# Properties

 Label 1690.2.l.g Level $1690$ Weight $2$ Character orbit 1690.l 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.l (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.4947179416$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - 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: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$171$$ $$677$$ $$\chi(n)$$ $$1 - \zeta_{12}^{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−0.866025 + 0.500000i 1.36603 + 2.36603i 0.500000 0.866025i 1.00000i −2.36603 1.36603i 2.59808 + 1.50000i 1.00000i −2.23205 + 3.86603i −0.500000 0.866025i
361.2 0.866025 0.500000i −0.366025 0.633975i 0.500000 0.866025i 1.00000i −0.633975 0.366025i −2.59808 1.50000i 1.00000i 1.23205 2.13397i −0.500000 0.866025i
1161.1 −0.866025 0.500000i 1.36603 2.36603i 0.500000 + 0.866025i 1.00000i −2.36603 + 1.36603i 2.59808 1.50000i 1.00000i −2.23205 3.86603i −0.500000 + 0.866025i
1161.2 0.866025 + 0.500000i −0.366025 + 0.633975i 0.500000 + 0.866025i 1.00000i −0.633975 + 0.366025i −2.59808 + 1.50000i 1.00000i 1.23205 + 2.13397i −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.e even 6 1 inner

## Twists

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.l.a 4 13.b even 2 1
130.2.l.a 4 13.c even 3 1
650.2.m.a 4 65.d even 2 1
650.2.m.a 4 65.n even 6 1
650.2.n.a 4 65.h odd 4 1
650.2.n.a 4 65.q odd 12 1
650.2.n.b 4 65.h odd 4 1
650.2.n.b 4 65.q odd 12 1
1040.2.da.a 4 52.b odd 2 1
1040.2.da.a 4 52.j odd 6 1
1170.2.bs.c 4 39.d odd 2 1
1170.2.bs.c 4 39.i odd 6 1
1690.2.a.j 2 13.f odd 12 1
1690.2.a.m 2 13.f odd 12 1
1690.2.d.f 4 13.c even 3 1
1690.2.d.f 4 13.e even 6 1
1690.2.e.l 4 13.d odd 4 1
1690.2.e.l 4 13.f odd 12 1
1690.2.e.n 4 13.d odd 4 1
1690.2.e.n 4 13.f odd 12 1
1690.2.l.g 4 1.a even 1 1 trivial
1690.2.l.g 4 13.e even 6 1 inner
8450.2.a.bf 2 65.s odd 12 1
8450.2.a.bm 2 65.s odd 12 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 $$T_{7}^{4} - 9T_{7}^{2} + 81$$ T7^4 - 9*T7^2 + 81

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{2} + 1$$
$3$ $$T^{4} - 2 T^{3} + 6 T^{2} + 4 T + 4$$
$5$ $$(T^{2} + 1)^{2}$$
$7$ $$T^{4} - 9T^{2} + 81$$
$11$ $$T^{4} - 9T^{2} + 81$$
$13$ $$T^{4}$$
$17$ $$T^{4} + 6 T^{3} + 54 T^{2} - 108 T + 324$$
$19$ $$T^{4} - 12 T^{3} + 51 T^{2} - 36 T + 9$$
$23$ $$T^{4} + 12 T^{3} + 120 T^{2} + \cdots + 576$$
$29$ $$T^{4} - 12 T^{3} + 120 T^{2} + \cdots + 576$$
$31$ $$T^{4} + 24T^{2} + 36$$
$37$ $$T^{4} + 18 T^{3} + 99 T^{2} - 162 T + 81$$
$41$ $$(T^{2} - 18 T + 108)^{2}$$
$43$ $$(T^{2} + 2 T + 4)^{2}$$
$47$ $$(T^{2} + 9)^{2}$$
$53$ $$(T^{2} + 6 T - 3)^{2}$$
$59$ $$(T^{2} + 18 T + 108)^{2}$$
$61$ $$T^{4} + 2 T^{3} + 30 T^{2} - 52 T + 676$$
$67$ $$T^{4}$$
$71$ $$T^{4} - 36T^{2} + 1296$$
$73$ $$T^{4} + 168T^{2} + 4356$$
$79$ $$(T^{2} - 2 T - 26)^{2}$$
$83$ $$T^{4} + 72T^{2} + 324$$
$89$ $$T^{4} - 18 T^{3} - 9 T^{2} + \cdots + 13689$$
$97$ $$T^{4} + 42 T^{3} + 726 T^{2} + \cdots + 19044$$