LMFDB - Embedded newform 2166.2.a.o.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2166,2,Mod(1,2166)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2166, 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("2166.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2166 = 2 \cdot 3 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2166.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:	$17.2955970778$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 114)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.347296$ of defining polynomial
Character	$\chi$	$=$	2166.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.22668 q^{5} +1.00000 q^{6} -0.652704 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.22668 q^{5} +1.00000 q^{6} -0.652704 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.22668 q^{10} -1.53209 q^{11} -1.00000 q^{12} -0.467911 q^{13} +0.652704 q^{14} -2.22668 q^{15} +1.00000 q^{16} -2.10607 q^{17} -1.00000 q^{18} +2.22668 q^{20} +0.652704 q^{21} +1.53209 q^{22} +5.90167 q^{23} +1.00000 q^{24} -0.0418891 q^{25} +0.467911 q^{26} -1.00000 q^{27} -0.652704 q^{28} -8.33275 q^{29} +2.22668 q^{30} -8.63816 q^{31} -1.00000 q^{32} +1.53209 q^{33} +2.10607 q^{34} -1.45336 q^{35} +1.00000 q^{36} -4.67499 q^{37} +0.467911 q^{39} -2.22668 q^{40} +3.47565 q^{41} -0.652704 q^{42} +10.2909 q^{43} -1.53209 q^{44} +2.22668 q^{45} -5.90167 q^{46} -4.63816 q^{47} -1.00000 q^{48} -6.57398 q^{49} +0.0418891 q^{50} +2.10607 q^{51} -0.467911 q^{52} -11.9513 q^{53} +1.00000 q^{54} -3.41147 q^{55} +0.652704 q^{56} +8.33275 q^{58} +14.2344 q^{59} -2.22668 q^{60} -9.12061 q^{61} +8.63816 q^{62} -0.652704 q^{63} +1.00000 q^{64} -1.04189 q^{65} -1.53209 q^{66} +0.248970 q^{67} -2.10607 q^{68} -5.90167 q^{69} +1.45336 q^{70} -4.44831 q^{71} -1.00000 q^{72} -9.09152 q^{73} +4.67499 q^{74} +0.0418891 q^{75} +1.00000 q^{77} -0.467911 q^{78} +12.0574 q^{79} +2.22668 q^{80} +1.00000 q^{81} -3.47565 q^{82} +3.70914 q^{83} +0.652704 q^{84} -4.68954 q^{85} -10.2909 q^{86} +8.33275 q^{87} +1.53209 q^{88} +16.7665 q^{89} -2.22668 q^{90} +0.305407 q^{91} +5.90167 q^{92} +8.63816 q^{93} +4.63816 q^{94} +1.00000 q^{96} +2.46791 q^{97} +6.57398 q^{98} -1.53209 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{6} - 3 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^6 - 3 * q^7 - 3 * q^8 + 3 * q^9	$ 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{6} - 3 q^{7} - 3 q^{8} + 3 q^{9} - 3 q^{12} - 6 q^{13} + 3 q^{14} + 3 q^{16} + 6 q^{17} - 3 q^{18} + 3 q^{21} + 6 q^{23} + 3 q^{24} + 3 q^{25} + 6 q^{26} - 3 q^{27} - 3 q^{28} - 6 q^{29} - 9 q^{31} - 3 q^{32} - 6 q^{34} + 9 q^{35} + 3 q^{36} - 9 q^{37} + 6 q^{39} - 9 q^{41} - 3 q^{42} + 15 q^{43} - 6 q^{46} + 3 q^{47} - 3 q^{48} - 12 q^{49} - 3 q^{50} - 6 q^{51} - 6 q^{52} + 3 q^{53} + 3 q^{54} + 3 q^{56} + 6 q^{58} + 12 q^{59} - 33 q^{61} + 9 q^{62} - 3 q^{63} + 3 q^{64} - 12 q^{67} + 6 q^{68} - 6 q^{69} - 9 q^{70} - 15 q^{71} - 3 q^{72} + 3 q^{73} + 9 q^{74} - 3 q^{75} + 3 q^{77} - 6 q^{78} - 15 q^{79} + 3 q^{81} + 9 q^{82} + 27 q^{83} + 3 q^{84} - 27 q^{85} - 15 q^{86} + 6 q^{87} + 15 q^{89} + 3 q^{91} + 6 q^{92} + 9 q^{93} - 3 q^{94} + 3 q^{96} + 12 q^{97} + 12 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^6 - 3 * q^7 - 3 * q^8 + 3 * q^9 - 3 * q^12 - 6 * q^13 + 3 * q^14 + 3 * q^16 + 6 * q^17 - 3 * q^18 + 3 * q^21 + 6 * q^23 + 3 * q^24 + 3 * q^25 + 6 * q^26 - 3 * q^27 - 3 * q^28 - 6 * q^29 - 9 * q^31 - 3 * q^32 - 6 * q^34 + 9 * q^35 + 3 * q^36 - 9 * q^37 + 6 * q^39 - 9 * q^41 - 3 * q^42 + 15 * q^43 - 6 * q^46 + 3 * q^47 - 3 * q^48 - 12 * q^49 - 3 * q^50 - 6 * q^51 - 6 * q^52 + 3 * q^53 + 3 * q^54 + 3 * q^56 + 6 * q^58 + 12 * q^59 - 33 * q^61 + 9 * q^62 - 3 * q^63 + 3 * q^64 - 12 * q^67 + 6 * q^68 - 6 * q^69 - 9 * q^70 - 15 * q^71 - 3 * q^72 + 3 * q^73 + 9 * q^74 - 3 * q^75 + 3 * q^77 - 6 * q^78 - 15 * q^79 + 3 * q^81 + 9 * q^82 + 27 * q^83 + 3 * q^84 - 27 * q^85 - 15 * q^86 + 6 * q^87 + 15 * q^89 + 3 * q^91 + 6 * q^92 + 9 * q^93 - 3 * q^94 + 3 * q^96 + 12 * q^97 + 12 * q^98

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	2.22668	0.995802	0.497901	−	0.867234i	$-0.334104\pi$
$5$	2.22668	0.995802	0.497901	+	0.867234i	$0.334104\pi$
$6$	1.00000	0.408248
$7$	−0.652704	−0.246699	−0.123349	−	0.992363i	$-0.539364\pi$
$7$	−0.652704	−0.246699	−0.123349	+	0.992363i	$0.539364\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−2.22668	−0.704139
$11$	−1.53209	−0.461942	−0.230971	−	0.972961i	$-0.574190\pi$
$11$	−1.53209	−0.461942	−0.230971	+	0.972961i	$0.574190\pi$
$12$	−1.00000	−0.288675
$13$	−0.467911	−0.129775	−0.0648876	−	0.997893i	$-0.520669\pi$
$13$	−0.467911	−0.129775	−0.0648876	+	0.997893i	$0.520669\pi$
$14$	0.652704	0.174442
$15$	−2.22668	−0.574927
$16$	1.00000	0.250000
$17$	−2.10607	−0.510796	−0.255398	−	0.966836i	$-0.582207\pi$
$17$	−2.10607	−0.510796	−0.255398	+	0.966836i	$0.582207\pi$
$18$	−1.00000	−0.235702
$19$	0	0
$19$	0	0
$20$	2.22668	0.497901
$21$	0.652704	0.142432
$22$	1.53209	0.326642
$23$	5.90167	1.23058	0.615292	−	0.788299i	$-0.289038\pi$
$23$	5.90167	1.23058	0.615292	+	0.788299i	$0.289038\pi$
$24$	1.00000	0.204124
$25$	−0.0418891	−0.00837781
$26$	0.467911	0.0917649
$27$	−1.00000	−0.192450
$28$	−0.652704	−0.123349
$29$	−8.33275	−1.54735	−0.773676	−	0.633581i	$-0.781584\pi$
$29$	−8.33275	−1.54735	−0.773676	+	0.633581i	$0.781584\pi$
$30$	2.22668	0.406535
$31$	−8.63816	−1.55146	−0.775729	−	0.631066i	$-0.782618\pi$
$31$	−8.63816	−1.55146	−0.775729	+	0.631066i	$0.782618\pi$
$32$	−1.00000	−0.176777
$33$	1.53209	0.266702
$34$	2.10607	0.361187
$35$	−1.45336	−0.245663
$36$	1.00000	0.166667
$37$	−4.67499	−0.768564	−0.384282	−	0.923216i	$-0.625551\pi$
$37$	−4.67499	−0.768564	−0.384282	+	0.923216i	$0.625551\pi$
$38$	0	0
$39$	0.467911	0.0749257
$40$	−2.22668	−0.352069
$41$	3.47565	0.542806	0.271403	−	0.962466i	$-0.412512\pi$
$41$	3.47565	0.542806	0.271403	+	0.962466i	$0.412512\pi$
$42$	−0.652704	−0.100714
$43$	10.2909	1.56934	0.784671	−	0.619913i	$-0.212832\pi$
