# Properties

 Label 13.8.e.a Level $13$ Weight $8$ Character orbit 13.e Analytic conductor $4.061$ Analytic rank $0$ Dimension $14$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [13,8,Mod(4,13)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(13, base_ring=CyclotomicField(6))

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("13.4");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$13$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 13.e (of order $$6$$, degree $$2$$, 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: $$4.06100533129$$ Analytic rank: $$0$$ Dimension: $$14$$ Relative dimension: $$7$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{14} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{14} + 1279 x^{12} + 629380 x^{10} + 148562016 x^{8} + 16872573312 x^{6} + 790180980480 x^{4} + \cdots + 4669637050368$$ x^14 + 1279*x^12 + 629380*x^10 + 148562016*x^8 + 16872573312*x^6 + 790180980480*x^4 + 10484997239808*x^2 + 4669637050368 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{10}\cdot 3^{3}\cdot 13^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{13}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + ( - \beta_{6} - \beta_{4} + 4 \beta_{3} + 4) q^{3} + (\beta_{7} + \beta_{6} - 55 \beta_{3} + \cdots + \beta_1) q^{4}+ \cdots + (2 \beta_{11} - \beta_{10} + \cdots + 17 \beta_1) q^{9}+O(q^{10})$$ q + b2 * q^2 + (-b6 - b4 + 4*b3 + 4) * q^3 + (b7 + b6 - 55*b3 + b2 + b1) * q^4 + (-b9 - b8 + 64*b3 + 32) * q^5 + (b10 - b7 + b6 - b5 + 3*b4 - 11*b3 + 17*b2 - 17*b1 - 22) * q^6 + (-b12 + b8 + 3*b6 - b5 + 5*b4 - 136*b3 - 20*b2 + 20*b1 - 272) * q^7 + (-b13 - b12 + b11 - b10 - b9 - b8 + 6*b7 + 10*b6 + 3*b5 + 5*b4 - 346*b3 + 40*b1 - 173) * q^8 + (2*b11 - b10 - 6*b9 - 3*b8 - 10*b6 + b5 + b4 + 496*b3 + 17*b2 + 17*b1) * q^9 $$q + \beta_{2} q^{2} + ( - \beta_{6} - \beta_{4} + 4 \beta_{3} + 4) q^{3} + (\beta_{7} + \beta_{6} - 55 \beta_{3} + \cdots + \beta_1) q^{4}+ \cdots + (3405 \beta_{13} + 3405 \beta_{12} + \cdots + 3596288) q^{99}+O(q^{100})$$ q + b2 * q^2 + (-b6 - b4 + 4*b3 + 4) * q^3 + (b7 + b6 - 55*b3 + b2 + b1) * q^4 + (-b9 - b8 + 64*b3 + 32) * q^5 + (b10 - b7 + b6 - b5 + 3*b4 - 11*b3 + 17*b2 - 17*b1 - 22) * q^6 + (-b12 + b8 + 3*b6 - b5 + 5*b4 - 136*b3 - 20*b2 + 20*b1 - 272) * q^7 + (-b13 - b12 + b11 - b10 - b9 - b8 + 6*b7 + 10*b6 + 3*b5 + 5*b4 - 346*b3 + 40*b1 - 173) * q^8 + (2*b11 - b10 - 6*b9 - 3*b8 - 10*b6 + b5 + b4 + 496*b3 + 17*b2 + 17*b1) * q^9 + (b13 + 2*b12 - 2*b11 + 4*b10 + b9 + 2*b8 - b7 - 13*b6 + 2*b5 - 10*b4 - 51*b3 + 45*b2 - 90*b1 - 51) * q^10 + (2*b13 - 5*b11 + 6*b9 - b6 - 7*b5 - 6*b4 - 298*b3 - 93*b2 + 298) * q^11 + (-3*b13 + 3*b12 - b11 - b10 + 9*b9 - 9*b8 - 8*b5 - 66*b4 + 122*b2 - 61*b1 + 2788) * q^12 + (4*b13 + 3*b12 + 7*b11 - 2*b10 + 4*b9 + 16*b8 - 8*b7 - 2*b6 - 28*b5 - 4*b4 + 880*b3 + 177*b2 - 106*b1 + 829) * q^13 + (-9*b11 - 9*b10 - 8*b9 + 8*b8 + 63*b5 - 27*b4 - 318*b2 + 159*b1 - 3453) * q^14 + (-3*b13 + 6*b11 + 21*b9 - 24*b7 - 112*b6 + 33*b5 + 121*b4 + 958*b3 - 150*b2 - 958) * q^15 + (-3*b13 - 6*b12 + 7*b11 - 14*b10 + 17*b9 + 34*b8 - 19*b7 + 59*b6 - 29*b5 + 49*b4 - 271*b3 - 350*b2 + 700*b1 - 271) * q^16 + (2*b13 + b12 + 10*b11 - 5*b10 - 26*b9 - 13*b8 - 48*b7 + 18*b6 + 4*b5 + 4*b4 + 5547*b3 - 183*b2 - 183*b1) * q^17 + (3*b13 + 3*b12 - 28*b11 + 28*b10 + 39*b9 + 39*b8 - 2*b7 + 302*b6 - b5 + 151*b4 - 7054*b3 - 779*b1 - 3527) * q^18 + (-5*b12 + 3*b10 - 85*b8 - 32*b7 + 139*b6 - 66*b5 + 276*b4 + 2072*b3 + 383*b2 - 383*b1 + 4144) * q^19 + (7*b12 + 19*b10 + 39*b8 - 99*b7 - 317*b6 - 172*b5 - 608*b4 + 4297*b3 - 10*b2 + 10*b1 + 8594) * q^20 + (3*b13 + 3*b12 + 4*b11 - 4*b10 - 54*b9 - 54*b8 + 416*b7 + 516*b6 + 208*b5 + 258*b4 - 16378*b3 - 64*b1 - 8189) * q^21 + (4*b13 + 2*b12 + 6*b11 - 3*b10 - 148*b9 - 74*b8 - 167*b7 - 761*b6 + b5 + b4 + 17943*b3 + 513*b2 + 513*b1) * q^22 + (-9*b13 - 18*b12 + 