# Properties

 Label 475.2.b.b 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: 6.0.1827904.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} + 9x^{4} + 14x^{2} + 1$$ x^6 + 9*x^4 + 14*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.

 $$f(q)$$ $$=$$ $$q + (\beta_{5} - \beta_{4}) q^{2} + (\beta_{4} + \beta_1) q^{3} + ( - \beta_{3} - 1) q^{4} + (2 \beta_{3} + \beta_{2} - 2) q^{6} + ( - \beta_{4} + \beta_1) q^{7} + ( - \beta_{5} + 2 \beta_{4} + 2 \beta_1) q^{8} + (\beta_{3} + \beta_{2} - 1) q^{9}+O(q^{10})$$ q + (b5 - b4) * q^2 + (b4 + b1) * q^3 + (-b3 - 1) * q^4 + (2*b3 + b2 - 2) * q^6 + (-b4 + b1) * q^7 + (-b5 + 2*b4 + 2*b1) * q^8 + (b3 + b2 - 1) * q^9 $$q + (\beta_{5} - \beta_{4}) q^{2} + (\beta_{4} + \beta_1) q^{3} + ( - \beta_{3} - 1) q^{4} + (2 \beta_{3} + \beta_{2} - 2) q^{6} + ( - \beta_{4} + \beta_1) q^{7} + ( - \beta_{5} + 2 \beta_{4} + 2 \beta_1) q^{8} + (\beta_{3} + \beta_{2} - 1) q^{9} + \beta_{2} q^{11} + (\beta_{5} - 3 \beta_{4} - \beta_1) q^{12} + (3 \beta_{5} - \beta_{4} - 3 \beta_1) q^{13} + ( - \beta_{2} - 2) q^{14} + (2 \beta_{3} + \beta_{2} - 3) q^{16} + ( - 5 \beta_{4} - \beta_1) q^{17} + ( - 3 \beta_{4} - \beta_1) q^{18} - q^{19} + (\beta_{3} - \beta_{2} - 2) q^{21} + ( - \beta_{5} - \beta_{4} + \beta_1) q^{22} + (3 \beta_{5} + \beta_{4} - 2 \beta_1) q^{23} - 5 q^{24} + ( - 4 \beta_{3} + 2 \beta_{2} - 3) q^{26} + ( - 2 \beta_{5} + 2 \beta_{4} + 3 \beta_1) q^{27} + ( - \beta_{5} + \beta_{4} + \beta_1) q^{28} + ( - \beta_{2} + 2) q^{29} + (\beta_{3} - 2 \beta_{2}) q^{31} + ( - 2 \beta_{5} + \beta_1) q^{32} + ( - \beta_{5} - 2 \beta_{4} + \beta_1) q^{33} + ( - 6 \beta_{3} - 5 \beta_{2} + 2) q^{34} + ( - 2 \beta_{3} - \beta_{2}) q^{36} + (2 \beta_{5} - 4 \beta_{4} - 5 \beta_1) q^{37} + ( - \beta_{5} + \beta_{4}) q^{38} + (3 \beta_{3} + 5 \beta_{2} + 1) q^{39} + ( - 5 \beta_{3} - 3 \beta_{2} + 3) q^{41} + (\beta_{5} - 3 \beta_1) q^{42} + ( - 6 \beta_{5} + \beta_{4} + 2 \beta_1) q^{43} + q^{44} + ( - \beta_{3} + 4 \beta_{2} - 5) q^{46} + (5 \beta_{5} - 5 \beta_{4} - \beta_1) q^{47} + ( - 3 \beta_{5} - \beta_{4} - 2 \beta_1) q^{48} + (\beta_{3} - 3 \beta_{2} + 3) q^{49} + ( - \beta_{3} - 5 \beta_{2} + 8) q^{51} + ( - 7 \beta_{5} + 11 \beta_{4} + 4 \beta_1) q^{52} + (2 \beta_{5} + 9 \beta_{4} - 2 \beta_1) q^{53} + 5 \beta_{3} q^{54} + (2 \beta_{3} - 2 \beta_{2} - 3) q^{56} + ( - \beta_{4} - \beta_1) q^{57} + (3 \beta_{5} - \beta_{4} - \beta_1) q^{58} + (2 \beta_{3} + 