Properties

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

Newform invariants

Self dual: no
Analytic conductor: \(62.7794529086\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.959512576.1
Defining polynomial: \(x^{8} + 7 x^{4} + 81\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
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.

\(f(q)\) \(=\) \( q + ( \beta_{3} - \beta_{5} ) q^{5} -2 q^{7} +O(q^{10})\) \( q + ( \beta_{3} - \beta_{5} ) q^{5} -2 q^{7} + ( -4 \beta_{3} + 2 \beta_{5} ) q^{11} + ( 4 \beta_{1} - \beta_{6} ) q^{13} + ( -\beta_{2} + \beta_{4} ) q^{17} + ( 4 \beta_{1} + 2 \beta_{6} ) q^{19} + ( -6 \beta_{2} - 2 \beta_{4} ) q^{23} + ( 21 - \beta_{7} ) q^{25} + ( 6 \beta_{3} + \beta_{5} ) q^{29} + ( 2 + 2 \beta_{7} ) q^{31} + ( -2 \beta_{3} + 2 \beta_{5} ) q^{35} -13 \beta_{1} q^{37} + ( 11 \beta_{2} - \beta_{4} ) q^{41} + ( -14 \beta_{1} + 4 \beta_{6} ) q^{43} + ( 38 \beta_{2} - 2 \beta_{4} ) q^{47} -45 q^{49} + ( -12 \beta_{3} + 3 \beta_{5} ) q^{53} + ( -100 + 4 \beta_{7} ) q^{55} -26 \beta_{3} q^{59} + ( 15 \beta_{1} + 4 \beta_{6} ) q^{61} + ( -52 \beta_{2} - 5 \beta_{4} ) q^{65} + ( 42 \beta_{1} + 2 \beta_{6} ) q^{67} + ( 22 \beta_{2} + 4 \beta_{4} ) q^{71} + ( 28 + 6 \beta_{7} ) q^{73} + ( 8 \beta_{3} - 4 \beta_{5} ) q^{77} + ( 38 - 2 \beta_{7} ) q^{79} + ( -6 \beta_{3} + 6 \beta_{5} ) q^{83} + ( -43 \beta_{1} + \beta_{6} ) q^{85} + ( 47 \beta_{2} + 6 \beta_{4} ) q^{89} + ( -8 \beta_{1} + 2 \beta_{6} ) q^{91} + ( 80 \beta_{2} - 2 \beta_{4} ) q^{95} + ( -88 + \beta_{7} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 16q^{7} + O(q^{10}) \) \( 8q - 16q^{7} + 168q^{25} + 16q^{31} - 360q^{49} - 800q^{55} + 224q^{73} + 304q^{79} - 704q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 7 x^{4} + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 2 \nu^{6} + 32 \nu^{2} \)\()/45\)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{7} - 9 \nu^{5} - 13 \nu^{3} - 9 \nu \)\()/135\)
\(\beta_{3}\)\(=\)\((\)\( -4 \nu^{7} - 18 \nu^{5} + 26 \nu^{3} - 18 \nu \)\()/135\)
\(\beta_{4}\)\(=\)\((\)\( 8 \nu^{4} + 28 \)\()/5\)
\(\beta_{5}\)\(=\)\((\)\( -2 \nu^{7} - 30 \nu^{6} - 9 \nu^{5} + 13 \nu^{3} + 60 \nu^{2} - 9 \nu \)\()/135\)
\(\beta_{6}\)\(=\)\((\)\( 16 \nu^{7} + 18 \nu^{5} + 166 \nu^{3} + 558 \nu \)\()/135\)
\(\beta_{7}\)\(=\)\((\)\( -32 \nu^{7} + 36 \nu^{5} - 332 \nu^{3} + 1116 \nu \)\()/135\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} + 2 \beta_{6} + 2 \beta_{3} + 4 \beta_{2}\)\()/16\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{5} - \beta_{3} + 10 \beta_{1}\)\()/8\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{7} + 2 \beta_{6} + 8 \beta_{3} - 16 \beta_{2}\)\()/8\)
\(\nu^{4}\)\(=\)\((\)\(5 \beta_{4} - 28\)\()/8\)
\(\nu^{5}\)\(=\)\((\)\(-\beta_{7} - 2 \beta_{6} - 62 \beta_{3} - 124 \beta_{2}\)\()/16\)
\(\nu^{6}\)\(=\)\((\)\(-8 \beta_{5} + 4 \beta_{3} + 5 \beta_{1}\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-13 \beta_{7} + 26 \beta_{6} - 166 \beta_{3} + 332 \beta_{2}\)\()/16\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2177.1
−1.52616 + 0.819051i
−1.52616 0.819051i
1.52616 + 0.819051i
1.52616 0.819051i
0.819051 + 1.52616i
0.819051 1.52616i
−0.819051 + 1.52616i
−0.819051 1.52616i
0 0 0 −8.04746 0 −2.00000 0 0 0
2177.2 0 0 0 −8.04746 0 −2.00000 0 0 0
2177.3 0 0 0 −5.21904 0 −2.00000 0 0 0
2177.4 0 0 0 −5.21904 0 −2.00000 0 0 0
2177.5 0 0 0 5.21904 0 −2.00000 0 0 0
2177.6 0 0 0 5.21904 0 −2.00000 0 0 0
2177.7 0 0 0 8.04746 0 −2.00000 0 0 0
2177.8 0 0 0 8.04746 0 −2.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2177.8
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.h.j 8
3.b odd 2 1 inner 2304.3.h.j 8
4.b odd 2 1 2304.3.h.l 8
8.b even 2 1 inner 2304.3.h.j 8
8.d odd 2 1 2304.3.h.l 8
12.b even 2 1 2304.3.h.l 8
16.e even 4 1 1152.3.e.e yes 4
16.e even 4 1 1152.3.e.g yes 4
16.f odd 4 1 1152.3.e.a 4
16.f odd 4 1 1152.3.e.c yes 4
24.f even 2 1 2304.3.h.l 8
24.h odd 2 1 inner 2304.3.h.j 8
48.i odd 4 1 1152.3.e.e yes 4
48.i odd 4 1 1152.3.e.g yes 4
48.k even 4 1 1152.3.e.a 4
48.k even 4 1 1152.3.e.c yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.3.e.a 4 16.f odd 4 1
1152.3.e.a 4 48.k even 4 1
1152.3.e.c yes 4 16.f odd 4 1
1152.3.e.c yes 4 48.k even 4 1
1152.3.e.e yes 4 16.e even 4 1
1152.3.e.e yes 4 48.i odd 4 1
1152.3.e.g yes 4 16.e even 4 1
1152.3.e.g yes 4 48.i odd 4 1
2304.3.h.j 8 1.a even 1 1 trivial
2304.3.h.j 8 3.b odd 2 1 inner
2304.3.h.j 8 8.b even 2 1 inner
2304.3.h.j 8 24.h odd 2 1 inner
2304.3.h.l 8 4.b odd 2 1
2304.3.h.l 8 8.d odd 2 1
2304.3.h.l 8 12.b even 2 1
2304.3.h.l 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 \(S_{3}^{\mathrm{new}}(2304, [\chi])\):

