LMFDB - Embedded newform 538.2.a.e.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [538,2,Mod(1,538)] 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("538.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(538, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 538 = 2 \cdot 269 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	538.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,7,1] 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:	$4.29595162874$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 15x^{5} + 16x^{4} + 49x^{3} - 53x^{2} - 44x + 48 $ x^7 - x^6 - 15x^5 + 16x^4 + 49x^3 - 53x^2 - 44*x + 48
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.19866$ of defining polynomial
Character	$\chi$	$=$	538.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +2.12827 q^{3} +1.00000 q^{4} +3.84433 q^{5} +2.12827 q^{6} -4.95403 q^{7} +1.00000 q^{8} +1.52953 q^{9} +3.84433 q^{10} -0.453515 q^{11} +2.12827 q^{12} -0.0945931 q^{13} -4.95403 q^{14} +8.18177 q^{15} +1.00000 q^{16} +1.41349 q^{17} +1.52953 q^{18} -2.49192 q^{19} +3.84433 q^{20} -10.5435 q^{21} -0.453515 q^{22} -2.25654 q^{23} +2.12827 q^{24} +9.77888 q^{25} -0.0945931 q^{26} -3.12955 q^{27} -4.95403 q^{28} -6.64864 q^{29} +8.18177 q^{30} +5.36736 q^{31} +1.00000 q^{32} -0.965201 q^{33} +1.41349 q^{34} -19.0449 q^{35} +1.52953 q^{36} +7.31117 q^{37} -2.49192 q^{38} -0.201320 q^{39} +3.84433 q^{40} -10.6653 q^{41} -10.5435 q^{42} -12.2765 q^{43} -0.453515 q^{44} +5.88003 q^{45} -2.25654 q^{46} -1.43212 q^{47} +2.12827 q^{48} +17.5424 q^{49} +9.77888 q^{50} +3.00829 q^{51} -0.0945931 q^{52} +9.96897 q^{53} -3.12955 q^{54} -1.74346 q^{55} -4.95403 q^{56} -5.30347 q^{57} -6.64864 q^{58} +7.89955 q^{59} +8.18177 q^{60} +0.950317 q^{61} +5.36736 q^{62} -7.57735 q^{63} +1.00000 q^{64} -0.363647 q^{65} -0.965201 q^{66} -8.06276 q^{67} +1.41349 q^{68} -4.80253 q^{69} -19.0449 q^{70} -3.06085 q^{71} +1.52953 q^{72} +12.8264 q^{73} +7.31117 q^{74} +20.8121 q^{75} -2.49192 q^{76} +2.24672 q^{77} -0.201320 q^{78} +13.7673 q^{79} +3.84433 q^{80} -11.2491 q^{81} -10.6653 q^{82} +10.1171 q^{83} -10.5435 q^{84} +5.43392 q^{85} -12.2765 q^{86} -14.1501 q^{87} -0.453515 q^{88} -1.77803 q^{89} +5.88003 q^{90} +0.468616 q^{91} -2.25654 q^{92} +11.4232 q^{93} -1.43212 q^{94} -9.57975 q^{95} +2.12827 q^{96} -12.9137 q^{97} +17.5424 q^{98} -0.693665 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{2} + q^{3} + 7 q^{4} + 7 q^{5} + q^{6} + 6 q^{7} + 7 q^{8} + 12 q^{9} + 7 q^{10} - 3 q^{11} + q^{12} - 9 q^{13} + 6 q^{14} + 8 q^{15} + 7 q^{16} + 8 q^{17} + 12 q^{18} - 11 q^{19} + 7 q^{20}+ \cdots - 37 q^{99}+O(q^{100}) $ 7 * q + 7 * q^2 + q^3 + 7 * q^4 + 7 * q^5 + q^6 + 6 * q^7 + 7 * q^8 + 12 * q^9 + 7 * q^10 - 3 * q^11 + q^12 - 9 * q^13 + 6 * q^14 + 8 * q^15 + 7 * q^16 + 8 * q^17 + 12 * q^18 - 11 * q^19 + 7 * q^20 - 6 * q^21 - 3 * q^22 + 12 * q^23 + q^24 + 22 * q^25 - 9 * q^26 - 14 * q^27 + 6 * q^28 - 5 * q^29 + 8 * q^30 + 14 * q^31 + 7 * q^32 - 4 * q^33 + 8 * q^34 - 4 * q^35 + 12 * q^36 + 13 * q^37 - 11 * q^38 - 18 * q^39 + 7 * q^40 + 12 * q^41 - 6 * q^42 - 11 * q^43 - 3 * q^44 + 3 * q^45 + 12 * q^46 + 2 * q^47 + q^48 + 15 * q^49 + 22 * q^50 - 26 * q^51 - 9 * q^52 + 19 * q^53 - 14 * q^54 - 40 * q^55 + 6 * q^56 - 12 * q^57 - 5 * q^58 - 9 * q^59 + 8 * q^60 - 3 * q^61 + 14 * q^62 - 26 * q^63 + 7 * q^64 - 10 * q^65 - 4 * q^66 - 33 * q^67 + 8 * q^68 - 64 * q^69 - 4 * q^70 + 28 * q^71 + 12 * q^72 - 14 * q^73 + 13 * q^74 - 45 * q^75 - 11 * q^76 + 10 * q^77 - 18 * q^78 + 2 * q^79 + 7 * q^80 + 15 * q^81 + 12 * q^82 - 7 * q^83 - 6 * q^84 - 16 * q^85 - 11 * q^86 + 16 * q^87 - 3 * q^88 + 18 * q^89 + 3 * q^90 - 26 * q^91 + 12 * q^92 + 6 * q^93 + 2 * q^94 - 34 * q^95 + q^96 - 4 * q^97 + 15 * q^98 - 37 * 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
$3$	2.12827	1.22876	0.614379	−	0.789011i	$-0.289407\pi$
$3$	2.12827	1.22876	0.614379	+	0.789011i	$0.289407\pi$
$4$	1.00000	0.500000
$5$	3.84433	1.71924	0.859619	−	0.510936i	$-0.170701\pi$
$5$	3.84433	1.71924	0.859619	+	0.510936i	$0.170701\pi$
$6$	2.12827	0.868863
$7$	−4.95403	−1.87245	−0.936223	−	0.351407i	$-0.885703\pi$
$7$	−4.95403	−1.87245	−0.936223	+	0.351407i	$0.885703\pi$
$8$	1.00000	0.353553
$9$	1.52953	0.509844
$10$	3.84433	1.21568
$11$	−0.453515	−0.136740	−0.0683699	−	0.997660i	$-0.521780\pi$
$11$	−0.453515	−0.136740	−0.0683699	+	0.997660i	$0.521780\pi$
$12$	2.12827	0.614379
$13$	−0.0945931	−0.0262354	−0.0131177	−	0.999914i	$-0.504176\pi$
$13$	−0.0945931	−0.0262354	−0.0131177	+	0.999914i	$0.504176\pi$
$14$	−4.95403	−1.32402
$15$	8.18177	2.11253
$16$	1.00000	0.250000
$17$	1.41349	0.342822	0.171411	−	0.985200i	$-0.445167\pi$
$17$	1.41349	0.342822	0.171411	+	0.985200i	$0.445167\pi$
$18$	1.52953	0.360514
$19$	−2.49192	−0.571685	−0.285842	−	0.958277i	$-0.592273\pi$
$19$	−2.49192	−0.571685	−0.285842	+	0.958277i	$0.592273\pi$
$20$	3.84433	0.859619
$21$	−10.5435	−2.30078
$22$	−0.453515	−0.0966896
$23$	−2.25654	−0.470521	−0.235261	−	0.971932i	$-0.575594\pi$
$23$	−2.25654	−0.470521	−0.235261	+	0.971932i	$0.575594\pi$
$24$	2.12827	0.434431
$25$	9.77888	1.95578
$26$	−0.0945931	−0.0185512
$27$	−3.12955	−0.602282
$28$	−4.95403	−0.936223
$29$	−6.64864	−1.23462	−0.617311	−	0.786720i	$-0.711778\pi$
$29$	−6.64864	−1.23462	−0.617311	+	0.786720i	$0.711778\pi$
$30$	8.18177	1.49378
$31$	5.36736	0.964006	0.482003	−	0.876170i	$-0.339909\pi$
$31$	5.36736	0.964006	0.482003	+	0.876170i	$0.339909\pi$
$32$	1.00000	0.176777
$33$	−0.965201	−0.168020
$34$	1.41349	0.242412
$35$	−19.0449	−3.21918
$36$	1.52953	0.254922
$37$	7.31117	1.20195	0.600974	−	0.799268i	$-0.294779\pi$
$37$	7.31117	1.20195	0.600974	+	0.799268i	$0.294779\pi$
$38$	−2.49192	−0.404242
$39$	−0.201320	−0.0322369
$40$	3.84433	0.607842
$41$	−10.6653	−1.66563	−0.832817	−	0.553548i	$-0.813274\pi$
$41$	−10.6653	−1.66563	−0.832817	+	0.553548i	$0.813274\pi$
$42$	−10.5435	−1.62690
$43$	−12.2765	−1.87215	−0.936076	−	0.351798i	$-0.885570\pi$
