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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,2,Mod(179,380)] 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("380.179"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.s (of order \(6\), 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: \(3.03431527681\)
Analytic rank: \(0\)
Dimension: \(112\)
Relative dimension: \(56\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 179.7
Character \(\chi\) \(=\) 380.179
Dual form 380.2.s.a.259.7

$q$-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+(-1.33018 + 0.480217i) q^{2} +(0.288375 + 0.166493i) q^{3} +(1.53878 - 1.27755i) q^{4} +(0.656216 - 2.13761i) q^{5} +(-0.463545 - 0.0829845i) q^{6} +2.90949 q^{7} +(-1.43336 + 2.43833i) q^{8} +(-1.44456 - 2.50205i) q^{9} +(0.153629 + 3.15854i) q^{10} -0.747872i q^{11} +(0.656451 - 0.112218i) q^{12} +(-0.956737 - 1.65712i) q^{13} +(-3.87016 + 1.39719i) q^{14} +(0.545134 - 0.507178i) q^{15} +(0.735709 - 3.93176i) q^{16} +(-3.29042 - 1.89973i) q^{17} +(3.12306 + 2.63449i) q^{18} +(0.569625 + 4.32152i) q^{19} +(-1.72114 - 4.12767i) q^{20} +(0.839024 + 0.484411i) q^{21} +(0.359141 + 0.994808i) q^{22} +(-2.45679 - 4.25529i) q^{23} +(-0.819313 + 0.464509i) q^{24} +(-4.13876 - 2.80547i) q^{25} +(2.06841 + 1.74483i) q^{26} -1.96100i q^{27} +(4.47707 - 3.71703i) q^{28} +(1.28167 - 0.739974i) q^{29} +(-0.481574 + 0.936423i) q^{30} +4.74438 q^{31} +(0.909468 + 5.58327i) q^{32} +(0.124516 - 0.215668i) q^{33} +(5.28915 + 0.946870i) q^{34} +(1.90925 - 6.21936i) q^{35} +(-5.41937 - 2.00461i) q^{36} +8.24780 q^{37} +(-2.83297 - 5.47488i) q^{38} -0.637162i q^{39} +(4.27161 + 4.66405i) q^{40} +(6.82049 + 3.93781i) q^{41} +(-1.34868 - 0.241442i) q^{42} +(0.852684 - 1.47689i) q^{43} +(-0.955448 - 1.15081i) q^{44} +(-6.29635 + 1.44602i) q^{45} +(5.31145 + 4.48053i) q^{46} +(4.43177 + 7.67605i) q^{47} +(0.866772 - 1.01133i) q^{48} +1.46513 q^{49} +(6.85255 + 1.74429i) q^{50} +(-0.632584 - 1.09567i) q^{51} +(-3.58927 - 1.32766i) q^{52} +(-6.43603 - 11.1475i) q^{53} +(0.941705 + 2.60849i) q^{54} +(-1.59866 - 0.490766i) q^{55} +(-4.17036 + 7.09430i) q^{56} +(-0.555239 + 1.34106i) q^{57} +(-1.34951 + 1.59978i) q^{58} +(5.50250 - 9.53060i) q^{59} +(0.190896 - 1.47688i) q^{60} +(3.23842 + 5.60910i) q^{61} +(-6.31090 + 2.27833i) q^{62} +(-4.20293 - 7.27969i) q^{63} +(-3.89094 - 6.99004i) q^{64} +(-4.17010 + 0.957705i) q^{65} +(-0.0620618 + 0.346673i) q^{66} +(2.15895 - 1.24647i) q^{67} +(-7.49025 + 1.28043i) q^{68} -1.63616i q^{69} +(0.446981 + 9.18975i) q^{70} +(-6.85048 + 11.8654i) q^{71} +(8.17141 + 0.0640297i) q^{72} +(2.13301 + 1.23149i) q^{73} +(-10.9711 + 3.96073i) q^{74} +(-0.726424 - 1.49810i) q^{75} +(6.39750 + 5.92216i) q^{76} -2.17593i q^{77} +(0.305976 + 0.847543i) q^{78} +(-2.97049 + 5.14504i) q^{79} +(-7.92179 - 4.15274i) q^{80} +(-4.00719 + 6.94065i) q^{81} +(-10.9635 - 1.96270i) q^{82} +5.90436 q^{83} +(1.90994 - 0.326496i) q^{84} +(-6.22010 + 5.78701i) q^{85} +(-0.424999 + 2.37401i) q^{86} +0.492803 q^{87} +(1.82356 + 1.07197i) q^{88} +(-11.3027 + 6.52563i) q^{89} +(7.68091 - 4.94709i) q^{90} +(-2.78362 - 4.82137i) q^{91} +(-9.21684 - 3.40928i) q^{92} +(1.36816 + 0.789907i) q^{93} +(-9.58124 - 8.08235i) q^{94} +(9.61152 + 1.61821i) q^{95} +(-0.667309 + 1.76150i) q^{96} +(0.986369 - 1.70844i) q^{97} +(-1.94889 + 0.703579i) q^{98} +(-1.87121 + 1.08035i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 112 q - 2 q^{5} - 8 q^{6} + 44 q^{9} + 6 q^{10} - 36 q^{14} - 4 q^{16} + 44 q^{20} - 48 q^{21} + 2 q^{24} - 2 q^{25} - 36 q^{26} - 12 q^{29} - 32 q^{30} - 30 q^{34} + 20 q^{36} - 24 q^{40} - 24 q^{41} - 14 q^{44}+ \cdots - 84 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.33018 + 0.480217i −0.940583 + 0.339565i
\(3\) 0.288375 + 0.166493i 0.166493 + 0.0961250i 0.580931 − 0.813953i \(-0.302688\pi\)
−0.414438 + 0.910078i \(0.636022\pi\)
\(4\) 1.53878 − 1.27755i 0.769392 − 0.638777i
\(5\) 0.656216 − 2.13761i 0.293469 − 0.955969i
\(6\) −0.463545 − 0.0829845i −0.189241 − 0.0338783i
\(7\) 2.90949 1.09968 0.549842 − 0.835269i \(-0.314688\pi\)
0.549842 + 0.835269i \(0.314688\pi\)
\(8\) −1.43336 + 2.43833i −0.506770 + 0.862081i
\(9\) −1.44456 − 2.50205i −0.481520 − 0.834017i
\(10\) 0.153629 + 3.15854i 0.0485816 + 0.998819i
\(11\) − 0.747872i − 0.225492i −0.993624 − 0.112746i \(-0.964035\pi\)
0.993624 − 0.112746i \(-0.0359646\pi\)
\(12\) 0.656451 − 0.112218i 0.189501 − 0.0323944i
\(13\) −0.956737 − 1.65712i −0.265351 − 0.459602i 0.702304 − 0.711877i \(-0.252155\pi\)
−0.967656 + 0.252275i \(0.918821\pi\)
\(14\) −3.87016 + 1.39719i −1.03434 + 0.373414i
\(15\) 0.545134 − 0.507178i 0.140753 − 0.130953i
\(16\) 0.735709 − 3.93176i 0.183927 − 0.982940i
\(17\) −3.29042 − 1.89973i −0.798044 − 0.460751i 0.0447425 − 0.998999i \(-0.485753\pi\)
−0.842787 + 0.538247i \(0.819087\pi\)
\(18\) 3.12306 + 2.63449i 0.736112 + 0.620955i
\(19\) 0.569625 + 4.32152i 0.130681 + 0.991424i
\(20\) −1.72114 − 4.12767i −0.384859 − 0.922975i
\(21\) 0.839024 + 0.484411i 0.183090 + 0.105707i
\(22\) 0.359141 + 0.994808i 0.0765691 + 0.212094i
\(23\) −2.45679 − 4.25529i −0.512277 − 0.887290i −0.999899 − 0.0142346i \(-0.995469\pi\)
0.487622 − 0.873055i \(-0.337864\pi\)
\(24\) −0.819313 + 0.464509i −0.167242 + 0.0948175i
\(25\) −4.13876 − 2.80547i −0.827752 − 0.561094i
\(26\) 2.06841 + 1.74483i 0.405649 + 0.342190i
