Properties

Label 1323.2.c.e
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: \(x^{16} - 12 x^{14} + 106 x^{12} - 384 x^{10} + 1005 x^{8} - 1200 x^{6} + 1030 x^{4} - 252 x^{2} + 49\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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.

\(f(q)\) \(=\) \( q -\beta_{7} q^{2} + ( -1 - \beta_{1} - \beta_{4} ) q^{4} + ( -\beta_{3} - \beta_{10} ) q^{5} + ( 2 \beta_{7} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{8} +O(q^{10})\) \( q -\beta_{7} q^{2} + ( -1 - \beta_{1} - \beta_{4} ) q^{4} + ( -\beta_{3} - \beta_{10} ) q^{5} + ( 2 \beta_{7} + \beta_{13} - \beta_{14} - \beta_{15} ) q^{8} + ( -2 \beta_{2} - \beta_{8} - \beta_{9} ) q^{10} + \beta_{15} q^{11} + ( -\beta_{2} - \beta_{9} + \beta_{12} ) q^{13} + ( 3 + 2 \beta_{1} + 3 \beta_{4} + 2 \beta_{6} ) q^{16} + ( -\beta_{5} - 2 \beta_{11} ) q^{17} + ( -2 \beta_{2} + \beta_{8} + \beta_{9} - \beta_{12} ) q^{19} + ( -\beta_{3} - 3 \beta_{5} - 2 \beta_{10} - \beta_{11} ) q^{20} + ( 1 - \beta_{1} ) q^{22} + ( -\beta_{7} + 2 \beta_{13} + \beta_{14} + \beta_{15} ) q^{23} + ( 2 - \beta_{1} - \beta_{4} + 2 \beta_{6} ) q^{25} + ( 2 \beta_{3} - 2 \beta_{5} - \beta_{10} - \beta_{11} ) q^{26} + ( -3 \beta_{7} - 2 \beta_{14} + \beta_{15} ) q^{29} + ( -\beta_{2} - 3 \beta_{8} - \beta_{9} - \beta_{12} ) q^{31} + ( -6 \beta_{7} - 3 \beta_{13} + 2 \beta_{14} ) q^{32} + ( 2 \beta_{2} + 4 \beta_{8} - \beta_{9} - \beta_{12} ) q^{34} + ( -1 - 4 \beta_{4} + \beta_{6} ) q^{37} + ( -2 \beta_{3} + \beta_{10} + 2 \beta_{11} ) q^{38} + ( 4 \beta_{2} + 3 \beta_{8} - 4 \beta_{12} ) q^{40} + ( 5 \beta_{3} + \beta_{5} - 3 \beta_{11} ) q^{41} + ( 2 - \beta_{1} + 3 \beta_{4} - 2 \beta_{6} ) q^{43} + ( \beta_{7} - \beta_{14} + \beta_{15} ) q^{44} + ( -5 - 3 \beta_{1} + 2 \beta_{4} + \beta_{6} ) q^{46} + ( 3 \beta_{3} + 2 \beta_{10} ) q^{47} + ( \beta_{7} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{50} + ( 6 \beta_{2} + 4 \beta_{8} + 4 \beta_{9} - 3 \beta_{12} ) q^{52} + ( 3 \beta_{7} + 3 \beta_{13} + 2 \beta_{14} ) q^{53} + ( \beta_{8} + 5 \beta_{9} + 3 \beta_{12} ) q^{55} + ( -6 - 2 \beta_{1} - \beta_{4} + 2 \beta_{6} ) q^{58} + ( 4 \beta_{3} - \beta_{5} + \beta_{10} - \beta_{11} ) q^{59} + ( -\beta_{2} - 5 \beta_{9} + \beta_{12} ) q^{61} + ( -3 \beta_{5} + \beta_{10} - 2 \beta_{11} ) q^{62} + ( -11 - 4 \beta_{1} - 8 \beta_{4} - \beta_{6} ) q^{64} + ( -3 \beta_{7} - 2 \beta_{13} + \beta_{15} ) q^{65} + ( 6 + 3 \beta_{4} + 