# Properties

 Label 120.6.f.a Level $120$ Weight $6$ Character orbit 120.f Analytic conductor $19.246$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$120 = 2^{3} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 120.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$19.2460583776$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.25787221056.1 Defining polynomial: $$x^{6} + 61x^{4} + 852x^{2} + 576$$ x^6 + 61*x^4 + 852*x^2 + 576 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}\cdot 5^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + 9 \beta_1 q^{3} + ( - 2 \beta_{5} + \beta_{3} - 9 \beta_1 + 8) q^{5} + ( - 3 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 38 \beta_1) q^{7} - 81 q^{9}+O(q^{10})$$ q + 9*b1 * q^3 + (-2*b5 + b3 - 9*b1 + 8) * q^5 + (-3*b5 + 3*b4 + 2*b3 + 2*b2 + 38*b1) * q^7 - 81 * q^9 $$q + 9 \beta_1 q^{3} + ( - 2 \beta_{5} + \beta_{3} - 9 \beta_1 + 8) q^{5} + ( - 3 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 38 \beta_1) q^{7} - 81 q^{9} + (9 \beta_{5} + 9 \beta_{4} + 4 \beta_{3} - 4 \beta_{2} - 102) q^{11} + ( - 2 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} - 3 \beta_{2} - 30 \beta_1) q^{13} + ( - 9 \beta_{5} - 18 \beta_{3} + 72 \beta_1 + 81) q^{15} + (16 \beta_{5} - 16 \beta_{4} - 23 \beta_{3} - 23 \beta_{2} - 398 \beta_1) q^{17} + (64 \beta_{5} + 64 \beta_{4} + 20 \beta_{3} - 20 \beta_{2} + 104) q^{19} + ( - 18 \beta_{5} - 18 \beta_{4} - 27 \beta_{3} + 27 \beta_{2} - 342) q^{21} + ( - 80 \beta_{5} + 80 \beta_{4} + 52 \beta_{3} + 52 \beta_{2} - 868 \beta_1) q^{23} + ( - 20 \beta_{5} + 100 \beta_{4} + 85 \beta_{3} + 75 \beta_{2} + \cdots + 655) q^{25}+ \cdots + ( - 729 \beta_{5} - 729 \beta_{4} - 324 \beta_{3} + 324 \beta_{2} + \cdots + 8262) q^{99}+O(q^{100})$$ q + 9*b1 * q^3 + (-2*b5 + b3 - 9*b1 + 8) * q^5 + (-3*b5 + 3*b4 + 2*b3 + 2*b2 + 38*b1) * q^7 - 81 * q^9 + (9*b5 + 9*b4 + 4*b3 - 4*b2 - 102) * q^11 + (-2*b5 + 2*b4 - 3*b3 - 3*b2 - 30*b1) * q^13 + (-9*b5 - 18*b3 + 72*b1 + 81) * q^15 + (16*b5 - 16*b4 - 23*b3 - 23*b2 - 398*b1) * q^17 + (64*b5 + 64*b4 + 20*b3 - 20*b2 + 104) * q^19 + (-18*b5 - 18*b4 - 27*b3 + 27*b2 - 342) * q^21 + (-80*b5 + 80*b4 + 52*b3 + 52*b2 - 868*b1) * q^23 + (-20*b5 + 100*b4 + 85*b3 + 75*b2 + 860*b1 + 655) * q^25 - 729*b1 * q^27 + (-6*b5 - 6*b4 - 115*b3 + 115*b2 - 396) * q^29 + (66*b5 + 66*b4 - 96*b3 + 96*b2 - 1728) * q^31 + (-36*b5 + 36*b4 + 