# Properties

 Label 475.2.j.a Level $475$ Weight $2$ Character orbit 475.j Analytic conductor $3.793$ Analytic rank $0$ Dimension $4$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,2,Mod(49,475)]

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([3, 4]))

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

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

chi := DirichletCharacter("475.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$475 = 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 475.j (of order $$6$$, degree $$2$$, not 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 95) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} - 2 \beta_{2} q^{4} + 2 \beta_{3} q^{7} + \beta_{2} q^{9}+O(q^{10})$$ q + b1 * q^3 - 2*b2 * q^4 + 2*b3 * q^7 + b2 * q^9 $$q + \beta_1 q^{3} - 2 \beta_{2} q^{4} + 2 \beta_{3} q^{7} + \beta_{2} q^{9} + 3 q^{11} - 2 \beta_{3} q^{12} + ( - \beta_{3} + \beta_1) q^{13} + (4 \beta_{2} - 4) q^{16} + 3 \beta_1 q^{17} + (3 \beta_{2} + 2) q^{19} + (8 \beta_{2} - 8) q^{21} - 2 \beta_{3} q^{27} + ( - 4 \beta_{3} + 4 \beta_1) q^{28} - 3 \beta_{2} q^{29} - 7 q^{31} + 3 \beta_1 q^{33} + ( - 2 \beta_{2} + 2) q^{36} - 4 \beta_{3} q^{37} + 4 q^{39} + ( - 6 \beta_{2} + 6) q^{41} + 2 \beta_1 q^{43} - 6 \beta_{2} q^{44} + (3 \beta_{3} - 3 \beta_1) q^{47} + (4 \beta_{3} - 4 \beta_1) q^{48} - 9 q^{49} + 12 \beta_{2} q^{51} - 2 \beta_1 q^{52} + (3 \beta_{3} - 3 \beta_1) q^{53} + (3 \beta_{3} + 2 \beta_1) q^{57} + (15 \beta_{2} - 15) q^{59} - 5 \beta_{2} q^{61} + (2 \beta_{3} - 2 \beta_1) q^{63} + 8 q^{64} + (\beta_{3} - \beta_1) q^{67} - 6 \beta_{3} q^{68} + ( - 3 \beta_{2} + 3) q^{71} - 4 \beta_1 q^{73} + ( - 10 \beta_{2} + 6) q^{76} + 6 \beta_{3} q^{77} + ( - 5 \beta_{2} + 5) q^{79} + ( - 11 \beta_{2} + 11) q^{81} + 6 \beta_{3} q^{83} + 16 q^{84} - 3 \beta_{3} q^{87} - 15 \beta_{2} q^{89} + 8 \beta_{2} q^{91} - 7 \beta_1 q^{93} + 4 \beta_1 q^{97} + 3 \beta_{2} q^{99}+O(q^{100})$$ q + b1 * q^3 - 2*b2 * q^4 + 2*b3 * q^7 + b2 * q^9 + 3 * q^11 - 2*b3 * q^12 + (-b3 + b1) * q^13 + (4*b2 - 4) * q^16 + 3*b1 * q^17 + (3*b2 + 2) * q^19 + (8*b2 - 8) * q^21 - 2*b3 * q^27 + (-4*b3 + 4*b1) * q^28 - 3*b2 * q^29 - 7 * q^31 + 3*b1 * q^33 + (-2*b2 + 2) * q^36 - 4*b3 * q^37 + 4 * q^39 + (-6*b2 + 6) * q^41 + 2*b1 * q^43 - 6*b2 * q^44 + (3*b3 - 3*b1) * q^47 + (4*b3 - 4*b1) * q^48 - 9 * q^49 + 12*b2 * q^51 - 2*b1 * q^52 + (3*b3 - 3*b1) * q^53 + (3*b3 + 2*b1) * q^57 + (15*b2 - 15) * q^59 - 5*b2 * q^61 + (2*b3 - 2*b1) * q^63 + 8 * q^64 + (b3 - b1) * q^67 - 6*b3 * q^68 + (-3*b2 + 3) * q^71 - 4*b1 * q^73 + (-10*b2 + 6) * q^76 + 6*b3 * q^77 + (-5*b2 + 5) * q^79 + (-11*b2 + 11) * q^81 + 6*b3 * q^83 + 16 * q^84 - 3*b3 * q^87 - 15*b2 * q^89 + 8*b2 * q^91 - 7*b1 * q^93 + 4*b1 * q^97 + 3*b2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} + 2 q^{9}+O(q^{10})$$ 4 * q - 4 * q^4 + 2 * q^9 $$4 q - 4 q^{4} + 2 q^{9} + 12 q^{11} - 8 q^{16} + 14 q^{19} - 16 q^{21} - 6 q^{29} - 28 q^{31} + 4 q^{36} + 16 q^{39} + 12 q^{41} - 12 q^{44} - 36 q^{49} + 24 q^{51} - 30 q^{59} - 10 q^{61} + 32 q^{64} + 6 q^{71} + 4 q^{76} + 10 q^{79} + 22 q^{81} + 64 q^{84} - 30 q^{89} + 16 q^{91} + 6 q^{99}+O(q^{100})$$ 4 * q - 4 * q^4 + 2 * q^9 + 12 * q^11 - 8 * q^16 + 14 * q^19 - 16 * q^21 - 6 * q^29 - 28 * q^31 + 4 * q^36 + 16 * q^39 + 12 * q^41 - 12 * q^44 - 36 * q^49 + 24 * q^51 - 30 * q^59 - 10 * q^61 + 32 * q^64 + 6 * q^71 + 4 * q^76 + 10 * q^79 + 22 * q^81 + 64 * q^84 - 30 * q^89 + 16 * q^91 + 6 * q^99

