Properties

Label 756.2.j.b
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
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
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.

\(f(q)\) \(=\) \( q + ( -\beta_{1} + \beta_{5} ) q^{5} + ( 1 - \beta_{4} ) q^{7} +O(q^{10})\) \( q + ( -\beta_{1} + \beta_{5} ) q^{5} + ( 1 - \beta_{4} ) q^{7} + ( 1 - 2 \beta_{2} - \beta_{4} - \beta_{5} ) q^{11} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{13} + ( -1 - 2 \beta_{1} + \beta_{3} ) q^{17} + ( -2 \beta_{1} - \beta_{3} ) q^{19} + ( -\beta_{1} - \beta_{2} - \beta_{3} + 4 \beta_{4} + \beta_{5} ) q^{23} + ( 2 + \beta_{2} - 2 \beta_{4} + \beta_{5} ) q^{25} + ( 1 + \beta_{2} - \beta_{4} + 3 \beta_{5} ) q^{29} + ( -3 \beta_{1} + 3 \beta_{5} ) q^{31} -\beta_{1} q^{35} + ( -2 - \beta_{1} - 2 \beta_{3} ) q^{37} + ( -\beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{41} + ( -4 \beta_{2} - \beta_{5} ) q^{43} + ( 8 - \beta_{2} - 8 \beta_{4} + 2 \beta_{5} ) q^{47} -\beta_{4} q^{49} + ( -2 + \beta_{1} - \beta_{3} ) q^{53} + ( 1 - 4 \beta_{1} + \beta_{3} ) q^{55} + ( \beta_{1} + \beta_{2} + \beta_{3} + 11 \beta_{4} - \beta_{5} ) q^{59} + ( -4 - \beta_{2} + 4 \beta_{4} - 7 \beta_{5} ) q^{61} + ( 4 - 2 \beta_{2} - 4 \beta_{4} - 5 \beta_{5} ) q^{65} + ( -2 \beta_{1} - \beta_{2} - \beta_{3} - 3 \beta_{4} + 2 \beta_{5} ) q^{67} + ( -3 + 5 \beta_{1} + 3 \beta_{3} ) q^{71} + ( 2 \beta_{1} + \beta_{3} ) q^{73} + ( \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{77} + ( 4 + \beta_{2} - 4 \beta_{4} + 4 \beta_{5} ) q^{79} + ( 7 + \beta_{2} - 7 \beta_{4} + 2 \beta_{5} ) q^{83} + ( -2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 5 \beta_{4} + 2 \beta_{5} ) q^{85} + ( -5 + \beta_{1} - 4 \beta_{3} ) q^{89} + ( -1 + 2 \beta_{1} - 2 \beta_{3} ) q^{91} + ( -\beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 7 \beta_{4} + \beta_{5} ) q^{95} + ( 1 + 5 \beta_{2} - \beta_{4} - \beta_{5} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + q^{5} + 3q^{7} + O(q^{10}) \) \( 6q + q^{5} + 3q^{7} + 2q^{11} - 3q^{13} - 4q^{17} + 6q^{19} + 14q^{23} + 6q^{25} + q^{29} + 3q^{31} + 2q^{35} - 6q^{37} - 3q^{43} + 21q^{47} - 3q^{49} - 12q^{53} + 12q^{55} + 31q^{59} - 6q^{61} + 15q^{65} - 6q^{67} - 34q^{71} - 6q^{73} - 2q^{77} + 9q^{79} + 20q^{83} + 15q^{85} - 24q^{89} - 6q^{91} + 20q^{95} + 9q^{97} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} - \nu + 2 \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{5} + \nu^{4} - 8 \nu^{3} + 5 \nu^{2} - 18 \nu + 6 \)\()/3\)
\(\beta_{3}\)\(=\)\( \nu^{4} - 2 \nu^{3} + 6 \nu^{2} - 5 \nu + 3 \)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{5} + 5 \nu^{4} - 16 \nu^{3} + 19 \nu^{2} - 21 \nu + 9 \)\()/3\)
\(\beta_{5}\)\(=\)\((\)\( 2 \nu^{5} - 5 \nu^{4} + 19 \nu^{3} - 22 \nu^{2} + 30 \nu - 9 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-2 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_{1} + 2\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-2 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + 4 \beta_{1} - 4\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(7 \beta_{5} + 5 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + \beta_{1} - 10\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(16 \beta_{5} + 11 \beta_{4} + 8 \beta_{3} + 10 \beta_{2} - 17 \beta_{1} + 5\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(-14 \beta_{5} - 16 \beta_{4} + 5 \beta_{3} - 5 \beta_{2} - 23 \beta_{1} + 47\)\()/3\)

