# Properties

 Label 637.2.g.k Level $637$ Weight $2$ Character orbit 637.g Analytic conductor $5.086$ Analytic rank $0$ Dimension $8$ 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(263,637)]

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, 2]))

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

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

chi := DirichletCharacter("637.263");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 637.g (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: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{7} + 7x^{6} + 38x^{4} - 16x^{3} + 15x^{2} + 3x + 1$$ x^8 - x^7 + 7*x^6 + 38*x^4 - 16*x^3 + 15*x^2 + 3*x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + \beta_{6} q^{3} + ( - \beta_{7} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 1) q^{4} + (2 \beta_{4} + \beta_{3} + \beta_1 + 2) q^{5} + ( - \beta_{4} + \beta_1) q^{6} + (\beta_{6} + 3 \beta_{3} + \beta_{2} - 1) q^{8} + ( - \beta_{6} + \beta_{2} + 2) q^{9}+O(q^{10})$$ q + b1 * q^2 + b6 * q^3 + (-b7 - b4 + b3 + b2 + b1 - 1) * q^4 + (2*b4 + b3 + b1 + 2) * q^5 + (-b4 + b1) * q^6 + (b6 + 3*b3 + b2 - 1) * q^8 + (-b6 + b2 + 2) * q^9 $$q + \beta_1 q^{2} + \beta_{6} q^{3} + ( - \beta_{7} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 1) q^{4} + (2 \beta_{4} + \beta_{3} + \beta_1 + 2) q^{5} + ( - \beta_{4} + \beta_1) q^{6} + (\beta_{6} + 3 \beta_{3} + \beta_{2} - 1) q^{8} + ( - \beta_{6} + \beta_{2} + 2) q^{9} + ( - \beta_{3} + \beta_{2} - 3) q^{10} - \beta_{6} q^{11} + ( - \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 3 \beta_{4} + 2 \beta_{3} + \beta_{2} + 2 \beta_1 - 3) q^{12} + ( - \beta_{7} - \beta_{5} - \beta_{4} + \beta_{2} + \beta_1) q^{13} + (2 \beta_{6} - 2 \beta_{5} - \beta_{4} + \beta_{3} + \beta_1 - 1) q^{15} + (\beta_{7} + \beta_{5} + 4 \beta_{4} - 4 \beta_1) q^{16} + (\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{17} + (\beta_{5} - \beta_{4} - 2 \beta_1) q^{18} + ( - 2 \beta_{6} - 3 \beta_{3} - \beta_{2}) q^{19} + ( - \beta_{7} + \beta_{5} - \beta_{4} - 3 \beta_1) q^{20} + (\beta_{4} - \beta_1) q^{22} + (\beta_{7} + \beta_{5} + \beta_1) q^{23} + (\beta_{6} + 4 \beta_{3} + 2 \beta_{2}) q^{24} + (\beta_{7} + 2 \beta_{4} + 3 \beta_1) q^{25} + ( - \beta_{7} + \beta_{6} - 2 \beta_{4} + 4 \beta_{3} + \beta_{2} + \beta_1) q^{26} + (3 \beta_{6} + \beta_{3} - 7) q^{27} + (\beta_{7} - \beta_{6} + \beta_{5} - \beta_{2}) q^{29} + (\beta_{2} - 1) q^{30} + ( - \beta_{7} - \beta_{4}) q^{31} + (2 \beta_{7} + \beta_{6} - \beta_{5} + 7 \beta_{4} - 4 \beta_{3} - 2 \beta_{2} - 4 \beta_1 + 7) q^{32} + (\beta_{6} - \beta_{2} - 5) q^{33} + ( - \beta_{6} - 4 \beta_{3} - \beta_{2}) q^{34} + ( - 2 \beta_{6} + 2 \beta_{5} + 9 \beta_{4} + 9) q^{36} + (\beta_{5} - 2 \beta_{4} + \beta_1) q^{37} + ( - 3 \beta_{7} - \beta_{5} - 5 \beta_{4} + 4 \beta_1) q^{38} + ( - 2 \beta_{7} + 3 \beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{39} + (\beta_{7} + \beta_{6} - \beta_{5} + 4 \beta_{4} - \beta_{2} + 4) q^{40} + ( - \beta_{7} + 6 \beta_{4} + 2 \beta_{3} + \beta_{2} + 2 \beta_1 + 6) q^{41} + (\beta_{4} + \beta_1) q^{43} + (\beta_{7} - 2 \beta_{6} + 2 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 3) q^{44} + ( - 2 \beta_{7} - 3 \beta_{6} + 3 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + \cdots + 3) q^{45}+ \cdots + ( - 6 \beta_{6} - \beta_{3} + 7) q^{99}+O(q^{100})$$ q + b1 * q^2 + b6 * q^3 + (-b7 - b4 + b3 + b2 + b1 - 1) * q^4 + (2*b4 + b3 + b1 + 2) * q^5 + (-b4 + b1) * q^6 + (b6 + 3*b3 + b2 - 1) * q^8 + (-b6 + b2 + 2) * q^9 + (-b3 + b2 - 3) * q^10 - b6 * q^11 + (-b7 + 2*b6 - 2*b5 - 3*b4 + 2*b3 + b2 + 2*b1 - 3) * q^12 + (-b7 - b5 - b4 + b2 + b1) * q^13 + (2*b6 - 2*b5 - b4 + b3 + b1 - 1) * q^15 + (b7 + b5 + 4*b4 - 4*b1) * q^16 + (b7 - b6 + b5 + b4 - b3 - b2 - b1 + 1) * q^17 + (b5 - b4 - 2*b1) * q^18 + (-2*b6 - 3*b3 - b2) * q^19 + (-b7 + b5 - b4 - 3*b1) * q^20 + (b4 - b1) * q^22 + (b7 + b5 + b1) * q^23 + (b6 + 4*b3 + 2*b2) * q^24 + (b7 + 2*b4 + 3*b1) * q^25 + (-b7 + b6 - 2*b4 + 4*b3 + b2 + b1) * q^26 + (3*b6 + b3 - 7) * q^27 + (b7 - b6 + b5 - b2) * q^29 + (b2 - 1) * q^30 + (-b7 - b4) * q^31 + (2*b7 + b6 - b5 + 7*b4 - 4*b3 - 2*b2 - 4*b1 + 7) * q^32 + (b6 - b2 - 5) * q^33 + (-b6 - 4*b3 - b2) * q^34 + (-2*b6 + 2*b5 + 9*b4 + 9) * q^36 + (b5 - 2*b4 + b1) * q^37 + (-3*b7 - b5 - 5*b4 + 4*b1) * q^38 + (-2*b7 + 3*b6 - b5 + 2*b4 + b3 + b2 + 2*b1 - 2) * q^39 + (b7 + b6 - b5 + 4*b4 - b2 + 4) * q^40 + (-b7 + 6*b4 + 2*b3 + b2 + 2*b1 + 6) * q^41 + (b4 + b1) * q^43 + (b7 - 2*b6 + 2*b5 + 3*b4 - 2*b3 - b2 - 2*b1 + 3) * q^44 + (-2*b7 - 3*b6 + 3*b5 + 3*b4 - 2*b3 + 2*b2 - 2*b1 + 3) * q^45 + (-b7 - b6 + b5 - 6*b4 - b3 + b2 - b1 - 6) * q^46 + (-b6 + b5 - b4 - 3*b3 - 3*b1 - 1) * q^47 + (2*b7 - 2*b5 + b4 - 5*b1) * q^48 + (-3*b7 - b6 + b5 - 11*b4 - 2*b3 + 3*b2 - 2*b1 - 11) * q^50 + (2*b7 - b6 + b5 - 2*b4 - 2*b3 - 2*b2 - 2*b1 - 2) * q^51 + (b7 - b6 + 8*b4 + 4*b3 + b2 + 1) * q^52 + (2*b7 + 4*b5 + b4 - 2*b1) * q^53 + (b7 - 5*b1) * q^54 + (-2*b6 + 