# Properties

 Label 120.3.c.a Level $120$ Weight $3$ Character orbit 120.c Analytic conductor $3.270$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$3.26976317232$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Defining polynomial: $$x^{12} + 34x^{10} + 305x^{8} + 616x^{6} + 305x^{4} + 34x^{2} + 1$$ x^12 + 34*x^10 + 305*x^8 + 616*x^6 + 305*x^4 + 34*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{15}\cdot 3^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} - \beta_{8} q^{5} - \beta_{9} q^{7} + ( - \beta_{4} + 1) q^{9}+O(q^{10})$$ q + b1 * q^3 - b8 * q^5 - b9 * q^7 + (-b4 + 1) * q^9 $$q + \beta_1 q^{3} - \beta_{8} q^{5} - \beta_{9} q^{7} + ( - \beta_{4} + 1) q^{9} + ( - \beta_{4} + \beta_{3}) q^{11} + (\beta_{11} + \beta_{10} - \beta_{9} - \beta_{5} - \beta_{2} + \beta_1) q^{13} + ( - \beta_{11} - \beta_{7} - \beta_{4} + \beta_{3} + 1) q^{15} + ( - \beta_{8} - \beta_{7} - \beta_{5}) q^{17} + (\beta_{11} - \beta_{10} + \beta_{6} + \beta_{4} - \beta_{3} + 1) q^{19} + ( - \beta_{11} + \beta_{10} - 2 \beta_{8} + 2 \beta_{7} - \beta_{3}) q^{21} + (2 \beta_{8} + 2 \beta_{7} + \beta_{5} + \beta_{2} + 3 \beta_1) q^{23} + ( - \beta_{9} - \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} - 3 \beta_1 + 2) q^{25} + ( - \beta_{11} - \beta_{10} + 3 \beta_{9} - \beta_{8} - \beta_{7} + 3 \beta_{5} + \beta_1) q^{27} + (\beta_{11} - \beta_{10} + 3 \beta_{8} - 3 \beta_{7} - \beta_{6} + \beta_{4} - 3 \beta_{3} + 1) q^{29} + (\beta_{11} - \beta_{10} - \beta_{6} + 3 \beta_{4} + \beta_{3} - 5) q^{31} + ( - \beta_{11} - \beta_{10} + 3 \beta_{9} + 2 \beta_{8} + 2 \beta_{7} - \beta_{2} + \beta_1) q^{33} + ( - \beta_{11} + \beta_{10} - 2 \beta_{8} + 4 \beta_{7} + \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + \cdots - 1) q^{35}+ \cdots + (4 \beta_{11} - 4 \beta_{10} - 2 \beta_{8} + 2 \beta_{7} - 2 \beta_{6} + 3 \beta_{4} + \cdots - 46) q^{99}+O(q^{100})$$ q + b1 * q^3 - b8 * q^5 - b9 * q^7 + (-b4 + 1) * q^9 + (-b4 + b3) * q^11 + (b11 + b10 - b9 - b5 - b2 + b1) * q^13 + (-b11 - b7 - b4 + b3 + 1) * q^15 + (-b8 - b7 - b5) * q^17 + (b11 - b10 + b6 + b4 - b3 + 1) * q^19 + (-b11 + b10 - 2*b8 + 2*b7 - b3) * q^21 + (2*b8 + 2*b7 + b5 + b2 + 3*b1) * q^23 + (-b9 - b6 + b4 + b3 + b2 - 3*b1 + 2) * q^25 + (-b11 - b10 + 3*b9 - b8 - b7 + 3*b5 + b1) * q^27 + (b11 - b10 + 3*b8 - 3*b7 - b6 + b4 - 3*b3 + 1) * q^29 + (b11 - b10 - b6 + 3*b4 + b3 - 5) * q^31 + (-b11 - b10 + 3*b9 + 2*b8 + 2*b7 - b2 + b1) * q^33 + (-b11 + b10 - 2*b8 + 4*b7 + b6 + 2*b5 + 2*b4 - b2 - 3*b1 - 1) * q^35 + (b11 + b10 + b9 - b5 + b2 - 5*b1) * q^37 + (5*b8 - 5*b7 + b6 - b4 - 3*b3 - 9) * q^39 + (-b11 + b10 + 2*b8 - 2*b7 + b6 + b4 + b3 - 1) * q^41 + (3*b2 - 9*b1) * q^43 + (-b11 + b10 + 3*b9 - b8 + 4*b7 - b6 - 3*b5 - b4 + 2*b3 + b2 + b1 - 7) * q^45 + (4*b8 + 4*b7 - 5*b5 - b2 - 3*b1) * q^47 + (3*b11 - 3*b10 + 3*b6 + 3*b4 - 3*b3 - 18) * q^49 + (-b11 + b10 + 2*b8 - 2*b7 - b6 - 2*b4 + 2*b3 + 3) * q^51 + (-5*b8 - 5*b7 + 2*b5 - 4*b2 - 12*b1) * q^53 + (2*b10 + b9 - 2*b6 - b5 + 2*b3 - 3*b2 + 7*b1 - 6) * q^55 + (-b11 - b10 - 3*b9 + 5*b8 + 5*b7 + 9*b5 + b2 + 3*b1) * q^57 + (2*b11 - 2*b10 + 2*b8 - 2*b7 - 2*b6 - b4 - 3*b3 + 2) * q^59 + (-b11 + b10 - b6 - b4 + b3 + 11) * q^61 + (-b11 - b10 - 7*b8 - 7*b7 - 3*b5 + 5*b2 - 3*b1) * q^63 + (-b11 + b10 + b8 - b7 + b6 + 7*b5 - 3*b4 + 5*b3 + 4*b2 + 12*b1 - 1) * q^65 + (2*b11 + 2*b10 - 4*b9 - 2*b5 - 3*b2 + 5*b1) * q^67 + (2*b11 - 2*b10 - 3*b8 + 3*b7 + b6 + b4 - 4*b3 + 25) * q^69 + (-12*b8 + 12*b7 + 2*b4 - 2*b3) * q^71 + (-4*b11 - 4*b10 - 2*b9 + 4*b5 - 2*b2 + 14*b1) * q^73 + (b11 + 3*b10 - 3*b9 - 3*b8 + b7 - 9*b5 + 3*b4 - b3 - 5*b2 + 2*b1 + 24) * q^75 + (-4*b8 - 4*b7 - 12*b5 + 4*b2 + 12*b1) * q^77 + (-b11 + b10 + 3*b6 - 5*b4 - 3*b3 + 39) * q^79 + (b11 - b10 - 2*b8 + 2*b7 + b6 - 3*b4 + 7*b3 - 16) * q^81 + (-2*b8 - 2*b7 + 2*b5 + 5*b2 + 15*b1) * q^83 + (2*b10 - 3*b9 - b6 - b5 - b4 + b3 + b2 - 5*b1 + 27) * q^85 + (4*b11 + 4*b10 - 3*b9 - 8*b8 - 8*b7 + 9*b5 - 5*b2 - b1) * q^87 + (12*b8 - 12*b7 - 6*b4 + 6*b3) * q^89 + (-4*b11 + 4*b10 - 8*b4) * q^91 + (3*b11 + 3*b10 - 9*b9 + 3*b8 + 3*b7 - 9*b5 - 9*b2 - 3*b1) * q^93 + (b11 - b10 + 6*b7 - b6 + 8*b5 - 7*b4 + 5*b3 - 4*b2 - 12*b1 + 1) * q^95 + (-4*b11 - 4*b10 + 4*b9 + 4*b5 + 8*b1) * q^97 + (4*b11 - 4*b10 - 2*b8 + 2*b7 - 2*b6 + 3*b4 + b3 - 46) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 8 q^{9}+O(q^{10})$$ 12 * q + 8 * q^9 $$12 q + 8 q^{9} + 16 q^{15} + 4 q^{21} + 36 q^{25} - 48 q^{31} - 128 q^{39} - 68 q^{45} - 252 q^{49} + 48 q^{51} - 48 q^{55} + 144 q^{61} + 