Properties

Label 2736.2.bm.r
Level $2736$
Weight $2$
Character orbit 2736.bm
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.bm (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 2 x^{7} - 30 x^{5} - 5 x^{4} + 114 x^{3} + 300 x^{2} + 116 x + 19\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 912)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( \beta_{3} + \beta_{6} - \beta_{7} ) q^{5} + ( -1 + \beta_{2} - \beta_{4} - 2 \beta_{6} ) q^{7} +O(q^{10})\) \( q + ( \beta_{3} + \beta_{6} - \beta_{7} ) q^{5} + ( -1 + \beta_{2} - \beta_{4} - 2 \beta_{6} ) q^{7} + ( -1 + \beta_{5} - \beta_{6} ) q^{11} + ( -\beta_{1} - \beta_{2} + \beta_{5} + \beta_{6} ) q^{13} + ( -1 - 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{17} + ( -\beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{19} + ( -2 - \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{23} + ( -\beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{25} + ( 2 + 2 \beta_{1} - 2 \beta_{5} ) q^{29} + ( -4 - 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{31} + ( -3 - \beta_{1} + \beta_{3} - 3 \beta_{4} + \beta_{7} ) q^{35} + ( -1 + 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{6} ) q^{37} + ( 1 + 2 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} ) q^{41} + ( -2 + \beta_{1} - \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} ) q^{43} + ( -1 - \beta_{2} - 2 \beta_{3} - \beta_{6} + \beta_{7} ) q^{47} + ( -1 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{49} + ( 1 + \beta_{2} ) q^{53} + ( \beta_{1} + \beta_{3} - 2 \beta_{6} + \beta_{7} ) q^{55} + ( -2 - \beta_{1} - 2 \beta_{3} + 2 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} ) q^{59} + ( -2 + \beta_{1} + \beta_{2} - 2 \beta_{4} + \beta_{5} - \beta_{6} ) q^{61} + ( -2 - 3 \beta_{2} + 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{65} + ( -1 + \beta_{2} - 2 \beta_{4} - 5 \beta_{6} + 4 \beta_{7} ) q^{67} + ( 1 + 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} ) q^{71} + ( -1 - 2 \beta_{3} - 3 \beta_{6} + 2 \beta_{7} ) q^{73} + ( -2 + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{77} + ( 2 - \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + \beta_{4} + 2 \beta_{5} + 7 \beta_{6} - 3 \beta_{7} ) q^{79} + ( 4 - \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} + 8 \beta_{6} + 2 \beta_{7} ) q^{83} + ( 2 - 2 \beta_{2} + 4 \beta_{4} + 4 \beta_{6} + 2 \beta_{7} ) q^{85} + ( -10 + 2 \beta_{2} + 2 \beta_{3} - 5 \beta_{6} - \beta_{7} ) q^{89} + ( \beta_{1} - \beta_{2} + 2 \beta_{4} + \beta_{5} - 6 \beta_{6} + \beta_{7} ) q^{91} + ( -7 - \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{95} + ( 5 - \beta_{1} - \beta_{4} - 5 \beta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 2q^{5} + O(q^{10}) \) \( 8q + 2q^{5} - 4q^{17} - 6q^{23} - 12q^{25} + 12q^{29} - 28q^{31} - 18q^{35} + 12q^{41} - 18q^{43} - 12q^{47} - 24q^{49} + 6q^{53} + 12q^{55} - 10q^{59} - 4q^{61} + 6q^{67} + 8q^{71} - 8q^{73} - 28q^{77} + 14q^{79} - 8q^{85} - 54q^{89} + 26q^{91} - 38q^{95} + 60q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} - 30 x^{5} - 5 x^{4} + 114 x^{3} + 300 x^{2} + 116 x + 19\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 44 \nu^{7} - 2059 \nu^{6} + 6647 \nu^{5} - 15007 \nu^{4} + 50938 \nu^{3} + 80360 \nu^{2} - 278981 \nu + 154035 \)\()/307713\)
\(\beta_{2}\)\(=\)\((\)\( -88 \nu^{7} + 4118 \nu^{6} - 13294 \nu^{5} + 30014 \nu^{4} - 101876 \nu^{3} + 146993 \nu^{2} + 250249 \nu - 308070 \)\()/307713\)
\(\beta_{3}\)\(=\)\((\)\( 3500 \nu^{7} - 15922 \nu^{6} + 21212 \nu^{5} - 114745 \nu^{4} + 251431 \nu^{3} + 373886 \nu^{2} + 131509 \nu - 1214655 \)\()/307713\)
\(\beta_{4}\)\(=\)\((\)\( -3772 \nu^{7} + 8669 \nu^{6} - 10351 \nu^{5} + 111605 \nu^{4} - 2846 \nu^{3} - 315175 \nu^{2} - 309125 \nu - 420924 \)\()/307713\)
\(\beta_{5}\)\(=\)\((\)\( 5106 \nu^{7} - 10484 \nu^{6} - 7253 \nu^{5} - 142319 \nu^{4} - 28670 \nu^{3} + 830669 \nu^{2} + 1282798 \nu + 414680 \)\()/307713\)
\(\beta_{6}\)\(=\)\((\)\( -1002 \nu^{7} + 2264 \nu^{6} - 178 \nu^{5} + 29375 \nu^{4} + 254 \nu^{3} - 124940 \nu^{2} - 273328 \nu - 78329 \)\()/43959\)
\(\beta_{7}\)\(=\)\((\)\( -32081 \nu^{7} + 65584 \nu^{6} - 5933 \nu^{5} + 958120 \nu^{4} + 99677 \nu^{3} - 3586874 \nu^{2} - 9255799 \nu - 2690259 \)\()/307713\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-2 \beta_{6} - 2 \beta_{5} + \beta_{4} + \beta_{2} + \beta_{1} + 1\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-2 \beta_{6} - 2 \beta_{5} + \beta_{4} + 4 \beta_{2} + 7 \beta_{1} + 1\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(-6 \beta_{7} + 35 \beta_{6} + 8 \beta_{5} + 5 \beta_{4} + 9 \beta_{3} + 5 \beta_{2} - \beta_{1} + 47\)\()/3\)
\(\nu^{4}\)\(=\)\(2 \beta_{7} - 15 \beta_{6} - 12 \beta_{5} - 2 \beta_{4} + 4 \beta_{3} + 14 \beta_{2} - 2 \beta_{1} + 35\)
\(\nu^{5}\)\(=\)\((\)\(-133 \beta_{6} - 187 \beta_{5} - 7 \beta_{4} + 146 \beta_{2} + 191 \beta_{1} + 56\)\()/3\)
\(\nu^{6}\)\(=\)\((\)\(-198 \beta_{7} + 992 \beta_{6} + 47 \beta_{5} + 38 \beta_{4} + 150 \beta_{3} + 335 \beta_{2} + 311 \beta_{1} + 797\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(-273 \beta_{7} + 1618 \beta_{6} - 119 \beta_{5} - 485 \beta_{4} + 693 \beta_{3} + 1192 \beta_{2} - 653 \beta_{1} + 4249\)\()/3\)

