LMFDB - Embedded newform 3381.2.a.bf.1.8

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 15x^{6} + 11x^{5} + 75x^{4} - 35x^{3} - 141x^{2} + 37x + 80 $ x^8 - x^7 - 15x^6 + 11x^5 + 75x^4 - 35x^3 - 141x^2 + 37x + 80
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 483)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$2.74585$ 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)

$f(q)$	$=$	$q+2.74585 q^{2} +1.00000 q^{3} +5.53968 q^{4} -1.91481 q^{5} +2.74585 q^{6} +9.71944 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.74585 q^{2} +1.00000 q^{3} +5.53968 q^{4} -1.91481 q^{5} +2.74585 q^{6} +9.71944 q^{8} +1.00000 q^{9} -5.25777 q^{10} +5.32332 q^{11} +5.53968 q^{12} -4.78352 q^{13} -1.91481 q^{15} +15.6087 q^{16} +2.61781 q^{17} +2.74585 q^{18} +4.46024 q^{19} -10.6074 q^{20} +14.6170 q^{22} -1.00000 q^{23} +9.71944 q^{24} -1.33352 q^{25} -13.1348 q^{26} +1.00000 q^{27} -8.86099 q^{29} -5.25777 q^{30} +5.90898 q^{31} +23.4204 q^{32} +5.32332 q^{33} +7.18812 q^{34} +5.53968 q^{36} -2.91929 q^{37} +12.2471 q^{38} -4.78352 q^{39} -18.6108 q^{40} +4.98968 q^{41} -6.45341 q^{43} +29.4895 q^{44} -1.91481 q^{45} -2.74585 q^{46} -6.94360 q^{47} +15.6087 q^{48} -3.66165 q^{50} +2.61781 q^{51} -26.4992 q^{52} +1.01228 q^{53} +2.74585 q^{54} -10.1931 q^{55} +4.46024 q^{57} -24.3309 q^{58} +6.15656 q^{59} -10.6074 q^{60} +2.24372 q^{61} +16.2252 q^{62} +33.0913 q^{64} +9.15951 q^{65} +14.6170 q^{66} +9.38847 q^{67} +14.5019 q^{68} -1.00000 q^{69} -10.2261 q^{71} +9.71944 q^{72} +3.85118 q^{73} -8.01593 q^{74} -1.33352 q^{75} +24.7083 q^{76} -13.1348 q^{78} +2.97781 q^{79} -29.8877 q^{80} +1.00000 q^{81} +13.7009 q^{82} -2.29460 q^{83} -5.01260 q^{85} -17.7201 q^{86} -8.86099 q^{87} +51.7397 q^{88} -10.0816 q^{89} -5.25777 q^{90} -5.53968 q^{92} +5.90898 q^{93} -19.0661 q^{94} -8.54048 q^{95} +23.4204 q^{96} +13.2317 q^{97} +5.32332 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{2} + 8 q^{3} + 15 q^{4} + 5 q^{5} + q^{6} + 9 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q + q^2 + 8 * q^3 + 15 * q^4 + 5 * q^5 + q^6 + 9 * q^8 + 8 * q^9	$ 8 q + q^{2} + 8 q^{3} + 15 q^{4} + 5 q^{5} + q^{6} + 9 q^{8} + 8 q^{9} - 3 q^{10} + 10 q^{11} + 15 q^{12} - 6 q^{13} + 5 q^{15} + 13 q^{16} + 21 q^{17} + q^{18} + 5 q^{19} - q^{20} + 18 q^{22} - 8 q^{23} + 9 q^{24} + 27 q^{25} + 3 q^{26} + 8 q^{27} + 2 q^{29} - 3 q^{30} - 13 q^{31} + 29 q^{32} + 10 q^{33} - 19 q^{34} + 15 q^{36} + 13 q^{37} - 6 q^{38} - 6 q^{39} + 7 q^{40} + 16 q^{41} + 15 q^{43} + 24 q^{44} + 5 q^{45} - q^{46} - q^{47} + 13 q^{48} + 16 q^{50} + 21 q^{51} - 19 q^{52} + 3 q^{53} + q^{54} - 10 q^{55} + 5 q^{57} - 40 q^{58} + 26 q^{59} - q^{60} - 14 q^{61} - 14 q^{62} + 49 q^{64} - 3 q^{65} + 18 q^{66} + 38 q^{67} + 43 q^{68} - 8 q^{69} + 9 q^{71} + 9 q^{72} - 6 q^{73} + 32 q^{74} + 27 q^{75} - 14 q^{76} + 3 q^{78} + 23 q^{79} - 17 q^{80} + 8 q^{81} + 20 q^{82} + 30 q^{83} - 37 q^{85} - 28 q^{86} + 2 q^{87} + 86 q^{88} + 12 q^{89} - 3 q^{90} - 15 q^{92} - 13 q^{93} - 45 q^{94} + 16 q^{95} + 29 q^{96} + 14 q^{97} + 10 q^{99}+O(q^{100}) $ 8 * q + q^2 + 8 * q^3 + 15 * q^4 + 5 * q^5 + q^6 + 9 * q^8 + 8 * q^9 - 3 * q^10 + 10 * q^11 + 15 * q^12 - 6 * q^13 + 5 * q^15 + 13 * q^16 + 21 * q^17 + q^18 + 5 * q^19 - q^20 + 18 * q^22 - 8 * q^23 + 9 * q^24 + 27 * q^25 + 3 * q^26 + 8 * q^27 + 2 * q^29 - 3 * q^30 - 13 * q^31 + 29 * q^32 + 10 * q^33 - 19 * q^34 + 15 * q^36 + 13 * q^37 - 6 * q^38 - 6 * q^39 + 7 * q^40 + 16 * q^41 + 15 * q^43 + 24 * q^44 + 5 * q^45 - q^46 - q^47 + 13 * q^48 + 16 * q^50 + 21 * q^51 - 19 * q^52 + 3 * q^53 + q^54 - 10 * q^55 + 5 * q^57 - 40 * q^58 + 26 * q^59 - q^60 - 14 * q^61 - 14 * q^62 + 49 * q^64 - 3 * q^65 + 18 * q^66 + 38 * q^67 + 43 * q^68 - 8 * q^69 + 9 * q^71 + 9 * q^72 - 6 * q^73 + 32 * q^74 + 27 * q^75 - 14 * q^76 + 3 * q^78 + 23 * q^79 - 17 * q^80 + 8 * q^81 + 20 * q^82 + 30 * q^83 - 37 * q^85 - 28 * q^86 + 2 * q^87 + 86 * q^88 + 12 * q^89 - 3 * q^90 - 15 * q^92 - 13 * q^93 - 45 * q^94 + 16 * q^95 + 29 * q^96 + 14 * q^97 + 10 * 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 $

$2$	2.74585	1.94161	0.970804	−	0.239874i	$-0.0771060\pi$
$2$	2.74585	1.94161	0.970804	+	0.239874i	$0.0771060\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	5.53968	2.76984
$5$	−1.91481	−0.856327	−0.428163	−	0.903701i	$-0.640839\pi$
$5$	−1.91481	−0.856327	−0.428163	+	0.903701i	$0.640839\pi$
$6$	2.74585	1.12099
$7$	0	0
$7$	0	0
$8$	9.71944	3.43634
$9$	1.00000	0.333333
$10$	−5.25777	−1.66265
$11$	5.32332	1.60504	0.802521	−	0.596624i	$-0.203492\pi$
$11$	5.32332	1.60504	0.802521	+	0.596624i	$0.203492\pi$
$12$	5.53968	1.59917
$13$	−4.78352	−1.32671	−0.663355	−	0.748305i	$-0.730868\pi$
$13$	−4.78352	−1.32671	−0.663355	+	0.748305i	$0.730868\pi$
$14$	0	0
$15$	−1.91481	−0.494401
$16$	15.6087	3.90218
$17$	2.61781	0.634913	0.317457	−	0.948273i	$-0.397171\pi$
$17$	2.61781	0.634913	0.317457	+	0.948273i	$0.397171\pi$
$18$	2.74585	0.647203
$19$	4.46024	1.02325	0.511624	−	0.859209i	$-0.329044\pi$
$19$	4.46024	1.02325	0.511624	+	0.859209i	$0.329044\pi$
$20$	−10.6074	−2.37189
$21$	0	0
$22$	14.6170	3.11636
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	9.71944	1.98397
$25$	−1.33352	−0.266704
$26$	−13.1348	−2.57595
$27$	1.00000	0.192450
$28$	0	0
$29$	−8.86099	−1.64544	−0.822722	−	0.568444i	$-0.807546\pi$
$29$	−8.86099	−1.64544	−0.822722	+	0.568444i	$0.807546\pi$
$30$	−5.25777	−0.959932
$31$	5.90898	1.06128	0.530642	−	0.847596i	$-0.321951\pi$
$31$	5.90898	1.06128	0.530642	+	0.847596i	$0.321951\pi$
$32$	23.4204	4.14017
$33$	5.32332	0.926671
$34$	7.18812	1.23275
$35$	0	0
$36$	5.53968	0.923281
$37$	−2.91929	−0.479929	−0.239964	−	0.970782i	$-0.577136\pi$
$37$	−2.91929	−0.479929	−0.239964	+	0.970782i	$0.577136\pi$
