# Properties

 Label 507.4.b.k Level $507$ Weight $4$ Character orbit 507.b Analytic conductor $29.914$ Analytic rank $0$ Dimension $18$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,4,Mod(337,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("507.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 507.b (of order $$2$$, degree $$1$$, 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: $$29.9139683729$$ Analytic rank: $$0$$ Dimension: $$18$$ Coefficient field: $$\mathbb{Q}[x]/(x^{18} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{18} + 112 x^{16} + 5026 x^{14} + 114847 x^{12} + 1397921 x^{10} + 8545747 x^{8} + 21033277 x^{6} + \cdots + 729$$ x^18 + 112*x^16 + 5026*x^14 + 114847*x^12 + 1397921*x^10 + 8545747*x^8 + 21033277*x^6 + 6703200*x^4 + 137781*x^2 + 729 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$13^{8}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{17}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{9} q^{2} + 3 q^{3} + (\beta_{2} - 5) q^{4} - \beta_{15} q^{5} - 3 \beta_{9} q^{6} + (\beta_{17} - \beta_{16} + \cdots + \beta_{9}) q^{7}+ \cdots + 9 q^{9}+O(q^{10})$$ q - b9 * q^2 + 3 * q^3 + (b2 - 5) * q^4 - b15 * q^5 - 3*b9 * q^6 + (b17 - b16 - b15 + b9) * q^7 + (-b16 - 3*b15 + b14 - 2*b13 + b12 - b11 + 4*b9) * q^8 + 9 * q^9 $$q - \beta_{9} q^{2} + 3 q^{3} + (\beta_{2} - 5) q^{4} - \beta_{15} q^{5} - 3 \beta_{9} q^{6} + (\beta_{17} - \beta_{16} + \cdots + \beta_{9}) q^{7}+ \cdots + (9 \beta_{15} + 9 \beta_{14} + \cdots - 45 \beta_{9}) q^{99}+O(q^{100})$$ q - b9 * q^2 + 3 * q^3 + (b2 - 5) * q^4 - b15 * q^5 - 3*b9 * q^6 + (b17 - b16 - b15 + b9) * q^7 + (-b16 - 3*b15 + b14 - 2*b13 + b12 - b11 + 4*b9) * q^8 + 9 * q^9 + (-b8 - b4 - b3 - 2*b2 - b1 + 6) * q^10 + (b15 + b14 + b11 + b10 - 5*b9) * q^11 + (3*b2 - 15) * q^12 + (2*b8 - 3*b7 - b6 + 2*b5 - b4 + 2*b3 - 2*b2 - b1 + 19) * q^14 - 3*b15 * q^15 + (-b8 - 3*b7 + 3*b6 - b5 - b4 + 2*b3 - 5*b2 - 4*b1 + 24) * q^16 + (-b8 - 3*b7 + 3*b5 - 3*b4 + b3 + 2*b2 - 4*b1 - 19) * q^17 - 9*b9 * q^18 + (4*b17 + 6*b16 - 2*b15 + 2*b14 + 2*b13 - 2*b12 + 8*b11 + 3*b10 + b9) * q^19 + (5*b17 - 3*b16 + 4*b15 - b14 + 2*b13 - 4*b12 + 3*b11 + 2*b10 - 16*b9) * q^20 + (3*b17 - 3*b16 - 3*b15 + 3*b9) * q^21 + (b8 - b7 + 3*b6 - 2*b5 + b3 + 5*b2 + 4*b1 - 70) * q^22 + (3*b8 - 2*b7 - 3*b6 - 3*b5 + 5*b4 - 5*b3 - 2*b2 + 2*b1 - 17) * q^23 + (-3*b16 - 9*b15 + 3*b14 - 6*b13 + 3*b12 - 3*b11 + 12*b9) * q^24 + (-4*b7 - 4*b6 + 3*b5 + 6*b3 + 3*b2 - b1 + 5) * q^25 + 27 * q^27 + (-2*b17 + 4*b16 - 2*b15 - 5*b14 + 12*b13 - 4*b12 + 7*b11 + 6*b10 - 35*b9) * q^28 + (9*b8 - 8*b6 + 2*b5 - 6*b4 + 5*b3 + 18*b2 - 4*b1 - 14) * q^29 + (-3*b8 - 3*b4 - 3*b3 - 6*b2 - 3*b1 + 18) * q^30 + (-6*b17 + 6*b16 + 12*b15 - 12*b14 + 2*b13 + 5*b12 + b11 - 4*b10 - 2*b9) * q^31 + (12*b17 + 15*b16 + 10*b15 + 5*b14 + 17*b13 - 12*b12 + 25*b11 + 14*b10 - 28*b9) * q^32 + (3*b15 + 3*b14 + 3*b11 + 3*b10 - 15*b9) * q^33 + (4*b17 + 6*b16 + 10*b15 + 6*b13 - 6*b12 + 21*b11 + 13*b10 + 32*b9) * q^34 + (-3*b8 + 3*b7 - 3*b6 - 5*b5 - 4*b4 - b3 - 8*b2 + 4*b1 - 46) * q^35 + (9*b2 - 45) * q^36 + (2*b17 - 12*b16 - 3*b15 - 13*b14 + 4*b13 + 9*b12 + 12*b11 + 2*b10 - 16*b9) * q^37 + (6*b8 - 12*b7 - 8*b6 - 3*b5 - 7*b4 - 16*b3 - 8*b2 + 12*b1 + 79) * q^38 + (5*b8 - 14*b7 - 8*b6 + 11*b5 - 4*b4 + 5*b3 + 12*b2 + 3*b1 - 159) * q^40 + (7*b17 + 6*b16 - 14*b15 + 18*b14 - 20*b13 + 7*b12 - 7*b11 + 12*b10 - 22*b9) * q^41 + (6*b8 - 9*b7 - 3*b6 + 6*b5 - 3*b4 + 6*b3 - 6*b2 - 3*b1 + 57) * q^42 + (11*b8 + 5*b7 - 3*b6 + 4*b5 - 8*b4 + 5*b3 - b2 - 7*b1 - 31) * q^43 + (-11*b17 - 8*b16 - 19*b15 + 3*b14 - 20*b13 + 15*b12 - 25*b11 - 14*b10 + 77*b9) * q^44 - 9*b15 * q^45 + (-4*b17 - 19*b16 - 14*b15 + 15*b14 + b13 + 2*b12 - 29*b11 + 6*b10 + 25*b9) * q^46 + (-8*b17 + 6*b16 + 8*b14 + 14*b13 - 20*b12 + 14*b11 + 5*b10 - 3*b9) * q^47 + (-3*b8 - 9*b7 + 9*b6 - 3*b5 - 3*b4 + 6*b3 - 15*b2 - 12*b1 + 72) * q^48 + (-9*b8 + 20*b7 - 6*b6 - 2*b5 - 4*b4 + 9*b3 + 14*b2 - 1) * q^49 + (-14*b17 + 15*b16 - 16*b15 + 17*b13 + 10*b12 + 15*b11 + 9*b10 - 7*b9) * q^50 + (-3*b8 - 9*b7 + 9*b5 - 9*b4 + 3*b3 + 6*b2 - 12*b1 - 57) * q^51 + (14*b8 - 8*b7 + 2*b6 + 9*b5 + 6*b4 - 10*b3 + 4*b2 + 6*b1 - 14) * q^53 - 27*b9 * q^54 + (b7 + 3*b6 + 7*b5 - 8*b4 - 6*b3 - 6*b2 - 6*b1 + 229) * q^55 + (-8*b8 + 7*b7 - 5*b6 + 7*b5 - 18*b4 - 16*b3 + 24*b2 + 9*b1 - 252) * q^56 + (12*b17 + 18*b16 - 6*b15 + 6*b14 + 6*b13 - 6*b12 + 24*b11 + 9*b10 + 3*b9) * q^57 + (-27*b17 + 6*b16 - 36*b15 + 15*b14 - 25*b13 - 6*b12 - 18*b11 + 25*b10 + 35*b9) * q^58 + (-18*b17 - 7*b16 - 5*b15 - 8*b14 + 10*b13 + b12 + 11*b11 - 3*b10 - 2*b9) * q^59 + (15*b17 - 9*b16 + 12*b15 - 3*b14 + 6*b13 - 12*b12 + 9*b11 + 6*b10 - 48*b9) * q^60 + (-21*b8 - 11*b7 - 15*b5 + 7*b4 + 9*b3 - 14*b2 - 143) * q^61 + (7*b8 + 29*b7 - 9*b6 - 30*b5 + 28*b4 - 26*b3 + 27*b2 + 5*b1 - 63) * q^62 + (9*b17 - 9*b16 - 9*b15 + 9*b9) * q^63 + (4*b8 - 38*b7 + 6*b6 - 2*b5 - 28*b4 - 17*b3 + 15*b2 + 31*b1 - 82) * q^64 + (3*b8 - 3*b7 + 9*b6 - 6*b5 + 3*b3 + 15*b2 + 12*b1 - 210) * q^66 + (37*b17 + 18*b16 - 3*b15 + 17*b14 + 34*b13 + 12*b12 + 9*b11 + 12*b10 - 32*b9) * q^67 + (8*b8 - 36*b7 + 6*b6 - 5*b5 - 13*b4 - 11*b3 - 14*b2 + 25*b1 + 326) * q^68 + (9*b8 - 6*b7 - 9*b6 - 9*b5 + 15*b4 - 15*b3 - 6*b2 + 6*b1 - 51) * q^69 + (-4*b17 - 30*b16 + 21*b15 - 15*b14 - 13*b13 + 9*b12 - 51*b11 - 28*b10 - 20*b9) * q^70 + (5*b17 + 16*b16 - 7*b15 + 13*b14 - 26*b13 - 20*b12 + 19*b11 + 23*b10 - 31*b9) * q^71 + (-9*b16 - 27*b15 + 9*b14 - 18*b13 + 9*b12 - 9*b11 + 36*b9) * q^72 + (-6*b17 - 3*b15 + 8*b14 - 18*b13 - 14*b12 + 38*b11 + 10*b10 + 6*b9) * q^73 + (38*b8 - 6*b7 + 8*b6 - 49*b5 + 17*b4 - 15*b3 + 6*b2 + 13*b1 - 154) * q^74 + (-12*b7 - 12*b6 + 9*b5 + 18*b3 + 9*b2 - 3*b1 + 15) * q^75 + (-24*b17 - 77*b16 - 43*b15 - 3*b14 - 70*b13 + 29*b12 - 143*b11 - 36*b10 - 31*b9) * q^76 + (14*b8 - 20*b7 - 2*b6 + 30*b5 - 14*b4 + 10*b3 + 5*b2 - 25*b1 + 244) * q^77 + (-3*b8 + 13*b7 - 24*b6 + 36*b5 - 11*b4 - 3*b3 - b2 - 27*b1 - 175) * q^79 + (-21*b17 - 42*b16 - 41*b15 - 18*b14 - 14*b13 + 15*b12 - 26*b11 + 6*b10 + 125*b9) * q^80 + 81 * q^81 + (-17*b8 - 21*b7 + 29*b6 + 2*b4 + 19*b3 - 44*b2 + 3*b1 - 153) * q^82 + (8*b17 + 31*b16 + 30*b15 + 35*b14 + 12*b13 - 2*b12 + 7*b11 - 33*b10 - 50*b9) * q^83 + (-6*b17 + 12*b16 - 6*b15 - 15*b14 + 36*b13 - 12*b12 + 21*b11 + 18*b10 - 105*b9) * q^84 + (15*b17 - 48*b16 + 32*b15 - 38*b14 + 34*b13 - 3*b12 - 3*b11 - 2*b10 + 76*b9) * q^85 + (-16*b17 + 51*b16 + 45*b15 - 9*b14 + 12*b13 - 49*b12 + 38*b11 + 33*b10 - 103*b9) * q^86 + (27*b8 - 24*b6 + 6*b5 - 18*b4 + 15*b3 + 54*b2 - 12*b1 - 42) * q^87 + (-12*b8 + 44*b6 - 21*b5 + b4 + 27*b3 - 87*b2 - 40*b1 + 403) * q^88 + (51*b17 + 22*b16 + 9*b15 + 37*b14 + 56*b13 + 14*b12 + 33*b11 + 26*b10 + 70*b9) * q^89 + (-9*b8 - 9*b4 - 9*b3 - 18*b2 - 9*b1 + 54) * q^90 + (-48*b8 + 32*b7 + 32*b6 + 32*b5 + 21*b3 - 48*b2 - 15*b1 + 67) * q^92 + (-18*b17 + 18*b16 + 36*b15 - 36*b14 + 6*b13 + 15*b12 + 3*b11 - 12*b10 - 6*b9) * q^93 + (-38*b8 + 4*b7 + 8*b6 + 13*b5 - 39*b4 - 20*b3 + 4*b2 + 24*b1 - 61) * q^94 + (6*b8 - 61*b7 - 9*b6 + 26*b5 + 8*b4 + 38*b3 - 16*b2 - 8*b1 - 430) * q^95 + (36*b17 + 45*b16 + 30*b15 + 15*b14 + 51*b13 - 36*b12 + 75*b11 + 42*b10 - 84*b9) * q^96 + (11*b17 - 10*b16 + 22*b15 - 75*b14 + 20*b13 - 4*b12 - 5*b11 - 10*b10 + 44*b9) * q^97 + (21*b17 + 2*b16 - 18*b15 + 5*b14 - 31*b13 + 16*b12 + 28*b11 - 11*b10 - 20*b9) * q^98 + (9*b15 + 9*b14 + 9*b11 + 9*b10 - 45*b9) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$18 q + 54 q^{3} - 88 q^{4} + 162 q^{9}+O(q^{10})$$ 18 * q + 54 * q^3 - 88 * q^4 + 162 * q^9 $$18 q + 54 q^{3} - 88 q^{4} + 162 q^{9} + 108 q^{10} - 264 q^{12} + 316 q^{14} + 432 q^{16} - 356 q^{17} - 1260 q^{22} - 300 q^{23} + 40 q^{25} + 486 q^{27} - 194 q^{29} + 324 q^{30} - 836 q^{35} - 792 q^{36} + 1320 q^{38} - 3012 q^{40} + 948 q^{42} - 484 q^{43} + 1296 q^{48} + 76 q^{49} - 1068 q^{51} - 302 q^{53} + 4128 q^{55} - 4552 q^{56} - 2680 q^{61} - 694 q^{62} - 1786 q^{64} - 3780 q^{66} + 5570 q^{68} - 900 q^{69} - 2382 q^{74} + 120 q^{75} + 4284 q^{77} - 3182 q^{79} + 1458 q^{81} - 3034 q^{82} - 582 q^{87} + 7432 q^{88} + 972 q^{90} + 1030 q^{92} - 1384 q^{94} - 8316 q^{95}+O(q^{100})$$ 18 * q + 54 * q^3 - 88 * q^4 + 162 * q^9 + 108 * q^10 - 264 * q^12 + 316 * q^14 + 432 * q^16 - 356 * q^17 - 1260 * q^22 - 300 * q^23 + 40 * q^25 + 486 * q^27 - 194 * q^29 + 324 * q^30 - 836 * q^35 - 792 * q^36 + 1320 * q^38 - 3012 * q^40 + 948 * q^42 - 484 * q^43 + 1296 * q^48 + 76 * q^49 - 1068 * q^51 - 302 * q^53 + 4128 * q^55 - 4552 * q^56 - 2680 * q^61 - 694 * q^62 - 1786 * q^64 - 3780 * q^66 + 5570 * q^68 - 900 * q^69 - 2382 * q^74 + 120 * q^75 + 4284 * q^77 - 3182 * q^79 + 1458 * q^81 - 3034 * q^82 - 582 * q^87 + 7432 * q^88 + 972 * q^90 + 1030 * q^92 - 1384 * q^94 - 8316 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{18} + 112 x^{16} + 5026 x^{14} + 114847 x^{12} + 1397921 x^{10} + 8545747 x^{8} + 21033277 x^{6} + \cdots + 729$$ :

