# Properties

 Label 242.6.a.d Level $242$ Weight $6$ Character orbit 242.a Self dual yes Analytic conductor $38.813$ Analytic rank $1$ 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] = [242,6,Mod(1,242)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(242, 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("242.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$242 = 2 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 242.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: $$38.8128843947$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 22) 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)

 $$f(q)$$ $$=$$ $$q + 4 q^{2} + q^{3} + 16 q^{4} - 51 q^{5} + 4 q^{6} + 166 q^{7} + 64 q^{8} - 242 q^{9}+O(q^{10})$$ q + 4 * q^2 + q^3 + 16 * q^4 - 51 * q^5 + 4 * q^6 + 166 * q^7 + 64 * q^8 - 242 * q^9 $$q + 4 q^{2} + q^{3} + 16 q^{4} - 51 q^{5} + 4 q^{6} + 166 q^{7} + 64 q^{8} - 242 q^{9} - 204 q^{10} + 16 q^{12} - 692 q^{13} + 664 q^{14} - 51 q^{15} + 256 q^{16} + 738 q^{17} - 968 q^{18} - 1424 q^{19} - 816 q^{20} + 166 q^{21} - 1779 q^{23} + 64 q^{24} - 524 q^{25} - 2768 q^{26} - 485 q^{27} + 2656 q^{28} + 2064 q^{29} - 204 q^{30} + 6245 q^{31} + 1024 q^{32} + 2952 q^{34} - 8466 q^{35} - 3872 q^{36} - 14785 q^{37} - 5696 q^{38} - 692 q^{39} - 3264 q^{40} - 5304 q^{41} + 664 q^{42} - 17798 q^{43} + 12342 q^{45} - 7116 q^{46} - 17184 q^{47} + 256 q^{48} + 10749 q^{49} - 2096 q^{50} + 738 q^{51} - 11072 q^{52} - 30726 q^{53} - 1940 q^{54} + 10624 q^{56} - 1424 q^{57} + 8256 q^{58} - 34989 q^{59} - 816 q^{60} + 45940 q^{61} + 24980 q^{62} - 40172 q^{63} + 4096 q^{64} + 35292 q^{65} + 25343 q^{67} + 11808 q^{68} - 1779 q^{69} - 33864 q^{70} + 13311 q^{71} - 15488 q^{72} + 53260 q^{73} - 59140 q^{74} - 524 q^{75} - 22784 q^{76} - 2768 q^{78} - 77234 q^{79} - 13056 q^{80} + 58321 q^{81} - 21216 q^{82} - 55014 q^{83} + 2656 q^{84} - 37638 q^{85} - 71192 q^{86} + 2064 q^{87} + 125415 q^{89} + 49368 q^{90} - 114872 q^{91} - 28464 q^{92} + 6245 q^{93} - 68736 q^{94} + 72624 q^{95} + 1024 q^{96} - 88807 q^{97} + 42996 q^{98}+O(q^{100})$$ q + 4 * q^2 + q^3 + 16 * q^4 - 51 * q^5 + 4 * q^6 + 166 * q^7 + 64 * q^8 - 242 * q^9 - 204 * q^10 + 16 * q^12 - 692 * q^13 + 664 * q^14 - 51 * q^15 + 256 * q^16 + 738 * q^17 - 968 * q^18 - 1424 * q^19 - 816 * q^20 + 166 * q^21 - 1779 * q^23 + 64 * q^24 - 524 * q^25 - 2768 * q^26 - 485 * q^27 + 2656 * q^28 + 2064 * q^29 - 204 * q^30 + 6245 * q^31 + 1024 * q^32 + 2952 * q^34 - 8466 * q^35 - 3872 * q^36 - 14785 * q^37 - 5696 * q^38 - 692 * q^39 - 3264 * q^40 - 5304 * q^41 + 664 * q^42 - 17798 * q^43 + 12342 * q^45 - 7116 * q^46 - 17184 * q^47 + 256 * q^48 + 10749 * q^49 - 2096 * q^50 + 738 * q^51 - 11072 * q^52 - 30726 * q^53 - 1940 * q^54 + 10624 * q^56 - 1424 * q^57 + 8256 * q^58 - 34989 * q^59 - 816 * q^60 + 45940 * q^61 + 24980 * q^62 - 40172 * q^63 + 4096 * q^64 + 35292 * q^65 + 25343 * q^67 + 11808 * q^68 - 1779 * q^69 - 33864 * q^70 + 13311 * q^71 - 15488 * q^72 + 53260 * q^73 - 59140 * q^74 - 524 * q^75 - 22784 * q^76 - 2768 * q^78 - 77234 * q^79 - 13056 * q^80 + 58321 * q^81 - 21216 * q^82 - 55014 * q^83 + 2656 * q^84 - 37638 * q^85 - 71192 * q^86 + 2064 * q^87 + 125415 * q^89 + 49368 * q^90 - 114872 * q^91 - 28464 * q^92 + 6245 * q^93 - 68736 * q^94 + 72624 * q^95 + 1024 * q^96 - 88807 * q^97 + 42996 * q^98

## 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 1.00000 16.0000 −51.0000 4.00000 166.000 64.0000 −242.000 −204.000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$11$$ $$-1$$

## 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 242.6.a.d 1
11.b odd 2 1 22.6.a.b 1
33.d even 2 1 198.6.a.i 1
44.c even 2 1 176.6.a.b 1
55.d odd 2 1 550.6.a.f 1
55.e even 4 2 550.6.b.f 2
77.b even 2 1 1078.6.a.a 1
88.b odd 2 1 704.6.a.e 1
88.g even 2 1 704.6.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.6.a.b 1 11.b odd 2 1
176.6.a.b 1 44.c even 2 1
198.6.a.i 1 33.d even 2 1
242.6.a.d 1 1.a even 1 1 trivial
550.6.a.f 1 55.d odd 2 1
550.6.b.f 2 55.e even 4 2
704.6.a.e 1 88.b odd 2 1
704.6.a.f 1 88.g even 2 1
1078.6.a.a 1 77.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(242))$$:

 $$T_{3} - 1$$ T3 - 1 $$T_{7} - 166$$ T7 - 166

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 4$$
$3$ $$T - 1$$
$5$ $$T + 51$$
$7$ $$T - 166$$
$11$ $$T$$
$13$ $$T + 692$$
$17$ $$T - 738$$
$19$ $$T + 1424$$
$23$ $$T + 1779$$
$29$ $$T - 2064$$
$31$ $$T - 6245$$
$37$ $$T + 14785$$
$41$ $$T + 5304$$
$43$ $$T + 17798$$
$47$ $$T + 17184$$
$53$ $$T + 30726$$
$59$ $$T + 34989$$
$61$ $$T - 45940$$
$67$ $$T - 25343$$
$71$ $$T - 13311$$
$73$ $$T - 53260$$
$79$ $$T + 77234$$
$83$ $$T + 55014$$
$89$ $$T - 125415$$
$97$ $$T + 88807$$