LMFDB - Embedded newform 1183.2.c.h.337.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1183,2,Mod(337,1183)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1183, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1183.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1183 = 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1183.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$9.44630255912$
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} + 16x^{10} + 96x^{8} + 266x^{6} + 332x^{4} + 141x^{2} + 1 $ x^12 + 16x^10 + 96x^8 + 266x^6 + 332x^4 + 141*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.12
Root			$-1.71083i$ of defining polynomial
Character	$\chi$	$=$	1183.337
Dual form			1183.2.c.h.337.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.63777i q^{2} -2.71083 q^{3} -4.95781 q^{4} -2.08281i q^{5} -7.15053i q^{6} +1.00000i q^{7} -7.80201i q^{8} +4.34860 q^{9} +O(q^{10})$	\(q+2.63777i q^{2} -2.71083 q^{3} -4.95781 q^{4} -2.08281i q^{5} -7.15053i q^{6} +1.00000i q^{7} -7.80201i q^{8} +4.34860 q^{9} +5.49396 q^{10} +3.60559i q^{11} +13.4398 q^{12} -2.63777 q^{14} +5.64614i q^{15} +10.6643 q^{16} +5.76942 q^{17} +11.4706i q^{18} +1.36230i q^{19} +10.3262i q^{20} -2.71083i q^{21} -9.51070 q^{22} +4.22817 q^{23} +21.1499i q^{24} +0.661912 q^{25} -3.65581 q^{27} -4.95781i q^{28} -8.29095 q^{29} -14.8932 q^{30} -0.734580i q^{31} +12.5258i q^{32} -9.77414i q^{33} +15.2184i q^{34} +2.08281 q^{35} -21.5595 q^{36} -2.16348i q^{37} -3.59344 q^{38} -16.2501 q^{40} -2.86719i q^{41} +7.15053 q^{42} -4.26879 q^{43} -17.8758i q^{44} -9.05729i q^{45} +11.1529i q^{46} +8.68467i q^{47} -28.9090 q^{48} -1.00000 q^{49} +1.74597i q^{50} -15.6399 q^{51} -10.9872 q^{53} -9.64317i q^{54} +7.50975 q^{55} +7.80201 q^{56} -3.69297i q^{57} -21.8696i q^{58} -6.75315i q^{59} -27.9925i q^{60} +10.5003 q^{61} +1.93765 q^{62} +4.34860i q^{63} -11.7116 q^{64} +25.7819 q^{66} +4.70356i q^{67} -28.6037 q^{68} -11.4619 q^{69} +5.49396i q^{70} +13.3023i q^{71} -33.9278i q^{72} +4.36010i q^{73} +5.70675 q^{74} -1.79433 q^{75} -6.75404i q^{76} -3.60559 q^{77} -1.73764 q^{79} -22.2116i q^{80} -3.13551 q^{81} +7.56297 q^{82} +5.74467i q^{83} +13.4398i q^{84} -12.0166i q^{85} -11.2601i q^{86} +22.4754 q^{87} +28.1308 q^{88} +8.41205i q^{89} +23.8910 q^{90} -20.9625 q^{92} +1.99132i q^{93} -22.9081 q^{94} +2.83742 q^{95} -33.9553i q^{96} +6.20363i q^{97} -2.63777i q^{98} +15.6792i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 8 q^{3} - 16 q^{4}+O(q^{10}) $ 12 * q - 8 * q^3 - 16 * q^4	$ 12 q - 8 q^{3} - 16 q^{4} + 28 q^{10} + 46 q^{12} - 4 q^{14} + 46 q^{17} - 8 q^{22} + 36 q^{23} + 20 q^{25} - 20 q^{27} - 30 q^{29} - 28 q^{30} - 4 q^{35} - 44 q^{36} + 22 q^{38} - 28 q^{40} + 16 q^{42} + 36 q^{43} - 22 q^{48} - 12 q^{49} - 28 q^{51} - 50 q^{53} + 6 q^{56} + 32 q^{61} + 18 q^{62} + 14 q^{64} + 32 q^{66} - 68 q^{68} + 2 q^{69} - 28 q^{74} - 30 q^{75} + 16 q^{77} + 4 q^{79} - 12 q^{81} + 20 q^{82} + 26 q^{87} + 96 q^{88} - 64 q^{92} - 28 q^{94} + 14 q^{95}+O(q^{100}) $ 12 * q - 8 * q^3 - 16 * q^4 + 28 * q^10 + 46 * q^12 - 4 * q^14 + 46 * q^17 - 8 * q^22 + 36 * q^23 + 20 * q^25 - 20 * q^27 - 30 * q^29 - 28 * q^30 - 4 * q^35 - 44 * q^36 + 22 * q^38 - 28 * q^40 + 16 * q^42 + 36 * q^43 - 22 * q^48 - 12 * q^49 - 28 * q^51 - 50 * q^53 + 6 * q^56 + 32 * q^61 + 18 * q^62 + 14 * q^64 + 32 * q^66 - 68 * q^68 + 2 * q^69 - 28 * q^74 - 30 * q^75 + 16 * q^77 + 4 * q^79 - 12 * q^81 + 20 * q^82 + 26 * q^87 + 96 * q^88 - 64 * q^92 - 28 * q^94 + 14 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times$.

$n$	$339$	$1016$
$\chi(n)$	$1$	$-1$

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 $

$2$	2.63777i	1.86518i	0.360935	+	0.932591i	$0.382458\pi$
$2$	2.63777i	1.86518i	−0.360935	+	0.932591i	$0.617542\pi$
$3$	−2.71083	−1.56510	−0.782549	−	0.622589i	$-0.786081\pi$
$3$	−2.71083	−1.56510	−0.782549	+	0.622589i	$0.786081\pi$
$4$	−4.95781	−2.47890
$5$	− 2.08281i	− 0.931460i	−0.884927	−	0.465730i	$-0.845792\pi$
$5$	− 2.08281i	− 0.931460i	0.884927	−	0.465730i	$-0.154208\pi$
$6$	− 7.15053i	− 2.91919i
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	− 7.80201i	− 2.75843i
$9$	4.34860	1.44953
$10$	5.49396	1.73734
$11$	3.60559i	1.08713i	0.839368	+	0.543563i	$0.182925\pi$
$11$	3.60559i	1.08713i	−0.839368	+	0.543563i	$0.817075\pi$
$12$	13.4398	3.87973
$13$	0	0
$13$	0	0
$14$	−2.63777	−0.704973
$15$	5.64614i	1.45783i
$16$	10.6643	2.66606
$17$	5.76942	1.39929	0.699645	−	0.714491i	$-0.253341\pi$
$17$	5.76942	1.39929	0.699645	+	0.714491i	$0.253341\pi$
$18$	11.4706i	2.70364i
$19$	1.36230i	0.312534i	0.987715	+	0.156267i	$0.0499460\pi$
$19$	1.36230i	0.312534i	−0.987715	+	0.156267i	$0.950054\pi$
$20$	10.3262i	2.30900i
$21$	− 2.71083i	− 0.591551i
$22$	−9.51070	−2.02769
$23$	4.22817	0.881635	0.440817	−	0.897597i	$-0.354689\pi$
$23$	4.22817	0.881635	0.440817	+	0.897597i	$0.354689\pi$
$24$	21.1499i	4.31721i
$25$	0.661912	0.132382
$26$	0	0
$27$	−3.65581	−0.703561
$28$	− 4.95781i	− 0.936938i
$29$	−8.29095	−1.53959	−0.769796	−	0.638290i	$-0.779642\pi$
$29$	−8.29095	−1.53959	−0.769796	+	0.638290i	$0.779642\pi$
$30$	−14.8932	−2.71911
$31$	− 0.734580i	− 0.131934i	−0.997822	−	0.0659672i	$-0.978987\pi$
$31$	− 0.734580i	− 0.131934i	0.997822	−	0.0659672i	$-0.0210133\pi$
$32$	12.5258i	2.21427i
$33$	− 9.77414i	− 1.70146i
$34$	15.2184i	2.60993i
$35$	2.08281	0.352059
$36$	−21.5595	−3.59325
$37$	− 2.16348i	− 0.355674i	−0.984060	−	0.177837i	$-0.943090\pi$
$37$	− 2.16348i	− 0.355674i	0.984060	−	0.177837i	$-0.0569099\pi$
$38$	−3.59344	−0.582933
$39$	0	0
$40$	−16.2501	−2.56936
$41$	− 2.86719i	− 0.447779i	−0.974614	−	0.223890i	$-0.928124\pi$
$41$	− 2.86719i	− 0.447779i	0.974614	−	0.223890i	$-0.0718755\pi$
$42$	7.15053	1.10335
$43$	−4.26879	−0.650984	−0.325492	−	0.945545i	$-0.605530\pi$
$43$	−4.26879	−0.650984	−0.325492	+	0.945545i	$0.605530\pi$
$44$	− 17.8758i	− 2.69488i
$45$	− 9.05729i	− 1.35018i
$46$	11.1529i	1.64441i
