# Properties

 Label 60.4.h.a Level $60$ Weight $4$ Character orbit 60.h Analytic conductor $3.540$ 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: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 60.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.54011460034$$ 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.

 $$f(q)$$ $$=$$ $$q + 2 \beta_{2} q^{2} + ( \beta_{1} - 4 \beta_{2} ) q^{3} -8 q^{4} -5 \beta_{3} q^{5} + ( 14 + 2 \beta_{3} ) q^{6} + ( 22 \beta_{1} - 11 \beta_{2} ) q^{7} -16 \beta_{2} q^{8} + ( -22 - 7 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + 2 \beta_{2} q^{2} + ( \beta_{1} - 4 \beta_{2} ) q^{3} -8 q^{4} -5 \beta_{3} q^{5} + ( 14 + 2 \beta_{3} ) q^{6} + ( 22 \beta_{1} - 11 \beta_{2} ) q^{7} -16 \beta_{2} q^{8} + ( -22 - 7 \beta_{3} ) q^{9} + ( 20 \beta_{1} - 10 \beta_{2} ) q^{10} + ( -8 \beta_{1} + 32 \beta_{2} ) q^{12} + 44 \beta_{3} q^{14} + ( -35 \beta_{1} + 5 \beta_{2} ) q^{15} + 64 q^{16} + ( 28 \beta_{1} - 58 \beta_{2} ) q^{18} + 40 \beta_{3} q^{20} + ( 55 - 77 \beta_{3} ) q^{21} + 67 \beta_{2} q^{23} + ( -112 - 16 \beta_{3} ) q^{24} -125 q^{25} + ( -71 \beta_{1} + 95 \beta_{2} ) q^{27} + ( -176 \beta_{1} + 88 \beta_{2} ) q^{28} -28 \beta_{3} q^{29} + ( 50 - 70 \beta_{3} ) q^{30} + 128 \beta_{2} q^{32} -275 \beta_{2} q^{35} + ( 176 + 56 \beta_{3} ) q^{36} + ( -160 \beta_{1} + 80 \beta_{2} ) q^{40} + 206 \beta_{3} q^{41} + ( 308 \beta_{1} - 44 \beta_{2} ) q^{42} + ( 238 \beta_{1} - 119 \beta_{2} ) q^{43} + ( -175 + 110 \beta_{3} ) q^{45} -268 q^{46} + 301 \beta_{2} q^{47} + ( 64 \beta_{1} - 256 \beta_{2} ) q^{48} + 867 q^{49} -250 \beta_{2} q^{50} + ( -238 - 142 \beta_{3} ) q^{54} -352 \beta_{3} q^{56} + ( 112 \beta_{1} - 56 \beta_{2} ) q^{58} + ( 280 \beta_{1} - 40 \beta_{2} ) q^{60} -952 q^{61} + ( -484 \beta_{1} - 143 \beta_{2} ) q^{63} -512 q^{64} + ( 490 \beta_{1} - 245 \beta_{2} ) q^{67} + ( 469 + 67 \beta_{3} ) q^{69} + 1100 q^{70} + ( -224 \beta_{1} + 464 \beta_{2} ) q^{72} + ( -125 \beta_{1} + 500 \beta_{2} ) q^{75} -320 \beta_{3} q^{80} + ( 239 + 308 \beta_{3} ) q^{81} + ( -824 \beta_{1} + 412 \beta_{2} ) q^{82} -77 \beta_{2} q^{83} + ( -440 + 616 \beta_{3} ) q^{84} + 476 \beta_{3} q^{86} + ( -196 \beta_{1} + 28 \beta_{2} ) q^{87} -424 \beta_{3} q^{89} + ( -440 \beta_{1} - 130 \beta_{2} ) q^{90} -536 \beta_{2} q^{92} -1204 q^{94} + ( 896 + 128 \beta_{3} ) q^{96} + 1734 \beta_{2} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 32q^{4} + 56q^{6} - 88q^{9} + O(q^{10})$$ $$4q - 32q^{4} + 56q^{6} - 88q^{9} + 256q^{16} + 220q^{21} - 448q^{24} - 500q^{25} + 200q^{30} + 704q^{36} - 700q^{45} - 1072q^{46} + 3468q^{49} - 952q^{54} - 3808q^{61} - 2048q^{64} + 1876q^{69} + 4400q^{70} + 956q^{81} - 1760q^{84} - 4816q^{94} + 3584q^{96} + O(q^{100})$$

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
2.82843i −1.58114 + 4.94975i −8.00000 11.1803i 14.0000 + 4.47214i −34.7851 22.6274i −22.0000 15.6525i −31.6228
59.2 2.82843i 1.58114 + 4.94975i −8.00000 11.1803i 14.0000 4.47214i 34.7851 22.6274i −22.0000 + 15.6525i 31.6228
59.3 2.82843i −1.58114 4.94975i −8.00000 11.1803i 14.0000 4.47214i −34.7851 22.6274i −22.0000 + 15.6525i −31.6228
59.4 2.82843i 1.58114 4.94975i −8.00000 11.1803i 14.0000 + 4.47214i 34.7851 22.6274i −22.0000 15.6525i 31.6228
 $$n$$: 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 60.4.h.a 4
3.b odd 2 1 inner 60.4.h.a 4
4.b odd 2 1 inner 60.4.h.a 4
5.b even 2 1 inner 60.4.h.a 4
12.b even 2 1 inner 60.4.h.a 4
15.d odd 2 1 inner 60.4.h.a 4
20.d odd 2 1 CM 60.4.h.a 4
60.h even 2 1 inner 60.4.h.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.4.h.a 4 1.a even 1 1 trivial
60.4.h.a 4 3.b odd 2 1 inner
60.4.h.a 4 4.b odd 2 1 inner
60.4.h.a 4 5.b even 2 1 inner
60.4.h.a 4 12.b even 2 1 inner
60.4.h.a 4 15.d odd 2 1 inner
60.4.h.a 4 20.d odd 2 1 CM
60.4.h.a 4 60.h even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} - 1210$$ acting on $$S_{4}^{\mathrm{new}}(60, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 8 + T^{2} )^{2}$$
$3$ $$729 + 44 T^{2} + T^{4}$$
$5$ $$( 125 + T^{2} )^{2}$$
$7$ $$( -1210 + T^{2} )^{2}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$( 8978 + T^{2} )^{2}$$
$29$ $$( 3920 + T^{2} )^{2}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$( 212180 + T^{2} )^{2}$$
$43$ $$( -141610 + T^{2} )^{2}$$
$47$ $$( 181202 + T^{2} )^{2}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$( 952 + T )^{4}$$
$67$ $$( -600250 + T^{2} )^{2}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$( 11858 + T^{2} )^{2}$$
$89$ $$( 898880 + T^{2} )^{2}$$
$97$ $$T^{4}$$