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Properties

Label 403.2.bz.a
Level $403$
Weight $2$
Character orbit 403.bz
Analytic conductor $3.218$
Analytic rank $0$
Dimension $288$
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] = [403,2,Mod(38,403)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(403, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([15, 28]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("403.38");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 403 = 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 403.bz (of order \(30\), degree \(8\), 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: \(3.21797120146\)
Analytic rank: \(0\)
Dimension: \(288\)
Relative dimension: \(36\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 288 q - 16 q^{3} + 60 q^{4} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 288 q - 16 q^{3} + 60 q^{4} + 16 q^{9} - 42 q^{10} - 58 q^{12} - 7 q^{13} - 26 q^{14} - 84 q^{16} - 30 q^{17} + 44 q^{22} + 44 q^{23} + 124 q^{25} - 21 q^{26} + 2 q^{27} - 44 q^{29} - 204 q^{30} - 34 q^{35} + 134 q^{36} + 38 q^{38} - 11 q^{39} - 84 q^{40} - 64 q^{42} + 60 q^{43} - 42 q^{48} - 50 q^{49} + 14 q^{51} - 13 q^{52} - 12 q^{53} - 92 q^{55} - 56 q^{56} + 52 q^{61} + 18 q^{62} + 174 q^{64} - 46 q^{65} - 128 q^{66} - 140 q^{68} - 226 q^{69} + 82 q^{74} + 46 q^{75} + 144 q^{77} - 95 q^{78} + 30 q^{79} + 104 q^{81} - 2 q^{82} + 30 q^{87} + 52 q^{88} - 328 q^{90} + 78 q^{91} + 152 q^{92} - 60 q^{94} - 150 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
38.1 −2.60476 + 0.846338i −2.83870 + 0.603383i 4.45046 − 3.23345i 2.15413 + 1.24368i 6.88346 − 3.97417i −0.647070 + 1.45334i −5.63613 + 7.75747i 4.95348 − 2.20543i −6.66356 − 1.41638i
38.2 −2.44210 + 0.793486i 0.151621 − 0.0322281i 3.71619 − 2.69997i −1.21524 − 0.701620i −0.344702 + 0.199014i 0.456028 − 1.02426i −3.91432 + 5.38760i −2.71869 + 1.21044i 3.52447 + 0.749148i
38.3 −2.39524 + 0.778262i 1.87376 − 0.398279i 3.51347 − 2.55268i 1.54681 + 0.893054i −4.17813 + 2.41225i 1.14729 − 2.57686i −3.46827 + 4.77366i 0.611696 − 0.272345i −4.40003 − 0.935255i
38.4 −2.29939 + 0.747117i 0.581957 − 0.123699i 3.11098 − 2.26026i 2.81265 + 1.62389i −1.24573 + 0.719222i −0.764399 + 1.71687i −2.62248 + 3.60953i −2.41726 + 1.07624i −7.68063 − 1.63257i
38.5 −2.19756 + 0.714031i −2.27192 + 0.482911i 2.70141 − 1.96269i −2.61081 − 1.50735i 4.64786 − 2.68345i 1.79222 − 4.02540i −1.81875 + 2.50330i 2.18776 − 0.974055i 6.81372 + 1.44830i
38.6 −2.08918 + 0.678815i 0.327283 − 0.0695662i 2.28584 − 1.66076i −2.01582 − 1.16383i −0.636531 + 0.367501i −1.69303 + 3.80261i −1.06581 + 1.46696i −2.63836 + 1.17467i 5.00143 + 1.06309i
38.7 −1.66585 + 0.541266i 3.00668 − 0.639090i 0.864039 − 0.627761i 1.65657 + 0.956423i −4.66275 + 2.69204i −0.104310 + 0.234285i 0.959529 − 1.32068i 5.89106 − 2.62287i −3.27727 − 0.696606i
38.8 −1.65648 + 0.538222i −2.26380 + 0.481186i 0.836200 − 0.607535i −0.549117 − 0.317033i 3.49095 − 2.01550i −1.65725 + 3.72223i 0.989361 − 1.36174i 2.15263 − 0.958411i 1.08023 + 0.229611i
