LMFDB - Embedded newform 7942.2.a.cd.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7942,2,Mod(1,7942)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7942, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7942.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7942 = 2 \cdot 11 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7942.a (trivial)

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:	yes
Analytic conductor:	$63.4171892853$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 6 x^{15} - 14 x^{14} + 128 x^{13} + 39 x^{12} - 1110 x^{11} + 348 x^{10} + 4996 x^{9} + \cdots - 359 $ x^16 - 6x^15 - 14x^14 + 128x^13 + 39x^12 - 1110x^11 + 348x^10 + 4996x^9 - 2932x^8 - 12256x^7 + 8993x^6 + 15430x^5 - 12896x^4 - 7688x^3 + 7371x^2 - 64*x - 359
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.78049$ of defining polynomial
Character	$\chi$	$=$	7942.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.78049 q^{3} +1.00000 q^{4} -0.579287 q^{5} -2.78049 q^{6} -2.06979 q^{7} +1.00000 q^{8} +4.73112 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.78049 q^{3} +1.00000 q^{4} -0.579287 q^{5} -2.78049 q^{6} -2.06979 q^{7} +1.00000 q^{8} +4.73112 q^{9} -0.579287 q^{10} -1.00000 q^{11} -2.78049 q^{12} -2.38101 q^{13} -2.06979 q^{14} +1.61070 q^{15} +1.00000 q^{16} -1.14111 q^{17} +4.73112 q^{18} -0.579287 q^{20} +5.75502 q^{21} -1.00000 q^{22} -1.18476 q^{23} -2.78049 q^{24} -4.66443 q^{25} -2.38101 q^{26} -4.81335 q^{27} -2.06979 q^{28} -4.05557 q^{29} +1.61070 q^{30} +4.19233 q^{31} +1.00000 q^{32} +2.78049 q^{33} -1.14111 q^{34} +1.19900 q^{35} +4.73112 q^{36} +7.04275 q^{37} +6.62037 q^{39} -0.579287 q^{40} -9.47254 q^{41} +5.75502 q^{42} -9.63251 q^{43} -1.00000 q^{44} -2.74067 q^{45} -1.18476 q^{46} +2.61227 q^{47} -2.78049 q^{48} -2.71599 q^{49} -4.66443 q^{50} +3.17284 q^{51} -2.38101 q^{52} +2.93059 q^{53} -4.81335 q^{54} +0.579287 q^{55} -2.06979 q^{56} -4.05557 q^{58} +1.98596 q^{59} +1.61070 q^{60} +1.21810 q^{61} +4.19233 q^{62} -9.79240 q^{63} +1.00000 q^{64} +1.37929 q^{65} +2.78049 q^{66} -8.48031 q^{67} -1.14111 q^{68} +3.29420 q^{69} +1.19900 q^{70} +8.58773 q^{71} +4.73112 q^{72} -7.47179 q^{73} +7.04275 q^{74} +12.9694 q^{75} +2.06979 q^{77} +6.62037 q^{78} +1.78218 q^{79} -0.579287 q^{80} -0.809886 q^{81} -9.47254 q^{82} +0.740015 q^{83} +5.75502 q^{84} +0.661029 q^{85} -9.63251 q^{86} +11.2765 q^{87} -1.00000 q^{88} +3.98775 q^{89} -2.74067 q^{90} +4.92818 q^{91} -1.18476 q^{92} -11.6567 q^{93} +2.61227 q^{94} -2.78049 q^{96} -0.250970 q^{97} -2.71599 q^{98} -4.73112 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{2} + 10 q^{3} + 16 q^{4} + 10 q^{6} + 6 q^{7} + 16 q^{8} + 20 q^{9}+O(q^{10}) $ 16 * q + 16 * q^2 + 10 * q^3 + 16 * q^4 + 10 * q^6 + 6 * q^7 + 16 * q^8 + 20 * q^9	$ 16 q + 16 q^{2} + 10 q^{3} + 16 q^{4} + 10 q^{6} + 6 q^{7} + 16 q^{8} + 20 q^{9} - 16 q^{11} + 10 q^{12} - 2 q^{13} + 6 q^{14} - 8 q^{15} + 16 q^{16} + 20 q^{18} + 6 q^{21} - 16 q^{22} - 6 q^{23} + 10 q^{24} + 18 q^{25} - 2 q^{26} + 46 q^{27} + 6 q^{28} + 28 q^{29} - 8 q^{30} + 8 q^{31} + 16 q^{32} - 10 q^{33} + 14 q^{35} + 20 q^{36} + 32 q^{37} + 24 q^{39} + 24 q^{41} + 6 q^{42} + 28 q^{43} - 16 q^{44} - 16 q^{45} - 6 q^{46} + 32 q^{47} + 10 q^{48} + 16 q^{49} + 18 q^{50} + 14 q^{51} - 2 q^{52} + 16 q^{53} + 46 q^{54} + 6 q^{56} + 28 q^{58} + 30 q^{59} - 8 q^{60} - 8 q^{61} + 8 q^{62} - 8 q^{63} + 16 q^{64} + 26 q^{65} - 10 q^{66} + 44 q^{67} + 16 q^{69} + 14 q^{70} + 36 q^{71} + 20 q^{72} + 10 q^{73} + 32 q^{74} + 76 q^{75} - 6 q^{77} + 24 q^{78} - 16 q^{79} + 48 q^{81} + 24 q^{82} + 12 q^{83} + 6 q^{84} - 4 q^{85} + 28 q^{86} + 24 q^{87} - 16 q^{88} + 32 q^{89} - 16 q^{90} - 24 q^{91} - 6 q^{92} - 14 q^{93} + 32 q^{94} + 10 q^{96} + 52 q^{97} + 16 q^{98} - 20 q^{99}+O(q^{100}) $ 16 * q + 16 * q^2 + 10 * q^3 + 16 * q^4 + 10 * q^6 + 6 * q^7 + 16 * q^8 + 20 * q^9 - 16 * q^11 + 10 * q^12 - 2 * q^13 + 6 * q^14 - 8 * q^15 + 16 * q^16 + 20 * q^18 + 6 * q^21 - 16 * q^22 - 6 * q^23 + 10 * q^24 + 18 * q^25 - 2 * q^26 + 46 * q^27 + 6 * q^28 + 28 * q^29 - 8 * q^30 + 8 * q^31 + 16 * q^32 - 10 * q^33 + 14 * q^35 + 20 * q^36 + 32 * q^37 + 24 * q^39 + 24 * q^41 + 6 * q^42 + 28 * q^43 - 16 * q^44 - 16 * q^45 - 6 * q^46 + 32 * q^47 + 10 * q^48 + 16 * q^49 + 18 * q^50 + 14 * q^51 - 2 * q^52 + 16 * q^53 + 46 * q^54 + 6 * q^56 + 28 * q^58 + 30 * q^59 - 8 * q^60 - 8 * q^61 + 8 * q^62 - 8 * q^63 + 16 * q^64 + 26 * q^65 - 10 * q^66 + 44 * q^67 + 16 * q^69 + 14 * q^70 + 36 * q^71 + 20 * q^72 + 10 * q^73 + 32 * q^74 + 76 * q^75 - 6 * q^77 + 24 * q^78 - 16 * q^79 + 48 * q^81 + 24 * q^82 + 12 * q^83 + 6 * q^84 - 4 * q^85 + 28 * q^86 + 24 * q^87 - 16 * q^88 + 32 * q^89 - 16 * q^90 - 24 * q^91 - 6 * q^92 - 14 * q^93 + 32 * q^94 + 10 * q^96 + 52 * q^97 + 16 * q^98 - 20 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−2.78049	−1.60532	−0.802658	−	0.596440i	$-0.796581\pi$
$3$	−2.78049	−1.60532	−0.802658	+	0.596440i	$0.796581\pi$
$4$	1.00000	0.500000
$5$	−0.579287	−0.259065	−0.129533	−	0.991575i	$-0.541348\pi$
$5$	−0.579287	−0.259065	−0.129533	+	0.991575i	$0.541348\pi$
$6$	−2.78049	−1.13513
$7$	−2.06979	−0.782306	−0.391153	−	0.920326i	$-0.627924\pi$
$7$	−2.06979	−0.782306	−0.391153	+	0.920326i	$0.627924\pi$
$8$	1.00000	0.353553
$9$	4.73112	1.57704
$10$	−0.579287	−0.183187
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−2.78049	−0.802658
$13$	−2.38101	−0.660373	−0.330187	−	0.943916i	$-0.607112\pi$
$13$	−2.38101	−0.660373	−0.330187	+	0.943916i	$0.607112\pi$
$14$	−2.06979	−0.553174
$15$	1.61070	0.415881
$16$	1.00000	0.250000
$17$	−1.14111	−0.276759	−0.138380	−	0.990379i	$-0.544189\pi$
$17$	−1.14111	−0.276759	−0.138380	+	0.990379i	$0.544189\pi$
$18$	4.73112	1.11513
$19$	0	0
$19$	0	0
$20$	−0.579287	−0.129533
$21$	5.75502	1.25585
$22$	−1.00000	−0.213201
$23$	−1.18476	−0.247039	−0.123519	−	0.992342i	$-0.539418\pi$
$23$	−1.18476	−0.247039	−0.123519	+	0.992342i	$0.539418\pi$
$24$	−2.78049	−0.567565
$25$	−4.66443	−0.932885
$26$	−2.38101	−0.466954
$27$	−4.81335	−0.926330
$28$	−2.06979	−0.391153
