LMFDB - Embedded newform 669.2.a.i.1.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [669,2,Mod(1,669)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("669.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(669, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 669 = 3 \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	669.a (trivial)

Newform invariants

comment:select newform

sage:traces = [14,0,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$5.34199189522$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 22 x^{12} - x^{11} + 187 x^{10} + 14 x^{9} - 774 x^{8} - 70 x^{7} + 1622 x^{6} + 168 x^{5} + \cdots - 11 $ x^14 - 22x^12 - x^11 + 187x^10 + 14x^9 - 774x^8 - 70x^7 + 1622x^6 + 168x^5 - 1617x^4 - 219x^3 + 604x^2 + 126*x - 11
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			1.10
Root			$-1.24204$ of defining polynomial
Character	$\chi$	$=$	669.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.24204 q^{2} +1.00000 q^{3} -0.457333 q^{4} -3.85024 q^{5} +1.24204 q^{6} +4.21468 q^{7} -3.05211 q^{8} +1.00000 q^{9} -4.78216 q^{10} +6.03056 q^{11} -0.457333 q^{12} +4.16401 q^{13} +5.23480 q^{14} -3.85024 q^{15} -2.87618 q^{16} -1.01812 q^{17} +1.24204 q^{18} +5.83769 q^{19} +1.76084 q^{20} +4.21468 q^{21} +7.49021 q^{22} -1.26052 q^{23} -3.05211 q^{24} +9.82438 q^{25} +5.17187 q^{26} +1.00000 q^{27} -1.92751 q^{28} -7.96642 q^{29} -4.78216 q^{30} -3.24626 q^{31} +2.53189 q^{32} +6.03056 q^{33} -1.26455 q^{34} -16.2275 q^{35} -0.457333 q^{36} -5.67490 q^{37} +7.25065 q^{38} +4.16401 q^{39} +11.7514 q^{40} +11.6923 q^{41} +5.23480 q^{42} +5.87065 q^{43} -2.75798 q^{44} -3.85024 q^{45} -1.56562 q^{46} -8.13978 q^{47} -2.87618 q^{48} +10.7635 q^{49} +12.2023 q^{50} -1.01812 q^{51} -1.90434 q^{52} +1.36163 q^{53} +1.24204 q^{54} -23.2191 q^{55} -12.8637 q^{56} +5.83769 q^{57} -9.89462 q^{58} -5.69357 q^{59} +1.76084 q^{60} +1.05393 q^{61} -4.03198 q^{62} +4.21468 q^{63} +8.89707 q^{64} -16.0325 q^{65} +7.49021 q^{66} -12.4984 q^{67} +0.465621 q^{68} -1.26052 q^{69} -20.1553 q^{70} +8.17083 q^{71} -3.05211 q^{72} -9.89152 q^{73} -7.04847 q^{74} +9.82438 q^{75} -2.66977 q^{76} +25.4169 q^{77} +5.17187 q^{78} -12.1893 q^{79} +11.0740 q^{80} +1.00000 q^{81} +14.5223 q^{82} -0.115461 q^{83} -1.92751 q^{84} +3.92002 q^{85} +7.29159 q^{86} -7.96642 q^{87} -18.4059 q^{88} -4.35186 q^{89} -4.78216 q^{90} +17.5500 q^{91} +0.576478 q^{92} -3.24626 q^{93} -10.1099 q^{94} -22.4765 q^{95} +2.53189 q^{96} +0.792677 q^{97} +13.3687 q^{98} +6.03056 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 14 q^{3} + 16 q^{4} + q^{5} + 14 q^{7} - 3 q^{8} + 14 q^{9} + 4 q^{10} + 13 q^{11} + 16 q^{12} + 10 q^{13} + q^{15} + 12 q^{16} + q^{17} + 32 q^{19} - 10 q^{20} + 14 q^{21} - 3 q^{22} - 7 q^{23}+ \cdots + 13 q^{99}+O(q^{100}) $ 14 * q + 14 * q^3 + 16 * q^4 + q^5 + 14 * q^7 - 3 * q^8 + 14 * q^9 + 4 * q^10 + 13 * q^11 + 16 * q^12 + 10 * q^13 + q^15 + 12 * q^16 + q^17 + 32 * q^19 - 10 * q^20 + 14 * q^21 - 3 * q^22 - 7 * q^23 - 3 * q^24 + 25 * q^25 - 3 * q^26 + 14 * q^27 + 28 * q^28 - q^29 + 4 * q^30 + 18 * q^31 - 16 * q^32 + 13 * q^33 + q^34 - 12 * q^35 + 16 * q^36 + 16 * q^37 - 31 * q^38 + 10 * q^39 + q^40 + 6 * q^41 + 15 * q^43 - 12 * q^44 + q^45 + 6 * q^46 - 21 * q^47 + 12 * q^48 + 28 * q^49 - 32 * q^50 + q^51 - 7 * q^52 - 17 * q^53 - q^55 - 33 * q^56 + 32 * q^57 - 27 * q^58 + 12 * q^59 - 10 * q^60 + 28 * q^61 - 16 * q^62 + 14 * q^63 + 3 * q^64 - 4 * q^65 - 3 * q^66 + 17 * q^67 + 5 * q^68 - 7 * q^69 - 14 * q^70 - 3 * q^71 - 3 * q^72 + 17 * q^73 - 16 * q^74 + 25 * q^75 + 14 * q^76 - 14 * q^77 - 3 * q^78 + 23 * q^79 - 36 * q^80 + 14 * q^81 + 5 * q^82 + 18 * q^83 + 28 * q^84 - 11 * q^85 + 9 * q^86 - q^87 - 35 * q^88 + 10 * q^89 + 4 * q^90 + 16 * q^91 - 45 * q^92 + 18 * q^93 - 43 * q^94 - 20 * q^95 - 16 * q^96 + 29 * q^97 - 19 * q^98 + 13 * q^99

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 $

$2$	1.24204	0.878256	0.439128	−	0.898425i	$-0.355287\pi$
$2$	1.24204	0.878256	0.439128	+	0.898425i	$0.355287\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−0.457333	−0.228667
$5$	−3.85024	−1.72188	−0.860941	−	0.508705i	$-0.830124\pi$
$5$	−3.85024	−1.72188	−0.860941	+	0.508705i	$0.830124\pi$
$6$	1.24204	0.507061
$7$	4.21468	1.59300	0.796499	−	0.604640i	$-0.206683\pi$
$7$	4.21468	1.59300	0.796499	+	0.604640i	$0.206683\pi$
$8$	−3.05211	−1.07908
$9$	1.00000	0.333333
$10$	−4.78216	−1.51225
$11$	6.03056	1.81828	0.909142	−	0.416487i	$-0.136739\pi$
$11$	6.03056	1.81828	0.909142	+	0.416487i	$0.136739\pi$
$12$	−0.457333	−0.132021
$13$	4.16401	1.15489	0.577445	−	0.816430i	$-0.304050\pi$
$13$	4.16401	1.15489	0.577445	+	0.816430i	$0.304050\pi$
$14$	5.23480	1.39906
$15$	−3.85024	−0.994129
$16$	−2.87618	−0.719045
$17$	−1.01812	−0.246931	−0.123465	−	0.992349i	$-0.539401\pi$
$17$	−1.01812	−0.246931	−0.123465	+	0.992349i	$0.539401\pi$
$18$	1.24204	0.292752
$19$	5.83769	1.33926	0.669629	−	0.742696i	$-0.266453\pi$
$19$	5.83769	1.33926	0.669629	+	0.742696i	$0.266453\pi$
$20$	1.76084	0.393737
$21$	4.21468	0.919718
$22$	7.49021	1.59692
$23$	−1.26052	−0.262837	−0.131418	−	0.991327i	$-0.541953\pi$
$23$	−1.26052	−0.262837	−0.131418	+	0.991327i	$0.541953\pi$
$24$	−3.05211	−0.623009
$25$	9.82438	1.96488
$26$	5.17187	1.01429
$27$	1.00000	0.192450
$28$	−1.92751	−0.364266
$29$	−7.96642	−1.47933	−0.739664	−	0.672977i	$-0.765015\pi$
$29$	−7.96642	−1.47933	−0.739664	+	0.672977i	$0.765015\pi$
$30$	−4.78216	−0.873099
$31$	−3.24626	−0.583045	−0.291522	−	0.956564i	$-0.594162\pi$
$31$	−3.24626	−0.583045	−0.291522	+	0.956564i	$0.594162\pi$
$32$	2.53189	0.447578
$33$	6.03056	1.04979
$34$	−1.26455	−0.216869
$35$	−16.2275	−2.74295
$36$	−0.457333	−0.0762222
$37$	−5.67490	−0.932949	−0.466474	−	0.884535i	$-0.654476\pi$
$37$	−5.67490	−0.932949	−0.466474	+	0.884535i	$0.654476\pi$
$38$	7.25065	1.17621
$39$	4.16401	0.666776
$40$	11.7514	1.85805
$41$	11.6923	1.82603	0.913016	−	0.407923i	$-0.133747\pi$
