# Properties

 Label 704.6.a.a Level $704$ Weight $6$ Character orbit 704.a Self dual yes Analytic conductor $112.910$ 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] = [704,6,Mod(1,704)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(704, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("704.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$704 = 2^{6} \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 704.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: $$112.910209148$$ 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)

 $$f(q)$$ $$=$$ $$q - 29 q^{3} + 31 q^{5} + 230 q^{7} + 598 q^{9}+O(q^{10})$$ q - 29 * q^3 + 31 * q^5 + 230 * q^7 + 598 * q^9 $$q - 29 q^{3} + 31 q^{5} + 230 q^{7} + 598 q^{9} + 121 q^{11} - 112 q^{13} - 899 q^{15} - 1142 q^{17} - 612 q^{19} - 6670 q^{21} + 1941 q^{23} - 2164 q^{25} - 10295 q^{27} - 1192 q^{29} + 1037 q^{31} - 3509 q^{33} + 7130 q^{35} - 8083 q^{37} + 3248 q^{39} - 10444 q^{41} + 58 q^{43} + 18538 q^{45} - 8656 q^{47} + 36093 q^{49} + 33118 q^{51} + 20318 q^{53} + 3751 q^{55} + 17748 q^{57} - 21351 q^{59} - 47044 q^{61} + 137540 q^{63} - 3472 q^{65} + 48093 q^{67} - 56289 q^{69} + 24967 q^{71} - 42288 q^{73} + 62756 q^{75} + 27830 q^{77} + 72410 q^{79} + 153241 q^{81} - 15806 q^{83} - 35402 q^{85} + 34568 q^{87} - 114761 q^{89} - 25760 q^{91} - 30073 q^{93} - 18972 q^{95} - 5159 q^{97} + 72358 q^{99}+O(q^{100})$$ q - 29 * q^3 + 31 * q^5 + 230 * q^7 + 598 * q^9 + 121 * q^11 - 112 * q^13 - 899 * q^15 - 1142 * q^17 - 612 * q^19 - 6670 * q^21 + 1941 * q^23 - 2164 * q^25 - 10295 * q^27 - 1192 * q^29 + 1037 * q^31 - 3509 * q^33 + 7130 * q^35 - 8083 * q^37 + 3248 * q^39 - 10444 * q^41 + 58 * q^43 + 18538 * q^45 - 8656 * q^47 + 36093 * q^49 + 33118 * q^51 + 20318 * q^53 + 3751 * q^55 + 17748 * q^57 - 21351 * q^59 - 47044 * q^61 + 137540 * q^63 - 3472 * q^65 + 48093 * q^67 - 56289 * q^69 + 24967 * q^71 - 42288 * q^73 + 62756 * q^75 + 27830 * q^77 + 72410 * q^79 + 153241 * q^81 - 15806 * q^83 - 35402 * q^85 + 34568 * q^87 - 114761 * q^89 - 25760 * q^91 - 30073 * q^93 - 18972 * q^95 - 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
0 −29.0000 0 31.0000 0 230.000 0 598.000 0
 $$n$$: 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 704.6.a.a 1
4.b odd 2 1 704.6.a.j 1
8.b even 2 1 176.6.a.e 1
8.d odd 2 1 22.6.a.c 1
24.f even 2 1 198.6.a.b 1
40.e odd 2 1 550.6.a.c 1
40.k even 4 2 550.6.b.a 2
56.e even 2 1 1078.6.a.f 1
88.g 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 8.d odd 2 1
176.6.a.e 1 8.b even 2 1
198.6.a.b 1 24.f even 2 1
242.6.a.a 1 88.g even 2 1
550.6.a.c 1 40.e odd 2 1
550.6.b.a 2 40.k even 4 2
704.6.a.a 1 1.a even 1 1 trivial
704.6.a.j 1 4.b odd 2 1
1078.6.a.f 1 56.e even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 29$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(704))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 29$$
$5$ $$T - 31$$
$7$ $$T - 230$$
$11$ $$T - 121$$
$13$ $$T + 112$$
$17$ $$T + 1142$$
$19$ $$T + 612$$
$23$ $$T - 1941$$
$29$ $$T + 1192$$
$31$ $$T - 1037$$
$37$ $$T + 8083$$
$41$ $$T + 10444$$
$43$ $$T - 58$$
$47$ $$T + 8656$$
$53$ $$T - 20318$$
$59$ $$T + 21351$$
$61$ $$T + 47044$$
$67$ $$T - 48093$$
$71$ $$T - 24967$$
$73$ $$T + 42288$$
$79$ $$T - 72410$$
$83$ $$T + 15806$$
$89$ $$T + 114761$$
$97$ $$T + 5159$$