LMFDB - Embedded newform 1083.2.a.p.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1083,2,Mod(1,1083)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1083.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1083, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1083 = 3 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1083.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$8.64779853890$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.6357609.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 9x^{4} - x^{3} + 18x^{2} - 8 $ x^6 - 9x^4 - x^3 + 18x^2 - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 57)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.842316$ of defining polynomial
Character	$\chi$	$=$	1083.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.842316 q^{2} -1.00000 q^{3} -1.29050 q^{4} +1.70747 q^{5} -0.842316 q^{6} +3.93033 q^{7} -2.77164 q^{8} +1.00000 q^{9} +1.43823 q^{10} +4.42535 q^{11} +1.29050 q^{12} -1.82462 q^{13} +3.31058 q^{14} -1.70747 q^{15} +0.246406 q^{16} -5.02644 q^{17} +0.842316 q^{18} -2.20349 q^{20} -3.93033 q^{21} +3.72755 q^{22} +3.45831 q^{23} +2.77164 q^{24} -2.08456 q^{25} -1.53691 q^{26} -1.00000 q^{27} -5.07211 q^{28} +1.45022 q^{29} -1.43823 q^{30} +2.86943 q^{31} +5.75084 q^{32} -4.42535 q^{33} -4.23385 q^{34} +6.71091 q^{35} -1.29050 q^{36} +4.83123 q^{37} +1.82462 q^{39} -4.73249 q^{40} +8.80804 q^{41} -3.31058 q^{42} +10.1017 q^{43} -5.71093 q^{44} +1.70747 q^{45} +2.91299 q^{46} +3.43531 q^{47} -0.246406 q^{48} +8.44752 q^{49} -1.75586 q^{50} +5.02644 q^{51} +2.35468 q^{52} +2.11234 q^{53} -0.842316 q^{54} +7.55614 q^{55} -10.8935 q^{56} +1.22154 q^{58} -6.40598 q^{59} +2.20349 q^{60} +0.0766957 q^{61} +2.41696 q^{62} +3.93033 q^{63} +4.35121 q^{64} -3.11548 q^{65} -3.72755 q^{66} +1.04715 q^{67} +6.48664 q^{68} -3.45831 q^{69} +5.65271 q^{70} +13.5488 q^{71} -2.77164 q^{72} +4.01120 q^{73} +4.06942 q^{74} +2.08456 q^{75} +17.3931 q^{77} +1.53691 q^{78} -16.9649 q^{79} +0.420729 q^{80} +1.00000 q^{81} +7.41916 q^{82} -13.0252 q^{83} +5.07211 q^{84} -8.58248 q^{85} +8.50883 q^{86} -1.45022 q^{87} -12.2655 q^{88} -12.3638 q^{89} +1.43823 q^{90} -7.17137 q^{91} -4.46296 q^{92} -2.86943 q^{93} +2.89362 q^{94} -5.75084 q^{96} -0.175254 q^{97} +7.11548 q^{98} +4.42535 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} + 6 q^{4} + 9 q^{5} + 9 q^{7} + 3 q^{8} + 6 q^{9} - 6 q^{10} + 9 q^{11} - 6 q^{12} - 3 q^{13} + 6 q^{14} - 9 q^{15} + 6 q^{16} + 15 q^{17} - 9 q^{20} - 9 q^{21} + 21 q^{22} + 6 q^{23} - 3 q^{24}+ \cdots + 9 q^{99}+O(q^{100}) $ 6 * q - 6 * q^3 + 6 * q^4 + 9 * q^5 + 9 * q^7 + 3 * q^8 + 6 * q^9 - 6 * q^10 + 9 * q^11 - 6 * q^12 - 3 * q^13 + 6 * q^14 - 9 * q^15 + 6 * q^16 + 15 * q^17 - 9 * q^20 - 9 * q^21 + 21 * q^22 + 6 * q^23 - 3 * q^24 + 15 * q^25 - 6 * q^27 - 3 * q^28 + 15 * q^29 + 6 * q^30 + 21 * q^32 - 9 * q^33 - 33 * q^34 - 3 * q^35 + 6 * q^36 + 6 * q^37 + 3 * q^39 - 39 * q^40 - 6 * q^41 - 6 * q^42 + 15 * q^43 - 6 * q^44 + 9 * q^45 + 27 * q^46 + 9 * q^47 - 6 * q^48 + 9 * q^49 + 9 * q^50 - 15 * q^51 - 18 * q^52 + 6 * q^53 - 6 * q^55 + 39 * q^56 - 12 * q^58 + 15 * q^59 + 9 * q^60 + 3 * q^61 + 27 * q^62 + 9 * q^63 + 21 * q^64 + 21 * q^65 - 21 * q^66 + 18 * q^67 + 21 * q^68 - 6 * q^69 - 9 * q^70 + 15 * q^71 + 3 * q^72 - 6 * q^73 + 9 * q^74 - 15 * q^75 + 48 * q^77 - 21 * q^79 - 21 * q^80 + 6 * q^81 - 15 * q^82 - 3 * q^83 + 3 * q^84 + 42 * q^85 + 30 * q^86 - 15 * q^87 + 42 * q^88 + 24 * q^89 - 6 * q^90 - 12 * q^91 + 42 * q^92 - 9 * q^94 - 21 * q^96 + 3 * q^97 + 3 * q^98 + 9 * q^99

Coefficient data

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

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

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

$2$	0.842316	0.595608	0.297804	−	0.954627i	$-0.403746\pi$
$2$	0.842316	0.595608	0.297804	+	0.954627i	$0.403746\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.29050	−0.645252
$5$	1.70747	0.763602	0.381801	−	0.924244i	$-0.375304\pi$
$5$	1.70747	0.763602	0.381801	+	0.924244i	$0.375304\pi$
$6$	−0.842316	−0.343874
$7$	3.93033	1.48553	0.742763	−	0.669554i	$-0.233515\pi$
$7$	3.93033	1.48553	0.742763	+	0.669554i	$0.233515\pi$
$8$	−2.77164	−0.979924
$9$	1.00000	0.333333
$10$	1.43823	0.454807
$11$	4.42535	1.33429	0.667147	−	0.744926i	$-0.267515\pi$
$11$	4.42535	1.33429	0.667147	+	0.744926i	$0.267515\pi$
$12$	1.29050	0.372536
$13$	−1.82462	−0.506059	−0.253030	−	0.967459i	$-0.581427\pi$
$13$	−1.82462	−0.506059	−0.253030	+	0.967459i	$0.581427\pi$
$14$	3.31058	0.884791
$15$	−1.70747	−0.440866
$16$	0.246406	0.0616014
$17$	−5.02644	−1.21909	−0.609545	−	0.792751i	$-0.708648\pi$
$17$	−5.02644	−1.21909	−0.609545	+	0.792751i	$0.708648\pi$
$18$	0.842316	0.198536
$19$	0	0
$19$	0	0
$20$	−2.20349	−0.492716
$21$	−3.93033	−0.857669
$22$	3.72755	0.794716
$23$	3.45831	0.721107	0.360553	−	0.932739i	$-0.382588\pi$
$23$	3.45831	0.721107	0.360553	+	0.932739i	$0.382588\pi$
$24$	2.77164	0.565760
$25$	−2.08456	−0.416912
$26$	−1.53691	−0.301413
$27$	−1.00000	−0.192450
$28$	−5.07211	−0.958538
$29$	1.45022	0.269299	0.134649	−	0.990893i	$-0.457009\pi$
$29$	1.45022	0.269299	0.134649	+	0.990893i	$0.457009\pi$
$30$	−1.43823	−0.262583
$31$	2.86943	0.515364	0.257682	−	0.966230i	$-0.417041\pi$
$31$	2.86943	0.515364	0.257682	+	0.966230i	$0.417041\pi$
$32$	5.75084	1.01661
$33$	−4.42535	−0.770355
$34$	−4.23385	−0.726100
$35$	6.71091	1.13435
$36$	−1.29050	−0.215084
$37$	4.83123	0.794249	0.397124	−	0.917765i	$-0.370008\pi$
$37$	4.83123	0.794249	0.397124	+	0.917765i	$0.370008\pi$
$38$	0	0
$39$	1.82462	0.292173
$40$	−4.73249	−0.748272
$41$	8.80804	1.37559	0.687793	−	0.725907i	$-0.258580\pi$
$41$	8.80804	1.37559	0.687793	+	0.725907i	$0.258580\pi$
$42$	−3.31058	−0.510834
$43$	10.1017	1.54050	0.770248	−	0.637745i	$-0.220132\pi$
$43$	10.1017	1.54050	0.770248	+	0.637745i	$0.220132\pi$
$44$	−5.71093	−0.860956
$45$	1.70747	0.254534