$43$	−12.2765	−1.87215	−0.936076	+	0.351798i	$0.885570\pi$
$44$	−0.453515	−0.0683699
$45$	5.88003	0.876543
$46$	−2.25654	−0.332709
$47$	−1.43212	−0.208897	−0.104448	−	0.994530i	$-0.533308\pi$
$47$	−1.43212	−0.208897	−0.104448	+	0.994530i	$0.533308\pi$
$48$	2.12827	0.307189
$49$	17.5424	2.50605
$50$	9.77888	1.38294
$51$	3.00829	0.421245
$52$	−0.0945931	−0.0131177
$53$	9.96897	1.36934	0.684672	−	0.728852i	$-0.259946\pi$
$53$	9.96897	1.36934	0.684672	+	0.728852i	$0.259946\pi$
$54$	−3.12955	−0.425878
$55$	−1.74346	−0.235088
$56$	−4.95403	−0.662010
$57$	−5.30347	−0.702462
$58$	−6.64864	−0.873009
$59$	7.89955	1.02843	0.514217	−	0.857660i	$-0.328083\pi$
$59$	7.89955	1.02843	0.514217	+	0.857660i	$0.328083\pi$
$60$	8.18177	1.05626
$61$	0.950317	0.121676	0.0608378	−	0.998148i	$-0.480623\pi$
$61$	0.950317	0.121676	0.0608378	+	0.998148i	$0.480623\pi$
$62$	5.36736	0.681655
$63$	−7.57735	−0.954656
$64$	1.00000	0.125000
$65$	−0.363647	−0.0451049
$66$	−0.965201	−0.118808
$67$	−8.06276	−0.985023	−0.492512	−	0.870306i	$-0.663921\pi$
$67$	−8.06276	−0.985023	−0.492512	+	0.870306i	$0.663921\pi$
$68$	1.41349	0.171411
$69$	−4.80253	−0.578156
$70$	−19.0449	−2.27630
$71$	−3.06085	−0.363256	−0.181628	−	0.983367i	$-0.558137\pi$
$71$	−3.06085	−0.363256	−0.181628	+	0.983367i	$0.558137\pi$
$72$	1.52953	0.180257
$73$	12.8264	1.50121	0.750607	−	0.660749i	$-0.229761\pi$
$73$	12.8264	1.50121	0.750607	+	0.660749i	$0.229761\pi$
$74$	7.31117	0.849906
$75$	20.8121	2.40317
$76$	−2.49192	−0.285842
$77$	2.24672	0.256038
$78$	−0.201320	−0.0227950
$79$	13.7673	1.54894	0.774469	−	0.632611i	$-0.218017\pi$
$79$	13.7673	1.54894	0.774469	+	0.632611i	$0.218017\pi$
$80$	3.84433	0.429809
$81$	−11.2491	−1.24990
$82$	−10.6653	−1.17778
$83$	10.1171	1.11049	0.555246	−	0.831686i	$-0.312624\pi$
$83$	10.1171	1.11049	0.555246	+	0.831686i	$0.312624\pi$
$84$	−10.5435	−1.15039
$85$	5.43392	0.589392
$86$	−12.2765	−1.32381
$87$	−14.1501	−1.51705
$88$	−0.453515	−0.0483448
$89$	−1.77803	−0.188471	−0.0942354	−	0.995550i	$-0.530041\pi$
$89$	−1.77803	−0.188471	−0.0942354	+	0.995550i	$0.530041\pi$
$90$	5.88003	0.619810
$91$	0.468616	0.0491244
$92$	−2.25654	−0.235261
$93$	11.4232	1.18453
$94$	−1.43212	−0.147712
$95$	−9.57975	−0.982862
$96$	2.12827	0.217216
$97$	−12.9137	−1.31119	−0.655593	−	0.755115i	$-0.727581\pi$
$97$	−12.9137	−1.31119	−0.655593	+	0.755115i	$0.727581\pi$
$98$	17.5424	1.77205
$99$	−0.693665	−0.0697160
$100$	9.77888	0.977888
$101$	−6.03448	−0.600453	−0.300226	−	0.953868i	$-0.597062\pi$
$101$	−6.03448	−0.600453	−0.300226	+	0.953868i	$0.597062\pi$
$102$	3.00829	0.297865
$103$	−1.09297	−0.107694	−0.0538468	−	0.998549i	$-0.517148\pi$
$103$	−1.09297	−0.107694	−0.0538468	+	0.998549i	$0.517148\pi$
$104$	−0.0945931	−0.00927561
$105$	−40.5327	−3.95559
$106$	9.96897	0.968272
$107$	5.58489	0.539911	0.269956	−	0.962873i	$-0.412991\pi$
$107$	5.58489	0.539911	0.269956	+	0.962873i	$0.412991\pi$
$108$	−3.12955	−0.301141
$109$	−6.76759	−0.648218	−0.324109	−	0.946020i	$-0.605064\pi$
$109$	−6.76759	−0.648218	−0.324109	+	0.946020i	$0.605064\pi$
$110$	−1.74346	−0.166232
$111$	15.5601	1.47690
$112$	−4.95403	−0.468111
$113$	−0.653864	−0.0615103	−0.0307552	−	0.999527i	$-0.509791\pi$
$113$	−0.653864	−0.0615103	−0.0307552	+	0.999527i	$0.509791\pi$
$114$	−5.30347	−0.496716
$115$	−8.67489	−0.808937
$116$	−6.64864	−0.617311
$117$	−0.144683	−0.0133760
$118$	7.89955	0.727213
$119$	−7.00247	−0.641915
$120$	8.18177	0.746890
$121$	−10.7943	−0.981302
$122$	0.950317	0.0860376
$123$	−22.6986	−2.04666
$124$	5.36736	0.482003
$125$	18.3716	1.64321
$126$	−7.57735	−0.675044
$127$	8.29619	0.736168	0.368084	−	0.929792i	$-0.380014\pi$
$127$	8.29619	0.736168	0.368084	+	0.929792i	$0.380014\pi$
$128$	1.00000	0.0883883
$129$	−26.1277	−2.30042
$130$	−0.363647	−0.0318940
$131$	−15.8736	−1.38688	−0.693440	−	0.720514i	$-0.743906\pi$
$131$	−15.8736	−1.38688	−0.693440	+	0.720514i	$0.743906\pi$
$132$	−0.965201	−0.0840100
$133$	12.3450	1.07045
$134$	−8.06276	−0.696517
$135$	−12.0310	−1.03547
$136$	1.41349	0.121206
$137$	−5.16032	−0.440876	−0.220438	−	0.975401i	$-0.570749\pi$
$137$	−5.16032	−0.440876	−0.220438	+	0.975401i	$0.570749\pi$
$138$	−4.80253	−0.408818
$139$	15.2369	1.29237	0.646187	−	0.763179i	$-0.276363\pi$
$139$	15.2369	1.29237	0.646187	+	0.763179i	$0.276363\pi$
$140$	−19.0449	−1.60959
$141$	−3.04794	−0.256683
$142$	−3.06085	−0.256861
$143$	0.0428993	0.00358742
$144$	1.52953	0.127461
$145$	−25.5596	−2.12261
$146$	12.8264	1.06152
$147$	37.3349	3.07933
$148$	7.31117	0.600974
$149$	16.0979	1.31879	0.659397	−	0.751795i	$-0.270812\pi$
$149$	16.0979	1.31879	0.659397	+	0.751795i	$0.270812\pi$
$150$	20.8121	1.69930
$151$	−7.57215	−0.616212	−0.308106	−	0.951352i	$-0.599695\pi$
$151$	−7.57215	−0.616212	−0.308106	+	0.951352i	$0.599695\pi$
$152$	−2.49192	−0.202121
$153$	2.16198	0.174786
$154$	2.24672	0.181046
$155$	20.6339	1.65735
$156$	−0.201320	−0.0161185
$157$	21.8948	1.74739	0.873696	−	0.486472i	$-0.161717\pi$
$157$	21.8948	1.74739	0.873696	+	0.486472i	$0.161717\pi$
$158$	13.7673	1.09527
$159$	21.2167	1.68259
$160$	3.84433	0.303921
$161$	11.1790	0.881025
$162$	−11.2491	−0.883815
$163$	−6.02116	−0.471614	−0.235807	−	0.971800i	$-0.575773\pi$
$163$	−6.02116	−0.471614	−0.235807	+	0.971800i	$0.575773\pi$
$164$	−10.6653	−0.832817
$165$	−3.71055	−0.288866
$166$	10.1171	0.785237
$167$	19.6784	1.52276	0.761382	−	0.648304i	$-0.224521\pi$
$167$	19.6784	1.52276	0.761382	+	0.648304i	$0.224521\pi$
$168$	−10.5435	−0.813449
$169$	−12.9911	−0.999312
$170$	5.43392	0.416763
$171$	−3.81147	−0.291470
$172$	−12.2765	−0.936076
$173$	12.4812	0.948931	0.474466	−	0.880274i	$-0.342641\pi$
$173$	12.4812	0.948931	0.474466	+	0.880274i	$0.342641\pi$
$174$	−14.1501	−1.07272
$175$	−48.4448	−3.66209
$176$	−0.453515	−0.0341849
$177$	16.8124	1.26370
$178$	−1.77803	−0.133269
$179$	7.84995	0.586733	0.293366	−	0.956000i	$-0.405224\pi$
$179$	7.84995	0.586733	0.293366	+	0.956000i	$0.405224\pi$
