Properties

Label 2736.2.s.ba
Level $2736$
Weight $2$
Character orbit 2736.s
Analytic conductor $21.847$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 2 x^{7} + 8 x^{6} + 21 x^{4} - 4 x^{3} + 28 x^{2} + 12 x + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1368)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{3} + \beta_{5} ) q^{5} + ( 1 - \beta_{6} ) q^{7} +O(q^{10})\) \( q + ( \beta_{3} + \beta_{5} ) q^{5} + ( 1 - \beta_{6} ) q^{7} + ( 1 - \beta_{1} ) q^{11} + ( \beta_{1} + \beta_{2} + \beta_{5} + \beta_{7} ) q^{13} + 2 \beta_{4} q^{17} + ( -\beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{19} + ( 1 - \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{23} + ( -\beta_{1} - \beta_{2} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{25} + ( -2 \beta_{2} - 2 \beta_{7} ) q^{29} + ( -\beta_{1} + \beta_{2} + \beta_{6} ) q^{31} + ( \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{35} + ( 1 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{6} ) q^{37} + ( 2 \beta_{4} - 2 \beta_{7} ) q^{41} + ( -3 \beta_{4} + \beta_{5} - \beta_{7} ) q^{43} + ( 2 - 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{47} + ( -1 + \beta_{1} + \beta_{2} - 2 \beta_{6} ) q^{49} + ( -5 + 3 \beta_{1} - 5 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} ) q^{53} + ( -3 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{7} ) q^{55} + ( -3 \beta_{3} - \beta_{5} + 2 \beta_{7} ) q^{59} + ( -2 - 3 \beta_{1} + \beta_{2} - 2 \beta_{3} - 3 \beta_{5} + \beta_{7} ) q^{61} + ( -5 - \beta_{1} - 2 \beta_{2} - 2 \beta_{6} ) q^{65} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{67} + ( 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{71} + ( \beta_{3} + 4 \beta_{5} ) q^{73} + ( 1 + \beta_{1} ) q^{77} + ( -6 \beta_{3} + \beta_{4} - 3 \beta_{5} - \beta_{7} ) q^{79} + ( 2 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{6} ) q^{83} + ( -4 \beta_{1} - 4 \beta_{4} - 4 \beta_{5} - 4 \beta_{6} ) q^{85} + ( 5 + \beta_{1} - 2 \beta_{2} + 5 \beta_{3} - 4 \beta_{4} + \beta_{5} - 4 \beta_{6} - 2 \beta_{7} ) q^{89} + ( 2 - \beta_{1} + \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{91} + ( 2 - 2 \beta_{1} - \beta_{3} - 2 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} ) q^{95} + ( -7 \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{7} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{5} + 4q^{7} + O(q^{10}) \) \( 8q - 4q^{5} + 4q^{7} + 8q^{11} - 4q^{17} - 8q^{19} + 8q^{23} - 4q^{25} + 4q^{31} + 16q^{37} - 4q^{41} + 6q^{43} + 4q^{47} - 16q^{49} - 16q^{53} + 16q^{55} + 12q^{59} - 8q^{61} - 48q^{65} - 2q^{67} - 4q^{71} - 4q^{73} + 8q^{77} + 22q^{79} + 8q^{83} - 8q^{85} + 12q^{89} + 2q^{91} + 32q^{95} + 24q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} + 8 x^{6} + 21 x^{4} - 4 x^{3} + 28 x^{2} + 12 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -16 \nu^{7} - 199 \nu^{6} - 84 \nu^{5} - 152 \nu^{4} - 4652 \nu^{3} - 272 \nu^{2} - 132 \nu + 3206 \)\()/4243\)
\(\beta_{2}\)\(=\)\((\)\( 152 \nu^{7} - 231 \nu^{6} + 798 \nu^{5} + 1444 \nu^{4} + 1764 \nu^{3} + 2584 \nu^{2} + 1254 \nu + 11973 \)\()/4243\)
\(\beta_{3}\)\(=\)\((\)\( 754 \nu^{7} - 1760 \nu^{6} + 6080 \nu^{5} - 1323 \nu^{4} + 13440 \nu^{3} - 12640 \nu^{2} + 16828 \nu - 5760 \)\()/12729\)
\(\beta_{4}\)\(=\)\((\)\( -815 \nu^{7} + 3388 \nu^{6} - 11704 \nu^{5} + 15594 \nu^{4} - 25872 \nu^{3} + 24332 \nu^{2} - 56579 \nu + 11088 \)\()/12729\)
\(\beta_{5}\)\(=\)\((\)\( 1052 \nu^{7} - 2827 \nu^{6} + 9766 \nu^{5} - 6978 \nu^{4} + 21588 \nu^{3} - 20303 \nu^{2} + 4436 \nu - 9252 \)\()/12729\)
\(\beta_{6}\)\(=\)\((\)\( -556 \nu^{7} + 510 \nu^{6} - 2919 \nu^{5} - 5282 \nu^{4} - 8909 \nu^{3} - 9452 \nu^{2} - 4587 \nu - 13760 \)\()/4243\)
\(\beta_{7}\)\(=\)\((\)\( 602 \nu^{7} - 1529 \nu^{6} + 5282 \nu^{5} - 2767 \nu^{4} + 11676 \nu^{3} - 10981 \nu^{2} + 15574 \nu - 5004 \)\()/4243\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} - \beta_{5} - \beta_{3}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{7} - 3 \beta_{3} + \beta_{2} - 3\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{6} + 7 \beta_{2} - 3 \beta_{1} - 11\)\()/2\)
\(\nu^{4}\)\(=\)\(-9 \beta_{7} + \beta_{5} - 2 \beta_{4} + 18 \beta_{3}\)
\(\nu^{5}\)\(=\)\((\)\(-53 \beta_{7} - 16 \beta_{6} + 13 \beta_{5} - 16 \beta_{4} + 89 \beta_{3} - 53 \beta_{2} + 13 \beta_{1} + 89\)\()/2\)
\(\nu^{6}\)\(=\)\(-20 \beta_{6} - 72 \beta_{2} + 11 \beta_{1} + 130\)
\(\nu^{7}\)\(=\)\((\)\(407 \beta_{7} - 79 \beta_{5} + 122 \beta_{4} - 699 \beta_{3}\)\()/2\)