4 \beta_{2}) q^{59} + ( - \beta_{3} - 2 \beta_{2} + 2) q^{61} + (4 \beta_{5} - \beta_{4} - 4 \beta_1) q^{62} - \beta_{4} q^{63} + (5 \beta_{3} - 2) q^{64} + ( - \beta_{3} - 3 \beta_{2} + 1) q^{66} + ( - 4 \beta_{5} + 6 \beta_{4} + \beta_1) q^{67} + ( - 5 \beta_{5} + 11 \beta_{4} + 5 \beta_1) q^{68} + (4 \beta_{3} + 7 \beta_{2} - 4) q^{69} + (2 \beta_{3} + 2 \beta_{2} + 1) q^{71} + ( - 3 \beta_{5} + \beta_{4} + \beta_1) q^{72} + ( - 3 \beta_{5} + 2 \beta_{4} + 2 \beta_1) q^{73} + ( - 9 \beta_{3} - 2 \beta_{2} + 4) q^{74} + (\beta_{3} + 1) q^{76} + ( - \beta_{5} - 2 \beta_{4} + 3 \beta_1) q^{77} + (2 \beta_{5} - 15 \beta_{4} - \beta_1) q^{78} + ( - 4 \beta_{3} + \beta_{2} + 7) q^{79} + (2 \beta_{3} + \beta_{2} - 8) q^{81} + ( - 4 \beta_{5} + 15 \beta_{4} + 7 \beta_1) q^{82} + (3 \beta_{5} + 2 \beta_{4} + 6 \beta_1) q^{83} + ( - \beta_{3} - \beta_{2} - 1) q^{84} + (3 \beta_{3} - 5 \beta_{2} + 14) q^{86} + (\beta_{5} + 4 \beta_{4} + \beta_1) q^{87} + ( - \beta_{5} - 3 \beta_{4} + 2 \beta_1) q^{88} + (5 \beta_{3} + 3 \beta_{2} + 4) q^{89} + ( - 3 \beta_{3} + 5 \beta_{2} + 5) q^{91} + ( - 5 \beta_{5} + 6 \beta_{4} + 2 \beta_1) q^{92} + (\beta_{5} + 6 \beta_{4} - 2 \beta_1) q^{93} + ( - 6 \beta_{3} - 13) q^{94} + ( - 3 \beta_{3} - 4 \beta_{2} + 3) q^{96} + (\beta_{5} + 5 \beta_{4} + 3 \beta_1) q^{97} + (8 \beta_{5} - 3 \beta_{4} - 5 \beta_1) q^{98} + ( - \beta_{3} - \beta_{2} + 2) q^{99}+O(q^{100})$$ q + (b5 - b4) * q^2 + (b4 + b1) * q^3 + (-b3 - 1) * q^4 + (2*b3 + b2 - 2) * q^6 + (-b4 + b1) * q^7 + (-b5 + 2*b4 + 2*b1) * q^8 + (b3 + b2 - 1) * q^9 + b2 * q^11 + (b5 - 3*b4 - b1) * q^12 + (3*b5 - b4 - 3*b1) * q^13 + (-b2 - 2) * q^14 + (2*b3 + b2 - 3) * q^16 + (-5*b4 - b1) * q^17 + (-3*b4 - b1) * q^18 - q^19 + (b3 - b2 - 2) * q^21 + (-b5 - b4 + b1) * q^22 + (3*b5 + b4 - 2*b1) * q^23 - 5 * q^24 + (-4*b3 + 2*b2 - 3) * q^26 + (-2*b5 + 2*b4 + 3*b1) * q^27 + (-b5 + b4 + b1) * q^28 + (-b2 + 2) * q^29 + (b3 - 2*b2) * q^31 + (-2*b5 + b1) * q^32 + (-b5 - 2*b4 + b1) * q^33 + (-6*b3 - 5*b2 + 2) * q^34 + (-2*b3 - b2) * q^36 + (2*b5 - 4*b4 - 5*b1) * q^37 + (-b5 + b4) * q^38 + (3*b3 + 5*b2 + 1) * q^39 + (-5*b3 - 3*b2 + 3) * q^41 + (b5 - 3*b1) * q^42 + (-6*b5 + b4 + 2*b1) * q^43 + q^44 + (-b3 + 4*b2 - 5) * q^46 + (5*b5 - 5*b4 - b1) * q^47 + (-3*b5 - b4 - 2*b1) * q^48 + (b3 - 3*b2 + 3) * q^49 + (-b3 - 5*b2 + 8) * q^51 + (-7*b5 + 11*b4 + 4*b1) * q^52 + (2*b5 + 9*b4 - 2*b1) * q^53 + 5*b3 * q^54 + (2*b3 - 2*b2 - 3) * q^56 + (-b4 - b1) * q^57 + (3*b5 - b4 - b1) * q^58 + (2*b3 + 4*b2) * q^59 + (-b3 - 2*b2 + 2) * q^61 + (4*b5 - b4 - 4*b1) * q^62 - b4 * q^63 + (5*b3 - 2) * q^64 + (-b3 - 3*b2 + 1) * q^66 + (-4*b5 + 6*b4 + b1) * q^67 + (-5*b5 + 11*b4 + 5*b1) * q^68 + (4*b3 + 7*b2 - 4) * q^69 + (2*b3 + 2*b2 + 1) * q^71 + (-3*b5 + b4 + b1) * q^72 + (-3*b5 + 2*b4 + 2*b1) * q^73 + (-9*b3 - 2*b2 + 4) * q^74 + (b3 + 1) * q^76 + (-b5 - 2*b4 + 3*b1) * q^77 + (2*b5 - 15*b4 - b1) * q^78 + (-4*b3 + b2 + 7) * q^79 + (2*b3 + b2 - 8) * q^81 + (-4*b5 + 15*b4 + 7*b1) * q^82 + (3*b5 + 2*b4 + 6*b1) * q^83 + (-b3 - b2 - 1) * q^84 + (3*b3 - 5*b2 + 14) * q^86 + (b5 + 4*b4 + b1) * q^87 + (-b5 - 3*b4 + 2*b1) * q^88 + (5*b3 + 3*b2 + 4) * q^89 + (-3*b3 + 5*b2 + 5) * q^91 + (-5*b5 + 6*b4 + 2*b1) * q^92 + (b5 + 6*b4 - 2*b1) * q^93 + (-6*b3 - 13) * q^94 + (-3*b3 - 4*b2 + 3) * q^96 + (b5 + 5*b4 + 3*b1) * q^97 + (8*b5 - 3*b4 - 5*b1) * q^98 + (-b3 - b2 + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 8 q^{4} - 6 q^{6} - 2 q^{9}+O(q^{10})$$ 6 * q - 8 * q^4 - 6 * q^6 - 2 * q^9 $$6 q - 8 q^{4} - 6 q^{6} - 2 q^{9} + 2 q^{11} - 14 q^{14} - 12 q^{16} - 6 q^{19} - 12 q^{21} - 30 q^{24} - 22 q^{26} + 10 q^{29} - 2 q^{31} - 10 q^{34} - 6 q^{36} + 22 q^{39} + 2 q^{41} + 6 q^{44} - 24 q^{46} + 14 q^{49} + 36 q^{51} + 10 q^{54} - 18 q^{56} + 12 q^{59} + 6 q^{61} - 2 q^{64} - 2 q^{66} - 2 q^{69} + 14 q^{71} + 2 q^{74} + 8 q^{76} + 36 q^{79} - 42 q^{81} - 10 q^{84} + 80 q^{86} + 40 q^{89} + 34 q^{91} - 90 q^{94} + 4 q^{96} + 8 q^{99}+O(q^{100})$$ 6 * q - 8 * q^4 - 6 * q^6 - 2 * q^9 + 2 * q^11 - 14 * q^14 - 12 * q^16 - 6 * q^19 - 12 * q^21 - 30 * q^24 - 22 * q^26 + 10 * q^29 - 2 * q^31 - 10 * q^34 - 6 * q^36 + 22 * q^39 + 2 * q^41 + 6 * q^44 - 24 * q^46 + 14 * q^49 + 36 * q^51 + 10 * q^54 - 18 * q^56 + 12 * q^59 + 6 * q^61 - 2 * q^64 - 2 * q^66 - 2 * q^69 + 14 * q^71 + 2 * q^74 + 8 * q^76 + 36 * q^79 - 42 * q^81 - 10 * q^84 + 80 * q^86 + 40 * q^89 + 34 * q^91 - 90 * q^94 + 4 * q^96 + 8 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 9x^{4} + 14x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{4} + 5\nu^{2} - 1 ) / 5$$ (v^4 + 5*v^2 - 1) / 5 $$\beta_{3}$$ $$=$$ $$( \nu^{4} + 10\nu^{2} + 14 ) / 5$$ (v^4 + 10*v^2 + 14) / 5 $$\beta_{4}$$ $$=$$ $$( \nu^{5} + 10\nu^{3} + 19\nu ) / 5$$ (v^5 + 10*v^3 + 19*v) / 5 $$\beta_{5}$$ $$=$$ $$( -3\nu^{5} - 25\nu^{3} - 27\nu ) / 5$$ (-3*v^5 - 25*v^3 - 27*v) / 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - \beta_{2} - 3$$ b3 - b2 - 3 $$\nu^{3}$$ $$=$$ $$\beta_{5} + 3\beta_{4} - 6\beta_1$$ b5 + 3*b4 - 6*b1 $$\nu^{4}$$ $$=$$ $$-5\beta_{3} + 10\beta_{2} + 16$$ -5*b3 + 10*b2 + 16 $$\nu^{5}$$ $$=$$ $$-10\beta_{5} - 25\beta_{4} + 41\beta_1$$ -10*b5 - 25*b4 + 41*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
