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Properties

Label 475.2.n.a
Level $475$
Weight $2$
Character orbit 475.n
Analytic conductor $3.793$
Analytic rank $0$
Dimension $80$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [475,2,Mod(39,475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(475, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("475.39");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.n (of order \(10\), degree \(4\), 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: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(20\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 16 q^{4} - 3 q^{5} + 6 q^{6} + 8 q^{9} - 36 q^{10} + 20 q^{11} + 45 q^{12} - 10 q^{14} - 20 q^{16} - 15 q^{17} + 20 q^{19} + 12 q^{20} + 16 q^{21} + 15 q^{23} + 72 q^{24} + 41 q^{25} - 84 q^{26}+ \cdots + 178 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
39.1 −1.60597 − 2.21043i −1.56040 − 0.507006i −1.68883 + 5.19768i 0.411675 − 2.19785i 1.38526 + 4.26341i − 3.04333i 9.00431 − 2.92568i −0.249248 − 0.181089i −5.51933 + 2.61970i
39.2 −1.39149 − 1.91522i 2.70917 + 0.880261i −1.11379 + 3.42788i 2.14567 + 0.629357i −2.08388 − 6.41351i − 1.72629i 3.61202 − 1.17362i 4.13767 + 3.00619i −1.78032 − 4.98517i
39.3 −1.21720 − 1.67534i −0.191192 − 0.0621221i −0.707134 + 2.17633i −2.22482 − 0.223992i 0.128644 + 0.395926i 3.63478i 0.567865 − 0.184511i −2.39436 − 1.73960i 2.33280 + 3.99997i
39.4 −1.12280 − 1.54541i 1.41655 + 0.460266i −0.509559 + 1.56826i −0.295058 − 2.21652i −0.879213 − 2.70594i − 3.52405i −0.637733 + 0.207212i −0.632270 − 0.459371i −3.09413 + 2.94469i
39.5 −1.02546 − 1.41142i −2.62954 − 0.854391i −0.322513 + 0.992591i 1.35649 + 1.77762i 1.49058 + 4.58753i 0.182993i −1.58676 + 0.515570i 3.75747 + 2.72996i 1.11795 − 3.73745i
39.6 −0.819032 − 1.12730i 0.641372 + 0.208394i 0.0180399 − 0.0555211i −0.501037 + 2.17921i −0.290381 − 0.893701i 0.0684183i −2.72781 + 0.886319i −2.05912 − 1.49604i 2.86699 − 1.22002i
39.7 −0.708585 − 0.975283i −1.48229 − 0.481625i 0.168949 − 0.519971i −1.59939 + 1.56267i 0.580606 + 1.78692i − 2.31635i −2.91986 + 0.948721i −0.461836 − 0.335543i 2.65735 + 0.452578i
39.8 −0.396268 − 0.545416i 2.29043 + 0.744207i 0.477583 − 1.46985i −1.89679 − 1.18414i −0.501723 − 1.54414i − 2.02861i −2.27328 + 0.738634i 2.26518 + 1.64575i 0.105787 + 1.50378i
39.9 −0.325781 − 0.448399i −0.381722 − 0.124029i 0.523105 − 1.60995i −0.781184 − 2.09517i 0.0687433 + 0.211570i 3.07504i −1.94657 + 0.632479i −2.29672 − 1.66867i −0.684979 + 1.03285i
39.10 0.0162182 + 0.0223225i 2.53599 + 0.823993i 0.617799 − 1.90139i 2.18453 − 0.477335i 0.0227357 + 0.0699732i 0.768841i 0.104947 − 0.0340992i 3.32522 + 2.41592i 0.0460844 + 0.0410225i
39.11 0.203802 + 0.280510i −1.02562 − 0.333243i 0.580884 − 1.78778i 1.88943 + 1.19586i −0.115545 − 0.355611i − 4.66677i 1.27939 − 0.415700i −1.48621 − 1.07980i 0.0496198 + 0.773721i
39.12 0.302513 + 0.416374i 2.54210 + 0.825977i 0.536181 − 1.65020i −1.61099 + 1.55071i 0.425102 + 1.30833i 4.22504i 1.82825 − 0.594036i 3.35296 + 2.43607i −1.13302 − 0.201663i
39.13 0.470002 + 0.646902i −1.99652 − 0.648707i 0.420454 − 1.29402i −0.877494 − 2.05670i −0.518716 − 1.59644i − 0.826548i 2.55568 − 0.830390i 1.13820 + 0.826953i 0.918057 − 1.53430i
39.14 0.494729 + 0.680936i −1.00441 − 0.326353i 0.399117 − 1.22836i 2.12122 − 0.707408i −0.274686 − 0.845396i 5.02465i 2.63486 − 0.856119i −1.52472 − 1.10777i 1.53113 + 1.09444i
39.15 0.636441 + 0.875986i 0.575766 + 0.187078i 0.255740 − 0.787086i −2.22529 − 0.219241i 0.202564 + 0.623427i − 2.44664i 2.91181 − 0.946103i −2.13054 − 1.54793i −1.22422 − 2.08886i
39.16 1.06731 + 1.46902i 1.88156 + 0.611355i −0.400851 + 1.23369i −0.796184 + 2.08952i 1.11011 + 3.41656i − 1.51173i 1.21373 − 0.394364i 0.739455 + 0.537245i −3.91933 + 1.06055i
39.17 1.10380 + 1.51924i −1.16023 − 0.376980i −0.471705 + 1.45176i 2.05956 − 0.870765i −0.707927 − 2.17878i − 1.60835i 0.845716 − 0.274790i −1.22304 − 0.888592i 3.59623 + 2.16782i
39.18 1.30771 + 1.79991i 1.66715 + 0.541690i −0.911532 + 2.80541i 0.941634 − 2.02813i 1.20516 + 3.70909i 2.10218i −2.00966 + 0.652977i 0.0589103 + 0.0428008i 4.88184 − 0.957357i
39.19 1.49224 + 2.05389i −0.805296 − 0.261657i −1.37366 + 4.22769i 0.616846 + 2.14930i −0.664281 − 2.04445i 1.89983i −5.90407 + 1.91835i −1.84701 − 1.34193i −3.49395 + 4.47421i
39.20 1.51783 + 2.08911i −2.90483 − 0.943838i −1.44255 + 4.43972i −2.22781 + 0.191942i −2.43726 − 7.50111i − 3.43848i −6.55283 + 2.12914i 5.12018 + 3.72003i −3.78243 − 4.36282i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 39.20
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.n.a ✓ 80
25.e even 10 1 inner 475.2.n.a ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
475.2.n.a ✓ 80 1.a even 1 1 trivial
475.2.n.a ✓ 80 25.e even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{80} - 28 T_{2}^{78} + 461 T_{2}^{76} - 5856 T_{2}^{74} + 40 T_{2}^{73} + 63429 T_{2}^{72} + \cdots + 625 \) acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\). Copy content Toggle raw display