# Properties

 Label 60.2.h.b Level $60$ Weight $2$ Character orbit 60.h Analytic conductor $0.479$ 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: $$k$$ $$=$$ $$2$$ 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: $$\Q(\sqrt{-3}, \sqrt{5})$$ Defining polynomial: $$x^{4} - x^{3} + 2x^{2} + x + 1$$ x^4 - x^3 + 2*x^2 + x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + (\beta_{2} - \beta_1) q^{3} + \beta_{3} q^{4} + (\beta_{2} + \beta_1) q^{5} + ( - \beta_{3} + 2) q^{6} + ( - \beta_{2} + 2 \beta_1) q^{8} - 3 q^{9}+O(q^{10})$$ q - b2 * q^2 + (b2 - b1) * q^3 + b3 * q^4 + (b2 + b1) * q^5 + (-b3 + 2) * q^6 + (-b2 + 2*b1) * q^8 - 3 * q^9 $$q - \beta_{2} q^{2} + (\beta_{2} - \beta_1) q^{3} + \beta_{3} q^{4} + (\beta_{2} + \beta_1) q^{5} + ( - \beta_{3} + 2) q^{6} + ( - \beta_{2} + 2 \beta_1) q^{8} - 3 q^{9} + ( - \beta_{3} - 2) q^{10} + ( - \beta_{2} - 2 \beta_1) q^{12} + (2 \beta_{3} - 1) q^{15} + (\beta_{3} - 4) q^{16} + ( - 2 \beta_{2} - 2 \beta_1) q^{17} + 3 \beta_{2} q^{18} + ( - 4 \beta_{3} + 2) q^{19} + (3 \beta_{2} - 2 \beta_1) q^{20} + ( - 2 \beta_{2} + 2 \beta_1) q^{23} + (\beta_{3} + 4) q^{24} + 5 q^{25} + ( - 3 \beta_{2} + 3 \beta_1) q^{27} + ( - \beta_{2} + 4 \beta_1) q^{30} + (4 \beta_{3} - 2) q^{31} + (3 \beta_{2} + 2 \beta_1) q^{32} + (2 \beta_{3} + 4) q^{34} - 3 \beta_{3} q^{36} + (2 \beta_{2} - 8 \beta_1) q^{38} + ( - 3 \beta_{3} + 4) q^{40} + ( - 3 \beta_{2} - 3 \beta_1) q^{45} + (2 \beta_{3} - 4) q^{46} + (6 \beta_{2} - 6 \beta_1) q^{47} + ( - 5 \beta_{2} + 2 \beta_1) q^{48} - 7 q^{49} - 5 \beta_{2} q^{50} + ( - 4 \beta_{3} + 2) q^{51} + (2 \beta_{2} + 2 \beta_1) q^{53} + (3 \beta_{3} - 6) q^{54} + (6 \beta_{2} + 6 \beta_1) q^{57} + (\beta_{3} - 8) q^{60} - 2 q^{61} + ( - 2 \beta_{2} + 8 \beta_1) q^{62} + ( - 3 \beta_{3} - 4) q^{64} + ( - 6 \beta_{2} + 4 \beta_1) q^{68} + 6 q^{69} + (3 \beta_{2} - 6 \beta_1) q^{72} + (5 \beta_{2} - 5 \beta_1) q^{75} + ( - 2 \beta_{3} + 16) q^{76} + (4 \beta_{3} - 2) q^{79} + ( - \beta_{2} - 6 \beta_1) q^{80} + 9 q^{81} + (2 \beta_{2} - 2 \beta_1) q^{83} - 10 q^{85} + (3 \beta_{3} + 6) q^{90} + (2 \beta_{2} + 4 \beta_1) q^{92} + ( - 6 \beta_{2} - 6 \beta_1) q^{93} + ( - 6 \beta_{3} + 12) q^{94} + ( - 10 \beta_{2} + 10 \beta_1) q^{95} + (5 \beta_{3} - 4) q^{96} + 7 \beta_{2} q^{98}+O(q^{100})$$ q - b2 * q^2 + (b2 - b1) * q^3 + b3 * q^4 + (b2 + b1) * q^5 + (-b3 + 2) * q^6 + (-b2 + 2*b1) * q^8 - 3 * q^9 + (-b3 - 2) * q^10 + (-b2 - 2*b1) * q^12 + (2*b3 - 1) * q^15 + (b3 - 4) * q^16 + (-2*b2 - 2*b1) * q^17 + 3*b2 * q^18 + (-4*b3 + 2) * q^19 + (3*b2 - 2*b1) * q^20 + (-2*b2 + 2*b1) * q^23 + (b3 + 4) * q^24 + 5 * q^25 + (-3*b2 + 3*b1) * q^27 + (-b2 + 4*b1) * q^30 + (4*b3 - 2) * q^31 + (3*b2 + 2*b1) * q^32 + (2*b3 + 4) * q^34 - 3*b3 * q^36 + (2*b2 - 8*b1) * q^38 + (-3*b3 + 4) * q^40 + (-3*b2 - 3*b1) * q^45 + (2*b3 - 4) * q^46 + (6*b2 - 6*b1) * q^47 + (-5*b2 + 2*b1) * q^48 - 7 * q^49 - 5*b2 * q^50 + (-4*b3 + 2) * q^51 + (2*b2 + 2*b1) * q^53 + (3*b3 - 6) * q^54 + (6*b2 + 6*b1) * q^57 + (b3 - 8) * q^60 - 2 * q^61 + (-2*b2 + 8*b1) * q^62 + (-3*b3 - 4) * q^64 + (-6*b2 + 4*b1) * q^68 + 6 * q^69 + (3*b2 - 6*b1) * q^72 + (5*b2 - 5*b1) * q^75 + (-2*b3 + 16) * q^76 + (4*b3 - 2) * q^79 + (-b2 - 6*b1) * q^80 + 9 * q^81 + (2*b2 - 2*b1) * q^83 - 10 * q^85 + (3*b3 + 6) * q^90 + (2*b2 + 4*b1) * q^92 + (-6*b2 - 6*b1) * q^93 + (-6*b3 + 12) * q^94 + (-10*b2 + 10*b1) * q^95 + (5*b3 - 4) * q^96 + 7*b2 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{4} + 6 q^{6} - 12 q^{9}+O(q^{10})$$ 4 * q + 2 * q^4 + 6 * q^6 - 12 * q^9 $$4 q + 2 q^{4} + 6 q^{6} - 12 q^{9} - 10 q^{10} - 14 q^{16} + 18 q^{24} + 20 q^{25} + 20 q^{34} - 6 q^{36} + 10 q^{40} - 12 q^{46} - 28 q^{49} - 18 q^{54} - 30 q^{60} - 8 q^{61} - 22 q^{64} + 24 q^{69} + 60 q^{76} + 36 q^{81} - 40 q^{85} + 30 q^{90} + 36 q^{94} - 6 q^{96}+O(q^{100})$$ 4 * q + 2 * q^4 + 6 * q^6 - 12 * q^9 - 10 * q^10 - 14 * q^16 + 18 * q^24 + 20 * q^25 + 20 * q^34 - 6 * q^36 + 10 * q^40 - 12 * q^46 - 28 * q^49 - 18 * q^54 - 30 * q^60 - 8 * q^61 - 22 * q^64 + 24 * q^69 + 60 * q^76 + 36 * q^81 - 40 * q^85 + 30 * q^90 + 36 * q^94 - 6 * q^96

