Properties

Label 2736.2.dc.c
Level $2736$
Weight $2$
Character orbit 2736.dc
Analytic conductor $21.847$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2736.dc (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 16 x^{14} + 174 x^{12} + 1012 x^{10} + 4243 x^{8} + 9708 x^{6} + 15858 x^{4} + 12150 x^{2} + 6561\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 171)
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.

\(f(q)\) \(=\) \( q + \beta_{10} q^{5} -\beta_{14} q^{7} +O(q^{10})\) \( q + \beta_{10} q^{5} -\beta_{14} q^{7} -\beta_{7} q^{11} + ( -2 - \beta_{4} - \beta_{5} ) q^{13} + ( \beta_{7} - \beta_{10} + \beta_{12} ) q^{17} + ( 1 + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{9} - \beta_{14} ) q^{19} + ( \beta_{6} + \beta_{15} ) q^{23} + ( 1 + \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{8} - \beta_{9} - \beta_{14} ) q^{25} -\beta_{13} q^{29} + ( -\beta_{3} - \beta_{8} + 2 \beta_{9} ) q^{31} + ( -\beta_{1} - 2 \beta_{7} - 2 \beta_{12} - 2 \beta_{13} - \beta_{15} ) q^{35} + ( 1 + 2 \beta_{2} - \beta_{3} + 2 \beta_{9} - \beta_{14} ) q^{37} + ( \beta_{1} - \beta_{7} + \beta_{10} + 2 \beta_{11} + \beta_{12} ) q^{41} + ( \beta_{5} - \beta_{8} + \beta_{9} ) q^{43} + ( -2 \beta_{1} + \beta_{6} - 2 \beta_{10} - 2 \beta_{11} + \beta_{12} - \beta_{13} + \beta_{15} ) q^{47} + ( 2 \beta_{5} + \beta_{8} ) q^{49} + ( -2 \beta_{7} - \beta_{12} ) q^{53} + ( 1 + \beta_{2} + \beta_{4} - \beta_{5} + \beta_{8} + \beta_{9} ) q^{55} + ( \beta_{1} - \beta_{7} + \beta_{12} ) q^{59} + ( 1 + \beta_{2} + \beta_{3} + 6 \beta_{4} - \beta_{5} - 2 \beta_{8} - \beta_{9} - \beta_{14} ) q^{61} + ( -\beta_{1} + \beta_{7} - 4 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{65} + ( 2 - 2 \beta_{2} + \beta_{3} + 2 \beta_{5} + \beta_{9} - 2 \beta_{14} ) q^{67} + ( -2 \beta_{6} - \beta_{7} + \beta_{12} - \beta_{15} ) q^{71} + ( -4 - 3 \beta_{2} - 4 \beta_{4} + 2 \beta_{5} - 2 \beta_{8} + \beta_{9} ) q^{73} + ( \beta_{6} - \beta_{7} - \beta_{11} ) q^{77} + ( 2 + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{8} - \beta_{9} ) q^{79} + ( -2 \beta_{6} + \beta_{7} + 3 \beta_{11} ) q^{83} + ( -2 - 2 \beta_{2} + 6 \beta_{4} - 2 \beta_{5} - 4 \beta_{8} + 2 \beta_{14} ) q^{85} + ( -\beta_{6} - 2 \beta_{7} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{15} ) q^{89} + ( 2 + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{9} + 2 \beta_{14} ) q^{91} + ( -\beta_{7} - 2 \beta_{10} - 3 \beta_{11} - 2 \beta_{13} - \beta_{15} ) q^{95} + ( 2 + 4 