Properties

Label 2736.2.f.g
Level $2736$
Weight $2$
Character orbit 2736.f
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.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \(x^{8} + 18 x^{6} + 37 x^{4} + 22 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 1368)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + \beta_{7} q^{5} -\beta_{4} q^{7} +O(q^{10})\) \( q + \beta_{7} q^{5} -\beta_{4} q^{7} + ( -\beta_{3} - \beta_{5} ) q^{11} + ( \beta_{2} + \beta_{3} - \beta_{5} ) q^{13} + ( -\beta_{3} + \beta_{5} ) q^{17} + ( -\beta_{3} - \beta_{6} - \beta_{7} ) q^{19} + ( \beta_{2} - \beta_{3} ) q^{23} + ( -1 - \beta_{1} - \beta_{4} + \beta_{6} ) q^{25} + ( -4 + \beta_{1} - \beta_{6} ) q^{29} + ( \beta_{2} - 3 \beta_{3} - \beta_{5} ) q^{31} + ( -\beta_{2} - \beta_{3} + 2 \beta_{5} - \beta_{7} ) q^{35} + ( -\beta_{2} + \beta_{3} - 3 \beta_{5} - 2 \beta_{7} ) q^{37} + ( 4 + 2 \beta_{4} - 2 \beta_{6} ) q^{41} + ( 3 + \beta_{1} + \beta_{4} ) q^{43} + ( -2 \beta_{3} + 2 \beta_{5} - \beta_{7} ) q^{47} + ( -\beta_{1} + \beta_{4} + 2 \beta_{6} ) q^{49} + ( -2 \beta_{1} - 2 \beta_{4} ) q^{53} + ( -3 + 2 \beta_{1} + \beta_{4} + \beta_{6} ) q^{55} + ( \beta_{1} - 2 \beta_{4} + \beta_{6} ) q^{59} + ( -2 \beta_{1} - 3 \beta_{4} ) q^{61} + ( 4 + \beta_{1} - 2 \beta_{4} - 3 \beta_{6} ) q^{65} + ( -2 \beta_{3} + 2 \beta_{7} ) q^{67} + ( -4 + \beta_{1} + 2 \beta_{4} + \beta_{6} ) q^{71} + ( -1 + \beta_{1} + 3 \beta_{4} + 2 \beta_{6} ) q^{73} -\beta_{3} q^{77} + ( \beta_{2} - \beta_{3} + 3 \beta_{5} - 2 \beta_{7} ) q^{79} + ( -2 \beta_{2} - 2 \beta_{3} + \beta_{5} - 2 \beta_{7} ) q^{83} + ( -1 + \beta_{4} + \beta_{6} ) q^{85} + ( -4 + 3 \beta_{1} + \beta_{6} ) q^{89} + ( -2 \beta_{2} - 2 \beta_{3} + 6 \beta_{5} - 4 \beta_{7} ) q^{91} + ( 4 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{95} + ( -2 \beta_{2} + 2 \beta_{5} - 2 \beta_{7} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 4q^{19} - 8q^{25} - 32q^{29} + 24q^{41} + 28q^{43} + 4q^{49} - 8q^{53} - 12q^{55} + 8q^{59} - 8q^{61} + 24q^{65} - 24q^{71} + 4q^{73} - 4q^{85} - 16q^{89} + 40q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 18 x^{6} + 37 x^{4} + 22 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{6} + 17 \nu^{4} + 20 \nu^{2} + 3 \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} + 18 \nu^{5} + 37 \nu^{3} + 24 \nu \)\()/2\)
\(\beta_{3}\)\(=\)\( -4 \nu^{7} - 70 \nu^{5} - 113 \nu^{3} - 32 \nu \)
\(\beta_{4}\)\(=\)\( 5 \nu^{6} + 88 \nu^{4} + 150 \nu^{2} + 50 \)
\(\beta_{5}\)\(=\)\((\)\( -9 \nu^{7} - 158 \nu^{5} - 263 \nu^{3} - 84 \nu \)\()/2\)
\(\beta_{6}\)\(=\)\( 7 \nu^{6} + 123 \nu^{4} + 206 \nu^{2} + 65 \)
\(\beta_{7}\)\(=\)\((\)\( 15 \nu^{7} + 264 \nu^{5} + 449 \nu^{3} + 144 \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} - \beta_{3} + \beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-3 \beta_{6} + 4 \beta_{4} + \beta_{1} - 8\)\()/2\)
\(\nu^{3}\)\(=\)\(-2 \beta_{7} - 12 \beta_{5} + 9 \beta_{3} - 6 \beta_{2}\)
\(\nu^{4}\)\(=\)\(25 \beta_{6} - 33 \beta_{4} - 10 \beta_{1} + 55\)
\(\nu^{5}\)\(=\)\(35 \beta_{7} + 194 \beta_{5} - 141 \beta_{3} + 93 \beta_{2}\)
\(\nu^{6}\)\(=\)\(-395 \beta_{6} + 521 \beta_{4} + 161 \beta_{1} - 858\)
\(\nu^{7}\)\(=\)\(-556 \beta_{7} - 3060 \beta_{5} + 2217 \beta_{3} - 1462 \beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1025.1
1.18847i
0.676777i
0.626815i
3.96694i
3.96694i
0.626815i
0.676777i
1.18847i
0 0 0 3.42865i 0 0.394100 0 0 0
1025.2 0 0 0 2.60938i 0 0.723074 0 0 0
1025.3 0 0 0 2.32945i 0 −4.34658 0 0 0
1025.4 0 0 0 0.0959660i 0 3.22941 0 0 0
1025.5 0 0 0 0.0959660i 0 3.22941 0 0 0
1025.6 0 0 0 2.32945i 0 −4.34658 0 0 0
1025.7 0 0 0 2.60938i 0 0.723074 0 0 0
1025.8 0 0 0 3.42865i 0 0.394100 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
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.g 8
3.b odd 2 1 2736.2.f.h 8
4.b odd 2 1 1368.2.f.c 8
12.b even 2 1 1368.2.f.d yes 8
19.b odd 2 1 2736.2.f.h 8
57.d even 2 1 inner 2736.2.f.g 8
76.d even 2 1 1368.2.f.d yes 8
228.b odd 2 1 1368.2.f.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1368.2.f.c 8 4.b odd 2 1
1368.2.f.c 8 228.b odd 2 1
1368.2.f.d yes 8 12.b even 2 1
1368.2.f.d yes 8 76.d even 2 1
2736.2.f.g 8 1.a even 1 1 trivial
2736.2.f.g 8 57.d even 2 1 inner
2736.2.f.h 8 3.b odd 2 1
2736.2.f.h 8 19.b odd 2 1

