# Properties

 Label 60.2.h.a Level $60$ Weight $2$ Character orbit 60.h Analytic conductor $0.479$ Analytic rank $0$ Dimension $4$ CM discriminant -20 Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$0.479102412128$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-5})$$ Defining polynomial: $$x^{4} - 4 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} -\beta_{1} q^{3} -2 q^{4} -\beta_{3} q^{5} + ( -1 + \beta_{3} ) q^{6} + ( 2 \beta_{1} - \beta_{2} ) q^{7} + 2 \beta_{2} q^{8} + ( 2 + \beta_{3} ) q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{2} -\beta_{1} q^{3} -2 q^{4} -\beta_{3} q^{5} + ( -1 + \beta_{3} ) q^{6} + ( 2 \beta_{1} - \beta_{2} ) q^{7} + 2 \beta_{2} q^{8} + ( 2 + \beta_{3} ) q^{9} + ( -2 \beta_{1} + \beta_{2} ) q^{10} + 2 \beta_{1} q^{12} -2 \beta_{3} q^{14} + ( -\beta_{1} + 3 \beta_{2} ) q^{15} + 4 q^{16} + ( 2 \beta_{1} - 3 \beta_{2} ) q^{18} + 2 \beta_{3} q^{20} + ( -5 - \beta_{3} ) q^{21} + \beta_{2} q^{23} + ( 2 - 2 \beta_{3} ) q^{24} -5 q^{25} + ( -\beta_{1} - 3 \beta_{2} ) q^{27} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{28} + 4 \beta_{3} q^{29} + ( 5 + \beta_{3} ) q^{30} -4 \beta_{2} q^{32} -5 \beta_{2} q^{35} + ( -4 - 2 \beta_{3} ) q^{36} + ( 4 \beta_{1} - 2 \beta_{2} ) q^{40} -2 \beta_{3} q^{41} + ( -2 \beta_{1} + 6 \beta_{2} ) q^{42} + ( 2 \beta_{1} - \beta_{2} ) q^{43} + ( 5 - 2 \beta_{3} ) q^{45} + 2 q^{46} + 7 \beta_{2} q^{47} -4 \beta_{1} q^{48} + 3 q^{49} + 5 \beta_{2} q^{50} + ( -7 + \beta_{3} ) q^{54} + 4 \beta_{3} q^{56} + ( 8 \beta_{1} - 4 \beta_{2} ) q^{58} + ( 2 \beta_{1} - 6 \beta_{2} ) q^{60} + 8 q^{61} + ( 4 \beta_{1} + 3 \beta_{2} ) q^{63} -8 q^{64} + ( -10 \beta_{1} + 5 \beta_{2} ) q^{67} + ( 1 - \beta_{3} ) q^{69} -10 q^{70} + ( -4 \beta_{1} + 6 \beta_{2} ) q^{72} + 5 \beta_{1} q^{75} -4 \beta_{3} q^{80} + ( -1 + 4 \beta_{3} ) q^{81} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{82} -11 \beta_{2} q^{83} + ( 10 + 2 \beta_{3} ) q^{84} -2 \beta_{3} q^{86} + ( 4 \beta_{1} - 12 \beta_{2} ) q^{87} -8 \beta_{3} q^{89} + ( -4 \beta_{1} - 3 \beta_{2} ) q^{90} -2 \beta_{2} q^{92} + 14 q^{94} + ( -4 + 4 \beta_{3} ) q^{96} -3 \beta_{2} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 8q^{4} - 4q^{6} + 8q^{9} + O(q^{10})$$ $$4q - 8q^{4} - 4q^{6} + 8q^{9} + 16q^{16} - 20q^{21} + 8q^{24} - 20q^{25} + 20q^{30} - 16q^{36} + 20q^{45} + 8q^{46} + 12q^{49} - 28q^{54} + 32q^{61} - 32q^{64} + 4q^{69} - 40q^{70} - 4q^{81} + 40q^{84} + 56q^{94} - 16q^{96} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 4 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} - \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} - 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 2$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{2} + \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