3*b11 - 6*b10 + b9 + 2*b8 + 264*b7 - 717*b6 + 252*b5 - 729*b4 - 9834*b3 - 831*b2 + 1662*b1 - 9834) * q^23 + (-24*b13 + 32*b11 + 12*b9 + 182*b7 + 34*b6 - 126*b5 + 22*b4 - 11030*b3 + 2866*b2 + 11030) * q^24 + (33*b13 - 33*b12 + 4*b11 + 4*b10 - 10*b9 + 10*b8 - 210*b5 - 594*b4 + 160*b2 - 80*b1 + 2915) * q^25 + (-45*b13 - 37*b12 - 30*b11 - 10*b10 + 59*b9 + 15*b8 - 144*b7 + 1472*b6 - 335*b5 + 305*b4 - 14060*b3 + 3771*b2 - 1375*b1 + 19667) * q^26 + (-3*b13 + 3*b12 + 56*b11 + 56*b10 - 15*b9 + 15*b8 + 358*b5 - 355*b4 + 4904*b2 - 2452*b1 - 29626) * q^27 + (31*b13 - 39*b11 - 37*b9 - 194*b7 - 1760*b6 + 124*b5 + 1690*b4 + 12838*b3 - 8639*b2 - 12838) * q^28 + (36*b13 + 72*b12 - 57*b11 + 114*b10 - 60*b9 - 120*b8 - 312*b7 + 732*b6 - 219*b5 + 825*b4 + 29649*b3 - 3919*b2 + 7838*b1 + 29649) * q^29 + (-12*b13 - 6*b12 - 116*b11 + 58*b10 + 180*b9 + 90*b8 - 414*b7 + 978*b6 - 52*b5 - 52*b4 + 24150*b3 - 4940*b2 - 4940*b1) * q^30 + (-32*b13 - 32*b12 + 237*b11 - 237*b10 - 136*b9 - 136*b8 + 432*b7 + 1536*b6 + 216*b5 + 768*b4 - 21164*b3 - 1451*b1 - 10582) * q^31 + (63*b12 - 69*b10 + 387*b8 + 141*b7 + 819*b6 + 276*b5 + 1632*b4 - 41271*b3 + 5262*b2 - 5262*b1 - 82542) * q^32 + (12*b12 - 179*b10 - 225*b8 + 152*b7 - 1422*b6 + 137*b5 - 3011*b4 + 9275*b3 - 12043*b2 + 12043*b1 + 18550) * q^33 + (54*b13 + 54*b12 - 72*b11 + 72*b10 + 210*b9 + 210*b8 - 604*b7 + 476*b6 - 302*b5 + 238*b4 + 64500*b3 - 11313*b1 + 32250) * q^34 + (-54*b13 - 27*b12 - 162*b11 + 81*b10 + 1058*b9 + 529*b8 - 48*b7 - 1074*b6 - 54*b5 - 54*b4 - 96684*b3 + 5755*b2 + 5755*b1) * q^35 + (15*b13 + 30*b12 + 89*b11 - 178*b10 + 39*b9 + 78*b8 - 303*b7 - 3925*b6 - 377*b5 - 3999*b4 + 90157*b3 - 3904*b2 + 7808*b1 + 90157) * q^36 + (113*b13 - 48*b11 - 95*b9 - 408*b7 - 1338*b6 + 247*b5 + 1177*b4 + 26247*b3 + 6332*b2 - 26247) * q^37 + (-128*b13 + 128*b12 + 11*b11 + 11*b10 - 132*b9 + 132*b8 + 401*b5 - 2761*b4 + 10030*b2 - 5015*b1 + 71181) * q^38 + (180*b13 + 174*b12 - 127*b11 + 235*b10 - 522*b9 - 450*b8 - 152*b7 + 1561*b6 + 950*b5 + 2485*b4 - 7434*b3 + 21420*b2 - 17003*b1 + 2920) * q^39 + (37*b13 - 37*b12 + 11*b11 + 11*b10 + 321*b9 - 321*b8 - 1385*b5 - 1271*b4 + 21700*b2 - 10850*b1 - 969) * q^40 + (-99*b13 + 15*b11 - 499*b9 + 600*b7 - 90*b6 - 486*b5 + 204*b4 - 77221*b3 - 29077*b2 + 77221) * q^41 + (-165*b13 - 330*b12 + 138*b11 - 276*b10 - 141*b9 - 282*b8 + 1107*b7 + 2331*b6 + 804*b5 + 2028*b4 + 16065*b3 - 30984*b2 + 61968*b1 + 16065) * q^42 + (-44*b13 - 22*b12 + 578*b11 - 289*b10 - 516*b9 - 258*b8 + 2928*b7 + 2385*b6 + 311*b5 + 311*b4 + 130850*b3 - 9223*b2 - 9223*b1) * q^43 + (109*b13 + 109*b12 - 589*b11 + 589*b10 - 343*b9 - 343*b8 - 1212*b7 + 320*b6 - 606*b5 + 160*b4 - 136572*b3 - 33445*b1 - 68286) * q^44 + (-315*b12 + 195*b10 + 162*b8 - 480*b7 + 730*b6 - 1080*b5 + 1340*b4 - 218164*b3 + 15375*b2 - 15375*b1 - 436328) * q^45 + (-258*b12 + 603*b10 + 66*b8 + 823*b7 + 3073*b6 + 1991*b5 + 6491*b4 - 160119*b3 - 27069*b2 + 27069*b1 - 320238) * q^46 + (-397*b13 - 397*b12 + 112*b11 - 112*b10 + 715*b9 + 715*b8 - 3216*b7 - 4460*b6 - 1608*b5 - 2230*b4 + 364380*b3 - 16020*b1 + 182190) * q^47 + (300*b13 + 150*b12 + 564*b11 - 282*b10 - 1620*b9 - 810*b8 + 3054*b7 + 1114*b6 + 132*b5 + 132*b4 - 169954*b3 + 22236*b2 + 22236*b1) * q^48 + (110*b13 + 220*b12 - 451*b11 + 902*b10 - 619*b9 - 1238*b8 - 2808*b7 + 2586*b6 - 2247*b5 + 3147*b4 + 379238*b3 + 4483*b2 - 8966*b1 + 379238) * q^49 + (-245*b13 + 80*b11 - 893*b9 - 945*b7 + 1915*b6 + 1270*b5 - 1590*b4 - 11899*b3 + 28980*b2 + 11899) * q^50 + (135*b13 - 135*b12 - 19*b11 - 19*b10 - 141*b9 + 141*b8 + 2476*b5 + 10345*b4 + 8642*b2 - 4321*b1 - 2892) * q^51 + (-197*b13 - 320*b12 + 913*b11 - 1052*b10 + 1051*b9 + 96*b8 + 4450*b7 - 6512*b6 + 2601*b5 - 4821*b4 - 406170*b3 + 39503*b2 + 8685*b1 + 160675) * q^52 + (-172*b13 + 172*b12 - 818*b11 - 818*b10 - 467*b9 + 467*b8 - 3620*b5 - 8252*b4 + 26444*b2 - 13222*b1 - 121062) * q^53 + (-42*b13 + 419*b11 + 2046*b9 + 2543*b7 + 9613*b6 - 2082*b5 - 9152*b4 - 452435*b3 - 45299*b2 + 452435) * q^54 + (307*b13 + 614*b12 + b11 - 2*b10 + 631*b9 + 1262*b8 + 2048*b7 - 6046*b6 + 2354*b5 - 5740*b4 + 337164*b3 - 36885*b2 + 73770*b1 + 337164) * q^55 + (556*b13 + 278*b12 - 1612*b11 + 806*b10 + 1172*b9 + 586*b8 - 5556*b7 - 13236*b6 - 1084*b5 - 1084*b4 + 1083768*b3 - 40654*b2 - 40654*b1) * q^56 + (15*b13 + 15*b12 - 232*b11 + 232*b10 + 2304*b9 + 2304*b8 - 3632*b7 - 14864*b6 - 1816*b5 - 7432*b4 - 590118*b3 - 41540*b1 - 295059) * q^57 + (726*b12 + 60*b10 - 2190*b8 + 2728*b7 - 7256*b6 + 6242*b5 - 13726*b4 - 705204*b3 + 30729*b2 - 30729*b1 - 1410408) * q^58 + (911*b12 - 754*b10 + 1459*b8 - 1056*b7 + 5609*b6 - 1955*b5 + 11375*b4 - 62046*b3 - 23798*b2 + 23798*b1 - 124092) * q^59 + (894*b13 + 894*b12 + 822*b11 - 822*b10 - 4122*b9 - 4122*b8 - 1236*b7 - 4268*b6 - 618*b5 - 2134*b4 + 1612748*b3 - 39540*b1 + 806374) * q^60 + (-836*b13 - 418*b12 + 102*b11 - 51*b10 - 952*b9 - 476*b8 - 3536*b7 + 20404*b6 + 469*b5 + 469*b4 - 564905*b3 + 14907*b2 + 14907*b1) * q^61 + (-522*b13 - 1044*b12 + 624*b11 - 1248*b10 + 2578*b9 + 5156*b8 + 1812*b7 + 35292*b6 + 666*b5 + 34146*b4 + 248856*b3 - 29858*b2 + 59716*b1 + 248856) * q^62 + (177*b13 - 1091*b11 + 5001*b9 - 80*b7 + 768*b6 - 1188*b5 - 2036*b4 - 366212*b3 - 17047*b2 + 366212) * q^63 + (345*b13 - 345*b12 - 445*b11 - 445*b10 + 2581*b9 - 2581*b8 - 4701*b5 + 3621*b4 - 43300*b2 + 21650*b1 + 1029827) * q^64 + (-574*b13 - 177*b12 - 1037*b11 + 1431*b10 + 193*b9 + 4659*b8 - 7848*b7 - 15664*b6 - 2209*b5 + 1341*b4 - 628314*b3 + 9070*b2 - 38635*b1 + 429450) * q^65 + (297*b13 - 297*b12 + 1920*b11 + 1920*b10 - 1755*b9 + 1755*b8 + 9747*b5 + 48483*b4 + 17004*b2 - 8502*b1 - 2290797) * q^66 + (915*b13 - 969*b11 - 2685*b9 - 7136*b7 + 15271*b6 + 5252*b5 - 17155*b4 - 226112*b3 - 24645*b2 + 226112) * q^67 + (54*b13 + 108*b12 - 198*b11 + 396*b10 + 550*b9 + 1100*b8 - 7545*b7 - 26085*b6 - 7293*b5 - 25833*b4 + 1387671*b3 + 38875*b2 - 77750*b1 + 1387671) * q^68 + (-1542*b13 - 771*b12 + 2604*b11 - 1302*b10 - 2544*b9 - 1272*b8 + 9144*b7 + 6474*b6 + 2073*b5 + 2073*b4 + 1946409*b3 + 46290*b2 + 46290*b1) * q^69 + (-972*b13 - 972*b12 + 2424*b11 - 2424*b10 - 3240*b9 - 3240*b8 + 13888*b7 - 1256*b6 + 6944*b5 - 628*b4 - 2048040*b3 + 119040*b1 - 1024020) * q^70 + (-442*b12 - 268*b10 - 2488*b8 - 8616*b7 - 11833*b6 - 17942*b5 - 24376*b4 - 1204022*b3 - 27244*b2 + 27244*b1 - 2408044) * q^71 + (-777*b12 + 287*b10 - 2541*b8 - 1367*b7 + 2495*b6 - 3224*b5 + 4500*b4 - 318331*b3 + 63166*b2 - 63166*b1 - 636662) * q^72 + (234*b13 + 234*b12 - 3003*b11 + 3003*b10 + 3141*b9 + 3141*b8 + 24416*b7 - 2788*b6 + 12208*b5 - 1394*b4 + 539572*b3 + 74637*b1 + 269786) * q^73 + (972*b13 + 486*b12 - 3960*b11 + 1980*b10 - 1988*b9 - 994*b8 - 4524*b7 - 1104*b6 - 2466*b5 - 2466*b4 - 1205628*b3 - 71191*b2 - 71191*b1) * q^74 + (462*b13 + 924*b12 + 471*b11 - 942*b10 - 4518*b9 - 9036*b8 + 15648*b7 + 589*b6 + 15639*b5 + 580*b4 + 1492478*b3 + 23025*b2 - 46050*b1 + 1492478) * q^75 + (197*b13 + 3399*b11 - 6575*b9 + 10176*b7 + 15006*b6 - 6974*b5 - 11804*b4 - 620096*b3 + 39857*b2 + 620096) * q^76 + (-655*b13 + 655*b12 + 1912*b11 + 1912*b10 - 4294*b9 + 4294*b8 - 5234*b5 + 12472*b4 + 143096*b2 - 71548*b1 + 1421697) * q^77 + (1446*b13 + 1494*b12 - 2754*b11 + 1799*b10 - 270*b9 - 5682*b8 - 5063*b7 - 23629*b6 - 21419*b5 - 56787*b4 - 816481*b3 - 85157*b2 + 15605*b1 + 3157324) * q^78 + (281*b13 - 281*b12 - 59*b11 - 59*b10 + 3537*b9 - 3537*b8 + 4376*b5 - 25906*b4 - 111550*b2 + 55775*b1 - 2205140) * q^79 + (-1577*b13 + 269*b11 + 2027*b9 - 891*b7 - 17093*b6 + 2737*b5 + 18939*b4 - 1410879*b3 + 195010*b2 + 1410879) * q^80 + (-1011*b13 - 2022*b12 - 735*b11 + 1470*b10 - 3987*b9 - 7974*b8 - 1392*b7 + 40030*b6 - 1668*b5 + 39754*b4 + 2022035*b3 + 115179*b2 - 230358*b1 + 2022035) * q^81 + (-64*b13 - 32*b12 - 1384*b11 + 692*b10 + 1880*b9 + 940*b8 - 23072*b7 - 27116*b6 - 660*b5 - 660*b4 + 5294220*b3 + 93401*b2 + 93401*b1) * q^82 + (2024*b13 + 2024*b12 + 826*b11 - 826*b10 + 444*b9 + 444*b8 - 1824*b7 - 22820*b6 - 912*b5 - 11410*b4 - 2516096*b3 + 62602*b1 - 1258048) * q^83 + (-1401*b12 - 2953*b10 + 14823*b8 + 15382*b7 + 16380*b6 + 26410*b5 + 28406*b4 - 4616246*b3 - 181001*b2 + 181001*b1 - 9232492) * q^84 + (-2670*b12 - 1680*b10 + 3267*b8 - 40*b7 + 8000*b6 - 4430*b5 + 11650*b4 - 1064004*b3 + 51120*b2 - 51120*b1 - 2128008) * q^85 + (-3822*b13 - 3822*b12 + 1101*b11 - 1101*b10 + 7110*b9 + 7110*b8 - 8982*b7 + 92118*b6 - 4491*b5 + 46059*b4 + 3309990*b3 + 118063*b1 + 1654995) * q^86 + (828*b13 + 414*b12 + 6536*b11 - 3268*b10 + 12960*b9 + 6480*b8 - 8808*b7 - 37851*b6 + 2854*b5 + 2854*b4 - 1776114*b3 - 98668*b2 - 98668*b1) * q^87 + (1762*b13 + 3524*b12 - 1346*b11 + 2692*b10 + 3550*b9 + 7100*b8 - 16364*b7 - 42476*b6 - 13256*b5 - 39368*b4 + 3897240*b3 + 69562*b2 - 139124*b1 + 3897240) * q^88 + (-785*b13 - 88*b11 - 2128*b9 - 5832*b7 - 28364*b6 + 6529*b5 + 29061*b4 - 62139*b3 - 232260*b2 + 62139) * q^89 + (-1023*b13 + 1023*b12 - 2314*b11 - 2314*b10 - 1203*b9 + 1203*b8 - 5*b5 - 67421*b4 - 316090*b2 + 158045*b1 + 2895207) * q^90 + (844*b13 - 927*b12 + 4558*b11 - 5726*b10 - 5708*b9 - 11795*b8 + 4448*b7 + 70805*b6 + 13397*b5 + 94355*b4 - 1944890*b3 - 453884*b2 + 199050*b1 + 1577372) * q^91 + (-1727*b13 + 1727*b12 - 3853*b11 - 3853*b10 + 4725*b9 - 4725*b8 + 16952*b5 + 11774*b4 - 412606*b2 + 206303*b1 - 3485604) * q^92 + (-930*b13 - 1012*b11 + 360*b9 - 3448*b7 - 20812*b6 + 3366*b5 + 20730*b4 - 1815066*b3 + 521452*b2 + 1815066) * q^93 + (1066*b13 + 2132*b12 + 788*b11 - 1576*b10 + 5586*b9 + 11172*b8 - 7132*b7 - 19000*b6 - 6854*b5 - 18722*b4 + 2904852*b3 + 396582*b2 - 793164*b1 + 2904852) * q^94 + (7630*b13 + 3815*b12 - 4720*b11 + 2360*b10 + 2862*b9 + 1431*b8 + 14040*b7 + 64140*b6 - 6175*b5 - 6175*b4 + 6178434*b3 + 28420*b2 + 28420*b1) * q^95 + (306*b13 + 306*b12 - 9642*b11 + 9642*b10 + 2250*b9 + 2250*b8 - 13524*b7 + 144852*b6 - 6762*b5 + 72426*b4 - 5482308*b3 + 146460*b1 - 2741154) * q^96 + (3461*b12 + 6684*b10 - 4520*b8 - 22168*b7 + 25064*b6 - 34191*b5 + 60273*b4 - 3292737*b3 - 81728*b2 + 81728*b1 - 6585474) * q^97 + (7041*b12 + 2448*b10 - 11835*b8 - 8679*b7 - 99039*b6 - 7869*b5 - 188589*b4 + 882807*b3 + 600179*b2 - 600179*b1 + 1765614) * q^98 + (3405*b13 + 3405*b12 + 8249*b11 - 8249*b10 - 87*b9 - 87*b8 + 3136*b7 - 143664*b6 + 1568*b5 - 71832*b4 + 7192576*b3 + 510037*b1 + 3596288) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$14 q - 3 q^{2} + 26 q^{3} + 383 q^{4} - 264 q^{6} - 2772 q^{7} - 3491 q^{9}+O(q^{10})$$ 14 * q - 3 * q^2 + 26 * q^3 + 383 * q^4 - 264 * q^6 - 2772 * q^7 - 3491 * q^9 $$14 q - 3 q^{2} + 26 q^{3} + 383 q^{4} - 264 q^{6} - 2772 q^{7} - 3491 q^{9} - 509 q^{10} + 6516 q^{11} + 38380 q^{12} + 5109 q^{13} - 47916 q^{14} - 19266 q^{15} - 633 q^{16} - 38403 q^{17} + 43254 q^{19} + 89409 q^{20} - 125882 q^{22} - 68550 q^{23} + 224454 q^{24} + 39380 q^{25} + 361959 q^{26} - 432400 q^{27} - 234780 q^{28} + 221583 q^{29} - 157890 q^{30} - 878067 q^{32} + 219756 q^{33} + 659616 q^{35} + 635675 q^{36} - 565347 q^{37} + 953640 q^{38} + 30082 q^{39} - 77182 q^{40} + 1716057 q^{41} + 206406 q^{42} - 882692 q^{43} - 4616853 q^{45} - 3267480 q^{46} + 1133630 q^{48} + 2653979 q^{49} + 145128 q^{50} - 38964 q^{51} + 4962178 q^{52} - 1791210 q^{53} + 9597234 q^{54} + 2453536 q^{55} - 7457952 q^{56} - 14978127 q^{58} - 1189032 q^{59} + 3858525 q^{61} + 1904142 q^{62} + 7718520 q^{63} + 14570922 q^{64} + 10424661 q^{65} - 31956396 q^{66} + 4674330 q^{67} + 9569619 q^{68} - 13743840 q^{69} - 25203126 q^{71} - 6851607 q^{72} + 8640105 q^{74} + 10316152 q^{75} + 12934506 q^{76} + 19599336 q^{77} + 50085750 q^{78} - 30688440 q^{79} + 29127963 q^{80} + 13886221 