# Properties

 Label 338.4.e.e Level $338$ Weight $4$ Character orbit 338.e Analytic conductor $19.943$ Analytic rank $0$ Dimension $8$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [338,4,Mod(23,338)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(338, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([5]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("338.23");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 338.e (of order $$6$$, degree $$2$$, 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: $$19.9426455819$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + 122x^{6} + 5305x^{4} + 97056x^{2} + 627264$$ x^8 + 122*x^6 + 5305*x^4 + 97056*x^2 + 627264 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 26) 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} + ( - \beta_{7} + \beta_{4} + 2 \beta_1 - 2) q^{3} + 4 \beta_1 q^{4} + (2 \beta_{7} + \beta_{6} + \beta_{5} + \cdots - 1) q^{5}+ \cdots + (3 \beta_{7} + 6 \beta_{3} + \cdots - 7 \beta_1) q^{9}+O(q^{10})$$ q + b3 * q^2 + (-b7 + b4 + 2*b1 - 2) * q^3 + 4*b1 * q^4 + (2*b7 + b6 + b5 - b4 + b3 + b2 + 2*b1 - 1) * q^5 + (-b6 - b4 + b2) * q^6 + (-b7 + b6 + 3*b4 - 2*b2 + 2*b1 - 4) * q^7 + (4*b3 + 4*b2) * q^8 + (3*b7 + 6*b3 + 3*b2 - 7*b1) * q^9 $$q + \beta_{3} q^{2} + ( - \beta_{7} + \beta_{4} + 2 \beta_1 - 2) q^{3} + 4 \beta_1 q^{4} + (2 \beta_{7} + \beta_{6} + \beta_{5} + \cdots - 1) q^{5}+ \cdots + ( - 52 \beta_{7} - 6 \beta_{6} + \cdots + 32) q^{99}+O(q^{100})$$ q + b3 * q^2 + (-b7 + b4 + 2*b1 - 2) * q^3 + 4*b1 * q^4 + (2*b7 + b6 + b5 - b4 + b3 + b2 + 2*b1 - 1) * q^5 + (-b6 - b4 + b2) * q^6 + (-b7 + b6 + 3*b4 - 2*b2 + 2*b1 - 4) * q^7 + (4*b3 + 4*b2) * q^8 + (3*b7 + 6*b3 + 3*b2 - 7*b1) * q^9 + (4*b7 + 2*b6 + b5 - 3*b4 + 2*b3 + 4*b2) * q^10 + (-b7 - 3*b5 + 2*b4 - 6*b3 + 2*b1 + 2) * q^11 + (4*b4 - 8) * q^12 + (-b6 + b5 - 6*b4 - b3 + b2 + 12) * q^14 + (-b5 + b4 + 20*b3 + 16*b1 + 16) * q^15 + (16*b1 - 16) * q^16 + (2*b7 - 4*b6 - 8*b5 + 4*b4 + 16*b3 + 8*b2 + 27*b1) * q^17 + (3*b6 + 3*b5 - 4*b3 - 4*b2 + 24*b1 - 12) * q^18 + (-b7 + 2*b4 + 18*b2 + 50*b1 - 100) * q^19 + (4*b7 + 4*b6 - 4*b4 + 4*b2 + 4*b1 - 8) * q^20 + (6*b7 - 4*b6 - 4*b5 - 3*b4 + 41*b3 + 41*b2 - 92*b1 + 46) * q^21 + (-12*b7 - b6 - 2*b5 + b4 + 2*b3 + b2 - 12*b1) * q^22 + (-17*b7 + 17*b4 + 12*b3 + 24*b2 + 66*b1 - 66) * q^23 + (4*b5 - 4*b4 - 4*b3) * q^24 + (4*b6 - 4*b5 - 5*b4 - 43*b3 + 43*b2 - 16) * q^25 + (-3*b6 + 3*b5 + 11*b4 - 12*b3 + 12*b2 + 50) * q^27 + (4*b7 - 4*b5 + 