# Properties

 Label 1155.2.a.b Level $1155$ Weight $2$ Character orbit 1155.a Self dual yes Analytic conductor $9.223$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1155,2,Mod(1,1155)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1155, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1155.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1155 = 3 \cdot 5 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1155.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$9.22272143346$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + q^{3} + 2 q^{4} - q^{5} - 2 q^{6} - q^{7} + q^{9}+O(q^{10})$$ q - 2 * q^2 + q^3 + 2 * q^4 - q^5 - 2 * q^6 - q^7 + q^9 $$q - 2 q^{2} + q^{3} + 2 q^{4} - q^{5} - 2 q^{6} - q^{7} + q^{9} + 2 q^{10} + q^{11} + 2 q^{12} - 2 q^{13} + 2 q^{14} - q^{15} - 4 q^{16} + q^{17} - 2 q^{18} - 7 q^{19} - 2 q^{20} - q^{21} - 2 q^{22} + 5 q^{23} + q^{25} + 4 q^{26} + q^{27} - 2 q^{28} + 3 q^{29} + 2 q^{30} - 4 q^{31} + 8 q^{32} + q^{33} - 2 q^{34} + q^{35} + 2 q^{36} - 2 q^{37} + 14 q^{38} - 2 q^{39} - 12 q^{41} + 2 q^{42} - q^{43} + 2 q^{44} - q^{45} - 10 q^{46} - 4 q^{47} - 4 q^{48} + q^{49} - 2 q^{50} + q^{51} - 4 q^{52} - q^{53} - 2 q^{54} - q^{55} - 7 q^{57} - 6 q^{58} + 9 q^{59} - 2 q^{60} - 11 q^{61} + 8 q^{62} - q^{63} - 8 q^{64} + 2 q^{65} - 2 q^{66} + 2 q^{67} + 2 q^{68} + 5 q^{69} - 2 q^{70} - 8 q^{71} + 8 q^{73} + 4 q^{74} + q^{75} - 14 q^{76} - q^{77} + 4 q^{78} + 2 q^{79} + 4 q^{80} + q^{81} + 24 q^{82} - 9 q^{83} - 2 q^{84} - q^{85} + 2 q^{86} + 3 q^{87} - 3 q^{89} + 2 q^{90} + 2 q^{91} + 10 q^{92} - 4 q^{93} + 8 q^{94} + 7 q^{95} + 8 q^{96} - 13 q^{97} - 2 q^{98} + q^{99}+O(q^{100})$$ q - 2 * q^2 + q^3 + 2 * q^4 - q^5 - 2 * q^6 - q^7 + q^9 + 2 * q^10 + q^11 + 2 * q^12 - 2 * q^13 + 2 * q^14 - q^15 - 4 * q^16 + q^17 - 2 * q^18 - 7 * q^19 - 2 * q^20 - q^21 - 2 * q^22 + 5 * q^23 + q^25 + 4 * q^26 + q^27 - 2 * q^28 + 3 * q^29 + 2 * q^30 - 4 * q^31 + 8 * q^32 + q^33 - 2 * q^34 + q^35 + 2 * q^36 - 2 * q^37 + 14 * q^38 - 2 * q^39 - 12 * q^41 + 2 * q^42 - q^43 + 2 * q^44 - q^45 - 10 * q^46 - 4 * q^47 - 4 * q^48 + q^49 - 2 * q^50 + q^51 - 4 * q^52 - q^53 - 2 * q^54 - q^55 - 7 * q^57 - 6 * q^58 + 9 * q^59 - 2 * q^60 - 11 * q^61 + 8 * q^62 - q^63 - 8 * q^64 + 2 * q^65 - 2 * q^66 + 2 * q^67 + 2 * q^68 + 5 * q^69 - 2 * q^70 - 8 * q^71 + 8 * q^73 + 4 * q^74 + q^75 - 14 * q^76 - q^77 + 4 * q^78 + 2 * q^79 + 4 * q^80 + q^81 + 24 * q^82 - 9 * q^83 - 2 * q^84 - q^85 + 2 * q^86 + 3 * q^87 - 3 * q^89 + 2 * q^90 + 2 * q^91 + 10 * q^92 - 4 * q^93 + 8 * q^94 + 7 * q^95 + 8 * q^96 - 13 * q^97 - 2 * q^98 + q^99

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
−2.00000 1.00000 2.00000 −1.00000 −2.00000 −1.00000 0 1.00000 2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$1$$
$$7$$ $$1$$
$$11$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1155.2.a.b 1
3.b odd 2 1 3465.2.a.t 1
5.b even 2 1 5775.2.a.z 1
7.b odd 2 1 8085.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.a.b 1 1.a even 1 1 trivial
3465.2.a.t 1 3.b odd 2 1
5775.2.a.z 1 5.b even 2 1
8085.2.a.c 1 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1155))$$:

 $$T_{2} + 2$$ T2 + 2 $$T_{13} + 2$$ T13 + 2 $$T_{17} - 1$$ T17 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T - 1$$
$5$ $$T + 1$$
$7$ $$T + 1$$
$11$ $$T - 1$$
$13$ $$T + 2$$
$17$ $$T - 1$$
$19$ $$T + 7$$
$23$ $$T - 5$$
$29$ $$T - 3$$
$31$ $$T + 4$$
$37$ $$T + 2$$
$41$ $$T + 12$$
$43$ $$T + 1$$
$47$ $$T + 4$$
$53$ $$T + 1$$
$59$ $$T - 9$$
$61$ $$T + 11$$
$67$ $$T - 2$$
$71$ $$T + 8$$
$73$ $$T - 8$$
$79$ $$T - 2$$
$83$ $$T + 9$$
$89$ $$T + 3$$
$97$ $$T + 13$$