# Properties

 Label 1155.1.e.b Level 1155 Weight 1 Character orbit 1155.e Analytic conductor 0.576 Analytic rank 0 Dimension 2 Projective image $$D_{2}$$ CM/RM discs -35, -231, 165 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 $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.576420089591$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{2}$$ Projective field Galois closure of $$\Q(\sqrt{-35}, \sqrt{165})$$ Artin image $D_4:C_2$ Artin field Galois closure of 8.0.1634180625.2

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -i q^{3} + q^{4} -i q^{5} -i q^{7} - q^{9} +O(q^{10})$$ $$q -i q^{3} + q^{4} -i q^{5} -i q^{7} - q^{9} + q^{11} -i q^{12} + 2 i q^{13} - q^{15} + q^{16} -i q^{20} - q^{21} - q^{25} + i q^{27} -i q^{28} -2 q^{29} -i q^{33} - q^{35} - q^{36} + 2 q^{39} + q^{44} + i q^{45} + 2 i q^{47} -i q^{48} - q^{49} + 2 i q^{52} -i q^{55} - q^{60} + i q^{63} + q^{64} + 2 q^{65} -2 i q^{73} + i q^{75} -i q^{77} -i q^{80} + q^{81} - q^{84} + 2 i q^{87} + 2 q^{91} - q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{4} - 2q^{9} + O(q^{10})$$ $$2q + 2q^{4} - 2q^{9} + 2q^{11} - 2q^{15} + 2q^{16} - 2q^{21} - 2q^{25} - 4q^{29} - 2q^{35} - 2q^{36} + 4q^{39} + 2q^{44} - 2q^{49} - 2q^{60} + 2q^{64} + 4q^{65} + 2q^{81} - 2q^{84} + 4q^{91} - 2q^{99} + O(q^{100})$$

## 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
 1.00000i − 1.00000i
0 1.00000i 1.00000 1.00000i 0 1.00000i 0 −1.00000 0
1154.2 0 1.00000i 1.00000 1.00000i 0 1.00000i 0 −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
35.c odd 2 1 CM by $$\Q(\sqrt{-35})$$
165.d even 2 1 RM by $$\Q(\sqrt{165})$$
231.h odd 2 1 CM by $$\Q(\sqrt{-231})$$
5.b even 2 1 inner
7.b odd 2 1 inner
33.d even 2 1 inner
1155.e odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1155.1.e.b yes 2
3.b odd 2 1 1155.1.e.a 2
5.b even 2 1 inner 1155.1.e.b yes 2
7.b odd 2 1 inner 1155.1.e.b yes 2
11.b odd 2 1 1155.1.e.a 2
15.d odd 2 1 1155.1.e.a 2
21.c even 2 1 1155.1.e.a 2
33.d even 2 1 inner 1155.1.e.b yes 2
35.c odd 2 1 CM 1155.1.e.b yes 2
55.d odd 2 1 1155.1.e.a 2
77.b even 2 1 1155.1.e.a 2
105.g even 2 1 1155.1.e.a 2
165.d even 2 1 RM 1155.1.e.b yes 2
231.h odd 2 1 CM 1155.1.e.b yes 2
385.h even 2 1 1155.1.e.a 2
1155.e odd 2 1 inner 1155.1.e.b yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.1.e.a 2 3.b odd 2 1
1155.1.e.a 2 11.b odd 2 1
1155.1.e.a 2 15.d odd 2 1
1155.1.e.a 2 21.c even 2 1
1155.1.e.a 2 55.d odd 2 1
1155.1.e.a 2 77.b even 2 1
1155.1.e.a 2 105.g even 2 1
1155.1.e.a 2 385.h even 2 1
1155.1.e.b yes 2 1.a even 1 1 trivial
1155.1.e.b yes 2 5.b even 2 1 inner
1155.1.e.b yes 2 7.b odd 2 1 inner
1155.1.e.b yes 2 33.d even 2 1 inner
1155.1.e.b yes 2 35.c odd 2 1 CM
1155.1.e.b yes 2 165.d even 2 1 RM
1155.1.e.b yes 2 231.h odd 2 1 CM
1155.1.e.b yes 2 1155.e odd 2 1 inner

## Hecke kernels

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

 $$T_{2}$$ $$T_{29} + 2$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$3$ $$1 + T^{2}$$
$5$ $$1 + T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$( 1 - T )^{2}$$
$13$ $$( 1 + T^{2} )^{2}$$
$17$ $$( 1 + T^{2} )^{2}$$
$19$ $$( 1 + T^{2} )^{2}$$
$23$ $$( 1 + T^{2} )^{2}$$
$29$ $$( 1 + T )^{4}$$
$31$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$37$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$41$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$43$ $$( 1 + T^{2} )^{2}$$
$47$ $$( 1 + T^{2} )^{2}$$
$53$ $$( 1 + T^{2} )^{2}$$
$59$ $$( 1 + T^{2} )^{2}$$
$61$ $$( 1 + T^{2} )^{2}$$
$67$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$71$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$73$ $$( 1 + T^{2} )^{2}$$
$79$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$83$ $$( 1 + T^{2} )^{2}$$
$89$ $$( 1 + T^{2} )^{2}$$
$97$ $$( 1 + T^{2} )^{2}$$