# Properties

 Label 169.8.b.e Level $169$ Weight $8$ Character orbit 169.b Analytic conductor $52.793$ Analytic rank $0$ Dimension $16$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,8,Mod(168,169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("169.168");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 169.b (of order $$2$$, degree $$1$$, not 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: $$52.7930693068$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} + 1591 x^{14} + 998837 x^{12} + 319862003 x^{10} + 57017400035 x^{8} + 5819167911653 x^{6} + \cdots + 11\!\cdots\!00$$ x^16 + 1591*x^14 + 998837*x^12 + 319862003*x^10 + 57017400035*x^8 + 5819167911653*x^6 + 333227942883751*x^4 + 9796730453852209*x^2 + 113340278793830400 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{15}\cdot 13^{8}$$ Twist minimal: no (minimal twist has level 13) 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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{5} q^{2} + (\beta_{3} + 4) q^{3} + (\beta_1 - 72) q^{4} + (\beta_{11} - 2 \beta_{8} + 4 \beta_{5}) q^{5} + ( - \beta_{12} + 3 \beta_{8} - 2 \beta_{5}) q^{6} + (\beta_{10} - 2 \beta_{8} - 5 \beta_{5}) q^{7} + ( - \beta_{14} - \beta_{12} + \cdots - 82 \beta_{5}) q^{8}+ \cdots + ( - \beta_{4} + 12 \beta_{3} + \cdots + 756) q^{9}+O(q^{10})$$ q + b5 * q^2 + (b3 + 4) * q^3 + (b1 - 72) * q^4 + (b11 - 2*b8 + 4*b5) * q^5 + (-b12 + 3*b8 - 2*b5) * q^6 + (b10 - 2*b8 - 5*b5) * q^7 + (-b14 - b12 + 21*b8 - 82*b5) * q^8 + (-b4 + 12*b3 + 5*b2 - 3*b1 + 756) * q^9 $$q + \beta_{5} q^{2} + (\beta_{3} + 4) q^{3} + (\beta_1 - 72) q^{4} + (\beta_{11} - 2 \beta_{8} + 4 \beta_{5}) q^{5} + ( - \beta_{12} + 3 \beta_{8} - 2 \beta_{5}) q^{6} + (\beta_{10} - 2 \beta_{8} - 5 \beta_{5}) q^{7} + ( - \beta_{14} - \beta_{12} + \cdots - 82 \beta_{5}) q^{8}+ \cdots + (2053 \beta_{15} + \cdots + 221880 \beta_{5}) q^{99}+O(q^{100})$$ q + b5 * q^2 + (b3 + 4) * q^3 + (b1 - 72) * q^4 + (b11 - 2*b8 + 4*b5) * q^5 + (-b12 + 3*b8 - 2*b5) * q^6 + (b10 - 2*b8 - 5*b5) * q^7 + (-b14 - b12 + 21*b8 - 82*b5) * q^8 + (-b4 + 12*b3 + 5*b2 - 3*b1 + 756) * q^9 + (-b7 + b4 - 13*b3 - 7*b2 + 3*b1 - 860) * q^10 + (-b15 - b14 - 2*b13 + 2*b12 + 3*b10 - 52*b8 - 51*b5) * q^11 + (-b9 + 2*b7 + b4 - 97*b3 + 9*b2 - 14*b1 + 941) * q^12 + (-b9 + b7 + 2*b6 + 2*b4 - 62*b3 + 3*b2 - 13*b1 + 874) * q^14 + (-5*b13 - 16*b12 + 30*b11 - 3*b10 - 4*b8 + 166*b5) * q^15 + (-b9 + 4*b7 - 2*b6 + 3*b4 - 155*b3 - 11*b2 - 125*b1 + 7573) * q^16 + (b9 + 4*b7 + 4*b6 - 3*b4 - 189*b3 - 9*b2 + 44*b1 - 2946) * q^17 + (-13*b15 + b14 + 16*b13 + 5*b12 + 86*b11 + 4*b10 - 1041*b8 + 1123*b5) * q^18 + (7*b15 - b14 - 27*b13 - 22*b12 + 14*b11 - 2*b10 + 680*b8 + 793*b5) * q^19 + (14*b15 - b14 - 2*b13 - 7*b12 + 16*b11 - 6*b10 + 1107*b8 - 706*b5) * q^20 + (-14*b15 + 14*b14 + 27*b13 + 8*b12 - 29*b11 + 5*b10 - 1219*b8 + 886*b5) * q^21 + (3*b9 - 5*b7 + 6*b6 + 8*b4 + 388*b3 - 221*b2 - 239*b1 + 9300) * q^22 + (-b9 + 2*b7 - 6*b6 - 13*b4 - 