# Properties

 Label 338.4.c.p Level $338$ Weight $4$ Character orbit 338.c Analytic conductor $19.943$ Analytic rank $0$ Dimension $12$ 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: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - x^{11} + 108 x^{10} - 63 x^{9} + 7831 x^{8} - 3348 x^{7} + 317885 x^{6} + 1680 x^{5} + \cdots + 1759886401$$ x^12 - x^11 + 108*x^10 - 63*x^9 + 7831*x^8 - 3348*x^7 + 317885*x^6 + 1680*x^5 + 9364467*x^4 + 988393*x^3 + 158096834*x^2 + 69596709*x + 1759886401 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$13^{4}$$ 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - 2 \beta_1 + 2) q^{2} + ( - \beta_{4} + \beta_{2} + 2 \beta_1 - 1) q^{3} - 4 \beta_1 q^{4} + (\beta_{11} + \beta_{9} - \beta_{4} - 3) q^{5} + (2 \beta_{2} + 4 \beta_1) q^{6} + (\beta_{8} + 2 \beta_{6} + \cdots + 5 \beta_1) q^{7}+ \cdots + (2 \beta_{8} + 2 \beta_{6} + \cdots - 18 \beta_1) q^{9}+O(q^{10})$$ q + (-2*b1 + 2) * q^2 + (-b4 + b2 + 2*b1 - 1) * q^3 - 4*b1 * q^4 + (b11 + b9 - b4 - 3) * q^5 + (2*b2 + 4*b1) * q^6 + (b8 + 2*b6 + b2 + 5*b1) * q^7 - 8 * q^8 + (2*b8 + 2*b6 - b5 - b3 - b2 - 18*b1) * q^9 $$q + ( - 2 \beta_1 + 2) q^{2} + ( - \beta_{4} + \beta_{2} + 2 \beta_1 - 1) q^{3} - 4 \beta_1 q^{4} + (\beta_{11} + \beta_{9} - \beta_{4} - 3) q^{5} + (2 \beta_{2} + 4 \beta_1) q^{6} + (\beta_{8} + 2 \beta_{6} + \cdots + 5 \beta_1) q^{7}+ \cdots + ( - 50 \beta_{11} + 80 \beta_{10} + \cdots + 323) q^{99}+O(q^{100})$$ q + (-2*b1 + 2) * q^2 + (-b4 + b2 + 2*b1 - 1) * q^3 - 4*b1 * q^4 + (b11 + b9 - b4 - 3) * q^5 + (2*b2 + 4*b1) * q^6 + (b8 + 2*b6 + b2 + 5*b1) * q^7 - 8 * q^8 + (2*b8 + 2*b6 - b5 - b3 - b2 - 18*b1) * q^9 + (2*b11 + 2*b9 + 2*b8 - 2*b5 - 2*b4 + 2*b2 + 8*b1 - 6) * q^10 + (-3*b10 - 2*b9 - 3*b6 + 2*b5 + 5*b4 - 5*b2 - 9*b1 + 4) * q^11 + (4*b4 + 4) * q^12 + (-2*b11 - 4*b10 + 2*b4 + 8) * q^14 + (-4*b11 + 9*b10 - 4*b8 + b7 + 9*b6 - 7*b4 + b3 + 7*b2 - 18*b1 + 24) * q^15 + (16*b1 - 16) * q^16 + (5*b8 - 5*b5 + 5*b3 + 3*b2 - 15*b1) * q^17 + (-4*b11 - 4*b10 - 2*b9 + 2*b7 - 2*b4 - 36) * q^18 + (-b8 + 2*b6 + 4*b5 + 3*b3 - 10*b2 - 21*b1) * q^19 + (4*b8 - 4*b5 + 4*b2 + 16*b1) * q^20 + (-9*b11 - 3*b10 - b9 - 5*b7 + 14*b4 + 3) * q^21 + (-6*b6 + 4*b5 - 10*b2 - 18*b1) * q^22 + (8*b11 - 4*b10 - 7*b9 + 8*b8 - 2*b7 - 4*b6 + 7*b5 - 2*b3 + 47*b1 - 45) * q^23 + (8*b4 - 8*b2 - 16*b1 + 8) * q^24 + (b10 - 16*b9 - 5*b7 + 11*b4 + 61) * q^25 + (3*b11 - 5*b10 - 12*b7 + 23*b4 + 105) * q^27 + (-4*b11 - 8*b10 - 4*b8 - 8*b6 + 4*b4 - 4*b2 - 20*b1 + 16) * q^28 + (-b11 - 8*b9 - b8 - 6*b7 + 8*b5 - 3*b4 - 6*b3 + 3*b2 - 17*b1 + 26) * q^29 + (-8*b8 + 18*b6 + 2*b3 + 14*b2 - 36*b1) * q^30 + (3*b11 + 8*b10 + 4*b9 + 4*b7 - 3*b4 + 102) * q^31 + 32*b1 * q^32 + (-19*b8 - b6 + 10*b5 - b3 - 15*b2 + 112*b1) * q^33 + (-10*b11 - 