# Properties

 Label 637.2.q.g Level $637$ Weight $2$ Character orbit 637.q Analytic conductor $5.086$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [637,2,Mod(491,637)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(637, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 5]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("637.491");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 637.q (of order $$6$$, degree $$2$$, 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: $$5.08647060876$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: 12.0.2346760387617129.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - 3 x^{11} + x^{10} + 10 x^{9} - 15 x^{8} - 10 x^{7} + 45 x^{6} - 20 x^{5} - 60 x^{4} + 80 x^{3} + 16 x^{2} - 96 x + 64$$ x^12 - 3*x^11 + x^10 + 10*x^9 - 15*x^8 - 10*x^7 + 45*x^6 - 20*x^5 - 60*x^4 + 80*x^3 + 16*x^2 - 96*x + 64 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{10} q^{2} + (\beta_{6} + \beta_{4} + \beta_1) q^{3} + (\beta_{11} - \beta_{7} + \beta_{4} + \beta_1) q^{4} + ( - \beta_{9} + \beta_{6} + \beta_{3} + \beta_1) q^{5} + (\beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} - \beta_{2} - 1) q^{6} + ( - \beta_{11} + \beta_{9} - \beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{8} + ( - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} - 1) q^{9}+O(q^{10})$$ q + b10 * q^2 + (b6 + b4 + b1) * q^3 + (b11 - b7 + b4 + b1) * q^4 + (-b9 + b6 + b3 + b1) * q^5 + (b7 - b6 + b5 - b3 - b2 - 1) * q^6 + (-b11 + b9 - b7 + b5 + b4 + b3 + b2 + b1) * q^8 + (-b7 - b6 + b5 - b3 + b2 - 1) * q^9 $$q + \beta_{10} q^{2} + (\beta_{6} + \beta_{4} + \beta_1) q^{3} + (\beta_{11} - \beta_{7} + \beta_{4} + \beta_1) q^{4} + ( - \beta_{9} + \beta_{6} + \beta_{3} + \beta_1) q^{5} + (\beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} - \beta_{2} - 1) q^{6} + ( - \beta_{11} + \beta_{9} - \beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{8} + ( - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} - 1) q^{9} + (\beta_{8} - \beta_{6} - 2 \beta_{4} - \beta_{3} - \beta_1) q^{10} + ( - \beta_{10} + \beta_{8} + \beta_{5} + \beta_{2} - \beta_1 - 1) q^{11} + (\beta_{11} - 2 \beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + \cdots + 1) q^{12}+ \cdots + (2 \beta_{11} + 4 \beta_{8} + 2 \beta_{7} + 4 \beta_{6} - 2 \beta_{5} + 8 \beta_{4} + \cdots + 5) q^{99}+O(q^{100})$$ q + b10 * q^2 + (b6 + b4 + b1) * q^3 + (b11 - b7 + b4 + b1) * q^4 + (-b9 + b6 + b3 + b1) * q^5 + (b7 - b6 + b5 - b3 - b2 - 1) * q^6 + (-b11 + b9 - b7 + b5 + b4 + b3 + b2 + b1) * q^8 + (-b7 - b6 + b5 - b3 + b2 - 1) * q^9 + (b8 - b6 - 2*b4 - b3 - b1) * q^10 + (-b10 + b8 + b5 + b2 - b1 - 1) * q^11 + (b11 - 2*b10 + b9 + b8 - b7 + b6 - b5 + b4 + b3 + b2 - b1 + 1) * q^12 + (-b11 + b10 - b9 - 2*b6 + 2*b5 - b4 - b3 - b2 + 2*b1 - 2) * q^13 + (2*b11 + b8 - 2*b7 + 2*b6 - b5 + b4 + b2 - b1 - 1) * q^15 + (-b11 - b10 + 2*b9 - b7 - 2*b6 + b5 - b1 - 1) * q^16 + (-b11 - b8 + b7 - b6 + 2*b4 + b3 + b1 + 3) * q^17 + (-b11 + b9 - b8 - b7 - b6 + b5 - b4 + b3 + b2 + b1 - 1) * q^18 + (b11 - b10 + b9 - b8 + b7 + b6 + b4) * q^19 + (b11 + b10 - b9 + 2*b8 - b6 - b5 + b4 + b2 - 2*b1 + 1) * q^20 + (-b10 - b9 - b8 - b6 - 3*b4 - b3 - 3) * q^22 + (b11 + b10 - 2*b9 - b8 + b7 + b5 - b4 - b3 - 2*b2 + b1 - 1) * q^23 + (b10 + 2*b6 - b5 - b4 + b3 - b2 - b1 - 1) * q^24 + (2*b11 - 2*b7 + 3*b6 - 2*b5 + 2*b4 + b3 + 2*b2 - b1 + 2) * q^25 + (b11 - b10 + 2*b9 - b8 - 3*b7 + b4 - b3 + 2*b2 + b1) * q^26 + (-b11 - 2*b8 - b7 + b6 - b5 - b4 - b2 + 2*b1) * q^27 + (2*b11 - b8 + 2*b7 + b6 - 3*b5 + b4 + 2*b3 + b2 - 2*b1 + 3) * q^29 + (-b11 + b8 + b7 + b6 + 2*b4 + 3*b3 + b1 + 3) * q^30 + (b11 - 2*b9 - b8 + b7 - 2*b6 - b5 - 3*b4 - 2*b3 - b2 - 2*b1 - 1) * q^31 + (-b10 + b9 - b8 + b7 + b5 + b3 - b2 - b1 - 1) * q^32 + (-b11 + b10 - b9 + b8 - 2*b7 - b5 - b4 + 3*b3 + b2 - 2*b1 + 1) * q^33 + (3*b9 - b7 - 2*b6 + b5 - 2*b4 - 2*b3 - b1 - 2) * q^34 + (-b10 + 2*b9 - 2*b8 + b5 + b4 + b3 - b2 + b1 - 1) * q^36 + (2*b11 - 3*b10 + 2*b8 - 2*b7 + 2*b6 - b5 + b4 + b3 + b2 - b1 - 1) * q^37 + (-b11 - 2*b10 + b9 - b8 + b7 - b6 + b5 - b4 - b3 - b2 + b1 - 4) * q^38 + (b11 - 2*b9 - 2*b7 + 4*b6 - b5 + b4 + 4*b3 + b1 + 1) * q^39 + (3*b11 + 2*b10 - b9 - 2*b7 + 2*b6 - 2*b5 + 3*b4 - b3 + 3*b2 + 1) * q^40 + (-2*b11 + b10 - b8 + 2*b7 + b6 + 2*b5 - b4 - b3 - 3*b1) * q^41 + (2*b11 - 2*b10 - 2*b9 + 3*b8 - 2*b7 + 3*b6 + 4*b4 + b3 + 2) * q^43 + (-b8 + 3*b7 - 3*b5 - 4*b4 + b3 - 2*b1 + 1) * q^44 + (b11 + b8 + b7 - b6 - b3 - b1 + 1) * q^45 + (b11 - 2*b10 + 2*b9 + b8 - 2*b7 + 2*b6 - 3*b5 + b4 + 3*b2 + b1 + 3) * q^46 + (-b8 - 2*b4 + b3 + b1 - 1) * q^47 + (-b11 - b8 + 3*b7 + b6 - 2*b5 + 2*b4 + 2*b3 - 2*b2 + 5) * q^48 + (-2*b11 + 2*b10 + 2*b8 + 2*b7 - b6 + b5 + 3*b3 - b2 + 3) * q^50 + (2*b10 - b9 - 5*b8 - b7 + 3*b6 - b5 - 2*b3 + 3*b1 - 2) * q^51 + (-b11 + 3*b9 + 2*b8 - b7 + 3*b5 + 3*b4 + 2*b3 + b2 - 2*b1 - 2) * q^52 + (2*b11 + 3*b8 - 2*b7 + b6 - 2*b5 + 2*b4 + 2*b3 + 2*b2 - 2*b1 + 3) * q^53 + (-3*b11 + b10 + b8 + 3*b7 + b6 + b5 - 2*b4 + 3*b3 - 2*b2 - 2*b1 + 1) * q^54 + (b11 - b10 + 2*b9 - b8 + b7 + b6 - 2*b5 + 3*b4 + 2*b3 + b2 - b1 + 2) * q^55 + (-2*b11 - b9 - 2*b7 - 3*b6 + 2*b5 - 4*b4 - b3 + 2*b2 - b1 - 3) * q^57 + (b11 + b10 - b9 + 3*b8 - 2*b6 - b5 - 2*b3 + b2 - b1 + 2) * q^58 + (4*b10 - 4*b9 - b8 + b6 + 3*b4 - 4*b3 + 5*b1 - 3) * q^59 + (b9 - 2*b8 + b7 - 2*b6 - b5 + 2*b4 - b1 + 2) * q^60 + (-2*b11 + b8 - b6 + 2*b5 + b4 + b3 + 2*b2 + 1) * q^61 + (b10 - 2*b9 + 3*b8 + 3*b6 - 5*b5 - 3*b4 + 2*b3 + 5*b2 - 2*b1 + 5) * q^62 + (b11 - 2*b10 + b9 - 4*b8 - b7 + 2*b6 - b5 + b4 - 3*b3 + b2 + 3*b1 - 1) * q^64 + (b11 - 2*b9 + 4*b8 - b7 + 4*b6 - b5 + 5*b4 + 4*b3 + 3*b2 + 2) * q^65 + (2*b8 - 2*b7 + 2*b6 - 2*b5 + 4*b3 - 2*b1 + 7) * q^66 + (-2*b11 - b10 - 5*b8 + 2*b7 - b6 + b5 - 3*b4 - 4*b3 - b2 - 3) * q^67 + (-b11 - 2*b8 - b7 + 4*b5 - b3 - 3*b2 + 4*b1 - 4) * q^68 + (2*b11 - 2*b10 - 2*b9 + 2*b8 - 3*b7 + b6 + b5 + 3*b4 + b2 + b1) * q^69 + (-3*b10 + 3*b9 + 2*b8 - 2*b6 - b4 - 3*b3 + b1 + 1) * q^71 + (-b11 - 4*b10 + 4*b9 - b8 + b7 - b6 + 2*b5 + 2*b4 - 2*b3 - 2*b2 + 2*b1 - 5) * q^72 + (b9 - 2*b7 + 2*b6 + 2*b5 - 4*b4 + 2*b3 + 4*b1 - 4) * q^73 + (-3*b11 + b8 + 3*b7 + b6 - 6*b4 + 2*b3 - 2*b1 - 3) * q^74 + (b11 - 2*b10 + 4*b9 + 3*b8 + b7 - 3*b6 - 4*b3 - b2 - 3*b1) * q^75 + (-2*b11 - 2*b10 - 2*b8 + 2*b7 - b5 - 5*b4 + b3 - 3*b2 + b1 - 5) * q^76 + (-2*b11 - 2*b10 + 3*b9 + 3*b8 + b6 - b5 - 3*b4 + 3*b3 - b2 - 2*b1 + 4) * q^78 + (-2*b10 + b9 - 3*b8 - b7 + 4*b6 - b5 + b3 + 6) * q^79 + (-2*b11 + 2*b10 - b8 + 2*b7 - 4*b6 + 2*b5 + 2*b4 - b3 + 2*b1 + 6) * q^80 + (-b11 - b10 + 2*b9 + b8 - b7 - b6 + 3*b5 - 3*b4 - 3*b3 - 2*b2 + 2*b1 - 3) * q^81 + (2*b11 - 4*b10 - 4*b9 - 2*b8 + b7 + b6 - 3*b5 - 3*b2 + b1 + 1) * q^82 + (3*b11 - 3*b9 + b8 - 2*b7 + 3*b6 + 2*b5 + b4 - b3 - 3*b2 + 4*b1 - 3) * q^83 + (5*b10 - 5*b9 - 6*b8 - b7 + 7*b6 - b5 + 3*b4 + b2 + 7*b1 - 2) * q^85 + (-b11 + 3*b9 + 2*b8 + 2*b7 + 2*b6 - 2*b5 - 5*b4 + b3 + b2 - 2*b1) * q^86 + (-b11 + b10 + b9 - 3*b8 - 4*b6 + b5 - 5*b3 + b2 - 2*b1) * q^87 + (2*b11 + 2*b10 - 4*b9 + 3*b8 + 2*b7 + b6 - 2*b5 + 5*b4 - 3*b3 - b1 + 2) * q^88 + (-3*b11 + 4*b10 - 3*b8 + 3*b7 - 2*b6 + 2*b5 - 5*b4 - 2*b3 - b2 - 6) * q^89 + (2*b11 + 2*b10 - b9 - b7 - b5 + 2*b4 - 2*b3 + 2*b2 + b1) * q^90 + (-b11 + 6*b10 - 3*b9 + 3*b8 + 2*b7 - 3*b6 + 2*b5 - b4 + b3 - b2 - 2*b1 - 5) * q^92 + (-b10 - b5 + 4*b4 + b3 - b2 + b1 + 9) * q^93 + (-b11 - b7 - b6 + b5 - b4 - 1) * q^94 + (b11 + b10 + b9 - b8 - b5 - b2 + b1) * q^95 + (-2*b9 - b6 + 2*b4 - b3 - b1 + 1) * q^96 + (-2*b11 - 2*b10 + 2*b9 - 2*b8 - b7 + b6 + b5 - 2*b4 + 3*b3 - b2 - 1) * q^97 + (2*b11 + 4*b8 + 2*b7 + 4*b6 - 2*b5 + 8*b4 - 2*b3 - 2*b2 - 2*b1 + 5) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 3 q^{3} + 4 q^{4} - 9 q^{6} - q^{9}+O(q^{10})$$ 12 * q - 3 * q^3 + 4 * q^4 - 9 * q^6 - q^9 $$12 q - 3 q^{3} + 4 q^{4} - 9 q^{6} - q^{9} + 12 q^{10} - 12 q^{11} + 2 q^{12} - 2 q^{13} - 12 q^{15} - 8 q^{16} + 17 q^{17} - 9 q^{19} - 3 q^{20} - 15 q^{22} + 3 q^{23} - 15 q^{24} + 10 q^{25} + 15 q^{26} + 12 q^{27} - q^{29} + 11 q^{30} - 18 q^{32} + 6 q^{33} - 13 q^{36} - 15 q^{37} - 38 q^{38} + 5 q^{39} + 2 q^{40} - 6 q^{41} + 11 q^{43} + 9 q^{45} + 30 q^{46} + 19 q^{48} + 18 q^{50} - 8 q^{51} - 40 q^{52} + 16 q^{53} - 6 q^{54} - 15 q^{55} + 24 q^{58} - 27 q^{59} + 5 q^{61} + 41 q^{62} + 2 q^{64} - 18 q^{65} + 68 q^{66} - 15 q^{67} - 11 q^{68} + 7 q^{69} + 30 q^{71} - 57 q^{72} - 33 q^{74} + q^{75} - 45 q^{76} + 44 q^{78} + 70 q^{79} + 63 q^{80} + 14 q^{81} + 5 q^{82} - 21 q^{85} + 10 q^{87} - 14 q^{88} - 48 q^{89} - 66 q^{92} + 81 q^{93} + q^{94} + 2 q^{95} - 3 q^{97}+O(q^{100})$$ 12 * q - 3 * q^3 + 4 * q^4 - 9 * q^6 - q^9 + 12 * q^10 - 12 * q^11 + 2 * q^12 - 2 * q^13 - 12 * q^15 - 8 * q^16 + 17 * q^17 - 9 * q^19 - 3 * q^20 - 15 * q^22 + 3 * q^23 - 15 * q^24 + 10 * q^25 + 15 * q^26 + 12 * q^27 - q^29 + 11 * q^30 - 18 * q^32 + 6 * q^33 - 13 * q^36 - 15 * q^37 - 38 * q^38 + 5 * q^39 + 2 * q^40 - 6 * q^41 + 11 * q^43 + 9 * q^45 + 30 * q^46 + 19 * q^48 + 18 * q^50 - 8 * q^51 - 40 * q^52 + 16 * q^53 - 6 * q^54 - 15 * q^55 + 24 * q^58 - 27 * q^59 + 5 * q^61 + 41 * q^62 + 2 * q^64 - 18 * q^65 + 68 * q^66 - 15 * q^67 - 11 * q^68 + 7 * q^69 + 30 * q^71 - 57 * q^72 - 33 * q^74 + q^75 - 45 * q^76 + 44 * q^78 + 70 * q^79 + 63 * q^80 + 14 * q^81 + 5 * q^82 - 21 * q^85 + 10 * q^87 - 14 * q^88 - 48 * q^89 - 66 * q^92 + 81 * q^93 + q^94 + 2 * q^95 - 3 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 3 x^{11} + x^{10} + 10 x^{9} - 15 x^{8} - 10 x^{7} + 45 x^{6} - 20 x^{5} - 60 x^{4} + 80 x^{3} + 16 x^{2} - 96 x + 64$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{11} - 13 \nu^{10} - 9 \nu^{9} + 72 \nu^{8} - 91 \nu^{7} - 164 \nu^{6} + 313 \nu^{5} + 42 \nu^{4} - 620 \nu^{3} + 344 \nu^{2} + 608 \nu - 800 ) / 224$$ (v^11 - 13*v^10 - 9*v^9 + 72*v^8 - 91*v^7 - 164*v^6 + 313*v^5 + 42*v^4 - 620*v^3 + 344*v^2 + 608*v - 800) / 224 $$\beta_{3}$$ $$=$$ $$( - 9 \nu^{11} + 5 \nu^{10} + 25 \nu^{9} - 32 \nu^{8} - 21 \nu^{7} + 132 \nu^{6} - 73 \nu^{5} - 154 \nu^{4} + 260 \nu^{3} + 40 \nu^{2} - 320 \nu + 256 ) / 224$$ (-9*v^11 + 5*v^10 + 25*v^9 - 32*v^8 - 