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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,3,Mod(227,380)] 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("380.227"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 2])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 380.j (of order \(4\), degree \(2\), 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: \(10.3542500457\)
Analytic rank: \(0\)
Dimension: \(232\)
Relative dimension: \(116\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 227.18
Character \(\chi\) \(=\) 380.227
Dual form 380.3.j.a.303.18

$q$-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+(-1.74987 + 0.968481i) q^{2} +(1.12028 - 1.12028i) q^{3} +(2.12409 - 3.38943i) q^{4} +(3.50045 - 3.57027i) q^{5} +(-0.875371 + 3.04530i) q^{6} +(8.74065 - 8.74065i) q^{7} +(-0.434280 + 7.98820i) q^{8} +6.48997i q^{9} +(-2.66760 + 9.63763i) q^{10} +15.3458i q^{11} +(-1.41753 - 6.17666i) q^{12} +(16.4027 + 16.4027i) q^{13} +(-6.82985 + 23.7602i) q^{14} +(-0.0782120 - 7.92115i) q^{15} +(-6.97649 - 14.3989i) q^{16} +(-7.11928 - 7.11928i) q^{17} +(-6.28541 - 11.3566i) q^{18} +(15.4584 - 11.0471i) q^{19} +(-4.66591 - 19.4481i) q^{20} -19.5839i q^{21} +(-14.8621 - 26.8531i) q^{22} +(10.8658 + 10.8658i) q^{23} +(8.46247 + 9.43550i) q^{24} +(-0.493643 - 24.9951i) q^{25} +(-44.5884 - 12.8169i) q^{26} +(17.3530 + 17.3530i) q^{27} +(-11.0599 - 48.1918i) q^{28} -37.4205 q^{29} +(7.80835 + 13.7852i) q^{30} +27.7673 q^{31} +(26.1530 + 18.4396i) q^{32} +(17.1915 + 17.1915i) q^{33} +(19.3527 + 5.56292i) q^{34} +(-0.610229 - 61.8027i) q^{35} +(21.9973 + 13.7853i) q^{36} +(-34.5457 + 34.5457i) q^{37} +(-16.3513 + 34.3021i) q^{38} +36.7512 q^{39} +(26.9999 + 29.5128i) q^{40} -53.3095i q^{41} +(18.9666 + 34.2692i) q^{42} +(-4.74516 - 4.74516i) q^{43} +(52.0135 + 32.5958i) q^{44} +(23.1709 + 22.7178i) q^{45} +(-29.5371 - 8.49043i) q^{46} +(-19.9047 + 19.9047i) q^{47} +(-23.9463 - 8.31515i) q^{48} -103.798i q^{49} +(25.0711 + 43.2601i) q^{50} -15.9511 q^{51} +(90.4369 - 20.7551i) q^{52} +(-25.1831 - 25.1831i) q^{53} +(-47.1716 - 13.5595i) q^{54} +(54.7886 + 53.7172i) q^{55} +(66.0262 + 73.6180i) q^{56} +(4.94187 - 29.6934i) q^{57} +(65.4810 - 36.2411i) q^{58} -24.6606i q^{59} +(-27.0143 - 16.5601i) q^{60} +23.0007 q^{61} +(-48.5892 + 26.8921i) q^{62} +(56.7266 + 56.7266i) q^{63} +(-63.6228 - 6.93823i) q^{64} +(115.979 - 1.14516i) q^{65} +(-46.7325 - 13.4332i) q^{66} +(-14.7953 - 14.7953i) q^{67} +(-39.2523 + 9.00832i) q^{68} +24.3454 q^{69} +(60.9226 + 107.556i) q^{70} +22.0799 q^{71} +(-51.8432 - 2.81846i) q^{72} +(-83.5805 + 83.5805i) q^{73} +(26.9936 - 93.9073i) q^{74} +(-28.5544 - 27.4484i) q^{75} +(-4.60836 - 75.8602i) q^{76} +(134.132 + 134.132i) q^{77} +(-64.3098 + 35.5928i) q^{78} -91.5393i q^{79} +(-75.8289 - 25.4948i) q^{80} -19.5294 q^{81} +(51.6292 + 93.2846i) q^{82} +(22.4132 + 22.4132i) q^{83} +(-66.3782 - 41.5979i) q^{84} +(-50.3384 + 0.497032i) q^{85} +(12.8990 + 3.70782i) q^{86} +(-41.9213 + 41.9213i) q^{87} +(-122.585 - 6.66436i) q^{88} -101.844 q^{89} +(-62.5479 - 17.3126i) q^{90} +286.741 q^{91} +(59.9089 - 13.7490i) q^{92} +(31.1070 - 31.1070i) q^{93} +(15.5533 - 54.1081i) q^{94} +(14.6703 - 93.8604i) q^{95} +(49.9560 - 8.64113i) q^{96} +(50.3855 - 50.3855i) q^{97} +(100.526 + 181.633i) q^{98} -99.5936 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 232 q - 8 q^{5} - 32 q^{6} - 64 q^{16} - 8 q^{17} + 80 q^{20} - 8 q^{25} + 40 q^{26} + 40 q^{28} - 104 q^{30} + 104 q^{36} + 184 q^{38} + 192 q^{42} + 192 q^{45} + 280 q^{58} + 112 q^{61} - 280 q^{62} + 216 q^{66}+ \cdots + 504 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{4}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.74987 + 0.968481i −0.874935 + 0.484240i
\(3\) 1.12028 − 1.12028i 0.373425 − 0.373425i −0.495298 − 0.868723i \(-0.664941\pi\)
0.868723 + 0.495298i \(0.164941\pi\)
\(4\) 2.12409 − 3.38943i 0.531022 − 0.847358i
\(5\) 3.50045 − 3.57027i 0.700091 − 0.714054i
\(6\) −0.875371 + 3.04530i −0.145895 + 0.507550i
\(7\) 8.74065 − 8.74065i 1.24866 − 1.24866i 0.292355 − 0.956310i \(-0.405561\pi\)
0.956310 − 0.292355i \(-0.0944389\pi\)
\(8\) −0.434280 + 7.98820i −0.0542850 + 0.998525i
\(9\) 6.48997i 0.721108i
\(10\) −2.66760 + 9.63763i −0.266760 + 0.963763i
\(11\) 15.3458i 1.39507i 0.716550 + 0.697535i \(0.245720\pi\)
−0.716550 + 0.697535i \(0.754280\pi\)
\(12\) −1.41753 − 6.17666i −0.118128 − 0.514722i
\(13\) 16.4027 + 16.4027i 1.26175 + 1.26175i 0.950245 + 0.311505i \(0.100833\pi\)
0.311505 + 0.950245i \(0.399167\pi\)
\(14\) −6.82985 + 23.7602i −0.487846 + 1.69715i
\(15\) −0.0782120 − 7.92115i −0.00521413 − 0.528077i
\(16\) −6.97649 − 14.3989i −0.436031 − 0.899932i
\(17\) −7.11928 − 7.11928i −0.418781 − 0.418781i 0.466002 − 0.884783i \(-0.345694\pi\)
−0.884783 + 0.466002i \(0.845694\pi\)
\(18\) −6.28541 − 11.3566i −0.349189 − 0.630922i
\(19\) 15.4584 − 11.0471i 0.813599 − 0.581426i
\(20\) −4.66591 − 19.4481i −0.233295 − 0.972406i
\(21\) − 19.5839i − 0.932565i
\(22\) −14.8621 − 26.8531i −0.675550 − 1.22060i
\(23\) 10.8658 + 10.8658i 0.472427 + 0.472427i 0.902699 − 0.430272i \(-0.141582\pi\)
−0.430272 + 0.902699i \(0.641582\pi\)
\(24\) 8.46247 + 9.43550i 0.352603 + 0.393146i
\(25\) −0.493643 − 24.9951i −0.0197457 − 0.999805i
