# Properties

 Label 475.2.a.f Level $475$ Weight $2$ Character orbit 475.a Self dual yes Analytic conductor $3.793$ Analytic rank $1$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.148.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + (\beta_{2} + \beta_1 - 1) q^{3} + (\beta_{2} + \beta_1) q^{4} + ( - \beta_{2} - \beta_1 - 1) q^{6} - 2 \beta_{2} q^{7} + ( - \beta_{2} - 1) q^{8} + ( - 2 \beta_{2} + 1) q^{9}+O(q^{10})$$ q - b1 * q^2 + (b2 + b1 - 1) * q^3 + (b2 + b1) * q^4 + (-b2 - b1 - 1) * q^6 - 2*b2 * q^7 + (-b2 - 1) * q^8 + (-2*b2 + 1) * q^9 $$q - \beta_1 q^{2} + (\beta_{2} + \beta_1 - 1) q^{3} + (\beta_{2} + \beta_1) q^{4} + ( - \beta_{2} - \beta_1 - 1) q^{6} - 2 \beta_{2} q^{7} + ( - \beta_{2} - 1) q^{8} + ( - 2 \beta_{2} + 1) q^{9} + ( - 2 \beta_1 - 2) q^{11} + ( - \beta_{2} + \beta_1 + 3) q^{12} + ( - \beta_{2} + \beta_1 - 3) q^{13} + (2 \beta_1 - 2) q^{14} + ( - 2 \beta_{2} - 1) q^{16} + (2 \beta_{2} - 2 \beta_1) q^{17} + (\beta_1 - 2) q^{18} - q^{19} + (4 \beta_{2} - 4) q^{21} + (2 \beta_{2} + 4 \beta_1 + 4) q^{22} + ( - 2 \beta_1 + 2) q^{23} + (\beta_{2} - \beta_1 - 1) q^{24} + ( - \beta_{2} + 3 \beta_1 - 3) q^{26} + (2 \beta_{2} - 2 \beta_1 - 2) q^{27} + (2 \beta_{2} - 4) q^{28} + (2 \beta_{2} + 2 \beta_1 - 4) q^{29} + 4 \beta_1 q^{31} + (2 \beta_{2} + 3 \beta_1) q^{32} + ( - 4 \beta_{2} - 4 \beta_1) q^{33} + (2 \beta_{2} + 6) q^{34} + (3 \beta_{2} + \beta_1 - 4) q^{36} + ( - \beta_{2} + \beta_1 - 7) q^{37} + \beta_1 q^{38} + ( - 2 \beta_1 + 2) q^{39} + (2 \beta_{2} - 2 \beta_1) q^{41} + 4 q^{42} + (6 \beta_{2} + 4 \beta_1) q^{43} + ( - 4 \beta_{2} - 6 \beta_1 - 2) q^{44} + (2 \beta_{2} + 4) q^{46} - 2 \beta_{2} q^{47} + (3 \beta_{2} - \beta_1 - 3) q^{48} + ( - 4 \beta_{2} - 4 \beta_1 + 5) q^{49} + ( - 6 \beta_{2} - 2 \beta_1 + 2) q^{51} + ( - \beta_{2} - \beta_1 - 1) q^{52} + (\beta_{2} - \beta_1 - 5) q^{53} + (2 \beta_{2} + 2 \beta_1 + 6) q^{54} + ( - 2 \beta_1 + 6) q^{56} + ( - \beta_{2} - \beta_1 + 1) q^{57} + ( - 2 \beta_{2} - 2) q^{58} + ( - 2 \beta_{2} - 2 \beta_1 - 6) q^{59} + ( - 2 \beta_{2} + 4 \beta_1 - 2) q^{61} + ( - 4 \beta_{2} - 4 \beta_1 - 8) q^{62} + ( - 6 \beta_{2} - 4 \beta_1 + 12) q^{63} + (\beta_{2} - 5 \beta_1 - 2) q^{64} + (4 \beta_{2} + 8 \beta_1 + 4) q^{66} + ( - \beta_{2} - 5 \beta_1 + 1) q^{67} + ( - 