LMFDB - Embedded newform 1682.2.a.q.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1682,2,Mod(1,1682)] 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("1682.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1682, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1682 = 2 \cdot 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1682.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,-6,-2,6,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$13.4308376200$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.13716913.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 13x^{4} + 17x^{3} + 52x^{2} - 32x - 64 $ x^6 - 2x^5 - 13x^4 + 17x^3 + 52x^2 - 32*x - 64
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 58)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.41744$ of defining polynomial
Character	$\chi$	$=$	1682.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.17047 q^{3} +1.00000 q^{4} -2.97240 q^{5} -1.17047 q^{6} +0.520906 q^{7} -1.00000 q^{8} -1.63001 q^{9} +2.97240 q^{10} +0.649559 q^{11} +1.17047 q^{12} +0.493598 q^{13} -0.520906 q^{14} -3.47909 q^{15} +1.00000 q^{16} +7.42032 q^{17} +1.63001 q^{18} -6.63001 q^{19} -2.97240 q^{20} +0.609702 q^{21} -0.649559 q^{22} +7.61793 q^{23} -1.17047 q^{24} +3.83518 q^{25} -0.493598 q^{26} -5.41927 q^{27} +0.520906 q^{28} +3.47909 q^{30} -5.98045 q^{31} -1.00000 q^{32} +0.760286 q^{33} -7.42032 q^{34} -1.54834 q^{35} -1.63001 q^{36} +2.48233 q^{37} +6.63001 q^{38} +0.577740 q^{39} +2.97240 q^{40} -7.82245 q^{41} -0.609702 q^{42} -0.649559 q^{43} +0.649559 q^{44} +4.84505 q^{45} -7.61793 q^{46} -8.79595 q^{47} +1.17047 q^{48} -6.72866 q^{49} -3.83518 q^{50} +8.68522 q^{51} +0.493598 q^{52} -1.15121 q^{53} +5.41927 q^{54} -1.93075 q^{55} -0.520906 q^{56} -7.76020 q^{57} -5.31686 q^{59} -3.47909 q^{60} -9.69702 q^{61} +5.98045 q^{62} -0.849083 q^{63} +1.00000 q^{64} -1.46717 q^{65} -0.760286 q^{66} +4.52091 q^{67} +7.42032 q^{68} +8.91652 q^{69} +1.54834 q^{70} -14.1448 q^{71} +1.63001 q^{72} -8.74045 q^{73} -2.48233 q^{74} +4.48894 q^{75} -6.63001 q^{76} +0.338359 q^{77} -0.577740 q^{78} -4.12847 q^{79} -2.97240 q^{80} -1.45303 q^{81} +7.82245 q^{82} +4.28153 q^{83} +0.609702 q^{84} -22.0562 q^{85} +0.649559 q^{86} -0.649559 q^{88} +1.19266 q^{89} -4.84505 q^{90} +0.257118 q^{91} +7.61793 q^{92} -6.99991 q^{93} +8.79595 q^{94} +19.7071 q^{95} -1.17047 q^{96} -11.7329 q^{97} +6.72866 q^{98} -1.05879 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{2} - 2 q^{3} + 6 q^{4} + 2 q^{6} - 3 q^{7} - 6 q^{8} + 12 q^{9} + q^{11} - 2 q^{12} - 3 q^{13} + 3 q^{14} - 27 q^{15} + 6 q^{16} + 6 q^{17} - 12 q^{18} - 18 q^{19} + 10 q^{21} - q^{22} - 14 q^{23}+ \cdots + 4 q^{99}+O(q^{100}) $ 6 * q - 6 * q^2 - 2 * q^3 + 6 * q^4 + 2 * q^6 - 3 * q^7 - 6 * q^8 + 12 * q^9 + q^11 - 2 * q^12 - 3 * q^13 + 3 * q^14 - 27 * q^15 + 6 * q^16 + 6 * q^17 - 12 * q^18 - 18 * q^19 + 10 * q^21 - q^22 - 14 * q^23 + 2 * q^24 + 4 * q^25 + 3 * q^26 - 23 * q^27 - 3 * q^28 + 27 * q^30 - 17 * q^31 - 6 * q^32 + 20 * q^33 - 6 * q^34 - 9 * q^35 + 12 * q^36 - 12 * q^37 + 18 * q^38 - 10 * q^39 + 15 * q^41 - 10 * q^42 - q^43 + q^44 + 18 * q^45 + 14 * q^46 - 8 * q^47 - 2 * q^48 + q^49 - 4 * q^50 - 11 * q^51 - 3 * q^52 + 3 * q^53 + 23 * q^54 - 18 * q^55 + 3 * q^56 - 19 * q^57 + 19 * q^59 - 27 * q^60 - 15 * q^61 + 17 * q^62 - 33 * q^63 + 6 * q^64 + 11 * q^65 - 20 * q^66 + 21 * q^67 + 6 * q^68 + 11 * q^69 + 9 * q^70 - 9 * q^71 - 12 * q^72 - 31 * q^73 + 12 * q^74 + q^75 - 18 * q^76 - 33 * q^77 + 10 * q^78 + 2 * q^79 + 2 * q^81 - 15 * q^82 - 2 * q^83 + 10 * q^84 - 19 * q^85 + q^86 - q^88 - 2 * q^89 - 18 * q^90 - 22 * q^91 - 14 * q^92 + q^93 + 8 * q^94 + 18 * q^95 + 2 * q^96 - 43 * q^97 - q^98 + 4 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.17047	0.675768	0.337884	−	0.941188i	$-0.390289\pi$
$3$	1.17047	0.675768	0.337884	+	0.941188i	$0.390289\pi$
$4$	1.00000	0.500000
$5$	−2.97240	−1.32930	−0.664649	−	0.747155i	$-0.731419\pi$
$5$	−2.97240	−1.32930	−0.664649	+	0.747155i	$0.731419\pi$
$6$	−1.17047	−0.477840
$7$	0.520906	0.196884	0.0984420	−	0.995143i	$-0.468614\pi$
$7$	0.520906	0.196884	0.0984420	+	0.995143i	$0.468614\pi$
$8$	−1.00000	−0.353553
$9$	−1.63001	−0.543337
$10$	2.97240	0.939956
$11$	0.649559	0.195849	0.0979247	−	0.995194i	$-0.468780\pi$
$11$	0.649559	0.195849	0.0979247	+	0.995194i	$0.468780\pi$
$12$	1.17047	0.337884
$13$	0.493598	0.136900	0.0684498	−	0.997655i	$-0.478195\pi$
$13$	0.493598	0.136900	0.0684498	+	0.997655i	$0.478195\pi$
$14$	−0.520906	−0.139218
$15$	−3.47909	−0.898298
$16$	1.00000	0.250000
$17$	7.42032	1.79969	0.899846	−	0.436208i	$-0.143679\pi$
$17$	7.42032	1.79969	0.899846	+	0.436208i	$0.143679\pi$
$18$	1.63001	0.384197
$19$	−6.63001	−1.52103	−0.760514	−	0.649321i	$-0.775053\pi$
$19$	−6.63001	−1.52103	−0.760514	+	0.649321i	$0.775053\pi$
$20$	−2.97240	−0.664649
$21$	0.609702	0.133048
$22$	−0.649559	−0.138486
$23$	7.61793	1.58845	0.794224	−	0.607625i	$-0.207878\pi$
$23$	7.61793	1.58845	0.794224	+	0.607625i	$0.207878\pi$
$24$	−1.17047	−0.238920
$25$	3.83518	0.767036
$26$	−0.493598	−0.0968026
$27$	−5.41927	−1.04294
$28$	0.520906	0.0984420
$29$	0	0
$29$	0	0
$30$	3.47909	0.635193
$31$	−5.98045	−1.07412	−0.537060	−	0.843544i	$-0.680465\pi$
$31$	−5.98045	−1.07412	−0.537060	+	0.843544i	$0.680465\pi$
$32$	−1.00000	−0.176777
$33$	0.760286	0.132349
$34$	−7.42032	−1.27257
$35$	−1.54834	−0.261718
$36$	−1.63001	−0.271669
$37$	2.48233	0.408092	0.204046	−	0.978961i	$-0.434591\pi$
$37$	2.48233	0.408092	0.204046	+	0.978961i	$0.434591\pi$
$38$	6.63001	1.07553
$39$	0.577740	0.0925124
$40$	2.97240	0.469978
$41$	−7.82245	−1.22166	−0.610830	−	0.791761i	$-0.709164\pi$
$41$	−7.82245	−1.22166	−0.610830	+	0.791761i	$0.709164\pi$
$42$	−0.609702	−0.0940791
$43$	−0.649559	−0.0990568	−0.0495284	−	0.998773i	$-0.515772\pi$
$43$	−0.649559	−0.0990568	−0.0495284	+	0.998773i	$0.515772\pi$
$44$	0.649559	0.0979247
$45$	4.84505	0.722257
