Properties

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

Newform invariants

Self dual: no
Analytic conductor: \(62.7794529086\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.5473632256.1
Defining polynomial: \(x^{8} + 49 x^{4} + 256\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: no (minimal twist has level 1152)
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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{5} + \beta_{5} q^{7} +O(q^{10})\) \( q + \beta_{6} q^{5} + \beta_{5} q^{7} -\beta_{3} q^{11} -\beta_{4} q^{13} -5 \beta_{2} q^{17} + \beta_{7} q^{19} + 5 \beta_{1} q^{23} -9 q^{25} + 5 \beta_{6} q^{29} -15 \beta_{5} q^{31} + \beta_{3} q^{35} -6 \beta_{4} q^{37} -19 \beta_{2} q^{41} -11 \beta_{1} q^{47} -41 q^{49} + 5 \beta_{6} q^{53} + 34 \beta_{5} q^{55} + 4 \beta_{3} q^{59} -10 \beta_{4} q^{61} -34 \beta_{2} q^{65} -5 \beta_{7} q^{67} + 25 \beta_{1} q^{71} -40 q^{73} -8 \beta_{6} q^{77} -45 \beta_{5} q^{79} -5 \beta_{3} q^{83} + 5 \beta_{4} q^{85} -81 \beta_{2} q^{89} -\beta_{7} q^{91} -34 \beta_{1} q^{95} -40 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 72q^{25} - 328q^{49} - 320q^{73} - 320q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 49 x^{4} + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{6} + 65 \nu^{2} \)\()/36\)
\(\beta_{2}\)\(=\)\((\)\( -5 \nu^{7} - 16 \nu^{5} - 181 \nu^{3} - 464 \nu \)\()/576\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{6} - 33 \nu^{2} \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( 4 \nu^{4} + 98 \)\()/9\)
\(\beta_{5}\)\(=\)\((\)\( -5 \nu^{7} + 16 \nu^{5} - 181 \nu^{3} + 464 \nu \)\()/288\)
\(\beta_{6}\)\(=\)\((\)\( -13 \nu^{7} - 16 \nu^{5} - 701 \nu^{3} - 1616 \nu \)\()/576\)
\(\beta_{7}\)\(=\)\((\)\( -13 \nu^{7} + 16 \nu^{5} - 701 \nu^{3} + 1616 \nu \)\()/144\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} - 4 \beta_{6} - 2 \beta_{5} + 4 \beta_{2}\)\()/16\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 9 \beta_{1}\)\()/8\)
\(\nu^{3}\)\(=\)\((\)\(-5 \beta_{7} - 20 \beta_{6} + 26 \beta_{5} + 52 \beta_{2}\)\()/16\)
\(\nu^{4}\)\(=\)\((\)\(9 \beta_{4} - 98\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-29 \beta_{7} + 116 \beta_{6} + 202 \beta_{5} - 404 \beta_{2}\)\()/16\)
\(\nu^{6}\)\(=\)\((\)\(-65 \beta_{3} - 297 \beta_{1}\)\()/8\)
\(\nu^{7}\)\(=\)\((\)\(181 \beta_{7} + 724 \beta_{6} - 1402 \beta_{5} - 2804 \beta_{2}\)\()/16\)

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
−1.10418 + 1.10418i
1.81129 + 1.81129i
1.10418 + 1.10418i
−1.81129 + 1.81129i
−1.10418 1.10418i
1.81129 1.81129i
1.10418 1.10418i
−1.81129 1.81129i
0 0 0 5.83095i 0 −2.82843 0 0 0
1025.2 0 0 0 5.83095i 0 −2.82843 0 0 0
1025.3 0 0 0 5.83095i 0 2.82843 0 0 0
1025.4 0 0 0 5.83095i 0 2.82843 0 0 0
1025.5 0 0 0 5.83095i 0 −2.82843 0 0 0
1025.6 0 0 0 5.83095i 0 −2.82843 0 0 0
1025.7 0 0 0 5.83095i 0 2.82843 0 0 0
1025.8 0 0 0 5.83095i 0 2.82843 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
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.e.m 8
3.b odd 2 1 inner 2304.3.e.m 8
4.b odd 2 1 inner 2304.3.e.m 8
8.b even 2 1 inner 2304.3.e.m 8
8.d odd 2 1 inner 2304.3.e.m 8
12.b even 2 1 inner 2304.3.e.m 8
16.e even 4 2 1152.3.h.f 8
16.f odd 4 2 1152.3.h.f 8
24.f even 2 1 inner 2304.3.e.m 8
24.h odd 2 1 inner 2304.3.e.m 8
48.i odd 4 2 1152.3.h.f 8
48.k even 4 2 1152.3.h.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.3.h.f 8 16.e even 4 2
1152.3.h.f 8 16.f odd 4 2
1152.3.h.f 8 48.i odd 4 2
1152.3.h.f 8 48.k even 4 2
2304.3.e.m 8 1.a even 1 1 trivial
2304.3.e.m 8 3.b odd 2 1 inner
2304.3.e.m 8 4.b odd 2 1 inner
2304.3.e.m 8 8.b even 2 1 inner
2304.3.e.m 8 8.d odd 2 1 inner
2304.3.e.m 8 12.b even 2 1 inner
2304.3.e.m 8 24.f even 2 1 inner
2304.3.e.m 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 \(S_{3}^{\mathrm{new}}(2304, [\chi])\):

\( T_{5}^{2} + 34 \)
\( T_{7}^{2} - 8 \)
\( T_{13}^{2} - 68 \)
\( T_{19}^{2} - 544 \)
\( T_{31}^{2} - 1800 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( 34 + T^{2} )^{4} \)
$7$ \( ( -8 + T^{2} )^{4} \)
$11$ \( ( 272 + T^{2} )^{4} \)
$13$ \( ( -68 + T^{2} )^{4} \)
$17$ \( ( 50 + T^{2} )^{4} \)
$19$ \( ( -544 + T^{2} )^{4} \)
$23$ \( ( 400 + T^{2} )^{4} \)
$29$ \( ( 850 + T^{2} )^{4} \)
$31$ \( ( -1800 + T^{2} )^{4} \)
$37$ \( ( -2448 + T^{2} )^{4} \)
$41$ \( ( 722 + T^{2} )^{4} \)
$43$ \( T^{8} \)
$47$ \( ( 1936 + T^{2} )^{4} \)
$53$ \( ( 850 + T^{2} )^{4} \)
$59$ \( ( 4352 + T^{2} )^{4} \)
$61$ \( ( -6800 + T^{2} )^{4} \)
$67$ \( ( -13600 + T^{2} )^{4} \)
$71$ \( ( 10000 + T^{2} )^{4} \)
$73$ \( ( 40 + T )^{8} \)
$79$ \( ( -16200 + T^{2} )^{4} \)
$83$ \( ( 6800 + T^{2} )^{4} \)
$89$ \( ( 13122 + T^{2} )^{4} \)
$97$ \( ( 40 + T )^{8} \)
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