Properties

Label 1323.2.f.g
Level $1323$
Weight $2$
Character orbit 1323.f
Analytic conductor $10.564$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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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: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 7 x^{10} + 37 x^{8} - 78 x^{6} + 123 x^{4} - 36 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{5} \)
Twist minimal: no (minimal twist has level 441)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + ( -\beta_{1} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{4} + ( \beta_{2} + \beta_{3} ) q^{5} + ( -2 - \beta_{1} - \beta_{5} ) q^{8} +O(q^{10})\) \( q + \beta_{6} q^{2} + ( -\beta_{1} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{4} + ( \beta_{2} + \beta_{3} ) q^{5} + ( -2 - \beta_{1} - \beta_{5} ) q^{8} + ( 2 \beta_{3} - \beta_{8} ) q^{10} + ( 1 + \beta_{4} + \beta_{7} ) q^{11} + ( \beta_{9} - \beta_{11} ) q^{13} + ( -2 \beta_{6} - \beta_{7} ) q^{16} + ( \beta_{3} - \beta_{8} + \beta_{9} ) q^{17} + ( \beta_{3} + \beta_{8} ) q^{19} + ( -3 \beta_{2} - 2 \beta_{10} - \beta_{11} ) q^{20} + \beta_{4} q^{22} + ( \beta_{1} - \beta_{4} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{23} + ( -1 - \beta_{4} - 3 \beta_{6} ) q^{25} + ( \beta_{8} - \beta_{9} ) q^{26} + ( 4 + 4 \beta_{4} + \beta_{6} - 2 \beta_{7} ) q^{29} + ( \beta_{2} + \beta_{3} - 2 \beta_{8} + 2 \beta_{10} ) q^{31} + ( -\beta_{1} - 3 \beta_{4} + \beta_{6} ) q^{32} + ( -3 \beta_{2} - 2 \beta_{11} ) q^{34} + ( -2 + 2 \beta_{1} - \beta_{5} ) q^{37} + ( -\beta_{2} - \beta_{10} + \beta_{11} ) q^{38} + ( -4 \beta_{2} - 4 \beta_{3} + 2 \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{40} + ( -\beta_{2} - \beta_{3} + 2 \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{41} + ( -2 - 2 \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{43} + ( 2 - \beta_{1} - 2 \beta_{5} ) q^{44} + ( 1 + 4 \beta_{1} + \beta_{5} ) q^{46} + ( -\beta_{2} - \beta_{10} - 2 \beta_{11} ) q^{47} + ( 4 \beta_{1} - 9 \beta_{4} + 3 \beta_{5} - 4 \beta_{6} + 3 \beta_{7} ) q^{50} + ( \beta_{2} - \beta_{10} ) q^{52} + ( -5 + 3 \beta_{1} + 2 \beta_{5} ) q^{53} + ( \beta_{3} + \beta_{8} + \beta_{9} ) q^{55} + ( -7 \beta_{1} + \beta_{4} - \beta_{5} + 7 \beta_{6} - \beta_{7} ) q^{58} + ( -2 \beta_{2} - 2 \beta_{3} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{59} + ( 2 \beta_{2} + \beta_{10} - 