$38$	12.2471	1.98675
$39$	−4.78352	−0.765976
$40$	−18.6108	−2.94263
$41$	4.98968	0.779258	0.389629	−	0.920972i	$-0.372603\pi$
$41$	4.98968	0.779258	0.389629	+	0.920972i	$0.372603\pi$
$42$	0	0
$43$	−6.45341	−0.984136	−0.492068	−	0.870557i	$-0.663759\pi$
$43$	−6.45341	−0.984136	−0.492068	+	0.870557i	$0.663759\pi$
$44$	29.4895	4.44571
$45$	−1.91481	−0.285442
$46$	−2.74585	−0.404853
$47$	−6.94360	−1.01283	−0.506414	−	0.862291i	$-0.669029\pi$
$47$	−6.94360	−1.01283	−0.506414	+	0.862291i	$0.669029\pi$
$48$	15.6087	2.25293
$49$	0	0
$50$	−3.66165	−0.517835
$51$	2.61781	0.366567
$52$	−26.4992	−3.67478
$53$	1.01228	0.139048	0.0695238	−	0.997580i	$-0.477852\pi$
$53$	1.01228	0.139048	0.0695238	+	0.997580i	$0.477852\pi$
$54$	2.74585	0.373663
$55$	−10.1931	−1.37444
$56$	0	0
$57$	4.46024	0.590773
$58$	−24.3309	−3.19481
$59$	6.15656	0.801515	0.400758	−	0.916184i	$-0.368747\pi$
$59$	6.15656	0.801515	0.400758	+	0.916184i	$0.368747\pi$
$60$	−10.6074	−1.36941
$61$	2.24372	0.287279	0.143639	−	0.989630i	$-0.454119\pi$
$61$	2.24372	0.287279	0.143639	+	0.989630i	$0.454119\pi$
$62$	16.2252	2.06060
$63$	0	0
$64$	33.0913	4.13641
$65$	9.15951	1.13610
$66$	14.6170	1.79923
$67$	9.38847	1.14698	0.573492	−	0.819211i	$-0.305588\pi$
$67$	9.38847	1.14698	0.573492	+	0.819211i	$0.305588\pi$
$68$	14.5019	1.75861
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−10.2261	−1.21362	−0.606810	−	0.794847i	$-0.707551\pi$
$71$	−10.2261	−1.21362	−0.606810	+	0.794847i	$0.707551\pi$
$72$	9.71944	1.14545
$73$	3.85118	0.450747	0.225373	−	0.974272i	$-0.427640\pi$
$73$	3.85118	0.450747	0.225373	+	0.974272i	$0.427640\pi$
$74$	−8.01593	−0.931833
$75$	−1.33352	−0.153982
$76$	24.7083	2.83424
$77$	0	0
$78$	−13.1348	−1.48723
$79$	2.97781	0.335030	0.167515	−	0.985870i	$-0.446426\pi$
$79$	2.97781	0.335030	0.167515	+	0.985870i	$0.446426\pi$
$80$	−29.8877	−3.34155
$81$	1.00000	0.111111
$82$	13.7009	1.51301
$83$	−2.29460	−0.251865	−0.125932	−	0.992039i	$-0.540192\pi$
$83$	−2.29460	−0.251865	−0.125932	+	0.992039i	$0.540192\pi$
$84$	0	0
$85$	−5.01260	−0.543693
$86$	−17.7201	−1.91081
$87$	−8.86099	−0.949998
$88$	51.7397	5.51547
$89$	−10.0816	−1.06865	−0.534325	−	0.845279i	$-0.679434\pi$
$89$	−10.0816	−1.06865	−0.534325	+	0.845279i	$0.679434\pi$
$90$	−5.25777	−0.554217
$91$	0	0
$92$	−5.53968	−0.577552
$93$	5.90898	0.612732
$94$	−19.0661	−1.96651
$95$	−8.54048	−0.876235
$96$	23.4204	2.39033
$97$	13.2317	1.34347	0.671735	−	0.740791i	$-0.265549\pi$
$97$	13.2317	1.34347	0.671735	+	0.740791i	$0.265549\pi$
$98$	0	0
$99$	5.32332	0.535014
$100$	−7.38728	−0.738728
$101$	−8.17507	−0.813450	−0.406725	−	0.913551i	$-0.633329\pi$
$101$	−8.17507	−0.813450	−0.406725	+	0.913551i	$0.633329\pi$
$102$	7.18812	0.711730
$103$	−12.1436	−1.19655	−0.598273	−	0.801293i	$-0.704146\pi$
$103$	−12.1436	−1.19655	−0.598273	+	0.801293i	$0.704146\pi$
$104$	−46.4931	−4.55903
$105$	0	0
$106$	2.77957	0.269976
$107$	11.6601	1.12722	0.563611	−	0.826040i	$-0.309412\pi$
$107$	11.6601	1.12722	0.563611	+	0.826040i	$0.309412\pi$
$108$	5.53968	0.533056
$109$	0.884608	0.0847301	0.0423650	−	0.999102i	$-0.486511\pi$
$109$	0.884608	0.0847301	0.0423650	+	0.999102i	$0.486511\pi$
$110$	−27.9888	−2.66863
$111$	−2.91929	−0.277087
$112$	0	0
$113$	−1.00924	−0.0949413	−0.0474706	−	0.998873i	$-0.515116\pi$
$113$	−1.00924	−0.0949413	−0.0474706	+	0.998873i	$0.515116\pi$
$114$	12.2471	1.14705
$115$	1.91481	0.178557
$116$	−49.0871	−4.55762
$117$	−4.78352	−0.442237
$118$	16.9050	1.55623
$119$	0	0
$120$	−18.6108	−1.69893
$121$	17.3378	1.57616
$122$	6.16091	0.557782
$123$	4.98968	0.449905
$124$	32.7339	2.93959
$125$	12.1275	1.08471
$126$	0	0
$127$	−1.94220	−0.172342	−0.0861711	−	0.996280i	$-0.527463\pi$
$127$	−1.94220	−0.172342	−0.0861711	+	0.996280i	$0.527463\pi$
$128$	44.0229	3.89111
$129$	−6.45341	−0.568191
$130$	25.1506	2.20586
$131$	−7.12385	−0.622414	−0.311207	−	0.950342i	$-0.600733\pi$
$131$	−7.12385	−0.622414	−0.311207	+	0.950342i	$0.600733\pi$
$132$	29.4895	2.56673
$133$	0	0
$134$	25.7793	2.22699
$135$	−1.91481	−0.164800
$136$	25.4437	2.18178
$137$	6.12064	0.522921	0.261461	−	0.965214i	$-0.415796\pi$
$137$	6.12064	0.522921	0.261461	+	0.965214i	$0.415796\pi$
$138$	−2.74585	−0.233742
$139$	−13.1313	−1.11378	−0.556892	−	0.830585i	$-0.688006\pi$
$139$	−13.1313	−1.11378	−0.556892	+	0.830585i	$0.688006\pi$
$140$	0	0
$141$	−6.94360	−0.584756
$142$	−28.0794	−2.35637
$143$	−25.4642	−2.12942
$144$	15.6087	1.30073
$145$	16.9671	1.40904
$146$	10.5748	0.875174
$147$	0	0
$148$	−16.1720	−1.32933
$149$	1.96204	0.160736	0.0803682	−	0.996765i	$-0.474390\pi$
$149$	1.96204	0.160736	0.0803682	+	0.996765i	$0.474390\pi$
$150$	−3.66165	−0.298972
$151$	−15.6637	−1.27469	−0.637345	−	0.770579i	$-0.719967\pi$
$151$	−15.6637	−1.27469	−0.637345	+	0.770579i	$0.719967\pi$
$152$	43.3510	3.51623
$153$	2.61781	0.211638
$154$	0	0
$155$	−11.3145	−0.908805
$156$	−26.4992	−2.12163
$157$	−17.4547	−1.39304	−0.696518	−	0.717539i	$-0.745269\pi$
$157$	−17.4547	−1.39304	−0.696518	+	0.717539i	$0.745269\pi$
$158$	8.17661	0.650496
$159$	1.01228	0.0802792
$160$	−44.8454	−3.54534
$161$	0	0
$162$	2.74585	0.215734
$163$	−11.8987	−0.931975	−0.465988	−	0.884791i	$-0.654301\pi$
$163$	−11.8987	−0.931975	−0.465988	+	0.884791i	$0.654301\pi$
$164$	27.6413	2.15842
$165$	−10.1931	−0.793534
$166$	−6.30062	−0.489023
$167$	−9.48883	−0.734267	−0.367134	−	0.930168i	$-0.619661\pi$
$167$	−9.48883	−0.734267	−0.367134	+	0.930168i	$0.619661\pi$
$168$	0	0
$169$	9.88206	0.760158
$170$	−13.7639	−1.05564
$171$	4.46024	0.341083
$172$	−35.7499	−2.72590
$173$	−5.46126	−0.415212	−0.207606	−	0.978213i	$-0.566567\pi$
$173$	−5.46126	−0.415212	−0.207606	+	0.978213i	$0.566567\pi$
$174$	−24.3309	−1.84452
$175$	0	0
$176$	83.0903	6.26317
$177$	6.15656	0.462755
$178$	−27.6826	−2.07490
