Properties

 Label 690.2.r.b 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.

 $$\operatorname{Tr}(f)(q) =$$ $$120q + 12q^{4} + 8q^{5} - 12q^{6} + 12q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$120q + 12q^{4} + 8q^{5} - 12q^{6} + 12q^{9} - 18q^{10} - 8q^{11} - 4q^{15} - 12q^{16} + 16q^{19} + 14q^{20} - 22q^{21} - 120q^{24} + 28q^{25} + 8q^{26} + 8q^{29} + 8q^{30} + 8q^{31} + 44q^{34} + 58q^{35} - 12q^{36} + 14q^{39} - 4q^{40} + 8q^{44} - 8q^{45} + 12q^{49} - 4q^{50} + 12q^{54} + 92q^{55} - 94q^{59} + 4q^{60} - 60q^{61} + 12q^{64} - 44q^{65} - 8q^{66} + 16q^{70} - 16q^{74} + 4q^{75} - 16q^{76} + 172q^{79} + 8q^{80} - 12q^{81} - 32q^{85} - 40q^{86} + 48q^{89} - 4q^{90} + 288q^{91} + 24q^{94} - 78q^{95} - 12q^{96} - 36q^{99} + O(q^{100})$$

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 −2.09099 0.792321i −0.959493 0.281733i −0.378014 0.588202i −0.909632 + 0.415415i 0.654861 + 0.755750i 2.18246 + 0.486678i
49.2 −0.989821 + 0.142315i 0.909632 + 0.415415i 0.959493 0.281733i −1.21290 + 1.87853i −0.959493 0.281733i 0.707520 + 1.10092i −0.909632 + 0.415415i 0.654861 + 0.755750i 0.933214 2.03202i
49.3 −0.989821 + 0.142315i 0.909632 + 0.415415i 0.959493 0.281733i −0.0689747 2.23500i −0.959493 0.281733i −1.33127 2.07150i −0.909632 + 0.415415i 0.654861 + 0.755750i 0.386347 + 2.20244i
49.4 −0.989821 + 0.142315i 0.909632 + 0.415415i 0.959493 0.281733i 0.934585 + 2.03139i −0.959493 0.281733i −2.47408 3.84974i −0.909632 + 0.415415i 0.654861 + 0.755750i −1.21417 1.87771i
49.5 −0.989821 + 0.142315i 0.909632 + 0.415415i 0.959493 0.281733i 0.981376 + 2.00920i −0.959493 0.281733i 2.34067 + 3.64216i −0.909632 + 0.415415i 0.654861 + 0.755750i −1.25733 1.84909i
49.6 −0.989821 + 0.142315i 0.909632 + 0.415415i 0.959493 0.281733i 2.01587 0.967617i −0.959493 0.281733i −0.0999940 0.155594i −0.909632 + 0.415415i 0.654861 + 0.755750i −1.85764 + 1.24466i
49.7 0.989821 0.142315i −0.909632 0.415415i 0.959493 0.281733i −2.14372 0.635975i −0.959493 0.281733i 2.47408 + 3.84974i 0.909632 0.415415i 0.654861 + 0.755750i −2.21241 0.324418i
49.8 0.989821 0.142315i −0.909632 0.415415i 0.959493 0.281733i −2.12842 0.685448i −0.959493 0.281733i −2.34067 3.64216i 0.909632 0.415415i 0.654861 + 0.755750i −2.20430 0.375565i
49.9 0.989821 0.142315i −0.909632 0.415415i 0.959493 0.281733i −1.68679 + 1.46790i −0.959493 0.281733i −0.707520 1.10092i 0.909632 0.415415i 0.654861 + 0.755750i −1.46072 + 1.69301i
49.10 0.989821 0.142315i −0.909632 0.415415i 0.959493 0.281733i 0.670880 2.13305i −0.959493 0.281733i 0.0999940 + 0.155594i 0.909632 0.415415i 0.654861 + 0.755750i 0.360487 2.20682i
49.11 0.989821 0.142315i −0.909632 0.415415i 0.959493 0.281733i 1.08184 + 1.95694i −0.959493 0.281733i 0.378014 + 0.588202i 0.909632 0.415415i 0.654861 + 0.755750i 1.34933 + 1.78306i
49.12 0.989821 0.142315i −0.909632 0.415415i 0.959493 0.281733i 2.22207 0.249802i −0.959493 0.281733i 1.33127 + 2.07150i 0.909632 0.415415i 0.654861 + 0.755750i 2.16390 0.563493i
169.1 −0.989821 0.142315i 0.909632 0.415415i 0.959493 + 0.281733i −2.09099 + 0.792321i −0.959493 + 0.281733i −0.378014 + 0.588202i −0.909632 0.415415i 0.654861 0.755750i 2.18246 0.486678i
169.2 −0.989821 0.142315i 0.909632 0.415415i 0.959493 + 0.281733i −1.21290 1.87853i −0.959493 + 0.281733i 0.707520 1.10092i −0.909632 0.415415i 0.654861 0.755750i 0.933214 + 2.03202i
169.3 −0.989821 0.142315i 0.909632 0.415415i 0.959493 + 0.281733i −0.0689747 + 2.23500i −0.959493 + 0.281733i −1.33127 + 2.07150i −0.909632 0.415415i 0.654861 0.755750i 0.386347 2.20244i
169.4 −0.989821 0.142315i 0.909632 0.415415i 0.959493 + 0.281733i 0.934585 2.03139i −0.959493 + 0.281733i −2.47408 + 3.84974i −0.909632 0.415415i 0.654861 0.755750i −1.21417 + 1.87771i
169.5 −0.989821 0.142315i 0.909632 0.415415i 0.959493 + 0.281733i 0.981376 2.00920i −0.959493 + 0.281733i 2.34067 3.64216i −0.909632 0.415415i 0.654861 0.755750i −1.25733 + 1.84909i
169.6 −0.989821 0.142315i 0.909632 0.415415i 0.959493 + 0.281733i 2.01587 + 0.967617i −0.959493 + 0.281733i −0.0999940 + 0.155594i −0.909632 0.415415i 0.654861 0.755750i −1.85764 1.24466i
169.7 0.989821 + 0.142315i −0.909632 + 0.415415i 0.959493 + 0.281733i −2.14372 + 0.635975i −0.959493 + 0.281733i 2.47408 3.84974i 0.909632 + 0.415415i 0.654861 0.755750i −2.21241 + 0.324418i
169.8 0.989821 + 0.142315i −0.909632 + 0.415415i 0.959493 + 0.281733i −2.12842 + 0.685448i −0.959493 + 0.281733i −2.34067 + 3.64216i 0.909632 + 0.415415i 0.654861 0.755750i −2.20430 + 0.375565i
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.b 120
5.b even 2 1 inner 690.2.r.b 120
23.c even 11 1 inner 690.2.r.b 120
115.j even 22 1 inner 690.2.r.b 120

