Properties

 Label 124.6.a.b Level $124$ Weight $6$ Character orbit 124.a Self dual yes Analytic conductor $19.888$ Analytic rank $0$ Dimension $6$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$124 = 2^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 124.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$19.8875936568$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ Defining polynomial: $$x^{6} - 2x^{5} - 847x^{4} + 1184x^{3} + 199815x^{2} - 13326x - 12452553$$ x^6 - 2*x^5 - 847*x^4 + 1184*x^3 + 199815*x^2 - 13326*x - 12452553 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_1 + 3) q^{3} + ( - \beta_{5} + \beta_1 + 4) q^{5} + ( - \beta_{5} - \beta_{4} + 2 \beta_{3} + 6) q^{7} + ( - \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + \beta_{2} + 5 \beta_1 + 48) q^{9}+O(q^{10})$$ q + (b1 + 3) * q^3 + (-b5 + b1 + 4) * q^5 + (-b5 - b4 + 2*b3 + 6) * q^7 + (-b5 + 2*b4 + 2*b3 + b2 + 5*b1 + 48) * q^9 $$q + (\beta_1 + 3) q^{3} + ( - \beta_{5} + \beta_1 + 4) q^{5} + ( - \beta_{5} - \beta_{4} + 2 \beta_{3} + 6) q^{7} + ( - \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + \beta_{2} + 5 \beta_1 + 48) q^{9} + (\beta_{5} - \beta_{4} - 6 \beta_{3} - \beta_{2} + 17 \beta_1 + 44) q^{11} + (3 \beta_{5} - 4 \beta_{4} - 11 \beta_{3} - 2 \beta_{2} + 22 \beta_1 + 201) q^{13} + (\beta_{5} + 11 \beta_{4} - 5 \beta_{2} + 10 \beta_1 + 311) q^{15} + (10 \beta_{5} - 7 \beta_{4} + 9 \beta_{3} + 3 \beta_{2} + 13 \beta_1 + 125) q^{17} + ( - 3 \beta_{5} + 3 \beta_{4} + 9 \beta_{3} + 29 \beta_{2} + 4 \beta_1 + 523) q^{19} + (14 \beta_{5} + 7 \beta_{4} + 39 \beta_{3} - 17 \beta_{2} - 29 \beta_1 + 163) q^{21} + ( - 15 \beta_{5} - 7 \beta_{4} - 16 \beta_{3} + 3 \beta_{2} - 58 \beta_1 + 1553) q^{23} + (4 \beta_{5} - 22 \beta_{4} - 33 \beta_{3} - 42 \beta_{2} - 26 \beta_1 + 1105) q^{25} + ( - 16 \beta_{5} + 22 \beta_{4} - 36 \beta_{2} - 94 \beta_1 + 1240) q^{27} + ( - 9 \beta_{5} - 42 \beta_{4} + \beta_{3} + 92 \beta_{2} - 26 \beta_1 + 2209) q^{29} - 961 q^{31} + ( - 20 \beta_{5} + 22 \beta_{4} - 29 \beta_{3} + 87 \beta_{2} + \cdots + 4149) q^{33}+ \cdots + ( - 47 \beta_{5} - 67 \beta_{4} - 1098 \beta_{3} - \beta_{2} + 2813 \beta_1 + 5318) q^{99}+O(q^{100})$$ q + (b1 + 3) * q^3 + (-b5 + b1 + 4) * q^5 + (-b5 - b4 + 2*b3 + 6) * q^7 + (-b5 + 2*b4 + 2*b3 + b2 + 5*b1 + 48) * q^9 + (b5 - b4 - 6*b3 - b2 + 17*b1 + 44) * q^11 + (3*b5 - 4*b4 - 11*b3 - 2*b2 + 22*b1 + 201) * q^13 + (b5 + 11*b4 - 5*b2 + 10*b1 + 311) * q^15 + (10*b5 - 7*b4 + 9*b3 + 3*b2 + 13*b1 + 125) * q^17 + (-3*b5 + 3*b4 + 9*b3 + 29*b2 + 4*b1 + 523) * q^19 + (14*b5 + 7*b4 + 39*b3 - 17*b2 - 29*b1 + 163) * q^21 + (-15*b5 - 7*b4 - 16*b3 + 3*b2 - 58*b1 + 1553) * q^23 + (4*b5 - 22*b4 - 33*b3 - 42*b2 - 26*b1 + 1105) * q^25 + (-16*b5 + 22*b4 - 36*b2 - 94*b1 + 1240) * q^27 + (-9*b5 - 42*b4 + b3 + 92*b2 - 26*b1 + 2209) * q^29 - 961 * q^31 + (-20*b5 + 22*b4 - 29*b3 + 87*b2 + 30*b1 + 4149) * q^33 + (-36*b5 + 13*b4 - 4*b3 - 153*b2 - 311*b1 + 3425) * q^35 + (-5*b5 + 67*b4 - 44*b3 - 99*b2 + 15*b1 + 3592) * q^37 + (-9*b5 + 4*b4 - 46*b3 + 173*b2 + 73*b1 + 5141) * q^39 + (68*b5 + 46*b4 + 210*b3 - 35*b2 + 10*b1 + 397) * q^41 + (-78*b5 - 44*b4 + 62*b3 + 110*b2 - 259*b1 + 6415) * q^43 + (96*b5 + 90*b4 + 33*b3 - 46*b2 + 364*b1 + 4170) * q^45 + (102*b5 - 232*b4 - 177*b3 - 83*b2 + 40*b1 + 5073) * q^47 + (-31*b5 - 18*b4 - 254*b3 - 102*b2 + 487*b1 + 5767) * q^49 + (61*b5 - 104*b4 + 187*b3 + 32*b2 - 89*b1 + 4038) * q^51 + (105*b5 - 61*b4 + 376*b3 + 185*b2 + 279*b1 - 1910) * q^53 + (-72*b5 + 58*b4 - 118*b3 + 120*b2 + 244*b1 + 3950) * q^55 + (126*b5 - 147*b4 - 519*b3 - 203*b2 + 1239*b1 + 2727) * q^57 + (16*b5 + 267*b4 + 212*b3 - 307*b2 - 543*b1 + 3361) * q^59 + (-265*b5 + 234*b4 + 151*b3 + 304*b2 + 192*b1 + 3479) * q^61 + (128*b5 + 245*b4 + 419*b3 - 294*b2 - 141*b1 - 3264) * q^63 + (-386*b5 + 37*b4 - 229*b3 + 207*b2 + 117*b1 + 811) * q^65 + (-135*b5 - 56*b4 + 23*b3 + 184*b2 - 399*b1 - 2644) * q^67 + (157*b5 - 46*b4 - 405*b3 + 36*b2 + 1347*b1 - 14200) * q^69 + (10*b5 - 49*b4 - 572*b3 + 239*b2 - 367*b1 + 7881) * q^71 + (-334*b5 - 618*b4 - 350*b3 - 328*b2 - 172*b1 - 26538) * q^73 + (-5*b5 + 85*b4 + 550*b3 + 575*b2 - 549*b1 - 8262) * q^75 + (100*b5 - 168*b4 + 794*b3 - 124*b2 - 1950*b1 - 23762) * q^77 + (-51*b5 + 561*b4 + 30*b3 - 873*b2 - 1764*b1 - 3821) * q^79 + (-31*b5 - 190*b4 - 112*b3 - 479*b2 - 85*b1 - 30120) * q^81 + (261*b5 - 307*b4 + 410*b3 + 587*b2 - 1575*b1 - 18550) * q^83 + (-264*b5 + 723*b4 + 261*b3 - 373*b2 - 1157*b1 - 38151) * q^85 + (925*b5 - 782*b4 - 1700*b3 - 71*b2 + 2653*b1 - 9303) * q^87 + (-258*b5 + 679*b4 + 521*b3 + 957*b2 - 1271*b1 - 33151) * q^89 + (-65*b5 - 719*b4 + 1144*b3 + 145*b2 - 2758*b1 - 30507) * q^91 + (-961*b1 - 2883) * q^93 + (638*b5 + 631*b4 - 1042*b3 - 641*b2 + 781*b1 - 2295) * q^95 + (-103*b5 - 874*b4 + 1175*b3 + 989*b2 - 1649*b1 - 46723) * q^97 + (-47*b5 - 67*b4 - 1098*b3 - b2 + 2813*b1 + 5318) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 20 q^{3} + 25 q^{5} + 39 q^{7} + 306 q^{9}+O(q^{10})$$ 6 * q + 20 * q^3 + 25 * q^5 + 39 * q^7 + 306 * q^9 $$6 q + 20 q^{3} + 25 q^{5} + 39 q^{7} + 306 q^{9} + 280 q^{11} + 1214 q^{13} + 1914 q^{15} + 796 