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Properties

Label 3700.1.cc.a.3351.2
Level $3700$
Weight $1$
Character 3700.3351
Analytic conductor $1.847$
Analytic rank $0$
Dimension $12$
Projective image $D_{18}$
CM discriminant -4
Inner twists $8$

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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] = [3700,1,Mod(151,3700)] 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("3700.151"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3700, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([9, 0, 13])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 3700 = 2^{2} \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3700.cc (of order \(18\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.84654054674\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{18})\)
Coefficient field: \(\Q(\zeta_{36})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{6} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 740)
Projective image: \(D_{18}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{18} - \cdots)\)

Embedding invariants

Embedding label 3351.2
Root \(-0.984808 - 0.173648i\) of defining polynomial
Character \(\chi\) \(=\) 3700.3351
Dual form 3700.1.cc.a.1251.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.342020 - 0.939693i) q^{2} +(-0.766044 - 0.642788i) q^{4} +(-0.866025 + 0.500000i) q^{8} +(0.766044 - 0.642788i) q^{9} +(0.642788 - 0.766044i) q^{13} +(0.173648 + 0.984808i) q^{16} +(-0.223238 - 0.266044i) q^{17} +(-0.342020 - 0.939693i) q^{18} +(-0.500000 - 0.866025i) q^{26} +(1.11334 - 0.642788i) q^{29} +(0.984808 + 0.173648i) q^{32} +(-0.326352 + 0.118782i) q^{34} -1.00000 q^{36} +(-0.642788 + 0.766044i) q^{37} +(0.266044 + 0.223238i) q^{41} +(-0.939693 - 0.342020i) q^{49} +(-0.984808 + 0.173648i) q^{52} +(-0.300767 - 1.70574i) q^{53} +(-0.223238 - 1.26604i) q^{58} +(0.826352 - 0.984808i) q^{61} +(0.500000 - 0.866025i) q^{64} +0.347296i q^{68} +(-0.342020 + 0.939693i) q^{72} -1.73205 q^{73} +(0.500000 + 0.866025i) q^{74} +(0.173648 - 0.984808i) q^{81} +(0.300767 - 0.173648i) q^{82} +(1.93969 - 0.342020i) q^{89} +(-1.32683 - 0.766044i) q^{97} +(-0.642788 + 0.766044i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{26} - 6 q^{34} - 12 q^{36} - 6 q^{41} + 12 q^{61} + 6 q^{64} + 6 q^{74} + 12 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3700\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1777\) \(1851\)
\(\chi(n)\) \(e\left(\frac{11}{18}\right)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.342020 − 0.939693i 0.342020 − 0.939693i
\(3\) 0 0 0.939693 − 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(4\) −0.766044 − 0.642788i −0.766044 − 0.642788i
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 0.173648 − 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(8\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(9\) 0.766044 − 0.642788i 0.766044 − 0.642788i
\(10\) 0 0
\(11\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(12\) 0 0
\(13\) 0.642788 − 0.766044i 0.642788 − 0.766044i −0.342020 − 0.939693i \(-0.611111\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.173648 + 0.984808i 0.173648 + 0.984808i
\(17\) −0.223238 − 0.266044i −0.223238 − 0.266044i 0.642788 − 0.766044i \(-0.277778\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(18\) −0.342020 − 0.939693i −0.342020 − 0.939693i
\(19\) 0 0 −0.342020 − 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(27\) 0 0
\(28\) 0 0