Character values

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

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(-1 - \beta_{3}\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
0.643668 + 1.11487i
−0.276205 0.478401i
1.39083 + 2.40898i
−0.758290 1.31340i
0.643668 1.11487i
−0.276205 + 0.478401i
1.39083 2.40898i
−0.758290 + 1.31340i
0 0 0 −1.95872 3.39260i 0 −2.04306 0 0 0
577.2 0 0 0 −0.795012 1.37700i 0 3.87834 0 0 0
577.3 0 0 0 −0.412855 0.715087i 0 0.703158 0 0 0
577.4 0 0 0 1.16659 + 2.02059i 0 −0.538445 0 0 0
1873.1 0 0 0 −1.95872 + 3.39260i 0 −2.04306 0 0 0
1873.2 0 0 0 −0.795012 + 1.37700i 0 3.87834 0 0 0
1873.3 0 0 0 −0.412855 + 0.715087i 0 0.703158 0 0 0
1873.4 0 0 0 1.16659 2.02059i 0 −0.538445 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1873.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.2.s.ba 8
3.b odd 2 1 2736.2.s.bc 8
4.b odd 2 1 1368.2.s.l 8
12.b even 2 1 1368.2.s.m yes 8
19.c even 3 1 inner 2736.2.s.ba 8
57.h odd 6 1 2736.2.s.bc 8
76.g odd 6 1 1368.2.s.l 8
228.m even 6 1 1368.2.s.m yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1368.2.s.l 8 4.b odd 2 1
1368.2.s.l 8 76.g odd 6 1
1368.2.s.m yes 8 12.b even 2 1
1368.2.s.m yes 8 228.m even 6 1
2736.2.s.ba 8 1.a even 1 1 trivial
2736.2.s.ba 8 19.c even 3 1 inner
2736.2.s.bc 8 3.b odd 2 1
2736.2.s.bc 8 57.h odd 6 1

