# Properties

 Label 5.4.a.a Level $5$ Weight $4$ Character orbit 5.a Self dual yes Analytic conductor $0.295$ Analytic rank $0$ 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] = [5,4,Mod(1,5)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("5.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 5.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: $$0.295009550029$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ 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)

 $$f(q)$$ $$=$$ $$q - 4 q^{2} + 2 q^{3} + 8 q^{4} - 5 q^{5} - 8 q^{6} + 6 q^{7} - 23 q^{9}+O(q^{10})$$ q - 4 * q^2 + 2 * q^3 + 8 * q^4 - 5 * q^5 - 8 * q^6 + 6 * q^7 - 23 * q^9 $$q - 4 q^{2} + 2 q^{3} + 8 q^{4} - 5 q^{5} - 8 q^{6} + 6 q^{7} - 23 q^{9} + 20 q^{10} + 32 q^{11} + 16 q^{12} - 38 q^{13} - 24 q^{14} - 10 q^{15} - 64 q^{16} + 26 q^{17} + 92 q^{18} + 100 q^{19} - 40 q^{20} + 12 q^{21} - 128 q^{22} - 78 q^{23} + 25 q^{25} + 152 q^{26} - 100 q^{27} + 48 q^{28} - 50 q^{29} + 40 q^{30} - 108 q^{31} + 256 q^{32} + 64 q^{33} - 104 q^{34} - 30 q^{35} - 184 q^{36} + 266 q^{37} - 400 q^{38} - 76 q^{39} + 22 q^{41} - 48 q^{42} + 442 q^{43} + 256 q^{44} + 115 q^{45} + 312 q^{46} - 514 q^{47} - 128 q^{48} - 307 q^{49} - 100 q^{50} + 52 q^{51} - 304 q^{52} + 2 q^{53} + 400 q^{54} - 160 q^{55} + 200 q^{57} + 200 q^{58} + 500 q^{59} - 80 q^{60} - 518 q^{61} + 432 q^{62} - 138 q^{63} - 512 q^{64} + 190 q^{65} - 256 q^{66} + 126 q^{67} + 208 q^{68} - 156 q^{69} + 120 q^{70} + 412 q^{71} - 878 q^{73} - 1064 q^{74} + 50 q^{75} + 800 q^{76} + 192 q^{77} + 304 q^{78} + 600 q^{79} + 320 q^{80} + 421 q^{81} - 88 q^{82} + 282 q^{83} + 96 q^{84} - 130 q^{85} - 1768 q^{86} - 100 q^{87} - 150 q^{89} - 460 q^{90} - 228 q^{91} - 624 q^{92} - 216 q^{93} + 2056 q^{94} - 500 q^{95} + 512 q^{96} + 386 q^{97} + 1228 q^{98} - 736 q^{99}+O(q^{100})$$ q - 4 * q^2 + 2 * q^3 + 8 * q^4 - 5 * q^5 - 8 * q^6 + 6 * q^7 - 23 * q^9 + 20 * q^10 + 32 * q^11 + 16 * q^12 - 38 * q^13 - 24 * q^14 - 10 * q^15 - 64 * q^16 + 26 * q^17 + 92 * q^18 + 100 * q^19 - 40 * q^20 + 12 * q^21 - 128 * q^22 - 78 * q^23 + 25 * q^25 + 152 * q^26 - 100 * q^27 + 48 * q^28 - 50 * q^29 + 40 * q^30 - 108 * q^31 + 256 * q^32 + 64 * q^33 - 104 * q^34 - 30 * q^35 - 184 * q^36 + 266 * q^37 - 400 * q^38 - 76 * q^39 + 22 * q^41 - 48 * q^42 + 442 * q^43 + 256 * q^44 + 115 * q^45 + 312 * q^46 - 514 * q^47 - 128 * q^48 - 307 * q^49 - 100 * q^50 + 52 * q^51 - 304 * q^52 + 2 * q^53 + 400 * q^54 - 160 * q^55 + 200 * q^57 + 200 * q^58 + 500 * q^59 - 80 * q^60 - 518 * q^61 + 432 * q^62 - 138 * q^63 - 512 * q^64 + 190 * q^65 - 256 * q^66 + 126 * q^67 + 208 * q^68 - 156 * q^69 + 120 * q^70 + 412 * q^71 - 878 * q^73 - 1064 * q^74 + 50 * q^75 + 800 * q^76 + 192 * q^77 + 304 * q^78 + 600 * q^79 + 320 * q^80 + 421 * q^81 - 88 * q^82 + 282 * q^83 + 96 * q^84 - 130 * q^85 - 1768 * q^86 - 100 * q^87 - 150 * q^89 - 460 * q^90 - 228 * q^91 - 624 * q^92 - 216 * q^93 + 2056 * q^94 - 500 * q^95 + 512 * q^96 + 386 * q^97 + 1228 * q^98 - 736 * q^99

