Properties

Label 2352.2.h.n
Level $2352$
Weight $2$
Character orbit 2352.h
Analytic conductor $18.781$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: 8.0.8275904784.2
Defining polynomial: \(x^{8} - 3 x^{7} + 5 x^{6} - 6 x^{5} + 4 x^{4} - 18 x^{3} + 45 x^{2} - 81 x + 81\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} -\beta_{7} q^{5} + ( -\beta_{2} + \beta_{7} ) q^{9} +O(q^{10})\) \( q + \beta_{3} q^{3} -\beta_{7} q^{5} + ( -\beta_{2} + \beta_{7} ) q^{9} + ( \beta_{1} + \beta_{3} + \beta_{5} ) q^{11} + 2 q^{13} + ( -\beta_{1} + \beta_{5} - \beta_{6} ) q^{15} + ( \beta_{2} - \beta_{4} - 2 \beta_{7} ) q^{17} + ( \beta_{3} - \beta_{5} ) q^{19} + ( -\beta_{3} - \beta_{5} ) q^{23} + ( -4 - 2 \beta_{2} - 2 \beta_{4} ) q^{25} + ( \beta_{1} + \beta_{3} - \beta_{5} - 2 \beta_{6} ) q^{27} + ( -\beta_{2} + \beta_{4} - \beta_{7} ) q^{29} + \beta_{6} q^{31} + ( 2 - 2 \beta_{2} - \beta_{4} - \beta_{7} ) q^{33} + ( -2 + \beta_{2} + \beta_{4} ) q^{37} + 2 \beta_{3} q^{39} + ( 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{7} ) q^{41} + ( -2 \beta_{3} + 2 \beta_{5} + 4 \beta_{6} ) q^{43} + ( 5 + \beta_{2} + 2 \beta_{4} + 2 \beta_{7} ) q^{45} + ( 3 \beta_{3} + 3 \beta_{5} ) q^{47} + ( -\beta_{1} - \beta_{3} + 4 \beta_{5} + 2 \beta_{6} ) q^{51} + ( -2 \beta_{2} + 2 \beta_{4} - \beta_{7} ) q^{53} + ( 4 \beta_{3} - 4 \beta_{5} + \beta_{6} ) q^{55} + ( -3 - \beta_{2} + \beta_{7} ) q^{57} + ( \beta_{1} - 3 \beta_{3} - 3 \beta_{5} ) q^{59} + ( -8 - \beta_{2} - \beta_{4} ) q^{61} -2 \beta_{7} q^{65} + ( -3 \beta_{3} + 3 \beta_{5} - 4 \beta_{6} ) q^{67} + ( -3 + \beta_{2} - \beta_{7} ) q^{69} + 2 \beta_{1} q^{71} + ( -4 - \beta_{2} - \beta_{4} ) q^{73} + ( 2 \beta_{1} - 2 \beta_{3} + 4 \beta_{5} - 4 \beta_{6} ) q^{75} -3 \beta_{6} q^{79} + ( -2 - 2 \beta_{2} + \beta_{4} - \beta_{7} ) q^{81} + \beta_{1} q^{83} + ( -10 - 3 \beta_{2} - 3 \beta_{4} ) q^{85} + ( -2 \beta_{1} + \beta_{3} - \beta_{5} - 5 \beta_{6} ) q^{87} + ( -3 \beta_{2} + 3 \beta_{4} + 4 \beta_{7} ) q^{89} + ( -1 - \beta_{4} ) q^{93} + ( -2 \beta_{1} + \beta_{3} + \beta_{5} ) q^{95} + ( 1 + \beta_{2} + \beta_{4} ) q^{97} + ( 4 \beta_{3} + 3 \beta_{5} - 6 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{9} + O(q^{10}) \) \( 8 q + 2 q^{9} + 16 q^{13} - 24 q^{25} + 22 q^{33} - 20 q^{37} + 34 q^{45} - 22 q^{57} - 60 q^{61} - 26 q^{69} - 28 q^{73} - 14 q^{81} - 68 q^{85} - 6 q^{93} + 4 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3 x^{7} + 5 x^{6} - 6 x^{5} + 4 x^{4} - 18 x^{3} + 45 x^{2} - 81 x + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 4 \nu^{7} - 3 \nu^{6} + 2 \nu^{5} - 6 \nu^{4} - 20 \nu^{3} - 90 \nu^{2} + 54 \nu - 81 \)\()/54\)
