LMFDB - Embedded newform 245.3.d.a.146.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [245,3,Mod(146,245)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(245, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("245.146");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 245 = 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	245.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$6.67576647683$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} + 19 x^{10} - 26 x^{9} + 244 x^{8} - 338 x^{7} + 1249 x^{6} - 986 x^{5} + 3532 x^{4} + \cdots + 441 $ x^12 - 2x^11 + 19x^10 - 26x^9 + 244x^8 - 338x^7 + 1249x^6 - 986x^5 + 3532x^4 - 3618x^3 + 3579x^2 - 1386*x + 441
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 35)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			146.11
Root			$1.77870 + 3.08079i$ of defining polynomial
Character	$\chi$	$=$	245.146
Dual form			245.3.d.a.146.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.55739 q^{2} -2.43315i q^{3} +8.65503 q^{4} -2.23607i q^{5} -8.65567i q^{6} +16.5598 q^{8} +3.07978 q^{9} +O(q^{10})$	\(q+3.55739 q^{2} -2.43315i q^{3} +8.65503 q^{4} -2.23607i q^{5} -8.65567i q^{6} +16.5598 q^{8} +3.07978 q^{9} -7.95457i q^{10} -13.9011 q^{11} -21.0590i q^{12} -0.702771i q^{13} -5.44069 q^{15} +24.2894 q^{16} +27.2652i q^{17} +10.9560 q^{18} -2.55941i q^{19} -19.3532i q^{20} -49.4517 q^{22} +32.9554 q^{23} -40.2924i q^{24} -5.00000 q^{25} -2.50003i q^{26} -29.3919i q^{27} +3.39850 q^{29} -19.3547 q^{30} +23.9635i q^{31} +20.1679 q^{32} +33.8235i q^{33} +96.9930i q^{34} +26.6556 q^{36} -23.8160 q^{37} -9.10483i q^{38} -1.70995 q^{39} -37.0288i q^{40} +25.1015i q^{41} -25.1201 q^{43} -120.315 q^{44} -6.88659i q^{45} +117.235 q^{46} +6.46334i q^{47} -59.0998i q^{48} -17.7870 q^{50} +66.3403 q^{51} -6.08250i q^{52} -36.0427 q^{53} -104.559i q^{54} +31.0838i q^{55} -6.22744 q^{57} +12.0898 q^{58} -36.9661i q^{59} -47.0893 q^{60} +35.0540i q^{61} +85.2476i q^{62} -25.4125 q^{64} -1.57144 q^{65} +120.323i q^{66} -45.7268 q^{67} +235.981i q^{68} -80.1854i q^{69} +80.4090 q^{71} +51.0004 q^{72} +89.8242i q^{73} -84.7229 q^{74} +12.1658i q^{75} -22.1518i q^{76} -6.08295 q^{78} +0.0831944 q^{79} -54.3128i q^{80} -43.7970 q^{81} +89.2958i q^{82} -95.3085i q^{83} +60.9668 q^{85} -89.3621 q^{86} -8.26907i q^{87} -230.199 q^{88} -121.025i q^{89} -24.4983i q^{90} +285.230 q^{92} +58.3068 q^{93} +22.9926i q^{94} -5.72302 q^{95} -49.0716i q^{96} -0.362083i q^{97} -42.8124 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{2} + 20 q^{4} - 4 q^{8} - 28 q^{9}+O(q^{10}) $ 12 * q + 4 * q^2 + 20 * q^4 - 4 * q^8 - 28 * q^9	$ 12 q + 4 q^{2} + 20 q^{4} - 4 q^{8} - 28 q^{9} + 28 q^{11} - 20 q^{15} + 44 q^{16} - 128 q^{18} - 88 q^{22} + 28 q^{23} - 60 q^{25} + 64 q^{29} - 40 q^{30} + 108 q^{32} + 156 q^{36} - 88 q^{37} + 48 q^{39} - 4 q^{43} - 12 q^{44} + 428 q^{46} - 20 q^{50} + 264 q^{51} - 392 q^{53} - 48 q^{57} - 316 q^{58} - 300 q^{60} - 140 q^{64} - 60 q^{65} + 276 q^{67} - 8 q^{71} + 392 q^{72} - 100 q^{74} - 312 q^{78} + 24 q^{79} + 620 q^{81} + 80 q^{86} - 1208 q^{88} + 732 q^{92} - 168 q^{93} - 120 q^{95} - 44 q^{99}+O(q^{100}) $ 12 * q + 4 * q^2 + 20 * q^4 - 4 * q^8 - 28 * q^9 + 28 * q^11 - 20 * q^15 + 44 * q^16 - 128 * q^18 - 88 * q^22 + 28 * q^23 - 60 * q^25 + 64 * q^29 - 40 * q^30 + 108 * q^32 + 156 * q^36 - 88 * q^37 + 48 * q^39 - 4 * q^43 - 12 * q^44 + 428 * q^46 - 20 * q^50 + 264 * q^51 - 392 * q^53 - 48 * q^57 - 316 * q^58 - 300 * q^60 - 140 * q^64 - 60 * q^65 + 276 * q^67 - 8 * q^71 + 392 * q^72 - 100 * q^74 - 312 * q^78 + 24 * q^79 + 620 * q^81 + 80 * q^86 - 1208 * q^88 + 732 * q^92 - 168 * q^93 - 120 * q^95 - 44 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/245\mathbb{Z}\right)^\times$.

$n$	$101$	$197$
$\chi(n)$	$-1$	$1$

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 $

$2$	3.55739	1.77870	0.889348	−	0.457231i	$-0.151159\pi$
$2$	3.55739	1.77870	0.889348	+	0.457231i	$0.151159\pi$
$3$	− 2.43315i	− 0.811050i	−0.914084	−	0.405525i	$-0.867089\pi$
$3$	− 2.43315i	− 0.811050i	0.914084	−	0.405525i	$-0.132911\pi$
$4$	8.65503	2.16376
$5$	− 2.23607i	− 0.447214i
$5$	− 2.23607i	− 0.447214i
$6$	− 8.65567i	− 1.44261i
$7$	0	0
$7$	0	0
$8$	16.5598	2.06997
$9$	3.07978	0.342198
$10$	− 7.95457i	− 0.795457i
$11$	−13.9011	−1.26374	−0.631869	−	0.775075i	$-0.717712\pi$
$11$	−13.9011	−1.26374	−0.631869	+	0.775075i	$0.717712\pi$
$12$	− 21.0590i	− 1.75492i
$13$	− 0.702771i	− 0.0540593i	−0.999635	−	0.0270296i	$-0.991395\pi$
$13$	− 0.702771i	− 0.0540593i	0.999635	−	0.0270296i	$-0.00860485\pi$
$14$	0	0
$15$	−5.44069	−0.362713
$16$	24.2894	1.51809
$17$	27.2652i	1.60384i	0.597435	+	0.801918i	$0.296187\pi$
$17$	27.2652i	1.60384i	−0.597435	+	0.801918i	$0.703813\pi$
$18$	10.9560	0.608665
$19$	− 2.55941i	− 0.134706i	−0.997729	−	0.0673530i	$-0.978545\pi$
$19$	− 2.55941i	− 0.134706i	0.997729	−	0.0673530i	$-0.0214554\pi$
$20$	− 19.3532i	− 0.967662i
$21$	0	0
$22$	−49.4517	−2.24781
$23$	32.9554	1.43284	0.716422	−	0.697668i	$-0.245779\pi$
$23$	32.9554	1.43284	0.716422	+	0.697668i	$0.245779\pi$
$24$	− 40.2924i	− 1.67885i
$25$	−5.00000	−0.200000
$26$	− 2.50003i	− 0.0961550i
$27$	− 29.3919i	− 1.08859i
$28$	0	0
$29$	3.39850	0.117190	0.0585949	−	0.998282i	$-0.481338\pi$
$29$	3.39850	0.117190	0.0585949	+	0.998282i	$0.481338\pi$
$30$	−19.3547	−0.645155
$31$	23.9635i	0.773016i	0.922286	+	0.386508i	$0.126319\pi$
$31$	23.9635i	0.773016i	−0.922286	+	0.386508i	$0.873681\pi$
$32$	20.1679	0.630248
$33$	33.8235i	1.02496i
$34$	96.9930i	2.85273i
$35$	0	0
$36$	26.6556	0.740433
$37$	−23.8160	−0.643677	−0.321838	−	0.946795i	$-0.604301\pi$
$37$	−23.8160	−0.643677	−0.321838	+	0.946795i	$0.604301\pi$
$38$	− 9.10483i	− 0.239601i
$39$	−1.70995	−0.0438448
$40$	− 37.0288i	− 0.925719i
$41$	25.1015i	0.612231i	0.951994	+	0.306116i	$0.0990295\pi$
$41$	25.1015i	0.612231i	−0.951994	+	0.306116i	$0.900971\pi$
$42$	0	0
$43$	−25.1201	−0.584189	−0.292094	−	0.956390i	$-0.594352\pi$
