Properties

Label 2352.3.m.h
Level $2352$
Weight $3$
Character orbit 2352.m
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.m (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(64.0873581775\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-19})\)
Defining polynomial: \(x^{4} - x^{3} - 4 x^{2} - 5 x + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + \beta_{2} q^{5} -3 q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + \beta_{2} q^{5} -3 q^{9} + ( -7 \beta_{1} - 5 \beta_{3} ) q^{11} + ( -7 + \beta_{2} ) q^{13} + ( -\beta_{1} - 3 \beta_{3} ) q^{15} + ( 12 - 4 \beta_{2} ) q^{17} + ( -8 \beta_{1} - 13 \beta_{3} ) q^{19} + 4 \beta_{3} q^{23} + ( -11 + \beta_{2} ) q^{25} -3 \beta_{1} q^{27} + ( -6 + 11 \beta_{2} ) q^{29} + ( 21 \beta_{1} + 12 \beta_{3} ) q^{31} + ( 16 - 5 \beta_{2} ) q^{33} + ( 13 - 7 \beta_{2} ) q^{37} + ( -8 \beta_{1} - 3 \beta_{3} ) q^{39} + ( 6 - 10 \beta_{2} ) q^{41} + ( 22 \beta_{1} - 5 \beta_{3} ) q^{43} -3 \beta_{2} q^{45} + 14 \beta_{1} q^{47} + ( 16 \beta_{1} + 12 \beta_{3} ) q^{51} + ( -82 - 5 \beta_{2} ) q^{53} + ( 27 \beta_{1} + 11 \beta_{3} ) q^{55} + ( 11 - 13 \beta_{2} ) q^{57} + ( -13 \beta_{1} - 17 \beta_{3} ) q^{59} + ( 50 - 4 \beta_{2} ) q^{61} + ( 14 - 6 \beta_{2} ) q^{65} + ( 6 \beta_{1} + 9 \beta_{3} ) q^{67} + ( 4 + 4 \beta_{2} ) q^{69} + ( 26 \beta_{1} + 40 \beta_{3} ) q^{71} + ( -95 - \beta_{2} ) q^{73} + ( -12 \beta_{1} - 3 \beta_{3} ) q^{75} + ( 19 \beta_{1} - 18 \beta_{3} ) q^{79} + 9 q^{81} + ( 47 \beta_{1} - 23 \beta_{3} ) q^{83} + ( -56 + 8 \beta_{2} ) q^{85} + ( -17 \beta_{1} - 33 \beta_{3} ) q^{87} + ( 60 + 2 \beta_{2} ) q^{89} + ( -51 + 12 \beta_{2} ) q^{93} + ( 60 \beta_{1} - 2 \beta_{3} ) q^{95} + ( -28 + 31 \beta_{2} ) q^{97} + ( 21 \beta_{1} + 15 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{5} - 12q^{9} + O(q^{10}) \) \( 4q + 2q^{5} - 12q^{9} - 26q^{13} + 40q^{17} - 42q^{25} - 2q^{29} + 54q^{33} + 38q^{37} + 4q^{41} - 6q^{45} - 338q^{53} + 18q^{57} + 192q^{61} + 44q^{65} + 24q^{69} - 382q^{73} + 36q^{81} - 208q^{85} + 244q^{89} - 180q^{93} - 50q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 4 x^{2} - 5 x + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 4 \nu^{2} - 4 \nu - 15 \)\()/10\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + \nu^{2} + 9 \nu + 5 \)\()/5\)
\(\beta_{3}\)\(=\)\((\)\( -3 \nu^{3} - 2 \nu^{2} + 2 \nu + 25 \)\()/10\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{3} + \beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + \beta_{2} + 5 \beta_{1} + 4\)\()/2\)
\(\nu^{3}\)\(=\)\(-4 \beta_{3} - 2 \beta_{1} + 7\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1471.1
−1.63746 + 1.52274i
2.13746 0.656712i
−1.63746 1.52274i
2.13746 + 0.656712i
0 1.73205i 0 −3.27492 0 0 0 −3.00000 0
1471.2 0 1.73205i 0 4.27492 0 0 0 −3.00000 0
1471.3 0 1.73205i 0 −3.27492 0 0 0 −3.00000 0
1471.4 0 1.73205i 0 4.27492 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
4.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.m.h 4
4.b odd 2 1 inner 2352.3.m.h 4
7.b odd 2 1 2352.3.m.g 4
7.d odd 6 1 336.3.be.a 4
7.d odd 6 1 336.3.be.b yes 4
21.g even 6 1 1008.3.cd.f 4
21.g even 6 1 1008.3.cd.g 4
28.d even 2 1 2352.3.m.g 4
28.f even 6 1 336.3.be.a 4
28.f even 6 1 336.3.be.b yes 4
84.j odd 6 1 1008.3.cd.f 4
84.j odd 6 1 1008.3.cd.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.3.be.a 4 7.d odd 6 1
336.3.be.a 4 28.f even 6 1
336.3.be.b yes 4 7.d odd 6 1
336.3.be.b yes 4 28.f even 6 1
1008.3.cd.f 4 21.g even 6 1
1008.3.cd.f 4 84.j odd 6 1
1008.3.cd.g 4 21.g even 6 1
1008.3.cd.g 4 84.j odd 6 1
2352.3.m.g 4 7.b odd 2 1
2352.3.m.g 4 28.d even 2 1
2352.3.m.h 4 1.a even 1 1 trivial
2352.3.m.h 4 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - T_{5} - 14 \) acting on \(S_{3}^{\mathrm{new}}(2352, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 3 + T^{2} )^{2} \)
$5$ \( ( -14 - T + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( 3364 + 359 T^{2} + T^{4} \)
$13$ \( ( 28 + 13 T + T^{2} )^{2} \)
$17$ \( ( -128 - 20 T + T^{2} )^{2} \)
$19$ \( 633616 + 1619 T^{2} + T^{4} \)
$23$ \( 4096 + 176 T^{2} + T^{4} \)
$29$ \( ( -1724 + T + T^{2} )^{2} \)
$31$ \( 81 + 2718 T^{2} + T^{4} \)
$37$ \( ( -608 - 19 T + T^{2} )^{2} \)
$41$ \( ( -1424 - 2 T + T^{2} )^{2} \)
$43$ \( 2829124 + 3839 T^{2} + T^{4} \)
$47$ \( ( 588 + T^{2} )^{2} \)
$53$ \( ( 6784 + 169 T + T^{2} )^{2} \)
$59$ \( 1721344 + 2867 T^{2} + T^{4} \)
$61$ \( ( 2076 - 96 T + T^{2} )^{2} \)
$67$ \( 142884 + 783 T^{2} + T^{4} \)
$71$ \( 56130064 + 15416 T^{2} + T^{4} \)
$73$ \( ( 9106 + 191 T + T^{2} )^{2} \)
$79$ \( 660969 + 7782 T^{2} + T^{4} \)
$83$ \( 60124516 + 25559 T^{2} + T^{4} \)
$89$ \( ( 3664 - 122 T + T^{2} )^{2} \)
$97$ \( ( -13538 + 25 T + T^{2} )^{2} \)
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