Properties

Label 1323.2.h.a
Level 1323
Weight 2
Character orbit 1323.h
Analytic conductor 10.564
Analytic rank 0
Dimension 2
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - q^{4} + ( 1 - \zeta_{6} ) q^{5} + 3 q^{8} +O(q^{10})\) \( q - q^{2} - q^{4} + ( 1 - \zeta_{6} ) q^{5} + 3 q^{8} + ( -1 + \zeta_{6} ) q^{10} + 5 \zeta_{6} q^{11} -5 \zeta_{6} q^{13} - q^{16} + ( -3 + 3 \zeta_{6} ) q^{17} + \zeta_{6} q^{19} + ( -1 + \zeta_{6} ) q^{20} -5 \zeta_{6} q^{22} + ( 3 - 3 \zeta_{6} ) q^{23} + 4 \zeta_{6} q^{25} + 5 \zeta_{6} q^{26} + ( -1 + \zeta_{6} ) q^{29} -5 q^{32} + ( 3 - 3 \zeta_{6} ) q^{34} -3 \zeta_{6} q^{37} -\zeta_{6} q^{38} + ( 3 - 3 \zeta_{6} ) q^{40} + 5 \zeta_{6} q^{41} + ( 1 - \zeta_{6} ) q^{43} -5 \zeta_{6} q^{44} + ( -3 + 3 \zeta_{6} ) q^{46} -4 \zeta_{6} q^{50} + 5 \zeta_{6} q^{52} + ( -9 + 9 \zeta_{6} ) q^{53} + 5 q^{55} + ( 1 - \zeta_{6} ) q^{58} + 14 q^{61} + 7 q^{64} -5 q^{65} + 4 q^{67} + ( 3 - 3 \zeta_{6} ) q^{68} + 12 q^{71} + ( 3 - 3 \zeta_{6} ) q^{73} + 3 \zeta_{6} q^{74} -\zeta_{6} q^{76} + 8 q^{79} + ( -1 + \zeta_{6} ) q^{80} -5 \zeta_{6} q^{82} + ( 9 - 9 \zeta_{6} ) q^{83} + 3 \zeta_{6} q^{85} + ( -1 + \zeta_{6} ) q^{86} + 15 \zeta_{6} q^{88} + 13 \zeta_{6} q^{89} + ( -3 + 3 \zeta_{6} ) q^{92} + q^{95} + ( -9 + 9 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 2q^{4} + q^{5} + 6q^{8} + O(q^{10}) \) \( 2q - 2q^{2} - 2q^{4} + q^{5} + 6q^{8} - q^{10} + 5q^{11} - 5q^{13} - 2q^{16} - 3q^{17} + q^{19} - q^{20} - 5q^{22} + 3q^{23} + 4q^{25} + 5q^{26} - q^{29} - 10q^{32} + 3q^{34} - 3q^{37} - q^{38} + 3q^{40} + 5q^{41} + q^{43} - 5q^{44} - 3q^{46} - 4q^{50} + 5q^{52} - 9q^{53} + 10q^{55} + q^{58} + 28q^{61} + 14q^{64} - 10q^{65} + 8q^{67} + 3q^{68} + 24q^{71} + 3q^{73} + 3q^{74} - q^{76} + 16q^{79} - q^{80} - 5q^{82} + 9q^{83} + 3q^{85} - q^{86} + 15q^{88} + 13q^{89} - 3q^{92} + 2q^{95} - 9q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1323\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(-\zeta_{6}\) \(-\zeta_{6}\)

