# Properties

 Label 1617.2.a.r 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: $$x^{3} - 4x - 1$$ x^3 - 4*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

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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{2} + \beta_1 + 1) q^{5} + \beta_1 q^{6} + q^{8} + q^{9}+O(q^{10})$$ q + b1 * q^2 + q^3 + (b2 + 1) * q^4 + (-b2 + b1 + 1) * q^5 + b1 * q^6 + q^8 + q^9 $$q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{2} + \beta_1 + 1) q^{5} + \beta_1 q^{6} + q^{8} + q^{9} + (\beta_{2} + 2) q^{10} + q^{11} + (\beta_{2} + 1) q^{12} + (\beta_{2} + \beta_1 + 1) q^{13} + ( - \beta_{2} + \beta_1 + 1) q^{15} + ( - 2 \beta_{2} + \beta_1 - 2) q^{16} + (2 \beta_{2} - 2 \beta_1 + 2) q^{17} + \beta_1 q^{18} + ( - \beta_{2} + \beta_1 + 1) q^{19} + (2 \beta_{2} + \beta_1 - 1) q^{20} + \beta_1 q^{22} + q^{24} + ( - 3 \beta_{2} + \beta_1) q^{25} + (\beta_{2} + 2 \beta_1 + 4) q^{26} + q^{27} + ( - 3 \beta_{2} - \beta_1 + 1) q^{29} + (\beta_{2} + 2) q^{30} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{31} + (\beta_{2} - 4 \beta_1 - 1) q^{32} + q^{33} + ( - 2 \beta_{2} + 4 \beta_1 - 4) q^{34} + (\beta_{2} + 1) q^{36} + ( - \beta_{2} - \beta_1 - 3) q^{37} + (\beta_{2} + 2) q^{38} + (\beta_{2} + \beta_1 + 1) q^{39} + ( - \beta_{2} + \beta_1 + 1) q^{40} + (2 \beta_{2} + 2 \beta_1 - 2) q^{41} + (2 \beta_{2} + 2 \beta_1 - 2) q^{43} + (\beta_{2} + 1) q^{44} + ( - \beta_{2} + \beta_1 + 1) q^{45} + ( - \beta_{2} + 3 \beta_1 + 3) q^{47} + ( - 2 \beta_{2} + \beta_1 - 2) q^{48} + (\beta_{2} - 3 \beta_1) q^{50} + (2 \beta_{2} - 2 \beta_1 + 2) q^{51} + (3 \beta_1 + 5) q^{52} + (2 \beta_{2} - 2 \beta_1 + 4) q^{53} + \beta_1 q^{54} + ( - \beta_{2} + \beta_1 + 1) q^{55} + ( - \beta_{2} + \beta_1 + 1) q^{57} + ( - \beta_{2} - 2 \beta_1 - 6) q^{58} + (3 \beta_{2} - 5 \beta_1 + 3) q^{59} + (2 \beta_{2} + \beta_1 - 1) q^{60} + (2 \beta_{2} + 2 \beta_1 + 6) q^{61} + ( - 2 \beta_{2} - 8) q^{62} + ( - 2 \beta_1 - 7) q^{64} + (3 \beta_{2} + \beta_1 + 1) q^{65} + \beta_1 q^{66} + ( - 5 \beta_{2} + 5 \beta_1 + 1) q^{67} + ( - 2 \beta_1 + 6) q^{68} + (6 \beta_{2} - 6 \beta_1 + 2) q^{71} + q^{72} + ( - 5 \beta_{2} - \beta_1 - 1) q^{73} + ( - \beta_{2} - 4 \beta_1 - 4) q^{74} + ( - 3 \beta_{2} + \beta_1) q^{75} + (2 \beta_{2} + \beta_1 - 1) q^{76} + (\beta_{2} + 2 \beta_1 + 4) q^{78} + ( - 4 \beta_1 - 4) q^{79} + ( - 3 \beta_{2} - 2 \beta_1 + 4) q^{80} + q^{81} + (2 \beta_{2} + 8) q^{82} + ( - 2 \beta_{2} - 2 \beta_1 - 2) q^{83} + (2 \beta_{2} + 2 \beta_1 - 6) q^{85} + (2 \beta_{2} + 8) q^{86} + ( - 3 \beta_{2} - \beta_1 + 1) q^{87} + q^{88} + (2 \beta_{2} - 2 \beta_1 + 6) q^{89} + (\beta_{2} + 2) q^{90} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{93} + (3 \beta_{2} + 2 \beta_1 + 8) q^{94} + ( - 3 \beta_{2} + \beta_1 + 5) q^{95} + (\beta_{2} - 4 \beta_1 - 1) q^{96} + ( - 2 \beta_{2} - 6 \beta_1 + 2) q^{97} + q^{99}+O(q^{100})$$ q + b1 * q^2 + q^3 + (b2 + 1) * q^4 + (-b2 + b1 + 1) * q^5 + b1 * q^6 + q^8 + q^9 + (b2 + 2) * q^10 + q^11 + (b2 + 1) * q^12 + (b2 + b1 + 1) * q^13 + (-b2 + b1 + 1) * q^15 + (-2*b2 + b1 - 2) * q^16 + (2*b2 - 2*b1 + 2) * q^17 + b1 * q^18 + (-b2 + b1 + 1) * q^19 + (2*b2 + b1 - 1) * q^20 + b1 * q^22 + q^24 + (-3*b2 + b1) * q^25 + (b2 + 2*b1 + 4) * q^26 + q^27 + (-3*b2 - b1 + 1) * q^29 + (b2 + 2) * q^30 + (-2*b2 - 2*b1 + 2) * q^31 + (b2 - 4*b1 - 1) * q^32 + q^33 + (-2*b2 + 4*b1 - 4) * q^34 + (b2 + 1) * q^36 + (-b2 - b1 - 3) * q^37 + (b2 + 2) * q^38 + (b2 + b1 + 1) * q^39 + (-b2 + b1 + 1) * q^40 + (2*b2 + 2*b1 - 2) * q^41 + (2*b2 + 2*b1 - 2) * q^43 + (b2 + 1) * q^44 + (-b2 + b1 + 1) * q^45 + (-b2 + 3*b1 + 3) * q^47 + (-2*b2 + b1 - 2) * q^48 + (b2 - 3*b1) * q^50 + (2*b2 - 2*b1 + 2) * q^51 + (3*b1 + 5) * q^52 + (2*b2 - 2*b1 + 4) * q^53 + b1 * q^54 + (-b2 + b1 + 1) * q^55 + (-b2 + b1 + 1) * q^57 + (-b2 - 2*b1 - 6) * q^58 + (3*b2 - 5*b1 + 3) * q^59 + (2*b2 + b1 - 1) * q^60 + (2*b2 + 2*b1 + 6) * q^61 + (-2*b2 - 8) * q^62 + (-2*b1 - 7) * q^64 + (3*b2 + b1 + 1) * q^65 + b1 * q^66 + (-5*b2 + 5*b1 + 1) * q^67 + (-2*b1 + 6) * q^68 + (6*b2 - 6*b1 + 2) * q^71 + q^72 + (-5*b2 - b1 - 1) * q^73 + (-b2 - 4*b1 - 4) * q^74 + (-3*b2 + b1) * q^75 + (2*b2 + b1 - 1) * q^76 + (b2 + 2*b1 + 4) * q^78 + (-4*b1 - 4) * q^79 + (-3*b2 - 2*b1 + 4) * q^80 + q^81 + (2*b2 + 8) * q^82 + (-2*b2 - 2*b1 - 2) * q^83 + (2*b2 + 2*b1 - 6) * q^85 + (2*b2 + 8) * q^86 + (-3*b2 - b1 + 1) * q^87 + q^88 + (2*b2 - 2*b1 + 6) * q^89 + (b2 + 2) * q^90 + (-2*b2 - 2*b1 + 2) * q^93 + (3*b2 + 2*b1 + 8) * q^94 + (-3*b2 + b1 + 5) * q^95 + (b2 - 4*b1 - 1) * q^96 + (-2*b2 - 6*b1 + 2) * q^97 + q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{3} + 2 q^{4} + 4 q^{5} + 3 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^3 + 2 * q^4 + 4 * q^5 + 3 * q^8 + 3 * q^9 $$3 q + 3 q^{3} + 2 q^{4} + 4 q^{5} + 3 q^{8} + 3 q^{9} + 5 q^{10} + 3 q^{11} + 2 q^{12} + 2 q^{13} + 4 q^{15} - 4 q^{16} + 4 q^{17} + 4 q^{19} - 5 q^{20} + 3 q^{24} + 3 q^{25} + 11 q^{26} + 3 q^{27} + 6 q^{29} + 5 q^{30} + 8 q^{31} - 4 q^{32} + 3 q^{33} - 10 q^{34} + 2 q^{36} - 8 q^{37} + 5 q^{38} + 2 q^{39} + 4 q^{40} - 8 q^{41} - 8 q^{43} + 2 q^{44} + 4 q^{45} + 10 q^{47} - 4 q^{48} - q^{50} + 4 q^{51} + 15 q^{52} + 10 q^{53} + 4 q^{55} + 4 q^{57} - 17 q^{58} + 6 q^{59} - 5 q^{60} + 16 q^{61} - 22 q^{62} - 21 q^{64} + 8 q^{67} + 18 q^{68} + 3 q^{72} + 2 q^{73} - 11 q^{74} + 3 q^{75} - 5 q^{76} + 11 q^{78} - 12 q^{79} + 15 q^{80} + 3 q^{81} + 22 q^{82} - 4 q^{83} - 20 q^{85} + 22 q^{86} + 6 q^{87} + 3 q^{88} + 16 