Properties

Label 2304.2.d.a
Level 2304
Weight 2
Character orbit 2304.d
Analytic conductor 18.398
Analytic rank 0
Dimension 2
CM discriminant -3
Inner twists 4

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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: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 36)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q -4 q^{7} +O(q^{10})\) \( q -4 q^{7} -2 i q^{13} + 8 i q^{19} + 5 q^{25} + 4 q^{31} -10 i q^{37} -8 i q^{43} + 9 q^{49} -14 i q^{61} -16 i q^{67} + 10 q^{73} + 4 q^{79} + 8 i q^{91} + 14 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 8q^{7} + O(q^{10}) \) \( 2q - 8q^{7} + 10q^{25} + 8q^{31} + 18q^{49} + 20q^{73} + 8q^{79} + 28q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1153.1
1.00000i
1.00000i
0 0 0 0 0 −4.00000 0 0 0
1153.2 0 0 0 0 0 −4.00000 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
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 2304.2.d.a 2
3.b odd 2 1 CM 2304.2.d.a 2
4.b odd 2 1 2304.2.d.q 2
8.b even 2 1 inner 2304.2.d.a 2
8.d odd 2 1 2304.2.d.q 2
12.b even 2 1 2304.2.d.q 2
16.e even 4 1 144.2.a.a 1
16.e even 4 1 576.2.a.f 1
16.f odd 4 1 36.2.a.a 1
16.f odd 4 1 576.2.a.e 1
24.f even 2 1 2304.2.d.q 2
24.h odd 2 1 inner 2304.2.d.a 2
48.i odd 4 1 144.2.a.a 1
48.i odd 4 1 576.2.a.f 1
48.k even 4 1 36.2.a.a 1
48.k even 4 1 576.2.a.e 1
80.i odd 4 1 3600.2.f.m 2
80.j even 4 1 900.2.d.b 2
80.k odd 4 1 900.2.a.g 1
80.q even 4 1 3600.2.a.e 1
80.s even 4 1 900.2.d.b 2
80.t odd 4 1 3600.2.f.m 2
112.j even 4 1 1764.2.a.e 1
112.l odd 4 1 7056.2.a.bb 1
112.u odd 12 2 1764.2.k.h 2
112.v even 12 2 1764.2.k.g 2
144.u even 12 2 324.2.e.c 2
144.v odd 12 2 324.2.e.c 2
144.w odd 12 2 1296.2.i.h 2
144.x even 12 2 1296.2.i.h 2
176.i even 4 1 4356.2.a.g 1
208.l even 4 1 6084.2.b.f 2
208.o odd 4 1 6084.2.a.i 1
208.s even 4 1 6084.2.b.f 2
240.t even 4 1 900.2.a.g 1
240.z odd 4 1 900.2.d.b 2
240.bb even 4 1 3600.2.f.m 2
240.bd odd 4 1 900.2.d.b 2
240.bf even 4 1 3600.2.f.m 2
240.bm odd 4 1 3600.2.a.e 1
336.v odd 4 1 1764.2.a.e 1
336.y even 4 1 7056.2.a.bb 1
336.br odd 12 2 1764.2.k.g 2
336.bu even 12 2 1764.2.k.h 2
528.s odd 4 1 4356.2.a.g 1
624.s odd 4 1 6084.2.b.f 2
624.v even 4 1 6084.2.a.i 1
624.bo odd 4 1 6084.2.b.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.a.a 1 16.f odd 4 1
36.2.a.a 1 48.k even 4 1
144.2.a.a 1 16.e even 4 1
144.2.a.a 1 48.i odd 4 1
324.2.e.c 2 144.u even 12 2
324.2.e.c 2 144.v odd 12 2
576.2.a.e 1 16.f odd 4 1
576.2.a.e 1 48.k even 4 1
576.2.a.f 1 16.e even 4 1
576.2.a.f 1 48.i odd 4 1
900.2.a.g 1 80.k odd 4 1
900.2.a.g 1 240.t even 4 1
900.2.d.b 2 80.j even 4 1
900.2.d.b 2 80.s even 4 1
900.2.d.b 2 240.z odd 4 1
900.2.d.b 2 240.bd odd 4 1
1296.2.i.h 2 144.w odd 12 2
1296.2.i.h 2 144.x even 12 2
1764.2.a.e 1 112.j even 4 1
1764.2.a.e 1 336.v odd 4 1
1764.2.k.g 2 112.v even 12 2
1764.2.k.g 2 336.br odd 12 2
1764.2.k.h 2 112.u odd 12 2
1764.2.k.h 2 336.bu even 12 2
2304.2.d.a 2 1.a even 1 1 trivial
2304.2.d.a 2 3.b odd 2 1 CM
2304.2.d.a 2 8.b even 2 1 inner
2304.2.d.a 2 24.h odd 2 1 inner
2304.2.d.q 2 4.b odd 2 1
2304.2.d.q 2 8.d odd 2 1
2304.2.d.q 2 12.b even 2 1
2304.2.d.q 2 24.f even 2 1
3600.2.a.e 1 80.q even 4 1
3600.2.a.e 1 240.bm odd 4 1
3600.2.f.m 2 80.i odd 4 1
3600.2.f.m 2 80.t odd 4 1
3600.2.f.m 2 240.bb even 4 1
3600.2.f.m 2 240.bf even 4 1
4356.2.a.g 1 176.i even 4 1
4356.2.a.g 1 528.s odd 4 1
6084.2.a.i 1 208.o odd 4 1
6084.2.a.i 1 624.v even 4 1
6084.2.b.f 2 208.l even 4 1
6084.2.b.f 2 208.s even 4 1
6084.2.b.f 2 624.s odd 4 1
6084.2.b.f 2 624.bo odd 4 1
7056.2.a.bb 1 112.l odd 4 1
7056.2.a.bb 1 336.y even 4 1

Hecke kernels

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

\( T_{5} \)
\( T_{7} + 4 \)
\( T_{11} \)
\( T_{17} \)
\( T_{23} \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( \)
$5$ \( ( 1 - 5 T^{2} )^{2} \)
$7$ \( ( 1 + 4 T + 7 T^{2} )^{2} \)
$11$ \( ( 1 - 11 T^{2} )^{2} \)
$13$ \( 1 - 22 T^{2} + 169 T^{4} \)
$17$ \( ( 1 + 17 T^{2} )^{2} \)
$19$ \( 1 + 26 T^{2} + 361 T^{4} \)
$23$ \( ( 1 + 23 T^{2} )^{2} \)
$29$ \( ( 1 - 29 T^{2} )^{2} \)
$31$ \( ( 1 - 4 T + 31 T^{2} )^{2} \)
$37$ \( 1 + 26 T^{2} + 1369 T^{4} \)
$41$ \( ( 1 + 41 T^{2} )^{2} \)
$43$ \( 1 - 22 T^{2} + 1849 T^{4} \)
$47$ \( ( 1 + 47 T^{2} )^{2} \)
$53$ \( ( 1 - 53 T^{2} )^{2} \)
$59$ \( ( 1 - 59 T^{2} )^{2} \)
$61$ \( 1 + 74 T^{2} + 3721 T^{4} \)
$67$ \( 1 + 122 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 10 T + 73 T^{2} )^{2} \)
$79$ \( ( 1 - 4 T + 79 T^{2} )^{2} \)
$83$ \( ( 1 - 83 T^{2} )^{2} \)
$89$ \( ( 1 + 89 T^{2} )^{2} \)
$97$ \( ( 1 - 14 T + 97 T^{2} )^{2} \)
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