4 \beta_{2} - 4 \beta_1 + 2) q^{68} - 4 q^{69} + ( - 4 \beta_{2} - 4 \beta_1) q^{71} + ( - \beta_{2} - 2 \beta_1 + 5) q^{72} + ( - 2 \beta_{2} - 2 \beta_1) q^{73} + ( - \beta_{2} + 7 \beta_1 - 3) q^{74} + ( - \beta_{2} - \beta_1) q^{76} + (4 \beta_{2} + 4 \beta_1 - 4) q^{77} + (2 \beta_{2} + 4) q^{78} + (6 \beta_{2} + 6 \beta_1 - 2) q^{79} + ( - 2 \beta_{2} - 4 \beta_1 + 1) q^{81} + (2 \beta_{2} + 6) q^{82} + (2 \beta_1 + 10) q^{83} + ( - 8 \beta_{2} - 4 \beta_1 + 8) q^{84} + ( - 4 \beta_{2} - 10 \beta_1 - 2) q^{86} + ( - 6 \beta_{2} - 2 \beta_1 + 10) q^{87} + (2 \beta_{2} + 4 \beta_1) q^{88} + (6 \beta_{2} + 2 \beta_1) q^{89} + (4 \beta_{2} - 4 \beta_1 + 8) q^{91} + ( - 2 \beta_1 - 2) q^{92} + (4 \beta_{2} + 4 \beta_1 + 4) q^{93} + (2 \beta_1 - 2) q^{94} + ( - \beta_{2} + 3 \beta_1 + 7) q^{96} + (5 \beta_{2} + 7 \beta_1 - 9) q^{97} + (4 \beta_{2} + 3 \beta_1 + 4) q^{98} + (4 \beta_{2} + 2 \beta_1 - 6) q^{99}+O(q^{100})$$ q - b1 * q^2 + (b2 + b1 - 1) * q^3 + (b2 + b1) * q^4 + (-b2 - b1 - 1) * q^6 - 2*b2 * q^7 + (-b2 - 1) * q^8 + (-2*b2 + 1) * q^9 + (-2*b1 - 2) * q^11 + (-b2 + b1 + 3) * q^12 + (-b2 + b1 - 3) * q^13 + (2*b1 - 2) * q^14 + (-2*b2 - 1) * q^16 + (2*b2 - 2*b1) * q^17 + (b1 - 2) * q^18 - q^19 + (4*b2 - 4) * q^21 + (2*b2 + 4*b1 + 4) * q^22 + (-2*b1 + 2) * q^23 + (b2 - b1 - 1) * q^24 + (-b2 + 3*b1 - 3) * q^26 + (2*b2 - 2*b1 - 2) * q^27 + (2*b2 - 4) * q^28 + (2*b2 + 2*b1 - 4) * q^29 + 4*b1 * q^31 + (2*b2 + 3*b1) * q^32 + (-4*b2 - 4*b1) * q^33 + (2*b2 + 6) * q^34 + (3*b2 + b1 - 4) * q^36 + (-b2 + b1 - 7) * q^37 + b1 * q^38 + (-2*b1 + 2) * q^39 + (2*b2 - 2*b1) * q^41 + 4 * q^42 + (6*b2 + 4*b1) * q^43 + (-4*b2 - 6*b1 - 2) * q^44 + (2*b2 + 4) * q^46 - 2*b2 * q^47 + (3*b2 - b1 - 3) * q^48 + (-4*b2 - 4*b1 + 5) * q^49 + (-6*b2 - 2*b1 + 2) * q^51 + (-b2 - b1 - 1) * q^52 + (b2 - b1 - 5) * q^53 + (2*b2 + 2*b1 + 6) * q^54 + (-2*b1 + 6) * q^56 + (-b2 - b1 + 1) * q^57 + (-2*b2 - 2) * q^58 + (-2*b2 - 2*b1 - 6) * q^59 + (-2*b2 + 4*b1 - 2) * q^61 + (-4*b2 - 4*b1 - 8) * q^62 + (-6*b2 - 4*b1 + 12) * q^63 + (b2 - 5*b1 - 2) * q^64 + (4*b2 + 8*b1 + 4) * q^66 + (-b2 - 5*b1 + 1) * q^67 + (-4*b2 - 4*b1 + 2) * q^68 - 4 * q^69 + (-4*b2 - 4*b1) * q^71 + (-b2 - 2*b1 + 5) * q^72 + (-2*b2 - 2*b1) * q^73 + (-b2 + 7*b1 - 3) * q^74 + (-b2 - b1) * q^76 + (4*b2 + 4*b1 - 