# Properties

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

# Related objects

## 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 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.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{5} + 5 \nu^{3} + \nu^{2} + 4 \nu + 2$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{4} + 8 \nu^{2} + 3$$ $$\beta_{4}$$ $$=$$ $$-\nu^{5} - 5 \nu^{3} + \nu^{2} - 4 \nu + 2$$ $$\beta_{5}$$ $$=$$ $$2 \nu^{5} + 12 \nu^{3} + 14 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{4} + \beta_{2} - 4$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} - \beta_{2} - 3 \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-4 \beta_{4} + \beta_{3} - 4 \beta_{2} + 13$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-5 \beta_{5} - 6 \beta_{4} + 6 \beta_{2} + 11 \beta_{1}$$$$)/2$$

## 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
 2.02852i − 2.02852i − 1.32132i 1.32132i 0.373087i − 0.373087i
0 1.00000 0 −2.11491 0.726062i 0 4.05705i 0 1.00000 0
49.2 0 1.00000 0 −2.11491 + 0.726062i 0 4.05705i 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 1.86081 1.23992i 0 0.746175i 0 1.00000 0
49.6 0 1.00000 0 1.86081 + 1.23992i 0 0.746175i 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.b 6
3.b odd 2 1 1440.2.d.f 6
4.b odd 2 1 120.2.d.b yes 6
5.b even 2 1 480.2.d.a 6
5.c odd 4 2 2400.2.k.f 12
8.b even 2 1 480.2.d.a 6
8.d odd 2 1 120.2.d.a 6
12.b even 2 1 360.2.d.e 6
15.d odd 2 1 1440.2.d.e 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.a 6
20.e even 4 2 600.2.k.f 12
24.f even 2 1 360.2.d.f 6
24.h odd 2 1 1440.2.d.e 6
40.e odd 2 1 120.2.d.b yes 6
40.f even 2 1 inner 480.2.d.b 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.f 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.f 6
120.m even 2 1 360.2.d.e 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 8.d odd 2 1
120.2.d.a 6 20.d odd 2 1
120.2.d.b yes 6 4.b odd 2 1
120.2.d.b yes 6 40.e odd 2 1
360.2.d.e 6 12.b even 2 1
360.2.d.e 6 120.m even 2 1
360.2.d.f 6 24.f even 2 1
360.2.d.f 6 60.h even 2 1
480.2.d.a 6 5.b even 2 1
480.2.d.a 6 8.b even 2 1
480.2.d.b 6 1.a even 1 1 trivial
480.2.d.b 6 40.f even 2 1 inner
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 15.d odd 2 1
1440.2.d.e 6 24.h odd 2 1
1440.2.d.f 6 3.b odd 2 1
1440.2.d.f 6 120.i 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 $$T_{13}^{3} - 4 T_{13}^{2} - 16 T_{13} + 56$$ acting on $$S_{2}^{\mathrm{new}}(480, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 - T )^{6}$$
$5$ $$1 - T^{2} + 8 T^{3} - 5 T^{4} + 125 T^{6}$$
$7$ $$1 - 18 T^{2} + 191 T^{4} - 1532 T^{6} + 9359 T^{8} - 43218 T^{10} + 117649 T^{12}$$
$11$ $$1 - 34 T^{2} + 503 T^{4} - 5436 T^{6} + 60863 T^{8} - 497794 T^{10} + 1771561 T^{12}$$
$13$ $$( 1 - 4 T + 23 T^{2} - 48 T^{3} + 299 T^{4} - 676 T^{5} + 2197 T^{6} )^{2}$$
$17$ $$1 - 66 T^{2} + 2255 T^{4} - 47324 T^{6} + 651695 T^{8} - 5512386 T^{10} + 24137569 T^{12}$$
$19$ $$1 - 54 T^{2} + 1367 T^{4} - 25652 T^{6} + 493487 T^{8} - 7037334 T^{10} + 47045881 T^{12}$$
$23$ $$1 - 46 T^{2} + 1775 T^{4} - 40932 T^{6} + 938975 T^{8} - 12872686 T^{10} + 148035889 T^{12}$$
$29$ $$1 - 66 T^{2} + 3207 T^{4} - 111228 T^{6} + 2697087 T^{8} - 46680546 T^{10} + 594823321 T^{12}$$
$31$ $$( 1 - 8 T + 89 T^{2} - 432 T^{3} + 2759 T^{4} - 7688 T^{5} + 29791 T^{6} )^{2}$$
$37$ $$( 1 - 8 T + 111 T^{2} - 584 T^{3} + 4107 T^{4} - 10952 T^{5} + 50653 T^{6} )^{2}$$
$41$ $$( 1 + 2 T + 23 T^{2} + 220 T^{3} + 943 T^{4} + 3362 T^{5} + 68921 T^{6} )^{2}$$
$43$ $$( 1 + 65 T^{2} + 64 T^{3} + 2795 T^{4} + 79507 T^{6} )^{2}$$
$47$ $$1 - 222 T^{2} + 22367 T^{4} - 1328324 T^{6} + 49408703 T^{8} - 1083289182 T^{10} + 10779215329 T^{12}$$
$53$ $$( 1 + 12 T + 191 T^{2} + 1264 T^{3} + 10123 T^{4} + 33708 T^{5} + 148877 T^{6} )^{2}$$
$59$ $$1 - 178 T^{2} + 20567 T^{4} - 1418652 T^{6} + 71593727 T^{8} - 2156890258 T^{10} + 42180533641 T^{12}$$
$61$ $$1 - 190 T^{2} + 20039 T^{4} - 1419204 T^{6} + 74565119 T^{8} - 2630709790 T^{10} + 51520374361 T^{12}$$
$67$ $$( 1 + 137 T^{2} - 64 T^{3} + 9179 T^{4} + 300763 T^{6} )^{2}$$
$71$ $$( 1 + 8 T + 133 T^{2} + 1008 T^{3} + 9443 T^{4} + 40328 T^{5} + 357911 T^{6} )^{2}$$
$73$ $$1 - 54 T^{2} + 2367 T^{4} - 531700 T^{6} + 12613743 T^{8} - 1533505014 T^{10} + 151334226289 T^{12}$$
$79$ $$( 1 + 8 T + 233 T^{2} + 1200 T^{3} + 18407 T^{4} + 49928 T^{5} + 493039 T^{6} )^{2}$$
$83$ $$( 1 + 8 T + 185 T^{2} + 880 T^{3} + 15355 T^{4} + 55112 T^{5} + 571787 T^{6} )^{2}$$
$89$ $$( 1 + 10 T + 103 T^{2} + 396 T^{3} + 9167 T^{4} + 79210 T^{5} + 704969 T^{6} )^{2}$$
$97$ $$1 - 246 T^{2} + 39183 T^{4} - 4535476 T^{6} + 368672847 T^{8} - 21778203126 T^{10} + 832972004929 T^{12}$$