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Properties

Label 3800.1.y.g.1443.2
Level $3800$
Weight $1$
Character 3800.1443
Analytic conductor $1.896$
Analytic rank $0$
Dimension $8$
Projective image $D_{6}$
CM discriminant -8
Inner twists $16$

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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(1443,3800)] 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("3800.1443"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3800, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 3, 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.y (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.89644704801\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.1371800000.3

Embedding invariants

Embedding label 1443.2
Root \(0.965926 + 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 3800.1443
Dual form 3800.1.y.g.2507.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.707107 - 0.707107i) q^{2} +(-0.707107 + 0.707107i) q^{3} +1.00000i q^{4} +1.00000 q^{6} +(0.707107 - 0.707107i) q^{8} +1.00000 q^{11} +(-0.707107 - 0.707107i) q^{12} -1.00000 q^{16} +(1.22474 - 1.22474i) q^{17} +(0.866025 + 0.500000i) q^{19} +(-0.707107 - 0.707107i) q^{22} +1.00000i q^{24} +(-0.707107 - 0.707107i) q^{27} +(0.707107 + 0.707107i) q^{32} +(-0.707107 + 0.707107i) q^{33} -1.73205 q^{34} +(-0.258819 - 0.965926i) q^{38} -1.73205i q^{41} +1.00000i q^{44} +(0.707107 - 0.707107i) q^{48} -1.00000i q^{49} +1.73205i q^{51} +1.00000i q^{54} +(-0.965926 + 0.258819i) q^{57} -1.00000i q^{64} +1.00000 q^{66} +(-0.707107 - 0.707107i) q^{67} +(1.22474 + 1.22474i) q^{68} +(-1.22474 - 1.22474i) q^{73} +(-0.500000 + 0.866025i) q^{76} +1.00000 q^{81} +(-1.22474 + 1.22474i) q^{82} +(1.22474 + 1.22474i) q^{83} +(0.707107 - 0.707107i) q^{88} +1.73205 q^{89} -1.00000 q^{96} +(1.41421 + 1.41421i) q^{97} +(-0.707107 + 0.707107i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{6} + 8 q^{11} - 8 q^{16} + 8 q^{66} - 4 q^{76} + 8 q^{81} - 8 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(401\) \(951\) \(1901\) \(1977\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.707107 − 0.707107i −0.707107 − 0.707107i
\(3\) −0.707107 + 0.707107i −0.707107 + 0.707107i −0.965926 − 0.258819i \(-0.916667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(4\) 1.00000i 1.00000i
\(5\) 0 0
\(6\) 1.00000 1.00000
\(7\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(8\) 0.707107 − 0.707107i 0.707107 − 0.707107i
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(12\) −0.707107 − 0.707107i −0.707107 − 0.707107i
\(13\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −1.00000 −1.00000
\(17\) 1.22474 − 1.22474i 1.22474 − 1.22474i 0.258819 − 0.965926i \(-0.416667\pi\)
0.965926 − 0.258819i \(-0.0833333\pi\)
\(18\) 0 0
\(19\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(20\) 0 0
\(21\) 0 0
\(22\) −0.707107 − 0.707107i −0.707107 − 0.707107i
\(23\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(24\) 1.00000i 1.00000i
\(25\) 0 0
\(26\) 0 0
\(27\) −0.707107 − 0.707107i −0.707107 − 0.707107i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(33\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(34\) −1.73205 −1.73205
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(38\) −0.258819 − 0.965926i −0.258819 − 0.965926i
\(39\) 0 0
\(40\) 0 0
\(41\) − 1.73205i − 1.73205i −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 − 0.866025i \(-0.333333\pi\)
\(42\) 0 0
\(43\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(44\) 1.00000i 1.00000i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(48\) 0.707107 − 0.707107i 0.707107 − 0.707107i
\(49\) − 1.00000i − 1.00000i
\(50\) 0 0
\(51\) 1.73205i 1.73205i
\(52\) 0 0
\(53\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(54\) 1.00000i 1.00000i
\(55\) 0 0
\(56\) 0 0
\(57\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) − 1.00000i − 1.00000i
\(65\) 0 0
\(66\) 1.00000 1.00000
\(67\) −0.707107 − 0.707107i −0.707107 − 0.707107i 0.258819 − 0.965926i \(-0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(68\) 1.22474 + 1.22474i 1.22474 + 1.22474i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −1.22474 − 1.22474i −1.22474 − 1.22474i −0.965926 − 0.258819i \(-0.916667\pi\)
−0.258819 − 0.965926i \(-0.583333\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 1.00000 1.00000
\(82\) −1.22474 + 1.22474i −1.22474 + 1.22474i
\(83\) 1.22474 + 1.22474i 1.22474 + 1.22474i 0.965926 + 0.258819i \(0.0833333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0.707107 − 0.707107i 0.707107 − 0.707107i
\(89\) 1.73205 1.73205 0.866025 − 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) −1.00000 −1.00000
\(97\) 1.41421 + 1.41421i 1.41421 + 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(98\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(99\) 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 3800.1.y.g.1443.2 yes 8
5.2 odd 4 inner 3800.1.y.g.2507.4 yes 8
5.3 odd 4 inner 3800.1.y.g.2507.1 yes 8
5.4 even 2 inner 3800.1.y.g.1443.3 yes 8
8.3 odd 2 CM 3800.1.y.g.1443.2 yes 8
19.18 odd 2 inner 3800.1.y.g.1443.4 yes 8
40.3 even 4 inner 3800.1.y.g.2507.1 yes 8
40.19 odd 2 inner 3800.1.y.g.1443.3 yes 8
40.27 even 4 inner 3800.1.y.g.2507.4 yes 8
95.18 even 4 inner 3800.1.y.g.2507.3 yes 8
95.37 even 4 inner 3800.1.y.g.2507.2 yes 8
95.94 odd 2 inner 3800.1.y.g.1443.1 ✓ 8
152.75 even 2 inner 3800.1.y.g.1443.4 yes 8
760.227 odd 4 inner 3800.1.y.g.2507.2 yes 8
760.379 even 2 inner 3800.1.y.g.1443.1 ✓ 8
760.683 odd 4 inner 3800.1.y.g.2507.3 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3800.1.y.g.1443.1 ✓ 8 95.94 odd 2 inner
3800.1.y.g.1443.1 ✓ 8 760.379 even 2 inner
3800.1.y.g.1443.2 yes 8 1.1 even 1 trivial
3800.1.y.g.1443.2 yes 8 8.3 odd 2 CM
3800.1.y.g.1443.3 yes 8 5.4 even 2 inner
3800.1.y.g.1443.3 yes 8 40.19 odd 2 inner
3800.1.y.g.1443.4 yes 8 19.18 odd 2 inner
3800.1.y.g.1443.4 yes 8 152.75 even 2 inner
3800.1.y.g.2507.1 yes 8 5.3 odd 4 inner
3800.1.y.g.2507.1 yes 8 40.3 even 4 inner
3800.1.y.g.2507.2 yes 8 95.37 even 4 inner
3800.1.y.g.2507.2 yes 8 760.227 odd 4 inner
3800.1.y.g.2507.3 yes 8 95.18 even 4 inner
3800.1.y.g.2507.3 yes 8 760.683 odd 4 inner
3800.1.y.g.2507.4 yes 8 5.2 odd 4 inner
3800.1.y.g.2507.4 yes 8 40.27 even 4 inner