Properties

Label 2736.2.d.a
Level $2736$
Weight $2$
Character orbit 2736.d
Analytic conductor $21.847$
Analytic rank $0$
Dimension $12$
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.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 12 x^{10} + 47 x^{8} - 44 x^{6} + 81 x^{4} + 848 x^{2} + 1600\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{5} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{4} + \beta_{9} ) q^{5} + ( \beta_{1} + \beta_{3} - \beta_{8} ) q^{7} +O(q^{10})\) \( q + ( -\beta_{4} + \beta_{9} ) q^{5} + ( \beta_{1} + \beta_{3} - \beta_{8} ) q^{7} + ( \beta_{5} + 2 \beta_{7} - \beta_{11} ) q^{11} -2 \beta_{2} q^{13} + ( \beta_{4} + \beta_{9} + \beta_{10} ) q^{17} -\beta_{1} q^{19} + ( 2 \beta_{7} + \beta_{11} ) q^{23} + ( 1 - \beta_{2} - 2 \beta_{6} ) q^{25} + ( -\beta_{4} - \beta_{10} ) q^{29} + ( 4 \beta_{1} - 2 \beta_{3} ) q^{31} + ( 3 \beta_{5} + 4 \beta_{7} + \beta_{11} ) q^{35} + ( 2 \beta_{2} + 2 \beta_{6} ) q^{37} + ( \beta_{4} + 2 \beta_{9} + \beta_{10} ) q^{41} + ( -2 \beta_{1} - \beta_{3} - 2 \beta_{8} ) q^{43} + ( -\beta_{5} + 6 \beta_{7} ) q^{47} + ( -4 + 3 \beta_{2} - \beta_{6} ) q^{49} + ( 3 \beta_{4} - 4 \beta_{9} + \beta_{10} ) q^{53} + ( 6 \beta_{1} - \beta_{3} + 2 \beta_{8} ) q^{55} + ( -4 \beta_{5} - 2 \beta_{11} ) q^{59} + ( 2 + \beta_{2} ) q^{61} -2 \beta_{9} q^{65} + ( 6 \beta_{1} - 2 \beta_{8} ) q^{67} + ( 4 \beta_{7} + 2 \beta_{11} ) q^{71} + ( -5 + \beta_{2} + \beta_{6} ) q^{73} + ( -5 \beta_{9} - 5 \beta_{10} ) q^{77} + ( 6 \beta_{1} + 4 \beta_{3} + 2 \beta_{8} ) q^{79} + ( 2 \beta_{7} + \beta_{11} ) q^{83} + ( -1 - \beta_{2} - \beta_{6} ) q^{85} + ( -3 \beta_{4} + 3 \beta_{10} ) q^{89} + ( 8 \beta_{1} + 6 \beta_{3} ) q^{91} + ( \beta_{5} + \beta_{7} ) q^{95} + ( 2 - 2 \beta_{2} + 2 \beta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + O(q^{10}) \) \( 12q + 4q^{25} + 8q^{37} - 52q^{49} + 24q^{61} - 56q^{73} - 16q^{85} + 32q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 12 x^{10} + 47 x^{8} - 44 x^{6} + 81 x^{4} + 848 x^{2} + 1600\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -191 \nu^{10} + 3596 \nu^{8} - 23761 \nu^{6} + 69052 \nu^{4} - 53263 \nu^{2} - 122584 \)\()/373088\)
\(\beta_{2}\)\(=\)\((\)\( 2904 \nu^{10} - 28060 \nu^{8} + 77483 \nu^{6} + 10540 \nu^{4} - 596096 \nu^{2} + 4361380 \)\()/2390095\)
\(\beta_{3}\)\(=\)\((\)\( 34469 \nu^{10} - 474620 \nu^{8} + 2736123 \nu^{6} - 6567820 \nu^{4} + 10577109 \nu^{2} + 18309320 \)\()/19120760\)