$180$	5.88003	0.438272
$181$	−17.9797	−1.33642	−0.668209	−	0.743973i	$-0.732939\pi$
$181$	−17.9797	−1.33642	−0.668209	+	0.743973i	$0.732939\pi$
$182$	0.468616	0.0347362
$183$	2.02253	0.149510
$184$	−2.25654	−0.166354
$185$	28.1065	2.06643
$186$	11.4232	0.837588
$187$	−0.641038	−0.0468774
$188$	−1.43212	−0.104448
$189$	15.5039	1.12774
$190$	−9.57975	−0.694988
$191$	−12.3054	−0.890386	−0.445193	−	0.895435i	$-0.646865\pi$
$191$	−12.3054	−0.890386	−0.445193	+	0.895435i	$0.646865\pi$
$192$	2.12827	0.153595
$193$	5.82207	0.419081	0.209541	−	0.977800i	$-0.432803\pi$
$193$	5.82207	0.419081	0.209541	+	0.977800i	$0.432803\pi$
$194$	−12.9137	−0.927148
$195$	−0.773939	−0.0554229
$196$	17.5424	1.25303
$197$	−21.6619	−1.54334	−0.771671	−	0.636021i	$-0.780579\pi$
$197$	−21.6619	−1.54334	−0.771671	+	0.636021i	$0.780579\pi$
$198$	−0.693665	−0.0492967
$199$	5.71843	0.405369	0.202684	−	0.979244i	$-0.435033\pi$
$199$	5.71843	0.405369	0.202684	+	0.979244i	$0.435033\pi$
$200$	9.77888	0.691471
$201$	−17.1597	−1.21035
$202$	−6.03448	−0.424584
$203$	32.9375	2.31176
$204$	3.00829	0.210622
$205$	−41.0008	−2.86362
$206$	−1.09297	−0.0761509
$207$	−3.45145	−0.239892
$208$	−0.0945931	−0.00655885
$209$	1.13012	0.0781721
$210$	−40.5327	−2.79702
$211$	−20.0839	−1.38264	−0.691318	−	0.722551i	$-0.742970\pi$
$211$	−20.0839	−1.38264	−0.691318	+	0.722551i	$0.742970\pi$
$212$	9.96897	0.684672
$213$	−6.51431	−0.446353
$214$	5.58489	0.381775
$215$	−47.1950	−3.21867
$216$	−3.12955	−0.212939
$217$	−26.5900	−1.80505
$218$	−6.76759	−0.458359
$219$	27.2980	1.84463
$220$	−1.74346	−0.117544
$221$	−0.133706	−0.00899406
$222$	15.5601	1.04433
$223$	−5.45166	−0.365070	−0.182535	−	0.983199i	$-0.558430\pi$
$223$	−5.45166	−0.365070	−0.182535	+	0.983199i	$0.558430\pi$
$224$	−4.95403	−0.331005
$225$	14.9571	0.997141
$226$	−0.653864	−0.0434944
$227$	14.8158	0.983360	0.491680	−	0.870776i	$-0.336383\pi$
$227$	14.8158	0.983360	0.491680	+	0.870776i	$0.336383\pi$
$228$	−5.30347	−0.351231
$229$	21.9863	1.45289	0.726447	−	0.687223i	$-0.241170\pi$
$229$	21.9863	1.45289	0.726447	+	0.687223i	$0.241170\pi$
$230$	−8.67489	−0.572005
$231$	4.78163	0.314608
$232$	−6.64864	−0.436504
$233$	17.7722	1.16429	0.582147	−	0.813084i	$-0.302213\pi$
$233$	17.7722	1.16429	0.582147	+	0.813084i	$0.302213\pi$
$234$	−0.144683	−0.00945824
$235$	−5.50555	−0.359143
$236$	7.89955	0.514217
$237$	29.3005	1.90327
$238$	−7.00247	−0.453902
$239$	−12.0820	−0.781517	−0.390759	−	0.920493i	$-0.627787\pi$
$239$	−12.0820	−0.781517	−0.390759	+	0.920493i	$0.627787\pi$
$240$	8.18177	0.528131
$241$	11.3215	0.729285	0.364642	−	0.931148i	$-0.381191\pi$
$241$	11.3215	0.729285	0.364642	+	0.931148i	$0.381191\pi$
$242$	−10.7943	−0.693885
$243$	−14.5525	−0.933545
$244$	0.950317	0.0608378
$245$	67.4387	4.30850
$246$	−22.6986	−1.44721
$247$	0.235718	0.0149984
$248$	5.36736	0.340827
$249$	21.5318	1.36453
$250$	18.3716	1.16192
$251$	8.60802	0.543333	0.271667	−	0.962391i	$-0.412425\pi$
$251$	8.60802	0.543333	0.271667	+	0.962391i	$0.412425\pi$
$252$	−7.57735	−0.477328
$253$	1.02337	0.0643390
$254$	8.29619	0.520550
$255$	11.5649	0.724219
$256$	1.00000	0.0625000
$257$	−14.3976	−0.898096	−0.449048	−	0.893508i	$-0.648237\pi$
$257$	−14.3976	−0.898096	−0.449048	+	0.893508i	$0.648237\pi$
$258$	−26.1277	−1.62664
$259$	−36.2197	−2.25058
$260$	−0.363647	−0.0225524
$261$	−10.1693	−0.629464
$262$	−15.8736	−0.980673
$263$	−22.3410	−1.37761	−0.688803	−	0.724949i	$-0.741863\pi$
$263$	−22.3410	−1.37761	−0.688803	+	0.724949i	$0.741863\pi$
$264$	−0.965201	−0.0594040
$265$	38.3240	2.35423
$266$	12.3450	0.756922
$267$	−3.78413	−0.231585
$268$	−8.06276	−0.492512
$269$	−1.00000	−0.0609711
$269$	−1.00000	−0.0609711
$270$	−12.0310	−0.732185
$271$	16.7534	1.01770	0.508848	−	0.860856i	$-0.330071\pi$
$271$	16.7534	1.01770	0.508848	+	0.860856i	$0.330071\pi$
$272$	1.41349	0.0857054
$273$	0.997342	0.0603619
$274$	−5.16032	−0.311747
$275$	−4.43487	−0.267432
$276$	−4.80253	−0.289078
$277$	14.7850	0.888345	0.444172	−	0.895941i	$-0.353498\pi$
$277$	14.7850	0.888345	0.444172	+	0.895941i	$0.353498\pi$
$278$	15.2369	0.913846
$279$	8.20955	0.491493
$280$	−19.0449	−1.13815
$281$	25.0439	1.49400	0.746998	−	0.664827i	$-0.231495\pi$
$281$	25.0439	1.49400	0.746998	+	0.664827i	$0.231495\pi$
$282$	−3.04794	−0.181502
$283$	24.6030	1.46249	0.731247	−	0.682113i	$-0.238939\pi$
$283$	24.6030	1.46249	0.731247	+	0.682113i	$0.238939\pi$
$284$	−3.06085	−0.181628
$285$	−20.3883	−1.20770
$286$	0.0428993	0.00253669
$287$	52.8360	3.11881
$288$	1.52953	0.0901286
$289$	−15.0020	−0.882473
$290$	−25.5596	−1.50091
$291$	−27.4838	−1.61113
$292$	12.8264	0.750607
$293$	−11.4618	−0.669607	−0.334804	−	0.942288i	$-0.608670\pi$
$293$	−11.4618	−0.669607	−0.334804	+	0.942288i	$0.608670\pi$
$294$	37.3349	2.17742
$295$	30.3685	1.76812
$296$	7.31117	0.424953
$297$	1.41930	0.0823560
$298$	16.0979	0.932528
$299$	0.213453	0.0123443
$300$	20.8121	1.20159
$301$	60.8182	3.50550
$302$	−7.57215	−0.435728
$303$	−12.8430	−0.737811
$304$	−2.49192	−0.142921
$305$	3.65333	0.209189
$306$	2.16198	0.123592
$307$	−17.7679	−1.01407	−0.507033	−	0.861927i	$-0.669258\pi$
$307$	−17.7679	−1.01407	−0.507033	+	0.861927i	$0.669258\pi$
$308$	2.24672	0.128019
$309$	−2.32614	−0.132329
$310$	20.6339	1.17193
$311$	7.62146	0.432173	0.216087	−	0.976374i	$-0.430671\pi$
$311$	7.62146	0.432173	0.216087	+	0.976374i	$0.430671\pi$
$312$	−0.201320	−0.0113975
$313$	7.97031	0.450509	0.225254	−	0.974300i	$-0.427679\pi$
$313$	7.97031	0.450509	0.225254	+	0.974300i	$0.427679\pi$
$314$	21.8948	1.23559
$315$	−29.1298	−1.64128
$316$	13.7673	0.774469
$317$	1.71640	0.0964028	0.0482014	−	0.998838i	$-0.484651\pi$
$317$	1.71640	0.0964028	0.0482014	+	0.998838i	$0.484651\pi$
$318$	21.2167	1.18977
$319$	3.01525	0.168822
$320$	3.84433	0.214905
$321$	11.8861	0.663420
$322$	11.1790	0.622979
$323$	−3.52230	−0.195986
$324$	−11.2491	−0.624952
$325$	−0.925014	−0.0513106
$326$	−6.02116	−0.333481
$327$	−14.4033	−0.796502
