Properties

Label 2352.3.m.q
Level $2352$
Weight $3$
Character orbit 2352.m
Analytic conductor $64.087$
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] = [2352,3,Mod(1471,2352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2352, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2352.1471");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2352.m (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: \(64.0873581775\)
Analytic rank: \(0\)
Dimension: \(8\)
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: \( 2^{8}\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + ( - \beta_{2} + \beta_1) q^{5} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} + ( - \beta_{2} + \beta_1) q^{5} - 3 q^{9} + ( - \beta_{7} - 2 \beta_{6}) q^{11} + ( - 3 \beta_{4} + \beta_{2} + \beta_1) q^{13} + (\beta_{6} + \beta_{5}) q^{15} + (3 \beta_{4} + 2 \beta_{2} - \beta_1) q^{17} + ( - \beta_{7} - 4 \beta_{6} + \beta_{5}) q^{19} + (\beta_{7} - 2 \beta_{6} - 2 \beta_{5} + 8 \beta_{3}) q^{23} + (2 \beta_{4} + 2 \beta_{2} - 12 \beta_1 + 1) q^{25} - 3 \beta_{3} q^{27} + (\beta_{4} + \beta_{2} + 12 \beta_1 - 12) q^{29} + ( - 3 \beta_{7} + 4 \beta_{6} - \beta_{5} - 4 \beta_{3}) q^{31} + ( - 3 \beta_{4} + 6 \beta_1) q^{33} + (2 \beta_{4} + 8 \beta_{2} - 12 \beta_1) q^{37} + (3 \beta_{7} + \beta_{6} - \beta_{5}) q^{39} + (3 \beta_{4} - 6 \beta_{2} - 11 \beta_1 - 24) q^{41} + ( - 2 \beta_{7} + 4 \beta_{6} - 4 \beta_{5} + 16 \beta_{3}) q^{43} + (3 \beta_{2} - 3 \beta_1) q^{45} + (5 \beta_{7} - 8 \beta_{6} - \beta_{5} - 4 \beta_{3}) q^{47} + ( - 3 \beta_{7} - \beta_{6} - 2 \beta_{5}) q^{51} + ( - 6 \beta_{4} + 6 \beta_{2} - 12 \beta_1 - 10) q^{53} + (\beta_{7} + 12 \beta_{6} + 3 \beta_{5} - 4 \beta_{3}) q^{55} + ( - 3 \beta_{4} + 3 \beta_{2} + 12 \beta_1) q^{57} + (3 \beta_{7} - 3 \beta_{5} - 8 \beta_{3}) q^{59} + ( - 3 \beta_{4} - \beta_{2} + 13 \beta_1 - 24) q^{61} + ( - 3 \beta_{4} + 3 \beta_{2} - 24 \beta_1 - 22) q^{65} + ( - 2 \beta_{7} - 16 \beta_{6} - 4 \beta_{5} + 8 \beta_{3}) q^{67} + (3 \beta_{4} - 6 \beta_{2} + 6 \beta_1 - 24) q^{69} + (5 \beta_{7} + 18 \beta_{6} + 2 \beta_{5} + 24 \beta_{3}) q^{71} + (6 \beta_{4} + 4 \beta_{2} - 13 \beta_1 - 72) q^{73} + ( - 2 \beta_{7} - 12 \beta_{6} - 2 \beta_{5} + \beta_{3}) q^{75} + (4 \beta_{7} - 16 \beta_{6} + 8 \beta_{5} - 8 \beta_{3}) q^{79} + 9 q^{81} + (2 \beta_{7} + 16 \beta_{6} - 2 \beta_{5} - 28 \beta_{3}) q^{83} + ( - 6 \beta_{2} + 60 \beta_1 - 50) q^{85} + ( - \beta_{7} + 12 \beta_{6} - \beta_{5} - 12 \beta_{3}) q^{87} + (3 \beta_{4} - 4 \beta_{2} + 11 \beta_1 - 24) q^{89} + ( - 9 \beta_{4} - 3 \beta_{2} - 12 \beta_1 + 12) q^{93} + (2 \beta_{7} + 4 \beta_{5} + 16 \beta_{3}) q^{95} + (18 \beta_{2} - 37 \beta_1 - 48) q^{97} + (3 \beta_{7} + 6 \beta_{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 24 q^{9} + 8 q^{25} - 96 q^{29} - 192 q^{41} - 80 q^{53} - 192 q^{61} - 176 q^{65} - 192 q^{69} - 576 q^{73} + 72 q^{81} - 400 q^{85} - 192 q^{89} + 96 q^{93} - 384 q^{97}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 20 ) / 14 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{7} + 7\nu^{5} - 28\nu^{3} + 30\nu ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{6} + 7\nu^{4} - 28\nu^{2} + 9 ) / 7 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4\nu^{7} - 21\nu^{5} + 70\nu^{3} - 74\nu ) / 7 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 6\nu^{7} - 21\nu^{5} + 84\nu^{3} - 6\nu ) / 7 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} + 4\nu^{4} - 12\nu^{2} + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -18\nu^{7} + 63\nu^{5} - 210\nu^{3} + 18\nu ) / 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + 3\beta_{2} ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} - 2\beta_{3} - \beta _1 + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} + 3\beta_{5} ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{6} - 3\beta_{3} + 2\beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2\beta_{7} + 5\beta_{5} - 6\beta_{4} - 15\beta_{2} ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 14\beta _1 - 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -7\beta_{7} - 17\beta_{5} - 21\beta_{4} - 51\beta_{2} ) / 6 \) Copy content Toggle raw display

