Properties

Label 2736.2.bm.m
Level $2736$
Weight $2$
Character orbit 2736.bm
Analytic conductor $21.847$
Analytic rank $0$
Dimension $6$
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.bm (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: 6.0.31726512.1
Defining polynomial: \(x^{6} - x^{5} + 10 x^{4} + 3 x^{3} + 84 x^{2} - 27 x + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 304)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} - \beta_{3} ) q^{5} + ( \beta_{4} - \beta_{5} ) q^{7} +O(q^{10})\) \( q + ( -1 + \beta_{1} - \beta_{3} ) q^{5} + ( \beta_{4} - \beta_{5} ) q^{7} + ( -1 - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{11} + ( 4 + 2 \beta_{3} ) q^{13} + ( 1 - \beta_{1} + \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{17} + ( 3 - \beta_{1} - \beta_{4} + \beta_{5} ) q^{19} -\beta_{4} q^{23} + ( -\beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{25} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{29} + ( 2 \beta_{2} + \beta_{4} + \beta_{5} ) q^{31} + ( 1 + \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{5} ) q^{35} + ( -1 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{37} + ( \beta_{1} - 2 \beta_{2} - 3 \beta_{5} ) q^{41} + ( 2 - 2 \beta_{3} ) q^{43} + ( 4 + 2 \beta_{3} + \beta_{4} ) q^{47} + ( \beta_{2} + \beta_{4} + \beta_{5} ) q^{49} + ( 8 + 4 \beta_{3} ) q^{53} + ( -2 - 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} - \beta_{5} ) q^{55} + ( -9 - 9 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{59} + ( \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{61} + ( -2 + 4 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{65} + ( 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{67} + ( -6 - 6 \beta_{3} ) q^{71} + ( -2 + \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{73} + ( 7 - \beta_{2} ) q^{77} + ( -6 + 2 \beta_{1} - 6 \beta_{3} ) q^{79} + ( 2 \beta_{1} - \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{83} + ( 4 \beta_{1} - 4 \beta_{2} - 10 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} ) q^{85} + ( 6 - \beta_{1} - \beta_{2} + 3 \beta_{3} + 3 \beta_{4} ) q^{89} + ( 2 \beta_{4} - 4 \beta_{5} ) q^{91} + ( -4 + 2 \beta_{1} + 2 \beta_{2} - 8 \beta_{3} - \beta_{4} + \beta_{5} ) q^{95} + ( 3 + 2 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 2q^{5} + O(q^{10}) \) \( 6q - 2q^{5} + 18q^{13} + 2q^{17} + 17q^{19} - 5q^{25} + 12q^{29} + 4q^{31} + 6q^{35} - 3q^{41} + 18q^{43} + 18q^{47} + 2q^{49} + 36q^{53} - 12q^{55} - 27q^{59} - 10q^{61} - 11q^{67} - 18q^{71} - 5q^{73} + 40q^{77} - 16q^{79} + 26q^{85} + 24q^{89} + 6q^{95} + 21q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} + 10 x^{4} + 3 x^{3} + 84 x^{2} - 27 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{5} + 10 \nu^{4} - 100 \nu^{3} + 84 \nu^{2} - 27 \nu + 270 \)\()/813\)
\(\beta_{3}\)\(=\)\((\)\( 30 \nu^{5} - 29 \nu^{4} + 290 \nu^{3} + 190 \nu^{2} + 2436 \nu - 783 \)\()/813\)
\(\beta_{4}\)\(=\)\((\)\( -161 \nu^{5} - 16 \nu^{4} - 1466 \nu^{3} - 1923 \nu^{2} - 14916 \nu - 5310 \)\()/2439\)
\(\beta_{5}\)\(=\)\((\)\( 191 \nu^{5} - 284 \nu^{4} + 2027 \nu^{3} - 597 \nu^{2} + 15726 \nu - 10107 \)\()/2439\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-2 \beta_{5} + \beta_{4} + 6 \beta_{3} - \beta_{2} + \beta_{1}\)
\(\nu^{3}\)\(=\)\(-\beta_{5} - \beta_{4} - 10 \beta_{2} - 3\)
\(\nu^{4}\)\(=\)\(10 \beta_{5} - 20 \beta_{4} - 57 \beta_{3} - 16 \beta_{1} - 57\)