$41$	11.6923	1.82603	0.913016	+	0.407923i	$0.133747\pi$
$42$	5.23480	0.807748
$43$	5.87065	0.895265	0.447633	−	0.894218i	$-0.352267\pi$
$43$	5.87065	0.895265	0.447633	+	0.894218i	$0.352267\pi$
$44$	−2.75798	−0.415781
$45$	−3.85024	−0.573960
$46$	−1.56562	−0.230838
$47$	−8.13978	−1.18731	−0.593654	−	0.804720i	$-0.702315\pi$
$47$	−8.13978	−1.18731	−0.593654	+	0.804720i	$0.702315\pi$
$48$	−2.87618	−0.415141
$49$	10.7635	1.53764
$50$	12.2023	1.72566
$51$	−1.01812	−0.142566
$52$	−1.90434	−0.264085
$53$	1.36163	0.187034	0.0935168	−	0.995618i	$-0.470189\pi$
$53$	1.36163	0.187034	0.0935168	+	0.995618i	$0.470189\pi$
$54$	1.24204	0.169020
$55$	−23.2191	−3.13087
$56$	−12.8637	−1.71898
$57$	5.83769	0.773221
$58$	−9.89462	−1.29923
$59$	−5.69357	−0.741240	−0.370620	−	0.928785i	$-0.620855\pi$
$59$	−5.69357	−0.741240	−0.370620	+	0.928785i	$0.620855\pi$
$60$	1.76084	0.227324
$61$	1.05393	0.134942	0.0674708	−	0.997721i	$-0.478507\pi$
$61$	1.05393	0.134942	0.0674708	+	0.997721i	$0.478507\pi$
$62$	−4.03198	−0.512062
$63$	4.21468	0.530999
$64$	8.89707	1.11213
$65$	−16.0325	−1.98858
$66$	7.49021	0.921981
$67$	−12.4984	−1.52692	−0.763461	−	0.645854i	$-0.776501\pi$
$67$	−12.4984	−1.52692	−0.763461	+	0.645854i	$0.776501\pi$
$68$	0.465621	0.0564649
$69$	−1.26052	−0.151749
$70$	−20.1553	−2.40901
$71$	8.17083	0.969699	0.484850	−	0.874598i	$-0.338874\pi$
$71$	8.17083	0.969699	0.484850	+	0.874598i	$0.338874\pi$
$72$	−3.05211	−0.359695
$73$	−9.89152	−1.15771	−0.578857	−	0.815429i	$-0.696501\pi$
$73$	−9.89152	−1.15771	−0.578857	+	0.815429i	$0.696501\pi$
$74$	−7.04847	−0.819367
$75$	9.82438	1.13442
$76$	−2.66977	−0.306244
$77$	25.4169	2.89652
$78$	5.17187	0.585600
$79$	−12.1893	−1.37140	−0.685701	−	0.727884i	$-0.740504\pi$
$79$	−12.1893	−1.37140	−0.685701	+	0.727884i	$0.740504\pi$
$80$	11.0740	1.23811
$81$	1.00000	0.111111
$82$	14.5223	1.60372
$83$	−0.115461	−0.0126735	−0.00633676	−	0.999980i	$-0.502017\pi$
$83$	−0.115461	−0.0126735	−0.00633676	+	0.999980i	$0.502017\pi$
$84$	−1.92751	−0.210309
$85$	3.92002	0.425186
$86$	7.29159	0.786272
$87$	−7.96642	−0.854090
$88$	−18.4059	−1.96208
$89$	−4.35186	−0.461296	−0.230648	−	0.973037i	$-0.574085\pi$
$89$	−4.35186	−0.461296	−0.230648	+	0.973037i	$0.574085\pi$
$90$	−4.78216	−0.504084
$91$	17.5500	1.83974
$92$	0.576478	0.0601020
$93$	−3.24626	−0.336621
$94$	−10.1099	−1.04276
$95$	−22.4765	−2.30604
$96$	2.53189	0.258409
$97$	0.792677	0.0804842	0.0402421	−	0.999190i	$-0.487187\pi$
$97$	0.792677	0.0804842	0.0402421	+	0.999190i	$0.487187\pi$
$98$	13.3687	1.35044
$99$	6.03056	0.606094
$100$	−4.49302	−0.449302
$101$	3.40873	0.339182	0.169591	−	0.985515i	$-0.445755\pi$
$101$	3.40873	0.339182	0.169591	+	0.985515i	$0.445755\pi$
$102$	−1.26455	−0.125209
$103$	−6.58517	−0.648856	−0.324428	−	0.945910i	$-0.605172\pi$
$103$	−6.58517	−0.648856	−0.324428	+	0.945910i	$0.605172\pi$
$104$	−12.7090	−1.24622
$105$	−16.2275	−1.58364
$106$	1.69120	0.164263
$107$	5.56837	0.538315	0.269157	−	0.963096i	$-0.413255\pi$
$107$	5.56837	0.538315	0.269157	+	0.963096i	$0.413255\pi$
$108$	−0.457333	−0.0440069
$109$	6.92395	0.663194	0.331597	−	0.943421i	$-0.392413\pi$
$109$	6.92395	0.663194	0.331597	+	0.943421i	$0.392413\pi$
$110$	−28.8391	−2.74970
$111$	−5.67490	−0.538638
$112$	−12.1222	−1.14544
$113$	4.36248	0.410388	0.205194	−	0.978721i	$-0.434217\pi$
$113$	4.36248	0.410388	0.205194	+	0.978721i	$0.434217\pi$
$114$	7.25065	0.679086
$115$	4.85331	0.452574
$116$	3.64331	0.338273
$117$	4.16401	0.384963
$118$	−7.07165	−0.650998
$119$	−4.29106	−0.393361
$120$	11.7514	1.07275
$121$	25.3677	2.30615
$122$	1.30902	0.118513
$123$	11.6923	1.05426
$124$	1.48462	0.133323
$125$	−18.5750	−1.66140
$126$	5.23480	0.466353
$127$	−15.7561	−1.39812	−0.699062	−	0.715061i	$-0.746399\pi$
$127$	−15.7561	−1.39812	−0.699062	+	0.715061i	$0.746399\pi$
$128$	5.98675	0.529159
$129$	5.87065	0.516882
$130$	−19.9130	−1.74648
$131$	4.54645	0.397225	0.198612	−	0.980078i	$-0.436357\pi$
$131$	4.54645	0.397225	0.198612	+	0.980078i	$0.436357\pi$
$132$	−2.75798	−0.240051
$133$	24.6040	2.13343
$134$	−15.5235	−1.34103
$135$	−3.85024	−0.331376
$136$	3.10742	0.266459
$137$	−22.0581	−1.88455	−0.942277	−	0.334835i	$-0.891319\pi$
$137$	−22.0581	−1.88455	−0.942277	+	0.334835i	$0.891319\pi$
$138$	−1.56562	−0.133274
$139$	4.87787	0.413736	0.206868	−	0.978369i	$-0.433673\pi$
$139$	4.87787	0.413736	0.206868	+	0.978369i	$0.433673\pi$
$140$	7.42139	0.627222
$141$	−8.13978	−0.685493
$142$	10.1485	0.851644
$143$	25.1113	2.09992
$144$	−2.87618	−0.239682
$145$	30.6727	2.54723
$146$	−12.2857	−1.01677
$147$	10.7635	0.887758
$148$	2.59532	0.213334
$149$	−6.70478	−0.549277	−0.274638	−	0.961548i	$-0.588558\pi$
$149$	−6.70478	−0.549277	−0.274638	+	0.961548i	$0.588558\pi$
$150$	12.2023	0.996312
$151$	12.7339	1.03627	0.518135	−	0.855299i	$-0.326627\pi$
$151$	12.7339	1.03627	0.518135	+	0.855299i	$0.326627\pi$
$152$	−17.8173	−1.44517
$153$	−1.01812	−0.0823103
$154$	31.5688	2.54389
$155$	12.4989	1.00393
$156$	−1.90434	−0.152469
$157$	17.3303	1.38311	0.691556	−	0.722323i	$-0.256926\pi$
$157$	17.3303	1.38311	0.691556	+	0.722323i	$0.256926\pi$
$158$	−15.1396	−1.20444
$159$	1.36163	0.107984
$160$	−9.74838	−0.770677
$161$	−5.31269	−0.418698
$162$	1.24204	0.0975840
$163$	−8.71687	−0.682758	−0.341379	−	0.939926i	$-0.610894\pi$
$163$	−8.71687	−0.682758	−0.341379	+	0.939926i	$0.610894\pi$
$164$	−5.34729	−0.417553
$165$	−23.2191	−1.80761
$166$	−0.143408	−0.0111306
$167$	25.4657	1.97059	0.985297	−	0.170853i	$-0.0546522\pi$
$167$	25.4657	1.97059	0.985297	+	0.170853i	$0.0546522\pi$
$168$	−12.8637	−0.992453
$169$	4.33900	0.333769
$170$	4.86883	0.373422
$171$	5.83769	0.446419
$172$	−2.68484	−0.204717
$173$	−4.53163	−0.344533	−0.172267	−	0.985050i	$-0.555109\pi$
$173$	−4.53163	−0.344533	−0.172267	+	0.985050i	$0.555109\pi$
$174$	−9.89462	−0.750109
$175$	41.4066	3.13004
$176$	−17.3450	−1.30743
$177$	−5.69357	−0.427955
$178$	−5.40518	−0.405136