2*b5 + b4 - b3 - b1 + 1) * q^55 + (-b6 - 4*b3 - 3*b2 - 5) * q^57 + (-b6 - 2*b3 - 3) * q^58 + (b7 + 2*b6 - 2*b5 + b4 - 2*b3 - b2 - 2*b1 + 1) * q^59 + (-3*b5 - 4*b4 - 2*b1) * q^60 + (-2*b6 - 2*b3 - b2 + 2) * q^61 + (b6 - b5 + 2*b4 + 4*b3 + 4*b1 + 2) * q^62 + (-10*b3 - 2*b2 + 1) * q^64 + (-2*b7 - 2*b6 + b5 - 2*b4 - b3 + b2 - 3*b1) * q^65 + (-b5 + b4 - b1) * q^66 + (4*b6 + 6*b3 + b2 - 1) * q^67 + (-2*b7 + b5 - 7*b4 + 4*b1) * q^68 + (2*b7 + 2*b5 - 4*b4) * q^69 + (-2*b7 - 2*b5 - 4*b4) * q^71 + (2*b6 - 3*b3) * q^72 + (3*b7 + 2*b5 - 2*b4 - 2*b1) * q^73 + (-b7 - 4*b4 + 4*b3 + b2 + 4*b1 - 4) * q^74 + (b7 + b5 - b4 + 2*b1) * q^75 + (-2*b7 - b6 + b5 - 5*b4 + 11*b3 + 2*b2 + 11*b1 - 5) * q^76 + (-b7 + 2*b6 - b5 - 3*b4 + 5*b3 + 2*b2 + 2*b1 - 1) * q^78 + (b7 + 3*b6 - 3*b5 + 6*b4 + b3 - b2 + b1 + 6) * q^79 + (b6 - 2*b3 - 2*b2 + 1) * q^80 + (-7*b6 + b3 + 8) * q^81 + (b6 - b3 + 2*b2 - 4) * q^82 + (-4*b3 - b2 - 1) * q^83 + (b7 + b5 + 2*b4 + 2*b1) * q^85 + (-b7 - 3*b4 + b2 - 3) * q^86 + (2*b7 - 2*b6 + 2*b5 - 3*b4 - b3 - 2*b2 - b1 - 3) * q^87 + (-b6 - 4*b3 - 2*b2) * q^88 + (-b7 + 2*b5 - b1) * q^89 + (2*b6 + 4*b3 - 2*b2 + 7) * q^90 + (3*b6 + 7*b3 + b2 + 4) * q^92 + (-b7 - 2*b5 - 2*b4 + b1) * q^93 + (-b3 - 3*b2 + 8) * q^94 + (-b7 - 3*b6 + 3*b5 - 5*b4 - 2*b3 + b2 - 2*b1 - 5) * q^95 + (b7 + 13*b4 - 6*b3 - b2 - 6*b1 + 13) * q^96 + (-b7 - 5*b5 - b4 - 2*b1) * q^97 + (-6*b6 - b3 + 7) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + q^{2} + 2 q^{3} - 5 q^{4} + 7 q^{5} + 5 q^{6} - 12 q^{8} + 14 q^{9}+O(q^{10})$$ 8 * q + q^2 + 2 * q^3 - 5 * q^4 + 7 * q^5 + 5 * q^6 - 12 * q^8 + 14 * q^9 $$8 q + q^{2} + 2 q^{3} - 5 q^{4} + 7 q^{5} + 5 q^{6} - 12 q^{8} + 14 q^{9} - 22 q^{10} - 2 q^{11} - 12 q^{12} + 4 q^{13} - 3 q^{15} - 19 q^{16} + 4 q^{17} + 3 q^{18} + 2 q^{19} + 2 q^{20} - 5 q^{22} + 2 q^{23} - 6 q^{24} - 5 q^{25} + 3 q^{26} - 52 q^{27} - q^{29} - 8 q^{30} + 4 q^{31} + 33 q^{32} - 38 q^{33} + 6 q^{34} + 34 q^{36} + 10 q^{37} + 23 q^{38} - 19 q^{39} + 17 q^{40} + 22 q^{41} - 3 q^{43} + 12 q^{44} + 11 q^{45} - 24 q^{46} - 2 q^{47} - 11 q^{48} - 43 q^{50} - 7 q^{51} - 34 q^{52} - 2 q^{53} - 5 q^{54} + 3 q^{55} - 34 q^{57} - 22 q^{58} + 8 q^{59} + 11 q^{60} + 16 q^{61} + 5 q^{62} + 28 q^{64} + 4 q^{65} - 6 q^{66} - 12 q^{67} + 33 q^{68} + 18 q^{69} + 14 q^{71} + 10 q^{72} + 8 q^{73} - 20 q^{74} + 7 q^{75} - 32 q^{76} - q^{78} + 26 q^{79} + 14 q^{80} + 48 q^{81} - 28 q^{82} - 5 q^{85} - 12 q^{86} - 13 q^{87} + 6 q^{88} + q^{89} + 52 q^{90} + 24 q^{92} + 7 q^{93} + 66 