# Properties

 Label 294.6.a.h Level $294$ Weight $6$ Character orbit 294.a Self dual yes Analytic conductor $47.153$ 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] = [294,6,Mod(1,294)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(294, 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("294.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$294 = 2 \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 294.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: $$47.1528430250$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 42) 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)

 $$f(q)$$ $$=$$ $$q - 4 q^{2} + 9 q^{3} + 16 q^{4} + 54 q^{5} - 36 q^{6} - 64 q^{8} + 81 q^{9}+O(q^{10})$$ q - 4 * q^2 + 9 * q^3 + 16 * q^4 + 54 * q^5 - 36 * q^6 - 64 * q^8 + 81 * q^9 $$q - 4 q^{2} + 9 q^{3} + 16 q^{4} + 54 q^{5} - 36 q^{6} - 64 q^{8} + 81 q^{9} - 216 q^{10} + 216 q^{11} + 144 q^{12} - 998 q^{13} + 486 q^{15} + 256 q^{16} - 1302 q^{17} - 324 q^{18} - 884 q^{19} + 864 q^{20} - 864 q^{22} - 2268 q^{23} - 576 q^{24} - 209 q^{25} + 3992 q^{26} + 729 q^{27} - 1482 q^{29} - 1944 q^{30} - 8360 q^{31} - 1024 q^{32} + 1944 q^{33} + 5208 q^{34} + 1296 q^{36} - 4714 q^{37} + 3536 q^{38} - 8982 q^{39} - 3456 q^{40} + 9786 q^{41} + 19436 q^{43} + 3456 q^{44} + 4374 q^{45} + 9072 q^{46} - 22200 q^{47} + 2304 q^{48} + 836 q^{50} - 11718 q^{51} - 15968 q^{52} + 26790 q^{53} - 2916 q^{54} + 11664 q^{55} - 7956 q^{57} + 5928 q^{58} - 28092 q^{59} + 7776 q^{60} + 38866 q^{61} + 33440 q^{62} + 4096 q^{64} - 53892 q^{65} - 7776 q^{66} + 23948 q^{67} - 20832 q^{68} - 20412 q^{69} - 20628 q^{71} - 5184 q^{72} - 290 q^{73} + 18856 q^{74} - 1881 q^{75} - 14144 q^{76} + 35928 q^{78} - 99544 q^{79} + 13824 q^{80} + 6561 q^{81} - 39144 q^{82} - 19308 q^{83} - 70308 q^{85} - 77744 q^{86} - 13338 q^{87} - 13824 q^{88} - 36390 q^{89} - 17496 q^{90} - 36288 q^{92} - 75240 q^{93} + 88800 q^{94} - 47736 q^{95} - 9216 q^{96} + 79078 q^{97} + 17496 q^{99}+O(q^{100})$$ q - 4 * q^2 + 9 * q^3 + 16 * q^4 + 54 * q^5 - 36 * q^6 - 64 * q^8 + 81 * q^9 - 216 * q^10 + 216 * q^11 + 144 * q^12 - 998 * q^13 + 486 * q^15 + 256 * q^16 - 1302 * q^17 - 324 * q^18 - 884 * q^19 + 864 * q^20 - 864 * q^22 - 2268 * q^23 - 576 * q^24 - 209 * q^25 + 3992 * q^26 + 729 * q^27 - 1482 * q^29 - 1944 * q^30 - 8360 * q^31 - 1024 * q^32 + 1944 * q^33 + 5208 * q^34 + 1296 * q^36 - 4714 * q^37 + 3536 * q^38 - 8982 * q^39 - 3456 * q^40 + 9786 * q^41 + 19436 * q^43 + 3456 * q^44 + 4374 * q^45 + 9072 * q^46 - 22200 * q^47 + 2304 * q^48 + 836 * q^50 - 11718 * q^51 - 15968 * q^52 + 26790 * q^53 - 2916 * q^54 + 11664 * q^55 - 7956 * q^57 + 5928 * q^58 - 28092 * q^59 + 7776 * q^60 + 38866 * q^61 + 33440 * q^62 + 4096 * q^64 - 53892 * q^65 - 7776 * q^66 + 23948 * q^67 - 20832 * q^68 - 20412 * q^69 - 20628 * q^71 - 5184 * q^72 - 290 * q^73 + 18856 * q^74 - 1881 * q^75 - 14144 * q^76 + 35928 * q^78 - 99544 * q^79 + 13824 * q^80 + 6561 * q^81 - 39144 * q^82 - 19308 * q^83 - 70308 * q^85 - 77744 * q^86 - 13338 * q^87 - 13824 * q^88 - 36390 * q^89 - 17496 * q^90 - 36288 * q^92 - 75240 * q^93 + 88800 * q^94 - 47736 * q^95 - 9216 * q^96 + 79078 * q^97 + 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 0 −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 294.6.a.h 1
3.b odd 2 1 882.6.a.o 1
7.b odd 2 1 42.6.a.a 1
7.c even 3 2 294.6.e.h 2
7.d odd 6 2 294.6.e.r 2
21.c even 2 1 126.6.a.k 1
28.d even 2 1 336.6.a.j 1
35.c odd 2 1 1050.6.a.n 1
35.f even 4 2 1050.6.g.o 2
84.h odd 2 1 1008.6.a.x 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.a.a 1 7.b odd 2 1
126.6.a.k 1 21.c even 2 1
294.6.a.h 1 1.a even 1 1 trivial
294.6.e.h 2 7.c even 3 2
294.6.e.r 2 7.d odd 6 2
336.6.a.j 1 28.d even 2 1
882.6.a.o 1 3.b odd 2 1
1008.6.a.x 1 84.h odd 2 1
1050.6.a.n 1 35.c odd 2 1
1050.6.g.o 2 35.f even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(294))$$:

 $$T_{5} - 54$$ T5 - 54 $$T_{11} - 216$$ T11 - 216

## Hecke characteristic polynomials

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