Properties

Label 2736.2.f.i
Level $2736$
Weight $2$
Character orbit 2736.f
Analytic conductor $21.847$
Analytic rank $0$
Dimension $8$
CM discriminant -19
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.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.691798081536.4
Defining polynomial: \(x^{8} - 14 x^{6} - 12 x^{5} + 127 x^{4} - 144 x^{3} - 282 x^{2} + 900 x + 1350\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 684)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{3} q^{5} + \beta_{2} q^{7} +O(q^{10})\) \( q -\beta_{3} q^{5} + \beta_{2} q^{7} + \beta_{6} q^{11} + ( \beta_{3} - \beta_{5} + \beta_{7} ) q^{17} + ( -\beta_{1} + \beta_{2} ) q^{19} + ( \beta_{3} + \beta_{6} - \beta_{7} ) q^{23} + ( -5 + 2 \beta_{1} + \beta_{2} ) q^{25} + ( \beta_{3} + \beta_{5} - 2 \beta_{7} ) q^{35} + ( -1 + \beta_{4} ) q^{43} + ( 2 \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{47} + ( 4 + \beta_{4} ) q^{49} + ( -3 - 2 \beta_{1} + 2 \beta_{2} - \beta_{4} ) q^{55} + ( -2 \beta_{1} - 3 \beta_{2} ) q^{61} + ( -5 - \beta_{4} ) q^{73} + ( \beta_{3} + 2 \beta_{5} - 3 \beta_{7} ) q^{77} + ( \beta_{3} - 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{83} + ( 7 - 4 \beta_{1} + 4 \beta_{2} - \beta_{4} ) q^{85} + ( -2 \beta_{3} + 2 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 40q^{25} - 4q^{43} + 36q^{49} - 28q^{55} - 44q^{73} + 52q^{85} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 14 x^{6} - 12 x^{5} + 127 x^{4} - 144 x^{3} - 282 x^{2} + 900 x + 1350\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 12960 \nu^{7} + 23083 \nu^{6} - 32796 \nu^{5} - 597740 \nu^{4} - 196776 \nu^{3} - 859412 \nu^{2} - 5392428 \nu - 11605830 \)\()/13725555\)
\(\beta_{2}\)\(=\)\((\)\( -24310 \nu^{7} + 8243 \nu^{6} + 544454 \nu^{5} + 90395 \nu^{4} - 5883646 \nu^{3} + 1377653 \nu^{2} + 27915822 \nu - 38703045 \)\()/13725555\)
\(\beta_{3}\)\(=\)\((\)\( -47546 \nu^{7} + 178954 \nu^{6} + 735426 \nu^{5} - 1249773 \nu^{4} - 7482925 \nu^{3} + 25205858 \nu^{2} - 16379547 \nu - 66840945 \)\()/13725555\)
\(\beta_{4}\)\(=\)\((\)\( 16 \nu^{7} + 54 \nu^{6} - 212 \nu^{5} - 567 \nu^{4} + 544 \nu^{3} - 1830 \nu^{2} - 9156 \nu + 32190 \)\()/3405\)
\(\beta_{5}\)\(=\)\((\)\( -200 \nu^{7} + 233 \nu^{6} + 1969 \nu^{5} - 2560 \nu^{4} - 21101 \nu^{3} + 71453 \nu^{2} - 13578 \nu - 147000 \)\()/32915\)
\(\beta_{6}\)\(=\)\((\)\( 108029 \nu^{7} - 113379 \nu^{6} - 1170618 \nu^{5} + 675757 \nu^{4} + 13107106 \nu^{3} - 37579915 \nu^{2} + 9710091 \nu + 83548605 \)\()/13725555\)
\(\beta_{7}\)\(=\)\((\)\( 123377 \nu^{7} - 86890 \nu^{6} - 1605973 \nu^{5} - 930214 \nu^{4} + 14155124 \nu^{3} - 23614798 \nu^{2} + 17283456 \nu + 75574110 \)\()/13725555\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} + 2 \beta_{2} - 4 \beta_{1}\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{7} - 4 \beta_{6} - 4 \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_{1} + 11\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(-3 \beta_{7} + 33 \beta_{6} + 25 \beta_{5} - 6 \beta_{4} + 15 \beta_{3} - 6 \beta_{2} - 12 \beta_{1} + 30\)\()/6\)
\(\nu^{4}\)\(=\)\((\)\(4 \beta_{7} - 32 \beta_{6} - 52 \beta_{5} + \beta_{4} + 16 \beta_{3} - 2 \beta_{2} - 56 \beta_{1} - 44\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(-49 \beta_{7} + 191 \beta_{6} + 49 \beta_{5} - 100 \beta_{4} + 161 \beta_{3} - 16 \beta_{2} + 248 \beta_{1} + 1220\)\()/6\)
\(\nu^{6}\)\(=\)\((\)\(-36 \beta_{7} + 276 \beta_{6} + 56 \beta_{5} + 120 \beta_{4} + 312 \beta_{3} + 120 \beta_{2} - 825 \beta_{1} - 1206\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(1009 \beta_{7} - 3065 \beta_{6} - 4607 \beta_{5} - 812 \beta_{4} + 1681 \beta_{3} - 44 \beta_{2} + 2776 \beta_{1} + 10612\)\()/6\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1025.1
3.04547 + 0.517638i
−1.31342 0.517638i
1.31342 + 1.93185i
−3.04547 1.93185i
−3.04547 + 1.93185i
1.31342 1.93185i
−1.31342 + 0.517638i
3.04547 0.517638i
0 0 0 4.46919i 0 −0.418627 0 0 0
1025.2 0 0 0 3.95155i 0 −4.77753 0 0 0
1025.3 0 0 0 2.09409i 0 4.77753 0 0 0
1025.4 0 0 0 0.162240i 0 0.418627 0 0 0
1025.5 0 0 0 0.162240i 0 0.418627 0 0 0
1025.6 0 0 0 2.09409i 0 4.77753 0 0 0
1025.7 0 0 0 3.95155i 0 −4.77753 0 0 0
1025.8 0 0 0 4.46919i 0 −0.418627 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1025.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)
3.b odd 2 1 inner
57.d 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.f.i 8
3.b odd 2 1 inner 2736.2.f.i 8
4.b odd 2 1 684.2.d.a 8
12.b even 2 1 684.2.d.a 8
19.b odd 2 1 CM 2736.2.f.i 8
57.d even 2 1 inner 2736.2.f.i 8
76.d even 2 1 684.2.d.a 8
228.b odd 2 1 684.2.d.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
684.2.d.a 8 4.b odd 2 1
684.2.d.a 8 12.b even 2 1
684.2.d.a 8 76.d even 2 1
684.2.d.a 8 228.b odd 2 1
2736.2.f.i 8 1.a even 1 1 trivial
2736.2.f.i 8 3.b odd 2 1 inner
2736.2.f.i 8 19.b odd 2 1 CM
2736.2.f.i 8 57.d even 2 1 inner