$179$	−0.840159	−0.0627964	−0.0313982	−	0.999507i	$-0.509996\pi$
$179$	−0.840159	−0.0627964	−0.0313982	+	0.999507i	$0.509996\pi$
$180$	−10.6074	−0.790630
$181$	17.2554	1.28258	0.641291	−	0.767298i	$-0.278399\pi$
$181$	17.2554	1.28258	0.641291	+	0.767298i	$0.278399\pi$
$182$	0	0
$183$	2.24372	0.165860
$184$	−9.71944	−0.716527
$185$	5.58988	0.410976
$186$	16.2252	1.18969
$187$	13.9355	1.01906
$188$	−38.4653	−2.80537
$189$	0	0
$190$	−23.4509	−1.70131
$191$	−13.7884	−0.997697	−0.498849	−	0.866689i	$-0.666244\pi$
$191$	−13.7884	−0.997697	−0.498849	+	0.866689i	$0.666244\pi$
$192$	33.0913	2.38816
$193$	−19.5816	−1.40951	−0.704756	−	0.709450i	$-0.748943\pi$
$193$	−19.5816	−1.40951	−0.704756	+	0.709450i	$0.748943\pi$
$194$	36.3321	2.60849
$195$	9.15951	0.655926
$196$	0	0
$197$	−18.8814	−1.34525	−0.672623	−	0.739986i	$-0.734832\pi$
$197$	−18.8814	−1.34525	−0.672623	+	0.739986i	$0.734832\pi$
$198$	14.6170	1.03879
$199$	−7.04079	−0.499108	−0.249554	−	0.968361i	$-0.580284\pi$
$199$	−7.04079	−0.499108	−0.249554	+	0.968361i	$0.580284\pi$
$200$	−12.9611	−0.916486
$201$	9.38847	0.662212
$202$	−22.4475	−1.57940
$203$	0	0
$204$	14.5019	1.01533
$205$	−9.55427	−0.667299
$206$	−33.3445	−2.32322
$207$	−1.00000	−0.0695048
$208$	−74.6647	−5.17707
$209$	23.7433	1.64236
$210$	0	0
$211$	−16.1639	−1.11277	−0.556386	−	0.830924i	$-0.687812\pi$
$211$	−16.1639	−1.11277	−0.556386	+	0.830924i	$0.687812\pi$
$212$	5.60772	0.385140
$213$	−10.2261	−0.700683
$214$	32.0168	2.18862
$215$	12.3570	0.842742
$216$	9.71944	0.661324
$217$	0	0
$218$	2.42900	0.164513
$219$	3.85118	0.260239
$220$	−56.4667	−3.80698
$221$	−12.5224	−0.842345
$222$	−8.01593	−0.537994
$223$	1.72721	0.115662	0.0578312	−	0.998326i	$-0.481581\pi$
$223$	1.72721	0.115662	0.0578312	+	0.998326i	$0.481581\pi$
$224$	0	0
$225$	−1.33352	−0.0889014
$226$	−2.77122	−0.184339
$227$	3.06888	0.203688	0.101844	−	0.994800i	$-0.467526\pi$
$227$	3.06888	0.203688	0.101844	+	0.994800i	$0.467526\pi$
$228$	24.7083	1.63635
$229$	2.87483	0.189974	0.0949872	−	0.995478i	$-0.469719\pi$
$229$	2.87483	0.189974	0.0949872	+	0.995478i	$0.469719\pi$
$230$	5.25777	0.346687
$231$	0	0
$232$	−86.1238	−5.65431
$233$	−9.06274	−0.593720	−0.296860	−	0.954921i	$-0.595939\pi$
$233$	−9.06274	−0.593720	−0.296860	+	0.954921i	$0.595939\pi$
$234$	−13.1348	−0.858650
$235$	13.2956	0.867312
$236$	34.1054	2.22007
$237$	2.97781	0.193429
$238$	0	0
$239$	28.5432	1.84630	0.923152	−	0.384436i	$-0.125604\pi$
$239$	28.5432	1.84630	0.923152	+	0.384436i	$0.125604\pi$
$240$	−29.8877	−1.92924
$241$	−10.3623	−0.667496	−0.333748	−	0.942662i	$-0.608313\pi$
$241$	−10.3623	−0.667496	−0.333748	+	0.942662i	$0.608313\pi$
$242$	47.6068	3.06028
$243$	1.00000	0.0641500
$244$	12.4295	0.795716
$245$	0	0
$246$	13.7009	0.873538
$247$	−21.3356	−1.35755
$248$	57.4319	3.64693
$249$	−2.29460	−0.145414
$250$	33.3002	2.10609
$251$	−13.6387	−0.860865	−0.430433	−	0.902623i	$-0.641639\pi$
$251$	−13.6387	−0.860865	−0.430433	+	0.902623i	$0.641639\pi$
$252$	0	0
$253$	−5.32332	−0.334674
$254$	−5.33298	−0.334621
$255$	−5.01260	−0.313901
$256$	54.6977	3.41861
$257$	10.9968	0.685962	0.342981	−	0.939342i	$-0.388563\pi$
$257$	10.9968	0.685962	0.342981	+	0.939342i	$0.388563\pi$
$258$	−17.7201	−1.10320
$259$	0	0
$260$	50.7408	3.14681
$261$	−8.86099	−0.548481
$262$	−19.5610	−1.20848
$263$	8.29171	0.511289	0.255644	−	0.966771i	$-0.417712\pi$
$263$	8.29171	0.511289	0.255644	+	0.966771i	$0.417712\pi$
$264$	51.7397	3.18436
$265$	−1.93832	−0.119070
$266$	0	0
$267$	−10.0816	−0.616986
$268$	52.0092	3.17697
$269$	−1.06056	−0.0646635	−0.0323318	−	0.999477i	$-0.510293\pi$
$269$	−1.06056	−0.0646635	−0.0323318	+	0.999477i	$0.510293\pi$
$270$	−5.25777	−0.319977
$271$	−22.3153	−1.35556	−0.677779	−	0.735266i	$-0.737057\pi$
$271$	−22.3153	−1.35556	−0.677779	+	0.735266i	$0.737057\pi$
$272$	40.8608	2.47755
$273$	0	0
$274$	16.8063	1.01531
$275$	−7.09876	−0.428071
$276$	−5.53968	−0.333450
$277$	24.4131	1.46684	0.733420	−	0.679776i	$-0.237923\pi$
$277$	24.4131	1.46684	0.733420	+	0.679776i	$0.237923\pi$
$278$	−36.0566	−2.16253
$279$	5.90898	0.353761
$280$	0	0
$281$	−0.764668	−0.0456163	−0.0228081	−	0.999740i	$-0.507261\pi$
$281$	−0.764668	−0.0456163	−0.0228081	+	0.999740i	$0.507261\pi$
$282$	−19.0661	−1.13537
$283$	2.82032	0.167650	0.0838252	−	0.996480i	$-0.473286\pi$
$283$	2.82032	0.167650	0.0838252	+	0.996480i	$0.473286\pi$
$284$	−56.6496	−3.36153
$285$	−8.54048	−0.505895
$286$	−69.9209	−4.13451
$287$	0	0
$288$	23.4204	1.38006
$289$	−10.1471	−0.596885
$290$	46.5890	2.73580
$291$	13.2317	0.775653
$292$	21.3343	1.24850
$293$	32.2644	1.88490	0.942452	−	0.334342i	$-0.108514\pi$
$293$	32.2644	1.88490	0.942452	+	0.334342i	$0.108514\pi$
$294$	0	0
$295$	−11.7886	−0.686359
$296$	−28.3739	−1.64920
$297$	5.32332	0.308890
$298$	5.38746	0.312087
$299$	4.78352	0.276638
$300$	−7.38728	−0.426505
$301$	0	0
$302$	−43.0100	−2.47495
$303$	−8.17507	−0.469645
$304$	69.6186	3.99290
$305$	−4.29628	−0.246004
$306$	7.18812	0.410918
$307$	−12.2320	−0.698118	−0.349059	−	0.937101i	$-0.613499\pi$
$307$	−12.2320	−0.698118	−0.349059	+	0.937101i	$0.613499\pi$
$308$	0	0
$309$	−12.1436	−0.690826
$310$	−31.0680	−1.76454
$311$	6.44397	0.365404	0.182702	−	0.983168i	$-0.441516\pi$
$311$	6.44397	0.365404	0.182702	+	0.983168i	$0.441516\pi$
$312$	−46.4931	−2.63215
$313$	−31.9752	−1.80735	−0.903673	−	0.428223i	$-0.859140\pi$
$313$	−31.9752	−1.80735	−0.903673	+	0.428223i	$0.859140\pi$
$314$	−47.9280	−2.70473
$315$	0	0
$316$	16.4961	0.927979
$317$	14.2714	0.801564	0.400782	−	0.916173i	$-0.368738\pi$
$317$	14.2714	0.801564	0.400782	+	0.916173i	$0.368738\pi$
$318$	2.77957	0.155871
$319$	−47.1699	−2.64101
$320$	−63.3633	−3.54212
$321$	11.6601	0.650802
$322$	0	0
$323$	11.6761	0.649674
$324$	5.53968	0.307760