81*b3 + 81*b2 - 918*b1) * q^33 + (-85*b5 + 175*b4 + 100*b2 + 1180*b1 - 3150) * q^35 + (-74*b5 + 74*b4 + 197*b3 + 197*b2 - 2422*b1) * q^37 + (27*b5 + 27*b4 - 18*b3 + 18*b2 + 270) * q^39 + (388*b5 + 388*b4 + 102*b3 - 102*b2 + 106) * q^41 + (-180*b5 + 180*b4 - 60*b3 - 60*b2 - 8316*b1) * q^43 + (162*b5 - 81*b3 + 729*b1 - 648) * q^45 + (-286*b5 + 286*b4 - 316*b3 - 316*b2 + 8456*b1) * q^47 + (24*b5 + 24*b4 - 318*b3 + 318*b2 + 4539) * q^49 + (207*b5 + 207*b4 + 144*b3 - 144*b2 + 3582) * q^51 + (-142*b5 + 142*b4 - 25*b3 - 25*b2 - 12230*b1) * q^53 + (337*b5 - 25*b4 - 166*b3 - 550*b2 + 5954*b1 - 7628) * q^55 + (-180*b5 + 180*b4 + 576*b3 + 576*b2 + 936*b1) * q^57 + (643*b5 + 643*b4 + 920*b3 - 920*b2 - 674) * q^59 + (-904*b5 - 904*b4 - 920*b3 + 920*b2 - 9902) * q^61 + (243*b5 - 243*b4 - 162*b3 - 162*b2 - 3078*b1) * q^63 + (-88*b5 - 100*b4 + 89*b3 + 175*b2 + 5804*b1 - 2338) * q^65 + (-286*b5 + 286*b4 + 112*b3 + 112*b2 + 10480*b1) * q^67 + (-468*b5 - 468*b4 - 720*b3 + 720*b2 + 7812) * q^69 + (-366*b5 - 366*b4 + 1120*b3 - 1120*b2 - 4212) * q^71 + (980*b5 - 980*b4 + 1038*b3 + 1038*b2 - 12540*b1) * q^73 + (-765*b5 - 675*b4 - 180*b3 + 900*b2 + 5895*b1 - 7740) * q^75 + (376*b5 - 376*b4 - 578*b3 - 578*b2 + 14964*b1) * q^77 + (-470*b5 - 470*b4 + 536*b3 - 536*b2 + 36744) * q^79 + 6561 * q^81 + (260*b5 - 260*b4 - 2476*b3 - 2476*b2 - 4364*b1) * q^83 + (402*b5 - 1550*b4 + 329*b3 - 225*b2 + 7684*b1 + 15782) * q^85 + (1035*b5 - 1035*b4 - 54*b3 - 54*b2 - 3564*b1) * q^87 + (-1296*b5 - 1296*b4 + 120*b3 - 120*b2 + 57786) * q^89 + (-38*b5 - 38*b4 - 280*b3 + 280*b2 - 3060) * q^91 + (864*b5 - 864*b4 + 594*b3 + 594*b2 - 15552*b1) * q^93 + (552*b5 - 600*b4 - 596*b3 - 3700*b2 + 25384*b1 - 57168) * q^95 + (2828*b5 - 2828*b4 + 204*b3 + 204*b2 - 3960*b1) * q^97 + (-729*b5 - 729*b4 - 324*b3 + 324*b2 + 8262) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 50 q^{5} - 486 q^{9}+O(q^{10})$$ 6 * q + 50 * q^5 - 486 * q^9 $$6 q + 50 q^{5} - 486 q^{9} - 664 q^{11} + 540 q^{15} + 288 q^{19} - 1872 q^{21} + 3750 q^{25} - 1892 q^{29} - 10248 q^{31} - 18880 q^{35} + 1584 q^{39} - 1324 q^{41} - 4050 q^{45} + 28410 q^{49} + 20088 q^{51} - 47160 q^{55} - 10296 q^{59} - 52116 q^{61} - 13480 q^{65} + 51624 q^{69} - 28288 q^{71} - 41400 q^{75} + 220200 q^{79} + 39366 q^{81} + 95880 q^{85} + 351420 q^{89} - 17088 q^{91} - 349120 q^{95} + 53784 q^{99}+O(q^{100})$$ 6 * q + 50 * q^5 - 486 * q^9 - 664 * q^11 + 540 * q^15 + 288 * q^19 - 1872 * q^21 + 3750 * q^25 - 1892 * q^29 - 10248 * q^31 - 18880 * q^35 + 1584 * q^39 - 1324 * q^41 - 4050 * q^45 + 28410 * q^49 + 20088 * q^51 - 47160 * q^55 - 10296 * q^59 - 52116 * q^61 - 13480 * q^65 + 51624 * q^69 - 28288 * q^71 - 41400 * q^75 + 220200 * q^79 + 39366 * q^81 + 95880 * q^85 + 351420 * q^89 - 17088 * q^91 - 349120 * q^95 + 53784 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 61x^{4} + 852x^{2} + 576$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{5} + 85\nu^{3} + 1596\nu ) / 1296$$ (v^5 + 85*v^3 + 1596*v) / 1296 $$\beta_{2}$$ $$=$$ $$( -7\nu^{5} - 28\nu^{4} - 379\nu^{3} - 1516\nu^{2} - 2748\nu - 11856 ) / 432$$ (-7*v^5 - 28*v^4 - 379*v^3 - 1516*v^2 - 2748*v - 11856) / 432 $$\beta_{3}$$ $$=$$ $$( -7\nu^{5} + 28\nu^{4} - 379\nu^{3} + 1516\nu^{2} - 2748\nu + 11856 ) / 432$$ (-7*v^5 + 28*v^4 - 379*v^3 + 1516*v^2 - 2748*v + 11856) / 432 $$\beta_{4}$$ $$=$$ $$( 65\nu^{5} - 108\nu^{4} + 3581\nu^{3} - 3996\nu^{2} + 40884\nu - 9072 ) / 1296$$ (65*v^5 - 108*v^4 + 3581*v^3 - 3996*v^2 + 40884*v - 9072) / 1296 $$\beta_{5}$$ $$=$$ $$( -65\nu^{5} - 108\nu^{4} - 3581\nu^{3} - 3996\nu^{2} - 40884\nu - 9072 ) / 1296$$ (-65*v^5 - 108*v^4 - 3581*v^3 - 3996*v^2 - 40884*v - 9072) / 1296
 $$\nu$$ $$=$$ $$( -\beta_{5} + \beta_{4} + 3\beta_{3} + 3\beta_{2} - 4\beta_1 ) / 20$$ (-b5 + b4 + 3*b3 + 3*b2 - 4*b1) / 20 $$\nu^{2}$$ $$=$$ $$( 7\beta_{5} + 7\beta_{4} + 9\beta_{3} - 9\beta_{2} - 396 ) / 20$$ (7*b5 + 7*b4 + 9*b3 - 9*b2 - 396) / 20 $$\nu^{3}$$ $$=$$ $$( 39\beta_{5} - 39\beta_{4} - 97\beta_{3} - 97\beta_{2} + 996\beta_1 ) / 20$$ (39*b5 - 39*b4 - 97*b3 - 97*b2 + 996*b1) / 20 $$\nu^{4}$$ $$=$$ $$( -379\beta_{5} - 379\beta_{4} - 333\beta_{3} + 333\beta_{2} + 12972 ) / 20$$ (-379*b5 - 379*b4 - 333*b3 + 333*b2 + 12972) / 20 $$\nu^{5}$$ $$=$$ $$( -1719\beta_{5} + 1719\beta_{4} + 3457\beta_{3} + 3457\beta_{2} - 52356\beta_1 ) / 20$$ (-1719*b5 + 1719*b4 + 3457*b3 + 3457*b2 - 52356*b1) / 20

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/120\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$41$$ $$61$$ $$97$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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
 − 4.49053i 6.33429i − 0.843753i 4.49053i − 6.33429i 0.843753i
0 9.00000i 0 −51.5439 + 21.6386i 0 21.3742i 0 −81.0000 0
49.2 0 9.00000i 0 33.8706 + 44.4723i 0 85.8595i 0 −81.0000 0