$46$	−7.61793	−1.12320
$47$	−8.79595	−1.28302	−0.641511	−	0.767114i	$-0.721692\pi$
$47$	−8.79595	−1.28302	−0.641511	+	0.767114i	$0.721692\pi$
$48$	1.17047	0.168942
$49$	−6.72866	−0.961237
$50$	−3.83518	−0.542376
$51$	8.68522	1.21617
$52$	0.493598	0.0684498
$53$	−1.15121	−0.158130	−0.0790652	−	0.996869i	$-0.525194\pi$
$53$	−1.15121	−0.158130	−0.0790652	+	0.996869i	$0.525194\pi$
$54$	5.41927	0.737469
$55$	−1.93075	−0.260342
$56$	−0.520906	−0.0696090
$57$	−7.76020	−1.02786
$58$	0	0
$59$	−5.31686	−0.692196	−0.346098	−	0.938198i	$-0.612493\pi$
$59$	−5.31686	−0.692196	−0.346098	+	0.938198i	$0.612493\pi$
$60$	−3.47909	−0.449149
$61$	−9.69702	−1.24158	−0.620788	−	0.783979i	$-0.713187\pi$
$61$	−9.69702	−1.24158	−0.620788	+	0.783979i	$0.713187\pi$
$62$	5.98045	0.759518
$63$	−0.849083	−0.106974
$64$	1.00000	0.125000
$65$	−1.46717	−0.181980
$66$	−0.760286	−0.0935848
$67$	4.52091	0.552317	0.276158	−	0.961112i	$-0.410939\pi$
$67$	4.52091	0.552317	0.276158	+	0.961112i	$0.410939\pi$
$68$	7.42032	0.899846
$69$	8.91652	1.07342
$70$	1.54834	0.185062
$71$	−14.1448	−1.67868	−0.839338	−	0.543610i	$-0.817057\pi$
$71$	−14.1448	−1.67868	−0.839338	+	0.543610i	$0.817057\pi$
$72$	1.63001	0.192099
$73$	−8.74045	−1.02299	−0.511496	−	0.859286i	$-0.670909\pi$
$73$	−8.74045	−1.02299	−0.511496	+	0.859286i	$0.670909\pi$
$74$	−2.48233	−0.288565
$75$	4.48894	0.518339
$76$	−6.63001	−0.760514
$77$	0.338359	0.0385596
$78$	−0.577740	−0.0654161
$79$	−4.12847	−0.464489	−0.232244	−	0.972657i	$-0.574607\pi$
$79$	−4.12847	−0.464489	−0.232244	+	0.972657i	$0.574607\pi$
$80$	−2.97240	−0.332325
$81$	−1.45303	−0.161448
$82$	7.82245	0.863845
$83$	4.28153	0.469958	0.234979	−	0.972000i	$-0.424498\pi$
$83$	4.28153	0.469958	0.234979	+	0.972000i	$0.424498\pi$
$84$	0.609702	0.0665240
$85$	−22.0562	−2.39233
$86$	0.649559	0.0700438
$87$	0	0
$88$	−0.649559	−0.0692432
$89$	1.19266	0.126421	0.0632107	−	0.998000i	$-0.479866\pi$
$89$	1.19266	0.126421	0.0632107	+	0.998000i	$0.479866\pi$
$90$	−4.84505	−0.510713
$91$	0.257118	0.0269533
$92$	7.61793	0.794224
$93$	−6.99991	−0.725857
$94$	8.79595	0.907233
$95$	19.7071	2.02190
$96$	−1.17047	−0.119460
$97$	−11.7329	−1.19129	−0.595647	−	0.803246i	$-0.703104\pi$
$97$	−11.7329	−1.19129	−0.595647	+	0.803246i	$0.703104\pi$
$98$	6.72866	0.679697
$99$	−1.05879	−0.106412
$100$	3.83518	0.383518
$101$	−9.58417	−0.953660	−0.476830	−	0.878995i	$-0.658214\pi$
$101$	−9.58417	−0.953660	−0.476830	+	0.878995i	$0.658214\pi$
$102$	−8.68522	−0.859965
$103$	6.47977	0.638470	0.319235	−	0.947676i	$-0.396574\pi$
$103$	6.47977	0.638470	0.319235	+	0.947676i	$0.396574\pi$
$104$	−0.493598	−0.0484013
$105$	−1.81228	−0.176861
$106$	1.15121	0.111815
$107$	9.78051	0.945517	0.472759	−	0.881192i	$-0.343258\pi$
$107$	9.78051	0.945517	0.472759	+	0.881192i	$0.343258\pi$
$108$	−5.41927	−0.521469
$109$	0.0280295	0.00268474	0.00134237	−	0.999999i	$-0.499573\pi$
$109$	0.0280295	0.00268474	0.00134237	+	0.999999i	$0.499573\pi$
$110$	1.93075	0.184090
$111$	2.90548	0.275776
$112$	0.520906	0.0492210
$113$	4.12883	0.388408	0.194204	−	0.980961i	$-0.437788\pi$
$113$	4.12883	0.388408	0.194204	+	0.980961i	$0.437788\pi$
$114$	7.76020	0.726809
$115$	−22.6436	−2.11152
$116$	0	0
$117$	−0.804571	−0.0743826
$118$	5.31686	0.489456
$119$	3.86529	0.354330
$120$	3.47909	0.317596
$121$	−10.5781	−0.961643
$122$	9.69702	0.877927
$123$	−9.15590	−0.825560
$124$	−5.98045	−0.537060
$125$	3.46232	0.309679
$126$	0.849083	0.0756423
$127$	−15.9153	−1.41226	−0.706128	−	0.708084i	$-0.749560\pi$
$127$	−15.9153	−1.41226	−0.706128	+	0.708084i	$0.749560\pi$
$128$	−1.00000	−0.0883883
$129$	−0.760286	−0.0669395
$130$	1.46717	0.128680
$131$	7.27202	0.635359	0.317679	−	0.948198i	$-0.397096\pi$
$131$	7.27202	0.635359	0.317679	+	0.948198i	$0.397096\pi$
$132$	0.760286	0.0661744
$133$	−3.45361	−0.299466
$134$	−4.52091	−0.390547
$135$	16.1082	1.38638
$136$	−7.42032	−0.636287
$137$	16.6994	1.42673	0.713365	−	0.700793i	$-0.247170\pi$
$137$	16.6994	1.42673	0.713365	+	0.700793i	$0.247170\pi$
$138$	−8.91652	−0.759025
$139$	16.9493	1.43763	0.718813	−	0.695204i	$-0.244686\pi$
$139$	16.9493	1.43763	0.718813	+	0.695204i	$0.244686\pi$
$140$	−1.54834	−0.130859
$141$	−10.2954	−0.867025
$142$	14.1448	1.18700
$143$	0.320621	0.0268117
$144$	−1.63001	−0.135834
$145$	0	0
$146$	8.74045	0.723365
$147$	−7.87566	−0.649573
$148$	2.48233	0.204046
$149$	16.5622	1.35683	0.678415	−	0.734679i	$-0.262667\pi$
$149$	16.5622	1.35683	0.678415	+	0.734679i	$0.262667\pi$
$150$	−4.48894	−0.366521
$151$	−7.51045	−0.611191	−0.305596	−	0.952161i	$-0.598856\pi$
$151$	−7.51045	−0.611191	−0.305596	+	0.952161i	$0.598856\pi$
$152$	6.63001	0.537765
$153$	−12.0952	−0.977839
$154$	−0.338359	−0.0272658
$155$	17.7763	1.42783
$156$	0.577740	0.0462562
$157$	1.40194	0.111887	0.0559436	−	0.998434i	$-0.482183\pi$
$157$	1.40194	0.111887	0.0559436	+	0.998434i	$0.482183\pi$
$158$	4.12847	0.328443
$159$	−1.34745	−0.106860
$160$	2.97240	0.234989
$161$	3.96823	0.312740
$162$	1.45303	0.114161
$163$	7.81821	0.612370	0.306185	−	0.951972i	$-0.400947\pi$
$163$	7.81821	0.612370	0.306185	+	0.951972i	$0.400947\pi$
$164$	−7.82245	−0.610830
$165$	−2.25988	−0.175931
$166$	−4.28153	−0.332311
$167$	−13.5098	−1.04542	−0.522709	−	0.852511i	$-0.675079\pi$
$167$	−13.5098	−1.04542	−0.522709	+	0.852511i	$0.675079\pi$
$168$	−0.609702	−0.0470396
$169$	−12.7564	−0.981259
$170$	22.0562	1.69163
$171$	10.8070	0.826431
$172$	−0.649559	−0.0495284
$173$	0.251083	0.0190895	0.00954476	−	0.999954i	$-0.496962\pi$
$173$	0.251083	0.0190895	0.00954476	+	0.999954i	$0.496962\pi$
$174$	0	0
$175$	1.99777	0.151017
$176$	0.649559	0.0489624
$177$	−6.22319	−0.467764
$178$	−1.19266	−0.0893935
$179$	12.7421	0.952386	0.476193	−	0.879341i	$-0.342016\pi$
$179$	12.7421	0.952386	0.476193	+	0.879341i	$0.342016\pi$
$180$	4.84505	0.361129
$181$	−6.89398	−0.512425	−0.256213	−	0.966620i	$-0.582475\pi$
$181$	−6.89398	−0.512425	−0.256213	+	0.966620i	$0.582475\pi$
$182$	−0.257118	−0.0190589