248*b3 - 105*b2 + 104*b1 - 9211) * q^23 + (-26*b15 + 14*b14 - 104*b13 + 40*b12 + 40*b11 - 28*b10 - 1880*b8 + 3344*b5) * q^24 + (7*b9 + 2*b7 + 2*b6 - 853*b3 + 38*b2 - 169*b1 - 4090) * q^25 + (9*b9 + 21*b7 - 3*b6 - 18*b4 + 1027*b3 + 279*b2 - 87*b1 + 23011) * q^27 + (-7*b15 + b14 - 90*b13 + 112*b12 - 26*b11 + 38*b10 - 1374*b8 + 2666*b5) * q^28 + (6*b9 - 47*b7 + b6 - 27*b4 - 171*b3 - 260*b2 - 309*b1 + 27517) * q^29 + (-8*b9 - 26*b7 - 6*b6 + 30*b4 - 3454*b3 - 98*b2 + 8*b1 - 34018) * q^30 + (-23*b15 - 15*b14 + 153*b13 + 86*b12 - 164*b11 - 51*b10 + 452*b8 + 8174*b5) * q^31 + (6*b15 + 15*b14 - 180*b13 + 177*b12 - 152*b11 + 24*b10 + 2203*b8 + 15034*b5) * q^32 + (-21*b15 + 15*b14 - 19*b13 - 42*b12 - 243*b11 - 15*b10 + 3085*b8 - 4794*b5) * q^33 + (28*b15 - 76*b14 + 84*b13 + 104*b12 + 332*b11 - 64*b10 + 2196*b8 - 8023*b5) * q^34 + (-29*b9 + 70*b7 - 34*b6 - 19*b4 + 193*b3 - 583*b2 - 54*b1 - 4105) * q^35 + (11*b9 + 38*b7 + 36*b6 + 69*b4 - 1013*b3 - 2335*b2 + 127*b1 - 146747) * q^36 + (64*b15 - 76*b14 - 231*b13 - 268*b12 - 28*b11 + 67*b10 + 3765*b8 + 1750*b5) * q^37 + (-7*b9 - 133*b7 - 20*b6 - 62*b4 - 2446*b3 + 1427*b2 + 903*b1 - 148310) * q^38 + (-27*b9 - 200*b7 - 42*b6 + 53*b4 - 1661*b3 + 1831*b2 + 113*b1 + 47815) * q^40 + (49*b15 - 15*b14 - 64*b13 - 90*b12 - 738*b11 - 192*b10 - 2879*b8 + 5794*b5) * q^41 + (4*b9 + 181*b7 + 66*b6 + 83*b4 - 2207*b3 - 2166*b2 + 2351*b1 - 196793) * q^42 + (14*b9 + 107*b7 - 101*b6 - 103*b4 + 998*b3 + 626*b2 - 1081*b1 + 49468) * q^43 + (251*b15 + 91*b14 - 86*b13 - 368*b12 - 1190*b11 + 258*b10 + 33054*b8 + 31270*b5) * q^44 + (-131*b15 - 91*b14 + 18*b13 - 442*b12 + 301*b11 - 106*b10 + 5552*b8 + 42034*b5) * q^45 + (-42*b15 - 90*b14 + 72*b13 + 225*b12 + 744*b11 + 196*b10 + 22541*b8 - 24362*b5) * q^46 + (56*b15 + 168*b14 + 43*b13 - 720*b12 - 1022*b11 + 315*b10 + 5556*b8 - 3104*b5) * q^47 + (96*b9 - 336*b7 + 24*b6 - 8*b4 + 1864*b3 - 4520*b2 + 4104*b1 - 573344) * q^48 + (-42*b9 - 266*b7 - 42*b6 - 175*b4 + 280*b3 - 2975*b2 - 2779*b1 - 139826) * q^49 + (159*b15 + 169*b14 + 408*b13 + 673*b12 - 214*b11 + 144*b10 - 13273*b8 + 24594*b5) * q^50 + (-61*b9 + 572*b7 + 84*b6 + 291*b4 + 2058*b3 + 665*b2 - 1544*b1 - 501645) * q^51 + (18*b9 + 469*b7 + 117*b6 - 20*b4 + 263*b3 + 1869*b2 + 1676*b1 + 208976) * q^53 + (-168*b15 + 90*b14 + 510*b13 - 967*b12 + 1836*b11 + 450*b10 - 52419*b8 + 30544*b5) * q^54 + (-97*b9 - 264*b7 - 224*b6 - 65*b4 + 6519*b3 - 87*b2 + 2272*b1 + 55843) * q^55 + (50*b9 - 456*b7 + 348*b6 + 54*b4 + 20874*b3 - 2622*b2 + 2632*b1 - 441110) * q^56 + (-40*b15 - 260*b14 - 511*b13 - 220*b12 + 2915*b11 - 599*b10 + 60017*b8 + 41932*b5) * q^57 + (162*b15 + 260*b14 + 1458*b13 + 84*b12 + 196*b11 + 154*b10 + 41052*b8 + 67877*b5) * q^58 + (-420*b15 + 244*b14 - 276*b13 + 352*b12 - 562*b11 - 429*b10 + 62768*b8 + 21581*b5) * q^59 + (78*b15 + 78*b14 - 1284*b13 + 1456*b12 + 1092*b11 - 612*b10 + 7404*b8 + 7508*b5) * q^60 + (-72*b9 + 687*b7 + 159*b6 + 117*b4 - 7275*b3 - 1734*b2 - 3335*b1 - 225357) * q^61 + (30*b9 + 828*b7 - 86*b6 + 122*b4 + 13006*b3 - 2156*b2 + 5134*b1 - 1618798) * q^62 + (-57*b15 + 63*b14 + 2794*b13 - 1126*b12 - 1566*b11 - 804*b10 - 62368*b8 + 20452*b5) * q^63 + (193*b9 - 620*b7 - 190*b6 - 323*b4 + 32539*b3 + 1027*b2 + 5213*b1 - 1996725) * q^64 + (34*b9 + 271*b7 + 42*b6 - 173*b4 - 6887*b3 + 2348*b2 - 3343*b1 + 1012243) * q^66 + (419*b15 - 245*b14 + 1095*b13 + 1858*b12 + 1102*b11 - 274*b10 - 85908*b8 - 29605*b5) * q^67 + (156*b9 + 368*b7 + 176*b6 - 132*b4 + 708*b3 - 556*b2 - 12459*b1 + 1263236) * q^68 + (-201*b9 + 1038*b7 - 114*b6 + 556*b4 + 3641*b3 + 1462*b2 - 2613*b1 - 575662) * q^69 + (-384*b15 + 196*b14 - 2688*b13 + 1182*b12 + 2636*b11 + 356*b10 + 120714*b8 - 9056*b5) * q^70 + (-212*b15 - 196*b14 + 509*b13 + 1640*b12 + 1362*b11 + 276*b10 + 83638*b8 - 78411*b5) * q^71 + (1666*b15 - 91*b14 + 1360*b13 - 2069*b12 + 2128*b11 - 388*b10 + 333553*b8 - 41314*b5) * q^72 + (-387*b15 - 1119*b14 + 405*b13 - 1278*b12 - 2541*b11 - 407*b10 + 19246*b8 - 61694*b5) * q^73 + (-232*b9 - 628*b7 - 146*b6 - 320*b4 - 37096*b3 + 9741*b2 - 9330*b1 - 291611) * q^74 + (234*b9 - 1251*b7 - 51*b6 + 455*b4 - 3256*b3 + 1526*b2 - 4623*b1 - 2730580) * q^75 + (-1409*b15 - 1029*b14 - 1182*b13 + 1004*b12 + 6370*b11 + 90*b10 - 195366*b8 - 137794*b5) * q^76 + (-147*b9 - 944*b7 + 240*b6 - 734*b4 + 13805*b3 - 2246*b2 - 15769*b1 - 2199478) * q^77 + (37*b9 - 2032*b7 - 184*b6 + 261*b4 + 12539*b3 - 17221*b2 + 4816*b1 + 791401) * q^79 + (-670*b15 + 105*b14 - 260*b13 + 3375*b12 - 2184*b11 - 960*b10 - 237787*b8 - 17946*b5) * q^80 + (369*b9 - 1782*b7 - 54*b6 + 215*b4 + 58053*b3 + 3587*b2 - 6978*b1 + 1194642) * q^81 + (68*b9 + 240*b7 - 512*b6 - 1424*b4 + 10992*b3 + 5577*b2 + 7320*b1 - 1160193) * q^82 + (-2336*b15 + 848*b14 + 5090*b13 + 4016*b12 - 6288*b11 + 1122*b10 + 98092*b8 - 120560*b5) * q^83 + (1525*b15 - 847*b14 + 362*b13 - 2204*b12 - 9134*b11 - 1066*b10 + 328794*b8 - 420562*b5) * q^84 + (-1022*b15 - 1182*b14 - 3024*b13 + 524*b12 - 9143*b11 + 2192*b10 + 368246*b8 + 92440*b5) * q^85 + (-1268*b15 + 1610*b14 + 182*b13 + 1175*b12 + 8456*b11 + 3846*b10 - 131625*b8 + 186364*b5) * q^86 + (246*b9 + 672*b7 - 576*b6 + 768*b4 + 73823*b3 + 18648*b2 - 13530*b1 - 608212) * q^87 + (-658*b9 - 72*b7 + 964*b6 - 1142*b4 - 12426*b3 + 48366*b2 + 21840*b1 - 4569546) * q^88 + (898*b15 + 1366*b14 - 967*b13 + 496*b12 - 3849*b11 + 4125*b10 - 166853*b8 + 75028*b5) * q^89 + (-92*b9 + 1273*b7 - 132*b6 + 1535*b4 - 102947*b3 - 13443*b2 + 16769*b1 - 8321390) * q^90 + (-87*b9 - 34*b7 - 472*b6 - 177*b4 - 5295*b3 - 3257*b2 - 27438*b1 + 4000571) * q^92 + (428*b15 + 1108*b14 - 3682*b13 - 10256*b12 - 15076*b11 + 2230*b10 - 263512*b8 - 215488*b5) * q^93 + (-1190*b9 + 2432*b7 + 854*b6 - 258*b4 - 178198*b3 + 30104*b2 + 15646*b1 + 676862) * q^94 + (543*b9 - 4823*b7 - 543*b6 + 460*b4 - 74164*b3 + 2685*b2 - 16447*b1 - 2069911) * q^95 + (3456*b15 - 2304*b14 - 2528*b13 - 10960*b12 - 15360*b11 - 2208*b10 + 729584*b8 - 775040*b5) * q^96 + (-1164*b15 + 2288*b14 - 9897*b13 - 540*b12 + 11113*b11 + 15*b10 + 168041*b8 - 206660*b5) * q^97 + (49*b15 + 2639*b14 + 3668*b13 + 3227*b12 + 2842*b11 + 168*b10 + 512057*b8 + 203237*b5) * q^98 + (2053*b15 + 1325*b14 + 4840*b13 + 2814*b12 - 1826*b11 - 4750*b10 + 150628*b8 + 221880*b5) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 56 q^{3} - 1154 q^{4} + 11976 q^{9}+O(q^{10})$$ 16 * q + 56 * q^3 - 1154 * q^4 + 11976 * q^9 $$16 q + 56 q^{3} - 1154 q^{4} + 11976 q^{9} - 13626 q^{10} + 15816 q^{12} + 14484 q^{14} + 122754 q^{16} - 45648 q^{17} + 147452 q^{22} - 144936 q^{23} - 58488 q^{25} + 358616 q^{27} + 443544 q^{29} - 516232 q^{30} - 63120 q^{35} - 2325986 q^{36} - 2364492 q^{38} + 766030 q^{40} - 3121908 q^{42} + 782984 q^{43} - 9171424 q^{48} - 2217560 q^{49} - 8040712 q^{51} + 3329328 q^{53} + 836432 q^{55} - 7218312 q^{56} - 3527096 q^{61} - 25996776 q^{62} - 32227074 q^{64} + 16245108 q^{66} + 20236182 q^{68} - 9236984 q^{69} - 4412442 q^{74} - 43669976 q^{75} - 35263992 q^{77} + 12644416 q^{79} + 18632544 q^{81} - 18695502 q^{82} - 10399976 q^{87} - 73352808 q^{88} - 132263562 q^{90} + 64127424 q^{92} + 12052256 q^{94} - 32534160 q^{95}+O(q^{100})$$ 16 * q + 56 * q^3 - 1154 * q^4 + 11976 * q^9 - 13626 * q^10 + 15816 * q^12 + 14484 * q^14 + 122754 * q^16 - 45648 * q^17 + 147452 * q^22 - 144936 * q^23 - 58488 * q^25 + 358616 * q^27 + 443544 * q^29 - 516232 * q^30 - 63120 * q^35 - 2325986 * q^36 - 2364492 * q^38 + 766030 * q^40 - 3121908 * q^42 + 782984 * q^43 - 9171424 * q^48 - 2217560 * q^49 - 8040712 * q^51 + 3329328 * q^53 + 836432 * q^55 - 7218312 * q^56 - 3527096 * q^61 - 25996776 * q^62 - 32227074 * q^64 + 16245108 * q^66 + 20236182 * q^68 - 9236984 * q^69 - 4412442 * q^74 - 43669976 * q^75 - 35263992 * q^77 + 12644416 * q^79 + 18632544 * q^81 - 18695502 * q^82 - 10399976 * q^87 - 73352808 * q^88 - 132263562 * q^90 + 64127424 * q^92 + 12052256 * q^94 - 32534160 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 1591 x^{14} + 998837 x^{12} + 319862003 x^{10} + 57017400035 x^{8} + 5819167911653 x^{6} + \cdots + 11\!\cdots\!00$$ :

 $$\beta_{1}$$ $$=$$ $$( - 2095593961 \nu^{14} - 3169890760198 \nu^{12} + \cdots - 27\!\cdots\!56 ) / 26\!\cdots\!36$$ (-2095593961*v^14 - 3169890760198*v^12 - 1845052138525815*v^10 - 526101710398367316*v^8 - 78487120466504551367*v^6 - 6118494553301365453766*v^4 - 228150751650104194799097*v^2 - 2732739322239963418976256) / 2631531056240149979136 $$\beta_{2}$$ $$=$$ $$( 2095593961 \nu^{14} + 3169890760198 \nu^{12} + \cdots + 32\!\cdots\!32 ) / 40\!\cdots\!44$$ (2095593961*v^14 + 3169890760198*v^12 + 1845052138525815*v^10 + 526101710398367316*v^8 + 78487120466504551367*v^6 + 6118494553301365453766*v^4 + 230782282706344344778233*v^2 + 3255604300568294757138432) / 404850931729253842944 $$\beta_{3}$$ $$=$$ $$( - 26\!\cdots\!73 \nu^{14} + \cdots - 28\!\cdots\!88 ) / 49\!