$43$	−25.1201	−0.584189	−0.292094	+	0.956390i	$0.594352\pi$
$44$	−120.315	−2.73442
$45$	− 6.88659i	− 0.153035i
$46$	117.235	2.54859
$47$	6.46334i	0.137518i	0.997633	+	0.0687590i	$0.0219039\pi$
$47$	6.46334i	0.137518i	−0.997633	+	0.0687590i	$0.978096\pi$
$48$	− 59.0998i	− 1.23125i
$49$	0	0
$50$	−17.7870	−0.355739
$51$	66.3403	1.30079
$52$	− 6.08250i	− 0.116971i
$53$	−36.0427	−0.680051	−0.340026	−	0.940416i	$-0.610436\pi$
$53$	−36.0427	−0.680051	−0.340026	+	0.940416i	$0.610436\pi$
$54$	− 104.559i	− 1.93627i
$55$	31.0838i	0.565161i
$56$	0	0
$57$	−6.22744	−0.109253
$58$	12.0898	0.208445
$59$	− 36.9661i	− 0.626543i	−0.949664	−	0.313272i	$-0.898575\pi$
$59$	− 36.9661i	− 0.626543i	0.949664	−	0.313272i	$-0.101425\pi$
$60$	−47.0893	−0.784822
$61$	35.0540i	0.574656i	0.957832	+	0.287328i	$0.0927670\pi$
$61$	35.0540i	0.574656i	−0.957832	+	0.287328i	$0.907233\pi$
$62$	85.2476i	1.37496i
$63$	0	0
$64$	−25.4125	−0.397070
$65$	−1.57144	−0.0241760
$66$	120.323i	1.82308i
$67$	−45.7268	−0.682490	−0.341245	−	0.939974i	$-0.610848\pi$
$67$	−45.7268	−0.682490	−0.341245	+	0.939974i	$0.610848\pi$
$68$	235.981i	3.47031i
$69$	− 80.1854i	− 1.16211i
$70$	0	0
$71$	80.4090	1.13252	0.566261	−	0.824226i	$-0.308389\pi$
$71$	80.4090	1.13252	0.566261	+	0.824226i	$0.308389\pi$
$72$	51.0004	0.708339
$73$	89.8242i	1.23047i	0.788344	+	0.615235i	$0.210939\pi$
$73$	89.8242i	1.23047i	−0.788344	+	0.615235i	$0.789061\pi$
$74$	−84.7229	−1.14490
$75$	12.1658i	0.162210i
$76$	− 22.1518i	− 0.291471i
$77$	0	0
$78$	−6.08295	−0.0779865
$79$	0.0831944	0.00105309	0.000526547	−	1.00000i	$-0.499832\pi$
$79$	0.0831944	0.00105309	0.000526547		1.00000i	$0.499832\pi$
$80$	− 54.3128i	− 0.678910i
$81$	−43.7970	−0.540703
$82$	89.2958i	1.08897i
$83$	− 95.3085i	− 1.14830i	−0.818752	−	0.574148i	$-0.805334\pi$
$83$	− 95.3085i	− 1.14830i	0.818752	−	0.574148i	$-0.194666\pi$
$84$	0	0
$85$	60.9668	0.717257
$86$	−89.3621	−1.03909
$87$	− 8.26907i	− 0.0950467i
$88$	−230.199	−2.61590
$89$	− 121.025i	− 1.35983i	−0.733289	−	0.679917i	$-0.762016\pi$
$89$	− 121.025i	− 1.35983i	0.733289	−	0.679917i	$-0.237984\pi$
$90$	− 24.4983i	− 0.272203i
$91$	0	0
$92$	285.230	3.10033
$93$	58.3068	0.626955
$94$	22.9926i	0.244603i
$95$	−5.72302	−0.0602423
$96$	− 49.0716i	− 0.511163i
$97$	− 0.362083i	− 0.00373282i	−0.999998	−	0.00186641i	$-0.999406\pi$
$97$	− 0.362083i	− 0.00373282i	0.999998	−	0.00186641i	$-0.000594097\pi$
$98$	0	0
$99$	−42.8124	−0.432448
$100$	−43.2752	−0.432752
$101$	− 160.658i	− 1.59067i	−0.606168	−	0.795336i	$-0.707294\pi$
$101$	− 160.658i	− 1.59067i	0.606168	−	0.795336i	$-0.292706\pi$
$102$	235.999	2.31371
$103$	− 143.453i	− 1.39275i	−0.717678	−	0.696375i	$-0.754795\pi$
$103$	− 143.453i	− 1.39275i	0.717678	−	0.696375i	$-0.245205\pi$
$104$	− 11.6377i	− 0.111901i
$105$	0	0
$106$	−128.218	−1.20960
$107$	−63.4633	−0.593115	−0.296558	−	0.955015i	$-0.595839\pi$
$107$	−63.4633	−0.593115	−0.296558	+	0.955015i	$0.595839\pi$
$108$	− 254.388i	− 2.35544i
$109$	56.7091	0.520267	0.260134	−	0.965573i	$-0.416233\pi$
$109$	56.7091	0.520267	0.260134	+	0.965573i	$0.416233\pi$
$110$	110.577i	1.00525i
$111$	57.9480i	0.522054i
$112$	0	0
$113$	53.1456	0.470315	0.235157	−	0.971957i	$-0.424439\pi$
$113$	53.1456	0.470315	0.235157	+	0.971957i	$0.424439\pi$
$114$	−22.1534	−0.194328
$115$	− 73.6905i	− 0.640787i
$116$	29.4141	0.253570
$117$	− 2.16438i	− 0.0184990i
$118$	− 131.503i	− 1.11443i
$119$	0	0
$120$	−90.0965	−0.750804
$121$	72.2411	0.597034
$122$	124.701i	1.02214i
$123$	61.0757	0.496550
$124$	207.405i	1.67262i
$125$	11.1803i	0.0894427i
$126$	0	0
$127$	−39.1961	−0.308630	−0.154315	−	0.988022i	$-0.549317\pi$
$127$	−39.1961	−0.308630	−0.154315	+	0.988022i	$0.549317\pi$
$128$	−171.074	−1.33651
$129$	61.1210i	0.473806i
$130$	−5.59024	−0.0430018
$131$	54.0978i	0.412961i	0.978451	+	0.206480i	$0.0662009\pi$
$131$	54.0978i	0.412961i	−0.978451	+	0.206480i	$0.933799\pi$
$132$	292.744i	2.21775i
$133$	0	0
$134$	−162.668	−1.21394
$135$	−65.7223	−0.486832
$136$	451.505i	3.31989i
$137$	−82.6681	−0.603417	−0.301708	−	0.953400i	$-0.597557\pi$
$137$	−82.6681	−0.603417	−0.301708	+	0.953400i	$0.597557\pi$
$138$	− 285.251i	− 2.06704i
$139$	− 98.5453i	− 0.708959i	−0.935064	−	0.354480i	$-0.884658\pi$
$139$	− 98.5453i	− 0.708959i	0.935064	−	0.354480i	$-0.115342\pi$
$140$	0	0
$141$	15.7263	0.111534
$142$	286.046	2.01441
$143$	9.76930i	0.0683168i
$144$	74.8061	0.519487
$145$	− 7.59928i	− 0.0524088i
$146$	319.540i	2.18863i
$147$	0	0
$148$	−206.129	−1.39276
$149$	75.1928	0.504650	0.252325	−	0.967643i	$-0.418805\pi$
$149$	75.1928	0.504650	0.252325	+	0.967643i	$0.418805\pi$
$150$	43.2783i	0.288522i
$151$	154.993	1.02644	0.513220	−	0.858257i	$-0.328452\pi$
$151$	154.993	1.02644	0.513220	+	0.858257i	$0.328452\pi$
$152$	− 42.3833i	− 0.278837i
$153$	83.9708i	0.548829i
$154$	0	0
$155$	53.5840	0.345703
$156$	−14.7996	−0.0948695
$157$	− 116.022i	− 0.738991i	−0.929233	−	0.369495i	$-0.879531\pi$
$157$	− 116.022i	− 0.738991i	0.929233	−	0.369495i	$-0.120469\pi$
$158$	0.295955	0.00187313
$159$	87.6973i	0.551556i
$160$	− 45.0969i	− 0.281856i
$161$	0	0
$162$	−155.803	−0.961746
$163$	−190.189	−1.16681	−0.583403	−	0.812183i	$-0.698279\pi$
$163$	−190.189	−1.16681	−0.583403	+	0.812183i	$0.698279\pi$
$164$	217.254i	1.32472i
$165$	75.6317	0.458374
$166$	− 339.050i	− 2.04247i
$167$	− 259.403i	− 1.55331i	−0.629925	−	0.776656i	$-0.716914\pi$
$167$	− 259.403i	− 1.55331i	0.629925	−	0.776656i	$-0.283086\pi$
$168$	0	0
$169$	168.506	0.997078
$170$	216.883	1.27578
$171$	− 7.88243i	− 0.0460961i
$172$	−217.415	−1.26404
$173$	226.530i	1.30942i	0.755880	+	0.654710i	$0.227209\pi$
$173$	226.530i	1.30942i	−0.755880	+	0.654710i	$0.772791\pi$
$174$	− 29.4163i	− 0.169059i
$175$	0	0
$176$	−337.650	−1.91847
$177$	−89.9440	−0.508158
$178$	− 430.534i	− 2.41873i
$179$	187.009	1.04474	0.522372	−	0.852718i	$-0.325047\pi$
$179$	187.009	1.04474	0.522372	+	0.852718i	$0.325047\pi$
$180$	− 59.6037i	− 0.331132i
$181$	183.025i	1.01119i	0.862771	+	0.505595i	$0.168727\pi$
