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Properties

Label 2112.2.b.u
Level $2112$
Weight $2$
Character orbit 2112.b
Analytic conductor $16.864$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2112,2,Mod(65,2112)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2112, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2112.65");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2112 = 2^{6} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2112.b (of order \(2\), degree \(1\), 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: \(16.8644049069\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1544804416.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 11x^{6} + 39x^{4} + 46x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 1056)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + (\beta_{7} - \beta_{5}) q^{5} + (\beta_{6} - \beta_{3} - \beta_{2}) q^{7} + (\beta_{5} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + (\beta_{7} - \beta_{5}) q^{5} + (\beta_{6} - \beta_{3} - \beta_{2}) q^{7} + (\beta_{5} + 1) q^{9} + (\beta_{6} - \beta_1) q^{11} + ( - \beta_{7} - \beta_{5} + \beta_{4}) q^{13} + ( - 2 \beta_{6} + \beta_{3} + \cdots + \beta_1) q^{15}+ \cdots + ( - 2 \beta_{6} + 2 \beta_{3} + \cdots - 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 10 q^{9} - 24 q^{17} - 12 q^{21} - 28 q^{25} - 8 q^{29} + 2 q^{33} + 4 q^{37} + 48 q^{41} + 34 q^{45} + 16 q^{49} - 28 q^{57} - 30 q^{69} - 16 q^{77} - 30 q^{81} - 34 q^{93} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 11x^{6} + 39x^{4} + 46x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{4} + 5\nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 8\nu^{4} + \nu^{3} + 14\nu^{2} + 4\nu - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} - 8\nu^{4} + \nu^{3} - 14\nu^{2} + 4\nu + 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{6} + 8\nu^{4} + 16\nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + \nu^{6} - 7\nu^{5} + 8\nu^{4} - 8\nu^{3} + 16\nu^{2} + 12\nu + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} + 9\nu^{5} + 23\nu^{3} + 16\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{6} + 2\nu^{5} + 8\nu^{4} + 15\nu^{3} + 16\nu^{2} + 24\nu + 4 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} + \beta_{6} + \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} + \beta_{3} - \beta_{2} - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{7} - 2\beta_{6} - 2\beta_{5} + \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -5\beta_{4} - 5\beta_{3} + 5\beta_{2} + 2\beta _1 + 24 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -16\beta_{7} + 18\beta_{6} + 18\beta_{5} - \beta_{4} - 15\beta_{3} - 15\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 13\beta_{4} + 12\beta_{3} - 12\beta_{2} - 8\beta _1 - 52 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 68\beta_{7} - 82\beta_{6} - 86\beta_{5} + 9\beta_{4} + 89\beta_{3} + 89\beta_{2} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2112\mathbb{Z}\right)^\times\).

\(n\) \(133\) \(1409\) \(1729\) \(2047\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
0.307077i
− 0.307077i
1.58495i
− 1.58495i
− 2.24711i
2.24711i
1.82870i
− 1.82870i
0 −1.62493 − 0.599676i 0 1.52162i 0 0.936426i 0 2.28078 + 1.94886i 0
65.2 0 −1.62493 + 0.599676i 0 − 1.52162i 0 − 0.936426i 0 2.28078 − 1.94886i 0
65.3 0 −1.26870 − 1.17915i 0 − 3.83206i 0 − 3.02045i 0 0.219224 + 2.99198i 0
65.4 0 −1.26870 + 1.17915i 0 3.83206i 0 3.02045i 0 0.219224 − 2.99198i 0
65.5 0 1.26870 − 1.17915i 0 3.83206i 0 − 3.02045i 0 0.219224 − 2.99198i 0
65.6 0 1.26870 + 1.17915i 0 − 3.83206i 0 3.02045i 0 0.219224 + 2.99198i 0
65.7 0 1.62493 − 0.599676i 0 − 1.52162i 0 0.936426i 0 2.28078 − 1.94886i 0
65.8 0 1.62493 + 0.599676i 0 1.52162i 0 − 0.936426i 0 2.28078 + 1.94886i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
33.d even 2 1 inner
132.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2112.2.b.u 8
3.b odd 2 1 2112.2.b.v 8
4.b odd 2 1 inner 2112.2.b.u 8
8.b even 2 1 1056.2.b.i ✓ 8
8.d odd 2 1 1056.2.b.i ✓ 8
11.b odd 2 1 2112.2.b.v 8
12.b even 2 1 2112.2.b.v 8
24.f even 2 1 1056.2.b.j yes 8
24.h odd 2 1 1056.2.b.j yes 8
33.d even 2 1 inner 2112.2.b.u 8
44.c even 2 1 2112.2.b.v 8
88.b odd 2 1 1056.2.b.j yes 8
88.g even 2 1 1056.2.b.j yes 8
132.d odd 2 1 inner 2112.2.b.u 8
264.m even 2 1 1056.2.b.i ✓ 8
264.p odd 2 1 1056.2.b.i ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1056.2.b.i ✓ 8 8.b even 2 1
1056.2.b.i ✓ 8 8.d odd 2 1
1056.2.b.i ✓ 8 264.m even 2 1
1056.2.b.i ✓ 8 264.p odd 2 1
1056.2.b.j yes 8 24.f even 2 1
1056.2.b.j yes 8 24.h odd 2 1
1056.2.b.j yes 8 88.b odd 2 1
1056.2.b.j yes 8 88.g even 2 1
2112.2.b.u 8 1.a even 1 1 trivial
2112.2.b.u 8 4.b odd 2 1 inner
2112.2.b.u 8 33.d even 2 1 inner
2112.2.b.u 8 132.d odd 2 1 inner
2112.2.b.v 8 3.b odd 2 1
2112.2.b.v 8 11.b odd 2 1
2112.2.b.v 8 12.b even 2 1
2112.2.b.v 8 44.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2112, [\chi])\):

\( T_{5}^{4} + 17T_{5}^{2} + 34 \) Copy content Toggle raw display
\( T_{7}^{4} + 10T_{7}^{2} + 8 \) Copy content Toggle raw display
\( T_{17}^{2} + 6T_{17} - 8 \) Copy content Toggle raw display
\( T_{29}^{2} + 2T_{29} - 16 \) Copy content Toggle raw display
\( T_{31}^{4} - 85T_{31}^{2} + 272 \) Copy content Toggle raw display
\( T_{83}^{4} - 272T_{83}^{2} + 17408 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 5 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( (T^{4} + 17 T^{2} + 34)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 10 T^{2} + 8)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 24 T^{6} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( (T^{4} + 34 T^{2} + 136)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 6 T - 8)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} + 28 T^{2} + 128)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 37 T^{2} + 338)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 2 T - 16)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 85 T^{2} + 272)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - T - 38)^{4} \) Copy content Toggle raw display
$41$ \( (T - 6)^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} + 184 T^{2} + 8192)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 58 T^{2} + 8)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 102 T^{2} + 2176)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 71 T^{2} + 32)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 102 T^{2} + 2176)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 85 T^{2} + 1088)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 29 T^{2} + 2)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 204 T^{2} + 8704)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 238 T^{2} + 9248)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 272 T^{2} + 17408)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 51 T^{2} + 544)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - T - 4)^{4} \) Copy content Toggle raw display
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