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(7,380)] 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("380.7"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 3, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.v (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.03431527681\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 7.1
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 380.7
Dual form 380.2.v.b.163.1

$q$-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+(1.36603 - 0.366025i) q^{2} +(-0.633975 + 2.36603i) q^{3} +(1.73205 - 1.00000i) q^{4} +(-2.23205 + 0.133975i) q^{5} +3.46410i q^{6} +(-3.46410 + 3.46410i) q^{7} +(2.00000 - 2.00000i) q^{8} +(-2.59808 - 1.50000i) q^{9} +(-3.00000 + 1.00000i) q^{10} +1.73205i q^{11} +(1.26795 + 4.73205i) q^{12} +(-0.366025 - 1.36603i) q^{13} +(-3.46410 + 6.00000i) q^{14} +(1.09808 - 5.36603i) q^{15} +(2.00000 - 3.46410i) q^{16} +(-1.09808 + 4.09808i) q^{17} +(-4.09808 - 1.09808i) q^{18} +(2.59808 + 3.50000i) q^{19} +(-3.73205 + 2.46410i) q^{20} +(-6.00000 - 10.3923i) q^{21} +(0.633975 + 2.36603i) q^{22} +(4.73205 - 1.26795i) q^{23} +(3.46410 + 6.00000i) q^{24} +(4.96410 - 0.598076i) q^{25} +(-1.00000 - 1.73205i) q^{26} +(-2.53590 + 9.46410i) q^{28} +(-0.866025 - 0.500000i) q^{29} +(-0.464102 - 7.73205i) q^{30} -8.66025i q^{31} +(1.46410 - 5.46410i) q^{32} +(-4.09808 - 1.09808i) q^{33} +6.00000i q^{34} +(7.26795 - 8.19615i) q^{35} -6.00000 q^{36} +(2.00000 + 2.00000i) q^{37} +(4.83013 + 3.83013i) q^{38} +3.46410 q^{39} +(-4.19615 + 4.73205i) q^{40} +(4.00000 + 6.92820i) q^{41} +(-12.0000 - 12.0000i) q^{42} +(-1.26795 + 4.73205i) q^{43} +(1.73205 + 3.00000i) q^{44} +(6.00000 + 3.00000i) q^{45} +(6.00000 - 3.46410i) q^{46} +(3.16987 + 11.8301i) q^{47} +(6.92820 + 6.92820i) q^{48} -17.0000i q^{49} +(6.56218 - 2.63397i) q^{50} +(-9.00000 - 5.19615i) q^{51} +(-2.00000 - 2.00000i) q^{52} +(-1.09808 - 4.09808i) q^{53} +(-0.232051 - 3.86603i) q^{55} +13.8564i q^{56} +(-9.92820 + 3.92820i) q^{57} +(-1.36603 - 0.366025i) q^{58} +(-0.866025 - 1.50000i) q^{59} +(-3.46410 - 10.3923i) q^{60} +(1.50000 - 2.59808i) q^{61} +(-3.16987 - 11.8301i) q^{62} +(14.1962 - 3.80385i) q^{63} -8.00000i q^{64} +(1.00000 + 3.00000i) q^{65} -6.00000 q^{66} +(-0.633975 - 2.36603i) q^{67} +(2.19615 + 8.19615i) q^{68} +12.0000i q^{69} +(6.92820 - 13.8564i) q^{70} +(1.50000 - 0.866025i) q^{71} +(-8.19615 + 2.19615i) q^{72} +(10.9282 + 2.92820i) q^{73} +(3.46410 + 2.00000i) q^{74} +(-1.73205 + 12.1244i) q^{75} +(8.00000 + 3.46410i) q^{76} +(-6.00000 - 6.00000i) q^{77} +(4.73205 - 1.26795i) q^{78} +(4.33013 + 7.50000i) q^{79} +(-4.00000 + 8.00000i) q^{80} +(-4.50000 - 7.79423i) q^{81} +(8.00000 + 8.00000i) q^{82} +(-6.92820 - 6.92820i) q^{83} +(-20.7846 - 12.0000i) q^{84} +(1.90192 - 9.29423i) q^{85} +6.92820i q^{86} +(1.73205 - 1.73205i) q^{87} +(3.46410 + 3.46410i) q^{88} +(0.866025 + 0.500000i) q^{89} +(9.29423 + 1.90192i) q^{90} +(6.00000 + 3.46410i) q^{91} +(6.92820 - 6.92820i) q^{92} +(20.4904 + 5.49038i) q^{93} +(8.66025 + 15.0000i) q^{94} +(-6.26795 - 7.46410i) q^{95} +(12.0000 + 6.92820i) q^{96} +(1.46410 - 5.46410i) q^{97} +(-6.22243 - 23.2224i) q^{98} +(2.59808 - 4.50000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 6 q^{3} - 2 q^{5} + 8 q^{8} - 12 q^{10} + 12 q^{12} + 2 q^{13} - 6 q^{15} + 8 q^{16} + 6 q^{17} - 6 q^{18} - 8 q^{20} - 24 q^{21} + 6 q^{22} + 12 q^{23} + 6 q^{25} - 4 q^{26} - 24 q^{28}+ \cdots + 34 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{4}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.36603 − 0.366025i 0.965926 − 0.258819i
\(3\) −0.633975 + 2.36603i −0.366025 + 1.36603i 0.500000 + 0.866025i \(0.333333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(4\) 1.73205 − 1.00000i 0.866025 − 0.500000i
\(5\) −2.23205 + 0.133975i −0.998203 + 0.0599153i
\(6\) 3.46410i 1.41421i
\(7\) −3.46410 + 3.46410i −1.30931 + 1.30931i −0.387392 + 0.921915i \(0.626624\pi\)
−0.921915 + 0.387392i \(0.873376\pi\)
\(8\) 2.00000 − 2.00000i 0.707107 − 0.707107i
\(9\) −2.59808 − 1.50000i −0.866025 − 0.500000i
\(10\) −3.00000 + 1.00000i −0.948683 + 0.316228i
\(11\) 1.73205i 0.522233i 0.965307 + 0.261116i \(0.0840907\pi\)
−0.965307 + 0.261116i \(0.915909\pi\)
\(12\) 1.26795 + 4.73205i 0.366025 + 1.36603i
\(13\) −0.366025 − 1.36603i −0.101517 − 0.378867i 0.896410 − 0.443227i \(-0.146166\pi\)
−0.997927 + 0.0643593i \(0.979500\pi\)
\(14\) −3.46410 + 6.00000i −0.925820 + 1.60357i
\(15\) 1.09808 − 5.36603i 0.283522 − 1.38550i
\(16\) 2.00000 − 3.46410i 0.500000 − 0.866025i
\(17\) −1.09808 + 4.09808i −0.266323 + 0.993929i 0.695113 + 0.718900i \(0.255354\pi\)
−0.961436 + 0.275029i \(0.911312\pi\)
\(18\) −4.09808 − 1.09808i −0.965926 − 0.258819i
\(19\) 2.59808 + 3.50000i 0.596040 + 0.802955i
\(20\) −3.73205 + 2.46410i −0.834512 + 0.550990i
\(21\) −6.00000 − 10.3923i −1.30931 − 2.26779i
\(22\) 0.633975 + 2.36603i 0.135164 + 0.504438i
\(23\) 4.73205 − 1.26795i 0.986701 − 0.264386i 0.270837 − 0.962625i \(-0.412700\pi\)
0.715864 + 0.698240i \(0.246033\pi\)
\(24\) 3.46410 + 6.00000i 0.707107 + 1.22474i
\(25\) 4.96410 − 0.598076i 0.992820 − 0.119615i
