# Properties

 Label 78.2.k.a Level $78$ Weight $2$ Character orbit 78.k Analytic conductor $0.623$ Analytic rank $0$ Dimension $16$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [78,2,Mod(11,78)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(78, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([6, 7]))

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

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

chi := DirichletCharacter("78.11");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$78 = 2 \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 78.k (of order $$12$$, 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: $$0.622833135766$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{12})$$ Coefficient field: 16.0.9349208943630483456.9 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + \cdots + 25$$ x^16 - 8*x^15 + 48*x^14 - 196*x^13 + 642*x^12 - 1668*x^11 + 3580*x^10 - 6328*x^9 + 9297*x^8 - 11276*x^7 + 11224*x^6 - 9024*x^5 + 5736*x^4 - 2780*x^3 + 972*x^2 - 220*x + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{15} - \beta_{7}) q^{2} + (\beta_{12} - \beta_{10} + \cdots - \beta_1) q^{3}+ \cdots + (2 \beta_{15} + \beta_{14} + \cdots + \beta_{6}) q^{9}+O(q^{10})$$ q + (-b15 - b7) * q^2 + (b12 - b10 - b4 - b3 - b1) * q^3 + (-b14 + b9) * q^4 + (-b13 - b12 + b10 - 2*b6 - b5 + b4 + b3 + b1) * q^5 + (-b11 + b8 - b4 - b1) * q^6 + (b14 - b11 - b9 - b3 - b2 + 1) * q^7 + b13 * q^8 + (2*b15 + b14 - b13 + b9 + b7 + b6) * q^9 $$q + ( - \beta_{15} - \beta_{7}) q^{2} + (\beta_{12} - \beta_{10} + \cdots - \beta_1) q^{3}+ \cdots + (\beta_{12} - 6 \beta_{11} + \cdots - 3 \beta_1) q^{99}+O(q^{100})$$ q + (-b15 - b7) * q^2 + (b12 - b10 - b4 - b3 - b1) * q^3 + (-b14 + b9) * q^4 + (-b13 - b12 + b10 - 2*b6 - b5 + b4 + b3 + b1) * q^5 + (-b11 + b8 - b4 - b1) * q^6 + (b14 - b11 - b9 - b3 - b2 + 1) * q^7 + b13 * q^8 + (2*b15 + b14 - b13 + b9 + b7 + b6) * q^9 + (b11 + b4 + b3 + b2 + b1 - 2) * q^10 + (b12 - b8 + 2*b5 - b3 - b1) * q^11 + (b12 + b10 + b8) * q^12 + (-b14 - b12 + 2*b11 + b4 + 2*b3 - b2 + b1 - 1) * q^13 + (-b15 - b13 - b12 + b11 - b8 - b5 + b3 + 2*b1) * q^14 + (-b14 + 2*b13 - b10 - b9 + b8 + b6 + b5 + b4 - b3 + 2*b2 - b1 - 1) * q^15 + b2 * q^16 + (-b15 + b13 - 2*b12 + 2*b11 + 2*b10 - 2*b8 + b7 + 2*b6 - 2*b5 + 2*b4 + 2*b3 + 4*b1) * q^17 + (b14 + b13 - 2*b9 + 2*b6 - 2*b2 + 1) * q^18 + (2*b14 - b11 - 2*b9 - 2*b4 - b3 + 2*b2 - 2*b1 - 2) * q^19 + (2*b15 - b11 + b7 + b5 - b1) * q^20 + (b15 - 2*b14 - 2*b10 + b9 - b8 + 2*b7 + b5 - b4 - b2 - b1 - 1) * q^21 + (-b12 - 2*b10 - 2*b8 - 2*b3) * q^22 + (b15 + 2*b13 - b11 - 2*b10 - b8 - b7 + b6 + 2*b5 - 2*b4 - b1) * q^23 + (-b10 - b8 + b5 - b4 - b3 - b1) * q^24 + (b14 + 4*b12 - 4*b11 + 4*b10 + 4*b8 - 2*b4 - 2*b2 - 2*b1 + 1) * q^25 + (b15 + b12 - b11 + b10 + b8 + 2*b7 - b6 + b5 + b4 - b3 - 2*b1) * q^26 + (-2*b12 + 3*b11 + b10 - 2*b8 - 3*b5 + 3*b4 + 3*b3 + 7*b1) * q^27 + (b12 + b9 - b4 - b2 - b1) * q^28 + (-b15 - b13 + 2*b12 - b11 - 2*b10 + 3*b8 - 2*b7 + b6 - 2*b4 - 2*b3 - 3*b1) * q^29 + (b15 - b13 - 2*b12 + 2*b10 - b7 - 2*b6 - b5 + 2*b4 + b3 + b2 + 2*b1 + 1) * q^30 + (-2*b14 - 3*b12 + b11 - 2*b10 + 2*b9 - 2*b8 - b3 + 2*b2) * q^31 - b7 * q^32 + (-4*b15 - b14 - 3*b13 + 2*b9 - 2*b7 - 3*b6 + b2 - 2) * q^33 + (b14 + 2*b11 - 2*b10 + b9 - 2*b8 - b2 + 2) * q^34 + (-4*b15 + 2*b13 - b12 + 2*b11 + 2*b10 + 2*b8 - 2*b7 - 2*b6 - 5*b5 + 2*b4 + b3 + 3*b1) * q^35 + (-b15 - 2*b13 + b7 - b6 - b2 + 2) * q^36 + (-b14 + 2*b12 + 3*b10 + b9 + 3*b8 + 2*b4 + 3*b3 + 2*b1 + 1) * q^37 + (2*b15 - 2*b13 + b12 + b11 + b8 - b5 - b3) * q^38 + (-2*b15 + 2*b14 + b13 + 2*b10 - b9 + b8 - 4*b7 - b6 + b5 + b4 + 2*b3 + 3) * q^39 + (b14 - 2*b9 + b4 + b1) * q^40 + (3*b15 - 4*b13 + 4*b12 - 4*b11 - 2*b10 + 4*b8 + 3*b7 - 2*b6 + 4*b5 - 2*b4 - 4*b3 - 8*b1) * q^41 + (b15 + 2*b14 + b13 + b12 - 2*b11 + b10 - b9 + 2*b8 + 2*b7 - b6 - b1) * q^42 + (-b12 + b11 - b10 - b8 - b4 - b1) * q^43 + (-2*b12 + b11 + b10 - 3*b5 + b4 + 2*b3 + 3*b1) * q^44 + (-3*b14 + b10 + 3*b9 - b8 - 3*b7 - b5 - b4 - 2*b3 - 3*b2 - 2*b1 + 3) * q^45 + (-b14 + 2*b12 - 2*b11 + b10 - b9 + b8 - b4 - b3 + b2 - b1 + 1) * q^46 + (-b15 - 3*b13 - 3*b11 - 3*b10 - 2*b7 + 3*b5 - 3*b4 - 3*b1) * q^47 + (-b5 - b3) * q^48 + (-2*b12 - 2*b10 + 2*b9 - 2*b8 + 2*b4 - 2*b3 - b2 + 2*b1 + 2) * q^49 + (-b15 + 2*b12 - 4*b10 - 2*b8 + b7 + b6 + 4*b5 - 4*b4 - 2*b3 - 2*b1) * q^50 + (-2*b15 + 2*b13 + 2*b12 - 3*b11 + b10 + 2*b7 - 2*b6 + b5 - 2*b4 - b3 + 4*b2 - 2*b1 - 2) * q^51 + (2*b14 - 2*b12 + b11 - b10 - b9 - b8 + 2*b4 + b2 + 2*b1 - 1) * q^52 + (b15 + b13 - 4*b12 + 4*b11 + 2*b10 - 3*b8 - b7 - b6 - 5*b5 + 2*b4 + 4*b3 + 8*b1) * q^53 + (-b10 - 2*b8 + b5 - 2*b4 + 2*b3 - b1) * q^54 + (-2*b14 + b12 + b11 - 3*b10 + b9 - 3*b8 - 2*b4 - 2*b3 - 3*b2 - 2*b1) * q^55 + (b13 + b12 - b11 - b10 + b8 + b7 + b6 + b5 - b4 - b3 - 2*b1) * q^56 + (-3*b15 + 2*b14 + 2*b13 - 4*b12 - b9 - 2*b8 + 4*b6 - 2*b5 + 2*b4 - b2 + 2*b1 - 1) * q^57 + (-2*b14 - 2*b11 + 3*b10 + b9 + 3*b8 - b4 + b3 - 2*b2 - b1 + 1) * q^58 + (4*b15 + 6*b13 + 2*b12 + b11 - b10 + b8 + 2*b7 + 6*b6 - b4 - 2*b3 - b1) * q^59 + (-b15 - b14 - b12 + 3*b11 - b9 - 2*b8 - 2*b7 - b5 + 2*b4 + b3 + b2 + 3*b1 - 2) * q^60 + (-4*b14 + b12 + 2*b10 - 4*b9 + 2*b8 + 2*b3 + 3*b2 - 3) * q^61 + (2*b13 - b12 + 2*b11 + 3*b10 + b8 - 2*b7 - 4*b5 + 3*b4 + b3 + 3*b1) * q^62 + (2*b15 + 2*b14 + 2*b12 + 3*b10 - b9 + b8 - 2*b7 + 2*b5 + 2*b4 + b3 + b2 - 2) * q^63 - b14 * q^64 + (4*b15 - b13 + 2*b12 - 2*b11 - 4*b10 - 2*b8 + 5*b7 + 4*b6 + 6*b5 - 4*b4 - 2*b3 - 4*b1) * q^65 + (2*b15 - 2*b14 + 2*b13 + 4*b9 + b7 + b6 - 3) * q^66 + (4*b14 - b12 + b4 - 4*b2 + b1 + 4) * q^67 + (-2*b15 + b13 - 2*b11 + 2*b8 - b7 + 2*b6 - 2*b1) * q^68 + (b15 + 3*b14 - b13 - 2*b12 + 3*b11 - b10 - 3*b9 - 3*b8 - b7 - 2*b6 - b5 - b4 + b3 - 2*b2 + 2*b1 - 2) * q^69 + (-2*b14 - b12 + 2*b11 + b10 + 4*b9 + b8 + 3*b3 + 4*b2 - 2) * q^70 + (b13 + b11 + 4*b10 + 2*b8 + 9*b7 - b6 - 3*b5 + 4*b4 + b1) * q^71 + (-2*b15 + b14 + b9 - b7 - b2 - 1) * q^72 + (6*b14 - 2*b12 + 2*b11 - 5*b9 + 4*b4 + 2*b3 + 5*b2 + 4*b1 + 1) * q^73 + (-b15 + b13 + b12 - 2*b11 - 2*b10 - 2*b8 - b7 + 5*b5 - 2*b4 - b3 - 3*b1) * q^74 + (-6*b13 - 2*b10 - 6*b9 - b8 + 6*b7 + 2*b5 - b4 + b3) * q^75 + (-b12 - 2*b9 - b4 - 2*b2 - b1) * q^76 + (b15 - b13 + b12 + b11 - 2*b8 - 4*b7 + 4*b6 + 2*b5 - b3) * q^77 + (-3*b15 - 4*b14 - b13 + b12 - b10 + 2*b9 - b8 - 3*b7 + b6 + 2*b5 + b4 - 2*b3 + 2*b2 - b1 - 1) * q^78 + (-b12 - b11 + 4*b4 - b3 + 4*b1 - 6) * q^79 + (-2*b13 - b12 - b8 - b6 + b3 + b1) * q^80 + (6*b15 + 6*b7 - 6*b6 + 3*b2) * q^81 + (3*b14 - 2*b12 - 2*b11 + 2*b10 - 3*b9 + 2*b8 + 2*b4 - 2*b2 + 2*b1 - 2) * q^82 + (6*b15 - 2*b13 + b12 - b10 - 4*b6 + b5 - b4 - b3 - b1) * q^83 + (2*b14 - b13 - b9 + b6 + b5 + b3 + 2*b2 + b1 - 1) * q^84 + (-b14 + 3*b11 + 2*b9 + 3*b3 + b2 - 2) * q^85 + (b12 + b10 + 2*b8 - b5 + b4 - b3 - b1) * q^86 + (5*b15 + b14 - b13 - b12 + 2*b10 + b9 + 2*b8 + b7 + 4*b6 - 3*b5 + 3*b3 - 6*b2 + 2*b1 + 6) * q^87 + (b12 + b11 + b10 + b8 + 2*b3) * q^88 + (-3*b15 - 2*b12 + 6*b10 + 2*b8 + 3*b7 - 3*b6 - 4*b5 + 6*b4 + 2*b3 + 2*b1) * q^89 + (-3*b15 - 3*b14 + 3*b13 - b12 + 3*b11 - 2*b10 - 2*b5 + b4 - b3 + b1) * q^90 + (-2*b14 + b12 + b11 + 6*b9 - 3*b4 + b3 + 2*b2 - 3*b1 - 2) * q^91 + (-b15 - b13 - 2*b10 - b8 - 2*b7 - 2*b6 + b5 - 2*b4) * q^92 + (-3*b15 + 6*b13 + 2*b12 + 2*b10 + 3*b9 + 2*b8 - 3*b7 + 3*b6 - 2*b5 - 2*b3 - 3*b2) * q^93 + (-2*b14 + 3*b12 - 3*b11 + 3*b10 + b9 + 3*b8 - 3*b2) * q^94 + (-3*b13 - b12 + b11 + b10 - b8 - 3*b7 - 3*b6 - b5 + b4 + b3 + 2*b1) * q^95 + (-b12 + b11 - b5 + b3 + b1) * q^96 + (2*b14 + 2*b11 - 6*b10 - 6*b8 - 2*b4 - 4*b3 + 2*b2 - 2*b1) * q^97 + (-2*b15 + 2*b13 - 4*b12 + 2*b11 + 2*b10 - 2*b8 - b7 + 2*b6 - 4*b5 + 2*b4 + 4*b3 + 6*b1) * q^98 + (b12 - 6*b11 + 3*b10 + 8*b8 + b5 + b4 - b3 - 3*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 8 q^{7}+O(q^{10})$$ 16 * q + 8 * q^7 $$16 q + 8 q^{7} - 24 q^{10} - 24 q^{13} + 8 q^{16} - 16 q^{19} - 24 q^{21} - 8 q^{28} + 24 q^{30} + 16 q^{31} - 24 q^{33} + 24 q^{34} + 24 q^{36} + 16 q^{37} + 48 q^{39} + 24 q^{45} + 24 q^{46} + 24 q^{49} - 8 q^{52} - 24 q^{55} - 24 q^{57} - 24 q^{60} - 24 q^{61} - 24 q^{63} - 48 q^{66} + 32 q^{67} - 48 q^{69} - 24 q^{72} + 56 q^{73} - 16 q^{76} - 96 q^{79} + 24 q^{81} - 48 q^{82} - 24 q^{85} + 48 q^{87} - 16 q^{91} - 24 q^{93} - 24 q^{94} + 16 q^{97}+O(q^{100})$$ 16 * q + 8 * q^7 - 24 * q^10 - 24 * q^13 + 8 * q^16 - 16 * q^19 - 24 * q^21 - 8 * q^28 + 24 * q^30 + 16 * q^31 - 24 * q^33 + 24 * q^34 + 24 * q^36 + 16 * q^37 + 48 * q^39 + 24 * q^45 + 24 * q^46 + 24 * q^49 - 8 * q^52 - 24 * q^55 - 24 * q^57 - 24 * q^60 - 24 * q^61 - 24 * q^63 - 48 * q^66 + 32 * q^67 - 48 * q^69 - 24 * q^72 + 56 * q^73 - 16 * q^76 - 96 * q^79 + 24 * q^81 - 48 * q^82 - 24 * q^85 + 48 * q^87 - 16 * q^91 - 24 * q^93 - 24 * q^94 + 16 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + \cdots + 25$$ :

 $$\beta_{1}$$ $$=$$ $$( 1548 \nu^{15} - 13188 \nu^{14} + 76652 \nu^{13} - 318997 \nu^{12} + 1019601 \nu^{11} - 2665197 \nu^{10} + \cdots - 204955 ) / 17095$$ (1548*v^15 - 13188*v^14 + 76652*v^13 - 318997*v^12 + 1019601*v^11 - 2665197*v^10 + 5595909*v^9 - 9912750*v^8 + 14273792*v^7 - 17359125*v^6 + 16940634*v^5 - 13645786*v^4 + 8382900*v^3 - 3952842*v^2 + 1197759*v - 204955) / 17095 $$\beta_{2}$$ $$=$$ $$( - 3456 \nu^{15} + 25920 \nu^{14} - 151876 \nu^{13} + 