# Properties

 Label 78.4.a.c Level $78$ Weight $4$ Character orbit 78.a Self dual yes Analytic conductor $4.602$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [78,4,Mod(1,78)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(78, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("78.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$78 = 2 \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 78.a (trivial)

## 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: yes Analytic conductor: $$4.60214898045$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + 3 q^{3} + 4 q^{4} + 10 q^{5} - 6 q^{6} - 8 q^{7} - 8 q^{8} + 9 q^{9}+O(q^{10})$$ q - 2 * q^2 + 3 * q^3 + 4 * q^4 + 10 * q^5 - 6 * q^6 - 8 * q^7 - 8 * q^8 + 9 * q^9 $$q - 2 q^{2} + 3 q^{3} + 4 q^{4} + 10 q^{5} - 6 q^{6} - 8 q^{7} - 8 q^{8} + 9 q^{9} - 20 q^{10} + 40 q^{11} + 12 q^{12} + 13 q^{13} + 16 q^{14} + 30 q^{15} + 16 q^{16} + 130 q^{17} - 18 q^{18} - 20 q^{19} + 40 q^{20} - 24 q^{21} - 80 q^{22} - 24 q^{24} - 25 q^{25} - 26 q^{26} + 27 q^{27} - 32 q^{28} - 18 q^{29} - 60 q^{30} - 184 q^{31} - 32 q^{32} + 120 q^{33} - 260 q^{34} - 80 q^{35} + 36 q^{36} - 74 q^{37} + 40 q^{38} + 39 q^{39} - 80 q^{40} - 362 q^{41} + 48 q^{42} + 76 q^{43} + 160 q^{44} + 90 q^{45} - 452 q^{47} + 48 q^{48} - 279 q^{49} + 50 q^{50} + 390 q^{51} + 52 q^{52} + 382 q^{53} - 54 q^{54} + 400 q^{55} + 64 q^{56} - 60 q^{57} + 36 q^{58} + 464 q^{59} + 120 q^{60} + 358 q^{61} + 368 q^{62} - 72 q^{63} + 64 q^{64} + 130 q^{65} - 240 q^{66} - 700 q^{67} + 520 q^{68} + 160 q^{70} - 748 q^{71} - 72 q^{72} + 1058 q^{73} + 148 q^{74} - 75 q^{75} - 80 q^{76} - 320 q^{77} - 78 q^{78} - 976 q^{79} + 160 q^{80} + 81 q^{81} + 724 q^{82} - 1008 q^{83} - 96 q^{84} + 1300 q^{85} - 152 q^{86} - 54 q^{87} - 320 q^{88} - 386 q^{89} - 180 q^{90} - 104 q^{91} - 552 q^{93} + 904 q^{94} - 200 q^{95} - 96 q^{96} - 614 q^{97} + 558 q^{98} + 360 q^{99}+O(q^{100})$$ q - 2 * q^2 + 3 * q^3 + 4 * q^4 + 10 * q^5 - 6 * q^6 - 8 * q^7 - 8 * q^8 + 9 * q^9 - 20 * q^10 + 40 * q^11 + 12 * q^12 + 13 * q^13 + 16 * q^14 + 30 * q^15 + 16 * q^16 + 130 * q^17 - 18 * q^18 - 20 * q^19 + 40 * q^20 - 24 * q^21 - 80 * q^22 - 24 * q^24 - 25 * q^25 - 26 * q^26 + 27 * q^27 - 32 * q^28 - 18 * q^29 - 60 * q^30 - 184 * q^31 - 32 * q^32 + 120 * q^33 - 260 * q^34 - 80 * q^35 + 36 * q^36 - 74 * q^37 + 40 * q^38 + 39 * q^39 - 80 * q^40 - 362 * q^41 + 48 * q^42 + 76 * q^43 + 160 * q^44 + 90 * q^45 - 452 * q^47 + 48 * q^48 - 279 * q^49 + 50 * q^50 + 390 * q^51 + 52 * q^52 + 382 * q^53 - 54 * q^54 + 400 * q^55 + 64 * q^56 - 60 * q^57 + 36 * q^58 + 464 * q^59 + 120 * q^60 + 358 * q^61 + 368 * q^62 - 72 * q^63 + 64 * q^64 + 130 * q^65 - 240 * q^66 - 700 * q^67 + 520 * q^68 + 160 * q^70 - 748 * q^71 - 72 * q^72 + 1058 * q^73 + 148 * q^74 - 75 * q^75 - 80 * q^76 - 320 * q^77 - 78 * q^78 - 976 * q^79 + 160 * q^80 + 81 * q^81 + 724 * q^82 - 1008 * q^83 - 96 * q^84 + 1300 * q^85 - 152 * q^86 - 54 * q^87 - 320 * q^88 - 386 * q^89 - 180 * q^90 - 104 * q^91 - 552 * q^93 + 904 * q^94 - 200 * q^95 - 96 * q^96 - 614 * q^97 + 558 * q^98 + 360 * q^99

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
−2.00000 3.00000 4.00000 10.0000 −6.00000 −8.00000 −8.00000 9.00000 −20.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$+1$$
$$3$$ $$-1$$
$$13$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 78.4.a.c 1
3.b odd 2 1 234.4.a.h 1
4.b odd 2 1 624.4.a.d 1
5.b even 2 1 1950.4.a.l 1
8.b even 2 1 2496.4.a.a 1
8.d odd 2 1 2496.4.a.j 1
12.b even 2 1 1872.4.a.d 1
13.b even 2 1 1014.4.a.j 1
13.d odd 4 2 1014.4.b.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.4.a.c 1 1.a even 1 1 trivial
234.4.a.h 1 3.b odd 2 1
624.4.a.d 1 4.b odd 2 1
1014.4.a.j 1 13.b even 2 1
1014.4.b.h 2 13.d odd 4 2
1872.4.a.d 1 12.b even 2 1
1950.4.a.l 1 5.b even 2 1
2496.4.a.a 1 8.b even 2 1
2496.4.a.j 1 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(78))$$:

 $$T_{5} - 10$$ T5 - 10 $$T_{7} + 8$$ T7 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T - 3$$
$5$ $$T - 10$$
$7$ $$T + 8$$
$11$ $$T - 40$$
$13$ $$T - 13$$
$17$ $$T - 130$$
$19$ $$T + 20$$
$23$ $$T$$
$29$ $$T + 18$$
$31$ $$T + 184$$
$37$ $$T + 74$$
$41$ $$T + 362$$
$43$ $$T - 76$$
$47$ $$T + 452$$
$53$ $$T - 382$$
$59$ $$T - 464$$
$61$ $$T - 358$$
$67$ $$T + 700$$
$71$ $$T + 748$$
$73$ $$T - 1058$$
$79$ $$T + 976$$
$83$ $$T + 1008$$
$89$ $$T + 386$$
$97$ $$T + 614$$
show more
show less