$183$	−11.3500	−0.839018
$184$	−7.61793	−0.561601
$185$	−7.37848	−0.542476
$186$	6.99991	0.513258
$187$	4.81994	0.352469
$188$	−8.79595	−0.641511
$189$	−2.82293	−0.205338
$190$	−19.7071	−1.42970
$191$	6.02931	0.436265	0.218133	−	0.975919i	$-0.430003\pi$
$191$	6.02931	0.436265	0.218133	+	0.975919i	$0.430003\pi$
$192$	1.17047	0.0844710
$193$	−8.90307	−0.640857	−0.320428	−	0.947273i	$-0.603827\pi$
$193$	−8.90307	−0.640857	−0.320428	+	0.947273i	$0.603827\pi$
$194$	11.7329	0.842372
$195$	−1.71727	−0.122977
$196$	−6.72866	−0.480618
$197$	4.71110	0.335652	0.167826	−	0.985817i	$-0.446325\pi$
$197$	4.71110	0.335652	0.167826	+	0.985817i	$0.446325\pi$
$198$	1.05879	0.0752448
$199$	−19.9563	−1.41466	−0.707332	−	0.706882i	$-0.750101\pi$
$199$	−19.9563	−1.41466	−0.707332	+	0.706882i	$0.750101\pi$
$200$	−3.83518	−0.271188
$201$	5.29156	0.373238
$202$	9.58417	0.674340
$203$	0	0
$204$	8.68522	0.608087
$205$	23.2515	1.62395
$206$	−6.47977	−0.451467
$207$	−12.4173	−0.863063
$208$	0.493598	0.0342249
$209$	−4.30658	−0.297893
$210$	1.81228	0.125059
$211$	4.19086	0.288511	0.144255	−	0.989540i	$-0.453921\pi$
$211$	4.19086	0.288511	0.144255	+	0.989540i	$0.453921\pi$
$212$	−1.15121	−0.0790652
$213$	−16.5560	−1.13440
$214$	−9.78051	−0.668582
$215$	1.93075	0.131676
$216$	5.41927	0.368734
$217$	−3.11525	−0.211477
$218$	−0.0280295	−0.00189840
$219$	−10.2304	−0.691306
$220$	−1.93075	−0.130171
$221$	3.66266	0.246377
$222$	−2.90548	−0.195003
$223$	6.29535	0.421568	0.210784	−	0.977533i	$-0.432398\pi$
$223$	6.29535	0.421568	0.210784	+	0.977533i	$0.432398\pi$
$224$	−0.520906	−0.0348045
$225$	−6.25139	−0.416759
$226$	−4.12883	−0.274646
$227$	−12.7672	−0.847390	−0.423695	−	0.905805i	$-0.639267\pi$
$227$	−12.7672	−0.847390	−0.423695	+	0.905805i	$0.639267\pi$
$228$	−7.76020	−0.513932
$229$	−11.9434	−0.789244	−0.394622	−	0.918843i	$-0.629125\pi$
$229$	−11.9434	−0.789244	−0.394622	+	0.918843i	$0.629125\pi$
$230$	22.6436	1.49307
$231$	0.396038	0.0260574
$232$	0	0
$233$	−11.7030	−0.766690	−0.383345	−	0.923605i	$-0.625228\pi$
$233$	−11.7030	−0.766690	−0.383345	+	0.923605i	$0.625228\pi$
$234$	0.804571	0.0525964
$235$	26.1451	1.70552
$236$	−5.31686	−0.346098
$237$	−4.83222	−0.313887
$238$	−3.86529	−0.250549
$239$	−16.7302	−1.08219	−0.541093	−	0.840963i	$-0.681989\pi$
$239$	−16.7302	−1.08219	−0.541093	+	0.840963i	$0.681989\pi$
$240$	−3.47909	−0.224575
$241$	−7.65233	−0.492930	−0.246465	−	0.969152i	$-0.579269\pi$
$241$	−7.65233	−0.492930	−0.246465	+	0.969152i	$0.579269\pi$
$242$	10.5781	0.679984
$243$	14.5571	0.933837
$244$	−9.69702	−0.620788
$245$	20.0003	1.27777
$246$	9.15590	0.583759
$247$	−3.27256	−0.208228
$248$	5.98045	0.379759
$249$	5.01138	0.317583
$250$	−3.46232	−0.218976
$251$	−1.58064	−0.0997694	−0.0498847	−	0.998755i	$-0.515885\pi$
$251$	−1.58064	−0.0997694	−0.0498847	+	0.998755i	$0.515885\pi$
$252$	−0.849083	−0.0534872
$253$	4.94830	0.311097
$254$	15.9153	0.998616
$255$	−25.8160	−1.61666
$256$	1.00000	0.0625000
$257$	−15.0284	−0.937446	−0.468723	−	0.883345i	$-0.655286\pi$
$257$	−15.0284	−0.937446	−0.468723	+	0.883345i	$0.655286\pi$
$258$	0.760286	0.0473334
$259$	1.29306	0.0803468
$260$	−1.46717	−0.0909902
$261$	0	0
$262$	−7.27202	−0.449267
$263$	−5.17364	−0.319020	−0.159510	−	0.987196i	$-0.550991\pi$
$263$	−5.17364	−0.319020	−0.159510	+	0.987196i	$0.550991\pi$
$264$	−0.760286	−0.0467924
$265$	3.42185	0.210203
$266$	3.45361	0.211755
$267$	1.39596	0.0854316
$268$	4.52091	0.276158
$269$	5.75602	0.350951	0.175475	−	0.984484i	$-0.443854\pi$
$269$	5.75602	0.350951	0.175475	+	0.984484i	$0.443854\pi$
$270$	−16.1082	−0.980317
$271$	−10.9361	−0.664319	−0.332159	−	0.943223i	$-0.607777\pi$
$271$	−10.9361	−0.664319	−0.332159	+	0.943223i	$0.607777\pi$
$272$	7.42032	0.449923
$273$	0.300948	0.0182142
$274$	−16.6994	−1.00885
$275$	2.49118	0.150224
$276$	8.91652	0.536712
$277$	−2.14237	−0.128722	−0.0643612	−	0.997927i	$-0.520501\pi$
$277$	−2.14237	−0.128722	−0.0643612	+	0.997927i	$0.520501\pi$
$278$	−16.9493	−1.01655
$279$	9.74820	0.583610
$280$	1.54834	0.0925312
$281$	9.52397	0.568153	0.284076	−	0.958802i	$-0.408313\pi$
$281$	9.52397	0.568153	0.284076	+	0.958802i	$0.408313\pi$
$282$	10.2954	0.613079
$283$	30.6436	1.82157	0.910785	−	0.412881i	$-0.135477\pi$
$283$	30.6436	1.82157	0.910785	+	0.412881i	$0.135477\pi$
$284$	−14.1448	−0.839338
$285$	23.0664	1.36634
$286$	−0.320621	−0.0189587
$287$	−4.07476	−0.240525
$288$	1.63001	0.0960493
$289$	38.0611	2.23889
$290$	0	0
$291$	−13.7329	−0.805039
$292$	−8.74045	−0.511496
$293$	−5.43961	−0.317785	−0.158893	−	0.987296i	$-0.550792\pi$
$293$	−5.43961	−0.317785	−0.158893	+	0.987296i	$0.550792\pi$
$294$	7.87566	0.459318
$295$	15.8038	0.920135
$296$	−2.48233	−0.144282
$297$	−3.52013	−0.204259
$298$	−16.5622	−0.959423
$299$	3.76020	0.217458
$300$	4.48894	0.259169
$301$	−0.338359	−0.0195027
$302$	7.51045	0.432178
$303$	−11.2179	−0.644453
$304$	−6.63001	−0.380257
$305$	28.8234	1.65043
$306$	12.0952	0.691437
$307$	−32.7348	−1.86828	−0.934138	−	0.356912i	$-0.883830\pi$
$307$	−32.7348	−1.86828	−0.934138	+	0.356912i	$0.883830\pi$
$308$	0.338359	0.0192798
$309$	7.58434	0.431458
$310$	−17.7763	−1.00963
$311$	−21.6580	−1.22811	−0.614057	−	0.789262i	$-0.710464\pi$
$311$	−21.6580	−1.22811	−0.614057	+	0.789262i	$0.710464\pi$
$312$	−0.577740	−0.0327081
$313$	−10.3916	−0.587370	−0.293685	−	0.955902i	$-0.594882\pi$
$313$	−10.3916	−0.587370	−0.293685	+	0.955902i	$0.594882\pi$
$314$	−1.40194	−0.0791161
$315$	2.52382	0.142201
$316$	−4.12847	−0.232244
$317$	25.0536	1.40715	0.703575	−	0.710621i	$-0.251586\pi$
$317$	25.0536	1.40715	0.703575	+	0.710621i	$0.251586\pi$
$318$	1.34745	0.0755611
$319$	0	0
$320$	−2.97240	−0.166162
$321$	11.4477	0.638951
$322$	−3.96823	−0.221141
$323$	−49.1968	−2.73738
$324$	−1.45303	−0.0807238
$325$	1.89304	0.105007
$326$	−7.81821	−0.433011
$327$	0.0328075	0.00181426
$328$	7.82245	0.431922
$329$	−4.58186	−0.252606
$330$	2.25988	0.124402
$331$	7.76962	0.427057	0.213528	−	0.976937i	$-0.431504\pi$
