# Properties

 Label 121.5.b.b Level $121$ Weight $5$ Character orbit 121.b Analytic conductor $12.508$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [121,5,Mod(120,121)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(121, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("121.120");

S:= CuspForms(chi, 5);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$121 = 11^{2}$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 121.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$12.5077655331$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} + 115x^{10} + 5030x^{8} + 102975x^{6} + 953170x^{4} + 2910655x^{2} + 73205$$ x^12 + 115*x^10 + 5030*x^8 + 102975*x^6 + 953170*x^4 + 2910655*x^2 + 73205 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}\cdot 11^{5}$$ Twist minimal: no (minimal twist has level 11) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + (\beta_{3} - 2) q^{3} + (\beta_{7} - 6) q^{4} + ( - \beta_{3} + \beta_1 - 2) q^{5} + ( - \beta_{11} - \beta_{10} - \beta_{8} + \beta_{6} - 2 \beta_{4} + 4 \beta_{2}) q^{6} + ( - \beta_{11} - \beta_{8} + \beta_{6} - 4 \beta_{2}) q^{7} + ( - 2 \beta_{11} - 2 \beta_{10} + \beta_{4}) q^{8} + (2 \beta_{9} - 3 \beta_{7} - 3 \beta_{5} - 4 \beta_{3} + 2 \beta_1 + 44) q^{9}+O(q^{10})$$ q - b2 * q^2 + (b3 - 2) * q^3 + (b7 - 6) * q^4 + (-b3 + b1 - 2) * q^5 + (-b11 - b10 - b8 + b6 - 2*b4 + 4*b2) * q^6 + (-b11 - b8 + b6 - 4*b2) * q^7 + (-2*b11 - 2*b10 + b4) * q^8 + (2*b9 - 3*b7 - 3*b5 - 4*b3 + 2*b1 + 44) * q^9 $$q - \beta_{2} q^{2} + (\beta_{3} - 2) q^{3} + (\beta_{7} - 6) q^{4} + ( - \beta_{3} + \beta_1 - 2) q^{5} + ( - \beta_{11} - \beta_{10} - \beta_{8} + \beta_{6} - 2 \beta_{4} + 4 \beta_{2}) q^{6} + ( - \beta_{11} - \beta_{8} + \beta_{6} - 4 \beta_{2}) q^{7} + ( - 2 \beta_{11} - 2 \beta_{10} + \beta_{4}) q^{8} + (2 \beta_{9} - 3 \beta_{7} - 3 \beta_{5} - 4 \beta_{3} + 2 \beta_1 + 44) q^{9} + (\beta_{11} + 3 \beta_{10} + 2 \beta_{8} - 3 \beta_{6} + 3 \beta_{4} + 4 \beta_{2}) q^{10} + ( - 2 \beta_{9} - 6 \beta_{7} + 9 \beta_{5} - 5 \beta_{3} + 7 \beta_1 + 49) q^{12} + ( - 4 \beta_{11} + 3 \beta_{8} - 3 \beta_{6} + 5 \beta_{2}) q^{13} + ( - 3 \beta_{9} - 4 \beta_{7} + 2 \beta_{5} - 5 \beta_{3} + 7 \beta_1 - 93) q^{14} + ( - 3 \beta_{9} + 