# Properties

 Label 2960.1.o.a 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

## Newspace parameters

 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

 Self dual: yes Analytic conductor: $$1.47723243739$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{16})^+$$ Defining polynomial: $$x^{4} - 4 x^{2} + 2$$ 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

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.

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

## 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.

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.a 4
4.b odd 2 1 inner 2960.1.o.a 4
5.b even 2 1 2960.1.o.b yes 4
20.d odd 2 1 2960.1.o.b yes 4
37.b even 2 1 2960.1.o.b yes 4
148.b odd 2 1 2960.1.o.b yes 4
185.d even 2 1 inner 2960.1.o.a 4
740.g odd 2 1 CM 2960.1.o.a 4

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

## 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

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