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{8} - 2 \beta_{9} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 8q^{7} + O(q^{10}) \) \( 16q - 8q^{7} - 24q^{13} + 12q^{19} + 20q^{25} + 4q^{55} - 44q^{61} + 24q^{67} - 20q^{73} + 48q^{79} - 56q^{85} + 24q^{91} + 48q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 16 x^{14} + 174 x^{12} + 1012 x^{10} + 4243 x^{8} + 9708 x^{6} + 15858 x^{4} + 12150 x^{2} + 6561\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\((\)\( -481967 \nu^{14} + 30766372 \nu^{12} + 473907666 \nu^{10} + 5119222054 \nu^{8} + 25255513579 \nu^{6} + 88135584891 \nu^{4} + 105715511190 \nu^{2} + 82286048988 \)\()/ 21904094763 \)
\(\beta_{3}\)\(=\)\((\)\( 326606 \nu^{14} + 4547720 \nu^{12} + 47592867 \nu^{10} + 231730940 \nu^{8} + 904757018 \nu^{6} + 962205174 \nu^{4} + 748162737 \nu^{2} - 7319943180 \)\()/ 2433788307 \)
\(\beta_{4}\)\(=\)\((\)\(9939136 \nu^{14} + 150207814 \nu^{12} + 1606621224 \nu^{10} + 8773398223 \nu^{8} + 35915018668 \nu^{6} + 72060692802 \nu^{4} + 131635278990 \nu^{2} + 34847824212\)\()/ 65712284289 \)
\(\beta_{5}\)\(=\)\((\)\( 3704024 \nu^{14} + 40373015 \nu^{12} + 381453396 \nu^{10} + 906590177 \nu^{8} + 887860574 \nu^{6} - 24565580241 \nu^{4} - 41971706802 \nu^{2} - 56770896384 \)\()/ 21904094763 \)
\(\beta_{6}\)\(=\)\((\)\(26555897 \nu^{15} + 62210210 \nu^{13} - 770314944 \nu^{11} - 30249558199 \nu^{9} - 191131571350 \nu^{7} - 937524731802 \nu^{5} - 1670986351080 \nu^{3} - 2199042758754 \nu\)\()/ 197136852867 \)
\(\beta_{7}\)\(=\)\((\)\(-32058956 \nu^{15} - 505462190 \nu^{13} - 5463091302 \nu^{11} - 31353520355 \nu^{9} - 130718214368 \nu^{7} - 308344384614 \nu^{5} - 552063322863 \nu^{3} - 700940621358 \nu\)\()/ 197136852867 \)
\(\beta_{8}\)\(=\)\((\)\(10005800 \nu^{14} + 163560617 \nu^{12} + 1774630644 \nu^{10} + 10542157166 \nu^{8} + 43490411282 \nu^{6} + 100433630874 \nu^{4} + 123833655000 \nu^{2} + 71014305663\)\()/ 21904094763 \)
\(\beta_{9}\)\(=\)\((\)\(-30938182 \nu^{14} - 478042816 \nu^{12} - 5141477487 \nu^{10} - 28836857512 \nu^{8} - 119231635186 \nu^{6} - 262263231510 \nu^{4} - 440628437772 \nu^{2} - 337029762708\)\()/ 65712284289 \)
\(\beta_{10}\)\(=\)\((\)\(-17666630 \nu^{15} - 328891601 \nu^{13} - 3662652030 \nu^{11} - 23457056468 \nu^{9} - 96537911477 \nu^{7} - 231158174538 \nu^{5} - 254108418858 \nu^{3} - 151450330770 \nu\)\()/ 65712284289 \)
\(\beta_{11}\)\(=\)\((\)\(-19112531 \nu^{15} - 236592485 \nu^{13} - 2240929032 \nu^{11} - 8099390306 \nu^{9} - 20771370740 \nu^{7} + 33248580135 \nu^{5} + 63038114712 \nu^{3} + 95407816194 \nu\)\()/ 65712284289 \)