$46$	2.91299	0.429497
$47$	3.43531	0.501092	0.250546	−	0.968105i	$-0.419390\pi$
$47$	3.43531	0.501092	0.250546	+	0.968105i	$0.419390\pi$
$48$	−0.246406	−0.0355656
$49$	8.44752	1.20679
$50$	−1.75586	−0.248316
$51$	5.02644	0.703842
$52$	2.35468	0.326536
$53$	2.11234	0.290152	0.145076	−	0.989421i	$-0.453657\pi$
$53$	2.11234	0.290152	0.145076	+	0.989421i	$0.453657\pi$
$54$	−0.842316	−0.114625
$55$	7.55614	1.01887
$56$	−10.8935	−1.45570
$57$	0	0
$58$	1.22154	0.160396
$59$	−6.40598	−0.833988	−0.416994	−	0.908909i	$-0.636916\pi$
$59$	−6.40598	−0.833988	−0.416994	+	0.908909i	$0.636916\pi$
$60$	2.20349	0.284470
$61$	0.0766957	0.00981988	0.00490994	−	0.999988i	$-0.498437\pi$
$61$	0.0766957	0.00981988	0.00490994	+	0.999988i	$0.498437\pi$
$62$	2.41696	0.306955
$63$	3.93033	0.495175
$64$	4.35121	0.543902
$65$	−3.11548	−0.386428
$66$	−3.72755	−0.458829
$67$	1.04715	0.127930	0.0639650	−	0.997952i	$-0.479625\pi$
$67$	1.04715	0.127930	0.0639650	+	0.997952i	$0.479625\pi$
$68$	6.48664	0.786620
$69$	−3.45831	−0.416331
$70$	5.65271	0.675628
$71$	13.5488	1.60794	0.803971	−	0.594668i	$-0.202717\pi$
$71$	13.5488	1.60794	0.803971	+	0.594668i	$0.202717\pi$
$72$	−2.77164	−0.326641
$73$	4.01120	0.469475	0.234738	−	0.972059i	$-0.424577\pi$
$73$	4.01120	0.469475	0.234738	+	0.972059i	$0.424577\pi$
$74$	4.06942	0.473061
$75$	2.08456	0.240704
$76$	0	0
$77$	17.3931	1.98213
$78$	1.53691	0.174021
$79$	−16.9649	−1.90870	−0.954348	−	0.298697i	$-0.903448\pi$
$79$	−16.9649	−1.90870	−0.954348	+	0.298697i	$0.903448\pi$
$80$	0.420729	0.0470390
$81$	1.00000	0.111111
$82$	7.41916	0.819309
$83$	−13.0252	−1.42970	−0.714850	−	0.699278i	$-0.753505\pi$
$83$	−13.0252	−1.42970	−0.714850	+	0.699278i	$0.753505\pi$
$84$	5.07211	0.553412
$85$	−8.58248	−0.930900
$86$	8.50883	0.917531
$87$	−1.45022	−0.155480
$88$	−12.2655	−1.30751
$89$	−12.3638	−1.31056	−0.655279	−	0.755387i	$-0.727449\pi$
$89$	−12.3638	−1.31056	−0.655279	+	0.755387i	$0.727449\pi$
$90$	1.43823	0.151602
$91$	−7.17137	−0.751764
$92$	−4.46296	−0.465295
$93$	−2.86943	−0.297546
$94$	2.89362	0.298454
$95$	0	0
$96$	−5.75084	−0.586943
$97$	−0.175254	−0.0177944	−0.00889718	−	0.999960i	$-0.502832\pi$
$97$	−0.175254	−0.0177944	−0.00889718	+	0.999960i	$0.502832\pi$
$98$	7.11548	0.718772
$99$	4.42535	0.444765
$100$	2.69013	0.269013
$101$	−11.2135	−1.11578	−0.557891	−	0.829914i	$-0.688389\pi$
$101$	−11.2135	−1.11578	−0.557891	+	0.829914i	$0.688389\pi$
$102$	4.23385	0.419214
$103$	−9.52010	−0.938043	−0.469022	−	0.883187i	$-0.655393\pi$
$103$	−9.52010	−0.938043	−0.469022	+	0.883187i	$0.655393\pi$
$104$	5.05720	0.495900
$105$	−6.71091	−0.654918
$106$	1.77925	0.172816
$107$	10.2690	0.992742	0.496371	−	0.868111i	$-0.334666\pi$
$107$	10.2690	0.992742	0.496371	+	0.868111i	$0.334666\pi$
$108$	1.29050	0.124179
$109$	2.04164	0.195554	0.0977770	−	0.995208i	$-0.468827\pi$
$109$	2.04164	0.195554	0.0977770	+	0.995208i	$0.468827\pi$
$110$	6.36466	0.606847
$111$	−4.83123	−0.458560
$112$	0.968456	0.0915105
$113$	13.9973	1.31675	0.658375	−	0.752690i	$-0.271244\pi$
$113$	13.9973	1.31675	0.658375	+	0.752690i	$0.271244\pi$
$114$	0	0
$115$	5.90494	0.550639
$116$	−1.87151	−0.173765
$117$	−1.82462	−0.168686
$118$	−5.39587	−0.496730
$119$	−19.7556	−1.81099
$120$	4.73249	0.432015
$121$	8.58375	0.780341
$122$	0.0646020	0.00584879
$123$	−8.80804	−0.794195
$124$	−3.70300	−0.332539
$125$	−12.0966	−1.08196
$126$	3.31058	0.294930
$127$	−1.95928	−0.173858	−0.0869288	−	0.996215i	$-0.527705\pi$
$127$	−1.95928	−0.173858	−0.0869288	+	0.996215i	$0.527705\pi$
$128$	−7.83658	−0.692662
$129$	−10.1017	−0.889406
$130$	−2.62422	−0.230159
$131$	−1.56004	−0.136302	−0.0681508	−	0.997675i	$-0.521710\pi$
$131$	−1.56004	−0.136302	−0.0681508	+	0.997675i	$0.521710\pi$
$132$	5.71093	0.497073
$133$	0	0
$134$	0.882034	0.0761961
$135$	−1.70747	−0.146955
$136$	13.9315	1.19462
$137$	15.3618	1.31244	0.656222	−	0.754568i	$-0.272154\pi$
$137$	15.3618	1.31244	0.656222	+	0.754568i	$0.272154\pi$
$138$	−2.91299	−0.247970
$139$	9.30803	0.789496	0.394748	−	0.918789i	$-0.370832\pi$
$139$	9.30803	0.789496	0.394748	+	0.918789i	$0.370832\pi$
$140$	−8.66046	−0.731942
$141$	−3.43531	−0.289306
$142$	11.4123	0.957703
$143$	−8.07460	−0.675232
$144$	0.246406	0.0205338
$145$	2.47620	0.205637
$146$	3.37870	0.279623
$147$	−8.44752	−0.696740
$148$	−6.23472	−0.512490
$149$	5.18414	0.424702	0.212351	−	0.977194i	$-0.431888\pi$
$149$	5.18414	0.424702	0.212351	+	0.977194i	$0.431888\pi$
$150$	1.75586	0.143365
$151$	−13.0922	−1.06543	−0.532713	−	0.846296i	$-0.678827\pi$
$151$	−13.0922	−1.06543	−0.532713	+	0.846296i	$0.678827\pi$
$152$	0	0
$153$	−5.02644	−0.406364
$154$	14.6505	1.18057
$155$	4.89945	0.393533
$156$	−2.35468	−0.188525
$157$	−7.36968	−0.588164	−0.294082	−	0.955780i	$-0.595014\pi$
$157$	−7.36968	−0.588164	−0.294082	+	0.955780i	$0.595014\pi$
$158$	−14.2898	−1.13683
$159$	−2.11234	−0.167519
$160$	9.81937	0.776289
$161$	13.5923	1.07122
$162$	0.842316	0.0661786
$163$	−16.0281	−1.25542	−0.627708	−	0.778449i	$-0.716007\pi$
$163$	−16.0281	−1.25542	−0.627708	+	0.778449i	$0.716007\pi$
$164$	−11.3668	−0.887599
$165$	−7.55614	−0.588245
$166$	−10.9713	−0.851540
$167$	13.1273	1.01582	0.507911	−	0.861409i	$-0.330418\pi$
$167$	13.1273	1.01582	0.507911	+	0.861409i	$0.330418\pi$
$168$	10.8935	0.840451
$169$	−9.67075	−0.743904
$170$	−7.22916	−0.554451
$171$	0	0
$172$	−13.0363	−0.994007
$173$	18.4720	1.40440	0.702201	−	0.711978i	$-0.252201\pi$
$173$	18.4720	1.40440	0.702201	+	0.711978i	$0.252201\pi$
$174$	−1.22154	−0.0926048
$175$	−8.19301	−0.619333
$176$	1.09043	0.0821944
$177$	6.40598	0.481503
$178$	−10.4142	−0.780578
$179$	−11.8588	−0.886370	−0.443185	−	0.896430i	$-0.646151\pi$
$179$	−11.8588	−0.886370	−0.443185	+	0.896430i	$0.646151\pi$
$180$	−2.20349	−0.164239
$181$	−0.273724	−0.0203457	−0.0101729	−	0.999948i	$-0.503238\pi$
$181$	−0.273724	−0.0203457	−0.0101729	+	0.999948i	$0.503238\pi$
$182$	−6.04056	−0.447756
