Properties

Label 1512.2.c.a
Level 1512
Weight 2
Character orbit 1512.c
Analytic conductor 12.073
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1512.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.0733807856\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( -1 - i ) q^{2} + 2 i q^{4} + 2 i q^{5} - q^{7} + ( 2 - 2 i ) q^{8} +O(q^{10})\) \( q + ( -1 - i ) q^{2} + 2 i q^{4} + 2 i q^{5} - q^{7} + ( 2 - 2 i ) q^{8} + ( 2 - 2 i ) q^{10} + 5 i q^{13} + ( 1 + i ) q^{14} -4 q^{16} + q^{17} + 4 i q^{19} -4 q^{20} -5 q^{23} + q^{25} + ( 5 - 5 i ) q^{26} -2 i q^{28} -9 i q^{29} -7 q^{31} + ( 4 + 4 i ) q^{32} + ( -1 - i ) q^{34} -2 i q^{35} + 2 i q^{37} + ( 4 - 4 i ) q^{38} + ( 4 + 4 i ) q^{40} + 2 q^{41} + 5 i q^{43} + ( 5 + 5 i ) q^{46} + q^{49} + ( -1 - i ) q^{50} -10 q^{52} -9 i q^{53} + ( -2 + 2 i ) q^{56} + ( -9 + 9 i ) q^{58} + i q^{59} + 6 i q^{61} + ( 7 + 7 i ) q^{62} -8 i q^{64} -10 q^{65} -9 i q^{67} + 2 i q^{68} + ( -2 + 2 i ) q^{70} -15 q^{71} + ( 2 - 2 i ) q^{74} -8 q^{76} -14 q^{79} -8 i q^{80} + ( -2 - 2 i ) q^{82} + 4 i q^{83} + 2 i q^{85} + ( 5 - 5 i ) q^{86} -13 q^{89} -5 i q^{91} -10 i q^{92} -8 q^{95} -14 q^{97} + ( -1 - i ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 2q^{7} + 4q^{8} + O(q^{10}) \) \( 2q - 2q^{2} - 2q^{7} + 4q^{8} + 4q^{10} + 2q^{14} - 8q^{16} + 2q^{17} - 8q^{20} - 10q^{23} + 2q^{25} + 10q^{26} - 14q^{31} + 8q^{32} - 2q^{34} + 8q^{38} + 8q^{40} + 4q^{41} + 10q^{46} + 2q^{49} - 2q^{50} - 20q^{52} - 4q^{56} - 18q^{58} + 14q^{62} - 20q^{65} - 4q^{70} - 30q^{71} + 4q^{74} - 16q^{76} - 28q^{79} - 4q^{82} + 10q^{86} - 26q^{89} - 16q^{95} - 28q^{97} - 2q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1081\) \(1135\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
757.1
1.00000i
1.00000i
−1.00000 1.00000i 0 2.00000i 2.00000i 0 −1.00000 2.00000 2.00000i 0 2.00000 2.00000i
757.2 −1.00000 + 1.00000i 0 2.00000i 2.00000i 0 −1.00000 2.00000 + 2.00000i 0 2.00000 + 2.00000i
\(n\): 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 1512.2.c.a 2
3.b odd 2 1 1512.2.c.b yes 2
4.b odd 2 1 6048.2.c.b 2
8.b even 2 1 inner 1512.2.c.a 2
8.d odd 2 1 6048.2.c.b 2
12.b even 2 1 6048.2.c.a 2
24.f even 2 1 6048.2.c.a 2
24.h odd 2 1 1512.2.c.b yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1512.2.c.a 2 1.a even 1 1 trivial
1512.2.c.a 2 8.b even 2 1 inner
1512.2.c.b yes 2 3.b odd 2 1
1512.2.c.b yes 2 24.h odd 2 1
6048.2.c.a 2 12.b even 2 1
6048.2.c.a 2 24.f even 2 1
6048.2.c.b 2 4.b odd 2 1
6048.2.c.b 2 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1512, [\chi])\):

\( T_{5}^{2} + 4 \)
\( T_{17} - 1 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 2 T^{2} \)
$3$ \( \)
$5$ \( ( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} ) \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( ( 1 - 11 T^{2} )^{2} \)
$13$ \( 1 - T^{2} + 169 T^{4} \)
$17$ \( ( 1 - T + 17 T^{2} )^{2} \)
$19$ \( 1 - 22 T^{2} + 361 T^{4} \)
$23$ \( ( 1 + 5 T + 23 T^{2} )^{2} \)
$29$ \( 1 + 23 T^{2} + 841 T^{4} \)
$31$ \( ( 1 + 7 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \)
$41$ \( ( 1 - 2 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 61 T^{2} + 1849 T^{4} \)
$47$ \( ( 1 + 47 T^{2} )^{2} \)
$53$ \( 1 - 25 T^{2} + 2809 T^{4} \)
$59$ \( 1 - 117 T^{2} + 3481 T^{4} \)
$61$ \( 1 - 86 T^{2} + 3721 T^{4} \)
$67$ \( 1 - 53 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 + 15 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 + 73 T^{2} )^{2} \)
$79$ \( ( 1 + 14 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 150 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 + 13 T + 89 T^{2} )^{2} \)
$97$ \( ( 1 + 14 T + 97 T^{2} )^{2} \)
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