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Properties

Label 2352.3.m.n
Level $2352$
Weight $3$
Character orbit 2352.m
Analytic conductor $64.087$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.0.1364138928.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 35x^{4} + 364x^{2} + 972 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 336)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{3} q^{5} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - \beta_{3} q^{5} - 3 q^{9} + \beta_{4} q^{11} + (\beta_{5} - 2 \beta_{3} - 7) q^{13} + ( - \beta_{4} - \beta_{2} + \beta_1) q^{15} + (\beta_{5} - 3 \beta_{3} - 2) q^{17} + (\beta_{4} + 3 \beta_{2} + 2 \beta_1) q^{19} + ( - 2 \beta_{4} + 3 \beta_{2} - 3 \beta_1) q^{23} + (\beta_{5} + 4 \beta_{3} + 3) q^{25} + 3 \beta_1 q^{27} + ( - \beta_{5} - 6 \beta_{3} + 8) q^{29} + (4 \beta_{4} + \beta_{2} - 6 \beta_1) q^{31} + ( - \beta_{5} - 2 \beta_{3} + 2) q^{33} + ( - 3 \beta_{5} - 1) q^{37} + ( - \beta_{4} - 4 \beta_{2} + 9 \beta_1) q^{39} + ( - 4 \beta_{5} + 2 \beta_{3} - 22) q^{41} + (3 \beta_{4} + 5 \beta_{2} - 4 \beta_1) q^{43} + 3 \beta_{3} q^{45} + ( - 2 \beta_{4} + \beta_{2} + 29 \beta_1) q^{47} + ( - 2 \beta_{4} - 5 \beta_{2} + 5 \beta_1) q^{51} + ( - \beta_{3} + 30) q^{53} + ( - 3 \beta_{4} - 4 \beta_{2} + 22 \beta_1) q^{55} + (2 \beta_{5} - 5 \beta_{3} + 11) q^{57} + (\beta_{4} - \beta_{2} + 7 \beta_1) q^{59} + ( - 4 \beta_{5} - 12 \beta_{3} + 22) q^{61} + (5 \beta_{5} + 11 \beta_{3} + 56) q^{65} + (\beta_{4} + \beta_{2} + 38 \beta_1) q^{67} + (5 \beta_{5} + \beta_{3} - 10) q^{69} + (2 \beta_{4} - 5 \beta_{2} + 11 \beta_1) q^{71} + ( - 3 \beta_{5} - 11) q^{73} + (5 \beta_{4} + 2 \beta_{2} - 7 \beta_1) q^{75} + (10 \beta_{4} + \beta_{2} - 4 \beta_1) q^{79} + 9 q^{81} + (7 \beta_{4} + 5 \beta_{2} - 5 \beta_1) q^{83} + (6 \beta_{5} + 10 \beta_{3} + 84) q^{85} + ( - 7 \beta_{4} - 4 \beta_{2} - 2 \beta_1) q^{87} + (12 \beta_{5} + 10 \beta_{3} + 60) q^{89} + ( - 3 \beta_{5} - 9 \beta_{3} - 9) q^{93} + ( - 4 \beta_{4} + 7 \beta_{2} + 47 \beta_1) q^{95} + ( - \beta_{5} - 6 \beta_{3} - 42) q^{97} - 3 \beta_{4} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{5} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{5} - 18 q^{9} - 44 q^{13} - 16 q^{17} + 28 q^{25} + 34 q^{29} + 6 q^{33} - 12 q^{37} - 136 q^{41} + 6 q^{45} + 178 q^{53} + 60 q^{57} + 100 q^{61} + 368 q^{65} - 48 q^{69} - 72 q^{73} + 54 q^{81} + 536 q^{85} + 404 q^{89} - 78 q^{93} - 266 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 35x^{4} + 364x^{2} + 972 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 17\nu^{3} + 22\nu ) / 36 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 17\nu^{3} + 94\nu ) / 36 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} + 23\nu^{3} + 106\nu ) / 6 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{4} + 20\nu^{2} + 68 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} - 7\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} - 20\beta_{3} + 172 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -17\beta_{4} + 108\beta_{2} + 30\beta_1 \) Copy content Toggle raw display