$29$	−4.05557	−0.753100	−0.376550	−	0.926396i	$-0.622890\pi$
$29$	−4.05557	−0.753100	−0.376550	+	0.926396i	$0.622890\pi$
$30$	1.61070	0.294072
$31$	4.19233	0.752964	0.376482	−	0.926424i	$-0.377134\pi$
$31$	4.19233	0.752964	0.376482	+	0.926424i	$0.377134\pi$
$32$	1.00000	0.176777
$33$	2.78049	0.484021
$34$	−1.14111	−0.195698
$35$	1.19900	0.202668
$36$	4.73112	0.788519
$37$	7.04275	1.15782	0.578910	−	0.815391i	$-0.303478\pi$
$37$	7.04275	1.15782	0.578910	+	0.815391i	$0.303478\pi$
$38$	0	0
$39$	6.62037	1.06011
$40$	−0.579287	−0.0915933
$41$	−9.47254	−1.47936	−0.739681	−	0.672957i	$-0.765024\pi$
$41$	−9.47254	−1.47936	−0.739681	+	0.672957i	$0.765024\pi$
$42$	5.75502	0.888018
$43$	−9.63251	−1.46894	−0.734472	−	0.678639i	$-0.762570\pi$
$43$	−9.63251	−1.46894	−0.734472	+	0.678639i	$0.762570\pi$
$44$	−1.00000	−0.150756
$45$	−2.74067	−0.408556
$46$	−1.18476	−0.174683
$47$	2.61227	0.381039	0.190519	−	0.981683i	$-0.438983\pi$
$47$	2.61227	0.381039	0.190519	+	0.981683i	$0.438983\pi$
$48$	−2.78049	−0.401329
$49$	−2.71599	−0.387998
$50$	−4.66443	−0.659650
$51$	3.17284	0.444286
$52$	−2.38101	−0.330187
$53$	2.93059	0.402547	0.201274	−	0.979535i	$-0.435492\pi$
$53$	2.93059	0.402547	0.201274	+	0.979535i	$0.435492\pi$
$54$	−4.81335	−0.655014
$55$	0.579287	0.0781110
$56$	−2.06979	−0.276587
$57$	0	0
$58$	−4.05557	−0.532522
$59$	1.98596	0.258550	0.129275	−	0.991609i	$-0.458735\pi$
$59$	1.98596	0.258550	0.129275	+	0.991609i	$0.458735\pi$
$60$	1.61070	0.207941
$61$	1.21810	0.155961	0.0779806	−	0.996955i	$-0.475153\pi$
$61$	1.21810	0.155961	0.0779806	+	0.996955i	$0.475153\pi$
$62$	4.19233	0.532426
$63$	−9.79240	−1.23373
$64$	1.00000	0.125000
$65$	1.37929	0.171080
$66$	2.78049	0.342254
$67$	−8.48031	−1.03603	−0.518017	−	0.855370i	$-0.673330\pi$
$67$	−8.48031	−1.03603	−0.518017	+	0.855370i	$0.673330\pi$
$68$	−1.14111	−0.138380
$69$	3.29420	0.396576
$70$	1.19900	0.143308
$71$	8.58773	1.01918	0.509588	−	0.860419i	$-0.329798\pi$
$71$	8.58773	1.01918	0.509588	+	0.860419i	$0.329798\pi$
$72$	4.73112	0.557567
$73$	−7.47179	−0.874507	−0.437254	−	0.899338i	$-0.644049\pi$
$73$	−7.47179	−0.874507	−0.437254	+	0.899338i	$0.644049\pi$
$74$	7.04275	0.818703
$75$	12.9694	1.49758
$76$	0	0
$77$	2.06979	0.235874
$78$	6.62037	0.749609
$79$	1.78218	0.200511	0.100256	−	0.994962i	$-0.468034\pi$
$79$	1.78218	0.200511	0.100256	+	0.994962i	$0.468034\pi$
$80$	−0.579287	−0.0647663
$81$	−0.809886	−0.0899873
$82$	−9.47254	−1.04607
$83$	0.740015	0.0812272	0.0406136	−	0.999175i	$-0.487069\pi$
$83$	0.740015	0.0812272	0.0406136	+	0.999175i	$0.487069\pi$
$84$	5.75502	0.627924
$85$	0.661029	0.0716986
$86$	−9.63251	−1.03870
$87$	11.2765	1.20896
$88$	−1.00000	−0.106600
$89$	3.98775	0.422701	0.211350	−	0.977410i	$-0.432214\pi$
$89$	3.98775	0.422701	0.211350	+	0.977410i	$0.432214\pi$
$90$	−2.74067	−0.288892
$91$	4.92818	0.516614
$92$	−1.18476	−0.123519
$93$	−11.6567	−1.20874
$94$	2.61227	0.269435
$95$	0	0
$96$	−2.78049	−0.283782
$97$	−0.250970	−0.0254822	−0.0127411	−	0.999919i	$-0.504056\pi$
$97$	−0.250970	−0.0254822	−0.0127411	+	0.999919i	$0.504056\pi$
$98$	−2.71599	−0.274356
$99$	−4.73112	−0.475495
$100$	−4.66443	−0.466443
$101$	−11.4348	−1.13780	−0.568900	−	0.822407i	$-0.692631\pi$
$101$	−11.4348	−1.13780	−0.568900	+	0.822407i	$0.692631\pi$
$102$	3.17284	0.314158
$103$	11.6147	1.14443	0.572216	−	0.820103i	$-0.306084\pi$
$103$	11.6147	1.14443	0.572216	+	0.820103i	$0.306084\pi$
$104$	−2.38101	−0.233477
$105$	−3.33381	−0.325346
$106$	2.93059	0.284644
$107$	−18.9854	−1.83538	−0.917692	−	0.397293i	$-0.869950\pi$
$107$	−18.9854	−1.83538	−0.917692	+	0.397293i	$0.869950\pi$
$108$	−4.81335	−0.463165
$109$	−7.66962	−0.734617	−0.367308	−	0.930099i	$-0.619721\pi$
$109$	−7.66962	−0.734617	−0.367308	+	0.930099i	$0.619721\pi$
$110$	0.579287	0.0552328
$111$	−19.5823	−1.85867
$112$	−2.06979	−0.195576
$113$	11.3727	1.06985	0.534927	−	0.844898i	$-0.320339\pi$
$113$	11.3727	1.06985	0.534927	+	0.844898i	$0.320339\pi$
$114$	0	0
$115$	0.686314	0.0639992
$116$	−4.05557	−0.376550
$117$	−11.2648	−1.04143
$118$	1.98596	0.182822
$119$	2.36185	0.216510
$120$	1.61070	0.147036
$121$	1.00000	0.0909091
$122$	1.21810	0.110281
$123$	26.3383	2.37484
$124$	4.19233	0.376482
$125$	5.59848	0.500743
$126$	−9.79240	−0.872376
$127$	17.4183	1.54563	0.772814	−	0.634633i	$-0.218849\pi$
$127$	17.4183	1.54563	0.772814	+	0.634633i	$0.218849\pi$
$128$	1.00000	0.0883883
$129$	26.7831	2.35812
$130$	1.37929	0.120972
$131$	−17.1293	−1.49659	−0.748297	−	0.663363i	$-0.769128\pi$
$131$	−17.1293	−1.49659	−0.748297	+	0.663363i	$0.769128\pi$
$132$	2.78049	0.242010
$133$	0	0
$134$	−8.48031	−0.732587
$135$	2.78831	0.239980
$136$	−1.14111	−0.0978491
$137$	−8.99038	−0.768100	−0.384050	−	0.923312i	$-0.625471\pi$
$137$	−8.99038	−0.768100	−0.384050	+	0.923312i	$0.625471\pi$
$138$	3.29420	0.280421
$139$	4.44283	0.376836	0.188418	−	0.982089i	$-0.439664\pi$
$139$	4.44283	0.376836	0.188418	+	0.982089i	$0.439664\pi$
$140$	1.19900	0.101334
$141$	−7.26338	−0.611687
$142$	8.58773	0.720666
$143$	2.38101	0.199110
$144$	4.73112	0.394260
$145$	2.34934	0.195102
$146$	−7.47179	−0.618370
$147$	7.55177	0.622859
$148$	7.04275	0.578910
$149$	−3.08841	−0.253013	−0.126506	−	0.991966i	$-0.540376\pi$
$149$	−3.08841	−0.253013	−0.126506	+	0.991966i	$0.540376\pi$
$150$	12.9694	1.05895
$151$	−22.1553	−1.80297	−0.901486	−	0.432808i	$-0.857523\pi$
$151$	−22.1553	−1.80297	−0.901486	+	0.432808i	$0.857523\pi$
$152$	0	0
$153$	−5.39871	−0.436460
$154$	2.06979	0.166788
$155$	−2.42856	−0.195067
$156$	6.62037	0.530054
$157$	20.2504	1.61616	0.808080	−	0.589072i	$-0.200507\pi$
$157$	20.2504	1.61616	0.808080	+	0.589072i	$0.200507\pi$
$158$	1.78218	0.141783
$159$	−8.14847	−0.646215
$160$	−0.579287	−0.0457967
$161$	2.45219	0.193260
$162$	−0.809886	−0.0636306
$163$	22.4331	1.75710	0.878549	−	0.477652i	$-0.158512\pi$
$163$	22.4331	1.75710	0.878549	+	0.477652i	$0.158512\pi$
$164$	−9.47254	−0.739681
$165$	−1.61070	−0.125393
$166$	0.740015	0.0574363
$167$	24.0828	1.86359	0.931793	−	0.362990i	$-0.118244\pi$
$167$	24.0828	1.86359	0.931793	+	0.362990i	$0.118244\pi$
$168$	5.75502	0.444009
