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Properties

Label 13.6.b.a
Level $13$
Weight $6$
Character orbit 13.b
Analytic conductor $2.085$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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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] = [13,6,Mod(12,13)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(13, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 6, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("13.12"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 13.b (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: \(2.08498965757\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 161x^{4} + 5856x^{2} + 18864 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 3) q^{3} + (\beta_{5} - \beta_{2} - 22) q^{4} + (\beta_{3} - \beta_1) q^{5} + (\beta_{4} - \beta_{3} + 9 \beta_1) q^{6} + ( - \beta_{4} - 2 \beta_{3} + 3 \beta_1) q^{7}+ \cdots + (605 \beta_{4} + 463 \beta_{3} - 5766 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 16 q^{3} - 130 q^{4} + 866 q^{9} + 482 q^{10} - 2470 q^{12} + 502 q^{13} - 1362 q^{14} + 3650 q^{16} - 1392 q^{17} + 8936 q^{22} - 9504 q^{23} + 2714 q^{25} - 16026 q^{26} - 4808 q^{27} + 15036 q^{29}+ \cdots - 163584 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 161x^{4} + 5856x^{2} + 18864 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} + 101\nu^{2} + 660 ) / 72 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} + 173\nu^{3} + 6492\nu ) / 144 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} + 125\nu^{3} + 2316\nu ) / 48 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{4} + 173\nu^{2} + 4548 ) / 72 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{2} - 54 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{4} + 3\beta_{3} - 87\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -101\beta_{5} + 173\beta_{2} + 4794 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 173\beta_{4} - 375\beta_{3} + 8559\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/13\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
12.1
− 10.4280i
− 6.97807i
− 1.88746i
1.88746i
6.97807i
10.4280i
− 10.4280i 23.8626 −76.7440 46.3074i − 248.840i 16.0658i 466.592i 326.425 482.896
12.2 − 6.97807i −23.2081 −16.6935 − 14.2993i 161.948i − 181.853i − 106.810i 295.618 −99.7813
12.3 − 1.88746i 7.34551 28.4375 − 75.2938i − 13.8644i 222.759i − 114.074i −189.043 −142.114
12.4 1.88746i 7.34551 28.4375 75.2938i 13.8644i − 222.759i 114.074i −189.043 −142.114
12.5 6.97807i −23.2081 −16.6935 14.2993i − 161.948i 181.853i 106.810i 295.618 −99.7813
12.6 10.4280i 23.8626 −76.7440 − 46.3074i 248.840i − 16.0658i − 466.592i 326.425 482.896
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 12.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 13.6.b.a ✓ 6
3.b odd 2 1 117.6.b.c 6
4.b odd 2 1 208.6.f.c 6
13.b even 2 1 inner 13.6.b.a ✓ 6
13.d odd 4 2 169.6.a.e 6
39.d odd 2 1 117.6.b.c 6
52.b odd 2 1 208.6.f.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.6.b.a ✓ 6 1.a even 1 1 trivial
13.6.b.a ✓ 6 13.b even 2 1 inner
117.6.b.c 6 3.b odd 2 1
117.6.b.c 6 39.d odd 2 1
169.6.a.e 6 13.d odd 4 2
208.6.f.c 6 4.b odd 2 1
208.6.f.c 6 52.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(13, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 161 T^{4} + \cdots + 18864 \) Copy content Toggle raw display
$3$ \( (T^{3} - 8 T^{2} + \cdots + 4068)^{2} \) Copy content Toggle raw display
$5$ \( T^{6} + \cdots + 2485690416 \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots + 423560602764 \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots + 7521473396736 \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 51\!\cdots\!57 \) Copy content Toggle raw display
$17$ \( (T^{3} + 696 T^{2} + \cdots + 227049966)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 43\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( (T^{3} + 4752 T^{2} + \cdots + 1744071264)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} - 7518 T^{2} + \cdots + 1625868288)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 52\!\cdots\!56 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 20\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 13\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( (T^{3} - 12792 T^{2} + \cdots + 191268739948)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 26\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( (T^{3} + \cdots - 1324868101272)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 14\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots - 21319109288576)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 98\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 22\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 30\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( (T^{3} + \cdots + 3525550887232)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 79\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 44\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 46\!\cdots\!36 \) Copy content Toggle raw display
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