\(\beta_{12}\)\(=\)\((\)\(-58514042 \nu^{15} - 873827510 \nu^{13} - 9318113529 \nu^{11} - 50123726495 \nu^{9} - 204003532826 \nu^{7} - 386283003708 \nu^{5} - 612664504560 \nu^{3} + 89111629089 \nu\)\()/ 197136852867 \)
\(\beta_{13}\)\(=\)\((\)\(-19878272 \nu^{15} - 300415628 \nu^{13} - 3213242448 \nu^{11} - 17546796446 \nu^{9} - 71830037336 \nu^{7} - 144121385604 \nu^{5} - 263270557980 \nu^{3} - 201120217002 \nu\)\()/ 65712284289 \)
\(\beta_{14}\)\(=\)\((\)\(-6210188 \nu^{14} - 89119619 \nu^{12} - 904945566 \nu^{10} - 4406204120 \nu^{8} - 15342294773 \nu^{6} - 18295668252 \nu^{4} - 14225798826 \nu^{2} + 11675287323\)\()/ 7301364921 \)
\(\beta_{15}\)\(=\)\((\)\(324198086 \nu^{15} + 5217299756 \nu^{13} + 56522041548 \nu^{11} + 328205077598 \nu^{9} + 1350470930357 \nu^{7} + 2986631978124 \nu^{5} + 4206283806420 \nu^{3} + 2317137727815 \nu\)\()/ 197136852867 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\(\beta_{9} + 4 \beta_{4} - \beta_{3}\)
\(\nu^{3}\)\(=\)\((\)\(-5 \beta_{13} + 4 \beta_{12} + 2 \beta_{7} - 5 \beta_{1}\)\()/2\)
\(\nu^{4}\)\(=\)\(-8 \beta_{9} - \beta_{8} + \beta_{5} - 23 \beta_{4} - 23\)
\(\nu^{5}\)\(=\)\((\)\(-2 \beta_{15} + 31 \beta_{13} - 20 \beta_{12} + 2 \beta_{11} - 2 \beta_{10} - 40 \beta_{7} + 2 \beta_{6}\)\()/2\)
\(\nu^{6}\)\(=\)\(-3 \beta_{14} - 13 \beta_{8} - 26 \beta_{5} + 58 \beta_{3} + 154\)
\(\nu^{7}\)\(=\)\((\)\(-26 \beta_{15} - 168 \beta_{12} - 64 \beta_{11} - 32 \beta_{10} + 168 \beta_{7} - 52 \beta_{6} + 209 \beta_{1}\)\()/2\)
\(\nu^{8}\)\(=\)\(48 \beta_{14} + 422 \beta_{9} + 252 \beta_{8} + 126 \beta_{5} + 1049 \beta_{4} - 422 \beta_{3} - 48 \beta_{2} - 48\)
\(\nu^{9}\)\(=\)\((\)\(504 \beta_{15} - 1471 \beta_{13} + 2696 \beta_{12} + 348 \beta_{11} + 696 \beta_{10} + 1348 \beta_{7} + 252 \beta_{6} - 1471 \beta_{1}\)\()/2\)
\(\nu^{10}\)\(=\)\(-3115 \beta_{9} - 1100 \beta_{8} + 1100 \beta_{5} - 7528 \beta_{4} + 522 \beta_{2} - 7528\)
\(\nu^{11}\)\(=\)\((\)\(-2200 \beta_{15} + 10643 \beta_{13} - 10630 \beta_{12} + 3244 \beta_{11} - 3244 \beta_{10} - 21260 \beta_{7} + 2200 \beta_{6}\)\()/2\)
\(\nu^{12}\)\(=\)\(-4866 \beta_{14} - 9137 \beta_{8} - 18274 \beta_{5} + 23288 \beta_{3} + 60083\)
\(\nu^{13}\)\(=\)\((\)\(-18274 \beta_{15} - 83124 \beta_{12} - 56012 \beta_{11} - 28006 \beta_{10} + 83124 \beta_{7} - 36548 \beta_{6} + 78505 \beta_{1}\)\()/2\)
\(\nu^{14}\)\(=\)\(42009 \beta_{14} + 175780 \beta_{9} + 147678 \beta_{8} + 73839 \beta_{5} + 411295 \beta_{4} - 175780 \beta_{3} - 42009 \beta_{2} - 42009\)
