Properties

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

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 756.be (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: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + 2 \beta_{2} q^{4} + 2 \beta_{1} q^{5} + ( -1 - 2 \beta_{2} ) q^{7} + 2 \beta_{3} q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + 2 \beta_{2} q^{4} + 2 \beta_{1} q^{5} + ( -1 - 2 \beta_{2} ) q^{7} + 2 \beta_{3} q^{8} + 4 \beta_{2} q^{10} + ( \beta_{1} + \beta_{3} ) q^{11} - q^{13} + ( -\beta_{1} - 2 \beta_{3} ) q^{14} + ( -4 + 4 \beta_{2} ) q^{16} + ( \beta_{1} - \beta_{3} ) q^{17} + ( 4 + 4 \beta_{2} ) q^{19} + 4 \beta_{3} q^{20} + ( -2 + 4 \beta_{2} ) q^{22} + ( -\beta_{1} + 2 \beta_{3} ) q^{23} + 3 \beta_{2} q^{25} -\beta_{1} q^{26} + ( 4 - 6 \beta_{2} ) q^{28} -5 \beta_{3} q^{29} + ( -6 + 3 \beta_{2} ) q^{31} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{32} + 2 q^{34} + ( -2 \beta_{1} - 4 \beta_{3} ) q^{35} + ( -5 + 5 \beta_{2} ) q^{37} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{38} + ( -8 + 8 \beta_{2} ) q^{40} -5 \beta_{3} q^{41} + ( 5 - 10 \beta_{2} ) q^{43} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{44} + ( -4 + 2 \beta_{2} ) q^{46} + ( 3 \beta_{1} - 6 \beta_{3} ) q^{47} + ( -3 + 8 \beta_{2} ) q^{49} + 3 \beta_{3} q^{50} -2 \beta_{2} q^{52} + ( 7 \beta_{1} - 7 \beta_{3} ) q^{53} + ( -4 + 8 \beta_{2} ) q^{55} + ( 4 \beta_{1} - 6 \beta_{3} ) q^{56} + ( 10 - 10 \beta_{2} ) q^{58} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{59} + ( -5 + 5 \beta_{2} ) q^{61} + ( -6 \beta_{1} + 3 \beta_{3} ) q^{62} -8 q^{64} -2 \beta_{1} q^{65} + ( 6 - 3 \beta_{2} ) q^{67} + 2 \beta_{1} q^{68} + ( 8 - 12 \beta_{2} ) q^{70} + ( -12 \beta_{1} + 6 \beta_{3} ) q^{71} + 4 \beta_{2} q^{73} + ( -5 \beta_{1} + 5 \beta_{3} ) q^{74} + ( -8 + 16 \beta_{2} ) q^{76} + ( \beta_{1} - 5 \beta_{3} ) q^{77} + ( -9 - 9 \beta_{2} ) q^{79} + ( -8 \beta_{1} + 8 \beta_{3} ) q^{80} + ( 10 - 10 \beta_{2} ) q^{82} + ( 2 \beta_{1} - \beta_{3} ) q^{83} + 4 q^{85} + ( 5 \beta_{1} - 10 \beta_{3} ) q^{86} + ( -8 + 4 \beta_{2} ) q^{88} -10 \beta_{1} q^{89} + ( 1 + 2 \beta_{2} ) q^{91} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{92} + ( 12 - 6 \beta_{2} ) q^{94} + ( 8 \beta_{1} + 8 \beta_{3} ) q^{95} + 5 q^{97} + ( -3 \beta_{1} + 8 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{4} - 8q^{7} + O(q^{10}) \) \( 4q + 4q^{4} - 8q^{7} + 8q^{10} - 4q^{13} - 8q^{16} + 24q^{19} + 6q^{25} + 4q^{28} - 18q^{31} + 8q^{34} - 10q^{37} - 16q^{40} - 12q^{46} + 4q^{49} - 4q^{52} + 20q^{58} - 10q^{61} - 32q^{64} + 18q^{67} + 8q^{70} + 8q^{73} - 54q^{79} + 20q^{82} + 16q^{85} - 24q^{88} + 8q^{91} + 36q^{94} + 20q^{97} + O(q^{100}) \)

