# Properties

 Label 5.6.a.a Level $5$ Weight $6$ Character orbit 5.a Self dual yes Analytic conductor $0.802$ 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] = [5,6,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, 6, names="a")

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

chi := DirichletCharacter("5.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5$$ Weight: $$k$$ $$=$$ $$6$$ 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: $$0.801919099065$$ 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{2} - 4 q^{3} - 28 q^{4} + 25 q^{5} - 8 q^{6} + 192 q^{7} - 120 q^{8} - 227 q^{9}+O(q^{10})$$ q + 2 * q^2 - 4 * q^3 - 28 * q^4 + 25 * q^5 - 8 * q^6 + 192 * q^7 - 120 * q^8 - 227 * q^9 $$q + 2 q^{2} - 4 q^{3} - 28 q^{4} + 25 q^{5} - 8 q^{6} + 192 q^{7} - 120 q^{8} - 227 q^{9} + 50 q^{10} - 148 q^{11} + 112 q^{12} + 286 q^{13} + 384 q^{14} - 100 q^{15} + 656 q^{16} - 1678 q^{17} - 454 q^{18} + 1060 q^{19} - 700 q^{20} - 768 q^{21} - 296 q^{22} + 2976 q^{23} + 480 q^{24} + 625 q^{25} + 572 q^{26} + 1880 q^{27} - 5376 q^{28} - 3410 q^{29} - 200 q^{30} - 2448 q^{31} + 5152 q^{32} + 592 q^{33} - 3356 q^{34} + 4800 q^{35} + 6356 q^{36} + 182 q^{37} + 2120 q^{38} - 1144 q^{39} - 3000 q^{40} - 9398 q^{41} - 1536 q^{42} - 1244 q^{43} + 4144 q^{44} - 5675 q^{45} + 5952 q^{46} - 12088 q^{47} - 2624 q^{48} + 20057 q^{49} + 1250 q^{50} + 6712 q^{51} - 8008 q^{52} + 23846 q^{53} + 3760 q^{54} - 3700 q^{55} - 23040 q^{56} - 4240 q^{57} - 6820 q^{58} - 20020 q^{59} + 2800 q^{60} + 32302 q^{61} - 4896 q^{62} - 43584 q^{63} - 10688 q^{64} + 7150 q^{65} + 1184 q^{66} + 60972 q^{67} + 46984 q^{68} - 11904 q^{69} + 9600 q^{70} - 32648 q^{71} + 27240 q^{72} - 38774 q^{73} + 364 q^{74} - 2500 q^{75} - 29680 q^{76} - 28416 q^{77} - 2288 q^{78} - 33360 q^{79} + 16400 q^{80} + 47641 q^{81} - 18796 q^{82} + 16716 q^{83} + 21504 q^{84} - 41950 q^{85} - 2488 q^{86} + 13640 q^{87} + 17760 q^{88} + 101370 q^{89} - 11350 q^{90} + 54912 q^{91} - 83328 q^{92} + 9792 q^{93} - 24176 q^{94} + 26500 q^{95} - 20608 q^{96} - 119038 q^{97} + 40114 q^{98} + 33596 q^{99}+O(q^{100})$$ q + 2 * q^2 - 4 * q^3 - 28 * q^4 + 25 * q^5 - 8 * q^6 + 192 * q^7 - 120 * q^8 - 227 * q^9 + 50 * q^10 - 148 * q^11 + 112 * q^12 + 286 * q^13 + 384 * q^14 - 100 * q^15 + 656 * q^16 - 1678 * q^17 - 454 * q^18 + 1060 * q^19 - 700 * q^20 - 768 * q^21 - 296 * q^22 + 2976 * q^23 + 480 * q^24 + 625 * q^25 + 572 * q^26 + 1880 * q^27 - 5376 * q^28 - 3410 * q^29 - 200 * q^30 - 2448 * q^31 + 5152 * q^32 + 