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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 2.5
Character \(\chi\) \(=\) 75.2
Dual form 75.6.l.a.38.5

$q$-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+(-1.52703 + 9.64126i) q^{2} +(-10.4038 + 11.6087i) q^{3} +(-60.1884 - 19.5564i) q^{4} +(42.5572 + 36.2475i) q^{5} +(-96.0355 - 118.032i) q^{6} +(-81.4490 - 81.4490i) q^{7} +(138.646 - 272.108i) q^{8} +(-26.5228 - 241.548i) q^{9} +(-414.458 + 354.954i) q^{10} +(-220.802 - 303.908i) q^{11} +(853.210 - 495.247i) q^{12} +(172.258 - 27.2831i) q^{13} +(909.646 - 660.897i) q^{14} +(-863.542 + 116.922i) q^{15} +(773.377 + 561.891i) q^{16} +(1247.33 + 635.544i) q^{17} +(2369.33 + 113.137i) q^{18} +(-1755.68 + 570.457i) q^{19} +(-1852.58 - 3013.94i) q^{20} +(1792.89 - 98.1380i) q^{21} +(3267.23 - 1664.74i) q^{22} +(-2873.57 - 455.130i) q^{23} +(1716.38 + 4440.46i) q^{24} +(497.233 + 3085.19i) q^{25} +1702.45i q^{26} +(3079.99 + 2205.12i) q^{27} +(3309.43 + 6495.13i) q^{28} +(-1627.50 + 5008.93i) q^{29} +(191.377 - 8504.18i) q^{30} +(-1817.88 - 5594.85i) q^{31} +(311.983 - 311.983i) q^{32} +(5825.14 + 598.571i) q^{33} +(-8032.15 + 11055.3i) q^{34} +(-513.917 - 6418.57i) q^{35} +(-3127.44 + 15057.1i) q^{36} +(-2245.86 - 14179.8i) q^{37} +(-2818.95 - 17798.1i) q^{38} +(-1475.42 + 2283.54i) q^{39} +(15763.7 - 6554.60i) q^{40} +(3487.72 - 4800.43i) q^{41} +(-1791.62 + 17435.6i) q^{42} +(10409.4 - 10409.4i) q^{43} +(7346.37 + 22609.8i) q^{44} +(7626.79 - 11241.0i) q^{45} +(8776.05 - 27009.9i) q^{46} +(-2702.87 - 5304.68i) q^{47} +(-14568.8 + 3132.09i) q^{48} -3539.12i q^{49} +(-30504.4 + 82.7917i) q^{50} +(-20354.7 + 7867.74i) q^{51} +(-10901.5 - 1726.63i) q^{52} +(-30289.9 + 15433.5i) q^{53} +(-25963.4 + 26327.8i) q^{54} +(1619.19 - 20937.0i) q^{55} +(-33455.6 + 10870.4i) q^{56} +(11643.5 - 26316.1i) q^{57} +(-45807.2 - 23339.9i) q^{58} +(10579.3 + 7686.28i) q^{59} +(54261.7 + 9850.42i) q^{60} +(-9231.57 + 6707.13i) q^{61} +(56717.4 - 8983.16i) q^{62} +(-17513.6 + 21834.1i) q^{63} +(20512.0 + 28232.4i) q^{64} +(8319.79 + 5082.85i) q^{65} +(-14666.1 + 55247.7i) q^{66} +(-10891.7 + 21376.1i) q^{67} +(-62645.5 - 62645.5i) q^{68} +(35179.5 - 28623.3i) q^{69} +(62667.9 + 4846.51i) q^{70} +(-23570.0 - 7658.36i) q^{71} +(-69404.6 - 26272.7i) q^{72} +(736.683 - 4651.23i) q^{73} +140141. q^{74} +(-40988.1 - 26325.4i) q^{75} +116828. q^{76} +(-6768.89 + 42737.1i) q^{77} +(-19763.2 - 17711.9i) q^{78} +(-74898.3 - 