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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 $4$

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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.

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

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 −2.05037 − 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:

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