Properties

Label 567.2.i.f
Level $567$
Weight $2$
Character orbit 567.i
Analytic conductor $4.528$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.i (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.52751779461\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 9 x^{10} + 59 x^{8} - 180 x^{6} + 403 x^{4} - 198 x^{2} + 81\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: no (minimal twist has level 189)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{11} q^{2} + ( -1 + \beta_{7} ) q^{4} -\beta_{6} q^{5} + ( 1 + \beta_{2} - \beta_{3} ) q^{7} + ( \beta_{8} - \beta_{10} + \beta_{11} ) q^{8} +O(q^{10})\) \( q -\beta_{11} q^{2} + ( -1 + \beta_{7} ) q^{4} -\beta_{6} q^{5} + ( 1 + \beta_{2} - \beta_{3} ) q^{7} + ( \beta_{8} - \beta_{10} + \beta_{11} ) q^{8} + ( -2 \beta_{2} - 2 \beta_{4} + \beta_{7} + \beta_{9} ) q^{10} + ( -\beta_{1} + \beta_{8} ) q^{11} + ( 2 - \beta_{3} + \beta_{4} - \beta_{7} ) q^{13} + ( -\beta_{1} + \beta_{5} + \beta_{8} ) q^{14} + ( 1 - \beta_{2} - 2 \beta_{4} + \beta_{9} ) q^{16} + ( -\beta_{1} + \beta_{11} ) q^{17} + ( 2 \beta_{7} - \beta_{9} ) q^{19} + ( -2 \beta_{8} + \beta_{10} ) q^{20} + ( -2 \beta_{2} + 3 \beta_{3} - \beta_{4} + \beta_{7} - 2 \beta_{9} ) q^{22} + ( \beta_{1} - 2 \beta_{5} - \beta_{6} + \beta_{10} + \beta_{11} ) q^{23} + ( -2 \beta_{2} - 4 \beta_{3} - \beta_{4} + \beta_{7} - \beta_{9} ) q^{25} + ( -2 \beta_{1} - \beta_{5} - \beta_{6} - 4 \beta_{11} ) q^{26} + ( 2 - 2 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{7} - 2 \beta_{9} ) q^{28} + ( \beta_{1} + \beta_{10} + \beta_{11} ) q^{29} + ( -1 + \beta_{2} + 2 \beta_{3} - 2 \beta_{7} + 3 \beta_{9} ) q^{31} + ( \beta_{5} + 2 \beta_{6} ) q^{32} + ( 3 + 3 \beta_{3} - \beta_{7} - \beta_{9} ) q^{34} + ( 3 \beta_{1} - \beta_{6} - \beta_{8} + 2 \beta_{10} + 3 \beta_{11} ) q^{35} + ( -2 \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{7} + \beta_{9} ) q^{37} + ( 2 \beta_{1} + \beta_{8} - 2 \beta_{10} + 4 \beta_{11} ) q^{38} + ( -\beta_{2} - \beta_{4} + 2 \beta_{7} + 2 \beta_{9} ) q^{40} + ( -\beta_{1} + \beta_{5} + \beta_{6} + \beta_{8} - 2 \beta_{10} - 2 \beta_{11} ) q^{41} + ( -1 + \beta_{2} + \beta_{3} - \beta_{4} ) q^{43} + ( 6 \beta_{1} - \beta_{5} + \beta_{6} - 2 \beta_{8} ) q^{44} + ( 3 + \beta_{2} - 3 \beta_{3} - \beta_{4} ) q^{46} + ( -4 \beta_{1} + \beta_{5} - 2 \beta_{11} ) q^{47} + ( -3 - \beta_{3} - \beta_{4} + 2 \beta_{7} - 2 \beta_{9} ) q^{49} + ( -\beta_{1} - \beta_{5} + \beta_{6} - 3 \beta_{8} ) q^{50} + ( -8 + 4 \beta_{3} + 2 \beta_{7} - \beta_{9} ) q^{52} + ( -2 \beta_{1} + 2 \beta_{5} + \beta_{6} - \beta_{10} - 2 \beta_{11} ) q^{53} + ( -\beta_{2} - \beta_{7} + 3 \beta_{9} ) q^{55} + ( 4 \beta_{1} + \beta_{5} + \beta_{6} - \beta_{8} - \beta_{10} ) q^{56} + ( 3 - \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{7} - \beta_{9} ) q^{58} + ( -2 \beta_{1} - \beta_{5} - \beta_{11} ) q^{59} + ( -1 + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{7} + 2 \beta_{9} ) q^{61} + ( -4 \beta_{1} + \beta_{5} + 2 \beta_{8} + 2 \beta_{10} - 2 \beta_{11} ) q^{62} + ( 2 - \beta_{7} ) q^{64} + ( -\beta_{5} - 2 \beta_{6} + 3 \beta_{8} - 3 \beta_{10} - 3 \beta_{11} ) q^{65} + ( -7 + 2 \beta_{2} + 4 \beta_{4} - \beta_{7} - 2 \beta_{9} ) q^{67} + ( 3 \beta_{1} - 2 \beta_{8} + \beta_{10} - 3 \beta_{11} ) q^{68} + ( 9 - 2 \beta_{2} - 9 \beta_{3} + \beta_{4} + \beta_{7} + \beta_{9} ) q^{70} + ( \beta_{5} + 2 \beta_{6} - \beta_{11} ) q^{71} + ( -3 + 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 2 \beta_{7} - 2 \beta_{9} ) q^{73} + ( \beta_{1} - \beta_{5} + \beta_{6} - \beta_{8} ) q^{74} + ( 12 - 6 \beta_{3} - 3 \beta_{4} - 3 \beta_{7} + 3 \beta_{9} ) q^{76} + ( 4 \beta_{1} + \beta_{6} + \beta_{8} - 2 \beta_{10} + 4 \beta_{11} ) q^{77} + ( 5 + 2 \beta_{2} + 4 \beta_{4} - \beta_{7} - 2 \beta_{9} ) q^{79} + ( -3 \beta_{1} + \beta_{6} - 2 \beta_{8} + \beta_{10} + 3 \beta_{11} ) q^{80} + ( -6 + 3 \beta_{3} - \beta_{4} - \beta_{7} + \beta_{9} ) q^{82} + ( -2 \beta_{1} + \beta_{6} + 2 \beta_{11} ) q^{83} + ( 4 \beta_{2} + 2 \beta_{4} - 2 \beta_{7} - \beta_{9} ) q^{85} + ( 2 \beta_{1} + 2 \beta_{5} + \beta_{6} + \beta_{10} + 2 \beta_{11} ) q^{86} + ( 4 \beta_{2} - 12 \beta_{3} + 2 \beta_{4} - 2 \beta_{7} + 3 \beta_{9} ) q^{88} + ( -2 \beta_{1} + \beta_{8} - 2 \beta_{10} - 4 \beta_{11} ) q^{89} + ( -2 + 4 \beta_{2} + 4 \beta_{3} + 3 \beta_{4} - 2 \beta_{7} ) q^{91} + ( -2 \beta_{5} - \beta_{6} + 3 \beta_{10} ) q^{92} + ( -6 - 2 \beta_{2} + 12 \beta_{3} + 3 \beta_{7} - 4 \beta_{9} ) q^{94} + ( -\beta_{5} - 2 \beta_{6} - 3 \beta_{8} + 3 \beta_{10} ) q^{95} + ( 4 - 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + 2 \beta_{7} + 2 \beta_{9} ) q^{97} + ( 4 \beta_{1} + \beta_{5} + \beta_{6} - \beta_{8} - \beta_{10} + 7 \beta_{11} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 16q^{4} + 4q^{7} + O(q^{10}) \) \( 12q - 16q^{4} + 4q^{7} - 6q^{10} + 24q^{13} + 8q^{16} - 6q^{19} + 20q^{22} - 24q^{25} + 28q^{28} + 60q^{34} + 8q^{37} - 12q^{40} - 10q^{43} + 14q^{46} - 48q^{49} - 78q^{52} + 20q^{58} + 28q^{64} - 72q^{67} + 54q^{70} - 42q^{73} + 108q^{76} + 72q^{79} - 54q^{82} + 6q^{85} - 74q^{88} + 6q^{91} + 60q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 9 x^{10} + 59 x^{8} - 180 x^{6} + 403 x^{4} - 198 x^{2} + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 81 \nu^{11} - 531 \nu^{9} + 3481 \nu^{7} - 3627 \nu^{5} + 1782 \nu^{3} + 76298 \nu \)\()/21995\)
