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Properties

Label 2156.2.q.a
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: 8.0.796594176.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 5x^{6} - 2x^{5} + 63x^{4} - 64x^{3} + 46x^{2} - 16x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + ( - 2 \beta_{5} - 2 \beta_{3}) q^{5} + (\beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + ( - 2 \beta_{5} - 2 \beta_{3}) q^{5} + (\beta_{2} - 1) q^{9} + (\beta_{6} - 2 \beta_{2}) q^{11} + \beta_1 q^{13} + 4 q^{15} + \beta_{4} q^{17} + (2 \beta_{4} + 2 \beta_1) q^{19} + (4 \beta_{2} - 4) q^{23} + 3 \beta_{2} q^{25} - 4 \beta_{5} q^{27} - \beta_{3} q^{31} + (2 \beta_{5} + \beta_{4} + \cdots + \beta_1) q^{33}+ \cdots + (\beta_{7} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{9} - 8 q^{11} + 32 q^{15} - 16 q^{23} + 12 q^{25} + 16 q^{37} + 16 q^{53} + 48 q^{67} + 64 q^{71} + 20 q^{81} + 8 q^{93} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 5x^{6} - 2x^{5} + 63x^{4} - 64x^{3} + 46x^{2} - 16x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 27\nu^{7} + 32\nu^{6} - 177\nu^{5} - 186\nu^{4} + 245\nu^{3} - 228\nu^{2} + 84\nu + 14516 ) / 3886 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -1816\nu^{7} + 3101\nu^{6} + 9746\nu^{5} + 7113\nu^{4} - 110750\nu^{3} + 85499\nu^{2} - 79052\nu + 27404 ) / 19430 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 5433 \nu^{7} + 14718 \nu^{6} + 22663 \nu^{5} - 14386 \nu^{4} - 368815 \nu^{3} + 583442 \nu^{2} + \cdots + 98912 ) / 19430 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 6733 \nu^{7} - 11738 \nu^{6} - 35503 \nu^{5} - 24794 \nu^{4} + 412275 \nu^{3} - 318082 \nu^{2} + \cdots - 102352 ) / 19430 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -117\nu^{7} + 132\nu^{6} + 767\nu^{5} + 806\nu^{4} - 7055\nu^{3} + 988\nu^{2} - 364\nu - 572 ) / 290 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 10112 \nu^{7} - 27667 \nu^{6} - 42542 \nu^{5} + 28569 \nu^{4} + 688330 \nu^{3} - 1090353 \nu^{2} + \cdots - 184948 ) / 19430 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 252\nu^{7} - 282\nu^{6} - 1652\nu^{5} - 1736\nu^{4} + 15240\nu^{3} - 2128\nu^{2} + 784\nu + 1237 ) / 335 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} + 2\beta_{5} + 2\beta_{3} - \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + 2\beta_{4} + 2\beta_{3} + 7\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{7} + 13\beta_{5} - 3\beta _1 + 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 15\beta_{7} + 15\beta_{6} + 28\beta_{5} + 12\beta_{4} + 28\beta_{3} + 45\beta_{2} + 12\beta _1 - 45 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -39\beta_{6} + 35\beta_{4} - 73\beta_{3} + 131\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 155\beta_{7} + 290\beta_{5} + 58\beta _1 - 217 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -147\beta_{7} - 147\beta_{6} - 275\beta_{5} + 329\beta_{4} - 275\beta_{3} + 1231\beta_{2} + 329\beta _1 - 1231 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(981\) \(1079\) \(1277\)
\(\chi(n)\) \(-1\) \(1\) \(1 - \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
901.1
0.329101 − 0.387344i
2.62039 + 0.935532i
−2.12039 − 1.80156i
0.170899 − 0.478682i
0.329101 + 0.387344i
2.62039 − 0.935532i
−2.12039 + 1.80156i
0.170899 + 0.478682i
