# Properties

 Label 338.4.a.j Level $338$ Weight $4$ Character orbit 338.a Self dual yes Analytic conductor $19.943$ Analytic rank $1$ 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] = [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: $$1$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 2x + 1$$ x^3 - x^2 - 2*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + ( - 3 \beta_{2} - 5) q^{3} + 4 q^{4} + ( - \beta_{2} + 8 \beta_1 + 1) q^{5} + (6 \beta_{2} + 10) q^{6} + ( - 5 \beta_{2} - 8 \beta_1 - 8) q^{7} - 8 q^{8} + (21 \beta_{2} + 9 \beta_1 + 7) q^{9}+O(q^{10})$$ q - 2 * q^2 + (-3*b2 - 5) * q^3 + 4 * q^4 + (-b2 + 8*b1 + 1) * q^5 + (6*b2 + 10) * q^6 + (-5*b2 - 8*b1 - 8) * q^7 - 8 * q^8 + (21*b2 + 9*b1 + 7) * q^9 $$q - 2 q^{2} + ( - 3 \beta_{2} - 5) q^{3} + 4 q^{4} + ( - \beta_{2} + 8 \beta_1 + 1) q^{5} + (6 \beta_{2} + 10) q^{6} + ( - 5 \beta_{2} - 8 \beta_1 - 8) q^{7} - 8 q^{8} + (21 \beta_{2} + 9 \beta_1 + 7) q^{9} + (2 \beta_{2} - 16 \beta_1 - 2) q^{10} + ( - \beta_{2} - 24 \beta_1 + 35) q^{11} + ( - 12 \beta_{2} - 20) q^{12} + (10 \beta_{2} + 16 \beta_1 + 16) q^{14} + ( - 25 \beta_{2} - 37 \beta_1 - 26) q^{15} + 16 q^{16} + (43 \beta_{2} - 5 \beta_1 - 14) q^{17} + ( - 42 \beta_{2} - 18 \beta_1 - 14) q^{18} + (56 \beta_{2} + 15 \beta_1 + 57) q^{19} + ( - 4 \beta_{2} + 32 \beta_1 + 4) q^{20} + (58 \beta_{2} + 55 \beta_1 + 79) q^{21} + (2 \beta_{2} + 48 \beta_1 - 70) q^{22} + (68 \beta_{2} - 102 \beta_1 + 63) q^{23} + (24 \beta_{2} + 40) q^{24} + (45 \beta_{2} + 17 \beta_1 - 11) q^{25} + ( - 9 \beta_{2} - 108 \beta_1 + 10) q^{27} + ( - 20 \beta_{2} - 32 \beta_1 - 32) q^{28} + ( - 179 \beta_{2} + 182 \beta_1 - 154) q^{29} + (50 \beta_{2} + 74 \beta_1 + 52) q^{30} + ( - 67 \beta_{2} + 68 \beta_1 + 128) q^{31} - 32 q^{32} + ( - 31 \beta_{2} + 123 \beta_1 - 100) q^{33} + ( - 86 \beta_{2} + 10 \beta_1 + 28) q^{34} + ( - 98 \beta_{2} - 67 \beta_1 - 163) q^{35} + (84 \beta_{2} + 36 \beta_1 + 28) q^{36} + (22 \beta_{2} - 59 \beta_1 + 55) q^{37} + ( - 112 \beta_{2} - 30 \beta_1 - 114) q^{38} + (8 \beta_{2} - 64 \beta_1 - 8) q^{40} + (276 \beta_{2} - 254 \beta_1 + 239) q^{41} + ( - 116 \beta_{2} - 110 \beta_1 - 158) q^{42} + (6 \beta_{2} - 29 \beta_1 - 390) q^{43} + ( - 4 \beta_{2} - 96 \beta_1 + 140) q^{44} + (266 \beta_{2} + 44 \beta_1 + 289) q^{45} + ( - 136 \beta_{2} + 204 \beta_1 - 126) q^{46} + ( - 263 \beta_{2} + 278 \beta_1 - 359) q^{47} + ( - 48 \beta_{2} - 80) q^{48} + (199 \beta_{2} + 153 \beta_1 - 46) q^{49} + ( - 90 \beta_{2} - 34 \beta_1 + 22) q^{50} + ( - 29 \beta_{2} - 104 \beta_1 - 44) q^{51} + (36 \beta_{2} - 63 \beta_1 - 332) q^{53} + (18 \beta_{2} + 216 \beta_1 - 20) q^{54} + ( - 213 \beta_{2} + 257 \beta_1 - 332) q^{55} + (40 \beta_{2} + 64 \beta_1 + 64) q^{56} + ( - 328 \beta_{2} - 243 \beta_1 - 498) q^{57} + (358 \beta_{2} - 364 \beta_1 + 308) q^{58} + (194 \beta_{2} - 166 \beta_1 - 351) q^{59} + ( - 100 \beta_{2} - 148 \beta_1 - 104) q^{60} + ( - 166 \beta_{2} + 19 \beta_1 - 431) q^{61} + (134 \beta_{2} - 136 \beta_1 - 256) q^{62} + ( - 383 \beta_{2} - 233 \beta_1 - 518) q^{63} + 64 q^{64} + (62 \beta_{2} - 246 \beta_1 + 200) q^{66} + (230 \beta_{2} - 79 \beta_1 - 432) q^{67} + (172 \beta_{2} - 20 \beta_1 - 56) q^{68} + ( - 19 \beta_{2} + 306 \beta_1 - 213) q^{69} + (196 \beta_{2} + 134 \beta_1 + 326) q^{70} + ( - 297 \beta_{2} - 129 \beta_1 - 359) q^{71} + ( - 168 \beta_{2} - 72 \beta_1 - 56) q^{72} + ( - 30 \beta_{2} + 623 \beta_1 - 122) q^{73} + ( - 44 \beta_{2} + 118 \beta_1 - 110) q^{74} + ( - 108 \beta_{2} - 220 \beta_1 - 131) q^{75} + (224 \beta_{2} + 60 \beta_1 + 228) q^{76} + (148 \beta_{2} - 83 \beta_1 + 237) q^{77} + ( - 95 \beta_{2} - 163 \beta_1 - 631) q^{79} + ( - 16 \beta_{2} + 128 \beta_1 + 16) q^{80} + ( - 255 \beta_{2} + 324 \beta_1 + 112) q^{81} + ( - 552 \beta_{2} + 508 \beta_1 - 478) q^{82} + (289 \beta_{2} - 141 \beta_1 + 207) q^{83} + (232 \beta_{2} + 220 \beta_1 + 316) q^{84} + (409 \beta_{2} - 160 \beta_1 + 212) q^{85} + ( - 12 \beta_{2} + 58 \beta_1 + 780) q^{86} + (274 \beta_{2} - 373 \beta_1 + 761) q^{87} + (8 \beta_{2} + 192 \beta_1 - 280) q^{88} + (217 \beta_{2} - 21 \beta_1 + 443) q^{89} + ( - 532 \beta_{2} - 88 \beta_1 - 578) q^{90} + (272 \beta_{2} - 408 \beta_1 + 252) q^{92} + ( - 454 \beta_{2} - 139 \beta_1 - 643) q^{93} + (526 \beta_{2} - 556 \beta_1 + 718) q^{94} + (608 \beta_{2} + 415 \beta_1 + 674) q^{95} + (96 \beta_{2} + 160) q^{96} + ( - 599 \beta_{2} + 674 \beta_1 - 740) q^{97} + ( - 398 \beta_{2} - 306 \beta_1 + 92) q^{98} + (20 \beta_{2} + 126 \beta_1 - 721) q^{99}+O(q^{100})$$ q - 2 * q^2 + (-3*b2 - 5) * q^3 + 4 * q^4 + (-b2 + 8*b1 + 1) * q^5 + (6*b2 + 10) * q^6 + (-5*b2 - 8*b1 - 8) * q^7 - 8 * q^8 + (21*b2 + 9*b1 + 7) * q^9 + (2*b2 - 16*b1 - 2) * q^10 + (-b2 - 24*b1 + 35) * q^11 + (-12*b2 - 20) * q^12 + (10*b2 + 16*b1 + 16) * q^14 + (-25*b2 - 37*b1 - 26) * q^15 + 16 * q^16 + (43*b2 - 5*b1 - 14) * q^17 + (-42*b2 - 18*b1 - 14) * q^18 + (56*b2 + 15*b1 + 57) * q^19 + (-4*b2 + 32*b1 + 4) * q^20 + (58*b2 + 55*b1 + 79) * q^21 + (2*b2 + 48*b1 - 70) * q^22 + (68*b2 - 102*b1 + 63) * q^23 + (24*b2 + 40) * q^24 + (45*b2 + 17*b1 - 11) * q^25 + (-9*b2 - 108*b1 + 10) * q^27 + (-20*b2 - 32*b1 - 32) * q^28 + (-179*b2 + 182*b1 - 154) * q^29 + (50*b2 + 74*b1 + 52) * q^30 + (-67*b2 + 68*b1 + 128) * q^31 - 32 * q^32 + (-31*b2 + 123*b1 - 100) * q^33 + (-86*b2 + 10*b1 + 28) * q^34 + (-98*b2 - 67*b1 - 163) * q^35 + (84*b2 + 36*b1 + 28) * q^36 + (22*b2 - 59*b1 + 55) * q^37 + (-112*b2 - 30*b1 - 114) * q^38 + (8*b2 - 64*b1 - 8) * q^40 + (276*b2 - 254*b1 + 239) * q^41 + (-116*b2 - 110*b1 - 158) * q^42 + (6*b2 - 29*b1 - 390) * q^43 + (-4*b2 - 96*b1 + 140) * q^44 + (266*b2 + 44*b1 + 289) * q^45 + (-136*b2 + 204*b1 - 126) * q^46 + (-263*b2 + 278*b1 - 359) * q^47 + (-48*b2 - 80) * q^48 + (199*b2 + 153*b1 - 46) * q^49 + (-90*b2 - 34*b1 + 22) * q^50 + (-29*b2 - 104*b1 - 44) * q^51 + (36*b2 - 63*b1 - 332) * q^53 + (18*b2 + 216*b1 - 20) * q^54 + (-213*b2 + 257*b1 - 332) * q^55 + (40*b2 + 64*b1 + 64) * q^56 + (-328*b2 - 243*b1 - 498) * q^57 + (358*b2 - 364*b1 + 308) * q^58 + (194*b2 - 166*b1 - 351) * q^59 + (-100*b2 - 148*b1 - 104) * q^60 + (-166*b2 + 19*b1 - 431) * q^61 + (134*b2 - 136*b1 - 256) * q^62 + (-383*b2 - 233*b1 - 518) * q^63 + 64 * q^64 + (62*b2 - 246*b1 + 200) * q^66 + (230*b2 - 79*b1 - 432) * q^67 + (172*b2 - 20*b1 - 56) * q^68 + (-19*b2 + 306*b1 - 213) * q^69 + (196*b2 + 134*b1 + 326) * q^70 + (-297*b2 - 129*b1 - 359) * q^71 + (-168*b2 - 72*b1 - 56) * q^72 + (-30*b2 + 623*b1 - 122) * q^73 + (-44*b2 + 118*b1 - 110) * q^74 + (-108*b2 - 220*b1 - 131) * q^75 + (224*b2 + 60*b1 + 228) * q^76 + (148*b2 - 83*b1 + 237) * q^77 + (-95*b2 - 163*b1 - 631) * q^79 + (-16*b2 + 128*b1 + 16) * q^80 + (-255*b2 + 324*b1 + 112) * q^81 + (-552*b2 + 508*b1 - 478) * q^82 + (289*b2 - 141*b1 + 207) * q^83 + (232*b2 + 220*b1 + 316) * q^84 + (409*b2 - 160*b1 + 212) * q^85 + (-12*b2 + 58*b1 + 780) * q^86 + (274*b2 - 373*b1 + 761) * q^87 + (8*b2 + 192*b1 - 280) * q^88 + (217*b2 - 21*b1 + 443) * q^89 + (-532*b2 - 88*b1 - 578) * q^90 + (272*b2 - 408*b1 + 252) * q^92 + (-454*b2 - 139*b1 - 643) * q^93 + (526*b2 - 556*b1 + 718) * q^94 + (608*b2 + 415*b1 + 674) * q^95 + (96*b2 + 160) * q^96 + (-599*b2 + 674*b1 - 740) * q^97 + (-398*b2 - 306*b1 + 92) * q^98 + (20*b2 + 126*b1 - 721) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 6 q^{2} - 12 q^{3} + 12 q^{4} + 12 q^{5} + 24 q^{6} - 27 q^{7} - 24 q^{8} + 9 q^{9}+O(q^{10})$$ 3 * q - 6 * q^2 - 12 * q^3 + 12 * q^4 + 12 * q^5 + 24 * q^6 - 27 * q^7 - 24 * q^8 + 9 * q^9 $$3 q - 6 q^{2} - 12 q^{3} + 12 q^{4} + 12 q^{5} + 24 q^{6} - 27 q^{7} - 24 q^{8} + 9 q^{9} - 24 q^{10} + 82 q^{11} - 48 q^{12} + 54 q^{14} - 90 q^{15} + 48 q^{16} - 90 q^{17} - 18 q^{18} + 130 q^{19} + 48 q^{20} + 234 q^{21} - 164 q^{22} + 19 q^{23} + 96 q^{24} - 61 q^{25} - 69 q^{27} - 108 q^{28} - 101 q^{29} + 180 q^{30} + 519 q^{31} - 96 q^{32} - 146 q^{33} + 180 q^{34} - 458 q^{35} + 36 q^{36} + 84 q^{37} - 260 q^{38} - 96 q^{40} + 187 q^{41} - 468 q^{42} - 1205 q^{43} + 328 q^{44} + 645 q^{45} - 38 q^{46} - 536 q^{47} - 192 q^{48} - 184 q^{49} + 122 q^{50} - 207 q^{51} - 1095 q^{53} + 138 q^{54} - 526 q^{55} + 216 q^{56} - 1409 q^{57} + 202 q^{58} - 1413 q^{59} - 360 q^{60} - 1108 q^{61} - 1038 q^{62} - 1404 q^{63} + 192 q^{64} + 292 q^{66} - 1605 q^{67} - 360 q^{68} - 314 q^{69} + 916 q^{70} - 909 q^{71} - 72 q^{72} + 287 q^{73} - 168 q^{74} - 505 q^{75} + 520 q^{76} + 480 q^{77} - 1961 q^{79} + 192 q^{80} + 915 q^{81} - 374 q^{82} + 191 q^{83} + 936 q^{84} + 67 q^{85} + 2410 q^{86} + 1636 q^{87} - 656 q^{88} + 1091 q^{89} - 1290 q^{90} + 76 q^{92} - 1614 q^{93} + 1072 q^{94} + 1829 q^{95} + 384 q^{96} - 947 q^{97} + 368 q^{98} - 2057 q^{99}+O(q^{100})$$ 3 * q - 6 * q^2 - 12 * q^3 + 12 * q^4 + 12 * q^5 + 24 * q^6 - 27 * q^7 - 24 * q^8 + 9 * q^9 - 24 * q^10 + 82 * q^11 - 48 * q^12 + 54 * q^14 - 90 * q^15 + 48 * q^16 - 90 * q^17 - 18 * q^18 + 130 * q^19 + 48 * q^20 + 234 * q^21 - 164 * q^22 + 19 * q^23 + 96 * q^24 - 61 * q^25 - 69 * q^27 - 108 * q^28 - 101 * q^29 + 180 * q^30 + 519 * q^31 - 96 * q^32 - 146 * q^33 + 180 * q^34 - 458 * q^35 + 36 * q^36 + 84 * q^37 - 260 * q^38 - 96 * q^40 + 187 * q^41 - 468 * q^42 - 1205 * q^43 + 328 * q^44 + 645 * q^45 - 38 * q^46 - 536 * q^47 - 192 * q^48 - 184 * q^49 + 122 * q^50 - 207 * q^51 - 1095 * q^53 + 138 * q^54 - 526 * q^55 + 216 * q^56 - 1409 * q^57 + 202 * q^58 - 1413 * q^59 - 360 * q^60 - 1108 * q^61 - 1038 * q^62 - 1404 * q^63 + 192 * q^64 + 292 * q^66 - 1605 * q^67 - 360 * q^68 - 314 * q^69 + 916 * q^70 - 909 * q^71 - 72 * q^72 + 287 * q^73 - 168 * q^74 - 505 * q^75 + 520 * q^76 + 480 * q^77 - 1961 * q^79 + 192 * q^80 + 915 * q^81 - 374 * q^82 + 191 * q^83 + 936 * q^84 + 67 * q^85 + 2410 * q^86 + 1636 * q^87 - 656 * q^88 + 1091 * q^89 - 1290 * q^90 + 76 * q^92 - 1614 * q^93 + 1072 * q^94 + 1829 * q^95 + 384 * q^96 - 947 * q^97 + 368 * q^98 - 2057 * q^99

