Properties

Label 1323.2.c.f
Level $1323$
Weight $2$
Character orbit 1323.c
Analytic conductor $10.564$
Analytic rank $0$
Dimension $16$
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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 24 x^{14} + 212 x^{12} + 872 x^{10} + 1815 x^{8} + 1928 x^{6} + 996 x^{4} + 200 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + ( -1 + \beta_{13} ) q^{4} -\beta_{2} q^{5} + ( -\beta_{1} - \beta_{5} + \beta_{12} - \beta_{15} ) q^{8} +O(q^{10})\) \( q + \beta_{5} q^{2} + ( -1 + \beta_{13} ) q^{4} -\beta_{2} q^{5} + ( -\beta_{1} - \beta_{5} + \beta_{12} - \beta_{15} ) q^{8} + ( -\beta_{4} - \beta_{6} - \beta_{10} - \beta_{14} ) q^{10} + ( -\beta_{5} - \beta_{12} - \beta_{15} ) q^{11} + ( \beta_{4} - \beta_{6} + \beta_{10} + \beta_{14} ) q^{13} + ( 3 + \beta_{7} + 3 \beta_{8} - \beta_{13} ) q^{16} + ( \beta_{9} + 2 \beta_{11} ) q^{17} + ( \beta_{4} - \beta_{6} + \beta_{10} ) q^{19} + ( 4 \beta_{2} + \beta_{3} - 3 \beta_{9} + 2 \beta_{11} ) q^{20} + ( 4 - \beta_{8} - 2 \beta_{13} ) q^{22} + ( -2 \beta_{1} - \beta_{12} + \beta_{15} ) q^{23} + ( -1 + \beta_{7} + \beta_{8} - \beta_{13} ) q^{25} + ( -4 \beta_{2} - 3 \beta_{3} - \beta_{9} - 2 \beta_{11} ) q^{26} + ( 2 \beta_{1} + \beta_{12} - 2 \beta_{15} ) q^{29} + ( 3 \beta_{4} + \beta_{6} - 2 \beta_{10} + 2 \beta_{14} ) q^{31} + ( 4 \beta_{1} + 2 \beta_{5} - 2 \beta_{12} - 2 \beta_{15} ) q^{32} + ( 2 \beta_{4} - \beta_{6} - 3 \beta_{10} ) q^{34} + ( 2 + 2 \beta_{7} + 3 \beta_{8} - \beta_{13} ) q^{37} + ( -\beta_{2} - 2 \beta_{3} - \beta_{9} - \beta_{11} ) q^{38} + ( 5 \beta_{4} + 4 \beta_{6} + 3 \beta_{10} + 2 \beta_{14} ) q^{40} + ( 2 \beta_{3} - \beta_{9} - \beta_{11} ) q^{41} + ( -1 - \beta_{7} + 4 \beta_{8} ) q^{43} + ( \beta_{1} + 6 \beta_{5} - 3 \beta_{12} ) q^{44} + ( 1 + 2 \beta_{7} + 3 \beta_{8} - \beta_{13} ) q^{46} + ( \beta_{2} - 2 \beta_{9} + 3 \beta_{11} ) q^{47} + ( 4 \beta_{1} - 2 \beta_{12} ) q^{50} + ( -7 \beta_{4} - 2 \beta_{6} + \beta_{10} - 2 \beta_{14} ) q^{52} + ( -2 \beta_{1} + 5 \beta_{5} + \beta_{12} - 2 \beta_{15} ) q^{53} + ( 2 \beta_{4} + \beta_{6} + 3 \beta_{10} + \beta_{14} ) q^{55} + ( -2 \beta_{7} - 3 \beta_{8} ) q^{58} + ( -3 \beta_{2} - 4 \beta_{3} + 2 \beta_{9} - 3 \beta_{11} ) q^{59} + ( -5 \beta_{4} - \beta_{6} - 4 \beta_{10} + \beta_{14} ) q^{61} + ( -8 \beta_{2} - 4 \beta_{3} - 5 \beta_{11} ) q^{62} + ( -2 - 2 \beta_{7} - 4 \beta_{8} + 2 \beta_{13} ) q^{64} + ( -2 \beta_{1} - 3 \beta_{5} - 2 \beta_{12} + 