$331$	7.76962	0.427057	0.213528	+	0.976937i	$0.431504\pi$
$332$	4.28153	0.234979
$333$	−4.04622	−0.221732
$334$	13.5098	0.739222
$335$	−13.4380	−0.734194
$336$	0.609702	0.0332620
$337$	20.0182	1.09046	0.545229	−	0.838287i	$-0.316443\pi$
$337$	20.0182	1.09046	0.545229	+	0.838287i	$0.316443\pi$
$338$	12.7564	0.693855
$339$	4.83265	0.262474
$340$	−22.0562	−1.19616
$341$	−3.88466	−0.210366
$342$	−10.8070	−0.584375
$343$	−7.15134	−0.386136
$344$	0.649559	0.0350219
$345$	−26.5035	−1.42690
$346$	−0.251083	−0.0134983
$347$	18.9324	1.01634	0.508172	−	0.861255i	$-0.330321\pi$
$347$	18.9324	1.01634	0.508172	+	0.861255i	$0.330321\pi$
$348$	0	0
$349$	−7.41998	−0.397182	−0.198591	−	0.980082i	$-0.563637\pi$
$349$	−7.41998	−0.397182	−0.198591	+	0.980082i	$0.563637\pi$
$350$	−1.99777	−0.106785
$351$	−2.67494	−0.142778
$352$	−0.649559	−0.0346216
$353$	19.6708	1.04697	0.523484	−	0.852035i	$-0.324632\pi$
$353$	19.6708	1.04697	0.523484	+	0.852035i	$0.324632\pi$
$354$	6.22319	0.330759
$355$	42.0440	2.23146
$356$	1.19266	0.0632107
$357$	4.52418	0.239445
$358$	−12.7421	−0.673439
$359$	−19.6856	−1.03896	−0.519482	−	0.854481i	$-0.673875\pi$
$359$	−19.6856	−1.03896	−0.519482	+	0.854481i	$0.673875\pi$
$360$	−4.84505	−0.255357
$361$	24.9571	1.31353
$362$	6.89398	0.362339
$363$	−12.3813	−0.649848
$364$	0.257118	0.0134767
$365$	25.9801	1.35986
$366$	11.3500	0.593275
$367$	−22.0074	−1.14878	−0.574388	−	0.818583i	$-0.694760\pi$
$367$	−22.0074	−1.14878	−0.574388	+	0.818583i	$0.694760\pi$
$368$	7.61793	0.397112
$369$	12.7507	0.663774
$370$	7.37848	0.383589
$371$	−0.599670	−0.0311333
$372$	−6.99991	−0.362928
$373$	23.8590	1.23537	0.617686	−	0.786425i	$-0.288070\pi$
$373$	23.8590	1.23537	0.617686	+	0.786425i	$0.288070\pi$
$374$	−4.81994	−0.249233
$375$	4.05252	0.209271
$376$	8.79595	0.453616
$377$	0	0
$378$	2.82293	0.145196
$379$	31.8334	1.63517	0.817586	−	0.575807i	$-0.195312\pi$
$379$	31.8334	1.63517	0.817586	+	0.575807i	$0.195312\pi$
$380$	19.7071	1.01095
$381$	−18.6283	−0.954359
$382$	−6.02931	−0.308486
$383$	21.2412	1.08537	0.542687	−	0.839935i	$-0.317407\pi$
$383$	21.2412	1.08537	0.542687	+	0.839935i	$0.317407\pi$
$384$	−1.17047	−0.0597300
$385$	−1.00574	−0.0512573
$386$	8.90307	0.453154
$387$	1.05879	0.0538213
$388$	−11.7329	−0.595647
$389$	−11.0403	−0.559764	−0.279882	−	0.960034i	$-0.590295\pi$
$389$	−11.0403	−0.559764	−0.279882	+	0.960034i	$0.590295\pi$
$390$	1.71727	0.0869576
$391$	56.5275	2.85872
$392$	6.72866	0.339848
$393$	8.51164	0.429355
$394$	−4.71110	−0.237342
$395$	12.2715	0.617444
$396$	−1.05879	−0.0532061
$397$	−21.7506	−1.09163	−0.545815	−	0.837906i	$-0.683780\pi$
$397$	−21.7506	−1.09163	−0.545815	+	0.837906i	$0.683780\pi$
$398$	19.9563	1.00032
$399$	−4.04233	−0.202370
$400$	3.83518	0.191759
$401$	35.6407	1.77981	0.889905	−	0.456146i	$-0.150771\pi$
$401$	35.6407	1.77981	0.889905	+	0.456146i	$0.150771\pi$
$402$	−5.29156	−0.263919
$403$	−2.95194	−0.147047
$404$	−9.58417	−0.476830
$405$	4.31899	0.214612
$406$	0	0
$407$	1.61242	0.0799246
$408$	−8.68522	−0.429983
$409$	15.5639	0.769584	0.384792	−	0.923003i	$-0.374273\pi$
$409$	15.5639	0.769584	0.384792	+	0.923003i	$0.374273\pi$
$410$	−23.2515	−1.14831
$411$	19.5461	0.964139
$412$	6.47977	0.319235
$413$	−2.76958	−0.136282
$414$	12.4173	0.610278
$415$	−12.7264	−0.624715
$416$	−0.493598	−0.0242006
$417$	19.8386	0.971502
$418$	4.30658	0.210642
$419$	−17.4900	−0.854442	−0.427221	−	0.904147i	$-0.640507\pi$
$419$	−17.4900	−0.854442	−0.427221	+	0.904147i	$0.640507\pi$
$420$	−1.81228	−0.0884303
$421$	4.66179	0.227202	0.113601	−	0.993526i	$-0.463762\pi$
$421$	4.66179	0.227202	0.113601	+	0.993526i	$0.463762\pi$
$422$	−4.19086	−0.204008
$423$	14.3375	0.697113
$424$	1.15121	0.0559075
$425$	28.4582	1.38043
$426$	16.5560	0.802139
$427$	−5.05123	−0.244446
$428$	9.78051	0.472759
$429$	0.375276	0.0181185
$430$	−1.93075	−0.0931091
$431$	−23.8990	−1.15118	−0.575588	−	0.817740i	$-0.695227\pi$
$431$	−23.8990	−1.15118	−0.575588	+	0.817740i	$0.695227\pi$
$432$	−5.41927	−0.260735
$433$	3.10772	0.149348	0.0746738	−	0.997208i	$-0.476208\pi$
$433$	3.10772	0.149348	0.0746738	+	0.997208i	$0.476208\pi$
$434$	3.11525	0.149537
$435$	0	0
$436$	0.0280295	0.00134237
$437$	−50.5070	−2.41608
$438$	10.2304	0.488827
$439$	32.9166	1.57102	0.785511	−	0.618848i	$-0.212400\pi$
$439$	32.9166	1.57102	0.785511	+	0.618848i	$0.212400\pi$
$440$	1.93075	0.0920450
$441$	10.9678	0.522276
$442$	−3.66266	−0.174215
$443$	−31.9332	−1.51719	−0.758596	−	0.651562i	$-0.774114\pi$
$443$	−31.9332	−1.51719	−0.758596	+	0.651562i	$0.774114\pi$
$444$	2.90548	0.137888
$445$	−3.54506	−0.168052
$446$	−6.29535	−0.298094
$447$	19.3855	0.916902
$448$	0.520906	0.0246105
$449$	14.0188	0.661588	0.330794	−	0.943703i	$-0.392684\pi$
$449$	14.0188	0.661588	0.330794	+	0.943703i	$0.392684\pi$
$450$	6.25139	0.294693
$451$	−5.08114	−0.239262
$452$	4.12883	0.194204
$453$	−8.79072	−0.413024
$454$	12.7672	0.599196
$455$	−0.764259	−0.0358290
$456$	7.76020	0.363405
$457$	1.20159	0.0562081	0.0281041	−	0.999605i	$-0.491053\pi$
$457$	1.20159	0.0562081	0.0281041	+	0.999605i	$0.491053\pi$
$458$	11.9434	0.558080
$459$	−40.2127	−1.87697
$460$	−22.6436	−1.05576
$461$	3.83255	0.178500	0.0892498	−	0.996009i	$-0.471553\pi$
$461$	3.83255	0.178500	0.0892498	+	0.996009i	$0.471553\pi$
$462$	−0.396038	−0.0184253
$463$	−5.53194	−0.257091	−0.128546	−	0.991704i	$-0.541031\pi$
$463$	−5.53194	−0.257091	−0.128546	+	0.991704i	$0.541031\pi$
$464$	0	0
$465$	20.8066	0.964881
$466$	11.7030	0.542132
$467$	17.2416	0.797846	0.398923	−	0.916984i	$-0.369384\pi$
$467$	17.2416	0.797846	0.398923	+	0.916984i	$0.369384\pi$
$468$	−0.804571	−0.0371913
$469$	2.35497	0.108742
$470$	−26.1451	−1.20598
$471$	1.64092	0.0756098
$472$	5.31686	0.244728
$473$	−0.421927	−0.0194002
$474$	4.83222	0.221951
$475$	−25.4273	−1.16668
$476$	3.86529	0.177165
$477$	1.87648	0.0859181
$478$	16.7302	0.765221
$479$	11.4349	0.522472	0.261236	−	0.965275i	$-0.415870\pi$
