# Properties

 Label 1155.2.c.d Level 1155 Weight 2 Character orbit 1155.c Analytic conductor 9.223 Analytic rank 0 Dimension 6 CM No Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: No Analytic conductor: $$9.22272143346$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.350464.1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$-\beta_{5} q^{2}$$ $$-\beta_{3} q^{3}$$ $$+ ( -1 + \beta_{1} - \beta_{2} ) q^{4}$$ $$+ ( -\beta_{2} + \beta_{5} ) q^{5}$$ $$+ \beta_{1} q^{6}$$ $$+ \beta_{3} q^{7}$$ $$+ 2 \beta_{3} q^{8}$$ $$- q^{9}$$ $$+O(q^{10})$$ $$q$$ $$-\beta_{5} q^{2}$$ $$-\beta_{3} q^{3}$$ $$+ ( -1 + \beta_{1} - \beta_{2} ) q^{4}$$ $$+ ( -\beta_{2} + \beta_{5} ) q^{5}$$ $$+ \beta_{1} q^{6}$$ $$+ \beta_{3} q^{7}$$ $$+ 2 \beta_{3} q^{8}$$ $$- q^{9}$$ $$+ ( 3 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{10}$$ $$+ q^{11}$$ $$+ ( \beta_{3} - \beta_{4} + \beta_{5} ) q^{12}$$ $$+ ( 3 \beta_{3} - \beta_{4} ) q^{13}$$ $$-\beta_{1} q^{14}$$ $$+ ( -\beta_{1} - \beta_{4} ) q^{15}$$ $$+ ( -2 - 2 \beta_{2} ) q^{16}$$ $$+ ( \beta_{4} + \beta_{5} ) q^{17}$$ $$+ \beta_{5} q^{18}$$ $$+ ( 3 - 2 \beta_{1} ) q^{19}$$ $$+ ( 1 + \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{5} ) q^{20}$$ $$+ q^{21}$$ $$-\beta_{5} q^{22}$$ $$+ ( 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{23}$$ $$+ 2 q^{24}$$ $$+ ( -1 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{25}$$ $$+ ( -1 - 3 \beta_{1} - \beta_{2} ) q^{26}$$ $$+ \beta_{3} q^{27}$$ $$+ ( -\beta_{3} + \beta_{4} - \beta_{5} ) q^{28}$$ $$+ ( 8 + \beta_{2} ) q^{29}$$ $$+ ( -1 - \beta_{2} - 3 \beta_{3} + \beta_{4} - \beta_{5} ) q^{30}$$ $$+ ( 2 - 3 \beta_{1} + 2 \beta_{2} ) q^{31}$$ $$+ ( 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{32}$$ $$-\beta_{3} q^{33}$$ $$+ ( 4 - \beta_{1} + 2 \beta_{2} ) q^{34}$$ $$+ ( \beta_{1} + \beta_{4} ) q^{35}$$ $$+ ( 1 - \beta_{1} + \beta_{2} ) q^{36}$$ $$+ ( 5 \beta_{3} - 3 \beta_{4} ) q^{37}$$ $$+ ( -6 \beta_{3} + 2 \beta_{4} - 5 \beta_{5} ) q^{38}$$ $$+ ( 3 + \beta_{2} ) q^{39}$$ $$+ ( 2 \beta_{1} + 2 \beta_{4} ) q^{40}$$ $$+ ( -5 - 2 \beta_{1} + \beta_{2} ) q^{41}$$ $$-\beta_{5} q^{42}$$ $$+ ( -3 \beta_{4} - 2 \beta_{5} ) q^{43}$$ $$+ ( -1 + \beta_{1} - \beta_{2} ) q^{44}$$ $$+ ( \beta_{2} - \beta_{5} ) q^{45}$$ $$+ ( -2 - \beta_{1} ) q^{46}$$ $$+ ( -4 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} ) q^{47}$$ $$+ ( 2 \beta_{3} - 2 \beta_{4} ) q^{48}$$ $$- q^{49}$$ $$+ ( -2 - 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + 3 \beta_{5} ) q^{50}$$ $$+ ( -\beta_{1} - \beta_{2} ) q^{51}$$ $$+ ( -4 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{52}$$ $$+ ( \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{53}$$ $$-\beta_{1} q^{54}$$ $$+ ( -\beta_{2} + \beta_{5} ) q^{55}$$ $$-2 q^{56}$$ $$+ ( -3 \beta_{3} - 2 \beta_{5} ) q^{57}$$ $$+ ( \beta_{3} - \beta_{4} - 8 \beta_{5} ) q^{58}$$ $$+ ( 1 - \beta_{1} + 4 \beta_{2} ) q^{59}$$ $$+ ( -2 + 2 \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{60}$$ $$+ ( -2 + \beta_{1} - 5 \beta_{2} ) q^{61}$$ $$+ ( -7 \beta_{3} + \beta_{4} - 5 \beta_{5} ) q^{62}$$ $$-\beta_{3} q^{63}$$ $$+ ( 4 - 4 \beta_{1} ) q^{64}$$ $$+ ( 1 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{65}$$ $$+ \beta_{1} q^{66}$$ $$+ ( 6 \beta_{3} - 8 \beta_{4} + 4 \beta_{5} ) q^{67}$$ $$+ ( -\beta_{3} + \beta_{4} - 3 \beta_{5} ) q^{68}$$ $$+ ( 2 + \beta_{1} - \beta_{2} ) q^{69}$$ $$+ ( 1 + \beta_{2} + 3 \beta_{3} - \beta_{4} + \beta_{5} ) q^{70}$$ $$+ ( -3 + 2 \beta_{1} - \beta_{2} ) q^{71}$$ $$-2 \beta_{3} q^{72}$$ $$+ ( 4 \beta_{3} + 2 \beta_{5} ) q^{73}$$ $$+ ( -3 - 5 \beta_{1} - 3 \beta_{2} ) q^{74}$$ $$+ ( 2 + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{75}$$ $$+ ( -7 + 7 \beta_{1} - 3 \beta_{2} ) q^{76}$$ $$+ \beta_{3} q^{77}$$ $$+ ( \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{78}$$ $$+ ( 6 - 3 \beta_{1} + 2 \beta_{2} ) q^{79}$$ $$+ ( 4 + 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{80}$$ $$+ q^{81}$$ $$+ ( -5 \beta_{3} + \beta_{4} + 3 \beta_{5} ) q^{82}$$ $$+ ( 4 \beta_{3} + \beta_{4} - \beta_{5} ) q^{83}$$ $$+ ( -1 + \beta_{1} - \beta_{2} ) q^{84}$$ $$+ ( -4 + \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - \beta_{5} ) q^{85}$$ $$+ ( -9 + 2 \beta_{1} - 5 \beta_{2} ) q^{86}$$ $$+ ( -8 \beta_{3} + \beta_{4} ) q^{87}$$ $$+ 2 \beta_{3} q^{88}$$ $$+ ( 6 - 6 \beta_{1} + 5 \beta_{2} ) q^{89}$$ $$+ ( -3 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{90}$$ $$+ ( -3 - \beta_{2} ) q^{91}$$ $$+ ( \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{92}$$ $$+ ( -2 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} ) q^{93}$$ $$+ ( -11 + 7 \beta_{1} - 5 \beta_{2} ) q^{94}$$ $$+ ( 2 - \beta_{2} + 6 \beta_{3} - 2 \beta_{4} + 5 \beta_{5} ) q^{95}$$ $$+ ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{96}$$ $$+ ( -5 \beta_{4} + 6 \beta_{5} ) q^{97}$$ $$+ \beta_{5} q^{98}$$ $$- q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q$$ $$\mathstrut -\mathstrut 4q^{4}$$ $$\mathstrut +\mathstrut 2q^{5}$$ $$\mathstrut -\mathstrut 6q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$6q$$ $$\mathstrut -\mathstrut 4q^{4}$$ $$\mathstrut +\mathstrut 2q^{5}$$ $$\mathstrut -\mathstrut 6q^{9}$$ $$\mathstrut +\mathstrut 16q^{10}$$ $$\mathstrut +\mathstrut 6q^{11}$$ $$\mathstrut -\mathstrut 8q^{16}$$ $$\mathstrut +\mathstrut 18q^{19}$$ $$\mathstrut +\mathstrut 8q^{20}$$ $$\mathstrut +\mathstrut 6q^{21}$$ $$\mathstrut +\mathstrut 12q^{24}$$ $$\mathstrut -\mathstrut 2q^{25}$$ $$\mathstrut -\mathstrut 4q^{26}$$ $$\mathstrut +\mathstrut 46q^{29}$$ $$\mathstrut -\mathstrut 4q^{30}$$ $$\mathstrut +\mathstrut 8q^{31}$$ $$\mathstrut +\mathstrut 20q^{34}$$ $$\mathstrut +\mathstrut 4q^{36}$$ $$\mathstrut +\mathstrut 16q^{39}$$ $$\mathstrut -\mathstrut 32q^{41}$$ $$\mathstrut -\mathstrut 4q^{44}$$ $$\mathstrut -\mathstrut 2q^{45}$$ $$\mathstrut -\mathstrut 12q^{46}$$ $$\mathstrut -\mathstrut 6q^{49}$$ $$\mathstrut -\mathstrut 8q^{50}$$ $$\mathstrut +\mathstrut 2q^{51}$$ $$\mathstrut +\mathstrut 2q^{55}$$ $$\mathstrut -\mathstrut 12q^{56}$$ $$\mathstrut -\mathstrut 2q^{59}$$ $$\mathstrut -\mathstrut 12q^{60}$$ $$\mathstrut -\mathstrut 2q^{61}$$ $$\mathstrut +\mathstrut 24q^{64}$$ $$\mathstrut +\mathstrut 4q^{65}$$ $$\mathstrut +\mathstrut 14q^{69}$$ $$\mathstrut +\mathstrut 4q^{70}$$ $$\mathstrut -\mathstrut 16q^{71}$$ $$\mathstrut -\mathstrut 12q^{74}$$ $$\mathstrut +\mathstrut 8q^{75}$$ $$\mathstrut -\mathstrut 36q^{76}$$ $$\mathstrut +\mathstrut 32q^{79}$$ $$\mathstrut +\mathstrut 24q^{80}$$ $$\mathstrut +\mathstrut 6q^{81}$$ $$\mathstrut -\mathstrut 4q^{84}$$ $$\mathstrut -\mathstrut 20q^{85}$$ $$\mathstrut -\mathstrut 44q^{86}$$ $$\mathstrut +\mathstrut 26q^{89}$$ $$\mathstrut -\mathstrut 16q^{90}$$ $$\mathstrut -\mathstrut 16q^{91}$$ $$\mathstrut -\mathstrut 56q^{94}$$ $$\mathstrut +\mathstrut 14q^{95}$$ $$\mathstrut +\mathstrut 16q^{96}$$ $$\mathstrut -\mathstrut 6q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6}\mathstrut -\mathstrut$$ $$2$$ $$x^{5}\mathstrut +\mathstrut$$ $$2$$ $$x^{4}\mathstrut +\mathstrut$$ $$2$$ $$x^{3}\mathstrut +\mathstrut$$ $$4$$ $$x^{2}\mathstrut -\mathstrut$$ $$4$$ $$x\mathstrut +\mathstrut$$ $$2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{5} + 8 \nu^{4} - 4 \nu^{3} - \nu^{2} + 2 \nu + 38$$$$)/23$$ $$\beta_{2}$$ $$=$$ $$($$$$-5 \nu^{5} + 17 \nu^{4} - 20 \nu^{3} - 5 \nu^{2} + 10 \nu + 29$$$$)/23$$ $$\beta_{3}$$ $$=$$ $$($$$$7 \nu^{5} - 10 \nu^{4} + 5 \nu^{3} + 30 \nu^{2} + 32 \nu - 13$$$$)/23$$ $$\beta_{4}$$ $$=$$ $$($$$$-11 \nu^{5} + 19 \nu^{4} - 21 \nu^{3} - 11 \nu^{2} - 70 \nu + 27$$$$)/23$$ $$\beta_{5}$$ $$=$$ $$($$$$-14 \nu^{5} + 20 \nu^{4} - 10 \nu^{3} - 37 \nu^{2} - 64 \nu + 26$$$$)/23$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5}\mathstrut -\mathstrut$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{3}$$ $$\nu^{3}$$ $$=$$ $$2$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$2$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$2$$ $$\nu^{4}$$ $$=$$ $$-$$$$\beta_{2}\mathstrut +\mathstrut$$ $$5$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$7$$ $$\nu^{5}$$ $$=$$ $$-$$$$8$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$3$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$9$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$3$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$8$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$9$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
694.1
 1.45161 + 1.45161i 0.403032 − 0.403032i −0.854638 + 0.854638i −0.854638 − 0.854638i 0.403032 + 0.403032i 1.45161 − 1.45161i
2.21432i 1.00000i −2.90321 0.311108 + 2.21432i −2.21432 1.00000i 2.00000i −1.00000 4.90321 0.688892i
694.2 1.67513i 1.00000i −0.806063 −1.48119 + 1.67513i 1.67513 1.00000i 2.00000i −1.00000 2.80606 + 2.48119i
694.3 0.539189i 1.00000i 1.70928 2.17009 + 0.539189i 0.539189 1.00000i 2.00000i −1.00000 0.290725 1.17009i
694.4 0.539189i 1.00000i 1.70928 2.17009 0.539189i 0.539189 1.00000i 2.00000i −1.00000 0.290725 + 1.17009i
694.5 1.67513i 1.00000i −0.806063 −1.48119 1.67513i 1.67513 1.00000i 2.00000i −1.00000 2.80606 2.48119i
694.6 2.21432i 1.00000i −2.90321 0.311108 2.21432i −2.21432 1.00000i 2.00000i −1.00000 4.90321 + 0.688892i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 694.6 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
5.b Even 1 yes

## Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1155, [\chi])$$:

 $$T_{2}^{6}$$ $$\mathstrut +\mathstrut 8 T_{2}^{4}$$ $$\mathstrut +\mathstrut 16 T_{2}^{2}$$ $$\mathstrut +\mathstrut 4$$ $$T_{13}^{6}$$ $$\mathstrut +\mathstrut 28 T_{13}^{4}$$ $$\mathstrut +\mathstrut 164 T_{13}^{2}$$ $$\mathstrut +\mathstrut 100$$