Properties

Label 1512.2.s.b
Level $1512$
Weight $2$
Character orbit 1512.s
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.s (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.0733807856\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
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.

\(f(q)\) \(=\) \( q -2 \zeta_{6} q^{5} + ( -1 - 2 \zeta_{6} ) q^{7} +O(q^{10})\) \( q -2 \zeta_{6} q^{5} + ( -1 - 2 \zeta_{6} ) q^{7} + 3 q^{13} + ( -4 + 4 \zeta_{6} ) q^{17} -4 \zeta_{6} q^{19} -2 \zeta_{6} q^{23} + ( 1 - \zeta_{6} ) q^{25} -4 q^{29} + ( -1 + \zeta_{6} ) q^{31} + ( -4 + 6 \zeta_{6} ) q^{35} + 3 \zeta_{6} q^{37} -8 q^{41} - q^{43} -8 \zeta_{6} q^{47} + ( -3 + 8 \zeta_{6} ) q^{49} + ( -14 + 14 \zeta_{6} ) q^{53} + ( -12 + 12 \zeta_{6} ) q^{59} -\zeta_{6} q^{61} -6 \zeta_{6} q^{65} + ( -3 + 3 \zeta_{6} ) q^{67} -6 q^{71} + ( -10 + 10 \zeta_{6} ) q^{73} -\zeta_{6} q^{79} -6 q^{83} + 8 q^{85} + 2 \zeta_{6} q^{89} + ( -3 - 6 \zeta_{6} ) q^{91} + ( -8 + 8 \zeta_{6} ) q^{95} + q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{5} - 4q^{7} + O(q^{10}) \) \( 2q - 2q^{5} - 4q^{7} + 6q^{13} - 4q^{17} - 4q^{19} - 2q^{23} + q^{25} - 8q^{29} - q^{31} - 2q^{35} + 3q^{37} - 16q^{41} - 2q^{43} - 8q^{47} + 2q^{49} - 14q^{53} - 12q^{59} - q^{61} - 6q^{65} - 3q^{67} - 12q^{71} - 10q^{73} - q^{79} - 12q^{83} + 16q^{85} + 2q^{89} - 12q^{91} - 8q^{95} + 2q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(757\) \(785\) \(1081\) \(1135\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
865.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 −1.00000 1.73205i 0 −2.00000 1.73205i 0 0 0
1297.1 0 0 0 −1.00000 + 1.73205i 0 −2.00000 + 1.73205i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1512.2.s.b 2
3.b odd 2 1 1512.2.s.f yes 2
7.c even 3 1 inner 1512.2.s.b 2
21.h odd 6 1 1512.2.s.f yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1512.2.s.b 2 1.a even 1 1 trivial
1512.2.s.b 2 7.c even 3 1 inner
1512.2.s.f yes 2 3.b odd 2 1
1512.2.s.f yes 2 21.h odd 6 1

Hecke kernels

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

\( T_{5}^{2} + 2 T_{5} + 4 \)
\( T_{11} \)
\( T_{13} - 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4} \)
$7$ \( 1 + 4 T + 7 T^{2} \)
$11$ \( 1 - 11 T^{2} + 121 T^{4} \)
$13$ \( ( 1 - 3 T + 13 T^{2} )^{2} \)
$17$ \( 1 + 4 T - T^{2} + 68 T^{3} + 289 T^{4} \)
$19$ \( 1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4} \)
$23$ \( 1 + 2 T - 19 T^{2} + 46 T^{3} + 529 T^{4} \)
$29$ \( ( 1 + 4 T + 29 T^{2} )^{2} \)
$31$ \( 1 + T - 30 T^{2} + 31 T^{3} + 961 T^{4} \)
$37$ \( 1 - 3 T - 28 T^{2} - 111 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 + 8 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 + T + 43 T^{2} )^{2} \)
$47$ \( 1 + 8 T + 17 T^{2} + 376 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 14 T + 143 T^{2} + 742 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4} \)
$61$ \( ( 1 - 13 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} ) \)
$67$ \( 1 + 3 T - 58 T^{2} + 201 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 + 6 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 7 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \)
$79$ \( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 + 6 T + 83 T^{2} )^{2} \)
$89$ \( 1 - 2 T - 85 T^{2} - 178 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 - T + 97 T^{2} )^{2} \)
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