\(\beta_{4}\)\(=\)\((\)\( 583 \nu^{11} - 8156 \nu^{9} + 40681 \nu^{7} - 43212 \nu^{5} - 118857 \nu^{3} + 694024 \nu \)\()/859360\)
\(\beta_{5}\)\(=\)\((\)\( 28103 \nu^{11} - 480268 \nu^{9} + 3361321 \nu^{7} - 11722716 \nu^{5} + 21325703 \nu^{3} + 19110872 \nu \)\()/38241520\)
\(\beta_{6}\)\(=\)\((\)\( -5772 \nu^{10} + 75525 \nu^{8} - 282399 \nu^{6} - 198725 \nu^{4} + 1500848 \nu^{2} - 2999625 \)\()/2390095\)
\(\beta_{7}\)\(=\)\((\)\( -1177 \nu^{11} + 16404 \nu^{9} - 79159 \nu^{7} + 142628 \nu^{5} - 362697 \nu^{3} - 181416 \nu \)\()/1167680\)
\(\beta_{8}\)\(=\)\((\)\( 258373 \nu^{10} - 3006820 \nu^{8} + 11106891 \nu^{6} - 15183860 \nu^{4} + 86189173 \nu^{2} + 114912200 \)\()/76483040\)
\(\beta_{9}\)\(=\)\((\)\( 25995 \nu^{11} - 288708 \nu^{9} + 997285 \nu^{7} - 523916 \nu^{5} + 2189915 \nu^{3} + 36395752 \nu \)\()/19120760\)
\(\beta_{10}\)\(=\)\((\)\( -31521 \nu^{11} + 417164 \nu^{9} - 1752127 \nu^{7} + 1740388 \nu^{5} + 3013479 \nu^{3} - 39405176 \nu \)\()/19120760\)
\(\beta_{11}\)\(=\)\((\)\( -80093 \nu^{11} + 1057684 \nu^{9} - 5355891 \nu^{7} + 12770548 \nu^{5} - 25705533 \nu^{3} - 15419336 \nu \)\()/38241520\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{11} + \beta_{9} + \beta_{5}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{8} + \beta_{3} - 3 \beta_{2} + 3 \beta_{1} + 4\)\()/2\)
\(\nu^{3}\)\(=\)\(2 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} - 4 \beta_{7} + \beta_{5}\)
\(\nu^{4}\)\(=\)\((\)\(-2 \beta_{8} - 7 \beta_{6} + 13 \beta_{3} - 15 \beta_{2} + 30 \beta_{1} + 19\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(19 \beta_{11} + 32 \beta_{10} + 17 \beta_{9} - 40 \beta_{7} - \beta_{5} + 44 \beta_{4}\)\()/2\)
\(\nu^{6}\)\(=\)\(-20 \beta_{8} - 12 \beta_{6} + 68 \beta_{3} - 27 \beta_{2} + 100 \beta_{1} + 32\)
\(\nu^{7}\)\(=\)\((\)\(38 \beta_{11} + 251 \beta_{10} + 101 \beta_{9} - 76 \beta_{7} + 11 \beta_{5} + 400 \beta_{4}\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(-294 \beta_{8} + 23 \beta_{6} + 985 \beta_{3} + 43 \beta_{2} + 1522 \beta_{1} - 51\)\()/2\)
\(\nu^{9}\)\(=\)\(-207 \beta_{11} + 868 \beta_{10} + 410 \beta_{9} + 360 \beta_{7} - 67 \beta_{5} + 1256 \beta_{4}\)
\(\nu^{10}\)\(=\)\((\)\(-1561 \beta_{8} + 888 \beta_{6} + 6047 \beta_{3} + 2941 \beta_{2} + 9877 \beta_{1} - 4452\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(-7410 \beta_{11} + 9959 \beta_{10} + 5309 \beta_{9} + 10780 \beta_{7} - 3499 \beta_{5} + 13440 \beta_{4}\)\()/2\)

Character values

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

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2015.1
−1.49887 1.07270i