\(\beta_{2}\)\(=\)\((\)\( 117 \nu^{10} - 767 \nu^{8} + 7472 \nu^{6} - 27234 \nu^{4} + 90554 \nu^{2} - 60864 \)\()/21995\)
\(\beta_{3}\)\(=\)\((\)\( -1298 \nu^{10} + 10953 \nu^{8} - 71803 \nu^{6} + 202311 \nu^{4} - 490451 \nu^{2} + 240966 \)\()/197955\)
\(\beta_{4}\)\(=\)\((\)\( 461 \nu^{10} - 5466 \nu^{8} + 28501 \nu^{6} - 98847 \nu^{4} + 142112 \nu^{2} - 186237 \)\()/65985\)
\(\beta_{5}\)\(=\)\((\)\( 1298 \nu^{11} - 10953 \nu^{9} + 71803 \nu^{7} - 202311 \nu^{5} + 490451 \nu^{3} - 438921 \nu \)\()/197955\)
\(\beta_{6}\)\(=\)\((\)\( 1298 \nu^{11} - 10953 \nu^{9} + 71803 \nu^{7} - 202311 \nu^{5} + 490451 \nu^{3} + 154944 \nu \)\()/197955\)
\(\beta_{7}\)\(=\)\((\)\( -288 \nu^{10} + 1888 \nu^{8} - 9933 \nu^{6} + 12896 \nu^{4} - 6336 \nu^{2} - 68439 \)\()/21995\)
\(\beta_{8}\)\(=\)\((\)\( -288 \nu^{11} + 1888 \nu^{9} - 9933 \nu^{7} + 12896 \nu^{5} - 6336 \nu^{3} - 90434 \nu \)\()/21995\)
\(\beta_{9}\)\(=\)\((\)\( 1138 \nu^{10} - 12348 \nu^{8} + 80948 \nu^{6} - 273351 \nu^{4} + 552916 \nu^{2} - 271656 \)\()/65985\)
\(\beta_{10}\)\(=\)\((\)\( 4712 \nu^{11} - 47997 \nu^{9} + 314647 \nu^{7} - 1022364 \nu^{5} + 2149199 \nu^{3} - 1055934 \nu \)\()/197955\)
\(\beta_{11}\)\(=\)\((\)\( -5192 \nu^{11} + 43812 \nu^{9} - 287212 \nu^{7} + 809244 \nu^{5} - 1763849 \nu^{3} + 172044 \nu \)\()/197955\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{6} - \beta_{5}\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-2 \beta_{9} + 2 \beta_{7} + \beta_{4} - 9 \beta_{3} - \beta_{2} + 9\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(3 \beta_{11} + 8 \beta_{6} + 4 \beta_{5}\)\()/3\)
\(\nu^{4}\)\(=\)\(-\beta_{9} + 2 \beta_{7} - 2 \beta_{4} - 12 \beta_{3} - 4 \beta_{2}\)
\(\nu^{5}\)\(=\)\((\)\(18 \beta_{11} + 3 \beta_{10} + 17 \beta_{6} + 34 \beta_{5} + 18 \beta_{1}\)\()/3\)
\(\nu^{6}\)\(=\)\((\)\(32 \beta_{9} - 5 \beta_{7} - 64 \beta_{4} - 32 \beta_{2} - 153\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(27 \beta_{8} - 74 \beta_{6} + 74 \beta_{5} + 96 \beta_{1}\)\()/3\)
\(\nu^{8}\)\(=\)\(51 \beta_{9} - 51 \beta_{7} - 55 \beta_{4} + 222 \beta_{3} + 55 \beta_{2} - 222\)
\(\nu^{9}\)\(=\)\((\)\(-495 \beta_{11} - 177 \beta_{10} + 177 \beta_{8} - 656 \beta_{6} - 328 \beta_{5}\)\()/3\)
\(\nu^{10}\)\(=\)\((\)\(-191 \beta_{9} - 835 \beta_{7} + 835 \beta_{4} + 2952 \beta_{3} + 1670 \beta_{2}\)\()/3\)