$328$	−10.6653	−0.588891
$329$	7.09477	0.391147
$330$	−3.71055	−0.204259
$331$	−13.7537	−0.755970	−0.377985	−	0.925812i	$-0.623383\pi$
$331$	−13.7537	−0.755970	−0.377985	+	0.925812i	$0.623383\pi$
$332$	10.1171	0.555246
$333$	11.1827	0.612806
$334$	19.6784	1.07676
$335$	−30.9959	−1.69349
$336$	−10.5435	−0.575195
$337$	18.7032	1.01883	0.509413	−	0.860522i	$-0.329863\pi$
$337$	18.7032	1.01883	0.509413	+	0.860522i	$0.329863\pi$
$338$	−12.9911	−0.706620
$339$	−1.39160	−0.0755812
$340$	5.43392	0.294696
$341$	−2.43417	−0.131818
$342$	−3.81147	−0.206101
$343$	−52.2272	−2.82000
$344$	−12.2765	−0.661906
$345$	−18.4625	−0.993988
$346$	12.4812	0.670996
$347$	12.9029	0.692662	0.346331	−	0.938112i	$-0.387427\pi$
$347$	12.9029	0.692662	0.346331	+	0.938112i	$0.387427\pi$
$348$	−14.1501	−0.758525
$349$	−4.29547	−0.229931	−0.114966	−	0.993369i	$-0.536676\pi$
$349$	−4.29547	−0.229931	−0.114966	+	0.993369i	$0.536676\pi$
$350$	−48.4448	−2.58949
$351$	0.296034	0.0158011
$352$	−0.453515	−0.0241724
$353$	−22.3707	−1.19067	−0.595337	−	0.803476i	$-0.702982\pi$
$353$	−22.3707	−1.19067	−0.595337	+	0.803476i	$0.702982\pi$
$354$	16.8124	0.893568
$355$	−11.7669	−0.624523
$356$	−1.77803	−0.0942354
$357$	−14.9031	−0.788758
$358$	7.84995	0.414883
$359$	32.0628	1.69221	0.846105	−	0.533016i	$-0.178942\pi$
$359$	32.0628	1.69221	0.846105	+	0.533016i	$0.178942\pi$
$360$	5.88003	0.309905
$361$	−12.7903	−0.673176
$362$	−17.9797	−0.944991
$363$	−22.9732	−1.20578
$364$	0.468616	0.0245622
$365$	49.3088	2.58094
$366$	2.02253	0.105719
$367$	−4.79576	−0.250337	−0.125168	−	0.992136i	$-0.539947\pi$
$367$	−4.79576	−0.250337	−0.125168	+	0.992136i	$0.539947\pi$
$368$	−2.25654	−0.117630
$369$	−16.3129	−0.849214
$370$	28.1065	1.46119
$371$	−49.3865	−2.56402
$372$	11.4232	0.592264
$373$	32.9108	1.70406	0.852028	−	0.523497i	$-0.175373\pi$
$373$	32.9108	1.70406	0.852028	+	0.523497i	$0.175373\pi$
$374$	−0.641038	−0.0331473
$375$	39.0997	2.01910
$376$	−1.43212	−0.0738561
$377$	0.628915	0.0323908
$378$	15.5039	0.797433
$379$	−8.98570	−0.461565	−0.230782	−	0.973005i	$-0.574129\pi$
$379$	−8.98570	−0.461565	−0.230782	+	0.973005i	$0.574129\pi$
$380$	−9.57975	−0.491431
$381$	17.6565	0.904572
$382$	−12.3054	−0.629598
$383$	0.787688	0.0402490	0.0201245	−	0.999797i	$-0.493594\pi$
$383$	0.787688	0.0402490	0.0201245	+	0.999797i	$0.493594\pi$
$384$	2.12827	0.108608
$385$	8.63715	0.440190
$386$	5.82207	0.296335
$387$	−18.7773	−0.954506
$388$	−12.9137	−0.655593
$389$	−4.15609	−0.210722	−0.105361	−	0.994434i	$-0.533600\pi$
$389$	−4.15609	−0.210722	−0.105361	+	0.994434i	$0.533600\pi$
$390$	−0.773939	−0.0391899
$391$	−3.18960	−0.161305
$392$	17.5424	0.886024
$393$	−33.7833	−1.70414
$394$	−21.6619	−1.09131
$395$	52.9259	2.66299
$396$	−0.693665	−0.0348580
$397$	1.90587	0.0956527	0.0478263	−	0.998856i	$-0.484771\pi$
$397$	1.90587	0.0956527	0.0478263	+	0.998856i	$0.484771\pi$
$398$	5.71843	0.286639
$399$	26.2735	1.31532
$400$	9.77888	0.488944
$401$	−11.0163	−0.550128	−0.275064	−	0.961426i	$-0.588699\pi$
$401$	−11.0163	−0.550128	−0.275064	+	0.961426i	$0.588699\pi$
$402$	−17.1597	−0.855850
$403$	−0.507715	−0.0252911
$404$	−6.03448	−0.300226
$405$	−43.2454	−2.14888
$406$	32.9375	1.63466
$407$	−3.31572	−0.164354
$408$	3.00829	0.148932
$409$	−5.30078	−0.262107	−0.131053	−	0.991375i	$-0.541836\pi$
$409$	−5.30078	−0.262107	−0.131053	+	0.991375i	$0.541836\pi$
$410$	−41.0008	−2.02489
$411$	−10.9826	−0.541730
$412$	−1.09297	−0.0538468
$413$	−39.1346	−1.92569
$414$	−3.45145	−0.169630
$415$	38.8934	1.90920
$416$	−0.0945931	−0.00463781
$417$	32.4281	1.58801
$418$	1.13012	0.0552760
$419$	−25.5590	−1.24864	−0.624319	−	0.781169i	$-0.714624\pi$
$419$	−25.5590	−1.24864	−0.624319	+	0.781169i	$0.714624\pi$
$420$	−40.5327	−1.97779
$421$	−24.8795	−1.21255	−0.606276	−	0.795254i	$-0.707337\pi$
$421$	−24.8795	−1.21255	−0.606276	+	0.795254i	$0.707337\pi$
$422$	−20.0839	−0.977671
$423$	−2.19048	−0.106505
$424$	9.96897	0.484136
$425$	13.8224	0.670483
$426$	−6.51431	−0.315619
$427$	−4.70789	−0.227831
$428$	5.58489	0.269956
$429$	0.0913014	0.00440807
$430$	−47.1950	−2.27595
$431$	19.4524	0.936988	0.468494	−	0.883467i	$-0.344797\pi$
$431$	19.4524	0.936988	0.468494	+	0.883467i	$0.344797\pi$
$432$	−3.12955	−0.150571
$433$	−16.7217	−0.803593	−0.401797	−	0.915729i	$-0.631614\pi$
$433$	−16.7217	−0.803593	−0.401797	+	0.915729i	$0.631614\pi$
$434$	−26.5900	−1.27636
$435$	−54.3977	−2.60817
$436$	−6.76759	−0.324109
$437$	5.62311	0.268990
$438$	27.2980	1.30435
$439$	−14.8215	−0.707390	−0.353695	−	0.935361i	$-0.615075\pi$
$439$	−14.8215	−0.707390	−0.353695	+	0.935361i	$0.615075\pi$
$440$	−1.74346	−0.0831162
$441$	26.8316	1.27770
$442$	−0.133706	−0.00635976
$443$	−24.8635	−1.18130	−0.590651	−	0.806927i	$-0.701129\pi$
$443$	−24.8635	−1.18130	−0.590651	+	0.806927i	$0.701129\pi$
$444$	15.5601	0.738451
$445$	−6.83533	−0.324026
$446$	−5.45166	−0.258144
$447$	34.2607	1.62048
$448$	−4.95403	−0.234056
$449$	22.0294	1.03963	0.519817	−	0.854278i	$-0.326000\pi$
$449$	22.0294	1.03963	0.519817	+	0.854278i	$0.326000\pi$
$450$	14.9571	0.705085
$451$	4.83685	0.227759
$452$	−0.653864	−0.0307552
$453$	−16.1156	−0.757175
$454$	14.8158	0.695340
$455$	1.80152	0.0844564
$456$	−5.30347	−0.248358
$457$	−27.1457	−1.26982	−0.634911	−	0.772585i	$-0.718963\pi$
$457$	−27.1457	−1.26982	−0.634911	+	0.772585i	$0.718963\pi$
$458$	21.9863	1.02735
$459$	−4.42359	−0.206475
$460$	−8.67489	−0.404469
$461$	27.6585	1.28819	0.644093	−	0.764947i	$-0.277235\pi$
$461$	27.6585	1.28819	0.644093	+	0.764947i	$0.277235\pi$
$462$	4.78163	0.222462
$463$	31.2336	1.45155	0.725774	−	0.687933i	$-0.241482\pi$
$463$	31.2336	1.45155	0.725774	+	0.687933i	$0.241482\pi$
$464$	−6.64864	−0.308655
$465$	43.9145	2.03649
$466$	17.7722	0.823280
$467$	−13.4254	−0.621255	−0.310628	−	0.950532i	$-0.600539\pi$
$467$	−13.4254	−0.621255	−0.310628	+	0.950532i	$0.600539\pi$
$468$	−0.144683	−0.00668798
$469$	39.9431	1.84440
$470$	−5.50555	−0.253952
$471$	46.5980	2.14712
