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 / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [13,6,Mod(12,13)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.08498965757\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
comment: defining polynomial
 
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 3) q^{3} + (\beta_{5} - \beta_{2} - 22) q^{4} + (\beta_{3} - \beta_1) q^{5} + (\beta_{4} - \beta_{3} + 9 \beta_1) q^{6} + ( - \beta_{4} - 2 \beta_{3} + 3 \beta_1) q^{7} + ( - \beta_{4} + 3 \beta_{3} - 23 \beta_1) q^{8} + ( - 6 \beta_{5} - \beta_{2} + 144) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 3) q^{3} + (\beta_{5} - \beta_{2} - 22) q^{4} + (\beta_{3} - \beta_1) q^{5} + (\beta_{4} - \beta_{3} + 9 \beta_1) q^{6} + ( - \beta_{4} - 2 \beta_{3} + 3 \beta_1) q^{7} + ( - \beta_{4} + 3 \beta_{3} - 23 \beta_1) q^{8} + ( - 6 \beta_{5} - \beta_{2} + 144) q^{9} + ( - 5 \beta_{5} + 11 \beta_{2} + 84) q^{10} + (\beta_{4} - 5 \beta_{3} - 30 \beta_1) q^{11} + (15 \beta_{5} - 43 \beta_{2} - 426) q^{12} + ( - 10 \beta_{5} + \beta_{4} + \cdots + 80) q^{13}+ \cdots + (605 \beta_{4} + 463 \beta_{3} - 5766 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 16 q^{3} - 130 q^{4} + 866 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 16 q^{3} - 130 q^{4} + 866 q^{9} + 482 q^{10} - 2470 q^{12} + 502 q^{13} - 1362 q^{14} + 3650 q^{16} - 1392 q^{17} + 8936 q^{22} - 9504 q^{23} + 2714 q^{25} - 16026 q^{26} - 4808 q^{27} + 15036 q^{29} + 25590 q^{30} + 26856 q^{35} - 70724 q^{36} + 15420 q^{38} - 25336 q^{39} - 63446 q^{40} + 73074 q^{42} + 25584 q^{43} + 184606 q^{48} - 65058 q^{49} - 76152 q^{51} - 77660 q^{52} - 65388 q^{53} + 69008 q^{55} - 3018 q^{56} + 172508 q^{61} - 163272 q^{62} - 37730 q^{64} - 828 q^{65} - 150816 q^{66} + 262482 q^{68} + 97848 q^{69} - 31170 q^{74} - 126136 q^{75} + 24216 q^{77} - 234102 q^{78} + 121904 q^{79} - 105370 q^{81} + 137312 q^{82} - 104784 q^{87} + 181600 q^{88} + 309996 q^{90} + 92352 q^{91} + 91260 q^{92} - 297298 q^{94} - 163584 q^{95}+O(q^{100}) \) 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
\(\nu\)\(=\) \( \beta_1 \) 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

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.

comment: embeddings in the coefficient field
 
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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