 $$\beta_{1}$$ $$=$$ $$( 16\!\cdots\!77 \nu^{16} + \cdots + 10\!\cdots\!68 ) / 30\!\cdots\!32$$ (1601044867583677*v^16 + 179305097519975323*v^14 + 8045612589142543945*v^12 + 183822566061610352491*v^10 + 2236951836672312074405*v^8 + 13667233608090341944987*v^6 + 33576707877832131502849*v^4 + 10467565459459412450652*v^2 + 104051653875419246568) / 3032421244108904232 $$\beta_{2}$$ $$=$$ $$( - 17\!\cdots\!82 \nu^{16} + \cdots - 86\!\cdots\!26 ) / 30\!\cdots\!32$$ (-1734137237833382*v^16 - 194201793863294741*v^14 - 8713432054116181529*v^12 - 199058877775579121987*v^10 - 2421951006453304882537*v^8 - 14793416297435623057127*v^6 - 36322397623835941473329*v^4 - 11259288060469626490641*v^2 - 86875928215438277826) / 3032421244108904232 $$\beta_{3}$$ $$=$$ $$( - 20\!\cdots\!10 \nu^{16} + \cdots - 11\!\cdots\!75 ) / 30\!\cdots\!32$$ (-2038071261700910*v^16 - 228234250530821033*v^14 - 10239866564043144446*v^12 - 233902954714980508178*v^10 - 2845151778625514796646*v^8 - 17366991252690389931998*v^6 - 42549134109079154978570*v^4 - 12862649157148374623424*v^2 - 113523248333235433275) / 3032421244108904232 $$\beta_{4}$$ $$=$$ $$( 10\!\cdots\!10 \nu^{16} + \cdots + 64\!\cdots\!52 ) / 15\!\cdots\!16$$ (1079366094781310*v^16 + 120873077291344265*v^14 + 5423019545415382019*v^12 + 123874522923848204201*v^10 + 1506817840103975516743*v^8 + 9198745336173102049853*v^6 + 22549257793644108016991*v^4 + 6869693527170214236975*v^2 + 64519121555398648152) / 1516210622054452116 $$\beta_{5}$$ $$=$$ $$( 98034109972 \nu^{16} + 10978375356709 \nu^{14} + 492560731054663 \nu^{12} + \cdots + 60\!\cdots\!96 ) / 50991629993928$$ (98034109972*v^16 + 10978375356709*v^14 + 492560731054663*v^12 + 11251941816685501*v^10 + 136887404185381487*v^8 + 835899628653827689*v^6 + 2050638752311140007*v^4 + 629456453820527145*v^2 + 6033608228060496) / 50991629993928 $$\beta_{6}$$ $$=$$ $$( 79\!\cdots\!43 \nu^{16} + \cdots + 55\!\cdots\!28 ) / 30\!\cdots\!32$$ (7948726537666543*v^16 + 890136922280136022*v^14 + 39937130751587450518*v^12 + 912311321808153371026*v^10 + 11098876671253416854978*v^8 + 67776203941121748833254*v^6 + 166290168946269976351726*v^4 + 51169954896152686256301*v^2 + 552995164470195684828) / 3032421244108904232 $$\beta_{7}$$ $$=$$ $$( 12\!\cdots\!58 \nu^{16} + \cdots + 83\!\cdots\!89 ) / 33\!\cdots\!48$$ (1202833801010558*v^16 + 134704700598988610*v^14 + 6044042621266104671*v^12 + 138079952831185821437*v^10 + 1680072585616760848183*v^8 + 10262367969749678587517*v^6 + 25198186392175608099059*v^4 + 7814023679691706406211*v^2 + 83114602924383017289) / 336935693789878248 $$\beta_{8}$$ $$=$$ $$( 13\!\cdots\!97 \nu^{16} + \cdots + 91\!\cdots\!10 ) / 30\!\cdots\!32$$ (13274576015701397*v^16 + 1486610281078533560*v^14 + 66702099923845469930*v^12 + 1523830694243708244014*v^10 + 18540464840465259889006*v^8 + 113242004905400576400818*v^6 + 277990265584479225693698*v^4 + 85994173135670544045135*v^2 + 910221237339627426210) / 3032421244108904232 $$\beta_{9}$$ $$=$$ $$( - 35\!\cdots\!60 \nu^{17} + \cdots - 24\!\cdots\!24 \nu ) / 11\!