$169$	−7.33080	−0.563907
$170$	0.661029	0.0506986
$171$	0	0
$172$	−9.63251	−0.734472
$173$	4.20083	0.319383	0.159692	−	0.987167i	$-0.448950\pi$
$173$	4.20083	0.319383	0.159692	+	0.987167i	$0.448950\pi$
$174$	11.2765	0.854866
$175$	9.65436	0.729801
$176$	−1.00000	−0.0753778
$177$	−5.52194	−0.415054
$178$	3.98775	0.298895
$179$	3.53598	0.264292	0.132146	−	0.991230i	$-0.457813\pi$
$179$	3.53598	0.264292	0.132146	+	0.991230i	$0.457813\pi$
$180$	−2.74067	−0.204278
$181$	9.26978	0.689017	0.344509	−	0.938783i	$-0.388046\pi$
$181$	9.26978	0.689017	0.344509	+	0.938783i	$0.388046\pi$
$182$	4.92818	0.365301
$183$	−3.38690	−0.250367
$184$	−1.18476	−0.0873415
$185$	−4.07977	−0.299951
$186$	−11.6567	−0.854712
$187$	1.14111	0.0834460
$188$	2.61227	0.190519
$189$	9.96260	0.724673
$190$	0	0
$191$	8.13310	0.588491	0.294245	−	0.955730i	$-0.404932\pi$
$191$	8.13310	0.588491	0.294245	+	0.955730i	$0.404932\pi$
$192$	−2.78049	−0.200664
$193$	−21.5031	−1.54783	−0.773915	−	0.633290i	$-0.781704\pi$
$193$	−21.5031	−1.54783	−0.773915	+	0.633290i	$0.781704\pi$
$194$	−0.250970	−0.0180186
$195$	−3.83509	−0.274637
$196$	−2.71599	−0.193999
$197$	−3.02887	−0.215798	−0.107899	−	0.994162i	$-0.534412\pi$
$197$	−3.02887	−0.215798	−0.107899	+	0.994162i	$0.534412\pi$
$198$	−4.73112	−0.336226
$199$	2.87487	0.203794	0.101897	−	0.994795i	$-0.467509\pi$
$199$	2.87487	0.203794	0.101897	+	0.994795i	$0.467509\pi$
$200$	−4.66443	−0.329825
$201$	23.5794	1.66316
$202$	−11.4348	−0.804546
$203$	8.39416	0.589154
$204$	3.17284	0.222143
$205$	5.48732	0.383251
$206$	11.6147	0.809235
$207$	−5.60522	−0.389590
$208$	−2.38101	−0.165093
$209$	0	0
$210$	−3.33381	−0.230054
$211$	−20.5064	−1.41172	−0.705858	−	0.708354i	$-0.749438\pi$
$211$	−20.5064	−1.41172	−0.705858	+	0.708354i	$0.749438\pi$
$212$	2.93059	0.201274
$213$	−23.8781	−1.63610
$214$	−18.9854	−1.29781
$215$	5.57999	0.380552
$216$	−4.81335	−0.327507
$217$	−8.67722	−0.589048
$218$	−7.66962	−0.519453
$219$	20.7752	1.40386
$220$	0.579287	0.0390555
$221$	2.71699	0.182764
$222$	−19.5823	−1.31428
$223$	25.8321	1.72984	0.864922	−	0.501906i	$-0.167368\pi$
$223$	25.8321	1.72984	0.864922	+	0.501906i	$0.167368\pi$
$224$	−2.06979	−0.138293
$225$	−22.0679	−1.47120
$226$	11.3727	0.756501
$227$	16.1427	1.07143	0.535716	−	0.844398i	$-0.320042\pi$
$227$	16.1427	1.07143	0.535716	+	0.844398i	$0.320042\pi$
$228$	0	0
$229$	−12.8061	−0.846252	−0.423126	−	0.906071i	$-0.639067\pi$
$229$	−12.8061	−0.846252	−0.423126	+	0.906071i	$0.639067\pi$
$230$	0.686314	0.0452542
$231$	−5.75502	−0.378652
$232$	−4.05557	−0.266261
$233$	1.86898	0.122441	0.0612206	−	0.998124i	$-0.480501\pi$
$233$	1.86898	0.122441	0.0612206	+	0.998124i	$0.480501\pi$
$234$	−11.2648	−0.736405
$235$	−1.51325	−0.0987138
$236$	1.98596	0.129275
$237$	−4.95534	−0.321884
$238$	2.36185	0.153096
$239$	−4.07877	−0.263834	−0.131917	−	0.991261i	$-0.542113\pi$
$239$	−4.07877	−0.263834	−0.131917	+	0.991261i	$0.542113\pi$
$240$	1.61070	0.103970
$241$	6.85830	0.441782	0.220891	−	0.975298i	$-0.429103\pi$
$241$	6.85830	0.441782	0.220891	+	0.975298i	$0.429103\pi$
$242$	1.00000	0.0642824
$243$	16.6919	1.07079
$244$	1.21810	0.0779806
$245$	1.57334	0.100517
$246$	26.3383	1.67927
$247$	0	0
$248$	4.19233	0.266213
$249$	−2.05760	−0.130395
$250$	5.59848	0.354079
$251$	8.85571	0.558967	0.279484	−	0.960150i	$-0.409837\pi$
$251$	8.85571	0.558967	0.279484	+	0.960150i	$0.409837\pi$
$252$	−9.79240	−0.616863
$253$	1.18476	0.0744851
$254$	17.4183	1.09292
$255$	−1.83798	−0.115099
$256$	1.00000	0.0625000
$257$	30.8719	1.92574	0.962869	−	0.269971i	$-0.0870140\pi$
$257$	30.8719	1.92574	0.962869	+	0.269971i	$0.0870140\pi$
$258$	26.7831	1.66744
$259$	−14.5770	−0.905770
$260$	1.37929	0.0855398
$261$	−19.1874	−1.18767
$262$	−17.1293	−1.05825
$263$	−0.765056	−0.0471754	−0.0235877	−	0.999722i	$-0.507509\pi$
$263$	−0.765056	−0.0471754	−0.0235877	+	0.999722i	$0.507509\pi$
$264$	2.78049	0.171127
$265$	−1.69765	−0.104286
$266$	0	0
$267$	−11.0879	−0.678568
$268$	−8.48031	−0.518017
$269$	4.65562	0.283858	0.141929	−	0.989877i	$-0.454669\pi$
$269$	4.65562	0.283858	0.141929	+	0.989877i	$0.454669\pi$
$270$	2.78831	0.169691
$271$	23.6377	1.43589	0.717943	−	0.696102i	$-0.245084\pi$
$271$	23.6377	1.43589	0.717943	+	0.696102i	$0.245084\pi$
$272$	−1.14111	−0.0691898
$273$	−13.7027	−0.829328
$274$	−8.99038	−0.543129
$275$	4.66443	0.281276
$276$	3.29420	0.198288
$277$	11.9486	0.717923	0.358962	−	0.933352i	$-0.383131\pi$
$277$	11.9486	0.717923	0.358962	+	0.933352i	$0.383131\pi$
$278$	4.44283	0.266463
$279$	19.8344	1.18745
$280$	1.19900	0.0716540
$281$	−12.5573	−0.749105	−0.374553	−	0.927206i	$-0.622204\pi$
$281$	−12.5573	−0.749105	−0.374553	+	0.927206i	$0.622204\pi$
$282$	−7.26338	−0.432528
$283$	−12.4632	−0.740860	−0.370430	−	0.928860i	$-0.620790\pi$
$283$	−12.4632	−0.740860	−0.370430	+	0.928860i	$0.620790\pi$
$284$	8.58773	0.509588
$285$	0	0
$286$	2.38101	0.140792
$287$	19.6061	1.15731
$288$	4.73112	0.278784
$289$	−15.6979	−0.923404
$290$	2.34934	0.137958
$291$	0.697820	0.0409069
$292$	−7.47179	−0.437254
$293$	28.9636	1.69207	0.846036	−	0.533125i	$-0.178982\pi$
$293$	28.9636	1.69207	0.846036	+	0.533125i	$0.178982\pi$
$294$	7.55177	0.440428
$295$	−1.15044	−0.0669812
$296$	7.04275	0.409351
$297$	4.81335	0.279299
$298$	−3.08841	−0.178907
$299$	2.82092	0.163138
$300$	12.9694	0.748788
$301$	19.9372	1.14916
$302$	−22.1553	−1.27489
$303$	31.7942	1.82653
$304$	0	0
$305$	−0.705627	−0.0404041
$306$	−5.39871	−0.308624
$307$	4.62712	0.264084	0.132042	−	0.991244i	$-0.457847\pi$
$307$	4.62712	0.264084	0.132042	+	0.991244i	$0.457847\pi$
$308$	2.06979	0.117937
$309$	−32.2946	−1.83717
$310$	−2.42856	−0.137933
$311$	−9.30319	−0.527535	−0.263768	−	0.964586i	$-0.584965\pi$
$311$	−9.30319	−0.527535	−0.263768	+	0.964586i	$0.584965\pi$
$312$	6.62037	0.374805
$313$	−6.07531	−0.343397	−0.171698	−	0.985150i	$-0.554925\pi$
$313$	−6.07531	−0.343397	−0.171698	+	0.985150i	$0.554925\pi$
$314$	20.2504	1.14280
$315$	5.67261	0.319615
$316$	1.78218	0.100256
$317$	22.5519	1.26664	0.633319	−	0.773891i	$-0.281692\pi$
$317$	22.5519	1.26664	0.633319	+	0.773891i	$0.281692\pi$