24335.9i) q^{79} +(12545.6 + 51945.5i) q^{80} +(-57642.1 + 12813.1i) q^{81} +(40956.4 + 40956.4i) q^{82} +(31524.3 - 61869.9i) q^{83} +(-109831. - 29155.7i) q^{84} +(30045.8 + 72259.4i) q^{85} +(84464.6 + 116256. i) q^{86} +(-41214.9 - 71004.9i) q^{87} +(-113309. + 17946.4i) q^{88} +(-89769.7 + 65221.5i) q^{89} +(96731.2 + 90697.2i) q^{90} +(-16252.5 - 11808.1i) q^{91} +(164055. + 83590.2i) q^{92} +(83861.7 + 37104.5i) q^{93} +(55271.2 - 17958.7i) q^{94} +(-95394.7 - 39362.2i) q^{95} +(375.908 + 6867.50i) q^{96} +(59843.8 - 30491.9i) q^{97} +(34121.6 + 5404.32i) q^{98} +(-67552.1 + 61394.8i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.52703 + 9.64126i −0.269943 + 1.70435i 0.364353 + 0.931261i \(0.381290\pi\)
−0.634296 + 0.773090i \(0.718710\pi\)
\(3\) −10.4038 + 11.6087i −0.667403 + 0.744697i
\(4\) −60.1884 − 19.5564i −1.88089 − 0.611137i
\(5\) 42.5572 + 36.2475i 0.761287 + 0.648416i
\(6\) −96.0355 − 118.032i −1.08906 − 1.33851i
\(7\) −81.4490 − 81.4490i −0.628262 − 0.628262i 0.319369 − 0.947631i \(-0.396529\pi\)
−0.947631 + 0.319369i \(0.896529\pi\)
\(8\) 138.646 − 272.108i 0.765919 − 1.50320i
\(9\) −26.5228 − 241.548i −0.109147 − 0.994026i
\(10\) −414.458 + 354.954i −1.31063 + 1.12246i
\(11\) −220.802 − 303.908i −0.550201 − 0.757287i 0.439838 − 0.898077i \(-0.355036\pi\)
−0.990039 + 0.140790i \(0.955036\pi\)
\(12\) 853.210 − 495.247i 1.71042 − 0.992816i
\(13\) 172.258 − 27.2831i 0.282698 − 0.0447749i −0.0134754 − 0.999909i \(-0.504289\pi\)
0.296173 + 0.955134i \(0.404289\pi\)
\(14\) 909.646 − 660.897i 1.24037 − 0.901184i
\(15\) −863.542 + 116.922i −0.990958 + 0.134174i
\(16\) 773.377 + 561.891i 0.755251 + 0.548722i
\(17\) 1247.33 + 635.544i 1.04678 + 0.533364i 0.890800 − 0.454396i \(-0.150145\pi\)
0.155985 + 0.987759i \(0.450145\pi\)
\(18\) 2369.33 + 113.137i 1.72363 + 0.0823044i
\(19\) −1755.68 + 570.457i −1.11574 + 0.362525i −0.808140 − 0.588991i \(-0.799525\pi\)
−0.307599 + 0.951516i \(0.599525\pi\)
\(20\) −1852.58 − 3013.94i −1.03562 − 1.68485i
\(21\) 1792.89 − 98.1380i 0.887168 − 0.0485611i
\(22\) 3267.23 − 1664.74i 1.43920 − 0.733312i
\(23\) −2873.57 − 455.130i −1.13267 − 0.179397i −0.438171 − 0.898892i \(-0.644373\pi\)
−0.694498 + 0.719495i \(0.744373\pi\)
\(24\) 1716.38 + 4440.46i 0.608253 + 1.57362i
\(25\) 497.233 + 3085.19i 0.159115 + 0.987260i
\(26\) 1702.45i 0.493903i
\(27\) 3079.99 + 2205.12i 0.813093 + 0.582133i
\(28\) 3309.43 + 6495.13i 0.797735 + 1.56564i
