Properties

Label 2736.3.o.n
Level $2736$
Weight $3$
Character orbit 2736.o
Analytic conductor $74.551$
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: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2736.o (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(74.5506003290\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.184143974400.3
Defining polynomial: \(x^{8} - 4 x^{7} - 22 x^{6} + 80 x^{5} + 215 x^{4} - 568 x^{3} - 1022 x^{2} + 1320 x + 2628\)
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 114)
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.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{1} - \beta_{3} ) q^{5} + ( 1 - \beta_{3} ) q^{7} +O(q^{10})\) \( q + ( -1 - \beta_{1} - \beta_{3} ) q^{5} + ( 1 - \beta_{3} ) q^{7} + ( 1 - \beta_{5} ) q^{11} + ( -\beta_{6} - \beta_{7} ) q^{13} + ( -1 + \beta_{1} - \beta_{3} ) q^{17} + ( 5 + \beta_{1} - \beta_{2} - \beta_{5} - \beta_{7} ) q^{19} + ( -8 + \beta_{1} - 3 \beta_{3} - \beta_{5} ) q^{23} + ( 18 + 4 \beta_{1} + 3 \beta_{3} + 2 \beta_{5} ) q^{25} + ( -\beta_{2} + \beta_{4} + 2 \beta_{6} - 2 \beta_{7} ) q^{29} + ( -2 \beta_{2} - \beta_{6} ) q^{31} + ( 29 + \beta_{1} - \beta_{3} ) q^{35} + ( -5 \beta_{6} - 2 \beta_{7} ) q^{37} + ( 3 \beta_{2} + \beta_{4} - 5 \beta_{6} + \beta_{7} ) q^{41} + ( -11 - 4 \beta_{1} + \beta_{3} - 2 \beta_{5} ) q^{43} + ( -23 + 2 \beta_{1} - \beta_{5} ) q^{47} + ( -6 + 2 \beta_{1} - 5 \beta_{3} - 2 \beta_{5} ) q^{49} + ( -5 \beta_{2} + \beta_{4} - 3 \beta_{6} + \beta_{7} ) q^{53} + ( -1 + 18 \beta_{1} + 5 \beta_{3} ) q^{55} + ( 2 \beta_{2} + 2 \beta_{4} + 3 \beta_{6} + 7 \beta_{7} ) q^{59} + ( -19 + 8 \beta_{1} + 5 \beta_{3} - 2 \beta_{5} ) q^{61} + ( 6 \beta_{2} - 2 \beta_{4} + \beta_{6} + 5 \beta_{7} ) q^{65} + ( -2 \beta_{4} + 2 \beta_{6} - 7 \beta_{7} ) q^{67} + ( 2 \beta_{2} + 2 \beta_{4} - 9 \beta_{6} + \beta_{7} ) q^{71} + ( -43 - 2 \beta_{1} + 5 \beta_{3} + 2 \beta_{5} ) q^{73} + ( -11 + 13 \beta_{1} - 5 \beta_{3} ) q^{77} + ( 6 \beta_{2} - 2 \beta_{4} - 12 \beta_{6} + 9 \beta_{7} ) q^{79} + ( 16 - 11 \beta_{1} - 3 \beta_{3} - \beta_{5} ) q^{83} + ( 19 + 2 \beta_{1} - \beta_{3} - 2 \beta_{5} ) q^{85} + ( -3 \beta_{2} - \beta_{4} + 11 \beta_{6} - 3 \beta_{7} ) q^{89} + ( 6 \beta_{2} - 2 \beta_{4} - 7 \beta_{6} - \beta_{7} ) q^{91} + ( -17 + 13 \beta_{1} + 4 \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 9 \beta_{6} + 5 \beta_{7} ) q^{95} + ( -4 \beta_{4} - 3 \beta_{6} - 3 \beta_{7} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{5} + 12q^{7} + O(q^{10}) \) \( 8q - 4q^{5} + 12q^{7} + 4q^{11} - 4q^{17} + 36q^{19} - 56q^{23} + 140q^{25} + 236q^{35} - 100q^{43} - 188q^{47} - 36q^{49} - 28q^{55} - 180q^{61} - 356q^{73} - 68q^{77} + 136q^{83} + 148q^{85} - 140q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{7} - 22 x^{6} + 80 x^{5} + 215 x^{4} - 568 x^{3} - 1022 x^{2} + 1320 x + 2628\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{6} - 3 \nu^{5} - 17 \nu^{4} + 39 \nu^{3} + 88 \nu^{2} - 108 \nu - 36 \)\()/45\)
