# Properties

 Label 2960.1.o.b Level $2960$ Weight $1$ Character orbit 2960.o Self dual yes Analytic conductor $1.477$ Analytic rank $0$ Dimension $4$ Projective image $D_{8}$ CM discriminant -740 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2960,1,Mod(2959,2960)]

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([1, 0, 1, 1]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("2960.2959");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2960 = 2^{4} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2960.o (of order $$2$$, degree $$1$$, 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: yes Analytic conductor: $$1.47723243739$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{16})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 4x^{2} + 2$$ x^4 - 4*x^2 + 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{8}$$ Projective field: Galois closure of 8.0.1620896000.2

## $q$-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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{3} + q^{5} + \beta_{3} q^{7} + (\beta_{2} + 1) q^{9}+O(q^{10})$$ q - b1 * q^3 + q^5 + b3 * q^7 + (b2 + 1) * q^9 $$q - \beta_1 q^{3} + q^{5} + \beta_{3} q^{7} + (\beta_{2} + 1) q^{9} - \beta_{2} q^{13} - \beta_1 q^{15} + \beta_{2} q^{17} + \beta_1 q^{19} - \beta_{2} q^{21} + q^{25} + ( - \beta_{3} - \beta_1) q^{27} - \beta_{3} q^{31} + \beta_{3} q^{35} - q^{37} + (\beta_{3} + \beta_1) q^{39} + (\beta_{2} + 1) q^{45} - \beta_{3} q^{47} + ( - \beta_{2} + 1) q^{49} + ( - \beta_{3} - \beta_1) q^{51} + ( - \beta_{2} - 2) q^{57} - \beta_{3} q^{59} + \beta_1 q^{63} - \beta_{2} q^{65} + \beta_1 q^{67} - \beta_1 q^{75} + \beta_{3} q^{79} + (\beta_{2} + 1) q^{81} + \beta_1 q^{83} + \beta_{2} q^{85} + (\beta_{3} - \beta_1) q^{91} + \beta_{2} q^{93} + \beta_1 q^{95}+O(q^{100})$$ q - b1 * q^3 + q^5 + b3 * q^7 + (b2 + 1) * q^9 - b2 * q^13 - b1 * q^15 + b2 * q^17 + b1 * q^19 - b2 * q^21 + q^25 + (-b3 - b1) * q^27 - b3 * q^31 + b3 * q^35 - q^37 + (b3 + b1) * q^39 + (b2 + 1) * q^45 - b3 * q^47 + (-b2 + 1) * q^49 + (-b3 - b1) * q^51 + (-b2 - 2) * q^57 - b3 * q^59 + b1 * q^63 - b2 * q^65 + b1 * q^67 - b1 * q^75 + b3 * q^79 + (b2 + 1) * q^81 + b1 * q^83 + b2 * q^85 + (b3 - b1) * q^91 + b2 * q^93 + b1 * q^95 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{5} + 4 q^{9}+O(q^{10})$$ 4 * q + 4 * q^5 + 4 * q^9 $$4 q + 4 q^{5} + 4 q^{9} + 4 q^{25} - 4 q^{37} + 4 q^{45} + 4 q^{49} - 8 q^{57} + 4 q^{81}+O(q^{100})$$ 4 * q + 4 * q^5 + 4 * q^9 + 4 * q^25 - 4 * q^37 + 4 * q^45 + 4 * q^49 - 8 * q^57 + 4 * q^81

Basis of coefficient ring in terms of $$\nu = \zeta_{16} + \zeta_{16}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 3\nu$$ v^3 - 3*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 3\beta_1$$ b3 + 3*b1

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2959.1
 1.84776 0.765367 −0.765367 −1.84776
0 −1.84776 0 1.00000 0 0.765367 0 2.41421 0
2959.2 0 −0.765367 0 1.00000 0 −1.84776 0 −0.414214 0
2959.3 0 0.765367 0 1.00000 0 1.84776 0 −0.414214 0
2959.4 0 1.84776 0 1.00000 0 −0.765367 0 2.41421 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
740.g odd 2 1 CM by $$\Q(\sqrt{-185})$$
4.b odd 2 1 inner
185.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2960.1.o.b yes 4
4.b odd 2 1 inner 2960.1.o.b yes 4
5.b even 2 1 2960.1.o.a 4
20.d odd 2 1 2960.1.o.a 4
37.b even 2 1 2960.1.o.a 4
148.b odd 2 1 2960.1.o.a 4
185.d even 2 1 inner 2960.1.o.b yes 4
740.g odd 2 1 CM 2960.1.o.b yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2960.1.o.a 4 5.b even 2 1
2960.1.o.a 4 20.d odd 2 1
2960.1.o.a 4 37.b even 2 1
2960.1.o.a 4 148.b odd 2 1
2960.1.o.b yes 4 1.a even 1 1 trivial
2960.1.o.b yes 4 4.b odd 2 1 inner
2960.1.o.b yes 4 185.d even 2 1 inner
2960.1.o.b yes 4 740.g odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{193} - 2$$ acting on $$S_{1}^{\mathrm{new}}(2960, [\chi])$$.

## Hecke characteristic polynomials

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