Properties

Label 756.2.t.d
Level $756$
Weight $2$
Character orbit 756.t
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.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.03669039281\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{5} + ( -2 + \beta_{2} ) q^{7} +O(q^{10})\) \( q + \beta_{1} q^{5} + ( -2 + \beta_{2} ) q^{7} + ( 2 + 4 \beta_{2} ) q^{13} + ( -\beta_{1} - \beta_{3} ) q^{17} + ( -1 + \beta_{2} ) q^{19} + ( \beta_{1} + 2 \beta_{3} ) q^{23} + 13 \beta_{2} q^{25} + ( -2 \beta_{1} - \beta_{3} ) q^{29} + ( -2 - \beta_{2} ) q^{31} + ( -2 \beta_{1} + \beta_{3} ) q^{35} + ( 8 + 8 \beta_{2} ) q^{37} -\beta_{3} q^{41} -5 q^{43} + \beta_{1} q^{47} + ( 3 - 5 \beta_{2} ) q^{49} + ( \beta_{1} - \beta_{3} ) q^{53} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{59} + ( -3 + 3 \beta_{2} ) q^{61} + ( 2 \beta_{1} + 4 \beta_{3} ) q^{65} + 2 \beta_{2} q^{67} + ( 4 \beta_{1} + 2 \beta_{3} ) q^{71} + ( -10 - 5 \beta_{2} ) q^{73} + ( 4 + 4 \beta_{2} ) q^{79} -4 \beta_{3} q^{83} + 18 q^{85} + 3 \beta_{1} q^{89} + ( -8 - 10 \beta_{2} ) q^{91} + ( -\beta_{1} + \beta_{3} ) q^{95} + ( 9 + 18 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{7} + O(q^{10}) \) \( 4 q - 10 q^{7} - 6 q^{19} - 26 q^{25} - 6 q^{31} + 16 q^{37} - 20 q^{43} + 22 q^{49} - 18 q^{61} - 4 q^{67} - 30 q^{73} + 8 q^{79} + 72 q^{85} - 12 q^{91} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 3 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( 3 \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/3\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)\(/3\)

Character values

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

\(n\) \(29\) \(325\) \(379\)
\(\chi(n)\) \(-1\) \(1 + \beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
269.1
−0.707107 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 1.22474i
0 0 0 −2.12132 3.67423i 0 −2.50000 + 0.866025i 0 0 0
269.2 0 0 0 2.12132 + 3.67423i 0 −2.50000 + 0.866025i 0 0 0
593.1 0 0 0 −2.12132 + 3.67423i 0 −2.50000 0.866025i 0 0 0
593.2 0 0 0 2.12132 3.67423i 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
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.2.t.d 4
3.b odd 2 1 inner 756.2.t.d 4
7.c even 3 1 5292.2.f.d 4
7.d odd 6 1 inner 756.2.t.d 4
7.d odd 6 1 5292.2.f.d 4
9.c even 3 1 2268.2.w.g 4
9.c even 3 1 2268.2.bm.h 4
9.d odd 6 1 2268.2.w.g 4
9.d odd 6 1 2268.2.bm.h 4
21.g even 6 1 inner 756.2.t.d 4
21.g even 6 1 5292.2.f.d 4
21.h odd 6 1 5292.2.f.d 4
63.i even 6 1 2268.2.bm.h 4
63.k odd 6 1 2268.2.w.g 4
63.s even 6 1 2268.2.w.g 4
63.t odd 6 1 2268.2.bm.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.t.d 4 1.a even 1 1 trivial
756.2.t.d 4 3.b odd 2 1 inner
756.2.t.d 4 7.d odd 6 1 inner
756.2.t.d 4 21.g even 6 1 inner
2268.2.w.g 4 9.c even 3 1
2268.2.w.g 4 9.d odd 6 1
2268.2.w.g 4 63.k odd 6 1
2268.2.w.g 4 63.s even 6 1
2268.2.bm.h 4 9.c even 3 1
2268.2.bm.h 4 9.d odd 6 1
2268.2.bm.h 4 63.i even 6 1
2268.2.bm.h 4 63.t odd 6 1
5292.2.f.d 4 7.c even 3 1
5292.2.f.d 4 7.d odd 6 1
5292.2.f.d 4 21.g even 6 1
5292.2.f.d 4 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(756, [\chi])\):

\( T_{5}^{4} + 18 T_{5}^{2} + 324 \)
\( T_{13}^{2} + 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 324 + 18 T^{2} + T^{4} \)
$7$ \( ( 7 + 5 T + T^{2} )^{2} \)
$11$ \( T^{4} \)
$13$ \( ( 12 + T^{2} )^{2} \)
$17$ \( 324 + 18 T^{2} + T^{4} \)
$19$ \( ( 3 + 3 T + T^{2} )^{2} \)
$23$ \( 2916 - 54 T^{2} + T^{4} \)
$29$ \( ( 54 + T^{2} )^{2} \)
$31$ \( ( 3 + 3 T + T^{2} )^{2} \)
$37$ \( ( 64 - 8 T + T^{2} )^{2} \)
$41$ \( ( -18 + T^{2} )^{2} \)
$43$ \( ( 5 + T )^{4} \)
$47$ \( 324 + 18 T^{2} + T^{4} \)
$53$ \( 2916 - 54 T^{2} + T^{4} \)
$59$ \( 5184 + 72 T^{2} + T^{4} \)
$61$ \( ( 27 + 9 T + T^{2} )^{2} \)
$67$ \( ( 4 + 2 T + T^{2} )^{2} \)
$71$ \( ( 216 + T^{2} )^{2} \)
$73$ \( ( 75 + 15 T + T^{2} )^{2} \)
$79$ \( ( 16 - 4 T + T^{2} )^{2} \)
$83$ \( ( -288 + T^{2} )^{2} \)
$89$ \( 26244 + 162 T^{2} + T^{4} \)
$97$ \( ( 243 + T^{2} )^{2} \)
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