# Properties

 Label 637.2.r.e Level $637$ Weight $2$ Character orbit 637.r Analytic conductor $5.086$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [637,2,Mod(116,637)]

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([4, 3]))

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

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

chi := DirichletCharacter("637.116");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 637.r (of order $$6$$, 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: $$5.08647060876$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: 12.0.89539436150784.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - 2x^{11} + 2x^{10} - 8x^{9} + 4x^{8} + 16x^{7} - 8x^{6} + 20x^{5} + 20x^{4} - 24x^{3} + 8x^{2} - 8x + 4$$ x^12 - 2*x^11 + 2*x^10 - 8*x^9 + 4*x^8 + 16*x^7 - 8*x^6 + 20*x^5 + 20*x^4 - 24*x^3 + 8*x^2 - 8*x + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{7} + \beta_{6} + \beta_{4} + \beta_{3}) q^{2} - \beta_{11} q^{3} + (\beta_{11} - \beta_{10} - \beta_{8} + 1) q^{4} + (\beta_{9} + \beta_{5} + \beta_{4} + \beta_{3}) q^{5} + (\beta_{9} - \beta_{7} - \beta_{4}) q^{6} + ( - 2 \beta_{9} + 2 \beta_{4}) q^{8} + ( - \beta_{11} - \beta_{10} - \beta_{2} + \beta_1) q^{9}+O(q^{10})$$ q + (b7 + b6 + b4 + b3) * q^2 - b11 * q^3 + (b11 - b10 - b8 + 1) * q^4 + (b9 + b5 + b4 + b3) * q^5 + (b9 - b7 - b4) * q^6 + (-2*b9 + 2*b4) * q^8 + (-b11 - b10 - b2 + b1) * q^9 $$q + (\beta_{7} + \beta_{6} + \beta_{4} + \beta_{3}) q^{2} - \beta_{11} q^{3} + (\beta_{11} - \beta_{10} - \beta_{8} + 1) q^{4} + (\beta_{9} + \beta_{5} + \beta_{4} + \beta_{3}) q^{5} + (\beta_{9} - \beta_{7} - \beta_{4}) q^{6} + ( - 2 \beta_{9} + 2 \beta_{4}) q^{8} + ( - \beta_{11} - \beta_{10} - \beta_{2} + \beta_1) q^{9} - \beta_{11} q^{10} + (\beta_{6} + 3 \beta_{3}) q^{11} + (2 \beta_{10} + 4 \beta_{8} + 2 \beta_{2}) q^{12} + (\beta_{7} + 2 \beta_{2} - \beta_1 + 2) q^{13} + (2 \beta_{9} + \beta_{7} + 3 \beta_{4}) q^{15} + ( - 2 \beta_{10} - 2 \beta_{8} - 2 \beta_{2}) q^{16} + ( - \beta_{11} - 2 \beta_{10} - 2 \beta_{8} + 2) q^{17} + (\beta_{6} - \beta_{3}) q^{18} + ( - 4 \beta_{9} - \beta_{7} - \beta_{6} - 4 \beta_{5}) q^{19} + ( - \beta_{9} - \beta_{7} - 3 \beta_{4}) q^{20} + (3 \beta_{2} + \beta_1 + 5) q^{22} + ( - \beta_{11} - 3 \beta_{10} - 2 \beta_{8} - 3 \beta_{2} + \beta_1) q^{23} + (2 \beta_{6} + 6 \beta_{3}) q^{24} + ( - 3 \beta_{11} - \beta_{10} + \beta_{8} - 1) q^{25} + (\beta_{11} - \beta_{9} - 2 \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} + 5 \beta_{4} + 5 \beta_{3} - \beta_1) q^{26} + ( - 