$43$	10.2909	1.56934	0.784671	+	0.619913i	$0.212832\pi$
$44$	−1.53209	−0.230971
$45$	2.22668	0.331934
$46$	−5.90167	−0.870154
$47$	−4.63816	−0.676545	−0.338272	−	0.941048i	$-0.609842\pi$
$47$	−4.63816	−0.676545	−0.338272	+	0.941048i	$0.609842\pi$
$48$	−1.00000	−0.144338
$49$	−6.57398	−0.939140
$50$	0.0418891	0.00592401
$51$	2.10607	0.294908
$52$	−0.467911	−0.0648876
$53$	−11.9513	−1.64164	−0.820819	−	0.571189i	$-0.806483\pi$
$53$	−11.9513	−1.64164	−0.820819	+	0.571189i	$0.806483\pi$
$54$	1.00000	0.136083
$55$	−3.41147	−0.460003
$56$	0.652704	0.0872212
$57$	0	0
$58$	8.33275	1.09414
$59$	14.2344	1.85316	0.926582	−	0.376093i	$-0.122733\pi$
$59$	14.2344	1.85316	0.926582	+	0.376093i	$0.122733\pi$
$60$	−2.22668	−0.287463
$61$	−9.12061	−1.16778	−0.583888	−	0.811835i	$-0.698469\pi$
$61$	−9.12061	−1.16778	−0.583888	+	0.811835i	$0.698469\pi$
$62$	8.63816	1.09705
$63$	−0.652704	−0.0822329
$64$	1.00000	0.125000
$65$	−1.04189	−0.129230
$66$	−1.53209	−0.188587
$67$	0.248970	0.0304166	0.0152083	−	0.999884i	$-0.495159\pi$
$67$	0.248970	0.0304166	0.0152083	+	0.999884i	$0.495159\pi$
$68$	−2.10607	−0.255398
$69$	−5.90167	−0.710478
$70$	1.45336	0.173710
$71$	−4.44831	−0.527917	−0.263959	−	0.964534i	$-0.585028\pi$
$71$	−4.44831	−0.527917	−0.263959	+	0.964534i	$0.585028\pi$
$72$	−1.00000	−0.117851
$73$	−9.09152	−1.06408	−0.532041	−	0.846719i	$-0.678575\pi$
$73$	−9.09152	−1.06408	−0.532041	+	0.846719i	$0.678575\pi$
$74$	4.67499	0.543457
$75$	0.0418891	0.00483693
$76$	0	0
$77$	1.00000	0.113961
$78$	−0.467911	−0.0529805
$79$	12.0574	1.35656	0.678280	−	0.734803i	$-0.262726\pi$
$79$	12.0574	1.35656	0.678280	+	0.734803i	$0.262726\pi$
$80$	2.22668	0.248951
$81$	1.00000	0.111111
$82$	−3.47565	−0.383822
$83$	3.70914	0.407131	0.203566	−	0.979061i	$-0.434747\pi$
$83$	3.70914	0.407131	0.203566	+	0.979061i	$0.434747\pi$
$84$	0.652704	0.0712158
$85$	−4.68954	−0.508652
$86$	−10.2909	−1.10969
$87$	8.33275	0.893364
$88$	1.53209	0.163321
$89$	16.7665	1.77725	0.888623	−	0.458638i	$-0.151662\pi$
$89$	16.7665	1.77725	0.888623	+	0.458638i	$0.151662\pi$
$90$	−2.22668	−0.234713
$91$	0.305407	0.0320154
$92$	5.90167	0.615292
$93$	8.63816	0.895735
$94$	4.63816	0.478389
$95$	0	0
$96$	1.00000	0.102062
$97$	2.46791	0.250578	0.125289	−	0.992120i	$-0.460014\pi$
$97$	2.46791	0.250578	0.125289	+	0.992120i	$0.460014\pi$
$98$	6.57398	0.664072
$99$	−1.53209	−0.153981
$100$	−0.0418891	−0.00418891
$101$	−5.73917	−0.571069	−0.285534	−	0.958368i	$-0.592171\pi$
$101$	−5.73917	−0.571069	−0.285534	+	0.958368i	$0.592171\pi$
$102$	−2.10607	−0.208532
$103$	−13.8084	−1.36058	−0.680291	−	0.732942i	$-0.738147\pi$
$103$	−13.8084	−1.36058	−0.680291	+	0.732942i	$0.738147\pi$
$104$	0.467911	0.0458825
$105$	1.45336	0.141834
$106$	11.9513	1.16081
$107$	1.91353	0.184988	0.0924941	−	0.995713i	$-0.470516\pi$
$107$	1.91353	0.184988	0.0924941	+	0.995713i	$0.470516\pi$
$108$	−1.00000	−0.0962250
$109$	−11.8152	−1.13169	−0.565846	−	0.824511i	$-0.691450\pi$
$109$	−11.8152	−1.13169	−0.565846	+	0.824511i	$0.691450\pi$
$110$	3.41147	0.325271
$111$	4.67499	0.443731
$112$	−0.652704	−0.0616747
$113$	5.39693	0.507700	0.253850	−	0.967244i	$-0.418303\pi$
$113$	5.39693	0.507700	0.253850	+	0.967244i	$0.418303\pi$
$114$	0	0
$115$	13.1411	1.22542
$116$	−8.33275	−0.773676
$117$	−0.467911	−0.0432584
$118$	−14.2344	−1.31038
$119$	1.37464	0.126013
$120$	2.22668	0.203267
$121$	−8.65270	−0.786609
$122$	9.12061	0.825742
$123$	−3.47565	−0.313389
$124$	−8.63816	−0.775729
$125$	−11.2267	−1.00414
$126$	0.652704	0.0581475
$127$	9.67499	0.858517	0.429258	−	0.903182i	$-0.358775\pi$
$127$	9.67499	0.858517	0.429258	+	0.903182i	$0.358775\pi$
$128$	−1.00000	−0.0883883
$129$	−10.2909	−0.906060
$130$	1.04189	0.0913797
$131$	−1.90167	−0.166150	−0.0830750	−	0.996543i	$-0.526474\pi$
$131$	−1.90167	−0.166150	−0.0830750	+	0.996543i	$0.526474\pi$
$132$	1.53209	0.133351
$133$	0	0
$134$	−0.248970	−0.0215078
$135$	−2.22668	−0.191642
$136$	2.10607	0.180594
$137$	−1.77837	−0.151936	−0.0759682	−	0.997110i	$-0.524205\pi$
$137$	−1.77837	−0.151936	−0.0759682	+	0.997110i	$0.524205\pi$
$138$	5.90167	0.502384
$139$	−0.751030	−0.0637015	−0.0318508	−	0.999493i	$-0.510140\pi$
$139$	−0.751030	−0.0637015	−0.0318508	+	0.999493i	$0.510140\pi$
$140$	−1.45336	−0.122832
$141$	4.63816	0.390603
$142$	4.44831	0.373294
$143$	0.716881	0.0599486
$144$	1.00000	0.0833333
$145$	−18.5544	−1.54086
$146$	9.09152	0.752419
$147$	6.57398	0.542213
$148$	−4.67499	−0.384282
$149$	−5.97771	−0.489713	−0.244857	−	0.969559i	$-0.578741\pi$
$149$	−5.97771	−0.489713	−0.244857	+	0.969559i	$0.578741\pi$
$150$	−0.0418891	−0.00342023
$151$	−16.5749	−1.34885	−0.674424	−	0.738345i	$-0.735608\pi$
$151$	−16.5749	−1.34885	−0.674424	+	0.738345i	$0.735608\pi$
$152$	0	0
$153$	−2.10607	−0.170265
$154$	−1.00000	−0.0805823
$155$	−19.2344	−1.54495
$156$	0.467911	0.0374629
$157$	−16.7023	−1.33299	−0.666496	−	0.745509i	$-0.732207\pi$
$157$	−16.7023	−1.33299	−0.666496	+	0.745509i	$0.732207\pi$
$158$	−12.0574	−0.959233
$159$	11.9513	0.947800
$160$	−2.22668	−0.176035
$161$	−3.85204	−0.303584
$162$	−1.00000	−0.0785674
$163$	−15.8571	−1.24202	−0.621012	−	0.783801i	$-0.713278\pi$
$163$	−15.8571	−1.24202	−0.621012	+	0.783801i	$0.713278\pi$
$164$	3.47565	0.271403
$165$	3.41147	0.265583
$166$	−3.70914	−0.287885
$167$	−2.26352	−0.175156	−0.0875781	−	0.996158i	$-0.527913\pi$
$167$	−2.26352	−0.175156	−0.0875781	+	0.996158i	$0.527913\pi$
$168$	−0.652704	−0.0503572
$169$	−12.7811	−0.983158
$170$	4.68954	0.359671
$171$	0	0
$172$	10.2909	0.784671
$173$	9.35504	0.711250	0.355625	−	0.934629i	$-0.384268\pi$
$173$	9.35504	0.711250	0.355625	+	0.934629i	$0.384268\pi$
$174$	−8.33275	−0.631704
$175$	0.0273411	0.00206680
$176$	−1.53209	−0.115486
$177$	−14.2344	−1.06992
$178$	−16.7665	−1.25670
$179$	−15.7784	−1.17933	−0.589665	−	0.807648i	$-0.700740\pi$
$179$	−15.7784	−1.17933	−0.589665	+	0.807648i	$0.700740\pi$
$180$	2.22668	0.165967
$181$	7.00269	0.520506	0.260253	−	0.965540i	$-0.416194\pi$
$181$	7.00269	0.520506	0.260253	+	0.965540i	$0.416194\pi$
