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Properties

Label 2352.3.m.m
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: 6.0.2682209403.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 56x^{4} + 784x^{2} + 2883 \) 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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + (\beta_{3} - 1) q^{5} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + (\beta_{3} - 1) q^{5} - 3 q^{9} + (\beta_{5} + \beta_{2}) q^{11} + ( - \beta_{4} - \beta_{3} - 1) q^{13} + ( - \beta_{5} + \beta_{2} - \beta_1) q^{15} + (2 \beta_{4} - 10) q^{17} + (\beta_{5} - 6 \beta_{2} + 2 \beta_1) q^{19} + ( - 4 \beta_{2} + 2 \beta_1) q^{23} + ( - 3 \beta_{4} - \beta_{3} + 21) q^{25} + 3 \beta_{2} q^{27} + (2 \beta_{4} + 3 \beta_{3} - 7) q^{29} + (4 \beta_{5} + \beta_{2} + \beta_1) q^{31} + ( - \beta_{4} + \beta_{3} + 4) q^{33} + ( - \beta_{4} - 5 \beta_{3} - 9) q^{37} + ( - \beta_{5} + 2 \beta_{2} + 2 \beta_1) q^{39} + ( - 2 \beta_{4} - 2 \beta_{3} - 30) q^{41} + ( - 3 \beta_{5} + 12 \beta_{2} - 4 \beta_1) q^{43} + ( - 3 \beta_{3} + 3) q^{45} + (8 \beta_{5} - 14 \beta_{2} + 2 \beta_1) q^{47} + (4 \beta_{5} + 8 \beta_{2} - 2 \beta_1) q^{51} + (2 \beta_{4} + 3 \beta_{3} + 17) q^{53} + (5 \beta_{5} + 17 \beta_{2} + 3 \beta_1) q^{55} + (\beta_{4} + 5 \beta_{3} - 19) q^{57} + ( - 3 \beta_{5} + 11 \beta_{2} + 8 \beta_1) q^{59} + (2 \beta_{4} - 4 \beta_{3} - 20) q^{61} + (4 \beta_{4} + 6 \beta_{3} - 32) q^{65} + (3 \beta_{5} + 40 \beta_{2} + 6 \beta_1) q^{67} + (2 \beta_{4} + 4 \beta_{3} - 14) q^{69} + (38 \beta_{2} + 2 \beta_1) q^{71} + ( - \beta_{4} + 13 \beta_{3} - 15) q^{73} + ( - 5 \beta_{5} - 18 \beta_{2} + 4 \beta_1) q^{75} + (2 \beta_{5} + 11 \beta_{2} - \beta_1) q^{79} + 9 q^{81} + ( - 9 \beta_{5} - \beta_{2} - 4 \beta_1) q^{83} + ( - 2 \beta_{4} - 24 \beta_{3} - 14) q^{85} + (\beta_{5} + 5 \beta_{2} - 5 \beta_1) q^{87} + (2 \beta_{3} - 86) q^{89} + ( - 3 \beta_{4} + 6 \beta_{3} + 6) q^{93} + (2 \beta_{5} + 72 \beta_{2} - 14 \beta_1) q^{95} + (7 \beta_{4} - 3 \beta_{3} - 60) q^{97} + ( - 3 \beta_{5} - 3 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{5} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 4 q^{5} - 18 q^{9} - 10 q^{13} - 56 q^{17} + 118 q^{25} - 32 q^{29} + 24 q^{33} - 66 q^{37} - 188 q^{41} + 12 q^{45} + 112 q^{53} - 102 q^{57} - 124 q^{61} - 172 q^{65} - 72 q^{69} - 66 q^{73} + 54 q^{81} - 136 q^{85} - 512 q^{89} + 42 q^{93} - 352 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 56x^{4} + 784x^{2} + 2883 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 28\nu ) / 31 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} + 59\nu^{2} + 589 ) / 31 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -4\nu^{4} - 174\nu^{2} - 1147 ) / 31 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} + 47\nu^{3} + 377\nu ) / 31 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} + 4\beta_{3} - 39 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 31\beta_{2} - 14\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -59\beta_{4} - 174\beta_{3} + 1123 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 62\beta_{5} - 2914\beta_{2} + 939\beta_1 ) / 2 \) 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
