Properties

Label 3024.2.cc.b
Level 3024
Weight 2
Character orbit 3024.cc
Analytic conductor 24.147
Analytic rank 0
Dimension 16
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3024.cc (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.1467615712\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 6 x^{14} + 9 x^{12} + 54 x^{10} - 288 x^{8} + 486 x^{6} + 729 x^{4} - 4374 x^{2} + 6561\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{6} \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -\beta_{2} - \beta_{9} + \beta_{15} ) q^{5} -\beta_{14} q^{7} +O(q^{10})\) \( q + ( -\beta_{2} - \beta_{9} + \beta_{15} ) q^{5} -\beta_{14} q^{7} + ( 1 + 2 \beta_{1} - 2 \beta_{4} - \beta_{6} - \beta_{10} + \beta_{12} - \beta_{14} ) q^{11} + ( -\beta_{3} - \beta_{9} ) q^{13} + ( -1 - \beta_{1} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{12} + \beta_{14} ) q^{17} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{10} - 2 \beta_{11} - \beta_{12} + \beta_{14} ) q^{19} + ( -5 - 2 \beta_{1} + \beta_{4} + 2 \beta_{6} + \beta_{7} + \beta_{12} - \beta_{14} ) q^{23} + ( -1 - \beta_{1} + \beta_{4} + \beta_{7} - \beta_{8} + \beta_{10} - \beta_{13} ) q^{25} + ( 1 - \beta_{4} + \beta_{6} + \beta_{7} + \beta_{12} - \beta_{14} ) q^{29} + ( -1 + \beta_{1} + \beta_{4} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{10} + 2 \beta_{11} + \beta_{12} - \beta_{14} ) q^{31} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{9} - \beta_{10} - 2 \beta_{12} + \beta_{15} ) q^{35} + ( \beta_{1} - 2 \beta_{4} + 2 \beta_{6} + \beta_{7} - \beta_{10} + 2 \beta_{12} - 2 \beta_{14} ) q^{37} + ( -1 - 2 \beta_{1} - 2 \beta_{3} + \beta_{4} - 3 \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{9} - \beta_{10} + 3 \beta_{11} + \beta_{12} - 2 \beta_{14} ) q^{41} + ( -1 + \beta_{6} - \beta_{10} + \beta_{12} - \beta_{14} ) q^{43} + ( -1 - 2 \beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{6} - \beta_{7} + 2 \beta_{10} + \beta_{12} + \beta_{14} + 2 \beta_{15} ) q^{47} + ( -1 - \beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{7} + 2 \beta_{9} - \beta_{11} - \beta_{12} + 2 \beta_{14} ) q^{49} + ( 2 + 3 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} - 2 \beta_{9} - \beta_{10} - 4 \beta_{11} - 2 \beta_{12} + 2 \beta_{14} + 2 \beta_{15} ) q^{55} + ( -1 - 2 \beta_{2} - 3 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} + 2 \beta_{15} ) q^{59} + ( 1 + 2 \beta_{1} - \beta_{3} - \beta_{4} + 4 