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Properties

Label 380.2.s.a.179.14
Level $380$
Weight $2$
Character 380.179
Analytic conductor $3.034$
Analytic rank $0$
Dimension $112$
Inner twists $8$

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Show commands: Magma / Pari/GP / SageMath

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 179.14
Character \(\chi\) \(=\) 380.179
Dual form 380.2.s.a.259.14

$q$-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+(-1.00970 - 0.990204i) q^{2} +(-1.08198 - 0.624679i) q^{3} +(0.0389922 + 1.99962i) q^{4} +(-0.0161872 + 2.23601i) q^{5} +(0.473912 + 1.70211i) q^{6} -0.151300 q^{7} +(1.94066 - 2.05763i) q^{8} +(-0.719553 - 1.24630i) q^{9} +(2.23045 - 2.24167i) q^{10} +2.04591i q^{11} +(1.20693 - 2.18790i) q^{12} +(-2.14755 - 3.71967i) q^{13} +(0.152768 + 0.149818i) q^{14} +(1.41430 - 2.40919i) q^{15} +(-3.99696 + 0.155939i) q^{16} +(-3.72170 - 2.14872i) q^{17} +(-0.507560 + 1.97090i) q^{18} +(0.185499 - 4.35495i) q^{19} +(-4.47180 + 0.0548188i) q^{20} +(0.163703 + 0.0945139i) q^{21} +(2.02587 - 2.06576i) q^{22} +(-1.14059 - 1.97555i) q^{23} +(-3.38510 + 1.01401i) q^{24} +(-4.99948 - 0.0723893i) q^{25} +(-1.51484 + 5.88226i) q^{26} +5.54603i q^{27} +(-0.00589952 - 0.302542i) q^{28} +(7.29773 - 4.21335i) q^{29} +(-3.81362 + 1.03212i) q^{30} -8.82532 q^{31} +(4.19015 + 3.80035i) q^{32} +(1.27804 - 2.21363i) q^{33} +(1.63013 + 5.85480i) q^{34} +(0.00244912 - 0.338308i) q^{35} +(2.46407 - 1.48743i) q^{36} -7.62939 q^{37} +(-4.49959 + 4.21352i) q^{38} +5.36611i q^{39} +(4.56946 + 4.37264i) q^{40} +(4.50780 + 2.60258i) q^{41} +(-0.0717029 - 0.257530i) q^{42} +(-3.03229 + 5.25208i) q^{43} +(-4.09105 + 0.0797747i) q^{44} +(2.79839 - 1.58875i) q^{45} +(-0.804550 + 3.12413i) q^{46} +(-1.85941 - 3.22060i) q^{47} +(4.42202 + 2.32809i) q^{48} -6.97711 q^{49} +(4.97630 + 5.02359i) q^{50} +(2.68452 + 4.64973i) q^{51} +(7.35418 - 4.43932i) q^{52} +(-2.82877 - 4.89958i) q^{53} +(5.49170 - 5.59983i) q^{54} +(-4.57468 - 0.0331175i) q^{55} +(-0.293622 + 0.311319i) q^{56} +(-2.92115 + 4.59607i) q^{57} +(-11.5406 - 2.97202i) q^{58} +(5.36468 - 9.29190i) q^{59} +(4.87262 + 2.73413i) q^{60} +(-1.82400 - 3.15926i) q^{61} +(8.91094 + 8.73887i) q^{62} +(0.108868 + 0.188566i) q^{63} +(-0.467669 - 7.98632i) q^{64} +(8.35197 - 4.74173i) q^{65} +(-3.48238 + 0.969583i) q^{66} +(12.1720 - 7.02749i) q^{67} +(4.15151 - 7.52576i) q^{68} +2.85000i q^{69} +(-0.337467 + 0.339165i) q^{70} +(-4.20876 + 7.28978i) q^{71} +(-3.96084 - 0.938078i) q^{72} +(-2.96398 - 1.71126i) q^{73} +(7.70340 + 7.55465i) q^{74} +(5.36409 + 3.20139i) q^{75} +(8.71548 + 0.201118i) q^{76} -0.309547i q^{77} +(5.31355 - 5.41817i) q^{78} +(4.22368 - 7.31563i) q^{79} +(-0.283982 - 8.93976i) q^{80} +(1.30583 - 2.26176i) q^{81} +(-1.97445 - 7.09147i) q^{82} -2.02124 q^{83} +(-0.182609 + 0.331029i) q^{84} +(4.86481 - 8.28696i) q^{85} +(8.26234 - 2.30045i) q^{86} -10.5280 q^{87} +(4.20973 + 3.97042i) q^{88} +(-6.49009 + 3.74705i) q^{89} +(-4.39873 - 1.16681i) q^{90} +(0.324924 + 0.562785i) q^{91} +(3.90588 - 2.35777i) q^{92} +(9.54878 + 5.51299i) q^{93} +(-1.31160 + 5.09304i) q^{94} +(9.73471 + 0.485271i) q^{95} +(-2.15963 - 6.72938i) q^{96} +(-4.45214 + 7.71132i) q^{97} +(7.04479 + 6.90876i) q^{98} +(2.54983 - 1.47214i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00970 − 0.990204i −0.713966 − 0.700180i
\(3\) −1.08198 − 0.624679i −0.624679 − 0.360658i 0.154010 − 0.988069i \(-0.450781\pi\)
−0.778688 + 0.627411i \(0.784115\pi\)
\(4\) 0.0389922 + 1.99962i 0.0194961 + 0.999810i
\(5\) −0.0161872 + 2.23601i −0.00723912 + 0.999974i
\(6\) 0.473912 + 1.70211i 0.193474 + 0.694885i
\(7\) −0.151300 −0.0571860 −0.0285930 − 0.999591i \(-0.509103\pi\)
−0.0285930 + 0.999591i \(0.509103\pi\)
\(8\) 1.94066 − 2.05763i 0.686127 − 0.727482i
\(9\) −0.719553 − 1.24630i −0.239851 − 0.415434i
\(10\) 2.23045 − 2.24167i 0.705330 − 0.708879i
\(11\) 2.04591i 0.616866i 0.951246 + 0.308433i \(0.0998046\pi\)
−0.951246 + 0.308433i \(0.900195\pi\)
\(12\) 1.20693 − 2.18790i 0.348411 − 0.631591i
\(13\) −2.14755 − 3.71967i −0.595623 − 1.03165i −0.993459 − 0.114193i \(-0.963572\pi\)
0.397836 − 0.917457i \(-0.369761\pi\)
\(14\) 0.152768 + 0.149818i 0.0408289 + 0.0400405i
\(15\) 1.41430 − 2.40919i 0.365171 − 0.622051i
\(16\) −3.99696 + 0.155939i −0.999240 + 0.0389848i
\(17\) −3.72170 − 2.14872i −0.902644 − 0.521142i −0.0245867 − 0.999698i \(-0.507827\pi\)
−0.878057 + 0.478556i \(0.841160\pi\)
\(18\) −0.507560 + 1.97090i −0.119633 + 0.464545i
\(19\) 0.185499 − 4.35495i 0.0425563 − 0.999094i
\(20\) −4.47180 + 0.0548188i −0.999925 + 0.0122579i
\(21\) 0.163703 + 0.0945139i 0.0357229 + 0.0206246i
\(22\) 2.02587 − 2.06576i 0.431917 − 0.440422i
\(23\) −1.14059 − 1.97555i −0.237829 − 0.411932i 0.722262 − 0.691619i \(-0.243102\pi\)
−0.960091 + 0.279688i \(0.909769\pi\)
\(24\) −3.38510 + 1.01401i −0.690981 + 0.206985i
\(25\) −4.99948 − 0.0723893i −0.999895 − 0.0144779i
\(26\) −1.51484 + 5.88226i −0.297085 + 1.15361i
\(27\) 5.54603i 1.06733i
\(28\) −0.00589952 − 0.302542i −0.00111490 − 0.0571752i
\(29\) 7.29773 − 4.21335i 1.35515 − 0.782399i 0.366188 − 0.930541i \(-0.380663\pi\)
0.988966 + 0.148142i \(0.0473292\pi\)