59.1
 1.58114 + 0.707107i −1.58114 + 0.707107i 1.58114 − 0.707107i −1.58114 − 0.707107i
1.41421i −1.58114 0.707107i −2.00000 2.23607i −1.00000 + 2.23607i 3.16228 2.82843i 2.00000 + 2.23607i −3.16228
59.2 1.41421i 1.58114 0.707107i −2.00000 2.23607i −1.00000 2.23607i −3.16228 2.82843i 2.00000 2.23607i 3.16228
59.3 1.41421i −1.58114 + 0.707107i −2.00000 2.23607i −1.00000 2.23607i 3.16228 2.82843i 2.00000 2.23607i −3.16228
59.4 1.41421i 1.58114 + 0.707107i −2.00000 2.23607i −1.00000 + 2.23607i −3.16228 2.82843i 2.00000 + 2.23607i 3.16228
 $$n$$: 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
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
15.d odd 2 1 inner
60.h even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 60.2.h.a 4
3.b odd 2 1 inner 60.2.h.a 4
4.b odd 2 1 inner 60.2.h.a 4
5.b even 2 1 inner 60.2.h.a 4
5.c odd 4 2 300.2.e.b 4
8.b even 2 1 960.2.o.c 4
8.d odd 2 1 960.2.o.c 4
12.b even 2 1 inner 60.2.h.a 4
15.d odd 2 1 inner 60.2.h.a 4
15.e even 4 2 300.2.e.b 4
20.d odd 2 1 CM 60.2.h.a 4
20.e even 4 2 300.2.e.b 4
24.f even 2 1 960.2.o.c 4
24.h odd 2 1 960.2.o.c 4
40.e odd 2 1 960.2.o.c 4
40.f even 2 1 960.2.o.c 4
60.h even 2 1 inner 60.2.h.a 4
60.l odd 4 2 300.2.e.b 4
120.i odd 2 1 960.2.o.c 4
120.m even 2 1 960.2.o.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.h.a 4 1.a even 1 1 trivial
60.2.h.a 4 3.b odd 2 1 inner
60.2.h.a 4 4.b odd 2 1 inner
60.2.h.a 4 5.b even 2 1 inner
60.2.h.a 4 12.b even 2 1 inner
60.2.h.a 4 15.d odd 2 1 inner
60.2.h.a 4 20.d odd 2 1 CM
60.2.h.a 4 60.h even 2 1 inner
300.2.e.b 4 5.c odd 4 2
300.2.e.b 4 15.e even 4 2
300.2.e.b 4 20.e even 4 2
300.2.e.b 4 60.l odd 4 2
960.2.o.c 4 8.b even 2 1
960.2.o.c 4 8.d odd 2 1
960.2.o.c 4 24.f even 2 1
960.2.o.c 4 24.h odd 2 1
960.2.o.c 4 40.e odd 2 1
960.2.o.c 4 40.f even 2 1
960.2.o.c 4 120.i odd 2 1
960.2.o.c 4 120.m even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} - 10$$ acting on $$S_{2}^{\mathrm{new}}(60, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 + T^{2} )^{2}$$
$3$ $$9 - 4 T^{2} + T^{4}$$
$5$ $$( 5 + T^{2} )^{2}$$
$7$ $$( -10 + T^{2} )^{2}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$( 2 + T^{2} )^{2}$$
$29$ $$( 80 + T^{2} )^{2}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$( 20 + T^{2} )^{2}$$
$43$ $$( -10 + T^{2} )^{2}$$
$47$ $$( 98 + T^{2} )^{2}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$( -8 + T )^{4}$$
$67$ $$( -250 + T^{2} )^{2}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$( 242 + T^{2} )^{2}$$
$89$ $$( 320 + T^{2} )^{2}$$
$97$ $$T^{4}$$