$479$	11.4349	0.522472	0.261236	+	0.965275i	$0.415870\pi$
$480$	3.47909	0.158798
$481$	1.22527	0.0558676
$482$	7.65233	0.348554
$483$	4.64467	0.211340
$484$	−10.5781	−0.480821
$485$	34.8749	1.58359
$486$	−14.5571	−0.660323
$487$	−29.3313	−1.32913	−0.664563	−	0.747232i	$-0.731382\pi$
$487$	−29.3313	−1.32913	−0.664563	+	0.747232i	$0.731382\pi$
$488$	9.69702	0.438963
$489$	9.15095	0.413820
$490$	−20.0003	−0.903521
$491$	0.558505	0.0252050	0.0126025	−	0.999921i	$-0.495988\pi$
$491$	0.558505	0.0252050	0.0126025	+	0.999921i	$0.495988\pi$
$492$	−9.15590	−0.412780
$493$	0	0
$494$	3.27256	0.147240
$495$	3.14715	0.141454
$496$	−5.98045	−0.268530
$497$	−7.36810	−0.330504
$498$	−5.01138	−0.224565
$499$	16.2520	0.727541	0.363771	−	0.931489i	$-0.381489\pi$
$499$	16.2520	0.727541	0.363771	+	0.931489i	$0.381489\pi$
$500$	3.46232	0.154840
$501$	−15.8127	−0.706460
$502$	1.58064	0.0705476
$503$	18.0975	0.806927	0.403463	−	0.914996i	$-0.367806\pi$
$503$	18.0975	0.806927	0.403463	+	0.914996i	$0.367806\pi$
$504$	0.849083	0.0378211
$505$	28.4880	1.26770
$506$	−4.94830	−0.219979
$507$	−14.9309	−0.663103
$508$	−15.9153	−0.706128
$509$	18.0875	0.801715	0.400857	−	0.916140i	$-0.368712\pi$
$509$	18.0875	0.801715	0.400857	+	0.916140i	$0.368712\pi$
$510$	25.8160	1.14315
$511$	−4.55295	−0.201411
$512$	−1.00000	−0.0441942
$513$	35.9298	1.58634
$514$	15.0284	0.662874
$515$	−19.2605	−0.848718
$516$	−0.760286	−0.0334697
$517$	−5.71349	−0.251279
$518$	−1.29306	−0.0568138
$519$	0.293884	0.0129001
$520$	1.46717	0.0643398
$521$	34.3541	1.50508	0.752541	−	0.658545i	$-0.228828\pi$
$521$	34.3541	1.50508	0.752541	+	0.658545i	$0.228828\pi$
$522$	0	0
$523$	44.4195	1.94233	0.971164	−	0.238412i	$-0.0766268\pi$
$523$	44.4195	1.94233	0.971164	+	0.238412i	$0.0766268\pi$
$524$	7.27202	0.317679
$525$	2.33832	0.102053
$526$	5.17364	0.225581
$527$	−44.3769	−1.93309
$528$	0.760286	0.0330872
$529$	35.0329	1.52317
$530$	−3.42185	−0.148636
$531$	8.66654	0.376096
$532$	−3.45361	−0.149733
$533$	−3.86115	−0.167245
$534$	−1.39596	−0.0604093
$535$	−29.0716	−1.25688
$536$	−4.52091	−0.195273
$537$	14.9141	0.643592
$538$	−5.75602	−0.248160
$539$	−4.37066	−0.188258
$540$	16.1082	0.693188
$541$	−19.9191	−0.856391	−0.428195	−	0.903686i	$-0.640850\pi$
$541$	−19.9191	−0.856391	−0.428195	+	0.903686i	$0.640850\pi$
$542$	10.9361	0.469744
$543$	−8.06916	−0.346281
$544$	−7.42032	−0.318143
$545$	−0.0833149	−0.00356882
$546$	−0.300948	−0.0128794
$547$	−2.94284	−0.125827	−0.0629134	−	0.998019i	$-0.520039\pi$
$547$	−2.94284	−0.125827	−0.0629134	+	0.998019i	$0.520039\pi$
$548$	16.6994	0.713365
$549$	15.8062	0.674594
$550$	−2.49118	−0.106224
$551$	0	0
$552$	−8.91652	−0.379512
$553$	−2.15054	−0.0914504
$554$	2.14237	0.0910205
$555$	−8.63625	−0.366588
$556$	16.9493	0.718813
$557$	−8.79820	−0.372792	−0.186396	−	0.982475i	$-0.559681\pi$
$557$	−8.79820	−0.372792	−0.186396	+	0.982475i	$0.559681\pi$
$558$	−9.74820	−0.412674
$559$	−0.320621	−0.0135608
$560$	−1.54834	−0.0654294
$561$	5.64157	0.238187
$562$	−9.52397	−0.401745
$563$	−9.54620	−0.402325	−0.201162	−	0.979558i	$-0.564472\pi$
$563$	−9.54620	−0.402325	−0.201162	+	0.979558i	$0.564472\pi$
$564$	−10.2954	−0.433513
$565$	−12.2726	−0.516310
$566$	−30.6436	−1.28804
$567$	−0.756892	−0.0317865
$568$	14.1448	0.593501
$569$	−14.4069	−0.603968	−0.301984	−	0.953313i	$-0.597649\pi$
$569$	−14.4069	−0.603968	−0.301984	+	0.953313i	$0.597649\pi$
$570$	−23.0664	−0.966147
$571$	−27.5591	−1.15331	−0.576656	−	0.816987i	$-0.695643\pi$
$571$	−27.5591	−1.15331	−0.576656	+	0.816987i	$0.695643\pi$
$572$	0.320621	0.0134058
$573$	7.05709	0.294814
$574$	4.07476	0.170077
$575$	29.2161	1.21840
$576$	−1.63001	−0.0679171
$577$	−0.178697	−0.00743926	−0.00371963	−	0.999993i	$-0.501184\pi$
$577$	−0.178697	−0.00743926	−0.00371963	+	0.999993i	$0.501184\pi$
$578$	−38.0611	−1.58313
$579$	−10.4207	−0.433071
$580$	0	0
$581$	2.23027	0.0925273
$582$	13.7329	0.569249
$583$	−0.747777	−0.0309698
$584$	8.74045	0.361682
$585$	2.39151	0.0988767
$586$	5.43961	0.224708
$587$	−14.5550	−0.600750	−0.300375	−	0.953821i	$-0.597112\pi$
$587$	−14.5550	−0.600750	−0.300375	+	0.953821i	$0.597112\pi$
$588$	−7.87566	−0.324787
$589$	39.6505	1.63377
$590$	−15.8038	−0.650634
$591$	5.51418	0.226823
$592$	2.48233	0.102023
$593$	36.7765	1.51023	0.755114	−	0.655594i	$-0.227582\pi$
$593$	36.7765	1.51023	0.755114	+	0.655594i	$0.227582\pi$
$594$	3.52013	0.144433
$595$	−11.4892	−0.471011
$596$	16.5622	0.678415
$597$	−23.3581	−0.955985
$598$	−3.76020	−0.153766
$599$	0.416973	0.0170371	0.00851853	−	0.999964i	$-0.497288\pi$
$599$	0.416973	0.0170371	0.00851853	+	0.999964i	$0.497288\pi$
$600$	−4.48894	−0.183260
$601$	9.06620	0.369818	0.184909	−	0.982756i	$-0.440801\pi$
$601$	9.06620	0.369818	0.184909	+	0.982756i	$0.440801\pi$
$602$	0.338359	0.0137905
$603$	−7.36913	−0.300094
$604$	−7.51045	−0.305596
$605$	31.4423	1.27831
$606$	11.2179	0.455697
$607$	−3.90564	−0.158525	−0.0792625	−	0.996854i	$-0.525257\pi$
$607$	−3.90564	−0.158525	−0.0792625	+	0.996854i	$0.525257\pi$
$608$	6.63001	0.268882
$609$	0	0
$610$	−28.8234	−1.16703
$611$	−4.34167	−0.175645
$612$	−12.0952	−0.488920
$613$	45.2712	1.82849	0.914244	−	0.405164i	$-0.132786\pi$
$613$	45.2712	1.82849	0.914244	+	0.405164i	$0.132786\pi$
$614$	32.7348	1.32107
$615$	27.2150	1.09742
$616$	−0.338359	−0.0136329
$617$	10.0310	0.403833	0.201917	−	0.979403i	$-0.435283\pi$
$617$	10.0310	0.403833	0.201917	+	0.979403i	$0.435283\pi$
$618$	−7.58434	−0.305087
$619$	6.63200	0.266563	0.133281	−	0.991078i	$-0.457449\pi$
$619$	6.63200	0.266563	0.133281	+	0.991078i	$0.457449\pi$
$620$	17.7763	0.713914
$621$	−41.2836	−1.65665
$622$	21.6580	0.868408
$623$	0.621263	0.0248904
$624$	0.577740	0.0231281
$625$	−29.4673	−1.17869
$626$	10.3916	0.415333
$627$	−5.04071	−0.201306
$628$	1.40194	0.0559436
$629$	18.4197	0.734440
$630$	−2.52382	−0.100551
$631$	11.8983	0.473663	0.236831	−	0.971551i	$-0.423891\pi$
$631$	11.8983	0.473663	0.236831	+	0.971551i	$0.423891\pi$