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - \beta_{12} + \beta_{14} + \beta_{15} ) q^{61} + ( -4 + 3 \beta_{1} + \beta_{4} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{12} - \beta_{13} + \beta_{14} ) q^{65} + ( -4 \beta_{1} + \beta_{6} + \beta_{7} - 4 \beta_{8} + \beta_{10} + 2 \beta_{13} ) q^{67} + ( -1 - 3 \beta_{1} - \beta_{8} + \beta_{10} - \beta_{12} + \beta_{14} ) q^{71} + ( 1 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{11} - \beta_{12} + \beta_{14} ) q^{73} + ( -6 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{9} + \beta_{10} - 2 \beta_{11} - \beta_{12} - \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{77} + ( 1 - 2 \beta_{4} - \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{10} - \beta_{12} - \beta_{13} + \beta_{14} ) q^{79} + ( 2 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} - 4 \beta_{7} + 2 \beta_{9} + 2 \beta_{10} - 6 \beta_{11} - 2 \beta_{12} + 4 \beta_{14} - 3 \beta_{15} ) q^{83} + ( 1 + 3 \beta_{1} - \beta_{4} - \beta_{6} - \beta_{10} + \beta_{12} - \beta_{14} ) q^{85} + ( 2 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{12} - 2 \beta_{14} ) q^{89} + ( -\beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} + \beta_{9} - \beta_{15} ) q^{91} + ( 2 \beta_{4} + \beta_{6} + \beta_{7} + \beta_{10} - \beta_{13} ) q^{95} + ( 4 + \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} - 3 \beta_{7} + 2 \beta_{9} + \beta_{10} - \beta_{11} - 4 \beta_{12} - 3 \beta_{14} - 2 \beta_{15} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 2q^{7} + O(q^{10}) \) \( 16q - 2q^{7} - 12q^{11} - 48q^{23} - 8q^{25} + 12q^{29} - 8q^{37} - 4q^{43} - 8q^{49} - 84q^{65} + 28q^{67} - 78q^{77} + 4q^{79} - 12q^{85} - 24q^{91} + 12q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 6 x^{14} + 9 x^{12} + 54 x^{10} - 288 x^{8} + 486 x^{6} + 729 x^{4} - 4374 x^{2} + 6561\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 2 \nu^{14} - 21 \nu^{12} + 18 \nu^{10} + 108 \nu^{8} - 576 \nu^{6} + 648 \nu^{4} + 972 \nu^{2} - 9477 \)\()/5832\)
\(\beta_{2}\)\(=\)\((\)\( -2 \nu^{15} - 9 \nu^{13} + 18 \nu^{11} - 396 \nu^{7} + 216 \nu^{5} - 324 \nu^{3} - 9477 \nu \)\()/5832\)
\(\beta_{3}\)\(=\)\((\)\( -7 \nu^{15} + 24 \nu^{13} - 36 \nu^{11} - 540 \nu^{9} + 1044 \nu^{7} - 1134 \nu^{5} - 8019 \nu^{3} + 13122 \nu \)\()/17496\)
\(\beta_{4}\)\(=\)\((\)\( -10 \nu^{14} + 15 \nu^{12} + 18 \nu^{10} - 216 \nu^{8} - 36 \nu^{6} - 648 \nu^{4} - 2916 \nu^{2} - 3645 \)\()/5832\)