8*b4 + 8*b3 - 8*b1 - 8) * q^28 + (14*b7 + 2*b6 + b5 - 13*b4 - 38*b3 - 76*b2 - 15*b1 + 15) * q^29 + (-4*b7 + 32*b3 + 16*b2 + 84*b1) * q^30 + (-20*b7 - 7*b6 - 7*b5 + 10*b4 - 70*b3 - 70*b2 + 64*b1 - 32) * q^31 + 16*b2 * q^32 + (-3*b7 + 6*b4 + 75*b2 - 2*b1 + 4) * q^33 + (-32*b7 + 2*b6 + 2*b5 + 16*b4 + 29*b3 + 29*b2 + 96*b1 - 48) * q^34 + (8*b7 + 12*b3 + 6*b2 - 36*b1) * q^35 + (12*b7 - 12*b4 + 12*b3 + 24*b2 - 28*b1 + 28) * q^36 + (14*b7 + 5*b5 + 9*b4 - 40*b3 + 57*b1 + 57) * q^37 + (-b6 + b5 - 2*b4 - 49*b3 + 49*b2 - 72) * q^38 + (4*b6 - 4*b5 - 8*b4 - 8*b3 + 8*b2) * q^40 + (-16*b7 + 4*b5 - 20*b4 + 46*b3 + 55*b1 + 55) * q^41 + (-16*b7 + 6*b6 + 3*b5 + 19*b4 - 43*b3 - 86*b2 + 180*b1 - 180) * q^42 + (-5*b7 + 13*b6 + 26*b5 - 13*b4 + 116*b3 + 58*b2 - 26*b1) * q^43 + (-8*b7 - 12*b6 - 12*b5 + 4*b4 - 24*b3 - 24*b2 + 16*b1 - 8) * q^44 + (11*b7 + 5*b6 - 17*b4 + 77*b2 + 47*b1 - 94) * q^45 + (-17*b6 - 17*b4 + 49*b2 + 48*b1 - 96) * q^46 + (52*b7 + 15*b6 + 15*b5 - 26*b4 + 168*b3 + 168*b2 - 136*b1 + 68) * q^47 + (16*b7 - 32*b1) * q^48 + (29*b7 - 16*b6 - 8*b5 - 37*b4 + 85*b3 + 170*b2 + 13*b1 - 13) * q^49 + (-16*b7 - 13*b5 - 3*b4 - 29*b3 - 156*b1 - 156) * q^50 + (-16*b6 + 16*b5 - 7*b4 - 134*b3 + 134*b2 + 150) * q^51 + (17*b6 - 17*b5 + 25*b4 - 35*b3 + 35*b2 + 9) * q^53 + (12*b7 + 17*b5 - 5*b4 + 67*b3 - 60*b1 - 60) * q^54 + (-24*b7 - 16*b6 - 8*b5 + 16*b4 - 86*b3 - 172*b2 - 300*b1 + 300) * q^55 + (-16*b7 + 4*b6 + 8*b5 - 4*b4 - 8*b3 - 4*b2 + 48*b1) * q^56 + (102*b7 + 18*b6 + 18*b5 - 51*b4 - 9*b3 - 9*b2 - 260*b1 + 130) * q^57 + (4*b7 + 14*b6 + 6*b4 - b2 - 156*b1 + 312) * q^58 + (-13*b7 - 23*b6 + 3*b4 + 112*b2 - 86*b1 + 172) * q^59 + (-4*b6 - 4*b5 + 80*b3 + 80*b2 + 128*b1 - 64) * q^60 + (-16*b7 + 9*b6 + 18*b5 - 9*b4 + 48*b3 + 24*b2 + 233*b1) * q^61 + (-28*b7 - 20*b6 - 10*b5 + 18*b4 + 22*b3 + 44*b2 - 252*b1 + 252) * q^62 + (-22*b7 + 22*b5 - 44*b4 - 134*b3 + 176*b1 + 176) * q^63 - 64 * q^64 + (-3*b6 + 3*b5 - 6*b4 + 5*b3 - 5*b2 - 300) * q^66 + (15*b7 + 10*b5 + 5*b4 + 214*b3 - 50*b1 - 50) * q^67 + (8*b7 - 32*b6 - 16*b5 - 24*b4 + 32*b3 + 64*b2 + 108*b1 - 108) * q^68 + (83*b7 + 12*b6 + 24*b5 - 12*b4 + 78*b3 + 39*b2 - 642*b1) * q^69 + (8*b6 + 8*b5 - 28*b3 - 28*b2 + 48*b1 - 24) * q^70 + (-65*b7 - 24*b6 + 106*b4 + 54*b2 - 90*b1 + 180) * q^71 + (12*b6 + 12*b4 - 16*b2 + 48*b1 - 96) * q^72 + (-86*b7 - 8*b6 - 8*b5 + 43*b4 + 199*b3 + 199*b2 + 202*b1 - 101) * q^73 + (20*b7 + 14*b6 + 28*b5 - 14*b4 + 142*b3 + 71*b2 - 180*b1) * q^74 + (3*b7 + 70*b6 + 35*b5 + 