\(\beta_{2}\)\(=\)\((\)\( 21 \nu^{7} - 316 \nu^{6} + 447 \nu^{5} + 5735 \nu^{4} - 8574 \nu^{3} - 48322 \nu^{2} + 43908 \nu + 141048 \)\()/4365\)
\(\beta_{3}\)\(=\)\((\)\( 85 \nu^{7} - 443 \nu^{6} - 491 \nu^{5} + 5536 \nu^{4} - 6602 \nu^{3} - 14624 \nu^{2} + 58884 \nu - 22482 \)\()/13095\)
\(\beta_{4}\)\(=\)\((\)\( 163 \nu^{7} + 157 \nu^{6} - 5399 \nu^{5} - 24725 \nu^{4} + 98488 \nu^{3} + 298294 \nu^{2} - 450156 \nu - 1097676 \)\()/26190\)
\(\beta_{5}\)\(=\)\((\)\( -205 \nu^{7} + 863 \nu^{6} + 3341 \nu^{5} - 13711 \nu^{4} - 25468 \nu^{3} + 71114 \nu^{2} + 192396 \nu - 106308 \)\()/26190\)
\(\beta_{6}\)\(=\)\((\)\( -32 \nu^{7} + 112 \nu^{6} + 760 \nu^{5} - 2180 \nu^{4} - 6224 \nu^{3} + 11572 \nu^{2} + 15792 \nu - 9900 \)\()/2619\)
\(\beta_{7}\)\(=\)\((\)\( -82 \nu^{7} + 287 \nu^{6} + 1511 \nu^{5} - 4495 \nu^{4} - 12457 \nu^{3} + 23324 \nu^{2} + 30864 \nu - 19476 \)\()/4365\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-5 \beta_{7} + 12 \beta_{5} - 3 \beta_{1} + 24\)\()/60\)
\(\nu^{2}\)\(=\)\((\)\(-16 \beta_{7} + 7 \beta_{6} + 12 \beta_{5} - 4 \beta_{4} - 20 \beta_{2} - 48 \beta_{1} + 444\)\()/60\)
\(\nu^{3}\)\(=\)\((\)\(-67 \beta_{7} + 39 \beta_{6} + 51 \beta_{5} - 3 \beta_{4} - 45 \beta_{3} - 15 \beta_{2} - 69 \beta_{1} + 282\)\()/30\)
\(\nu^{4}\)\(=\)\((\)\(-212 \beta_{7} + 119 \beta_{6} + 96 \beta_{5} - 68 \beta_{4} - 90 \beta_{3} - 160 \beta_{2} - 384 \beta_{1} + 1752\)\()/30\)
\(\nu^{5}\)\(=\)\((\)\(-1195 \beta_{7} + 1020 \beta_{6} + 429 \beta_{5} - 165 \beta_{4} - 585 \beta_{3} - 375 \beta_{2} - 1131 \beta_{1} + 3558\)\()/30\)
\(\nu^{6}\)\(=\)\((\)\(-2071 \beta_{7} + 1627 \beta_{6} + 525 \beta_{5} - 679 \beta_{4} - 765 \beta_{3} - 1190 \beta_{2} - 1965 \beta_{1} + 6150\)\()/15\)
\(\nu^{7}\)\(=\)\((\)\(-19531 \beta_{7} + 18732 \beta_{6} + 2535 \beta_{5} - 4179 \beta_{4} - 4365 \beta_{3} - 7035 \beta_{2} - 10455 \beta_{1} + 30270\)\()/30\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
721.1
2.51885 0.707107i
2.51885 + 0.707107i
−2.87998 0.707107i
−2.87998 + 0.707107i
−1.51885 0.707107i
−1.51885 + 0.707107i
3.87998 0.707107i
3.87998 + 0.707107i
0 0 0 −9.91350 0 −3.01452 0 0 0
721.2 0 0 0 −9.91350 0 −3.01452 0 0 0
721.3 0 0 0 −2.94975 0 −5.84873 0 0 0
721.4 0 0 0 −2.94975 0 −5.84873 0 0 0
721.5 0 0 0 4.01452 0 10.9135 0 0 0
721.6 0 0 0 4.01452 0 10.9135 0 0 0
721.7 0 0 0 6.84873 0 3.94975 0 0 0
