+	9567.65i	−0.832880	+	1.44259i	−0.895745	+	0.444569i	$0.853357\pi$
$354$		0			0
$355$	−1473.16	−	2551.60i	−0.220246	−	0.381478i
$356$	−1141.60			−0.169957
$357$		0			0
$358$	−751.268			−0.110910
$359$	−3781.82	−	6550.31i	−0.555980	−	0.962986i	−0.997827	−	0.0658953i	$-0.979010\pi$
$359$	−3781.82	−	6550.31i	−0.555980	−	0.962986i	0.441846	−	0.897091i	$-0.354324\pi$
$360$		0			0
$361$	−5700.14	+	9872.93i	−0.831045	+	1.43941i
$362$	2201.29	+	3812.74i	0.319605	+	0.553573i
$363$		0			0
$364$	2744.60	−	423.145i	0.395209	−	0.0609309i
$365$	4175.55			0.598790
$366$		0			0
$367$	1716.04	−	2972.26i	0.244077	−	0.422754i	−0.717794	−	0.696255i	$-0.754848\pi$
$367$	1716.04	−	2972.26i	0.244077	−	0.422754i	0.961872	+	0.273501i	$0.0881816\pi$
$368$	1539.18	−	2665.94i	0.218031	−	0.377641i
$369$		0			0
$370$	−2421.89			−0.340292
$371$	11015.8	−	1698.34i	1.54153	−	0.237664i
$372$		0			0
$373$	2636.84	+	4567.15i	0.366034	+	0.633989i	0.988941	−	0.148306i	$-0.0473822\pi$
$373$	2636.84	+	4567.15i	0.366034	+	0.633989i	−0.622908	+	0.782295i	$0.714049\pi$
$374$	−4519.70	+	7828.34i	−0.624887	+	1.08234i
$375$		0			0
$376$	−1379.96	−	2390.17i	−0.189272	−	0.327828i
$377$	4847.05			0.662164
$378$		0			0
$379$	8856.92			1.20039			0.600197	−	0.799852i	$-0.295089\pi$
$379$	8856.92			1.20039			0.600197	+	0.799852i	$0.295089\pi$
$380$	1197.63	+	2074.36i	0.161677	+	0.280032i
$381$		0			0
$382$	4353.28	−	7540.10i	0.583071	−	1.00991i
$383$	−520.852	−	902.142i	−0.0694890	−	0.120358i	0.829187	−	0.558971i	$-0.188804\pi$
$383$	−520.852	−	902.142i	−0.0694890	−	0.120358i	−0.898676	+	0.438612i	$0.855470\pi$
$384$		0			0
$385$	−2036.93	−	2536.64i	−0.269640	−	0.335790i
$386$	4452.84			0.587160
$387$		0			0
$388$	−480.618	+	832.454i	−0.0628857	+	0.108921i
$389$	4666.34	−	8082.33i	0.608207	−	1.05345i	−0.383328	−	0.923612i	$-0.625222\pi$
$389$	4666.34	−	8082.33i	0.608207	−	1.05345i	0.991536	−	0.129834i	$-0.0414445\pi$
$390$		0			0
$391$	21937.6			2.83743
$392$	592.555	−	2679.26i	0.0763483	−	0.345211i
$393$		0			0
$394$	−3746.39	−	6488.94i	−0.479037	−	0.829716i
$395$	7.36831	−	12.7623i	0.000938582	−	0.00162567i
$396$		0			0
$397$	−3411.16	−	5908.31i	−0.431238	−	0.746926i	0.565742	−	0.824582i	$-0.308590\pi$
$397$	−3411.16	−	5908.31i	−0.431238	−	0.746926i	−0.996980	+	0.0776562i	$0.975256\pi$
$398$	5728.08			0.721414
$399$		0			0
$400$	−1685.79			−0.210724
$401$	−4054.97	−	7023.42i	−0.504977	−	0.874646i	−0.999983	−	0.00575638i	$-0.998168\pi$
$401$	−4054.97	−	7023.42i	−0.504977	−	0.874646i	0.495007	−	0.868889i	$-0.335166\pi$
$402$		0			0
$403$	4584.22	−	7940.11i	0.566641	−	0.981452i
$404$	−891.284	−	1543.75i	−0.109760	−	0.190110i
$405$		0			0
$406$	1734.73	−	4464.20i	0.212053	−	0.545701i
$407$	10831.6			1.31917
$408$		0			0
$409$	−2216.52	+	3839.13i	−0.267970	+	0.464138i	−0.968338	−	0.249644i	$-0.919686\pi$
$409$	−2216.52	+	3839.13i	−0.267970	+	0.464138i	0.700367	+	0.713783i	$0.253020\pi$
$410$	1619.48	−	2805.03i	0.195075	−	0.337879i
$411$		0			0
$412$	−3384.76			−0.404745
$413$	−577.055	+	88.9667i	−0.0687530	+	0.0105999i
$414$		0			0
$415$	119.850	+	207.586i	0.0141764	+	0.0245542i
$416$	599.781	−	1038.85i	0.0706892	−	0.122437i