\(\beta_{5}\)\(=\)\((\)\( -4 \nu^{15} - 18 \nu^{14} + 24 \nu^{13} + 27 \nu^{12} - 90 \nu^{11} + 108 \nu^{9} - 486 \nu^{8} + 666 \nu^{7} - 648 \nu^{6} - 3402 \nu^{5} + 1458 \nu^{4} + 3888 \nu^{3} + 1458 \nu^{2} + 13122 \nu - 24057 \)\()/17496\)
\(\beta_{6}\)\(=\)\((\)\( -4 \nu^{14} + 9 \nu^{12} + 18 \nu^{10} - 216 \nu^{8} + 504 \nu^{6} + 432 \nu^{4} - 2754 \nu^{2} + 5589 \)\()/1944\)
\(\beta_{7}\)\(=\)\((\)\( -4 \nu^{15} + 6 \nu^{14} + 6 \nu^{13} + 18 \nu^{12} + 45 \nu^{11} - 27 \nu^{10} - 216 \nu^{9} + 81 \nu^{8} - 63 \nu^{7} + 1188 \nu^{6} + 1053 \nu^{5} - 1701 \nu^{4} - 3888 \nu^{3} - 2187 \nu^{2} + 2187 \nu + 26244 \)\()/8748\)
\(\beta_{8}\)\(=\)\((\)\( \nu^{14} - 3 \nu^{12} - 9 \nu^{10} + 81 \nu^{8} - 126 \nu^{6} - 135 \nu^{4} + 1458 \nu^{2} - 2187 \)\()/486\)
\(\beta_{9}\)\(=\)\((\)\( -4 \nu^{15} + 9 \nu^{13} + 18 \nu^{11} - 216 \nu^{9} + 504 \nu^{7} + 432 \nu^{5} - 2754 \nu^{3} + 9477 \nu \)\()/5832\)
\(\beta_{10}\)\(=\)\((\)\( 8 \nu^{15} + 18 \nu^{14} - 12 \nu^{13} - 27 \nu^{12} - 90 \nu^{11} + 432 \nu^{9} + 486 \nu^{8} + 126 \nu^{7} + 648 \nu^{6} - 2106 \nu^{5} - 1458 \nu^{4} + 7776 \nu^{3} - 1458 \nu^{2} - 4374 \nu + 24057 \)\()/17496\)
\(\beta_{11}\)\(=\)\((\)\( 4 \nu^{15} + 51 \nu^{14} - 24 \nu^{13} - 198 \nu^{12} - 72 \nu^{11} + 54 \nu^{10} + 378 \nu^{9} + 2268 \nu^{8} - 666 \nu^{7} - 7398 \nu^{6} - 972 \nu^{5} + 6804 \nu^{4} + 3402 \nu^{3} + 37179 \nu^{2} - 4374 \nu - 83106 \)\()/17496\)
\(\beta_{12}\)\(=\)\((\)\( 22 \nu^{15} + 3 \nu^{14} - 78 \nu^{13} + 36 \nu^{12} - 18 \nu^{11} - 216 \nu^{10} + 1026 \nu^{9} + 162 \nu^{8} - 2448 \nu^{7} + 594 \nu^{6} - 486 \nu^{5} - 5346 \nu^{4} + 15552 \nu^{3} + 729 \nu^{2} - 21870 \nu + 8748 \)\()/17496\)
\(\beta_{13}\)\(=\)\((\)\( 3 \nu^{14} - 10 \nu^{12} - 12 \nu^{10} + 180 \nu^{8} - 432 \nu^{6} - 198 \nu^{4} + 3483 \nu^{2} - 5832 \)\()/648\)
\(\beta_{14}\)\(=\)\((\)\( 14 \nu^{15} + 51 \nu^{14} - 66 \nu^{13} - 198 \nu^{12} + 72 \nu^{11} + 54 \nu^{10} + 594 \nu^{9} + 2268 \nu^{8} - 2574 \nu^{7} - 7398 \nu^{6} + 1620 \nu^{5} + 6804 \nu^{4} + 7776 \nu^{3} + 37179 \nu^{2} - 17496 \nu - 83106 \)\()/17496\)
\(\beta_{15}\)\(=\)\((\)\( -35 \nu^{15} + 138 \nu^{13} + 36 \nu^{11} - 2052 \nu^{9} + 6192 \nu^{7} - 2106 \nu^{5} - 37179 \nu^{3} + 87480 \nu \)\()/17496\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{12} + \beta_{11} + \beta_{9} + 2 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} + \beta_{1} - 1\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{13} - \beta_{8} + 3 \beta_{6} - 3 \beta_{1}\)\()/3\)
