# Properties

 Label 2960.2.p.d Level $2960$ Weight $2$ Character orbit 2960.p Analytic conductor $23.636$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2960,2,Mod(961,2960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2960, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2960.961");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2960 = 2^{4} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2960.p (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$23.6357189983$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 370) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - i q^{5} - 5 q^{7} - 3 q^{9} +O(q^{10})$$ q - i * q^5 - 5 * q^7 - 3 * q^9 $$q - i q^{5} - 5 q^{7} - 3 q^{9} - 3 q^{11} - 2 i q^{13} + i q^{17} + 2 i q^{19} + 6 i q^{23} - q^{25} - 5 i q^{29} + i q^{31} + 5 i q^{35} + (6 i + 1) q^{37} + 5 q^{41} - 11 i q^{43} + 3 i q^{45} + 8 q^{47} + 18 q^{49} - 9 q^{53} + 3 i q^{55} - 12 i q^{59} + 7 i q^{61} + 15 q^{63} - 2 q^{65} - 2 q^{67} - 2 q^{71} + 6 q^{73} + 15 q^{77} + 9 q^{81} + 12 q^{83} + q^{85} - 4 i q^{89} + 10 i q^{91} + 2 q^{95} + 11 i q^{97} + 9 q^{99} +O(q^{100})$$ q - i * q^5 - 5 * q^7 - 3 * q^9 - 3 * q^11 - 2*i * q^13 + i * q^17 + 2*i * q^19 + 6*i * q^23 - q^25 - 5*i * q^29 + i * q^31 + 5*i * q^35 + (6*i + 1) * q^37 + 5 * q^41 - 11*i * q^43 + 3*i * q^45 + 8 * q^47 + 18 * q^49 - 9 * q^53 + 3*i * q^55 - 12*i * q^59 + 7*i * q^61 + 15 * q^63 - 2 * q^65 - 2 * q^67 - 2 * q^71 + 6 * q^73 + 15 * q^77 + 9 * q^81 + 12 * q^83 + q^85 - 4*i * q^89 + 10*i * q^91 + 2 * q^95 + 11*i * q^97 + 9 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 10 q^{7} - 6 q^{9}+O(q^{10})$$ 2 * q - 10 * q^7 - 6 * q^9 $$2 q - 10 q^{7} - 6 q^{9} - 6 q^{11} - 2 q^{25} + 2 q^{37} + 10 q^{41} + 16 q^{47} + 36 q^{49} - 18 q^{53} + 30 q^{63} - 4 q^{65} - 4 q^{67} - 4 q^{71} + 12 q^{73} + 30 q^{77} + 18 q^{81} + 24 q^{83} + 2 q^{85} + 4 q^{95} + 18 q^{99}+O(q^{100})$$ 2 * q - 10 * q^7 - 6 * q^9 - 6 * q^11 - 2 * q^25 + 2 * q^37 + 10 * q^41 + 16 * q^47 + 36 * q^49 - 18 * q^53 + 30 * q^63 - 4 * q^65 - 4 * q^67 - 4 * q^71 + 12 * q^73 + 30 * q^77 + 18 * q^81 + 24 * q^83 + 2 * q^85 + 4 * q^95 + 18 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2960\mathbb{Z}\right)^\times$$.

 $$n$$ $$741$$ $$1777$$ $$2481$$ $$2591$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
961.1
 1.00000i − 1.00000i
0 0 0 1.00000i 0 −5.00000 0 −3.00000 0
961.2 0 0 0 1.00000i 0 −5.00000 0 −3.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2960.2.p.d 2
4.b odd 2 1 370.2.d.b 2
12.b even 2 1 3330.2.h.i 2
20.d odd 2 1 1850.2.d.a 2
20.e even 4 1 1850.2.c.b 2
20.e even 4 1 1850.2.c.e 2
37.b even 2 1 inner 2960.2.p.d 2
148.b odd 2 1 370.2.d.b 2
444.g even 2 1 3330.2.h.i 2
740.g odd 2 1 1850.2.d.a 2
740.m even 4 1 1850.2.c.b 2
740.m even 4 1 1850.2.c.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.d.b 2 4.b odd 2 1
370.2.d.b 2 148.b odd 2 1
1850.2.c.b 2 20.e even 4 1
1850.2.c.b 2 740.m even 4 1
1850.2.c.e 2 20.e even 4 1
1850.2.c.e 2 740.m even 4 1
1850.2.d.a 2 20.d odd 2 1
1850.2.d.a 2 740.g odd 2 1
2960.2.p.d 2 1.a even 1 1 trivial
2960.2.p.d 2 37.b even 2 1 inner
3330.2.h.i 2 12.b even 2 1
3330.2.h.i 2 444.g even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2960, [\chi])$$:

 $$T_{3}$$ T3 $$T_{7} + 5$$ T7 + 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 1$$
$7$ $$(T + 5)^{2}$$
$11$ $$(T + 3)^{2}$$
$13$ $$T^{2} + 4$$
$17$ $$T^{2} + 1$$
$19$ $$T^{2} + 4$$
$23$ $$T^{2} + 36$$
$29$ $$T^{2} + 25$$
$31$ $$T^{2} + 1$$
$37$ $$T^{2} - 2T + 37$$
$41$ $$(T - 5)^{2}$$
$43$ $$T^{2} + 121$$
$47$ $$(T - 8)^{2}$$
$53$ $$(T + 9)^{2}$$
$59$ $$T^{2} + 144$$
$61$ $$T^{2} + 49$$
$67$ $$(T + 2)^{2}$$
$71$ $$(T + 2)^{2}$$
$73$ $$(T - 6)^{2}$$
$79$ $$T^{2}$$
$83$ $$(T - 12)^{2}$$
$89$ $$T^{2} + 16$$
$97$ $$T^{2} + 121$$