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Properties

Label 451.2.bu.a
Level $451$
Weight $2$
Character orbit 451.bu
Analytic conductor $3.601$
Analytic rank $0$
Dimension $640$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [451,2,Mod(68,451)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(451, base_ring=CyclotomicField(40))
 
chi = DirichletCharacter(H, H._module([4, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("451.68");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 451 = 11 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 451.bu (of order \(40\), degree \(16\), 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: \(640\)
Relative dimension: \(40\) over \(\Q(\zeta_{40})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{40}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 640 q - 20 q^{2} - 20 q^{3} - 12 q^{5} - 20 q^{6} - 20 q^{7} - 20 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 640 q - 20 q^{2} - 20 q^{3} - 12 q^{5} - 20 q^{6} - 20 q^{7} - 20 q^{8} + 12 q^{9} - 16 q^{11} - 24 q^{12} - 40 q^{13} + 12 q^{14} - 12 q^{15} + 112 q^{16} - 20 q^{17} - 40 q^{18} - 20 q^{19} - 20 q^{20} - 116 q^{22} + 40 q^{24} - 28 q^{26} - 44 q^{27} - 20 q^{28} - 20 q^{29} - 40 q^{30} - 68 q^{33} - 96 q^{34} - 20 q^{35} + 24 q^{36} + 48 q^{37} - 52 q^{38} - 20 q^{39} - 20 q^{41} - 56 q^{42} + 56 q^{44} + 20 q^{46} + 40 q^{47} + 156 q^{48} + 60 q^{49} + 160 q^{50} - 120 q^{52} - 28 q^{53} - 72 q^{55} - 224 q^{56} - 200 q^{57} + 20 q^{58} - 24 q^{59} + 124 q^{60} - 100 q^{61} - 100 q^{62} + 140 q^{63} + 8 q^{67} - 20 q^{68} - 8 q^{69} - 72 q^{70} - 76 q^{71} - 204 q^{75} - 44 q^{77} - 128 q^{78} - 80 q^{79} + 148 q^{80} + 88 q^{82} + 80 q^{83} + 140 q^{84} - 260 q^{85} - 384 q^{88} - 64 q^{89} + 200 q^{90} - 240 q^{91} + 8 q^{92} + 32 q^{93} + 220 q^{94} + 220 q^{95} + 260 q^{96} - 40 q^{97} + 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
68.1 −1.27111 − 2.49469i −1.03584 + 1.69033i −3.43221 + 4.72403i −2.75997 − 1.40627i 5.53352 + 0.435498i −0.948106 + 3.94914i 10.6170 + 1.68156i −0.422291 − 0.828794i 8.67281i
68.2 −1.20534 − 2.36561i 1.67582 − 2.73469i −2.96771 + 4.08471i 1.97545 + 1.00654i −8.48916 − 0.668111i 0.297211 − 1.23797i 7.99533 + 1.26634i −3.30819 − 6.49269i − 5.88639i
68.3 −1.15012 − 2.25723i −1.02294 + 1.66929i −2.59676 + 3.57414i 3.19337 + 1.62710i 4.94448 + 0.389139i 0.194298 − 0.809310i 6.04993 + 0.958215i −0.378146 − 0.742154i − 9.07955i
68.4 −1.13594 − 2.22941i 0.0996230 − 0.162570i −2.50433 + 3.44692i 0.470995 + 0.239984i −0.475601 − 0.0374306i 0.409828 − 1.70706i 5.58672 + 0.884850i 1.34547 + 2.64063i − 1.32265i
68.5 −1.10687 − 2.17235i 0.686293 − 1.11993i −2.31837 + 3.19096i 0.450428 + 0.229504i −3.19251 − 0.251256i −0.927443 + 3.86308i 4.68187 + 0.741536i 0.578730 + 1.13582i − 1.23252i
68.6 −0.921979 − 1.80949i −1.38896 + 2.26658i −1.24862 + 1.71858i 1.08390 + 0.552276i 5.38193 + 0.423567i 0.222451 − 0.926576i 0.249293 + 0.0394841i −1.84619 − 3.62336i − 2.47049i
68.7 −0.894677 − 1.75590i −0.989893 + 1.61536i −1.10718 + 1.52390i −3.35209 − 1.70797i 3.72205 + 0.292932i 0.761169 − 3.17050i −0.226478 − 0.0358706i −0.267523 − 0.525043i 7.41403i