1.49887 1.07270i
−0.366740 1.25423i
0.366740 1.25423i
−2.57272 + 0.525570i
2.57272 + 0.525570i
2.57272 0.525570i
−2.57272 0.525570i
0.366740 + 1.25423i
−0.366740 + 1.25423i
1.49887 + 1.07270i
−1.49887 + 1.07270i
0 0 0 3.55961i 0 3.63675i 0 0 0
2015.2 0 0 0 3.55961i 0 3.63675i 0 0 0
2015.3 0 0 0 1.09425i 0 1.25511i 0 0 0
2015.4 0 0 0 1.09425i 0 1.25511i 0 0 0
2015.5 0 0 0 0.363074i 0 4.38164i 0 0 0
2015.6 0 0 0 0.363074i 0 4.38164i 0 0 0
2015.7 0 0 0 0.363074i 0 4.38164i 0 0 0
2015.8 0 0 0 0.363074i 0 4.38164i 0 0 0
2015.9 0 0 0 1.09425i 0 1.25511i 0 0 0
2015.10 0 0 0 1.09425i 0 1.25511i 0 0 0
2015.11 0 0 0 3.55961i 0 3.63675i 0 0 0
2015.12 0 0 0 3.55961i 0 3.63675i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2015.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.2.d.a 12
3.b odd 2 1 inner 2736.2.d.a 12
4.b odd 2 1 inner 2736.2.d.a 12
12.b even 2 1 inner 2736.2.d.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2736.2.d.a 12 1.a even 1 1 trivial
2736.2.d.a 12 3.b odd 2 1 inner
2736.2.d.a 12 4.b odd 2 1 inner
2736.2.d.a 12 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 14 T_{5}^{4} + 17 T_{5}^{2} + 2 \) acting on \(S_{2}^{\mathrm{new}}(2736, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \)
$3$ \( T^{12} \)
$5$ \( ( 2 + 17 T^{2} + 14 T^{4} + T^{6} )^{2} \)
$7$ \( ( 400 + 305 T^{2} + 34 T^{4} + T^{6} )^{2} \)
$11$ \( ( -5000 + 1025 T^{2} - 60 T^{4} + T^{6} )^{2} \)
$13$ \( ( -32 - 28 T + T^{3} )^{4} \)
$17$ \( ( 50 + 129 T^{2} + 46 T^{4} + T^{6} )^{2} \)
$19$ \( ( 1 + T^{2} )^{6} \)
$23$ \( ( -800 + 580 T^{2} - 52 T^{4} + T^{6} )^{2} \)
$29$ \( ( 32 + 112 T^{2} + 34 T^{4} + T^{6} )^{2} \)
$31$ \( ( 6400 + 2320 T^{2} + 104 T^{4} + T^{6} )^{2} \)
$37$ \( ( -32 - 64 T - 2 T^{2} + T^{3} )^{4} \)
$41$ \( ( 8192 + 1536 T^{2} + 82 T^{4} + T^{6} )^{2} \)
$43$ \( ( 40000 + 4425 T^{2} + 134 T^{4} + T^{6} )^{2} \)
$47$ \( ( -441800 + 19505 T^{2} - 252 T^{4} + T^{6} )^{2} \)
$53$ \( ( 199712 + 15536 T^{2} + 242 T^{4} + T^{6} )^{2} \)
$59$ \( ( -204800 + 19520 T^{2} - 272 T^{4} + T^{6} )^{2} \)
$61$ \( ( 10 + 5 T - 6 T^{2} + T^{3} )^{4} \)
$67$ \( ( 25600 + 6720 T^{2} + 176 T^{4} + T^{6} )^{2} \)
$71$ \( ( -51200 + 9280 T^{2} - 208 T^{4} + T^{6} )^{2} \)
$73$ \( ( 16 + 49 T + 14 T^{2} + T^{3} )^{4} \)
$79$ \( ( 2560000 + 65600 T^{2} + 480 T^{4} + T^{6} )^{2} \)
$83$ \( ( -800 + 580 T^{2} - 52 T^{4} + T^{6} )^{2} \)
$89$ \( ( 583200 + 24624 T^{2} + 306 T^{4} + T^{6} )^{2} \)
$97$ \( ( 464 - 60 T - 8 T^{2} + T^{3} )^{4} \)
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