Properties

Label 1152.2.i.b
Level $1152$
Weight $2$
Character orbit 1152.i
Analytic conductor $9.199$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(9.19876631285\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 - \zeta_{6} ) q^{3} + 2 \zeta_{6} q^{5} + ( -2 + 2 \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} +O(q^{10})\) \( q + ( -1 - \zeta_{6} ) q^{3} + 2 \zeta_{6} q^{5} + ( -2 + 2 \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} + ( -5 + 5 \zeta_{6} ) q^{11} -4 \zeta_{6} q^{13} + ( 2 - 4 \zeta_{6} ) q^{15} + q^{17} -5 q^{19} + ( 4 - 2 \zeta_{6} ) q^{21} -4 \zeta_{6} q^{23} + ( 1 - \zeta_{6} ) q^{25} + ( 3 - 6 \zeta_{6} ) q^{27} + ( 6 - 6 \zeta_{6} ) q^{29} + ( 10 - 5 \zeta_{6} ) q^{33} -4 q^{35} -10 q^{37} + ( -4 + 8 \zeta_{6} ) q^{39} + 3 \zeta_{6} q^{41} + ( 9 - 9 \zeta_{6} ) q^{43} + ( -6 + 6 \zeta_{6} ) q^{45} + ( 8 - 8 \zeta_{6} ) q^{47} + 3 \zeta_{6} q^{49} + ( -1 - \zeta_{6} ) q^{51} -12 q^{53} -10 q^{55} + ( 5 + 5 \zeta_{6} ) q^{57} -7 \zeta_{6} q^{59} + ( -4 + 4 \zeta_{6} ) q^{61} -6 q^{63} + ( 8 - 8 \zeta_{6} ) q^{65} + 7 \zeta_{6} q^{67} + ( -4 + 8 \zeta_{6} ) q^{69} -6 q^{71} -13 q^{73} + ( -2 + \zeta_{6} ) q^{75} -10 \zeta_{6} q^{77} + ( 2 - 2 \zeta_{6} ) q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} + ( -12 + 12 \zeta_{6} ) q^{83} + 2 \zeta_{6} q^{85} + ( -12 + 6 \zeta_{6} ) q^{87} + 10 q^{89} + 8 q^{91} -10 \zeta_{6} q^{95} + ( -13 + 13 \zeta_{6} ) q^{97} -15 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{3} + 2q^{5} - 2q^{7} + 3q^{9} + O(q^{10}) \) \( 2q - 3q^{3} + 2q^{5} - 2q^{7} + 3q^{9} - 5q^{11} - 4q^{13} + 2q^{17} - 10q^{19} + 6q^{21} - 4q^{23} + q^{25} + 6q^{29} + 15q^{33} - 8q^{35} - 20q^{37} + 3q^{41} + 9q^{43} - 6q^{45} + 8q^{47} + 3q^{49} - 3q^{51} - 24q^{53} - 20q^{55} + 15q^{57} - 7q^{59} - 4q^{61} - 12q^{63} + 8q^{65} + 7q^{67} - 12q^{71} - 26q^{73} - 3q^{75} - 10q^{77} + 2q^{79} - 9q^{81} - 12q^{83} + 2q^{85} - 18q^{87} + 20q^{89} + 16q^{91} - 10q^{95} - 13q^{97} - 30q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
385.1
0.500000 + 0.866025i
0.500000 0.866025i
0 −1.50000 0.866025i 0 1.00000 + 1.73205i 0 −1.00000 + 1.73205i 0 1.50000 + 2.59808i 0
769.1 0 −1.50000 + 0.866025i 0 1.00000 1.73205i 0 −1.00000 1.73205i 0 1.50000 2.59808i 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
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1152.2.i.b yes 2
3.b odd 2 1 3456.2.i.a 2
4.b odd 2 1 1152.2.i.d yes 2
8.b even 2 1 1152.2.i.c yes 2
8.d odd 2 1 1152.2.i.a 2
9.c even 3 1 inner 1152.2.i.b yes 2
9.d odd 6 1 3456.2.i.a 2
12.b even 2 1 3456.2.i.b 2
24.f even 2 1 3456.2.i.d 2
24.h odd 2 1 3456.2.i.c 2
36.f odd 6 1 1152.2.i.d yes 2
36.h even 6 1 3456.2.i.b 2
72.j odd 6 1 3456.2.i.c 2
72.l even 6 1 3456.2.i.d 2
72.n even 6 1 1152.2.i.c yes 2
72.p odd 6 1 1152.2.i.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.2.i.a 2 8.d odd 2 1
1152.2.i.a 2 72.p odd 6 1
1152.2.i.b yes 2 1.a even 1 1 trivial
1152.2.i.b yes 2 9.c even 3 1 inner
1152.2.i.c yes 2 8.b even 2 1
1152.2.i.c yes 2 72.n even 6 1
1152.2.i.d yes 2 4.b odd 2 1
1152.2.i.d yes 2 36.f odd 6 1
3456.2.i.a 2 3.b odd 2 1
3456.2.i.a 2 9.d odd 6 1
3456.2.i.b 2 12.b even 2 1
3456.2.i.b 2 36.h even 6 1
3456.2.i.c 2 24.h odd 2 1
3456.2.i.c 2 72.j odd 6 1
3456.2.i.d 2 24.f even 2 1
3456.2.i.d 2 72.l even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1152, [\chi])\):

\( T_{5}^{2} - 2 T_{5} + 4 \)
\( T_{7}^{2} + 2 T_{7} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 3 + 3 T + T^{2} \)
$5$ \( 4 - 2 T + T^{2} \)
$7$ \( 4 + 2 T + T^{2} \)
$11$ \( 25 + 5 T + T^{2} \)
$13$ \( 16 + 4 T + T^{2} \)
$17$ \( ( -1 + T )^{2} \)
$19$ \( ( 5 + T )^{2} \)
$23$ \( 16 + 4 T + T^{2} \)
$29$ \( 36 - 6 T + T^{2} \)
$31$ \( T^{2} \)
$37$ \( ( 10 + T )^{2} \)
$41$ \( 9 - 3 T + T^{2} \)
$43$ \( 81 - 9 T + T^{2} \)
$47$ \( 64 - 8 T + T^{2} \)
$53$ \( ( 12 + T )^{2} \)
$59$ \( 49 + 7 T + T^{2} \)
$61$ \( 16 + 4 T + T^{2} \)
$67$ \( 49 - 7 T + T^{2} \)
$71$ \( ( 6 + T )^{2} \)
$73$ \( ( 13 + T )^{2} \)
$79$ \( 4 - 2 T + T^{2} \)
$83$ \( 144 + 12 T + T^{2} \)
$89$ \( ( -10 + T )^{2} \)
$97$ \( 169 + 13 T + T^{2} \)
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