$183$	−0.0766957	−0.00566951
$184$	−9.58520	−0.706630
$185$	8.24916	0.606490
$186$	−2.41696	−0.177220
$187$	−22.2438	−1.62663
$188$	−4.43328	−0.323330
$189$	−3.93033	−0.285890
$190$	0	0
$191$	−12.1649	−0.880218	−0.440109	−	0.897944i	$-0.645060\pi$
$191$	−12.1649	−0.880218	−0.440109	+	0.897944i	$0.645060\pi$
$192$	−4.35121	−0.314022
$193$	−23.0762	−1.66106	−0.830532	−	0.556970i	$-0.811964\pi$
$193$	−23.0762	−1.66106	−0.830532	+	0.556970i	$0.811964\pi$
$194$	−0.147619	−0.0105985
$195$	3.11548	0.223104
$196$	−10.9016	−0.778682
$197$	6.46873	0.460878	0.230439	−	0.973087i	$-0.425984\pi$
$197$	6.46873	0.460878	0.230439	+	0.973087i	$0.425984\pi$
$198$	3.72755	0.264905
$199$	0.307088	0.0217689	0.0108845	−	0.999941i	$-0.496535\pi$
$199$	0.307088	0.0217689	0.0108845	+	0.999941i	$0.496535\pi$
$200$	5.77765	0.408542
$201$	−1.04715	−0.0738605
$202$	−9.44529	−0.664569
$203$	5.69984	0.400050
$204$	−6.48664	−0.454155
$205$	15.0394	1.05040
$206$	−8.01893	−0.558706
$207$	3.45831	0.240369
$208$	−0.449597	−0.0311740
$209$	0	0
$210$	−5.65271	−0.390074
$211$	−10.5856	−0.728741	−0.364371	−	0.931254i	$-0.618716\pi$
$211$	−10.5856	−0.728741	−0.364371	+	0.931254i	$0.618716\pi$
$212$	−2.72598	−0.187221
$213$	−13.5488	−0.928346
$214$	8.64975	0.591284
$215$	17.2483	1.17633
$216$	2.77164	0.188587
$217$	11.2778	0.765587
$218$	1.71971	0.116473
$219$	−4.01120	−0.271052
$220$	−9.75123	−0.657428
$221$	9.17135	0.616932
$222$	−4.06942	−0.273122
$223$	−7.43507	−0.497889	−0.248944	−	0.968518i	$-0.580084\pi$
$223$	−7.43507	−0.497889	−0.248944	+	0.968518i	$0.580084\pi$
$224$	22.6027	1.51021
$225$	−2.08456	−0.138971
$226$	11.7901	0.784267
$227$	−3.54389	−0.235216	−0.117608	−	0.993060i	$-0.537523\pi$
$227$	−3.54389	−0.235216	−0.117608	+	0.993060i	$0.537523\pi$
$228$	0	0
$229$	−11.3492	−0.749973	−0.374987	−	0.927030i	$-0.622353\pi$
$229$	−11.3492	−0.749973	−0.374987	+	0.927030i	$0.622353\pi$
$230$	4.97383	0.327965
$231$	−17.3931	−1.14438
$232$	−4.01949	−0.263892
$233$	−5.92767	−0.388335	−0.194167	−	0.980968i	$-0.562200\pi$
$233$	−5.92767	−0.388335	−0.194167	+	0.980968i	$0.562200\pi$
$234$	−1.53691	−0.100471
$235$	5.86568	0.382635
$236$	8.26695	0.538132
$237$	16.9649	1.10199
$238$	−16.6404	−1.07864
$239$	15.1684	0.981165	0.490583	−	0.871395i	$-0.336784\pi$
$239$	15.1684	0.981165	0.490583	+	0.871395i	$0.336784\pi$
$240$	−0.420729	−0.0271580
$241$	−8.86254	−0.570887	−0.285443	−	0.958396i	$-0.592141\pi$
$241$	−8.86254	−0.570887	−0.285443	+	0.958396i	$0.592141\pi$
$242$	7.23023	0.464777
$243$	−1.00000	−0.0641500
$244$	−0.0989760	−0.00633629
$245$	14.4239	0.921506
$246$	−7.41916	−0.473028
$247$	0	0
$248$	−7.95303	−0.505018
$249$	13.0252	0.825438
$250$	−10.1892	−0.644422
$251$	11.7071	0.738948	0.369474	−	0.929241i	$-0.379538\pi$
$251$	11.7071	0.738948	0.369474	+	0.929241i	$0.379538\pi$
$252$	−5.07211	−0.319513
$253$	15.3042	0.962169
$254$	−1.65033	−0.103551
$255$	8.58248	0.537456
$256$	−15.3033	−0.956457
$257$	9.91597	0.618541	0.309271	−	0.950974i	$-0.399915\pi$
$257$	9.91597	0.618541	0.309271	+	0.950974i	$0.399915\pi$
$258$	−8.50883	−0.529737
$259$	18.9883	1.17988
$260$	4.02054	0.249343
$261$	1.45022	0.0897662
$262$	−1.31405	−0.0811823
$263$	−6.73892	−0.415540	−0.207770	−	0.978178i	$-0.566621\pi$
$263$	−6.73892	−0.415540	−0.207770	+	0.978178i	$0.566621\pi$
$264$	12.2655	0.754890
$265$	3.60674	0.221560
$266$	0	0
$267$	12.3638	0.756651
$268$	−1.35135	−0.0825471
$269$	−0.718290	−0.0437949	−0.0218975	−	0.999760i	$-0.506971\pi$
$269$	−0.718290	−0.0437949	−0.0218975	+	0.999760i	$0.506971\pi$
$270$	−1.43823	−0.0875277
$271$	−28.8411	−1.75197	−0.875985	−	0.482339i	$-0.839787\pi$
$271$	−28.8411	−1.75197	−0.875985	+	0.482339i	$0.839787\pi$
$272$	−1.23854	−0.0750977
$273$	7.17137	0.434031
$274$	12.9395	0.781701
$275$	−9.22490	−0.556283
$276$	4.46296	0.268638
$277$	−3.83134	−0.230203	−0.115101	−	0.993354i	$-0.536719\pi$
$277$	−3.83134	−0.230203	−0.115101	+	0.993354i	$0.536719\pi$
$278$	7.84030	0.470230
$279$	2.86943	0.171788
$280$	−18.6003	−1.11158
$281$	22.0921	1.31791	0.658953	−	0.752184i	$-0.271000\pi$
$281$	22.0921	1.31791	0.658953	+	0.752184i	$0.271000\pi$
$282$	−2.89362	−0.172313
$283$	−25.1465	−1.49481	−0.747403	−	0.664371i	$-0.768699\pi$
$283$	−25.1465	−1.49481	−0.747403	+	0.664371i	$0.768699\pi$
$284$	−17.4847	−1.03753
$285$	0	0
$286$	−6.80136	−0.402173
$287$	34.6185	2.04347
$288$	5.75084	0.338872
$289$	8.26510	0.486182
$290$	2.08574	0.122479
$291$	0.175254	0.0102736
$292$	−5.17647	−0.302930
$293$	5.92712	0.346266	0.173133	−	0.984898i	$-0.444611\pi$
$293$	5.92712	0.346266	0.173133	+	0.984898i	$0.444611\pi$
$294$	−7.11548	−0.414983
$295$	−10.9380	−0.636835
$296$	−13.3904	−0.778304
$297$	−4.42535	−0.256785
$298$	4.36669	0.252955
$299$	−6.31010	−0.364923
$300$	−2.69013	−0.155315
$301$	39.7031	2.28845
$302$	−11.0277	−0.634575
$303$	11.2135	0.644197
$304$	0	0
$305$	0.130955	0.00749848
$306$	−4.23385	−0.242033
$307$	17.5842	1.00359	0.501793	−	0.864988i	$-0.332674\pi$
$307$	17.5842	1.00359	0.501793	+	0.864988i	$0.332674\pi$
$308$	−22.4459	−1.27897
$309$	9.52010	0.541579
$310$	4.12688	0.234391
$311$	−6.02866	−0.341854	−0.170927	−	0.985284i	$-0.554676\pi$
$311$	−6.02866	−0.341854	−0.170927	+	0.985284i	$0.554676\pi$
$312$	−5.05720	−0.286308
$313$	16.6980	0.943829	0.471914	−	0.881644i	$-0.343563\pi$
$313$	16.6980	0.943829	0.471914	+	0.881644i	$0.343563\pi$
$314$	−6.20760	−0.350315
$315$	6.71091	0.378117
$316$	21.8932	1.23159
$317$	2.19305	0.123174	0.0615870	−	0.998102i	$-0.480384\pi$
$317$	2.19305	0.123174	0.0615870	+	0.998102i	$0.480384\pi$
$318$	−1.77925	−0.0997756
$319$	6.41772	0.359323
$320$	7.42955	0.415325
$321$	−10.2690	−0.573160
$322$	11.4490	0.638029
$323$	0	0
$324$	−1.29050	−0.0716946
$325$	3.80353	0.210982
$326$	−13.5007	−0.747735
$327$	−2.04164	−0.112903
$328$	−24.4128	−1.34797
$329$	13.5019	0.744385
$330$	−6.36466	−0.350363
$331$	−15.5408	−0.854199	−0.427099	−	0.904205i	$-0.640465\pi$
