# Properties

 Label 338.4.a.n Level $338$ Weight $4$ Character orbit 338.a Self dual yes Analytic conductor $19.943$ Analytic rank $0$ Dimension $6$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [338,4,Mod(1,338)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(338, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("338.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 338.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: $$19.9426455819$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.6.6681389953.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{5} - 107x^{4} + 85x^{3} + 3703x^{2} - 1659x - 41951$$ x^6 - x^5 - 107*x^4 + 85*x^3 + 3703*x^2 - 1659*x - 41951 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$13^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + ( - \beta_1 + 1) q^{3} + 4 q^{4} + ( - \beta_{5} - \beta_{3} + \beta_1 - 3) q^{5} + (2 \beta_1 - 2) q^{6} + ( - \beta_{5} + 2 \beta_{4} + \beta_1 - 4) q^{7} - 8 q^{8} + ( - 2 \beta_{5} + 2 \beta_{4} + \cdots + 17) q^{9}+O(q^{10})$$ q - 2 * q^2 + (-b1 + 1) * q^3 + 4 * q^4 + (-b5 - b3 + b1 - 3) * q^5 + (2*b1 - 2) * q^6 + (-b5 + 2*b4 + b1 - 4) * q^7 - 8 * q^8 + (-2*b5 + 2*b4 - b3 + b2 - b1 + 17) * q^9 $$q - 2 q^{2} + ( - \beta_1 + 1) q^{3} + 4 q^{4} + ( - \beta_{5} - \beta_{3} + \beta_1 - 3) q^{5} + (2 \beta_1 - 2) q^{6} + ( - \beta_{5} + 2 \beta_{4} + \beta_1 - 4) q^{7} - 8 q^{8} + ( - 2 \beta_{5} + 2 \beta_{4} + \cdots + 17) q^{9}+ \cdots + (50 \beta_{5} + 80 \beta_{4} + \cdots + 345) q^{99}+O(q^{100})$$ q - 2 * q^2 + (-b1 + 1) * q^3 + 4 * q^4 + (-b5 - b3 + b1 - 3) * q^5 + (2*b1 - 2) * q^6 + (-b5 + 2*b4 + b1 - 4) * q^7 - 8 * q^8 + (-2*b5 + 2*b4 - b3 + b2 - b1 + 17) * q^9 + (2*b5 + 2*b3 - 2*b1 + 6) * q^10 + (3*b4 - 2*b3 + 5*b1 - 4) * q^11 + (-4*b1 + 4) * q^12 + (2*b5 - 4*b4 - 2*b1 + 8) * q^14 + (-4*b5 - 9*b4 + b2 - 7*b1 - 25) * q^15 + 16 * q^16 + (-5*b5 - 5*b3 - 5*b2 + 3*b1 + 18) * q^17 + (4*b5 - 4*b4 + 2*b3 - 2*b2 + 2*b1 - 34) * q^18 + (b5 + 2*b4 + 4*b3 - 3*b2 - 10*b1 + 11) * q^19 + (-4*b5 - 4*b3 + 4*b1 - 12) * q^20 + (9*b5 - 3*b4 + b3 + 5*b2 - 14*b1 - 2) * q^21 + (-6*b4 + 4*b3 - 10*b1 + 8) * q^22 + (8*b5 + 4*b4 - 7*b3 - 2*b2 + 47) * q^23 + (8*b1 - 8) * q^24 + (b4 + 16*b3 + 5*b2 - 11*b1 + 56) * q^25 + (-3*b5 - 5*b4 + 12*b2 - 23*b1 + 93) * q^27 + (-4*b5 + 8*b4 + 4*b1 - 16) * q^28 + (-b5 - 8*b3 - 6*b2 - 3*b1 - 20) * q^29 + (8*b5 + 18*b4 - 2*b2 + 14*b1 + 50) * q^30 + (-3*b5 + 8*b4 - 4*b3 - 4*b2 + 3*b1 + 106) * q^31 - 32 * q^32 + (19*b5 - b4 + 