# Properties

 Label 637.2.e.o Level $637$ Weight $2$ Character orbit 637.e Analytic conductor $5.086$ Analytic rank $0$ Dimension $12$ 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: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - 2 x^{11} + 9 x^{10} - 6 x^{9} + 34 x^{8} - 18 x^{7} + 85 x^{6} - 2 x^{5} + 92 x^{4} - 26 x^{3} + \cdots + 1$$ x^12 - 2*x^11 + 9*x^10 - 6*x^9 + 34*x^8 - 18*x^7 + 85*x^6 - 2*x^5 + 92*x^4 - 26*x^3 + 43*x^2 + 6*x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{11} - \beta_{3}) q^{2} + (\beta_{8} - \beta_{2} - \beta_1) q^{3} + (\beta_{9} - \beta_{8} - \beta_{7} + \cdots - \beta_1) q^{4}+ \cdots + ( - \beta_{11} + \beta_{3} - 3 \beta_1) q^{9}+O(q^{10})$$ q + (b11 - b3) * q^2 + (b8 - b2 - b1) * q^3 + (b9 - b8 - b7 - b5 - b4 - b2 - b1) * q^4 + (b9 - b8 + 1) * q^5 + (-b5 - b3 - b2 - 1) * q^6 + (-b6 - b5 - b2) * q^8 + (-b11 + b3 - 3*b1) * q^9 $$q + (\beta_{11} - \beta_{3}) q^{2} + (\beta_{8} - \beta_{2} - \beta_1) q^{3} + (\beta_{9} - \beta_{8} - \beta_{7} + \cdots - \beta_1) q^{4}+ \cdots + (\beta_{6} + \beta_{5} + \cdots - 5 \beta_{2}) q^{99}+O(q^{100})$$ q + (b11 - b3) * q^2 + (b8 - b2 - b1) * q^3 + (b9 - b8 - b7 - b5 - b4 - b2 - b1) * q^4 + (b9 - b8 + 1) * q^5 + (-b5 - b3 - b2 - 1) * q^6 + (-b6 - b5 - b2) * q^8 + (-b11 + b3 - 3*b1) * q^9 + (2*b11 + b8 + b2 + b1) * q^10 + (-2*b10 - b9 - 2*b6 + b4) * q^11 + (-b11 + b10 + b9 + b3 - b1) * q^12 - q^13 + (-b6 - b5 + 2*b4 - b3 - b2 + 2) * q^15 + (-b11 + b9 + b7 + b3) * q^16 + (b11 + b10 + b9 + 3*b8 + b7 + b6 + b5 - b4 + 2*b2 + 2*b1) * q^17 + (-b9 - 2*b7 - 2*b5 + b4 - 2*b2 - 2*b1) * q^18 + (-b11 - b10 - b9 + b7 + b3) * q^19 + (-b5 - b3 - b2 - 3) * q^20 + (2*b6 - 4*b5 - 2*b4 + b3 - b2 - 3) * q^22 + (-2*b10 - b9 + b7 + b1) * q^23 + (-2*b11 - b10 - 2*b9 + 2*b8 - b6 + 2*b4 - b2 - b1) * q^24 + (b9 + b8 + b7 + b5 - b4 + b2 + b1) * q^25 + (-b11 + b3) * q^26 + (b5 + 4*b3 + 4*b2 - 2) * q^27 + (-2*b5 - 2*b4 - 1) * q^29 + (3*b11 + b8 + 2*b7 - 3*b3 + 3*b1 - 1) * q^30 + (2*b11 - b10 + 3*b9 + b8 - b6 - 3*b4 - b2 - b1) * q^31 + (b11 - b10 - b9 + 4*b8 - b7 - b6 - b5 + b4 + b2 + b1) * q^32 + (-b11 - 3*b10 - 4*b9 + b8 + b7 + b3 + 2*b1 - 1) * q^33 + (3*b5 - 3*b3 + 3*b2 + 1) * q^34 + (-2*b6 - 2*b5 + b3 + b2 - 3) * q^36 + (-2*b11 - 2*b10 - b9 + 2*b7 + 2*b3 - 2*b1) * q^37 + (2*b10 + b8 - b7 + 2*b6 - b5 + b2 + b1) * q^38 + (-b8 + b2 + b1) * q^39 + (b11 + b10 + b9 - b3 + 3*b1) * q^40 + (b6 + 2*b5 + 2*b3 - 3*b2) * q^41 + (2*b6 + 2*b4 + 4*b2 + 1) * q^43 + (-3*b11 + 2*b10 - b9 - 2*b7 + 3*b3 + 4*b1) * q^44 + (-5*b11 - 3*b10 - 3*b9 + 2*b8 - 3*b7 - 3*b6 - 3*b5 + 3*b4 - 4*b2 - 4*b1) * q^45 + (b11 + 3*b10 + 2*b9 - 2*b8 - 3*b7 + 3*b6 - 3*b5 - 2*b4 + b2 + b1) * q^46 + (3*b11 + 3*b10 + b9 - 6*b8 - 3*b7 - 3*b3 + 6) * q^47 + (-b6 - b5 + 3*b4 - b3 + b2 + 2) * q^48 + (b6 + b5 + 3*b2 + 2) * q^50 + (5*b11 + 2*b10 + 4*b9 - b8 + 2*b7 - 5*b3 + b1 + 1) * q^51 + (-b9 + b8 + b7 + b5 + b4 + b2 + b1) * q^52 + (-4*b11 - b9 + 2*b8 + b7 + b5 + b4 - b2 - b1) * q^53 + (-2*b11 - b10 - 4*b9 + 8*b8 + 2*b3 - b1 - 8) * q^54 + (-2*b6 + 3*b5 - 3*b2 + 1) * q^55 + (2*b6 - 2*b4 + b3) * q^57 + (-3*b11 + 2*b10 - 2*b8 + 3*b3 + 2) * q^58 + (-b11 - 2*b10 - 2*b9 + 5*b8 + 3*b7 - 2*b6 + 3*b5 + 2*b4 + b2 + b1) * q^59 + (-3*b11 - b9 - 2*b8 - 2*b7 - 2*b5 + b4) * q^60 + (b11 + 3*b10 - b9 - 2*b8 - 2*b7 - b3 - 3*b1 + 2) * q^61 + (b6 - 5*b5 - 3*b4 + 3*b3 - 5) * q^62 + (-4*b5 - 2*b3 - 2*b2 - 4) * q^64 + (-b9 + b8 - 1) * q^65 + (-2*b11 + 4*b10 + 2*b9 - 2*b8 - 3*b7 + 4*b6 - 3*b5 - 2*b4) * q^66 + (b11 + 3*b10 + b9 - 4*b8 + b7 + 3*b6 + b5 - b4 - b2 - b1) * q^67 + (3*b11 - b10 + 5*b9 - 6*b8 - 4*b7 - 3*b3 - 5*b1 + 6) * q^68 + (3*b6 - 2*b5 - 3*b4 - b3 - 4*b2 + 1) * q^69 + (-3*b6 + 5*b5 + 5*b4 - 4*b3 - b2 + 2) * q^71 + (-5*b11 - 5*b9 + 4*b8 + 5*b3 - 2*b1 - 4) * q^72 + (b11 + 5*b10 + b9 - 2*b8 - b7 + 5*b6 - b5 - b4 + 2*b2 + 2*b1) * q^73 + (b11 + 4*b10 + b8 - 4*b7 + 4*b6 - 4*b5 + b2 + b1) * q^74 + (3*b11 + b10 + 3*b9 - 2*b8 + 2*b7 - 3*b3 + b1 + 2) * q^75 + (-b6 + 3*b5 - b3 + 3) * q^76 + (b5 + b3 + b2 + 1) * q^78 + (3*b11 - 3*b9 - 3*b8 - 3*b7 - 3*b3 + 2*b1 + 3) * q^79 + (-2*b11 - b10 - 4*b8 + 2*b7 - b6 + 2*b5 + b2 + b1) * q^80 + (6*b11 + b9 - 6*b8 + 5*b7 + 5*b5 - b4 + 5*b2 + 5*b1) * q^81 + (b11 - b10 - b9 + 8*b8 + 3*b7 - b3 + 3*b1 - 8) * q^82 + (-2*b6 - b4 + 2*b2 - 7) * q^83 + (2*b6 + 2*b5 + b4 + 4*b3 + 6*b2) * q^85 + (5*b11 + 2*b10 + 2*b9 - 4*b8 - 8*b7 - 5*b3 - 2*b1 + 4) * q^86 + (2*b10 + 2*b9 - 3*b8 + 2*b6 - 2*b4 + b2 + b1) * q^87 + (-b11 - b9 + 8*b8 + 3*b7 + 3*b5 + b4 + 2*b2 + 2*b1) * q^88 + (b11 - 2*b10 + 3*b9 - 4*b8 - b7 - b3 - b1 + 4) * q^89 + (-5*b5 + 2*b4 - 2*b3 - 5*b2 + 5) * q^90 + (-2*b6 + 4*b5 + b3 - 3*b2 + 3) * q^92 + (4*b11 + 2*b10 + 5*b9 + 2*b8 + 6*b7 - 4*b3 - 2) * q^93 + (4*b11 - 6*b10 - 5*b8 + 3*b7 - 6*b6 + 3*b5 - 5*b2 - 5*b1) * q^94 + (-2*b11 - 2*b10 + b8 + 2*b7 - 2*b6 + 2*b5 - 