2 \beta_{11} ) q^{61} + ( 4 \beta_{3} - \beta_{8} - 2 \beta_{9} ) q^{62} + ( -3 - 2 \beta_{1} + \beta_{5} ) q^{64} -3 \beta_{7} q^{65} + ( -3 \beta_{1} - 2 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} ) q^{67} + ( -4 \beta_{2} - 4 \beta_{3} + 3 \beta_{8} - 3 \beta_{10} ) q^{68} + ( -6 - \beta_{1} ) q^{71} + ( -\beta_{3} - \beta_{8} ) q^{73} + ( 7 + 7 \beta_{4} - \beta_{6} - 2 \beta_{7} ) q^{74} + ( -\beta_{2} - \beta_{3} + 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{76} + ( 1 + \beta_{4} - 3 \beta_{6} + 3 \beta_{7} ) q^{79} + ( -4 \beta_{3} + \beta_{8} - \beta_{9} ) q^{80} + ( -4 \beta_{3} + 2 \beta_{8} + \beta_{9} ) q^{82} + ( 3 \beta_{2} + \beta_{10} - \beta_{11} ) q^{83} + ( 6 \beta_{1} - 3 \beta_{4} - 6 \beta_{6} ) q^{85} + ( \beta_{1} + 7 \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{86} + ( 1 + \beta_{4} - \beta_{6} + \beta_{7} ) q^{88} + ( 3 \beta_{3} - 2 \beta_{8} - \beta_{9} ) q^{89} + ( 9 + 9 \beta_{4} + 4 \beta_{6} ) q^{92} + ( -3 \beta_{2} - 3 \beta_{3} + 3 \beta_{8} - \beta_{9} - 3 \beta_{10} + \beta_{11} ) q^{94} + ( -9 \beta_{4} - 3 \beta_{5} - 3 \beta_{7} ) q^{95} + ( 2 \beta_{2} + \beta_{10} + \beta_{11} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 2q^{2} - 6q^{4} - 24q^{8} + O(q^{10}) \) \( 12q + 2q^{2} - 6q^{4} - 24q^{8} + 8q^{11} - 6q^{16} - 6q^{22} + 4q^{23} - 12q^{25} + 22q^{29} + 16q^{32} - 12q^{37} - 6q^{43} + 28q^{44} + 24q^{46} + 56q^{50} - 56q^{53} - 18q^{58} - 48q^{64} - 6q^{65} - 76q^{71} + 36q^{74} + 6q^{79} + 30q^{85} - 36q^{86} + 6q^{88} + 62q^{92} + 60q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 7 x^{10} + 37 x^{8} - 78 x^{6} + 123 x^{4} - 36 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -49 \nu^{10} + 259 \nu^{8} - 1369 \nu^{6} + 861 \nu^{4} - 252 \nu^{2} - 7266 \)\()/4299\)
\(\beta_{2}\)\(=\)\((\)\( -148 \nu^{11} + 987 \nu^{9} - 5217 \nu^{7} + 10175 \nu^{5} - 17343 \nu^{3} - 3522 \nu \)\()/4299\)
\(\beta_{3}\)\(=\)\((\)\( -148 \nu^{11} + 987 \nu^{9} - 5217 \nu^{7} + 10175 \nu^{5} - 17343 \nu^{3} + 9375 \nu \)\()/4299\)
\(\beta_{4}\)\(=\)\((\)\( 148 \nu^{10} - 987 \nu^{8} + 5217 \nu^{6} - 10175 \nu^{4} + 17343 \nu^{2} - 5076 \)\()/4299\)
\(\beta_{5}\)\(=\)\((\)\( 161 \nu^{10} - 851 \nu^{8} + 3884 \nu^{6} - 2829 \nu^{4} + 828 \nu^{2} + 6678 \)\()/4299\)
\(\beta_{6}\)\(=\)\((\)\( -296 \nu^{10} + 1974 \nu^{8} - 10434 \nu^{6} + 20350 \nu^{4} - 30387 \nu^{2} + 1554 \)\()/4299\)