\(\nu^{5}\)\(=\)\(32 \beta_{5} - 16 \beta_{4} - 66 \beta_{3} + 103 \beta_{2} - 103 \beta_{1}\)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
559.1
−1.35887 + 2.35363i
0.162698 0.281802i
1.69617 2.93786i
−1.35887 2.35363i
0.162698 + 0.281802i
1.69617 + 2.93786i
0 0 0 −1.85887 + 3.21966i 0 2.36936i 0 0 0
559.2 0 0 0 −0.337302 + 0.584224i 0 3.59084i 0 0 0
559.3 0 0 0 1.19617 2.07183i 0 1.22147i 0 0 0
1855.1 0 0 0 −1.85887 3.21966i 0 2.36936i 0 0 0
1855.2 0 0 0 −0.337302 0.584224i 0 3.59084i 0 0 0
1855.3 0 0 0 1.19617 + 2.07183i 0 1.22147i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1855.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.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.bm.m 6
3.b odd 2 1 304.2.n.e yes 6
4.b odd 2 1 2736.2.bm.l 6
12.b even 2 1 304.2.n.d 6
19.d odd 6 1 2736.2.bm.l 6
24.f even 2 1 1216.2.n.e 6
24.h odd 2 1 1216.2.n.d 6
57.f even 6 1 304.2.n.d 6
76.f even 6 1 inner 2736.2.bm.m 6
228.n odd 6 1 304.2.n.e yes 6
456.s odd 6 1 1216.2.n.d 6
456.v even 6 1 1216.2.n.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
304.2.n.d 6 12.b even 2 1
304.2.n.d 6 57.f even 6 1
304.2.n.e yes 6 3.b odd 2 1
304.2.n.e yes 6 228.n odd 6 1
1216.2.n.d 6 24.h odd 2 1
1216.2.n.d 6 456.s odd 6 1
1216.2.n.e 6 24.f even 2 1
1216.2.n.e 6 456.v even 6 1
2736.2.bm.l 6 4.b odd 2 1
2736.2.bm.l 6 19.d odd 6 1
2736.2.bm.m 6 1.a even 1 1 trivial
2736.2.bm.m 6 76.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 \(S_{2}^{\mathrm{new}}(2736, [\chi])\):

\( T_{5}^{6} + 2 T_{5}^{5} + 12 T_{5}^{4} - 4 T_{5}^{3} + 76 T_{5}^{2} + 48 T_{5} + 36 \)
\( T_{7}^{6} + 20 T_{7}^{4} + 100 T_{7}^{2} + 108 \)
\( T_{11}^{6} + 29 T_{11}^{4} + 235 T_{11}^{2} + 507 \)
\( T_{23}^{6} - 10 T_{23}^{4} + 100 T_{23}^{2} + 180 T_{23} + 108 \)
\( T_{31}^{3} - 2 T_{31}^{2} - 54 T_{31} - 66 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( T^{6} \)
$5$ \( 36 + 48 T + 76 T^{2} - 4 T^{3} + 12 T^{4} + 2 T^{5} + T^{6} \)
$7$ \( 108 + 100 T^{2} + 20 T^{4} + T^{6} \)
$11$ \( 507 + 235 T^{2} + 29 T^{4} + T^{6} \)
$13$ \( ( 12 - 6 T + T^{2} )^{3} \)
$17$ \( 144 + 528 T + 1912 T^{2} + 112 T^{3} + 48 T^{4} - 2 T^{5} + T^{6} \)
$19$ \( 6859 - 6137 T + 2736 T^{2} - 775 T^{3} + 144 T^{4} - 17 T^{5} + T^{6} \)
$23$ \( 108 + 180 T + 100 T^{2} - 10 T^{4} + T^{6} \)
$29$ \( 2700 - 1080 T - 216 T^{2} + 144 T^{3} + 36 T^{4} - 12 T^{5} + T^{6} \)
$31$ \( ( -66 - 54 T - 2 T^{2} + T^{3} )^{2} \)
$37$ \( 2700 + 864 T^{2} + 72 T^{4} + T^{6} \)
$41$ \( 54675 + 40095 T + 9396 T^{2} - 297 T^{3} - 96 T^{4} + 3 T^{5} + T^{6} \)
$43$ \( ( 12 - 6 T + T^{2} )^{3} \)
$47$ \( 12 + 156 T + 640 T^{2} - 468 T^{3} + 134 T^{4} - 18 T^{5} + T^{6} \)
$53$ \( ( 48 - 12 T + T^{2} )^{3} \)
$59$ \( 263169 + 109269 T + 31518 T^{2} + 4725 T^{3} + 516 T^{4} + 27 T^{5} + T^{6} \)
$61$ \( 36 + 144 T + 516 T^{2} + 228 T^{3} + 76 T^{4} + 10 T^{5} + T^{6} \)
$67$ \( 245025 + 19305 T + 6966 T^{2} + 561 T^{3} + 160 T^{4} + 11 T^{5} + T^{6} \)
$71$ \( ( 36 + 6 T + T^{2} )^{3} \)
$73$ \( 289 + 425 T + 710 T^{2} - 91 T^{3} + 50 T^{4} + 5 T^{5} + T^{6} \)
$79$ \( 2304 - 2304 T + 3072 T^{2} + 864 T^{3} + 208 T^{4} + 16 T^{5} + T^{6} \)
$83$ \( 3 + 3031 T^{2} + 113 T^{4} + T^{6} \)
$89$ \( 97200 - 38880 T + 864 T^{2} + 1728 T^{3} + 120 T^{4} - 24 T^{5} + T^{6} \)
$97$ \( 151875 - 42525 T - 756 T^{2} + 1323 T^{3} + 84 T^{4} - 21 T^{5} + T^{6} \)
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