# Properties

 Label 3.6.a.a Level $3$ Weight $6$ Character orbit 3.a Self dual yes Analytic conductor $0.481$ 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] = [3,6,Mod(1,3)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3, 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("3.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 3.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: $$0.481151459439$$ 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)

 $$f(q)$$ $$=$$ $$q - 6 q^{2} + 9 q^{3} + 4 q^{4} + 6 q^{5} - 54 q^{6} - 40 q^{7} + 168 q^{8} + 81 q^{9}+O(q^{10})$$ q - 6 * q^2 + 9 * q^3 + 4 * q^4 + 6 * q^5 - 54 * q^6 - 40 * q^7 + 168 * q^8 + 81 * q^9 $$q - 6 q^{2} + 9 q^{3} + 4 q^{4} + 6 q^{5} - 54 q^{6} - 40 q^{7} + 168 q^{8} + 81 q^{9} - 36 q^{10} - 564 q^{11} + 36 q^{12} + 638 q^{13} + 240 q^{14} + 54 q^{15} - 1136 q^{16} + 882 q^{17} - 486 q^{18} - 556 q^{19} + 24 q^{20} - 360 q^{21} + 3384 q^{22} - 840 q^{23} + 1512 q^{24} - 3089 q^{25} - 3828 q^{26} + 729 q^{27} - 160 q^{28} + 4638 q^{29} - 324 q^{30} + 4400 q^{31} + 1440 q^{32} - 5076 q^{33} - 5292 q^{34} - 240 q^{35} + 324 q^{36} - 2410 q^{37} + 3336 q^{38} + 5742 q^{39} + 1008 q^{40} - 6870 q^{41} + 2160 q^{42} + 9644 q^{43} - 2256 q^{44} + 486 q^{45} + 5040 q^{46} - 18672 q^{47} - 10224 q^{48} - 15207 q^{49} + 18534 q^{50} + 7938 q^{51} + 2552 q^{52} + 33750 q^{53} - 4374 q^{54} - 3384 q^{55} - 6720 q^{56} - 5004 q^{57} - 27828 q^{58} - 18084 q^{59} + 216 q^{60} + 39758 q^{61} - 26400 q^{62} - 3240 q^{63} + 27712 q^{64} + 3828 q^{65} + 30456 q^{66} - 23068 q^{67} + 3528 q^{68} - 7560 q^{69} + 1440 q^{70} - 4248 q^{71} + 13608 q^{72} - 41110 q^{73} + 14460 q^{74} - 27801 q^{75} - 2224 q^{76} + 22560 q^{77} - 34452 q^{78} + 21920 q^{79} - 6816 q^{80} + 6561 q^{81} + 41220 q^{82} + 82452 q^{83} - 1440 q^{84} + 5292 q^{85} - 57864 q^{86} + 41742 q^{87} - 94752 q^{88} - 94086 q^{89} - 2916 q^{90} - 25520 q^{91} - 3360 q^{92} + 39600 q^{93} + 112032 q^{94} - 3336 q^{95} + 12960 q^{96} + 49442 q^{97} + 91242 q^{98} - 45684 q^{99}+O(q^{100})$$ q - 6 * q^2 + 9 * q^3 + 4 * q^4 + 6 * q^5 - 54 * q^6 - 40 * q^7 + 168 * q^8 + 81 * q^9 - 36 * q^10 - 564 * q^11 + 36 * q^12 + 638 * q^13 + 240 * q^14 + 54 * q^15 - 1136 * q^16 + 882 * q^17 - 486 * q^18 - 556 * q^19 + 24 * q^20 - 360 * q^21 + 3384 * q^22 - 840 * q^23 + 1512 * q^24 - 3089 * q^25 - 3828 * q^26 + 729 * q^27 - 160 * q^28 + 4638 * q^29 - 324 * q^30 + 4400 * q^31 + 1440 * q^32 - 5076 * q^33 - 5292 * q^34 - 240 * q^35 + 324 * q^36 - 2410 * q^37 + 3336 * q^38 + 5742 * q^39 + 1008 * q^40 - 6870 * q^41 + 2160 * q^42 + 9644 * q^43 - 2256 * q^44 + 486 * q^45 + 5040 * q^46 - 18672 * q^47 - 10224 * q^48 - 15207 * q^49 + 18534 * q^50 + 7938 * q^51 + 2552 * q^52 + 33750 * q^53 - 4374 * q^54 - 3384 * q^55 - 6720 * q^56 - 5004 * q^57 - 27828 * q^58 - 18084 * q^59 + 216 * q^60 + 39758 * q^61 - 26400 * q^62 - 3240 * q^63 + 27712 * q^64 + 3828 * q^65 + 30456 * q^66 - 23068 * q^67 + 3528 * q^68 - 7560 * q^69 + 1440 * q^70 - 4248 * q^71 + 13608 * q^72 - 41110 * q^73 + 14460 * q^74 - 27801 * q^75 - 2224 * q^76 + 22560 * q^77 - 34452 * q^78 + 21920 * q^79 - 6816 * q^80 + 6561 * q^81 + 41220 * q^82 + 82452 * q^83 - 1440 * q^84 + 5292 * q^85 - 57864 * q^86 + 41742 * q^87 - 94752 * q^88 - 94086 * q^89 - 2916 * q^90 - 25520 * q^91 - 3360 * q^92 + 39600 * q^93 + 112032 * q^94 - 3336 * q^95 + 12960 * q^96 + 49442 * q^97 + 91242 * q^98 - 45684 * q^99

