# Properties

 Label 1617.2.a.o Level $1617$ Weight $2$ Character orbit 1617.a Self dual yes Analytic conductor $12.912$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

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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: $$\Q(\sqrt{21})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 5$$ x^2 - x - 5 Coefficient ring: $$\Z[a_1, a_2]$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{21})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} + q^{3} + (\beta + 3) q^{4} - 3 q^{5} - \beta q^{6} + ( - 2 \beta - 5) q^{8} + q^{9} +O(q^{10})$$ q - b * q^2 + q^3 + (b + 3) * q^4 - 3 * q^5 - b * q^6 + (-2*b - 5) * q^8 + q^9 $$q - \beta q^{2} + q^{3} + (\beta + 3) q^{4} - 3 q^{5} - \beta q^{6} + ( - 2 \beta - 5) q^{8} + q^{9} + 3 \beta q^{10} - q^{11} + (\beta + 3) q^{12} - q^{13} - 3 q^{15} + (5 \beta + 4) q^{16} + (2 \beta - 4) q^{17} - \beta q^{18} + ( - 2 \beta + 3) q^{19} + ( - 3 \beta - 9) q^{20} + \beta q^{22} + (2 \beta - 2) q^{23} + ( - 2 \beta - 5) q^{24} + 4 q^{25} + \beta q^{26} + q^{27} + (4 \beta - 1) q^{29} + 3 \beta q^{30} + 2 \beta q^{31} + ( - 5 \beta - 15) q^{32} - q^{33} + (2 \beta - 10) q^{34} + (\beta + 3) q^{36} + q^{37} + ( - \beta + 10) q^{38} - q^{39} + (6 \beta + 15) q^{40} + ( - 4 \beta + 4) q^{41} + ( - 2 \beta - 2) q^{43} + ( - \beta - 3) q^{44} - 3 q^{45} - 10 q^{46} + ( - 2 \beta - 5) q^{47} + (5 \beta + 4) q^{48} - 4 \beta q^{50} + (2 \beta - 4) q^{51} + ( - \beta - 3) q^{52} + (2 \beta - 6) q^{53} - \beta q^{54} + 3 q^{55} + ( - 2 \beta + 3) q^{57} + ( - 3 \beta - 20) q^{58} + (2 \beta - 1) q^{59} + ( - 3 \beta - 9) q^{60} - 10 q^{61} + ( - 2 \beta - 10) q^{62} + (10 \beta + 17) q^{64} + 3 q^{65} + \beta q^{66} + ( - 2 \beta + 5) q^{67} + (4 \beta - 2) q^{68} + (2 \beta - 2) q^{69} + ( - 4 \beta + 4) q^{71} + ( - 2 \beta - 5) q^{72} - 7 q^{73} - \beta q^{74} + 4 q^{75} + ( - 5 \beta - 1) q^{76} + \beta q^{78} - 4 \beta q^{79} + ( - 15 \beta - 12) q^{80} + q^{81} + 20 q^{82} + ( - 2 \beta + 8) q^{83} + ( - 6 \beta + 12) q^{85} + (4 \beta + 10) q^{86} + (4 \beta - 1) q^{87} + (2 \beta + 5) q^{88} + (4 \beta - 2) q^{89} + 3 \beta q^{90} + (6 \beta + 4) q^{92} + 2 \beta q^{93} + (7 \beta + 10) q^{94} + (6 \beta - 9) q^{95} + ( - 5 \beta - 15) q^{96} + (2 \beta + 6) q^{97} - q^{99} +O(q^{100})$$ q - b * q^2 + q^3 + (b + 3) * q^4 - 3 * q^5 - b * q^6 + (-2*b - 5) * q^8 + q^9 + 3*b * q^10 - q^11 + (b + 3) * q^12 - q^13 - 3 * q^15 + (5*b + 4) * q^16 + (2*b - 4) * q^17 - b * q^18 + (-2*b + 3) * q^19 + (-3*b - 9) * q^20 + b * q^22 + (2*b - 2) * q^23 + (-2*b - 5) * q^24 + 4 * q^25 + b * q^26 + q^27 + (4*b - 1) * q^29 + 3*b * q^30 + 2*b * q^31 + (-5*b - 15) * q^32 - q^33 + (2*b - 10) * q^34 + (b + 3) * q^36 + q^37 + (-b + 10) * q^38 - q^39 + (6*b + 15) * q^40 + (-4*b + 4) * q^41 + (-2*b - 2) * q^43 + (-b - 3) * q^44 - 3 * q^45 - 10 * q^46 + (-2*b - 5) * q^47 + (5*b + 4) * q^48 - 4*b * q^50 + (2*b - 4) * q^51 + (-b - 3) * q^52 + (2*b - 6) * q^53 - b * q^54 + 3 * q^55 + (-2*b + 3) * q^57 + (-3*b - 20) * q^58 + (2*b - 1) * q^59 + (-3*b - 9) * q^60 - 10 * q^61 + (-2*b - 10) * q^62 + (10*b + 17) * q^64 + 3 * q^65 + b * q^66 + (-2*b + 5) * q^67 + (4*b - 2) * q^68 + (2*b - 2) * q^69 + (-4*b + 4) * q^71 + (-2*b - 5) * q^72 - 7 * q^73 - b * q^74 + 4 * q^75 + (-5*b - 1) * q^76 + b * q^78 - 4*b * q^79 + (-15*b - 12) * q^80 + q^81 + 20 * q^82 + (-2*b + 8) * q^83 + (-6*b + 12) * q^85 + (4*b + 10) * q^86 + (4*b - 1) * q^87 + (2*b + 5) * q^88 + (4*b - 2) * q^89 + 3*b * q^90 + (6*b + 4) * q^92 + 2*b * q^93 + (7*b + 10) * q^94 + (6*b - 9) * q^95 + (-5*b - 15) * q^96 + (2*b + 6) * q^97 - q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 2 q^{3} + 7 q^{4} - 6 q^{5} - q^{6} - 12 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - q^2 + 2 * q^3 + 7 * q^4 - 6 * q^5 - q^6 - 12 * q^8 + 2 * q^9 $$2 q - q^{2} + 2 q^{3} + 7 q^{4} - 6 q^{5} - q^{6} - 12 q^{8} + 2 q^{9} + 3 q^{10} - 2 q^{11} + 7 q^{12} - 2 q^{13} - 6 q^{15} + 13 q^{16} - 6 q^{17} - q^{18} + 4 q^{19} - 21 q^{20} + q^{22} - 2 q^{23} - 12 q^{24} + 8 q^{25} + q^{26} + 2 q^{27} + 2 q^{29} + 3 q^{30} + 2 q^{31} - 35 q^{32} - 2 q^{33} - 18 q^{34} + 7 q^{36} + 2 q^{37} + 19 q^{38} - 2 q^{39} + 36 q^{40} + 4 q^{41} - 6 q^{43} - 7 q^{44} - 6 q^{45} - 20 q^{46} - 12 q^{47} + 13 q^{48} - 4 q^{50} - 6 q^{51} - 7 q^{52} - 10 q^{53} - q^{54} + 6 q^{55} + 4 q^{57} - 43 q^{58} - 21 q^{60} - 20 q^{61} - 22 q^{62} + 44 q^{64} + 6 q^{65} + q^{66} + 8 q^{67} - 2 q^{69} + 4 q^{71} - 12 q^{72} - 14 q^{73} - q^{74} + 8 q^{75} - 7 q^{76} + q^{78} - 4 q^{79} - 39 q^{80} + 2 q^{81} + 40 q^{82} + 14 q^{83} + 18 q^{85} + 24 q^{86} + 2 q^{87} + 12 q^{88} + 3 q^{90} + 14 q^{92} + 2 q^{93} + 27 q^{94} - 12 q^{95} - 35 q^{96} + 14 q^{97} - 2 q^{99}+O(q^{100})$$ 2 * q - q^2 + 2 * q^3 + 7 * q^4 - 6 * q^5 - q^6 - 12 * q^8 + 2 * q^9 + 3 * q^10 - 2 * q^11 + 7 * q^12 - 2 * q^13 - 6 * q^15 + 13 * q^16 - 6 * q^17 - q^18 + 4 * q^19 - 21 * q^20 + q^22 - 2 * q^23 - 12 * q^24 + 8 * q^25 + q^26 + 2 * q^27 + 2 * q^29 + 3 * q^30 + 2 * q^31 - 35 * q^32 - 2 * q^33 - 18 * q^34 + 7 * q^36 + 2 * q^37 + 19 * q^38 - 2 * q^39 + 36 * q^40 + 4 * q^41 - 6 * q^43 - 7 * q^44 - 6 * q^45 - 20 * q^46 - 12 * q^47 + 13 * q^48 - 4 * q^50 - 6 * q^51 - 7 * q^52 - 10 * q^53 - q^54 + 6 * q^55 + 4 * q^57 - 43 * q^58 - 21 * q^60 - 20 * q^61 - 22 * q^62 + 44 * q^64 + 6 * q^65 + q^66 + 8 * q^67 - 2 * q^69 + 4 * q^71 - 12 * q^72 - 14 * q^73 - q^74 + 8 * q^75 - 7 * q^76 + q^78 - 4 * q^79 - 39 * q^80 + 2 * q^81 + 40 * q^82 + 14 * q^83 + 18 * q^85 + 24 * q^86 + 2 * q^87 + 12 * q^88 + 3 * q^90 + 14 * q^92 + 2 * q^93 + 27 * q^94 - 12 * q^95 - 35 * q^96 + 14 * 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
 2.79129 −1.79129
−2.79129 1.00000 5.79129 −3.00000 −2.79129 0 −10.5826 1.00000 8.37386
1.2 1.79129 1.00000 1.20871 −3.00000 1.79129 0 −1.41742 1.00000 −5.37386
 $$n$$: 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.o 2
3.b odd 2 1 4851.2.a.ba 2
7.b odd 2 1 231.2.a.b 2
21.c even 2 1 693.2.a.j 2
28.d even 2 1 3696.2.a.bl 2
35.c odd 2 1 5775.2.a.bn 2
77.b even 2 1 2541.2.a.z 2
231.h odd 2 1 7623.2.a.bf 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.a.b 2 7.b odd 2 1
693.2.a.j 2 21.c even 2 1
1617.2.a.o 2 1.a even 1 1 trivial
2541.2.a.z 2 77.b even 2 1
3696.2.a.bl 2 28.d even 2 1
4851.2.a.ba 2 3.b odd 2 1
5775.2.a.bn 2 35.c odd 2 1
7623.2.a.bf 2 231.h odd 2 1

