Properties

Label 756.2.k.e
Level $756$
Weight $2$
Character orbit 756.k
Analytic conductor $6.037$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
Defining polynomial: \(x^{6} - 3 x^{5} + 10 x^{4} - 15 x^{3} + 19 x^{2} - 12 x + 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} ) q^{5} + ( 1 + \beta_{2} + \beta_{4} ) q^{7} +O(q^{10})\) \( q + ( \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} ) q^{5} + ( 1 + \beta_{2} + \beta_{4} ) q^{7} + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{11} + ( -\beta_{1} + \beta_{3} ) q^{13} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{17} + ( 2 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{19} + ( \beta_{2} - \beta_{3} + 5 \beta_{4} ) q^{23} + ( -5 - \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 5 \beta_{4} + 2 \beta_{5} ) q^{25} + ( -2 - \beta_{2} - \beta_{5} ) q^{29} + ( -\beta_{2} - \beta_{3} + \beta_{5} ) q^{31} + ( -6 + \beta_{1} - 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{35} + ( -\beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{37} + ( 3 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} ) q^{41} + ( -4 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{5} ) q^{43} + ( -\beta_{1} + \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{47} + ( -1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{49} + ( 3 \beta_{2} + 3 \beta_{3} - 3 \beta_{5} ) q^{53} + ( 5 - 4 \beta_{1} - \beta_{2} + 4 \beta_{3} - \beta_{5} ) q^{55} + ( 2 - \beta_{1} + 2 \beta_{4} - \beta_{5} ) q^{59} + ( 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} ) q^{61} + ( 2 \beta_{1} + \beta_{2} + \beta_{3} + 9 \beta_{4} - 2 \beta_{5} ) q^{65} + ( 5 + \beta_{1} + 5 \beta_{4} + \beta_{5} ) q^{67} + ( 2 + 3 \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{5} ) q^{71} + ( 2 + \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{73} + ( -7 - 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - 7 \beta_{4} + \beta_{5} ) q^{77} + ( -2 \beta_{1} + \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} ) q^{79} + ( 5 + \beta_{1} - \beta_{3} ) q^{83} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} ) q^{85} + ( -2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} ) q^{89} + ( 4 + 3 \beta_{3} + 5 \beta_{4} - 2 \beta_{5} ) q^{91} + ( -14 - 5 \beta_{2} - 5 \beta_{3} - 14 \beta_{4} + 5 \beta_{5} ) q^{95} + ( 4 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - q^{5} + 2q^{7} + O(q^{10}) \) \( 6q - q^{5} + 2q^{7} - 5q^{11} - 4q^{13} + 4q^{17} - 3q^{19} - 14q^{23} - 10q^{25} - 10q^{29} + 2q^{31} - 26q^{35} + 24q^{41} - 18q^{43} + 9q^{47} - 6q^{53} + 16q^{55} + 5q^{59} - 7q^{61} - 24q^{65} + 16q^{67} + 22q^{71} + q^{73} - 31q^{77} + 8q^{79} + 34q^{83} + 10q^{85} + 3q^{89} + 5q^{91} - 32q^{95} + 28q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{5} + 10 x^{4} - 15 x^{3} + 19 x^{2} - 12 x + 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{5} + \nu^{4} - 5 \nu^{3} - \nu^{2} + 3 \nu - 6 \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{5} + \nu^{4} - 8 \nu^{3} + 8 \nu^{2} - 21 \nu + 12 \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{5} + 4 \nu^{4} - 11 \nu^{3} + 20 \nu^{2} - 15 \nu + 9 \)\()/3\)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{5} + 5 \nu^{4} - 16 \nu^{3} + 19 \nu^{2} - 21 \nu + 6 \)\()/3\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{5} - 4 \nu^{4} + 14 \nu^{3} - 20 \nu^{2} + 30 \nu - 9 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{4} + \beta_{3} + \beta_{1} + 1\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_{2} - 5\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(3 \beta_{5} + 5 \beta_{4} - 2 \beta_{3} - 5 \beta_{1} - 5\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(-\beta_{5} + 11 \beta_{4} - 9 \beta_{3} - 7 \beta_{2} - 7 \beta_{1} + 16\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(-17 \beta_{5} - 16 \beta_{4} + 2 \beta_{3} - 8 \beta_{2} + 12 \beta_{1} + 31\)\()/3\)