594074 \nu^{12} - 1879372 \nu^{11} + \cdots + 142555 ) / 17095$$ (-3456*v^15 + 25920*v^14 - 151876*v^13 + 594074*v^12 - 1879372*v^11 + 4666596*v^10 - 9554736*v^9 + 15945783*v^8 - 21928484*v^7 + 24527176*v^6 - 22138664*v^5 + 15739188*v^4 - 8598668*v^3 + 3418428*v^2 - 929924*v + 142555) / 17095 $$\beta_{3}$$ $$=$$ $$( 62 \nu^{14} - 434 \nu^{13} + 2541 \nu^{12} - 9604 \nu^{11} + 30137 \nu^{10} - 72992 \nu^{9} + \cdots + 5945 ) / 65$$ (62*v^14 - 434*v^13 + 2541*v^12 - 9604*v^11 + 30137*v^10 - 72992*v^9 + 147747*v^8 - 240948*v^7 + 326020*v^6 - 354283*v^5 + 310030*v^4 - 208129*v^3 + 103708*v^2 - 33855*v + 5945) / 65 $$\beta_{4}$$ $$=$$ $$( - 1548 \nu^{15} + 26338 \nu^{14} - 168702 \nu^{13} + 849994 \nu^{12} - 3008933 \nu^{11} + \cdots + 979490 ) / 17095$$ (-1548*v^15 + 26338*v^14 - 168702*v^13 + 849994*v^12 - 3008933*v^11 + 8792571*v^10 - 20191094*v^9 + 38713617*v^8 - 60027902*v^7 + 77090107*v^6 - 79360265*v^5 + 65574610*v^4 - 41344164*v^3 + 19292843*v^2 - 5858382*v + 979490) / 17095 $$\beta_{5}$$ $$=$$ $$( 15207 \nu^{15} - 93144 \nu^{14} + 525827 \nu^{13} - 1795712 \nu^{12} + 5254311 \nu^{11} + \cdots + 281130 ) / 17095$$ (15207*v^15 - 93144*v^14 + 525827*v^13 - 1795712*v^12 + 5254311*v^11 - 11330101*v^10 + 20458668*v^9 - 28148715*v^8 + 30952836*v^7 - 24081648*v^6 + 11893278*v^5 - 253940*v^4 - 4590360*v^3 + 3846345*v^2 - 1513502*v + 281130) / 17095 $$\beta_{6}$$ $$=$$ $$( 15977 \nu^{15} - 98393 \nu^{14} + 556086 \nu^{13} - 1902441 \nu^{12} + 5567337 \nu^{11} + \cdots + 610830 ) / 17095$$ (15977*v^15 - 98393*v^14 + 556086*v^13 - 1902441*v^12 + 5567337*v^11 - 11970872*v^10 + 21486123*v^9 - 29032660*v^8 + 30783293*v^7 - 21374907*v^6 + 6423019*v^5 + 7076841*v^4 - 11188252*v^3 + 8068367*v^2 - 3136623*v + 610830) / 17095 $$\beta_{7}$$ $$=$$ $$( 15977 \nu^{15} - 141262 \nu^{14} + 856169 \nu^{13} - 3642449 \nu^{12} + 12106306 \nu^{11} + \cdots - 1883725 ) / 17095$$ (15977*v^15 - 141262*v^14 + 856169*v^13 - 3642449*v^12 + 12106306*v^11 - 32236863*v^10 + 70027507*v^9 - 125576015*v^8 + 185376534*v^7 - 225133368*v^6 + 221440773*v^5 - 173497381*v^4 + 104335813*v^3 - 45658062*v^2 + 12999216*v - 1883725) / 17095 $$\beta_{8}$$ $$=$$ $$( 15207 \nu^{15} - 151267 \nu^{14} + 932688 \nu^{13} - 4151403 \nu^{12} + 14099264 \nu^{11} + \cdots - 2984015 ) / 17095$$ (15207*v^15 - 151267*v^14 + 932688*v^13 - 4151403*v^12 + 14099264*v^11 - 38692358*v^10 + 85888071*v^9 - 157951050*v^8 + 238257063*v^7 - 296346875*v^6 + 298087511*v^5 - 239467694*v^4 + 147674805*v^3 - 66643178*v^2 + 19588566*v - 2984015) / 17095 $$\beta_{9}$$ $$=$$ $$( 26630 \nu^{15} - 182630 \nu^{14} + 1056740 \nu^{13} - 3923413 \nu^{12} + 12097498 \nu^{11} + \cdots + 218885 ) / 17095$$ (26630*v^15 - 182630*v^14 + 1056740*v^13 - 3923413*v^12 + 12097498*v^11 - 28666706*v^10 + 56590650*v^9 - 89528958*v^8 + 116711850*v^7 - 121073088*v^6 + 99711814*v^5 - 61611091*v^4 + 27082796*v^3 - 6949514*v^2 + 510382*v + 218885) / 17095 $$\beta_{10}$$ $$=$$ $$( 49663 \nu^{15} - 358139 \nu^{14} + 2089429 \nu^{13} - 8006091 \nu^{12} + 25109764 \nu^{11} + \cdots - 1090565 ) / 17095$$ (49663*v^15 - 358139*v^14 + 2089429*v^13 - 8006091*v^12 + 25109764*v^11 - 61357045*v^10 + 124293850*v^9 - 204357372*v^8 + 277503791*v^7 - 305524016*v^6 + 271172221*v^5 - 188678893*v^4 + 99647214*v^3 - 37587574*v^2 + 9111403*v - 1090565) / 17095 $$\beta_{11}$$ $$=$$ $$( 46358 \nu^{15} - 350841 \nu^{14} + 2068305 \nu^{13} - 8157610 \nu^{12} + 26035188 \nu^{11} + \cdots - 2064105 ) / 17095$$ (46358*v^15 - 350841*v^14 + 2068305*v^13 - 8157610*v^12 + 26035188*v^11 - 65341929*v^10 + 135478704*v^9 - 229565910*v^8 + 321502876*v^7 - 367712403*v^6 + 340522371*v^5 - 249596834*v^4 + 140137971*v^3 - 57050598*v^2 + 15149982*v - 2064105) / 17095 $$\beta_{12}$$ $$=$$ $$( - 46358 \nu^{15} + 373985 \nu^{14} - 2230313 \nu^{13} + 9095994 \nu^{12} - 29559388 \nu^{11} + \cdots + 3439595 ) / 17095$$ (-46358*v^15 + 373985*v^14 - 2230313*v^13 + 9095994*v^12 - 29559388*v^11 + 76252747*v^10 - 161588818*v^9 + 281449761*v^8 - 404524612*v^7 + 477125400*v^6 - 456031708*v^5 + 346814258*v^4 - 202568648*v^3 + 86298565*v^2 - 24026495*v + 3439595) / 17095 $$\beta_{13}$$ $$=$$ $$( - 54184 \nu^{15} + 399016 \nu^{14} - 2335837 \nu^{13} + 9056462 \nu^{12} - 28575749 \nu^{11} + \cdots + 1312670 ) / 17095$$ (-54184*v^15 + 399016*v^14 - 2335837*v^13 + 9056462*v^12 - 28575749*v^11 + 70523459*v^10 - 143951776*v^9 + 239041170*v^8 - 327539991*v^7 + 364427829*v^6 - 326718848*v^5 + 229677473*v^4 - 122374471*v^3 + 46393946*v^2 - 11232929*v + 1312670) / 17095 $$\beta_{14}$$ $$=$$ $$( 10652 \nu^{15} - 79890 \nu^{14} + 470562 \nu^{13} - 1846988 \nu^{12} + 5882478 \nu^{11} + \cdots - 370592 ) / 3419$$ (10652*v^15 - 79890*v^14 + 470562*v^13 - 1846988*v^12 + 5882478*v^11 - 14702424*v^10 + 30390552*v^9 - 51252471*v^8 + 71445138*v^7 - 81160006*v^6 + 74558856*v^5 - 53991081*v^4 + 29816774*v^3 - 11804598*v^2 + 3003630*v - 370592) / 3419 $$\beta_{15}$$ $$=$$ $$( - 54184 \nu^{15} + 413744 \nu^{14} - 2438933 \nu^{13} + 9652683 \nu^{12} - 30812827 \nu^{11} + \cdots + 1951760 ) / 17095$$ (-54184*v^15 + 413744*v^14 - 2438933*v^13 + 9652683*v^12 - 30812827*v^11 + 77432206*v^10 - 160446084*v^9 + 271670265*v^8 - 379486173*v^7 + 432289456*v^6 - 397583146*v^5 + 288232582*v^4 - 159062971*v^3 + 62844859*v^2 - 15915907*v + 1951760) / 17095
 $$\nu$$ $$=$$ $$( \beta_{15} - \beta_{14} + \beta_{13} - \beta_{12} + \beta_{11} + 2 \beta_{10} - 2 \beta_{5} + 2 \beta_{4} + \cdots + 1 ) / 2$$ (b15 - b14 + b13 - b12 + b11 + 2*b10 - 2*b5 + 2*b4 + b3 + 2*b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{15} - \beta_{12} + \beta_{10} - \beta_{9} - \beta_{5} + 2\beta_{4} + \beta_{3} + 2\beta _1 - 2$$ b15 - b12 + b10 - b9 - b5 + 2*b4 + b3 + 2*b1 - 2 $$\nu^{3}$$ $$=$$ $$( - 2 \beta_{15} + 7 \beta_{14} - 5 \beta_{13} + 4 \beta_{12} - 7 \beta_{11} - 7 \beta_{10} - 3 \beta_{9} + \cdots - 5 ) / 2$$ (-2*b15 + 7*b14 - 5*b13 + 4*b12 - 7*b11 - 7*b10 - 3*b9 - b8 + 3*b7 + 3*b6 + 6*b5 - 7*b4 - 4*b3 - 3*b2 - 5*b1 - 5) / 2 $$\nu^{4}$$ $$=$$ $$- 10 \beta_{15} + 3 \beta_{14} + 2 \beta_{13} + 8 \beta_{12} - 4 \beta_{11} - 8 \beta_{10} + 6 \beta_{9} + \cdots + 9$$ -10*b15 + 3*b14 + 2*b13 + 8*b12 - 4*b11 - 