$417$		0			0
$418$	−5356.25	−	9277.30i	−0.626753	−	1.08557i
$419$	−6782.36			−0.790787			−0.395394	−	0.918512i	$-0.629392\pi$
$419$	−6782.36			−0.790787			−0.395394	+	0.918512i	$0.629392\pi$
$420$		0			0
$421$	−4798.62			−0.555512			−0.277756	−	0.960652i	$-0.589591\pi$
$421$	−4798.62			−0.555512			−0.277756	+	0.960652i	$0.589591\pi$
$422$	3729.13	+	6459.04i	0.430168	+	0.745073i
$423$		0			0
$424$	2407.29	−	4169.55i	0.275727	−	0.477573i
$425$	−6006.80	−	10404.1i	−0.685583	−	1.18746i
$426$		0			0
$427$	−5785.85	+	14889.4i	−0.655730	+	1.68747i
$428$	2249.25			0.254022
$429$		0			0
$430$	253.720	−	439.456i	0.0284546	−	0.0492848i
$431$	−6474.64	+	11214.4i	−0.723602	+	1.25332i	0.235945	+	0.971766i	$0.424182\pi$
$431$	−6474.64	+	11214.4i	−0.723602	+	1.25332i	−0.959547	+	0.281549i	$0.909152\pi$
$432$		0			0
$433$	937.240			0.104020			0.0520102	−	0.998647i	$-0.483437\pi$
$433$	937.240			0.104020			0.0520102	+	0.998647i	$0.483437\pi$
$434$	−5672.28	−	7063.86i	−0.627369	−	0.781281i
$435$		0			0
$436$	2231.27	+	3864.67i	0.245088	+	0.424505i
$437$	−12999.1	+	22515.0i	−1.42295	+	2.46462i
$438$		0			0
$439$	6800.15	+	11778.2i	0.739302	+	1.28051i	0.952810	+	0.303567i	$0.0981775\pi$
$439$	6800.15	+	11778.2i	0.739302	+	1.28051i	−0.213509	+	0.976941i	$0.568489\pi$
$440$	−1405.27			−0.152258
$441$		0			0
$442$	8548.56			0.919940
$443$	−2316.24	−	4011.84i	−0.248415	−	0.430267i	0.714671	−	0.699460i	$-0.246576\pi$
$443$	−2316.24	−	4011.84i	−0.248415	−	0.430267i	−0.963086	+	0.269193i	$0.913243\pi$
$444$		0			0
$445$	632.374	−	1095.30i	0.0673650	−	0.116680i
$446$	152.890	+	264.813i	0.0162322	+	0.0281149i
$447$		0			0
$448$	−742.139	−	924.207i	−0.0782651	−	0.0974658i
$449$	−254.426			−0.0267419			−0.0133709	−	0.999911i	$-0.504256\pi$
$449$	−254.426			−0.0267419			−0.0133709	+	0.999911i	$0.504256\pi$
$450$		0			0
$451$	−7242.94	+	12545.1i	−0.756224	+	1.30982i
$452$	−3036.63	+	5259.59i	−0.315998	+	0.547324i
$453$		0			0
$454$	−6772.16			−0.700073
$455$	−1114.35	+	2867.70i	−0.114817	+	0.295472i
$456$		0			0
$457$	−8205.25	−	14211.9i	−0.839881	−	1.45472i	−0.889994	−	0.455972i	$-0.849292\pi$
$457$	−8205.25	−	14211.9i	−0.839881	−	1.45472i	0.0501133	−	0.998744i	$-0.484042\pi$
$458$	−772.356	+	1337.76i	−0.0787988	+	0.136483i
$459$		0			0
$460$	1705.22	+	2953.53i	0.172840	+	0.299367i
$461$	5842.56			0.590272			0.295136	−	0.955455i	$-0.404635\pi$
$461$	5842.56			0.590272			0.295136	+	0.955455i	$0.404635\pi$
$462$		0			0
$463$	6699.12			0.672429			0.336215	−	0.941785i	$-0.390853\pi$
$463$	6699.12			0.672429			0.336215	+	0.941785i	$0.390853\pi$
$464$	−1034.41	−	1791.66i	−0.103495	−	0.179258i
$465$		0			0
$466$	−3563.85	+	6172.77i	−0.354275	+	0.613623i
$467$	−912.817	−	1581.05i	−0.0904500	−	0.156664i	0.817251	−	0.576282i	$-0.195497\pi$
$467$	−912.817	−	1581.05i	−0.0904500	−	0.156664i	−0.907701	+	0.419619i	$0.862164\pi$
$468$		0			0
$469$	−12494.1	+	1926.27i	−1.23012	+	0.189652i
$470$	3057.65			0.300083
$471$		0			0
$472$	−126.105	+	218.420i	−0.0122975	+	0.0212999i
$473$	−1134.73	+	1965.41i	−0.110307	+	0.191057i
$474$		0			0
$475$	14237.2			1.37526
$476$	3059.48	−	7873.35i	0.294603	−	0.758139i