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 0.866025i
0.500000 + 0.866025i
−1.00000 0 −1.00000 0.500000 + 0.866025i 0 0 3.00000 0 −0.500000 0.866025i
802.1 −1.00000 0 −1.00000 0.500000 0.866025i 0 0 3.00000 0 −0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
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.a 2
3.b odd 2 1 441.2.h.a 2
7.b odd 2 1 189.2.h.a 2
7.c even 3 1 1323.2.f.b 2
7.c even 3 1 1323.2.g.a 2
7.d odd 6 1 189.2.g.a 2
7.d odd 6 1 1323.2.f.a 2
9.c even 3 1 1323.2.g.a 2
9.d odd 6 1 441.2.g.a 2
21.c even 2 1 63.2.h.a yes 2
21.g even 6 1 63.2.g.a 2
21.g even 6 1 441.2.f.b 2
21.h odd 6 1 441.2.f.a 2
21.h odd 6 1 441.2.g.a 2
28.d even 2 1 3024.2.q.b 2
28.f even 6 1 3024.2.t.d 2
63.g even 3 1 1323.2.f.b 2
63.h even 3 1 inner 1323.2.h.a 2
63.h even 3 1 3969.2.a.a 1
63.i even 6 1 63.2.h.a yes 2
63.i even 6 1 3969.2.a.d 1
63.j odd 6 1 441.2.h.a 2
63.j odd 6 1 3969.2.a.f 1
63.k odd 6 1 567.2.e.b 2
63.k odd 6 1 1323.2.f.a 2
63.l odd 6 1 189.2.g.a 2
63.l odd 6 1 567.2.e.b 2
63.n odd 6 1 441.2.f.a 2
63.o even 6 1 63.2.g.a 2
63.o even 6 1 567.2.e.a 2
63.s even 6 1 441.2.f.b 2
63.s even 6 1 567.2.e.a 2
63.t odd 6 1 189.2.h.a 2
63.t odd 6 1 3969.2.a.c 1
84.h odd 2 1 1008.2.q.c 2
84.j odd 6 1 1008.2.t.d 2
252.r odd 6 1 1008.2.q.c 2
252.s odd 6 1 1008.2.t.d 2
252.bi even 6 1 3024.2.t.d 2
252.bj even 6 1 3024.2.q.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.g.a 2 21.g even 6 1
63.2.g.a 2 63.o even 6 1
63.2.h.a yes 2 21.c even 2 1
63.2.h.a yes 2 63.i even 6 1
189.2.g.a 2 7.d odd 6 1
189.2.g.a 2 63.l odd 6 1
189.2.h.a 2 7.b odd 2 1
189.2.h.a 2 63.t odd 6 1
441.2.f.a 2 21.h odd 6 1
441.2.f.a 2 63.n odd 6 1
441.2.f.b 2 21.g even 6 1
441.2.f.b 2 63.s even 6 1
441.2.g.a 2 9.d odd 6 1
441.2.g.a 2 21.h odd 6 1
441.2.h.a 2 3.b odd 2 1
441.2.h.a 2 63.j odd 6 1
567.2.e.a 2 63.o even 6 1
567.2.e.a 2 63.s even 6 1
567.2.e.b 2 63.k odd 6 1
567.2.e.b 2 63.l odd 6 1
1008.2.q.c 2 84.h odd 2 1
1008.2.q.c 2 252.r odd 6 1
1008.2.t.d 2 84.j odd 6 1
1008.2.t.d 2 252.s odd 6 1
1323.2.f.a 2 7.d odd 6 1
1323.2.f.a 2 63.k odd 6 1
1323.2.f.b 2 7.c even 3 1
1323.2.f.b 2 63.g even 3 1
1323.2.g.a 2 7.c even 3 1
1323.2.g.a 2 9.c even 3 1
1323.2.h.a 2 1.a even 1 1 trivial
1323.2.h.a 2 63.h even 3 1 inner
3024.2.q.b 2 28.d even 2 1
3024.2.q.b 2 252.bj even 6 1
3024.2.t.d 2 28.f even 6 1
3024.2.t.d 2 252.bi even 6 1
3969.2.a.a 1 63.h even 3 1
3969.2.a.c 1 63.t odd 6 1
3969.2.a.d 1 63.i even 6 1
3969.2.a.f 1 63.j odd 6 1

Hecke kernels

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

\( T_{2} + 1 \)
\( T_{5}^{2} - T_{5} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + 2 T^{2} )^{2} \)
$3$ 1
$5$ \( 1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4} \)
$7$ 1
$11$ \( 1 - 5 T + 14 T^{2} - 55 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - 2 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \)
$17$ \( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \)
$19$ \( ( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \)
$23$ \( 1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4} \)
$29$ \( 1 + T - 28 T^{2} + 29 T^{3} + 841 T^{4} \)
$31$ \( ( 1 + 31 T^{2} )^{2} \)
$37$ \( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} \)
$41$ \( 1 - 5 T - 16 T^{2} - 205 T^{3} + 1681 T^{4} \)
$43$ \( 1 - T - 42 T^{2} - 43 T^{3} + 1849 T^{4} \)
$47$ \( ( 1 + 47 T^{2} )^{2} \)
$53$ \( 1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4} \)
$59$ \( ( 1 + 59 T^{2} )^{2} \)
$61$ \( ( 1 - 14 T + 61 T^{2} )^{2} \)
$67$ \( ( 1 - 4 T + 67 T^{2} )^{2} \)
$71$ \( ( 1 - 12 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 3 T - 64 T^{2} - 219 T^{3} + 5329 T^{4} \)
$79$ \( ( 1 - 8 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 9 T - 2 T^{2} - 747 T^{3} + 6889 T^{4} \)
$89$ \( 1 - 13 T + 80 T^{2} - 1157 T^{3} + 7921 T^{4} \)
$97$ \( 1 + 9 T - 16 T^{2} + 873 T^{3} + 9409 T^{4} \)
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