$318$	−8.14847	−0.456943
$319$	4.05557	0.227068
$320$	−0.579287	−0.0323831
$321$	52.7886	2.94637
$322$	2.45219	0.136655
$323$	0	0
$324$	−0.809886	−0.0449937
$325$	11.1060	0.616052
$326$	22.4331	1.24246
$327$	21.3253	1.17929
$328$	−9.47254	−0.523034
$329$	−5.40684	−0.298089
$330$	−1.61070	−0.0886662
$331$	28.9152	1.58932	0.794661	−	0.607053i	$-0.207649\pi$
$331$	28.9152	1.58932	0.794661	+	0.607053i	$0.207649\pi$
$332$	0.740015	0.0406136
$333$	33.3201	1.82593
$334$	24.0828	1.31775
$335$	4.91253	0.268400
$336$	5.75502	0.313962
$337$	−2.23663	−0.121837	−0.0609184	−	0.998143i	$-0.519403\pi$
$337$	−2.23663	−0.121837	−0.0609184	+	0.998143i	$0.519403\pi$
$338$	−7.33080	−0.398743
$339$	−31.6217	−1.71745
$340$	0.661029	0.0358493
$341$	−4.19233	−0.227027
$342$	0	0
$343$	20.1100	1.08584
$344$	−9.63251	−0.519350
$345$	−1.90829	−0.102739
$346$	4.20083	0.225838
$347$	28.6968	1.54052	0.770262	−	0.637728i	$-0.220126\pi$
$347$	28.6968	1.54052	0.770262	+	0.637728i	$0.220126\pi$
$348$	11.2765	0.604482
$349$	−6.26178	−0.335185	−0.167593	−	0.985856i	$-0.553599\pi$
$349$	−6.26178	−0.335185	−0.167593	+	0.985856i	$0.553599\pi$
$350$	9.65436	0.516047
$351$	11.4606	0.611723
$352$	−1.00000	−0.0533002
$353$	−27.6978	−1.47420	−0.737102	−	0.675782i	$-0.763806\pi$
$353$	−27.6978	−1.47420	−0.737102	+	0.675782i	$0.763806\pi$
$354$	−5.52194	−0.293488
$355$	−4.97476	−0.264033
$356$	3.98775	0.211350
$357$	−6.56709	−0.347567
$358$	3.53598	0.186882
$359$	4.13875	0.218435	0.109217	−	0.994018i	$-0.465166\pi$
$359$	4.13875	0.218435	0.109217	+	0.994018i	$0.465166\pi$
$360$	−2.74067	−0.144446
$361$	0	0
$362$	9.26978	0.487209
$363$	−2.78049	−0.145938
$364$	4.92818	0.258307
$365$	4.32831	0.226554
$366$	−3.38690	−0.177036
$367$	13.8205	0.721424	0.360712	−	0.932677i	$-0.382534\pi$
$367$	13.8205	0.721424	0.360712	+	0.932677i	$0.382534\pi$
$368$	−1.18476	−0.0617597
$369$	−44.8157	−2.33301
$370$	−4.07977	−0.212097
$371$	−6.06569	−0.314915
$372$	−11.6567	−0.604372
$373$	15.0110	0.777239	0.388620	−	0.921398i	$-0.372952\pi$
$373$	15.0110	0.777239	0.388620	+	0.921398i	$0.372952\pi$
$374$	1.14111	0.0590053
$375$	−15.5665	−0.803851
$376$	2.61227	0.134717
$377$	9.65635	0.497327
$378$	9.96260	0.512421
$379$	−34.9272	−1.79409	−0.897046	−	0.441938i	$-0.854291\pi$
$379$	−34.9272	−1.79409	−0.897046	+	0.441938i	$0.854291\pi$
$380$	0	0
$381$	−48.4315	−2.48122
$382$	8.13310	0.416126
$383$	27.4305	1.40163	0.700817	−	0.713342i	$-0.252819\pi$
$383$	27.4305	1.40163	0.700817	+	0.713342i	$0.252819\pi$
$384$	−2.78049	−0.141891
$385$	−1.19900	−0.0611067
$386$	−21.5031	−1.09448
$387$	−45.5725	−2.31658
$388$	−0.250970	−0.0127411
$389$	−36.2675	−1.83884	−0.919418	−	0.393281i	$-0.871340\pi$
$389$	−36.2675	−1.83884	−0.919418	+	0.393281i	$0.871340\pi$
$390$	−3.83509	−0.194197
$391$	1.35194	0.0683703
$392$	−2.71599	−0.137178
$393$	47.6279	2.40251
$394$	−3.02887	−0.152592
$395$	−1.03240	−0.0519454
$396$	−4.73112	−0.237748
$397$	13.1364	0.659296	0.329648	−	0.944104i	$-0.393070\pi$
$397$	13.1364	0.659296	0.329648	+	0.944104i	$0.393070\pi$
$398$	2.87487	0.144104
$399$	0	0
$400$	−4.66443	−0.233221
$401$	−23.1434	−1.15572	−0.577862	−	0.816134i	$-0.696113\pi$
$401$	−23.1434	−1.15572	−0.577862	+	0.816134i	$0.696113\pi$
$402$	23.5794	1.17603
$403$	−9.98196	−0.497237
$404$	−11.4348	−0.568900
$405$	0.469156	0.0233126
$406$	8.39416	0.416595
$407$	−7.04275	−0.349096
$408$	3.17284	0.157079
$409$	9.23341	0.456563	0.228281	−	0.973595i	$-0.426689\pi$
$409$	9.23341	0.456563	0.228281	+	0.973595i	$0.426689\pi$
$410$	5.48732	0.270999
$411$	24.9977	1.23304
$412$	11.6147	0.572216
$413$	−4.11051	−0.202265
$414$	−5.60522	−0.275482
$415$	−0.428681	−0.0210431
$416$	−2.38101	−0.116739
$417$	−12.3532	−0.604941
$418$	0	0
$419$	38.4806	1.87990	0.939951	−	0.341310i	$-0.110871\pi$
$419$	38.4806	1.87990	0.939951	+	0.341310i	$0.110871\pi$
$420$	−3.33381	−0.162673
$421$	10.2075	0.497484	0.248742	−	0.968570i	$-0.419983\pi$
$421$	10.2075	0.497484	0.248742	+	0.968570i	$0.419983\pi$
$422$	−20.5064	−0.998234
$423$	12.3589	0.600913
$424$	2.93059	0.142322
$425$	5.32261	0.258185
$426$	−23.8781	−1.15690
$427$	−2.52120	−0.122009
$428$	−18.9854	−0.917692
$429$	−6.62037	−0.319634
$430$	5.57999	0.269091
$431$	17.3154	0.834056	0.417028	−	0.908894i	$-0.363072\pi$
$431$	17.3154	0.834056	0.417028	+	0.908894i	$0.363072\pi$
$432$	−4.81335	−0.231582
$433$	2.80814	0.134951	0.0674753	−	0.997721i	$-0.478506\pi$
$433$	2.80814	0.134951	0.0674753	+	0.997721i	$0.478506\pi$
$434$	−8.67722	−0.416520
$435$	−6.53231	−0.313200
$436$	−7.66962	−0.367308
$437$	0	0
$438$	20.7752	0.992679
$439$	−6.64344	−0.317074	−0.158537	−	0.987353i	$-0.550678\pi$
$439$	−6.64344	−0.317074	−0.158537	+	0.987353i	$0.550678\pi$
$440$	0.579287	0.0276164
$441$	−12.8496	−0.611888
$442$	2.71699	0.129234
$443$	−14.4952	−0.688688	−0.344344	−	0.938844i	$-0.611899\pi$
$443$	−14.4952	−0.688688	−0.344344	+	0.938844i	$0.611899\pi$
$444$	−19.5823	−0.929334
$445$	−2.31005	−0.109507
$446$	25.8321	1.22318
$447$	8.58730	0.406165
$448$	−2.06979	−0.0977882
$449$	13.0713	0.616873	0.308436	−	0.951245i	$-0.400194\pi$
$449$	13.0713	0.616873	0.308436	+	0.951245i	$0.400194\pi$
$450$	−22.0679	−1.04029
$451$	9.47254	0.446045
$452$	11.3727	0.534927
$453$	61.6025	2.89434
$454$	16.1427	0.757617
$455$	−2.85483	−0.133836
$456$	0	0
$457$	12.3869	0.579433	0.289717	−	0.957112i	$-0.406439\pi$
$457$	12.3869	0.579433	0.289717	+	0.957112i	$0.406439\pi$
$458$	−12.8061	−0.598391
$459$	5.49255	0.256370
$460$	0.686314	0.0319996
$461$	−11.4371	−0.532677	−0.266338	−	0.963880i	$-0.585814\pi$
$461$	−11.4371	−0.532677	−0.266338	+	0.963880i	$0.585814\pi$
$462$	−5.75502	−0.267748
$463$	23.0785	1.07255	0.536274	−	0.844044i	$-0.319831\pi$
$463$	23.0785	1.07255	0.536274	+	0.844044i	$0.319831\pi$
$464$	−4.05557	−0.188275
$465$	6.75258	0.313144
$466$	1.86898	0.0865789
$467$	−9.63721	−0.445957	−0.222979	−	0.974823i	$-0.571578\pi$
$467$	−9.63721	−0.445957	−0.222979	+	0.974823i	$0.571578\pi$
$468$	−11.2648	−0.520717
$469$	17.5524	0.810496
$470$	−1.51325	−0.0698012
$471$	−56.3061	−2.59445
