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Properties

Label 403.2.bp.a
Level $403$
Weight $2$
Character orbit 403.bp
Analytic conductor $3.218$
Analytic rank $0$
Dimension $280$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [403,2,Mod(10,403)]
 
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([25, 14]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("403.10");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 403 = 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 403.bp (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: \(280\)
Relative dimension: \(35\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 280 q - 9 q^{2} - 9 q^{3} - 35 q^{4} - 21 q^{6} - 3 q^{7} - 45 q^{8} - 63 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 280 q - 9 q^{2} - 9 q^{3} - 35 q^{4} - 21 q^{6} - 3 q^{7} - 45 q^{8} - 63 q^{9} - 24 q^{10} - 9 q^{11} - 8 q^{12} - 6 q^{13} + 4 q^{14} + 23 q^{16} - 21 q^{17} - 45 q^{18} - 27 q^{19} - 75 q^{20} + 76 q^{21} - 22 q^{22} - 10 q^{23} - 15 q^{24} + 96 q^{25} - 15 q^{26} - 24 q^{27} + 5 q^{28} + 13 q^{29} + 36 q^{30} - 2 q^{31} - 141 q^{32} - 3 q^{33} + 9 q^{34} + 32 q^{35} + 97 q^{36} - 49 q^{38} + 15 q^{39} - 75 q^{40} - 33 q^{41} - 16 q^{42} + 21 q^{43} - 18 q^{44} - 27 q^{45} + 51 q^{46} - 68 q^{48} - 24 q^{49} + 90 q^{50} + 47 q^{51} + 73 q^{52} + 12 q^{53} - 33 q^{54} - 65 q^{55} + 25 q^{56} + 105 q^{57} - 3 q^{58} + 12 q^{59} - 90 q^{60} - 57 q^{61} + 12 q^{62} + 201 q^{63} + 13 q^{64} + 11 q^{65} + 22 q^{66} - 45 q^{67} + 142 q^{68} - 139 q^{69} - 15 q^{71} - 15 q^{72} - 9 q^{73} + 4 q^{74} - 75 q^{75} - 80 q^{76} - 24 q^{77} - 104 q^{78} + 54 q^{79} - 21 q^{80} - 107 q^{81} + 43 q^{82} - 54 q^{83} - 15 q^{84} - 117 q^{85} - 84 q^{86} - 21 q^{87} + 49 q^{88} - 9 q^{89} + 11 q^{90} - 10 q^{91} + 266 q^{92} - 22 q^{93} + 33 q^{94} + 75 q^{95} - 204 q^{96} - 10 q^{97} + 168 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1 −2.76032 − 0.290121i −0.0839252 + 0.0609752i 5.57887 + 1.18583i −1.92798 + 1.11312i 0.249350 − 0.143962i −0.0225283 − 0.0202846i −9.77606 − 3.17644i −0.923726 + 2.84293i 5.64476 − 2.51321i
10.2 −2.69911 − 0.283688i −2.38863 + 1.73544i 5.24842 + 1.11559i 1.43529 − 0.828667i 6.93950 − 4.00652i −1.43143 − 1.28887i −8.68728 − 2.82267i 1.76675 − 5.43750i −4.10910 + 1.82949i
10.3 −2.60859 − 0.274174i 2.47351 − 1.79711i 4.77329 + 1.01459i 0.288360 − 0.166485i −6.94509 + 4.00975i 2.54091 + 2.28784i −7.18423 − 2.33430i 1.96159 − 6.03714i −0.797859 + 0.355230i
10.4 −2.19585 − 0.230793i −0.847098 + 0.615453i 2.81219 + 0.597750i 2.76986 − 1.59918i 2.00214 − 1.15594i 3.22981 + 2.90814i −1.83743 − 0.597018i −0.588258 + 1.81047i −6.45127 + 2.87229i
10.5 −2.18417 − 0.229566i 0.117173 − 0.0851309i 2.76161 + 0.586998i 0.0491735 − 0.0283903i −0.275468 + 0.159042i −2.29210 − 2.06382i −1.71965 − 0.558748i −0.920569 + 2.83322i −0.113921 + 0.0507208i
10.6 −2.06615 − 0.217161i −1.25606 + 0.912580i 2.26552 + 0.481551i −2.40278 + 1.38724i 2.79338 − 1.61276i 0.0422354 + 0.0380290i −0.624627 − 0.202953i −0.182170 + 0.560662i 5.26575 − 2.34446i
10.7 −2.02543 − 0.212881i 1.58944 − 1.15479i 2.10075 + 0.446527i 1.71112 − 0.987913i −3.46513 + 2.00059i −1.42691 − 1.28479i −0.286037 − 0.0929391i 0.265713 − 0.817780i −3.67605 + 1.63668i
