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Properties

Label 633.2.f.a
Level $633$
Weight $2$
Character orbit 633.f
Analytic conductor $5.055$
Analytic rank $0$
Dimension $68$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [633,2,Mod(55,633)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(633, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("633.55");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 633 = 3 \cdot 211 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 633.f (of order \(5\), degree \(4\), 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: \(5.05453044795\)
Analytic rank: \(0\)
Dimension: \(68\)
Relative dimension: \(17\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 68 q + 17 q^{3} - 16 q^{4} + 2 q^{5} - 8 q^{7} - 6 q^{8} - 17 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 68 q + 17 q^{3} - 16 q^{4} + 2 q^{5} - 8 q^{7} - 6 q^{8} - 17 q^{9} + 8 q^{10} + 14 q^{11} - 64 q^{12} + 6 q^{13} - 56 q^{14} + 8 q^{15} - 34 q^{16} - 20 q^{17} + 2 q^{19} + 5 q^{20} - 2 q^{21} - 22 q^{22} - 28 q^{23} + 6 q^{24} - 15 q^{25} - 14 q^{26} + 17 q^{27} + 22 q^{28} + 8 q^{29} - 8 q^{30} + 42 q^{31} + 18 q^{32} + 16 q^{33} - 22 q^{34} + 32 q^{35} - 16 q^{36} - 10 q^{37} + 28 q^{38} - 6 q^{39} - 96 q^{40} - 2 q^{41} - 19 q^{42} + 26 q^{43} - 7 q^{44} + 2 q^{45} + 66 q^{46} + q^{47} - 6 q^{48} - 49 q^{49} + 28 q^{50} - 15 q^{51} + 31 q^{52} - 22 q^{53} + 23 q^{55} + 22 q^{56} - 2 q^{57} - 10 q^{58} - 23 q^{59} - 12 q^{61} - 4 q^{62} + 12 q^{63} - 56 q^{64} + 47 q^{65} - 13 q^{66} + 46 q^{67} - 19 q^{68} + 28 q^{69} - 4 q^{70} + 38 q^{71} - 6 q^{72} + 6 q^{73} + 42 q^{74} + 15 q^{75} + 35 q^{76} + 27 q^{77} + 19 q^{78} - 22 q^{79} + 30 q^{80} - 17 q^{81} + 58 q^{82} + 10 q^{83} - 22 q^{84} + 26 q^{85} + 36 q^{86} + 22 q^{87} - 18 q^{88} - 26 q^{89} + 8 q^{90} + 80 q^{91} + 48 q^{92} + 13 q^{93} - 100 q^{94} + 27 q^{95} - 8 q^{96} - 10 q^{97} - 93 q^{98} + 14 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1 −0.811479 − 2.49748i 0.809017 + 0.587785i −3.96085 + 2.87773i −1.23830 + 0.899680i 0.811479 − 2.49748i 0.759729 − 2.33820i 6.15224 + 4.46986i 0.309017 + 0.951057i 3.25178 + 2.36256i
55.2 −0.680387 − 2.09402i 0.809017 + 0.587785i −2.30395 + 1.67391i −1.62327 + 1.17937i 0.680387 − 2.09402i 0.0269034 − 0.0828000i 1.51023 + 1.09724i 0.309017 + 0.951057i 3.57408 + 2.59672i
55.3 −0.636449 − 1.95879i 0.809017 + 0.587785i −1.81375 + 1.31777i 2.07667 − 1.50879i 0.636449 − 1.95879i −1.02422 + 3.15223i 0.403105 + 0.292873i 0.309017 + 0.951057i −4.27709 − 3.10749i
55.4 −0.562620 − 1.73157i 0.809017 + 0.587785i −1.06375 + 0.772858i 2.85170 − 2.07188i 0.562620 − 1.73157i 1.55671 − 4.79107i −1.00918 − 0.733209i 0.309017 + 0.951057i −5.19203 − 3.77223i
55.5 −0.529517 − 1.62969i 0.809017 + 0.587785i −0.757457 + 0.550325i −1.75952 + 1.27837i 0.529517 − 1.62969i 0.0131642 − 0.0405152i −1.47465 − 1.07139i 0.309017 + 0.951057i 3.01504 + 2.19055i
55.6 −0.287656 − 0.885314i 0.809017 + 0.587785i 0.916999 − 0.666239i −1.01211 + 0.735342i 0.287656 − 0.885314i −1.51874 + 4.67420i −2.35980 − 1.71449i 0.309017 + 0.951057i 0.942148 + 0.684510i