0 −1.22474 + 0.707107i 0 −2.44949 − 1.41421i 0 0 0 −0.500000 + 0.866025i 0
901.2 0 −1.22474 + 0.707107i 0 −2.44949 − 1.41421i 0 0 0 −0.500000 + 0.866025i 0
901.3 0 1.22474 − 0.707107i 0 2.44949 + 1.41421i 0 0 0 −0.500000 + 0.866025i 0
901.4 0 1.22474 − 0.707107i 0 2.44949 + 1.41421i 0 0 0 −0.500000 + 0.866025i 0
2089.1 0 −1.22474 − 0.707107i 0 −2.44949 + 1.41421i 0 0 0 −0.500000 − 0.866025i 0
2089.2 0 −1.22474 − 0.707107i 0 −2.44949 + 1.41421i 0 0 0 −0.500000 − 0.866025i 0
2089.3 0 1.22474 + 0.707107i 0 2.44949 − 1.41421i 0 0 0 −0.500000 − 0.866025i 0
2089.4 0 1.22474 + 0.707107i 0 2.44949 − 1.41421i 0 0 0 −0.500000 − 0.866025i 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.a 8
7.b odd 2 1 inner 2156.2.q.a 8
7.c even 3 1 308.2.c.b ✓ 4
7.c even 3 1 inner 2156.2.q.a 8
7.d odd 6 1 308.2.c.b ✓ 4
7.d odd 6 1 inner 2156.2.q.a 8
11.b odd 2 1 inner 2156.2.q.a 8
21.g even 6 1 2772.2.i.a 4
21.h odd 6 1 2772.2.i.a 4
28.f even 6 1 1232.2.e.d 4
28.g odd 6 1 1232.2.e.d 4
77.b even 2 1 inner 2156.2.q.a 8
77.h odd 6 1 308.2.c.b ✓ 4
77.h odd 6 1 inner 2156.2.q.a 8
77.i even 6 1 308.2.c.b ✓ 4
77.i even 6 1 inner 2156.2.q.a 8
231.k odd 6 1 2772.2.i.a 4
231.l even 6 1 2772.2.i.a 4
308.m odd 6 1 1232.2.e.d 4
308.n even 6 1 1232.2.e.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
308.2.c.b ✓ 4 7.c even 3 1
308.2.c.b ✓ 4 7.d odd 6 1
308.2.c.b ✓ 4 77.h odd 6 1
308.2.c.b ✓ 4 77.i even 6 1
1232.2.e.d 4 28.f even 6 1
1232.2.e.d 4 28.g odd 6 1
1232.2.e.d 4 308.m odd 6 1
1232.2.e.d 4 308.n even 6 1
2156.2.q.a 8 1.a even 1 1 trivial
2156.2.q.a 8 7.b odd 2 1 inner
2156.2.q.a 8 7.c even 3 1 inner
2156.2.q.a 8 7.d odd 6 1 inner
2156.2.q.a 8 11.b odd 2 1 inner
2156.2.q.a 8 77.b even 2 1 inner
2156.2.q.a 8 77.h odd 6 1 inner
2156.2.q.a 8 77.i even 6 1 inner
2772.2.i.a 4 21.g even 6 1
2772.2.i.a 4 21.h odd 6 1
2772.2.i.a 4 231.k odd 6 1
2772.2.i.a 4 231.l even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 2T_{3}^{2} + 4 \) 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
$3$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} - 8 T^{2} + 64)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + 4 T^{3} + \cdots + 121)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 14)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 14 T^{2} + 196)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 56 T^{2} + 3136)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 4 T + 16)^{4} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 4 T + 16)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 14)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 112)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 98 T^{2} + 9604)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 4 T + 16)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 126 T^{2} + 15876)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 12 T + 144)^{4} \) Copy content Toggle raw display
$71$ \( (T - 8)^{8} \) Copy content Toggle raw display
$73$ \( (T^{4} + 126 T^{2} + 15876)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 112 T^{2} + 12544)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 56)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} - 288 T^{2} + 82944)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 32)^{4} \) Copy content Toggle raw display
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