# Properties

 Label 480.2.d.a Level $480$ Weight $2$ Character orbit 480.d Analytic conductor $3.833$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$480 = 2^{5} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 480.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.83281929702$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.839056.1 Defining polynomial: $$x^{6} + 6x^{4} + 8x^{2} + 1$$ x^6 + 6*x^4 + 8*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{5}$$ Twist minimal: no (minimal twist has level 120) 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 6x^{4} + 8x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu$$ 2*v $$\beta_{2}$$ $$=$$ $$\nu^{5} + 5\nu^{3} + \nu^{2} + 4\nu + 2$$ v^5 + 5*v^3 + v^2 + 4*v + 2 $$\beta_{3}$$ $$=$$ $$2\nu^{4} + 8\nu^{2} + 3$$ 2*v^4 + 8*v^2 + 3 $$\beta_{4}$$ $$=$$ $$-\nu^{5} - 5\nu^{3} + \nu^{2} - 4\nu + 2$$ -v^5 - 5*v^3 + v^2 - 4*v + 2 $$\beta_{5}$$ $$=$$ $$2\nu^{5} + 12\nu^{3} + 14\nu$$ 2*v^5 + 12*v^3 + 14*v
 $$\nu$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{4} + \beta_{2} - 4 ) / 2$$ (b4 + b2 - 4) / 2 $$\nu^{3}$$ $$=$$ $$( \beta_{5} + \beta_{4} - \beta_{2} - 3\beta_1 ) / 2$$ (b5 + b4 - b2 - 3*b1) / 2 $$\nu^{4}$$ $$=$$ $$( -4\beta_{4} + \beta_{3} - 4\beta_{2} + 13 ) / 2$$ (-4*b4 + b3 - 4*b2 + 13) / 2 $$\nu^{5}$$ $$=$$ $$( -5\beta_{5} - 6\beta_{4} + 6\beta_{2} + 11\beta_1 ) / 2$$ (-5*b5 - 6*b4 + 6*b2 + 11*b1) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/480\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$97$$ $$161$$ $$421$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 0.373087i − 0.373087i − 1.32132i 1.32132i 2.02852i − 2.02852i
0 −1.00000 0 −1.86081 1.23992i 0 0.746175i 0 1.00000 0
49.2 0 −1.00000 0 −1.86081 + 1.23992i 0 0.746175i 0 1.00000 0
49.3 0 −1.00000 0 −0.254102 2.22158i 0 2.64265i 0 1.00000 0
49.4 0 −1.00000 0 −0.254102 + 2.22158i 0 2.64265i 0 1.00000 0
49.5 0 −1.00000 0 2.11491 0.726062i 0 4.05705i 0 1.00000 0
49.6 0 −1.00000 0 2.11491 + 0.726062i 0 4.05705i 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 49.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 480.2.d.a 6
3.b odd 2 1 1440.2.d.e 6
4.b odd 2 1 120.2.d.a 6
5.b even 2 1 480.2.d.b 6
5.c odd 4 2 2400.2.k.f 12
8.b even 2 1 480.2.d.b 6
8.d odd 2 1 120.2.d.b yes 6
12.b even 2 1 360.2.d.f 6
15.d odd 2 1 1440.2.d.f 6
15.e even 4 2 7200.2.k.u 12
16.e even 4 2 3840.2.f.m 12
16.f odd 4 2 3840.2.f.l 12
20.d odd 2 1 120.2.d.b yes 6
20.e even 4 2 600.2.k.f 12
24.f even 2 1 360.2.d.e 6
24.h odd 2 1 1440.2.d.f 6