Basis of coefficient ring in terms of $$\nu = \zeta_{14} + \zeta_{14}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2

## 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
 1.80194 −1.24698 0.445042
−2.00000 −8.74094 4.00000 14.1685 17.4819 −28.6504 −8.00000 49.4040 −28.3370
1.2 −2.00000 −3.66487 4.00000 −8.53079 7.32975 4.20105 −8.00000 −13.5687 17.0616
1.3 −2.00000 0.405813 4.00000 6.36227 −0.811626 −2.55065 −8.00000 −26.8353 −12.7245
 $$n$$: e.g. 2-40 or 990-1000 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.j 3
13.b even 2 1 338.4.a.k yes 3
13.c even 3 2 338.4.c.l 6
13.d odd 4 2 338.4.b.f 6
13.e even 6 2 338.4.c.k 6
13.f odd 12 4 338.4.e.h 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
338.4.a.j 3 1.a even 1 1 trivial
338.4.a.k yes 3 13.b even 2 1
338.4.b.f 6 13.d odd 4 2
338.4.c.k 6 13.e even 6 2
338.4.c.l 6 13.c even 3 2
338.4.e.h 12 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(338))$$:

 $$T_{3}^{3} + 12T_{3}^{2} + 27T_{3} - 13$$ T3^3 + 12*T3^2 + 27*T3 - 13 $$T_{5}^{3} - 12T_{5}^{2} - 85T_{5} + 769$$ T5^3 - 12*T5^2 - 85*T5 + 769

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 2)^{3}$$
$3$ $$T^{3} + 12 T^{2} + \cdots - 13$$
$5$ $$T^{3} - 12 T^{2} + \cdots + 769$$
$7$ $$T^{3} + 27 T^{2} + \cdots - 307$$
$11$ $$T^{3} - 82 T^{2} + \cdots + 16211$$
$13$ $$T^{3}$$
$17$ $$T^{3} + 90 T^{2} + \cdots - 77167$$
$19$ $$T^{3} - 130 T^{2} + \cdots + 76609$$
$23$ $$T^{3} - 19 T^{2} + \cdots - 604157$$
$29$ $$T^{3} + 101 T^{2} + \cdots - 3703349$$
$31$ $$T^{3} - 519 T^{2} + \cdots - 3401957$$
$37$ $$T^{3} - 84 T^{2} + \cdots - 30919$$
$41$ $$T^{3} - 187 T^{2} + \cdots + 20172347$$
$43$ $$T^{3} + 1205 T^{2} + \cdots + 64126453$$
$47$ $$T^{3} + 536 T^{2} + \cdots - 26128271$$
$53$ $$T^{3} + 1095 T^{2} + \cdots + 45870749$$
$59$ $$T^{3} + 1413 T^{2} + \cdots + 72818971$$
$61$ $$T^{3} + 1108 T^{2} + \cdots + 28377551$$
$67$ $$T^{3} + 1605 T^{2} + \cdots + 110488951$$
$71$ $$T^{3} + 909 T^{2} + \cdots - 7534771$$
$73$ $$T^{3} - 287 T^{2} + \cdots + 178534237$$
$79$ $$T^{3} + 1961 T^{2} + \cdots + 214064899$$
$83$ $$T^{3} - 191 T^{2} + \cdots + 29986853$$
$89$ $$T^{3} - 1091 T^{2} + \cdots - 10734907$$
$97$ $$T^{3} + 947 T^{2} + \cdots - 228842741$$