# Properties

 Label 480.2.k.b Level 480 Weight 2 Character orbit 480.k 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.k (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.83281929702$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.399424.1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{6}$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{3} + \beta_{2} q^{5} + ( -1 + \beta_{3} ) q^{7} - q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{3} + \beta_{2} q^{5} + ( -1 + \beta_{3} ) q^{7} - q^{9} + ( \beta_{2} - \beta_{4} - \beta_{5} ) q^{11} -\beta_{4} q^{13} + q^{15} + ( 2 - \beta_{1} ) q^{17} + ( -\beta_{2} + \beta_{5} ) q^{19} + ( \beta_{2} + \beta_{5} ) q^{21} + ( 1 + \beta_{3} ) q^{23} - q^{25} + \beta_{2} q^{27} -2 \beta_{2} q^{29} + ( 2 + \beta_{1} ) q^{31} + ( 1 + \beta_{1} + \beta_{3} ) q^{33} + ( -\beta_{2} - \beta_{5} ) q^{35} + ( -2 \beta_{2} - \beta_{4} - 2 \beta_{5} ) q^{37} + \beta_{1} q^{39} + ( -4 + 2 \beta_{3} ) q^{41} + ( 2 \beta_{2} + 2 \beta_{5} ) q^{43} -\beta_{2} q^{45} + ( -1 - 2 \beta_{1} - \beta_{3} ) q^{47} + ( 5 + 2 \beta_{1} ) q^{49} + ( -2 \beta_{2} - \beta_{4} ) q^{51} + 2 \beta_{2} q^{53} + ( -1 - \beta_{1} - \beta_{3} ) q^{55} + ( -1 - \beta_{3} ) q^{57} + ( 3 \beta_{2} + \beta_{4} + \beta_{5} ) q^{59} + ( -2 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} ) q^{61} + ( 1 - \beta_{3} ) q^{63} -\beta_{1} q^{65} + 4 \beta_{2} q^{67} + ( -\beta_{2} + \beta_{5} ) q^{69} + ( 2 - 2 \beta_{1} - 2 \beta_{3} ) q^{71} -6 q^{73} + \beta_{2} q^{75} + ( 4 \beta_{2} - 4 \beta_{5} ) q^{77} + ( -6 + \beta_{1} ) q^{79} + q^{81} + ( -6 \beta_{2} - 2 \beta_{5} ) q^{83} + ( 2 \beta_{2} + \beta_{4} ) q^{85} -2 q^{87} + ( -4 - 2 \beta_{3} ) q^{89} + ( -6 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{91} + ( -2 \beta_{2} + \beta_{4} ) q^{93} + ( 1 + \beta_{3} ) q^{95} + ( 6 - 2 \beta_{1} ) q^{97} + ( -\beta_{2} + \beta_{4} + \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 4q^{7} - 6q^{9} + O(q^{10})$$ $$6q - 4q^{7} - 6q^{9} + 6q^{15} + 12q^{17} + 8q^{23} - 6q^{25} + 12q^{31} + 8q^{33} - 20q^{41} - 8q^{47} + 30q^{49} - 8q^{55} - 8q^{57} + 4q^{63} + 8q^{71} - 36q^{73} - 36q^{79} + 6q^{81} - 12q^{87} - 28q^{89} + 8q^{95} + 36q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2 x^{5} + 3 x^{4} - 6 x^{3} + 6 x^{2} - 8 x + 8$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$-\nu^{4} + 2 \nu^{3} - \nu^{2} + 2 \nu - 2$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} - 3 \nu^{3} + 4 \nu^{2} - 2 \nu + 8$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{5} + 2 \nu^{4} - 3 \nu^{3} + 6 \nu^{2} - 2 \nu + 6$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{5} - 2 \nu^{4} + 3 \nu^{3} - 6 \nu^{2} + 10 \nu - 8$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$($$$$7 \nu^{5} - 4 \nu^{4} + 13 \nu^{3} - 16 \nu^{2} + 6 \nu - 32$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} + \beta_{3} + 1$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} + 2 \beta_{3} + 3 \beta_{2} + \beta_{1} - 2$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{4} + \beta_{3} - 4 \beta_{2} + 2 \beta_{1} + 5$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$-\beta_{5} + 2 \beta_{3} - 11 \beta_{2} - \beta_{1} + 6$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$4 \beta_{5} + \beta_{4} + 3 \beta_{3} + 8 \beta_{2} - 2 \beta_{1} + 7$$$$)/4$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
241.1
 −0.671462 − 1.24464i 0.264658 + 1.38923i 1.40680 − 0.144584i −0.671462 + 1.24464i 0.264658 − 1.38923i 1.40680 + 0.144584i
