# Properties

 Label 6.6.a.a Level $6$ Weight $6$ Character orbit 6.a Self dual yes Analytic conductor $0.962$ 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] = [6,6,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, 6, names="a")

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

chi := DirichletCharacter("6.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6 = 2 \cdot 3$$ Weight: $$k$$ $$=$$ $$6$$ 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: $$0.962302918878$$ 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)

 $$f(q)$$ $$=$$ $$q + 4 q^{2} - 9 q^{3} + 16 q^{4} - 66 q^{5} - 36 q^{6} + 176 q^{7} + 64 q^{8} + 81 q^{9}+O(q^{10})$$ q + 4 * q^2 - 9 * q^3 + 16 * q^4 - 66 * q^5 - 36 * q^6 + 176 * q^7 + 64 * q^8 + 81 * q^9 $$q + 4 q^{2} - 9 q^{3} + 16 q^{4} - 66 q^{5} - 36 q^{6} + 176 q^{7} + 64 q^{8} + 81 q^{9} - 264 q^{10} - 60 q^{11} - 144 q^{12} - 658 q^{13} + 704 q^{14} + 594 q^{15} + 256 q^{16} - 414 q^{17} + 324 q^{18} + 956 q^{19} - 1056 q^{20} - 1584 q^{21} - 240 q^{22} + 600 q^{23} - 576 q^{24} + 1231 q^{25} - 2632 q^{26} - 729 q^{27} + 2816 q^{28} + 5574 q^{29} + 2376 q^{30} - 3592 q^{31} + 1024 q^{32} + 540 q^{33} - 1656 q^{34} - 11616 q^{35} + 1296 q^{36} - 8458 q^{37} + 3824 q^{38} + 5922 q^{39} - 4224 q^{40} + 19194 q^{41} - 6336 q^{42} + 13316 q^{43} - 960 q^{44} - 5346 q^{45} + 2400 q^{46} - 19680 q^{47} - 2304 q^{48} + 14169 q^{49} + 4924 q^{50} + 3726 q^{51} - 10528 q^{52} - 31266 q^{53} - 2916 q^{54} + 3960 q^{55} + 11264 q^{56} - 8604 q^{57} + 22296 q^{58} + 26340 q^{59} + 9504 q^{60} - 31090 q^{61} - 14368 q^{62} + 14256 q^{63} + 4096 q^{64} + 43428 q^{65} + 2160 q^{66} - 16804 q^{67} - 6624 q^{68} - 5400 q^{69} - 46464 q^{70} + 6120 q^{71} + 5184 q^{72} - 25558 q^{73} - 33832 q^{74} - 11079 q^{75} + 15296 q^{76} - 10560 q^{77} + 23688 q^{78} + 74408 q^{79} - 16896 q^{80} + 6561 q^{81} + 76776 q^{82} - 6468 q^{83} - 25344 q^{84} + 27324 q^{85} + 53264 q^{86} - 50166 q^{87} - 3840 q^{88} - 32742 q^{89} - 21384 q^{90} - 115808 q^{91} + 9600 q^{92} + 32328 q^{93} - 78720 q^{94} - 63096 q^{95} - 9216 q^{96} + 166082 q^{97} + 56676 q^{98} - 4860 q^{99}+O(q^{100})$$ q + 4 * q^2 - 9 * q^3 + 16 * q^4 - 66 * q^5 - 36 * q^6 + 176 * q^7 + 64 * q^8 + 81 * q^9 - 264 * q^10 - 60 * q^11 - 144 * q^12 - 658 * q^13 + 704 * q^14 + 594 * q^15 + 256 * q^16 - 414 * q^17 + 324 * q^18 + 956 * q^19 - 1056 * q^20 - 1584 * q^21 - 240 * q^22 + 600 * q^23 - 576 * q^24 + 1231 * q^25 - 2632 * q^26 - 729 * q^27 + 2816 * q^28 + 5574 * q^29 + 2376 * q^30 - 3592 * q^31 + 1024 * q^32 + 540 * q^33 - 1656 * q^34 - 11616 * q^35 + 1296 * q^36 - 8458 * q^37 + 3824 * q^38 + 5922 * q^39 - 4224 * q^40 + 19194 * q^41 - 6336 * q^42 + 13316 * q^43 - 960 * q^44 - 5346 * q^45 + 2400 * q^46 - 19680 * q^47 - 2304 * q^48 + 14169 * q^49 + 4924 * q^50 + 3726 * q^51 - 10528 * q^52 - 31266 * q^53 - 2916 * q^54 + 3960 * q^55 + 11264 * q^56 - 8604 * q^57 + 22296 * q^58 + 26340 * q^59 + 9504 * q^60 - 31090 * q^61 - 14368 * q^62 + 14256 * q^63 + 4096 * q^64 + 43428 * q^65 + 2160 * q^66 - 16804 * q^67 - 6624 * q^68 - 5400 * q^69 - 46464 * q^70 + 6120 * q^71 + 5184 * q^72 - 25558 * q^73 - 33832 * q^74 - 11079 * q^75 + 15296 * q^76 - 10560 * q^77 + 23688 * q^78 + 74408 * q^79 - 16896 * q^80 + 6561 * q^81 + 76776 * q^82 - 6468 * q^83 - 25344 * q^84 + 27324 * q^85 + 53264 * q^86 - 50166 * q^87 - 3840 * q^88 - 32742 * q^89 - 21384 * q^90 - 115808 * q^91 + 9600 * q^92 + 32328 * q^93 - 78720 * q^94 - 63096 * q^95 - 9216 * q^96 + 166082 * q^97 + 56676 * q^98 - 4860 * 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
4.00000 −9.00000 16.0000 −66.0000 −36.0000 176.000 64.0000 81.0000 −264.000
 $$n$$: 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 6.6.a.a 1
3.b odd 2 1 18.6.a.b 1
4.b odd 2 1 48.6.a.c 1
5.b even 2 1 150.6.a.d 1
5.c odd 4 2 150.6.c.b 2
7.b odd 2 1 294.6.a.m 1
7.c even 3 2 294.6.e.g 2
7.d odd 6 2 294.6.e.a 2
8.b even 2 1 192.6.a.o 1
8.d odd 2 1 192.6.a.g 1
9.c even 3 2 162.6.c.e 2
9.d odd 6 2 162.6.c.h 2
11.b odd 2 1 726.6.a.a 1
12.b even 2 1 144.6.a.j 1
13.b even 2 1 1014.6.a.c 1
15.d odd 2 1 450.6.a.m 1
15.e even 4 2 450.6.c.j 2
16.e even 4 2 768.6.d.c 2
16.f odd 4 2 768.6.d.p 2
21.c even 2 1 882.6.a.a 1
24.f even 2 1 576.6.a.i 1
24.h odd 2 1 576.6.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.6.a.a 1 1.a even 1 1 trivial
18.6.a.b 1 3.b odd 2 1
48.6.a.c 1 4.b odd 2 1
144.6.a.j 1 12.b even 2 1
150.6.a.d 1 5.b even 2 1
150.6.c.b 2 5.c odd 4 2
162.6.c.e 2 9.c even 3 2
162.6.c.h 2 9.d odd 6 2
192.6.a.g 1 8.d odd 2 1
192.6.a.o 1 8.b even 2 1
294.6.a.m 1 7.b odd 2 1
294.6.e.a 2 7.d odd 6 2
294.6.e.g 2 7.c even 3 2
450.6.a.m 1 15.d odd 2 1
450.6.c.j 2 15.e even 4 2
576.6.a.i 1 24.f even 2 1
576.6.a.j 1 24.h odd 2 1
726.6.a.a 1 11.b odd 2 1
768.6.d.c 2 16.e even 4 2
768.6.d.p 2 16.f odd 4 2
882.6.a.a 1 21.c even 2 1
1014.6.a.c 1 13.b even 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 4$$
$3$ $$T + 9$$
$5$ $$T + 66$$
$7$ $$T - 176$$
$11$ $$T + 60$$
$13$ $$T + 658$$
$17$ $$T + 414$$
$19$ $$T - 956$$
$23$ $$T - 600$$
$29$ $$T - 5574$$
$31$ $$T + 3592$$
$37$ $$T + 8458$$
$41$ $$T - 19194$$
$43$ $$T - 13316$$
$47$ $$T + 19680$$
$53$ $$T + 31266$$
$59$ $$T - 26340$$
$61$ $$T + 31090$$
$67$ $$T + 16804$$
$71$ $$T - 6120$$
$73$ $$T + 25558$$
$79$ $$T - 74408$$
$83$ $$T + 6468$$
$89$ $$T + 32742$$
$97$ $$T - 166082$$