# Properties

 Label 637.2.e.i Level $637$ Weight $2$ Character orbit 637.e Analytic conductor $5.086$ Analytic rank $0$ Dimension $6$ 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(79,637)]

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([2, 0]))

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

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

chi := DirichletCharacter("637.79");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 637.e (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: $$5.08647060876$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.2696112.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{5} + 5x^{4} + 18x^{2} - 8x + 4$$ x^6 - x^5 + 5*x^4 + 18*x^2 - 8*x + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + (\beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{3} + (\beta_{5} - \beta_{4} - \beta_{3}) q^{4} + ( - \beta_{4} - \beta_1 + 1) q^{5} + ( - 2 \beta_{2} - 2) q^{6} + (\beta_{3} + 1) q^{8} + (3 \beta_{4} + 2 \beta_1 - 3) q^{9}+O(q^{10})$$ q - b1 * q^2 + (b5 - b4 - b3 - b2 - b1) * q^3 + (b5 - b4 - b3) * q^4 + (-b4 - b1 + 1) * q^5 + (-2*b2 - 2) * q^6 + (b3 + 1) * q^8 + (3*b4 + 2*b1 - 3) * q^9 $$q - \beta_1 q^{2} + (\beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{3} + (\beta_{5} - \beta_{4} - \beta_{3}) q^{4} + ( - \beta_{4} - \beta_1 + 1) q^{5} + ( - 2 \beta_{2} - 2) q^{6} + (\beta_{3} + 1) q^{8} + (3 \beta_{4} + 2 \beta_1 - 3) q^{9} + (\beta_{5} - 3 \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{10} + (\beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{11} + (4 \beta_{4} - 4) q^{12} - q^{13} + ( - \beta_{3} - 3 \beta_{2} - 3) q^{15} + (\beta_{5} - \beta_{4} - 2 \beta_1 + 1) q^{16} + ( - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - \beta_1) q^{17} + ( - 2 \beta_{5} + 6 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} + 3 \beta_1) q^{18} + (\beta_{4} - \beta_1 - 1) q^{19} - 2 \beta_{2} q^{20} + ( - 2 \beta_{2} - 2) q^{22} + (\beta_{5} + 4 \beta_{4} + 2 \beta_1 - 4) q^{23} - 4 \beta_{4} q^{24} + (\beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{25} + \beta_1 q^{26} + (4 \beta_{2} + 4) q^{27} + (\beta_{3} + 8) q^{29} + ( - 2 \beta_{5} + 8 \beta_{4} + 4 \beta_1 - 8) q^{30} + ( - 2 \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_1) q^{31} + ( - \beta_{5} - 3 \beta_{4} + \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{32} + (6 \beta_{4} + 2 \beta_1 - 6) q^{33} + ( - 