Properties

Label 756.2.j.a
Level $756$
Weight $2$
Character orbit 756.j
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.j (of order \(3\), degree \(2\), not 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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: no (minimal twist has level 252)
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.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} + \beta_{3} ) q^{5} -\beta_{3} q^{7} +O(q^{10})\) \( q + ( -1 + \beta_{1} + \beta_{3} ) q^{5} -\beta_{3} q^{7} + ( -\beta_{2} - 2 \beta_{3} ) q^{11} + ( 1 - \beta_{3} ) q^{13} + ( \beta_{1} + \beta_{2} + \beta_{4} ) q^{17} + ( 1 - \beta_{1} - \beta_{2} + \beta_{4} ) q^{19} + ( -2 + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{23} + ( 2 \beta_{2} - 2 \beta_{3} + \beta_{5} ) q^{25} + ( -5 \beta_{3} + \beta_{5} ) q^{29} + ( 1 + 3 \beta_{1} - \beta_{3} ) q^{31} + ( 1 - \beta_{1} - \beta_{2} ) q^{35} + ( -1 + 3 \beta_{1} + 3 \beta_{2} ) q^{37} + ( -2 - \beta_{1} + 2 \beta_{3} ) q^{41} + ( \beta_{2} - \beta_{3} + 2 \beta_{5} ) q^{43} + ( \beta_{2} - 5 \beta_{3} - \beta_{5} ) q^{47} + ( -1 + \beta_{3} ) q^{49} + ( 6 - 2 \beta_{1} - 2 \beta_{2} + \beta_{4} ) q^{53} + ( -4 - \beta_{1} - \beta_{2} + \beta_{4} ) q^{55} + ( -1 + 2 \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{59} + ( -2 \beta_{2} + 2 \beta_{3} - \beta_{5} ) q^{61} + ( -\beta_{2} + \beta_{3} ) q^{65} + ( 2 + \beta_{1} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{67} + ( 5 + 2 \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{71} + ( 3 + \beta_{1} + \beta_{2} - \beta_{4} ) q^{73} + ( -2 - \beta_{1} + 2 \beta_{3} ) q^{77} + ( -5 \beta_{2} - \beta_{3} - \beta_{5} ) q^{79} + ( -\beta_{2} - 6 \beta_{3} - \beta_{5} ) q^{83} + ( 5 - 4 \beta_{1} - 5 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{85} + ( -2 + 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{4} ) q^{89} - q^{91} + ( -8 - \beta_{1} + 8 \beta_{3} ) q^{95} + ( -2 \beta_{2} + 5 \beta_{3} - \beta_{5} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{5} - 3 q^{7} + O(q^{10}) \) \( 6 q - 3 q^{5} - 3 q^{7} - 6 q^{11} + 3 q^{13} + 6 q^{19} - 6 q^{23} - 6 q^{25} - 15 q^{29} + 3 q^{31} + 6 q^{35} - 6 q^{37} - 6 q^{41} - 3 q^{43} - 15 q^{47} - 3 q^{49} + 36 q^{53} - 24 q^{55} - 3 q^{59} + 6 q^{61} + 3 q^{65} + 6 q^{67} + 30 q^{71} + 18 q^{73} - 6 q^{77} - 3 q^{79} - 18 q^{83} + 15 q^{85} - 12 q^{89} - 6 q^{91} - 24 q^{95} + 15 q^{97} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{5} - \nu^{4} + 5 \nu^{3} - 2 \nu^{2} \)\()/3\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{5} + 4 \nu^{4} - 11 \nu^{3} + 17 \nu^{2} - 12 \nu + 3 \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{5} - 5 \nu^{4} + 16 \nu^{3} - 19 \nu^{2} + 21 \nu - 6 \)\()/3\)