$472$	1.98596	0.0914112
$473$	9.63251	0.442903
$474$	−4.95534	−0.227606
$475$	0	0
$476$	2.36185	0.108255
$477$	13.8650	0.634833
$478$	−4.07877	−0.186558
$479$	9.46510	0.432472	0.216236	−	0.976341i	$-0.430622\pi$
$479$	9.46510	0.432472	0.216236	+	0.976341i	$0.430622\pi$
$480$	1.61070	0.0735181
$481$	−16.7688	−0.764594
$482$	6.85830	0.312387
$483$	−6.81830	−0.310243
$484$	1.00000	0.0454545
$485$	0.145384	0.00660154
$486$	16.6919	0.757161
$487$	18.1199	0.821090	0.410545	−	0.911840i	$-0.365338\pi$
$487$	18.1199	0.821090	0.410545	+	0.911840i	$0.365338\pi$
$488$	1.21810	0.0551406
$489$	−62.3751	−2.82070
$490$	1.57334	0.0710761
$491$	−5.43289	−0.245183	−0.122591	−	0.992457i	$-0.539120\pi$
$491$	−5.43289	−0.245183	−0.122591	+	0.992457i	$0.539120\pi$
$492$	26.3383	1.18742
$493$	4.62784	0.208427
$494$	0	0
$495$	2.74067	0.123184
$496$	4.19233	0.188241
$497$	−17.7748	−0.797307
$498$	−2.05760	−0.0922034
$499$	37.3967	1.67410	0.837052	−	0.547123i	$-0.184277\pi$
$499$	37.3967	1.67410	0.837052	+	0.547123i	$0.184277\pi$
$500$	5.59848	0.250371
$501$	−66.9621	−2.99164
$502$	8.85571	0.395250
$503$	5.46251	0.243561	0.121781	−	0.992557i	$-0.461140\pi$
$503$	5.46251	0.243561	0.121781	+	0.992557i	$0.461140\pi$
$504$	−9.79240	−0.436188
$505$	6.62400	0.294764
$506$	1.18476	0.0526689
$507$	20.3832	0.905249
$508$	17.4183	0.772814
$509$	22.9526	1.01736	0.508678	−	0.860957i	$-0.330134\pi$
$509$	22.9526	1.01736	0.508678	+	0.860957i	$0.330134\pi$
$510$	−1.83798	−0.0813872
$511$	15.4650	0.684132
$512$	1.00000	0.0441942
$513$	0	0
$514$	30.8719	1.36170
$515$	−6.72825	−0.296482
$516$	26.7831	1.17906
$517$	−2.61227	−0.114887
$518$	−14.5770	−0.640476
$519$	−11.6804	−0.512711
$520$	1.37929	0.0604858
$521$	3.75032	0.164304	0.0821522	−	0.996620i	$-0.473821\pi$
$521$	3.75032	0.164304	0.0821522	+	0.996620i	$0.473821\pi$
$522$	−19.1874	−0.839808
$523$	−7.69315	−0.336398	−0.168199	−	0.985753i	$-0.553795\pi$
$523$	−7.69315	−0.336398	−0.168199	+	0.985753i	$0.553795\pi$
$524$	−17.1293	−0.748297
$525$	−26.8438	−1.17156
$526$	−0.765056	−0.0333580
$527$	−4.78389	−0.208390
$528$	2.78049	0.121005
$529$	−21.5964	−0.938972
$530$	−1.69765	−0.0737413
$531$	9.39580	0.407743
$532$	0	0
$533$	22.5542	0.976931
$534$	−11.0879	−0.479820
$535$	10.9980	0.475484
$536$	−8.48031	−0.366294
$537$	−9.83175	−0.424271
$538$	4.65562	0.200718
$539$	2.71599	0.116986
$540$	2.78831	0.119990
$541$	29.4637	1.26674	0.633371	−	0.773848i	$-0.281671\pi$
$541$	29.4637	1.26674	0.633371	+	0.773848i	$0.281671\pi$
$542$	23.6377	1.01532
$543$	−25.7745	−1.10609
$544$	−1.14111	−0.0489246
$545$	4.44291	0.190314
$546$	−13.7027	−0.586423
$547$	9.55757	0.408652	0.204326	−	0.978903i	$-0.434500\pi$
$547$	9.55757	0.408652	0.204326	+	0.978903i	$0.434500\pi$
$548$	−8.99038	−0.384050
$549$	5.76296	0.245957
$550$	4.66443	0.198892
$551$	0	0
$552$	3.29420	0.140211
$553$	−3.68874	−0.156861
$554$	11.9486	0.507648
$555$	11.3438	0.481516
$556$	4.44283	0.188418
$557$	−10.1070	−0.428248	−0.214124	−	0.976806i	$-0.568690\pi$
$557$	−10.1070	−0.428248	−0.214124	+	0.976806i	$0.568690\pi$
$558$	19.8344	0.839656
$559$	22.9351	0.970051
$560$	1.19900	0.0506670
$561$	−3.17284	−0.133957
$562$	−12.5573	−0.529698
$563$	−31.4489	−1.32541	−0.662706	−	0.748879i	$-0.730592\pi$
$563$	−31.4489	−1.32541	−0.662706	+	0.748879i	$0.730592\pi$
$564$	−7.26338	−0.305844
$565$	−6.58806	−0.277162
$566$	−12.4632	−0.523867
$567$	1.67629	0.0703976
$568$	8.58773	0.360333
$569$	−1.86413	−0.0781483	−0.0390742	−	0.999236i	$-0.512441\pi$
$569$	−1.86413	−0.0781483	−0.0390742	+	0.999236i	$0.512441\pi$
$570$	0	0
$571$	−0.430952	−0.0180348	−0.00901739	−	0.999959i	$-0.502870\pi$
$571$	−0.430952	−0.0180348	−0.00901739	+	0.999959i	$0.502870\pi$
$572$	2.38101	0.0995550
$573$	−22.6140	−0.944713
$574$	19.6061	0.818344
$575$	5.52621	0.230459
$576$	4.73112	0.197130
$577$	6.45213	0.268606	0.134303	−	0.990940i	$-0.457121\pi$
$577$	6.45213	0.268606	0.134303	+	0.990940i	$0.457121\pi$
$578$	−15.6979	−0.652945
$579$	59.7892	2.48476
$580$	2.34934	0.0975510
$581$	−1.53167	−0.0635445
$582$	0.697820	0.0289256
$583$	−2.93059	−0.121373
$584$	−7.47179	−0.309185
$585$	6.52557	0.269799
$586$	28.9636	1.19648
$587$	−17.6481	−0.728417	−0.364208	−	0.931317i	$-0.618660\pi$
$587$	−17.6481	−0.728417	−0.364208	+	0.931317i	$0.618660\pi$
$588$	7.55177	0.311430
$589$	0	0
$590$	−1.15044	−0.0473629
$591$	8.42174	0.346424
$592$	7.04275	0.289455
$593$	1.85105	0.0760135	0.0380068	−	0.999277i	$-0.487899\pi$
$593$	1.85105	0.0760135	0.0380068	+	0.999277i	$0.487899\pi$
$594$	4.81335	0.197494
$595$	−1.36819	−0.0560902
$596$	−3.08841	−0.126506
$597$	−7.99354	−0.327154
$598$	2.82092	0.115356
$599$	22.8230	0.932522	0.466261	−	0.884647i	$-0.345601\pi$
$599$	22.8230	0.932522	0.466261	+	0.884647i	$0.345601\pi$
$600$	12.9694	0.529473
$601$	5.62107	0.229288	0.114644	−	0.993407i	$-0.463427\pi$
$601$	5.62107	0.229288	0.114644	+	0.993407i	$0.463427\pi$
$602$	19.9372	0.812581
$603$	−40.1213	−1.63387
$604$	−22.1553	−0.901486
$605$	−0.579287	−0.0235514
$606$	31.7942	1.29155
$607$	−19.8879	−0.807225	−0.403613	−	0.914930i	$-0.632246\pi$
$607$	−19.8879	−0.807225	−0.403613	+	0.914930i	$0.632246\pi$
$608$	0	0
$609$	−23.3399	−0.945779
$610$	−0.705627	−0.0285700
$611$	−6.21984	−0.251628
$612$	−5.39871	−0.218230
$613$	−48.8745	−1.97402	−0.987012	−	0.160647i	$-0.948642\pi$
$613$	−48.8745	−1.97402	−0.987012	+	0.160647i	$0.948642\pi$
$614$	4.62712	0.186735
$615$	−15.2574	−0.615239
$616$	2.06979	0.0833940
$617$	21.8351	0.879047	0.439524	−	0.898231i	$-0.355147\pi$
$617$	21.8351	0.879047	0.439524	+	0.898231i	$0.355147\pi$
$618$	−32.2946	−1.29908
$619$	33.8871	1.36204	0.681019	−	0.732266i	$-0.261537\pi$
$619$	33.8871	1.36204	0.681019	+	0.732266i	$0.261537\pi$
$620$	−2.42856	−0.0975333
$621$	5.70265	0.228840
$622$	−9.30319	−0.373024
$623$	−8.25379	−0.330681
$624$	6.62037	0.265027
$625$	20.0790	0.803160
$626$	−6.07531	−0.242818
$627$	0	0
$628$	20.2504	0.808080
$629$	−8.03653	−0.320438
$630$	5.67261	0.226002
$631$	20.9381	0.833531	0.416765	−	0.909014i	$-0.363164\pi$
$631$	20.9381	0.833531	0.416765	+	0.909014i	$0.363164\pi$