2 \beta_{6} ) q^{67} + ( 5 \beta_{5} + \beta_{10} + \beta_{11} ) q^{68} + ( 3 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{71} + ( -\beta_{9} - 4 \beta_{12} ) q^{73} + ( 5 \beta_{7} + 3 \beta_{13} + \beta_{14} ) q^{74} + ( -3 \beta_{2} + \beta_{8} - \beta_{9} + \beta_{12} ) q^{76} + ( -4 + 2 \beta_{1} + \beta_{4} + \beta_{6} ) q^{79} + ( -6 \beta_{3} + 5 \beta_{5} + 5 \beta_{11} ) q^{80} + ( -2 \beta_{2} + \beta_{8} + 6 \beta_{9} - 4 \beta_{12} ) q^{82} + ( 2 \beta_{5} + 2 \beta_{10} + 2 \beta_{11} ) q^{83} + ( -2 - 3 \beta_{1} + 3 \beta_{4} - 2 \beta_{6} ) q^{85} + ( -3 \beta_{7} - \beta_{13} - 3 \beta_{14} - \beta_{15} ) q^{86} + ( 7 - \beta_{1} + 2 \beta_{4} + \beta_{6} ) q^{88} + ( -3 \beta_{3} + \beta_{5} + \beta_{10} + 2 \beta_{11} ) q^{89} + ( 7 \beta_{7} + \beta_{13} - \beta_{15} ) q^{92} + ( 5 \beta_{2} + 2 \beta_{8} + 4 \beta_{9} - \beta_{12} ) q^{94} + ( -3 \beta_{7} - 2 \beta_{13} - 5 \beta_{14} - \beta_{15} ) q^{95} + ( 2 \beta_{8} - 2 \beta_{9} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 16q^{4} + O(q^{10}) \) \( 16q - 16q^{4} + 48q^{16} + 16q^{22} + 32q^{25} - 16q^{37} + 32q^{43} - 80q^{46} - 96q^{58} - 176q^{64} + 96q^{67} - 64q^{79} - 32q^{85} + 112q^{88} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 12 x^{14} + 106 x^{12} - 384 x^{10} + 1005 x^{8} - 1200 x^{6} + 1030 x^{4} - 252 x^{2} + 49\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 878446 \nu^{14} - 9796819 \nu^{12} + 83290551 \nu^{10} - 251050620 \nu^{8} + 538652244 \nu^{6} - 279900636 \nu^{4} + 68773075 \nu^{2} - 272833414 \)\()/ 198602691 \)
\(\beta_{2}\)\(=\)\((\)\( 12378 \nu^{14} - 136713 \nu^{12} + 1187651 \nu^{10} - 3705756 \nu^{8} + 9677932 \nu^{6} - 9033702 \nu^{4} + 11793401 \nu^{2} - 1678362 \)\()/2797221\)
\(\beta_{3}\)\(=\)\((\)\( 514545 \nu^{15} - 6964488 \nu^{13} + 63470648 \nu^{11} - 275969081 \nu^{9} + 775230370 \nu^{7} - 1299818308 \nu^{5} + 1148028542 \nu^{3} - 498124634 \nu \)\()/ 198602691 \)
\(\beta_{4}\)\(=\)\((\)\( -5624 \nu^{14} + 62439 \nu^{12} - 533244 \nu^{10} + 1607280 \nu^{8} - 3592238 \nu^{6} + 1791984 \nu^{4} - 440300 \nu^{2} - 1259979 \)\()/932407\)
\(\beta_{5}\)\(=\)\((\)\( 878642 \nu^{15} - 9751721 \nu^{13} + 83806886 \nu^{11} - 257079648 \nu^{9} + 612892708 \nu^{7} - 460646739 \nu^{5} + 453052273 \nu^{3} - 195998558 \nu \)\()/ 198602691 \)
\(\beta_{6}\)\(=\)\((\)\( 1711368 \nu^{14} - 18874018 \nu^{12} + 162264708 \nu^{10} - 489090960 \nu^{8} + 1147767525 \nu^{6} - 545295888 \nu^{4} + 133982100 \nu^{2} + 618727571 \)\()/ 198602691 \)
