# Properties

 Label 17.2.a.a Level $17$ Weight $2$ Character orbit 17.a Self dual yes Analytic conductor $0.136$ 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] = [17,2,Mod(1,17)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(17, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

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

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

chi := DirichletCharacter("17.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 17.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: $$0.135745683436$$ 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)

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{4} - 2 q^{5} + 4 q^{7} + 3 q^{8} - 3 q^{9}+O(q^{10})$$ q - q^2 - q^4 - 2 * q^5 + 4 * q^7 + 3 * q^8 - 3 * q^9 $$q - q^{2} - q^{4} - 2 q^{5} + 4 q^{7} + 3 q^{8} - 3 q^{9} + 2 q^{10} - 2 q^{13} - 4 q^{14} - q^{16} + q^{17} + 3 q^{18} - 4 q^{19} + 2 q^{20} + 4 q^{23} - q^{25} + 2 q^{26} - 4 q^{28} + 6 q^{29} + 4 q^{31} - 5 q^{32} - q^{34} - 8 q^{35} + 3 q^{36} - 2 q^{37} + 4 q^{38} - 6 q^{40} - 6 q^{41} + 4 q^{43} + 6 q^{45} - 4 q^{46} + 9 q^{49} + q^{50} + 2 q^{52} + 6 q^{53} + 12 q^{56} - 6 q^{58} - 12 q^{59} - 10 q^{61} - 4 q^{62} - 12 q^{63} + 7 q^{64} + 4 q^{65} + 4 q^{67} - q^{68} + 8 q^{70} - 4 q^{71} - 9 q^{72} - 6 q^{73} + 2 q^{74} + 4 q^{76} + 12 q^{79} + 2 q^{80} + 9 q^{81} + 6 q^{82} - 4 q^{83} - 2 q^{85} - 4 q^{86} + 10 q^{89} - 6 q^{90} - 8 q^{91} - 4 q^{92} + 8 q^{95} + 2 q^{97} - 9 q^{98}+O(q^{100})$$ q - q^2 - q^4 - 2 * q^5 + 4 * q^7 + 3 * q^8 - 3 * q^9 + 2 * q^10 - 2 * q^13 - 4 * q^14 - q^16 + q^17 + 3 * q^18 - 4 * q^19 + 2 * q^20 + 4 * q^23 - q^25 + 2 * q^26 - 4 * q^28 + 6 * q^29 + 4 * q^31 - 5 * q^32 - q^34 - 8 * q^35 + 3 * q^36 - 2 * q^37 + 4 * q^38 - 6 * q^40 - 6 * q^41 + 4 * q^43 + 6 * q^45 - 4 * q^46 + 9 * q^49 + q^50 + 2 * q^52 + 6 * q^53 + 12 * q^56 - 6 * q^58 - 12 * q^59 - 10 * q^61 - 4 * q^62 - 12 * q^63 + 7 * q^64 + 4 * q^65 + 4 * q^67 - q^68 + 8 * q^70 - 4 * q^71 - 9 * q^72 - 6 * q^73 + 2 * q^74 + 4 * q^76 + 12 * q^79 + 2 * q^80 + 9 * q^81 + 6 * q^82 - 4 * q^83 - 2 * q^85 - 4 * q^86 + 10 * q^89 - 6 * q^90 - 8 * q^91 - 4 * q^92 + 8 * q^95 + 2 * q^97 - 9 * q^98

## 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 0 −1.00000 −2.00000 0 4.00000 3.00000 −3.00000 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
$$17$$ $$-1$$

## 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 17.2.a.a 1
3.b odd 2 1 153.2.a.c 1
4.b odd 2 1 272.2.a.b 1
5.b even 2 1 425.2.a.d 1
5.c odd 4 2 425.2.b.b 2
7.b odd 2 1 833.2.a.a 1
7.c even 3 2 833.2.e.b 2
7.d odd 6 2 833.2.e.a 2
8.b even 2 1 1088.2.a.i 1
8.d odd 2 1 1088.2.a.h 1
11.b odd 2 1 2057.2.a.e 1
12.b even 2 1 2448.2.a.o 1
13.b even 2 1 2873.2.a.c 1
15.d odd 2 1 3825.2.a.d 1
17.b even 2 1 289.2.a.a 1
17.c even 4 2 289.2.b.a 2
17.d even 8 4 289.2.c.a 4
17.e odd 16 8 289.2.d.d 8
19.b odd 2 1 6137.2.a.b 1
20.d odd 2 1 6800.2.a.n 1
21.c even 2 1 7497.2.a.l 1
23.b odd 2 1 8993.2.a.a 1
24.f even 2 1 9792.2.a.i 1
24.h odd 2 1 9792.2.a.n 1
51.c odd 2 1 2601.2.a.g 1
68.d odd 2 1 4624.2.a.d 1
85.c even 2 1 7225.2.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.2.a.a 1 1.a even 1 1 trivial
153.2.a.c 1 3.b odd 2 1
272.2.a.b 1 4.b odd 2 1
289.2.a.a 1 17.b even 2 1
289.2.b.a 2 17.c even 4 2
289.2.c.a 4 17.d even 8 4
289.2.d.d 8 17.e odd 16 8
425.2.a.d 1 5.b even 2 1
425.2.b.b 2 5.c odd 4 2
833.2.a.a 1 7.b odd 2 1
833.2.e.a 2 7.d odd 6 2
833.2.e.b 2 7.c even 3 2
1088.2.a.h 1 8.d odd 2 1
1088.2.a.i 1 8.b even 2 1
2057.2.a.e 1 11.b odd 2 1
2448.2.a.o 1 12.b even 2 1
2601.2.a.g 1 51.c odd 2 1
2873.2.a.c 1 13.b even 2 1
3825.2.a.d 1 15.d odd 2 1
4624.2.a.d 1 68.d odd 2 1
6137.2.a.b 1 19.b odd 2 1
6800.2.a.n 1 20.d odd 2 1
7225.2.a.g 1 85.c even 2 1
7497.2.a.l 1 21.c even 2 1
8993.2.a.a 1 23.b odd 2 1
9792.2.a.i 1 24.f even 2 1
9792.2.a.n 1 24.h odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(\Gamma_0(17))$$.

## Hecke characteristic polynomials

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