LMFDB - Embedded newform 3381.2.a.bl.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3381,2,Mod(1,3381)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3381, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3381.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3381 = 3 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3381.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$26.9974209234$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 4x^{9} - 6x^{8} + 36x^{7} + x^{6} - 100x^{5} + 26x^{4} + 108x^{3} - 33x^{2} - 36x + 14 $ x^10 - 4x^9 - 6x^8 + 36x^7 + x^6 - 100x^5 + 26x^4 + 108x^3 - 33x^2 - 36x + 14
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.28424$ of defining polynomial
Character	$\chi$	$=$	3381.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.28424 q^{2} +1.00000 q^{3} -0.350734 q^{4} -1.91754 q^{5} -1.28424 q^{6} +3.01890 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.28424 q^{2} +1.00000 q^{3} -0.350734 q^{4} -1.91754 q^{5} -1.28424 q^{6} +3.01890 q^{8} +1.00000 q^{9} +2.46257 q^{10} -4.55491 q^{11} -0.350734 q^{12} -6.28282 q^{13} -1.91754 q^{15} -3.17552 q^{16} -1.58249 q^{17} -1.28424 q^{18} +7.65943 q^{19} +0.672545 q^{20} +5.84959 q^{22} -1.00000 q^{23} +3.01890 q^{24} -1.32305 q^{25} +8.06864 q^{26} +1.00000 q^{27} +7.51029 q^{29} +2.46257 q^{30} -7.98032 q^{31} -1.95968 q^{32} -4.55491 q^{33} +2.03230 q^{34} -0.350734 q^{36} -6.74992 q^{37} -9.83652 q^{38} -6.28282 q^{39} -5.78885 q^{40} +6.77718 q^{41} -10.0274 q^{43} +1.59756 q^{44} -1.91754 q^{45} +1.28424 q^{46} -1.70737 q^{47} -3.17552 q^{48} +1.69912 q^{50} -1.58249 q^{51} +2.20360 q^{52} -8.10484 q^{53} -1.28424 q^{54} +8.73421 q^{55} +7.65943 q^{57} -9.64500 q^{58} +9.86090 q^{59} +0.672545 q^{60} +3.04974 q^{61} +10.2486 q^{62} +8.86773 q^{64} +12.0475 q^{65} +5.84959 q^{66} +0.0765035 q^{67} +0.555034 q^{68} -1.00000 q^{69} +7.63603 q^{71} +3.01890 q^{72} -0.159423 q^{73} +8.66850 q^{74} -1.32305 q^{75} -2.68642 q^{76} +8.06864 q^{78} +17.5583 q^{79} +6.08917 q^{80} +1.00000 q^{81} -8.70350 q^{82} +2.50979 q^{83} +3.03449 q^{85} +12.8776 q^{86} +7.51029 q^{87} -13.7508 q^{88} +3.99800 q^{89} +2.46257 q^{90} +0.350734 q^{92} -7.98032 q^{93} +2.19267 q^{94} -14.6872 q^{95} -1.95968 q^{96} +2.15733 q^{97} -4.55491 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 4 q^{2} + 10 q^{3} + 8 q^{4} + 4 q^{5} + 4 q^{6} + 12 q^{8} + 10 q^{9}+O(q^{10}) $ 10 * q + 4 * q^2 + 10 * q^3 + 8 * q^4 + 4 * q^5 + 4 * q^6 + 12 * q^8 + 10 * q^9	$ 10 q + 4 q^{2} + 10 q^{3} + 8 q^{4} + 4 q^{5} + 4 q^{6} + 12 q^{8} + 10 q^{9} + 8 q^{10} + 2 q^{11} + 8 q^{12} + 4 q^{15} + 4 q^{16} + 12 q^{17} + 4 q^{18} + 26 q^{19} + 24 q^{20} - 8 q^{22} - 10 q^{23} + 12 q^{24} - 2 q^{25} + 4 q^{26} + 10 q^{27} + 16 q^{29} + 8 q^{30} + 12 q^{31} + 8 q^{32} + 2 q^{33} + 28 q^{34} + 8 q^{36} - 8 q^{37} + 32 q^{38} + 4 q^{40} + 10 q^{41} - 4 q^{43} - 16 q^{44} + 4 q^{45} - 4 q^{46} + 2 q^{47} + 4 q^{48} - 8 q^{50} + 12 q^{51} + 24 q^{52} + 14 q^{53} + 4 q^{54} + 16 q^{55} + 26 q^{57} - 8 q^{58} + 38 q^{59} + 24 q^{60} + 14 q^{61} - 8 q^{62} + 8 q^{64} + 12 q^{65} - 8 q^{66} + 8 q^{68} - 10 q^{69} + 24 q^{71} + 12 q^{72} + 8 q^{73} - 8 q^{74} - 2 q^{75} + 64 q^{76} + 4 q^{78} - 16 q^{79} + 28 q^{80} + 10 q^{81} - 40 q^{82} + 28 q^{83} - 4 q^{85} + 20 q^{86} + 16 q^{87} - 68 q^{88} + 32 q^{89} + 8 q^{90} - 8 q^{92} + 12 q^{93} + 56 q^{94} + 8 q^{95} + 8 q^{96} + 4 q^{97} + 2 q^{99}+O(q^{100}) $ 10 * q + 4 * q^2 + 10 * q^3 + 8 * q^4 + 4 * q^5 + 4 * q^6 + 12 * q^8 + 10 * q^9 + 8 * q^10 + 2 * q^11 + 8 * q^12 + 4 * q^15 + 4 * q^16 + 12 * q^17 + 4 * q^18 + 26 * q^19 + 24 * q^20 - 8 * q^22 - 10 * q^23 + 12 * q^24 - 2 * q^25 + 4 * q^26 + 10 * q^27 + 16 * q^29 + 8 * q^30 + 12 * q^31 + 8 * q^32 + 2 * q^33 + 28 * q^34 + 8 * q^36 - 8 * q^37 + 32 * q^38 + 4 * q^40 + 10 * q^41 - 4 * q^43 - 16 * q^44 + 4 * q^45 - 4 * q^46 + 2 * q^47 + 4 * q^48 - 8 * q^50 + 12 * q^51 + 24 * q^52 + 14 * q^53 + 4 * q^54 + 16 * q^55 + 26 * q^57 - 8 * q^58 + 38 * q^59 + 24 * q^60 + 14 * q^61 - 8 * q^62 + 8 * q^64 + 12 * q^65 - 8 * q^66 + 8 * q^68 - 10 * q^69 + 24 * q^71 + 12 * q^72 + 8 * q^73 - 8 * q^74 - 2 * q^75 + 64 * q^76 + 4 * q^78 - 16 * q^79 + 28 * q^80 + 10 * q^81 - 40 * q^82 + 28 * q^83 - 4 * q^85 + 20 * q^86 + 16 * q^87 - 68 * q^88 + 32 * q^89 + 8 * q^90 - 8 * q^92 + 12 * q^93 + 56 * q^94 + 8 * q^95 + 8 * q^96 + 4 * q^97 + 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.28424	−0.908093	−0.454047	−	0.890978i	$-0.650020\pi$
$2$	−1.28424	−0.908093	−0.454047	+	0.890978i	$0.650020\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−0.350734	−0.175367
$5$	−1.91754	−0.857548	−0.428774	−	0.903412i	$-0.641054\pi$
$5$	−1.91754	−0.857548	−0.428774	+	0.903412i	$0.641054\pi$
$6$	−1.28424	−0.524288
$7$	0	0
$7$	0	0
$8$	3.01890	1.06734
$9$	1.00000	0.333333
$10$	2.46257	0.778734
$11$	−4.55491	−1.37336	−0.686679	−	0.726961i	$-0.740932\pi$
$11$	−4.55491	−1.37336	−0.686679	+	0.726961i	$0.740932\pi$
$12$	−0.350734	−0.101248
$13$	−6.28282	−1.74254	−0.871271	−	0.490803i	$-0.836704\pi$
$13$	−6.28282	−1.74254	−0.871271	+	0.490803i	$0.836704\pi$
$14$	0	0
$15$	−1.91754	−0.495106
$16$	−3.17552	−0.793880
$17$	−1.58249	−0.383811	−0.191906	−	0.981413i	$-0.561467\pi$
$17$	−1.58249	−0.383811	−0.191906	+	0.981413i	$0.561467\pi$
$18$	−1.28424	−0.302698
$19$	7.65943	1.75719	0.878596	−	0.477565i	$-0.158480\pi$
$19$	7.65943	1.75719	0.878596	+	0.477565i	$0.158480\pi$
$20$	0.672545	0.150386
$21$	0	0
$22$	5.84959	1.24714
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	3.01890	0.616231
$25$	−1.32305	−0.264611
$26$	8.06864	1.58239
$27$	1.00000	0.192450
$28$	0	0
$29$	7.51029	1.39463	0.697313	−	0.716767i	$-0.254379\pi$
$29$	7.51029	1.39463	0.697313	+	0.716767i	$0.254379\pi$
$30$	2.46257	0.449602
$31$	−7.98032	−1.43331	−0.716654	−	0.697429i	$-0.754327\pi$
