# Properties

 Label 60.4.d.a Level $60$ Weight $4$ Character orbit 60.d Analytic conductor $3.540$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$60 = 2^{2} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 60.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

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

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 i q^{3} + ( -10 + 5 i ) q^{5} + 22 i q^{7} -9 q^{9} +O(q^{10})$$ $$q + 3 i q^{3} + ( -10 + 5 i ) q^{5} + 22 i q^{7} -9 q^{9} -14 q^{11} + 30 i q^{13} + ( -15 - 30 i ) q^{15} + 62 i q^{17} + 120 q^{19} -66 q^{21} -188 i q^{23} + ( 75 - 100 i ) q^{25} -27 i q^{27} -96 q^{29} + 184 q^{31} -42 i q^{33} + ( -110 - 220 i ) q^{35} + 406 i q^{37} -90 q^{39} + 130 q^{41} -148 i q^{43} + ( 90 - 45 i ) q^{45} + 448 i q^{47} -141 q^{49} -186 q^{51} + 414 i q^{53} + ( 140 - 70 i ) q^{55} + 360 i q^{57} -266 q^{59} -838 q^{61} -198 i q^{63} + ( -150 - 300 i ) q^{65} + 248 i q^{67} + 564 q^{69} + 1020 q^{71} -484 i q^{73} + ( 300 + 225 i ) q^{75} -308 i q^{77} + 48 q^{79} + 81 q^{81} -548 i q^{83} + ( -310 - 620 i ) q^{85} -288 i q^{87} + 650 q^{89} -660 q^{91} + 552 i q^{93} + ( -1200 + 600 i ) q^{95} -1816 i q^{97} + 126 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 20q^{5} - 18q^{9} + O(q^{10})$$ $$2q - 20q^{5} - 18q^{9} - 28q^{11} - 30q^{15} + 240q^{19} - 132q^{21} + 150q^{25} - 192q^{29} + 368q^{31} - 220q^{35} - 180q^{39} + 260q^{41} + 180q^{45} - 282q^{49} - 372q^{51} + 280q^{55} - 532q^{59} - 1676q^{61} - 300q^{65} + 1128q^{69} + 2040q^{71} + 600q^{75} + 96q^{79} + 162q^{81} - 620q^{85} + 1300q^{89} - 1320q^{91} - 2400q^{95} + 252q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$31$$ $$37$$ $$41$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
 − 1.00000i 1.00000i
0 3.00000i 0 −10.0000 5.00000i 0 22.0000i 0 −9.00000 0
49.2 0 3.00000i 0 −10.0000 + 5.00000i 0 22.0000i 0 −9.00000 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
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 60.4.d.a 2
3.b odd 2 1 180.4.d.b 2
4.b odd 2 1 240.4.f.a 2
5.b even 2 1 inner 60.4.d.a 2
5.c odd 4 1 300.4.a.d 1
5.c odd 4 1 300.4.a.f 1
8.b even 2 1 960.4.f.j 2
8.d odd 2 1 960.4.f.i 2
12.b even 2 1 720.4.f.h 2
15.d odd 2 1 180.4.d.b 2
15.e even 4 1 900.4.a.d 1
15.e even 4 1 900.4.a.o 1
20.d odd 2 1 240.4.f.a 2
20.e even 4 1 1200.4.a.q 1
20.e even 4 1 1200.4.a.w 1
40.e odd 2 1 960.4.f.i 2
40.f even 2 1 960.4.f.j 2
60.h even 2 1 720.4.f.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.4.d.a 2 1.a even 1 1 trivial
60.4.d.a 2 5.b even 2 1 inner
180.4.d.b 2 3.b odd 2 1
180.4.d.b 2 15.d odd 2 1
240.4.f.a 2 4.b odd 2 1
240.4.f.a 2 20.d odd 2 1
300.4.a.d 1 5.c odd 4 1
300.4.a.f 1 5.c odd 4 1
720.4.f.h 2 12.b even 2 1
720.4.f.h 2 60.h even 2 1
900.4.a.d 1 15.e even 4 1
900.4.a.o 1 15.e even 4 1
960.4.f.i 2 8.d odd 2 1
960.4.f.i 2 40.e odd 2 1
960.4.f.j 2 8.b even 2 1
960.4.f.j 2 40.f even 2 1
1200.4.a.q 1 20.e even 4 1
1200.4.a.w 1 20.e even 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(60, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$9 + T^{2}$$
$5$ $$125 + 20 T + T^{2}$$
$7$ $$484 + T^{2}$$
$11$ $$( 14 + T )^{2}$$
$13$ $$900 + T^{2}$$
$17$ $$3844 + T^{2}$$
$19$ $$( -120 + T )^{2}$$
$23$ $$35344 + T^{2}$$
$29$ $$( 96 + T )^{2}$$
$31$ $$( -184 + T )^{2}$$
$37$ $$164836 + T^{2}$$
$41$ $$( -130 + T )^{2}$$
$43$ $$21904 + T^{2}$$
$47$ $$200704 + T^{2}$$
$53$ $$171396 + T^{2}$$
$59$ $$( 266 + T )^{2}$$
$61$ $$( 838 + T )^{2}$$
$67$ $$61504 + T^{2}$$
$71$ $$( -1020 + T )^{2}$$
$73$ $$234256 + T^{2}$$
$79$ $$( -48 + T )^{2}$$
$83$ $$300304 + T^{2}$$
$89$ $$( -650 + T )^{2}$$
$97$ $$3297856 + T^{2}$$