$325$	6.37892	0.353839
$326$	−32.6719	−1.80953
$327$	0.884608	0.0489189
$328$	48.4969	2.67779
$329$	0	0
$330$	−27.9888	−1.54073
$331$	5.19270	0.285416	0.142708	−	0.989765i	$-0.454419\pi$
$331$	5.19270	0.285416	0.142708	+	0.989765i	$0.454419\pi$
$332$	−12.7114	−0.697626
$333$	−2.91929	−0.159976
$334$	−26.0549	−1.42566
$335$	−17.9771	−0.982193
$336$	0	0
$337$	27.3455	1.48961	0.744803	−	0.667284i	$-0.232543\pi$
$337$	27.3455	1.48961	0.744803	+	0.667284i	$0.232543\pi$
$338$	27.1346	1.47593
$339$	−1.00924	−0.0548144
$340$	−27.7682	−1.50594
$341$	31.4554	1.70340
$342$	12.2471	0.662249
$343$	0	0
$344$	−62.7236	−3.38183
$345$	1.91481	0.103090
$346$	−14.9958	−0.806179
$347$	5.15907	0.276954	0.138477	−	0.990366i	$-0.455779\pi$
$347$	5.15907	0.276954	0.138477	+	0.990366i	$0.455779\pi$
$348$	−49.0871	−2.63134
$349$	15.1720	0.812140	0.406070	−	0.913842i	$-0.366899\pi$
$349$	15.1720	0.812140	0.406070	+	0.913842i	$0.366899\pi$
$350$	0	0
$351$	−4.78352	−0.255325
$352$	124.674	6.64515
$353$	4.97378	0.264728	0.132364	−	0.991201i	$-0.457743\pi$
$353$	4.97378	0.264728	0.132364	+	0.991201i	$0.457743\pi$
$354$	16.9050	0.898489
$355$	19.5811	1.03925
$356$	−55.8491	−2.95999
$357$	0	0
$358$	−2.30695	−0.121926
$359$	19.5574	1.03220	0.516099	−	0.856529i	$-0.327384\pi$
$359$	19.5574	1.03220	0.516099	+	0.856529i	$0.327384\pi$
$360$	−18.6108	−0.980877
$361$	0.893701	0.0470369
$362$	47.3807	2.49027
$363$	17.3378	0.909996
$364$	0	0
$365$	−7.37427	−0.385987
$366$	6.16091	0.322036
$367$	−11.6525	−0.608256	−0.304128	−	0.952631i	$-0.598365\pi$
$367$	−11.6525	−0.608256	−0.304128	+	0.952631i	$0.598365\pi$
$368$	−15.6087	−0.813662
$369$	4.98968	0.259753
$370$	15.3490	0.797954
$371$	0	0
$372$	32.7339	1.69717
$373$	−13.2588	−0.686516	−0.343258	−	0.939241i	$-0.611531\pi$
$373$	−13.2588	−0.686516	−0.343258	+	0.939241i	$0.611531\pi$
$374$	38.2647	1.97862
$375$	12.1275	0.626259
$376$	−67.4879	−3.48042
$377$	42.3867	2.18303
$378$	0	0
$379$	−5.81701	−0.298800	−0.149400	−	0.988777i	$-0.547734\pi$
$379$	−5.81701	−0.298800	−0.149400	+	0.988777i	$0.547734\pi$
$380$	−47.3116	−2.42703
$381$	−1.94220	−0.0995018
$382$	−37.8610	−1.93714
$383$	16.7857	0.857708	0.428854	−	0.903374i	$-0.358917\pi$
$383$	16.7857	0.857708	0.428854	+	0.903374i	$0.358917\pi$
$384$	44.0229	2.24653
$385$	0	0
$386$	−53.7680	−2.73672
$387$	−6.45341	−0.328045
$388$	73.2992	3.72120
$389$	3.08411	0.156370	0.0781852	−	0.996939i	$-0.475087\pi$
$389$	3.08411	0.156370	0.0781852	+	0.996939i	$0.475087\pi$
$390$	25.1506	1.27355
$391$	−2.61781	−0.132389
$392$	0	0
$393$	−7.12385	−0.359351
$394$	−51.8455	−2.61194
$395$	−5.70192	−0.286895
$396$	29.4895	1.48190
$397$	−20.6686	−1.03733	−0.518663	−	0.854979i	$-0.673570\pi$
$397$	−20.6686	−1.03733	−0.518663	+	0.854979i	$0.673570\pi$
$398$	−19.3329	−0.969073
$399$	0	0
$400$	−20.8146	−1.04073
$401$	27.3184	1.36422	0.682109	−	0.731251i	$-0.261063\pi$
$401$	27.3184	1.36422	0.682109	+	0.731251i	$0.261063\pi$
$402$	25.7793	1.28576
$403$	−28.2657	−1.40801
$404$	−45.2873	−2.25313
$405$	−1.91481	−0.0951474
$406$	0	0
$407$	−15.5403	−0.770306
$408$	25.4437	1.25965
$409$	10.0609	0.497477	0.248739	−	0.968571i	$-0.419984\pi$
$409$	10.0609	0.497477	0.248739	+	0.968571i	$0.419984\pi$
$410$	−26.2346	−1.29563
$411$	6.12064	0.301909
$412$	−67.2718	−3.31424
$413$	0	0
$414$	−2.74585	−0.134951
$415$	4.39371	0.215679
$416$	−112.032	−5.49281
$417$	−13.1313	−0.643044
$418$	65.1954	3.18881
$419$	−17.1206	−0.836396	−0.418198	−	0.908356i	$-0.637338\pi$
$419$	−17.1206	−0.836396	−0.418198	+	0.908356i	$0.637338\pi$
$420$	0	0
$421$	−20.4594	−0.997129	−0.498565	−	0.866853i	$-0.666139\pi$
$421$	−20.4594	−0.997129	−0.498565	+	0.866853i	$0.666139\pi$
$422$	−44.3837	−2.16057
$423$	−6.94360	−0.337609
$424$	9.83881	0.477815
$425$	−3.49091	−0.169334
$426$	−28.0794	−1.36045
$427$	0	0
$428$	64.5931	3.12223
$429$	−25.4642	−1.22942
$430$	33.9305	1.63628
$431$	18.0657	0.870194	0.435097	−	0.900383i	$-0.356714\pi$
$431$	18.0657	0.870194	0.435097	+	0.900383i	$0.356714\pi$
$432$	15.6087	0.750976
$433$	−14.9439	−0.718157	−0.359079	−	0.933307i	$-0.616909\pi$
$433$	−14.9439	−0.718157	−0.359079	+	0.933307i	$0.616909\pi$
$434$	0	0
$435$	16.9671	0.813509
$436$	4.90045	0.234689
$437$	−4.46024	−0.213362
$438$	10.5748	0.505282
$439$	−22.9749	−1.09653	−0.548265	−	0.836305i	$-0.684712\pi$
$439$	−22.9749	−1.09653	−0.548265	+	0.836305i	$0.684712\pi$
$440$	−99.0714	−4.72305
$441$	0	0
$442$	−34.3845	−1.63550
$443$	−13.6849	−0.650187	−0.325094	−	0.945682i	$-0.605396\pi$
$443$	−13.6849	−0.650187	−0.325094	+	0.945682i	$0.605396\pi$
$444$	−16.1720	−0.767487
$445$	19.3044	0.915114
$446$	4.74265	0.224571
$447$	1.96204	0.0928012
$448$	0	0
$449$	−19.1186	−0.902261	−0.451130	−	0.892458i	$-0.648979\pi$
$449$	−19.1186	−0.902261	−0.451130	+	0.892458i	$0.648979\pi$
$450$	−3.66165	−0.172612
$451$	26.5617	1.25074
$452$	−5.59087	−0.262972
$453$	−15.6637	−0.735943
$454$	8.42667	0.395483
$455$	0	0
$456$	43.3510	2.03010
$457$	−18.0657	−0.845076	−0.422538	−	0.906345i	$-0.638861\pi$
$457$	−18.0657	−0.845076	−0.422538	+	0.906345i	$0.638861\pi$
$458$	7.89386	0.368856
$459$	2.61781	0.122189
$460$	10.6074	0.494573
$461$	32.2584	1.50242	0.751211	−	0.660062i	$-0.229470\pi$
$461$	32.2584	1.50242	0.751211	+	0.660062i	$0.229470\pi$
$462$	0	0
$463$	24.3695	1.13255	0.566274	−	0.824217i	$-0.308385\pi$
$463$	24.3695	1.13255	0.566274	+	0.824217i	$0.308385\pi$
$464$	−138.309	−6.42083
$465$	−11.3145	−0.524699
$466$	−24.8849	−1.15277
$467$	−33.6645	−1.55781	−0.778903	−	0.627144i	$-0.784224\pi$
$467$	−33.6645	−1.55781	−0.778903	+	0.627144i	$0.784224\pi$
$468$	−26.4992	−1.22493
$469$	0	0
$470$	36.5078	1.68398
$471$	−17.4547	−0.804270
$472$	59.8383	2.75428
$473$	−34.3536	−1.57958
$474$	8.17661	0.375564
$475$	−5.94782	−0.272905
