Properties

 Label 1040.2.f.d Level $1040$ Weight $2$ Character orbit 1040.f Analytic conductor $8.304$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ Defining polynomial: $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ x^4 - x^3 - 2*x^2 - 3*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 130) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{3} + ( - \beta_1 + 1) q^{5} + ( - \beta_{3} - \beta_1) q^{7} + (\beta_{3} + \beta_1 - 1) q^{9}+O(q^{10})$$ q - b2 * q^3 + (-b1 + 1) * q^5 + (-b3 - b1) * q^7 + (b3 + b1 - 1) * q^9 $$q - \beta_{2} q^{3} + ( - \beta_1 + 1) q^{5} + ( - \beta_{3} - \beta_1) q^{7} + (\beta_{3} + \beta_1 - 1) q^{9} + (\beta_{3} - \beta_{2} - \beta_1) q^{11} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{13} + ( - \beta_{3} - 2 \beta_{2}) q^{15} + ( - \beta_{3} - 2 \beta_{2} + \beta_1) q^{17} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{19} + ( - \beta_{3} - 2 \beta_{2} + \beta_1) q^{21} + ( - \beta_{3} + 3 \beta_{2} + \beta_1) q^{23} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{25} + (\beta_{3} - \beta_1) q^{27} + ( - 2 \beta_{3} - 2 \beta_1 - 2) q^{29} + (\beta_{3} + \beta_{2} - \beta_1) q^{31} - 4 q^{33} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 + 4) q^{35} + ( - 3 \beta_{3} - 3 \beta_1 + 2) q^{37} + ( - 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 4) q^{39} - 4 \beta_{2} q^{41} + (2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{43} + (2 \beta_{3} - \beta_{2} - 5) q^{45} + (\beta_{3} + \beta_1 - 8) q^{47} + (\beta_{3} + \beta_1 + 1) q^{49} + (3 \beta_{3} + 3 \beta_1 - 8) q^{51} + 2 \beta_{2} q^{53} + ( - 3 \beta_{3} - \beta_{2} - \beta_1 - 4) q^{55} + (2 \beta_{3} + 2 \beta_1 - 4) q^{57} + ( - \beta_{3} + 3 \beta_{2} + \beta_1) q^{59} + (2 \beta_{3} + 2 \beta_1 - 2) q^{61} - 8 q^{63} + ( - \beta_{3} + 3 \beta_{2} - 2 \beta_1 - 3) q^{65} + 4 q^{67} + ( - 2 \beta_{3} - 2 \beta_1 + 12) q^{69} - 5 \beta_{2} q^{71} - 10 q^{73} + ( - 2 \beta_{3} - 4 \beta_{2} + \beta_1 + 4) q^{75} + ( - 4 \beta_{3} + 4 \beta_1) q^{77} + (2 \beta_{3} + 2 \beta_1) q^{79} + (2 \beta_{3} + 2 \beta_1 - 3) q^{81} + ( - 2 \beta_{3} - 2 \beta_1 + 4) q^{83} + ( - 5 \beta_{2} + \beta_1 + 4) q^{85} + ( - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{87} + ( - 2 \beta_{3} + 2 \beta_1) q^{89} + ( - 