Properties

 Label 1040.2.f.a Level $1040$ Weight $2$ Character orbit 1040.f Analytic conductor $8.304$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1040 = 2^{4} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1040.f (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$8.30444181021$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 65) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + ( - \beta - 1) q^{5} - q^{9}+O(q^{10})$$ q + b * q^3 + (-b - 1) * q^5 - q^9 $$q + \beta q^{3} + ( - \beta - 1) q^{5} - q^{9} + \beta q^{11} + (\beta + 3) q^{13} + ( - \beta + 4) q^{15} - 3 \beta q^{19} + 3 \beta q^{23} + (2 \beta - 3) q^{25} + 2 \beta q^{27} + 6 q^{29} + 3 \beta q^{31} - 4 q^{33} + 6 q^{37} + (3 \beta - 4) q^{39} + 4 \beta q^{41} - 3 \beta q^{43} + (\beta + 1) q^{45} + 8 q^{47} - 7 q^{49} + 6 \beta q^{53} + ( - \beta + 4) q^{55} + 12 q^{57} + \beta q^{59} + 6 q^{61} + ( - 4 \beta + 1) q^{65} - 12 q^{67} - 12 q^{69} - \beta q^{71} - 6 q^{73} + ( - 3 \beta - 8) q^{75} - 11 q^{81} + 4 q^{83} + 6 \beta q^{87} + 4 \beta q^{89} - 12 q^{93} + (3 \beta - 12) q^{95} - 6 q^{97} - \beta q^{99} +O(q^{100})$$ q + b * q^3 + (-b - 1) * q^5 - q^9 + b * q^11 + (b + 3) * q^13 + (-b + 4) * q^15 - 3*b * q^19 + 3*b * q^23 + (2*b - 3) * q^25 + 2*b * q^27 + 6 * q^29 + 3*b * q^31 - 4 * q^33 + 6 * q^37 + (3*b - 4) * q^39 + 4*b * q^41 - 3*b * q^43 + (b + 1) * q^45 + 8 * q^47 - 7 * q^49 + 6*b * q^53 + (-b + 4) * q^55 + 12 * q^57 + b * q^59 + 6 * q^61 + (-4*b + 1) * q^65 - 12 * q^67 - 12 * q^69 - b * q^71 - 6 * q^73 + (-3*b - 8) * q^75 - 11 * q^81 + 4 * q^83 + 6*b * q^87 + 4*b * q^89 - 12 * q^93 + (3*b - 12) * q^95 - 6 * q^97 - b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^5 - 2 * q^9 $$2 q - 2 q^{5} - 2 q^{9} + 6 q^{13} + 8 q^{15} - 6 q^{25} + 12 q^{29} - 8 q^{33} + 12 q^{37} - 8 q^{39} + 2 q^{45} + 16 q^{47} - 14 q^{49} + 8 q^{55} + 24 q^{57} + 12 q^{61} + 2 q^{65} - 24 q^{67} - 24 q^{69} - 12 q^{73} - 16 q^{75} - 22 q^{81} + 8 q^{83} - 24 q^{93} - 24 q^{95} - 12 q^{97}+O(q^{100})$$ 2 * q - 2 * q^5 - 2 * q^9 + 6 * q^13 + 8 * q^15 - 6 * q^25 + 12 * q^29 - 8 * q^33 + 12 * q^37 - 8 * q^39 + 2 * q^45 + 16 * q^47 - 14 * q^49 + 8 * q^55 + 24 * q^57 + 12 * q^61 + 2 * q^65 - 24 * q^67 - 24 * q^69 - 12 * q^73 - 16 * q^75 - 22 * q^81 + 8 * q^83 - 24 * q^93 - 24 * q^95 - 12 * q^97

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1040\mathbb{Z}\right)^\times$$.

