# Properties

 Label 1617.2.a.l Level $1617$ Weight $2$ Character orbit 1617.a Self dual yes Analytic conductor $12.912$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 1) q^{2} + q^{3} + 3 \beta q^{4} + (2 \beta - 1) q^{5} + ( - \beta - 1) q^{6} + ( - 4 \beta - 1) q^{8} + q^{9}+O(q^{10})$$ q + (-b - 1) * q^2 + q^3 + 3*b * q^4 + (2*b - 1) * q^5 + (-b - 1) * q^6 + (-4*b - 1) * q^8 + q^9 $$q + ( - \beta - 1) q^{2} + q^{3} + 3 \beta q^{4} + (2 \beta - 1) q^{5} + ( - \beta - 1) q^{6} + ( - 4 \beta - 1) q^{8} + q^{9} + ( - 3 \beta - 1) q^{10} - q^{11} + 3 \beta q^{12} + ( - 4 \beta + 3) q^{13} + (2 \beta - 1) q^{15} + (3 \beta + 5) q^{16} - 6 q^{17} + ( - \beta - 1) q^{18} + (2 \beta + 1) q^{19} + (3 \beta + 6) q^{20} + (\beta + 1) q^{22} + ( - 4 \beta + 4) q^{23} + ( - 4 \beta - 1) q^{24} + (5 \beta + 1) q^{26} + q^{27} + ( - 2 \beta - 7) q^{29} + ( - 3 \beta - 1) q^{30} + ( - 4 \beta - 2) q^{31} + ( - 3 \beta - 6) q^{32} - q^{33} + (6 \beta + 6) q^{34} + 3 \beta q^{36} + (4 \beta - 7) q^{37} + ( - 5 \beta - 3) q^{38} + ( - 4 \beta + 3) q^{39} + ( - 6 \beta - 7) q^{40} - 6 q^{41} + ( - 4 \beta + 6) q^{43} - 3 \beta q^{44} + (2 \beta - 1) q^{45} + 4 \beta q^{46} + ( - 4 \beta - 5) q^{47} + (3 \beta + 5) q^{48} - 6 q^{51} + ( - 3 \beta - 12) q^{52} - 8 \beta q^{53} + ( - \beta - 1) q^{54} + ( - 2 \beta + 1) q^{55} + (2 \beta + 1) q^{57} + (11 \beta + 9) q^{58} + (8 \beta - 1) q^{59} + (3 \beta + 6) q^{60} + (8 \beta - 6) q^{61} + (10 \beta + 6) q^{62} + (6 \beta - 1) q^{64} + (2 \beta - 11) q^{65} + (\beta + 1) q^{66} + (6 \beta - 9) q^{67} - 18 \beta q^{68} + ( - 4 \beta + 4) q^{69} + (4 \beta - 2) q^{71} + ( - 4 \beta - 1) q^{72} + q^{73} + ( - \beta + 3) q^{74} + (9 \beta + 6) q^{76} + (5 \beta + 1) q^{78} + ( - 4 \beta + 4) q^{79} + (13 \beta + 1) q^{80} + q^{81} + (6 \beta + 6) q^{82} + ( - 4 \beta - 6) q^{83} + ( - 12 \beta + 6) q^{85} + (2 \beta - 2) q^{86} + ( - 2 \beta - 7) q^{87} + (4 \beta + 1) q^{88} + (12 \beta - 6) q^{89} + ( - 3 \beta - 1) q^{90} - 12 q^{92} + ( - 4 \beta - 2) q^{93} + (13 \beta + 9) q^{94} + (4 \beta + 3) q^{95} + ( - 3 \beta - 6) q^{96} + (8 \beta - 6) q^{97} - q^{99} +O(q^{100})$$ q + (-b - 1) * q^2 + q^3 + 3*b * q^4 + (2*b - 1) * q^5 + (-b - 1) * q^6 + (-4*b - 1) * q^8 + q^9 + (-3*b - 1) * q^10 - q^11 + 