# Properties

 Label 60.4.j.a Level $60$ Weight $4$ Character orbit 60.j 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.j (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: 8.0.157351936.1 Defining polynomial: $$x^{8} + x^{4} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{2} + \beta_{4} ) q^{2} + 3 \beta_{5} q^{3} + ( 6 \beta_{3} + 2 \beta_{6} ) q^{4} + ( -3 - 2 \beta_{2} + 9 \beta_{3} + \beta_{7} ) q^{5} + ( 3 + 3 \beta_{1} ) q^{6} + ( -5 \beta_{1} - 6 \beta_{4} + 5 \beta_{6} ) q^{7} + ( 20 \beta_{5} + 4 \beta_{7} ) q^{8} + 9 \beta_{3} q^{9} +O(q^{10})$$ $$q + ( \beta_{2} + \beta_{4} ) q^{2} + 3 \beta_{5} q^{3} + ( 6 \beta_{3} + 2 \beta_{6} ) q^{4} + ( -3 - 2 \beta_{2} + 9 \beta_{3} + \beta_{7} ) q^{5} + ( 3 + 3 \beta_{1} ) q^{6} + ( -5 \beta_{1} - 6 \beta_{4} + 5 \beta_{6} ) q^{7} + ( 20 \beta_{5} + 4 \beta_{7} ) q^{8} + 9 \beta_{3} q^{9} + ( -7 + \beta_{1} - 3 \beta_{2} - 14 \beta_{3} - 3 \beta_{4} + 9 \beta_{5} - 2 \beta_{6} + 9 \beta_{7} ) q^{10} + ( -12 \beta_{1} + 5 \beta_{4} - 5 \beta_{5} ) q^{11} + ( 6 \beta_{2} - 18 \beta_{4} ) q^{12} + ( 40 - 40 \beta_{3} + 6 \beta_{7} ) q^{13} + ( -5 \beta_{2} + 6 \beta_{3} + 35 \beta_{4} + 35 \beta_{5} - 6 \beta_{6} - 5 \beta_{7} ) q^{14} + ( -6 \beta_{1} - 27 \beta_{4} - 9 \beta_{5} - 3 \beta_{6} ) q^{15} + ( -8 + 24 \beta_{1} ) q^{16} + ( 30 - 10 \beta_{2} + 30 \beta_{3} ) q^{17} + ( 9 \beta_{5} + 9 \beta_{7} ) q^{18} + ( -30 \beta_{4} - 30 \beta_{5} + 22 \beta_{6} ) q^{19} + ( -54 + 18 \beta_{1} - 6 \beta_{2} - 18 \beta_{3} - 14 \beta_{4} - 28 \beta_{5} - 6 \beta_{6} - 12 \beta_{7} ) q^{20} + ( -18 + 15 \beta_{2} - 15 \beta_{7} ) q^{21} + ( -5 - 5 \beta_{1} - 12 \beta_{2} - 5 \beta_{3} + 84 \beta_{4} + 5 \beta_{6} ) q^{22} + ( -30 \beta_{1} - 38 \beta_{5} - 30 \beta_{6} ) q^{23} + ( 60 \beta_{3} - 12 \beta_{6} ) q^{24} + ( -44 - 6 \beta_{2} - 33 \beta_{3} - 42 \beta_{7} ) q^{25} + ( -42 + 6 \beta_{1} + 40 \beta_{2} + 40 \beta_{4} - 40 \beta_{5} - 40 \beta_{7} ) q^{26} -27 \beta_{4} q^{27} + ( 70 + 30 \beta_{1} - 70 \beta_{3} - 36 \beta_{5} + 30 \beta_{6} + 12 \beta_{7} ) q^{28} + ( 35 \beta_{2} + 78 \beta_{3} + 35 \beta_{7} ) q^{29} + ( -9 - 9 \beta_{1} - 6 \beta_{2} + 27 \beta_{3} + 42 \beta_{4} - 21 \beta_{5} - 27 \beta_{6} + 3 \beta_{7} ) q^{30} + ( -12 \beta_{1} + 150 \beta_{4} - 150 \beta_{5} ) q^{31} + ( 16 \beta_{2} - 176 \beta_{4} ) q^{32} + ( 15 - 15 \beta_{3} - 36 \beta_{7} ) q^{33} + ( 30 \beta_{2} - 70 \beta_{3} + 30 \beta_{4} + 30 \beta_{5} - 10 \beta_{6} + 30 \beta_{7} ) q^{34} + ( 54 \beta_{1} - 87 \beta_{4} - 89 \beta_{5} + 42 \beta_{6} ) q^{35} + ( -54 + 18 \beta_{1} ) q^{36} + ( -80 - 66 \beta_{2} - 80 \beta_{3} ) q^{37} + ( -30 - 30 \beta_{1} + 30 \beta_{3} + 154 \beta_{5} - 30 \beta_{6} - 22 \beta_{7} ) q^{38} + ( 120 \beta_{4} + 120 \beta_{5} - 18 \beta_{6} ) q^{39} + ( 56 - 40 \beta_{1} - 36 \beta_{2} - 28 \beta_{3} - 180 \beta_{4} - 60 \beta_{5} - 20 \beta_{6} - 12 \beta_{7} ) q^{40} + ( 138 - 30 \beta_{2} + 30 \beta_{7} ) q^{41} + ( 105 - 15 \beta_{1} - 18 \beta_{2} + 105 \beta_{3} - 18 \beta_{4} + 15 \beta_{6} ) q^{42} + ( -10 \beta_{1} + 396 \beta_{5} - 10 \beta_{6} ) q^{43} + ( -10 \beta_{2} - 168 \beta_{3} + 30 \beta_{4} + 30 \beta_{5} + 72 \beta_{6} - 10 \beta_{7} ) q^{44} + ( -81 - 9 \beta_{2} - 27 \beta_{3} - 18 \beta_{7} ) q^{45} + ( -38 - 38 \beta_{1} - 30 \beta_{2} + 210 \beta_{4} - 210 \beta_{5} + 30 \beta_{7} ) q^{46} + ( 120 \beta_{1} - 70 \beta_{4} - 120 \beta_{6} ) q^{47} + ( -24 \beta_{5} + 72 \beta_{7} ) q^{48} + ( 60 \beta_{2} - 43 \beta_{3} + 60 \beta_{7} ) q^{49} + ( 294 - 42 \beta_{1} - 44 \beta_{2} - 42 \beta_{3} - 44 \beta_{4} - 33 \beta_{5} - 6 \beta_{6} - 33 \beta_{7} ) q^{50} + ( -30 \beta_{1} - 90 \beta_{4} + 90 \beta_{5} ) q^{51} + ( 240 - 80 \beta_{1} - 36 \beta_{2} + 240 \beta_{3} - 84 \beta_{4} + 80 \beta_{6} ) q^{52} + ( -150 + 150 \beta_{3} + 188 \beta_{7} ) q^{53} + ( 27 \beta_{3} - 27 \beta_{6} ) q^{54} + ( 51 \beta_{1} - 138 \beta_{4} + 144 \beta_{5} + 103 \beta_{6} ) q^{55} + ( -120 - 24 \beta_{1} + 100 \beta_{2} - 140 \beta_{4} + 140 \beta_{5} - 100 \beta_{7} ) q^{56} + ( -90 + 66 \beta_{2} - 90 \beta_{3} ) q^{57} + ( -245 + 35 \beta_{1} + 245 \beta_{3} + 78 \beta_{5} + 35 \beta_{6} + 78 \beta_{7} ) q^{58} + ( 255 \beta_{4} + 255 \beta_{5} + 108 \beta_{6} ) q^{59} + ( -42 - 18 \beta_{1} - 18 \beta_{2} - 84 \beta_{3} + 54 \beta_{4} - 162 \beta_{5} + 36 \beta_{6} + 54 \beta_{7} ) q^{60} + ( -34 + 150 \beta_{2} - 150 \beta_{7} ) q^{61} + ( -150 - 150 \beta_{1} - 12 \beta_{2} - 150 \beta_{3} + 84 \beta_{4} + 150 \beta_{6} ) q^{62} + ( 45 \beta_{1} - 54 \beta_{5} + 45 \beta_{6} ) q^{63} + ( 288 \beta_{3} - 160 \beta_{6} ) q^{64} + ( 324 - 94 \beta_{2} + 438 \beta_{3} + 102 \beta_{7} ) q^{65} + ( 252 - 36 \beta_{1} + 15 \beta_{2} + 15 \beta_{4} - 15 \beta_{5} - 15 \beta_{7} ) q^{66} + ( 190 \beta_{1} + 156 \beta_{4} - 190 \beta_{6} ) q^{67} + ( -180 + 60 \beta_{1} + 180 \beta_{3} - 140 \beta_{5} + 60 \beta_{6} - 60 \beta_{7} ) q^{68} + ( -90 \beta_{2} - 114 \beta_{3} - 90 \beta_{7} ) q^{69} + ( -89 - 