− 6.07006i
3.63982i
2.43024i
6.07006i
− 3.63982i
− 2.43024i
0 − 1.73205i 0 −8.33199 0 0 0 −3.00000 0
1471.2 0 − 1.73205i 0 −1.55264 0 0 0 −3.00000 0
1471.3 0 − 1.73205i 0 7.88463 0 0 0 −3.00000 0
1471.4 0 1.73205i 0 −8.33199 0 0 0 −3.00000 0
1471.5 0 1.73205i 0 −1.55264 0 0 0 −3.00000 0
1471.6 0 1.73205i 0 7.88463 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.m 6
4.b odd 2 1 inner 2352.3.m.m 6
7.b odd 2 1 2352.3.m.p 6
7.d odd 6 1 336.3.be.c ✓ 6
7.d odd 6 1 336.3.be.e yes 6
21.g even 6 1 1008.3.cd.l 6
21.g even 6 1 1008.3.cd.m 6
28.d even 2 1 2352.3.m.p 6
28.f even 6 1 336.3.be.c ✓ 6
28.f even 6 1 336.3.be.e yes 6
84.j odd 6 1 1008.3.cd.l 6
84.j odd 6 1 1008.3.cd.m 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.3.be.c ✓ 6 7.d odd 6 1
336.3.be.c ✓ 6 28.f even 6 1
336.3.be.e yes 6 7.d odd 6 1
336.3.be.e yes 6 28.f even 6 1
1008.3.cd.l 6 21.g even 6 1
1008.3.cd.l 6 84.j odd 6 1
1008.3.cd.m 6 21.g even 6 1
1008.3.cd.m 6 84.j odd 6 1
2352.3.m.m 6 1.a even 1 1 trivial
2352.3.m.m 6 4.b odd 2 1 inner
2352.3.m.p 6 7.b odd 2 1
2352.3.m.p 6 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{3} + 2T_{5}^{2} - 65T_{5} - 102 \) 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
$5$ \( (T^{3} + 2 T^{2} + \cdots - 102)^{2} \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + 194 T^{4} + \cdots + 8748 \) Copy content Toggle raw display
$13$ \( (T^{3} + 5 T^{2} + \cdots - 272)^{2} \) Copy content Toggle raw display
$17$ \( (T^{3} + 28 T^{2} + \cdots - 10752)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 1355 T^{4} + \cdots + 1997568 \) Copy content Toggle raw display
$23$ \( T^{6} + 1040 T^{4} + \cdots + 442368 \) Copy content Toggle raw display
$29$ \( (T^{3} + 16 T^{2} + \cdots + 2436)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 3105 T^{4} + \cdots + 217311363 \) Copy content Toggle raw display
$37$ \( (T^{3} + 33 T^{2} + \cdots - 31184)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} + 94 T^{2} + \cdots + 8352)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 1237244592 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 77964057792 \) Copy content Toggle raw display
$53$ \( (T^{3} - 56 T^{2} + \cdots + 18972)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 42160884912 \) Copy content Toggle raw display
$61$ \( (T^{3} + 62 T^{2} + \cdots - 1176)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 3610604592 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 63605068992 \) Copy content Toggle raw display
$73$ \( (T^{3} + 33 T^{2} + \cdots - 421532)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + 2169 T^{4} + \cdots + 24831387 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 67710764268 \) Copy content Toggle raw display
$89$ \( (T^{3} + 256 T^{2} + \cdots + 598272)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 176 T^{2} + \cdots - 489542)^{2} \) Copy content Toggle raw display
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