# Properties

 Label 9702.2.a.cu Level $9702$ Weight $2$ Character orbit 9702.a Self dual yes Analytic conductor $77.471$ Analytic rank $1$ 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] = [9702,2,Mod(1,9702)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9702, 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("9702.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9702.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: $$77.4708600410$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 154) 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 $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + (\beta + 1) q^{5} - q^{8}+O(q^{10})$$ q - q^2 + q^4 + (b + 1) * q^5 - q^8 $$q - q^{2} + q^{4} + (\beta + 1) q^{5} - q^{8} + ( - \beta - 1) q^{10} - q^{11} + ( - \beta + 1) q^{13} + q^{16} + ( - 2 \beta - 2) q^{17} + ( - \beta + 5) q^{19} + (\beta + 1) q^{20} + q^{22} - 4 q^{23} + (2 \beta + 1) q^{25} + (\beta - 1) q^{26} + 2 \beta q^{29} - 2 q^{31} - q^{32} + (2 \beta + 2) q^{34} + ( - 4 \beta - 2) q^{37} + (\beta - 5) q^{38} + ( - \beta - 1) q^{40} + (2 \beta + 2) q^{41} + (2 \beta - 6) q^{43} - q^{44} + 4 q^{46} - 2 q^{47} + ( - 2 \beta - 1) q^{50} + ( - \beta + 1) q^{52} + (2 \beta - 4) q^{53} + ( - \beta - 1) q^{55} - 2 \beta q^{58} + (\beta + 5) q^{59} + (\beta + 3) q^{61} + 2 q^{62} + q^{64} - 4 q^{65} + ( - 6 \beta - 2) q^{67} + ( - 2 \beta - 2) q^{68} + (2 \beta - 2) q^{71} + (4 \beta - 4) q^{73} + (4 \beta + 2) q^{74} + ( - \beta + 5) q^{76} + (\beta + 1) q^{80} + ( - 2 \beta - 2) q^{82} + (5 \beta - 1) q^{83} + ( - 4 \beta - 12) q^{85} + ( - 2 \beta + 6) q^{86} + q^{88} + 10 q^{89} - 4 q^{92} + 2 q^{94} + 4 \beta q^{95} + (2 \beta - 8) q^{97} +O(q^{100})$$ q - q^2 + q^4 + (b + 1) * q^5 - q^8 + (-b - 1) * q^10 - q^11 + (-b + 1) * q^13 + q^16 + (-2*b - 2) * q^17 + (-b + 5) * q^19 + (b + 1) * q^20 + q^22 - 4 * q^23 + (2*b + 1) * q^25 + (b - 1) * q^26 + 2*b * q^29 - 2 * q^31 - q^32 + (2*b + 2) * q^34 + (-4*b - 2) * q^37 + (b - 5) * q^38 + (-b - 1) * q^40 + (2*b + 2) * q^41 + (2*b - 6) * q^43 - q^44 + 4 * q^46 - 2 * q^47 + (-2*b - 1) * q^50 + (-b + 1) * q^52 + (2*b - 4) * q^53 + (-b - 1) * q^55 - 2*b * q^58 + (b + 5) * q^59 + (b + 3) * q^61 + 2 * q^62 + q^64 - 4 * q^65 + (-6*b - 2) * q^67 + (-2*b - 2) * q^68 + (2*b - 2) * q^71 + (4*b - 4) * q^73 + (4*b + 2) * q^74 + (-b + 5) * q^76 + (b + 1) * q^80 + (-2*b - 2) * q^82 + (5*b - 1) * q^83 + (-4*b - 12) * q^85 + (-2*b + 6) * q^86 + q^88 + 10 * q^89 - 4 * q^92 + 2 * q^94 + 4*b * q^95 + (2*b - 8) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^8 $$2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{8} - 2 q^{10} - 2 q^{11} + 2 q^{13} + 2 q^{16} - 4 q^{17} + 10 q^{19} + 2 q^{20} + 2 q^{22} - 8 q^{23} + 2 q^{25} - 2 q^{26} - 4 q^{31} - 2 q^{32} + 4 q^{34} - 4 q^{37} - 10 q^{38} - 2 q^{40} + 4 q^{41} - 12 q^{43} - 2 q^{44} + 8 q^{46} - 4 q^{47} - 2 q^{50} + 2 q^{52} - 8 q^{53} - 2 q^{55} + 10 q^{59} + 6 q^{61} + 4 q^{62} + 2 q^{64} - 8 q^{65} - 4 q^{67} - 4 q^{68} - 4 q^{71} - 8 q^{73} + 4 q^{74} + 10 q^{76} + 2 q^{80} - 4 q^{82} - 2 q^{83} - 24 q^{85} + 12 q^{86} + 2 q^{88} + 20 q^{89} - 8 q^{92} + 4 q^{94} - 16 q^{97}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^8 - 2 * q^10 - 2 * q^11 + 2 * q^13 + 2 * q^16 - 4 * q^17 + 10 * q^19 + 2 * q^20 + 2 * q^22 - 8 * q^23 + 2 * q^25 - 2 * q^26 - 4 * q^31 - 2 * q^32 + 4 * q^34 - 4 * q^37 - 10 * q^38 - 2 * q^40 + 4 * q^41 - 12 * q^43 - 2 * q^44 + 8 * q^46 - 4 * q^47 - 2 * q^50 + 2 * q^52 - 8 * q^53 - 2 * q^55 + 10 * q^59 + 6 * q^61 + 4 * q^62 + 2 * q^64 - 8 * q^65 - 4 * q^67 - 4 * q^68 - 4 * q^71 - 8 * q^73 + 4 * q^74 + 10 * q^76 + 2 * q^80 - 4 * q^82 - 2 * q^83 - 24 * q^85 + 12 * q^86 + 2 * q^88 + 20 * q^89 - 8 * q^92 + 4 * q^94 - 16 * q^97