## Expression as an eta quotient

$$f(z) = \eta(z)^{4}\eta(5z)^{4}=q\prod_{n=1}^\infty(1 - q^{n})^{4}(1 - q^{5n})^{4}$$

## 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
−4.00000 2.00000 8.00000 −5.00000 −8.00000 6.00000 0 −23.0000 20.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$

## 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 5.4.a.a 1
3.b odd 2 1 45.4.a.d 1
4.b odd 2 1 80.4.a.d 1
5.b even 2 1 25.4.a.c 1
5.c odd 4 2 25.4.b.a 2
7.b odd 2 1 245.4.a.a 1
7.c even 3 2 245.4.e.f 2
7.d odd 6 2 245.4.e.g 2
8.b even 2 1 320.4.a.g 1
8.d odd 2 1 320.4.a.h 1
9.c even 3 2 405.4.e.l 2
9.d odd 6 2 405.4.e.c 2
11.b odd 2 1 605.4.a.d 1
12.b even 2 1 720.4.a.u 1
13.b even 2 1 845.4.a.b 1
15.d odd 2 1 225.4.a.b 1
15.e even 4 2 225.4.b.c 2
16.e even 4 2 1280.4.d.e 2
16.f odd 4 2 1280.4.d.l 2
17.b even 2 1 1445.4.a.a 1
19.b odd 2 1 1805.4.a.h 1
20.d odd 2 1 400.4.a.m 1
20.e even 4 2 400.4.c.k 2
21.c even 2 1 2205.4.a.q 1
35.c odd 2 1 1225.4.a.k 1
40.e odd 2 1 1600.4.a.s 1
40.f even 2 1 1600.4.a.bi 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.4.a.a 1 1.a even 1 1 trivial
25.4.a.c 1 5.b even 2 1
25.4.b.a 2 5.c odd 4 2
45.4.a.d 1 3.b odd 2 1
80.4.a.d 1 4.b odd 2 1
225.4.a.b 1 15.d odd 2 1
225.4.b.c 2 15.e even 4 2
245.4.a.a 1 7.b odd 2 1
245.4.e.f 2 7.c even 3 2
245.4.e.g 2 7.d odd 6 2
320.4.a.g 1 8.b even 2 1
320.4.a.h 1 8.d odd 2 1
400.4.a.m 1 20.d odd 2 1
400.4.c.k 2 20.e even 4 2
405.4.e.c 2 9.d odd 6 2
405.4.e.l 2 9.c even 3 2
605.4.a.d 1 11.b odd 2 1
720.4.a.u 1 12.b even 2 1
845.4.a.b 1 13.b even 2 1
1225.4.a.k 1 35.c odd 2 1
1280.4.d.e 2 16.e even 4 2
1280.4.d.l 2 16.f odd 4 2
1445.4.a.a 1 17.b even 2 1
1600.4.a.s 1 40.e odd 2 1
1600.4.a.bi 1 40.f even 2 1
1805.4.a.h 1 19.b odd 2 1
2205.4.a.q 1 21.c even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(\Gamma_0(5))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 4$$
$3$ $$T - 2$$
$5$ $$T + 5$$
$7$ $$T - 6$$
$11$ $$T - 32$$
$13$ $$T + 38$$
$17$ $$T - 26$$
$19$ $$T - 100$$
$23$ $$T + 78$$
$29$ $$T + 50$$
$31$ $$T + 108$$
$37$ $$T - 266$$
$41$ $$T - 22$$
$43$ $$T - 442$$
$47$ $$T + 514$$
$53$ $$T - 2$$
$59$ $$T - 500$$
$61$ $$T + 518$$
$67$ $$T - 126$$
$71$ $$T - 412$$
$73$ $$T + 878$$
$79$ $$T - 600$$
$83$ $$T - 282$$
$89$ $$T + 150$$
$97$ $$T - 386$$