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Properties

Label 2352.3.f.k
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: 8.0.126303473664.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 9x^{6} + 2x^{5} + 92x^{4} + 14x^{3} - 441x^{2} - 686x + 2401 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 168)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + ( - \beta_{5} - \beta_{2}) q^{5} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + ( - \beta_{5} - \beta_{2}) q^{5} - 3 q^{9} + ( - \beta_{3} + \beta_1 + 3) q^{11} + ( - 3 \beta_{7} + \beta_{5} + \beta_{2}) q^{13} + (\beta_{4} - \beta_{3} + 2) q^{15} + (3 \beta_{7} + \beta_{6} - \beta_{5}) q^{17} + ( - 2 \beta_{7} + 3 \beta_{6} + 5 \beta_{2}) q^{19} + \beta_{3} q^{23} + ( - 4 \beta_{4} + 5 \beta_{3} - \beta_1) q^{25} - 3 \beta_{2} q^{27} + (\beta_{4} + 4 \beta_{3}) q^{29} + ( - \beta_{7} - 4 \beta_{6} + 4 \beta_{2}) q^{31} + ( - \beta_{7} + 2 \beta_{6} + \cdots + 2 \beta_{2}) q^{33}+ \cdots + (3 \beta_{3} - 3 \beta_1 - 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 24 q^{9} + 28 q^{11} + 12 q^{15} + 12 q^{25} - 4 q^{29} - 100 q^{37} - 12 q^{39} + 20 q^{43} - 24 q^{51} + 100 q^{53} - 156 q^{57} + 296 q^{65} + 68 q^{67} - 424 q^{71} - 80 q^{79} + 72 q^{81} - 232 q^{85} - 48 q^{93} + 288 q^{95} - 84 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 9x^{6} + 2x^{5} + 92x^{4} + 14x^{3} - 441x^{2} - 686x + 2401 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{6} + 2\nu^{5} + 9\nu^{4} - 2\nu^{3} - 43\nu^{2} - 112\nu + 245 ) / 49 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -99\nu^{7} - 75\nu^{6} + 800\nu^{5} + 1818\nu^{4} - 4950\nu^{3} - 16800\nu^{2} + 10731\nu + 100499 ) / 10976 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 13\nu^{7} - 12\nu^{6} - 96\nu^{5} - 198\nu^{4} + 440\nu^{3} + 1568\nu^{2} - 1029\nu - 9604 ) / 1372 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 29\nu^{7} + 12\nu^{6} - 352\nu^{5} - 670\nu^{4} + 2024\nu^{3} + 4200\nu^{2} - 7301\nu - 31556 ) / 2744 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 129\nu^{7} + 57\nu^{6} - 1056\nu^{5} - 1646\nu^{4} + 6226\nu^{3} + 15792\nu^{2} - 12985\nu - 106673 ) / 5488 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -367\nu^{7} - 407\nu^{6} + 2400\nu^{5} + 7330\nu^{4} - 12526\nu^{3} - 50624\nu^{2} + 22295\nu + 315903 ) / 10976 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -435\nu^{7} - 187\nu^{6} + 3040\nu^{5} + 7418\nu^{4} - 21750\nu^{3} - 54432\nu^{2} + 30331\nu + 391363 ) / 10976 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} + \beta_{6} + \beta_{5} - 2\beta_{4} + \beta_{3} + 2\beta_{2} - 2\beta _1 + 2 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} - 2\beta_{4} + \beta_{3} - 12\beta_{2} + 10 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{7} + 5\beta_{6} - 3\beta_{5} - 8\beta_{3} - 4\beta_{2} + 28 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4\beta_{7} + 3\beta_{6} + 12\beta_{5} - 8\beta_{4} - 4\beta_{3} - 11\beta_{2} + \beta _1 - 20 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -69\beta_{7} - 3\beta_{6} - 123\beta_{5} - 54\beta_{4} - 69\beta_{3} - 142\beta_{2} + 66\beta _1 + 22 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 15\beta_{7} - 29\beta_{6} + 43\beta_{5} - 112\beta_{3} + 36\beta_{2} + 22 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -57\beta_{7} - 447\beta_{6} + 505\beta_{5} - 562\beta_{4} + 57\beta_{3} + 1738\beta_{2} + 390\beta _1 + 786 ) / 8 \) 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
−2.38781 − 1.13946i
2.64247 + 0.131782i
2.18070 + 1.49818i
−1.43536 − 2.22255i
−1.43536 + 2.22255i
2.18070 − 1.49818i
2.64247 − 0.131782i
−2.38781 + 1.13946i
0 − 1.73205i 0 − 2.67601i 0 0 0 −3.00000 0
97.2 0 − 1.73205i 0 − 2.36835i 0 0 0 −3.00000 0
97.3 0 − 1.73205i 0 − 0.490921i 0 0 0 −3.00000 0
97.4 0 − 1.73205i 0 8.99938i 0 0 0 −3.00000 0
97.5 0 1.73205i 0 − 8.99938i 0 0 0 −3.00000 0
97.6 0 1.73205i 0 0.490921i 0 0 0 −3.00000 0
97.7 0 1.73205i 0 2.36835i 0 0 0 −3.00000 0
97.8 0 1.73205i 0 2.67601i 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.k 8
4.b odd 2 1 1176.3.f.a 8
7.b odd 2 1 inner 2352.3.f.k 8
7.c even 3 1 336.3.bh.h 8
7.d odd 6 1 336.3.bh.h 8
12.b even 2 1 3528.3.f.f 8
21.g even 6 1 1008.3.cg.n 8
21.h odd 6 1 1008.3.cg.n 8
28.d even 2 1 1176.3.f.a 8
28.f even 6 1 168.3.z.a ✓ 8
28.f even 6 1 1176.3.z.d 8
28.g odd 6 1 168.3.z.a ✓ 8
28.g odd 6 1 1176.3.z.d 8
84.h odd 2 1 3528.3.f.f 8
84.j odd 6 1 504.3.by.a 8
84.n even 6 1 504.3.by.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.3.z.a ✓ 8 28.f even 6 1
168.3.z.a ✓ 8 28.g odd 6 1
336.3.bh.h 8 7.c even 3 1
336.3.bh.h 8 7.d odd 6 1
504.3.by.a 8 84.j odd 6 1
504.3.by.a 8 84.n even 6 1
1008.3.cg.n 8 21.g even 6 1
1008.3.cg.n 8 21.h odd 6 1
1176.3.f.a 8 4.b odd 2 1
1176.3.f.a 8 28.d even 2 1
1176.3.z.d 8 28.f even 6 1
1176.3.z.d 8 28.g odd 6 1
2352.3.f.k 8 1.a even 1 1 trivial
2352.3.f.k 8 7.b odd 2 1 inner
3528.3.f.f 8 12.b even 2 1
3528.3.f.f 8 84.h odd 2 1

