Properties

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

Newform invariants

Self dual: no
Analytic conductor: \(62.7794529086\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.157351936.1
Defining polynomial: \(x^{8} + x^{4} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{18} \)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{5} + ( -\beta_{1} - \beta_{6} ) q^{7} +O(q^{10})\) \( q + \beta_{2} q^{5} + ( -\beta_{1} - \beta_{6} ) q^{7} + \beta_{4} q^{11} + ( 3 \beta_{1} + 2 \beta_{6} ) q^{13} + ( -\beta_{3} + \beta_{4} ) q^{17} + ( 12 + \beta_{7} ) q^{19} + ( -5 \beta_{2} - \beta_{5} ) q^{23} + ( -11 - 2 \beta_{7} ) q^{25} + ( -\beta_{2} - 2 \beta_{5} ) q^{29} + ( 9 \beta_{1} + \beta_{6} ) q^{31} + ( 8 \beta_{3} + 3 \beta_{4} ) q^{35} + ( -\beta_{1} + 2 \beta_{6} ) q^{37} + ( 7 \beta_{3} + \beta_{4} ) q^{41} + ( 20 - \beta_{7} ) q^{43} + ( -7 \beta_{2} + 5 \beta_{5} ) q^{47} + ( -11 - 2 \beta_{7} ) q^{49} + ( \beta_{2} + 6 \beta_{5} ) q^{53} + ( 28 \beta_{1} + 4 \beta_{6} ) q^{55} + ( -8 \beta_{3} + 2 \beta_{4} ) q^{59} + ( -7 \beta_{1} - 2 \beta_{6} ) q^{61} + ( -17 \beta_{3} - 7 \beta_{4} ) q^{65} + ( 24 - 6 \beta_{7} ) q^{67} + ( -8 \beta_{2} - 8 \beta_{5} ) q^{71} + ( 38 + 6 \beta_{7} ) q^{73} + ( -10 \beta_{2} - 6 \beta_{5} ) q^{77} + ( 29 \beta_{1} - 3 \beta_{6} ) q^{79} + ( 16 \beta_{3} + \beta_{4} ) q^{83} + 20 \beta_{1} q^{85} + ( 14 \beta_{3} - 6 \beta_{4} ) q^{89} + ( 124 + 5 \beta_{7} ) q^{91} + ( 25 \beta_{2} + 5 \beta_{5} ) q^{95} + ( 14 + 4 \beta_{7} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 96q^{19} - 88q^{25} + 160q^{43} - 88q^{49} + 192q^{67} + 304q^{73} + 992q^{91} + 112q^{97} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{6} + 5 \nu^{2} \)\()/6\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{7} + 4 \nu^{5} + 16 \nu^{4} + 7 \nu^{3} - 4 \nu + 8 \)\()/12\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} - 4 \nu^{5} + 7 \nu^{3} + 4 \nu \)\()/6\)
\(\beta_{4}\)\(=\)\( -\nu^{6} + 3 \nu^{2} \)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{7} + 12 \nu^{5} - 16 \nu^{4} + 21 \nu^{3} - 12 \nu - 8 \)\()/12\)
\(\beta_{6}\)\(=\)\((\)\( 5 \nu^{7} + 4 \nu^{5} + 13 \nu^{3} + 44 \nu \)\()/12\)
\(\beta_{7}\)\(=\)\((\)\( -5 \nu^{7} + 4 \nu^{5} - 13 \nu^{3} + 44 \nu \)\()/6\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(2 \beta_{7} + 4 \beta_{6} - \beta_{5} + 2 \beta_{3} - \beta_{2}\)\()/32\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{4} + 6 \beta_{1}\)\()/8\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{7} + 4 \beta_{6} + 5 \beta_{5} + 10 \beta_{3} + 5 \beta_{2}\)\()/32\)
\(\nu^{4}\)\(=\)\((\)\(-3 \beta_{5} + 9 \beta_{2} - 8\)\()/16\)
\(\nu^{5}\)\(=\)\((\)\(2 \beta_{7} + 4 \beta_{6} + 11 \beta_{5} - 22 \beta_{3} + 11 \beta_{2}\)\()/32\)
\(\nu^{6}\)\(=\)\((\)\(-5 \beta_{4} + 18 \beta_{1}\)\()/8\)
\(\nu^{7}\)\(=\)\((\)\(-14 \beta_{7} + 28 \beta_{6} - 13 \beta_{5} - 26 \beta_{3} - 13 \beta_{2}\)\()/32\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
0.581861 + 1.28897i
1.28897 0.581861i
−1.28897 + 0.581861i
−0.581861 1.28897i
−0.581861 + 1.28897i
−1.28897 0.581861i
1.28897 + 0.581861i
0.581861 1.28897i
0 0 0 8.11993i 0 9.48331i 0 0 0
127.2 0 0 0 8.11993i 0 9.48331i 0 0 0
127.3 0 0 0 2.46308i 0 5.48331i 0 0 0
127.4 0 0 0 2.46308i 0 5.48331i 0 0 0
127.5 0 0 0 2.46308i 0 5.48331i 0 0 0
127.6 0 0 0 2.46308i 0 5.48331i 0 0 0
127.7 0 0 0 8.11993i 0 9.48331i 0 0 0
127.8 0 0 0 8.11993i 0 9.48331i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.3.b.s 8
3.b odd 2 1 inner 2304.3.b.s 8
4.b odd 2 1 2304.3.b.r 8
8.b even 2 1 2304.3.b.r 8
8.d odd 2 1 inner 2304.3.b.s 8
12.b even 2 1 2304.3.b.r 8
16.e even 4 1 1152.3.g.d 8
16.e even 4 1 1152.3.g.e yes 8
16.f odd 4 1 1152.3.g.d 8
16.f odd 4 1 1152.3.g.e yes 8
24.f even 2 1 inner 2304.3.b.s 8
24.h odd 2 1 2304.3.b.r 8
48.i odd 4 1 1152.3.g.d 8
48.i odd 4 1 1152.3.g.e yes 8
48.k even 4 1 1152.3.g.d 8
48.k even 4 1 1152.3.g.e yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.3.g.d 8 16.e even 4 1
1152.3.g.d 8 16.f odd 4 1
1152.3.g.d 8 48.i odd 4 1
1152.3.g.d 8 48.k even 4 1
1152.3.g.e yes 8 16.e even 4 1
1152.3.g.e yes 8 16.f odd 4 1
1152.3.g.e yes 8 48.i odd 4 1
1152.3.g.e yes 8 48.k even 4 1
2304.3.b.r 8 4.b odd 2 1
2304.3.b.r 8 8.b even 2 1
2304.3.b.r 8 12.b even 2 1
2304.3.b.r 8 24.h odd 2 1
2304.3.b.s 8 1.a even 1 1 trivial
2304.3.b.s 8 3.b odd 2 1 inner
2304.3.b.s 8 8.d odd 2 1 inner
2304.3.b.s 8 24.f 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 \(S_{3}^{\mathrm{new}}(2304, [\chi])\):