$476$	0	0
$477$	1.01228	0.0463492
$478$	78.3752	3.58480
$479$	32.7675	1.49718	0.748592	−	0.663030i	$-0.230730\pi$
$479$	32.7675	1.49718	0.748592	+	0.663030i	$0.230730\pi$
$480$	−44.8454	−2.04690
$481$	13.9645	0.636726
$482$	−28.4534	−1.29602
$483$	0	0
$484$	96.0457	4.36571
$485$	−25.3360	−1.15045
$486$	2.74585	0.124554
$487$	21.7581	0.985954	0.492977	−	0.870042i	$-0.335909\pi$
$487$	21.7581	0.985954	0.492977	+	0.870042i	$0.335909\pi$
$488$	21.8077	0.987187
$489$	−11.8987	−0.538076
$490$	0	0
$491$	37.4957	1.69216	0.846078	−	0.533059i	$-0.178958\pi$
$491$	37.4957	1.69216	0.846078	+	0.533059i	$0.178958\pi$
$492$	27.6413	1.24616
$493$	−23.1964	−1.04471
$494$	−58.5844	−2.63584
$495$	−10.1931	−0.458147
$496$	92.2316	4.14132
$497$	0	0
$498$	−6.30062	−0.282338
$499$	7.62293	0.341249	0.170625	−	0.985336i	$-0.445421\pi$
$499$	7.62293	0.341249	0.170625	+	0.985336i	$0.445421\pi$
$500$	67.1823	3.00448
$501$	−9.48883	−0.423930
$502$	−37.4497	−1.67146
$503$	−18.3255	−0.817092	−0.408546	−	0.912738i	$-0.633964\pi$
$503$	−18.3255	−0.817092	−0.408546	+	0.912738i	$0.633964\pi$
$504$	0	0
$505$	15.6537	0.696579
$506$	−14.6170	−0.649806
$507$	9.88206	0.438878
$508$	−10.7592	−0.477361
$509$	37.3481	1.65543	0.827713	−	0.561152i	$-0.189642\pi$
$509$	37.3481	1.65543	0.827713	+	0.561152i	$0.189642\pi$
$510$	−13.7639	−0.609474
$511$	0	0
$512$	62.1458	2.74648
$513$	4.46024	0.196924
$514$	30.1956	1.33187
$515$	23.2526	1.02463
$516$	−35.7499	−1.57380
$517$	−36.9630	−1.62563
$518$	0	0
$519$	−5.46126	−0.239723
$520$	89.0253	3.90402
$521$	−26.6237	−1.16641	−0.583204	−	0.812326i	$-0.698201\pi$
$521$	−26.6237	−1.16641	−0.583204	+	0.812326i	$0.698201\pi$
$522$	−24.3309	−1.06494
$523$	40.8800	1.78756	0.893780	−	0.448506i	$-0.148044\pi$
$523$	40.8800	1.78756	0.893780	+	0.448506i	$0.148044\pi$
$524$	−39.4639	−1.72399
$525$	0	0
$526$	22.7678	0.992722
$527$	15.4686	0.673823
$528$	83.0903	3.61604
$529$	1.00000	0.0434783
$530$	−5.32234	−0.231188
$531$	6.15656	0.267172
$532$	0	0
$533$	−23.8682	−1.03385
$534$	−27.6826	−1.19794
$535$	−22.3268	−0.965270
$536$	91.2507	3.94143
$537$	−0.840159	−0.0362555
$538$	−2.91214	−0.125551
$539$	0	0
$540$	−10.6074	−0.456471
$541$	22.1375	0.951768	0.475884	−	0.879508i	$-0.342128\pi$
$541$	22.1375	0.951768	0.475884	+	0.879508i	$0.342128\pi$
$542$	−61.2744	−2.63196
$543$	17.2554	0.740499
$544$	61.3101	2.62865
$545$	−1.69385	−0.0725566
$546$	0	0
$547$	−24.7940	−1.06012	−0.530058	−	0.847962i	$-0.677830\pi$
$547$	−24.7940	−1.06012	−0.530058	+	0.847962i	$0.677830\pi$
$548$	33.9064	1.44841
$549$	2.24372	0.0957595
$550$	−19.4921	−0.831147
$551$	−39.5221	−1.68370
$552$	−9.71944	−0.413687
$553$	0	0
$554$	67.0346	2.84803
$555$	5.58988	0.237277
$556$	−72.7434	−3.08501
$557$	10.5365	0.446445	0.223223	−	0.974767i	$-0.428342\pi$
$557$	10.5365	0.446445	0.223223	+	0.974767i	$0.428342\pi$
$558$	16.2252	0.686865
$559$	30.8700	1.30566
$560$	0	0
$561$	13.9355	0.588356
$562$	−2.09966	−0.0885689
$563$	−4.12319	−0.173772	−0.0868859	−	0.996218i	$-0.527692\pi$
$563$	−4.12319	−0.173772	−0.0868859	+	0.996218i	$0.527692\pi$
$564$	−38.4653	−1.61968
$565$	1.93250	0.0813008
$566$	7.74416	0.325511
$567$	0	0
$568$	−99.3923	−4.17041
$569$	20.0857	0.842038	0.421019	−	0.907052i	$-0.361673\pi$
$569$	20.0857	0.842038	0.421019	+	0.907052i	$0.361673\pi$
$570$	−23.4509	−0.982249
$571$	39.2874	1.64413	0.822064	−	0.569396i	$-0.192823\pi$
$571$	39.2874	1.64413	0.822064	+	0.569396i	$0.192823\pi$
$572$	−141.064	−5.89817
$573$	−13.7884	−0.576021
$574$	0	0
$575$	1.33352	0.0556117
$576$	33.0913	1.37880
$577$	6.54033	0.272277	0.136139	−	0.990690i	$-0.456531\pi$
$577$	6.54033	0.272277	0.136139	+	0.990690i	$0.456531\pi$
$578$	−27.8623	−1.15892
$579$	−19.5816	−0.813782
$580$	93.9922	3.90281
$581$	0	0
$582$	36.3321	1.50601
$583$	5.38870	0.223177
$584$	37.4313	1.54892
$585$	9.15951	0.378699
$586$	88.5930	3.65974
$587$	45.0583	1.85976	0.929878	−	0.367868i	$-0.119912\pi$
$587$	45.0583	1.85976	0.929878	+	0.367868i	$0.119912\pi$
$588$	0	0
$589$	26.3554	1.08596
$590$	−32.3697	−1.33264
$591$	−18.8814	−0.776678
$592$	−45.5665	−1.87277
$593$	5.07341	0.208340	0.104170	−	0.994560i	$-0.466781\pi$
$593$	5.07341	0.208340	0.104170	+	0.994560i	$0.466781\pi$
$594$	14.6170	0.599744
$595$	0	0
$596$	10.8691	0.445214
$597$	−7.04079	−0.288160
$598$	13.1348	0.537123
$599$	45.3530	1.85307	0.926536	−	0.376206i	$-0.122771\pi$
$599$	45.3530	1.85307	0.926536	+	0.376206i	$0.122771\pi$
$600$	−12.9611	−0.529134
$601$	−4.71581	−0.192362	−0.0961810	−	0.995364i	$-0.530663\pi$
$601$	−4.71581	−0.192362	−0.0961810	+	0.995364i	$0.530663\pi$
$602$	0	0
$603$	9.38847	0.382328
$604$	−86.7717	−3.53069
$605$	−33.1984	−1.34971
$606$	−22.4475	−0.911867
$607$	−31.9679	−1.29754	−0.648769	−	0.760985i	$-0.724716\pi$
$607$	−31.9679	−1.29754	−0.648769	+	0.760985i	$0.724716\pi$
$608$	104.460	4.23642
$609$	0	0
$610$	−11.7969	−0.477644
$611$	33.2148	1.34373
$612$	14.5019	0.586203
$613$	17.4400	0.704396	0.352198	−	0.935926i	$-0.385434\pi$
$613$	17.4400	0.704396	0.352198	+	0.935926i	$0.385434\pi$
$614$	−33.5873	−1.35547
$615$	−9.55427	−0.385265
$616$	0	0
$617$	−2.50110	−0.100691	−0.0503453	−	0.998732i	$-0.516032\pi$
$617$	−2.50110	−0.100691	−0.0503453	+	0.998732i	$0.516032\pi$
$618$	−33.3445	−1.34131
$619$	16.1932	0.650858	0.325429	−	0.945566i	$-0.394491\pi$
$619$	16.1932	0.650858	0.325429	+	0.945566i	$0.394491\pi$
$620$	−62.6790	−2.51725
$621$	−1.00000	−0.0401286
$622$	17.6942	0.709471
$623$	0	0
$624$	−74.6647	−2.98898
$625$	−16.5541	−0.662165
$626$	−87.7991	−3.50916
$627$	23.7433	0.948215
$628$	−96.6935	−3.85849
$629$	−7.64216	−0.304713
$630$	0	0
$631$	−25.3726	−1.01007	−0.505033	−	0.863100i	$-0.668520\pi$
$631$	−25.3726	−1.01007	−0.505033	+	0.863100i	$0.668520\pi$
$632$	28.9426	1.15128
$633$	−16.1639	−0.642459
