# Properties

 Label 690.2.m.g Level $690$ Weight $2$ Character orbit 690.m Analytic conductor $5.510$ Analytic rank $0$ Dimension $30$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$690 = 2 \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 690.m (of order $$11$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.50967773947$$ Analytic rank: $$0$$ Dimension: $$30$$ Relative dimension: $$3$$ over $$\Q(\zeta_{11})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$30q - 3q^{2} - 3q^{3} - 3q^{4} - 3q^{5} - 3q^{6} - 8q^{7} - 3q^{8} - 3q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$30q - 3q^{2} - 3q^{3} - 3q^{4} - 3q^{5} - 3q^{6} - 8q^{7} - 3q^{8} - 3q^{9} - 3q^{10} + 8q^{11} - 3q^{12} - 5q^{13} - 8q^{14} - 3q^{15} - 3q^{16} + 4q^{17} - 3q^{18} + 6q^{19} - 3q^{20} + 3q^{21} + 8q^{22} + q^{23} + 30q^{24} - 3q^{25} - 5q^{26} - 3q^{27} - 8q^{28} - 10q^{29} - 3q^{30} - 10q^{31} - 3q^{32} - 14q^{33} - 7q^{34} + 3q^{35} - 3q^{36} - 12q^{37} - 5q^{38} - 5q^{39} - 3q^{40} + 5q^{41} + 3q^{42} + 2q^{43} + 8q^{44} + 30q^{45} - 21q^{46} + 96q^{47} - 3q^{48} - 43q^{49} - 3q^{50} + 15q^{51} - 16q^{52} + 12q^{53} - 3q^{54} + 8q^{55} - 8q^{56} + 17q^{57} + q^{58} - 9q^{59} - 3q^{60} + q^{61} - 32q^{62} + 3q^{63} - 3q^{64} - 16q^{65} - 3q^{66} - 28q^{67} + 4q^{68} + 23q^{69} + 14q^{70} + 3q^{71} - 3q^{72} - 27q^{73} - 12q^{74} - 3q^{75} - 16q^{76} + 47q^{77} + 6q^{78} + 2q^{79} - 3q^{80} - 3q^{81} + 27q^{82} + 11q^{83} + 3q^{84} - 7q^{85} + 2q^{86} - 32q^{87} - 3q^{88} + 25q^{89} - 3q^{90} - 90q^{91} - 10q^{92} + 56q^{93} - 25q^{94} - 5q^{95} - 3q^{96} - 7q^{97} - 32q^{98} - 14q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1 −0.142315 + 0.989821i 0.415415 + 0.909632i −0.959493 0.281733i −0.654861 0.755750i −0.959493 + 0.281733i −2.43922 1.56759i 0.415415 0.909632i −0.654861 + 0.755750i 0.841254 0.540641i
31.2 −0.142315 + 0.989821i 0.415415 + 0.909632i −0.959493 0.281733i −0.654861 0.755750i −0.959493 + 0.281733i −1.19214 0.766143i 0.415415 0.909632i −0.654861 + 0.755750i 0.841254 0.540641i
31.3 −0.142315 + 0.989821i 0.415415 + 0.909632i −0.959493 0.281733i −0.654861 0.755750i −0.959493 + 0.281733i 4.15900 + 2.67283i 0.415415 0.909632i −0.654861 + 0.755750i 0.841254 0.540641i
121.1 0.415415 0.909632i −0.959493 0.281733i −0.654861 0.755750i 0.841254 0.540641i −0.654861 + 0.755750i −0.498068 + 3.46414i −0.959493 + 0.281733i 0.841254 + 0.540641i −0.142315 0.989821i
121.2 0.415415 0.909632i −0.959493 0.281733i −0.654861 0.755750i 0.841254 0.540641i −0.654861 + 0.755750i −0.188781 + 1.31300i −0.959493 + 0.281733i 0.841254 + 0.540641i −0.142315 0.989821i
121.3 0.415415 0.909632i −0.959493 0.281733i −0.654861 0.755750i 0.841254 0.540641i −0.654861 + 0.755750i 0.743473 5.17097i −0.959493 + 0.281733i 0.841254 + 0.540641i −0.142315 0.989821i
151.1 −0.654861 0.755750i 0.841254 + 0.540641i −0.142315 + 0.989821i 0.415415 0.909632i −0.142315 0.989821i −2.83591 0.832699i 0.841254 0.540641i 0.415415 + 0.909632i −0.959493 + 0.281733i
