Properties

Label 756.2.f.d
Level $756$
Weight $2$
Character orbit 756.f
Analytic conductor $6.037$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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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: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
Defining polynomial: \(x^{4} + 4 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{5} + ( -1 + \beta_{1} ) q^{7} +O(q^{10})\) \( q -\beta_{2} q^{5} + ( -1 + \beta_{1} ) q^{7} + \beta_{3} q^{11} -\beta_{1} q^{13} -\beta_{2} q^{17} + \beta_{1} q^{19} + 2 \beta_{3} q^{23} -2 q^{25} + \beta_{3} q^{29} -3 \beta_{1} q^{31} + ( \beta_{2} + \beta_{3} ) q^{35} + q^{37} + \beta_{2} q^{41} + 7 q^{43} -7 \beta_{2} q^{47} + ( -5 - 2 \beta_{1} ) q^{49} + 3 \beta_{1} q^{55} -5 \beta_{2} q^{59} + \beta_{1} q^{61} -\beta_{3} q^{65} -10 q^{67} + 2 \beta_{3} q^{71} + 4 \beta_{1} q^{73} + ( 6 \beta_{2} - \beta_{3} ) q^{77} + 5 q^{79} + 7 \beta_{2} q^{83} + 3 q^{85} -6 \beta_{2} q^{89} + ( 6 + \beta_{1} ) q^{91} + \beta_{3} q^{95} -\beta_{1} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{7} + O(q^{10}) \) \( 4q - 4q^{7} - 8q^{25} + 4q^{37} + 28q^{43} - 20q^{49} - 40q^{67} + 20q^{79} + 12q^{85} + 24q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 4 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} + 5 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 2 \)
\(\beta_{3}\)\(=\)\( -3 \nu^{3} - 9 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + 3 \beta_{1}\)\()/6\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 2\)
\(\nu^{3}\)\(=\)\((\)\(-5 \beta_{3} - 9 \beta_{1}\)\()/6\)

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
0.517638i
0.517638i
1.93185i
1.93185i
0 0 0 −1.73205 0 −1.00000 2.44949i 0 0 0
377.2 0 0 0 −1.73205 0 −1.00000 + 2.44949i 0 0 0
377.3 0 0 0 1.73205 0 −1.00000 2.44949i 0 0 0
377.4 0 0 0 1.73205 0 −1.00000 + 2.44949i 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
3.b odd 2 1 inner
7.b odd 2 1 inner
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.d 4
3.b odd 2 1 inner 756.2.f.d 4
4.b odd 2 1 3024.2.k.h 4
7.b odd 2 1 inner 756.2.f.d 4
9.c even 3 2 2268.2.x.j 8
9.d odd 6 2 2268.2.x.j 8
12.b even 2 1 3024.2.k.h 4
21.c even 2 1 inner 756.2.f.d 4
28.d even 2 1 3024.2.k.h 4
63.l odd 6 2 2268.2.x.j 8
63.o even 6 2 2268.2.x.j 8
84.h odd 2 1 3024.2.k.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.f.d 4 1.a even 1 1 trivial
756.2.f.d 4 3.b odd 2 1 inner
756.2.f.d 4 7.b odd 2 1 inner
756.2.f.d 4 21.c even 2 1 inner
2268.2.x.j 8 9.c even 3 2
2268.2.x.j 8 9.d odd 6 2
2268.2.x.j 8 63.l odd 6 2
2268.2.x.j 8 63.o even 6 2
3024.2.k.h 4 4.b odd 2 1
3024.2.k.h 4 12.b even 2 1
3024.2.k.h 4 28.d even 2 1
3024.2.k.h 4 84.h odd 2 1

Hecke kernels

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