↑

Properties

Label 3800.1.cv.a.1651.1
Level $3800$
Weight $1$
Character 3800.1651
Analytic conductor $1.896$
Analytic rank $0$
Dimension $6$
Projective image $D_{9}$
CM discriminant -8
Inner twists $4$

Related objects

Downloads

Learn more

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 1651.1
Root \(0.939693 + 0.342020i\) of defining polynomial
Character \(\chi\) \(=\) 3800.1651
Dual form 3800.1.cv.a.2251.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.173648 + 0.984808i) q^{2} +(-1.17365 - 0.984808i) q^{3} +(-0.939693 - 0.342020i) q^{4} +(1.17365 - 0.984808i) q^{6} +(0.500000 - 0.866025i) q^{8} +(0.233956 + 1.32683i) q^{9} +(-0.173648 + 0.300767i) q^{11} +(0.766044 + 1.32683i) q^{12} +(0.766044 + 0.642788i) q^{16} +(0.173648 - 0.984808i) q^{17} -1.34730 q^{18} +(0.173648 + 0.984808i) q^{19} +(-0.266044 - 0.223238i) q^{22} +(-1.43969 + 0.524005i) q^{24} +(0.266044 - 0.460802i) q^{27} +(-0.766044 + 0.642788i) q^{32} +(0.500000 - 0.181985i) q^{33} +(0.939693 + 0.342020i) q^{34} +(0.233956 - 1.32683i) q^{36} -1.00000 q^{38} +(1.17365 + 0.984808i) q^{41} +(-0.939693 + 0.342020i) q^{43} +(0.266044 - 0.223238i) q^{44} +(-0.266044 - 1.50881i) q^{48} +(-0.500000 + 0.866025i) q^{49} +(-1.17365 + 0.984808i) q^{51} +(0.407604 + 0.342020i) q^{54} +(0.766044 - 1.32683i) q^{57} +(0.266044 - 1.50881i) q^{59} +(-0.500000 - 0.866025i) q^{64} +(0.0923963 + 0.524005i) q^{66} +(-0.266044 - 1.50881i) q^{67} +(-0.500000 + 0.866025i) q^{68} +(1.26604 + 0.460802i) q^{72} +(1.43969 + 1.20805i) q^{73} +(0.173648 - 0.984808i) q^{76} +(0.500000 - 0.181985i) q^{81} +(-1.17365 + 0.984808i) q^{82} +(0.173648 + 0.300767i) q^{83} +(-0.173648 - 0.984808i) q^{86} +(0.173648 + 0.300767i) q^{88} +(1.53209 - 1.28558i) q^{89} +1.53209 q^{96} +(0.326352 - 1.85083i) q^{97} +(-0.766044 - 0.642788i) q^{98} +(-0.439693 - 0.160035i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{3} + 6 q^{6} + 3 q^{8} + 6 q^{9} - 6 q^{18} + 3 q^{22} - 3 q^{24} - 3 q^{27} + 3 q^{33} + 6 q^{36} - 6 q^{38} + 6 q^{41} - 3 q^{44} + 3 q^{48} - 3 q^{49} - 6 q^{51} + 6 q^{54} - 3 q^{59} - 3 q^{64}+ \cdots + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(401\) \(951\) \(1901\) \(1977\)
\(\chi(n)\) \(e\left(\frac{5}{9}\right)\) \(-1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.173648 + 0.984808i −0.173648 + 0.984808i
\(3\) −1.17365 − 0.984808i −1.17365 − 0.984808i −0.173648 − 0.984808i \(-0.555556\pi\)
−1.00000 \(\pi\)
\(4\) −0.939693 − 0.342020i −0.939693 − 0.342020i
\(5\) 0 0
\(6\) 1.17365 − 0.984808i 1.17365 − 0.984808i
\(7\) 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
\(9\) 0.233956 + 1.32683i 0.233956 + 1.32683i
\(10\) 0 0
\(11\) −0.173648 + 0.300767i −0.173648 + 0.300767i −0.939693 − 0.342020i \(-0.888889\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(12\) 0.766044 + 1.32683i 0.766044 + 1.32683i
\(13\) 0 0 0.766044 − 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.766044 + 0.642788i 0.766044 + 0.642788i
\(17\) 0.173648 − 0.984808i 0.173648 − 0.984808i −0.766044 − 0.642788i \(-0.777778\pi\)
0.939693 − 0.342020i \(-0.111111\pi\)
\(18\) −1.34730 −1.34730
\(19\) 0.173648 + 0.984808i 0.173648 + 0.984808i
\(20\) 0 0
\(21\) 0 0
\(22\) −0.266044 − 0.223238i −0.266044 − 0.223238i
\(23\) 0 0 −0.939693 − 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(24\) −1.43969 + 0.524005i −1.43969 + 0.524005i
\(25\) 0 0
\(26\) 0 0
\(27\) 0.266044 − 0.460802i 0.266044 − 0.460802i
\(28\) 0 0
\(29\) 0 0 −0.173648 − 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(30\) 0 0
\(31\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(32\) −0.766044 + 0.642788i −0.766044 + 0.642788i
\(33\) 0.500000 − 0.181985i 0.500000 − 0.181985i
\(34\) 0.939693 + 0.342020i 0.939693 + 0.342020i
\(35\) 0 0
\(36\) 0.233956 − 1.32683i 0.233956 − 1.32683i
