# Properties

 Label 102.2.b.a Level $102$ Weight $2$ Character orbit 102.b Analytic conductor $0.814$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [102,2,Mod(67,102)]

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, 1]))

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

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

chi := DirichletCharacter("102.67");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$102 = 2 \cdot 3 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 102.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: $$0.814474100617$$ 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: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q + q^{2} + i q^{3} + q^{4} - 2 i q^{5} + i q^{6} + 2 i q^{7} + q^{8} - q^{9} +O(q^{10})$$ q + q^2 + i * q^3 + q^4 - 2*i * q^5 + i * q^6 + 2*i * q^7 + q^8 - q^9 $$q + q^{2} + i q^{3} + q^{4} - 2 i q^{5} + i q^{6} + 2 i q^{7} + q^{8} - q^{9} - 2 i q^{10} + i q^{12} - 6 q^{13} + 2 i q^{14} + 2 q^{15} + q^{16} + ( - 4 i - 1) q^{17} - q^{18} - 2 i q^{20} - 2 q^{21} - 6 i q^{23} + i q^{24} + q^{25} - 6 q^{26} - i q^{27} + 2 i q^{28} + 6 i q^{29} + 2 q^{30} + 10 i q^{31} + q^{32} + ( - 4 i - 1) q^{34} + 4 q^{35} - q^{36} + 2 i q^{37} - 6 i q^{39} - 2 i q^{40} - 2 q^{42} + 4 q^{43} + 2 i q^{45} - 6 i q^{46} + 8 q^{47} + i q^{48} + 3 q^{49} + q^{50} + ( - i + 4) q^{51} - 6 q^{52} - 6 q^{53} - i q^{54} + 2 i q^{56} + 6 i q^{58} + 2 q^{60} - 10 i q^{61} + 10 i q^{62} - 2 i q^{63} + q^{64} + 12 i q^{65} + 8 q^{67} + ( - 4 i - 1) q^{68} + 6 q^{69} + 4 q^{70} + 10 i q^{71} - q^{72} - 16 i q^{73} + 2 i q^{74} + i q^{75} - 6 i q^{78} + 6 i q^{79} - 2 i q^{80} + q^{81} - 16 q^{83} - 2 q^{84} + (2 i - 8) q^{85} + 4 q^{86} - 6 q^{87} + 10 q^{89} + 2 i q^{90} - 12 i q^{91} - 6 i q^{92} - 10 q^{93} + 8 q^{94} + i q^{96} + 12 i q^{97} + 3 q^{98} +O(q^{100})$$ q + q^2 + i * q^3 + q^4 - 2*i * q^5 + i * q^6 + 2*i * q^7 + q^8 - q^9 - 2*i * q^10 + i * q^12 - 6 * q^13 + 2*i * q^14 + 2 * q^15 + q^16 + (-4*i - 1) * q^17 - q^18 - 2*i * q^20 - 2 * q^21 - 6*i * q^23 + i * q^24 + q^25 - 6 * q^26 - i * q^27 + 2*i * q^28 + 6*i * q^29 + 2 * q^30 + 10*i * q^31 + q^32 + (-4*i - 1) * q^34 + 4 * q^35 - q^36 + 2*i * q^37 - 6*i * q^39 - 2*i * q^40 - 2 * q^42 + 4 * q^43 + 2*i * q^45 - 6*i * q^46 + 8 * q^47 + i * q^48 + 3 * q^49 + q^50 + (-i + 4) * q^51 - 6 * q^52 - 6 * q^53 - i * q^54 + 2*i * q^56 + 6*i * q^58 + 2 * q^60 - 10*i * q^61 + 10*i * q^62 - 2*i * q^63 + q^64 + 12*i * q^65 + 8 * q^67 + (-4*i - 1) * q^68 + 6 * q^69 + 4 * q^70 + 10*i * q^71 - q^72 - 16*i * q^73 + 2*i * q^74 + i * q^75 - 6*i * q^78 + 6*i * q^79 - 2*i * q^80 + q^81 - 16 * q^83 - 2 * q^84 + (2*i - 8) * q^85 + 4 * q^86 - 6 * q^87 + 10 * q^89 + 2*i * q^90 - 12*i * q^91 - 6*i * q^92 - 10 * q^93 + 8 * q^94 + i * q^96 + 12*i * q^97 + 3 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9 $$2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 2 q^{9} - 12 q^{13} + 4 q^{15} + 2 q^{16} - 2 q^{17} - 2 q^{18} - 4 q^{21} + 2 q^{25} - 12 q^{26} + 4 q^{30} + 2 q^{32} - 2 q^{34} + 8 q^{35} - 2 q^{36} - 4 q^{42} + 8 q^{43} + 16 q^{47} + 6 q^{49} + 2 q^{50} + 8 q^{51} - 12 q^{52} - 12 q^{53} + 4 q^{60} + 2 q^{64} + 16 q^{67} - 2 q^{68} + 12 q^{69} + 8 q^{70} - 2 q^{72} + 2 q^{81} - 32 q^{83} - 4 q^{84} - 16 q^{85} + 8 q^{86} - 12 q^{87} + 20 q^{89} - 20 q^{93} + 16 q^{94} + 6 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 - 2 * q^9 - 12 * q^13 + 4 * q^15 + 2 * q^16 - 2 * q^17 - 2 * q^18 - 4 * q^21 + 2 * q^25 - 12 * q^26 + 4 * q^30 + 2 * q^32 - 2 * q^34 + 8 * q^35 - 2 * q^36 - 4 * q^42 + 8 * q^43 + 16 * q^47 + 6 * q^49 + 2 * q^50 + 8 * q^51 - 12 * q^52 - 12 * q^53 + 4 * q^60 + 2 * q^64 + 16 * q^67 - 2 * q^68 + 12 * q^69 + 8 * q^70 - 2 * q^72 + 2 * q^81 - 32 * q^83 - 4 * q^84 - 16 * q^85 + 8 * q^86 - 12 * q^87 + 20 * q^89 - 20 * q^93 + 16 * q^94 + 6 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/102\mathbb{Z}\right)^\times$$.