Character values

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

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1471.1
1.60021 + 0.923880i
0.662827 + 0.382683i
−0.662827 0.382683i
−1.60021 0.923880i
1.60021 0.923880i
0.662827 0.382683i
−0.662827 + 0.382683i
−1.60021 + 0.923880i
0 1.73205i 0 −7.81504 0 0 0 −3.00000 0
1471.2 0 1.73205i 0 −1.23710 0 0 0 −3.00000 0
1471.3 0 1.73205i 0 4.06552 0 0 0 −3.00000 0
1471.4 0 1.73205i 0 4.98661 0 0 0 −3.00000 0
1471.5 0 1.73205i 0 −7.81504 0 0 0 −3.00000 0
1471.6 0 1.73205i 0 −1.23710 0 0 0 −3.00000 0
1471.7 0 1.73205i 0 4.06552 0 0 0 −3.00000 0
1471.8 0 1.73205i 0 4.98661 0 0 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1471.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2352.3.m.q 8
4.b odd 2 1 inner 2352.3.m.q 8
7.b odd 2 1 2352.3.m.r yes 8
28.d even 2 1 2352.3.m.r yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2352.3.m.q 8 1.a even 1 1 trivial
2352.3.m.q 8 4.b odd 2 1 inner
2352.3.m.r yes 8 7.b odd 2 1
2352.3.m.r yes 8 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 52T_{5}^{2} + 96T_{5} + 196 \) acting on \(S_{3}^{\mathrm{new}}(2352, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{4} \) Copy content Toggle raw display
$5$ \( (T^{4} - 52 T^{2} + 96 T + 196)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + 384 T^{6} + 36288 T^{4} + \cdots + 82944 \) Copy content Toggle raw display
$13$ \( (T^{4} - 484 T^{2} - 1344 T + 196)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 628 T^{2} - 672 T + 81988)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 960 T^{6} + \cdots + 65028096 \) Copy content Toggle raw display
$23$ \( T^{8} + 2304 T^{6} + \cdots + 199148544 \) Copy content Toggle raw display
$29$ \( (T^{4} + 48 T^{3} + 192 T^{2} - 6912 T + 8064)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 3456 T^{6} + \cdots + 75805507584 \) Copy content Toggle raw display
$37$ \( (T^{4} - 3840 T^{2} - 105984 T - 631296)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 96 T^{3} + 812 T^{2} + \cdots - 284156)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 9216 T^{6} + \cdots + 12556760776704 \) Copy content Toggle raw display
$47$ \( T^{8} + 9216 T^{6} + \cdots + 3186376704 \) Copy content Toggle raw display
$53$ \( (T^{4} + 40 T^{3} - 3432 T^{2} + \cdots + 1016848)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 5952 T^{6} + \cdots + 143647064064 \) Copy content Toggle raw display
$61$ \( (T^{4} + 96 T^{3} + 2300 T^{2} + \cdots - 385532)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 12672 T^{6} + \cdots + 1040449536 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 111454809670656 \) Copy content Toggle raw display
$73$ \( (T^{4} + 288 T^{3} + 27932 T^{2} + \cdots + 8516548)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 889409897496576 \) Copy content Toggle raw display
$83$ \( T^{8} + 17856 T^{6} + \cdots + 45478946340864 \) Copy content Toggle raw display
$89$ \( (T^{4} + 96 T^{3} + 1772 T^{2} + \cdots - 1024124)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 192 T^{3} - 7204 T^{2} + \cdots - 103232444)^{2} \) Copy content Toggle raw display
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