\cdots\!68$$ (-26826373825789573*v^14 - 39587201368684887238*v^12 - 22241128752957613787987*v^10 - 6031186549828366467355764*v^8 - 841893193346867977892734811*v^6 - 60894684614700435578075502054*v^4 - 2134554189150583319304020702701*v^2 - 28273262610744761704001218674688) / 4923913898660144415827591168 $$\beta_{4}$$ $$=$$ $$( 35\!\cdots\!81 \nu^{14} + \cdots + 69\!\cdots\!40 ) / 14\!\cdots\!04$$ (359364760365374581*v^14 + 563762665508213080870*v^12 + 343636104523239485815971*v^10 + 103503493101364585249881780*v^8 + 16288883713074149134767287531*v^6 + 1306382396820646784281734167366*v^4 + 48901878319725440263979595560733*v^2 + 690361009548557509804155524874240) / 14771741695980433247482773504 $$\beta_{5}$$ $$=$$ $$( 121102446247 \nu^{15} + 183841063433362 \nu^{13} + \cdots + 23\!\cdots\!08 \nu ) / 22\!\cdots\!80$$ (121102446247*v^15 + 183841063433362*v^13 + 107600514547780169*v^11 + 30959176260878942516*v^9 + 4687427913553165181705*v^7 + 373892256456907993404386*v^5 + 14565264498912586259608807*v^3 + 235844508380848105588574008*v) / 22183806804104464324116480 $$\beta_{6}$$ $$=$$ $$( 39\!\cdots\!29 \nu^{14} + \cdots + 90\!\cdots\!44 ) / 73\!\cdots\!52$$ (395162198847276829*v^14 + 620234694782662585230*v^12 + 379757259368320411122131*v^10 + 115912158030841958109167748*v^8 + 18813061461728885749280193395*v^6 + 1599260640351233256224408501646*v^4 + 64203183630609050347597023423805*v^2 + 907288213423680075502778149945344) / 7385870847990216623741386752 $$\beta_{7}$$ $$=$$ $$( - 33\!\cdots\!59 \nu^{14} + \cdots - 26\!\cdots\!12 ) / 28\!\cdots\!52$$ (-33875102754872459*v^14 - 49790545860058735874*v^12 - 27781648165267828849701*v^10 - 7439101889221766797615836*v^8 - 1013923891994056888910964133*v^6 - 70165211501451806230911657346*v^4 - 2276767436354414215520337956619*v^2 - 26900248081899185370408774746112) / 284071955691931408605437952 $$\beta_{8}$$ $$=$$ $$( - 121102446247 \nu^{15} - 183841063433362 \nu^{13} + \cdots - 21\!\cdots\!28 \nu ) / 17\!\cdots\!60$$ (-121102446247*v^15 - 183841063433362*v^13 - 107600514547780169*v^11 - 30959176260878942516*v^9 - 4687427913553165181705*v^7 - 373892256456907993404386*v^5 - 14565264498912586259608807*v^3 - 213660701576743641264457528*v) / 1706446677238804948008960 $$\beta_{9}$$ $$=$$ $$( 41\!\cdots\!79 \nu^{14} + \cdots + 53\!\cdots\!92 ) / 73\!\cdots\!52$$ (4102599392014344679*v^14 + 6112869675431836793674*v^12 + 3483493925858862923730345*v^10 + 964936774617793274181406284*v^8 + 138927413113720929553398731081*v^6 + 10468789899162998341130394952010*v^4 + 385856134524979244016829632227847*v^2 + 5398151817054690094281025178222592) / 7385870847990216623741386752 $$\beta_{10}$$ $$=$$ $$( 67\!\cdots\!51 \nu^{15} + \cdots + 66\!\cdots\!24 \nu ) / 62\!\cdots\!60$$ (67198961820066881651*v^15 + 94033453511006223465626*v^13 + 48664508476358318912771797*v^11 + 11589687078285808141881884188*v^9 + 1335708474798850692303689233645*v^7 + 80141675921619651069805206848218*v^5 + 3020238858096653799903266649540011*v^3 + 66268281926086016723442277033964624*v) / 62262891248557526138139890319360 $$\beta_{11}$$ $$=$$ $$( - 19\!