$31$	−7.98032	−1.43331	−0.716654	+	0.697429i	$0.754327\pi$
$32$	−1.95968	−0.346426
$33$	−4.55491	−0.792909
$34$	2.03230	0.348537
$35$	0	0
$36$	−0.350734	−0.0584556
$37$	−6.74992	−1.10968	−0.554840	−	0.831957i	$-0.687221\pi$
$37$	−6.74992	−1.10968	−0.554840	+	0.831957i	$0.687221\pi$
$38$	−9.83652	−1.59569
$39$	−6.28282	−1.00606
$40$	−5.78885	−0.915298
$41$	6.77718	1.05842	0.529208	−	0.848492i	$-0.322489\pi$
$41$	6.77718	1.05842	0.529208	+	0.848492i	$0.322489\pi$
$42$	0	0
$43$	−10.0274	−1.52917	−0.764586	−	0.644522i	$-0.777056\pi$
$43$	−10.0274	−1.52917	−0.764586	+	0.644522i	$0.777056\pi$
$44$	1.59756	0.240841
$45$	−1.91754	−0.285849
$46$	1.28424	0.189351
$47$	−1.70737	−0.249046	−0.124523	−	0.992217i	$-0.539740\pi$
$47$	−1.70737	−0.249046	−0.124523	+	0.992217i	$0.539740\pi$
$48$	−3.17552	−0.458347
$49$	0	0
$50$	1.69912	0.240291
$51$	−1.58249	−0.221594
$52$	2.20360	0.305584
$53$	−8.10484	−1.11328	−0.556642	−	0.830752i	$-0.687911\pi$
$53$	−8.10484	−1.11328	−0.556642	+	0.830752i	$0.687911\pi$
$54$	−1.28424	−0.174763
$55$	8.73421	1.17772
$56$	0	0
$57$	7.65943	1.01452
$58$	−9.64500	−1.26645
$59$	9.86090	1.28378	0.641890	−	0.766797i	$-0.278151\pi$
$59$	9.86090	1.28378	0.641890	+	0.766797i	$0.278151\pi$
$60$	0.672545	0.0868251
$61$	3.04974	0.390480	0.195240	−	0.980756i	$-0.437451\pi$
$61$	3.04974	0.390480	0.195240	+	0.980756i	$0.437451\pi$
$62$	10.2486	1.30158
$63$	0	0
$64$	8.86773	1.10847
$65$	12.0475	1.49431
$66$	5.84959	0.720035
$67$	0.0765035	0.00934639	0.00467320	−	0.999989i	$-0.498512\pi$
$67$	0.0765035	0.00934639	0.00467320	+	0.999989i	$0.498512\pi$
$68$	0.555034	0.0673078
$69$	−1.00000	−0.120386
$70$	0	0
$71$	7.63603	0.906230	0.453115	−	0.891452i	$-0.350313\pi$
$71$	7.63603	0.906230	0.453115	+	0.891452i	$0.350313\pi$
$72$	3.01890	0.355781
$73$	−0.159423	−0.0186591	−0.00932954	−	0.999956i	$-0.502970\pi$
$73$	−0.159423	−0.0186591	−0.00932954	+	0.999956i	$0.502970\pi$
$74$	8.66850	1.00769
$75$	−1.32305	−0.152773
$76$	−2.68642	−0.308153
$77$	0	0
$78$	8.06864	0.913593
$79$	17.5583	1.97546	0.987732	−	0.156161i	$-0.0499120\pi$
$79$	17.5583	1.97546	0.987732	+	0.156161i	$0.0499120\pi$
$80$	6.08917	0.680790
$81$	1.00000	0.111111
$82$	−8.70350	−0.961141
$83$	2.50979	0.275486	0.137743	−	0.990468i	$-0.456015\pi$
$83$	2.50979	0.275486	0.137743	+	0.990468i	$0.456015\pi$
$84$	0	0
$85$	3.03449	0.329137
$86$	12.8776	1.38863
$87$	7.51029	0.805187
$88$	−13.7508	−1.46584
$89$	3.99800	0.423787	0.211894	−	0.977293i	$-0.432037\pi$
$89$	3.99800	0.423787	0.211894	+	0.977293i	$0.432037\pi$
$90$	2.46257	0.259578
$91$	0	0
$92$	0.350734	0.0365665
$93$	−7.98032	−0.827521
$94$	2.19267	0.226157
$95$	−14.6872	−1.50688
$96$	−1.95968	−0.200009
$97$	2.15733	0.219044	0.109522	−	0.993984i	$-0.465068\pi$
$97$	2.15733	0.219044	0.109522	+	0.993984i	$0.465068\pi$
$98$	0	0
$99$	−4.55491	−0.457786
$100$	0.464040	0.0464040
$101$	−15.4997	−1.54228	−0.771138	−	0.636668i	$-0.780312\pi$
$101$	−15.4997	−1.54228	−0.771138	+	0.636668i	$0.780312\pi$
$102$	2.03230	0.201228
$103$	−8.57860	−0.845275	−0.422637	−	0.906299i	$-0.638896\pi$
$103$	−8.57860	−0.845275	−0.422637	+	0.906299i	$0.638896\pi$
$104$	−18.9672	−1.85989
$105$	0	0
$106$	10.4085	1.01097
$107$	3.26771	0.315901	0.157951	−	0.987447i	$-0.449511\pi$
$107$	3.26771	0.315901	0.157951	+	0.987447i	$0.449511\pi$
$108$	−0.350734	−0.0337494
$109$	9.71577	0.930602	0.465301	−	0.885152i	$-0.345946\pi$
$109$	9.71577	0.930602	0.465301	+	0.885152i	$0.345946\pi$
$110$	−11.2168	−1.06948
$111$	−6.74992	−0.640674
$112$	0	0
$113$	1.96980	0.185303	0.0926515	−	0.995699i	$-0.470466\pi$
$113$	1.96980	0.185303	0.0926515	+	0.995699i	$0.470466\pi$
$114$	−9.83652	−0.921275
$115$	1.91754	0.178811
$116$	−2.63411	−0.244571
$117$	−6.28282	−0.580847
$118$	−12.6637	−1.16579
$119$	0	0
$120$	−5.78885	−0.528447
$121$	9.74723	0.886112
$122$	−3.91660	−0.354592
$123$	6.77718	0.611077
$124$	2.79897	0.251355
$125$	12.1247	1.08446
$126$	0	0
$127$	−21.6822	−1.92399	−0.961994	−	0.273070i	$-0.911961\pi$
$127$	−21.6822	−1.92399	−0.961994	+	0.273070i	$0.911961\pi$
$128$	−7.46892	−0.660165
$129$	−10.0274	−0.882868
$130$	−15.4719	−1.35698
$131$	8.95496	0.782398	0.391199	−	0.920306i	$-0.372060\pi$
$131$	8.95496	0.782398	0.391199	+	0.920306i	$0.372060\pi$
$132$	1.59756	0.139050
$133$	0	0
$134$	−0.0982487	−0.00848739
$135$	−1.91754	−0.165035
$136$	−4.77740	−0.409658
$137$	12.2161	1.04369	0.521846	−	0.853040i	$-0.325244\pi$
$137$	12.2161	1.04369	0.521846	+	0.853040i	$0.325244\pi$
$138$	1.28424	0.109322
$139$	16.7758	1.42290	0.711451	−	0.702736i	$-0.248039\pi$
$139$	16.7758	1.42290	0.711451	+	0.702736i	$0.248039\pi$
$140$	0	0
$141$	−1.70737	−0.143787
$142$	−9.80648	−0.822942
$143$	28.6177	2.39313
$144$	−3.17552	−0.264627
$145$	−14.4013	−1.19596
$146$	0.204737	0.0169442
$147$	0	0
$148$	2.36742	0.194601
$149$	−7.14874	−0.585648	−0.292824	−	0.956166i	$-0.594595\pi$
$149$	−7.14874	−0.585648	−0.292824	+	0.956166i	$0.594595\pi$
$150$	1.69912	0.138732
$151$	−22.3685	−1.82032	−0.910162	−	0.414253i	$-0.864043\pi$
$151$	−22.3685	−1.82032	−0.910162	+	0.414253i	$0.864043\pi$
$152$	23.1230	1.87553
$153$	−1.58249	−0.127937
$154$	0	0
$155$	15.3026	1.22913
$156$	2.20360	0.176429
$157$	10.2145	0.815203	0.407602	−	0.913160i	$-0.366365\pi$
$157$	10.2145	0.815203	0.407602	+	0.913160i	$0.366365\pi$
$158$	−22.5490	−1.79390
$159$	−8.10484	−0.642755
$160$	3.75776	0.297077
$161$	0	0
$162$	−1.28424	−0.100899
$163$	−16.4072	−1.28511	−0.642556	−	0.766239i	$-0.722126\pi$
$163$	−16.4072	−1.28511	−0.642556	+	0.766239i	$0.722126\pi$
$164$	−2.37698	−0.185611
$165$	8.73421	0.679957
$166$	−3.22317	−0.250167
$167$	−18.8255	−1.45676	−0.728382	−	0.685171i	$-0.759727\pi$
$167$	−18.8255	−1.45676	−0.728382	+	0.685171i	$0.759727\pi$
$168$	0	0
$169$	26.4739	2.03645
$170$	−3.89701	−0.298887
$171$	7.65943	0.585731
$172$	3.51696	0.268166
$173$	−2.20415	−0.167579	−0.0837893	−	0.996483i	$-0.526702\pi$
