Properties

 Label 78.4.a.d Level $78$ Weight $4$ Character orbit 78.a Self dual yes Analytic conductor $4.602$ Analytic rank $1$ 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: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{2} - 3 q^{3} + 4 q^{4} - 20 q^{5} - 6 q^{6} - 32 q^{7} + 8 q^{8} + 9 q^{9}+O(q^{10})$$ q + 2 * q^2 - 3 * q^3 + 4 * q^4 - 20 * q^5 - 6 * q^6 - 32 * q^7 + 8 * q^8 + 9 * q^9 $$q + 2 q^{2} - 3 q^{3} + 4 q^{4} - 20 q^{5} - 6 q^{6} - 32 q^{7} + 8 q^{8} + 9 q^{9} - 40 q^{10} + 50 q^{11} - 12 q^{12} - 13 q^{13} - 64 q^{14} + 60 q^{15} + 16 q^{16} - 30 q^{17} + 18 q^{18} - 120 q^{19} - 80 q^{20} + 96 q^{21} + 100 q^{22} - 20 q^{23} - 24 q^{24} + 275 q^{25} - 26 q^{26} - 27 q^{27} - 128 q^{28} + 82 q^{29} + 120 q^{30} - 44 q^{31} + 32 q^{32} - 150 q^{33} - 60 q^{34} + 640 q^{35} + 36 q^{36} - 306 q^{37} - 240 q^{38} + 39 q^{39} - 160 q^{40} + 108 q^{41} + 192 q^{42} - 356 q^{43} + 200 q^{44} - 180 q^{45} - 40 q^{46} - 178 q^{47} - 48 q^{48} + 681 q^{49} + 550 q^{50} + 90 q^{51} - 52 q^{52} + 198 q^{53} - 54 q^{54} - 1000 q^{55} - 256 q^{56} + 360 q^{57} + 164 q^{58} + 94 q^{59} + 240 q^{60} - 62 q^{61} - 88 q^{62} - 288 q^{63} + 64 q^{64} + 260 q^{65} - 300 q^{66} - 140 q^{67} - 120 q^{68} + 60 q^{69} + 1280 q^{70} - 778 q^{71} + 72 q^{72} + 62 q^{73} - 612 q^{74} - 825 q^{75} - 480 q^{76} - 1600 q^{77} + 78 q^{78} - 1096 q^{79} - 320 q^{80} + 81 q^{81} + 216 q^{82} - 462 q^{83} + 384 q^{84} + 600 q^{85} - 712 q^{86} - 246 q^{87} + 400 q^{88} + 1224 q^{89} - 360 q^{90} + 416 q^{91} - 80 q^{92} + 132 q^{93} - 356 q^{94} + 2400 q^{95} - 96 q^{96} + 614 q^{97} + 1362 q^{98} + 450 q^{99}+O(q^{100})$$ q + 2 * q^2 - 3 * q^3 + 4 * q^4 - 20 * q^5 - 6 * q^6 - 32 * q^7 + 8 * q^8 + 9 * q^9 - 40 * q^10 + 50 * q^11 - 12 * q^12 - 13 * q^13 - 64 * q^14 + 60 * q^15 + 16 * q^16 - 30 * q^17 + 18 * q^18 - 120 * q^19 - 80 * q^20 + 96 * q^21 + 100 * q^22 - 20 * q^23 - 24 * q^24 + 275 * q^25 - 26 * q^26 - 27 * q^27 - 128 * q^28 + 82 * q^29 + 120 * q^30 - 44 * q^31 + 32 * q^32 - 150 * q^33 - 60 * q^34 + 640 * q^35 + 36 * q^36 - 306 * q^37 - 240 * q^38 + 39 * q^39 - 160 * q^40 + 108 * q^41 + 192 * q^42 - 356 * q^43 + 200 * q^44 - 180 * q^45 - 40 * q^46 - 178 * q^47 - 48 * q^48 + 681 * q^49 + 550 * q^50 + 90 * q^51 - 52 * q^52 + 198 * q^53 - 54 * q^54 - 1000 * q^55 - 256 * q^56 + 360 * q^57 + 164 * q^58 + 94 * q^59 + 240 * q^60 - 62 * q^61 - 88 * q^62 - 288 * q^63 + 64 * q^64 + 260 * q^65 - 300 * q^66 - 140 * q^67 - 120 * q^68 + 60 * q^69 + 1280 * q^70 - 778 * q^71 + 72 * q^72 + 62 * q^73 - 612 * q^74 - 825 * q^75 - 480 * q^76 - 1600 * q^77 + 78 * q^78 - 1096 * q^79 - 320 * q^80 + 81 * q^81 + 216 * q^82 - 462 * q^83 + 384 * q^84 + 600 * q^85 - 712 * q^86 - 246 * q^87 + 400 * q^88 + 1224 * q^89 - 360 * q^90 + 416 * q^91 - 80 * q^92 + 132 * q^93 - 356 * q^94 + 2400 * q^95 - 96 * q^96 + 614 * q^97 + 1362 * q^98 + 450 * 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 −20.0000 −6.00000 −32.0000 8.00000 9.00000 −40.0000
 $$n$$: 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.d 1
3.b odd 2 1 234.4.a.f 1
4.b odd 2 1 624.4.a.e 1
5.b even 2 1 1950.4.a.h 1
8.b even 2 1 2496.4.a.r 1
8.d odd 2 1 2496.4.a.i 1
12.b even 2 1 1872.4.a.r 1
13.b even 2 1 1014.4.a.d 1
13.d odd 4 2 1014.4.b.e 2

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

Hecke kernels

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

 $$T_{5} + 20$$ T5 + 20 $$T_{7} + 32$$ T7 + 32

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 2$$
$3$ $$T + 3$$
$5$ $$T + 20$$
$7$ $$T + 32$$
$11$ $$T - 50$$
$13$ $$T + 13$$
$17$ $$T + 30$$
$19$ $$T + 120$$
$23$ $$T + 20$$
$29$ $$T - 82$$
$31$ $$T + 44$$
$37$ $$T + 306$$
$41$ $$T - 108$$
$43$ $$T + 356$$
$47$ $$T + 178$$
$53$ $$T - 198$$
$59$ $$T - 94$$
$61$ $$T + 62$$
$67$ $$T + 140$$
$71$ $$T + 778$$
$73$ $$T - 62$$
$79$ $$T + 1096$$
$83$ $$T + 462$$
$89$ $$T - 1224$$
$97$ $$T - 614$$