# Properties

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

# Related objects

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)

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + 3 q^{3} + 4 q^{4} + 4 q^{5} + 6 q^{6} + 4 q^{7} + 8 q^{8} + 9 q^{9}+O(q^{10})$$ q + 2 * q^2 + 3 * q^3 + 4 * q^4 + 4 * q^5 + 6 * q^6 + 4 * q^7 + 8 * q^8 + 9 * q^9 $$q + 2 q^{2} + 3 q^{3} + 4 q^{4} + 4 q^{5} + 6 q^{6} + 4 q^{7} + 8 q^{8} + 9 q^{9} + 8 q^{10} + 2 q^{11} + 12 q^{12} - 13 q^{13} + 8 q^{14} + 12 q^{15} + 16 q^{16} - 6 q^{17} + 18 q^{18} - 36 q^{19} + 16 q^{20} + 12 q^{21} + 4 q^{22} - 20 q^{23} + 24 q^{24} - 109 q^{25} - 26 q^{26} + 27 q^{27} + 16 q^{28} - 14 q^{29} + 24 q^{30} - 152 q^{31} + 32 q^{32} + 6 q^{33} - 12 q^{34} + 16 q^{35} + 36 q^{36} - 258 q^{37} - 72 q^{38} - 39 q^{39} + 32 q^{40} + 84 q^{41} + 24 q^{42} - 188 q^{43} + 8 q^{44} + 36 q^{45} - 40 q^{46} + 254 q^{47} + 48 q^{48} - 327 q^{49} - 218 q^{50} - 18 q^{51} - 52 q^{52} + 366 q^{53} + 54 q^{54} + 8 q^{55} + 32 q^{56} - 108 q^{57} - 28 q^{58} + 550 q^{59} + 48 q^{60} - 14 q^{61} - 304 q^{62} + 36 q^{63} + 64 q^{64} - 52 q^{65} + 12 q^{66} + 448 q^{67} - 24 q^{68} - 60 q^{69} + 32 q^{70} + 926 q^{71} + 72 q^{72} + 254 q^{73} - 516 q^{74} - 327 q^{75} - 144 q^{76} + 8 q^{77} - 78 q^{78} + 1328 q^{79} + 64 q^{80} + 81 q^{81} + 168 q^{82} + 186 q^{83} + 48 q^{84} - 24 q^{85} - 376 q^{86} - 42 q^{87} + 16 q^{88} - 336 q^{89} + 72 q^{90} - 52 q^{91} - 80 q^{92} - 456 q^{93} + 508 q^{94} - 144 q^{95} + 96 q^{96} + 614 q^{97} - 654 q^{98} + 18 q^{99}+O(q^{100})$$ q + 2 * q^2 + 3 * q^3 + 4 * q^4 + 4 * q^5 + 6 * q^6 + 4 * q^7 + 8 * q^8 + 9 * q^9 + 8 * q^10 + 2 * q^11 + 12 * q^12 - 13 * q^13 + 8 * q^14 + 12 * q^15 + 16 * q^16 - 6 * q^17 + 18 * q^18 - 36 * q^19 + 16 * q^20 + 12 * q^21 + 4 * q^22 - 20 * q^23 + 24 * q^24 - 109 * q^25 - 26 * q^26 + 27 * q^27 + 16 * q^28 - 14 * q^29 + 24 * q^30 - 152 * q^31 + 32 * q^32 + 6 * q^33 - 12 * q^34 + 16 * q^35 + 36 * q^36 - 258 * q^37 - 72 * q^38 - 39 * q^39 + 32 * q^40 + 84 * q^41 + 24 * q^42 - 188 * q^43 + 8 * q^44 + 36 * q^45 - 40 * q^46 + 254 * q^47 + 48 * q^48 - 327 * q^49 - 218 * q^50 - 18 * q^51 - 52 * q^52 + 366 * q^53 + 54 * q^54 + 8 * q^55 + 32 * q^56 - 108 * q^57 - 28 * q^58 + 550 * q^59 + 48 * q^60 - 14 * q^61 - 304 * q^62 + 36 * q^63 + 64 * q^64 - 52 * q^65 + 12 * q^66 + 448 * q^67 - 24 * q^68 - 60 * q^69 + 32 * q^70 + 926 * q^71 + 72 * q^72 + 254 * q^73 - 516 * q^74 - 327 * q^75 - 144 * q^76 + 8 * q^77 - 78 * q^78 + 1328 * q^79 + 64 * q^80 + 81 * q^81 + 168 * q^82 + 186 * q^83 + 48 * q^84 - 24 * q^85 - 376 * q^86 - 42 * q^87 + 16 * q^88 - 336 * q^89 + 72 * q^90 - 52 * q^91 - 80 * q^92 - 456 * q^93 + 508 * q^94 - 144 * q^95 + 96 * q^96 + 614 * q^97 - 654 * q^98 + 18 * 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 4.00000 6.00000 4.00000 8.00000 9.00000 8.00000
 $$n$$: 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.f 1
3.b odd 2 1 234.4.a.c 1
4.b odd 2 1 624.4.a.c 1
5.b even 2 1 1950.4.a.a 1
8.b even 2 1 2496.4.a.c 1
8.d odd 2 1 2496.4.a.l 1
12.b even 2 1 1872.4.a.f 1
13.b even 2 1 1014.4.a.e 1
13.d odd 4 2 1014.4.b.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.4.a.f 1 1.a even 1 1 trivial
234.4.a.c 1 3.b odd 2 1
624.4.a.c 1 4.b odd 2 1
1014.4.a.e 1 13.b even 2 1
1014.4.b.g 2 13.d odd 4 2
1872.4.a.f 1 12.b even 2 1
1950.4.a.a 1 5.b even 2 1
2496.4.a.c 1 8.b even 2 1
2496.4.a.l 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(78))$$:

 $$T_{5} - 4$$ T5 - 4 $$T_{7} - 4$$ T7 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 2$$
$3$ $$T - 3$$
$5$ $$T - 4$$
$7$ $$T - 4$$
$11$ $$T - 2$$
$13$ $$T + 13$$
$17$ $$T + 6$$
$19$ $$T + 36$$
$23$ $$T + 20$$
$29$ $$T + 14$$
$31$ $$T + 152$$
$37$ $$T + 258$$
$41$ $$T - 84$$
$43$ $$T + 188$$
$47$ $$T - 254$$
$53$ $$T - 366$$
$59$ $$T - 550$$
$61$ $$T + 14$$
$67$ $$T - 448$$
$71$ $$T - 926$$
$73$ $$T - 254$$
$79$ $$T - 1328$$
$83$ $$T - 186$$
$89$ $$T + 336$$
$97$ $$T - 614$$