# Properties

 Label 27.7.b.c Level 27 Weight 7 Character orbit 27.b Analytic conductor 6.211 Analytic rank 0 Dimension 4 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$27 = 3^{3}$$ Weight: $$k$$ = $$7$$ Character orbit: $$[\chi]$$ = 27.b (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$6.21146025774$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{41})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3^{8}$$ 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.

 $$f(q)$$ $$=$$ $$q$$ $$-\beta_{1} q^{2}$$ $$+ ( -49 + \beta_{3} ) q^{4}$$ $$+ ( 7 \beta_{1} + \beta_{2} ) q^{5}$$ $$+ ( -171 + 4 \beta_{3} ) q^{7}$$ $$+ ( 49 \beta_{1} + 8 \beta_{2} ) q^{8}$$ $$+O(q^{10})$$ $$q$$ $$-\beta_{1} q^{2}$$ $$+ ( -49 + \beta_{3} ) q^{4}$$ $$+ ( 7 \beta_{1} + \beta_{2} ) q^{5}$$ $$+ ( -171 + 4 \beta_{3} ) q^{7}$$ $$+ ( 49 \beta_{1} + 8 \beta_{2} ) q^{8}$$ $$+ ( 821 - 13 \beta_{3} ) q^{10}$$ $$+ ( -133 \beta_{1} + 17 \beta_{2} ) q^{11}$$ $$+ ( -858 - 8 \beta_{3} ) q^{13}$$ $$+ ( 427 \beta_{1} + 32 \beta_{2} ) q^{14}$$ $$+ ( 2641 - 33 \beta_{3} ) q^{16}$$ $$+ ( 150 \beta_{1} - 14 \beta_{2} ) q^{17}$$ $$+ ( -976 + 120 \beta_{3} ) q^{19}$$ $$+ ( -1205 \beta_{1} - 40 \beta_{2} ) q^{20}$$ $$+ ( -14519 + 31 \beta_{3} ) q^{22}$$ $$+ ( -212 \beta_{1} - 196 \beta_{2} ) q^{23}$$ $$+ ( 4304 + 88 \beta_{3} ) q^{25}$$ $$+ ( 346 \beta_{1} - 64 \beta_{2} ) q^{26}$$ $$+ ( 38267 - 363 \beta_{3} ) q^{28}$$ $$+ ( 1322 \beta_{1} - 258 \beta_{2} ) q^{29}$$ $$+ ( -38241 - 140 \beta_{3} ) q^{31}$$ $$+ ( -1617 \beta_{1} + 248 \beta_{2} ) q^{32}$$ $$+ ( 16530 - 66 \beta_{3} ) q^{34}$$ $$+ ( -4645 \beta_{1} - 135 \beta_{2} ) q^{35}$$ $$+ ( 32180 + 120 \beta_{3} ) q^{37}$$ $$+ ( 8656 \beta_{1} + 960 \beta_{2} ) q^{38}$$ $$+ ( -84821 + 613 \beta_{3} ) q^{40}$$ $$+ ( -3068 \beta_{1} + 132 \beta_{2} ) q^{41}$$ $$+ ( 31458 + 88 \beta_{3} ) q^{43}$$ $$+ ( 7991 \beta_{1} + 1336 \beta_{2} ) q^{44}$$ $$+ ( -29836 + 1388 \beta_{3} ) q^{46}$$ $$+ ( 3182 \beta_{1} - 470 \beta_{2} ) q^{47}$$ $$+ ( 31144 - 1352 \beta_{3} ) q^{49}$$ $$+ ( 1328 \beta_{1} + 704 \beta_{2} ) q^{50}$$ $$+ ( -17734 - 474 \beta_{3} ) q^{52}$$ $$+ ( -11313 \beta_{1} - 1407 \beta_{2} ) q^{53}$$ $$+ ( 14435 - 1780 \beta_{3} ) q^{55}$$ $$+ ( -34171 \beta_{1} - 856 \beta_{2} ) q^{56}$$ $$+ ( 141646 + 226 \beta_{3} ) q^{58}$$ $$+ ( 18358 \beta_{1} - 2462 \beta_{2} ) q^{59}$$ $$+ ( 42324 - 800 \beta_{3} ) q^{61}$$ $$+ ( 29281 \beta_{1} - 1120 \beta_{2} ) q^{62}$$ $$+ ( -6257 - 1983 \beta_{3} ) q^{64}$$ $$+ ( 890 \beta_{1} - 930 \beta_{2} ) q^{65}$$ $$+ ( -2202 + 4408 \beta_{3} ) q^{67}$$ $$+ ( -11154 \beta_{1} - 1424 \beta_{2} ) q^{68}$$ $$+ ( -528935 + 5455 \beta_{3} ) q^{70}$$ $$+ ( -44958 \beta_{1} - 1058 \beta_{2} ) q^{71}$$ $$+ ( -472111 + 3384 \beta_{3} ) q^{73}$$ $$+ ( -24500 \beta_{1} + 960 \beta_{2} ) q^{74}$$ $$+ ( 944464 - 6736 \beta_{3} ) q^{76}$$ $$+ ( 28639 \beta_{1} + 5769 \beta_{2} ) q^{77}$$ $$+ ( 298446 - 2528 \beta_{3} ) q^{79}$$ $$+ ( 46933 \beta_{1} + 2344 \beta_{2} ) q^{80}$$ $$+ ( -342724 + 2276 \beta_{3} ) q^{82}$$ $$+ ( 53009 \beta_{1} + 4035 \beta_{2} ) q^{83}$$ $$+ ( -45114 + 1992 \beta_{3} ) q^{85}$$ $$+ ( -25826 \beta_{1} + 704 \beta_{2} ) q^{86}$$ $$+ ( 13847 - 14023 \beta_{3} ) q^{88}$$ $$+ ( 11838 \beta_{1} + 12338 \beta_{2} ) q^{89}$$ $$+ ( -92386 - 2096 \beta_{3} ) q^{91}$$ $$+ ( 105100 \beta_{1} - 1440 \beta_{2} ) q^{92}$$ $$+ ( 345466 - 362 \beta_{3} ) q^{94}$$ $$+ ( -110272 \beta_{1} + 104 \beta_{2} ) q^{95}$$ $$+ ( 238827 + 9952 \beta_{3} ) q^{97}$$ $$+ ( -117672 \beta_{1} - 10816 \beta_{2} ) q^{98}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q$$ $$\mathstrut -\mathstrut 194q^{4}$$ $$\mathstrut -\mathstrut 676q^{7}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$4q$$ $$\mathstrut -\mathstrut 194q^{4}$$ $$\mathstrut -\mathstrut 676q^{7}$$ $$\mathstrut +\mathstrut 3258q^{10}$$ $$\mathstrut -\mathstrut 3448q^{13}$$ $$\mathstrut +\mathstrut 10498q^{16}$$ $$\mathstrut -\mathstrut 3664q^{19}$$ $$\mathstrut -\mathstrut 58014q^{22}$$ $$\mathstrut +\mathstrut 17392q^{25}$$ $$\mathstrut +\mathstrut 152342q^{28}$$ $$\mathstrut -\mathstrut 153244q^{31}$$ $$\mathstrut +\mathstrut 65988q^{34}$$ $$\mathstrut +\mathstrut 128960q^{37}$$ $$\mathstrut -\mathstrut 338058q^{40}$$ $$\mathstrut +\mathstrut 126008q^{43}$$ $$\mathstrut -\mathstrut 116568q^{46}$$ $$\mathstrut +\mathstrut 121872q^{49}$$ $$\mathstrut -\mathstrut 71884q^{52}$$ $$\mathstrut +\mathstrut 54180q^{55}$$ $$\mathstrut +\mathstrut 567036q^{58}$$ $$\mathstrut +\mathstrut 167696q^{61}$$ $$\mathstrut -\mathstrut 28994q^{64}$$ $$\mathstrut +\mathstrut 8q^{67}$$ $$\mathstrut -\mathstrut 2104830q^{70}$$ $$\mathstrut -\mathstrut 1881676q^{73}$$ $$\mathstrut +\mathstrut 3764384q^{76}$$ $$\mathstrut +\mathstrut 1188728q^{79}$$ $$\mathstrut -\mathstrut 1366344q^{82}$$ $$\mathstrut -\mathstrut 176472q^{85}$$ $$\mathstrut +\mathstrut 27342q^{88}$$ $$\mathstrut -\mathstrut 373736q^{91}$$ $$\mathstrut +\mathstrut 1381140q^{94}$$ $$\mathstrut +\mathstrut 975212q^{97}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4}\mathstrut +\mathstrut$$ $$21$$ $$x^{2}\mathstrut +\mathstrut$$ $$100$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$3 \nu^{3} + 3 \nu$$$$)/10$$ $$\beta_{2}$$ $$=$$ $$($$$$36 \nu^{3} + 441 \nu$$$$)/5$$ $$\beta_{3}$$ $$=$$ $$27 \nu^{2} + 284$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2}\mathstrut -\mathstrut$$ $$24$$ $$\beta_{1}$$$$)/81$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3}\mathstrut -\mathstrut$$ $$284$$$$)/27$$ $$\nu^{3}$$ $$=$$ $$($$$$-$$$$\beta_{2}\mathstrut +\mathstrut$$ $$294$$ $$\beta_{1}$$$$)/81$$

## 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
 − 3.70156i − 2.70156i 2.70156i 3.70156i
14.1047i 0 −134.942 137.419i 0 −514.769 1000.62i 0 1938.25
26.2 5.10469i 0 37.9422 60.5813i 0 176.769 520.383i 0 −309.248
26.3 5.10469i 0 37.9422 60.5813i 0 176.769 520.383i 0 −309.248
26.4 14.1047i 0 −134.942 137.419i 0 −514.769 1000.62i 0 1938.25
 $$n$$: 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 $$T_{2}^{4}$$ $$\mathstrut +\mathstrut 225 T_{2}^{2}$$ $$\mathstrut +\mathstrut 5184$$ acting on $$S_{7}^{\mathrm{new}}(27, [\chi])$$.