# Properties

 Label 2960.1.bu.a Level $2960$ Weight $1$ Character orbit 2960.bu Analytic conductor $1.477$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -4 Inner twists $4$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$2960 = 2^{4} \cdot 5 \cdot 37$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2960.bu (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.47723243739$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.25326500.2

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + q^{5} -i q^{9} +O(q^{10})$$ $$q + q^{5} -i q^{9} + 2 i q^{13} + q^{25} + ( 1 + i ) q^{29} + q^{37} -i q^{45} -i q^{49} + ( -1 - i ) q^{53} + ( 1 - i ) q^{61} + 2 i q^{65} + ( -1 + i ) q^{73} - q^{81} + ( -1 - i ) q^{89} + 2 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{5} + O(q^{10})$$ $$2q + 2q^{5} + 2q^{25} + 2q^{29} + 2q^{37} - 2q^{53} + 2q^{61} - 2q^{73} - 2q^{81} - 2q^{89} + 4q^{97} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
783.1
 1.00000i − 1.00000i
0 0 0 1.00000 0 0 0 1.00000i 0
1807.1 0 0 0 1.00000 0 0 0 1.00000i 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
185.f even 4 1 inner
740.p odd 4 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(2960, [\chi])$$.

## Hecke characteristic polynomials

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