32*b4 + 38*b3 + 76*b2 + 214*b1 - 214) * q^75 + (4*b7 + 4*b4 - 72*b3 - 200*b1 - 200) * q^76 + (4*b6 - 4*b5 + 5*b4 - 79*b3 + 79*b2 + 294) * q^77 + (-26*b6 + 26*b5 - 56*b4 + 38*b3 - 38*b2 + 100) * q^79 + (-16*b7 - 16*b5 - 16*b3 - 16*b1 - 16) * q^80 + (48*b7 + 36*b6 + 18*b5 - 30*b4 + 30*b3 + 60*b2 - 491*b1 + 491) * q^81 + (16*b7 - 16*b6 - 32*b5 + 16*b4 + 78*b3 + 39*b2 + 168*b1) * q^82 + (-172*b7 - 26*b6 - 26*b5 + 86*b4 - 32*b3 - 32*b2 - 104*b1 + 52) * q^83 + (12*b7 - 16*b6 - 40*b4 + 164*b2 - 184*b1 + 368) * q^84 + (63*b7 + 41*b6 - 85*b4 - 97*b2 - 273*b1 + 546) * q^85 + (104*b7 - 5*b6 - 5*b5 - 52*b4 - 31*b3 - 31*b2 + 360*b1 - 180) * q^86 + (-29*b7 - 40*b6 - 80*b5 + 40*b4 + 52*b3 + 26*b2 + 414*b1) * q^87 + (-48*b7 - 8*b6 - 4*b5 + 44*b4 + 4*b3 + 8*b2 - 48*b1 + 48) * q^88 + (39*b7 - 30*b5 + 69*b4 + 87*b3 + 350*b1 + 350) * q^89 + (11*b6 - 11*b5 + 2*b4 - 58*b3 + 58*b2 - 288) * q^90 + (68*b4 - 48*b3 + 48*b2 - 264) * q^92 + (42*b7 - 56*b5 + 98*b4 - 50*b3 - 280*b1 - 280) * q^93 + (60*b7 + 52*b6 + 26*b5 - 34*b4 - 42*b3 - 84*b2 + 612*b1 - 612) * q^94 + (-216*b7 - 67*b6 - 134*b5 + 67*b4 - 128*b3 - 64*b2 - 96*b1) * q^95 + (16*b6 + 16*b5 - 16*b3 - 16*b2) * q^96 + (129*b7 + 42*b6 - 216*b4 - 3*b2 + 90*b1 - 180) * q^97 + (-32*b7 + 29*b6 + 93*b4 + 42*b2 + 372*b1 - 744) * q^98 + (-52*b7 - 6*b6 - 6*b5 + 26*b4 - 210*b3 - 210*b2 - 64*b1 + 32) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 6 q^{3} + 16 q^{4} - 18 q^{7} - 22 q^{9}+O(q^{10})$$ 8 * q - 6 * q^3 + 16 * q^4 - 18 * q^7 - 22 * q^9 $$8 q - 6 q^{3} + 16 q^{4} - 18 q^{7} - 22 q^{9} - 8 q^{10} + 18 q^{11} - 48 q^{12} + 80 q^{14} + 192 q^{15} - 64 q^{16} + 112 q^{17} - 594 q^{19} - 72 q^{20} - 72 q^{22} - 230 q^{23} - 180 q^{25} + 468 q^{27} - 72 q^{28} + 32 q^{29} + 328 q^{30} + 42 q^{33} - 128 q^{35} + 88 q^{36} + 768 q^{37} - 576 q^{38} - 64 q^{40} + 564 q^{41} - 688 q^{42} - 114 q^{43} - 630 q^{45} - 576 q^{46} - 96 q^{48} - 110 q^{49} - 1968 q^{50} + 1300 q^{51} + 36 q^{53} - 648 q^{54} + 1248 q^{55} + 160 q^{56} + 1848 q^{58} + 1110 q^{59} + 900 q^{61} + 1064 q^{62} + 1980 q^{63} - 512 q^{64} - 2400 q^{66} - 510 q^{67} - 448 q^{68} - 2402 q^{69} + 1470 q^{71} - 576 q^{72} - 680 q^{74} - 862 q^{75} - 2376 q^{76} + 2340 q^{77} + 784 q^{79} - 288 q^{80} + 1868 q^{81} + 704 q^{82} + 2136 q^{84} + 2898 q^{85} + 1598 q^{87} + 288 q^{88} + 4434 q^{89} - 2384 q^{90} - 1840 q^{92} - 3108 q^{93} - 2568 q^{94} - 816 q^{95} - 1854 q^{97} - 4272 q^{98}+O(q^{100})$$ 8 * q - 6 * q^3 + 