Properties

Label 2304.3.g.n
Level $2304$
Weight $3$
Character orbit 2304.g
Analytic conductor $62.779$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 384)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 q^{5} + ( 4 - 8 \zeta_{6} ) q^{7} +O(q^{10})\) \( q + 4 q^{5} + ( 4 - 8 \zeta_{6} ) q^{7} + ( 4 - 8 \zeta_{6} ) q^{11} + 18 q^{17} + ( -12 + 24 \zeta_{6} ) q^{19} + ( -24 + 48 \zeta_{6} ) q^{23} -9 q^{25} -4 q^{29} + ( -28 + 56 \zeta_{6} ) q^{31} + ( 16 - 32 \zeta_{6} ) q^{35} + 72 q^{37} + 18 q^{41} + ( -36 + 72 \zeta_{6} ) q^{43} + ( 24 - 48 \zeta_{6} ) q^{47} + q^{49} + 44 q^{53} + ( 16 - 32 \zeta_{6} ) q^{55} + ( 36 - 72 \zeta_{6} ) q^{59} -72 q^{61} + ( -12 + 24 \zeta_{6} ) q^{67} + ( 24 - 48 \zeta_{6} ) q^{71} + 82 q^{73} -48 q^{77} + ( 36 - 72 \zeta_{6} ) q^{79} + ( 76 - 152 \zeta_{6} ) q^{83} + 72 q^{85} + 126 q^{89} + ( -48 + 96 \zeta_{6} ) q^{95} + 110 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 8q^{5} + O(q^{10}) \) \( 2q + 8q^{5} + 36q^{17} - 18q^{25} - 8q^{29} + 144q^{37} + 36q^{41} + 2q^{49} + 88q^{53} - 144q^{61} + 164q^{73} - 96q^{77} + 144q^{85} + 252q^{89} + 220q^{97} + O(q^{100}) \)

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
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 4.00000 0 6.92820i 0 0 0
1279.2 0 0 0 4.00000 0 6.92820i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b 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.n 2
3.b odd 2 1 768.3.g.a 2
4.b odd 2 1 inner 2304.3.g.n 2
8.b even 2 1 2304.3.g.g 2
8.d odd 2 1 2304.3.g.g 2
12.b even 2 1 768.3.g.a 2
16.e even 4 2 1152.3.b.h 4
16.f odd 4 2 1152.3.b.h 4
24.f even 2 1 768.3.g.b 2
24.h odd 2 1 768.3.g.b 2
48.i odd 4 2 384.3.b.a 4
48.k even 4 2 384.3.b.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.3.b.a 4 48.i odd 4 2
384.3.b.a 4 48.k even 4 2
768.3.g.a 2 3.b odd 2 1
768.3.g.a 2 12.b even 2 1
768.3.g.b 2 24.f even 2 1
768.3.g.b 2 24.h odd 2 1
1152.3.b.h 4 16.e even 4 2
1152.3.b.h 4 16.f odd 4 2
2304.3.g.g 2 8.b even 2 1
2304.3.g.g 2 8.d odd 2 1
2304.3.g.n 2 1.a even 1 1 trivial
2304.3.g.n 2 4.b 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 \(S_{3}^{\mathrm{new}}(2304, [\chi])\):

\( T_{5} - 4 \)
\( T_{7}^{2} + 48 \)
\( T_{11}^{2} + 48 \)
\( T_{13} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( -4 + T )^{2} \)
$7$ \( 48 + T^{2} \)
$11$ \( 48 + T^{2} \)
$13$ \( T^{2} \)
$17$ \( ( -18 + T )^{2} \)
$19$ \( 432 + T^{2} \)
$23$ \( 1728 + T^{2} \)
$29$ \( ( 4 + T )^{2} \)
$31$ \( 2352 + T^{2} \)
$37$ \( ( -72 + T )^{2} \)
$41$ \( ( -18 + T )^{2} \)
$43$ \( 3888 + T^{2} \)
$47$ \( 1728 + T^{2} \)
$53$ \( ( -44 + T )^{2} \)
$59$ \( 3888 + T^{2} \)
$61$ \( ( 72 + T )^{2} \)
$67$ \( 432 + T^{2} \)
$71$ \( 1728 + T^{2} \)
$73$ \( ( -82 + T )^{2} \)
$79$ \( 3888 + T^{2} \)
$83$ \( 17328 + T^{2} \)
$89$ \( ( -126 + T )^{2} \)
$97$ \( ( -110 + T )^{2} \)
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