\(\beta_{7}\)\(=\)\((\)\( -1114268 \nu^{15} + 13051750 \nu^{13} - 114609720 \nu^{11} + 397588491 \nu^{9} - 1028539320 \nu^{7} + 1110627150 \nu^{5} - 1045904084 \nu^{3} + 57182020 \nu \)\()/ 198602691 \)
\(\beta_{8}\)\(=\)\((\)\( -2228536 \nu^{14} + 26103500 \nu^{12} - 229219440 \nu^{10} + 795176982 \nu^{8} - 2057078640 \nu^{6} + 2221254300 \nu^{4} - 2091808168 \nu^{2} + 312966731 \)\()/ 198602691 \)
\(\beta_{9}\)\(=\)\((\)\( -38598 \nu^{14} + 462014 \nu^{12} - 4073324 \nu^{10} + 14662382 \nu^{8} - 38003215 \nu^{6} + 44294824 \nu^{4} - 34225904 \nu^{2} + 5317207 \)\()/2797221\)
\(\beta_{10}\)\(=\)\((\)\( 1696759 \nu^{15} - 22910949 \nu^{13} + 209908028 \nu^{11} - 912484061 \nu^{9} + 2597720626 \nu^{7} - 4156802779 \nu^{5} + 3723863718 \nu^{3} - 1618480304 \nu \)\()/ 198602691 \)
\(\beta_{11}\)\(=\)\((\)\( -1992910 \nu^{15} + 22803471 \nu^{13} - 198416606 \nu^{11} + 654668139 \nu^{9} - 1641432028 \nu^{7} + 1571273889 \nu^{5} - 1498956357 \nu^{3} + 650385960 \nu \)\()/ 198602691 \)
\(\beta_{12}\)\(=\)\((\)\( -28 \nu^{14} + 343 \nu^{12} - 3036 \nu^{10} + 11298 \nu^{8} - 29154 \nu^{6} + 34818 \nu^{4} - 25084 \nu^{2} + 3955 \)\()/1491\)
\(\beta_{13}\)\(=\)\((\)\( 3578038 \nu^{15} - 42545475 \nu^{13} + 373599324 \nu^{11} - 1321216260 \nu^{9} + 3352783644 \nu^{7} - 3620369655 \nu^{5} + 2763402976 \nu^{3} - 186399234 \nu \)\()/ 198602691 \)
\(\beta_{14}\)\(=\)\((\)\( 8544409 \nu^{15} - 101186975 \nu^{13} + 888540684 \nu^{11} - 3127305423 \nu^{9} + 7974009804 \nu^{7} - 8610416355 \nu^{5} + 6475452802 \nu^{3} - 443317994 \nu \)\()/ 198602691 \)
\(\beta_{15}\)\(=\)\((\)\(-11651979 \nu^{15} + 136952000 \nu^{13} - 1202599680 \nu^{11} + 4191620109 \nu^{9} - 10792462080 \nu^{7} + 11653809600 \nu^{5} - 9788871639 \nu^{3} + 600010880 \nu\)\()/ 198602691 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{11} - \beta_{7} + \beta_{5}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{12} - 3 \beta_{8} + \beta_{4} - 3 \beta_{2} + \beta_{1} + 3\)\()/2\)
\(\nu^{3}\)\(=\)\(\beta_{15} + \beta_{14} - \beta_{13} - 6 \beta_{7}\)
\(\nu^{4}\)\(=\)\((\)\(9 \beta_{12} - 6 \beta_{9} - 17 \beta_{8} - 2 \beta_{6} - 9 \beta_{4} - 24 \beta_{2} - 8 \beta_{1} - 17\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(8 \beta_{15} + 10 \beta_{14} - 11 \beta_{13} - 26 \beta_{11} - 9 \beta_{10} - 42 \beta_{7} - 50 \beta_{5} + 15 \beta_{3}\)\()/2\)
