# Properties

 Label 60.4.i.b Level $60$ Weight $4$ Character orbit 60.i Analytic conductor $3.540$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$3.54011460034$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.370150560000.7 Defining polynomial: $$x^{8} + 3 x^{6} + 131 x^{4} + 705 x^{2} + 1600$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{5} q^{3} + ( \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{5} + ( -10 + 10 \beta_{1} + \beta_{3} - \beta_{6} ) q^{7} + ( 12 \beta_{1} + 3 \beta_{2} ) q^{9} +O(q^{10})$$ $$q -\beta_{5} q^{3} + ( \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{5} + ( -10 + 10 \beta_{1} + \beta_{3} - \beta_{6} ) q^{7} + ( 12 \beta_{1} + 3 \beta_{2} ) q^{9} + ( -5 \beta_{3} + 4 \beta_{4} - 5 \beta_{5} - 5 \beta_{6} + 5 \beta_{7} ) q^{11} + ( 15 + 15 \beta_{1} - 2 \beta_{5} - 2 \beta_{7} ) q^{13} + ( -15 - 15 \beta_{1} + 3 \beta_{2} + 9 \beta_{3} - 3 \beta_{4} + 4 \beta_{6} ) q^{15} + ( -5 \beta_{2} - 4 \beta_{3} - 5 \beta_{4} - 4 \beta_{6} ) q^{17} + ( -52 \beta_{1} - 5 \beta_{3} - 5 \beta_{5} + 5 \beta_{6} - 5 \beta_{7} ) q^{19} + ( 39 - 3 \beta_{4} + 10 \beta_{5} + 10 \beta_{6} ) q^{21} + ( -10 \beta_{2} + 10 \beta_{4} + 9 \beta_{5} - 9 \beta_{7} ) q^{23} + ( 5 + 10 \beta_{3} + 10 \beta_{5} - 10 \beta_{6} + 10 \beta_{7} ) q^{25} + ( 27 \beta_{3} + 24 \beta_{6} ) q^{27} + ( -14 \beta_{2} - 20 \beta_{3} + 20 \beta_{5} - 20 \beta_{6} - 20 \beta_{7} ) q^{29} + ( -56 - 20 \beta_{3} + 20 \beta_{5} + 20 \beta_{6} + 20 \beta_{7} ) q^{31} + ( -75 - 75 \beta_{1} + 15 \beta_{2} - 15 \beta_{4} + 16 \beta_{5} - 36 \beta_{7} ) q^{33} + ( -16 \beta_{2} + 20 \beta_{3} + 4 \beta_{4} - 13 \beta_{5} + 20 \beta_{6} + 13 \beta_{7} ) q^{35} + ( 75 - 75 \beta_{1} + 18 \beta_{3} - 18 \beta_{6} ) q^{37} + ( 78 \beta_{1} + 6 \beta_{2} - 15 \beta_{5} + 15 \beta_{6} ) q^{39} + ( -40 \beta_{3} - 8 \beta_{4} - 40 \beta_{5} - 40 \beta_{6} + 40 \beta_{7} ) q^{41} + ( 60 + 60 \beta_{1} - 37 \beta_{5} - 37 \beta_{7} ) q^{43} + ( 195 + 27 \beta_{3} + 12 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} + 27 \beta_{7} ) q^{45} + ( 30 \beta_{2} - 3 \beta_{3} + 30 \beta_{4} - 3 \beta_{6} ) q^{47} + ( 65 \beta_{1} - 20 \beta_{3} - 20 \beta_{5} + 20 \beta_{6} - 20 \beta_{7} ) q^{49} + ( -60 - 45 \beta_{3} - 12 \beta_{4} - 20 \beta_{5} - 20 \beta_{6} + 45 \beta_{7} ) q^{51} + ( 25 \beta_{2} - 25 \beta_{4} + 6 \beta_{5} - 6 \beta_{7} ) q^{53} + ( -300 + 260 \beta_{1} + 45 \beta_{3} + 5 \beta_{5} - 45 \beta_{6} + 5 \beta_{7} ) q^{55} + ( -195 + 195 \beta_{1} + 15 \beta_{2} + 15 \beta_{4} - 52 \beta_{6} ) q^{57} + 28 \beta_{2} q^{59} + ( -66 - 20 \beta_{3} + 20 \beta_{5} + 20 \beta_{6} + 20 \beta_{7} ) q^{61} + ( -120 - 120 \beta_{1} - 30 \beta_{2} + 30 \beta_{4} - 51 \beta_{5} + 27 \beta_{7} ) q^{63} + ( 27 \beta_{2} + 26 \beta_{3} + 3 \beta_{4} - 30 \beta_{5} + 26 \beta_{6} + 30 \beta_{7} ) q^{65} + ( 260 - 260 \beta_{1} - 9 \beta_{3} + 9 \beta_{6} ) q^{67} + ( 135 \beta_{1} - 27 \beta_{2} - 90 \beta_{3} + 40 \beta_{5} - 40 \beta_{6} - 90 \beta_{7} ) q^{69} + ( 25 \beta_{3} - 4 \beta_{4} + 25 \beta_{5} + 25 \beta_{6} - 25 \beta_{7} ) q^{71} + ( 325 + 325 \beta_{1} + 4 \beta_{5} + 4 \beta_{7} ) q^{73} + ( 390 - 390 \beta_{1} - 30 \beta_{2} - 30 \beta_{4} - 5 \beta_{5} ) q^{75} + ( -70 \beta_{2} + 152 \beta_{3} - 70 \beta_{4} + 152 \beta_{6} ) q^{77} + ( -636 \beta_{1} + 25 \beta_{3} + 25 \beta_{5} - 25 \beta_{6} + 25 \beta_{7} ) q^{79} + ( 441 + 72 \beta_{4} ) q^{81} + ( 143 \beta_{5} - 143 \beta_{7} ) q^{83} + ( -445 - 205 \beta_{1} - 50 \beta_{3} + 20 \beta_{5} + 50 \beta_{6} + 20 \beta_{7} ) q^{85} + ( -300 + 300 \beta_{1} - 60 \beta_{2} - 126 \beta_{3} - 60 \beta_{4} - 56 \beta_{6} ) q^{87} + ( 14 \beta_{2} + 60 \beta_{3} - 60 \beta_{5} + 60 \beta_{6} + 60 \beta_{7} ) q^{89} + ( -144 - 5 \beta_{3} + 5 \beta_{5} + 5 \beta_{6} + 5 \beta_{7} ) q^{91} + ( -780 - 780 \beta_{1} - 60 \beta_{2} + 60 \beta_{4} + 56 \beta_{5} ) q^{93} + ( 60 \beta_{2} + 13 \beta_{3} - 52 \beta_{4} + 117 \beta_{5} + 13 \beta_{6} - 117 \beta_{7} ) q^{95} + ( -15 + 15 \beta_{1} + 16 \beta_{3} - 16 \beta_{6} ) q^{97} + ( 780 \beta_{1} - 48 \beta_{2} + 135 \beta_{3} + 15 \beta_{5} - 15 \beta_{6} + 135 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 80q^{7} + O(q^{10})$$ $$8q - 80q^{7} + 120q^{13} - 120q^{15} + 312q^{21} + 40q^{25} - 448q^{31} - 600q^{33} + 600q^{37} + 480q^{43} + 1560q^{45} - 480q^{51} - 2400q^{55} - 1560q^{57} - 528q^{61} - 960q^{63} + 2080q^{67} + 2600q^{73} + 3120q^{75} + 3528q^{81} - 3560q^{85} - 2400q^{87} - 1152q^{91} - 6240q^{93} - 120q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 3 x^{6} + 131 x^{4} + 705 x^{2} + 1600$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-3 \nu^{7} - 14 \nu^{5} - 283 \nu^{3} - 2320 \nu$$$$)/3200$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{6} - 18 \nu^{4} - 81 \nu^{2} - 