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Properties

Label 229.2.k.a
Level $229$
Weight $2$
Character orbit 229.k
Analytic conductor $1.829$
Analytic rank $0$
Dimension $684$
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] = [229,2,Mod(5,229)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(229, base_ring=CyclotomicField(114))
 
chi = DirichletCharacter(H, H._module([49]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("229.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 229 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 229.k (of order \(114\), degree \(36\), 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: \(1.82857420629\)
Analytic rank: \(0\)
Dimension: \(684\)
Relative dimension: \(19\) over \(\Q(\zeta_{114})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{114}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 684 q - 38 q^{2} - 37 q^{3} + 14 q^{4} - 57 q^{5} - 41 q^{6} - 38 q^{7} - 38 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 684 q - 38 q^{2} - 37 q^{3} + 14 q^{4} - 57 q^{5} - 41 q^{6} - 38 q^{7} - 38 q^{8} - 12 q^{9} - 47 q^{10} - 22 q^{11} - 34 q^{12} - 19 q^{13} - 33 q^{14} - 22 q^{15} + 50 q^{16} - 42 q^{17} - 93 q^{18} - 34 q^{19} - 42 q^{20} - 38 q^{21} - 38 q^{22} + 8 q^{23} - 35 q^{24} - 12 q^{25} - 50 q^{26} - 52 q^{27} + 4 q^{28} - 35 q^{29} - 95 q^{30} + 123 q^{31} - 38 q^{32} - 5 q^{33} - 38 q^{34} - 19 q^{35} - 129 q^{36} - 21 q^{37} - 68 q^{38} - 53 q^{39} + 119 q^{40} - 23 q^{41} + 160 q^{42} - 74 q^{43} - 151 q^{44} + q^{45} + 37 q^{46} - 65 q^{47} + 243 q^{48} - 119 q^{49} - 8 q^{50} - 25 q^{51} - 38 q^{52} - 24 q^{53} + 19 q^{54} - 23 q^{55} - 296 q^{56} - 4 q^{57} - 30 q^{58} - 55 q^{59} - 136 q^{60} + 164 q^{61} - 26 q^{62} + 132 q^{63} + 66 q^{64} - 77 q^{65} + 222 q^{66} - 46 q^{67} - 70 q^{68} - 71 q^{69} - 55 q^{70} + 11 q^{71} + 24 q^{72} - 20 q^{73} + 153 q^{74} - 101 q^{75} + 251 q^{76} - 83 q^{77} + 484 q^{78} + 73 q^{79} - 369 q^{80} - 159 q^{81} - 227 q^{82} + 45 q^{83} + 114 q^{84} - 235 q^{85} - 38 q^{86} - 113 q^{87} - 38 q^{88} - 117 q^{89} + 138 q^{90} + 309 q^{91} + 31 q^{92} + 19 q^{93} - 173 q^{94} + 320 q^{95} + 188 q^{96} - 44 q^{97} + 9 q^{98} - 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −2.76732 − 0.229306i 0.117224 + 0.374785i 5.63273 + 0.939937i 2.02442 + 0.822247i −0.238454 − 1.06403i 2.82402 − 0.0778435i −9.98835 − 2.52939i 2.33861 − 1.62156i −5.41365 − 2.73963i
5.2 −2.68142 − 0.222189i −0.996334 − 3.18546i 5.16792 + 0.862373i −2.38020 − 0.966754i 1.96382 + 8.76292i −1.49413 + 0.0411854i −8.44919 − 2.13963i −6.68911 + 4.63813i 6.16751 + 3.12113i
5.3 −2.03435 − 0.168571i 0.770302 + 2.46279i 2.13743 + 0.356674i −0.121088 − 0.0491817i −1.15191 − 5.14003i 4.17515 − 0.115087i −0.330436 − 0.0836778i −3.00665 + 2.08477i 0.238044 + 0.120465i
5.4 −2.00988 − 0.166543i 0.103048 + 0.329464i 2.03916 + 0.340275i −4.03551 − 1.63908i −0.152245 − 0.679344i 0.208984 − 0.00576061i −0.131678 − 0.0333454i 2.36741 − 1.64152i 7.83790 + 3.96644i
5.5 −1.96658 − 0.162955i −0.100588 − 0.321599i 1.86815 + 0.311738i 0.246371 + 0.100067i 0.145409 + 0.648840i −2.78269 + 0.0767044i 0.202816 + 0.0513599i 2.37203 − 1.64473i −0.468201 − 0.236937i
