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Properties

Label 38.2.c.b
Level $38$
Weight $2$
Character orbit 38.c
Analytic conductor $0.303$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [38,2,Mod(7,38)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(38, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("38.7"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 38.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.303431527681\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{7})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{3} + \beta_1) q^{3} + ( - \beta_{2} - 1) q^{4} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{5} - \beta_1 q^{6} + ( - \beta_{3} + 1) q^{7} + q^{8} + ( - 4 \beta_{2} - 4) q^{9}+ \cdots + (8 \beta_{2} + 4 \beta_1 + 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 2 q^{4} - 2 q^{5} + 4 q^{7} + 4 q^{8} - 8 q^{9} - 2 q^{10} - 8 q^{11} - 4 q^{13} - 2 q^{14} + 14 q^{15} - 2 q^{16} + 16 q^{18} + 12 q^{19} + 4 q^{20} - 14 q^{21} + 4 q^{22} - 2 q^{23} - 6 q^{25}+ \cdots + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 7x^{2} + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 7 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 7\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 7\beta_{3} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/38\mathbb{Z}\right)^\times\).

\(n\) \(21\)
\(\chi(n)\) \(-1 - \beta_{2}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
1.32288 − 2.29129i
−1.32288 + 2.29129i
1.32288 + 2.29129i
−1.32288 − 2.29129i
−0.500000 − 0.866025i −1.32288 − 2.29129i −0.500000 + 0.866025i 0.822876 + 1.42526i −1.32288 + 2.29129i 3.64575 1.00000 −2.00000 + 3.46410i 0.822876 − 1.42526i
7.2 −0.500000 − 0.866025i 1.32288 + 2.29129i −0.500000 + 0.866025i −1.82288 − 3.15731i 1.32288 − 2.29129i −1.64575 1.00000 −2.00000 + 3.46410i −1.82288 + 3.15731i
11.1 −0.500000 + 0.866025i −1.32288 + 2.29129i −0.500000 − 0.866025i 0.822876 − 1.42526i −1.32288 − 2.29129i 3.64575 1.00000 −2.00000 − 3.46410i 0.822876 + 1.42526i
11.2 −0.500000 + 0.866025i 1.32288 − 2.29129i −0.500000 − 0.866025i −1.82288 + 3.15731i 1.32288 + 2.29129i −1.64575 1.00000 −2.00000 − 3.46410i −1.82288 − 3.15731i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 38.2.c.b ✓ 4
3.b odd 2 1 342.2.g.f 4
4.b odd 2 1 304.2.i.e 4
5.b even 2 1 950.2.e.k 4
5.c odd 4 2 950.2.j.g 8
8.b even 2 1 1216.2.i.l 4
8.d odd 2 1 1216.2.i.k 4
12.b even 2 1 2736.2.s.v 4
19.b odd 2 1 722.2.c.j 4
19.c even 3 1 inner 38.2.c.b ✓ 4
19.c even 3 1 722.2.a.j 2
19.d odd 6 1 722.2.a.g 2
19.d odd 6 1 722.2.c.j 4
19.e even 9 6 722.2.e.n 12
19.f odd 18 6 722.2.e.o 12
57.f even 6 1 6498.2.a.bg 2
57.h odd 6 1 342.2.g.f 4
57.h odd 6 1 6498.2.a.ba 2
76.f even 6 1 5776.2.a.z 2
76.g odd 6 1 304.2.i.e 4
76.g odd 6 1 5776.2.a.ba 2
95.i even 6 1 950.2.e.k 4
95.m odd 12 2 950.2.j.g 8
152.k odd 6 1 1216.2.i.k 4
152.p even 6 1 1216.2.i.l 4
228.m even 6 1 2736.2.s.v 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.c.b ✓ 4 1.a even 1 1 trivial
38.2.c.b ✓ 4 19.c even 3 1 inner
304.2.i.e 4 4.b odd 2 1
304.2.i.e 4 76.g odd 6 1
342.2.g.f 4 3.b odd 2 1
342.2.g.f 4 57.h odd 6 1
722.2.a.g 2 19.d odd 6 1
722.2.a.j 2 19.c even 3 1
722.2.c.j 4 19.b odd 2 1
722.2.c.j 4 19.d odd 6 1
722.2.e.n 12 19.e even 9 6
722.2.e.o 12 19.f odd 18 6
950.2.e.k 4 5.b even 2 1
950.2.e.k 4 95.i even 6 1
950.2.j.g 8 5.c odd 4 2
950.2.j.g 8 95.m odd 12 2
1216.2.i.k 4 8.d odd 2 1
1216.2.i.k 4 152.k odd 6 1
1216.2.i.l 4 8.b even 2 1
1216.2.i.l 4 152.p even 6 1
2736.2.s.v 4 12.b even 2 1
2736.2.s.v 4 228.m even 6 1
5776.2.a.z 2 76.f even 6 1
5776.2.a.ba 2 76.g odd 6 1
6498.2.a.ba 2 57.h odd 6 1
6498.2.a.bg 2 57.f even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 7T_{3}^{2} + 49 \) acting on \(S_{2}^{\mathrm{new}}(38, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 7T^{2} + 49 \) Copy content Toggle raw display
$5$ \( T^{4} + 2 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$7$ \( (T^{2} - 2 T - 6)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 4 T - 3)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} - 12 T^{3} + \cdots + 361 \) Copy content Toggle raw display
$23$ \( T^{4} + 2 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$29$ \( T^{4} - 2 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$31$ \( (T^{2} + 6 T + 2)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 6 T + 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 10 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$43$ \( T^{4} + 12 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$47$ \( T^{4} + 14 T^{3} + \cdots + 1764 \) Copy content Toggle raw display
$53$ \( T^{4} - 4 T^{3} + \cdots + 11664 \) Copy content Toggle raw display
$59$ \( T^{4} + 63T^{2} + 3969 \) Copy content Toggle raw display
$61$ \( T^{4} - 14 T^{3} + \cdots + 196 \) Copy content Toggle raw display
$67$ \( T^{4} - 4 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$71$ \( T^{4} - 16 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$73$ \( T^{4} + 14 T^{3} + \cdots + 441 \) Copy content Toggle raw display
$79$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 63)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} - 18 T^{3} + \cdots + 2809 \) Copy content Toggle raw display
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