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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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [667,2,Mod(16,667)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(667, base_ring=CyclotomicField(154))
 
chi = DirichletCharacter(H, H._module([56, 22]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("667.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 667 = 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 667.s (of order \(77\), degree \(60\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
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.

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 16.58
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 \(S_{2}^{\mathrm{new}}(667, [\chi])\).