$634$	39.1872	1.55632
$635$	3.71893	0.147581
$636$	5.60772	0.222361
$637$	0	0
$638$	−129.521	−5.12780
$639$	−10.2261	−0.404540
$640$	−84.2953	−3.33206
$641$	2.56881	0.101462	0.0507310	−	0.998712i	$-0.483845\pi$
$641$	2.56881	0.101462	0.0507310	+	0.998712i	$0.483845\pi$
$642$	32.0168	1.26360
$643$	32.1446	1.26766	0.633829	−	0.773473i	$-0.281482\pi$
$643$	32.1446	1.26766	0.633829	+	0.773473i	$0.281482\pi$
$644$	0	0
$645$	12.3570	0.486558
$646$	32.0607	1.26141
$647$	34.5119	1.35680	0.678400	−	0.734692i	$-0.262673\pi$
$647$	34.5119	1.35680	0.678400	+	0.734692i	$0.262673\pi$
$648$	9.71944	0.381816
$649$	32.7733	1.28647
$650$	17.5156	0.687017
$651$	0	0
$652$	−65.9148	−2.58142
$653$	20.6766	0.809137	0.404569	−	0.914508i	$-0.367422\pi$
$653$	20.6766	0.809137	0.404569	+	0.914508i	$0.367422\pi$
$654$	2.42900	0.0949814
$655$	13.6408	0.532990
$656$	77.8827	3.04081
$657$	3.85118	0.150249
$658$	0	0
$659$	−10.2297	−0.398491	−0.199246	−	0.979950i	$-0.563849\pi$
$659$	−10.2297	−0.398491	−0.199246	+	0.979950i	$0.563849\pi$
$660$	−56.4667	−2.19796
$661$	4.66774	0.181554	0.0907771	−	0.995871i	$-0.471065\pi$
$661$	4.66774	0.181554	0.0907771	+	0.995871i	$0.471065\pi$
$662$	14.2584	0.554167
$663$	−12.5224	−0.486328
$664$	−22.3022	−0.865494
$665$	0	0
$666$	−8.01593	−0.310611
$667$	8.86099	0.343099
$668$	−52.5651	−2.03381
$669$	1.72721	0.0667778
$670$	−49.3624	−1.90703
$671$	11.9440	0.461094
$672$	0	0
$673$	−36.2826	−1.39859	−0.699296	−	0.714832i	$-0.746503\pi$
$673$	−36.2826	−1.39859	−0.699296	+	0.714832i	$0.746503\pi$
$674$	75.0867	2.89223
$675$	−1.33352	−0.0513272
$676$	54.7435	2.10552
$677$	39.6435	1.52362	0.761811	−	0.647799i	$-0.224310\pi$
$677$	39.6435	1.52362	0.761811	+	0.647799i	$0.224310\pi$
$678$	−2.77122	−0.106428
$679$	0	0
$680$	−48.7197	−1.86832
$681$	3.06888	0.117600
$682$	86.3717	3.30734
$683$	−13.8778	−0.531019	−0.265510	−	0.964108i	$-0.585540\pi$
$683$	−13.8778	−0.531019	−0.265510	+	0.964108i	$0.585540\pi$
$684$	24.7083	0.944745
$685$	−11.7198	−0.447792
$686$	0	0
$687$	2.87483	0.109682
$688$	−100.730	−3.84028
$689$	−4.84227	−0.184476
$690$	5.25777	0.200160
$691$	37.3305	1.42012	0.710059	−	0.704142i	$-0.248668\pi$
$691$	37.3305	1.42012	0.710059	+	0.704142i	$0.248668\pi$
$692$	−30.2537	−1.15007
$693$	0	0
$694$	14.1660	0.537735
$695$	25.1439	0.953764
$696$	−86.1238	−3.26452
$697$	13.0621	0.494761
$698$	41.6601	1.57686
$699$	−9.06274	−0.342784
$700$	0	0
$701$	7.48359	0.282651	0.141326	−	0.989963i	$-0.454864\pi$
$701$	7.48359	0.282651	0.141326	+	0.989963i	$0.454864\pi$
$702$	−13.1348	−0.495742
$703$	−13.0207	−0.491086
$704$	176.155	6.63911
$705$	13.2956	0.500743
$706$	13.6572	0.513997
$707$	0	0
$708$	34.1054	1.28176
$709$	48.4901	1.82108	0.910542	−	0.413417i	$-0.135665\pi$
$709$	48.4901	1.82108	0.910542	+	0.413417i	$0.135665\pi$
$710$	53.7666	2.01783
$711$	2.97781	0.111677
$712$	−97.9878	−3.67225
$713$	−5.90898	−0.221293
$714$	0	0
$715$	48.7590	1.82348
$716$	−4.65421	−0.173936
$717$	28.5432	1.06596
$718$	53.7016	2.00412
$719$	−16.1138	−0.600943	−0.300471	−	0.953791i	$-0.597144\pi$
$719$	−16.1138	−0.600943	−0.300471	+	0.953791i	$0.597144\pi$
$720$	−29.8877	−1.11385
$721$	0	0
$722$	2.45397	0.0913272
$723$	−10.3623	−0.385379
$724$	95.5894	3.55255
$725$	11.8163	0.438847
$726$	47.6068	1.76686
$727$	16.2362	0.602167	0.301084	−	0.953598i	$-0.402652\pi$
$727$	16.2362	0.602167	0.301084	+	0.953598i	$0.402652\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−20.2486	−0.749435
$731$	−16.8938	−0.624841
$732$	12.4295	0.459407
$733$	1.39404	0.0514901	0.0257450	−	0.999669i	$-0.491804\pi$
$733$	1.39404	0.0514901	0.0257450	+	0.999669i	$0.491804\pi$
$734$	−31.9960	−1.18099
$735$	0	0
$736$	−23.4204	−0.863286
$737$	49.9778	1.84096
$738$	13.7009	0.504338
$739$	−8.19880	−0.301598	−0.150799	−	0.988564i	$-0.548185\pi$
$739$	−8.19880	−0.301598	−0.150799	+	0.988564i	$0.548185\pi$
$740$	30.9661	1.13834
$741$	−21.3356	−0.783784
$742$	0	0
$743$	−40.6236	−1.49034	−0.745168	−	0.666877i	$-0.767631\pi$
$743$	−40.6236	−1.49034	−0.745168	+	0.666877i	$0.767631\pi$
$744$	57.4319	2.10556
$745$	−3.75692	−0.137643
$746$	−36.4068	−1.33295
$747$	−2.29460	−0.0839550
$748$	77.1981	2.82264
$749$	0	0
$750$	33.3002	1.21595
$751$	31.7668	1.15919	0.579594	−	0.814905i	$-0.303211\pi$
$751$	31.7668	1.15919	0.579594	+	0.814905i	$0.303211\pi$
$752$	−108.381	−3.95224
$753$	−13.6387	−0.497021
$754$	116.387	4.23858
$755$	29.9928	1.09155
$756$	0	0
$757$	14.9385	0.542950	0.271475	−	0.962445i	$-0.412489\pi$
$757$	14.9385	0.542950	0.271475	+	0.962445i	$0.412489\pi$
$758$	−15.9726	−0.580152
$759$	−5.32332	−0.193224
$760$	−83.0087	−3.01104
$761$	46.6199	1.68997	0.844985	−	0.534790i	$-0.179609\pi$
$761$	46.6199	1.68997	0.844985	+	0.534790i	$0.179609\pi$
$762$	−5.33298	−0.193194
$763$	0	0
$764$	−76.3837	−2.76346
$765$	−5.01260	−0.181231
$766$	46.0909	1.66533
$767$	−29.4500	−1.06338
$768$	54.6977	1.97373
$769$	−32.7623	−1.18144	−0.590720	−	0.806876i	$-0.701156\pi$
$769$	−32.7623	−1.18144	−0.590720	+	0.806876i	$0.701156\pi$
$770$	0	0
$771$	10.9968	0.396040
$772$	−108.476	−3.90413
$773$	39.4800	1.42000	0.709998	−	0.704204i	$-0.248696\pi$
$773$	39.4800	1.42000	0.709998	+	0.704204i	$0.248696\pi$
$774$	−17.7201	−0.636936
$775$	−7.87974	−0.283049
$776$	128.604	4.61662
$777$	0	0
$778$	8.46849	0.303610
$779$	22.2552	0.797374
$780$	50.7408	1.81681
$781$	−54.4370	−1.94791
$782$	−7.18812	−0.257047
$783$	−8.86099	−0.316666
$784$	0	0
$785$	33.4224	1.19289
$786$	−19.5610	−0.697719
$787$	−20.5155	−0.731298	−0.365649	−	0.930753i	$-0.619153\pi$
$787$	−20.5155	−0.731298	−0.365649	+	0.930753i	$0.619153\pi$
$788$	−104.597	−3.72612
$789$	8.29171	0.295193
$790$	−15.6566	−0.557038
$791$	0	0
$792$	51.7397	1.83849
$793$	−10.7329	−0.381135
$794$	−56.7528	−2.01408