$632$	1.78218	0.0708914
$633$	57.0177	2.26625
$634$	22.5519	0.895649
$635$	−10.0902	−0.400418
$636$	−8.14847	−0.323108
$637$	6.46679	0.256223
$638$	4.05557	0.160561
$639$	40.6295	1.60728
$640$	−0.579287	−0.0228983
$641$	28.0089	1.10628	0.553142	−	0.833087i	$-0.313429\pi$
$641$	28.0089	1.10628	0.553142	+	0.833087i	$0.313429\pi$
$642$	52.7886	2.08340
$643$	−4.94532	−0.195024	−0.0975121	−	0.995234i	$-0.531088\pi$
$643$	−4.94532	−0.195024	−0.0975121	+	0.995234i	$0.531088\pi$
$644$	2.45219	0.0966300
$645$	−15.5151	−0.610906
$646$	0	0
$647$	21.5112	0.845692	0.422846	−	0.906202i	$-0.361031\pi$
$647$	21.5112	0.845692	0.422846	+	0.906202i	$0.361031\pi$
$648$	−0.809886	−0.0318153
$649$	−1.98596	−0.0779557
$650$	11.1060	0.435615
$651$	24.1269	0.945608
$652$	22.4331	0.878549
$653$	34.9056	1.36596	0.682981	−	0.730436i	$-0.260683\pi$
$653$	34.9056	1.36596	0.682981	+	0.730436i	$0.260683\pi$
$654$	21.3253	0.833885
$655$	9.92279	0.387715
$656$	−9.47254	−0.369841
$657$	−35.3499	−1.37913
$658$	−5.40684	−0.210780
$659$	32.2832	1.25758	0.628788	−	0.777577i	$-0.283551\pi$
$659$	32.2832	1.25758	0.628788	+	0.777577i	$0.283551\pi$
$660$	−1.61070	−0.0626964
$661$	−43.5462	−1.69375	−0.846875	−	0.531792i	$-0.821519\pi$
$661$	−43.5462	−1.69375	−0.846875	+	0.531792i	$0.821519\pi$
$662$	28.9152	1.12382
$663$	−7.55455	−0.293394
$664$	0.740015	0.0287181
$665$	0	0
$666$	33.3201	1.29113
$667$	4.80486	0.186045
$668$	24.0828	0.931793
$669$	−71.8258	−2.77695
$670$	4.91253	0.189788
$671$	−1.21810	−0.0470241
$672$	5.75502	0.222005
$673$	−40.5544	−1.56326	−0.781628	−	0.623744i	$-0.785611\pi$
$673$	−40.5544	−1.56326	−0.781628	+	0.623744i	$0.785611\pi$
$674$	−2.23663	−0.0861516
$675$	22.4515	0.864159
$676$	−7.33080	−0.281954
$677$	12.6233	0.485153	0.242577	−	0.970132i	$-0.422007\pi$
$677$	12.6233	0.485153	0.242577	+	0.970132i	$0.422007\pi$
$678$	−31.6217	−1.21442
$679$	0.519455	0.0199348
$680$	0.661029	0.0253493
$681$	−44.8847	−1.71999
$682$	−4.19233	−0.160532
$683$	−30.5401	−1.16859	−0.584293	−	0.811543i	$-0.698628\pi$
$683$	−30.5401	−1.16859	−0.584293	+	0.811543i	$0.698628\pi$
$684$	0	0
$685$	5.20801	0.198988
$686$	20.1100	0.767804
$687$	35.6073	1.35850
$688$	−9.63251	−0.367236
$689$	−6.97776	−0.265831
$690$	−1.90829	−0.0726473
$691$	13.3812	0.509045	0.254523	−	0.967067i	$-0.418082\pi$
$691$	13.3812	0.509045	0.254523	+	0.967067i	$0.418082\pi$
$692$	4.20083	0.159692
$693$	9.79240	0.371982
$694$	28.6968	1.08931
$695$	−2.57367	−0.0976250
$696$	11.2765	0.427433
$697$	10.8092	0.409427
$698$	−6.26178	−0.237012
$699$	−5.19668	−0.196557
$700$	9.65436	0.364901
$701$	−4.26910	−0.161242	−0.0806208	−	0.996745i	$-0.525690\pi$
$701$	−4.26910	−0.161242	−0.0806208	+	0.996745i	$0.525690\pi$
$702$	11.4606	0.432554
$703$	0	0
$704$	−1.00000	−0.0376889
$705$	4.20758	0.158467
$706$	−27.6978	−1.04242
$707$	23.6675	0.890107
$708$	−5.52194	−0.207527
$709$	28.0100	1.05194	0.525969	−	0.850504i	$-0.323703\pi$
$709$	28.0100	1.05194	0.525969	+	0.850504i	$0.323703\pi$
$710$	−4.97476	−0.186699
$711$	8.43171	0.316214
$712$	3.98775	0.149447
$713$	−4.96689	−0.186011
$714$	−6.56709	−0.245767
$715$	−1.37929	−0.0515824
$716$	3.53598	0.132146
$717$	11.3410	0.423536
$718$	4.13875	0.154457
$719$	−9.31494	−0.347389	−0.173694	−	0.984800i	$-0.555570\pi$
$719$	−9.31494	−0.347389	−0.173694	+	0.984800i	$0.555570\pi$
$720$	−2.74067	−0.102139
$721$	−24.0400	−0.895295
$722$	0	0
$723$	−19.0694	−0.709200
$724$	9.26978	0.344509
$725$	18.9169	0.702556
$726$	−2.78049	−0.103194
$727$	−26.2226	−0.972542	−0.486271	−	0.873808i	$-0.661643\pi$
$727$	−26.2226	−0.972542	−0.486271	+	0.873808i	$0.661643\pi$
$728$	4.92818	0.182650
$729$	−43.9821	−1.62897
$730$	4.32831	0.160198
$731$	10.9917	0.406544
$732$	−3.38690	−0.125184
$733$	39.2561	1.44996	0.724979	−	0.688771i	$-0.241849\pi$
$733$	39.2561	1.44996	0.724979	+	0.688771i	$0.241849\pi$
$734$	13.8205	0.510124
$735$	−4.37464	−0.161361
$736$	−1.18476	−0.0436707
$737$	8.48031	0.312376
$738$	−44.8157	−1.64969
$739$	43.7134	1.60802	0.804012	−	0.594613i	$-0.202695\pi$
$739$	43.7134	1.60802	0.804012	+	0.594613i	$0.202695\pi$
$740$	−4.07977	−0.149975
$741$	0	0
$742$	−6.06569	−0.222679
$743$	−19.3954	−0.711547	−0.355773	−	0.934572i	$-0.615783\pi$
$743$	−19.3954	−0.711547	−0.355773	+	0.934572i	$0.615783\pi$
$744$	−11.6567	−0.427356
$745$	1.78908	0.0655467
$746$	15.0110	0.549591
$747$	3.50110	0.128098
$748$	1.14111	0.0417230
$749$	39.2956	1.43583
$750$	−15.5665	−0.568408
$751$	40.3470	1.47228	0.736142	−	0.676828i	$-0.236646\pi$
$751$	40.3470	1.47228	0.736142	+	0.676828i	$0.236646\pi$
$752$	2.61227	0.0952596
$753$	−24.6232	−0.897319
$754$	9.65635	0.351663
$755$	12.8343	0.467087
$756$	9.96260	0.362336
$757$	−18.8686	−0.685792	−0.342896	−	0.939373i	$-0.611408\pi$
$757$	−18.8686	−0.685792	−0.342896	+	0.939373i	$0.611408\pi$
$758$	−34.9272	−1.26861
$759$	−3.29420	−0.119572
$760$	0	0
$761$	−2.95866	−0.107251	−0.0536257	−	0.998561i	$-0.517078\pi$
$761$	−2.95866	−0.107251	−0.0536257	+	0.998561i	$0.517078\pi$
$762$	−48.4315	−1.75449
$763$	15.8745	0.574695
$764$	8.13310	0.294245
$765$	3.12740	0.113072
$766$	27.4305	0.991104
$767$	−4.72859	−0.170739
$768$	−2.78049	−0.100332
$769$	−1.62375	−0.0585538	−0.0292769	−	0.999571i	$-0.509320\pi$
$769$	−1.62375	−0.0585538	−0.0292769	+	0.999571i	$0.509320\pi$
$770$	−1.19900	−0.0432090
$771$	−85.8390	−3.09142
$772$	−21.5031	−0.773915
$773$	−46.4995	−1.67247	−0.836236	−	0.548370i	$-0.815249\pi$
$773$	−46.4995	−1.67247	−0.836236	+	0.548370i	$0.815249\pi$
$774$	−45.5725	−1.63807
$775$	−19.5548	−0.702429
$776$	−0.250970	−0.00900931
$777$	40.5311	1.45405
$778$	−36.2675	−1.30025
$779$	0	0
$780$	−3.83509	−0.137318
$781$	−8.58773	−0.307293
$782$	1.35194	0.0483451
$783$	19.5209	0.697619
$784$	−2.71599	−0.0969995
$785$	−11.7308	−0.418691
$786$	47.6279	1.69883
$787$	−30.8974	−1.10137	−0.550686	−	0.834712i	$-0.685634\pi$
$787$	−30.8974	−1.10137	−0.550686	+	0.834712i	$0.685634\pi$
$788$	−3.02887	−0.107899
$789$	2.12723	0.0757314
$790$	−1.03240	−0.0367310
$791$	−23.5391	−0.836953
$792$	−4.73112	−0.168113
$793$	−2.90030	−0.102993
$794$	13.1364	0.466193