49.3 0 9.00000i 0 42.6733 36.1108i 0 168.485i 0 −81.0000 0
49.4 0 9.00000i 0 −51.5439 21.6386i 0 21.3742i 0 −81.0000 0
49.5 0 9.00000i 0 33.8706 44.4723i 0 85.8595i 0 −81.0000 0
49.6 0 9.00000i 0 42.6733 + 36.1108i 0 168.485i 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 120.6.f.a 6
3.b odd 2 1 360.6.f.a 6
4.b odd 2 1 240.6.f.e 6
5.b even 2 1 inner 120.6.f.a 6
5.c odd 4 1 600.6.a.r 3
5.c odd 4 1 600.6.a.s 3
12.b even 2 1 720.6.f.l 6
15.d odd 2 1 360.6.f.a 6
20.d odd 2 1 240.6.f.e 6
60.h even 2 1 720.6.f.l 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.6.f.a 6 1.a even 1 1 trivial
120.6.f.a 6 5.b even 2 1 inner
240.6.f.e 6 4.b odd 2 1
240.6.f.e 6 20.d odd 2 1
360.6.f.a 6 3.b odd 2 1
360.6.f.a 6 15.d odd 2 1
600.6.a.r 3 5.c odd 4 1
600.6.a.s 3 5.c odd 4 1
720.6.f.l 6 12.b even 2 1
720.6.f.l 6 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{6} + 36216T_{7}^{4} + 225603600T_{7}^{2} + 95604640000$$ acting on $$S_{6}^{\mathrm{new}}(120, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$(T^{2} + 81)^{3}$$
$5$ $$T^{6} - 50 T^{5} + \cdots + 30517578125$$
$7$ $$T^{6} + 36216 T^{4} + \cdots + 95604640000$$
$11$ $$(T^{3} + 332 T^{2} - 74644 T + 751600)^{2}$$
$13$ $$T^{6} + 62184 T^{4} + \cdots + 743071584256$$
$17$ $$T^{6} + 2125752 T^{4} + \cdots + 31\!\cdots\!04$$
$19$ $$(T^{3} - 144 T^{2} - 5516736 T + 2584131584)^{2}$$
$23$ $$T^{6} + 25770672 T^{4} + \cdots + 21\!\cdots\!24$$
$29$ $$(T^{3} + 946 T^{2} + \cdots + 37303134464)^{2}$$
$31$ $$(T^{3} + 5124 T^{2} + \cdots - 71486323200)^{2}$$
$37$ $$T^{6} + 125310888 T^{4} + \cdots + 54\!\cdots\!36$$
$41$ $$(T^{3} + 662 T^{2} + \cdots + 289547160040)^{2}$$
$43$ $$T^{6} + 379478448 T^{4} + \cdots + 40\!\cdots\!96$$
$47$ $$T^{6} + 1087347984 T^{4} + \cdots + 29\!\cdots\!00$$
$53$ $$T^{6} + 545716248 T^{4} + \cdots + 15\!\cdots\!00$$
$59$ $$(T^{3} + 5148 T^{2} + \cdots - 6990217203600)^{2}$$
$61$ $$(T^{3} + 26058 T^{2} + \cdots - 30927698302664)^{2}$$
$67$ $$T^{6} + 593605008 T^{4} + \cdots + 34\!\cdots\!56$$
$71$ $$(T^{3} + 14144 T^{2} + \cdots - 37186025753600)^{2}$$
$73$ $$T^{6} + 10300131936 T^{4} + \cdots + 20\!\cdots\!00$$
$79$ $$(T^{3} - 110100 T^{2} + \cdots + 4303737800704)^{2}$$
$83$ $$T^{6} + 17756024496 T^{4} + \cdots + 33\!\cdots\!04$$
$89$ $$(T^{3} - 175710 T^{2} + \cdots + 11306161037592)^{2}$$
$97$ $$T^{6} + 33401863872 T^{4} + \cdots + 34\!\cdots\!24$$