Properties

 Label 690.2.r.a Level $690$ Weight $2$ Character orbit 690.r Analytic conductor $5.510$ Analytic rank $0$ Dimension $120$ CM no Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$690 = 2 \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 690.r (of order $$22$$, degree $$10$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$5.50967773947$$ Analytic rank: $$0$$ Dimension: $$120$$ Relative dimension: $$12$$ over $$\Q(\zeta_{22})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$120q + 12q^{4} + 12q^{6} + 12q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$120q + 12q^{4} + 12q^{6} + 12q^{9} + 18q^{10} - 8q^{14} - 4q^{15} - 12q^{16} + 22q^{20} + 14q^{21} + 120q^{24} + 52q^{25} + 16q^{29} + 8q^{31} - 36q^{34} - 90q^{35} - 12q^{36} + 22q^{39} + 4q^{40} + 16q^{41} - 4q^{49} - 4q^{50} + 8q^{51} - 12q^{54} - 56q^{55} + 8q^{56} + 138q^{59} + 4q^{60} - 36q^{61} + 12q^{64} + 52q^{65} + 96q^{70} + 8q^{71} + 8q^{74} - 4q^{75} - 60q^{79} - 12q^{81} + 8q^{84} + 24q^{85} - 8q^{86} - 104q^{89} + 4q^{90} - 144q^{91} - 24q^{94} - 14q^{95} + 12q^{96} - 44q^{99} + O(q^{100})$$

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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1 −0.989821 + 0.142315i −0.909632 0.415415i 0.959493 0.281733i −2.05037 0.892171i 0.959493 + 0.281733i −0.0262635 0.0408667i −0.909632 + 0.415415i 0.654861 + 0.755750i 2.15647 + 0.591291i
49.2 −0.989821 + 0.142315i −0.909632 0.415415i 0.959493 0.281733i −1.41393 + 1.73228i 0.959493 + 0.281733i −0.692568 1.07766i −0.909632 + 0.415415i 0.654861 + 0.755750i 1.15301 1.91587i
49.3 −0.989821 + 0.142315i −0.909632 0.415415i 0.959493 0.281733i −0.967142 2.01609i 0.959493 + 0.281733i 0.684801 + 1.06557i −0.909632 + 0.415415i 0.654861 + 0.755750i 1.24422 + 1.85793i
49.4 −0.989821 + 0.142315i −0.909632 0.415415i 0.959493 0.281733i 1.14908 1.91823i 0.959493 + 0.281733i −1.79620 2.79494i −0.909632 + 0.415415i 0.654861 + 0.755750i −0.864395 + 2.06224i
49.5 −0.989821 + 0.142315i −0.909632 0.415415i 0.959493 0.281733i 1.87948 1.21142i 0.959493 + 0.281733i 2.74068 + 4.26458i −0.909632 + 0.415415i 0.654861 + 0.755750i −1.68795 + 1.46657i
49.6 −0.989821 + 0.142315i −0.909632 0.415415i 0.959493 0.281733i 1.93207 + 1.12565i 0.959493 + 0.281733i 0.0169454 + 0.0263676i −0.909632 + 0.415415i 0.654861 + 0.755750i −2.07260 0.839235i
49.7 0.989821 0.142315i 0.909632 + 0.415415i 0.959493 0.281733i −1.51343 + 1.64607i 0.959493 + 0.281733i 0.692568 + 1.07766i 0.909632 0.415415i 0.654861 + 0.755750i −1.26376 + 1.84470i
49.8 0.989821 0.142315i 0.909632 + 0.415415i 0.959493 0.281733i −1.38916 1.75221i 0.959493 + 0.281733i −0.0169454 0.0263676i 0.909632 0.415415i 0.654861 + 0.755750i −1.62439 1.53668i
49.9 0.989821 0.142315i 0.909632 + 0.415415i 0.959493 0.281733i 0.931614 2.03276i 0.959493 + 0.281733i −2.74068 4.26458i 0.909632 0.415415i 0.654861 + 0.755750i 0.632841 2.14465i
49.10 0.989821 0.142315i 0.909632 + 0.415415i 0.959493 0.281733i 1.17489 + 1.90253i 0.959493 + 0.281733i 0.0262635 + 0.0408667i 0.909632 0.415415i 0.654861 + 0.755750i 1.43369 + 1.71597i
49.11 0.989821 0.142315i 0.909632 + 0.415415i 0.959493 0.281733i 1.73517 1.41038i 0.959493 + 0.281733i 1.79620 + 2.79494i 0.909632 0.415415i 0.654861 + 0.755750i 1.51679 1.64297i
49.12 0.989821 0.142315i 0.909632 + 0.415415i 0.959493 0.281733i 2.13321 + 0.670378i 0.959493 + 0.281733i −0.684801 1.06557i 0.909632 0.415415i 0.654861 + 0.755750i 2.20690 + 0.359967i
169.1 −0.989821 0.142315i −0.909632 + 0.415415i 0.959493 + 0.281733i −2.05037 + 0.892171i 0.959493 0.281733i −0.0262635 + 0.0408667i −0.909632 0.415415i 0.654861 0.755750i 2.15647 0.591291i
169.2 −0.989821 0.142315i −0.909632 + 0.415415i 0.959493 + 0.281733i −1.41393 1.73228i 0.959493 0.281733i −0.692568 + 1.07766i −0.909632 0.415415i 0.654861 0.755750i 1.15301 + 1.91587i
169.3 −0.989821 0.142315i −0.909632 + 0.415415i 0.959493 + 0.281733i −0.967142 + 2.01609i 0.959493 0.281733i 0.684801 1.06557i −0.909632 0.415415i 0.654861 0.755750i 1.24422 1.85793i
169.4 −0.989821 0.142315i −0.909632 + 0.415415i 0.959493 + 0.281733i 1.14908 + 1.91823i 0.959493 0.281733i −1.79620 + 2.79494i −0.909632 0.415415i 0.654861 0.755750i −0.864395 2.06224i
169.5 −0.989821 0.142315i −0.909632 + 0.415415i 0.959493 + 0.281733i 1.87948 + 1.21142i 0.959493 0.281733i 2.74068 4.26458i −0.909632 0.415415i 0.654861 0.755750i −1.68795 1.46657i
169.6 −0.989821 0.142315i −0.909632 + 0.415415i 0.959493 + 0.281733i 1.93207 1.12565i 0.959493 0.281733i 0.0169454 0.0263676i −0.909632 0.415415i 0.654861 0.755750i −2.07260 + 0.839235i
169.7 0.989821 + 0.142315i 0.909632 0.415415i 0.959493 + 0.281733i −1.51343 1.64607i 0.959493 0.281733i 0.692568 1.07766i 0.909632 + 0.415415i 0.654861 0.755750i −1.26376 1.84470i
169.8 0.989821 + 0.142315i 0.909632 0.415415i 0.959493 + 0.281733i −1.38916 + 1.75221i 0.959493 0.281733i −0.0169454 + 0.0263676i 0.909632 + 0.415415i 0.654861 0.755750i −1.62439 + 1.53668i
See next 80 embeddings (of 120 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 679.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
23.c even 11 1 inner
115.j even 22 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 690.2.r.a 120
5.b even 2 1 inner 690.2.r.a 120
23.c even 11 1 inner 690.2.r.a 120
115.j even 22 1 inner 690.2.r.a 120

