Properties

Label 2352.2.h.j
Level $2352$
Weight $2$
Character orbit 2352.h
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.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.7808145554\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{11})\)
Defining polynomial: \(x^{4} - 5 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 336)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + ( -1 - 2 \beta_{3} ) q^{5} + ( 3 + \beta_{3} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + ( -1 - 2 \beta_{3} ) q^{5} + ( 3 + \beta_{3} ) q^{9} + ( -2 \beta_{1} + \beta_{2} ) q^{11} + 4 q^{13} + ( -\beta_{1} + 6 \beta_{2} ) q^{15} + ( 1 + 2 \beta_{3} ) q^{17} -7 \beta_{2} q^{19} + ( -2 \beta_{1} + \beta_{2} ) q^{23} -6 q^{25} + ( -2 \beta_{1} - 3 \beta_{2} ) q^{27} + ( -2 - 4 \beta_{3} ) q^{29} -3 \beta_{2} q^{31} + ( 6 + \beta_{3} ) q^{33} - q^{37} -4 \beta_{1} q^{39} + ( -2 - 4 \beta_{3} ) q^{41} -2 \beta_{2} q^{43} + ( 3 - 5 \beta_{3} ) q^{45} + ( 6 \beta_{1} - 3 \beta_{2} ) q^{47} + ( \beta_{1} - 6 \beta_{2} ) q^{51} + ( 1 + 2 \beta_{3} ) q^{53} + 11 \beta_{2} q^{55} + 7 \beta_{3} q^{57} + ( -2 \beta_{1} + \beta_{2} ) q^{59} -3 q^{61} + ( -4 - 8 \beta_{3} ) q^{65} + 9 \beta_{2} q^{67} + ( 6 + \beta_{3} ) q^{69} + ( -8 \beta_{1} + 4 \beta_{2} ) q^{71} -7 q^{73} + 6 \beta_{1} q^{75} + 9 \beta_{2} q^{79} + ( 6 + 5 \beta_{3} ) q^{81} + ( -8 \beta_{1} + 4 \beta_{2} ) q^{83} + 11 q^{85} + ( -2 \beta_{1} + 12 \beta_{2} ) q^{87} + ( 1 + 2 \beta_{3} ) q^{89} + 3 \beta_{3} q^{93} + ( -14 \beta_{1} + 7 \beta_{2} ) q^{95} -8 q^{97} + ( -5 \beta_{1} - 3 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 10q^{9} + O(q^{10}) \) \( 4q + 10q^{9} + 16q^{13} - 24q^{25} + 22q^{33} - 4q^{37} + 22q^{45} - 14q^{57} - 12q^{61} + 22q^{69} - 28q^{73} + 14q^{81} + 44q^{85} - 6q^{93} - 32q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 5 x^{2} + 9\):

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2255.1
1.65831 + 0.500000i
1.65831 0.500000i
−1.65831 + 0.500000i
−1.65831 0.500000i
0 −1.65831 0.500000i 0 3.31662i 0 0 0 2.50000 + 1.65831i 0
2255.2 0 −1.65831 + 0.500000i 0 3.31662i 0 0 0 2.50000 1.65831i 0
2255.3 0 1.65831 0.500000i 0 3.31662i 0 0 0 2.50000 1.65831i 0
2255.4 0 1.65831 + 0.500000i 0 3.31662i 0 0 0 2.50000 + 1.65831i 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
4.b odd 2 1 inner
12.b 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.h.j 4
3.b odd 2 1 inner 2352.2.h.j 4
4.b odd 2 1 inner 2352.2.h.j 4
7.b odd 2 1 2352.2.h.i 4
7.d odd 6 2 336.2.bj.f 8
12.b even 2 1 inner 2352.2.h.j 4
21.c even 2 1 2352.2.h.i 4
21.g even 6 2 336.2.bj.f 8
28.d even 2 1 2352.2.h.i 4
28.f even 6 2 336.2.bj.f 8
84.h odd 2 1 2352.2.h.i 4
84.j odd 6 2 336.2.bj.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.bj.f 8 7.d odd 6 2
336.2.bj.f 8 21.g even 6 2
336.2.bj.f 8 28.f even 6 2
336.2.bj.f 8 84.j odd 6 2
2352.2.h.i 4 7.b odd 2 1
2352.2.h.i 4 21.c even 2 1
2352.2.h.i 4 28.d even 2 1
2352.2.h.i 4 84.h odd 2 1
2352.2.h.j 4 1.a even 1 1 trivial
2352.2.h.j 4 3.b odd 2 1 inner
2352.2.h.j 4 4.b odd 2 1 inner
2352.2.h.j 4 12.b 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 \(S_{2}^{\mathrm{new}}(2352, [\chi])\):

\( T_{5}^{2} + 11 \)
\( T_{11}^{2} - 11 \)
\( T_{13} - 4 \)
\( T_{47}^{2} - 99 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 9 - 5 T^{2} + T^{4} \)
$5$ \( ( 11 + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( ( -11 + T^{2} )^{2} \)
$13$ \( ( -4 + T )^{4} \)
$17$ \( ( 11 + T^{2} )^{2} \)
$19$ \( ( 49 + T^{2} )^{2} \)
$23$ \( ( -11 + T^{2} )^{2} \)
$29$ \( ( 44 + T^{2} )^{2} \)
$31$ \( ( 9 + T^{2} )^{2} \)
$37$ \( ( 1 + T )^{4} \)
$41$ \( ( 44 + T^{2} )^{2} \)
$43$ \( ( 4 + T^{2} )^{2} \)
$47$ \( ( -99 + T^{2} )^{2} \)
$53$ \( ( 11 + T^{2} )^{2} \)
$59$ \( ( -11 + T^{2} )^{2} \)
$61$ \( ( 3 + T )^{4} \)
$67$ \( ( 81 + T^{2} )^{2} \)
$71$ \( ( -176 + T^{2} )^{2} \)
$73$ \( ( 7 + T )^{4} \)
$79$ \( ( 81 + T^{2} )^{2} \)
$83$ \( ( -176 + T^{2} )^{2} \)
$89$ \( ( 11 + T^{2} )^{2} \)
$97$ \( ( 8 + T )^{4} \)
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