# Properties

 Label 338.4.c.l Level $338$ Weight $4$ Character orbit 338.c Analytic conductor $19.943$ Analytic rank $0$ Dimension $6$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [338,4,Mod(191,338)]

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([4]))

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

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

chi := DirichletCharacter("338.191");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 338.c (of order $$3$$, 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: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.64827.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1$$ x^6 - x^5 + 3*x^4 + 5*x^2 - 2*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - 2 \beta_{5} + 2) q^{2} + ( - 5 \beta_{5} - 3 \beta_{4} + 5) q^{3} - 4 \beta_{5} q^{4} + (7 \beta_{3} - 8 \beta_{2} + 9) q^{5} + ( - 10 \beta_{5} - 6 \beta_{4} - 6 \beta_{3}) q^{6} + (16 \beta_{5} + 13 \beta_{4} + \cdots + 8 \beta_1) q^{7}+ \cdots + ( - 16 \beta_{5} - 30 \beta_{4} + \cdots - 9 \beta_1) q^{9}+O(q^{10})$$ q + (-2*b5 + 2) * q^2 + (-5*b5 - 3*b4 + 5) * q^3 - 4*b5 * q^4 + (7*b3 - 8*b2 + 9) * q^5 + (-10*b5 - 6*b4 - 6*b3) * q^6 + (16*b5 + 13*b4 + 13*b3 - 8*b2 + 8*b1) * q^7 - 8 * q^8 + (-16*b5 - 30*b4 - 30*b3 + 9*b2 - 9*b1) * q^9 $$q + ( - 2 \beta_{5} + 2) q^{2} + ( - 5 \beta_{5} - 3 \beta_{4} + 5) q^{3} - 4 \beta_{5} q^{4} + (7 \beta_{3} - 8 \beta_{2} + 9) q^{5} + ( - 10 \beta_{5} - 6 \beta_{4} - 6 \beta_{3}) q^{6} + (16 \beta_{5} + 13 \beta_{4} + \cdots + 8 \beta_1) q^{7}+ \cdots + (146 \beta_{3} - 126 \beta_{2} - 595) q^{99}+O(q^{100})$$ q + (-2*b5 + 2) * q^2 + (-5*b5 - 3*b4 + 5) * q^3 - 4*b5 * q^4 + (7*b3 - 8*b2 + 9) * q^5 + (-10*b5 - 6*b4 - 6*b3) * q^6 + (16*b5 + 13*b4 + 13*b3 - 8*b2 + 8*b1) * q^7 - 8 * q^8 + (-16*b5 - 30*b4 - 30*b3 + 9*b2 - 9*b1) * q^9 + (-18*b5 - 14*b4 - 16*b1 + 18) * q^10 + (11*b5 - 25*b4 - 24*b1 - 11) * q^11 + (-12*b3 - 20) * q^12 + (26*b3 - 16*b2 + 32) * q^14 + (-63*b5 - 62*b4 - 37*b1 + 63) * q^15 + (16*b5 - 16) * q^16 + (19*b5 - 38*b4 - 38*b3 - 5*b2 + 5*b1) * q^17 + (-60*b3 + 18*b2 - 32) * q^18 + (-72*b5 - 71*b4 - 71*b3 + 15*b2 - 15*b1) * q^19 + (-36*b5 - 28*b4 - 28*b3 + 32*b2 - 32*b1) * q^20 + (113*b3 - 55*b2 + 134) * q^21 + (22*b5 - 50*b4 - 50*b3 + 48*b2 - 48*b1) * q^22 + (-39*b5 - 34*b4 - 102*b1 + 39) * q^23 + (40*b5 + 24*b4 - 40) * q^24 + (62*b3 - 17*b2 + 6) * q^25 + (-117*b3 + 108*b2 - 98) * q^27 + (-64*b5 - 52*b4 - 32*b1 + 64) * q^28 + (28*b5 + 3*b4 + 182*b1 - 28) * q^29 + (-126*b5 - 124*b4 - 124*b3 + 74*b2 - 74*b1) * q^30 + (b3 - 68*b2 + 196) * q^31 + 32*b5 * q^32 + (-23*b5 - 92*b4 - 92*b3 + 123*b2 - 123*b1) * q^33 + (-76*b3 - 10*b2 + 38) * q^34 + (230*b5 + 165*b4 + 165*b3 - 67*b2 + 67*b1) * q^35 + (64*b5 + 120*b4 + 36*b1 - 64) * q^36 + (-4*b5 - 37*b4 - 59*b1 + 4) * q^37 + (-142*b3 + 30*b2 - 144) * q^38 + (-56*b3 + 64*b2 - 72) * q^40 + (-15*b5 + 22*b4 - 254*b1 + 15) * q^41 + (-268*b5 - 226*b4 - 110*b1 + 268) * q^42 + (419*b5 + 23*b4 + 23*b3 - 29*b2 + 29*b1) * q^43 + (-100*b3 + 96*b2 + 44) * q^44 + (-333*b5 - 310*b4 - 310*b3 + 44*b2 - 44*b1) * q^45 + (-78*b5 - 68*b4 - 68*b3 + 204*b2 - 204*b1) * q^46 + (15*b3 - 278*b2 - 81) * q^47 + (80*b5 + 48*b4 + 48*b3) * q^48 + (107*b5 + 352*b4 + 153*b1 - 107) * q^49 + (-12*b5 - 124*b4 - 34*b1 + 12) * q^50 + (-133*b3 + 104*b2 - 148) * q^51 + (-27*b3 + 63*b2 - 395) * q^53 + (196*b5 + 234*b4 + 216*b1 - 196) * q^54 + (-75*b5 + 44*b4 + 257*b1 + 75) * q^55 + (-128*b5 - 104*b4 - 104*b3 + 64*b2 - 64*b1) * q^56 + (-571*b3 + 243*b2 - 741) * q^57 + (56*b5 + 6*b4 + 6*b3 - 364*b2 + 364*b1) * q^58 + (517*b5 - 28*b4 - 28*b3 - 166*b2 + 166*b1) * q^59 + (-248*b3 + 148*b2 - 252) * q^60 + (412*b5 + 147*b4 + 147*b3 + 19*b2 - 19*b1) * q^61 + (-392*b5 - 2*b4 - 136*b1 + 392) * q^62 + (-751*b5 - 616*b4 - 233*b1 + 751) * q^63 + 64 * q^64 + (-184*b3 + 246*b2 - 46) * q^66 + (-511*b5 + 151*b4 - 79*b1 + 511) * q^67 + (-76*b5 + 152*b4 - 20*b1 + 76) * q^68 + (-93*b5 - 287*b4 - 287*b3 + 306*b2 - 306*b1) * q^69 + (330*b3 - 134*b2 + 460) * q^70 + (488*b5 + 426*b4 + 426*b3 - 129*b2 + 129*b1) * q^71 + (128*b5 + 240*b4 + 240*b3 - 72*b2 + 72*b1) * q^72 + (593*b3 - 623*b2 + 501) * q^73 + (-8*b5 - 74*b4 - 74*b3 + 118*b2 - 118*b1) * q^74 + (-351*b5 - 328*b4 - 220*b1 + 351) * q^75 + (288*b5 + 284*b4 + 60*b1 - 288) * q^76 + (65*b3 + 83*b2 + 154) * q^77 + (-258*b3 + 163*b2 - 794) * q^79 + (144*b5 + 112*b4 + 128*b1 - 144) * q^80 + (436*b5 + 69*b4 + 324*b1 - 436) * q^81 + (-30*b5 + 44*b4 + 44*b3 + 508*b2 - 508*b1) * q^82 + (148*b3 + 141*b2 + 66) * q^83 + (-536*b5 - 452*b4 - 452*b3 + 220*b2 - 220*b1) * q^84 + (-52*b5 - 249*b4 - 249*b3 - 160*b2 + 160*b1) * q^85 + (46*b3 - 58*b2 + 838) * q^86 + (-388*b5 + 99*b4 + 99*b3 - 373*b2 + 373*b1) * q^87 + (-88*b5 + 200*b4 + 192*b1 + 88) * q^88 + (422*b5 + 196*b4 - 21*b1 - 422) * q^89 + (-620*b3 + 88*b2 - 666) * q^90 + (-136*b3 + 408*b2 - 156) * q^92 + (-782*b5 - 593*b4 - 139*b1 + 782) * q^93 + (162*b5 - 30*b4 - 556*b1 - 162) * q^94 + (-1089*b5 - 1023*b4 - 1023*b3 + 415*b2 - 415*b1) * q^95 + (96*b3 + 160) * q^96 + (66*b5 - 75*b4 - 75*b3 + 674*b2 - 674*b1) * q^97 + (214*b5 + 704*b4 + 704*b3 - 306*b2 + 306*b1) * q^98 + (146*b3 - 126*b2 - 595) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 6 q^{2} + 12 q^{3} - 12 q^{4} + 24 q^{5} - 24 q^{6} + 27 q^{7} - 48 q^{8} - 9 q^{9}+O(q^{10})$$ 6 * q + 6 * q^2 + 12 * q^3 - 12 * q^4 + 24 * q^5 - 24 * q^6 + 27 * q^7 - 48 * q^8 - 9 * q^9 $$6 q + 6 q^{2} + 12 q^{3} - 12 q^{4} + 24 q^{5} - 24 q^{6} + 27 q^{7} - 48 q^{8} - 9 q^{9} + 24 q^{10} - 82 q^{11} - 96 q^{12} + 108 q^{14} + 90 q^{15} - 48 q^{16} + 90 q^{17} - 36 q^{18} - 130 q^{19} - 48 q^{20} + 468 q^{21} + 164 q^{22} - 19 q^{23} - 96 q^{24} - 122 q^{25} - 138 q^{27} + 108 q^{28} + 101 q^{29} - 180 q^{30} + 1038 q^{31} + 96 q^{32} + 146 q^{33} + 360 q^{34} + 458 q^{35} - 36 q^{36} - 84 q^{37} - 520 q^{38} - 192 q^{40} - 187 q^{41} + 468 q^{42} + 1205 q^{43} + 656 q^{44} - 645 q^{45} + 38 q^{46} - 1072 q^{47} + 192 q^{48} + 184 q^{49} - 122 q^{50} - 414 q^{51} - 2190 q^{53} - 138 q^{54} + 526 q^{55} - 216 q^{56} - 2818 q^{57} - 202 q^{58} + 1413 q^{59} - 720 q^{60} + 1108 q^{61} + 1038 q^{62} + 1404 q^{63} + 384 q^{64} + 584 q^{66} + 1605 q^{67} + 360 q^{68} + 314 q^{69} + 1832 q^{70} + 909 q^{71} + 72 q^{72} + 574 q^{73} + 168 q^{74} + 505 q^{75} - 520 q^{76} + 960 q^{77} - 3922 q^{79} - 192 q^{80} - 915 q^{81} + 374 q^{82} + 382 q^{83} - 936 q^{84} - 67 q^{85} + 4820 q^{86} - 1636 q^{87} + 656 q^{88} - 1091 q^{89} - 2580 q^{90} + 152 q^{92} + 1614 q^{93} - 1072 q^{94} - 1829 