# Properties

 Label 704.2.a.p Level $704$ Weight $2$ Character orbit 704.a Self dual yes Analytic conductor $5.621$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [704,2,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, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("704.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$704 = 2^{6} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ 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: $$5.62146830230$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 88) 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + (\beta - 2) q^{5} + 2 \beta q^{7} + (\beta + 1) q^{9}+O(q^{10})$$ q + b * q^3 + (b - 2) * q^5 + 2*b * q^7 + (b + 1) * q^9 $$q + \beta q^{3} + (\beta - 2) q^{5} + 2 \beta q^{7} + (\beta + 1) q^{9} - q^{11} + ( - 2 \beta + 2) q^{13} + ( - \beta + 4) q^{15} + 2 q^{17} - 4 q^{19} + (2 \beta + 8) q^{21} + ( - \beta - 4) q^{23} + ( - 3 \beta + 3) q^{25} + ( - \beta + 4) q^{27} + ( - 2 \beta + 2) q^{29} + ( - \beta + 4) q^{31} - \beta q^{33} + ( - 2 \beta + 8) q^{35} + ( - \beta + 6) q^{37} - 8 q^{39} + (2 \beta + 2) q^{41} + (2 \beta - 4) q^{43} + 2 q^{45} - 8 q^{47} + (4 \beta + 9) q^{49} + 2 \beta q^{51} + (4 \beta - 6) q^{53} + ( - \beta + 2) q^{55} - 4 \beta q^{57} - 5 \beta q^{59} + (2 \beta + 2) q^{61} + (4 \beta + 8) q^{63} + (4 \beta - 12) q^{65} + ( - \beta + 8) q^{67} + ( - 5 \beta - 4) q^{69} + ( - 3 \beta + 4) q^{71} + ( - 2 \beta + 2) q^{73} - 12 q^{75} - 2 \beta q^{77} + ( - 2 \beta + 8) q^{79} - 7 q^{81} + (2 \beta + 4) q^{83} + (2 \beta - 4) q^{85} - 8 q^{87} + ( - 3 \beta - 2) q^{89} - 16 q^{91} + (3 \beta - 4) q^{93} + ( - 4 \beta + 8) q^{95} + ( - \beta + 14) q^{97} + ( - \beta - 1) q^{99} +O(q^{100})$$ q + b * q^3 + (b - 2) * q^5 + 2*b * q^7 + (b + 1) * q^9 - q^11 + (-2*b + 2) * q^13 + (-b + 4) * q^15 + 2 * q^17 - 4 * q^19 + (2*b + 8) * q^21 + (-b - 4) * q^23 + (-3*b + 3) * q^25 + (-b + 4) * q^27 + (-2*b + 2) * q^29 + (-b + 4) * q^31 - b * q^33 + (-2*b + 8) * q^35 + (-b + 6) * q^37 - 8 * q^39 + (2*b + 2) * q^41 + (2*b - 4) * q^43 + 2 * q^45 - 8 * q^47 + (4*b + 9) * q^49 + 2*b * q^51 + (4*b - 6) * q^53 + (-b + 2) * q^55 - 4*b * q^57 - 5*b * q^59 + (2*b + 2) * q^61 + (4*b + 8) * q^63 + (4*b - 12) * q^65 + (-b + 8) * q^67 + (-5*b - 4) * q^69 + (-3*b + 4) * q^71 + (-2*b + 2) * q^73 - 12 * q^75 - 2*b * q^77 + (-2*b + 8) * q^79 - 7 * q^81 + (2*b + 4) * q^83 + (2*b - 4) * q^85 - 8 * q^87 + (-3*b - 2) * q^89 - 16 * q^91 + (3*b - 4) * q^93 + (-4*b + 8) * q^95 + (-b + 14) * q^97 + (-b - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10})$$ 2 * q + q^3 - 3 * q^5 + 2 * q^7 + 3 * q^9 $$2 q + q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9} - 2 q^{11} + 2 q^{13} + 7 q^{15} + 4 q^{17} - 8 q^{19} + 18 q^{21} - 9 q^{23} + 3 q^{25} + 7 q^{27} + 2 q^{29} + 7 q^{31} - q^{33} + 14 q^{35} + 11 q^{37} - 16 q^{39} + 6 q^{41} - 6 q^{43} + 4 q^{45} - 16 q^{47} + 22 q^{49} + 2 q^{51} - 8 q^{53} + 3 q^{55} - 4 q^{57} - 5 q^{59} + 6 q^{61} + 20 q^{63} - 20 q^{65} + 15 q^{67} - 13 q^{69} + 