# Properties

 Label 78.2.b.a Level $78$ Weight $2$ Character orbit 78.b Analytic conductor $0.623$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [78,2,Mod(25,78)]

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, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("78.25");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$78 = 2 \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 78.b (of order $$2$$, degree $$1$$, minimal)

## 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: no Analytic conductor: $$0.622833135766$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{2} + q^{3} - q^{4} + 2 i q^{5} + i q^{6} - 2 i q^{7} - i q^{8} + q^{9} +O(q^{10})$$ q + i * q^2 + q^3 - q^4 + 2*i * q^5 + i * q^6 - 2*i * q^7 - i * q^8 + q^9 $$q + i q^{2} + q^{3} - q^{4} + 2 i q^{5} + i q^{6} - 2 i q^{7} - i q^{8} + q^{9} - 2 q^{10} - q^{12} + ( - 2 i - 3) q^{13} + 2 q^{14} + 2 i q^{15} + q^{16} - 2 q^{17} + i q^{18} - 6 i q^{19} - 2 i q^{20} - 2 i q^{21} + 4 q^{23} - i q^{24} + q^{25} + ( - 3 i + 2) q^{26} + q^{27} + 2 i q^{28} - 10 q^{29} - 2 q^{30} + 10 i q^{31} + i q^{32} - 2 i q^{34} + 4 q^{35} - q^{36} + 8 i q^{37} + 6 q^{38} + ( - 2 i - 3) q^{39} + 2 q^{40} + 10 i q^{41} + 2 q^{42} + 4 q^{43} + 2 i q^{45} + 4 i q^{46} - 12 i q^{47} + q^{48} + 3 q^{49} + i q^{50} - 2 q^{51} + (2 i + 3) q^{52} - 6 q^{53} + i q^{54} - 2 q^{56} - 6 i q^{57} - 10 i q^{58} + 4 i q^{59} - 2 i q^{60} + 2 q^{61} - 10 q^{62} - 2 i q^{63} - q^{64} + ( - 6 i + 4) q^{65} - 2 i q^{67} + 2 q^{68} + 4 q^{69} + 4 i q^{70} - i q^{72} - 4 i q^{73} - 8 q^{74} + q^{75} + 6 i q^{76} + ( - 3 i + 2) q^{78} + 2 i q^{80} + q^{81} - 10 q^{82} - 4 i q^{83} + 2 i q^{84} - 4 i q^{85} + 4 i q^{86} - 10 q^{87} - 6 i q^{89} - 2 q^{90} + (6 i - 4) q^{91} - 4 q^{92} + 10 i q^{93} + 12 q^{94} + 12 q^{95} + i q^{96} - 12 i q^{97} + 3 i q^{98} +O(q^{100})$$ q + i * q^2 + q^3 - q^4 + 2*i * q^5 + i * q^6 - 2*i * q^7 - i * q^8 + q^9 - 2 * q^10 - q^12 + (-2*i - 3) * q^13 + 2 * q^14 + 2*i * q^15 + q^16 - 2 * q^17 + i * q^18 - 6*i * q^19 - 2*i * q^20 - 2*i * q^21 + 4 * q^23 - i * q^24 + q^25 + (-3*i + 2) * q^26 + q^27 + 2*i * q^28 - 10 * q^29 - 2 * q^30 + 10*i * q^31 + i * q^32 - 2*i * q^34 + 4 * q^35 - q^36 + 8*i * q^37 + 6 * q^38 + (-2*i - 3) * q^39 + 2 * q^40 + 10*i * q^41 + 2 * q^42 + 4 * q^43 + 2*i * q^45 + 4*i * q^46 - 12*i * q^47 + q^48 + 3 * q^49 + i * q^50 - 2 * q^51 + (2*i + 3) * q^52 - 6 * q^53 + i * q^54 - 2 * q^56 - 6*i * q^57 - 10*i * q^58 + 4*i * q^59 - 2*i * q^60 + 2 * q^61 - 10 * q^62 - 2*i * q^63 - q^64 + (-6*i + 4) * q^65 - 2*i * q^67 + 2 * q^68 + 4 * q^69 + 4*i * q^70 - i * q^72 - 4*i * q^73 - 8 * q^74 + q^75 + 6*i * q^76 + (-3*i + 2) * q^78 + 2*i * q^80 + q^81 - 10 * q^82 - 4*i * q^83 + 2*i * q^84 - 4*i * q^85 + 4*i * q^86 - 10 * q^87 - 6*i * q^89 - 2 * q^90 + (6*i - 4) * q^91 - 4 * q^92 + 10*i * q^93 + 12 * q^94 + 12 * q^95 + i * q^96 - 12*i * q^97 + 3*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 - 2 * q^4 + 2 * q^9 $$2 q + 2 q^{3} - 2 q^{4} + 2 q^{9} - 4 q^{10} - 2 q^{12} - 6 q^{13} + 4 q^{14} + 2 q^{16} - 4 q^{17} + 8 q^{23} + 2 q^{25} + 4 q^{26} + 2 q^{27} - 20 q^{29} - 4 q^{30} + 8 q^{35} - 2 q^{36} + 12 q^{38} - 6 q^{39} + 4 q^{40} + 4 q^{42} + 8 q^{43} + 2 q^{48} + 6 q^{49} - 4 q^{51} + 6 q^{52} - 12 q^{53} - 4 q^{56} + 4 q^{61} - 20 q^{62} - 2 q^{64} + 8 q^{65} + 4 q^{68} + 8 q^{69} - 16 q^{74} + 2 q^{75} + 4 q^{78} + 2 q^{81} - 20 q^{82} - 20 q^{87} - 4 q^{90} - 8 q^{91} - 8 q^{92} + 24 q^{94} + 24 q^{95}+O(q^{100})$$ 2 * q + 2 * q^3 - 2 * q^4 + 2 * q^9 - 4 * q^10 - 2 * q^12 - 6 * q^13 + 4 * q^14 + 2 * q^16 - 4 * q^17 + 8 * q^23 + 2 * q^25 + 4 * q^26 + 2 * q^27 - 20 * q^29 - 4 * q^30 + 8 * q^35 - 2 * q^36 + 12 * q^38 - 6 * q^39 + 4 * q^40 + 4 * q^42 + 8 * q^43 + 2 * q^48 + 6 * q^49 - 4 * q^51 + 6 * q^52 - 12 * q^53 - 4 * q^56 + 4 * q^61 - 20 * q^62 - 2 * q^64 + 8 * q^65 + 4 * q^68 + 8 * q^69 - 16 * q^74 + 2 * q^75 + 4 * q^78 + 2 * q^81 - 20 * q^82 - 20 * q^87 - 4 * q^90 - 8 * q^91 - 8 * q^92 + 24 * q^94 + 24 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/78\mathbb{Z}\right)^\times$$.