3*b * q^12 + (-4*b + 3) * q^13 + (2*b - 1) * q^15 + (3*b + 5) * q^16 - 6 * q^17 + (-b - 1) * q^18 + (2*b + 1) * q^19 + (3*b + 6) * q^20 + (b + 1) * q^22 + (-4*b + 4) * q^23 + (-4*b - 1) * q^24 + (5*b + 1) * q^26 + q^27 + (-2*b - 7) * q^29 + (-3*b - 1) * q^30 + (-4*b - 2) * q^31 + (-3*b - 6) * q^32 - q^33 + (6*b + 6) * q^34 + 3*b * q^36 + (4*b - 7) * q^37 + (-5*b - 3) * q^38 + (-4*b + 3) * q^39 + (-6*b - 7) * q^40 - 6 * q^41 + (-4*b + 6) * q^43 - 3*b * q^44 + (2*b - 1) * q^45 + 4*b * q^46 + (-4*b - 5) * q^47 + (3*b + 5) * q^48 - 6 * q^51 + (-3*b - 12) * q^52 - 8*b * q^53 + (-b - 1) * q^54 + (-2*b + 1) * q^55 + (2*b + 1) * q^57 + (11*b + 9) * q^58 + (8*b - 1) * q^59 + (3*b + 6) * q^60 + (8*b - 6) * q^61 + (10*b + 6) * q^62 + (6*b - 1) * q^64 + (2*b - 11) * q^65 + (b + 1) * q^66 + (6*b - 9) * q^67 - 18*b * q^68 + (-4*b + 4) * q^69 + (4*b - 2) * q^71 + (-4*b - 1) * q^72 + q^73 + (-b + 3) * q^74 + (9*b + 6) * q^76 + (5*b + 1) * q^78 + (-4*b + 4) * q^79 + (13*b + 1) * q^80 + q^81 + (6*b + 6) * q^82 + (-4*b - 6) * q^83 + (-12*b + 6) * q^85 + (2*b - 2) * q^86 + (-2*b - 7) * q^87 + (4*b + 1) * q^88 + (12*b - 6) * q^89 + (-3*b - 1) * q^90 - 12 * q^92 + (-4*b - 2) * q^93 + (13*b + 9) * q^94 + (4*b + 3) * q^95 + (-3*b - 6) * q^96 + (8*b - 6) * q^97 - q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} + 2 q^{3} + 3 q^{4} - 3 q^{6} - 6 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 3 * q^2 + 2 * q^3 + 3 * q^4 - 3 * q^6 - 6 * q^8 + 2 * q^9 $$2 q - 3 q^{2} + 2 q^{3} + 3 q^{4} - 3 q^{6} - 6 q^{8} + 2 q^{9} - 5 q^{10} - 2 q^{11} + 3 q^{12} + 2 q^{13} + 13 q^{16} - 12 q^{17} - 3 q^{18} + 4 q^{19} + 15 q^{20} + 3 q^{22} + 4 q^{23} - 6 q^{24} + 7 q^{26} + 2 q^{27} - 16 q^{29} - 5 q^{30} - 8 q^{31} - 15 q^{32} - 2 q^{33} + 18 q^{34} + 3 q^{36} - 10 q^{37} - 11 q^{38} + 2 q^{39} - 20 q^{40} - 12 q^{41} + 8 q^{43} - 3 q^{44} + 4 q^{46} - 14 q^{47} + 13 q^{48} - 12 q^{51} - 27 q^{52} - 8 q^{53} - 3 q^{54} + 4 q^{57} + 29 q^{58} + 6 q^{59} + 15 q^{60} - 4 q^{61} + 22 q^{62} + 4 q^{64} - 20 q^{65} + 3 q^{66} - 12 q^{67} - 18 q^{68} + 4 q^{69} - 6 q^{72} + 2 q^{73} + 5 q^{74} + 21 q^{76} + 7 q^{78} + 4 q^{79} + 15 q^{80} + 2 q^{81} + 18 q^{82} - 16 q^{83} - 2 q^{86} - 16 q^{87} + 6 q^{88} - 5 q^{90} - 24 q^{92} - 8 q^{93} + 31 q^{94} + 10 q^{95} - 15 q^{96} - 