# Properties

 Label 2320.2.a.k Level $2320$ Weight $2$ Character orbit 2320.a Self dual yes Analytic conductor $18.525$ 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] = [2320,2,Mod(1,2320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2320, 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("2320.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2320 = 2^{4} \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2320.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: $$18.5252932689$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 145) 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 $$\beta = 2\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 q^{3} + q^{5} + ( - \beta + 2) q^{7} + q^{9}+O(q^{10})$$ q + 2 * q^3 + q^5 + (-b + 2) * q^7 + q^9 $$q + 2 q^{3} + q^{5} + ( - \beta + 2) q^{7} + q^{9} + (\beta + 2) q^{11} - 2 q^{13} + 2 q^{15} - \beta q^{17} + ( - \beta + 2) q^{19} + ( - 2 \beta + 4) q^{21} + (\beta + 6) q^{23} + q^{25} - 4 q^{27} + q^{29} + (3 \beta + 2) q^{31} + (2 \beta + 4) q^{33} + ( - \beta + 2) q^{35} + 3 \beta q^{37} - 4 q^{39} - 6 q^{41} + 6 q^{43} + q^{45} + ( - 2 \beta + 6) q^{47} + ( - 4 \beta + 5) q^{49} - 2 \beta q^{51} + (2 \beta + 2) q^{53} + (\beta + 2) q^{55} + ( - 2 \beta + 4) q^{57} + (2 \beta + 2) q^{61} + ( - \beta + 2) q^{63} - 2 q^{65} + (3 \beta + 2) q^{67} + (2 \beta + 12) q^{69} + ( - 4 \beta + 4) q^{71} - 3 \beta q^{73} + 2 q^{75} - 4 q^{77} + ( - 3 \beta - 6) q^{79} - 11 q^{81} + ( - \beta - 10) q^{83} - \beta q^{85} + 2 q^{87} + (2 \beta - 2) q^{89} + (2 \beta - 4) q^{91} + (6 \beta + 4) q^{93} + ( - \beta + 2) q^{95} + (3 \beta - 4) q^{97} + (\beta + 2) q^{99} +O(q^{100})$$ q + 2 * q^3 + q^5 + (-b + 2) * q^7 + q^9 + (b + 2) * q^11 - 2 * q^13 + 2 * q^15 - b * q^17 + (-b + 2) * q^19 + (-2*b + 4) * q^21 + (b + 6) * q^23 + q^25 - 4 * q^27 + q^29 + (3*b + 2) * q^31 + (2*b + 4) * q^33 + (-b + 2) * q^35 + 3*b * q^37 - 4 * q^39 - 6 * q^41 + 6 * q^43 + q^45 + (-2*b + 6) * q^47 + (-4*b + 5) * q^49 - 2*b * q^51 + (2*b + 2) * q^53 + (b + 2) * q^55 + (-2*b + 4) * q^57 + (2*b + 2) * q^61 + (-b + 2) * q^63 - 2 * q^65 + (3*b + 2) * q^67 + (2*b + 12) * q^69 + (-4*b + 4) * q^71 - 3*b * q^73 + 2 * q^75 - 4 * q^77 + (-3*b - 6) * q^79 - 11 * q^81 + (-b - 10) * q^83 - b * q^85 + 2 * q^87 + (2*b - 2) * q^89 + (2*b - 4) * q^91 + (6*b + 4) * q^93 + (-b + 2) * q^95 + (3*b - 4) * q^97 + (b + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{3} + 2 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + 4 * q^3 + 2 * q^5 + 4 * q^7 + 2 * q^9 $$2 q + 4 q^{3} + 2 q^{5} + 4 q^{7} + 2 q^{9} + 4 q^{11} - 4 q^{13} + 4 q^{15} + 4 q^{19} + 8 q^{21} + 12 q^{23} + 2 q^{25} - 8 q^{27} + 2 q^{29} + 4 q^{31} + 8 q^{33} + 4 q^{35} - 8 q^{39} - 12 q^{41} + 12 q^{43} + 2 q^{45} + 12 q^{47} + 10 q^{49} + 4 q^{53} + 4 q^{55} + 8 q^{57} + 4 q^{61} + 4 q^{63} - 4 q^{65} + 4 q^{67} + 24 q^{69} + 8 q^{71} + 4 q^{75} - 8 q^{77} - 12 q^{79} - 22 q^{81} - 20 q^{83} + 4 q^{87} - 4 q^{89} - 8 q^{91} + 8 q^{93} + 4 q^{95} - 8 q^{97} + 4 q^{99}+O(q^{100})$$ 2 * q + 4 * q^3 + 2 * q^5 + 4 * q^7 + 2 * q^9 + 4 * q^11 - 4 * q^13 + 4 * q^15 + 4 * q^19 + 8 * q^21 + 12 * q^23 + 2 * q^25 - 8 * q^27 + 2 * q^29 + 4 * q^31 + 8 * q^33 + 4 * q^35 - 8 * q^39 - 12 * q^41 + 12 * q^43 + 2 * q^45 + 12 * q^47 + 10 * q^49 + 4 * q^53 + 4 * q^55 + 8 * q^57 + 4 * q^61 + 4 * q^63 - 4 * q^65 + 4 * q^67 + 24 * q^69 + 8 * q^71 + 4 * q^75 - 8 * q^77 - 12 * q^79 - 22 * q^81 - 20 * q^83 + 4 * q^87 - 4 * q^89 - 8 * q^91 + 8 * q^93 + 4 * q^95 - 8 * q^97 + 4 * 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
 1.41421 −1.41421
0 2.00000 0 1.00000 0 −0.828427 0 1.00000 0
1.2 0 2.00000 0 1.00000 0 4.82843 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
$$2$$ $$-1$$
$$5$$ $$-1$$
$$29$$ $$-1$$