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.2.r.a 120 1.a even 1 1 trivial
690.2.r.a 120 5.b even 2 1 inner
690.2.r.a 120 23.c even 11 1 inner
690.2.r.a 120 115.j even 22 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$18\!\cdots\!66$$$$T_{7}^{102} +$$$$37\!\cdots\!00$$$$T_{7}^{100} -$$$$62\!\cdots\!10$$$$T_{7}^{98} +$$$$89\!\cdots\!47$$$$T_{7}^{96} -$$$$11\!\cdots\!68$$$$T_{7}^{94} +$$$$14\!\cdots\!11$$$$T_{7}^{92} -$$$$21\!\cdots\!32$$$$T_{7}^{90} +$$$$32\!\cdots\!18$$$$T_{7}^{88} -$$$$44\!\cdots\!31$$$$T_{7}^{86} +$$$$53\!\cdots\!29$$$$T_{7}^{84} -$$$$57\!\cdots\!69$$$$T_{7}^{82} +$$$$55\!\cdots\!99$$$$T_{7}^{80} -$$$$46\!\cdots\!32$$$$T_{7}^{78} +$$$$32\!\cdots\!26$$$$T_{7}^{76} -$$$$20\!\cdots\!32$$$$T_{7}^{74} +$$$$15\!\cdots\!11$$$$T_{7}^{72} -$$$$14\!\cdots\!54$$$$T_{7}^{70} +$$$$13\!\cdots\!19$$$$T_{7}^{68} -$$$$84\!\cdots\!27$$$$T_{7}^{66} +$$$$35\!\cdots\!83$$$$T_{7}^{64} -$$$$93\!\cdots\!96$$$$T_{7}^{62} +$$$$16\!\cdots\!94$$$$T_{7}^{60} -$$$$30\!\cdots\!82$$$$T_{7}^{58} +$$$$10\!\cdots\!78$$$$T_{7}^{56} -$$$$30\!\cdots\!30$$$$T_{7}^{54} +$$$$68\!\cdots\!49$$$$T_{7}^{52} -$$$$13\!\cdots\!21$$$$T_{7}^{50} +$$$$28\!\cdots\!29$$$$T_{7}^{48} -$$$$47\!\cdots\!62$$$$T_{7}^{46} +$$$$83\!\cdots\!16$$$$T_{7}^{44} -$$$$10\!\cdots\!19$$$$T_{7}^{42} +$$$$20\!\cdots\!27$$$$T_{7}^{40} -$$$$26\!\cdots\!22$$$$T_{7}^{38} +$$$$50\!\cdots\!75$$$$T_{7}^{36} -$$$$60\!\cdots\!16$$$$T_{7}^{34} +$$$$80\!\cdots\!90$$$$T_{7}^{32} -$$$$90\!\cdots\!51$$$$T_{7}^{30} +$$$$10\!\cdots\!68$$$$T_{7}^{28} -$$$$52\!\cdots\!95$$$$T_{7}^{26} -$$$$32\!\cdots\!27$$$$T_{7}^{24} +$$$$85\!\cdots\!06$$$$T_{7}^{22} +$$$$32\!\cdots\!05$$$$T_{7}^{20} -$$$$52\!\cdots\!44$$$$T_{7}^{18} +$$$$19\!\cdots\!60$$$$T_{7}^{16} -$$$$18\!\cdots\!92$$$$T_{7}^{14} +$$$$16\!\cdots\!24$$$$T_{7}^{12} -$$$$88\!\cdots\!96$$$$T_{7}^{10} +$$$$16\!\cdots\!52$$$$T_{7}^{8} +$$$$44\!\cdots\!72$$$$T_{7}^{6} +$$$$14\!\cdots\!48$$$$T_{7}^{4} +$$$$11\!\cdots\!96$$$$T_{7}^{2} +$$$$10\!\cdots\!16$$">$$T_{7}^{120} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(690, [\chi])$$.