$179$	−16.5057	−1.23370	−0.616849	−	0.787082i	$-0.711591\pi$
$179$	−16.5057	−1.23370	−0.616849	+	0.787082i	$0.711591\pi$
$180$	1.76084	0.131246
$181$	12.4340	0.924212	0.462106	−	0.886825i	$-0.347094\pi$
$181$	12.4340	0.924212	0.462106	+	0.886825i	$0.347094\pi$
$182$	21.7978	1.61576
$183$	1.05393	0.0779086
$184$	3.84725	0.283623
$185$	21.8498	1.60643
$186$	−4.03198	−0.295639
$187$	−6.13985	−0.448990
$188$	3.72259	0.271498
$189$	4.21468	0.306573
$190$	−27.9168	−2.02530
$191$	−20.0815	−1.45305	−0.726523	−	0.687142i	$-0.758865\pi$
$191$	−20.0815	−1.45305	−0.726523	+	0.687142i	$0.758865\pi$
$192$	8.89707	0.642090
$193$	7.52880	0.541935	0.270967	−	0.962589i	$-0.412656\pi$
$193$	7.52880	0.541935	0.270967	+	0.962589i	$0.412656\pi$
$194$	0.984538	0.0706857
$195$	−16.0325	−1.14811
$196$	−4.92251	−0.351608
$197$	−7.97060	−0.567882	−0.283941	−	0.958842i	$-0.591642\pi$
$197$	−7.97060	−0.567882	−0.283941	+	0.958842i	$0.591642\pi$
$198$	7.49021	0.532306
$199$	−16.6235	−1.17841	−0.589203	−	0.807985i	$-0.700558\pi$
$199$	−16.6235	−1.17841	−0.589203	+	0.807985i	$0.700558\pi$
$200$	−29.9851	−2.12026
$201$	−12.4984	−0.881568
$202$	4.23379	0.297888
$203$	−33.5759	−2.35657
$204$	0.465621	0.0326000
$205$	−45.0183	−3.14421
$206$	−8.17906	−0.569862
$207$	−1.26052	−0.0876122
$208$	−11.9764	−0.830417
$209$	35.2045	2.43515
$210$	−20.1553	−1.39085
$211$	−10.4919	−0.722294	−0.361147	−	0.932509i	$-0.617615\pi$
$211$	−10.4919	−0.722294	−0.361147	+	0.932509i	$0.617615\pi$
$212$	−0.622717	−0.0427684
$213$	8.17083	0.559856
$214$	6.91615	0.472778
$215$	−22.6034	−1.54154
$216$	−3.05211	−0.207670
$217$	−13.6819	−0.928789
$218$	8.59983	0.582454
$219$	−9.89152	−0.668407
$220$	10.6189	0.715925
$221$	−4.23947	−0.285178
$222$	−7.04847	−0.473062
$223$	−1.00000	−0.0669650
$223$	−1.00000	−0.0669650
$224$	10.6711	0.712991
$225$	9.82438	0.654958
$226$	5.41838	0.360426
$227$	−11.2013	−0.743458	−0.371729	−	0.928341i	$-0.621235\pi$
$227$	−11.2013	−0.743458	−0.371729	+	0.928341i	$0.621235\pi$
$228$	−2.66977	−0.176810
$229$	−0.976014	−0.0644968	−0.0322484	−	0.999480i	$-0.510267\pi$
$229$	−0.976014	−0.0644968	−0.0322484	+	0.999480i	$0.510267\pi$
$230$	6.02801	0.397475
$231$	25.4169	1.67231
$232$	24.3144	1.59632
$233$	−22.5946	−1.48022	−0.740109	−	0.672487i	$-0.765226\pi$
$233$	−22.5946	−1.48022	−0.740109	+	0.672487i	$0.765226\pi$
$234$	5.17187	0.338096
$235$	31.3401	2.04440
$236$	2.60386	0.169497
$237$	−12.1893	−0.791779
$238$	−5.32967	−0.345471
$239$	−4.77427	−0.308822	−0.154411	−	0.988007i	$-0.549348\pi$
$239$	−4.77427	−0.308822	−0.154411	+	0.988007i	$0.549348\pi$
$240$	11.0740	0.714823
$241$	−10.6322	−0.684878	−0.342439	−	0.939540i	$-0.611253\pi$
$241$	−10.6322	−0.684878	−0.342439	+	0.939540i	$0.611253\pi$
$242$	31.5077	2.02539
$243$	1.00000	0.0641500
$244$	−0.481997	−0.0308567
$245$	−41.4421	−2.64764
$246$	14.5223	0.925911
$247$	24.3082	1.54669
$248$	9.90793	0.629154
$249$	−0.115461	−0.00731706
$250$	−23.0709	−1.45913
$251$	17.1203	1.08063	0.540313	−	0.841464i	$-0.318306\pi$
$251$	17.1203	1.08063	0.540313	+	0.841464i	$0.318306\pi$
$252$	−1.92751	−0.121422
$253$	−7.60165	−0.477911
$254$	−19.5697	−1.22791
$255$	3.92002	0.245481
$256$	−10.3583	−0.647396
$257$	15.3556	0.957853	0.478927	−	0.877855i	$-0.341026\pi$
$257$	15.3556	0.957853	0.478927	+	0.877855i	$0.341026\pi$
$258$	7.29159	0.453954
$259$	−23.9179	−1.48619
$260$	7.33218	0.454722
$261$	−7.96642	−0.493109
$262$	5.64687	0.348865
$263$	14.3234	0.883219	0.441610	−	0.897207i	$-0.354408\pi$
$263$	14.3234	0.883219	0.441610	+	0.897207i	$0.354408\pi$
$264$	−18.4059	−1.13281
$265$	−5.24259	−0.322050
$266$	30.5591	1.87370
$267$	−4.35186	−0.266329
$268$	5.71593	0.349156
$269$	−10.0480	−0.612637	−0.306319	−	0.951929i	$-0.599097\pi$
$269$	−10.0480	−0.612637	−0.306319	+	0.951929i	$0.599097\pi$
$270$	−4.78216	−0.291033
$271$	−7.44161	−0.452046	−0.226023	−	0.974122i	$-0.572572\pi$
$271$	−7.44161	−0.452046	−0.226023	+	0.974122i	$0.572572\pi$
$272$	2.92830	0.177554
$273$	17.5500	1.06217
$274$	−27.3971	−1.65512
$275$	59.2465	3.57270
$276$	0.576478	0.0346999
$277$	1.77535	0.106670	0.0533351	−	0.998577i	$-0.483015\pi$
$277$	1.77535	0.106670	0.0533351	+	0.998577i	$0.483015\pi$
$278$	6.05852	0.363366
$279$	−3.24626	−0.194348
$280$	49.5282	2.95988
$281$	−13.5754	−0.809839	−0.404919	−	0.914352i	$-0.632700\pi$
$281$	−13.5754	−0.809839	−0.404919	+	0.914352i	$0.632700\pi$
$282$	−10.1099	−0.602038
$283$	−7.44940	−0.442821	−0.221410	−	0.975181i	$-0.571066\pi$
$283$	−7.44940	−0.442821	−0.221410	+	0.975181i	$0.571066\pi$
$284$	−3.73679	−0.221738
$285$	−22.4765	−1.33139
$286$	31.1893	1.84426
$287$	49.2793	2.90887
$288$	2.53189	0.149193
$289$	−15.9634	−0.939025
$290$	38.0967	2.23712
$291$	0.792677	0.0464676
$292$	4.52372	0.264731
$293$	8.94256	0.522430	0.261215	−	0.965281i	$-0.415877\pi$
$293$	8.94256	0.522430	0.261215	+	0.965281i	$0.415877\pi$
$294$	13.3687	0.779679
$295$	21.9216	1.27633
$296$	17.3204	1.00673
$297$	6.03056	0.349929
$298$	−8.32761	−0.482406
$299$	−5.24882	−0.303547
$300$	−4.49302	−0.259404
$301$	24.7429	1.42616
$302$	15.8160	0.910109
$303$	3.40873	0.195827
$304$	−16.7902	−0.962986
$305$	−4.05788	−0.232354
$306$	−1.26455	−0.0722895
$307$	−6.95368	−0.396867	−0.198434	−	0.980114i	$-0.563585\pi$
$307$	−6.95368	−0.396867	−0.198434	+	0.980114i	$0.563585\pi$
$308$	−11.6240	−0.662338
$309$	−6.58517	−0.374617
$310$	15.5241	0.881711
$311$	−29.6158	−1.67936	−0.839678	−	0.543085i	$-0.817256\pi$
$311$	−29.6158	−1.67936	−0.839678	+	0.543085i	$0.817256\pi$
$312$	−12.7090	−0.719507
$313$	−4.34447	−0.245564	−0.122782	−	0.992434i	$-0.539182\pi$
$313$	−4.34447	−0.245564	−0.122782	+	0.992434i	$0.539182\pi$
$314$	21.5250	1.21473
$315$	−16.2275	−0.914318
$316$	5.57456	0.313594
$317$	1.94112	0.109024	0.0545122	−	0.998513i	$-0.482640\pi$
$317$	1.94112	0.109024	0.0545122	+	0.998513i	$0.482640\pi$
$318$	1.69120	0.0948375
$319$	−48.0420	−2.68984
$320$	−34.2559	−1.91496
$321$	5.56837	0.310796
