Properties

Label 2352.2.h.i
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: \( x^{4} - 5x^{2} + 9 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + (\beta_{2} - \beta_1) q^{3} + ( - 2 \beta_{3} - 1) q^{5} + ( - \beta_{3} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - \beta_1) q^{3} + ( - 2 \beta_{3} - 1) q^{5} + ( - \beta_{3} + 2) q^{9} + ( - \beta_{2} + 2 \beta_1) q^{11} - 4 q^{13} + (5 \beta_{2} + \beta_1) q^{15} + (2 \beta_{3} + 1) q^{17} + 7 \beta_{2} q^{19} + ( - \beta_{2} + 2 \beta_1) q^{23} - 6 q^{25} + (5 \beta_{2} - 2 \beta_1) q^{27} + (4 \beta_{3} + 2) q^{29} + 3 \beta_{2} q^{31} + (\beta_{3} - 5) q^{33} - q^{37} + ( - 4 \beta_{2} + 4 \beta_1) q^{39} + ( - 4 \beta_{3} - 2) q^{41} - 2 \beta_{2} q^{43} + ( - 5 \beta_{3} - 8) q^{45} + ( - 3 \beta_{2} + 6 \beta_1) q^{47} + ( - 5 \beta_{2} - \beta_1) q^{51} + ( - 2 \beta_{3} - 1) q^{53} - 11 \beta_{2} q^{55} + ( - 7 \beta_{3} - 7) q^{57} + (\beta_{2} - 2 \beta_1) q^{59} + 3 q^{61} + (8 \beta_{3} + 4) q^{65} + 9 \beta_{2} q^{67} + (\beta_{3} - 5) q^{69} + ( - 4 \beta_{2} + 8 \beta_1) q^{71} + 7 q^{73} + ( - 6 \beta_{2} + 6 \beta_1) q^{75} + 9 \beta_{2} q^{79} + ( - 5 \beta_{3} + 1) q^{81} + (4 \beta_{2} - 8 \beta_1) q^{83} + 11 q^{85} + ( - 10 \beta_{2} - 2 \beta_1) q^{87} + (2 \beta_{3} + 1) q^{89} + ( - 3 \beta_{3} - 3) q^{93} + ( - 7 \beta_{2} + 14 \beta_1) q^{95} + 8 q^{97} + ( - 8 \beta_{2} + 5 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 10 q^{9} - 16 q^{13} - 24 q^{25} - 22 q^{33} - 4 q^{37} - 22 q^{45} - 14 q^{57} + 12 q^{61} - 22 q^{69} + 28 q^{73} + 14 q^{81} + 44 q^{85} - 6 q^{93} + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 2\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{2} + 2\beta_1 \) Copy content Toggle raw display

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.i 4
3.b odd 2 1 inner 2352.2.h.i 4
4.b odd 2 1 inner 2352.2.h.i 4
7.b odd 2 1 2352.2.h.j 4
7.c even 3 2 336.2.bj.f 8
12.b even 2 1 inner 2352.2.h.i 4
21.c even 2 1 2352.2.h.j 4
21.h odd 6 2 336.2.bj.f 8
28.d even 2 1 2352.2.h.j 4
28.g odd 6 2 336.2.bj.f 8
84.h odd 2 1 2352.2.h.j 4
84.n even 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.c even 3 2
336.2.bj.f 8 21.h odd 6 2
336.2.bj.f 8 28.g odd 6 2
336.2.bj.f 8 84.n even 6 2
2352.2.h.i 4 1.a even 1 1 trivial
2352.2.h.i 4 3.b odd 2 1 inner
2352.2.h.i 4 4.b odd 2 1 inner
2352.2.h.i 4 12.b even 2 1 inner
2352.2.h.j 4 7.b odd 2 1
2352.2.h.j 4 21.c even 2 1
2352.2.h.j 4 28.d even 2 1
2352.2.h.j 4 84.h odd 2 1

Hecke kernels

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

\( T_{5}^{2} + 11 \) Copy content Toggle raw display
\( T_{11}^{2} - 11 \) Copy content Toggle raw display
\( T_{13} + 4 \) Copy content Toggle raw display
\( T_{47}^{2} - 99 \) Copy content Toggle raw display

Hecke characteristic polynomials

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