Properties

Label 2352.2.k.e
Level $2352$
Weight $2$
Character orbit 2352.k
Analytic conductor $18.781$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2352.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.7808145554\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{5} + 3 q^{9} +O(q^{10})\) \( q + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{3} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{5} + 3 q^{9} + 3 \zeta_{12}^{3} q^{11} + ( 2 - 4 \zeta_{12}^{2} ) q^{13} -3 q^{15} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{17} + ( -2 + 4 \zeta_{12}^{2} ) q^{19} + 6 \zeta_{12}^{3} q^{23} -2 q^{25} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{27} + 3 \zeta_{12}^{3} q^{29} + ( -1 + 2 \zeta_{12}^{2} ) q^{31} + ( -3 + 6 \zeta_{12}^{2} ) q^{33} -2 q^{37} -6 \zeta_{12}^{3} q^{39} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{41} + 8 q^{43} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{45} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{47} + 6 q^{51} -9 \zeta_{12}^{3} q^{53} + ( 3 - 6 \zeta_{12}^{2} ) q^{55} + 6 \zeta_{12}^{3} q^{57} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{59} + 6 \zeta_{12}^{3} q^{65} -2 q^{67} + ( -6 + 12 \zeta_{12}^{2} ) q^{69} + 12 \zeta_{12}^{3} q^{71} + ( -4 + 8 \zeta_{12}^{2} ) q^{73} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{75} + q^{79} + 9 q^{81} + ( 10 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{83} -6 q^{85} + ( -3 + 6 \zeta_{12}^{2} ) q^{87} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{89} + 3 \zeta_{12}^{3} q^{93} -6 \zeta_{12}^{3} q^{95} + ( 3 - 6 \zeta_{12}^{2} ) q^{97} + 9 \zeta_{12}^{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 12q^{9} + O(q^{10}) \) \( 4q + 12q^{9} - 12q^{15} - 8q^{25} - 8q^{37} + 32q^{43} + 24q^{51} - 8q^{67} + 4q^{79} + 36q^{81} - 24q^{85} + O(q^{100}) \)

