Properties

Label 756.2.f.a
Level $756$
Weight $2$
Character orbit 756.f
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.f (of order \(2\), degree \(1\), 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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q -3 q^{5} + ( 3 - 2 \zeta_{6} ) q^{7} +O(q^{10})\) \( q -3 q^{5} + ( 3 - 2 \zeta_{6} ) q^{7} + ( -3 + 6 \zeta_{6} ) q^{11} + ( 2 - 4 \zeta_{6} ) q^{13} + 6 q^{17} + ( 1 - 2 \zeta_{6} ) q^{19} + ( 3 - 6 \zeta_{6} ) q^{23} + 4 q^{25} + ( 6 - 12 \zeta_{6} ) q^{29} + ( 3 - 6 \zeta_{6} ) q^{31} + ( -9 + 6 \zeta_{6} ) q^{35} + q^{37} + 3 q^{41} + 10 q^{43} + 6 q^{47} + ( 5 - 8 \zeta_{6} ) q^{49} + ( 9 - 18 \zeta_{6} ) q^{55} -6 q^{59} + ( -8 + 16 \zeta_{6} ) q^{61} + ( -6 + 12 \zeta_{6} ) q^{65} + 2 q^{67} + ( 3 - 6 \zeta_{6} ) q^{71} + ( -2 + 4 \zeta_{6} ) q^{73} + ( 3 + 12 \zeta_{6} ) q^{77} + 14 q^{79} -6 q^{83} -18 q^{85} -9 q^{89} + ( -2 - 8 \zeta_{6} ) q^{91} + ( -3 + 6 \zeta_{6} ) q^{95} + ( -4 + 8 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 6q^{5} + 4q^{7} + O(q^{10}) \) \( 2q - 6q^{5} + 4q^{7} + 12q^{17} + 8q^{25} - 12q^{35} + 2q^{37} + 6q^{41} + 20q^{43} + 12q^{47} + 2q^{49} - 12q^{59} + 4q^{67} + 18q^{77} + 28q^{79} - 12q^{83} - 36q^{85} - 18q^{89} - 12q^{91} + O(q^{100}) \)

Character values

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

\(n\) \(29\) \(325\) \(379\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
377.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 −3.00000 0 2.00000 1.73205i 0 0 0
377.2 0 0 0 −3.00000 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
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.2.f.a 2
3.b odd 2 1 756.2.f.c yes 2
4.b odd 2 1 3024.2.k.a 2
7.b odd 2 1 756.2.f.c yes 2
9.c even 3 1 2268.2.x.g 2
9.c even 3 1 2268.2.x.h 2
9.d odd 6 1 2268.2.x.a 2
9.d odd 6 1 2268.2.x.b 2
12.b even 2 1 3024.2.k.d 2
21.c even 2 1 inner 756.2.f.a 2
28.d even 2 1 3024.2.k.d 2
63.l odd 6 1 2268.2.x.a 2
63.l odd 6 1 2268.2.x.b 2
63.o even 6 1 2268.2.x.g 2
63.o even 6 1 2268.2.x.h 2
84.h odd 2 1 3024.2.k.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.f.a 2 1.a even 1 1 trivial
756.2.f.a 2 21.c even 2 1 inner
756.2.f.c yes 2 3.b odd 2 1
756.2.f.c yes 2 7.b odd 2 1
2268.2.x.a 2 9.d odd 6 1
2268.2.x.a 2 63.l odd 6 1
2268.2.x.b 2 9.d odd 6 1
2268.2.x.b 2 63.l odd 6 1
2268.2.x.g 2 9.c even 3 1
2268.2.x.g 2 63.o even 6 1
2268.2.x.h 2 9.c even 3 1
2268.2.x.h 2 63.o even 6 1
3024.2.k.a 2 4.b odd 2 1
3024.2.k.a 2 84.h odd 2 1
3024.2.k.d 2 12.b even 2 1
3024.2.k.d 2 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 3 \) acting on \(S_{2}^{\mathrm{new}}(756, [\chi])\).