# Properties

 Label 1617.2.a.b Level $1617$ Weight $2$ Character orbit 1617.a Self dual yes Analytic conductor $12.912$ Analytic rank $1$ 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] = [1617,2,Mod(1,1617)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1617, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1617.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1617 = 3 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1617.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: $$12.9118100068$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 231) 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + q^{3} + 2 q^{4} - 2 q^{6} + q^{9}+O(q^{10})$$ q - 2 * q^2 + q^3 + 2 * q^4 - 2 * q^6 + q^9 $$q - 2 q^{2} + q^{3} + 2 q^{4} - 2 q^{6} + q^{9} - q^{11} + 2 q^{12} + 5 q^{13} - 4 q^{16} - 6 q^{17} - 2 q^{18} - 7 q^{19} + 2 q^{22} - 4 q^{23} - 5 q^{25} - 10 q^{26} + q^{27} - 2 q^{29} + 7 q^{31} + 8 q^{32} - q^{33} + 12 q^{34} + 2 q^{36} + 7 q^{37} + 14 q^{38} + 5 q^{39} - 4 q^{41} - 9 q^{43} - 2 q^{44} + 8 q^{46} - 6 q^{47} - 4 q^{48} + 10 q^{50} - 6 q^{51} + 10 q^{52} - 2 q^{53} - 2 q^{54} - 7 q^{57} + 4 q^{58} - 12 q^{59} + 2 q^{61} - 14 q^{62} - 8 q^{64} + 2 q^{66} + 7 q^{67} - 12 q^{68} - 4 q^{69} + 8 q^{71} + 5 q^{73} - 14 q^{74} - 5 q^{75} - 14 q^{76} - 10 q^{78} - 11 q^{79} + q^{81} + 8 q^{82} + 4 q^{83} + 18 q^{86} - 2 q^{87} - 6 q^{89} - 8 q^{92} + 7 q^{93} + 12 q^{94} + 8 q^{96} - 2 q^{97} - q^{99}+O(q^{100})$$ q - 2 * q^2 + q^3 + 2 * q^4 - 2 * q^6 + q^9 - q^11 + 2 * q^12 + 5 * q^13 - 4 * q^16 - 6 * q^17 - 2 * q^18 - 7 * q^19 + 2 * q^22 - 4 * q^23 - 5 * q^25 - 10 * q^26 + q^27 - 2 * q^29 + 7 * q^31 + 8 * q^32 - q^33 + 12 * q^34 + 2 * q^36 + 7 * q^37 + 14 * q^38 + 5 * q^39 - 4 * q^41 - 9 * q^43 - 2 * q^44 + 8 * q^46 - 6 * q^47 - 4 * q^48 + 10 * q^50 - 6 * q^51 + 10 * q^52 - 2 * q^53 - 2 * q^54 - 7 * q^57 + 4 * q^58 - 12 * q^59 + 2 * q^61 - 14 * q^62 - 8 * q^64 + 2 * q^66 + 7 * q^67 - 12 * q^68 - 4 * q^69 + 8 * q^71 + 5 * q^73 - 14 * q^74 - 5 * q^75 - 14 * q^76 - 10 * q^78 - 11 * q^79 + q^81 + 8 * q^82 + 4 * q^83 + 18 * q^86 - 2 * q^87 - 6 * q^89 - 8 * q^92 + 7 * q^93 + 12 * q^94 + 8 * q^96 - 2 * q^97 - 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
−2.00000 1.00000 2.00000 0 −2.00000 0 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$-1$$
$$11$$ $$+1$$

## 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 1617.2.a.b 1
3.b odd 2 1 4851.2.a.s 1
7.b odd 2 1 1617.2.a.a 1
7.d odd 6 2 231.2.i.c 2
21.c even 2 1 4851.2.a.r 1
21.g even 6 2 693.2.i.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.i.c 2 7.d odd 6 2
693.2.i.a 2 21.g even 6 2
1617.2.a.a 1 7.b odd 2 1
1617.2.a.b 1 1.a even 1 1 trivial
4851.2.a.r 1 21.c even 2 1
4851.2.a.s 1 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1617))$$:

 $$T_{2} + 2$$ T2 + 2 $$T_{5}$$ T5 $$T_{13} - 5$$ T13 - 5 $$T_{17} + 6$$ T17 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T + 1$$
$13$ $$T - 5$$
$17$ $$T + 6$$
$19$ $$T + 7$$
$23$ $$T + 4$$
$29$ $$T + 2$$
$31$ $$T - 7$$
$37$ $$T - 7$$
$41$ $$T + 4$$
$43$ $$T + 9$$
$47$ $$T + 6$$
$53$ $$T + 2$$
$59$ $$T + 12$$
$61$ $$T - 2$$
$67$ $$T - 7$$
$71$ $$T - 8$$
$73$ $$T - 5$$
$79$ $$T + 11$$
$83$ $$T - 4$$
$89$ $$T + 6$$
$97$ $$T + 2$$