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Properties

Label 345.2.s.a
Level $345$
Weight $2$
Character orbit 345.s
Analytic conductor $2.755$
Analytic rank $0$
Dimension $160$
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] = [345,2,Mod(11,345)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(345, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 0, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("345.11");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 345 = 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 345.s (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: \(2.75483886973\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 160 q + 14 q^{4} - 16 q^{5} - q^{6} + 22 q^{8} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 160 q + 14 q^{4} - 16 q^{5} - q^{6} + 22 q^{8} - 20 q^{9} + 8 q^{11} - 3 q^{12} + 12 q^{14} + 2 q^{16} - 36 q^{17} + 11 q^{18} + 14 q^{20} - 17 q^{21} - 4 q^{23} - 66 q^{24} - 16 q^{25} - 27 q^{27} - q^{30} + 8 q^{31} + 110 q^{32} - 26 q^{33} - 11 q^{34} - 29 q^{36} - 76 q^{38} + 62 q^{39} - 77 q^{40} - 22 q^{41} - 30 q^{42} - 44 q^{43} - 28 q^{44} + 2 q^{45} - 82 q^{46} + 23 q^{48} - 34 q^{49} - 6 q^{51} + 2 q^{52} + 10 q^{53} - 92 q^{54} - 14 q^{55} + 134 q^{56} - q^{57} - 2 q^{58} - 3 q^{60} + 22 q^{61} - 275 q^{62} + 155 q^{63} - 8 q^{64} + 94 q^{66} + 22 q^{67} + 16 q^{68} + 8 q^{69} + 12 q^{70} + 104 q^{72} + 10 q^{73} + 24 q^{74} + 44 q^{76} + 264 q^{77} - 302 q^{78} + 22 q^{79} + 13 q^{80} + 28 q^{81} + 70 q^{82} + 2 q^{83} + 16 q^{84} + 8 q^{85} - 186 q^{86} - 61 q^{87} - 44 q^{89} + 11 q^{90} + 40 q^{92} - 142 q^{93} - 30 q^{94} + 13 q^{96} + 88 q^{98} - 184 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −1.50149 − 2.33637i −1.48813 − 0.886262i −2.37331 + 5.19681i −0.959493 + 0.281733i 0.163786 + 4.80754i 1.95695 − 1.69571i 10.2072 − 1.46758i 1.42908 + 2.63775i 2.09890 + 1.81871i
11.2 −1.37512 − 2.13973i 1.34670 − 1.08922i −1.85666 + 4.06552i −0.959493 + 0.281733i −4.18252 − 1.38376i −2.74460 + 2.37821i 6.21702 − 0.893873i 0.627191 − 2.93371i 1.92225 + 1.66564i
11.3 −1.34961 − 2.10004i −0.0867854 + 1.72988i −1.75788 + 3.84922i −0.959493 + 0.281733i 3.74993 − 2.15241i −1.00507 + 0.870896i 5.51415 − 0.792815i −2.98494 − 0.300256i 1.88659 + 1.63474i
11.4 −1.07740 − 1.67647i 1.73204 − 0.00676729i −0.818925 + 1.79320i −0.959493 + 0.281733i −1.87744 − 2.89642i 3.35433 − 2.90655i −0.0565286 + 0.00812759i 2.99991 − 0.0234424i 1.50607 + 1.30502i
11.5 −0.885498 − 1.37786i −1.31641 + 1.12564i −0.283565 + 0.620921i −0.959493 + 0.281733i 2.71665 + 0.817077i 0.203828 − 0.176618i −2.13575 + 0.307075i 0.465867 − 2.96361i 1.23782 + 1.07257i
11.6 −0.649940 − 1.01133i −0.973085 − 1.43287i 0.230470 − 0.504660i −0.959493 + 0.281733i −0.816650 + 1.91538i −1.59910 + 1.38562i −3.04003 + 0.437090i −1.10621 + 2.78860i 0.908537 + 0.787252i
11.7 −0.337922 − 0.525816i 1.06489 − 1.36602i 0.668538 − 1.46389i −0.959493 + 0.281733i −1.07812 − 0.0983309i −0.135295 + 0.117234i −2.23301 + 0.321058i −0.732008 − 2.90932i 0.472373 + 0.409314i
