# Properties

 Label 475.2.a.j Level $475$ Weight $2$ Character orbit 475.a Self dual yes Analytic conductor $3.793$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# 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: $$6$$ Coefficient field: 6.6.66064384.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 9x^{4} + 13x^{2} - 1$$ x^6 - 9*x^4 + 13*x^2 - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{4} q^{2} + ( - \beta_{5} - \beta_1) q^{3} + ( - \beta_{3} + 1) q^{4} + \beta_{2} q^{6} + ( - \beta_{5} - \beta_{4}) q^{7} + ( - \beta_{5} - \beta_{4} + \beta_1) q^{8} + (2 \beta_{3} + 3) q^{9}+O(q^{10})$$ q - b4 * q^2 + (-b5 - b1) * q^3 + (-b3 + 1) * q^4 + b2 * q^6 + (-b5 - b4) * q^7 + (-b5 - b4 + b1) * q^8 + (2*b3 + 3) * q^9 $$q - \beta_{4} q^{2} + ( - \beta_{5} - \beta_1) q^{3} + ( - \beta_{3} + 1) q^{4} + \beta_{2} q^{6} + ( - \beta_{5} - \beta_{4}) q^{7} + ( - \beta_{5} - \beta_{4} + \beta_1) q^{8} + (2 \beta_{3} + 3) q^{9} + ( - \beta_{3} - \beta_{2}) q^{11} + (\beta_{5} - \beta_1) q^{12} + (\beta_{5} - \beta_1) q^{13} + ( - 2 \beta_{3} + 2) q^{14} + ( - \beta_{3} - \beta_{2} - 1) q^{16} + (\beta_{5} + \beta_{4} - 2 \beta_1) q^{17} + (2 \beta_{5} + \beta_{4} - 2 \beta_1) q^{18} - q^{19} + (2 \beta_{3} + 4) q^{21} + ( - 2 \beta_{4} + 4 \beta_1) q^{22} + 2 \beta_1 q^{23} + (2 \beta_{3} - \beta_{2} + 2) q^{24} + (2 \beta_{3} + \beta_{2} + 2) q^{26} - 4 \beta_{5} q^{27} + ( - 4 \beta_{4} + 2 \beta_1) q^{28} + 6 q^{29} - 2 \beta_{2} q^{31} + (2 \beta_{5} + \beta_{4} + 2 \beta_1) q^{32} + (2 \beta_{4} + 2 \beta_1) q^{33} + (4 \beta_{3} + 2 \beta_{2}) q^{34} + (\beta_{3} + 2 \beta_{2} - 5) q^{36} + (\beta_{5} + 2 \beta_{4} + \beta_1) q^{37} + \beta_{4} q^{38} + ( - 2 \beta_{3} + 2 \beta_{2} - 2) q^{39} + (2 \beta_{2} + 2) q^{41} + (2 \beta_{5} - 2 \beta_1) q^{42} + (\beta_{5} + \beta_{4}) q^{43} + ( - 4 \beta_{3} - 2 \beta_{2} + 2) q^{44} + ( - 2 \beta_{3} - 2 \beta_{2} - 2) q^{46} + ( - \beta_{5} + 3 \beta_{4} - 4 \beta_1) q^{47} + (\beta_{5} + 2 \beta_{4} + 3 \beta_1) q^{48} + ( - \beta_{3} - \beta_{2} - 1) q^{49} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{51} + ( - \beta_{5} + 2 \beta_{4} - 3 \beta_1) q^{52} + (\beta_{5} + 4 \beta_{4} - \beta_1) q^{53} + ( - 4 \beta_{3} - 4) q^{54} + ( - 2 \beta_{3} - 2 \beta_{2} + 6) q^{56} + (\beta_{5} + \beta_1) q^{57} - 6 \beta_{4} q^{58} + (2 \beta_{3} + 4) q^{59} + (\beta_{3} + 