# Properties

 Label 1155.2.i.a Level 1155 Weight 2 Character orbit 1155.i Analytic conductor 9.223 Analytic rank 0 Dimension 8 CM No Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$1155 = 3 \cdot 5 \cdot 7 \cdot 11$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 1155.i (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$9.22272143346$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.303595776.1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( \beta_{1} - \beta_{5} ) q^{2}$$ $$+ \beta_{4} q^{3}$$ $$+ ( -2 - \beta_{7} ) q^{4}$$ $$-\beta_{4} q^{5}$$ $$+ ( \beta_{3} - \beta_{6} ) q^{6}$$ $$+ ( -\beta_{1} + \beta_{3} - \beta_{6} ) q^{7}$$ $$+ ( -\beta_{1} + 3 \beta_{5} ) q^{8}$$ $$- q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( \beta_{1} - \beta_{5} ) q^{2}$$ $$+ \beta_{4} q^{3}$$ $$+ ( -2 - \beta_{7} ) q^{4}$$ $$-\beta_{4} q^{5}$$ $$+ ( \beta_{3} - \beta_{6} ) q^{6}$$ $$+ ( -\beta_{1} + \beta_{3} - \beta_{6} ) q^{7}$$ $$+ ( -\beta_{1} + 3 \beta_{5} ) q^{8}$$ $$- q^{9}$$ $$+ ( -\beta_{3} + \beta_{6} ) q^{10}$$ $$+ ( -2 \beta_{1} + \beta_{5} ) q^{11}$$ $$+ ( -\beta_{2} - 2 \beta_{4} ) q^{12}$$ $$+ ( 2 - \beta_{2} - 4 \beta_{4} ) q^{14}$$ $$+ q^{15}$$ $$+ ( 4 + \beta_{7} ) q^{16}$$ $$+ ( -2 \beta_{3} + 3 \beta_{6} ) q^{17}$$ $$+ ( -\beta_{1} + \beta_{5} ) q^{18}$$ $$+ ( -4 \beta_{3} + \beta_{6} ) q^{19}$$ $$+ ( \beta_{2} + 2 \beta_{4} ) q^{20}$$ $$+ ( -\beta_{1} + \beta_{5} + \beta_{6} ) q^{21}$$ $$+ ( 6 + \beta_{7} ) q^{22}$$ $$+ ( -1 - 3 \beta_{7} ) q^{23}$$ $$+ ( -3 \beta_{3} + \beta_{6} ) q^{24}$$ $$- q^{25}$$ $$-\beta_{4} q^{27}$$ $$+ ( -5 \beta_{3} - 2 \beta_{5} + 3 \beta_{6} ) q^{28}$$ $$+ ( -\beta_{1} - 2 \beta_{5} ) q^{29}$$ $$+ ( \beta_{1} - \beta_{5} ) q^{30}$$ $$-2 \beta_{2} q^{31}$$ $$+ ( 3 \beta_{1} - \beta_{5} ) q^{32}$$ $$+ ( -\beta_{3} + 2 \beta_{6} ) q^{33}$$ $$+ ( 2 \beta_{2} + 10 \beta_{4} ) q^{34}$$ $$+ ( \beta_{1} - \beta_{5} - \beta_{6} ) q^{35}$$ $$+ ( 2 + \beta_{7} ) q^{36}$$ $$+ ( 4 - 2 \beta_{7} ) q^{37}$$ $$+ ( 4 \beta_{2} + 10 \beta_{4} ) q^{38}$$ $$+ ( 3 \beta_{3} - \beta_{6} ) q^{40}$$ $$+ ( -2 \beta_{3} + 6 \beta_{6} ) q^{41}$$ $$+ ( 4 + 2 \beta_{4} + \beta_{7} ) q^{42}$$ $$+ ( -3 \beta_{1} + 2 \beta_{5} ) q^{43}$$ $$+ ( 3 \beta_{1} - 7 \beta_{5} ) q^{44}$$ $$+ \beta_{4} q^{45}$$ $$+ ( -4 \beta_{1} + 10 \beta_{5} ) q^{46}$$ $$+ ( 2 \beta_{2} - 6 \beta_{4} ) q^{47}$$ $$+ ( \beta_{2} + 4 \beta_{4} ) q^{48}$$ $$+ ( 1 + 4 \beta_{4} + 2 \beta_{7} ) q^{49}$$ $$+ ( -\beta_{1} + \beta_{5} ) q^{50}$$ $$+ ( 3 \beta_{1} - 2 \beta_{5} ) q^{51}$$ $$+ ( -5 - \beta_{7} ) q^{53}$$ $$+ ( -\beta_{3} + \beta_{6} ) q^{54}$$ $$+ ( \beta_{3} - 2 \beta_{6} ) q^{55}$$ $$+ ( 3 \beta_{2} + 8 \beta_{4} - 2 \beta_{7} ) q^{56}$$ $$+ ( \beta_{1} - 4 \beta_{5} ) q^{57}$$ $$+ ( -2 - 2 \beta_{7} ) q^{58}$$ $$+ ( -\beta_{2} - 5 \beta_{4} ) q^{59}$$ $$+ ( -2 - \beta_{7} ) q^{60}$$ $$+ ( -2 \beta_{3} - 3 \beta_{6} ) q^{61}$$ $$+ ( -6 \beta_{3} + 2 \beta_{6} ) q^{62}$$ $$+ ( \beta_{1} - \beta_{3} + \beta_{6} ) q^{63}$$ $$+ \beta_{7} q^{64}$$ $$+ ( \beta_{2} + 6 \beta_{4} ) q^{66}$$ $$+ ( 2 + 2 \beta_{7} ) q^{67}$$ $$+ ( 12 \beta_{3} - 6 \beta_{6} ) q^{68}$$ $$+ ( -3 \beta_{2} - \beta_{4} ) q^{69}$$ $$+ ( -4 - 2 \beta_{4} - \beta_{7} ) q^{70}$$ $$+ ( 10 + 2 \beta_{7} ) q^{71}$$ $$+ ( \beta_{1} - 3 \beta_{5} ) q^{72}$$ $$+ 4 \beta_{3} q^{73}$$ $$+ ( 2 \beta_{1} + 2 \beta_{5} ) q^{74}$$ $$-\beta_{4} q^{75}$$ $$+ ( 14 \beta_{3} - 12 \beta_{6} ) q^{76}$$ $$+ ( -5 + \beta_{2} + 6 \beta_{4} + \beta_{7} ) q^{77}$$ $$+ ( -4 \beta_{1} + 6 \beta_{5} ) q^{79}$$ $$+ ( -\beta_{2} - 4 \beta_{4} ) q^{80}$$ $$+ q^{81}$$ $$+ ( 2 \beta_{2} + 16 \beta_{4} ) q^{82}$$ $$+ ( -4 \beta_{3} - \beta_{6} ) q^{83}$$ $$+ ( 3 \beta_{1} + 2 \beta_{3} - 5 \beta_{5} ) q^{84}$$ $$+ ( -3 \beta_{1} + 2 \beta_{5} ) q^{85}$$ $$+ ( 10 + 2 \beta_{7} ) q^{86}$$ $$+ ( 2 \beta_{3} + \beta_{6} ) q^{87}$$ $$+ ( -8 - 5 \beta_{7} ) q^{88}$$ $$+ ( -\beta_{2} + 5 \beta_{4} ) q^{89}$$ $$+ ( \beta_{3} - \beta_{6} ) q^{90}$$ $$+ ( 26 + 4 \beta_{7} ) q^{92}$$ $$+ 2 \beta_{7} q^{93}$$ $$+ 4 \beta_{6} q^{94}$$ $$+ ( -\beta_{1} + 4 \beta_{5} ) q^{95}$$ $$+ ( \beta_{3} - 3 \beta_{6} ) q^{96}$$ $$+ ( \beta_{2} - 5 \beta_{4} ) q^{97}$$ $$+ ( 3 \beta_{1} + 4 \beta_{3} - 7 \beta_{5} - 4 \beta_{6} ) q^{98}$$ $$+ ( 2 \beta_{1} - \beta_{5} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q$$ $$\mathstrut -\mathstrut 12q^{4}$$ $$\mathstrut -\mathstrut 8q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$8q$$ $$\mathstrut -\mathstrut 12q^{4}$$ $$\mathstrut -\mathstrut 8q^{9}$$ $$\mathstrut +\mathstrut 16q^{14}$$ $$\mathstrut +\mathstrut 8q^{15}$$ $$\mathstrut +\mathstrut 28q^{16}$$ $$\mathstrut +\mathstrut 44q^{22}$$ $$\mathstrut +\mathstrut 4q^{23}$$ $$\mathstrut -\mathstrut 8q^{25}$$ $$\mathstrut +\mathstrut 12q^{36}$$ $$\mathstrut +\mathstrut 40q^{37}$$ $$\mathstrut +\mathstrut 28q^{42}$$ $$\mathstrut -\mathstrut 36q^{53}$$ $$\mathstrut +\mathstrut 8q^{56}$$ $$\mathstrut -\mathstrut 8q^{58}$$ $$\mathstrut -\mathstrut 12q^{60}$$ $$\mathstrut -\mathstrut 4q^{64}$$ $$\mathstrut +\mathstrut 8q^{67}$$ $$\mathstrut -\mathstrut 28q^{70}$$ $$\mathstrut +\mathstrut 72q^{71}$$ $$\mathstrut -\mathstrut 44q^{77}$$ $$\mathstrut +\mathstrut 8q^{81}$$ $$\mathstrut +\mathstrut 72q^{86}$$ $$\mathstrut -\mathstrut 44q^{88}$$ $$\mathstrut +\mathstrut 192q^{92}$$ $$\mathstrut -\mathstrut 8q^{93}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8}\mathstrut +\mathstrut$$ $$5$$ $$x^{6}\mathstrut +\mathstrut$$ $$16$$ $$x^{4}\mathstrut +\mathstrut$$ $$45$$ $$x^{2}\mathstrut +\mathstrut$$ $$81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{6} + 4 \nu^{4} + 20 \nu^{2} + 27$$$$)/36$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} + 8 \nu^{5} + 40 \nu^{3} + 165 \nu$$$$)/72$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} + 4 \nu^{5} + 2 \nu^{3} - 9 \nu$$$$)/54$$ $$\beta_{4}$$ $$=$$ $$($$$$5 \nu^{7} + 16 \nu^{5} + 8 \nu^{3} + 