10*b3 + b2 - 15*b1 - 127) * q^33 + (10*b5 + 10*b3 + 10*b2 - 6*b1 - 36) * q^34 + (-9*b5 - 9*b4 + 13*b3 - 3*b2 + 18*b1 + 112) * q^35 + (-8*b5 + 8*b4 - 4*b3 + 4*b2 - 4*b1 + 68) * q^36 + (7*b5 + 5*b4 - b3 - 9*b2 - 14*b1 - 46) * q^37 + (-2*b5 - 4*b4 - 8*b3 + 6*b2 + 20*b1 - 22) * q^38 + (8*b5 + 8*b3 - 8*b1 + 24) * q^40 + (2*b5 + 7*b4 + 8*b3 - 10*b2 + 2*b1 - 182) * q^41 + (-18*b5 + 6*b4 - 2*b3 - 10*b2 + 28*b1 + 4) * q^42 + (3*b5 + 6*b4 - 29*b3 - 13*b2 + 48*b1 + 98) * q^43 + (12*b4 - 8*b3 + 20*b1 - 16) * q^44 + (-46*b5 - 40*b4 + 5*b3 + 6*b1 + 191) * q^45 + (-16*b5 - 8*b4 + 14*b3 + 4*b2 - 94) * q^46 + (-9*b5 + 12*b4 - 5*b3 + 22*b2 + 7*b1 + 153) * q^47 + (-16*b1 + 16) * q^48 + (12*b5 + b4 - 11*b3 + 9*b2 + 51*b1 + 173) * q^49 + (-2*b4 - 32*b3 - 10*b2 + 22*b1 - 112) * q^50 + (-14*b5 - 46*b4 - 12*b3 - 8*b2 - 43*b1 + 24) * q^51 + (23*b5 + 9*b4 + 42*b3 + 25*b2 + 8*b1 + 21) * q^53 + (6*b5 + 10*b4 - 24*b2 + 46*b1 - 186) * q^54 + (4*b5 - 5*b4 + 42*b3 - 3*b2 + 39*b1 + 217) * q^55 + (8*b5 - 16*b4 - 8*b1 + 32) * q^56 + (11*b5 + 30*b4 - 16*b3 + 3*b2 + 10*b1 + 430) * q^57 + (2*b5 + 16*b3 + 12*b2 + 6*b1 + 40) * q^58 + (13*b5 - 16*b4 - 42*b3 + 14*b2 - 20*b1 - 201) * q^59 + (-16*b5 - 36*b4 + 4*b2 - 28*b1 - 100) * q^60 + (25*b5 + 21*b4 - 35*b3 - 17*b2 + 6*b1 + 298) * q^61 + (6*b5 - 16*b4 + 8*b3 + 8*b2 - 6*b1 - 212) * q^62 + (4*b5 + 33*b4 + 10*b3 + 5*b2 + 39*b1 + 557) * q^63 + 64 * q^64 + (-38*b5 + 2*b4 - 20*b3 - 2*b2 + 30*b1 + 254) * q^66 + (-10*b5 + 7*b4 - 13*b3 + 7*b2 + 44*b1 + 295) * q^67 + (-20*b5 - 20*b3 - 20*b2 + 12*b1 + 72) * q^68 + (27*b5 + 66*b4 + 23*b3 - 14*b2 - 7*b1 + 211) * q^69 + (18*b5 + 18*b4 - 26*b3 + 6*b2 - 36*b1 - 224) * q^70 + (-26*b5 - 5*b4 + 31*b3 - 31*b2 - 15*b1 + 208) * q^71 + (16*b5 - 16*b4 + 8*b3 - 8*b2 + 8*b1 - 136) * q^72 + (10*b5 - 21*b4 - 29*b3 - 27*b2 - 20*b1 + 261) * q^73 + (-14*b5 - 10*b4 + 2*b3 + 18*b2 + 28*b1 + 92) * q^74 + (21*b5 + 30*b4 - 16*b3 + 28*b2 - 54*b1 + 263) * q^75 + (4*b5 + 8*b4 + 16*b3 - 12*b2 - 40*b1 + 44) * q^76 + (-23*b5 + 21*b4 + 43*b3 - 7*b2 + 52*b1 + 494) * q^77 + (-16*b5 - 25*b4 - 17*b3 - 11*b2 - 59*b1 - 178) * q^79 + (-16*b5 - 16*b3 + 16*b1 - 48) * q^80 + (-50*b5 - 32*b4 + 17*b3 + 28*b2 - 125*b1 + 466) * q^81 + (-4*b5 - 14*b4 - 16*b3 + 20*b2 - 4*b1 + 364) * q^82 + (24*b5 - 66*b4 + 12*b3 - 7*b2 + 17*b1 + 43) * q^83 + (36*b5 - 12*b4 + 4*b3 + 20*b2 - 56*b1 - 8) * q^84 + (-19*b5 + 12*b4 + 64*b3 + 52*b2 - 33*b1 + 582) * q^85 + (-6*b5 - 12*b4 + 58*b3 + 26*b2 - 96*b1 - 196) * q^86 + (-21*b5 - 7*b4 - 9*b3 - 13*b2 + 