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Properties

Label 451.2.bo.a
Level $451$
Weight $2$
Character orbit 451.bo
Analytic conductor $3.601$
Analytic rank $0$
Dimension $320$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [451,2,Mod(20,451)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(451, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([12, 17]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("451.20");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 451 = 11 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 451.bo (of order \(20\), degree \(8\), 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: \(3.60125313116\)
Analytic rank: \(0\)
Dimension: \(320\)
Relative dimension: \(40\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 320 q - 10 q^{2} - 12 q^{3} + 74 q^{4} - 10 q^{5} - 8 q^{6} - 6 q^{7} - 10 q^{8} - 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 320 q - 10 q^{2} - 12 q^{3} + 74 q^{4} - 10 q^{5} - 8 q^{6} - 6 q^{7} - 10 q^{8} - 30 q^{9} - 8 q^{10} + 4 q^{11} - 18 q^{12} - 14 q^{13} - 6 q^{14} + 4 q^{15} - 86 q^{16} - 28 q^{17} - 30 q^{18} + 32 q^{19} - 20 q^{21} - 64 q^{22} - 20 q^{23} - 34 q^{24} + 62 q^{25} - 6 q^{26} - 6 q^{27} + 38 q^{28} - 18 q^{29} - 30 q^{30} + 8 q^{31} + 70 q^{33} - 4 q^{34} - 20 q^{35} - 16 q^{37} + 42 q^{38} + 22 q^{40} - 24 q^{41} - 20 q^{43} + 62 q^{44} - 84 q^{45} - 50 q^{46} + 6 q^{48} + 50 q^{49} - 20 q^{50} + 156 q^{51} - 98 q^{52} - 18 q^{53} - 154 q^{54} - 20 q^{55} - 46 q^{56} + 136 q^{57} + 34 q^{58} - 2 q^{59} - 80 q^{60} + 30 q^{61} - 50 q^{62} - 6 q^{63} + 86 q^{64} - 60 q^{65} + 38 q^{66} - 76 q^{67} - 76 q^{68} + 28 q^{69} - 18 q^{70} - 20 q^{71} + 6 q^{72} + 156 q^{75} - 6 q^{76} - 10 q^{77} + 60 q^{78} - 24 q^{79} + 130 q^{80} + 22 q^{81} + 86 q^{82} - 84 q^{83} - 530 q^{84} + 30 q^{85} - 32 q^{86} - 100 q^{87} + 110 q^{88} - 34 q^{89} + 30 q^{90} + 10 q^{91} - 84 q^{92} + 18 q^{93} + 226 q^{94} + 32 q^{95} - 302 q^{96} + 32 q^{97} - 100 q^{98} + 74 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
20.1 −1.56303 + 2.15133i 0.354098 − 2.23569i −1.56712 − 4.82310i −2.29952 − 0.747160i 4.25624 + 4.25624i 1.95155 + 3.83013i 7.76748 + 2.52381i −2.01974 − 0.656254i 5.20162 − 3.77920i
20.2 −1.54242 + 2.12297i −0.0230637 + 0.145618i −1.50987 − 4.64691i −1.85197 − 0.601743i −0.273569 − 0.273569i −0.843189 − 1.65485i 7.20271 + 2.34030i 2.83250 + 0.920334i 4.13401 − 3.00353i
20.3 −1.48338 + 2.04170i −0.493688 + 3.11702i −1.35008 − 4.15511i 1.03204 + 0.335331i −5.63169 − 5.63169i −2.18911 − 4.29637i 5.68584 + 1.84744i −6.61893 − 2.15062i −2.21555 + 1.60969i
20.4 −1.46132 + 2.01133i 0.354175 − 2.23617i −1.29197 − 3.97628i 3.14442 + 1.02168i 3.98013 + 3.98013i −0.659994 − 1.29531i 5.15668 + 1.67551i −2.02187 − 0.656944i −6.64994 + 4.83146i
20.5 −1.34773 + 1.85499i −0.133517 + 0.842995i −1.00658 − 3.09793i 1.20477 + 0.391453i −1.38380 − 1.38380i 0.812441 + 1.59450i 2.74188 + 0.890892i 2.16036 + 0.701942i −2.34984 + 1.70726i
20.6 −1.33025 + 1.83093i −0.435649 + 2.75058i −0.964711 − 2.96907i −0.427661 − 0.138955i −4.45660 − 4.45660i 1.90197 + 3.73282i 2.41470 + 0.784584i −4.52274 − 1.46953i 0.823313 − 0.598172i
