Properties

Label 1323.2.f.d
Level $1323$
Weight $2$
Character orbit 1323.f
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.f (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: \(\Q(\zeta_{18})\)
Defining polynomial: \(x^{6} - x^{3} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
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 primitive root of unity \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{2} + ( -1 + 2 \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{4} + ( -1 - \zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{3} ) q^{5} + ( -2 - \zeta_{18} - \zeta_{18}^{2} + 3 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{8} +O(q^{10})\) \( q + ( -\zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{2} + ( -1 + 2 \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{4} + ( -1 - \zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{3} ) q^{5} + ( -2 - \zeta_{18} - \zeta_{18}^{2} + 3 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{8} + ( \zeta_{18} + \zeta_{18}^{2} - 2 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{10} + ( \zeta_{18} - \zeta_{18}^{2} + 2 \zeta_{18}^{3} - \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{11} + ( -1 + 2 \zeta_{18} - 4 \zeta_{18}^{2} + \zeta_{18}^{3} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{13} + ( -3 \zeta_{18} + 3 \zeta_{18}^{2} - \zeta_{18}^{3} + 3 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{16} + ( 2 - \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{4} ) q^{17} + ( 1 + \zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{19} + ( \zeta_{18} - \zeta_{18}^{2} + 2 \zeta_{18}^{3} - \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{20} + ( -3 + 4 \zeta_{18} - 2 \zeta_{18}^{2} + 3 \zeta_{18}^{3} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{22} + ( 4 + \zeta_{18} + 2 \zeta_{18}^{2} - 4 \zeta_{18}^{3} - 3 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{23} + ( 2 \zeta_{18} - \zeta_{18}^{2} + 2 \zeta_{18}^{3} - \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{25} + ( -1 + \zeta_{18} + \zeta_{18}^{2} + 6 \zeta_{18}^{4} - 7 \zeta_{18}^{5} ) q^{26} + ( -4 \zeta_{18} + 5 \zeta_{18}^{2} + 3 \zeta_{18}^{3} + 5 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{29} + ( -1 + 3 \zeta_{18} + 3 \zeta_{18}^{2} + \zeta_{18}^{3} - 6 \zeta_{18}^{4} - 6 \zeta_{18}^{5} ) q^{31} + ( -3 \zeta_{18} + 3 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{32} + ( -\zeta_{18} - \zeta_{18}^{2} + 3 \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{34} + ( -1 + 6 \zeta_{18} + 6 \zeta_{18}^{2} - 3 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{37} + ( -3 \zeta_{18} + 2 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{38} + ( -3 + 2 \zeta_{18} + 2 \zeta_{18}^{2} + 3 \zeta_{18}^{3} - 4 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{40} + ( -\zeta_{18} - \zeta_{18}^{2} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{41} + ( \zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{43} + ( -5 + 3 \zeta_{18} + 3 \zeta_{18}^{2} + 4 \zeta_{18}^{4} - 7 \zeta_{18}^{5} ) q^{44} + ( -4 \zeta_{18} - 4 \zeta_{18}^{2} - \zeta_{18}^{4} + 5 \zeta_{18}^{5} ) q^{46} + ( -3 \zeta_{18} - 2 \zeta_{18}^{2} - \zeta_{18}^{3} - 2 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{47} + ( -2 + 5 \zeta_{18} - 3 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{50} + ( -5 \zeta_{18} + 10 \zeta_{18}^{2} - 7 \zeta_{18}^{3} + 10 \zeta_{18}^{4} - 5 \zeta_{18}^{5} ) q^{52} + ( -2 + \zeta_{18} + \zeta_{18}^{2} + 2 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{53} + ( -\zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{55} + ( 3 - 6 \zeta_{18} + 9 \zeta_{18}^{2} - 3 \zeta_{18}^{3} - 3 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{58} + ( 1 + 5 \zeta_{18} - \zeta_{18}^{3} - 5 \zeta_{18}^{4} - 5 \zeta_{18}^{5} ) q^{59} + ( 3 \zeta_{18}^{2} + 2 \zeta_{18}^{3} + 3 \zeta_{18}^{4} ) q^{61} + ( -10 + \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{5} ) q^{62} + ( 4 + 3 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{64} + ( -5 \zeta_{18} - \zeta_{18}^{2} + 5 \zeta_{18}^{3} - \zeta_{18}^{4} - 5 \zeta_{18}^{5} ) q^{65} + ( 4 - 3 \zeta_{18}^{2} - 4 \zeta_{18}^{3} + 3 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{67} + ( -2 + 3 \zeta_{18} - 2 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{68} + ( -3 + 3 \zeta_{18} + 3 \zeta_{18}^{2} - 6 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{71} + ( 7 - 5 \zeta_{18} - 5 \zeta_{18}^{2} + \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{73} + ( \zeta_{18}^{2} - 10 \zeta_{18}^{3} + \zeta_{18}^{4} ) q^{74} + ( 5 - 3 \zeta_{18} + 3 \zeta_{18}^{2} - 5 \zeta_{18}^{3} ) q^{76} + ( -3 \zeta_{18} + 7 \zeta_{18}^{3} - 3 \zeta_{18}^{5} ) q^{79} + ( -5 + 3 \zeta_{18} + 3 \zeta_{18}^{2} - 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{80} + 3 q^{82} + ( \zeta_{18} + 4 \zeta_{18}^{2} + 6 \zeta_{18}^{3} + 4 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{83} + ( -3 - 2 \zeta_{18} + 4 \zeta_{18}^{2} + 3 \zeta_{18}^{3} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{85} + ( 2 + \zeta_{18} - 2 \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{86} + ( -7 \zeta_{18} + 8 \zeta_{18}^{2} - 9 \zeta_{18}^{3} + 8 \zeta_{18}^{4} - 7 \zeta_{18}^{5} ) q^{88} + ( 4 + 4 \zeta_{18} + 4 \zeta_{18}^{2} - 7 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{89} + ( 2 \zeta_{18} - 8 \zeta_{18}^{2} + \zeta_{18}^{3} - 8 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{92} + ( -6 - 2 \zeta_{18} + \zeta_{18}^{2} + 6 \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{94} + ( -4 - \zeta_{18} + \zeta_{18}^{2} + 4 \zeta_{18}^{3} ) q^{95} + ( 8 \zeta_{18} - 7 \zeta_{18}^{2} - \zeta_{18}^{3} - 7 \zeta_{18}^{4} + 8 \zeta_{18}^{5} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 3q^{2} - 3q^{4} - 3q^{5} - 12q^{8} + O(q^{10}) \) \( 6q + 3q^{2} - 3q^{4} - 3q^{5} - 12q^{8} + 6q^{11} - 3q^{13} - 3q^{16} + 12q^{17} + 6q^{19} + 6q^{20} - 9q^{22} + 12q^{23} + 6q^{25} - 6q^{26} + 9q^{29} - 3q^{31} + 9q^{34} - 6q^{37} - 6q^{38} - 9q^{40} + 3q^{43} - 30q^{44} - 3q^{47} - 6q^{50} - 21q^{52} - 12q^{53} + 9q^{58} + 3q^{59} + 6q^{61} - 60q^{62} + 24q^{64} + 15q^{65} + 12q^{67} - 6q^{68} - 18q^{71} + 42q^{73} - 30q^{74} + 15q^{76} + 21q^{79} - 30q^{80} + 18q^{82} + 18q^{83} - 9q^{85} + 6q^{86} - 27q^{88} + 24q^{89} + 3q^{92} - 18q^{94} - 12q^{95} - 3q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(-1 + \zeta_{18}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
442.1
0.939693 + 0.342020i
−0.173648 0.984808i
−0.766044 + 0.642788i
0.939693 0.342020i
−0.173648 + 0.984808i
−0.766044 0.642788i
−0.439693 0.761570i 0 0.613341 1.06234i −0.673648 + 1.16679i 0 0 −2.83750 0 1.18479
442.2 0.673648 + 1.16679i 0 0.0923963 0.160035i −1.26604 + 2.19285i 0 0 2.94356 0 −3.41147
442.3 1.26604 + 2.19285i 0 −2.20574 + 3.82045i 0.439693 0.761570i 0 0 −6.10607 0 2.22668
883.1 −0.439693 + 0.761570i 0 0.613341 + 1.06234i −0.673648 1.16679i 0 0 −2.83750 0 1.18479
883.2 0.673648 1.16679i 0 0.0923963 + 0.160035i −1.26604 2.19285i 0 0 2.94356 0 −3.41147
883.3 1.26604 2.19285i 0 −2.20574 3.82045i 0.439693 + 0.761570i 0 0 −6.10607 0 2.22668
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 883.3
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 1323.2.f.d 6
3.b odd 2 1 441.2.f.c 6
7.b odd 2 1 189.2.f.b 6
7.c even 3 1 1323.2.g.e 6
7.c even 3 1 1323.2.h.b 6
7.d odd 6 1 1323.2.g.d 6
7.d odd 6 1 1323.2.h.c 6
9.c even 3 1 inner 1323.2.f.d 6
9.c even 3 1 3969.2.a.l 3
9.d odd 6 1 441.2.f.c 6
9.d odd 6 1 3969.2.a.q 3
21.c even 2 1 63.2.f.a 6
21.g even 6 1 441.2.g.c 6
21.g even 6 1 441.2.h.d 6
21.h odd 6 1 441.2.g.b 6
21.h odd 6 1 441.2.h.e 6
28.d even 2 1 3024.2.r.k 6
63.g even 3 1 1323.2.h.b 6
63.h even 3 1 1323.2.g.e 6
63.i even 6 1 441.2.g.c 6
63.j odd 6 1 441.2.g.b 6
63.k odd 6 1 1323.2.h.c 6
63.l odd 6 1 189.2.f.b 6
63.l odd 6 1 567.2.a.c 3
63.n odd 6 1 441.2.h.e 6
63.o even 6 1 63.2.f.a 6
63.o even 6 1 567.2.a.h 3
63.s even 6 1 441.2.h.d 6
63.t odd 6 1 1323.2.g.d 6
84.h odd 2 1 1008.2.r.h 6
252.s odd 6 1 1008.2.r.h 6
252.s odd 6 1 9072.2.a.ca 3
252.bi even 6 1 3024.2.r.k 6
252.bi even 6 1 9072.2.a.bs 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.f.a 6 21.c even 2 1
