# Properties

 Label 60.6.d.a Level $60$ Weight $6$ Character orbit 60.d Analytic conductor $9.623$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$9.62302918878$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} + \cdots)$$ Defining polynomial: $$x^{6} + 373 x^{4} + 33732 x^{2} + 186624$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}\cdot 3^{4}\cdot 5^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( -6 - \beta_{1} - \beta_{2} ) q^{5} + ( -3 \beta_{1} + \beta_{2} - \beta_{5} ) q^{7} -81 q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( -6 - \beta_{1} - \beta_{2} ) q^{5} + ( -3 \beta_{1} + \beta_{2} - \beta_{5} ) q^{7} -81 q^{9} + ( 48 + \beta_{1} + \beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{11} + ( 3 - \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 3 \beta_{5} ) q^{13} + ( 68 - 7 \beta_{1} - 2 \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{15} + ( 7 - 35 \beta_{1} - 9 \beta_{2} - 7 \beta_{3} - 5 \beta_{5} ) q^{17} + ( -998 - 12 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} ) q^{19} + ( 259 + 3 \beta_{1} + 9 \beta_{2} - \beta_{3} + 4 \beta_{4} + 3 \beta_{5} ) q^{21} + ( 2 - 36 \beta_{1} + 12 \beta_{2} - 2 \beta_{3} - 16 \beta_{5} ) q^{23} + ( 1010 + 65 \beta_{1} - \beta_{2} + \beta_{3} + 8 \beta_{4} - 11 \beta_{5} ) q^{25} -81 \beta_{1} q^{27} + ( -2661 + 3 \beta_{1} + 21 \beta_{2} - 3 \beta_{3} + 3 \beta_{5} ) q^{29} + ( 78 - 18 \beta_{1} - 78 \beta_{2} + 10 \beta_{3} - 16 \beta_{4} - 18 \beta_{5} ) q^{31} + ( 9 + 37 \beta_{1} - 45 \beta_{2} - 9 \beta_{3} + 27 \beta_{5} ) q^{33} + ( 3338 + 363 \beta_{1} - 5 \beta_{2} - 2 \beta_{3} - 6 \beta_{4} + 47 \beta_{5} ) q^{35} + ( -21 - 425 \beta_{1} + 61 \beta_{2} + 21 \beta_{3} - 19 \beta_{5} ) q^{37} + ( -24 + 9 \beta_{1} + 81 \beta_{2} - 12 \beta_{3} - 6 \beta_{4} + 9 \beta_{5} ) q^{39} + ( 2696 + 10 \beta_{1} + 94 \beta_{2} - 14 \beta_{3} - 8 \beta_{4} + 10 \beta_{5} ) q^{41} + ( -72 + 996 \beta_{1} + 80 \beta_{2} + 72 \beta_{3} + 64 \beta_{5} ) q^{43} + ( 486 + 81 \beta_{1} + 81 \beta_{2} ) q^{45} + ( -46 - 1414 \beta_{1} - 54 \beta_{2} + 46 \beta_{3} + 146 \beta_{5} ) q^{47} + ( -4499 - 54 \beta_{1} - 282 \beta_{2} + 38 \beta_{3} - 32 \beta_{4} - 54 \beta_{5} ) q^{49} + ( 2558 + 15 \beta_{1} + 171 \beta_{2} - 26 \beta_{3} - 22 \beta_{4} + 15 \beta_{5} ) q^{51} + ( 15 + 413 \beta_{1} + 15 \beta_{2} - 15 \beta_{3} - 45 \beta_{5} ) q^{53} + ( -4306 + 1849 \beta_{1} - 91 \beta_{2} + 90 \beta_{3} - 60 \beta_{4} - 65 \beta_{5} ) q^{55} + ( 54 - 1032 \beta_{1} - 108 \beta_{2} - 54 \beta_{3} ) q^{57} + ( 15328 + 31 \beta_{1} + 271 \beta_{2} - 40 \beta_{3} - 18 \beta_{4} + 31 \beta_{5} ) q^{59} + ( 1098 - 336 \beta_{2} + 56 \beta_{3} + 112 \beta_{4} ) q^{61} + ( 243 \beta_{1} - 81 \beta_{2} + 81 \beta_{5} ) q^{63} + ( -8705 + 2845 \beta_{1} - 45 \beta_{2} - 85 \beta_{3} + 40 \beta_{4} - 15 \beta_{5} ) q^{65} + ( -48 - 4622 \beta_{1} + 206 \beta_{2} + 48 \beta_{3} - 110 \beta_{5} ) q^{67} + ( 3070 + 48 \beta_{1} + 180 \beta_{2} - 22 \beta_{3} + 52 \beta_{4} + 48 \beta_{5} ) q^{69} + ( -26852 + 18 \beta_{1} + 186 \beta_{2} - 28 \beta_{3} - 20 \beta_{4} + 18 \beta_{5} ) q^{71} + ( 150 + 1594 \beta_{1} + 62 \beta_{2} - 150 \beta_{3} - 362 \beta_{5} ) q^{73} + ( -4900 + 990 \beta_{1} - 63 \beta_{2} - 92 \beta_{3} + 44 \beta_{4} + 87 \beta_{5} ) q^{75} + ( 254 - 8722 \beta_{1} - 54 \beta_{2} - 254 \beta_{3} - 454 \beta_{5} ) q^{77} + ( 21582 - 18 \beta_{1} - 414 \beta_{2} + 66 \beta_{3} + 96 \beta_{4} - 18 \beta_{5} ) q^{79} + 6561 q^{81} + ( 92 + 4564 \beta_{1} - 408 \beta_{2} - 92 \beta_{3} + 224 \beta_{5} ) q^{83} + ( -29617 + 6183 \beta_{1} - 95 \beta_{2} - 117 \beta_{3} + 144 \beta_{4} - 13 \beta_{5} ) q^{85} + ( -54 - 2643 \beta_{1} + 27 \beta_{2} + 54 \beta_{3} + 81 \beta_{5} ) q^{87} + ( 12686 + 32 \beta_{1} - 280 \beta_{2} + 52 \beta_{3} + 168 \beta_{4} + 32 \beta_{5} ) q^{89} + ( -16204 - 126 \beta_{1} - 294 \beta_{2} + 28 \beta_{3} - 196 \beta_{4} - 126 \beta_{5} ) q^{91} + ( 108 + 106 \beta_{1} + 270 \beta_{2} - 108 \beta_{3} - 486 \beta_{5} ) q^{93} + ( 38570 + 5320 \beta_{1} + 780 \beta_{2} + 190 \beta_{3} - 40 \beta_{4} - 240 \beta_{5} ) q^{95} + ( 84 - 12376 \beta_{1} - 424 \beta_{2} - 84 \beta_{3} + 256 \beta_{5} ) q^{97} + ( -3888 - 81 \beta_{1} - 81 \beta_{2} - 162 \beta_{4} - 81 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 38q^{5} - 486q^{9} + O(q^{10})$$ $$6q - 38q^{5} - 486q^{9} + 296q^{11} + 396q^{15} - 6000q^{19} + 1584q^{21} + 6054q^{25} - 15924q^{29} + 264q^{31} + 20096q^{35} + 16340q^{41} + 3078q^{45} - 27654q^{49} + 15624q^{51} - 26088q^{55} + 92456q^{59} + 6252q^{61} - 52440q^{65} + 18936q^{69} - 160800q^{71} - 29448q^{75} + 128952q^{79} + 39366q^{81} - 177864q^{85} + 76060q^{89} - 98400q^{91} + 232800q^{95} - 23976q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 373 x^{4} + 33732 x^{2} + 186624$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{5} - 805 \nu^{3} - 114516 \nu$$$$)/29664$$ $$\beta_{2}$$ $$=$$ $$($$$$95 \nu^{5} + 1944 \nu^{4} + 31979 \nu^{3} + 497016 \nu^{2} + 3092220 \nu + 15445728$$$$)/266976$$ $$\beta_{3}$$ $$=$$ $$($$$$-89 \nu^{5} - 1944 \nu^{4} - 27149 \nu^{3} - 497016 \nu^{2} + 264636 \nu - 15312240$$$$)/133488$$ $$\beta_{4}$$ $$=$$ $$($$$$187 \nu^{5} - 2592 \nu^{4} + 61543 \nu^{3} - 217728 \nu^{2} + 4506012 \nu + 34892208$$$$)/133488$$ $$\beta_{5}$$ $$=$$ $$($$$$-139 \nu^{5} + 324 \nu^{4} - 45151 \nu^{3} + 82836 \nu^{2} - 3347604 \nu + 2574288$$$$)/44496$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$3 \beta_{3} + 6 \beta_{2} + 4 \beta_{1} - 3$$$$)/60$$ $$\nu^{2}$$ $$=$$ $$($$$$4 \beta_{5} + 6 \beta_{4} - \beta_{3} + 10 \beta_{2} + 4 \beta_{1} - 2493$$$$)/20$$ $$\nu^{3}$$ $$=$$ $$($$$$12 \beta_{5} - 187 \beta_{3} - 386 \beta_{2} - 1488 \beta_{1} + 187$$$$)/20$$ $$\nu^{4}$$ $$=$$ $$($$$$-748 \beta_{5} - 1534 \beta_{4} - 19 \beta_{3} - 634 \beta_{2} - 748 \beta_{1} + 478745$$$$)/20$$ $$\nu^{5}$$ $$=$$ $$($$$$-9660 \beta_{5} + 36019 \beta_{3} + 81698 \beta_{2} + 451872 \beta_{1} - 36019$$$$)/20$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 2.43168i − 15.1546i 11.7229i − 2.43168i 15.1546i − 11.7229i