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

Character values

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

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

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
− 2.02297i
− 3.78298i
4.07390i
2.02297i
3.78298i
− 4.07390i
0 − 1.73205i 0 −7.90761 0 0 0 −3.00000 0
1471.2 0 − 1.73205i 0 2.31096 0 0 0 −3.00000 0
1471.3 0 − 1.73205i 0 4.59665 0 0 0 −3.00000 0
1471.4 0 1.73205i 0 −7.90761 0 0 0 −3.00000 0
1471.5 0 1.73205i 0 2.31096 0 0 0 −3.00000 0
1471.6 0 1.73205i 0 4.59665 0 0 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1471.6
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.n 6
4.b odd 2 1 inner 2352.3.m.n 6
7.b odd 2 1 2352.3.m.o 6
7.c even 3 1 336.3.be.d ✓ 6
7.c even 3 1 336.3.be.f yes 6
21.h odd 6 1 1008.3.cd.h 6
21.h odd 6 1 1008.3.cd.i 6
28.d even 2 1 2352.3.m.o 6
28.g odd 6 1 336.3.be.d ✓ 6
28.g odd 6 1 336.3.be.f yes 6
84.n even 6 1 1008.3.cd.h 6
84.n even 6 1 1008.3.cd.i 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.3.be.d ✓ 6 7.c even 3 1
336.3.be.d ✓ 6 28.g odd 6 1
336.3.be.f yes 6 7.c even 3 1
336.3.be.f yes 6 28.g odd 6 1
1008.3.cd.h 6 21.h odd 6 1
1008.3.cd.h 6 84.n even 6 1
1008.3.cd.i 6 21.h odd 6 1
1008.3.cd.i 6 84.n even 6 1
2352.3.m.n 6 1.a even 1 1 trivial
2352.3.m.n 6 4.b odd 2 1 inner
2352.3.m.o 6 7.b odd 2 1
2352.3.m.o 6 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{3} + T_{5}^{2} - 44T_{5} + 84 \) acting on \(S_{3}^{\mathrm{new}}(2352, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{3} \) Copy content Toggle raw display
$5$ \( (T^{3} + T^{2} - 44 T + 84)^{2} \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + 227 T^{4} + \cdots + 432 \) Copy content Toggle raw display
$13$ \( (T^{3} + 22 T^{2} + \cdots - 4312)^{2} \) Copy content Toggle raw display
$17$ \( (T^{3} + 8 T^{2} + \cdots - 4512)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 1262 T^{4} + \cdots + 1138368 \) Copy content Toggle raw display
$23$ \( T^{6} + 2768 T^{4} + \cdots + 66382848 \) Copy content Toggle raw display
$29$ \( (T^{3} - 17 T^{2} + \cdots + 35952)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 3537 T^{4} + \cdots + 3756483 \) Copy content Toggle raw display
$37$ \( (T^{3} + 6 T^{2} + \cdots - 13532)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} + 68 T^{2} + \cdots - 37152)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 4010 T^{4} + \cdots + 7660812 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 7468832448 \) Copy content Toggle raw display
$53$ \( (T^{3} - 89 T^{2} + \cdots - 24696)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 899 T^{4} + \cdots + 84672 \) Copy content Toggle raw display
$61$ \( (T^{3} - 50 T^{2} + \cdots + 271656)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 84665952108 \) Copy content Toggle raw display
$71$ \( T^{6} + 6164 T^{4} + \cdots + 615874752 \) Copy content Toggle raw display
$73$ \( (T^{3} + 36 T^{2} + \cdots - 26242)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + 21777 T^{4} + \cdots + 140726403 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 22314082608 \) Copy content Toggle raw display
$89$ \( (T^{3} - 202 T^{2} + \cdots + 3284544)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 133 T^{2} + \cdots + 35252)^{2} \) Copy content Toggle raw display
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