9 \beta_{7} + 12 \beta_{5} + \beta_{3} - 3 \beta_1 - 116) q^{15} + ( - 9 \beta_{7} + 8 \beta_{5} + 12 \beta_1 - 84) q^{16} + (2 \beta_{10} + 8 \beta_{8} - 9 \beta_{6} + \beta_{4} + 8 \beta_{2}) q^{17} + (5 \beta_{11} + 17 \beta_{10} + 5 \beta_{8} + 9 \beta_{6} + 2 \beta_{4} - 54 \beta_{2}) q^{18} + (7 \beta_{11} - 12 \beta_{10} + 4 \beta_{8} - \beta_{6} - 9 \beta_{4} - 35 \beta_{2}) q^{19} + (4 \beta_{9} + 4 \beta_{7} - 8 \beta_{5} + 14 \beta_{3} - 6 \beta_1 + 56) q^{20} + ( - 2 \beta_{11} + 10 \beta_{10} - 11 \beta_{8} - \beta_{6} - 16 \beta_{4} + \cdots - 31 \beta_{2}) q^{21}+ \cdots + (81 \beta_{11} + 11 \beta_{10} + 186 \beta_{8} - 263 \beta_{6} + 127 \beta_{4} + \cdots - 877 \beta_{2}) q^{98}+O(q^{100})$$ q - b2 * q^2 + (b3 - 2) * q^3 + (b7 - 6) * q^4 + (-b3 + b1 - 2) * q^5 + (-b11 - b10 - b8 + b6 - 2*b4 + 4*b2) * q^6 + (-b11 - b8 + b6 - 4*b2) * q^7 + (-2*b11 - 2*b10 + b4) * q^8 + (2*b9 - 3*b7 - 3*b5 - 4*b3 + 2*b1 + 44) * q^9 + (b11 + 3*b10 + 2*b8 - 3*b6 + 3*b4 + 4*b2) * q^10 + (-2*b9 - 6*b7 + 9*b5 - 5*b3 + 7*b1 + 49) * q^12 + (-4*b11 + 3*b8 - 3*b6 + 5*b2) * q^13 + (-3*b9 - 4*b7 + 2*b5 - 5*b3 + 7*b1 - 93) * q^14 + (-3*b9 + 9*b7 + 12*b5 + b3 - 3*b1 - 116) * q^15 + (-9*b7 + 8*b5 + 12*b1 - 84) * q^16 + (2*b10 + 8*b8 - 9*b6 + b4 + 8*b2) * q^17 + (5*b11 + 17*b10 + 5*b8 + 9*b6 + 2*b4 - 54*b2) * q^18 + (7*b11 - 12*b10 + 4*b8 - b6 - 9*b4 - 35*b2) * q^19 + (4*b9 + 4*b7 - 8*b5 + 14*b3 - 6*b1 + 56) * q^20 + (-2*b11 + 10*b10 - 11*b8 - b6 - 16*b4 - 31*b2) * q^21 + (8*b7 + 38*b5 + 20*b3 + 122) * q^23 + (12*b11 + 18*b10 - 9*b8 - 26*b6 - 10*b4 - 59*b2) * q^24 + (6*b9 - 7*b7 - 8*b5 + 9*b3 - 9*b1 - 312) * q^25 + (2*b9 - 23*b7 + 43*b5 - 13*b3 - 14*b1 + 76) * q^26 + (-5*b9 + 30*b5 + 23*b3 - 41*b1 - 346) * q^27 + (2*b11 + 20*b10 - 26*b6 + 18*b4 - 18*b2) * q^28 + (-2*b11 + 23*b10 - 9*b8 + 32*b6 - 9*b4 + 70*b2) * q^29 + (-4*b11 - 22*b10 - 10*b8 - 26*b6 + 19*b4 + 151*b2) * q^30 + (-23*b9 - 27*b7 + 9*b5 + 25*b3 + 12*b1 + 179) * q^31 + (-6*b11 + 18*b10 + 4*b8 - 32*b6 + 27*b4 + 26*b2) * q^32 + (16*b9 + 12*b7 + 36*b5 + 16*b3 - 57*b1 + 155) * q^34 + (-5*b11 + 20*b10 + 16*b8 - 30*b6 + 27*b4 + 96*b2) * q^35 + (2*b9 + 90*b7 - 21*b5 - 23*b3 - 25*b1 - 743) * q^36 + (16*b9 - 59*b7 - 16*b5 + 61*b3 + 75*b1 + 61) * q^37 + (21*b9 + 79*b7 + b5 - 40*b3 + 8*b1 - 690) * q^38 + (13*b11 - 2*b10 - 44*b8 - 32*b6 - 15*b4 - 194*b2) * q^39 + (-18*b11 + 18*b10 + 12*b8 + 14*b6 + 6*b4 + 96*b2) * q^40 + (-14*b11 - 27*b10 - 4*b8 + 36*b6 - 13*b4 + 123*b2) * q^41 + (-10*b9 + 129*b7 + 9*b5 - 147*b3 - 40*b1 - 762) * q^42 + (14*b11 + 34*b10 + 61*b8 + 6*b6 - 143*b2) * q^43 + (-4*b9 + 33*b7 - 12*b5 - 148*b3 + 32*b1 + 413) * q^45 + (2*b11 + 2*b10 - 58*b8 - 18*b6 + 6*b4 - 78*b2) * q^46 + (24*b9 - 150*b7 + 19*b5 + 80*b3 - 3*b1 + 809) * q^47 + (14*b9 + 174*b7 + 3*b5 - 121*b3 - 43*b1 - 337) * q^48 + (-28*b9 - 14*b7 - 42*b5 - 67*b3 + 21*b1 + 575) * q^49 + (-9*b11 - 3*b10 - 22*b8 + 77*b6 - 48*b4 + 282*b2) * q^50 + (b11 - 68*b10 - 26*b8 - 24*b6 + 28*b4 + 30*b2) * q^51 + (36*b11 + 80*b10 - 62*b6 + 30*b4 - 380*b2) * q^52 + (36*b9 - 46*b7 - 32*b5 + 99*b3 + 39*b1 + 798) * q^53 + (12*b11 - 90*b10 - 84*b8 + 40*b6 - 52*b4 + 133*b2) * q^54 + (-14*b9 - 18*b7 - 86*b5 + 72*b3 - 12*b1 - 1728) * q^56 + (-22*b11 - 88*b10 + 95*b8 + 145*b6 - 17*b4 + 103*b2) * q^57 + (-89*b9 + 14*b7 + 74*b5 - 89*b3 + 15*b1 + 1053) * q^58 + (-82*b9 - 69*b7 - 120*b5 + 43*b3 - 20*b1 - 216) * q^59 + (22*b9 - 198*b7 + 142*b3 + 34*b1 + 2116) * q^60 + (10*b11 - 74*b10 + 139*b8 + 19*b6 + 30*b4 - 177*b2) * q^61 + (61*b11 - 7*b10 + 24*b8 - 169*b6 - 33*b4 - 530*b2) * q^62 + (-44*b11 + 16*b10 + 55*b8 + 105*b6 - 11*b4 - 120*b2) * q^63 + (40*b9 - 233*b7 + 44*b5 + 196*b3 + 72*b1 - 588) * q^64 + (-34*b11 - 25*b10 - 27*b8 + 48*b6 + 59*b4 - 8*b2) * q^65 + (-66*b9 - 317*b7 - 161*b5 + 291*b3 + 231*b1 + 1341) * q^67 + (-20*b11 - 38*b10 - 13*b8 + 62*b6 - 41*b4 - 63*b2) * q^68 + (62*b9 + 120*b7 - 102*b5 - 100*b3 + 248*b1 + 1598) * q^69 + (35*b9 - 135*b7 - b5 + 212*b3 - 159*b1 + 2133) * q^70 + (-17*b9 + 93*b7 - 203*b5 + 129*b3 - 12*b1 + 759) * q^71 + (-100*b11 + 50*b10 + 95*b8 + 206*b6 + 120*b4 + 689*b2) * q^72 + (-8*b11 - 235*b10 + 34*b8 + 8*b6 - 75*b4 - 103*b2) * q^73 + (25*b11 + 239*b10 - 2*b8 + 39*b6 - 138*b4 - 85*b2) * q^74 + (39*b9 - 159*b7 - 93*b5 - 192*b3 - 48*b1 + 1941) * q^75 + (-26*b11 - 230*b10 + 69*b8 + 74*b6 + 3*b4 + 1017*b2) * q^76 + (79*b9 + 327*b7 - 405*b5 - 112*b3 + b1 - 3475) * q^78 + (56*b11 - 136*b10 - 73*b8 + 329*b6 - 195*b4 - 12*b2) * q^79 + (-100*b7 + 104*b5 + 212*b3 - 132*b1 + 