\(\nu^{3}\)\(=\)\(-\beta_{15} + \beta_{14} - \beta_{12} - 2 \beta_{11} + \beta_{10} + 2 \beta_{9} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{14} - 2 \beta_{13} - \beta_{12} - \beta_{10} + 2 \beta_{8} - 2 \beta_{7} - \beta_{6} - 2 \beta_{4} - 2 \beta_{1} + 2\)
\(\nu^{5}\)\(=\)\(-2 \beta_{15} - 5 \beta_{14} + 5 \beta_{11} - 4 \beta_{10} + 3 \beta_{9} - 4 \beta_{5} - 2 \beta_{3} - 3 \beta_{2}\)
\(\nu^{6}\)\(=\)\(3 \beta_{10} + 3 \beta_{8} + 3 \beta_{7} + 6 \beta_{6} + 3 \beta_{1} - 12\)
\(\nu^{7}\)\(=\)\(3 \beta_{15} - 3 \beta_{12} - 3 \beta_{11} + 6 \beta_{10} + 6 \beta_{9} - 18 \beta_{7} - 3 \beta_{6} - 9 \beta_{5} - 3 \beta_{4} - 9 \beta_{3} + 6 \beta_{2} - 15 \beta_{1} + 3\)
\(\nu^{8}\)\(=\)\(12 \beta_{14} - 18 \beta_{13} - 12 \beta_{12} + 15 \beta_{10} + 24 \beta_{8} + 3 \beta_{7} - 9 \beta_{6} + 18 \beta_{4} - 12 \beta_{1} - 3\)
\(\nu^{9}\)\(=\)\(18 \beta_{15} - 27 \beta_{14} + 15 \beta_{12} + 42 \beta_{11} - 9 \beta_{10} - 21 \beta_{9} + 21 \beta_{7} + 15 \beta_{6} - 3 \beta_{5} + 15 \beta_{4} - 72 \beta_{3} + 3 \beta_{2} + 6 \beta_{1} - 15\)
\(\nu^{10}\)\(=\)\(45 \beta_{14} + 30 \beta_{13} - 45 \beta_{12} + 90 \beta_{10} - 33 \beta_{8} + 45 \beta_{7} + 45 \beta_{6} + 108 \beta_{4} - 117 \beta_{1} - 153\)
\(\nu^{11}\)\(=\)\(63 \beta_{15} + 153 \beta_{14} - 36 \beta_{12} - 189 \beta_{11} + 72 \beta_{10} - 36 \beta_{9} - 54 \beta_{7} - 36 \beta_{6} + 54 \beta_{5} - 36 \beta_{4} - 117 \beta_{3} + 72 \beta_{2} - 18 \beta_{1} + 36\)
\(\nu^{12}\)\(=\)\(126 \beta_{14} - 72 \beta_{13} - 126 \beta_{12} + 36 \beta_{10} - 90 \beta_{7} - 180 \beta_{6} + 72 \beta_{4} - 684 \beta_{1} - 315\)
\(\nu^{13}\)\(=\)\(-36 \beta_{15} - 36 \beta_{14} - 189 \beta_{12} - 153 \beta_{11} - 234 \beta_{10} - 459 \beta_{9} + 216 \beta_{7} - 189 \beta_{6} + 171 \beta_{5} - 189 \beta_{4} - 144 \beta_{3} - 297 \beta_{2} + 405 \beta_{1} + 189\)
\(\nu^{14}\)\(=\)\(-54 \beta_{14} + 270 \beta_{13} + 54 \beta_{12} - 54 \beta_{10} - 621 \beta_{8} - 243 \beta_{6} - 540 \beta_{4} - 567 \beta_{1} - 1134\)
\(\nu^{15}\)\(=\)\(81 \beta_{15} + 837 \beta_{14} - 297 \beta_{12} - 1134 \beta_{11} - 81 \beta_{10} - 1026 \beta_{9} - 891 \beta_{7} - 297 \beta_{6} - 675 \beta_{5} - 297 \beta_{4} + 999 \beta_{3} - 864 \beta_{2} - 594 \beta_{1} + 297\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1135\) \(2593\)
\(\chi(n)\) \(1\) \(-\beta_{1}\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
881.1
−1.62181 0.608059i
1.40917 1.00709i
−1.69547 + 0.354107i
0.0967785 + 1.72934i
−0.0967785 1.72934i
1.69547 0.354107i
−1.40917 + 1.00709i
1.62181 + 0.608059i
