# Properties

 Label 390.2.t.b Level $390$ Weight $2$ Character orbit 390.t Analytic conductor $3.114$ Analytic rank $0$ Dimension $16$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [390,2,Mod(307,390)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(390, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("390.307");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$390 = 2 \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 390.t (of order $$4$$, degree $$2$$, 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: $$3.11416567883$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} + 32x^{14} + 396x^{12} + 2412x^{10} + 7716x^{8} + 12984x^{6} + 10756x^{4} + 3648x^{2} + 256$$ x^16 + 32*x^14 + 396*x^12 + 2412*x^10 + 7716*x^8 + 12984*x^6 + 10756*x^4 + 3648*x^2 + 256 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{2} + \beta_{8} q^{3} - q^{4} - \beta_{10} q^{5} + \beta_{5} q^{6} + (\beta_{15} - \beta_{14} + \cdots + 3 \beta_{3}) q^{7}+ \cdots - \beta_{4} q^{9}+O(q^{10})$$ q + b4 * q^2 + b8 * q^3 - q^4 - b10 * q^5 + b5 * q^6 + (b15 - b14 - b13 + 2*b12 - 2*b11 + b10 + b9 + b7 - b6 + 3*b3) * q^7 - b4 * q^8 - b4 * q^9 $$q + \beta_{4} q^{2} + \beta_{8} q^{3} - q^{4} - \beta_{10} q^{5} + \beta_{5} q^{6} + (\beta_{15} - \beta_{14} + \cdots + 3 \beta_{3}) q^{7}+ \cdots + ( - \beta_{13} + \beta_{6} - \beta_{4} + \cdots - 1) q^{99}+O(q^{100})$$ q + b4 * q^2 + b8 * q^3 - q^4 - b10 * q^5 + b5 * q^6 + (b15 - b14 - b13 + 2*b12 - 2*b11 + b10 + b9 + b7 - b6 + 3*b3) * q^7 - b4 * q^8 - b4 * q^9 - b14 * q^10 + (b15 - b10 + b5 - b4 + b2) * q^11 - b8 * q^12 + (b15 - b14 - b13 + b12 - 2*b11 + b8 + b7 + b5 + 2*b3 - 1) * q^13 + (b13 - b12 + b7 - b5 + b4 - b2 - b1) * q^14 + b9 * q^15 + q^16 + (-b14 + b10 + b9 + b8 + b7 + b3) * q^17 + q^18 + (b13 - b12 - b10 + b8 - b7 - b3) * q^19 + b10 * q^20 + (-b14 + b11 - b10 + b8 - b6 + b1) * q^21 + (b13 - b6 + b4 - b3 - b1 + 1) * q^22 + (b11 + b9 - b7 - b6 + b5 - b4 - b1 - 1) * q^23 - b5 * q^24 + (b15 - b14 - b13 + b12 - b11 + b10 + b9 - b8 + 2*b7 - b6 + b4 + b3 - b2 - b1 + 1) * q^25 + (b13 - b12 + b11 - b9 - b8 + b5 - b3 - b1) * q^26 - b5 * q^27 + (-b15 + b14 + b13 - 2*b12 + 2*b11 - b10 - b9 - b7 + b6 - 3*b3) * q^28 + (-b15 - b14 + b11 - b10 - b8 - b7 - b6 + 2*b5 - b4 + b2 + b1 + 1) * q^29 - b11 * q^30 + (2*b15 - 2*b14 - 3*b13 + 2*b12 - 3*b11 + 2*b10 + 2*b9 + 3*b7 + b6 + b4 + 5*b3 - 2*b2 + b1 - 1) * q^31 + b4 * q^32 + (-b15 + b14 + b13 - b12 + b11 - b10 + b8 - 2*b7 - b6 - b5 - 2*b3 + 1) * q^33 + (b14 - b12 + b5 - b3) * q^34 + (b14 + b11 - b9 + 2*b7 + b6 + 2*b4 + 2*b2 - b1 - 1) * q^35 + b4 * q^36 + (-b14 + b13 + b12 - 2*b10 - b9 + 3*b8 - b7 + 2*b5 - b3 - b2 + b1 - 2) * q^37 + (b13 - b9 + b5 - b3) * q^38 + (-b14 - b10 + b9 - b6 - b4 + 1) * q^39 + b14 * q^40 + (-b15 + 2*b14 + 3*b13 - 2*b12 + 2*b11 - b10 - 3*b9 - 2*b7 - 3*b5 - 4*b3 + b2 + 1) * q^41 + (b15 - b14 - b13 + b12 - b11 + b10 + b9 + b7 + b3 - b2 - 1) * q^42 + (-2*b15 + 2*b14 + b13 - 2*b12 + 2*b11 - 2*b10 - b9 - 2*b7 - 2*b3 + 2*b2 + 2) * q^43 + (-b15 + b10 - b5 + b4 - b2) * q^44 + b14 * q^45 + (-b15 + 2*b13 - b12 + b9 - b8 - b7 - b5 - b4 - b3 - b2 + 2) * q^46 + (-b15 + 2*b14 - b12 + b11 - b8 + b6 - b3 + b2 - b1 + 1) * q^47 + b8 * q^48 + (-b15 - b11 + b10 - 2*b9 - b8 - 2*b7 + b6 - 2*b5 + b3 - b2 + b1 + 4) * q^49 + (-b15 + b14 - b12 + b10 - b8 - b7 + b6 - 2*b5 + 2*b4 - b3 - b2 - b1) * q^50 + (b14 - b9 - b4 - b3) * q^51 + (-b15 + b14 + b13 - b12 + 2*b11 - b8 - b7 - b5 - 2*b3 + 1) * q^52 + (-b14 - b13 + b12 - 2*b11 + 4*b10 + 2*b9 - 4*b8 + 2*b7 + b6 + 3*b4 + 3*b3 - b1 - 3) * q^53 + b8 * q^54 + (2*b15 - 3*b13 + 4*b12 - 3*b11 + 2*b10 + b9 + b8 + 2*b7 + b6 + b5 - b4 + 3*b3 - b1 - 3) * q^55 + (-b13 + b12 - b7 + b5 - b4 + b2 + b1) * q^56 + (b14 - b12 - b10 - b7 - b4 - b3) * q^57 + (b15 - 2*b14 - b13 + b12 - b11 + b10 + b9 - b8 + b7 - b6 - 2*b5 + 2*b3 - b2 + b1) * q^58 + (b14 + b13 - b12 + b11 - 2*b9 - b7 + b6 + 2*b5 - 2*b4 + b3 + b1 - 2) * q^59 - b9 * q^60 + (-b15 + 3*b13 - 3*b12 + 2*b11 - 2*b10 - b9 - b8 - b7 + b6 - 2*b5 - 5*b3 - b2 + b1 - 1) * q^61 + (b15 + 2*b14 + b13 - 2*b12 - b10 - b9 - 2*b8 + 2*b6 + b5 + b4 - 2*b3 + b2 - 2*b1 - 2) * q^62 + (-b13 + b12 - b7 + b5 - b4 + b2 + b1) * q^63 - q^64 + (b15 + b14 - 2*b13 - b12 - b11 + 2*b10 + b9 - 3*b8 + 3*b7 + 2*b6 + 2*b4 - 2*b1 - 3) * q^65 + (-b14 + b12 - b11 + b8 + b3 - b2 + b1) * q^66 + (b15 + b14 + b12 + b10 - b9 - 3*b8 + b7 + b6 + 3*b5 - 2*b4 - 1) * q^67 + (b14 - b10 - b9 - b8 - b7 - b3) * q^68 + (-b14 - b13 - b11 + 2*b10 + b9 - b8 + b7 - 2*b5 + 2*b3 - b2 + b1 + 1) * q^69 + (-b15 - b13 - b12 + 2*b11 - b10 + b9 + 2*b8 - b7 - 2*b6 + b5 - b4 - b3 + b2 - 1) * q^70 + (-b15 + b14 + 2*b13 - b12 + 3*b11 - b10 - 3*b9 - 3*b7 + b6 - 3*b5 - b4 - b3 + b2 + b1) * q^71 - q^72 + (b15 - 2*b13 + 2*b12 - 4*b11 + 4*b10 + b9 + b8 + 2*b7 + b6 - 3*b5 + 3*b4 + b3 - 2*b2 - 2*b1 - 1) * q^73 + (b15 - 3*b14 - 2*b13 + 3*b12 - 2*b11 + 3*b10 + b9 - 3*b8 + 3*b7 + b6 + 3*b5 - 2*b4 + 4*b3 - 1) * q^74 + (b15 - 3*b14 - 2*b13 + 4*b12 - 3*b11 + 2*b10 + 2*b9 + 2*b8 + 2*b7 - b6 + b4 + 5*b3 - b2 + b1) * q^75 + (-b13 + b12 + b10 - b8 + b7 + b3) * q^76 + (b14 - b13 - b12 + 4*b11 - 2*b10 - 3*b9 + 2*b7 - b4 - 2*b3 + 1) * q^77 + (-b14 - b11 + b10 - b5 + b4 - b2 + 1) * q^78 + (-b15 + 3*b11 - 3*b10 + 3*b8 - 4*b7 - b6 - 6*b4 - b3 + 3*b2 + 3*b1 + 1) * q^79 - b10 * q^80 - q^81 + (-b13 + 2*b12 + b11 + 4*b8 + b7 - b6 + b3 + b1) * q^82 + (-b13 + 3*b12 - b11 - b10 - b9 - 2*b8 - b7 - 2*b5 + 2*b3) * q^83 + (b14 - b11 + b10 - b8 + b6 - b1) * q^84 + (-b15 + b12 + b9 + 4*b8 - 