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

Related objects

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: Complex embeddings Normalized embeddings Satake parameters Satake angles

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 $$S_{11}^{\mathrm{new}}(27, [\chi])$$.