\cdots\!48$$ (-351294310301276860*v^17 - 39341354801072342995*v^15 - 1765201224467671817689*v^13 - 40326974856989520487915*v^11 - 490667738706187857844865*v^9 - 2997036285041392095830743*v^7 - 7358110153966373216362609*v^5 - 2279231692555537478384199*v^3 - 24951778139040599323224*v) / 118264428520247265048 $$\beta_{10}$$ $$=$$ $$( 98\!\cdots\!55 \nu^{17} + \cdots + 47\!\cdots\!74 \nu ) / 11\!\cdots\!48$$ (983703339022442455*v^17 + 110161189351206053410*v^15 + 4942569832940758036828*v^13 + 112906948451625313607500*v^11 + 1373574158454111751451624*v^9 + 8387406494434905735659440*v^7 + 20573829758624817568974100*v^5 + 6307068476565465585596775*v^3 + 47261222379747912174174*v) / 118264428520247265048 $$\beta_{11}$$ $$=$$ $$( - 15\!\cdots\!78 \nu^{17} + \cdots - 99\!\cdots\!97 \nu ) / 11\!\cdots\!48$$ (-1571948875555483478*v^17 - 176040047976219790778*v^15 - 7898572866631670792183*v^13 - 180441933287432272993493*v^11 - 2195364852078138476531287*v^9 - 13407972960160536941644949*v^7 - 32907476482060537655524259*v^5 - 10155196761284993270910147*v^3 - 99536642989191731003697*v) / 118264428520247265048 $$\beta_{12}$$ $$=$$ $$( - 53\!\cdots\!73 \nu^{17} + \cdots - 44\!\cdots\!99 \nu ) / 39\!\cdots\!16$$ (-535048544355919573*v^17 - 59921178114515677318*v^15 - 2688677052085443470617*v^13 - 61427319708209783608723*v^11 - 747466609147348852486949*v^9 - 4566437600792833056717175*v^7 - 11217433878414679404604189*v^5 - 3496859371178284642113396*v^3 - 44429073271077606612399*v) / 39421476173415755016 $$\beta_{13}$$ $$=$$ $$( - 19\!\cdots\!02 \nu^{17} + \cdots - 14\!\cdots\!49 \nu ) / 13\!\cdots\!72$$ (-199825963018042202*v^17 - 22378415796312586260*v^15 - 1004091133388838528645*v^13 - 22938911069422955677215*v^11 - 279101082885132292202333*v^9 - 1704744450192583427024935*v^7 - 4185171196258535895870177*v^5 - 1295716017016055651501761*v^3 - 14126671221018042266649*v) / 13140492057805251672 $$\beta_{14}$$ $$=$$ $$( - 18\!\cdots\!32 \nu^{17} + \cdots - 11\!\cdots\!73 \nu ) / 11\!\cdots\!48$$ (-1894742639433313732*v^17 - 212189660851318789837*v^15 - 9520568872239458397820*v^13 - 217497551368298832831436*v^11 - 2646235448390013693745460*v^9 - 16162005282268721057812336*v^7 - 39669436928059992423501136*v^5 - 12250163265923155307477616*v^3 - 117854530825576418762373*v) / 118264428520247265048 $$\beta_{15}$$ $$=$$ $$( - 21\!\cdots\!56 \nu^{17} + \cdots - 15\!\cdots\!25 \nu ) / 11\!\cdots\!48$$ (-2183680095349438856*v^17 - 244550446460167690205*v^15 - 10972742991526896374708*v^13 - 250679917778232872609636*v^11 - 3050117193501780762086452*v^9 - 18630808445116862226400760*v^7 - 45744322168224785295664520*v^5 - 14181026290366663812397572*v^3 - 156186936516528005836725*v) / 118264428520247265048 $$\beta_{16}$$ $$=$$ $$( 94\!\cdots\!88 \nu^{17} + \cdots + 63\!\cdots\!22 \nu ) / 39\!\cdots\!16$$ (949112438926442588*v^17 + 106290075697754159453*v^15 + 4769061782192990779361*v^13 + 108949916379354111308963*v^11 + 1325578663983348867970777*v^9 + 8096217599600815029990719*v^7 + 19873518927273929084242097*v^5 + 6143032929765195340399791*v^3 + 63973440918298125700722*v) / 39421476173415755016 $$\beta_{17}$$ $$=$$ $$( 32\!\cdots\!29 \nu^{17} + \cdots + 21\!\cdots\!19 \nu ) / 11\!\cdots\!48$$ (3266697595312124629*v^17 + 365833804227819620518*v^15 + 16414355208445643925721*v^13 + 374988006387626869843771*v^11 + 4562419570010159402832821*v^9 + 27865711783016489158493239*v^7 + 68400103572511201807276405*v^5 + 21138611739620106672143964*v^3 + 214845478818258882970719*v) / 118264428520247265048
