# Properties

 Label 78.2.g.a Level $78$ Weight $2$ Character orbit 78.g Analytic conductor $0.623$ Analytic rank $0$ Dimension $12$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [78,2,Mod(5,78)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(78, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3]))

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

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

chi := DirichletCharacter("78.5");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$78 = 2 \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 78.g (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: $$0.622833135766$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(i)$$ Coefficient field: 12.0.58498535041007616.52 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - 12x^{9} + 72x^{6} - 324x^{3} + 729$$ x^12 - 12*x^9 + 72*x^6 - 324*x^3 + 729 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 12x^{9} + 72x^{6} - 324x^{3} + 729$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{9} - 3\nu^{6} + 18\nu^{3} - 81 ) / 81$$ (v^9 - 3*v^6 + 18*v^3 - 81) / 81 $$\beta_{3}$$ $$=$$ $$( \nu^{10} - 3\nu^{7} + 18\nu^{4} - 81\nu ) / 81$$ (v^10 - 3*v^7 + 18*v^4 - 81*v) / 81 $$\beta_{4}$$ $$=$$ $$( -\nu^{11} + 3\nu^{8} - 45\nu^{5} + 162\nu^{2} ) / 243$$ (-v^11 + 3*v^8 - 45*v^5 + 162*v^2) / 243 $$\beta_{5}$$ $$=$$ $$( -\nu^{11} + 12\nu^{8} - 72\nu^{5} + 324\nu^{2} ) / 243$$ (-v^11 + 12*v^8 - 72*v^5 + 324*v^2) / 243 $$\beta_{6}$$ $$=$$ $$( -2\nu^{9} + 15\nu^{6} - 63\nu^{3} + 243 ) / 81$$ (-2*v^9 + 15*v^6 - 63*v^3 + 243) / 81 $$\beta_{7}$$ $$=$$ $$( -2\nu^{10} + 15\nu^{7} - 63\nu^{4} + 243\nu ) / 81$$ (-2*v^10 + 15*v^7 - 63*v^4 + 243*v) / 81 $$\beta_{8}$$ $$=$$ $$( 2\nu^{9} - 15\nu^{6} + 90\nu^{3} - 324 ) / 81$$ (2*v^9 - 15*v^6 + 90*v^3 - 324) / 81 $$\beta_{9}$$ $$=$$ $$( 2\nu^{10} - 15\nu^{7} + 90\nu^{4} - 324\nu ) / 81$$ (2*v^10 - 15*v^7 + 90*v^4 - 324*v) / 81 $$\beta_{10}$$ $$=$$ $$( -\nu^{11} + 6\nu^{8} - 27\nu^{5} + 135\nu^{2} ) / 81$$ (-v^11 + 6*v^8 - 27*v^5 + 135*v^2) / 81 $$\beta_{11}$$ $$=$$ $$( 4\nu^{11} - 30\nu^{8} + 153\nu^{5} - 486\nu^{2} ) / 243$$ (4*v^11 - 30*v^8 + 153*v^5 - 486*v^2) / 243
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{11} + \beta_{10} + \beta_{5}$$ b11 + b10 + b5 $$\nu^{3}$$ $$=$$ $$3\beta_{8} + 3\beta_{6} + 3$$ 3*b8 + 3*b6 + 3 $$\nu^{4}$$ $$=$$ $$3\beta_{9} + 3\beta_{7} + 3\beta_1$$ 3*b9 + 3*b7 + 3*b1 $$\nu^{5}$$ $$=$$ $$3\beta_{11} + 6\beta_{10} - 6\beta_{4}$$ 3*b11 + 6*b10 - 6*b4 $$\nu^{6}$$ $$=$$ $$9\beta_{8} + 18\beta_{6} + 18\beta_{2}$$ 9*b8 + 18*b6 + 18*b2 $$\nu^{7}$$ $$=$$ $$9\beta_{9} + 18\beta_{7} + 18\beta_{3}$$ 9*b9 + 18*b7 + 18*b3 $$\nu^{8}$$ $$=$$ $$-9\beta_{11} + 9\beta_{5} - 45\beta_{4}$$ -9*b11 + 9*b5 - 45*b4 $$\nu^{9}$$ $$=$$ $$-27\beta_{8} + 135\beta_{2} + 27$$ -27*b8 + 135*b2 + 27 $$\nu^{10}$$ $$=$$ $$-27\beta_{9} + 135\beta_{3} + 27\beta_1$$ -27*b9 + 135*b3 + 27*b1 $$\nu^{11}$$ $$=$$ $$-108\beta_{10} + 189\beta_{5} - 108\beta_{4}$$ -108*b10 + 189*b5 - 108*b4