2*b2 - 2*b1) * q^95 + (-2*b10 - 4*b9 + 4*b8 - 4) * q^96 + (-2*b6 - 5*b5 + b4 + 3*b3 + 3*b2 + 4) * q^97 + (b6 + b5 - 7*b4 + 2*b3 - 5*b2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 8 q^{3} - 4 q^{4} + 6 q^{5} - 8 q^{6} - 6 q^{9}+O(q^{10})$$ 12 * q + 8 * q^3 - 4 * q^4 + 6 * q^5 - 8 * q^6 - 6 * q^9 $$12 q + 8 q^{3} - 4 q^{4} + 6 q^{5} - 8 q^{6} - 6 q^{9} + 4 q^{10} - 4 q^{11} - 4 q^{12} - 12 q^{13} + 24 q^{15} + 16 q^{17} + 4 q^{18} + 2 q^{19} - 32 q^{20} - 24 q^{22} + 6 q^{23} + 12 q^{24} + 4 q^{25} - 40 q^{27} - 12 q^{29} + 6 q^{31} + 20 q^{32} + 4 q^{33} - 48 q^{36} + 8 q^{38} - 8 q^{39} + 4 q^{40} + 16 q^{41} + 4 q^{43} + 4 q^{44} + 14 q^{45} - 8 q^{46} + 30 q^{47} + 16 q^{48} + 16 q^{50} + 4 q^{51} + 4 q^{52} + 14 q^{53} - 48 q^{54} + 16 q^{55} + 8 q^{57} + 8 q^{58} + 24 q^{59} - 12 q^{60} - 56 q^{62} - 40 q^{64} - 6 q^{65} - 4 q^{66} - 16 q^{67} + 28 q^{68} + 40 q^{69} + 16 q^{71} - 28 q^{72} - 6 q^{73} + 12 q^{74} + 12 q^{75} + 32 q^{76} + 8 q^{78} + 22 q^{79} - 28 q^{80} - 46 q^{81} - 40 q^{82} - 100 q^{83} - 16 q^{85} + 16 q^{86} - 16 q^{87} + 44 q^{88} + 26 q^{89} + 80 q^{90} + 40 q^{92} - 16 q^{93} - 32 q^{94} + 6 q^{95} - 20 q^{96} + 28 q^{97} + 24 q^{99}+O(q^{100})$$ 12 * q + 8 * q^3 - 4 * q^4 + 6 * q^5 - 8 * q^6 - 6 * q^9 + 4 * q^10 - 4 * q^11 - 4 * q^12 - 12 * q^13 + 24 * q^15 + 16 * q^17 + 4 * q^18 + 2 * q^19 - 32 * q^20 - 24 * q^22 + 6 * q^23 + 12 * q^24 + 4 * q^25 - 40 * q^27 - 12 * q^29 + 6 * q^31 + 20 * q^32 + 4 * q^33 - 48 * q^36 + 8 * q^38 - 8 * q^39 + 4 * q^40 + 16 * q^41 + 4 * q^43 + 4 * q^44 + 14 * q^45 - 8 * q^46 + 30 * q^47 + 16 * q^48 + 16 * q^50 + 4 * q^51 + 4 * q^52 + 14 * q^53 - 48 * q^54 + 16 * q^55 + 8 * q^57 + 8 * q^58 + 24 * q^59 - 12 * q^60 - 56 * q^62 - 40 * q^64 - 6 * q^65 - 4 * q^66 - 16 * q^67 + 28 * q^68 + 40 * q^69 + 16 * q^71 - 28 * q^72 - 6 * q^73 + 12 * q^74 + 12 * q^75 + 32 * q^76 + 8 * q^78 + 22 * q^79 - 28 * q^80 - 46 * q^81 - 40 * q^82 - 100 * q^83 - 16 * q^85 + 16 * q^86 - 16 * q^87 + 44 * q^88 + 26 * q^89 + 80 * q^90 + 40 * q^92 - 16 * q^93 - 32 * q^94 + 6 * q^95 - 20 * q^96 + 28 * q^97 + 24 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 2 x^{11} + 9 x^{10} - 6 x^{9} + 34 x^{8} - 18 x^{7} + 85 x^{6} - 2 x^{5} + 92 x^{4} - 26 x^{3} + \cdots + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 512378 \nu^{11} + 5336 \nu^{10} + 3065721 \nu^{9} + 4369060 \nu^{8} + 17395862 \nu^{7} + \cdots + 9713364 ) / 62842357$$ (512378*v^11 + 5336*v^10 + 3065721*v^9 + 4369060*v^8 + 17395862*v^7 + 17140164*v^6 + 44372910*v^5 + 58644524*v^4 + 109670152*v^3 + 40367568*v^2 + 5740174*v + 9713364) / 62842357 $$\beta_{3}$$ $$=$$ $$( 517714 \nu^{11} - 1551017 \nu^{10} + 4377607 \nu^{9} - 4394050 \nu^{8} + 8967106 \nu^{7} + \cdots + 115458972 ) / 62842357$$ (517714*v^11 - 1551017*v^10 + 4377607*v^9 - 4394050*v^8 + 8967106*v^7 - 16319384*v^6 + 15296370*v^5 + 3886852*v^4 - 55980756*v^3 + 6182709*v^2 + 898922*v + 115458972) / 62842357 $$\beta_{4}$$ $$=$$ $$( 2297118 \nu^{11} - 4668361 \nu^{10} + 18068728 \nu^{9} - 6968746 \nu^{8} + 52032662 \nu^{7} + \cdots + 103693971 ) / 62842357$$ (2297118*v^11 - 4668361*v^10 + 18068728*v^9 - 6968746*v^8 + 52032662*v^7 - 10138403*v^6 + 109881238*v^5 + 95992492*v^4 + 33981855*v^3 + 76649413*v^2 + 10958926*v + 103693971) / 62842357 $$\beta_{5}$$ $$=$$ $$( - 1878 \nu^{11} + 1119 \nu^{10} - 12173 \nu^{9} - 9570 \nu^{8} - 57462 \nu^{7} - 38216 \nu^{6} + \cdots - 66913 ) / 45971$$ (-1878*v^11 + 1119*v^10 - 12173*v^9 - 9570*v^8 - 57462*v^7 - 38216*v^6 - 141030*v^5 - 174444*v^4 - 233985*v^3 - 122643*v^2 - 17454*v - 66913) / 45971 $$\beta_{6}$$ $$=$$ $$( 6705029 \nu^{11} - 7815976 \nu^{10} + 48698198 \nu^{9} + 12543658 \nu^{8} + 184020371 \nu^{7} + \cdots + 159636402 ) / 62842357$$ (6705029*v^11 - 7815976*v^10 + 48698198*v^9 + 12543658*v^8 + 184020371*v^7 + 87381018*v^6 + 431005791*v^5 + 486894758*v^4 + 510469395*v^3 + 352920684*v^2 + 50284591*v + 159636402) / 62842357 $$\beta_{7}$$ $$=$$ $$( - 9197775 \nu^{11} + 20220925 \nu^{10} - 86127174 \nu^{9} + 69981075 \nu^{8} - 318884784 \nu^{7} + \cdots + 10820061 ) / 62842357$$ (-9197775*v^11 + 20220925*v^10 - 86127174*v^9 + 69981075*v^8 - 318884784*v^7 + 220945734*v^6 - 813418056*v^5 + 167410082*v^4 - 854628237*v^3 + 383580396*v^2 - 552502129*v + 10820061) / 62842357 $$\beta_{8}$$ $$=$$ $$( 9713364 \nu^{11} - 19939106 \nu^{10} + 87414940 \nu^{9} - 61345905 \nu^{8} + 325885316 \nu^{7} + \cdots + 52540010 ) / 62842357$$ (9713364*v^11 - 19939106*v^10 + 87414940*v^9 - 61345905*v^8 + 325885316*v^7 - 192236414*v^6 + 808495776*v^5 - 63799638*v^4 + 834984964*v^3 - 362217616*v^2 + 377307084*v + 52540010) / 62842357 $$\beta_{9}$$ $$=$$ $$( - 14791857 \nu^{11} + 31867988 \nu^{10} - 138901006 \nu^{9} + 113931690 \nu^{8} + \cdots + 17525090 ) / 62842357$$ (-14791857*v^11 + 31867988*v^10 - 