q^{94} - 21 q^{95} + 58 q^{96} - 3 q^{97} + 46 q^{99}+O(q^{100})$$ 8 * q + q^2 + 2 * q^3 - 5 * q^4 + 7 * q^5 + 5 * q^6 - 12 * q^8 + 14 * q^9 - 22 * q^10 - 2 * q^11 - 12 * q^12 + 4 * q^13 - 3 * q^15 - 19 * q^16 + 4 * q^17 + 3 * q^18 + 2 * q^19 + 2 * q^20 - 5 * q^22 + 2 * q^23 - 6 * q^24 - 5 * q^25 + 3 * q^26 - 52 * q^27 - q^29 - 8 * q^30 + 4 * q^31 + 33 * q^32 - 38 * q^33 + 6 * q^34 + 34 * q^36 + 10 * q^37 + 23 * q^38 - 19 * q^39 + 17 * q^40 + 22 * q^41 - 3 * q^43 + 12 * q^44 + 11 * q^45 - 24 * q^46 - 2 * q^47 - 11 * q^48 - 43 * q^50 - 7 * q^51 - 34 * q^52 - 2 * q^53 - 5 * q^54 + 3 * q^55 - 34 * q^57 - 22 * q^58 + 8 * q^59 + 11 * q^60 + 16 * q^61 + 5 * q^62 + 28 * q^64 + 4 * q^65 - 6 * q^66 - 12 * q^67 + 33 * q^68 + 18 * q^69 + 14 * q^71 + 10 * q^72 + 8 * q^73 - 20 * q^74 + 7 * q^75 - 32 * q^76 - q^78 + 26 * q^79 + 14 * q^80 + 48 * q^81 - 28 * q^82 - 5 * q^85 - 12 * q^86 - 13 * q^87 + 6 * q^88 + q^89 + 52 * q^90 + 24 * q^92 + 7 * q^93 + 66 * q^94 - 21 * q^95 + 58 * q^96 - 3 * q^97 + 46 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} + 7x^{6} + 38x^{4} - 16x^{3} + 15x^{2} + 3x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -12\nu^{7} - 7\nu^{6} - 76\nu^{5} - 44\nu^{4} - 602\nu^{3} - 36\nu^{2} - 8\nu + 1249 ) / 458$$ (-12*v^7 - 7*v^6 - 76*v^5 - 44*v^4 - 602*v^3 - 36*v^2 - 8*v + 1249) / 458 $$\beta_{3}$$ $$=$$ $$( 57\nu^{7} - 24\nu^{6} + 361\nu^{5} + 209\nu^{4} + 2287\nu^{3} + 171\nu^{2} + 38\nu + 193 ) / 916$$ (57*v^7 - 24*v^6 + 361*v^5 + 209*v^4 + 2287*v^3 + 171*v^2 + 38*v + 193) / 916 $$\beta_{4}$$ $$=$$ $$( 193\nu^{7} - 250\nu^{6} + 1375\nu^{5} - 361\nu^{4} + 7125\nu^{3} - 5375\nu^{2} + 2724\nu - 375 ) / 916$$ (193*v^7 - 250*v^6 + 1375*v^5 - 361*v^4 + 7125*v^3 - 5375*v^2 + 2724*v - 375) / 916 $$\beta_{5}$$ $$=$$ $$( 174\nu^{7} - 242\nu^{6} + 1331\nu^{5} - 507\nu^{4} + 6897\nu^{3} - 5203\nu^{2} + 5383\nu - 363 ) / 458$$ (174*v^7 - 242*v^6 + 1331*v^5 - 507*v^4 + 6897*v^3 - 5203*v^2 + 5383*v - 363) / 458 $$\beta_{6}$$ $$=$$ $$( -375\nu^{7} + 182\nu^{6} - 2375\nu^{5} - 1375\nu^{4} - 13889\nu^{3} - 1125\nu^{2} - 250\nu - 2933 ) / 916$$ (-375*v^7 + 182*v^6 - 2375*v^5 - 1375*v^4 - 13889*v^3 - 1125*v^2 - 250*v - 2933) / 916 $$\beta_{7}$$ $$=$$ $$( -273\nu^{7} + 356\nu^{6} - 1958\nu^{5} + 602\nu^{4} - 10146\nu^{3} + 7654\nu^{2} - 3617\nu + 534 ) / 458$$ (-273*v^7 + 356*v^6 - 1958*v^5 + 602*v^4 - 10146*v^3 + 7654*v^2 - 3617*v + 534) / 458
