Properties

Label 473.2.r.b
Level $473$
Weight $2$
Character orbit 473.r
Analytic conductor $3.777$
Analytic rank $0$
Dimension $216$
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] = [473,2,Mod(23,473)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(473, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([0, 16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("473.23");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 473 = 11 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 473.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: \(3.77692401561\)
Analytic rank: \(0\)
Dimension: \(216\)
Relative dimension: \(18\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 216 q + 2 q^{2} - q^{3} - 24 q^{4} - 2 q^{5} - 2 q^{6} + 25 q^{7} + 6 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 216 q + 2 q^{2} - q^{3} - 24 q^{4} - 2 q^{5} - 2 q^{6} + 25 q^{7} + 6 q^{8} + q^{9} + 40 q^{10} + 36 q^{11} + 36 q^{12} + q^{13} - 30 q^{14} - 48 q^{16} + 7 q^{18} + 2 q^{19} - 160 q^{20} - 24 q^{21} - 2 q^{22} - q^{23} + 22 q^{24} + 10 q^{25} + 23 q^{26} + 20 q^{27} - 6 q^{28} - 25 q^{29} + 2 q^{30} - 43 q^{31} - 46 q^{32} + q^{33} - 30 q^{34} - 4 q^{35} - 54 q^{36} + 93 q^{37} + 28 q^{38} + 5 q^{39} - 19 q^{40} - 24 q^{41} + 8 q^{42} - 118 q^{43} - 200 q^{44} + 2 q^{45} + 80 q^{46} + 30 q^{47} + 17 q^{48} - 91 q^{49} + 75 q^{50} - 16 q^{51} - 21 q^{52} + 93 q^{53} + 2 q^{54} + 2 q^{55} + 80 q^{56} - 60 q^{57} - 45 q^{58} - 22 q^{59} - 142 q^{60} - 15 q^{61} - 43 q^{62} - 199 q^{63} - 86 q^{64} - 34 q^{65} + 2 q^{66} - 23 q^{67} + 38 q^{68} + 42 q^{69} - 4 q^{70} - q^{71} + 148 q^{72} + 25 q^{73} - 73 q^{74} + 28 q^{75} + 81 q^{76} + 3 q^{77} + 154 q^{78} + 55 q^{79} + 151 q^{80} + 56 q^{81} + 32 q^{82} - 47 q^{83} + 221 q^{84} - 278 q^{85} - 203 q^{86} + 374 q^{87} - 6 q^{88} - 52 q^{89} + 172 q^{90} + 43 q^{91} + 24 q^{92} + 61 q^{93} + 129 q^{94} - 35 q^{95} + 87 q^{96} - 51 q^{97} + 167 q^{98} - 15 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −1.72094 2.15799i 0.802556 + 2.04488i −1.25025 + 5.47770i −0.246553 + 3.29002i 3.03168 5.25103i 1.87345 + 3.24491i 8.99879 4.33359i −1.33828 + 1.24174i 7.52415 5.12988i
23.2 −1.42694 1.78932i −1.22224 3.11421i −0.720483 + 3.15664i −0.221198 + 2.95168i −3.82827 + 6.63076i 0.202081 + 0.350015i 2.55236 1.22915i −6.00530 + 5.57210i 5.59714 3.81607i
23.3 −1.39482 1.74905i 0.461782 + 1.17660i −0.668609 + 2.92937i −0.0302423 + 0.403555i 1.41383 2.44883i −1.15712 2.00420i 2.02505 0.975214i 1.02801 0.953850i 0.748021 0.509992i
23.4 −1.38695 1.73919i −0.170874 0.435379i −0.656083 + 2.87449i 0.259261 3.45959i −0.520211 + 0.901031i 0.492664 + 0.853319i 1.90081 0.915381i 2.03880 1.89173i −6.37646 + 4.34740i
23.5 −1.12187 1.40678i −0.552247 1.40710i −0.275394 + 1.20658i −0.0144412 + 0.192705i −1.35993 + 2.35547i 2.29079 + 3.96777i −1.23595 + 0.595201i 0.524198 0.486384i 0.287294 0.195873i
23.6 −0.793381 0.994868i −0.717102 1.82715i 0.0847322 0.371236i −0.0370762 + 0.494748i −1.24883 + 2.16305i −2.08902 3.61829i −2.72949 + 1.31445i −0.625071 + 0.579981i 0.521625 0.355638i