\(26\) −44.5884 − 12.8169i −1.71494 − 0.492958i
\(27\) 17.3530 + 17.3530i 0.642705 + 0.642705i
\(28\) −11.0599 − 48.1918i −0.394997 − 1.72113i
\(29\) −37.4205 −1.29036 −0.645181 − 0.764030i \(-0.723218\pi\)
−0.645181 + 0.764030i \(0.723218\pi\)
\(30\) 7.80835 + 13.7852i 0.260278 + 0.459508i
\(31\) 27.7673 0.895720 0.447860 − 0.894104i \(-0.352186\pi\)
0.447860 + 0.894104i \(0.352186\pi\)
\(32\) 26.1530 + 18.4396i 0.817282 + 0.576238i
\(33\) 17.1915 + 17.1915i 0.520954 + 0.520954i
\(34\) 19.3527 + 5.56292i 0.569197 + 0.163615i
\(35\) −0.610229 − 61.8027i −0.0174351 − 1.76579i
\(36\) 21.9973 + 13.7853i 0.611036 + 0.382924i
\(37\) −34.5457 + 34.5457i −0.933667 + 0.933667i −0.997933 − 0.0642659i \(-0.979529\pi\)
0.0642659 + 0.997933i \(0.479529\pi\)
\(38\) −16.3513 + 34.3021i −0.430297 + 0.902688i
\(39\) 36.7512 0.942338
\(40\) 26.9999 + 29.5128i 0.674997 + 0.737821i
\(41\) − 53.3095i − 1.30023i −0.759835 − 0.650115i \(-0.774721\pi\)
0.759835 − 0.650115i \(-0.225279\pi\)
\(42\) 18.9666 + 34.2692i 0.451586 + 0.815934i
\(43\) −4.74516 − 4.74516i −0.110353 − 0.110353i 0.649774 − 0.760127i \(-0.274863\pi\)
−0.760127 + 0.649774i \(0.774863\pi\)
\(44\) 52.0135 + 32.5958i 1.18212 + 0.740814i
\(45\) 23.1709 + 22.7178i 0.514910 + 0.504841i
\(46\) −29.5371 − 8.49043i −0.642111 − 0.184575i
\(47\) −19.9047 + 19.9047i −0.423505 + 0.423505i −0.886409 − 0.462904i \(-0.846808\pi\)
0.462904 + 0.886409i \(0.346808\pi\)
\(48\) −23.9463 − 8.31515i −0.498882 − 0.173232i
\(49\) − 103.798i − 2.11833i
\(50\) 25.0711 + 43.2601i 0.501422 + 0.865203i
\(51\) −15.9511 −0.312767
\(52\) 90.4369 − 20.7551i 1.73917 − 0.399136i
\(53\) −25.1831 − 25.1831i −0.475153 − 0.475153i 0.428424 − 0.903578i \(-0.359069\pi\)
−0.903578 + 0.428424i \(0.859069\pi\)
\(54\) −47.1716 − 13.5595i −0.873548 − 0.251101i
\(55\) 54.7886 + 53.7172i 0.996156 + 0.976676i
\(56\) 66.0262 + 73.6180i 1.17904 + 1.31461i
\(57\) 4.94187 − 29.6934i 0.0866994 − 0.520937i
\(58\) 65.4810 − 36.2411i 1.12898 − 0.624846i
\(59\) − 24.6606i − 0.417976i −0.977918 − 0.208988i \(-0.932983\pi\)
0.977918 − 0.208988i \(-0.0670170\pi\)
\(60\) −27.0143 − 16.5601i −0.450239 − 0.276002i
\(61\) 23.0007 0.377061 0.188531 − 0.982067i \(-0.439627\pi\)
0.188531 + 0.982067i \(0.439627\pi\)
\(62\) −48.5892 + 26.8921i −0.783697 + 0.433744i
\(63\) 56.7266 + 56.7266i 0.900422 + 0.900422i
\(64\) −63.6228 − 6.93823i −0.994106 − 0.108410i
\(65\) 115.979 − 1.14516i 1.78430 − 0.0176178i
\(66\) −46.7325 − 13.4332i −0.708068 − 0.203534i
\(67\) −14.7953 − 14.7953i −0.220825 − 0.220825i 0.588021 − 0.808846i \(-0.299907\pi\)
−0.808846 + 0.588021i \(0.799907\pi\)