 $$n$$ $$261$$ $$417$$ $$561$$ $$911$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
129.1
 − 1.00000i 1.00000i
0 2.00000i 0 −1.00000 + 2.00000i 0 0 0 −1.00000 0
129.2 0 2.00000i 0 −1.00000 2.00000i 0 0 0 −1.00000 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
65.d even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1040.2.f.a 2
4.b odd 2 1 65.2.d.a 2
5.b even 2 1 1040.2.f.b 2
12.b even 2 1 585.2.h.c 2
13.b even 2 1 1040.2.f.b 2
20.d odd 2 1 65.2.d.b yes 2
20.e even 4 1 325.2.c.b 2
20.e even 4 1 325.2.c.e 2
52.b odd 2 1 65.2.d.b yes 2
52.f even 4 1 845.2.b.a 2
52.f even 4 1 845.2.b.b 2
52.i odd 6 2 845.2.l.a 4
52.j odd 6 2 845.2.l.b 4
52.l even 12 2 845.2.n.a 4
52.l even 12 2 845.2.n.b 4
60.h even 2 1 585.2.h.b 2
65.d even 2 1 inner 1040.2.f.a 2
156.h even 2 1 585.2.h.b 2
260.g odd 2 1 65.2.d.a 2
260.l odd 4 1 4225.2.a.k 1
260.l odd 4 1 4225.2.a.m 1
260.p even 4 1 325.2.c.b 2
260.p even 4 1 325.2.c.e 2
260.s odd 4 1 4225.2.a.e 1
260.s odd 4 1 4225.2.a.h 1
260.u even 4 1 845.2.b.a 2
260.u even 4 1 845.2.b.b 2
260.v odd 6 2 845.2.l.a 4
260.w odd 6 2 845.2.l.b 4
260.bc even 12 2 845.2.n.a 4
260.bc even 12 2 845.2.n.b 4
780.d even 2 1 585.2.h.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.d.a 2 4.b odd 2 1
65.2.d.a 2 260.g odd 2 1
65.2.d.b yes 2 20.d odd 2 1
65.2.d.b yes 2 52.b odd 2 1
325.2.c.b 2 20.e even 4 1
325.2.c.b 2 260.p even 4 1
325.2.c.e 2 20.e even 4 1
325.2.c.e 2 260.p even 4 1
585.2.h.b 2 60.h even 2 1
585.2.h.b 2 156.h even 2 1
585.2.h.c 2 12.b even 2 1
585.2.h.c 2 780.d even 2 1
845.2.b.a 2 52.f even 4 1
845.2.b.a 2 260.u even 4 1
845.2.b.b 2 52.f even 4 1
845.2.b.b 2 260.u even 4 1
845.2.l.a 4 52.i odd 6 2
845.2.l.a 4 260.v odd 6 2
845.2.l.b 4 52.j odd 6 2
845.2.l.b 4 260.w odd 6 2
845.2.n.a 4 52.l even 12 2
845.2.n.a 4 260.bc even 12 2
845.2.n.b 4 52.l even 12 2
845.2.n.b 4 260.bc even 12 2
1040.2.f.a 2 1.a even 1 1 trivial
1040.2.f.a 2 65.d even 2 1 inner
1040.2.f.b 2 5.b even 2 1
1040.2.f.b 2 13.b even 2 1
4225.2.a.e 1 260.s odd 4 1
4225.2.a.h 1 260.s odd 4 1
4225.2.a.k 1 260.l odd 4 1
4225.2.a.m 1 260.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1040, [\chi])$$:

 $$T_{3}^{2} + 4$$ T3^2 + 4 $$T_{7}$$ T7 $$T_{37} - 6$$ T37 - 6

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 4$$
$5$ $$T^{2} + 2T + 5$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 4$$
$13$ $$T^{2} - 6T + 13$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 36$$
$23$ $$T^{2} + 36$$
$29$ $$(T - 6)^{2}$$
$31$ $$T^{2} + 36$$
$37$ $$(T - 6)^{2}$$
$41$ $$T^{2} + 64$$
$43$ $$T^{2} + 36$$
$47$ $$(T - 8)^{2}$$
$53$ $$T^{2} + 144$$
$59$ $$T^{2} + 4$$
$61$ $$(T - 6)^{2}$$
$67$ $$(T + 12)^{2}$$
$71$ $$T^{2} + 4$$
$73$ $$(T + 6)^{2}$$
$79$ $$T^{2}$$
$83$ $$(T - 4)^{2}$$
$89$ $$T^{2} + 64$$
$97$ $$(T + 6)^{2}$$