Character values

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

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(-\beta_{6}\) \(1\) \(-1\) \(1\)

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
−0.213988 + 0.172868i
−0.654220 + 2.95767i
−1.27736 1.04884i
3.14556 0.349646i
−0.213988 0.172868i
−0.654220 2.95767i
−1.27736 + 1.04884i
3.14556 + 0.349646i
0 0 0 −2.00488 + 3.47255i 0 1.92982i 0 0 0
559.2 0 0 0 0.353597 0.612447i 0 0.0924751i 0 0 0
559.3 0 0 0 0.912850 1.58110i 0 4.99333i 0 0 0
559.4 0 0 0 1.73843 3.01105i 0 3.36658i 0 0 0
1855.1 0 0 0 −2.00488 3.47255i 0 1.92982i 0 0 0
1855.2 0 0 0 0.353597 + 0.612447i 0 0.0924751i 0 0 0
1855.3 0 0 0 0.912850 + 1.58110i 0 4.99333i 0 0 0
1855.4 0 0 0 1.73843 + 3.01105i 0 3.36658i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1855.4
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.r 8
3.b odd 2 1 912.2.bb.h yes 8
4.b odd 2 1 2736.2.bm.s 8
12.b even 2 1 912.2.bb.g 8
19.d odd 6 1 2736.2.bm.s 8
57.f even 6 1 912.2.bb.g 8
76.f even 6 1 inner 2736.2.bm.r 8
228.n odd 6 1 912.2.bb.h yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.2.bb.g 8 12.b even 2 1
912.2.bb.g 8 57.f even 6 1
912.2.bb.h yes 8 3.b odd 2 1
912.2.bb.h yes 8 228.n odd 6 1
2736.2.bm.r 8 1.a even 1 1 trivial
2736.2.bm.r 8 76.f even 6 1 inner
2736.2.bm.s 8 4.b odd 2 1
2736.2.bm.s 8 19.d odd 6 1