$182$	−0.305407	−0.0226383
$183$	9.12061	0.674215
$184$	−5.90167	−0.435077
$185$	−10.4097	−0.765338
$186$	−8.63816	−0.633380
$187$	3.22668	0.235958
$188$	−4.63816	−0.338272
$189$	0.652704	0.0474772
$190$	0	0
$191$	4.39187	0.317785	0.158892	−	0.987296i	$-0.449208\pi$
$191$	4.39187	0.317785	0.158892	+	0.987296i	$0.449208\pi$
$192$	−1.00000	−0.0721688
$193$	4.85710	0.349621	0.174811	−	0.984602i	$-0.444069\pi$
$193$	4.85710	0.349621	0.174811	+	0.984602i	$0.444069\pi$
$194$	−2.46791	−0.177186
$195$	1.04189	0.0746112
$196$	−6.57398	−0.469570
$197$	20.2841	1.44518	0.722589	−	0.691277i	$-0.242952\pi$
$197$	20.2841	1.44518	0.722589	+	0.691277i	$0.242952\pi$
$198$	1.53209	0.108881
$199$	−22.4611	−1.59223	−0.796113	−	0.605148i	$-0.793114\pi$
$199$	−22.4611	−1.59223	−0.796113	+	0.605148i	$0.793114\pi$
$200$	0.0418891	0.00296200
$201$	−0.248970	−0.0175610
$202$	5.73917	0.403807
$203$	5.43882	0.381730
$204$	2.10607	0.147454
$205$	7.73917	0.540527
$206$	13.8084	0.962077
$207$	5.90167	0.410195
$208$	−0.467911	−0.0324438
$209$	0	0
$210$	−1.45336	−0.100292
$211$	−9.95811	−0.685545	−0.342772	−	0.939419i	$-0.611366\pi$
$211$	−9.95811	−0.685545	−0.342772	+	0.939419i	$0.611366\pi$
$212$	−11.9513	−0.820819
$213$	4.44831	0.304793
$214$	−1.91353	−0.130806
$215$	22.9145	1.56275
$216$	1.00000	0.0680414
$217$	5.63816	0.382743
$218$	11.8152	0.800227
$219$	9.09152	0.614348
$220$	−3.41147	−0.230002
$221$	0.985452	0.0662887
$222$	−4.67499	−0.313765
$223$	1.92127	0.128658	0.0643290	−	0.997929i	$-0.479509\pi$
$223$	1.92127	0.128658	0.0643290	+	0.997929i	$0.479509\pi$
$224$	0.652704	0.0436106
$225$	−0.0418891	−0.00279260
$226$	−5.39693	−0.358998
$227$	0.901674	0.0598462	0.0299231	−	0.999552i	$-0.490474\pi$
$227$	0.901674	0.0598462	0.0299231	+	0.999552i	$0.490474\pi$
$228$	0	0
$229$	−0.354103	−0.0233998	−0.0116999	−	0.999932i	$-0.503724\pi$
$229$	−0.354103	−0.0233998	−0.0116999	+	0.999932i	$0.503724\pi$
$230$	−13.1411	−0.866502
$231$	−1.00000	−0.0657952
$232$	8.33275	0.547072
$233$	−16.7023	−1.09421	−0.547103	−	0.837065i	$-0.684269\pi$
$233$	−16.7023	−1.09421	−0.547103	+	0.837065i	$0.684269\pi$
$234$	0.467911	0.0305883
$235$	−10.3277	−0.673705
$236$	14.2344	0.926582
$237$	−12.0574	−0.783210
$238$	−1.37464	−0.0891045
$239$	10.0547	0.650383	0.325192	−	0.945648i	$-0.394571\pi$
$239$	10.0547	0.650383	0.325192	+	0.945648i	$0.394571\pi$
$240$	−2.22668	−0.143732
$241$	−23.0993	−1.48795	−0.743977	−	0.668205i	$-0.767063\pi$
$241$	−23.0993	−1.48795	−0.743977	+	0.668205i	$0.767063\pi$
$242$	8.65270	0.556217
$243$	−1.00000	−0.0641500
$244$	−9.12061	−0.583888
$245$	−14.6382	−0.935197
$246$	3.47565	0.221599
$247$	0	0
$248$	8.63816	0.548523
$249$	−3.70914	−0.235057
$250$	11.2267	0.710038
$251$	12.1625	0.767690	0.383845	−	0.923397i	$-0.374600\pi$
$251$	12.1625	0.767690	0.383845	+	0.923397i	$0.374600\pi$
$252$	−0.652704	−0.0411165
$253$	−9.04189	−0.568459
$254$	−9.67499	−0.607063
$255$	4.68954	0.293670
$256$	1.00000	0.0625000
$257$	−3.94625	−0.246160	−0.123080	−	0.992397i	$-0.539277\pi$
$257$	−3.94625	−0.246160	−0.123080	+	0.992397i	$0.539277\pi$
$258$	10.2909	0.640681
$259$	3.05138	0.189604
$260$	−1.04189	−0.0646152
$261$	−8.33275	−0.515784
$262$	1.90167	0.117486
$263$	9.06923	0.559233	0.279616	−	0.960112i	$-0.409793\pi$
$263$	9.06923	0.559233	0.279616	+	0.960112i	$0.409793\pi$
$264$	−1.53209	−0.0942936
$265$	−26.6117	−1.63475
$266$	0	0
$267$	−16.7665	−1.02609
$268$	0.248970	0.0152083
$269$	−28.0128	−1.70797	−0.853985	−	0.520298i	$-0.825821\pi$
$269$	−28.0128	−1.70797	−0.853985	+	0.520298i	$0.825821\pi$
$270$	2.22668	0.135512
$271$	−16.8357	−1.02270	−0.511349	−	0.859373i	$-0.670854\pi$
$271$	−16.8357	−1.02270	−0.511349	+	0.859373i	$0.670854\pi$
$272$	−2.10607	−0.127699
$273$	−0.305407	−0.0184841
$274$	1.77837	0.107435
$275$	0.0641778	0.00387007
$276$	−5.90167	−0.355239
$277$	−19.1506	−1.15065	−0.575325	−	0.817925i	$-0.695125\pi$
$277$	−19.1506	−1.15065	−0.575325	+	0.817925i	$0.695125\pi$
$278$	0.751030	0.0450438
$279$	−8.63816	−0.517153
$280$	1.45336	0.0868551
$281$	−3.95811	−0.236121	−0.118061	−	0.993006i	$-0.537668\pi$
$281$	−3.95811	−0.236121	−0.118061	+	0.993006i	$0.537668\pi$
$282$	−4.63816	−0.276198
$283$	−6.57667	−0.390942	−0.195471	−	0.980709i	$-0.562624\pi$
$283$	−6.57667	−0.390942	−0.195471	+	0.980709i	$0.562624\pi$
$284$	−4.44831	−0.263959
$285$	0	0
$286$	−0.716881	−0.0423901
$287$	−2.26857	−0.133909
$288$	−1.00000	−0.0589256
$289$	−12.5645	−0.739087
$290$	18.5544	1.08955
$291$	−2.46791	−0.144672
$292$	−9.09152	−0.532041
$293$	21.6955	1.26747	0.633733	−	0.773552i	$-0.281522\pi$
$293$	21.6955	1.26747	0.633733	+	0.773552i	$0.281522\pi$
$294$	−6.57398	−0.383402
$295$	31.6955	1.84538
$296$	4.67499	0.271728
$297$	1.53209	0.0889008
$298$	5.97771	0.346280
$299$	−2.76146	−0.159699
$300$	0.0418891	0.00241847
$301$	−6.71688	−0.387155
$302$	16.5749	0.953779
$303$	5.73917	0.329707
$304$	0	0
$305$	−20.3087	−1.16287
$306$	2.10607	0.120396
$307$	17.8007	1.01594	0.507969	−	0.861376i	$-0.330397\pi$
$307$	17.8007	1.01594	0.507969	+	0.861376i	$0.330397\pi$
$308$	1.00000	0.0569803
$309$	13.8084	0.785532
$310$	19.2344	1.09244
$311$	−1.62536	−0.0921659	−0.0460829	−	0.998938i	$-0.514674\pi$
$311$	−1.62536	−0.0921659	−0.0460829	+	0.998938i	$0.514674\pi$
$312$	−0.467911	−0.0264903
$313$	30.0729	1.69982	0.849909	−	0.526929i	$-0.176657\pi$
$313$	30.0729	1.69982	0.849909	+	0.526929i	$0.176657\pi$
$314$	16.7023	0.942567
$315$	−1.45336	−0.0818877
$316$	12.0574	0.678280
$317$	−11.3327	−0.636511	−0.318255	−	0.948005i	$-0.603097\pi$
$317$	−11.3327	−0.636511	−0.318255	+	0.948005i	$0.603097\pi$
$318$	−11.9513	−0.670196
$319$	12.7665	0.714787
$320$	2.22668	0.124475
$321$	−1.91353	−0.106803
$322$	3.85204	0.214666
$323$	0	0
$324$	1.00000	0.0555556
$325$	0.0196004	0.00108723
$326$	15.8571	0.878243
$327$	11.8152	0.653382
$328$	−3.47565	−0.191911
$329$	3.02734	0.166903
$330$	−3.41147	−0.187795
$331$	−32.1242	−1.76571	−0.882854	−	0.469648i	$-0.844381\pi$
$331$	−32.1242	−1.76571	−0.882854	+	0.469648i	$0.844381\pi$