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) −1.00000 −1.00000
\(39\) 0 0
\(40\) 0 0
\(41\) 1.17365 + 0.984808i 1.17365 + 0.984808i 1.00000 \(0\)
0.173648 + 0.984808i \(0.444444\pi\)
\(42\) 0 0
\(43\) −0.939693 + 0.342020i −0.939693 + 0.342020i −0.766044 − 0.642788i \(-0.777778\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(44\) 0.266044 − 0.223238i 0.266044 − 0.223238i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.173648 − 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(48\) −0.266044 − 1.50881i −0.266044 − 1.50881i
\(49\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(50\) 0 0
\(51\) −1.17365 + 0.984808i −1.17365 + 0.984808i
\(52\) 0 0
\(53\) 0 0 −0.939693 − 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(54\) 0.407604 + 0.342020i 0.407604 + 0.342020i
\(55\) 0 0
\(56\) 0 0
\(57\) 0.766044 − 1.32683i 0.766044 − 1.32683i
\(58\) 0 0
\(59\) 0.266044 − 1.50881i 0.266044 − 1.50881i −0.500000 − 0.866025i \(-0.666667\pi\)
0.766044 − 0.642788i \(-0.222222\pi\)
\(60\) 0 0
\(61\) 0 0 −0.939693 − 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(65\) 0 0
\(66\) 0.0923963 + 0.524005i 0.0923963 + 0.524005i
\(67\) −0.266044 − 1.50881i −0.266044 − 1.50881i −0.766044 − 0.642788i \(-0.777778\pi\)
0.500000 − 0.866025i \(-0.333333\pi\)
\(68\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.939693 − 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(72\) 1.26604 + 0.460802i 1.26604 + 0.460802i
\(73\) 1.43969 + 1.20805i 1.43969 + 1.20805i 0.939693 + 0.342020i \(0.111111\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0.173648 − 0.984808i 0.173648 − 0.984808i
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.766044 − 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(80\) 0 0
\(81\) 0.500000 − 0.181985i 0.500000 − 0.181985i
\(82\) −1.17365 + 0.984808i −1.17365 + 0.984808i
\(83\) 0.173648 + 0.300767i 0.173648 + 0.300767i 0.939693 − 0.342020i \(-0.111111\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −0.173648 − 0.984808i −0.173648 − 0.984808i
\(87\) 0 0
\(88\) 0.173648 + 0.300767i 0.173648 + 0.300767i
\(89\) 1.53209 − 1.28558i 1.53209 − 1.28558i 0.766044 − 0.642788i \(-0.222222\pi\)
0.766044 − 0.642788i \(-0.222222\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 1.53209 1.53209
\(97\) 0.326352 − 1.85083i 0.326352 − 1.85083i −0.173648 − 0.984808i \(-0.555556\pi\)
0.500000 − 0.866025i \(-0.333333\pi\)
\(98\) −0.766044 − 0.642788i −0.766044 − 0.642788i
\(99\) −0.439693 − 0.160035i −0.439693 − 0.160035i
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.1651.1 ✓ 6
5.2 odd 4 3800.1.cq.c.1499.1 12
5.3 odd 4 3800.1.cq.c.1499.2 12
5.4 even 2 3800.1.cv.e.1651.1 yes 6
8.3 odd 2 CM 3800.1.cv.a.1651.1 ✓ 6
19.9 even 9 inner 3800.1.cv.a.2251.1 yes 6
40.3 even 4 3800.1.cq.c.1499.2 12
40.19 odd 2 3800.1.cv.e.1651.1 yes 6
40.27 even 4 3800.1.cq.c.1499.1 12
95.9 even 18 3800.1.cv.e.2251.1 yes 6
95.28 odd 36 3800.1.cq.c.2099.1 12
95.47 odd 36 3800.1.cq.c.2099.2 12
152.123 odd 18 inner 3800.1.cv.a.2251.1 yes 6
760.123 even 36 3800.1.cq.c.2099.1 12
760.427 even 36 3800.1.cq.c.2099.2 12
760.579 odd 18 3800.1.cv.e.2251.1 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3800.1.cq.c.1499.1 12 5.2 odd 4
3800.1.cq.c.1499.1 12 40.27 even 4
3800.1.cq.c.1499.2 12 5.3 odd 4
3800.1.cq.c.1499.2 12 40.3 even 4
3800.1.cq.c.2099.1 12 95.28 odd 36
3800.1.cq.c.2099.1 12 760.123 even 36
3800.1.cq.c.2099.2 12 95.47 odd 36
3800.1.cq.c.2099.2 12 760.427 even 36
3800.1.cv.a.1651.1 ✓ 6 1.1 even 1 trivial
3800.1.cv.a.1651.1 ✓ 6 8.3 odd 2 CM
3800.1.cv.a.2251.1 yes 6 19.9 even 9 inner
3800.1.cv.a.2251.1 yes 6 152.123 odd 18 inner
3800.1.cv.e.1651.1 yes 6 5.4 even 2
3800.1.cv.e.1651.1 yes 6 40.19 odd 2
3800.1.cv.e.2251.1 yes 6 95.9 even 18
3800.1.cv.e.2251.1 yes 6 760.579 odd 18