Hecke kernels

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

\( T_{5}^{8} + 24 T_{5}^{6} + 181 T_{5}^{4} + 436 T_{5}^{2} + 4 \)
\( T_{7}^{4} - 15 T_{7}^{2} + 16 T_{7} - 4 \)
\( T_{11}^{8} + 32 T_{11}^{6} + 157 T_{11}^{4} + 68 T_{11}^{2} + 4 \)
\( T_{29}^{4} + 16 T_{29}^{3} + 60 T_{29}^{2} - 64 T_{29} - 320 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( 4 + 436 T^{2} + 181 T^{4} + 24 T^{6} + T^{8} \)
$7$ \( ( -4 + 16 T - 15 T^{2} + T^{4} )^{2} \)
$11$ \( 4 + 68 T^{2} + 157 T^{4} + 32 T^{6} + T^{8} \)
$13$ \( 16384 + 18432 T^{2} + 2368 T^{4} + 88 T^{6} + T^{8} \)
$17$ \( 100 + 452 T^{2} + 181 T^{4} + 24 T^{6} + T^{8} \)
$19$ \( 130321 + 27436 T + 1444 T^{2} - 1444 T^{3} - 362 T^{4} - 76 T^{5} + 4 T^{6} + 4 T^{7} + T^{8} \)
$23$ \( 16 + 4352 T^{2} + 1016 T^{4} + 64 T^{6} + T^{8} \)
$29$ \( ( -320 - 64 T + 60 T^{2} + 16 T^{3} + T^{4} )^{2} \)
$31$ \( 1048576 + 376832 T^{2} + 15424 T^{4} + 216 T^{6} + T^{8} \)
$37$ \( 65536 + 69632 T^{2} + 8256 T^{4} + 184 T^{6} + T^{8} \)
$41$ \( ( -3200 + 1248 T - 76 T^{2} - 12 T^{3} + T^{4} )^{2} \)
$43$ \( ( -16 - 16 T + 49 T^{2} - 14 T^{3} + T^{4} )^{2} \)
$47$ \( 58564 + 27588 T^{2} + 3613 T^{4} + 112 T^{6} + T^{8} \)
$53$ \( ( 1280 - 64 T - 92 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$59$ \( ( 3328 + 192 T - 128 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$61$ \( ( 4976 - 268 T - 139 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$67$ \( 409600 + 114688 T^{2} + 8512 T^{4} + 208 T^{6} + T^{8} \)
$71$ \( ( 256 - 320 T - 32 T^{2} + 12 T^{3} + T^{4} )^{2} \)
$73$ \( ( 1156 + 284 T - 219 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$79$ \( 15745024 + 1107968 T^{2} + 27456 T^{4} + 280 T^{6} + T^{8} \)
$83$ \( 512656 + 265152 T^{2} + 21464 T^{4} + 336 T^{6} + T^{8} \)
$89$ \( ( 5888 - 1152 T - 228 T^{2} + 8 T^{3} + T^{4} )^{2} \)
$97$ \( 16384 + 32768 T^{2} + 10048 T^{4} + 272 T^{6} + T^{8} \)
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