2 \beta_1 - 2) q^{27} + ( - 2 \beta_{2} - 2 \beta_1 - 3) q^{29} + ( - \beta_{11} - \beta_{10} - 3 \beta_{8} - \beta_{2} + \beta_1) q^{30} + (2 \beta_{6} - \beta_{5} - 3 \beta_{3}) q^{31} + ( - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{3}) q^{32} + (3 \beta_{9} - \beta_{7} - \beta_{6} + 3 \beta_{5} - \beta_{4} - \beta_{3}) q^{33} + ( - \beta_{9} + \beta_{7} + 5 \beta_{4}) q^{34} + (\beta_{2} - \beta_1 + 1) q^{36} + (3 \beta_{7} + 3 \beta_{6} - 3 \beta_{4} - 3 \beta_{3}) q^{37} + (3 \beta_{11} - 4 \beta_{10} - 2 \beta_{8} + 2) q^{38} + ( - 3 \beta_{11} + \beta_{10} - \beta_{8} + \beta_{6} + \beta_{3} + 1) q^{39} + (2 \beta_{11} + 2 \beta_{10} + 4 \beta_{8} + 2 \beta_{2} - 2 \beta_1) q^{40} + ( - 4 \beta_{7} - 2 \beta_{4}) q^{41} + ( - 2 \beta_{2} + 2 \beta_1 - 5) q^{43} + ( - 4 \beta_{9} + 4 \beta_{7} + 4 \beta_{6} - 4 \beta_{5} + 6 \beta_{4} + 6 \beta_{3}) q^{44} + ( - \beta_{6} - 2 \beta_{5} - 2 \beta_{3}) q^{45} + ( - \beta_{6} + 2 \beta_{5} - 7 \beta_{3}) q^{46} + (2 \beta_{9} + \beta_{7} + \beta_{6} + 2 \beta_{5} - 6 \beta_{4} - 6 \beta_{3}) q^{47} + (2 \beta_{2} + 2 \beta_1 + 2) q^{48} + (2 \beta_{9} - 4 \beta_{7} - 2 \beta_{4}) q^{50} + ( - 3 \beta_{11} + \beta_{10} - \beta_{8} + \beta_{2} + 3 \beta_1) q^{51} + (4 \beta_{11} - 2 \beta_{10} - 4 \beta_{8} - \beta_{6} + \beta_{5} - 3 \beta_{3} + 4) q^{52} + (\beta_{11} - \beta_{10}) q^{53} + (2 \beta_{9} - 4 \beta_{7} - 4 \beta_{6} + 2 \beta_{5} - 4 \beta_{4} - 4 \beta_{3}) q^{54} + ( - 3 \beta_1 + 2) q^{55} + ( - 4 \beta_{9} - 3 \beta_{7} - 11 \beta_{4}) q^{57} + (4 \beta_{9} - 5 \beta_{7} - 5 \beta_{6} + 4 \beta_{5} - 9 \beta_{4} - 9 \beta_{3}) q^{58} + (\beta_{6} + 3 \beta_{5} + \beta_{3}) q^{59} + (4 \beta_{5} + 2 \beta_{3}) q^{60} + ( - 2 \beta_{11} + 2 \beta_{10} - 4 \beta_{8} + 2 \beta_{2} + 2 \beta_1) q^{61} + ( - 2 \beta_{2} + 3 \beta_1 + 2) q^{62} + 4 \beta_{2} q^{64} + (4 \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} + 4 \beta_{5} + 3 \beta_{4} + 3 \beta_{3}) q^{65} + ( - 4 \beta_{11} + 4 \beta_{10} + 6 \beta_{8} - 6) q^{66} + ( - 2 \beta_{5} + 4 \beta_{3}) q^{67} + (4 \beta_{11} - 2 \beta_{10} - 4 \beta_{8} - 2 \beta_{2} - 4 \beta_1) q^{68} + (2 \beta_{2} + 3 \beta_1) q^{69} + ( - 2 \beta_{9} + 5 \beta_{7} - 3 \beta_{4}) q^{71} + ( - 2 \beta_{7} - 2 \beta_{6} + 4 \beta_{4} + 4 \beta_{3}) q^{72} + ( - 3 \beta_{6} + 4 \beta_{5} - 4 \beta_{3}) q^{73} + (3 \beta_{11} + 3 \beta_{10} - 3 \beta_{8} + 3) q^{74} + ( - 2 \beta_{11} - 2 \beta_{10} - 8 \beta_{8} - 2 \beta_{2} + 2 \beta_1) q^{75} + (\beta_{9} + 7 \beta_{7} + 13 \beta_{4}) q^{76} + (4 \beta_{9} - 2 \beta_{7} - 4 \beta_{4} + \beta_{2} + \beta_1 + 3) q^{78} + (4 \beta_{11} - 2 \beta_{10} - 5 \beta_{8} - 2 \beta_{2} - 4 \beta_1) q^{79} - 2 \beta_{5} q^{80} + ( - 3 \beta_{11} - 5 \beta_{10} - 6 \beta_{8} + 6) q^{81} + ( - 4 \beta_{11} + 2 \beta_{10} + 10 \beta_{8} + 2 \beta_{2} + 4 \beta_1) q^{82} + (5 \beta_{9} + 4 \beta_{7} + \beta_{4}) q^{83} + (4 \beta_{9} + \beta_{7} + 3 \beta_{4}) q^{85} + ( - 3 \beta_{7} - 3 \beta_{6} - 7 \beta_{4} - 7 \beta_{3}) q^{86} + (\beta_{11} - 4 \beta_{10} - 8 \beta_{8} + 8) q^{87} + (6 \beta_{11} - 4 \beta_{10} - 8 \beta_{8} + 8) q^{88} + ( - 2 \beta_{9} + 3 \beta_{7} + 3 \beta_{6} - 2 \beta_{5}) q^{89} + (\beta_1 - 2) q^{90} + ( - 3 \beta_{2} - 5 \beta_1 - 7) q^{92} + ( - 4 \beta_{9} - 3 \beta_{7} - 3 \beta_{6} - 4 \beta_{5} - 5 \beta_{4} - 5 \beta_{3}) q^{93} + ( - \beta_{11} + 8 \beta_{10} + 6 \beta_{8} - 6) q^{94} + (8 \beta_{11} + 4 \beta_{10} + 11 \beta_{8} - 11) q^{95} + ( - 4 \beta_{9} - 4 \beta_{5} - 4 \beta_{4} - 4 \beta_{3}) q^{96} + ( - 2 \beta_{9} - 9 \beta_{7} - 8 \beta_{4}) q^{97} + (2 \beta_{9} + \beta_{7} + \beta_{4}) q^{99}+O(q^{100})$$ q + (b7 + b6 + b4 + b3) * q^2 - b11 * q^3 + (b11 - b10 - b8 + 1) * q^4 + (b9 + b5 + b4 + b3) * q^5 + (b9 - b7 - b4) * q^6 + (-2*b9 + 2*b4) * q^8 + (-b11 - b10 - b2 + b1) * q^9 - b11 * q^10 + (b6 + 3*b3) * q^11 + (2*b10 + 4*b8 + 2*b2) * q^12 + (b7 + 2*b2 - b1 + 2) * q^13 + (2*b9 + b7 + 3*b4) * q^15 + (-2*b10 - 2*b8 - 2*b2) * q^16 + (-b11 - 2*b10 - 2*b8 + 2) * q^17 + (b6 - b3) * q^18 + (-4*b9 - b7 - b6 - 4*b5) * q^19 + (-b9 - b7 - 3*b4) * q^20 + (3*b2 + b1 + 5) * q^22 + (-b11 - 3*b10 - 2*b8 - 3*b2 + b1) * q^23 + (2*b6 + 6*b3) * q^24 + (-3*b11 - b10 + b8 - 1) * q^25 + (b11 - b9 - 2*b8 + b7 + b6 - b5 + 5*b4 + 5*b3 - b1) * q^26 + (-2*b1 - 2) * q^27 + (-2*b2 - 2*b1 - 3) * q^29 + (-b11 - b10 - 3*b8 - b2 + b1) * q^30 + (2*b6 - b5 - 3*b3) * q^31 + (-2*b6 - 2*b5 - 2*b3) * q^32 + (3*b9 - b7 - b6 + 3*b5 - b4 - b3) * q^33 + (-b9 + b7 + 5*b4) * q^34 + (b2 - b1 + 1) * q^36 + (3*b7 + 3*b6 - 3*b4 - 3*b3) * q^37 + (3*b11 - 4*b10 - 2*b8 + 2) * q^38 + (-3*b11 + b10 - b8 + b6 + b3 + 1) * q^39 + (2*b11 + 2*b10 + 4*b8 + 2*b2 - 2*b1) * q^40 + (-4*b7 - 2*b4) * q^41 + (-2*b2 + 2*b1 - 5) * q^43 + (-4*b9 + 4*b7 + 4*b6 - 4*b5 + 6*b4 + 6*b3) * q^44 + (-b6 - 2*b5 - 2*b3) * q^45 + (-b6 + 2*b5 - 7*b3) * q^46 + (2*b9 + b7 + b6 + 2*b5 - 6*b4 - 6*b3) * q^47 + (2*b2 + 2*b1 + 2) * q^48 + (2*b9 - 4*b7 - 2*b4) * q^50 + (-3*b11 + b10 - b8 + b2 + 3*b1) * q^51 + (4*b11 - 2*b10 - 4*b8 - b6 + b5 - 3*b3 + 4) * q^52 + (b11 - b10) * q^53 + (2*b9 - 4*b7 - 4*b6 + 2*b5 - 4*b4 - 4*b3) * q^54 + (-3*b1 + 2) * q^55 + (-4*b9 - 3*b7 - 11*b4) * q^57 + (4*b9 - 5*b7 - 5*b6 + 4*b5 - 9*b4 - 9*b3) * q^58 + (b6 + 3*b5 + b3) * q^59 + (4*b5 + 2*b3) * q^60 + (-2*b11 + 2*b10 - 4*b8 + 2*b2 + 2*b1) * q^61 + (-2*b2 + 3*b1 + 2) * q^62 + 4*b2 * q^64 + (4*b9 + b8 + b7 + b6 + 4*b5 + 3*b4 + 3*b3) * q^65 + (-4*b11 + 4*b10 + 6*b8 - 6) * q^66 + (-2*b5 + 4*b3) * q^67 + (4*b11 - 2*b10 - 4*b8 - 2*b2 - 4*b1) * q^68 + (2*b2 + 3*b1) * q^69 + (-2*b9 + 5*b7 - 3*b4) * q^71 + (-2*b7 - 2*b6 + 4*b4 + 4*b3) * q^72 + (-3*b6 + 4*b5 - 4*b3) * q^73 + (3*b11 + 3*b10 - 3*b8 + 3) * q^74 + (-2*b11 - 2*b10 - 8*b8 - 2*b2 + 2*b1) * q^75 + (b9 + 7*b7 + 13*b4) * q^76 + (4*b9 - 2*b7 - 4*b4 + b2 + b1 + 3) * q^78 + (4*b11 - 2*b10 - 5*b8 - 2*b2 - 4*b1) * q^79 - 2*b5 * q^80 + (-3*b11 - 5*b10 - 6*b8 + 6) * q^81 + (-4*b11 + 2*b10 + 10*b8 + 2*b2 + 4*b1) * q^82 + (5*b9 + 4*b7 + b4) * q^83 + (4*b9 + b7 + 3*b4) * q^85 + (-3*b7 - 3*b6 - 7*b4 - 7*b3) * q^86 + (b11 - 4*b10 - 8*b8 + 8) * q^87 + (6*b11 - 4*b10 - 8*b8 + 8) * q^88 + (-2*b9 + 3*b7 + 3*b6 - 2*b5) * q^89 + (b1 - 2) * q^90 + (-3*b2 - 5*b1 - 7) * q^92 + (-4*b9 - 3*b7 - 3*b6 - 4*b5 - 5*b4 - 5*b3) * q^93 + (-b11 + 8*b10 + 6*b8 - 6) * q^94 + (8*b11 + 4*b10 + 11*b8 - 11) * q^95 + (-4*b9 - 4*b5 - 4*b4 - 4*b3) * q^96 + (-2*b9 - 9*b7 - 8*b4) * q^97 + (2*b9 + b7 + b4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 4 q^{4} + 2 q^{9}+O(q^{10})$$ 12 * q + 4 * q^4 + 2 * q^9 $$12 q + 4 q^{4} + 2 q^{9} + 20 q^{12} + 16 q^{13} - 8 q^{16} + 8 q^{17} + 48 q^{22} - 6 q^{23} - 8 q^{25} - 12 q^{26} - 24 q^{27} - 28 q^{29} - 16 q^{30} + 8 q^{36} + 4 q^{38} + 8 q^{39} + 20 q^{40} - 52 q^{43} + 16 q^{48} - 8 q^{51} + 20 q^{52} - 2 q^{53} + 24 q^{55} - 28 q^{61} + 32 q^{62} - 16 q^{64} + 6 q^{65} - 28 q^{66} - 20 q^{68} - 8 q^{69} + 24 q^{74} - 44 q^{75} + 32 q^{78} - 26 q^{79} + 26 q^{81} + 56 q^{82} + 40 q^{87} + 40 q^{88} - 24 q^{90} - 72 q^{92} - 20 q^{94} - 58 q^{95}+O(q^{100})$$ 12 * q + 4 * q^4 + 2 * q^9 + 20 * q^12 + 16 * q^13 - 8 * q^16 + 8 * q^17 + 48 * q^22 - 6 * q^23 - 8 * q^25 - 12 * q^26 - 24 * q^27 - 28 * q^29 - 16 * q^30 + 8 * q^36 + 4 * q^38 + 8 * q^39 + 20 * q^40 - 52 * q^43 + 16 * q^48 - 8 * q^51 + 20 * q^52 - 2 * q^53 + 24 * q^55 - 28 * q^61 + 32 * q^62 - 16 * q^64 + 6 * q^65 - 28 * q^66 - 20 * q^68 - 8 * q^69 + 24 * q^74 - 44 * q^75 + 32 * q^78 - 26 * q^79 + 26 * q^81 + 56 * q^82 + 40 * q^87 + 40 * q^88 - 24 * q^90 - 72 * q^92 - 20 * q^94 - 58 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 2x^{11} + 2x^{10} - 8x^{9} + 4x^{8} + 16x^{7} - 8x^{6} + 20x^{5} + 20x^{4} - 24x^{3} + 8x^{2} - 8x + 4$$ :