10*b9 - 10*b7 + 6*b4 - 26) * q^34 + (9*b8 - 9*b6 + 13*b5 + 3*b3 + 18*b2 - 94*b1) * q^35 + (-8*b11 - 8*b10 - 4*b9 - 8*b8 + 4*b7 - 8*b6 + 4*b5 - 4*b4 + 4*b3 + 4*b2 + 72*b1 - 72) * q^36 + (7*b11 - 5*b10 - b9 + 7*b8 - 9*b7 - 5*b6 + b5 - 14*b4 - 9*b3 + 14*b2 - 32*b1 + 55) * q^37 + (2*b11 - 4*b10 + 8*b9 - 6*b7 - 20*b4 - 16) * q^38 + (-8*b11 - 8*b9 + 8*b4 + 24) * q^40 + (2*b11 - 7*b10 + 8*b9 + 2*b8 - 10*b7 - 7*b6 - 8*b5 + 2*b4 - 10*b3 - 2*b2 - 184*b1 + 192) * q^41 + (-18*b11 - 6*b10 - 2*b9 - 18*b8 - 10*b7 - 6*b6 + 2*b5 + 28*b4 - 10*b3 - 28*b2 - 24*b1 + 6) * q^42 + (-3*b8 + 6*b6 - 29*b5 + 13*b3 + 48*b2 - 50*b1) * q^43 + (12*b10 + 8*b9 - 20*b4 - 16) * q^44 + (46*b8 - 40*b6 + 5*b5 + 6*b2 - 185*b1) * q^45 + (16*b8 - 8*b6 + 14*b5 - 4*b3 + 94*b1) * q^46 + (9*b11 + 12*b10 + 5*b9 - 22*b7 - 7*b4 + 175) * q^47 + (-16*b2 - 32*b1) * q^48 + (12*b11 - b10 - 11*b9 + 12*b8 + 9*b7 - b6 + 11*b5 + 51*b4 + 9*b3 - 51*b2 + 122*b1 - 182) * q^49 + (2*b10 - 32*b9 - 10*b7 + 2*b6 + 32*b5 + 22*b4 - 10*b3 - 22*b2 - 134*b1 + 122) * q^50 + (14*b11 - 46*b10 + 12*b9 + 8*b7 + 43*b4 + 16) * q^51 + (-23*b11 + 9*b10 - 42*b9 - 25*b7 - 8*b4 + 46) * q^53 + (6*b11 - 10*b10 + 6*b8 - 24*b7 - 10*b6 + 46*b4 - 24*b3 - 46*b2 - 232*b1 + 210) * q^54 + (4*b11 + 5*b10 + 42*b9 + 4*b8 - 3*b7 + 5*b6 - 42*b5 + 39*b4 - 3*b3 - 39*b2 + 178*b1 - 214) * q^55 + (-8*b8 - 16*b6 - 8*b2 - 40*b1) * q^56 + (-11*b11 + 30*b10 + 16*b9 - 3*b7 - 10*b4 + 433) * q^57 + (-2*b8 + 16*b5 - 12*b3 + 6*b2 - 34*b1) * q^58 + (-13*b8 - 16*b6 - 42*b5 - 14*b3 - 20*b2 + 181*b1) * q^59 + (16*b11 - 36*b10 - 4*b7 + 28*b4 - 96) * q^60 + (-25*b8 + 21*b6 - 35*b5 + 17*b3 + 6*b2 - 292*b1) * q^61 + (6*b11 + 16*b10 + 8*b9 + 6*b8 + 8*b7 + 16*b6 - 8*b5 - 6*b4 + 8*b3 + 6*b2 - 206*b1 + 204) * q^62 + (4*b11 - 33*b10 + 10*b9 + 4*b8 + 5*b7 - 33*b6 - 10*b5 + 39*b4 + 5*b3 - 39*b2 + 518*b1 - 562) * q^63 + 64 * q^64 + (38*b11 + 2*b10 + 20*b9 + 2*b7 - 30*b4 + 252) * q^66 + (-10*b11 - 7*b10 - 13*b9 - 10*b8 + 7*b7 - 7*b6 + 13*b5 + 44*b4 + 7*b3 - 44*b2 + 251*b1 - 302) * q^67 + (-20*b11 - 20*b9 - 20*b8 - 20*b7 + 20*b5 + 12*b4 - 20*b3 - 12*b2 + 60*b1 - 52) * q^68 + (-27*b8 + 66*b6 + 23*b5 + 14*b3 - 7*b2 - 218*b1) * q^69 + (-18*b11 + 18*b10 + 26*b9 - 6*b7 + 36*b4 - 218) * q^70 + (26*b8 - 5*b6 + 31*b5 + 31*b3 - 15*b2 - 223*b1) * q^71 + (-16*b8 - 16*b6 + 8*b5 + 8*b3 + 8*b2 + 144*b1) * q^72 + (-10*b11 - 21*b10 + 29*b9 + 27*b7 + 20*b4 + 234) * q^73 + (14*b8 - 10*b6 + 2*b5 - 18*b3 + 28*b2 - 64*b1) * q^74 + (21*b11 - 30*b10 - 16*b9 + 21*b8 + 28*b7 - 30*b6 + 16*b5 - 54*b4 + 28*b3 + 54*b2 + 317*b1 - 291) * q^75 + (4*b11 - 8*b10 + 16*b9 + 4*b8 - 12*b7 - 8*b6 - 16*b5 - 40*b4 - 12*b3 + 40*b2 + 84*b1 - 32) * q^76 + (23*b11 + 21*b10 - 43*b9 + 7*b7 - 52*b4 + 487) * q^77 + (16*b11 - 25*b10 + 17*b9 + 11*b7 + 59*b4 - 189) * q^79 + (-16*b11 - 16*b9 - 16*b8 + 16*b5 + 16*b4 - 16*b2 - 64*b1 + 48) * q^80 + (-50*b11 + 32*b10 + 17*b9 - 50*b8 + 28*b7 + 32*b6 - 17*b5 - 125*b4 + 28*b3 + 125*b2 + 591*b1 - 494) * q^81 + (4*b8 - 14*b6 - 16*b5 - 20*b3 - 4*b2 - 368*b1) * q^82 + (-24*b11 - 66*b10 - 12*b9 + 7*b7 - 17*b4 + 36) * q^83 + (-36*b8 - 12*b6 + 4*b5 - 20*b3 - 56*b2 - 48*b1) * q^84 + (19*b8 + 12*b6 + 64*b5 - 52*b3 - 33*b2 - 615*b1) * q^85 + (6*b11 - 12*b10 - 58*b9 - 26*b7 + 96*b4 - 170) * q^86 + (21*b8 - 7*b6 - 9*b5 + 13*b3 + 26*b2 - 252*b1) * q^87 + (24*b10 + 16*b9 + 24*b6 - 16*b5 - 40*b4 + 40*b2 + 72*b1 - 32) * q^88 + (-43*b11 + 59*b10 - 48*b9 - 43*b8 - 7*b7 + 59*b6 + 48*b5 + b4 - 7*b3 - b2 + 43*b1 - 37) * q^89 + (-92*b11 + 80*b10 + 10*b9 + 12*b4 - 382) * q^90 + (-32*b11 + 16*b10 + 28*b9 + 8*b7 + 180) * q^92 + (33*b11 + 7*b10 + b9 + 33*b8 + 7*b7 + 7*b6 - b5 - 158*b4 + 7*b3 + 158*b2 + 398*b1 - 247) * q^93 + (18*b11 + 24*b10 + 10*b9 + 18*b8 - 44*b7 + 24*b6 - 10*b5 - 14*b4 - 44*b3 + 14*b2 - 292*b1 + 350) * q^94 + (33*b8 - 57*b6 - 38*b5 - 3*b3 + 4*b2 + 703*b1) * q^95 + (-32*b4 - 32) * q^96 + (-22*b8 + 51*b6 + b5 + 2*b3 - 17*b2 - 72*b1) * q^97 + (24*b8 - 2*b6 + 22*b5 + 18*b3 - 102*b2 + 244*b1) * q^98 + (-50*b11 + 80*b10 - 68*b9 + 22*b7 - 164*b4 + 323) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 12 q^{2} - 9 q^{3} - 24 q^{4} - 36 q^{5} + 18 q^{6} + 25 q^{7} - 96 q^{8} - 113 q^{9}+O(q^{10})$$ 12 * q + 12 * q^2 - 9 * q^3 - 24 * q^4 - 36 * q^5 + 18 * q^6 + 25 * q^7 - 96 * q^8 - 113 * q^9 $$12 q + 12 q^{2} - 9 q^{3} - 24 q^{4} - 36 q^{5} + 18 q^{6} + 25 q^{7} - 96 q^{8} - 113 q^{9} - 36 q^{10} + 37 q^{11} + 72 q^{12} + 100 q^{14} + 118 q^{15} - 96 q^{16} - 99 q^{17} - 452 q^{18} - 81 q^{19} + 72 q^{20} + 52 q^{21} - 74 q^{22} - 267 q^{23} + 72 q^{24} + 736 q^{25} + 1338 q^{27} + 100 q^{28} + 119 q^{29} - 236 q^{30} + 1250 q^{31} + 192 q^{32} + 762 q^{33} - 396 q^{34} - 614 q^{35} - 452 q^{36} + 274 q^{37} - 324 q^{38} + 288 q^{40} + 1140 q^{41} + 52 q^{42} - 428 q^{43} - 296 q^{44} - 1215 q^{45} + 534 q^{46} + 1972 q^{47} - 144 q^{48} - 899 q^{49} + 736 q^{50} + 578 q^{51} + 178 q^{53} + 1338 q^{54} - 1126 q^{55} - 200 q^{56} + 5106 q^{57} - 238 q^{58} + 1088 q^{59} - 944 q^{60} - 1704 q^{61} + 1250 q^{62} - 3222 q^{63} + 768 q^{64} + 3048 q^{66} - 1692 q^{67} - 396 q^{68} - 1168 q^{69} - 2456 q^{70} - 1221 q^{71} + 904 q^{72} + 3108 q^{73} - 548 q^{74} - 1798 q^{75} - 324 q^{76} + 5580 q^{77} - 1750 q^{79} + 288 q^{80} - 3338 q^{81} - 2280 q^{82} + 252 q^{83} - 104 q^{84} - 3721 q^{85} - 1712 q^{86} - 1602 q^{87} - 296 q^{88} - 374 q^{89} - 4860 q^{90} + 2136 q^{92} - 1868 q^{93} + 1972 q^{94} + 4093 q^{95} - 576 q^{96} - 330 q^{97} + 1798 q^{98} + 2688 q^{99}+O(q^{100})$$ 12 * q + 12 * q^2 - 9 * q^3 - 24 * q^4 - 36 * q^5 + 18 * q^6 + 25 * q^7 - 96 * q^8 - 113 * q^9 - 36 * q^10 + 37 * q^11 + 72 * q^12 + 100 * q^14 + 118 * q^15 - 96 * q^16 - 99 * q^17 - 452 * q^18 - 81 * q^19 + 72 * q^20 + 52 * q^21 - 74 * q^22 - 267 * q^23 + 72 * q^24 + 736 * q^25 + 1338 * q^27 + 100 * q^28 + 119 * q^29 - 236 * q^30 + 1250 * q^31 + 192 * q^32 + 762 * q^33 - 396 * q^34 - 614 * q^35 - 452 * q^36 + 274 * q^37 - 324 * q^38 + 288 * q^40 + 1140 * q^41 + 52 * q^42 - 428 * q^43 - 296 * q^44 - 1215 * q^45 + 534 * q^46 + 1972 * q^47 - 144 * q^48 - 899 * q^49 + 736 * q^50 + 578 * q^51 + 178 * q^53 + 1338 * q^54 - 1126 * q^55 - 200 * q^56 + 5106 * q^57 - 238 * q^58 + 1088 * q^59 - 944 * q^60 - 1704 * q^61 + 1250 * q^62 - 3222 * q^63 + 768 * q^64 + 3048 * q^66 - 1692 * q^67 - 396 * q^68 - 1168 * q^69 - 2456 * q^70 - 1221 * q^71 + 904 * q^72 + 3108 * q^73 - 548 * q^74 - 1798 * q^75 - 324 * q^76 + 5580 * q^77 - 1750 * q^79 + 288 * q^80 - 3338 * q^81 - 2280 * q^82 + 252 * q^83 - 104 * q^84 - 3721 * q^85 - 1712 * q^86 - 1602 * q^87 - 296 * q^88 - 374 * q^89 - 4860 * q^90 + 2136 * q^92 - 1868 * q^93 + 1972 * q^94 + 4093 * q^95 - 576 * q^96 - 330 * q^97 + 1798 * q^98 + 2688 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - x^{11} + 108 x^{10} - 63 x^{9} + 7831 x^{8} - 3348 x^{7} + 317885 x^{6} + 1680 x^{5} + \cdots + 1759886401$$ :

 $$\beta_{1}$$ $$=$$ $$( 78\!\cdots\!56 \nu^{11} + \cdots - 57\!\cdots\!19 ) / 10\!\cdots\!53$$ (785465789105492156*v^11 - 12583302160460081796*v^10 + 81639125628470322008*v^9 - 1117344864567355753028*v^8 + 5482839100737858520317*v^7 - 80542478732339622703916*v^6 + 196525802937295525301724*v^5 - 2705385180882866831036814*v^4 + 4345825383253100408110764*v^3 - 94837831006845500513879548*v^2 + 48502569645993746666183346*v - 575156369469953908988069719) / 1011894018356510709039836853 $$\beta_{2}$$ $$=$$ $$( - 16\!\cdots\!33 \nu^{11} + \cdots + 19\!\cdots\!38 ) / 23\!\cdots\!43$$ (-16553880023497608833*v^11 + 370883458201127290719*v^10 - 360457310158912898572*v^9 + 38639136423585897111815*v^8 - 12564013458437834262732*v^7 + 2435452139476923272423296*v^6 + 1836564128551304557170459*v^5 + 84105971544273066264214842*v^4 + 53673928845600740612192691*v^3 + 1817596045362211666346801671*v^2 + 1678901135017714428574662468*v + 19276064005550952937067932838) / 233514004236117855932270043 $$\beta_{3}$$ $$=$$ $$( - 21\!\cdots\!54 \nu^{11} + \cdots + 37\!\cdots\!22 ) / 23\!\cdots\!43$$ (-21585677710328996654*v^11 + 662205604624901393831*v^10 + 2155746170131529600372*v^9 + 70985147313081608539268*v^8 + 181502117720148545448712*v^7 + 4432638657272734227584464*v^6 + 13543951321796449818071754*v^5 + 156144862852435398181077138*v^4 + 344547982903838404395437019*v^3 + 3406938754609545570731676010*v^2 + 7251091337434180774275790892*v + 37596741784440733229790595322) / 233514004236117855932270043 $$\beta_{4}$$ $$=$$ $$( 11\!