# Properties

 Label 640.2.c.b Level $640$ Weight $2$ Character orbit 640.c Analytic conductor $5.110$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$640 = 2^{7} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 640.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.11042572936$$ 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: $$2^{4}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + (\beta_{3} - \beta_{2}) q^{5} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2}) q^{7} + ( - \beta_1 - 1) q^{9}+O(q^{10})$$ q + b2 * q^3 + (b3 - b2) * q^5 + (b5 - b4 - b3 + b2) * q^7 + (-b1 - 1) * q^9 $$q + \beta_{2} q^{3} + (\beta_{3} - \beta_{2}) q^{5} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2}) q^{7} + ( - \beta_1 - 1) q^{9} + ( - \beta_{5} - \beta_{3} - \beta_1) q^{11} + ( - \beta_{5} + 2 \beta_{4} + \beta_{3}) q^{13} + ( - \beta_{4} - \beta_{2} + \beta_1 + 2) q^{15} + 2 \beta_{2} q^{17} + (\beta_{5} + \beta_{3} - \beta_1) q^{19} + ( - \beta_{5} - \beta_{3} - 2 \beta_1 - 2) q^{21} + (\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2}) q^{23} + ( - 2 \beta_{5} - \beta_1 - 1) q^{25} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2}) q^{27} + 2 q^{29} + ( - 2 \beta_{5} - 2 \beta_{3} - 4) q^{31} + ( - 2 \beta_{5} + 4 \beta_{4} + 2 \beta_{3} - 2 \beta_{2}) q^{33} + (\beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 4) q^{35} + ( - \beta_{5} + \beta_{3} + 2 \beta_{2}) q^{37} + (2 \beta_{5} + 2 \beta_{3} + 2 \beta_1) q^{39} + (2 \beta_{5} + 2 \beta_{3} + \beta_1 + 2) q^{41} + ( - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 3 \beta_{2}) q^{43} + (\beta_{5} - 2 \beta_{4} + 3 \beta_{2} + 2) q^{45} + ( - \beta_{5} - \beta_{4} + \beta_{3} - 3 \beta_{2}) q^{47} + ( - \beta_1 - 1) q^{49} + ( - 2 \beta_1 - 8) q^{51} + (\beta_{5} + 2 \beta_{4} - \beta_{3}) q^{53} + (3 \beta_{5} - 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{55} + ( - 2 \beta_{5} + 2 \beta_{3} - 6 \beta_{2}) q^{57} + ( - \beta_{5} - \beta_{3} + \beta_1) q^{59} + ( - \beta_{5} - \beta_{3} - 2 \beta_1 - 4) q^{61} + ( - \beta_{5} + 3 \beta_{4} + \beta_{3} - 5 \beta_{2}) q^{63} + ( - 4 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - \beta_1 - 2) q^{65} + (2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 5 \beta_{2}) q^{67} + (\beta_{5} + \beta_{3} + 2) q^{69} + ( - 2 \beta_1 + 4) q^{71} + (2 \beta_{5} + 4 \beta_{4} - 2 \beta_{3} + 6 \beta_{2}) q^{73} + ( - 2 \beta_{5} + 4 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} - 4) q^{75} + ( - 2 \beta_{4} - 6 \beta_{2}) q^{77} + ( - 