\(\nu^{6}\)\(=\)\(-21 \beta_{6} - 74 \beta_{4} - 60 \beta_{1} - 117\)
\(\nu^{7}\)\(=\)\((\)\(-60 \beta_{15} - 81 \beta_{14} + 95 \beta_{13} - 191 \beta_{11} - 74 \beta_{10} + 311 \beta_{7} - 371 \beta_{5} + 137 \beta_{3}\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(-582 \beta_{12} + 528 \beta_{9} + 859 \beta_{8} - 176 \beta_{6} - 582 \beta_{4} + 1356 \beta_{2} - 452 \beta_{1} - 859\)\()/2\)
\(\nu^{9}\)\(=\)\(-452 \beta_{15} - 628 \beta_{14} + 758 \beta_{13} + 2345 \beta_{7}\)
\(\nu^{10}\)\(=\)\((\)\(-4489 \beta_{12} + 4158 \beta_{9} + 6453 \beta_{8} + 1386 \beta_{6} + 4489 \beta_{4} + 10275 \beta_{2} + 3425 \beta_{1} + 6453\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(-3425 \beta_{15} - 4811 \beta_{14} + 5875 \beta_{13} + 10942 \beta_{11} + 4489 \beta_{10} + 17792 \beta_{7} + 21217 \beta_{5} - 8647 \beta_{3}\)\()/2\)
\(\nu^{12}\)\(=\)\(10686 \beta_{6} + 34353 \beta_{4} + 26028 \beta_{1} + 48887\)
\(\nu^{13}\)\(=\)\((\)\(26028 \beta_{15} + 36714 \beta_{14} - 45039 \beta_{13} + 83240 \beta_{11} + 34353 \beta_{10} - 135296 \beta_{7} + 161324 \beta_{5} - 66411 \beta_{3}\)\()/2\)
\(\nu^{14}\)\(=\)\((\)\(262088 \beta_{12} - 245259 \beta_{9} - 371535 \beta_{8} + 81753 \beta_{6} + 262088 \beta_{4} - 594114 \beta_{2} + 198038 \beta_{1} + 371535\)\()/2\)
\(\nu^{15}\)\(=\)\(198038 \beta_{15} + 279791 \beta_{14} - 343841 \beta_{13} - 1029699 \beta_{7}\)

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
−2.38976 1.37973i
2.38976 1.37973i
−1.47769 0.853147i
1.47769 0.853147i
0.954123 0.550863i
−0.954123 0.550863i
−0.441700 0.255016i
0.441700 0.255016i
−0.441700 + 0.255016i
0.441700 + 0.255016i
0.954123 + 0.550863i
−0.954123 + 0.550863i
−1.47769 + 0.853147i
1.47769 + 0.853147i
−2.38976 + 1.37973i
2.38976 + 1.37973i
2.75946i 0 −5.61463 −2.24426 0 0 9.97442i 0 6.19294i
1322.2 2.75946i 0 −5.61463 2.24426 0 0 9.97442i 0 6.19294i
1322.3 1.70629i 0 −0.911441 −0.829297 0 0 1.85740i 0 1.41503i
1322.4 1.70629i 0 −0.911441 0.829297 0 0 1.85740i 0 1.41503i
1322.5 1.10173i 0 0.786199 −2.47687 0 0 3.06963i 0 2.72883i
1322.6 1.10173i 0 0.786199 2.47687 0 0 3.06963i 0 2.72883i
1322.7 0.510032i 0 1.73987 −4.01755 0 0 1.90745i 0 2.04908i
1322.8 0.510032i 0 1.73987 4.01755 0 0 1.90745i 0 2.04908i
1322.9 0.510032i 0 1.73987 −4.01755 0 0 1.90745i 0 2.04908i
1322.10 0.510032i 0 1.73987 4.01755 0 0 1.90745i 0 2.04908i
1322.11 1.10173i 0 0.786199 −2.47687 0 0 3.06963i 0 2.72883i
1322.12 1.10173i 0 0.786199 2.47687 0 0 3.06963i 0 2.72883i