\(\beta_{2}\)\(=\)\((\)\( 8 \nu^{7} - 3 \nu^{6} + 13 \nu^{5} + 3 \nu^{4} + 5 \nu^{3} - 87 \nu^{2} + 45 \nu - 270 \)\()/54\)
\(\beta_{3}\)\(=\)\((\)\( 4 \nu^{7} - 6 \nu^{6} + 11 \nu^{5} - 3 \nu^{4} + 7 \nu^{3} - 57 \nu^{2} + 99 \nu - 189 \)\()/18\)
\(\beta_{4}\)\(=\)\((\)\( -4 \nu^{7} + 6 \nu^{6} - 11 \nu^{5} + 3 \nu^{4} - 7 \nu^{3} + 57 \nu^{2} - 63 \nu + 171 \)\()/18\)
\(\beta_{5}\)\(=\)\((\)\( -4 \nu^{7} + 5 \nu^{6} - 11 \nu^{5} + 7 \nu^{4} - 7 \nu^{3} + 53 \nu^{2} - 75 \nu + 180 \)\()/18\)
\(\beta_{6}\)\(=\)\((\)\( 7 \nu^{7} - 9 \nu^{6} + 17 \nu^{5} - 9 \nu^{4} + 19 \nu^{3} - 105 \nu^{2} + 144 \nu - 297 \)\()/27\)
\(\beta_{7}\)\(=\)\((\)\( 13 \nu^{7} - 15 \nu^{6} + 38 \nu^{5} - 12 \nu^{4} + 25 \nu^{3} - 192 \nu^{2} + 225 \nu - 621 \)\()/27\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} + \beta_{3} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{7} - \beta_{6} - 2 \beta_{5} + \beta_{3} + \beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{3}\)\(=\)\(2 \beta_{6} + \beta_{5} - \beta_{3} - \beta_{1}\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{7} - \beta_{6} + 7 \beta_{5} - \beta_{4} + 4 \beta_{3} + 2 \beta_{2} - \beta_{1} + 2\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(6 \beta_{7} - 8 \beta_{6} - \beta_{5} + 2 \beta_{4} - 3 \beta_{2} - 2 \beta_{1} + 23\)\()/2\)
\(\nu^{6}\)\(=\)\(4 \beta_{7} + 10 \beta_{4} + 2 \beta_{2} + 7\)
\(\nu^{7}\)\(=\)\((\)\(-18 \beta_{7} - 24 \beta_{5} - \beta_{4} + 5 \beta_{3} + 30 \beta_{2} - 6 \beta_{1} + 29\)\()/2\)

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
−1.37009 + 1.05965i
−1.37009 1.05965i
0.906034 1.47618i
0.906034 + 1.47618i
1.73142 0.0465589i
1.73142 + 0.0465589i
0.232633 1.71636i
0.232633 + 1.71636i
0 −1.60273 0.656712i 0 0.670944i 0 0 0 2.13746 + 2.10506i 0
2255.2 0 −1.60273 + 0.656712i 0 0.670944i 0 0 0 2.13746 2.10506i 0
2255.3 0 −0.825391 1.52274i 0 3.94333i 0 0 0 −1.63746 + 2.51371i 0
2255.4 0 −0.825391 + 1.52274i 0 3.94333i 0 0 0 −1.63746 2.51371i 0
2255.5 0 0.825391 1.52274i 0 3.94333i 0 0 0 −1.63746 2.51371i 0
2255.6 0 0.825391 + 1.52274i 0 3.94333i 0 0 0 −1.63746 + 2.51371i 0
