# Properties

 Label 495.2.c.a Level $495$ Weight $2$ Character orbit 495.c Analytic conductor $3.953$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [495,2,Mod(199,495)]

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, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("495.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$495 = 3^{2} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 495.c (of order $$2$$, degree $$1$$, minimal)

## 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: no Analytic conductor: $$3.95259490005$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ x^4 - x^3 - 2*x^2 - 3*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 55) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} - \beta_{2} - \beta_1) q^{2} + (\beta_{3} + \beta_1 - 1) q^{4} + (\beta_{3} - \beta_{2} + 1) q^{5} - 2 \beta_{2} q^{7} + ( - \beta_{3} + 3 \beta_{2} + \beta_1) q^{8}+O(q^{10})$$ q + (b3 - b2 - b1) * q^2 + (b3 + b1 - 1) * q^4 + (b3 - b2 + 1) * q^5 - 2*b2 * q^7 + (-b3 + 3*b2 + b1) * q^8 $$q + (\beta_{3} - \beta_{2} - \beta_1) q^{2} + (\beta_{3} + \beta_1 - 1) q^{4} + (\beta_{3} - \beta_{2} + 1) q^{5} - 2 \beta_{2} q^{7} + ( - \beta_{3} + 3 \beta_{2} + \beta_1) q^{8} + (\beta_{3} + \beta_{2} + \beta_1 - 2) q^{10} + q^{11} + (2 \beta_{3} + 2 \beta_1 - 2) q^{14} + ( - \beta_{3} - \beta_1 + 3) q^{16} + ( - 2 \beta_{3} + 2 \beta_1) q^{17} - 4 q^{19} + ( - 3 \beta_{3} + 4 \beta_{2} + \cdots + 3) q^{20}+ \cdots + ( - 5 \beta_{3} + 5 \beta_{2} + 5 \beta_1) q^{98}+O(q^{100})$$ q + (b3 - b2 - b1) * q^2 + (b3 + b1 - 1) * q^4 + (b3 - b2 + 1) * q^5 - 2*b2 * q^7 + (-b3 + 3*b2 + b1) * q^8 + (b3 + b2 + b1 - 2) * q^10 + q^11 + (2*b3 + 2*b1 - 2) * q^14 + (-b3 - b1 + 3) * q^16 + (-2*b3 + 2*b1) * q^17 - 4 * q^19 + (-3*b3 + 4*b2 + 3*b1 + 3) * q^20 + (b3 - b2 - b1) * q^22 + (b3 - b1) * q^23 + (b2 + 3*b1 + 1) * q^25 + (-6*b3 + 6*b2 + 6*b1) * q^28 + (-2*b3 - 2*b1 + 2) * q^29 + (-b3 - b1) * q^31 + (3*b3 - b2 - 3*b1) * q^32 + 4 * q^34 + (-2*b3 + 4*b1 - 4) * q^35 + (3*b3 + 2*b2 - 3*b1) * q^37 + (-4*b3 + 4*b2 + 4*b1) * q^38 + (b3 - b2 - 5*b1 + 6) * q^40 + (2*b3 + 2*b1 - 2) * q^41 + 2*b2 * q^43 + (b3 + b1 - 1) * q^44 - 2 * q^46 + (4*b3 - 2*b2 - 4*b1) * q^47 - 5 * q^49 + (-3*b3 + 5*b2 + b1 + 4) * q^50 + (4*b3 - 4*b2 - 4*b1) * q^53 + (b3 - b2 + 1) * q^55 + (-2*b3 - 2*b1 + 14) * q^56 + (6*b3 - 10*b2 - 6*b1) * q^58 + (b3 + b1 - 4) * q^59 + (2*b3 + 2*b1 + 6) * q^61 + (2*b3 - 4*b2 - 2*b1) * q^62 + (-b3 - b1 - 1) * q^64 + (3*b3 - 4*b2 - 3*b1) * q^67 - 4*b2 * q^68 + (-6*b3 + 8*b2 + 6*b1 + 6) * q^70 + (-3*b3 - 3*b1) * q^71 - 4*b2 * q^73 + (-2*b3 - 2*b1 - 4) * q^74 + (-4*b3 - 4*b1 + 4) * q^76 - 2*b2 * q^77 + (-2*b3 - 2*b1 - 8) * q^79 + (5*b3 - 6*b2 - 3*b1 - 1) * q^80 + (-6*b3 + 10*b2 + 6*b1) * q^82 + (4*b3 - 2*b2 - 4*b1) * q^83 + (-4*b3 - 2*b2 + 2*b1) * q^85 + (-2*b3 - 2*b1 + 2) * q^86 + (-b3 + 3*b2 + b1) * q^88 + (-b3 - b1 - 2) * q^89 + 2*b2 * q^92 + (2*b3 + 2*b1 - 10) * q^94 + (-4*b3 + 4*b2 - 4) * q^95 + (3*b3 - 2*b2 - 3*b1) * q^97 + (-5*b3 + 5*b2 + 5*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 6 q^{4} + 3 q^{5}+O(q^{10})$$ 4 * q - 6 * q^4 + 3 * q^5 $$4 q - 6 q^{4} + 3 q^{5} - 10 q^{10} + 4 q^{11} - 12 q^{14} + 14 q^{16} - 16 q^{19} + 12 q^{20} + q^{25} + 12 q^{29} + 2 q^{31} + 16 q^{34} - 18 q^{35} + 28 q^{40} - 12 q^{41} - 6 q^{44} - 8 q^{46} - 20 q^{49} + 18 q^{50} + 3 q^{55} + 60 q^{56} - 18 q^{59} + 20 q^{61} - 2 q^{64} + 24 q^{70} + 6 q^{71} - 12 q^{74} + 24 q^{76} - 28 q^{79} - 6 q^{80} + 2 q^{85} + 12 q^{86} - 6 q^{89} - 44 q^{94} - 12 q^{95}+O(q^{100})$$ 4 * q - 6 * q^4 + 3 * q^5 - 10 * q^10 + 4 * q^11 - 12 * q^14 + 14 * q^16 - 16 * q^19 + 12 * q^20 + q^25 + 12 * q^29 + 2 * q^31 + 16 * q^34 - 18 * q^35 + 28 * q^40 - 12 * q^41 - 6 * q^44 - 8 * q^46 - 20 * q^49 + 18 * q^50 + 3 * q^55 + 60 * q^56 - 18 * q^59 + 20 * q^61 - 2 * q^64 + 24 * q^70 + 6 * q^71 - 12 * q^74 + 24 * q^76 - 28 * q^79 - 6 * q^80 + 2 * q^85 + 12 * q^86 - 6 * q^89 - 44 * q^94 - 12 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 2\nu^{2} + 4\nu - 9 ) / 6$$ (v^3 + 2*v^2 + 4*v - 9) / 6 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} - 2\nu^{2} + 2\nu + 6 ) / 3$$ (-v^3 - 2*v^2 + 2*v + 6) / 3 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 2\nu + 3 ) / 2$$ (-v^3 + 2*v + 3) / 2
 $$\nu$$ $$=$$ $$( \beta_{2} + 2\beta _1 + 1 ) / 2$$ (b2 + 2*b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( 2\beta_{3} - 3\beta_{2} + 3 ) / 2$$ (2*b3 - 3*b2 + 3) / 2 $$\nu^{3}$$ $$=$$ $$-2\beta_{3} + \beta_{2} + 2\beta _1 + 4$$ -2*b3 + b2 + 2*b1 + 4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/495\mathbb{Z}\right)^\times$$.

