Properties

Label 756.2.k.d
Level 756
Weight 2
Character orbit 756.k
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.k (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.03669039281\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{10})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{5} + ( 2 + 3 \beta_{2} ) q^{7} +O(q^{10})\) \( q + \beta_{1} q^{5} + ( 2 + 3 \beta_{2} ) q^{7} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{11} + ( \beta_{1} + \beta_{3} ) q^{17} + ( 7 + 7 \beta_{2} ) q^{19} -\beta_{1} q^{23} + 5 \beta_{2} q^{25} + \beta_{3} q^{29} + 3 \beta_{2} q^{31} + ( 2 \beta_{1} + 3 \beta_{3} ) q^{35} + ( 4 + 4 \beta_{2} ) q^{37} + 3 \beta_{3} q^{41} + 5 q^{43} + 3 \beta_{1} q^{47} + ( -5 + 3 \beta_{2} ) q^{49} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{53} + 20 q^{55} + ( -4 \beta_{1} - 4 \beta_{3} ) q^{59} + ( 3 + 3 \beta_{2} ) q^{61} + 10 \beta_{2} q^{67} -4 \beta_{3} q^{71} -5 \beta_{2} q^{73} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{77} + ( -12 - 12 \beta_{2} ) q^{79} + 2 \beta_{3} q^{83} -10 q^{85} -3 \beta_{1} q^{89} + ( 7 \beta_{1} + 7 \beta_{3} ) q^{95} -5 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{7} + O(q^{10}) \) \( 4q + 2q^{7} + 14q^{19} - 10q^{25} - 6q^{31} + 8q^{37} + 20q^{43} - 26q^{49} + 80q^{55} + 6q^{61} - 20q^{67} + 10q^{73} - 24q^{79} - 40q^{85} - 20q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 10 x^{2} + 100\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/10\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/10\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(10 \beta_{2}\)
\(\nu^{3}\)\(=\)\(10 \beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
−1.58114 2.73861i
1.58114 + 2.73861i
−1.58114 + 2.73861i
1.58114 2.73861i
0 0 0 −1.58114 2.73861i 0 0.500000 + 2.59808i 0 0 0
109.2 0 0 0 1.58114 + 2.73861i 0 0.500000 + 2.59808i 0 0 0
541.1 0 0 0 −1.58114 + 2.73861i 0 0.500000 2.59808i 0 0 0
541.2 0 0 0 1.58114 2.73861i 0 0.500000 2.59808i 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.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.2.k.d 4
3.b odd 2 1 inner 756.2.k.d 4
7.c even 3 1 inner 756.2.k.d 4
7.c even 3 1 5292.2.a.q 2
7.d odd 6 1 5292.2.a.r 2
9.c even 3 1 2268.2.i.i 4
9.c even 3 1 2268.2.l.i 4
9.d odd 6 1 2268.2.i.i 4
9.d odd 6 1 2268.2.l.i 4
21.g even 6 1 5292.2.a.r 2
21.h odd 6 1 inner 756.2.k.d 4
21.h odd 6 1 5292.2.a.q 2
63.g even 3 1 2268.2.i.i 4
63.h even 3 1 2268.2.l.i 4
63.j odd 6 1 2268.2.l.i 4
63.n odd 6 1 2268.2.i.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.k.d 4 1.a even 1 1 trivial
756.2.k.d 4 3.b odd 2 1 inner
756.2.k.d 4 7.c even 3 1 inner
756.2.k.d 4 21.h odd 6 1 inner
2268.2.i.i 4 9.c even 3 1
2268.2.i.i 4 9.d odd 6 1
2268.2.i.i 4 63.g even 3 1
2268.2.i.i 4 63.n odd 6 1
2268.2.l.i 4 9.c even 3 1
2268.2.l.i 4 9.d odd 6 1
2268.2.l.i 4 63.h even 3 1
2268.2.l.i 4 63.j odd 6 1
5292.2.a.q 2 7.c even 3 1
5292.2.a.q 2 21.h odd 6 1
5292.2.a.r 2 7.d odd 6 1
5292.2.a.r 2 21.g even 6 1

Hecke kernels

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

\( T_{5}^{4} + 10 T_{5}^{2} + 100 \)
\( T_{13} \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - 25 T^{4} + 625 T^{8} \)
$7$ \( ( 1 - T + 7 T^{2} )^{2} \)
$11$ \( 1 + 18 T^{2} + 203 T^{4} + 2178 T^{6} + 14641 T^{8} \)
$13$ \( ( 1 + 13 T^{2} )^{4} \)
$17$ \( 1 - 24 T^{2} + 287 T^{4} - 6936 T^{6} + 83521 T^{8} \)
$19$ \( ( 1 - 8 T + 19 T^{2} )^{2}( 1 + T + 19 T^{2} )^{2} \)
$23$ \( 1 - 36 T^{2} + 767 T^{4} - 19044 T^{6} + 279841 T^{8} \)
$29$ \( ( 1 + 48 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 + 3 T - 22 T^{2} + 93 T^{3} + 961 T^{4} )^{2} \)
$37$ \( ( 1 - 4 T - 21 T^{2} - 148 T^{3} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 - 8 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - 5 T + 43 T^{2} )^{4} \)
$47$ \( 1 - 4 T^{2} - 2193 T^{4} - 8836 T^{6} + 4879681 T^{8} \)
$53$ \( 1 - 16 T^{2} - 2553 T^{4} - 44944 T^{6} + 7890481 T^{8} \)
$59$ \( 1 + 42 T^{2} - 1717 T^{4} + 146202 T^{6} + 12117361 T^{8} \)
$61$ \( ( 1 - 3 T - 52 T^{2} - 183 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 + 10 T + 33 T^{2} + 670 T^{3} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 - 18 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 5 T - 48 T^{2} - 365 T^{3} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 + 12 T + 65 T^{2} + 948 T^{3} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 + 126 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( 1 - 88 T^{2} - 177 T^{4} - 697048 T^{6} + 62742241 T^{8} \)
$97$ \( ( 1 + 5 T + 97 T^{2} )^{4} \)
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