q^{81} - 37359719 q^{82} - 96398508 q^{84} - 22436313 q^{85} + 12769686 q^{87} + 27045536 q^{88} + 2123160 q^{89} + 41193546 q^{90} + 37205532 q^{91} - 47685588 q^{92} + 36638964 q^{93} + 19138810 q^{94} - 43420386 q^{95} - 68556288 q^{97} + 16173003 q^{98}+O(q^{100})$$ 14 * q - 3 * q^2 + 26 * q^3 + 383 * q^4 - 264 * q^6 - 2772 * q^7 - 3491 * q^9 - 509 * q^10 + 6516 * q^11 + 38380 * q^12 + 5109 * q^13 - 47916 * q^14 - 19266 * q^15 - 633 * q^16 - 38403 * q^17 + 43254 * q^19 + 89409 * q^20 - 125882 * q^22 - 68550 * q^23 + 224454 * q^24 + 39380 * q^25 + 361959 * q^26 - 432400 * q^27 - 234780 * q^28 + 221583 * q^29 - 157890 * q^30 - 878067 * q^32 + 219756 * q^33 + 659616 * q^35 + 635675 * q^36 - 565347 * q^37 + 953640 * q^38 + 30082 * q^39 - 77182 * q^40 + 1716057 * q^41 + 206406 * q^42 - 882692 * q^43 - 4616853 * q^45 - 3267480 * q^46 + 1133630 * q^48 + 2653979 * q^49 + 145128 * q^50 - 38964 * q^51 + 4962178 * q^52 - 1791210 * q^53 + 9597234 * q^54 + 2453536 * q^55 - 7457952 * q^56 - 14978127 * q^58 - 1189032 * q^59 + 3858525 * q^61 + 1904142 * q^62 + 7718520 * q^63 + 14570922 * q^64 + 10424661 * q^65 - 31956396 * q^66 + 4674330 * q^67 + 9569619 * q^68 - 13743840 * q^69 - 25203126 * q^71 - 6851607 * q^72 + 8640105 * q^74 + 10316152 * q^75 + 12934506 * q^76 + 19599336 * q^77 + 50085750 * q^78 - 30688440 * q^79 + 29127963 * q^80 + 13886221 * q^81 - 37359719 * q^82 - 96398508 * q^84 - 22436313 * q^85 + 12769686 * q^87 + 27045536 * q^88 + 2123160 * q^89 + 41193546 * q^90 + 37205532 * q^91 - 47685588 * q^92 + 36638964 * q^93 + 19138810 * q^94 - 43420386 * q^95 - 68556288 * q^97 + 16173003 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{14} + 1279 x^{12} + 629380 x^{10} + 148562016 x^{8} + 16872573312 x^{6} + 790180980480 x^{4} + \cdots + 4669637050368$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 1054513 \nu^{12} - 1163482171 \nu^{10} - 463409859688 \nu^{8} - 80046208978752 \nu^{6} + \cdots + 44\!\cdots\!68 ) / 16\!\cdots\!00$$ (-1054513*v^12 - 1163482171*v^10 - 463409859688*v^8 - 80046208978752*v^6 - 5575765377762432*v^4 - 113215875162598656*v^2 + 81430885066752000*v + 44747156707872768) / 162861770133504000 $$\beta_{3}$$ $$=$$ $$( - 124535171 \nu^{13} - 172984934657 \nu^{11} - 93500560218296 \nu^{9} + \cdots - 10\!\cdots\!00 ) / 21\!\cdots\!00$$ (-124535171*v^13 - 172984934657*v^11 - 93500560218296*v^9 - 24523670603169984*v^7 - 3141509334507817344*v^5 - 170867970374425028352*v^3 - 2777104437807149420544*v - 1058275782327508992000) / 2116551564655017984000 $$\beta_{4}$$ $$=$$ $$( 38514673 \nu^{12} + 38644774891 \nu^{10} + 13645656502648 \nu^{8} + \cdots + 46\!\cdots\!72 ) / 56\!\cdots\!00$$ (38514673*v^12 + 38644774891*v^10 + 13645656502648*v^8 + 2017240301188992*v^6 + 113412099177804672*v^4 + 1919534509330258176*v^2 + 4607758640046108672) / 56375228123136000 $$\beta_{5}$$ $$=$$ $$( - 255090683 \nu^{12} - 256426706561 \nu^{10} - 90782111635808 \nu^{8} + \cdots + 37\!\cdots\!88 ) / 36\!\cdots\!00$$ (-255090683*v^12 - 256426706561*v^10 - 90782111635808*v^8 - 13472269898132832*v^6 - 762269588855661312*v^4 - 12620006766077988096*v^2 + 37309264897355993088) / 366438982800384000 $$\beta_{6}$$ $$=$$ $$( 13570841413 \nu^{13} - 180749360389 \nu^{12} + 12086872800271 \nu^{11} + \cdots - 21\!\cdots\!96 ) / 52\!\cdots\!00$$ (13570841413*v^13 - 180749360389*v^12 + 12086872800271*v^11 - 181359928563463*v^10 + 3279266034305488*v^9 - 64039065966927064*v^8 + 176335921489001952*v^7 - 9466908733479939456*v^6 - 36882942671385946368*v^5 - 532242981441437325696*v^4 - 3269174288334486591744*v^3 - 9008375452286901619968*v^2 - 46338746170385255251968*v - 21624211297736387997696) / 529137891163754496000 $$\beta_{7}$$ $$=$$ $$( - 90777752893 \nu^{13} + 736701892504 \nu^{12} - 95124348537631 \nu^{11} + \cdots - 10\!\cdots\!44 ) / 21\!