$47$	8.68467i	1.26679i	0.773829	+	0.633395i	$0.218339\pi$
$47$	8.68467i	1.26679i	−0.773829	+	0.633395i	$0.781661\pi$
$48$	−28.9090	−4.17265
$49$	−1.00000	−0.142857
$50$	1.74597i	0.246917i
$51$	−15.6399	−2.19003
$52$	0	0
$53$	−10.9872	−1.50921	−0.754603	−	0.656182i	$-0.772171\pi$
$53$	−10.9872	−1.50921	−0.754603	+	0.656182i	$0.772171\pi$
$54$	− 9.64317i	− 1.31227i
$55$	7.50975	1.01261
$56$	7.80201	1.04259
$57$	− 3.69297i	− 0.489146i
$58$	− 21.8696i	− 2.87162i
$59$	− 6.75315i	− 0.879186i	−0.898197	−	0.439593i	$-0.855123\pi$
$59$	− 6.75315i	− 0.879186i	0.898197	−	0.439593i	$-0.144877\pi$
$60$	− 27.9925i	− 3.61381i
$61$	10.5003	1.34443	0.672216	−	0.740355i	$-0.265343\pi$
$61$	10.5003	1.34443	0.672216	+	0.740355i	$0.265343\pi$
$62$	1.93765	0.246082
$63$	4.34860i	0.547871i
$64$	−11.7116	−1.46395
$65$	0	0
$66$	25.7819	3.17353
$67$	4.70356i	0.574631i	0.957836	+	0.287316i	$0.0927629\pi$
$67$	4.70356i	0.574631i	−0.957836	+	0.287316i	$0.907237\pi$
$68$	−28.6037	−3.46871
$69$	−11.4619	−1.37984
$70$	5.49396i	0.656654i
$71$	13.3023i	1.57869i	0.613949	+	0.789345i	$0.289580\pi$
$71$	13.3023i	1.57869i	−0.613949	+	0.789345i	$0.710420\pi$
$72$	− 33.9278i	− 3.99843i
$73$	4.36010i	0.510312i	0.966900	+	0.255156i	$0.0821268\pi$
$73$	4.36010i	0.510312i	−0.966900	+	0.255156i	$0.917873\pi$
$74$	5.70675	0.663396
$75$	−1.79433	−0.207191
$76$	− 6.75404i	− 0.774742i
$77$	−3.60559	−0.410895
$78$	0	0
$79$	−1.73764	−0.195500	−0.0977499	−	0.995211i	$-0.531165\pi$
$79$	−1.73764	−0.195500	−0.0977499	+	0.995211i	$0.531165\pi$
$80$	− 22.2116i	− 2.48333i
$81$	−3.13551	−0.348390
$82$	7.56297	0.835190
$83$	5.74467i	0.630559i	0.948999	+	0.315280i	$0.102098\pi$
$83$	5.74467i	0.630559i	−0.948999	+	0.315280i	$0.897902\pi$
$84$	13.4398i	1.46640i
$85$	− 12.0166i	− 1.30338i
$86$	− 11.2601i	− 1.21420i
$87$	22.4754	2.40961
$88$	28.1308	2.99876
$89$	8.41205i	0.891675i	0.895114	+	0.445838i	$0.147094\pi$
$89$	8.41205i	0.891675i	−0.895114	+	0.445838i	$0.852906\pi$
$90$	23.8910	2.51833
$91$	0	0
$92$	−20.9625	−2.18549
$93$	1.99132i	0.206490i
$94$	−22.9081	−2.36279
$95$	2.83742	0.291113
$96$	− 33.9553i	− 3.46554i
$97$	6.20363i	0.629883i	0.949111	+	0.314942i	$0.101985\pi$
$97$	6.20363i	0.629883i	−0.949111	+	0.314942i	$0.898015\pi$
$98$	− 2.63777i	− 0.266455i
$99$	15.6792i	1.57582i
$100$	−3.28163	−0.328163
$101$	6.33082	0.629940	0.314970	−	0.949102i	$-0.398006\pi$
$101$	6.33082	0.629940	0.314970	+	0.949102i	$0.398006\pi$
$102$	− 41.2544i	− 4.08480i
$103$	−7.64462	−0.753247	−0.376623	−	0.926367i	$-0.622915\pi$
$103$	−7.64462	−0.753247	−0.376623	+	0.926367i	$0.622915\pi$
$104$	0	0
$105$	−5.64614	−0.551006
$106$	− 28.9816i	− 2.81494i
$107$	2.75373	0.266213	0.133107	−	0.991102i	$-0.457505\pi$
$107$	2.75373	0.266213	0.133107	+	0.991102i	$0.457505\pi$
$108$	18.1248	1.74406
$109$	4.89736i	0.469082i	0.972106	+	0.234541i	$0.0753587\pi$
$109$	4.89736i	0.469082i	−0.972106	+	0.234541i	$0.924641\pi$
$110$	19.8090i	1.88871i
$111$	5.86482i	0.556664i
$112$	10.6643i	1.00768i
$113$	−17.4120	−1.63798	−0.818991	−	0.573807i	$-0.805466\pi$
$113$	−17.4120	−1.63798	−0.818991	+	0.573807i	$0.805466\pi$
$114$	9.74120	0.912347
$115$	− 8.80647i	− 0.821207i
$116$	41.1050	3.81650
$117$	0	0
$118$	17.8132	1.63984
$119$	5.76942i	0.528882i
$120$	44.0512	4.02131
$121$	−2.00027	−0.181843
$122$	27.6975i	2.50761i
$123$	7.77245i	0.700819i
$124$	3.64191i	0.327053i
$125$	− 11.7927i	− 1.05477i
$126$	−11.4706	−1.02188
$127$	−11.0903	−0.984102	−0.492051	−	0.870566i	$-0.663753\pi$
$127$	−11.0903	−0.984102	−0.492051	+	0.870566i	$0.663753\pi$
$128$	− 5.84084i	− 0.516262i
$129$	11.5720	1.01885
$130$	0	0
$131$	15.2627	1.33351	0.666756	−	0.745276i	$-0.267682\pi$
$131$	15.2627	1.33351	0.666756	+	0.745276i	$0.267682\pi$
$132$	48.4583i	4.21775i
$133$	−1.36230	−0.118127
$134$	−12.4069	−1.07179
$135$	7.61435i	0.655339i
$136$	− 45.0131i	− 3.85984i
$137$	− 10.9701i	− 0.937236i	−0.883401	−	0.468618i	$-0.844752\pi$
$137$	− 10.9701i	− 0.937236i	0.883401	−	0.468618i	$-0.155248\pi$
$138$	− 30.2337i	− 2.57366i
$139$	−17.0445	−1.44570	−0.722849	−	0.691006i	$-0.757168\pi$
$139$	−17.0445	−1.44570	−0.722849	+	0.691006i	$0.757168\pi$
$140$	−10.3262	−0.872720
$141$	− 23.5427i	− 1.98265i
$142$	−35.0883	−2.94455
$143$	0	0
$144$	46.3745	3.86454
$145$	17.2685i	1.43407i
$146$	−11.5009	−0.951824
$147$	2.71083	0.223585
$148$	10.7261i	0.881681i
$149$	21.0025i	1.72059i	0.509795	+	0.860296i	$0.329721\pi$
$149$	21.0025i	1.72059i	−0.509795	+	0.860296i	$0.670279\pi$
$150$	− 4.73302i	− 0.386450i
$151$	9.14395i	0.744124i	0.928208	+	0.372062i	$0.121349\pi$
$151$	9.14395i	0.744124i	−0.928208	+	0.372062i	$0.878651\pi$
$152$	10.6287	0.862102
$153$	25.0889	2.02832
$154$	− 9.51070i	− 0.766394i
$155$	−1.52999	−0.122892
$156$	0	0
$157$	−19.4962	−1.55597	−0.777984	−	0.628284i	$-0.783758\pi$
$157$	−19.4962	−1.55597	−0.777984	+	0.628284i	$0.783758\pi$
$158$	− 4.58349i	− 0.364643i
$159$	29.7844	2.36206
$160$	26.0888	2.06250
$161$	4.22817i	0.333227i
$162$	− 8.27073i	− 0.649810i
$163$	21.0315i	1.64732i	0.567086	+	0.823659i	$0.308071\pi$
$163$	21.0315i	1.64732i	−0.567086	+	0.823659i	$0.691929\pi$
$164$	14.2150i	1.11000i
$165$	−20.3576	−1.58484
$166$	−15.1531	−1.17611
$167$	0.603845i	0.0467269i	0.999727	+	0.0233634i	$0.00743749\pi$
$167$	0.603845i	0.0467269i	−0.999727	+	0.0233634i	$0.992563\pi$
$168$	−21.1499	−1.63175
$169$	0	0
$170$	31.6970	2.43105
$171$	5.92411i	0.453028i
$172$	21.1638	1.61373
$173$	−13.4932	−1.02587	−0.512933	−	0.858429i	$-0.671441\pi$
$173$	−13.4932	−1.02587	−0.512933	+	0.858429i	$0.671441\pi$
$174$	59.2847i	4.49436i
$175$	0.661912i	0.0500358i
$176$	38.4509i	2.89835i
$177$	18.3066i	1.37601i
$178$	−22.1890	−1.66314
$179$	−1.74628	−0.130523	−0.0652614	−	0.997868i	$-0.520788\pi$
$179$	−1.74628	−0.130523	−0.0652614	+	0.997868i	$0.520788\pi$
$180$	44.9043i	3.34697i
$181$	−7.12074	−0.529280	−0.264640	−	0.964347i	$-0.585253\pi$
$181$	−7.12074	−0.529280	−0.264640	+	0.964347i	$0.585253\pi$
$182$	0	0
$183$	−28.4647	−2.10417
$184$	− 32.9882i	− 2.43192i
