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Properties

Label 441.3.bf.a
Level $441$
Weight $3$
Character orbit 441.bf
Analytic conductor $12.016$
Analytic rank $0$
Dimension $456$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [441,3,Mod(44,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([21, 8]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.44");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 441.bf (of order \(42\), 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: \(12.0163796583\)
Analytic rank: \(0\)
Dimension: \(456\)
Relative dimension: \(38\) over \(\Q(\zeta_{42})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 456 q - 80 q^{4} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 456 q - 80 q^{4} + 2 q^{7} + 16 q^{10} + 52 q^{13} + 176 q^{16} - 26 q^{19} - 184 q^{22} - 234 q^{25} - 504 q^{28} - 22 q^{31} - 32 q^{34} + 684 q^{37} + 1828 q^{40} - 76 q^{43} - 100 q^{46} - 226 q^{49} - 1148 q^{52} - 464 q^{55} - 1088 q^{58} - 1362 q^{61} + 472 q^{64} + 110 q^{67} - 472 q^{70} + 482 q^{73} + 84 q^{76} - 106 q^{79} + 2032 q^{82} - 768 q^{85} + 24 q^{88} + 338 q^{91} + 788 q^{94} - 568 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
44.1 −0.579739 − 3.84631i 0 −10.6357 + 3.28069i −8.14752 − 0.610572i 0 6.69072 + 2.05774i 12.0337 + 24.9883i 0 2.37498 + 31.6919i
44.2 −0.547236 − 3.63068i 0 −9.06005 + 2.79466i −0.655785 − 0.0491443i 0 6.91788 − 1.06907i 8.73214 + 18.1325i 0 0.180443 + 2.40784i
44.3 −0.538566 − 3.57315i 0 −8.65509 + 2.66974i 9.50336 + 0.712178i 0 −6.13341 + 3.37362i 7.92936 + 16.4655i 0 −2.57347 − 34.3405i
44.4 −0.515720 − 3.42158i 0 −7.61893 + 2.35013i 3.60801 + 0.270383i 0 0.0515124 − 6.99981i 5.96503 + 12.3865i 0 −0.935584 − 12.4845i
44.5 −0.473655 − 3.14250i 0 −5.82864 + 1.79790i 4.10372 + 0.307531i 0 −4.14534 + 5.64058i 2.89513 + 6.01180i 0 −0.977332 − 13.0416i
44.6 −0.470354 − 3.12059i 0 −5.69459 + 1.75655i −3.80858 − 0.285413i 0 −0.801904 − 6.95392i 2.68287 + 5.57104i 0 0.900720 + 12.0193i
44.7 −0.447992 − 2.97223i 0 −4.81117 + 1.48405i −5.44897 − 0.408344i 0 −6.92331 + 1.03333i 1.34962 + 2.80252i 0 1.22740 + 16.3785i
44.8 −0.439541 − 2.91616i 0 −4.48851 + 1.38452i 0.320724 + 0.0240349i 0 3.35816 + 6.14188i 0.892102 + 1.85247i 0 −0.0708814 − 0.945847i
44.9 −0.369074 − 2.44865i 0 −2.03737 + 0.628447i 5.33267 + 0.399629i 0 6.82314 + 1.56357i −2.00693 − 4.16743i 0 −0.989603 − 13.2053i
44.10 −0.318185 − 2.11102i 0 −0.532860 + 0.164366i 1.70285 + 0.127611i 0 −6.47976 − 2.64814i −3.18860 − 6.62120i 0 −0.272432 − 3.63536i
44.11 −0.293104 − 1.94462i 0 0.126670 − 0.0390726i −9.49401 − 0.711478i 0 −1.71115 + 6.78763i −3.52618 − 7.32219i 0 1.39918 + 18.6707i
44.12 −0.243590 − 1.61611i 0 1.26980 − 0.391683i −3.54724 − 0.265829i 0 3.81840 + 5.86684i −3.77882 − 7.84680i 0 0.434462 + 5.79750i
44.13 −0.220421 − 1.46240i 0 1.73226 − 0.534332i 9.03733 + 0.677254i 0 5.79829 − 3.92171i −3.72995 − 7.74532i 0 −1.00160 − 13.3655i
44.14 −0.205493 − 1.36336i 0 2.00578 − 0.618701i −5.63247 − 0.422095i 0 −0.599896 − 6.97425i −3.64856 − 7.57632i 0 0.581967 + 7.76580i
44.15 −0.191193 − 1.26848i 0 2.24981 − 0.693973i 7.28532 + 0.545959i 0 −5.10894 − 4.78526i −3.53679 − 7.34423i 0 −0.700360 − 9.34566i
44.16 −0.159699 − 1.05953i 0 2.72519 − 0.840609i −2.68245 − 0.201022i 0 5.57039 − 4.23919i −3.18548 − 6.61473i 0 0.215395 + 2.87424i
44.17 −0.0859245 − 0.570072i 0 3.50469 − 1.08105i 4.27529 + 0.320389i 0 −1.03452 + 6.92313i −1.91797 − 3.98271i 0 −0.184708 − 2.46475i
44.18 −0.0522334 − 0.346546i 0 3.70493 − 1.14282i 2.33681 + 0.175120i 0 −6.79471 + 1.68283i −1.19779 − 2.48725i 0 −0.0613726 − 0.818960i
44.19 −0.00565796 − 0.0375381i 0 3.82091 − 1.17860i −5.61319 − 0.420651i 0 6.67821 − 2.09798i −0.131745 − 0.273572i 0 0.0159688 + 0.213088i
44.20 0.00565796 + 0.0375381i 0 3.82091 − 1.17860i 5.61319 + 0.420651i 0 6.67821 − 2.09798i 0.131745 + 0.273572i 0 0.0159688 + 0.213088i
See next 80 embeddings (of 456 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 44.38
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
49.g even 21 1 inner
147.n odd 42 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.3.bf.a ✓ 456
3.b odd 2 1 inner 441.3.bf.a ✓ 456
49.g even 21 1 inner 441.3.bf.a ✓ 456
147.n odd 42 1 inner 441.3.bf.a ✓ 456
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.3.bf.a ✓ 456 1.a even 1 1 trivial
441.3.bf.a ✓ 456 3.b odd 2 1 inner
441.3.bf.a ✓ 456 49.g even 21 1 inner
441.3.bf.a ✓ 456 147.n odd 42 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(441, [\chi])\).