## 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 2320.2.a.k 2
4.b odd 2 1 145.2.a.b 2
8.b even 2 1 9280.2.a.w 2
8.d odd 2 1 9280.2.a.be 2
12.b even 2 1 1305.2.a.n 2
20.d odd 2 1 725.2.a.c 2
20.e even 4 2 725.2.b.c 4
28.d even 2 1 7105.2.a.e 2
60.h even 2 1 6525.2.a.p 2
116.d odd 2 1 4205.2.a.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.a.b 2 4.b odd 2 1
725.2.a.c 2 20.d odd 2 1
725.2.b.c 4 20.e even 4 2
1305.2.a.n 2 12.b even 2 1
2320.2.a.k 2 1.a even 1 1 trivial
4205.2.a.d 2 116.d odd 2 1
6525.2.a.p 2 60.h even 2 1
7105.2.a.e 2 28.d even 2 1
9280.2.a.w 2 8.b even 2 1
9280.2.a.be 2 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2320))$$:

 $$T_{3} - 2$$ T3 - 2 $$T_{7}^{2} - 4T_{7} - 4$$ T7^2 - 4*T7 - 4 $$T_{11}^{2} - 4T_{11} - 4$$ T11^2 - 4*T11 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 2)^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} - 4T - 4$$
$11$ $$T^{2} - 4T - 4$$
$13$ $$(T + 2)^{2}$$
$17$ $$T^{2} - 8$$
$19$ $$T^{2} - 4T - 4$$
$23$ $$T^{2} - 12T + 28$$
$29$ $$(T - 1)^{2}$$
$31$ $$T^{2} - 4T - 68$$
$37$ $$T^{2} - 72$$
$41$ $$(T + 6)^{2}$$
$43$ $$(T - 6)^{2}$$
$47$ $$T^{2} - 12T + 4$$
$53$ $$T^{2} - 4T - 28$$
$59$ $$T^{2}$$
$61$ $$T^{2} - 4T - 28$$
$67$ $$T^{2} - 4T - 68$$
$71$ $$T^{2} - 8T - 112$$
$73$ $$T^{2} - 72$$
$79$ $$T^{2} + 12T - 36$$
$83$ $$T^{2} + 20T + 92$$
$89$ $$T^{2} + 4T - 28$$
$97$ $$T^{2} + 8T - 56$$