\cdots\!00$$ (-90777752893*v^13 + 736701892504*v^12 - 95124348537631*v^11 + 740560328548168*v^10 - 36250141193675368*v^9 + 262178738404213504*v^8 - 6233455938223975872*v^7 + 38907915465807618816*v^6 - 499827084378787354752*v^5 + 2201434572615149869056*v^4 - 19663494938794965954816*v^3 + 36446579540433229621248*v^2 - 325448420735574335434752*v - 107749157023564108038144) / 2116551564655017984000 $$\beta_{8}$$ $$=$$ $$( 117312010813 \nu^{13} + 1183029064020 \nu^{12} + 136138707608671 \nu^{11} + \cdots - 70\!\cdots\!20 ) / 21\!\cdots\!00$$ (117312010813*v^13 + 1183029064020*v^12 + 136138707608671*v^11 + 1174496727557340*v^10 + 59784767126965288*v^9 + 406809975172698720*v^8 + 12422156529451445952*v^7 + 57498756909241812480*v^6 + 1232622039639896802432*v^5 + 2723340132446828705280*v^4 + 50980791208834444697856*v^3 - 190314307915092495360*v^2 + 621198086996247499026432*v - 704033130027672789073920) / 2116551564655017984000 $$\beta_{9}$$ $$=$$ $$( 117312010813 \nu^{13} - 1183029064020 \nu^{12} + 136138707608671 \nu^{11} + \cdots + 70\!\cdots\!20 ) / 21\!\cdots\!00$$ (117312010813*v^13 - 1183029064020*v^12 + 136138707608671*v^11 - 1174496727557340*v^10 + 59784767126965288*v^9 - 406809975172698720*v^8 + 12422156529451445952*v^7 - 57498756909241812480*v^6 + 1232622039639896802432*v^5 - 2723340132446828705280*v^4 + 50980791208834444697856*v^3 + 190314307915092495360*v^2 + 621198086996247499026432*v + 704033130027672789073920) / 2116551564655017984000 $$\beta_{10}$$ $$=$$ $$( 288283218065 \nu^{13} + 8453701114932 \nu^{12} + 290032520116955 \nu^{11} + \cdots - 21\!\cdots\!52 ) / 10\!\cdots\!00$$ (288283218065*v^13 + 8453701114932*v^12 + 290032520116955*v^11 + 8438304811421244*v^10 + 102880276187204840*v^9 + 2947217330901040032*v^8 + 15329537466552452160*v^7 + 424354697025076124928*v^6 + 871060004696459871360*v^5 + 22033050227793985909248*v^4 + 13783578174785955790080*v^3 + 278514715400524435497984*v^2 - 35193577254354394583040*v - 21944147639292329213952) / 1058275782327508992000 $$\beta_{11}$$ $$=$$ $$( - 288283218065 \nu^{13} + 8453701114932 \nu^{12} - 290032520116955 \nu^{11} + \cdots - 21\!\cdots\!52 ) / 10\!\cdots\!00$$ (-288283218065*v^13 + 8453701114932*v^12 - 290032520116955*v^11 + 8438304811421244*v^10 - 102880276187204840*v^9 + 2947217330901040032*v^8 - 15329537466552452160*v^7 + 424354697025076124928*v^6 - 871060004696459871360*v^5 + 22033050227793985909248*v^4 - 13783578174785955790080*v^3 + 278514715400524435497984*v^2 + 35193577254354394583040*v - 21944147639292329213952) / 1058275782327508992000 $$\beta_{12}$$ $$=$$ $$( - 673250292779 \nu^{13} - 12171829886856 \nu^{12} - 729912943754393 \nu^{11} + \cdots + 14\!\cdots\!16 ) / 21\!\cdots\!00$$ (-673250292779*v^13 - 12171829886856*v^12 - 729912943754393*v^11 - 11901202106377752*v^10 - 292534825478526104*v^9 - 4007447292765632256*v^8 - 54012285833099831616*v^7 - 535047413047945300224*v^6 - 4668401040727045136256*v^5 - 22286621773682313283584*v^4 - 172303483484377947080448*v^3 - 51455518676628787279872*v^2 - 1660242418792387309817856*v + 1452758486216417941487616) / 2116551564655017984000 $$\beta_{13}$$ $$=$$ $$( - 673250292779 \nu^{13} + 12171829886856 \nu^{12} - 729912943754393 \nu^{11} + \cdots - 14\!\cdots\!16 ) / 21\!\cdots\!00$$ (-673250292779*v^13 + 12171829886856*v^12 - 729912943754393*v^11 + 11901202106377752*v^10 - 292534825478526104*v^9 + 4007447292765632256*v^8 - 54012285833099831616*v^7 + 535047413047945300224*v^6 - 4668401040727045136256*v^5 + 22286621773682313283584*v^4 - 172303483484377947080448*v^3 + 51455518676628787279872*v^2 - 1660242418792387309817856*v - 1452758486216417941487616) / 2116551564655017984000
