# Properties

 Label 13.6.a.b Level $13$ Weight $6$ Character orbit 13.a Self dual yes Analytic conductor $2.085$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [13,6,Mod(1,13)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(13, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("13.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$13$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 13.a (trivial)

## 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: yes Analytic conductor: $$2.08498965757$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.168897.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 100x + 256$$ x^3 - x^2 - 100*x + 256 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 + 2) q^{2} + ( - \beta_{2} - \beta_1 + 3) q^{3} + (4 \beta_{2} + \beta_1 + 40) q^{4} + ( - \beta_{2} - 7 \beta_1 + 21) q^{5} + ( - 10 \beta_{2} - \beta_1 - 66) q^{6} + ( - 3 \beta_{2} - 3 \beta_1 - 19) q^{7} + (28 \beta_{2} + 27 \beta_1 + 100) q^{8} + (\beta_{2} + 7 \beta_1 - 66) q^{9}+O(q^{10})$$ q + (b1 + 2) * q^2 + (-b2 - b1 + 3) * q^3 + (4*b2 + b1 + 40) * q^4 + (-b2 - 7*b1 + 21) * q^5 + (-10*b2 - b1 - 66) * q^6 + (-3*b2 - 3*b1 - 19) * q^7 + (28*b2 + 27*b1 + 100) * q^8 + (b2 + 7*b1 - 66) * q^9 $$q + (\beta_1 + 2) q^{2} + ( - \beta_{2} - \beta_1 + 3) q^{3} + (4 \beta_{2} + \beta_1 + 40) q^{4} + ( - \beta_{2} - 7 \beta_1 + 21) q^{5} + ( - 10 \beta_{2} - \beta_1 - 66) q^{6} + ( - 3 \beta_{2} - 3 \beta_1 - 19) q^{7} + (28 \beta_{2} + 27 \beta_1 + 100) q^{8} + (\beta_{2} + 7 \beta_1 - 66) q^{9} + ( - 34 \beta_{2} + 23 \beta_1 - 438) q^{10} + (30 \beta_{2} - 26 \beta_1 + 194) q^{11} + ( - 32 \beta_{2} - 83 \beta_1 - 336) q^{12} + 169 q^{13} + ( - 30 \beta_{2} - 31 \beta_1 - 254) q^{14} + (31 \beta_{2} - 17 \beta_1 + 663) q^{15} + (148 \beta_{2} + 181 \beta_1 + 868) q^{16} + ( - 93 \beta_{2} + 77 \beta_1 + 277) q^{17} + (34 \beta_{2} - 68 \beta_1 + 348) q^{18} + ( - 134 \beta_{2} + 178 \beta_1 - 10) q^{19} + ( - 80 \beta_{2} - 407 \beta_1 - 120) q^{20} + (31 \beta_{2} + 49 \beta_1 + 447) q^{21} + (76 \beta_{2} + 370 \beta_1 - 1260) q^{22} + (152 \beta_{2} - 72 \beta_1 + 1232) q^{23} + ( - 204 \beta_{2} - 381 \beta_1 - 4332) q^{24} + (205 \beta_{2} - 365 \beta_1 + 796) q^{25} + (169 \beta_1 + 338) q^{26} + (257 \beta_{2} + 305 \beta_1 - 1527) q^{27} + ( - 208 \beta_{2} - 277 \beta_1 - 2128) q^{28} + ( - 584 \beta_{2} + 56 \beta_1 - 2938) q^{29} + (118 \beta_{2} + 835 \beta_1 + 294) q^{30} + ( - 