\cdots\!07 \nu^{15} + \cdots - 11\!\cdots\!80 \nu ) / 12\!\cdots\!72$$ (-19033912172945629807*v^15 - 27878519808022909363666*v^13 - 15461127746840404936669673*v^11 - 4093393673184697738435466156*v^9 - 545607618154055235217865219921*v^7 - 36130592410581154637659531122386*v^5 - 1083044899551066230073651751403191*v^3 - 11619493213452554698539062381259280*v) / 12452578249711505227627978063872 $$\beta_{12}$$ $$=$$ $$( 14\!\cdots\!77 \nu^{15} + \cdots + 15\!\cdots\!08 \nu ) / 31\!\cdots\!80$$ (146509088144634924077*v^15 + 215909611162649022796742*v^13 + 121064454786749259094516579*v^11 + 32733702050691350532146331196*v^9 + 4550607557507092860086090006755*v^7 + 327526006369434918345237723208246*v^5 + 11424434346291067633536090185222957*v^3 + 150623146981901027468393087872552808*v) / 31131445624278763069069945159680 $$\beta_{13}$$ $$=$$ $$( 80\!\cdots\!75 \nu^{15} + \cdots + 75\!\cdots\!00 \nu ) / 12\!\cdots\!72$$ (80559532995626212675*v^15 + 118717973173522471364170*v^13 + 66533575373791366246492757*v^11 + 17957775231450284159150792252*v^9 + 2484390902860590674840261359037*v^7 + 176749541018099307002526203542346*v^5 + 6012239829169352154517504718070667*v^3 + 75563269274905810016147765704368400*v) / 12452578249711505227627978063872 $$\beta_{14}$$ $$=$$ $$( - 35\!\cdots\!67 \nu^{15} + \cdots - 47\!\cdots\!48 \nu ) / 51\!\cdots\!80$$ (-35498425483608845567*v^15 - 52839009155340118298542*v^13 - 30072279885111018241725849*v^11 - 8316688153181249831201752456*v^9 - 1195075114327251591076712381665*v^7 - 89836298793540982131575298279206*v^5 - 3304782871922206764323343467898087*v^3 - 47694665405114434975182347509663748*v) / 5188574270713127178178324193280 $$\beta_{15}$$ $$=$$ $$( 33\!\cdots\!01 \nu^{15} + \cdots + 62\!\cdots\!64 \nu ) / 31\!\cdots\!80$$ (332401356068878691401*v^15 + 504059278115513429573686*v^13 + 294784113490499069320490447*v^11 + 84858803817728956347714162908*v^9 + 12903329885988433013275520643095*v^7 + 1041649920914212293973154038203398*v^5 + 41447924729196980478101046138499441*v^3 + 622363120851441425509544556142192264*v) / 31131445624278763069069945159680
 $$\nu$$ $$=$$ $$( \beta_{8} + 13\beta_{5} ) / 13$$ (b8 + 13*b5) / 13 $$\nu^{2}$$ $$=$$ $$( 2\beta_{2} + 13\beta _1 - 2583 ) / 13$$ (2*b2 + 13*b1 - 2583) / 13 $$\nu^{3}$$ $$=$$ $$( -3\beta_{15} - 13\beta_{14} - 10\beta_{12} + 6\beta_{11} - 325\beta_{8} - 4349\beta_{5} ) / 13$$ (-3*b15 - 13*b14 - 10*b12 + 6*b11 - 325*b8 - 4349*b5) / 13 $$\nu^{4}$$ $$=$$ $$( -\beta_{9} + 52\beta_{7} - 18\beta_{6} + 67\beta_{4} - 1563\beta_{3} - 1503\beta_{2} - 6723\beta _1 + 864902 ) / 13$$ (-b9 + 52*b7 - 18*b6 + 67*b4 - 1563*b3 - 1503*b2 - 6723*b1 + 864902) / 13 $$\nu^{5}$$ $$=$$ $$( 2498 \beta_{15} + 7046 \beta_{14} - 1920 \beta_{13} + 5692 \beta_{12} - 8636 \beta_{11} + \cdots + 1787549 \beta_{5} ) / 13$$ (2498*b15 + 7046*b14 - 1920*b13 + 5692*b12 - 8636*b11 + 312*b10 + 227249*b8 + 1787549*b5) / 13 $$\nu^{6}$$ $$=$$ $$( 614 \beta_{9} - 35232 \beta_{7} + 8892 \beta_{6} - 50290 \beta_{4} + 1138178 \beta_{3} + \cdots - 355199513 ) / 13$$ (614*b9 - 35232*b7 + 8892*b6 - 50290*b4 + 1138178*b3 + 949056*b2 + 3147177*b1 - 355199513) / 13 $$\nu^{7}$$ $$=$$ $$( - 1565725 \beta_{15} - 3357835 \beta_{14} + 1345984 \beta_{13} - 2650006 \beta_{12} + \cdots - 789907117 \beta_{5} ) / 13$$ (-1565725*b15 - 3357835*b14 + 1345984*b13 - 2650006*b12 + 5773986*b11 - 109096*b10 - 151673029*b8 - 789907117*b5) / 13 $$\nu^{8}$$ $$=$$ $$( 77041 \beta_{9} + 18533444 \beta_{7} - 3480702 \beta_{6} + 28700653 \beta_{4} - 607070229 \beta_{3} + \cdots + 156793010544 ) / 13$$ (77041*b9 + 18533444*b7 - 3480702*b6 + 28700653*b4 - 607070229*b3 - 544475845*b2 - 1464427839*b1 + 156793010544) / 13 $$\nu^{9}$$ $$=$$ $$( 886481140 \beta_{15} + 1575202876 \beta_{14} - 709523200 \beta_{13} + 1137492760 \beta_{12} + \cdots + 359691594733 \beta_{5} ) / 13$$ (886481140*b15 + 1575202876*b14 - 709523200*b13 + 1137492760*b12 - 3257072440*b11 + 28336464*b10 + 90520039681*b8 + 359691594733*b5) / 13 $$\nu^{10}$$ $$=$$ $$( - 350132260 \beta_{9} - 8931498272 \beta_{7} + 1303760088 \beta_{6} - 15009735188 \beta_{4} + \cdots - 71332230924843 ) / 13$$ (-350132260*b9 - 8931498272*b7 + 1303760088*b6 - 15009735188*b4 + 286644976820*b3 + 295983963662*b2 + 685085566981*b1 - 71332230924843) / 13 $$\nu^{11}$$ $$=$$ $$( - 478095208231 \beta_{15} - 740641622201 \beta_{14} + 338842305664 \beta_{13} + \cdots - 166326435048989 \beta_{5} ) / 13$$ (-478095208231*b15 - 740641622201*b14 + 338842305664*b13 - 462236933570*b12 + 1731016812286*b11 - 7857121584*b10 - 50490636599269*b8 - 166326435048989*b5) / 13 $$\nu^{12}$$ $$=$$ $$( 348642011475 \beta_{9} + 4132838704692 \beta_{7} - 485859842634 \beta_{6} + 7574372804679 \beta_{4} + \cdots + 32\!\cdots\!30 ) / 13$$ (348642011475*b9 + 4132838704692*b7 - 485859842634*b6 + 7574372804679*b4 - 126956846005823*b3 - 155996290925499*b2 - 322334188822779*b1 + 32960695123690730) / 13 $$\nu^{13}$$ $$=$$ $$( 251000441048214 \beta_{15} + 349852894181154 \beta_{14} - 154716697982080 \beta_{13} + \cdots + 77\!\cdots\!57 \beta_{5} ) / 13$$ (251000441048214*b15 + 349852894181154*b14 - 154716697982080*b13 + 178632842938004*b12 - 895648436890740*b11 + 3243986328648*b10 + 27089931682184177*b8 + 77630305984616957*b5) / 13 $$\nu^{14}$$ $$=$$ $$( - 258518026429598 \beta_{9} + \cdots - 15\!\cdots\!17 ) / 13$$ (-258518026429598*b9 - 1872952196805248*b7 + 180400400469588*b6 - 3759542634119142*b4 + 54006638080604950*b3 + 80686505013233004*b2 + 152369773003423521*b1 - 15374501341026205517) / 13 $$\nu^{15}$$ $$=$$ $$( - 12\!\cdots\!33 \beta_{15} + \cdots - 36\!\cdots\!93 \beta_{5} ) / 13$$ (-129614949498521633*b15 - 165945275814786455*b14 + 69020899878911552*b13 - 65067131131683470*b12 + 456731814071631130*b11 - 1928085767866648*b10 - 14197579593467688613*b8 - 36462804378049423693*b5) / 13

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/169\mathbb{Z}\right)^\times$$.