2480) * q^80 + (-50*b9 + 168*b7 - 330*b5 - 354*b3 + 40*b1 - 1080) * q^81 + (-59*b9 - 290*b7 + 167*b5 - 164*b3 + 216*b1 + 2520) * q^82 + (-36*b11 - 268*b10 - 166*b8 - 75*b6 - 53*b4 + 377*b2) * q^83 + (-124*b11 - 52*b10 - 58*b8 - 162*b6 + 146*b4 + 1014*b2) * q^84 + (-6*b11 - 67*b10 - 81*b8 + 156*b6 - 15*b4 - 500*b2) * q^85 + (-32*b9 + 418*b7 + 375*b5 + 117*b3 - 315*b1 - 4113) * q^86 + (56*b11 + 74*b10 - 121*b8 + 109*b6 - 113*b4 - 818*b2) * q^87 + (12*b9 + 162*b7 + 79*b5 - 281*b3 - 359*b1 + 147) * q^89 + (74*b11 + 122*b10 + 200*b8 - 228*b6 + 353*b4 - 255*b2) * q^90 + (-49*b9 + 189*b7 + 140*b5 - 37*b3 - 441*b1 - 1466) * q^91 + (36*b9 + 156*b7 + 130*b5 + 318*b3 + 122*b1 + 1066) * q^92 + (78*b9 + 489*b7 - 444*b5 - 368*b3 - 138*b1 + 277) * q^93 + (215*b11 + 305*b10 - 150*b8 + 235*b6 - 318*b4 - 2031*b2) * q^94 + (143*b11 + 4*b10 + 38*b8 - 108*b6 + 39*b4 + 402*b2) * q^95 + (-46*b11 + 20*b10 - 97*b8 - 356*b6 + 202*b4 + 811*b2) * q^96 + (-180*b9 + 380*b7 - 291*b5 - 348*b3 + 528*b1 - 539) * q^97 + (81*b11 + 11*b10 + 186*b8 - 263*b6 + 127*b4 - 877*b2) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 26 q^{3} - 68 q^{4} - 28 q^{5} + 486 q^{9}+O(q^{10})$$ 12 * q - 26 * q^3 - 68 * q^4 - 28 * q^5 + 486 * q^9 $$12 q - 26 q^{3} - 68 q^{4} - 28 q^{5} + 486 q^{9} + 594 q^{12} - 1140 q^{14} - 1256 q^{15} - 1068 q^{16} + 632 q^{20} + 1684 q^{23} - 3808 q^{25} + 1180 q^{26} - 3752 q^{27} + 2064 q^{31} + 2370 q^{34} - 8494 q^{36} - 236 q^{37} - 8010 q^{38} - 8000 q^{42} + 5136 q^{45} + 8984 q^{47} - 2886 q^{48} + 6712 q^{49} + 8624 q^{53} - 21340 q^{56} + 13580 q^{58} - 3226 q^{59} + 24024 q^{60} - 8708 q^{64} + 12154 q^{67} + 17508 q^{69} + 25440 q^{70} + 8144 q^{71} + 22614 q^{75} - 42920 q^{78} + 30352 q^{80} - 13600 q^{81} + 29350 q^{82} - 43650 q^{86} + 5554 q^{89} - 13080 q^{91} + 12684 q^{92} + 3868 q^{93} - 8446 q^{97}+O(q^{100})$$ 12 * q - 26 * q^3 - 68 * q^4 - 28 * q^5 + 486 * q^9 + 594 * q^12 - 1140 * q^14 - 1256 * q^15 - 1068 * q^16 + 632 * q^20 + 1684 * q^23 - 3808 * q^25 + 1180 * q^26 - 3752 * q^27 + 2064 * q^31 + 2370 * q^34 - 8494 * q^36 - 236 * q^37 - 8010 * q^38 - 8000 * q^42 + 5136 * q^45 + 8984 * q^47 - 2886 * q^48 + 6712 * q^49 + 8624 * q^53 - 21340 * q^56 + 13580 * q^58 - 3226 * q^59 + 24024 * q^60 - 8708 * q^64 + 12154 * q^67 + 17508 * q^69 + 25440 * q^70 + 8144 * q^71 + 22614 * q^75 - 42920 * q^78 + 30352 * q^80 - 13600 * q^81 + 29350 * q^82 - 43650 * q^86 + 5554 * q^89 - 13080 * q^91 + 12684 * q^92 + 3868 * q^93 - 8446 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 115x^{10} + 5030x^{8} + 102975x^{6} + 953170x^{4} + 2910655x^{2} + 73205$$ :

 $$\beta_{1}$$ $$=$$ $$( 6428\nu^{10} + 537997\nu^{8} + 15377352\nu^{6} + 177671499\nu^{4} + 787262006\nu^{2} + 1398748747 ) / 60901236$$ (6428*v^10 + 537997*v^8 + 15377352*v^6 + 177671499*v^4 + 787262006*v^2 + 1398748747) / 60901236 $$\beta_{2}$$ $$=$$ $$( 31452 \nu^{11} + 2737673 \nu^{9} + 79586472 \nu^{7} + 827426403 \nu^{5} + 1134660378 \nu^{3} - 7303391689 \nu ) / 669913596$$ (31452*v^11 + 2737673*v^9 + 79586472*v^7 + 827426403*v^5 + 1134660378*v^3 - 7303391689*v) / 669913596 $$\beta_{3}$$ $$=$$ $$( - 66971 \nu^{10} - 5921271 \nu^{8} - 178739757 \nu^{6} - 2088065409 \nu^{4} - 6870421673 \nu^{2} + 477411429 ) / 60901236$$ (-66971*v^10 - 5921271*v^8 - 178739757*v^6 - 2088065409*v^4 - 6870421673*v^2 + 477411429) / 60901236 $$\beta_{4}$$ $$=$$ $$( - 86029 \nu^{11} - 7196850 \nu^{9} - 195308019 \nu^{7} - 1742959254 \nu^{5} + 418754123 \nu^{3} + 22453083312 \nu ) / 669913596$$ (-86029*v^11 - 7196850*v^9 - 195308019*v^7 - 1742959254*v^5 + 418754123*v^3 + 22453083312*v) / 669913596 $$\beta_{5}$$ $$=$$ $$( 344\nu^{10} + 30727\nu^{8} + 942126\nu^{6} + 11311125\nu^{4} + 40042370\nu^{2} + 3293983 ) / 178596$$ (344*v^10 + 30727*v^8 + 942126*v^6 + 11311125*v^4 + 40042370*v^2 + 3293983) / 178596 $$\beta_{6}$$ $$=$$ $$( - 6526 \nu^{11} - 609888 \nu^{9} - 20067903 \nu^{7} - 274692837 \nu^{5} - 1411940227 \nu^{3} - 2508246510 \nu ) / 30450618$$ (-6526*v^11 - 609888*v^9 - 20067903*v^7 - 274692837*v^5 - 1411940227*v^3 - 2508246510*v) / 30450618 $$\beta_{7}$$ $$=$$ $$( - 170538 \nu^{10} - 15246553 \nu^{8} - 467631960 \nu^{6} - 5595088959 \nu^{4} - 19274817024 \nu^{2} + 669602747 ) / 60901236$$ (-170538*v^10 - 15246553*v^8 - 467631960*v^6 - 5595088959*v^4 - 19274817024*v^2 + 669602747) / 60901236 $$\beta_{8}$$ $$=$$ $$( 127386 \nu^{11} + 11038024 \nu^{9} + 317228919 \nu^{7} + 3171684078 \nu^{5} + 1902331650 \nu^{3} - 48594396785 \nu ) / 334956798$$ (127386*v^11 + 11038024*v^9 + 317228919*v^7 + 3171684078*v^5 + 1902331650*v^3 - 48594396785*v) / 334956798 $$\beta_{9}$$ $$=$$ $$( 235513 \nu^{10} + 21188149 \nu^{8} + 654204021 \nu^{6} + 7872498921 \nu^{4} + 27304989541 \nu^{2} + 1997452633 ) / 60901236$$ (235513*v^10 + 21188149*v^8 + 654204021*v^6 + 7872498921*v^4 + 27304989541*v^2 + 1997452633) / 60901236 $$\beta_{10}$$ $$=$$ $$( - 157260 \nu^{11} - 13688365 \nu^{9} - 397932360 \nu^{7} - 4137132015 \nu^{5} - 5673301890 \nu^{3} + 40201483223 \nu ) / 334956798$$ (-157260*v^11 - 13688365*v^9 - 397932360*v^7 - 4137132015*v^5 - 5673301890*v^3 + 40201483223*v) / 334956798 $$\beta_{11}$$ $$=$$ $$( - 760831 \nu^{11} - 69558041 \nu^{9} - 2226629937 \nu^{7} - 29377144047 \nu^{5} - 140005058425 \nu^{3} - 179167525229 \nu ) / 669913596$$ (-760831*v^11 - 69558041*v^9 - 2226629937*v^7 - 29377144047*v^5 - 140005058425*v^3 - 179167525229*v) / 669913596
 $$\nu$$ $$=$$ $$( \beta_{10} + 10\beta_{2} ) / 11$$ (b10 + 10*b2) / 11 $$\nu^{2}$$ $$=$$ $$( 2\beta_{9} + \beta_{7} - 3\beta_{5} + 8\beta _1 - 205 ) / 11$$ (2*b9 + b7 - 3*b5 + 8*b1 - 205) / 11 $$\nu^{3}$$ $$=$$ $$( \beta_{11} - 38\beta_{10} - 12\beta_{8} - 2\beta_{6} + \beta_{4} - 265\beta_{2} ) / 11$$ (b11 - 38*b10 - 12*b8 - 2*b6 + b4 - 265*b2) / 11 $$\nu^{4}$$ $$=$$ $$( -64\beta_{9} + 45\beta_{7} + 199\beta_{5} - 22\beta_{3} - 322\beta _1 + 5502 ) / 11$$ (-64*b9 + 45*b7 + 199*b5 - 22*b3 - 322*b1 + 5502) / 11 $$\nu^{5}$$ $$=$$ $$( -27\beta_{11} + 1349\beta_{10} + 566\beta_{8} - 28\beta_{6} - 203\beta_{4} + 7569\beta_{2} ) / 11$$ (-27*b11 + 1349*b10 + 566*b8 - 28*b6 - 203*b4 + 7569*b2) / 11 $$\nu^{6}$$ $$=$$ $$( 1812\beta_{9} - 3824\beta_{7} - 8702\beta_{5} + 2068\beta_{3} + 12506\beta _1 - 160331 ) / 11$$ (1812*b9 - 3824*b7 - 8702*b5 + 2068*b3 + 12506*b1 - 160331) / 11 $$\nu^{7}$$ $$=$$ $$( 274\beta_{11} - 48065\beta_{10} - 23418\beta_{8} + 5288\beta_{6} + 11802\beta_{4} - 227908\beta_{2} ) / 11$$ (274*b11 - 48065*b10 - 23418*b8 + 5288*b6 + 11802*b4 - 227908*b2) / 11 $$\nu^{8}$$ $$=$$ $$( -49812\beta_{9} + 194643\beta_{7} + 348285\beta_{5} - 106854\beta_{3} - 480078\beta _1 + 4933807 ) / 11$$ (-49812*b9 + 194643*b7 + 348285*b5 - 106854*b3 - 480078*b1 + 4933807) / 11 $$\nu^{9}$$ $$=$$ $$( 15075\beta_{11} + 1734170\beta_{10} + 