138901006*v^9 + 113931690*v^8 - 526956324*v^7 + 359867408*v^6 - 1313722872*v^5 + 273803355*v^4 - 1381879675*v^3 + 621612052*v^2 - 609066000*v + 17525090) / 62842357 $$\beta_{10}$$ $$=$$ $$( 15232235 \nu^{11} - 36300822 \nu^{10} + 151870530 \nu^{9} - 148661200 \nu^{8} + 577528481 \nu^{7} + \cdots - 20852300 ) / 62842357$$ (15232235*v^11 - 36300822*v^10 + 151870530*v^9 - 148661200*v^8 + 577528481*v^7 - 473129740*v^6 + 1478901160*v^5 - 492235424*v^4 + 1591586825*v^3 - 747084060*v^2 + 943653761*v - 20852300) / 62842357 $$\beta_{11}$$ $$=$$ $$( 18914350 \nu^{11} - 39883548 \nu^{10} + 171764159 \nu^{9} - 127060870 \nu^{8} + 634374770 \nu^{7} + \cdots + 95366656 ) / 62842357$$ (18914350*v^11 - 39883548*v^10 + 171764159*v^9 - 127060870*v^8 + 634374770*v^7 - 401612992*v^6 + 1572618642*v^5 - 186243800*v^4 + 1560299776*v^3 - 701960443*v^2 + 686031637*v + 95366656) / 62842357
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{11} - 2\beta_{8} + \beta_{2} + \beta_1$$ b11 - 2*b8 + b2 + b1 $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{3} + 4\beta_{2} - 1$$ b5 + b3 + 4*b2 - 1 $$\nu^{4}$$ $$=$$ $$-5\beta_{11} - \beta_{9} + 7\beta_{8} - \beta_{7} + 5\beta_{3} - 6\beta _1 - 7$$ -5*b11 - b9 + 7*b8 - b7 + 5*b3 - 6*b1 - 7 $$\nu^{5}$$ $$=$$ $$- 8 \beta_{11} - \beta_{10} - 2 \beta_{9} + 8 \beta_{8} - 6 \beta_{7} - \beta_{6} - 6 \beta_{5} + \cdots - 18 \beta_1$$ -8*b11 - b10 - 2*b9 + 8*b8 - 6*b7 - b6 - 6*b5 + 2*b4 - 18*b2 - 18*b1 $$\nu^{6}$$ $$=$$ $$-2\beta_{6} - 9\beta_{5} + 9\beta_{4} - 26\beta_{3} - 33\beta_{2} + 30$$ -2*b6 - 9*b5 + 9*b4 - 26*b3 - 33*b2 + 30 $$\nu^{7}$$ $$=$$ $$51\beta_{11} + 9\beta_{10} + 20\beta_{9} - 49\beta_{8} + 33\beta_{7} - 51\beta_{3} + 87\beta _1 + 49$$ 51*b11 + 9*b10 + 20*b9 - 49*b8 + 33*b7 - 51*b3 + 87*b1 + 49 $$\nu^{8}$$ $$=$$ $$140 \beta_{11} + 20 \beta_{10} + 62 \beta_{9} - 143 \beta_{8} + 62 \beta_{7} + 20 \beta_{6} + \cdots + 178 \beta_1$$ 140*b11 + 20*b10 + 62*b9 - 143*b8 + 62*b7 + 20*b6 + 62*b5 - 62*b4 + 178*b2 + 178*b1 $$\nu^{9}$$ $$=$$ $$62\beta_{6} + 182\beta_{5} - 144\beta_{4} + 302\beta_{3} + 441\beta_{2} - 278$$ 62*b6 + 182*b5 - 144*b4 + 302*b3 + 441*b2 - 278 $$\nu^{10}$$ $$=$$ $$-767\beta_{11} - 144\beta_{10} - 388\beta_{9} + 724\beta_{8} - 384\beta_{7} + 767\beta_{3} - 959\beta _1 - 724$$ -767*b11 - 144*b10 - 388*b9 + 724*b8 - 384*b7 + 767*b3 - 959*b1 - 724 $$\nu^{11}$$ $$=$$ $$- 1731 \beta_{11} - 388 \beta_{10} - 916 \beta_{9} + 1539 \beta_{8} - 1011 \beta_{7} + \cdots - 2306 \beta_1$$ -1731*b11 - 388*b10 - 916*b9 + 1539*b8 - 1011*b7 - 388*b6 - 1011*b5 + 916*b4 - 2306*b2 - 2306*b1

