Properties

Label 2352.3.f.f
Level $2352$
Weight $3$
Character orbit 2352.f
Analytic conductor $64.087$
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: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2352.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(64.0873581775\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{65})\)
Defining polynomial: \(x^{4} - x^{3} + 17 x^{2} + 16 x + 256\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 84)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( 2 \beta_{1} + \beta_{2} ) q^{5} -3 q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( 2 \beta_{1} + \beta_{2} ) q^{5} -3 q^{9} + ( 8 - \beta_{3} ) q^{11} + ( \beta_{1} + \beta_{2} ) q^{13} + ( -4 - \beta_{3} ) q^{15} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{17} + ( -13 \beta_{1} + \beta_{2} ) q^{19} + 24 q^{23} + ( -29 - 3 \beta_{3} ) q^{25} -3 \beta_{1} q^{27} + ( 2 - \beta_{3} ) q^{29} + ( -11 \beta_{1} - 6 \beta_{2} ) q^{31} + ( 6 \beta_{1} - 3 \beta_{2} ) q^{33} + ( -37 + \beta_{3} ) q^{37} + ( -1 - \beta_{3} ) q^{39} + ( -14 \beta_{1} - 4 \beta_{2} ) q^{41} + ( -19 + 3 \beta_{3} ) q^{43} + ( -6 \beta_{1} - 3 \beta_{2} ) q^{45} + ( -14 \beta_{1} + 2 \beta_{2} ) q^{47} + ( 8 + 2 \beta_{3} ) q^{51} + ( -50 + \beta_{3} ) q^{53} + ( -36 \beta_{1} + 3 \beta_{2} ) q^{55} + ( 41 - \beta_{3} ) q^{57} + ( -40 \beta_{1} + \beta_{2} ) q^{59} + ( 36 \beta_{1} + 8 \beta_{2} ) q^{61} + ( -50 - 2 \beta_{3} ) q^{65} + ( -1 - 5 \beta_{3} ) q^{67} + 24 \beta_{1} q^{69} + ( 54 - 6 \beta_{3} ) q^{71} + ( -11 \beta_{1} - 7 \beta_{2} ) q^{73} + ( -35 \beta_{1} - 9 \beta_{2} ) q^{75} + ( 29 - 2 \beta_{3} ) q^{79} + 9 q^{81} + ( 2 \beta_{1} - 17 \beta_{2} ) q^{83} + ( 108 + 6 \beta_{3} ) q^{85} -3 \beta_{2} q^{87} + ( -36 \beta_{1} - 6 \beta_{2} ) q^{89} + ( 21 + 6 \beta_{3} ) q^{93} + ( 6 + 12 \beta_{3} ) q^{95} + ( 12 \beta_{1} + 7 \beta_{2} ) q^{97} + ( -24 + 3 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 12q^{9} + O(q^{10}) \) \( 4q - 12q^{9} + 30q^{11} - 18q^{15} + 96q^{23} - 122q^{25} + 6q^{29} - 146q^{37} - 6q^{39} - 70q^{43} + 36q^{51} - 198q^{53} + 162q^{57} - 204q^{65} - 14q^{67} + 204q^{71} + 112q^{79} + 36q^{81} + 444q^{85} + 96q^{93} + 48q^{95} - 90q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 17 x^{2} + 16 x + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{3} + 17 \nu^{2} - 17 \nu + 120 \)\()/136\)
\(\beta_{2}\)\(=\)\((\)\( 9 \nu^{3} - 17 \nu^{2} + 289 \nu + 8 \)\()/136\)
\(\beta_{3}\)\(=\)\((\)\( -3 \nu^{3} - 65 \)\()/17\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + 3 \beta_{2} + 3 \beta_{1} + 1\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + 3 \beta_{2} + 51 \beta_{1} - 49\)\()/6\)
