# Properties

 Label 6.4.a.a Level $6$ Weight $4$ Character orbit 6.a Self dual yes Analytic conductor $0.354$ 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] = [6,4,Mod(1,6)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6, 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("6.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6 = 2 \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 6.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: $$0.354011460034$$ 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{2} - 3 q^{3} + 4 q^{4} + 6 q^{5} + 6 q^{6} - 16 q^{7} - 8 q^{8} + 9 q^{9}+O(q^{10})$$ q - 2 * q^2 - 3 * q^3 + 4 * q^4 + 6 * q^5 + 6 * q^6 - 16 * q^7 - 8 * q^8 + 9 * q^9 $$q - 2 q^{2} - 3 q^{3} + 4 q^{4} + 6 q^{5} + 6 q^{6} - 16 q^{7} - 8 q^{8} + 9 q^{9} - 12 q^{10} + 12 q^{11} - 12 q^{12} + 38 q^{13} + 32 q^{14} - 18 q^{15} + 16 q^{16} - 126 q^{17} - 18 q^{18} + 20 q^{19} + 24 q^{20} + 48 q^{21} - 24 q^{22} + 168 q^{23} + 24 q^{24} - 89 q^{25} - 76 q^{26} - 27 q^{27} - 64 q^{28} + 30 q^{29} + 36 q^{30} - 88 q^{31} - 32 q^{32} - 36 q^{33} + 252 q^{34} - 96 q^{35} + 36 q^{36} + 254 q^{37} - 40 q^{38} - 114 q^{39} - 48 q^{40} + 42 q^{41} - 96 q^{42} - 52 q^{43} + 48 q^{44} + 54 q^{45} - 336 q^{46} - 96 q^{47} - 48 q^{48} - 87 q^{49} + 178 q^{50} + 378 q^{51} + 152 q^{52} + 198 q^{53} + 54 q^{54} + 72 q^{55} + 128 q^{56} - 60 q^{57} - 60 q^{58} - 660 q^{59} - 72 q^{60} - 538 q^{61} + 176 q^{62} - 144 q^{63} + 64 q^{64} + 228 q^{65} + 72 q^{66} + 884 q^{67} - 504 q^{68} - 504 q^{69} + 192 q^{70} + 792 q^{71} - 72 q^{72} + 218 q^{73} - 508 q^{74} + 267 q^{75} + 80 q^{76} - 192 q^{77} + 228 q^{78} - 520 q^{79} + 96 q^{80} + 81 q^{81} - 84 q^{82} - 492 q^{83} + 192 q^{84} - 756 q^{85} + 104 q^{86} - 90 q^{87} - 96 q^{88} + 810 q^{89} - 108 q^{90} - 608 q^{91} + 672 q^{92} + 264 q^{93} + 192 q^{94} + 120 q^{95} + 96 q^{96} + 1154 q^{97} + 174 q^{98} + 108 q^{99}+O(q^{100})$$ q - 2 * q^2 - 3 * q^3 + 4 * q^4 + 6 * q^5 + 6 * q^6 - 16 * q^7 - 8 * q^8 + 9 * q^9 - 12 * q^10 + 12 * q^11 - 12 * q^12 + 38 * q^13 + 32 * q^14 - 18 * q^15 + 16 * q^16 - 126 * q^17 - 18 * q^18 + 20 * q^19 + 24 * q^20 + 48 * q^21 - 24 * q^22 + 168 * q^23 + 24 * q^24 - 89 * q^25 - 76 * q^26 - 27 * q^27 - 64 * q^28 + 30 * q^29 + 36 * q^30 - 88 * q^31 - 32 * q^32 - 36 * q^33 + 252 * q^34 - 96 * q^35 + 36 * q^36 + 254 * q^37 - 40 * q^38 - 114 * q^39 - 48 * q^40 + 42 * q^41 - 96 * q^42 - 52 * q^43 + 48 * q^44 + 54 * q^45 - 336 * q^46 - 96 * q^47 - 48 * q^48 - 87 * q^49 + 178 * q^50 + 378 * q^51 + 152 * q^52 + 198 * q^53 + 54 * q^54 + 72 * q^55 + 128 * q^56 - 60 * q^57 - 60 * q^58 - 660 * q^59 - 72 * q^60 - 538 * q^61 + 176 * q^62 - 144 * q^63 + 64 * q^64 + 228 * q^65 + 72 * q^66 + 884 * q^67 - 504 * q^68 - 504 * q^69 + 192 * q^70 + 792 * q^71 - 72 * q^72 + 218 * q^73 - 508 * q^74 + 267 * q^75 + 80 * q^76 - 192 * q^77 + 228 * q^78 - 520 * q^79 + 96 * q^80 + 81 * q^81 - 84 * q^82 - 492 * q^83 + 192 * q^84 - 756 * q^85 + 104 * q^86 - 90 * q^87 - 96 * q^88 + 810 * q^89 - 108 * q^90 - 608 * q^91 + 672 * q^92 + 264 * q^93 + 192 * q^94 + 120 * q^95 + 96 * q^96 + 1154 * q^97 + 174 * q^98 + 108 * q^99

