Properties

Label 756.2.l.a
Level $756$
Weight $2$
Character orbit 756.l
Analytic conductor $6.037$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 756.l (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.03669039281\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 252)
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.

\(f(q)\) \(=\) \( q -2 q^{5} + ( 3 - 2 \zeta_{6} ) q^{7} +O(q^{10})\) \( q -2 q^{5} + ( 3 - 2 \zeta_{6} ) q^{7} -4 q^{11} + ( -3 + 3 \zeta_{6} ) q^{13} + ( 7 - 7 \zeta_{6} ) q^{17} -5 \zeta_{6} q^{19} -4 q^{23} - q^{25} -\zeta_{6} q^{29} + 3 \zeta_{6} q^{31} + ( -6 + 4 \zeta_{6} ) q^{35} -11 \zeta_{6} q^{37} + ( -9 + 9 \zeta_{6} ) q^{41} -5 \zeta_{6} q^{43} + ( 3 - 3 \zeta_{6} ) q^{47} + ( 5 - 8 \zeta_{6} ) q^{49} + ( 3 - 3 \zeta_{6} ) q^{53} + 8 q^{55} -7 \zeta_{6} q^{59} + ( -3 + 3 \zeta_{6} ) q^{61} + ( 6 - 6 \zeta_{6} ) q^{65} -13 \zeta_{6} q^{67} + 8 q^{71} + ( -7 + 7 \zeta_{6} ) q^{73} + ( -12 + 8 \zeta_{6} ) q^{77} + ( 9 - 9 \zeta_{6} ) q^{79} + \zeta_{6} q^{83} + ( -14 + 14 \zeta_{6} ) q^{85} + 15 \zeta_{6} q^{89} + ( -3 + 9 \zeta_{6} ) q^{91} + 10 \zeta_{6} q^{95} + 17 \zeta_{6} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{5} + 4q^{7} + O(q^{10}) \) \( 2q - 4q^{5} + 4q^{7} - 8q^{11} - 3q^{13} + 7q^{17} - 5q^{19} - 8q^{23} - 2q^{25} - q^{29} + 3q^{31} - 8q^{35} - 11q^{37} - 9q^{41} - 5q^{43} + 3q^{47} + 2q^{49} + 3q^{53} + 16q^{55} - 7q^{59} - 3q^{61} + 6q^{65} - 13q^{67} + 16q^{71} - 7q^{73} - 16q^{77} + 9q^{79} + q^{83} - 14q^{85} + 15q^{89} + 3q^{91} + 10q^{95} + 17q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/756\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(325\) \(379\)
\(\chi(n)\) \(-1 + \zeta_{6}\) \(-\zeta_{6}\) \(1\)

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 −2.00000 0 2.00000 + 1.73205i 0 0 0
361.1 0 0 0 −2.00000 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
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 756.2.l.a 2
3.b odd 2 1 252.2.l.a yes 2
4.b odd 2 1 3024.2.t.b 2
7.b odd 2 1 5292.2.l.b 2
7.c even 3 1 756.2.i.a 2
7.c even 3 1 5292.2.j.c 2
7.d odd 6 1 5292.2.i.b 2
7.d odd 6 1 5292.2.j.b 2
9.c even 3 1 756.2.i.a 2
9.c even 3 1 2268.2.k.b 2
9.d odd 6 1 252.2.i.a 2
9.d odd 6 1 2268.2.k.a 2
12.b even 2 1 1008.2.t.b 2
21.c even 2 1 1764.2.l.b 2
21.g even 6 1 1764.2.i.b 2
21.g even 6 1 1764.2.j.a 2
21.h odd 6 1 252.2.i.a 2
21.h odd 6 1 1764.2.j.c 2
28.g odd 6 1 3024.2.q.e 2
36.f odd 6 1 3024.2.q.e 2
36.h even 6 1 1008.2.q.f 2
63.g even 3 1 inner 756.2.l.a 2
63.h even 3 1 2268.2.k.b 2
63.h even 3 1 5292.2.j.c 2
63.i even 6 1 1764.2.j.a 2
63.j odd 6 1 1764.2.j.c 2
63.j odd 6 1 2268.2.k.a 2
63.k odd 6 1 5292.2.l.b 2
63.l odd 6 1 5292.2.i.b 2
63.n odd 6 1 252.2.l.a yes 2
63.o even 6 1 1764.2.i.b 2
63.s even 6 1 1764.2.l.b 2
63.t odd 6 1 5292.2.j.b 2
84.n even 6 1 1008.2.q.f 2
252.o even 6 1 1008.2.t.b 2
252.bl odd 6 1 3024.2.t.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.i.a 2 9.d odd 6 1
252.2.i.a 2 21.h odd 6 1
252.2.l.a yes 2 3.b odd 2 1
252.2.l.a yes 2 63.n odd 6 1
756.2.i.a 2 7.c even 3 1
756.2.i.a 2 9.c even 3 1
756.2.l.a 2 1.a even 1 1 trivial
756.2.l.a 2 63.g even 3 1 inner
1008.2.q.f 2 36.h even 6 1
1008.2.q.f 2 84.n even 6 1
1008.2.t.b 2 12.b even 2 1
1008.2.t.b 2 252.o even 6 1
1764.2.i.b 2 21.g even 6 1
1764.2.i.b 2 63.o even 6 1
1764.2.j.a 2 21.g even 6 1
1764.2.j.a 2 63.i even 6 1
1764.2.j.c 2 21.h odd 6 1
1764.2.j.c 2 63.j odd 6 1
1764.2.l.b 2 21.c even 2 1
1764.2.l.b 2 63.s even 6 1
2268.2.k.a 2 9.d odd 6 1
2268.2.k.a 2 63.j odd 6 1
2268.2.k.b 2 9.c even 3 1
2268.2.k.b 2 63.h even 3 1
3024.2.q.e 2 28.g odd 6 1
3024.2.q.e 2 36.f odd 6 1
3024.2.t.b 2 4.b odd 2 1
3024.2.t.b 2 252.bl odd 6 1
5292.2.i.b 2 7.d odd 6 1
5292.2.i.b 2 63.l odd 6 1
5292.2.j.b 2 7.d odd 6 1
5292.2.j.b 2 63.t odd 6 1
5292.2.j.c 2 7.c even 3 1
5292.2.j.c 2 63.h even 3 1
5292.2.l.b 2 7.b odd 2 1
5292.2.l.b 2 63.k odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 2 \) acting on \(S_{2}^{\mathrm{new}}(756, [\chi])\).