# Properties

 Label 475.2.b.c Level $475$ Weight $2$ Character orbit 475.b 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(324,475)]

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([1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("475.324");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$475 = 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 475.b (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$3.79289409601$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.153664.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} + 5x^{4} + 6x^{2} + 1$$ x^6 + 5*x^4 + 6*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{5} + \beta_{3}) q^{2} + ( - \beta_{3} + \beta_1) q^{3} + (3 \beta_{4} + 2 \beta_{2} - 3) q^{4} + ( - \beta_{2} + 2) q^{6} + ( - \beta_{3} - \beta_1) q^{7} + (4 \beta_{5} - \beta_{3} - 4 \beta_1) q^{8} + ( - \beta_{4} + \beta_{2} + 1) q^{9}+O(q^{10})$$ q + (-b5 + b3) * q^2 + (-b3 + b1) * q^3 + (3*b4 + 2*b2 - 3) * q^4 + (-b2 + 2) * q^6 + (-b3 - b1) * q^7 + (4*b5 - b3 - 4*b1) * q^8 + (-b4 + b2 + 1) * q^9 $$q + ( - \beta_{5} + \beta_{3}) q^{2} + ( - \beta_{3} + \beta_1) q^{3} + (3 \beta_{4} + 2 \beta_{2} - 3) q^{4} + ( - \beta_{2} + 2) q^{6} + ( - \beta_{3} - \beta_1) q^{7} + (4 \beta_{5} - \beta_{3} - 4 \beta_1) q^{8} + ( - \beta_{4} + \beta_{2} + 1) q^{9} + (2 \beta_{4} - \beta_{2}) q^{11} + ( - 2 \beta_{5} + \beta_1) q^{12} + (2 \beta_{5} - \beta_1) q^{13} + ( - 4 \beta_{4} - \beta_{2} + 4) q^{14} + ( - 8 \beta_{4} - \beta_{2} + 9) q^{16} + ( - 2 \beta_{5} - 3 \beta_{3} + \beta_1) q^{17} + ( - 2 \beta_{5} + \beta_{3} + 3 \beta_1) q^{18} + q^{19} + (\beta_{4} - \beta_{2}) q^{21} + (2 \beta_{5} - 5 \beta_1) q^{22} + (3 \beta_{5} + \beta_{3}) q^{23} + (4 \beta_{4} - 1) q^{24} + ( - 4 \beta_{4} - 2 \beta_{2} + 5) q^{26} + ( - \beta_{5} - 3 \beta_{3} + \beta_1) q^{27} + ( - 8 \beta_{5} + 2 \beta_{3} + 5 \beta_1) q^{28} + ( - 5 \beta_{2} + 4) q^{29} + (5 \beta_{4} + 2 \beta_{2} - 4) q^{31} + ( - 9 \beta_{5} + 7 \beta_{3} + 7 \beta_1) q^{32} + (\beta_{5} - 2 \beta_{3} + 5 \beta_1) q^{33} + ( - 2 \beta_{4} - \beta_{2} + 4) q^{34} + (8 \beta_{4} + 5 \beta_{2} - 8) q^{36} + ( - 3 \beta_{5} - 3 \beta_{3} + \beta_1) q^{37} + ( - \beta_{5} + \beta_{3}) q^{38} + (\beta_{4} + \beta_{2} - 1) q^{39} + ( - 5 \beta_{4} + 3 \beta_{2} + 1) q^{41} + (\beta_{5} - 3 \beta_1) q^{42} + (5 \beta_{5} + 4 \beta_{3} - 6 \beta_1) q^{43} + ( - 8 \beta_{4} - 4 \beta_{2} + 9) q^{44} + ( - \beta_{4} - 2 \beta_{2} + 3) q^{46} + (2 \beta_{3} - \beta_1) q^{47} + (\beta_{5} - \beta_{3} - 6 \beta_1) q^{48} + (3 \beta_{4} + \beta_{2} + 1) q^{49} + ( - \beta_{4} - \beta_{2} - 2) q^{51} + ( - 5 \beta_{5} + 5 \beta_{3} + 4 \beta_1) q^{52} + (3 \beta_{5} - 6 \beta_{3} + 4 \beta_1) q^{53} + ( - 3 \beta_{4} - 2 \beta_{2} + 6) q^{54} + (14 \beta_{4} + 8 \beta_{2} - 19) q^{56} + ( - \beta_{3} + \beta_1) q^{57} + ( - 4 \beta_{5} + 4 \beta_{3} - 5 \beta_1) q^{58} + ( - 6 \beta_{4} - 4 \beta_{2}) q^{59} + (3 \beta_{4} + 4 \beta_{2} - 8) q^{61} + (9 \beta_{5} - 4 \beta_{3} - 8 \beta_1) q^{62} + (\beta_{5} - 4 \beta_{3}) q^{63} + (21 \beta_{4} + 14 \beta_{2} - 28) q^{64} + (5 \beta_{4} - 3 \beta_{2} + 3) q^{66} + (5 \beta_{5} + 9 \beta_{3} - 5 \beta_1) q^{67} + ( - 10 \beta_{5} - 2 \beta_{3} + 5 \beta_1) q^{68} + (3 \beta_{2} - 2) q^{69} + ( - 6 \beta_{4} - 4 \beta_{2} - 3) q^{71} + (12 \beta_{5} - 6 \beta_{3} - 5 \beta_1) q^{72} + (2 \beta_{5} - \beta_{3} - 8 \beta_1) q^{73} + ( - \beta_{4} + 2) q^{74} + (3 \beta_{4} + 2 \beta_{2} - 3) q^{76} + ( - 3 \beta_{5} + 4 \beta_{3} - \beta_1) q^{77} + (2 \beta_{5} - \beta_{3} - \beta_1) q^{78} + (3 \beta_{2} - 7) q^{79} + ( - 4 \beta_{4} + 3 \beta_{2}) q^{81} + ( - 6 \beta_{5} + \beta_{3} + 13 \beta_1) q^{82} + (6 \beta_{5} + 5 \beta_{3}) q^{83} + ( - 5 \beta_{4} - 3 \beta_{2} + 5) q^{84} + ( - 9 \beta_{4} - \beta_{2} + 4) q^{86} + (5 \beta_{5} - 4 \beta_{3} + 9 \beta_1) q^{87} + ( - 13 \beta_{5} + 9 \beta_{3} + 2 \beta_1) q^{88} + ( - 7 \beta_{4} - \beta_{2} + 2) q^{89} + (5 \beta_{4} + 3 \beta_{2} - 5) q^{91} + (2 \beta_{5} + 5 \beta_{3}) q^{92} + ( - 2 \beta_{5} - \beta_{3} + 4 \beta_1) q^{93} + (2 \beta_{4} + 2 \beta_{2} - 5) q^{94} + ( - 7 \beta_{4} - 2 \beta_{2} + 9) q^{96} + ( - 2 \beta_{5} + \beta_1) q^{97} + (2 \beta_{5} + \beta_{3} - 5 \beta_1) q^{98} + (\beta_{4} + 3 \beta_{2} - 8) q^{99}+O(q^{100})$$ q + (-b5 + b3) * q^2 + (-b3 + b1) * q^3 + (3*b4 + 2*b2 - 3) * q^4 + (-b2 + 2) * q^6 + (-b3 - b1) * q^7 + (4*b5 - b3 - 4*b1) * q^8 + (-b4 + b2 + 1) * q^9 + (2*b4 - b2) * q^11 + (-2*b5 + b1) * q^12 + (2*b5 - b1) * q^13 + (-4*b4 - b2 + 4) * q^14 + (-8*b4 - b2 + 9) * q^16 + (-2*b5 - 3*b3 + b1) * q^17 + (-2*b5 + b3 + 3*b1) * q^18 + q^19 + (b4 - b2) * q^21 + (2*b5 - 5*b1) * q^22 + (3*b5 + b3) * q^23 + (4*b4 - 1) * q^24 + (-4*b4 - 2*b2 + 5) * q^26 + (-b5 - 3*b3 + b1) * q^27 + (-8*b5 + 2*b3 + 5*b1) * q^28 + (-5*b2 + 4) * q^29 + (5*b4 + 2*b2 - 4) * q^31 + (-9*b5 + 7*b3 + 7*b1) * q^32 + (b5 - 2*b3 + 5*b1) * q^33 + (-2*b4 - b2 + 4) * q^34 + (8*b4 + 5*b2 - 8) * q^36 + (-3*b5 - 3*b3 + b1) * q^37 + (-b5 + b3) * q^38 + (b4 + b2 - 1) * q^39 + (-5*b4 + 3*b2 + 