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{7} - 3\beta_{4} + \beta_{3} + \beta_{2} + \beta _1 - 3$$ -b7 - 3*b4 + b3 + b2 + b1 - 3 $$\nu^{3}$$ $$=$$ $$\beta_{6} + 7\beta_{3} + \beta_{2} - 1$$ b6 + 7*b3 + b2 - 1 $$\nu^{4}$$ $$=$$ $$7\beta_{7} + \beta_{5} + 18\beta_{4} - 10\beta_1$$ 7*b7 + b5 + 18*b4 - 10*b1 $$\nu^{5}$$ $$=$$ $$10\beta_{7} - 7\beta_{6} + 7\beta_{5} + 15\beta_{4} - 48\beta_{3} - 10\beta_{2} - 48\beta _1 + 15$$ 10*b7 - 7*b6 + 7*b5 + 15*b4 - 48*b3 - 10*b2 - 48*b1 + 15 $$\nu^{6}$$ $$=$$ $$-10\beta_{6} - 86\beta_{3} - 48\beta_{2} + 117$$ -10*b6 - 86*b3 - 48*b2 + 117 $$\nu^{7}$$ $$=$$ $$-86\beta_{7} - 48\beta_{5} - 152\beta_{4} + 337\beta_1$$ -86*b7 - 48*b5 - 152*b4 + 337*b1

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
263.1
 −1.11000 + 1.92258i −0.115680 + 0.200364i 0.355143 − 0.615126i 1.37054 − 2.37385i −1.11000 − 1.92258i −0.115680 − 0.200364i 0.355143 + 0.615126i 1.37054 + 2.37385i
−1.11000 + 1.92258i 0.549551 −1.46422 2.53610i 2.11000 + 3.65463i −0.610004 + 1.05656i 0 2.06113 −2.69799 −9.36845
263.2 −0.115680 + 0.200364i −3.32225 0.973236 + 1.68569i 1.11568 + 1.93242i 0.384320 0.665661i 0 −0.913059 8.03736 −0.516249
263.3 0.355143 0.615126i 2.40788 0.747746 + 1.29513i 0.644857 + 1.11692i 0.855143 1.48115i 0 2.48280 2.79790 0.916066
263.4 1.37054 2.37385i 1.36482 −2.75677 4.77486i −0.370541 0.641796i 1.87054 3.23987i 0 −9.63087 −1.13727 −2.03137
373.1 −1.11000 1.92258i 0.549551 −1.46422 + 2.53610i 2.11000 3.65463i −0.610004 1.05656i 0 2.06113 −2.69799 −9.36845
373.2 −0.115680 0.200364i −3.32225 0.973236 1.68569i 1.11568 1.93242i 0.384320 + 0.665661i 0 −0.913059 8.03736 −0.516249
373.3 0.355143 + 0.615126i 2.40788 0.747746 1.29513i 0.644857 1.11692i 0.855143 + 1.48115i 0 2.48280 2.79790 0.916066
373.4 1.37054 + 2.37385i 1.36482 −2.75677 + 4.77486i −0.370541 + 0.641796i 1.87054 + 3.23987i 0 −9.63087 −1.13727 −2.03137
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 263.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.g 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.g.k 8
7.b odd 2 1 637.2.g.j 8
7.c even 3 1 91.2.f.c 8
7.c even 3 1 637.2.h.h 8
7.d odd 6 1 637.2.f.i 8
7.d odd 6 1 637.2.h.i 8
13.c even 3 1 637.2.h.h 8
21.h odd 6 1 819.2.o.h 8
28.g odd 6 1 1456.2.s.q 8
91.g even 3 1 inner 637.2.g.k 8
91.g even 3 1 1183.2.a.k 4
91.h even 3 1 91.2.f.c 8
91.m odd 6 1 637.2.g.j 8
91.m odd 6 1 8281.2.a.bp 4
91.n odd 6 1 637.2.h.i 8
91.p odd 6 1 8281.2.a.bt 4
91.u even 6 1 1183.2.a.l 4
91.v odd 6 1 637.2.f.i 8
91.bd odd 12 2 1183.2.c.g 8
273.s odd 6 1 819.2.o.h 8
