Properties

Label 27.11.b.c
Level 27
Weight 11
Character orbit 27.b
Analytic conductor 17.155
Analytic rank 0
Dimension 4
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 27 = 3^{3} \)
Weight: \( k \) = \( 11 \)
Character orbit: \([\chi]\) = 27.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(17.1546458222\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{9} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{2} \) \( + ( -137 - \beta_{3} ) q^{4} \) \( + ( -43 \beta_{1} - \beta_{2} ) q^{5} \) \( + ( 5129 + 44 \beta_{3} ) q^{7} \) \( + ( -570 \beta_{1} + 4 \beta_{2} ) q^{8} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{2} \) \( + ( -137 - \beta_{3} ) q^{4} \) \( + ( -43 \beta_{1} - \beta_{2} ) q^{5} \) \( + ( 5129 + 44 \beta_{3} ) q^{7} \) \( + ( -570 \beta_{1} + 4 \beta_{2} ) q^{8} \) \( + ( -49950 - 254 \beta_{3} ) q^{10} \) \( + ( -4541 \beta_{1} + 25 \beta_{2} ) q^{11} \) \( + ( -65605 + 548 \beta_{3} ) q^{13} \) \( + ( -19077 \beta_{1} - 176 \beta_{2} ) q^{14} \) \( + ( -801950 - 750 \beta_{3} ) q^{16} \) \( + ( -17015 \beta_{1} + 299 \beta_{2} ) q^{17} \) \( + ( -2058379 + 1176 \beta_{3} ) q^{19} \) \( + ( 86436 \beta_{1} - 8 \beta_{2} ) q^{20} \) \( + ( -5271426 + 734 \beta_{3} ) q^{22} \) \( + ( 190053 \beta_{1} + 367 \beta_{2} ) q^{23} \) \( + ( -11800895 - 3280 \beta_{3} ) q^{25} \) \( + ( -108111 \beta_{1} - 2192 \beta_{2} ) q^{26} \) \( + ( -16901053 - 11157 \beta_{3} ) q^{28} \) \( + ( 272954 \beta_{1} - 498 \beta_{2} ) q^{29} \) \( + ( -11748730 + 10544 \beta_{3} ) q^{31} \) \( + ( 456020 \beta_{1} + 7096 \beta_{2} ) q^{32} \) \( + ( -19746342 + 46074 \beta_{3} ) q^{34} \) \( + ( -3764527 \beta_{1} + 1251 \beta_{2} ) q^{35} \) \( + ( 27021611 + 25692 \beta_{3} ) q^{37} \) \( + ( 1685587 \beta_{1} - 4704 \beta_{2} ) q^{38} \) \( + ( 49203180 - 175348 \beta_{3} ) q^{40} \) \( + ( -1710170 \beta_{1} - 32622 \beta_{2} ) q^{41} \) \( + ( 19562750 - 130576 \beta_{3} ) q^{43} \) \( + ( 388764 \beta_{1} + 22664 \beta_{2} ) q^{44} \) \( + ( 220661442 + 267490 \beta_{3} ) q^{46} \) \( + ( 4810635 \beta_{1} + 46913 \beta_{2} ) q^{47} \) \( + ( 456560112 + 451352 \beta_{3} ) q^{49} \) \( + ( 12840655 \beta_{1} + 13120 \beta_{2} ) q^{50} \) \( + ( -192755575 - 9471 \beta_{3} ) q^{52} \) \( + ( -4070082 \beta_{1} - 43206 \beta_{2} ) q^{53} \) \( + ( 258643800 - 1344464 \beta_{3} ) q^{55} \) \( + ( 902974 \beta_{1} - 135596 \beta_{2} ) q^{56} \) \( + ( 316886148 + 167876 \beta_{3} ) q^{58} \) \( + ( -19660783 \beta_{1} + 111299 \beta_{2} ) q^{59} \) \( + ( -805597381 + 1061660 \beta_{3} ) q^{61} \) \( + ( 8406282 \beta_{1} - 42176 \beta_{2} ) q^{62} \) \( + ( -291565988 + 1185276 \beta_{3} ) q^{64} \) \( + ( -41317645 \beta_{1} + 145065 \beta_{2} ) q^{65} \) \( + ( 493071629 - 1570768 \beta_{3} ) q^{67} \) \( + ( -12282476 \beta_{1} + 121880 \beta_{2} ) q^{68} \) \( + ( -4370582070 - 3500566 \beta_{3} ) q^{70} \) \( + ( 52068820 \beta_{1} - 20644 \beta_{2} ) q^{71} \) \( + ( 1019399735 + 4099752 \beta_{3} ) q^{73} \) \( + ( -35165975 \beta_{1} - 102768 \beta_{2} ) q^{74} \) \( + ( -150940597 + 1897267 \beta_{3} ) q^{76} \) \( + ( -13023257 \beta_{1} - 1019691 \beta_{2} ) q^{77} \) \( + ( -2600695063 + 3878228 \beta_{3} ) q^{79} \) \( + ( 94892600 \beta_{1} + 693200 \beta_{2} ) q^{80} \) \( + ( -1986388164 - 8593412 \beta_{3} ) q^{82} \) \( + ( 92538182 \beta_{1} + 503826 \beta_{2} ) q^{83} \) \( + ( 4956283080 - 6606768 \beta_{3} ) q^{85} \) \( + ( 21829842 \beta_{1} + 522304 \beta_{2} ) q^{86} \) \( + ( -4945973292 + 5922484 \beta_{3} ) q^{88} \) \( + ( 63090053 \beta_{1} - 597281 \beta_{2} ) q^{89} \) \( + ( 8540224195 - 75928 \beta_{3} ) q^{91} \) \( + ( -110841500 \beta_{1} - 694152 \beta_{2} ) q^{92} \) \( + ( 5586413886 + 14709278 \beta_{3} ) q^{94} \) \( + ( -6210623 \beta_{1} + 2228899 \beta_{2} ) q^{95} \) \( + ( 3242449367 - 14713960 \beta_{3} ) q^{97} \) \( + ( -599638696 \beta_{1} - 1805408 \beta_{2} ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 548q^{4} \) \(\mathstrut +\mathstrut 20516q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 548q^{4} \) \(\mathstrut +\mathstrut 20516q^{7} \) \(\mathstrut -\mathstrut 199800q^{10} \) \(\mathstrut -\mathstrut 262420q^{13} \) \(\mathstrut -\mathstrut 3207800q^{16} \) \(\mathstrut -\mathstrut 8233516q^{19} \) \(\mathstrut -\mathstrut 21085704q^{22} \) \(\mathstrut -\mathstrut 47203580q^{25} \) \(\mathstrut -\mathstrut 67604212q^{28} \) \(\mathstrut -\mathstrut 46994920q^{31} \) \(\mathstrut -\mathstrut 78985368q^{34} \) \(\mathstrut +\mathstrut 108086444q^{37} \) \(\mathstrut +\mathstrut 196812720q^{40} \) \(\mathstrut +\mathstrut 78251000q^{43} \) \(\mathstrut +\mathstrut 882645768q^{46} \) \(\mathstrut +\mathstrut 1826240448q^{49} \) \(\mathstrut -\mathstrut 771022300q^{52} \) \(\mathstrut +\mathstrut 1034575200q^{55} \) \(\mathstrut +\mathstrut 1267544592q^{58} \) \(\mathstrut -\mathstrut 3222389524q^{61} \) \(\mathstrut -\mathstrut 1166263952q^{64} \) \(\mathstrut +\mathstrut 1972286516q^{67} \) \(\mathstrut -\mathstrut 17482328280q^{70} \) \(\mathstrut +\mathstrut 4077598940q^{73} \) \(\mathstrut -\mathstrut 603762388q^{76} \) \(\mathstrut -\mathstrut 10402780252q^{79} \) \(\mathstrut -\mathstrut 7945552656q^{82} \) \(\mathstrut +\mathstrut 19825132320q^{85} \) \(\mathstrut -\mathstrut 19783893168q^{88} \) \(\mathstrut +\mathstrut 34160896780q^{91} \) \(\mathstrut +\mathstrut 22345655544q^{94} \) \(\mathstrut +\mathstrut 12969797468q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut +\mathstrut \) \(188\) \(x^{2}\mathstrut +\mathstrut \) \(756\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 170 \nu \)\()/8\)
\(\beta_{2}\)\(=\)\((\)\( 49 \nu^{3} + 12218 \nu \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( 27 \nu^{2} + 2538 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2}\mathstrut -\mathstrut \) \(49\) \(\beta_{1}\)\()/486\)
\(\nu^{2}\)\(=\)\((\)\(4\) \(\beta_{3}\mathstrut -\mathstrut \) \(2538\)\()/27\)
\(\nu^{3}\)\(=\)\((\)\(-\)\(85\) \(\beta_{2}\mathstrut +\mathstrut \) \(6109\) \(\beta_{1}\)\()/243\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
2.02760i
13.5606i
13.5606i
2.02760i
42.0446i 0 −743.750 4853.52i 0 31826.0 11783.0i 0 −204064.
26.2 23.5425i 0 469.750 4424.52i 0 −21568.0 35166.6i 0 104164.
26.3 23.5425i 0 469.750 4424.52i 0 −21568.0 35166.6i 0 104164.
26.4 42.0446i 0 −743.750 4853.52i 0 31826.0 11783.0i 0 −204064.
\(n\): 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

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{4} \) \(\mathstrut +\mathstrut 2322 T_{2}^{2} \) \(\mathstrut +\mathstrut 979776 \) acting on \(S_{11}^{\mathrm{new}}(27, [\chi])\).