$185$	−4.50611	−0.331296
$186$	−5.25264	−0.385142
$187$	20.8022i	1.52120i
$188$	− 43.0569i	− 3.14025i
$189$	− 3.65581i	− 0.265921i
$190$	7.48444i	0.542978i
$191$	10.4609	0.756927	0.378464	−	0.925616i	$-0.376452\pi$
$191$	10.4609	0.756927	0.378464	+	0.925616i	$0.376452\pi$
$192$	31.7481	2.29122
$193$	6.15874i	0.443316i	0.975124	+	0.221658i	$0.0711469\pi$
$193$	6.15874i	0.443316i	−0.975124	+	0.221658i	$0.928853\pi$
$194$	−16.3637	−1.17485
$195$	0	0
$196$	4.95781	0.354129
$197$	− 14.7190i	− 1.04869i	−0.851507	−	0.524343i	$-0.824311\pi$
$197$	− 14.7190i	− 1.04869i	0.851507	−	0.524343i	$-0.175689\pi$
$198$	−41.3582	−2.93920
$199$	9.37716	0.664729	0.332365	−	0.943151i	$-0.392154\pi$
$199$	9.37716	0.664729	0.332365	+	0.943151i	$0.392154\pi$
$200$	− 5.16424i	− 0.365167i
$201$	− 12.7506i	− 0.899354i
$202$	16.6992i	1.17495i
$203$	− 8.29095i	− 0.581911i
$204$	77.5397	5.42887
$205$	−5.97180	−0.417089
$206$	− 20.1647i	− 1.40494i
$207$	18.3866	1.27796
$208$	0	0
$209$	−4.91191	−0.339764
$210$	− 14.8932i	− 1.02773i
$211$	−11.3257	−0.779692	−0.389846	−	0.920880i	$-0.627472\pi$
$211$	−11.3257	−0.779692	−0.389846	+	0.920880i	$0.627472\pi$
$212$	54.4724	3.74118
$213$	− 36.0602i	− 2.47081i
$214$	7.26370i	0.496536i
$215$	8.89107i	0.606366i
$216$	28.5227i	1.94072i
$217$	0.734580	0.0498665
$218$	−12.9181	−0.874923
$219$	− 11.8195i	− 0.798688i
$220$	−37.2319	−2.51017
$221$	0	0
$222$	−15.4700	−1.03828
$223$	13.1511i	0.880663i	0.897835	+	0.440331i	$0.145139\pi$
$223$	13.1511i	0.880663i	−0.897835	+	0.440331i	$0.854861\pi$
$224$	−12.5258	−0.836914
$225$	2.87839	0.191892
$226$	− 45.9287i	− 3.05513i
$227$	− 6.15435i	− 0.408479i	−0.978921	−	0.204239i	$-0.934528\pi$
$227$	− 6.15435i	− 0.408479i	0.978921	−	0.204239i	$-0.0654721\pi$
$228$	18.3091i	1.21255i
$229$	− 17.2159i	− 1.13766i	−0.822455	−	0.568831i	$-0.807396\pi$
$229$	− 17.2159i	− 1.13766i	0.822455	−	0.568831i	$-0.192604\pi$
$230$	23.2294	1.53170
$231$	9.77414	0.643091
$232$	64.6861i	4.24685i
$233$	19.1225	1.25276	0.626378	−	0.779520i	$-0.284537\pi$
$233$	19.1225	1.25276	0.626378	+	0.779520i	$0.284537\pi$
$234$	0	0
$235$	18.0885	1.17996
$236$	33.4808i	2.17942i
$237$	4.71044	0.305976
$238$	−15.2184	−0.986461
$239$	− 8.98811i	− 0.581393i	−0.956815	−	0.290696i	$-0.906113\pi$
$239$	− 8.98811i	− 0.581393i	0.956815	−	0.290696i	$-0.0938869\pi$
$240$	60.2118i	3.88666i
$241$	8.52326i	0.549032i	0.961583	+	0.274516i	$0.0885176\pi$
$241$	8.52326i	0.549032i	−0.961583	+	0.274516i	$0.911482\pi$
$242$	− 5.27625i	− 0.339170i
$243$	19.4673	1.24882
$244$	−52.0587	−3.33272
$245$	2.08281i	0.133066i
$246$	−20.5019	−1.30715
$247$	0	0
$248$	−5.73120	−0.363932
$249$	− 15.5728i	− 0.986887i
$250$	31.1063	1.96734
$251$	18.4820	1.16657	0.583287	−	0.812266i	$-0.301766\pi$
$251$	18.4820	1.16657	0.583287	+	0.812266i	$0.301766\pi$
$252$	− 21.5595i	− 1.35812i
$253$	15.2450i	0.958448i
$254$	− 29.2535i	− 1.83553i
$255$	32.5749i	2.03992i
$256$	−8.01639	−0.501024
$257$	28.5202	1.77904	0.889520	−	0.456896i	$-0.151039\pi$
$257$	28.5202	1.77904	0.889520	+	0.456896i	$0.151039\pi$
$258$	30.5241i	1.90035i
$259$	2.16348	0.134432
$260$	0	0
$261$	−36.0540	−2.23169
$262$	40.2596i	2.48724i
$263$	−11.9385	−0.736162	−0.368081	−	0.929794i	$-0.619985\pi$
$263$	−11.9385	−0.736162	−0.368081	+	0.929794i	$0.619985\pi$
$264$	−76.2579	−4.69335
$265$	22.8842i	1.40577i
$266$	− 3.59344i	− 0.220328i
$267$	− 22.8036i	− 1.39556i
$268$	− 23.3194i	− 1.42446i
$269$	−1.05171	−0.0641239	−0.0320619	−	0.999486i	$-0.510207\pi$
$269$	−1.05171	−0.0641239	−0.0320619	+	0.999486i	$0.510207\pi$
$270$	−20.0849	−1.22233
$271$	4.18149i	0.254008i	0.991902	+	0.127004i	$0.0405360\pi$
$271$	4.18149i	0.254008i	−0.991902	+	0.127004i	$0.959464\pi$
$272$	61.5266	3.73060
$273$	0	0
$274$	28.9365	1.74811
$275$	2.38658i	0.143916i
$276$	56.8257	3.42050
$277$	1.79552	0.107882	0.0539412	−	0.998544i	$-0.482822\pi$
$277$	1.79552	0.107882	0.0539412	+	0.998544i	$0.482822\pi$
$278$	− 44.9594i	− 2.69649i
$279$	− 3.19439i	− 0.191243i
$280$	− 16.2501i	− 0.971128i
$281$	− 21.2449i	− 1.26736i	−0.773593	−	0.633682i	$-0.781543\pi$
$281$	− 21.2449i	− 1.26736i	0.773593	−	0.633682i	$-0.218457\pi$
$282$	62.1000	3.69800
$283$	−27.3633	−1.62658	−0.813291	−	0.581857i	$-0.802326\pi$
$283$	−27.3633	−1.62658	−0.813291	+	0.581857i	$0.802326\pi$
$284$	− 65.9502i	− 3.91342i
$285$	−7.69175	−0.455620
$286$	0	0
$287$	2.86719	0.169245
$288$	54.4696i	3.20965i
$289$	16.2862	0.958013
$290$	−45.5502	−2.67480
$291$	− 16.8170i	− 0.985829i
$292$	− 21.6166i	− 1.26501i
$293$	25.9939i	1.51858i	0.650751	+	0.759291i	$0.274454\pi$
$293$	25.9939i	1.51858i	−0.650751	+	0.759291i	$0.725546\pi$
$294$	7.15053i	0.417028i
$295$	−14.0655	−0.818926
$296$	−16.8795	−0.981100
$297$	− 13.1814i	− 0.764860i
$298$	−55.3997	−3.20922
$299$	0	0
$300$	8.89595	0.513608
$301$	− 4.26879i	− 0.246049i
$302$	−24.1196	−1.38793
$303$	−17.1618	−0.985918
$304$	14.5279i	0.833235i
$305$	− 21.8702i	− 1.25228i
$306$	66.1786i	3.78318i
$307$	− 6.54230i	− 0.373389i	−0.982418	−	0.186694i	$-0.940223\pi$
$307$	− 6.54230i	− 0.373389i	0.982418	−	0.186694i	$-0.0597774\pi$
$308$	17.8758	1.01857
$309$	20.7233	1.17890
$310$	− 4.03575i	− 0.229215i
$311$	14.7482	0.836292	0.418146	−	0.908380i	$-0.362680\pi$
$311$	14.7482	0.836292	0.418146	+	0.908380i	$0.362680\pi$
$312$	0	0
$313$	−26.8881	−1.51980	−0.759902	−	0.650038i	$-0.774753\pi$
$313$	−26.8881	−1.51980	−0.759902	+	0.650038i	$0.774753\pi$
$314$	− 51.4265i	− 2.90217i
$315$	9.05729	0.510320
$316$	8.61489	0.484625
$317$	19.6995i	1.10644i	0.833036	+	0.553218i	$0.186601\pi$
$317$	19.6995i	1.10644i	−0.833036	+	0.553218i	$0.813399\pi$
$318$	78.5642i	4.40566i
$319$	− 29.8938i	− 1.67373i
$320$	24.3930i	1.36361i
$321$	−7.46490	−0.416650
$322$	−11.1529	−0.621528
$323$	7.85970i	0.437326i
$324$	15.5452	0.863624
$325$	0	0
$326$	−55.4763	−3.07255
$327$	− 13.2759i	− 0.734159i
$328$	−22.3698	−1.23517
$329$	−8.68467	−0.478801
$330$	− 53.6987i	− 2.95602i
$331$	27.2645i	1.49859i	0.662234	+	0.749297i	$0.269609\pi$
