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Properties

Label 2352.3.f.l
Level $2352$
Weight $3$
Character orbit 2352.f
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(97,2352)]
 
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([0, 0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2352.97");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2352.f (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: \(64.0873581775\)
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: \( 2^{4}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 147)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + (\beta_{7} - \beta_{6}) q^{5} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + (\beta_{7} - \beta_{6}) q^{5} - 3 q^{9} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} + 8) q^{11} + ( - 2 \beta_{7} - 4 \beta_{6} + 5 \beta_1) q^{13} + (2 \beta_{5} - \beta_{4}) q^{15} + (\beta_{7} - 5 \beta_{6} + \cdots + 2 \beta_1) q^{17}+ \cdots + (3 \beta_{5} - 6 \beta_{4} + 3 \beta_{3} - 24) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 24 q^{9} + 64 q^{11} + 144 q^{23} - 56 q^{25} + 16 q^{29} + 64 q^{37} - 16 q^{43} - 96 q^{51} + 112 q^{53} + 192 q^{57} + 112 q^{65} - 192 q^{67} + 160 q^{71} + 176 q^{79} + 72 q^{81} - 544 q^{85} + 288 q^{93} + 16 q^{95} - 192 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( ( 2\nu^{7} - 7\nu^{5} + 14\nu^{3} - 2\nu ) / 14 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{6} + 7\nu^{4} - 28\nu^{2} + 9 ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4\nu^{7} + \nu^{6} - 14\nu^{5} + 56\nu^{3} - 60\nu + 20 ) / 14 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4\nu^{7} + \nu^{6} - 21\nu^{5} + 70\nu^{3} - 74\nu + 20 ) / 14 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} + \nu^{6} + 14\nu^{5} - 49\nu^{3} + 52\nu + 20 ) / 7 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -2\nu^{7} - 7\nu^{6} + 7\nu^{5} + 28\nu^{4} - 28\nu^{3} - 84\nu^{2} + 2\nu + 28 ) / 14 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 10\nu^{7} - 35\nu^{5} + 119\nu^{3} - 10\nu ) / 7 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{5} + \beta_{4} - 3\beta_{3} - 3\beta_1 ) / 14 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + 7\beta_{6} - 2\beta_{5} - 2\beta_{4} - \beta_{3} - 14\beta_{2} - 3\beta _1 + 14 ) / 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} - 10\beta_1 ) / 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{7} + 14\beta_{6} + 4\beta_{5} + 4\beta_{4} + 2\beta_{3} - 21\beta_{2} - 6\beta _1 - 21 ) / 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{7} - \beta_{5} - 15\beta_{4} + 17\beta_{3} - 17\beta_1 ) / 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4\beta_{5} + 4\beta_{4} + 2\beta_{3} - 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -3\beta_{7} - 3\beta_{5} - 52\beta_{4} + 58\beta_{3} + 58\beta_1 ) / 7 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
−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 − 1.73205i 0 − 4.62452i 0 0 0 −3.00000 0
97.2 0 − 1.73205i 0 − 3.84891i 0 0 0 −3.00000 0
97.3 0 − 1.73205i 0 − 1.05007i 0 0 0 −3.00000 0
97.4 0 − 1.73205i 0 9.52350i 0 0 0 −3.00000 0
97.5 0 1.73205i 0 − 9.52350i 0 0 0 −3.00000 0
97.6 0 1.73205i 0 1.05007i 0 0 0 −3.00000 0
97.7 0 1.73205i 0 3.84891i 0 0 0 −3.00000 0
97.8 0 1.73205i 0 4.62452i 0 0 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.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.f.l 8
4.b odd 2 1 147.3.d.d ✓ 8
7.b odd 2 1 inner 2352.3.f.l 8
12.b even 2 1 441.3.d.h 8
28.d even 2 1 147.3.d.d ✓ 8
28.f even 6 1 147.3.f.f 8
28.f even 6 1 147.3.f.g 8
28.g odd 6 1 147.3.f.f 8
28.g odd 6 1 147.3.f.g 8
84.h odd 2 1 441.3.d.h 8
84.j odd 6 1 441.3.m.j 8
84.j odd 6 1 441.3.m.k 8
84.n even 6 1 441.3.m.j 8
84.n even 6 1 441.3.m.k 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.3.d.d ✓ 8 4.b odd 2 1
147.3.d.d ✓ 8 28.d even 2 1
147.3.f.f 8 28.f even 6 1
147.3.f.f 8 28.g odd 6 1
147.3.f.g 8 28.f even 6 1
147.3.f.g 8 28.g odd 6 1
441.3.d.h 8 12.b even 2 1
441.3.d.h 8 84.h odd 2 1
441.3.m.j 8 84.j odd 6 1
441.3.m.j 8 84.n even 6 1
441.3.m.k 8 84.j odd 6 1
441.3.m.k 8 84.n even 6 1
2352.3.f.l 8 1.a even 1 1 trivial
2352.3.f.l 8 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(2352, [\chi])\):

\( T_{5}^{8} + 128T_{5}^{6} + 3740T_{5}^{4} + 32704T_{5}^{2} + 31684 \) Copy content Toggle raw display
\( T_{11}^{4} - 32T_{11}^{3} + 260T_{11}^{2} - 16T_{11} - 932 \) 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
$5$ \( T^{8} + 128 T^{6} + \cdots + 31684 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 32 T^{3} + \cdots - 932)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 904 T^{6} + \cdots + 10640644 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 1039546564 \) Copy content Toggle raw display
$19$ \( T^{8} + 1264 T^{6} + \cdots + 85082176 \) Copy content Toggle raw display
$23$ \( (T^{4} - 72 T^{3} + \cdots + 67228)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 8 T^{3} + \cdots + 301468)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 3376 T^{6} + \cdots + 851238976 \) Copy content Toggle raw display
$37$ \( (T^{4} - 32 T^{3} + \cdots - 12092)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 1143056063044 \) Copy content Toggle raw display
$43$ \( (T^{4} + 8 T^{3} + \cdots + 1217296)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 36741733758016 \) Copy content Toggle raw display
$53$ \( (T^{4} - 56 T^{3} + \cdots + 5392)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 68212902976 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 10776263729284 \) Copy content Toggle raw display
$67$ \( (T^{4} + 96 T^{3} + \cdots - 23831424)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 80 T^{3} + \cdots + 1658524)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 25117918721284 \) Copy content Toggle raw display
$79$ \( (T^{4} - 88 T^{3} + \cdots + 32034832)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 7398574081024 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 7946907588676 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 20\!\cdots\!44 \) Copy content Toggle raw display
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