$173$	−2.20415	−0.167579	−0.0837893	+	0.996483i	$0.526702\pi$
$174$	−9.64500	−0.731185
$175$	0	0
$176$	14.4642	1.09028
$177$	9.86090	0.741191
$178$	−5.13439	−0.384838
$179$	25.1880	1.88264	0.941318	−	0.337520i	$-0.109588\pi$
$179$	25.1880	1.88264	0.941318	+	0.337520i	$0.109588\pi$
$180$	0.672545	0.0501285
$181$	24.9888	1.85740	0.928702	−	0.370827i	$-0.120926\pi$
$181$	24.9888	1.85740	0.928702	+	0.370827i	$0.120926\pi$
$182$	0	0
$183$	3.04974	0.225444
$184$	−3.01890	−0.222556
$185$	12.9432	0.951604
$186$	10.2486	0.751466
$187$	7.20813	0.527110
$188$	0.598833	0.0436744
$189$	0	0
$190$	18.8619	1.36839
$191$	−1.38554	−0.100254	−0.0501272	−	0.998743i	$-0.515963\pi$
$191$	−1.38554	−0.100254	−0.0501272	+	0.998743i	$0.515963\pi$
$192$	8.86773	0.639974
$193$	−5.21684	−0.375516	−0.187758	−	0.982215i	$-0.560122\pi$
$193$	−5.21684	−0.375516	−0.187758	+	0.982215i	$0.560122\pi$
$194$	−2.77053	−0.198912
$195$	12.0475	0.862742
$196$	0	0
$197$	8.15087	0.580725	0.290363	−	0.956917i	$-0.406224\pi$
$197$	8.15087	0.580725	0.290363	+	0.956917i	$0.406224\pi$
$198$	5.84959	0.415712
$199$	9.22963	0.654271	0.327136	−	0.944977i	$-0.393917\pi$
$199$	9.22963	0.654271	0.327136	+	0.944977i	$0.393917\pi$
$200$	−3.99417	−0.282430
$201$	0.0765035	0.00539614
$202$	19.9053	1.40053
$203$	0	0
$204$	0.555034	0.0388602
$205$	−12.9955	−0.907644
$206$	11.0170	0.767588
$207$	−1.00000	−0.0695048
$208$	19.9512	1.38337
$209$	−34.8880	−2.41325
$210$	0	0
$211$	−14.6092	−1.00574	−0.502871	−	0.864362i	$-0.667723\pi$
$211$	−14.6092	−1.00574	−0.502871	+	0.864362i	$0.667723\pi$
$212$	2.84264	0.195233
$213$	7.63603	0.523212
$214$	−4.19651	−0.286868
$215$	19.2280	1.31134
$216$	3.01890	0.205410
$217$	0	0
$218$	−12.4774	−0.845073
$219$	−0.159423	−0.0107728
$220$	−3.06338	−0.206533
$221$	9.94253	0.668807
$222$	8.66850	0.581792
$223$	14.6121	0.978501	0.489251	−	0.872143i	$-0.337270\pi$
$223$	14.6121	0.978501	0.489251	+	0.872143i	$0.337270\pi$
$224$	0	0
$225$	−1.32305	−0.0882036
$226$	−2.52969	−0.168272
$227$	25.0807	1.66466	0.832332	−	0.554277i	$-0.187005\pi$
$227$	25.0807	1.66466	0.832332	+	0.554277i	$0.187005\pi$
$228$	−2.68642	−0.177912
$229$	−7.98892	−0.527923	−0.263961	−	0.964533i	$-0.585029\pi$
$229$	−7.98892	−0.527923	−0.263961	+	0.964533i	$0.585029\pi$
$230$	−2.46257	−0.162377
$231$	0	0
$232$	22.6728	1.48854
$233$	14.4056	0.943741	0.471871	−	0.881668i	$-0.343579\pi$
$233$	14.4056	0.943741	0.471871	+	0.881668i	$0.343579\pi$
$234$	8.06864	0.527463
$235$	3.27395	0.213569
$236$	−3.45855	−0.225132
$237$	17.5583	1.14053
$238$	0	0
$239$	−0.892671	−0.0577421	−0.0288710	−	0.999583i	$-0.509191\pi$
$239$	−0.892671	−0.0577421	−0.0288710	+	0.999583i	$0.509191\pi$
$240$	6.08917	0.393054
$241$	3.23488	0.208377	0.104188	−	0.994558i	$-0.466775\pi$
$241$	3.23488	0.208377	0.104188	+	0.994558i	$0.466775\pi$
$242$	−12.5178	−0.804672
$243$	1.00000	0.0641500
$244$	−1.06965	−0.0684772
$245$	0	0
$246$	−8.70350	−0.554915
$247$	−48.1228	−3.06198
$248$	−24.0918	−1.52983
$249$	2.50979	0.159052
$250$	−15.5710	−0.984795
$251$	19.8482	1.25281	0.626405	−	0.779498i	$-0.284526\pi$
$251$	19.8482	1.25281	0.626405	+	0.779498i	$0.284526\pi$
$252$	0	0
$253$	4.55491	0.286365
$254$	27.8452	1.74716
$255$	3.03449	0.190027
$256$	−8.14360	−0.508975
$257$	−0.0229517	−0.00143169	−0.000715844	−	1.00000i	$-0.500228\pi$
$257$	−0.0229517	−0.00143169	−0.000715844		1.00000i	$0.500228\pi$
$258$	12.8776	0.801726
$259$	0	0
$260$	−4.22548	−0.262053
$261$	7.51029	0.464875
$262$	−11.5003	−0.710491
$263$	25.8111	1.59158	0.795791	−	0.605571i	$-0.207055\pi$
$263$	25.8111	1.59158	0.795791	+	0.605571i	$0.207055\pi$
$264$	−13.7508	−0.846305
$265$	15.5413	0.954695
$266$	0	0
$267$	3.99800	0.244674
$268$	−0.0268324	−0.00163905
$269$	23.2678	1.41866	0.709331	−	0.704876i	$-0.248997\pi$
$269$	23.2678	1.41866	0.709331	+	0.704876i	$0.248997\pi$
$270$	2.46257	0.149867
$271$	9.30029	0.564952	0.282476	−	0.959274i	$-0.408844\pi$
$271$	9.30029	0.564952	0.282476	+	0.959274i	$0.408844\pi$
$272$	5.02524	0.304700
$273$	0	0
$274$	−15.6884	−0.947769
$275$	6.02640	0.363405
$276$	0.350734	0.0211117
$277$	−6.76040	−0.406193	−0.203096	−	0.979159i	$-0.565101\pi$
$277$	−6.76040	−0.406193	−0.203096	+	0.979159i	$0.565101\pi$
$278$	−21.5441	−1.29213
$279$	−7.98032	−0.477769
$280$	0	0
$281$	0.713658	0.0425733	0.0212866	−	0.999773i	$-0.493224\pi$
$281$	0.713658	0.0425733	0.0212866	+	0.999773i	$0.493224\pi$
$282$	2.19267	0.130572
$283$	−23.2461	−1.38184	−0.690919	−	0.722932i	$-0.742794\pi$
$283$	−23.2461	−1.38184	−0.690919	+	0.722932i	$0.742794\pi$
$284$	−2.67821	−0.158923
$285$	−14.6872	−0.869996
$286$	−36.7519	−2.17319
$287$	0	0
$288$	−1.95968	−0.115475
$289$	−14.4957	−0.852689
$290$	18.4946	1.08604
$291$	2.15733	0.126465
$292$	0.0559151	0.00327218
$293$	28.2669	1.65137	0.825685	−	0.564132i	$-0.190789\pi$
$293$	28.2669	1.65137	0.825685	+	0.564132i	$0.190789\pi$
$294$	0	0
$295$	−18.9086	−1.10090
$296$	−20.3773	−1.18441
$297$	−4.55491	−0.264303
$298$	9.18068	0.531823
$299$	6.28282	0.363345
$300$	0.464040	0.0267913
$301$	0	0
$302$	28.7265	1.65302
$303$	−15.4997	−0.890433
$304$	−24.3227	−1.39500
$305$	−5.84799	−0.334855
$306$	2.03230	0.116179
$307$	16.8156	0.959718	0.479859	−	0.877346i	$-0.340688\pi$
$307$	16.8156	0.959718	0.479859	+	0.877346i	$0.340688\pi$
$308$	0	0
$309$	−8.57860	−0.488020
$310$	−19.6521	−1.11617
$311$	10.4865	0.594635	0.297317	−	0.954779i	$-0.403908\pi$
$311$	10.4865	0.594635	0.297317	+	0.954779i	$0.403908\pi$
$312$	−18.9672	−1.07381
$313$	26.1063	1.47562	0.737809	−	0.675010i	$-0.235861\pi$
$313$	26.1063	1.47562	0.737809	+	0.675010i	$0.235861\pi$
$314$	−13.1178	−0.740280
$315$	0	0
$316$	−6.15829	−0.346431
$317$	29.4405	1.65354	0.826772	−	0.562538i	$-0.190175\pi$
$317$	29.4405	1.65354	0.826772	+	0.562538i	$0.190175\pi$
$318$	10.4085	0.583682
$319$	−34.2087	−1.91532
$320$	−17.0042	−0.950564
