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Properties

Label 633.2.r.a
Level $633$
Weight $2$
Character orbit 633.r
Analytic conductor $5.055$
Analytic rank $0$
Dimension $204$
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(34,633)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(633, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([0, 40]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("633.34");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 633 = 3 \cdot 211 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 633.r (of order \(21\), degree \(12\), 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: \(204\)
Relative dimension: \(17\) over \(\Q(\zeta_{21})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 204 q - 4 q^{2} - 17 q^{3} + 26 q^{4} + 4 q^{5} - 10 q^{6} + 4 q^{7} - 10 q^{8} + 17 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 204 q - 4 q^{2} - 17 q^{3} + 26 q^{4} + 4 q^{5} - 10 q^{6} + 4 q^{7} - 10 q^{8} + 17 q^{9} - 18 q^{10} + 12 q^{11} + 45 q^{12} + 12 q^{13} + 16 q^{14} - 12 q^{15} - 12 q^{16} + 9 q^{17} - 6 q^{18} + 29 q^{19} - q^{20} + 24 q^{21} - 12 q^{22} - 12 q^{23} - 5 q^{24} - 18 q^{25} + 23 q^{26} + 34 q^{27} - 138 q^{28} - 15 q^{29} - 66 q^{30} - q^{31} - 26 q^{32} + 20 q^{33} + 56 q^{34} - 22 q^{35} + 12 q^{36} - 5 q^{37} + 5 q^{38} - 22 q^{39} + 86 q^{40} - 24 q^{41} - 10 q^{42} + 5 q^{43} - 79 q^{44} - 2 q^{45} + 9 q^{46} + 15 q^{47} + 47 q^{48} + 67 q^{49} - 62 q^{50} - 9 q^{51} - 11 q^{52} - 38 q^{53} - 3 q^{54} - 8 q^{55} + 72 q^{56} - q^{57} + 42 q^{58} - 69 q^{59} - 2 q^{60} - 56 q^{61} + 41 q^{62} - 8 q^{63} - 48 q^{64} - 42 q^{65} - 30 q^{66} + 4 q^{67} + 149 q^{68} + 8 q^{69} + 101 q^{70} - 102 q^{71} - 16 q^{72} + 10 q^{73} + 28 q^{74} - 23 q^{75} - 2 q^{76} + 26 q^{77} + 12 q^{78} - 88 q^{79} - 305 q^{80} + 17 q^{81} + 25 q^{82} - 5 q^{83} + 22 q^{84} + 77 q^{85} - 118 q^{86} + 12 q^{87} - 123 q^{88} + 26 q^{89} + 29 q^{90} + 108 q^{91} + 35 q^{92} + q^{93} + 202 q^{94} + 2 q^{95} - 38 q^{96} - 12 q^{97} + 87 q^{98} - 20 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
34.1 −2.75805 + 0.415709i 0.988831 − 0.149042i 5.52287 − 1.70358i 0.350103 − 0.439016i −2.66529 + 0.822132i −1.11293 − 2.83569i −9.49819 + 4.57409i 0.955573 − 0.294755i −0.783099 + 1.35637i
34.2 −2.41322 + 0.363735i 0.988831 − 0.149042i 3.78019 − 1.16603i 1.35866 − 1.70370i −2.33206 + 0.719344i 1.33599 + 3.40406i −4.30072 + 2.07112i 0.955573 − 0.294755i −2.65904 + 4.60560i
34.3 −2.32646 + 0.350658i 0.988831 − 0.149042i 3.37832 − 1.04207i −0.972362 + 1.21930i −2.24822 + 0.693483i 0.158630 + 0.404183i −3.25464 + 1.56735i 0.955573 − 0.294755i 1.83461 − 3.17763i
34.4 −1.86995 + 0.281849i 0.988831 − 0.149042i 1.50611 − 0.464573i −2.42909 + 3.04598i −1.80705 + 0.557402i −0.501360 − 1.27744i 0.722181 − 0.347784i 0.955573 − 0.294755i 3.68376 − 6.38045i
34.5 −1.69878 + 0.256049i 0.988831 − 0.149042i 0.909132 − 0.280430i 2.28365 − 2.86361i −1.64164 + 0.506379i −1.04874 − 2.67214i 1.62306 − 0.781623i 0.955573 − 0.294755i −3.14619 + 5.44936i
34.6 −1.50829 + 0.227338i 0.988831 − 0.149042i 0.312114 − 0.0962743i −0.503679 + 0.631594i −1.45756 + 0.449598i 1.16287 + 2.96294i 2.29967 − 1.10746i 0.955573 − 0.294755i 0.616110 − 1.06713i