10.8 −1.94065 − 0.203970i 1.97372 − 1.43400i 1.76822 + 0.375846i −2.63433 + 1.52093i −4.12280 + 2.38030i −1.07378 − 0.966833i 0.356842 + 0.115945i 0.912197 − 2.80745i 5.42253 − 2.41427i
10.9 −1.83649 − 0.193022i −2.43858 + 1.77173i 1.37913 + 0.293142i −0.615405 + 0.355305i 4.82040 − 2.78306i 1.21282 + 1.09203i 1.03628 + 0.336709i 1.88059 − 5.78785i 1.19876 − 0.533725i
10.10 −1.48870 − 0.156468i −1.43929 + 1.04571i 0.235442 + 0.0500448i 3.43058 − 1.98065i 2.30629 − 1.33154i −1.81022 − 1.62993i 2.50460 + 0.813793i 0.0510066 − 0.156982i −5.41700 + 2.41181i
10.11 −1.34554 − 0.141422i 1.22820 − 0.892341i −0.165811 − 0.0352442i 0.772095 − 0.445769i −1.77880 + 1.02699i 2.87973 + 2.59292i 2.79159 + 0.907044i −0.214843 + 0.661218i −1.10193 + 0.490610i
10.12 −1.17237 − 0.123221i −1.03818 + 0.754283i −0.597037 − 0.126904i −2.09944 + 1.21211i 1.31007 − 0.756370i 2.47362 + 2.22725i 2.92656 + 0.950898i −0.418173 + 1.28701i 2.61067 − 1.16234i
10.13 −1.06147 − 0.111565i 1.27207 − 0.924209i −0.842029 − 0.178979i −2.77463 + 1.60194i −1.45336 + 0.839101i 0.448540 + 0.403867i 2.90397 + 0.943557i −0.163064 + 0.501860i 3.12390 − 1.39085i
10.14 −0.794088 − 0.0834620i −0.0739898 + 0.0537568i −1.33269 − 0.283271i 0.698098 − 0.403047i 0.0632411 − 0.0365123i −2.33566 − 2.10304i 2.55339 + 0.829648i −0.924466 + 2.84521i −0.587991 + 0.261790i
10.15 −0.501537 − 0.0527137i −0.192422 + 0.139802i −1.70753 − 0.362948i 0.660141 − 0.381132i 0.103876 − 0.0599729i −1.06122 − 0.955523i 1.79650 + 0.583717i −0.909570 + 2.79937i −0.351176 + 0.156354i
10.16 −0.349700 − 0.0367550i 1.55724 − 1.13140i −1.83536 − 0.390117i 3.40354 − 1.96504i −0.586153 + 0.338415i 1.19593 + 1.07682i 1.29632 + 0.421200i 0.217879 − 0.670563i −1.26245 + 0.562077i
10.17 −0.228252 − 0.0239903i −2.32883 + 1.69199i −1.90477 − 0.404872i −2.58167 + 1.49053i 0.572152 − 0.330332i −2.78454 − 2.50721i 0.861608 + 0.279953i 1.63355 − 5.02756i 0.625030 − 0.278281i
10.18 0.0702218 + 0.00738061i 2.77994 − 2.01975i −1.95142 − 0.414787i −0.986518 + 0.569566i 0.210119 − 0.121312i 2.56944 + 2.31354i −0.268276 − 0.0871682i 2.72165 − 8.37638i −0.0734788 + 0.0327149i
10.19 0.0782606 + 0.00822552i −1.50523 + 1.09361i −1.95024 − 0.414536i −0.722573 + 0.417178i −0.126795 + 0.0732053i −0.267898 − 0.241217i −0.298897 − 0.0971177i 0.142670 − 0.439093i −0.0599805 + 0.0267050i
10.20 0.342148 + 0.0359612i −2.36245 + 1.71642i −1.84052 − 0.391215i 2.15569 − 1.24459i −0.870032 + 0.502313i 3.18311 + 2.86608i −1.27005 − 0.412665i 1.70801 − 5.25672i 0.782322 − 0.348312i
See next 80 embeddings (of 280 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.35
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
403.bp 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.bp.a ✓ 280
13.e even 6 1 403.2.bt.a yes 280
31.g even 15 1 403.2.bt.a yes 280
403.bp even 30 1 inner 403.2.bp.a ✓ 280
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.bp.a ✓ 280 1.a even 1 1 trivial
403.2.bp.a ✓ 280 403.bp even 30 1 inner
403.2.bt.a yes 280 13.e even 6 1
403.2.bt.a yes 280 31.g even 15 1

Hecke kernels

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