Properties

Label 2736.2.bm.t
Level $2736$
Weight $2$
Character orbit 2736.bm
Analytic conductor $21.847$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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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: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 28 x^{14} + 542 x^{12} + 5488 x^{10} + 40451 x^{8} + 151312 x^{6} + 395134 x^{4} + 52164 x^{2} + 6561\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{8} q^{5} + ( \beta_{3} + \beta_{5} ) q^{7} +O(q^{10})\) \( q -\beta_{8} q^{5} + ( \beta_{3} + \beta_{5} ) q^{7} + ( \beta_{11} - \beta_{12} ) q^{11} + ( \beta_{1} + \beta_{6} ) q^{13} + ( \beta_{8} - \beta_{9} - \beta_{10} ) q^{17} + \beta_{7} q^{19} -\beta_{15} q^{23} + ( -\beta_{1} + \beta_{6} ) q^{25} + 2 \beta_{10} q^{29} + ( \beta_{3} - \beta_{4} ) q^{31} + ( -\beta_{14} + \beta_{15} ) q^{35} + ( 1 + 2 \beta_{2} ) q^{37} + ( \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{13} ) q^{41} + ( \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{43} + ( \beta_{11} - \beta_{15} ) q^{47} + ( -1 - \beta_{1} ) q^{49} + ( 2 \beta_{8} + \beta_{10} - 2 \beta_{13} ) q^{53} + ( \beta_{3} - \beta_{4} + \beta_{7} ) q^{55} + ( -2 \beta_{11} + \beta_{12} ) q^{59} + ( -2 + \beta_{1} - 2 \beta_{2} - \beta_{6} ) q^{61} + ( -4 \beta_{8} + 3 \beta_{9} - 2 \beta_{13} ) q^{65} + ( \beta_{3} + 2 \beta_{7} ) q^{67} + ( 2 \beta_{11} - \beta_{12} + \beta_{14} + \beta_{15} ) q^{71} + ( -3 \beta_{2} - 4 \beta_{6} ) q^{73} + ( 2 \beta_{9} - 4 \beta_{10} - \beta_{13} ) q^{77} + ( 5 \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{7} ) q^{79} + ( -\beta_{11} + \beta_{12} + \beta_{14} ) q^{83} + ( 2 + 4 \beta_{1} + 2 \beta_{2} - 4 \beta_{6} ) q^{85} + ( -\beta_{8} + 2 \beta_{10} + \beta_{13} ) q^{89} + ( 2 \beta_{3} + \beta_{5} + 3 \beta_{7} ) q^{91} + ( -\beta_{11} - \beta_{12} + \beta_{15} ) q^{95} + ( -6 - 6 \beta_{1} - 3 \beta_{2} + 3 \beta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q - 16q^{49} - 16q^{61} + 24q^{73} + 16q^{85} - 72q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 28 x^{14} + 542 x^{12} + 5488 x^{10} + 40451 x^{8} + 151312 x^{6} + 395134 x^{4} + 52164 x^{2} + 6561\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -10164 \nu^{14} - 255509 \nu^{12} - 4754946 \nu^{10} - 41215272 \nu^{8} - 266357336 \nu^{6} - 462486696 \nu^{4} - 61069302 \nu^{2} + 4905120953 \)\()/ 1856785928 \)
\(\beta_{2}\)\(=\)\((\)\(-3806323948 \nu^{14} - 105837336802 \nu^{12} - 2044580895560 \nu^{10} - 20543041875661 \nu^{8} - 150969971356232 \nu^{6} - 555207640209094 \nu^{4} - 1470348324238444 \nu^{2} - 194108471710191\)\()/ 184747182558867 \)
\(\beta_{3}\)\(=\)\((\)\(-96213205709 \nu^{14} - 2754784164545 \nu^{12} - 53992628470114 \nu^{10} - 561812335230251 \nu^{8} - 4226571858284230 \nu^{6} - 16618957602281825 \nu^{4} - 44301633706386407 \nu^{2} - 11417555800756728\)\()/ 2955954920941872 \)
\(\beta_{4}\)\(=\)\((\)\(-120384307763 \nu^{14} - 3391806056156 \nu^{12} - 65300409366145 \nu^{10} - 659826753342743 \nu^{8} - 4778910298762015 \nu^{6} - 17718801475523981 \nu^{4} - 44446863176182604 \nu^{2} - 15031274091976611\)\()/ 2955954920941872 \)
\(\beta_{5}\)\(=\)\((\)\(-132220047635 \nu^{14} - 3686953004252 \nu^{12} - 70837432581553 \nu^{10} - 707820971971799 \nu^{8} - 5110667882932927 \nu^{6} - 18257356397613389 \nu^{4} - 44517976947595100 \nu^{2} + 5260837691851245\)\()/ 2955954920941872 \)
\(\beta_{6}\)\(=\)\((\)\(77189178416 \nu^{14} + 2176005166862 \nu^{12} + 42180465284398 \nu^{10} + 430495021839779 \nu^{8} + 3182021455043788 \nu^{6} + 12113792985422063 \nu^{4} + 31169326351674140 \nu^{2} + 4114868769414801\)\()/ 1477977460470936 \)
\(\beta_{7}\)\(=\)\((\)\(-65958342944 \nu^{14} - 1804014851009 \nu^{12} - 34720999750351 \nu^{10} - 343729127623994 \nu^{8} - 2523799016893609 \nu^{6} - 9102313901174438 \nu^{4} - 24301148913260783 \nu^{2} + 1702751241319065\)\()/ 985318306980624 \)
\(\beta_{8}\)\(=\)\((\)\(-990175670 \nu^{15} - 25195594535 \nu^{13} - 463226273255 \nu^{11} - 4015186891660 \nu^{9} - 25480512543977 \nu^{7} - 45055398866380 \nu^{5} - 5949364130685 \nu^{3} + 831702838632193 \nu\)\()/ 109479811886736 \)