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 0.379209 − 0.656810i −0.0731214 + 0.126650i 0.954516 − 1.65327i −0.602377 + 1.04335i 1.17550 − 2.03602i −0.833726 + 1.44406i 0.379209 + 0.656810i −0.0731214 − 0.126650i 0.954516 + 1.65327i −0.602377 − 1.04335i 1.17550 + 2.03602i −0.833726 − 1.44406i
−1.09161 + 1.89072i 0.879209 + 1.52284i −1.38322 2.39581i 1.05533 1.82788i −3.83901 0 1.67333 −0.0460183 + 0.0797060i 2.30401 + 3.99066i
79.2 −0.916185 + 1.58688i 0.426879 + 0.739375i −0.678791 1.17570i 1.31278 2.27379i −1.56440 0 −1.17715 1.13555 1.96683i 2.40549 + 4.16643i
79.3 −0.132313 + 0.229173i 1.45452 + 2.51930i 0.964986 + 1.67141i −0.717577 + 1.24288i −0.769807 0 −1.03998 −2.73124 + 4.73064i −0.189890 0.328899i
79.4 0.328092 0.568272i −0.102377 0.177322i 0.784711 + 1.35916i −0.679981 + 1.17776i −0.134356 0 2.34220 1.47904 2.56177i 0.446193 + 0.772828i
79.5 0.588093 1.01861i 1.67550 + 2.90205i 0.308293 + 0.533979i 1.57431 2.72679i 3.94140 0 3.07759 −4.11459 + 7.12667i −1.85168 3.20721i
79.6 1.22392 2.11990i −0.333726 0.578030i −1.99598 3.45713i 0.455143 0.788331i −1.63382 0 −4.87599 1.27725 2.21227i −1.11412 1.92971i
508.1 −1.09161 1.89072i 0.879209 1.52284i −1.38322 + 2.39581i 1.05533 + 1.82788i −3.83901 0 1.67333 −0.0460183 0.0797060i 2.30401 3.99066i
508.2 −0.916185 1.58688i 0.426879 0.739375i −0.678791 + 1.17570i 1.31278 + 2.27379i −1.56440 0 −1.17715 1.13555 + 1.96683i 2.40549 4.16643i
508.3 −0.132313 0.229173i 1.45452 2.51930i 0.964986 1.67141i −0.717577 1.24288i −0.769807 0 −1.03998 −2.73124 4.73064i −0.189890 + 0.328899i
508.4 0.328092 + 0.568272i −0.102377 + 0.177322i 0.784711 1.35916i −0.679981 1.17776i −0.134356 0 2.34220 1.47904 + 2.56177i 0.446193 0.772828i
508.5 0.588093 + 1.01861i 1.67550 2.90205i 0.308293 0.533979i 1.57431 + 2.72679i 3.94140 0 3.07759 −4.11459 7.12667i −1.85168 + 3.20721i
508.6 1.22392 + 2.11990i −0.333726 + 0.578030i −1.99598 + 3.45713i 0.455143 + 0.788331i −1.63382 0 −4.87599 1.27725 + 2.21227i −1.11412 + 1.92971i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 79.6 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.e.o 12
7.b odd 2 1 637.2.e.n 12
7.c even 3 1 637.2.a.m 6
7.c even 3 1 inner 637.2.e.o 12
7.d odd 6 1 637.2.a.n yes 6