$632$	4.12847	0.164222
$633$	4.90526	0.194967
$634$	−25.0536	−0.995006
$635$	47.3068	1.87731
$636$	−1.34745	−0.0534298
$637$	−3.32125	−0.131593
$638$	0	0
$639$	23.0561	0.912087
$640$	2.97240	0.117495
$641$	−42.8604	−1.69288	−0.846442	−	0.532482i	$-0.821260\pi$
$641$	−42.8604	−1.69288	−0.846442	+	0.532482i	$0.821260\pi$
$642$	−11.4477	−0.451806
$643$	5.02778	0.198276	0.0991381	−	0.995074i	$-0.468391\pi$
$643$	5.02778	0.198276	0.0991381	+	0.995074i	$0.468391\pi$
$644$	3.96823	0.156370
$645$	2.25988	0.0889826
$646$	49.1968	1.93562
$647$	−38.1546	−1.50001	−0.750007	−	0.661430i	$-0.769950\pi$
$647$	−38.1546	−1.50001	−0.750007	+	0.661430i	$0.769950\pi$
$648$	1.45303	0.0570804
$649$	−3.45361	−0.135566
$650$	−1.89304	−0.0742511
$651$	−3.64630	−0.142910
$652$	7.81821	0.306185
$653$	−13.4657	−0.526954	−0.263477	−	0.964666i	$-0.584869\pi$
$653$	−13.4657	−0.526954	−0.263477	+	0.964666i	$0.584869\pi$
$654$	−0.0328075	−0.00128288
$655$	−21.6154	−0.844582
$656$	−7.82245	−0.305415
$657$	14.2470	0.555830
$658$	4.58186	0.178620
$659$	−26.0874	−1.01622	−0.508111	−	0.861292i	$-0.669656\pi$
$659$	−26.0874	−1.01622	−0.508111	+	0.861292i	$0.669656\pi$
$660$	−2.25988	−0.0879656
$661$	26.2718	1.02186	0.510928	−	0.859624i	$-0.329302\pi$
$661$	26.2718	1.02186	0.510928	+	0.859624i	$0.329302\pi$
$662$	−7.76962	−0.301975
$663$	4.28701	0.166494
$664$	−4.28153	−0.166155
$665$	10.2655	0.398080
$666$	4.04622	0.156788
$667$	0	0
$668$	−13.5098	−0.522709
$669$	7.36849	0.284882
$670$	13.4380	0.519153
$671$	−6.29879	−0.243162
$672$	−0.609702	−0.0235198
$673$	−43.0043	−1.65770	−0.828848	−	0.559474i	$-0.811003\pi$
$673$	−43.0043	−1.65770	−0.828848	+	0.559474i	$0.811003\pi$
$674$	−20.0182	−0.771071
$675$	−20.7839	−0.799971
$676$	−12.7564	−0.490629
$677$	2.26895	0.0872028	0.0436014	−	0.999049i	$-0.486117\pi$
$677$	2.26895	0.0872028	0.0436014	+	0.999049i	$0.486117\pi$
$678$	−4.83265	−0.185597
$679$	−6.11173	−0.234547
$680$	22.0562	0.845816
$681$	−14.9436	−0.572640
$682$	3.88466	0.148751
$683$	31.5204	1.20610	0.603048	−	0.797705i	$-0.293953\pi$
$683$	31.5204	1.20610	0.603048	+	0.797705i	$0.293953\pi$
$684$	10.8070	0.413216
$685$	−49.6375	−1.89655
$686$	7.15134	0.273039
$687$	−13.9794	−0.533346
$688$	−0.649559	−0.0247642
$689$	−0.568234	−0.0216480
$690$	26.5035	1.00897
$691$	−12.7666	−0.485664	−0.242832	−	0.970068i	$-0.578076\pi$
$691$	−12.7666	−0.485664	−0.242832	+	0.970068i	$0.578076\pi$
$692$	0.251083	0.00954476
$693$	−0.551529	−0.0209509
$694$	−18.9324	−0.718664
$695$	−50.3803	−1.91103
$696$	0	0
$697$	−58.0450	−2.19861
$698$	7.41998	0.280850
$699$	−13.6980	−0.518105
$700$	1.99777	0.0755085
$701$	−41.1901	−1.55573	−0.777865	−	0.628432i	$-0.783697\pi$
$701$	−41.1901	−1.55573	−0.777865	+	0.628432i	$0.783697\pi$
$702$	2.67494	0.100959
$703$	−16.4579	−0.620720
$704$	0.649559	0.0244812
$705$	30.6019	1.15254
$706$	−19.6708	−0.740319
$707$	−4.99245	−0.187760
$708$	−6.22319	−0.233882
$709$	−30.2095	−1.13454	−0.567271	−	0.823531i	$-0.692001\pi$
$709$	−30.2095	−1.13454	−0.567271	+	0.823531i	$0.692001\pi$
$710$	−42.0440	−1.57788
$711$	6.72944	0.252374
$712$	−1.19266	−0.0446967
$713$	−45.5587	−1.70619
$714$	−4.52418	−0.169313
$715$	−0.953016	−0.0356408
$716$	12.7421	0.476193
$717$	−19.5821	−0.731307
$718$	19.6856	0.734658
$719$	−34.0336	−1.26924	−0.634620	−	0.772824i	$-0.718843\pi$
$719$	−34.0336	−1.26924	−0.634620	+	0.772824i	$0.718843\pi$
$720$	4.84505	0.180564
$721$	3.37535	0.125705
$722$	−24.9571	−0.928805
$723$	−8.95679	−0.333106
$724$	−6.89398	−0.256213
$725$	0	0
$726$	12.3813	0.459512
$727$	−19.1878	−0.711634	−0.355817	−	0.934556i	$-0.615797\pi$
$727$	−19.1878	−0.711634	−0.355817	+	0.934556i	$0.615797\pi$
$728$	−0.257118	−0.00952944
$729$	21.3976	0.792505
$730$	−25.9801	−0.961568
$731$	−4.81994	−0.178272
$732$	−11.3500	−0.419509
$733$	−3.94792	−0.145820	−0.0729098	−	0.997339i	$-0.523229\pi$
$733$	−3.94792	−0.145820	−0.0729098	+	0.997339i	$0.523229\pi$
$734$	22.0074	0.812307
$735$	23.4096	0.863477
$736$	−7.61793	−0.280801
$737$	2.93660	0.108171
$738$	−12.7507	−0.469359
$739$	−45.9525	−1.69039	−0.845195	−	0.534457i	$-0.820516\pi$
$739$	−45.9525	−1.69039	−0.845195	+	0.534457i	$0.820516\pi$
$740$	−7.37848	−0.271238
$741$	−3.83042	−0.140714
$742$	0.599670	0.0220146
$743$	21.3552	0.783446	0.391723	−	0.920083i	$-0.371879\pi$
$743$	21.3552	0.783446	0.391723	+	0.920083i	$0.371879\pi$
$744$	6.99991	0.256629
$745$	−49.2296	−1.80363
$746$	−23.8590	−0.873540
$747$	−6.97894	−0.255346
$748$	4.81994	0.176234
$749$	5.09472	0.186157
$750$	−4.05252	−0.147977
$751$	14.5350	0.530390	0.265195	−	0.964195i	$-0.414564\pi$
$751$	14.5350	0.530390	0.265195	+	0.964195i	$0.414564\pi$
$752$	−8.79595	−0.320755
$753$	−1.85009	−0.0674210
$754$	0	0
$755$	22.3241	0.812456
$756$	−2.82293	−0.102669
$757$	−37.9668	−1.37993	−0.689963	−	0.723844i	$-0.742373\pi$
$757$	−37.9668	−1.37993	−0.689963	+	0.723844i	$0.742373\pi$
$758$	−31.8334	−1.15624
$759$	5.79181	0.210229
$760$	−19.7071	−0.714850
$761$	−20.5739	−0.745804	−0.372902	−	0.927871i	$-0.621637\pi$
$761$	−20.5739	−0.745804	−0.372902	+	0.927871i	$0.621637\pi$
$762$	18.6283	0.674833
$763$	0.0146007	0.000528582 0
$764$	6.02931	0.218133
$765$	35.9518	1.29984
$766$	−21.2412	−0.767475
$767$	−2.62439	−0.0947613
$768$	1.17047	0.0422355
$769$	6.26299	0.225849	0.112925	−	0.993604i	$-0.463978\pi$
$769$	6.26299	0.225849	0.112925	+	0.993604i	$0.463978\pi$
$770$	1.00574	0.0362444
$771$	−17.5902	−0.633496
$772$	−8.90307	−0.320428
$773$	2.65347	0.0954385	0.0477193	−	0.998861i	$-0.484805\pi$
$773$	2.65347	0.0954385	0.0477193	+	0.998861i	$0.484805\pi$
$774$	−1.05879	−0.0380574
$775$	−22.9361	−0.823889
$776$	11.7329	0.421186
$777$	1.51348	0.0542958
$778$	11.0403	0.395813
$779$	51.8629	1.85818
$780$	−1.71727	−0.0614883
$781$	−9.18786	−0.328768
$782$	−56.5275	−2.02142
$783$	0	0
$784$	−6.72866	−0.240309
$785$	−4.16714	−0.148731
$786$	−8.51164	−0.303600
$787$	3.79800	0.135384	0.0676920	−	0.997706i	$-0.478436\pi$