\(29\) 1.11334 − 0.642788i 1.11334 − 0.642788i 0.173648 − 0.984808i \(-0.444444\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0.984808 + 0.173648i 0.984808 + 0.173648i
\(33\) 0 0
\(34\) −0.326352 + 0.118782i −0.326352 + 0.118782i
\(35\) 0 0
\(36\) −1.00000 −1.00000
\(37\) −0.642788 + 0.766044i −0.642788 + 0.766044i
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.266044 + 0.223238i 0.266044 + 0.223238i 0.766044 − 0.642788i \(-0.222222\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(48\) 0 0
\(49\) −0.939693 − 0.342020i −0.939693 − 0.342020i
\(50\) 0 0
\(51\) 0 0
\(52\) −0.984808 + 0.173648i −0.984808 + 0.173648i
\(53\) −0.300767 − 1.70574i −0.300767 − 1.70574i −0.642788 − 0.766044i \(-0.722222\pi\)
0.342020 − 0.939693i \(-0.388889\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −0.223238 − 1.26604i −0.223238 − 1.26604i
\(59\) 0 0 0.984808 − 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(60\) 0 0
\(61\) 0.826352 − 0.984808i 0.826352 − 0.984808i −0.173648 − 0.984808i \(-0.555556\pi\)
1.00000 \(0\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0.500000 − 0.866025i 0.500000 − 0.866025i
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 0.173648 − 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(68\) 0.347296i 0.347296i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.939693 − 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(72\) −0.342020 + 0.939693i −0.342020 + 0.939693i
\(73\) −1.73205 −1.73205 −0.866025 − 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(74\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.984808 − 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(80\) 0 0
\(81\) 0.173648 − 0.984808i 0.173648 − 0.984808i
\(82\) 0.300767 − 0.173648i 0.300767 − 0.173648i
\(83\) 0 0 0.766044 − 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.93969 − 0.342020i 1.93969 − 0.342020i 0.939693 − 0.342020i \(-0.111111\pi\)
1.00000 \(0\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.32683 − 0.766044i −1.32683 − 0.766044i −0.342020 − 0.939693i \(-0.611111\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(98\) −0.642788 + 0.766044i −0.642788 + 0.766044i
\(99\) 0 0
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 3700.1.cc.a.3351.2 12
4.3 odd 2 CM 3700.1.cc.a.3351.2 12
5.2 odd 4 740.1.bu.b.539.1 yes 6
5.3 odd 4 740.1.bu.a.539.1 ✓ 6
5.4 even 2 inner 3700.1.cc.a.3351.1 12
20.3 even 4 740.1.bu.a.539.1 ✓ 6
20.7 even 4 740.1.bu.b.539.1 yes 6
20.19 odd 2 inner 3700.1.cc.a.3351.1 12
37.30 even 18 inner 3700.1.cc.a.1251.2 12
148.67 odd 18 inner 3700.1.cc.a.1251.2 12
185.67 odd 36 740.1.bu.a.659.1 yes 6
185.104 even 18 inner 3700.1.cc.a.1251.1 12
185.178 odd 36 740.1.bu.b.659.1 yes 6
740.67 even 36 740.1.bu.a.659.1 yes 6
740.363 even 36 740.1.bu.b.659.1 yes 6
740.659 odd 18 inner 3700.1.cc.a.1251.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
740.1.bu.a.539.1 ✓ 6 5.3 odd 4
740.1.bu.a.539.1 ✓ 6 20.3 even 4
740.1.bu.a.659.1 yes 6 185.67 odd 36
740.1.bu.a.659.1 yes 6 740.67 even 36
740.1.bu.b.539.1 yes 6 5.2 odd 4
740.1.bu.b.539.1 yes 6 20.7 even 4
740.1.bu.b.659.1 yes 6 185.178 odd 36
740.1.bu.b.659.1 yes 6 740.363 even 36
3700.1.cc.a.1251.1 12 185.104 even 18 inner
3700.1.cc.a.1251.1 12 740.659 odd 18 inner
3700.1.cc.a.1251.2 12 37.30 even 18 inner
3700.1.cc.a.1251.2 12 148.67 odd 18 inner
3700.1.cc.a.3351.1 12 5.4 even 2 inner
3700.1.cc.a.3351.1 12 20.19 odd 2 inner
3700.1.cc.a.3351.2 12 1.1 even 1 trivial
3700.1.cc.a.3351.2 12 4.3 odd 2 CM