$477$		0			0
$478$	5832.92	+	10102.9i	0.558141	+	0.966729i
$479$	5518.07	−	9557.58i	0.526362	−	0.911685i	−0.473167	−	0.880973i	$-0.656889\pi$
$479$	5518.07	−	9557.58i	0.526362	−	0.911685i	0.999528	−	0.0307121i	$-0.00977749\pi$
$480$		0			0
$481$	−5121.73	−	8871.10i	−0.485511	−	0.840930i
$482$	−7386.43			−0.698014
$483$		0			0
$484$	960.900			0.0902423
$485$	−532.464	−	922.255i	−0.0498515	−	0.0863453i
$486$		0			0
$487$	−4092.78	+	7088.90i	−0.380824	+	0.659607i	−0.991180	−	0.132520i	$-0.957693\pi$
$487$	−4092.78	+	7088.90i	−0.380824	+	0.659607i	0.610356	+	0.792127i	$0.291026\pi$
$488$	3450.08	+	5975.71i	0.320036	+	0.554319i
$489$		0			0
$490$	2242.37	+	2052.67i	0.206735	+	0.189245i
$491$	−19006.2			−1.74692			−0.873459	−	0.486897i	$-0.838129\pi$
$491$	−19006.2			−1.74692			−0.873459	+	0.486897i	$0.838129\pi$
$492$		0			0
$493$	7371.65	−	12768.1i	0.673433	−	1.16642i
$494$	−5065.41	+	8773.55i	−0.461343	+	0.799070i
$495$		0			0
$496$	−3913.30			−0.354258
$497$	−7709.66	−	9601.07i	−0.695826	−	0.866533i
$498$		0			0
$499$	−8246.79	−	14283.9i	−0.739834	−	1.28143i	−0.952570	−	0.304320i	$-0.901571\pi$
$499$	−8246.79	−	14283.9i	−0.739834	−	1.28143i	0.212736	−	0.977110i	$-0.431763\pi$
$500$	2041.70	−	3536.32i	0.182615	−	0.316298i
$501$		0			0
$502$	−1571.78	−	2722.41i	−0.139745	−	0.242046i
$503$	−10368.0			−0.919059			−0.459530	−	0.888162i	$-0.651982\pi$
$503$	−10368.0			−0.919059			−0.459530	+	0.888162i	$0.651982\pi$
$504$		0			0
$505$	1974.86			0.174020
$506$	−7626.39	−	13209.3i	−0.670029	−	1.16052i
$507$		0			0
$508$	1606.16	−	2781.95i	0.140279	−	0.242970i
$509$	3254.37	+	5636.74i	0.283394	+	0.490853i	0.972218	−	0.234075i	$-0.0752063\pi$
$509$	3254.37	+	5636.74i	0.283394	+	0.490853i	−0.688825	+	0.724928i	$0.741873\pi$
$510$		0			0
$511$	17246.8	−	2659.01i	1.49306	−	0.230191i
$512$	−512.000			−0.0441942
$513$		0			0
$514$	−3652.28	+	6325.94i	−0.313415	+	0.542851i
$515$	1874.94	−	3247.50i	0.160427	−	0.277868i
$516$		0			0
$517$	−13675.0			−1.16330
$518$	−10003.4	+	1542.27i	−0.848506	+	0.130817i
$519$		0			0
$520$	664.483	+	1150.92i	0.0560375	+	0.0970598i
$521$	6759.54	−	11707.9i	0.568409	−	0.984513i	−0.428315	−	0.903630i	$-0.640893\pi$
$521$	6759.54	−	11707.9i	0.568409	−	0.984513i	0.996724	−	0.0808831i	$-0.0257741\pi$
$522$		0			0
$523$	−929.020	−	1609.11i	−0.0776734	−	0.134534i	0.824572	−	0.565757i	$-0.191416\pi$
$523$	−929.020	−	1609.11i	−0.0776734	−	0.134534i	−0.902246	+	0.431222i	$0.858082\pi$
$524$	3560.85			0.296864
$525$		0			0
$526$	−1421.07			−0.117798
$527$	−13943.9	−	24151.5i	−1.15257	−	1.99631i
$528$		0			0
$529$	−12424.9	+	21520.6i	−1.02120	+	1.76877i
$530$	2666.98	+	4619.34i	0.218577	+	0.378587i
$531$		0			0
$532$	6267.68	+	7805.32i	0.510787	+	0.636097i
$533$	13699.3			1.11329
$534$		0			0
$535$	−1245.94	+	2158.03i	−0.100686	+	0.174392i
$536$	−2730.36	+	4729.13i	−0.220026	+	0.381095i
$537$		0			0
$538$	16.5329			0.00132487
$539$	−10028.7	−	9180.30i	−0.801424	−	0.733624i
$540$		0			0
$541$	−6846.27	−	11858.1i	−0.544075	−	0.942365i	−0.998665	−	0.0516640i	$-0.983548\pi$
$541$	−6846.27	−	11858.1i	−0.544075	−	0.942365i	0.454590	−	0.890701i	$-0.349786\pi$
