# Properties

 Label 1040.2.d.c Level $1040$ Weight $2$ Character orbit 1040.d Analytic conductor $8.304$ Analytic rank $0$ Dimension $6$ 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.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.30444181021$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.350464.1 Defining polynomial: $$x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2$$ x^6 - 2*x^5 + 2*x^4 + 2*x^3 + 4*x^2 - 4*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{4} + \beta_{3}) q^{3} + (\beta_{4} + \beta_1) q^{5} + ( - \beta_{5} + \beta_{4} - \beta_{3}) q^{7} + (3 \beta_{2} - \beta_1) q^{9}+O(q^{10})$$ q + (b4 + b3) * q^3 + (b4 + b1) * q^5 + (-b5 + b4 - b3) * q^7 + (3*b2 - b1) * q^9 $$q + (\beta_{4} + \beta_{3}) q^{3} + (\beta_{4} + \beta_1) q^{5} + ( - \beta_{5} + \beta_{4} - \beta_{3}) q^{7} + (3 \beta_{2} - \beta_1) q^{9} + (\beta_1 + 2) q^{11} + \beta_{3} q^{13} + ( - \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} - \beta_1 - 2) q^{15} + (2 \beta_{5} + 2 \beta_{3}) q^{17} - \beta_1 q^{19} + (2 \beta_{2} - 2 \beta_1) q^{21} + ( - \beta_{4} + 5 \beta_{3}) q^{23} + (2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 1) q^{25} + (4 \beta_{5} - 4 \beta_{4} - 2 \beta_{3}) q^{27} + ( - 3 \beta_{2} + 3 \beta_1 - 3) q^{29} + (2 \beta_{2} - \beta_1 + 4) q^{31} + ( - \beta_{5} + 3 \beta_{4} + \beta_{3}) q^{33} + (2 \beta_{5} + 2 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{35} + (\beta_{5} - 3 \beta_{4} + \beta_{3}) q^{37} + (\beta_{2} - 1) q^{39} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{41} + (2 \beta_{5} + 3 \beta_{4} + \beta_{3}) q^{43} + (3 \beta_{5} - 4 \beta_{4} - 5 \beta_{3} + 2 \beta_{2} + \beta_1) q^{45} + (\beta_{5} - \beta_{4} - 3 \beta_{3}) q^{47} + (2 \beta_1 + 3) q^{49} + (2 \beta_1 - 4) q^{51} + (4 \beta_{5} - 2 \beta_{4} - 2 \beta_{3}) q^{53} + (3 \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 3) q^{55} + (\beta_{5} - \beta_{4} + \beta_{3}) q^{57} + (2 \beta_{2} + 3 \beta_1 - 2) q^{59} + ( - 3 \beta_{2} + \beta_1 + 1) q^{61} + (\beta_{5} - 3 \beta_{4} - 5 \beta_{3}) q^{63} + ( - \beta_{5} + \beta_{2}) q^{65} + (5 \beta_{5} - \beta_{4} - 3 \beta_{3}) q^{67} + (3 \beta_{2} + \beta_1 - 3) q^{69} + ( - 6 \beta_{2} - \beta_1 + 2) q^{71} + (\beta_{5} - 3 \beta_{4} + 9 \beta_{3}) q^{73} + (4 \beta_{5} - 5 \beta_{4} - \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 2) q^{75} + 2 \beta_{4} q^{77} + ( - 2 \beta_{2} + 