2 \beta_{3} + 2 \beta_{2} - 4) q^{34} + (\beta_{3} + 4 \beta_{2} + 1) q^{36} + ( - \beta_{5} - \beta_{4} - 3 \beta_1 + 1) q^{37} + (\beta_{5} - 3 \beta_{4} - \beta_{3} + \beta_{2} + \beta_1) q^{38} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{39} - 2 \beta_1 q^{40} + (2 \beta_{3} + 2 \beta_{2}) q^{41} + ( - 3 \beta_{3} + 2 \beta_{2} + 4) q^{43} + (4 \beta_{4} - 4) q^{44} + ( - 2 \beta_{5} + 9 \beta_{4} + 2 \beta_{3} + 5 \beta_{2} + 5 \beta_1) q^{45} + ( - 3 \beta_{5} + 7 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} + 3 \beta_1) q^{46} + ( - 4 \beta_{5} + 3 \beta_{4} + \beta_1 - 3) q^{47} + ( - 4 \beta_{2} - 8) q^{48} + ( - \beta_{3} - 5) q^{50} + ( - 2 \beta_{4} + 2 \beta_1 + 2) q^{51} + ( - \beta_{5} + \beta_{4} + \beta_{3}) q^{52} + ( - 3 \beta_{5} - 2 \beta_{4} + 3 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{53} + (4 \beta_{5} - 12 \beta_{4} - 4 \beta_1 + 12) q^{54} + ( - \beta_{3} - 3 \beta_{2} - 3) q^{55} + (\beta_{3} - \beta_{2} - 1) q^{57} + ( - \beta_{5} + \beta_{4} - 9 \beta_1 - 1) q^{58} + ( - 4 \beta_{5} - 2 \beta_{4} + 4 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{59} + (4 \beta_{4} + 4 \beta_{2} + 4 \beta_1) q^{60} + (2 \beta_{4} - 2) q^{61} + ( - \beta_{3} + \beta_{2} + 1) q^{62} + ( - \beta_{3} + 2 \beta_{2} - 5) q^{64} + (\beta_{4} + \beta_1 - 1) q^{65} + ( - 2 \beta_{5} + 6 \beta_{4} + 2 \beta_{3} + 6 \beta_{2} + 6 \beta_1) q^{66} + (4 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} - 6 \beta_{2} - 6 \beta_1) q^{67} + (2 \beta_{5} - 6 \beta_{4} + 4 \beta_1 + 6) q^{68} + (5 \beta_{3} + 9 \beta_{2} + 5) q^{69} + ( - \beta_{3} - 3 \beta_{2} - 3) q^{71} + ( - \beta_{5} + \beta_{4} + 4 \beta_1 - 1) q^{72} + (4 \beta_{5} - 3 \beta_{4} - 4 \beta_{3} + \beta_{2} + \beta_1) q^{73} + (4 \beta_{5} - 10 \beta_{4} - 4 \beta_{3}) q^{74} + (2 \beta_{5} + 6 \beta_{4} + 2 \beta_1 - 6) q^{75} + (2 \beta_{3} - 2 \beta_{2} + 2) q^{76} + (2 \beta_{2} + 2) q^{78} + (\beta_{5} - 6 \beta_{4} - 4 \beta_1 + 6) q^{79} + (2 \beta_{5} - 6 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} - 4 \beta_1) q^{80} + (4 \beta_{5} - 3 \beta_{4} - 4 \beta_{3} - 6 \beta_{2} - 6 \beta_1) q^{81} + ( - 4 \beta_{4} - 2 \beta_1 + 4) q^{82} + ( - 4 \beta_{3} - 9 \beta_{2} + 1) q^{83} + ( - \beta_{3} + \beta_{2} - 3) q^{85} + (5 \beta_{5} - 9 \beta_{4} - \beta_1 + 9) q^{86} + (7 \beta_{5} - 11 \beta_{4} - 7 \beta_{3} - 7 \beta_{2} - 7 \beta_1) q^{87} - 4 \beta_{4} q^{88} + ( - 