16 * q^4 - 18 * q^7 - 22 * q^9 - 8 * q^10 + 18 * q^11 - 48 * q^12 + 80 * q^14 + 192 * q^15 - 64 * q^16 + 112 * q^17 - 594 * q^19 - 72 * q^20 - 72 * q^22 - 230 * q^23 - 180 * q^25 + 468 * q^27 - 72 * q^28 + 32 * q^29 + 328 * q^30 + 42 * q^33 - 128 * q^35 + 88 * q^36 + 768 * q^37 - 576 * q^38 - 64 * q^40 + 564 * q^41 - 688 * q^42 - 114 * q^43 - 630 * q^45 - 576 * q^46 - 96 * q^48 - 110 * q^49 - 1968 * q^50 + 1300 * q^51 + 36 * q^53 - 648 * q^54 + 1248 * q^55 + 160 * q^56 + 1848 * q^58 + 1110 * q^59 + 900 * q^61 + 1064 * q^62 + 1980 * q^63 - 512 * q^64 - 2400 * q^66 - 510 * q^67 - 448 * q^68 - 2402 * q^69 + 1470 * q^71 - 576 * q^72 - 680 * q^74 - 862 * q^75 - 2376 * q^76 + 2340 * q^77 + 784 * q^79 - 288 * q^80 + 1868 * q^81 + 704 * q^82 + 2136 * q^84 + 2898 * q^85 + 1598 * q^87 + 288 * q^88 + 4434 * q^89 - 2384 * q^90 - 1840 * q^92 - 3108 * q^93 - 2568 * q^94 - 816 * q^95 - 1854 * q^97 - 4272 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 122x^{6} + 5305x^{4} + 97056x^{2} + 627264$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{7} + 122\nu^{5} + 4513\nu^{3} + 48744\nu + 9504 ) / 19008$$ (v^7 + 122*v^5 + 4513*v^3 + 48744*v + 9504) / 19008 $$\beta_{2}$$ $$=$$ $$( 29\nu^{7} + 132\nu^{6} + 2746\nu^{5} + 12144\nu^{4} + 80981\nu^{3} + 325644\nu^{2} + 737208\nu + 2414016 ) / 123552$$ (29*v^7 + 132*v^6 + 2746*v^5 + 12144*v^4 + 80981*v^3 + 325644*v^2 + 737208*v + 2414016) / 123552 $$\beta_{3}$$ $$=$$ $$( 29\nu^{7} - 132\nu^{6} + 2746\nu^{5} - 12144\nu^{4} + 80981\nu^{3} - 325644\nu^{2} + 737208\nu - 2414016 ) / 123552$$ (29*v^7 - 132*v^6 + 2746*v^5 - 12144*v^4 + 80981*v^3 - 325644*v^2 + 737208*v - 2414016) / 123552 $$\beta_{4}$$ $$=$$ $$( \nu^{6} + 92\nu^{4} + 2623\nu^{2} + 23124 ) / 156$$ (v^6 + 92*v^4 + 2623*v^2 + 23124) / 156 $$\beta_{5}$$ $$=$$ $$( \nu^{6} + 105\nu^{4} + 3416\nu^{2} + 156\nu + 33420 ) / 156$$ (v^6 + 105*v^4 + 3416*v^2 + 156*v + 33420) / 156 $$\beta_{6}$$ $$=$$ $$( -\nu^{6} - 105\nu^{4} - 3416\nu^{2} + 156\nu - 33420 ) / 156$$ (-v^6 - 105*v^4 - 3416*v^2 + 156*v - 33420) / 156 $$\beta_{7}$$ $$=$$ $$( - 119 \nu^{7} + 792 \nu^{6} - 10558 \nu^{5} + 72864 \nu^{4} - 266975 \nu^{3} + 2077416 \nu^{2} + \cdots + 18314208 ) / 247104$$ (-119*v^7 + 792*v^6 - 10558*v^5 + 72864*v^4 - 266975*v^3 + 2077416*v^2 - 1780344*v + 18314208) / 247104
