Properties

Label 2304.3.g.z
Level $2304$
Weight $3$
Character orbit 2304.g
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.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(62.7794529086\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.22581504.2
Defining polynomial: \(x^{8} - 4 x^{7} + 5 x^{6} + 2 x^{5} - 11 x^{4} + 4 x^{3} + 20 x^{2} - 32 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{24} \)
Twist minimal: no (minimal twist has level 24)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{5} -\beta_{7} q^{7} +O(q^{10})\) \( q -\beta_{1} q^{5} -\beta_{7} q^{7} + \beta_{6} q^{11} + ( \beta_{1} - \beta_{4} ) q^{13} + ( 2 + \beta_{5} ) q^{17} + ( -\beta_{3} - \beta_{6} ) q^{19} -2 \beta_{7} q^{23} + ( 11 - 2 \beta_{5} ) q^{25} + ( -\beta_{1} - 2 \beta_{4} ) q^{29} -\beta_{2} q^{31} + ( -\beta_{3} + 3 \beta_{6} ) q^{35} + ( -3 \beta_{1} + \beta_{4} ) q^{37} + ( 10 + 3 \beta_{5} ) q^{41} + ( 3 \beta_{3} + \beta_{6} ) q^{43} + ( -\beta_{2} - \beta_{7} ) q^{47} + ( -11 - 4 \beta_{5} ) q^{49} + ( 3 \beta_{1} - 2 \beta_{4} ) q^{53} + ( -\beta_{2} - 3 \beta_{7} ) q^{55} + ( 3 \beta_{3} + 4 \beta_{6} ) q^{59} + ( -7 \beta_{1} + \beta_{4} ) q^{61} + ( -24 + 5 \beta_{5} ) q^{65} + ( 3 \beta_{3} + 8 \beta_{6} ) q^{67} + ( -\beta_{2} + 7 \beta_{7} ) q^{71} + ( -50 - 4 \beta_{5} ) q^{73} + ( 4 \beta_{1} + 4 \beta_{4} ) q^{77} + ( \beta_{2} - 6 \beta_{7} ) q^{79} + ( 4 \beta_{3} + 5 \beta_{6} ) q^{83} + ( 10 \beta_{1} - 4 \beta_{4} ) q^{85} + ( -50 + 6 \beta_{5} ) q^{89} + ( -14 \beta_{3} + 9 \beta_{6} ) q^{91} + ( 2 \beta_{2} + 2 \beta_{7} ) q^{95} + ( 14 - 6 \beta_{5} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 16q^{17} + 88q^{25} + 80q^{41} - 88q^{49} - 192q^{65} - 400q^{73} - 400q^{89} + 112q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{7} + 5 x^{6} + 2 x^{5} - 11 x^{4} + 4 x^{3} + 20 x^{2} - 32 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{7} - 2 \nu^{6} - \nu^{5} + 4 \nu^{4} - \nu^{3} - 6 \nu^{2} + 10 \nu - 2 \)
\(\beta_{2}\)\(=\)\( -10 \nu^{7} + 18 \nu^{6} + 2 \nu^{5} - 38 \nu^{4} + 22 \nu^{3} + 70 \nu^{2} - 84 \nu + 22 \)
\(\beta_{3}\)\(=\)\( -6 \nu^{7} + 14 \nu^{6} - 6 \nu^{5} - 22 \nu^{4} + 30 \nu^{3} + 22 \nu^{2} - 80 \nu + 60 \)
\(\beta_{4}\)\(=\)\( -5 \nu^{7} + 15 \nu^{6} - 9 \nu^{5} - 19 \nu^{4} + 33 \nu^{3} + 15 \nu^{2} - 76 \nu + 66 \)
\(\beta_{5}\)\(=\)\( -6 \nu^{7} + 18 \nu^{6} - 10 \nu^{5} - 26 \nu^{4} + 42 \nu^{3} + 26 \nu^{2} - 108 \nu + 80 \)
\(\beta_{6}\)\(=\)\( 8 \nu^{7} - 22 \nu^{6} + 12 \nu^{5} + 34 \nu^{4} - 48 \nu^{3} - 30 \nu^{2} + 124 \nu - 96 \)
\(\beta_{7}\)\(=\)\( 10 \nu^{7} - 26 \nu^{6} + 14 \nu^{5} + 38 \nu^{4} - 54 \nu^{3} - 38 \nu^{2} + 148 \nu - 114 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(3 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + 4 \beta_{4} + \beta_{2} + 4 \beta_{1} + 32\)\()/64\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{7} + 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{3} + \beta_{2} + 4 \beta_{1} + 24\)\()/32\)
\(\nu^{3}\)\(=\)\((\)\(11 \beta_{7} + 2 \beta_{6} + 6 \beta_{5} + 4 \beta_{4} + 12 \beta_{3} + \beta_{2} + 12 \beta_{1} - 16\)\()/64\)
\(\nu^{4}\)\(=\)\((\)\(-2 \beta_{7} + 6 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + \beta_{3} - 4\)\()/16\)
\(\nu^{5}\)\(=\)\((\)\(9 \beta_{7} + 4 \beta_{6} + 6 \beta_{5} + 4 \beta_{4} + 10 \beta_{3} - \beta_{2} - 16 \beta_{1} + 56\)\()/32\)
\(\nu^{6}\)\(=\)\((\)\(\beta_{6} + 4 \beta_{4} - 3 \beta_{3} - 6 \beta_{1}\)\()/8\)
\(\nu^{7}\)\(=\)\((\)\(19 \beta_{7} - 26 \beta_{6} + 30 \beta_{5} + 4 \beta_{4} - 56 \beta_{3} + \beta_{2} - 44 \beta_{1} + 256\)\()/64\)