11.8 −0.0422559 − 0.0657515i 0.361230 + 1.69396i 0.828292 − 1.81371i −0.959493 + 0.281733i 0.0961165 − 0.0953314i 2.31213 − 2.00347i −0.308981 + 0.0444248i −2.73903 + 1.22382i 0.0590686 + 0.0511832i
11.9 0.0518555 + 0.0806888i −1.55854 + 0.755616i 0.827008 − 1.81090i −0.959493 + 0.281733i −0.141789 − 0.0865738i −1.36836 + 1.18569i 0.378881 − 0.0544749i 1.85809 − 2.35531i −0.0724877 − 0.0628109i
11.10 0.390950 + 0.608331i −0.266780 − 1.71138i 0.613606 − 1.34361i −0.959493 + 0.281733i 0.936789 − 0.831356i 2.49752 − 2.16411i 2.48878 − 0.357832i −2.85766 + 0.913124i −0.546501 − 0.473546i
11.11 0.561180 + 0.873212i 1.60263 + 0.656934i 0.383253 − 0.839206i −0.959493 + 0.281733i 0.325722 + 1.76810i 2.04970 − 1.77608i 3.00273 − 0.431727i 2.13687 + 2.10565i −0.784460 − 0.679739i
11.12 0.628776 + 0.978395i 0.00845989 + 1.73203i 0.268933 − 0.588880i −0.959493 + 0.281733i −1.68929 + 1.09734i −3.68436 + 3.19251i 3.04762 − 0.438182i −2.99986 + 0.0293056i −0.878952 − 0.761617i
11.13 0.790269 + 1.22968i −1.57809 − 0.713880i −0.0567635 + 0.124295i −0.959493 + 0.281733i −0.369272 − 2.50471i −0.148193 + 0.128410i 2.69599 − 0.387625i 1.98075 + 2.25314i −1.10470 − 0.957227i
11.14 1.15818 + 1.80216i 1.01039 − 1.40681i −1.07557 + 2.35516i −0.959493 + 0.281733i 3.70550 + 0.191559i 1.78777 − 1.54912i −1.24921 + 0.179610i −0.958208 − 2.84286i −1.61899 − 1.40286i
11.15 1.20534 + 1.87554i 1.71979 + 0.205747i −1.23399 + 2.70207i −0.959493 + 0.281733i 1.68704 + 3.47353i −2.85846 + 2.47687i −2.14168 + 0.307927i 2.91534 + 0.707681i −1.68492 − 1.45999i
11.16 1.36242 + 2.11997i −1.57831 + 0.713397i −1.80726 + 3.95735i −0.959493 + 0.281733i −3.66271 − 2.37403i −0.618805 + 0.536197i −5.86299 + 0.842970i 1.98213 − 2.25193i −1.90450 − 1.65026i
56.1 −1.82549 + 1.58180i −1.55227 − 0.768404i 0.545705 − 3.79546i 0.415415 − 0.909632i 4.04912 − 1.05267i 1.12638 − 3.83611i 2.39566 + 3.72772i 1.81911 + 2.38555i 0.680516 + 2.31763i
56.2 −1.81340 + 1.57132i −0.872817 + 1.49606i 0.534743 − 3.71922i 0.415415 − 0.909632i −0.768019 − 4.08443i −0.188775 + 0.642910i 2.27988 + 3.54756i −1.47638 − 2.61157i 0.676009 + 2.30228i
56.3 −1.46765 + 1.27172i 1.70217 − 0.320351i 0.252078 − 1.75324i 0.415415 − 0.909632i −2.09079 + 2.63485i −1.13212 + 3.85565i −0.240145 − 0.373673i 2.79475 − 1.09058i 0.547118 + 1.86331i
56.4 −1.14799 + 0.994739i 1.62234 + 0.606648i 0.0437456 − 0.304257i 0.415415 − 0.909632i −2.46588 + 0.917376i 1.22043 − 4.15641i −1.39004 − 2.16294i 2.26396 + 1.96838i 0.427954 + 1.45748i
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 345.2.s.a ✓ 160
3.b odd 2 1 345.2.s.b yes 160
23.d odd 22 1 345.2.s.b yes 160
69.g even 22 1 inner 345.2.s.a ✓ 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
345.2.s.a ✓ 160 1.a even 1 1 trivial
345.2.s.a ✓ 160 69.g even 22 1 inner
345.2.s.b yes 160 3.b odd 2 1
345.2.s.b yes 160 23.d odd 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{160} - 23 T_{2}^{158} - 22 T_{2}^{157} + 317 T_{2}^{156} + 484 T_{2}^{155} - 3282 T_{2}^{154} + \cdots + 8068889929 \) acting on \(S_{2}^{\mathrm{new}}(345, [\chi])\). Copy content Toggle raw display