Properties

Label 1323.2.c.d
Level $1323$
Weight $2$
Character orbit 1323.c
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.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.5642081874\)
Analytic rank: \(0\)
Dimension: \(12\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 189)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{8} q^{2} + ( -1 - \beta_{6} ) q^{4} + \beta_{1} q^{5} + ( -\beta_{8} + \beta_{10} ) q^{8} +O(q^{10})\) \( q + \beta_{8} q^{2} + ( -1 - \beta_{6} ) q^{4} + \beta_{1} q^{5} + ( -\beta_{8} + \beta_{10} ) q^{8} + ( \beta_{5} + \beta_{11} ) q^{10} + ( \beta_{8} - \beta_{10} ) q^{11} + ( \beta_{5} + \beta_{7} ) q^{13} + ( \beta_{6} + \beta_{9} ) q^{16} -\beta_{3} q^{17} + ( -\beta_{5} + \beta_{11} ) q^{19} -\beta_{2} q^{20} + ( -2 - 3 \beta_{6} - \beta_{9} ) q^{22} + ( \beta_{4} + \beta_{8} + \beta_{10} ) q^{23} + ( 5 - 2 \beta_{6} - \beta_{9} ) q^{25} + ( -\beta_{1} - 2 \beta_{3} ) q^{26} + ( \beta_{8} + \beta_{10} ) q^{29} + ( -\beta_{5} + \beta_{7} + 2 \beta_{11} ) q^{31} + \beta_{4} q^{32} + ( -\beta_{5} - 3 \beta_{7} + \beta_{11} ) q^{34} + ( -1 - \beta_{9} ) q^{37} + ( -\beta_{2} + 2 \beta_{3} ) q^{38} + ( 2 \beta_{5} - \beta_{11} ) q^{40} + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{41} + ( 2 - \beta_{9} ) q^{43} + ( -\beta_{4} - 6 \beta_{8} + 2 \beta_{10} ) q^{44} + ( -2 - \beta_{9} ) q^{46} + ( -\beta_{1} + 2 \beta_{3} ) q^{47} + ( -\beta_{4} + \beta_{8} + 3 \beta_{10} ) q^{50} + ( -\beta_{5} - 4 \beta_{7} + \beta_{11} ) q^{52} + ( -\beta_{4} - 2 \beta_{8} - \beta_{10} ) q^{53} + ( -2 \beta_{5} + \beta_{11} ) q^{55} + ( -4 + \beta_{6} + \beta_{9} ) q^{58} + ( \beta_{1} + \beta_{3} ) q^{59} + ( \beta_{7} + 2 \beta_{11} ) q^{61} + ( -\beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{62} + ( 2 + \beta_{6} ) q^{64} + ( -\beta_{4} + 3 \beta_{8} + 3 \beta_{10} ) q^{65} + ( -5 - \beta_{6} - 2 \beta_{9} ) q^{67} + ( -\beta_{2} + 3 \beta_{3} ) q^{68} + ( \beta_{4} + \beta_{8} ) q^{71} + ( -2 \beta_{5} + 3 \beta_{7} - \beta_{11} ) q^{73} + ( -\beta_{4} - \beta_{8} + \beta_{10} ) q^{74} + ( 6 \beta_{7} - 3 \beta_{11} ) q^{76} + ( 7 - \beta_{6} - 2 \beta_{9} ) q^{79} + ( -\beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{80} + ( -3 \beta_{7} - \beta_{11} ) q^{82} + ( -\beta_{1} - 2 \beta_{3} ) q^{83} + ( -2 + \beta_{6} + 2 \beta_{9} ) q^{85} + ( -\beta_{4} + 2 \beta_{8} + \beta_{10} ) q^{86} + ( 10 + 5 \beta_{6} + 2 \beta_{9} ) q^{88} + ( -\beta_{2} - 2 \beta_{3} ) q^{89} + ( \beta_{4} + 3 \beta_{10} ) q^{92} + ( \beta_{5} + 6 \beta_{7} - 3 \beta_{11} ) q^{94} + ( -\beta_{4} - 3 \beta_{10} ) q^{95} + ( 2 \beta_{5} - 4 \beta_{7} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 16q^{4} + O(q^{10}) \) \( 12q - 16q^{4} + 8q^{16} - 40q^{22} + 48q^{25} - 16q^{37} + 20q^{43} - 28q^{46} - 40q^{58} + 28q^{64} - 72q^{67} + 72q^{79} - 12q^{85} + 148q^{88} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 1298 \nu^{11} - 10953 \nu^{9} + 71803 \nu^{7} - 202311 \nu^{5} + 490451 \nu^{3} - 438921 \nu \)\()/197955\)
