Properties

Label 27.10.a.d
Level 27
Weight 10
Character orbit 27.a
Self dual Yes
Analytic conductor 13.906
Analytic rank 0
Dimension 4
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 27 = 3^{3} \)
Weight: \( k \) = \( 10 \)
Character orbit: \([\chi]\) = 27.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(13.9059675764\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{8} \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \( + ( 559 + \beta_{3} ) q^{4} \) \( + ( -10 \beta_{1} - \beta_{2} ) q^{5} \) \( + ( 2963 + 4 \beta_{3} ) q^{7} \) \( + ( 818 \beta_{1} + 10 \beta_{2} ) q^{8} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{2} \) \( + ( 559 + \beta_{3} ) q^{4} \) \( + ( -10 \beta_{1} - \beta_{2} ) q^{5} \) \( + ( 2963 + 4 \beta_{3} ) q^{7} \) \( + ( 818 \beta_{1} + 10 \beta_{2} ) q^{8} \) \( + ( -10440 - 40 \beta_{3} ) q^{10} \) \( + ( 802 \beta_{1} - 35 \beta_{2} ) q^{11} \) \( + ( 44921 - 44 \beta_{3} ) q^{13} \) \( + ( 6047 \beta_{1} + 40 \beta_{2} ) q^{14} \) \( + ( 587170 + 606 \beta_{3} ) q^{16} \) \( + ( -10062 \beta_{1} + 5 \beta_{2} ) q^{17} \) \( + ( 252857 - 816 \beta_{3} ) q^{19} \) \( + ( -36160 \beta_{1} + 112 \beta_{2} ) q^{20} \) \( + ( 868392 - 248 \beta_{3} ) q^{22} \) \( + ( 3050 \beta_{1} - 335 \beta_{2} ) q^{23} \) \( + ( 638515 - 1640 \beta_{3} ) q^{25} \) \( + ( 10997 \beta_{1} - 440 \beta_{2} ) q^{26} \) \( + ( 4948481 + 5199 \beta_{3} ) q^{28} \) \( + ( -94832 \beta_{1} + 1320 \beta_{2} ) q^{29} \) \( + ( 222284 + 3568 \beta_{3} ) q^{31} \) \( + ( 635580 \beta_{1} + 940 \beta_{2} ) q^{32} \) \( + ( -10777752 - 9912 \beta_{3} ) q^{34} \) \( + ( -151910 \beta_{1} - 279 \beta_{2} ) q^{35} \) \( + ( -951289 - 3300 \beta_{3} ) q^{37} \) \( + ( -376279 \beta_{1} - 8160 \beta_{2} ) q^{38} \) \( + ( -33412320 - 12320 \beta_{3} ) q^{40} \) \( + ( -412192 \beta_{1} + 1200 \beta_{2} ) q^{41} \) \( + ( 2896256 + 33208 \beta_{3} ) q^{43} \) \( + ( 266560 \beta_{1} + 15440 \beta_{2} ) q^{44} \) \( + ( 3357000 - 7000 \beta_{3} ) q^{46} \) \( + ( -965582 \beta_{1} + 9605 \beta_{2} ) q^{47} \) \( + ( -18405582 + 23704 \beta_{3} ) q^{49} \) \( + ( -625925 \beta_{1} - 16400 \beta_{2} ) q^{50} \) \( + ( -11102965 + 20325 \beta_{3} ) q^{52} \) \( + ( -1036332 \beta_{1} - 53790 \beta_{2} ) q^{53} \) \( + ( 78680520 - 103480 \beta_{3} ) q^{55} \) \( + ( 5860846 \beta_{1} + 31510 \beta_{2} ) q^{56} \) \( + ( -101921472 - 55232 \beta_{3} ) q^{58} \) \( + ( 388466 \beta_{1} + 56165 \beta_{2} ) q^{59} \) \( + ( 54970643 - 81836 \beta_{3} ) q^{61} \) \( + ( 2973212 \beta_{1} + 35680 \beta_{2} ) q^{62} \) \( + ( 379821340 + 353508 \beta_{3} ) q^{64} \) \( + ( 895870 \beta_{1} - 74445 \beta_{2} ) q^{65} \) \( + ( 97640309 + 129184 \beta_{3} ) q^{67} \) \( + ( -13268160 \beta_{1} - 101680 \beta_{2} ) q^{68} \) \( + ( -162620280 - 160280 \beta_{3} ) q^{70} \) \( + ( 5495364 \beta_{1} + 134330 \beta_{2} ) q^{71} \) \( + ( 299487377 - 156456 \beta_{3} ) q^{73} \) \( + ( -3495589 \beta_{1} - 33000 \beta_{2} ) q^{74} \) \( + ( -530254393 - 203287 \beta_{3} ) q^{76} \) \( + ( 1649294 \beta_{1} + 36315 \beta_{2} ) q^{77} \) \( + ( 12453581 + 25804 \beta_{3} ) q^{79} \) \( + ( -24397120 \beta_{1} - 180544 \beta_{2} ) q^{80} \) \( + ( -441781632 - 376192 \beta_{3} ) q^{82} \) \( + ( -1514636 \beta_{1} + 124530 \beta_{2} ) q^{83} \) \( + ( 92611080 + 412680 \beta_{3} ) q^{85} \) \( + ( 28499624 \beta_{1} + 332080 \beta_{2} ) q^{86} \) \( + ( -163299744 + 856736 \beta_{3} ) q^{88} \) \( + ( 24107850 \beta_{1} - 212255 \beta_{2} ) q^{89} \) \( + ( -11754293 + 49312 \beta_{3} ) q^{91} \) \( + ( -3601600 \beta_{1} + 101520 \beta_{2} ) q^{92} \) \( + ( -1036731672 - 677432 \beta_{3} ) q^{94} \) \( + ( 22416550 \beta_{1} - 800393 \beta_{2} ) q^{95} \) \( + ( 483933629 + 532528 \beta_{3} ) q^{97} \) \( + ( -129798 \beta_{1} + 237040 \beta_{2} ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut +\mathstrut 2236q^{4} \) \(\mathstrut +\mathstrut 11852q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut +\mathstrut 2236q^{4} \) \(\mathstrut +\mathstrut 11852q^{7} \) \(\mathstrut -\mathstrut 41760q^{10} \) \(\mathstrut +\mathstrut 179684q^{13} \) \(\mathstrut +\mathstrut 2348680q^{16} \) \(\mathstrut +\mathstrut 1011428q^{19} \) \(\mathstrut +\mathstrut 3473568q^{22} \) \(\mathstrut +\mathstrut 2554060q^{25} \) \(\mathstrut +\mathstrut 19793924q^{28} \) \(\mathstrut +\mathstrut 889136q^{31} \) \(\mathstrut -\mathstrut 43111008q^{34} \) \(\mathstrut -\mathstrut 3805156q^{37} \) \(\mathstrut -\mathstrut 133649280q^{40} \) \(\mathstrut +\mathstrut 11585024q^{43} \) \(\mathstrut +\mathstrut 13428000q^{46} \) \(\mathstrut -\mathstrut 73622328q^{49} \) \(\mathstrut -\mathstrut 44411860q^{52} \) \(\mathstrut +\mathstrut 314722080q^{55} \) \(\mathstrut -\mathstrut 407685888q^{58} \) \(\mathstrut +\mathstrut 219882572q^{61} \) \(\mathstrut +\mathstrut 1519285360q^{64} \) \(\mathstrut +\mathstrut 390561236q^{67} \) \(\mathstrut -\mathstrut 650481120q^{70} \) \(\mathstrut +\mathstrut 1197949508q^{73} \) \(\mathstrut -\mathstrut 2121017572q^{76} \) \(\mathstrut +\mathstrut 49814324q^{79} \) \(\mathstrut -\mathstrut 1767126528q^{82} \) \(\mathstrut +\mathstrut 370444320q^{85} \) \(\mathstrut -\mathstrut 653198976q^{88} \) \(\mathstrut -\mathstrut 47017172q^{91} \) \(\mathstrut -\mathstrut 4146926688q^{94} \) \(\mathstrut +\mathstrut 1935734516q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut -\mathstrut \) \(83\) \(x^{2}\mathstrut +\mathstrut \) \(1440\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} - 35 \nu \)\()/4\)
\(\beta_{2}\)\(=\)\((\)\( -21 \nu^{3} + 1383 \nu \)\()/2\)
\(\beta_{3}\)\(=\)\( 54 \nu^{2} - 2241 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2}\mathstrut +\mathstrut \) \(42\) \(\beta_{1}\)\()/324\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(2241\)\()/54\)
\(\nu^{3}\)\(=\)\((\)\(35\) \(\beta_{2}\mathstrut +\mathstrut \) \(2766\) \(\beta_{1}\)\()/324\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−7.63546
4.96988
−4.96988
7.63546
−44.4771 0 1466.22 1050.62 0 6591.86 −42440.8 0 −46728.6
1.2 −12.7978 0 −348.216 −2019.77 0 −665.864 11008.9 0 25848.6
1.3 12.7978 0 −348.216 2019.77 0 −665.864 −11008.9 0 25848.6
1.4 44.4771 0 1466.22 −1050.62 0 6591.86 42440.8 0 −46728.6
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{4} \) \(\mathstrut -\mathstrut 2142 T_{2}^{2} \) \(\mathstrut +\mathstrut 324000 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(27))\).