 $$\nu$$ $$=$$ $$( -\beta_{16} + \beta_{12} - \beta_{11} - \beta_{10} - 11\beta_{9} ) / 13$$ (-b16 + b12 - b11 - b10 - 11*b9) / 13 $$\nu^{2}$$ $$=$$ $$( 4\beta_{8} - 2\beta_{7} + 3\beta_{5} - 4\beta_{4} + 4\beta_{3} + 13\beta_{2} - 6\beta _1 - 164 ) / 13$$ (4*b8 - 2*b7 + 3*b5 - 4*b4 + 4*b3 + 13*b2 - 6*b1 - 164) / 13 $$\nu^{3}$$ $$=$$ $$( - 12 \beta_{17} + 38 \beta_{16} + 5 \beta_{15} + \beta_{14} + \beta_{13} - 28 \beta_{12} + \cdots + 218 \beta_{9} ) / 13$$ (-12*b17 + 38*b16 + 5*b15 + b14 + b13 - 28*b12 + 37*b11 + 36*b10 + 218*b9) / 13 $$\nu^{4}$$ $$=$$ $$( - 133 \beta_{8} + 86 \beta_{7} + 25 \beta_{6} - 119 \beta_{5} + 112 \beta_{4} - 134 \beta_{3} + \cdots + 3635 ) / 13$$ (-133*b8 + 86*b7 + 25*b6 - 119*b5 + 112*b4 - 134*b3 - 311*b2 + 172*b1 + 3635) / 13 $$\nu^{5}$$ $$=$$ $$( 471 \beta_{17} - 1120 \beta_{16} - 108 \beta_{15} + 29 \beta_{14} - 127 \beta_{13} + \cdots - 4973 \beta_{9} ) / 13$$ (471*b17 - 1120*b16 - 108*b15 + 29*b14 - 127*b13 + 718*b12 - 1029*b11 - 987*b10 - 4973*b9) / 13 $$\nu^{6}$$ $$=$$ $$( 3951 \beta_{8} - 2869 \beta_{7} - 881 \beta_{6} + 3542 \beta_{5} - 2825 \beta_{4} + 4097 \beta_{3} + \cdots - 88251 ) / 13$$ (3951*b8 - 2869*b7 - 881*b6 + 3542*b5 - 2825*b4 + 4097*b3 + 7797*b2 - 4414*b1 - 88251) / 13 $$\nu^{7}$$ $$=$$ $$( - 15422 \beta_{17} + 31613 \beta_{16} + 830 \beta_{15} - 1824 \beta_{14} + 5066 \beta_{13} + \cdots + 122423 \beta_{9} ) / 13$$ (-15422*b17 + 31613*b16 + 830*b15 - 1824*b14 + 5066*b13 - 17616*b12 + 27096*b11 + 25574*b10 + 122423*b9) / 13 $$\nu^{8}$$ $$=$$ $$( - 112659 \beta_{8} + 87424 \beta_{7} + 26613 \beta_{6} - 101131 \beta_{5} + 69455 \beta_{4} + \cdots + 2241983 ) / 13$$ (-112659*b8 + 87424*b7 + 26613*b6 - 101131*b5 + 69455*b4 - 121120*b3 - 203673*b2 + 110654*b1 + 2241983) / 13 $$\nu^{9}$$ $$=$$ $$( 472480 \beta_{17} - 880613 \beta_{16} + 49184 \beta_{15} + 66984 \beta_{14} - 164273 \beta_{13} + \cdots - 3144277 \beta_{9} ) / 13$$ (472480*b17 - 880613*b16 + 49184*b15 + 66984*b14 - 164273*b13 + 422747*b12 - 715006*b11 - 657831*b10 - 3144277*b9) / 13 $$\nu^{10}$$ $$=$$ $$( 3151008 \beta_{8} - 2549244 \beta_{7} - 780149 \beta_{6} + 2870585 \beta_{5} - 1683812 \beta_{4} + \cdots - 58513476 ) / 13$$ (3151008*b8 - 2549244*b7 - 780149*b6 + 2870585*b5 - 1683812*b4 + 3534216*b3 + 5471940*b2 - 2772413*b1 - 58513476) / 13 $$\nu^{11}$$ $$=$$ $$( - 13985706 \beta_{17} + 24482374 \beta_{16} - 3433113 \beta_{15} - 2071596 \beta_{14} + \cdots + 82894537 \beta_{9} ) / 13$$ (-13985706*b17 + 24482374*b16 - 3433113*b15 - 2071596*b14 + 4997674*b13 - 10008957*b12 + 19183436*b11 + 17030993*b10 + 82894537*b9) / 13 $$\nu^{12}$$ $$=$$ $$- 6717553 \beta_{8} + 5587839 \beta_{7} + 1747024 \beta_{6} - 6267832 \beta_{5} + 3091124 \beta_{4} + \cdots + 119635062$$ -6717553*b8 + 5587839*b7 + 1747024*b6 - 6267832*b5 + 3091124*b4 - 7887469*b3 - 11521438*b2 + 5376733*b1 + 119635062 $$\nu^{13}$$ $$=$$ $$( 405866305 \beta_{17} - 682002933 \beta_{16} + 151906638 \beta_{15} + 58422208 \beta_{14} + \cdots - 2224629301 \beta_{9} ) / 13$$ (405866305*b17 - 682002933*b16 + 151906638*b15 + 58422208*b14 - 148945251*b13 + 234290681*b12 - 523892539*b11 - 445631226*b10 - 2224629301*b9) / 13 $$\nu^{14}$$ $$=$$ $$( 2411900560 \beta_{8} - 2044996748 \beta_{7} - 660895040 \beta_{6} + 2314331308 \beta_{5} + \cdots - 41900390969 ) / 13$$ (2411900560*b8 - 2044996748*b7 - 660895040*b6 + 2314331308*b5 - 938935300*b4 + 2966243492*b3 + 4152803408*b2 - 1776487056*b1 - 41900390969) / 13 $$\nu^{15}$$ $$=$$ $$( - 11641827508 \beta_{17} + 19058011049 \beta_{16} - 5749856744 \beta_{15} + \cdots + 60489950895 \beta_{9} ) / 13$$ (-11641827508*b17 + 19058011049*b16 - 5749856744*b15 - 1543210564*b14 + 4412106180*b13 - 5414805541*b12 + 14520351545*b11 + 11796507681*b10 + 60489950895*b9) / 13 $$\nu^{16}$$ $$=$$ $$( - 66610447868 \beta_{8} + 57224896958 \beta_{7} + 19271064448 \beta_{6} - 65768623299 \beta_{5} + \cdots + 1140922643492 ) / 13$$ (-66610447868*b8 + 57224896958*b7 + 19271064448*b6 - 65768623299*b5 + 21293864236*b4 - 85666003544*b3 - 116220339065*b2 + 45522752382*b1 + 1140922643492) / 13 $$\nu^{17}$$ $$=$$ $$( 331693568820 \beta_{17} - 534304309710 \beta_{16} + 201173682407 \beta_{15} + \cdots - 1661682958306 \beta_{9} ) / 13$$ (331693568820*b17 - 534304309710*b16 + 201173682407*b15 + 38458587547*b14 - 130512482617*b13 + 123072569376*b12 - 406993842361*b11 - 315859269400*b10 - 1661682958306*b9) / 13

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$$.

 $$n$$ $$170$$ $$340$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 − 3.27560i − 5.39246i − 4.14324i − 4.83218i − 2.37150i − 5.06791i − 0.100291i 0.107680i 0.588238i − 0.588238i − 0.107680i 0.100291i 5.06791i 2.37150i 4.83218i 4.14324i 5.39246i 3.27560i
5.52257i 3.00000 −22.4988 6.08065i 16.5677i 20.2718i 80.0709i 9.00000 33.5808
337.2 4.83750i 3.00000 −15.4014 21.1983i 14.5125i 16.2806i 35.8043i 9.00000 102.547
337.3 4.69820i 3.00000 −14.0731 4.47249i 14.0946i 27.2096i 28.5326i 9.00000 −21.0127
337.4 4.03025i 3.00000 −8.24289 8.08864i 12.0907i 5.95078i 0.978887i 9.00000 −32.5992
337.5 3.17344i 3.00000 −2.07074 6.74147i 9.52033i 14.1726i 18.8162i 9.00000 21.3937
337.6 2.82093i 3.00000 0.0423641 3.41089i 8.46278i 13.3442i 22.6869i 9.00000 −9.62187
337.7 2.34727i 3.00000 2.49032 15.3991i 7.04181i 10.1317i 24.6236i 9.00000 −36.1458
337.8 0.447278i 3.00000 7.79994 1.93073i 1.34183i 8.14537i 7.06697i 9.00000 −0.863573
337.9 0.213700i 3.00000 7.95433 15.3391i 0.641100i 32.3928i 3.40944i 9.00000 −3.27797
337.10 0.213700i 3.00000 7.95433 15.3391i 0.641100i 32.3928i 3.40944i 9.00000 −3.27797
