# Properties

 Label 475.2.a.i Level $475$ Weight $2$ Character orbit 475.a Self dual yes Analytic conductor $3.793$ Analytic rank $0$ Dimension $4$ 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: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.11344.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 4x^{2} + 4x + 3$$ x^4 - 2*x^3 - 4*x^2 + 4*x + 3 Coefficient ring: $$\Z[a_1, a_2]$$ 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)

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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + \beta_{3} q^{3} + ( - \beta_{2} + \beta_1 + 1) q^{4} + (2 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{6} + (2 \beta_1 - 2) q^{7} + ( - \beta_{3} - \beta_{2} + 2) q^{8} + ( - 2 \beta_{2} + 1) q^{9}+O(q^{10})$$ q - b2 * q^2 + b3 * q^3 + (-b2 + b1 + 1) * q^4 + (2*b3 + b2 - b1 + 2) * q^6 + (2*b1 - 2) * q^7 + (-b3 - b2 + 2) * q^8 + (-2*b2 + 1) * q^9 $$q - \beta_{2} q^{2} + \beta_{3} q^{3} + ( - \beta_{2} + \beta_1 + 1) q^{4} + (2 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{6} + (2 \beta_1 - 2) q^{7} + ( - \beta_{3} - \beta_{2} + 2) q^{8} + ( - 2 \beta_{2} + 1) q^{9} + 2 \beta_1 q^{11} + (3 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{12} + ( - \beta_{3} - 2 \beta_1) q^{13} + ( - 2 \beta_{3} - 2 \beta_1 - 2) q^{14} + ( - 2 \beta_{3} - 2 \beta_{2} - 1) q^{16} + ( - 2 \beta_{3} - 2) q^{17} + ( - 3 \beta_{2} + 2 \beta_1 + 6) q^{18} + q^{19} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{21} + ( - 2 \beta_{3} - 2 \beta_{2} + \cdots - 2) q^{22}+ \cdots + ( - 4 \beta_{3} - 4 \beta_{2} + \cdots - 4) q^{99}+O(q^{100})$$ q - b2 * q^2 + b3 * q^3 + (-b2 + b1 + 1) * q^4 + (2*b3 + b2 - b1 + 2) * q^6 + (2*b1 - 2) * q^7 + (-b3 - b2 + 2) * q^8 + (-2*b2 + 1) * q^9 + 2*b1 * q^11 + (3*b3 + 2*b2 - 2*b1 + 2) * q^12 + (-b3 - 2*b1) * q^13 + (-2*b3 - 2*b1 - 2) * q^14 + (-2*b3 - 2*b2 - 1) * q^16 + (-2*b3 - 2) * q^17 + (-3*b2 + 2*b1 + 6) * q^18 + q^19 + (-2*b3 + 2*b2 - 2*b1) * q^21 + (-2*b3 - 2*b2 - 2*b1 - 2) * q^22 + (2*b3 + 2*b1 + 2) * q^23 + (4*b3 + 3*b2 - b1 - 2) * q^24 + (b2 + 3*b1) * q^26 + (2*b3 + 2*b2 - 2*b1 + 4) * q^27 + (-2*b3 + 2*b2 + 2) * q^28 + (2*b3 + 2) * q^29 + (2*b3 + 2*b2 - 2*b1 + 4) * q^31 + (-2*b3 - b2 + 4*b1 - 2) * q^32 + (2*b2 - 2*b1) * q^33 + (-4*b3 + 2*b1 - 4) * q^34 + (-2*b3 - 7*b2 + b1 + 5) * q^36 + (-b3 - 2*b2) * q^37 - b2 * q^38 + (2*b1 - 4) * q^39 + (2*b2 - 2*b1 + 6) * q^41 + (-2*b3 + 2*b2 + 2*b1 - 8) * q^42 + (2*b1 - 2) * q^43 + (-2*b3 + 2*b1 + 4) * q^44 + (2*b3 - 2*b2 - 4*b1 + 2) * q^46 + (4*b2 - 2*b1 + 6) * q^47 + (3*b3 + 6*b2 - 2*b1 - 4) * q^48 + (4*b2 - 4*b1 + 9) * q^49 + (-2*b3 + 4*b2 - 8) * q^51 + (-b3 - 2*b2 - 6) * q^52 + (b3 + 2*b2 + 4) * q^53 + (6*b3 + 2*b2 - 2*b1) * q^54 + (-2*b2 + 4*b1 - 6) * q^56 + b3 * q^57 + (4*b3 - 