# Properties

 Label 9408.2.a.dw Level $9408$ Weight $2$ Character orbit 9408.a Self dual yes Analytic conductor $75.123$ 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] = [9408,2,Mod(1,9408)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9408, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("9408.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9408 = 2^{6} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9408.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: $$75.1232582216$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{57})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 14$$ x^2 - x - 14 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 168) 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 $$\beta = \frac{1}{2}(1 + \sqrt{57})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} - \beta q^{5} + q^{9} +O(q^{10})$$ q + q^3 - b * q^5 + q^9 $$q + q^{3} - \beta q^{5} + q^{9} - \beta q^{11} + ( - \beta + 3) q^{13} - \beta q^{15} + 4 q^{17} + ( - \beta + 3) q^{19} - 4 q^{23} + (\beta + 9) q^{25} + q^{27} + (\beta - 2) q^{29} - q^{31} - \beta q^{33} + ( - \beta - 1) q^{37} + ( - \beta + 3) q^{39} + ( - 2 \beta - 2) q^{41} + ( - \beta - 3) q^{43} - \beta q^{45} - 6 q^{47} + 4 q^{51} + (\beta - 6) q^{53} + (\beta + 14) q^{55} + ( - \beta + 3) q^{57} + ( - \beta - 2) q^{59} + 10 q^{61} + ( - 2 \beta + 14) q^{65} + (\beta + 3) q^{67} - 4 q^{69} - 2 q^{71} + ( - \beta + 1) q^{73} + (\beta + 9) q^{75} + (2 \beta - 5) q^{79} + q^{81} + (\beta - 4) q^{83} - 4 \beta q^{85} + (\beta - 2) q^{87} + ( - 2 \beta + 4) q^{89} - q^{93} + ( - 2 \beta + 14) q^{95} + ( - \beta - 12) q^{97} - \beta q^{99} +O(q^{100})$$ q + q^3 - b * q^5 + q^9 - b * q^11 + (-b + 3) * q^13 - b * q^15 + 4 * q^17 + (-b + 3) * q^19 - 4 * q^23 + (b + 9) * q^25 + q^27 + (b - 2) * q^29 - q^31 - b * q^33 + (-b - 1) * q^37 + (-b + 3) * q^39 + (-2*b - 2) * q^41 + (-b - 3) * q^43 - b * q^45 - 6 * q^47 + 4 * q^51 + (b - 6) * q^53 + (b + 14) * q^55 + (-b + 3) * q^57 + (-b - 2) * q^59 + 10 * q^61 + (-2*b + 14) * q^65 + (b + 3) * q^67 - 4 * q^69 - 2 * q^71 + (-b + 1) * q^73 + (b + 9) * q^75 + (2*b - 5) * q^79 + q^81 + (b - 4) * q^83 - 4*b * q^85 + (b - 2) * q^87 + (-2*b + 4) * q^89 - q^93 + (-2*b + 14) * q^95 + (-b - 12) * q^97 - b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - q^{5} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 - q^5 + 2 * q^9 $$2 q + 2 q^{3} - q^{5} + 2 q^{9} - q^{11} + 5 q^{13} - q^{15} + 8 q^{17} + 5 q^{19} - 8 q^{23} + 19 q^{25} + 2 q^{27} - 3 q^{29} - 2 q^{31} - q^{33} - 3 q^{37} + 5 q^{39} - 6 q^{41} - 7 q^{43} - q^{45} - 12 q^{47} + 8 q^{51} - 11 q^{53} + 29 q^{55} + 5 q^{57} - 5 q^{59} + 20 q^{61} + 26 q^{65} + 7 q^{67} - 8 q^{69} - 4 q^{71} + q^{73} + 19 q^{75} - 8 q^{79} + 2 q^{81} - 7 q^{83} - 4 q^{85} - 3 q^{87} + 6 q^{89} - 2 q^{93} + 26 q^{95} - 25 q^{97} - q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 - q^5 + 2 * q^9 - q^11 + 5 * q^13 - q^15 + 8 * q^17 + 5 * q^19 - 8 * q^23 + 19 * q^25 + 2 * q^27 - 3 * q^29 - 2 * q^31 - q^33 - 3 * q^37 + 5 * q^39 - 6 * q^41 - 7 * q^43 - q^45 - 12 * q^47 + 8 * q^51 - 11 * q^53 + 29 * q^55 + 5 * q^57 - 5 * q^59 + 20 * q^61 + 26 * q^65 + 7 * q^67 - 8 * q^69 - 4 * q^71 + q^73 + 19 * q^75 - 8 * q^79 + 2 * q^81 - 7 * q^83 - 4 * q^85 - 3 * q^87 + 6 * q^89 - 2 * q^93 + 26 * q^95 - 25 * q^97 - 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
 4.27492 −3.27492
0 1.00000 0 −4.27492 0 0 0 1.00000 0
1.2 0 1.00000 0 3.27492 0 0 0 1.00000 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$-1$$