\(30\) −3.81362 + 1.03212i −0.696268 + 0.188438i
\(31\) −8.82532 −1.58507 −0.792537 − 0.609823i \(-0.791240\pi\)
−0.792537 + 0.609823i \(0.791240\pi\)
\(32\) 4.19015 + 3.80035i 0.740720 + 0.671814i
\(33\) 1.27804 − 2.21363i 0.222478 − 0.385343i
\(34\) 1.63013 + 5.85480i 0.279564 + 1.00409i
\(35\) 0.00244912 − 0.338308i 0.000413976 − 0.0571845i
\(36\) 2.46407 − 1.48743i 0.410679 − 0.247905i
\(37\) −7.62939 −1.25426 −0.627132 − 0.778913i \(-0.715771\pi\)
−0.627132 + 0.778913i \(0.715771\pi\)
\(38\) −4.49959 + 4.21352i −0.729929 + 0.683523i
\(39\) 5.36611i 0.859266i
\(40\) 4.56946 + 4.37264i 0.722495 + 0.691376i
\(41\) 4.50780 + 2.60258i 0.704001 + 0.406455i 0.808836 − 0.588035i \(-0.200098\pi\)
−0.104835 + 0.994490i \(0.533431\pi\)
\(42\) −0.0717029 − 0.257530i −0.0110640 − 0.0397377i
\(43\) −3.03229 + 5.25208i −0.462420 + 0.800935i −0.999081 − 0.0428629i \(-0.986352\pi\)
0.536661 + 0.843798i \(0.319685\pi\)
\(44\) −4.09105 + 0.0797747i −0.616749 + 0.0120265i
\(45\) 2.79839 − 1.58875i 0.417160 − 0.236837i
\(46\) −0.804550 + 3.12413i −0.118624 + 0.460628i
\(47\) −1.85941 − 3.22060i −0.271223 − 0.469773i 0.697952 − 0.716145i \(-0.254095\pi\)
−0.969175 + 0.246372i \(0.920762\pi\)
\(48\) 4.42202 + 2.32809i 0.638264 + 0.336031i
\(49\) −6.97711 −0.996730
\(50\) 4.97630 + 5.02359i 0.703754 + 0.710443i
\(51\) 2.68452 + 4.64973i 0.375908 + 0.651092i
\(52\) 7.35418 − 4.43932i 1.01984 − 0.615623i
\(53\) −2.82877 − 4.89958i −0.388562 − 0.673009i 0.603694 − 0.797216i \(-0.293695\pi\)
−0.992256 + 0.124207i \(0.960361\pi\)
\(54\) 5.49170 − 5.59983i 0.747326 − 0.762041i
\(55\) −4.57468 − 0.0331175i −0.616850 − 0.00446556i
\(56\) −0.293622 + 0.311319i −0.0392369 + 0.0416018i
\(57\) −2.92115 + 4.59607i −0.386916 + 0.608764i
\(58\) −11.5406 − 2.97202i −1.51535 − 0.390246i
\(59\) 5.36468 − 9.29190i 0.698422 − 1.20970i −0.270592 − 0.962694i \(-0.587219\pi\)
0.969013 − 0.247008i \(-0.0794474\pi\)
\(60\) 4.87262 + 2.73413i 0.629053 + 0.352974i
\(61\) −1.82400 − 3.15926i −0.233540 − 0.404502i 0.725308 − 0.688425i \(-0.241698\pi\)
−0.958847 + 0.283922i \(0.908364\pi\)
\(62\) 8.91094 + 8.73887i 1.13169 + 1.10984i
\(63\) 0.108868 + 0.188566i 0.0137161 + 0.0237570i
\(64\) −0.467669 − 7.98632i −0.0584587 − 0.998290i
\(65\) 8.35197 − 4.74173i 1.03593 − 0.588139i
\(66\) −3.48238 + 0.969583i −0.428651 + 0.119347i
\(67\) 12.1720 − 7.02749i 1.48704 − 0.858545i 0.487153 − 0.873317i \(-0.338036\pi\)
0.999891 + 0.0147718i \(0.00470218\pi\)
\(68\) 4.15151 − 7.52576i 0.503444 − 0.912632i
\(69\) 2.85000i 0.343100i
\(70\) −0.337467 + 0.339165i −0.0403350 + 0.0405380i
