# Properties

 Label 1960.2.q.u 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: $$\Q(\sqrt{-3}, \sqrt{17})$$ Defining polynomial: $$x^{4} - x^{3} + 5 x^{2} + 4 x + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( 1 - \beta_{2} ) q^{5} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( 1 - \beta_{2} ) q^{5} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} + \beta_{1} q^{11} + ( -2 - 3 \beta_{3} ) q^{13} -\beta_{3} q^{15} + ( -\beta_{1} + 6 \beta_{2} ) q^{17} + ( -4 - 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{19} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{23} -\beta_{2} q^{25} + ( -4 - \beta_{3} ) q^{27} + ( 2 - \beta_{3} ) q^{29} -4 \beta_{1} q^{31} + ( -4 + \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{33} + ( 2 + 4 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{37} + ( \beta_{1} + 12 \beta_{2} ) q^{39} + ( -2 + 2 \beta_{3} ) q^{41} + ( -4 - 2 \beta_{3} ) q^{43} + ( \beta_{1} + \beta_{2} ) q^{45} + ( 4 - \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{47} + ( 4 + 5 \beta_{1} - 4 \beta_{2} + 5 \beta_{3} ) q^{51} + ( -2 \beta_{1} + 10 \beta_{2} ) q^{53} -\beta_{3} q^{55} + ( 8 + 2 \beta_{3} ) q^{57} -4 \beta_{2} q^{59} + ( -10 + 2 \beta_{1} + 10 \beta_{2} + 2 \beta_{3} ) q^{61} + ( -2 - 3 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{65} + ( 4 \beta_{1} + 4 \beta_{2} ) q^{67} + ( 8 - 2 \beta_{3} ) q^{69} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{73} + ( -\beta_{1} - \beta_{3} ) q^{75} + ( 4 - 3 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} ) q^{79} + 7 \beta_{2} q^{81} -12 q^{83} + ( 6 + \beta_{3} ) q^{85} + ( 3 \beta_{1} + 4 \beta_{2} ) q^{87} + ( 2 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{89} + ( 16 - 4 \beta_{1} - 16 \beta_{2} - 4 \beta_{3} ) q^{93} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{95} + ( -6 + 3 \beta_{3} ) q^{97} + ( -4 + 2 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + q^{3} + 2q^{5} - 3q^{9} + O(q^{10})$$ $$4q + q^{3} + 2q^{5} - 3q^{9} + q^{11} - 2q^{13} + 2q^{15} + 11q^{17} - 6q^{19} + 2q^{23} - 2q^{25} - 14q^{27} + 10q^{29} - 4q^{31} - 9q^{33} + 25q^{39} - 12q^{41} - 12q^{43} + 3q^{45} + 9q^{47} + 3q^{51} + 18q^{53} + 2q^{55} + 28q^{57} - 8q^{59} - 22q^{61} - q^{65} + 12q^{67} + 36q^{69} + 8q^{73} + q^{75} + 11q^{79} + 14q^{81} - 48q^{83} + 22q^{85} + 11q^{87} + 2q^{89} + 36q^{93} + 6q^{95} - 30q^{97} - 20q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 5 x^{2} + 4 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 5 \nu^{2} - 5 \nu + 16$$$$)/20$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 4$$$$)/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 4 \beta_{2} + \beta_{1} - 4$$ $$\nu^{3}$$ $$=$$ $$5 \beta_{3} - 4$$