Hecke kernels

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

\(T_{5}^{8} + \cdots\)
\( T_{7}^{4} - 2 T_{7}^{3} - 8 T_{7}^{2} + 2 T_{7} + 3 \)
\( T_{11}^{4} - 4 T_{11}^{3} - 4 T_{11}^{2} + 12 T_{11} - 4 \)
\( T_{13}^{8} + 30 T_{13}^{6} - 16 T_{13}^{5} + 719 T_{13}^{4} - 240 T_{13}^{3} + 5494 T_{13}^{2} + 1448 T_{13} + 32761 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( 144 + 240 T + 352 T^{2} + 176 T^{3} + 108 T^{4} + 24 T^{5} + 20 T^{6} + 4 T^{7} + T^{8} \)
$7$ \( ( 3 + 2 T - 8 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$11$ \( ( -4 + 12 T - 4 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$13$ \( 32761 + 1448 T + 5494 T^{2} - 240 T^{3} + 719 T^{4} - 16 T^{5} + 30 T^{6} + T^{8} \)
$17$ \( 4096 + 8192 T + 14336 T^{2} + 4608 T^{3} + 1600 T^{4} + 128 T^{5} + 48 T^{6} + 4 T^{7} + T^{8} \)
$19$ \( 130321 + 54872 T + 10108 T^{2} + 1672 T^{3} + 326 T^{4} + 88 T^{5} + 28 T^{6} + 8 T^{7} + T^{8} \)
$23$ \( 400 - 9360 T + 217824 T^{2} - 28400 T^{3} + 7364 T^{4} - 456 T^{5} + 124 T^{6} - 8 T^{7} + T^{8} \)
$29$ \( 36864 + 43008 T + 37888 T^{2} + 14336 T^{3} + 4288 T^{4} + 448 T^{5} + 64 T^{6} + T^{8} \)
$31$ \( ( 27 + 18 T - 24 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$37$ \( ( 197 + 320 T - 78 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$41$ \( 409600 + 368640 T + 260096 T^{2} + 69632 T^{3} + 15488 T^{4} + 704 T^{5} + 128 T^{6} + 4 T^{7} + T^{8} \)
$43$ \( 2455489 + 435626 T + 196376 T^{2} - 2324 T^{3} + 5877 T^{4} - 100 T^{5} + 112 T^{6} - 6 T^{7} + T^{8} \)
$47$ \( 409600 - 327680 T + 210944 T^{2} - 46080 T^{3} + 9088 T^{4} - 704 T^{5} + 96 T^{6} - 4 T^{7} + T^{8} \)
$53$ \( 26998416 + 8001840 T + 2142976 T^{2} + 234032 T^{3} + 31772 T^{4} + 2376 T^{5} + 300 T^{6} + 16 T^{7} + T^{8} \)
$59$ \( 15376 + 11408 T + 9952 T^{2} + 1872 T^{3} + 1124 T^{4} - 40 T^{5} + 156 T^{6} - 12 T^{7} + T^{8} \)
$61$ \( 299209 + 323824 T + 312174 T^{2} + 50192 T^{3} + 10183 T^{4} + 624 T^{5} + 134 T^{6} + 8 T^{7} + T^{8} \)
$67$ \( 2002225 - 319790 T + 226536 T^{2} + 22364 T^{3} + 14413 T^{4} + 204 T^{5} + 128 T^{6} + 2 T^{7} + T^{8} \)
$71$ \( 11505664 + 217088 T + 546816 T^{2} - 37376 T^{3} + 21952 T^{4} - 768 T^{5} + 176 T^{6} + 4 T^{7} + T^{8} \)
$73$ \( 25281 + 90948 T + 302698 T^{2} + 89360 T^{3} + 26163 T^{4} + 528 T^{5} + 170 T^{6} + 4 T^{7} + T^{8} \)
$79$ \( 139129 - 306606 T + 696572 T^{2} + 29620 T^{3} + 21593 T^{4} - 2876 T^{5} + 428 T^{6} - 22 T^{7} + T^{8} \)
$83$ \( ( 3392 - 64 T - 160 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$89$ \( 39388176 + 5999856 T + 1692160 T^{2} + 32080 T^{3} + 20572 T^{4} - 424 T^{5} + 268 T^{6} - 12 T^{7} + T^{8} \)
$97$ \( 25600 - 43520 T + 50944 T^{2} - 31488 T^{3} + 14048 T^{4} - 2912 T^{5} + 432 T^{6} - 24 T^{7} + T^{8} \)
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