$322$	−6.59858	−0.367724
$323$	−5.94348	−0.330704
$324$	−0.457333	−0.0254074
$325$	40.9088	2.26921
$326$	−10.8267	−0.599636
$327$	6.92395	0.382895
$328$	−35.6862	−1.97044
$329$	−34.3065	−1.89138
$330$	−28.8391	−1.58754
$331$	4.00127	0.219930	0.109965	−	0.993935i	$-0.464926\pi$
$331$	4.00127	0.219930	0.109965	+	0.993935i	$0.464926\pi$
$332$	0.0528043	0.00289801
$333$	−5.67490	−0.310983
$334$	31.6294	1.73068
$335$	48.1218	2.62918
$336$	−12.1222	−0.661318
$337$	−24.3150	−1.32452	−0.662261	−	0.749274i	$-0.730403\pi$
$337$	−24.3150	−1.32452	−0.662261	+	0.749274i	$0.730403\pi$
$338$	5.38921	0.293134
$339$	4.36248	0.236938
$340$	−1.79276	−0.0972258
$341$	−19.5767	−1.06014
$342$	7.25065	0.392070
$343$	15.8619	0.856464
$344$	−17.9179	−0.966066
$345$	4.85331	0.261293
$346$	−5.62847	−0.302588
$347$	34.9216	1.87469	0.937345	−	0.348402i	$-0.113275\pi$
$347$	34.9216	1.87469	0.937345	+	0.348402i	$0.113275\pi$
$348$	3.64331	0.195302
$349$	−3.56263	−0.190703	−0.0953517	−	0.995444i	$-0.530398\pi$
$349$	−3.56263	−0.190703	−0.0953517	+	0.995444i	$0.530398\pi$
$350$	51.4287	2.74898
$351$	4.16401	0.222259
$352$	15.2687	0.813824
$353$	5.83845	0.310749	0.155375	−	0.987856i	$-0.450342\pi$
$353$	5.83845	0.310749	0.155375	+	0.987856i	$0.450342\pi$
$354$	−7.07165	−0.375854
$355$	−31.4597	−1.66971
$356$	1.99025	0.105483
$357$	−4.29106	−0.227107
$358$	−20.5008	−1.08350
$359$	7.51374	0.396560	0.198280	−	0.980145i	$-0.436464\pi$
$359$	7.51374	0.396560	0.198280	+	0.980145i	$0.436464\pi$
$360$	11.7514	0.619351
$361$	15.0786	0.793610
$362$	15.4435	0.811694
$363$	25.3677	1.33146
$364$	−8.02618	−0.420686
$365$	38.0848	1.99345
$366$	1.30902	0.0684237
$367$	26.2868	1.37216	0.686081	−	0.727525i	$-0.259330\pi$
$367$	26.2868	1.37216	0.686081	+	0.727525i	$0.259330\pi$
$368$	3.62548	0.188991
$369$	11.6923	0.608678
$370$	27.1383	1.41085
$371$	5.73881	0.297944
$372$	1.48462	0.0769740
$373$	5.89586	0.305276	0.152638	−	0.988282i	$-0.451223\pi$
$373$	5.89586	0.305276	0.152638	+	0.988282i	$0.451223\pi$
$374$	−7.62595	−0.394328
$375$	−18.5750	−0.959210
$376$	24.8435	1.28121
$377$	−33.1723	−1.70846
$378$	5.23480	0.269249
$379$	14.7995	0.760199	0.380100	−	0.924946i	$-0.375890\pi$
$379$	14.7995	0.760199	0.380100	+	0.924946i	$0.375890\pi$
$380$	10.2793	0.527315
$381$	−15.7561	−0.807207
$382$	−24.9421	−1.27615
$383$	−4.68617	−0.239452	−0.119726	−	0.992807i	$-0.538202\pi$
$383$	−4.68617	−0.239452	−0.119726	+	0.992807i	$0.538202\pi$
$384$	5.98675	0.305510
$385$	−97.8611	−4.98747
$386$	9.35108	0.475957
$387$	5.87065	0.298422
$388$	−0.362518	−0.0184041
$389$	24.9398	1.26450	0.632250	−	0.774765i	$-0.282132\pi$
$389$	24.9398	1.26450	0.632250	+	0.774765i	$0.282132\pi$
$390$	−19.9130	−1.00833
$391$	1.28336	0.0649025
$392$	−32.8514	−1.65925
$393$	4.54645	0.229338
$394$	−9.89981	−0.498745
$395$	46.9317	2.36139
$396$	−2.75798	−0.138594
$397$	21.6004	1.08409	0.542046	−	0.840349i	$-0.317650\pi$
$397$	21.6004	1.08409	0.542046	+	0.840349i	$0.317650\pi$
$398$	−20.6470	−1.03494
$399$	24.6040	1.23174
$400$	−28.2567	−1.41283
$401$	−22.9795	−1.14754	−0.573771	−	0.819016i	$-0.694520\pi$
$401$	−22.9795	−1.14754	−0.573771	+	0.819016i	$0.694520\pi$
$402$	−15.5235	−0.774243
$403$	−13.5174	−0.673352
$404$	−1.55893	−0.0775596
$405$	−3.85024	−0.191320
$406$	−41.7026	−2.06967
$407$	−34.2229	−1.69636
$408$	3.10742	0.153840
$409$	1.08253	0.0535275	0.0267637	−	0.999642i	$-0.491480\pi$
$409$	1.08253	0.0535275	0.0267637	+	0.999642i	$0.491480\pi$
$410$	−55.9145	−2.76142
$411$	−22.0581	−1.08805
$412$	3.01162	0.148372
$413$	−23.9966	−1.18079
$414$	−1.56562	−0.0769459
$415$	0.444554	0.0218223
$416$	10.5428	0.516903
$417$	4.87787	0.238870
$418$	43.7255	2.13868
$419$	29.4075	1.43665	0.718326	−	0.695707i	$-0.244909\pi$
$419$	29.4075	1.43665	0.718326	+	0.695707i	$0.244909\pi$
$420$	7.42139	0.362127
$421$	−3.06238	−0.149251	−0.0746257	−	0.997212i	$-0.523776\pi$
$421$	−3.06238	−0.149251	−0.0746257	+	0.997212i	$0.523776\pi$
$422$	−13.0314	−0.634359
$423$	−8.13978	−0.395770
$424$	−4.15583	−0.201825
$425$	−10.0024	−0.485189
$426$	10.1485	0.491697
$427$	4.44197	0.214962
$428$	−2.54660	−0.123095
$429$	25.1113	1.21239
$430$	−28.0744	−1.35387
$431$	−29.7458	−1.43280	−0.716402	−	0.697688i	$-0.754212\pi$
$431$	−29.7458	−1.43280	−0.716402	+	0.697688i	$0.754212\pi$
$432$	−2.87618	−0.138380
$433$	−8.53399	−0.410117	−0.205059	−	0.978750i	$-0.565739\pi$
$433$	−8.53399	−0.410117	−0.205059	+	0.978750i	$0.565739\pi$
$434$	−16.9935	−0.815714
$435$	30.6727	1.47064
$436$	−3.16655	−0.151650
$437$	−7.35852	−0.352006
$438$	−12.2857	−0.587032
$439$	39.8526	1.90206	0.951030	−	0.309099i	$-0.100028\pi$
$439$	39.8526	1.90206	0.951030	+	0.309099i	$0.100028\pi$
$440$	70.8673	3.37847
$441$	10.7635	0.512548
$442$	−5.26560	−0.250459
$443$	−21.0487	−1.00005	−0.500027	−	0.866010i	$-0.666677\pi$
$443$	−21.0487	−1.00005	−0.500027	+	0.866010i	$0.666677\pi$
$444$	2.59532	0.123169
$445$	16.7557	0.794297
$446$	−1.24204	−0.0588124
$447$	−6.70478	−0.317125
$448$	37.4983	1.77163
$449$	22.5601	1.06468	0.532339	−	0.846531i	$-0.321313\pi$
$449$	22.5601	1.06468	0.532339	+	0.846531i	$0.321313\pi$
$450$	12.2023	0.575221
$451$	70.5112	3.32024
$452$	−1.99511	−0.0938420
$453$	12.7339	0.598290
$454$	−13.9125	−0.652947
$455$	−67.5716	−3.16781
$456$	−17.8173	−0.834370
$457$	4.96027	0.232032	0.116016	−	0.993247i	$-0.462988\pi$
$457$	4.96027	0.232032	0.116016	+	0.993247i	$0.462988\pi$
$458$	−1.21225	−0.0566447
$459$	−1.01812	−0.0475219
$460$	−2.21958	−0.103489
$461$	7.09912	0.330639	0.165319	−	0.986240i	$-0.447134\pi$
$461$	7.09912	0.330639	0.165319	+	0.986240i	$0.447134\pi$
$462$	31.5688	1.46871
$463$	4.80537	0.223324	0.111662	−	0.993746i	$-0.464383\pi$
$463$	4.80537	0.223324	0.111662	+	0.993746i	$0.464383\pi$
$464$	22.9129	1.06370
$465$	12.4989	0.579621
$466$	−28.0634	−1.30001
$467$	−8.71811	−0.403426	−0.201713	−	0.979445i	$-0.564651\pi$
$467$	−8.71811	−0.403426	−0.201713	+	0.979445i	$0.564651\pi$
