Properties

Label 2352.2.q.k
Level $2352$
Weight $2$
Character orbit 2352.q
Analytic conductor $18.781$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Self dual: no
Analytic conductor: \(18.7808145554\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 588)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + ( -1 + \zeta_{6} ) q^{3} + 2 \zeta_{6} q^{5} -\zeta_{6} q^{9} +O(q^{10})\) \( q + ( -1 + \zeta_{6} ) q^{3} + 2 \zeta_{6} q^{5} -\zeta_{6} q^{9} + ( 2 - 2 \zeta_{6} ) q^{11} + 4 q^{13} -2 q^{15} + ( 6 - 6 \zeta_{6} ) q^{17} -8 \zeta_{6} q^{19} -6 \zeta_{6} q^{23} + ( 1 - \zeta_{6} ) q^{25} + q^{27} -10 q^{29} + ( -4 + 4 \zeta_{6} ) q^{31} + 2 \zeta_{6} q^{33} -6 \zeta_{6} q^{37} + ( -4 + 4 \zeta_{6} ) q^{39} + 6 q^{41} -4 q^{43} + ( 2 - 2 \zeta_{6} ) q^{45} -8 \zeta_{6} q^{47} + 6 \zeta_{6} q^{51} + ( -2 + 2 \zeta_{6} ) q^{53} + 4 q^{55} + 8 q^{57} + ( 4 - 4 \zeta_{6} ) q^{59} -8 \zeta_{6} q^{61} + 8 \zeta_{6} q^{65} + ( -8 + 8 \zeta_{6} ) q^{67} + 6 q^{69} + 10 q^{71} + ( 4 - 4 \zeta_{6} ) q^{73} + \zeta_{6} q^{75} + 4 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} + 12 q^{83} + 12 q^{85} + ( 10 - 10 \zeta_{6} ) q^{87} -14 \zeta_{6} q^{89} -4 \zeta_{6} q^{93} + ( 16 - 16 \zeta_{6} ) q^{95} -4 q^{97} -2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} + 2q^{5} - q^{9} + O(q^{10}) \) \( 2q - q^{3} + 2q^{5} - q^{9} + 2q^{11} + 8q^{13} - 4q^{15} + 6q^{17} - 8q^{19} - 6q^{23} + q^{25} + 2q^{27} - 20q^{29} - 4q^{31} + 2q^{33} - 6q^{37} - 4q^{39} + 12q^{41} - 8q^{43} + 2q^{45} - 8q^{47} + 6q^{51} - 2q^{53} + 8q^{55} + 16q^{57} + 4q^{59} - 8q^{61} + 8q^{65} - 8q^{67} + 12q^{69} + 20q^{71} + 4q^{73} + q^{75} + 4q^{79} - q^{81} + 24q^{83} + 24q^{85} + 10q^{87} - 14q^{89} - 4q^{93} + 16q^{95} - 8q^{97} - 4q^{99} + O(q^{100}) \)

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.

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.k 2
4.b odd 2 1 588.2.i.g 2
7.b odd 2 1 2352.2.q.p 2
7.c even 3 1 2352.2.a.p 1
7.c even 3 1 inner 2352.2.q.k 2
7.d odd 6 1 2352.2.a.j 1
7.d odd 6 1 2352.2.q.p 2
12.b even 2 1 1764.2.k.c 2
21.g even 6 1 7056.2.a.n 1
21.h odd 6 1 7056.2.a.bu 1
28.d even 2 1 588.2.i.a 2
28.f even 6 1 588.2.a.e yes 1
28.f even 6 1 588.2.i.a 2
28.g odd 6 1 588.2.a.b 1
28.g odd 6 1 588.2.i.g 2
56.j odd 6 1 9408.2.a.ca 1
56.k odd 6 1 9408.2.a.cu 1
56.m even 6 1 9408.2.a.l 1
56.p even 6 1 9408.2.a.bf 1
84.h odd 2 1 1764.2.k.i 2
84.j odd 6 1 1764.2.a.b 1
84.j odd 6 1 1764.2.k.i 2
84.n even 6 1 1764.2.a.i 1
84.n even 6 1 1764.2.k.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.2.a.b 1 28.g odd 6 1
588.2.a.e yes 1 28.f even 6 1
588.2.i.a 2 28.d even 2 1
588.2.i.a 2 28.f even 6 1
588.2.i.g 2 4.b odd 2 1
588.2.i.g 2 28.g odd 6 1
1764.2.a.b 1 84.j odd 6 1
1764.2.a.i 1 84.n even 6 1
1764.2.k.c 2 12.b even 2 1
1764.2.k.c 2 84.n even 6 1
1764.2.k.i 2 84.h odd 2 1
1764.2.k.i 2 84.j odd 6 1
2352.2.a.j 1 7.d odd 6 1
2352.2.a.p 1 7.c even 3 1
2352.2.q.k 2 1.a even 1 1 trivial
2352.2.q.k 2 7.c even 3 1 inner
2352.2.q.p 2 7.b odd 2 1
2352.2.q.p 2 7.d odd 6 1
7056.2.a.n 1 21.g even 6 1
7056.2.a.bu 1 21.h odd 6 1
9408.2.a.l 1 56.m even 6 1
9408.2.a.bf 1 56.p even 6 1
9408.2.a.ca 1 56.j odd 6 1
9408.2.a.cu 1 56.k odd 6 1

Hecke kernels

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

\( T_{5}^{2} - 2 T_{5} + 4 \)
\( T_{11}^{2} - 2 T_{11} + 4 \)
\( T_{13} - 4 \)
\( T_{17}^{2} - 6 T_{17} + 36 \)