Properties

Label 756.2.b.b
Level 756
Weight 2
Character orbit 756.b
Analytic conductor 6.037
Analytic rank 0
Dimension 4
CM discriminant -84
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.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.03669039281\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{7})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} -2 q^{4} + ( \beta_{1} - \beta_{2} ) q^{5} + \beta_{3} q^{7} -2 \beta_{2} q^{8} +O(q^{10})\) \( q + \beta_{2} q^{2} -2 q^{4} + ( \beta_{1} - \beta_{2} ) q^{5} + \beta_{3} q^{7} -2 \beta_{2} q^{8} + ( 3 - \beta_{3} ) q^{10} + ( -3 \beta_{1} - \beta_{2} ) q^{11} + ( 2 \beta_{1} + \beta_{2} ) q^{14} + 4 q^{16} + ( -2 \beta_{1} - \beta_{2} ) q^{17} + ( -6 - \beta_{3} ) q^{19} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{20} + ( -1 + 3 \beta_{3} ) q^{22} + ( -3 \beta_{1} - 4 \beta_{2} ) q^{23} + ( -3 + 3 \beta_{3} ) q^{25} -2 \beta_{3} q^{28} + ( 3 + 2 \beta_{3} ) q^{31} + 4 \beta_{2} q^{32} + 2 \beta_{3} q^{34} + ( -3 \beta_{1} + 2 \beta_{2} ) q^{35} + ( -4 + 3 \beta_{3} ) q^{37} + ( -2 \beta_{1} - 7 \beta_{2} ) q^{38} + ( -6 + 2 \beta_{3} ) q^{40} + ( \beta_{1} + 8 \beta_{2} ) q^{41} + ( 6 \beta_{1} + 2 \beta_{2} ) q^{44} + ( 5 + 3 \beta_{3} ) q^{46} + 7 q^{49} + 6 \beta_{1} q^{50} + ( 12 - 5 \beta_{3} ) q^{55} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{56} + ( 4 \beta_{1} + 5 \beta_{2} ) q^{62} -8 q^{64} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{68} + ( -7 + 3 \beta_{3} ) q^{70} + ( -3 \beta_{1} - 7 \beta_{2} ) q^{71} + ( 6 \beta_{1} - \beta_{2} ) q^{74} + ( 12 + 2 \beta_{3} ) q^{76} + ( \beta_{1} - 10 \beta_{2} ) q^{77} + ( 4 \beta_{1} - 4 \beta_{2} ) q^{80} + ( -15 - \beta_{3} ) q^{82} + ( 7 - 3 \beta_{3} ) q^{85} + ( 2 - 6 \beta_{3} ) q^{88} + ( -5 \beta_{1} - 4 \beta_{2} ) q^{89} + ( 6 \beta_{1} + 8 \beta_{2} ) q^{92} + ( -3 \beta_{1} + 4 \beta_{2} ) q^{95} + 7 \beta_{2} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{4} + O(q^{10}) \) \( 4q - 8q^{4} + 12q^{10} + 16q^{16} - 24q^{19} - 4q^{22} - 12q^{25} + 12q^{31} - 16q^{37} - 24q^{40} + 20q^{46} + 28q^{49} + 48q^{55} - 32q^{64} - 28q^{70} + 48q^{76} - 60q^{82} + 28q^{85} + 8q^{88} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 8 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 5 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\( \nu^{2} + 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 4\)
\(\nu^{3}\)\(=\)\(3 \beta_{2} - 5 \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
1.16372i
2.57794i
2.57794i
1.16372i
1.41421i 0 −2.00000 0.250492i 0 2.64575 2.82843i 0 0.354249
55.2 1.41421i 0 −2.00000 3.99215i 0 −2.64575 2.82843i 0 5.64575
55.3 1.41421i 0 −2.00000 3.99215i 0 −2.64575 2.82843i 0 5.64575
55.4 1.41421i 0 −2.00000 0.250492i 0 2.64575 2.82843i 0 0.354249
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
84.h odd 2 1 CM by \(\Q(\sqrt{-21}) \)
3.b odd 2 1 inner
28.d 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.b.b yes 4
3.b odd 2 1 inner 756.2.b.b yes 4
4.b odd 2 1 756.2.b.a 4
7.b odd 2 1 756.2.b.a 4
12.b even 2 1 756.2.b.a 4
21.c even 2 1 756.2.b.a 4
28.d even 2 1 inner 756.2.b.b yes 4
84.h odd 2 1 CM 756.2.b.b yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.b.a 4 4.b odd 2 1
756.2.b.a 4 7.b odd 2 1
756.2.b.a 4 12.b even 2 1
756.2.b.a 4 21.c even 2 1
756.2.b.b yes 4 1.a even 1 1 trivial
756.2.b.b yes 4 3.b odd 2 1 inner
756.2.b.b yes 4 28.d even 2 1 inner
756.2.b.b yes 4 84.h odd 2 1 CM

Hecke kernels

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

\( T_{5}^{4} + 16 T_{5}^{2} + 1 \)
\( T_{19}^{2} + 12 T_{19} + 29 \)
\( T_{47} \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 2 T^{2} )^{2} \)
$3$ 1
$5$ \( 1 - 4 T^{2} - 9 T^{4} - 100 T^{6} + 625 T^{8} \)
$7$ \( ( 1 - 7 T^{2} )^{2} \)
$11$ \( 1 + 20 T^{2} + 279 T^{4} + 2420 T^{6} + 14641 T^{8} \)
$13$ \( ( 1 - 13 T^{2} )^{4} \)
$17$ \( ( 1 - 20 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 + 12 T + 67 T^{2} + 228 T^{3} + 361 T^{4} )^{2} \)
$23$ \( 1 - 4 T^{2} - 513 T^{4} - 2116 T^{6} + 279841 T^{8} \)
$29$ \( ( 1 + 29 T^{2} )^{4} \)
$31$ \( ( 1 - 6 T + 43 T^{2} - 186 T^{3} + 961 T^{4} )^{2} \)
$37$ \( ( 1 + 8 T + 27 T^{2} + 296 T^{3} + 1369 T^{4} )^{2} \)
$41$ \( 1 + 68 T^{2} + 2943 T^{4} + 114308 T^{6} + 2825761 T^{8} \)
$43$ \( ( 1 - 43 T^{2} )^{4} \)
$47$ \( ( 1 + 47 T^{2} )^{4} \)
$53$ \( ( 1 + 53 T^{2} )^{4} \)
$59$ \( ( 1 + 59 T^{2} )^{4} \)
$61$ \( ( 1 - 61 T^{2} )^{4} \)
$67$ \( ( 1 - 67 T^{2} )^{4} \)
$71$ \( 1 - 100 T^{2} + 4959 T^{4} - 504100 T^{6} + 25411681 T^{8} \)
$73$ \( ( 1 - 73 T^{2} )^{4} \)
$79$ \( ( 1 - 79 T^{2} )^{4} \)
$83$ \( ( 1 + 83 T^{2} )^{4} \)
$89$ \( 1 - 172 T^{2} + 21663 T^{4} - 1362412 T^{6} + 62742241 T^{8} \)
$97$ \( ( 1 - 97 T^{2} )^{4} \)
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