26*b1 + 278) * q^87 + (-24*b4 + 16*b3 - 40*b1 + 32) * q^88 + (-43*b5 - 59*b4 - 48*b3 - 7*b2 + b1 + 44) * q^89 + (92*b5 + 80*b4 - 10*b3 - 12*b1 - 382) * q^90 + (32*b5 + 16*b4 - 28*b3 - 8*b2 + 188) * q^92 + (33*b5 - 7*b4 + b3 + 7*b2 - 158*b1 + 240) * q^93 + (18*b5 - 24*b4 + 10*b3 - 44*b2 - 14*b1 - 306) * q^94 + (-33*b5 - 57*b4 - 38*b3 + 3*b2 + 4*b1 - 699) * q^95 + (32*b1 - 32) * q^96 + (22*b5 + 51*b4 + b3 - 2*b2 - 17*b1 + 55) * q^97 + (-24*b5 - 2*b4 + 22*b3 - 18*b2 - 102*b1 - 346) * q^98 + (50*b5 + 80*b4 + 68*b3 - 22*b2 + 164*b1 + 345) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 12 q^{2} + 9 q^{3} + 24 q^{4} - 18 q^{5} - 18 q^{6} - 25 q^{7} - 48 q^{8} + 113 q^{9}+O(q^{10})$$ 6 * q - 12 * q^2 + 9 * q^3 + 24 * q^4 - 18 * q^5 - 18 * q^6 - 25 * q^7 - 48 * q^8 + 113 * q^9 $$6 q - 12 q^{2} + 9 q^{3} + 24 q^{4} - 18 q^{5} - 18 q^{6} - 25 q^{7} - 48 q^{8} + 113 q^{9} + 36 q^{10} - 37 q^{11} + 36 q^{12} + 50 q^{14} - 118 q^{15} + 96 q^{16} + 99 q^{17} - 226 q^{18} + 81 q^{19} - 72 q^{20} + 26 q^{21} + 74 q^{22} + 267 q^{23} - 72 q^{24} + 368 q^{25} + 669 q^{27} - 100 q^{28} - 119 q^{29} + 236 q^{30} + 625 q^{31} - 192 q^{32} - 762 q^{33} - 198 q^{34} + 614 q^{35} + 452 q^{36} - 274 q^{37} - 162 q^{38} + 144 q^{40} - 1140 q^{41} - 52 q^{42} + 428 q^{43} - 148 q^{44} + 1215 q^{45} - 534 q^{46} + 986 q^{47} + 144 q^{48} + 899 q^{49} - 736 q^{50} + 289 q^{51} + 89 q^{53} - 1338 q^{54} + 1126 q^{55} + 200 q^{56} + 2553 q^{57} + 238 q^{58} - 1088 q^{59} - 472 q^{60} + 1704 q^{61} - 1250 q^{62} + 3222 q^{63} + 384 q^{64} + 1524 q^{66} + 1692 q^{67} + 396 q^{68} + 1168 q^{69} - 1228 q^{70} + 1221 q^{71} - 904 q^{72} + 1554 q^{73} + 548 q^{74} + 1798 q^{75} + 324 q^{76} + 2790 q^{77} - 875 q^{79} - 288 q^{80} + 3338 q^{81} + 2280 q^{82} + 126 q^{83} + 104 q^{84} + 3721 q^{85} - 856 q^{86} + 1602 q^{87} + 296 q^{88} + 374 q^{89} - 2430 q^{90} + 1068 q^{92} + 1868 q^{93} - 1972 q^{94} - 4093 q^{95} - 288 q^{96} + 330 q^{97} - 1798 q^{98} + 1344 q^{99}+O(q^{100})$$ 6 * q - 12 * q^2 + 9 * q^3 + 24 * q^4 - 18 * q^5 - 18 * q^6 - 25 * q^7 - 48 * q^8 + 113 * q^9 + 36 * q^10 - 37 * q^11 + 36 * q^12 + 50 * q^14 - 118 * q^15 + 96 * q^16 + 99 * q^17 - 226 * q^18 + 81 * q^19 - 72 * q^20 + 26 * q^21 + 74 * q^22 + 267 * q^23 - 72 * q^24 + 368 * q^25 + 669 * q^27 - 100 * q^28 - 119 * q^29 + 236 * q^30 + 625 * q^31 - 192 * q^32 - 762 * q^33 - 198 * q^34 + 614 * q^35 + 452 * q^36 - 274 * q^37 - 162 * q^38 + 144 * q^40 - 1140 * q^41 - 52 * q^42 + 428 * q^43 - 148 * q^44 + 1215 * q^45 - 534 * q^46 + 986 * q^47 + 144 * q^48 + 899 * q^49 - 736 * q^50 + 289 * q^51 + 89 * q^53 - 1338 * q^54 + 1126 * q^55 + 200 * q^56 + 2553 * q^57 + 238 * q^58 - 1088 * q^59 - 472 * q^60 + 1704 * q^61 - 1250 * q^62 + 3222 * q^63 + 384 * q^64 + 1524 * q^66 + 1692 * q^67 + 396 * q^68 + 1168 * q^69 - 1228 * q^70 + 1221 * q^71 - 904 * q^72 + 1554 * q^73 + 548 * q^74 + 1798 * q^75 + 324 * q^76 + 2790 * q^77 - 875 * q^79 - 288 * q^80 + 3338 * q^81 + 2280 * q^82 + 126 * q^83 + 104 * q^84 + 3721 * q^85 - 856 * q^86 + 1602 * q^87 + 296 * q^88 + 374 * q^89 - 2430 * q^90 + 1068 * q^92 + 1868 * q^93 - 1972 * q^94 - 4093 * q^95 - 288 * q^96 + 330 * q^97 - 1798 * q^98 + 1344 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} - 107x^{4} + 85x^{3} + 3703x^{2} - 1659x - 41951$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{5} + 7\nu^{4} - 77\nu^{3} - 531\nu^{2} + 1431\nu + 9815 ) / 39$$ (v^5 + 7*v^4 - 77*v^3 - 531*v^2 + 1431*v + 9815) / 39 $$\beta_{2}$$ $$=$$ $$( -22\nu^{5} - 102\nu^{4} + 1655\nu^{3} + 7574\nu^{2} - 30520\nu - 139594 ) / 273$$ (-22*v^5 - 102*v^4 + 1655*v^3 + 7574*v^2 - 30520*v - 139594) / 273 $$\beta_{3}$$ $$=$$ $$( 50\nu^{5} + 207\nu^{4} - 3902\nu^{3} - 16163\nu^{2} + 73955\nu + 314496 ) / 273$$ (50*v^5 + 207*v^4 - 3902*v^3 - 16163*v^2 + 73955*v + 314496) / 273 $$\beta_{4}$$ $$=$$ $$( 131\nu^{5} + 657\nu^{4} - 10347\nu^{3} - 51114\nu^{2} + 200123\nu + 982345 ) / 273$$ (131*v^5 + 657*v^4 - 10347*v^3 - 51114*v^2 + 200123*v + 982345) / 273 $$\beta_{5}$$ $$=$$ $$( 172\nu^{5} + 814\nu^{4} - 13543\nu^{3} - 62888\nu^{2} + 260211\nu + 1204567 ) / 273$$ (172*v^5 + 814*v^4 - 13543*v^3 - 62888*v^2 + 260211*v + 1204567) / 273
 $$\nu$$ $$=$$ $$( -\beta_{5} + 2\beta_{4} - 2\beta_{3} - 3\beta_{2} - 8\beta _1 - 1 ) / 13$$ (-b5 + 2*b4 - 2*b3 - 3*b2 - 8*b1 - 1) / 13 $$\nu^{2}$$ $$=$$ $$( 6\beta_{5} - 5\beta_{4} - 8\beta_{3} - 2\beta_{2} - 3\beta _1 + 466 ) / 13$$ (6*b5 - 5*b4 - 8*b3 - 2*b2 - 3*b1 + 466) / 13 $$\nu^{3}$$ $$=$$ $$( -66\beta_{5} + 92\beta_{4} - 40\beta_{3} - 145\beta_{2} - 270\beta _1 + 54 ) / 13$$ (-66*b5 + 92*b4 - 40*b3 - 145*b2 - 270*b1 + 54) / 13 $$\nu^{4}$$ $$=$$ $$( 443\beta_{5} - 363\beta_{4} - 625\beta_{3} - 143\beta_{2} - 77\beta _1 + 17789 ) / 13$$ (443*b5 - 363*b4 - 625*b3 - 143*b2 - 77*b1 + 17789) / 13 $$\nu^{5}$$ $$=$$ $$( -3566\beta_{5} + 4108\beta_{4} - 91\beta_{3} - 6933\beta_{2} - 9889\beta _1 + 917 ) / 13$$ (-3566*b5 + 4108*b4 - 91*b3 - 6933*b2 - 9889*b1 + 917) / 13