Character values

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

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

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 + 0.224437i
0.500000 1.41036i
0.500000 + 2.05195i
0.500000 0.224437i
0.500000 + 1.41036i
0.500000 2.05195i
0 0 0 −0.849814 + 1.47192i 0 0.500000 + 0.866025i 0 0 0
253.2 0 0 0 0.119562 0.207087i 0 0.500000 + 0.866025i 0 0 0
253.3 0 0 0 1.23025 2.13086i 0 0.500000 + 0.866025i 0 0 0
505.1 0 0 0 −0.849814 1.47192i 0 0.500000 0.866025i 0 0 0
505.2 0 0 0 0.119562 + 0.207087i 0 0.500000 0.866025i 0 0 0
505.3 0 0 0 1.23025 + 2.13086i 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.b 6
3.b odd 2 1 252.2.j.a 6
4.b odd 2 1 3024.2.r.j 6
7.b odd 2 1 5292.2.j.d 6
7.c even 3 1 5292.2.i.f 6
7.c even 3 1 5292.2.l.e 6
7.d odd 6 1 5292.2.i.e 6
7.d odd 6 1 5292.2.l.f 6
9.c even 3 1 inner 756.2.j.b 6
9.c even 3 1 2268.2.a.h 3
9.d odd 6 1 252.2.j.a 6
9.d odd 6 1 2268.2.a.i 3
12.b even 2 1 1008.2.r.j 6
21.c even 2 1 1764.2.j.e 6
21.g even 6 1 1764.2.i.d 6
21.g even 6 1 1764.2.l.f 6
21.h odd 6 1 1764.2.i.g 6
21.h odd 6 1 1764.2.l.e 6
36.f odd 6 1 3024.2.r.j 6
36.f odd 6 1 9072.2.a.bv 3
36.h even 6 1 1008.2.r.j 6
36.h even 6 1 9072.2.a.by 3
63.g even 3 1 5292.2.i.f 6
63.h even 3 1 5292.2.l.e 6
63.i even 6 1 1764.2.l.f 6
63.j odd 6 1 1764.2.l.e 6
63.k odd 6 1 5292.2.i.e 6
63.l odd 6 1 5292.2.j.d 6
63.n odd 6 1 1764.2.i.g 6
63.o even 6 1 1764.2.j.e 6
63.s even 6 1 1764.2.i.d 6
63.t odd 6 1 5292.2.l.f 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.j.a 6 3.b odd 2 1
252.2.j.a 6 9.d odd 6 1
756.2.j.b 6 1.a even 1 1 trivial
756.2.j.b 6 9.c even 3 1 inner
1008.2.r.j 6 12.b even 2 1
1008.2.r.j 6 36.h even 6 1
1764.2.i.d 6 21.g even 6 1
1764.2.i.d 6 63.s even 6 1
1764.2.i.g 6 21.h odd 6 1
1764.2.i.g 6 63.n odd 6 1
1764.2.j.e 6 21.c even 2 1
1764.2.j.e 6 63.o even 6 1
1764.2.l.e 6 21.h odd 6 1
1764.2.l.e 6 63.j odd 6 1
1764.2.l.f 6 21.g even 6 1
1764.2.l.f 6 63.i even 6 1
2268.2.a.h 3 9.c even 3 1
2268.2.a.i 3 9.d odd 6 1
3024.2.r.j 6 4.b odd 2 1
3024.2.r.j 6 36.f odd 6 1
5292.2.i.e 6 7.d odd 6 1
5292.2.i.e 6 63.k odd 6 1
5292.2.i.f 6 7.c even 3 1
5292.2.i.f 6 63.g even 3 1
5292.2.j.d 6 7.b odd 2 1
5292.2.j.d 6 63.l odd 6 1
5292.2.l.e 6 7.c even 3 1
5292.2.l.e 6 63.h even 3 1
5292.2.l.f 6 7.d odd 6 1
5292.2.l.f 6 63.t odd 6 1
9072.2.a.bv 3 36.f odd 6 1
9072.2.a.by 3 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} - T_{5}^{5} + 5 T_{5}^{4} + 2 T_{5}^{3} + 17 T_{5}^{2} - 4 T_{5} + 1 \) acting on \(S_{2}^{\mathrm{new}}(756, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - T - 10 T^{2} + 7 T^{3} + 57 T^{4} - 14 T^{5} - 299 T^{6} - 70 T^{7} + 1425 T^{8} + 875 T^{9} - 6250 T^{10} - 3125 T^{11} + 15625 T^{12} \)
$7$ \( ( 1 - T + T^{2} )^{3} \)
$11$ \( 1 - 2 T - 4 T^{2} - 46 T^{3} + 6 T^{4} + 230 T^{5} + 1699 T^{6} + 2530 T^{7} + 726 T^{8} - 61226 T^{9} - 58564 T^{10} - 322102 T^{11} + 1771561 T^{12} \)
$13$ \( 1 + 3 T + 3 T^{2} - 84 T^{3} - 195 T^{4} + 345 T^{5} + 5006 T^{6} + 4485 T^{7} - 32955 T^{8} - 184548 T^{9} + 85683 T^{10} + 1113879 T^{11} + 4826809 T^{12} \)
