# Properties

 Label 60.6.h.b Level $60$ Weight $6$ Character orbit 60.h Analytic conductor $9.623$ Analytic rank $0$ Dimension $4$ CM discriminant -15 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$60 = 2^{2} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 60.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.62302918878$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{5})$$ Defining polynomial: $$x^{4} - x^{3} + 2 x^{2} + x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}\cdot 5^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} + 9 \beta_{1} q^{3} + ( 31 - \beta_{3} ) q^{4} + ( -5 \beta_{1} - 10 \beta_{2} ) q^{5} + ( -9 - 9 \beta_{3} ) q^{6} + ( 32 \beta_{1} - 29 \beta_{2} ) q^{8} -243 q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{2} + 9 \beta_{1} q^{3} + ( 31 - \beta_{3} ) q^{4} + ( -5 \beta_{1} - 10 \beta_{2} ) q^{5} + ( -9 - 9 \beta_{3} ) q^{6} + ( 32 \beta_{1} - 29 \beta_{2} ) q^{8} -243 q^{9} + ( 315 - 5 \beta_{3} ) q^{10} + ( 288 \beta_{1} + 27 \beta_{2} ) q^{12} + ( 45 - 90 \beta_{3} ) q^{15} + ( 867 - 61 \beta_{3} ) q^{16} + ( 58 \beta_{1} + 116 \beta_{2} ) q^{17} + 243 \beta_{2} q^{18} + ( 118 - 236 \beta_{3} ) q^{19} + ( 160 \beta_{1} - 305 \beta_{2} ) q^{20} -2818 \beta_{1} q^{23} + ( -1125 - 261 \beta_{3} ) q^{24} + 3125 q^{25} -2187 \beta_{1} q^{27} + ( 2880 \beta_{1} + 135 \beta_{2} ) q^{30} + ( -358 + 716 \beta_{3} ) q^{31} + ( 1952 \beta_{1} - 745 \beta_{2} ) q^{32} + ( -3654 + 58 \beta_{3} ) q^{34} + ( -7533 + 243 \beta_{3} ) q^{36} + ( 7552 \beta_{1} + 354 \beta_{2} ) q^{38} + ( 9295 - 465 \beta_{3} ) q^{40} + ( 1215 \beta_{1} + 2430 \beta_{2} ) q^{45} + ( 2818 + 2818 \beta_{3} ) q^{46} -16026 \beta_{1} q^{47} + ( 8352 \beta_{1} + 1647 \beta_{2} ) q^{48} -16807 q^{49} -3125 \beta_{2} q^{50} + ( -522 + 1044 \beta_{3} ) q^{51} + ( -3658 \beta_{1} - 7316 \beta_{2} ) q^{53} + ( 2187 + 2187 \beta_{3} ) q^{54} + ( 3186 \beta_{1} + 6372 \beta_{2} ) q^{57} + ( -7065 - 2745 \beta_{3} ) q^{60} -34802 q^{61} + ( -22912 \beta_{1} - 1074 \beta_{2} ) q^{62} + ( 21143 - 2697 \beta_{3} ) q^{64} + ( -1856 \beta_{1} + 3538 \beta_{2} ) q^{68} + 76086 q^{69} + ( -7776 \beta_{1} + 7047 \beta_{2} ) q^{72} + 28125 \beta_{1} q^{75} + ( -18526 - 7198 \beta_{3} ) q^{76} + ( 4442 - 8884 \beta_{3} ) q^{79} + ( 14880 \beta_{1} - 8365 \beta_{2} ) q^{80} + 59049 q^{81} + 59218 \beta_{1} q^{83} -36250 q^{85} + ( -76545 + 1215 \beta_{3} ) q^{90} + ( -90176 \beta_{1} - 8454 \beta_{2} ) q^{92} + ( -9666 \beta_{1} - 19332 \beta_{2} ) q^{93} + ( 16026 + 16026 \beta_{3} ) q^{94} + 73750 \beta_{1} q^{95} + ( -59409 - 6705 \beta_{3} ) q^{96} + 16807 \beta_{2} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 122q^{4} - 54q^{6} - 972q^{9} + O(q^{10})$$ $$4q + 122q^{4} - 54q^{6} - 972q^{9} + 1250q^{10} + 3346q^{16} - 5022q^{24} + 12500q^{25} - 14500q^{34} - 29646q^{36} + 36250q^{40} + 16908q^{46} - 67228q^{49} + 13122q^{54} - 33750q^{60} - 139208q^{61} + 79178q^{64} + 304344q^{69} - 88500q^{76} + 236196q^{81} - 145000q^{85} - 303750q^{90} + 96156q^{94} - 251046q^{96} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 2 x^{2} + x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$-\nu^{3} + 2 \nu^{2} - 2 \nu$$ $$\beta_{2}$$ $$=$$ $$-2 \nu^{3} - \nu^{2} + \nu - 5$$ $$\beta_{3}$$ $$=$$ $$5 \nu^{3} - 5 \nu^{2} + 15 \nu + 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} + 3 \beta_{1} + 2$$$$)/10$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - \beta_{2} + 7 \beta_{1} - 8$$$$)/10$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{2} - \beta_{1} - 10$$$$)/5$$

## 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
 0.809017 − 1.40126i 0.809017 + 1.40126i −0.309017 + 0.535233i −0.309017 − 0.535233i
−5.59017 0.866025i 15.5885i 30.5000 + 9.68246i −55.9017 −13.5000 + 87.1421i 0 −162.115 80.5404i −243.000 312.500 + 48.4123i
59.2 −5.59017 + 0.866025i 15.5885i 30.5000 9.68246i −55.9017 −13.5000 87.1421i 0 −162.115 + 80.5404i −243.000 312.500 48.4123i
59.3 5.59017 0.866025i 15.5885i 30.5000 9.68246i 55.9017 −13.5000 87.1421i 0 162.115 80.5404i −243.000 312.500 48.4123i
59.4 5.59017 + 0.866025i 15.5885i 30.5000 + 9.68246i 55.9017 −13.5000 + 87.1421i 0 162.115 + 80.5404i −243.000 312.500 + 48.4123i
 $$n$$: 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
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
20.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 60.6.h.b 4
3.b odd 2 1 inner 60.6.h.b 4
4.b odd 2 1 inner 60.6.h.b 4
5.b even 2 1 inner 60.6.h.b 4
12.b even 2 1 inner 60.6.h.b 4
15.d odd 2 1 CM 60.6.h.b 4
20.d odd 2 1 inner 60.6.h.b 4
60.h even 2 1 inner 60.6.h.b 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}$$ acting on $$S_{6}^{\mathrm{new}}(60, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1024 - 61 T^{2} + T^{4}$$
$3$ $$( 243 + T^{2} )^{2}$$
$5$ $$( -3125 + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$( -420500 + T^{2} )^{2}$$
$19$ $$( 5221500 + T^{2} )^{2}$$
$23$ $$( 23823372 + T^{2} )^{2}$$
$29$ $$T^{4}$$
$31$ $$( 48061500 + T^{2} )^{2}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$( 770498028 + T^{2} )^{2}$$
$53$ $$( -1672620500 + T^{2} )^{2}$$
$59$ $$T^{4}$$
$61$ $$( 34802 + T )^{4}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$( 7399261500 + T^{2} )^{2}$$
$83$ $$( 10520314572 + T^{2} )^{2}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$