$542$	−2723.70	+	4717.58i	−0.215854	+	0.373870i
$543$		0			0
$544$	−1824.36	−	3159.88i	−0.143784	−	0.249042i
$545$	−4943.94			−0.388578
$546$		0			0
$547$	−9054.00			−0.707717			−0.353858	−	0.935299i	$-0.615131\pi$
$547$	−9054.00			−0.707717			−0.353858	+	0.935299i	$0.615131\pi$
$548$	−4089.13	−	7082.58i	−0.318757	−	0.552104i
$549$		0			0
$550$	−4176.40	+	7233.74i	−0.323786	+	0.560814i
$551$	8736.07	+	15131.3i	0.675443	+	1.16990i
$552$		0			0
$553$	22.3072	−	57.4059i	0.00171537	−	0.00441437i
$554$	−2481.14			−0.190277
$555$		0			0
$556$	−1174.94	+	2035.06i	−0.0896198	+	0.155226i
$557$	4717.22	−	8170.46i	0.358842	−	0.621533i	−0.628926	−	0.777465i	$-0.716505\pi$
$557$	4717.22	−	8170.46i	0.358842	−	0.621533i	0.987768	+	0.155933i	$0.0498383\pi$
$558$		0			0
$559$	2146.23			0.162390
$560$	1297.83	−	200.091i	0.0979343	−	0.0150989i
$561$		0			0
$562$	7346.62	+	12724.7i	0.551421	+	0.955088i
$563$	−3205.10	+	5551.39i	−0.239927	+	0.415565i	−0.960693	−	0.277613i	$-0.910457\pi$
$563$	−3205.10	+	5551.39i	−0.239927	+	0.415565i	0.720766	+	0.693178i	$0.243790\pi$
$564$		0			0
$565$	−3364.20	−	5826.97i	−0.250501	−	0.433880i
$566$	−8235.46			−0.611594
$567$		0			0
$568$	−5318.88			−0.392914
$569$	−2432.99	−	4214.06i	−0.179255	−	0.310479i	0.762370	−	0.647141i	$-0.224035\pi$
$569$	−2432.99	−	4214.06i	−0.179255	−	0.310479i	−0.941626	+	0.336662i	$0.890702\pi$
$570$		0			0
$571$	−9349.38	+	16193.6i	−0.685218	+	1.18683i	0.288150	+	0.957585i	$0.406960\pi$
$571$	−9349.38	+	16193.6i	−0.685218	+	1.18683i	−0.973368	+	0.229247i	$0.926374\pi$
$572$	−2971.82	−	5147.34i	−0.217234	−	0.376261i
$573$		0			0
$574$	4902.91	−	12617.3i	0.356522	−	0.917482i
$575$	20271.4			1.47022
$576$		0			0
$577$	9174.11	−	15890.0i	0.661912	−	1.14647i	−0.318200	−	0.948023i	$-0.603079\pi$
$577$	9174.11	−	15890.0i	0.661912	−	1.14647i	0.980113	−	0.198442i	$-0.0635882\pi$
$578$	8088.10	−	14009.0i	0.582042	−	1.00813i
$579$		0			0
$580$	2292.00			0.164087
$581$	627.223	+	781.099i	0.0447876	+	0.0557753i
$582$		0			0
$583$	−11927.7	−	20659.4i	−0.847334	−	1.46763i
$584$	3768.98	−	6528.06i	0.267057	−	0.462556i
$585$		0			0
$586$	−2410.10	−	4174.42i	−0.169898	−	0.294273i
$587$	−17246.6			−1.21268			−0.606340	−	0.795206i	$-0.707363\pi$
$587$	−17246.6			−1.21268			−0.606340	+	0.795206i	$0.707363\pi$
$588$		0			0
$589$	33049.5			2.31202
$590$	−139.708	−	241.982i	−0.00974863	−	0.0168851i
$591$		0			0
$592$	−2186.07	+	3786.38i	−0.151768	+	0.262870i
$593$	−8806.40	−	15253.1i	−0.609841	−	1.05628i	−0.991266	−	0.131875i	$-0.957900\pi$
$593$	−8806.40	−	15253.1i	−0.609841	−	1.05628i	0.381426	−	0.924400i	$-0.375433\pi$
$594$		0			0
$595$	5859.30	+	7296.76i	0.403711	+	0.502753i
$596$	10853.3			0.745918
$597$		0			0
$598$	−7212.29	+	12492.0i	−0.493198	+	0.854244i
$599$	−6533.22	+	11315.9i	−0.445643	+	0.771876i	−0.998097	−	0.0616672i	$-0.980358\pi$
$599$	−6533.22	+	11315.9i	−0.445643	+	0.771876i	0.552454	+	0.833544i	$0.313692\pi$
$600$		0			0
$601$	−10232.8			−0.694516			−0.347258	−	0.937770i	$-0.612887\pi$
$601$	−10232.8			−0.694516			−0.347258	+	0.937770i	$0.612887\pi$
$602$	768.125	−	1976.71i	0.0520041	−	0.133828i
$603$		0			0