$331$	27.2645i	1.49859i	−0.662234	+	0.749297i	$0.730391\pi$
$332$	− 28.4810i	− 1.56310i
$333$	− 9.40809i	− 0.515560i
$334$	−1.59280	−0.0871542
$335$	9.79661	0.535246
$336$	− 28.9090i	− 1.57711i
$337$	11.9935	0.653326	0.326663	−	0.945141i	$-0.394076\pi$
$337$	11.9935	0.653326	0.326663	+	0.945141i	$0.394076\pi$
$338$	0	0
$339$	47.2009	2.56360
$340$	59.5760i	3.23096i
$341$	2.64859	0.143429
$342$	−15.6264	−0.844979
$343$	− 1.00000i	− 0.0539949i
$344$	33.3051i	1.79569i
$345$	23.8728i	1.28527i
$346$	− 35.5918i	− 1.91343i
$347$	0.180455	0.00968735	0.00484368	−	0.999988i	$-0.498458\pi$
$347$	0.180455	0.00968735	0.00484368	+	0.999988i	$0.498458\pi$
$348$	−111.429	−5.97320
$349$	35.4438i	1.89726i	0.316381	+	0.948632i	$0.397532\pi$
$349$	35.4438i	1.89726i	−0.316381	+	0.948632i	$0.602468\pi$
$350$	−1.74597	−0.0933259
$351$	0	0
$352$	−45.1628	−2.40719
$353$	21.7983i	1.16021i	0.814543	+	0.580103i	$0.196988\pi$
$353$	21.7983i	1.16021i	−0.814543	+	0.580103i	$0.803012\pi$
$354$	−48.2886	−2.56651
$355$	27.7061	1.47049
$356$	− 41.7053i	− 2.21038i
$357$	− 15.6399i	− 0.827752i
$358$	− 4.60627i	− 0.243449i
$359$	− 2.89545i	− 0.152816i	−0.997077	−	0.0764079i	$-0.975655\pi$
$359$	− 2.89545i	− 0.152816i	0.997077	−	0.0764079i	$-0.0243451\pi$
$360$	−70.6650	−3.72437
$361$	17.1441	0.902323
$362$	− 18.7828i	− 0.987204i
$363$	5.42240	0.284602
$364$	0	0
$365$	9.08126	0.475335
$366$	− 75.0831i	− 3.92466i
$367$	−25.7297	−1.34308	−0.671540	−	0.740968i	$-0.734367\pi$
$367$	−25.7297	−1.34308	−0.671540	+	0.740968i	$0.734367\pi$
$368$	45.0903	2.35049
$369$	− 12.4682i	− 0.649070i
$370$	− 11.8861i	− 0.617927i
$371$	− 10.9872i	− 0.570426i
$372$	− 9.87259i	− 0.511870i
$373$	−12.7012	−0.657643	−0.328821	−	0.944392i	$-0.606651\pi$
$373$	−12.7012	−0.657643	−0.328821	+	0.944392i	$0.606651\pi$
$374$	−54.8712	−2.83732
$375$	31.9679i	1.65082i
$376$	67.7579	3.49435
$377$	0	0
$378$	9.64317	0.495991
$379$	15.1801i	0.779748i	0.920868	+	0.389874i	$0.127482\pi$
$379$	15.1801i	0.779748i	−0.920868	+	0.389874i	$0.872518\pi$
$380$	−14.0674	−0.721641
$381$	30.0638	1.54022
$382$	27.5935i	1.41181i
$383$	− 23.4381i	− 1.19763i	−0.800887	−	0.598816i	$-0.795638\pi$
$383$	− 23.4381i	− 1.19763i	0.800887	−	0.598816i	$-0.204362\pi$
$384$	15.8335i	0.808001i
$385$	7.50975i	0.382732i
$386$	−16.2453	−0.826865
$387$	−18.5632	−0.943622
$388$	− 30.7564i	− 1.56142i
$389$	−0.0433907	−0.00220000	−0.00110000	−	0.999999i	$-0.500350\pi$
$389$	−0.0433907	−0.00220000	−0.00110000	+	0.999999i	$0.500350\pi$
$390$	0	0
$391$	24.3941	1.23366
$392$	7.80201i	0.394061i
$393$	−41.3747	−2.08708
$394$	38.8253	1.95599
$395$	3.61917i	0.182100i
$396$	− 77.7347i	− 3.90632i
$397$	3.70335i	0.185866i	0.995672	+	0.0929328i	$0.0296242\pi$
$397$	3.70335i	0.185866i	−0.995672	+	0.0929328i	$0.970376\pi$
$398$	24.7348i	1.23984i
$399$	3.69297	0.184880
$400$	7.05879	0.352940
$401$	30.2889i	1.51256i	0.654250	+	0.756279i	$0.272985\pi$
$401$	30.2889i	1.51256i	−0.654250	+	0.756279i	$0.727015\pi$
$402$	33.6330	1.67746
$403$	0	0
$404$	−31.3870	−1.56156
$405$	6.53066i	0.324511i
$406$	21.8696	1.08537
$407$	7.80062	0.386662
$408$	122.023i	6.04103i
$409$	− 7.03924i	− 0.348068i	−0.984740	−	0.174034i	$-0.944320\pi$
$409$	− 7.03924i	− 0.348068i	0.984740	−	0.174034i	$-0.0556802\pi$
$410$	− 15.7522i	− 0.777946i
$411$	29.7380i	1.46687i
$412$	37.9006	1.86723
$413$	6.75315	0.332301
$414$	48.4996i	2.38362i
$415$	11.9650	0.587341
$416$	0	0
$417$	46.2048	2.26266
$418$	− 12.9565i	− 0.633721i
$419$	15.9101	0.777258	0.388629	−	0.921394i	$-0.372949\pi$
$419$	15.9101	0.777258	0.388629	+	0.921394i	$0.372949\pi$
$420$	27.9925	1.36589
$421$	− 9.91446i	− 0.483201i	−0.970376	−	0.241601i	$-0.922328\pi$
$421$	− 9.91446i	− 0.483201i	0.970376	−	0.241601i	$-0.0776724\pi$
$422$	− 29.8745i	− 1.45427i
$423$	37.7661i	1.83625i
$424$	85.7221i	4.16303i
$425$	3.81885	0.185241
$426$	95.1184	4.60850
$427$	10.5003i	0.508147i
$428$	−13.6525	−0.659917
$429$	0	0
$430$	−23.4525	−1.13098
$431$	18.3787i	0.885270i	0.896702	+	0.442635i	$0.145956\pi$
$431$	18.3787i	0.885270i	−0.896702	+	0.442635i	$0.854044\pi$
$432$	−38.9865	−1.87574
$433$	11.4478	0.550147	0.275074	−	0.961423i	$-0.411298\pi$
$433$	11.4478	0.550147	0.275074	+	0.961423i	$0.411298\pi$
$434$	1.93765i	0.0930102i
$435$	− 46.8119i	− 2.24446i
$436$	− 24.2802i	− 1.16281i
$437$	5.76005i	0.275541i
$438$	31.1771	1.48970
$439$	21.1811	1.01092	0.505459	−	0.862851i	$-0.331323\pi$
$439$	21.1811	1.01092	0.505459	+	0.862851i	$0.331323\pi$
$440$	− 58.5911i	− 2.79322i
$441$	−4.34860	−0.207076
$442$	0	0
$443$	35.4959	1.68646	0.843231	−	0.537552i	$-0.180651\pi$
$443$	35.4959	1.68646	0.843231	+	0.537552i	$0.180651\pi$
$444$	− 29.0767i	− 1.37992i
$445$	17.5207	0.830560
$446$	−34.6895	−1.64260
$447$	− 56.9342i	− 2.69289i
$448$	− 11.7116i	− 0.553320i
$449$	− 9.93940i	− 0.469069i	−0.972108	−	0.234535i	$-0.924643\pi$
$449$	− 9.93940i	− 0.469069i	0.972108	−	0.234535i	$-0.0753566\pi$
$450$	7.59251i	0.357914i
$451$	10.3379	0.486793
$452$	86.3253	4.06040
$453$	− 24.7877i	− 1.16463i
$454$	16.2337	0.761887
$455$	0	0
$456$	−28.8126	−1.34927
$457$	6.51624i	0.304817i	0.988318	+	0.152409i	$0.0487029\pi$
$457$	6.51624i	0.304817i	−0.988318	+	0.152409i	$0.951297\pi$
$458$	45.4116	2.12195
$459$	−21.0919	−0.984486
$460$	43.6608i	2.03569i
$461$	− 17.9564i	− 0.836314i	−0.908375	−	0.418157i	$-0.862676\pi$
$461$	− 17.9564i	− 0.836314i	0.908375	−	0.418157i	$-0.137324\pi$
$462$	25.7819i	1.19948i
$463$	27.0873i	1.25885i	0.777059	+	0.629427i	$0.216710\pi$
$463$	27.0873i	1.25885i	−0.777059	+	0.629427i	$0.783290\pi$
$464$	−88.4168	−4.10465
$465$	4.14754	0.192338
$466$	50.4406i	2.33662i
$467$	−24.6940	−1.14270	−0.571351	−	0.820706i	$-0.693581\pi$
$467$	−24.6940	−1.14270	−0.571351	+	0.820706i	$0.693581\pi$
$468$	0	0
$469$	−4.70356	−0.217190
$470$	47.7132i	2.20085i
$471$	52.8510	2.43524
$472$	−52.6881	−2.42517
$473$	− 15.3915i	− 0.707702i
$474$	12.4250i	0.570701i
$475$	0.901725i	0.0413740i
$476$	− 28.6037i	− 1.31105i
$477$	−47.7788	−2.18764