2 \beta_{5} - 2 \beta_{3} + 4) q^{79} + (2 \beta_{5} + 2 \beta_{3} + \beta_1 + 1) q^{81} + (2 \beta_{5} - 6 \beta_{4} - 2 \beta_{3} + \beta_{2}) q^{83} + ( - 2 \beta_{4} - 2 \beta_{2} + 2 \beta_1 + 4) q^{85} + 2 \beta_{2} q^{87} + ( - 2 \beta_{5} - 2 \beta_{3} + 2 \beta_1 + 2) q^{89} + ( - 2 \beta_{5} - 2 \beta_{3} + 8) q^{91} + 4 \beta_{4} q^{93} + ( - \beta_{5} - 2 \beta_{4} + \beta_{3} + 2 \beta_{2} - \beta_1 + 6) q^{95} + ( - 4 \beta_{4} + 2 \beta_{2}) q^{97} + (\beta_{5} + \beta_{3} + 3 \beta_1 + 8) q^{99}+O(q^{100})$$ q + b2 * q^3 + (b3 - b2) * q^5 + (b5 - b4 - b3 + b2) * q^7 + (-b1 - 1) * q^9 + (-b5 - b3 - b1) * q^11 + (-b5 + 2*b4 + b3) * q^13 + (-b4 - b2 + b1 + 2) * q^15 + 2*b2 * q^17 + (b5 + b3 - b1) * q^19 + (-b5 - b3 - 2*b1 - 2) * q^21 + (b5 + b4 - b3 + b2) * q^23 + (-2*b5 - b1 - 1) * q^25 + (-2*b5 + 2*b4 + 2*b3 - 2*b2) * q^27 + 2 * q^29 + (-2*b5 - 2*b3 - 4) * q^31 + (-2*b5 + 4*b4 + 2*b3 - 2*b2) * q^33 + (b5 + 2*b4 + b3 + b2 + b1 + 4) * q^35 + (-b5 + b3 + 2*b2) * q^37 + (2*b5 + 2*b3 + 2*b1) * q^39 + (2*b5 + 2*b3 + b1 + 2) * q^41 + (-2*b5 - 2*b4 + 2*b3 - 3*b2) * q^43 + (b5 - 2*b4 + 3*b2 + 2) * q^45 + (-b5 - b4 + b3 - 3*b2) * q^47 + (-b1 - 1) * q^49 + (-2*b1 - 8) * q^51 + (b5 + 2*b4 - b3) * q^53 + (3*b5 - 2*b4 + b3 + 2*b2 + b1 - 2) * q^55 + (-2*b5 + 2*b3 - 6*b2) * q^57 + (-b5 - b3 + b1) * q^59 + (-b5 - b3 - 2*b1 - 4) * q^61 + (-b5 + 3*b4 + b3 - 5*b2) * q^63 + (-4*b4 - 2*b3 - 2*b2 - b1 - 2) * q^65 + (2*b5 + 2*b4 - 2*b3 + 5*b2) * q^67 + (b5 + b3 + 2) * q^69 + (-2*b1 + 4) * q^71 + (2*b5 + 4*b4 - 2*b3 + 6*b2) * q^73 + (-2*b5 + 4*b4 + 2*b3 - 3*b2 - 4) * q^75 + (-2*b4 - 6*b2) * q^77 + (-2*b5 - 2*b3 + 4) * q^79 + (2*b5 + 2*b3 + b1 + 1) * q^81 + (2*b5 - 6*b4 - 2*b3 + b2) * q^83 + (-2*b4 - 2*b2 + 2*b1 + 4) * q^85 + 2*b2 * q^87 + (-2*b5 - 2*b3 + 2*b1 + 2) * q^89 + (-2*b5 - 2*b3 + 8) * q^91 + 4*b4 * q^93 + (-b5 - 2*b4 + b3 + 2*b2 - b1 + 6) * q^95 + (-4*b4 + 2*b2) * q^97 + (b5 + b3 + 3*b1 + 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 2 q^{5} - 6 q^{9}+O(q^{10})$$ 6 * q - 2 * q^5 - 6 * q^9 $$6 q - 2 q^{5} - 6 q^{9} + 4 q^{11} + 12 q^{15} - 4 q^{19} - 8 q^{21} - 2 q^{25} + 12 q^{29} - 16 q^{31} + 20 q^{35} - 8 q^{39} + 4 q^{41} + 10 q^{45} - 6 q^{49} - 48 q^{51} - 20 q^{55} + 4 q^{59} - 20 q^{61} - 8 q^{65} + 8 q^{69} + 24 q^{71} - 24 q^{75} + 32 q^{79} - 2 q^{81} + 24 q^{85} + 20 q^{89} + 56 q^{91} + 36 q^{95} + 44 q^{99}+O(q^{100})$$ 6 * q - 2 * q^5 - 6 * q^9 + 4 * q^11 + 12 * q^15 - 4 * q^19 - 8 * q^21 - 2 * q^25 + 12 * q^29 - 16 * q^31 + 20 * q^35 - 8 * q^39 + 4 * q^41 + 10 * q^45 - 6 * q^49 - 48 * q^51 - 20 * q^55 + 4 * q^59 - 20 * q^61 - 8 * q^65 + 8 * q^69 + 24 * q^71 - 24 * q^75 + 32 * q^79 - 2 * q^81 + 24 * q^85 + 20 * q^89 + 56 * q^91 + 36 * q^95 + 44 * q^99