63.2.f.a 6 63.o even 6 1
189.2.f.b 6 7.b odd 2 1
189.2.f.b 6 63.l odd 6 1
441.2.f.c 6 3.b odd 2 1
441.2.f.c 6 9.d odd 6 1
441.2.g.b 6 21.h odd 6 1
441.2.g.b 6 63.j odd 6 1
441.2.g.c 6 21.g even 6 1
441.2.g.c 6 63.i even 6 1
441.2.h.d 6 21.g even 6 1
441.2.h.d 6 63.s even 6 1
441.2.h.e 6 21.h odd 6 1
441.2.h.e 6 63.n odd 6 1
567.2.a.c 3 63.l odd 6 1
567.2.a.h 3 63.o even 6 1
1008.2.r.h 6 84.h odd 2 1
1008.2.r.h 6 252.s odd 6 1
1323.2.f.d 6 1.a even 1 1 trivial
1323.2.f.d 6 9.c even 3 1 inner
1323.2.g.d 6 7.d odd 6 1
1323.2.g.d 6 63.t odd 6 1
1323.2.g.e 6 7.c even 3 1
1323.2.g.e 6 63.h even 3 1
1323.2.h.b 6 7.c even 3 1
1323.2.h.b 6 63.g even 3 1
1323.2.h.c 6 7.d odd 6 1
1323.2.h.c 6 63.k odd 6 1
3024.2.r.k 6 28.d even 2 1
3024.2.r.k 6 252.bi even 6 1
3969.2.a.l 3 9.c even 3 1
3969.2.a.q 3 9.d odd 6 1
9072.2.a.bs 3 252.bi even 6 1
9072.2.a.ca 3 252.s odd 6 1

Hecke kernels

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

\( T_{2}^{6} - 3 T_{2}^{5} + 9 T_{2}^{4} - 6 T_{2}^{3} + 9 T_{2}^{2} + 9 \)
\( T_{5}^{6} + 3 T_{5}^{5} + 9 T_{5}^{4} + 6 T_{5}^{3} + 9 T_{5}^{2} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 9 + 9 T^{2} - 6 T^{3} + 9 T^{4} - 3 T^{5} + T^{6} \)
$3$ \( T^{6} \)
$5$ \( 9 + 9 T^{2} + 6 T^{3} + 9 T^{4} + 3 T^{5} + T^{6} \)
$7$ \( T^{6} \)
$11$ \( 9 - 27 T + 63 T^{2} - 48 T^{3} + 27 T^{4} - 6 T^{5} + T^{6} \)
$13$ \( 11449 + 3531 T + 1410 T^{2} + 115 T^{3} + 42 T^{4} + 3 T^{5} + T^{6} \)
$17$ \( ( -3 + 9 T - 6 T^{2} + T^{3} )^{2} \)
$19$ \( ( 17 - 6 T - 3 T^{2} + T^{3} )^{2} \)
$23$ \( 9 + 81 T + 765 T^{2} - 330 T^{3} + 117 T^{4} - 12 T^{5} + T^{6} \)
$29$ \( 110889 - 11988 T + 4293 T^{2} - 342 T^{3} + 117 T^{4} - 9 T^{5} + T^{6} \)
$31$ \( 104329 + 25194 T + 7053 T^{2} + 412 T^{3} + 87 T^{4} + 3 T^{5} + T^{6} \)
$37$ \( ( -323 - 78 T + 3 T^{2} + T^{3} )^{2} \)
$41$ \( 81 - 81 T + 81 T^{2} - 18 T^{3} + 9 T^{4} + T^{6} \)
$43$ \( 1 + 6 T + 33 T^{2} + 20 T^{3} + 15 T^{4} - 3 T^{5} + T^{6} \)
$47$ \( 2601 - 2754 T + 2763 T^{2} - 264 T^{3} + 63 T^{4} + 3 T^{5} + T^{6} \)
$53$ \( ( 3 - 9 T + 6 T^{2} + T^{3} )^{2} \)
$59$ \( 2601 + 3672 T + 5031 T^{2} + 318 T^{3} + 81 T^{4} - 3 T^{5} + T^{6} \)
$61$ \( 361 - 285 T + 339 T^{2} + 52 T^{3} + 51 T^{4} - 6 T^{5} + T^{6} \)
$67$ \( 289 + 357 T + 645 T^{2} - 286 T^{3} + 123 T^{4} - 12 T^{5} + T^{6} \)
$71$ \( ( 27 - 54 T + 9 T^{2} + T^{3} )^{2} \)
$73$ \( ( 269 + 84 T - 21 T^{2} + T^{3} )^{2} \)
$79$ \( 32761 - 21720 T + 10599 T^{2} - 2158 T^{3} + 321 T^{4} - 21 T^{5} + T^{6} \)
$83$ \( 81 - 405 T + 1863 T^{2} - 792 T^{3} + 279 T^{4} - 18 T^{5} + T^{6} \)
$89$ \( ( 813 - 63 T - 12 T^{2} + T^{3} )^{2} \)
$97$ \( 104329 + 54264 T + 29193 T^{2} + 142 T^{3} + 177 T^{4} + 3 T^{5} + T^{6} \)
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