## Hecke kernels

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

 $$T_{2}^{2} + T_{2} - 5$$ T2^2 + T2 - 5 $$T_{5} + 3$$ T5 + 3 $$T_{13} + 1$$ T13 + 1 $$T_{17}^{2} + 6T_{17} - 12$$ T17^2 + 6*T17 - 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T - 5$$
$3$ $$(T - 1)^{2}$$
$5$ $$(T + 3)^{2}$$
$7$ $$T^{2}$$
$11$ $$(T + 1)^{2}$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} + 6T - 12$$
$19$ $$T^{2} - 4T - 17$$
$23$ $$T^{2} + 2T - 20$$
$29$ $$T^{2} - 2T - 83$$
$31$ $$T^{2} - 2T - 20$$
$37$ $$(T - 1)^{2}$$
$41$ $$T^{2} - 4T - 80$$
$43$ $$T^{2} + 6T - 12$$
$47$ $$T^{2} + 12T + 15$$
$53$ $$T^{2} + 10T + 4$$
$59$ $$T^{2} - 21$$
$61$ $$(T + 10)^{2}$$
$67$ $$T^{2} - 8T - 5$$
$71$ $$T^{2} - 4T - 80$$
$73$ $$(T + 7)^{2}$$
$79$ $$T^{2} + 4T - 80$$
$83$ $$T^{2} - 14T + 28$$
$89$ $$T^{2} - 84$$
$97$ $$T^{2} - 14T + 28$$
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