 $$\beta_{1}$$ $$=$$ $$( - \nu^{11} + \nu^{10} - 4 \nu^{9} + 28 \nu^{8} - 18 \nu^{7} + 22 \nu^{6} - 94 \nu^{5} - 146 \nu^{4} + 144 \nu^{3} - 48 \nu^{2} + 48 \nu + 748 ) / 460$$ (-v^11 + v^10 - 4*v^9 + 28*v^8 - 18*v^7 + 22*v^6 - 94*v^5 - 146*v^4 + 144*v^3 - 48*v^2 + 48*v + 748) / 460 $$\beta_{2}$$ $$=$$ $$( - 5 \nu^{11} + 5 \nu^{10} - 20 \nu^{9} + 94 \nu^{8} - 44 \nu^{7} + 64 \nu^{6} - 286 \nu^{5} - 454 \nu^{4} + 444 \nu^{3} - 148 \nu^{2} + 148 \nu + 612 ) / 460$$ (-5*v^11 + 5*v^10 - 20*v^9 + 94*v^8 - 44*v^7 + 64*v^6 - 286*v^5 - 454*v^4 + 444*v^3 - 148*v^2 + 148*v + 612) / 460 $$\beta_{3}$$ $$=$$ $$( 12 \nu^{11} - 43 \nu^{10} + 43 \nu^{9} - 103 \nu^{8} + 166 \nu^{7} + 264 \nu^{6} - 414 \nu^{5} + 110 \nu^{4} - 50 \nu^{3} - 1370 \nu^{2} + 12 \nu - 12 ) / 460$$ (12*v^11 - 43*v^10 + 43*v^9 - 103*v^8 + 166*v^7 + 264*v^6 - 414*v^5 + 110*v^4 - 50*v^3 - 1370*v^2 + 12*v - 12) / 460 $$\beta_{4}$$ $$=$$ $$( 31 \nu^{11} - 31 \nu^{10} + 9 \nu^{9} - 201 \nu^{8} - 109 \nu^{7} + 560 \nu^{6} + 246 \nu^{5} + 524 \nu^{4} + 1148 \nu^{3} + 154 \nu^{2} - 154 \nu - 142 ) / 230$$ (31*v^11 - 31*v^10 + 9*v^9 - 201*v^8 - 109*v^7 + 560*v^6 + 246*v^5 + 524*v^4 + 1148*v^3 + 154*v^2 - 154*v - 142) / 230 $$\beta_{5}$$ $$=$$ $$( - 12 \nu^{11} + 43 \nu^{10} - 43 \nu^{9} + 103 \nu^{8} - 166 \nu^{7} - 264 \nu^{6} + 414 \nu^{5} - 110 \nu^{4} + 50 \nu^{3} + 1140 \nu^{2} - 12 \nu + 12 ) / 230$$ (-12*v^11 + 43*v^10 - 43*v^9 + 103*v^8 - 166*v^7 - 264*v^6 + 414*v^5 - 110*v^4 + 50*v^3 + 1140*v^2 - 12*v + 12) / 230 $$\beta_{6}$$ $$=$$ $$( 32 \nu^{11} - 107 \nu^{10} + 107 \nu^{9} - 267 \nu^{8} + 412 \nu^{7} + 658 \nu^{6} - 1058 \nu^{5} + 278 \nu^{4} - 118 \nu^{3} - 1982 \nu^{2} + 32 \nu - 32 ) / 460$$ (32*v^11 - 107*v^10 + 107*v^9 - 267*v^8 + 412*v^7 + 658*v^6 - 1058*v^5 + 278*v^4 - 118*v^3 - 1982*v^2 + 32*v - 32) / 460 $$\beta_{7}$$ $$=$$ $$( 81 \nu^{11} - 81 \nu^{10} + 48 \nu^{9} - 566 \nu^{8} - 244 \nu^{7} + 1208 \nu^{6} + 806 \nu^{5} + 1614 \nu^{4} + 3424 \nu^{3} + 484 \nu^{2} - 484 \nu - 420 ) / 460$$ (81*v^11 - 81*v^10 + 48*v^9 - 566*v^8 - 244*v^7 + 1208*v^6 + 806*v^5 + 1614*v^4 + 3424*v^3 + 484*v^2 - 484*v - 420) / 460 $$\beta_{8}$$ $$=$$ $$( 3 \nu^{11} - 5 \nu^{10} + 5 \nu^{9} - 23 \nu^{8} + 4 \nu^{7} + 46 \nu^{6} - 12 \nu^{5} + 74 \nu^{4} + 86 \nu^{3} - 38 \nu^{2} + 32 \nu - 12 ) / 20$$ (3*v^11 - 5*v^10 + 5*v^9 - 23*v^8 + 4*v^7 + 46*v^6 - 12*v^5 + 74*v^4 + 86*v^3 - 38*v^2 + 32*v - 12) / 20 $$\beta_{9}$$ $$=$$ $$( - 101 \nu^{11} + 101 \nu^{10} - 36 \nu^{9} + 666 \nu^{8} + 344 \nu^{7} - 1734 \nu^{6} - 