# Properties

 Label 1155.2.a.j Level $1155$ Weight $2$ Character orbit 1155.a Self dual yes Analytic conductor $9.223$ Analytic rank $0$ 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: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} - q^{4} - q^{5} - q^{6} + q^{7} - 3 q^{8} + q^{9}+O(q^{10})$$ q + q^2 - q^3 - q^4 - q^5 - q^6 + q^7 - 3 * q^8 + q^9 $$q + q^{2} - q^{3} - q^{4} - q^{5} - q^{6} + q^{7} - 3 q^{8} + q^{9} - q^{10} - q^{11} + q^{12} - 2 q^{13} + q^{14} + q^{15} - q^{16} + 6 q^{17} + q^{18} + 4 q^{19} + q^{20} - q^{21} - q^{22} - 4 q^{23} + 3 q^{24} + q^{25} - 2 q^{26} - q^{27} - q^{28} + 10 q^{29} + q^{30} - 4 q^{31} + 5 q^{32} + q^{33} + 6 q^{34} - q^{35} - q^{36} - 2 q^{37} + 4 q^{38} + 2 q^{39} + 3 q^{40} + 10 q^{41} - q^{42} + 12 q^{43} + q^{44} - q^{45} - 4 q^{46} + q^{48} + q^{49} + q^{50} - 6 q^{51} + 2 q^{52} + 10 q^{53} - q^{54} + q^{55} - 3 q^{56} - 4 q^{57} + 10 q^{58} - 12 q^{59} - q^{60} - 2 q^{61} - 4 q^{62} + q^{63} + 7 q^{64} + 2 q^{65} + q^{66} + 4 q^{67} - 6 q^{68} + 4 q^{69} - q^{70} + 8 q^{71} - 3 q^{72} - 14 q^{73} - 2 q^{74} - q^{75} - 4 q^{76} - q^{77} + 2 q^{78} + 4 q^{79} + q^{80} + q^{81} + 10 q^{82} + 8 q^{83} + q^{84} - 6 q^{85} + 12 q^{86} - 10 q^{87} + 3 q^{88} + 6 q^{89} - q^{90} - 2 q^{91} + 4 q^{92} + 4 q^{93} - 4 q^{95} - 5 q^{96} - 10 q^{97} + q^{98} - q^{99}+O(q^{100})$$ q + q^2 - q^3 - q^4 - q^5 - q^6 + q^7 - 3 * q^8 + q^9 - q^10 - q^11 + q^12 - 2 * q^13 + q^14 + q^15 - q^16 + 6 * q^17 + q^18 + 4 * q^19 + q^20 - q^21 - q^22 - 4 * q^23 + 3 * q^24 + q^25 - 2 * q^26 - q^27 - q^28 + 10 * q^29 + q^30 - 4 * q^31 + 5 * q^32 + q^33 + 6 * q^34 - q^35 - q^36 - 2 * q^37 + 4 * q^38 + 2 * q^39 + 3 * q^40 + 10 * q^41 - q^42 + 12 * q^43 + q^44 - q^45 - 4 * q^46 + q^48 + q^49 + q^50 - 6 * q^51 + 2 * q^52 + 10 * q^53 - q^54 + q^55 - 3 * q^56 - 4 * q^57 + 10 * q^58 - 12 * q^59 - q^60 - 2 * q^61 - 4 * q^62 + q^63 + 7 * q^64 + 2 * q^65 + q^66 + 4 * q^67 - 6 * q^68 + 4 * q^69 - q^70 + 8 * q^71 - 3 * q^72 - 14 * q^73 - 2 * q^74 - q^75 - 4 * q^76 - q^77 + 2 * q^78 + 4 * q^79 + q^80 + q^81 + 10 * q^82 + 8 * q^83 + q^84 - 6 * q^85 + 12 * q^86 - 10 * q^87 + 3 * q^88 + 6 * q^89 - q^90 - 2 * q^91 + 4 * q^92 + 4 * q^93 - 4 * q^95 - 5 * q^96 - 10 * q^97 + 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
1.00000 −1.00000 −1.00000 −1.00000 −1.00000 1.00000 −3.00000 1.00000 −1.00000
 $$n$$: 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.j 1
3.b odd 2 1 3465.2.a.g 1
5.b even 2 1 5775.2.a.f 1
7.b odd 2 1 8085.2.a.w 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.a.j 1 1.a even 1 1 trivial
3465.2.a.g 1 3.b odd 2 1
5775.2.a.f 1 5.b even 2 1
8085.2.a.w 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1155))$$:

 $$T_{2} - 1$$ T2 - 1 $$T_{13} + 2$$ T13 + 2 $$T_{17} - 6$$ T17 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T + 1$$
$5$ $$T + 1$$
$7$ $$T - 1$$
$11$ $$T + 1$$
$13$ $$T + 2$$
$17$ $$T - 6$$
$19$ $$T - 4$$
$23$ $$T + 4$$
$29$ $$T - 10$$
$31$ $$T + 4$$
$37$ $$T + 2$$
$41$ $$T - 10$$
$43$ $$T - 12$$
$47$ $$T$$
$53$ $$T - 10$$
$59$ $$T + 12$$
$61$ $$T + 2$$
$67$ $$T - 4$$
$71$ $$T - 8$$
$73$ $$T + 14$$
$79$ $$T - 4$$
$83$ $$T - 8$$
$89$ $$T - 6$$
$97$ $$T + 10$$