1) * q^41 + (b5 - 3*b1) * q^42 + (5*b5 + 4*b3 - 6*b1) * q^43 + (-8*b4 - 4*b2 + 9) * q^44 + (-b4 - 2*b2 + 3) * q^46 + (2*b3 - b1) * q^47 + (b5 - b3 - 6*b1) * q^48 + (3*b4 + b2 + 1) * q^49 + (-b4 - b2 - 2) * q^51 + (-5*b5 + 5*b3 + 4*b1) * q^52 + (3*b5 - 6*b3 + 4*b1) * q^53 + (-3*b4 - 2*b2 + 6) * q^54 + (14*b4 + 8*b2 - 19) * q^56 + (-b3 + b1) * q^57 + (-4*b5 + 4*b3 - 5*b1) * q^58 + (-6*b4 - 4*b2) * q^59 + (3*b4 + 4*b2 - 8) * q^61 + (9*b5 - 4*b3 - 8*b1) * q^62 + (b5 - 4*b3) * q^63 + (21*b4 + 14*b2 - 28) * q^64 + (5*b4 - 3*b2 + 3) * q^66 + (5*b5 + 9*b3 - 5*b1) * q^67 + (-10*b5 - 2*b3 + 5*b1) * q^68 + (3*b2 - 2) * q^69 + (-6*b4 - 4*b2 - 3) * q^71 + (12*b5 - 6*b3 - 5*b1) * q^72 + (2*b5 - b3 - 8*b1) * q^73 + (-b4 + 2) * q^74 + (3*b4 + 2*b2 - 3) * q^76 + (-3*b5 + 4*b3 - b1) * q^77 + (2*b5 - b3 - b1) * q^78 + (3*b2 - 7) * q^79 + (-4*b4 + 3*b2) * q^81 + (-6*b5 + b3 + 13*b1) * q^82 + (6*b5 + 5*b3) * q^83 + (-5*b4 - 3*b2 + 5) * q^84 + (-9*b4 - b2 + 4) * q^86 + (5*b5 - 4*b3 + 9*b1) * q^87 + (-13*b5 + 9*b3 + 2*b1) * q^88 + (-7*b4 - b2 + 2) * q^89 + (5*b4 + 3*b2 - 5) * q^91 + (2*b5 + 5*b3) * q^92 + (-2*b5 - b3 + 4*b1) * q^93 + (2*b4 + 2*b2 - 5) * q^94 + (-7*b4 - 2*b2 + 9) * q^96 + (-2*b5 + b1) * q^97 + (2*b5 + b3 - 5*b1) * q^98 + (b4 + 3*b2 - 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 8 q^{4} + 10 q^{6} + 6 q^{9}+O(q^{10})$$ 6 * q - 8 * q^4 + 10 * q^6 + 6 * q^9 $$6 q - 8 q^{4} + 10 q^{6} + 6 q^{9} + 2 q^{11} + 14 q^{14} + 36 q^{16} + 6 q^{19} + 2 q^{24} + 18 q^{26} + 14 q^{29} - 10 q^{31} + 18 q^{34} - 22 q^{36} - 2 q^{39} + 2 q^{41} + 30 q^{44} + 12 q^{46} + 14 q^{49} - 16 q^{51} + 26 q^{54} - 70 q^{56} - 20 q^{59} - 34 q^{61} - 98 q^{64} + 22 q^{66} - 6 q^{69} - 38 q^{71} + 10 q^{74} - 8 q^{76} - 36 q^{79} - 2 q^{81} + 14 q^{84} + 4 q^{86} - 4 q^{89} - 14 q^{91} - 22 q^{94} + 36 q^{96} - 40 q^{99}+O(q^{100})$$ 6 * q - 8 * q^4 + 10 * q^6 + 6 * q^9 + 2 * q^11 + 14 * q^14 + 36 * q^16 + 6 * q^19 + 2 * q^24 + 18 * q^26 + 14 * q^29 - 10 * q^31 + 18 * q^34 - 22 * q^36 - 2 * q^39 + 2 * q^41 + 30 * q^44 + 12 * q^46 + 14 * q^49 - 16 * q^51 + 26 * q^54 - 70 * q^56 - 20 * q^59 - 34 * q^61 - 98 * q^64 + 22 * q^66 - 6 * q^69 - 38 * q^71 + 10 * q^74 - 8 * q^76 - 36 * q^79 - 2 * q^81 + 14 * q^84 + 4 * q^86 - 4 * q^89 - 14 * q^91 - 22 * q^94 + 36 * q^96 - 40 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 5x^{4} + 6x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + 2$$ v^2 + 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} + 3\nu$$ v^3 + 3*v $$\beta_{4}$$ $$=$$ $$\nu^{4} + 3\nu^{2} + 1$$ v^4 + 3*v^2 + 1 $$\beta_{5}$$ $$=$$ $$\nu^{5} + 4\nu^{3} + 3\nu$$ v^5 + 4*v^3 + 3*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 2$$ b2 - 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 3\beta_1$$ b3 - 3*b1 $$\nu^{4}$$ $$=$$ $$\beta_{4} - 3\beta_{2} + 5$$ b4 - 3*b2 + 5 $$\nu^{5}$$ $$=$$ $$\beta_{5} - 4\beta_{3} + 9\beta_1$$ b5 - 4*b3 + 9*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
324.1
 − 1.24698i 1.80194i − 0.445042i 0.445042i − 1.80194i 1.24698i
2.80194i 0.554958i −5.85086 0 1.55496 3.04892i 10.7899i 2.69202 0
324.2 1.44504i 2.24698i −0.0881460 0 3.24698 1.35690i 2.76271i −2.04892 0
324.3 0.246980i 0.801938i 1.93900 0 0.198062 1.69202i 0.972853i 2.35690 0
324.4 0.246980i 0.801938i 1.93900 0 0.198062 1.69202i 0.972853i 2.35690 0
324.5 1.44504i 2.24698i −0.0881460 0 3.24698 1.35690i 2.76271i −2.04892 0
324.6 2.80194i 0.554958i −5.85086 0 1.55496 3.04892i 10.7899i 2.69202 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 324.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.b.c 6
5.b even 2 1 inner 475.2.b.c 6
5.c odd 4 1 475.2.a.d 3
5.c odd 4 1 475.2.a.h yes 3
15.e even 4 1 4275.2.a.z 3
15.e even 4 1 4275.2.a.bn 3
20.e even 4 1 7600.2.a.bn 3
20.e even 4 1 7600.2.a.bw 3
95.g even 4 1 9025.2.a.w 3
95.g even 4 1 9025.2.a.be 3

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + 10T_{2}^{4} + 17T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(475, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 10 T^{4} + 17 T^{2} + 1$$
$3$ $$T^{6} + 6 T^{4} + 5 T^{2} + 1$$
$5$ $$T^{6}$$
$7$ $$T^{6} + 14 T^{4} + 49 T^{2} + 49$$
$11$ $$(T^{3} - T^{2} - 16 T - 13)^{2}$$
$13$ $$T^{6} + 13 T^{4} + 26 T^{2} + 1$$
$17$ $$T^{6} + 34 T^{4} + 173 T^{2} + \cdots + 169$$
$19$ $$(T - 1)^{6}$$
$23$ $$T^{6} + 26 T^{4} + 153 T^{2} + \cdots + 169$$
$29$ $$(T^{3} - 7 T^{2} - 42 T + 91)^{2}$$
$31$ $$(T^{3} + 5 T^{2} - 36 T + 43)^{2}$$
$37$ $$T^{6} + 41 T^{4} + 54 T^{2} + 1$$
$41$ $$(T^{3} - T^{2} - 114 T + 421)^{2}$$
$43$ $$T^{6} + 139 T^{4} + 6179 T^{2} + \cdots + 85849$$
$47$ $$T^{6} + 17 T^{4} + 94 T^{2} + \cdots + 169$$
$53$ $$T^{6} + 251 T^{4} + 14691 T^{2} + \cdots + 94249$$
$59$ $$(T^{3} + 10 T^{2} - 32 T - 328)^{2}$$
$61$ $$(T^{3} + 17 T^{2} + 66 T - 41)^{2}$$
$67$ $$T^{6} + 285 T^{4} + 19046 T^{2} + \cdots + 312481$$
$71$ $$(T^{3} + 19 T^{2} + 55 T - 307)^{2}$$
$73$ $$T^{6} + 341 T^{4} + 29826 T^{2} + \cdots + 214369$$
$79$ $$(T^{3} + 18 T^{2} + 87 T + 97)^{2}$$
$83$ $$T^{6} + 173 T^{4} + 3618 T^{2} + \cdots + 19321$$
$89$ $$(T^{3} + 2 T^{2} - 99 T - 281)^{2}$$
$97$ $$T^{6} + 13 T^{4} + 26 T^{2} + 1$$