3 \beta_{3} + 4 \beta_{2} + \beta_1) q^{91} + ( - 2 \beta_{3} - 2 \beta_1 + 4) q^{93} + (\beta_{3} - 3 \beta_{2} + \beta_1 + 4) q^{95} + (2 \beta_{3} + 2 \beta_1 - 2) q^{97} + (3 \beta_{3} + \beta_{2} - 3 \beta_1) q^{99}+O(q^{100})$$ q - b2 * q^3 + (-b1 + 1) * q^5 + (-b3 - b1) * q^7 + (b3 + b1 - 1) * q^9 + (b3 - b2 - b1) * q^11 + (b3 + b2 - b1 + 1) * q^13 + (-b3 - 2*b2) * q^15 + (-b3 - 2*b2 + b1) * q^17 + (-b3 - b2 + b1) * q^19 + (-b3 - 2*b2 + b1) * q^21 + (-b3 + 3*b2 + b1) * q^23 + (-2*b3 + b2 - b1 + 1) * q^25 + (b3 - b1) * q^27 + (-2*b3 - 2*b1 - 2) * q^29 + (b3 + b2 - b1) * q^31 - 4 * q^33 + (-2*b3 + b2 + b1 + 4) * q^35 + (-3*b3 - 3*b1 + 2) * q^37 + (-2*b3 - b2 - 2*b1 + 4) * q^39 - 4*b2 * q^41 + (2*b3 - b2 - 2*b1) * q^43 + (2*b3 - b2 - 5) * q^45 + (b3 + b1 - 8) * q^47 + (b3 + b1 + 1) * q^49 + (3*b3 + 3*b1 - 8) * q^51 + 2*b2 * q^53 + (-3*b3 - b2 - b1 - 4) * q^55 + (2*b3 + 2*b1 - 4) * q^57 + (-b3 + 3*b2 + b1) * q^59 + (2*b3 + 2*b1 - 2) * q^61 - 8 * q^63 + (-b3 + 3*b2 - 2*b1 - 3) * q^65 + 4 * q^67 + (-2*b3 - 2*b1 + 12) * q^69 - 5*b2 * q^71 - 10 * q^73 + (-2*b3 - 4*b2 + b1 + 4) * q^75 + (-4*b3 + 4*b1) * q^77 + (2*b3 + 2*b1) * q^79 + (2*b3 + 2*b1 - 3) * q^81 + (-2*b3 - 2*b1 + 4) * q^83 + (-5*b2 + b1 + 4) * q^85 + (-2*b3 - 2*b2 + 2*b1) * q^87 + (-2*b3 + 2*b1) * q^89 + (-3*b3 + 4*b2 + b1) * q^91 + (-2*b3 - 2*b1 + 4) * q^93 + (b3 - 3*b2 + b1 + 4) * q^95 + (2*b3 + 2*b1 - 2) * q^97 + (3*b3 + b2 - 3*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 3 q^{5} - 2 q^{7} - 2 q^{9}+O(q^{10})$$ 4 * q + 3 * q^5 - 2 * q^7 - 2 * q^9 $$4 q + 3 q^{5} - 2 q^{7} - 2 q^{9} + 4 q^{13} - q^{15} + q^{25} - 12 q^{29} - 16 q^{33} + 15 q^{35} + 2 q^{37} + 12 q^{39} - 18 q^{45} - 30 q^{47} + 6 q^{49} - 26 q^{51} - 20 q^{55} - 12 q^{57} - 4 q^{61} - 32 q^{63} - 15 q^{65} + 16 q^{67} + 44 q^{69} - 40 q^{73} + 15 q^{75} + 4 q^{79} - 8 q^{81} + 12 q^{83} + 17 q^{85} - 2 q^{91} + 12 q^{93} + 18 q^{95} - 4 q^{97}+O(q^{100})$$ 4 * q + 3 * q^5 - 2 * q^7 - 2 * q^9 + 4 * q^13 - q^15 + q^25 - 12 * q^29 - 16 * q^33 + 15 * q^35 + 2 * q^37 + 12 * q^39 - 18 * q^45 - 30 * q^47 + 6 * q^49 - 26 * q^51 - 20 * q^55 - 12 * q^57 - 4 * q^61 - 32 * q^63 - 15 * q^65 + 16 * q^67 + 44 * q^69 - 40 * q^73 + 15 * q^75 + 4 * q^79 - 8 * q^81 + 12 * q^83 + 17 * q^85 - 2 * q^91 + 12 * q^93 + 18 * q^95 - 4 * q^97