2 \beta_{15} ) q^{65} + ( 3 - 2 \beta_{7} + \beta_{8} + \beta_{13} ) q^{67} + ( 2 \beta_{2} - 3 \beta_{3} - 3 \beta_{9} + 2 \beta_{11} ) q^{68} + ( \beta_{1} - 3 \beta_{5} + 5 \beta_{12} - \beta_{15} ) q^{71} + ( \beta_{4} - 2 \beta_{6} + \beta_{10} - \beta_{14} ) q^{73} + ( 8 \beta_{1} + 2 \beta_{5} - 4 \beta_{12} - \beta_{15} ) q^{74} + ( -2 \beta_{4} + 3 \beta_{10} - \beta_{14} ) q^{76} + ( -4 - \beta_{7} + 3 \beta_{8} + 3 \beta_{13} ) q^{79} + ( -10 \beta_{2} - \beta_{3} + 5 \beta_{9} - 3 \beta_{11} ) q^{80} + ( \beta_{4} - \beta_{6} + 2 \beta_{10} ) q^{82} + 5 \beta_{3} q^{83} + ( 4 + 2 \beta_{7} - \beta_{8} - \beta_{13} ) q^{85} + ( 2 \beta_{1} - 4 \beta_{12} + \beta_{15} ) q^{86} + ( -11 - \beta_{7} - 7 \beta_{8} + 3 \beta_{13} ) q^{88} + ( 2 \beta_{2} - 3 \beta_{3} - 5 \beta_{9} - 2 \beta_{11} ) q^{89} + ( 4 \beta_{1} + \beta_{5} - 6 \beta_{12} + \beta_{15} ) q^{92} + ( 4 \beta_{4} + 3 \beta_{6} + \beta_{14} ) q^{94} -\beta_{12} q^{95} + ( 4 \beta_{4} + 5 \beta_{6} - 6 \beta_{10} + 2 \beta_{14} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 16q^{4} + O(q^{10}) \) \( 16q - 16q^{4} + 48q^{16} + 64q^{22} - 16q^{25} + 32q^{37} - 16q^{43} + 16q^{46} - 32q^{64} + 48q^{67} - 64q^{79} + 64q^{85} - 176q^{88} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 24 x^{14} + 212 x^{12} + 872 x^{10} + 1815 x^{8} + 1928 x^{6} + 996 x^{4} + 200 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -3566 \nu^{15} - 82183 \nu^{13} - 677054 \nu^{11} - 2451006 \nu^{9} - 4029525 \nu^{7} - 2653934 \nu^{5} - 400750 \nu^{3} + 102636 \nu \)\()/7231\)
\(\beta_{2}\)\(=\)\((\)\( 4109 \nu^{14} + 95961 \nu^{12} + 809596 \nu^{10} + 3069810 \nu^{8} + 5537952 \nu^{6} + 4485192 \nu^{4} + 1280801 \nu^{2} - 9938 \)\()/7231\)
\(\beta_{3}\)\(=\)\((\)\( -4213 \nu^{14} - 98695 \nu^{12} - 836669 \nu^{10} - 3196410 \nu^{8} - 5829935 \nu^{6} - 4816716 \nu^{4} - 1459060 \nu^{2} - 8680 \)\()/7231\)
\(\beta_{4}\)\(=\)\((\)\( 11880 \nu^{15} + 277026 \nu^{13} + 2330454 \nu^{11} + 8786075 \nu^{9} + 15693672 \nu^{7} + 12631032 \nu^{5} + 3918733 \nu^{3} + 206322 \nu \)\()/7231\)
\(\beta_{5}\)\(=\)\((\)\( 12411 \nu^{15} + 289535 \nu^{13} + 2437027 \nu^{11} + 9191538 \nu^{9} + 16395526 \nu^{7} + 13061987 \nu^{5} + 3821459 \nu^{3} + 89795 \nu \)\()/7231\)
\(\beta_{6}\)\(=\)\((\)\( -12661 \nu^{15} - 294796 \nu^{13} - 2472050 \nu^{11} - 9248819 \nu^{9} - 16190613 \nu^{7} - 12258167 \nu^{5} - 2969225 \nu^{3} + 184528 \nu \)\()/7231\)