5.6 −1.66258 − 0.137765i −0.735295 − 2.35087i 0.772468 + 0.128902i 2.59191 + 1.05274i 0.898619 + 4.00981i 2.83416 − 0.0781229i 1.96793 + 0.498346i −2.52060 + 1.74774i −4.16423 − 2.10735i
5.7 −1.28830 − 0.106751i 0.574689 + 1.83738i −0.324413 − 0.0541349i 3.76545 + 1.52939i −0.544226 − 2.42844i −1.75231 + 0.0483020i 2.91847 + 0.739057i −0.580376 + 0.402423i −4.68774 − 2.37227i
5.8 −0.787241 − 0.0652326i −0.675632 − 2.16012i −1.35723 − 0.226482i −0.568489 − 0.230900i 0.390975 + 1.74461i −2.18634 + 0.0602660i 2.58523 + 0.654668i −1.74429 + 1.20947i 0.432475 + 0.218858i
5.9 −0.604294 − 0.0500732i 0.191271 + 0.611528i −1.61006 − 0.268671i −1.62298 − 0.659198i −0.0849629 − 0.379121i 0.768620 − 0.0211869i 2.13512 + 0.540685i 2.12795 − 1.47549i 0.947750 + 0.479617i
5.10 −0.00190287 0.000157676i 0.834890 + 2.66929i −1.97272 − 0.329189i −1.58493 − 0.643743i −0.00116780 − 0.00521096i −3.15846 + 0.0870623i 0.00740386 + 0.00187491i −3.96274 + 2.74771i 0.00291442 + 0.00147487i
5.11 0.137188 + 0.0113678i 0.186758 + 0.597097i −1.95403 − 0.326070i 0.319523 + 0.129779i 0.0188333 + 0.0840378i 4.23273 − 0.116674i −0.531256 − 0.134532i 2.14369 − 1.48640i 0.0423596 + 0.0214365i
5.12 0.544227 + 0.0450959i −0.463965 − 1.48338i −1.67857 − 0.280104i 3.38873 + 1.37638i −0.185608 − 0.828218i 0.953342 − 0.0262787i −1.95966 − 0.496252i 0.480182 − 0.332951i 1.78217 + 0.901882i
5.13 1.14933 + 0.0952364i −0.415711 − 1.32910i −0.660828 − 0.110273i −2.26746 − 0.920964i −0.351211 − 1.56717i −4.04821 + 0.111588i −2.98497 − 0.755898i 0.871636 − 0.604379i −2.51836 − 1.27444i
5.14 1.15021 + 0.0953093i −0.832740 − 2.66242i −0.658819 − 0.109937i −1.67172 − 0.678996i −0.704074 − 3.14171i 2.79600 − 0.0770712i −2.98498 − 0.755899i −3.92967 + 2.72478i −1.85812 − 0.940320i
5.15 1.57919 + 0.130855i 0.534999 + 1.71049i 0.503985 + 0.0841003i 1.79129 + 0.727557i 0.621037 + 2.77119i −1.02188 + 0.0281679i −2.28734 − 0.579233i −0.174210 + 0.120795i 2.73357 + 1.38335i
5.16 2.08649 + 0.172892i 0.875144 + 2.79799i 2.35084 + 0.392286i −3.59698 − 1.46097i 1.34223 + 5.98929i 3.98286 − 0.109787i 0.778028 + 0.197023i −4.59755 + 3.18787i −7.25249 − 3.67019i
5.17 2.12420 + 0.176017i −0.274981 − 0.879164i 2.50854 + 0.418601i −0.805405 − 0.327127i −0.429369 − 1.91592i 1.95593 − 0.0539148i 1.12244 + 0.284241i 1.76802 − 1.22592i −1.65326 − 0.836650i
5.18 2.36933 + 0.196328i −0.975188 − 3.11785i 3.60246 + 0.601144i 2.76693 + 1.12383i −1.69842 − 7.57867i −3.57084 + 0.0984295i 3.80799 + 0.964314i −6.30466 + 4.37156i 6.33512 + 3.20595i
5.19 2.67732 + 0.221849i 0.211967 + 0.677695i 5.14608 + 0.858729i −1.50200 − 0.610059i 0.417156 + 1.86143i −4.60451 + 0.126923i 8.37861 + 2.12175i 2.05099 − 1.42213i −3.88598 − 1.96654i
12.1 −2.71928 − 0.225326i 1.22387 + 0.274276i 5.37096 + 0.896255i −0.604655 − 4.36046i −3.26625 − 1.02160i −0.610804 − 0.993666i −9.11298 − 2.30772i −1.29043 − 0.608970i 0.661702 + 11.9935i
See next 80 embeddings (of 684 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.19
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
229.k even 114 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 229.2.k.a ✓ 684
229.k even 114 1 inner 229.2.k.a ✓ 684
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
229.2.k.a ✓ 684 1.a even 1 1 trivial
229.2.k.a ✓ 684 229.k even 114 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(229, [\chi])\).