Properties

Label 2352.2.b.j
Level $2352$
Weight $2$
Character orbit 2352.b
Analytic conductor $18.781$
Analytic rank $0$
Dimension $4$
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.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.7808145554\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.2048.2
Defining polynomial: \(x^{4} + 4 x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + q^{3} + \beta_{1} q^{5} + q^{9} +O(q^{10})\) \( q + q^{3} + \beta_{1} q^{5} + q^{9} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{11} + ( -2 \beta_{1} + \beta_{3} ) q^{13} + \beta_{1} q^{15} + ( -2 \beta_{1} - \beta_{3} ) q^{17} + 4 \beta_{2} q^{19} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{23} + ( 3 + \beta_{2} ) q^{25} + q^{27} + ( -4 + \beta_{2} ) q^{29} + ( -4 + 4 \beta_{2} ) q^{31} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{33} -\beta_{2} q^{37} + ( -2 \beta_{1} + \beta_{3} ) q^{39} + ( 3 \beta_{1} + 2 \beta_{3} ) q^{41} + 4 \beta_{3} q^{43} + \beta_{1} q^{45} + ( 4 + 4 \beta_{2} ) q^{47} + ( -2 \beta_{1} - \beta_{3} ) q^{51} + ( -4 + 4 \beta_{2} ) q^{53} + ( 4 - 4 \beta_{2} ) q^{55} + 4 \beta_{2} q^{57} + 4 \beta_{2} q^{59} + ( 2 \beta_{1} + 3 \beta_{3} ) q^{61} + ( 4 - 3 \beta_{2} ) q^{65} + ( -4 \beta_{1} + 8 \beta_{3} ) q^{67} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{69} + ( 6 \beta_{1} + 6 \beta_{3} ) q^{71} + ( -\beta_{1} + 4 \beta_{3} ) q^{73} + ( 3 + \beta_{2} ) q^{75} + ( -8 \beta_{1} - 4 \beta_{3} ) q^{79} + q^{81} + ( 12 - 4 \beta_{2} ) q^{83} + ( 4 - \beta_{2} ) q^{85} + ( -4 + \beta_{2} ) q^{87} + ( 2 \beta_{1} - 7 \beta_{3} ) q^{89} + ( -4 + 4 \beta_{2} ) q^{93} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{95} -5 \beta_{1} q^{97} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} + 4q^{9} + 12q^{25} + 4q^{27} - 16q^{29} - 16q^{31} + 16q^{47} - 16q^{53} + 16q^{55} + 16q^{65} + 12q^{75} + 4q^{81} + 48q^{83} + 16q^{85} - 16q^{87} - 16q^{93} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + 3 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 2\)
\(\nu^{3}\)\(=\)\(\beta_{3} - 3 \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1567.1
1.84776i
0.765367i
0.765367i
1.84776i
0 1.00000 0 1.84776i 0 0 0 1.00000 0
1567.2 0 1.00000 0 0.765367i 0 0 0 1.00000 0
1567.3 0 1.00000 0 0.765367i 0 0 0 1.00000 0
1567.4 0 1.00000 0 1.84776i 0 0 0 1.00000 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
28.d 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.b.j yes 4
3.b odd 2 1 7056.2.b.t 4
4.b odd 2 1 2352.2.b.i 4
7.b odd 2 1 2352.2.b.i 4
7.c even 3 2 2352.2.bl.p 8
7.d odd 6 2 2352.2.bl.s 8
12.b even 2 1 7056.2.b.u 4
21.c even 2 1 7056.2.b.u 4
28.d even 2 1 inner 2352.2.b.j yes 4
28.f even 6 2 2352.2.bl.p 8
28.g odd 6 2 2352.2.bl.s 8
84.h odd 2 1 7056.2.b.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2352.2.b.i 4 4.b odd 2 1
2352.2.b.i 4 7.b odd 2 1
2352.2.b.j yes 4 1.a even 1 1 trivial
2352.2.b.j yes 4 28.d even 2 1 inner
2352.2.bl.p 8 7.c even 3 2
2352.2.bl.p 8 28.f even 6 2
2352.2.bl.s 8 7.d odd 6 2
2352.2.bl.s 8 28.g odd 6 2
7056.2.b.t 4 3.b odd 2 1
7056.2.b.t 4 84.h odd 2 1
7056.2.b.u 4 12.b even 2 1
7056.2.b.u 4 21.c even 2 1

Hecke kernels

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

\( T_{5}^{4} + 4 T_{5}^{2} + 2 \)
\( T_{11}^{4} + 32 T_{11}^{2} + 128 \)
\( T_{13}^{4} + 20 T_{13}^{2} + 2 \)
\( T_{19}^{2} - 32 \)
\( T_{31}^{2} + 8 T_{31} - 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( 2 + 4 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( 128 + 32 T^{2} + T^{4} \)
$13$ \( 2 + 20 T^{2} + T^{4} \)
$17$ \( 98 + 20 T^{2} + T^{4} \)
$19$ \( ( -32 + T^{2} )^{2} \)
$23$ \( 128 + 32 T^{2} + T^{4} \)
$29$ \( ( 14 + 8 T + T^{2} )^{2} \)
$31$ \( ( -16 + 8 T + T^{2} )^{2} \)
$37$ \( ( -2 + T^{2} )^{2} \)
$41$ \( 578 + 52 T^{2} + T^{4} \)
$43$ \( 512 + 64 T^{2} + T^{4} \)
$47$ \( ( -16 - 8 T + T^{2} )^{2} \)
$53$ \( ( -16 + 8 T + T^{2} )^{2} \)
$59$ \( ( -32 + T^{2} )^{2} \)
$61$ \( 98 + 52 T^{2} + T^{4} \)
$67$ \( 25088 + 320 T^{2} + T^{4} \)
$71$ \( 10368 + 288 T^{2} + T^{4} \)
$73$ \( 1058 + 68 T^{2} + T^{4} \)
$79$ \( 25088 + 320 T^{2} + T^{4} \)
$83$ \( ( 112 - 24 T + T^{2} )^{2} \)
$89$ \( 10658 + 212 T^{2} + T^{4} \)
$97$ \( 1250 + 100 T^{2} + T^{4} \)
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