846 \nu^{5} - 1774 \nu^{4} - 3856 \nu^{3} - 524 \nu^{2} + 524 \nu + 476 ) / 460$$ (-101*v^11 + 101*v^10 - 36*v^9 + 666*v^8 + 344*v^7 - 1734*v^6 - 846*v^5 - 1774*v^4 - 3856*v^3 - 524*v^2 + 524*v + 476) / 460 $$\beta_{10}$$ $$=$$ $$( 117 \nu^{11} - 195 \nu^{10} + 195 \nu^{9} - 941 \nu^{8} + 250 \nu^{7} + 1700 \nu^{6} - 92 \nu^{5} + 2510 \nu^{4} + 2790 \nu^{3} - 1294 \nu^{2} + 1060 \nu - 1060 ) / 460$$ (117*v^11 - 195*v^10 + 195*v^9 - 941*v^8 + 250*v^7 + 1700*v^6 - 92*v^5 + 2510*v^4 + 2790*v^3 - 1294*v^2 + 1060*v - 1060) / 460 $$\beta_{11}$$ $$=$$ $$( - 60 \nu^{11} + 100 \nu^{10} - 100 \nu^{9} + 469 \nu^{8} - 94 \nu^{7} - 906 \nu^{6} + 184 \nu^{5} - 1424 \nu^{4} - 1636 \nu^{3} + 732 \nu^{2} - 612 \nu + 612 ) / 230$$ (-60*v^11 + 100*v^10 - 100*v^9 + 469*v^8 - 94*v^7 - 906*v^6 + 184*v^5 - 1424*v^4 - 1636*v^3 + 732*v^2 - 612*v + 612) / 230
 $$\nu$$ $$=$$ $$( \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 ) / 2$$ (b11 + b10 - b9 + b8 - b7 - b6 - b5 - b4 - b3 + b2 - b1) / 2 $$\nu^{2}$$ $$=$$ $$-\beta_{5} - 2\beta_{3}$$ -b5 - 2*b3 $$\nu^{3}$$ $$=$$ $$2\beta_{9} + \beta_{7} + 2\beta_{4} + \beta_{2} - 2\beta _1 + 2$$ 2*b9 + b7 + 2*b4 + b2 - 2*b1 + 2 $$\nu^{4}$$ $$=$$ $$5\beta_{11} + \beta_{10} + 7\beta_{8} + \beta_{2} - 5\beta_1$$ 5*b11 + b10 + 7*b8 + b2 - 5*b1 $$\nu^{5}$$ $$=$$ $$8\beta_{11} + 3\beta_{10} + 9\beta_{8} - 3\beta_{6} - 8\beta_{5} - 9\beta_{3} - 9$$ 8*b11 + 3*b10 + 9*b8 - 3*b6 - 8*b5 - 9*b3 - 9 $$\nu^{6}$$ $$=$$ $$22\beta_{9} + 6\beta_{7} + 28\beta_{4}$$ 22*b9 + 6*b7 + 28*b4 $$\nu^{7}$$ $$=$$ $$33 \beta_{11} + 11 \beta_{10} + 33 \beta_{9} + 39 \beta_{8} + 11 \beta_{7} + 11 \beta_{6} + 33 \beta_{5} + 39 \beta_{4} + 39 \beta_{3} + 11 \beta_{2} - 33 \beta_1$$ 33*b11 + 11*b10 + 33*b9 + 39*b8 + 11*b7 + 11*b6 + 33*b5 + 39*b4 + 39*b3 + 11*b2 - 33*b1 $$\nu^{8}$$ $$=$$ $$94\beta_{11} + 28\beta_{10} + 116\beta_{8} - 116$$ 94*b11 + 28*b10 + 116*b8 - 116 $$\nu^{9}$$ $$=$$ $$138\beta_{9} + 44\beta_{7} + 166\beta_{4} - 44\beta_{2} + 138\beta _1 - 166$$ 138*b9 + 44*b7 + 166*b4 - 44*b2 + 138*b1 - 166 $$\nu^{10}$$ $$=$$ $$398\beta_{9} + 122\beta_{7} + 122\beta_{6} + 398\beta_{5} + 486\beta_{4} + 486\beta_{3}$$ 398*b9 + 122*b7 + 122*b6 + 398*b5 + 486*b4 + 486*b3 $$\nu^{11}$$ $$=$$ $$580\beta_{11} + 182\beta_{10} + 702\beta_{8} + 182\beta_{6} + 580\beta_{5} + 702\beta_{3} - 702$$ 580*b11 + 182*b10 + 702*b8 + 182*b6 + 580*b5 + 702*b3 - 702

## Character values

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

 $$n$$ $$197$$ $$248$$ $$\chi(n)$$ $$-1$$ $$-\beta_{8}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
