# Properties

 Label 42.6.a.d 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)

 $$f(q)$$ $$=$$ $$q - 4 q^{2} + 9 q^{3} + 16 q^{4} + 26 q^{5} - 36 q^{6} - 49 q^{7} - 64 q^{8} + 81 q^{9}+O(q^{10})$$ q - 4 * q^2 + 9 * q^3 + 16 * q^4 + 26 * q^5 - 36 * q^6 - 49 * q^7 - 64 * q^8 + 81 * q^9 $$q - 4 q^{2} + 9 q^{3} + 16 q^{4} + 26 q^{5} - 36 q^{6} - 49 q^{7} - 64 q^{8} + 81 q^{9} - 104 q^{10} + 664 q^{11} + 144 q^{12} + 318 q^{13} + 196 q^{14} + 234 q^{15} + 256 q^{16} + 1582 q^{17} - 324 q^{18} + 236 q^{19} + 416 q^{20} - 441 q^{21} - 2656 q^{22} + 2212 q^{23} - 576 q^{24} - 2449 q^{25} - 1272 q^{26} + 729 q^{27} - 784 q^{28} - 4954 q^{29} - 936 q^{30} - 7128 q^{31} - 1024 q^{32} + 5976 q^{33} - 6328 q^{34} - 1274 q^{35} + 1296 q^{36} + 4358 q^{37} - 944 q^{38} + 2862 q^{39} - 1664 q^{40} + 10542 q^{41} + 1764 q^{42} - 8452 q^{43} + 10624 q^{44} + 2106 q^{45} - 8848 q^{46} + 5352 q^{47} + 2304 q^{48} + 2401 q^{49} + 9796 q^{50} + 14238 q^{51} + 5088 q^{52} - 33354 q^{53} - 2916 q^{54} + 17264 q^{55} + 3136 q^{56} + 2124 q^{57} + 19816 q^{58} - 15436 q^{59} + 3744 q^{60} - 36762 q^{61} + 28512 q^{62} - 3969 q^{63} + 4096 q^{64} + 8268 q^{65} - 23904 q^{66} + 40972 q^{67} + 25312 q^{68} + 19908 q^{69} + 5096 q^{70} - 9092 q^{71} - 5184 q^{72} - 73454 q^{73} - 17432 q^{74} - 22041 q^{75} + 3776 q^{76} - 32536 q^{77} - 11448 q^{78} + 89400 q^{79} + 6656 q^{80} + 6561 q^{81} - 42168 q^{82} - 6428 q^{83} - 7056 q^{84} + 41132 q^{85} + 33808 q^{86} - 44586 q^{87} - 42496 q^{88} - 122658 q^{89} - 8424 q^{90} - 15582 q^{91} + 35392 q^{92} - 64152 q^{93} - 21408 q^{94} + 6136 q^{95} - 9216 q^{96} + 21370 q^{97} - 9604 q^{98} + 53784 q^{99}+O(q^{100})$$ q - 4 * q^2 + 9 * q^3 + 16 * q^4 + 26 * q^5 - 36 * q^6 - 49 * q^7 - 64 * q^8 + 81 * q^9 - 104 * q^10 + 664 * q^11 + 144 * q^12 + 318 * q^13 + 196 * q^14 + 234 * q^15 + 256 * q^16 + 1582 * q^17 - 324 * q^18 + 236 * q^19 + 416 * q^20 - 441 * q^21 - 2656 * q^22 + 2212 * q^23 - 576 * q^24 - 2449 * q^25 - 1272 * q^26 + 729 * q^27 - 784 * q^28 - 4954 * q^29 - 936 * q^30 - 7128 * q^31 - 1024 * q^32 + 5976 * q^33 - 6328 * q^34 - 1274 * q^35 + 1296 * q^36 + 4358 * q^37 - 944 * q^38 + 2862 * q^39 - 1664 * q^40 + 10542 * q^41 + 1764 * q^42 - 8452 * q^43 + 10624 * q^44 + 2106 * q^45 - 8848 * q^46 + 5352 * q^47 + 2304 * q^48 + 2401 * q^49 + 9796 * q^50 + 14238 * q^51 + 5088 * q^52 - 33354 * q^53 - 2916 * q^54 + 17264 * q^55 + 3136 * q^56 + 2124 * q^57 + 19816 * q^58 - 15436 * q^59 + 3744 * q^60 - 36762 * q^61 + 28512 * q^62 - 3969 * q^63 + 4096 * q^64 + 8268 * q^65 - 23904 * q^66 + 40972 * q^67 + 25312 * q^68 + 19908 * q^69 + 5096 * q^70 - 9092 * q^71 - 5184 * q^72 - 73454 * q^73 - 17432 * q^74 - 22041 * q^75 + 3776 * q^76 - 32536 * q^77 - 11448 * q^78 + 89400 * q^79 + 6656 * q^80 + 6561 * q^81 - 42168 * q^82 - 6428 * q^83 - 7056 * q^84 + 41132 * q^85 + 33808 * q^86 - 44586 * q^87 - 42496 * q^88 - 122658 * q^89 - 8424 * q^90 - 15582 * q^91 + 35392 * q^92 - 64152 * q^93 - 21408 * q^94 + 6136 * q^95 - 9216 * q^96 + 21370 * q^97 - 9604 * q^98 + 53784 * 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 26.0000 −36.0000 −49.0000 −64.0000 81.0000 −104.000
 $$n$$: 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.d 1
3.b odd 2 1 126.6.a.i 1
4.b odd 2 1 336.6.a.h 1
5.b even 2 1 1050.6.a.k 1
5.c odd 4 2 1050.6.g.i 2
7.b odd 2 1 294.6.a.b 1
7.c even 3 2 294.6.e.i 2
7.d odd 6 2 294.6.e.p 2
12.b even 2 1 1008.6.a.j 1
21.c even 2 1 882.6.a.s 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 26$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(42))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 4$$
$3$ $$T - 9$$
$5$ $$T - 26$$
$7$ $$T + 49$$
$11$ $$T - 664$$
$13$ $$T - 318$$
$17$ $$T - 1582$$
$19$ $$T - 236$$
$23$ $$T - 2212$$
$29$ $$T + 4954$$
$31$ $$T + 7128$$
$37$ $$T - 4358$$
$41$ $$T - 10542$$
$43$ $$T + 8452$$
$47$ $$T - 5352$$
$53$ $$T + 33354$$
$59$ $$T + 15436$$
$61$ $$T + 36762$$
$67$ $$T - 40972$$
$71$ $$T + 9092$$
$73$ $$T + 73454$$
$79$ $$T - 89400$$
$83$ $$T + 6428$$
$89$ $$T + 122658$$
$97$ $$T - 21370$$