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
1.20036 + 0.747754i
1.20036 0.747754i
1.40994 + 0.109843i
1.40994 0.109843i
−1.27597 + 0.609843i
−1.27597 0.609843i
0.665665 + 1.24775i
0.665665 1.24775i
0 0 0 −7.98203 0 2.13878i 0 0 0
1279.2 0 0 0 −7.98203 0 2.13878i 0 0 0
1279.3 0 0 0 −2.87875 0 10.7436i 0 0 0
1279.4 0 0 0 −2.87875 0 10.7436i 0 0 0
1279.5 0 0 0 2.87875 0 10.7436i 0 0 0
1279.6 0 0 0 2.87875 0 10.7436i 0 0 0
1279.7 0 0 0 7.98203 0 2.13878i 0 0 0
1279.8 0 0 0 7.98203 0 2.13878i 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
4.b odd 2 1 inner
8.b even 2 1 inner
8.d 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.z 8
3.b odd 2 1 768.3.g.h 8
4.b odd 2 1 inner 2304.3.g.z 8
8.b even 2 1 inner 2304.3.g.z 8
8.d odd 2 1 inner 2304.3.g.z 8
12.b even 2 1 768.3.g.h 8
16.e even 4 1 72.3.b.b 4
16.e even 4 1 288.3.b.b 4
16.f odd 4 1 72.3.b.b 4
16.f odd 4 1 288.3.b.b 4
24.f even 2 1 768.3.g.h 8
24.h odd 2 1 768.3.g.h 8
48.i odd 4 1 24.3.b.a 4
48.i odd 4 1 96.3.b.a 4
48.k even 4 1 24.3.b.a 4
48.k even 4 1 96.3.b.a 4
240.t even 4 1 600.3.g.a 4
240.t even 4 1 2400.3.g.a 4
240.z odd 4 1 600.3.p.a 8
240.z odd 4 1 2400.3.p.a 8
240.bb even 4 1 600.3.p.a 8
240.bb even 4 1 2400.3.p.a 8
240.bd odd 4 1 600.3.p.a 8
240.bd odd 4 1 2400.3.p.a 8
240.bf even 4 1 600.3.p.a 8
240.bf even 4 1 2400.3.p.a 8
240.bm odd 4 1 600.3.g.a 4
240.bm odd 4 1 2400.3.g.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.3.b.a 4 48.i odd 4 1
24.3.b.a 4 48.k even 4 1
72.3.b.b 4 16.e even 4 1
72.3.b.b 4 16.f odd 4 1
96.3.b.a 4 48.i odd 4 1
96.3.b.a 4 48.k even 4 1
288.3.b.b 4 16.e even 4 1
288.3.b.b 4 16.f odd 4 1
600.3.g.a 4 240.t even 4 1
600.3.g.a 4 240.bm odd 4 1
600.3.p.a 8 240.z odd 4 1
600.3.p.a 8 240.bb even 4 1
600.3.p.a 8 240.bd odd 4 1
600.3.p.a 8 240.bf even 4 1
768.3.g.h 8 3.b odd 2 1
768.3.g.h 8 12.b even 2 1
768.3.g.h 8 24.f even 2 1
768.3.g.h 8 24.h odd 2 1
2304.3.g.z 8 1.a even 1 1 trivial
2304.3.g.z 8 4.b odd 2 1 inner
2304.3.g.z 8 8.b even 2 1 inner
2304.3.g.z 8 8.d odd 2 1 inner
2400.3.g.a 4 240.t even 4 1
2400.3.g.a 4 240.bm odd 4 1
2400.3.p.a 8 240.z odd 4 1
2400.3.p.a 8 240.bb even 4 1
2400.3.p.a 8 240.bd odd 4 1
2400.3.p.a 8 240.bf even 4 1