68.8 −0.883817 − 1.73459i 1.40438 − 2.29175i −1.05210 + 1.44809i −3.85435 − 1.96389i −5.21646 − 0.410544i −0.225234 + 0.938166i −0.403917 − 0.0639742i −1.91783 − 3.76396i 8.42143i
68.9 −0.835224 − 1.63922i 0.449055 − 0.732791i −0.813871 + 1.12020i −0.926292 − 0.471969i −1.57627 − 0.124055i −0.412910 + 1.71989i −1.11816 − 0.177100i 1.02664 + 2.01489i 1.91260i
68.10 −0.738608 − 1.44960i 1.47473 − 2.40654i −0.380226 + 0.523336i 0.0575152 + 0.0293055i −4.57777 − 0.360278i 0.387504 − 1.61407i −2.17432 − 0.344379i −2.25465 − 4.42499i − 0.105019i
68.11 −0.714332 − 1.40196i 0.532482 − 0.868932i −0.279638 + 0.384889i 3.34975 + 1.70678i −1.59857 − 0.125810i 1.00998 − 4.20689i −2.36881 − 0.375182i 0.890466 + 1.74764i − 5.91541i
68.12 −0.687795 − 1.34987i −1.70499 + 2.78230i −0.173527 + 0.238839i −0.350584 − 0.178631i 4.92844 + 0.387876i −0.967624 + 4.03044i −2.55094 − 0.404029i −3.47220 − 6.81459i 0.596105i
68.13 −0.581355 − 1.14097i −0.271087 + 0.442374i 0.211724 − 0.291413i −2.27145 − 1.15736i 0.662335 + 0.0521269i −0.672473 + 2.80105i −2.98514 − 0.472800i 1.23976 + 2.43318i 3.26450i
68.14 −0.563880 − 1.10668i −0.455308 + 0.742995i 0.268798 − 0.369969i −0.344499 − 0.175531i 1.07899 + 0.0849186i 0.708345 − 2.95047i −3.01453 − 0.477454i 1.01724 + 1.99644i 0.480227i
68.15 −0.541693 − 1.06313i −0.272663 + 0.444946i 0.338752 − 0.466252i 2.20689 + 1.12446i 0.620736 + 0.0488530i −0.212540 + 0.885293i −3.03617 − 0.480882i 1.23834 + 2.43038i − 2.95532i
68.16 −0.393610 − 0.772503i 1.28404 − 2.09537i 0.733739 − 1.00990i 2.68807 + 1.36964i −2.12409 − 0.167169i −0.652148 + 2.71639i −2.78161 − 0.440564i −1.37982 − 2.70805i − 2.61564i
68.17 −0.258390 − 0.507120i −1.53382 + 2.50297i 0.985166 − 1.35596i 0.0484184 + 0.0246704i 1.66563 + 0.131088i 0.779099 − 3.24518i −2.06649 − 0.327299i −2.55026 − 5.00517i − 0.0309285i
68.18 −0.169519 − 0.332700i 0.834773 − 1.36223i 1.09362 − 1.50524i −2.27121 − 1.15724i −0.594723 − 0.0468057i 0.784942 − 3.26952i −1.42378 − 0.225505i 0.203159 + 0.398723i 0.951806i
68.19 −0.116819 − 0.229270i −0.239098 + 0.390172i 1.13665 − 1.56447i −1.56356 − 0.796672i 0.117386 + 0.00923846i 0.0368148 − 0.153345i −0.999762 − 0.158347i 1.26690 + 2.48644i 0.451542i
68.20 −0.113445 − 0.222649i −0.908609 + 1.48272i 1.13887 − 1.56752i 2.85301 + 1.45368i 0.433202 + 0.0340937i −0.760246 + 3.16665i −0.971821 − 0.153921i −0.0109026 − 0.0213975i − 0.800132i
See next 80 embeddings (of 640 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 68.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner
41.e odd 8 1 inner
451.bu even 40 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 451.2.bu.a ✓ 640
11.d odd 10 1 inner 451.2.bu.a ✓ 640
41.e odd 8 1 inner 451.2.bu.a ✓ 640
451.bu even 40 1 inner 451.2.bu.a ✓ 640
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
451.2.bu.a ✓ 640 1.a even 1 1 trivial
451.2.bu.a ✓ 640 11.d odd 10 1 inner
451.2.bu.a ✓ 640 41.e odd 8 1 inner
451.2.bu.a ✓ 640 451.bu even 40 1 inner

Hecke kernels

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