\(\beta_{7}\)\(=\)\((\)\( -120 \nu^{10} + 839 \nu^{8} - 4230 \nu^{6} + 8250 \nu^{4} - 10034 \nu^{2} + 630 \)\()/1433\)
\(\beta_{8}\)\(=\)\((\)\( -494 \nu^{11} + 3430 \nu^{9} - 18130 \nu^{7} + 38978 \nu^{5} - 64569 \nu^{3} + 34836 \nu \)\()/4299\)
\(\beta_{9}\)\(=\)\((\)\( 532 \nu^{11} - 4245 \nu^{9} + 23052 \nu^{7} - 58070 \nu^{5} + 93015 \nu^{3} - 50082 \nu \)\()/4299\)
\(\beta_{10}\)\(=\)\((\)\( 641 \nu^{11} - 4207 \nu^{9} + 22237 \nu^{7} - 41561 \nu^{5} + 65325 \nu^{3} + 12756 \nu \)\()/4299\)
\(\beta_{11}\)\(=\)\((\)\( -1162 \nu^{11} + 7575 \nu^{9} - 38811 \nu^{7} + 69140 \nu^{5} - 96255 \nu^{3} - 17544 \nu \)\()/4299\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} - \beta_{2}\)\()/3\)
\(\nu^{2}\)\(=\)\(\beta_{6} + 2 \beta_{4} + 2\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{10} + \beta_{8} - 4 \beta_{3} - 8 \beta_{2}\)\()/3\)
\(\nu^{4}\)\(=\)\(-\beta_{7} + 5 \beta_{6} - \beta_{5} + 7 \beta_{4} - 5 \beta_{1}\)
\(\nu^{5}\)\(=\)\((\)\(-\beta_{11} - 6 \beta_{10} + 2 \beta_{9} + 12 \beta_{8} - 34 \beta_{3} - 17 \beta_{2}\)\()/3\)
\(\nu^{6}\)\(=\)\(-7 \beta_{5} - 23 \beta_{1} - 28\)
\(\nu^{7}\)\(=\)\((\)\(7 \beta_{11} + 30 \beta_{10} + 7 \beta_{9} + 30 \beta_{8} - 74 \beta_{3} + 74 \beta_{2}\)\()/3\)
\(\nu^{8}\)\(=\)\(37 \beta_{7} - 104 \beta_{6} - 118 \beta_{4} - 118\)
\(\nu^{9}\)\(=\)\((\)\(74 \beta_{11} + 282 \beta_{10} - 37 \beta_{9} - 141 \beta_{8} + 326 \beta_{3} + 652 \beta_{2}\)\()/3\)
\(\nu^{10}\)\(=\)\(178 \beta_{7} - 467 \beta_{6} + 178 \beta_{5} - 511 \beta_{4} + 467 \beta_{1}\)
\(\nu^{11}\)\(=\)\((\)\(178 \beta_{11} + 645 \beta_{10} - 356 \beta_{9} - 1290 \beta_{8} + 2890 \beta_{3} + 1445 \beta_{2}\)\()/3\)

Character values

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

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

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
−0.474636 0.274031i
0.474636 + 0.274031i
−1.29589 0.748185i
1.29589 + 0.748185i
−1.82904 1.05600i
1.82904 + 1.05600i
−0.474636 + 0.274031i
0.474636 0.274031i
−1.29589 + 0.748185i
1.29589 0.748185i
−1.82904 + 1.05600i
1.82904 1.05600i
−0.849814 1.47192i 0 −0.444368 + 0.769668i −0.474636 + 0.822093i 0 0 −1.88874 0 1.61341
442.2 −0.849814 1.47192i 0 −0.444368 + 0.769668i 0.474636 0.822093i 0 0 −1.88874 0 −1.61341
442.3 0.119562 + 0.207087i 0 0.971410 1.68253i −1.29589 + 2.24456i 0 0 0.942820 0 −0.619757
