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Properties

Label 633.2.f.b
Level $633$
Weight $2$
Character orbit 633.f
Analytic conductor $5.055$
Analytic rank $0$
Dimension $76$
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: \(76\)
Relative dimension: \(19\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 76 q - 4 q^{2} - 19 q^{3} - 30 q^{4} - 2 q^{5} - 4 q^{6} + 10 q^{7} + 2 q^{8} - 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 76 q - 4 q^{2} - 19 q^{3} - 30 q^{4} - 2 q^{5} - 4 q^{6} + 10 q^{7} + 2 q^{8} - 19 q^{9} + 4 q^{10} + 10 q^{11} + 80 q^{12} - 14 q^{13} + 32 q^{14} + 8 q^{15} + 16 q^{17} + 6 q^{18} + 20 q^{19} - 17 q^{20} - 10 q^{21} + 2 q^{22} - 8 q^{24} - 33 q^{25} - 2 q^{26} - 19 q^{27} + 36 q^{28} - 4 q^{29} - 16 q^{30} + 14 q^{31} + 50 q^{32} - 30 q^{34} - 30 q^{36} + 20 q^{37} + 32 q^{38} - 14 q^{39} + 48 q^{40} - 30 q^{41} - 13 q^{42} + 46 q^{43} - 25 q^{44} - 2 q^{45} + 18 q^{46} - 15 q^{47} - 40 q^{48} + 23 q^{49} - 68 q^{50} - 9 q^{51} - 29 q^{52} - 16 q^{53} - 4 q^{54} - 21 q^{55} - 54 q^{56} - 10 q^{57} + 6 q^{58} + 29 q^{59} + 18 q^{60} - 12 q^{61} - 18 q^{62} - 22 q^{64} - 45 q^{65} - 3 q^{66} + 26 q^{67} + q^{68} + 8 q^{70} + 24 q^{71} - 8 q^{72} - 22 q^{73} - 26 q^{74} - 13 q^{75} + 13 q^{76} + 7 q^{77} + 3 q^{78} - 34 q^{79} - 40 q^{80} - 19 q^{81} - 18 q^{82} - 14 q^{83} + 36 q^{84} - 14 q^{85} - 52 q^{86} - 4 q^{87} + 22 q^{88} - 10 q^{89} - 16 q^{90} - 88 q^{91} + 82 q^{92} + 9 q^{93} + 44 q^{94} - 17 q^{95} + 10 q^{96} + 40 q^{97} + 13 q^{98} + 10 q^{99}+O(q^{100}) \) 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 −0.852418 − 2.62347i −0.809017 − 0.587785i −4.53795 + 3.29702i 2.94373 − 2.13875i −0.852418 + 2.62347i −0.00811746 + 0.0249830i 8.05454 + 5.85197i 0.309017 + 0.951057i −8.12022 − 5.89969i
55.2 −0.795003 − 2.44677i −0.809017 − 0.587785i −3.73661 + 2.71481i −2.19832 + 1.59717i −0.795003 + 2.44677i −0.438190 + 1.34861i 5.45043 + 3.95997i 0.309017 + 0.951057i 5.65559 + 4.10902i
55.3 −0.731043 − 2.24992i −0.809017 − 0.587785i −2.90968 + 2.11400i 0.0298937 − 0.0217190i −0.731043 + 2.24992i 0.553754 − 1.70428i 3.05565 + 2.22006i 0.309017 + 0.951057i −0.0707196 − 0.0513808i
55.4 −0.610744 − 1.87968i −0.809017 − 0.587785i −1.54214 + 1.12043i −0.850559 + 0.617967i −0.610744 + 1.87968i 1.09186 − 3.36041i −0.150003 − 0.108983i 0.309017 + 0.951057i 1.68105 + 1.22135i
55.5 −0.581450 − 1.78952i −0.809017 − 0.587785i −1.24626 + 0.905464i 1.14098 − 0.828968i −0.581450 + 1.78952i −1.17940 + 3.62983i −0.699528 − 0.508237i 0.309017 + 0.951057i −2.14688 − 1.55980i
55.6 −0.544678 − 1.67635i −0.809017 − 0.587785i −0.895428 + 0.650566i −3.15655 + 2.29337i −0.544678 + 1.67635i −0.438603 + 1.34988i −1.27368 − 0.925380i 0.309017 + 0.951057i 5.56379 + 4.04233i
55.7 −0.447011 − 1.37576i −0.809017 − 0.587785i −0.0748604 + 0.0543893i 3.34958 − 2.43361i −0.447011 + 1.37576i 0.247824 − 0.762723i −2.23229 − 1.62185i 0.309017 + 0.951057i −4.84536 − 3.52036i