## 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
 −4.85211 −7.06450 −5.40469 6.65405 5.81752 5.84973
−2.00000 −8.14801 4.00000 12.6646 16.2960 28.7705 −8.00000 39.3901 −25.3292
1.2 −2.00000 −3.92743 4.00000 0.152827 7.85487 −33.9913 −8.00000 −11.5753 −0.305653
1.3 −2.00000 −1.24585 4.00000 −20.5002 2.49171 −21.2545 −8.00000 −25.4478 41.0004
1.4 −2.00000 3.35815 4.00000 −6.80407 −6.71631 −15.5380 −8.00000 −15.7228 13.6081
1.5 −2.00000 8.95458 4.00000 −17.3267 −17.9092 16.7510 −8.00000 53.1845 34.6535
1.6 −2.00000 10.0086 4.00000 13.8136 −20.0171 0.262285 −8.00000 73.1713 −27.6271
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$+1$$
$$13$$ $$+1$$

## 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 338.4.a.n 6
13.b even 2 1 338.4.a.o yes 6
13.c even 3 2 338.4.c.p 12
13.d odd 4 2 338.4.b.h 12
13.e even 6 2 338.4.c.o 12
13.f odd 12 4 338.4.e.i 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
338.4.a.n 6 1.a even 1 1 trivial
338.4.a.o yes 6 13.b even 2 1
338.4.b.h 12 13.d odd 4 2
338.4.c.o 12 13.e even 6 2
338.4.c.p 12 13.c even 3 2
338.4.e.i 24 13.f odd 12 4

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(338))$$:

 $$T_{3}^{6} - 9T_{3}^{5} - 97T_{3}^{4} + 731T_{3}^{3} + 2313T_{3}^{2} - 8047T_{3} - 11999$$ T3^6 - 9*T3^5 - 97*T3^4 + 731*T3^3 + 2313*T3^2 - 8047*T3 - 11999 $$T_{5}^{6} + 18T_{5}^{5} - 397T_{5}^{4} - 5935T_{5}^{3} + 44090T_{5}^{2} + 416208T_{5} - 64616$$ T5^6 + 18*T5^5 - 397*T5^4 - 5935*T5^3 + 44090*T5^2 + 416208*T5 - 64616

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 2)^{6}$$
$3$ $$T^{6} - 9 T^{5} + \cdots - 11999$$
$5$ $$T^{6} + 18 T^{5} + \cdots - 64616$$
$7$ $$T^{6} + 25 T^{5} + \cdots - 1418984$$
$11$ $$T^{6} + 37 T^{5} + \cdots - 284930113$$
$13$ $$T^{6}$$
$17$ $$T^{6} + \cdots - 137132126111$$
$19$ $$T^{6} + \cdots + 98183802067$$
$23$ $$T^{6} + \cdots - 920633860328$$
$29$ $$T^{6} + \cdots + 83240818856$$
$31$ $$T^{6} + \cdots + 254759240632$$
$37$ $$T^{6} + \cdots + 19945258052936$$
$41$ $$T^{6} + \cdots - 6202922352809$$
$43$ $$T^{6} + \cdots - 12\!\cdots\!89$$
$47$ $$T^{6} + \cdots + 549423790361944$$
$53$ $$T^{6} + \cdots + 14\!\cdots\!88$$
$59$ $$T^{6} + \cdots + 35\!\cdots\!77$$
$61$ $$T^{6} + \cdots + 833240079095872$$
$67$ $$T^{6} + \cdots - 2729196191063$$
$71$ $$T^{6} + \cdots + 12\!\cdots\!24$$
$73$ $$T^{6} + \cdots - 52281997669273$$
$79$ $$T^{6} + \cdots + 79\!\cdots\!96$$
$83$ $$T^{6} + \cdots - 99\!\cdots\!91$$
$89$ $$T^{6} + \cdots - 95\!\cdots\!09$$
$97$ $$T^{6} + \cdots + 15227749501399$$