89 \beta_{1} + 54 \beta_{2} + 87 \beta_{3} - 378 \beta_{4} + 294 \beta_{5} - 87 \beta_{6} - 42 \beta_{7} ) q^{70} + ( -60 \beta_{1} + 320 \beta_{4} - 320 \beta_{5} ) q^{71} + ( -36 \beta_{2} - 180 \beta_{4} ) q^{72} + ( 55 - 55 \beta_{3} - 48 \beta_{7} ) q^{73} + ( -80 \beta_{2} - 462 \beta_{3} - 80 \beta_{4} - 80 \beta_{5} - 66 \beta_{6} - 80 \beta_{7} ) q^{74} + ( -18 \beta_{1} + 99 \beta_{4} - 132 \beta_{5} + 126 \beta_{6} ) q^{75} + ( 308 + 132 \beta_{1} - 60 \beta_{2} + 180 \beta_{4} - 180 \beta_{5} + 60 \beta_{7} ) q^{76} + ( -390 + 22 \beta_{2} - 390 \beta_{3} ) q^{77} + ( 120 + 120 \beta_{1} - 120 \beta_{3} - 126 \beta_{5} + 120 \beta_{6} + 18 \beta_{7} ) q^{78} + ( -330 \beta_{4} - 330 \beta_{5} - 32 \beta_{6} ) q^{79} + ( 24 - 72 \beta_{1} + 16 \beta_{2} - 72 \beta_{3} + 336 \beta_{4} - 168 \beta_{5} - 216 \beta_{6} - 8 \beta_{7} ) q^{80} -81 q^{81} + ( -210 + 30 \beta_{1} + 138 \beta_{2} - 210 \beta_{3} + 138 \beta_{4} - 30 \beta_{6} ) q^{82} + ( -180 \beta_{1} - 26 \beta_{5} - 180 \beta_{6} ) q^{83} + ( 90 \beta_{2} - 108 \beta_{3} + 210 \beta_{4} + 210 \beta_{5} - 36 \beta_{6} + 90 \beta_{7} ) q^{84} + ( -290 - 60 \beta_{2} + 320 \beta_{3} - 120 \beta_{7} ) q^{85} + ( 396 + 396 \beta_{1} - 10 \beta_{2} + 70 \beta_{4} - 70 \beta_{5} + 10 \beta_{7} ) q^{86} + ( 105 \beta_{1} - 234 \beta_{4} - 105 \beta_{6} ) q^{87} + ( 100 + 20 \beta_{1} - 100 \beta_{3} + 336 \beta_{5} + 20 \beta_{6} - 240 \beta_{7} ) q^{88} + ( -100 \beta_{2} - 366 \beta_{3} - 100 \beta_{7} ) q^{89} + ( 126 - 18 \beta_{1} - 81 \beta_{2} - 63 \beta_{3} - 81 \beta_{4} - 27 \beta_{5} - 9 \beta_{6} - 27 \beta_{7} ) q^{90} + ( -436 \beta_{1} - 450 \beta_{4} + 450 \beta_{5} ) q^{91} + ( -420 - 180 \beta_{1} - 76 \beta_{2} - 420 \beta_{3} + 228 \beta_{4} + 180 \beta_{6} ) q^{92} + ( 450 - 450 \beta_{3} - 36 \beta_{7} ) q^{93} + ( 120 \beta_{2} + 70 \beta_{3} - 840 \beta_{4} - 840 \beta_{5} - 70 \beta_{6} + 120 \beta_{7} ) q^{94} + ( 228 \beta_{1} + 206 \beta_{4} - 488 \beta_{5} + 24 \beta_{6} ) q^{95} + ( -528 + 48 \beta_{1} ) q^{96} + ( -5 + 48 \beta_{2} - 5 \beta_{3} ) q^{97} + ( -420 + 60 \beta_{1} + 420 \beta_{3} - 43 \beta_{5} + 60 \beta_{6} - 43 \beta_{7} ) q^{98} + ( 45 \beta_{4} + 45 \beta_{5} + 108 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 24q^{5} + 24q^{6} + O(q^{10})$$ $$8q - 24q^{5} + 24q^{6} - 56q^{10} + 320q^{13} - 64q^{16} + 240q^{17} - 432q^{20} - 144q^{21} - 40q^{22} - 352q^{25} - 336q^{26} + 560q^{28} - 72q^{30} + 120q^{33} - 432q^{36} - 640q^{37} - 240q^{38} + 