\(26\) −1.00000 − 1.73205i −0.196116 − 0.339683i
\(27\) 0 0
\(28\) −2.53590 + 9.46410i −0.479240 + 1.78855i
\(29\) −0.866025 − 0.500000i −0.160817 − 0.0928477i 0.417432 − 0.908708i \(-0.362930\pi\)
−0.578249 + 0.815861i \(0.696264\pi\)
\(30\) −0.464102 − 7.73205i −0.0847330 − 1.41167i
\(31\) − 8.66025i − 1.55543i −0.628619 − 0.777714i \(-0.716379\pi\)
0.628619 − 0.777714i \(-0.283621\pi\)
\(32\) 1.46410 − 5.46410i 0.258819 − 0.965926i
\(33\) −4.09808 − 1.09808i −0.713384 − 0.191151i
\(34\) 6.00000i 1.02899i
\(35\) 7.26795 − 8.19615i 1.22851 − 1.38540i
\(36\) −6.00000 −1.00000
\(37\) 2.00000 + 2.00000i 0.328798 + 0.328798i 0.852129 − 0.523331i \(-0.175311\pi\)
−0.523331 + 0.852129i \(0.675311\pi\)
\(38\) 4.83013 + 3.83013i 0.783550 + 0.621329i
\(39\) 3.46410 0.554700
\(40\) −4.19615 + 4.73205i −0.663470 + 0.748203i
\(41\) 4.00000 + 6.92820i 0.624695 + 1.08200i 0.988600 + 0.150567i \(0.0481100\pi\)
−0.363905 + 0.931436i \(0.618557\pi\)
\(42\) −12.0000 − 12.0000i −1.85164 − 1.85164i
\(43\) −1.26795 + 4.73205i −0.193360 + 0.721631i 0.799325 + 0.600899i \(0.205191\pi\)
−0.992685 + 0.120732i \(0.961476\pi\)
\(44\) 1.73205 + 3.00000i 0.261116 + 0.452267i
\(45\) 6.00000 + 3.00000i 0.894427 + 0.447214i
\(46\) 6.00000 − 3.46410i 0.884652 − 0.510754i
\(47\) 3.16987 + 11.8301i 0.462373 + 1.72560i 0.665454 + 0.746439i \(0.268238\pi\)
−0.203080 + 0.979162i \(0.565095\pi\)
\(48\) 6.92820 + 6.92820i 1.00000 + 1.00000i
\(49\) − 17.0000i − 2.42857i
\(50\) 6.56218 − 2.63397i 0.928032 − 0.372500i
\(51\) −9.00000 − 5.19615i −1.26025 − 0.727607i
\(52\) −2.00000 − 2.00000i −0.277350 − 0.277350i
\(53\) −1.09808 − 4.09808i −0.150832 − 0.562914i −0.999426 − 0.0338693i \(-0.989217\pi\)
0.848594 − 0.529045i \(-0.177450\pi\)
\(54\) 0 0
\(55\) −0.232051 − 3.86603i −0.0312897 − 0.521295i
\(56\) 13.8564i 1.85164i
\(57\) −9.92820 + 3.92820i −1.31502 + 0.520303i
\(58\) −1.36603 − 0.366025i −0.179368 − 0.0480615i
\(59\) −0.866025 − 1.50000i −0.112747 − 0.195283i 0.804130 − 0.594454i \(-0.202632\pi\)
−0.916877 + 0.399170i \(0.869298\pi\)
\(60\) −3.46410 − 10.3923i −0.447214 − 1.34164i
\(61\) 1.50000 − 2.59808i 0.192055 − 0.332650i −0.753876 − 0.657017i \(-0.771818\pi\)
0.945931 + 0.324367i \(0.105151\pi\)
\(62\) −3.16987 − 11.8301i −0.402574 − 1.50243i
\(63\) 14.1962 − 3.80385i 1.78855 − 0.479240i
\(64\) − 8.00000i − 1.00000i
\(65\) 1.00000 + 3.00000i 0.124035 + 0.372104i
\(66\) −6.00000 −0.738549
\(67\) −0.633975 − 2.36603i −0.0774523 − 0.289056i 0.916326 − 0.400433i \(-0.131140\pi\)
−0.993778 + 0.111377i \(0.964474\pi\)
\(68\) 2.19615 + 8.19615i 0.266323 + 0.993929i