1322.13 1.70629i 0 −0.911441 −0.829297 0 0 1.85740i 0 1.41503i
1322.14 1.70629i 0 −0.911441 0.829297 0 0 1.85740i 0 1.41503i
1322.15 2.75946i 0 −5.61463 −2.24426 0 0 9.97442i 0 6.19294i
1322.16 2.75946i 0 −5.61463 2.24426 0 0 9.97442i 0 6.19294i
\(n\): 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.e 16
3.b odd 2 1 inner 1323.2.c.e 16
7.b odd 2 1 inner 1323.2.c.e 16
21.c even 2 1 inner 1323.2.c.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1323.2.c.e 16 1.a even 1 1 trivial
1323.2.c.e 16 3.b odd 2 1 inner
1323.2.c.e 16 7.b odd 2 1 inner
1323.2.c.e 16 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 12 T_{2}^{6} + 38 T_{2}^{4} + 36 T_{2}^{2} + 7 \) acting on \(S_{2}^{\mathrm{new}}(1323, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 7 + 36 T^{2} + 38 T^{4} + 12 T^{6} + T^{8} )^{2} \)
$3$ \( T^{16} \)
$5$ \( ( 343 - 644 T^{2} + 230 T^{4} - 28 T^{6} + T^{8} )^{2} \)
$7$ \( T^{16} \)
$11$ \( ( 7 + 204 T^{2} + 326 T^{4} + 36 T^{6} + T^{8} )^{2} \)
$13$ \( ( 16 + 608 T^{2} + 392 T^{4} + 40 T^{6} + T^{8} )^{2} \)
$17$ \( ( 67228 - 22960 T^{2} + 2180 T^{4} - 80 T^{6} + T^{8} )^{2} \)
$19$ \( ( 9409 + 8084 T^{2} + 1442 T^{4} + 76 T^{6} + T^{8} )^{2} \)
$23$ \( ( 55447 + 40420 T^{2} + 4122 T^{4} + 124 T^{6} + T^{8} )^{2} \)
$29$ \( ( 887152 + 159840 T^{2} + 8936 T^{4} + 168 T^{6} + T^{8} )^{2} \)
$31$ \( ( 1681 + 94388 T^{2} + 6710 T^{4} + 148 T^{6} + T^{8} )^{2} \)
$37$ \( ( 487 - 244 T - 70 T^{2} + 4 T^{3} + T^{4} )^{4} \)
$41$ \( ( 22104103 - 1755260 T^{2} + 39290 T^{4} - 340 T^{6} + T^{8} )^{2} \)
$43$ \( ( -98 + 280 T - 72 T^{2} - 8 T^{3} + T^{4} )^{4} \)
$47$ \( ( 263452 - 135152 T^{2} + 8036 T^{4} - 160 T^{6} + T^{8} )^{2} \)
$53$ \( ( 4171888 + 419168 T^{2} + 14232 T^{4} + 200 T^{6} + T^{8} )^{2} \)
$59$ \( ( 3294172 - 519104 T^{2} + 20924 T^{4} - 280 T^{6} + T^{8} )^{2} \)
$61$ \( ( 2972176 + 437600 T^{2} + 16904 T^{4} + 232 T^{6} + T^{8} )^{2} \)
$67$ \( ( -3708 + 432 T + 132 T^{2} - 24 T^{3} + T^{4} )^{4} \)
$71$ \( ( 15463 + 171756 T^{2} + 19382 T^{4} + 276 T^{6} + T^{8} )^{2} \)
$73$ \( ( 78039556 + 3466000 T^{2} + 56084 T^{4} + 392 T^{6} + T^{8} )^{2} \)
$79$ \( ( -1538 - 424 T + 32 T^{2} + 16 T^{3} + T^{4} )^{4} \)
$83$ \( ( 351232 - 112640 T^{2} + 7616 T^{4} - 160 T^{6} + T^{8} )^{2} \)
$89$ \( ( 794983 - 267620 T^{2} + 14378 T^{4} - 220 T^{6} + T^{8} )^{2} \)
$97$ \( ( 256 + 4864 T^{2} + 1568 T^{4} + 80 T^{6} + T^{8} )^{2} \)
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