\(29\) −1627.50 + 5008.93i −0.359357 + 1.10599i 0.594083 + 0.804404i \(0.297515\pi\)
−0.953440 + 0.301583i \(0.902485\pi\)
\(30\) 191.377 − 8504.18i 0.0388228 − 1.72516i
\(31\) −1817.88 − 5594.85i −0.339751 − 1.04565i −0.964334 − 0.264687i \(-0.914731\pi\)
0.624583 − 0.780958i \(-0.285269\pi\)
\(32\) 311.983 − 311.983i 0.0538587 − 0.0538587i
\(33\) 5825.14 + 598.571i 0.931155 + 0.0956821i
\(34\) −8032.15 + 11055.3i −1.19161 + 1.64011i
\(35\) −513.917 − 6418.57i −0.0709126 − 0.885662i
\(36\) −3127.44 + 15057.1i −0.402192 + 1.93635i
\(37\) −2245.86 − 14179.8i −0.269699 − 1.70281i −0.635485 − 0.772113i \(-0.719200\pi\)
0.365787 − 0.930699i \(-0.380800\pi\)
\(38\) −2818.95 − 17798.1i −0.316685 − 1.99947i
\(39\) −1475.42 + 2283.54i −0.155329 + 0.240407i
\(40\) 15763.7 − 6554.60i 1.55778 − 0.647733i
\(41\) 3487.72 − 4800.43i 0.324027 − 0.445985i −0.615664 − 0.788009i \(-0.711112\pi\)
0.939691 + 0.342023i \(0.111112\pi\)
\(42\) −1791.62 + 17435.6i −0.156719 + 1.52516i
\(43\) 10409.4 − 10409.4i 0.858531 − 0.858531i −0.132634 − 0.991165i \(-0.542344\pi\)
0.991165 + 0.132634i \(0.0423436\pi\)
\(44\) 7346.37 + 22609.8i 0.572060 + 1.76062i
\(45\) 7626.79 − 11241.0i 0.561449 − 0.827511i
\(46\) 8776.05 − 27009.9i 0.611511 − 1.88204i
\(47\) −2702.87 − 5304.68i −0.178476 − 0.350279i 0.784386 − 0.620273i \(-0.212978\pi\)
−0.962862 + 0.269994i \(0.912978\pi\)
\(48\) −14568.8 + 3132.09i −0.912688 + 0.196215i
\(49\) − 3539.12i − 0.210574i
\(50\) −30504.4 + 82.7917i −1.72559 + 0.00468341i
\(51\) −20354.7 + 7867.74i −1.09582 + 0.423569i
\(52\) −10901.5 − 1726.63i −0.559086 − 0.0885505i
\(53\) −30289.9 + 15433.5i −1.48118 + 0.754699i −0.993008 − 0.118049i \(-0.962336\pi\)
−0.488172 + 0.872748i \(0.662336\pi\)
\(54\) −25963.4 + 26327.8i −1.21165 + 1.22865i
\(55\) 1619.19 − 20937.0i 0.0721758 − 0.933271i
\(56\) −33455.6 + 10870.4i −1.42560 + 0.463206i
\(57\) 11643.5 − 26316.1i 0.474675 − 1.07284i
\(58\) −45807.2 − 23339.9i −1.78798 − 0.911023i
\(59\) 10579.3 + 7686.28i 0.395663 + 0.287466i 0.767772 − 0.640723i \(-0.221365\pi\)
−0.372109 + 0.928189i \(0.621365\pi\)
\(60\) 54261.7 + 9850.42i 1.94588 + 0.353246i
\(61\) −9231.57 + 6707.13i −0.317652 + 0.230787i −0.735173 − 0.677880i \(-0.762899\pi\)
0.417521 + 0.908667i \(0.362899\pi\)
\(62\) 56717.4 − 8983.16i 1.87386 − 0.296790i
\(63\) −17513.6 + 21834.1i −0.555935 + 0.693082i
\(64\) 20512.0 + 28232.4i 0.625977 + 0.861583i
\(65\) 8319.79 + 5082.85i 0.244247 + 0.149219i
\(66\) −14666.1 + 55247.7i −0.414434 + 1.56119i
