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Properties

Label 473.2.bb.a
Level $473$
Weight $2$
Character orbit 473.bb
Analytic conductor $3.777$
Analytic rank $0$
Dimension $1008$
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] = [473,2,Mod(2,473)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(473, base_ring=CyclotomicField(70))
 
chi = DirichletCharacter(H, H._module([7, 45]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("473.2");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 473 = 11 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 473.bb (of order \(70\), degree \(24\), 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: \(3.77692401561\)
Analytic rank: \(0\)
Dimension: \(1008\)
Relative dimension: \(42\) over \(\Q(\zeta_{70})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{70}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1008 q - 35 q^{2} - 21 q^{3} + 21 q^{4} - 21 q^{5} - 50 q^{6} - 105 q^{8} - 65 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1008 q - 35 q^{2} - 21 q^{3} + 21 q^{4} - 21 q^{5} - 50 q^{6} - 105 q^{8} - 65 q^{9} - 24 q^{11} - 84 q^{12} - 20 q^{13} + 19 q^{14} + 17 q^{15} - 39 q^{16} - 35 q^{17} - 35 q^{18} + 70 q^{19} + 7 q^{20} - 49 q^{22} - 58 q^{23} + 30 q^{24} - 65 q^{25} - 7 q^{26} - 21 q^{27} - 35 q^{28} - 35 q^{29} - 35 q^{30} - 13 q^{31} - 126 q^{33} - 14 q^{34} + 65 q^{35} + 176 q^{36} - 77 q^{38} - 35 q^{39} + 5 q^{40} - 25 q^{41} + 332 q^{44} + 98 q^{45} - 35 q^{46} - 51 q^{47} + 63 q^{48} - 178 q^{49} - 35 q^{51} + 95 q^{52} - 75 q^{53} - 98 q^{55} - 62 q^{56} - 55 q^{57} - 95 q^{58} - 113 q^{59} + 39 q^{60} - 35 q^{62} - 35 q^{63} + 17 q^{64} - 206 q^{66} - 150 q^{67} - 470 q^{68} + 147 q^{69} - 140 q^{70} - 7 q^{71} - 245 q^{72} - 35 q^{73} - 135 q^{74} + 56 q^{75} + 70 q^{77} - 258 q^{78} + 202 q^{81} + 105 q^{82} - 50 q^{83} - 10 q^{84} + 187 q^{86} + 161 q^{88} - 42 q^{89} - 295 q^{90} + 77 q^{91} + 108 q^{92} + 385 q^{94} - 195 q^{95} - 105 q^{96} + 152 q^{97} - 83 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −0.372583 − 2.75052i −0.0959763 − 0.528873i −5.49860 + 1.51752i 0.152095 + 1.68992i −1.41892 + 0.461033i 3.09656 + 2.24978i 4.04085 + 9.45405i 2.53821 − 0.952606i 4.59148 − 1.04798i
2.2 −0.362749 − 2.67792i 0.513347 + 2.82877i −5.11175 + 1.41075i −0.142181 − 1.57976i 7.38901 − 2.40084i −0.688627 − 0.500317i 3.50796 + 8.20728i −4.92973 + 1.85016i −4.17889 + 0.953805i
2.3 −0.342918 − 2.53152i −0.506523 − 2.79117i −4.36309 + 1.20414i −0.105428 − 1.17140i −6.89222 + 2.23942i −0.320535 − 0.232882i 2.53641 + 5.93422i −4.72536 + 1.77346i −2.92926 + 0.668585i
2.4 −0.322780 − 2.38286i 0.110576 + 0.609324i −3.64591 + 1.00621i −0.279544 − 3.10599i 1.41624 − 0.460165i −0.654933 − 0.475837i 1.68432 + 3.94066i 2.44966 − 0.919371i −7.31091 + 1.66867i
2.5 −0.307106 − 2.26715i 0.000236830 0.00130504i −3.11771 + 0.860434i 0.264418 + 2.93792i 0.00288598 0.000937712i −1.62388 − 1.17981i 1.10983 + 2.59658i 2.80870 − 1.05412i 6.57950 − 1.50173i
2.6 −0.295635 − 2.18246i 0.255543 + 1.40816i −2.74783 + 0.758352i 0.228913 + 2.54343i 2.99770 − 0.974013i −2.21775 − 1.61129i 0.736234 + 1.72251i 0.891103 − 0.334437i 5.48328 − 1.25152i
