Properties

Label 2304.3.g.m
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: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 128)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 q^{5} -4 \beta q^{7} +O(q^{10})\) \( q + 4 q^{5} -4 \beta q^{7} -5 \beta q^{11} -20 q^{13} + 10 q^{17} -5 \beta q^{19} -4 \beta q^{23} -9 q^{25} + 20 q^{29} -16 \beta q^{35} -20 q^{37} -30 q^{41} + \beta q^{43} + 24 \beta q^{47} -79 q^{49} -60 q^{53} -20 \beta q^{55} + 15 \beta q^{59} + 28 q^{61} -80 q^{65} -29 \beta q^{67} + 20 \beta q^{71} + 10 q^{73} -160 q^{77} + 40 \beta q^{79} + 9 \beta q^{83} + 40 q^{85} + 22 q^{89} + 80 \beta q^{91} -20 \beta q^{95} + 150 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 8q^{5} + O(q^{10}) \) \( 2q + 8q^{5} - 40q^{13} + 20q^{17} - 18q^{25} + 40q^{29} - 40q^{37} - 60q^{41} - 158q^{49} - 120q^{53} + 56q^{61} - 160q^{65} + 20q^{73} - 320q^{77} + 80q^{85} + 44q^{89} + 300q^{97} + O(q^{100}) \)

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
1.41421i
1.41421i
0 0 0 4.00000 0 11.3137i 0 0 0
1279.2 0 0 0 4.00000 0 11.3137i 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.m 2
3.b odd 2 1 256.3.c.c 2
4.b odd 2 1 inner 2304.3.g.m 2
8.b even 2 1 2304.3.g.h 2
8.d odd 2 1 2304.3.g.h 2
12.b even 2 1 256.3.c.c 2
16.e even 4 2 1152.3.b.g 4
16.f odd 4 2 1152.3.b.g 4
24.f even 2 1 256.3.c.f 2
24.h odd 2 1 256.3.c.f 2
48.i odd 4 2 128.3.d.c 4
48.k even 4 2 128.3.d.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.3.d.c 4 48.i odd 4 2
128.3.d.c 4 48.k even 4 2
256.3.c.c 2 3.b odd 2 1
256.3.c.c 2 12.b even 2 1
256.3.c.f 2 24.f even 2 1
256.3.c.f 2 24.h odd 2 1
1152.3.b.g 4 16.e even 4 2
1152.3.b.g 4 16.f odd 4 2
2304.3.g.h 2 8.b even 2 1
2304.3.g.h 2 8.d odd 2 1
2304.3.g.m 2 1.a even 1 1 trivial
2304.3.g.m 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 \(S_{3}^{\mathrm{new}}(2304, [\chi])\):

\( T_{5} - 4 \)
\( T_{7}^{2} + 128 \)
\( T_{11}^{2} + 200 \)
\( T_{13} + 20 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( -4 + T )^{2} \)
$7$ \( 128 + T^{2} \)
$11$ \( 200 + T^{2} \)
$13$ \( ( 20 + T )^{2} \)
$17$ \( ( -10 + T )^{2} \)
$19$ \( 200 + T^{2} \)
$23$ \( 128 + T^{2} \)
$29$ \( ( -20 + T )^{2} \)
$31$ \( T^{2} \)
$37$ \( ( 20 + T )^{2} \)
$41$ \( ( 30 + T )^{2} \)
$43$ \( 8 + T^{2} \)
$47$ \( 4608 + T^{2} \)
$53$ \( ( 60 + T )^{2} \)
$59$ \( 1800 + T^{2} \)
$61$ \( ( -28 + T )^{2} \)
$67$ \( 6728 + T^{2} \)
$71$ \( 3200 + T^{2} \)
$73$ \( ( -10 + T )^{2} \)
$79$ \( 12800 + T^{2} \)
$83$ \( 648 + T^{2} \)
$89$ \( ( -22 + T )^{2} \)
$97$ \( ( -150 + T )^{2} \)
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