# Properties

 Label 68.1.d.a Level $68$ Weight $1$ Character orbit 68.d Self dual yes Analytic conductor $0.034$ Analytic rank $0$ Dimension $1$ Projective image $D_{2}$ CM/RM discs -4, -68, 17 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [68,1,Mod(67,68)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(68, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("68.67");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$68 = 2^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 68.d (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: yes Analytic conductor: $$0.0339364208590$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(i, \sqrt{17})$$ Artin image: $D_4$ Artin field: Galois closure of 4.0.272.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} - q^{8} - q^{9}+O(q^{10})$$ q - q^2 + q^4 - q^8 - q^9 $$q - q^{2} + q^{4} - q^{8} - q^{9} - 2 q^{13} + q^{16} + q^{17} + q^{18} + q^{25} + 2 q^{26} - q^{32} - q^{34} - q^{36} - q^{49} - q^{50} - 2 q^{52} + 2 q^{53} + q^{64} + q^{68} + q^{72} + q^{81} - 2 q^{89} + q^{98}+O(q^{100})$$ q - q^2 + q^4 - q^8 - q^9 - 2 * q^13 + q^16 + q^17 + q^18 + q^25 + 2 * q^26 - q^32 - q^34 - q^36 - q^49 - q^50 - 2 * q^52 + 2 * q^53 + q^64 + q^68 + q^72 + q^81 - 2 * q^89 + q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/68\mathbb{Z}\right)^\times$$.

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

## 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
 0
−1.00000 0 1.00000 0 0 0 −1.00000 −1.00000 0
 $$n$$: 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
17.b even 2 1 RM by $$\Q(\sqrt{17})$$
68.d odd 2 1 CM by $$\Q(\sqrt{-17})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 68.1.d.a 1
3.b odd 2 1 612.1.e.a 1
4.b odd 2 1 CM 68.1.d.a 1
5.b even 2 1 1700.1.h.d 1
5.c odd 4 2 1700.1.d.b 2
7.b odd 2 1 3332.1.g.a 1
7.c even 3 2 3332.1.o.c 2
7.d odd 6 2 3332.1.o.d 2
8.b even 2 1 1088.1.g.a 1
8.d odd 2 1 1088.1.g.a 1
12.b even 2 1 612.1.e.a 1
17.b even 2 1 RM 68.1.d.a 1
17.c even 4 2 1156.1.c.a 1
17.d even 8 4 1156.1.f.a 2
17.e odd 16 8 1156.1.g.a 4
20.d odd 2 1 1700.1.h.d 1
20.e even 4 2 1700.1.d.b 2
28.d even 2 1 3332.1.g.a 1
28.f even 6 2 3332.1.o.d 2
28.g odd 6 2 3332.1.o.c 2
51.c odd 2 1 612.1.e.a 1
68.d odd 2 1 CM 68.1.d.a 1
68.f odd 4 2 1156.1.c.a 1
68.g odd 8 4 1156.1.f.a 2
68.i even 16 8 1156.1.g.a 4
85.c even 2 1 1700.1.h.d 1
85.g odd 4 2 1700.1.d.b 2
119.d odd 2 1 3332.1.g.a 1
119.h odd 6 2 3332.1.o.d 2
119.j even 6 2 3332.1.o.c 2
136.e odd 2 1 1088.1.g.a 1
136.h even 2 1 1088.1.g.a 1
204.h even 2 1 612.1.e.a 1
340.d odd 2 1 1700.1.h.d 1
340.r even 4 2 1700.1.d.b 2
476.e even 2 1 3332.1.g.a 1
476.o odd 6 2 3332.1.o.c 2
476.q even 6 2 3332.1.o.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.1.d.a 1 1.a even 1 1 trivial
68.1.d.a 1 4.b odd 2 1 CM
68.1.d.a 1 17.b even 2 1 RM
68.1.d.a 1 68.d odd 2 1 CM
612.1.e.a 1 3.b odd 2 1
612.1.e.a 1 12.b even 2 1
612.1.e.a 1 51.c odd 2 1
612.1.e.a 1 204.h even 2 1
1088.1.g.a 1 8.b even 2 1
1088.1.g.a 1 8.d odd 2 1
1088.1.g.a 1 136.e odd 2 1
1088.1.g.a 1 136.h even 2 1
1156.1.c.a 1 17.c even 4 2
1156.1.c.a 1 68.f odd 4 2
1156.1.f.a 2 17.d even 8 4
1156.1.f.a 2 68.g odd 8 4
1156.1.g.a 4 17.e odd 16 8
1156.1.g.a 4 68.i even 16 8
1700.1.d.b 2 5.c odd 4 2
1700.1.d.b 2 20.e even 4 2
1700.1.d.b 2 85.g odd 4 2
1700.1.d.b 2 340.r even 4 2
1700.1.h.d 1 5.b even 2 1
1700.1.h.d 1 20.d odd 2 1
1700.1.h.d 1 85.c even 2 1
1700.1.h.d 1 340.d odd 2 1
3332.1.g.a 1 7.b odd 2 1
3332.1.g.a 1 28.d even 2 1
3332.1.g.a 1 119.d odd 2 1
3332.1.g.a 1 476.e even 2 1
3332.1.o.c 2 7.c even 3 2
3332.1.o.c 2 28.g odd 6 2
3332.1.o.c 2 119.j even 6 2
3332.1.o.c 2 476.o odd 6 2
3332.1.o.d 2 7.d odd 6 2
3332.1.o.d 2 28.f even 6 2
3332.1.o.d 2 119.h odd 6 2
3332.1.o.d 2 476.q even 6 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(68, [\chi])$$.

## Hecke characteristic polynomials

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