2*b7 + b5 + 3*b4 + b1 - 2) * q^85 + (-2*b14 + b11 - 2*b10 + 2*b8 - b7 - 2*b6 + 2*b1) * q^86 + (-b15 + b14 - b12 + b11 - b10 - b9 - b7 + b6 + b4 - 2*b3 + b2 + b1 + 2) * q^87 + (-b13 + b6 - b4 + b3 + b1 - 1) * q^88 + (-b15 + 3*b14 + 2*b13 - 3*b12 + 3*b11 - b10 - 2*b9 - 3*b7 + 3*b5 - b4 - 4*b3 + b2) * q^89 - b10 * q^90 + (-b14 - 3*b13 + 2*b11 - b10 - 2*b9 + 2*b7 + 2*b3 + b2 + 3*b1 + 4) * q^91 + (-b11 - b9 + b7 + b6 - b5 + b4 + b1 + 1) * q^92 + (2*b15 - b14 - 2*b13 + 2*b12 - b11 + b10 + 2*b9 + b8 + 2*b7 - 2*b6 + 2*b5 + 3*b3 + b2 - b1) * q^93 + (-b15 - b12 + b11 - 2*b10 + b8 - 2*b7 - b6 - b3 + b2 + b1 + 1) * q^94 + (b15 - 2*b14 - 2*b13 + 2*b12 - 3*b11 + 3*b10 + 3*b9 + 2*b8 + 3*b7 - 3*b5 + 4*b3 - 2*b2 - b1 - 3) * q^95 + b5 * q^96 + (2*b14 + 2*b13 + b11 + b10 + 2*b9 - b8 - 3*b7 - b4 - 2*b3 - b2 - b1) * q^97 + (b15 + b13 + 2*b11 - b10 - b9 + b8 + b7 + b6 + 3*b4 + b3 + b2 + b1 - 1) * q^98 + (-b13 + b6 - b4 + b3 + b1 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 16 q^{4}+O(q^{10})$$ 16 * q - 16 * q^4 $$16 q - 16 q^{4} + 4 q^{11} - 8 q^{13} - 4 q^{15} + 16 q^{16} - 4 q^{17} + 16 q^{18} - 4 q^{19} - 8 q^{21} + 4 q^{22} - 16 q^{23} + 16 q^{25} + 4 q^{26} - 4 q^{30} + 12 q^{31} - 4 q^{34} - 12 q^{35} - 32 q^{37} - 4 q^{38} + 4 q^{39} + 4 q^{41} - 8 q^{42} + 16 q^{43} - 4 q^{44} + 16 q^{46} + 16 q^{47} + 80 q^{49} + 8 q^{50} + 8 q^{52} - 44 q^{53} - 20 q^{55} - 12 q^{59} + 4 q^{60} - 32 q^{61} - 12 q^{62} - 16 q^{64} - 28 q^{65} + 4 q^{68} + 16 q^{69} - 36 q^{70} + 16 q^{71} - 16 q^{72} + 4 q^{76} + 32 q^{77} + 16 q^{78} - 16 q^{81} - 4 q^{82} + 16 q^{83} + 8 q^{84} - 40 q^{85} - 16 q^{86} + 28 q^{87} - 4 q^{88} - 4 q^{89} + 76 q^{91} + 16 q^{92} - 40 q^{95} - 4 q^{99}+O(q^{100})$$ 16 * q - 16 * q^4 + 4 * q^11 - 8 * q^13 - 4 * q^15 + 16 * q^16 - 4 * q^17 + 16 * q^18 - 4 * q^19 - 8 * q^21 + 4 * q^22 - 16 * q^23 + 16 * q^25 + 4 * q^26 - 4 * q^30 + 12 * q^31 - 4 * q^34 - 12 * q^35 - 32 * q^37 - 4 * q^38 + 4 * q^39 + 4 * q^41 - 8 * q^42 + 16 * q^43 - 4 * q^44 + 16 * q^46 + 16 * q^47 + 80 * q^49 + 8 * q^50 + 8 * q^52 - 44 * q^53 - 20 * q^55 - 12 * q^59 + 4 * q^60 - 32 * q^61 - 12 * q^62 - 16 * q^64 - 28 * q^65 + 4 * q^68 + 16 * q^69 - 36 * q^70 + 16 * q^71 - 16 * q^72 + 4 * q^76 + 32 * q^77 + 16 * q^78 - 16 * q^81 - 4 * q^82 + 16 * q^83 + 8 * q^84 - 40 * q^85 - 16 * q^86 + 28 * q^87 - 4 * q^88 - 4 * q^89 + 76 * q^91 + 16 * q^92 - 40 * q^95 - 4 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 32x^{14} + 396x^{12} + 2412x^{10} + 7716x^{8} + 12984x^{6} + 10756x^{4} + 3648x^{2} + 256$$ :

