# Properties

 Label 136.2.n.c Level $136$ Weight $2$ Character orbit 136.n Analytic conductor $1.086$ 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] = [136,2,Mod(9,136)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(136, base_ring=CyclotomicField(8))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("136.9");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$136 = 2^{3} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 136.n (of order $$8$$, degree $$4$$, 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: $$1.08596546749$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$3$$ over $$\Q(\zeta_{8})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} + 28x^{10} + 258x^{8} + 880x^{6} + 1033x^{4} + 132x^{2} + 4$$ x^12 + 28*x^10 + 258*x^8 + 880*x^6 + 1033*x^4 + 132*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + \beta_{8} q^{5} + (\beta_{11} - \beta_{9} - \beta_{8} - \beta_{6} - \beta_{3} + 1) q^{7} + ( - \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} - \beta_{5} + \beta_{4}) q^{9}+O(q^{10})$$ q + b2 * q^3 + b8 * q^5 + (b11 - b9 - b8 - b6 - b3 + 1) * q^7 + (-b11 + b10 - b9 - b8 + b7 - b5 + b4) * q^9 $$q + \beta_{2} q^{3} + \beta_{8} q^{5} + (\beta_{11} - \beta_{9} - \beta_{8} - \beta_{6} - \beta_{3} + 1) q^{7} + ( - \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} - \beta_{5} + \beta_{4}) q^{9} + (\beta_{11} + \beta_{8} + \beta_{7} - \beta_{6} - \beta_{4} + \beta_1 - 1) q^{11} + ( - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} - 2 \beta_1) q^{13} + ( - \beta_{11} + \beta_{9} - \beta_{5} - \beta_{4} + \beta_1 + 1) q^{15} + (\beta_{9} + \beta_{7} + 2 \beta_{6} + \beta_{5} + \beta_{3}) q^{17} + ( - \beta_{11} + \beta_{9} + \beta_{3} - \beta_{2} - \beta_1 - 1) q^{19} + (\beta_{11} - \beta_{10} - \beta_{7} + \beta_{6} + \beta_{4} - \beta_{2} + 2 \beta_1) q^{21} + ( - \beta_{10} - 2 \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{2} - 2 \beta_1 - 1) q^{23} + ( - \beta_{10} + \beta_{8} - \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{25} + ( - 2 \beta_{6} + \beta_{5} + 2 \beta_{3} - \beta_{2} + 2) q^{27} + ( - \beta_{8} - \beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} - \beta_1 + 1) q^{29} + ( - \beta_{11} - 3 \beta_{7} - \beta_{4} - \beta_{3} - \beta_{2} - 3) q^{31} + ( - \beta_{5} - 2 \beta_{4} - \beta_{3} - 2 \beta_{2} - 2) q^{33} + (\beta_{11} + \beta_{10} + 3 \beta_{7} + 3 \beta_{6} + \beta_{5} + \beta_{3} - 2) q^{35} + ( - \beta_{11} + \beta_{10} + \beta_{9} + 2 \beta_{6} - 2 \beta_1) q^{37} + ( - 2 \beta_{8} + 4 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 4 \beta_1) q^{39} + ( - \beta_{11} + 2 \beta_{9} + \beta_{8} + 3 \beta_{7} + 2 \beta_{5} - 2 \beta_{2} - 3 \beta_1) q^{41} + (\beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + 2 \beta_{7} - \beta_{3} - \beta_{2}) q^{43} + ( - \beta_{11} + \beta_{10} - \beta_{8} + \beta_{7} - 5 \beta_{6} + \beta_{5} + \beta_{2} + \beta_1 - 5) q^{45} + ( - \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + 2 \beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{4} + \cdots - 2 \beta_1) q^{47}+ \cdots + (2 \beta_{11} - \beta_{10} - \beta_{9} - 5 \beta_{7} - 7 \beta_{6} - \beta_{4} - \beta_{3} - \beta_{2} + \cdots - 5) q^{99}+O(q^{100})$$ q + b2 * q^3 + b8 * q^5 + (b11 - b9 - b8 - b6 - b3 + 1) * q^7 + (-b11 + b10 - b9 - b8 + b7 - b5 + b4) * q^9 + (b11 + b8 + b7 - b6 - b4 + b1 - 1) * q^11 + (-b7 + b6 + b5 - b3 - 2*b1) * q^13 + (-b11 + b9 - b5 - b4 + b1 + 1) * q^15 + (b9 + b7 + 2*b6 + b5 + b3) * q^17 + (-b11 + b9 + b3 - b2 - b1 - 1) * q^19 + (b11 - b10 - b7 + b6 + b4 - b2 + 2*b1) * q^21 + (-b10 - 2*b7 - b6 + b5 + b4 + b2 - 2*b1 - 1) * q^23 + (-b10 + b8 - b7 - b5 + b4 + b3 + b2 + 2*b1 - 2) * q^25 + (-2*b6 + b5 + 2*b3 - b2 + 2) * q^27 + (-b8 - b7 - b6 - b4 + b3 - b1 + 1) * q^29 + (-b11 - 3*b7 - b4 - b3 - b2 - 3) * q^31 + (-b5 - 2*b4 - b3 - 2*b2 - 2) * q^33 + (b11 + b10 + 3*b7 + 3*b6 + b5 + b3 - 2) * q^35 + (-b11 + b10 + b9 + 2*b6 - 2*b1) * q^37 + (-2*b8 + 4*b6 - 2*b5 + 2*b4 - 2*b3 + 4*b1) * q^39 + (-b11 + 2*b9 + b8 + 3*b7 + 2*b5 - 2*b2 - 3*b1) * q^41 + (b11 - b10 + b9 + b8 + 2*b7 - b3 - b2) * q^43 + (-b11 + b10 - b8 + b7 - 5*b6 + b5 + b2 + b1 - 5) * q^45 + (-b11 + b10 - b9 + b8 + 2*b7 - 2*b6 - b5 + b4 + b3 - b2 - 2*b1) * q^47 + (2*b11 - b10 - 2*b9 - b8 - 5*b6 - b5 - b4 - 2*b3 + 2*b2 + 5*b1 + 5) * q^49 + (b11 - b10 - 2*b9 + b7 + 3*b6 + b5 + b4 - b3 + 2*b2 - 4*b1) * q^51 + (b11 - b9 - b5 - b4 + b3 - b2) * q^53 + (-b11 + b10 - 5*b7 + 5*b6 - b5 + b3 + 4*b1) * q^55 + (b11 + b8 - 3*b7 - b6 + b5 - 2*b4 + b2 - 3*b1 - 1) * q^57 + (-b11 + b10 - b9 - b8 + 2*b7 + b3 + b2 - 2*b1 + 2) * q^59 + (-b9 - 2*b7 - 4*b6 - 2*b5 + 2*b2 + 2*b1 + 4) * q^61 + (b8 - 8*b7 - b6 + b5 + b4 - b3 - b1 + 8) * q^63 + (2*b11 - b7 - b6 + b4 + b3 + 2*b2 + b1 - 1) * q^65 + (b9 + b8 + 3*b7 + 3*b6 + b5 + b4 + b3 + b2 - 4) * q^67 + (-b11 - b10 + 3*b7 + 3*b6 - 2*b5 + b4 - 2*b3 + b2 + 4) * q^69 + (b11 + 3*b7 - 2*b6 - b4 - b3 + b2 + 2*b1 + 3) * q^71 + (b10 - b9 - 2*b8 - 2*b7 - b6 + 2*b5 + 2*b4 - 2*b3 - b1 + 2) * q^73 + (-3*b11 + 2*b9 + 3*b8 + 3*b7 - 5*b6 + b5 + b3 - b2 - 3*b1 + 5) * q^75 + (2*b11 - 3*b10 + 2*b9 + 3*b8 + 10*b7 + b5 - b4 + 2*b3 + 2*b2 + 3*b1 - 3) * q^77 + (-2*b11 - b10 - 2*b8 + 2*b7 - b6 - b5 + b4 - b2 + 2*b1 - 1) * q^79 + (b11 - b10 + b9 - b8 - 4*b7 + 4*b6 + b5 - 2*b4 - b3 + 2*b2 - 5*b1) * q^81 + (b11 + b10 - b9 + b8 + 2*b6 - b5 - b4 + 4*b1 + 4) * q^83 + (2*b10 - 2*b9 - b8 + 3*b7 + 3*b6 - b5 - 2*b4 - b2 + b1 + 5) * q^85 + (3*b11 - 3*b9 - b5 - b4 - 2*b3 + 2*b2 - 5*b1 - 5) * q^87 + (-3*b11 + 3*b10 - 4*b7 + 4*b6 + b4 - b2 + 2*b1) * q^89 + (-2*b11 + 4*b10 - 2*b8 - 4*b7 - 2*b6 - 2*b5 + 2*b4 - 2*b2 - 4*b1 - 2) * q^91 + (b11 + 2*b10 + b9 - 2*b8 - 4*b7 - 3*b3 - 3*b2 + 5*b1 - 5) * q^93 + (b11 - b8 + 3*b7 - 3*b6 + b5 - 2*b3 - b2 - 3*b1 + 3) * q^95 + (2*b8 + 2*b7 - 3*b6 - 2*b5 - b4 + b3 - 3*b1 - 2) * q^97 + (2*b11 - b10 - b9 - 5*b7 - 7*b6 - b4 - b3 - b2 + 7*b1 - 5) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 4 q^{5} - 8 q^{9}+O(q^{10})$$ 12 * q + 4 * q^5 - 8 * q^9 $$12 q + 4 q^{5} - 8 q^{9} - 12 q^{11} + 20 q^{15} + 4 q^{17} - 4 q^{19} - 8 q^{23} - 16 q^{25} + 24 q^{27} + 8 q^{29} - 32 q^{31} - 24 q^{33} - 32 q^{35} + 4 q^{37} - 8 q^{39} + 16 q^{41} + 8 q^{43} - 64 q^{45} + 44 q^{49} - 8 q^{51} - 8 q^{53} - 12 q^{57} + 16 q^{59} + 44 q^{61} + 100 q^{63} - 20 q^{65} - 40 q^{67} + 56 q^{69} + 32 q^{71} + 8 q^{73} + 92 q^{75} - 12 q^{77} - 8 q^{79} + 40 q^{83} + 40 q^{85} - 84 q^{87} - 40 q^{91} - 76 q^{93} + 28 q^{95} - 16 q^{97} - 68 q^{99}+O(q^{100})$$ 12 * q + 4 * q^5 - 8 * q^9 - 12 * q^11 + 20 * q^15 + 4 * q^17 - 4 * q^19 - 8 * q^23 - 16 * q^25 + 24 * q^27 + 8 * q^29 - 32 * q^31 - 24 * q^33 - 32 * q^35 + 4 * q^37 - 8 * q^39 + 16 * q^41 + 8 * q^43 - 64 * q^45 + 44 * q^49 - 8 * q^51 - 8 * q^53 - 12 * q^57 + 16 * q^59 + 44 * q^61 + 100 * q^63 - 20 * q^65 - 40 * q^67 + 56 * q^69 + 32 * q^71 + 8 * q^73 + 92 * q^75 - 12 * q^77 - 8 * q^79 + 40 * q^83 + 40 * q^85 - 84 * q^87 - 40 * q^91 - 76 * q^93 + 28 * q^95 - 16 * q^97 - 68 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 28x^{10} + 258x^{8} + 880x^{6} + 1033x^{4} + 132x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$( -9\nu^{11} - 250\nu^{9} - 2274\nu^{7} - 7596\nu^{5} - 8901\nu^{3} - 1658\nu ) / 272$$ (-9*v^11 - 250*v^9 - 2274*v^7 - 7596*v^5 - 8901*v^3 - 1658*v) / 272 $$\beta_{2}$$ $$=$$ $$( - 39 \nu^{11} + 17 \nu^{10} - 1089 \nu^{9} + 459 \nu^{8} - 9973 \nu^{7} + 3927 \nu^{6} - 33443 \nu^{5} + 11305 \nu^{4} - 37228 \nu^{3} + 10064 \nu^{2} - 3524 \nu + 612 ) / 1088$$ (-39*v^11 + 17*v^10 - 1089*v^9 + 