$468$	−1.90434	−0.0880282
$469$	−52.6767	−2.43238
$470$	38.9257	1.79551
$471$	17.3303	0.798540
$472$	17.3774	0.799860
$473$	35.4033	1.62785
$474$	−15.1396	−0.695384
$475$	57.3516	2.63147
$476$	1.96244	0.0899485
$477$	1.36163	0.0623446
$478$	−5.92984	−0.271224
$479$	−4.02989	−0.184131	−0.0920653	−	0.995753i	$-0.529347\pi$
$479$	−4.02989	−0.184131	−0.0920653	+	0.995753i	$0.529347\pi$
$480$	−9.74838	−0.444950
$481$	−23.6304	−1.07745
$482$	−13.2056	−0.601498
$483$	−5.31269	−0.241736
$484$	−11.6015	−0.527340
$485$	−3.05200	−0.138584
$486$	1.24204	0.0563401
$487$	−14.1758	−0.642368	−0.321184	−	0.947017i	$-0.604081\pi$
$487$	−14.1758	−0.642368	−0.321184	+	0.947017i	$0.604081\pi$
$488$	−3.21670	−0.145613
$489$	−8.71687	−0.394190
$490$	−51.4728	−2.32530
$491$	29.1330	1.31475	0.657377	−	0.753561i	$-0.271666\pi$
$491$	29.1330	1.31475	0.657377	+	0.753561i	$0.271666\pi$
$492$	−5.34729	−0.241074
$493$	8.11079	0.365292
$494$	30.1918	1.35839
$495$	−23.2191	−1.04362
$496$	9.33681	0.419235
$497$	34.4374	1.54473
$498$	−0.143408	−0.00642625
$499$	11.6823	0.522970	0.261485	−	0.965208i	$-0.415788\pi$
$499$	11.6823	0.522970	0.261485	+	0.965208i	$0.415788\pi$
$500$	8.49498	0.379907
$501$	25.4657	1.13772
$502$	21.2642	0.949067
$503$	−16.1726	−0.721099	−0.360549	−	0.932740i	$-0.617411\pi$
$503$	−16.1726	−0.721099	−0.360549	+	0.932740i	$0.617411\pi$
$504$	−12.8637	−0.572993
$505$	−13.1245	−0.584031
$506$	−9.44156	−0.419729
$507$	4.33900	0.192702
$508$	7.20577	0.319704
$509$	−3.63261	−0.161012	−0.0805062	−	0.996754i	$-0.525654\pi$
$509$	−3.63261	−0.161012	−0.0805062	+	0.996754i	$0.525654\pi$
$510$	4.86883	0.215595
$511$	−41.6895	−1.84424
$512$	−24.8390	−1.09774
$513$	5.83769	0.257740
$514$	19.0722	0.841240
$515$	25.3545	1.11725
$516$	−2.68484	−0.118194
$517$	−49.0874	−2.15886
$518$	−29.7070	−1.30525
$519$	−4.53163	−0.198916
$520$	48.9328	2.14585
$521$	13.2882	0.582168	0.291084	−	0.956697i	$-0.405984\pi$
$521$	13.2882	0.582168	0.291084	+	0.956697i	$0.405984\pi$
$522$	−9.89462	−0.433076
$523$	−1.52769	−0.0668013	−0.0334006	−	0.999442i	$-0.510634\pi$
$523$	−1.52769	−0.0668013	−0.0334006	+	0.999442i	$0.510634\pi$
$524$	−2.07924	−0.0908321
$525$	41.4066	1.80713
$526$	17.7903	0.775692
$527$	3.30509	0.143972
$528$	−17.3450	−0.754843
$529$	−21.4111	−0.930917
$530$	−6.51151	−0.282842
$531$	−5.69357	−0.247080
$532$	−11.2522	−0.487845
$533$	48.6869	2.10887
$534$	−5.40518	−0.233905
$535$	−21.4396	−0.926914
$536$	38.1465	1.64768
$537$	−16.5057	−0.712276
$538$	−12.4800	−0.538052
$539$	64.9100	2.79587
$540$	1.76084	0.0757747
$541$	−30.3869	−1.30644	−0.653218	−	0.757170i	$-0.726582\pi$
$541$	−30.3869	−1.30644	−0.653218	+	0.757170i	$0.726582\pi$
$542$	−9.24279	−0.397012
$543$	12.4340	0.533594
$544$	−2.57777	−0.110521
$545$	−26.6589	−1.14194
$546$	21.7978	0.932859
$547$	−25.6247	−1.09563	−0.547816	−	0.836599i	$-0.684541\pi$
$547$	−25.6247	−1.09563	−0.547816	+	0.836599i	$0.684541\pi$
$548$	10.0879	0.430935
$549$	1.05393	0.0449806
$550$	73.5866	3.13774
$551$	−46.5055	−1.98120
$552$	3.84725	0.163750
$553$	−51.3739	−2.18464
$554$	2.20505	0.0936838
$555$	21.8498	0.927471
$556$	−2.23081	−0.0946075
$557$	−33.7751	−1.43110	−0.715548	−	0.698564i	$-0.753823\pi$
$557$	−33.7751	−1.43110	−0.715548	+	0.698564i	$0.753823\pi$
$558$	−4.03198	−0.170687
$559$	24.4454	1.03393
$560$	46.6733	1.97231
$561$	−6.13985	−0.259225
$562$	−16.8612	−0.711246
$563$	35.0432	1.47690	0.738448	−	0.674310i	$-0.235559\pi$
$563$	35.0432	1.47690	0.738448	+	0.674310i	$0.235559\pi$
$564$	3.72259	0.156749
$565$	−16.7966	−0.706639
$566$	−9.25246	−0.388910
$567$	4.21468	0.177000
$568$	−24.9383	−1.04639
$569$	−10.0273	−0.420367	−0.210183	−	0.977662i	$-0.567406\pi$
$569$	−10.0273	−0.420367	−0.210183	+	0.977662i	$0.567406\pi$
$570$	−27.9168	−1.16930
$571$	−31.0401	−1.29899	−0.649493	−	0.760367i	$-0.725019\pi$
$571$	−31.0401	−1.29899	−0.649493	+	0.760367i	$0.725019\pi$
$572$	−11.4843	−0.480181
$573$	−20.0815	−0.838917
$574$	61.2070	2.55473
$575$	−12.3838	−0.516441
$576$	8.89707	0.370711
$577$	22.7440	0.946845	0.473422	−	0.880835i	$-0.343018\pi$
$577$	22.7440	0.946845	0.473422	+	0.880835i	$0.343018\pi$
$578$	−19.8272	−0.824704
$579$	7.52880	0.312886
$580$	−14.0276	−0.582466
$581$	−0.486632	−0.0201889
$582$	0.984538	0.0408104
$583$	8.21137	0.340080
$584$	30.1900	1.24927
$585$	−16.0325	−0.662861
$586$	11.1070	0.458827
$587$	18.0123	0.743449	0.371725	−	0.928343i	$-0.378767\pi$
$587$	18.0123	0.743449	0.371725	+	0.928343i	$0.378767\pi$
$588$	−4.92251	−0.203001
$589$	−18.9506	−0.780847
$590$	27.2276	1.12094
$591$	−7.97060	−0.327867
$592$	16.3220	0.670832
$593$	−25.7411	−1.05706	−0.528531	−	0.848914i	$-0.677257\pi$
$593$	−25.7411	−1.05706	−0.528531	+	0.848914i	$0.677257\pi$
$594$	7.49021	0.307327
$595$	16.5216	0.677320
$596$	3.06632	0.125601
$597$	−16.6235	−0.680353
$598$	−6.51925	−0.266592
$599$	6.60009	0.269672	0.134836	−	0.990868i	$-0.456949\pi$
$599$	6.60009	0.269672	0.134836	+	0.990868i	$0.456949\pi$
$600$	−29.9851	−1.22414
$601$	23.5244	0.959582	0.479791	−	0.877383i	$-0.340712\pi$
$601$	23.5244	0.959582	0.479791	+	0.877383i	$0.340712\pi$
$602$	30.7317	1.25253
$603$	−12.4984	−0.508974
$604$	−5.82363	−0.236960
$605$	−97.6718	−3.97092
$606$	4.23379	0.171986
$607$	−15.8131	−0.641833	−0.320917	−	0.947107i	$-0.603991\pi$
$607$	−15.8131	−0.641833	−0.320917	+	0.947107i	$0.603991\pi$
$608$	14.7804	0.599423
$609$	−33.5759	−1.36056
$610$	−5.04005	−0.204066
$611$	−33.8941	−1.37121
$612$	0.465621	0.0188216
$613$	26.8396	1.08404	0.542021	−	0.840365i	$-0.317659\pi$
$613$	26.8396	1.08404	0.542021	+	0.840365i	$0.317659\pi$
$614$	−8.63676	−0.348551
$615$	−45.0183	−1.81531
$616$	−77.5751	−3.12559
$617$	7.53575	0.303378	0.151689	−	0.988428i	$-0.451529\pi$
$617$	7.53575	0.303378	0.151689	+	0.988428i	$0.451529\pi$
$618$	−8.17906	−0.329010
$619$	−5.57969	−0.224267	−0.112133	−	0.993693i	$-0.535768\pi$
$619$	−5.57969	−0.224267	−0.112133	+	0.993693i	$0.535768\pi$