442.4 0.119562 + 0.207087i 0 0.971410 1.68253i 1.29589 2.24456i 0 0 0.942820 0 0.619757
442.5 1.23025 + 2.13086i 0 −2.02704 + 3.51094i −1.82904 + 3.16799i 0 0 −5.05408 0 −9.00071
442.6 1.23025 + 2.13086i 0 −2.02704 + 3.51094i 1.82904 3.16799i 0 0 −5.05408 0 9.00071
883.1 −0.849814 + 1.47192i 0 −0.444368 0.769668i −0.474636 0.822093i 0 0 −1.88874 0 1.61341
883.2 −0.849814 + 1.47192i 0 −0.444368 0.769668i 0.474636 + 0.822093i 0 0 −1.88874 0 −1.61341
883.3 0.119562 0.207087i 0 0.971410 + 1.68253i −1.29589 2.24456i 0 0 0.942820 0 −0.619757
883.4 0.119562 0.207087i 0 0.971410 + 1.68253i 1.29589 + 2.24456i 0 0 0.942820 0 0.619757
883.5 1.23025 2.13086i 0 −2.02704 3.51094i −1.82904 3.16799i 0 0 −5.05408 0 −9.00071
883.6 1.23025 2.13086i 0 −2.02704 3.51094i 1.82904 + 3.16799i 0 0 −5.05408 0 9.00071
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 883.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
9.c even 3 1 inner
63.l odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1323.2.f.g 12
3.b odd 2 1 441.2.f.g 12
7.b odd 2 1 inner 1323.2.f.g 12
7.c even 3 1 1323.2.g.g 12
7.c even 3 1 1323.2.h.g 12
7.d odd 6 1 1323.2.g.g 12
7.d odd 6 1 1323.2.h.g 12
9.c even 3 1 inner 1323.2.f.g 12
9.c even 3 1 3969.2.a.bd 6
9.d odd 6 1 441.2.f.g 12
9.d odd 6 1 3969.2.a.be 6
21.c even 2 1 441.2.f.g 12
21.g even 6 1 441.2.g.g 12
21.g even 6 1 441.2.h.g 12
21.h odd 6 1 441.2.g.g 12
21.h odd 6 1 441.2.h.g 12
63.g even 3 1 1323.2.h.g 12
63.h even 3 1 1323.2.g.g 12
63.i even 6 1 441.2.g.g 12
63.j odd 6 1 441.2.g.g 12
63.k odd 6 1 1323.2.h.g 12
63.l odd 6 1 inner 1323.2.f.g 12
63.l odd 6 1 3969.2.a.bd 6
63.n odd 6 1 441.2.h.g 12
63.o even 6 1 441.2.f.g 12
63.o even 6 1 3969.2.a.be 6
63.s even 6 1 441.2.h.g 12
63.t odd 6 1 1323.2.g.g 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.f.g 12 3.b odd 2 1
441.2.f.g 12 9.d odd 6 1
441.2.f.g 12 21.c even 2 1
441.2.f.g 12 63.o even 6 1
441.2.g.g 12 21.g even 6 1
441.2.g.g 12 21.h odd 6 1
441.2.g.g 12 63.i even 6 1
441.2.g.g 12 63.j odd 6 1
441.2.h.g 12 21.g even 6 1
441.2.h.g 12 21.h odd 6 1
441.2.h.g 12 63.n odd 6 1
441.2.h.g 12 63.s even 6 1
1323.2.f.g 12 1.a even 1 1 trivial
1323.2.f.g 12 7.b odd 2 1 inner
1323.2.f.g 12 9.c even 3 1 inner
1323.2.f.g 12 63.l odd 6 1 inner
1323.2.g.g 12 7.c even 3 1
1323.2.g.g 12 7.d odd 6 1
1323.2.g.g 12 63.h even 3 1
1323.2.g.g 12 63.t odd 6 1
1323.2.h.g 12 7.c even 3 1
1323.2.h.g 12 7.d odd 6 1
1323.2.h.g 12 63.g even 3 1
1323.2.h.g 12 63.k odd 6 1
