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Properties

Label 451.2.cc.a
Level $451$
Weight $2$
Character orbit 451.cc
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(54,451)]
 
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([20, 31]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("451.54");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 451 = 11 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 451.cc (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.

\(\operatorname{Tr}(f)(q) = \) \( 640 q - 40 q^{3} - 40 q^{4} - 32 q^{5} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 640 q - 40 q^{3} - 40 q^{4} - 32 q^{5} - 8 q^{9} - 16 q^{11} - 24 q^{12} - 48 q^{14} - 32 q^{15} + 112 q^{16} - 80 q^{20} + 4 q^{22} - 40 q^{23} - 40 q^{25} - 48 q^{26} - 64 q^{27} - 28 q^{33} - 96 q^{34} + 144 q^{36} - 112 q^{37} - 72 q^{38} + 224 q^{42} - 124 q^{44} - 40 q^{45} + 80 q^{47} - 224 q^{48} - 80 q^{49} - 8 q^{53} - 72 q^{55} - 24 q^{56} - 24 q^{59} - 176 q^{60} - 40 q^{64} + 60 q^{66} + 128 q^{67} - 128 q^{69} + 8 q^{70} - 96 q^{71} + 96 q^{75} + 36 q^{77} + 312 q^{78} - 312 q^{80} - 472 q^{82} + 160 q^{86} + 136 q^{88} + 96 q^{89} - 120 q^{91} - 152 q^{92} - 208 q^{93} + 68 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
54.1 −1.24049 + 2.43459i −0.631895 + 1.52553i −3.21286 − 4.42212i 4.26513 + 0.675531i −2.93018 − 3.43080i 0.181574 − 0.212596i 9.35402 − 1.48153i 0.193372 + 0.193372i −6.93548 + 9.54587i
54.2 −1.20014 + 2.35540i 0.247620 − 0.597808i −2.93202 − 4.03558i −1.87560 − 0.297066i 1.11090 + 1.30070i −2.68350 + 3.14198i 7.80226 − 1.23576i 1.82526 + 1.82526i 2.95069 − 4.06127i
54.3 −1.15567 + 2.26813i −0.651728 + 1.57341i −2.63326 − 3.62437i −1.76356 − 0.279321i −2.81551 − 3.29654i 2.59026 − 3.03281i 6.23521 − 0.987561i 0.0704482 + 0.0704482i 2.67163 − 3.67718i
54.4 −1.12901 + 2.21580i 1.00112 − 2.41692i −2.45955 − 3.38528i 1.91364 + 0.303090i 4.22514 + 4.94701i 0.311364 − 0.364560i 5.36550 − 0.849811i −2.71793 − 2.71793i −2.83210 + 3.89805i
54.5 −1.07597 + 2.11171i 0.721177 − 1.74108i −2.12604 − 2.92624i −3.35638 − 0.531599i 2.90068 + 3.39626i 1.44520 − 1.69211i 3.78523 − 0.599522i −0.389929 − 0.389929i 4.73395 − 6.51573i
54.6 −1.07457 + 2.10896i −1.21188 + 2.92574i −2.11743 − 2.91440i −1.97890 − 0.313427i −4.86802 − 5.69972i −2.17500 + 2.54660i 3.74607 − 0.593318i −4.97000 − 4.97000i 2.78746 − 3.83662i
54.7 −0.956812 + 1.87785i −0.0838687 + 0.202477i −1.43526 − 1.97546i 1.03527 + 0.163971i −0.299975 − 0.351225i −2.19399 + 2.56883i 0.919671 − 0.145662i 2.08736 + 2.08736i −1.29848 + 1.78720i
54.8 −0.849259 + 1.66676i 0.494462 − 1.19374i −0.881292 − 1.21300i 2.85514 + 0.452210i 1.56975 + 1.83794i 1.41362 − 1.65514i −0.925023 + 0.146509i 0.940803 + 0.940803i −3.17848 + 4.37481i