$787$	3.79800	0.135384	0.0676920	+	0.997706i	$0.478436\pi$
$788$	4.71110	0.167826
$789$	−6.05556	−0.215584
$790$	−12.2715	−0.436599
$791$	2.15073	0.0764713
$792$	1.05879	0.0376224
$793$	−4.78643	−0.169971
$794$	21.7506	0.771899
$795$	4.00516	0.142048
$796$	−19.9563	−0.707332
$797$	2.13424	0.0755988	0.0377994	−	0.999285i	$-0.487965\pi$
$797$	2.13424	0.0755988	0.0377994	+	0.999285i	$0.487965\pi$
$798$	4.04233	0.143097
$799$	−65.2687	−2.30904
$800$	−3.83518	−0.135594
$801$	−1.94405	−0.0686895
$802$	−35.6407	−1.25852
$803$	−5.67744	−0.200352
$804$	5.29156	0.186619
$805$	−11.7952	−0.415725
$806$	2.95194	0.103978
$807$	6.73722	0.237161
$808$	9.58417	0.337170
$809$	−28.4708	−1.00098	−0.500491	−	0.865742i	$-0.666847\pi$
$809$	−28.4708	−1.00098	−0.500491	+	0.865742i	$0.666847\pi$
$810$	−4.31899	−0.151754
$811$	−22.8054	−0.800804	−0.400402	−	0.916340i	$-0.631130\pi$
$811$	−22.8054	−0.800804	−0.400402	+	0.916340i	$0.631130\pi$
$812$	0	0
$813$	−12.8003	−0.448926
$814$	−1.61242	−0.0565152
$815$	−23.2389	−0.814023
$816$	8.68522	0.304044
$817$	4.30658	0.150668
$818$	−15.5639	−0.544178
$819$	−0.419106	−0.0146447
$820$	23.2515	0.811976
$821$	−52.3625	−1.82746	−0.913731	−	0.406319i	$-0.866812\pi$
$821$	−52.3625	−1.82746	−0.913731	+	0.406319i	$0.866812\pi$
$822$	−19.5461	−0.681749
$823$	43.0940	1.50216	0.751081	−	0.660211i	$-0.229533\pi$
$823$	43.0940	1.50216	0.751081	+	0.660211i	$0.229533\pi$
$824$	−6.47977	−0.225733
$825$	2.91583	0.101516
$826$	2.76958	0.0963661
$827$	−31.4897	−1.09500	−0.547501	−	0.836805i	$-0.684421\pi$
$827$	−31.4897	−1.09500	−0.547501	+	0.836805i	$0.684421\pi$
$828$	−12.4173	−0.431531
$829$	26.6287	0.924853	0.462427	−	0.886658i	$-0.346979\pi$
$829$	26.6287	0.924853	0.462427	+	0.886658i	$0.346979\pi$
$830$	12.7264	0.441740
$831$	−2.50757	−0.0869866
$832$	0.493598	0.0171124
$833$	−49.9288	−1.72993
$834$	−19.8386	−0.686955
$835$	40.1565	1.38967
$836$	−4.30658	−0.148946
$837$	32.4097	1.12024
$838$	17.4900	0.604181
$839$	35.7731	1.23502	0.617512	−	0.786562i	$-0.288141\pi$
$839$	35.7731	1.23502	0.617512	+	0.786562i	$0.288141\pi$
$840$	1.81228	0.0625296
$841$	0	0
$842$	−4.66179	−0.160656
$843$	11.1475	0.383940
$844$	4.19086	0.144255
$845$	37.9170	1.30439
$846$	−14.3375	−0.492933
$847$	−5.51018	−0.189332
$848$	−1.15121	−0.0395326
$849$	35.8672	1.23096
$850$	−28.4582	−0.976110
$851$	18.9102	0.648233
$852$	−16.5560	−0.567198
$853$	40.5771	1.38933	0.694666	−	0.719332i	$-0.255552\pi$
$853$	40.5771	1.38933	0.694666	+	0.719332i	$0.255552\pi$
$854$	5.05123	0.172850
$855$	−32.1227	−1.09857
$856$	−9.78051	−0.334291
$857$	9.07270	0.309918	0.154959	−	0.987921i	$-0.450476\pi$
$857$	9.07270	0.309918	0.154959	+	0.987921i	$0.450476\pi$
$858$	−0.375276	−0.0128117
$859$	−46.9839	−1.60307	−0.801536	−	0.597947i	$-0.795983\pi$
$859$	−46.9839	−1.60307	−0.801536	+	0.597947i	$0.795983\pi$
$860$	1.93075	0.0658381
$861$	−4.76936	−0.162539
$862$	23.8990	0.814004
$863$	−3.77534	−0.128514	−0.0642570	−	0.997933i	$-0.520468\pi$
$863$	−3.77534	−0.128514	−0.0642570	+	0.997933i	$0.520468\pi$
$864$	5.41927	0.184367
$865$	−0.746321	−0.0253757
$866$	−3.10772	−0.105605
$867$	44.5492	1.51297
$868$	−3.11525	−0.105739
$869$	−2.68168	−0.0909698
$870$	0	0
$871$	2.23151	0.0756119
$872$	−0.0280295	−0.000949198 0
$873$	19.1247	0.647274
$874$	50.5070	1.70842
$875$	1.80354	0.0609708
$876$	−10.2304	−0.345653
$877$	46.6597	1.57559	0.787793	−	0.615941i	$-0.211224\pi$
$877$	46.6597	1.57559	0.787793	+	0.615941i	$0.211224\pi$
$878$	−32.9166	−1.11088
$879$	−6.36687	−0.214749
$880$	−1.93075	−0.0650856
$881$	30.7060	1.03451	0.517255	−	0.855831i	$-0.326954\pi$
$881$	30.7060	1.03451	0.517255	+	0.855831i	$0.326954\pi$
$882$	−10.9678	−0.369305
$883$	45.3506	1.52617	0.763084	−	0.646299i	$-0.223684\pi$
$883$	45.3506	1.52617	0.763084	+	0.646299i	$0.223684\pi$
$884$	3.66266	0.123188
$885$	18.4978	0.621798
$886$	31.9332	1.07282
$887$	27.2539	0.915096	0.457548	−	0.889185i	$-0.348728\pi$
$887$	27.2539	0.915096	0.457548	+	0.889185i	$0.348728\pi$
$888$	−2.90548	−0.0975015
$889$	−8.29039	−0.278051
$890$	3.54506	0.118831
$891$	−0.943828	−0.0316194
$892$	6.29535	0.210784
$893$	58.3172	1.95151
$894$	−19.3855	−0.648348
$895$	−37.8745	−1.26601
$896$	−0.520906	−0.0174022
$897$	4.40118	0.146951
$898$	−14.0188	−0.467813
$899$	0	0
$900$	−6.25139	−0.208380
$901$	−8.54232	−0.284586
$902$	5.08114	0.169184
$903$	−0.396038	−0.0131793
$904$	−4.12883	−0.137323
$905$	20.4917	0.681166
$906$	8.79072	0.292052
$907$	6.49496	0.215662	0.107831	−	0.994169i	$-0.465610\pi$
$907$	6.49496	0.215662	0.107831	+	0.994169i	$0.465610\pi$
$908$	−12.7672	−0.423695
$909$	15.6223	0.518159
$910$	0.764259	0.0253349
$911$	9.49135	0.314462	0.157231	−	0.987562i	$-0.449743\pi$
$911$	9.49135	0.314462	0.157231	+	0.987562i	$0.449743\pi$
$912$	−7.76020	−0.256966
$913$	2.78110	0.0920411
$914$	−1.20159	−0.0397452
$915$	33.7368	1.11531
$916$	−11.9434	−0.394622
$917$	3.78804	0.125092
$918$	40.2127	1.32722
$919$	6.19520	0.204361	0.102180	−	0.994766i	$-0.467418\pi$
$919$	6.19520	0.204361	0.102180	+	0.994766i	$0.467418\pi$
$920$	22.6436	0.746536
$921$	−38.3150	−1.26252
$922$	−3.83255	−0.126218
$923$	−6.98183	−0.229810
$924$	0.396038	0.0130287
$925$	9.52017	0.313021
$926$	5.53194	0.181791
$927$	−10.5621	−0.346905
$928$	0	0
$929$	17.9637	0.589369	0.294684	−	0.955595i	$-0.404785\pi$
$929$	17.9637	0.589369	0.294684	+	0.955595i	$0.404785\pi$
$930$	−20.8066	−0.682274
$931$	44.6111	1.46207
$932$	−11.7030	−0.383345
$933$	−25.3500	−0.829921
$934$	−17.2416	−0.564163
$935$	−14.3268	−0.468536
$936$	0.804571	0.0262982
$937$	17.3905	0.568122	0.284061	−	0.958806i	$-0.408318\pi$
$937$	17.3905	0.568122	0.284061	+	0.958806i	$0.408318\pi$
$938$	−2.35497	−0.0768924
$939$	−12.1630	−0.396926
$940$	26.1451	0.852759
$941$	−4.23399	−0.138024	−0.0690120	−	0.997616i	$-0.521985\pi$
$941$	−4.23399	−0.138024	−0.0690120	+	0.997616i	$0.521985\pi$
$942$	−1.64092	−0.0534642
$943$	−59.5909	−1.94055