4) * q^77 + (2*b2 + 4) * q^78 + (6*b2 + 6*b1 - 2) * q^79 + (-2*b2 - 4*b1 + 1) * q^81 + (2*b2 + 6) * q^82 + (2*b1 + 10) * q^83 + (-8*b2 - 4*b1 + 8) * q^84 + (-4*b2 - 10*b1 - 2) * q^86 + (-6*b2 - 2*b1 + 10) * q^87 + (2*b2 + 4*b1) * q^88 + (6*b2 + 2*b1) * q^89 + (4*b2 - 4*b1 + 8) * q^91 + (-2*b1 - 2) * q^92 + (4*b2 + 4*b1 + 4) * q^93 + (2*b1 - 2) * q^94 + (-b2 + 3*b1 + 7) * q^96 + (5*b2 + 7*b1 - 9) * q^97 + (4*b2 + 3*b1 + 4) * q^98 + (4*b2 + 2*b1 - 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{2} - 2 q^{3} + q^{4} - 4 q^{6} - 3 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q - q^2 - 2 * q^3 + q^4 - 4 * q^6 - 3 * q^8 + 3 * q^9 $$3 q - q^{2} - 2 q^{3} + q^{4} - 4 q^{6} - 3 q^{8} + 3 q^{9} - 8 q^{11} + 10 q^{12} - 8 q^{13} - 4 q^{14} - 3 q^{16} - 2 q^{17} - 5 q^{18} - 3 q^{19} - 12 q^{21} + 16 q^{22} + 4 q^{23} - 4 q^{24} - 6 q^{26} - 8 q^{27} - 12 q^{28} - 10 q^{29} + 4 q^{31} + 3 q^{32} - 4 q^{33} + 18 q^{34} - 11 q^{36} - 20 q^{37} + q^{38} + 4 q^{39} - 2 q^{41} + 12 q^{42} + 4 q^{43} - 12 q^{44} + 12 q^{46} - 10 q^{48} + 11 q^{49} + 4 q^{51} - 4 q^{52} - 16 q^{53} + 20 q^{54} + 16 q^{56} + 2 q^{57} - 6 q^{58} - 20 q^{59} - 2 q^{61} - 28 q^{62} + 32 q^{63} - 11 q^{64} + 20 q^{66} - 2 q^{67} + 2 q^{68} - 12 q^{69} - 4 q^{71} + 13 q^{72} - 2 q^{73} - 2 q^{74} - q^{76} - 8 q^{77} + 12 q^{78} - q^{81} + 18 q^{82} + 32 q^{83} + 20 q^{84} - 16 q^{86} + 28 q^{87} + 4 q^{88} + 2 q^{89} + 20 q^{91} - 8 q^{92} + 16 q^{93} - 4 q^{94} + 24 q^{96} - 20 q^{97} + 15 q^{98} - 16 q^{99}+O(q^{100})$$ 3 * q - q^2 - 2 * q^3 + q^4 - 4 * q^6 - 3 * q^8 + 3 * q^9 - 8 * q^11 + 10 * q^12 - 8 * q^13 - 4 * q^14 - 3 * q^16 - 2 * q^17 - 5 * q^18 - 3 * q^19 - 12 * q^21 + 16 * q^22 + 4 * q^23 - 4 * q^24 - 6 * q^26 - 8 * q^27 - 12 * q^28 - 10 * q^29 + 4 * q^31 + 3 * q^32 - 4 * q^33 + 18 * q^34 - 11 * q^36 - 20 * q^37 + q^38 + 4 * q^39 - 2 * q^41 + 12 * q^42 + 4 * q^43 - 12 * q^44 + 12 * q^46 - 10 * q^48 + 11 * q^49 + 4 * q^51 - 4 * q^52 - 16 * q^53 + 20 * q^54 + 16 * q^56 + 2 * q^57 - 6 * q^58 - 20 * q^59 - 2 * q^61 - 28 * q^62 + 32 * q^63 - 11 * q^64 + 20 * q^66 - 2 * q^67 + 2 * q^68 - 12 * q^69 - 4 * q^71 + 13 * q^72 - 2 * q^73 - 2 * q^74 - q^76 - 8 * q^77 + 12 * q^78 - q^81 + 18 * q^82 + 32 * q^83 + 20 * q^84 - 16 * q^86 + 28 * q^87 + 4 * q^88 + 2 * q^89 + 20 * q^91 - 8 * q^92 + 16 * q^93 - 4 * q^94 + 24 * q^96 - 20 * q^97 + 15 * q^98 - 16 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 3x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ v^2 - v - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 2$$ b2 + b1 + 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.17009 0.311108 −1.48119
