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Properties

Label 2352.2.q.bd
Level $2352$
Weight $2$
Character orbit 2352.q
Analytic conductor $18.781$
Analytic rank $0$
Dimension $4$
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] = [2352,2,Mod(961,2352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2352, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2352.961");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2352.q (of order \(3\), degree \(2\), 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: \(18.7808145554\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 147)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + 1) q^{3} + ( - \beta_{3} + 2 \beta_{2} - \beta_1) q^{5} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 1) q^{3} + ( - \beta_{3} + 2 \beta_{2} - \beta_1) q^{5} + \beta_{2} q^{9} + ( - 2 \beta_{2} - 2) q^{11} + ( - \beta_{3} + 4) q^{13} + ( - \beta_{3} - 2) q^{15} + ( - 2 \beta_{2} - 3 \beta_1 - 2) q^{17} + ( - 2 \beta_{3} - 2 \beta_1) q^{19} + ( - 4 \beta_{3} + 2 \beta_{2} - 4 \beta_1) q^{23} + ( - \beta_{2} + 4 \beta_1 - 1) q^{25} - q^{27} + ( - 2 \beta_{3} - 4) q^{29} + ( - 4 \beta_{2} - 2 \beta_1 - 4) q^{31} - 2 \beta_{2} q^{33} - 4 \beta_{2} q^{37} + (4 \beta_{2} + \beta_1 + 4) q^{39} + (3 \beta_{3} + 2) q^{41} - 4 \beta_{3} q^{43} + ( - 2 \beta_{2} + \beta_1 - 2) q^{45} + ( - 2 \beta_{3} - 2 \beta_1) q^{47} + ( - 3 \beta_{3} - 2 \beta_{2} - 3 \beta_1) q^{51} + (2 \beta_{2} + 2) q^{53} + (2 \beta_{3} + 4) q^{55} - 2 \beta_{3} q^{57} + ( - 4 \beta_{2} - 2 \beta_1 - 4) q^{59} + ( - 3 \beta_{3} + 8 \beta_{2} - 3 \beta_1) q^{61} + ( - 2 \beta_{3} + 6 \beta_{2} - 2 \beta_1) q^{65} + 4 \beta_1 q^{67} + ( - 4 \beta_{3} - 2) q^{69} + ( - 8 \beta_{3} + 2) q^{71} + ( - 4 \beta_{2} + 7 \beta_1 - 4) q^{73} + (4 \beta_{3} - \beta_{2} + 4 \beta_1) q^{75} + (4 \beta_{3} - 8 \beta_{2} + 4 \beta_1) q^{79} + ( - \beta_{2} - 1) q^{81} + ( - 8 \beta_{3} + 4) q^{83} + ( - 4 \beta_{3} - 2) q^{85} + ( - 4 \beta_{2} + 2 \beta_1 - 4) q^{87} + (3 \beta_{3} - 10 \beta_{2} + 3 \beta_1) q^{89} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{93} + ( - 4 \beta_{2} + 4 \beta_1 - 4) q^{95} + ( - \beta_{3} + 4) q^{97} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 4 q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} - 4 q^{5} - 2 q^{9} - 4 q^{11} + 16 q^{13} - 8 q^{15} - 4 q^{17} - 4 q^{23} - 2 q^{25} - 4 q^{27} - 16 q^{29} - 8 q^{31} + 4 q^{33} + 8 q^{37} + 8 q^{39} + 8 q^{41} - 4 q^{45} + 4 q^{51} + 4 q^{53} + 16 q^{55} - 8 q^{59} - 16 q^{61} - 12 q^{65} - 8 q^{69} + 8 q^{71} - 8 q^{73} + 2 q^{75} + 16 q^{79} - 2 q^{81} + 16 q^{83} - 8 q^{85} - 8 q^{87} + 20 q^{89} + 8 q^{93} - 8 q^{95} + 16 q^{97} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
961.1
−0.707107 − 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 − 1.22474i
0 0.500000 + 0.866025i 0 −1.70711 + 2.95680i 0 0 0 −0.500000 + 0.866025i 0
961.2 0 0.500000 + 0.866025i 0 −0.292893 + 0.507306i 0 0 0 −0.500000 + 0.866025i 0
1537.1 0 0.500000 − 0.866025i 0 −1.70711 − 2.95680i 0 0 0 −0.500000 − 0.866025i 0
1537.2 0 0.500000 − 0.866025i 0 −0.292893 − 0.507306i 0 0 0 −0.500000 − 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2352.2.q.bd 4