 $$\beta_{1}$$ $$=$$ $$( - 217 \nu^{15} + 584 \nu^{14} - 5720 \nu^{13} + 17472 \nu^{12} - 49324 \nu^{11} + 194656 \nu^{10} + \cdots - 701696 ) / 113152$$ (-217*v^15 + 584*v^14 - 5720*v^13 + 17472*v^12 - 49324*v^11 + 194656*v^10 - 115244*v^9 + 992800*v^8 + 423900*v^7 + 2279200*v^6 + 2233256*v^5 + 1779008*v^4 + 2981212*v^3 - 625248*v^2 + 1114080*v - 701696) / 113152 $$\beta_{2}$$ $$=$$ $$( 210 \nu^{15} + 93 \nu^{14} + 7072 \nu^{13} + 4056 \nu^{12} + 92728 \nu^{11} + 67196 \nu^{10} + \cdots + 205728 ) / 56576$$ (210*v^15 + 93*v^14 + 7072*v^13 + 4056*v^12 + 92728*v^11 + 67196*v^10 + 596824*v^9 + 530524*v^8 + 1948360*v^7 + 2031924*v^6 + 2967984*v^5 + 3398648*v^4 + 1607112*v^3 + 1963572*v^2 + 8960*v + 205728) / 56576 $$\beta_{3}$$ $$=$$ $$( 12 \nu^{15} - 483 \nu^{14} + 208 \nu^{13} - 15080 \nu^{12} - 656 \nu^{11} - 179204 \nu^{10} + \cdots - 58464 ) / 56576$$ (12*v^15 - 483*v^14 + 208*v^13 - 15080*v^12 - 656*v^11 - 179204*v^10 - 34512*v^9 - 1017348*v^8 - 264016*v^7 - 2858828*v^6 - 828064*v^5 - 3718344*v^4 - 1054352*v^3 - 1684748*v^2 - 314816*v - 58464) / 56576 $$\beta_{4}$$ $$=$$ $$( - 725 \nu^{15} - 22776 \nu^{13} - 273788 \nu^{11} - 1589116 \nu^{9} - 4674452 \nu^{7} + \cdots - 697504 \nu ) / 113152$$ (-725*v^15 - 22776*v^13 - 273788*v^11 - 1589116*v^9 - 4674452*v^7 - 6751736*v^5 - 4099732*v^3 - 697504*v) / 113152 $$\beta_{5}$$ $$=$$ $$( 247 \nu^{15} - 527 \nu^{14} + 7384 \nu^{13} - 15912 \nu^{12} + 82420 \nu^{11} - 180404 \nu^{10} + \cdots - 147424 ) / 56576$$ (247*v^15 - 527*v^14 + 7384*v^13 - 15912*v^12 + 82420*v^11 - 180404*v^10 + 426868*v^9 - 957780*v^8 + 1058460*v^7 - 2462076*v^6 + 1238120*v^5 - 2955688*v^4 + 668252*v^3 - 1409980*v^2 + 183456*v - 147424) / 56576 $$\beta_{6}$$ $$=$$ $$( - 182 \nu^{15} + 147 \nu^{14} - 5200 \nu^{13} + 4472 \nu^{12} - 53976 \nu^{11} + 51348 \nu^{10} + \cdots + 44128 ) / 28288$$ (-182*v^15 + 147*v^14 - 5200*v^13 + 4472*v^12 - 53976*v^11 + 51348*v^10 - 246584*v^9 + 279540*v^8 - 487448*v^7 + 759820*v^6 - 403728*v^5 + 1028936*v^4 - 155896*v^3 + 588588*v^2 + 6656*v + 44128) / 28288 $$\beta_{7}$$ $$=$$ $$( 515 \nu^{15} - 37 \nu^{14} + 15704 \nu^{13} - 312 \nu^{12} + 181060 \nu^{11} + 10308 \nu^{10} + \cdots - 4768 ) / 56576$$ (515*v^15 - 37*v^14 + 15704*v^13 - 312*v^12 + 181060*v^11 + 10308*v^10 + 992292*v^9 + 169956*v^8 + 2726092*v^7 + 889900*v^6 + 3783752*v^5 + 1729864*v^4 + 2464332*v^3 + 967148*v^2 + 518816*v - 4768) / 56576 $$\beta_{8}$$ $$=$$ $$( - 247 \nu^{15} - 527 \nu^{14} - 7384 \nu^{13} - 15912 \nu^{12} - 82420 \nu^{11} - 180404 \nu^{10} + \cdots - 147424 ) / 56576$$ (-247*v^15 - 527*v^14 - 7384*v^13 - 15912*v^12 - 82420*v^11 - 180404*v^10 - 426868*v^9 - 957780*v^8 - 1058460*v^7 - 2462076*v^6 - 1238120*v^5 - 2955688*v^4 - 668252*v^3 - 1409980*v^2 - 183456*v - 147424) / 56576 $$\beta_{9}$$ $$=$$ $$( - 515 \nu^{15} - 37 \nu^{14} - 15704 \nu^{13} - 312 \nu^{12} - 181060 \nu^{11} + 10308 \nu^{10} + \cdots - 4768 ) / 56576$$ (-515*v^15 - 37*v^14 - 15704*v^13 - 312*v^12 - 181060*v^11 + 10308*v^10 - 992292*v^9 + 169956*v^8 - 2726092*v^7 + 889900*v^6 - 3783752*v^5 + 1729864*v^4 - 2464332*v^3 + 967148*v^2 - 518816*v - 4768) / 56576 $$\beta_{10}$$ $$=$$ $$( - 527 \nu^{15} + 144 \nu^{14} - 15912 \nu^{13} + 4160 \nu^{12} - 180404 \nu^{11} + 43296 \nu^{10} + \cdots - 190208 ) / 56576$$ (-527*v^15 + 144*v^14 - 15912*v^13 + 4160*v^12 - 180404*v^11 + 43296*v^10 - 957780*v^9 + 189056*v^8 - 2462076*v^7 + 260480*v^6 - 2955688*v^5 - 278080*v^4 - 1409980*v^3 - 719680*v^2 - 147424*v - 190208) / 56576 $$\beta_{11}$$ $$=$$ $$( - 515 \nu^{15} + 249 \nu^{14} - 15704 \nu^{13} + 6968 \nu^{12} - 181060 \nu^{11} + 69484 \nu^{10} + \cdots + 97568 ) / 56576$$ (-515*v^15 + 249*v^14 - 15704*v^13 + 6968*v^12 - 181060*v^11 + 69484*v^10 - 992292*v^9 + 289868*v^8 - 2726092*v^7 + 440932*v^6 - 3783752*v^5 + 119320*v^4 - 2464332*v^3 + 6500*v^2 - 462240*v + 97568) / 56576 $$\beta_{12}$$ $$=$$ $$( - 527 \nu^{15} - 144 \nu^{14} - 15912 \nu^{13} - 4160 \nu^{12} - 180404 \nu^{11} - 43296 \nu^{10} + \cdots + 190208 ) / 56576$$ (-527*v^15 - 144*v^14 - 15912*v^13 - 4160*v^12 - 180404*v^11 - 43296*v^10 - 957780*v^9 - 189056*v^8 - 2462076*v^7 - 260480*v^6 - 2955688*v^5 + 278080*v^4 - 1409980*v^3 + 719680*v^2 - 147424*v + 190208) / 56576 $$\beta_{13}$$ $$=$$ $$( - 515 \nu^{15} - 249 \nu^{14} - 15704 \nu^{13} - 6968 \nu^{12} - 181060 \nu^{11} - 69484 \nu^{10} + \cdots - 97568 ) / 56576$$ (-515*v^15 - 249*v^14 - 15704*v^13 - 6968*v^12 - 181060*v^11 - 69484*v^10 - 992292*v^9 - 289868*v^8 - 2726092*v^7 - 440932*v^6 - 3783752*v^5 - 119320*v^4 - 2464332*v^3 - 6500*v^2 - 462240*v - 97568) / 56576 $$\beta_{14}$$ $$=$$ $$( - 87 \nu^{15} - 80 \nu^{14} - 2664 \nu^{13} - 2368 \nu^{12} - 30740 \nu^{11} - 25984 \nu^{10} + \cdots - 9728 ) / 8704$$ (-87*v^15 - 80*v^14 - 2664*v^13 - 2368*v^12 - 30740*v^11 - 25984*v^10 - 166356*v^9 - 130368*v^8 - 429980*v^7 - 302912*v^6 - 475304*v^5 - 305920*v^4 - 120732*v^3 - 110400*v^2 + 56096*v - 9728) / 8704 $$\beta_{15}$$ $$=$$ $$( - 1859 \nu^{15} + 452 \nu^{14} - 55432 \nu^{13} + 12896 \nu^{12} - 615524 \nu^{11} + 132400 \nu^{10} + \cdots + 63104 ) / 113152$$ (-1859*v^15 + 452*v^14 - 55432*v^13 + 12896*v^12 - 615524*v^11 + 132400*v^10 - 3148964*v^9 + 576624*v^8 - 7539532*v^7 + 898576*v^6 - 7793864*v^5 - 138784*v^4 - 2193100*v^3 - 919152*v^2 + 755872*v + 63104) / 113152