\(\beta_{2}\)\(=\)\((\)\( 8 \nu^{11} - 117 \nu^{9} + 850 \nu^{7} - 3420 \nu^{5} + 7895 \nu^{3} - 7056 \nu \)\()/747\)
\(\beta_{3}\)\(=\)\((\)\( -3734 \nu^{11} + 34254 \nu^{9} - 224554 \nu^{7} + 743958 \nu^{5} - 1731773 \nu^{3} + 1545408 \nu \)\()/197955\)
\(\beta_{4}\)\(=\)\((\)\( 1298 \nu^{11} - 10953 \nu^{9} + 71803 \nu^{7} - 202311 \nu^{5} + 490451 \nu^{3} - 43011 \nu \)\()/65985\)
\(\beta_{5}\)\(=\)\((\)\( 787 \nu^{10} - 10047 \nu^{8} + 58532 \nu^{6} - 191649 \nu^{4} + 281254 \nu^{2} - 89064 \)\()/65985\)
\(\beta_{6}\)\(=\)\((\)\( 288 \nu^{10} - 1888 \nu^{8} + 9933 \nu^{6} - 12896 \nu^{4} + 6336 \nu^{2} + 68439 \)\()/21995\)
\(\beta_{7}\)\(=\)\((\)\( 2596 \nu^{10} - 21906 \nu^{8} + 143606 \nu^{6} - 404622 \nu^{4} + 980902 \nu^{2} - 283977 \)\()/197955\)
\(\beta_{8}\)\(=\)\((\)\( 5192 \nu^{11} - 43812 \nu^{9} + 287212 \nu^{7} - 809244 \nu^{5} + 1763849 \nu^{3} - 172044 \nu \)\()/197955\)
\(\beta_{9}\)\(=\)\((\)\( -333 \nu^{10} + 2183 \nu^{8} - 9423 \nu^{6} + 14911 \nu^{4} - 7326 \nu^{2} + 48026 \)\()/21995\)
\(\beta_{10}\)\(=\)\((\)\( -88 \nu^{11} + 783 \nu^{9} - 4868 \nu^{7} + 13716 \nu^{5} - 26581 \nu^{3} + 2916 \nu \)\()/2385\)
\(\beta_{11}\)\(=\)\((\)\( -2353 \nu^{10} + 20313 \nu^{8} - 133163 \nu^{6} + 393741 \nu^{4} - 843586 \nu^{2} + 248931 \)\()/65985\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} - 3 \beta_{1}\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(3 \beta_{11} - \beta_{9} + 9 \beta_{7} - 2 \beta_{6} + 10\)\()/6\)
\(\nu^{3}\)\(=\)\((\)\(-3 \beta_{8} + 4 \beta_{4}\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(5 \beta_{11} + 2 \beta_{9} + 12 \beta_{7} + 3 \beta_{6} + \beta_{5} - 14\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-3 \beta_{10} - 18 \beta_{8} + 17 \beta_{4} + 18 \beta_{3} + 3 \beta_{2} + 51 \beta_{1}\)\()/6\)
\(\nu^{6}\)\(=\)\((\)\(32 \beta_{9} + 37 \beta_{6} - 185\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(27 \beta_{10} + 96 \beta_{8} - 74 \beta_{4} + 96 \beta_{3} + 27 \beta_{2} + 222 \beta_{1}\)\()/6\)
\(\nu^{8}\)\(=\)\((\)\(-106 \beta_{11} + 55 \beta_{9} - 222 \beta_{7} + 51 \beta_{6} - 59 \beta_{5} - 277\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(177 \beta_{10} + 495 \beta_{8} - 328 \beta_{4}\)\()/3\)
\(\nu^{10}\)\(=\)\((\)\(-1479 \beta_{11} - 835 \beta_{9} - 2952 \beta_{7} - 644 \beta_{6} - 1026 \beta_{5} + 3787\)\()/6\)
\(\nu^{11}\)\(=\)\((\)\(1026 \beta_{10} + 2505 \beta_{8} - 1477 \beta_{4} - 2505 \beta_{3} - 1026 \beta_{2} - 4431 \beta_{1}\)\()/6\)

