# Properties

 Label 78.2.i.b Level $78$ Weight $2$ Character orbit 78.i Analytic conductor $0.623$ Analytic rank $0$ Dimension $4$ 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(43,78)]

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, 1]))

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

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

chi := DirichletCharacter("78.43");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$78 = 2 \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 78.i (of order $$6$$, 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$53$$ $$67$$ $$\chi(n)$$ $$1$$ $$\zeta_{12}^{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
43.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−0.866025 0.500000i 0.500000 0.866025i 0.500000 + 0.866025i 1.73205i −0.866025 + 0.500000i 1.09808 0.633975i 1.00000i −0.500000 0.866025i −0.866025 + 1.50000i
43.2 0.866025 + 0.500000i 0.500000 0.866025i 0.500000 + 0.866025i 1.73205i 0.866025 0.500000i −4.09808 + 2.36603i 1.00000i −0.500000 0.866025i 0.866025 1.50000i
49.1 −0.866025 + 0.500000i 0.500000 + 0.866025i 0.500000 0.866025i 1.73205i −0.866025 0.500000i 1.09808 + 0.633975i 1.00000i −0.500000 + 0.866025i −0.866025 1.50000i
49.2 0.866025 0.500000i 0.500000 + 0.866025i 0.500000 0.866025i 1.73205i 0.866025 + 0.500000i −4.09808 2.36603i 1.00000i −0.500000 + 0.866025i 0.866025 + 1.50000i
 $$n$$: 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.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 78.2.i.b 4
3.b odd 2 1 234.2.l.a 4
4.b odd 2 1 624.2.bv.d 4
5.b even 2 1 1950.2.bc.c 4
5.c odd 4 1 1950.2.y.a 4
5.c odd 4 1 1950.2.y.h 4
12.b even 2 1 1872.2.by.k 4
13.b even 2 1 1014.2.i.f 4
13.c even 3 1 1014.2.b.d 4
13.c even 3 1 1014.2.i.f 4
13.d odd 4 1 1014.2.e.h 4
13.d odd 4 1 1014.2.e.j 4
13.e even 6 1 inner 78.2.i.b 4
13.e even 6 1 1014.2.b.d 4
13.f odd 12 1 1014.2.a.h 2
13.f odd 12 1 1014.2.a.j 2
13.f odd 12 1 1014.2.e.h 4
13.f odd 12 1 1014.2.e.j 4
39.h odd 6 1 234.2.l.a 4
39.h odd 6 1 3042.2.b.l 4
39.i odd 6 1 3042.2.b.l 4
39.k even 12 1 3042.2.a.s 2
39.k even 12 1 3042.2.a.v 2
52.i odd 6 1 624.2.bv.d 4
52.l even 12 1 8112.2.a.bq 2
52.l even 12 1 8112.2.a.bx 2
65.l even 6 1 1950.2.bc.c 4
65.r odd 12 1 1950.2.y.a 4
65.r odd 12 1 1950.2.y.h 4
156.r even 6 1 1872.2.by.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.i.b 4 1.a even 1 1 trivial
78.2.i.b 4 13.e even 6 1 inner
234.2.l.a 4 3.b odd 2 1
234.2.l.a 4 39.h odd 6 1
624.2.bv.d 4 4.b odd 2 1
624.2.bv.d 4 52.i odd 6 1
1014.2.a.h 2 13.f odd 12 1
1014.2.a.j 2 13.f odd 12 1
1014.2.b.d 4 13.c even 3 1
1014.2.b.d 4 13.e even 6 1
1014.2.e.h 4 13.d odd 4 1
1014.2.e.h 4 13.f odd 12 1
1014.2.e.j 4 13.d odd 4 1
1014.2.e.j 4 13.f odd 12 1
1014.2.i.f 4 13.b even 2 1
1014.2.i.f 4 13.c even 3 1
1872.2.by.k 4 12.b even 2 1
1872.2.by.k 4 156.r even 6 1
1950.2.y.a 4 5.c odd 4 1
1950.2.y.a 4 65.r odd 12 1
1950.2.y.h 4 5.c odd 4 1
1950.2.y.h 4 65.r odd 12 1
1950.2.bc.c 4 5.b even 2 1
1950.2.bc.c 4 65.l even 6 1
3042.2.a.s 2 39.k even 12 1
3042.2.a.v 2 39.k even 12 1
3042.2.b.l 4 39.h odd 6 1
3042.2.b.l 4 39.i odd 6 1
8112.2.a.bq 2 52.l even 12 1
8112.2.a.bx 2 52.l even 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 3$$ acting on $$S_{2}^{\mathrm{new}}(78, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{2} + 1$$
$3$ $$(T^{2} - T + 1)^{2}$$
$5$ $$(T^{2} + 3)^{2}$$
$7$ $$T^{4} + 6 T^{3} + \cdots + 36$$
$11$ $$T^{4} - 6 T^{3} + \cdots + 36$$
$13$ $$T^{4} + 4 T^{3} + \cdots + 169$$
$17$ $$T^{4} + 27T^{2} + 729$$
$19$ $$T^{4} + 6 T^{3} + \cdots + 36$$
$23$ $$T^{4} + 6 T^{3} + \cdots + 324$$
$29$ $$(T^{2} - 3 T + 9)^{2}$$
$31$ $$T^{4} + 96T^{2} + 576$$
$37$ $$T^{4} - 9T^{2} + 81$$
$41$ $$T^{4} - 12 T^{3} + \cdots + 9$$
$43$ $$T^{4} - 2 T^{3} + \cdots + 676$$
$47$ $$T^{4} + 24T^{2} + 36$$
$53$ $$(T - 3)^{4}$$
$59$ $$(T^{2} + 24 T + 192)^{2}$$
$61$ $$T^{4} + 20 T^{3} + \cdots + 5329$$
$67$ $$T^{4} + 6 T^{3} + \cdots + 6084$$
$71$ $$T^{4} - 18 T^{3} + \cdots + 324$$
$73$ $$(T^{2} + 147)^{2}$$
$79$ $$(T^{2} + 4 T - 104)^{2}$$
$83$ $$T^{4} + 168T^{2} + 4356$$
$89$ $$T^{4} + 12 T^{3} + \cdots + 576$$
$97$ $$T^{4} - 36T^{2} + 1296$$