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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,6,Mod(2,75)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("75.2"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([10, 1])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 75.l (of order \(20\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.0287864860\)
Analytic rank: \(0\)
Dimension: \(384\)
Relative dimension: \(48\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label 2.8
Character \(\chi\) \(=\) 75.2
Dual form 75.6.l.a.38.8

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.26293 + 7.97384i) q^{2} +(1.88797 + 15.4737i) q^{3} +(-31.5534 - 10.2523i) q^{4} +(-26.5989 + 49.1680i) q^{5} +(-125.769 - 4.48788i) q^{6} +(164.867 + 164.867i) q^{7} +(4.31459 - 8.46787i) q^{8} +(-235.871 + 58.4278i) q^{9} +(-358.466 - 274.192i) q^{10} +(-66.4958 - 91.5237i) q^{11} +(99.0695 - 507.604i) q^{12} +(201.417 - 31.9013i) q^{13} +(-1522.84 + 1106.41i) q^{14} +(-811.030 - 318.756i) q^{15} +(-796.837 - 578.936i) q^{16} +(822.974 + 419.326i) q^{17} +(-168.004 - 1954.59i) q^{18} +(2811.11 - 913.385i) q^{19} +(1343.37 - 1278.72i) q^{20} +(-2239.84 + 2862.37i) q^{21} +(813.775 - 414.639i) q^{22} +(3296.31 + 522.085i) q^{23} +(139.175 + 50.7757i) q^{24} +(-1709.99 - 2615.64i) q^{25} +1646.36i q^{26} +(-1349.41 - 3539.49i) q^{27} +(-3511.84 - 6892.38i) q^{28} +(-446.004 + 1372.66i) q^{29} +(3565.99 - 6064.46i) q^{30} +(-1296.06 - 3988.88i) q^{31} +(5837.74 - 5837.74i) q^{32} +(1290.67 - 1201.73i) q^{33} +(-4383.00 + 6032.69i) q^{34} +(-12491.5 + 3720.90i) q^{35} +(8041.55 + 574.632i) q^{36} +(821.477 + 5186.60i) q^{37} +(3732.94 + 23568.9i) q^{38} +(873.901 + 3056.44i) q^{39} +(301.585 + 437.376i) q^{40} +(7576.50 - 10428.2i) q^{41} +(-19995.3 - 21475.1i) q^{42} +(-5374.96 + 5374.96i) q^{43} +(1159.84 + 3569.62i) q^{44} +(3401.14 - 13151.4i) q^{45} +(-8326.05 + 25624.9i) q^{46} +(-4708.77 - 9241.49i) q^{47} +(7453.88 - 13423.0i) q^{48} +37555.3i q^{49} +(23016.3 - 10331.8i) q^{50} +(-4934.78 + 13526.1i) q^{51} +(-6682.45 - 1058.40i) q^{52} +(-33507.2 + 17072.8i) q^{53} +(29927.6 - 6289.86i) q^{54} +(6268.76 - 835.038i) q^{55} +(2107.41 - 684.738i) q^{56} +(19440.7 + 41773.8i) q^{57} +(-10382.1 - 5289.95i) q^{58} +(11662.9 + 8473.60i) q^{59} +(22322.7 + 18372.8i) q^{60} +(-11306.0 + 8214.30i) q^{61} +(33443.5 - 5296.93i) q^{62} +(-48520.2 - 29254.6i) q^{63} +(20650.6 + 28423.1i) q^{64} +(-3788.96 + 10751.8i) q^{65} +(7952.39 + 11809.3i) q^{66} +(17931.8 - 35193.2i) q^{67} +(-21668.6 - 21668.6i) q^{68} +(-1855.25 + 51991.9i) q^{69} +(-13894.0 - 104304. i) q^{70} +(24962.2 + 8110.71i) q^{71} +(-522.930 + 