\(\nu^{15}\)\(=\)\((\)\(295356 \beta_{15} - 587075 \beta_{13} + 1293832 \beta_{12} + 231696 \beta_{11} + 463392 \beta_{10} + 646916 \beta_{7} + 147678 \beta_{6} - 587075 \beta_{1}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(-\beta_{4}\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
0.733987 + 1.27130i
−1.13921 1.97317i
1.38883 + 2.40552i
−0.484374 0.838961i
0.484374 + 0.838961i
−1.38883 2.40552i
1.13921 + 1.97317i
−0.733987 1.27130i
0.733987 1.27130i
−1.13921 + 1.97317i
1.38883 2.40552i
−0.484374 + 0.838961i
0.484374 0.838961i
−1.38883 + 2.40552i
1.13921 1.97317i
−0.733987 + 1.27130i
0 0 0 −3.12320 + 1.80318i 0 3.25512 0 0 0
449.2 0 0 0 −2.90154 + 1.67520i 0 −3.54697 0 0 0
449.3 0 0 0 −2.00395 + 1.15698i 0 0.442911 0 0 0
449.4 0 0 0 −0.557544 + 0.321898i 0 −2.15106 0 0 0
449.5 0 0 0 0.557544 0.321898i 0 −2.15106 0 0 0
449.6 0 0 0 2.00395 1.15698i 0 0.442911 0 0 0
449.7 0 0 0 2.90154 1.67520i 0 −3.54697 0 0 0
449.8 0 0 0 3.12320 1.80318i 0 3.25512 0 0 0
1889.1 0 0 0 −3.12320 1.80318i 0 3.25512 0 0 0
1889.2 0 0 0 −2.90154 1.67520i 0 −3.54697 0 0 0
1889.3 0 0 0 −2.00395 1.15698i 0 0.442911 0 0 0
1889.4 0 0 0 −0.557544 0.321898i 0 −2.15106 0 0 0
1889.5 0 0 0 0.557544 + 0.321898i 0 −2.15106 0 0 0
1889.6 0 0 0 2.00395 + 1.15698i 0 0.442911 0 0 0
1889.7 0 0 0 2.90154 + 1.67520i 0 −3.54697 0 0 0
1889.8 0 0 0 3.12320 + 1.80318i 0 3.25512 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1889.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
19.d odd 6 1 inner
57.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.2.dc.c 16
3.b odd 2 1 inner 2736.2.dc.c 16
4.b odd 2 1 171.2.m.a 16
12.b even 2 1 171.2.m.a 16
19.d odd 6 1 inner 2736.2.dc.c 16
57.f even 6 1 inner 2736.2.dc.c 16
76.f even 6 1 171.2.m.a 16
228.n odd 6 1 171.2.m.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.2.m.a 16 4.b odd 2 1
171.2.m.a 16 12.b even 2 1
171.2.m.a 16 76.f even 6 1
171.2.m.a 16 228.n odd 6 1
2736.2.dc.c 16 1.a even 1 1 trivial
2736.2.dc.c 16 3.b odd 2 1 inner
2736.2.dc.c 16 19.d odd 6 1 inner
2736.2.dc.c 16 57.f even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2736, [\chi])\):

\(T_{5}^{16} - \cdots\)
\(T_{17}^{16} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( T^{16} \)
$5$ \( 104976 - 290304 T^{2} + 709504 T^{4} - 238608 T^{6} + 55740 T^{8} - 6848 T^{10} + 612 T^{12} - 30 T^{14} + T^{16} \)
$7$ \( ( 11 - 20 T - 12 T^{2} + 2 T^{3} + T^{4} )^{4} \)
$11$ \( ( 36 + 584 T^{2} + 240 T^{4} + 30 T^{6} + T^{8} )^{2} \)
