# Properties

 Label 1617.2.a.t Level $1617$ Weight $2$ Character orbit 1617.a Self dual yes Analytic conductor $12.912$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1617 = 3 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1617.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$12.9118100068$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.229.1 Defining polynomial: $$x^{3} - 4 x - 1$$ 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

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{2} ) q^{2} - q^{3} + ( 2 + \beta_{1} ) q^{4} + ( -1 + \beta_{1} + \beta_{2} ) q^{5} + ( -1 - \beta_{2} ) q^{6} + ( 1 + 2 \beta_{1} ) q^{8} + q^{9} +O(q^{10})$$ $$q + ( 1 + \beta_{2} ) q^{2} - q^{3} + ( 2 + \beta_{1} ) q^{4} + ( -1 + \beta_{1} + \beta_{2} ) q^{5} + ( -1 - \beta_{2} ) q^{6} + ( 1 + 2 \beta_{1} ) q^{8} + q^{9} + ( 3 + 3 \beta_{1} - 2 \beta_{2} ) q^{10} - q^{11} + ( -2 - \beta_{1} ) q^{12} + ( 1 + 3 \beta_{1} - \beta_{2} ) q^{13} + ( 1 - \beta_{1} - \beta_{2} ) q^{15} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{16} + ( -2 - 4 \beta_{1} + 2 \beta_{2} ) q^{17} + ( 1 + \beta_{2} ) q^{18} + ( 3 - \beta_{1} + \beta_{2} ) q^{19} + ( 2 + 2 \beta_{1} + 3 \beta_{2} ) q^{20} + ( -1 - \beta_{2} ) q^{22} + ( 4 + 2 \beta_{2} ) q^{23} + ( -1 - 2 \beta_{1} ) q^{24} + ( 4 + \beta_{1} - 3 \beta_{2} ) q^{25} + ( 1 + 5 \beta_{1} + 2 \beta_{2} ) q^{26} - q^{27} + ( -1 - 3 \beta_{1} + \beta_{2} ) q^{29} + ( -3 - 3 \beta_{1} + 2 \beta_{2} ) q^{30} + ( 2 - 2 \beta_{1} + 4 \beta_{2} ) q^{31} + ( 2 + \beta_{1} - 2 \beta_{2} ) q^{32} + q^{33} + ( -6 \beta_{1} - 4 \beta_{2} ) q^{34} + ( 2 + \beta_{1} ) q^{36} + ( 1 - \beta_{1} + 3 \beta_{2} ) q^{37} + ( 5 - \beta_{1} + 2 \beta_{2} ) q^{38} + ( -1 - 3 \beta_{1} + \beta_{2} ) q^{39} + ( 7 + \beta_{1} + 3 \beta_{2} ) q^{40} + ( -4 - 2 \beta_{1} + 2 \beta_{2} ) q^{41} + ( -4 + 4 \beta_{1} + 2 \beta_{2} ) q^{43} + ( -2 - \beta_{1} ) q^{44} + ( -1 + \beta_{1} + \beta_{2} ) q^{45} + ( 10 + 2 \beta_{1} + 2 \beta_{2} ) q^{46} + ( -1 - \beta_{1} - 3 \beta_{2} ) q^{47} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{48} + ( -4 - \beta_{1} + 7 \beta_{2} ) q^{50} + ( 2 + 4 \beta_{1} - 2 \beta_{2} ) q^{51} + ( 10 + 6 \beta_{1} + \beta_{2} ) q^{52} -2 \beta_{1} q^{53} + ( -1 - \beta_{2} ) q^{54} + ( 1 - \beta_{1} - \beta_{2} ) q^{55} + ( -3 + \beta_{1} - \beta_{2} ) q^{57} + ( -1 - 5 \beta_{1} - 2 \beta_{2} ) q^{58} + ( -1 + 3 \beta_{1} - 3 \beta_{2} ) q^{59} + ( -2 - 2 \beta_{1} - 3 \beta_{2} ) q^{60} + 2 q^{61} + ( 12 - 2 \beta_{2} ) q^{62} + ( -1 - 4 \beta_{1} + 2 \beta_{2} ) q^{64} + ( 7 - \beta_{1} + 7 \beta_{2} ) q^{65} + ( 1 + \beta_{2} ) q^{66} + ( -1 - 5 \beta_{1} + \beta_{2} ) q^{67} + ( -14 - 8 \beta_{1} ) q^{68} + ( -4 - 2 \beta_{2} ) q^{69} + ( -4 + 4 \beta_{1} ) q^{71} + ( 1 + 2 \beta_{1} ) q^{72} + ( 7 - 3 \beta_{1} + \beta_{2} ) q^{73} + ( 9 + \beta_{1} - 2 \beta_{2} ) q^{74} + ( -4 - \beta_{1} + 3 \beta_{2} ) q^{75} + ( 4 + 2 \beta_{1} + \beta_{2} ) q^{76} + ( -1 - 5 \beta_{1} - 2 \beta_{2} ) q^{78} + ( 4 - 4 \beta_{1} ) q^{79} + ( 13 + \beta_{1} - 2 \beta_{2} ) q^{80} + q^{81} + ( -2 \beta_{1} - 6 \beta_{2} ) q^{82} + ( -2 + 6 \beta_{1} ) q^{83} + ( -6 + 2 \beta_{1} - 12 \beta_{2} ) q^{85} + ( 6 + 10 \beta_{1} - 6 \beta_{2} ) q^{86} + ( 1 + 3 \beta_{1} - \beta_{2} ) q^{87} + ( -1 - 2 \beta_{1} ) q^{88} + ( -10 - 4 \beta_{2} ) q^{89} + ( 3 + 3 \beta_{1} - 2 \beta_{2} ) q^{90} + ( 10 + 6 \beta_{1} + 4 \beta_{2} ) q^{92} + ( -2 + 2 \beta_{1} - 4 \beta_{2} ) q^{93} + ( -11 - 5 \beta_{1} + 2 \beta_{2} ) q^{94} + ( -3 + 5 \beta_{1} - \beta_{2} ) q^{95} + ( -2 - \beta_{1} + 2 \beta_{2} ) q^{96} + ( -2 \beta_{1} - 4 \beta_{2} ) q^{97} - q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{2} - 3 q^{3} + 6 q^{4} - 4 q^{5} - 2 q^{6} + 3 q^{8} + 3 q^{9} + O(q^{10})$$ $$3 q + 2 q^{2} - 3 q^{3} + 6 q^{4} - 4 q^{5} - 2 q^{6} + 3 q^{8} + 3 q^{9} + 11 q^{10} - 3 q^{11} - 6 q^{12} + 4 q^{13} + 4 q^{15} - 4 q^{16} - 8 q^{17} + 2 q^{18} + 8 q^{19} + 3 q^{20} - 2 q^{22} + 10 q^{23} - 3 q^{24} + 15 q^{25} + q^{26} - 3 q^{27} - 4 q^{29} - 11 q^{30} + 2 q^{31} + 8 q^{32} + 3 q^{33} + 4 q^{34} + 6 q^{36} + 13 q^{38} - 4 q^{39} + 18 q^{40} - 14 q^{41} - 14 q^{43} - 6 q^{44} - 4 q^{45} + 28 q^{46} + 4 q^{48} - 19 q^{50} + 8 q^{51} + 29 q^{52} - 2 q^{54} + 4 q^{55} - 8 q^{57} - q^{58} - 3 q^{60} + 6 q^{61} + 38 q^{62} - 5 q^{64} + 14 q^{65} + 2 q^{66} - 4 q^{67} - 42 q^{68} - 10 q^{69} - 12 q^{71} + 3 q^{72} + 20 q^{73} + 29 q^{74} - 15 