# Properties

 Label 78.4.a.a 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{2} - 3 q^{3} + 4 q^{4} - 16 q^{5} + 6 q^{6} + 28 q^{7} - 8 q^{8} + 9 q^{9}+O(q^{10})$$ q - 2 * q^2 - 3 * q^3 + 4 * q^4 - 16 * q^5 + 6 * q^6 + 28 * q^7 - 8 * q^8 + 9 * q^9 $$q - 2 q^{2} - 3 q^{3} + 4 q^{4} - 16 q^{5} + 6 q^{6} + 28 q^{7} - 8 q^{8} + 9 q^{9} + 32 q^{10} + 34 q^{11} - 12 q^{12} - 13 q^{13} - 56 q^{14} + 48 q^{15} + 16 q^{16} + 138 q^{17} - 18 q^{18} + 108 q^{19} - 64 q^{20} - 84 q^{21} - 68 q^{22} - 52 q^{23} + 24 q^{24} + 131 q^{25} + 26 q^{26} - 27 q^{27} + 112 q^{28} - 190 q^{29} - 96 q^{30} - 176 q^{31} - 32 q^{32} - 102 q^{33} - 276 q^{34} - 448 q^{35} + 36 q^{36} + 342 q^{37} - 216 q^{38} + 39 q^{39} + 128 q^{40} + 240 q^{41} + 168 q^{42} - 140 q^{43} + 136 q^{44} - 144 q^{45} + 104 q^{46} + 454 q^{47} - 48 q^{48} + 441 q^{49} - 262 q^{50} - 414 q^{51} - 52 q^{52} + 198 q^{53} + 54 q^{54} - 544 q^{55} - 224 q^{56} - 324 q^{57} + 380 q^{58} - 154 q^{59} + 192 q^{60} + 34 q^{61} + 352 q^{62} + 252 q^{63} + 64 q^{64} + 208 q^{65} + 204 q^{66} - 656 q^{67} + 552 q^{68} + 156 q^{69} + 896 q^{70} + 550 q^{71} - 72 q^{72} + 614 q^{73} - 684 q^{74} - 393 q^{75} + 432 q^{76} + 952 q^{77} - 78 q^{78} + 8 q^{79} - 256 q^{80} + 81 q^{81} - 480 q^{82} + 762 q^{83} - 336 q^{84} - 2208 q^{85} + 280 q^{86} + 570 q^{87} - 272 q^{88} - 444 q^{89} + 288 q^{90} - 364 q^{91} - 208 q^{92} + 528 q^{93} - 908 q^{94} - 1728 q^{95} + 96 q^{96} + 1022 q^{97} - 882 q^{98} + 306 q^{99}+O(q^{100})$$ q - 2 * q^2 - 3 * q^3 + 4 * q^4 - 16 * q^5 + 6 * q^6 + 28 * q^7 - 8 * q^8 + 9 * q^9 + 32 * q^10 + 34 * q^11 - 12 * q^12 - 13 * q^13 - 56 * q^14 + 48 * q^15 + 16 * q^16 + 138 * q^17 - 18 * q^18 + 108 * q^19 - 64 * q^20 - 84 * q^21 - 68 * q^22 - 52 * q^23 + 24 * q^24 + 131 * q^25 + 26 * q^26 - 27 * q^27 + 112 * q^28 - 190 * q^29 - 96 * q^30 - 176 * q^31 - 32 * q^32 - 102 * q^33 - 276 * q^34 - 448 * q^35 + 36 * q^36 + 342 * q^37 - 216 * q^38 + 39 * q^39 + 128 * q^40 + 240 * q^41 + 168 * q^42 - 140 * q^43 + 136 * q^44 - 144 * q^45 + 104 * q^46 + 454 * q^47 - 48 * q^48 + 441 * q^49 - 262 * q^50 - 414 * q^51 - 52 * q^52 + 198 * q^53 + 54 * q^54 - 544 * q^55 - 224 * q^56 - 324 * q^57 + 380 * q^58 - 154 * q^59 + 192 * q^60 + 34 * q^61 + 352 * q^62 + 252 * q^63 + 64 * q^64 + 208 * q^65 + 204 * q^66 - 656 * q^67 + 552 * q^68 + 156 * q^69 + 896 * q^70 + 550 * q^71 - 72 * q^72 + 614 * q^73 - 684 * q^74 - 393 * q^75 + 432 * q^76 + 952 * q^77 - 78 * q^78 + 8 * q^79 - 256 * q^80 + 81 * q^81 - 480 * q^82 + 762 * q^83 - 336 * q^84 - 2208 * q^85 + 280 * q^86 + 570 * q^87 - 272 * q^88 - 444 * q^89 + 288 * q^90 - 364 * q^91 - 208 * q^92 + 528 * q^93 - 908 * q^94 - 1728 * q^95 + 96 * q^96 + 1022 * q^97 - 882 * q^98 + 306 * 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 −16.0000 6.00000 28.0000 −8.00000 9.00000 32.0000
 $$n$$: 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.a 1
3.b odd 2 1 234.4.a.k 1
4.b odd 2 1 624.4.a.f 1
5.b even 2 1 1950.4.a.o 1
8.b even 2 1 2496.4.a.q 1
8.d odd 2 1 2496.4.a.g 1
12.b even 2 1 1872.4.a.o 1
13.b even 2 1 1014.4.a.i 1
13.d odd 4 2 1014.4.b.a 2

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

 $$T_{5} + 16$$ T5 + 16 $$T_{7} - 28$$ T7 - 28

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T + 3$$
$5$ $$T + 16$$
$7$ $$T - 28$$
$11$ $$T - 34$$
$13$ $$T + 13$$
$17$ $$T - 138$$
$19$ $$T - 108$$
$23$ $$T + 52$$
$29$ $$T + 190$$
$31$ $$T + 176$$
$37$ $$T - 342$$
$41$ $$T - 240$$
$43$ $$T + 140$$
$47$ $$T - 454$$
$53$ $$T - 198$$
$59$ $$T + 154$$
$61$ $$T - 34$$
$67$ $$T + 656$$
$71$ $$T - 550$$
$73$ $$T - 614$$
$79$ $$T - 8$$
$83$ $$T - 762$$
$89$ $$T + 444$$
$97$ $$T - 1022$$