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
168.1
 − 20.7945i − 22.0918i − 16.2034i − 9.24529i − 11.1141i − 10.7397i − 5.78595i − 7.08346i 7.08346i 5.78595i 10.7397i 11.1141i 9.24529i 16.2034i 22.0918i 20.7945i
21.7945i −67.8320 −346.998 157.804i 1478.36i 842.108i 4772.94i 2414.17 −3439.24
168.2 21.0918i 5.55059 −316.864 126.804i 117.072i 29.3160i 3983.49i −2156.19 2674.54
168.3 17.2034i 86.0173 −167.957 289.904i 1479.79i 968.548i 687.398i 5211.97 −4987.34
168.4 10.2453i 9.69718 23.0341 294.418i 99.3504i 1430.82i 1547.39i −2092.96 3016.39
168.5 10.1141i −70.4106 25.7042 163.930i 712.143i 246.938i 1554.59i 2770.65 −1658.01
168.6 9.73969i 69.5263 33.1385 459.709i 677.164i 861.089i 1569.44i 2646.91 −4477.42
168.7 6.78595i −30.1769 81.9509 93.2351i 204.779i 1618.45i 1424.72i −1276.35 −632.688
168.8 6.08346i 25.6281 90.9916 442.310i 155.908i 761.546i 1332.23i −1530.20 2690.77
168.9 6.08346i 25.6281 90.9916 442.310i 155.908i 761.546i 1332.23i −1530.20 2690.77
168.10 6.78595i −30.1769 81.9509 93.2351i 204.779i 1618.45i 1424.72i −1276.35 −632.688
168.11 9.73969i 69.5263 33.1385 459.709i 677.164i 861.089i 1569.44i 2646.91 −4477.42
168.12 10.1141i −70.4106 25.7042 163.930i 712.143i 246.938i 1554.59i 2770.65 −1658.01
168.13 10.2453i 9.69718 23.0341 294.418i 99.3504i 1430.82i 1547.39i −2092.96 3016.39
168.14 17.2034i 86.0173 −167.957 289.904i 1479.79i 968.548i 687.398i 5211.97 −4987.34
168.15 21.0918i 5.55059 −316.864 126.804i 117.072i 29.3160i 3983.49i −2156.19 2674.54
168.16 21.7945i −67.8320 −346.998 157.804i 1478.36i 842.108i 4772.94i 2414.17 −3439.24
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 168.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.8.b.e 16
13.b even 2 1 inner 169.8.b.e 16
13.d odd 4 1 169.8.a.e 8
13.d odd 4 1 169.8.a.f 8
13.f odd 12 2 13.8.c.a 16
39.k even 12 2 117.8.g.d 16
52.l even 12 2 208.8.i.d 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.8.c.a 16 13.f odd 12 2
117.8.g.d 16 39.k even 12 2
169.8.a.e 8 13.d odd 4 1
169.8.a.f 8 13.d odd 4 1
169.8.b.e 16 1.a even 1 1 trivial
169.8.b.e 16 13.b even 2 1 inner
208.8.i.d 16 52.l even 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{16} + 1601 T_{2}^{14} + 1009056 T_{2}^{12} + 322391392 T_{2}^{10} + 56816563968 T_{2}^{8} + \cdots + 10\!\cdots\!04$$ acting on $$S_{8}^{\mathrm{new}}(169, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} + \cdots + 10\!\cdots\!04$$
$3$ $$(T^{8} + \cdots - 1189005516288)^{2}$$
$5$ $$T^{16} + \cdots + 28\!\cdots\!00$$
$7$ $$T^{16} + \cdots + 80\!\cdots\!00$$
$11$ $$T^{16} + \cdots + 10\!\cdots\!56$$
$13$ $$T^{16}$$
$17$ $$(T^{8} + \cdots + 18\!\cdots\!33)^{2}$$
$19$ $$T^{16} + \cdots + 50\!\cdots\!00$$
$23$ $$(T^{8} + \cdots + 21\!\cdots\!32)^{2}$$
$29$ $$(T^{8} + \cdots - 50\!\cdots\!43)^{2}$$
$31$ $$T^{16} + \cdots + 35\!\cdots\!00$$
$37$ $$T^{16} + \cdots + 85\!\cdots\!69$$
$41$ $$T^{16} + \cdots + 50\!\cdots\!25$$
$43$ $$(T^{8} + \cdots - 81\!\cdots\!72)^{2}$$
$47$ $$T^{16} + \cdots + 70\!\cdots\!00$$
$53$ $$(T^{8} + \cdots + 57\!\cdots\!00)^{2}$$
$59$ $$T^{16} + \cdots + 62\!\cdots\!44$$
$61$ $$(T^{8} + \cdots - 12\!\cdots\!75)^{2}$$
$67$ $$T^{16} + \cdots + 45\!\cdots\!44$$
$71$ $$T^{16} + \cdots + 29\!\cdots\!16$$
$73$ $$T^{16} + \cdots + 52\!\cdots\!00$$
$79$ $$(T^{8} + \cdots + 20\!\cdots\!00)^{2}$$
$83$ $$T^{16} + \cdots + 85\!\cdots\!00$$
$89$ $$T^{16} + \cdots + 16\!\cdots\!44$$
$97$ $$T^{16} + \cdots + 11\!\cdots\!44$$