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{5} + \beta_{4} - 2\beta_{2} + \beta _1 - 183$$ b5 + b4 - 2*b2 + b1 - 183 $$\nu^{3}$$ $$=$$ $$\beta_{13} + \beta_{12} - \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} - 6 \beta_{7} - 10 \beta_{6} + \cdots + 173$$ b13 + b12 - b11 + b10 + b9 + b8 - 6*b7 - 10*b6 - 3*b5 - 5*b4 + 346*b3 - 296*b1 + 173 $$\nu^{4}$$ $$=$$ $$- 3 \beta_{13} + 3 \beta_{12} + 7 \beta_{11} + 7 \beta_{10} + 17 \beta_{9} - 17 \beta_{8} + \cdots + 54159$$ -3*b13 + 3*b12 + 7*b11 + 7*b10 + 17*b9 - 17*b8 - 345*b5 - 423*b4 + 1468*b2 - 734*b1 + 54159 $$\nu^{5}$$ $$=$$ $$- 449 \beta_{13} - 449 \beta_{12} + 581 \beta_{11} - 581 \beta_{10} - 125 \beta_{9} - 125 \beta_{8} + \cdots - 129847$$ -449*b13 - 449*b12 + 581*b11 - 581*b10 - 125*b9 - 125*b8 + 3354*b7 + 6758*b6 + 1677*b5 + 3379*b4 - 259694*b3 + 97138*b1 - 129847 $$\nu^{6}$$ $$=$$ $$2265 \beta_{13} - 2265 \beta_{12} - 4925 \beta_{11} - 4925 \beta_{10} - 8299 \beta_{9} + 8299 \beta_{8} + \cdots - 17739453$$ 2265*b13 - 2265*b12 - 4925*b11 - 4925*b10 - 8299*b9 + 8299*b8 + 117795*b5 + 176037*b4 - 786212*b2 + 393106*b1 - 17739453 $$\nu^{7}$$ $$=$$ $$179827 \beta_{13} + 179827 \beta_{12} - 267391 \beta_{11} + 267391 \beta_{10} - 51833 \beta_{9} + \cdots + 70177157$$ 179827*b13 + 179827*b12 - 267391*b11 + 267391*b10 - 51833*b9 - 51833*b8 - 1629774*b7 - 3398962*b6 - 814887*b5 - 1699481*b4 + 140354314*b3 - 33777398*b1 + 70177157 $$\nu^{8}$$ $$=$$ $$- 1221675 \beta_{13} + 1221675 \beta_{12} + 2522455 \beta_{11} + 2522455 \beta_{10} + 3425633 \beta_{9} + \cdots + 6155642823$$ -1221675*b13 + 1221675*b12 + 2522455*b11 + 2522455*b10 + 3425633*b9 - 3425633*b8 - 41718297*b5 - 71738271*b4 + 378399628*b2 - 189199814*b1 + 6155642823 $$\nu^{9}$$ $$=$$ $$- 70953161 \beta_{13} - 70953161 \beta_{12} + 114440429 \beta_{11} - 114440429 \beta_{10} + \cdots - 33942873535$$ -70953161*b13 - 70953161*b12 + 114440429*b11 - 114440429*b10 + 49175419*b9 + 49175419*b8 + 741503370*b7 + 1542768374*b6 + 370751685*b5 + 771384187*b4 - 67885747070*b3 + 12260783074*b1 - 33942873535 $$\nu^{10}$$ $$=$$ $$577854801 \beta_{13} - 577854801 \beta_{12} - 1153547669 \beta_{11} - 1153547669 \beta_{10} + \cdots - 2230336988517$$ 577854801*b13 - 577854801*b12 - 1153547669*b11 - 1153547669*b10 - 1363055635*b9 + 1363055635*b8 + 15319970235*b5 + 28996716909*b4 - 172076735204*b2 + 86038367602*b1 - 2230336988517 $$\nu^{11}$$ $$=$$ $$28059663691 \beta_{13} + 28059663691 \beta_{12} - 47572113271 \beta_{11} + 47572113271 \beta_{10} + \cdots + 15480788538317$$ 28059663691*b13 + 28059663691*b12 - 47572113271*b11 + 47572113271*b10 - 27336567617*b9 - 27336567617*b8 - 324693153534*b7 - 668343012994*b6 - 162346576767*b5 - 334171506497*b4 + 30961577076634*b3 - 4600210934150*b1 + 15480788538317 $$\nu^{12}$$ $$=$$ $$- 256767707235 \beta_{13} + 256767707235 \beta_{12} + 501083296495 \beta_{11} + 501083296495 \beta_{10} + \cdots + 835582948516959$$ -256767707235*b13 + 256767707235*b12 + 501083296495*b11 + 501083296495*b10 + 538574961545*b9 - 538574961545*b8 - 5794582025985*b5 - 11700853999095*b4 + 75547415153260*b2 - 37773707576630*b1 + 835582948516959 $$\nu^{13}$$ $$=$$ $$- 11162205575345 \beta_{13} - 11162205575345 \beta_{12} + 19529218262645 \beta_{11} + \cdots - 68\!\cdots\!75$$ -11162205575345*b13 - 11162205575345*b12 + 19529218262645*b11 - 19529218262645*b10 + 13039218072115*b9 + 13039218072115*b8 + 138857718329370*b7 + 282626578454630*b6 + 69428859164685*b5 + 141313289227315*b4 - 13618158632994350*b3 + 1769482174317874*b1 - 6809079316497175

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/13\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-\beta_{3}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 − 16.7657i − 16.7213i − 8.41902i 0.679146i 4.51724i 14.7644i 20.2132i 16.7657i 16.7213i 8.41902i − 0.679146i − 4.51724i − 14.7644i − 20.2132i
−14.5196 8.38287i 40.5114 70.1679i 76.5449 + 132.580i 223.318i −1176.42 + 679.204i −577.559 + 333.454i 420.649i −2188.85 3791.20i −1872.05 + 3242.48i
