# Properties

 Label 1617.2.a.j Level $1617$ Weight $2$ Character orbit 1617.a Self dual yes Analytic conductor $12.912$ Analytic rank $0$ 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) 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 $$q + q^{2} + q^{3} - q^{4} + 2 q^{5} + q^{6} - 3 q^{8} + q^{9} + 2 q^{10} + q^{11} - q^{12} + 2 q^{13} + 2 q^{15} - q^{16} + 2 q^{17} + q^{18} - 2 q^{20} + q^{22} + 8 q^{23} - 3 q^{24} - q^{25} + 2 q^{26} + q^{27} - 6 q^{29} + 2 q^{30} + 8 q^{31} + 5 q^{32} + q^{33} + 2 q^{34} - q^{36} + 6 q^{37} + 2 q^{39} - 6 q^{40} + 2 q^{41} - q^{44} + 2 q^{45} + 8 q^{46} - 8 q^{47} - q^{48} - q^{50} + 2 q^{51} - 2 q^{52} + 6 q^{53} + q^{54} + 2 q^{55} - 6 q^{58} + 4 q^{59} - 2 q^{60} - 6 q^{61} + 8 q^{62} + 7 q^{64} + 4 q^{65} + q^{66} - 4 q^{67} - 2 q^{68} + 8 q^{69} - 3 q^{72} + 14 q^{73} + 6 q^{74} - q^{75} + 2 q^{78} - 4 q^{79} - 2 q^{80} + q^{81} + 2 q^{82} - 12 q^{83} + 4 q^{85} - 6 q^{87} - 3 q^{88} + 6 q^{89} + 2 q^{90} - 8 q^{92} + 8 q^{93} - 8 q^{94} + 5 q^{96} - 2 q^{97} + q^{99}+O(q^{100})$$ q + q^2 + q^3 - q^4 + 2 * q^5 + q^6 - 3 * q^8 + q^9 + 2 * q^10 + q^11 - q^12 + 2 * q^13 + 2 * q^15 - q^16 + 2 * q^17 + q^18 - 2 * q^20 + q^22 + 8 * q^23 - 3 * q^24 - q^25 + 2 * q^26 + q^27 - 6 * q^29 + 2 * q^30 + 8 * q^31 + 5 * q^32 + q^33 + 2 * q^34 - q^36 + 6 * q^37 + 2 * q^39 - 6 * q^40 + 2 * q^41 - q^44 + 2 * q^45 + 8 * q^46 - 8 * q^47 - q^48 - q^50 + 2 * q^51 - 2 * q^52 + 6 * q^53 + q^54 + 2 * q^55 - 6 * q^58 + 4 * q^59 - 2 * q^60 - 6 * q^61 + 8 * q^62 + 7 * q^64 + 4 * q^65 + q^66 - 4 * q^67 - 2 * q^68 + 8 * q^69 - 3 * q^72 + 14 * q^73 + 6 * q^74 - q^75 + 2 * q^78 - 4 * q^79 - 2 * q^80 + q^81 + 2 * q^82 - 12 * q^83 + 4 * q^85 - 6 * q^87 - 3 * q^88 + 6 * q^89 + 2 * q^90 - 8 * q^92 + 8 * q^93 - 8 * q^94 + 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.j 1
3.b odd 2 1 4851.2.a.b 1
7.b odd 2 1 33.2.a.a 1
21.c even 2 1 99.2.a.b 1
28.d even 2 1 528.2.a.g 1
35.c odd 2 1 825.2.a.a 1
35.f even 4 2 825.2.c.a 2
56.e even 2 1 2112.2.a.j 1
56.h odd 2 1 2112.2.a.bb 1
63.l odd 6 2 891.2.e.e 2
63.o even 6 2 891.2.e.g 2
77.b even 2 1 363.2.a.b 1
77.j odd 10 4 363.2.e.e 4
77.l even 10 4 363.2.e.g 4
84.h odd 2 1 1584.2.a.o 1
91.b odd 2 1 5577.2.a.a 1
105.g even 2 1 2475.2.a.g 1
105.k odd 4 2 2475.2.c.d 2
119.d odd 2 1 9537.2.a.m 1
168.e odd 2 1 6336.2.a.n 1
168.i even 2 1 6336.2.a.x 1
231.h odd 2 1 1089.2.a.j 1
308.g odd 2 1 5808.2.a.t 1
385.h even 2 1 9075.2.a.q 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.a.a 1 7.b odd 2 1
99.2.a.b 1 21.c even 2 1
363.2.a.b 1 77.b even 2 1
363.2.e.e 4 77.j odd 10 4
363.2.e.g 4 77.l even 10 4
528.2.a.g 1 28.d even 2 1
825.2.a.a 1 35.c odd 2 1
825.2.c.a 2 35.f even 4 2
891.2.e.e 2 63.l odd 6 2
891.2.e.g 2 63.o even 6 2
1089.2.a.j 1 231.h odd 2 1
1584.2.a.o 1 84.h odd 2 1
1617.2.a.j 1 1.a even 1 1 trivial
2112.2.a.j 1 56.e even 2 1
2112.2.a.bb 1 56.h odd 2 1
2475.2.a.g 1 105.g even 2 1
2475.2.c.d 2 105.k odd 4 2
4851.2.a.b 1 3.b odd 2 1
5577.2.a.a 1 91.b odd 2 1
5808.2.a.t 1 308.g odd 2 1
6336.2.a.n 1 168.e odd 2 1
6336.2.a.x 1 168.i even 2 1
9075.2.a.q 1 385.h even 2 1
9537.2.a.m 1 119.d 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 $$T_{13} - 2$$ T13 - 2 $$T_{17} - 2$$ T17 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T - 1$$
$5$ $$T - 2$$
$7$ $$T$$
$11$ $$T - 1$$
$13$ $$T - 2$$
$17$ $$T - 2$$
$19$ $$T$$
$23$ $$T - 8$$
$29$ $$T + 6$$
$31$ $$T - 8$$
$37$ $$T - 6$$
$41$ $$T - 2$$
$43$ $$T$$
$47$ $$T + 8$$
$53$ $$T - 6$$
$59$ $$T - 4$$
$61$ $$T + 6$$
$67$ $$T + 4$$
$71$ $$T$$
$73$ $$T - 14$$
$79$ $$T + 4$$
$83$ $$T + 12$$
$89$ $$T - 6$$
$97$ $$T + 2$$