38.9 −1.60645 + 0.521968i 1.69082 − 0.359396i 0.690208 − 0.501465i −1.67367 − 0.966293i −2.52864 + 1.45991i 0.582041 − 1.30729i 1.13865 − 1.56721i −0.0109161 + 0.00486016i 3.19305 + 0.678703i
38.10 −1.52508 + 0.495529i −2.41071 + 0.512413i 0.462291 − 0.335874i −0.561077 − 0.323938i 3.42262 − 1.97605i −0.0399925 + 0.0898246i 1.34651 − 1.85331i 2.80834 − 1.25035i 1.01621 + 0.216002i
38.11 −1.09973 + 0.357324i 0.0227929 − 0.00484479i −0.536308 + 0.389651i 2.94283 + 1.69904i −0.0233349 + 0.0134724i −1.47906 + 3.32203i 1.80990 − 2.49112i −2.74014 + 1.21999i −3.84343 − 0.816946i
38.12 −1.08301 + 0.351891i −0.146913 + 0.0312274i −0.568950 + 0.413366i 1.21706 + 0.702668i 0.148120 − 0.0855171i 1.11912 − 2.51358i 1.80939 − 2.49042i −2.72003 + 1.21103i −1.56535 − 0.332725i
38.13 −0.802861 + 0.260866i −0.963024 + 0.204697i −1.04150 + 0.756693i −3.54704 − 2.04789i 0.719776 − 0.415563i 0.345954 − 0.777024i 1.63118 − 2.24512i −1.85512 + 0.825954i 3.38201 + 0.718868i
38.14 −0.697365 + 0.226588i 0.705648 − 0.149990i −1.18306 + 0.859542i −0.658335 − 0.380090i −0.458108 + 0.264489i 0.753519 − 1.69243i 1.49225 − 2.05391i −2.26519 + 1.00853i 0.545224 + 0.115891i
38.15 −0.614391 + 0.199628i −3.29662 + 0.700717i −1.28041 + 0.930272i 3.41239 + 1.97014i 1.88553 − 1.08861i 0.675482 − 1.51716i 1.36039 − 1.87242i 7.63604 − 3.39978i −2.48984 − 0.529231i
38.16 −0.245789 + 0.0798617i −1.67915 + 0.356913i −1.56400 + 1.13631i −0.442541 − 0.255501i 0.384212 − 0.221825i −0.641731 + 1.44135i 0.597478 − 0.822358i −0.0484946 + 0.0215912i 0.129176 + 0.0274573i
38.17 −0.181578 + 0.0589982i 2.72617 − 0.579466i −1.58854 + 1.15415i −2.54353 − 1.46851i −0.460825 + 0.266057i 0.826174 − 1.85562i 0.444794 − 0.612207i 4.35561 − 1.93924i 0.548488 + 0.116585i
38.18 −0.0579861 + 0.0188408i 1.89240 − 0.402242i −1.61503 + 1.17339i −2.24387 − 1.29550i −0.102154 + 0.0589787i −2.03436 + 4.56926i 0.143216 − 0.197120i 0.678735 − 0.302192i 0.154522 + 0.0328446i
38.19 0.0579861 − 0.0188408i 1.89240 − 0.402242i −1.61503 + 1.17339i 2.24387 + 1.29550i 0.102154 − 0.0589787i 2.03436 − 4.56926i −0.143216 + 0.197120i 0.678735 − 0.302192i 0.154522 + 0.0328446i
38.20 0.181578 − 0.0589982i 2.72617 − 0.579466i −1.58854 + 1.15415i 2.54353 + 1.46851i 0.460825 − 0.266057i −0.826174 + 1.85562i −0.444794 + 0.612207i 4.35561 − 1.93924i 0.548488 + 0.116585i
See next 80 embeddings (of 288 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 38.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
31.g even 15 1 inner
403.bz even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.2.bz.a ✓ 288
13.b even 2 1 inner 403.2.bz.a ✓ 288
31.g even 15 1 inner 403.2.bz.a ✓ 288
403.bz even 30 1 inner 403.2.bz.a ✓ 288
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.bz.a ✓ 288 1.a even 1 1 trivial
403.2.bz.a ✓ 288 13.b even 2 1 inner
403.2.bz.a ✓ 288 31.g even 15 1 inner
403.2.bz.a ✓ 288 403.bz even 30 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(403, [\chi])\).