21*v^7 + 132*v^6 - 73*v^5 - 154*v^4 + 260*v^3 + 40*v^2 - 320*v + 256) / 224 $$\beta_{4}$$ $$=$$ $$( - 11 \nu^{11} + 17 \nu^{10} + 29 \nu^{9} - 78 \nu^{8} + 21 \nu^{7} + 166 \nu^{6} - 167 \nu^{5} - 140 \nu^{4} + 380 \nu^{3} - 88 \nu^{2} - 304 \nu + 288 ) / 224$$ (-11*v^11 + 17*v^10 + 29*v^9 - 78*v^8 + 21*v^7 + 166*v^6 - 167*v^5 - 140*v^4 + 380*v^3 - 88*v^2 - 304*v + 288) / 224 $$\beta_{5}$$ $$=$$ $$( - 13 \nu^{11} + 29 \nu^{10} + 5 \nu^{9} - 96 \nu^{8} + 91 \nu^{7} + 200 \nu^{6} - 289 \nu^{5} - 126 \nu^{4} + 584 \nu^{3} - 160 \nu^{2} - 512 \nu + 544 ) / 224$$ (-13*v^11 + 29*v^10 + 5*v^9 - 96*v^8 + 91*v^7 + 200*v^6 - 289*v^5 - 126*v^4 + 584*v^3 - 160*v^2 - 512*v + 544) / 224 $$\beta_{6}$$ $$=$$ $$( 8 \nu^{11} - 13 \nu^{10} - 9 \nu^{9} + 51 \nu^{8} - 42 \nu^{7} - 101 \nu^{6} + 194 \nu^{5} + 7 \nu^{4} - 340 \nu^{3} + 260 \nu^{2} + 216 \nu - 464 ) / 112$$ (8*v^11 - 13*v^10 - 9*v^9 + 51*v^8 - 42*v^7 - 101*v^6 + 194*v^5 + 7*v^4 - 340*v^3 + 260*v^2 + 216*v - 464) / 112 $$\beta_{7}$$ $$=$$ $$( 13 \nu^{11} - 57 \nu^{10} - 5 \nu^{9} + 208 \nu^{8} - 231 \nu^{7} - 396 \nu^{6} + 821 \nu^{5} + 42 \nu^{4} - 1452 \nu^{3} + 720 \nu^{2} + 1184 \nu - 1664 ) / 224$$ (13*v^11 - 57*v^10 - 5*v^9 + 208*v^8 - 231*v^7 - 396*v^6 + 821*v^5 + 42*v^4 - 1452*v^3 + 720*v^2 + 1184*v - 1664) / 224 $$\beta_{8}$$ $$=$$ $$( 2 \nu^{11} - 5 \nu^{10} - 4 \nu^{9} + 18 \nu^{8} - 7 \nu^{7} - 41 \nu^{6} + 45 \nu^{5} + 35 \nu^{4} - 99 \nu^{3} + 16 \nu^{2} + 96 \nu - 88 ) / 28$$ (2*v^11 - 5*v^10 - 4*v^9 + 18*v^8 - 7*v^7 - 41*v^6 + 45*v^5 + 35*v^4 - 99*v^3 + 16*v^2 + 96*v - 88) / 28 $$\beta_{9}$$ $$=$$ $$( 3 \nu^{11} - 4 \nu^{10} - 6 \nu^{9} + 20 \nu^{8} - 44 \nu^{6} + 43 \nu^{5} + 56 \nu^{4} - 82 \nu^{3} + 3 \nu^{2} + 102 \nu - 48 ) / 28$$ (3*v^11 - 4*v^10 - 6*v^9 + 20*v^8 - 44*v^6 + 43*v^5 + 56*v^4 - 82*v^3 + 3*v^2 + 102*v - 48) / 28 $$\beta_{10}$$ $$=$$ $$( - 15 \nu^{11} + 20 \nu^{10} + 30 \nu^{9} - 121 \nu^{8} + 21 \nu^{7} + 269 \nu^{6} - 271 \nu^{5} - 273 \nu^{4} + 634 \nu^{3} - 64 \nu^{2} - 664 \nu + 464 ) / 112$$ (-15*v^11 + 20*v^10 + 30*v^9 - 121*v^8 + 21*v^7 + 269*v^6 - 271*v^5 - 273*v^4 + 634*v^3 - 64*v^2 - 664*v + 464) / 112 $$\beta_{11}$$ $$=$$ $$( - 17 \nu^{11} + 39 \nu^{10} + 13 \nu^{9} - 160 \nu^{8} + 133 \nu^{7} + 310 \nu^{6} - 547 \nu^{5} - 168 \nu^{4} + 1062 \nu^{3} - 500 \nu^{2} - 872 \nu + 1056 ) / 112$$ (-17*v^11 + 39*v^10 + 13*v^9 - 160*v^8 + 133*v^7 + 310*v^6 - 547*v^5 - 168*v^4 + 1062*v^3 - 500*v^2 - 872*v + 1056) / 112