$181$	183.025i	1.01119i	−0.862771	+	0.505595i	$0.831273\pi$
$182$	0	0
$183$	85.2917	0.466075
$184$	545.733	2.96594
$185$	53.2543i	0.287861i
$186$	207.420	1.11516
$187$	− 379.017i	− 2.02683i
$188$	55.9404i	0.297555i
$189$	0	0
$190$	−20.3590	−0.107153
$191$	−197.530	−1.03419	−0.517093	−	0.855929i	$-0.672986\pi$
$191$	−197.530	−1.03419	−0.517093	+	0.855929i	$0.672986\pi$
$192$	61.8323i	0.322043i
$193$	−84.6529	−0.438616	−0.219308	−	0.975656i	$-0.570380\pi$
$193$	−84.6529	−0.438616	−0.219308	+	0.975656i	$0.570380\pi$
$194$	− 1.28807i	− 0.00663955i
$195$	3.82356i	0.0196080i
$196$	0	0
$197$	−247.330	−1.25548	−0.627742	−	0.778421i	$-0.716021\pi$
$197$	−247.330	−1.25548	−0.627742	+	0.778421i	$0.716021\pi$
$198$	−152.300	−0.769194
$199$	− 122.514i	− 0.615650i	−0.951443	−	0.307825i	$-0.900399\pi$
$199$	− 122.514i	− 0.615650i	0.951443	−	0.307825i	$-0.0996012\pi$
$200$	−82.7988	−0.413994
$201$	111.260i	0.553533i
$202$	− 571.523i	− 2.82932i
$203$	0	0
$204$	574.178	2.81460
$205$	56.1286	0.273798
$206$	− 510.319i	− 2.47728i
$207$	101.495	0.490315
$208$	− 17.0699i	− 0.0820668i
$209$	35.5787i	0.170233i
$210$	0	0
$211$	410.851	1.94716	0.973580	−	0.228348i	$-0.0733325\pi$
$211$	410.851	1.94716	0.973580	+	0.228348i	$0.0733325\pi$
$212$	−311.951	−1.47147
$213$	− 195.647i	− 0.918532i
$214$	−225.764	−1.05497
$215$	56.1703i	0.261257i
$216$	− 486.723i	− 2.25335i
$217$	0	0
$218$	201.737	0.925397
$219$	218.556	0.997972
$220$	269.032i	1.22287i
$221$	19.1612	0.0867022
$222$	206.144i	0.928575i
$223$	69.8903i	0.313409i	0.987646	+	0.156705i	$0.0500871\pi$
$223$	69.8903i	0.313409i	−0.987646	+	0.156705i	$0.949913\pi$
$224$	0	0
$225$	−15.3989	−0.0684395
$226$	189.060	0.836547
$227$	− 103.620i	− 0.456477i	−0.973605	−	0.228238i	$-0.926703\pi$
$227$	− 103.620i	− 0.456477i	0.973605	−	0.228238i	$-0.0732965\pi$
$228$	−53.8987	−0.236398
$229$	303.360i	1.32472i	0.749187	+	0.662358i	$0.230444\pi$
$229$	303.360i	1.32472i	−0.749187	+	0.662358i	$0.769556\pi$
$230$	− 262.146i	− 1.13976i
$231$	0	0
$232$	56.2784	0.242579
$233$	428.443	1.83881	0.919406	−	0.393310i	$-0.128670\pi$
$233$	428.443	1.83881	0.919406	+	0.393310i	$0.128670\pi$
$234$	− 7.69954i	− 0.0329040i
$235$	14.4525	0.0614999
$236$	− 319.942i	− 1.35569i
$237$	− 0.202425i	0 0.000854112i
$238$	0	0
$239$	109.615	0.458641	0.229320	−	0.973351i	$-0.426350\pi$
$239$	109.615	0.458641	0.229320	+	0.973351i	$0.426350\pi$
$240$	−132.151	−0.550630
$241$	285.930i	1.18643i	0.805044	+	0.593215i	$0.202142\pi$
$241$	285.930i	1.18643i	−0.805044	+	0.593215i	$0.797858\pi$
$242$	256.990	1.06194
$243$	− 157.963i	− 0.650052i
$244$	303.394i	1.24342i
$245$	0	0
$246$	217.270	0.883212
$247$	−1.79868	−0.00728211
$248$	396.830i	1.60012i
$249$	−231.900	−0.931325
$250$	39.7728i	0.159091i
$251$	− 101.215i	− 0.403248i	−0.979463	−	0.201624i	$-0.935378\pi$
$251$	− 101.215i	− 0.403248i	0.979463	−	0.201624i	$-0.0646219\pi$
$252$	0	0
$253$	−458.117	−1.81074
$254$	−139.436	−0.548960
$255$	− 148.341i	− 0.581731i
$256$	−506.927	−1.98018
$257$	− 506.428i	− 1.97054i	−0.171010	−	0.985269i	$-0.554703\pi$
$257$	− 506.428i	− 1.97054i	0.171010	−	0.985269i	$-0.445297\pi$
$258$	217.431i	0.842757i
$259$	0	0
$260$	−13.6009	−0.0523111
$261$	10.4666	0.0401020
$262$	192.447i	0.734531i
$263$	−54.5785	−0.207523	−0.103761	−	0.994602i	$-0.533088\pi$
$263$	−54.5785	−0.207523	−0.103761	+	0.994602i	$0.533088\pi$
$264$	560.109i	2.12163i
$265$	80.5940i	0.304128i
$266$	0	0
$267$	−294.472	−1.10289
$268$	−395.767	−1.47674
$269$	489.240i	1.81874i	0.415992	+	0.909368i	$0.363434\pi$
$269$	489.240i	1.81874i	−0.415992	+	0.909368i	$0.636566\pi$
$270$	−233.800	−0.865926
$271$	487.452i	1.79871i	0.437214	+	0.899357i	$0.355965\pi$
$271$	487.452i	1.79871i	−0.437214	+	0.899357i	$0.644035\pi$
$272$	662.256i	2.43477i
$273$	0	0
$274$	−294.083	−1.07329
$275$	69.5056	0.252748
$276$	− 694.007i	− 2.51452i
$277$	−304.975	−1.10099	−0.550497	−	0.834837i	$-0.685562\pi$
$277$	−304.975	−1.10099	−0.550497	+	0.834837i	$0.685562\pi$
$278$	− 350.564i	− 1.26102i
$279$	73.8023i	0.264524i
$280$	0	0
$281$	−110.177	−0.392089	−0.196044	−	0.980595i	$-0.562810\pi$
$281$	−110.177	−0.392089	−0.196044	+	0.980595i	$0.562810\pi$
$282$	55.9445	0.198385
$283$	266.231i	0.940746i	0.882468	+	0.470373i	$0.155881\pi$
$283$	266.231i	0.940746i	−0.882468	+	0.470373i	$0.844119\pi$
$284$	695.943	2.45050
$285$	13.9250i	0.0488596i
$286$	34.7532i	0.121515i
$287$	0	0
$288$	62.1128	0.215669
$289$	−454.391	−1.57229
$290$	− 27.0336i	− 0.0932193i
$291$	−0.881003	−0.00302750
$292$	777.432i	2.66244i
$293$	8.79423i	0.0300145i	0.999887	+	0.0150072i	$0.00477713\pi$
$293$	8.79423i	0.0300145i	−0.999887	+	0.0150072i	$0.995223\pi$
$294$	0	0
$295$	−82.6586	−0.280199
$296$	−394.388	−1.33239
$297$	408.581i	1.37569i
$298$	267.490	0.897618
$299$	− 23.1601i	− 0.0774585i
$300$	105.295i	0.350983i
$301$	0	0
$302$	551.369	1.82572
$303$	−390.905	−1.29012
$304$	− 62.1667i	− 0.204496i
$305$	78.3832	0.256994
$306$	298.717i	0.976199i
$307$	447.196i	1.45667i	0.685224	+	0.728333i	$0.259704\pi$
$307$	447.196i	1.45667i	−0.685224	+	0.728333i	$0.740296\pi$
$308$	0	0
$309$	−349.043	−1.12959
$310$	190.619	0.614901
$311$	− 225.045i	− 0.723618i	−0.932252	−	0.361809i	$-0.882159\pi$
$311$	− 225.045i	− 0.723618i	0.932252	−	0.361809i	$-0.117841\pi$
$312$	−28.3163	−0.0907574
$313$	336.098i	1.07380i	0.843647	+	0.536898i	$0.180404\pi$
$313$	336.098i	1.07380i	−0.843647	+	0.536898i	$0.819596\pi$
$314$	− 412.734i	− 1.31444i
$315$	0	0
$316$	0.720050	0.00227864
$317$	−82.4347	−0.260046	−0.130023	−	0.991511i	$-0.541505\pi$
$317$	−82.4347	−0.260046	−0.130023	+	0.991511i	$0.541505\pi$
$318$	311.974i	0.981049i
$319$	−47.2430	−0.148097
$320$	56.8240i	0.177575i
$321$	154.416i	0.481046i
$322$	0	0
$323$	69.7829	0.216046
$324$	−379.064	−1.16995
$325$	3.51385i	0.0108119i
$326$	−676.578	−2.07539
$327$	− 137.982i	− 0.421963i
$328$	415.675i	1.26730i
$329$	0	0
$330$	269.051	0.815307
$331$	−94.7636	−0.286295	−0.143147	−	0.989701i	$-0.545722\pi$