2 \beta_{5} + \beta_{4} + 5 \beta_1 - 1) q^{89} + (3 \beta_{3} + 11 \beta_{2} + 13) q^{90} + (2 \beta_{3} + 6 \beta_{2} - 2) q^{92} + ( - 3 \beta_{5} - 7 \beta_{4} + \beta_1 + 7) q^{93} + (3 \beta_{5} - \beta_{4} - 3 \beta_{3} + 7 \beta_{2} + 7 \beta_1) q^{94} + (\beta_{5} - 2 \beta_{4} - \beta_{3}) q^{95} + ( - 4 \beta_{5} + 4 \beta_{4} + 8 \beta_1 - 4) q^{96} + ( - \beta_{2} + 3) q^{97} + (3 \beta_{3} + 7 \beta_{2} + 7) q^{99}+O(q^{100})$$ q - b1 * q^2 + (b5 - b4 - b3 - b2 - b1) * q^3 + (b5 - b4 - b3) * q^4 + (-b4 - b1 + 1) * q^5 + (-2*b2 - 2) * q^6 + (b3 + 1) * q^8 + (3*b4 + 2*b1 - 3) * q^9 + (b5 - 3*b4 - b3 - b2 - b1) * q^10 + (b5 - b4 - b3 - b2 - b1) * q^11 + (4*b4 - 4) * q^12 - q^13 + (-b3 - 3*b2 - 3) * q^15 + (b5 - b4 - 2*b1 + 1) * q^16 + (-b5 + b4 + b3 - b2 - b1) * q^17 + (-2*b5 + 6*b4 + 2*b3 + 3*b2 + 3*b1) * q^18 + (b4 - b1 - 1) * q^19 - 2*b2 * q^20 + (-2*b2 - 2) * q^22 + (b5 + 4*b4 + 2*b1 - 4) * q^23 - 4*b4 * q^24 + (b5 + b4 - b3 - 2*b2 - 2*b1) * q^25 + b1 * q^26 + (4*b2 + 4) * q^27 + (b3 + 8) * q^29 + (-2*b5 + 8*b4 + 4*b1 - 8) * q^30 + (-2*b5 - b4 + 2*b3 + b2 + b1) * q^31 + (-b5 - 3*b4 + b3 - 2*b2 - 2*b1) * q^32 + (6*b4 + 2*b1 - 6) * q^33 + (-2*b3 + 2*b2 - 4) * q^34 + (b3 + 4*b2 + 1) * q^36 + (-b5 - b4 - 3*b1 + 1) * q^37 + (b5 - 3*b4 - b3 + b2 + b1) * q^38 + (-b5 + b4 + b3 + b2 + b1) * q^39 - 2*b1 * q^40 + (2*b3 + 2*b2) * q^41 + (-3*b3 + 2*b2 + 4) * q^43 + (4*b4 - 4) * q^44 + (-2*b5 + 9*b4 + 2*b3 + 5*b2 + 5*b1) * q^45 + (-3*b5 + 7*b4 + 3*b3 + 3*b2 + 3*b1) * q^46 + (-4*b5 + 3*b4 + b1 - 3) * q^47 + (-4*b2 - 8) * q^48 + (-b3 - 5) * q^50 + (-2*b4 + 2*b1 + 2) * q^51 + (-b5 + b4 + b3) * q^52 + (-3*b5 - 2*b4 + 3*b3 + 2*b2 + 2*b1) * q^53 + (4*b5 - 12*b4 - 4*b1 + 12) * q^54 + (-b3 - 3*b2 - 3) * q^55 + (b3 - b2 - 1) * q^57 + (-b5 + b4 - 9*b1 - 1) * q^58 + (-4*b5 - 2*b4 + 4*b3 - 2*b2 - 2*b1) * q^59 + (4*b4 + 4*b2 + 4*b1) * q^60 + (2*b4 - 2) * q^61 + (-b3 + b2 + 1) * q^62 + (-b3 + 2*b2 - 5) * q^64 + (b4 + b1 - 1) * q^65 + (-2*b5 + 6*b4 + 2*b3 + 6*b2 + 6*b1) * q^66 + (4*b5 + 2*b4 - 4*b3 - 6*b2 - 6*b1) * q^67 + (2*b5 - 6*b4 + 4*b1 + 6) * q^68 + (5*b3 + 9*b2 + 5) * q^69 + (-b3 - 3*b2 - 3) * q^71 + (-b5 + b4 + 4*b1 - 1) * q^72 + (4*b5 - 3*b4 - 4*b3 + b2 + b1) * q^73 + (4*b5 - 10*b4 - 4*b3) * q^74 + (2*b5 + 6*b4 + 2*b1 - 6) * q^75 + (2*b3 - 2*b2 + 2) * q^76 + (2*b2 + 2) * q^78 + (b5 - 6*b4 - 4*b1 + 6) * q^79 + (2*b5 - 6*b4 - 2*b3 - 4*b2 - 4*b1) * q^80 + (4*b5 - 3*b4 - 4*b3 - 6*b2 - 