4 \beta_1 - 6) q^{79} + ( - 5 \beta_{2} + 5 \beta_1 + 6) q^{81} + (\beta_{5} + \beta_{4} + 7 \beta_{3}) q^{83} + ( - 4 \beta_{5} + 2 \beta_{4} - 6 \beta_{3} - 2) q^{85} + ( - 6 \beta_{5} + 6 \beta_{4}) q^{87} + (4 \beta_{2} - 4 \beta_1 - 2) q^{89} + (\beta_{2} - \beta_1 + 1) q^{91} + (3 \beta_{5} - \beta_{4} + \beta_{3}) q^{93} + ( - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 3) q^{95} + ( - 6 \beta_{5} + 4 \beta_{4} - 6 \beta_{3}) q^{97} + (8 \beta_{2} - \beta_1) q^{99}+O(q^{100})$$ q + (b4 + b3) * q^3 + (b4 + b1) * q^5 + (-b5 + b4 - b3) * q^7 + (3*b2 - b1) * q^9 + (b1 + 2) * q^11 + b3 * q^13 + (-b5 + b4 - b3 + 2*b2 - b1 - 2) * q^15 + (2*b5 + 2*b3) * q^17 - b1 * q^19 + (2*b2 - 2*b1) * q^21 + (-b4 + 5*b3) * q^23 + (2*b4 - 2*b3 + 2*b2 - 2*b1 + 1) * q^25 + (4*b5 - 4*b4 - 2*b3) * q^27 + (-3*b2 + 3*b1 - 3) * q^29 + (2*b2 - b1 + 4) * q^31 + (-b5 + 3*b4 + b3) * q^33 + (2*b5 + 2*b3 + b2 - b1 - 1) * q^35 + (b5 - 3*b4 + b3) * q^37 + (b2 - 1) * q^39 + (-2*b2 - 2*b1 - 2) * q^41 + (2*b5 + 3*b4 + b3) * q^43 + (3*b5 - 4*b4 - 5*b3 + 2*b2 + b1) * q^45 + (b5 - b4 - 3*b3) * q^47 + (2*b1 + 3) * q^49 + (2*b1 - 4) * q^51 + (4*b5 - 2*b4 - 2*b3) * q^53 + (3*b4 - b3 + b2 + b1 + 3) * q^55 + (b5 - b4 + b3) * q^57 + (2*b2 + 3*b1 - 2) * q^59 + (-3*b2 + b1 + 1) * q^61 + (b5 - 3*b4 - 5*b3) * q^63 + (-b5 + b2) * q^65 + (5*b5 - b4 - 3*b3) * q^67 + (3*b2 + b1 - 3) * q^69 + (-6*b2 - b1 + 2) * q^71 + (b5 - 3*b4 + 9*b3) * q^73 + (4*b5 - 5*b4 - b3 + 2*b2 - 2*b1 - 2) * q^75 + 2*b4 * q^77 + (-2*b2 + 4*b1 - 6) * q^79 + (-5*b2 + 5*b1 + 6) * q^81 + (b5 + b4 + 7*b3) * q^83 + (-4*b5 + 2*b4 - 6*b3 - 2) * q^85 + (-6*b5 + 6*b4) * q^87 + (4*b2 - 4*b1 - 2) * q^89 + (b2 - b1 + 1) * q^91 + (3*b5 - b4 + b3) * q^93 + (-b4 + b3 - b2 + b1 - 3) * q^95 + (-6*b5 + 4*b4 - 6*b3) * q^97 + (8*b2 - b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{9}+O(q^{10})$$ 6 * q - 6 * q^9 $$6 q - 6 q^{9} + 12 q^{11} - 16 q^{15} - 4 q^{21} + 2 q^{25} - 12 q^{29} + 20 q^{31} - 8 q^{35} - 8 q^{39} - 8 q^{41} - 4 q^{45} + 18 q^{49} - 24 q^{51} + 16 q^{55} - 16 q^{59} + 12 q^{61} - 2 q^{65} - 24 q^{69} + 24 q^{71} - 16 q^{75} - 32 q^{79} + 46 q^{81} - 12 q^{85} - 20 q^{89} + 4 q^{91} - 16 q^{95} - 16 q^{99}+O(q^{100})$$ 6 * q - 6 * q^9 + 12 * q^11 - 16 * q^15 - 4 * q^21 + 2 * q^25 - 12 * q^29 + 20 * q^31 - 8 * q^35 - 8 * q^39 - 8 * q^41 - 4 * q^45 + 18 * q^49 - 24 * q^51 + 16 * q^55 - 16 * q^59 + 12 * q^61 - 2 * q^65 - 24 * q^69 + 24 * q^71 - 16 * q^75 - 32 * q^79 + 46 * q^81 - 12 * q^85 - 20 * q^89 + 4 * q^91 - 16 * q^95 - 16 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{5} + 8\nu^{4} - 4\nu^{3} - \nu^{2} + 2\nu + 38 ) / 23$$ (-v^5 + 8*v^4 - 4*v^3 - v^2 + 2*v + 38) / 23 $$\beta_{2}$$ $$=$$ $$( -5\nu^{5} + 17\nu^{4} - 20\nu^{3} - 5\nu^{2} + 10\nu + 29 ) / 23$$ (-5*v^5 + 17*v^4 - 20*v^3 - 5*v^2 + 10*v + 29) / 23 $$\beta_{3}$$ $$=$$ $$( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23$$ (7*v^5 - 10*v^4 + 5*v^3 + 30*v^2 + 32*v - 13) / 23 $$\beta_{4}$$ $$=$$ $$( -11\nu^{5} + 19\nu^{4} - 21\nu^{3} - 11\nu^{2} - 70\nu + 27 ) / 23$$ (-11*v^5 + 19*v^4 - 21*v^3 - 11*v^2 - 70*v + 27) / 23 $$\beta_{5}$$ $$=$$ $$( -14\nu^{5} + 20\nu^{4} - 10\nu^{3} - 37\nu^{2} - 64\nu + 26 ) / 23$$ (-14*v^5 + 20*v^4 - 10*v^3 - 37*v^2 - 64*v + 26) / 23
 $$\nu$$ $$=$$ $$( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2$$ (b5 - b4 + b3 + b2 - b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{5} + 2\beta_{3}$$ b5 + 2*b3 $$\nu^{3}$$ $$=$$ $$2\beta_{5} - \beta_{4} + 2\beta_{3} - \beta_{2} + 2\beta _1 - 2$$ 2*b5 - b4 + 2*b3 - b2 + 2*b1 - 2 $$\nu^{4}$$ $$=$$ $$-\beta_{2} + 5\beta _1 - 7$$ -b2 + 5*b1 - 7 $$\nu^{5}$$ $$=$$ $$-8\beta_{5} + 3\beta_{4} - 9\beta_{3} - 3\beta_{2} + 8\beta _1 - 9$$ -8*b5 + 3*b4 - 9*b3 - 3*b2 + 8*b1 - 9

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
209.1
 −0.854638 + 0.854638i 1.45161 − 1.45161i 0.403032 + 0.403032i 0.403032 − 0.403032i 1.45161 + 1.45161i −0.854638 − 0.854638i
0 3.17009i 0 0.539189 2.17009i 0 1.70928i 0 −7.04945 0
209.2 0 1.31111i 0 −2.21432 0.311108i 0 2.90321i 0 1.28100 0
209.3 0 0.481194i 0 1.67513 1.48119i 0 0.806063i 0 2.76845 0
209.4 0 0.481194i 0 1.67513 + 1.48119i 0 0.806063i 0 2.76845 0
209.5 0 1.31111i 0 −2.21432 + 0.311108i 0 2.90321i 0 1.28100 0
209.6 0 3.17009i 0 0.539189 + 2.17009i 0 1.70928i 0 −7.04945 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 209.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b 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.d.c 6
4.b odd 2 1 65.2.b.a 6
5.b even 2 1 inner 1040.2.d.c 6
5.c odd 4 1 5200.2.a.cb 3
5.c odd 4 1 5200.2.a.cj 3
12.b even 2 1 585.2.c.b 6
20.d odd 2 1 65.2.b.a 6
20.e even 4 1 325.2.a.j 3
20.e even 4 1 325.2.a.k 3
52.b odd 2 1 845.2.b.c 6
52.f even 4 1 845.2.d.a 6
52.f even 4 1 845.2.d.b 6
52.i odd 6 2 845.2.n.g 12
52.j odd 6 2 845.2.n.f 12
52.l even 12 2 845.2.l.d 12
52.l even 12 2 845.2.l.e 12
60.h even 2 1 585.2.c.b 6