3 \beta_{2} - 2) q^{61} + (2 \beta_{5} + 6 \beta_1) q^{62} + ( - 5 \beta_{5} + 3 \beta_{4} - 4 \beta_1) q^{63} + (3 \beta_{3} - 1) q^{64} + ( - 2 \beta_{2} - 8) q^{66} + (5 \beta_{5} + 2 \beta_{4} + 3 \beta_1) q^{67} + (6 \beta_{4} - 6 \beta_1) q^{68} + ( - 2 \beta_{2} - 4) q^{69} + ( - 2 \beta_{3} + 8) q^{71} + ( - 5 \beta_{5} + 5 \beta_{4} - 3 \beta_1) q^{72} + (3 \beta_{5} - \beta_{4} + 2 \beta_1) q^{73} + (2 \beta_{3} - \beta_{2} - 6) q^{74} + (\beta_{3} - 1) q^{76} + ( - \beta_{5} - 3 \beta_{4} + 4 \beta_1) q^{77} + ( - 4 \beta_{5} - 2 \beta_{4} - 4 \beta_1) q^{78} + ( - 2 \beta_{2} - 4) q^{79} + (2 \beta_{3} - 4 \beta_{2} + 7) q^{81} + ( - 2 \beta_{5} - 2 \beta_{4} - 6 \beta_1) q^{82} + ( - 4 \beta_{4} - 2 \beta_1) q^{83} + (2 \beta_{2} - 4) q^{84} + (2 \beta_{3} - 2) q^{86} + ( - 6 \beta_{5} - 6 \beta_1) q^{87} + ( - 2 \beta_{5} - 6 \beta_{4} + 2 \beta_1) q^{88} + ( - 6 \beta_{3} + 2 \beta_{2} + 2) q^{89} + (2 \beta_{2} - 4) q^{91} + ( - 2 \beta_{4} + 4 \beta_1) q^{92} + ( - 4 \beta_{5} + 4 \beta_{4} + 4 \beta_1) q^{93} + (6 \beta_{3} + 4 \beta_{2} - 6) q^{94} + ( - 4 \beta_{3} - \beta_{2} - 12) q^{96} + (\beta_{5} - \beta_1) q^{97} + ( - \beta_{4} + 4 \beta_1) q^{98} + (3 \beta_{3} - \beta_{2} - 4) q^{99}+O(q^{100})$$ q - b4 * q^2 + (-b5 - b1) * q^3 + (-b3 + 1) * q^4 + b2 * q^6 + (-b5 - b4) * q^7 + (-b5 - b4 + b1) * q^8 + (2*b3 + 3) * q^9 + (-b3 - b2) * q^11 + (b5 - b1) * q^12 + (b5 - b1) * q^13 + (-2*b3 + 2) * q^14 + (-b3 - b2 - 1) * q^16 + (b5 + b4 - 2*b1) * q^17 + (2*b5 + b4 - 2*b1) * q^18 - q^19 + (2*b3 + 4) * q^21 + (-2*b4 + 4*b1) * q^22 + 2*b1 * q^23 + (2*b3 - b2 + 2) * q^24 + (2*b3 + b2 + 2) * q^26 - 4*b5 * q^27 + (-4*b4 + 2*b1) * q^28 + 6 * q^29 - 2*b2 * q^31 + (2*b5 + b4 + 2*b1) * q^32 + (2*b4 + 2*b1) * q^33 + (4*b3 + 2*b2) * q^34 + (b3 + 2*b2 - 5) * q^36 + (b5 + 2*b4 + b1) * q^37 + b4 * q^38 + (-2*b3 + 2*b2 - 2) * q^39 + (2*b2 + 2) * q^41 + (2*b5 - 2*b1) * q^42 + (b5 + b4) * q^43 + (-4*b3 - 2*b2 + 2) * q^44 + (-2*b3 - 2*b2 - 2) * q^46 + (-b5 + 3*b4 - 4*b1) * q^47 + (b5 + 2*b4 + 3*b1) * q^48 + (-b3 - b2 - 1) * q^49 + (-2*b3 + 2*b2) * q^51 + (-b5 + 2*b4 - 3*b1) * q^52 + (b5 + 4*b4 - b1) * q^53 + (-4*b3 - 4) * q^54 + (-2*b3 - 2*b2 + 6) * q^56 + (b5 + b1) * q^57 - 6*b4 * q^58 + (2*b3 + 4) * q^59 + (b3 + 3*b2 - 2) * q^61 + (2*b5 + 6*b1) * q^62 + (-5*b5 + 3*b4 - 4*b1) * q^63 + (3*b3 - 1) * q^64 + (-2*b2 - 