\( T_{5}^{4} - 92 T_{5}^{2} + 1764 \)
\( T_{7} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( 1764 - 92 T^{2} + T^{4} )^{2} \)
$7$ \( ( 2 + T )^{8} \)
$11$ \( ( 10816 - 496 T^{2} + T^{4} )^{2} \)
$13$ \( ( 576 + 304 T^{2} + T^{4} )^{2} \)
$17$ \( ( 30276 + 356 T^{2} + T^{4} )^{2} \)
$19$ \( ( 82944 + 832 T^{2} + T^{4} )^{2} \)
$23$ \( ( 399424 + 1552 T^{2} + T^{4} )^{2} \)
$29$ \( ( 86436 - 764 T^{2} + T^{4} )^{2} \)
$31$ \( ( -1404 - 4 T + T^{2} )^{4} \)
$37$ \( ( 676 + T^{2} )^{4} \)
$41$ \( ( 4356 + 836 T^{2} + T^{4} )^{2} \)
$43$ \( ( 389376 + 4384 T^{2} + T^{4} )^{2} \)
$47$ \( ( 4769856 + 7184 T^{2} + T^{4} )^{2} \)
$53$ \( ( 236196 - 2556 T^{2} + T^{4} )^{2} \)
$59$ \( ( -5408 + T^{2} )^{4} \)
$61$ \( ( 258064 + 4616 T^{2} + T^{4} )^{2} \)
$67$ \( ( 44943616 + 14816 T^{2} + T^{4} )^{2} \)
$71$ \( ( 3415104 + 7568 T^{2} + T^{4} )^{2} \)
$73$ \( ( -11888 - 56 T + T^{2} )^{4} \)
$79$ \( ( 36 - 76 T + T^{2} )^{4} \)
$83$ \( ( 2286144 - 3312 T^{2} + T^{4} )^{2} \)
$89$ \( ( 3678724 + 21508 T^{2} + T^{4} )^{2} \)
$97$ \( ( 7392 + 176 T + T^{2} )^{4} \)
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