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 83.1
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 380.83
Dual form 380.2.v.b.87.1

$q$-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+(-0.366025 - 1.36603i) q^{2} +(-2.36603 - 0.633975i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(1.23205 - 1.86603i) 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} +(4.73205 - 1.26795i) q^{12} +(1.36603 - 0.366025i) q^{13} +(3.46410 - 6.00000i) q^{14} +(-4.09808 + 3.63397i) q^{15} +(2.00000 - 3.46410i) q^{16} +(4.09808 + 1.09808i) q^{17} +(1.09808 - 4.09808i) q^{18} +(-2.59808 - 3.50000i) q^{19} +(-0.267949 + 4.46410i) q^{20} +(-6.00000 - 10.3923i) q^{21} +(2.36603 - 0.633975i) q^{22} +(1.26795 + 4.73205i) q^{23} +(-3.46410 - 6.00000i) q^{24} +(-1.96410 - 4.59808i) q^{25} +(-1.00000 - 1.73205i) q^{26} +(-9.46410 - 2.53590i) q^{28} +(0.866025 + 0.500000i) q^{29} +(6.46410 + 4.26795i) q^{30} -8.66025i q^{31} +(-5.46410 - 1.46410i) q^{32} +(1.09808 - 4.09808i) q^{33} -6.00000i q^{34} +(10.7321 - 2.19615i) q^{35} -6.00000 q^{36} +(2.00000 - 2.00000i) q^{37} +(-3.83013 + 4.83013i) q^{38} -3.46410 q^{39} +(6.19615 - 1.26795i) q^{40} +(4.00000 + 6.92820i) q^{41} +(-12.0000 + 12.0000i) q^{42} +(-4.73205 - 1.26795i) q^{43} +(-1.73205 - 3.00000i) q^{44} +(6.00000 - 3.00000i) q^{45} +(6.00000 - 3.46410i) q^{46} +(11.8301 - 3.16987i) q^{47} +(-6.92820 + 6.92820i) q^{48} +17.0000i q^{49} +(-5.56218 + 4.36603i) q^{50} +(-9.00000 - 5.19615i) q^{51} +(-2.00000 + 2.00000i) q^{52} +(4.09808 - 1.09808i) q^{53} +(3.23205 + 2.13397i) q^{55} +13.8564i q^{56} +(3.92820 + 9.92820i) q^{57} +(0.366025 - 1.36603i) q^{58} +(0.866025 + 1.50000i) q^{59} +(3.46410 - 10.3923i) q^{60} +(1.50000 - 2.59808i) q^{61} +(-11.8301 + 3.16987i) q^{62} +(3.80385 + 14.1962i) q^{63} +8.00000i q^{64} +(1.00000 - 3.00000i) q^{65} -6.00000 q^{66} +(-2.36603 + 0.633975i) q^{67} +(-8.19615 + 2.19615i) q^{68} -12.0000i q^{69} +(-6.92820 - 13.8564i) q^{70} +(1.50000 - 0.866025i) q^{71} +(2.19615 + 8.19615i) q^{72} +(-2.92820 + 10.9282i) 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} +(1.26795 + 4.73205i) 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} +(7.09808 - 6.29423i) q^{85} +6.92820i q^{86} +(-1.73205 - 1.73205i) q^{87} +(-3.46410 + 3.46410i) q^{88} +(-0.866025 - 0.500000i) q^{89} +(-6.29423 - 7.09808i) q^{90} +(6.00000 + 3.46410i) q^{91} +(-6.92820 - 6.92820i) q^{92} +(-5.49038 + 20.4904i) q^{93} +(-8.66025 - 15.0000i) q^{94} +(-9.73205 + 0.535898i) q^{95} +(12.0000 + 6.92820i) q^{96} +(-5.46410 - 1.46410i) q^{97} +(23.2224 - 6.22243i) 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 \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.366025 − 1.36603i −0.258819 − 0.965926i