$331$	−94.7636	−0.286295	−0.143147	+	0.989701i	$0.545722\pi$
$332$	− 824.898i	− 2.48463i
$333$	−73.3481	−0.220265
$334$	− 922.798i	− 2.76287i
$335$	102.248i	0.305219i
$336$	0	0
$337$	−362.615	−1.07601	−0.538005	−	0.842942i	$-0.680822\pi$
$337$	−362.615	−1.07601	−0.538005	+	0.842942i	$0.680822\pi$
$338$	599.442	1.77350
$339$	− 129.311i	− 0.381449i
$340$	527.670	1.55197
$341$	− 333.120i	− 0.976890i
$342$	− 28.0409i	− 0.0819908i
$343$	0	0
$344$	−415.983	−1.20925
$345$	−179.300	−0.519710
$346$	805.855i	2.32906i
$347$	387.992	1.11813	0.559066	−	0.829123i	$-0.311160\pi$
$347$	387.992	1.11813	0.559066	+	0.829123i	$0.311160\pi$
$348$	− 71.5690i	− 0.205658i
$349$	− 445.741i	− 1.27719i	−0.769541	−	0.638597i	$-0.779515\pi$
$349$	− 445.741i	− 1.27719i	0.769541	−	0.638597i	$-0.220485\pi$
$350$	0	0
$351$	−20.6558	−0.0588484
$352$	−280.357	−0.796469
$353$	286.056i	0.810358i	0.914237	+	0.405179i	$0.132791\pi$
$353$	286.056i	0.810358i	−0.914237	+	0.405179i	$0.867209\pi$
$354$	−319.966	−0.903858
$355$	− 179.800i	− 0.506479i
$356$	− 1047.48i	− 2.94235i
$357$	0	0
$358$	665.265	1.85828
$359$	430.179	1.19827	0.599135	−	0.800648i	$-0.295511\pi$
$359$	430.179	1.19827	0.599135	+	0.800648i	$0.295511\pi$
$360$	− 114.040i	− 0.316779i
$361$	354.449	0.981854
$362$	651.093i	1.79860i
$363$	− 175.774i	− 0.484225i
$364$	0	0
$365$	200.853	0.550283
$366$	303.416	0.829005
$367$	− 333.918i	− 0.909859i	−0.890527	−	0.454930i	$-0.849664\pi$
$367$	− 333.918i	− 0.909859i	0.890527	−	0.454930i	$-0.150336\pi$
$368$	800.468	2.17518
$369$	77.3070i	0.209504i
$370$	189.446i	0.512017i
$371$	0	0
$372$	504.647	1.35658
$373$	−49.2810	−0.132121	−0.0660603	−	0.997816i	$-0.521043\pi$
$373$	−49.2810	−0.132121	−0.0660603	+	0.997816i	$0.521043\pi$
$374$	− 1348.31i	− 3.60511i
$375$	27.2035	0.0725425
$376$	107.031i	0.284658i
$377$	− 2.38837i	− 0.00633519i
$378$	0	0
$379$	−635.496	−1.67677	−0.838385	−	0.545078i	$-0.816500\pi$
$379$	−635.496	−1.67677	−0.838385	+	0.545078i	$0.816500\pi$
$380$	−49.5329	−0.130350
$381$	95.3699i	0.250315i
$382$	−702.690	−1.83950
$383$	97.5347i	0.254660i	0.991860	+	0.127330i	$0.0406407\pi$
$383$	97.5347i	0.254660i	−0.991860	+	0.127330i	$0.959359\pi$
$384$	416.248i	1.08398i
$385$	0	0
$386$	−301.143	−0.780164
$387$	−77.3644	−0.199908
$388$	− 3.13384i	− 0.00807691i
$389$	−670.861	−1.72458	−0.862289	−	0.506416i	$-0.830970\pi$
$389$	−670.861	−1.72458	−0.862289	+	0.506416i	$0.830970\pi$
$390$	13.6019i	0.0348766i
$391$	898.535i	2.29804i
$392$	0	0
$393$	131.628	0.334932
$394$	−879.851	−2.23312
$395$	− 0.186028i	0 0.000470958i
$396$	−370.542	−0.935713
$397$	557.352i	1.40391i	0.712222	+	0.701954i	$0.247689\pi$
$397$	557.352i	1.40391i	−0.712222	+	0.701954i	$0.752311\pi$
$398$	− 435.832i	− 1.09505i
$399$	0	0
$400$	−121.447	−0.303618
$401$	595.009	1.48381	0.741906	−	0.670504i	$-0.233922\pi$
$401$	595.009	1.48381	0.741906	+	0.670504i	$0.233922\pi$
$402$	395.796i	0.984567i
$403$	16.8408	0.0417887
$404$	− 1390.50i	− 3.44183i
$405$	97.9330i	0.241810i
$406$	0	0
$407$	331.070	0.813439
$408$	1098.58	2.69260
$409$	455.902i	1.11467i	0.830286	+	0.557337i	$0.188177\pi$
$409$	455.902i	1.11467i	−0.830286	+	0.557337i	$0.811823\pi$
$410$	199.672	0.487004
$411$	201.144i	0.489401i
$412$	− 1241.59i	− 3.01357i
$413$	0	0
$414$	361.058	0.872122
$415$	−213.116	−0.513533
$416$	− 14.1734i	− 0.0340708i
$417$	−239.776	−0.575001
$418$	126.567i	0.302793i
$419$	120.662i	0.287977i	0.989579	+	0.143989i	$0.0459929\pi$
$419$	120.662i	0.287977i	−0.989579	+	0.143989i	$0.954007\pi$
$420$	0	0
$421$	−206.898	−0.491444	−0.245722	−	0.969340i	$-0.579025\pi$
$421$	−206.898	−0.491444	−0.245722	+	0.969340i	$0.579025\pi$
$422$	1461.56	3.46340
$423$	19.9057i	0.0470583i
$424$	−596.859	−1.40769
$425$	− 136.326i	− 0.320767i
$426$	− 695.994i	− 1.63379i
$427$	0	0
$428$	−549.277	−1.28336
$429$	23.7702	0.0554083
$430$	199.820i	0.464697i
$431$	−269.858	−0.626120	−0.313060	−	0.949733i	$-0.601354\pi$
$431$	−269.858	−0.626120	−0.313060	+	0.949733i	$0.601354\pi$
$432$	− 713.913i	− 1.65258i
$433$	8.67846i	0.0200426i	0.999950	+	0.0100213i	$0.00318994\pi$
$433$	8.67846i	0.0200426i	−0.999950	+	0.0100213i	$0.996810\pi$
$434$	0	0
$435$	−18.4902	−0.0425062
$436$	490.819	1.12573
$437$	− 84.3465i	− 0.193013i
$438$	777.489	1.77509
$439$	− 3.59352i	− 0.00818569i	−0.999992	−	0.00409284i	$-0.998697\pi$
$439$	− 3.59352i	− 0.00818569i	0.999992	−	0.00409284i	$-0.00130280\pi$
$440$	514.741i	1.16987i
$441$	0	0
$442$	68.1638	0.154217
$443$	404.403	0.912873	0.456437	−	0.889756i	$-0.349126\pi$
$443$	404.403	0.912873	0.456437	+	0.889756i	$0.349126\pi$
$444$	501.542i	1.12960i
$445$	−270.620	−0.608136
$446$	248.627i	0.557460i
$447$	− 182.955i	− 0.409296i
$448$	0	0
$449$	339.672	0.756508	0.378254	−	0.925702i	$-0.376525\pi$
$449$	339.672	0.756508	0.378254	+	0.925702i	$0.376525\pi$
$450$	−54.7799	−0.121733
$451$	− 348.939i	− 0.773700i
$452$	459.977	1.01765
$453$	− 377.120i	− 0.832495i
$454$	− 368.618i	− 0.811933i
$455$	0	0
$456$	−103.125	−0.226151
$457$	−636.278	−1.39229	−0.696147	−	0.717900i	$-0.745104\pi$
$457$	−636.278	−1.39229	−0.696147	+	0.717900i	$0.745104\pi$
$458$	1079.17i	2.35627i
$459$	801.377	1.74592
$460$	− 637.793i	− 1.38651i
$461$	− 483.724i	− 1.04929i	−0.851321	−	0.524646i	$-0.824198\pi$
$461$	− 483.724i	− 1.04929i	0.851321	−	0.524646i	$-0.175802\pi$
$462$	0	0
$463$	409.737	0.884961	0.442480	−	0.896778i	$-0.354099\pi$
$463$	409.737	0.884961	0.442480	+	0.896778i	$0.354099\pi$
$464$	82.5477	0.177904
$465$	− 130.378i	− 0.280383i
$466$	1524.14	3.27069
$467$	− 366.607i	− 0.785025i	−0.919747	−	0.392512i	$-0.871606\pi$
$467$	− 366.607i	− 0.785025i	0.919747	−	0.392512i	$-0.128394\pi$
$468$	− 18.7328i	− 0.0400273i
$469$	0	0
$470$	51.4131	0.109390
$471$	−282.298	−0.599358
$472$	− 612.149i	− 1.29693i
$473$	349.198	0.738261
$474$	− 0.720103i	− 0.00151921i
$475$	12.7971i	0.0269412i
$476$	0	0
$477$	−111.004	−0.232712
$478$	389.944	0.815782
$479$	− 84.9170i	− 0.177280i	−0.996064	−	0.0886399i	$-0.971748\pi$
