Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3800,1,Mod(1451,3800)] 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("3800.1451"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3800, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 0, 2])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3800.bd (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-1,-2,-1,0,-2,0,2,-3,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.89644704801\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.72200.1
Artin image: $C_3\times S_3$
Artin field: Galois closure of 6.0.115520000.2

Embedding invariants

Embedding label 1451.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 3800.1451
Dual form 3800.1.bd.a.3051.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{2} +(-1.00000 - 1.73205i) q^{3} +(-0.500000 + 0.866025i) q^{4} +(-1.00000 + 1.73205i) q^{6} +1.00000 q^{8} +(-1.50000 + 2.59808i) q^{9} -1.00000 q^{11} +2.00000 q^{12} +(-0.500000 - 0.866025i) q^{16} +(0.500000 + 0.866025i) q^{17} +3.00000 q^{18} +1.00000 q^{19} +(0.500000 + 0.866025i) q^{22} +(-1.00000 - 1.73205i) q^{24} +4.00000 q^{27} +(-0.500000 + 0.866025i) q^{32} +(1.00000 + 1.73205i) q^{33} +(0.500000 - 0.866025i) q^{34} +(-1.50000 - 2.59808i) q^{36} +(-0.500000 - 0.866025i) q^{38} +(0.500000 + 0.866025i) q^{41} +(-1.00000 - 1.73205i) q^{43} +(0.500000 - 0.866025i) q^{44} +(-1.00000 + 1.73205i) q^{48} +1.00000 q^{49} +(1.00000 - 1.73205i) q^{51} +(-2.00000 - 3.46410i) q^{54} +(-1.00000 - 1.73205i) q^{57} +(0.500000 + 0.866025i) q^{59} +1.00000 q^{64} +(1.00000 - 1.73205i) q^{66} +(0.500000 - 0.866025i) q^{67} -1.00000 q^{68} +(-1.50000 + 2.59808i) q^{72} +(-1.00000 - 1.73205i) q^{73} +(-0.500000 + 0.866025i) q^{76} +(-2.50000 - 4.33013i) q^{81} +(0.500000 - 0.866025i) q^{82} +2.00000 q^{83} +(-1.00000 + 1.73205i) q^{86} -1.00000 q^{88} +(0.500000 - 0.866025i) q^{89} +2.00000 q^{96} +(0.500000 + 0.866025i) q^{97} +(-0.500000 - 0.866025i) q^{98} +(1.50000 - 2.59808i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - 2 q^{3} - q^{4} - 2 q^{6} + 2 q^{8} - 3 q^{9} - 2 q^{11} + 4 q^{12} - q^{16} + q^{17} + 6 q^{18} + 2 q^{19} + q^{22} - 2 q^{24} + 8 q^{27} - q^{32} + 2 q^{33} + q^{34} - 3 q^{36} - q^{38}+ \cdots + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(951\) \(1901\) \(1977\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(-1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(3\) −1.00000 − 1.73205i −1.00000 − 1.73205i −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 − 0.866025i \(-0.666667\pi\)
\(4\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(5\) 0 0
\(6\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 1.00000 1.00000
\(9\) −1.50000 + 2.59808i −1.50000 + 2.59808i
\(10\) 0 0
\(11\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(12\) 2.00000 2.00000
\(13\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(17\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(18\) 3.00000 3.00000
\(19\) 1.00000 1.00000
\(20\) 0 0
\(21\) 0 0
\(22\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(23\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) −1.00000 − 1.73205i −1.00000 − 1.73205i
\(25\) 0 0
\(26\) 0 0
\(27\) 4.00000 4.00000
\(28\) 0 0
\(29\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(33\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(34\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(35\) 0 0
\(36\) −1.50000 − 2.59808i −1.50000 − 2.59808i
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(39\) 0 0
\(40\) 0 0
\(41\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(42\) 0 0
\(43\) −1.00000 − 1.73205i −1.00000 − 1.73205i −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 − 0.866025i \(-0.666667\pi\)
\(44\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(48\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(49\) 1.00000 1.00000
\(50\) 0 0
\(51\) 1.00000 − 1.73205i 1.00000 − 1.73205i
\(52\) 0 0
\(53\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(54\) −2.00000 − 3.46410i −2.00000 − 3.46410i
\(55\) 0 0
\(56\) 0 0
\(57\) −1.00000 − 1.73205i −1.00000 − 1.73205i
\(58\) 0 0
\(59\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(60\) 0 0
\(61\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 1.00000
\(65\) 0 0
\(66\) 1.00000 − 1.73205i 1.00000 − 1.73205i
\(67\) 0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(68\) −1.00000 −1.00000
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(72\) −1.50000 + 2.59808i −1.50000 + 2.59808i
\(73\) −1.00000 − 1.73205i −1.00000 − 1.73205i −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 − 0.866025i \(-0.666667\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(80\) 0 0
\(81\) −2.50000 − 4.33013i −2.50000 − 4.33013i
\(82\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(83\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(87\) 0 0
\(88\) −1.00000 −1.00000
\(89\) 0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 − 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 2.00000 2.00000
\(97\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(98\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(99\) 1.50000 − 2.59808i 1.50000 − 2.59808i
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 3800.1.bd.a.1451.1 ✓ 2
5.2 odd 4 3800.1.bn.a.1299.2 4
5.3 odd 4 3800.1.bn.a.1299.1 4
5.4 even 2 3800.1.bd.e.1451.1 yes 2
8.3 odd 2 CM 3800.1.bd.a.1451.1 ✓ 2
19.11 even 3 inner 3800.1.bd.a.3051.1 yes 2
40.3 even 4 3800.1.bn.a.1299.1 4
40.19 odd 2 3800.1.bd.e.1451.1 yes 2
40.27 even 4 3800.1.bn.a.1299.2 4
95.49 even 6 3800.1.bd.e.3051.1 yes 2
95.68 odd 12 3800.1.bn.a.2899.2 4
95.87 odd 12 3800.1.bn.a.2899.1 4
152.11 odd 6 inner 3800.1.bd.a.3051.1 yes 2
760.163 even 12 3800.1.bn.a.2899.2 4
760.467 even 12 3800.1.bn.a.2899.1 4
760.619 odd 6 3800.1.bd.e.3051.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3800.1.bd.a.1451.1 ✓ 2 1.1 even 1 trivial
3800.1.bd.a.1451.1 ✓ 2 8.3 odd 2 CM
3800.1.bd.a.3051.1 yes 2 19.11 even 3 inner
3800.1.bd.a.3051.1 yes 2 152.11 odd 6 inner
3800.1.bd.e.1451.1 yes 2 5.4 even 2
3800.1.bd.e.1451.1 yes 2 40.19 odd 2
3800.1.bd.e.3051.1 yes 2 95.49 even 6
3800.1.bd.e.3051.1 yes 2 760.619 odd 6
3800.1.bn.a.1299.1 4 5.3 odd 4
3800.1.bn.a.1299.1 4 40.3 even 4
3800.1.bn.a.1299.2 4 5.2 odd 4
3800.1.bn.a.1299.2 4 40.27 even 4
3800.1.bn.a.2899.1 4 95.87 odd 12
3800.1.bn.a.2899.1 4 760.467 even 12
3800.1.bn.a.2899.2 4 95.68 odd 12
3800.1.bn.a.2899.2 4 760.163 even 12