$332$	3.70914	0.203566
$333$	−4.67499	−0.256188
$334$	2.26352	0.123854
$335$	0.554378	0.0302889
$336$	0.652704	0.0356079
$337$	30.3337	1.65238	0.826190	−	0.563391i	$-0.190503\pi$
$337$	30.3337	1.65238	0.826190	+	0.563391i	$0.190503\pi$
$338$	12.7811	0.695198
$339$	−5.39693	−0.293121
$340$	−4.68954	−0.254326
$341$	13.2344	0.716684
$342$	0	0
$343$	8.85978	0.478383
$344$	−10.2909	−0.554846
$345$	−13.1411	−0.707496
$346$	−9.35504	−0.502930
$347$	4.30541	0.231126	0.115563	−	0.993300i	$-0.463133\pi$
$347$	4.30541	0.231126	0.115563	+	0.993300i	$0.463133\pi$
$348$	8.33275	0.446682
$349$	−2.66044	−0.142410	−0.0712052	−	0.997462i	$-0.522685\pi$
$349$	−2.66044	−0.142410	−0.0712052	+	0.997462i	$0.522685\pi$
$350$	−0.0273411	−0.00146145
$351$	0.467911	0.0249752
$352$	1.53209	0.0816606
$353$	36.1925	1.92633	0.963167	−	0.268904i	$-0.0866613\pi$
$353$	36.1925	1.92633	0.963167	+	0.268904i	$0.0866613\pi$
$354$	14.2344	0.756551
$355$	−9.90497	−0.525701
$356$	16.7665	0.888623
$357$	−1.37464	−0.0727535
$358$	15.7784	0.833913
$359$	1.87164	0.0987816	0.0493908	−	0.998780i	$-0.484272\pi$
$359$	1.87164	0.0987816	0.0493908	+	0.998780i	$0.484272\pi$
$360$	−2.22668	−0.117356
$361$	0	0
$362$	−7.00269	−0.368053
$363$	8.65270	0.454149
$364$	0.305407	0.0160077
$365$	−20.2439	−1.05962
$366$	−9.12061	−0.476742
$367$	3.53983	0.184778	0.0923888	−	0.995723i	$-0.470550\pi$
$367$	3.53983	0.184778	0.0923888	+	0.995723i	$0.470550\pi$
$368$	5.90167	0.307646
$369$	3.47565	0.180935
$370$	10.4097	0.541176
$371$	7.80066	0.404990
$372$	8.63816	0.447868
$373$	16.5185	0.855294	0.427647	−	0.903946i	$-0.359343\pi$
$373$	16.5185	0.855294	0.427647	+	0.903946i	$0.359343\pi$
$374$	−3.22668	−0.166848
$375$	11.2267	0.579743
$376$	4.63816	0.239195
$377$	3.89899	0.200808
$378$	−0.652704	−0.0335715
$379$	−3.56893	−0.183323	−0.0916617	−	0.995790i	$-0.529218\pi$
$379$	−3.56893	−0.183323	−0.0916617	+	0.995790i	$0.529218\pi$
$380$	0	0
$381$	−9.67499	−0.495665
$382$	−4.39187	−0.224708
$383$	−12.0642	−0.616451	−0.308225	−	0.951313i	$-0.599735\pi$
$383$	−12.0642	−0.616451	−0.308225	+	0.951313i	$0.599735\pi$
$384$	1.00000	0.0510310
$385$	2.22668	0.113482
$386$	−4.85710	−0.247220
$387$	10.2909	0.523114
$388$	2.46791	0.125289
$389$	18.8949	0.958008	0.479004	−	0.877813i	$-0.340998\pi$
$389$	18.8949	0.958008	0.479004	+	0.877813i	$0.340998\pi$
$390$	−1.04189	−0.0527581
$391$	−12.4293	−0.628578
$392$	6.57398	0.332036
$393$	1.90167	0.0959268
$394$	−20.2841	−1.02190
$395$	26.8479	1.35087
$396$	−1.53209	−0.0769904
$397$	18.2422	0.915548	0.457774	−	0.889069i	$-0.348647\pi$
$397$	18.2422	0.915548	0.457774	+	0.889069i	$0.348647\pi$
$398$	22.4611	1.12587
$399$	0	0
$400$	−0.0418891	−0.00209445
$401$	33.8854	1.69215	0.846077	−	0.533060i	$-0.178958\pi$
$401$	33.8854	1.69215	0.846077	+	0.533060i	$0.178958\pi$
$402$	0.248970	0.0124175
$403$	4.04189	0.201341
$404$	−5.73917	−0.285534
$405$	2.22668	0.110645
$406$	−5.43882	−0.269924
$407$	7.16250	0.355032
$408$	−2.10607	−0.104266
$409$	−0.172933	−0.00855098	−0.00427549	−	0.999991i	$-0.501361\pi$
$409$	−0.172933	−0.00855098	−0.00427549	+	0.999991i	$0.501361\pi$
$410$	−7.73917	−0.382210
$411$	1.77837	0.0877206
$412$	−13.8084	−0.680291
$413$	−9.29086	−0.457173
$414$	−5.90167	−0.290051
$415$	8.25908	0.405422
$416$	0.467911	0.0229412
$417$	0.751030	0.0367781
$418$	0	0
$419$	−36.5800	−1.78705	−0.893524	−	0.449015i	$-0.851775\pi$
$419$	−36.5800	−1.78705	−0.893524	+	0.449015i	$0.851775\pi$
$420$	1.45336	0.0709169
$421$	36.8631	1.79660	0.898298	−	0.439386i	$-0.144804\pi$
$421$	36.8631	1.79660	0.898298	+	0.439386i	$0.144804\pi$
$422$	9.95811	0.484753
$423$	−4.63816	−0.225515
$424$	11.9513	0.580407
$425$	0.0882212	0.00427936
$426$	−4.44831	−0.215521
$427$	5.95306	0.288089
$428$	1.91353	0.0924941
$429$	−0.716881	−0.0346114
$430$	−22.9145	−1.10503
$431$	−18.9172	−0.911207	−0.455604	−	0.890183i	$-0.650577\pi$
$431$	−18.9172	−0.911207	−0.455604	+	0.890183i	$0.650577\pi$
$432$	−1.00000	−0.0481125
$433$	22.2668	1.07007	0.535037	−	0.844828i	$-0.320297\pi$
$433$	22.2668	1.07007	0.535037	+	0.844828i	$0.320297\pi$
$434$	−5.63816	−0.270640
$435$	18.5544	0.889614
$436$	−11.8152	−0.565846
$437$	0	0
$438$	−9.09152	−0.434410
$439$	33.4388	1.59595	0.797974	−	0.602692i	$-0.205905\pi$
$439$	33.4388	1.59595	0.797974	+	0.602692i	$0.205905\pi$
$440$	3.41147	0.162636
$441$	−6.57398	−0.313047
$442$	−0.985452	−0.0468732
$443$	34.8949	1.65791	0.828953	−	0.559319i	$-0.188937\pi$
$443$	34.8949	1.65791	0.828953	+	0.559319i	$0.188937\pi$
$444$	4.67499	0.221865
$445$	37.3337	1.76979
$446$	−1.92127	−0.0909750
$447$	5.97771	0.282736
$448$	−0.652704	−0.0308373
$449$	−8.18304	−0.386181	−0.193091	−	0.981181i	$-0.561851\pi$
$449$	−8.18304	−0.386181	−0.193091	+	0.981181i	$0.561851\pi$
$450$	0.0418891	0.00197467
$451$	−5.32501	−0.250745
$452$	5.39693	0.253850
$453$	16.5749	0.778757
$454$	−0.901674	−0.0423177
$455$	0.680045	0.0318810
$456$	0	0
$457$	−41.0259	−1.91911	−0.959556	−	0.281519i	$-0.909162\pi$
$457$	−41.0259	−1.91911	−0.959556	+	0.281519i	$0.909162\pi$
$458$	0.354103	0.0165462
$459$	2.10607	0.0983028
$460$	13.1411	0.612709
$461$	−4.62267	−0.215299	−0.107650	−	0.994189i	$-0.534332\pi$
$461$	−4.62267	−0.215299	−0.107650	+	0.994189i	$0.534332\pi$
$462$	1.00000	0.0465242
$463$	5.51249	0.256187	0.128094	−	0.991762i	$-0.459114\pi$
$463$	5.51249	0.256187	0.128094	+	0.991762i	$0.459114\pi$
$464$	−8.33275	−0.386838
$465$	19.2344	0.891975
$466$	16.7023	0.773721
$467$	11.1693	0.516854	0.258427	−	0.966031i	$-0.416796\pi$
$467$	11.1693	0.516854	0.258427	+	0.966031i	$0.416796\pi$
$468$	−0.467911	−0.0216292
$469$	−0.162504	−0.00750373
$470$	10.3277	0.476381
$471$	16.7023	0.769603
$472$	−14.2344	−0.655192
$473$	−15.7665	−0.724945
$474$	12.0574	0.553813
$475$	0	0
$476$	1.37464	0.0630064
$477$	−11.9513	−0.547213
$478$	−10.0547	−0.459890
$479$	8.66456	0.395894	0.197947	−	0.980213i	$-0.436573\pi$
$479$	8.66456	0.395894	0.197947	+	0.980213i	$0.436573\pi$
$480$	2.22668	0.101634
$481$	2.18748	0.0997405
$482$	23.0993	1.05214