\(\beta_{9}\)\(=\)\((\)\(-2834183765 \nu^{15} - 78805865495 \nu^{13} - 1518941639794 \nu^{11} - 15250342530713 \nu^{9} - 111711718597870 \nu^{7} - 414448034891111 \nu^{5} - 1093376918915915 \nu^{3} - 281697264791856 \nu\)\()/ 83923010373744 \)
\(\beta_{10}\)\(=\)\((\)\(909972638167 \nu^{15} + 25176125585668 \nu^{13} + 486901477983041 \nu^{11} + 4881138908066497 \nu^{9} + 36022974087467963 \nu^{7} + 131904960593157655 \nu^{5} + 346669798424140396 \nu^{3} - 41317391409520995 \nu\)\()/ 26603594288476848 \)
\(\beta_{11}\)\(=\)\((\)\(-1890306552809 \nu^{15} - 52424270976620 \nu^{13} - 1009971126798727 \nu^{11} - 10090741797306707 \nu^{9} - 73463991194917621 \nu^{7} - 263761122113038781 \nu^{5} - 655064719310154740 \nu^{3} + 77578071609638889 \nu\)\()/ 26603594288476848 \)
\(\beta_{12}\)\(=\)\((\)\(-3394373004184 \nu^{15} - 95065139318341 \nu^{13} - 1839250515565853 \nu^{11} - 18615270407097682 \nu^{9} - 136919912259057455 \nu^{7} - 509947381143146086 \nu^{5} - 1307808756467713693 \nu^{3} - 90623477496149271 \nu\)\()/ 26603594288476848 \)
\(\beta_{13}\)\(=\)\((\)\(46212447101 \nu^{15} + 1289258286323 \nu^{13} + 24906077016940 \nu^{11} + 250804054500443 \nu^{9} + 1839041801676268 \nu^{7} + 6763265898390281 \nu^{5} + 17340653478163769 \nu^{3} + 114034611732426 \nu\)\()/ 286060153639536 \)
\(\beta_{14}\)\(=\)\((\)\(-78658668889 \nu^{15} - 2212467122437 \nu^{13} - 42911401938458 \nu^{11} - 437082575176471 \nu^{9} - 3236508022255694 \nu^{7} - 12307020335183269 \nu^{5} - 32608540003616059 \nu^{3} - 8402395795060824 \nu\)\()/ 286060153639536 \)
\(\beta_{15}\)\(=\)\((\)\(2811623585401 \nu^{15} + 78035033220436 \nu^{13} + 1504847016559811 \nu^{11} + 15062914749592717 \nu^{9} + 110062989712353245 \nu^{7} + 398499357853747327 \nu^{5} + 1013107101355984660 \nu^{3} - 120301723739013183 \nu\)\()/ 8867864762825616 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{15} - \beta_{14} + \beta_{12} - 2 \beta_{11} + \beta_{10} + \beta_{9} - 3 \beta_{8}\)\()/6\)
\(\nu^{2}\)\(=\)\(\beta_{7} + \beta_{5} + \beta_{3} - 7 \beta_{2} - 7\)
\(\nu^{3}\)\(=\)\((\)\(22 \beta_{15} - 11 \beta_{14} - 45 \beta_{13} + 5 \beta_{12} + 5 \beta_{11} - 34 \beta_{10} + 17 \beta_{9}\)\()/6\)
\(\nu^{4}\)\(=\)\(-14 \beta_{7} - 8 \beta_{6} - 14 \beta_{4} - 14 \beta_{3} + 75 \beta_{2}\)
\(\nu^{5}\)\(=\)\((\)\(-139 \beta_{15} + 278 \beta_{14} + 633 \beta_{13} - 14 \beta_{12} + 7 \beta_{11} + 223 \beta_{10} - 446 \beta_{9} + 633 \beta_{8}\)\()/6\)
\(\nu^{6}\)\(=\)\(-197 \beta_{5} + 181 \beta_{4} + 16 \beta_{3} + 168 \beta_{1} + 889\)
\(\nu^{7}\)\(=\)\((\)\(-1829 \beta_{15} - 1829 \beta_{14} - 361 \beta_{12} + 722 \beta_{11} + 2819 \beta_{10} + 2819 \beta_{9} - 8955 \beta_{8}\)\()/6\)
\(\nu^{8}\)\(=\)\(2324 \beta_{7} + 2768 \beta_{6} + 3220 \beta_{5} + 2772 \beta_{3} - 11169 \beta_{2} - 2768 \beta_{1} - 11169\)
\(\nu^{9}\)\(=\)\((\)\(49154 \beta_{15} - 24577 \beta_{14} - 126291 \beta_{13} - 8663 \beta_{12} - 8663 \beta_{11} - 71666 \beta_{10} + 35833 \beta_{9}\)\()/6\)
\(\nu^{10}\)\(=\)\(-30205 \beta_{7} - 42000 \beta_{6} - 8672 \beta_{5} - 30205 \beta_{4} - 47549 \beta_{3} + 145327 \beta_{2}\)
\(\nu^{11}\)\(=\)\((\)\(-334163 \beta_{15} + 668326 \beta_{14} + 1773045 \beta_{13} + 302198 \beta_{12} - 151099 \beta_{11} + 463361 \beta_{10} - 926722 \beta_{9} + 1773045 \beta_{8}\)\()/6\)
\(\nu^{12}\)\(=\)\(-543466 \beta_{5} + 398762 \beta_{4} + 144704 \beta_{3} + 613688 \beta_{1} + 1932699\)
\(\nu^{13}\)\(=\)\((\)\(-4575547 \beta_{15} - 4575547 \beta_{14} - 2367377 \beta_{12} + 4734754 \beta_{11} + 6099895 \beta_{10} + 6099895 \beta_{9} - 24798585 \beta_{8}\)\()/6\)
\(\nu^{14}\)\(=\)\(5337721 \beta_{7} + 8788248 \beta_{6} + 9818329 \beta_{5} + 7578025 \beta_{3} - 26067265 \beta_{2} - 8788248 \beta_{1} - 26067265\)
\(\nu^{15}\)\(=\)\((\)\(125854618 \beta_{15} - 62927309 \beta_{14} - 345889539 \beta_{13} - 35270329 \beta_{12} - 35270329 \beta_{11} - 163023622 \beta_{10} + 81511811 \beta_{9}\)\()/6\)

