# Properties

 Label 2960.2.p.c Level $2960$ Weight $2$ Character orbit 2960.p Analytic conductor $23.636$ Analytic rank $1$ 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: $$1$$ 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 185) 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.

 $$f(q)$$ $$=$$ $$q - q^{3} + i q^{5} - q^{7} - 2 q^{9}+O(q^{10})$$ q - q^3 + i * q^5 - q^7 - 2 * q^9 $$q - q^{3} + i q^{5} - q^{7} - 2 q^{9} + 3 q^{11} - i q^{15} - 6 i q^{17} - 6 i q^{19} + q^{21} + 6 i q^{23} - q^{25} + 5 q^{27} + 6 i q^{29} + 6 i q^{31} - 3 q^{33} - i q^{35} + (6 i + 1) q^{37} - 3 q^{41} + 6 i q^{43} - 2 i q^{45} + 3 q^{47} - 6 q^{49} + 6 i q^{51} - 9 q^{53} + 3 i q^{55} + 6 i q^{57} - 12 i q^{59} - 6 i q^{61} + 2 q^{63} - 4 q^{67} - 6 i q^{69} - 9 q^{71} + 7 q^{73} + q^{75} - 3 q^{77} - 12 i q^{79} + q^{81} - 15 q^{83} + 6 q^{85} - 6 i q^{87} - 6 i q^{93} + 6 q^{95} - 6 i q^{97} - 6 q^{99} +O(q^{100})$$ q - q^3 + i * q^5 - q^7 - 2 * q^9 + 3 * q^11 - i * q^15 - 6*i * q^17 - 6*i * q^19 + q^21 + 6*i * q^23 - q^25 + 5 * q^27 + 6*i * q^29 + 6*i * q^31 - 3 * q^33 - i * q^35 + (6*i + 1) * q^37 - 3 * q^41 + 6*i * q^43 - 2*i * q^45 + 3 * q^47 - 6 * q^49 + 6*i * q^51 - 9 * q^53 + 3*i * q^55 + 6*i * q^57 - 12*i * q^59 - 6*i * q^61 + 2 * q^63 - 4 * q^67 - 6*i * q^69 - 9 * q^71 + 7 * q^73 + q^75 - 3 * q^77 - 12*i * q^79 + q^81 - 15 * q^83 + 6 * q^85 - 6*i * q^87 - 6*i * q^93 + 6 * q^95 - 6*i * q^97 - 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 2 q^{7} - 4 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 - 2 * q^7 - 4 * q^9 $$2 q - 2 q^{3} - 2 q^{7} - 4 q^{9} + 6 q^{11} + 2 q^{21} - 2 q^{25} + 10 q^{27} - 6 q^{33} + 2 q^{37} - 6 q^{41} + 6 q^{47} - 12 q^{49} - 18 q^{53} + 4 q^{63} - 8 q^{67} - 18 q^{71} + 14 q^{73} + 2 q^{75} - 6 q^{77} + 2 q^{81} - 30 q^{83} + 12 q^{85} + 12 q^{95} - 12 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 - 2 * q^7 - 4 * q^9 + 6 * q^11 + 2 * q^21 - 2 * q^25 + 10 * q^27 - 6 * q^33 + 2 * q^37 - 6 * q^41 + 6 * q^47 - 12 * q^49 - 18 * q^53 + 4 * q^63 - 8 * q^67 - 18 * q^71 + 14 * q^73 + 2 * q^75 - 6 * q^77 + 2 * q^81 - 30 * q^83 + 12 * q^85 + 12 * q^95 - 12 * 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 −1.00000 0 1.00000i 0 −1.00000 0 −2.00000 0
961.2 0 −1.00000 0 1.00000i 0 −1.00000 0 −2.00000 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.c 2
4.b odd 2 1 185.2.c.a 2
12.b even 2 1 1665.2.e.b 2
20.d odd 2 1 925.2.c.a 2
20.e even 4 1 925.2.d.b 2
20.e even 4 1 925.2.d.c 2
37.b even 2 1 inner 2960.2.p.c 2
148.b odd 2 1 185.2.c.a 2
148.g even 4 1 6845.2.a.c 1
148.g even 4 1 6845.2.a.d 1
444.g even 2 1 1665.2.e.b 2
740.g odd 2 1 925.2.c.a 2
740.m even 4 1 925.2.d.b 2
740.m even 4 1 925.2.d.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
185.2.c.a 2 4.b odd 2 1
185.2.c.a 2 148.b odd 2 1
925.2.c.a 2 20.d odd 2 1
925.2.c.a 2 740.g odd 2 1
925.2.d.b 2 20.e even 4 1
925.2.d.b 2 740.m even 4 1
925.2.d.c 2 20.e even 4 1
925.2.d.c 2 740.m even 4 1
1665.2.e.b 2 12.b even 2 1
1665.2.e.b 2 444.g even 2 1
2960.2.p.c 2 1.a even 1 1 trivial
2960.2.p.c 2 37.b even 2 1 inner
6845.2.a.c 1 148.g even 4 1
6845.2.a.d 1 148.g even 4 1

## Hecke kernels

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

 $$T_{3} + 1$$ T3 + 1 $$T_{7} + 1$$ T7 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2} + 1$$
$7$ $$(T + 1)^{2}$$
$11$ $$(T - 3)^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 36$$
$19$ $$T^{2} + 36$$
$23$ $$T^{2} + 36$$
$29$ $$T^{2} + 36$$
$31$ $$T^{2} + 36$$
$37$ $$T^{2} - 2T + 37$$
$41$ $$(T + 3)^{2}$$
$43$ $$T^{2} + 36$$
$47$ $$(T - 3)^{2}$$
$53$ $$(T + 9)^{2}$$
$59$ $$T^{2} + 144$$
$61$ $$T^{2} + 36$$
$67$ $$(T + 4)^{2}$$
$71$ $$(T + 9)^{2}$$
$73$ $$(T - 7)^{2}$$
$79$ $$T^{2} + 144$$
$83$ $$(T + 15)^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 36$$