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$16\!\cdots\!82$$$$T_{7}^{102} +$$$$39\!\cdots\!24$$$$T_{7}^{100} -$$$$81\!\cdots\!82$$$$T_{7}^{98} +$$$$15\!\cdots\!91$$$$T_{7}^{96} -$$$$28\!\cdots\!48$$$$T_{7}^{94} +$$$$51\!\cdots\!95$$$$T_{7}^{92} -$$$$88\!\cdots\!40$$$$T_{7}^{90} +$$$$14\!\cdots\!90$$$$T_{7}^{88} -$$$$24\!\cdots\!99$$$$T_{7}^{86} +$$$$39\!\cdots\!13$$$$T_{7}^{84} -$$$$57\!\cdots\!37$$$$T_{7}^{82} +$$$$75\!\cdots\!79$$$$T_{7}^{80} -$$$$84\!\cdots\!20$$$$T_{7}^{78} +$$$$80\!\cdots\!42$$$$T_{7}^{76} -$$$$64\!\cdots\!88$$$$T_{7}^{74} +$$$$44\!\cdots\!15$$$$T_{7}^{72} -$$$$26\!\cdots\!38$$$$T_{7}^{70} +$$$$15\!\cdots\!83$$$$T_{7}^{68} -$$$$83\!\cdots\!27$$$$T_{7}^{66} +$$$$43\!\cdots\!19$$$$T_{7}^{64} -$$$$22\!\cdots\!96$$$$T_{7}^{62} +$$$$12\!\cdots\!06$$$$T_{7}^{60} -$$$$65\!\cdots\!26$$$$T_{7}^{58} +$$$$29\!\cdots\!22$$$$T_{7}^{56} -$$$$11\!\cdots\!98$$$$T_{7}^{54} +$$$$32\!\cdots\!97$$$$T_{7}^{52} -$$$$62\!\cdots\!25$$$$T_{7}^{50} +$$$$10\!\cdots\!49$$$$T_{7}^{48} -$$$$22\!\cdots\!10$$$$T_{7}^{46} +$$$$38\!\cdots\!24$$$$T_{7}^{44} -$$$$77\!\cdots\!27$$$$T_{7}^{42} +$$$$16\!\cdots\!51$$$$T_{7}^{40} -$$$$22\!\cdots\!86$$$$T_{7}^{38} +$$$$46\!\cdots\!19$$$$T_{7}^{36} -$$$$57\!\cdots\!28$$$$T_{7}^{34} +$$$$63\!\cdots\!66$$$$T_{7}^{32} -$$$$43\!\cdots\!31$$$$T_{7}^{30} +$$$$25\!\cdots\!80$$$$T_{7}^{28} -$$$$17\!\cdots\!75$$$$T_{7}^{26} +$$$$28\!\cdots\!33$$$$T_{7}^{24} -$$$$36\!\cdots\!82$$$$T_{7}^{22} +$$$$24\!\cdots\!21$$$$T_{7}^{20} -$$$$10\!\cdots\!44$$$$T_{7}^{18} +$$$$50\!\cdots\!96$$$$T_{7}^{16} -$$$$21\!\cdots\!48$$$$T_{7}^{14} +$$$$67\!\cdots\!76$$$$T_{7}^{12} +$$$$76\!\cdots\!08$$$$T_{7}^{10} +$$$$25\!\cdots\!80$$$$T_{7}^{8} -$$$$42\!\cdots\!24$$$$T_{7}^{6} +$$$$16\!\cdots\!08$$$$T_{7}^{4} +$$$$43\!\cdots\!08$$$$T_{7}^{2} +$$$$23\!\cdots\!16$$">$$T_{7}^{120} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(690, [\chi])$$.