$479$	− 84.9170i	− 0.177280i	0.996064	−	0.0886399i	$-0.0282520\pi$
$480$	−109.728	−0.228599
$481$	16.7372i	0.0347967i
$482$	1017.16i	2.11030i
$483$	0	0
$484$	625.249	1.29184
$485$	−0.809643	−0.00166937
$486$	− 561.935i	− 1.15624i
$487$	−32.7456	−0.0672395	−0.0336197	−	0.999435i	$-0.510704\pi$
$487$	−32.7456	−0.0672395	−0.0336197	+	0.999435i	$0.510704\pi$
$488$	580.486i	1.18952i
$489$	462.760i	0.946339i
$490$	0	0
$491$	522.932	1.06503	0.532517	−	0.846419i	$-0.321246\pi$
$491$	522.932	1.06503	0.532517	+	0.846419i	$0.321246\pi$
$492$	528.612	1.07441
$493$	92.6608i	0.187953i
$494$	−6.39861	−0.0129527
$495$	95.7314i	0.193397i
$496$	582.060i	1.17351i
$497$	0	0
$498$	−824.959	−1.65654
$499$	−60.9163	−0.122077	−0.0610384	−	0.998135i	$-0.519441\pi$
$499$	−60.9163	−0.122077	−0.0610384	+	0.998135i	$0.519441\pi$
$500$	96.7662i	0.193532i
$501$	−631.167	−1.25981
$502$	− 360.062i	− 0.717255i
$503$	− 523.122i	− 1.04000i	−0.854165	−	0.520002i	$-0.825931\pi$
$503$	− 523.122i	− 1.04000i	0.854165	−	0.520002i	$-0.174069\pi$
$504$	0	0
$505$	−359.242	−0.711370
$506$	−1629.70	−3.22075
$507$	− 410.001i	− 0.808680i
$508$	−339.243	−0.667802
$509$	12.6189i	0.0247916i	0.999923	+	0.0123958i	$0.00394580\pi$
$509$	12.6189i	0.0247916i	−0.999923	+	0.0123958i	$0.996054\pi$
$510$	− 527.709i	− 1.03472i
$511$	0	0
$512$	−1119.04	−2.18563
$513$	−75.2261	−0.146640
$514$	− 1801.56i	− 3.50499i
$515$	−320.771	−0.622856
$516$	529.004i	1.02520i
$517$	− 89.8477i	− 0.173787i
$518$	0	0
$519$	551.181	1.06201
$520$	−26.0227	−0.0500437
$521$	− 250.394i	− 0.480603i	−0.970698	−	0.240302i	$-0.922754\pi$
$521$	− 250.394i	− 0.480603i	0.970698	−	0.240302i	$-0.0772463\pi$
$522$	37.2339	0.0713293
$523$	350.066i	0.669341i	0.942335	+	0.334671i	$0.108625\pi$
$523$	350.066i	0.669341i	−0.942335	+	0.334671i	$0.891375\pi$
$524$	468.218i	0.893547i
$525$	0	0
$526$	−194.157	−0.369120
$527$	−653.370	−1.23979
$528$	821.554i	1.55597i
$529$	557.058	1.05304
$530$	286.704i	0.540951i
$531$	− 113.847i	− 0.214402i
$532$	0	0
$533$	17.6406	0.0330968
$534$	−1047.55	−1.96171
$535$	141.908i	0.265249i
$536$	−757.225	−1.41273
$537$	− 455.022i	− 0.847340i
$538$	1740.42i	3.23498i
$539$	0	0
$540$	−568.829	−1.05339
$541$	409.346	0.756648	0.378324	−	0.925673i	$-0.376501\pi$
$541$	409.346	0.756648	0.378324	+	0.925673i	$0.376501\pi$
$542$	1734.06i	3.19937i
$543$	445.328	0.820125
$544$	549.883i	1.01081i
$545$	− 126.805i	− 0.232671i
$546$	0	0
$547$	189.589	0.346598	0.173299	−	0.984869i	$-0.444557\pi$
$547$	189.589	0.346598	0.173299	+	0.984869i	$0.444557\pi$
$548$	−715.495	−1.30565
$549$	107.959i	0.196646i
$550$	247.259	0.449561
$551$	− 8.69817i	− 0.0157862i
$552$	− 1327.85i	− 2.40553i
$553$	0	0
$554$	−1084.92	−1.95833
$555$	129.576	0.233470
$556$	− 852.913i	− 1.53402i
$557$	123.398	0.221541	0.110770	−	0.993846i	$-0.464668\pi$
$557$	123.398	0.221541	0.110770	+	0.993846i	$0.464668\pi$
$558$	262.544i	0.470508i
$559$	17.6537i	0.0315808i
$560$	0	0
$561$	−922.205	−1.64386
$562$	−391.942	−0.697406
$563$	− 431.235i	− 0.765958i	−0.923757	−	0.382979i	$-0.874898\pi$
$563$	− 431.235i	− 0.765958i	0.923757	−	0.382979i	$-0.125102\pi$
$564$	136.111	0.241332
$565$	− 118.837i	− 0.210331i
$566$	947.088i	1.67330i
$567$	0	0
$568$	1331.55	2.34429
$569$	861.519	1.51409	0.757046	−	0.653361i	$-0.226642\pi$
$569$	861.519	1.51409	0.757046	+	0.653361i	$0.226642\pi$
$570$	49.5366i	0.0869063i
$571$	428.938	0.751206	0.375603	−	0.926781i	$-0.377436\pi$
$571$	428.938	0.751206	0.375603	+	0.926781i	$0.377436\pi$
$572$	84.5536i	0.147821i
$573$	480.619i	0.838777i
$574$	0	0
$575$	−164.777	−0.286569
$576$	−78.2647	−0.135876
$577$	− 181.156i	− 0.313962i	−0.987602	−	0.156981i	$-0.949824\pi$
$577$	− 181.156i	− 0.313962i	0.987602	−	0.156981i	$-0.0501762\pi$
$578$	−1616.45	−2.79662
$579$	205.973i	0.355740i
$580$	− 65.7720i	− 0.113400i
$581$	0	0
$582$	−3.13407	−0.00538501
$583$	501.034	0.859407
$584$	1487.47i	2.54703i
$585$	−4.83970	−0.00827298
$586$	31.2845i	0.0533866i
$587$	− 104.998i	− 0.178872i	−0.995993	−	0.0894359i	$-0.971494\pi$
$587$	− 104.998i	− 0.178872i	0.995993	−	0.0894359i	$-0.0285064\pi$
$588$	0	0
$589$	61.3325	0.104130
$590$	−294.049	−0.498388
$591$	601.792i	1.01826i
$592$	−578.478	−0.977159
$593$	31.7968i	0.0536202i	0.999641	+	0.0268101i	$0.00853494\pi$
$593$	31.7968i	0.0536202i	−0.999641	+	0.0268101i	$0.991465\pi$
$594$	1453.48i	2.44694i
$595$	0	0
$596$	650.796	1.09194
$597$	−298.096	−0.499323
$598$	− 82.3895i	− 0.137775i
$599$	−117.835	−0.196720	−0.0983600	−	0.995151i	$-0.531360\pi$
$599$	−117.835	−0.196720	−0.0983600	+	0.995151i	$0.531360\pi$
$600$	201.462i	0.335770i
$601$	− 1044.07i	− 1.73722i	−0.495494	−	0.868611i	$-0.665013\pi$
$601$	− 1044.07i	− 1.73722i	0.495494	−	0.868611i	$-0.334987\pi$
$602$	0	0
$603$	−140.828	−0.233546
$604$	1341.46	2.22097
$605$	− 161.536i	− 0.267002i
$606$	−1390.60	−2.29472
$607$	290.747i	0.478990i	0.970898	+	0.239495i	$0.0769818\pi$
$607$	290.747i	0.478990i	−0.970898	+	0.239495i	$0.923018\pi$
$608$	− 51.6181i	− 0.0848982i
$609$	0	0
$610$	278.840	0.457114
$611$	4.54225	0.00743412
$612$	726.770i	1.18753i
$613$	−186.125	−0.303629	−0.151815	−	0.988409i	$-0.548512\pi$
$613$	−186.125	−0.303629	−0.151815	+	0.988409i	$0.548512\pi$
$614$	1590.85i	2.59096i
$615$	− 136.569i	− 0.222064i
$616$	0	0
$617$	386.307	0.626105	0.313053	−	0.949736i	$-0.398648\pi$
$617$	386.307	0.626105	0.313053	+	0.949736i	$0.398648\pi$
$618$	−1241.68	−2.00920
$619$	35.7797i	0.0578024i	0.999582	+	0.0289012i	$0.00920082\pi$
$619$	35.7797i	0.0578024i	−0.999582	+	0.0289012i	$0.990799\pi$
$620$	463.771	0.748018
$621$	− 968.622i	− 1.55978i
$622$	− 800.574i	− 1.28710i
$623$	0	0
$624$	−41.5336	−0.0665603
$625$	25.0000	0.0400000
$626$	1195.63i	1.90996i
$627$	86.5684	0.138068
$628$	− 1004.17i	− 1.59900i
$629$	− 649.349i	− 1.03235i
$630$	0	0
$631$	888.207	1.40762	0.703809	−	0.710389i	$-0.251481\pi$
$631$	888.207	1.40762	0.703809	+	0.710389i	$0.251481\pi$
$632$	1.37768	0.00217987
$633$	− 999.661i	− 1.57924i
$634$	−293.252	−0.462543