\cdots\!40 \nu^{11} + \cdots + 30\!\cdots\!60 ) / 94\!\cdots\!17$$ (118779499945057540*v^11 + 1380439302722201040*v^10 + 13263849694404390308*v^9 + 104952837180862680404*v^8 + 824965424785263333648*v^7 + 6151332868493844664957*v^6 + 30128822630137545551904*v^5 + 150094921467699215219508*v^4 + 653535778798831295391792*v^3 + 2810597205734286840839128*v^2 + 7872496886500894622824668*v + 306810975560983459453160) / 940713373123499264679117 $$\beta_{5}$$ $$=$$ $$( - 32\!\cdots\!43 \nu^{11} + \cdots + 93\!\cdots\!08 ) / 23\!\cdots\!43$$ (-32543747968994249543*v^11 + 1612939893338604608466*v^10 + 6854506390437765525912*v^9 + 168963634970536967934149*v^8 + 558920393202987221307600*v^7 + 10844844836850643398436320*v^6 + 38380493932766711282375634*v^5 + 383746570303263853533259356*v^4 + 981706590846123048913960170*v^3 + 8676000795474715156164944317*v^2 + 20098339592542933161910639848*v + 93396345214209992717424369708) / 233514004236117855932270043 $$\beta_{6}$$ $$=$$ $$( - 14\!\cdots\!73 \nu^{11} + \cdots + 38\!\cdots\!33 ) / 30\!\cdots\!59$$ (-1408501136748026708073*v^11 + 67791883095162409510727*v^10 + 193960943125496453338604*v^9 + 7076515734275849069698161*v^8 + 16633654861900269500103211*v^7 + 452978094399140824718682172*v^6 + 1229969438088894418849935315*v^5 + 15927943438447614103950248808*v^4 + 32897560185353703329768712987*v^3 + 355258316904570559440617176464*v^2 + 701701719144131101890350452778*v + 3818525423460866069703592491833) / 3035682055069532127119510559 $$\beta_{7}$$ $$=$$ $$( - 36\!\cdots\!88 \nu^{11} + \cdots - 21\!\cdots\!67 ) / 72\!\cdots\!09$$ (-36441248282711188*v^11 - 208051372983284096*v^10 - 3955477144668104300*v^9 - 16502464841369367482*v^8 - 246617712778301315944*v^7 - 976245571346147074786*v^6 - 8783537493841911655020*v^5 - 26905299755797297373124*v^4 - 191069674216334861449215*v^3 - 534897999294615283500628*v^2 - 2170522917299941447493978*v - 2131068887893052316448967) / 72362567163346097283009 $$\beta_{8}$$ $$=$$ $$( 15\!\cdots\!38 \nu^{11} + \cdots - 46\!\cdots\!79 ) / 30\!\cdots\!59$$ (1567713134615326810138*v^11 - 81041261232142686640927*v^10 - 287905399078775041966524*v^9 - 8507334162374758105745287*v^8 - 23966883038624541289446569*v^7 - 543194182633713017598948468*v^6 - 1689568938471963009029341746*v^5 - 19160541668389774481256298836*v^4 - 44305816393080176063283570003*v^3 - 427821423981795590337044482238*v^2 - 923987966388239320405503341494*v - 4628475536259828756959162786079) / 3035682055069532127119510559 $$\beta_{9}$$ $$=$$ $$( 12\!\cdots\!21 \nu^{11} + \cdots + 50\!\cdots\!26 ) / 94\!