\(67\) −10891.7 + 21376.1i −0.296420 + 0.581757i −0.990399 − 0.138238i \(-0.955856\pi\)
0.693979 + 0.719995i \(0.255856\pi\)
\(68\) −62645.5 − 62645.5i −1.64293 − 1.64293i
\(69\) 35179.5 − 28623.3i 0.889543 − 0.723765i
\(70\) 62667.9 + 4846.51i 1.52862 + 0.118218i
\(71\) −23570.0 − 7658.36i −0.554899 − 0.180298i 0.0181259 − 0.999836i \(-0.494230\pi\)
−0.573025 + 0.819538i \(0.694230\pi\)
\(72\) −69404.6 − 26272.7i −1.57782 − 0.597273i
\(73\) 736.683 − 4651.23i 0.0161798 − 0.102155i −0.978277 − 0.207303i \(-0.933531\pi\)
0.994457 + 0.105148i \(0.0335315\pi\)
\(74\) 140141. 2.97499
\(75\) −40988.1 − 26325.4i −0.841403 − 0.540408i
\(76\) 116828. 2.32013
\(77\) −6768.89 + 42737.1i −0.130104 + 0.821445i
\(78\) −19763.2 − 17711.9i −0.367808 − 0.329632i
\(79\) −74898.3 − 24335.9i −1.35022 − 0.438713i −0.457453 − 0.889234i \(-0.651238\pi\)
−0.892766 + 0.450521i \(0.851238\pi\)
\(80\) 12545.6 + 51945.5i 0.219162 + 0.907451i
\(81\) −57642.1 + 12813.1i −0.976174 + 0.216991i
\(82\) 40956.4 + 40956.4i 0.672647 + 0.672647i
\(83\) 31524.3 − 61869.9i 0.502285 − 0.985789i −0.491116 − 0.871094i \(-0.663411\pi\)
0.993401 − 0.114695i \(-0.0365891\pi\)
\(84\) −109831. − 29155.7i −1.69834 − 0.450843i
\(85\) 30045.8 + 72259.4i 0.451062 + 1.08479i
\(86\) 84464.6 + 116256.i 1.23148 + 1.69499i
\(87\) −41214.9 − 71004.9i −0.583789 − 1.00575i
\(88\) −113309. + 17946.4i −1.55976 + 0.247042i
\(89\) −89769.7 + 65221.5i −1.20131 + 0.872802i −0.994413 − 0.105560i \(-0.966336\pi\)
−0.206897 + 0.978363i \(0.566336\pi\)
\(90\) 96731.2 + 90697.2i 1.25881 + 1.18029i
\(91\) −16252.5 − 11808.1i −0.205739 − 0.149478i
\(92\) 164055. + 83590.2i 2.02078 + 1.02964i
\(93\) 83861.7 + 37104.5i 1.00544 + 0.444855i
\(94\) 55271.2 − 17958.7i 0.645177 − 0.209631i
\(95\) −95394.7 − 39362.2i −1.08446 − 0.447477i
\(96\) 375.908 + 6867.50i 0.00416297 + 0.0760538i
\(97\) 59843.8 − 30491.9i 0.645788 − 0.329045i −0.100237 − 0.994964i \(-0.531960\pi\)
0.746025 + 0.665918i \(0.231960\pi\)
\(98\) 34121.6 + 5404.32i 0.358892 + 0.0568429i
\(99\) −67552.1 + 61394.8i −0.692709 + 0.629570i
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.2.5 ✓ 384
3.2 odd 2 inner 75.6.l.a.2.44 yes 384
25.13 odd 20 inner 75.6.l.a.38.44 yes 384
75.38 even 20 inner 75.6.l.a.38.5 yes 384
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.6.l.a.2.5 ✓ 384 1.1 even 1 trivial
75.6.l.a.2.44 yes 384 3.2 odd 2 inner
75.6.l.a.38.5 yes 384 75.38 even 20 inner
75.6.l.a.38.44 yes 384 25.13 odd 20 inner