# Properties

 Label 475.2.a.a Level $475$ Weight $2$ Character orbit 475.a Self dual yes Analytic conductor $3.793$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,2,Mod(1,475)]

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([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("475.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$475 = 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 475.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: $$3.79289409601$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 95) 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)

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{4} + 2 q^{7} + 3 q^{8} - 3 q^{9}+O(q^{10})$$ q - q^2 - q^4 + 2 * q^7 + 3 * q^8 - 3 * q^9 $$q - q^{2} - q^{4} + 2 q^{7} + 3 q^{8} - 3 q^{9} - 4 q^{11} - 2 q^{13} - 2 q^{14} - q^{16} + 4 q^{17} + 3 q^{18} + q^{19} + 4 q^{22} - 6 q^{23} + 2 q^{26} - 2 q^{28} - 6 q^{29} - 4 q^{31} - 5 q^{32} - 4 q^{34} + 3 q^{36} - 10 q^{37} - q^{38} - 10 q^{41} + 2 q^{43} + 4 q^{44} + 6 q^{46} - 6 q^{47} - 3 q^{49} + 2 q^{52} + 10 q^{53} + 6 q^{56} + 6 q^{58} + 2 q^{61} + 4 q^{62} - 6 q^{63} + 7 q^{64} + 8 q^{67} - 4 q^{68} + 4 q^{71} - 9 q^{72} + 4 q^{73} + 10 q^{74} - q^{76} - 8 q^{77} + 4 q^{79} + 9 q^{81} + 10 q^{82} - 18 q^{83} - 2 q^{86} - 12 q^{88} - 2 q^{89} - 4 q^{91} + 6 q^{92} + 6 q^{94} + 6 q^{97} + 3 q^{98} + 12 q^{99}+O(q^{100})$$ q - q^2 - q^4 + 2 * q^7 + 3 * q^8 - 3 * q^9 - 4 * q^11 - 2 * q^13 - 2 * q^14 - q^16 + 4 * q^17 + 3 * q^18 + q^19 + 4 * q^22 - 6 * q^23 + 2 * q^26 - 2 * q^28 - 6 * q^29 - 4 * q^31 - 5 * q^32 - 4 * q^34 + 3 * q^36 - 10 * q^37 - q^38 - 10 * q^41 + 2 * q^43 + 4 * q^44 + 6 * q^46 - 6 * q^47 - 3 * q^49 + 2 * q^52 + 10 * q^53 + 6 * q^56 + 6 * q^58 + 2 * q^61 + 4 * q^62 - 6 * q^63 + 7 * q^64 + 8 * q^67 - 4 * q^68 + 4 * q^71 - 9 * q^72 + 4 * q^73 + 10 * q^74 - q^76 - 8 * q^77 + 4 * q^79 + 9 * q^81 + 10 * q^82 - 18 * q^83 - 2 * q^86 - 12 * q^88 - 2 * q^89 - 4 * q^91 + 6 * q^92 + 6 * q^94 + 6 * q^97 + 3 * q^98 + 12 * 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
 0
−1.00000 0 −1.00000 0 0 2.00000 3.00000 −3.00000 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
$$5$$ $$-1$$
$$19$$ $$-1$$

## 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 475.2.a.a 1
3.b odd 2 1 4275.2.a.p 1
4.b odd 2 1 7600.2.a.i 1
5.b even 2 1 475.2.a.c 1
5.c odd 4 2 95.2.b.a 2
15.d odd 2 1 4275.2.a.e 1
15.e even 4 2 855.2.c.b 2
19.b odd 2 1 9025.2.a.h 1
20.d odd 2 1 7600.2.a.l 1
20.e even 4 2 1520.2.d.b 2
95.d odd 2 1 9025.2.a.c 1
95.g even 4 2 1805.2.b.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.b.a 2 5.c odd 4 2
475.2.a.a 1 1.a even 1 1 trivial
475.2.a.c 1 5.b even 2 1
855.2.c.b 2 15.e even 4 2
1520.2.d.b 2 20.e even 4 2
1805.2.b.c 2 95.g even 4 2
4275.2.a.e 1 15.d odd 2 1
4275.2.a.p 1 3.b odd 2 1
7600.2.a.i 1 4.b odd 2 1
7600.2.a.l 1 20.d odd 2 1
9025.2.a.c 1 95.d odd 2 1
9025.2.a.h 1 19.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(475))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T - 2$$
$11$ $$T + 4$$
$13$ $$T + 2$$
$17$ $$T - 4$$
$19$ $$T - 1$$
$23$ $$T + 6$$
$29$ $$T + 6$$
$31$ $$T + 4$$
$37$ $$T + 10$$
$41$ $$T + 10$$
$43$ $$T - 2$$
$47$ $$T + 6$$
$53$ $$T - 10$$
$59$ $$T$$
$61$ $$T - 2$$
$67$ $$T - 8$$
$71$ $$T - 4$$
$73$ $$T - 4$$
$79$ $$T - 4$$
$83$ $$T + 18$$
$89$ $$T + 2$$
$97$ $$T - 6$$