8*b10 + 6*b9 + 6*b6 + 6*b5 - 16*b4 - 10*b3 - 3*b2 - 14*b1 + 9 $$\nu^{5}$$ $$=$$ $$( - 21 \beta_{15} - 46 \beta_{14} + 44 \beta_{13} - 19 \beta_{12} + 44 \beta_{11} + 19 \beta_{10} + \cdots + 36 ) / 2$$ (-21*b15 - 46*b14 + 44*b13 - 19*b12 + 44*b11 + 19*b10 + 35*b9 + 3*b8 - 35*b7 - 5*b6 - 26*b5 + 19*b4 - b3 + 25*b2 + 3*b1 + 36) / 2 $$\nu^{6}$$ $$=$$ $$60 \beta_{15} - 43 \beta_{14} - 5 \beta_{13} - 70 \beta_{12} + 55 \beta_{11} + 49 \beta_{10} - 30 \beta_{9} + \cdots - 50$$ 60*b15 - 43*b14 - 5*b13 - 70*b12 + 55*b11 + 49*b10 - 30*b9 - 10*b8 - 15*b7 - 60*b6 - 40*b5 + 122*b4 + 69*b3 + 45*b2 + 93*b1 - 50 $$\nu^{7}$$ $$=$$ $$( 321 \beta_{15} + 267 \beta_{14} - 365 \beta_{13} + 17 \beta_{12} - 213 \beta_{11} - 28 \beta_{10} + \cdots - 307 ) / 2$$ (321*b15 + 267*b14 - 365*b13 + 17*b12 - 213*b11 - 28*b10 - 336*b9 - 60*b8 + 294*b7 - 126*b6 + 136*b5 + 84*b4 + 179*b3 - 126*b2 + 162*b1 - 307) / 2 $$\nu^{8}$$ $$=$$ $$- 256 \beta_{15} + 446 \beta_{14} - 108 \beta_{13} + 544 \beta_{12} - 528 \beta_{11} - 304 \beta_{10} + \cdots + 267$$ -256*b15 + 446*b14 - 108*b13 + 544*b12 - 528*b11 - 304*b10 + 74*b9 + 84*b8 + 280*b7 + 420*b6 + 316*b5 - 868*b4 - 392*b3 - 469*b2 - 572*b1 + 267 $$\nu^{9}$$ $$=$$ $$( - 3026 \beta_{15} - 1225 \beta_{14} + 2701 \beta_{13} + 906 \beta_{12} + 525 \beta_{11} - 297 \beta_{10} + \cdots + 2725 ) / 2$$ (-3026*b15 - 1225*b14 + 2701*b13 + 906*b12 + 525*b11 - 297*b10 + 2835*b9 + 795*b8 - 1899*b7 + 2007*b6 - 606*b5 - 2265*b4 - 2202*b3 + 9*b2 - 2277*b1 + 2725) / 2 $$\nu^{10}$$ $$=$$ $$274 \beta_{15} - 4001 \beta_{14} + 2070 \beta_{13} - 3737 \beta_{12} + 4287 \beta_{11} + 1922 \beta_{10} + \cdots - 932$$ 274*b15 - 4001*b14 + 2070*b13 - 3737*b12 + 4287*b11 + 1922*b10 + 846*b9 - 300*b8 - 3285*b7 - 2250*b6 - 2609*b5 + 5530*b4 + 1694*b3 + 3870*b2 + 3067*b1 - 932 $$\nu^{11}$$ $$=$$ $$( 23673 \beta_{15} + 1926 \beta_{14} - 17346 \beta_{13} - 14155 \beta_{12} + 4640 \beta_{11} + \cdots - 22758 ) / 2$$ (23673*b15 + 1926*b14 - 17346*b13 - 14155*b12 + 4640*b11 + 5251*b10 - 20779*b9 - 7509*b8 + 8569*b7 - 20801*b6 + 506*b5 + 28021*b4 + 20095*b3 + 8261*b2 + 22843*b1 - 22758) / 2 $$\nu^{12}$$ $$=$$ $$10280 \beta_{15} + 31947 \beta_{14} - 24436 \beta_{13} + 21968 \beta_{12} - 30552 \beta_{11} + \cdots - 3138$$ 10280*b15 + 31947*b14 - 24436*b13 + 21968*b12 - 30552*b11 - 11620*b10 - 17265*b9 - 1650*b8 + 30690*b7 + 6798*b6 + 20444*b5 - 28982*b4 - 2254*b3 - 26565*b2 - 11868*b1 - 3138 $$\nu^{13}$$ $$=$$ $$( - 160655 \beta_{15} + 46695 \beta_{14} + 88659 \beta_{13} + 152123 \beta_{12} - 96769 \beta_{11} + \cdots + 171575 ) / 2$$ (-160655*b15 + 46695*b14 + 88659*b13 + 152123*b12 - 96769*b11 - 59234*b10 + 128544*b9 + 57480*b8 - 4992*b7 + 177762*b6 + 32656*b5 - 273318*b4 - 157219*b3 - 120978*b2 - 196060*b1 + 171575) / 2 $$\nu^{14}$$ $$=$$ $$- 160979 \beta_{15} - 225841 \beta_{14} + 234065 \beta_{13} - 95851 \beta_{12} + 187408 \beta_{11} + \cdots + 106657$$ -160979*b15 - 225841*b14 + 234065*b13 - 95851*b12 + 187408*b11 + 60319*b10 + 200859*b9 + 43732*b8 - 244699*b7 + 38129*b6 - 144445*b5 + 90048*b4 - 62947*b3 + 146874*b2 - 5512*b1 + 106657 $$\nu^{15}$$ $$=$$ $$( 914760 \beta_{15} - 806121 \beta_{14} - 233687 \beta_{13} - 1368388 \beta_{12} + 1121513 \beta_{11} + \cdots - 1132093 ) / 2$$ (914760*b15 - 806121*b14 - 233687*b13 - 1368388*b12 + 1121513*b11 + 561279*b10 - 600297*b9 - 364797*b8 - 458997*b7 - 1313487*b6 - 528672*b5 + 2298683*b4 + 1078190*b3 + 1251033*b2 + 1491905*b1 - 1132093) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/78\mathbb{Z}\right)^\times$$.