268 q^{69} + 304 q^{75} + 432 q^{79} - 188 q^{81} + 336 q^{85} - 560 q^{99}+O(q^{100})$$ 12 * q + 8 * q^9 + 16 * q^15 + 4 * q^21 + 36 * q^25 - 48 * q^31 - 128 * q^39 - 68 * q^45 - 252 * q^49 + 48 * q^51 - 48 * q^55 + 144 * q^61 + 268 * q^69 + 304 * q^75 + 432 * q^79 - 188 * q^81 + 336 * q^85 - 560 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 34x^{10} + 305x^{8} + 616x^{6} + 305x^{4} + 34x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{11} - \nu^{10} + 35 \nu^{9} - 35 \nu^{8} + 340 \nu^{7} - 340 \nu^{6} + 956 \nu^{5} - 956 \nu^{4} + 1261 \nu^{3} - 1261 \nu^{2} + 1295 \nu - 647 ) / 216$$ (v^11 - v^10 + 35*v^9 - 35*v^8 + 340*v^7 - 340*v^6 + 956*v^5 - 956*v^4 + 1261*v^3 - 1261*v^2 + 1295*v - 647) / 216 $$\beta_{2}$$ $$=$$ $$( - \nu^{11} - \nu^{10} - 35 \nu^{9} - 35 \nu^{8} - 340 \nu^{7} - 340 \nu^{6} - 956 \nu^{5} - 956 \nu^{4} - 1261 \nu^{3} - 1261 \nu^{2} - 1295 \nu - 647 ) / 72$$ (-v^11 - v^10 - 35*v^9 - 35*v^8 - 340*v^7 - 340*v^6 - 956*v^5 - 956*v^4 - 1261*v^3 - 1261*v^2 - 1295*v - 647) / 72 $$\beta_{3}$$ $$=$$ $$( \nu^{11} - 4 \nu^{10} + 25 \nu^{9} - 138 \nu^{8} + 4 \nu^{7} - 1288 \nu^{6} - 1964 \nu^{5} - 3072 \nu^{4} - 3875 \nu^{3} - 2380 \nu^{2} - 851 \nu - 282 ) / 36$$ (v^11 - 4*v^10 + 25*v^9 - 138*v^8 + 4*v^7 - 1288*v^6 - 1964*v^5 - 3072*v^4 - 3875*v^3 - 2380*v^2 - 851*v - 282) / 36 $$\beta_{4}$$ $$=$$ $$( 5 \nu^{11} - 4 \nu^{10} + 173 \nu^{9} - 138 \nu^{8} + 1628 \nu^{7} - 1288 \nu^{6} + 4028 \nu^{5} - 3072 \nu^{4} + 3641 \nu^{3} - 2380 \nu^{2} + 1289 \nu - 282 ) / 36$$ (5*v^11 - 4*v^10 + 173*v^9 - 138*v^8 + 1628*v^7 - 1288*v^6 + 4028*v^5 - 3072*v^4 + 3641*v^3 - 2380*v^2 + 1289*v - 282) / 36 $$\beta_{5}$$ $$=$$ $$( -35\nu^{10} - 1189\nu^{8} - 10640\nu^{6} - 21220\nu^{4} - 9755\nu^{2} - 577 ) / 27$$ (-35*v^10 - 1189*v^8 - 10640*v^6 - 21220*v^4 - 9755*v^2 - 577) / 27 $$\beta_{6}$$ $$=$$ $$( 3 \nu^{11} + 64 \nu^{10} + 99 \nu^{9} + 2172 \nu^{8} + 816 \nu^{7} + 19384 \nu^{6} + 1032 \nu^{5} + 38208 \nu^{4} - 117 \nu^{3} + 17128 \nu^{2} + 219 \nu + 1038 ) / 18$$ (3*v^11 + 64*v^10 + 99*v^9 + 2172*v^8 + 816*v^7 + 19384*v^6 + 1032*v^5 + 38208*v^4 - 117*v^3 + 17128*v^2 + 219*v + 1038) / 18 $$\beta_{7}$$ $$=$$ $$( - 137 \nu^{11} - 26 \nu^{10} - 4649 \nu^{9} - 880 \nu^{8} - 41480 \nu^{7} - 7796 \nu^{6} - 81680 \nu^{5} - 14860 \nu^{4} - 36521 \nu^{3} - 5966 \nu^{2} - 2393 \nu - 280 ) / 36$$ (-137*v^11 - 26*v^10 - 4649*v^9 - 880*v^8 - 41480*v^7 - 7796*v^6 - 81680*v^5 - 14860*v^4 - 36521*v^3 - 5966*v^2 - 2393*v - 280) / 36 $$\beta_{8}$$ $$=$$ $$( 137 \nu^{11} - 26 \nu^{10} + 4649 \nu^{9} - 880 \nu^{8} + 41480 \nu^{7} - 7796 \nu^{6} + 81680 \nu^{5} - 14860 \nu^{4} + 36521 \nu^{3} - 5966 \nu^{2} + 2393 \nu - 280 ) / 36$$ (137*v^11 - 26*v^10 + 4649*v^9 - 880*v^8 + 41480*v^7 - 7796*v^6 + 81680*v^5 - 14860*v^4 + 36521*v^3 - 5966*v^2 + 2393*v - 280) / 36 $$\beta_{9}$$ $$=$$ $$( -655\nu^{11} - 22205\nu^{9} - 197572\nu^{7} - 383900\nu^{5} - 162043\nu^{3} - 7337\nu ) / 108$$ (-655*v^11 - 22205*v^9 - 197572*v^7 - 383900*v^5 - 162043*v^3 - 7337*v) / 108 $$\beta_{10}$$ $$=$$ $$( - 713 \nu^{11} - 167 \nu^{10} - 24191 \nu^{9} - 5665 \nu^{8} - 215732 \nu^{7} - 50492 \nu^{6} - 423692 \nu^{5} - 98884 \nu^{4} - 186485 \nu^{3} - 42923 \nu^{2} - 10235 \nu - 2461 ) / 72$$ (-713*v^11 - 167*v^10 - 24191*v^9 - 5665*v^8 - 215732*v^7 - 50492*v^6 - 423692*v^5 - 98884*v^4 - 186485*v^3 - 42923*v^2 - 10235*v - 2461) / 72 $$\beta_{11}$$ $$=$$ $$( - 733 \nu^{11} + 73 \nu^{10} - 24883 \nu^{9} + 2471 \nu^{8} - 222244 \nu^{7} + 21892 \nu^{6} - 439804 \nu^{5} + 41660 \nu^{4} - 201049 \nu^{3} + 16069 \nu^{2} - 15391 \nu + 491 ) / 72$$ (-733*v^11 + 73*v^10 - 24883*v^9 + 2471*v^8 - 222244*v^7 + 21892*v^6 - 439804*v^5 + 41660*v^4 - 201049*v^3 + 16069*v^2 - 15391*v + 491) / 72
 $$\nu$$ $$=$$ $$( - 2 \beta_{11} - 4 \beta_{10} + 6 \beta_{9} - 3 \beta_{8} + 3 \beta_{7} - \beta_{6} + 3 \beta_{5} - 6 \beta_{4} + 4 \beta_{3} - 4 \beta_{2} + 18 \beta _1 + 1 ) / 48$$ (-2*b11 - 4*b10 + 6*b9 - 3*b8 + 3*b7 - b6 + 3*b5 - 6*b4 + 4*b3 - 4*b2 + 18*b1 + 1) / 48 $$\nu^{2}$$ $$=$$ $$( \beta_{11} - \beta_{10} - 4\beta_{6} - 9\beta_{5} + 6\beta_{4} + 4\beta_{3} - 12\beta_{2} - 36\beta _1 - 140 ) / 24$$ (b11 - b10 - 4*b6 - 9*b5 + 6*b4 + 4*b3 - 12*b2 - 36*b1 - 140) / 24 $$\nu^{3}$$ $$=$$ $$( 64 \beta_{11} + 86 \beta_{10} - 114 \beta_{9} + 105 \beta_{8} - 105 \beta_{7} + 11 \beta_{6} - 75 \beta_{5} + 102 \beta_{4} - 80 \beta_{3} + 16 \beta_{2} - 198 \beta _1 - 11 ) / 48$$ (64*b11 + 86*b10 - 114*b9 + 105*b8 - 105*b7 + 11*b6 - 75*b5 + 102*b4 - 80*b3 + 16*b2 - 198*b1 - 11) / 48 $$\nu^{4}$$ $$=$$ $$( 6\beta_{8} + 6\beta_{7} + 18\beta_{6} + 45\beta_{5} - 18\beta_{4} - 18\beta_{3} + 36\beta_{2} + 108\beta _1 + 382 ) / 4$$ (6*b8 + 6*b7 + 18*b6 + 45*b5 - 18*b4 - 18*b3 + 36*b2 + 108*b1 + 382) / 4 $$\nu^{5}$$ $$=$$ $$( - 1466 \beta_{11} - 1744 \beta_{10} + 1950 \beta_{9} - 2649 \beta_{8} + 2649 \beta_{7} - 139 \beta_{6} + 1605 \beta_{5} - 1734 \beta_{4} + 1456 \beta_{3} - 64 \beta_{2} + 3402 \beta _1 + 139 ) / 48$$ (-1466*b11 - 1744*b10 + 1950*b9 - 2649*b8 + 2649*b7 - 139*b6 + 1605*b5 - 1734*b4 + 1456*b3 - 64*b2 + 3402*b1 + 139) / 48 $$\nu^{6}$$ $$=$$ $$( - 305 \beta_{11} + 305 \beta_{10} - 1296 \beta_{8} - 1296 \beta_{7} - 2452 \beta_{6} - 6345 \beta_{5} + 1842 \beta_{4} + 2452 \beta_{3} - 4140 \beta_{2} - 12420 \beta _1 - 42620 ) / 24$$ (-305*b11 + 305*b10 - 1296*b8 - 1296*b7 - 2452*b6 - 6345*b5 + 1842*b4 + 2452*b3 - 4140*b2 - 12420*b1 - 42620) / 24 $$\nu^{7}$$ $$=$$ $$( 31516 \beta_{11} + 35630 \beta_{10} - 35322 \beta_{9} + 59883 \beta_{8} - 59883 \beta_{7} + 2057 \beta_{6} - 33573 \beta_{5} + 31638 \beta_{4} - 27524 \beta_{3} - 980 \beta_{2} - 64206 \beta _1 - 2057 ) / 48$$ (31516*b11 + 35630*b10 - 35322*b9 + 59883*b8 - 59883*b7 + 2057*b6 - 33573*b5 + 31638*b4 - 27524*b3 - 980*b2 - 64206*b1 - 2057) / 48 $$\nu^{8}$$ $$=$$ $$405 \beta_{11} - 405 \beta_{10} + 1380 \beta_{8} + 1380 \beta_{7} + 2205 \beta_{6} + 5787 \beta_{5} - 1395 \beta_{4} - 2205 \beta_{3} + 3420 \beta_{2} + 10260 \beta _1 + 34642$$ 405*b11 - 405*b10 + 1380*b8 + 1380*b7 + 2205*b6 + 5787*b5 - 1395*b4 - 2205*b3 + 3420*b2 + 10260*b1 + 34642 $$\nu^{9}$$ $$=$$ $$( - 663278 \beta_{11} - 731260 \beta_{10} + 674682 \beta_{9} - 1291845 \beta_{8} + 1291845 \beta_{7} - 33991 \beta_{6} + 697269 \beta_{5} - 607722 \beta_{4} + 539740 \beta_{3} + \cdots + 33991 ) / 48$$ (-663278*b11 - 731260*b10 + 674682*b9 - 1291845*b8 + 1291845*b7 - 33991*b6 + 697269*b5 - 607722*b4 + 539740*b3 + 44084*b2 + 1262286*b1 + 33991) / 48 $$\nu^{10}$$ $$=$$ $$( - 237761 \beta_{11} + 237761 \beta_{10} - 752976 \beta_{8} - 752976 \beta_{7} - 1116724 \beta_{6} - 2950551 \beta_{5} + 641202 \beta_{4} + 1116724 \beta_{3} - 1657428 \beta_{2} + \cdots - 16638620 ) / 24$$ (-237761*b11 + 237761*b10 - 752976*b8 - 752976*b7 - 1116724*b6 - 2950551*b5 + 641202*b4 + 1116724*b3 - 1657428*b2 - 4972284*b1 - 16638620) / 24 $$\nu^{11}$$ $$=$$ $$( 13822672 \beta_{11} + 15043898 \beta_{10} - 13332606 \beta_{9} + 27258279 \beta_{8} - 27258279 \beta_{7} + 610613 \beta_{6} - 14433285 \beta_{5} + 12050202 \beta_{4} + \cdots - 610613 ) / 48$$ (13822672*b11 + 15043898*b10 - 13332606*b9 + 27258279*b8 - 27258279*b7 + 610613*b6 - 14433285*b5 + 12050202*b4 - 10828976*b3 - 1165280*b2 - 25370730*b1 - 610613) / 48

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
89.1
 − 0.220185i 0.220185i − 0.304307i 0.304307i − 0.723561i 0.723561i − 1.38205i 1.38205i − 3.28615i 3.28615i − 4.54164i 4.54164i
0 −2.72256 1.26002i 0 0.689011 + 4.95230i 0 0.735748i 0 5.82469 + 6.86097i 0
89.2 0 −2.72256 + 1.26002i 0 0.689011 4.95230i 0 0.735748i 0 5.82469 6.86097i 0
89.3 0 −2.49147 1.67109i 0 −4.19906 2.71439i 0 12.7692i 0 3.41489 + 8.32698i 0
89.4 0 −2.49147 + 1.67109i 0 −4.19906 + 2.71439i 0 12.7692i 0 3.41489 8.32698i 0
89.5 0 −0.938195 2.84952i 0 4.88807 + 1.05205i 0 6.81219i 0 −7.23958 + 5.34682i 0
89.6 0 −0.938195 + 2.84952i 0 4.88807 1.05205i 0 6.81219i 0 −7.23958 5.34682i 0
89.7 0 0.938195 2.84952i 0 −4.88807 1.05205i 0 6.81219i 0 −7.23958 5.34682i 0
89.8 0 0.938195 + 2.84952i 0 −4.88807 + 1.05205i 0 6.81219i 0 −7.23958 + 5.34682i 0
89.9 0 2.49147 1.67109i 0 4.19906 + 2.71439i 0 12.7692i 0 3.41489 8.32698i 0
89.10 0 2.49147 + 1.67109i 0 4.19906 2.71439i 0 12.7692i 0 3.41489 + 8.32698i 0
89.11 0 2.72256 1.26002i 0 −0.689011 4.95230i 0 0.735748i 0 5.82469 6.86097i 0
89.12 0 2.72256 + 1.26002i 0 −0.689011 + 4.95230i 0 0.735748i 0 5.82469 + 6.86097i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 89.12 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.b even 2 1 inner
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 120.3.c.a 12
3.b odd 2 1 inner 120.3.c.a 12
4.b odd 2 1 240.3.c.e 12
5.b even 2 1 inner 120.3.c.a 12
5.c odd 4 2 600.3.l.g 12
8.b even 2 1 960.3.c.k 12
8.d odd 2 1 960.3.c.j 12
12.b even 2 1 240.3.c.e 12
15.d odd 2 1 inner 120.3.c.a 12
15.e even 4 2 600.3.l.g 12
20.d odd 2 1 240.3.c.e 12
20.e even 4 2 1200.3.l.y 12
24.f even 2 1 960.3.c.j 12
24.h odd 2 1 960.3.c.k 12
40.e odd 2 1 960.3.c.j 12
40.f even 2 1 960.3.c.k 12