54.9 −0.821767 + 1.61281i −0.263586 + 0.636354i −0.750278 − 1.03267i 1.37676 + 0.218057i −0.809710 − 0.948048i 2.85434 − 3.34200i −1.29357 + 0.204882i 1.78585 + 1.78585i −1.48306 + 2.04125i
54.10 −0.756927 + 1.48555i −0.308393 + 0.744527i −0.458360 − 0.630879i −1.21958 − 0.193163i −0.872604 − 1.02169i 0.0889468 − 0.104143i −2.00935 + 0.318249i 1.66211 + 1.66211i 1.21009 − 1.66554i
54.11 −0.700281 + 1.37438i 1.16522 − 2.81308i −0.222955 − 0.306871i −1.85506 − 0.293813i 3.05026 + 3.57140i 0.0558545 − 0.0653972i −2.46914 + 0.391073i −4.43438 − 4.43438i 1.70288 − 2.34381i
54.12 −0.495073 + 0.971636i −1.09126 + 2.63455i 0.476591 + 0.655971i 2.45519 + 0.388864i −2.01956 − 2.36461i 1.27024 − 1.48726i −3.02745 + 0.479501i −3.62865 − 3.62865i −1.59333 + 2.19303i
54.13 −0.482559 + 0.947075i −1.14870 + 2.77321i 0.511483 + 0.703996i −2.19114 − 0.347043i −2.07212 − 2.42615i 1.50072 − 1.75712i −3.01324 + 0.477250i −4.24988 − 4.24988i 1.38603 − 1.90771i
54.14 −0.474018 + 0.930313i −0.510670 + 1.23287i 0.534781 + 0.736063i −3.99366 − 0.632534i −0.904885 − 1.05948i −1.80194 + 2.10981i −3.00079 + 0.475278i 0.862144 + 0.862144i 2.48152 − 3.41552i
54.15 −0.393241 + 0.771778i −0.769919 + 1.85875i 0.734567 + 1.01104i 2.73162 + 0.432646i −1.13178 − 1.32514i −2.80833 + 3.28814i −2.78021 + 0.440342i −0.740849 − 0.740849i −1.40809 + 1.93807i
54.16 −0.376551 + 0.739023i 0.699901 − 1.68971i 0.771206 + 1.06147i −1.11258 − 0.176215i 0.985188 + 1.15351i −1.16644 + 1.36572i −2.71328 + 0.429741i −0.243943 − 0.243943i 0.549169 − 0.755866i
54.17 −0.272725 + 0.535253i 0.304747 − 0.735725i 0.963453 + 1.32608i 1.88037 + 0.297822i 0.310687 + 0.363768i 0.436507 − 0.511084i −2.15921 + 0.341986i 1.67290 + 1.67290i −0.672234 + 0.925251i
54.18 −0.173351 + 0.340220i 1.20164 − 2.90101i 1.08987 + 1.50008i 2.60714 + 0.412930i 0.778677 + 0.911713i 2.97738 − 3.48607i −1.45356 + 0.230221i −4.85060 − 4.85060i −0.592436 + 0.815418i
54.19 −0.121086 + 0.237645i 0.455405 − 1.09945i 1.13376 + 1.56048i −3.40898 − 0.539929i 0.206135 + 0.241353i 3.09987 − 3.62948i −1.03499 + 0.163926i 1.11993 + 1.11993i 0.541092 − 0.744749i
54.20 −0.0855810 + 0.167962i −0.494261 + 1.19325i 1.15468 + 1.58929i −0.315871 − 0.0500291i −0.158122 − 0.185137i 1.28264 − 1.50178i −0.738134 + 0.116909i 0.941766 + 0.941766i 0.0354356 − 0.0487729i
See next 80 embeddings (of 640 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 54.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
41.h odd 40 1 inner
451.cc 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.cc.a ✓ 640
11.b odd 2 1 inner 451.2.cc.a ✓ 640
41.h odd 40 1 inner 451.2.cc.a ✓ 640
451.cc even 40 1 inner 451.2.cc.a ✓ 640
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
451.2.cc.a ✓ 640 1.a even 1 1 trivial
451.2.cc.a ✓ 640 11.b odd 2 1 inner
451.2.cc.a ✓ 640 41.h odd 40 1 inner
451.2.cc.a ✓ 640 451.cc even 40 1 inner

Hecke kernels

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