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Properties

Label 2352.2.bl.r
Level $2352$
Weight $2$
Character orbit 2352.bl
Analytic conductor $18.781$
Analytic rank $0$
Dimension $8$
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] = [2352,2,Mod(31,2352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2352, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2352.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2352.bl (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: \(18.7808145554\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.339738624.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} + (\beta_{6} + \beta_{3} - \beta_{2}) q^{5} + ( - \beta_{4} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{3} + (\beta_{6} + \beta_{3} - \beta_{2}) q^{5} + ( - \beta_{4} - 1) q^{9} + (\beta_{7} - \beta_{6} + \cdots - 2 \beta_{2}) q^{11}+ \cdots + ( - \beta_{7} + 2 \beta_{6} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{3} - 4 q^{9} - 24 q^{23} + 12 q^{25} - 8 q^{27} + 16 q^{29} - 16 q^{31} - 8 q^{47} - 8 q^{53} + 64 q^{55} + 24 q^{59} + 48 q^{61} + 8 q^{65} + 48 q^{67} + 48 q^{73} - 12 q^{75} + 24 q^{79} - 4 q^{81} - 64 q^{85} + 8 q^{87} + 48 q^{89} + 16 q^{93} - 72 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 20 ) / 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 34\nu ) / 14 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{6} + 7\nu^{4} - 28\nu^{2} + 2 ) / 14 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{7} + 7\nu^{5} - 28\nu^{3} + 16\nu ) / 14 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -2\nu^{6} + 7\nu^{4} - 21\nu^{2} + 2 ) / 7 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -6\nu^{7} + 21\nu^{5} - 70\nu^{3} + 6\nu ) / 14 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - 2\beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - 3\beta_{5} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{6} - 6\beta_{4} + 4\beta_{2} - 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{7} - 10\beta_{5} + 4\beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 14\beta_{2} - 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 14\beta_{3} - 34\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1 + \beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
−0.662827 + 0.382683i
0.662827 − 0.382683i
1.60021 − 0.923880i
−1.60021 + 0.923880i
−0.662827 − 0.382683i
0.662827 + 0.382683i
1.60021 + 0.923880i
−1.60021 − 0.923880i
0 0.500000 − 0.866025i 0 −3.72153 + 2.14862i 0 0 0 −0.500000 − 0.866025i 0
31.2 0 0.500000 − 0.866025i 0 −0.521114 + 0.300865i 0 0 0 −0.500000 − 0.866025i 0
31.3 0 0.500000 − 0.866025i 0 1.45849 − 0.842061i 0 0 0 −0.500000 − 0.866025i 0
31.4 0 0.500000 − 0.866025i 0 2.78415 − 1.60743i 0 0 0 −0.500000 − 0.866025i 0
607.1 0 0.500000 + 0.866025i 0 −3.72153 − 2.14862i 0 0 0 −0.500000 + 0.866025i 0
607.2 0 0.500000 + 0.866025i 0 −0.521114 − 0.300865i 0 0 0 −0.500000 + 0.866025i 0
607.3 0 0.500000 + 0.866025i 0 1.45849 + 0.842061i 0 0 0 −0.500000 + 0.866025i 0
607.4 0 0.500000 + 0.866025i 0 2.78415 + 1.60743i 0 0 0 −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2352.2.bl.r 8
4.b odd 2 1 2352.2.bl.q 8
7.b odd 2 1 2352.2.bl.o 8
7.c even 3 1 2352.2.b.k ✓ 8
7.c even 3 1 2352.2.bl.t 8
7.d odd 6 1 2352.2.b.l yes 8
7.d odd 6 1 2352.2.bl.q 8