$472$	7.89955	0.363606
$473$	5.56758	0.255998
$474$	29.3005	1.34581
$475$	−24.3682	−1.11809
$476$	−7.00247	−0.320958
$477$	15.2479	0.698152
$478$	−12.0820	−0.552616
$479$	2.09566	0.0957533	0.0478767	−	0.998853i	$-0.484755\pi$
$479$	2.09566	0.0957533	0.0478767	+	0.998853i	$0.484755\pi$
$480$	8.18177	0.373445
$481$	−0.691586	−0.0315336
$482$	11.3215	0.515682
$483$	23.7918	1.08257
$484$	−10.7943	−0.490651
$485$	−49.6445	−2.25424
$486$	−14.5525	−0.660116
$487$	−38.1661	−1.72947	−0.864736	−	0.502226i	$-0.832515\pi$
$487$	−38.1661	−1.72947	−0.864736	+	0.502226i	$0.832515\pi$
$488$	0.950317	0.0430188
$489$	−12.8147	−0.579499
$490$	67.4387	3.04657
$491$	6.36732	0.287353	0.143676	−	0.989625i	$-0.454108\pi$
$491$	6.36732	0.287353	0.143676	+	0.989625i	$0.454108\pi$
$492$	−22.6986	−1.02333
$493$	−9.39778	−0.423255
$494$	0.235718	0.0106055
$495$	−2.66668	−0.119858
$496$	5.36736	0.241001
$497$	15.1635	0.680176
$498$	21.5318	0.964865
$499$	33.6939	1.50835	0.754173	−	0.656676i	$-0.228038\pi$
$499$	33.6939	1.50835	0.754173	+	0.656676i	$0.228038\pi$
$500$	18.3716	0.821603
$501$	41.8810	1.87111
$502$	8.60802	0.384195
$503$	−29.6505	−1.32205	−0.661026	−	0.750363i	$-0.729879\pi$
$503$	−29.6505	−1.32205	−0.661026	+	0.750363i	$0.729879\pi$
$504$	−7.57735	−0.337522
$505$	−23.1985	−1.03232
$506$	1.02337	0.0454945
$507$	−27.6485	−1.22791
$508$	8.29619	0.368084
$509$	−3.41765	−0.151485	−0.0757423	−	0.997127i	$-0.524133\pi$
$509$	−3.41765	−0.151485	−0.0757423	+	0.997127i	$0.524133\pi$
$510$	11.5649	0.512100
$511$	−63.5422	−2.81094
$512$	1.00000	0.0441942
$513$	7.79858	0.344316
$514$	−14.3976	−0.635050
$515$	−4.20174	−0.185151
$516$	−26.1277	−1.15021
$517$	0.649488	0.0285645
$518$	−36.2197	−1.59140
$519$	26.5635	1.16601
$520$	−0.363647	−0.0159470
$521$	28.2730	1.23866	0.619330	−	0.785130i	$-0.287404\pi$
$521$	28.2730	1.23866	0.619330	+	0.785130i	$0.287404\pi$
$522$	−10.1693	−0.445099
$523$	−17.2424	−0.753956	−0.376978	−	0.926222i	$-0.623037\pi$
$523$	−17.2424	−0.753956	−0.376978	+	0.926222i	$0.623037\pi$
$524$	−15.8736	−0.693440
$525$	−103.104	−4.49981
$526$	−22.3410	−0.974114
$527$	7.58670	0.330482
$528$	−0.965201	−0.0420050
$529$	−17.9080	−0.778610
$530$	38.3240	1.66469
$531$	12.0826	0.524341
$532$	12.3450	0.535225
$533$	1.00886	0.0436986
$534$	−3.78413	−0.163755
$535$	21.4702	0.928236
$536$	−8.06276	−0.348258
$537$	16.7068	0.720952
$538$	−1.00000	−0.0431131
$539$	−7.95572	−0.342677
$540$	−12.0310	−0.517733
$541$	−2.32570	−0.0999896	−0.0499948	−	0.998749i	$-0.515920\pi$
$541$	−2.32570	−0.0999896	−0.0499948	+	0.998749i	$0.515920\pi$
$542$	16.7534	0.719620
$543$	−38.2656	−1.64213
$544$	1.41349	0.0606029
$545$	−26.0169	−1.11444
$546$	0.997342	0.0426823
$547$	26.5955	1.13714	0.568570	−	0.822635i	$-0.307497\pi$
$547$	26.5955	1.13714	0.568570	+	0.822635i	$0.307497\pi$
$548$	−5.16032	−0.220438
$549$	1.45354	0.0620356
$550$	−4.43487	−0.189103
$551$	16.5679	0.705814
$552$	−4.80253	−0.204409
$553$	−68.2034	−2.90030
$554$	14.7850	0.628155
$555$	59.8183	2.53915
$556$	15.2369	0.646187
$557$	19.7222	0.835658	0.417829	−	0.908526i	$-0.362791\pi$
$557$	19.7222	0.835658	0.417829	+	0.908526i	$0.362791\pi$
$558$	8.20955	0.347538
$559$	1.16127	0.0491166
$560$	−19.0449	−0.804795
$561$	−1.36430	−0.0576009
$562$	25.0439	1.05641
$563$	−25.9388	−1.09319	−0.546595	−	0.837397i	$-0.684076\pi$
$563$	−25.9388	−1.09319	−0.546595	+	0.837397i	$0.684076\pi$
$564$	−3.04794	−0.128342
$565$	−2.51367	−0.105751
$566$	24.6030	1.03414
$567$	55.7285	2.34038
$568$	−3.06085	−0.128430
$569$	−44.9748	−1.88544	−0.942720	−	0.333584i	$-0.891742\pi$
$569$	−44.9748	−1.88544	−0.942720	+	0.333584i	$0.891742\pi$
$570$	−20.3883	−0.853972
$571$	−44.4599	−1.86059	−0.930294	−	0.366814i	$-0.880448\pi$
$571$	−44.4599	−1.86059	−0.930294	+	0.366814i	$0.880448\pi$
$572$	0.0428993	0.00179371
$573$	−26.1892	−1.09407
$574$	52.8360	2.20533
$575$	−22.0664	−0.920234
$576$	1.52953	0.0637305
$577$	−36.6191	−1.52447	−0.762236	−	0.647299i	$-0.775898\pi$
$577$	−36.6191	−1.52447	−0.762236	+	0.647299i	$0.775898\pi$
$578$	−15.0020	−0.624003
$579$	12.3909	0.514949
$580$	−25.5596	−1.06130
$581$	−50.1202	−2.07934
$582$	−27.4838	−1.13924
$583$	−4.52107	−0.187244
$584$	12.8264	0.530759
$585$	−0.556210	−0.0229965
$586$	−11.4618	−0.473484
$587$	20.5256	0.847184	0.423592	−	0.905853i	$-0.360769\pi$
$587$	20.5256	0.847184	0.423592	+	0.905853i	$0.360769\pi$
$588$	37.3349	1.53967
$589$	−13.3750	−0.551108
$590$	30.3685	1.25025
$591$	−46.1023	−1.89639
$592$	7.31117	0.300487
$593$	−1.99866	−0.0820753	−0.0410377	−	0.999158i	$-0.513066\pi$
$593$	−1.99866	−0.0820753	−0.0410377	+	0.999158i	$0.513066\pi$
$594$	1.41930	0.0582345
$595$	−26.9198	−1.10360
$596$	16.0979	0.659397
$597$	12.1704	0.498100
$598$	0.213453	0.00872874
$599$	21.2473	0.868141	0.434070	−	0.900879i	$-0.357077\pi$
$599$	21.2473	0.868141	0.434070	+	0.900879i	$0.357077\pi$
$600$	20.8121	0.849650
$601$	−22.2574	−0.907899	−0.453949	−	0.891028i	$-0.649985\pi$
$601$	−22.2574	−0.907899	−0.453949	+	0.891028i	$0.649985\pi$
$602$	60.8182	2.47876
$603$	−12.3323	−0.502208
$604$	−7.57215	−0.308106
$605$	−41.4970	−1.68709
$606$	−12.8430	−0.521711
$607$	7.20720	0.292531	0.146266	−	0.989245i	$-0.453275\pi$
$607$	7.20720	0.292531	0.146266	+	0.989245i	$0.453275\pi$
$608$	−2.49192	−0.101061
$609$	70.0999	2.84059
$610$	3.65333	0.147919
$611$	0.135469	0.00548048
$612$	2.16198	0.0873928
$613$	15.9436	0.643956	0.321978	−	0.946747i	$-0.395652\pi$
$613$	15.9436	0.643956	0.321978	+	0.946747i	$0.395652\pi$
$614$	−17.7679	−0.717052
$615$	−87.2608	−3.51870
$616$	2.24672	0.0905231
$617$	18.0019	0.724729	0.362364	−	0.932037i	$-0.381970\pi$
$617$	18.0019	0.724729	0.362364	+	0.932037i	$0.381970\pi$
$618$	−2.32614	−0.0935709
$619$	−17.9006	−0.719487	−0.359744	−	0.933051i	$-0.617136\pi$
$619$	−17.9006	−0.719487	−0.359744	+	0.933051i	$0.617136\pi$
$620$	20.6339	0.828677
$621$	7.06196	0.283387
$622$	7.62146	0.305593
$623$	8.80840	0.352901
$624$	−0.201320	−0.00805923