\(\beta_{4}\)\(=\)\( \nu^{4} - 2 \nu^{3} + 8 \nu^{2} - 7 \nu + 8 \)
\(\beta_{5}\)\(=\)\((\)\( 7 \nu^{5} - 16 \nu^{4} + 62 \nu^{3} - 68 \nu^{2} + 99 \nu - 30 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} - \beta_{1} + 1\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{4} + \beta_{3} - 2 \beta_{1} - 6\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{5} + \beta_{4} - 6 \beta_{3} - 4 \beta_{2} + \beta_{1} - 6\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(2 \beta_{5} - 3 \beta_{4} - 13 \beta_{3} - \beta_{2} + 11 \beta_{1} + 19\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(-3 \beta_{5} - 6 \beta_{4} + 19 \beta_{3} + 19 \beta_{2} + 11 \beta_{1} + 37\)\()/3\)

Character values

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

\(n\) \(29\) \(325\) \(379\)
\(\chi(n)\) \(-1 + \beta_{3}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
253.1
0.500000 1.41036i
0.500000 + 0.224437i
0.500000 + 2.05195i
0.500000 + 1.41036i
0.500000 0.224437i
0.500000 2.05195i
0 0 0 −1.97141 + 3.41458i 0 −0.500000 0.866025i 0 0 0
253.2 0 0 0 −0.555632 + 0.962383i 0 −0.500000 0.866025i 0 0 0
253.3 0 0 0 1.02704 1.77889i 0 −0.500000 0.866025i 0 0 0
505.1 0 0 0 −1.97141 3.41458i 0 −0.500000 + 0.866025i 0 0 0
505.2 0 0 0 −0.555632 0.962383i 0 −0.500000 + 0.866025i 0 0 0
505.3 0 0 0 1.02704 + 1.77889i 0 −0.500000 + 0.866025i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 505.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.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.j.a 6
3.b odd 2 1 252.2.j.b 6
4.b odd 2 1 3024.2.r.i 6
7.b odd 2 1 5292.2.j.e 6
7.c even 3 1 5292.2.i.d 6
7.c even 3 1 5292.2.l.g 6
7.d odd 6 1 5292.2.i.g 6
7.d odd 6 1 5292.2.l.d 6
9.c even 3 1 inner 756.2.j.a 6
9.c even 3 1 2268.2.a.j 3
9.d odd 6 1 252.2.j.b 6
9.d odd 6 1 2268.2.a.g 3
12.b even 2 1 1008.2.r.g 6
21.c even 2 1 1764.2.j.d 6
21.g even 6 1 1764.2.i.e 6
21.g even 6 1 1764.2.l.g 6
21.h odd 6 1 1764.2.i.f 6
21.h odd 6 1 1764.2.l.d 6
36.f odd 6 1 3024.2.r.i 6
36.f odd 6 1 9072.2.a.bz 3
36.h even 6 1 1008.2.r.g 6
36.h even 6 1 9072.2.a.bt 3
63.g even 3 1 5292.2.i.d 6
63.h even 3 1 5292.2.l.g 6
63.i even 6 1 1764.2.l.g 6
63.j odd 6 1 1764.2.l.d 6
63.k odd 6 1 5292.2.i.g 6
63.l odd 6 1 5292.2.j.e 6
63.n odd 6 1 1764.2.i.f 6
63.o even 6 1 1764.2.j.d 6
63.s even 6 1 1764.2.i.e 6
63.t odd 6 1 5292.2.l.d 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.j.b 6 3.b odd 2 1
