# Properties

 Label 495.2.a.g Level $495$ Weight $2$ Character orbit 495.a Self dual yes Analytic conductor $3.953$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [495,2,Mod(1,495)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(495, 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("495.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$495 = 3^{2} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 495.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: $$3.95259490005$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.48704.2 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 6x^{2} + 4x + 6$$ x^4 - 2*x^3 - 6*x^2 + 4*x + 6 Coefficient ring: $$\Z[a_1, a_2]$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 1) q^{2} + (\beta_{2} + 2) q^{4} + q^{5} + (\beta_{3} + \beta_1 + 1) q^{7} + ( - \beta_{3} - 2 \beta_1 + 2) q^{8}+O(q^{10})$$ q + (-b1 + 1) * q^2 + (b2 + 2) * q^4 + q^5 + (b3 + b1 + 1) * q^7 + (-b3 - 2*b1 + 2) * q^8 $$q + ( - \beta_1 + 1) q^{2} + (\beta_{2} + 2) q^{4} + q^{5} + (\beta_{3} + \beta_1 + 1) q^{7} + ( - \beta_{3} - 2 \beta_1 + 2) q^{8} + ( - \beta_1 + 1) q^{10} - q^{11} + ( - \beta_{3} + \beta_1 + 1) q^{13} + (2 \beta_{3} - \beta_{2} - 1) q^{14} + ( - 2 \beta_{3} - 2 \beta_1 + 3) q^{16} + (\beta_{3} + 2 \beta_{2} + \beta_1 + 1) q^{17} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{19} + (\beta_{2} + 2) q^{20} + (\beta_1 - 1) q^{22} + ( - 2 \beta_{2} - 2) q^{23} + q^{25} + ( - 2 \beta_{3} - \beta_{2} - 4 \beta_1 - 3) q^{26} + (3 \beta_{3} + 5 \beta_1 - 3) q^{28} + ( - 2 \beta_{3} - 2 \beta_{2} - 2) q^{29} + (2 \beta_{3} + 2 \beta_1) q^{31} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots + 3) q^{32}+ \cdots + (2 \beta_{3} + 2 \beta_{2} - 5 \beta_1 + 19) q^{98}+O(q^{100})$$ q + (-b1 + 1) * q^2 + (b2 + 2) * q^4 + q^5 + (b3 + b1 + 1) * q^7 + (-b3 - 2*b1 + 2) * q^8 + (-b1 + 1) * q^10 - q^11 + (-b3 + b1 + 1) * q^13 + (2*b3 - b2 - 1) * q^14 + (-2*b3 - 2*b1 + 3) * q^16 + (b3 + 2*b2 + b1 + 1) * q^17 + (-2*b2 - 2*b1 + 2) * q^19 + (b2 + 2) * q^20 + (b1 - 1) * q^22 + (-2*b2 - 2) * q^23 + q^25 + (-2*b3 - b2 - 4*b1 - 3) * q^26 + (3*b3 + 5*b1 - 3) * q^28 + (-2*b3 - 2*b2 - 2) * q^29 + (2*b3 + 2*b1) * q^31 + (-2*b3 + 2*b2 - b1 + 3) * q^32 + (-b2 - 4*b1 + 3) * q^34 + (b3 + b1 + 1) * q^35 + (-2*b3 - 2*b2 - 2*b1 + 2) * q^37 + (2*b3 + 2*b2 + 4*b1 + 4) * q^38 + (-b3 - 2*b1 + 2) * q^40 + (2*b3 + 2*b2 + 4*b1 - 2) * q^41 + (-b3 - b1 + 3) * q^43 + (-b2 - 2) * q^44 + (2*b3 + 6*b1 - 6) * q^46 + (-2*b3 + 2*b1 - 2) * q^47 + (2*b3 + 2*b2 - 2*b1 + 7) * q^49 + (-b1 + 1) * q^50 + (-b3 + 4*b2 + 3*b1 + 3) * q^52 + (-2*b2 + 4) * q^53 - q^55 + (2*b3 - 3*b2 + 4*b1 - 13) * q^56 + (-2*b3 + 2*b1 - 8) * q^58 + (2*b2 + 4*b1 - 8) * q^59 + (-2*b3 + 2*b1) * q^61 + (4*b3 - 2*b2 + 2*b1 - 4) * q^62 + (-2*b3 + b2 - 6*b1 + 2) * q^64 + (-b3 + b1 + 1) * q^65 + (2*b2 - 4*b1 + 2) * q^67 + (-b3 + b1 + 11) * q^68 + (2*b3 - b2 - 1) * q^70 + (2*b2 - 4) * q^71 + (-b3 + b1 + 1) * q^73 + (-2*b3 + 2*b2 + 2) * q^74 + (2*b3 - 4*b1 - 6) * q^76 + (-b3 - b1 - 1) * q^77 + (2*b2 + 2*b1 + 2) * q^79 + (-2*b3 - 2*b1 + 3) * q^80 + (2*b3 - 4*b2 - 2*b1 - 8) * q^82 + (-b3 + 2*b2 + b1 + 1) * q^83 + (b3 + 2*b2 + b1 + 1) * q^85 + (-2*b3 + b2 - 4*b1 + 5) * q^86 + (b3 + 2*b1 - 2) * q^88 + (-4*b1 - 2) * q^89 + (-2*b3 + 6*b1 - 8) * q^91 + (4*b3 - 2*b2 + 4*b1 - 18) * q^92 + (-4*b3 - 2*b2 - 4*b1 - 10) * q^94 + (-2*b2 - 2*b1 + 2) * q^95 + (-2*b2 - 4*b1 + 4) * q^97 + (2*b3 + 2*b2 - 5*b1 + 19) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} + 8 q^{4} + 4 q^{5} + 4 q^{7} + 6 q^{8}+O(q^{10})$$ 4 * q + 2 * q^2 + 8 * q^4 + 4 * q^5 + 4 * q^7 + 6 * q^8 $$4 q + 2 q^{2} + 8 q^{4} + 4 q^{5} + 4 q^{7} + 6 q^{8} + 2 q^{10} - 4 q^{11} + 8 q^{13} - 8 q^{14} + 12 q^{16} + 4 q^{17} + 4 q^{19} + 8 q^{20} - 2 q^{22} - 8 q^{23} + 4 q^{25} - 16 q^{26} - 8 q^{28} - 4 q^{29} + 14 q^{32} + 4 q^{34} + 4 q^{35} + 8 q^{37} + 20 q^{38} + 6 q^{40} - 4 q^{41} + 12 q^{43} - 8 q^{44} - 16 q^{46} + 20 q^{49} + 2 q^{50} + 20 q^{52} + 16 q^{53} - 4 q^{55} - 48 q^{56} - 24 q^{58} - 24 q^{59} + 8 q^{61} - 20 q^{62} + 8 q^{65} + 48 q^{68} - 8 q^{70} - 16 q^{71} + 8 q^{73} + 12 q^{74} - 36 q^{76} - 4 q^{77} + 12 q^{79} + 12 q^{80} - 40 q^{82} + 8 q^{83} + 4 q^{85} + 16 q^{86} - 6 q^{88} - 16 q^{89} - 16 q^{91} - 72 q^{92} - 40 q^{94} + 4 q^{95} + 8 q^{97} + 62 q^{98}+O(q^{100})$$ 4 * q + 2 * q^2 + 8 * q^4 + 4 * q^5 + 4 * q^7 + 6 * q^8 + 2 * q^10 - 4 * q^11 + 8 * q^13 - 8 * q^14 + 12 * q^16 + 4 * q^17 + 4 * q^19 + 8 * q^20 - 2 * q^22 - 8 * q^23 + 4 * q^25 - 16 * q^26 - 8 * q^28 - 4 * q^29 + 14 * q^32 + 4 * q^34 + 4 * q^35 + 8 * q^37 + 20 * q^38 + 6 * q^40 - 4 * q^41 + 12 * q^43 - 8 * q^44 - 16 * q^46 + 20 * q^49 + 2 * q^50 + 20 * q^52 + 16 * q^53 - 4 * q^55 - 48 * q^56 - 24 * q^58 - 24 * q^59 + 8 * q^61 - 20 * q^62 + 8 * q^65 + 48 * q^68 - 8 * q^70 - 16 * q^71 + 8 * q^73 + 12 * q^74 - 36 * q^76 - 4 * q^77 + 12 * q^79 + 12 * q^80 - 40 * q^82 + 8 * q^83 + 4 * q^85 + 16 * q^86 - 6 * q^88 - 16 * q^89 - 16 * q^91 - 72 * q^92 - 40 * q^94 + 4 * q^95 + 8 * q^97 + 62 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - 6x^{2} + 4x + 6$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2\nu - 3$$ v^2 - 2*v - 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 3\nu^{2} - 3\nu + 5$$ v^3 - 3*v^2 - 3*v + 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2\beta _1 + 3$$ b2 + 2*b1 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 3\beta_{2} + 9\beta _1 + 4$$ b3 + 3*b2 + 9*b1 + 4