$625$	21.7321	0.869285
$626$	7.97031	0.318558
$627$	2.40520	0.0960545
$628$	21.8948	0.873696
$629$	10.3343	0.412054
$630$	−29.1298	−1.16056
$631$	25.6888	1.02265	0.511327	−	0.859386i	$-0.329154\pi$
$631$	25.6888	1.02265	0.511327	+	0.859386i	$0.329154\pi$
$632$	13.7673	0.547633
$633$	−42.7441	−1.69892
$634$	1.71640	0.0681671
$635$	31.8933	1.26565
$636$	21.2167	0.841295
$637$	−1.65939	−0.0657473
$638$	3.01525	0.119375
$639$	−4.68166	−0.185204
$640$	3.84433	0.151961
$641$	25.4553	1.00542	0.502712	−	0.864454i	$-0.332336\pi$
$641$	25.4553	1.00542	0.502712	+	0.864454i	$0.332336\pi$
$642$	11.8861	0.469109
$643$	22.7411	0.896823	0.448411	−	0.893827i	$-0.351990\pi$
$643$	22.7411	0.896823	0.448411	+	0.893827i	$0.351990\pi$
$644$	11.1790	0.440513
$645$	−100.444	−3.95497
$646$	−3.52230	−0.138583
$647$	28.5975	1.12428	0.562142	−	0.827041i	$-0.309977\pi$
$647$	28.5975	1.12428	0.562142	+	0.827041i	$0.309977\pi$
$648$	−11.2491	−0.441907
$649$	−3.58256	−0.140628
$650$	−0.925014	−0.0362820
$651$	−56.5907	−2.21797
$652$	−6.02116	−0.235807
$653$	−5.54791	−0.217107	−0.108553	−	0.994091i	$-0.534622\pi$
$653$	−5.54791	−0.217107	−0.108553	+	0.994091i	$0.534622\pi$
$654$	−14.4033	−0.563212
$655$	−61.0233	−2.38438
$656$	−10.6653	−0.416409
$657$	19.6184	0.765385
$658$	7.09477	0.276583
$659$	−27.7914	−1.08260	−0.541299	−	0.840830i	$-0.682067\pi$
$659$	−27.7914	−1.08260	−0.541299	+	0.840830i	$0.682067\pi$
$660$	−3.71055	−0.144433
$661$	1.92881	0.0750221	0.0375111	−	0.999296i	$-0.488057\pi$
$661$	1.92881	0.0750221	0.0375111	+	0.999296i	$0.488057\pi$
$662$	−13.7537	−0.534552
$663$	−0.284563	−0.0110515
$664$	10.1171	0.392618
$665$	47.4584	1.84036
$666$	11.1827	0.433320
$667$	15.0029	0.580915
$668$	19.6784	0.761382
$669$	−11.6026	−0.448583
$670$	−30.9959	−1.19748
$671$	−0.430982	−0.0166379
$672$	−10.5435	−0.406725
$673$	19.2283	0.741195	0.370597	−	0.928794i	$-0.379153\pi$
$673$	19.2283	0.741195	0.370597	+	0.928794i	$0.379153\pi$
$674$	18.7032	0.720419
$675$	−30.6035	−1.17793
$676$	−12.9911	−0.499656
$677$	3.56493	0.137011	0.0685057	−	0.997651i	$-0.478177\pi$
$677$	3.56493	0.137011	0.0685057	+	0.997651i	$0.478177\pi$
$678$	−1.39160	−0.0534440
$679$	63.9747	2.45512
$680$	5.43392	0.208381
$681$	31.5320	1.20831
$682$	−2.43417	−0.0932094
$683$	36.0055	1.37771	0.688856	−	0.724899i	$-0.258113\pi$
$683$	36.0055	1.37771	0.688856	+	0.724899i	$0.258113\pi$
$684$	−3.81147	−0.145735
$685$	−19.8380	−0.757971
$686$	−52.2272	−1.99404
$687$	46.7927	1.78525
$688$	−12.2765	−0.468038
$689$	−0.942995	−0.0359253
$690$	−18.4625	−0.702855
$691$	6.11836	0.232754	0.116377	−	0.993205i	$-0.462872\pi$
$691$	6.11836	0.232754	0.116377	+	0.993205i	$0.462872\pi$
$692$	12.4812	0.474466
$693$	3.43644	0.130539
$694$	12.9029	0.489786
$695$	58.5755	2.22190
$696$	−14.1501	−0.536358
$697$	−15.0752	−0.571016
$698$	−4.29547	−0.162586
$699$	37.8240	1.43063
$700$	−48.4448	−1.83104
$701$	−27.4930	−1.03840	−0.519198	−	0.854654i	$-0.673769\pi$
$701$	−27.4930	−1.03840	−0.519198	+	0.854654i	$0.673769\pi$
$702$	0.296034	0.0111731
$703$	−18.2188	−0.687136
$704$	−0.453515	−0.0170925
$705$	−11.7173	−0.441299
$706$	−22.3707	−0.841934
$707$	29.8949	1.12432
$708$	16.8124	0.631848
$709$	−41.8276	−1.57087	−0.785433	−	0.618946i	$-0.787560\pi$
$709$	−41.8276	−1.57087	−0.785433	+	0.618946i	$0.787560\pi$
$710$	−11.7669	−0.441604
$711$	21.0575	0.789717
$712$	−1.77803	−0.0666345
$713$	−12.1117	−0.453585
$714$	−14.9031	−0.557736
$715$	0.164919	0.00616763
$716$	7.84995	0.293366
$717$	−25.7137	−0.960295
$718$	32.0628	1.19657
$719$	−36.4204	−1.35825	−0.679126	−	0.734022i	$-0.737641\pi$
$719$	−36.4204	−1.35825	−0.679126	+	0.734022i	$0.737641\pi$
$720$	5.88003	0.219136
$721$	5.41461	0.201650
$722$	−12.7903	−0.476008
$723$	24.0953	0.896114
$724$	−17.9797	−0.668209
$725$	−65.0162	−2.41464
$726$	−22.9732	−0.852617
$727$	1.23081	0.0456481	0.0228241	−	0.999739i	$-0.492734\pi$
$727$	1.23081	0.0456481	0.0228241	+	0.999739i	$0.492734\pi$
$728$	0.468616	0.0173681
$729$	2.77568	0.102803
$730$	49.3088	1.82500
$731$	−17.3527	−0.641814
$732$	2.02253	0.0747549
$733$	27.5812	1.01873	0.509367	−	0.860550i	$-0.329880\pi$
$733$	27.5812	1.01873	0.509367	+	0.860550i	$0.329880\pi$
$734$	−4.79576	−0.177015
$735$	143.528	5.29410
$736$	−2.25654	−0.0831772
$737$	3.65658	0.134692
$738$	−16.3129	−0.600485
$739$	19.3896	0.713257	0.356629	−	0.934246i	$-0.383926\pi$
$739$	19.3896	0.713257	0.356629	+	0.934246i	$0.383926\pi$
$740$	28.1065	1.03322
$741$	0.501672	0.0184294
$742$	−49.3865	−1.81304
$743$	−3.02696	−0.111048	−0.0555242	−	0.998457i	$-0.517683\pi$
$743$	−3.02696	−0.111048	−0.0555242	+	0.998457i	$0.517683\pi$
$744$	11.4232	0.418794
$745$	61.8858	2.26732
$746$	32.9108	1.20495
$747$	15.4744	0.566178
$748$	−0.641038	−0.0234387
$749$	−27.6677	−1.01096
$750$	39.0997	1.42772
$751$	5.10910	0.186434	0.0932168	−	0.995646i	$-0.470285\pi$
$751$	5.10910	0.186434	0.0932168	+	0.995646i	$0.470285\pi$
$752$	−1.43212	−0.0522241
$753$	18.3202	0.667625
$754$	0.628915	0.0229037
$755$	−29.1098	−1.05942
$756$	15.5039	0.563871
$757$	−27.3805	−0.995161	−0.497580	−	0.867418i	$-0.665778\pi$
$757$	−27.3805	−0.995161	−0.497580	+	0.867418i	$0.665778\pi$
$758$	−8.98570	−0.326375
$759$	2.17802	0.0790570
$760$	−9.57975	−0.347494
$761$	−49.3210	−1.78789	−0.893943	−	0.448181i	$-0.852072\pi$
$761$	−49.3210	−1.78789	−0.893943	+	0.448181i	$0.852072\pi$
$762$	17.6565	0.639629
$763$	33.5268	1.21375
$764$	−12.3054	−0.445193
$765$	8.31136	0.300498
$766$	0.787688	0.0284603
$767$	−0.747243	−0.0269814
$768$	2.12827	0.0767973
$769$	11.8488	0.427277	0.213639	−	0.976913i	$-0.431468\pi$
$769$	11.8488	0.427277	0.213639	+	0.976913i	$0.431468\pi$
$770$	8.63715	0.311261
$771$	−30.6419	−1.10354
$772$	5.82207	0.209541
$773$	14.0516	0.505401	0.252700	−	0.967545i	$-0.418681\pi$
$773$	14.0516	0.505401	0.252700	+	0.967545i	$0.418681\pi$
$774$	−18.7773	−0.674937
$775$	52.4867	1.88538
$776$	−12.9137	−0.463574
$777$	−77.0853	−2.76542