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

 $$\beta_{1}$$ $$=$$ $$( 2\nu^{5} - 16\nu^{4} + 8\nu^{3} + 2\nu^{2} - 4\nu - 76 ) / 23$$ (2*v^5 - 16*v^4 + 8*v^3 + 2*v^2 - 4*v - 76) / 23 $$\beta_{2}$$ $$=$$ $$( 4\nu^{5} - 9\nu^{4} + 16\nu^{3} + 4\nu^{2} + 38\nu - 14 ) / 23$$ (4*v^5 - 9*v^4 + 16*v^3 + 4*v^2 + 38*v - 14) / 23 $$\beta_{3}$$ $$=$$ $$( 13\nu^{5} - 12\nu^{4} + 6\nu^{3} + 36\nu^{2} + 112\nu - 11 ) / 23$$ (13*v^5 - 12*v^4 + 6*v^3 + 36*v^2 + 112*v - 11) / 23 $$\beta_{4}$$ $$=$$ $$( -14\nu^{5} + 20\nu^{4} - 10\nu^{3} - 60\nu^{2} - 64\nu + 26 ) / 23$$ (-14*v^5 + 20*v^4 - 10*v^3 - 60*v^2 - 64*v + 26) / 23 $$\beta_{5}$$ $$=$$ $$-\nu^{5} + 2\nu^{4} - 2\nu^{3} - 2\nu^{2} - 4\nu + 3$$ -v^5 + 2*v^4 - 2*v^3 - 2*v^2 - 4*v + 3
 $$\nu$$ $$=$$ $$( \beta_{5} + \beta_{3} + 2\beta_{2} + \beta _1 + 2 ) / 4$$ (b5 + b3 + 2*b2 + b1 + 2) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{5} - 2\beta_{4} - \beta_{3} + 2\beta_{2} ) / 2$$ (b5 - 2*b4 - b3 + 2*b2) / 2 $$\nu^{3}$$ $$=$$ $$( -\beta_{4} - 2\beta_{3} + 4\beta_{2} - 2\beta _1 - 4 ) / 2$$ (-b4 - 2*b3 + 4*b2 - 2*b1 - 4) / 2 $$\nu^{4}$$ $$=$$ $$( -\beta_{5} - \beta_{3} - 5\beta _1 - 14 ) / 2$$ (-b5 - b3 - 5*b1 - 14) / 2 $$\nu^{5}$$ $$=$$ $$-4\beta_{5} + 3\beta_{4} + \beta_{3} - 8\beta_{2} - 4\beta _1 - 9$$ -4*b5 + 3*b4 + b3 - 8*b2 - 4*b1 - 9

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/640\mathbb{Z}\right)^\times$$.

 $$n$$ $$257$$ $$261$$ $$511$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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
 1.45161 − 1.45161i −0.854638 − 0.854638i 0.403032 − 0.403032i 0.403032 + 0.403032i −0.854638 + 0.854638i 1.45161 + 1.45161i
0 2.90321i 0 −0.311108 + 2.21432i 0 3.52543i 0 −5.42864 0
129.2 0 1.70928i 0 −2.17009 + 0.539189i 0 2.63090i 0 0.0783777 0
129.3 0 0.806063i 0 1.48119 1.67513i 0 2.15633i 0 2.35026 0
129.4 0 0.806063i 0 1.48119 + 1.67513i 0 2.15633i 0 2.35026 0
129.5 0 1.70928i 0 −2.17009 0.539189i 0 2.63090i 0 0.0783777 0
129.6 0 2.90321i 0 −0.311108 2.21432i 0 3.52543i 0 −5.42864 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 129.6 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 640.2.c.b yes 6
4.b odd 2 1 640.2.c.a 6
5.b even 2 1 inner 640.2.c.b yes 6
5.c odd 4 1 3200.2.a.br 3
5.c odd 4 1 3200.2.a.bt 3
8.b even 2 1 640.2.c.c yes 6
8.d odd 2 1 640.2.c.d yes 6
16.e even 4 1 1280.2.f.j 6
