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Properties

Label 353.2.g.a
Level $353$
Weight $2$
Character orbit 353.g
Analytic conductor $2.819$
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] = [353,2,Mod(16,353)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(353, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([19]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("353.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 353 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 353.g (of order \(22\), degree \(10\), 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: \(2.81871919135\)
Analytic rank: \(0\)
Dimension: \(280\)
Relative dimension: \(28\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 280 q - 7 q^{2} - 11 q^{3} - 39 q^{4} - 11 q^{5} - 26 q^{8} + 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 280 q - 7 q^{2} - 11 q^{3} - 39 q^{4} - 11 q^{5} - 26 q^{8} + 19 q^{9} - 10 q^{11} + 11 q^{12} - 11 q^{13} + 11 q^{14} + 4 q^{15} - 39 q^{16} + q^{17} - 7 q^{18} + 6 q^{19} + 11 q^{20} + 7 q^{21} + 3 q^{22} - 21 q^{23} + 88 q^{24} + 43 q^{25} - 44 q^{26} - 11 q^{27} + 33 q^{28} - 11 q^{29} + 59 q^{30} - 88 q^{31} + 9 q^{32} + 44 q^{33} - 43 q^{34} + 61 q^{35} - 234 q^{36} - 66 q^{37} + 21 q^{38} - 56 q^{39} + 121 q^{40} - 31 q^{41} + 30 q^{42} - 170 q^{43} - 21 q^{44} - 55 q^{45} + 31 q^{46} + 51 q^{47} + 77 q^{48} - 186 q^{49} + 81 q^{50} - 11 q^{51} - 11 q^{52} - 11 q^{53} - 11 q^{55} + 176 q^{56} - 99 q^{57} - 61 q^{58} - 64 q^{60} + 69 q^{61} - 11 q^{62} + 66 q^{63} - 40 q^{64} - 11 q^{65} + 55 q^{66} - q^{68} - 55 q^{69} + 28 q^{70} - 11 q^{71} - 78 q^{72} - 54 q^{73} - 99 q^{74} - 11 q^{75} - 27 q^{76} + 66 q^{77} - 127 q^{78} - 55 q^{79} - 154 q^{80} - 19 q^{81} + 119 q^{82} + 28 q^{83} - 144 q^{84} - 55 q^{85} + 131 q^{86} + 165 q^{87} + 179 q^{88} - 77 q^{89} - 132 q^{90} + 52 q^{91} - 61 q^{92} + 122 q^{93} - 6 q^{94} + 77 q^{95} + 55 q^{96} + 28 q^{97} + 48 q^{98} - 193 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −1.08986 + 2.38645i 0.333766 − 0.152426i −3.19764 − 3.69027i 0.797295 − 0.364113i 0.962637i − 3.07327i 7.25709 − 2.13087i −1.87642 + 2.16550i 2.29954i
16.2 −1.04784 + 2.29446i −0.717799 + 0.327808i −2.85683 − 3.29696i 1.59230 − 0.727179i − 1.99045i 4.71986i 5.71780 − 1.67890i −1.55680 + 1.79665i 4.41543i
16.3 −1.04616 + 2.29076i 2.92866 − 1.33747i −2.84343 − 3.28149i 0.527731 − 0.241007i 8.10806i 0.271339i 5.65913 − 1.66167i 4.82361 − 5.56674i 1.46104i
16.4 −0.908019 + 1.98829i 1.17580 − 0.536971i −1.81906 − 2.09930i −2.66947 + 1.21911i 2.82541i − 0.973459i 1.63121 − 0.478967i −0.870407 + 1.00450i − 6.41465i
16.5 −0.901095 + 1.97312i −2.63399 + 1.20290i −1.77152 − 2.04445i 1.21180 − 0.553410i − 6.28111i − 1.51383i 1.46770 − 0.430954i 3.52633 − 4.06960i 2.88970i
