# Properties

 Label 1690.2.d.f Level $1690$ Weight $2$ Character orbit 1690.d Analytic conductor $13.495$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + (\beta_{3} - 1) q^{3} - q^{4} + \beta_1 q^{5} + ( - \beta_{2} + \beta_1) q^{6} - 3 \beta_1 q^{7} + \beta_1 q^{8} + ( - 2 \beta_{3} + 1) q^{9}+O(q^{10})$$ q - b1 * q^2 + (b3 - 1) * q^3 - q^4 + b1 * q^5 + (-b2 + b1) * q^6 - 3*b1 * q^7 + b1 * q^8 + (-2*b3 + 1) * q^9 $$q - \beta_1 q^{2} + (\beta_{3} - 1) q^{3} - q^{4} + \beta_1 q^{5} + ( - \beta_{2} + \beta_1) q^{6} - 3 \beta_1 q^{7} + \beta_1 q^{8} + ( - 2 \beta_{3} + 1) q^{9} + q^{10} + 3 \beta_1 q^{11} + ( - \beta_{3} + 1) q^{12} - 3 q^{14} + (\beta_{2} - \beta_1) q^{15} + q^{16} + (3 \beta_{3} + 3) q^{17} + (2 \beta_{2} - \beta_1) q^{18} + (2 \beta_{2} - 3 \beta_1) q^{19} - \beta_1 q^{20} + ( - 3 \beta_{2} + 3 \beta_1) q^{21} + 3 q^{22} + (2 \beta_{3} + 6) q^{23} + (\beta_{2} - \beta_1) q^{24} - q^{25} - 4 q^{27} + 3 \beta_1 q^{28} + (2 \beta_{3} - 6) q^{29} + (\beta_{3} - 1) q^{30} + ( - \beta_{2} - 3 \beta_1) q^{31} - \beta_1 q^{32} + (3 \beta_{2} - 3 \beta_1) q^{33} + ( - 3 \beta_{2} - 3 \beta_1) q^{34} + 3 q^{35} + (2 \beta_{3} - 1) q^{36} + (3 \beta_{2} - 6 \beta_1) q^{37} + (2 \beta_{3} - 3) q^{38} - q^{40} - 6 \beta_{2} q^{41} + ( - 3 \beta_{3} + 3) q^{42} + 2 q^{43} - 3 \beta_1 q^{44} + ( - 2 \beta_{2} + \beta_1) q^{45} + ( - 2 \beta_{2} - 6 \beta_1) q^{46} - 3 \beta_1 q^{47} + (\beta_{3} - 1) q^{48} - 2 q^{49} + \beta_1 q^{50} + 6 q^{51} + (2 \beta_{3} - 3) q^{53} + 4 \beta_1 q^{54} - 3 q^{55} + 3 q^{56} + ( - 5 \beta_{2} + 9 \beta_1) q^{57} + ( - 2 \beta_{2} + 6 \beta_1) q^{58} - 6 \beta_{2} q^{59} + ( - \beta_{2} + \beta_1) q^{60} + (3 \beta_{3} + 1) q^{61} + ( - \beta_{3} - 3) q^{62} + (6 \beta_{2} - 3 \beta_1) q^{63} - q^{64} + (3 \beta_{3} - 3) q^{66} + ( - 3 \beta_{3} - 3) q^{68} + 4 \beta_{3} q^{69} - 3 \beta_1 q^{70} + 6 \beta_1 q^{71} + ( - 2 \beta_{2} + \beta_1) q^{72} + ( - 5 \beta_{2} - 3 \beta_1) q^{73} + (3 \beta_{3} - 6) q^{74} + ( - \beta_{3} + 1) q^{75} + ( - 2 \beta_{2} + 3 \beta_1) q^{76} + 9 q^{77} + ( - 3 \beta_{3} + 1) q^{79} + \beta_1 q^{80} + (2 \beta_{3} + 1) q^{81} - 6 \beta_{3} q^{82} + (3 \beta_{2} + 3 \beta_1) q^{83} + (3 \beta_{2} - 3 \beta_1) q^{84} + (3 \beta_{2} + 3 \beta_1) q^{85} - 2 \beta_1 q^{86} + ( - 8 \beta_{3} + 12) q^{87} - 3 q^{88} + ( - 3 \beta_{2} + 12 \beta_1) q^{89} + ( - 2 \beta_{3} + 1) q^{90} + ( - 2 \beta_{3} - 6) q^{92} - 2 \beta_{2} q^{93} - 3 q^{94} + ( - 2 \beta_{3} + 3) q^{95} + ( - \beta_{2} + \beta_1) q^{96} + ( - 7 \beta_{2} + 3 \beta_1) q^{97} + 2 \beta_1 q^{98} + ( - 6 \beta_{2} + 3 \beta_1) q^{99}+O(q^{100})$$ q - b1 * q^2 + (b3 - 1) * q^3 - q^4 + b1 * q^5 + (-b2 + b1) * q^6 - 3*b1 * q^7 + b1 * q^8 + (-2*b3 + 1) * q^9 + q^10 + 3*b1 * q^11 + (-b3 + 1) * q^12 - 3 * q^14 + (b2 - b1) * q^15 + q^16 + (3*b3 + 3) * q^17 + (2*b2 - b1) * q^18 + (2*b2 - 3*b1) * q^19 - b1 * q^20 + (-3*b2 + 3*b1) * q^21 + 3 * q^22 + (2*b3 + 6) * q^23 + (b2 - b1) * q^24 - q^25 - 4 * q^27 + 3*b1 * q^28 + (2*b3 - 6) * q^29 + (b3 - 1) * q^30 + (-b2 - 3*b1) * q^31 - b1 * q^32 + (3*b2 - 3*b1) * q^33 + (-3*b2 - 3*b1) * q^34 + 3 * q^35 + (2*b3 - 1) * q^36 + (3*b2 - 6*b1) * q^37 + (2*b3 - 3) * q^38 - q^40 - 6*b2 * q^41 + (-3*b3 + 3) * q^42 + 2 * q^43 - 3*b1 * q^44 + (-2*b2 + b1) * q^45 + (-2*b2 - 6*b1) * q^46 - 3*b1 * q^47 + (b3 - 1) * q^48 - 2 * q^49 + b1 * q^50 + 6 * q^51 + (2*b3 - 3) * q^53 + 4*b1 * q^54 - 3 * q^55 + 3 * q^56 + (-5*b2 + 9*b1) * q^57 + (-2*b2 + 6*b1) * q^58 - 6*b2 * q^59 + (-b2 + b1) * q^60 + (3*b3 + 1) * q^61 + (-b3 - 3) * q^62 + (6*b2 - 3*b1) * q^63 - q^64 + (3*b3 - 3) * q^66 + (-3*b3 - 3) * q^68 + 4*b3 * q^69 - 3*b1 * q^70 + 6*b1 * q^71 + (-2*b2 + b1) * q^72 + (-5*b2 - 3*b1) * q^73 + (3*b3 - 6) * q^74 + (-b3 + 1) * q^75 + (-2*b2 + 3*b1) * q^76 + 9 * q^77 + (-3*b3 + 1) * q^79 + b1 * q^80 + (2*b3 + 1) * q^81 - 6*b3 * q^82 + (3*b2 + 3*b1) * q^83 + (3*b2 - 3*b1) * q^84 + (3*b2 + 3*b1) * q^85 - 2*b1 * q^86 + (-8*b3 + 12) * q^87 - 3 * q^88 + (-3*b2 + 12*b1) * q^89 + (-2*b3 + 1) * q^90 + (-2*b3 - 6) * q^92 - 2*b2 * q^93 - 3 * q^94 + (-2*b3 + 3) * q^95 + (-b2 + b1) * q^96 + (-7*b2 + 3*b1) * q^97 + 2*b1 * q^98 + (-6*b2 + 3*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3} - 4 q^{4} + 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^3 - 4 * q^4 + 4 * q^9 $$4 q - 4 q^{3} - 4 q^{4} + 4 q^{9} + 4 q^{10} + 4 q^{12} - 12 q^{14} + 4 q^{16} + 12 q^{17} + 12 q^{22} + 24 q^{23} - 4 q^{25} - 16 q^{27} - 24 q^{29} - 4 q^{30} + 12 q^{35} - 4 q^{36} - 12 q^{38} - 4 q^{40} + 12 q^{42} + 8 q^{43} - 4 q^{48} - 8 q^{49} + 24 q^{51} - 12 q^{53} - 12 q^{55} + 12 q^{56} + 4 q^{61} - 12 q^{62} - 4 q^{64} - 12 q^{66} - 12 q^{68} - 24 q^{74} + 4 q^{75} + 36 q^{77} + 4 q^{79} + 4 q^{81} + 48 q^{87} - 12 q^{88} + 4 q^{90} - 24 q^{92} - 12 q^{94} + 12 q^{95}+O(q^{100})$$ 4 * q - 4 * q^3 - 4 * q^4 + 4 * q^9 + 4 * q^10 + 4 * q^12 - 12 * q^14 + 4 * q^16 + 12 * q^17 + 12 * q^22 + 24 * q^23 - 4 * q^25 - 16 * q^27 - 24 * q^29 - 4 * q^30 + 12 * q^35 - 4 * q^36 - 12 * q^38 - 4 * q^40 + 12 * q^42 + 8 * q^43 - 4 * q^48 - 8 * q^49 + 24 * q^51 - 12 * q^53 - 12 * q^55 + 12 * q^56 + 4 * q^61 - 12 * q^62 - 4 * q^64 - 12 * q^66 - 12 * q^68 - 24 * q^74 + 4 * q^75 + 36 * q^77 + 4 * q^79 + 4 * q^81 + 48 * q^87 - 12 * q^88 + 4 * q^90 - 24 * q^92 - 12 * q^94 + 12 * q^95

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{3}$$ v^3 $$\beta_{2}$$ $$=$$ $$2\zeta_{12}^{2} - 1$$ 2*v^2 - 1 $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}$$ -v^3 + 2*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 2$$ (b3 + b1) / 2 $$\zeta_{12}^{2}$$ $$=$$ $$( \beta_{2} + 1 ) / 2$$ (b2 + 1) / 2 $$\zeta_{12}^{3}$$ $$=$$ $$\beta_1$$ b1