21.g even 6 1 7056.2.b.w 8
21.h odd 6 1 7056.2.b.x 8
28.d even 2 1 2352.2.bl.t 8
28.f even 6 1 2352.2.b.k ✓ 8
28.f even 6 1 inner 2352.2.bl.r 8
28.g odd 6 1 2352.2.b.l yes 8
28.g odd 6 1 2352.2.bl.o 8
84.j odd 6 1 7056.2.b.x 8
84.n even 6 1 7056.2.b.w 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2352.2.b.k ✓ 8 7.c even 3 1
2352.2.b.k ✓ 8 28.f even 6 1
2352.2.b.l yes 8 7.d odd 6 1
2352.2.b.l yes 8 28.g odd 6 1
2352.2.bl.o 8 7.b odd 2 1
2352.2.bl.o 8 28.g odd 6 1
2352.2.bl.q 8 4.b odd 2 1
2352.2.bl.q 8 7.d odd 6 1
2352.2.bl.r 8 1.a even 1 1 trivial
2352.2.bl.r 8 28.f even 6 1 inner
2352.2.bl.t 8 7.c even 3 1
2352.2.bl.t 8 28.d even 2 1
7056.2.b.w 8 21.g even 6 1
7056.2.b.w 8 84.n even 6 1
7056.2.b.x 8 21.h odd 6 1
7056.2.b.x 8 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2352, [\chi])\):

\( T_{5}^{8} - 16T_{5}^{6} + 242T_{5}^{4} - 384T_{5}^{3} - 32T_{5}^{2} + 336T_{5} + 196 \) Copy content Toggle raw display
\( T_{11}^{8} - 20T_{11}^{6} + 404T_{11}^{4} - 960T_{11}^{3} + 848T_{11}^{2} - 192T_{11} + 16 \) Copy content Toggle raw display
\( T_{17}^{8} - 32T_{17}^{6} + 1106T_{17}^{4} - 5376T_{17}^{3} + 12032T_{17}^{2} - 13776T_{17} + 6724 \) Copy content Toggle raw display
\( T_{19}^{8} + 40T_{19}^{6} - 192T_{19}^{5} + 1656T_{19}^{4} - 3840T_{19}^{3} + 6976T_{19}^{2} - 5376T_{19} + 3136 \) Copy content Toggle raw display
\( T_{31}^{8} + 16 T_{31}^{7} + 200 T_{31}^{6} + 1216 T_{31}^{5} + 6520 T_{31}^{4} + 17408 T_{31}^{3} + \cdots + 678976 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} - 16 T^{6} + \cdots + 196 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - 20 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$13$ \( (T^{4} + 20 T^{2} + 98)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 32 T^{6} + \cdots + 6724 \) Copy content Toggle raw display
$19$ \( T^{8} + 40 T^{6} + \cdots + 3136 \) Copy content Toggle raw display
$23$ \( T^{8} + 24 T^{7} + \cdots + 38416 \) Copy content Toggle raw display
$29$ \( (T^{4} - 8 T^{3} - 4 T^{2} + \cdots + 28)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 16 T^{7} + \cdots + 678976 \) Copy content Toggle raw display
$37$ \( T^{8} + 100 T^{6} + \cdots + 929296 \) Copy content Toggle raw display
$41$ \( T^{8} + 80 T^{6} + \cdots + 6724 \) Copy content Toggle raw display
$43$ \( T^{8} + 368 T^{6} + \cdots + 20214016 \) Copy content Toggle raw display
$47$ \( T^{8} + 8 T^{7} + \cdots + 153664 \) Copy content Toggle raw display
$53$ \( T^{8} + 8 T^{7} + \cdots + 614656 \) Copy content Toggle raw display
$59$ \( T^{8} - 24 T^{7} + \cdots + 71503936 \) Copy content Toggle raw display
$61$ \( T^{8} - 48 T^{7} + \cdots + 9604 \) Copy content Toggle raw display
$67$ \( T^{8} - 48 T^{7} + \cdots + 802816 \) Copy content Toggle raw display
$71$ \( T^{8} + 360 T^{6} + \cdots + 10265616 \) Copy content Toggle raw display
$73$ \( T^{8} - 48 T^{7} + \cdots + 4866436 \) Copy content Toggle raw display
$79$ \( T^{8} - 24 T^{7} + \cdots + 430336 \) Copy content Toggle raw display
$83$ \( (T^{4} - 256 T^{2} + \cdots + 12256)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 1766857156 \) Copy content Toggle raw display
$97$ \( T^{8} + 584 T^{6} + \cdots + 325658116 \) Copy content Toggle raw display
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