## 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
 3.28632 1.26270 −0.852061 −1.69696
−2.28632 0 3.22727 1.00000 0 2.51962 −2.80595 0 −2.28632
1.2 −0.262696 0 −1.93099 1.00000 0 0.704647 1.03266 0 −0.262696
1.3 1.85206 0 1.43013 1.00000 0 4.90749 −1.05543 0 1.85206
1.4 2.69696 0 5.27358 1.00000 0 −4.13176 8.82872 0 2.69696
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$5$$ $$-1$$
$$11$$ $$1$$

## 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 495.2.a.g yes 4
3.b odd 2 1 495.2.a.f 4
4.b odd 2 1 7920.2.a.cn 4
5.b even 2 1 2475.2.a.bf 4
5.c odd 4 2 2475.2.c.s 8
11.b odd 2 1 5445.2.a.bh 4
12.b even 2 1 7920.2.a.cm 4
15.d odd 2 1 2475.2.a.bj 4
15.e even 4 2 2475.2.c.t 8
33.d even 2 1 5445.2.a.bs 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
495.2.a.f 4 3.b odd 2 1
495.2.a.g yes 4 1.a even 1 1 trivial
2475.2.a.bf 4 5.b even 2 1
2475.2.a.bj 4 15.d odd 2 1
2475.2.c.s 8 5.c odd 4 2
2475.2.c.t 8 15.e even 4 2
5445.2.a.bh 4 11.b odd 2 1
5445.2.a.bs 4 33.d even 2 1
7920.2.a.cm 4 12.b even 2 1
7920.2.a.cn 4 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 2T_{2}^{3} - 6T_{2}^{2} + 10T_{2} + 3$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(495))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2 T^{3} + \cdots + 3$$
$3$ $$T^{4}$$
$5$ $$(T - 1)^{4}$$
$7$ $$T^{4} - 4 T^{3} + \cdots - 36$$
$11$ $$(T + 1)^{4}$$
$13$ $$T^{4} - 8 T^{3} + \cdots - 292$$
$17$ $$T^{4} - 4 T^{3} + \cdots - 324$$
$19$ $$T^{4} - 4 T^{3} + \cdots + 288$$
$23$ $$T^{4} + 8 T^{3} + \cdots - 192$$
$29$ $$T^{4} + 4 T^{3} + \cdots - 144$$
$31$ $$T^{4} - 88 T^{2} + \cdots + 144$$
$37$ $$T^{4} - 8 T^{3} + \cdots + 976$$
$41$ $$T^{4} + 4 T^{3} + \cdots + 2160$$
$43$ $$T^{4} - 12 T^{3} + \cdots - 36$$
$47$ $$T^{4} - 128 T^{2} + \cdots - 576$$
$53$ $$T^{4} - 16 T^{3} + \cdots - 240$$
$59$ $$T^{4} + 24 T^{3} + \cdots - 8496$$
$61$ $$T^{4} - 8 T^{3} + \cdots - 2224$$
$67$ $$T^{4} - 224 T^{2} + \cdots + 6208$$
$71$ $$T^{4} + 16 T^{3} + \cdots - 240$$
$73$ $$T^{4} - 8 T^{3} + \cdots - 292$$
$79$ $$T^{4} - 12 T^{3} + \cdots + 160$$
$83$ $$T^{4} - 8 T^{3} + \cdots + 1836$$
$89$ $$T^{4} + 16 T^{3} + \cdots + 720$$
$97$ $$T^{4} - 8 T^{3} + \cdots - 2864$$