Properties

Label 1152.4.d.g
Level $1152$
Weight $4$
Character orbit 1152.d
Analytic conductor $67.970$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1152.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(67.9702003266\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q + 2 \beta q^{5} + 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta q^{5} + 12 q^{7} + 3 \beta q^{11} - 5 \beta q^{13} - 62 q^{17} - 27 \beta q^{19} + 72 q^{23} + 61 q^{25} - 32 \beta q^{29} - 204 q^{31} + 24 \beta q^{35} - 57 \beta q^{37} + 22 q^{41} - 51 \beta q^{43} + 600 q^{47} - 199 q^{49} - 64 \beta q^{53} - 96 q^{55} + 207 \beta q^{59} + 21 \beta q^{61} + 160 q^{65} - 87 \beta q^{67} - 456 q^{71} + 822 q^{73} + 36 \beta q^{77} - 1356 q^{79} + 27 \beta q^{83} - 124 \beta q^{85} + 938 q^{89} - 60 \beta q^{91} + 864 q^{95} + 1278 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 24 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 24 q^{7} - 124 q^{17} + 144 q^{23} + 122 q^{25} - 408 q^{31} + 44 q^{41} + 1200 q^{47} - 398 q^{49} - 192 q^{55} + 320 q^{65} - 912 q^{71} + 1644 q^{73} - 2712 q^{79} + 1876 q^{89} + 1728 q^{95} + 2556 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 8.00000i 0 12.0000 0 0 0
577.2 0 0 0 8.00000i 0 12.0000 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 1152.4.d.g 2
3.b odd 2 1 384.4.d.b yes 2
4.b odd 2 1 1152.4.d.b 2
8.b even 2 1 inner 1152.4.d.g 2
8.d odd 2 1 1152.4.d.b 2
12.b even 2 1 384.4.d.a 2
16.e even 4 1 2304.4.a.e 1
16.e even 4 1 2304.4.a.k 1
16.f odd 4 1 2304.4.a.f 1
16.f odd 4 1 2304.4.a.l 1
24.f even 2 1 384.4.d.a 2
24.h odd 2 1 384.4.d.b yes 2
48.i odd 4 1 768.4.a.b 1
48.i odd 4 1 768.4.a.c 1
48.k even 4 1 768.4.a.a 1
48.k even 4 1 768.4.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.d.a 2 12.b even 2 1
384.4.d.a 2 24.f even 2 1
384.4.d.b yes 2 3.b odd 2 1
384.4.d.b yes 2 24.h odd 2 1
768.4.a.a 1 48.k even 4 1
768.4.a.b 1 48.i odd 4 1
768.4.a.c 1 48.i odd 4 1
768.4.a.d 1 48.k even 4 1
1152.4.d.b 2 4.b odd 2 1
1152.4.d.b 2 8.d odd 2 1
1152.4.d.g 2 1.a even 1 1 trivial
1152.4.d.g 2 8.b even 2 1 inner
2304.4.a.e 1 16.e even 4 1
2304.4.a.f 1 16.f odd 4 1
2304.4.a.k 1 16.e even 4 1
2304.4.a.l 1 16.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1152, [\chi])\):

\( T_{5}^{2} + 64 \) Copy content Toggle raw display
\( T_{7} - 12 \) Copy content Toggle raw display
\( T_{17} + 62 \) 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
$5$ \( T^{2} + 64 \) Copy content Toggle raw display
$7$ \( (T - 12)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 144 \) Copy content Toggle raw display
$13$ \( T^{2} + 400 \) Copy content Toggle raw display
$17$ \( (T + 62)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 11664 \) Copy content Toggle raw display
$23$ \( (T - 72)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 16384 \) Copy content Toggle raw display
$31$ \( (T + 204)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 51984 \) Copy content Toggle raw display
$41$ \( (T - 22)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 41616 \) Copy content Toggle raw display
$47$ \( (T - 600)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 65536 \) Copy content Toggle raw display
$59$ \( T^{2} + 685584 \) Copy content Toggle raw display
$61$ \( T^{2} + 7056 \) Copy content Toggle raw display
$67$ \( T^{2} + 121104 \) Copy content Toggle raw display
$71$ \( (T + 456)^{2} \) Copy content Toggle raw display
$73$ \( (T - 822)^{2} \) Copy content Toggle raw display
$79$ \( (T + 1356)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 11664 \) Copy content Toggle raw display
$89$ \( (T - 938)^{2} \) Copy content Toggle raw display
$97$ \( (T - 1278)^{2} \) Copy content Toggle raw display
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