Properties

Label 2304.3.e.n
Level $2304$
Weight $3$
Character orbit 2304.e
Analytic conductor $62.779$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2304.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(62.7794529086\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.540942598144.16
Defining polynomial: \(x^{8} + 16 x^{6} + 71 x^{4} + 56 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 72)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{5} + \beta_{3} q^{7} +O(q^{10})\) \( q + \beta_{2} q^{5} + \beta_{3} q^{7} + ( \beta_{2} + \beta_{4} ) q^{11} + \beta_{5} q^{13} + ( -\beta_{1} - 2 \beta_{6} ) q^{17} + ( -\beta_{5} - \beta_{7} ) q^{19} + ( -3 \beta_{1} - \beta_{6} ) q^{23} + ( -5 - 4 \beta_{3} ) q^{25} + ( -\beta_{2} + 2 \beta_{4} ) q^{29} + ( -16 - 3 \beta_{3} ) q^{31} + ( 7 \beta_{2} - \beta_{4} ) q^{35} + ( -\beta_{5} - \beta_{7} ) q^{37} + ( \beta_{1} - 2 \beta_{6} ) q^{41} + ( 2 \beta_{5} - 2 \beta_{7} ) q^{43} + ( -7 \beta_{1} + 3 \beta_{6} ) q^{47} + 3 q^{49} + ( -5 \beta_{2} - 2 \beta_{4} ) q^{53} + ( -32 - 2 \beta_{3} ) q^{55} + ( 4 \beta_{2} - 4 \beta_{4} ) q^{59} + ( -3 \beta_{5} + \beta_{7} ) q^{61} + 2 \beta_{6} q^{65} + ( 3 \beta_{5} - \beta_{7} ) q^{67} + ( -15 \beta_{1} + 3 \beta_{6} ) q^{71} + ( 20 + 8 \beta_{3} ) q^{73} + ( 4 \beta_{2} - 8 \beta_{4} ) q^{77} + ( -48 + 11 \beta_{3} ) q^{79} + ( -7 \beta_{2} + \beta_{4} ) q^{83} + ( 2 \beta_{5} + 5 \beta_{7} ) q^{85} + ( -7 \beta_{1} + 4 \beta_{6} ) q^{89} + ( -7 \beta_{5} + \beta_{7} ) q^{91} + ( -34 \beta_{1} - 14 \beta_{6} ) q^{95} + ( -24 - 4 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 40q^{25} - 128q^{31} + 24q^{49} - 256q^{55} + 160q^{73} - 384q^{79} - 192q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 16 x^{6} + 71 x^{4} + 56 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{5} + 9 \nu^{3} + 11 \nu \)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{7} + 29 \nu^{5} + 115 \nu^{3} + 73 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\( 4 \nu^{4} + 32 \nu^{2} + 14 \)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{7} - 35 \nu^{5} - 157 \nu^{3} - 43 \nu \)\()/3\)
\(\beta_{5}\)\(=\)\( 2 \nu^{6} + 24 \nu^{4} + 68 \nu^{2} + 16 \)
\(\beta_{6}\)\(=\)\((\)\( 4 \nu^{7} + 61 \nu^{5} + 245 \nu^{3} + 107 \nu \)\()/3\)
\(\beta_{7}\)\(=\)\( 2 \nu^{6} + 24 \nu^{4} + 76 \nu^{2} + 48 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{6} + \beta_{4} - \beta_{2} + \beta_{1}\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{7} - \beta_{5} - 32\)\()/8\)
\(\nu^{3}\)\(=\)\((\)\(-4 \beta_{6} - 3 \beta_{4} + 5 \beta_{2} - 2 \beta_{1}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(-4 \beta_{7} + 4 \beta_{5} + \beta_{3} + 114\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(61 \beta_{6} + 43 \beta_{4} - 79 \beta_{2} + 33 \beta_{1}\)\()/8\)