$635$	87.6451i	0.138024i
$636$	759.023i	1.19343i
$637$	0	0
$638$	−168.062	−0.263420
$639$	247.642	0.387546
$640$	382.533i	0.597707i
$641$	644.933	1.00614	0.503068	−	0.864247i	$-0.332205\pi$
$641$	644.933	1.00614	0.503068	+	0.864247i	$0.332205\pi$
$642$	549.318i	0.855635i
$643$	− 466.160i	− 0.724976i	−0.931988	−	0.362488i	$-0.881927\pi$
$643$	− 466.160i	− 0.724976i	0.931988	−	0.362488i	$-0.118073\pi$
$644$	0	0
$645$	136.671	0.211893
$646$	248.245	0.384280
$647$	− 527.798i	− 0.815762i	−0.913035	−	0.407881i	$-0.866268\pi$
$647$	− 527.798i	− 0.815762i	0.913035	−	0.407881i	$-0.133732\pi$
$648$	−725.267	−1.11924
$649$	513.870i	0.791787i
$650$	12.5001i	0.0192310i
$651$	0	0
$652$	−1646.10	−2.52469
$653$	−441.120	−0.675529	−0.337764	−	0.941231i	$-0.609671\pi$
$653$	−441.120	−0.675529	−0.337764	+	0.941231i	$0.609671\pi$
$654$	− 490.855i	− 0.750543i
$655$	120.966	0.184682
$656$	609.701i	0.929422i
$657$	276.639i	0.421064i
$658$	0	0
$659$	884.354	1.34196	0.670982	−	0.741474i	$-0.265873\pi$
$659$	884.354	1.34196	0.670982	+	0.741474i	$0.265873\pi$
$660$	654.595	0.991810
$661$	728.875i	1.10268i	0.834279	+	0.551342i	$0.185884\pi$
$661$	728.875i	1.10268i	−0.834279	+	0.551342i	$0.814116\pi$
$662$	−337.111	−0.509231
$663$	− 46.6220i	− 0.0703198i
$664$	− 1578.29i	− 2.37694i
$665$	0	0
$666$	−260.928	−0.391784
$667$	111.999	0.167914
$668$	− 2245.14i	− 3.36099i
$669$	170.054	0.254191
$670$	363.737i	0.542891i
$671$	− 487.290i	− 0.726215i
$672$	0	0
$673$	1042.57	1.54914	0.774571	−	0.632487i	$-0.217966\pi$
$673$	1042.57	1.54914	0.774571	+	0.632487i	$0.217966\pi$
$674$	−1289.96	−1.91389
$675$	146.960i	0.217718i
$676$	1458.43	2.15743
$677$	815.613i	1.20475i	0.798215	+	0.602373i	$0.205778\pi$
$677$	815.613i	1.20475i	−0.798215	+	0.602373i	$0.794222\pi$
$678$	− 460.011i	− 0.678482i
$679$	0	0
$680$	1009.60	1.48470
$681$	−252.124	−0.370225
$682$	− 1185.04i	− 1.73759i
$683$	712.143	1.04267	0.521334	−	0.853353i	$-0.325434\pi$
$683$	712.143	1.04267	0.521334	+	0.853353i	$0.325434\pi$
$684$	− 68.2226i	− 0.0997407i
$685$	184.852i	0.269856i
$686$	0	0
$687$	738.121	1.07441
$688$	−610.153	−0.886850
$689$	25.3298i	0.0367631i
$690$	−637.840	−0.924406
$691$	− 1010.69i	− 1.46265i	−0.682028	−	0.731326i	$-0.738902\pi$
$691$	− 1010.69i	− 1.46265i	0.682028	−	0.731326i	$-0.261098\pi$
$692$	1960.62i	2.83327i
$693$	0	0
$694$	1380.24	1.98882
$695$	−220.354	−0.317056
$696$	− 136.934i	− 0.196744i
$697$	−684.397	−0.981918
$698$	− 1585.67i	− 2.27174i
$699$	− 1042.47i	− 1.49137i
$700$	0	0
$701$	60.6890	0.0865748	0.0432874	−	0.999063i	$-0.486217\pi$
$701$	60.6890	0.0865748	0.0432874	+	0.999063i	$0.486217\pi$
$702$	−73.4807	−0.104673
$703$	60.9551i	0.0867071i
$704$	353.262	0.501792
$705$	− 35.1650i	− 0.0498795i
$706$	1017.61i	1.44138i
$707$	0	0
$708$	−778.468	−1.09953
$709$	649.002	0.915377	0.457688	−	0.889113i	$-0.348678\pi$
$709$	649.002	0.915377	0.457688	+	0.889113i	$0.348678\pi$
$710$	− 639.619i	− 0.900872i
$711$	0.256220	0.000360366 0
$712$	− 2004.15i	− 2.81481i
$713$	789.727i	1.10761i
$714$	0	0
$715$	21.8448	0.0305522
$716$	1618.57	2.26057
$717$	− 266.710i	− 0.371981i
$718$	1530.31	2.13136
$719$	210.187i	0.292332i	0.989260	+	0.146166i	$0.0466934\pi$
$719$	210.187i	0.292332i	−0.989260	+	0.146166i	$0.953307\pi$
$720$	− 167.271i	− 0.232321i
$721$	0	0
$722$	1260.92	1.74642
$723$	695.710	0.962254
$724$	1584.09i	2.18797i
$725$	−16.9925	−0.0234379
$726$	− 625.295i	− 0.861288i
$727$	743.893i	1.02324i	0.859213	+	0.511618i	$0.170954\pi$
$727$	743.893i	1.02324i	−0.859213	+	0.511618i	$0.829046\pi$
$728$	0	0
$729$	−778.520	−1.06793
$730$	714.513	0.978785
$731$	− 684.905i	− 0.936942i
$732$	738.202	1.00847
$733$	354.004i	0.482952i	0.970407	+	0.241476i	$0.0776315\pi$
$733$	354.004i	0.482952i	−0.970407	+	0.241476i	$0.922369\pi$
$734$	− 1187.88i	− 1.61836i
$735$	0	0
$736$	664.643	0.903047
$737$	635.654	0.862488
$738$	275.011i	0.372644i
$739$	−1239.21	−1.67688	−0.838438	−	0.544997i	$-0.816531\pi$
$739$	−1239.21	−1.67688	−0.838438	+	0.544997i	$0.816531\pi$
$740$	460.917i	0.622861i
$741$	4.37646i	0.00590615i
$742$	0	0
$743$	−1060.01	−1.42666	−0.713328	−	0.700830i	$-0.752813\pi$
$743$	−1060.01	−1.42666	−0.713328	+	0.700830i	$0.752813\pi$
$744$	965.547	1.29778
$745$	− 168.136i	− 0.225686i
$746$	−175.312	−0.235002
$747$	− 293.529i	− 0.392944i
$748$	− 3280.40i	− 4.38556i
$749$	0	0
$750$	96.7733	0.129031
$751$	−1019.98	−1.35817	−0.679084	−	0.734061i	$-0.737623\pi$
$751$	−1019.98	−1.35817	−0.679084	+	0.734061i	$0.737623\pi$
$752$	156.991i	0.208765i
$753$	−246.272	−0.327054
$754$	− 8.49636i	− 0.0112684i
$755$	− 346.574i	− 0.459038i
$756$	0	0
$757$	1023.43	1.35196	0.675978	−	0.736921i	$-0.263721\pi$
$757$	1023.43	1.35196	0.675978	+	0.736921i	$0.263721\pi$
$758$	−2260.71	−2.98246
$759$	1114.67i	1.46860i
$760$	−94.7719	−0.124700
$761$	448.176i	0.588930i	0.955662	+	0.294465i	$0.0951414\pi$
$761$	448.176i	0.588930i	−0.955662	+	0.294465i	$0.904859\pi$
$762$	339.268i	0.445234i
$763$	0	0
$764$	−1709.62	−2.23773
$765$	187.764	0.245444
$766$	346.969i	0.452962i
$767$	−25.9787	−0.0338705
$768$	1233.43i	1.60603i
$769$	1383.52i	1.79912i	0.436798	+	0.899560i	$0.356112\pi$
$769$	1383.52i	1.79912i	−0.436798	+	0.899560i	$0.643888\pi$
$770$	0	0
$771$	−1232.22	−1.59821
$772$	−732.673	−0.949059
$773$	305.717i	0.395494i	0.980253	+	0.197747i	$0.0633625\pi$
$773$	305.717i	0.395494i	−0.980253	+	0.197747i	$0.936638\pi$
$774$	−275.215	−0.355575
$775$	− 119.818i	− 0.154603i
$776$	− 5.99601i	− 0.00772682i
$777$	0	0
$778$	−2386.52	−3.06750
$779$	64.2451	0.0824712
$780$	33.0930i	0.0424269i
$781$	−1117.78	−1.43121
$782$	3196.44i	4.08752i
$783$	− 99.8885i	− 0.127571i
$784$	0	0
$785$	−259.432	−0.330487
$786$	468.253	0.595742
$787$	562.016i	0.714125i	0.934081	+	0.357062i	$0.116222\pi$
$787$	562.016i	0.714125i	−0.934081	+	0.357062i	$0.883778\pi$
$788$	−2140.65	−2.71656
$789$	132.798i	0.168311i
$790$	− 0.661776i	0 0.000837691i
$791$	0	0
$792$	−708.963	−0.895155
$793$	24.6349	0.0310655
$794$	1982.72i	2.49713i