60.l odd 4 1 2925.2.a.bf 3
60.l odd 4 1 2925.2.a.bj 3
260.g odd 2 1 845.2.b.c 6
260.p even 4 1 4225.2.a.ba 3
260.p even 4 1 4225.2.a.bh 3
260.u even 4 1 845.2.d.a 6
260.u even 4 1 845.2.d.b 6
260.v odd 6 2 845.2.n.f 12
260.w odd 6 2 845.2.n.g 12
260.bc even 12 2 845.2.l.d 12
260.bc even 12 2 845.2.l.e 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.b.a 6 4.b odd 2 1
65.2.b.a 6 20.d odd 2 1
325.2.a.j 3 20.e even 4 1
325.2.a.k 3 20.e even 4 1
585.2.c.b 6 12.b even 2 1
585.2.c.b 6 60.h even 2 1
845.2.b.c 6 52.b odd 2 1
845.2.b.c 6 260.g odd 2 1
845.2.d.a 6 52.f even 4 1
845.2.d.a 6 260.u even 4 1
845.2.d.b 6 52.f even 4 1
845.2.d.b 6 260.u even 4 1
845.2.l.d 12 52.l even 12 2
845.2.l.d 12 260.bc even 12 2
845.2.l.e 12 52.l even 12 2
845.2.l.e 12 260.bc even 12 2
845.2.n.f 12 52.j odd 6 2
845.2.n.f 12 260.v odd 6 2
845.2.n.g 12 52.i odd 6 2
845.2.n.g 12 260.w odd 6 2
1040.2.d.c 6 1.a even 1 1 trivial
1040.2.d.c 6 5.b even 2 1 inner
2925.2.a.bf 3 60.l odd 4 1
2925.2.a.bj 3 60.l odd 4 1
4225.2.a.ba 3 260.p even 4 1
4225.2.a.bh 3 260.p even 4 1
5200.2.a.cb 3 5.c odd 4 1
5200.2.a.cj 3 5.c odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} + 12T_{3}^{4} + 20T_{3}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(1040, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} + 12 T^{4} + 20 T^{2} + 4$$
$5$ $$T^{6} - T^{4} + 16 T^{3} - 5 T^{2} + \cdots + 125$$
$7$ $$T^{6} + 12 T^{4} + 32 T^{2} + 16$$
$11$ $$(T^{3} - 6 T^{2} + 8 T + 2)^{2}$$
$13$ $$(T^{2} + 1)^{3}$$
$17$ $$T^{6} + 44 T^{4} + 112 T^{2} + \cdots + 64$$
$19$ $$(T^{3} - 4 T - 2)^{2}$$
$23$ $$T^{6} + 72 T^{4} + 1436 T^{2} + \cdots + 7396$$
$29$ $$(T^{3} + 6 T^{2} - 36 T - 108)^{2}$$
$31$ $$(T^{3} - 10 T^{2} + 20 T + 26)^{2}$$
$37$ $$T^{6} + 56 T^{4} + 784 T^{2} + \cdots + 2704$$
$41$ $$(T^{3} + 4 T^{2} - 32 T + 32)^{2}$$
$43$ $$T^{6} + 128 T^{4} + 5452 T^{2} + \cdots + 77284$$
$47$ $$T^{6} + 44 T^{4} + 384 T^{2} + \cdots + 400$$
$53$ $$T^{6} + 144 T^{4} + 6464 T^{2} + \cdots + 92416$$
$59$ $$(T^{3} + 8 T^{2} - 40 T - 262)^{2}$$
$61$ $$(T^{3} - 6 T^{2} - 16 T - 4)^{2}$$
$67$ $$T^{6} + 220 T^{4} + 15680 T^{2} + \cdots + 364816$$
$71$ $$(T^{3} - 12 T^{2} - 88 T + 754)^{2}$$
$73$ $$T^{6} + 248 T^{4} + 15568 T^{2} + \cdots + 55696$$
$79$ $$(T^{3} + 16 T^{2} + 24 T - 16)^{2}$$
$83$ $$T^{6} + 180 T^{4} + 9200 T^{2} + \cdots + 99856$$
$89$ $$(T^{3} + 10 T^{2} - 52 T - 200)^{2}$$
$97$ $$T^{6} + 364 T^{4} + 12656 T^{2} + \cdots + 40000$$