$604$	332.864	+	576.537i	0.0224239	+	0.0388393i
$605$	−532.279	+	921.934i	−0.0357689	+	0.0619536i
$606$		0			0
$607$	5116.84	+	8862.63i	0.342152	+	0.592625i	0.984832	−	0.173510i	$-0.0555109\pi$
$607$	5116.84	+	8862.63i	0.342152	+	0.592625i	−0.642680	+	0.766135i	$0.722178\pi$
$608$	4324.06			0.288427
$609$		0			0
$610$	−7644.51			−0.507405
$611$	6466.22	+	11199.8i	0.428143	+	0.741565i
$612$		0			0
$613$	1435.08	−	2485.63i	0.0945550	−	0.163774i	−0.814868	−	0.579647i	$-0.803191\pi$
$613$	1435.08	−	2485.63i	0.0945550	−	0.163774i	0.909423	+	0.415873i	$0.136524\pi$
$614$	−5172.84	−	8959.63i	−0.339998	−	0.588894i
$615$		0			0
$616$	−5804.37	+	894.882i	−0.379651	+	0.0585322i
$617$	743.774			0.0485304			0.0242652	−	0.999706i	$-0.492275\pi$
$617$	743.774			0.0485304			0.0242652	+	0.999706i	$0.492275\pi$
$618$		0			0
$619$	11435.6	−	19807.0i	0.742543	−	1.28612i	−0.208790	−	0.977960i	$-0.566953\pi$
$619$	11435.6	−	19807.0i	0.742543	−	1.28612i	0.951334	−	0.308162i	$-0.0997140\pi$
$620$	2167.72	−	3754.60i	0.140416	−	0.243207i
$621$		0			0
$622$	−6856.94			−0.442023
$623$	1914.48	−	4926.78i	0.123117	−	0.316833i
$624$		0			0
$625$	−4323.17	−	7487.95i	−0.276683	−	0.479228i
$626$	−4584.06	+	7939.82i	−0.292677	+	0.506931i
$627$		0			0
$628$	1071.14	+	1855.28i	0.0680626	+	0.117888i
$629$	−31157.6			−1.97509
$630$		0			0
$631$	−16861.8			−1.06380			−0.531901	−	0.846807i	$-0.678522\pi$
$631$	−16861.8			−1.06380			−0.531901	+	0.846807i	$0.678522\pi$
$632$	−13.3017	−	23.0392i	−0.000837204	−	0.00145008i
$633$		0			0
$634$	148.134	−	256.576i	0.00927943	−	0.0160724i
$635$	1779.42	+	3082.05i	0.111203	+	0.192610i
$636$		0			0
$637$	−2776.59	+	12554.4i	−0.172704	+	0.780887i
$638$	−10250.7			−0.636096
$639$		0			0
$640$	283.616	−	491.237i	0.0175170	−	0.0303404i
$641$	−4259.46	+	7377.61i	−0.262463	+	0.454599i	−0.966896	−	0.255172i	$-0.917868\pi$
$641$	−4259.46	+	7377.61i	−0.262463	+	0.454599i	0.704433	+	0.709771i	$0.251201\pi$
$642$		0			0
$643$	−12045.2			−0.738748			−0.369374	−	0.929281i	$-0.620428\pi$
$643$	−12045.2			−0.738748			−0.369374	+	0.929281i	$0.620428\pi$
$644$	8924.11	+	11113.5i	0.546055	+	0.680018i
$645$		0			0
$646$	15407.5	+	26686.5i	0.938389	+	1.62534i
$647$	−2078.82	+	3600.63i	−0.126317	+	0.218787i	−0.922247	−	0.386601i	$-0.873649\pi$
$647$	−2078.82	+	3600.63i	−0.126317	+	0.218787i	0.795930	+	0.605389i	$0.206982\pi$
$648$		0			0
$649$	624.828	+	1082.23i	0.0377914	+	0.0654566i
$650$	7899.26			0.476668
$651$		0			0
$652$	4811.03			0.288979
$653$	−15174.8	−	26283.5i	−0.909394	−	1.57512i	−0.814907	−	0.579591i	$-0.803212\pi$
$653$	−15174.8	−	26283.5i	−0.909394	−	1.57512i	−0.0944872	−	0.995526i	$-0.530121\pi$
$654$		0			0
$655$	−1972.49	+	3416.45i	−0.117667	+	0.203804i
$656$	−2923.59	−	5063.80i	−0.174004	−	0.301384i
$657$		0			0
$658$	12629.4	−	1947.13i	0.748246	−	0.115360i
$659$	−14482.7			−0.856093			−0.428046	−	0.903757i	$-0.640798\pi$
$659$	−14482.7			−0.856093			−0.428046	+	0.903757i	$0.640798\pi$
$660$		0			0
$661$	3262.70	−	5651.16i	0.191988	−	0.332533i	−0.753921	−	0.656965i	$-0.771840\pi$
$661$	3262.70	−	5651.16i	0.191988	−	0.332533i	0.945909	+	0.324432i	$0.105173\pi$