448q^{40} + 1104q^{41} + 840q^{42} - 648q^{45} - 304q^{46} + 2352q^{50} + 1920q^{52} - 1200q^{53} - 960q^{56} - 720q^{57} - 1960q^{58} - 336q^{60} - 272q^{61} - 1200q^{62} + 2592q^{65} + 2016q^{66} - 1440q^{68} - 712q^{70} + 440q^{73} + 2464q^{76} - 3120q^{77} + 960q^{78} + 192q^{80} - 648q^{81} - 1680q^{82} - 2320q^{85} + 3168q^{86} + 800q^{88} + 1008q^{90} - 3360q^{92} + 3600q^{93} - 4224q^{96} - 40q^{97} - 3360q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + x^{4} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$2 \nu^{4} + 1$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} + 11 \nu$$$$)/6$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} + 5 \nu^{2}$$$$)/12$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{5} + \nu$$$$)/6$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} + 7 \nu^{3}$$$$)/24$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{6} + 3 \nu^{2}$$$$)/4$$ $$\beta_{7}$$ $$=$$ $$($$$$5 \nu^{7} + 13 \nu^{3}$$$$)/24$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} + \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} + 3 \beta_{3}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{7} + 5 \beta_{5}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$3 \beta_{1} - 1$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-11 \beta_{4} + \beta_{2}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-5 \beta_{6} + 9 \beta_{3}$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$7 \beta_{7} - 13 \beta_{5}$$$$)/2$$

## Character values

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

 $$n$$ $$31$$ $$37$$ $$41$$ $$\chi(n)$$ $$-1$$ $$\beta_{3}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 −1.28897 − 0.581861i −0.581861 − 1.28897i 0.581861 + 1.28897i 1.28897 + 0.581861i −1.28897 + 0.581861i −0.581861 + 1.28897i 0.581861 − 1.28897i 1.28897 − 0.581861i
−2.57794 1.16372i −2.12132 2.12132i 5.29150 + 6.00000i 2.61249 + 10.8708i 3.00000 + 7.93725i 17.4714 17.4714i −6.65882 21.6255i 9.00000i 5.91580 31.0645i
7.2 −1.16372 2.57794i 2.12132 + 2.12132i −5.29150 + 6.00000i 2.61249 + 10.8708i 3.00000 7.93725i −17.4714 + 17.4714i 21.6255 + 6.65882i 9.00000i 24.9841 19.3854i
7.3 1.16372 + 2.57794i −2.12132 2.12132i −5.29150 + 6.00000i −8.61249 + 7.12917i 3.00000 7.93725i −8.98612 + 8.98612i −21.6255 6.65882i 9.00000i −28.4011 13.9061i
7.4 2.57794 + 1.16372i 2.12132 + 2.12132i 5.29150 + 6.00000i −8.61249 + 7.12917i 3.00000 + 7.93725i 8.98612 8.98612i 6.65882 + 21.6255i 9.00000i −30.4988 + 8.35600i