60.h even 2 1 240.3.c.e 12
60.l odd 4 2 1200.3.l.y 12
120.i odd 2 1 960.3.c.k 12
120.m even 2 1 960.3.c.j 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.3.c.a 12 1.a even 1 1 trivial
120.3.c.a 12 3.b odd 2 1 inner
120.3.c.a 12 5.b even 2 1 inner
120.3.c.a 12 15.d odd 2 1 inner
240.3.c.e 12 4.b odd 2 1
240.3.c.e 12 12.b even 2 1
240.3.c.e 12 20.d odd 2 1
240.3.c.e 12 60.h even 2 1
600.3.l.g 12 5.c odd 4 2
600.3.l.g 12 15.e even 4 2
960.3.c.j 12 8.d odd 2 1
960.3.c.j 12 24.f even 2 1
960.3.c.j 12 40.e odd 2 1
960.3.c.j 12 120.m even 2 1
960.3.c.k 12 8.b even 2 1
960.3.c.k 12 24.h odd 2 1
960.3.c.k 12 40.f even 2 1
960.3.c.k 12 120.i odd 2 1
1200.3.l.y 12 20.e even 4 2
1200.3.l.y 12 60.l odd 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(120, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12} - 4 T^{10} + 55 T^{8} + \cdots + 531441$$
$5$ $$T^{12} - 18 T^{10} + \cdots + 244140625$$
$7$ $$(T^{6} + 210 T^{4} + 7680 T^{2} + \cdots + 4096)^{2}$$
$11$ $$(T^{6} + 336 T^{4} + 34944 T^{2} + \cdots + 1083392)^{2}$$
$13$ $$(T^{6} + 768 T^{4} + 157824 T^{2} + \cdots + 6553600)^{2}$$
$17$ $$(T^{6} - 264 T^{4} + 13968 T^{2} + \cdots - 165888)^{2}$$
$19$ $$(T^{3} - 780 T + 5648)^{4}$$
$23$ $$(T^{6} - 1050 T^{4} + 289440 T^{2} + \cdots - 13148192)^{2}$$
$29$ $$(T^{6} + 4416 T^{4} + 4295808 T^{2} + \cdots + 807698432)^{2}$$
$31$ $$(T^{3} + 12 T^{2} - 1692 T - 31104)^{4}$$
$37$ $$(T^{6} + 2592 T^{4} + 1828992 T^{2} + \cdots + 237899776)^{2}$$
$41$ $$(T^{6} + 4356 T^{4} + 4826112 T^{2} + \cdots + 772087808)^{2}$$
$43$ $$(T^{6} + 4050 T^{4} + \cdots + 1224440064)^{2}$$
$47$ $$(T^{6} - 5754 T^{4} + \cdots - 2393766432)^{2}$$
$53$ $$(T^{6} - 11400 T^{4} + \cdots - 21525635072)^{2}$$
$59$ $$(T^{6} + 11856 T^{4} + \cdots + 1313998848)^{2}$$
$61$ $$(T^{3} - 36 T^{2} - 348 T + 1984)^{4}$$
$67$ $$(T^{6} + 5106 T^{4} + 4443264 T^{2} + \cdots + 1763584)^{2}$$
$71$ $$(T^{6} + 19200 T^{4} + \cdots + 124856041472)^{2}$$
$73$ $$(T^{6} + 26400 T^{4} + \cdots + 238331428864)^{2}$$
$79$ $$(T^{3} - 108 T^{2} - 3516 T + 234400)^{4}$$
$83$ $$(T^{6} - 14586 T^{4} + \cdots - 387755552)^{2}$$
$89$ $$(T^{6} + 27648 T^{4} + \cdots + 278628139008)^{2}$$
$97$ $$(T^{6} + 13344 T^{4} + 1572864 T^{2} + \cdots + 16777216)^{2}$$