0 9.00000i 0 −53.1009 17.4726i 0 222.092i 0 −81.0000 0
49.2 0 9.00000i 0 −20.3651 + 52.0602i 0 121.510i 0 −81.0000 0
49.3 0 9.00000i 0 54.4661 12.5876i 0 12.5817i 0 −81.0000 0
49.4 0 9.00000i 0 −53.1009 + 17.4726i 0 222.092i 0 −81.0000 0
49.5 0 9.00000i 0 −20.3651 52.0602i 0 121.510i 0 −81.0000 0
49.6 0 9.00000i 0 54.4661 + 12.5876i 0 12.5817i 0 −81.0000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 49.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b 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.d.a 6
3.b odd 2 1 180.6.d.d 6
4.b odd 2 1 240.6.f.d 6
5.b even 2 1 inner 60.6.d.a 6
5.c odd 4 1 300.6.a.i 3
5.c odd 4 1 300.6.a.j 3
12.b even 2 1 720.6.f.m 6
15.d odd 2 1 180.6.d.d 6
15.e even 4 1 900.6.a.w 3
15.e even 4 1 900.6.a.x 3
20.d odd 2 1 240.6.f.d 6
60.h even 2 1 720.6.f.m 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.6.d.a 6 1.a even 1 1 trivial
60.6.d.a 6 5.b even 2 1 inner
180.6.d.d 6 3.b odd 2 1
180.6.d.d 6 15.d odd 2 1
240.6.f.d 6 4.b odd 2 1
240.6.f.d 6 20.d odd 2 1
300.6.a.i 3 5.c odd 4 1
300.6.a.j 3 5.c odd 4 1
720.6.f.m 6 12.b even 2 1
720.6.f.m 6 60.h even 2 1
900.6.a.w 3 15.e even 4 1
900.6.a.x 3 15.e even 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{6}^{\mathrm{new}}(60, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$( 81 + T^{2} )^{3}$$
$5$ $$30517578125 + 371093750 T - 7203125 T^{2} - 233700 T^{3} - 2305 T^{4} + 38 T^{5} + T^{6}$$
$7$ $$115284695296 + 738417168 T^{2} + 64248 T^{4} + T^{6}$$
$11$ $$( 78696304 - 480532 T - 148 T^{2} + T^{3} )^{2}$$
$13$ $$23386973184000000 + 292572810000 T^{2} + 1081800 T^{4} + T^{6}$$
$17$ $$5027980972101888256 + 10587775398288 T^{2} + 5928408 T^{4} + T^{6}$$
$19$ $$( -829440000 + 316800 T + 3000 T^{2} + T^{3} )^{2}$$
$23$ $$15367082902947106816 + 32380420670208 T^{2} + 13854768 T^{4} + T^{6}$$
$29$ $$( 12509144064 + 18394848 T + 7962 T^{2} + T^{3} )^{2}$$
$31$ $$( 233735426816 - 71304192 T - 132 T^{2} + T^{3} )^{2}$$
$37$ $$13\!\cdots\!64$$$$+ 9075483172965648 T^{2} + 171675912 T^{4} + T^{6}$$
$41$ $$( 323508673000 - 38366500 T - 8170 T^{2} + T^{3} )^{2}$$
$43$ $$15\!\cdots\!96$$$$+ 212408511395021568 T^{2} + 836774448 T^{4} + T^{6}$$
$47$ $$66\!\cdots\!36$$$$+ 617480270927950848 T^{2} + 1500491088 T^{4} + T^{6}$$
$53$ $$53\!\cdots\!44$$$$+ 5163713112409488 T^{2} + 138197592 T^{4} + T^{6}$$
$59$ $$( 7848174247024 + 229958828 T - 46228 T^{2} + T^{3} )^{2}$$
$61$ $$( 20932589363512 - 2100371508 T - 3126 T^{2} + T^{3} )^{2}$$
$67$ $$13\!\cdots\!56$$$$+ 11341956370837567488 T^{2} + 6887557008 T^{4} + T^{6}$$
$71$ $$( 13864753152000 + 1901473200 T + 80400 T^{2} + T^{3} )^{2}$$
$73$ $$57\!\cdots\!24$$$$+ 13497205135724521728 T^{2} + 7132671072 T^{4} + T^{6}$$
$79$ $$( 33510082240512 - 679079808 T - 64476 T^{2} + T^{3} )^{2}$$
$83$ $$88\!\cdots\!56$$$$+ 28614113929846149888 T^{2} + 11536133808 T^{4} + T^{6}$$
$89$ $$( 129133988519000 - 3253345300 T - 38030 T^{2} + T^{3} )^{2}$$
$97$ $$97\!\cdots\!04$$$$+$$$$43\!\cdots\!68$$$$T^{2} + 43563893952 T^{4} + T^{6}$$