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Properties

Label 156.2.q.b
Level $156$
Weight $2$
Character orbit 156.q
Analytic conductor $1.246$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [156,2,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, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("156.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 156 = 2^{2} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
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: \(1.24566627153\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-43})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 10x^{2} - 11x + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} - 1) q^{3} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{5} + ( - \beta_{3} - 1) q^{7} - \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - 1) q^{3} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{5} + ( - \beta_{3} - 1) q^{7} - \beta_{2} q^{9} + (2 \beta_{2} + 2) q^{11} + (\beta_{3} - \beta_1 + 2) q^{13} + ( - \beta_1 + 1) q^{15} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{17} + (2 \beta_{2} - 4) q^{19} + (\beta_{3} - \beta_{2} - \beta_1 + 1) q^{21} + ( - 2 \beta_{2} + 2) q^{23} + (\beta_{3} + \beta_{2} + \beta_1 - 7) q^{25} + q^{27} + (2 \beta_{3} - 2 \beta_{2} - \beta_1 + 3) q^{29} + (\beta_{3} - 5 \beta_{2} - \beta_1 + 3) q^{31} + (2 \beta_{2} - 4) q^{33} - 10 \beta_{2} q^{35} + (2 \beta_{2} - \beta_1 + 3) q^{37} + (2 \beta_{2} + \beta_1 - 2) q^{39} + (2 \beta_{2} + \beta_1 + 1) q^{41} + (\beta_{3} + 6 \beta_{2} - 2 \beta_1 + 1) q^{43} + (\beta_{3} + \beta_{2} - 1) q^{45} + (12 \beta_{2} - 6) q^{47} + (2 \beta_{3} - 4 \beta_{2} - \beta_1 + 5) q^{49} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{51} + ( - \beta_{3} - \beta_{2} - \beta_1 - 4) q^{53} + ( - 4 \beta_{3} - 4 \beta_{2} + \cdots + 2) q^{55}+ \cdots + ( - 4 \beta_{2} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} - 3 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} - 3 q^{7} - 2 q^{9} + 12 q^{11} + 6 q^{13} + 3 q^{15} + q^{17} - 12 q^{19} + 4 q^{23} - 26 q^{25} + 4 q^{27} + 5 q^{29} - 12 q^{33} - 20 q^{35} + 15 q^{37} - 3 q^{39} + 9 q^{41} + 13 q^{43} - 3 q^{45} + 9 q^{49} - 2 q^{51} - 18 q^{53} + 6 q^{55} - 18 q^{59} - 10 q^{61} + 3 q^{63} + 43 q^{65} + 3 q^{67} + 4 q^{69} - 12 q^{71} + 13 q^{75} - 12 q^{77} + 30 q^{79} - 2 q^{81} - 63 q^{85} + 5 q^{87} + 24 q^{89} + 17 q^{91} + 15 q^{93} + 6 q^{95} + 27 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 10x^{2} - 11x + 121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 10\nu^{2} - 10\nu - 11 ) / 110 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 10\nu + 11 ) / 10 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 11\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -10\beta_{3} + 10\beta _1 + 11 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(53\) \(79\) \(145\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{2}\)

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
3.08945 − 1.20635i
−2.58945 + 2.07237i
−2.58945 − 2.07237i
3.08945 + 1.20635i
0 −0.500000 − 0.866025i 0 − 2.41269i 0 −3.58945 − 2.07237i 0 −0.500000 + 0.866025i 0
49.2 0 −0.500000 − 0.866025i 0 4.14474i 0 2.08945 + 1.20635i 0 −0.500000 + 0.866025i 0
121.1 0 −0.500000 + 0.866025i 0 − 4.14474i 0 2.08945 − 1.20635i 0 −0.500000 − 0.866025i 0
121.2 0 −0.500000 + 0.866025i 0 2.41269i 0 −3.58945 + 2.07237i 0 −0.500000 − 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
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 156.2.q.b ✓ 4
3.b odd 2 1 468.2.t.d 4
4.b odd 2 1 624.2.bv.f 4
5.b even 2 1 3900.2.cd.i 4
5.c odd 4 2 3900.2.bw.j 8
12.b even 2 1 1872.2.by.j 4
13.b even 2 1 2028.2.q.f 4
13.c even 3 1 2028.2.b.e 4
13.c even 3 1 2028.2.q.f 4
13.d odd 4 2 2028.2.i.n 8
13.e even 6 1 inner 156.2.q.b ✓ 4
13.e even 6 1 2028.2.b.e 4
13.f odd 12 2 2028.2.a.m 4
13.f odd 12 2 2028.2.i.n 8
39.h odd 6 1 468.2.t.d 4
39.h odd 6 1 6084.2.b.o 4
39.i odd 6 1 6084.2.b.o 4
39.k even 12 2 6084.2.a.bd 4
52.i odd 6 1 624.2.bv.f 4
52.l even 12 2 8112.2.a.cr 4
65.l even 6 1 3900.2.cd.i 4
65.r odd 12 2 3900.2.bw.j 8
156.r even 6 1 1872.2.by.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.q.b ✓ 4 1.a even 1 1 trivial
156.2.q.b ✓ 4 13.e even 6 1 inner
468.2.t.d 4 3.b odd 2 1
468.2.t.d 4 39.h odd 6 1
624.2.bv.f 4 4.b odd 2 1
624.2.bv.f 4 52.i odd 6 1
1872.2.by.j 4 12.b even 2 1
1872.2.by.j 4 156.r even 6 1
2028.2.a.m 4 13.f odd 12 2
2028.2.b.e 4 13.c even 3 1
2028.2.b.e 4 13.e even 6 1
2028.2.i.n 8 13.d odd 4 2
2028.2.i.n 8 13.f odd 12 2
2028.2.q.f 4 13.b even 2 1
2028.2.q.f 4 13.c even 3 1
3900.2.bw.j 8 5.c odd 4 2
3900.2.bw.j 8 65.r odd 12 2
3900.2.cd.i 4 5.b even 2 1
3900.2.cd.i 4 65.l even 6 1
6084.2.a.bd 4 39.k even 12 2
6084.2.b.o 4 39.h odd 6 1
6084.2.b.o 4 39.i odd 6 1
8112.2.a.cr 4 52.l even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 23T_{5}^{2} + 100 \) acting on \(S_{2}^{\mathrm{new}}(156, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 23T^{2} + 100 \) Copy content Toggle raw display
$7$ \( T^{4} + 3 T^{3} + \cdots + 100 \) Copy content Toggle raw display
$11$ \( (T^{2} - 6 T + 12)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 3 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - T^{3} + \cdots + 1024 \) Copy content Toggle raw display
$19$ \( (T^{2} + 6 T + 12)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 5 T^{3} + \cdots + 676 \) Copy content Toggle raw display
$31$ \( T^{4} + 59T^{2} + 64 \) Copy content Toggle raw display
$37$ \( T^{4} - 15 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$41$ \( T^{4} - 9 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$43$ \( T^{4} - 13 T^{3} + \cdots + 100 \) Copy content Toggle raw display
$47$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 9 T - 12)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 18 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$61$ \( (T^{2} + 5 T + 25)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 3 T^{3} + \cdots + 100 \) Copy content Toggle raw display
$71$ \( (T^{2} + 6 T + 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 350 T^{2} + 28561 \) Copy content Toggle raw display
$79$ \( (T^{2} - 15 T + 24)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 140T^{2} + 256 \) Copy content Toggle raw display
$89$ \( (T^{2} - 12 T + 48)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 27 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
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