# Properties

 Label 704.6.a.d 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 44) 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)

 $$f(q)$$ $$=$$ $$q - 7 q^{3} + 79 q^{5} - 50 q^{7} - 194 q^{9}+O(q^{10})$$ q - 7 * q^3 + 79 * q^5 - 50 * q^7 - 194 * q^9 $$q - 7 q^{3} + 79 q^{5} - 50 q^{7} - 194 q^{9} - 121 q^{11} + 380 q^{13} - 553 q^{15} - 1154 q^{17} + 1824 q^{19} + 350 q^{21} + 3591 q^{23} + 3116 q^{25} + 3059 q^{27} - 8032 q^{29} - 2945 q^{31} + 847 q^{33} - 3950 q^{35} - 6979 q^{37} - 2660 q^{39} - 520 q^{41} + 2486 q^{43} - 15326 q^{45} - 6920 q^{47} - 14307 q^{49} + 8078 q^{51} + 13718 q^{53} - 9559 q^{55} - 12768 q^{57} + 31779 q^{59} - 34156 q^{61} + 9700 q^{63} + 30020 q^{65} + 61503 q^{67} - 25137 q^{69} - 14971 q^{71} - 36444 q^{73} - 21812 q^{75} + 6050 q^{77} - 28538 q^{79} + 25729 q^{81} - 77482 q^{83} - 91166 q^{85} + 56224 q^{87} + 36271 q^{89} - 19000 q^{91} + 20615 q^{93} + 144096 q^{95} - 49799 q^{97} + 23474 q^{99}+O(q^{100})$$ q - 7 * q^3 + 79 * q^5 - 50 * q^7 - 194 * q^9 - 121 * q^11 + 380 * q^13 - 553 * q^15 - 1154 * q^17 + 1824 * q^19 + 350 * q^21 + 3591 * q^23 + 3116 * q^25 + 3059 * q^27 - 8032 * q^29 - 2945 * q^31 + 847 * q^33 - 3950 * q^35 - 6979 * q^37 - 2660 * q^39 - 520 * q^41 + 2486 * q^43 - 15326 * q^45 - 6920 * q^47 - 14307 * q^49 + 8078 * q^51 + 13718 * q^53 - 9559 * q^55 - 12768 * q^57 + 31779 * q^59 - 34156 * q^61 + 9700 * q^63 + 30020 * q^65 + 61503 * q^67 - 25137 * q^69 - 14971 * q^71 - 36444 * q^73 - 21812 * q^75 + 6050 * q^77 - 28538 * q^79 + 25729 * q^81 - 77482 * q^83 - 91166 * q^85 + 56224 * q^87 + 36271 * q^89 - 19000 * q^91 + 20615 * q^93 + 144096 * q^95 - 49799 * q^97 + 23474 * 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 −7.00000 0 79.0000 0 −50.0000 0 −194.000 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.d 1
4.b odd 2 1 704.6.a.g 1
8.b even 2 1 44.6.a.a 1
8.d odd 2 1 176.6.a.a 1
24.h odd 2 1 396.6.a.e 1
40.f even 2 1 1100.6.a.a 1
40.i odd 4 2 1100.6.b.a 2
88.b odd 2 1 484.6.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.6.a.a 1 8.b even 2 1
176.6.a.a 1 8.d odd 2 1
396.6.a.e 1 24.h odd 2 1
484.6.a.b 1 88.b odd 2 1
704.6.a.d 1 1.a even 1 1 trivial
704.6.a.g 1 4.b odd 2 1
1100.6.a.a 1 40.f even 2 1
1100.6.b.a 2 40.i odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 7$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(704))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 7$$
$5$ $$T - 79$$
$7$ $$T + 50$$
$11$ $$T + 121$$
$13$ $$T - 380$$
$17$ $$T + 1154$$
$19$ $$T - 1824$$
$23$ $$T - 3591$$
$29$ $$T + 8032$$
$31$ $$T + 2945$$
$37$ $$T + 6979$$
$41$ $$T + 520$$
$43$ $$T - 2486$$
$47$ $$T + 6920$$
$53$ $$T - 13718$$
$59$ $$T - 31779$$
$61$ $$T + 34156$$
$67$ $$T - 61503$$
$71$ $$T + 14971$$
$73$ $$T + 36444$$
$79$ $$T + 28538$$
$83$ $$T + 77482$$
$89$ $$T - 36271$$
$97$ $$T + 49799$$