Properties

Label 2304.3.b.f
Level $2304$
Weight $3$
Character orbit 2304.b
Analytic conductor $62.779$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2304.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(62.7794529086\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 16)
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.

\(f(q)\) \(=\) \( q + 6 i q^{5} +O(q^{10})\) \( q + 6 i q^{5} -10 i q^{13} + 30 q^{17} -11 q^{25} + 42 i q^{29} -70 i q^{37} + 18 q^{41} + 49 q^{49} -90 i q^{53} + 22 i q^{61} + 60 q^{65} + 110 q^{73} + 180 i q^{85} -78 q^{89} + 130 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q + 60q^{17} - 22q^{25} + 36q^{41} + 98q^{49} + 120q^{65} + 220q^{73} - 156q^{89} + 260q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
1.00000i
1.00000i
0 0 0 6.00000i 0 0 0 0 0
127.2 0 0 0 6.00000i 0 0 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 CM by \(\Q(\sqrt{-1}) \)
8.b even 2 1 inner
8.d 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.b.f 2
3.b odd 2 1 256.3.d.b 2
4.b odd 2 1 CM 2304.3.b.f 2
8.b even 2 1 inner 2304.3.b.f 2
8.d odd 2 1 inner 2304.3.b.f 2
12.b even 2 1 256.3.d.b 2
16.e even 4 1 144.3.g.a 1
16.e even 4 1 576.3.g.b 1
16.f odd 4 1 144.3.g.a 1
16.f odd 4 1 576.3.g.b 1
24.f even 2 1 256.3.d.b 2
24.h odd 2 1 256.3.d.b 2
48.i odd 4 1 16.3.c.a 1
48.i odd 4 1 64.3.c.a 1
48.k even 4 1 16.3.c.a 1
48.k even 4 1 64.3.c.a 1
80.i odd 4 1 3600.3.j.a 2
80.j even 4 1 3600.3.j.a 2
80.k odd 4 1 3600.3.e.c 1
80.q even 4 1 3600.3.e.c 1
80.s even 4 1 3600.3.j.a 2
80.t odd 4 1 3600.3.j.a 2
144.u even 12 2 1296.3.o.o 2
144.v odd 12 2 1296.3.o.b 2
144.w odd 12 2 1296.3.o.o 2
144.x even 12 2 1296.3.o.b 2
240.t even 4 1 400.3.b.a 1
240.t even 4 1 1600.3.b.b 1
240.z odd 4 1 400.3.h.a 2
240.z odd 4 1 1600.3.h.b 2
240.bb even 4 1 400.3.h.a 2
240.bb even 4 1 1600.3.h.b 2
240.bd odd 4 1 400.3.h.a 2
240.bd odd 4 1 1600.3.h.b 2
240.bf even 4 1 400.3.h.a 2
240.bf even 4 1 1600.3.h.b 2
240.bm odd 4 1 400.3.b.a 1
240.bm odd 4 1 1600.3.b.b 1
336.v odd 4 1 784.3.d.b 1
336.y even 4 1 784.3.d.b 1
336.bo even 12 2 784.3.r.d 2
336.br odd 12 2 784.3.r.d 2
336.bt odd 12 2 784.3.r.e 2
336.bu even 12 2 784.3.r.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.3.c.a 1 48.i odd 4 1
16.3.c.a 1 48.k even 4 1
64.3.c.a 1 48.i odd 4 1
64.3.c.a 1 48.k even 4 1
144.3.g.a 1 16.e even 4 1
144.3.g.a 1 16.f odd 4 1
256.3.d.b 2 3.b odd 2 1
256.3.d.b 2 12.b even 2 1
256.3.d.b 2 24.f even 2 1
256.3.d.b 2 24.h odd 2 1
400.3.b.a 1 240.t even 4 1
400.3.b.a 1 240.bm odd 4 1
400.3.h.a 2 240.z odd 4 1
400.3.h.a 2 240.bb even 4 1
400.3.h.a 2 240.bd odd 4 1
400.3.h.a 2 240.bf even 4 1
576.3.g.b 1 16.e even 4 1
576.3.g.b 1 16.f odd 4 1
784.3.d.b 1 336.v odd 4 1
784.3.d.b 1 336.y even 4 1
784.3.r.d 2 336.bo even 12 2
784.3.r.d 2 336.br odd 12 2
784.3.r.e 2 336.bt odd 12 2
784.3.r.e 2 336.bu even 12 2
1296.3.o.b 2 144.v odd 12 2
1296.3.o.b 2 144.x even 12 2
1296.3.o.o 2 144.u even 12 2
1296.3.o.o 2 144.w odd 12 2
1600.3.b.b 1 240.t even 4 1
1600.3.b.b 1 240.bm odd 4 1
1600.3.h.b 2 240.z odd 4 1
1600.3.h.b 2 240.bb even 4 1
1600.3.h.b 2 240.bd odd 4 1
1600.3.h.b 2 240.bf even 4 1
2304.3.b.f 2 1.a even 1 1 trivial
2304.3.b.f 2 4.b odd 2 1 CM
2304.3.b.f 2 8.b even 2 1 inner
2304.3.b.f 2 8.d odd 2 1 inner
3600.3.e.c 1 80.k odd 4 1
3600.3.e.c 1 80.q even 4 1
3600.3.j.a 2 80.i odd 4 1
3600.3.j.a 2 80.j even 4 1
3600.3.j.a 2 80.s even 4 1
3600.3.j.a 2 80.t odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(2304, [\chi])\):

\( T_{5}^{2} + 36 \)
\( T_{7} \)
\( T_{11} \)
\( T_{17} - 30 \)
\( T_{19} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 36 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( 100 + T^{2} \)
$17$ \( ( -30 + T )^{2} \)
$19$ \( T^{2} \)
$23$ \( T^{2} \)
$29$ \( 1764 + T^{2} \)
$31$ \( T^{2} \)
$37$ \( 4900 + T^{2} \)
$41$ \( ( -18 + T )^{2} \)
$43$ \( T^{2} \)
$47$ \( T^{2} \)
$53$ \( 8100 + T^{2} \)
$59$ \( T^{2} \)
$61$ \( 484 + T^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( -110 + T )^{2} \)
$79$ \( T^{2} \)
$83$ \( T^{2} \)
$89$ \( ( 78 + T )^{2} \)
$97$ \( ( -130 + T )^{2} \)
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