$331$	−15.5408	−0.854199	−0.427099	+	0.904205i	$0.640465\pi$
$332$	16.8091	0.922516
$333$	4.83123	0.264750
$334$	11.0574	0.605032
$335$	1.78798	0.0976877
$336$	−0.968456	−0.0528336
$337$	−12.3767	−0.674200	−0.337100	−	0.941469i	$-0.609446\pi$
$337$	−12.3767	−0.674200	−0.337100	+	0.941469i	$0.609446\pi$
$338$	−8.14583	−0.443075
$339$	−13.9973	−0.760226
$340$	11.0757	0.600665
$341$	12.6982	0.687647
$342$	0	0
$343$	5.68923	0.307189
$344$	−27.9983	−1.50957
$345$	−5.90494	−0.317911
$346$	15.5593	0.836473
$347$	−22.2056	−1.19206	−0.596030	−	0.802962i	$-0.703256\pi$
$347$	−22.2056	−1.19206	−0.596030	+	0.802962i	$0.703256\pi$
$348$	1.87151	0.100323
$349$	−19.9292	−1.06679	−0.533393	−	0.845867i	$-0.679083\pi$
$349$	−19.9292	−1.06679	−0.533393	+	0.845867i	$0.679083\pi$
$350$	−6.90110	−0.368879
$351$	1.82462	0.0973911
$352$	25.4495	1.35646
$353$	−5.20604	−0.277089	−0.138545	−	0.990356i	$-0.544242\pi$
$353$	−5.20604	−0.277089	−0.138545	+	0.990356i	$0.544242\pi$
$354$	5.39587	0.286787
$355$	23.1341	1.22783
$356$	15.9555	0.845640
$357$	19.7556	1.04558
$358$	−9.98887	−0.527928
$359$	−5.16826	−0.272771	−0.136385	−	0.990656i	$-0.543549\pi$
$359$	−5.16826	−0.272771	−0.136385	+	0.990656i	$0.543549\pi$
$360$	−4.73249	−0.249424
$361$	0	0
$362$	−0.230562	−0.0121181
$363$	−8.58375	−0.450530
$364$	9.25468	0.485077
$365$	6.84899	0.358493
$366$	−0.0646020	−0.00337680
$367$	−6.06224	−0.316447	−0.158223	−	0.987403i	$-0.550577\pi$
$367$	−6.06224	−0.316447	−0.158223	+	0.987403i	$0.550577\pi$
$368$	0.852146	0.0444212
$369$	8.80804	0.458528
$370$	6.94840	0.361230
$371$	8.30218	0.431028
$372$	3.70300	0.191992
$373$	−9.98545	−0.517027	−0.258514	−	0.966008i	$-0.583233\pi$
$373$	−9.98545	−0.517027	−0.258514	+	0.966008i	$0.583233\pi$
$374$	−18.7363	−0.968830
$375$	12.0966	0.624668
$376$	−9.52147	−0.491032
$377$	−2.64610	−0.136281
$378$	−3.31058	−0.170278
$379$	13.7799	0.707826	0.353913	−	0.935278i	$-0.384851\pi$
$379$	13.7799	0.707826	0.353913	+	0.935278i	$0.384851\pi$
$380$	0	0
$381$	1.95928	0.100377
$382$	−10.2467	−0.524265
$383$	−16.3124	−0.833524	−0.416762	−	0.909016i	$-0.636835\pi$
$383$	−16.3124	−0.833524	−0.416762	+	0.909016i	$0.636835\pi$
$384$	7.83658	0.399909
$385$	29.6982	1.51356
$386$	−19.4375	−0.989343
$387$	10.1017	0.513499
$388$	0.226166	0.0114818
$389$	−11.4713	−0.581617	−0.290808	−	0.956781i	$-0.593924\pi$
$389$	−11.4713	−0.581617	−0.290808	+	0.956781i	$0.593924\pi$
$390$	2.62422	0.132883
$391$	−17.3830	−0.879095
$392$	−23.4135	−1.18256
$393$	1.56004	0.0786938
$394$	5.44871	0.274502
$395$	−28.9669	−1.45748
$396$	−5.71093	−0.286985
$397$	−0.448109	−0.0224899	−0.0112450	−	0.999937i	$-0.503579\pi$
$397$	−0.448109	−0.0224899	−0.0112450	+	0.999937i	$0.503579\pi$
$398$	0.258665	0.0129657
$399$	0	0
$400$	−0.513647	−0.0256823
$401$	−14.5064	−0.724416	−0.362208	−	0.932097i	$-0.617977\pi$
$401$	−14.5064	−0.724416	−0.362208	+	0.932097i	$0.617977\pi$
$402$	−0.882034	−0.0439919
$403$	−5.23562	−0.260805
$404$	14.4710	0.719961
$405$	1.70747	0.0848447
$406$	4.80106	0.238273
$407$	21.3799	1.05976
$408$	−13.9315	−0.689712
$409$	13.1990	0.652647	0.326323	−	0.945258i	$-0.394190\pi$
$409$	13.1990	0.652647	0.326323	+	0.945258i	$0.394190\pi$
$410$	12.6680	0.625626
$411$	−15.3618	−0.757739
$412$	12.2857	0.605274
$413$	−25.1777	−1.23891
$414$	2.91299	0.143166
$415$	−22.2401	−1.09172
$416$	−10.4931	−0.514467
$417$	−9.30803	−0.455816
$418$	0	0
$419$	11.4229	0.558047	0.279024	−	0.960284i	$-0.409989\pi$
$419$	11.4229	0.558047	0.279024	+	0.960284i	$0.409989\pi$
$420$	8.66046	0.422587
$421$	16.5837	0.808240	0.404120	−	0.914706i	$-0.367578\pi$
$421$	16.5837	0.808240	0.404120	+	0.914706i	$0.367578\pi$
$422$	−8.91641	−0.434044
$423$	3.43531	0.167031
$424$	−5.85464	−0.284327
$425$	10.4779	0.508253
$426$	−11.4123	−0.552930
$427$	0.301440	0.0145877
$428$	−13.2522	−0.640568
$429$	8.07460	0.389845
$430$	14.5285	0.700629
$431$	−28.4987	−1.37274	−0.686368	−	0.727255i	$-0.740796\pi$
$431$	−28.4987	−1.37274	−0.686368	+	0.727255i	$0.740796\pi$
$432$	−0.246406	−0.0118552
$433$	−13.9975	−0.672677	−0.336339	−	0.941741i	$-0.609189\pi$
$433$	−13.9975	−0.672677	−0.336339	+	0.941741i	$0.609189\pi$
$434$	9.49947	0.455989
$435$	−2.47620	−0.118725
$436$	−2.63475	−0.126182
$437$	0	0
$438$	−3.37870	−0.161440
$439$	26.8291	1.28048	0.640242	−	0.768174i	$-0.278834\pi$
$439$	26.8291	1.28048	0.640242	+	0.768174i	$0.278834\pi$
$440$	−20.9429	−0.998416
$441$	8.44752	0.402263
$442$	7.72518	0.367449
$443$	−13.9183	−0.661277	−0.330638	−	0.943758i	$-0.607264\pi$
$443$	−13.9183	−0.661277	−0.330638	+	0.943758i	$0.607264\pi$
$444$	6.23472	0.295886
$445$	−21.1107	−1.00075
$446$	−6.26268	−0.296546
$447$	−5.18414	−0.245202
$448$	17.1017	0.807981
$449$	−21.2841	−1.00446	−0.502229	−	0.864735i	$-0.667487\pi$
$449$	−21.2841	−1.00446	−0.502229	+	0.864735i	$0.667487\pi$
$450$	−1.75586	−0.0827719
$451$	38.9787	1.83544
$452$	−18.0635	−0.849636
$453$	13.0922	0.615124
$454$	−2.98507	−0.140097
$455$	−12.2449	−0.574049
$456$	0	0
$457$	−0.745738	−0.0348842	−0.0174421	−	0.999848i	$-0.505552\pi$
$457$	−0.745738	−0.0348842	−0.0174421	+	0.999848i	$0.505552\pi$
$458$	−9.55958	−0.446690
$459$	5.02644	0.234614
$460$	−7.62035	−0.355301
$461$	28.1942	1.31314	0.656568	−	0.754267i	$-0.272007\pi$
$461$	28.1942	1.31314	0.656568	+	0.754267i	$0.272007\pi$
$462$	−14.6505	−0.681603
$463$	8.85705	0.411622	0.205811	−	0.978592i	$-0.434017\pi$
$463$	8.85705	0.411622	0.205811	+	0.978592i	$0.434017\pi$
$464$	0.357342	0.0165892
$465$	−4.89945	−0.227206
$466$	−4.99298	−0.231295
$467$	37.2776	1.72500	0.862500	−	0.506057i	$-0.168898\pi$
$467$	37.2776	1.72500	0.862500	+	0.506057i	$0.168898\pi$
$468$	2.35468	0.108845
$469$	4.11566	0.190044
$470$	4.94076	0.227900
$471$	7.36968	0.339577
$472$	17.7551	0.817245
$473$	44.7036	2.05547
$474$	14.2898	0.656351
$475$	0	0
$476$	25.4946	1.16855
$477$	2.11234	0.0967172
$478$	12.7766	0.584389