43.1 −2.57794 + 1.16372i −2.12132 + 2.12132i 5.29150 6.00000i 2.61249 10.8708i 3.00000 7.93725i 17.4714 + 17.4714i −6.65882 + 21.6255i 9.00000i 5.91580 + 31.0645i
43.2 −1.16372 + 2.57794i 2.12132 2.12132i −5.29150 6.00000i 2.61249 10.8708i 3.00000 + 7.93725i −17.4714 17.4714i 21.6255 6.65882i 9.00000i 24.9841 + 19.3854i
43.3 1.16372 2.57794i −2.12132 + 2.12132i −5.29150 6.00000i −8.61249 7.12917i 3.00000 + 7.93725i −8.98612 8.98612i −21.6255 + 6.65882i 9.00000i −28.4011 + 13.9061i
43.4 2.57794 1.16372i 2.12132 2.12132i 5.29150 6.00000i −8.61249 7.12917i 3.00000 7.93725i 8.98612 + 8.98612i 6.65882 21.6255i 9.00000i −30.4988 8.35600i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 43.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.c odd 4 1 inner
20.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.j.a 8
3.b odd 2 1 180.4.k.d 8
4.b odd 2 1 inner 60.4.j.a 8
5.c odd 4 1 inner 60.4.j.a 8
12.b even 2 1 180.4.k.d 8
15.e even 4 1 180.4.k.d 8
20.e even 4 1 inner 60.4.j.a 8
60.l odd 4 1 180.4.k.d 8

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{8} + 398792 T_{7}^{4} + 9721171216$$ acting on $$S_{4}^{\mathrm{new}}(60, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4096 + 16 T^{4} + T^{8}$$
$3$ $$( 81 + T^{4} )^{2}$$
$5$ $$( 15625 + 1500 T + 160 T^{2} + 12 T^{3} + T^{4} )^{2}$$
$7$ $$9721171216 + 398792 T^{4} + T^{8}$$
$11$ $$( 917764 + 2116 T^{2} + T^{4} )^{2}$$
$13$ $$( 8690704 - 471680 T + 12800 T^{2} - 160 T^{3} + T^{4} )^{2}$$
$17$ $$( 1210000 - 132000 T + 7200 T^{2} - 120 T^{3} + T^{4} )^{2}$$
$19$ $$( 2521744 - 10376 T^{2} + T^{4} )^{2}$$
$23$ $$15489379570544896 + 540023072 T^{4} + T^{8}$$
$29$ $$( 122456356 + 46468 T^{2} + T^{4} )^{2}$$
$31$ $$( 1935296064 + 92016 T^{2} + T^{4} )^{2}$$
$37$ $$( 313006864 - 5661440 T + 51200 T^{2} + 320 T^{3} + T^{4} )^{2}$$
$41$ $$( 6444 - 276 T + T^{2} )^{4}$$
$43$ $$58\!\cdots\!36$$$$+ 51820944512 T^{4} + T^{8}$$
$47$ $$14\!\cdots\!00$$$$+ 93187220000 T^{4} + T^{8}$$
$53$ $$( 40968998464 - 121444800 T + 180000 T^{2} + 600 T^{3} + T^{4} )^{2}$$
$59$ $$( 2342753604 - 423396 T^{2} + T^{4} )^{2}$$
$61$ $$( -313844 + 68 T + T^{2} )^{4}$$
$67$ $$53\!\cdots\!16$$$$+ 659635774592 T^{4} + T^{8}$$
$71$ $$( 32256160000 + 460000 T^{2} + T^{4} )^{2}$$
$73$ $$( 101566084 + 2217160 T + 24200 T^{2} - 220 T^{3} + T^{4} )^{2}$$
$79$ $$( 44365839424 - 449936 T^{2} + T^{4} )^{2}$$
$83$ $$42\!\cdots\!76$$$$+ 415186437152 T^{4} + T^{8}$$
$89$ $$( 36529936 + 547912 T^{2} + T^{4} )^{2}$$
$97$ $$( 258502084 - 321560 T + 200 T^{2} + 20 T^{3} + T^{4} )^{2}$$