# Properties

 Label 1960.2.q.r 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: $$\Q(\sqrt{-3}, \sqrt{-11})$$ Defining polynomial: $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 280) 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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} + \beta_{3} ) q^{3} + ( -1 + \beta_{1} ) q^{5} + ( -5 + 6 \beta_{1} + \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} + \beta_{3} ) q^{3} + ( -1 + \beta_{1} ) q^{5} + ( -5 + 6 \beta_{1} + \beta_{2} - \beta_{3} ) q^{9} + ( -3 \beta_{1} - \beta_{3} ) q^{11} + ( -2 - \beta_{2} ) q^{13} -\beta_{2} q^{15} + ( 3 \beta_{1} - \beta_{3} ) q^{17} + ( -2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{19} + ( 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{23} -\beta_{1} q^{25} + ( 8 - 3 \beta_{2} ) q^{27} + ( -2 - \beta_{2} ) q^{29} -8 \beta_{1} q^{31} + ( 8 - 5 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{33} + ( 2 - 2 \beta_{1} ) q^{37} + ( -7 \beta_{1} - \beta_{3} ) q^{39} + ( -2 - 2 \beta_{2} ) q^{41} + ( -4 - 2 \beta_{2} ) q^{43} + ( -6 \beta_{1} + \beta_{3} ) q^{45} + ( -3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{47} + ( 8 - 11 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{51} + ( -4 \beta_{1} - 2 \beta_{3} ) q^{53} + ( 4 + \beta_{2} ) q^{55} + ( -16 + 2 \beta_{2} ) q^{57} + 8 \beta_{1} q^{59} + ( 2 - 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{61} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{65} + 4 \beta_{1} q^{67} + ( 16 - 2 \beta_{2} ) q^{69} + 8 q^{71} -6 \beta_{1} q^{73} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{75} + ( -8 + 5 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{79} + ( -17 \beta_{1} + 8 \beta_{3} ) q^{81} + 4 \beta_{2} q^{83} + ( -2 + \beta_{2} ) q^{85} + ( -7 \beta_{1} - \beta_{3} ) q^{87} + ( 10 - 8 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{89} + ( 8 \beta_{1} + 8 \beta_{2} - 8 \beta_{3} ) q^{93} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{95} + ( -2 + 5 \beta_{2} ) q^{97} + ( 12 + 2 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - q^{3} - 2q^{5} - 11q^{9} + O(q^{10})$$ $$4q - q^{3} - 2q^{5} - 11q^{9} - 7q^{11} - 6q^{13} + 2q^{15} + 5q^{17} + 2q^{19} - 2q^{23} - 2q^{25} + 38q^{27} - 6q^{29} - 16q^{31} + 13q^{33} + 4q^{37} - 15q^{39} - 4q^{41} - 12q^{43} - 11q^{45} + 3q^{47} + 19q^{51} - 10q^{53} + 14q^{55} - 68q^{57} + 16q^{59} + 6q^{61} + 3q^{65} + 8q^{67} + 68q^{69} + 32q^{71} - 12q^{73} - q^{75} - 13q^{79} - 26q^{81} - 8q^{83} - 10q^{85} - 15q^{87} + 18q^{89} - 8q^{93} + 2q^{95} - 18q^{97} + 44q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 2 \nu^{2} - 2 \nu - 3$$$$)/6$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + \nu^{2} + 5 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{3} + \nu^{2} + 2 \nu - 9$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} - 2 \beta_{1} + 2$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{3} + 2 \beta_{2} + 8 \beta_{1} + 1$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$4 \beta_{3} - 2 \beta_{2} - 2 \beta_{1} + 11$$$$)/3$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 −1.18614 + 1.26217i 1.68614 − 0.396143i −1.18614 − 1.26217i 1.68614 + 0.396143i
0 −1.68614 + 2.92048i 0 −0.500000 0.866025i 0 0 0 −4.18614 7.25061i 0
361.2 0 1.18614 2.05446i 0 −0.500000 0.866025i 0 0 0 −1.31386 2.27567i 0
961.1 0 −1.68614 2.92048i 0 −0.500000 + 0.866025i 0 0 0 −4.18614 + 7.25061i 0
961.2 0 1.18614 + 2.05446i 0 −0.500000 + 0.866025i 0 0 0 −1.31386 + 2.27567i 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.r 4
7.b odd 2 1 1960.2.q.t 4
7.c even 3 1 1960.2.a.s 2
7.c even 3 1 inner 1960.2.q.r 4
7.d odd 6 1 280.2.a.c 2
7.d odd 6 1 1960.2.q.t 4
21.g even 6 1 2520.2.a.x 2
28.f even 6 1 560.2.a.h 2
28.g odd 6 1 3920.2.a.bt 2
35.i odd 6 1 1400.2.a.r 2
35.j even 6 1 9800.2.a.bu 2
35.k even 12 2 1400.2.g.i 4
56.j odd 6 1 2240.2.a.bk 2
56.m even 6 1 2240.2.a.bg 2
84.j odd 6 1 5040.2.a.by 2
140.s even 6 1 2800.2.a.bk 2
140.x odd 12 2 2800.2.g.r 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.a.c 2 7.d odd 6 1
560.2.a.h 2 28.f even 6 1
1400.2.a.r 2 35.i odd 6 1
1400.2.g.i 4 35.k even 12 2
1960.2.a.s 2 7.c even 3 1
1960.2.q.r 4 1.a even 1 1 trivial
1960.2.q.r 4 7.c even 3 1 inner
1960.2.q.t 4 7.b odd 2 1
1960.2.q.t 4 7.d odd 6 1
2240.2.a.bg 2 56.m even 6 1
2240.2.a.bk 2 56.j odd 6 1
2520.2.a.x 2 21.g even 6 1
2800.2.a.bk 2 140.s even 6 1
2800.2.g.r 4 140.x odd 12 2
3920.2.a.bt 2 28.g odd 6 1
5040.2.a.by 2 84.j odd 6 1
9800.2.a.bu 2 35.j even 6 1

