Properties

Label 2352.2.h.o
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: 8.0.56070144.2
Defining polynomial: \(x^{8} - 4 x^{7} + 16 x^{6} - 34 x^{5} + 63 x^{4} - 74 x^{3} + 70 x^{2} - 38 x + 13\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + \beta_{2} q^{5} + ( 1 - \beta_{5} + \beta_{7} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + \beta_{2} q^{5} + ( 1 - \beta_{5} + \beta_{7} ) q^{9} + ( 2 \beta_{3} + \beta_{4} ) q^{11} + ( 1 - 3 \beta_{5} ) q^{13} + ( -\beta_{1} + \beta_{3} + 3 \beta_{4} - \beta_{6} ) q^{15} + 2 \beta_{7} q^{17} + ( -\beta_{1} - \beta_{6} ) q^{19} + ( 2 \beta_{1} - 2 \beta_{6} ) q^{23} + ( -3 + 2 \beta_{5} ) q^{25} + ( -2 \beta_{1} - 2 \beta_{3} - \beta_{6} ) q^{27} + 2 \beta_{7} q^{29} + ( -2 \beta_{1} + 4 \beta_{4} - 2 \beta_{6} ) q^{31} + ( 1 - \beta_{2} + 3 \beta_{5} + 2 \beta_{7} ) q^{33} + ( 8 + 2 \beta_{5} ) q^{37} + ( -\beta_{1} - 3 \beta_{3} + 3 \beta_{6} ) q^{39} + ( -2 \beta_{2} - 2 \beta_{7} ) q^{41} -\beta_{4} q^{43} + ( 1 + 2 \beta_{2} + 3 \beta_{5} + 2 \beta_{7} ) q^{45} + ( -2 \beta_{1} - 2 \beta_{3} - 4 \beta_{6} ) q^{51} + ( -2 \beta_{2} - 2 \beta_{7} ) q^{53} + ( 6 \beta_{1} - 2 \beta_{4} + 6 \beta_{6} ) q^{55} + ( -2 - \beta_{5} + \beta_{7} ) q^{57} + ( \beta_{1} + 4 \beta_{3} + 2 \beta_{4} - \beta_{6} ) q^{59} + ( 3 - \beta_{5} ) q^{61} + ( 4 \beta_{2} + 6 \beta_{7} ) q^{65} + ( -2 \beta_{1} + \beta_{4} - 2 \beta_{6} ) q^{67} + ( -8 + 2 \beta_{5} - 2 \beta_{7} ) q^{69} + ( -2 \beta_{1} - 2 \beta_{3} - \beta_{4} + 2 \beta_{6} ) q^{71} + ( 8 + 2 \beta_{5} ) q^{73} + ( 3 \beta_{1} + 2 \beta_{3} - 2 \beta_{6} ) q^{75} -6 \beta_{4} q^{79} + ( -1 + 2 \beta_{2} - 4 \beta_{5} ) q^{81} + ( \beta_{1} + 4 \beta_{3} + 2 \beta_{4} - \beta_{6} ) q^{83} + ( 2 + 6 \beta_{5} ) q^{85} + ( -2 \beta_{1} - 2 \beta_{3} - 4 \beta_{6} ) q^{87} + ( 2 \beta_{2} + 6 \beta_{7} ) q^{89} + ( 4 \beta_{2} + 2 \beta_{5} + 2 \beta_{7} ) q^{93} + ( 2 \beta_{3} + \beta_{4} ) q^{95} + ( -2 + 4 \beta_{5} ) q^{97} + ( -2 \beta_{1} - 3 \beta_{4} - 6 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{9} + O(q^{10}) \) \( 8 q + 8 q^{9} + 8 q^{13} - 24 q^{25} + 8 q^{33} + 64 q^{37} + 8 q^{45} - 16 q^{57} + 24 q^{61} - 64 q^{69} + 64 q^{73} - 8 q^{81} + 16 q^{85} - 16 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{7} + 16 x^{6} - 34 x^{5} + 63 x^{4} - 74 x^{3} + 70 x^{2} - 38 x + 13\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 5 \nu^{7} + \nu^{6} + 25 \nu^{5} + 46 \nu^{4} + 5 \nu^{3} + 95 \nu^{2} + 9 \nu - 19 \)\()/37\)
\(\beta_{2}\)\(=\)\((\)\( -6 \nu^{7} + 21 \nu^{6} - 67 \nu^{5} + 115 \nu^{4} - 117 \nu^{3} + 71 \nu^{2} + 115 \nu - 66 \)\()/37\)
\(\beta_{3}\)\(=\)\((\)\( 6 \nu^{7} - 21 \nu^{6} + 67 \nu^{5} - 78 \nu^{4} + 43 \nu^{3} + 188 \nu^{2} - 263 \nu + 251 \)\()/37\)
\(\beta_{4}\)\(=\)\((\)\( -12 \nu^{7} + 42 \nu^{6} - 134 \nu^{5} + 230 \nu^{4} - 234 \nu^{3} + 142 \nu^{2} + 82 \nu - 58 \)\()/37\)
\(\beta_{5}\)\(=\)\( \nu^{6} - 3 \nu^{5} + 11 \nu^{4} - 17 \nu^{3} + 25 \nu^{2} - 17 \nu + 9 \)
\(\beta_{6}\)\(=\)\((\)\( 5 \nu^{7} - 36 \nu^{6} + 136 \nu^{5} - 361 \nu^{4} + 634 \nu^{3} - 756 \nu^{2} + 564 \nu - 167 \)\()/37\)
\(\beta_{7}\)\(=\)\((\)\( -42 \nu^{7} + 147 \nu^{6} - 543 \nu^{5} + 990 \nu^{4} - 1559 \nu^{3} + 1422 \nu^{2} - 971 \nu + 278 \)\()/37\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{4} + 2 \beta_{2} + 2\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{6} + 2 \beta_{5} - \beta_{4} + 2 \beta_{2} - 2 \beta_{1} - 8\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{7} + 3 \beta_{5} + 7 \beta_{4} - 5 \beta_{2} - 6 \beta_{1} - 13\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(-4 \beta_{7} - 14 \beta_{6} - 8 \beta_{5} + 17 \beta_{4} + 4 \beta_{3} - 12 \beta_{2} + 2 \beta_{1} + 18\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(8 \beta_{7} - 10 \beta_{6} - 25 \beta_{5} - 16 \beta_{4} + 10 \beta_{3} + 11 \beta_{2} + 40 \beta_{1} + 67\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(17 \beta_{7} + 37 \beta_{6} + 9 \beta_{5} - 54 \beta_{4} - 7 \beta_{3} + 32 \beta_{2} + 23 \beta_{1} - 10\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-8 \beta_{7} + 126 \beta_{6} + 154 \beta_{5} - 41 \beta_{4} - 84 \beta_{3} + 6 \beta_{2} - 154 \beta_{1} - 320\)\()/4\)

