# Properties

 Label 1960.2.q.p Level $1960$ Weight $2$ Character orbit 1960.q Analytic conductor $15.651$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1960 = 2^{3} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1960.q (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.6506787962$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} - \beta_{2} ) q^{3} -\beta_{2} q^{5} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{1} - \beta_{2} ) q^{3} -\beta_{2} q^{5} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{9} + ( 1 + \beta_{2} ) q^{11} + ( 1 - \beta_{3} ) q^{13} + ( -1 - \beta_{3} ) q^{15} + ( -1 + \beta_{1} - \beta_{2} ) q^{17} + 2 \beta_{2} q^{19} + ( -3 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{23} + ( -1 - \beta_{2} ) q^{25} + ( 1 - \beta_{3} ) q^{27} + q^{29} + ( -6 + 3 \beta_{1} - 6 \beta_{2} ) q^{31} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{33} + ( -7 \beta_{1} + 2 \beta_{2} - 7 \beta_{3} ) q^{37} + ( 1 + \beta_{2} ) q^{39} + ( 6 - \beta_{3} ) q^{41} + ( -6 - 4 \beta_{3} ) q^{43} -2 \beta_{1} q^{45} + ( \beta_{1} + 9 \beta_{2} + \beta_{3} ) q^{47} + ( -2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{51} + ( -8 - \beta_{1} - 8 \beta_{2} ) q^{53} + q^{55} + ( 2 + 2 \beta_{3} ) q^{57} + ( -6 - 3 \beta_{1} - 6 \beta_{2} ) q^{59} + ( 2 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{61} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{65} -\beta_{1} q^{67} + ( 8 + 5 \beta_{3} ) q^{69} + ( 6 - 6 \beta_{3} ) q^{71} + ( 8 - 2 \beta_{1} + 8 \beta_{2} ) q^{73} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{75} + ( -4 \beta_{1} - 9 \beta_{2} - 4 \beta_{3} ) q^{79} + ( 1 - 6 \beta_{1} + \beta_{2} ) q^{81} -8 \beta_{3} q^{83} + ( -1 - \beta_{3} ) q^{85} + ( -1 + \beta_{1} - \beta_{2} ) q^{87} + ( -4 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{89} + ( -9 \beta_{1} + 12 \beta_{2} - 9 \beta_{3} ) q^{93} + ( 2 + 2 \beta_{2} ) q^{95} + ( 7 + 5 \beta_{3} ) q^{97} -2 \beta_{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{3} + 2q^{5} + O(q^{10})$$ $$4q - 2q^{3} + 2q^{5} + 2q^{11} + 4q^{13} - 4q^{15} - 2q^{17} - 4q^{19} - 4q^{23} - 2q^{25} + 4q^{27} + 4q^{29} - 12q^{31} + 2q^{33} - 4q^{37} + 2q^{39} + 24q^{41} - 24q^{43} - 18q^{47} - 6q^{51} - 16q^{53} + 4q^{55} + 8q^{57} - 12q^{59} + 8q^{61} + 2q^{65} + 32q^{69} + 24q^{71} + 16q^{73} - 2q^{75} + 18q^{79} + 2q^{81} - 4q^{85} - 2q^{87} - 8q^{89} - 24q^{93} + 4q^{95} + 28q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1960\mathbb{Z}\right)^\times$$.

 $$n$$ $$981$$ $$1081$$ $$1177$$ $$1471$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 −0.707107 + 1.22474i 0.707107 − 1.22474i −0.707107 − 1.22474i 0.707107 + 1.22474i
0 −1.20711 + 2.09077i 0 0.500000 + 0.866025i 0 0 0 −1.41421 2.44949i 0
361.2 0 0.207107 0.358719i 0 0.500000 + 0.866025i 0 0 0 1.41421 + 2.44949i 0
961.1 0 −1.20711 2.09077i 0 0.500000 0.866025i 0 0 0 −1.41421 + 2.44949i 0
961.2 0 0.207107 + 0.358719i 0 0.500000 0.866025i 0 0 0 1.41421 2.44949i 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1960.2.q.p 4
7.b odd 2 1 1960.2.q.v 4
7.c even 3 1 1960.2.a.u yes 2
7.c even 3 1 inner 1960.2.q.p 4
7.d odd 6 1 1960.2.a.q 2
7.d odd 6 1 1960.2.q.v 4
28.f even 6 1 3920.2.a.by 2
28.g odd 6 1 3920.2.a.bn 2
35.i odd 6 1 9800.2.a.ca 2
35.j even 6 1 9800.2.a.bs 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1960.2.a.q 2 7.d odd 6 1
1960.2.a.u yes 2 7.c even 3 1
1960.2.q.p 4 1.a even 1 1 trivial
1960.2.q.p 4 7.c even 3 1 inner
1960.2.q.v 4 7.b odd 2 1
1960.2.q.v 4 7.d odd 6 1
3920.2.a.bn 2 28.g odd 6 1
3920.2.a.by 2 28.f even 6 1
9800.2.a.bs 2 35.j even 6 1
9800.2.a.ca 2 35.i odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1960, [\chi])$$:

 $$T_{3}^{4} + 2 T_{3}^{3} + 5 T_{3}^{2} - 2 T_{3} + 1$$ $$T_{11}^{2} - T_{11} + 1$$ $$T_{13}^{2} - 2 T_{13} - 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$1 - 2 T + 5 T^{2} + 2 T^{3} + T^{4}$$
$5$ $$( 1 - T + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$( 1 - T + T^{2} )^{2}$$
$13$ $$( -1 - 2 T + T^{2} )^{2}$$
$17$ $$1 - 2 T + 5 T^{2} + 2 T^{3} + T^{4}$$
$19$ $$( 4 + 2 T + T^{2} )^{2}$$
$23$ $$196 - 56 T + 30 T^{2} + 4 T^{3} + T^{4}$$
$29$ $$( -1 + T )^{4}$$
$31$ $$324 + 216 T + 126 T^{2} + 12 T^{3} + T^{4}$$
$37$ $$8836 - 376 T + 110 T^{2} + 4 T^{3} + T^{4}$$
$41$ $$( 34 - 12 T + T^{2} )^{2}$$
$43$ $$( 4 + 12 T + T^{2} )^{2}$$
$47$ $$6241 + 1422 T + 245 T^{2} + 18 T^{3} + T^{4}$$
$53$ $$3844 + 992 T + 194 T^{2} + 16 T^{3} + T^{4}$$
$59$ $$324 + 216 T + 126 T^{2} + 12 T^{3} + T^{4}$$
$61$ $$64 - 64 T + 56 T^{2} - 8 T^{3} + T^{4}$$
$67$ $$4 + 2 T^{2} + T^{4}$$
$71$ $$( -36 - 12 T + T^{2} )^{2}$$
$73$ $$3136 - 896 T + 200 T^{2} - 16 T^{3} + T^{4}$$
$79$ $$2401 - 882 T + 275 T^{2} - 18 T^{3} + T^{4}$$
$83$ $$( -128 + T^{2} )^{2}$$
$89$ $$256 - 128 T + 80 T^{2} + 8 T^{3} + T^{4}$$
$97$ $$( -1 - 14 T + T^{2} )^{2}$$