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{8} - \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + 1$$ b8 - b7 + b6 + b4 + b3 + b2 + 1 $$\nu^{3}$$ $$=$$ $$\beta_{11} + \beta_{9} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2}$$ b11 + b9 + b6 - b5 + b4 + b3 + b2 $$\nu^{4}$$ $$=$$ $$-\beta_{11} + \beta_{10} + \beta_{9} - \beta_{7} - \beta_{6} + \beta_{2} - \beta _1 - 1$$ -b11 + b10 + b9 - b7 - b6 + b2 - b1 - 1 $$\nu^{5}$$ $$=$$ $$\beta_{10} + 2\beta_{9} - 2\beta_{8} + 2\beta_{7} - \beta_{6} + \beta_{5} - 2\beta_{3} - \beta_{2} - \beta_1$$ b10 + 2*b9 - 2*b8 + 2*b7 - b6 + b5 - 2*b3 - b2 - b1 $$\nu^{6}$$ $$=$$ $$- 4 \beta_{11} + 2 \beta_{10} - 3 \beta_{8} + \beta_{7} - 5 \beta_{6} + 4 \beta_{5} - 7 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} + 3 \beta _1 - 6$$ -4*b11 + 2*b10 - 3*b8 + b7 - 5*b6 + 4*b5 - 7*b4 - 2*b3 - 4*b2 + 3*b1 - 6 $$\nu^{7}$$ $$=$$ $$- \beta_{11} - \beta_{10} - \beta_{9} + 3 \beta_{8} + \beta_{7} + \beta_{6} + 6 \beta_{5} + 4 \beta_{4} - \beta_{3} - 4 \beta_{2} + \beta_1$$ -b11 - b10 - b9 + 3*b8 + b7 + b6 + 6*b5 + 4*b4 - b3 - 4*b2 + b1 $$\nu^{8}$$ $$=$$ $$-4\beta_{10} - 2\beta_{9} - \beta_{8} + 2\beta_{5} - 4\beta_{4} + 8\beta_{3} - 2\beta_{2} + 3\beta _1 - 6$$ -4*b10 - 2*b9 - b8 + 2*b5 - 4*b4 + 8*b3 - 2*b2 + 3*b1 - 6 $$\nu^{9}$$ $$=$$ $$2 \beta_{11} - 6 \beta_{10} - 2 \beta_{9} + 6 \beta_{8} - 3 \beta_{7} + 7 \beta_{6} - 4 \beta_{5} + 21 \beta_{4} + 6 \beta_{3} - 3 \beta _1 + 4$$ 2*b11 - 6*b10 - 2*b9 + 6*b8 - 3*b7 + 7*b6 - 4*b5 + 21*b4 + 6*b3 - 3*b1 + 4 $$\nu^{10}$$ $$=$$ $$5 \beta_{11} - 9 \beta_{10} + \beta_{9} - 16 \beta_{8} + \beta_{7} + 3 \beta_{6} - 8 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 7 \beta_{2} - 6 \beta _1 + 1$$ 5*b11 - 9*b10 + b9 - 16*b8 + b7 + 3*b6 - 8*b5 + 2*b4 + 2*b3 + 7*b2 - 6*b1 + 1 $$\nu^{11}$$ $$=$$ $$- 2 \beta_{11} - \beta_{10} - 19 \beta_{8} + \beta_{7} + 4 \beta_{6} - 15 \beta_{5} - 5 \beta_{4} - 13 \beta_{3} - 14 \beta_{2} + 9 \beta _1 - 5$$ -2*b11 - b10 - 19*b8 + b7 + 4*b6 - 15*b5 - 5*b4 - 13*b3 - 14*b2 + 9*b1 - 5

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/637\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$248$$ $$\chi(n)$$ $$1 + \beta_{4}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
491.1
 1.21245 + 0.727987i 0.874681 − 1.11128i −1.18541 + 0.771231i 0.655911 + 1.25291i −1.38488 − 0.286553i 1.32725 − 0.488273i 1.21245 − 0.727987i 0.874681 + 1.11128i −1.18541 − 0.771231i 0.655911 − 1.25291i −1.38488 + 0.286553i 1.32725 + 0.488273i
