# Properties

 Label 5775.2.a.bn Level $5775$ Weight $2$ Character orbit 5775.a Self dual yes Analytic conductor $46.114$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{21})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 5$$ x^2 - x - 5 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 231) 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{21})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + q^{3} + (\beta + 3) q^{4} + \beta q^{6} - q^{7} + (2 \beta + 5) q^{8} + q^{9}+O(q^{10})$$ q + b * q^2 + q^3 + (b + 3) * q^4 + b * q^6 - q^7 + (2*b + 5) * q^8 + q^9 $$q + \beta q^{2} + q^{3} + (\beta + 3) q^{4} + \beta q^{6} - q^{7} + (2 \beta + 5) q^{8} + q^{9} - q^{11} + (\beta + 3) q^{12} - q^{13} - \beta q^{14} + (5 \beta + 4) q^{16} + (2 \beta - 4) q^{17} + \beta q^{18} + (2 \beta - 3) q^{19} - q^{21} - \beta q^{22} + ( - 2 \beta + 2) q^{23} + (2 \beta + 5) q^{24} - \beta q^{26} + q^{27} + ( - \beta - 3) q^{28} + (4 \beta - 1) q^{29} - 2 \beta q^{31} + (5 \beta + 15) q^{32} - q^{33} + ( - 2 \beta + 10) q^{34} + (\beta + 3) q^{36} - q^{37} + ( - \beta + 10) q^{38} - q^{39} + (4 \beta - 4) q^{41} - \beta q^{42} + (2 \beta + 2) q^{43} + ( - \beta - 3) q^{44} - 10 q^{46} + ( - 2 \beta - 5) q^{47} + (5 \beta + 4) q^{48} + q^{49} + (2 \beta - 4) q^{51} + ( - \beta - 3) q^{52} + ( - 2 \beta + 6) q^{53} + \beta q^{54} + ( - 2 \beta - 5) q^{56} + (2 \beta - 3) q^{57} + (3 \beta + 20) q^{58} + ( - 2 \beta + 1) q^{59} + 10 q^{61} + ( - 2 \beta - 10) q^{62} - q^{63} + (10 \beta + 17) q^{64} - \beta q^{66} + (2 \beta - 5) q^{67} + (4 \beta - 2) q^{68} + ( - 2 \beta + 2) q^{69} + ( - 4 \beta + 4) q^{71} + (2 \beta + 5) q^{72} - 7 q^{73} - \beta q^{74} + (5 \beta + 1) q^{76} + q^{77} - \beta q^{78} - 4 \beta q^{79} + q^{81} + 20 q^{82} + ( - 2 \beta + 8) q^{83} + ( - \beta - 3) q^{84} + (4 \beta + 10) q^{86} + (4 \beta - 1) q^{87} + ( - 2 \beta - 5) q^{88} + ( - 4 \beta + 2) q^{89} + q^{91} + ( - 6 \beta - 4) q^{92} - 2 \beta q^{93} + ( - 7 \beta - 10) q^{94} + (5 \beta + 15) q^{96} + (2 \beta + 6) q^{97} + \beta q^{98} - q^{99} +O(q^{100})$$ q + b * q^2 + q^3 + (b + 3) * q^4 + b * q^6 - q^7 + (2*b + 5) * q^8 + q^9 - q^11 + (b + 3) * q^12 - q^13 - b * q^14 + (5*b + 4) * q^16 + (2*b - 4) * q^17 + b * q^18 + (2*b - 3) * q^19 - q^21 - b * q^22 + (-2*b + 2) * q^23 + (2*b + 5) * q^24 - b * q^26 + q^27 + (-b - 3) * q^28 + (4*b - 1) * q^29 - 2*b * q^31 + (5*b + 15) * q^32 - q^33 + (-2*b + 10) * q^34 + (b + 3) * q^36 - q^37 + (-b + 10) * q^38 - q^39 + (4*b - 4) * q^41 - b * q^42 + (2*b + 2) * q^43 + (-b - 3) * q^44 - 10 * q^46 + (-2*b - 5) * q^47 + (5*b + 4) * q^48 + q^49 + (2*b - 4) * q^51 + (-b - 3) * q^52 + (-2*b + 6) * q^53 + b * q^54 + (-2*b - 5) * q^56 + (2*b - 3) * q^57 + (3*b + 20) * q^58 + (-2*b + 1) * q^59 + 10 * q^61 + (-2*b - 10) * q^62 - q^63 + (10*b + 17) * q^64 - b * q^66 + (2*b - 5) * q^67 + (4*b - 2) * q^68 + (-2*b + 2) * q^69 + (-4*b + 4) * q^71 + (2*b + 5) * q^72 - 7 * q^73 - b * q^74 + (5*b + 1) * q^76 + q^77 - b * q^78 - 4*b * q^79 + q^81 + 20 * q^82 + (-2*b + 8) * q^83 + (-b - 3) * q^84 + (4*b + 10) * q^86 + (4*b - 1) * q^87 + (-2*b - 5) * q^88 + (-4*b + 2) * q^89 + q^91 + (-6*b - 4) * q^92 - 2*b * q^93 + (-7*b - 10) * q^94 + (5*b + 15) * q^96 + (2*b + 6) * q^97 + b * q^98 - q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + 2 q^{3} + 7 q^{4} + q^{6} - 2 q^{7} + 12 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + q^2 + 2 * q^3 + 7 * q^4 + q^6 - 2 * q^7 + 12 * q^8 + 2 * q^9 $$2 q + q^{2} + 2 q^{3} + 7 q^{4} + q^{6} - 2 q^{7} + 12 q^{8} + 2 q^{9} - 2 q^{11} + 7 q^{12} - 2 q^{13} - q^{14} + 13 q^{16} - 6 q^{17} + q^{18} - 4 q^{19} - 2 q^{21} - q^{22} + 2 q^{23} + 12 q^{24} - q^{26} + 2 q^{27} - 7 q^{28} + 2 q^{29} - 2 q^{31} + 35 q^{32} - 2 q^{33} + 18 q^{34} + 7 q^{36} - 2 q^{37} + 19 q^{38} - 2 q^{39} - 4 q^{41} - q^{42} + 6 q^{43} - 7 q^{44} - 20 q^{46} - 12 q^{47} + 13 q^{48} + 2 q^{49} - 6 q^{51} - 7 q^{52} + 10 q^{53} + q^{54} - 12 q^{56} - 4 q^{57} + 43 q^{58} + 20 q^{61} - 22 q^{62} - 2 q^{63} + 44 q^{64} - q^{66} - 8 q^{67} + 2 q^{69} + 4 q^{71} + 12 q^{72} - 14 q^{73} - q^{74} + 7 q^{76} + 2 q^{77} - q^{78} - 4 q^{79} + 2 q^{81} + 40 q^{82} + 14 q^{83} - 7 q^{84} + 24 q^{86} + 2 q^{87} - 12 q^{88} + 2 q^{91} - 14 q^{92} - 2 q^{93} - 27 q^{94} + 35 q^{96} + 14 q^{97} + q^{98} - 2 q^{99}+O(q^{100})$$ 2 * q + q^2 + 2 * q^3 + 7 * q^4 + q^6 - 2 * q^7 + 12 * q^8 + 2 * q^9 - 2 * q^11 + 7 * q^12 - 2 * q^13 - q^14 + 13 * q^16 - 6 * q^17 + q^18 - 4 * q^19 - 2 * q^21 - q^22 + 2 * q^23 + 12 * q^24 - q^26 + 2 * q^27 - 7 * q^28 + 2 * q^29 - 2 * q^31 + 35 * q^32 - 2 * q^33 + 18 * q^34 + 7 * q^36 - 2 * q^37 + 19 * q^38 - 2 * q^39 - 4 * q^41 - q^42 + 6 * q^43 - 7 * q^44 - 20 * q^46 - 12 * q^47 + 13 * q^48 + 2 * q^49 - 6 * q^51 - 7 * q^52 + 10 * q^53 + q^54 - 12 * q^56 - 4 * q^57 + 43 * q^58 + 20 * q^61 - 22 * q^62 - 2 * q^63 + 44 * q^64 - q^66 - 8 * q^67 + 2 * q^69 + 4 * q^71 + 12 * q^72 - 14 * q^73 - q^74 + 7 * q^76 + 2 * q^77 - q^78 - 4 * q^79 + 2 * q^81 + 40 * q^82 + 14 * q^83 - 7 * q^84 + 24 * q^86 + 2 * q^87 - 12 * q^88 + 2 * q^91 - 14 * q^92 - 2 * q^93 - 27 * q^94 + 35 * q^96 + 14 * q^97 + q^98 - 2 * 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
 −1.79129 2.79129
−1.79129 1.00000 1.20871 0 −1.79129 −1.00000 1.41742 1.00000 0
1.2 2.79129 1.00000 5.79129 0 2.79129 −1.00000 10.5826 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.bn 2
5.b even 2 1 231.2.a.b 2
15.d odd 2 1 693.2.a.j 2
20.d odd 2 1 3696.2.a.bl 2
35.c odd 2 1 1617.2.a.o 2
55.d odd 2 1 2541.2.a.z 2
105.g even 2 1 4851.2.a.ba 2
165.d even 2 1 7623.2.a.bf 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.a.b 2 5.b even 2 1
693.2.a.j 2 15.d odd 2 1
1617.2.a.o 2 35.c odd 2 1
2541.2.a.z 2 55.d odd 2 1
3696.2.a.bl 2 20.d odd 2 1
4851.2.a.ba 2 105.g even 2 1
5775.2.a.bn 2 1.a even 1 1 trivial
7623.2.a.bf 2 165.d even 2 1

