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Properties

Label 633.2.k.a
Level $633$
Weight $2$
Character orbit 633.k
Analytic conductor $5.055$
Analytic rank $0$
Dimension $272$
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] = [633,2,Mod(23,633)]
 
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([5, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("633.23");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 633 = 3 \cdot 211 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 633.k (of order \(10\), 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: \(272\)
Relative dimension: \(68\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 272 q - 10 q^{3} - 74 q^{4} + 9 q^{6} - 30 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 272 q - 10 q^{3} - 74 q^{4} + 9 q^{6} - 30 q^{7} - 6 q^{9} - 20 q^{10} + 6 q^{13} - 46 q^{16} - 20 q^{18} + 6 q^{19} - 3 q^{21} - 10 q^{22} + 23 q^{24} + 50 q^{25} - 40 q^{27} + 80 q^{28} - q^{30} + 12 q^{34} + 13 q^{36} - 42 q^{37} + 25 q^{39} - 10 q^{42} + 40 q^{43} - 23 q^{45} - 2 q^{46} - 15 q^{48} + 30 q^{49} - 11 q^{51} - 34 q^{52} + 128 q^{54} + 14 q^{55} + 20 q^{57} + 60 q^{58} + 30 q^{60} - 10 q^{61} - 108 q^{64} - 73 q^{66} + 40 q^{69} + 70 q^{70} - 105 q^{72} - 20 q^{73} - 10 q^{75} - 12 q^{76} - 82 q^{78} - 30 q^{79} - 42 q^{81} - 106 q^{82} + 121 q^{84} - 110 q^{85} - 11 q^{87} + 60 q^{90} - 40 q^{91} - 45 q^{93} - 114 q^{96} - 80 q^{97} - 18 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −0.856290 + 2.63539i 0.668987 + 1.59764i −4.59400 − 3.33774i 1.28944 − 1.77477i −4.78325 + 0.394998i −3.18493 + 1.03484i 8.24645 − 5.99140i −2.10491 + 2.13760i 3.57306 + 4.91790i
23.2 −0.820916 + 2.52652i −1.67832 − 0.428070i −4.09137 − 2.97255i −0.423424 + 0.582793i 2.45929 − 3.88890i 1.45633 − 0.473190i 6.57052 − 4.77376i 2.63351 + 1.43688i −1.12484 − 1.54821i
23.3 −0.800735 + 2.46441i −0.0806239 − 1.73017i −3.81410 − 2.77111i −2.36373 + 3.25340i 4.32841 + 1.18672i −3.44751 + 1.12016i 5.69052 − 4.13440i −2.98700 + 0.278987i −6.12498 − 8.43031i
23.4 −0.799825 + 2.46161i 1.66620 − 0.473050i −3.80176 − 2.76214i 0.419853 − 0.577878i −0.168205 + 4.47989i −2.14759 + 0.697796i 5.65212 − 4.10650i 2.55245 − 1.57639i 1.08670 + 1.49572i
23.5 −0.789413 + 2.42956i 1.14347 − 1.30095i −3.66157 − 2.66028i −0.731975 + 1.00748i 2.25807 + 3.80512i 4.04934 − 1.31571i 5.22040 − 3.79284i −0.384944 − 2.97520i −1.86990 − 2.57369i
23.6 −0.774372 + 2.38327i −0.564785 − 1.63738i −3.46230 − 2.51551i 2.13481 − 2.93832i 4.33968 − 0.0780937i 0.315520 − 0.102519i 4.62158 − 3.35777i −2.36204 + 1.84954i 5.34967 + 7.36319i
23.7 −0.718944 + 2.21268i 1.57202 + 0.727145i −2.76105 − 2.00602i −2.27009 + 3.12451i −2.73914 + 2.95561i −0.855199 + 0.277871i 2.65928 − 1.93208i 1.94252 + 2.28618i −5.28148 − 7.26933i
23.8 −0.711089 + 2.18851i −1.25181 + 1.19707i −2.66588 − 1.93687i −0.411283 + 0.566082i −1.72964 − 3.59082i −3.59487 + 1.16804i 2.41123 − 1.75186i 0.134063 − 2.99700i −0.946415 − 1.30263i