\(68\) −39.2523 + 9.00832i −0.577239 + 0.132475i
\(69\) 24.3454 0.352832
\(70\) 60.9226 + 107.556i 0.870323 + 1.53651i
\(71\) 22.0799 0.310984 0.155492 − 0.987837i \(-0.450304\pi\)
0.155492 + 0.987837i \(0.450304\pi\)
\(72\) −51.8432 − 2.81846i −0.720044 − 0.0391453i
\(73\) −83.5805 + 83.5805i −1.14494 + 1.14494i −0.157404 + 0.987534i \(0.550313\pi\)
−0.987534 + 0.157404i \(0.949687\pi\)
\(74\) 26.9936 − 93.9073i 0.364778 − 1.26902i
\(75\) −28.5544 − 27.4484i −0.380726 − 0.365979i
\(76\) −4.60836 − 75.8602i −0.0606363 − 0.998160i
\(77\) 134.132 + 134.132i 1.74198 + 1.74198i
\(78\) −64.3098 + 35.5928i −0.824484 + 0.456318i
\(79\) − 91.5393i − 1.15873i −0.815070 − 0.579363i \(-0.803302\pi\)
0.815070 − 0.579363i \(-0.196698\pi\)
\(80\) −75.8289 − 25.4948i −0.947861 − 0.318685i
\(81\) −19.5294 −0.241104
\(82\) 51.6292 + 93.2846i 0.629624 + 1.13762i
\(83\) 22.4132 + 22.4132i 0.270039 + 0.270039i 0.829116 − 0.559077i \(-0.188844\pi\)
−0.559077 + 0.829116i \(0.688844\pi\)
\(84\) −66.3782 − 41.5979i −0.790217 − 0.495213i
\(85\) −50.3384 + 0.497032i −0.592217 + 0.00584744i
\(86\) 12.8990 + 3.70782i 0.149988 + 0.0431141i
\(87\) −41.9213 + 41.9213i −0.481854 + 0.481854i
\(88\) −122.585 − 6.66436i −1.39301 − 0.0757314i
\(89\) −101.844 −1.14432 −0.572160 − 0.820142i \(-0.693894\pi\)
−0.572160 + 0.820142i \(0.693894\pi\)
\(90\) −62.5479 − 17.3126i −0.694977 − 0.192363i
\(91\) 286.741 3.15100
\(92\) 59.9089 − 13.7490i 0.651184 − 0.149445i
\(93\) 31.1070 − 31.1070i 0.334484 − 0.334484i
\(94\) 15.5533 − 54.1081i 0.165461 − 0.575618i
\(95\) 14.6703 − 93.8604i 0.154424 − 0.988005i
\(96\) 49.9560 − 8.64113i 0.520375 − 0.0900118i
\(97\) 50.3855 − 50.3855i 0.519438 − 0.519438i −0.397963 − 0.917401i \(-0.630283\pi\)
0.917401 + 0.397963i \(0.130283\pi\)
\(98\) 100.526 + 181.633i 1.02578 + 1.85340i
\(99\) −99.5936 −1.00600
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 380.3.j.a.227.18 ✓ 232
4.3 odd 2 inner 380.3.j.a.227.41 yes 232
5.3 odd 4 inner 380.3.j.a.303.76 yes 232
19.18 odd 2 inner 380.3.j.a.227.99 yes 232
20.3 even 4 inner 380.3.j.a.303.99 yes 232
76.75 even 2 inner 380.3.j.a.227.76 yes 232
95.18 even 4 inner 380.3.j.a.303.41 yes 232
380.303 odd 4 inner 380.3.j.a.303.18 yes 232
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.3.j.a.227.18 ✓ 232 1.1 even 1 trivial
380.3.j.a.227.41 yes 232 4.3 odd 2 inner
380.3.j.a.227.76 yes 232 76.75 even 2 inner
380.3.j.a.227.99 yes 232 19.18 odd 2 inner
380.3.j.a.303.18 yes 232 380.303 odd 4 inner
380.3.j.a.303.41 yes 232 95.18 even 4 inner
380.3.j.a.303.76 yes 232 5.3 odd 4 inner
380.3.j.a.303.99 yes 232 20.3 even 4 inner