# Properties

 Label 1155.2.l.c 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 $$2$$, degree $$1$$, minimal)

## 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: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1121.1
 −0.707107 + 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i 0.707107 − 0.707107i
−1.41421 1.00000 1.41421i 0 1.00000i −1.41421 + 2.00000i 1.00000i 2.82843 −1.00000 2.82843i 1.41421i
1121.2 −1.41421 1.00000 + 1.41421i 0 1.00000i −1.41421 2.00000i 1.00000i 2.82843 −1.00000 + 2.82843i 1.41421i
1121.3 1.41421 1.00000 1.41421i 0 1.00000i 1.41421 2.00000i 1.00000i −2.82843 −1.00000 2.82843i 1.41421i
1121.4 1.41421 1.00000 + 1.41421i 0 1.00000i 1.41421 + 2.00000i 1.00000i −2.82843 −1.00000 + 2.82843i 1.41421i
 $$n$$: 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
33.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1155.2.l.c yes 4
3.b odd 2 1 1155.2.l.b 4
11.b odd 2 1 1155.2.l.b 4
33.d even 2 1 inner 1155.2.l.c yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.l.b 4 3.b odd 2 1
1155.2.l.b 4 11.b odd 2 1
1155.2.l.c yes 4 1.a even 1 1 trivial
1155.2.l.c yes 4 33.d even 2 1 inner

## Hecke kernels

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

 $$T_{2}^{2} - 2$$ $$T_{17}^{2} - 6 T_{17} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 2 T^{2} + 4 T^{4} )^{2}$$
$3$ $$( 1 - 2 T + 3 T^{2} )^{2}$$
$5$ $$( 1 + T^{2} )^{2}$$
$7$ $$( 1 + T^{2} )^{2}$$
$11$ $$( 1 - 6 T + 11 T^{2} )^{2}$$
$13$ $$1 - 8 T^{2} + 66 T^{4} - 1352 T^{6} + 28561 T^{8}$$
$17$ $$( 1 - 6 T + 35 T^{2} - 102 T^{3} + 289 T^{4} )^{2}$$
$19$ $$1 - 38 T^{2} + 1011 T^{4} - 13718 T^{6} + 130321 T^{8}$$
$23$ $$1 - 38 T^{2} + 771 T^{4} - 20102 T^{6} + 279841 T^{8}$$
$29$ $$( 1 + 6 T + 59 T^{2} + 174 T^{3} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 8 T + 60 T^{2} - 248 T^{3} + 961 T^{4} )^{2}$$
$37$ $$( 1 - 8 T + 72 T^{2} - 296 T^{3} + 1369 T^{4} )^{2}$$
$41$ $$( 1 - 12 T + 116 T^{2} - 492 T^{3} + 1681 T^{4} )^{2}$$
$43$ $$1 - 134 T^{2} + 8115 T^{4} - 247766 T^{6} + 3418801 T^{8}$$
$47$ $$1 - 16 T^{2} - 2718 T^{4} - 35344 T^{6} + 4879681 T^{8}$$
$53$ $$1 - 190 T^{2} + 14571 T^{4} - 533710 T^{6} + 7890481 T^{8}$$
$59$ $$1 - 22 T^{2} + 3555 T^{4} - 76582 T^{6} + 12117361 T^{8}$$
$61$ $$1 - 158 T^{2} + 11883 T^{4} - 587918 T^{6} + 13845841 T^{8}$$
$67$ $$( 1 + 8 T + 78 T^{2} + 536 T^{3} + 4489 T^{4} )^{2}$$
$71$ $$1 - 176 T^{2} + 15234 T^{4} - 887216 T^{6} + 25411681 T^{8}$$
$73$ $$1 - 140 T^{2} + 14406 T^{4} - 746060 T^{6} + 28398241 T^{8}$$
$79$ $$1 - 272 T^{2} + 30690 T^{4} - 1697552 T^{6} + 38950081 T^{8}$$
$83$ $$( 1 + 6 T - 25 T^{2} + 498 T^{3} + 6889 T^{4} )^{2}$$
$89$ $$1 + 98 T^{2} + 16443 T^{4} + 776258 T^{6} + 62742241 T^{8}$$
$97$ $$( 1 + 7 T + 97 T^{2} )^{4}$$