20.7 −1.23756 + 1.70336i 0.417391 − 2.63530i −0.751839 − 2.31392i −0.508062 − 0.165079i 3.97232 + 3.97232i −0.702522 − 1.37878i 0.867054 + 0.281723i −3.91743 − 1.27285i 0.909949 − 0.661117i
20.8 −1.02530 + 1.41120i 0.0666472 − 0.420794i −0.322218 − 0.991687i 1.07503 + 0.349298i 0.525492 + 0.525492i −1.67643 − 3.29017i −1.58809 − 0.516003i 2.68054 + 0.870961i −1.59516 + 1.15895i
20.9 −0.983780 + 1.35406i −0.0822230 + 0.519136i −0.247613 − 0.762076i −0.719584 − 0.233807i −0.622050 − 0.622050i 0.942762 + 1.85027i −1.90809 − 0.619975i 2.59043 + 0.841681i 1.02450 − 0.744343i
20.10 −0.978152 + 1.34631i −0.271597 + 1.71480i −0.237737 − 0.731679i 4.12669 + 1.34084i −2.04299 − 2.04299i −0.357410 − 0.701458i −1.94776 − 0.632864i −0.0135972 − 0.00441801i −5.84172 + 4.24426i
20.11 −0.933697 + 1.28512i 0.144270 − 0.910882i −0.161718 − 0.497717i −2.68067 − 0.871001i 1.03589 + 1.03589i 0.483018 + 0.947975i −2.23088 − 0.724858i 2.04428 + 0.664226i 3.62227 − 2.63173i
20.12 −0.847159 + 1.16601i −0.295334 + 1.86466i −0.0238766 − 0.0734845i −3.44502 − 1.11935i −1.92403 − 1.92403i −1.04058 − 2.04225i −2.63555 − 0.856343i −0.536584 − 0.174347i 4.22366 − 3.06867i
20.13 −0.740079 + 1.01863i 0.493114 − 3.11340i 0.128141 + 0.394378i 0.154129 + 0.0500795i 2.80646 + 2.80646i −0.332570 − 0.652705i −2.89151 − 0.939507i −6.59694 − 2.14348i −0.165080 + 0.119938i
20.14 −0.645181 + 0.888015i 0.263429 − 1.66322i 0.245721 + 0.756252i 2.07537 + 0.674328i 1.30701 + 1.30701i 1.52127 + 2.98565i −2.91795 − 0.948099i 0.156253 + 0.0507697i −1.93780 + 1.40790i
20.15 −0.416093 + 0.572702i −0.325002 + 2.05198i 0.463179 + 1.42552i −0.568505 − 0.184719i −1.03994 − 1.03994i −1.42038 − 2.78765i −2.35563 − 0.765390i −1.25184 − 0.406746i 0.342339 − 0.248724i
20.16 −0.405108 + 0.557583i 0.210083 − 1.32641i 0.471248 + 1.45035i −3.21608 − 1.04497i 0.654477 + 0.654477i 0.848952 + 1.66616i −2.31055 − 0.750744i 1.13794 + 0.369740i 1.88552 − 1.36991i
20.17 −0.302403 + 0.416222i −0.166137 + 1.04895i 0.536241 + 1.65038i 1.65187 + 0.536727i −0.386355 − 0.386355i 1.77400 + 3.48166i −1.82768 − 0.593850i 1.78048 + 0.578513i −0.722929 + 0.525239i
20.18 −0.268350 + 0.369353i −0.499685 + 3.15488i 0.553625 + 1.70388i 0.669680 + 0.217592i −1.03117 − 1.03117i 0.0853913 + 0.167590i −1.64630 − 0.534915i −6.85044 − 2.22584i −0.260077 + 0.188957i
20.19 −0.131312 + 0.180736i 0.362448 − 2.28841i 0.602611 + 1.85465i −2.18271 − 0.709207i 0.366003 + 0.366003i −2.18016 − 4.27881i −0.839266 − 0.272694i −2.25227 − 0.731808i 0.414796 − 0.301367i
20.20 −0.0641715 + 0.0883245i 0.196345 − 1.23967i 0.614351 + 1.89078i 3.60294 + 1.17067i 0.0968936 + 0.0968936i −0.812648 − 1.59491i −0.414089 − 0.134546i 1.35493 + 0.440245i −0.334605 + 0.243105i
See next 80 embeddings (of 320 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 20.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
451.bo even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 451.2.bo.a yes 320
11.c even 5 1 451.2.bg.a ✓ 320
41.g even 20 1 451.2.bg.a ✓ 320
451.bo even 20 1 inner 451.2.bo.a yes 320
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
451.2.bg.a ✓ 320 11.c even 5 1
451.2.bg.a ✓ 320 41.g even 20 1
451.2.bo.a yes 320 1.a even 1 1 trivial
451.2.bo.a yes 320 451.bo even 20 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(451, [\chi])\).