$479$	−29.8552	−1.36412	−0.682059	−	0.731297i	$-0.738915\pi$
$479$	−29.8552	−1.36412	−0.682059	+	0.731297i	$0.738915\pi$
$480$	−9.81937	−0.448191
$481$	−8.81517	−0.401937
$482$	−7.46506	−0.340024
$483$	−13.5923	−0.618471
$484$	−11.0774	−0.503516
$485$	−0.299241	−0.0135878
$486$	−0.842316	−0.0382082
$487$	−30.8808	−1.39934	−0.699671	−	0.714465i	$-0.746670\pi$
$487$	−30.8808	−1.39934	−0.699671	+	0.714465i	$0.746670\pi$
$488$	−0.212573	−0.00962273
$489$	16.0281	0.724815
$490$	12.1494	0.548856
$491$	10.1276	0.457051	0.228525	−	0.973538i	$-0.426610\pi$
$491$	10.1276	0.457051	0.228525	+	0.973538i	$0.426610\pi$
$492$	11.3668	0.512455
$493$	−7.28943	−0.328299
$494$	0	0
$495$	7.55614	0.339623
$496$	0.707042	0.0317471
$497$	53.2512	2.38864
$498$	10.9713	0.491637
$499$	8.59460	0.384747	0.192374	−	0.981322i	$-0.438381\pi$
$499$	8.59460	0.384747	0.192374	+	0.981322i	$0.438381\pi$
$500$	15.6108	0.698135
$501$	−13.1273	−0.586486
$502$	9.86111	0.440123
$503$	−18.0671	−0.805570	−0.402785	−	0.915295i	$-0.631958\pi$
$503$	−18.0671	−0.805570	−0.402785	+	0.915295i	$0.631958\pi$
$504$	−10.8935	−0.485234
$505$	−19.1466	−0.852014
$506$	12.8910	0.573075
$507$	9.67075	0.429493
$508$	2.52845	0.112182
$509$	−31.4464	−1.39384	−0.696919	−	0.717150i	$-0.745446\pi$
$509$	−31.4464	−1.39384	−0.696919	+	0.717150i	$0.745446\pi$
$510$	7.22916	0.320113
$511$	15.7654	0.697418
$512$	2.78294	0.122990
$513$	0	0
$514$	8.35239	0.368408
$515$	−16.2553	−0.716292
$516$	13.0363	0.573890
$517$	15.2025	0.668604
$518$	15.9942	0.702744
$519$	−18.4720	−0.810832
$520$	8.63501	0.378670
$521$	−19.9501	−0.874030	−0.437015	−	0.899454i	$-0.643964\pi$
$521$	−19.9501	−0.874030	−0.437015	+	0.899454i	$0.643964\pi$
$522$	1.22154	0.0534654
$523$	−3.16211	−0.138269	−0.0691346	−	0.997607i	$-0.522024\pi$
$523$	−3.16211	−0.138269	−0.0691346	+	0.997607i	$0.522024\pi$
$524$	2.01324	0.0879488
$525$	8.19301	0.357572
$526$	−5.67630	−0.247499
$527$	−14.4230	−0.628275
$528$	−1.09043	−0.0474550
$529$	−11.0401	−0.480005
$530$	3.03802	0.131963
$531$	−6.40598	−0.277996
$532$	0	0
$533$	−16.0714	−0.696128
$534$	10.4142	0.450667
$535$	17.5340	0.758060
$536$	−2.90234	−0.125362
$537$	11.8588	0.511746
$538$	−0.605028	−0.0260846
$539$	37.3832	1.61021
$540$	2.20349	0.0948232
$541$	0.594406	0.0255555	0.0127778	−	0.999918i	$-0.495933\pi$
$541$	0.594406	0.0255555	0.0127778	+	0.999918i	$0.495933\pi$
$542$	−24.2933	−1.04349
$543$	0.273724	0.0117466
$544$	−28.9063	−1.23935
$545$	3.48604	0.149325
$546$	6.04056	0.258512
$547$	−26.5738	−1.13622	−0.568108	−	0.822954i	$-0.692324\pi$
$547$	−26.5738	−1.13622	−0.568108	+	0.822954i	$0.692324\pi$
$548$	−19.8244	−0.846856
$549$	0.0766957	0.00327329
$550$	−7.77029	−0.331326
$551$	0	0
$552$	9.58520	0.407973
$553$	−66.6776	−2.83542
$554$	−3.22720	−0.137111
$555$	−8.24916	−0.350157
$556$	−12.0120	−0.509424
$557$	34.1718	1.44791	0.723953	−	0.689849i	$-0.242323\pi$
$557$	34.1718	1.44791	0.723953	+	0.689849i	$0.242323\pi$
$558$	2.41696	0.102318
$559$	−18.4318	−0.779582
$560$	1.65361	0.0698776
$561$	22.2438	0.939133
$562$	18.6086	0.784955
$563$	30.0716	1.26737	0.633683	−	0.773593i	$-0.281543\pi$
$563$	30.0716	1.26737	0.633683	+	0.773593i	$0.281543\pi$
$564$	4.43328	0.186675
$565$	23.8998	1.00547
$566$	−21.1813	−0.890317
$567$	3.93033	0.165058
$568$	−37.5524	−1.57566
$569$	24.7511	1.03762	0.518809	−	0.854890i	$-0.326376\pi$
$569$	24.7511	1.03762	0.518809	+	0.854890i	$0.326376\pi$
$570$	0	0
$571$	31.5869	1.32187	0.660934	−	0.750444i	$-0.270160\pi$
$571$	31.5869	1.32187	0.660934	+	0.750444i	$0.270160\pi$
$572$	10.4203	0.435694
$573$	12.1649	0.508194
$574$	29.1598	1.21711
$575$	−7.20904	−0.300638
$576$	4.35121	0.181301
$577$	−30.8109	−1.28267	−0.641337	−	0.767260i	$-0.721620\pi$
$577$	−30.8109	−1.28267	−0.641337	+	0.767260i	$0.721620\pi$
$578$	6.96182	0.289574
$579$	23.0762	0.959016
$580$	−3.19554	−0.132688
$581$	−51.1933	−2.12386
$582$	0.147619	0.00611902
$583$	9.34783	0.387148
$584$	−11.1176	−0.460050
$585$	−3.11548	−0.128809
$586$	4.99251	0.206239
$587$	−14.1246	−0.582986	−0.291493	−	0.956573i	$-0.594152\pi$
$587$	−14.1246	−0.582986	−0.291493	+	0.956573i	$0.594152\pi$
$588$	10.9016	0.449572
$589$	0	0
$590$	−9.21326	−0.379304
$591$	−6.46873	−0.266088
$592$	1.19044	0.0489268
$593$	37.7151	1.54877	0.774387	−	0.632712i	$-0.218058\pi$
$593$	37.7151	1.54877	0.774387	+	0.632712i	$0.218058\pi$
$594$	−3.72755	−0.152943
$595$	−33.7320	−1.38288
$596$	−6.69015	−0.274039
$597$	−0.307088	−0.0125683
$598$	−5.31510	−0.217351
$599$	−0.0927993	−0.00379167	−0.00189584	−	0.999998i	$-0.500603\pi$
$599$	−0.0927993	−0.00379167	−0.00189584	+	0.999998i	$0.500603\pi$
$600$	−5.77765	−0.235872
$601$	23.0380	0.939740	0.469870	−	0.882736i	$-0.344301\pi$
$601$	23.0380	0.939740	0.469870	+	0.882736i	$0.344301\pi$
$602$	33.4425	1.36302
$603$	1.04715	0.0426434
$604$	16.8955	0.687468
$605$	14.6565	0.595870
$606$	9.44529	0.383689
$607$	−31.9363	−1.29625	−0.648127	−	0.761532i	$-0.724447\pi$
$607$	−31.9363	−1.29625	−0.648127	+	0.761532i	$0.724447\pi$
$608$	0	0
$609$	−5.69984	−0.230969
$610$	0.110306	0.00446615
$611$	−6.26815	−0.253582
$612$	6.48664	0.262207
$613$	43.7113	1.76548	0.882741	−	0.469859i	$-0.155695\pi$
$613$	43.7113	1.76548	0.882741	+	0.469859i	$0.155695\pi$
$614$	14.8115	0.597743
$615$	−15.0394	−0.606449
$616$	−48.2075	−1.94234
$617$	−13.4123	−0.539957	−0.269978	−	0.962866i	$-0.587017\pi$
$617$	−13.4123	−0.539957	−0.269978	+	0.962866i	$0.587017\pi$
$618$	8.01893	0.322569
$619$	22.9340	0.921797	0.460899	−	0.887453i	$-0.347527\pi$
$619$	22.9340	0.921797	0.460899	+	0.887453i	$0.347527\pi$
$620$	−6.32275	−0.253928
$621$	−3.45831	−0.138777
$622$	−5.07804	−0.203611
$623$	−48.5938	−1.94687
$624$	0.449597	0.0179983
$625$	−10.2318	−0.409273
$626$	14.0650	0.562152
$627$	0	0
$628$	9.51059	0.379514
$629$	−24.2839	−0.968261
$630$	5.65271	0.225209
$631$	−21.3573	−0.850219	−0.425109	−	0.905142i	$-0.639764\pi$
