Properties

Label 667.2.s.a
Level $667$
Weight $2$
Character orbit 667.s
Analytic conductor $5.326$
Analytic rank $0$
Dimension $3480$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 667 = 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 667.s (of order \(77\), degree \(60\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.32602181482\)
Analytic rank: \(0\)
Dimension: \(3480\)
Relative dimension: \(58\) over \(\Q(\zeta_{77})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{77}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 3480q - 49q^{2} - 45q^{3} + 7q^{4} - 49q^{5} - 39q^{6} - 49q^{7} - 50q^{8} + q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 3480q - 49q^{2} - 45q^{3} + 7q^{4} - 49q^{5} - 39q^{6} - 49q^{7} - 50q^{8} + q^{9} - 53q^{10} - 25q^{11} - 134q^{12} - 25q^{13} - 65q^{14} - 65q^{15} - 111q^{16} - 112q^{17} - 91q^{18} - 35q^{19} - 81q^{20} - 7q^{21} - 78q^{22} + 2q^{23} + 72q^{24} - 93q^{25} - 43q^{26} + 15q^{27} - 156q^{28} - 63q^{29} - 140q^{30} - 59q^{31} + 41q^{32} - 13q^{33} - 97q^{34} - 9q^{35} + 153q^{36} - 67q^{37} + q^{38} + 9q^{39} - 121q^{40} - 132q^{41} - 137q^{42} + 15q^{43} + 95q^{44} - 36q^{45} + 44q^{46} - 120q^{47} + 222q^{48} - 233q^{49} + 57q^{50} - 231q^{51} + 51q^{52} - 89q^{53} + 4q^{54} - 241q^{55} + 155q^{56} - 14q^{57} - 12q^{58} - 164q^{59} - 59q^{60} - 29q^{61} + 139q^{62} + 25q^{63} + 76q^{64} - 89q^{65} + 169q^{66} - 49q^{67} - 104q^{68} - 48q^{69} - 180q^{70} - 31q^{71} + 35q^{72} - 57q^{73} - 478q^{74} - 212q^{75} - 69q^{76} - 266q^{77} + 97q^{78} + 35q^{79} - 39q^{80} - 95q^{81} - 209q^{82} - 11q^{83} + 147q^{84} + 137q^{85} - 600q^{86} + 72q^{87} + 40q^{88} + 61q^{89} - 31q^{90} + 48q^{91} - 43q^{92} - 764q^{93} - 15q^{94} + 55q^{95} - 70q^{96} + 43q^{97} + 117q^{98} + 88q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −2.56954 + 1.11070i 0.106208 1.03764i 3.99861 4.25113i −0.0480349 + 2.35433i 0.879600 + 2.78423i −1.35731 + 3.32469i −3.64997 + 10.0994i 1.87237 + 0.387351i −2.49152 6.10290i
16.2 −2.56467 + 1.10859i −0.235859 + 2.30432i 3.97829 4.22953i −0.0423129 + 2.07388i −1.94965 6.17130i 0.883000 2.16289i −3.61488 + 10.0023i −2.31648 0.479227i −2.19056 5.36572i
16.3 −2.50540 + 1.08297i 0.310483 3.03340i 3.73393 3.96973i 0.0136512 0.669086i 2.50719 + 7.93612i 1.55184 3.80118i −3.20047 + 8.85561i −6.16731 1.27588i 0.690398 + 1.69111i
16.4 −2.29489 + 0.991977i −0.162496 + 1.58757i 2.91223 3.09614i 0.0749307 3.67257i −1.20193 3.80450i 0.954585 2.33823i −1.91243 + 5.29164i 0.443809 + 0.0918140i 3.47115 + 8.50248i
16.5 −2.21235 + 0.956296i 0.00599793 0.0585993i 2.60969 2.77449i 0.0144875 0.710072i 0.0427688 + 0.135378i −0.132450 + 0.324432i −1.48190 + 4.10038i 2.93439 + 0.607060i 0.646988 + 1.58478i
16.6 −2.12539 + 0.918708i 0.0467407 0.456653i 2.30296 2.44840i 0.0386222 1.89299i 0.320189 + 1.01351i 1.08088 2.64759i −1.07133 + 2.96435i 2.73145 + 0.565074i 1.65702 + 4.05882i
16.7 −2.10118 + 0.908245i 0.241745 2.36183i 2.21977 2.35996i 0.00893255 0.437810i 1.63717 + 5.18221i −1.25928 + 3.08457i −0.964662 + 2.66920i −2.58203 0.534163i 0.378870 + 0.928032i