 $$n$$ $$53$$ $$67$$ $$\chi(n)$$ $$-1$$ $$-\beta_{9} + \beta_{14}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 0.5 − 1.74530i 0.5 + 1.33108i 0.5 − 0.331082i 0.5 + 2.74530i 0.5 − 0.410882i 0.5 − 2.00333i 0.5 − 0.589118i 0.5 + 1.00333i 0.5 + 0.410882i 0.5 + 2.00333i 0.5 + 0.589118i 0.5 − 1.00333i 0.5 + 1.74530i 0.5 − 1.33108i 0.5 + 0.331082i 0.5 − 2.74530i
−0.965926 0.258819i −1.73022 + 0.0795432i 0.866025 + 0.500000i 2.76293 2.76293i 1.69185 + 0.370982i −0.657464 2.45369i −0.707107 0.707107i 2.98735 0.275255i −3.38389 + 1.95369i
11.2 −0.965926 0.258819i 1.73022 0.0795432i 0.866025 + 0.500000i −0.313444 + 0.313444i −1.69185 0.370982i −0.0745867 0.278362i −0.707107 0.707107i 2.98735 0.275255i 0.383889 0.221638i
11.3 0.965926 + 0.258819i −0.933998 + 1.45865i 0.866025 + 0.500000i 0.313444 0.313444i −1.27970 + 1.16721i −0.0745867 0.278362i 0.707107 + 0.707107i −1.25529 2.72474i 0.383889 0.221638i
11.4 0.965926 + 0.258819i 0.933998 1.45865i 0.866025 + 0.500000i −2.76293 + 2.76293i 1.27970 1.16721i −0.657464 2.45369i 0.707107 + 0.707107i −1.25529 2.72474i −3.38389 + 1.95369i
41.1 −0.258819 + 0.965926i −0.0795432 1.73022i −0.866025 0.500000i 2.02097 + 2.02097i 1.69185 + 0.370982i 3.46723 0.929042i 0.707107 0.707107i −2.98735 + 0.275255i −2.47517 + 1.42904i
41.2 −0.258819 + 0.965926i 0.0795432 + 1.73022i −0.866025 0.500000i 0.428520 + 0.428520i −1.69185 0.370982i −0.735180 + 0.196991i 0.707107 0.707107i −2.98735 + 0.275255i −0.524827 + 0.303009i
41.3 0.258819 0.965926i −1.45865 0.933998i −0.866025 0.500000i −2.02097 2.02097i −1.27970 + 1.16721i 3.46723 0.929042i −0.707107 + 0.707107i 1.25529 + 2.72474i −2.47517 + 1.42904i
41.4 0.258819 0.965926i 1.45865 + 0.933998i −0.866025 0.500000i −0.428520 0.428520i 1.27970 1.16721i −0.735180 + 0.196991i −0.707107 + 0.707107i 1.25529 + 2.72474i −0.524827 + 0.303009i
59.1 −0.258819 0.965926i −0.0795432 + 1.73022i −0.866025 + 0.500000i 2.02097 2.02097i 1.69185 0.370982i 3.46723 + 0.929042i 0.707107 + 0.707107i −2.98735 0.275255i −2.47517 1.42904i
59.2 −0.258819 0.965926i 0.0795432 1.73022i −0.866025 + 0.500000i 0.428520 0.428520i −1.69185 + 0.370982i −0.735180 0.196991i 0.707107 + 0.707107i −2.98735 0.275255i −0.524827 0.303009i
59.3 0.258819 + 0.965926i −1.45865 + 0.933998i −0.866025 + 0.500000i −2.02097 + 2.02097i −1.27970 1.16721i 3.46723 + 0.929042i −0.707107 0.707107i 1.25529 2.72474i −2.47517 1.42904i
59.4 0.258819 + 0.965926i 1.45865 0.933998i −0.866025 + 0.500000i −0.428520 + 0.428520i 1.27970 + 1.16721i −0.735180 0.196991i −0.707107 0.707107i 1.25529 2.72474i −0.524827 0.303009i