\(\nu^{3}\)\(=\)\((\)\(-17 \beta_{3} - 65\)\()/3\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
2.26556 3.92407i
−1.76556 + 3.05805i
−1.76556 3.05805i
2.26556 + 3.92407i
0 1.73205i 0 9.58020i 0 0 0 −3.00000 0
97.2 0 1.73205i 0 4.38404i 0 0 0 −3.00000 0
97.3 0 1.73205i 0 4.38404i 0 0 0 −3.00000 0
97.4 0 1.73205i 0 9.58020i 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
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2352.3.f.f 4
4.b odd 2 1 588.3.d.b 4
7.b odd 2 1 inner 2352.3.f.f 4
7.c even 3 1 336.3.bh.f 4
7.d odd 6 1 336.3.bh.f 4
12.b even 2 1 1764.3.d.f 4
21.g even 6 1 1008.3.cg.m 4
21.h odd 6 1 1008.3.cg.m 4
28.d even 2 1 588.3.d.b 4
28.f even 6 1 84.3.m.b 4
28.f even 6 1 588.3.m.d 4
28.g odd 6 1 84.3.m.b 4
28.g odd 6 1 588.3.m.d 4
84.h odd 2 1 1764.3.d.f 4
84.j odd 6 1 252.3.z.e 4
84.j odd 6 1 1764.3.z.h 4
84.n even 6 1 252.3.z.e 4
84.n even 6 1 1764.3.z.h 4
140.p odd 6 1 2100.3.bd.f 4
140.s even 6 1 2100.3.bd.f 4
140.w even 12 2 2100.3.be.d 8
140.x odd 12 2 2100.3.be.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.3.m.b 4 28.f even 6 1
84.3.m.b 4 28.g odd 6 1
252.3.z.e 4 84.j odd 6 1
252.3.z.e 4 84.n even 6 1
336.3.bh.f 4 7.c even 3 1
336.3.bh.f 4 7.d odd 6 1
588.3.d.b 4 4.b odd 2 1
588.3.d.b 4 28.d even 2 1
588.3.m.d 4 28.f even 6 1
588.3.m.d 4 28.g odd 6 1
1008.3.cg.m 4 21.g even 6 1
1008.3.cg.m 4 21.h odd 6 1
1764.3.d.f 4 12.b even 2 1
1764.3.d.f 4 84.h odd 2 1
1764.3.z.h 4 84.j odd 6 1
1764.3.z.h 4 84.n even 6 1
2100.3.bd.f 4 140.p odd 6 1
2100.3.bd.f 4 140.s even 6 1
2100.3.be.d 8 140.w even 12 2
2100.3.be.d 8 140.x odd 12 2
2352.3.f.f 4 1.a even 1 1 trivial
2352.3.f.f 4 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(2352, [\chi])\):

\( T_{5}^{4} + 111 T_{5}^{2} + 1764 \)
\( T_{11}^{2} - 15 T_{11} - 90 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 3 + T^{2} )^{2} \)
$5$ \( 1764 + 111 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( -90 - 15 T + T^{2} )^{2} \)
$13$ \( 2304 + 99 T^{2} + T^{4} \)
$17$ \( 28224 + 444 T^{2} + T^{4} \)
$19$ \( 248004 + 1191 T^{2} + T^{4} \)
$23$ \( ( -24 + T )^{4} \)
$29$ \( ( -144 - 3 T + T^{2} )^{2} \)
$31$ \( 2442969 + 3894 T^{2} + T^{4} \)
$37$ \( ( 1186 + 73 T + T^{2} )^{2} \)
$41$ \( 121104 + 2424 T^{2} + T^{4} \)
$43$ \( ( -1010 + 35 T + T^{2} )^{2} \)
$47$ \( 230400 + 1740 T^{2} + T^{4} \)
$53$ \( ( 2304 + 99 T + T^{2} )^{2} \)
$59$ \( 23736384 + 9939 T^{2} + T^{4} \)
$61$ \( 2304 + 12384 T^{2} + T^{4} \)
$67$ \( ( -3644 + 7 T + T^{2} )^{2} \)
$71$ \( ( -2664 - 102 T + T^{2} )^{2} \)
$73$ \( 4928400 + 5115 T^{2} + T^{4} \)
$79$ \( ( 199 - 56 T + T^{2} )^{2} \)
$83$ \( 189282564 + 28839 T^{2} + T^{4} \)
$89$ \( 2286144 + 10044 T^{2} + T^{4} \)
$97$ \( 4717584 + 5211 T^{2} + T^{4} \)
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