592 * q^33 - 3356 * q^34 + 4800 * q^35 + 6356 * q^36 + 182 * q^37 + 2120 * q^38 - 1144 * q^39 - 3000 * q^40 - 9398 * q^41 - 1536 * q^42 - 1244 * q^43 + 4144 * q^44 - 5675 * q^45 + 5952 * q^46 - 12088 * q^47 - 2624 * q^48 + 20057 * q^49 + 1250 * q^50 + 6712 * q^51 - 8008 * q^52 + 23846 * q^53 + 3760 * q^54 - 3700 * q^55 - 23040 * q^56 - 4240 * q^57 - 6820 * q^58 - 20020 * q^59 + 2800 * q^60 + 32302 * q^61 - 4896 * q^62 - 43584 * q^63 - 10688 * q^64 + 7150 * q^65 + 1184 * q^66 + 60972 * q^67 + 46984 * q^68 - 11904 * q^69 + 9600 * q^70 - 32648 * q^71 + 27240 * q^72 - 38774 * q^73 + 364 * q^74 - 2500 * q^75 - 29680 * q^76 - 28416 * q^77 - 2288 * q^78 - 33360 * q^79 + 16400 * q^80 + 47641 * q^81 - 18796 * q^82 + 16716 * q^83 + 21504 * q^84 - 41950 * q^85 - 2488 * q^86 + 13640 * q^87 + 17760 * q^88 + 101370 * q^89 - 11350 * q^90 + 54912 * q^91 - 83328 * q^92 + 9792 * q^93 - 24176 * q^94 + 26500 * q^95 - 20608 * q^96 - 119038 * q^97 + 40114 * q^98 + 33596 * 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
2.00000 −4.00000 −28.0000 25.0000 −8.00000 192.000 −120.000 −227.000 50.0000
 $$n$$: 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 5.6.a.a 1
3.b odd 2 1 45.6.a.b 1
4.b odd 2 1 80.6.a.e 1
5.b even 2 1 25.6.a.a 1
5.c odd 4 2 25.6.b.a 2
7.b odd 2 1 245.6.a.b 1
8.b even 2 1 320.6.a.j 1
8.d odd 2 1 320.6.a.g 1
11.b odd 2 1 605.6.a.a 1
12.b even 2 1 720.6.a.a 1
13.b even 2 1 845.6.a.b 1
15.d odd 2 1 225.6.a.f 1
15.e even 4 2 225.6.b.e 2
20.d odd 2 1 400.6.a.g 1
20.e even 4 2 400.6.c.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.6.a.a 1 1.a even 1 1 trivial
25.6.a.a 1 5.b even 2 1
25.6.b.a 2 5.c odd 4 2
45.6.a.b 1 3.b odd 2 1
80.6.a.e 1 4.b odd 2 1
225.6.a.f 1 15.d odd 2 1
225.6.b.e 2 15.e even 4 2
245.6.a.b 1 7.b odd 2 1
320.6.a.g 1 8.d odd 2 1
320.6.a.j 1 8.b even 2 1
400.6.a.g 1 20.d odd 2 1
400.6.c.j 2 20.e even 4 2
605.6.a.a 1 11.b odd 2 1
720.6.a.a 1 12.b even 2 1
845.6.a.b 1 13.b even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{6}^{\mathrm{new}}(\Gamma_0(5))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 2$$
$3$ $$T + 4$$
$5$ $$T - 25$$
$7$ $$T - 192$$
$11$ $$T + 148$$
$13$ $$T - 286$$
$17$ $$T + 1678$$
$19$ $$T - 1060$$
$23$ $$T - 2976$$
$29$ $$T + 3410$$
$31$ $$T + 2448$$
$37$ $$T - 182$$
$41$ $$T + 9398$$
$43$ $$T + 1244$$
$47$ $$T + 12088$$
$53$ $$T - 23846$$
$59$ $$T + 20020$$
$61$ $$T - 32302$$
$67$ $$T - 60972$$
$71$ $$T + 32648$$
$73$ $$T + 38774$$
$79$ $$T + 33360$$
$83$ $$T - 16716$$
$89$ $$T - 101370$$
$97$ $$T + 119038$$