## 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 9408.2.a.dw 2
4.b odd 2 1 9408.2.a.dj 2
7.b odd 2 1 9408.2.a.dp 2
7.d odd 6 2 1344.2.q.x 4
8.b even 2 1 2352.2.a.ba 2
8.d odd 2 1 1176.2.a.n 2
24.f even 2 1 3528.2.a.bd 2
24.h odd 2 1 7056.2.a.ch 2
28.d even 2 1 9408.2.a.ec 2
28.f even 6 2 1344.2.q.w 4
56.e even 2 1 1176.2.a.k 2
56.h odd 2 1 2352.2.a.bf 2
56.j odd 6 2 336.2.q.g 4
56.k odd 6 2 1176.2.q.l 4
56.m even 6 2 168.2.q.c 4
56.p even 6 2 2352.2.q.bf 4
168.e odd 2 1 3528.2.a.bk 2
168.i even 2 1 7056.2.a.cu 2
168.v even 6 2 3528.2.s.bk 4
168.ba even 6 2 1008.2.s.r 4
168.be odd 6 2 504.2.s.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.q.c 4 56.m even 6 2
336.2.q.g 4 56.j odd 6 2
504.2.s.i 4 168.be odd 6 2
1008.2.s.r 4 168.ba even 6 2
1176.2.a.k 2 56.e even 2 1
1176.2.a.n 2 8.d odd 2 1
1176.2.q.l 4 56.k odd 6 2
1344.2.q.w 4 28.f even 6 2
1344.2.q.x 4 7.d odd 6 2
2352.2.a.ba 2 8.b even 2 1
2352.2.a.bf 2 56.h odd 2 1
2352.2.q.bf 4 56.p even 6 2
3528.2.a.bd 2 24.f even 2 1
3528.2.a.bk 2 168.e odd 2 1
3528.2.s.bk 4 168.v even 6 2
7056.2.a.ch 2 24.h odd 2 1
7056.2.a.cu 2 168.i even 2 1
9408.2.a.dj 2 4.b odd 2 1
9408.2.a.dp 2 7.b odd 2 1
9408.2.a.dw 2 1.a even 1 1 trivial
9408.2.a.ec 2 28.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(9408))$$:

 $$T_{5}^{2} + T_{5} - 14$$ T5^2 + T5 - 14 $$T_{11}^{2} + T_{11} - 14$$ T11^2 + T11 - 14 $$T_{13}^{2} - 5T_{13} - 8$$ T13^2 - 5*T13 - 8 $$T_{17} - 4$$ T17 - 4 $$T_{19}^{2} - 5T_{19} - 8$$ T19^2 - 5*T19 - 8 $$T_{31} + 1$$ T31 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2} + T - 14$$
$7$ $$T^{2}$$
$11$ $$T^{2} + T - 14$$
$13$ $$T^{2} - 5T - 8$$
$17$ $$(T - 4)^{2}$$
$19$ $$T^{2} - 5T - 8$$
$23$ $$(T + 4)^{2}$$
$29$ $$T^{2} + 3T - 12$$
$31$ $$(T + 1)^{2}$$
$37$ $$T^{2} + 3T - 12$$
$41$ $$T^{2} + 6T - 48$$
$43$ $$T^{2} + 7T - 2$$
$47$ $$(T + 6)^{2}$$
$53$ $$T^{2} + 11T + 16$$
$59$ $$T^{2} + 5T - 8$$
$61$ $$(T - 10)^{2}$$
$67$ $$T^{2} - 7T - 2$$
$71$ $$(T + 2)^{2}$$
$73$ $$T^{2} - T - 14$$
$79$ $$T^{2} + 8T - 41$$
$83$ $$T^{2} + 7T - 2$$
$89$ $$T^{2} - 6T - 48$$
$97$ $$T^{2} + 25T + 142$$