Hecke kernels

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

\(T_{5}^{8} - \cdots\)
\( T_{7}^{8} + 40 T_{7}^{6} + 418 T_{7}^{4} + 1056 T_{7}^{2} + 9 \)
\( T_{11}^{8} + 40 T_{11}^{6} + 520 T_{11}^{4} + 2376 T_{11}^{2} + 2916 \)
\(T_{23}^{8} + \cdots\)
\( T_{31}^{4} + 14 T_{31}^{3} + 10 T_{31}^{2} - 282 T_{31} - 45 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( T^{8} \)
$5$ \( 324 - 648 T + 1044 T^{2} - 576 T^{3} + 286 T^{4} - 44 T^{5} + 18 T^{6} - 2 T^{7} + T^{8} \)
$7$ \( 9 + 1056 T^{2} + 418 T^{4} + 40 T^{6} + T^{8} \)
$11$ \( 2916 + 2376 T^{2} + 520 T^{4} + 40 T^{6} + T^{8} \)
$13$ \( 225 - 1080 T + 2118 T^{2} - 1872 T^{3} + 691 T^{4} - 26 T^{6} + T^{8} \)
$17$ \( 129600 + 95040 T + 53856 T^{2} + 14496 T^{3} + 3352 T^{4} + 352 T^{5} + 60 T^{6} + 4 T^{7} + T^{8} \)
$19$ \( 130321 - 1444 T^{2} - 1368 T^{3} + 258 T^{4} - 72 T^{5} - 4 T^{6} + T^{8} \)
$23$ \( 12996 + 2736 T - 4140 T^{2} - 912 T^{3} + 1378 T^{4} - 228 T^{5} - 26 T^{6} + 6 T^{7} + T^{8} \)
$29$ \( 746496 - 746496 T + 297216 T^{2} - 48384 T^{3} + 544 T^{4} + 672 T^{5} - 8 T^{6} - 12 T^{7} + T^{8} \)
$31$ \( ( -45 - 282 T + 10 T^{2} + 14 T^{3} + T^{4} )^{2} \)
$37$ \( 81225 + 42372 T^{2} + 6046 T^{4} + 148 T^{6} + T^{8} \)
$41$ \( 129600 - 233280 T + 161568 T^{2} - 38880 T^{3} + 1368 T^{4} + 720 T^{5} - 12 T^{6} - 12 T^{7} + T^{8} \)
$43$ \( 1946025 - 1330830 T + 211302 T^{2} + 62964 T^{3} - 2763 T^{4} - 1188 T^{5} + 42 T^{6} + 18 T^{7} + T^{8} \)
$47$ \( 576 - 9792 T + 53856 T^{2} + 27744 T^{3} + 2968 T^{4} - 816 T^{5} - 20 T^{6} + 12 T^{7} + T^{8} \)
$53$ \( 36 + 288 T + 684 T^{2} - 672 T^{3} + 94 T^{4} + 84 T^{5} - 2 T^{6} - 6 T^{7} + T^{8} \)
$59$ \( 14152644 - 2302344 T + 743220 T^{2} - 15264 T^{3} + 11962 T^{4} + 244 T^{5} + 198 T^{6} + 10 T^{7} + T^{8} \)
$61$ \( 1590121 - 630500 T + 424018 T^{2} + 58912 T^{3} + 19783 T^{4} + 448 T^{5} + 154 T^{6} + 4 T^{7} + T^{8} \)
$67$ \( 693889 - 1154538 T + 1749398 T^{2} - 295512 T^{3} + 51585 T^{4} - 1536 T^{5} + 242 T^{6} - 6 T^{7} + T^{8} \)
$71$ \( 627264 + 95040 T + 68256 T^{2} + 4512 T^{3} + 4792 T^{4} + 304 T^{5} + 132 T^{6} - 8 T^{7} + T^{8} \)
$73$ \( 393129 + 240768 T + 123630 T^{2} + 24624 T^{3} + 5143 T^{4} + 464 T^{5} + 102 T^{6} + 8 T^{7} + T^{8} \)
$79$ \( 5331481 - 2683058 T + 1225558 T^{2} - 127400 T^{3} + 21493 T^{4} - 1568 T^{5} + 250 T^{6} - 14 T^{7} + T^{8} \)
$83$ \( 13483584 + 2229120 T^{2} + 72288 T^{4} + 504 T^{6} + T^{8} \)
$89$ \( 61496964 + 14774328 T - 400932 T^{2} - 380568 T^{3} + 14734 T^{4} + 10908 T^{5} + 1174 T^{6} + 54 T^{7} + T^{8} \)
$97$ \( 10890000 - 12038400 T + 5835168 T^{2} - 1546752 T^{3} + 249436 T^{4} - 25440 T^{5} + 1624 T^{6} - 60 T^{7} + T^{8} \)
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