\(\beta_{7}\)\(=\)\((\)\( -15611 \nu^{14} - 362454 \nu^{12} - 3026016 \nu^{10} - 11245322 \nu^{8} - 19531408 \nu^{6} - 14787914 \nu^{4} - 3904397 \nu^{2} + 16978 \)\()/7231\)
\(\beta_{8}\)\(=\)\((\)\( -2432 \nu^{14} - 56782 \nu^{12} - 478539 \nu^{10} - 1808300 \nu^{8} - 3232355 \nu^{6} - 2573576 \nu^{4} - 736177 \nu^{2} - 5170 \)\()/1033\)
\(\beta_{9}\)\(=\)\((\)\( 20 \nu^{14} + 467 \nu^{12} + 3936 \nu^{10} + 14872 \nu^{8} + 26564 \nu^{6} + 21096 \nu^{4} + 5980 \nu^{2} + 18 \)\()/7\)
\(\beta_{10}\)\(=\)\((\)\( -20975 \nu^{15} - 489639 \nu^{13} - 4125450 \nu^{11} - 15583888 \nu^{9} - 27854760 \nu^{7} - 22235265 \nu^{5} - 6487208 \nu^{3} - 109968 \nu \)\()/7231\)
\(\beta_{11}\)\(=\)\((\)\( 24141 \nu^{14} + 562578 \nu^{12} + 4726427 \nu^{10} + 17764742 \nu^{8} + 31483261 \nu^{6} + 24752870 \nu^{4} + 7015238 \nu^{2} + 62386 \)\()/7231\)
\(\beta_{12}\)\(=\)\((\)\( 25023 \nu^{15} + 585089 \nu^{13} + 4943611 \nu^{11} + 18772057 \nu^{9} + 33876484 \nu^{7} + 27549033 \nu^{5} + 8361354 \nu^{3} + 246543 \nu \)\()/7231\)
\(\beta_{13}\)\(=\)\((\)\( -25833 \nu^{14} - 602052 \nu^{12} - 5058697 \nu^{10} - 19017892 \nu^{8} - 33715265 \nu^{6} - 26500974 \nu^{4} - 7440426 \nu^{2} - 19311 \)\()/7231\)
\(\beta_{14}\)\(=\)\((\)\( 25926 \nu^{15} + 603881 \nu^{13} + 5068077 \nu^{11} + 18998083 \nu^{9} + 33405881 \nu^{7} + 25519016 \nu^{5} + 6225156 \nu^{3} - 437333 \nu \)\()/7231\)
\(\beta_{15}\)\(=\)\((\)\( 33297 \nu^{15} + 775702 \nu^{13} + 6512858 \nu^{11} + 24444196 \nu^{9} + 43160577 \nu^{7} + 33528527 \nu^{5} + 8990368 \nu^{3} - 173250 \nu \)\()/7231\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{10} - \beta_{6} + \beta_{4} + \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{11} + \beta_{7} + 3 \beta_{3} + \beta_{2} - 6\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{15} - 5 \beta_{10} + 6 \beta_{6} + 6 \beta_{5} - 8 \beta_{4} - 7 \beta_{1}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(2 \beta_{13} - 9 \beta_{11} - 3 \beta_{9} - 6 \beta_{8} - 10 \beta_{7} - 27 \beta_{3} - 10 \beta_{2} + 38\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(5 \beta_{15} + 3 \beta_{14} - \beta_{12} + 33 \beta_{10} - 47 \beta_{6} - 62 \beta_{5} + 71 \beta_{4} + 55 \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-18 \beta_{13} + 86 \beta_{11} + 42 \beta_{9} + 83 \beta_{8} + 90 \beta_{7} + 229 \beta_{3} + 91 \beta_{2} - 294\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-11 \beta_{15} - 51 \beta_{14} + 34 \beta_{12} - 219 \beta_{10} + 385 \beta_{6} + 559 \beta_{5} - 630 \beta_{4} - 467 \beta_{1}\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(113 \beta_{13} - 820 \beta_{11} - 476 \beta_{9} - 909 \beta_{8} - 787 \beta_{7} - 1948 \beta_{3} - 832 \beta_{2} + 2417\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(-145 \beta_{15} + 630 \beta_{14} - 552 \beta_{12} + 1387 \beta_{10} - 3204 \beta_{6} - 4908 \beta_{5} + 5582 \beta_{4} + 4085 \beta_{1}\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(-514 \beta_{13} + 7743 \beta_{11} + 4970 \beta_{9} + 9227 \beta_{8} + 6847 \beta_{7} + 16682 \beta_{3} + 7634 \beta_{2} - 20336\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(3077 \beta_{15} - 6896 \beta_{14} + 6970 \beta_{12} - 8084 \beta_{10} + 26945 \beta_{6} + 42906 \beta_{5} - 49477 \beta_{4} - 36116 \beta_{1}\)\()/2\)
\(\nu^{12}\)\(=\)\(252 \beta_{13} - 36212 \beta_{11} - 24831 \beta_{9} - 45170 \beta_{8} - 29832 \beta_{7} - 71870 \beta_{3} - 34986 \beta_{2} + 86566\)
\(\nu^{13}\)\(=\)\((\)\(-40723 \beta_{15} + 70975 \beta_{14} - 78066 \beta_{12} + 39791 \beta_{10} - 228507 \beta_{6} - 375611 \beta_{5} + 439060 \beta_{4} + 320677 \beta_{1}\)\()/2\)
\(\nu^{14}\)\(=\)\((\)\(27158 \beta_{13} + 671999 \beta_{11} + 482846 \beta_{9} + 865586 \beta_{8} + 521589 \beta_{7} + 1245269 \beta_{3} + 639533 \beta_{2} - 1485588\)\()/2\)
\(\nu^{15}\)\(=\)\((\)\(462786 \beta_{15} - 704028 \beta_{14} + 817509 \beta_{12} - 112000 \beta_{10} + 1951782 \beta_{6} + 3297345 \beta_{5} - 3901487 \beta_{4} - 2853438 \beta_{1}\)\()/2\)

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
1.54124i
1.70654i
0.829521i
2.99366i
0.964652i
0.0716223i
2.73296i
0.810833i
2.73296i
0.810833i
0.964652i
0.0716223i
0.829521i
2.99366i
1.54124i
1.70654i
2.67093i 0 −5.13386 −3.20318 0 0 8.37032i 0 8.55547i
1322.2 2.67093i 0 −5.13386 3.20318 0 0 8.37032i 0 8.55547i
1322.3 2.02428i 0 −2.09771 −0.182734 0 0 0.197790i 0 0.369905i
1322.4 2.02428i 0 −2.09771 0.182734 0 0 0.197790i 0 0.369905i
1322.5 0.698626i 0 1.51192 −2.26676 0 0 2.45352i 0 1.58361i
1322.6 0.698626i 0 1.51192 2.26676 0 0 2.45352i 0 1.58361i
1322.7 0.529484i 0 1.71965 −0.753692 0 0 1.96949i 0 0.399068i
1322.8 0.529484i 0 1.71965 0.753692 0 0 1.96949i 0 0.399068i
1322.9 0.529484i 0 1.71965 −0.753692 0 0 1.96949i 0 0.399068i
1322.10 0.529484i 0 1.71965 0.753692 0 0 1.96949i 0 0.399068i
1322.11 0.698626i 0 1.51192 −2.26676 0 0 2.45352i 0 1.58361i
1322.12 0.698626i 0 1.51192 2.26676 0 0 2.45352i 0 1.58361i
1322.13 2.02428i 0 −2.09771 −0.182734 0 0 0.197790i 0 0.369905i