\(27\) − 1.96100i − 0.377395i
\(28\) 4.47707 − 3.71703i 0.846087 − 0.702453i
\(29\) 1.28167 − 0.739974i 0.238001 − 0.137410i −0.376257 − 0.926515i \(-0.622789\pi\)
0.614258 + 0.789106i \(0.289456\pi\)
\(30\) −0.481574 + 0.936423i −0.0879230 + 0.170967i
\(31\) 4.74438 0.852115 0.426057 − 0.904696i \(-0.359902\pi\)
0.426057 + 0.904696i \(0.359902\pi\)
\(32\) 0.909468 + 5.58327i 0.160773 + 0.986991i
\(33\) 0.124516 − 0.215668i 0.0216754 − 0.0375429i
\(34\) 5.28915 + 0.946870i 0.907082 + 0.162387i
\(35\) 1.90925 − 6.21936i 0.322723 − 1.05126i
\(36\) −5.41937 − 2.00461i −0.903229 − 0.334102i
\(37\) 8.24780 1.35593 0.677965 − 0.735094i \(-0.262862\pi\)
0.677965 + 0.735094i \(0.262862\pi\)
\(38\) −2.83297 − 5.47488i −0.459569 − 0.888142i
\(39\) − 0.637162i − 0.102028i
\(40\) 4.27161 + 4.66405i 0.675401 + 0.737450i
\(41\) 6.82049 + 3.93781i 1.06518 + 0.614983i 0.926861 − 0.375405i \(-0.122496\pi\)
0.138320 + 0.990388i \(0.455830\pi\)
\(42\) −1.34868 − 0.241442i −0.208106 − 0.0372554i
\(43\) 0.852684 − 1.47689i 0.130033 − 0.225224i −0.793656 − 0.608367i \(-0.791825\pi\)
0.923689 + 0.383143i \(0.125158\pi\)
\(44\) −0.955448 − 1.15081i −0.144039 − 0.173492i
\(45\) −6.29635 + 1.44602i −0.938605 + 0.215560i
\(46\) 5.31145 + 4.48053i 0.783131 + 0.660618i
\(47\) 4.43177 + 7.67605i 0.646440 + 1.11967i 0.983967 + 0.178351i \(0.0570762\pi\)
−0.337527 + 0.941316i \(0.609590\pi\)
\(48\) 0.866772 − 1.01133i 0.125108 − 0.145973i
\(49\) 1.46513 0.209304
\(50\) 6.85255 + 1.74429i 0.969097 + 0.246680i
\(51\) −0.632584 − 1.09567i −0.0885794 − 0.153424i
\(52\) −3.58927 − 1.32766i −0.497742 − 0.184113i
\(53\) −6.43603 − 11.1475i −0.884057 − 1.53123i −0.846791 − 0.531926i \(-0.821468\pi\)
−0.0372664 − 0.999305i \(-0.511865\pi\)
\(54\) 0.941705 + 2.60849i 0.128150 + 0.354971i
\(55\) −1.59866 − 0.490766i −0.215563 − 0.0661748i
\(56\) −4.17036 + 7.09430i −0.557287 + 0.948016i
\(57\) −0.555239 + 1.34106i −0.0735432 + 0.177627i
\(58\) −1.34951 + 1.59978i −0.177200 + 0.210062i
\(59\) 5.50250 − 9.53060i 0.716364 − 1.24078i −0.246067 − 0.969253i \(-0.579138\pi\)
0.962431 − 0.271526i \(-0.0875283\pi\)
\(60\) 0.190896 − 1.47688i 0.0246446 − 0.190664i
\(61\) 3.23842 + 5.60910i 0.414637 + 0.718172i 0.995390 − 0.0959077i \(-0.0305754\pi\)
−0.580754 + 0.814079i \(0.697242\pi\)
\(62\) −6.31090 + 2.27833i −0.801485 + 0.289348i
\(63\) −4.20293 − 7.27969i −0.529520 − 0.917155i
\(64\) −3.89094 − 6.99004i −0.486367 − 0.873754i
\(65\) −4.17010 + 0.957705i −0.517237 + 0.118789i
\(66\) −0.0620618 + 0.346673i −0.00763928 + 0.0426724i
\(67\) 2.15895 − 1.24647i 0.263758 − 0.152281i −0.362290 − 0.932065i \(-0.618005\pi\)
0.626047 + 0.779785i \(0.284672\pi\)
\(68\) −7.49025 + 1.28043i −0.908326 + 0.155274i
