# Properties

 Label 5577.2.a.a Level $5577$ Weight $2$ Character orbit 5577.a Self dual yes Analytic conductor $44.533$ 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] = [5577,2,Mod(1,5577)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5577, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5577.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5577 = 3 \cdot 11 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5577.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: $$44.5325692073$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) 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)

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} - q^{4} + 2 q^{5} + q^{6} - 4 q^{7} + 3 q^{8} + q^{9}+O(q^{10})$$ q - q^2 - q^3 - q^4 + 2 * q^5 + q^6 - 4 * q^7 + 3 * q^8 + q^9 $$q - q^{2} - q^{3} - q^{4} + 2 q^{5} + q^{6} - 4 q^{7} + 3 q^{8} + q^{9} - 2 q^{10} - q^{11} + q^{12} + 4 q^{14} - 2 q^{15} - q^{16} - 2 q^{17} - q^{18} - 2 q^{20} + 4 q^{21} + q^{22} + 8 q^{23} - 3 q^{24} - q^{25} - q^{27} + 4 q^{28} - 6 q^{29} + 2 q^{30} + 8 q^{31} - 5 q^{32} + q^{33} + 2 q^{34} - 8 q^{35} - q^{36} - 6 q^{37} + 6 q^{40} + 2 q^{41} - 4 q^{42} + q^{44} + 2 q^{45} - 8 q^{46} - 8 q^{47} + q^{48} + 9 q^{49} + q^{50} + 2 q^{51} + 6 q^{53} + q^{54} - 2 q^{55} - 12 q^{56} + 6 q^{58} + 4 q^{59} + 2 q^{60} + 6 q^{61} - 8 q^{62} - 4 q^{63} + 7 q^{64} - q^{66} + 4 q^{67} + 2 q^{68} - 8 q^{69} + 8 q^{70} + 3 q^{72} + 14 q^{73} + 6 q^{74} + q^{75} + 4 q^{77} - 4 q^{79} - 2 q^{80} + q^{81} - 2 q^{82} - 12 q^{83} - 4 q^{84} - 4 q^{85} + 6 q^{87} - 3 q^{88} + 6 q^{89} - 2 q^{90} - 8 q^{92} - 8 q^{93} + 8 q^{94} + 5 q^{96} - 2 q^{97} - 9 q^{98} - q^{99}+O(q^{100})$$ q - q^2 - q^3 - q^4 + 2 * q^5 + q^6 - 4 * q^7 + 3 * q^8 + q^9 - 2 * q^10 - q^11 + q^12 + 4 * q^14 - 2 * q^15 - q^16 - 2 * q^17 - q^18 - 2 * q^20 + 4 * q^21 + q^22 + 8 * q^23 - 3 * q^24 - q^25 - q^27 + 4 * q^28 - 6 * q^29 + 2 * q^30 + 8 * q^31 - 5 * q^32 + q^33 + 2 * q^34 - 8 * q^35 - q^36 - 6 * q^37 + 6 * q^40 + 2 * q^41 - 4 * q^42 + q^44 + 2 * q^45 - 8 * q^46 - 8 * q^47 + q^48 + 9 * q^49 + q^50 + 2 * q^51 + 6 * q^53 + q^54 - 2 * q^55 - 12 * q^56 + 6 * q^58 + 4 * q^59 + 2 * q^60 + 6 * q^61 - 8 * q^62 - 4 * q^63 + 7 * q^64 - q^66 + 4 * q^67 + 2 * q^68 - 8 * q^69 + 8 * q^70 + 3 * q^72 + 14 * q^73 + 6 * q^74 + q^75 + 4 * q^77 - 4 * q^79 - 2 * q^80 + q^81 - 2 * q^82 - 12 * q^83 - 4 * q^84 - 4 * q^85 + 6 * q^87 - 3 * q^88 + 6 * q^89 - 2 * q^90 - 8 * q^92 - 8 * q^93 + 8 * q^94 + 5 * q^96 - 2 * q^97 - 9 * 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 2.00000 1.00000 −4.00000 3.00000 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
$$3$$ $$+1$$
$$11$$ $$+1$$
$$13$$ $$+1$$

## 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 5577.2.a.a 1
13.b even 2 1 33.2.a.a 1
39.d odd 2 1 99.2.a.b 1
52.b odd 2 1 528.2.a.g 1
65.d even 2 1 825.2.a.a 1
65.h odd 4 2 825.2.c.a 2
91.b odd 2 1 1617.2.a.j 1
104.e even 2 1 2112.2.a.bb 1
104.h odd 2 1 2112.2.a.j 1
117.n odd 6 2 891.2.e.g 2
117.t even 6 2 891.2.e.e 2
143.d odd 2 1 363.2.a.b 1
143.l odd 10 4 363.2.e.g 4
143.n even 10 4 363.2.e.e 4
156.h even 2 1 1584.2.a.o 1
195.e odd 2 1 2475.2.a.g 1
195.s even 4 2 2475.2.c.d 2
221.b even 2 1 9537.2.a.m 1
273.g even 2 1 4851.2.a.b 1
312.b odd 2 1 6336.2.a.x 1
312.h even 2 1 6336.2.a.n 1
429.e even 2 1 1089.2.a.j 1
572.b even 2 1 5808.2.a.t 1
715.c odd 2 1 9075.2.a.q 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.a.a 1 13.b even 2 1
99.2.a.b 1 39.d odd 2 1
363.2.a.b 1 143.d odd 2 1
363.2.e.e 4 143.n even 10 4
363.2.e.g 4 143.l odd 10 4
528.2.a.g 1 52.b odd 2 1
825.2.a.a 1 65.d even 2 1
825.2.c.a 2 65.h odd 4 2
891.2.e.e 2 117.t even 6 2
891.2.e.g 2 117.n odd 6 2
1089.2.a.j 1 429.e even 2 1
1584.2.a.o 1 156.h even 2 1
1617.2.a.j 1 91.b odd 2 1
2112.2.a.j 1 104.h odd 2 1
2112.2.a.bb 1 104.e even 2 1
2475.2.a.g 1 195.e odd 2 1
2475.2.c.d 2 195.s even 4 2
4851.2.a.b 1 273.g even 2 1
5577.2.a.a 1 1.a even 1 1 trivial
5808.2.a.t 1 572.b even 2 1
6336.2.a.n 1 312.h even 2 1
6336.2.a.x 1 312.b odd 2 1
9075.2.a.q 1 715.c odd 2 1
9537.2.a.m 1 221.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5577))$$:

 $$T_{2} + 1$$ T2 + 1 $$T_{5} - 2$$ T5 - 2 $$T_{7} + 4$$ T7 + 4

## Hecke characteristic polynomials

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