Properties

Label 1323.2.h.d
Level $1323$
Weight $2$
Character orbit 1323.h
Analytic conductor $10.564$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1323.h (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.5642081874\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 63)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( 1 - \beta_{1} + \beta_{3} ) q^{4} + ( -\beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{5} + ( 2 - \beta_{1} + \beta_{3} ) q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( 1 - \beta_{1} + \beta_{3} ) q^{4} + ( -\beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{5} + ( 2 - \beta_{1} + \beta_{3} ) q^{8} + ( 3 \beta_{1} + \beta_{4} - 3 \beta_{5} ) q^{10} + ( -1 - \beta_{2} + \beta_{4} - 2 \beta_{5} ) q^{11} + ( -1 + \beta_{4} ) q^{13} + ( -2 \beta_{1} - \beta_{3} ) q^{16} + ( -\beta_{1} + \beta_{2} + \beta_{3} - 4 \beta_{4} + \beta_{5} ) q^{17} + ( 1 - \beta_{2} - \beta_{4} - \beta_{5} ) q^{19} + ( 2 \beta_{1} - \beta_{2} - \beta_{3} - 5 \beta_{4} - 2 \beta_{5} ) q^{20} + ( 5 - 2 \beta_{2} - 5 \beta_{4} - 2 \beta_{5} ) q^{22} + ( \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{23} + ( -3 + 2 \beta_{2} + 3 \beta_{4} - \beta_{5} ) q^{25} + \beta_{5} q^{26} + ( -\beta_{1} + \beta_{5} ) q^{29} + ( -2 - 2 \beta_{1} - \beta_{3} ) q^{31} + ( 3 + \beta_{1} ) q^{32} + ( 2 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{34} + 3 \beta_{2} q^{37} + ( 2 - \beta_{2} - 2 \beta_{4} - 3 \beta_{5} ) q^{38} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 7 \beta_{4} - 2 \beta_{5} ) q^{40} + ( -7 - \beta_{2} + 7 \beta_{4} ) q^{41} + ( -4 \beta_{1} + \beta_{2} + \beta_{3} + 4 \beta_{5} ) q^{43} + ( 6 - 6 \beta_{4} - 5 \beta_{5} ) q^{44} + ( -2 \beta_{1} - \beta_{2} - \beta_{3} - 5 \beta_{4} + 2 \beta_{5} ) q^{46} + ( 3 - 3 \beta_{1} + 3 \beta_{3} ) q^{47} + ( 5 - \beta_{2} - 5 \beta_{4} + 4 \beta_{5} ) q^{50} + ( -1 + \beta_{2} + \beta_{4} + \beta_{5} ) q^{52} + ( \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 5 \beta_{4} - \beta_{5} ) q^{53} + ( 5 \beta_{1} + \beta_{3} ) q^{55} + ( -\beta_{1} + \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{58} + ( 4 + \beta_{1} + 2 \beta_{3} ) q^{59} + ( -1 + \beta_{1} + 2 \beta_{3} ) q^{61} + ( 7 + \beta_{1} + 2 \beta_{3} ) q^{62} + ( -3 + 2 \beta_{1} + \beta_{3} ) q^{64} + ( 2 + \beta_{3} ) q^{65} + ( 2 + 7 \beta_{1} - \beta_{3} ) q^{67} + ( \beta_{1} - 4 \beta_{2} - 4 \beta_{3} + \beta_{4} - \beta_{5} ) q^{68} + ( -2 + \beta_{1} + 2 \beta_{3} ) q^{71} + ( \beta_{1} + 5 \beta_{2} + 5 \beta_{3} + \beta_{4} - \beta_{5} ) q^{73} + ( 3 - 3 \beta_{4} + 3 \beta_{5} ) q^{74} + ( 6 - \beta_{2} - 6 \beta_{4} - 4 \beta_{5} ) q^{76} + ( 3 - 5 \beta_{1} - \beta_{3} ) q^{79} + ( 7 \beta_{1} + 6 \beta_{4} - 7 \beta_{5} ) q^{80} + ( -1 + \beta_{4} + 6 \beta_{5} ) q^{82} + ( -\beta_{1} + \beta_{2} + \beta_{3} - 4 \beta_{4} + \beta_{5} ) q^{83} + ( -5 + 4 \beta_{2} + 5 \beta_{4} - 2 \beta_{5} ) q^{85} + ( -5 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} + 11 \beta_{4} + 5 \beta_{5} ) q^{86} + ( 5 - \beta_{2} - 5 \beta_{4} - 7 \beta_{5} ) q^{88} + ( 1 + \beta_{2} - \beta_{4} + 6 \beta_{5} ) q^{89} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 5 \beta_{4} - 2 \beta_{5} ) q^{92} + ( 6 - 9 \beta_{1} + 3 \beta_{3} ) q^{94} + ( -5 + 2 \beta_{1} - \beta_{3} ) q^{95} + ( -5 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 5 \beta_{5} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 2q^{2} + 6q^{4} - 5q^{5} + 12q^{8} + O(q^{10}) \) \( 6q + 2q^{2} + 6q^{4} - 5q^{5} + 12q^{8} - 2q^{11} - 3q^{13} + 6q^{16} - 12q^{17} + 3q^{19} - 16q^{20} + 15q^{22} - 6q^{25} - q^{26} + q^{29} - 6q^{31} + 16q^{32} + 3q^{34} + 3q^{37} + 8q^{38} - 21q^{40} - 22q^{41} + 3q^{43} + 23q^{44} - 12q^{46} + 18q^{47} + 10q^{50} - 3q^{52} - 18q^{53} - 12q^{55} + 9q^{58} + 18q^{59} - 12q^{61} + 36q^{62} - 24q^{64} + 10q^{65} + 6q^{68} - 18q^{71} - 3q^{73} + 6q^{74} + 21q^{76} + 30q^{79} + 11q^{80} - 9q^{82} - 12q^{83} - 9q^{85} + 34q^{86} + 21q^{88} - 2q^{89} + 15q^{92} + 48q^{94} - 32q^{95} - 3q^{97} + O(q^{100}) \)

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 \(\left(\mathbb{Z}/1323\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(-1 + \beta_{4}\) \(-1 + \beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
226.1
0.500000 0.224437i
0.500000 + 1.41036i
0.500000 2.05195i
0.500000 + 0.224437i
0.500000 1.41036i
0.500000 + 2.05195i
−1.69963 0 0.888736 −1.79418 3.10761i 0 0 1.88874 0 3.04944 + 5.28179i
226.2 0.239123 0 −1.94282 0.590972 + 1.02359i 0 0 −0.942820 0 0.141315 + 0.244765i
226.3 2.46050 0 4.05408 −1.29679 2.24611i 0 0 5.05408 0 −3.19076 5.52655i
802.1 −1.69963 0 0.888736 −1.79418 + 3.10761i 0 0 1.88874 0 3.04944 5.28179i
802.2 0.239123 0 −1.94282 0.590972 1.02359i 0 0 −0.942820 0 0.141315 0.244765i
802.3 2.46050 0 4.05408 −1.29679 + 2.24611i 0 0 5.05408 0 −3.19076 + 5.52655i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 802.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1323.2.h.d 6
3.b odd 2 1 441.2.h.c 6
7.b odd 2 1 1323.2.h.e 6
7.c even 3 1 189.2.f.a 6
7.c even 3 1 1323.2.g.c 6
7.d odd 6 1 1323.2.f.c 6
7.d odd 6 1 1323.2.g.b 6
9.c even 3 1 1323.2.g.c 6
9.d odd 6 1 441.2.g.e 6
21.c even 2 1 441.2.h.b 6
21.g even 6 1 441.2.f.d 6
21.g even 6 1 441.2.g.d 6
21.h odd 6 1 63.2.f.b 6
21.h odd 6 1 441.2.g.e 6
28.g odd 6 1 3024.2.r.g 6
63.g even 3 1 189.2.f.a 6
63.h even 3 1 567.2.a.g 3
63.h even 3 1 inner 1323.2.h.d 6
63.i even 6 1 441.2.h.b 6
63.i even 6 1 3969.2.a.m 3
63.j odd 6 1 441.2.h.c 6
63.j odd 6 1 567.2.a.d 3
63.k odd 6 1 1323.2.f.c 6
63.l odd 6 1 1323.2.g.b 6
63.n odd 6 1 63.2.f.b 6
63.o even 6 1 441.2.g.d 6
63.s even 6 1 441.2.f.d 6
63.t odd 6 1 1323.2.h.e 6
63.t odd 6 1 3969.2.a.p 3
84.n even 6 1 1008.2.r.k 6