−2.17009 1.70928 2.70928 0 −3.70928 −1.07838 −1.53919 −0.0783777 0
1.2 −0.311108 −2.90321 −1.90321 0 0.903212 4.42864 1.21432 5.42864 0
1.3 1.48119 −0.806063 0.193937 0 −1.19394 −3.35026 −2.67513 −2.35026 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.f 3
3.b odd 2 1 4275.2.a.bk 3
4.b odd 2 1 7600.2.a.bx 3
5.b even 2 1 95.2.a.a 3
5.c odd 4 2 475.2.b.d 6
15.d odd 2 1 855.2.a.i 3
19.b odd 2 1 9025.2.a.bb 3
20.d odd 2 1 1520.2.a.p 3
35.c odd 2 1 4655.2.a.u 3
40.e odd 2 1 6080.2.a.by 3
40.f even 2 1 6080.2.a.bo 3
95.d odd 2 1 1805.2.a.f 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.a.a 3 5.b even 2 1
475.2.a.f 3 1.a even 1 1 trivial
475.2.b.d 6 5.c odd 4 2
855.2.a.i 3 15.d odd 2 1
1520.2.a.p 3 20.d odd 2 1
1805.2.a.f 3 95.d odd 2 1
4275.2.a.bk 3 3.b odd 2 1
4655.2.a.u 3 35.c odd 2 1
6080.2.a.bo 3 40.f even 2 1
6080.2.a.by 3 40.e odd 2 1
7600.2.a.bx 3 4.b odd 2 1
9025.2.a.bb 3 19.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} + T_{2}^{2} - 3T_{2} - 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(475))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + T^{2} - 3T - 1$$
$3$ $$T^{3} + 2 T^{2} - 4 T - 4$$
$5$ $$T^{3}$$
$7$ $$T^{3} - 16T - 16$$
$11$ $$T^{3} + 8 T^{2} + 8 T - 16$$
$13$ $$T^{3} + 8 T^{2} + 12 T + 4$$
$17$ $$T^{3} + 2 T^{2} - 36 T - 104$$
$19$ $$(T + 1)^{3}$$
$23$ $$T^{3} - 4 T^{2} - 8 T + 16$$
$29$ $$T^{3} + 10 T^{2} + 12 T - 40$$
$31$ $$T^{3} - 4 T^{2} - 48 T + 64$$
$37$ $$T^{3} + 20 T^{2} + 124 T + 244$$
$41$ $$T^{3} + 2 T^{2} - 36 T - 104$$
$43$ $$T^{3} - 4 T^{2} - 144 T + 592$$
$47$ $$T^{3} - 16T - 16$$
$53$ $$T^{3} + 16 T^{2} + 76 T + 92$$
$59$ $$T^{3} + 20 T^{2} + 112 T + 160$$
$61$ $$T^{3} + 2 T^{2} - 84 T + 232$$
$67$ $$T^{3} + 2 T^{2} - 76 T + 116$$
$71$ $$T^{3} + 4 T^{2} - 80 T - 64$$
$73$ $$T^{3} + 2 T^{2} - 20 T - 8$$
$79$ $$T^{3} - 192T - 160$$
$83$ $$T^{3} - 32 T^{2} + 328 T - 1072$$
$89$ $$T^{3} - 2 T^{2} - 132 T + 680$$
$97$ $$T^{3} + 20 T^{2} - 60 T - 1748$$