 $$n$$ $$53$$ $$67$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1
 − 1.00000i 1.00000i
1.00000i 1.00000 −1.00000 2.00000i 1.00000i 2.00000i 1.00000i 1.00000 −2.00000
25.2 1.00000i 1.00000 −1.00000 2.00000i 1.00000i 2.00000i 1.00000i 1.00000 −2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 78.2.b.a 2
3.b odd 2 1 234.2.b.a 2
4.b odd 2 1 624.2.c.a 2
5.b even 2 1 1950.2.b.c 2
5.c odd 4 1 1950.2.f.d 2
5.c odd 4 1 1950.2.f.g 2
7.b odd 2 1 3822.2.c.d 2
8.b even 2 1 2496.2.c.f 2
8.d odd 2 1 2496.2.c.m 2
12.b even 2 1 1872.2.c.b 2
13.b even 2 1 inner 78.2.b.a 2
13.c even 3 2 1014.2.i.c 4
13.d odd 4 1 1014.2.a.b 1
13.d odd 4 1 1014.2.a.g 1
13.e even 6 2 1014.2.i.c 4
13.f odd 12 2 1014.2.e.b 2
13.f odd 12 2 1014.2.e.e 2
39.d odd 2 1 234.2.b.a 2
39.f even 4 1 3042.2.a.c 1
39.f even 4 1 3042.2.a.n 1
52.b odd 2 1 624.2.c.a 2
52.f even 4 1 8112.2.a.g 1
52.f even 4 1 8112.2.a.j 1
65.d even 2 1 1950.2.b.c 2
65.h odd 4 1 1950.2.f.d 2
65.h odd 4 1 1950.2.f.g 2
91.b odd 2 1 3822.2.c.d 2
104.e even 2 1 2496.2.c.f 2
104.h odd 2 1 2496.2.c.m 2
156.h even 2 1 1872.2.c.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.b.a 2 1.a even 1 1 trivial
78.2.b.a 2 13.b even 2 1 inner
234.2.b.a 2 3.b odd 2 1
234.2.b.a 2 39.d odd 2 1
624.2.c.a 2 4.b odd 2 1
624.2.c.a 2 52.b odd 2 1
1014.2.a.b 1 13.d odd 4 1
1014.2.a.g 1 13.d odd 4 1
1014.2.e.b 2 13.f odd 12 2
1014.2.e.e 2 13.f odd 12 2
1014.2.i.c 4 13.c even 3 2
1014.2.i.c 4 13.e even 6 2
1872.2.c.b 2 12.b even 2 1
1872.2.c.b 2 156.h even 2 1
1950.2.b.c 2 5.b even 2 1
1950.2.b.c 2 65.d even 2 1
1950.2.f.d 2 5.c odd 4 1
1950.2.f.d 2 65.h odd 4 1
1950.2.f.g 2 5.c odd 4 1
1950.2.f.g 2 65.h odd 4 1
2496.2.c.f 2 8.b even 2 1
2496.2.c.f 2 104.e even 2 1
2496.2.c.m 2 8.d odd 2 1
2496.2.c.m 2 104.h odd 2 1
3042.2.a.c 1 39.f even 4 1
3042.2.a.n 1 39.f even 4 1
3822.2.c.d 2 7.b odd 2 1
3822.2.c.d 2 91.b odd 2 1
8112.2.a.g 1 52.f even 4 1
8112.2.a.j 1 52.f even 4 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2} + 4$$
$7$ $$T^{2} + 4$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 6T + 13$$
$17$ $$(T + 2)^{2}$$
$19$ $$T^{2} + 36$$
$23$ $$(T - 4)^{2}$$
$29$ $$(T + 10)^{2}$$
$31$ $$T^{2} + 100$$
$37$ $$T^{2} + 64$$
$41$ $$T^{2} + 100$$
$43$ $$(T - 4)^{2}$$
$47$ $$T^{2} + 144$$
$53$ $$(T + 6)^{2}$$
$59$ $$T^{2} + 16$$
$61$ $$(T - 2)^{2}$$
$67$ $$T^{2} + 4$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 16$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 16$$
$89$ $$T^{2} + 36$$
$97$ $$T^{2} + 144$$