# Properties

 Label 60.4.i.a Level $60$ Weight $4$ Character orbit 60.i Analytic conductor $3.540$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$60 = 2^{2} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 60.i (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.54011460034$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$5^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - 5 \zeta_{8} + \zeta_{8}^{2} ) q^{3} + ( 5 \zeta_{8} + 10 \zeta_{8}^{3} ) q^{5} + ( 23 + 23 \zeta_{8}^{2} ) q^{7} + ( 10 \zeta_{8} + 23 \zeta_{8}^{2} - 10 \zeta_{8}^{3} ) q^{9} +O(q^{10})$$ $$q + ( -1 - 5 \zeta_{8} + \zeta_{8}^{2} ) q^{3} + ( 5 \zeta_{8} + 10 \zeta_{8}^{3} ) q^{5} + ( 23 + 23 \zeta_{8}^{2} ) q^{7} + ( 10 \zeta_{8} + 23 \zeta_{8}^{2} - 10 \zeta_{8}^{3} ) q^{9} + ( -5 \zeta_{8} - 5 \zeta_{8}^{3} ) q^{11} + ( -24 + 24 \zeta_{8}^{2} ) q^{13} + ( 50 - 15 \zeta_{8} - 25 \zeta_{8}^{2} - 5 \zeta_{8}^{3} ) q^{15} + 110 \zeta_{8}^{3} q^{17} -68 \zeta_{8}^{2} q^{19} + ( -46 - 115 \zeta_{8} - 115 \zeta_{8}^{3} ) q^{21} + 140 \zeta_{8} q^{23} + ( -100 - 75 \zeta_{8}^{2} ) q^{25} + ( -73 - 73 \zeta_{8}^{2} - 95 \zeta_{8}^{3} ) q^{27} + ( 95 \zeta_{8} - 95 \zeta_{8}^{3} ) q^{29} + 94 q^{31} + ( -25 + 10 \zeta_{8} + 25 \zeta_{8}^{2} ) q^{33} + ( -115 \zeta_{8} + 345 \zeta_{8}^{3} ) q^{35} + ( -66 - 66 \zeta_{8}^{2} ) q^{37} + ( 120 \zeta_{8} - 48 \zeta_{8}^{2} - 120 \zeta_{8}^{3} ) q^{39} + ( -140 \zeta_{8} - 140 \zeta_{8}^{3} ) q^{41} + ( 126 - 126 \zeta_{8}^{2} ) q^{43} + ( -50 - 230 \zeta_{8} + 150 \zeta_{8}^{2} + 115 \zeta_{8}^{3} ) q^{45} + 120 \zeta_{8}^{3} q^{47} + 715 \zeta_{8}^{2} q^{49} + ( 550 - 110 \zeta_{8} - 110 \zeta_{8}^{3} ) q^{51} + 460 \zeta_{8} q^{53} + ( 75 + 25 \zeta_{8}^{2} ) q^{55} + ( 68 + 68 \zeta_{8}^{2} + 340 \zeta_{8}^{3} ) q^{57} + ( 35 \zeta_{8} - 35 \zeta_{8}^{3} ) q^{59} -126 q^{61} + ( -529 + 460 \zeta_{8} + 529 \zeta_{8}^{2} ) q^{63} + ( -360 \zeta_{8} - 120 \zeta_{8}^{3} ) q^{65} + ( 68 + 68 \zeta_{8}^{2} ) q^{67} + ( -140 \zeta_{8} - 700 \zeta_{8}^{2} + 140 \zeta_{8}^{3} ) q^{69} + ( -670 \zeta_{8} - 670 \zeta_{8}^{3} ) q^{71} + ( 403 - 403 \zeta_{8}^{2} ) q^{73} + ( 175 + 500 \zeta_{8} - 25 \zeta_{8}^{2} + 375 \zeta_{8}^{3} ) q^{75} -230 \zeta_{8}^{3} q^{77} -234 \zeta_{8}^{2} q^{79} + ( -329 + 460 \zeta_{8} + 460 \zeta_{8}^{3} ) q^{81} -1020 \zeta_{8} q^{83} + ( -550 - 1100 \zeta_{8}^{2} ) q^{85} + ( -475 - 475 \zeta_{8}^{2} + 190 \zeta_{8}^{3} ) q^{87} + ( 730 \zeta_{8} - 730 \zeta_{8}^{3} ) q^{89} -1104 q^{91} + ( -94 - 470 \zeta_{8} + 94 \zeta_{8}^{2} ) q^{93} + ( 680 \zeta_{8} - 340 \zeta_{8}^{3} ) q^{95} + ( 723 + 723 \zeta_{8}^{2} ) q^{97} + ( 115 \zeta_{8} - 100 \zeta_{8}^{2} - 115 \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{3} + 92q^{7} + O(q^{10})$$ $$4q - 4q^{3} + 92q^{7} - 96q^{13} + 200q^{15} - 184q^{21} - 400q^{25} - 292q^{27} + 376q^{31} - 100q^{33} - 264q^{37} + 504q^{43} - 200q^{45} + 2200q^{51} + 300q^{55} + 272q^{57} - 504q^{61} - 2116q^{63} + 272q^{67} + 1612q^{73} + 700q^{75} - 1316q^{81} - 2200q^{85} - 1900q^{87} - 4416q^{91} - 376q^{93} + 2892q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/60\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$37$$ $$41$$ $$\chi(n)$$ $$1$$ $$\zeta_{8}^{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 0.707107 + 0.707107i −0.707107 − 0.707107i 0.707107 − 0.707107i −0.707107 + 0.707107i
0 −4.53553 2.53553i 0 −3.53553 + 10.6066i 0 23.0000 + 23.0000i 0 14.1421 + 23.0000i 0
17.2 0 2.53553 + 4.53553i 0 3.53553 10.6066i 0 23.0000 + 23.0000i 0 −14.1421 + 23.0000i 0
53.1 0 −4.53553 + 2.53553i 0 −3.53553 10.6066i 0 23.0000 23.0000i 0 14.1421 23.0000i 0
53.2 0 2.53553 4.53553i 0 3.53553 + 10.6066i 0 23.0000 23.0000i 0 −14.1421 23.0000i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 60.4.i.a 4
3.b odd 2 1 inner 60.4.i.a 4
4.b odd 2 1 240.4.v.a 4
5.b even 2 1 300.4.i.c 4
5.c odd 4 1 inner 60.4.i.a 4
5.c odd 4 1 300.4.i.c 4
12.b even 2 1 240.4.v.a 4
15.d odd 2 1 300.4.i.c 4
15.e even 4 1 inner 60.4.i.a 4
15.e even 4 1 300.4.i.c 4
20.e even 4 1 240.4.v.a 4
60.l odd 4 1 240.4.v.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.4.i.a 4 1.a even 1 1 trivial
60.4.i.a 4 3.b odd 2 1 inner
60.4.i.a 4 5.c odd 4 1 inner
60.4.i.a 4 15.e even 4 1 inner
240.4.v.a 4 4.b odd 2 1
240.4.v.a 4 12.b even 2 1
240.4.v.a 4 20.e even 4 1
240.4.v.a 4 60.l odd 4 1
300.4.i.c 4 5.b even 2 1
300.4.i.c 4 5.c odd 4 1
300.4.i.c 4 15.d odd 2 1
300.4.i.c 4 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} - 46 T_{7} + 1058$$ acting on $$S_{4}^{\mathrm{new}}(60, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$729 + 108 T + 8 T^{2} + 4 T^{3} + T^{4}$$
$5$ $$15625 + 200 T^{2} + T^{4}$$
$7$ $$( 1058 - 46 T + T^{2} )^{2}$$
$11$ $$( 50 + T^{2} )^{2}$$
$13$ $$( 1152 + 48 T + T^{2} )^{2}$$
$17$ $$146410000 + T^{4}$$
$19$ $$( 4624 + T^{2} )^{2}$$
$23$ $$384160000 + T^{4}$$
$29$ $$( -18050 + T^{2} )^{2}$$
$31$ $$( -94 + T )^{4}$$
$37$ $$( 8712 + 132 T + T^{2} )^{2}$$
$41$ $$( 39200 + T^{2} )^{2}$$
$43$ $$( 31752 - 252 T + T^{2} )^{2}$$
$47$ $$207360000 + T^{4}$$
$53$ $$44774560000 + T^{4}$$
$59$ $$( -2450 + T^{2} )^{2}$$
$61$ $$( 126 + T )^{4}$$
$67$ $$( 9248 - 136 T + T^{2} )^{2}$$
$71$ $$( 897800 + T^{2} )^{2}$$
$73$ $$( 324818 - 806 T + T^{2} )^{2}$$
$79$ $$( 54756 + T^{2} )^{2}$$
$83$ $$1082432160000 + T^{4}$$
$89$ $$( -1065800 + T^{2} )^{2}$$
$97$ $$( 1045458 - 1446 T + T^{2} )^{2}$$