$321$	3.26771	0.182386
$322$	0	0
$323$	−12.1210	−0.674431
$324$	−0.350734	−0.0194852
$325$	8.31251	0.461095
$326$	21.0707	1.16700
$327$	9.71577	0.537283
$328$	20.4596	1.12969
$329$	0	0
$330$	−11.2168	−0.617465
$331$	8.48076	0.466145	0.233072	−	0.972459i	$-0.425122\pi$
$331$	8.48076	0.466145	0.233072	+	0.972459i	$0.425122\pi$
$332$	−0.880269	−0.0483111
$333$	−6.74992	−0.369893
$334$	24.1765	1.32288
$335$	−0.146698	−0.00801498
$336$	0	0
$337$	−24.8097	−1.35147	−0.675735	−	0.737145i	$-0.736173\pi$
$337$	−24.8097	−1.35147	−0.675735	+	0.737145i	$0.736173\pi$
$338$	−33.9987	−1.84929
$339$	1.96980	0.106985
$340$	−1.06430	−0.0577197
$341$	36.3497	1.96844
$342$	−9.83652	−0.531898
$343$	0	0
$344$	−30.2719	−1.63215
$345$	1.91754	0.103237
$346$	2.83066	0.152177
$347$	−18.4810	−0.992114	−0.496057	−	0.868290i	$-0.665219\pi$
$347$	−18.4810	−0.992114	−0.496057	+	0.868290i	$0.665219\pi$
$348$	−2.63411	−0.141203
$349$	−8.21835	−0.439918	−0.219959	−	0.975509i	$-0.570592\pi$
$349$	−8.21835	−0.439918	−0.219959	+	0.975509i	$0.570592\pi$
$350$	0	0
$351$	−6.28282	−0.335352
$352$	8.92617	0.475767
$353$	3.47302	0.184850	0.0924251	−	0.995720i	$-0.470538\pi$
$353$	3.47302	0.184850	0.0924251	+	0.995720i	$0.470538\pi$
$354$	−12.6637	−0.673070
$355$	−14.6424	−0.777136
$356$	−1.40223	−0.0743183
$357$	0	0
$358$	−32.3473	−1.70961
$359$	7.88893	0.416362	0.208181	−	0.978090i	$-0.433246\pi$
$359$	7.88893	0.416362	0.208181	+	0.978090i	$0.433246\pi$
$360$	−5.78885	−0.305099
$361$	39.6668	2.08773
$362$	−32.0916	−1.68670
$363$	9.74723	0.511597
$364$	0	0
$365$	0.305700	0.0160011
$366$	−3.91660	−0.204724
$367$	17.5616	0.916708	0.458354	−	0.888770i	$-0.348439\pi$
$367$	17.5616	0.916708	0.458354	+	0.888770i	$0.348439\pi$
$368$	3.17552	0.165535
$369$	6.77718	0.352806
$370$	−16.6222	−0.864145
$371$	0	0
$372$	2.79897	0.145120
$373$	−10.5960	−0.548641	−0.274321	−	0.961638i	$-0.588453\pi$
$373$	−10.5960	−0.548641	−0.274321	+	0.961638i	$0.588453\pi$
$374$	−9.25695	−0.478665
$375$	12.1247	0.626116
$376$	−5.15439	−0.265817
$377$	−47.1858	−2.43019
$378$	0	0
$379$	−10.5939	−0.544172	−0.272086	−	0.962273i	$-0.587714\pi$
$379$	−10.5939	−0.544172	−0.272086	+	0.962273i	$0.587714\pi$
$380$	5.15131	0.264256
$381$	−21.6822	−1.11082
$382$	1.77937	0.0910403
$383$	20.9536	1.07068	0.535341	−	0.844636i	$-0.320183\pi$
$383$	20.9536	1.07068	0.535341	+	0.844636i	$0.320183\pi$
$384$	−7.46892	−0.381146
$385$	0	0
$386$	6.69966	0.341004
$387$	−10.0274	−0.509724
$388$	−0.756649	−0.0384131
$389$	−12.9987	−0.659060	−0.329530	−	0.944145i	$-0.606890\pi$
$389$	−12.9987	−0.659060	−0.329530	+	0.944145i	$0.606890\pi$
$390$	−15.4719	−0.783450
$391$	1.58249	0.0800302
$392$	0	0
$393$	8.95496	0.451718
$394$	−10.4677	−0.527353
$395$	−33.6687	−1.69406
$396$	1.59756	0.0802805
$397$	22.7115	1.13986	0.569928	−	0.821694i	$-0.306971\pi$
$397$	22.7115	1.13986	0.569928	+	0.821694i	$0.306971\pi$
$398$	−11.8530	−0.594139
$399$	0	0
$400$	4.20138	0.210069
$401$	1.40232	0.0700286	0.0350143	−	0.999387i	$-0.488852\pi$
$401$	1.40232	0.0700286	0.0350143	+	0.999387i	$0.488852\pi$
$402$	−0.0982487	−0.00490020
$403$	50.1389	2.49760
$404$	5.43626	0.270464
$405$	−1.91754	−0.0952831
$406$	0	0
$407$	30.7453	1.52399
$408$	−4.77740	−0.236516
$409$	−2.22027	−0.109785	−0.0548927	−	0.998492i	$-0.517482\pi$
$409$	−2.22027	−0.109785	−0.0548927	+	0.998492i	$0.517482\pi$
$410$	16.6893	0.824225
$411$	12.2161	0.602576
$412$	3.00880	0.148233
$413$	0	0
$414$	1.28424	0.0631168
$415$	−4.81262	−0.236242
$416$	12.3123	0.603661
$417$	16.7758	0.821512
$418$	44.8045	2.19146
$419$	3.50708	0.171332	0.0856660	−	0.996324i	$-0.472698\pi$
$419$	3.50708	0.171332	0.0856660	+	0.996324i	$0.472698\pi$
$420$	0	0
$421$	−0.461625	−0.0224982	−0.0112491	−	0.999937i	$-0.503581\pi$
$421$	−0.461625	−0.0224982	−0.0112491	+	0.999937i	$0.503581\pi$
$422$	18.7617	0.913307
$423$	−1.70737	−0.0830153
$424$	−24.4677	−1.18826
$425$	2.09373	0.101561
$426$	−9.80648	−0.475126
$427$	0	0
$428$	−1.14609	−0.0553986
$429$	28.6177	1.38168
$430$	−24.6933	−1.19082
$431$	−20.0253	−0.964587	−0.482293	−	0.876010i	$-0.660196\pi$
$431$	−20.0253	−0.964587	−0.482293	+	0.876010i	$0.660196\pi$
$432$	−3.17552	−0.152782
$433$	2.50142	0.120211	0.0601054	−	0.998192i	$-0.480856\pi$
$433$	2.50142	0.120211	0.0601054	+	0.998192i	$0.480856\pi$
$434$	0	0
$435$	−14.4013	−0.690487
$436$	−3.40765	−0.163197
$437$	−7.65943	−0.366400
$438$	0.204737	0.00978273
$439$	7.32969	0.349827	0.174913	−	0.984584i	$-0.444035\pi$
$439$	7.32969	0.349827	0.174913	+	0.984584i	$0.444035\pi$
$440$	26.3677	1.25703
$441$	0	0
$442$	−12.7686	−0.607339
$443$	−28.4630	−1.35232	−0.676160	−	0.736755i	$-0.736357\pi$
$443$	−28.4630	−1.35232	−0.676160	+	0.736755i	$0.736357\pi$
$444$	2.36742	0.112353
$445$	−7.66632	−0.363418
$446$	−18.7655	−0.888570
$447$	−7.14874	−0.338124
$448$	0	0
$449$	13.3958	0.632188	0.316094	−	0.948728i	$-0.397629\pi$
$449$	13.3958	0.632188	0.316094	+	0.948728i	$0.397629\pi$
$450$	1.69912	0.0800971
$451$	−30.8694	−1.45359
$452$	−0.690874	−0.0324960
$453$	−22.3685	−1.05096
$454$	−32.2096	−1.51167
$455$	0	0
$456$	23.1230	1.08284
$457$	−12.4360	−0.581732	−0.290866	−	0.956764i	$-0.593943\pi$
$457$	−12.4360	−0.581732	−0.290866	+	0.956764i	$0.593943\pi$
$458$	10.2597	0.479403
$459$	−1.58249	−0.0738645
$460$	−0.672545	−0.0313575
$461$	0.633635	0.0295113	0.0147557	−	0.999891i	$-0.495303\pi$
$461$	0.633635	0.0295113	0.0147557	+	0.999891i	$0.495303\pi$
$462$	0	0
$463$	−2.53977	−0.118033	−0.0590167	−	0.998257i	$-0.518797\pi$
$463$	−2.53977	−0.118033	−0.0590167	+	0.998257i	$0.518797\pi$
$464$	−23.8491	−1.10716
$465$	15.3026	0.709639
$466$	−18.5002	−0.857005
$467$	−13.7068	−0.634276	−0.317138	−	0.948379i	$-0.602722\pi$
$467$	−13.7068	−0.634276	−0.317138	+	0.948379i	$0.602722\pi$
$468$	2.20360	0.101861
$469$	0	0
$470$	−4.20453	−0.193940
$471$	10.2145	0.470658
$472$	29.7691	1.37023
$473$	45.6742	2.10010