q^{95} + 768 q^{96} + 947 q^{97} - 368 q^{98} - 4114 q^{99}+O(q^{100})$$ 6 * q + 6 * q^2 + 12 * q^3 - 12 * q^4 + 24 * q^5 - 24 * q^6 + 27 * q^7 - 48 * q^8 - 9 * q^9 + 24 * q^10 - 82 * q^11 - 96 * q^12 + 108 * q^14 + 90 * q^15 - 48 * q^16 + 90 * q^17 - 36 * q^18 - 130 * q^19 - 48 * q^20 + 468 * q^21 + 164 * q^22 - 19 * q^23 - 96 * q^24 - 122 * q^25 - 138 * q^27 + 108 * q^28 + 101 * q^29 - 180 * q^30 + 1038 * q^31 + 96 * q^32 + 146 * q^33 + 360 * q^34 + 458 * q^35 - 36 * q^36 - 84 * q^37 - 520 * q^38 - 192 * q^40 - 187 * q^41 + 468 * q^42 + 1205 * q^43 + 656 * q^44 - 645 * q^45 + 38 * q^46 - 1072 * q^47 + 192 * q^48 + 184 * q^49 - 122 * q^50 - 414 * q^51 - 2190 * q^53 - 138 * q^54 + 526 * q^55 - 216 * q^56 - 2818 * q^57 - 202 * q^58 + 1413 * q^59 - 720 * q^60 + 1108 * q^61 + 1038 * q^62 + 1404 * q^63 + 384 * q^64 + 584 * q^66 + 1605 * q^67 + 360 * q^68 + 314 * q^69 + 1832 * q^70 + 909 * q^71 + 72 * q^72 + 574 * q^73 + 168 * q^74 + 505 * q^75 - 520 * q^76 + 960 * q^77 - 3922 * q^79 - 192 * q^80 - 915 * q^81 + 374 * q^82 + 382 * q^83 - 936 * q^84 - 67 * q^85 + 4820 * q^86 - 1636 * q^87 + 656 * q^88 - 1091 * q^89 - 2580 * q^90 + 152 * q^92 + 1614 * q^93 - 1072 * q^94 - 1829 * q^95 + 768 * q^96 + 947 * q^97 - 368 * q^98 - 4114 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{5} + 3\nu^{4} - 9\nu^{3} + 5\nu^{2} - 2\nu + 6 ) / 13$$ (-v^5 + 3*v^4 - 9*v^3 + 5*v^2 - 2*v + 6) / 13 $$\beta_{3}$$ $$=$$ $$( -3\nu^{5} + 9\nu^{4} - 14\nu^{3} + 15\nu^{2} - 6\nu + 18 ) / 13$$ (-3*v^5 + 9*v^4 - 14*v^3 + 15*v^2 - 6*v + 18) / 13 $$\beta_{4}$$ $$=$$ $$( -4\nu^{5} - \nu^{4} - 10\nu^{3} - 6\nu^{2} - 34\nu - 2 ) / 13$$ (-4*v^5 - v^4 - 10*v^3 - 6*v^2 - 34*v - 2) / 13 $$\beta_{5}$$ $$=$$ $$( -6\nu^{5} + 5\nu^{4} - 15\nu^{3} - 9\nu^{2} - 25\nu + 10 ) / 13$$ (-6*v^5 + 5*v^4 - 15*v^3 - 9*v^2 - 25*v + 10) / 13
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1$$ -b5 + b4 + b3 - b2 + b1 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 3\beta_{2}$$ b3 - 3*b2 $$\nu^{4}$$ $$=$$ $$2\beta_{5} - 3\beta_{4} - 4\beta _1 - 2$$ 2*b5 - 3*b4 - 4*b1 - 2 $$\nu^{5}$$ $$=$$ $$\beta_{5} - 4\beta_{4} - 4\beta_{3} + 9\beta_{2} - 9\beta_1$$ b5 - 4*b4 - 4*b3 + 9*b2 - 9*b1