$483$	3.85204	0.175274
$484$	−8.65270	−0.393305
$485$	5.49525	0.249527
$486$	1.00000	0.0453609
$487$	−0.0942073	−0.00426894	−0.00213447	−	0.999998i	$-0.500679\pi$
$487$	−0.0942073	−0.00426894	−0.00213447	+	0.999998i	$0.500679\pi$
$488$	9.12061	0.412871
$489$	15.8571	0.717083
$490$	14.6382	0.661284
$491$	−27.9959	−1.26344	−0.631718	−	0.775198i	$-0.717650\pi$
$491$	−27.9959	−1.26344	−0.631718	+	0.775198i	$0.717650\pi$
$492$	−3.47565	−0.156694
$493$	17.5493	0.790382
$494$	0	0
$495$	−3.41147	−0.153334
$496$	−8.63816	−0.387865
$497$	2.90343	0.130237
$498$	3.70914	0.166211
$499$	−36.9796	−1.65543	−0.827717	−	0.561146i	$-0.810361\pi$
$499$	−36.9796	−1.65543	−0.827717	+	0.561146i	$0.810361\pi$
$500$	−11.2267	−0.502072
$501$	2.26352	0.101127
$502$	−12.1625	−0.542839
$503$	−22.5945	−1.00744	−0.503720	−	0.863867i	$-0.668036\pi$
$503$	−22.5945	−1.00744	−0.503720	+	0.863867i	$0.668036\pi$
$504$	0.652704	0.0290737
$505$	−12.7793	−0.568672
$506$	9.04189	0.401961
$507$	12.7811	0.567627
$508$	9.67499	0.429258
$509$	−44.4502	−1.97022	−0.985110	−	0.171927i	$-0.945001\pi$
$509$	−44.4502	−1.97022	−0.985110	+	0.171927i	$0.945001\pi$
$510$	−4.68954	−0.207656
$511$	5.93407	0.262508
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	3.94625	0.174062
$515$	−30.7469	−1.35487
$516$	−10.2909	−0.453030
$517$	7.10607	0.312524
$518$	−3.05138	−0.134070
$519$	−9.35504	−0.410641
$520$	1.04189	0.0456899
$521$	−24.2645	−1.06304	−0.531522	−	0.847044i	$-0.678380\pi$
$521$	−24.2645	−1.06304	−0.531522	+	0.847044i	$0.678380\pi$
$522$	8.33275	0.364714
$523$	−24.2104	−1.05865	−0.529323	−	0.848420i	$-0.677554\pi$
$523$	−24.2104	−1.05865	−0.529323	+	0.848420i	$0.677554\pi$
$524$	−1.90167	−0.0830750
$525$	−0.0273411	−0.00119327
$526$	−9.06923	−0.395437
$527$	18.1925	0.792479
$528$	1.53209	0.0666756
$529$	11.8298	0.514337
$530$	26.6117	1.15594
$531$	14.2344	0.617721
$532$	0	0
$533$	−1.62630	−0.0704427
$534$	16.7665	0.725558
$535$	4.26083	0.184212
$536$	−0.248970	−0.0107539
$537$	15.7784	0.680887
$538$	28.0128	1.20772
$539$	10.0719	0.433828
$540$	−2.22668	−0.0958211
$541$	34.2303	1.47168	0.735838	−	0.677158i	$-0.236789\pi$
$541$	34.2303	1.47168	0.735838	+	0.677158i	$0.236789\pi$
$542$	16.8357	0.723157
$543$	−7.00269	−0.300514
$544$	2.10607	0.0902969
$545$	−26.3087	−1.12694
$546$	0.305407	0.0130702
$547$	25.2935	1.08147	0.540737	−	0.841192i	$-0.318145\pi$
$547$	25.2935	1.08147	0.540737	+	0.841192i	$0.318145\pi$
$548$	−1.77837	−0.0759682
$549$	−9.12061	−0.389258
$550$	−0.0641778	−0.00273655
$551$	0	0
$552$	5.90167	0.251192
$553$	−7.86989	−0.334662
$554$	19.1506	0.813633
$555$	10.4097	0.441868
$556$	−0.751030	−0.0318508
$557$	−10.8939	−0.461591	−0.230795	−	0.973002i	$-0.574133\pi$
$557$	−10.8939	−0.461591	−0.230795	+	0.973002i	$0.574133\pi$
$558$	8.63816	0.365682
$559$	−4.81521	−0.203662
$560$	−1.45336	−0.0614158
$561$	−3.22668	−0.136231
$562$	3.95811	0.166963
$563$	17.2867	0.728549	0.364275	−	0.931292i	$-0.381317\pi$
$563$	17.2867	0.728549	0.364275	+	0.931292i	$0.381317\pi$
$564$	4.63816	0.195302
$565$	12.0172	0.505569
$566$	6.57667	0.276438
$567$	−0.652704	−0.0274110
$568$	4.44831	0.186647
$569$	6.47834	0.271586	0.135793	−	0.990737i	$-0.456642\pi$
$569$	6.47834	0.271586	0.135793	+	0.990737i	$0.456642\pi$
$570$	0	0
$571$	38.5476	1.61317	0.806583	−	0.591121i	$-0.201314\pi$
$571$	38.5476	1.61317	0.806583	+	0.591121i	$0.201314\pi$
$572$	0.716881	0.0299743
$573$	−4.39187	−0.183473
$574$	2.26857	0.0946883
$575$	−0.247216	−0.0103096
$576$	1.00000	0.0416667
$577$	−11.4551	−0.476883	−0.238441	−	0.971157i	$-0.576636\pi$
$577$	−11.4551	−0.476883	−0.238441	+	0.971157i	$0.576636\pi$
$578$	12.5645	0.522614
$579$	−4.85710	−0.201854
$580$	−18.5544	−0.770429
$581$	−2.42097	−0.100439
$582$	2.46791	0.102298
$583$	18.3105	0.758342
$584$	9.09152	0.376210
$585$	−1.04189	−0.0430768
$586$	−21.6955	−0.896234
$587$	42.7478	1.76439	0.882196	−	0.470882i	$-0.156064\pi$
$587$	42.7478	1.76439	0.882196	+	0.470882i	$0.156064\pi$
$588$	6.57398	0.271106
$589$	0	0
$590$	−31.6955	−1.30488
$591$	−20.2841	−0.834374
$592$	−4.67499	−0.192141
$593$	32.7151	1.34345	0.671725	−	0.740801i	$-0.265554\pi$
$593$	32.7151	1.34345	0.671725	+	0.740801i	$0.265554\pi$
$594$	−1.53209	−0.0628624
$595$	3.06088	0.125484
$596$	−5.97771	−0.244857
$597$	22.4611	0.919272
$598$	2.76146	0.112924
$599$	9.21482	0.376507	0.188254	−	0.982120i	$-0.439717\pi$
$599$	9.21482	0.376507	0.188254	+	0.982120i	$0.439717\pi$
$600$	−0.0418891	−0.00171011
$601$	9.67768	0.394761	0.197380	−	0.980327i	$-0.436757\pi$
$601$	9.67768	0.394761	0.197380	+	0.980327i	$0.436757\pi$
$602$	6.71688	0.273760
$603$	0.248970	0.0101389
$604$	−16.5749	−0.674424
$605$	−19.2668	−0.783307
$606$	−5.73917	−0.233138
$607$	36.4766	1.48054	0.740269	−	0.672310i	$-0.234698\pi$
$607$	36.4766	1.48054	0.740269	+	0.672310i	$0.234698\pi$
$608$	0	0
$609$	−5.43882	−0.220392
$610$	20.3087	0.822275
$611$	2.17024	0.0877987
$612$	−2.10607	−0.0851327
$613$	−11.6895	−0.472136	−0.236068	−	0.971737i	$-0.575859\pi$
$613$	−11.6895	−0.472136	−0.236068	+	0.971737i	$0.575859\pi$
$614$	−17.8007	−0.718376
$615$	−7.73917	−0.312073
$616$	−1.00000	−0.0402911
$617$	17.8135	0.717143	0.358571	−	0.933502i	$-0.383264\pi$
$617$	17.8135	0.717143	0.358571	+	0.933502i	$0.383264\pi$
$618$	−13.8084	−0.555455
$619$	27.9864	1.12487	0.562434	−	0.826842i	$-0.309865\pi$
$619$	27.9864	1.12487	0.562434	+	0.826842i	$0.309865\pi$
$620$	−19.2344	−0.772473
$621$	−5.90167	−0.236826
$622$	1.62536	0.0651711
$623$	−10.9436	−0.438445
$624$	0.467911	0.0187314
$625$	−24.7888	−0.991552
$626$	−30.0729	−1.20195
$627$	0	0
$628$	−16.7023	−0.666496
$629$	9.84585	0.392580
$630$	1.45336	0.0579034
$631$	−3.76146	−0.149741	−0.0748707	−	0.997193i	$-0.523854\pi$
$631$	−3.76146	−0.149741	−0.0748707	+	0.997193i	$0.523854\pi$
$632$	−12.0574	−0.479616
$633$	9.95811	0.395799
$634$	11.3327	0.450081
$635$	21.5431	0.854913
$636$	11.9513	0.473900
$637$	3.07604	0.121877
$638$	−12.7665	−0.505431
$639$	−4.44831	−0.175972
$640$	−2.22668	−0.0880173