$631$	−21.3573	−0.850219	−0.425109	+	0.905142i	$0.639764\pi$
$632$	47.0206	1.87038
$633$	10.5856	0.420739
$634$	1.84724	0.0733634
$635$	−3.34540	−0.132758
$636$	2.72598	0.108092
$637$	−15.4135	−0.610706
$638$	5.40575	0.214016
$639$	13.5488	0.535981
$640$	−13.3807	−0.528919
$641$	29.5018	1.16525	0.582625	−	0.812741i	$-0.302026\pi$
$641$	29.5018	1.16525	0.582625	+	0.812741i	$0.302026\pi$
$642$	−8.64975	−0.341378
$643$	6.83309	0.269471	0.134735	−	0.990882i	$-0.456982\pi$
$643$	6.83309	0.269471	0.134735	+	0.990882i	$0.456982\pi$
$644$	−17.5409	−0.691209
$645$	−17.2483	−0.679152
$646$	0	0
$647$	−18.9452	−0.744812	−0.372406	−	0.928070i	$-0.621467\pi$
$647$	−18.9452	−0.744812	−0.372406	+	0.928070i	$0.621467\pi$
$648$	−2.77164	−0.108880
$649$	−28.3487	−1.11279
$650$	3.20378	0.125662
$651$	−11.2778	−0.442012
$652$	20.6843	0.810060
$653$	27.1399	1.06207	0.531033	−	0.847351i	$-0.321804\pi$
$653$	27.1399	1.06207	0.531033	+	0.847351i	$0.321804\pi$
$654$	−1.71971	−0.0672460
$655$	−2.66372	−0.104080
$656$	2.17035	0.0847380
$657$	4.01120	0.156492
$658$	11.3729	0.443361
$659$	39.9075	1.55457	0.777287	−	0.629146i	$-0.216595\pi$
$659$	39.9075	1.55457	0.777287	+	0.629146i	$0.216595\pi$
$660$	9.75123	0.379566
$661$	50.3466	1.95826	0.979129	−	0.203241i	$-0.0651476\pi$
$661$	50.3466	1.95826	0.979129	+	0.203241i	$0.0651476\pi$
$662$	−13.0903	−0.508767
$663$	−9.17135	−0.356186
$664$	36.1012	1.40100
$665$	0	0
$666$	4.06942	0.157687
$667$	5.01530	0.194193
$668$	−16.9409	−0.655461
$669$	7.43507	0.287456
$670$	1.50604	0.0581835
$671$	0.339405	0.0131026
$672$	−22.6027	−0.871919
$673$	47.8104	1.84295	0.921477	−	0.388432i	$-0.126983\pi$
$673$	47.8104	1.84295	0.921477	+	0.388432i	$0.126983\pi$
$674$	−10.4251	−0.401559
$675$	2.08456	0.0802347
$676$	12.4801	0.480005
$677$	−41.2610	−1.58579	−0.792894	−	0.609359i	$-0.791427\pi$
$677$	−41.2610	−1.58579	−0.792894	+	0.609359i	$0.791427\pi$
$678$	−11.7901	−0.452797
$679$	−0.688807	−0.0264340
$680$	23.7876	0.912212
$681$	3.54389	0.135802
$682$	10.6959	0.409568
$683$	17.8805	0.684178	0.342089	−	0.939668i	$-0.388866\pi$
$683$	17.8805	0.684178	0.342089	+	0.939668i	$0.388866\pi$
$684$	0	0
$685$	26.2297	1.00218
$686$	4.79213	0.182964
$687$	11.3492	0.432997
$688$	2.48912	0.0948967
$689$	−3.85421	−0.146834
$690$	−4.97383	−0.189350
$691$	−11.3331	−0.431132	−0.215566	−	0.976489i	$-0.569160\pi$
$691$	−11.3331	−0.431132	−0.215566	+	0.976489i	$0.569160\pi$
$692$	−23.8382	−0.906193
$693$	17.3931	0.660710
$694$	−18.7042	−0.710000
$695$	15.8931	0.602861
$696$	4.01949	0.152358
$697$	−44.2731	−1.67696
$698$	−16.7867	−0.635386
$699$	5.92767	0.224205
$700$	10.5731	0.399626
$701$	6.89362	0.260368	0.130184	−	0.991490i	$-0.458443\pi$
$701$	6.89362	0.260368	0.130184	+	0.991490i	$0.458443\pi$
$702$	1.53691	0.0580069
$703$	0	0
$704$	19.2557	0.725725
$705$	−5.86568	−0.220914
$706$	−4.38513	−0.165037
$707$	−44.0727	−1.65752
$708$	−8.26695	−0.310691
$709$	−50.6485	−1.90214	−0.951072	−	0.308971i	$-0.900015\pi$
$709$	−50.6485	−1.90214	−0.951072	+	0.308971i	$0.900015\pi$
$710$	19.4862	0.731304
$711$	−16.9649	−0.636232
$712$	34.2680	1.28425
$713$	9.92335	0.371632
$714$	16.6404	0.622753
$715$	−13.7871	−0.515609
$716$	15.3038	0.571931
$717$	−15.1684	−0.566476
$718$	−4.35331	−0.162464
$719$	19.6429	0.732556	0.366278	−	0.930505i	$-0.380632\pi$
$719$	19.6429	0.732556	0.366278	+	0.930505i	$0.380632\pi$
$720$	0.420729	0.0156797
$721$	−37.4172	−1.39349
$722$	0	0
$723$	8.86254	0.329602
$724$	0.353242	0.0131281
$725$	−3.02306	−0.112274
$726$	−7.23023	−0.268339
$727$	−17.8693	−0.662734	−0.331367	−	0.943502i	$-0.607510\pi$
$727$	−17.8693	−0.662734	−0.331367	+	0.943502i	$0.607510\pi$
$728$	19.8765	0.736672
$729$	1.00000	0.0370370
$730$	5.76902	0.213521
$731$	−50.7756	−1.87800
$732$	0.0989760	0.00365826
$733$	39.1472	1.44594	0.722968	−	0.690882i	$-0.242777\pi$
$733$	39.1472	1.44594	0.722968	+	0.690882i	$0.242777\pi$
$734$	−5.10633	−0.188478
$735$	−14.4239	−0.532032
$736$	19.8882	0.733088
$737$	4.63402	0.170696
$738$	7.41916	0.273103
$739$	30.9676	1.13916	0.569581	−	0.821935i	$-0.307105\pi$
$739$	30.9676	1.13916	0.569581	+	0.821935i	$0.307105\pi$
$740$	−10.6456	−0.391339
$741$	0	0
$742$	6.99306	0.256723
$743$	12.2426	0.449136	0.224568	−	0.974458i	$-0.427903\pi$
$743$	12.2426	0.449136	0.224568	+	0.974458i	$0.427903\pi$
$744$	7.95303	0.291572
$745$	8.85175	0.324303
$746$	−8.41091	−0.307945
$747$	−13.0252	−0.476567
$748$	28.7057	1.04958
$749$	40.3606	1.47474
$750$	10.1892	0.372057
$751$	44.9133	1.63891	0.819455	−	0.573143i	$-0.194276\pi$
$751$	44.9133	1.63891	0.819455	+	0.573143i	$0.194276\pi$
$752$	0.846480	0.0308680
$753$	−11.7071	−0.426632
$754$	−2.22885	−0.0811700
$755$	−22.3544	−0.813561
$756$	5.07211	0.184471
$757$	−15.2301	−0.553549	−0.276775	−	0.960935i	$-0.589266\pi$
$757$	−15.2301	−0.553549	−0.276775	+	0.960935i	$0.589266\pi$
$758$	11.6070	0.421587
$759$	−15.3042	−0.555508
$760$	0	0
$761$	−32.2300	−1.16834	−0.584169	−	0.811632i	$-0.698579\pi$
$761$	−32.2300	−1.16834	−0.584169	+	0.811632i	$0.698579\pi$
$762$	1.65033	0.0597851
$763$	8.02434	0.290501
$764$	15.6988	0.567962
$765$	−8.58248	−0.310300
$766$	−13.7402	−0.496453
$767$	11.6885	0.422047
$768$	15.3033	0.552211
$769$	10.7275	0.386843	0.193422	−	0.981116i	$-0.438041\pi$
$769$	10.7275	0.386843	0.193422	+	0.981116i	$0.438041\pi$
$770$	25.0152	0.901487
$771$	−9.91597	−0.357115
$772$	29.7800	1.07180
$773$	−0.826686	−0.0297338	−0.0148669	−	0.999889i	$-0.504732\pi$
$773$	−0.826686	−0.0297338	−0.0148669	+	0.999889i	$0.504732\pi$
$774$	8.50883	0.305844
$775$	−5.98148	−0.214861
$776$	0.485742	0.0174371
$777$	−18.9883	−0.681203
$778$	−9.66244	−0.346415
$779$	0	0
$780$	−4.02054	−0.143958
$781$	59.9581	2.14547
$782$	−14.6420	−0.523595
$783$	−1.45022	−0.0518265
$784$	2.08152	0.0743398
$785$	−12.5835	−0.449124
$786$	1.31405	0.0468706
$787$	13.0266	0.464348	0.232174	−	0.972674i	$-0.425416\pi$
$787$	13.0266	0.464348	0.232174	+	0.972674i	$0.425416\pi$