$795$	4.72030	0.167412
$796$	2.87487	0.101897
$797$	−52.3499	−1.85433	−0.927164	−	0.374655i	$-0.877761\pi$
$797$	−52.3499	−1.85433	−0.927164	+	0.374655i	$0.877761\pi$
$798$	0	0
$799$	−2.98088	−0.105456
$800$	−4.66443	−0.164912
$801$	18.8665	0.666616
$802$	−23.1434	−0.817221
$803$	7.47179	0.263674
$804$	23.5794	0.831581
$805$	−1.42052	−0.0500669
$806$	−9.98196	−0.351600
$807$	−12.9449	−0.455682
$808$	−11.4348	−0.402273
$809$	22.3630	0.786242	0.393121	−	0.919487i	$-0.371395\pi$
$809$	22.3630	0.786242	0.393121	+	0.919487i	$0.371395\pi$
$810$	0.469156	0.0164845
$811$	−30.2471	−1.06212	−0.531059	−	0.847335i	$-0.678206\pi$
$811$	−30.2471	−1.06212	−0.531059	+	0.847335i	$0.678206\pi$
$812$	8.39416	0.294577
$813$	−65.7242	−2.30505
$814$	−7.04275	−0.246848
$815$	−12.9952	−0.455203
$816$	3.17284	0.111071
$817$	0	0
$818$	9.23341	0.322839
$819$	23.3158	0.814720
$820$	5.48732	0.191626
$821$	−51.0189	−1.78057	−0.890287	−	0.455401i	$-0.849496\pi$
$821$	−51.0189	−1.78057	−0.890287	+	0.455401i	$0.849496\pi$
$822$	24.9977	0.871894
$823$	−33.2359	−1.15853	−0.579265	−	0.815139i	$-0.696660\pi$
$823$	−33.2359	−1.15853	−0.579265	+	0.815139i	$0.696660\pi$
$824$	11.6147	0.404617
$825$	−12.9694	−0.451536
$826$	−4.11051	−0.143023
$827$	29.2485	1.01707	0.508535	−	0.861041i	$-0.330187\pi$
$827$	29.2485	1.01707	0.508535	+	0.861041i	$0.330187\pi$
$828$	−5.60522	−0.194795
$829$	32.1382	1.11621	0.558103	−	0.829772i	$-0.311529\pi$
$829$	32.1382	1.11621	0.558103	+	0.829772i	$0.311529\pi$
$830$	−0.428681	−0.0148797
$831$	−33.2230	−1.15249
$832$	−2.38101	−0.0825466
$833$	3.09923	0.107382
$834$	−12.3532	−0.427758
$835$	−13.9509	−0.482790
$836$	0	0
$837$	−20.1791	−0.697493
$838$	38.4806	1.32929
$839$	−24.1776	−0.834705	−0.417352	−	0.908745i	$-0.637042\pi$
$839$	−24.1776	−0.834705	−0.417352	+	0.908745i	$0.637042\pi$
$840$	−3.33381	−0.115027
$841$	−12.5524	−0.432840
$842$	10.2075	0.351775
$843$	34.9154	1.20255
$844$	−20.5064	−0.705858
$845$	4.24663	0.146089
$846$	12.3589	0.424909
$847$	−2.06979	−0.0711187
$848$	2.93059	0.100637
$849$	34.6538	1.18931
$850$	5.32261	0.182564
$851$	−8.34395	−0.286027
$852$	−23.8781	−0.818049
$853$	−38.4971	−1.31812	−0.659058	−	0.752092i	$-0.729045\pi$
$853$	−38.4971	−1.31812	−0.659058	+	0.752092i	$0.729045\pi$
$854$	−2.52120	−0.0862737
$855$	0	0
$856$	−18.9854	−0.648906
$857$	−27.5568	−0.941324	−0.470662	−	0.882314i	$-0.655985\pi$
$857$	−27.5568	−0.941324	−0.470662	+	0.882314i	$0.655985\pi$
$858$	−6.62037	−0.226016
$859$	−34.7203	−1.18464	−0.592320	−	0.805702i	$-0.701788\pi$
$859$	−34.7203	−1.18464	−0.592320	+	0.805702i	$0.701788\pi$
$860$	5.57999	0.190276
$861$	−54.5146	−1.85785
$862$	17.3154	0.589766
$863$	37.8223	1.28748	0.643742	−	0.765242i	$-0.277381\pi$
$863$	37.8223	1.28748	0.643742	+	0.765242i	$0.277381\pi$
$864$	−4.81335	−0.163753
$865$	−2.43349	−0.0827411
$866$	2.80814	0.0954245
$867$	43.6478	1.48236
$868$	−8.67722	−0.294524
$869$	−1.78218	−0.0604564
$870$	−6.53231	−0.221466
$871$	20.1917	0.684169
$872$	−7.66962	−0.259726
$873$	−1.18737	−0.0401864
$874$	0	0
$875$	−11.5876	−0.391734
$876$	20.7752	0.701930
$877$	−26.4071	−0.891704	−0.445852	−	0.895107i	$-0.647099\pi$
$877$	−26.4071	−0.891704	−0.445852	+	0.895107i	$0.647099\pi$
$878$	−6.64344	−0.224205
$879$	−80.5330	−2.71631
$880$	0.579287	0.0195278
$881$	45.4155	1.53009	0.765044	−	0.643978i	$-0.222717\pi$
$881$	45.4155	1.53009	0.765044	+	0.643978i	$0.222717\pi$
$882$	−12.8496	−0.432670
$883$	0.0877881	0.00295431	0.00147715	−	0.999999i	$-0.499530\pi$
$883$	0.0877881	0.00295431	0.00147715	+	0.999999i	$0.499530\pi$
$884$	2.71699	0.0913822
$885$	3.19879	0.107526
$886$	−14.4952	−0.486976
$887$	−3.33031	−0.111821	−0.0559105	−	0.998436i	$-0.517806\pi$
$887$	−3.33031	−0.111821	−0.0559105	+	0.998436i	$0.517806\pi$
$888$	−19.5823	−0.657138
$889$	−36.0522	−1.20915
$890$	−2.31005	−0.0774331
$891$	0.809886	0.0271322
$892$	25.8321	0.864922
$893$	0	0
$894$	8.58730	0.287202
$895$	−2.04835	−0.0684687
$896$	−2.06979	−0.0691467
$897$	−7.84353	−0.261888
$898$	13.0713	0.436195
$899$	−17.0023	−0.567057
$900$	−22.0679	−0.735598
$901$	−3.34412	−0.111409
$902$	9.47254	0.315401
$903$	−55.4352	−1.84477
$904$	11.3727	0.378251
$905$	−5.36986	−0.178500
$906$	61.6025	2.04661
$907$	19.3509	0.642536	0.321268	−	0.946988i	$-0.395891\pi$
$907$	19.3509	0.642536	0.321268	+	0.946988i	$0.395891\pi$
$908$	16.1427	0.535716
$909$	−54.0991	−1.79436
$910$	−2.85483	−0.0946367
$911$	−36.2476	−1.20094	−0.600469	−	0.799648i	$-0.705019\pi$
$911$	−36.2476	−1.20094	−0.600469	+	0.799648i	$0.705019\pi$
$912$	0	0
$913$	−0.740015	−0.0244909
$914$	12.3869	0.409721
$915$	1.96199	0.0648614
$916$	−12.8061	−0.423126
$917$	35.4540	1.17079
$918$	5.49255	0.181281
$919$	−21.0035	−0.692842	−0.346421	−	0.938079i	$-0.612603\pi$
$919$	−21.0035	−0.692842	−0.346421	+	0.938079i	$0.612603\pi$
$920$	0.686314	0.0226271
$921$	−12.8657	−0.423938
$922$	−11.4371	−0.376659
$923$	−20.4475	−0.673036
$924$	−5.75502	−0.189326
$925$	−32.8504	−1.08011
$926$	23.0785	0.758406
$927$	54.9505	1.80481
$928$	−4.05557	−0.133131
$929$	0.828481	0.0271816	0.0135908	−	0.999908i	$-0.495674\pi$
$929$	0.828481	0.0271816	0.0135908	+	0.999908i	$0.495674\pi$
$930$	6.75258	0.221426
$931$	0	0
$932$	1.86898	0.0612206
$933$	25.8674	0.846861
$934$	−9.63721	−0.315339
$935$	−0.661029	−0.0216179
$936$	−11.2648	−0.368203
$937$	−33.7525	−1.10265	−0.551324	−	0.834291i	$-0.685877\pi$
$937$	−33.7525	−1.10265	−0.551324	+	0.834291i	$0.685877\pi$
$938$	17.5524	0.573107
$939$	16.8923	0.551260
$940$	−1.51325	−0.0493569
$941$	−18.5753	−0.605539	−0.302769	−	0.953064i	$-0.597911\pi$
$941$	−18.5753	−0.605539	−0.302769	+	0.953064i	$0.597911\pi$
$942$	−56.3061	−1.83455
$943$	11.2227	0.365460
$944$	1.98596	0.0646375
$945$	−5.77121	−0.187737
$946$	9.63251	0.313180
$947$	−54.4140	−1.76822	−0.884109	−	0.467280i	$-0.845234\pi$
$947$	−54.4140	−1.76822	−0.884109	+	0.467280i	$0.845234\pi$
$948$	−4.95534	−0.160942
$949$	17.7904	0.577501
$950$	0	0
$951$	−62.7052	−2.03335
$952$	2.36185	0.0765479
$953$	2.19350	0.0710545	0.0355272	−	0.999369i	$-0.488689\pi$
$953$	2.19350	0.0710545	0.0355272	+	0.999369i	$0.488689\pi$
