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Properties

Label 156.2.c.c
Level $156$
Weight $2$
Character orbit 156.c
Analytic conductor $1.246$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [156,2,Mod(131,156)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(156, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("156.131");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 156 = 2^{2} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 156.c (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: \(1.24566627153\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.121550625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 4x^{6} - 9x^{5} + 23x^{4} + 18x^{3} - 16x^{2} + 8x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + ( - \beta_{6} + \beta_1 + 1) q^{3} + (\beta_{6} - \beta_{2} - \beta_1) q^{4} + (2 \beta_{6} + 2 \beta_{4} - \beta_{2} + \cdots - 1) q^{5} + (\beta_{6} + \beta_{4} - \beta_{3} - 1) q^{6}+ \cdots + ( - \beta_{7} + 2 \beta_{6} - \beta_{5} + \cdots - 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4} - 5 q^{6} + 2 q^{9} + 6 q^{10} + 15 q^{12} + 8 q^{13} - 28 q^{16} + 3 q^{18} + 22 q^{21} + 5 q^{24} - 36 q^{25} - 30 q^{28} - 25 q^{30} - 6 q^{34} + q^{36} - 4 q^{37} + 18 q^{40} - 3 q^{42}+ \cdots + 40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 4x^{6} - 9x^{5} + 23x^{4} + 18x^{3} - 16x^{2} + 8x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -19\nu^{7} + 73\nu^{6} + 26\nu^{5} - 89\nu^{4} - 1043\nu^{3} + 704\nu^{2} + 1696\nu - 520 ) / 816 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 71\nu^{7} - 33\nu^{6} - 566\nu^{5} - 827\nu^{4} + 1811\nu^{3} + 5132\nu^{2} - 1320\nu - 920 ) / 2448 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -49\nu^{7} + 165\nu^{6} + 76\nu^{5} + 157\nu^{4} - 2323\nu^{3} + 1574\nu^{2} + 2316\nu - 296 ) / 1224 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 131\nu^{7} - 81\nu^{6} - 530\nu^{5} - 1319\nu^{4} + 2603\nu^{3} + 2168\nu^{2} + 432\nu + 808 ) / 2448 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 37\nu^{7} - 84\nu^{6} - 73\nu^{5} - 181\nu^{4} + 1114\nu^{3} - 767\nu^{2} - 606\nu + 1460 ) / 612 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 20\nu^{7} - 33\nu^{6} - 56\nu^{5} - 164\nu^{4} + 536\nu^{3} + 32\nu^{2} - 96\nu + 355 ) / 153 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -205\nu^{7} + 249\nu^{6} + 676\nu^{5} + 1681\nu^{4} - 4831\nu^{3} - 1858\nu^{2} + 3228\nu - 2504 ) / 1224 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} - \beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} - 3\beta_{5} + 3\beta_{4} - \beta_{3} - \beta_{2} - 2\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{7} + 2\beta_{6} + \beta_{5} + 2\beta_{4} + 7\beta_{3} - 8\beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{7} + 12\beta_{6} - 20\beta_{5} - 7\beta_{4} + 7\beta_{3} - 11\beta_{2} - 12\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4\beta_{7} - \beta_{6} - 28\beta_{5} + 44\beta_{4} - 4\beta_{3} - 20\beta_{2} - 24\beta _1 + 39 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -7\beta_{7} + 30\beta_{6} - 12\beta_{5} - 7\beta_{4} + 98\beta_{3} - 37\beta_{2} - 86\beta _1 - 98 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/156\mathbb{Z}\right)^\times\).

