# Properties

 Label 495.2.a.d Level $495$ Weight $2$ Character orbit 495.a Self dual yes Analytic conductor $3.953$ 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] = [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: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 165) 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.41421 1.41421
−0.414214 0 −1.82843 1.00000 0 −4.82843 1.58579 0 −0.414214
1.2 2.41421 0 3.82843 1.00000 0 0.828427 4.41421 0 2.41421
 $$n$$: 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.d 2
3.b odd 2 1 165.2.a.a 2
4.b odd 2 1 7920.2.a.cg 2
5.b even 2 1 2475.2.a.m 2
5.c odd 4 2 2475.2.c.m 4
11.b odd 2 1 5445.2.a.m 2
12.b even 2 1 2640.2.a.bb 2
15.d odd 2 1 825.2.a.g 2
15.e even 4 2 825.2.c.e 4
21.c even 2 1 8085.2.a.ba 2
33.d even 2 1 1815.2.a.k 2
165.d even 2 1 9075.2.a.v 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.a.a 2 3.b odd 2 1
495.2.a.d 2 1.a even 1 1 trivial
825.2.a.g 2 15.d odd 2 1
825.2.c.e 4 15.e even 4 2
1815.2.a.k 2 33.d even 2 1
2475.2.a.m 2 5.b even 2 1
2475.2.c.m 4 5.c odd 4 2
2640.2.a.bb 2 12.b even 2 1
5445.2.a.m 2 11.b odd 2 1
7920.2.a.cg 2 4.b odd 2 1
8085.2.a.ba 2 21.c even 2 1
9075.2.a.v 2 165.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 2T_{2} - 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(495))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T - 1$$
$3$ $$T^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} + 4T - 4$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} - 32$$
$17$ $$T^{2} - 8T + 8$$
$19$ $$T^{2} + 8T + 8$$
$23$ $$(T - 4)^{2}$$
$29$ $$T^{2} - 4T - 4$$
$31$ $$T^{2}$$
$37$ $$T^{2} - 12T + 4$$
$41$ $$T^{2} + 4T - 4$$
$43$ $$T^{2} + 12T + 28$$
$47$ $$(T - 4)^{2}$$
$53$ $$T^{2} - 4T - 124$$
$59$ $$(T - 4)^{2}$$
$61$ $$T^{2} + 12T + 4$$
$67$ $$T^{2} - 32$$
$71$ $$T^{2} + 16T + 32$$
$73$ $$T^{2} - 128$$
$79$ $$T^{2} - 72$$
$83$ $$(T - 10)^{2}$$
$89$ $$T^{2} - 4T - 28$$
$97$ $$T^{2} - 12T + 4$$