# Properties

 Label 475.2.e.b Level $475$ Weight $2$ Character orbit 475.e Analytic conductor $3.793$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,2,Mod(26,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

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

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

chi := DirichletCharacter("475.26");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$475 = 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 475.e (of order $$3$$, degree $$2$$, 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: $$3.79289409601$$ 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: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 95) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{6} - 2) q^{3} + 2 \zeta_{6} q^{4} + 4 q^{7} - \zeta_{6} q^{9} +O(q^{10})$$ q + (2*z - 2) * q^3 + 2*z * q^4 + 4 * q^7 - z * q^9 $$q + (2 \zeta_{6} - 2) q^{3} + 2 \zeta_{6} q^{4} + 4 q^{7} - \zeta_{6} q^{9} + 3 q^{11} - 4 q^{12} + 2 \zeta_{6} q^{13} + (4 \zeta_{6} - 4) q^{16} + ( - 6 \zeta_{6} + 6) q^{17} + ( - 3 \zeta_{6} - 2) q^{19} + (8 \zeta_{6} - 8) q^{21} - 4 q^{27} + 8 \zeta_{6} q^{28} + 3 \zeta_{6} q^{29} - 7 q^{31} + (6 \zeta_{6} - 6) q^{33} + ( - 2 \zeta_{6} + 2) q^{36} - 8 q^{37} - 4 q^{39} + ( - 6 \zeta_{6} + 6) q^{41} + (4 \zeta_{6} - 4) q^{43} + 6 \zeta_{6} q^{44} + 6 \zeta_{6} q^{47} - 8 \zeta_{6} q^{48} + 9 q^{49} + 12 \zeta_{6} q^{51} + (4 \zeta_{6} - 4) q^{52} - 6 \zeta_{6} q^{53} + ( - 4 \zeta_{6} + 10) q^{57} + ( - 15 \zeta_{6} + 15) q^{59} - 5 \zeta_{6} q^{61} - 4 \zeta_{6} q^{63} - 8 q^{64} + 2 \zeta_{6} q^{67} + 12 q^{68} + ( - 3 \zeta_{6} + 3) q^{71} + ( - 8 \zeta_{6} + 8) q^{73} + ( - 10 \zeta_{6} + 6) q^{76} + 12 q^{77} + (5 \zeta_{6} - 5) q^{79} + ( - 11 \zeta_{6} + 11) q^{81} - 12 q^{83} - 16 q^{84} - 6 q^{87} + 15 \zeta_{6} q^{89} + 8 \zeta_{6} q^{91} + ( - 14 \zeta_{6} + 14) q^{93} + ( - 8 \zeta_{6} + 8) q^{97} - 3 \zeta_{6} q^{99} +O(q^{100})$$ q + (2*z - 2) * q^3 + 2*z * q^4 + 4 * q^7 - z * q^9 + 3 * q^11 - 4 * q^12 + 2*z * q^13 + (4*z - 4) * q^16 + (-6*z + 6) * q^17 + (-3*z - 2) * q^19 + (8*z - 8) * q^21 - 4 * q^27 + 8*z * q^28 + 3*z * q^29 - 7 * q^31 + (6*z - 6) * q^33 + (-2*z + 2) * q^36 - 8 * q^37 - 4 * q^39 + (-6*z + 6) * q^41 + (4*z - 4) * q^43 + 6*z * q^44 + 6*z * q^47 - 8*z * q^48 + 9 * q^49 + 12*z * q^51 + (4*z - 4) * q^52 - 6*z * q^53 + (-4*z + 10) * q^57 + (-15*z + 15) * q^59 - 5*z * q^61 - 4*z * q^63 - 8 * q^64 + 2*z * q^67 + 12 * q^68 + (-3*z + 3) * q^71 + (-8*z + 8) * q^73 + (-10*z + 6) * q^76 + 12 * q^77 + (5*z - 5) * q^79 + (-11*z + 11) * q^81 - 12 * q^83 - 16 * q^84 - 6 * q^87 + 15*z * q^89 + 8*z * q^91 + (-14*z + 14) * q^93 + (-8*z + 8) * q^97 - 3*z * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{4} + 8 q^{7} - q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 2 * q^4 + 8 * q^7 - q^9 $$2 q - 2 q^{3} + 2 q^{4} + 8 q^{7} - q^{9} + 6 