q^{17} + 3147 q^{19} + 1082 q^{21} + 9122 q^{23} + 6481 q^{25} + 7316 q^{27} + 13020 q^{29} - 5766 q^{31} + 24804 q^{33} + 20059 q^{35} + 21678 q^{37} + 30680 q^{39} + 3227 q^{41} + 37882 q^{43} + 26169 q^{45} + 29708 q^{47} + 34849 q^{49} + 24432 q^{51} - 9976 q^{53} + 23758 q^{55} + 17318 q^{57} + 20573 q^{59} + 21610 q^{61} - 17697 q^{63} + 3894 q^{65} - 17024 q^{67} - 83692 q^{69} + 44509 q^{71} - 161864 q^{73} - 49430 q^{75} - 144202 q^{77} - 24420 q^{79} - 181158 q^{81} - 114160 q^{83} - 228882 q^{85} - 56180 q^{87} - 199742 q^{89} - 186774 q^{91} - 19220 q^{93} - 12793 q^{95} - 282951 q^{97} + 34060 q^{99}+O(q^{100})$$ 6 * q + 20 * q^3 + 25 * q^5 + 39 * q^7 + 306 * q^9 + 280 * q^11 + 1214 * q^13 + 1914 * q^15 + 796 * q^17 + 3147 * q^19 + 1082 * q^21 + 9122 * q^23 + 6481 * q^25 + 7316 * q^27 + 13020 * q^29 - 5766 * q^31 + 24804 * q^33 + 20059 * q^35 + 21678 * q^37 + 30680 * q^39 + 3227 * q^41 + 37882 * q^43 + 26169 * q^45 + 29708 * q^47 + 34849 * q^49 + 24432 * q^51 - 9976 * q^53 + 23758 * q^55 + 17318 * q^57 + 20573 * q^59 + 21610 * q^61 - 17697 * q^63 + 3894 * q^65 - 17024 * q^67 - 83692 * q^69 + 44509 * q^71 - 161864 * q^73 - 49430 * q^75 - 144202 * q^77 - 24420 * q^79 - 181158 * q^81 - 114160 * q^83 - 228882 * q^85 - 56180 * q^87 - 199742 * q^89 - 186774 * q^91 - 19220 * q^93 - 12793 * q^95 - 282951 * q^97 + 34060 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} - 847x^{4} + 1184x^{3} + 199815x^{2} - 13326x - 12452553$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -527\nu^{5} - 4605\nu^{4} + 205184\nu^{3} + 2426160\nu^{2} + 6013815\nu - 230250843 ) / 5613600$$ (-527*v^5 - 4605*v^4 + 205184*v^3 + 2426160*v^2 + 6013815*v - 230250843) / 5613600 $$\beta_{3}$$ $$=$$ $$( 667\nu^{5} + 14705\nu^{4} - 376864\nu^{3} - 7771760\nu^{2} + 40226685\nu + 691203303 ) / 5613600$$ (667*v^5 + 14705*v^4 - 376864*v^3 - 7771760*v^2 + 40226685*v + 691203303) / 5613600 $$\beta_{4}$$ $$=$$ $$( 303\nu^{5} - 11555\nu^{4} - 211176\nu^{3} + 8091560\nu^{2} + 26940465\nu - 1024004973 ) / 2806800$$ (303*v^5 - 11555*v^4 - 211176*v^3 + 8091560*v^2 + 26940465*v - 1024004973) / 2806800 $$\beta_{5}$$ $$=$$ $$( 2019\nu^{5} - 21415\nu^{4} - 1393248\nu^{3} + 13635280\nu^{2} + 188615445\nu - 1360828929 ) / 5613600$$ (2019*v^5 - 21415*v^4 - 1393248*v^3 + 13635280*v^2 + 188615445*v - 1360828929) / 5613600