Character values

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

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1322.1
0.617942 0.356769i
−0.617942 0.356769i
1.90412 + 1.09935i
−1.90412 + 1.09935i
1.65604 0.956115i
−1.65604 0.956115i
1.65604 + 0.956115i
−1.65604 + 0.956115i
1.90412 1.09935i
−1.90412 1.09935i
0.617942 + 0.356769i
−0.617942 + 0.356769i
2.49086i 0 −4.20440 −1.23588 0 0 5.49086i 0 3.07842i
1322.2 2.49086i 0 −4.20440 1.23588 0 0 5.49086i 0 3.07842i
1322.3 1.83424i 0 −1.36445 −3.80824 0 0 1.16576i 0 6.98525i
1322.4 1.83424i 0 −1.36445 3.80824 0 0 1.16576i 0 6.98525i
1322.5 0.656620i 0 1.56885 −3.31208 0 0 2.34338i 0 2.17478i
1322.6 0.656620i 0 1.56885 3.31208 0 0 2.34338i 0 2.17478i
1322.7 0.656620i 0 1.56885 −3.31208 0 0 2.34338i 0 2.17478i
1322.8 0.656620i 0 1.56885 3.31208 0 0 2.34338i 0 2.17478i
1322.9 1.83424i 0 −1.36445 −3.80824 0 0 1.16576i 0 6.98525i
1322.10 1.83424i 0 −1.36445 3.80824 0 0 1.16576i 0 6.98525i
1322.11 2.49086i 0 −4.20440 −1.23588 0 0 5.49086i 0 3.07842i
1322.12 2.49086i 0 −4.20440 1.23588 0 0 5.49086i 0 3.07842i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1322.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 10 T_{2}^{4} + 25 T_{2}^{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(1323, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 9 + 25 T^{2} + 10 T^{4} + T^{6} )^{2} \)
$3$ \( T^{12} \)
$5$ \( ( -243 + 198 T^{2} - 27 T^{4} + T^{6} )^{2} \)
$7$ \( T^{12} \)
$11$ \( ( 225 + 214 T^{2} + 37 T^{4} + T^{6} )^{2} \)
$13$ \( ( 675 + 369 T^{2} + 42 T^{4} + T^{6} )^{2} \)
$17$ \( ( -243 + 225 T^{2} - 30 T^{4} + T^{6} )^{2} \)
$19$ \( ( 243 + 630 T^{2} + 51 T^{4} + T^{6} )^{2} \)
$23$ \( ( 729 + 517 T^{2} + 94 T^{4} + T^{6} )^{2} \)
$29$ \( ( 81 + 322 T^{2} + 37 T^{4} + T^{6} )^{2} \)
$31$ \( ( 64827 + 5733 T^{2} + 138 T^{4} + T^{6} )^{2} \)
$37$ \( ( -67 - 19 T + 4 T^{2} + T^{3} )^{4} \)
$41$ \( ( -243 + 1953 T^{2} - 114 T^{4} + T^{6} )^{2} \)
$43$ \( ( -1 - 16 T - 5 T^{2} + T^{3} )^{4} \)
$47$ \( ( -19683 + 3870 T^{2} - 135 T^{4} + T^{6} )^{2} \)
$53$ \( ( 9 + 1717 T^{2} + 118 T^{4} + T^{6} )^{2} \)
$59$ \( ( -2187 + 738 T^{2} - 63 T^{4} + T^{6} )^{2} \)
$61$ \( ( 49923 + 4923 T^{2} + 129 T^{4} + T^{6} )^{2} \)
$67$ \( ( -677 + 15 T + 18 T^{2} + T^{3} )^{4} \)
$71$ \( ( 19881 + 2290 T^{2} + 85 T^{4} + T^{6} )^{2} \)
$73$ \( ( 177147 + 10782 T^{2} + 195 T^{4} + T^{6} )^{2} \)
$79$ \( ( 7 + 15 T - 18 T^{2} + T^{3} )^{4} \)
$83$ \( ( -6075 + 5814 T^{2} - 159 T^{4} + T^{6} )^{2} \)
$89$ \( ( -87723 + 9675 T^{2} - 201 T^{4} + T^{6} )^{2} \)
$97$ \( ( 1728 + 3744 T^{2} + 204 T^{4} + T^{6} )^{2} \)
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