23.9 −0.706109 + 2.17318i −0.517600 + 1.65290i −2.60608 − 1.89343i 1.39731 − 1.92323i −3.22657 − 2.29197i 3.70895 − 1.20511i 2.25771 − 1.64032i −2.46418 − 1.71108i 3.19287 + 4.39461i
23.10 −0.699689 + 2.15342i 0.826205 + 1.52230i −2.52962 − 1.83788i −0.414257 + 0.570176i −3.85623 + 0.714033i 1.18044 − 0.383549i 2.06405 − 1.49962i −1.63477 + 2.51546i −0.937977 − 1.29102i
23.11 −0.643573 + 1.98072i −1.36348 + 1.06814i −1.89101 − 1.37390i −1.09574 + 1.50816i −1.23818 − 3.38809i 2.08360 − 0.677003i 0.568519 − 0.413053i 0.718149 − 2.91278i −2.28205 − 3.14097i
23.12 −0.632074 + 1.94532i 1.49433 − 0.875766i −1.76673 − 1.28360i 1.78459 − 2.45627i 0.759119 + 3.46051i 0.100785 − 0.0327470i 0.304146 − 0.220975i 1.46607 − 2.61737i 3.65025 + 5.02414i
23.13 −0.584372 + 1.79851i −1.68750 − 0.390325i −1.27512 − 0.926429i 1.54917 − 2.13225i 1.68813 − 2.80689i −3.81267 + 1.23881i −0.648473 + 0.471143i 2.69529 + 1.31734i 2.92959 + 4.03223i
23.14 −0.575669 + 1.77173i −0.528679 − 1.64939i −1.18959 − 0.864285i −0.832701 + 1.14612i 3.22662 + 0.0128300i 2.59824 − 0.844220i −0.798157 + 0.579895i −2.44100 + 1.74400i −1.55124 − 2.13510i
23.15 −0.571191 + 1.75794i 1.54042 + 0.791900i −1.14608 − 0.832674i 1.51180 − 2.08081i −2.27199 + 2.25565i 1.92528 − 0.625561i −0.872371 + 0.633815i 1.74579 + 2.43972i 2.79443 + 3.84620i
23.16 −0.513669 + 1.58091i −1.69150 − 0.372591i −0.617385 − 0.448557i −2.18888 + 3.01274i 1.45790 − 2.48272i −0.712998 + 0.231667i −1.66334 + 1.20849i 2.72235 + 1.26048i −3.63851 − 5.00797i
23.17 −0.508191 + 1.56405i −1.72629 + 0.141104i −0.569965 − 0.414104i 2.00069 − 2.75371i 0.656594 − 2.77172i 1.24486 − 0.404480i −1.72359 + 1.25226i 2.96018 − 0.487173i 3.29021 + 4.52859i
23.18 −0.504007 + 1.55117i 0.690180 − 1.58860i −0.534085 − 0.388035i 0.00566708 − 0.00780006i 2.11634 + 1.87126i −3.86122 + 1.25459i −1.76792 + 1.28447i −2.04730 − 2.19284i 0.00924301 + 0.0127219i
23.19 −0.489132 + 1.50539i 0.372318 − 1.69156i −0.408927 − 0.297103i 0.626907 − 0.862864i 2.36435 + 1.38788i −2.01574 + 0.654954i −1.91385 + 1.39050i −2.72276 − 1.25960i 0.992309 + 1.36580i
23.20 −0.470539 + 1.44817i 0.398365 + 1.68562i −0.257758 − 0.187272i −1.72018 + 2.36762i −2.62851 − 0.216249i −0.393213 + 0.127763i −2.07129 + 1.50488i −2.68261 + 1.34298i −2.61931 − 3.60518i
See next 80 embeddings (of 272 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.68
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
211.g odd 10 1 inner
633.k even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 633.2.k.a ✓ 272
3.b odd 2 1 inner 633.2.k.a ✓ 272
211.g odd 10 1 inner 633.2.k.a ✓ 272
633.k even 10 1 inner 633.2.k.a ✓ 272
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
633.2.k.a ✓ 272 1.a even 1 1 trivial
633.2.k.a ✓ 272 3.b odd 2 1 inner
633.2.k.a ✓ 272 211.g odd 10 1 inner
633.2.k.a ✓ 272 633.k even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(633, [\chi])\).