## Character values

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

 $$n$$ $$53$$ $$67$$ $$\chi(n)$$ $$-1$$ $$-\beta_{8}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 1.54662 + 0.779723i −1.44857 + 0.949550i −0.0980500 − 1.72927i −0.779723 − 1.54662i −0.949550 + 1.44857i 1.72927 + 0.0980500i 1.54662 − 0.779723i −1.44857 − 0.949550i −0.0980500 + 1.72927i −0.779723 + 1.54662i −0.949550 − 1.44857i 1.72927 − 0.0980500i
−0.707107 0.707107i −1.64497 0.542278i 1.00000i −2.32634 2.32634i 0.779723 + 1.54662i −1.76690 1.76690i 0.707107 0.707107i 2.41187 + 1.78406i 3.28995i
5.2 −0.707107 0.707107i 0.352860 + 1.69573i 1.00000i 0.499019 + 0.499019i 0.949550 1.44857i 1.39812 + 1.39812i 0.707107 0.707107i −2.75098 + 1.19671i 0.705720i
5.3 −0.707107 0.707107i 1.29211 1.15345i 1.00000i 1.82732 + 1.82732i −1.72927 0.0980500i −2.63122 2.63122i 0.707107 0.707107i 0.339111 2.98077i 2.58423i
5.4 0.707107 + 0.707107i −1.64497 + 0.542278i 1.00000i 2.32634 + 2.32634i −1.54662 0.779723i −1.76690 1.76690i −0.707107 + 0.707107i 2.41187 1.78406i 3.28995i
5.5 0.707107 + 0.707107i 0.352860 1.69573i 1.00000i −0.499019 0.499019i 1.44857 0.949550i 1.39812 + 1.39812i −0.707107 + 0.707107i −2.75098 1.19671i 0.705720i
5.6 0.707107 + 0.707107i 1.29211 + 1.15345i 1.00000i −1.82732 1.82732i 0.0980500 + 1.72927i −2.63122 2.63122i −0.707107 + 0.707107i 0.339111 + 2.98077i 2.58423i
47.1 −0.707107 + 0.707107i −1.64497 + 0.542278i 1.00000i −2.32634 + 2.32634i 0.779723 1.54662i −1.76690 + 1.76690i 0.707107 + 0.707107i 2.41187 1.78406i 3.28995i
47.2 −0.707107 + 0.707107i 0.352860 1.69573i 1.00000i 0.499019 0.499019i 0.949550 + 1.44857i 1.39812 1.39812i 0.707107 + 0.707107i −2.75098 1.19671i 0.705720i
47.3 −0.707107 + 0.707107i 1.29211 + 1.15345i 1.00000i 1.82732 1.82732i −1.72927 + 0.0980500i −2.63122 + 2.63122i 0.707107 + 0.707107i 0.339111 + 2.98077i 2.58423i
47.4 0.707107 0.707107i −1.64497 0.542278i 1.00000i 2.32634 2.32634i −1.54662 + 0.779723i −1.76690 + 1.76690i −0.707107 0.707107i 2.41187 + 1.78406i 3.28995i
47.5 0.707107 0.707107i 0.352860 + 1.69573i 1.00000i −0.499019 + 0.499019i 1.44857 + 0.949550i 1.39812 1.39812i −0.707107 0.707107i −2.75098 + 1.19671i 0.705720i
