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Properties

Label 380.1.g.a.189.1
Level $380$
Weight $1$
Character 380.189
Analytic conductor $0.190$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -19
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] = [380,1,Mod(189,380)] 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("380.189"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 380.g (of order \(2\), degree \(1\), 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: \(0.189644704801\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.722000.1

Embedding invariants

Embedding label 189.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 380.189
Dual form 380.1.g.a.189.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{5} -1.73205i q^{7} -1.00000 q^{9} +1.00000 q^{11} +1.73205i q^{17} +1.00000 q^{19} +(-0.500000 + 0.866025i) q^{25} +(-1.50000 + 0.866025i) q^{35} -1.73205i q^{43} +(0.500000 + 0.866025i) q^{45} +1.73205i q^{47} -2.00000 q^{49} +(-0.500000 - 0.866025i) q^{55} +1.00000 q^{61} +1.73205i q^{63} +1.73205i q^{73} -1.73205i q^{77} +1.00000 q^{81} +(1.50000 - 0.866025i) q^{85} +(-0.500000 - 0.866025i) q^{95} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{5} - 2 q^{9} + 2 q^{11} + 2 q^{19} - q^{25} - 3 q^{35} + q^{45} - 4 q^{49} - q^{55} + 2 q^{61} + 2 q^{81} + 3 q^{85} - q^{95} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 0
\(3\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 0 0
\(5\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(6\) 0 0
\(7\) − 1.73205i − 1.73205i −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 − 0.866025i \(-0.333333\pi\)
\(8\) 0 0
\(9\) −1.00000 −1.00000
\(10\) 0 0
\(11\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.73205i 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(18\) 0 0
\(19\) 1.00000 1.00000
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.50000 + 0.866025i −1.50000 + 0.866025i
\(36\) 0 0
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) − 1.73205i − 1.73205i −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 − 0.866025i \(-0.333333\pi\)
\(44\) 0 0
\(45\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(46\) 0 0
\(47\) 1.73205i 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(48\) 0 0
\(49\) −2.00000 −2.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(62\) 0 0
\(63\) 1.73205i 1.73205i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 1.73205i 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 1.73205i − 1.73205i
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 1.00000 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 1.50000 − 0.866025i 1.50000 − 0.866025i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −0.500000 − 0.866025i −0.500000 − 0.866025i
\(96\) 0 0
\(97\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) −1.00000 −1.00000
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 380.1.g.a.189.1 ✓ 2
3.2 odd 2 3420.1.h.a.2089.2 2
4.3 odd 2 1520.1.m.c.1329.1 2
5.2 odd 4 1900.1.e.b.1101.2 2
5.3 odd 4 1900.1.e.b.1101.1 2
5.4 even 2 inner 380.1.g.a.189.2 yes 2
15.14 odd 2 3420.1.h.a.2089.1 2
19.18 odd 2 CM 380.1.g.a.189.1 ✓ 2
20.19 odd 2 1520.1.m.c.1329.2 2
57.56 even 2 3420.1.h.a.2089.2 2
76.75 even 2 1520.1.m.c.1329.1 2
95.18 even 4 1900.1.e.b.1101.1 2
95.37 even 4 1900.1.e.b.1101.2 2
95.94 odd 2 inner 380.1.g.a.189.2 yes 2
285.284 even 2 3420.1.h.a.2089.1 2
380.379 even 2 1520.1.m.c.1329.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.1.g.a.189.1 ✓ 2 1.1 even 1 trivial
380.1.g.a.189.1 ✓ 2 19.18 odd 2 CM
380.1.g.a.189.2 yes 2 5.4 even 2 inner
380.1.g.a.189.2 yes 2 95.94 odd 2 inner
1520.1.m.c.1329.1 2 4.3 odd 2
1520.1.m.c.1329.1 2 76.75 even 2
1520.1.m.c.1329.2 2 20.19 odd 2
1520.1.m.c.1329.2 2 380.379 even 2
1900.1.e.b.1101.1 2 5.3 odd 4
1900.1.e.b.1101.1 2 95.18 even 4
1900.1.e.b.1101.2 2 5.2 odd 4
1900.1.e.b.1101.2 2 95.37 even 4
3420.1.h.a.2089.1 2 15.14 odd 2
3420.1.h.a.2089.1 2 285.284 even 2
3420.1.h.a.2089.2 2 3.2 odd 2
3420.1.h.a.2089.2 2 57.56 even 2