$478$	23.7085	1.08440
$479$	− 22.5936i	− 1.03233i	−0.856490	−	0.516163i	$-0.827360\pi$
$479$	− 22.5936i	− 1.03233i	0.856490	−	0.516163i	$-0.172640\pi$
$480$	−70.7223	−3.22802
$481$	0	0
$482$	−22.4824	−1.02404
$483$	− 11.4619i	− 0.521532i
$484$	9.91697	0.450771
$485$	12.9210	0.586711
$486$	51.3501i	2.32929i
$487$	20.6049i	0.933696i	0.884338	+	0.466848i	$0.154610\pi$
$487$	20.6049i	0.933696i	−0.884338	+	0.466848i	$0.845390\pi$
$488$	− 81.9238i	− 3.70852i
$489$	− 57.0129i	− 2.57821i
$490$	−5.49396	−0.248192
$491$	−2.14396	−0.0967555	−0.0483778	−	0.998829i	$-0.515405\pi$
$491$	−2.14396	−0.0967555	−0.0483778	+	0.998829i	$0.515405\pi$
$492$	− 38.5343i	− 1.73726i
$493$	−47.8340	−2.15434
$494$	0	0
$495$	32.6569	1.46782
$496$	− 7.83375i	− 0.351746i
$497$	−13.3023	−0.596689
$498$	41.0774	1.84072
$499$	− 0.962977i	− 0.0431088i	−0.999768	−	0.0215544i	$-0.993138\pi$
$499$	− 0.962977i	− 0.0431088i	0.999768	−	0.0215544i	$-0.00686151\pi$
$500$	58.4658i	2.61467i
$501$	− 1.63692i	− 0.0731322i
$502$	48.7512i	2.17587i
$503$	−14.0985	−0.628623	−0.314311	−	0.949320i	$-0.601774\pi$
$503$	−14.0985	−0.628623	−0.314311	+	0.949320i	$0.601774\pi$
$504$	33.9278	1.51126
$505$	− 13.1859i	− 0.586764i
$506$	−40.2129	−1.78768
$507$	0	0
$508$	54.9834	2.43950
$509$	14.6607i	0.649823i	0.945744	+	0.324912i	$0.105335\pi$
$509$	14.6607i	0.649823i	−0.945744	+	0.324912i	$0.894665\pi$
$510$	−85.9251	−3.80483
$511$	−4.36010	−0.192880
$512$	− 32.8270i	− 1.45076i
$513$	− 4.98032i	− 0.219887i
$514$	75.2295i	3.31823i
$515$	15.9223i	0.701619i
$516$	−57.3715	−2.52564
$517$	−31.3134	−1.37716
$518$	5.70675i	0.250740i
$519$	36.5777	1.60558
$520$	0	0
$521$	34.0043	1.48976	0.744878	−	0.667200i	$-0.232507\pi$
$521$	34.0043	1.48976	0.744878	+	0.667200i	$0.232507\pi$
$522$	− 95.1020i	− 4.16250i
$523$	−32.7639	−1.43266	−0.716332	−	0.697759i	$-0.754181\pi$
$523$	−32.7639	−1.43266	−0.716332	+	0.697759i	$0.754181\pi$
$524$	−75.6698	−3.30565
$525$	− 1.79433i	− 0.0783110i
$526$	− 31.4911i	− 1.37308i
$527$	− 4.23810i	− 0.184615i
$528$	− 104.234i	− 4.53620i
$529$	−5.12256	−0.222720
$530$	−60.3632	−2.62201
$531$	− 29.3667i	− 1.27441i
$532$	6.75404	0.292825
$533$	0	0
$534$	60.1506	2.60297
$535$	− 5.73550i	− 0.247967i
$536$	36.6972	1.58508
$537$	4.73386	0.204281
$538$	− 2.77416i	− 0.119603i
$539$	− 3.60559i	− 0.155304i
$540$	− 37.7505i	− 1.62452i
$541$	− 9.58117i	− 0.411926i	−0.978560	−	0.205963i	$-0.933967\pi$
$541$	− 9.58117i	− 0.411926i	0.978560	−	0.205963i	$-0.0660327\pi$
$542$	−11.0298	−0.473770
$543$	19.3031	0.828376
$544$	72.2665i	3.09840i
$545$	10.2003	0.436931
$546$	0	0
$547$	−22.1667	−0.947779	−0.473889	−	0.880584i	$-0.657150\pi$
$547$	−22.1667	−0.947779	−0.473889	+	0.880584i	$0.657150\pi$
$548$	54.3875i	2.32332i
$549$	45.6618	1.94880
$550$	−6.29525	−0.268430
$551$	− 11.2948i	− 0.481174i
$552$	89.4255i	3.80620i
$553$	− 1.73764i	− 0.0738920i
$554$	4.73616i	0.201220i
$555$	12.2153	0.518510
$556$	84.5035	3.58375
$557$	18.6947i	0.792118i	0.918225	+	0.396059i	$0.129622\pi$
$557$	18.6947i	0.792118i	−0.918225	+	0.396059i	$0.870378\pi$
$558$	8.42606	0.356703
$559$	0	0
$560$	22.2116	0.938611
$561$	− 56.3911i	− 2.38083i
$562$	56.0391	2.36387
$563$	−0.984350	−0.0414854	−0.0207427	−	0.999785i	$-0.506603\pi$
$563$	−0.984350	−0.0414854	−0.0207427	+	0.999785i	$0.506603\pi$
$564$	116.720i	4.91480i
$565$	36.2658i	1.52571i
$566$	− 72.1781i	− 3.03387i
$567$	− 3.13551i	− 0.131679i
$568$	103.785	4.35470
$569$	−20.4218	−0.856128	−0.428064	−	0.903748i	$-0.640804\pi$
$569$	−20.4218	−0.856128	−0.428064	+	0.903748i	$0.640804\pi$
$570$	− 20.2890i	− 0.849814i
$571$	29.6858	1.24231	0.621157	−	0.783686i	$-0.286663\pi$
$571$	29.6858	1.24231	0.621157	+	0.783686i	$0.286663\pi$
$572$	0	0
$573$	−28.3578	−1.18467
$574$	7.56297i	0.315672i
$575$	2.79868	0.116713
$576$	−50.9289	−2.12204
$577$	11.8652i	0.493957i	0.969021	+	0.246978i	$0.0794376\pi$
$577$	11.8652i	0.493957i	−0.969021	+	0.246978i	$0.920562\pi$
$578$	42.9592i	1.78687i
$579$	− 16.6953i	− 0.693833i
$580$	− 85.6137i	− 3.55492i
$581$	−5.74467	−0.238329
$582$	44.3593	1.83875
$583$	− 39.6153i	− 1.64070i
$584$	34.0176	1.40766
$585$	0	0
$586$	−68.5659	−2.83243
$587$	− 48.3763i	− 1.99670i	−0.0573787	−	0.998352i	$-0.518274\pi$
$587$	− 48.3763i	− 1.99670i	0.0573787	−	0.998352i	$-0.481726\pi$
$588$	−13.4398	−0.554247
$589$	1.00072	0.0412340
$590$	− 37.1015i	− 1.52745i
$591$	39.9007i	1.64130i
$592$	− 23.0719i	− 0.948248i
$593$	− 34.3228i	− 1.40947i	−0.709471	−	0.704734i	$-0.751066\pi$
$593$	− 34.3228i	− 1.40947i	0.709471	−	0.704734i	$-0.248934\pi$
$594$	34.7693	1.42660
$595$	12.0166	0.492632
$596$	− 104.126i	− 4.26518i
$597$	−25.4199	−1.04037
$598$	0	0
$599$	−22.8086	−0.931933	−0.465967	−	0.884802i	$-0.654293\pi$
$599$	−22.8086	−0.931933	−0.465967	+	0.884802i	$0.654293\pi$
$600$	13.9994i	0.571522i
$601$	15.6032	0.636467	0.318233	−	0.948012i	$-0.396910\pi$
$601$	15.6032	0.636467	0.318233	+	0.948012i	$0.396910\pi$
$602$	11.2601	0.458926
$603$	20.4539i	0.832946i
$604$	− 45.3339i	− 1.84461i
$605$	4.16618i	0.169379i
$606$	− 45.2687i	− 1.83892i
$607$	−5.67517	−0.230348	−0.115174	−	0.993345i	$-0.536743\pi$
$607$	−5.67517	−0.230348	−0.115174	+	0.993345i	$0.536743\pi$
$608$	−17.0639	−0.692033
$609$	22.4754i	0.910748i
$610$	57.6885	2.33574
$611$	0	0
$612$	−124.386	−5.02800
$613$	− 28.5664i	− 1.15379i	−0.816819	−	0.576894i	$-0.804265\pi$
$613$	− 28.5664i	− 1.15379i	0.816819	−	0.576894i	$-0.195735\pi$
$614$	17.2570	0.696438
$615$	16.1885	0.652784
$616$	28.1308i	1.13342i
$617$	− 12.3751i	− 0.498201i	−0.968478	−	0.249101i	$-0.919865\pi$
$617$	− 12.3751i	− 0.498201i	0.968478	−	0.249101i	$-0.0801350\pi$
$618$	54.6631i	2.19887i
$619$	− 23.9715i	− 0.963494i	−0.876310	−	0.481747i	$-0.840002\pi$
$619$	− 23.9715i	− 0.963494i	0.876310	−	0.481747i	$-0.159998\pi$
$620$	7.58539	0.304637
$621$	−15.4574	−0.620284
$622$	38.9022i	1.55984i
$623$	−8.41205	−0.337022
$624$	0	0
$625$	−21.2523	−0.850093
$626$	− 70.9244i	− 2.83471i
$627$	13.3153	0.531763
$628$	96.6586	3.85710