933246\beta_{8} - 332016\beta_{6} - 530877\beta_{4} + 7179089\beta_{2} ) / 11$$ (15075*b11 + 1734170*b10 + 933246*b8 - 332016*b6 - 530877*b4 + 7179089*b2) / 11 $$\nu^{10}$$ $$=$$ $$( 1358338\beta_{9} - 8509153\beta_{7} - 13465683\beta_{5} + 4604160\beta_{3} + 18287710\beta _1 - 158751284 ) / 11$$ (1358338*b9 - 8509153*b7 - 13465683*b5 + 4604160*b3 + 18287710*b1 - 158751284) / 11 $$\nu^{11}$$ $$=$$ $$( - 1331275 \beta_{11} - 63208837 \beta_{10} - 36432462 \beta_{8} + 16327586 \beta_{6} + 21649529 \beta_{4} - 235193531 \beta_{2} ) / 11$$ (-1331275*b11 - 63208837*b10 - 36432462*b8 + 16327586*b6 + 21649529*b4 - 235193531*b2) / 11

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/121\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
120.1
 5.08417i 4.56289i 6.10049i 5.04186i 2.38108i − 0.159251i 0.159251i − 2.38108i − 5.04186i − 6.10049i − 4.56289i − 5.08417i
6.98629i −13.4457 −32.8082 12.4278 93.9353i 25.0373i 117.427i 99.7859 86.8239i
120.2 5.73846i 12.8600 −16.9299 −30.2132 73.7966i 57.1862i 5.33646i 84.3794 173.377i
120.3 4.19838i −7.32262 −1.62639 −18.2108 30.7431i 50.4691i 60.3459i −27.3793 76.4560i
120.4 3.86629i 2.03418 1.05183 −5.71842 7.86472i 4.30133i 65.9273i −76.8621 22.1090i
120.5 3.55665i −15.8040 3.35025 17.7513 56.2093i 33.4062i 68.8220i 168.766 63.1351i
120.6 1.74286i 8.67811 12.9624 9.96342 15.1248i 58.9175i 50.4775i −5.69033 17.3649i
120.7 1.74286i 8.67811 12.9624 9.96342 15.1248i 58.9175i 50.4775i −5.69033 17.3649i
120.8 3.55665i −15.8040 3.35025 17.7513 56.2093i 33.4062i 68.8220i 168.766 63.1351i
120.9 3.86629i 2.03418 1.05183 −5.71842 7.86472i 4.30133i 65.9273i −76.8621 22.1090i
120.10 4.19838i −7.32262 −1.62639 −18.2108 30.7431i 50.4691i 60.3459i −27.3793 76.4560i
120.11 5.73846i 12.8600 −16.9299 −30.2132 73.7966i 57.1862i 5.33646i 84.3794 173.377i
120.12 6.98629i −13.4457 −32.8082 12.4278 93.9353i 25.0373i 117.427i 99.7859 86.8239i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 120.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 121.5.b.b 12
11.b odd 2 1 inner 121.5.b.b 12
11.c even 5 1 11.5.d.a 12
11.c even 5 1 121.5.d.c 12
11.c even 5 1 121.5.d.d 12
11.c even 5 1 121.5.d.e 12
11.d odd 10 1 11.5.d.a 12
11.d odd 10 1 121.5.d.c 12
11.d odd 10 1 121.5.d.d 12
