# Properties

 Label 27.4.c.a Level 27 Weight 4 Character orbit 27.c Analytic conductor 1.593 Analytic rank 0 Dimension 4 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$27 = 3^{3}$$ Weight: $$k$$ = $$4$$ Character orbit: $$[\chi]$$ = 27.c (of order $$3$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$1.59305157016$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( \beta_{1} + \beta_{3} ) q^{2}$$ $$+ ( -4 + \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{4}$$ $$+ ( 7 - 8 \beta_{1} - \beta_{2} + \beta_{3} ) q^{5}$$ $$+ ( -2 \beta_{1} - 3 \beta_{3} ) q^{7}$$ $$+ ( -16 + \beta_{2} ) q^{8}$$ $$+O(q^{10})$$ $$q$$ $$+ ( \beta_{1} + \beta_{3} ) q^{2}$$ $$+ ( -4 + \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{4}$$ $$+ ( 7 - 8 \beta_{1} - \beta_{2} + \beta_{3} ) q^{5}$$ $$+ ( -2 \beta_{1} - 3 \beta_{3} ) q^{7}$$ $$+ ( -16 + \beta_{2} ) q^{8}$$ $$+ ( 6 + 6 \beta_{2} ) q^{10}$$ $$+ ( 37 \beta_{1} - 8 \beta_{3} ) q^{11}$$ $$+ ( 13 + 2 \beta_{1} + 15 \beta_{2} - 15 \beta_{3} ) q^{13}$$ $$+ ( 34 - 26 \beta_{1} + 8 \beta_{2} - 8 \beta_{3} ) q^{14}$$ $$+ ( -\beta_{1} + 9 \beta_{3} ) q^{16}$$ $$+ ( -54 - 9 \beta_{2} ) q^{17}$$ $$+ ( -52 - 27 \beta_{2} ) q^{19}$$ $$+ ( -16 \beta_{1} + 20 \beta_{3} ) q^{20}$$ $$+ ( 6 - 27 \beta_{1} - 21 \beta_{2} + 21 \beta_{3} ) q^{22}$$ $$+ ( 7 - 26 \beta_{1} - 19 \beta_{2} + 19 \beta_{3} ) q^{23}$$ $$+ ( 53 \beta_{1} + 15 \beta_{3} ) q^{25}$$ $$+ ( 146 + 28 \beta_{2} ) q^{26}$$ $$+ ( 92 + 18 \beta_{2} ) q^{28}$$ $$+ ( -26 \beta_{1} + \beta_{3} ) q^{29}$$ $$+ ( -23 + 20 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{31}$$ $$+ ( -216 + 207 \beta_{1} - 9 \beta_{2} + 9 \beta_{3} ) q^{32}$$ $$+ ( -117 \beta_{1} - 63 \beta_{3} ) q^{34}$$ $$+ ( -11 - 19 \beta_{2} ) q^{35}$$ $$+ ( 2 + 54 \beta_{2} ) q^{37}$$ $$+ ( -241 \beta_{1} - 79 \beta_{3} ) q^{38}$$ $$+ ( -120 + 144 \beta_{1} + 24 \beta_{2} - 24 \beta_{3} ) q^{40}$$ $$+ ( 115 - 17 \beta_{1} + 98 \beta_{2} - 98 \beta_{3} ) q^{41}$$ $$+ ( -47 \beta_{1} + 6 \beta_{3} ) q^{43}$$ $$+ ( 76 - 79 \beta_{2} ) q^{44}$$ $$+ ( -138 - 12 \beta_{2} ) q^{46}$$ $$+ ( 154 \beta_{1} + 91 \beta_{3} ) q^{47}$$ $$+ ( 246 - 267 \beta_{1} - 21 \beta_{2} + 21 \beta_{3} ) q^{49}$$ $$+ ( -256 + 173 \beta_{1} - 83 \beta_{2} + 83 \beta_{3} ) q^{50}$$ $$+ ( 358 \beta_{1} + 54 \beta_{3} ) q^{52}$$ $$+ ( 54 + 162 \beta_{2} ) q^{53}$$ $$+ ( 267 - 93 \beta_{2} ) q^{55}$$ $$+ ( 10 \beta_{1} + 46 \beta_{3} ) q^{56}$$ $$+ ( 42 - 18 \beta_{1} + 24 \beta_{2} - 24 \beta_{3} ) q^{58}$$ $$+ ( 331 - 467 \beta_{1} - 136 \beta_{2} + 136 \beta_{3} ) q^{59}$$ $$+ ( -272 \beta_{1} + 105 \beta_{3} ) q^{61}$$ $$+ ( -70 - 26 \beta_{2} ) q^{62}$$ $$+ ( -440 - 153 \beta_{2} ) q^{64}$$ $$+ ( 136 \beta_{1} - 107 \beta_{3} ) q^{65}$$ $$+ ( -527 + 461 \beta_{1} - 66 \beta_{2} + 66 \beta_{3} ) q^{67}$$ $$+ ( 432 - 261 \beta_{1} + 171 \beta_{2} - 171 \beta_{3} ) q^{68}$$ $$+ ( -144 \beta_{1} - 30 \beta_{3} ) q^{70}$$ $$+ ( -756 - 144 \beta_{2} ) q^{71}$$ $$+ ( -106 + 243 \beta_{2} ) q^{73}$$ $$+ ( 380 \beta_{1} + 56 \beta_{3} ) q^{74}$$ $$+ ( 856 - 673 \beta_{1} + 183 \beta_{2} - 183 \beta_{3} ) q^{76}$$ $$+ ( -47 + 118 \beta_{1} + 71 \beta_{2} - 71 \beta_{3} ) q^{77}$$ $$+ ( 556 \beta_{1} - 309 \beta_{3} ) q^{79}$$ $$+ ( -16 + 64 \beta_{2} ) q^{80}$$ $$+ ( 1014 + 213 \beta_{2} ) q^{82}$$ $$+ ( 460 \beta_{1} - 107 \beta_{3} ) q^{83}$$ $$+ ( -306 + 288 \beta_{1} - 18 \beta_{2} + 18 \beta_{3} ) q^{85}$$ $$+ ( 34 + \beta_{1} + 35 \beta_{2} - 35 \beta_{3} ) q^{86}$$ $$+ ( -693 \beta_{1} + 165 \beta_{3} ) q^{88}$$ $$+ ( 162 - 72 \beta_{2} ) q^{89}$$ $$+ ( -425 - 69 \beta_{2} ) q^{91}$$ $$+ ( -430 \beta_{1} + 2 \beta_{3} ) q^{92}$$ $$+ ( -1218 + 882 \beta_{1} - 336 \beta_{2} + 336 \beta_{3} ) q^{94}$$ $$+ ( -148 - 16 \beta_{1} - 164 \beta_{2} + 164 \beta_{3} ) q^{95}$$ $$+ ( -317 \beta_{1} - 102 \beta_{3} ) q^{97}$$ $$+ ( 324 + 225 \beta_{2} ) q^{98}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q$$ $$\mathstrut +\mathstrut 3q^{2}$$ $$\mathstrut -\mathstrut 5q^{4}$$ $$\mathstrut +\mathstrut 15q^{5}$$ $$\mathstrut -\mathstrut 7q^{7}$$ $$\mathstrut -\mathstrut 66q^{8}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$4q$$ $$\mathstrut +\mathstrut 3q^{2}$$ $$\mathstrut -\mathstrut 5q^{4}$$ $$\mathstrut +\mathstrut 15q^{5}$$ $$\mathstrut -\mathstrut 7q^{7}$$ $$\mathstrut -\mathstrut 66q^{8}$$ $$\mathstrut +\mathstrut 12q^{10}$$ $$\mathstrut +\mathstrut 66q^{11}$$ $$\mathstrut +\mathstrut 11q^{13}$$ $$\mathstrut +\mathstrut 60q^{14}$$ $$\mathstrut +\mathstrut 7q^{16}$$ $$\mathstrut -\mathstrut 198q^{17}$$ $$\mathstrut -\mathstrut 154q^{19}$$ $$\mathstrut -\mathstrut 12q^{20}$$ $$\mathstrut +\mathstrut 33q^{22}$$ $$\mathstrut +\mathstrut 33q^{23}$$ $$\mathstrut +\mathstrut 121q^{25}$$ $$\mathstrut +\mathstrut 528q^{26}$$ $$\mathstrut +\mathstrut 332q^{28}$$ $$\mathstrut -\mathstrut 51q^{29}$$ $$\mathstrut -\mathstrut 43q^{31}$$ $$\mathstrut -\mathstrut 423q^{32}$$ $$\mathstrut -\mathstrut 297q^{34}$$ $$\mathstrut -\mathstrut 6q^{35}$$ $$\mathstrut -\mathstrut 100q^{37}$$ $$\mathstrut -\mathstrut 561q^{38}$$ $$\mathstrut -\mathstrut 264q^{40}$$ $$\mathstrut +\mathstrut 132q^{41}$$ $$\mathstrut -\mathstrut 88q^{43}$$ $$\mathstrut +\mathstrut 462q^{44}$$ $$\mathstrut -\mathstrut 528q^{46}$$ $$\mathstrut +\mathstrut 399q^{47}$$ $$\mathstrut +\mathstrut 513q^{49}$$ $$\mathstrut -\mathstrut 429q^{50}$$ $$\mathstrut +\mathstrut 770q^{52}$$ $$\mathstrut -\mathstrut 108q^{53}$$ $$\mathstrut +\mathstrut 1254q^{55}$$ $$\mathstrut +\mathstrut 66q^{56}$$ $$\mathstrut +\mathstrut 60q^{58}$$ $$\mathstrut +\mathstrut 798q^{59}$$ $$\mathstrut -\mathstrut 439q^{61}$$ $$\mathstrut -\mathstrut 228q^{62}$$ $$\mathstrut -\mathstrut 1454q^{64}$$ $$\mathstrut +\mathstrut 165q^{65}$$ $$\mathstrut -\mathstrut 988q^{67}$$ $$\mathstrut +\mathstrut 693q^{68}$$ $$\mathstrut -\mathstrut 318q^{70}$$ $$\mathstrut -\mathstrut 2736q^{71}$$ $$\mathstrut -\mathstrut 910q^{73}$$ $$\mathstrut +\mathstrut 816q^{74}$$ $$\mathstrut +\mathstrut 1529q^{76}$$ $$\mathstrut -\mathstrut 165q^{77}$$ $$\mathstrut +\mathstrut 803q^{79}$$ $$\mathstrut -\mathstrut 192q^{80}$$ $$\mathstrut +\mathstrut 3630q^{82}$$ $$\mathstrut +\mathstrut 813q^{83}$$ $$\mathstrut -\mathstrut 594q^{85}$$ $$\mathstrut +\mathstrut 33q^{86}$$ $$\mathstrut -\mathstrut 1221q^{88}$$ $$\mathstrut +\mathstrut 792q^{89}$$ $$\mathstrut -\mathstrut 1562q^{91}$$ $$\mathstrut -\mathstrut 858q^{92}$$ $$\mathstrut -\mathstrut 2100q^{94}$$ $$\mathstrut -\mathstrut 132q^{95}$$ $$\mathstrut -\mathstrut 736q^{97}$$ $$\mathstrut +\mathstrut 846q^{98}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4}\mathstrut -\mathstrut$$ $$x^{3}\mathstrut -\mathstrut$$ $$2$$ $$x^{2}\mathstrut -\mathstrut$$ $$3$$ $$x\mathstrut +\mathstrut$$ $$9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 2 \nu^{2} - 2 \nu - 3$$$$)/6$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + \nu^{2} + 5 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{3} + \nu^{2} + 2 \nu - 9$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3}\mathstrut +\mathstrut$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$2$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-$$$$\beta_{3}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$8$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$1$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$4$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$2$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$11$$$$)/3$$

## Character Values

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

 $$n$$ $$2$$ $$\chi(n)$$ $$-1 + \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
10.1
 −1.18614 − 1.26217i 1.68614 + 0.396143i −1.18614 + 1.26217i 1.68614 − 0.396143i
−0.686141 1.18843i 0 3.05842 5.29734i 5.18614 8.98266i 0 2.55842 + 4.43132i −19.3723 0 −14.2337
10.2 2.18614 + 3.78651i 0 −5.55842 + 9.62747i 2.31386 4.00772i 0 −6.05842 10.4935i −13.6277 0 20.2337
19.1 −0.686141 + 1.18843i 0 3.05842 + 5.29734i 5.18614 + 8.98266i 0 2.55842 4.43132i −19.3723 0 −14.2337
19.2 2.18614 3.78651i 0 −5.55842 9.62747i 2.31386 + 4.00772i 0 −6.05842 + 10.4935i −13.6277 0 20.2337
 $$n$$: 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
9.c Even 1 yes

## Hecke kernels

There are no other newforms in $$S_{4}^{\mathrm{new}}(27, [\chi])$$.