$795$	196.097	0.246663
$796$	− 1060.37i	− 1.33212i
$797$	1187.30i	1.48971i	0.667226	+	0.744855i	$0.267482\pi$
$797$	1187.30i	1.48971i	−0.667226	+	0.744855i	$0.732518\pi$
$798$	0	0
$799$	−176.224	−0.220556
$800$	−100.840	−0.126050
$801$	− 372.731i	− 0.465332i
$802$	2116.68	2.63925
$803$	− 1248.66i	− 1.55499i
$804$	962.960i	1.19771i
$805$	0	0
$806$	59.9095	0.0743294
$807$	1190.39	1.47509
$808$	− 2660.46i	− 3.29265i
$809$	−109.659	−0.135549	−0.0677743	−	0.997701i	$-0.521590\pi$
$809$	−109.659	−0.135549	−0.0677743	+	0.997701i	$0.521590\pi$
$810$	348.386i	0.430106i
$811$	− 343.153i	− 0.423123i	−0.977365	−	0.211561i	$-0.932145\pi$
$811$	− 343.153i	− 0.423123i	0.977365	−	0.211561i	$-0.0678549\pi$
$812$	0	0
$813$	1186.04	1.45885
$814$	1177.74	1.44686
$815$	425.277i	0.521812i
$816$	1611.37	1.97472
$817$	64.2927i	0.0786937i
$818$	1621.82i	1.98267i
$819$	0	0
$820$	485.795	0.592433
$821$	−472.563	−0.575595	−0.287797	−	0.957691i	$-0.592923\pi$
$821$	−472.563	−0.575595	−0.287797	+	0.957691i	$0.592923\pi$
$822$	715.548i	0.870496i
$823$	−1534.93	−1.86505	−0.932523	−	0.361111i	$-0.882398\pi$
$823$	−1534.93	−1.86505	−0.932523	+	0.361111i	$0.882398\pi$
$824$	− 2375.55i	− 2.88295i
$825$	− 169.118i	− 0.204991i
$826$	0	0
$827$	−421.191	−0.509299	−0.254650	−	0.967033i	$-0.581960\pi$
$827$	−421.191	−0.509299	−0.254650	+	0.967033i	$0.581960\pi$
$828$	878.445	1.06092
$829$	− 1357.03i	− 1.63695i	−0.574539	−	0.818477i	$-0.694819\pi$
$829$	− 1357.03i	− 1.63695i	0.574539	−	0.818477i	$-0.305181\pi$
$830$	−758.138	−0.913419
$831$	742.050i	0.892961i
$832$	17.8591i	0.0214653i
$833$	0	0
$834$	−852.976	−1.02275
$835$	−580.043	−0.694662
$836$	307.935i	0.368343i
$837$	704.333	0.841497
$838$	429.243i	0.512224i
$839$	− 45.8593i	− 0.0546595i	−0.999626	−	0.0273297i	$-0.991300\pi$
$839$	− 45.8593i	− 0.0546595i	0.999626	−	0.0273297i	$-0.00870041\pi$
$840$	0	0
$841$	−829.450	−0.986267
$842$	−736.017	−0.874130
$843$	268.077i	0.318004i
$844$	3555.92	4.21318
$845$	− 376.791i	− 0.445907i
$846$	70.8122i	0.0837024i
$847$	0	0
$848$	−875.457	−1.03238
$849$	647.780	0.762992
$850$	− 484.965i	− 0.570547i
$851$	−784.867	−0.922288
$852$	− 1693.33i	− 1.98748i
$853$	− 713.413i	− 0.836358i	−0.908365	−	0.418179i	$-0.862668\pi$
$853$	− 713.413i	− 0.836358i	0.908365	−	0.418179i	$-0.137332\pi$
$854$	0	0
$855$	−17.6256	−0.0206148
$856$	−1050.94	−1.22773
$857$	172.083i	0.200797i	0.994947	+	0.100399i	$0.0320118\pi$
$857$	172.083i	0.200797i	−0.994947	+	0.100399i	$0.967988\pi$
$858$	84.5598	0.0985546
$859$	− 260.716i	− 0.303511i	−0.988418	−	0.151756i	$-0.951507\pi$
$859$	− 260.716i	− 0.303511i	0.988418	−	0.151756i	$-0.0484927\pi$
$860$	486.155i	0.565297i
$861$	0	0
$862$	−959.990	−1.11368
$863$	−1090.10	−1.26315	−0.631576	−	0.775314i	$-0.717592\pi$
$863$	−1090.10	−1.26315	−0.631576	+	0.775314i	$0.717592\pi$
$864$	− 592.775i	− 0.686082i
$865$	506.536	0.585591
$866$	30.8727i	0.0356497i
$867$	1105.60i	1.27520i
$868$	0	0
$869$	−1.15650	−0.00133083
$870$	−65.7768	−0.0756056
$871$	32.1355i	0.0368949i
$872$	939.090	1.07694
$873$	− 1.11514i	− 0.00127736i
$874$	− 300.053i	− 0.343310i
$875$	0	0
$876$	1891.61	2.15937
$877$	−651.118	−0.742437	−0.371219	−	0.928545i	$-0.621060\pi$
$877$	−651.118	−0.742437	−0.371219	+	0.928545i	$0.621060\pi$
$878$	− 12.7835i	− 0.0145598i
$879$	21.3977	0.0243432
$880$	755.009i	0.857965i
$881$	− 1649.45i	− 1.87225i	−0.351671	−	0.936123i	$-0.614387\pi$
$881$	− 1649.45i	− 1.87225i	0.351671	−	0.936123i	$-0.385613\pi$
$882$	0	0
$883$	−1487.78	−1.68492	−0.842460	−	0.538759i	$-0.818893\pi$
$883$	−1487.78	−1.68492	−0.842460	+	0.538759i	$0.818893\pi$
$884$	165.841	0.187602
$885$	201.121i	0.227255i
$886$	1438.62	1.62372
$887$	− 153.911i	− 0.173518i	−0.996229	−	0.0867592i	$-0.972349\pi$
$887$	− 153.911i	− 0.173518i	0.996229	−	0.0867592i	$-0.0276511\pi$
$888$	959.605i	1.08064i
$889$	0	0
$890$	−962.703	−1.08169
$891$	608.827	0.683307
$892$	604.902i	0.678142i
$893$	16.5424	0.0185245
$894$	− 650.844i	− 0.728014i
$895$	− 418.165i	− 0.467224i
$896$	0	0
$897$	−56.3520	−0.0628227
$898$	1208.35	1.34560
$899$	81.4400i	0.0905895i
$900$	−133.278	−0.148087
$901$	− 982.712i	− 1.09069i
$902$	− 1241.31i	− 1.37618i
$903$	0	0
$904$	880.078	0.973538
$905$	409.257	0.452218
$906$	− 1341.56i	− 1.48075i
$907$	1051.91	1.15977	0.579883	−	0.814700i	$-0.303098\pi$
$907$	1051.91	1.15977	0.579883	+	0.814700i	$0.303098\pi$
$908$	− 896.836i	− 0.987705i
$909$	− 494.791i	− 0.544324i
$910$	0	0
$911$	−1305.28	−1.43279	−0.716397	−	0.697693i	$-0.754210\pi$
$911$	−1305.28	−1.43279	−0.716397	+	0.697693i	$0.754210\pi$
$912$	−151.261	−0.165856
$913$	1324.89i	1.45114i
$914$	−2263.49	−2.47647
$915$	− 190.718i	− 0.208435i
$916$	2625.59i	2.86636i
$917$	0	0
$918$	2850.81	3.10546
$919$	−1709.70	−1.86039	−0.930194	−	0.367069i	$-0.880361\pi$
$919$	−1709.70	−1.86039	−0.930194	+	0.367069i	$0.880361\pi$
$920$	− 1220.30i	− 1.32641i
$921$	1088.10	1.18143
$922$	− 1720.79i	− 1.86637i
$923$	− 56.5091i	− 0.0612233i
$924$	0	0
$925$	119.080	0.128735
$926$	1457.59	1.57408
$927$	− 441.804i	− 0.476596i
$928$	68.5408	0.0738586
$929$	812.099i	0.874164i	0.899422	+	0.437082i	$0.143988\pi$
$929$	812.099i	0.874164i	−0.899422	+	0.437082i	$0.856012\pi$
$930$	− 463.805i	− 0.498716i
$931$	0	0
$932$	3708.19	3.97874
$933$	−547.569	−0.586890
$934$	− 1304.16i	− 1.39632i
$935$	−847.507	−0.906425
$936$	− 35.8416i	− 0.0382923i
$937$	− 996.727i	− 1.06374i	−0.846825	−	0.531871i	$-0.821489\pi$
$937$	− 996.727i	− 1.06374i	0.846825	−	0.531871i	$-0.178511\pi$
$938$	0	0
$939$	817.777	0.870902
$940$	125.087	0.133071
$941$	1535.84i	1.63214i	0.577956	+	0.816068i	$0.303850\pi$
$941$	1535.84i	1.63214i	−0.577956	+	0.816068i	$0.696150\pi$
$942$	−1004.24	−1.06608
$943$	827.229i	0.877232i
$944$	− 897.884i	− 0.951149i
$945$	0	0
$946$	1242.23	1.31314
$947$	1195.73	1.26265	0.631323	−	0.775520i	$-0.282512\pi$
$947$	1195.73	1.26265	0.631323	+	0.775520i	$0.282512\pi$
$948$	− 1.75199i	− 0.00184809i
$949$	63.1258	0.0665183
$950$	45.5242i	0.0479202i