$474$	−22.5490	−1.03571
$475$	−10.1338	−0.464972
$476$	0	0
$477$	−8.10484	−0.371095
$478$	1.14640	0.0524352
$479$	−40.5159	−1.85122	−0.925610	−	0.378479i	$-0.876447\pi$
$479$	−40.5159	−1.85122	−0.925610	+	0.378479i	$0.876447\pi$
$480$	3.75776	0.171517
$481$	42.4086	1.93366
$482$	−4.15436	−0.189226
$483$	0	0
$484$	−3.41868	−0.155395
$485$	−4.13676	−0.187841
$486$	−1.28424	−0.0582542
$487$	−2.40832	−0.109131	−0.0545656	−	0.998510i	$-0.517377\pi$
$487$	−2.40832	−0.109131	−0.0545656	+	0.998510i	$0.517377\pi$
$488$	9.20687	0.416776
$489$	−16.4072	−0.741959
$490$	0	0
$491$	11.5892	0.523015	0.261507	−	0.965201i	$-0.415780\pi$
$491$	11.5892	0.523015	0.261507	+	0.965201i	$0.415780\pi$
$492$	−2.37698	−0.107163
$493$	−11.8850	−0.535273
$494$	61.8011	2.78056
$495$	8.73421	0.392574
$496$	25.3417	1.13787
$497$	0	0
$498$	−3.22317	−0.144434
$499$	−0.392538	−0.0175724	−0.00878622	−	0.999961i	$-0.502797\pi$
$499$	−0.392538	−0.0175724	−0.00878622	+	0.999961i	$0.502797\pi$
$500$	−4.25254	−0.190179
$501$	−18.8255	−0.841063
$502$	−25.4899	−1.13767
$503$	−9.41915	−0.419979	−0.209990	−	0.977704i	$-0.567343\pi$
$503$	−9.41915	−0.419979	−0.209990	+	0.977704i	$0.567343\pi$
$504$	0	0
$505$	29.7212	1.32258
$506$	−5.84959	−0.260046
$507$	26.4739	1.17575
$508$	7.60469	0.337404
$509$	−21.3471	−0.946195	−0.473098	−	0.881010i	$-0.656864\pi$
$509$	−21.3471	−0.946195	−0.473098	+	0.881010i	$0.656864\pi$
$510$	−3.89701	−0.172562
$511$	0	0
$512$	25.3962	1.12236
$513$	7.65943	0.338172
$514$	0.0294754	0.00130011
$515$	16.4498	0.724864
$516$	3.51696	0.154826
$517$	7.77693	0.342029
$518$	0	0
$519$	−2.20415	−0.0967515
$520$	36.3703	1.59494
$521$	1.50073	0.0657482	0.0328741	−	0.999460i	$-0.489534\pi$
$521$	1.50073	0.0657482	0.0328741	+	0.999460i	$0.489534\pi$
$522$	−9.64500	−0.422150
$523$	13.3169	0.582308	0.291154	−	0.956676i	$-0.405961\pi$
$523$	13.3169	0.582308	0.291154	+	0.956676i	$0.405961\pi$
$524$	−3.14081	−0.137207
$525$	0	0
$526$	−33.1476	−1.44530
$527$	12.6288	0.550120
$528$	14.4642	0.629474
$529$	1.00000	0.0434783
$530$	−19.9587	−0.866952
$531$	9.86090	0.427927
$532$	0	0
$533$	−42.5798	−1.84434
$534$	−5.13439	−0.222187
$535$	−6.26595	−0.270900
$536$	0.230956	0.00997580
$537$	25.1880	1.08694
$538$	−29.8814	−1.28828
$539$	0	0
$540$	0.672545	0.0289417
$541$	−6.19737	−0.266446	−0.133223	−	0.991086i	$-0.542533\pi$
$541$	−6.19737	−0.266446	−0.133223	+	0.991086i	$0.542533\pi$
$542$	−11.9438	−0.513029
$543$	24.9888	1.07237
$544$	3.10118	0.132962
$545$	−18.6303	−0.798036
$546$	0	0
$547$	31.6413	1.35289	0.676443	−	0.736495i	$-0.263520\pi$
$547$	31.6413	1.35289	0.676443	+	0.736495i	$0.263520\pi$
$548$	−4.28460	−0.183029
$549$	3.04974	0.130160
$550$	−7.73933	−0.330006
$551$	57.5245	2.45063
$552$	−3.01890	−0.128493
$553$	0	0
$554$	8.68196	0.368861
$555$	12.9432	0.549409
$556$	−5.88382	−0.249530
$557$	28.2471	1.19687	0.598433	−	0.801173i	$-0.295790\pi$
$557$	28.2471	1.19687	0.598433	+	0.801173i	$0.295790\pi$
$558$	10.2486	0.433859
$559$	63.0007	2.66464
$560$	0	0
$561$	7.20813	0.304327
$562$	−0.916507	−0.0386605
$563$	6.41281	0.270268	0.135134	−	0.990827i	$-0.456854\pi$
$563$	6.41281	0.270268	0.135134	+	0.990827i	$0.456854\pi$
$564$	0.598833	0.0252154
$565$	−3.77716	−0.158906
$566$	29.8535	1.25484
$567$	0	0
$568$	23.0524	0.967258
$569$	−18.8305	−0.789415	−0.394708	−	0.918807i	$-0.629154\pi$
$569$	−18.8305	−0.789415	−0.394708	+	0.918807i	$0.629154\pi$
$570$	18.8619	0.790038
$571$	3.47196	0.145297	0.0726485	−	0.997358i	$-0.476855\pi$
$571$	3.47196	0.145297	0.0726485	+	0.997358i	$0.476855\pi$
$572$	−10.0372	−0.419676
$573$	−1.38554	−0.0578819
$574$	0	0
$575$	1.32305	0.0551752
$576$	8.86773	0.369489
$577$	−10.5914	−0.440924	−0.220462	−	0.975396i	$-0.570757\pi$
$577$	−10.5914	−0.440924	−0.220462	+	0.975396i	$0.570757\pi$
$578$	18.6159	0.774321
$579$	−5.21684	−0.216805
$580$	5.05100	0.209732
$581$	0	0
$582$	−2.77053	−0.114842
$583$	36.9168	1.52894
$584$	−0.481283	−0.0199156
$585$	12.0475	0.498104
$586$	−36.3014	−1.49960
$587$	−17.5776	−0.725504	−0.362752	−	0.931886i	$-0.618163\pi$
$587$	−17.5776	−0.725504	−0.362752	+	0.931886i	$0.618163\pi$
$588$	0	0
$589$	−61.1247	−2.51860
$590$	24.2832	0.999723
$591$	8.15087	0.335282
$592$	21.4345	0.880953
$593$	−37.7386	−1.54974	−0.774869	−	0.632122i	$-0.782184\pi$
$593$	−37.7386	−1.54974	−0.774869	+	0.632122i	$0.782184\pi$
$594$	5.84959	0.240012
$595$	0	0
$596$	2.50730	0.102703
$597$	9.22963	0.377744
$598$	−8.06864	−0.329951
$599$	−0.966053	−0.0394719	−0.0197359	−	0.999805i	$-0.506283\pi$
$599$	−0.966053	−0.0394719	−0.0197359	+	0.999805i	$0.506283\pi$
$600$	−3.99417	−0.163061
$601$	12.6088	0.514323	0.257162	−	0.966368i	$-0.417213\pi$
$601$	12.6088	0.514323	0.257162	+	0.966368i	$0.417213\pi$
$602$	0	0
$603$	0.0765035	0.00311546
$604$	7.84539	0.319224
$605$	−18.6907	−0.759884
$606$	19.9053	0.808596
$607$	−3.65034	−0.148163	−0.0740813	−	0.997252i	$-0.523602\pi$
$607$	−3.65034	−0.148163	−0.0740813	+	0.997252i	$0.523602\pi$
$608$	−15.0100	−0.608737
$609$	0	0
$610$	7.51022	0.304080
$611$	10.7271	0.433973
$612$	0.555034	0.0224359
$613$	26.8292	1.08362	0.541811	−	0.840500i	$-0.317739\pi$
$613$	26.8292	1.08362	0.541811	+	0.840500i	$0.317739\pi$
$614$	−21.5953	−0.871513
$615$	−12.9955	−0.524028
$616$	0	0
$617$	−14.0527	−0.565739	−0.282869	−	0.959158i	$-0.591286\pi$
$617$	−14.0527	−0.565739	−0.282869	+	0.959158i	$0.591286\pi$
$618$	11.0170	0.443167
$619$	14.4565	0.581057	0.290528	−	0.956866i	$-0.406169\pi$
$619$	14.4565	0.581057	0.290528	+	0.956866i	$0.406169\pi$
$620$	−5.36712	−0.215549
$621$	−1.00000	−0.0401286
$622$	−13.4672	−0.539984
$623$	0	0
$624$	19.9512	0.798688
$625$	−16.6343	−0.665370
$626$	−33.5267	−1.34000
$627$	−34.8880	−1.39329
$628$	−3.58256	−0.142960
$629$	10.6817	0.425908
$630$	0	0
$631$	−43.3382	−1.72527	−0.862633	−	0.505830i	$-0.831186\pi$
$631$	−43.3382	−1.72527	−0.862633	+	0.505830i	$0.831186\pi$