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

 $$\beta_{1}$$ $$=$$ $$\nu^{2} - \nu + 1$$ v^2 - v + 1 $$\beta_{2}$$ $$=$$ $$\nu^{3} - \nu^{2} + \nu + 1$$ v^3 - v^2 + v + 1 $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} + 3\nu + 1$$ v^3 - v^2 + 3*v + 1
 $$\nu$$ $$=$$ $$( \beta_{3} - \beta_{2} ) / 2$$ (b3 - b2) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - \beta_{2} + 2\beta _1 - 2 ) / 2$$ (b3 - b2 + 2*b1 - 2) / 2 $$\nu^{3}$$ $$=$$ $$\beta_{2} + \beta _1 - 2$$ b2 + b1 - 2

## 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
 −0.309017 + 0.535233i −0.309017 − 0.535233i 0.809017 − 1.40126i 0.809017 + 1.40126i
−1.11803 0.866025i 1.73205i 0.500000 + 1.93649i 2.23607 1.50000 1.93649i 0 1.11803 2.59808i −3.00000 −2.50000 1.93649i
59.2 −1.11803 + 0.866025i 1.73205i 0.500000 1.93649i 2.23607 1.50000 + 1.93649i 0 1.11803 + 2.59808i −3.00000 −2.50000 + 1.93649i
59.3 1.11803 0.866025i 1.73205i 0.500000 1.93649i −2.23607 1.50000 + 1.93649i 0 −1.11803 2.59808i −3.00000 −2.50000 + 1.93649i
59.4 1.11803 + 0.866025i 1.73205i 0.500000 + 1.93649i −2.23607 1.50000 1.93649i 0 −1.11803 + 2.59808i −3.00000 −2.50000 1.93649i
 $$n$$: 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 60.2.h.b 4
3.b odd 2 1 inner 60.2.h.b 4
4.b odd 2 1 inner 60.2.h.b 4
5.b even 2 1 inner 60.2.h.b 4
5.c odd 4 2 300.2.e.a 4
8.b even 2 1 960.2.o.a 4
8.d odd 2 1 960.2.o.a 4
12.b even 2 1 inner 60.2.h.b 4
15.d odd 2 1 CM 60.2.h.b 4
15.e even 4 2 300.2.e.a 4
20.d odd 2 1 inner 60.2.h.b 4
20.e even 4 2 300.2.e.a 4
24.f even 2 1 960.2.o.a 4
24.h odd 2 1 960.2.o.a 4
40.e odd 2 1 960.2.o.a 4
40.f even 2 1 960.2.o.a 4
60.h even 2 1 inner 60.2.h.b 4
60.l odd 4 2 300.2.e.a 4
120.i odd 2 1 960.2.o.a 4
120.m even 2 1 960.2.o.a 4

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{2} + 4$$
$3$ $$(T^{2} + 3)^{2}$$
$5$ $$(T^{2} - 5)^{2}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$(T^{2} - 20)^{2}$$
$19$ $$(T^{2} + 60)^{2}$$
$23$ $$(T^{2} + 12)^{2}$$
$29$ $$T^{4}$$
$31$ $$(T^{2} + 60)^{2}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$(T^{2} + 108)^{2}$$
$53$ $$(T^{2} - 20)^{2}$$
$59$ $$T^{4}$$
$61$ $$(T + 2)^{4}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$(T^{2} + 60)^{2}$$
$83$ $$(T^{2} + 12)^{2}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$