Properties

Label 1152.2.d.d
Level $1152$
Weight $2$
Character orbit 1152.d
Analytic conductor $9.199$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1152.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.19876631285\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 128)
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.

\(f(q)\) \(=\) \( q + 4 i q^{5} +O(q^{10})\) \( q + 4 i q^{5} + 4 i q^{13} + 2 q^{17} -11 q^{25} -4 i q^{29} + 12 i q^{37} -10 q^{41} -7 q^{49} + 4 i q^{53} -12 i q^{61} -16 q^{65} + 6 q^{73} + 8 i q^{85} + 10 q^{89} -18 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q + 4q^{17} - 22q^{25} - 20q^{41} - 14q^{49} - 32q^{65} + 12q^{73} + 20q^{89} - 36q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
1.00000i
1.00000i
0 0 0 4.00000i 0 0 0 0 0
577.2 0 0 0 4.00000i 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 1152.2.d.d 2
3.b odd 2 1 128.2.b.b 2
4.b odd 2 1 CM 1152.2.d.d 2
8.b even 2 1 inner 1152.2.d.d 2
8.d odd 2 1 inner 1152.2.d.d 2
12.b even 2 1 128.2.b.b 2
15.d odd 2 1 3200.2.d.e 2
15.e even 4 1 3200.2.f.c 2
15.e even 4 1 3200.2.f.d 2
16.e even 4 1 2304.2.a.a 1
16.e even 4 1 2304.2.a.p 1
16.f odd 4 1 2304.2.a.a 1
16.f odd 4 1 2304.2.a.p 1
24.f even 2 1 128.2.b.b 2
24.h odd 2 1 128.2.b.b 2
48.i odd 4 1 256.2.a.b 1
48.i odd 4 1 256.2.a.c 1
48.k even 4 1 256.2.a.b 1
48.k even 4 1 256.2.a.c 1
60.h even 2 1 3200.2.d.e 2
60.l odd 4 1 3200.2.f.c 2
60.l odd 4 1 3200.2.f.d 2
96.o even 8 4 1024.2.e.k 4
96.p odd 8 4 1024.2.e.k 4
120.i odd 2 1 3200.2.d.e 2
120.m even 2 1 3200.2.d.e 2
120.q odd 4 1 3200.2.f.c 2
120.q odd 4 1 3200.2.f.d 2
120.w even 4 1 3200.2.f.c 2
120.w even 4 1 3200.2.f.d 2
240.t even 4 1 6400.2.a.l 1
240.t even 4 1 6400.2.a.m 1
240.bm odd 4 1 6400.2.a.l 1
240.bm odd 4 1 6400.2.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.2.b.b 2 3.b odd 2 1
128.2.b.b 2 12.b even 2 1
128.2.b.b 2 24.f even 2 1
128.2.b.b 2 24.h odd 2 1
256.2.a.b 1 48.i odd 4 1
256.2.a.b 1 48.k even 4 1
256.2.a.c 1 48.i odd 4 1
256.2.a.c 1 48.k even 4 1
1024.2.e.k 4 96.o even 8 4
1024.2.e.k 4 96.p odd 8 4
1152.2.d.d 2 1.a even 1 1 trivial
1152.2.d.d 2 4.b odd 2 1 CM
1152.2.d.d 2 8.b even 2 1 inner
1152.2.d.d 2 8.d odd 2 1 inner
2304.2.a.a 1 16.e even 4 1
2304.2.a.a 1 16.f odd 4 1
2304.2.a.p 1 16.e even 4 1
2304.2.a.p 1 16.f odd 4 1
3200.2.d.e 2 15.d odd 2 1
3200.2.d.e 2 60.h even 2 1
3200.2.d.e 2 120.i odd 2 1
3200.2.d.e 2 120.m even 2 1
3200.2.f.c 2 15.e even 4 1
3200.2.f.c 2 60.l odd 4 1
3200.2.f.c 2 120.q odd 4 1
3200.2.f.c 2 120.w even 4 1
3200.2.f.d 2 15.e even 4 1
3200.2.f.d 2 60.l odd 4 1
3200.2.f.d 2 120.q odd 4 1
3200.2.f.d 2 120.w even 4 1
6400.2.a.l 1 240.t even 4 1
6400.2.a.l 1 240.bm odd 4 1
6400.2.a.m 1 240.t even 4 1
6400.2.a.m 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 \(S_{2}^{\mathrm{new}}(1152, [\chi])\):

\( T_{5}^{2} + 16 \)
\( T_{7} \)
\( T_{17} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 16 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( 16 + T^{2} \)
$17$ \( ( -2 + T )^{2} \)
$19$ \( T^{2} \)
$23$ \( T^{2} \)
$29$ \( 16 + T^{2} \)
$31$ \( T^{2} \)
$37$ \( 144 + T^{2} \)
$41$ \( ( 10 + T )^{2} \)
$43$ \( T^{2} \)
$47$ \( T^{2} \)
$53$ \( 16 + T^{2} \)
$59$ \( T^{2} \)
$61$ \( 144 + T^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( -6 + T )^{2} \)
$79$ \( T^{2} \)
$83$ \( T^{2} \)
$89$ \( ( -10 + T )^{2} \)
$97$ \( ( 18 + T )^{2} \)
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