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Properties

Label 403.2.s.a
Level $403$
Weight $2$
Character orbit 403.s
Analytic conductor $3.218$
Analytic rank $0$
Dimension $70$
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(160,403)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(403, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("403.160");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 403 = 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 403.s (of order \(6\), degree \(2\), 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: \(70\)
Relative dimension: \(35\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 70 q - 6 q^{2} - 2 q^{3} + 30 q^{4} - 29 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 70 q - 6 q^{2} - 2 q^{3} + 30 q^{4} - 29 q^{9} + 2 q^{10} + 13 q^{12} + q^{13} - 14 q^{14} - 15 q^{15} - 28 q^{16} - 12 q^{17} - 3 q^{20} - 9 q^{21} + 4 q^{22} + 10 q^{23} + 18 q^{24} + 19 q^{25} + 6 q^{26} + 34 q^{27} - 33 q^{28} - 18 q^{29} - 31 q^{30} - 2 q^{31} + 36 q^{32} - 12 q^{33} + 9 q^{34} - 12 q^{35} - 16 q^{36} - 18 q^{37} - 21 q^{38} - 30 q^{39} + 5 q^{40} + 98 q^{42} - 38 q^{43} + 42 q^{44} - 6 q^{46} + 54 q^{48} - 18 q^{49} - 51 q^{50} - 7 q^{51} + 41 q^{52} - 22 q^{53} + 18 q^{54} - 15 q^{55} - 50 q^{56} + 15 q^{57} - 12 q^{58} - 13 q^{61} - 23 q^{62} - 6 q^{63} - 38 q^{64} - 12 q^{65} - 52 q^{66} - 44 q^{68} + 32 q^{69} + 27 q^{71} - 15 q^{72} - 9 q^{73} + 38 q^{74} - 50 q^{75} + 126 q^{76} + 34 q^{77} + 14 q^{78} + 6 q^{79} - 11 q^{81} + 39 q^{82} - 54 q^{83} + 15 q^{84} - 33 q^{85} - 24 q^{86} + 28 q^{87} - 32 q^{88} - 6 q^{89} - 11 q^{90} - 70 q^{91} - 6 q^{92} + 14 q^{93} - 43 q^{94} + 25 q^{95} + 36 q^{96} - 75 q^{97} + 93 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
160.1 −2.41811 − 1.39609i −0.822614 − 1.42481i 2.89816 + 5.01976i −1.92977 + 1.11415i 4.59379i − 3.90079i − 10.6000i 0.146612 − 0.253939i 6.22185
160.2 −2.31373 − 1.33583i 0.915310 + 1.58536i 2.56889 + 4.44945i 2.88237 − 1.66414i − 4.89080i 0.418923i − 8.38310i −0.175583 + 0.304119i −8.89202
160.3 −2.12624 − 1.22758i 0.398773 + 0.690696i 2.01392 + 3.48821i −3.14114 + 1.81354i − 1.95811i 4.93931i − 4.97868i 1.18196 − 2.04721i 8.90508
160.4 −2.01769 − 1.16491i −1.03272 − 1.78873i 1.71404 + 2.96880i 0.697719 − 0.402828i 4.81213i 3.53851i − 3.32715i −0.633039 + 1.09646i −1.87704
160.5 −1.90799 − 1.10158i 0.706059 + 1.22293i 1.42696 + 2.47156i −1.04549 + 0.603615i − 3.11112i − 0.408769i − 1.88131i 0.502961 − 0.871154i 2.65972
160.6 −1.88902 − 1.09062i 0.0723809 + 0.125367i 1.37892 + 2.38836i 1.56313 − 0.902471i − 0.315761i − 2.55122i − 1.65304i 1.48952 − 2.57993i −3.93702
160.7 −1.86211 − 1.07509i −1.53180 − 2.65315i 1.31164 + 2.27183i 2.43528 − 1.40601i 6.58728i − 2.16759i − 1.34017i −3.19279 + 5.53008i −6.04634
160.8 −1.77854 − 1.02684i 1.56768 + 2.71530i 1.10881 + 1.92051i 0.188573 − 0.108873i − 6.43902i 0.507266i − 0.446916i −3.41522 + 5.91533i −0.447180
160.9 −1.42222 − 0.821121i −0.972983 − 1.68526i 0.348479 + 0.603584i −3.04464 + 1.75782i 3.19575i − 0.764751i 2.13991i −0.393391 + 0.681372i 5.77355
160.10 −1.06311 − 0.613789i −0.443615 − 0.768363i −0.246526 − 0.426995i 0.237568 − 0.137160i 1.08914i 1.55840i 3.06042i 1.10641 − 1.91636i −0.336749
160.11 −1.01226 − 0.584427i 0.506588 + 0.877437i −0.316890 − 0.548870i 2.14676 − 1.23943i − 1.18426i 4.28486i 3.07850i 0.986736 − 1.70908i −2.89744
160.12 −1.00051 − 0.577645i −1.19453 − 2.06899i −0.332651 − 0.576169i −0.697454 + 0.402675i 2.76006i 1.78753i 3.07920i −1.35381 + 2.34486i 0.930413
160.13 −0.918845 − 0.530496i 0.853477 + 1.47826i −0.437149 − 0.757164i 0.822265 − 0.474735i − 1.81106i − 4.11244i 3.04960i 0.0431555 − 0.0747476i −1.00738
160.14 −0.793251 − 0.457984i 0.684833 + 1.18617i −0.580502 − 1.00546i −3.20188 + 1.84861i − 1.25457i − 1.52463i 2.89538i 0.562006 − 0.973423i 3.38653
160.15 −0.700842 − 0.404632i −0.535618 − 0.927717i −0.672547 − 1.16488i 3.37908 − 1.95091i 0.866912i − 2.38491i 2.70706i 0.926227 − 1.60427i −3.15760
160.16 −0.605874 − 0.349801i 1.45071 + 2.51270i −0.755278 − 1.30818i −1.79550 + 1.03663i − 2.02984i 0.152727i 2.45599i −2.70911 + 4.69232i 1.45046
160.17 −0.135075 − 0.0779857i −1.60395 − 2.77812i −0.987836 − 1.71098i −1.02567 + 0.592170i 0.500340i − 3.75932i 0.620091i −3.64529 + 6.31383i 0.184723
160.18 −0.0898111 − 0.0518525i 0.233773 + 0.404906i −0.994623 − 1.72274i −1.88817 + 1.09014i − 0.0484867i 2.79921i 0.413704i 1.39070 − 2.40876i 0.226105
160.19 0.163021 + 0.0941205i 1.52391 + 2.63948i −0.982283 − 1.70136i 2.94789 − 1.70197i 0.573723i 1.34890i − 0.746294i −3.14458 + 5.44658i 0.640760
160.20 0.388678 + 0.224403i −0.0472394 − 0.0818210i −0.899286 − 1.55761i 0.0856400 − 0.0494443i − 0.0424027i 2.60744i − 1.70483i 1.49554 − 2.59035i 0.0443818
See all 70 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 160.35
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
403.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.2.s.a ✓ 70
13.e even 6 1 403.2.v.a yes 70
31.c even 3 1 403.2.v.a yes 70
403.s even 6 1 inner 403.2.s.a ✓ 70
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.s.a ✓ 70 1.a even 1 1 trivial
403.2.s.a ✓ 70 403.s even 6 1 inner
403.2.v.a yes 70 13.e even 6 1
403.2.v.a yes 70 31.c even 3 1

Hecke kernels

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