\(\nu^{11}\)\(=\)\((\)\(-2505 \beta_{11} - 1026 \beta_{10} - 1477 \beta_{6} - 2954 \beta_{5} - 2505 \beta_{1}\)\()/3\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/567\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(\beta_{3}\) \(\beta_{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
215.1
−0.617942 0.356769i
1.90412 + 1.09935i
−1.65604 0.956115i
1.65604 + 0.956115i
−1.90412 1.09935i
0.617942 + 0.356769i
0.617942 0.356769i
−1.90412 + 1.09935i
1.65604 0.956115i
−1.65604 + 0.956115i
1.90412 1.09935i
−0.617942 + 0.356769i
2.49086i 0 −4.20440 0.617942 + 1.07031i 0 −1.10220 + 2.40523i 5.49086i 0 2.66599 1.53921i
215.2 1.83424i 0 −1.36445 −1.90412 3.29804i 0 0.317776 2.62660i 1.16576i 0 −6.04940 + 3.49262i
215.3 0.656620i 0 1.56885 1.65604 + 2.86834i 0 1.78442 + 1.95341i 2.34338i 0 1.88341 1.08739i
215.4 0.656620i 0 1.56885 −1.65604 2.86834i 0 1.78442 + 1.95341i 2.34338i 0 1.88341 1.08739i
215.5 1.83424i 0 −1.36445 1.90412 + 3.29804i 0 0.317776 2.62660i 1.16576i 0 −6.04940 + 3.49262i
215.6 2.49086i 0 −4.20440 −0.617942 1.07031i 0 −1.10220 + 2.40523i 5.49086i 0 2.66599 1.53921i
269.1 2.49086i 0 −4.20440 −0.617942 + 1.07031i 0 −1.10220 2.40523i 5.49086i 0 2.66599 + 1.53921i
269.2 1.83424i 0 −1.36445 1.90412 3.29804i 0 0.317776 + 2.62660i 1.16576i 0 −6.04940 3.49262i
269.3 0.656620i 0 1.56885 −1.65604 + 2.86834i 0 1.78442 1.95341i 2.34338i 0 1.88341 + 1.08739i
269.4 0.656620i 0 1.56885 1.65604 2.86834i 0 1.78442 1.95341i 2.34338i 0 1.88341 + 1.08739i
269.5 1.83424i 0 −1.36445 −1.90412 + 3.29804i 0 0.317776 + 2.62660i 1.16576i 0 −6.04940 3.49262i
269.6 2.49086i 0 −4.20440 0.617942 1.07031i 0 −1.10220 2.40523i 5.49086i 0 2.66599 + 1.53921i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 269.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
63.i even 6 1 inner
63.t odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 567.2.i.f 12
3.b odd 2 1 inner 567.2.i.f 12
7.d odd 6 1 567.2.s.f 12
9.c even 3 1 189.2.p.d 12
9.c even 3 1 567.2.s.f 12
9.d odd 6 1 189.2.p.d 12
9.d odd 6 1 567.2.s.f 12
21.g even 6 1 567.2.s.f 12
63.h even 3 1 1323.2.c.d 12
63.i even 6 1 inner 567.2.i.f 12
63.i even 6 1 1323.2.c.d 12
63.j odd 6 1 1323.2.c.d 12
63.k odd 6 1 189.2.p.d 12
63.s even 6 1 189.2.p.d 12
63.t odd 6 1 inner 567.2.i.f 12
63.t odd 6 1 1323.2.c.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.p.d 12 9.c even 3 1
189.2.p.d 12 9.d odd 6 1
189.2.p.d 12 63.k odd 6 1
189.2.p.d 12 63.s even 6 1
567.2.i.f 12 1.a even 1 1 trivial
567.2.i.f 12 3.b odd 2 1 inner
567.2.i.f 12 63.i even 6 1 inner
567.2.i.f 12 63.t odd 6 1 inner
