# Properties

 Label 1078.6.a.f Level $1078$ Weight $6$ Character orbit 1078.a Self dual yes Analytic conductor $172.894$ 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] = [1078,6,Mod(1,1078)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1078, 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("1078.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1078 = 2 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 1078.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: $$172.893757758$$ Analytic rank: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q + 4 q^{2} + 29 q^{3} + 16 q^{4} + 31 q^{5} + 116 q^{6} + 64 q^{8} + 598 q^{9}+O(q^{10})$$ q + 4 * q^2 + 29 * q^3 + 16 * q^4 + 31 * q^5 + 116 * q^6 + 64 * q^8 + 598 * q^9 $$q + 4 q^{2} + 29 q^{3} + 16 q^{4} + 31 q^{5} + 116 q^{6} + 64 q^{8} + 598 q^{9} + 124 q^{10} + 121 q^{11} + 464 q^{12} - 112 q^{13} + 899 q^{15} + 256 q^{16} + 1142 q^{17} + 2392 q^{18} + 612 q^{19} + 496 q^{20} + 484 q^{22} - 1941 q^{23} + 1856 q^{24} - 2164 q^{25} - 448 q^{26} + 10295 q^{27} + 1192 q^{29} + 3596 q^{30} + 1037 q^{31} + 1024 q^{32} + 3509 q^{33} + 4568 q^{34} + 9568 q^{36} + 8083 q^{37} + 2448 q^{38} - 3248 q^{39} + 1984 q^{40} + 10444 q^{41} + 58 q^{43} + 1936 q^{44} + 18538 q^{45} - 7764 q^{46} - 8656 q^{47} + 7424 q^{48} - 8656 q^{50} + 33118 q^{51} - 1792 q^{52} - 20318 q^{53} + 41180 q^{54} + 3751 q^{55} + 17748 q^{57} + 4768 q^{58} + 21351 q^{59} + 14384 q^{60} - 47044 q^{61} + 4148 q^{62} + 4096 q^{64} - 3472 q^{65} + 14036 q^{66} + 48093 q^{67} + 18272 q^{68} - 56289 q^{69} - 24967 q^{71} + 38272 q^{72} + 42288 q^{73} + 32332 q^{74} - 62756 q^{75} + 9792 q^{76} - 12992 q^{78} - 72410 q^{79} + 7936 q^{80} + 153241 q^{81} + 41776 q^{82} + 15806 q^{83} + 35402 q^{85} + 232 q^{86} + 34568 q^{87} + 7744 q^{88} + 114761 q^{89} + 74152 q^{90} - 31056 q^{92} + 30073 q^{93} - 34624 q^{94} + 18972 q^{95} + 29696 q^{96} + 5159 q^{97} + 72358 q^{99}+O(q^{100})$$ q + 4 * q^2 + 29 * q^3 + 16 * q^4 + 31 * q^5 + 116 * q^6 + 64 * q^8 + 598 * q^9 + 124 * q^10 + 121 * q^11 + 464 * q^12 - 112 * q^13 + 899 * q^15 + 256 * q^16 + 1142 * q^17 + 2392 * q^18 + 612 * q^19 + 496 * q^20 + 484 * q^22 - 1941 * q^23 + 1856 * q^24 - 2164 * q^25 - 448 * q^26 + 10295 * q^27 + 1192 * q^29 + 3596 * q^30 + 1037 * q^31 + 1024 * q^32 + 3509 * q^33 + 4568 * q^34 + 9568 * q^36 + 8083 * q^37 + 2448 * q^38 - 3248 * q^39 + 1984 * q^40 + 10444 * q^41 + 58 * q^43 + 1936 * q^44 + 18538 * q^45 - 7764 * q^46 - 8656 * q^47 + 7424 * q^48 - 8656 * q^50 + 33118 * q^51 - 1792 * q^52 - 20318 * q^53 + 41180 * q^54 + 3751 * q^55 + 17748 * q^57 + 4768 * q^58 + 21351 * q^59 + 14384 * q^60 - 47044 * q^61 + 4148 * q^62 + 4096 * q^64 - 3472 * q^65 + 14036 * q^66 + 48093 * q^67 + 18272 * q^68 - 56289 * q^69 - 24967 * q^71 + 38272 * q^72 + 42288 * q^73 + 32332 * q^74 - 62756 * q^75 + 9792 * q^76 - 12992 * q^78 - 72410 * q^79 + 7936 * q^80 + 153241 * q^81 + 41776 * q^82 + 15806 * q^83 + 35402 * q^85 + 232 * q^86 + 34568 * q^87 + 7744 * q^88 + 114761 * q^89 + 74152 * q^90 - 31056 * q^92 + 30073 * q^93 - 34624 * q^94 + 18972 * q^95 + 29696 * q^96 + 5159 * q^97 + 72358 * 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 29.0000 16.0000 31.0000 116.000 0 64.0000 598.000 124.000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$7$$ $$-1$$
$$11$$ $$-1$$

## 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 1078.6.a.f 1
7.b odd 2 1 22.6.a.c 1
21.c even 2 1 198.6.a.b 1
28.d even 2 1 176.6.a.e 1
35.c odd 2 1 550.6.a.c 1
35.f even 4 2 550.6.b.a 2
56.e even 2 1 704.6.a.a 1
56.h odd 2 1 704.6.a.j 1
77.b even 2 1 242.6.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.6.a.c 1 7.b odd 2 1
176.6.a.e 1 28.d even 2 1
198.6.a.b 1 21.c even 2 1
242.6.a.a 1 77.b even 2 1
550.6.a.c 1 35.c odd 2 1
550.6.b.a 2 35.f even 4 2
704.6.a.a 1 56.e even 2 1
704.6.a.j 1 56.h odd 2 1
1078.6.a.f 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 29$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(1078))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 4$$
$3$ $$T - 29$$
$5$ $$T - 31$$
$7$ $$T$$
$11$ $$T - 121$$
$13$ $$T + 112$$
$17$ $$T - 1142$$
$19$ $$T - 612$$
$23$ $$T + 1941$$
$29$ $$T - 1192$$
$31$ $$T - 1037$$
$37$ $$T - 8083$$
$41$ $$T - 10444$$
$43$ $$T - 58$$
$47$ $$T + 8656$$
$53$ $$T + 20318$$
$59$ $$T - 21351$$
$61$ $$T + 47044$$
$67$ $$T - 48093$$
$71$ $$T + 24967$$
$73$ $$T - 42288$$
$79$ $$T + 72410$$
$83$ $$T - 15806$$
$89$ $$T - 114761$$
$97$ $$T - 5159$$