# Properties

 Label 78.2.a.a Level $78$ Weight $2$ Character orbit 78.a Self dual yes Analytic conductor $0.623$ 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] = [78,2,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, 2, names="a")

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

chi := DirichletCharacter("78.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$78 = 2 \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ 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: $$0.622833135766$$ 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)

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} + q^{4} + 2 q^{5} + q^{6} + 4 q^{7} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 - q^3 + q^4 + 2 * q^5 + q^6 + 4 * q^7 - q^8 + q^9 $$q - q^{2} - q^{3} + q^{4} + 2 q^{5} + q^{6} + 4 q^{7} - q^{8} + q^{9} - 2 q^{10} - 4 q^{11} - q^{12} + q^{13} - 4 q^{14} - 2 q^{15} + q^{16} + 2 q^{17} - q^{18} - 8 q^{19} + 2 q^{20} - 4 q^{21} + 4 q^{22} + q^{24} - q^{25} - q^{26} - q^{27} + 4 q^{28} + 6 q^{29} + 2 q^{30} - 4 q^{31} - q^{32} + 4 q^{33} - 2 q^{34} + 8 q^{35} + q^{36} - 2 q^{37} + 8 q^{38} - q^{39} - 2 q^{40} - 10 q^{41} + 4 q^{42} + 4 q^{43} - 4 q^{44} + 2 q^{45} + 8 q^{47} - q^{48} + 9 q^{49} + q^{50} - 2 q^{51} + q^{52} - 10 q^{53} + q^{54} - 8 q^{55} - 4 q^{56} + 8 q^{57} - 6 q^{58} + 4 q^{59} - 2 q^{60} - 2 q^{61} + 4 q^{62} + 4 q^{63} + q^{64} + 2 q^{65} - 4 q^{66} - 16 q^{67} + 2 q^{68} - 8 q^{70} - 8 q^{71} - q^{72} + 2 q^{73} + 2 q^{74} + q^{75} - 8 q^{76} - 16 q^{77} + q^{78} + 8 q^{79} + 2 q^{80} + q^{81} + 10 q^{82} + 12 q^{83} - 4 q^{84} + 4 q^{85} - 4 q^{86} - 6 q^{87} + 4 q^{88} + 14 q^{89} - 2 q^{90} + 4 q^{91} + 4 q^{93} - 8 q^{94} - 16 q^{95} + q^{96} + 10 q^{97} - 9 q^{98} - 4 q^{99}+O(q^{100})$$ q - q^2 - q^3 + q^4 + 2 * q^5 + q^6 + 4 * q^7 - q^8 + q^9 - 2 * q^10 - 4 * q^11 - q^12 + q^13 - 4 * q^14 - 2 * q^15 + q^16 + 2 * q^17 - q^18 - 8 * q^19 + 2 * q^20 - 4 * q^21 + 4 * q^22 + q^24 - q^25 - q^26 - q^27 + 4 * q^28 + 6 * q^29 + 2 * q^30 - 4 * q^31 - q^32 + 4 * q^33 - 2 * q^34 + 8 * q^35 + q^36 - 2 * q^37 + 8 * q^38 - q^39 - 2 * q^40 - 10 * q^41 + 4 * q^42 + 4 * q^43 - 4 * q^44 + 2 * q^45 + 8 * q^47 - q^48 + 9 * q^49 + q^50 - 2 * q^51 + q^52 - 10 * q^53 + q^54 - 8 * q^55 - 4 * q^56 + 8 * q^57 - 6 * q^58 + 4 * q^59 - 2 * q^60 - 2 * q^61 + 4 * q^62 + 4 * q^63 + q^64 + 2 * q^65 - 4 * q^66 - 16 * q^67 + 2 * q^68 - 8 * q^70 - 8 * q^71 - q^72 + 2 * q^73 + 2 * q^74 + q^75 - 8 * q^76 - 16 * q^77 + q^78 + 8 * q^79 + 2 * q^80 + q^81 + 10 * q^82 + 12 * q^83 - 4 * q^84 + 4 * q^85 - 4 * q^86 - 6 * q^87 + 4 * q^88 + 14 * q^89 - 2 * q^90 + 4 * q^91 + 4 * q^93 - 8 * q^94 - 16 * q^95 + q^96 + 10 * q^97 - 9 * q^98 - 4 * 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
−1.00000 −1.00000 1.00000 2.00000 1.00000 4.00000 −1.00000 1.00000 −2.00000
 $$n$$: 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 78.2.a.a 1
3.b odd 2 1 234.2.a.c 1
4.b odd 2 1 624.2.a.h 1
5.b even 2 1 1950.2.a.w 1
5.c odd 4 2 1950.2.e.i 2
7.b odd 2 1 3822.2.a.j 1
8.b even 2 1 2496.2.a.t 1
8.d odd 2 1 2496.2.a.b 1
9.c even 3 2 2106.2.e.q 2
9.d odd 6 2 2106.2.e.j 2
11.b odd 2 1 9438.2.a.t 1
12.b even 2 1 1872.2.a.c 1
13.b even 2 1 1014.2.a.d 1
13.c even 3 2 1014.2.e.f 2
13.d odd 4 2 1014.2.b.b 2
13.e even 6 2 1014.2.e.c 2
13.f odd 12 4 1014.2.i.d 4
15.d odd 2 1 5850.2.a.d 1
15.e even 4 2 5850.2.e.bb 2
24.f even 2 1 7488.2.a.bk 1
24.h odd 2 1 7488.2.a.bz 1
39.d odd 2 1 3042.2.a.f 1
39.f even 4 2 3042.2.b.g 2
52.b odd 2 1 8112.2.a.v 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.a.a 1 1.a even 1 1 trivial
234.2.a.c 1 3.b odd 2 1
624.2.a.h 1 4.b odd 2 1
1014.2.a.d 1 13.b even 2 1
1014.2.b.b 2 13.d odd 4 2
1014.2.e.c 2 13.e even 6 2
1014.2.e.f 2 13.c even 3 2
1014.2.i.d 4 13.f odd 12 4
1872.2.a.c 1 12.b even 2 1
1950.2.a.w 1 5.b even 2 1
1950.2.e.i 2 5.c odd 4 2
2106.2.e.j 2 9.d odd 6 2
2106.2.e.q 2 9.c even 3 2
2496.2.a.b 1 8.d odd 2 1
2496.2.a.t 1 8.b even 2 1
3042.2.a.f 1 39.d odd 2 1
3042.2.b.g 2 39.f even 4 2
3822.2.a.j 1 7.b odd 2 1
5850.2.a.d 1 15.d odd 2 1
5850.2.e.bb 2 15.e even 4 2
7488.2.a.bk 1 24.f even 2 1
7488.2.a.bz 1 24.h odd 2 1
8112.2.a.v 1 52.b odd 2 1
9438.2.a.t 1 11.b odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(\Gamma_0(78))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T + 1$$
$5$ $$T - 2$$
$7$ $$T - 4$$
$11$ $$T + 4$$
$13$ $$T - 1$$
$17$ $$T - 2$$
$19$ $$T + 8$$
$23$ $$T$$
$29$ $$T - 6$$
$31$ $$T + 4$$
$37$ $$T + 2$$
$41$ $$T + 10$$
$43$ $$T - 4$$
$47$ $$T - 8$$
$53$ $$T + 10$$
$59$ $$T - 4$$
$61$ $$T + 2$$
$67$ $$T + 16$$
$71$ $$T + 8$$
$73$ $$T - 2$$
$79$ $$T - 8$$
$83$ $$T - 12$$
$89$ $$T - 14$$
$97$ $$T - 10$$