 $$\nu$$ $$=$$ $$( \beta_{13} + \beta_{11} - \beta_{9} + \beta_{7} ) / 2$$ (b13 + b11 - b9 + b7) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{15} - \beta_{14} - \beta_{13} + \beta_{12} + \beta_{9} + \beta_{7} - \beta_{6} + \beta_{3} - 4$$ b15 - b14 - b13 + b12 + b9 + b7 - b6 + b3 - 4 $$\nu^{3}$$ $$=$$ $$\beta_{14} - 2 \beta_{13} - \beta_{12} - 3 \beta_{11} + 4 \beta_{9} - \beta_{8} - 4 \beta_{7} + \cdots - \beta_1$$ b14 - 2*b13 - b12 - 3*b11 + 4*b9 - b8 - 4*b7 - 4*b4 - b3 - b2 - b1 $$\nu^{4}$$ $$=$$ $$- 9 \beta_{15} + 10 \beta_{14} + 12 \beta_{13} - 10 \beta_{12} - \beta_{11} - \beta_{10} - 12 \beta_{9} + \cdots + 29$$ -9*b15 + 10*b14 + 12*b13 - 10*b12 - b11 - b10 - 12*b9 - b8 - 12*b7 + 9*b6 - 11*b3 + b2 - b1 + 29 $$\nu^{5}$$ $$=$$ $$- 10 \beta_{14} + 13 \beta_{13} + 11 \beta_{12} + 27 \beta_{11} - 3 \beta_{10} - 36 \beta_{9} + \cdots + 14 \beta_1$$ -10*b14 + 13*b13 + 11*b12 + 27*b11 - 3*b10 - 36*b9 + 7*b8 + 32*b7 + 7*b5 + 48*b4 + 10*b3 + 14*b2 + 14*b1 $$\nu^{6}$$ $$=$$ $$79 \beta_{15} - 90 \beta_{14} - 128 \beta_{13} + 94 \beta_{12} + 15 \beta_{11} + 19 \beta_{10} + \cdots - 251$$ 79*b15 - 90*b14 - 128*b13 + 94*b12 + 15*b11 + 19*b10 + 123*b9 + 10*b8 + 123*b7 - 79*b6 - b5 + 113*b3 - 11*b2 + 11*b1 - 251 $$\nu^{7}$$ $$=$$ $$8 \beta_{15} + 86 \beta_{14} - 111 \beta_{13} - 106 \beta_{12} - 273 \beta_{11} + 56 \beta_{10} + \cdots - 8$$ 8*b15 + 86*b14 - 111*b13 - 106*b12 - 273*b11 + 56*b10 + 339*b9 - 46*b8 - 255*b7 + 8*b6 - 116*b5 - 484*b4 - 78*b3 - 162*b2 - 162*b1 - 8 $$\nu^{8}$$ $$=$$ $$- 716 \beta_{15} + 822 \beta_{14} + 1318 \beta_{13} - 898 \beta_{12} - 154 \beta_{11} - 266 \beta_{10} + \cdots + 2302$$ -716*b15 + 822*b14 + 1318*b13 - 898*b12 - 154*b11 - 266*b10 - 1230*b9 - 74*b8 - 1230*b7 + 716*b6 + 32*b5 - 1164*b3 + 106*b2 - 106*b1 + 2302 $$\nu^{9}$$ $$=$$ $$- 160 \beta_{15} - 738 \beta_{14} + 1034 \beta_{13} + 1036 \beta_{12} + 2772 \beta_{11} - 702 \beta_{10} + \cdots + 160$$ -160*b15 - 738*b14 + 1034*b13 + 1036*b12 + 2772*b11 - 702*b10 - 3240*b9 + 280*b8 + 2080*b7 - 160*b6 + 1458*b5 + 4696*b4 + 578*b3 + 1738*b2 + 1738*b1 + 160 $$\nu^{10}$$ $$=$$ $$6676 \beta_{15} - 7712 \beta_{14} - 13368 \beta_{13} + 8788 \beta_{12} + 1308 \beta_{11} + 3272 \beta_{10} + \cdots - 21604$$ 6676*b15 - 7712*b14 - 13368*b13 + 8788*b12 + 1308*b11 + 3272*b10 + 12258*b9 + 414*b8 + 12258*b7 - 6676*b6 - 622*b5 + 12060*b3 - 1036*b2 + 1036*b1 - 21604 $$\nu^{11}$$ $$=$$ $$2236 \beta_{15} + 6552 \beta_{14} - 9828 \beta_{13} - 10446 \beta_{12} - 27776 \beta_{11} + 7502 \beta_{10} + \cdots - 2236$$ 2236*b15 + 6552*b14 - 9828*b13 - 10446*b12 - 27776*b11 + 7502*b10 + 31106*b9 - 1326*b8 - 17474*b7 + 2236*b6 - 16622*b5 - 45104*b4 - 4316*b3 - 17948*b2 - 17948*b1 - 2236 $$\nu^{12}$$ $$=$$ $$- 63498 \beta_{15} + 73944 \beta_{14} + 134552 \beta_{13} - 87536 \beta_{12} - 9346 \beta_{11} + \cdots + 205078$$ -63498*b15 + 73944*b14 + 134552*b13 - 87536*b12 - 9346*b11 - 37670*b10 - 122170*b9 - 808*b8 - 122170*b7 + 63498*b6 + 9638*b5 - 125206*b3 + 10446*b2 - 10446*b1 + 205078 $$\nu^{13}$$ $$=$$ $$- 27224 \beta_{15} - 60312 \beta_{14} + 93686 \beta_{13} + 107620 \beta_{12} + 275254 \beta_{11} + \cdots + 27224$$ -27224*b15 - 60312*b14 + 93686*b13 + 107620*b12 + 275254*b11 - 73948*b10 - 299146*b9 + 580*b8 + 150666*b7 - 27224*b6 + 180988*b5 + 432032*b4 + 33088*b3 + 181568*b2 + 181568*b1 + 27224 $$\nu^{14}$$ $$=$$ $$611880 \beta_{15} - 719500 \beta_{14} - 1348420 \beta_{13} + 880988 \beta_{12} + 49844 \beta_{11} + \cdots - 1959780$$ 611880*b15 - 719500*b14 - 1348420*b13 + 880988*b12 + 49844*b11 + 417588*b10 + 1217968*b9 - 24000*b8 + 1217968*b7 - 611880*b6 - 131620*b5 + 1298576*b3 - 107620*b2 + 107620*b1 - 1959780 $$\nu^{15}$$ $$=$$ $$309968 \beta_{15} + 571020 \beta_{14} - 892700 \beta_{13} - 1120228 \beta_{12} - 2708552 \beta_{11} + \cdots - 309968$$ 309968*b15 + 571020*b14 - 892700*b13 - 1120228*b12 - 2708552*b11 + 695624*b10 + 2880012*b9 + 105852*b8 - 1325212*b7 + 309968*b6 - 1921704*b5 - 4136384*b4 - 261052*b3 - 1815852*b2 - 1815852*b1 - 309968

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/390\mathbb{Z}\right)^\times$$.

 $$n$$ $$131$$ $$157$$ $$301$$ $$\chi(n)$$ $$1$$ $$\beta_{4}$$ $$\beta_{4}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
307.1
 − 1.09662i 3.03462i − 1.63314i − 0.304860i 0.821039i − 2.42515i 3.14581i − 1.54171i 1.09662i − 3.03462i 1.63314i 0.304860i − 0.821039i 2.42515i − 3.14581i 1.54171i
1.00000i −0.707107 + 0.707107i −1.00000 −2.09731 0.775429i −0.707107 0.707107i −0.946681 1.00000i 1.00000i 0.775429 2.09731i
307.2 1.00000i −0.707107 + 0.707107i −1.00000 −0.628922 + 2.14580i −0.707107 0.707107i 5.15192 1.00000i 1.00000i −2.14580 0.628922i
307.3 1.00000i −0.707107 + 0.707107i −1.00000 1.91479 1.15480i −0.707107 0.707107i −4.24302 1.00000i 1.00000i 1.15480 + 1.91479i
307.4 1.00000i −0.707107 + 0.707107i −1.00000 2.22565 0.215569i −0.707107 0.707107i 2.86620 1.00000i 1.00000i 0.215569 + 2.22565i
307.5 1.00000i 0.707107 0.707107i −1.00000 −2.15939 0.580562i 0.707107 + 0.707107i 3.80685 1.00000i 1.00000i 0.580562 2.15939i
307.6 1.00000i 0.707107 0.707107i −1.00000 −1.43504 + 1.71484i 0.707107 + 0.707107i −3.30195 1.00000i 1.00000i −1.71484 1.43504i
307.7 1.00000i 0.707107 0.707107i −1.00000 0.227886 2.22443i 0.707107 + 0.707107i −4.05336 1.00000i 1.00000i 2.22443 + 0.227886i
307.8 1.00000i 0.707107 0.707107i −1.00000 1.95232 + 1.09015i 0.707107 + 0.707107i 0.720033 1.00000i 1.00000i −1.09015 + 1.95232i
343.1 1.00000i −0.707107 0.707107i −1.00000 −2.09731 + 0.775429i −0.707107 + 0.707107i −0.946681 1.00000i 1.00000i 0.775429 + 2.09731i
343.2 1.00000i −0.707107 0.707107i −1.00000 −0.628922 2.14580i −0.707107 + 0.707107i 5.15192 1.00000i 1.00000i −2.14580 + 0.628922i
343.3 1.00000i −0.707107 0.707107i −1.00000 1.91479 + 1.15480i −0.707107 + 0.707107i −4.24302 1.00000i 1.00000i 1.15480 1.91479i