252.2.j.b 6 9.d odd 6 1
756.2.j.a 6 1.a even 1 1 trivial
756.2.j.a 6 9.c even 3 1 inner
1008.2.r.g 6 12.b even 2 1
1008.2.r.g 6 36.h even 6 1
1764.2.i.e 6 21.g even 6 1
1764.2.i.e 6 63.s even 6 1
1764.2.i.f 6 21.h odd 6 1
1764.2.i.f 6 63.n odd 6 1
1764.2.j.d 6 21.c even 2 1
1764.2.j.d 6 63.o even 6 1
1764.2.l.d 6 21.h odd 6 1
1764.2.l.d 6 63.j odd 6 1
1764.2.l.g 6 21.g even 6 1
1764.2.l.g 6 63.i even 6 1
2268.2.a.g 3 9.d odd 6 1
2268.2.a.j 3 9.c even 3 1
3024.2.r.i 6 4.b odd 2 1
3024.2.r.i 6 36.f odd 6 1
5292.2.i.d 6 7.c even 3 1
5292.2.i.d 6 63.g even 3 1
5292.2.i.g 6 7.d odd 6 1
5292.2.i.g 6 63.k odd 6 1
5292.2.j.e 6 7.b odd 2 1
5292.2.j.e 6 63.l odd 6 1
5292.2.l.d 6 7.d odd 6 1
5292.2.l.d 6 63.t odd 6 1
5292.2.l.g 6 7.c even 3 1
5292.2.l.g 6 63.h even 3 1
9072.2.a.bt 3 36.h even 6 1
9072.2.a.bz 3 36.f odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 3 T_{5}^{5} + 15 T_{5}^{4} + 63 T_{5}^{2} + 54 T_{5} + 81 \) acting on \(S_{2}^{\mathrm{new}}(756, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( T^{6} \)
$5$ \( 81 + 54 T + 63 T^{2} + 15 T^{4} + 3 T^{5} + T^{6} \)
$7$ \( ( 1 + T + T^{2} )^{3} \)
$11$ \( 81 - 27 T + 63 T^{2} + 36 T^{3} + 33 T^{4} + 6 T^{5} + T^{6} \)
$13$ \( ( 1 - T + T^{2} )^{3} \)
$17$ \( ( -9 - 33 T + T^{3} )^{2} \)
$19$ \( ( 49 - 36 T - 3 T^{2} + T^{3} )^{2} \)
$23$ \( 9801 + 1485 T + 819 T^{2} + 108 T^{3} + 51 T^{4} + 6 T^{5} + T^{6} \)
$29$ \( 3969 - 3024 T + 3249 T^{2} + 846 T^{3} + 177 T^{4} + 15 T^{5} + T^{6} \)
$31$ \( 2809 - 4134 T + 6243 T^{2} + 128 T^{3} + 87 T^{4} - 3 T^{5} + T^{6} \)
$37$ \( ( -107 - 78 T + 3 T^{2} + T^{3} )^{2} \)
$41$ \( 81 - 27 T + 63 T^{2} + 36 T^{3} + 33 T^{4} + 6 T^{5} + T^{6} \)
$43$ \( 214369 + 50004 T + 13053 T^{2} + 602 T^{3} + 117 T^{4} + 3 T^{5} + T^{6} \)
$47$ \( 6561 - 2916 T + 2511 T^{2} + 702 T^{3} + 189 T^{4} + 15 T^{5} + T^{6} \)
$53$ \( ( 387 + 39 T - 18 T^{2} + T^{3} )^{2} \)
$59$ \( 6561 - 4374 T + 2673 T^{2} - 324 T^{3} + 63 T^{4} + 3 T^{5} + T^{6} \)
$61$ \( 961 + 1395 T + 1839 T^{2} + 332 T^{3} + 81 T^{4} - 6 T^{5} + T^{6} \)
$67$ \( 3481 - 1593 T + 1083 T^{2} + 44 T^{3} + 63 T^{4} - 6 T^{5} + T^{6} \)
$71$ \( ( 297 + 18 T - 15 T^{2} + T^{3} )^{2} \)
$73$ \( ( 79 - 12 T - 9 T^{2} + T^{3} )^{2} \)
$79$ \( 1857769 + 318942 T + 58845 T^{2} + 2024 T^{3} + 243 T^{4} + 3 T^{5} + T^{6} \)
$83$ \( 729 + 2025 T + 5139 T^{2} + 1296 T^{3} + 249 T^{4} + 18 T^{5} + T^{6} \)
$89$ \( ( -1089 - 195 T + 6 T^{2} + T^{3} )^{2} \)
$97$ \( 529 + 414 T + 669 T^{2} - 316 T^{3} + 207 T^{4} - 15 T^{5} + T^{6} \)
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