2249.42i) q^{72} +(-7230.02 + 45648.5i) q^{73} -42394.6 q^{74} +(37245.2 - 31398.2i) q^{75} -98064.3 q^{76} +(4126.27 - 26052.2i) q^{77} +(-25475.3 + 3108.27i) q^{78} +(-55010.5 - 17874.0i) q^{79} +(49660.1 - 23779.8i) q^{80} +(52221.4 - 27562.8i) q^{81} +(73583.9 + 73583.9i) q^{82} +(11694.0 - 22950.8i) q^{83} +(100020. - 67353.8i) q^{84} +(-42507.7 + 29310.4i) q^{85} +(-36070.9 - 49647.3i) q^{86} +(-22082.2 - 4309.80i) q^{87} +(-1061.91 + 168.190i) q^{88} +(55506.1 - 40327.6i) q^{89} +(100572. + 43729.6i) q^{90} +(38466.5 + 27947.6i) q^{91} +(-98657.3 - 50268.4i) q^{92} +(59275.8 - 27585.8i) q^{93} +(79637.0 - 25875.6i) q^{94} +(-29863.2 + 162512. i) q^{95} +(101353. + 79310.0i) q^{96} +(-48713.2 + 24820.6i) q^{97} +(-299460. - 47429.8i) q^{98} +(21032.0 + 17702.6i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 384 q - 10 q^{3} - 20 q^{4} - 6 q^{6} + 60 q^{7} - 10 q^{9} - 1080 q^{10} - 430 q^{12} + 2100 q^{13} + 2990 q^{15} + 21492 q^{16} - 6010 q^{18} - 8800 q^{19} - 6 q^{21} + 6040 q^{22} + 23880 q^{25} + 12950 q^{27}+ \cdots + 715720 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{20}\right)\)

Coefficient data

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



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.26293 + 7.97384i −0.223257 + 1.40959i 0.580320 + 0.814389i \(0.302928\pi\)
−0.803577 + 0.595201i \(0.797072\pi\)
\(3\) 1.88797 + 15.4737i 0.121113 + 0.992639i
\(4\) −31.5534 − 10.2523i −0.986043 − 0.320385i
\(5\) −26.5989 + 49.1680i −0.475816 + 0.879545i
\(6\) −125.769 − 4.48788i −1.42625 − 0.0508936i
\(7\) 164.867 + 164.867i 1.27171 + 1.27171i 0.945189 + 0.326523i \(0.105877\pi\)
0.326523 + 0.945189i \(0.394123\pi\)
\(8\) 4.31459 − 8.46787i 0.0238350 − 0.0467788i
\(9\) −235.871 + 58.4278i −0.970663 + 0.240443i
\(10\) −358.466 − 274.192i −1.13357 − 0.867070i
\(11\) −66.4958 − 91.5237i −0.165696 − 0.228061i 0.718092 − 0.695948i \(-0.245016\pi\)
−0.883789 + 0.467887i \(0.845016\pi\)
\(12\) 99.0695 − 507.604i 0.198603 − 1.01759i
\(13\) 201.417 − 31.9013i 0.330551 − 0.0523541i 0.0110466 − 0.999939i \(-0.496484\pi\)
0.319504 + 0.947585i \(0.396484\pi\)
\(14\) −1522.84 + 1106.41i −2.07651 + 1.50867i
\(15\) −811.030 − 318.756i −0.930698 − 0.365789i
\(16\) −796.837 − 578.936i −0.778161 − 0.565367i
\(17\) 822.974 + 419.326i 0.690659 + 0.351909i 0.763836 − 0.645410i \(-0.223314\pi\)
−0.0731765 + 0.997319i \(0.523314\pi\)
\(18\) −168.004 − 1954.59i −0.122219 − 1.42192i
\(19\) 2811.11 − 913.385i 1.78646 − 0.580457i 0.787122 − 0.616797i \(-0.211570\pi\)
0.999339 + 0.0363408i \(0.0115702\pi\)
\(20\) 1343.37 − 1278.72i 0.750968 − 0.714825i
\(21\) −2239.84 + 2862.37i −1.10833 + 1.41637i