4.b odd 2 1 147.2.e.d 4
7.b odd 2 1 2352.2.q.bb 4
7.c even 3 1 2352.2.a.bc 2
7.c even 3 1 inner 2352.2.q.bd 4
7.d odd 6 1 2352.2.a.be 2
7.d odd 6 1 2352.2.q.bb 4
12.b even 2 1 441.2.e.g 4
21.g even 6 1 7056.2.a.cv 2
21.h odd 6 1 7056.2.a.cf 2
28.d even 2 1 147.2.e.e 4
28.f even 6 1 147.2.a.d ✓ 2
28.f even 6 1 147.2.e.e 4
28.g odd 6 1 147.2.a.e yes 2
28.g odd 6 1 147.2.e.d 4
56.j odd 6 1 9408.2.a.dq 2
56.k odd 6 1 9408.2.a.di 2
56.m even 6 1 9408.2.a.ef 2
56.p even 6 1 9408.2.a.dt 2
84.h odd 2 1 441.2.e.f 4
84.j odd 6 1 441.2.a.j 2
84.j odd 6 1 441.2.e.f 4
84.n even 6 1 441.2.a.i 2
84.n even 6 1 441.2.e.g 4
140.p odd 6 1 3675.2.a.bd 2
140.s even 6 1 3675.2.a.bf 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.2.a.d ✓ 2 28.f even 6 1
147.2.a.e yes 2 28.g odd 6 1
147.2.e.d 4 4.b odd 2 1
147.2.e.d 4 28.g odd 6 1
147.2.e.e 4 28.d even 2 1
147.2.e.e 4 28.f even 6 1
441.2.a.i 2 84.n even 6 1
441.2.a.j 2 84.j odd 6 1
441.2.e.f 4 84.h odd 2 1
441.2.e.f 4 84.j odd 6 1
441.2.e.g 4 12.b even 2 1
441.2.e.g 4 84.n even 6 1
2352.2.a.bc 2 7.c even 3 1
2352.2.a.be 2 7.d odd 6 1
2352.2.q.bb 4 7.b odd 2 1
2352.2.q.bb 4 7.d odd 6 1
2352.2.q.bd 4 1.a even 1 1 trivial
2352.2.q.bd 4 7.c even 3 1 inner
3675.2.a.bd 2 140.p odd 6 1
3675.2.a.bf 2 140.s even 6 1
7056.2.a.cf 2 21.h odd 6 1
7056.2.a.cv 2 21.g even 6 1
9408.2.a.di 2 56.k odd 6 1
9408.2.a.dq 2 56.j odd 6 1
9408.2.a.dt 2 56.p even 6 1
9408.2.a.ef 2 56.m even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2352, [\chi])\):

\( T_{5}^{4} + 4T_{5}^{3} + 14T_{5}^{2} + 8T_{5} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} + 2T_{11} + 4 \) Copy content Toggle raw display
\( T_{13}^{2} - 8T_{13} + 14 \) Copy content Toggle raw display
\( T_{17}^{4} + 4T_{17}^{3} + 30T_{17}^{2} - 56T_{17} + 196 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 4 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 8 T + 14)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 4 T^{3} + \cdots + 196 \) Copy content Toggle raw display
$19$ \( T^{4} + 8T^{2} + 64 \) Copy content Toggle raw display
$23$ \( T^{4} + 4 T^{3} + \cdots + 784 \) Copy content Toggle raw display
$29$ \( (T^{2} + 8 T + 8)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 8 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$37$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 4 T - 14)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 8T^{2} + 64 \) Copy content Toggle raw display
$53$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 8 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$61$ \( T^{4} + 16 T^{3} + \cdots + 2116 \) Copy content Toggle raw display
$67$ \( T^{4} + 32T^{2} + 1024 \) Copy content Toggle raw display
$71$ \( (T^{2} - 4 T - 124)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 8 T^{3} + \cdots + 6724 \) Copy content Toggle raw display
$79$ \( T^{4} - 16 T^{3} + \cdots + 1024 \) Copy content Toggle raw display
$83$ \( (T^{2} - 8 T - 112)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 20 T^{3} + \cdots + 6724 \) Copy content Toggle raw display
$97$ \( (T^{2} - 8 T + 14)^{2} \) Copy content Toggle raw display
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