Hecke kernels

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

\( T_{5}^{8} + 94T_{5}^{6} + 1097T_{5}^{4} + 3512T_{5}^{2} + 784 \) Copy content Toggle raw display
\( T_{11}^{4} - 14T_{11}^{3} - 67T_{11}^{2} + 1484T_{11} - 4508 \) 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
$5$ \( T^{8} + 94 T^{6} + \cdots + 784 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 14 T^{3} + \cdots - 4508)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 4781999104 \) Copy content Toggle raw display
$17$ \( T^{8} + 1400 T^{6} + \cdots + 2166784 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 123878657296 \) Copy content Toggle raw display
$23$ \( (T^{2} - 32)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} + 2 T^{3} + \cdots + 278272)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 4336090801 \) Copy content Toggle raw display
$37$ \( (T^{4} + 50 T^{3} + \cdots - 82076)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 22503866142976 \) Copy content Toggle raw display
$43$ \( (T^{4} - 10 T^{3} + \cdots + 994948)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 107206475776 \) Copy content Toggle raw display
$53$ \( (T^{4} - 50 T^{3} + \cdots + 786208)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 562746188841984 \) Copy content Toggle raw display
$61$ \( (T^{2} + 1200)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 34 T^{3} + \cdots + 763888)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 212 T^{3} + \cdots - 11002304)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 97154396416 \) Copy content Toggle raw display
$79$ \( (T^{4} + 40 T^{3} + \cdots + 21235057)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 13\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 53944676847616 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 656906741956864 \) Copy content Toggle raw display
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