# Properties

 Label 475.2.a.g Level $475$ Weight $2$ Character orbit 475.a Self dual yes Analytic conductor $3.793$ Analytic rank $0$ 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: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.169.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 4x - 1$$ x^3 - x^2 - 4*x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 1) q^{2} + (\beta_{2} + 1) q^{3} + (\beta_{2} - \beta_1 + 2) q^{4} + (\beta_{2} - 2 \beta_1) q^{6} + ( - \beta_{2} + 1) q^{7} + (2 \beta_{2} - \beta_1 + 2) q^{8} + \beta_1 q^{9}+O(q^{10})$$ q + (-b1 + 1) * q^2 + (b2 + 1) * q^3 + (b2 - b1 + 2) * q^4 + (b2 - 2*b1) * q^6 + (-b2 + 1) * q^7 + (2*b2 - b1 + 2) * q^8 + b1 * q^9 $$q + ( - \beta_1 + 1) q^{2} + (\beta_{2} + 1) q^{3} + (\beta_{2} - \beta_1 + 2) q^{4} + (\beta_{2} - 2 \beta_1) q^{6} + ( - \beta_{2} + 1) q^{7} + (2 \beta_{2} - \beta_1 + 2) q^{8} + \beta_1 q^{9} - \beta_{2} q^{11} + (\beta_{2} - \beta_1 + 3) q^{12} + ( - 3 \beta_{2} + 3 \beta_1 - 1) q^{13} + ( - \beta_{2} + 2) q^{14} + (\beta_{2} - 2 \beta_1 - 1) q^{16} + (\beta_{2} + 5) q^{17} + ( - \beta_{2} - 3) q^{18} + q^{19} + (2 \beta_{2} - \beta_1 - 1) q^{21} + ( - \beta_{2} + \beta_1 + 1) q^{22} + ( - 2 \beta_{2} + 3 \beta_1 + 1) q^{23} + 5 q^{24} + ( - 6 \beta_{2} + 4 \beta_1 - 7) q^{26} + ( - 3 \beta_{2} + 2 \beta_1 - 2) q^{27} + (\beta_{2} - \beta_1 + 1) q^{28} + ( - \beta_{2} - 2) q^{29} + (3 \beta_{2} - \beta_1 + 1) q^{31} + ( - \beta_{2} + 2 \beta_1) q^{32} + (\beta_{2} - \beta_1 - 2) q^{33} + (\beta_{2} - 6 \beta_1 + 4) q^{34} + ( - \beta_{2} + 2 \beta_1 - 2) q^{36} + (5 \beta_{2} - 2 \beta_1 + 4) q^{37} + ( - \beta_1 + 1) q^{38} + (2 \beta_{2} + 3 \beta_1 - 4) q^{39} + ( - 2 \beta_{2} + 5 \beta_1 - 2) q^{41} + (3 \beta_{2} - \beta_1) q^{42} + (2 \beta_{2} - 6 \beta_1 + 1) q^{43} - q^{44} + ( - 5 \beta_{2} + \beta_1 - 6) q^{46} + (\beta_{2} - 5 \beta_1 + 5) q^{47} + ( - 2 \beta_{2} - 3 \beta_1 - 1) q^{48} + ( - 4 \beta_{2} + \beta_1 - 4) q^{49} + (4 \beta_{2} + \beta_1 + 7) q^{51} + ( - 4 \beta_{2} + 7 \beta_1 - 11) q^{52} + ( - 2 \beta_{2} + 2 \beta_1 + 9) q^{53} + ( - 5 \beta_{2} + 5 \beta_1 - 5) q^{54} + (4 \beta_{2} - 2 \beta_1 - 1) q^{56} + (\beta_{2} + 1) q^{57} + ( - \beta_{2} + 3 \beta_1 - 1) q^{58} + (2 \beta_{2} + 2 \beta_1 - 2) q^{59} + (\beta_{2} + \beta_1 + 1) q^{61} + (4 \beta_{2} - 4 \beta_1 + 1) q^{62} - q^{63} + ( - 5 \beta_{2} + 5 \beta_1 - 3) q^{64} + (2 \beta_{2} + \beta_1) q^{66} + ( - \beta_{2} + 4 \beta_1 - 6) q^{67} + (5 \beta_{2} - 5 \beta_1 + 11) q^{68} + (3 \beta_{2} + 4 \beta_1) q^{69} + ( - 2 \beta_1 + 3) q^{71} + ( - \beta_{2} + 3 \beta_1 - 1) q^{72} + (2 \beta_{2} - 3 \beta_1 + 2) q^{73} + (7 \beta_{2} - 9 \beta_1 + 5) q^{74} + (\beta_{2} - \beta_1 + 2) q^{76} + ( - 3 \beta_{2} + \beta_1 + 2) q^{77} + ( - \beta_{2} + 2 \beta_1 - 15) q^{78} + (5 \beta_{2} - 4 \beta_1 - 3) q^{79} + (\beta_{2} - 2 \beta_1 - 6) q^{81} + ( - 7 \beta_{2} + 4 \beta_1 - 15) q^{82} + (6 \beta_{2} + 3 \beta_1 + 2) q^{83} + ( - \beta_1 + 2) q^{84} + (8 \beta_{2} - 3 \beta_1 + 17) q^{86} + ( - \beta_{2} - \beta_1 - 4) q^{87} + (2 \beta_{2} - \beta_1 - 3) q^{88} + ( - 2 \beta_{2} + 5 \beta_1 - 9) q^{89} + ( - 8 \beta_{2} + 3 \beta_1 + 2) q^{91} + ( - 2 \beta_{2} + 5 \beta_1 - 6) q^{92} + ( - 2 \beta_{2} + \beta_1 + 6) q^{93} + (6 \beta_{2} - 6 \beta_1 + 19) q^{94} + (\beta_{2} + 3 \beta_1) q^{96} + ( - 3 \beta_{2} - \beta_1 - 5) q^{97} + ( - 5 \beta_{2} + 8 \beta_1 - 3) q^{98} + ( - \beta_1 - 1) q^{99}+O(q^{100})$$ q + (-b1 + 1) * q^2 + (b2 + 1) * q^3 + (b2 - b1 + 2) * q^4 + (b2 - 2*b1) * q^6 + (-b2 + 1) * q^7 + (2*b2 - b1 + 2) * q^8 + b1 * q^9 - b2 * q^11 + (b2 - b1 + 3) * q^12 + (-3*b2 + 3*b1 - 1) * q^13 + (-b2 + 2) * q^14 + (b2 - 2*b1 - 1) * q^16 + (b2 + 5) * q^17 + (-b2 - 3) * q^18 + q^19 + (2*b2 - b1 - 1) * q^21 + (-b2 + b1 + 1) * q^22 + (-2*b2 + 3*b1 + 1) * q^23 + 5 * q^24 + (-6*b2 + 4*b1 - 7) * q^26 + (-3*b2 + 2*b1 - 2) * q^27 + (b2 - b1 + 1) * q^28 + (-b2 - 2) * q^29 + (3*b2 - b1 + 1) * q^31 + (-b2 + 2*b1) * q^32 + (b2 - b1 - 2) * q^33 + (b2 - 6*b1 + 4) * q^34 + (-b2 + 2*b1 - 2) * q^36 + (5*b2 - 2*b1 + 4) * q^37 + (-b1 + 1) * q^38 + (2*b2 + 3*b1 - 4) * q^39 + (-2*b2 + 5*b1 - 2) * q^41 + (3*b2 - b1) * q^42 + (2*b2 - 6*b1 + 1) * q^43 - q^44 + (-5*b2 + b1 - 6) * q^46 + (b2 - 5*b1 + 5) * q^47 + (-2*b2 - 3*b1 - 1) * q^48 + (-4*b2 + b1 - 4) * q^49 + (4*b2 + b1 + 7) * q^51 + (-4*b2 + 7*b1 - 11) * q^52 + (-2*b2 + 2*b1 + 9) * q^53 + (-5*b2 + 5*b1 - 5) * q^54 + (4*b2 - 2*b1 - 1) * q^56 + (b2 + 1) * q^57 + (-b2 + 3*b1 - 1) * q^58 + (2*b2 + 2*b1 - 2) * q^59 + (b2 + b1 + 1) * q^61 + (4*b2 - 4*b1 + 1) * q^62 - q^63 + (-5*b2 + 5*b1 - 3) * q^64 + (2*b2 + b1) * q^66 + (-b2 + 4*b1 - 6) * q^67 + (5*b2 - 5*b1 + 11) * q^68 + (3*b2 + 4*b1) * q^69 + (-2*b1 + 3) * q^71 + (-b2 + 3*b1 - 1) * q^72 + (2*b2 - 3*b1 + 2) * q^73 + (7*b2 - 9*b1 + 5) * q^74 + (b2 - b1 + 2) * q^76 + (-3*b2 + b1 + 2) * q^77 + (-b2 + 2*b1 - 15) * q^78 + (5*b2 - 4*b1 - 3) * q^79 + (b2 - 2*b1 - 6) * q^81 + (-7*b2 + 4*b1 - 15) * q^82 + (6*b2 + 3*b1 + 2) * q^83 + (-b1 + 2) * q^84 + (8*b2 - 3*b1 + 17) * q^86 + (-b2 - b1 - 4) * q^87 + (2*b2 - b1 - 3) * q^88 + (-2*b2 + 5*b1 - 9) * q^89 + (-8*b2 + 3*b1 + 2) * q^91 + (-2*b2 + 5*b1 - 6) * q^92 + (-2*b2 + b1 + 6) * q^93 + (6*b2 - 6*b1 + 19) * q^94 + (b2 + 3*b1) * q^96 + (-3*b2 - b1 - 5) * q^97 + (-5*b2 + 8*b1 - 3) * q^98 + (-b1 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{2} + 2 q^{3} + 4 q^{4} - 3 q^{6} + 4 q^{7} + 3 q^{8} + q^{9}+O(q^{10})$$ 3 * q + 2 * q^2 + 2 * q^3 + 4 * q^4 - 3 * q^6 + 4 * q^7 + 3 * q^8 + q^9 $$3 q + 2 q^{2} + 2 q^{3} + 4 q^{4} - 3 q^{6} + 4 q^{7} + 3 q^{8} + q^{9} + q^{11} + 7 q^{12} + 3 q^{13} + 7 q^{14} - 6 q^{16} + 14 q^{17} - 8 q^{18} + 3 q^{19} - 6 q^{21} + 5 q^{22} + 8 q^{23} + 15 q^{24} - 11 q^{26} - q^{27} + q^{28} - 5 q^{29} - q^{31} + 3 q^{32} - 8 q^{33} + 5 q^{34} - 3 q^{36} + 5 q^{37} + 2 q^{38} - 11 q^{39} + q^{41} - 4 q^{42} - 5 q^{43} - 3 q^{44} - 12 q^{46} + 9 q^{47} - 4 q^{48} - 7 q^{49} + 18 q^{51} - 22 q^{52} + 31 q^{53} - 5 q^{54} - 9 q^{56} + 2 q^{57} + q^{58} - 6 q^{59} + 3 q^{61} - 5 q^{62} - 3 q^{63} + q^{64} - q^{66} - 13 q^{67} + 23 q^{68} + q^{69} + 7 q^{71} + q^{72} + q^{73} - q^{74} + 4 q^{76} + 10 q^{77} - 42 q^{78} - 18 q^{79} - 21 q^{81} - 34 q^{82} + 3 q^{83} + 5 q^{84} + 40 q^{86} - 12 q^{87} - 12 q^{88} - 20 q^{89} + 17 q^{91} - 11 q^{92} + 21 q^{93} + 45 q^{94} + 2 q^{96} - 13 q^{97} + 4 q^{98} - 4 q^{99}+O(q^{100})$$ 3 * q + 2 * q^2 + 2 * q^3 + 4 * q^4 - 3 * q^6 + 4 * q^7 + 3 * q^8 + q^9 + q^11 + 7 * q^12 + 3 * q^13 + 7 * q^14 - 6 * q^16 + 14 * q^17 - 8 * q^18 + 3 * q^19 - 6 * q^21 + 5 * q^22 + 8 * q^23 + 15 * q^24 - 11 * q^26 - q^27 + q^28 - 5 * q^29 - q^31 + 3 * q^32 - 8 * q^33 + 5 * q^34 - 3 * q^36 + 5 * q^37 + 2 * q^38 - 11 * q^39 + q^41 - 4 * q^42 - 5 * q^43 - 3 * q^44 - 12 * q^46 + 9 * q^47 - 4 * q^48 - 7 * q^49 + 18 * q^51 - 22 * q^52 + 31 * q^53 - 5 * q^54 - 9 * q^56 + 2 * q^57 + q^58 - 6 * q^59 + 3 * q^61 - 5 * q^62 - 3 * q^63 + q^64 - q^66 - 13 * q^67 + 23 * q^68 + q^69 + 7 * q^71 + q^72 + q^73 - q^74 + 4 * q^76 + 10 * q^77 - 42 * q^78 - 18 * q^79 - 21 * q^81 - 34 * q^82 + 3 * q^83 + 5 * q^84 + 40 * q^86 - 12 * q^87 - 12 * q^88 - 20 * q^89 + 17 * q^91 - 11 * q^92 + 21 * q^93 + 45 * q^94 + 2 * q^96 - 13 * q^97 + 4 * q^98 - 4 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 4x - 1$$ :