81 \nu$$$$)/216$$ $$\beta_{5}$$ $$=$$ $$($$$$5 \nu^{6} + 16 \nu^{4} + 80 \nu^{2} + 153$$$$)/72$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} + 5 \nu^{5} + 16 \nu^{3} + 18 \nu$$$$)/27$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{6} + 5 \nu^{4} + 7 \nu^{2} + 18$$$$)/9$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-$$$$\beta_{6}\mathstrut +\mathstrut$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$\beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-$$$$\beta_{7}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$3$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$2$$ $$\beta_{6}\mathstrut -\mathstrut$$ $$4$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$\beta_{3}$$ $$\nu^{4}$$ $$=$$ $$($$$$5$$ $$\beta_{7}\mathstrut -\mathstrut$$ $$6$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$5$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$1$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-$$$$\beta_{6}\mathstrut +\mathstrut$$ $$15$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$16$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$\beta_{2}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$8$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$16$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$5$$ $$\nu^{7}$$ $$=$$ $$($$$$13$$ $$\beta_{6}\mathstrut +\mathstrut$$ $$35$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$48$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$13$$ $$\beta_{2}$$$$)/2$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1155\mathbb{Z}\right)^\times$$.

 $$n$$ $$211$$ $$232$$ $$386$$ $$661$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
76.1
 −0.396143 − 1.68614i 0.396143 + 1.68614i −1.26217 + 1.18614i 1.26217 − 1.18614i 1.26217 + 1.18614i −1.26217 − 1.18614i 0.396143 − 1.68614i −0.396143 + 1.68614i
2.52434i 1.00000i −4.37228 1.00000i −2.52434 −2.52434 + 0.792287i 5.98844i −1.00000 2.52434
76.2 2.52434i 1.00000i −4.37228 1.00000i 2.52434 2.52434 + 0.792287i 5.98844i −1.00000 −2.52434
76.3 0.792287i 1.00000i 1.37228 1.00000i −0.792287 −0.792287 + 2.52434i 2.67181i −1.00000 0.792287
76.4 0.792287i 1.00000i 1.37228 1.00000i 0.792287 0.792287 + 2.52434i 2.67181i −1.00000 −0.792287
76.5 0.792287i 1.00000i 1.37228 1.00000i 0.792287 0.792287 2.52434i 2.67181i −1.00000 −0.792287
76.6 0.792287i 1.00000i 1.37228 1.00000i −0.792287 −0.792287 2.52434i 2.67181i −1.00000 0.792287
76.7 2.52434i 1.00000i −4.37228 1.00000i 2.52434 2.52434 0.792287i 5.98844i −1.00000 −2.52434
76.8 2.52434i 1.00000i −4.37228 1.00000i −2.52434 −2.52434 0.792287i 5.98844i −1.00000 2.52434
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 76.8 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
7.b Odd 1 yes
11.b Odd 1 yes
77.b Even 1 yes

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{2}^{4}$$ $$\mathstrut +\mathstrut 7 T_{2}^{2}$$ $$\mathstrut +\mathstrut 4$$ acting on $$S_{2}^{\mathrm{new}}(1155, [\chi])$$.