# Properties

 Label 5775.2.a.f Level $5775$ Weight $2$ Character orbit 5775.a Self dual yes Analytic conductor $46.114$ 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] = [5775,2,Mod(1,5775)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5775, 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("5775.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5775 = 3 \cdot 5^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5775.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: $$46.1136071673$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1155) 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)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} - q^{4} - q^{6} - q^{7} + 3 q^{8} + q^{9}+O(q^{10})$$ q - q^2 + q^3 - q^4 - q^6 - q^7 + 3 * q^8 + q^9 $$q - q^{2} + q^{3} - q^{4} - q^{6} - q^{7} + 3 q^{8} + q^{9} - q^{11} - q^{12} + 2 q^{13} + q^{14} - q^{16} - 6 q^{17} - q^{18} + 4 q^{19} - q^{21} + q^{22} + 4 q^{23} + 3 q^{24} - 2 q^{26} + q^{27} + q^{28} + 10 q^{29} - 4 q^{31} - 5 q^{32} - q^{33} + 6 q^{34} - q^{36} + 2 q^{37} - 4 q^{38} + 2 q^{39} + 10 q^{41} + q^{42} - 12 q^{43} + q^{44} - 4 q^{46} - q^{48} + q^{49} - 6 q^{51} - 2 q^{52} - 10 q^{53} - q^{54} - 3 q^{56} + 4 q^{57} - 10 q^{58} - 12 q^{59} - 2 q^{61} + 4 q^{62} - q^{63} + 7 q^{64} + q^{66} - 4 q^{67} + 6 q^{68} + 4 q^{69} + 8 q^{71} + 3 q^{72} + 14 q^{73} - 2 q^{74} - 4 q^{76} + q^{77} - 2 q^{78} + 4 q^{79} + q^{81} - 10 q^{82} - 8 q^{83} + q^{84} + 12 q^{86} + 10 q^{87} - 3 q^{88} + 6 q^{89} - 2 q^{91} - 4 q^{92} - 4 q^{93} - 5 q^{96} + 10 q^{97} - q^{98} - q^{99}+O(q^{100})$$ q - q^2 + q^3 - q^4 - q^6 - q^7 + 3 * q^8 + q^9 - q^11 - q^12 + 2 * q^13 + q^14 - q^16 - 6 * q^17 - q^18 + 4 * q^19 - q^21 + q^22 + 4 * q^23 + 3 * q^24 - 2 * q^26 + q^27 + q^28 + 10 * q^29 - 4 * q^31 - 5 * q^32 - q^33 + 6 * q^34 - q^36 + 2 * q^37 - 4 * q^38 + 2 * q^39 + 10 * q^41 + q^42 - 12 * q^43 + q^44 - 4 * q^46 - q^48 + q^49 - 6 * q^51 - 2 * q^52 - 10 * q^53 - q^54 - 3 * q^56 + 4 * q^57 - 10 * q^58 - 12 * q^59 - 2 * q^61 + 4 * q^62 - q^63 + 7 * q^64 + q^66 - 4 * q^67 + 6 * q^68 + 4 * q^69 + 8 * q^71 + 3 * q^72 + 14 * q^73 - 2 * q^74 - 4 * q^76 + q^77 - 2 * q^78 + 4 * q^79 + q^81 - 10 * q^82 - 8 * q^83 + q^84 + 12 * q^86 + 10 * q^87 - 3 * q^88 + 6 * q^89 - 2 * q^91 - 4 * q^92 - 4 * q^93 - 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 0 −1.00000 −1.00000 3.00000 1.00000 0
 $$n$$: 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 5775.2.a.f 1
5.b even 2 1 1155.2.a.j 1
15.d odd 2 1 3465.2.a.g 1
35.c 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 5.b even 2 1
3465.2.a.g 1 15.d odd 2 1
5775.2.a.f 1 1.a even 1 1 trivial
8085.2.a.w 1 35.c odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5775))$$:

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

## Hecke characteristic polynomials

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