−1.62181 + 0.608059i
1.40917 + 1.00709i
−1.69547 0.354107i
0.0967785 1.72934i
−0.0967785 + 1.72934i
1.69547 + 0.354107i
−1.40917 1.00709i
1.62181 0.608059i
0 0 0 −1.94556 + 3.36980i 0 −0.343982 2.62329i 0 0 0
881.2 0 0 0 −1.17468 + 2.03460i 0 −1.55364 + 2.14154i 0 0 0
881.3 0 0 0 −0.895175 + 1.55049i 0 −0.0213944 + 2.64566i 0 0 0
881.4 0 0 0 −0.183299 + 0.317483i 0 0.624224 2.57106i 0 0 0
881.5 0 0 0 0.183299 0.317483i 0 −2.53871 0.744936i 0 0 0
881.6 0 0 0 0.895175 1.55049i 0 2.30191 + 1.30430i 0 0 0
881.7 0 0 0 1.17468 2.03460i 0 2.63145 0.274725i 0 0 0
881.8 0 0 0 1.94556 3.36980i 0 −2.09985 1.60954i 0 0 0
2897.1 0 0 0 −1.94556 3.36980i 0 −0.343982 + 2.62329i 0 0 0
2897.2 0 0 0 −1.17468 2.03460i 0 −1.55364 2.14154i 0 0 0
2897.3 0 0 0 −0.895175 1.55049i 0 −0.0213944 2.64566i 0 0 0
2897.4 0 0 0 −0.183299 0.317483i 0 0.624224 + 2.57106i 0 0 0
2897.5 0 0 0 0.183299 + 0.317483i 0 −2.53871 + 0.744936i 0 0 0
2897.6 0 0 0 0.895175 + 1.55049i 0 2.30191 1.30430i 0 0 0
2897.7 0 0 0 1.17468 + 2.03460i 0 2.63145 + 0.274725i 0 0 0
2897.8 0 0 0 1.94556 + 3.36980i 0 −2.09985 + 1.60954i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2897.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
9.d odd 6 1 inner
63.o even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3024.2.cc.b 16
3.b odd 2 1 1008.2.cc.b 16
4.b odd 2 1 378.2.m.a 16
7.b odd 2 1 inner 3024.2.cc.b 16
9.c even 3 1 1008.2.cc.b 16
9.d odd 6 1 inner 3024.2.cc.b 16
12.b even 2 1 126.2.m.a 16
21.c even 2 1 1008.2.cc.b 16
28.d even 2 1 378.2.m.a 16
28.f even 6 1 2646.2.l.b 16
28.f even 6 1 2646.2.t.a 16
28.g odd 6 1 2646.2.l.b 16
28.g odd 6 1 2646.2.t.a 16
36.f odd 6 1 126.2.m.a 16
36.f odd 6 1 1134.2.d.a 16
36.h even 6 1 378.2.m.a 16
36.h even 6 1 1134.2.d.a 16
63.l odd 6 1 1008.2.cc.b 16
63.o even 6 1 inner 3024.2.cc.b 16
84.h odd 2 1 126.2.m.a 16
84.j odd 6 1 882.2.l.a 16
84.j odd 6 1 882.2.t.b 16
84.n even 6 1 882.2.l.a 16
84.n even 6 1 882.2.t.b 16
252.n even 6 1 882.2.l.a 16
252.o even 6 1 2646.2.l.b 16
252.r odd 6 1 2646.2.t.a 16
252.s odd 6 1 378.2.m.a 16
252.s odd 6 1 1134.2.d.a 16
252.u odd 6 1 882.2.t.b 16
252.bb even 6 1 2646.2.t.a 16
252.bi even 6 1 126.2.m.a 16
252.bi even 6 1 1134.2.d.a 16
252.bj even 6 1 882.2.t.b 16
252.bl odd 6 1 882.2.l.a 16
252.bn odd 6 1 2646.2.l.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.m.a 16 12.b even 2 1
126.2.m.a 16 36.f odd 6 1
126.2.m.a 16 84.h odd 2 1
126.2.m.a 16 252.bi even 6 1
378.2.m.a 16 4.b odd 2 1
378.2.m.a 16 28.d even 2 1