$17$ \( ( 1 + 2 T + 32 T^{2} + 21 T^{3} + 544 T^{4} + 578 T^{5} + 4913 T^{6} )^{2} \)
$19$ \( ( 1 - 3 T + 33 T^{2} - 35 T^{3} + 627 T^{4} - 1083 T^{5} + 6859 T^{6} )^{2} \)
$23$ \( 1 - 14 T + 74 T^{2} - 358 T^{3} + 2628 T^{4} - 11188 T^{5} + 33943 T^{6} - 257324 T^{7} + 1390212 T^{8} - 4355786 T^{9} + 20708234 T^{10} - 90108802 T^{11} + 148035889 T^{12} \)
$29$ \( 1 - T - 46 T^{2} - 149 T^{3} + 897 T^{4} + 4282 T^{5} - 13523 T^{6} + 124178 T^{7} + 754377 T^{8} - 3633961 T^{9} - 32534926 T^{10} - 20511149 T^{11} + 594823321 T^{12} \)
$31$ \( 1 - 3 T - 48 T^{2} + 147 T^{3} + 1005 T^{4} - 1344 T^{5} - 24505 T^{6} - 41664 T^{7} + 965805 T^{8} + 4379277 T^{9} - 44329008 T^{10} - 85887453 T^{11} + 887503681 T^{12} \)
$37$ \( ( 1 + 3 T + 81 T^{2} + 199 T^{3} + 2997 T^{4} + 4107 T^{5} + 50653 T^{6} )^{2} \)
$41$ \( 1 - 90 T^{2} - 18 T^{3} + 4410 T^{4} + 810 T^{5} - 194177 T^{6} + 33210 T^{7} + 7413210 T^{8} - 1240578 T^{9} - 254318490 T^{10} + 4750104241 T^{12} \)
$43$ \( 1 + 3 T - 24 T^{2} - 979 T^{3} - 1947 T^{4} + 14820 T^{5} + 386067 T^{6} + 637260 T^{7} - 3600003 T^{8} - 77837353 T^{9} - 82051224 T^{10} + 441025329 T^{11} + 6321363049 T^{12} \)
$47$ \( 1 - 21 T + 180 T^{2} - 1119 T^{3} + 10053 T^{4} - 100416 T^{5} + 788551 T^{6} - 4719552 T^{7} + 22207077 T^{8} - 116177937 T^{9} + 878342580 T^{10} - 4816245147 T^{11} + 10779215329 T^{12} \)
$53$ \( ( 1 + 6 T + 162 T^{2} + 627 T^{3} + 8586 T^{4} + 16854 T^{5} + 148877 T^{6} )^{2} \)
$59$ \( 1 - 31 T + 476 T^{2} - 5741 T^{3} + 62553 T^{4} - 587576 T^{5} + 4781851 T^{6} - 34666984 T^{7} + 217746993 T^{8} - 1179080839 T^{9} + 5767863836 T^{10} - 22162653269 T^{11} + 42180533641 T^{12} \)
$61$ \( 1 + 6 T + 48 T^{2} + 642 T^{3} + 3018 T^{4} + 35394 T^{5} + 438671 T^{6} + 2159034 T^{7} + 11229978 T^{8} + 145721802 T^{9} + 664600368 T^{10} + 5067577806 T^{11} + 51520374361 T^{12} \)
$67$ \( 1 + 6 T - 150 T^{2} - 506 T^{3} + 17268 T^{4} + 28236 T^{5} - 1220289 T^{6} + 1891812 T^{7} + 77516052 T^{8} - 152186078 T^{9} - 3022668150 T^{10} + 8100750642 T^{11} + 90458382169 T^{12} \)
$71$ \( ( 1 + 17 T + 119 T^{2} + 507 T^{3} + 8449 T^{4} + 85697 T^{5} + 357911 T^{6} )^{2} \)
$73$ \( ( 1 + 3 T + 195 T^{2} + 359 T^{3} + 14235 T^{4} + 15987 T^{5} + 389017 T^{6} )^{2} \)
$79$ \( 1 - 9 T - 114 T^{2} + 351 T^{3} + 13143 T^{4} + 15786 T^{5} - 1414609 T^{6} + 1247094 T^{7} + 82025463 T^{8} + 173056689 T^{9} - 4440309234 T^{10} - 27693507591 T^{11} + 243087455521 T^{12} \)
$83$ \( 1 - 20 T + 38 T^{2} - 346 T^{3} + 32058 T^{4} - 183754 T^{5} - 606869 T^{6} - 15251582 T^{7} + 220847562 T^{8} - 197838302 T^{9} + 1803416198 T^{10} - 78780812860 T^{11} + 326940373369 T^{12} \)
$89$ \( ( 1 + 12 T + 216 T^{2} + 1425 T^{3} + 19224 T^{4} + 95052 T^{5} + 704969 T^{6} )^{2} \)
$97$ \( 1 - 9 T - 66 T^{2} + 2023 T^{3} - 7707 T^{4} - 73950 T^{5} + 1766073 T^{6} - 7173150 T^{7} - 72515163 T^{8} + 1846337479 T^{9} - 5842932546 T^{10} - 77286062313 T^{11} + 832972004929 T^{12} \)
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