Properties

Label 2352.2.bl.s
Level $2352$
Weight $2$
Character orbit 2352.bl
Analytic conductor $18.781$
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] = [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: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + (\beta_{4} + 1) q^{3} + ( - \beta_{7} - \beta_{3}) q^{5} + \beta_{4} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} + 1) q^{3} + ( - \beta_{7} - \beta_{3}) q^{5} + \beta_{4} q^{9} + (2 \beta_{3} + 2 \beta_1) q^{11} + ( - 2 \beta_{7} - \beta_{5} + \beta_1) q^{13} - \beta_{7} q^{15} + ( - 2 \beta_{3} + \beta_1) q^{17} + 4 \beta_{6} q^{19} + ( - 2 \beta_{7} + 2 \beta_{5} - 2 \beta_{3}) q^{23} + (\beta_{6} - 3 \beta_{4} + \beta_{2} - 3) q^{25} - q^{27} + ( - \beta_{2} - 4) q^{29} + ( - 4 \beta_{6} - 4 \beta_{4} - 4 \beta_{2} - 4) q^{31} + (2 \beta_{7} + 2 \beta_{5} + 2 \beta_{3}) q^{33} + \beta_{6} q^{37} + (2 \beta_{3} + \beta_1) q^{39} + (3 \beta_{7} - 2 \beta_{5} + 2 \beta_1) q^{41} + (4 \beta_{5} - 4 \beta_1) q^{43} + \beta_{3} q^{45} + (4 \beta_{6} - 4 \beta_{4}) q^{47} + ( - 2 \beta_{7} + \beta_{5} - 2 \beta_{3}) q^{51} + (4 \beta_{6} + 4 \beta_{4} + 4 \beta_{2} + 4) q^{53} + ( - 4 \beta_{2} - 4) q^{55} - 4 \beta_{2} q^{57} + ( - 4 \beta_{6} - 4 \beta_{2}) q^{59} + ( - 2 \beta_{7} + 3 \beta_{5} - 2 \beta_{3}) q^{61} + (3 \beta_{6} + 4 \beta_{4}) q^{65} + (4 \beta_{3} + 8 \beta_1) q^{67} + ( - 2 \beta_{7} + 2 \beta_{5} - 2 \beta_1) q^{69} + ( - 6 \beta_{7} + 6 \beta_{5} - 6 \beta_1) q^{71} + ( - \beta_{3} - 4 \beta_1) q^{73} + (\beta_{6} - 3 \beta_{4}) q^{75} + ( - 8 \beta_{7} + 4 \beta_{5} - 8 \beta_{3}) q^{79} + ( - \beta_{4} - 1) q^{81} + ( - 4 \beta_{2} - 12) q^{83} + (\beta_{2} + 4) q^{85} + ( - \beta_{6} - 4 \beta_{4} - \beta_{2} - 4) q^{87} + ( - 2 \beta_{7} - 7 \beta_{5} - 2 \beta_{3}) q^{89} + ( - 4 \beta_{6} - 4 \beta_{4}) q^{93} + (4 \beta_{3} + 4 \beta_1) q^{95} - 5 \beta_{7} q^{97} + (2 \beta_{7} + 2 \beta_{5} - 2 \beta_1) q^{99}+O(q^{100}) \) 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 \( 8 q + 4 q^{3} - 4 q^{9} - 12 q^{25} - 8 q^{27} - 32 q^{29} - 16 q^{31} + 16 q^{47} + 16 q^{53} - 32 q^{55} - 16 q^{65} + 12 q^{75} - 4 q^{81} - 96 q^{83} + 32 q^{85} - 16 q^{87} + 16 q^{93}+O(q^{100}) \) Copy content Toggle raw display

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
\(\nu\)\(=\) \( \beta_1 \) 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