$620$	−5.71615	−0.229566
$621$	−1.26052	−0.0505829
$622$	−36.7840	−1.47490
$623$	−18.3417	−0.734843
$624$	−11.9764	−0.479442
$625$	22.3965	0.895860
$626$	−5.39601	−0.215668
$627$	35.2045	1.40593
$628$	−7.92574	−0.316272
$629$	5.77775	0.230374
$630$	−20.1553	−0.803005
$631$	−30.2830	−1.20555	−0.602774	−	0.797912i	$-0.705938\pi$
$631$	−30.2830	−1.20555	−0.602774	+	0.797912i	$0.705938\pi$
$632$	37.2030	1.47986
$633$	−10.4919	−0.417017
$634$	2.41096	0.0957513
$635$	60.6646	2.40740
$636$	−0.622717	−0.0246923
$637$	44.8193	1.77581
$638$	−59.6701	−2.36236
$639$	8.17083	0.323233
$640$	−23.0505	−0.911149
$641$	46.2374	1.82627	0.913133	−	0.407662i	$-0.133656\pi$
$641$	46.2374	1.82627	0.913133	+	0.407662i	$0.133656\pi$
$642$	6.91615	0.272958
$643$	−23.7884	−0.938123	−0.469061	−	0.883166i	$-0.655408\pi$
$643$	−23.7884	−0.938123	−0.469061	+	0.883166i	$0.655408\pi$
$644$	2.42967	0.0957424
$645$	−22.6034	−0.890009
$646$	−7.38205	−0.290443
$647$	6.97536	0.274230	0.137115	−	0.990555i	$-0.456217\pi$
$647$	6.97536	0.274230	0.137115	+	0.990555i	$0.456217\pi$
$648$	−3.05211	−0.119898
$649$	−34.3354	−1.34778
$650$	50.8104	1.99295
$651$	−13.6819	−0.536237
$652$	3.98651	0.156124
$653$	22.9930	0.899784	0.449892	−	0.893083i	$-0.351462\pi$
$653$	22.9930	0.899784	0.449892	+	0.893083i	$0.351462\pi$
$654$	8.59983	0.336280
$655$	−17.5049	−0.683974
$656$	−33.6292	−1.31300
$657$	−9.89152	−0.385905
$658$	−42.6101	−1.66112
$659$	22.9604	0.894410	0.447205	−	0.894431i	$-0.352419\pi$
$659$	22.9604	0.894410	0.447205	+	0.894431i	$0.352419\pi$
$660$	10.6189	0.413340
$661$	−19.5894	−0.761940	−0.380970	−	0.924587i	$-0.624410\pi$
$661$	−19.5894	−0.761940	−0.380970	+	0.924587i	$0.624410\pi$
$662$	4.96975	0.193155
$663$	−4.23947	−0.164648
$664$	0.352400	0.0136758
$665$	−94.7313	−3.67352
$666$	−7.04847	−0.273122
$667$	10.0418	0.388821
$668$	−11.6463	−0.450609
$669$	−1.00000	−0.0386622
$670$	59.7693	2.30909
$671$	6.35578	0.245362
$672$	10.6711	0.411646
$673$	37.5925	1.44908	0.724542	−	0.689231i	$-0.242051\pi$
$673$	37.5925	1.44908	0.724542	+	0.689231i	$0.242051\pi$
$674$	−30.2002	−1.16327
$675$	9.82438	0.378140
$676$	−1.98437	−0.0763218
$677$	−7.02104	−0.269841	−0.134920	−	0.990856i	$-0.543078\pi$
$677$	−7.02104	−0.269841	−0.134920	+	0.990856i	$0.543078\pi$
$678$	5.41838	0.208092
$679$	3.34088	0.128211
$680$	−11.9643	−0.458811
$681$	−11.2013	−0.429236
$682$	−24.3151	−0.931074
$683$	13.7069	0.524478	0.262239	−	0.965003i	$-0.415539\pi$
$683$	13.7069	0.524478	0.262239	+	0.965003i	$0.415539\pi$
$684$	−2.66977	−0.102081
$685$	84.9292	3.24498
$686$	19.7012	0.752194
$687$	−0.976014	−0.0372373
$688$	−16.8850	−0.643736
$689$	5.66983	0.216003
$690$	6.02801	0.229483
$691$	48.0004	1.82602	0.913010	−	0.407936i	$-0.133751\pi$
$691$	48.0004	1.82602	0.913010	+	0.407936i	$0.133751\pi$
$692$	2.07246	0.0787833
$693$	25.4169	0.965507
$694$	43.3741	1.64646
$695$	−18.7810	−0.712404
$696$	24.3144	0.921635
$697$	−11.9042	−0.450904
$698$	−4.42494	−0.167486
$699$	−22.5946	−0.854605
$700$	−18.9366	−0.715736
$701$	−8.92817	−0.337212	−0.168606	−	0.985684i	$-0.553927\pi$
$701$	−8.92817	−0.337212	−0.168606	+	0.985684i	$0.553927\pi$
$702$	5.17187	0.195200
$703$	−33.1283	−1.24946
$704$	53.6543	2.02217
$705$	31.3401	1.18034
$706$	7.25160	0.272917
$707$	14.3667	0.540316
$708$	2.60386	0.0978590
$709$	−10.4365	−0.391950	−0.195975	−	0.980609i	$-0.562787\pi$
$709$	−10.4365	−0.391950	−0.195975	+	0.980609i	$0.562787\pi$
$710$	−39.0742	−1.46643
$711$	−12.1893	−0.457134
$712$	13.2823	0.497777
$713$	4.09197	0.153246
$714$	−5.32967	−0.199458
$715$	−96.6848	−3.61580
$716$	7.54863	0.282106
$717$	−4.77427	−0.178298
$718$	9.33237	0.348281
$719$	−15.0662	−0.561875	−0.280938	−	0.959726i	$-0.590645\pi$
$719$	−15.0662	−0.561875	−0.280938	+	0.959726i	$0.590645\pi$
$720$	11.0740	0.412703
$721$	−27.7544	−1.03363
$722$	18.7282	0.696993
$723$	−10.6322	−0.395414
$724$	−5.68648	−0.211336
$725$	−78.2651	−2.90669
$726$	31.5077	1.16936
$727$	29.1221	1.08008	0.540039	−	0.841640i	$-0.318410\pi$
$727$	29.1221	1.08008	0.540039	+	0.841640i	$0.318410\pi$
$728$	−53.5644	−1.98523
$729$	1.00000	0.0370370
$730$	47.3028	1.75076
$731$	−5.97704	−0.221069
$732$	−0.481997	−0.0178151
$733$	−30.4951	−1.12636	−0.563181	−	0.826333i	$-0.690423\pi$
$733$	−30.4951	−1.12636	−0.563181	+	0.826333i	$0.690423\pi$
$734$	32.6493	1.20511
$735$	−41.4421	−1.52861
$736$	−3.19149	−0.117640
$737$	−75.3723	−2.77638
$738$	14.5223	0.534575
$739$	−38.6662	−1.42236	−0.711179	−	0.703011i	$-0.751839\pi$
$739$	−38.6662	−1.42236	−0.711179	+	0.703011i	$0.751839\pi$
$740$	−9.99263	−0.367336
$741$	24.3082	0.892984
$742$	7.12784	0.261671
$743$	−30.4187	−1.11595	−0.557977	−	0.829856i	$-0.688422\pi$
$743$	−30.4187	−1.11595	−0.557977	+	0.829856i	$0.688422\pi$
$744$	9.90793	0.363242
$745$	25.8150	0.945790
$746$	7.32291	0.268111
$747$	−0.115461	−0.00422450
$748$	2.80796	0.102669
$749$	23.4689	0.857534
$750$	−23.0709	−0.842432
$751$	25.3352	0.924494	0.462247	−	0.886751i	$-0.347043\pi$
$751$	25.3352	0.924494	0.462247	+	0.886751i	$0.347043\pi$
$752$	23.4115	0.853728
$753$	17.1203	0.623900
$754$	−41.2013	−1.50046
$755$	−49.0286	−1.78433
$756$	−1.92751	−0.0701029
$757$	13.5289	0.491717	0.245858	−	0.969306i	$-0.420930\pi$
$757$	13.5289	0.491717	0.245858	+	0.969306i	$0.420930\pi$
$758$	18.3816	0.667650
$759$	−7.60165	−0.275922
$760$	68.6008	2.48841
$761$	18.7406	0.679346	0.339673	−	0.940544i	$-0.389684\pi$
$761$	18.7406	0.679346	0.339673	+	0.940544i	$0.389684\pi$
$762$	−19.5697	−0.708934
$763$	29.1822	1.05647
$764$	9.18394	0.332263
$765$	3.92002	0.141729
$766$	−5.82041	−0.210300
$767$	−23.7081	−0.856050
$768$	−10.3583	−0.373774
$769$	−9.27248	−0.334374	−0.167187	−	0.985925i	$-0.553468\pi$
$769$	−9.27248	−0.334374	−0.167187	+	0.985925i	$0.553468\pi$
$770$	−121.548	−4.38027
$771$	15.3556	0.553017
$772$	−3.44317	−0.123922
$773$	1.56721	0.0563688	0.0281844	−	0.999603i	$-0.491027\pi$
$773$	1.56721	0.0563688	0.0281844	+	0.999603i	$0.491027\pi$