55.8 −0.346212 − 1.06553i −0.809017 − 0.587785i 0.602541 − 0.437772i −1.41360 + 1.02704i −0.346212 + 1.06553i 1.39508 − 4.29362i −2.48786 − 1.80753i 0.309017 + 0.951057i 1.58375 + 1.15066i
55.9 −0.217807 − 0.670342i −0.809017 − 0.587785i 1.21612 − 0.883559i 0.0532600 − 0.0386957i −0.217807 + 0.670342i −1.19823 + 3.68778i −1.99762 − 1.45136i 0.309017 + 0.951057i −0.0375398 − 0.0272743i
55.10 −0.0511819 − 0.157522i −0.809017 − 0.587785i 1.59584 − 1.15945i 1.44876 − 1.05258i −0.0511819 + 0.157522i 0.273722 − 0.842430i −0.532308 − 0.386744i 0.309017 + 0.951057i −0.239955 − 0.174338i
55.11 −0.0427059 − 0.131435i −0.809017 − 0.587785i 1.60258 − 1.16434i −2.76609 + 2.00968i −0.0427059 + 0.131435i −0.278748 + 0.857898i −0.445086 − 0.323374i 0.309017 + 0.951057i 0.382271 + 0.277736i
55.12 0.125400 + 0.385943i −0.809017 − 0.587785i 1.48481 − 1.07878i −0.273130 + 0.198441i 0.125400 − 0.385943i −0.931369 + 2.86646i 1.25915 + 0.914824i 0.309017 + 0.951057i −0.110837 − 0.0805281i
55.13 0.281031 + 0.864925i −0.809017 − 0.587785i 0.948918 − 0.689429i 2.41086 − 1.75159i 0.281031 − 0.864925i 0.725503 − 2.23287i 2.33448 + 1.69610i 0.309017 + 0.951057i 2.19252 + 1.59296i
55.14 0.323724 + 0.996320i −0.809017 − 0.587785i 0.730178 − 0.530505i −1.66692 + 1.21109i 0.323724 − 0.996320i 0.824454 − 2.53741i 2.45997 + 1.78727i 0.309017 + 0.951057i −1.74625 − 1.26873i
55.15 0.563370 + 1.73388i −0.809017 − 0.587785i −1.07090 + 0.778058i −1.31561 + 0.955846i 0.563370 − 1.73388i 0.162836 − 0.501158i 0.997474 + 0.724707i 0.309017 + 0.951057i −2.39849 − 1.74261i
55.16 0.603160 + 1.85634i −0.809017 − 0.587785i −1.46415 + 1.06376i 1.18930 − 0.864079i 0.603160 − 1.85634i −0.119512 + 0.367820i 0.300370 + 0.218231i 0.309017 + 0.951057i 2.32136 + 1.68657i
55.17 0.712801 + 2.19378i −0.809017 − 0.587785i −2.68654 + 1.95188i 1.26571 − 0.919591i 0.712801 − 2.19378i −0.915307 + 2.81702i −2.46469 − 1.79070i 0.309017 + 0.951057i 2.91957 + 2.12120i
55.18 0.745807 + 2.29536i −0.809017 − 0.587785i −3.09441 + 2.24822i −3.22805 + 2.34531i 0.745807 − 2.29536i 0.113639 − 0.349745i −3.56320 − 2.58881i 0.309017 + 0.951057i −7.79084 − 5.66037i
55.19 0.864959 + 2.66207i −0.809017 − 0.587785i −4.72043 + 3.42960i 1.41873 − 1.03077i 0.864959 − 2.66207i 1.50078 − 4.61891i −8.68383 − 6.30917i 0.309017 + 0.951057i 3.97113 + 2.88519i
493.1 −2.27854 + 1.65546i 0.309017 + 0.951057i 1.83318 − 5.64196i 0.0526232 − 0.161958i −2.27854 − 1.65546i −1.44229 − 1.04789i 3.42239 + 10.5330i −0.809017 + 0.587785i 0.148210 + 0.456143i
See all 76 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 55.19
Significant digits:
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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.b ✓ 76
211.d even 5 1 inner 633.2.f.b ✓ 76
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
633.2.f.b ✓ 76 1.a even 1 1 trivial
633.2.f.b ✓ 76 211.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{76} + 4 T_{2}^{75} + 42 T_{2}^{74} + 138 T_{2}^{73} + 879 T_{2}^{72} + 2518 T_{2}^{71} + \cdots + 6561 \) acting on \(S_{2}^{\mathrm{new}}(633, [\chi])\). Copy content Toggle raw display