q^{11} - 8 q^{12} + 2 q^{13} - 4 q^{16} + 6 q^{17} - 7 q^{19} - 8 q^{21} - 8 q^{27} + 8 q^{28} + 3 q^{29} - 14 q^{31} - 6 q^{33} + 2 q^{36} - 16 q^{37} - 8 q^{39} + 6 q^{41} - 4 q^{43} + 6 q^{44} + 6 q^{47} - 8 q^{48} + 18 q^{49} + 12 q^{51} - 4 q^{52} - 6 q^{53} + 16 q^{57} + 15 q^{59} - 5 q^{61} - 4 q^{63} - 16 q^{64} + 2 q^{67} + 24 q^{68} + 3 q^{71} + 8 q^{73} + 2 q^{76} + 24 q^{77} - 5 q^{79} + 11 q^{81} - 24 q^{83} - 32 q^{84} - 12 q^{87} + 15 q^{89} + 8 q^{91} + 14 q^{93} + 8 q^{97} - 3 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^4 + 8 * q^7 - q^9 + 6 * q^11 - 8 * q^12 + 2 * q^13 - 4 * q^16 + 6 * q^17 - 7 * q^19 - 8 * q^21 - 8 * q^27 + 8 * q^28 + 3 * q^29 - 14 * q^31 - 6 * q^33 + 2 * q^36 - 16 * q^37 - 8 * q^39 + 6 * q^41 - 4 * q^43 + 6 * q^44 + 6 * q^47 - 8 * q^48 + 18 * q^49 + 12 * q^51 - 4 * q^52 - 6 * q^53 + 16 * q^57 + 15 * q^59 - 5 * q^61 - 4 * q^63 - 16 * q^64 + 2 * q^67 + 24 * q^68 + 3 * q^71 + 8 * q^73 + 2 * q^76 + 24 * q^77 - 5 * q^79 + 11 * q^81 - 24 * q^83 - 32 * q^84 - 12 * q^87 + 15 * q^89 + 8 * q^91 + 14 * q^93 + 8 * q^97 - 3 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
26.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 −1.00000 1.73205i 1.00000 1.73205i 0 0 4.00000 0 −0.500000 + 0.866025i 0
201.1 0 −1.00000 + 1.73205i 1.00000 + 1.73205i 0 0 4.00000 0 −0.500000 0.866025i 0
 $$n$$: 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
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.e.b 2
5.b even 2 1 95.2.e.a 2
5.c odd 4 2 475.2.j.a 4
15.d odd 2 1 855.2.k.b 2
19.c even 3 1 inner 475.2.e.b 2
19.c even 3 1 9025.2.a.g 1
19.d odd 6 1 9025.2.a.e 1
20.d odd 2 1 1520.2.q.c 2
95.h odd 6 1 1805.2.a.b 1
95.i even 6 1 95.2.e.a 2
95.i even 6 1 1805.2.a.a 1
95.m odd 12 2 475.2.j.a 4
285.n odd 6 1 855.2.k.b 2
380.p odd 6 1 1520.2.q.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.e.a 2 5.b even 2 1
95.2.e.a 2 95.i even 6 1
475.2.e.b 2 1.a even 1 1 trivial
475.2.e.b 2 19.c even 3 1 inner
475.2.j.a 4 5.c odd 4 2
475.2.j.a 4 95.m odd 12 2
855.2.k.b 2 15.d odd 2 1
855.2.k.b 2 285.n odd 6 1
1520.2.q.c 2 20.d odd 2 1
1520.2.q.c 2 380.p odd 6 1
1805.2.a.a 1 95.i even 6 1
1805.2.a.b 1 95.h odd 6 1
9025.2.a.e 1 19.d odd 6 1
9025.2.a.g 1 19.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{2}^{\mathrm{new}}(475, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 2T + 4$$
$5$ $$T^{2}$$
$7$ $$(T - 4)^{2}$$
$11$ $$(T - 3)^{2}$$
$13$ $$T^{2} - 2T + 4$$
$17$ $$T^{2} - 6T + 36$$
$19$ $$T^{2} + 7T + 19$$
$23$ $$T^{2}$$
$29$ $$T^{2} - 3T + 9$$
$31$ $$(T + 7)^{2}$$
$37$ $$(T + 8)^{2}$$
$41$ $$T^{2} - 6T + 36$$
$43$ $$T^{2} + 4T + 16$$
$47$ $$T^{2} - 6T + 36$$
$53$ $$T^{2} + 6T + 36$$
$59$ $$T^{2} - 15T + 225$$
$61$ $$T^{2} + 5T + 25$$
$67$ $$T^{2} - 2T + 4$$
$71$ $$T^{2} - 3T + 9$$
$73$ $$T^{2} - 8T + 64$$
$79$ $$T^{2} + 5T + 25$$
$83$ $$(T + 12)^{2}$$
$89$ $$T^{2} - 15T + 225$$
$97$ $$T^{2} - 8T + 64$$