$662$	−7786.61	+	13486.8i	−0.457153	+	0.791812i
$663$		0			0
$664$	432.720			0.0252903
$665$	−10960.7	+	1689.85i	−0.639155	+	0.0985409i
$666$		0			0
$667$	12438.7	+	21544.4i	0.722081	+	1.25068i
$668$	−6934.27	+	12010.5i	−0.401639	+	0.695660i
$669$		0			0
$670$	−3024.90	−	5239.28i	−0.174421	−	0.302106i
$671$	34189.1			1.96700
$672$		0			0
$673$	1453.70			0.0832629			0.0416314	−	0.999133i	$-0.486744\pi$
$673$	1453.70			0.0832629			0.0416314	+	0.999133i	$0.486744\pi$
$674$	906.591	+	1570.26i	0.0518110	+	0.0897392i
$675$		0			0
$676$	1583.55	−	2742.79i	0.0900972	−	0.156053i
$677$	7385.35	+	12791.8i	0.419265	+	0.726187i	0.995866	−	0.0908383i	$-0.0289546\pi$
$677$	7385.35	+	12791.8i	0.419265	+	0.726187i	−0.576601	+	0.817026i	$0.695621\pi$
$678$		0			0
$679$	−2786.60	−	3470.23i	−0.157496	−	0.196135i
$680$	4042.32			0.227965
$681$		0			0
$682$	−9694.87	+	16792.0i	−0.544334	+	0.942814i
$683$	4539.15	−	7862.04i	0.254298	−	0.440457i	−0.710406	−	0.703792i	$-0.751489\pi$
$683$	4539.15	−	7862.04i	0.254298	−	0.440457i	0.964705	+	0.263334i	$0.0848222\pi$
$684$		0			0
$685$	9060.49			0.505377
$686$	10569.1	+	7050.44i	0.588236	+	0.392401i
$687$		0			0
$688$	−458.030	−	793.331i	−0.0253811	−	0.0439614i
$689$	−11280.1	+	19537.6i	−0.623710	+	1.08030i
$690$		0			0
$691$	−2234.84	−	3870.86i	−0.123035	−	0.213104i	0.797928	−	0.602753i	$-0.205930\pi$
$691$	−2234.84	−	3870.86i	−0.123035	−	0.213104i	−0.920963	+	0.389649i	$0.872596\pi$
$692$	1664.77			0.0914524
$693$		0			0
$694$	−8392.65			−0.459050
$695$	−1301.69	−	2254.59i	−0.0710444	−	0.123053i
$696$		0			0
$697$	20834.6	−	36086.6i	1.13224	−	1.96109i
$698$	−5970.17	−	10340.6i	−0.323745	−	0.560743i
$699$		0			0
$700$	2827.10	−	7275.33i	0.152649	−	0.392831i
$701$	29024.1			1.56380			0.781901	−	0.623403i	$-0.214250\pi$
$701$	29024.1			1.56380			0.781901	+	0.623403i	$0.214250\pi$
$702$		0			0
$703$	18462.3	−	31977.6i	0.990496	−	1.71559i
$704$	−1268.44	+	2197.00i	−0.0679063	+	0.117617i
$705$		0			0
$706$	−22095.5			−1.17787
$707$	8157.03	−	1257.60i	0.433913	−	0.0668980i
$708$		0			0
$709$	33.0540	+	57.2511i	0.00175087	+	0.00303260i	0.866900	−	0.498483i	$-0.166109\pi$
$709$	33.0540	+	57.2511i	0.00175087	+	0.00303260i	−0.865149	+	0.501516i	$0.832776\pi$
$710$	2946.33	−	5103.19i	0.155738	−	0.269746i
$711$		0			0
$712$	−1141.60	−	1977.31i	−0.0600888	−	0.104077i
$713$	47056.8			2.47166
$714$		0			0
$715$	6584.81			0.344417
$716$	−751.268	−	1301.23i	−0.0392126	−	0.0679182i
$717$		0			0
$718$	7563.64	−	13100.6i	0.393137	−	0.680934i
$719$	14833.0	+	25691.4i	0.769368	+	1.33259i	0.937906	+	0.346890i	$0.112762\pi$
$719$	14833.0	+	25691.4i	0.769368	+	1.33259i	−0.168537	+	0.985695i	$0.553904\pi$
$720$		0			0
$721$	5676.30	−	14607.5i	0.293199	−	0.754525i
$722$	−22800.5			−1.17527
$723$		0			0
$724$	−4402.58	+	7625.49i	−0.225995	+	0.391435i
$725$	6811.74	−	11798.3i	0.348940	−	0.604382i
$726$		0			0
$727$	23619.7			1.20496			0.602481	−	0.798133i	$-0.294179\pi$
$727$	23619.7			1.20496			0.602481	+	0.798133i	$0.294179\pi$
$728$	3477.51	+	4330.64i	0.177040	+	0.220473i
$729$		0			0
$730$	4175.55	+	7232.27i	0.211704	+	0.366683i
$731$	3264.10	−	5653.59i	0.165154	−	0.286054i