8) * q^66 + (5*b5 + 2*b4 + 3*b1) * q^67 + (6*b4 - 6*b1) * q^68 + (-2*b2 - 4) * q^69 + (-2*b3 + 8) * q^71 + (-5*b5 + 5*b4 - 3*b1) * q^72 + (3*b5 - b4 + 2*b1) * q^73 + (2*b3 - b2 - 6) * q^74 + (b3 - 1) * q^76 + (-b5 - 3*b4 + 4*b1) * q^77 + (-4*b5 - 2*b4 - 4*b1) * q^78 + (-2*b2 - 4) * q^79 + (2*b3 - 4*b2 + 7) * q^81 + (-2*b5 - 2*b4 - 6*b1) * q^82 + (-4*b4 - 2*b1) * q^83 + (2*b2 - 4) * q^84 + (2*b3 - 2) * q^86 + (-6*b5 - 6*b1) * q^87 + (-2*b5 - 6*b4 + 2*b1) * q^88 + (-6*b3 + 2*b2 + 2) * q^89 + (2*b2 - 4) * q^91 + (-2*b4 + 4*b1) * q^92 + (-4*b5 + 4*b4 + 4*b1) * q^93 + (6*b3 + 4*b2 - 6) * q^94 + (-4*b3 - b2 - 12) * q^96 + (b5 - b1) * q^97 + (-b4 + 4*b1) * q^98 + (3*b3 - b2 - 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 8 q^{4} + 14 q^{9}+O(q^{10})$$ 6 * q + 8 * q^4 + 14 * q^9 $$6 q + 8 q^{4} + 14 q^{9} + 2 q^{11} + 16 q^{14} - 4 q^{16} - 6 q^{19} + 20 q^{21} + 8 q^{24} + 8 q^{26} + 36 q^{29} - 8 q^{34} - 32 q^{36} - 8 q^{39} + 12 q^{41} + 20 q^{44} - 8 q^{46} - 4 q^{49} + 4 q^{51} - 16 q^{54} + 40 q^{56} + 20 q^{59} - 14 q^{61} - 12 q^{64} - 48 q^{66} - 24 q^{69} + 52 q^{71} - 40 q^{74} - 8 q^{76} - 24 q^{79} + 38 q^{81} - 24 q^{84} - 16 q^{86} + 24 q^{89} - 24 q^{91} - 48 q^{94} - 64 q^{96} - 30 q^{99}+O(q^{100})$$ 6 * q + 8 * q^4 + 14 * q^9 + 2 * q^11 + 16 * q^14 - 4 * q^16 - 6 * q^19 + 20 * q^21 + 8 * q^24 + 8 * q^26 + 36 * q^29 - 8 * q^34 - 32 * q^36 - 8 * q^39 + 12 * q^41 + 20 * q^44 - 8 * q^46 - 4 * q^49 + 4 * q^51 - 16 * q^54 + 40 * q^56 + 20 * q^59 - 14 * q^61 - 12 * q^64 - 48 * q^66 - 24 * q^69 + 52 * q^71 - 40 * q^74 - 8 * q^76 - 24 * q^79 + 38 * q^81 - 24 * q^84 - 16 * q^86 + 24 * q^89 - 24 * q^91 - 48 * q^94 - 64 * q^96 - 30 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 9x^{4} + 13x^{2} - 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{3}$$ $$=$$ $$( \nu^{4} - 8\nu^{2} + 5 ) / 2$$ (v^4 - 8*v^2 + 5) / 2 $$\beta_{4}$$ $$=$$ $$( \nu^{5} - 8\nu^{3} + 7\nu ) / 2$$ (v^5 - 8*v^3 + 7*v) / 2 $$\beta_{5}$$ $$=$$ $$-\nu^{5} + 9\nu^{3} - 13\nu$$ -v^5 + 9*v^3 - 13*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{5} + 2\beta_{4} + 6\beta_1$$ b5 + 2*b4 + 6*b1 $$\nu^{4}$$ $$=$$ $$2\beta_{3} + 8\beta_{2} + 19$$ 2*b3 + 8*b2 + 19 $$\nu^{5}$$ $$=$$ $$8\beta_{5} + 18\beta_{4} + 41\beta_1$$ 8*b5 + 18*b4 + 41*b1

## 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
 −1.30397 2.68667 0.285442 −0.285442 −2.68667 1.30397