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
0.500000 + 2.19293i
0.500000 2.19293i
0.500000 1.56488i
0.500000 + 1.56488i
0.500000 + 0.564882i
0.500000 0.564882i
0.500000 1.19293i
0.500000 + 1.19293i
0 −1.69293 0.366025i 0 3.38587i 0 0 0 2.73205 + 1.23931i 0
2255.2 0 −1.69293 + 0.366025i 0 3.38587i 0 0 0 2.73205 1.23931i 0
2255.3 0 −1.06488 1.36603i 0 2.12976i 0 0 0 −0.732051 + 2.90931i 0
2255.4 0 −1.06488 + 1.36603i 0 2.12976i 0 0 0 −0.732051 2.90931i 0
2255.5 0 1.06488 1.36603i 0 2.12976i 0 0 0 −0.732051 2.90931i 0
2255.6 0 1.06488 + 1.36603i 0 2.12976i 0 0 0 −0.732051 + 2.90931i 0
2255.7 0 1.69293 0.366025i 0 3.38587i 0 0 0 2.73205 1.23931i 0
2255.8 0 1.69293 + 0.366025i 0 3.38587i 0 0 0 2.73205 + 1.23931i 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.o 8
3.b odd 2 1 inner 2352.2.h.o 8
4.b odd 2 1 inner 2352.2.h.o 8
7.b odd 2 1 336.2.h.b 8
12.b even 2 1 inner 2352.2.h.o 8
21.c even 2 1 336.2.h.b 8
28.d even 2 1 336.2.h.b 8
56.e even 2 1 1344.2.h.g 8
56.h odd 2 1 1344.2.h.g 8
84.h odd 2 1 336.2.h.b 8
168.e odd 2 1 1344.2.h.g 8
168.i even 2 1 1344.2.h.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.h.b 8 7.b odd 2 1
336.2.h.b 8 21.c even 2 1
336.2.h.b 8 28.d even 2 1
336.2.h.b 8 84.h odd 2 1
1344.2.h.g 8 56.e even 2 1
1344.2.h.g 8 56.h odd 2 1
1344.2.h.g 8 168.e odd 2 1
1344.2.h.g 8 168.i even 2 1
2352.2.h.o 8 1.a even 1 1 trivial
2352.2.h.o 8 3.b odd 2 1 inner
2352.2.h.o 8 4.b odd 2 1 inner
2352.2.h.o 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 \(S_{2}^{\mathrm{new}}(2352, [\chi])\):

\( T_{5}^{4} + 16 T_{5}^{2} + 52 \)
\( T_{11}^{4} - 40 T_{11}^{2} + 208 \)
\( T_{13}^{2} - 2 T_{13} - 26 \)
\( T_{47} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( 81 - 36 T^{2} + 10 T^{4} - 4 T^{6} + T^{8} \)
$5$ \( ( 52 + 16 T^{2} + T^{4} )^{2} \)
$7$ \( T^{8} \)
$11$ \( ( 208 - 40 T^{2} + T^{4} )^{2} \)
$13$ \( ( -26 - 2 T + T^{2} )^{4} \)
$17$ \( ( 208 + 40 T^{2} + T^{4} )^{2} \)
$19$ \( ( 4 + 8 T^{2} + T^{4} )^{2} \)
$23$ \( ( 832 - 64 T^{2} + T^{4} )^{2} \)
$29$ \( ( 208 + 40 T^{2} + T^{4} )^{2} \)
$31$ \( ( 576 + 96 T^{2} + T^{4} )^{2} \)
$37$ \( ( 52 - 16 T + T^{2} )^{4} \)
$41$ \( ( 208 + 88 T^{2} + T^{4} )^{2} \)
$43$ \( ( 4 + T^{2} )^{4} \)
$47$ \( T^{8} \)
$53$ \( ( 208 + 88 T^{2} + T^{4} )^{2} \)
$59$ \( ( 6292 - 160 T^{2} + T^{4} )^{2} \)
$61$ \( ( 6 - 6 T + T^{2} )^{4} \)
$67$ \( ( 12 + T^{2} )^{4} \)
$71$ \( ( 208 - 88 T^{2} + T^{4} )^{2} \)
$73$ \( ( 52 - 16 T + T^{2} )^{4} \)
$79$ \( ( 144 + T^{2} )^{4} \)
$83$ \( ( 6292 - 160 T^{2} + T^{4} )^{2} \)
$89$ \( ( 35152 + 376 T^{2} + T^{4} )^{2} \)
$97$ \( ( -44 + 4 T + T^{2} )^{4} \)
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