# Properties

 Label 475.2.a.d 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: $$\Q(\zeta_{14})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 2x + 1$$ x^3 - x^2 - 2*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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 - 1) q^{2} + (\beta_{2} - \beta_1) q^{3} + (\beta_{2} + 2 \beta_1 + 1) q^{4} + ( - \beta_{2} + \beta_1 + 1) q^{6} + (\beta_{2} + \beta_1) q^{7} + ( - 4 \beta_{2} - \beta_1 - 4) q^{8} + ( - 2 \beta_{2} + \beta_1 - 2) q^{9}+O(q^{10})$$ q + (-b1 - 1) * q^2 + (b2 - b1) * q^3 + (b2 + 2*b1 + 1) * q^4 + (-b2 + b1 + 1) * q^6 + (b2 + b1) * q^7 + (-4*b2 - b1 - 4) * q^8 + (-2*b2 + b1 - 2) * q^9 $$q + ( - \beta_1 - 1) q^{2} + (\beta_{2} - \beta_1) q^{3} + (\beta_{2} + 2 \beta_1 + 1) q^{4} + ( - \beta_{2} + \beta_1 + 1) q^{6} + (\beta_{2} + \beta_1) q^{7} + ( - 4 \beta_{2} - \beta_1 - 4) q^{8} + ( - 2 \beta_{2} + \beta_1 - 2) q^{9} + ( - 3 \beta_{2} + \beta_1 - 1) q^{11} + ( - \beta_{2} - 2) q^{12} + ( - \beta_{2} - 2) q^{13} + ( - 3 \beta_{2} - \beta_1 - 3) q^{14} + (7 \beta_{2} + \beta_1 + 8) q^{16} + ( - \beta_{2} + 3 \beta_1 - 2) q^{17} + (3 \beta_{2} + \beta_1 + 2) q^{18} - q^{19} + ( - 2 \beta_{2} + \beta_1 - 1) q^{21} + (5 \beta_{2} + 2) q^{22} + (\beta_1 - 3) q^{23} + (4 \beta_{2} + 1) q^{24} + (2 \beta_{2} + 2 \beta_1 + 3) q^{26} + ( - \beta_{2} + 3 \beta_1 - 1) q^{27} + (5 \beta_{2} + 2 \beta_1 + 8) q^{28} + (5 \beta_{2} - 5 \beta_1 + 1) q^{29} + ( - 3 \beta_{2} - 2 \beta_1 - 2) q^{31} + ( - 7 \beta_{2} - 7 \beta_1 - 9) q^{32} + (5 \beta_{2} - 2 \beta_1 - 1) q^{33} + ( - \beta_{2} - \beta_1 - 3) q^{34} + ( - 3 \beta_{2} - 5 \beta_1 - 3) q^{36} + ( - \beta_{2} + 3 \beta_1 - 3) q^{37} + (\beta_1 + 1) q^{38} + \beta_1 q^{39} + (8 \beta_{2} - 3 \beta_1 + 4) q^{41} + (3 \beta_{2} + 1) q^{42} + ( - 6 \beta_{2} + 4 \beta_1 - 5) q^{43} + ( - 4 \beta_{2} - 4 \beta_1 - 5) q^{44} + ( - \beta_{2} + 2 \beta_1 + 1) q^{46} + (\beta_{2} - 2 \beta_1) q^{47} + ( - 6 \beta_{2} - \beta_1 - 1) q^{48} + (2 \beta_{2} + \beta_1 - 2) q^{49} + (\beta_1 - 3) q^{51} + ( - 4 \beta_{2} - 5 \beta_1 - 5) q^{52} + (4 \beta_{2} - 6 \beta_1 - 3) q^{53} + ( - \beta_{2} - 2 \beta_1 - 4) q^{54} + ( - 6 \beta_{2} - 8 \beta_1 - 11) q^{56} + ( - \beta_{2} + \beta_1) q^{57} + ( - 5 \beta_{2} + 4 \beta_1 + 4) q^{58} + ( - 2 \beta_{2} - 4 \beta_1 + 4) q^{59} + (\beta_{2} - 