16.e even 4 1 1280.2.f.k 6
16.f odd 4 1 1280.2.f.i 6
16.f odd 4 1 1280.2.f.l 6
20.d odd 2 1 640.2.c.a 6
20.e even 4 1 3200.2.a.bq 3
20.e even 4 1 3200.2.a.bs 3
40.e odd 2 1 640.2.c.d yes 6
40.f even 2 1 640.2.c.c yes 6
40.i odd 4 1 3200.2.a.bo 3
40.i odd 4 1 3200.2.a.bu 3
40.k even 4 1 3200.2.a.bp 3
40.k even 4 1 3200.2.a.bv 3
80.k odd 4 1 1280.2.f.i 6
80.k odd 4 1 1280.2.f.l 6
80.q even 4 1 1280.2.f.j 6
80.q even 4 1 1280.2.f.k 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.2.c.a 6 4.b odd 2 1
640.2.c.a 6 20.d odd 2 1
640.2.c.b yes 6 1.a even 1 1 trivial
640.2.c.b yes 6 5.b even 2 1 inner
640.2.c.c yes 6 8.b even 2 1
640.2.c.c yes 6 40.f even 2 1
640.2.c.d yes 6 8.d odd 2 1
640.2.c.d yes 6 40.e odd 2 1
1280.2.f.i 6 16.f odd 4 1
1280.2.f.i 6 80.k odd 4 1
1280.2.f.j 6 16.e even 4 1
1280.2.f.j 6 80.q even 4 1
1280.2.f.k 6 16.e even 4 1
1280.2.f.k 6 80.q even 4 1
1280.2.f.l 6 16.f odd 4 1
1280.2.f.l 6 80.k odd 4 1
3200.2.a.bo 3 40.i odd 4 1
3200.2.a.bp 3 40.k even 4 1
3200.2.a.bq 3 20.e even 4 1
3200.2.a.br 3 5.c odd 4 1
3200.2.a.bs 3 20.e even 4 1
3200.2.a.bt 3 5.c odd 4 1
3200.2.a.bu 3 40.i odd 4 1
3200.2.a.bv 3 40.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(640, [\chi])$$:

 $$T_{11}^{3} - 2T_{11}^{2} - 20T_{11} + 8$$ T11^3 - 2*T11^2 - 20*T11 + 8 $$T_{29} - 2$$ T29 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} + 12 T^{4} + 32 T^{2} + 16$$
$5$ $$T^{6} + 2 T^{5} + 3 T^{4} + 12 T^{3} + \cdots + 125$$
$7$ $$T^{6} + 24 T^{4} + 176 T^{2} + \cdots + 400$$
$11$ $$(T^{3} - 2 T^{2} - 20 T + 8)^{2}$$
$13$ $$T^{6} + 48 T^{4} + 320 T^{2} + \cdots + 256$$
$17$ $$T^{6} + 48 T^{4} + 512 T^{2} + \cdots + 1024$$
$19$ $$(T^{3} + 2 T^{2} - 36 T - 104)^{2}$$
$23$ $$T^{6} + 40 T^{4} + 80 T^{2} + 16$$
$29$ $$(T - 2)^{6}$$
$31$ $$(T^{3} + 8 T^{2} - 32 T - 128)^{2}$$
$37$ $$T^{6} + 128 T^{4} + 5376 T^{2} + \cdots + 73984$$
$41$ $$(T^{3} - 2 T^{2} - 52 T + 184)^{2}$$
$43$ $$T^{6} + 156 T^{4} + 3200 T^{2} + \cdots + 10000$$
$47$ $$T^{6} + 72 T^{4} + 1712 T^{2} + \cdots + 13456$$
$53$ $$T^{6} + 112 T^{4} + 2624 T^{2} + \cdots + 6400$$
$59$ $$(T^{3} - 2 T^{2} - 36 T + 104)^{2}$$
$61$ $$(T^{3} + 10 T^{2} - 28 T - 8)^{2}$$
$67$ $$T^{6} + 220 T^{4} + 15680 T^{2} + \cdots + 364816$$
$71$ $$(T^{3} - 12 T^{2} - 16 T + 320)^{2}$$
$73$ $$T^{6} + 400 T^{4} + 47360 T^{2} + \cdots + 1401856$$
$79$ $$(T^{3} - 16 T^{2} + 32 T + 128)^{2}$$
$83$ $$T^{6} + 380 T^{4} + 36160 T^{2} + \cdots + 274576$$
$89$ $$(T^{3} - 10 T^{2} - 116 T + 1096)^{2}$$
$97$ $$T^{6} + 304 T^{4} + 23552 T^{2} + \cdots + 369664$$