# Properties

 Label 1690.2.d.a Level $1690$ Weight $2$ Character orbit 1690.d Analytic conductor $13.495$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1690 = 2 \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1690.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.4947179416$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 130) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1690\mathbb{Z}\right)^\times$$.

 $$n$$ $$171$$ $$677$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1351.1
 − 1.00000i 1.00000i
1.00000i −2.00000 −1.00000 1.00000i 2.00000i 1.00000i 1.00000i 1.00000 −1.00000
1351.2 1.00000i −2.00000 −1.00000 1.00000i 2.00000i 1.00000i 1.00000i 1.00000 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{3} + 2$$ T3 + 2 $$T_{7}^{2} + 1$$ T7^2 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$(T + 2)^{2}$$
$5$ $$T^{2} + 1$$
$7$ $$T^{2} + 1$$
$11$ $$T^{2} + 9$$
$13$ $$T^{2}$$
$17$ $$(T - 6)^{2}$$
$19$ $$T^{2} + 25$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2} + 16$$
$37$ $$T^{2} + 121$$
$41$ $$T^{2} + 36$$
$43$ $$(T + 2)^{2}$$
$47$ $$T^{2} + 9$$
$53$ $$(T + 9)^{2}$$
$59$ $$T^{2}$$
$61$ $$(T - 8)^{2}$$
$67$ $$T^{2} + 256$$
$71$ $$T^{2} + 36$$
$73$ $$T^{2} + 196$$
$79$ $$(T + 16)^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$T^{2} + 81$$
$97$ $$T^{2} + 100$$