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Properties

Label 3800.1.cv.a.651.1
Level $3800$
Weight $1$
Character 3800.651
Analytic conductor $1.896$
Analytic rank $0$
Dimension $6$
Projective image $D_{9}$
CM discriminant -8
Inner twists $4$

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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] = [3800,1,Mod(251,3800)] 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("3800.251"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3800, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([9, 9, 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.cv (of order \(18\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,-6,0,0,6,0,3,6,0,0,0,0,0,0,0,0,-6,0,0,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(22)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.89644704801\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{9}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{9} - \cdots)\)

Embedding invariants

Embedding label 651.1
Root \(-0.766044 + 0.642788i\) of defining polynomial
Character \(\chi\) \(=\) 3800.651
Dual form 3800.1.cv.a.251.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.939693 - 0.342020i) q^{2} +(-0.0603074 + 0.342020i) q^{3} +(0.766044 - 0.642788i) q^{4} +(0.0603074 + 0.342020i) q^{6} +(0.500000 - 0.866025i) q^{8} +(0.826352 + 0.300767i) q^{9} +(0.939693 - 1.62760i) q^{11} +(0.173648 + 0.300767i) q^{12} +(0.173648 - 0.984808i) q^{16} +(-0.939693 + 0.342020i) q^{17} +0.879385 q^{18} +(-0.939693 - 0.342020i) q^{19} +(0.326352 - 1.85083i) q^{22} +(0.266044 + 0.223238i) q^{24} +(-0.326352 + 0.565258i) q^{27} +(-0.173648 - 0.984808i) q^{32} +(0.500000 + 0.419550i) q^{33} +(-0.766044 + 0.642788i) q^{34} +(0.826352 - 0.300767i) q^{36} -1.00000 q^{38} +(0.0603074 - 0.342020i) q^{41} +(0.766044 + 0.642788i) q^{43} +(-0.326352 - 1.85083i) q^{44} +(0.326352 + 0.118782i) q^{48} +(-0.500000 + 0.866025i) q^{49} +(-0.0603074 - 0.342020i) q^{51} +(-0.113341 + 0.642788i) q^{54} +(0.173648 - 0.300767i) q^{57} +(-0.326352 + 0.118782i) q^{59} +(-0.500000 - 0.866025i) q^{64} +(0.613341 + 0.223238i) q^{66} +(0.326352 + 0.118782i) q^{67} +(-0.500000 + 0.866025i) q^{68} +(0.673648 - 0.565258i) q^{72} +(-0.266044 + 1.50881i) q^{73} +(-0.939693 + 0.342020i) q^{76} +(0.500000 + 0.419550i) q^{81} +(-0.0603074 - 0.342020i) q^{82} +(-0.939693 - 1.62760i) q^{83} +(0.939693 + 0.342020i) q^{86} +(-0.939693 - 1.62760i) q^{88} +(0.347296 + 1.96962i) q^{89} +0.347296 q^{96} +(1.43969 - 0.524005i) q^{97} +(-0.173648 + 0.984808i) q^{98} +(1.26604 - 1.06234i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{3} + 6 q^{6} + 3 q^{8} + 6 q^{9} - 6 q^{18} + 3 q^{22} - 3 q^{24} - 3 q^{27} + 3 q^{33} + 6 q^{36} - 6 q^{38} + 6 q^{41} - 3 q^{44} + 3 q^{48} - 3 q^{49} - 6 q^{51} + 6 q^{54} - 3 q^{59} - 3 q^{64}+ \cdots + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(401\) \(951\) \(1901\) \(1977\)
\(\chi(n)\) \(e\left(\frac{8}{9}\right)\) \(-1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.939693 − 0.342020i 0.939693 − 0.342020i
\(3\) −0.0603074 + 0.342020i −0.0603074 + 0.342020i 0.939693 + 0.342020i \(0.111111\pi\)
−1.00000 \(\pi\)
\(4\) 0.766044 − 0.642788i 0.766044 − 0.642788i
\(5\) 0 0
\(6\) 0.0603074 + 0.342020i 0.0603074 + 0.342020i
\(7\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(8\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(9\) 0.826352 + 0.300767i 0.826352 + 0.300767i
\(10\) 0 0
\(11\) 0.939693 − 1.62760i 0.939693 − 1.62760i 0.173648 − 0.984808i \(-0.444444\pi\)
0.766044 − 0.642788i \(-0.222222\pi\)
\(12\) 0.173648 + 0.300767i 0.173648 + 0.300767i
\(13\) 0 0 −0.173648 − 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.173648 − 0.984808i 0.173648 − 0.984808i
\(17\) −0.939693 + 0.342020i −0.939693 + 0.342020i −0.766044 − 0.642788i \(-0.777778\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(18\) 0.879385 0.879385
\(19\) −0.939693 − 0.342020i −0.939693 − 0.342020i
\(20\) 0 0
\(21\) 0 0
\(22\) 0.326352 − 1.85083i 0.326352 − 1.85083i
\(23\) 0 0 0.766044 − 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(24\) 0.266044 + 0.223238i 0.266044 + 0.223238i
\(25\) 0 0
\(26\) 0 0