$629$	− 12.4820i	− 0.497691i
$630$	23.8910i	0.951840i
$631$	3.79928i	0.151247i	0.997136	+	0.0756236i	$0.0240947\pi$
$631$	3.79928i	0.151247i	−0.997136	+	0.0756236i	$0.975905\pi$
$632$	13.5571i	0.539272i
$633$	30.7020	1.22029
$634$	−51.9628	−2.06371
$635$	23.0989i	0.916652i
$636$	−147.665	−5.85531
$637$	0	0
$638$	78.8528	3.12181
$639$	57.8462i	2.28836i
$640$	−12.1653	−0.480878
$641$	−2.98792	−0.118016	−0.0590080	−	0.998258i	$-0.518794\pi$
$641$	−2.98792	−0.118016	−0.0590080	+	0.998258i	$0.518794\pi$
$642$	− 19.6907i	− 0.777128i
$643$	34.0839i	1.34414i	0.740489	+	0.672069i	$0.234594\pi$
$643$	34.0839i	1.34414i	−0.740489	+	0.672069i	$0.765406\pi$
$644$	− 20.9625i	− 0.826037i
$645$	− 24.1022i	− 0.949022i
$646$	−20.7321	−0.815692
$647$	−1.79132	−0.0704240	−0.0352120	−	0.999380i	$-0.511211\pi$
$647$	−1.79132	−0.0704240	−0.0352120	+	0.999380i	$0.511211\pi$
$648$	24.4632i	0.961007i
$649$	24.3491	0.955786
$650$	0	0
$651$	−1.99132	−0.0780460
$652$	− 104.270i	− 4.08354i
$653$	−23.6436	−0.925245	−0.462623	−	0.886555i	$-0.653092\pi$
$653$	−23.6436	−0.925245	−0.462623	+	0.886555i	$0.653092\pi$
$654$	35.0187	1.36934
$655$	− 31.7894i	− 1.24211i
$656$	− 30.5764i	− 1.19381i
$657$	18.9603i	0.739713i
$658$	− 22.9081i	− 0.893052i
$659$	14.8861	0.579878	0.289939	−	0.957045i	$-0.406365\pi$
$659$	14.8861	0.579878	0.289939	+	0.957045i	$0.406365\pi$
$660$	100.929	3.92867
$661$	− 23.7725i	− 0.924643i	−0.886712	−	0.462322i	$-0.847017\pi$
$661$	− 23.7725i	− 0.924643i	0.886712	−	0.462322i	$-0.152983\pi$
$662$	−71.9175	−2.79515
$663$	0	0
$664$	44.8199	1.73935
$665$	2.83742i	0.110030i
$666$	24.8163	0.961614
$667$	−35.0556	−1.35736
$668$	− 2.99375i	− 0.115832i
$669$	− 35.6504i	− 1.37832i
$670$	25.8412i	0.998332i
$671$	37.8599i	1.46157i
$672$	33.9553	1.30985
$673$	−0.690661	−0.0266230	−0.0133115	−	0.999911i	$-0.504237\pi$
$673$	−0.690661	−0.0266230	−0.0133115	+	0.999911i	$0.504237\pi$
$674$	31.6360i	1.21857i
$675$	−2.41982	−0.0931391
$676$	0	0
$677$	24.4126	0.938252	0.469126	−	0.883131i	$-0.344569\pi$
$677$	24.4126	0.938252	0.469126	+	0.883131i	$0.344569\pi$
$678$	124.505i	4.78158i
$679$	−6.20363	−0.238073
$680$	−93.7536	−3.59529
$681$	16.6834i	0.639309i
$682$	6.98637i	0.267522i
$683$	34.4983i	1.32004i	0.751248	+	0.660020i	$0.229452\pi$
$683$	34.4983i	1.32004i	−0.751248	+	0.660020i	$0.770548\pi$
$684$	− 29.3706i	− 1.12301i
$685$	−22.8485	−0.872997
$686$	2.63777	0.100710
$687$	46.6695i	1.78055i
$688$	−45.5234	−1.73556
$689$	0	0
$690$	−62.9709	−2.39726
$691$	− 44.2084i	− 1.68177i	−0.541217	−	0.840883i	$-0.682036\pi$
$691$	− 44.2084i	− 1.68177i	0.541217	−	0.840883i	$-0.317964\pi$
$692$	66.8965	2.54302
$693$	−15.6792	−0.595605
$694$	0.475999i	0.0180687i
$695$	35.5005i	1.34661i
$696$	− 175.353i	− 6.64674i
$697$	− 16.5420i	− 0.626573i
$698$	−93.4925	−3.53874
$699$	−51.8378	−1.96068
$700$	− 3.28163i	− 0.124034i
$701$	−20.6216	−0.778866	−0.389433	−	0.921055i	$-0.627329\pi$
$701$	−20.6216	−0.778866	−0.389433	+	0.921055i	$0.627329\pi$
$702$	0	0
$703$	2.94731	0.111160
$704$	− 42.2271i	− 1.59150i
$705$	−49.0348	−1.84676
$706$	−57.4987	−2.16399
$707$	6.33082i	0.238095i
$708$	− 90.7608i	− 3.41100i
$709$	− 18.9972i	− 0.713456i	−0.934208	−	0.356728i	$-0.883892\pi$
$709$	− 18.9972i	− 0.713456i	0.934208	−	0.356728i	$-0.116108\pi$
$710$	73.0822i	2.74273i
$711$	−7.55629	−0.283383
$712$	65.6308	2.45962
$713$	− 3.10593i	− 0.116318i
$714$	41.2544	1.54391
$715$	0	0
$716$	8.65771	0.323554
$717$	24.3652i	0.909937i
$718$	7.63751	0.285029
$719$	34.2107	1.27584	0.637921	−	0.770101i	$-0.279794\pi$
$719$	34.2107	1.27584	0.637921	+	0.770101i	$0.279794\pi$
$720$	− 96.5892i	− 3.59967i
$721$	− 7.64462i	− 0.284700i
$722$	45.2222i	1.68300i
$723$	− 23.1051i	− 0.859288i
$724$	35.3033	1.31204
$725$	−5.48788	−0.203815
$726$	14.3030i	0.530835i
$727$	−21.3777	−0.792855	−0.396427	−	0.918066i	$-0.629750\pi$
$727$	−21.3777	−0.792855	−0.396427	+	0.918066i	$0.629750\pi$
$728$	0	0
$729$	−43.3659	−1.60614
$730$	23.9542i	0.886586i
$731$	−24.6284	−0.910916
$732$	141.122	5.21603
$733$	− 5.24309i	− 0.193658i	−0.995301	−	0.0968290i	$-0.969130\pi$
$733$	− 5.24309i	− 0.193658i	0.995301	−	0.0968290i	$-0.0308700\pi$
$734$	− 67.8690i	− 2.50509i
$735$	− 5.64614i	− 0.208261i
$736$	52.9612i	1.95217i
$737$	−16.9591	−0.624697
$738$	32.8883	1.21063
$739$	− 30.1907i	− 1.11058i	−0.831656	−	0.555291i	$-0.812607\pi$
$739$	− 30.1907i	− 1.11058i	0.831656	−	0.555291i	$-0.187393\pi$
$740$	22.3404	0.821251
$741$	0	0
$742$	28.9816	1.06395
$743$	17.0073i	0.623937i	0.950092	+	0.311969i	$0.100988\pi$
$743$	17.0073i	0.623937i	−0.950092	+	0.311969i	$0.899012\pi$
$744$	15.5363	0.569589
$745$	43.7442	1.60266
$746$	− 33.5028i	− 1.22662i
$747$	24.9812i	0.914016i
$748$	− 103.133i	− 3.77092i
$749$	2.75373i	0.100619i
$750$	−84.3239	−3.07907
$751$	−36.9314	−1.34765	−0.673823	−	0.738893i	$-0.735349\pi$
$751$	−36.9314	−1.34765	−0.673823	+	0.738893i	$0.735349\pi$
$752$	92.6155i	3.37734i
$753$	−50.1016	−1.82580
$754$	0	0
$755$	19.0451	0.693121
$756$	18.1248i	0.659193i
$757$	32.6360	1.18618	0.593088	−	0.805138i	$-0.297909\pi$
$757$	32.6360	1.18618	0.593088	+	0.805138i	$0.297909\pi$
$758$	−40.0415	−1.45437
$759$	− 41.3267i	− 1.50007i
$760$	− 22.1375i	− 0.803013i
$761$	29.2956i	1.06196i	0.847383	+	0.530982i	$0.178177\pi$
$761$	29.2956i	1.06196i	−0.847383	+	0.530982i	$0.821823\pi$
$762$	79.3013i	2.87278i
$763$	−4.89736	−0.177296
$764$	−51.8634	−1.87635
$765$	− 52.2553i	− 1.88929i
$766$	61.8243	2.23380
$767$	0	0
$768$	21.7311	0.784152
$769$	− 31.3796i	− 1.13158i	−0.824551	−	0.565788i	$-0.808572\pi$
$769$	− 31.3796i	− 1.13158i	0.824551	−	0.565788i	$-0.191428\pi$
$770$	−19.8090	−0.713865
$771$	−77.3133	−2.78437
$772$	− 30.5339i	− 1.09894i
$773$	− 12.6440i	− 0.454774i	−0.973804	−	0.227387i	$-0.926982\pi$
$773$	− 12.6440i	− 0.454774i	0.973804	−	0.227387i	$-0.0730182\pi$
$774$	− 48.9655i	− 1.76003i
$775$	− 0.486227i	− 0.0174658i
$776$	48.4008	1.73749
$777$	−5.86482	−0.210399
$778$	− 0.114455i	− 0.00410340i
