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Properties

Label 156.4.q.c
Level $156$
Weight $4$
Character orbit 156.q
Analytic conductor $9.204$
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] = [156,4,Mod(49,156)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(156, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("156.49");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 156 = 2^{2} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 156.q (of order \(6\), degree \(2\), 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: \(9.20429796090\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 443x^{4} + 53128x^{2} + 1862832 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (3 \beta_1 + 3) q^{3} + ( - \beta_{2} - 3 \beta_1 - 1) q^{5} + (\beta_{5} + 2 \beta_{4} - \beta_{3} + \cdots + 1) q^{7} + 9 \beta_1 q^{9} + ( - \beta_{5} + \beta_{4} - \beta_{3} + \cdots - 14) q^{11}+ \cdots + (18 \beta_{5} + 9 \beta_{4} + \cdots - 126) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 9 q^{3} + 9 q^{7} - 27 q^{9} - 120 q^{11} - 77 q^{13} + 36 q^{15} + 32 q^{17} - 24 q^{19} + 104 q^{23} - 166 q^{25} - 162 q^{27} + 220 q^{29} - 360 q^{33} - 88 q^{35} - 684 q^{37} - 66 q^{39} + 576 q^{41}+ \cdots - 873 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 443x^{4} + 53128x^{2} + 1862832 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 345\nu^{3} + 122596\nu - 447584 ) / 895168 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 345\nu^{3} + 772572\nu + 447584 ) / 895168 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{4} - 223\nu^{2} + 568\nu - 3500 ) / 1136 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} + 365\nu^{2} - 142\nu + 23948 ) / 284 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 159\nu^{5} - 985\nu^{4} + 57041\nu^{3} - 331551\nu^{2} + 3893500\nu - 19336732 ) / 447584 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{4} + 8\beta_{3} - 3\beta_{2} - 3\beta _1 - 144 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{5} + 2\beta_{4} - 2\beta_{3} - 207\beta_{2} + 1065\beta _1 + 634 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -446\beta_{4} - 2920\beta_{3} + 1237\beta_{2} + 1237\beta _1 + 28612 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 1380\beta_{5} + 690\beta_{4} - 690\beta_{3} + 51181\beta_{2} - 405147\beta _1 - 228854 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(53\) \(79\) \(145\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
10.1667i
8.11290i
− 16.5475i
16.5475i
− 8.11290i
− 10.1667i
0 1.50000 + 2.59808i 0 − 11.8987i 0 −19.2504 − 11.1142i 0 −4.50000 + 7.79423i 0
49.2 0 1.50000 + 2.59808i 0 − 9.84495i 0 20.4668 + 11.8165i 0 −4.50000 + 7.79423i 0
49.3 0 1.50000 + 2.59808i 0 14.8155i 0 3.28368 + 1.89583i 0 −4.50000 + 7.79423i 0
121.1 0 1.50000 − 2.59808i 0 − 14.8155i 0 3.28368 − 1.89583i 0 −4.50000 − 7.79423i 0
121.2 0 1.50000 − 2.59808i 0 9.84495i 0 20.4668 − 11.8165i 0 −4.50000 − 7.79423i 0
121.3 0 1.50000 − 2.59808i 0 11.8987i 0 −19.2504 + 11.1142i 0 −4.50000 − 7.79423i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 156.4.q.c ✓ 6
3.b odd 2 1 468.4.t.f 6
4.b odd 2 1 624.4.bv.f 6
13.c even 3 1 2028.4.b.h 6
13.e even 6 1 inner 156.4.q.c ✓ 6
13.e even 6 1 2028.4.b.h 6
13.f odd 12 2 2028.4.a.o 6
39.h odd 6 1 468.4.t.f 6
52.i odd 6 1 624.4.bv.f 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.4.q.c ✓ 6 1.a even 1 1 trivial
156.4.q.c ✓ 6 13.e even 6 1 inner
468.4.t.f 6 3.b odd 2 1
468.4.t.f 6 39.h odd 6 1
624.4.bv.f 6 4.b odd 2 1
624.4.bv.f 6 52.i odd 6 1
2028.4.a.o 6 13.f odd 12 2
2028.4.b.h 6 13.c even 3 1
2028.4.b.h 6 13.e even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 458T_{5}^{4} + 66073T_{5}^{2} + 3012012 \) acting on \(S_{4}^{\mathrm{new}}(156, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3 T + 9)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} + 458 T^{4} + \cdots + 3012012 \) Copy content Toggle raw display
$7$ \( T^{6} - 9 T^{5} + \cdots + 3967500 \) Copy content Toggle raw display
$11$ \( T^{6} + 120 T^{5} + \cdots + 3736368 \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 10604499373 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 25825775616 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 21832364592 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 180975369744 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots + 45581396004 \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 33708194689728 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 605859508992 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 32622734707632 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 951754954476676 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 642983529610800 \) Copy content Toggle raw display
$53$ \( (T^{3} - 366 T^{2} + \cdots + 113725620)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 192663934225 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 87\!\cdots\!32 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 109003263187248 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 27\!\cdots\!75 \) Copy content Toggle raw display
$79$ \( (T^{3} - 1111 T^{2} + \cdots - 29342760)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 64\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 25\!\cdots\!52 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 11\!\cdots\!72 \) Copy content Toggle raw display
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