Properties

Label 1512.2.s.e
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: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q -2 \zeta_{6} q^{5} + ( 2 + \zeta_{6} ) q^{7} +O(q^{10})\) \( q -2 \zeta_{6} q^{5} + ( 2 + \zeta_{6} ) q^{7} + ( -3 + 3 \zeta_{6} ) q^{11} + 6 q^{13} + ( -4 + 4 \zeta_{6} ) q^{17} -4 \zeta_{6} q^{19} + 4 \zeta_{6} q^{23} + ( 1 - \zeta_{6} ) q^{25} + 5 q^{29} + ( -7 + 7 \zeta_{6} ) q^{31} + ( 2 - 6 \zeta_{6} ) q^{35} -2 q^{41} + 8 q^{43} -2 \zeta_{6} q^{47} + ( 3 + 5 \zeta_{6} ) q^{49} + ( 10 - 10 \zeta_{6} ) q^{53} + 6 q^{55} + ( -9 + 9 \zeta_{6} ) q^{59} + 8 \zeta_{6} q^{61} -12 \zeta_{6} q^{65} + ( 6 - 6 \zeta_{6} ) q^{67} + 12 q^{71} + ( 11 - 11 \zeta_{6} ) q^{73} + ( -9 + 6 \zeta_{6} ) q^{77} -\zeta_{6} q^{79} + 15 q^{83} + 8 q^{85} -10 \zeta_{6} q^{89} + ( 12 + 6 \zeta_{6} ) q^{91} + ( -8 + 8 \zeta_{6} ) q^{95} -5 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{5} + 5q^{7} + O(q^{10}) \) \( 2q - 2q^{5} + 5q^{7} - 3q^{11} + 12q^{13} - 4q^{17} - 4q^{19} + 4q^{23} + q^{25} + 10q^{29} - 7q^{31} - 2q^{35} - 4q^{41} + 16q^{43} - 2q^{47} + 11q^{49} + 10q^{53} + 12q^{55} - 9q^{59} + 8q^{61} - 12q^{65} + 6q^{67} + 24q^{71} + 11q^{73} - 12q^{77} - q^{79} + 30q^{83} + 16q^{85} - 10q^{89} + 30q^{91} - 8q^{95} - 10q^{97} + O(q^{100}) \)

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 2.50000 + 0.866025i 0 0 0
1297.1 0 0 0 −1.00000 + 1.73205i 0 2.50000 0.866025i 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.e 2
3.b odd 2 1 1512.2.s.i yes 2
7.c even 3 1 inner 1512.2.s.e 2
21.h odd 6 1 1512.2.s.i yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1512.2.s.e 2 1.a even 1 1 trivial
1512.2.s.e 2 7.c even 3 1 inner
1512.2.s.i yes 2 3.b odd 2 1
1512.2.s.i 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 \(S_{2}^{\mathrm{new}}(1512, [\chi])\):

\( T_{5}^{2} + 2 T_{5} + 4 \)
\( T_{11}^{2} + 3 T_{11} + 9 \)
\( T_{13} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4} \)
$7$ \( 1 - 5 T + 7 T^{2} \)
$11$ \( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - 6 T + 13 T^{2} )^{2} \)
$17$ \( 1 + 4 T - T^{2} + 68 T^{3} + 289 T^{4} \)
$19$ \( 1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4} \)
$23$ \( 1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4} \)
$29$ \( ( 1 - 5 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 - 4 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \)
$37$ \( 1 - 37 T^{2} + 1369 T^{4} \)
$41$ \( ( 1 + 2 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 - 8 T + 43 T^{2} )^{2} \)
$47$ \( 1 + 2 T - 43 T^{2} + 94 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 10 T + 47 T^{2} - 530 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 9 T + 22 T^{2} + 531 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 8 T + 3 T^{2} - 488 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 6 T - 31 T^{2} - 402 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 - 12 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 11 T + 48 T^{2} - 803 T^{3} + 5329 T^{4} \)
$79$ \( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 - 15 T + 83 T^{2} )^{2} \)
$89$ \( 1 + 10 T + 11 T^{2} + 890 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 + 5 T + 97 T^{2} )^{2} \)
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