364.bi odd 6 1 1456.2.s.q 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.f.c 8 7.c even 3 1
91.2.f.c 8 91.h even 3 1
637.2.f.i 8 7.d odd 6 1
637.2.f.i 8 91.v odd 6 1
637.2.g.j 8 7.b odd 2 1
637.2.g.j 8 91.m odd 6 1
637.2.g.k 8 1.a even 1 1 trivial
637.2.g.k 8 91.g even 3 1 inner
637.2.h.h 8 7.c even 3 1
637.2.h.h 8 13.c even 3 1
637.2.h.i 8 7.d odd 6 1
637.2.h.i 8 91.n odd 6 1
819.2.o.h 8 21.h odd 6 1
819.2.o.h 8 273.s odd 6 1
1183.2.a.k 4 91.g even 3 1
1183.2.a.l 4 91.u even 6 1
1183.2.c.g 8 91.bd odd 12 2
1456.2.s.q 8 28.g odd 6 1
1456.2.s.q 8 364.bi odd 6 1
8281.2.a.bp 4 91.m odd 6 1
8281.2.a.bt 4 91.p odd 6 1

## Hecke kernels

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

 $$T_{2}^{8} - T_{2}^{7} + 7T_{2}^{6} + 38T_{2}^{4} - 16T_{2}^{3} + 15T_{2}^{2} + 3T_{2} + 1$$ T2^8 - T2^7 + 7*T2^6 + 38*T2^4 - 16*T2^3 + 15*T2^2 + 3*T2 + 1 $$T_{3}^{4} - T_{3}^{3} - 9T_{3}^{2} + 16T_{3} - 6$$ T3^4 - T3^3 - 9*T3^2 + 16*T3 - 6 $$T_{5}^{8} - 7T_{5}^{7} + 37T_{5}^{6} - 86T_{5}^{5} + 160T_{5}^{4} - 114T_{5}^{3} + 109T_{5}^{2} - 9T_{5} + 81$$ T5^8 - 7*T5^7 + 37*T5^6 - 86*T5^5 + 160*T5^4 - 114*T5^3 + 109*T5^2 - 9*T5 + 81

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - T^{7} + 7 T^{6} + 38 T^{4} + \cdots + 1$$
$3$ $$(T^{4} - T^{3} - 9 T^{2} + 16 T - 6)^{2}$$
$5$ $$T^{8} - 7 T^{7} + 37 T^{6} - 86 T^{5} + \cdots + 81$$
$7$ $$T^{8}$$
$11$ $$(T^{4} + T^{3} - 9 T^{2} - 16 T - 6)^{2}$$
$13$ $$T^{8} - 4 T^{7} + 16 T^{6} + \cdots + 28561$$
$17$ $$T^{8} - 4 T^{7} + 28 T^{6} + \cdots + 2809$$
$19$ $$(T^{4} - T^{3} - 55 T^{2} + 500)^{2}$$
$23$ $$T^{8} - 2 T^{7} + 42 T^{6} + \cdots + 5184$$
$29$ $$T^{8} + T^{7} + 23 T^{6} - 64 T^{5} + \cdots + 25$$
$31$ $$T^{8} - 4 T^{7} + 29 T^{6} + \cdots + 2916$$
$37$ $$T^{8} - 10 T^{7} + 83 T^{6} + \cdots + 256$$
$41$ $$T^{8} - 22 T^{7} + 335 T^{6} + \cdots + 318096$$
$43$ $$T^{8} + 3 T^{7} + 12 T^{6} + 7 T^{5} + \cdots + 4$$
$47$ $$T^{8} + 2 T^{7} + 55 T^{6} + \cdots + 10000$$
$53$ $$T^{8} + 2 T^{7} + 144 T^{6} + \cdots + 1929321$$
$59$ $$T^{8} - 8 T^{7} + 141 T^{6} + \cdots + 498436$$
$61$ $$(T^{4} - 8 T^{3} - 15 T^{2} + 176 T - 100)^{2}$$
$67$ $$(T^{4} + 6 T^{3} - 211 T^{2} - 634 T + 11010)^{2}$$
$71$ $$T^{8} - 14 T^{7} + 212 T^{6} + \cdots + 4129024$$
$73$ $$T^{8} - 8 T^{7} + 183 T^{6} + \cdots + 3139984$$
$79$ $$T^{8} - 26 T^{7} + 542 T^{6} + \cdots + 58982400$$
$83$ $$(T^{4} - 97 T^{2} - 442 T - 426)^{2}$$
$89$ $$T^{8} - T^{7} + 72 T^{6} - 297 T^{5} + \cdots + 11664$$
$97$ $$T^{8} + 3 T^{7} + 314 T^{6} + \cdots + 246238864$$