## Expression as an eta quotient

$$f(z) = \eta(z)^{2}\eta(2z)^{2}\eta(3z)^{2}\eta(6z)^{2}=q\prod_{n=1}^\infty(1 - q^{n})^{2}(1 - q^{2n})^{2}(1 - q^{3n})^{2}(1 - q^{6n})^{2}$$

## 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 −3.00000 4.00000 6.00000 6.00000 −16.0000 −8.00000 9.00000 −12.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$1$$

## 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 6.4.a.a 1
3.b odd 2 1 18.4.a.a 1
4.b odd 2 1 48.4.a.c 1
5.b even 2 1 150.4.a.i 1
5.c odd 4 2 150.4.c.d 2
7.b odd 2 1 294.4.a.e 1
7.c even 3 2 294.4.e.h 2
7.d odd 6 2 294.4.e.g 2
8.b even 2 1 192.4.a.i 1
8.d odd 2 1 192.4.a.c 1
9.c even 3 2 162.4.c.f 2
9.d odd 6 2 162.4.c.c 2
11.b odd 2 1 726.4.a.f 1
12.b even 2 1 144.4.a.c 1
13.b even 2 1 1014.4.a.g 1
13.d odd 4 2 1014.4.b.d 2
15.d odd 2 1 450.4.a.h 1
15.e even 4 2 450.4.c.e 2
16.e even 4 2 768.4.d.n 2
16.f odd 4 2 768.4.d.c 2
17.b even 2 1 1734.4.a.d 1
19.b odd 2 1 2166.4.a.i 1
20.d odd 2 1 1200.4.a.b 1
20.e even 4 2 1200.4.f.j 2
21.c even 2 1 882.4.a.n 1
21.g even 6 2 882.4.g.f 2
21.h odd 6 2 882.4.g.i 2
24.f even 2 1 576.4.a.r 1
24.h odd 2 1 576.4.a.q 1
28.d even 2 1 2352.4.a.e 1
33.d even 2 1 2178.4.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.4.a.a 1 1.a even 1 1 trivial
18.4.a.a 1 3.b odd 2 1
48.4.a.c 1 4.b odd 2 1
144.4.a.c 1 12.b even 2 1
150.4.a.i 1 5.b even 2 1
150.4.c.d 2 5.c odd 4 2
162.4.c.c 2 9.d odd 6 2
162.4.c.f 2 9.c even 3 2
192.4.a.c 1 8.d odd 2 1
192.4.a.i 1 8.b even 2 1
294.4.a.e 1 7.b odd 2 1
294.4.e.g 2 7.d odd 6 2
294.4.e.h 2 7.c even 3 2
450.4.a.h 1 15.d odd 2 1
450.4.c.e 2 15.e even 4 2
576.4.a.q 1 24.h odd 2 1
576.4.a.r 1 24.f even 2 1
726.4.a.f 1 11.b odd 2 1
768.4.d.c 2 16.f odd 4 2
768.4.d.n 2 16.e even 4 2
882.4.a.n 1 21.c even 2 1
882.4.g.f 2 21.g even 6 2
882.4.g.i 2 21.h odd 6 2
1014.4.a.g 1 13.b even 2 1
1014.4.b.d 2 13.d odd 4 2
1200.4.a.b 1 20.d odd 2 1
1200.4.f.j 2 20.e even 4 2
1734.4.a.d 1 17.b even 2 1
2166.4.a.i 1 19.b odd 2 1
2178.4.a.e 1 33.d even 2 1
2352.4.a.e 1 28.d even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(\Gamma_0(6))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T + 3$$
$5$ $$T - 6$$
$7$ $$T + 16$$
$11$ $$T - 12$$
$13$ $$T - 38$$
$17$ $$T + 126$$
$19$ $$T - 20$$
$23$ $$T - 168$$
$29$ $$T - 30$$
$31$ $$T + 88$$
$37$ $$T - 254$$
$41$ $$T - 42$$
$43$ $$T + 52$$
$47$ $$T + 96$$
$53$ $$T - 198$$
$59$ $$T + 660$$
$61$ $$T + 538$$
$67$ $$T - 884$$
$71$ $$T - 792$$
$73$ $$T - 218$$
$79$ $$T + 520$$
$83$ $$T + 492$$
$89$ $$T - 810$$
$97$ $$T - 1154$$