16.8 −2.01330 + 0.870257i 0.169590 1.65688i 1.92574 2.04736i −0.0877619 + 4.30147i 1.10048 + 3.48338i 0.176453 0.432216i −0.604387 + 1.67232i 0.221308 + 0.0457837i −3.56669 8.73651i
16.9 −1.98648 + 0.858663i 0.240879 2.35336i 1.83850 1.95461i 0.0694953 3.40617i 1.54225 + 4.88173i −0.823769 + 2.01780i −0.502672 + 1.39088i −2.54251 0.525987i 2.78670 + 6.82594i
16.10 −1.94761 + 0.841863i −0.0985846 + 0.963163i 1.71417 1.82242i −0.0626912 + 3.07268i −0.618847 1.95886i 0.801093 1.96226i −0.361971 + 1.00156i 2.01983 + 0.417857i −2.46468 6.03716i
16.11 −1.93014 + 0.834310i −0.255525 + 2.49645i 1.65907 1.76385i −0.0578815 + 2.83694i −1.58962 5.03169i −0.855892 + 2.09648i −0.301249 + 0.833550i −3.22920 0.668048i −2.25517 5.52398i
16.12 −1.66649 + 0.720349i −0.320321 + 3.12952i 0.888013 0.944093i 0.0140387 0.688078i −1.72053 5.44606i 0.237956 0.582866i 0.434358 1.20186i −6.75347 1.39714i 0.472261 + 1.15679i
16.13 −1.53669 + 0.664239i −0.0152959 + 0.149440i 0.549907 0.584635i −0.00415421 + 0.203610i −0.0757589 0.239803i −0.850325 + 2.08285i 0.681322 1.88520i 2.91569 + 0.603191i −0.128862 0.315644i
16.14 −1.39248 + 0.601904i −0.128696 + 1.25735i 0.206419 0.219454i −0.0390564 + 1.91427i −0.577598 1.82829i −0.826752 + 2.02511i 0.875879 2.42354i 1.37342 + 0.284130i −1.09782 2.68908i
16.15 −1.35911 + 0.587481i 0.166831 1.62993i 0.131762 0.140083i −0.0219552 + 1.07609i 0.730809 + 2.31326i 1.70928 4.18684i 0.909729 2.51720i 0.308966 + 0.0639181i −0.602341 1.47542i
16.16 −1.34692 + 0.582212i −0.226539 + 2.21327i 0.104937 0.111564i 0.0257547 1.26231i −0.983460 3.11298i 1.79332 4.39269i 0.921096 2.54865i −1.90943 0.395018i 0.700245 + 1.71523i
16.17 −1.33775 + 0.578247i 0.303737 2.96749i 0.0849137 0.0902761i −0.0545287 + 2.67261i 1.30962 + 4.14538i −0.813080 + 1.99162i 0.929300 2.57135i −5.77594 1.19491i −1.47248 3.60681i
16.18 −1.32024 + 0.570680i −0.196646 + 1.92121i 0.0470787 0.0500518i 0.0871976 4.27381i −0.836777 2.64868i −1.16167 + 2.84549i 0.944136 2.61240i −0.714590 0.147832i 2.32386 + 5.69222i
16.19 −1.18250 + 0.511141i 0.157151 1.53535i −0.233243 + 0.247973i 0.0758402 3.71715i 0.598949 + 1.89588i 0.924146 2.26367i 1.02478 2.83554i 0.605190 + 0.125200i 1.81030 + 4.43429i
16.20 −1.16206 + 0.502303i 0.00409113 0.0399700i −0.272219 + 0.289411i 0.0343445 1.68333i 0.0153230 + 0.0485024i −0.164829 + 0.403743i 1.03154 2.85425i 2.93621 + 0.607436i 0.805630 + 1.97337i
See next 80 embeddings (of 3480 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 662.58
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner
29.d even 7 1 inner
667.s even 77 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 667.2.s.a 3480
23.c even 11 1 inner 667.2.s.a 3480
29.d even 7 1 inner 667.2.s.a 3480
667.s even 77 1 inner 667.2.s.a 3480
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
667.2.s.a 3480 1.a even 1 1 trivial
667.2.s.a 3480 23.c even 11 1 inner
667.2.s.a 3480 29.d even 7 1 inner
667.2.s.a 3480 667.s even 77 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(667, [\chi])\).