q^{75} + 11 q^{76} - q^{78} + 12 q^{79} + 41 q^{80} + 3 q^{81} + 6 q^{82} - 6 q^{83} - 6 q^{85} + 24 q^{86} + 4 q^{87} - 3 q^{88} - 26 q^{89} + 11 q^{90} + 26 q^{92} - 2 q^{93} - 35 q^{94} - 8 q^{95} - 8 q^{96} + 4 q^{97} - 3 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 4 x - 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −0.254102 −1.86081 2.11491
−1.93543 −1.00000 1.74590 −4.18953 1.93543 0 0.491797 1.00000 8.10856
1.2 1.46260 −1.00000 0.139194 −2.39821 −1.46260 0 −2.72161 1.00000 −3.50761
1.3 2.47283 −1.00000 4.11491 2.58774 −2.47283 0 5.22982 1.00000 6.39905
 $$n$$: 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.t 3
3.b odd 2 1 4851.2.a.bi 3
7.b odd 2 1 231.2.a.e 3
21.c even 2 1 693.2.a.l 3
28.d even 2 1 3696.2.a.bo 3
35.c odd 2 1 5775.2.a.bp 3
77.b even 2 1 2541.2.a.bg 3
231.h odd 2 1 7623.2.a.cd 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.a.e 3 7.b odd 2 1
693.2.a.l 3 21.c even 2 1
1617.2.a.t 3 1.a even 1 1 trivial
2541.2.a.bg 3 77.b even 2 1
3696.2.a.bo 3 28.d even 2 1
4851.2.a.bi 3 3.b odd 2 1
5775.2.a.bp 3 35.c odd 2 1
7623.2.a.cd 3 231.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1617))$$:

 $$T_{2}^{3} - 2 T_{2}^{2} - 4 T_{2} + 7$$ $$T_{5}^{3} + 4 T_{5}^{2} - 7 T_{5} - 26$$ $$T_{13}^{3} - 4 T_{13}^{2} - 27 T_{13} + 94$$ $$T_{17}^{3} + 8 T_{17}^{2} - 40 T_{17} - 328$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$7 - 4 T - 2 T^{2} + T^{3}$$
$3$ $$( 1 + T )^{3}$$
$5$ $$-26 - 7 T + 4 T^{2} + T^{3}$$
$7$ $$T^{3}$$
$11$ $$( 1 + T )^{3}$$
$13$ $$94 - 27 T - 4 T^{2} + T^{3}$$
$17$ $$-328 - 40 T + 8 T^{2} + T^{3}$$
$19$ $$-4 + 15 T - 8 T^{2} + T^{3}$$
$23$ $$64 + 12 T - 10 T^{2} + T^{3}$$
$29$ $$-94 - 27 T + 4 T^{2} + T^{3}$$
$31$ $$256 - 76 T - 2 T^{2} + T^{3}$$
$37$ $$106 - 43 T + T^{3}$$
$41$ $$-32 + 40 T + 14 T^{2} + T^{3}$$
$43$ $$-848 - 44 T + 14 T^{2} + T^{3}$$
$47$ $$-32 - 61 T + T^{3}$$
$53$ $$8 - 16 T + T^{3}$$
$59$ $$52 - 57 T + T^{3}$$
$61$ $$( -2 + T )^{3}$$
$67$ $$-236 - 85 T + 4 T^{2} + T^{3}$$
$71$ $$-256 - 16 T + 12 T^{2} + T^{3}$$
$73$ $$-134 + 101 T - 20 T^{2} + T^{3}$$
$79$ $$256 - 16 T - 12 T^{2} + T^{3}$$
$83$ $$-496 - 132 T + 6 T^{2} + T^{3}$$
$89$ $$-328 + 140 T + 26 T^{2} + T^{3}$$
$97$ $$232 - 120 T - 4 T^{2} + T^{3}$$