Properties

Label 2352.2.h.g
Level $2352$
Weight $2$
Character orbit 2352.h
Analytic conductor $18.781$
Analytic rank $0$
Dimension $4$
CM discriminant -84
Inner twists $8$

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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: \(\Q(\sqrt{-3}, \sqrt{-14})\)
Defining polynomial: \(x^{4} - 14 x^{2} + 196\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + \beta_{1} q^{5} -3 q^{9} +O(q^{10})\) \( q + \beta_{2} q^{3} + \beta_{1} q^{5} -3 q^{9} -\beta_{3} q^{11} -\beta_{3} q^{15} + \beta_{1} q^{17} -4 \beta_{2} q^{19} + \beta_{3} q^{23} -9 q^{25} -3 \beta_{2} q^{27} + 2 \beta_{2} q^{31} -3 \beta_{1} q^{33} -8 q^{37} -\beta_{1} q^{41} -3 \beta_{1} q^{45} -\beta_{3} q^{51} -14 \beta_{2} q^{55} + 12 q^{57} + 3 \beta_{1} q^{69} + \beta_{3} q^{71} -9 \beta_{2} q^{75} + 9 q^{81} -14 q^{85} + 5 \beta_{1} q^{89} -6 q^{93} + 4 \beta_{3} q^{95} + 3 \beta_{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 12q^{9} + O(q^{10}) \) \( 4q - 12q^{9} - 36q^{25} - 32q^{37} + 48q^{57} + 36q^{81} - 56q^{85} - 24q^{93} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 14 x^{2} + 196\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} \)\(/14\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{2} - 7 \)\()/7\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 28 \nu \)\()/14\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\(7 \beta_{2} + 7\)
\(\nu^{3}\)\(=\)\(14 \beta_{1}\)

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
3.24037 1.87083i
−3.24037 + 1.87083i
−3.24037 1.87083i
3.24037 + 1.87083i
0 1.73205i 0 3.74166i 0 0 0 −3.00000 0
2255.2 0 1.73205i 0 3.74166i 0 0 0 −3.00000 0
2255.3 0 1.73205i 0 3.74166i 0 0 0 −3.00000 0
2255.4 0 1.73205i 0 3.74166i 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
84.h odd 2 1 CM by \(\Q(\sqrt{-21}) \)
3.b odd 2 1 inner
4.b odd 2 1 inner
7.b odd 2 1 inner
12.b even 2 1 inner
21.c even 2 1 inner
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.h.g 4
3.b odd 2 1 inner 2352.2.h.g 4
4.b odd 2 1 inner 2352.2.h.g 4
7.b odd 2 1 inner 2352.2.h.g 4
12.b even 2 1 inner 2352.2.h.g 4
21.c even 2 1 inner 2352.2.h.g 4
28.d even 2 1 inner 2352.2.h.g 4
84.h odd 2 1 CM 2352.2.h.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2352.2.h.g 4 1.a even 1 1 trivial
2352.2.h.g 4 3.b odd 2 1 inner
2352.2.h.g 4 4.b odd 2 1 inner
2352.2.h.g 4 7.b odd 2 1 inner
2352.2.h.g 4 12.b even 2 1 inner
2352.2.h.g 4 21.c even 2 1 inner
2352.2.h.g 4 28.d even 2 1 inner
2352.2.h.g 4 84.h odd 2 1 CM

Hecke kernels

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

\( T_{5}^{2} + 14 \)
\( T_{11}^{2} - 42 \)
\( T_{13} \)
\( T_{47} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 3 + T^{2} )^{2} \)
$5$ \( ( 14 + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( ( -42 + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( ( 14 + T^{2} )^{2} \)
$19$ \( ( 48 + T^{2} )^{2} \)
$23$ \( ( -42 + T^{2} )^{2} \)
$29$ \( T^{4} \)
$31$ \( ( 12 + T^{2} )^{2} \)
$37$ \( ( 8 + T )^{4} \)
$41$ \( ( 14 + T^{2} )^{2} \)
$43$ \( T^{4} \)
$47$ \( T^{4} \)
$53$ \( T^{4} \)
$59$ \( T^{4} \)
$61$ \( T^{4} \)
$67$ \( T^{4} \)
$71$ \( ( -42 + T^{2} )^{2} \)
$73$ \( T^{4} \)
$79$ \( T^{4} \)
$83$ \( T^{4} \)
$89$ \( ( 350 + T^{2} )^{2} \)
$97$ \( T^{4} \)
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