\(3\) −2.36603 − 0.633975i −1.36603 − 0.366025i −0.500000 − 0.866025i \(-0.666667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(4\) −1.73205 + 1.00000i −0.866025 + 0.500000i
\(5\) 1.23205 − 1.86603i 0.550990 − 0.834512i
\(6\) 3.46410i 1.41421i
\(7\) 3.46410 + 3.46410i 1.30931 + 1.30931i 0.921915 + 0.387392i \(0.126624\pi\)
0.387392 + 0.921915i \(0.373376\pi\)
\(8\) 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 \(0.915909\pi\)
\(12\) 4.73205 − 1.26795i 1.36603 − 0.366025i
\(13\) 1.36603 − 0.366025i 0.378867 − 0.101517i −0.0643593 − 0.997927i \(-0.520500\pi\)
0.443227 + 0.896410i \(0.353834\pi\)
\(14\) 3.46410 − 6.00000i 0.925820 − 1.60357i
\(15\) −4.09808 + 3.63397i −1.05812 + 0.938288i
\(16\) 2.00000 − 3.46410i 0.500000 − 0.866025i
\(17\) 4.09808 + 1.09808i 0.993929 + 0.266323i 0.718900 − 0.695113i \(-0.244646\pi\)
0.275029 + 0.961436i \(0.411312\pi\)
\(18\) 1.09808 − 4.09808i 0.258819 − 0.965926i
\(19\) −2.59808 − 3.50000i −0.596040 − 0.802955i
\(20\) −0.267949 + 4.46410i −0.0599153 + 0.998203i
\(21\) −6.00000 − 10.3923i −1.30931 − 2.26779i
\(22\) 2.36603 − 0.633975i 0.504438 − 0.135164i
\(23\) 1.26795 + 4.73205i 0.264386 + 0.986701i 0.962625 + 0.270837i \(0.0873003\pi\)
−0.698240 + 0.715864i \(0.746033\pi\)
\(24\) −3.46410 − 6.00000i −0.707107 − 1.22474i
\(25\) −1.96410 − 4.59808i −0.392820 − 0.919615i
\(26\) −1.00000 − 1.73205i −0.196116 − 0.339683i
\(27\) 0 0
\(28\) −9.46410 − 2.53590i −1.78855 − 0.479240i
\(29\) 0.866025 + 0.500000i 0.160817 + 0.0928477i 0.578249 − 0.815861i \(-0.303736\pi\)
−0.417432 + 0.908708i \(0.637070\pi\)
\(30\) 6.46410 + 4.26795i 1.18018 + 0.779217i
\(31\) − 8.66025i − 1.55543i −0.628619 − 0.777714i \(-0.716379\pi\)
0.628619 − 0.777714i \(-0.283621\pi\)
\(32\) −5.46410 − 1.46410i −0.965926 − 0.258819i
\(33\) 1.09808 − 4.09808i 0.191151 − 0.713384i
\(34\) − 6.00000i − 1.02899i
\(35\) 10.7321 − 2.19615i 1.81405 − 0.371218i
\(36\) −6.00000 −1.00000
\(37\) 2.00000 − 2.00000i 0.328798 − 0.328798i −0.523331 − 0.852129i \(-0.675311\pi\)
0.852129 + 0.523331i \(0.175311\pi\)
\(38\) −3.83013 + 4.83013i −0.621329 + 0.783550i
\(39\) −3.46410 −0.554700
\(40\) 6.19615 − 1.26795i 0.979698 − 0.200480i
\(41\) 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
\(43\) −4.73205 − 1.26795i −0.721631 − 0.193360i −0.120732 − 0.992685i \(-0.538524\pi\)
−0.600899 + 0.799325i \(0.705191\pi\)
\(44\) −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
\(47\) 11.8301 − 3.16987i 1.72560 − 0.462373i 0.746439 − 0.665454i \(-0.231762\pi\)
0.979162 + 0.203080i \(0.0650952\pi\)
\(48\) −6.92820 + 6.92820i −1.00000 + 1.00000i
\(49\) 17.0000i 2.42857i
\(50\) −5.56218 + 4.36603i −0.786611 + 0.617449i
\(51\) −9.00000 − 5.19615i −1.26025 − 0.727607i
\(52\) −2.00000 + 2.00000i −0.277350 + 0.277350i
\(53\) 4.09808 − 1.09808i 0.562914 − 0.150832i 0.0338693 − 0.999426i \(-0.489217\pi\)