3969.2.a.bd 6 9.c even 3 1
3969.2.a.bd 6 63.l odd 6 1
3969.2.a.be 6 9.d odd 6 1
3969.2.a.be 6 63.o even 6 1

Hecke kernels

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

\( T_{2}^{6} - T_{2}^{5} + 5 T_{2}^{4} + 2 T_{2}^{3} + 17 T_{2}^{2} - 4 T_{2} + 1 \)
\( T_{5}^{12} + 21 T_{5}^{10} + 333 T_{5}^{8} + 2106 T_{5}^{6} + 9963 T_{5}^{4} + 8748 T_{5}^{2} + 6561 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 4 T + 17 T^{2} + 2 T^{3} + 5 T^{4} - T^{5} + T^{6} )^{2} \)
$3$ \( T^{12} \)
$5$ \( 6561 + 8748 T^{2} + 9963 T^{4} + 2106 T^{6} + 333 T^{8} + 21 T^{10} + T^{12} \)
$7$ \( T^{12} \)
$11$ \( ( 1 - T + 5 T^{2} + 2 T^{3} + 17 T^{4} - 4 T^{5} + T^{6} )^{2} \)
$13$ \( 6561 + 28431 T^{2} + 120042 T^{4} + 13527 T^{6} + 1170 T^{8} + 39 T^{10} + T^{12} \)
$17$ \( ( -3969 + 1593 T^{2} - 84 T^{4} + T^{6} )^{2} \)
$19$ \( ( -3969 + 1296 T^{2} - 75 T^{4} + T^{6} )^{2} \)
$23$ \( ( 3481 - 1475 T + 743 T^{2} - 68 T^{3} + 29 T^{4} - 2 T^{5} + T^{6} )^{2} \)
$29$ \( ( 7921 + 1246 T + 1175 T^{2} - 332 T^{3} + 107 T^{4} - 11 T^{5} + T^{6} )^{2} \)
$31$ \( 6059221281 + 428748228 T^{2} + 20296575 T^{4} + 554850 T^{6} + 11133 T^{8} + 129 T^{10} + T^{12} \)
$37$ \( ( 27 - 24 T + 3 T^{2} + T^{3} )^{4} \)
$41$ \( 43046721 + 39858075 T^{2} + 35842743 T^{4} + 971028 T^{6} + 20169 T^{8} + 162 T^{10} + T^{12} \)
$43$ \( ( 729 - 648 T + 495 T^{2} - 126 T^{3} + 33 T^{4} + 3 T^{5} + T^{6} )^{2} \)
$47$ \( 37822859361 + 2058386904 T^{2} + 76431033 T^{4} + 1547910 T^{6} + 22905 T^{8} + 183 T^{10} + T^{12} \)
$53$ \( ( -263 + 11 T + 14 T^{2} + T^{3} )^{4} \)
$59$ \( 22430753361 + 1415317050 T^{2} + 61894773 T^{4} + 1429812 T^{6} + 24039 T^{8} + 183 T^{10} + T^{12} \)
$61$ \( 311374044081 + 12007795671 T^{2} + 315752985 T^{4} + 4564998 T^{6} + 48177 T^{8} + 264 T^{10} + T^{12} \)
$67$ \( ( 124609 + 39183 T + 12321 T^{2} + 706 T^{3} + 111 T^{4} + T^{6} )^{2} \)
$71$ \( ( 227 + 116 T + 19 T^{2} + T^{3} )^{4} \)
$73$ \( ( -3969 + 1296 T^{2} - 75 T^{4} + T^{6} )^{2} \)
$79$ \( ( 11449 - 8346 T + 6405 T^{2} + 20 T^{3} + 87 T^{4} - 3 T^{5} + T^{6} )^{2} \)
$83$ \( 51769445841 + 3040925085 T^{2} + 126746613 T^{4} + 2592162 T^{6} + 38619 T^{8} + 228 T^{10} + T^{12} \)
$89$ \( ( -3969 + 5751 T^{2} - 246 T^{4} + T^{6} )^{2} \)
$97$ \( 96059601 + 19582398 T^{2} + 2904093 T^{4} + 202176 T^{6} + 10323 T^{8} + 111 T^{10} + T^{12} \)
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