 $$n$$ $$46$$ $$56$$ $$397$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 −1.18614 + 1.26217i 1.68614 + 0.396143i 1.68614 − 0.396143i −1.18614 − 1.26217i
2.52434i 0 −4.37228 −0.686141 2.12819i 0 3.46410i 5.98844i 0 −5.37228 + 1.73205i
199.2 0.792287i 0 1.37228 2.18614 + 0.469882i 0 3.46410i 2.67181i 0 0.372281 1.73205i
199.3 0.792287i 0 1.37228 2.18614 0.469882i 0 3.46410i 2.67181i 0 0.372281 + 1.73205i
199.4 2.52434i 0 −4.37228 −0.686141 + 2.12819i 0 3.46410i 5.98844i 0 −5.37228 1.73205i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 495.2.c.a 4
3.b odd 2 1 55.2.b.a 4
5.b even 2 1 inner 495.2.c.a 4
5.c odd 4 2 2475.2.a.bi 4
12.b even 2 1 880.2.b.h 4
15.d odd 2 1 55.2.b.a 4
15.e even 4 2 275.2.a.h 4
33.d even 2 1 605.2.b.c 4
33.f even 10 4 605.2.j.j 16
33.h odd 10 4 605.2.j.i 16
60.h even 2 1 880.2.b.h 4
60.l odd 4 2 4400.2.a.cc 4
165.d even 2 1 605.2.b.c 4
165.l odd 4 2 3025.2.a.ba 4
165.o odd 10 4 605.2.j.i 16
165.r even 10 4 605.2.j.j 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.b.a 4 3.b odd 2 1
55.2.b.a 4 15.d odd 2 1
275.2.a.h 4 15.e even 4 2
495.2.c.a 4 1.a even 1 1 trivial
495.2.c.a 4 5.b even 2 1 inner
605.2.b.c 4 33.d even 2 1
605.2.b.c 4 165.d even 2 1
605.2.j.i 16 33.h odd 10 4
605.2.j.i 16 165.o odd 10 4
605.2.j.j 16 33.f even 10 4
605.2.j.j 16 165.r even 10 4
880.2.b.h 4 12.b even 2 1
880.2.b.h 4 60.h even 2 1
2475.2.a.bi 4 5.c odd 4 2
3025.2.a.ba 4 165.l odd 4 2
4400.2.a.cc 4 60.l odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(495, [\chi])$$:

 $$T_{2}^{4} + 7T_{2}^{2} + 4$$ T2^4 + 7*T2^2 + 4 $$T_{29}^{2} - 6T_{29} - 24$$ T29^2 - 6*T29 - 24

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 7T^{2} + 4$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 3 T^{3} + \cdots + 25$$
$7$ $$(T^{2} + 12)^{2}$$
$11$ $$(T - 1)^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4} + 28T^{2} + 64$$
$19$ $$(T + 4)^{4}$$
$23$ $$T^{4} + 7T^{2} + 4$$
$29$ $$(T^{2} - 6 T - 24)^{2}$$
$31$ $$(T^{2} - T - 8)^{2}$$
$37$ $$T^{4} + 123T^{2} + 144$$
$41$ $$(T^{2} + 6 T - 24)^{2}$$
$43$ $$(T^{2} + 12)^{2}$$
$47$ $$(T^{2} + 44)^{2}$$
$53$ $$T^{4} + 112T^{2} + 1024$$
$59$ $$(T^{2} + 9 T + 12)^{2}$$
$61$ $$(T^{2} - 10 T - 8)^{2}$$
$67$ $$T^{4} + 87T^{2} + 36$$
$71$ $$(T^{2} - 3 T - 72)^{2}$$
$73$ $$(T^{2} + 48)^{2}$$
$79$ $$(T^{2} + 14 T + 16)^{2}$$
$83$ $$(T^{2} + 44)^{2}$$
$89$ $$(T^{2} + 3 T - 6)^{2}$$
$97$ $$T^{4} + 51T^{2} + 576$$