\cdots\!17$$ (1281063312026774521*v^11 + 6061373028285428532*v^10 + 135504534310075166493*v^9 + 488884385220460981769*v^8 + 8631984668079910044948*v^7 + 29822213380320885968568*v^6 + 306104840457371077650240*v^5 + 834552231455356677908133*v^4 + 6747744465048076813006794*v^3 + 16900795603491844568120926*v^2 + 74880331681636947878026251*v + 50406377248758977276038026) / 940713373123499264679117 $$\beta_{10}$$ $$=$$ $$( - 33\!\cdots\!60 \nu^{11} + \cdots - 10\!\cdots\!16 ) / 94\!\cdots\!17$$ (-3320134982517473760*v^11 - 19324942268917584080*v^10 - 350373502300904812013*v^9 - 1530448459709666126754*v^8 - 22480269046232079650248*v^7 - 92511467515689182284411*v^6 - 801050335007919946951524*v^5 - 2484154246677028301944608*v^4 - 18044865063865613223048270*v^3 - 49295649792029985252707988*v^2 - 198174471796536925367291978*v - 104702573589709474254373316) / 940713373123499264679117 $$\beta_{11}$$ $$=$$ $$( 44\!\cdots\!10 \nu^{11} + \cdots + 17\!\cdots\!89 ) / 94\!\cdots\!17$$ (4406532938458813010*v^11 + 23559060514356012640*v^10 + 464985088477127685876*v^9 + 1879026713206351187653*v^8 + 29773319536745936462036*v^7 + 113598931874616947088732*v^6 + 1058706413154373608851958*v^5 + 3111378222416234153055306*v^4 + 23662124379472948752050409*v^3 + 62251355149229957922672986*v^2 + 260646964814713140894443941*v + 176306249811122561453038989) / 940713373123499264679117
 $$\nu$$ $$=$$ $$( \beta_{11} + 2 \beta_{10} + 2 \beta_{9} + \beta_{8} + 3 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + \cdots - 4 ) / 13$$ (b11 + 2*b10 + 2*b9 + b8 + 3*b7 + 2*b6 - 2*b5 + 8*b4 + 3*b3 - 8*b2 - 7*b1 - 4) / 13 $$\nu^{2}$$ $$=$$ $$( -6\beta_{8} - 5\beta_{6} - 8\beta_{5} + 2\beta_{3} - 3\beta_{2} - 469\beta_1 ) / 13$$ (-6*b8 - 5*b6 - 8*b5 + 2*b3 - 3*b2 - 469*b1) / 13 $$\nu^{3}$$ $$=$$ $$( -66\beta_{11} - 92\beta_{10} - 40\beta_{9} - 145\beta_{7} - 270\beta_{4} + 91 ) / 13$$ (-66*b11 - 92*b10 - 40*b9 - 145*b7 - 270*b4 + 91) / 13 $$\nu^{4}$$ $$=$$ $$( 443 \beta_{11} + 363 \beta_{10} - 625 \beta_{9} + 443 \beta_{8} - 143 \beta_{7} + 363 \beta_{6} + \cdots - 17646 ) / 13$$ (443*b11 + 363*b10 - 625*b9 + 443*b8 - 143*b7 + 363*b6 + 625*b5 - 77*b4 - 143*b3 + 77*b2 + 17866*b1 - 17646) / 13 $$\nu^{5}$$ $$=$$ $$( -3566\beta_{8} - 4108\beta_{6} + 91\beta_{5} - 6933\beta_{3} + 9889\beta_{2} + 10806\beta_1 ) / 13$$ (-3566*b8 - 4108*b6 + 91*b5 - 6933*b3 + 9889*b2 + 10806*b1) / 13 $$\nu^{6}$$ $$=$$ $$( -25568\beta_{11} - 20720\beta_{10} + 37260\beta_{9} + 7480\beta_{7} - 2659\beta_{4} + 710376 ) / 13$$ (-25568*b11 - 20720*b10 + 37260*b9 + 7480*b7 - 2659*b4 + 710376) / 13 $$\nu^{7}$$ $$=$$ $$( 181248 \beta_{11} + 184622 \beta_{10} - 57074 \beta_{9} + 181248 \beta_{8} + 329392 \beta_{7} + \cdots - 494301 ) / 13$$ (181248*b11 + 184622*b10 - 57074*b9 + 181248*b8 + 329392*b7 + 184622*b6 + 57074*b5 + 389694*b4 + 329392*b3 - 389694*b2 - 224785*b1 - 494301) / 13 $$\nu^{8}$$ $$=$$ $$( - 1359421 \beta_{8} - 1094536 \beta_{6} - 2009604 \beta_{5} + 335375 \beta_{3} + 446732 \beta_{2} - 29887091 \beta_1 ) / 13$$ (-1359421*b8 - 1094536*b6 - 2009604*b5 + 335375*b3 + 446732*b2 - 29887091*b1) / 13 $$\nu^{9}$$ $$=$$ $$( - 9036326 \beta_{11} - 8466569 \beta_{10} + 4886472 \beta_{9} - 15591752 \beta_{7} - 16312087 \beta_{4} + 35539298 ) / 13$$ (-9036326*b11 - 8466569*b10 + 4886472*b9 - 15591752*b7 - 16312087*b4 + 35539298) / 13 $$\nu^{10}$$ $$=$$ $$( 69817796 \beta_{11} + 56028534 \beta_{10} - 103389010 \beta_{9} + 69817796 \beta_{8} + \cdots - 1316074697 ) / 13$$ (69817796*b11 + 56028534*b10 - 103389010*b9 + 69817796*b8 - 13280957*b7 + 56028534*b6 + 103389010*b5 + 35129872*b4 - 13280957*b3 - 35129872*b2 + 1294225782*b1 - 1316074697) / 13 $$\nu^{11}$$ $$=$$ $$( - 448638481 \beta_{8} - 397234871 \beta_{6} - 313302931 \beta_{5} - 736608573 \beta_{3} + \cdots - 827715592 \beta_1 ) / 13$$ (-448638481*b8 - 397234871*b6 - 313302931*b5 - 736608573*b3 + 715640933*b2 - 827715592*b1) / 13

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
191.1
 2.92486 − 5.06601i 2.90876 − 5.03812i 3.32703 − 5.76258i −2.70234 + 4.68060i −3.53225 + 6.11803i −2.42606 + 4.20205i 2.92486 + 5.06601i 2.90876 + 5.03812i 3.32703 + 5.76258i −2.70234 − 4.68060i −3.53225 − 6.11803i −2.42606 − 4.20205i
1.00000 1.73205i −5.00428 + 8.66767i −2.00000 3.46410i 13.8136 10.0086 + 17.3353i −0.131143 0.227146i −8.00000 −36.5857 63.3682i 13.8136 23.9258i
191.2 1.00000 1.73205i −4.47729 + 7.75489i −2.00000 3.46410i −17.3267 8.95458 + 15.5098i −8.37550 14.5068i −8.00000 −26.5923 46.0591i −17.3267 + 30.0108i
191.3 1.00000 1.73205i −1.67908 + 2.90825i −2.00000 3.46410i −6.80407 3.35815 + 5.81649i 7.76902 + 13.4563i −8.00000 7.86140 + 13.6163i −6.80407 + 11.7850i
191.4 1.00000 1.73205i 0.622927 1.07894i −2.00000 3.46410i −20.5002 −1.24585 2.15788i 10.6273 + 18.4070i −8.00000 12.7239 + 22.0385i −20.5002 + 35.5074i
191.5 1.00000 1.73205i 1.96372 3.40126i −2.00000 3.46410i 0.152827 −3.92743 6.80251i 16.9956 + 29.4373i −8.00000 5.78763 + 10.0245i 0.152827 0.264703i
191.6 1.00000 1.73205i 4.07401 7.05638i −2.00000 3.46410i 12.6646 −8.14801 14.1128i −14.3853 24.9160i −8.00000 −19.6950 34.1128i 12.6646 21.9357i
315.1 1.00000 + 1.73205i −5.00428 8.66767i −2.00000 + 3.46410i 13.8136 10.0086 17.3353i −0.131143 + 0.227146i −8.00000 −36.5857 + 63.3682i 13.8136 + 23.9258i
315.2 1.00000 + 1.73205i −4.47729 7.75489i −2.00000 + 3.46410i −17.3267 8.95458 15.5098i −8.37550 + 14.5068i −8.00000 −26.5923 + 46.0591i −17.3267 30.0108i
315.3 1.00000 + 1.73205i −1.67908 2.90825i −2.00000 + 3.46410i −6.80407 3.35815 5.81649i 7.76902 13.4563i −8.00000 7.86140 13.6163i −6.80407 11.7850i