71.1 −0.965926 + 0.258819i −1.73022 0.0795432i 0.866025 0.500000i 2.76293 + 2.76293i 1.69185 0.370982i −0.657464 + 2.45369i −0.707107 + 0.707107i 2.98735 + 0.275255i −3.38389 1.95369i
71.2 −0.965926 + 0.258819i 1.73022 + 0.0795432i 0.866025 0.500000i −0.313444 0.313444i −1.69185 + 0.370982i −0.0745867 + 0.278362i −0.707107 + 0.707107i 2.98735 + 0.275255i 0.383889 + 0.221638i
71.3 0.965926 0.258819i −0.933998 1.45865i 0.866025 0.500000i 0.313444 + 0.313444i −1.27970 1.16721i −0.0745867 + 0.278362i 0.707107 0.707107i −1.25529 + 2.72474i 0.383889 + 0.221638i
71.4 0.965926 0.258819i 0.933998 + 1.45865i 0.866025 0.500000i −2.76293 2.76293i 1.27970 + 1.16721i −0.657464 + 2.45369i 0.707107 0.707107i −1.25529 + 2.72474i −3.38389 1.95369i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 11.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.f odd 12 1 inner
39.k even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 78.2.k.a 16
3.b odd 2 1 inner 78.2.k.a 16
4.b odd 2 1 624.2.cn.d 16
12.b even 2 1 624.2.cn.d 16
13.c even 3 1 1014.2.g.c 16
13.e even 6 1 1014.2.g.d 16
13.f odd 12 1 inner 78.2.k.a 16
13.f odd 12 1 1014.2.g.c 16
13.f odd 12 1 1014.2.g.d 16
39.h odd 6 1 1014.2.g.d 16
39.i odd 6 1 1014.2.g.c 16
39.k even 12 1 inner 78.2.k.a 16
39.k even 12 1 1014.2.g.c 16
39.k even 12 1 1014.2.g.d 16
52.l even 12 1 624.2.cn.d 16
156.v odd 12 1 624.2.cn.d 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.k.a 16 1.a even 1 1 trivial
78.2.k.a 16 3.b odd 2 1 inner
78.2.k.a 16 13.f odd 12 1 inner
78.2.k.a 16 39.k even 12 1 inner
624.2.cn.d 16 4.b odd 2 1
624.2.cn.d 16 12.b even 2 1
624.2.cn.d 16 52.l even 12 1
624.2.cn.d 16 156.v odd 12 1
1014.2.g.c 16 13.c even 3 1
1014.2.g.c 16 13.f odd 12 1
1014.2.g.c 16 39.i odd 6 1
1014.2.g.c 16 39.k even 12 1
1014.2.g.d 16 13.e even 6 1
1014.2.g.d 16 13.f odd 12 1
1014.2.g.d 16 39.h odd 6 1
1014.2.g.d 16 39.k even 12 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(78, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{8} - T^{4} + 1)^{2}$$
$3$ $$T^{16} - 6 T^{12} + \cdots + 6561$$
$5$ $$T^{16} + 300 T^{12} + \cdots + 81$$
$7$ $$(T^{8} - 4 T^{7} + 2 T^{6} + \cdots + 4)^{2}$$
$11$ $$(T^{8} - 30 T^{6} + \cdots + 36)^{2}$$
$13$ $$(T^{8} + 12 T^{7} + \cdots + 28561)^{2}$$
$17$ $$T^{16} + \cdots + 151807041$$
$19$ $$(T^{8} + 8 T^{7} + \cdots + 5476)^{2}$$
$23$ $$T^{16} + 60 T^{14} + \cdots + 18974736$$
$29$ $$T^{16} + \cdots + 33871089681$$
$31$ $$(T^{8} - 8 T^{7} + \cdots + 7744)^{2}$$
$37$ $$(T^{8} - 8 T^{7} + \cdots + 375769)^{2}$$
$41$ $$T^{16} + \cdots + 65697655057281$$
$43$ $$(T^{8} - 18 T^{6} + \cdots + 2916)^{2}$$
$47$ $$T^{16} + \cdots + 41006250000$$
$53$ $$(T^{8} + 156 T^{6} + \cdots + 522729)^{2}$$
$59$ $$T^{16} + \cdots + 43489065701376$$
$61$ $$(T^{8} + 12 T^{7} + \cdots + 47961)^{2}$$
$67$ $$(T^{8} - 16 T^{7} + \cdots + 676)^{2}$$
$71$ $$T^{16} + \cdots + 5143987297296$$
$73$ $$(T^{8} - 28 T^{7} + \cdots + 16834609)^{2}$$
$79$ $$(T^{4} + 24 T^{3} + \cdots + 312)^{4}$$
$83$ $$T^{16} + \cdots + 36804120336$$
$89$ $$T^{16} + \cdots + 59\!\cdots\!56$$
$97$ $$(T^{8} - 8 T^{7} + \cdots + 43264)^{2}$$