$641$	16.7769	0.662649	0.331325	−	0.943517i	$-0.392504\pi$
$641$	16.7769	0.662649	0.331325	+	0.943517i	$0.392504\pi$
$642$	1.91353	0.0755211
$643$	−22.1771	−0.874578	−0.437289	−	0.899321i	$-0.644061\pi$
$643$	−22.1771	−0.874578	−0.437289	+	0.899321i	$0.644061\pi$
$644$	−3.85204	−0.151792
$645$	−22.9145	−0.902256
$646$	0	0
$647$	16.6759	0.655598	0.327799	−	0.944747i	$-0.393693\pi$
$647$	16.6759	0.655598	0.327799	+	0.944747i	$0.393693\pi$
$648$	−1.00000	−0.0392837
$649$	−21.8084	−0.856055
$650$	−0.0196004	−0.000768789 0
$651$	−5.63816	−0.220977
$652$	−15.8571	−0.621012
$653$	−26.8066	−1.04903	−0.524513	−	0.851403i	$-0.675752\pi$
$653$	−26.8066	−1.04903	−0.524513	+	0.851403i	$0.675752\pi$
$654$	−11.8152	−0.462011
$655$	−4.23442	−0.165453
$656$	3.47565	0.135701
$657$	−9.09152	−0.354694
$658$	−3.02734	−0.118018
$659$	−17.6777	−0.688625	−0.344312	−	0.938855i	$-0.611888\pi$
$659$	−17.6777	−0.688625	−0.344312	+	0.938855i	$0.611888\pi$
$660$	3.41147	0.132791
$661$	8.67230	0.337314	0.168657	−	0.985675i	$-0.446057\pi$
$661$	8.67230	0.337314	0.168657	+	0.985675i	$0.446057\pi$
$662$	32.1242	1.24854
$663$	−0.985452	−0.0382718
$664$	−3.70914	−0.143943
$665$	0	0
$666$	4.67499	0.181152
$667$	−49.1772	−1.90415
$668$	−2.26352	−0.0875781
$669$	−1.92127	−0.0742808
$670$	−0.554378	−0.0214175
$671$	13.9736	0.539445
$672$	−0.652704	−0.0251786
$673$	2.16487	0.0834495	0.0417248	−	0.999129i	$-0.486715\pi$
$673$	2.16487	0.0834495	0.0417248	+	0.999129i	$0.486715\pi$
$674$	−30.3337	−1.16841
$675$	0.0418891	0.00161231
$676$	−12.7811	−0.491579
$677$	12.7537	0.490165	0.245083	−	0.969502i	$-0.421185\pi$
$677$	12.7537	0.490165	0.245083	+	0.969502i	$0.421185\pi$
$678$	5.39693	0.207268
$679$	−1.61081	−0.0618174
$680$	4.68954	0.179836
$681$	−0.901674	−0.0345522
$682$	−13.2344	−0.506772
$683$	−25.2608	−0.966579	−0.483289	−	0.875461i	$-0.660558\pi$
$683$	−25.2608	−0.966579	−0.483289	+	0.875461i	$0.660558\pi$
$684$	0	0
$685$	−3.95987	−0.151299
$686$	−8.85978	−0.338268
$687$	0.354103	0.0135099
$688$	10.2909	0.392335
$689$	5.59215	0.213044
$690$	13.1411	0.500275
$691$	15.2959	0.581884	0.290942	−	0.956741i	$-0.406031\pi$
$691$	15.2959	0.581884	0.290942	+	0.956741i	$0.406031\pi$
$692$	9.35504	0.355625
$693$	1.00000	0.0379869
$694$	−4.30541	−0.163431
$695$	−1.67230	−0.0634341
$696$	−8.33275	−0.315852
$697$	−7.31996	−0.277263
$698$	2.66044	0.100699
$699$	16.7023	0.631740
$700$	0.0273411	0.00103340
$701$	39.8631	1.50561	0.752804	−	0.658245i	$-0.228701\pi$
$701$	39.8631	1.50561	0.752804	+	0.658245i	$0.228701\pi$
$702$	−0.467911	−0.0176602
$703$	0	0
$704$	−1.53209	−0.0577428
$705$	10.3277	0.388964
$706$	−36.1925	−1.36212
$707$	3.74598	0.140882
$708$	−14.2344	−0.534962
$709$	23.8749	0.896642	0.448321	−	0.893873i	$-0.352022\pi$
$709$	23.8749	0.896642	0.448321	+	0.893873i	$0.352022\pi$
$710$	9.90497	0.371727
$711$	12.0574	0.452187
$712$	−16.7665	−0.628352
$713$	−50.9796	−1.90920
$714$	1.37464	0.0514445
$715$	1.59627	0.0596970
$716$	−15.7784	−0.589665
$717$	−10.0547	−0.375499
$718$	−1.87164	−0.0698492
$719$	−36.5066	−1.36147	−0.680734	−	0.732531i	$-0.738339\pi$
$719$	−36.5066	−1.36147	−0.680734	+	0.732531i	$0.738339\pi$
$720$	2.22668	0.0829835
$721$	9.01279	0.335654
$722$	0	0
$723$	23.0993	0.859071
$724$	7.00269	0.260253
$725$	0.349051	0.0129634
$726$	−8.65270	−0.321132
$727$	32.2249	1.19516	0.597578	−	0.801811i	$-0.296130\pi$
$727$	32.2249	1.19516	0.597578	+	0.801811i	$0.296130\pi$
$728$	−0.305407	−0.0113191
$729$	1.00000	0.0370370
$730$	20.2439	0.749261
$731$	−21.6732	−0.801614
$732$	9.12061	0.337108
$733$	−6.97535	−0.257640	−0.128820	−	0.991668i	$-0.541119\pi$
$733$	−6.97535	−0.257640	−0.128820	+	0.991668i	$0.541119\pi$
$734$	−3.53983	−0.130657
$735$	14.6382	0.539937
$736$	−5.90167	−0.217539
$737$	−0.381445	−0.0140507
$738$	−3.47565	−0.127941
$739$	24.3696	0.896450	0.448225	−	0.893921i	$-0.352056\pi$
$739$	24.3696	0.896450	0.448225	+	0.893921i	$0.352056\pi$
$740$	−10.4097	−0.382669
$741$	0	0
$742$	−7.80066	−0.286371
$743$	−38.0574	−1.39619	−0.698095	−	0.716005i	$-0.745969\pi$
$743$	−38.0574	−1.39619	−0.698095	+	0.716005i	$0.745969\pi$
$744$	−8.63816	−0.316690
$745$	−13.3105	−0.487658
$746$	−16.5185	−0.604784
$747$	3.70914	0.135710
$748$	3.22668	0.117979
$749$	−1.24897	−0.0456364
$750$	−11.2267	−0.409940
$751$	−2.93582	−0.107130	−0.0535648	−	0.998564i	$-0.517058\pi$
$751$	−2.93582	−0.107130	−0.0535648	+	0.998564i	$0.517058\pi$
$752$	−4.63816	−0.169136
$753$	−12.1625	−0.443226
$754$	−3.89899	−0.141993
$755$	−36.9071	−1.34319
$756$	0.652704	0.0237386
$757$	11.3655	0.413085	0.206542	−	0.978438i	$-0.433779\pi$
$757$	11.3655	0.413085	0.206542	+	0.978438i	$0.433779\pi$
$758$	3.56893	0.129629
$759$	9.04189	0.328200
$760$	0	0
$761$	10.4976	0.380538	0.190269	−	0.981732i	$-0.439064\pi$
$761$	10.4976	0.380538	0.190269	+	0.981732i	$0.439064\pi$
$762$	9.67499	0.350488
$763$	7.71183	0.279187
$764$	4.39187	0.158892
$765$	−4.68954	−0.169551
$766$	12.0642	0.435896
$767$	−6.66044	−0.240495
$768$	−1.00000	−0.0360844
$769$	7.74153	0.279167	0.139583	−	0.990210i	$-0.455424\pi$
$769$	7.74153	0.279167	0.139583	+	0.990210i	$0.455424\pi$
$770$	−2.22668	−0.0802440
$771$	3.94625	0.142121
$772$	4.85710	0.174811
$773$	−6.15333	−0.221320	−0.110660	−	0.993858i	$-0.535296\pi$
$773$	−6.15333	−0.221320	−0.110660	+	0.993858i	$0.535296\pi$
$774$	−10.2909	−0.369897
$775$	0.361844	0.0129978
$776$	−2.46791	−0.0885928
$777$	−3.05138	−0.109468
$778$	−18.8949	−0.677414
$779$	0	0
$780$	1.04189	0.0373056
$781$	6.81521	0.243867
$782$	12.4293	0.444472
$783$	8.33275	0.297788
$784$	−6.57398	−0.234785
$785$	−37.1908	−1.32740
$786$	−1.90167	−0.0678305
$787$	−8.05199	−0.287023	−0.143511	−	0.989649i	$-0.545839\pi$
$787$	−8.05199	−0.287023	−0.143511	+	0.989649i	$0.545839\pi$
$788$	20.2841	0.722589
$789$	−9.06923	−0.322873
$790$	−26.8479	−0.955206
$791$	−3.52259	−0.125249
$792$	1.53209	0.0544404
$793$	4.26764	0.151548
$794$	−18.2422	−0.647390
$795$	26.6117	0.943821
$796$	−22.4611	−0.796113
$797$	10.2139	0.361794	0.180897	−	0.983502i	$-0.442100\pi$