11.d odd 10 1 121.5.d.e 12
33.f even 10 1 99.5.k.a 12
33.h odd 10 1 99.5.k.a 12
44.g even 10 1 176.5.n.a 12
44.h odd 10 1 176.5.n.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.5.d.a 12 11.c even 5 1
11.5.d.a 12 11.d odd 10 1
99.5.k.a 12 33.f even 10 1
99.5.k.a 12 33.h odd 10 1
121.5.b.b 12 1.a even 1 1 trivial
121.5.b.b 12 11.b odd 2 1 inner
121.5.d.c 12 11.c even 5 1
121.5.d.c 12 11.d odd 10 1
121.5.d.d 12 11.c even 5 1
121.5.d.d 12 11.d odd 10 1
121.5.d.e 12 11.c even 5 1
121.5.d.e 12 11.d odd 10 1
176.5.n.a 12 44.g even 10 1
176.5.n.a 12 44.h odd 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} + 130T_{2}^{10} + 6365T_{2}^{8} + 149400T_{2}^{6} + 1756840T_{2}^{4} + 9482560T_{2}^{2} + 16272080$$ acting on $$S_{5}^{\mathrm{new}}(121, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} + 130 T^{10} + \cdots + 16272080$$
$3$ $$(T^{6} + 13 T^{5} - 280 T^{4} + \cdots - 353241)^{2}$$
$5$ $$(T^{6} + 14 T^{5} - 825 T^{4} + \cdots - 6915664)^{2}$$
$7$ $$T^{12} + 11050 T^{10} + \cdots + 37\!\cdots\!00$$
$11$ $$T^{12}$$
$13$ $$T^{12} + 157270 T^{10} + \cdots + 10\!\cdots\!80$$
$17$ $$T^{12} + 315855 T^{10} + \cdots + 16\!\cdots\!05$$
$19$ $$T^{12} + 854815 T^{10} + \cdots + 31\!\cdots\!05$$
$23$ $$(T^{6} - 842 T^{5} + \cdots - 10\!\cdots\!96)^{2}$$
$29$ $$T^{12} + 3050950 T^{10} + \cdots + 61\!\cdots\!80$$
$31$ $$(T^{6} - 1032 T^{5} + \cdots + 64\!\cdots\!44)^{2}$$
$37$ $$(T^{6} + 118 T^{5} + \cdots - 70\!\cdots\!96)^{2}$$
$41$ $$T^{12} + 6127575 T^{10} + \cdots + 72\!\cdots\!05$$
$43$ $$T^{12} + 17669305 T^{10} + \cdots + 16\!\cdots\!00$$
$47$ $$(T^{6} - 4492 T^{5} + \cdots - 92\!\cdots\!36)^{2}$$
$53$ $$(T^{6} - 4312 T^{5} + \cdots - 34\!\cdots\!76)^{2}$$
$59$ $$(T^{6} + 1613 T^{5} + \cdots - 61\!\cdots\!21)^{2}$$
$61$ $$T^{12} + 68134630 T^{10} + \cdots + 15\!\cdots\!80$$
$67$ $$(T^{6} - 6077 T^{5} + \cdots + 15\!\cdots\!84)^{2}$$
$71$ $$(T^{6} - 4072 T^{5} + \cdots + 23\!\cdots\!84)^{2}$$
$73$ $$T^{12} + 90452295 T^{10} + \cdots + 14\!\cdots\!05$$
$79$ $$T^{12} + 252190570 T^{10} + \cdots + 36\!\cdots\!80$$
$83$ $$T^{12} + 243117675 T^{10} + \cdots + 18\!\cdots\!05$$
$89$ $$(T^{6} - 2777 T^{5} + \cdots + 66\!\cdots\!44)^{2}$$
$97$ $$(T^{6} + 4223 T^{5} + \cdots - 13\!\cdots\!01)^{2}$$