116.1
 0.550552 + 0.147520i 0.312819 − 1.16746i 1.98293 + 0.531325i −0.531325 + 1.98293i −1.16746 − 0.312819i −0.147520 + 0.550552i 0.550552 − 0.147520i 0.312819 + 1.16746i 1.98293 − 0.531325i −0.531325 − 1.98293i −1.16746 + 0.312819i −0.147520 − 0.550552i
−2.14878 + 1.24060i −0.837565 + 1.45071i 2.07816 3.59948i 0.584680 0.337565i 4.15633i 0 5.35026i 0.0969683 + 0.167954i −0.837565 + 1.45071i
116.2 −1.01332 + 0.585043i −0.269594 + 0.466951i −0.315449 + 0.546373i 0.399074 0.230406i 0.630898i 0 3.07838i 1.35464 + 2.34630i −0.269594 + 0.466951i
116.3 −0.596598 + 0.344446i 1.10716 1.91766i −0.762714 + 1.32106i −2.78368 + 1.60716i 1.52543i 0 2.42864i −0.951606 1.64823i 1.10716 1.91766i
116.4 0.596598 0.344446i 1.10716 1.91766i −0.762714 + 1.32106i 2.78368 1.60716i 1.52543i 0 2.42864i −0.951606 1.64823i 1.10716 1.91766i
116.5 1.01332 0.585043i −0.269594 + 0.466951i −0.315449 + 0.546373i −0.399074 + 0.230406i 0.630898i 0 3.07838i 1.35464 + 2.34630i −0.269594 + 0.466951i
116.6 2.14878 1.24060i −0.837565 + 1.45071i 2.07816 3.59948i −0.584680 + 0.337565i 4.15633i 0 5.35026i 0.0969683 + 0.167954i −0.837565 + 1.45071i
324.1 −2.14878 1.24060i −0.837565 1.45071i 2.07816 + 3.59948i 0.584680 + 0.337565i 4.15633i 0 5.35026i 0.0969683 0.167954i −0.837565 1.45071i
324.2 −1.01332 0.585043i −0.269594 0.466951i −0.315449 0.546373i 0.399074 + 0.230406i 0.630898i 0 3.07838i 1.35464 2.34630i −0.269594 0.466951i
324.3 −0.596598 0.344446i 1.10716 + 1.91766i −0.762714 1.32106i −2.78368 1.60716i 1.52543i 0 2.42864i −0.951606 + 1.64823i 1.10716 + 1.91766i
324.4 0.596598 + 0.344446i 1.10716 + 1.91766i −0.762714 1.32106i 2.78368 + 1.60716i 1.52543i 0 2.42864i −0.951606 + 1.64823i 1.10716 + 1.91766i
324.5 1.01332 + 0.585043i −0.269594 0.466951i −0.315449 0.546373i −0.399074 0.230406i 0.630898i 0 3.07838i 1.35464 2.34630i −0.269594 0.466951i
324.6 2.14878 + 1.24060i −0.837565 1.45071i 2.07816 + 3.59948i −0.584680 0.337565i 4.15633i 0 5.35026i 0.0969683 0.167954i −0.837565 1.45071i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 116.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
13.b even 2 1 inner
91.r 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.r.e 12
7.b odd 2 1 637.2.r.d 12
7.c even 3 1 91.2.c.a 6
7.c even 3 1 inner 637.2.r.e 12
7.d odd 6 1 637.2.c.d 6
7.d odd 6 1 637.2.r.d 12
13.b even 2 1 inner 637.2.r.e 12
21.h odd 6 1 819.2.c.b 6
28.g odd 6 1 1456.2.k.c 6
91.b odd 2 1 637.2.r.d 12
91.r even 6 1 91.2.c.a 6
91.r even 6 1 inner 637.2.r.e 12
91.s odd 6 1 637.2.c.d 6
91.s odd 6 1 637.2.r.d 12
91.z odd 12 1 1183.2.a.h 3