$795$	−1.93832	−0.0687452
$796$	−39.0038	−1.38245
$797$	11.1887	0.396325	0.198162	−	0.980169i	$-0.436503\pi$
$797$	11.1887	0.396325	0.198162	+	0.980169i	$0.436503\pi$
$798$	0	0
$799$	−18.1770	−0.643058
$800$	−31.2315	−1.10420
$801$	−10.0816	−0.356217
$802$	75.0123	2.64878
$803$	20.5011	0.723468
$804$	52.0092	1.83422
$805$	0	0
$806$	−77.6133	−2.73381
$807$	−1.06056	−0.0373335
$808$	−79.4571	−2.79529
$809$	8.83213	0.310521	0.155260	−	0.987874i	$-0.450378\pi$
$809$	8.83213	0.310521	0.155260	+	0.987874i	$0.450378\pi$
$810$	−5.25777	−0.184739
$811$	40.0206	1.40531	0.702657	−	0.711529i	$-0.251997\pi$
$811$	40.0206	1.40531	0.702657	+	0.711529i	$0.251997\pi$
$812$	0	0
$813$	−22.3153	−0.782632
$814$	−42.6714	−1.49563
$815$	22.7836	0.798076
$816$	40.8608	1.43041
$817$	−28.7837	−1.00702
$818$	27.6256	0.965906
$819$	0	0
$820$	−52.9277	−1.84831
$821$	−33.6641	−1.17488	−0.587442	−	0.809266i	$-0.699865\pi$
$821$	−33.6641	−1.17488	−0.587442	+	0.809266i	$0.699865\pi$
$822$	16.8063	0.586189
$823$	12.3738	0.431323	0.215662	−	0.976468i	$-0.430809\pi$
$823$	12.3738	0.431323	0.215662	+	0.976468i	$0.430809\pi$
$824$	−118.029	−4.11174
$825$	−7.09876	−0.247147
$826$	0	0
$827$	8.19126	0.284838	0.142419	−	0.989806i	$-0.454512\pi$
$827$	8.19126	0.284838	0.142419	+	0.989806i	$0.454512\pi$
$828$	−5.53968	−0.192517
$829$	−8.00301	−0.277956	−0.138978	−	0.990295i	$-0.544382\pi$
$829$	−8.00301	−0.277956	−0.138978	+	0.990295i	$0.544382\pi$
$830$	12.0645	0.418764
$831$	24.4131	0.846881
$832$	−158.293	−5.48781
$833$	0	0
$834$	−36.0566	−1.24854
$835$	18.1693	0.628773
$836$	131.530	4.54907
$837$	5.90898	0.204244
$838$	−47.0106	−1.62395
$839$	−30.7825	−1.06273	−0.531365	−	0.847143i	$-0.678321\pi$
$839$	−30.7825	−1.06273	−0.531365	+	0.847143i	$0.678321\pi$
$840$	0	0
$841$	49.5171	1.70749
$842$	−56.1784	−1.93603
$843$	−0.764668	−0.0263366
$844$	−89.5431	−3.08220
$845$	−18.9222	−0.650944
$846$	−19.0661	−0.655505
$847$	0	0
$848$	15.8004	0.542589
$849$	2.82032	0.0967930
$850$	−9.58551	−0.328780
$851$	2.91929	0.100072
$852$	−56.6496	−1.94078
$853$	37.1025	1.27037	0.635183	−	0.772362i	$-0.280925\pi$
$853$	37.1025	1.27037	0.635183	+	0.772362i	$0.280925\pi$
$854$	0	0
$855$	−8.54048	−0.292078
$856$	113.329	3.87352
$857$	39.2181	1.33966	0.669832	−	0.742513i	$-0.266366\pi$
$857$	39.2181	1.33966	0.669832	+	0.742513i	$0.266366\pi$
$858$	−69.9209	−2.38706
$859$	−26.5529	−0.905972	−0.452986	−	0.891518i	$-0.649641\pi$
$859$	−26.5529	−0.905972	−0.452986	+	0.891518i	$0.649641\pi$
$860$	68.4540	2.33426
$861$	0	0
$862$	49.6057	1.68958
$863$	−43.5713	−1.48318	−0.741591	−	0.670852i	$-0.765929\pi$
$863$	−43.5713	−1.48318	−0.741591	+	0.670852i	$0.765929\pi$
$864$	23.4204	0.796777
$865$	10.4572	0.355557
$866$	−41.0337	−1.39438
$867$	−10.1471	−0.344612
$868$	0	0
$869$	15.8518	0.537737
$870$	46.5890	1.57951
$871$	−44.9099	−1.52171
$872$	8.59789	0.291161
$873$	13.2317	0.447824
$874$	−12.2471	−0.414265
$875$	0	0
$876$	21.3343	0.720821
$877$	−56.6084	−1.91153	−0.955765	−	0.294133i	$-0.904969\pi$
$877$	−56.6084	−1.91153	−0.955765	+	0.294133i	$0.904969\pi$
$878$	−63.0855	−2.12903
$879$	32.2644	1.08825
$880$	−159.102	−5.36332
$881$	12.0366	0.405524	0.202762	−	0.979228i	$-0.435008\pi$
$881$	12.0366	0.405524	0.202762	+	0.979228i	$0.435008\pi$
$882$	0	0
$883$	2.42054	0.0814577	0.0407289	−	0.999170i	$-0.487032\pi$
$883$	2.42054	0.0814577	0.0407289	+	0.999170i	$0.487032\pi$
$884$	−69.3699	−2.33316
$885$	−11.7886	−0.396270
$886$	−37.5765	−1.26241
$887$	17.2377	0.578786	0.289393	−	0.957210i	$-0.406547\pi$
$887$	17.2377	0.578786	0.289393	+	0.957210i	$0.406547\pi$
$888$	−28.3739	−0.952165
$889$	0	0
$890$	53.0069	1.77679
$891$	5.32332	0.178338
$892$	9.56819	0.320367
$893$	−30.9701	−1.03637
$894$	5.38746	0.180184
$895$	1.60874	0.0537743
$896$	0	0
$897$	4.78352	0.159717
$898$	−52.4967	−1.75184
$899$	−52.3594	−1.74628
$900$	−7.38728	−0.246243
$901$	2.64997	0.0882832
$902$	72.9344	2.42845
$903$	0	0
$904$	−9.80924	−0.326251
$905$	−33.0407	−1.09831
$906$	−43.0100	−1.42891
$907$	−23.1026	−0.767110	−0.383555	−	0.923518i	$-0.625300\pi$
$907$	−23.1026	−0.767110	−0.383555	+	0.923518i	$0.625300\pi$
$908$	17.0006	0.564185
$909$	−8.17507	−0.271150
$910$	0	0
$911$	−22.5226	−0.746209	−0.373104	−	0.927789i	$-0.621707\pi$
$911$	−22.5226	−0.746209	−0.373104	+	0.927789i	$0.621707\pi$
$912$	69.6186	2.30530
$913$	−12.2149	−0.404254
$914$	−49.6056	−1.64081
$915$	−4.29628	−0.142031
$916$	15.9257	0.526199
$917$	0	0
$918$	7.18812	0.237243
$919$	−15.1590	−0.500049	−0.250025	−	0.968239i	$-0.580439\pi$
$919$	−15.1590	−0.500049	−0.250025	+	0.968239i	$0.580439\pi$
$920$	18.6108	0.613581
$921$	−12.2320	−0.403059
$922$	88.5766	2.91712
$923$	48.9169	1.61012
$924$	0	0
$925$	3.89294	0.127999
$926$	66.9150	2.19896
$927$	−12.1436	−0.398848
$928$	−207.527	−6.81242
$929$	47.7655	1.56713	0.783567	−	0.621307i	$-0.213398\pi$
$929$	47.7655	1.56713	0.783567	+	0.621307i	$0.213398\pi$
$930$	−31.0680	−1.01876
$931$	0	0
$932$	−50.2047	−1.64451
$933$	6.44397	0.210966
$934$	−92.4376	−3.02465
$935$	−26.6837	−0.872650
$936$	−46.4931	−1.51968
$937$	−55.2717	−1.80565	−0.902824	−	0.430011i	$-0.858510\pi$
$937$	−55.2717	−1.80565	−0.902824	+	0.430011i	$0.858510\pi$
$938$	0	0
$939$	−31.9752	−1.04347
$940$	73.6536	2.40232
$941$	9.61902	0.313571	0.156786	−	0.987633i	$-0.449887\pi$
$941$	9.61902	0.313571	0.156786	+	0.987633i	$0.449887\pi$
$942$	−47.9280	−1.56158
$943$	−4.98968	−0.162486
$944$	96.0961	3.12766
$945$	0	0
$946$	−94.3298	−3.06693
$947$	−40.9406	−1.33039	−0.665195	−	0.746670i	$-0.731652\pi$
$947$	−40.9406	−1.33039	−0.665195	+	0.746670i	$0.731652\pi$
$948$	16.4961	0.535769
$949$	−18.4222	−0.598010
$950$	−16.3318	−0.529874
$951$	14.2714	0.462783
$952$	0	0
$953$	−40.0752	−1.29816	−0.649081	−	0.760719i	$-0.724846\pi$
