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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 $4$

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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.

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

Display 100 coefficients 10 coefficients

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:

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 \(S_{2}^{\mathrm{new}}(690, [\chi])\).