−1.99469 + 1.15163i 0.736680 + 1.27597i 1.65252 2.86225i 0.847292i −2.93889 1.69677i 0 3.00585i 0.414604 0.718115i 0.975769 + 1.69008i
491.2 −1.16500 + 0.672613i −1.02505 1.77544i −0.0951832 + 0.164862i 3.56778i 2.38837 + 1.37893i 0 2.94654i −0.601462 + 1.04176i 2.39973 + 4.15646i
491.3 −0.433001 + 0.249993i −0.424801 0.735776i −0.875007 + 1.51556i 1.04248i 0.367878 + 0.212395i 0 1.87496i 1.13909 1.97296i −0.260612 0.451393i
491.4 0.156598 0.0904119i 0.913006 + 1.58137i −0.983651 + 1.70373i 2.68664i 0.285950 + 0.165093i 0 0.717383i −0.167162 + 0.289532i 0.242904 + 0.420723i
491.5 1.19430 0.689527i −1.44060 2.49520i −0.0491037 + 0.0850501i 0.805948i −3.44101 1.98667i 0 2.89354i −2.65067 + 4.59109i 0.555723 + 0.962541i
491.6 2.24179 1.29430i −0.259233 0.449005i 2.35043 4.07106i 1.61205i −1.16229 0.671051i 0 6.99143i 1.36560 2.36528i 2.08648 + 3.61389i
589.1 −1.99469 1.15163i 0.736680 1.27597i 1.65252 + 2.86225i 0.847292i −2.93889 + 1.69677i 0 3.00585i 0.414604 + 0.718115i 0.975769 1.69008i
589.2 −1.16500 0.672613i −1.02505 + 1.77544i −0.0951832 0.164862i 3.56778i 2.38837 1.37893i 0 2.94654i −0.601462 1.04176i 2.39973 4.15646i
589.3 −0.433001 0.249993i −0.424801 + 0.735776i −0.875007 1.51556i 1.04248i 0.367878 0.212395i 0 1.87496i 1.13909 + 1.97296i −0.260612 + 0.451393i
589.4 0.156598 + 0.0904119i 0.913006 1.58137i −0.983651 1.70373i 2.68664i 0.285950 0.165093i 0 0.717383i −0.167162 0.289532i 0.242904 0.420723i
589.5 1.19430 + 0.689527i −1.44060 + 2.49520i −0.0491037 0.0850501i 0.805948i −3.44101 + 1.98667i 0 2.89354i −2.65067 4.59109i 0.555723 0.962541i
589.6 2.24179 + 1.29430i −0.259233 + 0.449005i 2.35043 + 4.07106i 1.61205i −1.16229 + 0.671051i 0 6.99143i 1.36560 + 2.36528i 2.08648 3.61389i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 589.6 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.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.q.g 12
7.b odd 2 1 637.2.q.i 12
7.c even 3 1 91.2.k.b 12
7.c even 3 1 91.2.u.b yes 12
7.d odd 6 1 637.2.k.i 12
7.d odd 6 1 637.2.u.g 12
13.e even 6 1 inner 637.2.q.g 12
13.f odd 12 2 8281.2.a.cp 12
21.h odd 6 1 819.2.bm.f 12
21.h odd 6 1 819.2.do.e 12
91.k even 6 1 91.2.u.b yes 12
91.l odd 6 1 637.2.u.g 12
91.p odd 6 1 637.2.k.i 12
91.t odd 6 1 637.2.q.i 12
91.u even 6 1 91.2.k.b 12
91.x odd 12 2 1183.2.e.j 24
91.bc even 12 2 8281.2.a.co 12
91.bd odd 12 2 1183.2.e.j 24
273.x odd 6 1 819.2.bm.f 12
273.bp odd 6 1 819.2.do.e 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.k.b 12 7.c even 3 1
91.2.k.b 12 91.u even 6 1
91.2.u.b yes 12 7.c even 3 1