34.7 −1.10780 + 0.166974i 0.988831 − 0.149042i −0.711807 + 0.219563i 1.55979 − 1.95591i −1.07054 + 0.330218i 0.0809571 + 0.206275i 2.77061 − 1.33426i 0.955573 − 0.294755i −1.40134 + 2.42720i
34.8 −0.571492 + 0.0861386i 0.988831 − 0.149042i −1.59196 + 0.491055i −0.377481 + 0.473347i −0.552271 + 0.170353i −0.130444 − 0.332367i 1.90892 − 0.919288i 0.955573 − 0.294755i 0.174954 − 0.303030i
34.9 −0.132590 + 0.0199847i 0.988831 − 0.149042i −1.89397 + 0.584211i −1.61874 + 2.02984i −0.128130 + 0.0395229i −1.02218 − 2.60448i 0.481062 − 0.231667i 0.955573 − 0.294755i 0.174063 − 0.301485i
34.10 0.0825653 − 0.0124447i 0.988831 − 0.149042i −1.90448 + 0.587455i 2.19517 − 2.75266i 0.0797883 − 0.0246114i 0.628499 + 1.60139i −0.300392 + 0.144661i 0.955573 − 0.294755i 0.146989 − 0.254592i
34.11 0.814828 − 0.122816i 0.988831 − 0.149042i −1.26228 + 0.389363i −1.37255 + 1.72112i 0.787422 − 0.242888i 1.19596 + 3.04725i −2.46558 + 1.18736i 0.955573 − 0.294755i −0.907012 + 1.57099i
34.12 0.977455 − 0.147328i 0.988831 − 0.149042i −0.977432 + 0.301498i 1.51123 − 1.89502i 0.944580 − 0.291364i −1.29266 − 3.29366i −2.69219 + 1.29649i 0.955573 − 0.294755i 1.19797 − 2.07494i
34.13 1.13665 − 0.171322i 0.988831 − 0.149042i −0.648526 + 0.200044i 0.272040 − 0.341128i 1.09842 − 0.338817i 1.34545 + 3.42816i −2.77418 + 1.33597i 0.955573 − 0.294755i 0.250772 − 0.434349i
34.14 1.53910 − 0.231982i 0.988831 − 0.149042i 0.403868 − 0.124577i −1.75817 + 2.20467i 1.48733 − 0.458782i −0.0868590 − 0.221313i −2.21200 + 1.06524i 0.955573 − 0.294755i −2.19455 + 3.80108i
34.15 2.28670 − 0.344665i 0.988831 − 0.149042i 3.19907 − 0.986783i 0.510016 − 0.639540i 2.20979 − 0.681631i 0.549172 + 1.39927i 2.80818 − 1.35235i 0.955573 − 0.294755i 0.945828 − 1.63822i
34.16 2.29548 − 0.345987i 0.988831 − 0.149042i 3.23836 − 0.998901i 0.261645 − 0.328092i 2.21827 − 0.684246i −0.947534 − 2.41428i 2.90494 − 1.39895i 0.955573 − 0.294755i 0.487084 − 0.843654i
34.17 2.57107 − 0.387526i 0.988831 − 0.149042i 4.54907 − 1.40320i −2.51720 + 3.15647i 2.48459 − 0.766396i 1.14655 + 2.92136i 6.46696 − 3.11432i 0.955573 − 0.294755i −5.24869 + 9.09099i
43.1 −0.985636 − 2.51136i −0.365341 − 0.930874i −3.86935 + 3.59023i 0.464810 + 0.582854i −1.97767 + 1.83501i 2.14871 − 0.323865i 7.96877 + 3.83756i −0.733052 + 0.680173i 1.00562 − 1.74179i
43.2 −0.888696 − 2.26436i −0.365341 − 0.930874i −2.87145 + 2.66431i −1.66490 − 2.08772i −1.78316 + 1.65453i −1.07368 + 0.161831i 4.20158 + 2.02337i −0.733052 + 0.680173i −3.24777 + 5.62530i
43.3 −0.741161 − 1.88845i −0.365341 − 0.930874i −1.55081 + 1.43894i 2.22609 + 2.79142i −1.48713 + 1.37986i −0.616758 + 0.0929613i 0.211213 + 0.101715i −0.733052 + 0.680173i 3.62157 − 6.27274i
See next 80 embeddings (of 204 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 34.17
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
211.j even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 633.2.r.a ✓ 204
211.j even 21 1 inner 633.2.r.a ✓ 204
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
633.2.r.a ✓ 204 1.a even 1 1 trivial
633.2.r.a ✓ 204 211.j even 21 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{204} + 4 T_{2}^{203} - 22 T_{2}^{202} - 98 T_{2}^{201} + 217 T_{2}^{200} + 1118 T_{2}^{199} + \cdots + 1179745717921 \) acting on \(S_{2}^{\mathrm{new}}(633, [\chi])\). Copy content Toggle raw display