# Properties

 Label 1155.2.a.f Level $1155$ Weight $2$ Character orbit 1155.a Self dual yes Analytic conductor $9.223$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Learn more

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.a.f 1 1.a even 1 1 trivial
3465.2.a.n 1 3.b odd 2 1
5775.2.a.q 1 5.b even 2 1
8085.2.a.f 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 $$T_{17} - 2$$ T17 - 2

## Hecke characteristic polynomials

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