$632$	53.0068	2.10850
$633$	−14.6092	−0.580665
$634$	−37.8086	−1.50157
$635$	41.5765	1.64991
$636$	2.84264	0.112718
$637$	0	0
$638$	43.9321	1.73929
$639$	7.63603	0.302077
$640$	14.3219	0.566123
$641$	−4.47690	−0.176827	−0.0884135	−	0.996084i	$-0.528180\pi$
$641$	−4.47690	−0.176827	−0.0884135	+	0.996084i	$0.528180\pi$
$642$	−4.19651	−0.165623
$643$	−10.2005	−0.402267	−0.201134	−	0.979564i	$-0.564463\pi$
$643$	−10.2005	−0.402267	−0.201134	+	0.979564i	$0.564463\pi$
$644$	0	0
$645$	19.2280	0.757102
$646$	15.5662	0.612446
$647$	−20.5202	−0.806731	−0.403366	−	0.915039i	$-0.632160\pi$
$647$	−20.5202	−0.806731	−0.403366	+	0.915039i	$0.632160\pi$
$648$	3.01890	0.118594
$649$	−44.9155	−1.76309
$650$	−10.6752	−0.418718
$651$	0	0
$652$	5.75456	0.225366
$653$	−27.9440	−1.09353	−0.546767	−	0.837285i	$-0.684142\pi$
$653$	−27.9440	−1.09353	−0.546767	+	0.837285i	$0.684142\pi$
$654$	−12.4774	−0.487903
$655$	−17.1715	−0.670945
$656$	−21.5210	−0.840256
$657$	−0.159423	−0.00621969
$658$	0	0
$659$	−13.5006	−0.525908	−0.262954	−	0.964808i	$-0.584697\pi$
$659$	−13.5006	−0.525908	−0.262954	+	0.964808i	$0.584697\pi$
$660$	−3.06338	−0.119242
$661$	−6.96611	−0.270950	−0.135475	−	0.990781i	$-0.543256\pi$
$661$	−6.96611	−0.270950	−0.135475	+	0.990781i	$0.543256\pi$
$662$	−10.8913	−0.423303
$663$	9.94253	0.386136
$664$	7.57682	0.294038
$665$	0	0
$666$	8.66850	0.335898
$667$	−7.51029	−0.290800
$668$	6.60275	0.255468
$669$	14.6121	0.564938
$670$	0.188395	0.00727835
$671$	−13.8913	−0.536268
$672$	0	0
$673$	−5.49327	−0.211750	−0.105875	−	0.994379i	$-0.533764\pi$
$673$	−5.49327	−0.211750	−0.105875	+	0.994379i	$0.533764\pi$
$674$	31.8615	1.22726
$675$	−1.32305	−0.0509244
$676$	−9.28527	−0.357126
$677$	19.7081	0.757443	0.378721	−	0.925511i	$-0.376364\pi$
$677$	19.7081	0.757443	0.378721	+	0.925511i	$0.376364\pi$
$678$	−2.52969	−0.0971521
$679$	0	0
$680$	9.16083	0.351302
$681$	25.0807	0.961095
$682$	−46.6816	−1.78753
$683$	−13.4820	−0.515874	−0.257937	−	0.966162i	$-0.583043\pi$
$683$	−13.4820	−0.515874	−0.257937	+	0.966162i	$0.583043\pi$
$684$	−2.68642	−0.102718
$685$	−23.4248	−0.895016
$686$	0	0
$687$	−7.98892	−0.304796
$688$	31.8423	1.21398
$689$	50.9212	1.93994
$690$	−2.46257	−0.0937485
$691$	−18.4208	−0.700760	−0.350380	−	0.936608i	$-0.613947\pi$
$691$	−18.4208	−0.700760	−0.350380	+	0.936608i	$0.613947\pi$
$692$	0.773070	0.0293877
$693$	0	0
$694$	23.7340	0.900932
$695$	−32.1681	−1.22021
$696$	22.6728	0.859411
$697$	−10.7248	−0.406233
$698$	10.5543	0.399487
$699$	14.4056	0.544869
$700$	0	0
$701$	−15.6151	−0.589775	−0.294887	−	0.955532i	$-0.595282\pi$
$701$	−15.6151	−0.589775	−0.294887	+	0.955532i	$0.595282\pi$
$702$	8.06864	0.304531
$703$	−51.7005	−1.94992
$704$	−40.3917	−1.52232
$705$	3.27395	0.123304
$706$	−4.46018	−0.167861
$707$	0	0
$708$	−3.45855	−0.129980
$709$	−13.7547	−0.516569	−0.258284	−	0.966069i	$-0.583157\pi$
$709$	−13.7547	−0.516569	−0.258284	+	0.966069i	$0.583157\pi$
$710$	18.8043	0.705712
$711$	17.5583	0.658488
$712$	12.0696	0.452326
$713$	7.98032	0.298865
$714$	0	0
$715$	−54.8755	−2.05223
$716$	−8.83426	−0.330152
$717$	−0.892671	−0.0333374
$718$	−10.1313	−0.378095
$719$	−3.05165	−0.113807	−0.0569037	−	0.998380i	$-0.518123\pi$
$719$	−3.05165	−0.113807	−0.0569037	+	0.998380i	$0.518123\pi$
$720$	6.08917	0.226930
$721$	0	0
$722$	−50.9416	−1.89585
$723$	3.23488	0.120307
$724$	−8.76442	−0.325727
$725$	−9.93652	−0.369033
$726$	−12.5178	−0.464578
$727$	−40.5156	−1.50264	−0.751319	−	0.659939i	$-0.770582\pi$
$727$	−40.5156	−1.50264	−0.751319	+	0.659939i	$0.770582\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−0.392591	−0.0145305
$731$	15.8684	0.586913
$732$	−1.06965	−0.0395353
$733$	13.9465	0.515126	0.257563	−	0.966261i	$-0.417080\pi$
$733$	13.9465	0.515126	0.257563	+	0.966261i	$0.417080\pi$
$734$	−22.5533	−0.832456
$735$	0	0
$736$	1.95968	0.0722348
$737$	−0.348467	−0.0128359
$738$	−8.70350	−0.320380
$739$	25.1613	0.925573	0.462786	−	0.886470i	$-0.346850\pi$
$739$	25.1613	0.925573	0.462786	+	0.886470i	$0.346850\pi$
$740$	−4.53962	−0.166880
$741$	−48.1228	−1.76784
$742$	0	0
$743$	−26.4397	−0.969978	−0.484989	−	0.874520i	$-0.661176\pi$
$743$	−26.4397	−0.969978	−0.484989	+	0.874520i	$0.661176\pi$
$744$	−24.0918	−0.883248
$745$	13.7080	0.502221
$746$	13.6078	0.498217
$747$	2.50979	0.0918286
$748$	−2.52813	−0.0924377
$749$	0	0
$750$	−15.5710	−0.568572
$751$	16.4689	0.600958	0.300479	−	0.953788i	$-0.402853\pi$
$751$	16.4689	0.600958	0.300479	+	0.953788i	$0.402853\pi$
$752$	5.42179	0.197712
$753$	19.8482	0.723310
$754$	60.5978	2.20684
$755$	42.8924	1.56102
$756$	0	0
$757$	21.3892	0.777402	0.388701	−	0.921364i	$-0.372924\pi$
$757$	21.3892	0.777402	0.388701	+	0.921364i	$0.372924\pi$
$758$	13.6051	0.494159
$759$	4.55491	0.165333
$760$	−44.3393	−1.60835
$761$	1.43675	0.0520823	0.0260411	−	0.999661i	$-0.491710\pi$
$761$	1.43675	0.0520823	0.0260411	+	0.999661i	$0.491710\pi$
$762$	27.8452	1.00872
$763$	0	0
$764$	0.485957	0.0175813
$765$	3.03449	0.109712
$766$	−26.9095	−0.972278
$767$	−61.9543	−2.23704
$768$	−8.14360	−0.293857
$769$	−10.0891	−0.363824	−0.181912	−	0.983315i	$-0.558229\pi$
$769$	−10.0891	−0.363824	−0.181912	+	0.983315i	$0.558229\pi$
$770$	0	0
$771$	−0.0229517	−0.000826586 0
$772$	1.82972	0.0658531
$773$	−1.89676	−0.0682219	−0.0341109	−	0.999418i	$-0.510860\pi$
$773$	−1.89676	−0.0682219	−0.0341109	+	0.999418i	$0.510860\pi$
$774$	12.8776	0.462877
$775$	10.5584	0.379269
$776$	6.51277	0.233795
$777$	0	0
$778$	16.6934	0.598488
$779$	51.9093	1.85984
$780$	−4.22548	−0.151296
$781$	−34.7815	−1.24458
$782$	−2.03230	−0.0726749
$783$	7.51029	0.268396
$784$	0	0
$785$	−19.5866	−0.699076
$786$	−11.5003	−0.410202
$787$	−12.0981	−0.431249	−0.215625	−	0.976476i	$-0.569179\pi$
$787$	−12.0981	−0.431249	−0.215625	+	0.976476i	$0.569179\pi$
$788$	−2.85878	−0.101840
$789$	25.8111	0.918900
$790$	43.2386	1.53836
