Properties

Label 1512.2.t.a
Level 1512
Weight 2
Character orbit 1512.t
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.t (of order \(3\), degree \(2\), not 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: no (minimal twist has level 504)
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.

\(f(q)\) \(=\) \( q - q^{5} + ( -3 + \zeta_{6} ) q^{7} +O(q^{10})\) \( q - q^{5} + ( -3 + \zeta_{6} ) q^{7} + 3 q^{11} + ( -1 + \zeta_{6} ) q^{13} + ( 3 - 3 \zeta_{6} ) q^{17} -5 \zeta_{6} q^{19} - q^{23} -4 q^{25} + 9 \zeta_{6} q^{29} -4 \zeta_{6} q^{31} + ( 3 - \zeta_{6} ) q^{35} -5 \zeta_{6} q^{37} + ( 7 - 7 \zeta_{6} ) q^{41} -3 \zeta_{6} q^{43} + ( 8 - 8 \zeta_{6} ) q^{47} + ( 8 - 5 \zeta_{6} ) q^{49} + ( 9 - 9 \zeta_{6} ) q^{53} -3 q^{55} -4 \zeta_{6} q^{59} + ( -2 + 2 \zeta_{6} ) q^{61} + ( 1 - \zeta_{6} ) q^{65} -12 \zeta_{6} q^{67} -8 q^{71} + ( 13 - 13 \zeta_{6} ) q^{73} + ( -9 + 3 \zeta_{6} ) q^{77} + ( -8 + 8 \zeta_{6} ) q^{79} -13 \zeta_{6} q^{83} + ( -3 + 3 \zeta_{6} ) q^{85} -9 \zeta_{6} q^{89} + ( 2 - 3 \zeta_{6} ) q^{91} + 5 \zeta_{6} q^{95} + 17 \zeta_{6} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{5} - 5q^{7} + O(q^{10}) \) \( 2q - 2q^{5} - 5q^{7} + 6q^{11} - q^{13} + 3q^{17} - 5q^{19} - 2q^{23} - 8q^{25} + 9q^{29} - 4q^{31} + 5q^{35} - 5q^{37} + 7q^{41} - 3q^{43} + 8q^{47} + 11q^{49} + 9q^{53} - 6q^{55} - 4q^{59} - 2q^{61} + q^{65} - 12q^{67} - 16q^{71} + 13q^{73} - 15q^{77} - 8q^{79} - 13q^{83} - 3q^{85} - 9q^{89} + q^{91} + 5q^{95} + 17q^{97} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 −1.00000 0 −2.50000 0.866025i 0 0 0
361.1 0 0 0 −1.00000 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
63.g 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.t.a 2
3.b odd 2 1 504.2.t.a yes 2
4.b odd 2 1 3024.2.t.c 2
7.c even 3 1 1512.2.q.b 2
9.c even 3 1 1512.2.q.b 2
9.d odd 6 1 504.2.q.a 2
12.b even 2 1 1008.2.t.e 2
21.h odd 6 1 504.2.q.a 2
28.g odd 6 1 3024.2.q.d 2
36.f odd 6 1 3024.2.q.d 2
36.h even 6 1 1008.2.q.b 2
63.g even 3 1 inner 1512.2.t.a 2
63.n odd 6 1 504.2.t.a yes 2
84.n even 6 1 1008.2.q.b 2
252.o even 6 1 1008.2.t.e 2
252.bl odd 6 1 3024.2.t.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.q.a 2 9.d odd 6 1
504.2.q.a 2 21.h odd 6 1
504.2.t.a yes 2 3.b odd 2 1
504.2.t.a yes 2 63.n odd 6 1
1008.2.q.b 2 36.h even 6 1
1008.2.q.b 2 84.n even 6 1
1008.2.t.e 2 12.b even 2 1
1008.2.t.e 2 252.o even 6 1
1512.2.q.b 2 7.c even 3 1
1512.2.q.b 2 9.c even 3 1
1512.2.t.a 2 1.a even 1 1 trivial
1512.2.t.a 2 63.g even 3 1 inner
3024.2.q.d 2 28.g odd 6 1
3024.2.q.d 2 36.f odd 6 1
3024.2.t.c 2 4.b odd 2 1
3024.2.t.c 2 252.bl odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 1 \) acting on \(S_{2}^{\mathrm{new}}(1512, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 + T + 5 T^{2} )^{2} \)
$7$ \( 1 + 5 T + 7 T^{2} \)
$11$ \( ( 1 - 3 T + 11 T^{2} )^{2} \)
$13$ \( 1 + T - 12 T^{2} + 13 T^{3} + 169 T^{4} \)
$17$ \( 1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4} \)
$19$ \( 1 + 5 T + 6 T^{2} + 95 T^{3} + 361 T^{4} \)
$23$ \( ( 1 + T + 23 T^{2} )^{2} \)
$29$ \( 1 - 9 T + 52 T^{2} - 261 T^{3} + 841 T^{4} \)
$31$ \( ( 1 - 7 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \)
$37$ \( 1 + 5 T - 12 T^{2} + 185 T^{3} + 1369 T^{4} \)
$41$ \( 1 - 7 T + 8 T^{2} - 287 T^{3} + 1681 T^{4} \)
$43$ \( 1 + 3 T - 34 T^{2} + 129 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 8 T + 17 T^{2} - 376 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 9 T + 28 T^{2} - 477 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 4 T - 43 T^{2} + 236 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 12 T + 77 T^{2} + 804 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 + 8 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 13 T + 96 T^{2} - 949 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 13 T + 86 T^{2} + 1079 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 9 T - 8 T^{2} + 801 T^{3} + 7921 T^{4} \)
$97$ \( 1 - 17 T + 192 T^{2} - 1649 T^{3} + 9409 T^{4} \)
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