$778$	−4.15609	−0.149003
$779$	26.5770	0.952218
$780$	−0.773939	−0.0277115
$781$	1.38814	0.0496715
$782$	−3.18960	−0.114060
$783$	20.8073	0.743591
$784$	17.5424	0.626513
$785$	84.1707	3.00418
$786$	−33.7833	−1.20501
$787$	−22.7651	−0.811489	−0.405744	−	0.913987i	$-0.632988\pi$
$787$	−22.7651	−0.811489	−0.405744	+	0.913987i	$0.632988\pi$
$788$	−21.6619	−0.771671
$789$	−47.5477	−1.69274
$790$	52.9259	1.88302
$791$	3.23926	0.115175
$792$	−0.693665	−0.0246483
$793$	−0.0898934	−0.00319221
$794$	1.90587	0.0676367
$795$	81.5638	2.89277
$796$	5.71843	0.202684
$797$	32.9141	1.16588	0.582938	−	0.812517i	$-0.301903\pi$
$797$	32.9141	1.16588	0.582938	+	0.812517i	$0.301903\pi$
$798$	26.2735	0.930073
$799$	−2.02429	−0.0716143
$800$	9.77888	0.345736
$801$	−2.71955	−0.0960907
$802$	−11.0163	−0.388999
$803$	−5.81695	−0.205276
$804$	−17.1597	−0.605177
$805$	42.9756	1.51469
$806$	−0.507715	−0.0178835
$807$	−2.12827	−0.0749186
$808$	−6.03448	−0.212292
$809$	−35.8700	−1.26112	−0.630560	−	0.776140i	$-0.717175\pi$
$809$	−35.8700	−1.26112	−0.630560	+	0.776140i	$0.717175\pi$
$810$	−43.2454	−1.51949
$811$	−16.6343	−0.584109	−0.292055	−	0.956402i	$-0.594339\pi$
$811$	−16.6343	−0.584109	−0.292055	+	0.956402i	$0.594339\pi$
$812$	32.9375	1.15588
$813$	35.6558	1.25050
$814$	−3.31572	−0.116216
$815$	−23.1473	−0.810816
$816$	3.00829	0.105311
$817$	30.5921	1.07028
$818$	−5.30078	−0.185337
$819$	0.716764	0.0250458
$820$	−41.0008	−1.43181
$821$	32.4924	1.13399	0.566997	−	0.823720i	$-0.308105\pi$
$821$	32.4924	1.13399	0.566997	+	0.823720i	$0.308105\pi$
$822$	−10.9826	−0.383061
$823$	−29.9885	−1.04533	−0.522666	−	0.852537i	$-0.675063\pi$
$823$	−29.9885	−1.04533	−0.522666	+	0.852537i	$0.675063\pi$
$824$	−1.09297	−0.0380754
$825$	−9.43859	−0.328610
$826$	−39.1346	−1.36167
$827$	31.1200	1.08215	0.541075	−	0.840975i	$-0.318018\pi$
$827$	31.1200	1.08215	0.541075	+	0.840975i	$0.318018\pi$
$828$	−3.45145	−0.119946
$829$	−15.8365	−0.550025	−0.275013	−	0.961441i	$-0.588682\pi$
$829$	−15.8365	−0.550025	−0.275013	+	0.961441i	$0.588682\pi$
$830$	38.8934	1.35001
$831$	31.4665	1.09156
$832$	−0.0945931	−0.00327942
$833$	24.7960	0.859129
$834$	32.4281	1.12289
$835$	75.6504	2.61799
$836$	1.13012	0.0390860
$837$	−16.7974	−0.580604
$838$	−25.5590	−0.882921
$839$	−12.2519	−0.422981	−0.211491	−	0.977380i	$-0.567832\pi$
$839$	−12.2519	−0.422981	−0.211491	+	0.977380i	$0.567832\pi$
$840$	−40.5327	−1.39851
$841$	15.2044	0.524289
$842$	−24.8795	−0.857403
$843$	53.3002	1.83576
$844$	−20.0839	−0.691318
$845$	−49.9419	−1.71805
$846$	−2.19048	−0.0753102
$847$	53.4754	1.83744
$848$	9.96897	0.342336
$849$	52.3617	1.79705
$850$	13.8224	0.474103
$851$	−16.4979	−0.565542
$852$	−6.51431	−0.223176
$853$	−9.41699	−0.322431	−0.161216	−	0.986919i	$-0.551541\pi$
$853$	−9.41699	−0.322431	−0.161216	+	0.986919i	$0.551541\pi$
$854$	−4.70789	−0.161101
$855$	−14.6525	−0.501107
$856$	5.58489	0.190888
$857$	19.2781	0.658529	0.329264	−	0.944238i	$-0.393199\pi$
$857$	19.2781	0.658529	0.329264	+	0.944238i	$0.393199\pi$
$858$	0.0913014	0.00311698
$859$	−48.7953	−1.66487	−0.832437	−	0.554120i	$-0.813055\pi$
$859$	−48.7953	−1.66487	−0.832437	+	0.554120i	$0.813055\pi$
$860$	−47.1950	−1.60934
$861$	112.449	3.83226
$862$	19.4524	0.662551
$863$	27.3052	0.929481	0.464741	−	0.885447i	$-0.346148\pi$
$863$	27.3052	0.929481	0.464741	+	0.885447i	$0.346148\pi$
$864$	−3.12955	−0.106469
$865$	47.9820	1.63144
$866$	−16.7217	−0.568226
$867$	−31.9284	−1.08435
$868$	−26.5900	−0.902524
$869$	−6.24366	−0.211802
$870$	−54.3977	−1.84425
$871$	0.762681	0.0258425
$872$	−6.76759	−0.229180
$873$	−19.7519	−0.668500
$874$	5.62311	0.190205
$875$	−91.0134	−3.07681
$876$	27.2980	0.922314
$877$	−14.3551	−0.484737	−0.242368	−	0.970184i	$-0.577924\pi$
$877$	−14.3551	−0.484737	−0.242368	+	0.970184i	$0.577924\pi$
$878$	−14.8215	−0.500201
$879$	−24.3939	−0.822785
$880$	−1.74346	−0.0587720
$881$	−20.4566	−0.689201	−0.344600	−	0.938749i	$-0.611986\pi$
$881$	−20.4566	−0.689201	−0.344600	+	0.938749i	$0.611986\pi$
$882$	26.8316	0.903468
$883$	13.2088	0.444510	0.222255	−	0.974989i	$-0.428658\pi$
$883$	13.2088	0.444510	0.222255	+	0.974989i	$0.428658\pi$
$884$	−0.133706	−0.00449703
$885$	64.6323	2.17259
$886$	−24.8635	−0.835307
$887$	−21.4469	−0.720115	−0.360058	−	0.932930i	$-0.617243\pi$
$887$	−21.4469	−0.720115	−0.360058	+	0.932930i	$0.617243\pi$
$888$	15.5601	0.522164
$889$	−41.0996	−1.37844
$890$	−6.83533	−0.229121
$891$	5.10164	0.170911
$892$	−5.45166	−0.182535
$893$	3.56873	0.119423
$894$	34.2607	1.14585
$895$	30.1778	1.00873
$896$	−4.95403	−0.165502
$897$	0.454286	0.0151682
$898$	22.0294	0.735132
$899$	−35.6856	−1.19018
$900$	14.9571	0.498571
$901$	14.0910	0.469441
$902$	4.83685	0.161050
$903$	129.438	4.30741
$904$	−0.653864	−0.0217472
$905$	−69.1198	−2.29762
$906$	−16.1156	−0.535404
$907$	−53.7295	−1.78406	−0.892029	−	0.451977i	$-0.850719\pi$
$907$	−53.7295	−1.78406	−0.892029	+	0.451977i	$0.850719\pi$
$908$	14.8158	0.491680
$909$	−9.22993	−0.306137
$910$	1.80152	0.0597197
$911$	−17.7091	−0.586728	−0.293364	−	0.956001i	$-0.594775\pi$
$911$	−17.7091	−0.586728	−0.293364	+	0.956001i	$0.594775\pi$
$912$	−5.30347	−0.175615
$913$	−4.58824	−0.151848
$914$	−27.1457	−0.897900
$915$	7.77528	0.257043
$916$	21.9863	0.726447
$917$	78.6381	2.59686
$918$	−4.42359	−0.146000
$919$	59.7641	1.97144	0.985718	−	0.168406i	$-0.0538619\pi$
$919$	59.7641	1.97144	0.985718	+	0.168406i	$0.0538619\pi$
$920$	−8.67489	−0.286003
$921$	−37.8148	−1.24604
$922$	27.6585	0.910885
$923$	0.289535	0.00953015
$924$	4.78163	0.157304
$925$	71.4950	2.35074
$926$	31.2336	1.02640
$927$	−1.67173	−0.0549070
$928$	−6.64864	−0.218252
$929$	−53.9545	−1.77019	−0.885095	−	0.465410i	$-0.845907\pi$
$929$	−53.9545	−1.77019	−0.885095	+	0.465410i	$0.845907\pi$
$930$	43.9145	1.44001
$931$	−43.7141	−1.43267
$932$	17.7722	0.582147
$933$	16.2205	0.531036
$934$	−13.4254	−0.439294
$935$	−2.46436	−0.0805933
$936$	−0.144683	−0.00472912
$937$	−23.7929	−0.777280	−0.388640	−	0.921390i	$-0.627055\pi$