## Character values

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

 $$n$$ $$171$$ $$\chi(n)$$ $$-\beta_{5}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
191.1
 −0.623490 + 1.07992i 0.900969 − 1.56052i 0.222521 − 0.385418i −0.623490 − 1.07992i 0.900969 + 1.56052i 0.222521 + 0.385418i
1.00000 1.73205i −0.202907 + 0.351445i −2.00000 3.46410i 6.36227 0.405813 + 0.702889i 1.27532 + 2.20892i −8.00000 13.4177 + 23.2401i 6.36227 11.0198i
191.2 1.00000 1.73205i 1.83244 3.17387i −2.00000 3.46410i −8.53079 −3.66487 6.34775i −2.10052 3.63821i −8.00000 6.78435 + 11.7508i −8.53079 + 14.7758i
191.3 1.00000 1.73205i 4.37047 7.56988i −2.00000 3.46410i 14.1685 −8.74094 15.1398i 14.3252 + 24.8120i −8.00000 −24.7020 42.7851i 14.1685 24.5406i
315.1 1.00000 + 1.73205i −0.202907 0.351445i −2.00000 + 3.46410i 6.36227 0.405813 0.702889i 1.27532 2.20892i −8.00000 13.4177 23.2401i 6.36227 + 11.0198i
315.2 1.00000 + 1.73205i 1.83244 + 3.17387i −2.00000 + 3.46410i −8.53079 −3.66487 + 6.34775i −2.10052 + 3.63821i −8.00000 6.78435 11.7508i −8.53079 14.7758i
315.3 1.00000 + 1.73205i 4.37047 + 7.56988i −2.00000 + 3.46410i 14.1685 −8.74094 + 15.1398i 14.3252 24.8120i −8.00000 −24.7020 + 42.7851i 14.1685 + 24.5406i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 191.3 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 338.4.c.l 6
13.b even 2 1 338.4.c.k 6
13.c even 3 1 338.4.a.j 3
13.c even 3 1 inner 338.4.c.l 6
13.d odd 4 2 338.4.e.h 12
13.e even 6 1 338.4.a.k yes 3
13.e even 6 1 338.4.c.k 6
13.f odd 12 2 338.4.b.f 6
13.f odd 12 2 338.4.e.h 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
338.4.a.j 3 13.c even 3 1
338.4.a.k yes 3 13.e even 6 1
338.4.b.f 6 13.f odd 12 2
338.4.c.k 6 13.b even 2 1
338.4.c.k 6 13.e even 6 1
338.4.c.l 6 1.a even 1 1 trivial
338.4.c.l 6 13.c even 3 1 inner
338.4.e.h 12 13.d odd 4 2
338.4.e.h 12 13.f odd 12 2

## Hecke kernels

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

 $$T_{3}^{6} - 12T_{3}^{5} + 117T_{3}^{4} - 350T_{3}^{3} + 885T_{3}^{2} + 351T_{3} + 169$$ T3^6 - 12*T3^5 + 117*T3^4 - 350*T3^3 + 885*T3^2 + 351*T3 + 169 $$T_{5}^{3} - 12T_{5}^{2} - 85T_{5} + 769$$ T5^3 - 12*T5^2 - 85*T5 + 769

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2 T + 4)^{3}$$
$3$ $$T^{6} - 12 T^{5} + \cdots + 169$$
$5$ $$(T^{3} - 12 T^{2} + \cdots + 769)^{2}$$
$7$ $$T^{6} - 27 T^{5} + \cdots + 94249$$
$11$ $$T^{6} + 82 T^{5} + \cdots + 262796521$$
$13$ $$T^{6}$$
$17$ $$T^{6} + \cdots + 5954745889$$
$19$ $$T^{6} + \cdots + 5868938881$$
$23$ $$T^{6} + \cdots + 365005680649$$
$29$ $$T^{6} + \cdots + 13714793815801$$
$31$ $$(T^{3} - 519 T^{2} + \cdots - 3401957)^{2}$$
$37$ $$T^{6} + 84 T^{5} + \cdots + 955984561$$
$41$ $$T^{6} + \cdots + 406923583488409$$
$43$ $$T^{6} + \cdots + 41\!\cdots\!09$$
$47$ $$(T^{3} + 536 T^{2} + \cdots - 26128271)^{2}$$
$53$ $$(T^{3} + 1095 T^{2} + \cdots + 45870749)^{2}$$
$59$ $$T^{6} + \cdots + 53\!\cdots\!41$$
$61$ $$T^{6} + \cdots + 805285400757601$$
$67$ $$T^{6} + \cdots + 12\!\cdots\!01$$
$71$ $$T^{6} + \cdots + 56772774022441$$
$73$ $$(T^{3} - 287 T^{2} + \cdots + 178534237)^{2}$$
$79$ $$(T^{3} + 1961 T^{2} + \cdots + 214064899)^{2}$$
$83$ $$(T^{3} - 191 T^{2} + \cdots + 29986853)^{2}$$
$89$ $$T^{6} + \cdots + 115238228298649$$
$97$ $$T^{6} + \cdots + 52\!\cdots\!81$$