$13$ \( ( 225 + 1080 T + 1698 T^{2} - 144 T^{3} - 269 T^{4} + 24 T^{5} + 50 T^{6} + 12 T^{7} + T^{8} )^{2} \)
$17$ \( 26873856 - 19408896 T^{2} + 9787392 T^{4} - 2474496 T^{6} + 451008 T^{8} - 38208 T^{10} + 2320 T^{12} - 56 T^{14} + T^{16} \)
$19$ \( ( 130321 - 41154 T + 8664 T^{2} + 2166 T^{3} - 662 T^{4} + 114 T^{5} + 24 T^{6} - 6 T^{7} + T^{8} )^{2} \)
$23$ \( 1296 - 103968 T^{2} + 8302096 T^{4} - 3079632 T^{6} + 949980 T^{8} - 64712 T^{10} + 3288 T^{12} - 66 T^{14} + T^{16} \)
$29$ \( 429981696 + 199065600 T^{2} + 64954368 T^{4} + 9940992 T^{6} + 1086208 T^{8} + 64768 T^{10} + 2784 T^{12} + 64 T^{14} + T^{16} \)
$31$ \( ( 9801 + 9948 T^{2} + 1798 T^{4} + 80 T^{6} + T^{8} )^{2} \)
$37$ \( ( 793881 + 149796 T^{2} + 8910 T^{4} + 180 T^{6} + T^{8} )^{2} \)
$41$ \( 16796160000 + 23377766400 T^{2} + 31211283456 T^{4} + 1799439360 T^{6} + 71537344 T^{8} + 1523392 T^{10} + 23616 T^{12} + 184 T^{14} + T^{16} \)
$43$ \( ( 39601 - 8358 T + 10520 T^{2} + 1848 T^{3} + 1737 T^{4} + 84 T^{5} + 44 T^{6} + T^{8} )^{2} \)
$47$ \( 857889103585536 - 50068122711552 T^{2} + 1886624680576 T^{4} - 42154191360 T^{6} + 687138864 T^{8} - 7611008 T^{10} + 61992 T^{12} - 312 T^{14} + T^{16} \)
$53$ \( 8503056 + 45979488 T^{2} + 242331264 T^{4} + 33534000 T^{6} + 3243564 T^{8} + 162864 T^{10} + 5940 T^{12} + 90 T^{14} + T^{16} \)
$59$ \( 13680577296 + 4937752224 T^{2} + 1362055968 T^{4} + 126843504 T^{6} + 8310604 T^{8} + 296320 T^{10} + 7644 T^{12} + 106 T^{14} + T^{16} \)
$61$ \( ( 18879025 + 3328270 T + 951736 T^{2} + 126836 T^{3} + 28253 T^{4} + 3380 T^{5} + 400 T^{6} + 22 T^{7} + T^{8} )^{2} \)
$67$ \( ( 45369 + 228762 T + 346578 T^{2} - 191172 T^{3} + 27175 T^{4} + 2136 T^{5} - 130 T^{6} - 12 T^{7} + T^{8} )^{2} \)
$71$ \( 892616806656 + 916697461248 T^{2} + 909388238976 T^{4} + 32223619584 T^{6} + 799781040 T^{8} + 10267776 T^{10} + 95688 T^{12} + 360 T^{14} + T^{16} \)
$73$ \( ( 2152089 - 2807838 T + 3974400 T^{2} + 376428 T^{3} + 62617 T^{4} + 1708 T^{5} + 312 T^{6} + 10 T^{7} + T^{8} )^{2} \)
$79$ \( ( 154449 - 106110 T + 11724 T^{2} + 8640 T^{3} - 743 T^{4} - 768 T^{5} + 224 T^{6} - 24 T^{7} + T^{8} )^{2} \)
$83$ \( ( 627264 + 448544 T^{2} + 50256 T^{4} + 456 T^{6} + T^{8} )^{2} \)
$89$ \( 34867844010000 + 7622139398400 T^{2} + 1368386409456 T^{4} + 59930903376 T^{6} + 1972507788 T^{8} + 19509336 T^{10} + 141408 T^{12} + 438 T^{14} + T^{16} \)
$97$ \( ( 944784 + 1831248 T + 1109280 T^{2} - 143184 T^{3} - 10268 T^{4} + 1824 T^{5} + 116 T^{6} - 24 T^{7} + T^{8} )^{2} \)
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