\(27\) −0.326352 + 0.565258i −0.326352 + 0.565258i
\(28\) 0 0
\(29\) 0 0 −0.939693 − 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(30\) 0 0
\(31\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(32\) −0.173648 − 0.984808i −0.173648 − 0.984808i
\(33\) 0.500000 + 0.419550i 0.500000 + 0.419550i
\(34\) −0.766044 + 0.642788i −0.766044 + 0.642788i
\(35\) 0 0
\(36\) 0.826352 − 0.300767i 0.826352 − 0.300767i
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) −1.00000 −1.00000
\(39\) 0 0
\(40\) 0 0
\(41\) 0.0603074 − 0.342020i 0.0603074 − 0.342020i −0.939693 − 0.342020i \(-0.888889\pi\)
1.00000 \(0\)
\(42\) 0 0
\(43\) 0.766044 + 0.642788i 0.766044 + 0.642788i 0.939693 − 0.342020i \(-0.111111\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(44\) −0.326352 − 1.85083i −0.326352 − 1.85083i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.939693 − 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(48\) 0.326352 + 0.118782i 0.326352 + 0.118782i
\(49\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(50\) 0 0
\(51\) −0.0603074 − 0.342020i −0.0603074 − 0.342020i
\(52\) 0 0
\(53\) 0 0 0.766044 − 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(54\) −0.113341 + 0.642788i −0.113341 + 0.642788i
\(55\) 0 0
\(56\) 0 0
\(57\) 0.173648 − 0.300767i 0.173648 − 0.300767i
\(58\) 0 0
\(59\) −0.326352 + 0.118782i −0.326352 + 0.118782i −0.500000 − 0.866025i \(-0.666667\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(60\) 0 0
\(61\) 0 0 0.766044 − 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(65\) 0 0
\(66\) 0.613341 + 0.223238i 0.613341 + 0.223238i
\(67\) 0.326352 + 0.118782i 0.326352 + 0.118782i 0.500000 − 0.866025i \(-0.333333\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(68\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.766044 − 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(72\) 0.673648 − 0.565258i 0.673648 − 0.565258i
\(73\) −0.266044 + 1.50881i −0.266044 + 1.50881i 0.500000 + 0.866025i \(0.333333\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −0.939693 + 0.342020i −0.939693 + 0.342020i
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.173648 − 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(80\) 0 0
\(81\) 0.500000 + 0.419550i 0.500000 + 0.419550i
\(82\) −0.0603074 − 0.342020i −0.0603074 − 0.342020i
\(83\) −0.939693 − 1.62760i −0.939693 − 1.62760i −0.766044 − 0.642788i \(-0.777778\pi\)
−0.173648 − 0.984808i \(-0.555556\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0.939693 + 0.342020i 0.939693 + 0.342020i
\(87\) 0 0
\(88\) −0.939693 − 1.62760i −0.939693 − 1.62760i
\(89\) 0.347296 + 1.96962i 0.347296 + 1.96962i 0.173648 + 0.984808i \(0.444444\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0.347296 0.347296
\(97\) 1.43969 − 0.524005i 1.43969 − 0.524005i 0.500000 − 0.866025i \(-0.333333\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(98\) −0.173648 + 0.984808i −0.173648 + 0.984808i
\(99\) 1.26604 − 1.06234i 1.26604 − 1.06234i
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.cv.a.651.1 yes 6
5.2 odd 4 3800.1.cq.c.499.2 12
5.3 odd 4 3800.1.cq.c.499.1 12
5.4 even 2 3800.1.cv.e.651.1 yes 6
8.3 odd 2 CM 3800.1.cv.a.651.1 yes 6
19.4 even 9 inner 3800.1.cv.a.251.1 ✓ 6
40.3 even 4 3800.1.cq.c.499.1 12
40.19 odd 2 3800.1.cv.e.651.1 yes 6
40.27 even 4 3800.1.cq.c.499.2 12
95.4 even 18 3800.1.cv.e.251.1 yes 6
95.23 odd 36 3800.1.cq.c.99.2 12
95.42 odd 36 3800.1.cq.c.99.1 12
152.99 odd 18 inner 3800.1.cv.a.251.1 ✓ 6
760.99 odd 18 3800.1.cv.e.251.1 yes 6
760.403 even 36 3800.1.cq.c.99.2 12
760.707 even 36 3800.1.cq.c.99.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3800.1.cq.c.99.1 12 95.42 odd 36
3800.1.cq.c.99.1 12 760.707 even 36
3800.1.cq.c.99.2 12 95.23 odd 36
3800.1.cq.c.99.2 12 760.403 even 36
3800.1.cq.c.499.1 12 5.3 odd 4
3800.1.cq.c.499.1 12 40.3 even 4
3800.1.cq.c.499.2 12 5.2 odd 4
3800.1.cq.c.499.2 12 40.27 even 4
3800.1.cv.a.251.1 ✓ 6 19.4 even 9 inner
3800.1.cv.a.251.1 ✓ 6 152.99 odd 18 inner
3800.1.cv.a.651.1 yes 6 1.1 even 1 trivial
3800.1.cv.a.651.1 yes 6 8.3 odd 2 CM
3800.1.cv.e.251.1 yes 6 95.4 even 18
3800.1.cv.e.251.1 yes 6 760.99 odd 18
3800.1.cv.e.651.1 yes 6 5.4 even 2
3800.1.cv.e.651.1 yes 6 40.19 odd 2