252.o even 6 1 1008.2.r.k 6
252.u odd 6 1 9072.2.a.cd 3
252.bb even 6 1 9072.2.a.bq 3
252.bl odd 6 1 3024.2.r.g 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.f.b 6 21.h odd 6 1
63.2.f.b 6 63.n odd 6 1
189.2.f.a 6 7.c even 3 1
189.2.f.a 6 63.g even 3 1
441.2.f.d 6 21.g even 6 1
441.2.f.d 6 63.s even 6 1
441.2.g.d 6 21.g even 6 1
441.2.g.d 6 63.o even 6 1
441.2.g.e 6 9.d odd 6 1
441.2.g.e 6 21.h odd 6 1
441.2.h.b 6 21.c even 2 1
441.2.h.b 6 63.i even 6 1
441.2.h.c 6 3.b odd 2 1
441.2.h.c 6 63.j odd 6 1
567.2.a.d 3 63.j odd 6 1
567.2.a.g 3 63.h even 3 1
1008.2.r.k 6 84.n even 6 1
1008.2.r.k 6 252.o even 6 1
1323.2.f.c 6 7.d odd 6 1
1323.2.f.c 6 63.k odd 6 1
1323.2.g.b 6 7.d odd 6 1
1323.2.g.b 6 63.l odd 6 1
1323.2.g.c 6 7.c even 3 1
1323.2.g.c 6 9.c even 3 1
1323.2.h.d 6 1.a even 1 1 trivial
1323.2.h.d 6 63.h even 3 1 inner
1323.2.h.e 6 7.b odd 2 1
1323.2.h.e 6 63.t odd 6 1
3024.2.r.g 6 28.g odd 6 1
3024.2.r.g 6 252.bl odd 6 1
3969.2.a.m 3 63.i even 6 1
3969.2.a.p 3 63.t odd 6 1
9072.2.a.bq 3 252.bb even 6 1
9072.2.a.cd 3 252.u odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1323, [\chi])\):

\( T_{2}^{3} - T_{2}^{2} - 4 T_{2} + 1 \)
\( T_{5}^{6} + 5 T_{5}^{5} + 23 T_{5}^{4} + 32 T_{5}^{3} + 59 T_{5}^{2} - 22 T_{5} + 121 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 4 T - T^{2} + T^{3} )^{2} \)
$3$ \( T^{6} \)
$5$ \( 121 - 22 T + 59 T^{2} + 32 T^{3} + 23 T^{4} + 5 T^{5} + T^{6} \)
$7$ \( T^{6} \)
$11$ \( 2209 + 893 T + 455 T^{2} + 56 T^{3} + 23 T^{4} + 2 T^{5} + T^{6} \)
$13$ \( ( 1 + T + T^{2} )^{3} \)
$17$ \( 729 + 1053 T + 1197 T^{2} + 414 T^{3} + 105 T^{4} + 12 T^{5} + T^{6} \)
$19$ \( 49 - 42 T + 57 T^{2} + 4 T^{3} + 15 T^{4} - 3 T^{5} + T^{6} \)
$23$ \( 81 + 297 T + 1089 T^{2} + 18 T^{3} + 33 T^{4} + T^{6} \)
$29$ \( 1 - 4 T + 17 T^{2} + 2 T^{3} + 5 T^{4} - T^{5} + T^{6} \)
$31$ \( ( 27 - 24 T + 3 T^{2} + T^{3} )^{2} \)
$37$ \( 6561 + 4374 T + 2673 T^{2} + 324 T^{3} + 63 T^{4} - 3 T^{5} + T^{6} \)
$41$ \( 124609 + 54715 T + 16259 T^{2} + 2704 T^{3} + 329 T^{4} + 22 T^{5} + T^{6} \)
$43$ \( 14641 + 7986 T + 3993 T^{2} + 440 T^{3} + 75 T^{4} - 3 T^{5} + T^{6} \)
$47$ \( ( 189 - 54 T - 9 T^{2} + T^{3} )^{2} \)
$53$ \( 81 + 675 T + 5463 T^{2} + 1332 T^{3} + 249 T^{4} + 18 T^{5} + T^{6} \)
$59$ \( ( 63 - 6 T - 9 T^{2} + T^{3} )^{2} \)
$61$ \( ( -67 - 21 T + 6 T^{2} + T^{3} )^{2} \)
$67$ \( ( 683 - 207 T + T^{3} )^{2} \)
$71$ \( ( -81 - 6 T + 9 T^{2} + T^{3} )^{2} \)
$73$ \( 59049 - 40824 T + 27495 T^{2} - 990 T^{3} + 177 T^{4} + 3 T^{5} + T^{6} \)
$79$ \( ( 769 - 48 T - 15 T^{2} + T^{3} )^{2} \)
$83$ \( 729 + 1053 T + 1197 T^{2} + 414 T^{3} + 105 T^{4} + 12 T^{5} + T^{6} \)
$89$ \( 143641 - 57229 T + 22043 T^{2} - 1060 T^{3} + 155 T^{4} + 2 T^{5} + T^{6} \)
$97$ \( 363609 + 68742 T + 14805 T^{2} + 864 T^{3} + 123 T^{4} + 3 T^{5} + T^{6} \)
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