0.529045 + 0.848594i \(0.322550\pi\)
\(54\) 0 0
\(55\) 3.23205 + 2.13397i 0.435810 + 0.287745i
\(56\) 13.8564i 1.85164i
\(57\) 3.92820 + 9.92820i 0.520303 + 1.31502i
\(58\) 0.366025 − 1.36603i 0.0480615 − 0.179368i
\(59\) 0.866025 + 1.50000i 0.112747 + 0.195283i 0.916877 − 0.399170i \(-0.130702\pi\)
−0.804130 + 0.594454i \(0.797368\pi\)
\(60\) 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 \(0.105151\pi\)
\(62\) −11.8301 + 3.16987i −1.50243 + 0.402574i
\(63\) 3.80385 + 14.1962i 0.479240 + 1.78855i
\(64\) 8.00000i 1.00000i
\(65\) 1.00000 − 3.00000i 0.124035 − 0.372104i
\(66\) −6.00000 −0.738549
\(67\) −2.36603 + 0.633975i −0.289056 + 0.0774523i −0.400433 − 0.916326i \(-0.631140\pi\)
0.111377 + 0.993778i \(0.464474\pi\)
\(68\) −8.19615 + 2.19615i −0.993929 + 0.266323i
\(69\) − 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 \(0.300560\pi\)
\(72\) 2.19615 + 8.19615i 0.258819 + 0.965926i
\(73\) −2.92820 + 10.9282i −0.342720 + 1.27905i 0.552533 + 0.833491i \(0.313661\pi\)
−0.895253 + 0.445558i \(0.853005\pi\)
\(74\) −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
\(78\) 1.26795 + 4.73205i 0.143567 + 0.535799i
\(79\) −4.33013 − 7.50000i −0.487177 − 0.843816i 0.512714 − 0.858559i \(-0.328640\pi\)
−0.999891 + 0.0147436i \(0.995307\pi\)
\(80\) −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 \(-0.569281\pi\)
0.976407 + 0.215938i \(0.0692809\pi\)
\(84\) 20.7846 + 12.0000i 2.26779 + 1.30931i
\(85\) 7.09808 − 6.29423i 0.769894 − 0.682705i
\(86\) 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 0.453398 − 0.891308i \(-0.350212\pi\)
−0.545197 + 0.838308i \(0.683545\pi\)
\(90\) −6.29423 − 7.09808i −0.663470 − 0.748203i
\(91\) 6.00000 + 3.46410i 0.628971 + 0.363137i
\(92\) −6.92820 − 6.92820i −0.722315 − 0.722315i
\(93\) −5.49038 + 20.4904i −0.569326 + 2.12475i
\(94\) −8.66025 − 15.0000i −0.893237 − 1.54713i
\(95\) −9.73205 + 0.535898i −0.998487 + 0.0549820i
\(96\) 12.0000 + 6.92820i 1.22474 + 0.707107i
\(97\) −5.46410 − 1.46410i −0.554795 − 0.148657i −0.0294813 − 0.999565i \(-0.509386\pi\)
−0.525314 + 0.850908i \(0.676052\pi\)
\(98\) 23.2224 − 6.22243i 2.34582 − 0.628561i
\(99\) −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.83.1 yes 4
4.3 odd 2 380.2.v.a.83.1 yes 4
5.2 odd 4 inner 380.2.v.b.7.1 yes 4
19.11 even 3 380.2.v.a.163.1 yes 4
20.7 even 4 380.2.v.a.7.1 ✓ 4
76.11 odd 6 inner 380.2.v.b.163.1 yes 4
95.87 odd 12 380.2.v.a.87.1 yes 4
380.87 even 12 inner 380.2.v.b.87.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.v.a.7.1 ✓ 4 20.7 even 4
380.2.v.a.83.1 yes 4 4.3 odd 2
380.2.v.a.87.1 yes 4 95.87 odd 12
380.2.v.a.163.1 yes 4 19.11 even 3
380.2.v.b.7.1 yes 4 5.2 odd 4 inner
380.2.v.b.83.1 yes 4 1.1 even 1 trivial
380.2.v.b.87.1 yes 4 380.87 even 12 inner
380.2.v.b.163.1 yes 4 76.11 odd 6 inner