567.2.s.f 12 7.d odd 6 1
567.2.s.f 12 9.c even 3 1
567.2.s.f 12 9.d odd 6 1
567.2.s.f 12 21.g even 6 1
1323.2.c.d 12 63.h even 3 1
1323.2.c.d 12 63.i even 6 1
1323.2.c.d 12 63.j odd 6 1
1323.2.c.d 12 63.t odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(567, [\chi])\):

\( T_{2}^{6} + 10 T_{2}^{4} + 25 T_{2}^{2} + 9 \)
\( T_{11}^{12} - 37 T_{11}^{10} + 1155 T_{11}^{8} - 7468 T_{11}^{6} + 37471 T_{11}^{4} - 48150 T_{11}^{2} + 50625 \)
\( T_{13}^{6} - 12 T_{13}^{5} + 51 T_{13}^{4} - 36 T_{13}^{3} - 171 T_{13}^{2} + 135 T_{13} + 675 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 9 + 25 T^{2} + 10 T^{4} + T^{6} )^{2} \)
$3$ \( T^{12} \)
$5$ \( 59049 + 48114 T^{2} + 32643 T^{4} + 4860 T^{6} + 531 T^{8} + 27 T^{10} + T^{12} \)
$7$ \( ( 343 - 98 T + 98 T^{2} - 23 T^{3} + 14 T^{4} - 2 T^{5} + T^{6} )^{2} \)
$11$ \( 50625 - 48150 T^{2} + 37471 T^{4} - 7468 T^{6} + 1155 T^{8} - 37 T^{10} + T^{12} \)
$13$ \( ( 675 + 135 T - 171 T^{2} - 36 T^{3} + 51 T^{4} - 12 T^{5} + T^{6} )^{2} \)
$17$ \( 59049 + 54675 T^{2} + 43335 T^{4} + 6264 T^{6} + 675 T^{8} + 30 T^{10} + T^{12} \)
$19$ \( ( 243 + 648 T + 549 T^{2} - 72 T^{3} - 21 T^{4} + 3 T^{5} + T^{6} )^{2} \)
$23$ \( 531441 - 376893 T^{2} + 198763 T^{4} - 47140 T^{6} + 8319 T^{8} - 94 T^{10} + T^{12} \)
$29$ \( 6561 - 26082 T^{2} + 100687 T^{4} - 11752 T^{6} + 1047 T^{8} - 37 T^{10} + T^{12} \)
$31$ \( ( 64827 + 5733 T^{2} + 138 T^{4} + T^{6} )^{2} \)
$37$ \( ( 4489 - 1273 T + 629 T^{2} - 58 T^{3} + 35 T^{4} - 4 T^{5} + T^{6} )^{2} \)
$41$ \( 59049 + 474579 T^{2} + 3786507 T^{4} + 222156 T^{6} + 11043 T^{8} + 114 T^{10} + T^{12} \)
$43$ \( ( 1 - 16 T + 251 T^{2} - 82 T^{3} + 41 T^{4} + 5 T^{5} + T^{6} )^{2} \)
$47$ \( ( -19683 + 3870 T^{2} - 135 T^{4} + T^{6} )^{2} \)
$53$ \( 81 - 15453 T^{2} + 2947027 T^{4} - 202588 T^{6} + 12207 T^{8} - 118 T^{10} + T^{12} \)
$59$ \( ( -2187 + 738 T^{2} - 63 T^{4} + T^{6} )^{2} \)
$61$ \( ( 49923 + 4923 T^{2} + 129 T^{4} + T^{6} )^{2} \)
$67$ \( ( -677 + 15 T + 18 T^{2} + T^{3} )^{4} \)
$71$ \( ( 19881 + 2290 T^{2} + 85 T^{4} + T^{6} )^{2} \)
$73$ \( ( 177147 + 17496 T - 4527 T^{2} - 504 T^{3} + 123 T^{4} + 21 T^{5} + T^{6} )^{2} \)
$79$ \( ( 7 + 15 T - 18 T^{2} + T^{3} )^{4} \)
$83$ \( 36905625 + 35320050 T^{2} + 32836671 T^{4} + 912276 T^{6} + 19467 T^{8} + 159 T^{10} + T^{12} \)
$89$ \( 7695324729 + 848720025 T^{2} + 75973302 T^{4} + 1769229 T^{6} + 30726 T^{8} + 201 T^{10} + T^{12} \)
$97$ \( ( 1728 + 3456 T + 1584 T^{2} - 1440 T^{3} + 348 T^{4} - 30 T^{5} + T^{6} )^{2} \)
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