# Properties

 Label 495.2.a.c 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: $$\Q(\sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.73205 1.73205
−1.73205 0 1.00000 1.00000 0 2.00000 1.73205 0 −1.73205
1.2 1.73205 0 1.00000 1.00000 0 2.00000 −1.73205 0 1.73205
 $$n$$: 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.c 2
3.b odd 2 1 165.2.a.b 2
4.b odd 2 1 7920.2.a.bz 2
5.b even 2 1 2475.2.a.r 2
5.c odd 4 2 2475.2.c.n 4
11.b odd 2 1 5445.2.a.s 2
12.b even 2 1 2640.2.a.x 2
15.d odd 2 1 825.2.a.e 2
15.e even 4 2 825.2.c.c 4
21.c even 2 1 8085.2.a.bd 2
33.d even 2 1 1815.2.a.i 2
165.d even 2 1 9075.2.a.bh 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.a.b 2 3.b odd 2 1
495.2.a.c 2 1.a even 1 1 trivial
825.2.a.e 2 15.d odd 2 1
825.2.c.c 4 15.e even 4 2
1815.2.a.i 2 33.d even 2 1
2475.2.a.r 2 5.b even 2 1
2475.2.c.n 4 5.c odd 4 2
2640.2.a.x 2 12.b even 2 1
5445.2.a.s 2 11.b odd 2 1
7920.2.a.bz 2 4.b odd 2 1
8085.2.a.bd 2 21.c even 2 1
9075.2.a.bh 2 165.d even 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 3$$
$3$ $$T^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$(T - 2)^{2}$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} - 4T - 8$$
$17$ $$T^{2}$$
$19$ $$T^{2} - 4T - 8$$
$23$ $$T^{2} - 48$$
$29$ $$T^{2} - 12$$
$31$ $$T^{2} + 8T - 32$$
$37$ $$T^{2} - 4T - 44$$
$41$ $$T^{2} - 12$$
$43$ $$T^{2} - 4T - 44$$
$47$ $$T^{2} - 48$$
$53$ $$T^{2} - 12T - 12$$
$59$ $$T^{2} - 48$$
$61$ $$(T - 2)^{2}$$
$67$ $$(T - 8)^{2}$$
$71$ $$T^{2} - 192$$
$73$ $$T^{2} - 4T - 104$$
$79$ $$T^{2} + 20T + 88$$
$83$ $$T^{2} + 24T + 132$$
$89$ $$T^{2} - 12T - 12$$
$97$ $$(T + 10)^{2}$$