$797$	10.2139	0.361794	0.180897	+	0.983502i	$0.442100\pi$
$798$	0	0
$799$	9.76827	0.345576
$800$	0.0418891	0.00148100
$801$	16.7665	0.592416
$802$	−33.8854	−1.19653
$803$	13.9290	0.491544
$804$	−0.248970	−0.00878051
$805$	−8.57728	−0.302309
$806$	−4.04189	−0.142369
$807$	28.0128	0.986097
$808$	5.73917	0.201903
$809$	47.2995	1.66296	0.831482	−	0.555552i	$-0.187493\pi$
$809$	47.2995	1.66296	0.831482	+	0.555552i	$0.187493\pi$
$810$	−2.22668	−0.0782376
$811$	52.6792	1.84982	0.924909	−	0.380189i	$-0.124141\pi$
$811$	52.6792	1.84982	0.924909	+	0.380189i	$0.124141\pi$
$812$	5.43882	0.190865
$813$	16.8357	0.590455
$814$	−7.16250	−0.251046
$815$	−35.3087	−1.23681
$816$	2.10607	0.0737271
$817$	0	0
$818$	0.172933	0.00604646
$819$	0.305407	0.0106718
$820$	7.73917	0.270264
$821$	−55.1516	−1.92480	−0.962402	−	0.271630i	$-0.912437\pi$
$821$	−55.1516	−1.92480	−0.962402	+	0.271630i	$0.912437\pi$
$822$	−1.77837	−0.0620278
$823$	−24.6435	−0.859020	−0.429510	−	0.903062i	$-0.641314\pi$
$823$	−24.6435	−0.859020	−0.429510	+	0.903062i	$0.641314\pi$
$824$	13.8084	0.481038
$825$	−0.0641778	−0.00223438
$826$	9.29086	0.323270
$827$	37.5562	1.30596	0.652979	−	0.757376i	$-0.273519\pi$
$827$	37.5562	1.30596	0.652979	+	0.757376i	$0.273519\pi$
$828$	5.90167	0.205097
$829$	−13.0000	−0.451509	−0.225754	−	0.974184i	$-0.572485\pi$
$829$	−13.0000	−0.451509	−0.225754	+	0.974184i	$0.572485\pi$
$830$	−8.25908	−0.286677
$831$	19.1506	0.664328
$832$	−0.467911	−0.0162219
$833$	13.8452	0.479709
$834$	−0.751030	−0.0260060
$835$	−5.04013	−0.174421
$836$	0	0
$837$	8.63816	0.298578
$838$	36.5800	1.26363
$839$	−18.8425	−0.650517	−0.325259	−	0.945625i	$-0.605451\pi$
$839$	−18.8425	−0.650517	−0.325259	+	0.945625i	$0.605451\pi$
$840$	−1.45336	−0.0501458
$841$	40.4347	1.39430
$842$	−36.8631	−1.27039
$843$	3.95811	0.136325
$844$	−9.95811	−0.342772
$845$	−28.4593	−0.979031
$846$	4.63816	0.159463
$847$	5.64765	0.194056
$848$	−11.9513	−0.410409
$849$	6.57667	0.225711
$850$	−0.0882212	−0.00302596
$851$	−27.5903	−0.945783
$852$	4.44831	0.152397
$853$	−11.8271	−0.404951	−0.202476	−	0.979287i	$-0.564899\pi$
$853$	−11.8271	−0.404951	−0.202476	+	0.979287i	$0.564899\pi$
$854$	−5.95306	−0.203709
$855$	0	0
$856$	−1.91353	−0.0654032
$857$	19.7864	0.675892	0.337946	−	0.941166i	$-0.390268\pi$
$857$	19.7864	0.675892	0.337946	+	0.941166i	$0.390268\pi$
$858$	0.716881	0.0244739
$859$	32.1976	1.09857	0.549284	−	0.835636i	$-0.314901\pi$
$859$	32.1976	1.09857	0.549284	+	0.835636i	$0.314901\pi$
$860$	22.9145	0.781377
$861$	2.26857	0.0773127
$862$	18.9172	0.644321
$863$	−38.2300	−1.30136	−0.650682	−	0.759350i	$-0.725517\pi$
$863$	−38.2300	−1.30136	−0.650682	+	0.759350i	$0.725517\pi$
$864$	1.00000	0.0340207
$865$	20.8307	0.708265
$866$	−22.2668	−0.756657
$867$	12.5645	0.426712
$868$	5.63816	0.191371
$869$	−18.4730	−0.626652
$870$	−18.5544	−0.629052
$871$	−0.116496	−0.00394732
$872$	11.8152	0.400113
$873$	2.46791	0.0835261
$874$	0	0
$875$	7.32770	0.247721
$876$	9.09152	0.307174
$877$	−29.9249	−1.01049	−0.505246	−	0.862975i	$-0.668598\pi$
$877$	−29.9249	−1.01049	−0.505246	+	0.862975i	$0.668598\pi$
$878$	−33.4388	−1.12851
$879$	−21.6955	−0.731772
$880$	−3.41147	−0.115001
$881$	−40.8316	−1.37565	−0.687826	−	0.725875i	$-0.741435\pi$
$881$	−40.8316	−1.37565	−0.687826	+	0.725875i	$0.741435\pi$
$882$	6.57398	0.221357
$883$	−24.5776	−0.827102	−0.413551	−	0.910481i	$-0.635712\pi$
$883$	−24.5776	−0.827102	−0.413551	+	0.910481i	$0.635712\pi$
$884$	0.985452	0.0331443
$885$	−31.6955	−1.06543
$886$	−34.8949	−1.17232
$887$	24.0624	0.807937	0.403969	−	0.914773i	$-0.367631\pi$
$887$	24.0624	0.807937	0.403969	+	0.914773i	$0.367631\pi$
$888$	−4.67499	−0.156882
$889$	−6.31490	−0.211795
$890$	−37.3337	−1.25143
$891$	−1.53209	−0.0513269
$892$	1.92127	0.0643290
$893$	0	0
$894$	−5.97771	−0.199925
$895$	−35.1334	−1.17438
$896$	0.652704	0.0218053
$897$	2.76146	0.0922024
$898$	8.18304	0.273072
$899$	71.9796	2.40065
$900$	−0.0418891	−0.00139630
$901$	25.1702	0.838542
$902$	5.32501	0.177303
$903$	6.71688	0.223524
$904$	−5.39693	−0.179499
$905$	15.5928	0.518321
$906$	−16.5749	−0.550665
$907$	0.502059	0.0166706	0.00833530	−	0.999965i	$-0.497347\pi$
$907$	0.502059	0.0166706	0.00833530	+	0.999965i	$0.497347\pi$
$908$	0.901674	0.0299231
$909$	−5.73917	−0.190356
$910$	−0.680045	−0.0225433
$911$	−11.2243	−0.371878	−0.185939	−	0.982561i	$-0.559533\pi$
$911$	−11.2243	−0.371878	−0.185939	+	0.982561i	$0.559533\pi$
$912$	0	0
$913$	−5.68273	−0.188071
$914$	41.0259	1.35702
$915$	20.3087	0.671385
$916$	−0.354103	−0.0116999
$917$	1.24123	0.0409890
$918$	−2.10607	−0.0695106
$919$	23.5398	0.776507	0.388254	−	0.921553i	$-0.373078\pi$
$919$	23.5398	0.776507	0.388254	+	0.921553i	$0.373078\pi$
$920$	−13.1411	−0.433251
$921$	−17.8007	−0.586552
$922$	4.62267	0.152240
$923$	2.08141	0.0685106
$924$	−1.00000	−0.0328976
$925$	0.195831	0.00643889
$926$	−5.51249	−0.181152
$927$	−13.8084	−0.453527
$928$	8.33275	0.273536
$929$	−35.0978	−1.15152	−0.575761	−	0.817618i	$-0.695294\pi$
$929$	−35.0978	−1.15152	−0.575761	+	0.817618i	$0.695294\pi$
$930$	−19.2344	−0.630722
$931$	0	0
$932$	−16.7023	−0.547103
$933$	1.62536	0.0532120
$934$	−11.1693	−0.365471
$935$	7.18479	0.234968
$936$	0.467911	0.0152942
$937$	−44.8120	−1.46394	−0.731972	−	0.681334i	$-0.761400\pi$
$937$	−44.8120	−1.46394	−0.731972	+	0.681334i	$0.761400\pi$
$938$	0.162504	0.00530594
$939$	−30.0729	−0.981390
$940$	−10.3277	−0.336852
$941$	−12.4902	−0.407169	−0.203584	−	0.979057i	$-0.565259\pi$
$941$	−12.4902	−0.407169	−0.203584	+	0.979057i	$0.565259\pi$
$942$	−16.7023	−0.544191
$943$	20.5122	0.667968
$944$	14.2344	0.463291
$945$	1.45336	0.0472779
$946$	15.7665	0.512613
$947$	51.7110	1.68038	0.840191	−	0.542291i	$-0.182443\pi$
$947$	51.7110	1.68038	0.840191	+	0.542291i	$0.182443\pi$
$948$	−12.0574	−0.391605
$949$	4.25402	0.138091
$950$	0	0
$951$	11.3327	0.367490
$952$	−1.37464	−0.0445523
$953$	39.6769	1.28526	0.642630	−	0.766177i	$-0.277843\pi$
$953$	39.6769	1.28526	0.642630	+	0.766177i	$0.277843\pi$
$954$	11.9513	0.386938
$955$	9.77930	0.316451
