Properties

 Label 1155.1.e.c Level 1155 Weight 1 Character orbit 1155.e Analytic conductor 0.576 Analytic rank 0 Dimension 4 Projective image $$D_{6}$$ CM disc. -231 Inner twists 8

Related objects

Newspace parameters

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

Newform invariants

 Self dual: No Analytic conductor: $$0.576420089591$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Projective image $$D_{6}$$ Projective field Galois closure of 6.0.46690875.1

$q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q$$ $$+ ( \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{2}$$ $$+ \zeta_{12}^{3} q^{3}$$ $$+ ( -1 - \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{4}$$ $$-\zeta_{12} q^{5}$$ $$+ ( -\zeta_{12} + \zeta_{12}^{5} ) q^{6}$$ $$-\zeta_{12}^{3} q^{7}$$ $$+ ( -\zeta_{12}^{2} - \zeta_{12}^{4} ) q^{8}$$ $$- q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ ( \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{2}$$ $$+ \zeta_{12}^{3} q^{3}$$ $$+ ( -1 - \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{4}$$ $$-\zeta_{12} q^{5}$$ $$+ ( -\zeta_{12} + \zeta_{12}^{5} ) q^{6}$$ $$-\zeta_{12}^{3} q^{7}$$ $$+ ( -\zeta_{12}^{2} - \zeta_{12}^{4} ) q^{8}$$ $$- q^{9}$$ $$+ ( -\zeta_{12}^{3} - \zeta_{12}^{5} ) q^{10}$$ $$- q^{11}$$ $$+ ( -\zeta_{12} - \zeta_{12}^{3} - \zeta_{12}^{5} ) q^{12}$$ $$-\zeta_{12}^{3} q^{13}$$ $$+ ( \zeta_{12} - \zeta_{12}^{5} ) q^{14}$$ $$-\zeta_{12}^{4} q^{15}$$ $$+ q^{16}$$ $$+ ( -\zeta_{12}^{2} - \zeta_{12}^{4} ) q^{18}$$ $$+ ( -\zeta_{12} + \zeta_{12}^{5} ) q^{19}$$ $$+ ( \zeta_{12} + \zeta_{12}^{3} - \zeta_{12}^{5} ) q^{20}$$ $$+ q^{21}$$ $$+ ( -\zeta_{12}^{2} - \zeta_{12}^{4} ) q^{22}$$ $$+ ( \zeta_{12} - \zeta_{12}^{5} ) q^{24}$$ $$+ \zeta_{12}^{2} q^{25}$$ $$+ ( \zeta_{12} - \zeta_{12}^{5} ) q^{26}$$ $$-\zeta_{12}^{3} q^{27}$$ $$+ ( \zeta_{12} + \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{28}$$ $$- q^{29}$$ $$+ ( 1 + \zeta_{12}^{2} ) q^{30}$$ $$-\zeta_{12}^{3} q^{33}$$ $$+ \zeta_{12}^{4} q^{35}$$ $$+ ( 1 + \zeta_{12}^{2} - \zeta_{12}^{4} ) q^{36}$$ $$+ ( \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{37}$$ $$+ ( -\zeta_{12} - 2 \zeta_{12}^{3} - \zeta_{12}^{5} ) q^{38}$$ $$+ q^{39}$$ $$+ ( \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{40}$$ $$+ ( \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{42}$$ $$+ ( 1 + \zeta_{12}^{2} - \zeta_{12}^{4} ) q^{44}$$ $$+ \zeta_{12} q^{45}$$ $$+ \zeta_{12}^{3} q^{47}$$ $$+ \zeta_{12}^{3} q^{48}$$ $$- q^{49}$$ $$+ ( -1 + \zeta_{12}^{4} ) q^{50}$$ $$+ ( \zeta_{12} + \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{52}$$ $$+ ( \zeta_{12} - \zeta_{12}^{5} ) q^{54}$$ $$+ \zeta_{12} q^{55}$$ $$+ ( -\zeta_{12} + \zeta_{12}^{5} ) q^{56}$$ $$+ ( -\zeta_{12}^{2} - \zeta_{12}^{4} ) q^{57}$$ $$+ ( -\zeta_{12}^{2} - \zeta_{12}^{4} ) q^{58}$$ $$+ ( -\zeta_{12} + \zeta_{12}^{5} ) q^{59}$$ $$+ ( -1 + \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{60}$$ $$+ \zeta_{12}^{3} q^{63}$$ $$+ q^{64}$$ $$+ \zeta_{12}^{4} q^{65}$$ $$+ ( \zeta_{12} - \zeta_{12}^{5} ) q^{66}$$ $$+ ( -\zeta_{12}^{2} - \zeta_{12}^{4} ) q^{67}$$ $$+ ( -1 - \zeta_{12}^{2} ) q^{70}$$ $$+ ( \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{72}$$ $$+ \zeta_{12}^{3} q^{73}$$ $$+ ( -2 - \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{74}$$ $$+ \zeta_{12}^{5} q^{75}$$ $$+ ( 2 \zeta_{12} - 2 \zeta_{12}^{5} ) q^{76}$$ $$+ \zeta_{12}^{3} q^{77}$$ $$+ ( \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{78}$$ $$-\zeta_{12} q^{80}$$ $$+ q^{81}$$ $$+ ( -1 - \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{84}$$ $$-\zeta_{12}^{3} q^{87}$$ $$+ ( \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{88}$$ $$+ ( \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{90}$$ $$- q^{91}$$ $$+ ( -\zeta_{12} + \zeta_{12}^{5} ) q^{94}$$ $$+ ( 1 + \zeta_{12}^{2} ) q^{95}$$ $$+ ( -\zeta_{12}^{2} - \zeta_{12}^{4} ) q^{98}$$ $$+ q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q$$ $$\mathstrut -\mathstrut 8q^{4}$$ $$\mathstrut -\mathstrut 4q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$4q$$ $$\mathstrut -\mathstrut 8q^{4}$$ $$\mathstrut -\mathstrut 4q^{9}$$ $$\mathstrut -\mathstrut 4q^{11}$$ $$\mathstrut +\mathstrut 2q^{15}$$ $$\mathstrut +\mathstrut 4q^{16}$$ $$\mathstrut +\mathstrut 4q^{21}$$ $$\mathstrut +\mathstrut 2q^{25}$$ $$\mathstrut -\mathstrut 4q^{29}$$ $$\mathstrut +\mathstrut 6q^{30}$$ $$\mathstrut -\mathstrut 2q^{35}$$ $$\mathstrut +\mathstrut 8q^{36}$$ $$\mathstrut +\mathstrut 4q^{39}$$ $$\mathstrut +\mathstrut 8q^{44}$$ $$\mathstrut -\mathstrut 4q^{49}$$ $$\mathstrut -\mathstrut 6q^{50}$$ $$\mathstrut -\mathstrut 4q^{60}$$ $$\mathstrut +\mathstrut 4q^{64}$$ $$\mathstrut -\mathstrut 2q^{65}$$ $$\mathstrut -\mathstrut 6q^{70}$$ $$\mathstrut -\mathstrut 12q^{74}$$ $$\mathstrut +\mathstrut 4q^{81}$$ $$\mathstrut -\mathstrut 8q^{84}$$ $$\mathstrut -\mathstrut 4q^{91}$$ $$\mathstrut +\mathstrut 6q^{95}$$ $$\mathstrut +\mathstrut 4q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1154.1
 0.866025 − 0.500000i −0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
1.73205i 1.00000i −2.00000 −0.866025 + 0.500000i −1.73205 1.00000i 1.73205i −1.00000 0.866025 + 1.50000i
1154.2 1.73205i 1.00000i −2.00000 0.866025 0.500000i 1.73205 1.00000i 1.73205i −1.00000 −0.866025 1.50000i
1154.3 1.73205i 1.00000i −2.00000 0.866025 + 0.500000i 1.73205 1.00000i 1.73205i −1.00000 −0.866025 + 1.50000i
1154.4 1.73205i 1.00000i −2.00000 −0.866025 0.500000i −1.73205 1.00000i 1.73205i −1.00000 0.866025 1.50000i
 $$n$$: 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
231.h Odd 1 CM by $$\Q(\sqrt{-231})$$ yes
5.b Even 1 yes
7.b Odd 1 yes
33.d Even 1 yes
35.c Odd 1 yes
165.d Even 1 yes
1155.e Odd 1 yes

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(1155, [\chi])$$:

 $$T_{2}^{2}$$ $$\mathstrut +\mathstrut 3$$ $$T_{29}$$ $$\mathstrut +\mathstrut 1$$