91.z odd 12 1 1183.2.a.j 3
91.bb even 12 1 8281.2.a.be 3
91.bb even 12 1 8281.2.a.bi 3
273.w odd 6 1 819.2.c.b 6
364.bl odd 6 1 1456.2.k.c 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.c.a 6 7.c even 3 1
91.2.c.a 6 91.r even 6 1
637.2.c.d 6 7.d odd 6 1
637.2.c.d 6 91.s odd 6 1
637.2.r.d 12 7.b odd 2 1
637.2.r.d 12 7.d odd 6 1
637.2.r.d 12 91.b odd 2 1
637.2.r.d 12 91.s odd 6 1
637.2.r.e 12 1.a even 1 1 trivial
637.2.r.e 12 7.c even 3 1 inner
637.2.r.e 12 13.b even 2 1 inner
637.2.r.e 12 91.r even 6 1 inner
819.2.c.b 6 21.h odd 6 1
819.2.c.b 6 273.w odd 6 1
1183.2.a.h 3 91.z odd 12 1
1183.2.a.j 3 91.z odd 12 1
1456.2.k.c 6 28.g odd 6 1
1456.2.k.c 6 364.bl odd 6 1
8281.2.a.be 3 91.bb even 12 1
8281.2.a.bi 3 91.bb even 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(637, [\chi])$$:

 $$T_{2}^{12} - 8T_{2}^{10} + 52T_{2}^{8} - 88T_{2}^{6} + 112T_{2}^{4} - 48T_{2}^{2} + 16$$ T2^12 - 8*T2^10 + 52*T2^8 - 88*T2^6 + 112*T2^4 - 48*T2^2 + 16 $$T_{3}^{6} + 4T_{3}^{4} + 4T_{3}^{3} + 16T_{3}^{2} + 8T_{3} + 4$$ T3^6 + 4*T3^4 + 4*T3^3 + 16*T3^2 + 8*T3 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - 8 T^{10} + 52 T^{8} - 88 T^{6} + \cdots + 16$$
$3$ $$(T^{6} + 4 T^{4} + 4 T^{3} + 16 T^{2} + 8 T + 4)^{2}$$
$5$ $$T^{12} - 11 T^{10} + 114 T^{8} - 75 T^{6} + \cdots + 1$$
$7$ $$T^{12}$$
$11$ $$T^{12} - 28 T^{10} + 620 T^{8} + \cdots + 10000$$
$13$ $$(T^{6} - 8 T^{5} + 7 T^{4} + 64 T^{3} + \cdots + 2197)^{2}$$
$17$ $$(T^{6} - 4 T^{5} + 24 T^{4} - 36 T^{3} + \cdots + 1156)^{2}$$
$19$ $$T^{12} - 119 T^{10} + \cdots + 1387488001$$
$23$ $$(T^{6} + 3 T^{5} + 34 T^{4} + 83 T^{3} + \cdots + 6241)^{2}$$
$29$ $$(T^{3} + 7 T^{2} - 21 T + 5)^{4}$$
$31$ $$T^{12} - 83 T^{10} + 5098 T^{8} + \cdots + 17850625$$
$37$ $$T^{12} - 108 T^{10} + 10044 T^{8} + \cdots + 8503056$$
$41$ $$(T^{6} + 108 T^{4} + 2864 T^{2} + \cdots + 1600)^{2}$$
$43$ $$(T^{3} + 13 T^{2} + 35 T - 17)^{4}$$
$47$ $$T^{12} - 151 T^{10} + \cdots + 352275361$$
$53$ $$(T^{6} + T^{5} + 10 T^{4} + 17 T^{3} + \cdots + 169)^{2}$$
$59$ $$T^{12} - 68 T^{10} + 3808 T^{8} + \cdots + 7311616$$
$61$ $$(T^{6} + 14 T^{5} + 168 T^{4} + \cdots + 23104)^{2}$$
$67$ $$T^{12} - 80 T^{10} + 4992 T^{8} + \cdots + 65536$$
$71$ $$(T^{6} + 304 T^{4} + 27836 T^{2} + \cdots + 792100)^{2}$$
$73$ $$T^{12} - 263 T^{10} + 61446 T^{8} + \cdots + 923521$$
$79$ $$(T^{6} + 13 T^{5} + 206 T^{4} + \cdots + 34225)^{2}$$
$83$ $$(T^{6} + 227 T^{4} + 13095 T^{2} + \cdots + 26569)^{2}$$
$89$ $$T^{12} - 119 T^{10} + \cdots + 2655237841$$
$97$ $$(T^{6} + 575 T^{4} + 96115 T^{2} + \cdots + 4765489)^{2}$$