$732$		0			0
$733$	−10767.9	−	18650.5i	−0.542593	−	0.939798i	−0.998754	−	0.0499012i	$-0.984109\pi$
$733$	−10767.9	−	18650.5i	−0.542593	−	0.939798i	0.456161	−	0.889897i	$-0.349224\pi$
$734$	6864.15			0.345177
$735$		0			0
$736$	6156.73			0.308342
$737$	13528.5	+	23432.0i	0.676158	+	1.17114i
$738$		0			0
$739$	−5373.78	+	9307.66i	−0.267494	+	0.463312i	−0.968214	−	0.250124i	$-0.919529\pi$
$739$	−5373.78	+	9307.66i	−0.267494	+	0.463312i	0.700720	+	0.713436i	$0.252862\pi$
$740$	−2421.89	−	4194.84i	−0.120311	−	0.208386i
$741$		0			0
$742$	13957.4	+	17381.5i	0.690554	+	0.859966i
$743$	−19106.9			−0.943426			−0.471713	−	0.881752i	$-0.656364\pi$
$743$	−19106.9			−0.943426			−0.471713	+	0.881752i	$0.656364\pi$
$744$		0			0
$745$	−6012.03	+	10413.1i	−0.295656	+	0.512091i
$746$	−5273.69	+	9134.29i	−0.258825	+	0.448298i
$747$		0			0
$748$	−18078.8			−0.883724
$749$	−3772.03	+	9707.03i	−0.184015	+	0.473547i
$750$		0			0
$751$	2371.18	+	4107.00i	0.115214	+	0.199556i	0.917865	−	0.396892i	$-0.129911\pi$
$751$	2371.18	+	4107.00i	0.115214	+	0.199556i	−0.802651	+	0.596448i	$0.796578\pi$
$752$	2759.93	−	4780.33i	0.133835	−	0.231810i
$753$		0			0
$754$	4847.05	+	8395.34i	0.234110	+	0.405491i
$755$	−737.543			−0.0355522
$756$		0			0
$757$	−3717.28			−0.178476			−0.0892382	−	0.996010i	$-0.528443\pi$
$757$	−3717.28			−0.178476			−0.0892382	+	0.996010i	$0.528443\pi$
$758$	8856.92	+	15340.6i	0.424403	+	0.735088i
$759$		0			0
$760$	−2395.26	+	4148.71i	−0.114323	+	0.198013i
$761$	2047.43	+	3546.26i	0.0975288	+	0.168925i	0.910661	−	0.413154i	$-0.135573\pi$
$761$	2047.43	+	3546.26i	0.0975288	+	0.168925i	−0.813132	+	0.582079i	$0.802240\pi$
$762$		0			0
$763$	−20420.6	+	3148.32i	−0.968905	+	0.149380i
$764$	17413.1			0.824587
$765$		0			0
$766$	1041.70	−	1804.28i	0.0491361	−	0.0851063i
$767$	590.900	−	1023.47i	0.0278177	−	0.0481816i
$768$		0			0
$769$	16871.2			0.791145			0.395573	−	0.918435i	$-0.370546\pi$
$769$	16871.2			0.791145			0.395573	+	0.918435i	$0.370546\pi$
$770$	2356.67	−	6064.70i	0.110297	−	0.283840i
$771$		0			0
$772$	4452.84	+	7712.54i	0.207592	+	0.359560i
$773$	14015.1	−	24274.9i	0.652121	−	1.12951i	−0.330487	−	0.943811i	$-0.607213\pi$
$773$	14015.1	−	24274.9i	0.652121	−	1.12951i	0.982607	−	0.185696i	$-0.0594538\pi$
$774$		0			0
$775$	−12884.7	−	22317.0i	−0.597205	−	1.03439i
$776$	−1922.47			−0.0889339
$777$		0			0
$778$	18665.4			0.860135
$779$	24690.9	+	42766.0i	1.13562	+	1.96694i
$780$		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:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	378.g (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(-2.21575 - 3.83779i) q^{5} +(-11.5959 - 14.4407i) q^{7} -8.00000 q^{8} +O(q^{10})\)	\(q+(1.00000 + 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(-2.21575 - 3.83779i) q^{5} +(-11.5959 - 14.4407i) q^{7} -8.00000 q^{8} +(4.43150 - 7.67558i) q^{10} +(-19.8193 + 34.3281i) q^{11} +37.4863 q^{13} +(13.4162 - 34.5255i) q^{14} +(-8.00000 - 13.8564i) q^{16} +(57.0112 - 98.7463i) q^{17} +(67.5634 + 117.023i) q^{19} +17.7260 q^{20} -79.2774 q^{22} +(96.1989 + 166.621i) q^{23} +(52.6809 - 91.2460i) q^{25} +(37.4863 + 64.9282i) q^{26} +(73.2160 - 11.2880i) q^{28} +129.302 q^{29} +(122.291 - 211.813i) q^{31} +(16.0000 - 27.7128i) q^{32} +228.045 q^{34} +(-29.7269 + 76.4998i) q^{35} +(-136.629 - 236.649i) q^{37} +(-135.127 + 234.047i) q^{38} +(17.7260 + 30.7023i) q^{40} +365.448 q^{41} +57.2537 q^{43} +(-79.2774 - 137.312i) q^{44} +(-192.398 + 333.243i) q^{46} +(172.495 + 298.771i) q^{47} +(-74.0693 + 334.907i) q^{49} +210.724 q^{50} +(-74.9727 + 129.856i) q^{52} +(-300.911 + 521.193i) q^{53} +175.659 q^{55} +(92.7674 + 115.526i) q^{56} +(129.302 + 223.957i) q^{58} +(15.7631 - 27.3024i) q^{59} +(-431.259 - 746.963i) q^{61} +489.162 q^{62} +64.0000 q^{64} +(-83.0604 - 143.865i) q^{65} +(341.295 - 591.141i) q^{67} +(228.045 + 394.985i) q^{68} +(-162.228 + 25.0114i) q^{70} +664.860 q^{71} +(-471.122 + 816.007i) q^{73} +(273.259 - 473.298i) q^{74} -540.507 q^{76} +(725.546 - 111.860i) q^{77} +(1.66271 + 2.87990i) q^{79} +(-35.4520 + 61.4047i) q^{80} +(365.448 + 632.975i) q^{82} -54.0900 q^{83} -505.290 q^{85} +(57.2537 + 99.1664i) q^{86} +(158.555 - 274.625i) q^{88} +(142.700 + 247.163i) q^{89} +(-434.689 - 541.330i) q^{91} -769.591 q^{92} +(-344.991 + 597.542i) q^{94} +(299.407 - 518.589i) q^{95} +240.309 q^{97} +(-654.145 + 206.615i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} - 16 q^{4} - 4 q^{5} + 25 q^{7} - 64 q^{8}+O(q^{10}) \) 8 * q + 8 * q^2 - 16 * q^4 - 4 * q^5 + 25 * q^7 - 64 * q^8	\( 8 q + 8 q^{2} - 16 q^{4} - 4 q^{5} + 25 q^{7} - 64 q^{8} + 8 q^{10} - 56 q^{11} - 18 q^{13} + 22 q^{14} - 64 q^{16} + 118 q^{17} + 37 q^{19} + 32 q^{20} - 224 q^{22} - 200 q^{23} - 104 q^{25} - 18 q^{26} - 56 q^{28} + 524 q^{29} + 276 q^{31} + 128 q^{32} + 472 q^{34} - 290 q^{35} - 185 q^{37} - 74 q^{38} + 32 q^{40} - 60 q^{41} - 1556 q^{43} - 224 q^{44} + 400 q^{46} - 30 q^{47} - 1159 q^{49} - 416 q^{50} + 36 q^{52} - 480 q^{53} + 1456 q^{55} - 200 q^{56} + 524 q^{58} + 296 q^{59} + 474 q^{61} + 1104 q^{62} + 512 q^{64} - 1542 q^{65} + 1319 q^{67} + 472 q^{68} - 32 q^{70} + 1852 q^{71} - 1423 q^{73} + 370 q^{74} - 296 q^{76} - 1228 q^{77} + 765 q^{79} - 64 q^{80} - 60 q^{82} + 1660 q^{83} - 584 q^{85} - 1556 q^{86} + 448 q^{88} + 864 q^{89} - 738 q^{91} + 1600 q^{92} + 60 q^{94} - 1766 q^{95} + 1088 q^{97} - 2704 q^{98}+O(q^{100}) \) 8 * q + 8 * q^2 - 16 * q^4 - 4 * q^5 + 25 * q^7 - 64 * q^8 + 8 * q^10 - 56 * q^11 - 18 * q^13 + 22 * q^14 - 64 * q^16 + 118 * q^17 + 37 * q^19 + 32 * q^20 - 224 * q^22 - 200 * q^23 - 104 * q^25 - 18 * q^26 - 56 * q^28 + 524 * q^29 + 276 * q^31 + 128 * q^32 + 472 * q^34 - 290 * q^35 - 185 * q^37 - 74 * q^38 + 32 * q^40 - 60 * q^41 - 1556 * q^43 - 224 * q^44 + 400 * q^46 - 30 * q^47 - 1159 * q^49 - 416 * q^50 + 36 * q^52 - 480 * q^53 + 1456 * q^55 - 200 * q^56 + 524 * q^58 + 296 * q^59 + 474 * q^61 + 1104 * q^62 + 512 * q^64 - 1542 * q^65 + 1319 * q^67 + 472 * q^68 - 32 * q^70 + 1852 * q^71 - 1423 * q^73 + 370 * q^74 - 296 * q^76 - 1228 * q^77 + 765 * q^79 - 64 * q^80 - 60 * q^82 + 1660 * q^83 - 584 * q^85 - 1556 * q^86 + 448 * q^88 + 864 * q^89 - 738 * q^91 + 1600 * q^92 + 60 * q^94 - 1766 * q^95 + 1088 * q^97 - 2704 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more