378.2.m.a 16 36.h even 6 1
378.2.m.a 16 252.s odd 6 1
882.2.l.a 16 84.j odd 6 1
882.2.l.a 16 84.n even 6 1
882.2.l.a 16 252.n even 6 1
882.2.l.a 16 252.bl odd 6 1
882.2.t.b 16 84.j odd 6 1
882.2.t.b 16 84.n even 6 1
882.2.t.b 16 252.u odd 6 1
882.2.t.b 16 252.bj even 6 1
1008.2.cc.b 16 3.b odd 2 1
1008.2.cc.b 16 9.c even 3 1
1008.2.cc.b 16 21.c even 2 1
1008.2.cc.b 16 63.l odd 6 1
1134.2.d.a 16 36.f odd 6 1
1134.2.d.a 16 36.h even 6 1
1134.2.d.a 16 252.s odd 6 1
1134.2.d.a 16 252.bi even 6 1
2646.2.l.b 16 28.f even 6 1
2646.2.l.b 16 28.g odd 6 1
2646.2.l.b 16 252.o even 6 1
2646.2.l.b 16 252.bn odd 6 1
2646.2.t.a 16 28.f even 6 1
2646.2.t.a 16 28.g odd 6 1
2646.2.t.a 16 252.r odd 6 1
2646.2.t.a 16 252.bb even 6 1
3024.2.cc.b 16 1.a even 1 1 trivial
3024.2.cc.b 16 7.b odd 2 1 inner
3024.2.cc.b 16 9.d odd 6 1 inner
3024.2.cc.b 16 63.o even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{5}^{16} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(3024, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - 16 T^{2} + 123 T^{4} - 584 T^{6} + 1481 T^{8} + 1416 T^{10} - 59414 T^{12} + 576968 T^{14} - 3477114 T^{16} + 14424200 T^{18} - 37133750 T^{20} + 22125000 T^{22} + 578515625 T^{24} - 5703125000 T^{26} + 30029296875 T^{28} - 97656250000 T^{30} + 152587890625 T^{32} \)
$7$ \( 1 + 2 T + 6 T^{2} - 8 T^{3} - 58 T^{4} - 222 T^{5} - 104 T^{6} + 662 T^{7} + 3483 T^{8} + 4634 T^{9} - 5096 T^{10} - 76146 T^{11} - 139258 T^{12} - 134456 T^{13} + 705894 T^{14} + 1647086 T^{15} + 5764801 T^{16} \)
$11$ \( ( 1 + 6 T + 35 T^{2} + 138 T^{3} + 481 T^{4} + 1512 T^{5} + 3854 T^{6} + 13116 T^{7} + 37618 T^{8} + 144276 T^{9} + 466334 T^{10} + 2012472 T^{11} + 7042321 T^{12} + 22225038 T^{13} + 62004635 T^{14} + 116923026 T^{15} + 214358881 T^{16} )^{2} \)
$13$ \( 1 + 68 T^{2} + 2376 T^{4} + 57352 T^{6} + 1082018 T^{8} + 16951644 T^{10} + 232506496 T^{12} + 2987085740 T^{14} + 38351015667 T^{16} + 504817490060 T^{18} + 6640618032256 T^{20} + 81822347823996 T^{22} + 882635323274978 T^{24} + 7906460224523848 T^{26} + 55356250251014856 T^{28} + 267741594227551652 T^{30} + 665416609183179841 T^{32} \)
$17$ \( ( 1 + 94 T^{2} + 4285 T^{4} + 124198 T^{6} + 2503180 T^{8} + 35893222 T^{10} + 357887485 T^{12} + 2268931486 T^{14} + 6975757441 T^{16} )^{2} \)
$19$ \( ( 1 - 98 T^{2} + 4546 T^{4} - 134984 T^{6} + 2932423 T^{8} - 48729224 T^{10} + 592439266 T^{12} - 4610496338 T^{14} + 16983563041 T^{16} )^{2} \)
