# Properties

 Label 475.2.b.a Level $475$ Weight $2$ Character orbit 475.b 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(324,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("475.324");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$475 = 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 475.b (of order $$2$$, degree $$1$$, 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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 19) 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
324.1
 − 1.00000i 1.00000i
0 2.00000i 2.00000 0 0 1.00000i 0 −1.00000 0
324.2 0 2.00000i 2.00000 0 0 1.00000i 0 −1.00000 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.b.a 2
5.b even 2 1 inner 475.2.b.a 2
5.c odd 4 1 19.2.a.a 1
5.c odd 4 1 475.2.a.b 1
15.e even 4 1 171.2.a.b 1
15.e even 4 1 4275.2.a.i 1
20.e even 4 1 304.2.a.f 1
20.e even 4 1 7600.2.a.c 1
35.f even 4 1 931.2.a.a 1
35.k even 12 2 931.2.f.b 2
35.l odd 12 2 931.2.f.c 2
40.i odd 4 1 1216.2.a.o 1
40.k even 4 1 1216.2.a.b 1
55.e even 4 1 2299.2.a.b 1
60.l odd 4 1 2736.2.a.c 1
65.h odd 4 1 3211.2.a.a 1
85.g odd 4 1 5491.2.a.b 1
95.g even 4 1 361.2.a.b 1
95.g even 4 1 9025.2.a.d 1
95.l even 12 2 361.2.c.a 2
95.m odd 12 2 361.2.c.c 2
95.q odd 36 6 361.2.e.d 6
95.r even 36 6 361.2.e.e 6
105.k odd 4 1 8379.2.a.j 1
285.j odd 4 1 3249.2.a.d 1
380.j odd 4 1 5776.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.2.a.a 1 5.c odd 4 1
171.2.a.b 1 15.e even 4 1
304.2.a.f 1 20.e even 4 1
361.2.a.b 1 95.g even 4 1
361.2.c.a 2 95.l even 12 2
361.2.c.c 2 95.m odd 12 2
361.2.e.d 6 95.q odd 36 6
361.2.e.e 6 95.r even 36 6
475.2.a.b 1 5.c odd 4 1
475.2.b.a 2 1.a even 1 1 trivial
475.2.b.a 2 5.b even 2 1 inner
931.2.a.a 1 35.f even 4 1
931.2.f.b 2 35.k even 12 2
931.2.f.c 2 35.l odd 12 2
1216.2.a.b 1 40.k even 4 1
1216.2.a.o 1 40.i odd 4 1
2299.2.a.b 1 55.e even 4 1
2736.2.a.c 1 60.l odd 4 1
3211.2.a.a 1 65.h odd 4 1
3249.2.a.d 1 285.j odd 4 1
4275.2.a.i 1 15.e even 4 1
5491.2.a.b 1 85.g odd 4 1
5776.2.a.c 1 380.j odd 4 1
7600.2.a.c 1 20.e even 4 1
8379.2.a.j 1 105.k odd 4 1
9025.2.a.d 1 95.g even 4 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

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 4$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 1$$
$11$ $$(T - 3)^{2}$$
$13$ $$T^{2} + 16$$
$17$ $$T^{2} + 9$$
$19$ $$(T + 1)^{2}$$
$23$ $$T^{2}$$
$29$ $$(T + 6)^{2}$$
$31$ $$(T + 4)^{2}$$
$37$ $$T^{2} + 4$$
$41$ $$(T + 6)^{2}$$
$43$ $$T^{2} + 1$$
$47$ $$T^{2} + 9$$
$53$ $$T^{2} + 144$$
$59$ $$(T - 6)^{2}$$
$61$ $$(T + 1)^{2}$$
$67$ $$T^{2} + 16$$
$71$ $$(T - 6)^{2}$$
$73$ $$T^{2} + 49$$
$79$ $$(T + 8)^{2}$$
$83$ $$T^{2} + 144$$
$89$ $$(T + 12)^{2}$$
$97$ $$T^{2} + 64$$