# Properties

 Label 476.2.a.b Level $476$ Weight $2$ Character orbit 476.a Self dual yes Analytic conductor $3.801$ Analytic rank $1$ 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] = [476,2,Mod(1,476)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(476, 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("476.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$476 = 2^{2} \cdot 7 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 476.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: $$3.80087913621$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{3} + (\beta - 1) q^{5} - q^{7} + (\beta - 2) q^{9} +O(q^{10})$$ q - b * q^3 + (b - 1) * q^5 - q^7 + (b - 2) * q^9 $$q - \beta q^{3} + (\beta - 1) q^{5} - q^{7} + (\beta - 2) q^{9} + (2 \beta - 4) q^{11} + (2 \beta - 2) q^{13} - q^{15} + q^{17} + ( - 4 \beta - 2) q^{19} + \beta q^{21} + ( - 6 \beta + 2) q^{23} + ( - \beta - 3) q^{25} + (4 \beta - 1) q^{27} + ( - 6 \beta + 4) q^{29} + (7 \beta - 5) q^{31} + (2 \beta - 2) q^{33} + ( - \beta + 1) q^{35} + (4 \beta - 6) q^{37} - 2 q^{39} + ( - 5 \beta + 8) q^{41} + ( - 5 \beta - 4) q^{43} + ( - 2 \beta + 3) q^{45} + ( - 4 \beta - 2) q^{47} + q^{49} - \beta q^{51} + (11 \beta - 7) q^{53} + ( - 4 \beta + 6) q^{55} + (6 \beta + 4) q^{57} + ( - 7 \beta + 4) q^{61} + ( - \beta + 2) q^{63} + ( - 2 \beta + 4) q^{65} + ( - 5 \beta - 5) q^{67} + (4 \beta + 6) q^{69} + (8 \beta - 2) q^{71} + ( - 3 \beta + 12) q^{73} + (4 \beta + 1) q^{75} + ( - 2 \beta + 4) q^{77} + (8 \beta - 10) q^{79} + ( - 6 \beta + 2) q^{81} + (12 \beta - 4) q^{83} + (\beta - 1) q^{85} + (2 \beta + 6) q^{87} + 2 q^{89} + ( - 2 \beta + 2) q^{91} + ( - 2 \beta - 7) q^{93} + ( - 2 \beta - 2) q^{95} + (5 \beta + 7) q^{97} + ( - 6 \beta + 10) q^{99} +O(q^{100})$$ q - b * q^3 + (b - 1) * q^5 - q^7 + (b - 2) * q^9 + (2*b - 4) * q^11 + (2*b - 2) * q^13 - q^15 + q^17 + (-4*b - 2) * q^19 + b * q^21 + (-6*b + 2) * q^23 + (-b - 3) * q^25 + (4*b - 1) * q^27 + (-6*b + 4) * q^29 + (7*b - 5) * q^31 + (2*b - 2) * q^33 + (-b + 1) * q^35 + (4*b - 6) * q^37 - 2 * q^39 + (-5*b + 8) * q^41 + (-5*b - 4) * q^43 + (-2*b + 3) * q^45 + (-4*b - 2) * q^47 + q^49 - b * q^51 + (11*b - 7) * q^53 + (-4*b + 6) * q^55 + (6*b + 4) * q^57 + (-7*b + 4) * q^61 + (-b + 2) * q^63 + (-2*b + 4) * q^65 + (-5*b - 5) * q^67 + (4*b + 6) * q^69 + (8*b - 2) * q^71 + (-3*b + 12) * q^73 + (4*b + 1) * q^75 + (-2*b + 4) * q^77 + (8*b - 10) * q^79 + (-6*b + 2) * q^81 + (12*b - 4) * q^83 + (b - 1) * q^85 + (2*b + 6) * q^87 + 2 * q^89 + (-2*b + 2) * q^91 + (-2*b - 7) * q^93 + (-2*b - 2) * q^95 + (5*b + 7) * q^97 + (-6*b + 10) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} - q^{5} - 2 q^{7} - 3 q^{9}+O(q^{10})$$ 2 * q - q^3 - q^5 - 2 * q^7 - 3 * q^9 $$2 q - q^{3} - q^{5} - 2 q^{7} - 3 q^{9} - 6 q^{11} - 2 q^{13} - 2 q^{15} + 2 q^{17} - 8 q^{19} + q^{21} - 2 q^{23} - 7 q^{25} + 2 q^{27} + 2 q^{29} - 3 q^{31} - 2 q^{33} + q^{35} - 8 q^{37} - 4 q^{39} + 11 q^{41} - 13 q^{43} + 4 q^{45} - 8 q^{47} + 2 q^{49} - q^{51} - 3 q^{53} + 8 q^{55} + 14 q^{57} + q^{61} + 3 q^{63} + 6 q^{65} - 15 q^{67} + 16 q^{69} + 4 q^{71} + 21 q^{73} + 6 q^{75} + 6 q^{77} - 12 q^{79} - 2 q^{81} + 4 q^{83} - q^{85} + 14 q^{87} + 4 q^{89} + 2 q^{91} - 16 q^{93} - 6 q^{95} + 19 q^{97} + 14 q^{99}+O(q^{100})$$ 2 * q - q^3 - q^5 - 2 * q^7 - 3 * q^9 - 6 * q^11 - 2 * q^13 - 2 * q^15 + 2 * q^17 - 8 * q^19 + q^21 - 2 * q^23 - 7 * q^25 + 2 * q^27 + 2 * q^29 - 3 * q^31 - 2 * q^33 + q^35 - 8 * q^37 - 4 * q^39 + 11 * q^41 - 13 * q^43 + 4 * q^45 - 8 * q^47 + 2 * q^49 - q^51 - 3 * q^53 + 8 * q^55 + 14 * q^57 + q^61 + 3 * q^63 + 6 * q^65 - 15 * q^67 + 16 * q^69 + 4 * q^71 + 21 * q^73 + 6 * q^75 + 6 * q^77 - 12 * q^79 - 2 * q^81 + 4 * q^83 - q^85 + 14 * q^87 + 4 * q^89 + 2 * q^91 - 16 * q^93 - 6 * q^95 + 19 * q^97 + 14 * 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
 1.61803 −0.618034
0 −1.61803 0 0.618034 0 −1.00000 0 −0.381966 0
1.2 0 0.618034 0 −1.61803 0 −1.00000 0 −2.61803 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
$$2$$ $$-1$$
$$7$$ $$1$$
$$17$$ $$-1$$

## 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 476.2.a.b 2
3.b odd 2 1 4284.2.a.m 2
4.b odd 2 1 1904.2.a.j 2
7.b odd 2 1 3332.2.a.l 2
8.b even 2 1 7616.2.a.u 2
8.d odd 2 1 7616.2.a.p 2
17.b even 2 1 8092.2.a.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
476.2.a.b 2 1.a even 1 1 trivial
1904.2.a.j 2 4.b odd 2 1
3332.2.a.l 2 7.b odd 2 1
4284.2.a.m 2 3.b odd 2 1
7616.2.a.p 2 8.d odd 2 1
7616.2.a.u 2 8.b even 2 1
8092.2.a.m 2 17.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + T_{3} - 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(476))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + T - 1$$
$5$ $$T^{2} + T - 1$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2} + 6T + 4$$
$13$ $$T^{2} + 2T - 4$$
$17$ $$(T - 1)^{2}$$
$19$ $$T^{2} + 8T - 4$$
$23$ $$T^{2} + 2T - 44$$
$29$ $$T^{2} - 2T - 44$$
$31$ $$T^{2} + 3T - 59$$
$37$ $$T^{2} + 8T - 4$$
$41$ $$T^{2} - 11T - 1$$
$43$ $$T^{2} + 13T + 11$$
$47$ $$T^{2} + 8T - 4$$
$53$ $$T^{2} + 3T - 149$$
$59$ $$T^{2}$$
$61$ $$T^{2} - T - 61$$
$67$ $$T^{2} + 15T + 25$$
$71$ $$T^{2} - 4T - 76$$
$73$ $$T^{2} - 21T + 99$$
$79$ $$T^{2} + 12T - 44$$
$83$ $$T^{2} - 4T - 176$$
$89$ $$(T - 2)^{2}$$
$97$ $$T^{2} - 19T + 59$$