4.2 −14.4811 8.36065i −21.2782 + 36.8549i 75.8010 + 131.291i 94.5127i 616.262 355.799i 1225.76 707.691i 394.655i 187.977 + 325.587i −790.187 + 1368.64i
4.3 −7.29108 4.20951i 0.573744 0.993754i −28.5601 49.4675i 399.024i −8.36643 + 4.83036i −817.728 + 472.116i 1558.53i 1092.84 + 1892.86i 1679.70 2909.32i
4.4 0.588158 + 0.339573i −31.2267 + 54.0862i −63.7694 110.452i 439.155i −36.7324 + 21.2075i −1128.14 + 651.329i 173.548i −856.712 1483.87i 149.125 258.293i
4.5 3.91205 + 2.25862i 21.9195 37.9656i −53.7973 93.1796i 49.0275i 171.500 99.0156i 736.478 425.206i 1064.24i 132.573 + 229.623i 110.735 191.798i
4.6 12.7863 + 7.38219i −22.6982 + 39.3144i 44.9933 + 77.9308i 248.787i −580.452 + 335.124i 497.017 286.953i 561.243i 63.0878 + 109.271i −1836.59 + 3181.07i
4.7 17.5052 + 10.1066i 25.1984 43.6449i 140.288 + 242.985i 228.046i 882.204 509.341i −1321.83 + 763.159i 3084.03i −176.416 305.562i 2304.77 3991.98i
10.1 −14.5196 + 8.38287i 40.5114 + 70.1679i 76.5449 132.580i 223.318i −1176.42 679.204i −577.559 333.454i 420.649i −2188.85 + 3791.20i −1872.05 3242.48i
10.2 −14.4811 + 8.36065i −21.2782 36.8549i 75.8010 131.291i 94.5127i 616.262 + 355.799i 1225.76 + 707.691i 394.655i 187.977 325.587i −790.187 1368.64i
10.3 −7.29108 + 4.20951i 0.573744 + 0.993754i −28.5601 + 49.4675i 399.024i −8.36643 4.83036i −817.728 472.116i 1558.53i 1092.84 1892.86i 1679.70 + 2909.32i
10.4 0.588158 0.339573i −31.2267 54.0862i −63.7694 + 110.452i 439.155i −36.7324 21.2075i −1128.14 651.329i 173.548i −856.712 + 1483.87i 149.125 + 258.293i
10.5 3.91205 2.25862i 21.9195 + 37.9656i −53.7973 + 93.1796i 49.0275i 171.500 + 99.0156i 736.478 + 425.206i 1064.24i 132.573 229.623i 110.735 + 191.798i
10.6 12.7863 7.38219i −22.6982 39.3144i 44.9933 77.9308i 248.787i −580.452 335.124i 497.017 + 286.953i 561.243i 63.0878 109.271i −1836.59 3181.07i
10.7 17.5052 10.1066i 25.1984 + 43.6449i 140.288 242.985i 228.046i 882.204 + 509.341i −1321.83 763.159i 3084.03i −176.416 + 305.562i 2304.77 + 3991.98i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 4.7 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.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 13.8.e.a 14
3.b odd 2 1 117.8.q.b 14
4.b odd 2 1 208.8.w.a 14
13.c even 3 1 169.8.b.d 14
13.e even 6 1 inner 13.8.e.a 14
13.e even 6 1 169.8.b.d 14
13.f odd 12 2 169.8.a.g 14
39.h odd 6 1 117.8.q.b 14
52.i odd 6 1 208.8.w.a 14

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.8.e.a 14 1.a even 1 1 trivial
13.8.e.a 14 13.e even 6 1 inner
117.8.q.b 14 3.b odd 2 1
117.8.q.b 14 39.h odd 6 1
169.8.a.g 14 13.f odd 12 2
169.8.b.d 14 13.c even 3 1
169.8.b.d 14 13.e even 6 1
208.8.w.a 14 4.b odd 2 1
208.8.w.a 14 52.i odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{8}^{\mathrm{new}}(13, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{14} + \cdots + 4669637050368$$
$3$ $$T^{14} + \cdots + 61\!\cdots\!44$$
$5$ $$T^{14} + \cdots + 10\!\cdots\!00$$
$7$ $$T^{14} + \cdots + 74\!\cdots\!00$$
$11$ $$T^{14} + \cdots + 48\!\cdots\!00$$
$13$ $$T^{14} + \cdots + 38\!\cdots\!73$$
$17$ $$T^{14} + \cdots + 30\!\cdots\!29$$
$19$ $$T^{14} + \cdots + 27\!\cdots\!72$$
$23$ $$T^{14} + \cdots + 19\!\cdots\!56$$
$29$ $$T^{14} + \cdots + 79\!\cdots\!81$$
$31$ $$T^{14} + \cdots + 20\!\cdots\!00$$
$37$ $$T^{14} + \cdots + 25\!\cdots\!75$$
$41$ $$T^{14} + \cdots + 15\!\cdots\!83$$
$43$ $$T^{14} + \cdots + 66\!\cdots\!00$$
$47$ $$T^{14} + \cdots + 12\!\cdots\!48$$
$53$ $$(T^{7} + \cdots - 32\!\cdots\!76)^{2}$$
$59$ $$T^{14} + \cdots + 15\!\cdots\!28$$
$61$ $$T^{14} + \cdots + 23\!\cdots\!25$$
$67$ $$T^{14} + \cdots + 29\!\cdots\!00$$
$71$ $$T^{14} + \cdots + 65\!\cdots\!00$$
$73$ $$T^{14} + \cdots + 13\!\cdots\!00$$
$79$ $$(T^{7} + \cdots - 60\!\cdots\!00)^{2}$$
$83$ $$T^{14} + \cdots + 20\!\cdots\!48$$
$89$ $$T^{14} + \cdots + 35\!\cdots\!68$$
$97$ $$T^{14} + \cdots + 28\!\cdots\!28$$