459*v^8 - 9973*v^7 + 3927*v^6 - 33443*v^5 + 11305*v^4 - 37228*v^3 + 10064*v^2 - 3524*v + 612) / 1088 $$\beta_{3}$$ $$=$$ $$( 39 \nu^{11} - 25 \nu^{10} + 1089 \nu^{9} - 651 \nu^{8} + 9973 \nu^{7} - 5223 \nu^{6} + 33443 \nu^{5} - 12889 \nu^{4} + 37228 \nu^{3} - 8184 \nu^{2} + 2436 \nu - 756 ) / 1088$$ (39*v^11 - 25*v^10 + 1089*v^9 - 651*v^8 + 9973*v^7 - 5223*v^6 + 33443*v^5 - 12889*v^4 + 37228*v^3 - 8184*v^2 + 2436*v - 756) / 1088 $$\beta_{4}$$ $$=$$ $$( 39 \nu^{11} + 17 \nu^{10} + 1089 \nu^{9} + 459 \nu^{8} + 9973 \nu^{7} + 3927 \nu^{6} + 33443 \nu^{5} + 11305 \nu^{4} + 37228 \nu^{3} + 10064 \nu^{2} + 3524 \nu + 612 ) / 1088$$ (39*v^11 + 17*v^10 + 1089*v^9 + 459*v^8 + 9973*v^7 + 3927*v^6 + 33443*v^5 + 11305*v^4 + 37228*v^3 + 10064*v^2 + 3524*v + 612) / 1088 $$\beta_{5}$$ $$=$$ $$( - 39 \nu^{11} - 25 \nu^{10} - 1089 \nu^{9} - 651 \nu^{8} - 9973 \nu^{7} - 5223 \nu^{6} - 33443 \nu^{5} - 12889 \nu^{4} - 37228 \nu^{3} - 8184 \nu^{2} - 2436 \nu - 756 ) / 1088$$ (-39*v^11 - 25*v^10 - 1089*v^9 - 651*v^8 - 9973*v^7 - 5223*v^6 - 33443*v^5 - 12889*v^4 - 37228*v^3 - 8184*v^2 - 2436*v - 756) / 1088 $$\beta_{6}$$ $$=$$ $$( 153 \nu^{11} - 39 \nu^{10} + 4267 \nu^{9} - 1089 \nu^{8} + 39015 \nu^{7} - 9973 \nu^{6} + 130713 \nu^{5} - 33443 \nu^{4} + 146744 \nu^{3} - 37228 \nu^{2} + 10132 \nu - 2436 ) / 1088$$ (153*v^11 - 39*v^10 + 4267*v^9 - 1089*v^8 + 39015*v^7 - 9973*v^6 + 130713*v^5 - 33443*v^4 + 146744*v^3 - 37228*v^2 + 10132*v - 2436) / 1088 $$\beta_{7}$$ $$=$$ $$( - 153 \nu^{11} - 39 \nu^{10} - 4267 \nu^{9} - 1089 \nu^{8} - 39015 \nu^{7} - 9973 \nu^{6} - 130713 \nu^{5} - 33443 \nu^{4} - 146744 \nu^{3} - 37228 \nu^{2} - 10132 \nu - 2436 ) / 1088$$ (-153*v^11 - 39*v^10 - 4267*v^9 - 1089*v^8 - 39015*v^7 - 9973*v^6 - 130713*v^5 - 33443*v^4 - 146744*v^3 - 37228*v^2 - 10132*v - 2436) / 1088 $$\beta_{8}$$ $$=$$ $$( 119 \nu^{11} - 27 \nu^{10} + 3315 \nu^{9} - 767 \nu^{8} + 30243 \nu^{7} - 7247 \nu^{6} + 100725 \nu^{5} - 25865 \nu^{4} + 110670 \nu^{3} - 31582 \nu^{2} + 3808 \nu - 2016 ) / 544$$ (119*v^11 - 27*v^10 + 3315*v^9 - 767*v^8 + 30243*v^7 - 7247*v^6 + 100725*v^5 - 25865*v^4 + 110670*v^3 - 31582*v^2 + 3808*v - 2016) / 544 $$\beta_{9}$$ $$=$$ $$( - 119 \nu^{11} - 27 \nu^{10} - 3315 \nu^{9} - 767 \nu^{8} - 30243 \nu^{7} - 7247 \nu^{6} - 100725 \nu^{5} - 25865 \nu^{4} - 110670 \nu^{3} - 31582 \nu^{2} - 3808 \nu - 2016 ) / 544$$ (-119*v^11 - 27*v^10 - 3315*v^9 - 767*v^8 - 30243*v^7 - 7247*v^6 - 100725*v^5 - 25865*v^4 - 110670*v^3 - 31582*v^2 - 3808*v - 2016) / 544 $$\beta_{10}$$ $$=$$ $$( 123 \nu^{11} - 41 \nu^{10} + 3445 \nu^{9} - 1137 \nu^{8} + 31741 \nu^{7} - 10297 \nu^{6} + 107943 \nu^{5} - 33839 \nu^{4} + 123568 \nu^{3} - 36486 \nu^{2} + 8300 \nu - 1384 ) / 544$$ (123*v^11 - 41*v^10 + 3445*v^9 - 1137*v^8 + 31741*v^7 - 10297*v^6 + 107943*v^5 - 33839*v^4 + 123568*v^3 - 36486*v^2 + 8300*v - 1384) / 544 $$\beta_{11}$$ $$=$$ $$( - 123 \nu^{11} - 41 \nu^{10} - 3445 \nu^{9} - 1137 \nu^{8} - 31741 \nu^{7} - 10297 \nu^{6} - 107943 \nu^{5} - 33839 \nu^{4} - 123568 \nu^{3} - 36486 \nu^{2} - 8300 \nu - 1384 ) / 544$$ (-123*v^11 - 41*v^10 - 3445*v^9 - 1137*v^8 - 31741*v^7 - 10297*v^6 - 107943*v^5 - 33839*v^4 - 123568*v^3 - 36486*v^2 - 8300*v - 1384) / 544
 $$\nu$$ $$=$$ $$( \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} ) / 2$$ (b5 + b4 - b3 - b2) / 2 $$\nu^{2}$$ $$=$$ $$( 2\beta_{11} + 2\beta_{10} - 4\beta_{7} - 4\beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - 8 ) / 2$$ (2*b11 + 2*b10 - 4*b7 - 4*b6 - b5 - b4 - b3 - b2 - 8) / 2 $$\nu^{3}$$ $$=$$ $$( 2\beta_{7} - 2\beta_{6} - 13\beta_{5} - 7\beta_{4} + 13\beta_{3} + 7\beta_{2} - 4\beta_1 ) / 2$$ (2*b7 - 2*b6 - 13*b5 - 7*b4 + 13*b3 + 7*b2 - 4*b1) / 2 $$\nu^{4}$$ $$=$$ $$( - 20 \beta_{11} - 20 \beta_{10} - 6 \beta_{9} - 6 \beta_{8} + 52 \beta_{7} + 52 \beta_{6} + 9 \beta_{5} + 17 \beta_{4} + 9 \beta_{3} + 17 \beta_{2} + 80 ) / 2$$ (-20*b11 - 20*b10 - 6*b9 - 6*b8 + 52*b7 + 52*b6 + 9*b5 + 17*b4 + 9*b3 + 17*b2 + 80) / 2 $$\nu^{5}$$ $$=$$ $$( 2 \beta_{11} - 2 \beta_{10} + 6 \beta_{9} - 6 \beta_{8} - 42 \beta_{7} + 42 \beta_{6} + 155 \beta_{5} + 69 \beta_{4} - 155 \beta_{3} - 69 \beta_{2} + 64 \beta_1 ) / 2$$ (2*b11 - 2*b10 + 6*b9 - 6*b8 - 42*b7 + 42*b6 + 155*b5 + 69*b4 - 155*b3 - 69*b2 + 64*b1) / 2 $$\nu^{6}$$ $$=$$ $$( 218 \beta_{11} + 218 \beta_{10} + 84 \beta_{9} + 84 \beta_{8} - 612 \beta_{7} - 612 \beta_{6} - 107 \beta_{5} - 243 \beta_{4} - 107 \beta_{3} - 243 \beta_{2} - 884 ) / 2$$ (218*b11 + 218*b10 + 84*b9 + 84*b8 - 612*b7 - 612*b6 - 107*b5 - 243*b4 - 107*b3 - 243*b2 - 884) / 2 $$\nu^{7}$$ $$=$$ $$( - 52 \beta_{11} + 52 \beta_{10} - 132 \beta_{9} + 132 \beta_{8} + 670 \beta_{7} - 670 \beta_{6} - 1817 \beta_{5} - 767 \beta_{4} + 1817 \beta_{3} + 767 \beta_{2} - 964 \beta_1 ) / 2$$ (-52*b11 + 52*b10 - 132*b9 + 132*b8 + 670*b7 - 670*b6 - 1817*b5 - 767*b4 + 1817*b3 + 767*b2 - 964*b1) / 2 $$\nu^{8}$$ $$=$$ $$( - 2452 \beta_{11} - 2452 \beta_{10} - 998 \beta_{9} - 998 \beta_{8} + 7084 \beta_{7} + 7084 \beta_{6} + 1381 \beta_{5} + 3285 \beta_{4} + 1381 \beta_{3} + 3285 \beta_{2} + 10072 ) / 2$$ (-2452*b11 - 2452*b10 - 998*b9 - 998*b8 + 7084*b7 + 7084*b6 + 1381*b5 + 3285*b4 + 1381*b3 + 3285*b2 + 10072) / 2 $$\nu^{9}$$ $$=$$ $$( 906 \beta_{11} - 906 \beta_{10} + 2214 \beta_{9} - 2214 \beta_{8} - 9690 \beta_{7} + 9690 \beta_{6} + 21307 \beta_{5} + 8869 \beta_{4} - 21307 \beta_{3} - 8869 \beta_{2} + 13760 \beta_1 ) / 2$$ (906*b11 - 906*b10 + 2214*b9 - 2214*b8 - 9690*b7 + 9690*b6 + 21307*b5 + 8869*b4 - 21307*b3 - 8869*b2 + 13760*b1) / 2 $$\nu^{10}$$ $$=$$ $$( 27962 \beta_{11} + 27962 \beta_{10} + 11532 \beta_{9} + 11532 \beta_{8} - 82108 \beta_{7} - 82108 \beta_{6} - 17963 \beta_{5} - 43211 \beta_{4} - 17963 \beta_{3} - 43211 \beta_{2} - 116276 ) / 2$$ (27962*b11 + 27962*b10 + 11532*b9 + 11532*b8 - 82108*b7 - 82108*b6 - 17963*b5 - 43211*b4 - 17963*b3 - 43211*b2 - 116276) / 2 $$\nu^{11}$$ $$=$$ $$( - 13716 \beta_{11} + 13716 \beta_{10} - 33212 \beta_{9} + 33212 \beta_{8} + 133350 \beta_{7} - 133350 \beta_{6} - 250913 \beta_{5} - 104063 \beta_{4} + 250913 \beta_{3} + 104063 \beta_{2} - 188772 \beta_1 ) / 2$$ (-13716*b11 + 13716*b10 - 33212*b9 + 33212*b8 + 133350*b7 - 133350*b6 - 250913*b5 - 104063*b4 + 250913*b3 + 104063*b2 - 188772*b1) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/136\mathbb{Z}\right)^\times$$.

 $$n$$ $$69$$ $$103$$ $$105$$ $$\chi(n)$$ $$1$$ $$1$$ $$\beta_{7}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1
 3.18938i 0.306239i − 3.49562i − 1.75800i 0.216105i 1.54190i 1.75800i − 0.216105i − 1.54190i − 3.18938i − 0.306239i 3.49562i
0 −2.25523 0.934148i 0 −1.35305 + 3.26655i 0 0.666590 + 1.60929i 0 2.09212 + 2.09212i 0
9.2 0 −0.216544 0.0896953i 0 1.33137 3.21420i 0 0.934059 + 2.25502i 0 −2.08247 2.08247i 0
9.3 0 2.47178 + 1.02384i 0 −0.392531 + 0.947653i 0 −0.893542 2.15720i 0 2.94010 + 2.94010i 0
25.1 0 −1.24310 + 3.00110i 0 0.194339 + 0.0804980i 0 −1.76317 + 0.730328i 0 −5.33998 5.33998i 0
25.2 0 0.152809 0.368914i 0 −1.09698 0.454383i 0 4.72436 1.95689i 0 2.00857 + 2.00857i 0
25.3 0 1.09029 2.63218i 0 3.31685 + 1.37389i 0 −3.66830 + 1.51946i 0 −3.61834 3.61834i 0
49.1 0 −1.24310 3.00110i 0 0.194339 0.0804980i 0 −1.76317 0.730328i 0 −5.33998 + 5.33998i 0
49.2 0 0.152809 + 0.368914i 0 −1.09698 + 0.454383i 0 4.72436 + 1.95689i 0 2.00857 2.00857i 0
49.3 0 1.09029 + 2.63218i 0 3.31685 1.37389i 0 −3.66830 1.51946i 0 −3.61834 + 3.61834i 0
121.1 0 −2.25523 + 0.934148i 0 −1.35305 3.26655i 0 0.666590 1.60929i 0 2.09212 2.09212i 0
121.2 0 −0.216544 + 0.0896953i 0 1.33137 + 3.21420i 0 0.934059 2.25502i 0 −2.08247 + 2.08247i 0
