# Properties

 Label 1690.2.e.l Level $1690$ Weight $2$ Character orbit 1690.e 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.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.4947179416$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ 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: $$3$$ Twist minimal: no (minimal twist has level 130) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

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

Basis of coefficient ring

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

## Character values

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

 $$n$$ $$171$$ $$677$$ $$\chi(n)$$ $$-1 + \beta_{1}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
191.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
−0.500000 + 0.866025i −0.366025 + 0.633975i −0.500000 0.866025i −1.00000 −0.366025 0.633975i 1.50000 + 2.59808i 1.00000 1.23205 + 2.13397i 0.500000 0.866025i
191.2 −0.500000 + 0.866025i 1.36603 2.36603i −0.500000 0.866025i −1.00000 1.36603 + 2.36603i 1.50000 + 2.59808i 1.00000 −2.23205 3.86603i 0.500000 0.866025i
991.1 −0.500000 0.866025i −0.366025 0.633975i −0.500000 + 0.866025i −1.00000 −0.366025 + 0.633975i 1.50000 2.59808i 1.00000 1.23205 2.13397i 0.500000 + 0.866025i
991.2 −0.500000 0.866025i 1.36603 + 2.36603i −0.500000 + 0.866025i −1.00000 1.36603 2.36603i 1.50000 2.59808i 1.00000 −2.23205 + 3.86603i 0.500000 + 0.866025i
 $$n$$: 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.c even 3 1 inner

## Twists

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

## Hecke kernels

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

 $$T_{3}^{4} - 2T_{3}^{3} + 6T_{3}^{2} + 4T_{3} + 4$$ T3^4 - 2*T3^3 + 6*T3^2 + 4*T3 + 4 $$T_{7}^{2} - 3T_{7} + 9$$ T7^2 - 3*T7 + 9 $$T_{11}^{2} + 3T_{11} + 9$$ T11^2 + 3*T11 + 9 $$T_{19}^{4} + 6T_{19}^{3} + 39T_{19}^{2} - 18T_{19} + 9$$ T19^4 + 6*T19^3 + 39*T19^2 - 18*T19 + 9 $$T_{31}^{2} - 6T_{31} + 6$$ T31^2 - 6*T31 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + T + 1)^{2}$$
$3$ $$T^{4} - 2 T^{3} + 6 T^{2} + 4 T + 4$$
$5$ $$(T + 1)^{4}$$
$7$ $$(T^{2} - 3 T + 9)^{2}$$
$11$ $$(T^{2} + 3 T + 9)^{2}$$
$13$ $$T^{4}$$
$17$ $$T^{4} - 6 T^{3} + 54 T^{2} + 108 T + 324$$
$19$ $$T^{4} + 6 T^{3} + 39 T^{2} - 18 T + 9$$
$23$ $$T^{4} - 12 T^{3} + 120 T^{2} + \cdots + 576$$
$29$ $$T^{4} - 12 T^{3} + 120 T^{2} + \cdots + 576$$
$31$ $$(T^{2} - 6 T + 6)^{2}$$
$37$ $$T^{4} - 12 T^{3} + 135 T^{2} + \cdots + 81$$
$41$ $$T^{4} + 108 T^{2} + 11664$$
$43$ $$(T^{2} - 2 T + 4)^{2}$$
$47$ $$(T + 3)^{4}$$
$53$ $$(T^{2} + 6 T - 3)^{2}$$
$59$ $$T^{4} + 108 T^{2} + 11664$$
$61$ $$T^{4} + 2 T^{3} + 30 T^{2} - 52 T + 676$$
$67$ $$T^{4}$$
$71$ $$(T^{2} - 6 T + 36)^{2}$$
$73$ $$(T^{2} + 6 T - 66)^{2}$$
$79$ $$(T^{2} - 2 T - 26)^{2}$$
$83$ $$(T^{2} + 6 T - 18)^{2}$$
$89$ $$T^{4} + 24 T^{3} + 459 T^{2} + \cdots + 13689$$
$97$ $$T^{4} - 6 T^{3} + 174 T^{2} + \cdots + 19044$$