$791$	0	0
$792$	−13.7508	−0.488614
$793$	−19.1610	−0.680427
$794$	−29.1669	−1.03510
$795$	15.5413	0.551194
$796$	−3.23714	−0.114737
$797$	−39.0780	−1.38421	−0.692107	−	0.721795i	$-0.743317\pi$
$797$	−39.0780	−1.38421	−0.692107	+	0.721795i	$0.743317\pi$
$798$	0	0
$799$	2.70191	0.0955867
$800$	2.59276	0.0916681
$801$	3.99800	0.141262
$802$	−1.80091	−0.0635925
$803$	0.726159	0.0256256
$804$	−0.0268324	−0.000946304 0
$805$	0	0
$806$	−64.3903	−2.26805
$807$	23.2678	0.819065
$808$	−46.7920	−1.64614
$809$	−39.6918	−1.39549	−0.697745	−	0.716346i	$-0.745813\pi$
$809$	−39.6918	−1.39549	−0.697745	+	0.716346i	$0.745813\pi$
$810$	2.46257	0.0865260
$811$	−37.1125	−1.30320	−0.651599	−	0.758564i	$-0.725901\pi$
$811$	−37.1125	−1.30320	−0.651599	+	0.758564i	$0.725901\pi$
$812$	0	0
$813$	9.30029	0.326175
$814$	−39.4843	−1.38392
$815$	31.4614	1.10204
$816$	5.02524	0.175919
$817$	−76.8045	−2.68705
$818$	2.85136	0.0996954
$819$	0	0
$820$	4.55795	0.159171
$821$	−18.5286	−0.646653	−0.323326	−	0.946287i	$-0.604801\pi$
$821$	−18.5286	−0.646653	−0.323326	+	0.946287i	$0.604801\pi$
$822$	−15.6884	−0.547195
$823$	37.3190	1.30086	0.650429	−	0.759567i	$-0.274589\pi$
$823$	37.3190	1.30086	0.650429	+	0.759567i	$0.274589\pi$
$824$	−25.8979	−0.902198
$825$	6.02640	0.209812
$826$	0	0
$827$	19.2547	0.669552	0.334776	−	0.942298i	$-0.391339\pi$
$827$	19.2547	0.669552	0.334776	+	0.942298i	$0.391339\pi$
$828$	0.350734	0.0121888
$829$	−8.07326	−0.280396	−0.140198	−	0.990123i	$-0.544774\pi$
$829$	−8.07326	−0.280396	−0.140198	+	0.990123i	$0.544774\pi$
$830$	6.18055	0.214530
$831$	−6.76040	−0.234516
$832$	−55.7144	−1.93155
$833$	0	0
$834$	−21.5441	−0.746010
$835$	36.0987	1.24925
$836$	12.2364	0.423205
$837$	−7.98032	−0.275840
$838$	−4.50392	−0.155585
$839$	−16.3156	−0.563277	−0.281639	−	0.959521i	$-0.590878\pi$
$839$	−16.3156	−0.563277	−0.281639	+	0.959521i	$0.590878\pi$
$840$	0	0
$841$	27.4044	0.944981
$842$	0.592836	0.0204305
$843$	0.713658	0.0245797
$844$	5.12395	0.176374
$845$	−50.7646	−1.74635
$846$	2.19267	0.0753856
$847$	0	0
$848$	25.7371	0.883814
$849$	−23.2461	−0.797805
$850$	−2.68884	−0.0922265
$851$	6.74992	0.231384
$852$	−2.67821	−0.0917541
$853$	4.23896	0.145139	0.0725696	−	0.997363i	$-0.476880\pi$
$853$	4.23896	0.145139	0.0725696	+	0.997363i	$0.476880\pi$
$854$	0	0
$855$	−14.6872	−0.502293
$856$	9.86488	0.337175
$857$	52.6691	1.79914	0.899571	−	0.436774i	$-0.143879\pi$
$857$	52.6691	1.79914	0.899571	+	0.436774i	$0.143879\pi$
$858$	−36.7519	−1.25469
$859$	18.4724	0.630270	0.315135	−	0.949047i	$-0.397950\pi$
$859$	18.4724	0.630270	0.315135	+	0.949047i	$0.397950\pi$
$860$	−6.74391	−0.229965
$861$	0	0
$862$	25.7173	0.875935
$863$	19.5401	0.665153	0.332577	−	0.943076i	$-0.392082\pi$
$863$	19.5401	0.665153	0.332577	+	0.943076i	$0.392082\pi$
$864$	−1.95968	−0.0666697
$865$	4.22654	0.143707
$866$	−3.21242	−0.109163
$867$	−14.4957	−0.492300
$868$	0	0
$869$	−79.9765	−2.71302
$870$	18.4946	0.627027
$871$	−0.480658	−0.0162865
$872$	29.3310	0.993271
$873$	2.15733	0.0730147
$874$	9.83652	0.332725
$875$	0	0
$876$	0.0559151	0.00188920
$877$	−45.9461	−1.55149	−0.775744	−	0.631048i	$-0.782625\pi$
$877$	−45.9461	−1.55149	−0.775744	+	0.631048i	$0.782625\pi$
$878$	−9.41306	−0.317675
$879$	28.2669	0.953418
$880$	−27.7356	−0.934969
$881$	59.2280	1.99544	0.997721	−	0.0674737i	$-0.0214939\pi$
$881$	59.2280	1.99544	0.997721	+	0.0674737i	$0.0214939\pi$
$882$	0	0
$883$	11.5678	0.389288	0.194644	−	0.980874i	$-0.437645\pi$
$883$	11.5678	0.389288	0.194644	+	0.980874i	$0.437645\pi$
$884$	−3.48718	−0.117287
$885$	−18.9086	−0.635607
$886$	36.5533	1.22803
$887$	6.07633	0.204023	0.102012	−	0.994783i	$-0.467472\pi$
$887$	6.07633	0.204023	0.102012	+	0.994783i	$0.467472\pi$
$888$	−20.3773	−0.683819
$889$	0	0
$890$	9.84537	0.330018
$891$	−4.55491	−0.152595
$892$	−5.12497	−0.171597
$893$	−13.0775	−0.437622
$894$	9.18068	0.307048
$895$	−48.2988	−1.61445
$896$	0	0
$897$	6.28282	0.209777
$898$	−17.2034	−0.574085
$899$	−59.9345	−1.99893
$900$	0.464040	0.0154680
$901$	12.8259	0.427291
$902$	39.6437	1.31999
$903$	0	0
$904$	5.94662	0.197782
$905$	−47.9170	−1.59281
$906$	28.7265	0.954373
$907$	−11.1894	−0.371537	−0.185768	−	0.982594i	$-0.559477\pi$
$907$	−11.1894	−0.371537	−0.185768	+	0.982594i	$0.559477\pi$
$908$	−8.79665	−0.291927
$909$	−15.4997	−0.514092
$910$	0	0
$911$	−38.1310	−1.26334	−0.631669	−	0.775238i	$-0.717630\pi$
$911$	−38.1310	−1.26334	−0.631669	+	0.775238i	$0.717630\pi$
$912$	−24.3227	−0.805403
$913$	−11.4319	−0.378341
$914$	15.9708	0.528267
$915$	−5.84799	−0.193329
$916$	2.80198	0.0925801
$917$	0	0
$918$	2.03230	0.0670759
$919$	12.1881	0.402047	0.201024	−	0.979586i	$-0.435573\pi$
$919$	12.1881	0.402047	0.201024	+	0.979586i	$0.435573\pi$
$920$	5.78885	0.190853
$921$	16.8156	0.554093
$922$	−0.813738	−0.0267990
$923$	−47.9758	−1.57914
$924$	0	0
$925$	8.93051	0.293633
$926$	3.26167	0.107185
$927$	−8.57860	−0.281758
$928$	−14.7178	−0.483134
$929$	−38.5488	−1.26474	−0.632372	−	0.774665i	$-0.717919\pi$
$929$	−38.5488	−1.26474	−0.632372	+	0.774665i	$0.717919\pi$
$930$	−19.6521	−0.644418
$931$	0	0
$932$	−5.05252	−0.165501
$933$	10.4865	0.343313
$934$	17.6028	0.575982
$935$	−13.8218	−0.452023
$936$	−18.9672	−0.619963
$937$	−34.7326	−1.13466	−0.567332	−	0.823489i	$-0.692024\pi$
$937$	−34.7326	−1.13466	−0.567332	+	0.823489i	$0.692024\pi$
$938$	0	0
$939$	26.1063	0.851948
$940$	−1.14828	−0.0374529
$941$	−7.06750	−0.230394	−0.115197	−	0.993343i	$-0.536750\pi$
$941$	−7.06750	−0.230394	−0.115197	+	0.993343i	$0.536750\pi$
$942$	−13.1178	−0.427401
$943$	−6.77718	−0.220695
$944$	−31.3135	−1.01917
$945$	0	0
$946$	−58.6565	−1.90709
$947$	57.7173	1.87556	0.937779	−	0.347232i	$-0.112878\pi$
$947$	57.7173	1.87556	0.937779	+	0.347232i	$0.112878\pi$
$948$	−6.15829	−0.200012
$949$	1.00163	0.0325142
$950$	13.0143	0.422238
$951$	29.4405	0.954674
$952$	0	0