$956$	10.0547	0.325192
$957$	−12.7665	−0.412683
$958$	−8.66456	−0.279939
$959$	1.16075	0.0374825
$960$	−2.22668	−0.0718658
$961$	43.6177	1.40702
$962$	−2.18748	−0.0705272
$963$	1.91353	0.0616628
$964$	−23.0993	−0.743977
$965$	10.8152	0.348154
$966$	−3.85204	−0.123937
$967$	27.4587	0.883014	0.441507	−	0.897258i	$-0.354444\pi$
$967$	27.4587	0.883014	0.441507	+	0.897258i	$0.354444\pi$
$968$	8.65270	0.278108
$969$	0	0
$970$	−5.49525	−0.176442
$971$	−23.3019	−0.747793	−0.373897	−	0.927470i	$-0.621979\pi$
$971$	−23.3019	−0.747793	−0.373897	+	0.927470i	$0.621979\pi$
$972$	−1.00000	−0.0320750
$973$	0.490200	0.0157151
$974$	0.0942073	0.00301860
$975$	−0.0196004	−0.000627714 0
$976$	−9.12061	−0.291944
$977$	13.4602	0.430629	0.215315	−	0.976545i	$-0.430922\pi$
$977$	13.4602	0.430629	0.215315	+	0.976545i	$0.430922\pi$
$978$	−15.8571	−0.507054
$979$	−25.6878	−0.820985
$980$	−14.6382	−0.467599
$981$	−11.8152	−0.377231
$982$	27.9959	0.893384
$983$	−45.7665	−1.45973	−0.729863	−	0.683594i	$-0.760416\pi$
$983$	−45.7665	−1.45973	−0.729863	+	0.683594i	$0.760416\pi$
$984$	3.47565	0.110800
$985$	45.1661	1.43911
$986$	−17.5493	−0.558884
$987$	−3.02734	−0.0963613
$988$	0	0
$989$	60.7333	1.93121
$990$	3.41147	0.108424
$991$	25.0256	0.794964	0.397482	−	0.917610i	$-0.369884\pi$
$991$	25.0256	0.794964	0.397482	+	0.917610i	$0.369884\pi$
$992$	8.63816	0.274262
$993$	32.1242	1.01943
$994$	−2.90343	−0.0920912
$995$	−50.0137	−1.58554
$996$	−3.70914	−0.117529
$997$	−26.7187	−0.846191	−0.423096	−	0.906085i	$-0.639057\pi$
$997$	−26.7187	−0.846191	−0.423096	+	0.906085i	$0.639057\pi$
$998$	36.9796	1.17057
$999$	4.67499	0.147910

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2166.2.a.o.1.3		3
3.2	odd	2		6498.2.a.bs.1.1		3
19.4	even	9		114.2.i.d.73.1	yes	6
19.5	even	9		114.2.i.d.25.1	✓	6
19.18	odd	2		2166.2.a.u.1.3		3
57.5	odd	18		342.2.u.a.253.1		6
57.23	odd	18		342.2.u.a.73.1		6
57.56	even	2		6498.2.a.bn.1.1		3
76.23	odd	18		912.2.bo.f.529.1		6
76.43	odd	18		912.2.bo.f.481.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.2.i.d.25.1	✓	6	19.5	even	9
114.2.i.d.73.1	yes	6	19.4	even	9
342.2.u.a.73.1		6	57.23	odd	18
342.2.u.a.253.1		6	57.5	odd	18
912.2.bo.f.481.1		6	76.43	odd	18
912.2.bo.f.529.1		6	76.23	odd	18
2166.2.a.o.1.3		3	1.1	even	1	trivial
2166.2.a.u.1.3		3	19.18	odd	2
6498.2.a.bn.1.1		3	57.56	even	2
6498.2.a.bs.1.1		3	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2166 = 2 \cdot 3 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2166.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.22668 q^{5} +1.00000 q^{6} -0.652704 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.22668 q^{5} +1.00000 q^{6} -0.652704 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.22668 q^{10} -1.53209 q^{11} -1.00000 q^{12} -0.467911 q^{13} +0.652704 q^{14} -2.22668 q^{15} +1.00000 q^{16} -2.10607 q^{17} -1.00000 q^{18} +2.22668 q^{20} +0.652704 q^{21} +1.53209 q^{22} +5.90167 q^{23} +1.00000 q^{24} -0.0418891 q^{25} +0.467911 q^{26} -1.00000 q^{27} -0.652704 q^{28} -8.33275 q^{29} +2.22668 q^{30} -8.63816 q^{31} -1.00000 q^{32} +1.53209 q^{33} +2.10607 q^{34} -1.45336 q^{35} +1.00000 q^{36} -4.67499 q^{37} +0.467911 q^{39} -2.22668 q^{40} +3.47565 q^{41} -0.652704 q^{42} +10.2909 q^{43} -1.53209 q^{44} +2.22668 q^{45} -5.90167 q^{46} -4.63816 q^{47} -1.00000 q^{48} -6.57398 q^{49} +0.0418891 q^{50} +2.10607 q^{51} -0.467911 q^{52} -11.9513 q^{53} +1.00000 q^{54} -3.41147 q^{55} +0.652704 q^{56} +8.33275 q^{58} +14.2344 q^{59} -2.22668 q^{60} -9.12061 q^{61} +8.63816 q^{62} -0.652704 q^{63} +1.00000 q^{64} -1.04189 q^{65} -1.53209 q^{66} +0.248970 q^{67} -2.10607 q^{68} -5.90167 q^{69} +1.45336 q^{70} -4.44831 q^{71} -1.00000 q^{72} -9.09152 q^{73} +4.67499 q^{74} +0.0418891 q^{75} +1.00000 q^{77} -0.467911 q^{78} +12.0574 q^{79} +2.22668 q^{80} +1.00000 q^{81} -3.47565 q^{82} +3.70914 q^{83} +0.652704 q^{84} -4.68954 q^{85} -10.2909 q^{86} +8.33275 q^{87} +1.53209 q^{88} +16.7665 q^{89} -2.22668 q^{90} +0.305407 q^{91} +5.90167 q^{92} +8.63816 q^{93} +4.63816 q^{94} +1.00000 q^{96} +2.46791 q^{97} +6.57398 q^{98} -1.53209 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{6} - 3 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^6 - 3 * q^7 - 3 * q^8 + 3 * q^9	\( 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{6} - 3 q^{7} - 3 q^{8} + 3 q^{9} - 3 q^{12} - 6 q^{13} + 3 q^{14} + 3 q^{16} + 6 q^{17} - 3 q^{18} + 3 q^{21} + 6 q^{23} + 3 q^{24} + 3 q^{25} + 6 q^{26} - 3 q^{27} - 3 q^{28} - 6 q^{29} - 9 q^{31} - 3 q^{32} - 6 q^{34} + 9 q^{35} + 3 q^{36} - 9 q^{37} + 6 q^{39} - 9 q^{41} - 3 q^{42} + 15 q^{43} - 6 q^{46} + 3 q^{47} - 3 q^{48} - 12 q^{49} - 3 q^{50} - 6 q^{51} - 6 q^{52} + 3 q^{53} + 3 q^{54} + 3 q^{56} + 6 q^{58} + 12 q^{59} - 33 q^{61} + 9 q^{62} - 3 q^{63} + 3 q^{64} - 12 q^{67} + 6 q^{68} - 6 q^{69} - 9 q^{70} - 15 q^{71} - 3 q^{72} + 3 q^{73} + 9 q^{74} - 3 q^{75} + 3 q^{77} - 6 q^{78} - 15 q^{79} + 3 q^{81} + 9 q^{82} + 27 q^{83} + 3 q^{84} - 27 q^{85} - 15 q^{86} + 6 q^{87} + 15 q^{89} + 3 q^{91} + 6 q^{92} + 9 q^{93} - 3 q^{94} + 3 q^{96} + 12 q^{97} + 12 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^6 - 3 * q^7 - 3 * q^8 + 3 * q^9 - 3 * q^12 - 6 * q^13 + 3 * q^14 + 3 * q^16 + 6 * q^17 - 3 * q^18 + 3 * q^21 + 6 * q^23 + 3 * q^24 + 3 * q^25 + 6 * q^26 - 3 * q^27 - 3 * q^28 - 6 * q^29 - 9 * q^31 - 3 * q^32 - 6 * q^34 + 9 * q^35 + 3 * q^36 - 9 * q^37 + 6 * q^39 - 9 * q^41 - 3 * q^42 + 15 * q^43 - 6 * q^46 + 3 * q^47 - 3 * q^48 - 12 * q^49 - 3 * q^50 - 6 * q^51 - 6 * q^52 + 3 * q^53 + 3 * q^54 + 3 * q^56 + 6 * q^58 + 12 * q^59 - 33 * q^61 + 9 * q^62 - 3 * q^63 + 3 * q^64 - 12 * q^67 + 6 * q^68 - 6 * q^69 - 9 * q^70 - 15 * q^71 - 3 * q^72 + 3 * q^73 + 9 * q^74 - 3 * q^75 + 3 * q^77 - 6 * q^78 - 15 * q^79 + 3 * q^81 + 9 * q^82 + 27 * q^83 + 3 * q^84 - 27 * q^85 - 15 * q^86 + 6 * q^87 + 15 * q^89 + 3 * q^91 + 6 * q^92 + 9 * q^93 - 3 * q^94 + 3 * q^96 + 12 * q^97 + 12 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more