121.3 0 2.47178 1.02384i 0 −0.392531 0.947653i 0 −0.893542 + 2.15720i 0 2.94010 2.94010i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 121.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.d even 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 136.2.n.c 12
3.b odd 2 1 1224.2.bq.c 12
4.b odd 2 1 272.2.v.f 12
17.d even 8 1 inner 136.2.n.c 12
17.e odd 16 2 2312.2.a.w 12
17.e odd 16 2 2312.2.b.n 12
51.g odd 8 1 1224.2.bq.c 12
68.g odd 8 1 272.2.v.f 12
68.i even 16 2 4624.2.a.bt 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.2.n.c 12 1.a even 1 1 trivial
136.2.n.c 12 17.d even 8 1 inner
272.2.v.f 12 4.b odd 2 1
272.2.v.f 12 68.g odd 8 1
1224.2.bq.c 12 3.b odd 2 1
1224.2.bq.c 12 51.g odd 8 1
2312.2.a.w 12 17.e odd 16 2
2312.2.b.n 12 17.e odd 16 2
4624.2.a.bt 12 68.i even 16 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{12} + 4 T_{3}^{10} - 8 T_{3}^{9} + 8 T_{3}^{8} + 40 T_{3}^{7} - 224 T_{3}^{6} + 96 T_{3}^{5} + 3652 T_{3}^{4} + 464 T_{3}^{3} + 304 T_{3}^{2} + 192 T_{3} + 32$$ acting on $$S_{2}^{\mathrm{new}}(136, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12} + 4 T^{10} - 8 T^{9} + 8 T^{8} + \cdots + 32$$
$5$ $$T^{12} - 4 T^{11} + 16 T^{10} - 56 T^{9} + \cdots + 128$$
$7$ $$T^{12} - 22 T^{10} - 12 T^{9} + \cdots + 147968$$
$11$ $$T^{12} + 12 T^{11} + 48 T^{10} + \cdots + 16928$$
$13$ $$T^{12} + 92 T^{10} + 3060 T^{8} + \cdots + 262144$$
$17$ $$T^{12} - 4 T^{11} - 40 T^{10} + \cdots + 24137569$$
$19$ $$T^{12} + 4 T^{11} + 8 T^{10} + 8 T^{9} + \cdots + 1024$$
$23$ $$T^{12} + 8 T^{11} - 6 T^{10} + \cdots + 43655168$$
$29$ $$T^{12} - 8 T^{11} + 20 T^{10} + \cdots + 118210688$$
$31$ $$T^{12} + 32 T^{11} + \cdots + 103219712$$
$37$ $$T^{12} - 4 T^{11} + 56 T^{10} - 328 T^{9} + \cdots + 512$$
$41$ $$T^{12} - 16 T^{11} + \cdots + 591542408$$
$43$ $$T^{12} - 8 T^{11} + 32 T^{10} + \cdots + 148254976$$
$47$ $$T^{12} + 360 T^{10} + \cdots + 440664064$$
$53$ $$T^{12} + 8 T^{11} + 32 T^{10} + \cdots + 25080064$$
$59$ $$T^{12} - 16 T^{11} + 128 T^{10} + \cdots + 9339136$$
$61$ $$T^{12} - 44 T^{11} + \cdots + 118949888$$
$67$ $$(T^{6} + 20 T^{5} + 58 T^{4} - 880 T^{3} + \cdots + 28928)^{2}$$
$71$ $$T^{12} - 32 T^{11} + 578 T^{10} + \cdots + 10913792$$
$73$ $$T^{12} - 8 T^{11} + 158 T^{10} + \cdots + 545424392$$
$79$ $$T^{12} + 8 T^{11} + \cdots + 1130596352$$
$83$ $$T^{12} - 40 T^{11} + 800 T^{10} + \cdots + 256$$
$89$ $$T^{12} + 552 T^{10} + \cdots + 12558340096$$
$97$ $$T^{12} + 16 T^{11} + 334 T^{10} + \cdots + 30451208$$