$944$	−5.31686	−0.173049
$945$	8.39088	0.272955
$946$	0.421927	0.0137180
$947$	−40.9467	−1.33059	−0.665294	−	0.746582i	$-0.731694\pi$
$947$	−40.9467	−1.33059	−0.665294	+	0.746582i	$0.731694\pi$
$948$	−4.83222	−0.156943
$949$	−4.31427	−0.140047
$950$	25.4273	0.824970
$951$	29.3244	0.950908
$952$	−3.86529	−0.125275
$953$	14.7024	0.476256	0.238128	−	0.971234i	$-0.423466\pi$
$953$	14.7024	0.476256	0.238128	+	0.971234i	$0.423466\pi$
$954$	−1.87648	−0.0607533
$955$	−17.9215	−0.579927
$956$	−16.7302	−0.541093
$957$	0	0
$958$	−11.4349	−0.369443
$959$	8.69884	0.280900
$960$	−3.47909	−0.112287
$961$	4.76581	0.153736
$962$	−1.22527	−0.0395044
$963$	−15.9423	−0.513735
$964$	−7.65233	−0.246465
$965$	26.4635	0.851890
$966$	−4.64467	−0.149440
$967$	5.99979	0.192940	0.0964701	−	0.995336i	$-0.469245\pi$
$967$	5.99979	0.192940	0.0964701	+	0.995336i	$0.469245\pi$
$968$	10.5781	0.339992
$969$	−57.5831	−1.84984
$970$	−34.8749	−1.11976
$971$	22.7421	0.729829	0.364915	−	0.931041i	$-0.381098\pi$
$971$	22.7421	0.729829	0.364915	+	0.931041i	$0.381098\pi$
$972$	14.5571	0.466919
$973$	8.82902	0.283045
$974$	29.3313	0.939834
$975$	2.21573	0.0709603
$976$	−9.69702	−0.310394
$977$	0.625854	0.0200228	0.0100114	−	0.999950i	$-0.496813\pi$
$977$	0.625854	0.0200228	0.0100114	+	0.999950i	$0.496813\pi$
$978$	−9.15095	−0.292615
$979$	0.774702	0.0247596
$980$	20.0003	0.638885
$981$	−0.0456884	−0.00145872
$982$	−0.558505	−0.0178226
$983$	−13.0976	−0.417749	−0.208875	−	0.977942i	$-0.566980\pi$
$983$	−13.0976	−0.417749	−0.208875	+	0.977942i	$0.566980\pi$
$984$	9.15590	0.291879
$985$	−14.0033	−0.446182
$986$	0	0
$987$	−5.36291	−0.170703
$988$	−3.27256	−0.104114
$989$	−4.94830	−0.157347
$990$	−3.14715	−0.100023
$991$	−51.4636	−1.63480	−0.817398	−	0.576074i	$-0.804584\pi$
$991$	−51.4636	−1.63480	−0.817398	+	0.576074i	$0.804584\pi$
$992$	5.98045	0.189880
$993$	9.09407	0.288592
$994$	7.36810	0.233702
$995$	59.3181	1.88051
$996$	5.01138	0.158792
$997$	10.5977	0.335634	0.167817	−	0.985818i	$-0.446328\pi$
$997$	10.5977	0.335634	0.167817	+	0.985818i	$0.446328\pi$
$998$	−16.2520	−0.514449
$999$	−13.4524	−0.425615

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1682.2.a.q.1.4		6
29.9	even	14		58.2.d.b.23.2	✓	12
29.12	odd	4		1682.2.b.i.1681.4		12
29.13	even	14		58.2.d.b.53.2	yes	12
29.17	odd	4		1682.2.b.i.1681.9		12
29.28	even	2		1682.2.a.t.1.3		6
87.38	odd	14		522.2.k.h.487.2		12
87.71	odd	14		522.2.k.h.343.2		12
116.67	odd	14		464.2.u.h.81.1		12
116.71	odd	14		464.2.u.h.401.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
58.2.d.b.23.2	✓	12	29.9	even	14
58.2.d.b.53.2	yes	12	29.13	even	14
464.2.u.h.81.1		12	116.67	odd	14
464.2.u.h.401.1		12	116.71	odd	14
522.2.k.h.343.2		12	87.71	odd	14
522.2.k.h.487.2		12	87.38	odd	14
1682.2.a.q.1.4		6	1.1	even	1	trivial
1682.2.a.t.1.3		6	29.28	even	2
1682.2.b.i.1681.4		12	29.12	odd	4
1682.2.b.i.1681.9		12	29.17	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1682 = 2 \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1682.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.17047 q^{3} +1.00000 q^{4} -2.97240 q^{5} -1.17047 q^{6} +0.520906 q^{7} -1.00000 q^{8} -1.63001 q^{9} +2.97240 q^{10} +0.649559 q^{11} +1.17047 q^{12} +0.493598 q^{13} -0.520906 q^{14} -3.47909 q^{15} +1.00000 q^{16} +7.42032 q^{17} +1.63001 q^{18} -6.63001 q^{19} -2.97240 q^{20} +0.609702 q^{21} -0.649559 q^{22} +7.61793 q^{23} -1.17047 q^{24} +3.83518 q^{25} -0.493598 q^{26} -5.41927 q^{27} +0.520906 q^{28} +3.47909 q^{30} -5.98045 q^{31} -1.00000 q^{32} +0.760286 q^{33} -7.42032 q^{34} -1.54834 q^{35} -1.63001 q^{36} +2.48233 q^{37} +6.63001 q^{38} +0.577740 q^{39} +2.97240 q^{40} -7.82245 q^{41} -0.609702 q^{42} -0.649559 q^{43} +0.649559 q^{44} +4.84505 q^{45} -7.61793 q^{46} -8.79595 q^{47} +1.17047 q^{48} -6.72866 q^{49} -3.83518 q^{50} +8.68522 q^{51} +0.493598 q^{52} -1.15121 q^{53} +5.41927 q^{54} -1.93075 q^{55} -0.520906 q^{56} -7.76020 q^{57} -5.31686 q^{59} -3.47909 q^{60} -9.69702 q^{61} +5.98045 q^{62} -0.849083 q^{63} +1.00000 q^{64} -1.46717 q^{65} -0.760286 q^{66} +4.52091 q^{67} +7.42032 q^{68} +8.91652 q^{69} +1.54834 q^{70} -14.1448 q^{71} +1.63001 q^{72} -8.74045 q^{73} -2.48233 q^{74} +4.48894 q^{75} -6.63001 q^{76} +0.338359 q^{77} -0.577740 q^{78} -4.12847 q^{79} -2.97240 q^{80} -1.45303 q^{81} +7.82245 q^{82} +4.28153 q^{83} +0.609702 q^{84} -22.0562 q^{85} +0.649559 q^{86} -0.649559 q^{88} +1.19266 q^{89} -4.84505 q^{90} +0.257118 q^{91} +7.61793 q^{92} -6.99991 q^{93} +8.79595 q^{94} +19.7071 q^{95} -1.17047 q^{96} -11.7329 q^{97} +6.72866 q^{98} -1.05879 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{2} - 2 q^{3} + 6 q^{4} + 2 q^{6} - 3 q^{7} - 6 q^{8} + 12 q^{9} + q^{11} - 2 q^{12} - 3 q^{13} + 3 q^{14} - 27 q^{15} + 6 q^{16} + 6 q^{17} - 12 q^{18} - 18 q^{19} + 10 q^{21} - q^{22} - 14 q^{23}+ \cdots + 4 q^{99}+O(q^{100}) \) 6 * q - 6 * q^2 - 2 * q^3 + 6 * q^4 + 2 * q^6 - 3 * q^7 - 6 * q^8 + 12 * q^9 + q^11 - 2 * q^12 - 3 * q^13 + 3 * q^14 - 27 * q^15 + 6 * q^16 + 6 * q^17 - 12 * q^18 - 18 * q^19 + 10 * q^21 - q^22 - 14 * q^23 + 2 * q^24 + 4 * q^25 + 3 * q^26 - 23 * q^27 - 3 * q^28 + 27 * q^30 - 17 * q^31 - 6 * q^32 + 20 * q^33 - 6 * q^34 - 9 * q^35 + 12 * q^36 - 12 * q^37 + 18 * q^38 - 10 * q^39 + 15 * q^41 - 10 * q^42 - q^43 + q^44 + 18 * q^45 + 14 * q^46 - 8 * q^47 - 2 * q^48 + q^49 - 4 * q^50 - 11 * q^51 - 3 * q^52 + 3 * q^53 + 23 * q^54 - 18 * q^55 + 3 * q^56 - 19 * q^57 + 19 * q^59 - 27 * q^60 - 15 * q^61 + 17 * q^62 - 33 * q^63 + 6 * q^64 + 11 * q^65 - 20 * q^66 + 21 * q^67 + 6 * q^68 + 11 * q^69 + 9 * q^70 - 9 * q^71 - 12 * q^72 - 31 * q^73 + 12 * q^74 + q^75 - 18 * q^76 - 33 * q^77 + 10 * q^78 + 2 * q^79 + 2 * q^81 - 15 * q^82 - 2 * q^83 + 10 * q^84 - 19 * q^85 + q^86 - q^88 - 2 * q^89 - 18 * q^90 - 22 * q^91 - 14 * q^92 + q^93 + 8 * q^94 + 18 * q^95 + 2 * q^96 - 43 * q^97 - q^98 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more