40.e odd 2 1 120.2.d.a 6
40.f even 2 1 inner 480.2.d.a 6
40.i odd 4 2 2400.2.k.f 12
40.k even 4 2 600.2.k.f 12
60.h even 2 1 360.2.d.e 6
60.l odd 4 2 1800.2.k.u 12
80.k odd 4 2 3840.2.f.l 12
80.q even 4 2 3840.2.f.m 12
120.i odd 2 1 1440.2.d.e 6
120.m even 2 1 360.2.d.f 6
120.q odd 4 2 1800.2.k.u 12
120.w even 4 2 7200.2.k.u 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.d.a 6 4.b odd 2 1
120.2.d.a 6 40.e odd 2 1
120.2.d.b yes 6 8.d odd 2 1
120.2.d.b yes 6 20.d odd 2 1
360.2.d.e 6 24.f even 2 1
360.2.d.e 6 60.h even 2 1
360.2.d.f 6 12.b even 2 1
360.2.d.f 6 120.m even 2 1
480.2.d.a 6 1.a even 1 1 trivial
480.2.d.a 6 40.f even 2 1 inner
480.2.d.b 6 5.b even 2 1
480.2.d.b 6 8.b even 2 1
600.2.k.f 12 20.e even 4 2
600.2.k.f 12 40.k even 4 2
1440.2.d.e 6 3.b odd 2 1
1440.2.d.e 6 120.i odd 2 1
1440.2.d.f 6 15.d odd 2 1
1440.2.d.f 6 24.h odd 2 1
1800.2.k.u 12 60.l odd 4 2
1800.2.k.u 12 120.q odd 4 2
2400.2.k.f 12 5.c odd 4 2
2400.2.k.f 12 40.i odd 4 2
3840.2.f.l 12 16.f odd 4 2
3840.2.f.l 12 80.k odd 4 2
3840.2.f.m 12 16.e even 4 2
3840.2.f.m 12 80.q even 4 2
7200.2.k.u 12 15.e even 4 2
7200.2.k.u 12 120.w even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{13}^{3} + 4T_{13}^{2} - 16T_{13} - 56$$ acting on $$S_{2}^{\mathrm{new}}(480, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$(T + 1)^{6}$$
$5$ $$T^{6} - T^{4} - 8 T^{3} - 5 T^{2} + \cdots + 125$$
$7$ $$T^{6} + 24 T^{4} + 128 T^{2} + \cdots + 64$$
$11$ $$T^{6} + 32 T^{4} + 96 T^{2} + 64$$
$13$ $$(T^{3} + 4 T^{2} - 16 T - 56)^{2}$$
$17$ $$T^{6} + 36 T^{4} + 368 T^{2} + \cdots + 1024$$
$19$ $$T^{6} + 60 T^{4} + 512 T^{2} + \cdots + 1024$$
$23$ $$T^{6} + 92 T^{4} + 2304 T^{2} + \cdots + 16384$$
$29$ $$T^{6} + 108 T^{4} + 3120 T^{2} + \cdots + 12544$$
$31$ $$(T^{3} - 8 T^{2} - 4 T + 64)^{2}$$
$37$ $$(T^{3} + 8 T^{2} - 8)^{2}$$
$41$ $$(T^{3} + 2 T^{2} - 100 T + 56)^{2}$$
$43$ $$(T^{3} - 64 T - 64)^{2}$$
$47$ $$T^{6} + 60 T^{4} + 512 T^{2} + \cdots + 1024$$
$53$ $$(T^{3} - 12 T^{2} + 32 T + 8)^{2}$$
$59$ $$T^{6} + 176 T^{4} + 9888 T^{2} + \cdots + 179776$$
$61$ $$T^{6} + 176 T^{4} + 7168 T^{2} + \cdots + 65536$$
$67$ $$(T^{3} - 64 T + 64)^{2}$$
$71$ $$(T^{3} + 8 T^{2} - 80 T - 128)^{2}$$
$73$ $$T^{6} + 384 T^{4} + 34560 T^{2} + \cdots + 16384$$
$79$ $$(T^{3} + 8 T^{2} - 4 T - 64)^{2}$$
$83$ $$(T^{3} - 8 T^{2} - 64 T + 448)^{2}$$
$89$ $$(T^{3} + 10 T^{2} - 164 T - 1384)^{2}$$
$97$ $$T^{6} + 336 T^{4} + 28416 T^{2} + \cdots + 262144$$
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