4 q^{97} - 2 q^{99}+O(q^{100})$$ 2 * q - 3 * q^2 + 2 * q^3 + 3 * q^4 - 3 * q^6 - 6 * q^8 + 2 * q^9 - 5 * q^10 - 2 * q^11 + 3 * q^12 + 2 * q^13 + 13 * q^16 - 12 * q^17 - 3 * q^18 + 4 * q^19 + 15 * q^20 + 3 * q^22 + 4 * q^23 - 6 * q^24 + 7 * q^26 + 2 * q^27 - 16 * q^29 - 5 * q^30 - 8 * q^31 - 15 * q^32 - 2 * q^33 + 18 * q^34 + 3 * q^36 - 10 * q^37 - 11 * q^38 + 2 * q^39 - 20 * q^40 - 12 * q^41 + 8 * q^43 - 3 * q^44 + 4 * q^46 - 14 * q^47 + 13 * q^48 - 12 * q^51 - 27 * q^52 - 8 * q^53 - 3 * q^54 + 4 * q^57 + 29 * q^58 + 6 * q^59 + 15 * q^60 - 4 * q^61 + 22 * q^62 + 4 * q^64 - 20 * q^65 + 3 * q^66 - 12 * q^67 - 18 * q^68 + 4 * q^69 - 6 * q^72 + 2 * q^73 + 5 * q^74 + 21 * q^76 + 7 * q^78 + 4 * q^79 + 15 * q^80 + 2 * q^81 + 18 * q^82 - 16 * q^83 - 2 * q^86 - 16 * q^87 + 6 * q^88 - 5 * q^90 - 24 * q^92 - 8 * q^93 + 31 * q^94 + 10 * q^95 - 15 * q^96 - 4 * q^97 - 2 * 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
 1.61803 −0.618034
−2.61803 1.00000 4.85410 2.23607 −2.61803 0 −7.47214 1.00000 −5.85410
1.2 −0.381966 1.00000 −1.85410 −2.23607 −0.381966 0 1.47214 1.00000 0.854102
 $$n$$: 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.l yes 2
3.b odd 2 1 4851.2.a.bg 2
7.b odd 2 1 1617.2.a.k 2
21.c even 2 1 4851.2.a.bh 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1617.2.a.k 2 7.b odd 2 1
1617.2.a.l yes 2 1.a even 1 1 trivial
4851.2.a.bg 2 3.b odd 2 1
4851.2.a.bh 2 21.c even 2 1

## Hecke kernels

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

 $$T_{2}^{2} + 3T_{2} + 1$$ T2^2 + 3*T2 + 1 $$T_{5}^{2} - 5$$ T5^2 - 5 $$T_{13}^{2} - 2T_{13} - 19$$ T13^2 - 2*T13 - 19 $$T_{17} + 6$$ T17 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 3T + 1$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2} - 5$$
$7$ $$T^{2}$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2} - 2T - 19$$
$17$ $$(T + 6)^{2}$$
$19$ $$T^{2} - 4T - 1$$
$23$ $$T^{2} - 4T - 16$$
$29$ $$T^{2} + 16T + 59$$
$31$ $$T^{2} + 8T - 4$$
$37$ $$T^{2} + 10T + 5$$
$41$ $$(T + 6)^{2}$$
$43$ $$T^{2} - 8T - 4$$
$47$ $$T^{2} + 14T + 29$$
$53$ $$T^{2} + 8T - 64$$
$59$ $$T^{2} - 6T - 71$$
$61$ $$T^{2} + 4T - 76$$
$67$ $$T^{2} + 12T - 9$$
$71$ $$T^{2} - 20$$
$73$ $$(T - 1)^{2}$$
$79$ $$T^{2} - 4T - 16$$
$83$ $$T^{2} + 16T + 44$$
$89$ $$T^{2} - 180$$
$97$ $$T^{2} + 4T - 76$$