Hecke kernels

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

\( T_{5}^{8} + 40 T_{5}^{6} + 469 T_{5}^{4} + 1380 T_{5}^{2} + 36 \)
\( T_{7}^{4} - 23 T_{7}^{2} + 4 \)
\( T_{11}^{8} + 88 T_{11}^{6} + 2653 T_{11}^{4} + 31548 T_{11}^{2} + 125316 \)
\( T_{29} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( 36 + 1380 T^{2} + 469 T^{4} + 40 T^{6} + T^{8} \)
$7$ \( ( 4 - 23 T^{2} + T^{4} )^{2} \)
$11$ \( 125316 + 31548 T^{2} + 2653 T^{4} + 88 T^{6} + T^{8} \)
$13$ \( T^{8} \)
$17$ \( 412164 + 102612 T^{2} + 6133 T^{4} + 136 T^{6} + T^{8} \)
$19$ \( ( -19 + T^{2} )^{4} \)
$23$ \( ( 900 + 92 T^{2} + T^{4} )^{2} \)
$29$ \( T^{8} \)
$31$ \( T^{8} \)
$37$ \( T^{8} \)
$41$ \( T^{8} \)
$43$ \( ( -128 + T + T^{2} )^{4} \)
$47$ \( 1004004 + 1434252 T^{2} + 42973 T^{4} + 376 T^{6} + T^{8} \)
$53$ \( T^{8} \)
$59$ \( T^{8} \)
$61$ \( ( 26896 - 347 T^{2} + T^{4} )^{2} \)
$67$ \( T^{8} \)
$71$ \( T^{8} \)
$73$ \( ( -98 + 11 T + T^{2} )^{4} \)
$79$ \( T^{8} \)
$83$ \( ( 8100 + 332 T^{2} + T^{4} )^{2} \)
$89$ \( T^{8} \)
$97$ \( T^{8} \)
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