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Properties

Label 2156.2.q.b
Level $2156$
Weight $2$
Character orbit 2156.q
Analytic conductor $17.216$
Analytic rank $0$
Dimension $8$
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2156,2,Mod(901,2156)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2156, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2156.901");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2156 = 2^{2} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2156.q (of order \(6\), degree \(2\), not 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: \(17.2157466758\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.3317760000.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 308)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{6} q^{3} + \beta_1 q^{5} + 2 \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{3} + \beta_1 q^{5} + 2 \beta_{3} q^{9} + (\beta_{7} - \beta_{3} + 1) q^{11} + 2 \beta_{2} q^{13} - 5 q^{15} - 4 \beta_{4} q^{17} + (\beta_{4} + \beta_{2}) q^{19} + 5 \beta_{3} q^{23} + (\beta_{6} - \beta_1) q^{27} + (3 \beta_{7} + 3 \beta_{5}) q^{29} - \beta_{6} q^{31} + (5 \beta_{4} + 5 \beta_{2} - \beta_1) q^{33} - 11 \beta_{3} q^{37} + 2 \beta_{7} q^{39} + 7 \beta_{2} q^{41} + (\beta_{7} + \beta_{5}) q^{43} + 2 \beta_{6} q^{45} + 2 \beta_1 q^{47} - 4 \beta_{5} q^{51} + (8 \beta_{3} - 8) q^{53} + ( - \beta_{6} - 5 \beta_{2} + \beta_1) q^{55} + (\beta_{7} + \beta_{5}) q^{57} + \beta_{6} q^{59} + (3 \beta_{4} + 3 \beta_{2}) q^{61} + 2 \beta_{5} q^{65} + (9 \beta_{3} - 9) q^{67} + ( - 5 \beta_{6} + 5 \beta_1) q^{69} - q^{71} - 3 \beta_{4} q^{73} + 2 \beta_{5} q^{79} + ( - 11 \beta_{3} + 11) q^{81} - \beta_{2} q^{83} + (4 \beta_{7} + 4 \beta_{5}) q^{85} + 15 \beta_{4} q^{87} - 3 \beta_1 q^{89} + 5 \beta_{3} q^{93} - \beta_{7} q^{95} + ( - 7 \beta_{6} + 7 \beta_1) q^{97} + (2 \beta_{7} + 2 \beta_{5} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{9} + 4 q^{11} - 40 q^{15} + 20 q^{23} - 44 q^{37} - 32 q^{53} - 36 q^{67} - 8 q^{71} + 44 q^{81} + 20 q^{93} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 14\nu^{4} + 7\nu^{2} - 36 ) / 63 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4\nu^{7} + 7\nu^{5} + 35\nu^{3} + 81\nu ) / 189 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{6} + 7\nu^{4} - 28\nu^{2} + 144 ) / 63 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{7} - 7\nu^{5} - 35\nu^{3} + 180\nu ) / 189 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 13\nu ) / 21 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -8\nu^{6} + 14\nu^{4} + 7\nu^{2} + 162 ) / 63 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 19\nu^{7} - 49\nu^{5} + 133\nu^{3} - 684\nu ) / 189 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - 2\beta_{3} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} + \beta_{5} + 7\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{3} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -5\beta_{7} - 19\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -7\beta_{6} + 7\beta _1 + 22 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 29\beta_{5} - 13\beta_{4} - 13\beta_{2} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2156\mathbb{Z}\right)^\times\).