$954$	13.8650	0.448894
$955$	−4.71140	−0.152457
$956$	−4.07877	−0.131917
$957$	−11.2765	−0.364516
$958$	9.46510	0.305804
$959$	18.6082	0.600889
$960$	1.61070	0.0519851
$961$	−13.4244	−0.433045
$962$	−16.7688	−0.540649
$963$	−89.8219	−2.89447
$964$	6.85830	0.220891
$965$	12.4565	0.400989
$966$	−6.81830	−0.219375
$967$	−27.7151	−0.891258	−0.445629	−	0.895218i	$-0.647020\pi$
$967$	−27.7151	−0.891258	−0.445629	+	0.895218i	$0.647020\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	0.145384	0.00466799
$971$	−57.3361	−1.84000	−0.920001	−	0.391915i	$-0.871813\pi$
$971$	−57.3361	−1.84000	−0.920001	+	0.391915i	$0.871813\pi$
$972$	16.6919	0.535394
$973$	−9.19571	−0.294801
$974$	18.1199	0.580598
$975$	−30.8802	−0.988959
$976$	1.21810	0.0389903
$977$	12.5635	0.401942	0.200971	−	0.979597i	$-0.435590\pi$
$977$	12.5635	0.401942	0.200971	+	0.979597i	$0.435590\pi$
$978$	−62.3751	−1.99453
$979$	−3.98775	−0.127449
$980$	1.57334	0.0502584
$981$	−36.2859	−1.15852
$982$	−5.43289	−0.173370
$983$	24.2101	0.772184	0.386092	−	0.922460i	$-0.373825\pi$
$983$	24.2101	0.772184	0.386092	+	0.922460i	$0.373825\pi$
$984$	26.3383	0.839634
$985$	1.75458	0.0559057
$986$	4.62784	0.147380
$987$	15.0336	0.478526
$988$	0	0
$989$	11.4122	0.362886
$990$	2.74067	0.0871043
$991$	22.0581	0.700697	0.350349	−	0.936619i	$-0.386063\pi$
$991$	22.0581	0.700697	0.350349	+	0.936619i	$0.386063\pi$
$992$	4.19233	0.133106
$993$	−80.3983	−2.55136
$994$	−17.7748	−0.563781
$995$	−1.66537	−0.0527959
$996$	−2.05760	−0.0651976
$997$	−42.9228	−1.35938	−0.679689	−	0.733500i	$-0.737885\pi$
$997$	−42.9228	−1.35938	−0.679689	+	0.733500i	$0.737885\pi$
$998$	37.3967	1.18377
$999$	−33.8992	−1.07252

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7942.2.a.cd.1.1	yes	16
19.18	odd	2		7942.2.a.cc.1.16	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7942.2.a.cc.1.16	✓	16	19.18	odd	2
7942.2.a.cd.1.1	yes	16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7942 = 2 \cdot 11 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7942.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.78049 q^{3} +1.00000 q^{4} -0.579287 q^{5} -2.78049 q^{6} -2.06979 q^{7} +1.00000 q^{8} +4.73112 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.78049 q^{3} +1.00000 q^{4} -0.579287 q^{5} -2.78049 q^{6} -2.06979 q^{7} +1.00000 q^{8} +4.73112 q^{9} -0.579287 q^{10} -1.00000 q^{11} -2.78049 q^{12} -2.38101 q^{13} -2.06979 q^{14} +1.61070 q^{15} +1.00000 q^{16} -1.14111 q^{17} +4.73112 q^{18} -0.579287 q^{20} +5.75502 q^{21} -1.00000 q^{22} -1.18476 q^{23} -2.78049 q^{24} -4.66443 q^{25} -2.38101 q^{26} -4.81335 q^{27} -2.06979 q^{28} -4.05557 q^{29} +1.61070 q^{30} +4.19233 q^{31} +1.00000 q^{32} +2.78049 q^{33} -1.14111 q^{34} +1.19900 q^{35} +4.73112 q^{36} +7.04275 q^{37} +6.62037 q^{39} -0.579287 q^{40} -9.47254 q^{41} +5.75502 q^{42} -9.63251 q^{43} -1.00000 q^{44} -2.74067 q^{45} -1.18476 q^{46} +2.61227 q^{47} -2.78049 q^{48} -2.71599 q^{49} -4.66443 q^{50} +3.17284 q^{51} -2.38101 q^{52} +2.93059 q^{53} -4.81335 q^{54} +0.579287 q^{55} -2.06979 q^{56} -4.05557 q^{58} +1.98596 q^{59} +1.61070 q^{60} +1.21810 q^{61} +4.19233 q^{62} -9.79240 q^{63} +1.00000 q^{64} +1.37929 q^{65} +2.78049 q^{66} -8.48031 q^{67} -1.14111 q^{68} +3.29420 q^{69} +1.19900 q^{70} +8.58773 q^{71} +4.73112 q^{72} -7.47179 q^{73} +7.04275 q^{74} +12.9694 q^{75} +2.06979 q^{77} +6.62037 q^{78} +1.78218 q^{79} -0.579287 q^{80} -0.809886 q^{81} -9.47254 q^{82} +0.740015 q^{83} +5.75502 q^{84} +0.661029 q^{85} -9.63251 q^{86} +11.2765 q^{87} -1.00000 q^{88} +3.98775 q^{89} -2.74067 q^{90} +4.92818 q^{91} -1.18476 q^{92} -11.6567 q^{93} +2.61227 q^{94} -2.78049 q^{96} -0.250970 q^{97} -2.71599 q^{98} -4.73112 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{2} + 10 q^{3} + 16 q^{4} + 10 q^{6} + 6 q^{7} + 16 q^{8} + 20 q^{9}+O(q^{10}) \) 16 * q + 16 * q^2 + 10 * q^3 + 16 * q^4 + 10 * q^6 + 6 * q^7 + 16 * q^8 + 20 * q^9	\( 16 q + 16 q^{2} + 10 q^{3} + 16 q^{4} + 10 q^{6} + 6 q^{7} + 16 q^{8} + 20 q^{9} - 16 q^{11} + 10 q^{12} - 2 q^{13} + 6 q^{14} - 8 q^{15} + 16 q^{16} + 20 q^{18} + 6 q^{21} - 16 q^{22} - 6 q^{23} + 10 q^{24} + 18 q^{25} - 2 q^{26} + 46 q^{27} + 6 q^{28} + 28 q^{29} - 8 q^{30} + 8 q^{31} + 16 q^{32} - 10 q^{33} + 14 q^{35} + 20 q^{36} + 32 q^{37} + 24 q^{39} + 24 q^{41} + 6 q^{42} + 28 q^{43} - 16 q^{44} - 16 q^{45} - 6 q^{46} + 32 q^{47} + 10 q^{48} + 16 q^{49} + 18 q^{50} + 14 q^{51} - 2 q^{52} + 16 q^{53} + 46 q^{54} + 6 q^{56} + 28 q^{58} + 30 q^{59} - 8 q^{60} - 8 q^{61} + 8 q^{62} - 8 q^{63} + 16 q^{64} + 26 q^{65} - 10 q^{66} + 44 q^{67} + 16 q^{69} + 14 q^{70} + 36 q^{71} + 20 q^{72} + 10 q^{73} + 32 q^{74} + 76 q^{75} - 6 q^{77} + 24 q^{78} - 16 q^{79} + 48 q^{81} + 24 q^{82} + 12 q^{83} + 6 q^{84} - 4 q^{85} + 28 q^{86} + 24 q^{87} - 16 q^{88} + 32 q^{89} - 16 q^{90} - 24 q^{91} - 6 q^{92} - 14 q^{93} + 32 q^{94} + 10 q^{96} + 52 q^{97} + 16 q^{98} - 20 q^{99}+O(q^{100}) \) 16 * q + 16 * q^2 + 10 * q^3 + 16 * q^4 + 10 * q^6 + 6 * q^7 + 16 * q^8 + 20 * q^9 - 16 * q^11 + 10 * q^12 - 2 * q^13 + 6 * q^14 - 8 * q^15 + 16 * q^16 + 20 * q^18 + 6 * q^21 - 16 * q^22 - 6 * q^23 + 10 * q^24 + 18 * q^25 - 2 * q^26 + 46 * q^27 + 6 * q^28 + 28 * q^29 - 8 * q^30 + 8 * q^31 + 16 * q^32 - 10 * q^33 + 14 * q^35 + 20 * q^36 + 32 * q^37 + 24 * q^39 + 24 * q^41 + 6 * q^42 + 28 * q^43 - 16 * q^44 - 16 * q^45 - 6 * q^46 + 32 * q^47 + 10 * q^48 + 16 * q^49 + 18 * q^50 + 14 * q^51 - 2 * q^52 + 16 * q^53 + 46 * q^54 + 6 * q^56 + 28 * q^58 + 30 * q^59 - 8 * q^60 - 8 * q^61 + 8 * q^62 - 8 * q^63 + 16 * q^64 + 26 * q^65 - 10 * q^66 + 44 * q^67 + 16 * q^69 + 14 * q^70 + 36 * q^71 + 20 * q^72 + 10 * q^73 + 32 * q^74 + 76 * q^75 - 6 * q^77 + 24 * q^78 - 16 * q^79 + 48 * q^81 + 24 * q^82 + 12 * q^83 + 6 * q^84 - 4 * q^85 + 28 * q^86 + 24 * q^87 - 16 * q^88 + 32 * q^89 - 16 * q^90 - 24 * q^91 - 6 * q^92 - 14 * q^93 + 32 * q^94 + 10 * q^96 + 52 * q^97 + 16 * q^98 - 20 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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