# Properties

 Label 42.6.a.a Level $42$ Weight $6$ Character orbit 42.a Self dual yes Analytic conductor $6.736$ 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] = [42,6,Mod(1,42)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(42, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("42.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$42 = 2 \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 42.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: $$6.73612043215$$ 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)

 $$f(q)$$ $$=$$ $$q - 4 q^{2} - 9 q^{3} + 16 q^{4} - 54 q^{5} + 36 q^{6} + 49 q^{7} - 64 q^{8} + 81 q^{9}+O(q^{10})$$ q - 4 * q^2 - 9 * q^3 + 16 * q^4 - 54 * q^5 + 36 * q^6 + 49 * q^7 - 64 * q^8 + 81 * q^9 $$q - 4 q^{2} - 9 q^{3} + 16 q^{4} - 54 q^{5} + 36 q^{6} + 49 q^{7} - 64 q^{8} + 81 q^{9} + 216 q^{10} + 216 q^{11} - 144 q^{12} + 998 q^{13} - 196 q^{14} + 486 q^{15} + 256 q^{16} + 1302 q^{17} - 324 q^{18} + 884 q^{19} - 864 q^{20} - 441 q^{21} - 864 q^{22} - 2268 q^{23} + 576 q^{24} - 209 q^{25} - 3992 q^{26} - 729 q^{27} + 784 q^{28} - 1482 q^{29} - 1944 q^{30} + 8360 q^{31} - 1024 q^{32} - 1944 q^{33} - 5208 q^{34} - 2646 q^{35} + 1296 q^{36} - 4714 q^{37} - 3536 q^{38} - 8982 q^{39} + 3456 q^{40} - 9786 q^{41} + 1764 q^{42} + 19436 q^{43} + 3456 q^{44} - 4374 q^{45} + 9072 q^{46} + 22200 q^{47} - 2304 q^{48} + 2401 q^{49} + 836 q^{50} - 11718 q^{51} + 15968 q^{52} + 26790 q^{53} + 2916 q^{54} - 11664 q^{55} - 3136 q^{56} - 7956 q^{57} + 5928 q^{58} + 28092 q^{59} + 7776 q^{60} - 38866 q^{61} - 33440 q^{62} + 3969 q^{63} + 4096 q^{64} - 53892 q^{65} + 7776 q^{66} + 23948 q^{67} + 20832 q^{68} + 20412 q^{69} + 10584 q^{70} - 20628 q^{71} - 5184 q^{72} + 290 q^{73} + 18856 q^{74} + 1881 q^{75} + 14144 q^{76} + 10584 q^{77} + 35928 q^{78} - 99544 q^{79} - 13824 q^{80} + 6561 q^{81} + 39144 q^{82} + 19308 q^{83} - 7056 q^{84} - 70308 q^{85} - 77744 q^{86} + 13338 q^{87} - 13824 q^{88} + 36390 q^{89} + 17496 q^{90} + 48902 q^{91} - 36288 q^{92} - 75240 q^{93} - 88800 q^{94} - 47736 q^{95} + 9216 q^{96} - 79078 q^{97} - 9604 q^{98} + 17496 q^{99}+O(q^{100})$$ q - 4 * q^2 - 9 * q^3 + 16 * q^4 - 54 * q^5 + 36 * q^6 + 49 * q^7 - 64 * q^8 + 81 * q^9 + 216 * q^10 + 216 * q^11 - 144 * q^12 + 998 * q^13 - 196 * q^14 + 486 * q^15 + 256 * q^16 + 1302 * q^17 - 324 * q^18 + 884 * q^19 - 864 * q^20 - 441 * q^21 - 864 * q^22 - 2268 * q^23 + 576 * q^24 - 209 * q^25 - 3992 * q^26 - 729 * q^27 + 784 * q^28 - 1482 * q^29 - 1944 * q^30 + 8360 * q^31 - 1024 * q^32 - 1944 * q^33 - 5208 * q^34 - 2646 * q^35 + 1296 * q^36 - 4714 * q^37 - 3536 * q^38 - 8982 * q^39 + 3456 * q^40 - 9786 * q^41 + 1764 * q^42 + 19436 * q^43 + 3456 * q^44 - 4374 * q^45 + 9072 * q^46 + 22200 * q^47 - 2304 * q^48 + 2401 * q^49 + 836 * q^50 - 11718 * q^51 + 15968 * q^52 + 26790 * q^53 + 2916 * q^54 - 11664 * q^55 - 3136 * q^56 - 7956 * q^57 + 5928 * q^58 + 28092 * q^59 + 7776 * q^60 - 38866 * q^61 - 33440 * q^62 + 3969 * q^63 + 4096 * q^64 - 53892 * q^65 + 7776 * q^66 + 23948 * q^67 + 20832 * q^68 + 20412 * q^69 + 10584 * q^70 - 20628 * q^71 - 5184 * q^72 + 290 * q^73 + 18856 * q^74 + 1881 * q^75 + 14144 * q^76 + 10584 * q^77 + 35928 * q^78 - 99544 * q^79 - 13824 * q^80 + 6561 * q^81 + 39144 * q^82 + 19308 * q^83 - 7056 * q^84 - 70308 * q^85 - 77744 * q^86 + 13338 * q^87 - 13824 * q^88 + 36390 * q^89 + 17496 * q^90 + 48902 * q^91 - 36288 * q^92 - 75240 * q^93 - 88800 * q^94 - 47736 * q^95 + 9216 * q^96 - 79078 * q^97 - 9604 * q^98 + 17496 * 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 −54.0000 36.0000 49.0000 −64.0000 81.0000 216.000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$+1$$
$$3$$ $$+1$$
$$7$$ $$-1$$

## 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 42.6.a.a 1
3.b odd 2 1 126.6.a.k 1
4.b odd 2 1 336.6.a.j 1
5.b even 2 1 1050.6.a.n 1
5.c odd 4 2 1050.6.g.o 2
7.b odd 2 1 294.6.a.h 1
7.c even 3 2 294.6.e.r 2
7.d odd 6 2 294.6.e.h 2
12.b even 2 1 1008.6.a.x 1
21.c even 2 1 882.6.a.o 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.a.a 1 1.a even 1 1 trivial
126.6.a.k 1 3.b odd 2 1
294.6.a.h 1 7.b odd 2 1
294.6.e.h 2 7.d odd 6 2
294.6.e.r 2 7.c even 3 2
336.6.a.j 1 4.b odd 2 1
882.6.a.o 1 21.c even 2 1
1008.6.a.x 1 12.b even 2 1
1050.6.a.n 1 5.b even 2 1
1050.6.g.o 2 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 54$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(42))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 4$$
$3$ $$T + 9$$
$5$ $$T + 54$$
$7$ $$T - 49$$
$11$ $$T - 216$$
$13$ $$T - 998$$
$17$ $$T - 1302$$
$19$ $$T - 884$$
$23$ $$T + 2268$$
$29$ $$T + 1482$$
$31$ $$T - 8360$$
$37$ $$T + 4714$$
$41$ $$T + 9786$$
$43$ $$T - 19436$$
$47$ $$T - 22200$$
$53$ $$T - 26790$$
$59$ $$T - 28092$$
$61$ $$T + 38866$$
$67$ $$T - 23948$$
$71$ $$T + 20628$$
$73$ $$T - 290$$
$79$ $$T + 99544$$
$83$ $$T - 19308$$
$89$ $$T - 36390$$
$97$ $$T + 79078$$