1440$$$$)/160$$ $$\beta_{3}$$ $$=$$ $$($$$$6 \nu^{7} - 55 \nu^{6} + 28 \nu^{5} + 10 \nu^{4} + 966 \nu^{3} - 6655 \nu^{2} + 1840 \nu - 17600$$$$)/3200$$ $$\beta_{4}$$ $$=$$ $$($$$$3 \nu^{7} + 14 \nu^{5} + 443 \nu^{3} + 3760 \nu$$$$)/640$$ $$\beta_{5}$$ $$=$$ $$($$$$-6 \nu^{7} - 55 \nu^{6} - 28 \nu^{5} + 10 \nu^{4} - 966 \nu^{3} - 6655 \nu^{2} - 1840 \nu - 17600$$$$)/3200$$ $$\beta_{6}$$ $$=$$ $$($$$$-13 \nu^{7} + 10 \nu^{6} + 6 \nu^{5} - 20 \nu^{4} - 1693 \nu^{3} + 610 \nu^{2} - 3120 \nu + 1600$$$$)/1600$$ $$\beta_{7}$$ $$=$$ $$($$$$-13 \nu^{7} - 10 \nu^{6} + 6 \nu^{5} + 20 \nu^{4} - 1693 \nu^{3} - 610 \nu^{2} - 3120 \nu - 1600$$$$)/1600$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} - \beta_{3} + \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$5 \beta_{7} - 5 \beta_{6} - 2 \beta_{5} - 2 \beta_{3} + \beta_{2} - 3$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-9 \beta_{5} + 7 \beta_{4} + 9 \beta_{3} + 71 \beta_{1}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$11 \beta_{7} - 11 \beta_{6} + 2 \beta_{5} + 2 \beta_{3} - 33 \beta_{2} - 253$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$48 \beta_{7} + 48 \beta_{6} - 167 \beta_{5} - 55 \beta_{4} + 167 \beta_{3} - 439 \beta_{1}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$-603 \beta_{7} + 603 \beta_{6} + 126 \beta_{5} + 126 \beta_{3} - 127 \beta_{2} - 963$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$-224 \beta_{7} - 224 \beta_{6} + 855 \beta_{5} - 1177 \beta_{4} - 855 \beta_{3} - 9689 \beta_{1}$$$$)/4$$

## Character values

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

 $$n$$ $$31$$ $$37$$ $$41$$ $$\chi(n)$$ $$1$$ $$-\beta_{1}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 2.52950 − 2.26556i 0.593004 + 1.76556i −0.593004 + 1.76556i −2.52950 − 2.26556i 2.52950 + 2.26556i 0.593004 − 1.76556i −0.593004 − 1.76556i −2.52950 + 2.26556i
0 −5.05899 + 1.18601i 0 8.06226 7.74597i 0 −16.2450 16.2450i 0 24.1868 12.0000i 0
17.2 0 −1.18601 + 5.05899i 0 −8.06226 + 7.74597i 0 −16.2450 16.2450i 0 −24.1868 12.0000i 0
17.3 0 1.18601 5.05899i 0 −8.06226 7.74597i 0 −3.75500 3.75500i 0 −24.1868 12.0000i 0
17.4 0 5.05899 1.18601i 0 8.06226 + 7.74597i 0 −3.75500 3.75500i 0 24.1868 12.0000i 0
53.1 0 −5.05899 1.18601i 0 8.06226 + 7.74597i 0 −16.2450 + 16.2450i 0 24.1868 + 12.0000i 0
53.2 0 −1.18601 5.05899i 0 −8.06226 7.74597i 0 −16.2450 + 16.2450i 0 −24.1868 + 12.0000i 0
