# Properties

 Label 78.2.e.b Level $78$ Weight $2$ Character orbit 78.e Analytic conductor $0.623$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [78,2,Mod(55,78)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(78, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

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

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

chi := DirichletCharacter("78.55");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$78 = 2 \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 78.e (of order $$3$$, 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ 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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$53$$ $$67$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
55.1
 0.5 − 0.866025i 0.5 + 0.866025i
0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i −1.00000 −0.500000 + 0.866025i 1.00000 1.73205i −1.00000 −0.500000 + 0.866025i −0.500000 0.866025i
61.1 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i −1.00000 −0.500000 0.866025i 1.00000 + 1.73205i −1.00000 −0.500000 0.866025i −0.500000 + 0.866025i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 78.2.e.b 2
3.b odd 2 1 234.2.h.b 2
4.b odd 2 1 624.2.q.b 2
5.b even 2 1 1950.2.i.b 2
5.c odd 4 2 1950.2.z.b 4
12.b even 2 1 1872.2.t.i 2
13.b even 2 1 1014.2.e.d 2
13.c even 3 1 inner 78.2.e.b 2
13.c even 3 1 1014.2.a.a 1
13.d odd 4 2 1014.2.i.e 4
13.e even 6 1 1014.2.a.e 1
13.e even 6 1 1014.2.e.d 2
13.f odd 12 2 1014.2.b.a 2
13.f odd 12 2 1014.2.i.e 4
39.h odd 6 1 3042.2.a.d 1
39.i odd 6 1 234.2.h.b 2
39.i odd 6 1 3042.2.a.m 1
39.k even 12 2 3042.2.b.d 2
52.i odd 6 1 8112.2.a.bb 1
52.j odd 6 1 624.2.q.b 2
52.j odd 6 1 8112.2.a.x 1
65.n even 6 1 1950.2.i.b 2
65.q odd 12 2 1950.2.z.b 4
156.p even 6 1 1872.2.t.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.e.b 2 1.a even 1 1 trivial
78.2.e.b 2 13.c even 3 1 inner
234.2.h.b 2 3.b odd 2 1
234.2.h.b 2 39.i odd 6 1
624.2.q.b 2 4.b odd 2 1
624.2.q.b 2 52.j odd 6 1
1014.2.a.a 1 13.c even 3 1
1014.2.a.e 1 13.e even 6 1
1014.2.b.a 2 13.f odd 12 2
1014.2.e.d 2 13.b even 2 1
1014.2.e.d 2 13.e even 6 1
1014.2.i.e 4 13.d odd 4 2
1014.2.i.e 4 13.f odd 12 2
1872.2.t.i 2 12.b even 2 1
1872.2.t.i 2 156.p even 6 1
1950.2.i.b 2 5.b even 2 1
1950.2.i.b 2 65.n even 6 1
1950.2.z.b 4 5.c odd 4 2
1950.2.z.b 4 65.q odd 12 2
3042.2.a.d 1 39.h odd 6 1
3042.2.a.m 1 39.i odd 6 1
3042.2.b.d 2 39.k even 12 2
8112.2.a.x 1 52.j odd 6 1
8112.2.a.bb 1 52.i odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 1$$ acting on $$S_{2}^{\mathrm{new}}(78, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2} - T + 1$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} - 2T + 4$$
$11$ $$T^{2} + 2T + 4$$
$13$ $$T^{2} - 5T + 13$$
$17$ $$T^{2} + 5T + 25$$
$19$ $$T^{2} - 2T + 4$$
$23$ $$T^{2} + 6T + 36$$
$29$ $$T^{2} - 9T + 81$$
$31$ $$(T + 4)^{2}$$
$37$ $$T^{2} - 11T + 121$$
$41$ $$T^{2} + 5T + 25$$
$43$ $$T^{2} + 10T + 100$$
$47$ $$(T - 2)^{2}$$
$53$ $$(T + 1)^{2}$$
$59$ $$T^{2} - 8T + 64$$
$61$ $$T^{2} - 11T + 121$$
$67$ $$T^{2} + 2T + 4$$
$71$ $$T^{2} - 14T + 196$$
$73$ $$(T + 13)^{2}$$
$79$ $$(T + 4)^{2}$$
$83$ $$(T - 6)^{2}$$
$89$ $$T^{2} + 2T + 4$$
$97$ $$T^{2} - 2T + 4$$