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Properties

Label 451.2.ca.a
Level $451$
Weight $2$
Character orbit 451.ca
Analytic conductor $3.601$
Analytic rank $0$
Dimension $640$
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(6,451)]
 
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([36, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("451.6");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 451 = 11 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 451.ca (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.

\(\operatorname{Tr}(f)(q) = \) \( 640 q - 20 q^{2} - 20 q^{4} - 12 q^{5} - 20 q^{6} - 20 q^{7} - 20 q^{8} - 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 640 q - 20 q^{2} - 20 q^{4} - 12 q^{5} - 20 q^{6} - 20 q^{7} - 20 q^{8} - 48 q^{9} - 16 q^{11} + 36 q^{12} - 60 q^{13} - 28 q^{14} - 12 q^{15} + 132 q^{16} + 40 q^{17} - 20 q^{18} - 20 q^{19} - 60 q^{20} + 60 q^{21} + 24 q^{22} - 40 q^{23} - 140 q^{24} - 20 q^{25} - 28 q^{26} - 24 q^{27} - 20 q^{28} - 20 q^{29} + 60 q^{30} - 40 q^{31} - 40 q^{32} + 32 q^{33} - 96 q^{34} - 20 q^{35} + 24 q^{36} - 32 q^{37} - 52 q^{38} - 20 q^{39} - 240 q^{40} - 20 q^{41} - 136 q^{42} - 4 q^{44} - 40 q^{45} + 60 q^{46} + 40 q^{47} + 76 q^{48} - 60 q^{49} - 60 q^{50} - 20 q^{52} - 28 q^{53} + 140 q^{54} - 72 q^{55} + 56 q^{56} + 20 q^{58} - 4 q^{59} + 4 q^{60} - 20 q^{61} + 60 q^{62} - 20 q^{63} - 20 q^{64} + 120 q^{65} - 20 q^{66} - 152 q^{67} + 200 q^{68} + 32 q^{69} + 168 q^{70} - 76 q^{71} - 20 q^{72} + 60 q^{73} + 280 q^{74} - 44 q^{75} + 100 q^{76} - 44 q^{77} - 208 q^{78} + 40 q^{79} - 12 q^{80} - 140 q^{81} + 28 q^{82} + 100 q^{83} + 460 q^{84} - 20 q^{85} - 220 q^{86} - 160 q^{87} - 64 q^{88} - 4 q^{89} - 120 q^{90} - 180 q^{91} + 8 q^{92} - 28 q^{93} - 20 q^{94} + 60 q^{95} - 680 q^{96} - 100 q^{97} + 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1 −0.419785 + 2.65042i 0.169024 − 0.704037i −4.94638 − 1.60718i 0.705708 + 0.359576i 1.79504 + 0.743529i 1.63571 + 1.91517i 3.89958 − 7.65335i 2.20592 + 1.12397i −1.24927 + 1.71948i
6.2 −0.408546 + 2.57946i 0.202876 − 0.845041i −4.58458 − 1.48962i 0.161215 + 0.0821431i 2.09686 + 0.868549i −2.41217 − 2.82429i 3.34413 − 6.56322i 2.00008 + 1.01909i −0.277748 + 0.382288i
6.3 −0.392049 + 2.47530i −0.669355 + 2.78806i −4.07128 − 1.32284i −0.523374 − 0.266672i −6.63887 − 2.74991i −1.47752 − 1.72996i 2.59503 − 5.09303i −4.65224 − 2.37043i 0.865281 − 1.19096i
6.4 −0.385476 + 2.43380i 0.638580 − 2.65988i −3.87267 − 1.25831i −3.03814 − 1.54801i 6.22745 + 2.57949i 2.60803 + 3.05362i 2.31790 − 4.54913i −3.99414 − 2.03511i 4.93867 − 6.79750i
6.5 −0.374239 + 2.36285i −0.426459 + 1.77633i −3.54091 − 1.15051i 2.83043 + 1.44218i −4.03760 − 1.67243i 1.25003 + 1.46360i 1.87146 − 3.67296i −0.300455 − 0.153089i −4.46691 + 6.14818i
6.6 −0.357833 + 2.25927i 0.752799 − 3.13563i −3.07412 − 0.998844i 0.951458 + 0.484792i 6.81485 + 2.82281i −2.67627 − 3.13351i 1.27973 − 2.51162i −6.59247 − 3.35903i −1.43574 + 1.97612i
6.7 −0.339812 + 2.14549i 0.0150461 − 0.0626713i −2.58552 − 0.840088i −3.62639 − 1.84774i 0.129348 + 0.0535775i −0.837792 − 0.980928i 0.708644 − 1.39079i 2.66932 + 1.36009i 5.19659 − 7.15249i