$951$	200.576i	0.210911i
$952$	0	0
$953$	770.228	0.808214	0.404107	−	0.914712i	$-0.367582\pi$
$953$	770.228	0.808214	0.404107	+	0.914712i	$0.367582\pi$
$954$	−394.883	−0.413924
$955$	441.689i	0.462502i
$956$	948.723	0.992388
$957$	114.949i	0.120114i
$958$	− 302.083i	− 0.315327i
$959$	0	0
$960$	138.261	0.144022
$961$	386.750	0.402446
$962$	59.5408i	0.0618927i
$963$	−195.453	−0.202963
$964$	2474.73i	2.56715i
$965$	189.290i	0.196155i
$966$	0	0
$967$	−628.771	−0.650229	−0.325114	−	0.945675i	$-0.605403\pi$
$967$	−628.771	−0.650229	−0.325114	+	0.945675i	$0.605403\pi$
$968$	1196.30	1.23584
$969$	− 169.792i	− 0.175224i
$970$	−2.88022	−0.00296930
$971$	674.788i	0.694941i	0.937691	+	0.347471i	$0.112959\pi$
$971$	674.788i	0.694941i	−0.937691	+	0.347471i	$0.887041\pi$
$972$	− 1367.17i	− 1.40656i
$973$	0	0
$974$	−116.489	−0.119599
$975$	8.54973	0.00876896
$976$	851.442i	0.872379i
$977$	−1155.00	−1.18220	−0.591098	−	0.806600i	$-0.701305\pi$
$977$	−1155.00	−1.18220	−0.591098	+	0.806600i	$0.701305\pi$
$978$	1646.22i	1.68325i
$979$	1682.39i	1.71847i
$980$	0	0
$981$	174.652	0.178034
$982$	1860.27	1.89437
$983$	239.246i	0.243383i	0.992568	+	0.121692i	$0.0388319\pi$
$983$	239.246i	0.243383i	−0.992568	+	0.121692i	$0.961168\pi$
$984$	1011.40	1.02784
$985$	553.048i	0.561470i
$986$	329.631i	0.334311i
$987$	0	0
$988$	−15.5676	−0.0157567
$989$	−827.843	−0.837051
$990$	340.554i	0.343994i
$991$	−712.460	−0.718930	−0.359465	−	0.933159i	$-0.617041\pi$
$991$	−712.460	−0.718930	−0.359465	+	0.933159i	$0.617041\pi$
$992$	483.295i	0.487192i
$993$	230.574i	0.232199i
$994$	0	0
$995$	−273.951	−0.275327
$996$	−2007.10	−2.01516
$997$	− 1395.61i	− 1.39981i	−0.714235	−	0.699906i	$-0.753225\pi$
$997$	− 1395.61i	− 1.39981i	0.714235	−	0.699906i	$-0.246775\pi$
$998$	−216.703	−0.217137
$999$	699.999i	0.700700i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.3.d.a.146.11		12
7.2	even	3		245.3.h.c.31.1		12
7.3	odd	6		245.3.h.c.166.1		12
7.4	even	3		35.3.h.a.26.1	✓	12
7.5	odd	6		35.3.h.a.31.1	yes	12
7.6	odd	2	inner	245.3.d.a.146.12		12
21.5	even	6		315.3.w.c.136.6		12
21.11	odd	6		315.3.w.c.271.6		12
28.11	odd	6		560.3.bx.c.481.5		12
28.19	even	6		560.3.bx.c.241.5		12
35.4	even	6		175.3.i.d.26.6		12
35.12	even	12		175.3.j.b.24.12		24
35.18	odd	12		175.3.j.b.124.12		24
35.19	odd	6		175.3.i.d.101.6		12
35.32	odd	12		175.3.j.b.124.1		24
35.33	even	12		175.3.j.b.24.1		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.3.h.a.26.1	✓	12	7.4	even	3
35.3.h.a.31.1	yes	12	7.5	odd	6
175.3.i.d.26.6		12	35.4	even	6
175.3.i.d.101.6		12	35.19	odd	6
175.3.j.b.24.1		24	35.33	even	12
175.3.j.b.24.12		24	35.12	even	12
175.3.j.b.124.1		24	35.32	odd	12
175.3.j.b.124.12		24	35.18	odd	12
245.3.d.a.146.11		12	1.1	even	1	trivial
245.3.d.a.146.12		12	7.6	odd	2	inner
245.3.h.c.31.1		12	7.2	even	3
245.3.h.c.166.1		12	7.3	odd	6
315.3.w.c.136.6		12	21.5	even	6
315.3.w.c.271.6		12	21.11	odd	6
560.3.bx.c.241.5		12	28.19	even	6
560.3.bx.c.481.5		12	28.11	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 245 = 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	245.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.55739 q^{2} -2.43315i q^{3} +8.65503 q^{4} -2.23607i q^{5} -8.65567i q^{6} +16.5598 q^{8} +3.07978 q^{9} +O(q^{10})\)	\(q+3.55739 q^{2} -2.43315i q^{3} +8.65503 q^{4} -2.23607i q^{5} -8.65567i q^{6} +16.5598 q^{8} +3.07978 q^{9} -7.95457i q^{10} -13.9011 q^{11} -21.0590i q^{12} -0.702771i q^{13} -5.44069 q^{15} +24.2894 q^{16} +27.2652i q^{17} +10.9560 q^{18} -2.55941i q^{19} -19.3532i q^{20} -49.4517 q^{22} +32.9554 q^{23} -40.2924i q^{24} -5.00000 q^{25} -2.50003i q^{26} -29.3919i q^{27} +3.39850 q^{29} -19.3547 q^{30} +23.9635i q^{31} +20.1679 q^{32} +33.8235i q^{33} +96.9930i q^{34} +26.6556 q^{36} -23.8160 q^{37} -9.10483i q^{38} -1.70995 q^{39} -37.0288i q^{40} +25.1015i q^{41} -25.1201 q^{43} -120.315 q^{44} -6.88659i q^{45} +117.235 q^{46} +6.46334i q^{47} -59.0998i q^{48} -17.7870 q^{50} +66.3403 q^{51} -6.08250i q^{52} -36.0427 q^{53} -104.559i q^{54} +31.0838i q^{55} -6.22744 q^{57} +12.0898 q^{58} -36.9661i q^{59} -47.0893 q^{60} +35.0540i q^{61} +85.2476i q^{62} -25.4125 q^{64} -1.57144 q^{65} +120.323i q^{66} -45.7268 q^{67} +235.981i q^{68} -80.1854i q^{69} +80.4090 q^{71} +51.0004 q^{72} +89.8242i q^{73} -84.7229 q^{74} +12.1658i q^{75} -22.1518i q^{76} -6.08295 q^{78} +0.0831944 q^{79} -54.3128i q^{80} -43.7970 q^{81} +89.2958i q^{82} -95.3085i q^{83} +60.9668 q^{85} -89.3621 q^{86} -8.26907i q^{87} -230.199 q^{88} -121.025i q^{89} -24.4983i q^{90} +285.230 q^{92} +58.3068 q^{93} +22.9926i q^{94} -5.72302 q^{95} -49.0716i q^{96} -0.362083i q^{97} -42.8124 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{2} + 20 q^{4} - 4 q^{8} - 28 q^{9}+O(q^{10}) \) 12 * q + 4 * q^2 + 20 * q^4 - 4 * q^8 - 28 * q^9	\( 12 q + 4 q^{2} + 20 q^{4} - 4 q^{8} - 28 q^{9} + 28 q^{11} - 20 q^{15} + 44 q^{16} - 128 q^{18} - 88 q^{22} + 28 q^{23} - 60 q^{25} + 64 q^{29} - 40 q^{30} + 108 q^{32} + 156 q^{36} - 88 q^{37} + 48 q^{39} - 4 q^{43} - 12 q^{44} + 428 q^{46} - 20 q^{50} + 264 q^{51} - 392 q^{53} - 48 q^{57} - 316 q^{58} - 300 q^{60} - 140 q^{64} - 60 q^{65} + 276 q^{67} - 8 q^{71} + 392 q^{72} - 100 q^{74} - 312 q^{78} + 24 q^{79} + 620 q^{81} + 80 q^{86} - 1208 q^{88} + 732 q^{92} - 168 q^{93} - 120 q^{95} - 44 q^{99}+O(q^{100}) \) 12 * q + 4 * q^2 + 20 * q^4 - 4 * q^8 - 28 * q^9 + 28 * q^11 - 20 * q^15 + 44 * q^16 - 128 * q^18 - 88 * q^22 + 28 * q^23 - 60 * q^25 + 64 * q^29 - 40 * q^30 + 108 * q^32 + 156 * q^36 - 88 * q^37 + 48 * q^39 - 4 * q^43 - 12 * q^44 + 428 * q^46 - 20 * q^50 + 264 * q^51 - 392 * q^53 - 48 * q^57 - 316 * q^58 - 300 * q^60 - 140 * q^64 - 60 * q^65 + 276 * q^67 - 8 * q^71 + 392 * q^72 - 100 * q^74 - 312 * q^78 + 24 * q^79 + 620 * q^81 + 80 * q^86 - 1208 * q^88 + 732 * q^92 - 168 * q^93 - 120 * q^95 - 44 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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