$788$	−8.34791	−0.297382
$789$	6.73892	0.239912
$790$	−24.3993	−0.868089
$791$	55.0139	1.95607
$792$	−12.2655	−0.435836
$793$	−0.139941	−0.00496944
$794$	−0.377449	−0.0133952
$795$	−3.60674	−0.127918
$796$	−0.396298	−0.0140464
$797$	47.0808	1.66769	0.833844	−	0.552000i	$-0.186135\pi$
$797$	47.0808	1.66769	0.833844	+	0.552000i	$0.186135\pi$
$798$	0	0
$799$	−17.2674	−0.610877
$800$	−11.9880	−0.423838
$801$	−12.3638	−0.436853
$802$	−12.2190	−0.431468
$803$	17.7510	0.626418
$804$	1.35135	0.0476586
$805$	23.2084	0.817988
$806$	−4.41004	−0.155337
$807$	0.718290	0.0252850
$808$	31.0798	1.09338
$809$	−10.0150	−0.352107	−0.176053	−	0.984381i	$-0.556333\pi$
$809$	−10.0150	−0.352107	−0.176053	+	0.984381i	$0.556333\pi$
$810$	1.43823	0.0505341
$811$	31.5550	1.10804	0.554022	−	0.832502i	$-0.313092\pi$
$811$	31.5550	1.10804	0.554022	+	0.832502i	$0.313092\pi$
$812$	−7.35566	−0.258133
$813$	28.8411	1.01150
$814$	18.0086	0.631202
$815$	−27.3674	−0.958639
$816$	1.23854	0.0433577
$817$	0	0
$818$	11.1177	0.388721
$819$	−7.17137	−0.250588
$820$	−19.4085	−0.677772
$821$	−38.3465	−1.33830	−0.669151	−	0.743126i	$-0.733342\pi$
$821$	−38.3465	−1.33830	−0.669151	+	0.743126i	$0.733342\pi$
$822$	−12.9395	−0.451315
$823$	49.1374	1.71282	0.856410	−	0.516296i	$-0.172689\pi$
$823$	49.1374	1.71282	0.856410	+	0.516296i	$0.172689\pi$
$824$	26.3863	0.919211
$825$	9.22490	0.321170
$826$	−21.2075	−0.737905
$827$	−16.0238	−0.557201	−0.278600	−	0.960407i	$-0.589870\pi$
$827$	−16.0238	−0.557201	−0.278600	+	0.960407i	$0.589870\pi$
$828$	−4.46296	−0.155098
$829$	−34.4952	−1.19807	−0.599033	−	0.800724i	$-0.704448\pi$
$829$	−34.4952	−1.19807	−0.599033	+	0.800724i	$0.704448\pi$
$830$	−18.7332	−0.650238
$831$	3.83134	0.132908
$832$	−7.93932	−0.275247
$833$	−42.4609	−1.47118
$834$	−7.84030	−0.271487
$835$	22.4145	0.775685
$836$	0	0
$837$	−2.86943	−0.0991818
$838$	9.62173	0.332377
$839$	17.0321	0.588013	0.294007	−	0.955803i	$-0.405011\pi$
$839$	17.0321	0.588013	0.294007	+	0.955803i	$0.405011\pi$
$840$	18.6003	0.641770
$841$	−26.8969	−0.927478
$842$	13.9687	0.481394
$843$	−22.0921	−0.760894
$844$	13.6607	0.470222
$845$	−16.5125	−0.568047
$846$	2.89362	0.0994847
$847$	33.7370	1.15922
$848$	0.520491	0.0178737
$849$	25.1465	0.863026
$850$	8.82571	0.302719
$851$	16.7079	0.572738
$852$	17.4847	0.599017
$853$	34.2140	1.17146	0.585732	−	0.810505i	$-0.300807\pi$
$853$	34.2140	1.17146	0.585732	+	0.810505i	$0.300807\pi$
$854$	0.253907	0.00868853
$855$	0	0
$856$	−28.4620	−0.972812
$857$	−8.26690	−0.282392	−0.141196	−	0.989982i	$-0.545095\pi$
$857$	−8.26690	−0.282392	−0.141196	+	0.989982i	$0.545095\pi$
$858$	6.80136	0.232195
$859$	−50.4943	−1.72284	−0.861421	−	0.507891i	$-0.830425\pi$
$859$	−50.4943	−1.72284	−0.861421	+	0.507891i	$0.830425\pi$
$860$	−22.2590	−0.759026
$861$	−34.6185	−1.17980
$862$	−24.0049	−0.817611
$863$	33.0070	1.12357	0.561785	−	0.827283i	$-0.310115\pi$
$863$	33.0070	1.12357	0.561785	+	0.827283i	$0.310115\pi$
$864$	−5.75084	−0.195648
$865$	31.5404	1.07241
$866$	−11.7903	−0.400652
$867$	−8.26510	−0.280697
$868$	−14.5540	−0.493996
$869$	−75.0755	−2.54676
$870$	−2.08574	−0.0707133
$871$	−1.91066	−0.0647402
$872$	−5.65871	−0.191628
$873$	−0.175254	−0.00593145
$874$	0	0
$875$	−47.5438	−1.60728
$876$	5.17647	0.174897
$877$	−2.11604	−0.0714535	−0.0357267	−	0.999362i	$-0.511375\pi$
$877$	−2.11604	−0.0714535	−0.0357267	+	0.999362i	$0.511375\pi$
$878$	22.5986	0.762666
$879$	−5.92712	−0.199917
$880$	1.86188	0.0627638
$881$	−53.0406	−1.78698	−0.893492	−	0.449080i	$-0.851752\pi$
$881$	−53.0406	−1.78698	−0.893492	+	0.449080i	$0.851752\pi$
$882$	7.11548	0.239591
$883$	17.1836	0.578275	0.289137	−	0.957288i	$-0.406632\pi$
$883$	17.1836	0.578275	0.289137	+	0.957288i	$0.406632\pi$
$884$	−11.8357	−0.398076
$885$	10.9380	0.367677
$886$	−11.7236	−0.393861
$887$	3.89561	0.130802	0.0654008	−	0.997859i	$-0.479167\pi$
$887$	3.89561	0.130802	0.0654008	+	0.997859i	$0.479167\pi$
$888$	13.3904	0.449354
$889$	−7.70060	−0.258270
$890$	−17.7819	−0.596051
$891$	4.42535	0.148255
$892$	9.59498	0.321264
$893$	0	0
$894$	−4.36669	−0.146044
$895$	−20.2485	−0.676834
$896$	−30.8004	−1.02897
$897$	6.31010	0.210688
$898$	−17.9279	−0.598263
$899$	4.16129	0.138787
$900$	2.69013	0.0896710
$901$	−10.6175	−0.353721
$902$	32.8324	1.09320
$903$	−39.7031	−1.32124
$904$	−38.7954	−1.29032
$905$	−0.467375	−0.0155361
$906$	11.0277	0.366372
$907$	8.84164	0.293582	0.146791	−	0.989168i	$-0.453106\pi$
$907$	8.84164	0.293582	0.146791	+	0.989168i	$0.453106\pi$
$908$	4.57340	0.151774
$909$	−11.2135	−0.371928
$910$	−10.3141	−0.341908
$911$	−18.4416	−0.610997	−0.305498	−	0.952193i	$-0.598823\pi$
$911$	−18.4416	−0.610997	−0.305498	+	0.952193i	$0.598823\pi$
$912$	0	0
$913$	−57.6411	−1.90764
$914$	−0.628147	−0.0207773
$915$	−0.130955	−0.00432925
$916$	14.6461	0.483922
$917$	−6.13149	−0.202480
$918$	4.23385	0.139738
$919$	−45.5362	−1.50210	−0.751051	−	0.660245i	$-0.770453\pi$
$919$	−45.5362	−1.50210	−0.751051	+	0.660245i	$0.770453\pi$
$920$	−16.3664	−0.539584
$921$	−17.5842	−0.579421
$922$	23.7485	0.782114
$923$	−24.7214	−0.813714
$924$	22.4459	0.738415
$925$	−10.0710	−0.331132
$926$	7.46044	0.245165
$927$	−9.52010	−0.312681
$928$	8.33997	0.273773
$929$	−1.51634	−0.0497494	−0.0248747	−	0.999691i	$-0.507919\pi$
$929$	−1.51634	−0.0497494	−0.0248747	+	0.999691i	$0.507919\pi$
$930$	−4.12688	−0.135326
$931$	0	0
$932$	7.64968	0.250574
$933$	6.02866	0.197370
$934$	31.3995	1.02742
$935$	−37.9805	−1.24209
$936$	5.05720	0.165300
$937$	7.13697	0.233155	0.116577	−	0.993182i	$-0.462808\pi$
$937$	7.13697	0.233155	0.116577	+	0.993182i	$0.462808\pi$
$938$	3.46669	0.113191
$939$	−16.6980	−0.544920
$940$	−7.56968	−0.246896
$941$	−49.7958	−1.62330	−0.811648	−	0.584147i	$-0.801429\pi$
$941$	−49.7958	−1.62330	−0.811648	+	0.584147i	$0.801429\pi$
$942$	6.20760	0.202254
$943$	30.4609	0.991944
$944$	−1.57847	−0.0513748
$945$	−6.71091	−0.218306
