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Properties

Label 2156.2.c.a
Level $2156$
Weight $2$
Character orbit 2156.c
Analytic conductor $17.216$
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] = [2156,2,Mod(1077,2156)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2156, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2156.1077");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2156 = 2^{2} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2156.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: \(17.2157466758\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.10485760000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 8x^{6} + 20x^{4} - 16x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_1 q^{5} + ( - \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_1 q^{5} + ( - \beta_{2} + 1) q^{9} + (\beta_{6} - 1) q^{11} - \beta_{4} q^{13} + ( - \beta_{2} - 2) q^{15} + \beta_{7} q^{17} + ( - \beta_{7} - \beta_{4}) q^{19} + 3 \beta_{2} q^{23} + ( - \beta_{2} + 3) q^{25} + ( - \beta_{5} + 3 \beta_1) q^{27} - \beta_{3} q^{29} + ( - \beta_{5} - 4 \beta_1) q^{31} + ( - \beta_{4} - \beta_1) q^{33} + ( - 2 \beta_{2} - 4) q^{37} + ( - 2 \beta_{6} - \beta_{3}) q^{39} + (\beta_{7} - 2 \beta_{4}) q^{41} + (2 \beta_{6} - \beta_{3}) q^{43} - \beta_{5} q^{45} + (4 \beta_{5} + \beta_1) q^{47} + \beta_{3} q^{51} + 5 \beta_{2} q^{53} + ( - \beta_{4} - \beta_1) q^{55} + ( - 2 \beta_{6} - 2 \beta_{3}) q^{57} + ( - 2 \beta_{5} + 3 \beta_1) q^{59} - \beta_{7} q^{61} + ( - 2 \beta_{6} - \beta_{3}) q^{65} + ( - 5 \beta_{2} - 4) q^{67} + (3 \beta_{5} + 3 \beta_1) q^{69} + ( - 3 \beta_{2} + 8) q^{71} + (2 \beta_{7} + \beta_{4}) q^{73} + ( - \beta_{5} + 2 \beta_1) q^{75} + 3 \beta_{3} q^{79} + ( - 5 \beta_{2} - 3) q^{81} + 2 \beta_{7} q^{83} + \beta_{3} q^{85} + (\beta_{7} + \beta_{4}) q^{87} + (\beta_{5} + 2 \beta_1) q^{89} + (5 \beta_{2} + 8) q^{93} + ( - 2 \beta_{6} - 2 \beta_{3}) q^{95} - 7 \beta_{5} q^{97} + (\beta_{6} - \beta_{3} + \beta_{2} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{9} - 8 q^{11} - 16 q^{15} + 24 q^{25} - 32 q^{37} - 32 q^{67} + 64 q^{71} - 24 q^{81} + 64 q^{93} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 8x^{6} + 20x^{4} - 16x^{2} + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{5} - 5\nu^{3} - 5\nu ) / 21 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} + 6\nu^{4} - \nu^{2} - 14 ) / 21 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{4} - 8\nu^{2} + 4 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 3\nu^{5} + 17\nu^{3} - 17\nu ) / 21 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 7\nu^{5} + 14\nu^{3} + 6\nu ) / 21 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} - 6\nu^{4} + 15\nu^{2} - 14 ) / 7 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} + 11\nu^{5} - 45\nu^{3} + 89\nu ) / 21 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + 2\beta_{5} + \beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + 3\beta_{2} + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} + 6\beta_{5} + 5\beta_{4} + 8\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4\beta_{6} + 3\beta_{3} + 12\beta_{2} + 12 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5\beta_{7} + 20\beta_{5} + 15\beta_{4} + 62\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 23\beta_{6} + 18\beta_{3} + 27\beta_{2} + 40 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 15\beta_{7} + 128\beta_{5} + 29\beta_{4} + 322\beta_1 ) / 4 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(981\) \(1079\) \(1277\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1077.1
2.06586 − 0.382683i
−2.06586 − 0.382683i
−0.855706 + 0.923880i
0.855706 + 0.923880i
0.855706 − 0.923880i
−0.855706 − 0.923880i
−2.06586 + 0.382683i
2.06586 + 0.382683i
0 − 1.84776i 0 − 1.84776i 0 0 0 −0.414214 0
1077.2 0 − 1.84776i 0 − 1.84776i 0 0 0 −0.414214 0
1077.3 0 − 0.765367i 0 − 0.765367i 0 0 0 2.41421 0
1077.4 0 − 0.765367i 0 − 0.765367i 0 0 0 2.41421 0
1077.5 0 0.765367i 0 0.765367i 0 0 0 2.41421 0
1077.6 0 0.765367i 0 0.765367i 0 0 0 2.41421 0
1077.7 0 1.84776i 0 1.84776i 0 0 0 −0.414214 0
1077.8 0 1.84776i 0 1.84776i 0 0 0 −0.414214 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1077.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
11.b odd 2 1 inner
77.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2156.2.c.a ✓ 8
7.b odd 2 1 inner 2156.2.c.a ✓ 8
7.c even 3 2 2156.2.q.d 16
7.d odd 6 2 2156.2.q.d 16
11.b odd 2 1 inner 2156.2.c.a ✓ 8
77.b even 2 1 inner 2156.2.c.a ✓ 8
77.h odd 6 2 2156.2.q.d 16
77.i even 6 2 2156.2.q.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2156.2.c.a ✓ 8 1.a even 1 1 trivial
2156.2.c.a ✓ 8 7.b odd 2 1 inner
2156.2.c.a ✓ 8 11.b odd 2 1 inner
2156.2.c.a ✓ 8 77.b even 2 1 inner
2156.2.q.d 16 7.c even 3 2
2156.2.q.d 16 7.d odd 6 2
2156.2.q.d 16 77.h odd 6 2
2156.2.q.d 16 77.i even 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 4T_{3}^{2} + 2 \) 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
$3$ \( (T^{4} + 4 T^{2} + 2)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + 4 T^{2} + 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{2} + 2 T + 11)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} - 40 T^{2} + 200)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 40 T^{2} + 200)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 80 T^{2} + 800)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 18)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 20)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 68 T^{2} + 98)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 8 T + 8)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 200 T^{2} + 9800)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 120 T^{2} + 400)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 68 T^{2} + 1058)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 50)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} + 52 T^{2} + 578)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 40 T^{2} + 200)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 8 T - 34)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} - 16 T + 46)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} - 200 T^{2} + 9800)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 180)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 160 T^{2} + 3200)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 20 T^{2} + 2)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 196 T^{2} + 4802)^{2} \) Copy content Toggle raw display
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