# Properties

 Label 2320.2.a.i Level $2320$ Weight $2$ Character orbit 2320.a Self dual yes Analytic conductor $18.525$ 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] = [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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 290) 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 $$\beta = \frac{1}{2}(1 + \sqrt{13})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{3} - q^{5} + (\beta - 3) q^{7} + \beta q^{9} +O(q^{10})$$ q - b * q^3 - q^5 + (b - 3) * q^7 + b * q^9 $$q - \beta q^{3} - q^{5} + (\beta - 3) q^{7} + \beta q^{9} + ( - 2 \beta + 2) q^{11} + (\beta + 4) q^{13} + \beta q^{15} + ( - 3 \beta + 3) q^{17} + (2 \beta - 4) q^{19} + (2 \beta - 3) q^{21} + (\beta - 4) q^{23} + q^{25} + (2 \beta - 3) q^{27} + q^{29} + (3 \beta + 1) q^{31} + 6 q^{33} + ( - \beta + 3) q^{35} + (6 \beta - 4) q^{37} + ( - 5 \beta - 3) q^{39} + ( - 2 \beta - 4) q^{41} + ( - \beta - 1) q^{43} - \beta q^{45} + ( - 5 \beta + 5) q^{49} + 9 q^{51} + ( - 3 \beta + 3) q^{53} + (2 \beta - 2) q^{55} + (2 \beta - 6) q^{57} + (3 \beta - 6) q^{59} + (3 \beta - 10) q^{61} + ( - 2 \beta + 3) q^{63} + ( - \beta - 4) q^{65} + 4 \beta q^{67} + (3 \beta - 3) q^{69} + (5 \beta - 9) q^{73} - \beta q^{75} + (6 \beta - 12) q^{77} + (3 \beta - 2) q^{79} + ( - 2 \beta - 6) q^{81} + (2 \beta + 4) q^{83} + (3 \beta - 3) q^{85} - \beta q^{87} + ( - 2 \beta - 10) q^{89} + (2 \beta - 9) q^{91} + ( - 4 \beta - 9) q^{93} + ( - 2 \beta + 4) q^{95} + ( - 3 \beta + 8) q^{97} - 6 q^{99} +O(q^{100})$$ q - b * q^3 - q^5 + (b - 3) * q^7 + b * q^9 + (-2*b + 2) * q^11 + (b + 4) * q^13 + b * q^15 + (-3*b + 3) * q^17 + (2*b - 4) * q^19 + (2*b - 3) * q^21 + (b - 4) * q^23 + q^25 + (2*b - 3) * q^27 + q^29 + (3*b + 1) * q^31 + 6 * q^33 + (-b + 3) * q^35 + (6*b - 4) * q^37 + (-5*b - 3) * q^39 + (-2*b - 4) * q^41 + (-b - 1) * q^43 - b * q^45 + (-5*b + 5) * q^49 + 9 * q^51 + (-3*b + 3) * q^53 + (2*b - 2) * q^55 + (2*b - 6) * q^57 + (3*b - 6) * q^59 + (3*b - 10) * q^61 + (-2*b + 3) * q^63 + (-b - 4) * q^65 + 4*b * q^67 + (3*b - 3) * q^69 + (5*b - 9) * q^73 - b * q^75 + (6*b - 12) * q^77 + (3*b - 2) * q^79 + (-2*b - 6) * q^81 + (2*b + 4) * q^83 + (3*b - 3) * q^85 - b * q^87 + (-2*b - 10) * q^89 + (2*b - 9) * q^91 + (-4*b - 9) * q^93 + (-2*b + 4) * q^95 + (-3*b + 8) * q^97 - 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} - 2 q^{5} - 5 q^{7} + q^{9}+O(q^{10})$$ 2 * q - q^3 - 2 * q^5 - 5 * q^7 + q^9 $$2 q - q^{3} - 2 q^{5} - 5 q^{7} + q^{9} + 2 q^{11} + 9 q^{13} + q^{15} + 3 q^{17} - 6 q^{19} - 4 q^{21} - 7 q^{23} + 2 q^{25} - 4 q^{27} + 2 q^{29} + 5 q^{31} + 12 q^{33} + 5 q^{35} - 2 q^{37} - 11 q^{39} - 10 q^{41} - 3 q^{43} - q^{45} + 5 q^{49} + 18 q^{51} + 3 q^{53} - 2 q^{55} - 10 q^{57} - 9 q^{59} - 17 q^{61} + 4 q^{63} - 9 q^{65} + 4 q^{67} - 3 q^{69} - 13 q^{73} - q^{75} - 18 q^{77} - q^{79} - 14 q^{81} + 10 q^{83} - 3 q^{85} - q^{87} - 22 q^{89} - 16 q^{91} - 22 q^{93} + 6 q^{95} + 13 q^{97} - 12 q^{99}+O(q^{100})$$ 2 * q - q^3 - 2 * q^5 - 5 * q^7 + q^9 + 2 * q^11 + 9 * q^13 + q^15 + 3 * q^17 - 6 * q^19 - 4 * q^21 - 7 * q^23 + 2 * q^25 - 4 * q^27 + 2 * q^29 + 5 * q^31 + 12 * q^33 + 5 * q^35 - 2 * q^37 - 11 * q^39 - 10 * q^41 - 3 * q^43 - q^45 + 5 * q^49 + 18 * q^51 + 3 * q^53 - 2 * q^55 - 10 * q^57 - 9 * q^59 - 17 * q^61 + 4 * q^63 - 9 * q^65 + 4 * q^67 - 3 * q^69 - 13 * q^73 - q^75 - 18 * q^77 - q^79 - 14 * q^81 + 10 * q^83 - 3 * q^85 - q^87 - 22 * q^89 - 16 * q^91 - 22 * q^93 + 6 * q^95 + 13 * q^97 - 12 * 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
 2.30278 −1.30278
0 −2.30278 0 −1.00000 0 −0.697224 0 2.30278 0
1.2 0 1.30278 0 −1.00000 0 −4.30278 0 −1.30278 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.i 2
4.b odd 2 1 290.2.a.b 2
8.b even 2 1 9280.2.a.bc 2
8.d odd 2 1 9280.2.a.z 2
12.b even 2 1 2610.2.a.v 2
20.d odd 2 1 1450.2.a.m 2
20.e even 4 2 1450.2.b.g 4
116.d odd 2 1 8410.2.a.r 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
290.2.a.b 2 4.b odd 2 1
1450.2.a.m 2 20.d odd 2 1
1450.2.b.g 4 20.e even 4 2
2320.2.a.i 2 1.a even 1 1 trivial
2610.2.a.v 2 12.b even 2 1
8410.2.a.r 2 116.d odd 2 1
9280.2.a.z 2 8.d odd 2 1
9280.2.a.bc 2 8.b even 2 1

