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Properties

Label 2352.2.q.v
Level $2352$
Weight $2$
Character orbit 2352.q
Analytic conductor $18.781$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{3} + 2 \zeta_{6} q^{5} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} + 1) q^{3} + 2 \zeta_{6} q^{5} - \zeta_{6} q^{9} + (6 \zeta_{6} - 6) q^{11} + 3 q^{13} + 2 q^{15} + ( - 4 \zeta_{6} + 4) q^{17} + 5 \zeta_{6} q^{19} - 4 \zeta_{6} q^{23} + ( - \zeta_{6} + 1) q^{25} - q^{27} - 4 q^{29} + (7 \zeta_{6} - 7) q^{31} + 6 \zeta_{6} q^{33} + 9 \zeta_{6} q^{37} + ( - 3 \zeta_{6} + 3) q^{39} + 2 q^{41} + q^{43} + ( - 2 \zeta_{6} + 2) q^{45} - 2 \zeta_{6} q^{47} - 4 \zeta_{6} q^{51} + (8 \zeta_{6} - 8) q^{53} - 12 q^{55} + 5 q^{57} + 10 \zeta_{6} q^{61} + 6 \zeta_{6} q^{65} + (15 \zeta_{6} - 15) q^{67} - 4 q^{69} + 6 q^{71} + (11 \zeta_{6} - 11) q^{73} - \zeta_{6} q^{75} + \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{81} + 6 q^{83} + 8 q^{85} + (4 \zeta_{6} - 4) q^{87} - 8 \zeta_{6} q^{89} + 7 \zeta_{6} q^{93} + (10 \zeta_{6} - 10) q^{95} + 14 q^{97} + 6 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 2 q^{5} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} + 2 q^{5} - q^{9} - 6 q^{11} + 6 q^{13} + 4 q^{15} + 4 q^{17} + 5 q^{19} - 4 q^{23} + q^{25} - 2 q^{27} - 8 q^{29} - 7 q^{31} + 6 q^{33} + 9 q^{37} + 3 q^{39} + 4 q^{41} + 2 q^{43} + 2 q^{45} - 2 q^{47} - 4 q^{51} - 8 q^{53} - 24 q^{55} + 10 q^{57} + 10 q^{61} + 6 q^{65} - 15 q^{67} - 8 q^{69} + 12 q^{71} - 11 q^{73} - q^{75} + q^{79} - q^{81} + 12 q^{83} + 16 q^{85} - 4 q^{87} - 8 q^{89} + 7 q^{93} - 10 q^{95} + 28 q^{97} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-\zeta_{6}\)

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
0.500000 − 0.866025i
0.500000 + 0.866025i
0 0.500000 + 0.866025i 0 1.00000 − 1.73205i 0 0 0 −0.500000 + 0.866025i 0
1537.1 0 0.500000 − 0.866025i 0 1.00000 + 1.73205i 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.v 2
4.b odd 2 1 1176.2.q.e 2
7.b odd 2 1 336.2.q.a 2
7.c even 3 1 2352.2.a.e 1
7.c even 3 1 inner 2352.2.q.v 2
7.d odd 6 1 336.2.q.a 2
7.d odd 6 1 2352.2.a.x 1
12.b even 2 1 3528.2.s.d 2
21.c even 2 1 1008.2.s.m 2
21.g even 6 1 1008.2.s.m 2
21.g even 6 1 7056.2.a.i 1
21.h odd 6 1 7056.2.a.bn 1
28.d even 2 1 168.2.q.b ✓ 2
28.f even 6 1 168.2.q.b ✓ 2
28.f even 6 1 1176.2.a.d 1
28.g odd 6 1 1176.2.a.e 1
28.g odd 6 1 1176.2.q.e 2
56.e even 2 1 1344.2.q.i 2
56.h odd 2 1 1344.2.q.t 2
56.j odd 6 1 1344.2.q.t 2
56.j odd 6 1 9408.2.a.f 1
56.k odd 6 1 9408.2.a.bk 1
56.m even 6 1 1344.2.q.i 2
56.m even 6 1 9408.2.a.cd 1
56.p even 6 1 9408.2.a.cs 1
84.h odd 2 1 504.2.s.g 2
84.j odd 6 1 504.2.s.g 2
84.j odd 6 1 3528.2.a.f 1
84.n even 6 1 3528.2.a.y 1
84.n even 6 1 3528.2.s.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.q.b ✓ 2 28.d even 2 1
168.2.q.b ✓ 2 28.f even 6 1
336.2.q.a 2 7.b odd 2 1
336.2.q.a 2 7.d odd 6 1
504.2.s.g 2 84.h odd 2 1
504.2.s.g 2 84.j odd 6 1
1008.2.s.m 2 21.c even 2 1
1008.2.s.m 2 21.g even 6 1
1176.2.a.d 1 28.f even 6 1
1176.2.a.e 1 28.g odd 6 1
1176.2.q.e 2 4.b odd 2 1
1176.2.q.e 2 28.g odd 6 1
1344.2.q.i 2 56.e even 2 1
1344.2.q.i 2 56.m even 6 1
1344.2.q.t 2 56.h odd 2 1
1344.2.q.t 2 56.j odd 6 1
2352.2.a.e 1 7.c even 3 1
2352.2.a.x 1 7.d odd 6 1
2352.2.q.v 2 1.a even 1 1 trivial
2352.2.q.v 2 7.c even 3 1 inner
3528.2.a.f 1 84.j odd 6 1
3528.2.a.y 1 84.n even 6 1
3528.2.s.d 2 12.b even 2 1
3528.2.s.d 2 84.n even 6 1
7056.2.a.i 1 21.g even 6 1
7056.2.a.bn 1 21.h odd 6 1
9408.2.a.f 1 56.j odd 6 1
9408.2.a.bk 1 56.k odd 6 1
9408.2.a.cd 1 56.m even 6 1
9408.2.a.cs 1 56.p even 6 1

Hecke kernels

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

\( T_{5}^{2} - 2T_{5} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} + 6T_{11} + 36 \) Copy content Toggle raw display
\( T_{13} - 3 \) Copy content Toggle raw display
\( T_{17}^{2} - 4T_{17} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$13$ \( (T - 3)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$19$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$23$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$29$ \( (T + 4)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$37$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$41$ \( (T - 2)^{2} \) Copy content Toggle raw display
$43$ \( (T - 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$53$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$67$ \( T^{2} + 15T + 225 \) Copy content Toggle raw display
$71$ \( (T - 6)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$79$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$83$ \( (T - 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$97$ \( (T - 14)^{2} \) Copy content Toggle raw display
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