Hecke kernels

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

\( T_{5}^{4} - 72 T_{5}^{2} + 528 \)
\( T_{7}^{4} + 120 T_{7}^{2} + 528 \)
\( T_{11}^{2} + 64 \)
\( T_{13}^{4} - 384 T_{13}^{2} + 33792 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( 528 - 72 T^{2} + T^{4} )^{2} \)
$7$ \( ( 528 + 120 T^{2} + T^{4} )^{2} \)
$11$ \( ( 64 + T^{2} )^{4} \)
$13$ \( ( 33792 - 384 T^{2} + T^{4} )^{2} \)
$17$ \( ( -188 - 4 T + T^{2} )^{4} \)
$19$ \( ( 256 + 224 T^{2} + T^{4} )^{2} \)
$23$ \( ( 8448 + 480 T^{2} + T^{4} )^{2} \)
$29$ \( ( 528 - 1608 T^{2} + T^{4} )^{2} \)
$31$ \( ( 279312 + 3384 T^{2} + T^{4} )^{2} \)
$37$ \( ( 76032 - 864 T^{2} + T^{4} )^{2} \)
$41$ \( ( -1628 - 20 T + T^{2} )^{4} \)
$43$ \( ( 135424 + 992 T^{2} + T^{4} )^{2} \)
$47$ \( ( 8448 + 3552 T^{2} + T^{4} )^{2} \)
$53$ \( ( 803088 - 1800 T^{2} + T^{4} )^{2} \)
$59$ \( ( 350464 + 2912 T^{2} + T^{4} )^{2} \)
$61$ \( ( 8448 - 3552 T^{2} + T^{4} )^{2} \)
$67$ \( ( 13424896 + 9056 T^{2} + T^{4} )^{2} \)
$71$ \( ( 12849408 + 8928 T^{2} + T^{4} )^{2} \)
$73$ \( ( -572 + 100 T + T^{2} )^{4} \)
$79$ \( ( 10797072 + 7416 T^{2} + T^{4} )^{2} \)
$83$ \( ( 692224 + 4736 T^{2} + T^{4} )^{2} \)
$89$ \( ( -4412 + 100 T + T^{2} )^{4} \)
$97$ \( ( -6716 - 28 T + T^{2} )^{4} \)
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