6*b1) * q^81 + (-4*b4 - 2*b1 + 4) * q^82 + (-4*b3 - 9*b2 + 1) * q^83 + (-b3 + b2 - 3) * q^85 + (5*b5 - 9*b4 - b1 + 9) * q^86 + (7*b5 - 11*b4 - 7*b3 - 7*b2 - 7*b1) * q^87 - 4*b4 * q^88 + (-2*b5 + b4 + 5*b1 - 1) * q^89 + (3*b3 + 11*b2 + 13) * q^90 + (2*b3 + 6*b2 - 2) * q^92 + (-3*b5 - 7*b4 + b1 + 7) * q^93 + (3*b5 - b4 - 3*b3 + 7*b2 + 7*b1) * q^94 + (b5 - 2*b4 - b3) * q^95 + (-4*b5 + 4*b4 + 8*b1 - 4) * q^96 + (-b2 + 3) * q^97 + (3*b3 + 7*b2 + 7) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - q^{2} - 2 q^{3} - 3 q^{4} + 2 q^{5} - 8 q^{6} + 6 q^{8} - 7 q^{9}+O(q^{10})$$ 6 * q - q^2 - 2 * q^3 - 3 * q^4 + 2 * q^5 - 8 * q^6 + 6 * q^8 - 7 * q^9 $$6 q - q^{2} - 2 q^{3} - 3 q^{4} + 2 q^{5} - 8 q^{6} + 6 q^{8} - 7 q^{9} - 8 q^{10} - 2 q^{11} - 12 q^{12} - 6 q^{13} - 12 q^{15} + q^{16} + 4 q^{17} + 15 q^{18} - 4 q^{19} + 4 q^{20} - 8 q^{22} - 10 q^{23} - 12 q^{24} + 5 q^{25} + q^{26} + 16 q^{27} + 48 q^{29} - 20 q^{30} - 4 q^{31} - 7 q^{32} - 16 q^{33} - 28 q^{34} - 2 q^{36} - 10 q^{38} + 2 q^{39} - 2 q^{40} - 4 q^{41} + 20 q^{43} - 12 q^{44} + 22 q^{45} + 18 q^{46} - 8 q^{47} - 40 q^{48} - 30 q^{50} + 8 q^{51} + 3 q^{52} - 8 q^{53} + 32 q^{54} - 12 q^{55} - 4 q^{57} - 12 q^{58} - 4 q^{59} + 8 q^{60} - 6 q^{61} + 4 q^{62} - 34 q^{64} - 2 q^{65} + 12 q^{66} + 12 q^{67} + 22 q^{68} + 12 q^{69} - 12 q^{71} + q^{72} - 10 q^{73} - 30 q^{74} - 16 q^{75} + 16 q^{76} + 8 q^{78} + 14 q^{79} - 14 q^{80} - 3 q^{81} + 10 q^{82} + 24 q^{83} - 20 q^{85} + 26 q^{86} - 26 q^{87} - 12 q^{88} + 2 q^{89} + 56 q^{90} - 24 q^{92} + 22 q^{93} - 10 q^{94} - 6 q^{95} - 4 q^{96} + 20 q^{97} + 28 q^{99}+O(q^{100})$$ 6 * q - q^2 - 2 * q^3 - 3 * q^4 + 2 * q^5 - 8 * q^6 + 6 * q^8 - 7 * q^9 - 8 * q^10 - 2 * q^11 - 12 * q^12 - 6 * q^13 - 12 * q^15 + q^16 + 4 * q^17 + 15 * q^18 - 4 * q^19 + 4 * q^20 - 8 * q^22 - 10 * q^23 - 12 * q^24 + 5 * q^25 + q^26 + 16 * q^27 + 48 * q^29 - 20 * q^30 - 4 * q^31 - 7 * q^32 - 16 * q^33 - 28 * q^34 - 2 * q^36 - 10 * q^38 + 2 * q^39 - 2 * q^40 - 4 * q^41 + 20 * q^43 - 12 * q^44 + 22 * q^45 + 18 * q^46 - 8 * q^47 - 40 * q^48 - 30 * q^50 + 8 * q^51 + 3 * q^52 - 8 * q^53 + 32 * q^54 - 12 * q^55 - 4 * q^57 - 12 * q^58 - 4 * q^59 + 8 * q^60 - 6 * q^61 + 4 * q^62 - 34 * q^64 - 2 * q^65 + 12 * q^66 + 12 * q^67 + 22 * q^68 + 12 * q^69 - 12 * q^71 + q^72 - 10 * q^73 - 30 * q^74 - 16 * q^75 + 16 * q^76 + 8 * q^78 + 14 * q^79 - 14 * q^80 - 3 * q^81 + 10 * q^82 + 24 * q^83 - 20 * q^85 + 26 * q^86 - 26 * q^87 - 12 * q^88 + 2 * q^89 + 56 * q^90 - 24 * q^92 + 22 * q^93 - 10 * q^94 - 6 * q^95 - 4 * q^96 + 20 * q^97 + 28 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 5x^{4} + 18x^{2} - 8x + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{5} - 5\nu^{4} + 25\nu^{3} - 18\nu^{2} + 8\nu - 40 ) / 82$$ (v^5 - 5*v^4 + 25*v^3 - 18*v^2 + 8*v - 40) / 82 $$\beta_{3}$$ $$=$$ $$( 2\nu^{5} - 10\nu^{4} + 9\nu^{3} - 36\nu^{2} + 16\nu - 121 ) / 41$$ (2*v^5 - 10*v^4 + 9*v^3 - 36*v^2 + 16*v - 121) / 41 $$\beta_{4}$$ $$=$$ $$( -10\nu^{5} + 9\nu^{4} - 45\nu^{3} - 25\nu^{2} - 162\nu + 72 ) / 82$$ (-10*v^5 + 9*v^4 - 45*v^3 - 25*v^2 - 162*v + 72) / 82 $$\beta_{5}$$ $$=$$ $$( -26\nu^{5} + 7\nu^{4} - 117\nu^{3} - 65\nu^{2} - 454\nu - 26 ) / 82$$ (-26*v^5 + 7*v^4 - 117*v^3 - 65*v^2 - 454*v - 26) / 82
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{5} - 3\beta_{4} - \beta_{3}$$ b5 - 3*b4 - b3 $$\nu^{3}$$ $$=$$ $$-\beta_{3} + 4\beta_{2} - 1$$ -b3 + 4*b2 - 1 $$\nu^{4}$$ $$=$$ $$-5\beta_{5} + 13\beta_{4} - 2\beta _1 - 13$$ -5*b5 + 13*b4 - 2*b1 - 13 $$\nu^{5}$$ $$=$$ $$-7\beta_{5} + 11\beta_{4} + 7\beta_{3} - 18\beta_{2} - 18\beta_1$$ -7*b5 + 11*b4 + 7*b3 - 18*b2 - 18*b1

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 1.17146 − 2.02903i 0.235342 − 0.407624i −0.906803 + 1.57063i 1.17146 + 2.02903i 0.235342 + 0.407624i −0.906803 − 1.57063i
−1.17146 + 2.02903i −0.573183 0.992782i −1.74464 3.02181i −0.671462 + 1.16301i 2.68585 0 3.48929 0.842923 1.45999i −1.57318 2.72483i
79.2 −0.235342 + 0.407624i 1.12457 + 1.94781i 0.889229 + 1.54019i 0.264658 0.458402i −1.05863 0 −1.77846 −1.02932 + 1.78283i 0.124570 + 0.215762i
79.3 0.906803 1.57063i −1.55139 2.68708i −0.644584 1.11645i 1.40680 2.43665i −5.62721 0 1.28917 −3.31361 + 5.73933i −2.55139 4.41913i
508.1 −1.17146 2.02903i −0.573183 + 0.992782i −1.74464 + 3.02181i −0.671462 1.16301i 2.68585 0 3.48929 0.842923 + 1.45999i −1.57318 + 2.72483i
508.2 −0.235342 0.407624i 1.12457 1.94781i 0.889229 1.54019i 0.264658 + 0.458402i −1.05863 0 −1.77846 −1.02932 1.78283i 0.124570 0.215762i
508.3 0.906803 + 1.57063i −1.55139 + 2.68708i −0.644584 + 1.11645i 1.40680 + 2.43665i −5.62721 0 1.28917 −3.31361 5.73933i −2.55139 + 4.41913i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 508.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.e.i 6
7.b odd 2 1 637.2.e.j 6
7.c even 3 1 637.2.a.j 3
7.c even 3 1 inner 637.2.e.i 6
7.d odd 6 1 91.2.a.d 3
7.d odd 6 1 637.2.e.j 6