\( T_{5}^{4} + 72 T_{5}^{2} + 400 \)
\( T_{7}^{4} + 120 T_{7}^{2} + 2704 \)
\( T_{11}^{2} - 112 \)
\( T_{17}^{4} - 288 T_{17}^{2} + 6400 \)
\( T_{19}^{2} - 24 T_{19} - 80 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( 400 + 72 T^{2} + T^{4} )^{2} \)
$7$ \( ( 2704 + 120 T^{2} + T^{4} )^{2} \)
$11$ \( ( -112 + T^{2} )^{4} \)
$13$ \( ( 35344 + 520 T^{2} + T^{4} )^{2} \)
$17$ \( ( 6400 - 288 T^{2} + T^{4} )^{2} \)
$19$ \( ( -80 - 24 T + T^{2} )^{4} \)
$23$ \( ( 4096 + 1920 T^{2} + T^{4} )^{2} \)
$29$ \( ( 132496 + 840 T^{2} + T^{4} )^{2} \)
$31$ \( ( 71824 + 760 T^{2} + T^{4} )^{2} \)
$37$ \( ( 48400 + 456 T^{2} + T^{4} )^{2} \)
$41$ \( ( 2119936 - 3360 T^{2} + T^{4} )^{2} \)
$43$ \( ( 176 - 40 T + T^{2} )^{4} \)
$47$ \( ( 12390400 + 9088 T^{2} + T^{4} )^{2} \)
$53$ \( ( 4787344 + 7176 T^{2} + T^{4} )^{2} \)
$59$ \( ( 2560000 - 4992 T^{2} + T^{4} )^{2} \)
$61$ \( ( 784 + 840 T^{2} + T^{4} )^{2} \)
$67$ \( ( -7488 - 48 T + T^{2} )^{4} \)
$71$ \( ( 8192 + T^{2} )^{4} \)
$73$ \( ( -6620 - 76 T + T^{2} )^{4} \)
$79$ \( ( 8179600 + 7736 T^{2} + T^{4} )^{2} \)
$83$ \( ( 65286400 - 16608 T^{2} + T^{4} )^{2} \)
$89$ \( ( 5017600 - 20608 T^{2} + T^{4} )^{2} \)
$97$ \( ( -3388 - 28 T + T^{2} )^{4} \)
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