$23$ \( ( 1 + 24 T + 317 T^{2} + 3000 T^{3} + 22111 T^{4} + 134028 T^{5} + 704756 T^{6} + 3411156 T^{7} + 16228318 T^{8} + 78456588 T^{9} + 372815924 T^{10} + 1630718676 T^{11} + 6187564351 T^{12} + 19309029000 T^{13} + 46927376813 T^{14} + 81715810728 T^{15} + 78310985281 T^{16} )^{2} \)
$29$ \( ( 1 - 6 T + 98 T^{2} - 516 T^{3} + 4846 T^{4} - 25650 T^{5} + 193448 T^{6} - 972210 T^{7} + 6347347 T^{8} - 28194090 T^{9} + 162689768 T^{10} - 625577850 T^{11} + 3427483726 T^{12} - 10583752884 T^{13} + 58292685458 T^{14} - 103499257854 T^{15} + 500246412961 T^{16} )^{2} \)
$31$ \( 1 + 104 T^{2} + 3888 T^{4} + 96880 T^{6} + 4455362 T^{8} + 148421160 T^{10} + 1870813504 T^{12} + 70884338648 T^{14} + 4079738375235 T^{16} + 68119849440728 T^{18} + 1727735558027584 T^{20} + 131724325838289960 T^{22} + 3799938318355208642 T^{24} + 79405588442700000880 T^{26} + \)\(30\!\cdots\!68\)\( T^{28} + \)\(78\!\cdots\!84\)\( T^{30} + \)\(72\!\cdots\!81\)\( T^{32} \)
$37$ \( ( 1 + 2 T + 46 T^{2} + 38 T^{3} + 2002 T^{4} + 1406 T^{5} + 62974 T^{6} + 101306 T^{7} + 1874161 T^{8} )^{4} \)
$41$ \( 1 - 70 T^{2} - 1569 T^{4} + 150526 T^{6} + 5171081 T^{8} - 280356900 T^{10} - 8583026738 T^{12} + 255096492224 T^{14} + 9422146980954 T^{16} + 428817203428544 T^{18} - 24253582218197618 T^{20} - 1331724499683612900 T^{22} + 41290695138728249801 T^{24} + \)\(20\!\cdots\!26\)\( T^{26} - \)\(35\!\cdots\!89\)\( T^{28} - \)\(26\!\cdots\!70\)\( T^{30} + \)\(63\!\cdots\!41\)\( T^{32} \)
$43$ \( ( 1 + 2 T - 129 T^{2} + 46 T^{3} + 9833 T^{4} - 11184 T^{5} - 521114 T^{6} + 232628 T^{7} + 22298490 T^{8} + 10003004 T^{9} - 963539786 T^{10} - 889206288 T^{11} + 33617070233 T^{12} + 6762388378 T^{13} - 815455833321 T^{14} + 543637222214 T^{15} + 11688200277601 T^{16} )^{2} \)
$47$ \( 1 - 136 T^{2} + 8016 T^{4} - 225584 T^{6} - 1533310 T^{8} + 489880632 T^{10} - 30253322048 T^{12} + 1486359308360 T^{14} - 68731587628605 T^{16} + 3283367712167240 T^{18} - 147626560784506688 T^{20} + 5280528817834607928 T^{22} - 36510083951344758910 T^{24} - \)\(11\!\cdots\!16\)\( T^{26} + \)\(93\!\cdots\!56\)\( T^{28} - \)\(34\!\cdots\!84\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \)
$53$ \( ( 1 - 53 T^{2} )^{16} \)
$59$ \( 1 - 178 T^{2} + 20643 T^{4} - 1496150 T^{6} + 80456981 T^{8} - 3515943660 T^{10} + 180649052698 T^{12} - 13219132050040 T^{14} + 857467356385554 T^{16} - 46015798666189240 T^{18} + 2188989785849689978 T^{20} - \)\(14\!\cdots\!60\)\( T^{22} + \)\(11\!\cdots\!01\)\( T^{24} - \)\(76\!\cdots\!50\)\( T^{26} + \)\(36\!\cdots\!83\)\( T^{28} - \)\(11\!\cdots\!58\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \)