4 \beta_1 - 4) q^{61} + (8 \beta_{2} + 4 \beta_1 + 9) q^{62} + ( - 4 \beta_1 - 1) q^{63} + (7 \beta_{2} + 14 \beta_1 + 14) q^{64} + ( - 8 \beta_{2} + 3 \beta_1) q^{66} + (5 \beta_{2} - 9 \beta_1 + 5) q^{67} + (5 \beta_{2} - 2 \beta_1 + 10) q^{68} + ( - 3 \beta_{2} + 3 \beta_1 - 1) q^{69} + (2 \beta_{2} + 4 \beta_1 - 7) q^{71} + (5 \beta_{2} + 6 \beta_1 + 12) q^{72} + ( - 8 \beta_{2} - \beta_1 - 2) q^{73} + ( - \beta_{2} - 2) q^{74} + ( - \beta_{2} - 2 \beta_1 - 1) q^{76} + (\beta_{2} - 4 \beta_1 - 3) q^{77} + ( - \beta_{2} - \beta_1 - 2) q^{78} + ( - 3 \beta_{2} + 3 \beta_1 + 4) q^{79} + (7 \beta_{2} - 3 \beta_1 + 3) q^{81} + ( - 13 \beta_{2} - \beta_1 - 6) q^{82} + (5 \beta_1 - 6) q^{83} + ( - 2 \beta_{2} - 3 \beta_1 - 2) q^{84} + (8 \beta_{2} + \beta_1 + 3) q^{86} + ( - 9 \beta_{2} + 4 \beta_1 + 5) q^{87} + (2 \beta_{2} + 9 \beta_1 + 13) q^{88} + ( - 6 \beta_{2} - \beta_1 - 1) q^{89} + ( - 2 \beta_{2} - 3 \beta_1 - 2) q^{91} + ( - 5 \beta_1 + 2) q^{92} + (4 \beta_{2} - \beta_1 + 2) q^{93} + (2 \beta_1 + 3) q^{94} + (5 \beta_{2} + 2 \beta_1 + 7) q^{96} + ( - \beta_{2} - 2) q^{97} + ( - 5 \beta_{2} + \beta_1 - 2) q^{98} + ( - 2 \beta_{2} + 3 \beta_1 + 5) q^{99}+O(q^{100})$$ q + (-b1 - 1) * q^2 + (b2 - b1) * q^3 + (b2 + 2*b1 + 1) * q^4 + (-b2 + b1 + 1) * q^6 + (b2 + b1) * q^7 + (-4*b2 - b1 - 4) * q^8 + (-2*b2 + b1 - 2) * q^9 + (-3*b2 + b1 - 1) * q^11 + (-b2 - 2) * q^12 + (-b2 - 2) * q^13 + (-3*b2 - b1 - 3) * q^14 + (7*b2 + b1 + 8) * q^16 + (-b2 + 3*b1 - 2) * q^17 + (3*b2 + b1 + 2) * q^18 - q^19 + (-2*b2 + b1 - 1) * q^21 + (5*b2 + 2) * q^22 + (b1 - 3) * q^23 + (4*b2 + 1) * q^24 + (2*b2 + 2*b1 + 3) * q^26 + (-b2 + 3*b1 - 1) * q^27 + (5*b2 + 2*b1 + 8) * q^28 + (5*b2 - 5*b1 + 1) * q^29 + (-3*b2 - 2*b1 - 2) * q^31 + (-7*b2 - 7*b1 - 9) * q^32 + (5*b2 - 2*b1 - 1) * q^33 + (-b2 - b1 - 3) * q^34 + (-3*b2 - 5*b1 - 3) * q^36 + (-b2 + 3*b1 - 3) * q^37 + (b1 + 1) * q^38 + b1 * q^39 + (8*b2 - 3*b1 + 4) * q^41 + (3*b2 + 1) * q^42 + (-6*b2 + 4*b1 - 5) * q^43 + (-4*b2 - 4*b1 - 5) * q^44 + (-b2 + 2*b1 + 1) * q^46 + (b2 - 2*b1) * q^47 + (-6*b2 - b1 - 1) * q^48 + (2*b2 + b1 - 2) * q^49 + (b1 - 3) * q^51 + (-4*b2 - 5*b1 - 5) * q^52 + (4*b2 - 6*b1 - 3) * q^53 + (-b2 - 2*b1 - 4) * q^54 + (-6*b2 - 8*b1 - 11) * q^56 + (-b2 + b1) * q^57 + (-5*b2 + 4*b1 + 4) * q^58 + (-2*b2 - 4*b1 + 4) * q^59 + (b2 - 4*b1 - 4) * q^61 + (8*b2 + 4*b1 + 9) * q^62 + (-4*b1 - 1) * q^63 + (7*b2 + 14*b1 + 14) * q^64 + (-8*b2 + 3*b1) * q^66 + (5*b2 - 9*b1 + 5) * q^67 + (5*b2 - 2*b1 + 10) * q^68 + (-3*b2 + 3*b1 - 1) * q^69 + (2*b2 + 4*b1 - 7) * q^71 + (5*b2 + 6*b1 + 12) * q^72 + (-8*b2 - b1 - 2) * q^73 + (-b2 - 2) * q^74 + (-b2 - 2*b1 - 1) * q^76 + (b2 - 4*b1 - 3) * q^77 + (-b2 - b1 - 2) * q^78 + (-3*b2 + 3*b1 + 4) * q^79 + (7*b2 - 3*b1 + 3) * q^81 + (-13*b2 - b1 - 6) * q^82 + (5*b1 - 6) * q^83 + (-2*b2 - 3*b1 - 2) * q^84 + (8*b2 + b1 + 3) * q^86 + (-9*b2 + 4*b1 + 5) * q^87 + (2*b2 + 9*b1 + 13) * q^88 + (-6*b2 - b1 - 1) * q^89 + (-2*b2 - 3*b1 - 2) * q^91 + (-5*b1 + 2) * q^92 + (4*b2 - b1 + 2) * q^93 + (2*b1 + 3) * q^94 + (5*b2 + 2*b1 + 7) * q^96 + (-b2 - 2) * q^97 + (-5*b2 + b1 - 2) * q^98 + (-2*b2 + 3*b1 + 5) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 4 q^{2} - 2 q^{3} + 4 q^{4} + 5 q^{6} - 9 q^{8} - 3 q^{9}+O(q^{10})$$ 3 * q - 4 * q^2 - 2 * q^3 + 4 * q^4 + 5 * q^6 - 9 * q^8 - 3 * q^9 $$3 q - 4 q^{2} - 2 q^{3} + 4 q^{4} + 5 q^{6} - 9 q^{8} - 3 q^{9} + q^{11} - 5 q^{12} - 5 q^{13} - 7 q^{14} + 18 q^{16} - 2 q^{17} + 4 q^{18} - 3 q^{19} + q^{22} - 8 q^{23} - q^{24} + 9 q^{26} + q^{27} + 21 q^{28} - 7 q^{29} - 5 q^{31} - 27 q^{32} - 10 q^{33} - 9 q^{34} - 11 q^{36} - 5 q^{37} + 4 q^{38} + q^{39} + q^{41} - 5 q^{43} - 15 q^{44} + 6 q^{46} - 3 q^{47} + 2 q^{48} - 7 q^{49} - 8 q^{51} - 16 q^{52} - 19 q^{53} - 13 q^{54} - 35 q^{56} + 2 q^{57} + 21 q^{58} + 10 q^{59} - 17 q^{61} + 23 q^{62} - 7 q^{63} + 49 q^{64} + 11 q^{66} + q^{67} + 23 q^{68} + 3 q^{69} - 19 q^{71} + 37 q^{72} + q^{73} - 5 q^{74} - 4 q^{76} - 14 q^{77} - 6 q^{78} + 18 q^{79} - q^{81} - 6 q^{82} - 13 q^{83} - 7 q^{84} + 2 q^{86} + 28 q^{87} + 46 q^{88} + 2 q^{89} - 7 q^{91} + q^{92} + q^{93} + 11 q^{94} + 18 q^{96} - 5 q^{97} + 20 q^{99}+O(q^{100})$$ 3 * q - 4 * q^2 - 2 * q^3 + 4 * q^4 + 5 * q^6 - 9 * q^8 - 3 * q^9 + q^11 - 5 * q^12 - 5 * q^13 - 7 * q^14 + 18 * q^16 - 2 * q^17 + 4 * q^18 - 3 * q^19 + q^22 - 8 * q^23 - q^24 + 9 * q^26 + q^27 + 21 * q^28 - 7 * q^29 - 5 * q^31 - 27 * q^32 - 10 * q^33 - 9 * q^34 - 11 * q^36 - 5 * q^37 + 4 * q^38 + q^39 + q^41 - 5 * q^43 - 15 * q^44 + 6 * q^46 - 3 * q^47 + 2 * q^48 - 7 * q^49 - 8 * q^51 - 16 * q^52 - 19 * q^53 - 13 * q^54 - 35 * q^56 + 2 * q^57 + 21 * q^58 + 10 * q^59 - 17 * q^61 + 23 * q^62 - 7 * q^63 + 49 * q^64 + 11 * q^66 + q^67 + 23 * q^68 + 3 * q^69 - 19 * q^71 + 37 * q^72 + q^73 - 5 * q^74 - 4 * q^76 - 14 * q^77 - 6 * q^78 + 18 * q^79 - q^81 - 6 * q^82 - 13 * q^83 - 7 * q^84 + 2 * q^86 + 28 * q^87 + 46 * q^88 + 2 * q^89 - 7 * q^91 + q^92 + q^93 + 11 * q^94 + 18 * q^96 - 5 * q^97 + 20 * q^99