## Character values

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

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

## 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
 −0.780776 + 1.35234i 1.28078 − 2.21837i −0.780776 − 1.35234i 1.28078 + 2.21837i
0 −0.780776 + 1.35234i 0 0.500000 + 0.866025i 0 0 0 0.280776 + 0.486319i 0
361.2 0 1.28078 2.21837i 0 0.500000 + 0.866025i 0 0 0 −1.78078 3.08440i 0
961.1 0 −0.780776 1.35234i 0 0.500000 0.866025i 0 0 0 0.280776 0.486319i 0
961.2 0 1.28078 + 2.21837i 0 0.500000 0.866025i 0 0 0 −1.78078 + 3.08440i 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.u 4
7.b odd 2 1 1960.2.q.s 4
7.c even 3 1 1960.2.a.r 2
7.c even 3 1 inner 1960.2.q.u 4
7.d odd 6 1 280.2.a.d 2
7.d odd 6 1 1960.2.q.s 4
21.g even 6 1 2520.2.a.w 2
28.f even 6 1 560.2.a.g 2
28.g odd 6 1 3920.2.a.bu 2
35.i odd 6 1 1400.2.a.p 2
35.j even 6 1 9800.2.a.by 2
35.k even 12 2 1400.2.g.k 4
56.j odd 6 1 2240.2.a.be 2
56.m even 6 1 2240.2.a.bi 2
84.j odd 6 1 5040.2.a.bq 2
140.s even 6 1 2800.2.a.bn 2
140.x odd 12 2 2800.2.g.u 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.a.d 2 7.d odd 6 1
560.2.a.g 2 28.f even 6 1
1400.2.a.p 2 35.i odd 6 1
1400.2.g.k 4 35.k even 12 2
1960.2.a.r 2 7.c even 3 1
1960.2.q.s 4 7.b odd 2 1
1960.2.q.s 4 7.d odd 6 1
1960.2.q.u 4 1.a even 1 1 trivial
1960.2.q.u 4 7.c even 3 1 inner
2240.2.a.be 2 56.j odd 6 1
2240.2.a.bi 2 56.m even 6 1
2520.2.a.w 2 21.g even 6 1
2800.2.a.bn 2 140.s even 6 1
2800.2.g.u 4 140.x odd 12 2
3920.2.a.bu 2 28.g odd 6 1
5040.2.a.bq 2 84.j odd 6 1
9800.2.a.by 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 $$S_{2}^{\mathrm{new}}(1960, [\chi])$$:

 $$T_{3}^{4} - T_{3}^{3} + 5 T_{3}^{2} + 4 T_{3} + 16$$ $$T_{11}^{4} - T_{11}^{3} + 5 T_{11}^{2} + 4 T_{11} + 16$$ $$T_{13}^{2} + T_{13} - 38$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$16 + 4 T + 5 T^{2} - T^{3} + T^{4}$$
$5$ $$( 1 - T + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$16 + 4 T + 5 T^{2} - T^{3} + T^{4}$$
$13$ $$( -38 + T + T^{2} )^{2}$$
$17$ $$676 - 286 T + 95 T^{2} - 11 T^{3} + T^{4}$$
$19$ $$64 - 48 T + 44 T^{2} + 6 T^{3} + T^{4}$$
$23$ $$256 + 32 T + 20 T^{2} - 2 T^{3} + T^{4}$$
$29$ $$( 2 - 5 T + T^{2} )^{2}$$
$31$ $$4096 - 256 T + 80 T^{2} + 4 T^{3} + T^{4}$$
$37$ $$4624 + 68 T^{2} + T^{4}$$
$41$ $$( -8 + 6 T + T^{2} )^{2}$$
$43$ $$( -8 + 6 T + T^{2} )^{2}$$
$47$ $$256 - 144 T + 65 T^{2} - 9 T^{3} + T^{4}$$
$53$ $$4096 - 1152 T + 260 T^{2} - 18 T^{3} + T^{4}$$
$59$ $$( 16 + 4 T + T^{2} )^{2}$$
$61$ $$10816 + 2288 T + 380 T^{2} + 22 T^{3} + T^{4}$$
$67$ $$1024 + 384 T + 176 T^{2} - 12 T^{3} + T^{4}$$
$71$ $$T^{4}$$
$73$ $$2704 + 416 T + 116 T^{2} - 8 T^{3} + T^{4}$$
$79$ $$64 + 88 T + 129 T^{2} - 11 T^{3} + T^{4}$$
$83$ $$( 12 + T )^{4}$$
$89$ $$256 + 32 T + 20 T^{2} - 2 T^{3} + T^{4}$$
$97$ $$( 18 + 15 T + T^{2} )^{2}$$