7.d odd 6 1 637.2.e.n 12
21.g even 6 1 5733.2.a.br 6
21.h odd 6 1 5733.2.a.bu 6
91.r even 6 1 8281.2.a.cc 6
91.s odd 6 1 8281.2.a.cd 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.a.m 6 7.c even 3 1
637.2.a.n yes 6 7.d odd 6 1
637.2.e.n 12 7.b odd 2 1
637.2.e.n 12 7.d odd 6 1
637.2.e.o 12 1.a even 1 1 trivial
637.2.e.o 12 7.c even 3 1 inner
5733.2.a.br 6 21.g even 6 1
5733.2.a.bu 6 21.h odd 6 1
8281.2.a.cc 6 91.r even 6 1
8281.2.a.cd 6 91.s odd 6 1

## Hecke kernels

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

 $$T_{2}^{12} + 8T_{2}^{10} + 50T_{2}^{8} - 4T_{2}^{7} + 108T_{2}^{6} - 64T_{2}^{5} + 180T_{2}^{4} - 56T_{2}^{3} + 44T_{2}^{2} + 8T_{2} + 4$$ T2^12 + 8*T2^10 + 50*T2^8 - 4*T2^7 + 108*T2^6 - 64*T2^5 + 180*T2^4 - 56*T2^3 + 44*T2^2 + 8*T2 + 4 $$T_{3}^{12} - 8 T_{3}^{11} + 44 T_{3}^{10} - 136 T_{3}^{9} + 316 T_{3}^{8} - 424 T_{3}^{7} + 452 T_{3}^{6} + \cdots + 4$$ T3^12 - 8*T3^11 + 44*T3^10 - 136*T3^9 + 316*T3^8 - 424*T3^7 + 452*T3^6 - 192*T3^5 + 200*T3^4 - 48*T3^3 + 88*T3^2 + 16*T3 + 4 $$T_{5}^{12} - 6 T_{5}^{11} + 31 T_{5}^{10} - 78 T_{5}^{9} + 200 T_{5}^{8} - 278 T_{5}^{7} + 637 T_{5}^{6} + \cdots + 961$$ T5^12 - 6*T5^11 + 31*T5^10 - 78*T5^9 + 200*T5^8 - 278*T5^7 + 637*T5^6 - 670*T5^5 + 1430*T5^4 - 682*T5^3 + 1637*T5^2 - 806*T5 + 961

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} + 8 T^{10} + \cdots + 4$$
$3$ $$T^{12} - 8 T^{11} + \cdots + 4$$
$5$ $$T^{12} - 6 T^{11} + \cdots + 961$$
$7$ $$T^{12}$$
$11$ $$T^{12} + 4 T^{11} + \cdots + 315844$$
$13$ $$(T + 1)^{12}$$
$17$ $$T^{12} - 16 T^{11} + \cdots + 28026436$$
$19$ $$T^{12} - 2 T^{11} + \cdots + 5329$$
$23$ $$T^{12} - 6 T^{11} + \cdots + 279841$$
$29$ $$(T^{6} + 6 T^{5} + \cdots + 529)^{2}$$
$31$ $$T^{12} + \cdots + 1957974001$$
$37$ $$T^{12} + 82 T^{10} + \cdots + 64516$$
$41$ $$(T^{6} - 8 T^{5} + \cdots + 28784)^{2}$$
$43$ $$(T^{6} - 2 T^{5} + \cdots + 35153)^{2}$$
$47$ $$T^{12} + \cdots + 18391970689$$
$53$ $$T^{12} - 14 T^{11} + \cdots + 1739761$$
$59$ $$T^{12} - 24 T^{11} + \cdots + 2347024$$
$61$ $$T^{12} + \cdots + 46908629056$$
$67$ $$T^{12} + 16 T^{11} + \cdots + 37356544$$
$71$ $$(T^{6} - 8 T^{5} + \cdots - 1206162)^{2}$$
$73$ $$T^{12} + \cdots + 20351019649$$
$79$ $$T^{12} - 22 T^{11} + \cdots + 62615569$$
$83$ $$(T^{6} + 50 T^{5} + \cdots - 167041)^{2}$$
$89$ $$T^{12} - 26 T^{11} + \cdots + 99181681$$
$97$ $$(T^{6} - 14 T^{5} + \cdots + 217287)^{2}$$