# Properties

 Label 704.6.a.f Level $704$ Weight $6$ Character orbit 704.a Self dual yes Analytic conductor $112.910$ 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] = [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: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{3} + 51 q^{5} + 166 q^{7} - 242 q^{9}+O(q^{10})$$ q + q^3 + 51 * q^5 + 166 * q^7 - 242 * q^9 $$q + q^{3} + 51 q^{5} + 166 q^{7} - 242 q^{9} - 121 q^{11} - 692 q^{13} + 51 q^{15} - 738 q^{17} + 1424 q^{19} + 166 q^{21} + 1779 q^{23} - 524 q^{25} - 485 q^{27} + 2064 q^{29} - 6245 q^{31} - 121 q^{33} + 8466 q^{35} + 14785 q^{37} - 692 q^{39} + 5304 q^{41} + 17798 q^{43} - 12342 q^{45} + 17184 q^{47} + 10749 q^{49} - 738 q^{51} + 30726 q^{53} - 6171 q^{55} + 1424 q^{57} - 34989 q^{59} + 45940 q^{61} - 40172 q^{63} - 35292 q^{65} + 25343 q^{67} + 1779 q^{69} - 13311 q^{71} - 53260 q^{73} - 524 q^{75} - 20086 q^{77} - 77234 q^{79} + 58321 q^{81} + 55014 q^{83} - 37638 q^{85} + 2064 q^{87} + 125415 q^{89} - 114872 q^{91} - 6245 q^{93} + 72624 q^{95} - 88807 q^{97} + 29282 q^{99}+O(q^{100})$$ q + q^3 + 51 * q^5 + 166 * q^7 - 242 * q^9 - 121 * q^11 - 692 * q^13 + 51 * q^15 - 738 * q^17 + 1424 * q^19 + 166 * q^21 + 1779 * q^23 - 524 * q^25 - 485 * q^27 + 2064 * q^29 - 6245 * q^31 - 121 * q^33 + 8466 * q^35 + 14785 * q^37 - 692 * q^39 + 5304 * q^41 + 17798 * q^43 - 12342 * q^45 + 17184 * q^47 + 10749 * q^49 - 738 * q^51 + 30726 * q^53 - 6171 * q^55 + 1424 * q^57 - 34989 * q^59 + 45940 * q^61 - 40172 * q^63 - 35292 * q^65 + 25343 * q^67 + 1779 * q^69 - 13311 * q^71 - 53260 * q^73 - 524 * q^75 - 20086 * q^77 - 77234 * q^79 + 58321 * q^81 + 55014 * q^83 - 37638 * q^85 + 2064 * q^87 + 125415 * q^89 - 114872 * q^91 - 6245 * q^93 + 72624 * q^95 - 88807 * q^97 + 29282 * 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 1.00000 0 51.0000 0 166.000 0 −242.000 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.f 1
4.b odd 2 1 704.6.a.e 1
8.b even 2 1 176.6.a.b 1
8.d odd 2 1 22.6.a.b 1
24.f even 2 1 198.6.a.i 1
40.e odd 2 1 550.6.a.f 1
40.k even 4 2 550.6.b.f 2
56.e even 2 1 1078.6.a.a 1
88.g even 2 1 242.6.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.6.a.b 1 8.d odd 2 1
176.6.a.b 1 8.b even 2 1
198.6.a.i 1 24.f even 2 1
242.6.a.d 1 88.g even 2 1
550.6.a.f 1 40.e odd 2 1
550.6.b.f 2 40.k even 4 2
704.6.a.e 1 4.b odd 2 1
704.6.a.f 1 1.a even 1 1 trivial
1078.6.a.a 1 56.e even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 1$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(704))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T - 51$$
$7$ $$T - 166$$
$11$ $$T + 121$$
$13$ $$T + 692$$
$17$ $$T + 738$$
$19$ $$T - 1424$$
$23$ $$T - 1779$$
$29$ $$T - 2064$$
$31$ $$T + 6245$$
$37$ $$T - 14785$$
$41$ $$T - 5304$$
$43$ $$T - 17798$$
$47$ $$T - 17184$$
$53$ $$T - 30726$$
$59$ $$T + 34989$$
$61$ $$T - 45940$$
$67$ $$T - 25343$$
$71$ $$T + 13311$$
$73$ $$T + 53260$$
$79$ $$T + 77234$$
$83$ $$T - 55014$$
$89$ $$T - 125415$$
$97$ $$T + 88807$$