2.7 −0.273013 − 2.01546i 0.524432 + 2.88986i −2.05962 + 0.568418i 0.168353 + 1.87055i 5.68122 − 1.84594i 3.23413 + 2.34973i 0.109204 + 0.255496i −5.26755 + 1.97694i 3.72406 − 0.849993i
2.8 −0.269715 − 1.99111i −0.481201 − 2.65164i −1.96386 + 0.541990i 0.239058 + 2.65615i −5.14992 + 1.67331i −1.86622 − 1.35589i 0.0294362 + 0.0688694i −3.99091 + 1.49781i 5.22421 − 1.19239i
2.9 −0.260060 − 1.91984i −0.361699 − 1.99313i −1.69023 + 0.466475i −0.260769 − 2.89738i −3.73242 + 1.21274i 0.918059 + 0.667009i −0.187755 − 0.439274i −1.03303 + 0.387701i −5.49469 + 1.25413i
2.10 −0.250506 − 1.84931i −0.126174 − 0.695275i −1.42926 + 0.394451i 0.127317 + 1.41460i −1.25417 + 0.407504i 2.71157 + 1.97007i −0.379425 − 0.887710i 2.34122 − 0.878674i 2.58414 − 0.589814i
2.11 −0.228008 − 1.68322i 0.417127 + 2.29856i −0.853321 + 0.235502i −0.0721221 − 0.801342i 3.77387 − 1.22621i −2.75063 − 1.99845i −0.744215 − 1.74118i −2.30067 + 0.863456i −1.33239 + 0.304110i
2.12 −0.182674 − 1.34855i −0.0794355 − 0.437725i 0.142699 − 0.0393825i −0.260312 − 2.89230i −0.575785 + 0.187084i −3.11194 − 2.26095i −1.14889 − 2.68796i 2.62341 − 0.984583i −3.85287 + 0.879392i
2.13 −0.169153 − 1.24874i 0.347366 + 1.91414i 0.397191 − 0.109618i −0.220300 − 2.44773i 2.33151 − 0.757552i 3.34902 + 2.43321i −1.19461 − 2.79492i −0.734577 + 0.275691i −3.01932 + 0.689139i
2.14 −0.159508 − 1.17754i −0.219944 − 1.21199i 0.566778 − 0.156421i 0.130059 + 1.44508i −1.39208 + 0.452314i −0.616162 − 0.447668i −1.20865 − 2.82778i 1.38816 − 0.520986i 1.68088 − 0.383651i
2.15 −0.125050 − 0.923153i 0.226607 + 1.24871i 1.09135 − 0.301194i 0.0615785 + 0.684193i 1.12441 − 0.365343i 0.261946 + 0.190315i −1.14679 − 2.68306i 1.30079 − 0.488194i 0.623915 − 0.142404i
2.16 −0.104678 − 0.772764i −0.507692 − 2.79761i 1.34172 − 0.370291i 0.246212 + 2.73564i −2.10875 + 0.685174i 4.00390 + 2.90901i −1.03957 − 2.43221i −4.76018 + 1.78652i 2.08823 − 0.476626i
2.17 −0.0774585 − 0.571822i 0.541596 + 2.98444i 1.60695 − 0.443489i 0.295074 + 3.27854i 1.66462 − 0.540867i −0.530962 − 0.385767i −0.831654 − 1.94575i −5.80485 + 2.17860i 1.85189 − 0.422681i
2.18 −0.0740138 − 0.546392i −0.0159061 − 0.0876497i 1.63486 − 0.451193i −0.0741801 − 0.824208i −0.0467138 + 0.0151782i 1.61609 + 1.17415i −0.800944 − 1.87390i 2.80128 − 1.05134i −0.444850 + 0.101534i
2.19 −0.0602079 − 0.444473i −0.557399 − 3.07152i 1.73399 − 0.478552i −0.0730657 − 0.811827i −1.33165 + 0.432679i −2.21122 − 1.60654i −0.669672 − 1.56678i −6.31486 + 2.37001i −0.356435 + 0.0813541i
2.20 −0.00163409 − 0.0120634i 0.398539 + 2.19613i 1.92778 − 0.532034i −0.324212 − 3.60229i 0.0258415 − 0.00839641i −1.70529 − 1.23896i −0.0191373 − 0.0447740i −1.85546 + 0.696365i −0.0429259 + 0.00979757i
See next 80 embeddings (of 1008 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.42
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner
43.f odd 14 1 inner
473.bb even 70 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 473.2.bb.a ✓ 1008
11.d odd 10 1 inner 473.2.bb.a ✓ 1008
43.f odd 14 1 inner 473.2.bb.a ✓ 1008
473.bb even 70 1 inner 473.2.bb.a ✓ 1008
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
473.2.bb.a ✓ 1008 1.a even 1 1 trivial
473.2.bb.a ✓ 1008 11.d odd 10 1 inner
473.2.bb.a ✓ 1008 43.f odd 14 1 inner
473.2.bb.a ✓ 1008 473.bb even 70 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(473, [\chi])\).