# Properties

 Label 690.2.m.h 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.

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

## 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 −3.23498 2.07900i −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 0.208237 + 0.133826i −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 2.49910 + 1.60607i −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 −0.588355 + 4.09210i 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 −0.0949893 + 0.660665i 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 0.626720 4.35894i 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 −4.28731 1.25887i −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 1.94609 + 0.571424i −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 4.34479 + 1.27575i −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 −0.588355 4.09210i 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 −0.0949893 0.660665i 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 0.626720 + 4.35894i 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 −1.89055 4.13973i 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 0.0700250 + 0.153333i 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 1.63201 + 3.57361i 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 −1.50479 + 1.73662i 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 1.32702 1.53147i 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 2.94698 3.40099i 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 −1.89055 + 4.13973i 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 0.0700250 0.153333i 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.h 30
23.c even 11 1 inner 690.2.m.h 30

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$19\!\cdots\!21$$$$T_{7}^{10} -$$$$48\!\cdots\!72$$$$T_{7}^{9} +$$$$12\!\cdots\!76$$$$T_{7}^{8} -$$$$20\!\cdots\!00$$$$T_{7}^{7} +$$$$21\!\cdots\!12$$$$T_{7}^{6} -$$$$13\!\cdots\!80$$$$T_{7}^{5} +$$$$92\!\cdots\!36$$$$T_{7}^{4} -$$$$34\!\cdots\!84$$$$T_{7}^{3} + 790923217920 T_{7}^{2} - 101206088192 T_{7} + 7930971136$$">$$T_{7}^{30} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(690, [\chi])$$.