$946$	37.6546	1.22426
$947$	−0.349552	−0.0113589	−0.00567945	−	0.999984i	$-0.501808\pi$
$947$	−0.349552	−0.0113589	−0.00567945	+	0.999984i	$0.501808\pi$
$948$	−21.8932	−0.711058
$949$	−7.31892	−0.237582
$950$	0	0
$951$	−2.19305	−0.0711146
$952$	54.7554	1.77463
$953$	−21.0086	−0.680534	−0.340267	−	0.940329i	$-0.610517\pi$
$953$	−21.0086	−0.680534	−0.340267	+	0.940329i	$0.610517\pi$
$954$	1.77925	0.0576055
$955$	−20.7711	−0.672137
$956$	−19.5749	−0.633099
$957$	−6.41772	−0.207456
$958$	−25.1475	−0.812479
$959$	60.3768	1.94967
$960$	−7.42955	−0.239788
$961$	−22.7664	−0.734400
$962$	−7.42516	−0.239397
$963$	10.2690	0.330914
$964$	11.4371	0.368366
$965$	−39.4019	−1.26839
$966$	−11.4490	−0.368366
$967$	−36.8410	−1.18473	−0.592364	−	0.805670i	$-0.701805\pi$
$967$	−36.8410	−1.18473	−0.592364	+	0.805670i	$0.701805\pi$
$968$	−23.7911	−0.764675
$969$	0	0
$970$	−0.252055	−0.00809300
$971$	−5.39119	−0.173012	−0.0865058	−	0.996251i	$-0.527570\pi$
$971$	−5.39119	−0.173012	−0.0865058	+	0.996251i	$0.527570\pi$
$972$	1.29050	0.0413929
$973$	36.5836	1.17282
$974$	−26.0114	−0.833459
$975$	−3.80353	−0.121810
$976$	0.0188982	0.000604918 0
$977$	40.7182	1.30269	0.651345	−	0.758782i	$-0.274205\pi$
$977$	40.7182	1.30269	0.651345	+	0.758782i	$0.274205\pi$
$978$	13.5007	0.431705
$979$	−54.7141	−1.74867
$980$	−18.6140	−0.594603
$981$	2.04164	0.0651847
$982$	8.53061	0.272223
$983$	−43.9907	−1.40309	−0.701543	−	0.712627i	$-0.747505\pi$
$983$	−43.9907	−1.40309	−0.701543	+	0.712627i	$0.747505\pi$
$984$	24.4128	0.778251
$985$	11.0451	0.351927
$986$	−6.14000	−0.195538
$987$	−13.5019	−0.429771
$988$	0	0
$989$	34.9348	1.11086
$990$	6.36466	0.202282
$991$	−33.2649	−1.05669	−0.528347	−	0.849029i	$-0.677188\pi$
$991$	−33.2649	−1.05669	−0.528347	+	0.849029i	$0.677188\pi$
$992$	16.5016	0.523926
$993$	15.5408	0.493172
$994$	44.8543	1.42269
$995$	0.524343	0.0166228
$996$	−16.8091	−0.532615
$997$	27.0800	0.857633	0.428817	−	0.903392i	$-0.358931\pi$
$997$	27.0800	0.857633	0.428817	+	0.903392i	$0.358931\pi$
$998$	7.23937	0.229158
$999$	−4.83123	−0.152853

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1083.2.a.p.1.4		6
3.2	odd	2		3249.2.a.bg.1.3		6
19.2	odd	18		57.2.i.b.4.2	✓	12
19.10	odd	18		57.2.i.b.43.2	yes	12
19.18	odd	2		1083.2.a.q.1.3		6
57.2	even	18		171.2.u.e.118.1		12
57.29	even	18		171.2.u.e.100.1		12
57.56	even	2		3249.2.a.bh.1.4		6
76.59	even	18		912.2.bo.j.289.1		12
76.67	even	18		912.2.bo.j.385.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.2.i.b.4.2	✓	12	19.2	odd	18
57.2.i.b.43.2	yes	12	19.10	odd	18
171.2.u.e.100.1		12	57.29	even	18
171.2.u.e.118.1		12	57.2	even	18
912.2.bo.j.289.1		12	76.59	even	18
912.2.bo.j.385.1		12	76.67	even	18
1083.2.a.p.1.4		6	1.1	even	1	trivial
1083.2.a.q.1.3		6	19.18	odd	2
3249.2.a.bg.1.3		6	3.2	odd	2
3249.2.a.bh.1.4		6	57.56	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1083 = 3 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1083.a (trivial)

\(f(q)\)	\(=\)	\(q+0.842316 q^{2} -1.00000 q^{3} -1.29050 q^{4} +1.70747 q^{5} -0.842316 q^{6} +3.93033 q^{7} -2.77164 q^{8} +1.00000 q^{9} +1.43823 q^{10} +4.42535 q^{11} +1.29050 q^{12} -1.82462 q^{13} +3.31058 q^{14} -1.70747 q^{15} +0.246406 q^{16} -5.02644 q^{17} +0.842316 q^{18} -2.20349 q^{20} -3.93033 q^{21} +3.72755 q^{22} +3.45831 q^{23} +2.77164 q^{24} -2.08456 q^{25} -1.53691 q^{26} -1.00000 q^{27} -5.07211 q^{28} +1.45022 q^{29} -1.43823 q^{30} +2.86943 q^{31} +5.75084 q^{32} -4.42535 q^{33} -4.23385 q^{34} +6.71091 q^{35} -1.29050 q^{36} +4.83123 q^{37} +1.82462 q^{39} -4.73249 q^{40} +8.80804 q^{41} -3.31058 q^{42} +10.1017 q^{43} -5.71093 q^{44} +1.70747 q^{45} +2.91299 q^{46} +3.43531 q^{47} -0.246406 q^{48} +8.44752 q^{49} -1.75586 q^{50} +5.02644 q^{51} +2.35468 q^{52} +2.11234 q^{53} -0.842316 q^{54} +7.55614 q^{55} -10.8935 q^{56} +1.22154 q^{58} -6.40598 q^{59} +2.20349 q^{60} +0.0766957 q^{61} +2.41696 q^{62} +3.93033 q^{63} +4.35121 q^{64} -3.11548 q^{65} -3.72755 q^{66} +1.04715 q^{67} +6.48664 q^{68} -3.45831 q^{69} +5.65271 q^{70} +13.5488 q^{71} -2.77164 q^{72} +4.01120 q^{73} +4.06942 q^{74} +2.08456 q^{75} +17.3931 q^{77} +1.53691 q^{78} -16.9649 q^{79} +0.420729 q^{80} +1.00000 q^{81} +7.41916 q^{82} -13.0252 q^{83} +5.07211 q^{84} -8.58248 q^{85} +8.50883 q^{86} -1.45022 q^{87} -12.2655 q^{88} -12.3638 q^{89} +1.43823 q^{90} -7.17137 q^{91} -4.46296 q^{92} -2.86943 q^{93} +2.89362 q^{94} -5.75084 q^{96} -0.175254 q^{97} +7.11548 q^{98} +4.42535 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} + 6 q^{4} + 9 q^{5} + 9 q^{7} + 3 q^{8} + 6 q^{9} - 6 q^{10} + 9 q^{11} - 6 q^{12} - 3 q^{13} + 6 q^{14} - 9 q^{15} + 6 q^{16} + 15 q^{17} - 9 q^{20} - 9 q^{21} + 21 q^{22} + 6 q^{23} - 3 q^{24}+ \cdots + 9 q^{99}+O(q^{100}) \) 6 * q - 6 * q^3 + 6 * q^4 + 9 * q^5 + 9 * q^7 + 3 * q^8 + 6 * q^9 - 6 * q^10 + 9 * q^11 - 6 * q^12 - 3 * q^13 + 6 * q^14 - 9 * q^15 + 6 * q^16 + 15 * q^17 - 9 * q^20 - 9 * q^21 + 21 * q^22 + 6 * q^23 - 3 * q^24 + 15 * q^25 - 6 * q^27 - 3 * q^28 + 15 * q^29 + 6 * q^30 + 21 * q^32 - 9 * q^33 - 33 * q^34 - 3 * q^35 + 6 * q^36 + 6 * q^37 + 3 * q^39 - 39 * q^40 - 6 * q^41 - 6 * q^42 + 15 * q^43 - 6 * q^44 + 9 * q^45 + 27 * q^46 + 9 * q^47 - 6 * q^48 + 9 * q^49 + 9 * q^50 - 15 * q^51 - 18 * q^52 + 6 * q^53 - 6 * q^55 + 39 * q^56 - 12 * q^58 + 15 * q^59 + 9 * q^60 + 3 * q^61 + 27 * q^62 + 9 * q^63 + 21 * q^64 + 21 * q^65 - 21 * q^66 + 18 * q^67 + 21 * q^68 - 6 * q^69 - 9 * q^70 + 15 * q^71 + 3 * q^72 - 6 * q^73 + 9 * q^74 - 15 * q^75 + 48 * q^77 - 21 * q^79 - 21 * q^80 + 6 * q^81 - 15 * q^82 - 3 * q^83 + 3 * q^84 + 42 * q^85 + 30 * q^86 - 15 * q^87 + 42 * q^88 + 24 * q^89 - 6 * q^90 - 12 * q^91 + 42 * q^92 - 9 * q^94 - 21 * q^96 + 3 * q^97 + 3 * q^98 + 9 * q^99

Introduction

L-functions

Modular forms

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Database

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