268 \beta_{2} - 148 \beta_1 - 820) q^{31} + (716 \beta_{2} + 563 \beta_1 + 11436) q^{32} + (134 \beta_{2} - 550 \beta_1 - 426) q^{33} + ( - 250 \beta_{2} - 265 \beta_1 + 5418) q^{34} + (121 \beta_{2} + 145 \beta_1 + 1401) q^{35} + ( - 100 \beta_{2} + 362 \beta_1 - 1680) q^{36} + (419 \beta_{2} - 451 \beta_1 - 6727) q^{37} + ( - 92 \beta_{2} - 858 \beta_1 + 11548) q^{38} + ( - 169 \beta_{2} - 169 \beta_1 + 507) q^{39} + ( - 1020 \beta_{2} - 849 \beta_1 - 14220) q^{40} + ( - 250 \beta_{2} + 858 \beta_1 - 3952) q^{41} + (382 \beta_{2} + 553 \beta_1 + 4350) q^{42} + (697 \beta_{2} - 431 \beta_1 + 821) q^{43} + (976 \beta_{2} - 418 \beta_1 + 16736) q^{44} + ( - 160 \beta_{2} + 680 \beta_1 - 4866) q^{45} + (624 \beta_{2} + 2064 \beta_1 - 1824) q^{46} + ( - 807 \beta_{2} + 33 \beta_1 + 11409) q^{47} + ( - 1724 \beta_{2} - 2315 \beta_1 - 24636) q^{48} + (177 \beta_{2} + 231 \beta_1 - 14934) q^{49} + ( - 230 \beta_{2} + 2186 \beta_1 - 22408) q^{50} + ( - 1265 \beta_{2} + 823 \beta_1 + 4215) q^{51} + (676 \beta_{2} + 169 \beta_1 + 6760) q^{52} + (2154 \beta_{2} + 822 \beta_1 - 4464) q^{53} + (2762 \beta_{2} - 547 \beta_1 + 18714) q^{54} + (578 \beta_{2} - 3550 \beta_1 + 12954) q^{55} + ( - 1396 \beta_{2} - 1899 \beta_1 - 15796) q^{56} + ( - 1950 \beta_{2} + 1662 \beta_1 + 18) q^{57} + ( - 3280 \beta_{2} - 5914 \beta_1 - 4404) q^{58} + ( - 950 \beta_{2} + 1458 \beta_1 + 20830) q^{59} + (3056 \beta_{2} + 593 \beta_1 + 36624) q^{60} + (2330 \beta_{2} + 1910 \beta_1 - 4888) q^{61} + ( - 2200 \beta_{2} - 2012 \beta_1 - 12776) q^{62} + (14 \beta_{2} - 10 \beta_1 - 546) q^{63} + (1812 \beta_{2} + 8661 \beta_1 + 36244) q^{64} + ( - 169 \beta_{2} - 1183 \beta_1 + 3549) q^{65} + ( - 1396 \beta_{2} + 794 \beta_1 - 37716) q^{66} + ( - 1666 \beta_{2} + 1478 \beta_1 + 18218) q^{67} + (416 \beta_{2} + 1969 \beta_1 - 17048) q^{68} + ( - 48 \beta_{2} - 2976 \beta_1 - 5712) q^{69} + (1306 \beta_{2} + 1861 \beta_1 + 13146) q^{70} + (3127 \beta_{2} - 633 \beta_1 + 26071) q^{71} + ( - 240 \beta_{2} - 366 \beta_1 + 9720) q^{72} + ( - 2568 \beta_{2} - 2712 \beta_1 - 13438) q^{73} + (710 \beta_{2} - 4181 \beta_1 - 42446) q^{74} + (2944 \beta_{2} - 3416 \beta_1 + 8988) q^{75} + (304 \beta_{2} + 6250 \beta_1 - 35296) q^{76} + ( - 438 \beta_{2} - 922 \beta_1 - 6710) q^{77} + ( - 1690 \beta_{2} - 169 \beta_1 - 11154) q^{78} + ( - 3576 \beta_{2} - 2856 \beta_1 - 19672) q^{79} + ( - 6956 \beta_{2} - 5447 \beta_1 - 86412) q^{80} + ( - 128 \beta_{2} - 2696 \beta_1 - 35175) q^{81} + (1932 \beta_{2} - 6060 \beta_1 + 49440) q^{82} + (4840 \beta_{2} + 3808 \beta_1 + 31428) q^{83} + (3512 \beta_{2} + 4139 \beta_1 + 33528) q^{84} + ( - 2555 \beta_{2} + 4723 \beta_1 - 19983) q^{85} + (2458 \beta_{2} + 4737 \beta_1 - 24878) q^{86} + (154 \beta_{2} + 9418 \beta_1 + 43218) q^{87} + (1752 \beta_{2} + 10194 \beta_1 + 49272) q^{88} + ( - 432 \beta_{2} - 8864 \beta_1 + 14186) q^{89} + (1760 \beta_{2} - 6346 \beta_1 + 35868) q^{90} + ( - 507 \beta_{2} - 507 \beta_1 - 3211) q^{91} + (7136 \beta_{2} + 1536 \beta_1 + 99776) q^{92} + (932 \beta_{2} + 3620 \beta_1 + 33924) q^{93} + ( - 4710 \beta_{2} + 7341 \beta_1 + 21834) q^{94} + ( - 5418 \beta_{2} + 12150 \beta_1 - 69570) q^{95} + ( - 13076 \beta_{2} - 18749 \beta_1 - 74964) q^{96} + (500 \beta_{2} - 1396 \beta_1 + 25910) q^{97} + (1986 \beta_{2} - 14280 \beta_1 - 13452) q^{98} + ( - 1928 \beta_{2} + 4720 \beta_1 - 21684) q^{99}+O(q^{100})$$ q + (b1 + 2) * q^2 + (-b2 - b1 + 3) * q^3 + (4*b2 + b1 + 40) * q^4 + (-b2 - 7*b1 + 21) * q^5 + (-10*b2 - b1 - 66) * q^6 + (-3*b2 - 3*b1 - 19) * q^7 + (28*b2 + 27*b1 + 100) * q^8 + (b2 + 7*b1 - 66) * q^9 + (-34*b2 + 23*b1 - 438) * q^10 + (30*b2 - 26*b1 + 194) * q^11 + (-32*b2 - 83*b1 - 336) * q^12 + 169 * q^13 + (-30*b2 - 31*b1 - 254) * q^14 + (31*b2 - 17*b1 + 663) * q^15 + (148*b2 + 181*b1 + 868) * q^16 + (-93*b2 + 77*b1 + 277) * q^17 + (34*b2 - 68*b1 + 348) * q^18 + (-134*b2 + 178*b1 - 10) * q^19 + (-80*b2 - 407*b1 - 120) * q^20 + (31*b2 + 49*b1 + 447) * q^21 + (76*b2 + 370*b1 - 1260) * q^22 + (152*b2 - 72*b1 + 1232) * q^23 + (-204*b2 - 381*b1 - 4332) * q^24 + (205*b2 - 365*b1 + 796) * q^25 + (169*b1 + 338) * q^26 + (257*b2 + 305*b1 - 1527) * q^27 + (-208*b2 - 277*b1 - 2128) * q^28 + (-584*b2 + 56*b1 - 2938) * q^29 + (118*b2 + 835*b1 + 294) * q^30 + (-268*b2 - 148*b1 - 820) * q^31 + (716*b2 + 563*b1 + 11436) * q^32 + (134*b2 - 550*b1 - 426) * q^33 + (-250*b2 - 265*b1 + 5418) * q^34 + (121*b2 + 145*b1 + 1401) * q^35 + (-100*b2 + 362*b1 - 1680) * q^36 + (419*b2 - 451*b1 - 6727) * q^37 + (-92*b2 - 858*b1 + 11548) * q^38 + (-169*b2 - 169*b1 + 507) * q^39 + (-1020*b2 - 849*b1 - 14220) * q^40 + (-250*b2 + 858*b1 - 3952) * q^41 + (382*b2 + 553*b1 + 4350) * q^42 + (697*b2 - 431*b1 + 821) * q^43 + (976*b2 - 418*b1 + 16736) * q^44 + (-160*b2 + 680*b1 - 4866) * q^45 + (624*b2 + 2064*b1 - 1824) * q^46 + (-807*b2 + 33*b1 + 11409) * q^47 + (-1724*b2 - 2315*b1 - 24636) * q^48 + (177*b2 + 231*b1 - 14934) * q^49 + (-230*b2 + 2186*b1 - 22408) * q^50 + (-1265*b2 + 823*b1 + 4215) * q^51 + (676*b2 + 169*b1 + 6760) * q^52 + (2154*b2 + 822*b1 - 4464) * q^53 + (2762*b2 - 547*b1 + 18714) * q^54 + (578*b2 - 3550*b1 + 12954) * q^55 + (-1396*b2 - 1899*b1 - 15796) * q^56 + (-1950*b2 + 1662*b1 + 18) * q^57 + (-3280*b2 - 5914*b1 - 4404) * q^58 + (-950*b2 + 1458*b1 + 20830) * q^59 + (3056*b2 + 593*b1 + 36624) * q^60 + (2330*b2 + 1910*b1 - 4888) * q^61 + (-2200*b2 - 2012*b1 - 12776) * q^62 + (14*b2 - 10*b1 - 546) * q^63 + (1812*b2 + 8661*b1 + 36244) * q^64 + (-169*b2 - 1183*b1 + 3549) * q^65 + (-1396*b2 + 794*b1 - 37716) * q^66 + (-1666*b2 + 1478*b1 + 18218) * q^67 + (416*b2 + 1969*b1 - 17048) * q^68 + (-48*b2 - 2976*b1 - 5712) * q^69 + (1306*b2 + 1861*b1 + 13146) * q^70 + (3127*b2 - 633*b1 + 26071) * q^71 + (-240*b2 - 366*b1 + 9720) * q^72 + (-2568*b2 - 2712*b1 - 13438) * q^73 + (710*b2 - 4181*b1 - 42446) * q^74 + (2944*b2 - 3416*b1 + 8988) * q^75 + (304*b2 + 6250*b1 - 35296) * q^76 + (-438*b2 - 922*b1 - 6710) * q^77 + (-1690*b2 - 169*b1 - 11154) * q^78 + (-3576*b2 - 2856*b1 - 19672) * q^79 + (-6956*b2 - 5447*b1 - 86412) * q^80 + (-128*b2 - 2696*b1 - 35175) * q^81 + (1932*b2 - 6060*b1 + 49440) * q^82 + (4840*b2 + 3808*b1 + 31428) * q^83 + (3512*b2 + 4139*b1 + 33528) * q^84 + (-2555*b2 + 4723*b1 - 19983) * q^85 + (2458*b2 + 4737*b1 - 24878) * q^86 + (154*b2 + 9418*b1 + 43218) * q^87 + (1752*b2 + 10194*b1 + 49272) * q^88 + (-432*b2 - 8864*b1 + 14186) * q^89 + (1760*b2 - 6346*b1 + 35868) * q^90 + (-507*b2 - 507*b1 - 3211) * q^91 + (7136*b2 + 1536*b1 + 99776) * q^92 + (932*b2 + 3620*b1 + 33924) * q^93 + (-4710*b2 + 7341*b1 + 21834) * q^94 + (-5418*b2 + 12150*b1 - 69570) * q^95 + (-13076*b2 - 18749*b1 - 74964) * q^96 + (500*b2 - 1396*b1 + 25910) * q^97 + (1986*b2 - 14280*b1 - 13452) * q^98 + (-1928*b2 + 4720*b1 - 21684) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 7 q^{2} + 8 q^{3} + 121 q^{4} + 56 q^{5} - 199 q^{6} - 60 q^{7} + 327 q^{8} - 191 q^{9}+O(q^{10})$$ 3 * q + 7 * q^2 + 8 * q^3 + 121 * q^4 + 56 * q^5 - 199 * q^6 - 60 * q^7 + 327 * q^8 - 191 * q^9 $$3 q + 7 q^{2} + 8 q^{3} + 121 q^{4} + 56 q^{5} - 199 q^{6} - 60 q^{7} + 327 q^{8} - 191 q^{9} - 1291 q^{10} + 556 q^{11} - 1091 q^{12} + 507 q^{13} - 793 q^{14} + 1972 q^{15} + 2785 q^{16} + 908 q^{17} + 976 q^{18} + 148 q^{19} - 767 q^{20} + 1390 q^{21} - 3410 q^{22} + 3624 q^{23} - 13377 q^{24} + 2023 q^{25} + 1183 q^{26} - 4276 q^{27} - 6661 