721.8 0 0 0 6.84873 0 3.94975 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 721.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.3.o.n 8
3.b odd 2 1 912.3.o.d 8
4.b odd 2 1 342.3.d.b 8
12.b even 2 1 114.3.d.a 8
19.b odd 2 1 inner 2736.3.o.n 8
57.d even 2 1 912.3.o.d 8
76.d even 2 1 342.3.d.b 8
228.b odd 2 1 114.3.d.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.3.d.a 8 12.b even 2 1
114.3.d.a 8 228.b odd 2 1
342.3.d.b 8 4.b odd 2 1
342.3.d.b 8 76.d even 2 1
912.3.o.d 8 3.b odd 2 1
912.3.o.d 8 57.d even 2 1
2736.3.o.n 8 1.a even 1 1 trivial
2736.3.o.n 8 19.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(2736, [\chi])\):

\( T_{5}^{4} + 2 T_{5}^{3} - 83 T_{5}^{2} + 36 T_{5} + 804 \)
\( T_{7}^{4} - 6 T_{7}^{3} - 71 T_{7}^{2} + 120 T_{7} + 760 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( ( 804 + 36 T - 83 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$7$ \( ( 760 + 120 T - 71 T^{2} - 6 T^{3} + T^{4} )^{2} \)
$11$ \( ( 28650 - 60 T - 389 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$13$ \( ( 576 + 240 T^{2} + T^{4} )^{2} \)
$17$ \( ( -60 - 540 T - 179 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$19$ \( 16983563041 - 1693651716 T + 35447312 T^{2} - 454860 T^{3} + 95646 T^{4} - 1260 T^{5} + 272 T^{6} - 36 T^{7} + T^{8} \)
$23$ \( ( -302040 - 36600 T - 914 T^{2} + 28 T^{3} + T^{4} )^{2} \)
$29$ \( 30611001600 + 1631327040 T^{2} + 4588164 T^{4} + 3924 T^{6} + T^{8} \)
$31$ \( 6046617600 + 483356160 T^{2} + 2039184 T^{4} + 2616 T^{6} + T^{8} \)
$37$ \( ( 831744 + 2976 T^{2} + T^{4} )^{2} \)
$41$ \( 4246237209600 + 24032972160 T^{2} + 27965124 T^{4} + 9924 T^{6} + T^{8} \)
$43$ \( ( -1616000 - 128800 T - 1935 T^{2} + 50 T^{3} + T^{4} )^{2} \)
$47$ \( ( 89850 + 29940 T + 2779 T^{2} + 94 T^{3} + T^{4} )^{2} \)
$53$ \( 492631164527616 + 458261981568 T^{2} + 148406724 T^{4} + 20292 T^{6} + T^{8} \)
$59$ \( 652283427225600 + 577716572160 T^{2} + 178910784 T^{4} + 22704 T^{6} + T^{8} \)
$61$ \( ( 250000 - 15000 T - 1775 T^{2} + 90 T^{3} + T^{4} )^{2} \)
$67$ \( 292222411776 + 15853674240 T^{2} + 96643152 T^{4} + 22440 T^{6} + T^{8} \)
$71$ \( 110381078937600 + 320548700160 T^{2} + 164948544 T^{4} + 25584 T^{6} + T^{8} \)
$73$ \( ( 577600 + 122320 T + 8001 T^{2} + 178 T^{3} + T^{4} )^{2} \)
$79$ \( 27650847555993600 + 8768517027840 T^{2} + 1026425664 T^{4} + 52656 T^{6} + T^{8} \)
$83$ \( ( -57240 + 259560 T - 3794 T^{2} - 68 T^{3} + T^{4} )^{2} \)
$89$ \( 109746576000000 + 266438160000 T^{2} + 163296900 T^{4} + 26340 T^{6} + T^{8} \)
$97$ \( 2593798080270336 + 2658749349888 T^{2} + 534420864 T^{4} + 39552 T^{6} + T^{8} \)
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