\(22\) 813.775 − 414.639i 0.358466 − 0.182647i
\(23\) 3296.31 + 522.085i 1.29930 + 0.205789i 0.767482 − 0.641071i \(-0.221509\pi\)
0.531817 + 0.846859i \(0.321509\pi\)
\(24\) 139.175 + 50.7757i 0.0493212 + 0.0179940i
\(25\) −1709.99 − 2615.64i −0.547198 − 0.837003i
\(26\) 1646.36i 0.477629i
\(27\) −1349.41 − 3539.49i −0.356234 − 0.934397i
\(28\) −3511.84 − 6892.38i −0.846526 − 1.66140i
\(29\) −446.004 + 1372.66i −0.0984791 + 0.303087i −0.988145 − 0.153525i \(-0.950937\pi\)
0.889666 + 0.456613i \(0.150937\pi\)
\(30\) 3565.99 − 6064.46i 0.723397 − 1.23024i
\(31\) −1296.06 − 3988.88i −0.242227 − 0.745498i −0.996080 − 0.0884550i \(-0.971807\pi\)
0.753853 − 0.657043i \(-0.228193\pi\)
\(32\) 5837.74 − 5837.74i 1.00779 − 1.00779i
\(33\) 1290.67 − 1201.73i 0.206315 − 0.192098i
\(34\) −4383.00 + 6032.69i −0.650241 + 0.894980i
\(35\) −12491.5 + 3720.90i −1.72363 + 0.513426i
\(36\) 8041.55 + 574.632i 1.03415 + 0.0738982i
\(37\) 821.477 + 5186.60i 0.0986486 + 0.622843i 0.986632 + 0.162967i \(0.0521063\pi\)
−0.887983 + 0.459876i \(0.847894\pi\)
\(38\) 3732.94 + 23568.9i 0.419365 + 2.64777i
\(39\) 873.901 + 3056.44i 0.0920028 + 0.321777i
\(40\) 301.585 + 437.376i 0.0298030 + 0.0432220i
\(41\) 7576.50 − 10428.2i 0.703897 − 0.968831i −0.296010 − 0.955185i \(-0.595656\pi\)
0.999907 − 0.0136461i \(-0.00434383\pi\)
\(42\) −19995.3 − 21475.1i −1.74906 − 1.87850i
\(43\) −5374.96 + 5374.96i −0.443306 + 0.443306i −0.893122 − 0.449815i \(-0.851490\pi\)
0.449815 + 0.893122i \(0.351490\pi\)
\(44\) 1159.84 + 3569.62i 0.0903163 + 0.277965i
\(45\) 3401.14 − 13151.4i 0.250377 − 0.968149i
\(46\) −8326.05 + 25624.9i −0.580155 + 1.78553i
\(47\) −4708.77 − 9241.49i −0.310930 − 0.610235i 0.681670 − 0.731659i \(-0.261254\pi\)
−0.992601 + 0.121424i \(0.961254\pi\)
\(48\) 7453.88 − 13423.0i 0.466959 − 0.840906i
\(49\) 37555.3i 2.23450i
\(50\) 23016.3 − 10331.8i 1.30200 − 0.584457i
\(51\) −4934.78 + 13526.1i −0.265670 + 0.728196i
\(52\) −6682.45 − 1058.40i −0.342711 − 0.0542800i
\(53\) −33507.2 + 17072.8i −1.63851 + 0.834861i −0.640762 + 0.767739i \(0.721382\pi\)
−0.997745 + 0.0671220i \(0.978618\pi\)
\(54\) 29927.6 − 6289.86i 1.39665 − 0.293533i
\(55\) 6268.76 − 835.038i 0.279431 − 0.0372220i
\(56\) 2107.41 − 684.738i 0.0898004 − 0.0291779i
\(57\) 19440.7 + 41773.8i 0.792548 + 1.70301i
\(58\) −10382.1 − 5289.95i −0.405243 − 0.206482i
\(59\) 11662.9 + 8473.60i 0.436191 + 0.316912i 0.784120 − 0.620610i \(-0.213115\pi\)
−0.347928 + 0.937521i \(0.613115\pi\)
\(60\) 22322.7 + 18372.8i 0.800515 + 0.658865i
\(61\) −11306.0 + 8214.30i −0.389032 + 0.282648i −0.765059 − 0.643961i \(-0.777290\pi\)
0.376027 + 0.926609i \(0.377290\pi\)
\(62\) 33443.5 − 5296.93i 1.10492 − 0.175003i
