# Properties

 Label 1155.2.l.d Level 1155 Weight 2 Character orbit 1155.l Analytic conductor 9.223 Analytic rank 0 Dimension 4 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.l (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$9.22272143346$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\zeta_{8}^{2}$$ $$\beta_{2}$$ $$=$$ $$\zeta_{8}^{3} + \zeta_{8}$$ $$\beta_{3}$$ $$=$$ $$-\zeta_{8}^{3} + \zeta_{8}$$
 $$1$$ $$=$$ $$\beta_0$$ $$\zeta_{8}$$ $$=$$ $$($$$$\beta_{3}\mathstrut +\mathstrut$$ $$\beta_{2}$$$$)/2$$ $$\zeta_{8}^{2}$$ $$=$$ $$\beta_{1}$$ $$\zeta_{8}^{3}$$ $$=$$ $$($$$$-$$$$\beta_{3}\mathstrut +\mathstrut$$ $$\beta_{2}$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1121.1
 0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i
−0.414214 1.00000 1.41421i −1.82843 1.00000i −0.414214 + 0.585786i 1.00000i 1.58579 −1.00000 2.82843i 0.414214i
1121.2 −0.414214 1.00000 + 1.41421i −1.82843 1.00000i −0.414214 0.585786i 1.00000i 1.58579 −1.00000 + 2.82843i 0.414214i
1121.3 2.41421 1.00000 1.41421i 3.82843 1.00000i 2.41421 3.41421i 1.00000i 4.41421 −1.00000 2.82843i 2.41421i
1121.4 2.41421 1.00000 + 1.41421i 3.82843 1.00000i 2.41421 + 3.41421i 1.00000i 4.41421 −1.00000 + 2.82843i 2.41421i
 $$n$$: 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
33.d Even 1 yes

## Hecke kernels

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

 $$T_{2}^{2}$$ $$\mathstrut -\mathstrut 2 T_{2}$$ $$\mathstrut -\mathstrut 1$$ $$T_{17}^{2}$$ $$\mathstrut +\mathstrut 8 T_{17}$$ $$\mathstrut +\mathstrut 8$$