Properties

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

Hecke kernels

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

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