# Properties

 Label 102.2.a.c Level $102$ Weight $2$ Character orbit 102.a Self dual yes Analytic conductor $0.814$ 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] = [102,2,Mod(1,102)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(102, 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("102.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$102 = 2 \cdot 3 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 102.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: $$0.814474100617$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} - 2 q^{5} + q^{6} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 + q^3 + q^4 - 2 * q^5 + q^6 + q^8 + q^9 $$q + q^{2} + q^{3} + q^{4} - 2 q^{5} + q^{6} + q^{8} + q^{9} - 2 q^{10} - 4 q^{11} + q^{12} - 2 q^{13} - 2 q^{15} + q^{16} + q^{17} + q^{18} + 4 q^{19} - 2 q^{20} - 4 q^{22} + q^{24} - q^{25} - 2 q^{26} + q^{27} - 10 q^{29} - 2 q^{30} + 8 q^{31} + q^{32} - 4 q^{33} + q^{34} + q^{36} - 2 q^{37} + 4 q^{38} - 2 q^{39} - 2 q^{40} + 10 q^{41} + 12 q^{43} - 4 q^{44} - 2 q^{45} + q^{48} - 7 q^{49} - q^{50} + q^{51} - 2 q^{52} + 6 q^{53} + q^{54} + 8 q^{55} + 4 q^{57} - 10 q^{58} + 12 q^{59} - 2 q^{60} - 10 q^{61} + 8 q^{62} + q^{64} + 4 q^{65} - 4 q^{66} - 12 q^{67} + q^{68} + q^{72} + 10 q^{73} - 2 q^{74} - q^{75} + 4 q^{76} - 2 q^{78} - 8 q^{79} - 2 q^{80} + q^{81} + 10 q^{82} + 4 q^{83} - 2 q^{85} + 12 q^{86} - 10 q^{87} - 4 q^{88} - 6 q^{89} - 2 q^{90} + 8 q^{93} - 8 q^{95} + q^{96} - 14 q^{97} - 7 q^{98} - 4 q^{99}+O(q^{100})$$ q + q^2 + q^3 + q^4 - 2 * q^5 + q^6 + q^8 + q^9 - 2 * q^10 - 4 * q^11 + q^12 - 2 * q^13 - 2 * q^15 + q^16 + q^17 + q^18 + 4 * q^19 - 2 * q^20 - 4 * q^22 + q^24 - q^25 - 2 * q^26 + q^27 - 10 * q^29 - 2 * q^30 + 8 * q^31 + q^32 - 4 * q^33 + q^34 + q^36 - 2 * q^37 + 4 * q^38 - 2 * q^39 - 2 * q^40 + 10 * q^41 + 12 * q^43 - 4 * q^44 - 2 * q^45 + q^48 - 7 * q^49 - q^50 + q^51 - 2 * q^52 + 6 * q^53 + q^54 + 8 * q^55 + 4 * q^57 - 10 * q^58 + 12 * q^59 - 2 * q^60 - 10 * q^61 + 8 * q^62 + q^64 + 4 * q^65 - 4 * q^66 - 12 * q^67 + q^68 + q^72 + 10 * q^73 - 2 * q^74 - q^75 + 4 * q^76 - 2 * q^78 - 8 * q^79 - 2 * q^80 + q^81 + 10 * q^82 + 4 * q^83 - 2 * q^85 + 12 * q^86 - 10 * q^87 - 4 * q^88 - 6 * q^89 - 2 * q^90 + 8 * q^93 - 8 * q^95 + q^96 - 14 * q^97 - 7 * 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 0 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$17$$ $$-1$$

## 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 102.2.a.c 1
3.b odd 2 1 306.2.a.b 1
4.b odd 2 1 816.2.a.b 1
5.b even 2 1 2550.2.a.c 1
5.c odd 4 2 2550.2.d.m 2
7.b odd 2 1 4998.2.a.be 1
8.b even 2 1 3264.2.a.m 1
8.d odd 2 1 3264.2.a.bc 1
12.b even 2 1 2448.2.a.p 1
15.d odd 2 1 7650.2.a.ca 1
17.b even 2 1 1734.2.a.j 1
17.c even 4 2 1734.2.b.b 2
17.d even 8 4 1734.2.f.e 4
24.f even 2 1 9792.2.a.l 1
24.h odd 2 1 9792.2.a.k 1
51.c odd 2 1 5202.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
102.2.a.c 1 1.a even 1 1 trivial
306.2.a.b 1 3.b odd 2 1
816.2.a.b 1 4.b odd 2 1
1734.2.a.j 1 17.b even 2 1
1734.2.b.b 2 17.c even 4 2
1734.2.f.e 4 17.d even 8 4
2448.2.a.p 1 12.b even 2 1
2550.2.a.c 1 5.b even 2 1
2550.2.d.m 2 5.c odd 4 2
3264.2.a.m 1 8.b even 2 1
3264.2.a.bc 1 8.d odd 2 1
4998.2.a.be 1 7.b odd 2 1
5202.2.a.c 1 51.c odd 2 1
7650.2.a.ca 1 15.d odd 2 1
9792.2.a.k 1 24.h odd 2 1
9792.2.a.l 1 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 2$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(102))$$.

## Hecke characteristic polynomials

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