## 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
 −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i
1.00000i −2.73205 −1.00000 1.00000i 2.73205i 3.00000i 1.00000i 4.46410 1.00000
1351.2 1.00000i 0.732051 −1.00000 1.00000i 0.732051i 3.00000i 1.00000i −2.46410 1.00000
1351.3 1.00000i −2.73205 −1.00000 1.00000i 2.73205i 3.00000i 1.00000i 4.46410 1.00000
1351.4 1.00000i 0.732051 −1.00000 1.00000i 0.732051i 3.00000i 1.00000i −2.46410 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.f 4
13.b even 2 1 inner 1690.2.d.f 4
13.c even 3 1 130.2.l.a 4
13.c even 3 1 1690.2.l.g 4
13.d odd 4 1 1690.2.a.j 2
13.d odd 4 1 1690.2.a.m 2
13.e even 6 1 130.2.l.a 4
13.e even 6 1 1690.2.l.g 4
13.f odd 12 2 1690.2.e.l 4
13.f odd 12 2 1690.2.e.n 4
39.h odd 6 1 1170.2.bs.c 4
39.i odd 6 1 1170.2.bs.c 4
52.i odd 6 1 1040.2.da.a 4
52.j odd 6 1 1040.2.da.a 4
65.g odd 4 1 8450.2.a.bf 2
65.g odd 4 1 8450.2.a.bm 2
65.l even 6 1 650.2.m.a 4
65.n even 6 1 650.2.m.a 4
65.q odd 12 1 650.2.n.a 4
65.q odd 12 1 650.2.n.b 4
65.r odd 12 1 650.2.n.a 4
65.r odd 12 1 650.2.n.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.l.a 4 13.c even 3 1
130.2.l.a 4 13.e even 6 1
650.2.m.a 4 65.l even 6 1
650.2.m.a 4 65.n even 6 1
650.2.n.a 4 65.q odd 12 1
650.2.n.a 4 65.r odd 12 1
650.2.n.b 4 65.q odd 12 1
650.2.n.b 4 65.r odd 12 1
1040.2.da.a 4 52.i odd 6 1
1040.2.da.a 4 52.j odd 6 1
1170.2.bs.c 4 39.h odd 6 1
1170.2.bs.c 4 39.i odd 6 1
1690.2.a.j 2 13.d odd 4 1
1690.2.a.m 2 13.d odd 4 1
1690.2.d.f 4 1.a even 1 1 trivial
1690.2.d.f 4 13.b even 2 1 inner
1690.2.e.l 4 13.f odd 12 2
1690.2.e.n 4 13.f odd 12 2
1690.2.l.g 4 13.c even 3 1
1690.2.l.g 4 13.e even 6 1
8450.2.a.bf 2 65.g odd 4 1
8450.2.a.bm 2 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 $$S_{2}^{\mathrm{new}}(1690, [\chi])$$:

 $$T_{3}^{2} + 2T_{3} - 2$$ T3^2 + 2*T3 - 2 $$T_{7}^{2} + 9$$ T7^2 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$(T^{2} + 2 T - 2)^{2}$$
$5$ $$(T^{2} + 1)^{2}$$
$7$ $$(T^{2} + 9)^{2}$$
$11$ $$(T^{2} + 9)^{2}$$
$13$ $$T^{4}$$
$17$ $$(T^{2} - 6 T - 18)^{2}$$
$19$ $$T^{4} + 42T^{2} + 9$$
$23$ $$(T^{2} - 12 T + 24)^{2}$$
$29$ $$(T^{2} + 12 T + 24)^{2}$$
$31$ $$T^{4} + 24T^{2} + 36$$
$37$ $$T^{4} + 126T^{2} + 81$$
$41$ $$(T^{2} + 108)^{2}$$
$43$ $$(T - 2)^{4}$$
$47$ $$(T^{2} + 9)^{2}$$
$53$ $$(T^{2} + 6 T - 3)^{2}$$
$59$ $$(T^{2} + 108)^{2}$$
$61$ $$(T^{2} - 2 T - 26)^{2}$$
$67$ $$T^{4}$$
$71$ $$(T^{2} + 36)^{2}$$
$73$ $$T^{4} + 168T^{2} + 4356$$
$79$ $$(T^{2} - 2 T - 26)^{2}$$
$83$ $$T^{4} + 72T^{2} + 324$$
$89$ $$T^{4} + 342 T^{2} + 13689$$
$97$ $$T^{4} + 312 T^{2} + 19044$$