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$2\zeta_{12}$$ 2*v $$\beta_{2}$$ $$=$$ $$\zeta_{12}^{2}$$ v^2 $$\beta_{3}$$ $$=$$ $$2\zeta_{12}^{3}$$ 2*v^3
 $$\zeta_{12}$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\zeta_{12}^{2}$$ $$=$$ $$\beta_{2}$$ b2 $$\zeta_{12}^{3}$$ $$=$$ $$( \beta_{3} ) / 2$$ (b3) / 2

## Character values

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

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$-1$$ $$-\beta_{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
0 −1.73205 1.00000i −1.00000 1.73205i 0 0 4.00000i 0 0.500000 + 0.866025i 0
49.2 0 1.73205 + 1.00000i −1.00000 1.73205i 0 0 4.00000i 0 0.500000 + 0.866025i 0
349.1 0 −1.73205 + 1.00000i −1.00000 + 1.73205i 0 0 4.00000i 0 0.500000 0.866025i 0
349.2 0 1.73205 1.00000i −1.00000 + 1.73205i 0 0 4.00000i 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
5.b even 2 1 inner
19.c even 3 1 inner
95.i even 6 1 inner

## Twists

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

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

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 4T^{2} + 16$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 16)^{2}$$
$11$ $$(T - 3)^{4}$$
$13$ $$T^{4} - 4T^{2} + 16$$
$17$ $$T^{4} - 36T^{2} + 1296$$
$19$ $$(T^{2} - 7 T + 19)^{2}$$
$23$ $$T^{4}$$
$29$ $$(T^{2} + 3 T + 9)^{2}$$
$31$ $$(T + 7)^{4}$$
$37$ $$(T^{2} + 64)^{2}$$
$41$ $$(T^{2} - 6 T + 36)^{2}$$
$43$ $$T^{4} - 16T^{2} + 256$$
$47$ $$T^{4} - 36T^{2} + 1296$$
$53$ $$T^{4} - 36T^{2} + 1296$$
$59$ $$(T^{2} + 15 T + 225)^{2}$$
$61$ $$(T^{2} + 5 T + 25)^{2}$$
$67$ $$T^{4} - 4T^{2} + 16$$
$71$ $$(T^{2} - 3 T + 9)^{2}$$
$73$ $$T^{4} - 64T^{2} + 4096$$
$79$ $$(T^{2} - 5 T + 25)^{2}$$
$83$ $$(T^{2} + 144)^{2}$$
$89$ $$(T^{2} + 15 T + 225)^{2}$$
$97$ $$T^{4} - 64T^{2} + 4096$$