Character values

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

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(1 + \beta_{2}\) \(1\) \(-1\) \(1\)

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
−1.86197 + 3.22503i
0.181830 0.314939i
−1.09560 + 1.89764i
1.51646 2.62659i
1.09560 1.89764i
−1.51646 + 2.62659i
1.86197 3.22503i
−0.181830 + 0.314939i
0.181830 + 0.314939i
−1.86197 3.22503i
1.51646 + 2.62659i
−1.09560 1.89764i
−1.51646 2.62659i
1.09560 + 1.89764i
−0.181830 0.314939i
1.86197 + 3.22503i
0 0 0 −1.38255 + 2.39464i 0 3.26278i 0 0 0
559.2 0 0 0 −1.38255 + 2.39464i 0 3.26278i 0 0 0
559.3 0 0 0 −0.767178 + 1.32879i 0 2.31392i 0 0 0
559.4 0 0 0 −0.767178 + 1.32879i 0 2.31392i 0 0 0
559.5 0 0 0 0.767178 1.32879i 0 2.31392i 0 0 0
559.6 0 0 0 0.767178 1.32879i 0 2.31392i 0 0 0
559.7 0 0 0 1.38255 2.39464i 0 3.26278i 0 0 0
559.8 0 0 0 1.38255 2.39464i 0 3.26278i 0 0 0
1855.1 0 0 0 −1.38255 2.39464i 0 3.26278i 0 0 0
1855.2 0 0 0 −1.38255 2.39464i 0 3.26278i 0 0 0
1855.3 0 0 0 −0.767178 1.32879i 0 2.31392i 0 0 0
1855.4 0 0 0 −0.767178 1.32879i 0 2.31392i 0 0 0
1855.5 0 0 0 0.767178 + 1.32879i 0 2.31392i 0 0 0
1855.6 0 0 0 0.767178 + 1.32879i 0 2.31392i 0 0 0
1855.7 0 0 0 1.38255 + 2.39464i 0 3.26278i 0 0 0
1855.8 0 0 0 1.38255 + 2.39464i 0 3.26278i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1855.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
19.d odd 6 1 inner
57.f even 6 1 inner
76.f even 6 1 inner
228.n odd 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.t 16
3.b odd 2 1 inner 2736.2.bm.t 16
4.b odd 2 1 inner 2736.2.bm.t 16
12.b even 2 1 inner 2736.2.bm.t 16
19.d odd 6 1 inner 2736.2.bm.t 16
57.f even 6 1 inner 2736.2.bm.t 16
76.f even 6 1 inner 2736.2.bm.t 16
228.n odd 6 1 inner 2736.2.bm.t 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2736.2.bm.t 16 1.a even 1 1 trivial
2736.2.bm.t 16 3.b odd 2 1 inner
2736.2.bm.t 16 4.b odd 2 1 inner
2736.2.bm.t 16 12.b even 2 1 inner
2736.2.bm.t 16 19.d odd 6 1 inner
2736.2.bm.t 16 57.f even 6 1 inner
2736.2.bm.t 16 76.f even 6 1 inner
2736.2.bm.t 16 228.n odd 6 1 inner