## Hecke kernels

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

 $$T_{3}^{4} + T_{3}^{3} + 9 T_{3}^{2} - 8 T_{3} + 64$$ $$T_{11}^{4} + 7 T_{11}^{3} + 45 T_{11}^{2} + 28 T_{11} + 16$$ $$T_{13}^{2} + 3 T_{13} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$64 - 8 T + 9 T^{2} + T^{3} + T^{4}$$
$5$ $$( 1 + T + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$16 + 28 T + 45 T^{2} + 7 T^{3} + T^{4}$$
$13$ $$( -6 + 3 T + T^{2} )^{2}$$
$17$ $$4 + 10 T + 27 T^{2} - 5 T^{3} + T^{4}$$
$19$ $$1024 + 64 T + 36 T^{2} - 2 T^{3} + T^{4}$$
$23$ $$1024 - 64 T + 36 T^{2} + 2 T^{3} + T^{4}$$
$29$ $$( -6 + 3 T + T^{2} )^{2}$$
$31$ $$( 64 + 8 T + T^{2} )^{2}$$
$37$ $$( 4 - 2 T + T^{2} )^{2}$$
$41$ $$( -32 + 2 T + T^{2} )^{2}$$
$43$ $$( -24 + 6 T + T^{2} )^{2}$$
$47$ $$5184 + 216 T + 81 T^{2} - 3 T^{3} + T^{4}$$
$53$ $$64 - 80 T + 108 T^{2} + 10 T^{3} + T^{4}$$
$59$ $$( 64 - 8 T + T^{2} )^{2}$$
$61$ $$576 + 144 T + 60 T^{2} - 6 T^{3} + T^{4}$$
$67$ $$( 16 - 4 T + T^{2} )^{2}$$
$71$ $$( -8 + T )^{4}$$
$73$ $$( 36 + 6 T + T^{2} )^{2}$$
$79$ $$1024 - 416 T + 201 T^{2} + 13 T^{3} + T^{4}$$
$83$ $$( -128 + 4 T + T^{2} )^{2}$$
$89$ $$2304 - 864 T + 276 T^{2} - 18 T^{3} + T^{4}$$
$97$ $$( -186 + 9 T + T^{2} )^{2}$$