\(63\) −48520.2 − 29254.6i −1.54018 − 0.928629i
\(64\) 20650.6 + 28423.1i 0.630206 + 0.867405i
\(65\) −3788.96 + 10751.8i −0.111234 + 0.315645i
\(66\) 7952.39 + 11809.3i 0.224718 + 0.333706i
\(67\) 17931.8 − 35193.2i 0.488020 − 0.957794i −0.507356 − 0.861737i \(-0.669377\pi\)
0.995376 − 0.0960570i \(-0.0306231\pi\)
\(68\) −21668.6 − 21668.6i −0.568274 − 0.568274i
\(69\) −1855.25 + 51991.9i −0.0469115 + 1.31466i
\(70\) −13894.0 − 104304.i −0.338908 − 2.54424i
\(71\) 24962.2 + 8110.71i 0.587675 + 0.190947i 0.587736 − 0.809053i \(-0.300019\pi\)
−6.10365e−5 1.00000i \(0.500019\pi\)
\(72\) −522.930 + 2249.42i −0.0118881 + 0.0511374i
\(73\) −7230.02 + 45648.5i −0.158793 + 1.00258i 0.771624 + 0.636079i \(0.219445\pi\)
−0.930417 + 0.366502i \(0.880555\pi\)
\(74\) −42394.6 −0.899977
\(75\) 37245.2 − 31398.2i 0.764569 − 0.644542i
\(76\) −98064.3 −1.94750
\(77\) 4126.27 − 26052.2i 0.0793104 − 0.500746i
\(78\) −25475.3 + 3108.27i −0.474113 + 0.0578472i
\(79\) −55010.5 − 17874.0i −0.991694 − 0.322221i −0.232152 − 0.972679i \(-0.574577\pi\)
−0.759542 + 0.650459i \(0.774577\pi\)
\(80\) 49660.1 − 23779.8i 0.867527 − 0.415416i
\(81\) 52221.4 − 27562.8i 0.884374 − 0.466779i
\(82\) 73583.9 + 73583.9i 1.20850 + 1.20850i
\(83\) 11694.0 − 22950.8i 0.186324 − 0.365681i −0.778883 − 0.627169i \(-0.784213\pi\)
0.965207 + 0.261489i \(0.0842134\pi\)
\(84\) 100020. − 67353.8i 1.54664 − 1.04151i
\(85\) −42507.7 + 29310.4i −0.638146 + 0.440022i
\(86\) −36070.9 − 49647.3i −0.525909 − 0.723852i
\(87\) −22082.2 − 4309.80i −0.312783 − 0.0610462i
\(88\) −1061.91 + 168.190i −0.0146178 + 0.00231523i
\(89\) 55506.1 − 40327.6i 0.742790 − 0.539668i −0.150794 − 0.988565i \(-0.548183\pi\)
0.893584 + 0.448897i \(0.148183\pi\)
\(90\) 100572. + 43729.6i 1.30879 + 0.569074i
\(91\) 38466.5 + 27947.6i 0.486945 + 0.353786i
\(92\) −98657.3 − 50268.4i −1.21523 − 0.619192i
\(93\) 59275.8 − 27585.8i 0.710673 − 0.330733i
\(94\) 79637.0 − 25875.6i 0.929599 − 0.302045i
\(95\) −29863.2 + 162512.i −0.339490 + 1.84746i
\(96\) 101353. + 79310.0i 1.12243 + 0.878314i
\(97\) −48713.2 + 24820.6i −0.525676 + 0.267845i −0.696626 − 0.717435i \(-0.745316\pi\)
0.170950 + 0.985280i \(0.445316\pi\)
\(98\) −299460. − 47429.8i −3.14973 − 0.498869i
\(99\) 21032.0 + 17702.6i 0.215671 + 0.181530i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 75.6.l.a.2.8 ✓ 384
3.2 odd 2 inner 75.6.l.a.2.41 yes 384
25.13 odd 20 inner 75.6.l.a.38.41 yes 384
75.38 even 20 inner 75.6.l.a.38.8 yes 384
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.6.l.a.2.8 ✓ 384 1.1 even 1 trivial
75.6.l.a.2.41 yes 384 3.2 odd 2 inner
75.6.l.a.38.8 yes 384 75.38 even 20 inner
75.6.l.a.38.41 yes 384 25.13 odd 20 inner