Properties

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

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(62.7794529086\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.3317760000.5
Defining polynomial: \(x^{8} - 7 x^{4} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{22} \)
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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{5} + \beta_{5} q^{7} +O(q^{10})\) \( q -\beta_{2} q^{5} + \beta_{5} q^{7} + \beta_{4} q^{11} + \beta_{1} q^{13} + \beta_{7} q^{17} -\beta_{3} q^{19} + \beta_{6} q^{23} - q^{25} + 5 \beta_{2} q^{29} + \beta_{5} q^{31} -3 \beta_{4} q^{35} + 3 \beta_{1} q^{37} -\beta_{7} q^{41} -5 \beta_{3} q^{43} + \beta_{6} q^{47} -11 q^{49} -3 \beta_{2} q^{53} -8 \beta_{5} q^{55} -2 \beta_{4} q^{59} -\beta_{1} q^{61} -3 \beta_{7} q^{65} -10 \beta_{3} q^{67} + 10 q^{73} -20 \beta_{2} q^{77} -7 \beta_{5} q^{79} + 11 \beta_{4} q^{83} -8 \beta_{1} q^{85} + 2 \beta_{7} q^{89} -15 \beta_{3} q^{91} + \beta_{6} q^{95} + 50 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 8q^{25} - 88q^{49} + 80q^{73} + 400q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 7 x^{4} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( -\nu^{6} + 11 \nu^{2} \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} - 4 \nu^{5} + \nu^{3} + 20 \nu \)\()/4\)
\(\beta_{3}\)\(=\)\( -2 \nu^{6} + 6 \nu^{2} \)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{7} + 4 \nu^{5} - 13 \nu^{3} - 4 \nu \)\()/2\)
\(\beta_{5}\)\(=\)\( 4 \nu^{4} - 14 \)
\(\beta_{6}\)\(=\)\( -2 \nu^{7} - 8 \nu^{5} - 2 \nu^{3} + 40 \nu \)
\(\beta_{7}\)\(=\)\( -3 \nu^{7} + 4 \nu^{5} + 13 \nu^{3} - 4 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(2 \beta_{7} + \beta_{6} + 4 \beta_{4} + 8 \beta_{2}\)\()/64\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + 2 \beta_{1}\)\()/16\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{7} - 3 \beta_{6} - 4 \beta_{4} + 24 \beta_{2}\)\()/64\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{5} + 14\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(10 \beta_{7} + \beta_{6} + 20 \beta_{4} + 8 \beta_{2}\)\()/64\)
\(\nu^{6}\)\(=\)\((\)\(-11 \beta_{3} + 6 \beta_{1}\)\()/16\)
\(\nu^{7}\)\(=\)\((\)\(-2 \beta_{7} - 13 \beta_{6} + 4 \beta_{4} + 104 \beta_{2}\)\()/64\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1279.1
−0.178197 1.40294i
1.40294 0.178197i
−0.178197 + 1.40294i
1.40294 + 0.178197i
0.178197 + 1.40294i
−1.40294 + 0.178197i
0.178197 1.40294i
−1.40294 0.178197i
0 0 0 −4.89898 0 7.74597i 0 0 0
1279.2 0 0 0 −4.89898 0 7.74597i 0 0 0
1279.3 0 0 0 −4.89898 0 7.74597i 0 0 0
1279.4 0 0 0 −4.89898 0 7.74597i 0 0 0
1279.5 0 0 0 4.89898 0 7.74597i 0 0 0
1279.6 0 0 0 4.89898 0 7.74597i 0 0 0
1279.7 0 0 0 4.89898 0 7.74597i 0 0 0
1279.8 0 0 0 4.89898 0 7.74597i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1279.8
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
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f 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.g.y 8
3.b odd 2 1 inner 2304.3.g.y 8
4.b odd 2 1 inner 2304.3.g.y 8
8.b even 2 1 inner 2304.3.g.y 8
8.d odd 2 1 inner 2304.3.g.y 8
12.b even 2 1 inner 2304.3.g.y 8
16.e even 4 1 72.3.b.c 4
16.e even 4 1 288.3.b.c 4
16.f odd 4 1 72.3.b.c 4
16.f odd 4 1 288.3.b.c 4
24.f even 2 1 inner 2304.3.g.y 8
24.h odd 2 1 inner 2304.3.g.y 8
48.i odd 4 1 72.3.b.c 4
48.i odd 4 1 288.3.b.c 4
48.k even 4 1 72.3.b.c 4
48.k even 4 1 288.3.b.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.3.b.c 4 16.e even 4 1
72.3.b.c 4 16.f odd 4 1
72.3.b.c 4 48.i odd 4 1
72.3.b.c 4 48.k even 4 1
288.3.b.c 4 16.e even 4 1
288.3.b.c 4 16.f odd 4 1
288.3.b.c 4 48.i odd 4 1
288.3.b.c 4 48.k even 4 1
2304.3.g.y 8 1.a even 1 1 trivial
2304.3.g.y 8 3.b odd 2 1 inner
2304.3.g.y 8 4.b odd 2 1 inner
2304.3.g.y 8 8.b even 2 1 inner
2304.3.g.y 8 8.d odd 2 1 inner
2304.3.g.y 8 12.b even 2 1 inner
2304.3.g.y 8 24.f even 2 1 inner
2304.3.g.y 8 24.h 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 \(S_{3}^{\mathrm{new}}(2304, [\chi])\):

\( T_{5}^{2} - 24 \)
\( T_{7}^{2} + 60 \)
\( T_{11}^{2} + 160 \)
\( T_{13}^{2} - 240 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( -24 + T^{2} )^{4} \)
$7$ \( ( 60 + T^{2} )^{4} \)
$11$ \( ( 160 + T^{2} )^{4} \)
$13$ \( ( -240 + T^{2} )^{4} \)
$17$ \( ( -640 + T^{2} )^{4} \)
$19$ \( ( 64 + T^{2} )^{4} \)
$23$ \( ( 1536 + T^{2} )^{4} \)
$29$ \( ( -600 + T^{2} )^{4} \)
$31$ \( ( 60 + T^{2} )^{4} \)
$37$ \( ( -2160 + T^{2} )^{4} \)
$41$ \( ( -640 + T^{2} )^{4} \)
$43$ \( ( 1600 + T^{2} )^{4} \)
$47$ \( ( 1536 + T^{2} )^{4} \)
$53$ \( ( -216 + T^{2} )^{4} \)
$59$ \( ( 640 + T^{2} )^{4} \)
$61$ \( ( -240 + T^{2} )^{4} \)
$67$ \( ( 6400 + T^{2} )^{4} \)
$71$ \( T^{8} \)
$73$ \( ( -10 + T )^{8} \)
$79$ \( ( 2940 + T^{2} )^{4} \)
$83$ \( ( 19360 + T^{2} )^{4} \)
$89$ \( ( -2560 + T^{2} )^{4} \)
$97$ \( ( -50 + T )^{8} \)
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