## Expression as an eta quotient

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

## 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
−6.00000 9.00000 4.00000 6.00000 −54.0000 −40.0000 168.000 81.0000 −36.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$

## 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 3.6.a.a 1
3.b odd 2 1 9.6.a.a 1
4.b odd 2 1 48.6.a.a 1
5.b even 2 1 75.6.a.e 1
5.c odd 4 2 75.6.b.b 2
7.b odd 2 1 147.6.a.a 1
7.c even 3 2 147.6.e.h 2
7.d odd 6 2 147.6.e.k 2
8.b even 2 1 192.6.a.d 1
8.d odd 2 1 192.6.a.l 1
9.c even 3 2 81.6.c.c 2
9.d odd 6 2 81.6.c.a 2
11.b odd 2 1 363.6.a.d 1
12.b even 2 1 144.6.a.f 1
13.b even 2 1 507.6.a.b 1
15.d odd 2 1 225.6.a.a 1
15.e even 4 2 225.6.b.b 2
16.e even 4 2 768.6.d.k 2
16.f odd 4 2 768.6.d.h 2
17.b even 2 1 867.6.a.a 1
19.b odd 2 1 1083.6.a.c 1
21.c even 2 1 441.6.a.i 1
24.f even 2 1 576.6.a.t 1
24.h odd 2 1 576.6.a.s 1
33.d even 2 1 1089.6.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.6.a.a 1 1.a even 1 1 trivial
9.6.a.a 1 3.b odd 2 1
48.6.a.a 1 4.b odd 2 1
75.6.a.e 1 5.b even 2 1
75.6.b.b 2 5.c odd 4 2
81.6.c.a 2 9.d odd 6 2
81.6.c.c 2 9.c even 3 2
144.6.a.f 1 12.b even 2 1
147.6.a.a 1 7.b odd 2 1
147.6.e.h 2 7.c even 3 2
147.6.e.k 2 7.d odd 6 2
192.6.a.d 1 8.b even 2 1
192.6.a.l 1 8.d odd 2 1
225.6.a.a 1 15.d odd 2 1
225.6.b.b 2 15.e even 4 2
363.6.a.d 1 11.b odd 2 1
441.6.a.i 1 21.c even 2 1
507.6.a.b 1 13.b even 2 1
576.6.a.s 1 24.h odd 2 1
576.6.a.t 1 24.f even 2 1
768.6.d.h 2 16.f odd 4 2
768.6.d.k 2 16.e even 4 2
867.6.a.a 1 17.b even 2 1
1083.6.a.c 1 19.b odd 2 1
1089.6.a.b 1 33.d even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{6}^{\mathrm{new}}(\Gamma_0(3))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 6$$
$3$ $$T - 9$$
$5$ $$T - 6$$
$7$ $$T + 40$$
$11$ $$T + 564$$
$13$ $$T - 638$$
$17$ $$T - 882$$
$19$ $$T + 556$$
$23$ $$T + 840$$
$29$ $$T - 4638$$
$31$ $$T - 4400$$
$37$ $$T + 2410$$
$41$ $$T + 6870$$
$43$ $$T - 9644$$
$47$ $$T + 18672$$
$53$ $$T - 33750$$
$59$ $$T + 18084$$
$61$ $$T - 39758$$
$67$ $$T + 23068$$
$71$ $$T + 4248$$
$73$ $$T + 41110$$
$79$ $$T - 21920$$
$83$ $$T - 82452$$
$89$ $$T + 94086$$
$97$ $$T - 49442$$