## Hecke kernels

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

 $$T_{2}^{2} - T_{2} - 5$$ T2^2 - T2 - 5 $$T_{13} + 1$$ T13 + 1 $$T_{17}^{2} + 6T_{17} - 12$$ T17^2 + 6*T17 - 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T - 5$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2}$$
$7$ $$(T + 1)^{2}$$
$11$ $$(T + 1)^{2}$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} + 6T - 12$$
$19$ $$T^{2} + 4T - 17$$
$23$ $$T^{2} - 2T - 20$$
$29$ $$T^{2} - 2T - 83$$
$31$ $$T^{2} + 2T - 20$$
$37$ $$(T + 1)^{2}$$
$41$ $$T^{2} + 4T - 80$$
$43$ $$T^{2} - 6T - 12$$
$47$ $$T^{2} + 12T + 15$$
$53$ $$T^{2} - 10T + 4$$
$59$ $$T^{2} - 21$$
$61$ $$(T - 10)^{2}$$
$67$ $$T^{2} + 8T - 5$$
$71$ $$T^{2} - 4T - 80$$
$73$ $$(T + 7)^{2}$$
$79$ $$T^{2} + 4T - 80$$
$83$ $$T^{2} - 14T + 28$$
$89$ $$T^{2} - 84$$
$97$ $$T^{2} - 14T + 28$$