\(71\) −4.20876 + 7.28978i −0.499487 + 0.865137i −1.00000 0.000591848i \(-0.999812\pi\)
0.500512 + 0.865729i \(0.333145\pi\)
\(72\) −3.96084 − 0.938078i −0.466789 − 0.110554i
\(73\) −2.96398 − 1.71126i −0.346908 − 0.200287i 0.316415 − 0.948621i \(-0.397521\pi\)
−0.663323 + 0.748334i \(0.730854\pi\)
\(74\) 7.70340 + 7.55465i 0.895502 + 0.878210i
\(75\) 5.36409 + 3.20139i 0.619392 + 0.369665i
\(76\) 8.71548 + 0.201118i 0.999734 + 0.0230698i
\(77\) − 0.309547i − 0.0352761i
\(78\) 5.31355 − 5.41817i 0.601641 − 0.613487i
\(79\) 4.22368 − 7.31563i 0.475201 − 0.823072i −0.524395 − 0.851475i \(-0.675709\pi\)
0.999597 + 0.0284024i \(0.00904198\pi\)
\(80\) −0.283982 − 8.93976i −0.0317502 − 0.999496i
\(81\) 1.30583 − 2.26176i 0.145092 − 0.251306i
\(82\) −1.97445 − 7.09147i −0.218041 − 0.783122i
\(83\) −2.02124 −0.221860 −0.110930 − 0.993828i \(-0.535383\pi\)
−0.110930 + 0.993828i \(0.535383\pi\)
\(84\) −0.182609 + 0.331029i −0.0199242 + 0.0361182i
\(85\) 4.86481 − 8.28696i 0.527662 − 0.898847i
\(86\) 8.26234 − 2.30045i 0.890951 − 0.248064i
\(87\) −10.5280 −1.12871
\(88\) 4.20973 + 3.97042i 0.448759 + 0.423249i
\(89\) −6.49009 + 3.74705i −0.687948 + 0.397187i −0.802843 − 0.596191i \(-0.796680\pi\)
0.114895 + 0.993378i \(0.463347\pi\)
\(90\) −4.39873 − 1.16681i −0.463667 − 0.122993i
\(91\) 0.324924 + 0.562785i 0.0340613 + 0.0589959i
\(92\) 3.90588 − 2.35777i 0.407217 − 0.245815i
\(93\) 9.54878 + 5.51299i 0.990162 + 0.571671i
\(94\) −1.31160 + 5.09304i −0.135281 + 0.525307i
\(95\) 9.73471 + 0.485271i 0.998760 + 0.0497878i
\(96\) −2.15963 − 6.72938i −0.220417 − 0.686815i
\(97\) −4.45214 + 7.71132i −0.452046 + 0.782966i −0.998513 − 0.0545141i \(-0.982639\pi\)
0.546467 + 0.837481i \(0.315972\pi\)
\(98\) 7.04479 + 6.90876i 0.711632 + 0.697890i
\(99\) 2.54983 − 1.47214i 0.256267 − 0.147956i
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.179.14 yes 112
4.3 odd 2 inner 380.2.s.a.179.52 yes 112
5.4 even 2 inner 380.2.s.a.179.43 yes 112
19.12 odd 6 inner 380.2.s.a.259.5 yes 112
20.19 odd 2 inner 380.2.s.a.179.5 ✓ 112
76.31 even 6 inner 380.2.s.a.259.43 yes 112
95.69 odd 6 inner 380.2.s.a.259.52 yes 112
380.259 even 6 inner 380.2.s.a.259.14 yes 112
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.s.a.179.5 ✓ 112 20.19 odd 2 inner
380.2.s.a.179.14 yes 112 1.1 even 1 trivial
380.2.s.a.179.43 yes 112 5.4 even 2 inner
380.2.s.a.179.52 yes 112 4.3 odd 2 inner
380.2.s.a.259.5 yes 112 19.12 odd 6 inner
380.2.s.a.259.14 yes 112 380.259 even 6 inner
380.2.s.a.259.43 yes 112 76.31 even 6 inner
380.2.s.a.259.52 yes 112 95.69 odd 6 inner