 $$\nu$$ $$=$$ $$( \beta_{6} + \beta_{5} ) / 2$$ (b6 + b5) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{4} + 3\beta_{3} - 3\beta_{2} - 31$$ b4 + 3*b3 - 3*b2 - 31 $$\nu^{3}$$ $$=$$ $$( 24\beta_{7} - 31\beta_{6} - 31\beta_{5} - 12\beta_{4} + 30\beta_{3} + 30\beta_{2} - 48\beta _1 + 24 ) / 2$$ (24*b7 - 31*b6 - 31*b5 - 12*b4 + 30*b3 + 30*b2 - 48*b1 + 24) / 2 $$\nu^{4}$$ $$=$$ $$-6\beta_{6} + 6\beta_{5} - 73\beta_{4} - 183\beta_{3} + 183\beta_{2} + 1099$$ -6*b6 + 6*b5 - 73*b4 - 183*b3 + 183*b2 + 1099 $$\nu^{5}$$ $$=$$ $$( - 1512 \beta_{7} + 1099 \beta_{6} + 1099 \beta_{5} + 756 \beta_{4} - 2046 \beta_{3} - 2046 \beta_{2} + \cdots - 2208 ) / 2$$ (-1512*b7 + 1099*b6 + 1099*b5 + 756*b4 - 2046*b3 - 2046*b2 + 4416*b1 - 2208) / 2 $$\nu^{6}$$ $$=$$ $$552\beta_{6} - 552\beta_{5} + 4249\beta_{4} + 8967\beta_{3} - 8967\beta_{2} - 42919$$ 552*b6 - 552*b5 + 4249*b4 + 8967*b3 - 8967*b2 - 42919 $$\nu^{7}$$ $$=$$ $$( 76152 \beta_{7} - 42919 \beta_{6} - 42919 \beta_{5} - 38076 \beta_{4} + 114222 \beta_{3} + 114222 \beta_{2} + \cdots + 142056 ) / 2$$ (76152*b7 - 42919*b6 - 42919*b5 - 38076*b4 + 114222*b3 + 114222*b2 - 284112*b1 + 142056) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/338\mathbb{Z}\right)^\times$$.

 $$n$$ $$171$$ $$\chi(n)$$ $$\beta_{1}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
23.1
 − 6.87513i 5.87513i 4.95620i − 3.95620i 6.87513i − 5.87513i − 4.95620i 3.95620i
−1.73205 + 1.00000i −3.93757 6.82006i 2.00000 3.46410i 3.30629i 13.6401 + 7.87513i −27.1849 15.6952i 8.00000i −17.5089 + 30.3262i 3.30629 + 5.72666i
23.2 −1.73205 + 1.00000i 2.43757 + 4.22199i 2.00000 3.46410i 0.110135i −8.44398 4.87513i 14.0246 + 8.09712i 8.00000i 1.61655 2.79994i −0.110135 0.190760i
23.3 1.73205 1.00000i −2.97810 5.15822i 2.00000 3.46410i 13.0327i −10.3164 5.95620i 3.11419 + 1.79798i 8.00000i −4.23814 + 7.34068i 13.0327 + 22.5733i
23.4 1.73205 1.00000i 1.47810 + 2.56014i 2.00000 3.46410i 20.2288i 5.12028 + 2.95620i 1.04606 + 0.603945i 8.00000i 9.13045 15.8144i −20.2288 35.0374i
147.1 −1.73205 1.00000i −3.93757 + 6.82006i 2.00000 + 3.46410i 3.30629i 13.6401 7.87513i −27.1849 + 15.6952i 8.00000i −17.5089 30.3262i 3.30629 5.72666i
147.2 −1.73205 1.00000i 2.43757 4.22199i 2.00000 + 3.46410i 0.110135i −8.44398 + 4.87513i 14.0246 8.09712i 8.00000i 1.61655 + 2.79994i −0.110135 + 0.190760i
147.3 1.73205 + 1.00000i −2.97810 + 5.15822i 2.00000 + 3.46410i 13.0327i −10.3164 + 5.95620i 3.11419 1.79798i 8.00000i −4.23814 7.34068i 13.0327 22.5733i
147.4 1.73205 + 1.00000i 1.47810 2.56014i 2.00000 + 3.46410i 20.2288i 5.12028 2.95620i 1.04606 0.603945i 8.00000i 9.13045 + 15.8144i −20.2288 + 35.0374i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 23.4 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 338.4.e.e 8
