# Properties

 Label 42.6.a.f 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)

 $$f(q)$$ $$=$$ $$q + 4 q^{2} + 9 q^{3} + 16 q^{4} + 24 q^{5} + 36 q^{6} + 49 q^{7} + 64 q^{8} + 81 q^{9}+O(q^{10})$$ q + 4 * q^2 + 9 * q^3 + 16 * q^4 + 24 * q^5 + 36 * q^6 + 49 * q^7 + 64 * q^8 + 81 * q^9 $$q + 4 q^{2} + 9 q^{3} + 16 q^{4} + 24 q^{5} + 36 q^{6} + 49 q^{7} + 64 q^{8} + 81 q^{9} + 96 q^{10} + 66 q^{11} + 144 q^{12} + 98 q^{13} + 196 q^{14} + 216 q^{15} + 256 q^{16} - 216 q^{17} + 324 q^{18} - 340 q^{19} + 384 q^{20} + 441 q^{21} + 264 q^{22} - 1038 q^{23} + 576 q^{24} - 2549 q^{25} + 392 q^{26} + 729 q^{27} + 784 q^{28} - 2490 q^{29} + 864 q^{30} - 7048 q^{31} + 1024 q^{32} + 594 q^{33} - 864 q^{34} + 1176 q^{35} + 1296 q^{36} - 12238 q^{37} - 1360 q^{38} + 882 q^{39} + 1536 q^{40} + 6468 q^{41} + 1764 q^{42} - 15412 q^{43} + 1056 q^{44} + 1944 q^{45} - 4152 q^{46} + 20604 q^{47} + 2304 q^{48} + 2401 q^{49} - 10196 q^{50} - 1944 q^{51} + 1568 q^{52} + 32490 q^{53} + 2916 q^{54} + 1584 q^{55} + 3136 q^{56} - 3060 q^{57} - 9960 q^{58} + 34224 q^{59} + 3456 q^{60} + 35654 q^{61} - 28192 q^{62} + 3969 q^{63} + 4096 q^{64} + 2352 q^{65} + 2376 q^{66} + 12680 q^{67} - 3456 q^{68} - 9342 q^{69} + 4704 q^{70} - 42642 q^{71} + 5184 q^{72} + 33734 q^{73} - 48952 q^{74} - 22941 q^{75} - 5440 q^{76} + 3234 q^{77} + 3528 q^{78} - 85108 q^{79} + 6144 q^{80} + 6561 q^{81} + 25872 q^{82} - 106764 q^{83} + 7056 q^{84} - 5184 q^{85} - 61648 q^{86} - 22410 q^{87} + 4224 q^{88} + 34884 q^{89} + 7776 q^{90} + 4802 q^{91} - 16608 q^{92} - 63432 q^{93} + 82416 q^{94} - 8160 q^{95} + 9216 q^{96} + 18662 q^{97} + 9604 q^{98} + 5346 q^{99}+O(q^{100})$$ q + 4 * q^2 + 9 * q^3 + 16 * q^4 + 24 * q^5 + 36 * q^6 + 49 * q^7 + 64 * q^8 + 81 * q^9 + 96 * q^10 + 66 * q^11 + 144 * q^12 + 98 * q^13 + 196 * q^14 + 216 * q^15 + 256 * q^16 - 216 * q^17 + 324 * q^18 - 340 * q^19 + 384 * q^20 + 441 * q^21 + 264 * q^22 - 1038 * q^23 + 576 * q^24 - 2549 * q^25 + 392 * q^26 + 729 * q^27 + 784 * q^28 - 2490 * q^29 + 864 * q^30 - 7048 * q^31 + 1024 * q^32 + 594 * q^33 - 864 * q^34 + 1176 * q^35 + 1296 * q^36 - 12238 * q^37 - 1360 * q^38 + 882 * q^39 + 1536 * q^40 + 6468 * q^41 + 1764 * q^42 - 15412 * q^43 + 1056 * q^44 + 1944 * q^45 - 4152 * q^46 + 20604 * q^47 + 2304 * q^48 + 2401 * q^49 - 10196 * q^50 - 1944 * q^51 + 1568 * q^52 + 32490 * q^53 + 2916 * q^54 + 1584 * q^55 + 3136 * q^56 - 3060 * q^57 - 9960 * q^58 + 34224 * q^59 + 3456 * q^60 + 35654 * q^61 - 28192 * q^62 + 3969 * q^63 + 4096 * q^64 + 2352 * q^65 + 2376 * q^66 + 12680 * q^67 - 3456 * q^68 - 9342 * q^69 + 4704 * q^70 - 42642 * q^71 + 5184 * q^72 + 33734 * q^73 - 48952 * q^74 - 22941 * q^75 - 5440 * q^76 + 3234 * q^77 + 3528 * q^78 - 85108 * q^79 + 6144 * q^80 + 6561 * q^81 + 25872 * q^82 - 106764 * q^83 + 7056 * q^84 - 5184 * q^85 - 61648 * q^86 - 22410 * q^87 + 4224 * q^88 + 34884 * q^89 + 7776 * q^90 + 4802 * q^91 - 16608 * q^92 - 63432 * q^93 + 82416 * q^94 - 8160 * q^95 + 9216 * q^96 + 18662 * q^97 + 9604 * q^98 + 5346 * 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 24.0000 36.0000 49.0000 64.0000 81.0000 96.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## 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.f 1
3.b odd 2 1 126.6.a.b 1
4.b odd 2 1 336.6.a.g 1
5.b even 2 1 1050.6.a.a 1
5.c odd 4 2 1050.6.g.m 2
7.b odd 2 1 294.6.a.i 1
7.c even 3 2 294.6.e.b 2
7.d odd 6 2 294.6.e.f 2
12.b even 2 1 1008.6.a.k 1
21.c even 2 1 882.6.a.i 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 24$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(42))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 4$$
$3$ $$T - 9$$
$5$ $$T - 24$$
$7$ $$T - 49$$
$11$ $$T - 66$$
$13$ $$T - 98$$
$17$ $$T + 216$$
$19$ $$T + 340$$
$23$ $$T + 1038$$
$29$ $$T + 2490$$
$31$ $$T + 7048$$
$37$ $$T + 12238$$
$41$ $$T - 6468$$
$43$ $$T + 15412$$
$47$ $$T - 20604$$
$53$ $$T - 32490$$
$59$ $$T - 34224$$
$61$ $$T - 35654$$
$67$ $$T - 12680$$
$71$ $$T + 42642$$
$73$ $$T - 33734$$
$79$ $$T + 85108$$
$83$ $$T + 106764$$
$89$ $$T - 34884$$
$97$ $$T - 18662$$