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
−1.22474 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i
1.22474 0.707107i
−1.22474 0.707107i 0 1.00000 + 1.73205i −2.44949 1.41421i 0 −2.00000 1.73205i 2.82843i 0 2.00000 + 3.46410i
107.2 1.22474 + 0.707107i 0 1.00000 + 1.73205i 2.44949 + 1.41421i 0 −2.00000 1.73205i 2.82843i 0 2.00000 + 3.46410i
431.1 −1.22474 + 0.707107i 0 1.00000 1.73205i −2.44949 + 1.41421i 0 −2.00000 + 1.73205i 2.82843i 0 2.00000 3.46410i
431.2 1.22474 0.707107i 0 1.00000 1.73205i 2.44949 1.41421i 0 −2.00000 + 1.73205i 2.82843i 0 2.00000 3.46410i
\(n\): 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
28.g odd 6 1 inner
84.n 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.be.a 4
3.b odd 2 1 inner 756.2.be.a 4
4.b odd 2 1 756.2.be.b yes 4
7.c even 3 1 756.2.be.b yes 4
12.b even 2 1 756.2.be.b yes 4
21.h odd 6 1 756.2.be.b yes 4
28.g odd 6 1 inner 756.2.be.a 4
84.n even 6 1 inner 756.2.be.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.be.a 4 1.a even 1 1 trivial
756.2.be.a 4 3.b odd 2 1 inner
756.2.be.a 4 28.g odd 6 1 inner
756.2.be.a 4 84.n even 6 1 inner
756.2.be.b yes 4 4.b odd 2 1
756.2.be.b yes 4 7.c even 3 1
756.2.be.b yes 4 12.b even 2 1
756.2.be.b yes 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 \(S_{2}^{\mathrm{new}}(756, [\chi])\):

\( T_{5}^{4} - 8 T_{5}^{2} + 64 \)
\( T_{19}^{2} - 12 T_{19} + 48 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T^{2} + 4 T^{4} \)
$3$ 1
$5$ \( 1 + 2 T^{2} - 21 T^{4} + 50 T^{6} + 625 T^{8} \)
$7$ \( ( 1 + 4 T + 7 T^{2} )^{2} \)
$11$ \( 1 - 16 T^{2} + 135 T^{4} - 1936 T^{6} + 14641 T^{8} \)
$13$ \( ( 1 + T + 13 T^{2} )^{4} \)
$17$ \( 1 + 32 T^{2} + 735 T^{4} + 9248 T^{6} + 83521 T^{8} \)
$19$ \( ( 1 - 12 T + 67 T^{2} - 228 T^{3} + 361 T^{4} )^{2} \)
$23$ \( 1 - 40 T^{2} + 1071 T^{4} - 21160 T^{6} + 279841 T^{8} \)
$29$ \( ( 1 - 8 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 + 9 T + 58 T^{2} + 279 T^{3} + 961 T^{4} )^{2} \)
$37$ \( ( 1 + 5 T - 12 T^{2} + 185 T^{3} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 - 32 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - 11 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( 1 - 40 T^{2} - 609 T^{4} - 88360 T^{6} + 4879681 T^{8} \)
$53$ \( 1 + 8 T^{2} - 2745 T^{4} + 22472 T^{6} + 7890481 T^{8} \)
$59$ \( 1 - 64 T^{2} + 615 T^{4} - 222784 T^{6} + 12117361 T^{8} \)
$61$ \( ( 1 + 5 T - 36 T^{2} + 305 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 - 9 T + 94 T^{2} - 603 T^{3} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 - 74 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 4 T - 57 T^{2} - 292 T^{3} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 + 27 T + 322 T^{2} + 2133 T^{3} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 + 160 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( 1 - 22 T^{2} - 7437 T^{4} - 174262 T^{6} + 62742241 T^{8} \)
$97$ \( ( 1 - 5 T + 97 T^{2} )^{4} \)
show more
show less