151.2 −0.654861 0.755750i 0.841254 + 0.540641i −0.142315 + 0.989821i 0.415415 0.909632i −0.142315 0.989821i −1.69063 0.496413i 0.841254 0.540641i 0.415415 + 0.909632i −0.959493 + 0.281733i
151.3 −0.654861 0.755750i 0.841254 + 0.540641i −0.142315 + 0.989821i 0.415415 0.909632i −0.142315 0.989821i 2.52297 + 0.740810i 0.841254 0.540641i 0.415415 + 0.909632i −0.959493 + 0.281733i
211.1 0.415415 + 0.909632i −0.959493 + 0.281733i −0.654861 + 0.755750i 0.841254 + 0.540641i −0.654861 0.755750i −0.498068 3.46414i −0.959493 0.281733i 0.841254 0.540641i −0.142315 + 0.989821i
211.2 0.415415 + 0.909632i −0.959493 + 0.281733i −0.654861 + 0.755750i 0.841254 + 0.540641i −0.654861 0.755750i −0.188781 1.31300i −0.959493 0.281733i 0.841254 0.540641i −0.142315 + 0.989821i
211.3 0.415415 + 0.909632i −0.959493 + 0.281733i −0.654861 + 0.755750i 0.841254 + 0.540641i −0.654861 0.755750i 0.743473 + 5.17097i −0.959493 0.281733i 0.841254 0.540641i −0.142315 + 0.989821i
271.1 −0.959493 0.281733i −0.654861 + 0.755750i 0.841254 + 0.540641i −0.142315 + 0.989821i 0.841254 0.540641i −1.37681 3.01480i −0.654861 0.755750i −0.142315 0.989821i 0.415415 0.909632i
271.2 −0.959493 0.281733i −0.654861 + 0.755750i 0.841254 + 0.540641i −0.142315 + 0.989821i 0.841254 0.540641i −0.117211 0.256656i −0.654861 0.755750i −0.142315 0.989821i 0.415415 0.909632i
271.3 −0.959493 0.281733i −0.654861 + 0.755750i 0.841254 + 0.540641i −0.142315 + 0.989821i 0.841254 0.540641i 1.68254 + 3.68425i −0.654861 0.755750i −0.142315 0.989821i 0.415415 0.909632i
301.1 0.841254 + 0.540641i −0.142315 0.989821i 0.415415 + 0.909632i −0.959493 0.281733i 0.415415 0.909632i −2.82773 + 3.26337i −0.142315 + 0.989821i −0.959493 + 0.281733i −0.654861 0.755750i
301.2 0.841254 + 0.540641i −0.142315 0.989821i 0.415415 + 0.909632i −0.959493 0.281733i 0.415415 0.909632i −1.70152 + 1.96366i −0.142315 + 0.989821i −0.959493 + 0.281733i −0.654861 0.755750i
301.3 0.841254 + 0.540641i −0.142315 0.989821i 0.415415 + 0.909632i −0.959493 0.281733i 0.415415 0.909632i 1.76004 2.03119i −0.142315 + 0.989821i −0.959493 + 0.281733i −0.654861 0.755750i
331.1 −0.959493 + 0.281733i −0.654861 0.755750i 0.841254 0.540641i −0.142315 0.989821i 0.841254 + 0.540641i −1.37681 + 3.01480i −0.654861 + 0.755750i −0.142315 + 0.989821i 0.415415 + 0.909632i
331.2 −0.959493 + 0.281733i −0.654861 0.755750i 0.841254 0.540641i −0.142315 0.989821i 0.841254 + 0.540641i −0.117211 + 0.256656i −0.654861 + 0.755750i −0.142315 + 0.989821i 0.415415 + 0.909632i
See all 30 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 601.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 690.2.m.g 30
23.c even 11 1 inner 690.2.m.g 30

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.m.g 30 1.a even 1 1 trivial
690.2.m.g 30 23.c even 11 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$43\!\cdots\!84$$$$T_{7}^{8} +$$$$13\!\cdots\!88$$$$T_{7}^{7} +$$$$28\!\cdots\!88$$$$T_{7}^{6} +$$$$42\!\cdots\!88$$$$T_{7}^{5} +$$$$45\!\cdots\!76$$$$T_{7}^{4} +$$$$33\!\cdots\!36$$$$T_{7}^{3} +$$$$15\!\cdots\!16$$$$T_{7}^{2} +$$$$35\!\cdots\!64$$$$T_{7} + 593713398784$$">$$T_{7}^{30} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(690, [\chi])$$.