\(n\) \(981\) \(1079\) \(1277\)
\(\chi(n)\) \(-1\) \(1\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
901.1
1.72286 + 0.178197i
−1.72286 − 0.178197i
1.01575 + 1.40294i
−1.01575 − 1.40294i
1.72286 − 0.178197i
−1.72286 + 0.178197i
1.01575 − 1.40294i
−1.01575 + 1.40294i
0 −1.93649 + 1.11803i 0 1.93649 + 1.11803i 0 0 0 1.00000 − 1.73205i 0
901.2 0 −1.93649 + 1.11803i 0 1.93649 + 1.11803i 0 0 0 1.00000 − 1.73205i 0
901.3 0 1.93649 − 1.11803i 0 −1.93649 − 1.11803i 0 0 0 1.00000 − 1.73205i 0
901.4 0 1.93649 − 1.11803i 0 −1.93649 − 1.11803i 0 0 0 1.00000 − 1.73205i 0
2089.1 0 −1.93649 − 1.11803i 0 1.93649 − 1.11803i 0 0 0 1.00000 + 1.73205i 0
2089.2 0 −1.93649 − 1.11803i 0 1.93649 − 1.11803i 0 0 0 1.00000 + 1.73205i 0
2089.3 0 1.93649 + 1.11803i 0 −1.93649 + 1.11803i 0 0 0 1.00000 + 1.73205i 0
2089.4 0 1.93649 + 1.11803i 0 −1.93649 + 1.11803i 0 0 0 1.00000 + 1.73205i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 901.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
11.b odd 2 1 inner
77.b even 2 1 inner
77.h odd 6 1 inner
77.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2156.2.q.b 8
7.b odd 2 1 inner 2156.2.q.b 8
7.c even 3 1 308.2.c.a ✓ 4
7.c even 3 1 inner 2156.2.q.b 8
7.d odd 6 1 308.2.c.a ✓ 4
7.d odd 6 1 inner 2156.2.q.b 8
11.b odd 2 1 inner 2156.2.q.b 8
21.g even 6 1 2772.2.i.c 4
21.h odd 6 1 2772.2.i.c 4
28.f even 6 1 1232.2.e.b 4
28.g odd 6 1 1232.2.e.b 4
77.b even 2 1 inner 2156.2.q.b 8
77.h odd 6 1 308.2.c.a ✓ 4
77.h odd 6 1 inner 2156.2.q.b 8
77.i even 6 1 308.2.c.a ✓ 4
77.i even 6 1 inner 2156.2.q.b 8
231.k odd 6 1 2772.2.i.c 4
231.l even 6 1 2772.2.i.c 4
308.m odd 6 1 1232.2.e.b 4
308.n even 6 1 1232.2.e.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
308.2.c.a ✓ 4 7.c even 3 1
308.2.c.a ✓ 4 7.d odd 6 1
308.2.c.a ✓ 4 77.h odd 6 1
308.2.c.a ✓ 4 77.i even 6 1
1232.2.e.b 4 28.f even 6 1
1232.2.e.b 4 28.g odd 6 1
1232.2.e.b 4 308.m odd 6 1
1232.2.e.b 4 308.n even 6 1
2156.2.q.b 8 1.a even 1 1 trivial
2156.2.q.b 8 7.b odd 2 1 inner
2156.2.q.b 8 7.c even 3 1 inner
2156.2.q.b 8 7.d odd 6 1 inner
2156.2.q.b 8 11.b odd 2 1 inner
2156.2.q.b 8 77.b even 2 1 inner
2156.2.q.b 8 77.h odd 6 1 inner
2156.2.q.b 8 77.i even 6 1 inner
2772.2.i.c 4 21.g even 6 1
2772.2.i.c 4 21.h odd 6 1
2772.2.i.c 4 231.k odd 6 1
2772.2.i.c 4 231.l even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 5T_{3}^{2} + 25 \) acting on \(S_{2}^{\mathrm{new}}(2156, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 5 T^{2} + 25)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} - 5 T^{2} + 25)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 2 T^{3} + \cdots + 121)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 8)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 32 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 5 T + 25)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 90)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 5 T^{2} + 25)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 11 T + 121)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 98)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 10)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 20 T^{2} + 400)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 8 T + 64)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} - 5 T^{2} + 25)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 18 T^{2} + 324)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 9 T + 81)^{4} \) Copy content Toggle raw display
$71$ \( (T + 1)^{8} \) Copy content Toggle raw display
$73$ \( (T^{4} + 18 T^{2} + 324)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 40 T^{2} + 1600)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 2)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} - 45 T^{2} + 2025)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 245)^{4} \) Copy content Toggle raw display
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