Character values

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

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

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
881.1
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
0 −1.73205 0 1.73205 0 0 0 3.00000 0
881.2 0 −1.73205 0 1.73205 0 0 0 3.00000 0
881.3 0 1.73205 0 −1.73205 0 0 0 3.00000 0
881.4 0 1.73205 0 −1.73205 0 0 0 3.00000 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
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2352.2.k.e 4
3.b odd 2 1 inner 2352.2.k.e 4
4.b odd 2 1 294.2.d.a 4
7.b odd 2 1 inner 2352.2.k.e 4
7.c even 3 1 336.2.bc.e 4
7.d odd 6 1 336.2.bc.e 4
12.b even 2 1 294.2.d.a 4
21.c even 2 1 inner 2352.2.k.e 4
21.g even 6 1 336.2.bc.e 4
21.h odd 6 1 336.2.bc.e 4
28.d even 2 1 294.2.d.a 4
28.f even 6 1 42.2.f.a 4
28.f even 6 1 294.2.f.a 4
28.g odd 6 1 42.2.f.a 4
28.g odd 6 1 294.2.f.a 4
84.h odd 2 1 294.2.d.a 4
84.j odd 6 1 42.2.f.a 4
84.j odd 6 1 294.2.f.a 4
84.n even 6 1 42.2.f.a 4
84.n even 6 1 294.2.f.a 4
140.p odd 6 1 1050.2.s.b 4
140.s even 6 1 1050.2.s.b 4
140.w even 12 1 1050.2.u.a 4
140.w even 12 1 1050.2.u.d 4
140.x odd 12 1 1050.2.u.a 4
140.x odd 12 1 1050.2.u.d 4
252.n even 6 1 1134.2.t.d 4
252.o even 6 1 1134.2.t.d 4
252.r odd 6 1 1134.2.l.c 4
252.u odd 6 1 1134.2.l.c 4
252.bb even 6 1 1134.2.l.c 4
252.bj even 6 1 1134.2.l.c 4
252.bl odd 6 1 1134.2.t.d 4
252.bn odd 6 1 1134.2.t.d 4
420.ba even 6 1 1050.2.s.b 4
420.be odd 6 1 1050.2.s.b 4
420.bp odd 12 1 1050.2.u.a 4
420.bp odd 12 1 1050.2.u.d 4
420.br even 12 1 1050.2.u.a 4
420.br even 12 1 1050.2.u.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.f.a 4 28.f even 6 1
42.2.f.a 4 28.g odd 6 1
42.2.f.a 4 84.j odd 6 1
42.2.f.a 4 84.n even 6 1
294.2.d.a 4 4.b odd 2 1
294.2.d.a 4 12.b even 2 1
294.2.d.a 4 28.d even 2 1
294.2.d.a 4 84.h odd 2 1
294.2.f.a 4 28.f even 6 1
294.2.f.a 4 28.g odd 6 1
294.2.f.a 4 84.j odd 6 1
294.2.f.a 4 84.n even 6 1
336.2.bc.e 4 7.c even 3 1
336.2.bc.e 4 7.d odd 6 1
336.2.bc.e 4 21.g even 6 1
336.2.bc.e 4 21.h odd 6 1
1050.2.s.b 4 140.p odd 6 1
1050.2.s.b 4 140.s even 6 1
1050.2.s.b 4 420.ba even 6 1
1050.2.s.b 4 420.be odd 6 1
1050.2.u.a 4 140.w even 12 1
1050.2.u.a 4 140.x odd 12 1
1050.2.u.a 4 420.bp odd 12 1
1050.2.u.a 4 420.br even 12 1
1050.2.u.d 4 140.w even 12 1
1050.2.u.d 4 140.x odd 12 1
1050.2.u.d 4 420.bp odd 12 1
1050.2.u.d 4 420.br even 12 1
1134.2.l.c 4 252.r odd 6 1
1134.2.l.c 4 252.u odd 6 1
1134.2.l.c 4 252.bb even 6 1
1134.2.l.c 4 252.bj even 6 1
1134.2.t.d 4 252.n even 6 1
1134.2.t.d 4 252.o even 6 1
1134.2.t.d 4 252.bl odd 6 1
1134.2.t.d 4 252.bn odd 6 1
2352.2.k.e 4 1.a even 1 1 trivial
2352.2.k.e 4 3.b odd 2 1 inner
2352.2.k.e 4 7.b odd 2 1 inner
2352.2.k.e 4 21.c even 2 1 inner

Hecke kernels

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

\( T_{5}^{2} - 3 \)
\( T_{13}^{2} + 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -3 + T^{2} )^{2} \)
$5$ \( ( -3 + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( ( 9 + T^{2} )^{2} \)
$13$ \( ( 12 + T^{2} )^{2} \)
$17$ \( ( -12 + T^{2} )^{2} \)
$19$ \( ( 12 + T^{2} )^{2} \)
$23$ \( ( 36 + T^{2} )^{2} \)
$29$ \( ( 9 + T^{2} )^{2} \)
$31$ \( ( 3 + T^{2} )^{2} \)
$37$ \( ( 2 + T )^{4} \)
$41$ \( ( -48 + T^{2} )^{2} \)
$43$ \( ( -8 + T )^{4} \)
$47$ \( ( -48 + T^{2} )^{2} \)
$53$ \( ( 81 + T^{2} )^{2} \)
$59$ \( ( -3 + T^{2} )^{2} \)
$61$ \( T^{4} \)
$67$ \( ( 2 + T )^{4} \)
$71$ \( ( 144 + T^{2} )^{2} \)
$73$ \( ( 48 + T^{2} )^{2} \)
$79$ \( ( -1 + T )^{4} \)
$83$ \( ( -75 + T^{2} )^{2} \)
$89$ \( ( -108 + T^{2} )^{2} \)
$97$ \( ( 27 + T^{2} )^{2} \)
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