5 q^{71} + 2 q^{73} - 24 q^{75} - 2 q^{77} + 14 q^{79} - 14 q^{81} + 10 q^{83} - 6 q^{85} - 16 q^{87} - 7 q^{89} - 32 q^{91} - 5 q^{93} + 12 q^{95} + 27 q^{97} - 3 q^{99}+O(q^{100})$$ 2 * q + q^3 - 3 * q^5 + 2 * q^7 + 3 * q^9 - 2 * q^11 + 2 * q^13 + 7 * q^15 + 4 * q^17 - 8 * q^19 + 18 * q^21 - 9 * q^23 + 3 * q^25 + 7 * q^27 + 2 * q^29 + 7 * q^31 - q^33 + 14 * q^35 + 11 * q^37 - 16 * q^39 + 6 * q^41 - 6 * q^43 + 4 * q^45 - 16 * q^47 + 22 * q^49 + 2 * q^51 - 8 * q^53 + 3 * q^55 - 4 * q^57 - 5 * q^59 + 6 * q^61 + 20 * q^63 - 20 * q^65 + 15 * q^67 - 13 * q^69 + 5 * q^71 + 2 * q^73 - 24 * q^75 - 2 * q^77 + 14 * q^79 - 14 * q^81 + 10 * q^83 - 6 * q^85 - 16 * q^87 - 7 * q^89 - 32 * q^91 - 5 * q^93 + 12 * q^95 + 27 * q^97 - 3 * 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
 −1.56155 2.56155
0 −1.56155 0 −3.56155 0 −3.12311 0 −0.561553 0
1.2 0 2.56155 0 0.561553 0 5.12311 0 3.56155 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 704.2.a.p 2
3.b odd 2 1 6336.2.a.cx 2
4.b odd 2 1 704.2.a.m 2
8.b even 2 1 176.2.a.d 2
8.d odd 2 1 88.2.a.b 2
11.b odd 2 1 7744.2.a.cl 2
12.b even 2 1 6336.2.a.cu 2
16.e even 4 2 2816.2.c.p 4
16.f odd 4 2 2816.2.c.w 4
24.f even 2 1 792.2.a.h 2
24.h odd 2 1 1584.2.a.t 2
40.e odd 2 1 2200.2.a.o 2
40.f even 2 1 4400.2.a.bp 2
40.i odd 4 2 4400.2.b.v 4
40.k even 4 2 2200.2.b.g 4
44.c even 2 1 7744.2.a.by 2
56.e even 2 1 4312.2.a.n 2
56.h odd 2 1 8624.2.a.cb 2
88.b odd 2 1 1936.2.a.r 2
88.g even 2 1 968.2.a.j 2
88.k even 10 4 968.2.i.q 8
88.l odd 10 4 968.2.i.r 8
264.p odd 2 1 8712.2.a.bb 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.2.a.b 2 8.d odd 2 1
176.2.a.d 2 8.b even 2 1
704.2.a.m 2 4.b odd 2 1
704.2.a.p 2 1.a even 1 1 trivial
792.2.a.h 2 24.f even 2 1
968.2.a.j 2 88.g even 2 1
968.2.i.q 8 88.k even 10 4
968.2.i.r 8 88.l odd 10 4
1584.2.a.t 2 24.h odd 2 1
1936.2.a.r 2 88.b odd 2 1
2200.2.a.o 2 40.e odd 2 1
2200.2.b.g 4 40.k even 4 2
2816.2.c.p 4 16.e even 4 2
2816.2.c.w 4 16.f odd 4 2
4312.2.a.n 2 56.e even 2 1
4400.2.a.bp 2 40.f even 2 1
4400.2.b.v 4 40.i odd 4 2
6336.2.a.cu 2 12.b even 2 1
6336.2.a.cx 2 3.b odd 2 1
7744.2.a.by 2 44.c even 2 1
7744.2.a.cl 2 11.b odd 2 1
8624.2.a.cb 2 56.h odd 2 1
8712.2.a.bb 2 264.p odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(704))$$:

 $$T_{3}^{2} - T_{3} - 4$$ T3^2 - T3 - 4 $$T_{5}^{2} + 3T_{5} - 2$$ T5^2 + 3*T5 - 2 $$T_{7}^{2} - 2T_{7} - 16$$ T7^2 - 2*T7 - 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T - 4$$
$5$ $$T^{2} + 3T - 2$$
$7$ $$T^{2} - 2T - 16$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2} - 2T - 16$$
$17$ $$(T - 2)^{2}$$
$19$ $$(T + 4)^{2}$$
$23$ $$T^{2} + 9T + 16$$
$29$ $$T^{2} - 2T - 16$$
$31$ $$T^{2} - 7T + 8$$
$37$ $$T^{2} - 11T + 26$$
$41$ $$T^{2} - 6T - 8$$
$43$ $$T^{2} + 6T - 8$$
$47$ $$(T + 8)^{2}$$
$53$ $$T^{2} + 8T - 52$$
$59$ $$T^{2} + 5T - 100$$
$61$ $$T^{2} - 6T - 8$$
$67$ $$T^{2} - 15T + 52$$
$71$ $$T^{2} - 5T - 32$$
$73$ $$T^{2} - 2T - 16$$
$79$ $$T^{2} - 14T + 32$$
$83$ $$T^{2} - 10T + 8$$
$89$ $$T^{2} + 7T - 26$$
$97$ $$T^{2} - 27T + 178$$