\(69\) − 1.63616i − 0.196970i
\(70\) 0.446981 + 9.18975i 0.0534244 + 1.09839i
\(71\) −6.85048 + 11.8654i −0.813002 + 1.40816i 0.0977513 + 0.995211i \(0.468835\pi\)
−0.910754 + 0.412950i \(0.864498\pi\)
\(72\) 8.17141 + 0.0640297i 0.963010 + 0.00754598i
\(73\) 2.13301 + 1.23149i 0.249650 + 0.144136i 0.619604 − 0.784915i \(-0.287293\pi\)
−0.369954 + 0.929050i \(0.620627\pi\)
\(74\) −10.9711 + 3.96073i −1.27536 + 0.460426i
\(75\) −0.726424 − 1.49810i −0.0838802 − 0.172986i
\(76\) 6.39750 + 5.92216i 0.733844 + 0.679318i
\(77\) − 2.17593i − 0.247970i
\(78\) 0.305976 + 0.847543i 0.0346449 + 0.0959653i
\(79\) −2.97049 + 5.14504i −0.334206 + 0.578863i −0.983332 − 0.181819i \(-0.941802\pi\)
0.649126 + 0.760681i \(0.275135\pi\)
\(80\) −7.92179 − 4.15274i −0.885683 − 0.464291i
\(81\) −4.00719 + 6.94065i −0.445243 + 0.771183i
\(82\) −10.9635 − 1.96270i −1.21072 − 0.216744i
\(83\) 5.90436 0.648088 0.324044 − 0.946042i \(-0.394957\pi\)
0.324044 + 0.946042i \(0.394957\pi\)
\(84\) 1.90994 − 0.326496i 0.208391 − 0.0356236i
\(85\) −6.22010 + 5.78701i −0.674665 + 0.627690i
\(86\) −0.424999 + 2.37401i −0.0458288 + 0.255996i
\(87\) 0.492803 0.0528341
\(88\) 1.82356 + 1.07197i 0.194392 + 0.114273i
\(89\) −11.3027 + 6.52563i −1.19809 + 0.691716i −0.960128 − 0.279560i \(-0.909811\pi\)
−0.237958 + 0.971275i \(0.576478\pi\)
\(90\) 7.68091 − 4.94709i 0.809639 − 0.521469i
\(91\) −2.78362 − 4.82137i −0.291802 − 0.505416i
\(92\) −9.21684 − 3.40928i −0.960922 − 0.355442i
\(93\) 1.36816 + 0.789907i 0.141872 + 0.0819096i
\(94\) −9.58124 − 8.08235i −0.988229 − 0.833631i
\(95\) 9.61152 + 1.61821i 0.986122 + 0.166025i
\(96\) −0.667309 + 1.76150i −0.0681070 + 0.179782i
\(97\) 0.986369 − 1.70844i 0.100151 − 0.173466i −0.811596 − 0.584219i \(-0.801401\pi\)
0.911747 + 0.410753i \(0.134734\pi\)
\(98\) −1.94889 + 0.703579i −0.196868 + 0.0710722i
\(99\) −1.87121 + 1.08035i −0.188064 + 0.108579i
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 380.2.s.a.179.7 ✓ 112
4.3 odd 2 inner 380.2.s.a.179.32 yes 112
5.4 even 2 inner 380.2.s.a.179.50 yes 112
19.12 odd 6 inner 380.2.s.a.259.25 yes 112
20.19 odd 2 inner 380.2.s.a.179.25 yes 112
76.31 even 6 inner 380.2.s.a.259.50 yes 112
95.69 odd 6 inner 380.2.s.a.259.32 yes 112
380.259 even 6 inner 380.2.s.a.259.7 yes 112
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.s.a.179.7 ✓ 112 1.1 even 1 trivial
380.2.s.a.179.25 yes 112 20.19 odd 2 inner
380.2.s.a.179.32 yes 112 4.3 odd 2 inner
380.2.s.a.179.50 yes 112 5.4 even 2 inner
380.2.s.a.259.7 yes 112 380.259 even 6 inner
380.2.s.a.259.25 yes 112 19.12 odd 6 inner
380.2.s.a.259.32 yes 112 95.69 odd 6 inner
380.2.s.a.259.50 yes 112 76.31 even 6 inner