$779$	3.90598	0.139946
$780$	0	0
$781$	−47.9626	−1.71624
$782$	64.3459i	2.30101i
$783$	30.3102	1.08320
$784$	−10.6643	−0.380866
$785$	40.6069i	1.44932i
$786$	− 109.137i	− 3.89278i
$787$	47.8788i	1.70669i	0.521343	+	0.853347i	$0.325431\pi$
$787$	47.8788i	1.70669i	−0.521343	+	0.853347i	$0.674569\pi$
$788$	72.9740i	2.59959i
$789$	32.3634	1.15217
$790$	−9.54652	−0.339650
$791$	− 17.4120i	− 0.619099i
$792$	122.330	4.34679
$793$	0	0
$794$	−9.76856	−0.346673
$795$	− 62.0352i	− 2.20016i
$796$	−46.4902	−1.64780
$797$	0.670274	0.0237423	0.0118712	−	0.999930i	$-0.496221\pi$
$797$	0.670274	0.0237423	0.0118712	+	0.999930i	$0.496221\pi$
$798$	9.74120i	0.344835i
$799$	50.1055i	1.77261i
$800$	8.29096i	0.293130i
$801$	36.5806i	1.29251i
$802$	−79.8951	−2.82120
$803$	−15.7207	−0.554773
$804$	63.2148i	2.22941i
$805$	8.80647	0.310387
$806$	0	0
$807$	2.85101	0.100360
$808$	− 49.3931i	− 1.73764i
$809$	−4.18768	−0.147231	−0.0736155	−	0.997287i	$-0.523454\pi$
$809$	−4.18768	−0.147231	−0.0736155	+	0.997287i	$0.523454\pi$
$810$	−17.2263	−0.605272
$811$	35.4890i	1.24619i	0.782147	+	0.623094i	$0.214125\pi$
$811$	35.4890i	1.24619i	−0.782147	+	0.623094i	$0.785875\pi$
$812$	41.1050i	1.44250i
$813$	− 11.3353i	− 0.397547i
$814$	20.5762i	0.721195i
$815$	43.8046	1.53441
$816$	−166.788	−5.83875
$817$	− 5.81539i	− 0.203455i
$818$	18.5679	0.649210
$819$	0	0
$820$	29.6070	1.03392
$821$	47.5099i	1.65811i	0.559170	+	0.829053i	$0.311120\pi$
$821$	47.5099i	1.65811i	−0.559170	+	0.829053i	$0.688880\pi$
$822$	−78.4418	−2.73597
$823$	−1.31971	−0.0460023	−0.0230011	−	0.999735i	$-0.507322\pi$
$823$	−1.31971	−0.0460023	−0.0230011	+	0.999735i	$0.507322\pi$
$824$	59.6434i	2.07777i
$825$	− 6.46962i	− 0.225243i
$826$	17.8132i	0.619802i
$827$	− 32.9254i	− 1.14493i	−0.819930	−	0.572464i	$-0.805988\pi$
$827$	− 32.9254i	− 1.14493i	0.819930	−	0.572464i	$-0.194012\pi$
$828$	−91.1573	−3.16793
$829$	−38.3442	−1.33175	−0.665874	−	0.746064i	$-0.731941\pi$
$829$	−38.3442	−1.33175	−0.665874	+	0.746064i	$0.731941\pi$
$830$	31.5610i	1.09550i
$831$	−4.86735	−0.168846
$832$	0	0
$833$	−5.76942	−0.199899
$834$	121.877i	4.22027i
$835$	1.25769	0.0435242
$836$	24.3523	0.842242
$837$	2.68549i	0.0928240i
$838$	41.9670i	1.44973i
$839$	− 49.6084i	− 1.71267i	−0.516420	−	0.856335i	$-0.672736\pi$
$839$	− 49.6084i	− 1.71267i	0.516420	−	0.856335i	$-0.327264\pi$
$840$	44.0512i	1.51991i
$841$	39.7399	1.37034
$842$	26.1520	0.901258
$843$	57.5913i	1.98355i
$844$	56.1506	1.93278
$845$	0	0
$846$	−99.6182	−3.42494
$847$	− 2.00027i	− 0.0687302i
$848$	−117.170	−4.02364
$849$	74.1773	2.54576
$850$	10.0732i	0.345509i
$851$	− 9.14756i	− 0.313574i
$852$	178.780i	6.12489i
$853$	− 3.85470i	− 0.131982i	−0.997820	−	0.0659912i	$-0.978979\pi$
$853$	− 3.85470i	− 0.131982i	0.997820	−	0.0659912i	$-0.0210209\pi$
$854$	−27.6975	−0.947788
$855$	12.3388	0.421977
$856$	− 21.4846i	− 0.734330i
$857$	−37.5468	−1.28257	−0.641287	−	0.767301i	$-0.721599\pi$
$857$	−37.5468	−1.28257	−0.641287	+	0.767301i	$0.721599\pi$
$858$	0	0
$859$	4.89387	0.166977	0.0834883	−	0.996509i	$-0.473394\pi$
$859$	4.89387	0.166977	0.0834883	+	0.996509i	$0.473394\pi$
$860$	− 44.0802i	− 1.50312i
$861$	−7.77245	−0.264885
$862$	−48.4787	−1.65119
$863$	17.9815i	0.612097i	0.952016	+	0.306049i	$0.0990070\pi$
$863$	17.9815i	0.612097i	−0.952016	+	0.306049i	$0.900993\pi$
$864$	− 45.7919i	− 1.55787i
$865$	28.1037i	0.955553i
$866$	30.1967i	1.02612i
$867$	−44.1492	−1.49938
$868$	−3.64191	−0.123614
$869$	− 6.26521i	− 0.212533i
$870$	123.479	4.18632
$871$	0	0
$872$	38.2092	1.29393
$873$	26.9771i	0.913036i
$874$	−15.1937	−0.513934
$875$	11.7927	0.398665
$876$	58.5988i	1.97987i
$877$	− 21.6582i	− 0.731347i	−0.930743	−	0.365674i	$-0.880839\pi$
$877$	− 21.6582i	− 0.731347i	0.930743	−	0.365674i	$-0.119161\pi$
$878$	55.8708i	1.88555i
$879$	− 70.4651i	− 2.37673i
$880$	80.0858	2.69969
$881$	4.96810	0.167379	0.0836897	−	0.996492i	$-0.473330\pi$
$881$	4.96810	0.167379	0.0836897	+	0.996492i	$0.473330\pi$
$882$	− 11.4706i	− 0.386234i
$883$	−9.13265	−0.307338	−0.153669	−	0.988122i	$-0.549109\pi$
$883$	−9.13265	−0.307338	−0.153669	+	0.988122i	$0.549109\pi$
$884$	0	0
$885$	38.1292	1.28170
$886$	93.6299i	3.14556i
$887$	17.4343	0.585388	0.292694	−	0.956206i	$-0.405448\pi$
$887$	17.4343	0.585388	0.292694	+	0.956206i	$0.405448\pi$
$888$	45.7574	1.53552
$889$	− 11.0903i	− 0.371956i
$890$	46.2154i	1.54915i
$891$	− 11.3053i	− 0.378743i
$892$	− 65.2006i	− 2.18308i
$893$	−11.8312	−0.395915
$894$	150.179	5.02274
$895$	3.63716i	0.121577i
$896$	5.84084	0.195129
$897$	0	0
$898$	26.2178	0.874900
$899$	6.09037i	0.203125i
$900$	−14.2705	−0.475683
$901$	−63.3897	−2.11182
$902$	27.2690i	0.907957i
$903$	11.5720i	0.385091i
$904$	135.848i	4.51825i
$905$	14.8311i	0.493003i
$906$	65.3841	2.17224
$907$	−2.02670	−0.0672954	−0.0336477	−	0.999434i	$-0.510712\pi$
$907$	−2.02670	−0.0672954	−0.0336477	+	0.999434i	$0.510712\pi$
$908$	30.5121i	1.01258i
$909$	27.5302	0.913118
$910$	0	0
$911$	11.4856	0.380535	0.190268	−	0.981732i	$-0.439064\pi$
$911$	11.4856	0.380535	0.190268	+	0.981732i	$0.439064\pi$
$912$	− 39.3828i	− 1.30409i
$913$	−20.7129	−0.685497
$914$	−17.1883	−0.568539
$915$	59.2864i	1.95995i
$916$	85.3533i	2.82015i
$917$	15.2627i	0.504020i
$918$	− 55.6355i	− 1.83625i
$919$	−21.7918	−0.718846	−0.359423	−	0.933175i	$-0.617027\pi$
$919$	−21.7918	−0.718846	−0.359423	+	0.933175i	$0.617027\pi$
$920$	−68.7081	−2.26524
$921$	17.7350i	0.584390i
$922$	47.3649	1.55988
$923$	0	0
$924$	−48.4583	−1.59416
$925$	− 1.43203i	− 0.0470849i
$926$	−71.4500	−2.34799
$927$	−33.2433	−1.09185
$928$	− 103.851i	− 3.40907i
$929$	− 39.9819i	− 1.31176i	−0.754863	−	0.655882i	$-0.772297\pi$
$929$	− 39.9819i	− 1.31176i	0.754863	−	0.655882i	$-0.227703\pi$
$930$	10.9402i	0.358745i
$931$	− 1.36230i	− 0.0446477i
$932$	−94.8056	−3.10546
$933$	−39.9798	−1.30888
$934$	− 65.1370i	− 2.13135i
$935$	43.3269	1.41694
$936$	0	0
$937$	−37.2739	−1.21768	−0.608842	−	0.793292i	$-0.708366\pi$
$937$	−37.2739	−1.21768	−0.608842	+	0.793292i	$0.708366\pi$