Hecke kernels

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

\( T_{5}^{8} + 10 T_{5}^{6} + 82 T_{5}^{4} + 180 T_{5}^{2} + 324 \)
\( T_{7}^{4} + 16 T_{7}^{2} + 57 \)
\( T_{11}^{4} + 34 T_{11}^{2} + 114 \)
\( T_{23}^{8} - 94 T_{23}^{6} + 7810 T_{23}^{4} - 96444 T_{23}^{2} + 1052676 \)
\( T_{31}^{4} - 40 T_{31}^{2} + 57 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( T^{16} \)
$5$ \( ( 324 + 180 T^{2} + 82 T^{4} + 10 T^{6} + T^{8} )^{2} \)
$7$ \( ( 57 + 16 T^{2} + T^{4} )^{4} \)
$11$ \( ( 114 + 34 T^{2} + T^{4} )^{4} \)
$13$ \( ( 441 - 21 T^{2} + T^{4} )^{4} \)
$17$ \( ( 419904 + 33696 T^{2} + 2056 T^{4} + 52 T^{6} + T^{8} )^{2} \)
$19$ \( ( 130321 + 8664 T^{2} + 418 T^{4} + 24 T^{6} + T^{8} )^{2} \)
$23$ \( ( 1052676 - 96444 T^{2} + 7810 T^{4} - 94 T^{6} + T^{8} )^{2} \)
$29$ \( ( 82944 - 20736 T^{2} + 4896 T^{4} - 72 T^{6} + T^{8} )^{2} \)
$31$ \( ( 57 - 40 T^{2} + T^{4} )^{4} \)
$37$ \( ( 3 + T^{2} )^{8} \)
$41$ \( ( 419904 - 38880 T^{2} + 2952 T^{4} - 60 T^{6} + T^{8} )^{2} \)
$43$ \( ( 263169 - 61560 T^{2} + 13887 T^{4} - 120 T^{6} + T^{8} )^{2} \)
$47$ \( ( 207936 - 56544 T^{2} + 14920 T^{4} - 124 T^{6} + T^{8} )^{2} \)
$53$ \( ( 42224004 - 1052676 T^{2} + 19746 T^{4} - 162 T^{6} + T^{8} )^{2} \)
$59$ \( ( 1052676 + 104652 T^{2} + 9378 T^{4} + 102 T^{6} + T^{8} )^{2} \)
$61$ \( ( 9 - 12 T + 19 T^{2} + 4 T^{3} + T^{4} )^{4} \)
$67$ \( ( 21316689 + 627912 T^{2} + 13879 T^{4} + 136 T^{6} + T^{8} )^{2} \)
$71$ \( ( 1364268096 + 14626656 T^{2} + 119880 T^{4} + 396 T^{6} + T^{8} )^{2} \)
$73$ \( ( 10609 + 618 T + 139 T^{2} - 6 T^{3} + T^{4} )^{4} \)
$79$ \( ( 423412929 + 6255408 T^{2} + 71839 T^{4} + 304 T^{6} + T^{8} )^{2} \)
$83$ \( ( 456 + 124 T^{2} + T^{4} )^{4} \)
$89$ \( ( 2125764 - 113724 T^{2} + 4626 T^{4} - 78 T^{6} + T^{8} )^{2} \)
$97$ \( ( 26244 - 2916 T - 54 T^{2} + 18 T^{3} + T^{4} )^{4} \)
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