\(n\) \(53\) \(79\) \(145\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
131.1
−0.862555 + 0.141174i
0.553538 − 0.676408i
−0.862555 − 0.141174i
0.553538 + 0.676408i
2.25820 − 0.369600i
−1.44918 + 1.77086i
2.25820 + 0.369600i
−1.44918 − 1.77086i
−1.11803 − 0.866025i −0.586627 − 1.62968i 0.500000 + 1.93649i 3.82407i −0.755479 + 2.33008i 3.25937i 1.11803 − 2.59808i −2.31174 + 1.91203i 3.31174 − 4.27543i
131.2 −1.11803 − 0.866025i 1.70466 − 0.306808i 0.500000 + 1.93649i − 2.09201i −2.17157 − 1.13326i 0.613616i 1.11803 − 2.59808i 2.81174 − 1.04601i −1.81174 + 2.33894i
131.3 −1.11803 + 0.866025i −0.586627 + 1.62968i 0.500000 − 1.93649i − 3.82407i −0.755479 − 2.33008i − 3.25937i 1.11803 + 2.59808i −2.31174 − 1.91203i 3.31174 + 4.27543i
131.4 −1.11803 + 0.866025i 1.70466 + 0.306808i 0.500000 − 1.93649i 2.09201i −2.17157 + 1.13326i − 0.613616i 1.11803 + 2.59808i 2.81174 + 1.04601i −1.81174 − 2.33894i
131.5 1.11803 − 0.866025i −1.70466 + 0.306808i 0.500000 − 1.93649i − 2.09201i −1.64017 + 1.81930i − 0.613616i −1.11803 − 2.59808i 2.81174 − 1.04601i −1.81174 − 2.33894i
131.6 1.11803 − 0.866025i 0.586627 + 1.62968i 0.500000 − 1.93649i 3.82407i 2.06722 + 1.31401i − 3.25937i −1.11803 − 2.59808i −2.31174 + 1.91203i 3.31174 + 4.27543i
131.7 1.11803 + 0.866025i −1.70466 − 0.306808i 0.500000 + 1.93649i 2.09201i −1.64017 − 1.81930i 0.613616i −1.11803 + 2.59808i 2.81174 + 1.04601i −1.81174 + 2.33894i
131.8 1.11803 + 0.866025i 0.586627 − 1.62968i 0.500000 + 1.93649i − 3.82407i 2.06722 − 1.31401i 3.25937i −1.11803 + 2.59808i −2.31174 − 1.91203i 3.31174 − 4.27543i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 131.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 156.2.c.c ✓ 8
3.b odd 2 1 inner 156.2.c.c ✓ 8
4.b odd 2 1 inner 156.2.c.c ✓ 8
8.b even 2 1 2496.2.d.m 8
8.d odd 2 1 2496.2.d.m 8
12.b even 2 1 inner 156.2.c.c ✓ 8
24.f even 2 1 2496.2.d.m 8
24.h odd 2 1 2496.2.d.m 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.c.c ✓ 8 1.a even 1 1 trivial
156.2.c.c ✓ 8 3.b odd 2 1 inner
156.2.c.c ✓ 8 4.b odd 2 1 inner
156.2.c.c ✓ 8 12.b even 2 1 inner
2496.2.d.m 8 8.b even 2 1
2496.2.d.m 8 8.d odd 2 1
2496.2.d.m 8 24.f even 2 1
2496.2.d.m 8 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(156, [\chi])\):

\( T_{5}^{4} + 19T_{5}^{2} + 64 \) Copy content Toggle raw display
\( T_{23}^{4} - 52T_{23}^{2} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - T^{6} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( (T^{4} + 19 T^{2} + 64)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 11 T^{2} + 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( (T - 1)^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} + 19 T^{2} + 64)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 28)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 52 T^{2} + 256)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 48)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 28)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + T - 26)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 124 T^{2} + 64)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 179 T^{2} + 6724)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 13 T^{2} + 16)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 124 T^{2} + 64)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( (T + 2)^{8} \) Copy content Toggle raw display
$67$ \( (T^{2} + 28)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} - 13 T^{2} + 16)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 6 T - 96)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 44 T^{2} + 64)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 208 T^{2} + 4096)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 220 T^{2} + 1600)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 10 T - 80)^{4} \) Copy content Toggle raw display
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