# Properties

 Label 1155.2.c.a Level 1155 Weight 2 Character orbit 1155.c Analytic conductor 9.223 Analytic rank 0 Dimension 2 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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 − 1.00000i 1.00000i
1.00000i 1.00000i 1.00000 −2.00000 1.00000i −1.00000 1.00000i 3.00000i −1.00000 −1.00000 + 2.00000i
694.2 1.00000i 1.00000i 1.00000 −2.00000 + 1.00000i −1.00000 1.00000i 3.00000i −1.00000 −1.00000 2.00000i
 $$n$$: 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
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 $$S_{2}^{\mathrm{new}}(1155, [\chi])$$:

 $$T_{2}^{2}$$ $$\mathstrut +\mathstrut 1$$ $$T_{13}^{2}$$ $$\mathstrut +\mathstrut 16$$