\(\nu^{6}\)\(=\)\((\)\(31 \beta_{7} - 29 \beta_{5} - 12 \beta_{3} - 856\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(-461 \beta_{6} - 315 \beta_{4} + 619 \beta_{2} - 285 \beta_{1}\)\()/8\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1025.1
2.78961i
0.467011i
0.854013i
2.69642i
0.854013i
2.69642i
2.78961i
0.467011i
0 0 0 7.67101i 0 7.21110 0 0 0
1025.2 0 0 0 7.67101i 0 7.21110 0 0 0
1025.3 0 0 0 1.07498i 0 −7.21110 0 0 0
1025.4 0 0 0 1.07498i 0 −7.21110 0 0 0
1025.5 0 0 0 1.07498i 0 −7.21110 0 0 0
1025.6 0 0 0 1.07498i 0 −7.21110 0 0 0
1025.7 0 0 0 7.67101i 0 7.21110 0 0 0
1025.8 0 0 0 7.67101i 0 7.21110 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1025.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.3.e.n 8
3.b odd 2 1 inner 2304.3.e.n 8
4.b odd 2 1 2304.3.e.o 8
8.b even 2 1 inner 2304.3.e.n 8
8.d odd 2 1 2304.3.e.o 8
12.b even 2 1 2304.3.e.o 8
16.e even 4 2 72.3.h.a 8
16.f odd 4 2 288.3.h.a 8
24.f even 2 1 2304.3.e.o 8
24.h odd 2 1 inner 2304.3.e.n 8
48.i odd 4 2 72.3.h.a 8
48.k even 4 2 288.3.h.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.3.h.a 8 16.e even 4 2
72.3.h.a 8 48.i odd 4 2
288.3.h.a 8 16.f odd 4 2
288.3.h.a 8 48.k even 4 2
2304.3.e.n 8 1.a even 1 1 trivial
2304.3.e.n 8 3.b odd 2 1 inner
2304.3.e.n 8 8.b even 2 1 inner
2304.3.e.n 8 24.h odd 2 1 inner
2304.3.e.o 8 4.b odd 2 1
2304.3.e.o 8 8.d odd 2 1
2304.3.e.o 8 12.b even 2 1
2304.3.e.o 8 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(2304, [\chi])\):

\( T_{5}^{4} + 60 T_{5}^{2} + 68 \)
\( T_{7}^{2} - 52 \)
\( T_{13}^{4} - 472 T_{13}^{2} + 2448 \)
\( T_{19}^{4} - 1504 T_{19}^{2} + 352512 \)
\( T_{31}^{2} + 32 T_{31} - 212 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( 68 + 60 T^{2} + T^{4} )^{2} \)
$7$ \( ( -52 + T^{2} )^{4} \)
$11$ \( ( 9792 + 304 T^{2} + T^{4} )^{2} \)
$13$ \( ( 2448 - 472 T^{2} + T^{4} )^{2} \)
$17$ \( ( 171396 + 836 T^{2} + T^{4} )^{2} \)
$19$ \( ( 352512 - 1504 T^{2} + T^{4} )^{2} \)
$23$ \( ( 576 + 464 T^{2} + T^{4} )^{2} \)
$29$ \( ( 103428 + 988 T^{2} + T^{4} )^{2} \)
$31$ \( ( -212 + 32 T + T^{2} )^{4} \)
$37$ \( ( 352512 - 1504 T^{2} + T^{4} )^{2} \)
$41$ \( ( 133956 + 932 T^{2} + T^{4} )^{2} \)
$43$ \( ( 10027008 - 6400 T^{2} + T^{4} )^{2} \)
$47$ \( ( 46656 + 4176 T^{2} + T^{4} )^{2} \)
$53$ \( ( 1591812 + 2524 T^{2} + T^{4} )^{2} \)
$59$ \( ( 4456448 + 4608 T^{2} + T^{4} )^{2} \)
$61$ \( ( 3172608 - 5472 T^{2} + T^{4} )^{2} \)
$67$ \( ( 3172608 - 5472 T^{2} + T^{4} )^{2} \)
$71$ \( ( 13483584 + 11088 T^{2} + T^{4} )^{2} \)
$73$ \( ( -2928 - 40 T + T^{2} )^{4} \)
$79$ \( ( -3988 + 96 T + T^{2} )^{4} \)
$83$ \( ( 183872 + 3120 T^{2} + T^{4} )^{2} \)
$89$ \( ( 171396 + 5828 T^{2} + T^{4} )^{2} \)
$97$ \( ( -256 + 48 T + T^{2} )^{4} \)
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