23.7 −0.767971 0.963006i 1.01559 + 2.58768i 0.107442 0.470734i 0.266362 3.55435i 1.71201 2.96528i 2.33563 + 4.04542i −2.75533 + 1.32690i −3.46551 + 3.21552i −3.62742 + 2.47313i
23.8 −0.251971 0.315962i 0.494965 + 1.26115i 0.408699 1.79063i −0.282298 + 3.76700i 0.273759 0.474164i −0.816165 1.41364i −1.39697 + 0.672745i 0.853646 0.792068i 1.26136 0.859981i
23.9 −0.125794 0.157741i −0.0118994 0.0303191i 0.435984 1.91017i 0.0924840 1.23411i −0.00328568 + 0.00569096i 0.165685 + 0.286975i −0.719710 + 0.346594i 2.19838 2.03980i −0.206304 + 0.140655i
23.10 0.0685627 + 0.0859749i −0.619315 1.57799i 0.442351 1.93807i 0.294853 3.93453i 0.0932055 0.161437i −0.767294 1.32899i 0.395106 0.190273i 0.0926591 0.0859751i 0.358487 0.244412i
23.11 0.279204 + 0.350110i 0.346563 + 0.883028i 0.400419 1.75435i −0.0265570 + 0.354379i −0.212396 + 0.367880i 1.98882 + 3.44473i 1.53294 0.738224i 1.53952 1.42847i −0.131487 + 0.0896460i
23.12 0.458566 + 0.575023i 1.19884 + 3.05458i 0.324673 1.42248i −0.0138241 + 0.184470i −1.20671 + 2.09008i 0.106488 + 0.184443i 2.29214 1.10384i −5.69411 + 5.28336i −0.112414 + 0.0766425i
23.13 0.984171 + 1.23411i −0.618559 1.57606i −0.109397 + 0.479298i −0.182040 + 2.42915i 1.33627 2.31449i 0.374993 + 0.649507i 2.14516 1.03306i 0.0977958 0.0907412i −3.17700 + 2.16604i
23.14 0.999056 + 1.25278i −1.04783 2.66984i −0.126295 + 0.553333i 0.142046 1.89548i 2.29787 3.98002i 0.104383 + 0.180797i 2.06798 0.995886i −3.83093 + 3.55458i 2.51652 1.71574i
23.15 1.00822 + 1.26427i 0.243777 + 0.621133i −0.136823 + 0.599461i 0.140295 1.87211i −0.539498 + 0.934438i −2.29484 3.97479i 2.01801 0.971823i 1.87278 1.73768i 2.50830 1.71013i
23.16 1.30329 + 1.63428i 0.873208 + 2.22490i −0.527249 + 2.31003i 0.0204355 0.272693i −2.49806 + 4.32676i 0.601730 + 1.04223i −0.695765 + 0.335063i −1.98852 + 1.84508i 0.472290 0.322002i
23.17 1.56017 + 1.95640i 0.254139 + 0.647536i −0.948303 + 4.15479i −0.242018 + 3.22950i −0.870337 + 1.50747i −1.45912 2.52727i −5.09890 + 2.45550i 1.84444 1.71139i −6.69578 + 4.56511i
23.18 1.70590 + 2.13913i −1.09669 2.79431i −1.22075 + 5.34845i −0.0689497 + 0.920070i 4.10657 7.11279i 0.873097 + 1.51225i −8.59332 + 4.13832i −4.40631 + 4.08846i −2.08577 + 1.42206i
56.1 −0.586117 + 2.56795i −2.10263 + 1.95096i −4.44888 2.14247i 1.69476 + 0.255443i −3.77757 6.54294i 0.715890 1.23996i 4.82478 6.05008i 0.390637 5.21268i −1.64929 + 4.20232i
56.2 −0.576843 + 2.52732i 0.912126 0.846329i −4.25264 2.04796i −1.32344 0.199476i 1.61279 + 2.79343i 1.41640 2.45328i 4.39640 5.51291i −0.108490 + 1.44769i 1.26756 3.22968i
See next 80 embeddings (of 216 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.g even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 473.2.r.b 216
43.g even 21 1 inner 473.2.r.b 216
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
473.2.r.b 216 1.a even 1 1 trivial
473.2.r.b 216 43.g even 21 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{216} - 2 T_{2}^{215} + 50 T_{2}^{214} - 102 T_{2}^{213} + 1399 T_{2}^{212} - 2892 T_{2}^{211} + 28839 T_{2}^{210} - 59524 T_{2}^{209} + 486847 T_{2}^{208} - 992249 T_{2}^{207} + 7115853 T_{2}^{206} + \cdots + 23\!\cdots\!61 \) acting on \(S_{2}^{\mathrm{new}}(473, [\chi])\). Copy content Toggle raw display