315.4 1.00000 + 1.73205i 0.622927 + 1.07894i −2.00000 + 3.46410i −20.5002 −1.24585 + 2.15788i 10.6273 18.4070i −8.00000 12.7239 22.0385i −20.5002 35.5074i
315.5 1.00000 + 1.73205i 1.96372 + 3.40126i −2.00000 + 3.46410i 0.152827 −3.92743 + 6.80251i 16.9956 29.4373i −8.00000 5.78763 10.0245i 0.152827 + 0.264703i
315.6 1.00000 + 1.73205i 4.07401 + 7.05638i −2.00000 + 3.46410i 12.6646 −8.14801 + 14.1128i −14.3853 + 24.9160i −8.00000 −19.6950 + 34.1128i 12.6646 + 21.9357i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 191.6 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.p 12
13.b even 2 1 338.4.c.o 12
13.c even 3 1 338.4.a.n 6
13.c even 3 1 inner 338.4.c.p 12
13.d odd 4 2 338.4.e.i 24
13.e even 6 1 338.4.a.o yes 6
13.e even 6 1 338.4.c.o 12
13.f odd 12 2 338.4.b.h 12
13.f odd 12 2 338.4.e.i 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
338.4.a.n 6 13.c even 3 1
338.4.a.o yes 6 13.e even 6 1
338.4.b.h 12 13.f odd 12 2
338.4.c.o 12 13.b even 2 1
338.4.c.o 12 13.e even 6 1
338.4.c.p 12 1.a even 1 1 trivial
338.4.c.p 12 13.c even 3 1 inner
338.4.e.i 24 13.d odd 4 2
338.4.e.i 24 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 $$S_{4}^{\mathrm{new}}(338, [\chi])$$:

 $$T_{3}^{12} + 9 T_{3}^{11} + 178 T_{3}^{10} + 589 T_{3}^{9} + 13675 T_{3}^{8} + 37320 T_{3}^{7} + \cdots + 143976001$$ T3^12 + 9*T3^11 + 178*T3^10 + 589*T3^9 + 13675*T3^8 + 37320*T3^7 + 662301*T3^6 - 237676*T3^5 + 10068423*T3^4 + 1070173*T3^3 + 92507896*T3^2 - 96555953*T3 + 143976001 $$T_{5}^{6} + 18T_{5}^{5} - 397T_{5}^{4} - 5935T_{5}^{3} + 44090T_{5}^{2} + 416208T_{5} - 64616$$ T5^6 + 18*T5^5 - 397*T5^4 - 5935*T5^3 + 44090*T5^2 + 416208*T5 - 64616

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2 T + 4)^{6}$$
$3$ $$T^{12} + \cdots + 143976001$$
$5$ $$(T^{6} + 18 T^{5} + \cdots - 64616)^{2}$$
$7$ $$T^{12} + \cdots + 2013515592256$$
$11$ $$T^{12} + \cdots + 81\!\cdots\!69$$
$13$ $$T^{12}$$
$17$ $$T^{12} + \cdots + 18\!\cdots\!21$$
$19$ $$T^{12} + \cdots + 96\!\cdots\!89$$
$23$ $$T^{12} + \cdots + 84\!\cdots\!84$$
$29$ $$T^{12} + \cdots + 69\!\cdots\!36$$
$31$ $$(T^{6} - 625 T^{5} + \cdots + 254759240632)^{2}$$
$37$ $$T^{12} + \cdots + 39\!\cdots\!96$$
$41$ $$T^{12} + \cdots + 38\!\cdots\!81$$
$43$ $$T^{12} + \cdots + 15\!\cdots\!21$$
$47$ $$(T^{6} + \cdots + 549423790361944)^{2}$$
$53$ $$(T^{6} + \cdots + 14\!\cdots\!88)^{2}$$
$59$ $$T^{12} + \cdots + 12\!\cdots\!29$$
$61$ $$T^{12} + \cdots + 69\!\cdots\!84$$
$67$ $$T^{12} + \cdots + 74\!\cdots\!69$$
$71$ $$T^{12} + \cdots + 14\!\cdots\!76$$
$73$ $$(T^{6} + \cdots - 52281997669273)^{2}$$
$79$ $$(T^{6} + \cdots + 79\!\cdots\!96)^{2}$$
$83$ $$(T^{6} + \cdots - 99\!\cdots\!91)^{2}$$
$89$ $$T^{12} + \cdots + 91\!\cdots\!81$$
$97$ $$T^{12} + \cdots + 23\!\cdots\!01$$