Properties

Label 2304.2.d.m
Level $2304$
Weight $2$
Character orbit 2304.d
Analytic conductor $18.398$
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: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2304.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.3975326257\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{7} - 2 \beta q^{11} - 3 \beta q^{13} - 6 q^{17} - 4 q^{23} + 5 q^{25} + 2 \beta q^{29} - 10 q^{31} + \beta q^{37} - 2 q^{41} - 4 \beta q^{43} - 12 q^{47} - 3 q^{49} + 6 \beta q^{53} - 2 \beta q^{59} - \beta q^{61} + 2 \beta q^{67} + 4 q^{71} + 10 q^{73} - 4 \beta q^{77} + 6 q^{79} - 6 \beta q^{83} + 2 q^{89} - 6 \beta q^{91} - 6 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{7} - 12 q^{17} - 8 q^{23} + 10 q^{25} - 20 q^{31} - 4 q^{41} - 24 q^{47} - 6 q^{49} + 8 q^{71} + 20 q^{73} + 12 q^{79} + 4 q^{89} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1153.1
1.00000i
1.00000i
0 0 0 0 0 2.00000 0 0 0
1153.2 0 0 0 0 0 2.00000 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
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.2.d.m 2
3.b odd 2 1 768.2.d.g 2
4.b odd 2 1 2304.2.d.d 2
8.b even 2 1 inner 2304.2.d.m 2
8.d odd 2 1 2304.2.d.d 2
12.b even 2 1 768.2.d.b 2
16.e even 4 1 1152.2.a.i 1
16.e even 4 1 1152.2.a.j 1
16.f odd 4 1 1152.2.a.k 1
16.f odd 4 1 1152.2.a.l 1
24.f even 2 1 768.2.d.b 2
24.h odd 2 1 768.2.d.g 2
48.i odd 4 1 384.2.a.b 1
48.i odd 4 1 384.2.a.f yes 1
48.k even 4 1 384.2.a.c yes 1
48.k even 4 1 384.2.a.g yes 1
240.t even 4 1 9600.2.a.h 1
240.t even 4 1 9600.2.a.bh 1
240.bm odd 4 1 9600.2.a.w 1
240.bm odd 4 1 9600.2.a.bw 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.2.a.b 1 48.i odd 4 1
384.2.a.c yes 1 48.k even 4 1
384.2.a.f yes 1 48.i odd 4 1
384.2.a.g yes 1 48.k even 4 1
768.2.d.b 2 12.b even 2 1
768.2.d.b 2 24.f even 2 1
768.2.d.g 2 3.b odd 2 1
768.2.d.g 2 24.h odd 2 1
1152.2.a.i 1 16.e even 4 1
1152.2.a.j 1 16.e even 4 1
1152.2.a.k 1 16.f odd 4 1
1152.2.a.l 1 16.f odd 4 1
2304.2.d.d 2 4.b odd 2 1
2304.2.d.d 2 8.d odd 2 1
2304.2.d.m 2 1.a even 1 1 trivial
2304.2.d.m 2 8.b even 2 1 inner
9600.2.a.h 1 240.t even 4 1
9600.2.a.w 1 240.bm odd 4 1
9600.2.a.bh 1 240.t even 4 1
9600.2.a.bw 1 240.bm odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2304, [\chi])\):

\( T_{5} \) Copy content Toggle raw display
\( T_{7} - 2 \) Copy content Toggle raw display
\( T_{11}^{2} + 16 \) Copy content Toggle raw display
\( T_{17} + 6 \) Copy content Toggle raw display
\( T_{23} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T - 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 16 \) Copy content Toggle raw display
$13$ \( T^{2} + 36 \) Copy content Toggle raw display
$17$ \( (T + 6)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( (T + 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 16 \) Copy content Toggle raw display
$31$ \( (T + 10)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( (T + 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 64 \) Copy content Toggle raw display
$47$ \( (T + 12)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 144 \) Copy content Toggle raw display
$59$ \( T^{2} + 16 \) Copy content Toggle raw display
$61$ \( T^{2} + 4 \) Copy content Toggle raw display
$67$ \( T^{2} + 16 \) Copy content Toggle raw display
$71$ \( (T - 4)^{2} \) Copy content Toggle raw display
$73$ \( (T - 10)^{2} \) Copy content Toggle raw display
$79$ \( (T - 6)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 144 \) Copy content Toggle raw display
$89$ \( (T - 2)^{2} \) Copy content Toggle raw display
$97$ \( (T + 6)^{2} \) Copy content Toggle raw display
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