Basis of coefficient ring in terms of $$\nu = \zeta_{14} + \zeta_{14}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 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
 1.80194 0.445042 −1.24698
−2.80194 −0.554958 5.85086 0 1.55496 3.04892 −10.7899 −2.69202 0
1.2 −1.44504 −2.24698 0.0881460 0 3.24698 −1.35690 2.76271 2.04892 0
1.3 0.246980 0.801938 −1.93900 0 0.198062 −1.69202 −0.972853 −2.35690 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.d 3
3.b odd 2 1 4275.2.a.bn 3
4.b odd 2 1 7600.2.a.bw 3
5.b even 2 1 475.2.a.h yes 3
5.c odd 4 2 475.2.b.c 6
15.d odd 2 1 4275.2.a.z 3
19.b odd 2 1 9025.2.a.be 3
20.d odd 2 1 7600.2.a.bn 3
95.d odd 2 1 9025.2.a.w 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
475.2.a.d 3 1.a even 1 1 trivial
475.2.a.h yes 3 5.b even 2 1
475.2.b.c 6 5.c odd 4 2
4275.2.a.z 3 15.d odd 2 1
4275.2.a.bn 3 3.b odd 2 1
7600.2.a.bn 3 20.d odd 2 1
7600.2.a.bw 3 4.b odd 2 1
9025.2.a.w 3 95.d odd 2 1
9025.2.a.be 3 19.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} + 4T_{2}^{2} + 3T_{2} - 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(475))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} + 4 T^{2} + \cdots - 1$$
$3$ $$T^{3} + 2T^{2} - T - 1$$
$5$ $$T^{3}$$
$7$ $$T^{3} - 7T - 7$$
$11$ $$T^{3} - T^{2} + \cdots - 13$$
$13$ $$T^{3} + 5 T^{2} + \cdots + 1$$
$17$ $$T^{3} + 2 T^{2} + \cdots + 13$$
$19$ $$(T + 1)^{3}$$
$23$ $$T^{3} + 8 T^{2} + \cdots + 13$$
$29$ $$T^{3} + 7 T^{2} + \cdots - 91$$
$31$ $$T^{3} + 5 T^{2} + \cdots + 43$$
$37$ $$T^{3} + 5 T^{2} + \cdots + 1$$
$41$ $$T^{3} - T^{2} + \cdots + 421$$
$43$ $$T^{3} + 5 T^{2} + \cdots - 293$$
$47$ $$T^{3} + 3 T^{2} + \cdots - 13$$
$53$ $$T^{3} + 19 T^{2} + \cdots - 307$$
$59$ $$T^{3} - 10 T^{2} + \cdots + 328$$
$61$ $$T^{3} + 17 T^{2} + \cdots - 41$$
$67$ $$T^{3} - T^{2} + \cdots - 559$$
$71$ $$T^{3} + 19 T^{2} + \cdots - 307$$
$73$ $$T^{3} - T^{2} + \cdots + 463$$
$79$ $$T^{3} - 18 T^{2} + \cdots - 97$$
$83$ $$T^{3} + 13 T^{2} + \cdots - 139$$
$89$ $$T^{3} - 2 T^{2} + \cdots + 281$$
$97$ $$T^{3} + 5 T^{2} + \cdots + 1$$