16.6 −0.795666 + 1.74226i −2.56503 + 1.17141i −1.09268 − 1.26102i −2.97048 + 1.35657i − 5.40102i 3.58314i −0.609089 + 0.178845i 3.24261 − 3.74217i − 6.25474i
16.7 −0.704006 + 1.54156i −1.04413 + 0.476840i −0.571054 − 0.659032i 1.68930 − 0.771478i − 1.94529i − 3.50836i −1.83415 + 0.538556i −1.10174 + 1.27148i 3.14728i
16.8 −0.618848 + 1.35509i 2.05859 − 0.940124i −0.143570 − 0.165689i 3.23421 − 1.47701i 3.37136i 1.69618i −2.54536 + 0.747386i 1.38936 − 1.60340i 5.29668i
16.9 −0.554456 + 1.21409i 0.949093 − 0.433436i 0.143131 + 0.165182i −1.02988 + 0.470329i 1.39260i 2.95193i −2.84118 + 0.834246i −1.25167 + 1.44451i − 1.51114i
16.10 −0.448659 + 0.982427i −0.785332 + 0.358649i 0.545854 + 0.629949i 2.41621 − 1.10344i − 0.932442i 1.16978i −2.93634 + 0.862187i −1.47647 + 1.70393i 2.86882i
16.11 −0.366719 + 0.803004i 2.10949 − 0.963372i 0.799390 + 0.922545i −0.0819113 + 0.0374076i 2.04722i − 4.55356i −2.72800 + 0.801013i 1.55728 − 1.79720i − 0.0794931i
16.12 −0.336573 + 0.736992i −1.73555 + 0.792598i 0.879846 + 1.01540i −2.93421 + 1.34001i − 1.54585i − 4.38002i −2.59925 + 0.763209i 0.419334 − 0.483937i − 2.61350i
16.13 −0.178714 + 0.391329i 2.97940 − 1.36065i 1.18852 + 1.37163i −3.93830 + 1.79856i 1.40909i 3.51867i −1.57472 + 0.462380i 5.06091 − 5.84060i − 1.86260i
16.14 −0.0760492 + 0.166525i −0.0716929 + 0.0327411i 1.28777 + 1.48617i −0.799903 + 0.365304i − 0.0144286i 0.932064i −0.696723 + 0.204576i −1.96051 + 2.26255i − 0.160985i
16.15 −0.0194765 + 0.0426475i −2.37701 + 1.08554i 1.30828 + 1.50984i −0.956706 + 0.436913i − 0.122516i 3.02555i −0.179842 + 0.0528064i 2.50719 − 2.89345i − 0.0493107i
16.16 0.0712686 − 0.156056i 1.74962 − 0.799026i 1.29045 + 1.48926i 1.49308 − 0.681867i − 0.329986i − 0.474839i 0.653598 − 0.191914i 0.458157 − 0.528742i − 0.281601i
16.17 0.101102 − 0.221382i −2.84244 + 1.29810i 1.27093 + 1.46673i 2.50983 − 1.14620i 0.760507i − 2.60408i 0.920237 − 0.270206i 4.42984 − 5.11230i − 0.671514i
16.18 0.188124 − 0.411935i −0.471930 + 0.215523i 1.17542 + 1.35651i 3.43964 − 1.57083i 0.234949i − 0.951728i 1.64895 − 0.484175i −1.78831 + 2.06383i − 1.71242i
16.19 0.266764 − 0.584132i −0.372443 + 0.170089i 1.03967 + 1.19985i −3.00853 + 1.37395i 0.262930i − 0.860539i 2.21052 − 0.649067i −1.85480 + 2.14055i 2.12390i
16.20 0.496152 − 1.08642i 2.22339 − 1.01539i 0.375578 + 0.433441i −1.44167 + 0.658387i − 2.91933i − 3.23863i 2.94919 − 0.865959i 1.94788 − 2.24797i 1.89292i
See next 80 embeddings (of 280 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
353.g even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 353.2.g.a ✓ 280
353.g even 22 1 inner 353.2.g.a ✓ 280
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
353.2.g.a ✓ 280 1.a even 1 1 trivial
353.2.g.a ✓ 280 353.g even 22 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(353, [\chi])\).