$953$	−40.0752	−1.29816	−0.649081	+	0.760719i	$0.724846\pi$
$954$	2.77957	0.0899920
$955$	26.4022	0.854355
$956$	158.120	5.11397
$957$	−47.1699	−1.52479
$958$	89.9746	2.90695
$959$	0	0
$960$	−63.3633	−2.04504
$961$	3.91599	0.126322
$962$	38.3444	1.23627
$963$	11.6601	0.375741
$964$	−57.4040	−1.84886
$965$	37.4949	1.20700
$966$	0	0
$967$	45.8542	1.47457	0.737285	−	0.675582i	$-0.236107\pi$
$967$	45.8542	1.47457	0.737285	+	0.675582i	$0.236107\pi$
$968$	168.513	5.41622
$969$	11.6761	0.375089
$970$	−69.5689	−2.23372
$971$	−30.7243	−0.985991	−0.492996	−	0.870032i	$-0.664098\pi$
$971$	−30.7243	−0.985991	−0.492996	+	0.870032i	$0.664098\pi$
$972$	5.53968	0.177685
$973$	0	0
$974$	59.7445	1.91434
$975$	6.37892	0.204289
$976$	35.0216	1.12101
$977$	0.268936	0.00860402	0.00430201	−	0.999991i	$-0.498631\pi$
$977$	0.268936	0.00860402	0.00430201	+	0.999991i	$0.498631\pi$
$978$	−32.6719	−1.04473
$979$	−53.6678	−1.71523
$980$	0	0
$981$	0.884608	0.0282434
$982$	102.957	3.28550
$983$	−8.59485	−0.274133	−0.137067	−	0.990562i	$-0.543767\pi$
$983$	−8.59485	−0.274133	−0.137067	+	0.990562i	$0.543767\pi$
$984$	48.4969	1.54603
$985$	36.1542	1.15197
$986$	−63.6939	−2.02843
$987$	0	0
$988$	−118.193	−3.76021
$989$	6.45341	0.205207
$990$	−27.9888	−0.889542
$991$	24.2310	0.769722	0.384861	−	0.922975i	$-0.374249\pi$
$991$	24.2310	0.769722	0.384861	+	0.922975i	$0.374249\pi$
$992$	138.390	4.39390
$993$	5.19270	0.164785
$994$	0	0
$995$	13.4817	0.427400
$996$	−12.7114	−0.402775
$997$	−37.9955	−1.20333	−0.601665	−	0.798749i	$-0.705496\pi$
$997$	−37.9955	−1.20333	−0.601665	+	0.798749i	$0.705496\pi$
$998$	20.9314	0.662572
$999$	−2.91929	−0.0923623

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3381.2.a.bf.1.8		8
7.2	even	3		483.2.i.g.277.1	✓	16
7.4	even	3		483.2.i.g.415.1	yes	16
7.6	odd	2		3381.2.a.be.1.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.i.g.277.1	✓	16	7.2	even	3
483.2.i.g.415.1	yes	16	7.4	even	3
3381.2.a.be.1.8		8	7.6	odd	2
3381.2.a.bf.1.8		8	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)

\(f(q)\)	\(=\)	\(q+2.74585 q^{2} +1.00000 q^{3} +5.53968 q^{4} -1.91481 q^{5} +2.74585 q^{6} +9.71944 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.74585 q^{2} +1.00000 q^{3} +5.53968 q^{4} -1.91481 q^{5} +2.74585 q^{6} +9.71944 q^{8} +1.00000 q^{9} -5.25777 q^{10} +5.32332 q^{11} +5.53968 q^{12} -4.78352 q^{13} -1.91481 q^{15} +15.6087 q^{16} +2.61781 q^{17} +2.74585 q^{18} +4.46024 q^{19} -10.6074 q^{20} +14.6170 q^{22} -1.00000 q^{23} +9.71944 q^{24} -1.33352 q^{25} -13.1348 q^{26} +1.00000 q^{27} -8.86099 q^{29} -5.25777 q^{30} +5.90898 q^{31} +23.4204 q^{32} +5.32332 q^{33} +7.18812 q^{34} +5.53968 q^{36} -2.91929 q^{37} +12.2471 q^{38} -4.78352 q^{39} -18.6108 q^{40} +4.98968 q^{41} -6.45341 q^{43} +29.4895 q^{44} -1.91481 q^{45} -2.74585 q^{46} -6.94360 q^{47} +15.6087 q^{48} -3.66165 q^{50} +2.61781 q^{51} -26.4992 q^{52} +1.01228 q^{53} +2.74585 q^{54} -10.1931 q^{55} +4.46024 q^{57} -24.3309 q^{58} +6.15656 q^{59} -10.6074 q^{60} +2.24372 q^{61} +16.2252 q^{62} +33.0913 q^{64} +9.15951 q^{65} +14.6170 q^{66} +9.38847 q^{67} +14.5019 q^{68} -1.00000 q^{69} -10.2261 q^{71} +9.71944 q^{72} +3.85118 q^{73} -8.01593 q^{74} -1.33352 q^{75} +24.7083 q^{76} -13.1348 q^{78} +2.97781 q^{79} -29.8877 q^{80} +1.00000 q^{81} +13.7009 q^{82} -2.29460 q^{83} -5.01260 q^{85} -17.7201 q^{86} -8.86099 q^{87} +51.7397 q^{88} -10.0816 q^{89} -5.25777 q^{90} -5.53968 q^{92} +5.90898 q^{93} -19.0661 q^{94} -8.54048 q^{95} +23.4204 q^{96} +13.2317 q^{97} +5.32332 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{2} + 8 q^{3} + 15 q^{4} + 5 q^{5} + q^{6} + 9 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q + q^2 + 8 * q^3 + 15 * q^4 + 5 * q^5 + q^6 + 9 * q^8 + 8 * q^9	\( 8 q + q^{2} + 8 q^{3} + 15 q^{4} + 5 q^{5} + q^{6} + 9 q^{8} + 8 q^{9} - 3 q^{10} + 10 q^{11} + 15 q^{12} - 6 q^{13} + 5 q^{15} + 13 q^{16} + 21 q^{17} + q^{18} + 5 q^{19} - q^{20} + 18 q^{22} - 8 q^{23} + 9 q^{24} + 27 q^{25} + 3 q^{26} + 8 q^{27} + 2 q^{29} - 3 q^{30} - 13 q^{31} + 29 q^{32} + 10 q^{33} - 19 q^{34} + 15 q^{36} + 13 q^{37} - 6 q^{38} - 6 q^{39} + 7 q^{40} + 16 q^{41} + 15 q^{43} + 24 q^{44} + 5 q^{45} - q^{46} - q^{47} + 13 q^{48} + 16 q^{50} + 21 q^{51} - 19 q^{52} + 3 q^{53} + q^{54} - 10 q^{55} + 5 q^{57} - 40 q^{58} + 26 q^{59} - q^{60} - 14 q^{61} - 14 q^{62} + 49 q^{64} - 3 q^{65} + 18 q^{66} + 38 q^{67} + 43 q^{68} - 8 q^{69} + 9 q^{71} + 9 q^{72} - 6 q^{73} + 32 q^{74} + 27 q^{75} - 14 q^{76} + 3 q^{78} + 23 q^{79} - 17 q^{80} + 8 q^{81} + 20 q^{82} + 30 q^{83} - 37 q^{85} - 28 q^{86} + 2 q^{87} + 86 q^{88} + 12 q^{89} - 3 q^{90} - 15 q^{92} - 13 q^{93} - 45 q^{94} + 16 q^{95} + 29 q^{96} + 14 q^{97} + 10 q^{99}+O(q^{100}) \) 8 * q + q^2 + 8 * q^3 + 15 * q^4 + 5 * q^5 + q^6 + 9 * q^8 + 8 * q^9 - 3 * q^10 + 10 * q^11 + 15 * q^12 - 6 * q^13 + 5 * q^15 + 13 * q^16 + 21 * q^17 + q^18 + 5 * q^19 - q^20 + 18 * q^22 - 8 * q^23 + 9 * q^24 + 27 * q^25 + 3 * q^26 + 8 * q^27 + 2 * q^29 - 3 * q^30 - 13 * q^31 + 29 * q^32 + 10 * q^33 - 19 * q^34 + 15 * q^36 + 13 * q^37 - 6 * q^38 - 6 * q^39 + 7 * q^40 + 16 * q^41 + 15 * q^43 + 24 * q^44 + 5 * q^45 - q^46 - q^47 + 13 * q^48 + 16 * q^50 + 21 * q^51 - 19 * q^52 + 3 * q^53 + q^54 - 10 * q^55 + 5 * q^57 - 40 * q^58 + 26 * q^59 - q^60 - 14 * q^61 - 14 * q^62 + 49 * q^64 - 3 * q^65 + 18 * q^66 + 38 * q^67 + 43 * q^68 - 8 * q^69 + 9 * q^71 + 9 * q^72 - 6 * q^73 + 32 * q^74 + 27 * q^75 - 14 * q^76 + 3 * q^78 + 23 * q^79 - 17 * q^80 + 8 * q^81 + 20 * q^82 + 30 * q^83 - 37 * q^85 - 28 * q^86 + 2 * q^87 + 86 * q^88 + 12 * q^89 - 3 * q^90 - 15 * q^92 - 13 * q^93 - 45 * q^94 + 16 * q^95 + 29 * q^96 + 14 * q^97 + 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more