21.g even 6 1 819.2.a.i 3
21.h odd 6 1 5733.2.a.x 3
28.f even 6 1 1456.2.a.t 3
35.i odd 6 1 2275.2.a.m 3
56.j odd 6 1 5824.2.a.by 3
56.m even 6 1 5824.2.a.bs 3
91.r even 6 1 8281.2.a.bg 3
91.s odd 6 1 1183.2.a.i 3
91.bb even 12 2 1183.2.c.f 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.a.d 3 7.d odd 6 1
637.2.a.j 3 7.c even 3 1
637.2.e.i 6 1.a even 1 1 trivial
637.2.e.i 6 7.c even 3 1 inner
637.2.e.j 6 7.b odd 2 1
637.2.e.j 6 7.d odd 6 1
819.2.a.i 3 21.g even 6 1
1183.2.a.i 3 91.s odd 6 1
1183.2.c.f 6 91.bb even 12 2
1456.2.a.t 3 28.f even 6 1
2275.2.a.m 3 35.i odd 6 1
5733.2.a.x 3 21.h odd 6 1
5824.2.a.bs 3 56.m even 6 1
5824.2.a.by 3 56.j odd 6 1
8281.2.a.bg 3 91.r even 6 1

## Hecke kernels

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

 $$T_{2}^{6} + T_{2}^{5} + 5T_{2}^{4} + 18T_{2}^{2} + 8T_{2} + 4$$ T2^6 + T2^5 + 5*T2^4 + 18*T2^2 + 8*T2 + 4 $$T_{3}^{6} + 2T_{3}^{5} + 10T_{3}^{4} + 4T_{3}^{3} + 52T_{3}^{2} + 48T_{3} + 64$$ T3^6 + 2*T3^5 + 10*T3^4 + 4*T3^3 + 52*T3^2 + 48*T3 + 64 $$T_{5}^{6} - 2T_{5}^{5} + 7T_{5}^{4} + 2T_{5}^{3} + 13T_{5}^{2} - 6T_{5} + 4$$ T5^6 - 2*T5^5 + 7*T5^4 + 2*T5^3 + 13*T5^2 - 6*T5 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + T^{5} + 5 T^{4} + 18 T^{2} + \cdots + 4$$
$3$ $$T^{6} + 2 T^{5} + 10 T^{4} + 4 T^{3} + \cdots + 64$$
$5$ $$T^{6} - 2 T^{5} + 7 T^{4} + 2 T^{3} + \cdots + 4$$
$7$ $$T^{6}$$
$11$ $$T^{6} + 2 T^{5} + 10 T^{4} + 4 T^{3} + \cdots + 64$$
$13$ $$(T + 1)^{6}$$
$17$ $$T^{6} - 4 T^{5} + 26 T^{4} + 48 T^{3} + \cdots + 16$$
$19$ $$T^{6} + 4 T^{5} + 15 T^{4} + 12 T^{3} + \cdots + 16$$
$23$ $$T^{6} + 10 T^{5} + 99 T^{4} + \cdots + 18496$$
$29$ $$(T^{3} - 24 T^{2} + 185 T - 454)^{2}$$
$31$ $$T^{6} + 4 T^{5} + 35 T^{4} - 108 T^{3} + \cdots + 256$$
$37$ $$T^{6} + 58 T^{4} - 248 T^{3} + \cdots + 15376$$
$41$ $$(T^{3} + 2 T^{2} - 28 T + 8)^{2}$$
$43$ $$(T^{3} - 10 T^{2} - 71 T + 628)^{2}$$
$47$ $$T^{6} + 8 T^{5} + 143 T^{4} + \cdots + 295936$$
$53$ $$T^{6} + 8 T^{5} + 99 T^{4} - 324 T^{3} + \cdots + 484$$
$59$ $$T^{6} + 4 T^{5} + 172 T^{4} + \cdots + 473344$$
$61$ $$(T^{2} + 2 T + 4)^{3}$$
$67$ $$T^{6} - 12 T^{5} + 268 T^{4} + \cdots + 952576$$
$71$ $$(T^{3} + 6 T^{2} - 22 T + 16)^{2}$$
$73$ $$T^{6} + 10 T^{5} + 199 T^{4} + \cdots + 75076$$
$79$ $$T^{6} - 14 T^{5} + 191 T^{4} + \cdots + 256$$
$83$ $$(T^{3} - 12 T^{2} - 271 T + 3268)^{2}$$
$89$ $$T^{6} - 2 T^{5} + 99 T^{4} + \cdots + 178084$$
$97$ $$(T^{3} - 10 T^{2} + 29 T - 22)^{2}$$