13.b even 2 1 26.4.e.a 8
13.c even 3 1 26.4.e.a 8
13.c even 3 1 338.4.b.g 8
13.d odd 4 1 338.4.c.m 8
13.d odd 4 1 338.4.c.n 8
13.e even 6 1 338.4.b.g 8
13.e even 6 1 inner 338.4.e.e 8
13.f odd 12 1 338.4.a.l 4
13.f odd 12 1 338.4.a.m 4
13.f odd 12 1 338.4.c.m 8
13.f odd 12 1 338.4.c.n 8
39.d odd 2 1 234.4.l.b 8
39.i odd 6 1 234.4.l.b 8
52.b odd 2 1 208.4.w.d 8
52.j odd 6 1 208.4.w.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.4.e.a 8 13.b even 2 1
26.4.e.a 8 13.c even 3 1
208.4.w.d 8 52.b odd 2 1
208.4.w.d 8 52.j odd 6 1
234.4.l.b 8 39.d odd 2 1
234.4.l.b 8 39.i odd 6 1
338.4.a.l 4 13.f odd 12 1
338.4.a.m 4 13.f odd 12 1
338.4.b.g 8 13.c even 3 1
338.4.b.g 8 13.e even 6 1
338.4.c.m 8 13.d odd 4 1
338.4.c.m 8 13.f odd 12 1
338.4.c.n 8 13.d odd 4 1
338.4.c.n 8 13.f odd 12 1
338.4.e.e 8 1.a even 1 1 trivial
338.4.e.e 8 13.e even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(338, [\chi])$$:

 $$T_{3}^{8} + 6T_{3}^{7} + 83T_{3}^{6} + 54T_{3}^{5} + 2541T_{3}^{4} - 216T_{3}^{3} + 59996T_{3}^{2} - 113568T_{3} + 456976$$ T3^8 + 6*T3^7 + 83*T3^6 + 54*T3^5 + 2541*T3^4 - 216*T3^3 + 59996*T3^2 - 113568*T3 + 456976 $$T_{5}^{8} + 590T_{5}^{6} + 75841T_{5}^{4} + 760704T_{5}^{2} + 9216$$ T5^8 + 590*T5^6 + 75841*T5^4 + 760704*T5^2 + 9216

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} - 4 T^{2} + 16)^{2}$$
$3$ $$T^{8} + 6 T^{7} + \cdots + 456976$$
$5$ $$T^{8} + 590 T^{6} + \cdots + 9216$$
$7$ $$T^{8} + 18 T^{7} + \cdots + 4875264$$
$11$ $$T^{8} + \cdots + 482146919424$$
$13$ $$T^{8}$$
$17$ $$T^{8} + \cdots + 132748349592609$$
$19$ $$T^{8} + \cdots + 12\!\cdots\!96$$
$23$ $$T^{8} + \cdots + 21\!\cdots\!36$$
$29$ $$T^{8} + \cdots + 24\!\cdots\!01$$
$31$ $$T^{8} + \cdots + 14\!\cdots\!56$$
$37$ $$T^{8} + \cdots + 46\!\cdots\!41$$
$41$ $$T^{8} + \cdots + 11\!\cdots\!89$$
$43$ $$T^{8} + \cdots + 30\!\cdots\!16$$
$47$ $$T^{8} + \cdots + 99\!\cdots\!16$$
$53$ $$(T^{4} - 18 T^{3} + \cdots + 7709429988)^{2}$$
$59$ $$T^{8} + \cdots + 15\!\cdots\!44$$
$61$ $$T^{8} + \cdots + 28\!\cdots\!69$$
$67$ $$T^{8} + \cdots + 33\!\cdots\!76$$
$71$ $$T^{8} + \cdots + 66\!\cdots\!76$$
$73$ $$T^{8} + \cdots + 79\!\cdots\!96$$
$79$ $$(T^{4} - 392 T^{3} + \cdots + 40629076224)^{2}$$
$83$ $$T^{8} + \cdots + 77\!\cdots\!56$$
$89$ $$T^{8} + \cdots + 11\!\cdots\!04$$
$97$ $$T^{8} + \cdots + 15\!\cdots\!84$$