$938$	− 12.4069i	− 0.405099i
$939$	72.8890	2.37864
$940$	−89.6793	−2.92502
$941$	− 12.8373i	− 0.418484i	−0.977864	−	0.209242i	$-0.932900\pi$
$941$	− 12.8373i	− 0.418484i	0.977864	−	0.209242i	$-0.0670997\pi$
$942$	139.408i	4.54217i
$943$	− 12.1230i	− 0.394778i
$944$	− 72.0173i	− 2.34396i
$945$	−7.61435	−0.247695
$946$	40.5992	1.31999
$947$	− 17.6547i	− 0.573702i	−0.957975	−	0.286851i	$-0.907392\pi$
$947$	− 17.6547i	− 0.573702i	0.957975	−	0.286851i	$-0.0926084\pi$
$948$	−23.3535	−0.758486
$949$	0	0
$950$	−2.37854	−0.0771700
$951$	− 53.4021i	− 1.73168i
$952$	45.0131	1.45888
$953$	19.3254	0.626011	0.313005	−	0.949751i	$-0.398664\pi$
$953$	19.3254	0.626011	0.313005	+	0.949751i	$0.398664\pi$
$954$	− 126.029i	− 4.08035i
$955$	− 21.7881i	− 0.705048i
$956$	44.5613i	1.44122i
$957$	81.0369i	2.61955i
$958$	59.5965	1.92548
$959$	10.9701	0.354242
$960$	− 66.1252i	− 2.13418i
$961$	30.4604	0.982593
$962$	0	0
$963$	11.9749	0.385885
$964$	− 42.2567i	− 1.36100i
$965$	12.8275	0.412931
$966$	30.2337	0.972753
$967$	47.1716i	1.51693i	0.651711	+	0.758467i	$0.274052\pi$
$967$	47.1716i	1.51693i	−0.651711	+	0.758467i	$0.725948\pi$
$968$	15.6061i	0.501600i
$969$	− 21.3063i	− 0.684457i
$970$	34.0825i	1.09432i
$971$	17.7036	0.568136	0.284068	−	0.958804i	$-0.408316\pi$
$971$	17.7036	0.568136	0.284068	+	0.958804i	$0.408316\pi$
$972$	−96.5149	−3.09572
$973$	− 17.0445i	− 0.546422i
$974$	−54.3508	−1.74151
$975$	0	0
$976$	111.978	3.58434
$977$	− 47.7932i	− 1.52904i	−0.644600	−	0.764520i	$-0.722976\pi$
$977$	− 47.7932i	− 1.52904i	0.644600	−	0.764520i	$-0.277024\pi$
$978$	150.387	4.80884
$979$	−30.3304	−0.969363
$980$	− 10.3262i	− 0.329857i
$981$	21.2966i	0.679949i
$982$	− 5.65526i	− 0.180467i
$983$	7.81325i	0.249204i	0.992207	+	0.124602i	$0.0397654\pi$
$983$	7.81325i	0.249204i	−0.992207	+	0.124602i	$0.960235\pi$
$984$	60.6407	1.93316
$985$	−30.6569	−0.976808
$986$	− 126.175i	− 4.01823i
$987$	23.5427	0.749371
$988$	0	0
$989$	−18.0492	−0.573930
$990$	86.1411i	2.73775i
$991$	9.77365	0.310470	0.155235	−	0.987878i	$-0.450386\pi$
$991$	9.77365	0.310470	0.155235	+	0.987878i	$0.450386\pi$
$992$	9.20119	0.292138
$993$	− 73.9095i	− 2.34545i
$994$	− 35.0883i	− 1.11293i
$995$	− 19.5308i	− 0.619169i
$996$	77.2070i	2.44640i
$997$	18.3134	0.579990	0.289995	−	0.957028i	$-0.406346\pi$
$997$	18.3134	0.579990	0.289995	+	0.957028i	$0.406346\pi$
$998$	2.54011	0.0804057
$999$	7.90927i	0.250238i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1183.2.c.h.337.12		12
13.5	odd	4		1183.2.a.o.1.6	yes	6
13.8	odd	4		1183.2.a.n.1.1	✓	6
13.12	even	2	inner	1183.2.c.h.337.1		12
91.34	even	4		8281.2.a.cb.1.1		6
91.83	even	4		8281.2.a.cg.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1183.2.a.n.1.1	✓	6	13.8	odd	4
1183.2.a.o.1.6	yes	6	13.5	odd	4
1183.2.c.h.337.1		12	13.12	even	2	inner
1183.2.c.h.337.12		12	1.1	even	1	trivial
8281.2.a.cb.1.1		6	91.34	even	4
8281.2.a.cg.1.6		6	91.83	even	4

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:	\( N \)	\(=\)	\( 1183 = 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1183.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.63777i q^{2} -2.71083 q^{3} -4.95781 q^{4} -2.08281i q^{5} -7.15053i q^{6} +1.00000i q^{7} -7.80201i q^{8} +4.34860 q^{9} +O(q^{10})\)	\(q+2.63777i q^{2} -2.71083 q^{3} -4.95781 q^{4} -2.08281i q^{5} -7.15053i q^{6} +1.00000i q^{7} -7.80201i q^{8} +4.34860 q^{9} +5.49396 q^{10} +3.60559i q^{11} +13.4398 q^{12} -2.63777 q^{14} +5.64614i q^{15} +10.6643 q^{16} +5.76942 q^{17} +11.4706i q^{18} +1.36230i q^{19} +10.3262i q^{20} -2.71083i q^{21} -9.51070 q^{22} +4.22817 q^{23} +21.1499i q^{24} +0.661912 q^{25} -3.65581 q^{27} -4.95781i q^{28} -8.29095 q^{29} -14.8932 q^{30} -0.734580i q^{31} +12.5258i q^{32} -9.77414i q^{33} +15.2184i q^{34} +2.08281 q^{35} -21.5595 q^{36} -2.16348i q^{37} -3.59344 q^{38} -16.2501 q^{40} -2.86719i q^{41} +7.15053 q^{42} -4.26879 q^{43} -17.8758i q^{44} -9.05729i q^{45} +11.1529i q^{46} +8.68467i q^{47} -28.9090 q^{48} -1.00000 q^{49} +1.74597i q^{50} -15.6399 q^{51} -10.9872 q^{53} -9.64317i q^{54} +7.50975 q^{55} +7.80201 q^{56} -3.69297i q^{57} -21.8696i q^{58} -6.75315i q^{59} -27.9925i q^{60} +10.5003 q^{61} +1.93765 q^{62} +4.34860i q^{63} -11.7116 q^{64} +25.7819 q^{66} +4.70356i q^{67} -28.6037 q^{68} -11.4619 q^{69} +5.49396i q^{70} +13.3023i q^{71} -33.9278i q^{72} +4.36010i q^{73} +5.70675 q^{74} -1.79433 q^{75} -6.75404i q^{76} -3.60559 q^{77} -1.73764 q^{79} -22.2116i q^{80} -3.13551 q^{81} +7.56297 q^{82} +5.74467i q^{83} +13.4398i q^{84} -12.0166i q^{85} -11.2601i q^{86} +22.4754 q^{87} +28.1308 q^{88} +8.41205i q^{89} +23.8910 q^{90} -20.9625 q^{92} +1.99132i q^{93} -22.9081 q^{94} +2.83742 q^{95} -33.9553i q^{96} +6.20363i q^{97} -2.63777i q^{98} +15.6792i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 8 q^{3} - 16 q^{4}+O(q^{10}) \) 12 * q - 8 * q^3 - 16 * q^4	\( 12 q - 8 q^{3} - 16 q^{4} + 28 q^{10} + 46 q^{12} - 4 q^{14} + 46 q^{17} - 8 q^{22} + 36 q^{23} + 20 q^{25} - 20 q^{27} - 30 q^{29} - 28 q^{30} - 4 q^{35} - 44 q^{36} + 22 q^{38} - 28 q^{40} + 16 q^{42} + 36 q^{43} - 22 q^{48} - 12 q^{49} - 28 q^{51} - 50 q^{53} + 6 q^{56} + 32 q^{61} + 18 q^{62} + 14 q^{64} + 32 q^{66} - 68 q^{68} + 2 q^{69} - 28 q^{74} - 30 q^{75} + 16 q^{77} + 4 q^{79} - 12 q^{81} + 20 q^{82} + 26 q^{87} + 96 q^{88} - 64 q^{92} - 28 q^{94} + 14 q^{95}+O(q^{100}) \) 12 * q - 8 * q^3 - 16 * q^4 + 28 * q^10 + 46 * q^12 - 4 * q^14 + 46 * q^17 - 8 * q^22 + 36 * q^23 + 20 * q^25 - 20 * q^27 - 30 * q^29 - 28 * q^30 - 4 * q^35 - 44 * q^36 + 22 * q^38 - 28 * q^40 + 16 * q^42 + 36 * q^43 - 22 * q^48 - 12 * q^49 - 28 * q^51 - 50 * q^53 + 6 * q^56 + 32 * q^61 + 18 * q^62 + 14 * q^64 + 32 * q^66 - 68 * q^68 + 2 * q^69 - 28 * q^74 - 30 * q^75 + 16 * q^77 + 4 * q^79 - 12 * q^81 + 20 * q^82 + 26 * q^87 + 96 * q^88 - 64 * q^92 - 28 * q^94 + 14 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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