$774$	7.29159	0.262091
$775$	−31.8924	−1.14561
$776$	−2.41934	−0.0868492
$777$	−23.9179	−0.858049
$778$	30.9763	1.11055
$779$	68.2561	2.44553
$780$	7.33218	0.262534
$781$	49.2747	1.76319
$782$	1.59399	0.0570010
$783$	−7.96642	−0.284697
$784$	−30.9578	−1.10563
$785$	−66.7260	−2.38155
$786$	5.64687	0.201417
$787$	8.99683	0.320703	0.160351	−	0.987060i	$-0.448737\pi$
$787$	8.99683	0.320703	0.160351	+	0.987060i	$0.448737\pi$
$788$	3.64522	0.129856
$789$	14.3234	0.509927
$790$	58.2911	2.07390
$791$	18.3865	0.653747
$792$	−18.4059	−0.654027
$793$	4.38857	0.155843
$794$	26.8286	0.952110
$795$	−5.24259	−0.185936
$796$	7.60246	0.269462
$797$	19.0662	0.675358	0.337679	−	0.941261i	$-0.390358\pi$
$797$	19.0662	0.675358	0.337679	+	0.941261i	$0.390358\pi$
$798$	30.5591	1.08178
$799$	8.28729	0.293183
$800$	24.8742	0.879436
$801$	−4.35186	−0.153765
$802$	−28.5415	−1.00784
$803$	−59.6514	−2.10505
$804$	5.71593	0.201585
$805$	20.4551	0.720949
$806$	−16.7892	−0.591375
$807$	−10.0480	−0.353706
$808$	−10.4038	−0.366005
$809$	20.1912	0.709884	0.354942	−	0.934888i	$-0.384501\pi$
$809$	20.1912	0.709884	0.354942	+	0.934888i	$0.384501\pi$
$810$	−4.78216	−0.168028
$811$	15.4880	0.543857	0.271929	−	0.962317i	$-0.412339\pi$
$811$	15.4880	0.543857	0.271929	+	0.962317i	$0.412339\pi$
$812$	15.3554	0.538868
$813$	−7.44161	−0.260989
$814$	−42.5062	−1.48984
$815$	33.5621	1.17563
$816$	2.92830	0.102511
$817$	34.2710	1.19899
$818$	1.34454	0.0470108
$819$	17.5500	0.613245
$820$	20.5884	0.718977
$821$	24.6436	0.860069	0.430034	−	0.902813i	$-0.358501\pi$
$821$	24.6436	0.860069	0.430034	+	0.902813i	$0.358501\pi$
$822$	−27.3971	−0.955584
$823$	−25.0087	−0.871749	−0.435875	−	0.900007i	$-0.643561\pi$
$823$	−25.0087	−0.871749	−0.435875	+	0.900007i	$0.643561\pi$
$824$	20.0987	0.700170
$825$	59.2465	2.06270
$826$	−29.8047	−1.03704
$827$	9.12496	0.317306	0.158653	−	0.987334i	$-0.449285\pi$
$827$	9.12496	0.317306	0.158653	+	0.987334i	$0.449285\pi$
$828$	0.576478	0.0200340
$829$	47.0763	1.63503	0.817514	−	0.575909i	$-0.195352\pi$
$829$	47.0763	1.63503	0.817514	+	0.575909i	$0.195352\pi$
$830$	0.552154	0.0191656
$831$	1.77535	0.0615861
$832$	37.0475	1.28439
$833$	−10.9586	−0.379692
$834$	6.05852	0.209789
$835$	−98.0491	−3.39313
$836$	−16.1002	−0.556837
$837$	−3.24626	−0.112207
$838$	36.5254	1.26175
$839$	−28.1790	−0.972848	−0.486424	−	0.873723i	$-0.661699\pi$
$839$	−28.1790	−0.972848	−0.486424	+	0.873723i	$0.661699\pi$
$840$	49.5282	1.70889
$841$	34.4639	1.18841
$842$	−3.80360	−0.131081
$843$	−13.5754	−0.467561
$844$	4.79831	0.165165
$845$	−16.7062	−0.574710
$846$	−10.1099	−0.347587
$847$	106.917	3.67370
$848$	−3.91628	−0.134486
$849$	−7.44940	−0.255663
$850$	−12.4234	−0.426120
$851$	7.15333	0.245213
$852$	−3.73679	−0.128020
$853$	−13.6133	−0.466110	−0.233055	−	0.972464i	$-0.574872\pi$
$853$	−13.6133	−0.466110	−0.233055	+	0.972464i	$0.574872\pi$
$854$	5.51711	0.188791
$855$	−22.4765	−0.768681
$856$	−16.9953	−0.580887
$857$	2.34790	0.0802029	0.0401014	−	0.999196i	$-0.487232\pi$
$857$	2.34790	0.0802029	0.0401014	+	0.999196i	$0.487232\pi$
$858$	31.1893	1.06479
$859$	20.3279	0.693577	0.346789	−	0.937943i	$-0.387272\pi$
$859$	20.3279	0.693577	0.346789	+	0.937943i	$0.387272\pi$
$860$	10.3373	0.352499
$861$	49.2793	1.67944
$862$	−36.9455	−1.25837
$863$	9.80337	0.333711	0.166855	−	0.985981i	$-0.446639\pi$
$863$	9.80337	0.333711	0.166855	+	0.985981i	$0.446639\pi$
$864$	2.53189	0.0861365
$865$	17.4479	0.593245
$866$	−10.5996	−0.360188
$867$	−15.9634	−0.542146
$868$	6.25720	0.212383
$869$	−73.5082	−2.49360
$870$	38.0967	1.29160
$871$	−52.0434	−1.76342
$872$	−21.1326	−0.715642
$873$	0.792677	0.0268281
$874$	−9.13959	−0.309151
$875$	−78.2877	−2.64661
$876$	4.52372	0.152842
$877$	5.95102	0.200952	0.100476	−	0.994940i	$-0.467964\pi$
$877$	5.95102	0.200952	0.100476	+	0.994940i	$0.467964\pi$
$878$	49.4985	1.67049
$879$	8.94256	0.301625
$880$	66.7824	2.25123
$881$	−18.9288	−0.637728	−0.318864	−	0.947800i	$-0.603301\pi$
$881$	−18.9288	−0.637728	−0.318864	+	0.947800i	$0.603301\pi$
$882$	13.3687	0.450148
$883$	−37.7029	−1.26880	−0.634402	−	0.773004i	$-0.718753\pi$
$883$	−37.7029	−1.26880	−0.634402	+	0.773004i	$0.718753\pi$
$884$	1.93885	0.0652107
$885$	21.9216	0.736888
$886$	−26.1434	−0.878303
$887$	−38.6759	−1.29861	−0.649305	−	0.760528i	$-0.724940\pi$
$887$	−38.6759	−1.29861	−0.649305	+	0.760528i	$0.724940\pi$
$888$	17.3204	0.581236
$889$	−66.4067	−2.22721
$890$	20.8113	0.697596
$891$	6.03056	0.202031
$892$	0.457333	0.0153127
$893$	−47.5175	−1.59011
$894$	−8.32761	−0.278517
$895$	63.5511	2.12428
$896$	25.2322	0.842949
$897$	−5.24882	−0.175253
$898$	28.0206	0.935060
$899$	25.8610	0.862514
$900$	−4.49302	−0.149767
$901$	−1.38630	−0.0461844
$902$	87.5779	2.91602
$903$	24.7429	0.823391
$904$	−13.3148	−0.442843
$905$	−47.8739	−1.59138
$906$	15.8160	0.525452
$907$	−27.5035	−0.913238	−0.456619	−	0.889662i	$-0.650940\pi$
$907$	−27.5035	−0.913238	−0.456619	+	0.889662i	$0.650940\pi$
$908$	5.12274	0.170004
$909$	3.40873	0.113061
$910$	−83.9268	−2.78215
$911$	18.9730	0.628603	0.314301	−	0.949323i	$-0.398230\pi$
$911$	18.9730	0.628603	0.314301	+	0.949323i	$0.398230\pi$
$912$	−16.7902	−0.555980
$913$	−0.696296	−0.0230440
$914$	6.16086	0.203783
$915$	−4.05788	−0.134149
$916$	0.446364	0.0147483
$917$	19.1618	0.632778
$918$	−1.26455	−0.0417364
$919$	−21.1146	−0.696507	−0.348254	−	0.937400i	$-0.613225\pi$
$919$	−21.1146	−0.696507	−0.348254	+	0.937400i	$0.613225\pi$
$920$	−14.8128	−0.488365
$921$	−6.95368	−0.229131
$922$	8.81740	0.290386
$923$	34.0234	1.11990
$924$	−11.6240	−0.382401
$925$	−55.7524	−1.83313
$926$	5.96847	0.196136
$927$	−6.58517	−0.216285
$928$	−20.1701	−0.662115
$929$	23.3514	0.766134	0.383067	−	0.923721i	$-0.374868\pi$
$929$	23.3514	0.766134	0.383067	+	0.923721i	$0.374868\pi$
$930$	15.5241	0.509056
$931$	62.8339	2.05930
$932$	10.3332	0.338477
$933$	−29.6158	−0.969577
$934$	−10.8282	−0.354311
$935$	23.6399	0.773108
