# Properties

 Label 2960.2.p.b 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 1480) 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.

 $$f(q)$$ $$=$$ $$q - 2 q^{3} + i q^{5} - 4 q^{7} + q^{9}+O(q^{10})$$ q - 2 * q^3 + i * q^5 - 4 * q^7 + q^9 $$q - 2 q^{3} + i q^{5} - 4 q^{7} + q^{9} - 4 q^{11} - 2 i q^{13} - 2 i q^{15} + 6 i q^{17} + 8 i q^{19} + 8 q^{21} + 4 i q^{23} - q^{25} + 4 q^{27} + 6 i q^{29} - 2 i q^{31} + 8 q^{33} - 4 i q^{35} + (i - 6) q^{37} + 4 i q^{39} - 2 q^{41} + 4 i q^{43} + i q^{45} - 12 q^{47} + 9 q^{49} - 12 i q^{51} - 12 q^{53} - 4 i q^{55} - 16 i q^{57} - 8 i q^{59} - 2 i q^{61} - 4 q^{63} + 2 q^{65} + 2 q^{67} - 8 i q^{69} + 8 q^{71} + 6 q^{73} + 2 q^{75} + 16 q^{77} - 6 i q^{79} - 11 q^{81} - 14 q^{83} - 6 q^{85} - 12 i q^{87} - 4 i q^{89} + 8 i q^{91} + 4 i q^{93} - 8 q^{95} + 2 i q^{97} - 4 q^{99} +O(q^{100})$$ q - 2 * q^3 + i * q^5 - 4 * q^7 + q^9 - 4 * q^11 - 2*i * q^13 - 2*i * q^15 + 6*i * q^17 + 8*i * q^19 + 8 * q^21 + 4*i * q^23 - q^25 + 4 * q^27 + 6*i * q^29 - 2*i * q^31 + 8 * q^33 - 4*i * q^35 + (i - 6) * q^37 + 4*i * q^39 - 2 * q^41 + 4*i * q^43 + i * q^45 - 12 * q^47 + 9 * q^49 - 12*i * q^51 - 12 * q^53 - 4*i * q^55 - 16*i * q^57 - 8*i * q^59 - 2*i * q^61 - 4 * q^63 + 2 * q^65 + 2 * q^67 - 8*i * q^69 + 8 * q^71 + 6 * q^73 + 2 * q^75 + 16 * q^77 - 6*i * q^79 - 11 * q^81 - 14 * q^83 - 6 * q^85 - 12*i * q^87 - 4*i * q^89 + 8*i * q^91 + 4*i * q^93 - 8 * q^95 + 2*i * q^97 - 4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{3} - 8 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q - 4 * q^3 - 8 * q^7 + 2 * q^9 $$2 q - 4 q^{3} - 8 q^{7} + 2 q^{9} - 8 q^{11} + 16 q^{21} - 2 q^{25} + 8 q^{27} + 16 q^{33} - 12 q^{37} - 4 q^{41} - 24 q^{47} + 18 q^{49} - 24 q^{53} - 8 q^{63} + 4 q^{65} + 4 q^{67} + 16 q^{71} + 12 q^{73} + 4 q^{75} + 32 q^{77} - 22 q^{81} - 28 q^{83} - 12 q^{85} - 16 q^{95} - 8 q^{99}+O(q^{100})$$ 2 * q - 4 * q^3 - 8 * q^7 + 2 * q^9 - 8 * q^11 + 16 * q^21 - 2 * q^25 + 8 * q^27 + 16 * q^33 - 12 * q^37 - 4 * q^41 - 24 * q^47 + 18 * q^49 - 24 * q^53 - 8 * q^63 + 4 * q^65 + 4 * q^67 + 16 * q^71 + 12 * q^73 + 4 * q^75 + 32 * q^77 - 22 * q^81 - 28 * q^83 - 12 * q^85 - 16 * q^95 - 8 * 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 −2.00000 0 1.00000i 0 −4.00000 0 1.00000 0
961.2 0 −2.00000 0 1.00000i 0 −4.00000 0 1.00000 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.b 2
4.b odd 2 1 1480.2.p.a 2
37.b even 2 1 inner 2960.2.p.b 2
148.b odd 2 1 1480.2.p.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1480.2.p.a 2 4.b odd 2 1
1480.2.p.a 2 148.b odd 2 1
2960.2.p.b 2 1.a even 1 1 trivial
2960.2.p.b 2 37.b even 2 1 inner

## Hecke kernels

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

 $$T_{3} + 2$$ T3 + 2 $$T_{7} + 4$$ T7 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 2)^{2}$$
$5$ $$T^{2} + 1$$
$7$ $$(T + 4)^{2}$$
$11$ $$(T + 4)^{2}$$
$13$ $$T^{2} + 4$$
$17$ $$T^{2} + 36$$
$19$ $$T^{2} + 64$$
$23$ $$T^{2} + 16$$
$29$ $$T^{2} + 36$$
$31$ $$T^{2} + 4$$
$37$ $$T^{2} + 12T + 37$$
$41$ $$(T + 2)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$(T + 12)^{2}$$
$53$ $$(T + 12)^{2}$$
$59$ $$T^{2} + 64$$
$61$ $$T^{2} + 4$$
$67$ $$(T - 2)^{2}$$
$71$ $$(T - 8)^{2}$$
$73$ $$(T - 6)^{2}$$
$79$ $$T^{2} + 36$$
$83$ $$(T + 14)^{2}$$
$89$ $$T^{2} + 16$$
$97$ $$T^{2} + 4$$