91.2.u.b yes 12 91.k even 6 1
637.2.k.i 12 7.d odd 6 1
637.2.k.i 12 91.p odd 6 1
637.2.q.g 12 1.a even 1 1 trivial
637.2.q.g 12 13.e even 6 1 inner
637.2.q.i 12 7.b odd 2 1
637.2.q.i 12 91.t odd 6 1
637.2.u.g 12 7.d odd 6 1
637.2.u.g 12 91.l odd 6 1
819.2.bm.f 12 21.h odd 6 1
819.2.bm.f 12 273.x odd 6 1
819.2.do.e 12 21.h odd 6 1
819.2.do.e 12 273.bp odd 6 1
1183.2.e.j 24 91.x odd 12 2
1183.2.e.j 24 91.bd odd 12 2
8281.2.a.co 12 91.bc even 12 2
8281.2.a.cp 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 $$S_{2}^{\mathrm{new}}(637, [\chi])$$:

 $$T_{2}^{12} - 8T_{2}^{10} + 52T_{2}^{8} + 18T_{2}^{7} - 91T_{2}^{6} - 36T_{2}^{5} + 130T_{2}^{4} + 72T_{2}^{3} - 6T_{2} + 1$$ T2^12 - 8*T2^10 + 52*T2^8 + 18*T2^7 - 91*T2^6 - 36*T2^5 + 130*T2^4 + 72*T2^3 - 6*T2 + 1 $$T_{3}^{12} + 3 T_{3}^{11} + 14 T_{3}^{10} + 17 T_{3}^{9} + 69 T_{3}^{8} + 75 T_{3}^{7} + 233 T_{3}^{6} + 147 T_{3}^{5} + 355 T_{3}^{4} + 300 T_{3}^{3} + 333 T_{3}^{2} + 133 T_{3} + 49$$ T3^12 + 3*T3^11 + 14*T3^10 + 17*T3^9 + 69*T3^8 + 75*T3^7 + 233*T3^6 + 147*T3^5 + 355*T3^4 + 300*T3^3 + 333*T3^2 + 133*T3 + 49

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - 8 T^{10} + 52 T^{8} + 18 T^{7} + \cdots + 1$$
$3$ $$T^{12} + 3 T^{11} + 14 T^{10} + 17 T^{9} + \cdots + 49$$
$5$ $$T^{12} + 25 T^{10} + 201 T^{8} + \cdots + 121$$
$7$ $$T^{12}$$
$11$ $$T^{12} + 12 T^{11} + 41 T^{10} + \cdots + 85849$$
$13$ $$T^{12} + 2 T^{11} - 18 T^{10} + \cdots + 4826809$$
$17$ $$T^{12} - 17 T^{11} + 193 T^{10} + \cdots + 361$$
$19$ $$T^{12} + 9 T^{11} + T^{10} - 234 T^{9} + \cdots + 1$$
$23$ $$T^{12} - 3 T^{11} + 59 T^{10} + \cdots + 628849$$
$29$ $$T^{12} + T^{11} + 87 T^{10} + \cdots + 16072081$$
$31$ $$T^{12} + 232 T^{10} + \cdots + 241274089$$
$37$ $$T^{12} + 15 T^{11} + 39 T^{10} + \cdots + 123201$$
$41$ $$T^{12} + 6 T^{11} - 159 T^{10} + \cdots + 389707081$$
$43$ $$T^{12} - 11 T^{11} + \cdots + 418898089$$
$47$ $$T^{12} + 41 T^{10} + 509 T^{8} + \cdots + 121$$
$53$ $$(T^{6} - 8 T^{5} - 38 T^{4} + 404 T^{3} + \cdots - 17)^{2}$$
$59$ $$T^{12} + 27 T^{11} + \cdots + 35582408689$$
$61$ $$T^{12} - 5 T^{11} + 100 T^{10} + \cdots + 3157729$$
$67$ $$T^{12} + 15 T^{11} + \cdots + 5708255809$$
$71$ $$T^{12} - 30 T^{11} + \cdots + 639230089$$
$73$ $$T^{12} + 400 T^{10} + \cdots + 484396081$$
$79$ $$(T^{6} - 35 T^{5} + 344 T^{4} + \cdots + 255121)^{2}$$
$83$ $$T^{12} + 463 T^{10} + \cdots + 402363481$$
$89$ $$T^{12} + 48 T^{11} + \cdots + 145033849$$
$97$ $$T^{12} + 3 T^{11} - 176 T^{10} + \cdots + 1681$$