53.3 0 1.18601 + 5.05899i 0 −8.06226 + 7.74597i 0 −3.75500 + 3.75500i 0 −24.1868 + 12.0000i 0
53.4 0 5.05899 + 1.18601i 0 8.06226 7.74597i 0 −3.75500 + 3.75500i 0 24.1868 + 12.0000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 53.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 60.4.i.b 8
3.b odd 2 1 inner 60.4.i.b 8
4.b odd 2 1 240.4.v.b 8
5.b even 2 1 300.4.i.f 8
5.c odd 4 1 inner 60.4.i.b 8
5.c odd 4 1 300.4.i.f 8
12.b even 2 1 240.4.v.b 8
15.d odd 2 1 300.4.i.f 8
15.e even 4 1 inner 60.4.i.b 8
15.e even 4 1 300.4.i.f 8
20.e even 4 1 240.4.v.b 8
60.l odd 4 1 240.4.v.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.4.i.b 8 1.a even 1 1 trivial
60.4.i.b 8 3.b odd 2 1 inner
60.4.i.b 8 5.c odd 4 1 inner
60.4.i.b 8 15.e even 4 1 inner
240.4.v.b 8 4.b odd 2 1
240.4.v.b 8 12.b even 2 1
240.4.v.b 8 20.e even 4 1
240.4.v.b 8 60.l odd 4 1
300.4.i.f 8 5.b even 2 1
300.4.i.f 8 5.c odd 4 1
300.4.i.f 8 15.d odd 2 1
300.4.i.f 8 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} + 40 T_{7}^{3} + 800 T_{7}^{2} + 4880 T_{7} + 14884$$ acting on $$S_{4}^{\mathrm{new}}(60, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$531441 - 882 T^{4} + T^{8}$$
$5$ $$( 15625 - 10 T^{2} + T^{4} )^{2}$$
$7$ $$( 14884 + 4880 T + 800 T^{2} + 40 T^{3} + T^{4} )^{2}$$
$11$ $$( 211600 + 5080 T^{2} + T^{4} )^{2}$$
$13$ $$( 19044 - 8280 T + 1800 T^{2} - 60 T^{3} + T^{4} )^{2}$$
$17$ $$58873394410000 + 40305800 T^{4} + T^{8}$$
$19$ $$( 1430416 + 13208 T^{2} + T^{4} )^{2}$$
$23$ $$12482453280010000 + 728889800 T^{4} + T^{8}$$
$29$ $$( 126787600 - 73480 T^{2} + T^{4} )^{2}$$
$31$ $$( -59264 + 112 T + T^{2} )^{4}$$
$37$ $$( 196616484 + 4206600 T + 45000 T^{2} - 300 T^{3} + T^{4} )^{2}$$
$41$ $$( 8434585600 + 200320 T^{2} + T^{4} )^{2}$$
$43$ $$( 9916574724 + 23899680 T + 28800 T^{2} - 240 T^{3} + T^{4} )^{2}$$
$47$ $$18\!\cdots\!00$$$$+ 27757225800 T^{4} + T^{8}$$
$53$ $$41309271332995210000 + 14258457800 T^{4} + T^{8}$$
$59$ $$( -50960 + T^{2} )^{4}$$
$61$ $$( -58044 + 132 T + T^{2} )^{4}$$
$67$ $$( 16610569924 - 134037280 T + 540800 T^{2} - 1040 T^{3} + T^{4} )^{2}$$
$71$ $$( 1329331600 + 77080 T^{2} + T^{4} )^{2}$$
$73$ $$( 44100840004 - 273002600 T + 845000 T^{2} - 1300 T^{3} + T^{4} )^{2}$$
$79$ $$( 94246544016 + 1003992 T^{2} + T^{4} )^{2}$$
$83$ $$( 376345440900 + T^{4} )^{2}$$
$89$ $$( 41314627600 - 457480 T^{2} + T^{4} )^{2}$$
$97$ $$( 380952324 - 1171080 T + 1800 T^{2} + 60 T^{3} + T^{4} )^{2}$$