343.4 1.00000i −0.707107 0.707107i −1.00000 2.22565 + 0.215569i −0.707107 + 0.707107i 2.86620 1.00000i 1.00000i 0.215569 2.22565i
343.5 1.00000i 0.707107 + 0.707107i −1.00000 −2.15939 + 0.580562i 0.707107 0.707107i 3.80685 1.00000i 1.00000i 0.580562 + 2.15939i
343.6 1.00000i 0.707107 + 0.707107i −1.00000 −1.43504 1.71484i 0.707107 0.707107i −3.30195 1.00000i 1.00000i −1.71484 + 1.43504i
343.7 1.00000i 0.707107 + 0.707107i −1.00000 0.227886 + 2.22443i 0.707107 0.707107i −4.05336 1.00000i 1.00000i 2.22443 0.227886i
343.8 1.00000i 0.707107 + 0.707107i −1.00000 1.95232 1.09015i 0.707107 0.707107i 0.720033 1.00000i 1.00000i −1.09015 1.95232i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 307.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.f even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.t.b yes 16
3.b odd 2 1 1170.2.w.i 16
5.b even 2 1 1950.2.t.e 16
5.c odd 4 1 390.2.j.b 16
5.c odd 4 1 1950.2.j.e 16
13.d odd 4 1 390.2.j.b 16
15.e even 4 1 1170.2.m.i 16
39.f even 4 1 1170.2.m.i 16
65.f even 4 1 inner 390.2.t.b yes 16
65.g odd 4 1 1950.2.j.e 16
65.k even 4 1 1950.2.t.e 16
195.u odd 4 1 1170.2.w.i 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.j.b 16 5.c odd 4 1
390.2.j.b 16 13.d odd 4 1
390.2.t.b yes 16 1.a even 1 1 trivial
390.2.t.b yes 16 65.f even 4 1 inner
1170.2.m.i 16 15.e even 4 1
1170.2.m.i 16 39.f even 4 1
1170.2.w.i 16 3.b odd 2 1
1170.2.w.i 16 195.u odd 4 1
1950.2.j.e 16 5.c odd 4 1
1950.2.j.e 16 65.g odd 4 1
1950.2.t.e 16 5.b even 2 1
1950.2.t.e 16 65.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{8} - 48T_{7}^{6} - 12T_{7}^{5} + 728T_{7}^{4} + 224T_{7}^{3} - 3652T_{7}^{2} - 768T_{7} + 2176$$ acting on $$S_{2}^{\mathrm{new}}(390, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{8}$$
$3$ $$(T^{4} + 1)^{4}$$
$5$ $$T^{16} - 8 T^{14} + \cdots + 390625$$
$7$ $$(T^{8} - 48 T^{6} + \cdots + 2176)^{2}$$
$11$ $$T^{16} - 4 T^{15} + \cdots + 58003456$$
$13$ $$T^{16} + \cdots + 815730721$$
$17$ $$T^{16} + 4 T^{15} + \cdots + 2534464$$
$19$ $$T^{16} + 4 T^{15} + \cdots + 1024$$
$23$ $$T^{16} + \cdots + 77275104256$$
$29$ $$T^{16} + \cdots + 21484523776$$
$31$ $$T^{16} + \cdots + 837825347584$$
$37$ $$(T^{8} + 16 T^{7} + \cdots - 1045376)^{2}$$
$41$ $$T^{16} + \cdots + 79370665984$$
$43$ $$T^{16} + \cdots + 349241344$$
$47$ $$(T^{8} - 8 T^{7} + \cdots - 1024)^{2}$$
$53$ $$T^{16} + \cdots + 50182602625024$$
$59$ $$T^{16} + \cdots + 48729781571584$$
$61$ $$(T^{8} + 16 T^{7} + \cdots - 3982336)^{2}$$
$67$ $$T^{16} + \cdots + 48809181184$$
$71$ $$T^{16} + \cdots + 846046756864$$
$73$ $$T^{16} + \cdots + 8806459834624$$
$79$ $$T^{16} + \cdots + 604860841984$$
$83$ $$(T^{8} - 8 T^{7} + \cdots - 713728)^{2}$$
$89$ $$T^{16} + \cdots + 17644340224$$
$97$ $$T^{16} + \cdots + 157829170008064$$