47.6 0.707107 0.707107i 1.29211 1.15345i 1.00000i −1.82732 + 1.82732i 0.0980500 1.72927i −2.63122 + 2.63122i −0.707107 0.707107i 0.339111 2.98077i 2.58423i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 5.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.d odd 4 1 inner
39.f even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 78.2.g.a 12
3.b odd 2 1 inner 78.2.g.a 12
4.b odd 2 1 624.2.bf.f 12
12.b even 2 1 624.2.bf.f 12
13.b even 2 1 1014.2.g.b 12
13.d odd 4 1 inner 78.2.g.a 12
13.d odd 4 1 1014.2.g.b 12
39.d odd 2 1 1014.2.g.b 12
39.f even 4 1 inner 78.2.g.a 12
39.f even 4 1 1014.2.g.b 12
52.f even 4 1 624.2.bf.f 12
156.l odd 4 1 624.2.bf.f 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.g.a 12 1.a even 1 1 trivial
78.2.g.a 12 3.b odd 2 1 inner
78.2.g.a 12 13.d odd 4 1 inner
78.2.g.a 12 39.f even 4 1 inner
624.2.bf.f 12 4.b odd 2 1
624.2.bf.f 12 12.b even 2 1
624.2.bf.f 12 52.f even 4 1
624.2.bf.f 12 156.l odd 4 1
1014.2.g.b 12 13.b even 2 1
1014.2.g.b 12 13.d odd 4 1
1014.2.g.b 12 39.d odd 2 1
1014.2.g.b 12 39.f even 4 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{4} + 1)^{3}$$
$3$ $$(T^{6} + 6 T^{3} + 27)^{2}$$
$5$ $$T^{12} + 162 T^{8} + \cdots + 1296$$
$7$ $$(T^{6} + 6 T^{5} + \cdots + 338)^{2}$$
$11$ $$T^{12} + 648 T^{8} + \cdots + 331776$$
$13$ $$(T^{6} + 21 T^{4} + \cdots + 2197)^{2}$$
$17$ $$(T^{6} - 54 T^{4} + \cdots - 1152)^{2}$$
$19$ $$(T^{6} + 6 T^{5} + \cdots + 1568)^{2}$$
$23$ $$(T^{6} - 36 T^{4} + \cdots - 288)^{2}$$
$29$ $$(T^{6} + 36 T^{4} + \cdots + 288)^{2}$$
$31$ $$(T^{6} - 6 T^{5} + \cdots + 512)^{2}$$
$37$ $$(T^{6} - 6 T^{5} + \cdots + 4802)^{2}$$
$41$ $$T^{12} + 18576 T^{8} + \cdots + 5308416$$
$43$ $$(T^{6} + 162 T^{4} + \cdots + 324)^{2}$$
$47$ $$T^{12} + 15714 T^{8} + \cdots + 1679616$$
$53$ $$(T^{6} + 324 T^{4} + \cdots + 1152)^{2}$$
$59$ $$T^{12} + 2592 T^{8} + \cdots + 5308416$$
$61$ $$(T^{3} - 72 T + 192)^{4}$$
$67$ $$(T^{6} + 6 T^{5} + \cdots + 392)^{2}$$
$71$ $$T^{12} + 13986 T^{8} + \cdots + 20736$$
$73$ $$(T^{6} + 6 T^{5} + \cdots + 1568)^{2}$$
$79$ $$(T^{3} - 18 T^{2} + \cdots + 12)^{4}$$
$83$ $$T^{12} + \cdots + 21743271936$$
$89$ $$T^{12} + \cdots + 27710263296$$
$97$ $$(T^{6} + 30 T^{5} + \cdots + 392)^{2}$$