q^{89} + 5 q^{90} + 8 q^{93} + 21 q^{94} + 18 q^{95} - 4 q^{96} + 8 q^{97} + 3 q^{99}+O(q^{100})$$ 3 * q + 3 * q^3 + 2 * q^4 + 4 * q^5 + 3 * q^8 + 3 * q^9 + 5 * q^10 + 3 * q^11 + 2 * q^12 + 2 * q^13 + 4 * q^15 - 4 * q^16 + 4 * q^17 + 4 * q^19 - 5 * q^20 + 3 * q^24 + 3 * q^25 + 11 * q^26 + 3 * q^27 + 6 * q^29 + 5 * q^30 + 8 * q^31 - 4 * q^32 + 3 * q^33 - 10 * q^34 + 2 * q^36 - 8 * q^37 + 5 * q^38 + 2 * q^39 + 4 * q^40 - 8 * q^41 - 8 * q^43 + 2 * q^44 + 4 * q^45 + 10 * q^47 - 4 * q^48 - q^50 + 4 * q^51 + 15 * q^52 + 10 * q^53 + 4 * q^55 + 4 * q^57 - 17 * q^58 + 6 * q^59 - 5 * q^60 + 16 * q^61 - 22 * q^62 - 21 * q^64 + 8 * q^67 + 18 * q^68 + 3 * q^72 + 2 * q^73 - 11 * q^74 + 3 * q^75 - 5 * q^76 + 11 * q^78 - 12 * q^79 + 15 * q^80 + 3 * q^81 + 22 * q^82 - 4 * q^83 - 20 * q^85 + 22 * q^86 + 6 * q^87 + 3 * q^88 + 16 * q^89 + 5 * q^90 + 8 * q^93 + 21 * q^94 + 18 * q^95 - 4 * q^96 + 8 * q^97 + 3 * q^99

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 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
 −1.86081 −0.254102 2.11491
−1.86081 1.00000 1.46260 −1.32340 −1.86081 0 1.00000 1.00000 2.46260
1.2 −0.254102 1.00000 −1.93543 3.68133 −0.254102 0 1.00000 1.00000 −0.935432
1.3 2.11491 1.00000 2.47283 1.64207 2.11491 0 1.00000 1.00000 3.47283
 $$n$$: 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.r yes 3
3.b odd 2 1 4851.2.a.bl 3
7.b odd 2 1 1617.2.a.q 3
21.c even 2 1 4851.2.a.bm 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1617.2.a.q 3 7.b odd 2 1
1617.2.a.r yes 3 1.a even 1 1 trivial
4851.2.a.bl 3 3.b odd 2 1
4851.2.a.bm 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1617))$$:

 $$T_{2}^{3} - 4T_{2} - 1$$ T2^3 - 4*T2 - 1 $$T_{5}^{3} - 4T_{5}^{2} - T_{5} + 8$$ T5^3 - 4*T5^2 - T5 + 8 $$T_{13}^{3} - 2T_{13}^{2} - 11T_{13} - 4$$ T13^3 - 2*T13^2 - 11*T13 - 4 $$T_{17}^{3} - 4T_{17}^{2} - 20T_{17} + 16$$ T17^3 - 4*T17^2 - 20*T17 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - 4T - 1$$
$3$ $$(T - 1)^{3}$$
$5$ $$T^{3} - 4T^{2} - T + 8$$
$7$ $$T^{3}$$
$11$ $$(T - 1)^{3}$$
$13$ $$T^{3} - 2 T^{2} - 11 T - 4$$
$17$ $$T^{3} - 4 T^{2} - 20 T + 16$$
$19$ $$T^{3} - 4T^{2} - T + 8$$
$23$ $$T^{3}$$
$29$ $$T^{3} - 6 T^{2} - 49 T + 82$$
$31$ $$T^{3} - 8 T^{2} - 28 T + 208$$
$37$ $$T^{3} + 8 T^{2} + 9 T - 2$$
$41$ $$T^{3} + 8 T^{2} - 28 T - 208$$
$43$ $$T^{3} + 8 T^{2} - 28 T - 208$$
$47$ $$T^{3} - 10 T^{2} + T + 124$$
$53$ $$T^{3} - 10 T^{2} + 8 T + 32$$
$59$ $$T^{3} - 6 T^{2} - 91 T - 196$$
$61$ $$T^{3} - 16 T^{2} + 36 T + 16$$
$67$ $$T^{3} - 8 T^{2} - 137 T + 644$$
$71$ $$T^{3} - 228T - 416$$
$73$ $$T^{3} - 2 T^{2} - 151 T - 212$$
$79$ $$T^{3} + 12 T^{2} - 16 T - 128$$
$83$ $$T^{3} + 4 T^{2} - 44 T + 32$$
$89$ $$T^{3} - 16 T^{2} + 60 T - 32$$
$97$ $$T^{3} - 8 T^{2} - 180 T + 1568$$