1322.14 2.02428i 0 −2.09771 0.182734 0 0 0.197790i 0 0.369905i
1322.15 2.67093i 0 −5.13386 −3.20318 0 0 8.37032i 0 8.55547i
1322.16 2.67093i 0 −5.13386 3.20318 0 0 8.37032i 0 8.55547i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1322.16
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.f 16
3.b odd 2 1 inner 1323.2.c.f 16
7.b odd 2 1 inner 1323.2.c.f 16
21.c even 2 1 inner 1323.2.c.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1323.2.c.f 16 1.a even 1 1 trivial
1323.2.c.f 16 3.b odd 2 1 inner
1323.2.c.f 16 7.b odd 2 1 inner
1323.2.c.f 16 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 12 T_{2}^{6} + 38 T_{2}^{4} + 24 T_{2}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(1323, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 + 24 T^{2} + 38 T^{4} + 12 T^{6} + T^{8} )^{2} \)
$3$ \( T^{16} \)
$5$ \( ( 1 - 32 T^{2} + 62 T^{4} - 16 T^{6} + T^{8} )^{2} \)
$7$ \( T^{16} \)
$11$ \( ( 2116 + 7128 T^{2} + 1142 T^{4} + 60 T^{6} + T^{8} )^{2} \)
$13$ \( ( 20164 + 12920 T^{2} + 1790 T^{4} + 76 T^{6} + T^{8} )^{2} \)
$17$ \( ( 134689 - 36124 T^{2} + 2918 T^{4} - 92 T^{6} + T^{8} )^{2} \)
$19$ \( ( 4 + 152 T^{2} + 206 T^{4} + 28 T^{6} + T^{8} )^{2} \)
$23$ \( ( 148996 + 33712 T^{2} + 2676 T^{4} + 88 T^{6} + T^{8} )^{2} \)
$29$ \( ( 4 + 10392 T^{2} + 2774 T^{4} + 108 T^{6} + T^{8} )^{2} \)
$31$ \( ( 9604 + 176792 T^{2} + 9662 T^{4} + 172 T^{6} + T^{8} )^{2} \)
$37$ \( ( 391 + 476 T - 88 T^{2} - 8 T^{3} + T^{4} )^{4} \)
$41$ \( ( 529 - 1664 T^{2} + 878 T^{4} - 112 T^{6} + T^{8} )^{2} \)
$43$ \( ( 943 + 16 T - 96 T^{2} + 4 T^{3} + T^{4} )^{4} \)
$47$ \( ( 18769 - 56036 T^{2} + 6806 T^{4} - 196 T^{6} + T^{8} )^{2} \)
$53$ \( ( 6791236 + 1315376 T^{2} + 37044 T^{4} + 344 T^{6} + T^{8} )^{2} \)
$59$ \( ( 24990001 - 1458692 T^{2} + 31286 T^{4} - 292 T^{6} + T^{8} )^{2} \)
$61$ \( ( 5827396 + 3061832 T^{2} + 63062 T^{4} + 436 T^{6} + T^{8} )^{2} \)
$67$ \( ( 612 + 504 T - 42 T^{2} - 12 T^{3} + T^{4} )^{4} \)
$71$ \( ( 68029504 + 3723648 T^{2} + 63824 T^{4} + 432 T^{6} + T^{8} )^{2} \)
$73$ \( ( 103684 + 43216 T^{2} + 3572 T^{4} + 104 T^{6} + T^{8} )^{2} \)
$79$ \( ( -6167 - 1744 T - 58 T^{2} + 16 T^{3} + T^{4} )^{4} \)
$83$ \( ( 390625 - 500000 T^{2} + 38750 T^{4} - 400 T^{6} + T^{8} )^{2} \)
$89$ \( ( 2155024 - 1096496 T^{2} + 109580 T^{4} - 652 T^{6} + T^{8} )^{2} \)
$97$ \( ( 36409156 + 7276072 T^{2} + 113798 T^{4} + 596 T^{6} + T^{8} )^{2} \)
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