# Properties

 Label 475.4.a.e Level $475$ Weight $4$ Character orbit 475.a Self dual yes Analytic conductor $28.026$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,4,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, 4, names="a")

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

chi := DirichletCharacter("475.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$475 = 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$28.0259072527$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 19) 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{2} + 5 q^{3} + q^{4} + 15 q^{6} - 11 q^{7} - 21 q^{8} - 2 q^{9}+O(q^{10})$$ q + 3 * q^2 + 5 * q^3 + q^4 + 15 * q^6 - 11 * q^7 - 21 * q^8 - 2 * q^9 $$q + 3 q^{2} + 5 q^{3} + q^{4} + 15 q^{6} - 11 q^{7} - 21 q^{8} - 2 q^{9} - 54 q^{11} + 5 q^{12} - 11 q^{13} - 33 q^{14} - 71 q^{16} + 93 q^{17} - 6 q^{18} + 19 q^{19} - 55 q^{21} - 162 q^{22} - 183 q^{23} - 105 q^{24} - 33 q^{26} - 145 q^{27} - 11 q^{28} - 249 q^{29} + 56 q^{31} - 45 q^{32} - 270 q^{33} + 279 q^{34} - 2 q^{36} + 250 q^{37} + 57 q^{38} - 55 q^{39} + 240 q^{41} - 165 q^{42} + 196 q^{43} - 54 q^{44} - 549 q^{46} + 168 q^{47} - 355 q^{48} - 222 q^{49} + 465 q^{51} - 11 q^{52} - 435 q^{53} - 435 q^{54} + 231 q^{56} + 95 q^{57} - 747 q^{58} + 195 q^{59} - 358 q^{61} + 168 q^{62} + 22 q^{63} + 433 q^{64} - 810 q^{66} + 961 q^{67} + 93 q^{68} - 915 q^{69} - 246 q^{71} + 42 q^{72} - 353 q^{73} + 750 q^{74} + 19 q^{76} + 594 q^{77} - 165 q^{78} - 34 q^{79} - 671 q^{81} + 720 q^{82} - 234 q^{83} - 55 q^{84} + 588 q^{86} - 1245 q^{87} + 1134 q^{88} - 168 q^{89} + 121 q^{91} - 183 q^{92} + 280 q^{93} + 504 q^{94} - 225 q^{96} - 758 q^{97} - 666 q^{98} + 108 q^{99}+O(q^{100})$$ q + 3 * q^2 + 5 * q^3 + q^4 + 15 * q^6 - 11 * q^7 - 21 * q^8 - 2 * q^9 - 54 * q^11 + 5 * q^12 - 11 * q^13 - 33 * q^14 - 71 * q^16 + 93 * q^17 - 6 * q^18 + 19 * q^19 - 55 * q^21 - 162 * q^22 - 183 * q^23 - 105 * q^24 - 33 * q^26 - 145 * q^27 - 11 * q^28 - 249 * q^29 + 56 * q^31 - 45 * q^32 - 270 * q^33 + 279 * q^34 - 2 * q^36 + 250 * q^37 + 57 * q^38 - 55 * q^39 + 240 * q^41 - 165 * q^42 + 196 * q^43 - 54 * q^44 - 549 * q^46 + 168 * q^47 - 355 * q^48 - 222 * q^49 + 465 * q^51 - 11 * q^52 - 435 * q^53 - 435 * q^54 + 231 * q^56 + 95 * q^57 - 747 * q^58 + 195 * q^59 - 358 * q^61 + 168 * q^62 + 22 * q^63 + 433 * q^64 - 810 * q^66 + 961 * q^67 + 93 * q^68 - 915 * q^69 - 246 * q^71 + 42 * q^72 - 353 * q^73 + 750 * q^74 + 19 * q^76 + 594 * q^77 - 165 * q^78 - 34 * q^79 - 671 * q^81 + 720 * q^82 - 234 * q^83 - 55 * q^84 + 588 * q^86 - 1245 * q^87 + 1134 * q^88 - 168 * q^89 + 121 * q^91 - 183 * q^92 + 280 * q^93 + 504 * q^94 - 225 * q^96 - 758 * q^97 - 666 * q^98 + 108 * q^99

## 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
 0
3.00000 5.00000 1.00000 0 15.0000 −11.0000 −21.0000 −2.00000 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 475.4.a.e 1
5.b even 2 1 19.4.a.a 1
5.c odd 4 2 475.4.b.c 2
15.d odd 2 1 171.4.a.d 1
20.d odd 2 1 304.4.a.b 1
35.c odd 2 1 931.4.a.a 1
40.e odd 2 1 1216.4.a.a 1
40.f even 2 1 1216.4.a.f 1
55.d odd 2 1 2299.4.a.b 1
95.d odd 2 1 361.4.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.4.a.a 1 5.b even 2 1
171.4.a.d 1 15.d odd 2 1
304.4.a.b 1 20.d odd 2 1
361.4.a.b 1 95.d odd 2 1
475.4.a.e 1 1.a even 1 1 trivial
475.4.b.c 2 5.c odd 4 2
931.4.a.a 1 35.c odd 2 1
1216.4.a.a 1 40.e odd 2 1
1216.4.a.f 1 40.f even 2 1
2299.4.a.b 1 55.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(475))$$:

 $$T_{2} - 3$$ T2 - 3 $$T_{3} - 5$$ T3 - 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 3$$
$3$ $$T - 5$$
$5$ $$T$$
$7$ $$T + 11$$
$11$ $$T + 54$$
$13$ $$T + 11$$
$17$ $$T - 93$$
$19$ $$T - 19$$
$23$ $$T + 183$$
$29$ $$T + 249$$
$31$ $$T - 56$$
$37$ $$T - 250$$
$41$ $$T - 240$$
$43$ $$T - 196$$
$47$ $$T - 168$$
$53$ $$T + 435$$
$59$ $$T - 195$$
$61$ $$T + 358$$
$67$ $$T - 961$$
$71$ $$T + 246$$
$73$ $$T + 353$$
$79$ $$T + 34$$
$83$ $$T + 234$$
$89$ $$T + 168$$
$97$ $$T + 758$$