$61$ \( 1 + 248 T^{2} + 28191 T^{4} + 2052448 T^{6} + 122525357 T^{8} + 7198089144 T^{10} + 457346362462 T^{12} + 32095759051208 T^{14} + 2131627513941198 T^{16} + 119428319429544968 T^{18} + 6332345016577220542 T^{20} + \)\(37\!\cdots\!84\)\( T^{22} + \)\(23\!\cdots\!17\)\( T^{24} + \)\(14\!\cdots\!48\)\( T^{26} + \)\(74\!\cdots\!11\)\( T^{28} + \)\(24\!\cdots\!68\)\( T^{30} + \)\(36\!\cdots\!61\)\( T^{32} \)
$67$ \( ( 1 - 14 T + 39 T^{2} + 110 T^{3} - 2659 T^{4} + 53760 T^{5} - 92054 T^{6} - 2669060 T^{7} + 22240746 T^{8} - 178827020 T^{9} - 413230406 T^{10} + 16169018880 T^{11} - 53581830739 T^{12} + 148513761770 T^{13} + 3527876904591 T^{14} - 84849962474522 T^{15} + 406067677556641 T^{16} )^{2} \)
$71$ \( ( 1 - 478 T^{2} + 105553 T^{4} - 13986142 T^{6} + 1213269316 T^{8} - 70504141822 T^{10} + 2682279164593 T^{12} - 61231935714238 T^{14} + 645753531245761 T^{16} )^{2} \)
$73$ \( ( 1 - 362 T^{2} + 64045 T^{4} - 7316714 T^{6} + 612211324 T^{8} - 38990768906 T^{10} + 1818765344845 T^{12} - 54782989916618 T^{14} + 806460091894081 T^{16} )^{2} \)
$79$ \( ( 1 - 2 T - 183 T^{2} - 982 T^{3} + 19715 T^{4} + 144312 T^{5} - 491612 T^{6} - 7480148 T^{7} - 8945118 T^{8} - 590931692 T^{9} - 3068150492 T^{10} + 71151444168 T^{11} + 767900846915 T^{12} - 3021669383818 T^{13} - 44485004360343 T^{14} - 38407817972318 T^{15} + 1517108809906561 T^{16} )^{2} \)
$83$ \( 1 + 44 T^{2} - 13368 T^{4} - 595880 T^{6} + 87112226 T^{8} + 3596762100 T^{10} - 160886956928 T^{12} - 11841264313180 T^{14} - 673218489607821 T^{16} - 81574469853497020 T^{18} - 7635424846602197888 T^{20} + \)\(11\!\cdots\!00\)\( T^{22} + \)\(19\!\cdots\!66\)\( T^{24} - \)\(92\!\cdots\!20\)\( T^{26} - \)\(14\!\cdots\!48\)\( T^{28} + \)\(32\!\cdots\!76\)\( T^{30} + \)\(50\!\cdots\!81\)\( T^{32} \)
$89$ \( ( 1 + 496 T^{2} + 119404 T^{4} + 18272464 T^{6} + 1934931814 T^{8} + 144736187344 T^{10} + 7491674544364 T^{12} + 246502720316656 T^{14} + 3936588805702081 T^{16} )^{2} \)
$97$ \( 1 + 74 T^{2} + 11511 T^{4} - 803858 T^{6} - 150210775 T^{8} - 25062425316 T^{10} - 114467134418 T^{12} + 73367970993632 T^{14} + 27832786456667274 T^{16} + 690319239079083488 T^{18} - 10133693108155893458 T^{20} - \)\(20\!\cdots\!64\)\( T^{22} - \)\(11\!\cdots\!75\)\( T^{24} - \)\(59\!\cdots\!42\)\( T^{26} + \)\(79\!\cdots\!51\)\( T^{28} + \)\(48\!\cdots\!06\)\( T^{30} + \)\(61\!\cdots\!21\)\( T^{32} \)
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