2*b1 + 4) * q^58 + (2*b2 + 2*b1) * q^59 + (-2*b2 + 4*b1 + 2) * q^61 + (6*b3 + 2*b2 - 2*b1) * q^62 + (-4*b3 - 2*b1 - 6) * q^63 + (-4*b3 - b2 - b1 - 3) * q^64 + (2*b3 + 4*b2 - 4) * q^66 + (3*b3 - 2*b2 + 2*b1 + 4) * q^67 + (-6*b3 - 2*b2 + 2*b1 - 6) * q^68 + (2*b3 - 2*b2 - 2*b1 + 8) * q^69 + (2*b3 - 2*b2 - 2*b1 - 4) * q^71 + (-5*b3 - 9*b2 + 4*b1 + 4) * q^72 + (2*b3 - 6) * q^73 + (-2*b3 - 3*b2 + 3*b1 + 4) * q^74 + (-b2 + b1 + 1) * q^76 + (4*b2 + 12) * q^77 + (-2*b3 + 2*b2 - 2*b1 - 2) * q^78 + (2*b2 + 2*b1 - 4) * q^79 + (-2*b2 + 4*b1 + 1) * q^81 + (2*b3 - 2*b2 - 4) * q^82 + (-2*b3 + 2*b1 - 2) * q^83 + (-2*b3 + 2*b2 + 2*b1 - 12) * q^84 + (-2*b3 - 2*b1 - 2) * q^86 + (2*b3 - 4*b2 + 8) * q^87 + (-2*b3 - 4*b2 + 4*b1 - 2) * q^88 + (-2*b3 - 4*b2 - 2) * q^89 + (2*b3 - 6*b2 + 2*b1 - 12) * q^91 + (4*b3 + 2*b2 + 10) * q^92 + (-8*b2 + 4*b1 + 4) * q^93 + (2*b3 - 2*b1 - 10) * q^94 + (9*b2 - 5*b1 - 6) * q^96 + (b3 + 2*b2 - 4*b1 - 4) * q^97 + (4*b3 - b2 - 8) * q^98 + (-4*b3 - 4*b2 - 2*b1 - 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} - 2 q^{3} + 8 q^{4} - 4 q^{7} + 12 q^{8} + 8 q^{9}+O(q^{10})$$ 4 * q + 2 * q^2 - 2 * q^3 + 8 * q^4 - 4 * q^7 + 12 * q^8 + 8 * q^9 $$4 q + 2 q^{2} - 2 q^{3} + 8 q^{4} - 4 q^{7} + 12 q^{8} + 8 q^{9} + 4 q^{11} - 6 q^{12} - 2 q^{13} - 8 q^{14} + 4 q^{16} - 4 q^{17} + 34 q^{18} + 4 q^{19} - 4 q^{21} - 4 q^{22} + 8 q^{23} - 24 q^{24} + 4 q^{26} + 4 q^{27} + 8 q^{28} + 4 q^{29} + 4 q^{31} + 6 q^{32} - 8 q^{33} - 4 q^{34} + 40 q^{36} + 6 q^{37} + 2 q^{38} - 12 q^{39} + 16 q^{41} - 28 q^{42} - 4 q^{43} + 24 q^{44} + 12 q^{47} - 38 q^{48} + 20 q^{49} - 36 q^{51} - 18 q^{52} + 10 q^{53} - 20 q^{54} - 12 q^{56} - 2 q^{57} + 4 q^{58} + 20 q^{61} - 20 q^{62} - 20 q^{63} - 4 q^{64} - 28 q^{66} + 18 q^{67} - 4 q^{68} + 28 q^{69} - 20 q^{71} + 52 q^{72} - 28 q^{73} + 32 q^{74} + 8 q^{76} + 40 q^{77} - 12 q^{78} - 16 q^{79} + 16 q^{81} - 16 q^{82} - 44 q^{84} - 8 q^{86} + 36 q^{87} + 12 q^{88} + 4 q^{89} - 36 q^{91} + 28 q^{92} + 40 q^{93} - 48 q^{94} - 52 q^{96} - 30 q^{97} - 38 q^{98} - 4 q^{99}+O(q^{100})$$ 4 * q + 2 * q^2 - 2 * q^3 + 8 * q^4 - 4 * q^7 + 12 * q^8 + 8 * q^9 + 4 * q^11 - 6 * q^12 - 2 * q^13 - 8 * q^14 + 4 * q^16 - 4 * q^17 + 34 * q^18 + 4 * q^19 - 4 * q^21 - 4 * q^22 + 8 * q^23 - 24 * q^24 + 4 * q^26 + 4 * q^27 + 8 * q^28 + 4 * q^29 + 4 * q^31 + 6 * q^32 - 8 * q^33 - 4 * q^34 + 40 * q^36 + 6 * q^37 + 2 * q^38 - 12 * q^39 + 16 * q^41 - 28 * q^42 - 4 * q^43 + 24 * q^44 + 12 * q^47 - 38 * q^48 + 20 * q^49 - 36 * q^51 - 18 * q^52 + 10 * q^53 - 20 * q^54 - 12 * q^56 - 2 * q^57 + 4 * q^58 + 20 * q^61 - 20 * q^62 - 20 * q^63 - 4 * q^64 - 28 * q^66 + 18 * q^67 - 4 * q^68 + 28 * q^69 - 20 * q^71 + 52 * q^72 - 28 * q^73 + 32 * q^74 + 8 * q^76 + 40 * q^77 - 12 * q^78 - 16 * q^79 + 16 * q^81 - 16 * q^82 - 44 * q^84 - 8 * q^86 + 36 * q^87 + 12 * q^88 + 4 * q^89 - 36 * q^91 + 28 * q^92 + 40 * q^93 - 48 * q^94 - 52 * q^96 - 30 * q^97 - 38 * q^98 - 4 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - 4x^{2} + 4x + 3$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ v^2 - v - 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 2\nu^{2} - 3\nu + 2$$ v^3 - 2*v^2 - 3*v + 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 3$$ b2 + b1 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 2\beta_{2} + 5\beta _1 + 4$$ b3 + 2*b2 + 5*b1 + 4

## 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
 2.78165 −1.51658 −0.552409 1.28734
−1.95594 −0.296842 1.82571 0 0.580605 3.56331 0.340899 −2.91188 0
1.2 −0.816594 −1.53844 −1.33317 0 1.25628 −5.03316 2.72185 −0.633188 0
1.3 2.14243 2.87834 2.59002 0 6.16666 −3.10482 1.26409 5.28487 0
1.4 2.63010 −3.04306 4.91744 0 −8.00355 0.574672 7.67316 6.26020 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.i 4
3.b odd 2 1 4275.2.a.bo 4
4.b odd 2 1 7600.2.a.cf 4
5.b even 2 1 95.2.a.b 4
5.c odd 4 2 475.2.b.e 8
15.d odd 2 1 855.2.a.m 4
19.b odd 2 1 9025.2.a.bf 4
20.d odd 2 1 1520.2.a.t 4
35.c odd 2 1 4655.2.a.y 4
40.e odd 2 1 6080.2.a.ch 4
40.f even 2 1 6080.2.a.cc 4
95.d odd 2 1 1805.2.a.p 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.a.b 4 5.b even 2 1
475.2.a.i 4 1.a even 1 1 trivial
475.2.b.e 8 5.c odd 4 2
855.2.a.m 4 15.d odd 2 1
1520.2.a.t 4 20.d odd 2 1
1805.2.a.p 4 95.d odd 2 1
4275.2.a.bo 4 3.b odd 2 1
4655.2.a.y 4 35.c odd 2 1
6080.2.a.cc 4 40.f even 2 1
6080.2.a.ch 4 40.e odd 2 1
7600.2.a.cf 4 4.b odd 2 1
9025.2.a.bf 4 19.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 2T_{2}^{3} - 6T_{2}^{2} + 8T_{2} + 9$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(475))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2 T^{3} + \cdots + 9$$
$3$ $$T^{4} + 2 T^{3} + \cdots - 4$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 4 T^{3} + \cdots + 32$$
$11$ $$T^{4} - 4 T^{3} + \cdots + 48$$
$13$ $$T^{4} + 2 T^{3} + \cdots + 20$$
$17$ $$T^{4} + 4 T^{3} + \cdots + 48$$
$19$ $$(T - 1)^{4}$$
$23$ $$T^{4} - 8 T^{3} + \cdots + 288$$
$29$ $$T^{4} - 4 T^{3} + \cdots + 48$$
$31$ $$T^{4} - 4 T^{3} + \cdots - 640$$
$37$ $$T^{4} - 6 T^{3} + \cdots + 4$$
$41$ $$T^{4} - 16 T^{3} + \cdots - 240$$
$43$ $$T^{4} + 4 T^{3} + \cdots + 32$$
$47$ $$T^{4} - 12 T^{3} + \cdots + 1056$$
$53$ $$T^{4} - 10 T^{3} + \cdots - 348$$
$59$ $$T^{4} - 64 T^{2} + \cdots - 192$$
$61$ $$T^{4} - 20 T^{3} + \cdots - 2656$$
$67$ $$T^{4} - 18 T^{3} + \cdots - 1076$$
$71$ $$T^{4} + 20 T^{3} + \cdots - 4224$$
$73$ $$T^{4} + 28 T^{3} + \cdots + 176$$
$79$ $$T^{4} + 16 T^{3} + \cdots - 1856$$
$83$ $$T^{4} - 72 T^{2} + \cdots + 480$$
$89$ $$T^{4} - 4 T^{3} + \cdots + 240$$
$97$ $$T^{4} + 30 T^{3} + \cdots - 1388$$