 0.273891i − 1.37720i − 2.65109i 2.65109i 1.37720i − 0.273891i
2.37720i 1.27389i −3.65109 0 3.02830 0.726109i 3.92498i 1.37720 0
324.2 1.65109i 2.37720i −0.726109 0 −3.92498 0.377203i 2.10331i −2.65109 0
324.3 1.27389i 1.65109i 0.377203 0 −2.10331 3.65109i 3.02830i 0.273891 0
324.4 1.27389i 1.65109i 0.377203 0 −2.10331 3.65109i 3.02830i 0.273891 0
324.5 1.65109i 2.37720i −0.726109 0 −3.92498 0.377203i 2.10331i −2.65109 0
324.6 2.37720i 1.27389i −3.65109 0 3.02830 0.726109i 3.92498i 1.37720 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.b 6
5.b even 2 1 inner 475.2.b.b 6
5.c odd 4 1 475.2.a.e 3
5.c odd 4 1 475.2.a.g yes 3
15.e even 4 1 4275.2.a.ba 3
15.e even 4 1 4275.2.a.bm 3
20.e even 4 1 7600.2.a.bh 3
20.e even 4 1 7600.2.a.cc 3
95.g even 4 1 9025.2.a.y 3
95.g even 4 1 9025.2.a.bc 3

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + 10T_{2}^{4} + 29T_{2}^{2} + 25$$ acting on $$S_{2}^{\mathrm{new}}(475, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 10 T^{4} + 29 T^{2} + 25$$
$3$ $$T^{6} + 10 T^{4} + 29 T^{2} + 25$$
$5$ $$T^{6}$$
$7$ $$T^{6} + 14 T^{4} + 9 T^{2} + 1$$
$11$ $$(T^{3} - T^{2} - 4 T - 1)^{2}$$
$13$ $$T^{6} + 81 T^{4} + 1914 T^{2} + \cdots + 10609$$
$17$ $$T^{6} + 74 T^{4} + 1509 T^{2} + \cdots + 6241$$
$19$ $$(T + 1)^{6}$$
$23$ $$T^{6} + 82 T^{4} + 2081 T^{2} + \cdots + 15625$$
$29$ $$(T^{3} - 5 T^{2} + 4 T + 5)^{2}$$
$31$ $$(T^{3} + T^{2} - 30 T + 53)^{2}$$
$37$ $$T^{6} + 173 T^{4} + 9426 T^{2} + \cdots + 156025$$
$41$ $$(T^{3} - T^{2} - 82 T + 155)^{2}$$
$43$ $$T^{6} + 251 T^{4} + 15939 T^{2} + \cdots + 100489$$
$47$ $$T^{6} + 209 T^{4} + 9694 T^{2} + \cdots + 96721$$
$53$ $$T^{6} + 355 T^{4} + 35699 T^{2} + \cdots + 819025$$
$59$ $$(T^{3} - 6 T^{2} - 40 T + 200)^{2}$$
$61$ $$(T^{3} - 3 T^{2} - 10 T - 1)^{2}$$
$67$ $$T^{6} + 169 T^{4} + 4394 T^{2} + \cdots + 28561$$
$71$ $$(T^{3} - 7 T^{2} - T + 47)^{2}$$
$73$ $$T^{6} + 61 T^{4} + 794 T^{2} + \cdots + 2809$$
$79$ $$(T^{3} - 18 T^{2} + 17 T + 395)^{2}$$
$83$ $$T^{6} + 549 T^{4} + 72114 T^{2} + \cdots + 17161$$
$89$ $$(T^{3} - 20 T^{2} + 51 T + 125)^{2}$$
$97$ $$T^{6} + 169 T^{4} + 4394 T^{2} + \cdots + 28561$$