$936$	−12.7090	−0.415407
$937$	−12.6313	−0.412645	−0.206323	−	0.978484i	$-0.566150\pi$
$937$	−12.6313	−0.412645	−0.206323	+	0.978484i	$0.566150\pi$
$938$	−65.4266	−2.13625
$939$	−4.34447	−0.141776
$940$	−14.3329	−0.467487
$941$	53.4167	1.74134	0.870668	−	0.491872i	$-0.163687\pi$
$941$	53.4167	1.74134	0.870668	+	0.491872i	$0.163687\pi$
$942$	21.5250	0.701322
$943$	−14.7384	−0.479948
$944$	16.3757	0.532985
$945$	−16.2275	−0.527882
$946$	43.9724	1.42967
$947$	58.5627	1.90303	0.951516	−	0.307598i	$-0.0995253\pi$
$947$	58.5627	1.90303	0.951516	+	0.307598i	$0.0995253\pi$
$948$	5.57456	0.181053
$949$	−41.1884	−1.33703
$950$	71.2331	2.31111
$951$	1.94112	0.0629453
$952$	13.0968	0.424469
$953$	−10.2980	−0.333587	−0.166793	−	0.985992i	$-0.553341\pi$
$953$	−10.2980	−0.333587	−0.166793	+	0.985992i	$0.553341\pi$
$954$	1.69120	0.0547545
$955$	77.3187	2.50197
$956$	2.18343	0.0706172
$957$	−48.0420	−1.55298
$958$	−5.00530	−0.161714
$959$	−92.9679	−3.00209
$960$	−34.2559	−1.10560
$961$	−20.4618	−0.660059
$962$	−29.3499	−0.946279
$963$	5.56837	0.179438
$964$	4.86244	0.156609
$965$	−28.9877	−0.933147
$966$	−6.59858	−0.212306
$967$	−22.2539	−0.715636	−0.357818	−	0.933791i	$-0.616479\pi$
$967$	−22.2539	−0.715636	−0.357818	+	0.933791i	$0.616479\pi$
$968$	−77.4250	−2.48853
$969$	−5.94348	−0.190932
$970$	−3.79071	−0.121712
$971$	−1.22732	−0.0393865	−0.0196932	−	0.999806i	$-0.506269\pi$
$971$	−1.22732	−0.0393865	−0.0196932	+	0.999806i	$0.506269\pi$
$972$	−0.457333	−0.0146690
$973$	20.5586	0.659080
$974$	−17.6070	−0.564164
$975$	40.9088	1.31013
$976$	−3.03129	−0.0970291
$977$	−10.0999	−0.323125	−0.161562	−	0.986862i	$-0.551653\pi$
$977$	−10.0999	−0.323125	−0.161562	+	0.986862i	$0.551653\pi$
$978$	−10.8267	−0.346200
$979$	−26.2441	−0.838766
$980$	18.9529	0.605427
$981$	6.92395	0.221065
$982$	36.1844	1.15469
$983$	15.0078	0.478675	0.239337	−	0.970936i	$-0.423070\pi$
$983$	15.0078	0.478675	0.239337	+	0.970936i	$0.423070\pi$
$984$	−35.6862	−1.13764
$985$	30.6887	0.977825
$986$	10.0739	0.320820
$987$	−34.3065	−1.09199
$988$	−11.1170	−0.353677
$989$	−7.40007	−0.235309
$990$	−28.8391	−0.916568
$991$	−7.47991	−0.237607	−0.118804	−	0.992918i	$-0.537906\pi$
$991$	−7.47991	−0.237607	−0.118804	+	0.992918i	$0.537906\pi$
$992$	−8.21915	−0.260958
$993$	4.00127	0.126977
$994$	42.7727	1.35667
$995$	64.0044	2.02908
$996$	0.0528043	0.00167317
$997$	8.57285	0.271505	0.135753	−	0.990743i	$-0.456655\pi$
$997$	8.57285	0.271505	0.135753	+	0.990743i	$0.456655\pi$
$998$	14.5099	0.459302
$999$	−5.67490	−0.179546

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	669.2.a.i.1.10	✓	14
3.2	odd	2		2007.2.a.n.1.5		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
669.2.a.i.1.10	✓	14	1.1	even	1	trivial
2007.2.a.n.1.5		14	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 669 = 3 \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	669.a (trivial)

\(f(q)\)	\(=\)	\(q+1.24204 q^{2} +1.00000 q^{3} -0.457333 q^{4} -3.85024 q^{5} +1.24204 q^{6} +4.21468 q^{7} -3.05211 q^{8} +1.00000 q^{9} -4.78216 q^{10} +6.03056 q^{11} -0.457333 q^{12} +4.16401 q^{13} +5.23480 q^{14} -3.85024 q^{15} -2.87618 q^{16} -1.01812 q^{17} +1.24204 q^{18} +5.83769 q^{19} +1.76084 q^{20} +4.21468 q^{21} +7.49021 q^{22} -1.26052 q^{23} -3.05211 q^{24} +9.82438 q^{25} +5.17187 q^{26} +1.00000 q^{27} -1.92751 q^{28} -7.96642 q^{29} -4.78216 q^{30} -3.24626 q^{31} +2.53189 q^{32} +6.03056 q^{33} -1.26455 q^{34} -16.2275 q^{35} -0.457333 q^{36} -5.67490 q^{37} +7.25065 q^{38} +4.16401 q^{39} +11.7514 q^{40} +11.6923 q^{41} +5.23480 q^{42} +5.87065 q^{43} -2.75798 q^{44} -3.85024 q^{45} -1.56562 q^{46} -8.13978 q^{47} -2.87618 q^{48} +10.7635 q^{49} +12.2023 q^{50} -1.01812 q^{51} -1.90434 q^{52} +1.36163 q^{53} +1.24204 q^{54} -23.2191 q^{55} -12.8637 q^{56} +5.83769 q^{57} -9.89462 q^{58} -5.69357 q^{59} +1.76084 q^{60} +1.05393 q^{61} -4.03198 q^{62} +4.21468 q^{63} +8.89707 q^{64} -16.0325 q^{65} +7.49021 q^{66} -12.4984 q^{67} +0.465621 q^{68} -1.26052 q^{69} -20.1553 q^{70} +8.17083 q^{71} -3.05211 q^{72} -9.89152 q^{73} -7.04847 q^{74} +9.82438 q^{75} -2.66977 q^{76} +25.4169 q^{77} +5.17187 q^{78} -12.1893 q^{79} +11.0740 q^{80} +1.00000 q^{81} +14.5223 q^{82} -0.115461 q^{83} -1.92751 q^{84} +3.92002 q^{85} +7.29159 q^{86} -7.96642 q^{87} -18.4059 q^{88} -4.35186 q^{89} -4.78216 q^{90} +17.5500 q^{91} +0.576478 q^{92} -3.24626 q^{93} -10.1099 q^{94} -22.4765 q^{95} +2.53189 q^{96} +0.792677 q^{97} +13.3687 q^{98} +6.03056 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 14 q^{3} + 16 q^{4} + q^{5} + 14 q^{7} - 3 q^{8} + 14 q^{9} + 4 q^{10} + 13 q^{11} + 16 q^{12} + 10 q^{13} + q^{15} + 12 q^{16} + q^{17} + 32 q^{19} - 10 q^{20} + 14 q^{21} - 3 q^{22} - 7 q^{23}+ \cdots + 13 q^{99}+O(q^{100}) \) 14 * q + 14 * q^3 + 16 * q^4 + q^5 + 14 * q^7 - 3 * q^8 + 14 * q^9 + 4 * q^10 + 13 * q^11 + 16 * q^12 + 10 * q^13 + q^15 + 12 * q^16 + q^17 + 32 * q^19 - 10 * q^20 + 14 * q^21 - 3 * q^22 - 7 * q^23 - 3 * q^24 + 25 * q^25 - 3 * q^26 + 14 * q^27 + 28 * q^28 - q^29 + 4 * q^30 + 18 * q^31 - 16 * q^32 + 13 * q^33 + q^34 - 12 * q^35 + 16 * q^36 + 16 * q^37 - 31 * q^38 + 10 * q^39 + q^40 + 6 * q^41 + 15 * q^43 - 12 * q^44 + q^45 + 6 * q^46 - 21 * q^47 + 12 * q^48 + 28 * q^49 - 32 * q^50 + q^51 - 7 * q^52 - 17 * q^53 - q^55 - 33 * q^56 + 32 * q^57 - 27 * q^58 + 12 * q^59 - 10 * q^60 + 28 * q^61 - 16 * q^62 + 14 * q^63 + 3 * q^64 - 4 * q^65 - 3 * q^66 + 17 * q^67 + 5 * q^68 - 7 * q^69 - 14 * q^70 - 3 * q^71 - 3 * q^72 + 17 * q^73 - 16 * q^74 + 25 * q^75 + 14 * q^76 - 14 * q^77 - 3 * q^78 + 23 * q^79 - 36 * q^80 + 14 * q^81 + 5 * q^82 + 18 * q^83 + 28 * q^84 - 11 * q^85 + 9 * q^86 - q^87 - 35 * q^88 + 10 * q^89 + 4 * q^90 + 16 * q^91 - 45 * q^92 + 18 * q^93 - 43 * q^94 - 20 * q^95 - 16 * q^96 + 29 * q^97 - 19 * q^98 + 13 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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