−2.41987 0.537080 3.85577 0 −1.29966 −3.18676 −4.49073 −2.71155 0
1.2 −1.82254 −2.31446 1.32164 0 4.21819 −1.45033 1.23634 2.35673 0
1.3 −0.906968 3.21789 −1.17741 0 −2.91852 2.59637 2.88181 7.35482 0
1.4 0.906968 −3.21789 −1.17741 0 −2.91852 −2.59637 −2.88181 7.35482 0
1.5 1.82254 2.31446 1.32164 0 4.21819 1.45033 −1.23634 2.35673 0
1.6 2.41987 −0.537080 3.85577 0 −1.29966 3.18676 4.49073 −2.71155 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$-1$$
$$19$$ $$1$$

## 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.a.j 6
3.b odd 2 1 4275.2.a.br 6
4.b odd 2 1 7600.2.a.ck 6
5.b even 2 1 inner 475.2.a.j 6
5.c odd 4 2 95.2.b.b 6
15.d odd 2 1 4275.2.a.br 6
15.e even 4 2 855.2.c.d 6
19.b odd 2 1 9025.2.a.bx 6
20.d odd 2 1 7600.2.a.ck 6
20.e even 4 2 1520.2.d.h 6
95.d odd 2 1 9025.2.a.bx 6
95.g even 4 2 1805.2.b.e 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.b.b 6 5.c odd 4 2
475.2.a.j 6 1.a even 1 1 trivial
475.2.a.j 6 5.b even 2 1 inner
855.2.c.d 6 15.e even 4 2
1520.2.d.h 6 20.e even 4 2
1805.2.b.e 6 95.g even 4 2
4275.2.a.br 6 3.b odd 2 1
4275.2.a.br 6 15.d odd 2 1
7600.2.a.ck 6 4.b odd 2 1
7600.2.a.ck 6 20.d odd 2 1
9025.2.a.bx 6 19.b odd 2 1
9025.2.a.bx 6 95.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} - 10T_{2}^{4} + 27T_{2}^{2} - 16$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(475))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - 10 T^{4} + \cdots - 16$$
$3$ $$T^{6} - 16 T^{4} + \cdots - 16$$
$5$ $$T^{6}$$
$7$ $$T^{6} - 19 T^{4} + \cdots - 144$$
$11$ $$(T^{3} - T^{2} - 16 T + 12)^{2}$$
$13$ $$T^{6} - 28 T^{4} + \cdots - 576$$
$17$ $$T^{6} - 59 T^{4} + \cdots - 5184$$
$19$ $$(T + 1)^{6}$$
$23$ $$T^{6} - 36 T^{4} + \cdots - 64$$
$29$ $$(T - 6)^{6}$$
$31$ $$(T^{3} - 56 T + 128)^{2}$$
$37$ $$T^{6} - 56 T^{4} + \cdots - 1296$$
$41$ $$(T^{3} - 6 T^{2} - 44 T - 24)^{2}$$
$43$ $$T^{6} - 19 T^{4} + \cdots - 144$$
$47$ $$T^{6} - 187 T^{4} + \cdots - 85264$$
$53$ $$T^{6} - 156 T^{4} + \cdots - 64$$
$59$ $$(T^{3} - 10 T^{2} + \cdots + 48)^{2}$$
$61$ $$(T^{3} + 7 T^{2} + \cdots - 776)^{2}$$
$67$ $$T^{6} - 340 T^{4} + \cdots - 484416$$
$71$ $$(T^{3} - 26 T^{2} + \cdots - 432)^{2}$$
$73$ $$T^{6} - 131 T^{4} + \cdots - 5184$$
$79$ $$(T^{3} + 12 T^{2} + \cdots - 32)^{2}$$
$83$ $$T^{6} - 228 T^{4} + \cdots - 141376$$
$89$ $$(T^{3} - 12 T^{2} + \cdots + 3456)^{2}$$
$97$ $$T^{6} - 28 T^{4} + \cdots - 576$$