55.7 −0.219011 − 0.674047i 0.809017 + 0.587785i 1.21166 − 0.880323i −2.86895 + 2.08442i 0.219011 − 0.674047i 1.24219 − 3.82307i −2.00550 − 1.45708i 0.309017 + 0.951057i 2.03333 + 1.47730i
55.8 −0.198338 − 0.610420i 0.809017 + 0.587785i 1.28476 − 0.933432i 0.241867 − 0.175727i 0.198338 − 0.610420i 0.611938 − 1.88335i −1.86311 − 1.35363i 0.309017 + 0.951057i −0.155238 − 0.112787i
55.9 −0.109950 − 0.338392i 0.809017 + 0.587785i 1.51561 − 1.10116i 2.54447 − 1.84866i 0.109950 − 0.338392i −0.256532 + 0.789523i −1.11497 − 0.810075i 0.309017 + 0.951057i −0.905338 − 0.657767i
55.10 0.154555 + 0.475671i 0.809017 + 0.587785i 1.41566 − 1.02854i 1.51613 − 1.10153i −0.154555 + 0.475671i 0.693129 − 2.13323i 1.51730 + 1.10238i 0.309017 + 0.951057i 0.758294 + 0.550932i
55.11 0.184517 + 0.567885i 0.809017 + 0.587785i 1.32959 − 0.966002i −0.590143 + 0.428764i −0.184517 + 0.567885i −0.681500 + 2.09744i 1.76005 + 1.27875i 0.309017 + 0.951057i −0.352380 − 0.256019i
55.12 0.418913 + 1.28928i 0.809017 + 0.587785i 0.131273 − 0.0953757i 2.18291 − 1.58597i −0.418913 + 1.28928i −1.08421 + 3.33686i 2.37142 + 1.72293i 0.309017 + 0.951057i 2.95922 + 2.15000i
55.13 0.423584 + 1.30366i 0.809017 + 0.587785i 0.0979324 − 0.0711520i −2.66987 + 1.93977i −0.423584 + 1.30366i 0.240770 − 0.741013i 2.35216 + 1.70894i 0.309017 + 0.951057i −3.65972 − 2.65894i
55.14 0.600861 + 1.84926i 0.809017 + 0.587785i −1.44070 + 1.04673i 1.18494 − 0.860908i −0.600861 + 1.84926i 1.26392 − 3.88994i 0.344822 + 0.250528i 0.309017 + 0.951057i 2.30403 + 1.67397i
55.15 0.709005 + 2.18209i 0.809017 + 0.587785i −2.64081 + 1.91866i −1.99668 + 1.45068i −0.709005 + 2.18209i −1.34793 + 4.14849i −2.34665 − 1.70494i 0.309017 + 0.951057i −4.58117 − 3.32842i
55.16 0.755851 + 2.32627i 0.809017 + 0.587785i −3.22218 + 2.34105i 3.14459 − 2.28468i −0.755851 + 2.32627i −0.467518 + 1.43887i −3.92373 − 2.85076i 0.309017 + 0.951057i 7.69162 + 5.58829i
55.17 0.788121 + 2.42559i 0.809017 + 0.587785i −3.64431 + 2.64774i −0.366383 + 0.266193i −0.788121 + 2.42559i 0.208263 − 0.640967i −5.16784 − 3.75465i 0.309017 + 0.951057i −0.934430 − 0.678903i
493.1 −2.16619 + 1.57383i −0.309017 − 0.951057i 1.59740 − 4.91628i 0.443168 − 1.36393i 2.16619 + 1.57383i 3.54429 + 2.57508i 2.62230 + 8.07060i −0.809017 + 0.587785i 1.18661 + 3.65200i
493.2 −2.14095 + 1.55549i −0.309017 − 0.951057i 1.54608 − 4.75834i −0.990036 + 3.04702i 2.14095 + 1.55549i −2.33865 − 1.69913i 2.45594 + 7.55860i −0.809017 + 0.587785i −2.61999 − 8.06350i
493.3 −1.61746 + 1.17515i −0.309017 − 0.951057i 0.617159 − 1.89942i 0.255865 − 0.787470i 1.61746 + 1.17515i 1.52166 + 1.10555i −0.00174873 − 0.00538204i −0.809017 + 0.587785i 0.511548 + 1.57438i
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 55.17
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
211.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 633.2.f.a ✓ 68
211.d even 5 1 inner 633.2.f.a ✓ 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
633.2.f.a ✓ 68 1.a even 1 1 trivial
633.2.f.a ✓ 68 211.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{68} + 25 T_{2}^{66} + 2 T_{2}^{65} + 379 T_{2}^{64} + 8 T_{2}^{63} + 4544 T_{2}^{62} + \cdots + 6974881 \) acting on \(S_{2}^{\mathrm{new}}(633, [\chi])\). Copy content Toggle raw display