$937$	−23.7929	−0.777280	−0.388640	+	0.921390i	$0.627055\pi$
$938$	39.9431	1.30419
$939$	16.9630	0.553566
$940$	−5.50555	−0.179571
$941$	−30.7434	−1.00221	−0.501103	−	0.865388i	$-0.667072\pi$
$941$	−30.7434	−1.00221	−0.501103	+	0.865388i	$0.667072\pi$
$942$	46.5980	1.51824
$943$	24.0666	0.783716
$944$	7.89955	0.257108
$945$	59.6020	1.93885
$946$	5.56758	0.181018
$947$	44.2821	1.43897	0.719487	−	0.694506i	$-0.244377\pi$
$947$	44.2821	1.43897	0.719487	+	0.694506i	$0.244377\pi$
$948$	29.3005	0.951635
$949$	−1.21329	−0.0393849
$950$	−24.3682	−0.790608
$951$	3.65297	0.118456
$952$	−7.00247	−0.226951
$953$	−9.40636	−0.304702	−0.152351	−	0.988326i	$-0.548684\pi$
$953$	−9.40636	−0.304702	−0.152351	+	0.988326i	$0.548684\pi$
$954$	15.2479	0.493668
$955$	−47.3059	−1.53078
$956$	−12.0820	−0.390759
$957$	6.41727	0.207441
$958$	2.09566	0.0677078
$959$	25.5644	0.825517
$960$	8.18177	0.264066
$961$	−2.19148	−0.0706930
$962$	−0.691586	−0.0222976
$963$	8.54227	0.275271
$964$	11.3215	0.364642
$965$	22.3819	0.720500
$966$	23.7918	0.765490
$967$	−56.8679	−1.82875	−0.914373	−	0.404872i	$-0.867316\pi$
$967$	−56.8679	−1.82875	−0.914373	+	0.404872i	$0.867316\pi$
$968$	−10.7943	−0.346943
$969$	−7.49640	−0.240819
$970$	−49.6445	−1.59399
$971$	−21.1723	−0.679452	−0.339726	−	0.940524i	$-0.610334\pi$
$971$	−21.1723	−0.679452	−0.339726	+	0.940524i	$0.610334\pi$
$972$	−14.5525	−0.466773
$973$	−75.4838	−2.41990
$974$	−38.1661	−1.22292
$975$	−1.96868	−0.0630482
$976$	0.950317	0.0304189
$977$	6.11162	0.195528	0.0977641	−	0.995210i	$-0.468831\pi$
$977$	6.11162	0.195528	0.0977641	+	0.995210i	$0.468831\pi$
$978$	−12.8147	−0.409768
$979$	0.806362	0.0257714
$980$	67.4387	2.15425
$981$	−10.3513	−0.330490
$982$	6.36732	0.203189
$983$	42.4508	1.35397	0.676985	−	0.735997i	$-0.263286\pi$
$983$	42.4508	1.35397	0.676985	+	0.735997i	$0.263286\pi$
$984$	−22.6986	−0.723604
$985$	−83.2753	−2.65337
$986$	−9.39778	−0.299286
$987$	15.0996	0.480625
$988$	0.235718	0.00749919
$989$	27.7025	0.880887
$990$	−2.66668	−0.0847526
$991$	−52.9269	−1.68128	−0.840640	−	0.541594i	$-0.817821\pi$
$991$	−52.9269	−1.68128	−0.840640	+	0.541594i	$0.817821\pi$
$992$	5.36736	0.170414
$993$	−29.2715	−0.928904
$994$	15.1635	0.480957
$995$	21.9835	0.696925
$996$	21.5318	0.682263
$997$	−43.4208	−1.37515	−0.687575	−	0.726113i	$-0.741325\pi$
$997$	−43.4208	−1.37515	−0.687575	+	0.726113i	$0.741325\pi$
$998$	33.6939	1.06656
$999$	−22.8807	−0.723912

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	538.2.a.e.1.6	✓	7
3.2	odd	2		4842.2.a.n.1.2		7
4.3	odd	2		4304.2.a.h.1.2		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
538.2.a.e.1.6	✓	7	1.1	even	1	trivial
4304.2.a.h.1.2		7	4.3	odd	2
4842.2.a.n.1.2		7	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:	\( N \)	\(=\)	\( 538 = 2 \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	538.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.12827 q^{3} +1.00000 q^{4} +3.84433 q^{5} +2.12827 q^{6} -4.95403 q^{7} +1.00000 q^{8} +1.52953 q^{9} +3.84433 q^{10} -0.453515 q^{11} +2.12827 q^{12} -0.0945931 q^{13} -4.95403 q^{14} +8.18177 q^{15} +1.00000 q^{16} +1.41349 q^{17} +1.52953 q^{18} -2.49192 q^{19} +3.84433 q^{20} -10.5435 q^{21} -0.453515 q^{22} -2.25654 q^{23} +2.12827 q^{24} +9.77888 q^{25} -0.0945931 q^{26} -3.12955 q^{27} -4.95403 q^{28} -6.64864 q^{29} +8.18177 q^{30} +5.36736 q^{31} +1.00000 q^{32} -0.965201 q^{33} +1.41349 q^{34} -19.0449 q^{35} +1.52953 q^{36} +7.31117 q^{37} -2.49192 q^{38} -0.201320 q^{39} +3.84433 q^{40} -10.6653 q^{41} -10.5435 q^{42} -12.2765 q^{43} -0.453515 q^{44} +5.88003 q^{45} -2.25654 q^{46} -1.43212 q^{47} +2.12827 q^{48} +17.5424 q^{49} +9.77888 q^{50} +3.00829 q^{51} -0.0945931 q^{52} +9.96897 q^{53} -3.12955 q^{54} -1.74346 q^{55} -4.95403 q^{56} -5.30347 q^{57} -6.64864 q^{58} +7.89955 q^{59} +8.18177 q^{60} +0.950317 q^{61} +5.36736 q^{62} -7.57735 q^{63} +1.00000 q^{64} -0.363647 q^{65} -0.965201 q^{66} -8.06276 q^{67} +1.41349 q^{68} -4.80253 q^{69} -19.0449 q^{70} -3.06085 q^{71} +1.52953 q^{72} +12.8264 q^{73} +7.31117 q^{74} +20.8121 q^{75} -2.49192 q^{76} +2.24672 q^{77} -0.201320 q^{78} +13.7673 q^{79} +3.84433 q^{80} -11.2491 q^{81} -10.6653 q^{82} +10.1171 q^{83} -10.5435 q^{84} +5.43392 q^{85} -12.2765 q^{86} -14.1501 q^{87} -0.453515 q^{88} -1.77803 q^{89} +5.88003 q^{90} +0.468616 q^{91} -2.25654 q^{92} +11.4232 q^{93} -1.43212 q^{94} -9.57975 q^{95} +2.12827 q^{96} -12.9137 q^{97} +17.5424 q^{98} -0.693665 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{2} + q^{3} + 7 q^{4} + 7 q^{5} + q^{6} + 6 q^{7} + 7 q^{8} + 12 q^{9} + 7 q^{10} - 3 q^{11} + q^{12} - 9 q^{13} + 6 q^{14} + 8 q^{15} + 7 q^{16} + 8 q^{17} + 12 q^{18} - 11 q^{19} + 7 q^{20}+ \cdots - 37 q^{99}+O(q^{100}) \) 7 * q + 7 * q^2 + q^3 + 7 * q^4 + 7 * q^5 + q^6 + 6 * q^7 + 7 * q^8 + 12 * q^9 + 7 * q^10 - 3 * q^11 + q^12 - 9 * q^13 + 6 * q^14 + 8 * q^15 + 7 * q^16 + 8 * q^17 + 12 * q^18 - 11 * q^19 + 7 * q^20 - 6 * q^21 - 3 * q^22 + 12 * q^23 + q^24 + 22 * q^25 - 9 * q^26 - 14 * q^27 + 6 * q^28 - 5 * q^29 + 8 * q^30 + 14 * q^31 + 7 * q^32 - 4 * q^33 + 8 * q^34 - 4 * q^35 + 12 * q^36 + 13 * q^37 - 11 * q^38 - 18 * q^39 + 7 * q^40 + 12 * q^41 - 6 * q^42 - 11 * q^43 - 3 * q^44 + 3 * q^45 + 12 * q^46 + 2 * q^47 + q^48 + 15 * q^49 + 22 * q^50 - 26 * q^51 - 9 * q^52 + 19 * q^53 - 14 * q^54 - 40 * q^55 + 6 * q^56 - 12 * q^57 - 5 * q^58 - 9 * q^59 + 8 * q^60 - 3 * q^61 + 14 * q^62 - 26 * q^63 + 7 * q^64 - 10 * q^65 - 4 * q^66 - 33 * q^67 + 8 * q^68 - 64 * q^69 - 4 * q^70 + 28 * q^71 + 12 * q^72 - 14 * q^73 + 13 * q^74 - 45 * q^75 - 11 * q^76 + 10 * q^77 - 18 * q^78 + 2 * q^79 + 7 * q^80 + 15 * q^81 + 12 * q^82 - 7 * q^83 - 6 * q^84 - 16 * q^85 - 11 * q^86 + 16 * q^87 - 3 * q^88 + 18 * q^89 + 3 * q^90 - 26 * q^91 + 12 * q^92 + 6 * q^93 + 2 * q^94 - 34 * q^95 + q^96 - 4 * q^97 + 15 * q^98 - 37 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more