Character values

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

\(n\) \(29\) \(325\) \(379\)
\(\chi(n)\) \(1\) \(\beta_{4}\) \(1\)

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
0.500000 + 0.224437i
0.500000 + 2.05195i
0.500000 1.41036i
0.500000 0.224437i
0.500000 2.05195i
0.500000 + 1.41036i
0 0 0 −2.14400 3.71351i 0 1.40545 2.24159i 0 0 0
109.2 0 0 0 0.433463 + 0.750780i 0 −2.25729 1.38008i 0 0 0
109.3 0 0 0 1.21053 + 2.09671i 0 1.85185 + 1.88962i 0 0 0
541.1 0 0 0 −2.14400 + 3.71351i 0 1.40545 + 2.24159i 0 0 0
541.2 0 0 0 0.433463 0.750780i 0 −2.25729 + 1.38008i 0 0 0
541.3 0 0 0 1.21053 2.09671i 0 1.85185 1.88962i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 541.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

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

Hecke kernels

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

\( T_{5}^{6} + T_{5}^{5} + 13 T_{5}^{4} - 30 T_{5}^{3} + 135 T_{5}^{2} - 108 T_{5} + 81 \)
\( T_{13}^{3} + 2 T_{13}^{2} - 11 T_{13} - 21 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( T^{6} \)
$5$ \( 81 - 108 T + 135 T^{2} - 30 T^{3} + 13 T^{4} + T^{5} + T^{6} \)
$7$ \( 343 - 98 T + 14 T^{2} + 19 T^{3} + 2 T^{4} - 2 T^{5} + T^{6} \)
$11$ \( 3969 + 756 T + 459 T^{2} + 66 T^{3} + 37 T^{4} + 5 T^{5} + T^{6} \)
$13$ \( ( -21 - 11 T + 2 T^{2} + T^{3} )^{2} \)
$17$ \( 81 + 135 T + 189 T^{2} + 78 T^{3} + 31 T^{4} - 4 T^{5} + T^{6} \)
$19$ \( 2401 + 1764 T + 1443 T^{2} - 10 T^{3} + 45 T^{4} + 3 T^{5} + T^{6} \)
$23$ \( 3969 - 2457 T + 2403 T^{2} + 672 T^{3} + 157 T^{4} + 14 T^{5} + T^{6} \)
$29$ \( ( -63 - 12 T + 5 T^{2} + T^{3} )^{2} \)
$31$ \( 441 - 231 T + 163 T^{2} - 20 T^{3} + 15 T^{4} - 2 T^{5} + T^{6} \)
$37$ \( 18769 - 7809 T + 3249 T^{2} - 274 T^{3} + 57 T^{4} + T^{6} \)
$41$ \( ( 81 + 9 T - 12 T^{2} + T^{3} )^{2} \)
$43$ \( ( -281 - 30 T + 9 T^{2} + T^{3} )^{2} \)
$47$ \( 59049 - 13122 T + 5103 T^{2} + 135 T^{4} - 9 T^{5} + T^{6} \)
$53$ \( 321489 + 56133 T + 13203 T^{2} + 540 T^{3} + 135 T^{4} + 6 T^{5} + T^{6} \)
$59$ \( 6561 - 1458 T + 729 T^{2} - 72 T^{3} + 43 T^{4} - 5 T^{5} + T^{6} \)
$61$ \( 321489 + 50463 T + 11890 T^{2} + 511 T^{3} + 138 T^{4} + 7 T^{5} + T^{6} \)
$67$ \( 2809 - 3127 T + 2633 T^{2} - 838 T^{3} + 197 T^{4} - 16 T^{5} + T^{6} \)
$71$ \( ( 189 - 72 T - 11 T^{2} + T^{3} )^{2} \)
$73$ \( 25921 + 13202 T + 6563 T^{2} + 404 T^{3} + 83 T^{4} - T^{5} + T^{6} \)
$79$ \( 762129 - 135315 T + 31009 T^{2} - 506 T^{3} + 219 T^{4} - 8 T^{5} + T^{6} \)
$83$ \( ( -99 + 84 T - 17 T^{2} + T^{3} )^{2} \)
$89$ \( 1750329 - 297675 T + 54594 T^{2} - 1971 T^{3} + 234 T^{4} - 3 T^{5} + T^{6} \)
$97$ \( ( 56 - 40 T - 14 T^{2} + T^{3} )^{2} \)
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