0 1.00000i 0 1.00000i 0 −4.68585 0 −1.00000 0
241.2 0 1.00000i 0 1.00000i 0 −0.941367 0 −1.00000 0
241.3 0 1.00000i 0 1.00000i 0 3.62721 0 −1.00000 0
241.4 0 1.00000i 0 1.00000i 0 −4.68585 0 −1.00000 0
241.5 0 1.00000i 0 1.00000i 0 −0.941367 0 −1.00000 0
241.6 0 1.00000i 0 1.00000i 0 3.62721 0 −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 241.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b 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.k.b 6
3.b odd 2 1 1440.2.k.f 6
4.b odd 2 1 120.2.k.b 6
5.b even 2 1 2400.2.k.c 6
5.c odd 4 1 2400.2.d.e 6
5.c odd 4 1 2400.2.d.f 6
8.b even 2 1 inner 480.2.k.b 6
8.d odd 2 1 120.2.k.b 6
12.b even 2 1 360.2.k.f 6
15.d odd 2 1 7200.2.k.p 6
15.e even 4 1 7200.2.d.q 6
15.e even 4 1 7200.2.d.r 6
16.e even 4 1 3840.2.a.bo 3
16.e even 4 1 3840.2.a.br 3
16.f odd 4 1 3840.2.a.bp 3
16.f odd 4 1 3840.2.a.bq 3
20.d odd 2 1 600.2.k.c 6
20.e even 4 1 600.2.d.e 6
20.e even 4 1 600.2.d.f 6
24.f even 2 1 360.2.k.f 6
24.h odd 2 1 1440.2.k.f 6
40.e odd 2 1 600.2.k.c 6
40.f even 2 1 2400.2.k.c 6
40.i odd 4 1 2400.2.d.e 6
40.i odd 4 1 2400.2.d.f 6
40.k even 4 1 600.2.d.e 6
40.k even 4 1 600.2.d.f 6
60.h even 2 1 1800.2.k.p 6
60.l odd 4 1 1800.2.d.q 6
60.l odd 4 1 1800.2.d.r 6
120.i odd 2 1 7200.2.k.p 6
120.m even 2 1 1800.2.k.p 6
120.q odd 4 1 1800.2.d.q 6
120.q odd 4 1 1800.2.d.r 6
120.w even 4 1 7200.2.d.q 6
120.w even 4 1 7200.2.d.r 6

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{3} + 2 T_{7}^{2} - 16 T_{7} - 16$$ acting on $$S_{2}^{\mathrm{new}}(480, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 + T^{2} )^{3}$$
$5$ $$( 1 + T^{2} )^{3}$$
$7$ $$( 1 + 2 T + 5 T^{2} + 12 T^{3} + 35 T^{4} + 98 T^{5} + 343 T^{6} )^{2}$$
$11$ $$1 - 2 T^{2} + 87 T^{4} + 4 T^{6} + 10527 T^{8} - 29282 T^{10} + 1771561 T^{12}$$
$13$ $$1 - 22 T^{2} + 407 T^{4} - 7284 T^{6} + 68783 T^{8} - 628342 T^{10} + 4826809 T^{12}$$
$17$ $$( 1 - 6 T + 35 T^{2} - 172 T^{3} + 595 T^{4} - 1734 T^{5} + 4913 T^{6} )^{2}$$
$19$ $$1 - 74 T^{2} + 2647 T^{4} - 60620 T^{6} + 955567 T^{8} - 9643754 T^{10} + 47045881 T^{12}$$
$23$ $$( 1 - 4 T + 57 T^{2} - 168 T^{3} + 1311 T^{4} - 2116 T^{5} + 12167 T^{6} )^{2}$$
$29$ $$( 1 - 54 T^{2} + 841 T^{4} )^{3}$$
$31$ $$( 1 - 6 T + 77 T^{2} - 308 T^{3} + 2387 T^{4} - 5766 T^{5} + 29791 T^{6} )^{2}$$
$37$ $$1 - 86 T^{2} + 6055 T^{4} - 248372 T^{6} + 8289295 T^{8} - 161177846 T^{10} + 2565726409 T^{12}$$
$41$ $$( 1 + 10 T + 87 T^{2} + 588 T^{3} + 3567 T^{4} + 16810 T^{5} + 68921 T^{6} )^{2}$$
$43$ $$1 - 114 T^{2} + 8087 T^{4} - 416540 T^{6} + 14952863 T^{8} - 389743314 T^{10} + 6321363049 T^{12}$$
$47$ $$( 1 + 4 T + 49 T^{2} - 120 T^{3} + 2303 T^{4} + 8836 T^{5} + 103823 T^{6} )^{2}$$
$53$ $$( 1 - 102 T^{2} + 2809 T^{4} )^{3}$$
$59$ $$1 - 274 T^{2} + 33911 T^{4} - 2503644 T^{6} + 118044191 T^{8} - 3320156914 T^{10} + 42180533641 T^{12}$$
$61$ $$1 - 110 T^{2} + 10759 T^{4} - 685796 T^{6} + 40034239 T^{8} - 1523042510 T^{10} + 51520374361 T^{12}$$
$67$ $$( 1 - 118 T^{2} + 4489 T^{4} )^{3}$$
$71$ $$( 1 - 4 T + 101 T^{2} - 632 T^{3} + 7171 T^{4} - 20164 T^{5} + 357911 T^{6} )^{2}$$
$73$ $$( 1 + 6 T + 73 T^{2} )^{6}$$
$79$ $$( 1 + 18 T + 317 T^{2} + 2908 T^{3} + 25043 T^{4} + 112338 T^{5} + 493039 T^{6} )^{2}$$
$83$ $$1 - 274 T^{2} + 37415 T^{4} - 3513756 T^{6} + 257751935 T^{8} - 13003579954 T^{10} + 326940373369 T^{12}$$
$89$ $$( 1 + 14 T + 263 T^{2} + 2308 T^{3} + 23407 T^{4} + 110894 T^{5} + 704969 T^{6} )^{2}$$
$97$ $$( 1 - 18 T + 287 T^{2} - 3164 T^{3} + 27839 T^{4} - 169362 T^{5} + 912673 T^{6} )^{2}$$