6.8 −0.291949 + 1.84330i −0.682621 + 2.84332i −1.41039 − 0.458265i −2.79494 − 1.42409i −5.04179 − 2.08838i 2.59818 + 3.04208i −0.438059 + 0.859739i −4.94548 − 2.51985i 3.44100 − 4.73614i
6.9 −0.271350 + 1.71324i 0.238050 − 0.991549i −0.959445 − 0.311743i 3.06080 + 1.55956i 1.63417 + 0.676893i 0.423506 + 0.495862i −0.780544 + 1.53190i 1.74652 + 0.889896i −3.50244 + 4.82070i
6.10 −0.267465 + 1.68871i 0.0319543 − 0.133099i −0.878092 − 0.285309i −1.45435 − 0.741031i 0.216219 + 0.0895609i −1.41505 − 1.65682i −0.835766 + 1.64028i 2.65633 + 1.35347i 1.64038 − 2.25778i
6.11 −0.266590 + 1.68318i −0.248481 + 1.03500i −0.859921 − 0.279405i −0.246450 − 0.125573i −1.67585 − 0.694159i 1.83358 + 2.14684i −0.847812 + 1.66392i 1.66354 + 0.847617i 0.277063 − 0.381344i
6.12 −0.264842 + 1.67215i 0.604412 − 2.51756i −0.823822 − 0.267676i 2.34204 + 1.19333i 4.04965 + 1.67742i 1.56604 + 1.83359i −0.871427 + 1.71027i −3.29976 − 1.68131i −2.61570 + 3.60020i
6.13 −0.165716 + 1.04629i −0.424286 + 1.76728i 0.834847 + 0.271258i 1.35101 + 0.688376i −1.77878 − 0.736795i −0.296339 − 0.346969i −1.38402 + 2.71629i −0.270240 − 0.137694i −0.944127 + 1.29948i
6.14 −0.151247 + 0.954938i 0.550255 − 2.29198i 1.01308 + 0.329170i −1.78391 − 0.908949i 2.10547 + 0.872115i −1.22379 − 1.43288i −1.34544 + 2.64057i −2.27735 − 1.16037i 1.13780 − 1.56605i
6.15 −0.129937 + 0.820388i −0.793896 + 3.30682i 1.24596 + 0.404837i 3.15889 + 1.60953i −2.60972 − 1.08098i −1.17870 − 1.38008i −1.24820 + 2.44973i −7.63174 − 3.88857i −1.73090 + 2.38238i
6.16 −0.128137 + 0.809023i 0.157196 − 0.654769i 1.26401 + 0.410703i 2.00419 + 1.02119i 0.509580 + 0.211075i −2.72830 − 3.19443i −1.23797 + 2.42965i 2.26901 + 1.15612i −1.08297 + 1.49059i
6.17 −0.126080 + 0.796037i −0.541922 + 2.25727i 1.28433 + 0.417305i −2.71247 − 1.38207i −1.72854 − 0.715986i −2.13713 − 2.50225i −1.22592 + 2.40600i −2.12855 − 1.08455i 1.44217 − 1.98497i
6.18 −0.114813 + 0.724902i 0.298436 − 1.24308i 1.38981 + 0.451577i −1.27514 − 0.649716i 0.866844 + 0.359059i 2.88705 + 3.38030i −1.15332 + 2.26352i 1.21685 + 0.620014i 0.617384 − 0.849756i
6.19 −0.0302716 + 0.191127i 0.502148 − 2.09160i 1.86650 + 0.606463i 0.694305 + 0.353766i 0.384560 + 0.159290i 1.26972 + 1.48665i −0.348116 + 0.683217i −1.44960 − 0.738608i −0.0886320 + 0.121991i
6.20 0.00643644 − 0.0406381i 0.0398564 − 0.166014i 1.90050 + 0.617511i −3.01876 − 1.53814i −0.00648995 − 0.00268822i 1.75025 + 2.04928i 0.0746855 − 0.146579i 2.64705 + 1.34874i −0.0819370 + 0.112777i
See next 80 embeddings (of 640 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 6.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
451.ca 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.ca.a yes 640
11.d odd 10 1 451.2.bt.a ✓ 640
41.h odd 40 1 451.2.bt.a ✓ 640
451.ca even 40 1 inner 451.2.ca.a yes 640
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
451.2.bt.a ✓ 640 11.d odd 10 1
451.2.bt.a ✓ 640 41.h odd 40 1
451.2.ca.a yes 640 1.a even 1 1 trivial
451.2.ca.a yes 640 451.ca even 40 1 inner

Hecke kernels

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