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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [238,2,Mod(169,238)] 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("238.169"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(238, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 238 = 2 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 238.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6] 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: \(1.90043956811\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + (\beta_{4} + \beta_1) q^{3} + q^{4} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_1) q^{5} + (\beta_{4} + \beta_1) q^{6} - \beta_{4} q^{7} + q^{8} + (\beta_{5} + \beta_{3} - 2 \beta_{2} - 1) q^{9}+ \cdots + (\beta_{5} + 9 \beta_{4} + \cdots + 5 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} + 6 q^{4} + 6 q^{8} - 6 q^{9} - 8 q^{13} + 6 q^{16} - 2 q^{17} - 6 q^{18} + 4 q^{19} + 4 q^{21} - 10 q^{25} - 8 q^{26} + 6 q^{32} + 8 q^{33} - 2 q^{34} - 4 q^{35} - 6 q^{36} + 4 q^{38} + 4 q^{42}+ \cdots - 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -3\nu^{5} + \nu^{4} + 11\nu^{3} - 26\nu^{2} + 6\nu - 1 ) / 23 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4\nu^{5} + 9\nu^{4} - 16\nu^{3} - 4\nu^{2} + 8\nu - 9 ) / 23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6\nu^{5} - 2\nu^{4} + \nu^{3} + 6\nu^{2} + 80\nu + 2 ) / 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -16\nu^{5} + 36\nu^{4} - 41\nu^{3} - 16\nu^{2} - 60\nu + 56 ) / 23 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} + \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 4\beta_{4} - \beta_{3} + 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + 2\beta_{4} - 2\beta_{2} + 2\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} + 2\beta_{3} - 5\beta_{2} - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -9\beta_{4} + 5\beta_{3} - 8\beta_{2} - 8\beta _1 - 9 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/238\mathbb{Z}\right)^\times\).

\(n\) \(71\) \(171\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
169.1
1.45161 − 1.45161i
−0.854638 − 0.854638i
0.403032 − 0.403032i
0.403032 + 0.403032i
−0.854638 + 0.854638i
1.45161 + 1.45161i
1.00000 − 2.90321i 1.00000 − 1.52543i − 2.90321i 1.00000i 1.00000 −5.42864 − 1.52543i
169.2 1.00000 − 1.70928i 1.00000 0.630898i − 1.70928i − 1.00000i 1.00000 0.0783777 0.630898i
169.3 1.00000 − 0.806063i 1.00000 4.15633i − 0.806063i 1.00000i 1.00000 2.35026 4.15633i
169.4 1.00000 0.806063i 1.00000 − 4.15633i 0.806063i − 1.00000i 1.00000 2.35026 − 4.15633i
169.5 1.00000 1.70928i 1.00000 − 0.630898i 1.70928i 1.00000i 1.00000 0.0783777 − 0.630898i
169.6 1.00000 2.90321i 1.00000 1.52543i 2.90321i − 1.00000i 1.00000 −5.42864 1.52543i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 169.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 238.2.b.b ✓ 6
3.b odd 2 1 2142.2.b.g 6
4.b odd 2 1 1904.2.c.e 6
7.b odd 2 1 1666.2.b.m 6
17.b even 2 1 inner 238.2.b.b ✓ 6
17.c even 4 1 4046.2.a.x 3
17.c even 4 1 4046.2.a.ba 3
51.c odd 2 1 2142.2.b.g 6
68.d odd 2 1 1904.2.c.e 6
119.d odd 2 1 1666.2.b.m 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
238.2.b.b ✓ 6 1.a even 1 1 trivial
238.2.b.b ✓ 6 17.b even 2 1 inner
1666.2.b.m 6 7.b odd 2 1
1666.2.b.m 6 119.d odd 2 1
1904.2.c.e 6 4.b odd 2 1
1904.2.c.e 6 68.d odd 2 1
2142.2.b.g 6 3.b odd 2 1
2142.2.b.g 6 51.c odd 2 1
4046.2.a.x 3 17.c even 4 1
4046.2.a.ba 3 17.c even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 12T_{3}^{4} + 32T_{3}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(238, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 12 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{6} + 20 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{6} + 24 T^{4} + \cdots + 400 \) Copy content Toggle raw display
$13$ \( (T^{3} + 4 T^{2} - 8 T - 16)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + 2 T^{5} + \cdots + 4913 \) Copy content Toggle raw display
$19$ \( (T^{3} - 2 T^{2} - 52 T + 40)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 16)^{3} \) Copy content Toggle raw display
$29$ \( T^{6} + 72 T^{4} + \cdots + 13456 \) Copy content Toggle raw display
$31$ \( T^{6} + 96 T^{4} + \cdots + 25600 \) Copy content Toggle raw display
$37$ \( T^{6} + 56 T^{4} + \cdots + 2704 \) Copy content Toggle raw display
$41$ \( T^{6} + 128 T^{4} + \cdots + 6400 \) Copy content Toggle raw display
$43$ \( (T^{3} + 8 T^{2} + \cdots - 256)^{2} \) Copy content Toggle raw display
$47$ \( (T^{3} + 4 T^{2} - 88 T + 16)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} - 2 T^{2} + \cdots + 104)^{2} \) Copy content Toggle raw display
$59$ \( (T^{3} + 18 T^{2} + \cdots - 40)^{2} \) Copy content Toggle raw display
$61$ \( T^{6} + 244 T^{4} + \cdots + 336400 \) Copy content Toggle raw display
$67$ \( (T^{3} + 4 T^{2} - 48 T - 64)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} + 96 T^{4} + \cdots + 25600 \) Copy content Toggle raw display
$73$ \( T^{6} + 240 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$79$ \( T^{6} + 224 T^{4} + \cdots + 173056 \) Copy content Toggle raw display
$83$ \( (T^{3} - 6 T^{2} + \cdots + 1528)^{2} \) Copy content Toggle raw display
$89$ \( (T + 10)^{6} \) Copy content Toggle raw display
$97$ \( T^{6} + 160 T^{4} + \cdots + 43264 \) Copy content Toggle raw display
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