2255.7 0 1.60273 0.656712i 0 0.670944i 0 0 0 2.13746 2.10506i 0
2255.8 0 1.60273 + 0.656712i 0 0.670944i 0 0 0 2.13746 + 2.10506i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2255.8
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.n 8
3.b odd 2 1 inner 2352.2.h.n 8
4.b odd 2 1 inner 2352.2.h.n 8
7.b odd 2 1 2352.2.h.m 8
7.c even 3 1 336.2.bj.e 8
7.c even 3 1 336.2.bj.g yes 8
12.b even 2 1 inner 2352.2.h.n 8
21.c even 2 1 2352.2.h.m 8
21.h odd 6 1 336.2.bj.e 8
21.h odd 6 1 336.2.bj.g yes 8
28.d even 2 1 2352.2.h.m 8
28.g odd 6 1 336.2.bj.e 8
28.g odd 6 1 336.2.bj.g yes 8
84.h odd 2 1 2352.2.h.m 8
84.n even 6 1 336.2.bj.e 8
84.n even 6 1 336.2.bj.g yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.bj.e 8 7.c even 3 1
336.2.bj.e 8 21.h odd 6 1
336.2.bj.e 8 28.g odd 6 1
336.2.bj.e 8 84.n even 6 1
336.2.bj.g yes 8 7.c even 3 1
336.2.bj.g yes 8 21.h odd 6 1
336.2.bj.g yes 8 28.g odd 6 1
336.2.bj.g yes 8 84.n even 6 1
2352.2.h.m 8 7.b odd 2 1
2352.2.h.m 8 21.c even 2 1
2352.2.h.m 8 28.d even 2 1
2352.2.h.m 8 84.h odd 2 1
2352.2.h.n 8 1.a even 1 1 trivial
2352.2.h.n 8 3.b odd 2 1 inner
2352.2.h.n 8 4.b odd 2 1 inner
2352.2.h.n 8 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 \(S_{2}^{\mathrm{new}}(2352, [\chi])\):

\( T_{5}^{4} + 16 T_{5}^{2} + 7 \)
\( T_{11}^{4} - 40 T_{11}^{2} + 343 \)
\( T_{13} - 2 \)
\( T_{47}^{4} - 117 T_{47}^{2} + 2268 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( 81 - 9 T^{2} + 4 T^{4} - T^{6} + T^{8} \)
$5$ \( ( 7 + 16 T^{2} + T^{4} )^{2} \)
$7$ \( T^{8} \)
$11$ \( ( 343 - 40 T^{2} + T^{4} )^{2} \)
$13$ \( ( -2 + T )^{8} \)
$17$ \( ( 448 + 43 T^{2} + T^{4} )^{2} \)
$19$ \( ( 16 + 11 T^{2} + T^{4} )^{2} \)
$23$ \( ( 28 - 13 T^{2} + T^{4} )^{2} \)
$29$ \( ( 1792 + 85 T^{2} + T^{4} )^{2} \)
$31$ \( ( 3 + T^{2} )^{4} \)
$37$ \( ( -8 + 5 T + T^{2} )^{4} \)
$41$ \( ( 448 + 100 T^{2} + T^{4} )^{2} \)
$43$ \( ( 64 + 92 T^{2} + T^{4} )^{2} \)
$47$ \( ( 2268 - 117 T^{2} + T^{4} )^{2} \)
$53$ \( ( 12943 + 232 T^{2} + T^{4} )^{2} \)
$59$ \( ( 5887 - 160 T^{2} + T^{4} )^{2} \)
$61$ \( ( 42 + 15 T + T^{2} )^{4} \)
$67$ \( ( 2304 + 267 T^{2} + T^{4} )^{2} \)
$71$ \( ( 1792 - 124 T^{2} + T^{4} )^{2} \)
$73$ \( ( -2 + 7 T + T^{2} )^{4} \)
$79$ \( ( 27 + T^{2} )^{4} \)
$83$ \( ( 112 - 31 T^{2} + T^{4} )^{2} \)
$89$ \( ( 10108 + 247 T^{2} + T^{4} )^{2} \)
$97$ \( ( -14 - T + T^{2} )^{4} \)
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