## 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
 −0.618034 1.61803
−1.00000 0 1.00000 −1.23607 0 0 −1.00000 0 1.23607
1.2 −1.00000 0 1.00000 3.23607 0 0 −1.00000 0 −3.23607
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$7$$ $$-1$$
$$11$$ $$1$$

## 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 9702.2.a.cu 2
3.b odd 2 1 1078.2.a.w 2
7.b odd 2 1 1386.2.a.m 2
12.b even 2 1 8624.2.a.bf 2
21.c even 2 1 154.2.a.d 2
21.g even 6 2 1078.2.e.q 4
21.h odd 6 2 1078.2.e.n 4
84.h odd 2 1 1232.2.a.p 2
105.g even 2 1 3850.2.a.bj 2
105.k odd 4 2 3850.2.c.q 4
168.e odd 2 1 4928.2.a.bk 2
168.i even 2 1 4928.2.a.bt 2
231.h odd 2 1 1694.2.a.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.a.d 2 21.c even 2 1
1078.2.a.w 2 3.b odd 2 1
1078.2.e.n 4 21.h odd 6 2
1078.2.e.q 4 21.g even 6 2
1232.2.a.p 2 84.h odd 2 1
1386.2.a.m 2 7.b odd 2 1
1694.2.a.l 2 231.h odd 2 1
3850.2.a.bj 2 105.g even 2 1
3850.2.c.q 4 105.k odd 4 2
4928.2.a.bk 2 168.e odd 2 1
4928.2.a.bt 2 168.i even 2 1
8624.2.a.bf 2 12.b even 2 1
9702.2.a.cu 2 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9702))$$:

 $$T_{5}^{2} - 2T_{5} - 4$$ T5^2 - 2*T5 - 4 $$T_{13}^{2} - 2T_{13} - 4$$ T13^2 - 2*T13 - 4 $$T_{17}^{2} + 4T_{17} - 16$$ T17^2 + 4*T17 - 16 $$T_{19}^{2} - 10T_{19} + 20$$ T19^2 - 10*T19 + 20 $$T_{23} + 4$$ T23 + 4 $$T_{29}^{2} - 20$$ T29^2 - 20

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 2T - 4$$
$7$ $$T^{2}$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2} - 2T - 4$$
$17$ $$T^{2} + 4T - 16$$
$19$ $$T^{2} - 10T + 20$$
$23$ $$(T + 4)^{2}$$
$29$ $$T^{2} - 20$$
$31$ $$(T + 2)^{2}$$
$37$ $$T^{2} + 4T - 76$$
$41$ $$T^{2} - 4T - 16$$
$43$ $$T^{2} + 12T + 16$$
$47$ $$(T + 2)^{2}$$
$53$ $$T^{2} + 8T - 4$$
$59$ $$T^{2} - 10T + 20$$
$61$ $$T^{2} - 6T + 4$$
$67$ $$T^{2} + 4T - 176$$
$71$ $$T^{2} + 4T - 16$$
$73$ $$T^{2} + 8T - 64$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 2T - 124$$
$89$ $$(T - 10)^{2}$$
$97$ $$T^{2} + 16T + 44$$