337.11 0.447278i 3.00000 7.79994 1.93073i 1.34183i 8.14537i 7.06697i 9.00000 −0.863573
337.12 2.34727i 3.00000 2.49032 15.3991i 7.04181i 10.1317i 24.6236i 9.00000 −36.1458
337.13 2.82093i 3.00000 0.0423641 3.41089i 8.46278i 13.3442i 22.6869i 9.00000 −9.62187
337.14 3.17344i 3.00000 −2.07074 6.74147i 9.52033i 14.1726i 18.8162i 9.00000 21.3937
337.15 4.03025i 3.00000 −8.24289 8.08864i 12.0907i 5.95078i 0.978887i 9.00000 −32.5992
337.16 4.69820i 3.00000 −14.0731 4.47249i 14.0946i 27.2096i 28.5326i 9.00000 −21.0127
337.17 4.83750i 3.00000 −15.4014 21.1983i 14.5125i 16.2806i 35.8043i 9.00000 102.547
337.18 5.52257i 3.00000 −22.4988 6.08065i 16.5677i 20.2718i 80.0709i 9.00000 33.5808
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 337.18 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.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.4.b.k 18
13.b even 2 1 inner 507.4.b.k 18
13.d odd 4 1 507.4.a.o 9
13.d odd 4 1 507.4.a.p yes 9
39.f even 4 1 1521.4.a.bf 9
39.f even 4 1 1521.4.a.bi 9

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.4.a.o 9 13.d odd 4 1
507.4.a.p yes 9 13.d odd 4 1
507.4.b.k 18 1.a even 1 1 trivial
507.4.b.k 18 13.b even 2 1 inner
1521.4.a.bf 9 39.f even 4 1
1521.4.a.bi 9 39.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(507, [\chi])$$:

 $$T_{2}^{18} + 116 T_{2}^{16} + 5516 T_{2}^{14} + 138863 T_{2}^{12} + 1992090 T_{2}^{10} + 16267053 T_{2}^{8} + \cdots + 1032256$$ T2^18 + 116*T2^16 + 5516*T2^14 + 138863*T2^12 + 1992090*T2^10 + 16267053*T2^8 + 70428381*T2^6 + 129478420*T2^4 + 28371296*T2^2 + 1032256 $$T_{5}^{18} + 1105 T_{5}^{16} + 449620 T_{5}^{14} + 86264149 T_{5}^{12} + 8372407387 T_{5}^{10} + \cdots + 23\!\cdots\!01$$ T5^18 + 1105*T5^16 + 449620*T5^14 + 86264149*T5^12 + 8372407387*T5^10 + 433913382945*T5^8 + 12132347525883*T5^6 + 174948957367794*T5^4 + 1145975316696272*T5^2 + 2391268778161201

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{18} + 116 T^{16} + \cdots + 1032256$$
$3$ $$(T - 3)^{18}$$
$5$ $$T^{18} + \cdots + 23\!\cdots\!01$$
$7$ $$T^{18} + \cdots + 72\!\cdots\!21$$
$11$ $$T^{18} + \cdots + 16\!\cdots\!21$$
$13$ $$T^{18}$$
$17$ $$(T^{9} + \cdots + 25\!\cdots\!76)^{2}$$
$19$ $$T^{18} + \cdots + 30\!\cdots\!76$$
$23$ $$(T^{9} + \cdots - 60\!\cdots\!44)^{2}$$
$29$ $$(T^{9} + \cdots + 34\!\cdots\!27)^{2}$$
$31$ $$T^{18} + \cdots + 16\!\cdots\!09$$
$37$ $$T^{18} + \cdots + 64\!\cdots\!36$$
$41$ $$T^{18} + \cdots + 13\!\cdots\!24$$
$43$ $$(T^{9} + \cdots - 31\!\cdots\!88)^{2}$$
$47$ $$T^{18} + \cdots + 14\!\cdots\!44$$
$53$ $$(T^{9} + \cdots + 65\!\cdots\!51)^{2}$$
$59$ $$T^{18} + \cdots + 21\!\cdots\!76$$
$61$ $$(T^{9} + \cdots - 61\!\cdots\!68)^{2}$$
$67$ $$T^{18} + \cdots + 66\!\cdots\!64$$
$71$ $$T^{18} + \cdots + 51\!\cdots\!44$$
$73$ $$T^{18} + \cdots + 37\!\cdots\!29$$
$79$ $$(T^{9} + \cdots + 11\!\cdots\!83)^{2}$$
$83$ $$T^{18} + \cdots + 40\!\cdots\!09$$
$89$ $$T^{18} + \cdots + 30\!\cdots\!84$$
$97$ $$T^{18} + \cdots + 99\!\cdots\!21$$