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{5} + 2\beta_{4} + 2\beta_{3} + \beta_{2} - \beta _1 + 282$$ -b5 + 2*b4 + 2*b3 + b2 - b1 + 282 $$\nu^{3}$$ $$=$$ $$-7\beta_{5} + 4\beta_{4} - 18\beta_{3} - 45\beta_{2} + 374\beta _1 + 133$$ -7*b5 + 4*b4 - 18*b3 - 45*b2 + 374*b1 + 133 $$\nu^{4}$$ $$=$$ $$-622\beta_{5} + 1112\beta_{4} + 1454\beta_{3} + 736\beta_{2} - 982\beta _1 + 106065$$ -622*b5 + 1112*b4 + 1454*b3 + 736*b2 - 982*b1 + 106065 $$\nu^{5}$$ $$=$$ $$-1894\beta_{5} + 1048\beta_{4} - 10506\beta_{3} - 30000\beta_{2} + 161003\beta _1 - 13688$$ -1894*b5 + 1048*b4 - 10506*b3 - 30000*b2 + 161003*b1 - 13688

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −22.5479 −13.5834 −11.0199 9.40536 18.2423 21.5036
0 −19.5479 0 −16.3712 0 −21.6934 0 139.120 0
1.2 0 −10.5834 0 −84.7805 0 −158.720 0 −130.991 0
1.3 0 −8.01991 0 93.2607 0 174.764 0 −178.681 0
1.4 0 12.4054 0 −65.1867 0 116.704 0 −89.1069 0
1.5 0 21.2423 0 45.9749 0 −213.890 0 208.235 0
1.6 0 24.5036 0 52.1028 0 141.835 0 357.424 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$31$$ $$1$$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 124.6.a.b 6
4.b odd 2 1 496.6.a.f 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.6.a.b 6 1.a even 1 1 trivial
496.6.a.f 6 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} - 20T_{3}^{5} - 682T_{3}^{4} + 10628T_{3}^{3} + 145176T_{3}^{2} - 1091040T_{3} - 10713600$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(124))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} - 20 T^{5} - 682 T^{4} + \cdots - 10713600$$
$5$ $$T^{6} - 25 T^{5} + \cdots - 20212306536$$
$7$ $$T^{6} - 39 T^{5} + \cdots - 2130457113216$$
$11$ $$T^{6} - 280 T^{5} + \cdots - 1676525942016$$
$13$ $$T^{6} - 1214 T^{5} + \cdots + 15\!\cdots\!84$$
$17$ $$T^{6} - 796 T^{5} + \cdots - 58\!\cdots\!00$$
$19$ $$T^{6} - 3147 T^{5} + \cdots - 23\!\cdots\!20$$
$23$ $$T^{6} - 9122 T^{5} + \cdots + 43\!\cdots\!36$$
$29$ $$T^{6} - 13020 T^{5} + \cdots + 20\!\cdots\!76$$
$31$ $$(T + 961)^{6}$$
$37$ $$T^{6} - 21678 T^{5} + \cdots + 98\!\cdots\!88$$
$41$ $$T^{6} - 3227 T^{5} + \cdots + 28\!\cdots\!76$$
$43$ $$T^{6} - 37882 T^{5} + \cdots + 22\!\cdots\!40$$
$47$ $$T^{6} - 29708 T^{5} + \cdots - 97\!\cdots\!56$$
$53$ $$T^{6} + 9976 T^{5} + \cdots - 36\!\cdots\!92$$
$59$ $$T^{6} - 20573 T^{5} + \cdots - 63\!\cdots\!04$$
$61$ $$T^{6} - 21610 T^{5} + \cdots - 21\!\cdots\!24$$
$67$ $$T^{6} + 17024 T^{5} + \cdots + 49\!\cdots\!24$$
$71$ $$T^{6} - 44509 T^{5} + \cdots + 12\!\cdots\!00$$
$73$ $$T^{6} + 161864 T^{5} + \cdots - 11\!\cdots\!76$$
$79$ $$T^{6} + 24420 T^{5} + \cdots + 44\!\cdots\!80$$
$83$ $$T^{6} + 114160 T^{5} + \cdots + 12\!\cdots\!72$$
$89$ $$T^{6} + 199742 T^{5} + \cdots + 65\!\cdots\!28$$
$97$ $$T^{6} + 282951 T^{5} + \cdots - 27\!\cdots\!20$$