$953$	8.14786	0.263935	0.131968	−	0.991254i	$-0.457871\pi$
$953$	8.14786	0.263935	0.131968	+	0.991254i	$0.457871\pi$
$954$	10.4085	0.336989
$955$	2.65683	0.0859730
$956$	0.313090	0.0101260
$957$	−34.2087	−1.10581
$958$	52.0321	1.68108
$959$	0	0
$960$	−17.0042	−0.548808
$961$	32.6855	1.05437
$962$	−54.4627	−1.75595
$963$	3.26771	0.105300
$964$	−1.13458	−0.0365424
$965$	10.0035	0.322023
$966$	0	0
$967$	50.4806	1.62335	0.811673	−	0.584112i	$-0.198557\pi$
$967$	50.4806	1.62335	0.811673	+	0.584112i	$0.198557\pi$
$968$	29.4259	0.945785
$969$	−12.1210	−0.389383
$970$	5.31259	0.170577
$971$	−23.7670	−0.762719	−0.381360	−	0.924427i	$-0.624544\pi$
$971$	−23.7670	−0.762719	−0.381360	+	0.924427i	$0.624544\pi$
$972$	−0.350734	−0.0112498
$973$	0	0
$974$	3.09285	0.0991014
$975$	8.31251	0.266214
$976$	−9.68452	−0.309994
$977$	35.8925	1.14830	0.574151	−	0.818749i	$-0.305332\pi$
$977$	35.8925	1.14830	0.574151	+	0.818749i	$0.305332\pi$
$978$	21.0707	0.673768
$979$	−18.2106	−0.582012
$980$	0	0
$981$	9.71577	0.310201
$982$	−14.8833	−0.474946
$983$	10.3744	0.330892	0.165446	−	0.986219i	$-0.447094\pi$
$983$	10.3744	0.330892	0.165446	+	0.986219i	$0.447094\pi$
$984$	20.4596	0.652229
$985$	−15.6296	−0.498000
$986$	15.2632	0.486078
$987$	0	0
$988$	16.8783	0.536970
$989$	10.0274	0.318854
$990$	−11.2168	−0.356493
$991$	−30.3762	−0.964930	−0.482465	−	0.875915i	$-0.660259\pi$
$991$	−30.3762	−0.964930	−0.482465	+	0.875915i	$0.660259\pi$
$992$	15.6389	0.496535
$993$	8.48076	0.269129
$994$	0	0
$995$	−17.6982	−0.561069
$996$	−0.880269	−0.0278924
$997$	−10.6389	−0.336936	−0.168468	−	0.985707i	$-0.553882\pi$
$997$	−10.6389	−0.336936	−0.168468	+	0.985707i	$0.553882\pi$
$998$	0.504113	0.0159574
$999$	−6.74992	−0.213558

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3381.2.a.bl.1.2	yes	10
7.6	odd	2		3381.2.a.bk.1.2	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3381.2.a.bk.1.2	✓	10	7.6	odd	2
3381.2.a.bl.1.2	yes	10	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3381.a (trivial)

\(f(q)\)	\(=\)	\(q-1.28424 q^{2} +1.00000 q^{3} -0.350734 q^{4} -1.91754 q^{5} -1.28424 q^{6} +3.01890 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.28424 q^{2} +1.00000 q^{3} -0.350734 q^{4} -1.91754 q^{5} -1.28424 q^{6} +3.01890 q^{8} +1.00000 q^{9} +2.46257 q^{10} -4.55491 q^{11} -0.350734 q^{12} -6.28282 q^{13} -1.91754 q^{15} -3.17552 q^{16} -1.58249 q^{17} -1.28424 q^{18} +7.65943 q^{19} +0.672545 q^{20} +5.84959 q^{22} -1.00000 q^{23} +3.01890 q^{24} -1.32305 q^{25} +8.06864 q^{26} +1.00000 q^{27} +7.51029 q^{29} +2.46257 q^{30} -7.98032 q^{31} -1.95968 q^{32} -4.55491 q^{33} +2.03230 q^{34} -0.350734 q^{36} -6.74992 q^{37} -9.83652 q^{38} -6.28282 q^{39} -5.78885 q^{40} +6.77718 q^{41} -10.0274 q^{43} +1.59756 q^{44} -1.91754 q^{45} +1.28424 q^{46} -1.70737 q^{47} -3.17552 q^{48} +1.69912 q^{50} -1.58249 q^{51} +2.20360 q^{52} -8.10484 q^{53} -1.28424 q^{54} +8.73421 q^{55} +7.65943 q^{57} -9.64500 q^{58} +9.86090 q^{59} +0.672545 q^{60} +3.04974 q^{61} +10.2486 q^{62} +8.86773 q^{64} +12.0475 q^{65} +5.84959 q^{66} +0.0765035 q^{67} +0.555034 q^{68} -1.00000 q^{69} +7.63603 q^{71} +3.01890 q^{72} -0.159423 q^{73} +8.66850 q^{74} -1.32305 q^{75} -2.68642 q^{76} +8.06864 q^{78} +17.5583 q^{79} +6.08917 q^{80} +1.00000 q^{81} -8.70350 q^{82} +2.50979 q^{83} +3.03449 q^{85} +12.8776 q^{86} +7.51029 q^{87} -13.7508 q^{88} +3.99800 q^{89} +2.46257 q^{90} +0.350734 q^{92} -7.98032 q^{93} +2.19267 q^{94} -14.6872 q^{95} -1.95968 q^{96} +2.15733 q^{97} -4.55491 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 4 q^{2} + 10 q^{3} + 8 q^{4} + 4 q^{5} + 4 q^{6} + 12 q^{8} + 10 q^{9}+O(q^{10}) \) 10 * q + 4 * q^2 + 10 * q^3 + 8 * q^4 + 4 * q^5 + 4 * q^6 + 12 * q^8 + 10 * q^9	\( 10 q + 4 q^{2} + 10 q^{3} + 8 q^{4} + 4 q^{5} + 4 q^{6} + 12 q^{8} + 10 q^{9} + 8 q^{10} + 2 q^{11} + 8 q^{12} + 4 q^{15} + 4 q^{16} + 12 q^{17} + 4 q^{18} + 26 q^{19} + 24 q^{20} - 8 q^{22} - 10 q^{23} + 12 q^{24} - 2 q^{25} + 4 q^{26} + 10 q^{27} + 16 q^{29} + 8 q^{30} + 12 q^{31} + 8 q^{32} + 2 q^{33} + 28 q^{34} + 8 q^{36} - 8 q^{37} + 32 q^{38} + 4 q^{40} + 10 q^{41} - 4 q^{43} - 16 q^{44} + 4 q^{45} - 4 q^{46} + 2 q^{47} + 4 q^{48} - 8 q^{50} + 12 q^{51} + 24 q^{52} + 14 q^{53} + 4 q^{54} + 16 q^{55} + 26 q^{57} - 8 q^{58} + 38 q^{59} + 24 q^{60} + 14 q^{61} - 8 q^{62} + 8 q^{64} + 12 q^{65} - 8 q^{66} + 8 q^{68} - 10 q^{69} + 24 q^{71} + 12 q^{72} + 8 q^{73} - 8 q^{74} - 2 q^{75} + 64 q^{76} + 4 q^{78} - 16 q^{79} + 28 q^{80} + 10 q^{81} - 40 q^{82} + 28 q^{83} - 4 q^{85} + 20 q^{86} + 16 q^{87} - 68 q^{88} + 32 q^{89} + 8 q^{90} - 8 q^{92} + 12 q^{93} + 56 q^{94} + 8 q^{95} + 8 q^{96} + 4 q^{97} + 2 q^{99}+O(q^{100}) \) 10 * q + 4 * q^2 + 10 * q^3 + 8 * q^4 + 4 * q^5 + 4 * q^6 + 12 * q^8 + 10 * q^9 + 8 * q^10 + 2 * q^11 + 8 * q^12 + 4 * q^15 + 4 * q^16 + 12 * q^17 + 4 * q^18 + 26 * q^19 + 24 * q^20 - 8 * q^22 - 10 * q^23 + 12 * q^24 - 2 * q^25 + 4 * q^26 + 10 * q^27 + 16 * q^29 + 8 * q^30 + 12 * q^31 + 8 * q^32 + 2 * q^33 + 28 * q^34 + 8 * q^36 - 8 * q^37 + 32 * q^38 + 4 * q^40 + 10 * q^41 - 4 * q^43 - 16 * q^44 + 4 * q^45 - 4 * q^46 + 2 * q^47 + 4 * q^48 - 8 * q^50 + 12 * q^51 + 24 * q^52 + 14 * q^53 + 4 * q^54 + 16 * q^55 + 26 * q^57 - 8 * q^58 + 38 * q^59 + 24 * q^60 + 14 * q^61 - 8 * q^62 + 8 * q^64 + 12 * q^65 - 8 * q^66 + 8 * q^68 - 10 * q^69 + 24 * q^71 + 12 * q^72 + 8 * q^73 - 8 * q^74 - 2 * q^75 + 64 * q^76 + 4 * q^78 - 16 * q^79 + 28 * q^80 + 10 * q^81 - 40 * q^82 + 28 * q^83 - 4 * q^85 + 20 * q^86 + 16 * q^87 - 68 * q^88 + 32 * q^89 + 8 * q^90 - 8 * q^92 + 12 * q^93 + 56 * q^94 + 8 * q^95 + 8 * q^96 + 4 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

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Representations

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Database

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