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

## 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
 2.65109 −0.273891 −1.37720
−1.65109 2.37720 0.726109 0 −3.92498 −0.377203 2.10331 2.65109 0
1.2 1.27389 −1.65109 −0.377203 0 −2.10331 3.65109 −3.02830 −0.273891 0
1.3 2.37720 1.27389 3.65109 0 3.02830 0.726109 3.92498 −1.37720 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.g yes 3
3.b odd 2 1 4275.2.a.ba 3
4.b odd 2 1 7600.2.a.bh 3
5.b even 2 1 475.2.a.e 3
5.c odd 4 2 475.2.b.b 6
15.d odd 2 1 4275.2.a.bm 3
19.b odd 2 1 9025.2.a.y 3
20.d odd 2 1 7600.2.a.cc 3
95.d odd 2 1 9025.2.a.bc 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
475.2.a.e 3 5.b even 2 1
475.2.a.g yes 3 1.a even 1 1 trivial
475.2.b.b 6 5.c odd 4 2
4275.2.a.ba 3 3.b odd 2 1
4275.2.a.bm 3 15.d odd 2 1
7600.2.a.bh 3 4.b odd 2 1
7600.2.a.cc 3 20.d odd 2 1
9025.2.a.y 3 19.b odd 2 1
9025.2.a.bc 3 95.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - 2T_{2}^{2} - 3T_{2} + 5$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(475))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - 2 T^{2} + \cdots + 5$$
$3$ $$T^{3} - 2 T^{2} + \cdots + 5$$
$5$ $$T^{3}$$
$7$ $$T^{3} - 4T^{2} + T + 1$$
$11$ $$T^{3} - T^{2} - 4T - 1$$
$13$ $$T^{3} - 3 T^{2} + \cdots + 103$$
$17$ $$T^{3} - 14 T^{2} + \cdots - 79$$
$19$ $$(T - 1)^{3}$$
$23$ $$T^{3} - 8 T^{2} + \cdots + 125$$
$29$ $$T^{3} + 5 T^{2} + \cdots - 5$$
$31$ $$T^{3} + T^{2} + \cdots + 53$$
$37$ $$T^{3} - 5 T^{2} + \cdots + 395$$
$41$ $$T^{3} - T^{2} + \cdots + 155$$
$43$ $$T^{3} + 5 T^{2} + \cdots - 317$$
$47$ $$T^{3} - 9 T^{2} + \cdots + 311$$
$53$ $$T^{3} - 31 T^{2} + \cdots - 905$$
$59$ $$T^{3} + 6 T^{2} + \cdots - 200$$
$61$ $$T^{3} - 3 T^{2} + \cdots - 1$$
$67$ $$T^{3} + 13T^{2} - 169$$
$71$ $$T^{3} - 7T^{2} - T + 47$$
$73$ $$T^{3} - T^{2} + \cdots - 53$$
$79$ $$T^{3} + 18 T^{2} + \cdots - 395$$
$83$ $$T^{3} - 3 T^{2} + \cdots - 131$$
$89$ $$T^{3} + 20 T^{2} + \cdots - 125$$
$97$ $$T^{3} + 13T^{2} - 169$$