\(69\) 12.0000i 1.44463i
\(70\) 6.92820 − 13.8564i 0.828079 − 1.65616i
\(71\) 1.50000 − 0.866025i 0.178017 − 0.102778i −0.408344 − 0.912828i \(-0.633893\pi\)
0.586361 + 0.810050i \(0.300560\pi\)
\(72\) −8.19615 + 2.19615i −0.965926 + 0.258819i
\(73\) 10.9282 + 2.92820i 1.27905 + 0.342720i 0.833491 − 0.552533i \(-0.186339\pi\)
0.445558 + 0.895253i \(0.353005\pi\)
\(74\) 3.46410 + 2.00000i 0.402694 + 0.232495i
\(75\) −1.73205 + 12.1244i −0.200000 + 1.40000i
\(76\) 8.00000 + 3.46410i 0.917663 + 0.397360i
\(77\) −6.00000 − 6.00000i −0.683763 − 0.683763i
\(78\) 4.73205 − 1.26795i 0.535799 − 0.143567i
\(79\) 4.33013 + 7.50000i 0.487177 + 0.843816i 0.999891 − 0.0147436i \(-0.00469319\pi\)
−0.512714 + 0.858559i \(0.671360\pi\)
\(80\) −4.00000 + 8.00000i −0.447214 + 0.894427i
\(81\) −4.50000 − 7.79423i −0.500000 − 0.866025i
\(82\) 8.00000 + 8.00000i 0.883452 + 0.883452i
\(83\) −6.92820 − 6.92820i −0.760469 − 0.760469i 0.215938 − 0.976407i \(-0.430719\pi\)
−0.976407 + 0.215938i \(0.930719\pi\)
\(84\) −20.7846 − 12.0000i −2.26779 − 1.30931i
\(85\) 1.90192 − 9.29423i 0.206293 − 1.00810i
\(86\) 6.92820i 0.747087i
\(87\) 1.73205 − 1.73205i 0.185695 − 0.185695i
\(88\) 3.46410 + 3.46410i 0.369274 + 0.369274i
\(89\) 0.866025 + 0.500000i 0.0917985 + 0.0529999i 0.545197 − 0.838308i \(-0.316455\pi\)
−0.453398 + 0.891308i \(0.649788\pi\)
\(90\) 9.29423 + 1.90192i 0.979698 + 0.200480i
\(91\) 6.00000 + 3.46410i 0.628971 + 0.363137i
\(92\) 6.92820 − 6.92820i 0.722315 − 0.722315i
\(93\) 20.4904 + 5.49038i 2.12475 + 0.569326i
\(94\) 8.66025 + 15.0000i 0.893237 + 1.54713i
\(95\) −6.26795 − 7.46410i −0.643078 − 0.765801i
\(96\) 12.0000 + 6.92820i 1.22474 + 0.707107i
\(97\) 1.46410 − 5.46410i 0.148657 − 0.554795i −0.850908 − 0.525314i \(-0.823948\pi\)
0.999565 − 0.0294813i \(-0.00938554\pi\)
\(98\) −6.22243 − 23.2224i −0.628561 − 2.34582i
\(99\) 2.59808 − 4.50000i 0.261116 − 0.452267i
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.v.b.7.1 yes 4
4.3 odd 2 380.2.v.a.7.1 ✓ 4
5.3 odd 4 inner 380.2.v.b.83.1 yes 4
19.11 even 3 380.2.v.a.87.1 yes 4
20.3 even 4 380.2.v.a.83.1 yes 4
76.11 odd 6 inner 380.2.v.b.87.1 yes 4
95.68 odd 12 380.2.v.a.163.1 yes 4
380.163 even 12 inner 380.2.v.b.163.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.v.a.7.1 ✓ 4 4.3 odd 2
380.2.v.a.83.1 yes 4 20.3 even 4
380.2.v.a.87.1 yes 4 19.11 even 3
380.2.v.a.163.1 yes 4 95.68 odd 12
380.2.v.b.7.1 yes 4 1.1 even 1 trivial
380.2.v.b.83.1 yes 4 5.3 odd 4 inner
380.2.v.b.87.1 yes 4 76.11 odd 6 inner
380.2.v.b.163.1 yes 4 380.163 even 12 inner