Properties

 Label 1617.2.a.e 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

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: $$1$$ 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

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

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
−1.00000 1.00000 −1.00000 2.00000 −1.00000 0 3.00000 1.00000 −2.00000
 $$n$$: 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.e 1
3.b odd 2 1 4851.2.a.p 1
7.b odd 2 1 231.2.a.a 1
21.c even 2 1 693.2.a.d 1
28.d even 2 1 3696.2.a.t 1
35.c odd 2 1 5775.2.a.t 1
77.b even 2 1 2541.2.a.h 1
231.h odd 2 1 7623.2.a.f 1

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T - 1$$
$5$ $$T - 2$$
$7$ $$T$$
$11$ $$T + 1$$
$13$ $$T + 6$$
$17$ $$T + 2$$
$19$ $$T + 4$$
$23$ $$T$$
$29$ $$T + 2$$
$31$ $$T + 8$$
$37$ $$T - 6$$
$41$ $$T + 10$$
$43$ $$T + 4$$
$47$ $$T - 8$$
$53$ $$T - 6$$
$59$ $$T + 4$$
$61$ $$T - 10$$
$67$ $$T + 12$$
$71$ $$T$$
$73$ $$T + 2$$
$79$ $$T - 16$$
$83$ $$T + 4$$
$89$ $$T + 18$$
$97$ $$T + 2$$