## Hecke kernels

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

 $$T_{3}^{2} + T_{3} - 3$$ T3^2 + T3 - 3 $$T_{7}^{2} + 5T_{7} + 3$$ T7^2 + 5*T7 + 3 $$T_{11}^{2} - 2T_{11} - 12$$ T11^2 - 2*T11 - 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + T - 3$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} + 5T + 3$$
$11$ $$T^{2} - 2T - 12$$
$13$ $$T^{2} - 9T + 17$$
$17$ $$T^{2} - 3T - 27$$
$19$ $$T^{2} + 6T - 4$$
$23$ $$T^{2} + 7T + 9$$
$29$ $$(T - 1)^{2}$$
$31$ $$T^{2} - 5T - 23$$
$37$ $$T^{2} + 2T - 116$$
$41$ $$T^{2} + 10T + 12$$
$43$ $$T^{2} + 3T - 1$$
$47$ $$T^{2}$$
$53$ $$T^{2} - 3T - 27$$
$59$ $$T^{2} + 9T - 9$$
$61$ $$T^{2} + 17T + 43$$
$67$ $$T^{2} - 4T - 48$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 13T - 39$$
$79$ $$T^{2} + T - 29$$
$83$ $$T^{2} - 10T + 12$$
$89$ $$T^{2} + 22T + 108$$
$97$ $$T^{2} - 13T + 13$$