 $$n$$ $$35$$ $$37$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 − 1.00000i 1.00000i
1.00000 1.00000i 1.00000 2.00000i 1.00000i 2.00000i 1.00000 −1.00000 2.00000i
67.2 1.00000 1.00000i 1.00000 2.00000i 1.00000i 2.00000i 1.00000 −1.00000 2.00000i
 $$n$$: 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
17.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 102.2.b.a 2
3.b odd 2 1 306.2.b.a 2
4.b odd 2 1 816.2.c.a 2
5.b even 2 1 2550.2.c.f 2
5.c odd 4 1 2550.2.f.e 2
5.c odd 4 1 2550.2.f.j 2
8.b even 2 1 3264.2.c.j 2
8.d odd 2 1 3264.2.c.i 2
12.b even 2 1 2448.2.c.a 2
17.b even 2 1 inner 102.2.b.a 2
17.c even 4 1 1734.2.a.d 1
17.c even 4 1 1734.2.a.e 1
17.d even 8 4 1734.2.f.h 4
51.c odd 2 1 306.2.b.a 2
51.f odd 4 1 5202.2.a.h 1
51.f odd 4 1 5202.2.a.n 1
68.d odd 2 1 816.2.c.a 2
85.c even 2 1 2550.2.c.f 2
85.g odd 4 1 2550.2.f.e 2
85.g odd 4 1 2550.2.f.j 2
136.e odd 2 1 3264.2.c.i 2
136.h even 2 1 3264.2.c.j 2
204.h even 2 1 2448.2.c.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
102.2.b.a 2 1.a even 1 1 trivial
102.2.b.a 2 17.b even 2 1 inner
306.2.b.a 2 3.b odd 2 1
306.2.b.a 2 51.c odd 2 1
816.2.c.a 2 4.b odd 2 1
816.2.c.a 2 68.d odd 2 1
1734.2.a.d 1 17.c even 4 1
1734.2.a.e 1 17.c even 4 1
1734.2.f.h 4 17.d even 8 4
2448.2.c.a 2 12.b even 2 1
2448.2.c.a 2 204.h even 2 1
2550.2.c.f 2 5.b even 2 1
2550.2.c.f 2 85.c even 2 1
2550.2.f.e 2 5.c odd 4 1
2550.2.f.e 2 85.g odd 4 1
2550.2.f.j 2 5.c odd 4 1
2550.2.f.j 2 85.g odd 4 1
3264.2.c.i 2 8.d odd 2 1
3264.2.c.i 2 136.e odd 2 1
3264.2.c.j 2 8.b even 2 1
3264.2.c.j 2 136.h even 2 1
5202.2.a.h 1 51.f odd 4 1
5202.2.a.n 1 51.f odd 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(102, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2} + 4$$
$7$ $$T^{2} + 4$$
$11$ $$T^{2}$$
$13$ $$(T + 6)^{2}$$
$17$ $$T^{2} + 2T + 17$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 36$$
$29$ $$T^{2} + 36$$
$31$ $$T^{2} + 100$$
$37$ $$T^{2} + 4$$
$41$ $$T^{2}$$
$43$ $$(T - 4)^{2}$$
$47$ $$(T - 8)^{2}$$
$53$ $$(T + 6)^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 100$$
$67$ $$(T - 8)^{2}$$
$71$ $$T^{2} + 100$$
$73$ $$T^{2} + 256$$
$79$ $$T^{2} + 36$$
$83$ $$(T + 16)^{2}$$
$89$ $$(T - 10)^{2}$$
$97$ $$T^{2} + 144$$