q^{28} - 8758 q^{29} + 1717 q^{30} - 2608 q^{31} + 34871 q^{32} - 1828 q^{33} + 15989 q^{34} + 4348 q^{35} - 4678 q^{36} - 20632 q^{37} + 33786 q^{38} + 1352 q^{39} - 43509 q^{40} - 10998 q^{41} + 13603 q^{42} + 2032 q^{43} + 49790 q^{44} - 13918 q^{45} - 3408 q^{46} + 34260 q^{47} - 76223 q^{48} - 44571 q^{49} - 65038 q^{50} + 13468 q^{51} + 20449 q^{52} - 12570 q^{53} + 55595 q^{54} + 35312 q^{55} - 49287 q^{56} + 1716 q^{57} - 19126 q^{58} + 63948 q^{59} + 110465 q^{60} - 12754 q^{61} - 40340 q^{62} - 1648 q^{63} + 117393 q^{64} + 9464 q^{65} - 112354 q^{66} + 56132 q^{67} - 49175 q^{68} - 20112 q^{69} + 41299 q^{70} + 77580 q^{71} + 28794 q^{72} - 43026 q^{73} - 131519 q^{74} + 23548 q^{75} - 99638 q^{76} - 21052 q^{77} - 33631 q^{78} - 61872 q^{79} - 264683 q^{80} - 108221 q^{81} + 142260 q^{82} + 98092 q^{83} + 104723 q^{84} - 55226 q^{85} - 69897 q^{86} + 139072 q^{87} + 158010 q^{88} + 33694 q^{89} + 101258 q^{90} - 10140 q^{91} + 300864 q^{92} + 105392 q^{93} + 72843 q^{94} - 196560 q^{95} - 243641 q^{96} + 76334 q^{97} - 54636 q^{98} - 60332 q^{99}+O(q^{100})$$ 3 * q + 7 * q^2 + 8 * q^3 + 121 * q^4 + 56 * q^5 - 199 * q^6 - 60 * q^7 + 327 * q^8 - 191 * q^9 - 1291 * q^10 + 556 * q^11 - 1091 * q^12 + 507 * q^13 - 793 * q^14 + 1972 * q^15 + 2785 * q^16 + 908 * q^17 + 976 * q^18 + 148 * q^19 - 767 * q^20 + 1390 * q^21 - 3410 * q^22 + 3624 * q^23 - 13377 * q^24 + 2023 * q^25 + 1183 * q^26 - 4276 * q^27 - 6661 * q^28 - 8758 * q^29 + 1717 * q^30 - 2608 * q^31 + 34871 * q^32 - 1828 * q^33 + 15989 * q^34 + 4348 * q^35 - 4678 * q^36 - 20632 * q^37 + 33786 * q^38 + 1352 * q^39 - 43509 * q^40 - 10998 * q^41 + 13603 * q^42 + 2032 * q^43 + 49790 * q^44 - 13918 * q^45 - 3408 * q^46 + 34260 * q^47 - 76223 * q^48 - 44571 * q^49 - 65038 * q^50 + 13468 * q^51 + 20449 * q^52 - 12570 * q^53 + 55595 * q^54 + 35312 * q^55 - 49287 * q^56 + 1716 * q^57 - 19126 * q^58 + 63948 * q^59 + 110465 * q^60 - 12754 * q^61 - 40340 * q^62 - 1648 * q^63 + 117393 * q^64 + 9464 * q^65 - 112354 * q^66 + 56132 * q^67 - 49175 * q^68 - 20112 * q^69 + 41299 * q^70 + 77580 * q^71 + 28794 * q^72 - 43026 * q^73 - 131519 * q^74 + 23548 * q^75 - 99638 * q^76 - 21052 * q^77 - 33631 * q^78 - 61872 * q^79 - 264683 * q^80 - 108221 * q^81 + 142260 * q^82 + 98092 * q^83 + 104723 * q^84 - 55226 * q^85 - 69897 * q^86 + 139072 * q^87 + 158010 * q^88 + 33694 * q^89 + 101258 * q^90 - 10140 * q^91 + 300864 * q^92 + 105392 * q^93 + 72843 * q^94 - 196560 * q^95 - 243641 * q^96 + 76334 * q^97 - 54636 * q^98 - 60332 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 100x + 256$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} + 3\nu - 68 ) / 4$$ (v^2 + 3*v - 68) / 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$4\beta_{2} - 3\beta _1 + 68$$ 4*b2 - 3*b1 + 68

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −10.6486 2.68079 8.96778
−8.64858 10.2870 42.7979 92.1784 −88.9676 2.86088 −93.3863 −137.178 −797.212
1.2 4.68079 13.5120 −10.0902 15.4272 63.2466 12.5359 −197.015 −60.4272 72.2115
1.3 10.9678 −15.7989 88.2923 −51.6056 −173.279 −75.3967 617.402 6.60562 −565.999
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$13$$ $$-1$$

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 13.6.a.b 3
3.b odd 2 1 117.6.a.d 3
4.b odd 2 1 208.6.a.j 3
5.b even 2 1 325.6.a.c 3
5.c odd 4 2 325.6.b.c 6
7.b odd 2 1 637.6.a.b 3
8.b even 2 1 832.6.a.s 3
8.d odd 2 1 832.6.a.t 3
13.b even 2 1 169.6.a.b 3
13.d odd 4 2 169.6.b.b 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.6.a.b 3 1.a even 1 1 trivial
117.6.a.d 3 3.b odd 2 1
169.6.a.b 3 13.b even 2 1
169.6.b.b 6 13.d odd 4 2
208.6.a.j 3 4.b odd 2 1
325.6.a.c 3 5.b even 2 1
325.6.b.c 6 5.c odd 4 2
637.6.a.b 3 7.b odd 2 1
832.6.a.s 3 8.b even 2 1
832.6.a.t 3 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - 7T_{2}^{2} - 84T_{2} + 444$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(13))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - 7 T^{2} + \cdots + 444$$
$3$ $$T^{3} - 8 T^{2} + \cdots + 2196$$
$5$ $$T^{3} - 56 T^{2} + \cdots + 73386$$
$7$ $$T^{3} + 60 T^{2} + \cdots + 2704$$
$11$ $$T^{3} - 556 T^{2} + \cdots + 39698256$$
$13$ $$(T - 169)^{3}$$
$17$ $$T^{3} - 908 T^{2} + \cdots + 77884638$$
$19$ $$T^{3} + \cdots + 1415854512$$
$23$ $$T^{3} + \cdots + 5045833728$$
$29$ $$T^{3} + \cdots - 221025174456$$
$31$ $$T^{3} + \cdots - 1607044480$$
$37$ $$T^{3} + \cdots + 46212896426$$
$41$ $$T^{3} + \cdots + 29456898048$$
$43$ $$T^{3} + \cdots + 281385762060$$
$47$ $$T^{3} + \cdots - 696870885384$$
$53$ $$T^{3} + \cdots - 4415410372608$$
$59$ $$T^{3} + \cdots - 1932677407728$$
$61$ $$T^{3} + \cdots - 18650455523968$$
$67$ $$T^{3} + \cdots + 2080268535536$$
$71$ $$T^{3} + \cdots + 37395101110464$$
$73$ $$T^{3} + \cdots + 5649650834008$$
$79$ $$T^{3} + \cdots - 2044988893184$$
$83$ $$T^{3} + \cdots + 17971240920768$$
$89$ $$T^{3} + \cdots - 28887869991912$$
$97$ $$T^{3} + \cdots - 12102379894216$$