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

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 2\nu^{2} + \nu - 6 ) / 3$$ (v^3 + 2*v^2 + v - 6) / 3 $$\beta_{2}$$ $$=$$ $$( \nu^{3} - \nu^{2} + \nu - 3 ) / 3$$ (v^3 - v^2 + v - 3) / 3 $$\beta_{3}$$ $$=$$ $$( -2\nu^{3} - \nu^{2} + 4\nu + 9 ) / 3$$ (-2*v^3 - v^2 + 4*v + 9) / 3
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} + \beta_1 ) / 2$$ (b3 + b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$-\beta_{2} + \beta _1 + 1$$ -b2 + b1 + 1 $$\nu^{3}$$ $$=$$ $$( -\beta_{3} + 3\beta_{2} + \beta _1 + 8 ) / 2$$ (-b3 + 3*b2 + b1 + 8) / 2

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
 −1.18614 + 1.26217i 1.68614 + 0.396143i 1.68614 − 0.396143i −1.18614 − 1.26217i
0 2.52434i 0 2.18614 + 0.469882i 0 2.37228 0 −3.37228 0
129.2 0 0.792287i 0 −0.686141 2.12819i 0 −3.37228 0 2.37228 0
129.3 0 0.792287i 0 −0.686141 + 2.12819i 0 −3.37228 0 2.37228 0
129.4 0 2.52434i 0 2.18614 0.469882i 0 2.37228 0 −3.37228 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.d 4
4.b odd 2 1 130.2.c.a 4
5.b even 2 1 1040.2.f.c 4
12.b even 2 1 1170.2.f.b 4
13.b even 2 1 1040.2.f.c 4
20.d odd 2 1 130.2.c.b yes 4
20.e even 4 2 650.2.d.e 8
52.b odd 2 1 130.2.c.b yes 4
52.f even 4 2 1690.2.b.d 8
60.h even 2 1 1170.2.f.a 4
65.d even 2 1 inner 1040.2.f.d 4
156.h even 2 1 1170.2.f.a 4
260.g odd 2 1 130.2.c.a 4
260.l odd 4 2 8450.2.a.cn 4
260.p even 4 2 650.2.d.e 8
260.s odd 4 2 8450.2.a.cj 4
260.u even 4 2 1690.2.b.d 8
780.d even 2 1 1170.2.f.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.c.a 4 4.b odd 2 1
130.2.c.a 4 260.g odd 2 1
130.2.c.b yes 4 20.d odd 2 1
130.2.c.b yes 4 52.b odd 2 1
650.2.d.e 8 20.e even 4 2
650.2.d.e 8 260.p even 4 2
1040.2.f.c 4 5.b even 2 1
1040.2.f.c 4 13.b even 2 1
1040.2.f.d 4 1.a even 1 1 trivial
1040.2.f.d 4 65.d even 2 1 inner
1170.2.f.a 4 60.h even 2 1
1170.2.f.a 4 156.h even 2 1
1170.2.f.b 4 12.b even 2 1
1170.2.f.b 4 780.d even 2 1
1690.2.b.d 8 52.f even 4 2
1690.2.b.d 8 260.u even 4 2
8450.2.a.cj 4 260.s odd 4 2
8450.2.a.cn 4 260.l odd 4 2

Hecke kernels

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

 $$T_{3}^{4} + 7T_{3}^{2} + 4$$ T3^4 + 7*T3^2 + 4 $$T_{7}^{2} + T_{7} - 8$$ T7^2 + T7 - 8 $$T_{37}^{2} - T_{37} - 74$$ T37^2 - T37 - 74

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 7T^{2} + 4$$
$5$ $$T^{4} - 3 T^{3} + 4 T^{2} - 15 T + 25$$
$7$ $$(T^{2} + T - 8)^{2}$$
$11$ $$T^{4} + 28T^{2} + 64$$
$13$ $$(T^{2} - 2 T + 13)^{2}$$
$17$ $$T^{4} + 43T^{2} + 256$$
$19$ $$(T^{2} + 12)^{2}$$
$23$ $$(T^{2} + 44)^{2}$$
$29$ $$(T^{2} + 6 T - 24)^{2}$$
$31$ $$(T^{2} + 12)^{2}$$
$37$ $$(T^{2} - T - 74)^{2}$$
$41$ $$T^{4} + 112T^{2} + 1024$$
$43$ $$T^{4} + 87T^{2} + 36$$
$47$ $$(T^{2} + 15 T + 48)^{2}$$
$53$ $$T^{4} + 28T^{2} + 64$$
$59$ $$(T^{2} + 44)^{2}$$
$61$ $$(T^{2} + 2 T - 32)^{2}$$
$67$ $$(T - 4)^{4}$$
$71$ $$T^{4} + 175T^{2} + 2500$$
$73$ $$(T + 10)^{4}$$
$79$ $$(T^{2} - 2 T - 32)^{2}$$
$83$ $$(T^{2} - 6 T - 24)^{2}$$
$89$ $$T^{4} + 76T^{2} + 256$$
$97$ $$(T^{2} + 2 T - 32)^{2}$$