Character values

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

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

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
−1.60021 0.923880i
1.60021 + 0.923880i
−0.662827 0.382683i
0.662827 0.382683i
−1.60021 + 0.923880i
1.60021 0.923880i
−0.662827 + 0.382683i
0 0.500000 0.866025i 0 −1.60021 + 0.923880i 0 0 0 −0.500000 0.866025i 0
31.2 0 0.500000 0.866025i 0 −0.662827 + 0.382683i 0 0 0 −0.500000 0.866025i 0
31.3 0 0.500000 0.866025i 0 0.662827 0.382683i 0 0 0 −0.500000 0.866025i 0
31.4 0 0.500000 0.866025i 0 1.60021 0.923880i 0 0 0 −0.500000 0.866025i 0
607.1 0 0.500000 + 0.866025i 0 −1.60021 0.923880i 0 0 0 −0.500000 + 0.866025i 0
607.2 0 0.500000 + 0.866025i 0 −0.662827 0.382683i 0 0 0 −0.500000 + 0.866025i 0
607.3 0 0.500000 + 0.866025i 0 0.662827 + 0.382683i 0 0 0 −0.500000 + 0.866025i 0
607.4 0 0.500000 + 0.866025i 0 1.60021 + 0.923880i 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
7.c even 3 1 inner
28.d even 2 1 inner
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.s 8
4.b odd 2 1 2352.2.bl.p 8
7.b odd 2 1 2352.2.bl.p 8
7.c even 3 1 2352.2.b.i 4
7.c even 3 1 inner 2352.2.bl.s 8
7.d odd 6 1 2352.2.b.j yes 4
7.d odd 6 1 2352.2.bl.p 8
21.g even 6 1 7056.2.b.t 4
21.h odd 6 1 7056.2.b.u 4
28.d even 2 1 inner 2352.2.bl.s 8
28.f even 6 1 2352.2.b.i 4
28.f even 6 1 inner 2352.2.bl.s 8
28.g odd 6 1 2352.2.b.j yes 4
28.g odd 6 1 2352.2.bl.p 8
84.j odd 6 1 7056.2.b.u 4
84.n even 6 1 7056.2.b.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2352.2.b.i 4 7.c even 3 1
2352.2.b.i 4 28.f even 6 1
2352.2.b.j yes 4 7.d odd 6 1
2352.2.b.j yes 4 28.g odd 6 1
2352.2.bl.p 8 4.b odd 2 1
2352.2.bl.p 8 7.b odd 2 1
2352.2.bl.p 8 7.d odd 6 1
2352.2.bl.p 8 28.g odd 6 1
2352.2.bl.s 8 1.a even 1 1 trivial
2352.2.bl.s 8 7.c even 3 1 inner
2352.2.bl.s 8 28.d even 2 1 inner
2352.2.bl.s 8 28.f even 6 1 inner
7056.2.b.t 4 21.g even 6 1
7056.2.b.t 4 84.n even 6 1
7056.2.b.u 4 21.h odd 6 1
7056.2.b.u 4 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 \(S_{2}^{\mathrm{new}}(2352, [\chi])\):

\( T_{5}^{8} - 4T_{5}^{6} + 14T_{5}^{4} - 8T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{8} - 32T_{11}^{6} + 896T_{11}^{4} - 4096T_{11}^{2} + 16384 \) Copy content Toggle raw display
\( T_{17}^{8} - 20T_{17}^{6} + 302T_{17}^{4} - 1960T_{17}^{2} + 9604 \) Copy content Toggle raw display
\( T_{19}^{4} + 32T_{19}^{2} + 1024 \) Copy content Toggle raw display
\( T_{31}^{4} + 8T_{31}^{3} + 80T_{31}^{2} - 128T_{31} + 256 \) 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
$5$ \( T^{8} - 4 T^{6} + 14 T^{4} - 8 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - 32 T^{6} + 896 T^{4} + \cdots + 16384 \) Copy content Toggle raw display
$13$ \( (T^{4} + 20 T^{2} + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 20 T^{6} + 302 T^{4} + \cdots + 9604 \) Copy content Toggle raw display
$19$ \( (T^{4} + 32 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 32 T^{6} + 896 T^{4} + \cdots + 16384 \) Copy content Toggle raw display
$29$ \( (T^{2} + 8 T + 14)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 8 T^{3} + 80 T^{2} - 128 T + 256)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 52 T^{2} + 578)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 64 T^{2} + 512)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 8 T^{3} + 80 T^{2} + 128 T + 256)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 8 T^{3} + 80 T^{2} + 128 T + 256)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 32 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} - 52 T^{6} + 2606 T^{4} + \cdots + 9604 \) Copy content Toggle raw display
$67$ \( T^{8} - 320 T^{6} + \cdots + 629407744 \) Copy content Toggle raw display
$71$ \( (T^{4} + 288 T^{2} + 10368)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 68 T^{6} + 3566 T^{4} + \cdots + 1119364 \) Copy content Toggle raw display
$79$ \( T^{8} - 320 T^{6} + \cdots + 629407744 \) Copy content Toggle raw display
$83$ \( (T^{2} + 24 T + 112)^{4} \) Copy content Toggle raw display
$89$ \( T^{8} - 212 T^{6} + \cdots + 113592964 \) Copy content Toggle raw display
$97$ \( (T^{4} + 100 T^{2} + 1250)^{2} \) Copy content Toggle raw display
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