# Properties

 Label 1690.2.a.a Level $1690$ Weight $2$ Character orbit 1690.a Self dual yes Analytic conductor $13.495$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1690 = 2 \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1690.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$13.4947179416$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 130) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} - 2 q^{3} + q^{4} - q^{5} + 2 q^{6} - q^{7} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 - 2 * q^3 + q^4 - q^5 + 2 * q^6 - q^7 - q^8 + q^9 $$q - q^{2} - 2 q^{3} + q^{4} - q^{5} + 2 q^{6} - q^{7} - q^{8} + q^{9} + q^{10} + 3 q^{11} - 2 q^{12} + q^{14} + 2 q^{15} + q^{16} - 6 q^{17} - q^{18} + 5 q^{19} - q^{20} + 2 q^{21} - 3 q^{22} + 2 q^{24} + q^{25} + 4 q^{27} - q^{28} - 2 q^{30} - 4 q^{31} - q^{32} - 6 q^{33} + 6 q^{34} + q^{35} + q^{36} + 11 q^{37} - 5 q^{38} + q^{40} + 6 q^{41} - 2 q^{42} + 2 q^{43} + 3 q^{44} - q^{45} - 3 q^{47} - 2 q^{48} - 6 q^{49} - q^{50} + 12 q^{51} - 9 q^{53} - 4 q^{54} - 3 q^{55} + q^{56} - 10 q^{57} + 2 q^{60} + 8 q^{61} + 4 q^{62} - q^{63} + q^{64} + 6 q^{66} - 16 q^{67} - 6 q^{68} - q^{70} + 6 q^{71} - q^{72} + 14 q^{73} - 11 q^{74} - 2 q^{75} + 5 q^{76} - 3 q^{77} - 16 q^{79} - q^{80} - 11 q^{81} - 6 q^{82} - 6 q^{83} + 2 q^{84} + 6 q^{85} - 2 q^{86} - 3 q^{88} + 9 q^{89} + q^{90} + 8 q^{93} + 3 q^{94} - 5 q^{95} + 2 q^{96} - 10 q^{97} + 6 q^{98} + 3 q^{99}+O(q^{100})$$ q - q^2 - 2 * q^3 + q^4 - q^5 + 2 * q^6 - q^7 - q^8 + q^9 + q^10 + 3 * q^11 - 2 * q^12 + q^14 + 2 * q^15 + q^16 - 6 * q^17 - q^18 + 5 * q^19 - q^20 + 2 * q^21 - 3 * q^22 + 2 * q^24 + q^25 + 4 * q^27 - q^28 - 2 * q^30 - 4 * q^31 - q^32 - 6 * q^33 + 6 * q^34 + q^35 + q^36 + 11 * q^37 - 5 * q^38 + q^40 + 6 * q^41 - 2 * q^42 + 2 * q^43 + 3 * q^44 - q^45 - 3 * q^47 - 2 * q^48 - 6 * q^49 - q^50 + 12 * q^51 - 9 * q^53 - 4 * q^54 - 3 * q^55 + q^56 - 10 * q^57 + 2 * q^60 + 8 * q^61 + 4 * q^62 - q^63 + q^64 + 6 * q^66 - 16 * q^67 - 6 * q^68 - q^70 + 6 * q^71 - q^72 + 14 * q^73 - 11 * q^74 - 2 * q^75 + 5 * q^76 - 3 * q^77 - 16 * q^79 - q^80 - 11 * q^81 - 6 * q^82 - 6 * q^83 + 2 * q^84 + 6 * q^85 - 2 * q^86 - 3 * q^88 + 9 * q^89 + q^90 + 8 * q^93 + 3 * q^94 - 5 * q^95 + 2 * q^96 - 10 * q^97 + 6 * q^98 + 3 * 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 −2.00000 1.00000 −1.00000 2.00000 −1.00000 −1.00000 1.00000 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$5$$ $$1$$
$$13$$ $$1$$

## 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 1690.2.a.a 1
5.b even 2 1 8450.2.a.w 1
13.b even 2 1 1690.2.a.g 1
13.c even 3 2 130.2.e.b 2
13.d odd 4 2 1690.2.d.a 2
13.e even 6 2 1690.2.e.e 2
13.f odd 12 4 1690.2.l.i 4
39.i odd 6 2 1170.2.i.f 2
52.j odd 6 2 1040.2.q.c 2
65.d even 2 1 8450.2.a.k 1
65.n even 6 2 650.2.e.a 2
65.q odd 12 4 650.2.o.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.e.b 2 13.c even 3 2
650.2.e.a 2 65.n even 6 2
650.2.o.b 4 65.q odd 12 4
1040.2.q.c 2 52.j odd 6 2
1170.2.i.f 2 39.i odd 6 2
1690.2.a.a 1 1.a even 1 1 trivial
1690.2.a.g 1 13.b even 2 1
1690.2.d.a 2 13.d odd 4 2
1690.2.e.e 2 13.e even 6 2
1690.2.l.i 4 13.f odd 12 4
8450.2.a.k 1 65.d even 2 1
8450.2.a.w 1 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1690))$$:

 $$T_{3} + 2$$ T3 + 2 $$T_{7} + 1$$ T7 + 1 $$T_{11} - 3$$ T11 - 3 $$T_{31} + 4$$ T31 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T + 2$$
$5$ $$T + 1$$
$7$ $$T + 1$$
$11$ $$T - 3$$
$13$ $$T$$
$17$ $$T + 6$$
$19$ $$T - 5$$
$23$ $$T$$
$29$ $$T$$
$31$ $$T + 4$$
$37$ $$T - 11$$
$41$ $$T - 6$$
$43$ $$T - 2$$
$47$ $$T + 3$$
$53$ $$T + 9$$
$59$ $$T$$
$61$ $$T - 8$$
$67$ $$T + 16$$
$71$ $$T - 6$$
$73$ $$T - 14$$
$79$ $$T + 16$$
$83$ $$T + 6$$
$89$ $$T - 9$$
$97$ $$T + 10$$