Properties

Label 1035.2.i.f
Level $1035$
Weight $2$
Character orbit 1035.i
Analytic conductor $8.265$
Analytic rank $0$
Dimension $50$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1035,2,Mod(346,1035)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1035.346"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1035, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1035 = 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1035.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [50,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.26451660920\)
Analytic rank: \(0\)
Dimension: \(50\)
Relative dimension: \(25\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 50 q - 4 q^{2} - q^{3} - 28 q^{4} - 25 q^{5} + 5 q^{6} + 5 q^{7} + 18 q^{8} + q^{9} + 8 q^{10} - 12 q^{11} - 11 q^{12} - 13 q^{14} - q^{15} - 38 q^{16} + 36 q^{17} + 8 q^{19} - 28 q^{20} + 7 q^{21} - 4 q^{22}+ \cdots - 57 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
346.1 −1.33759 − 2.31677i 1.66454 + 0.478863i −2.57828 + 4.46572i −0.500000 + 0.866025i −1.11705 − 4.49688i −1.42303 − 2.46476i 8.44438 2.54138 + 1.59417i 2.67518
346.2 −1.33070 − 2.30485i −1.58178 + 0.705663i −2.54155 + 4.40209i −0.500000 + 0.866025i 3.73133 + 2.70674i −0.741099 − 1.28362i 8.20539 2.00408 − 2.23241i 2.66141
346.3 −1.31498 − 2.27760i −0.333048 + 1.69973i −2.45832 + 4.25794i −0.500000 + 0.866025i 4.30926 − 1.47655i 2.48869 + 4.31055i 7.67062 −2.77816 − 1.13218i 2.62995
346.4 −1.22327 − 2.11876i −0.972779 − 1.43307i −1.99277 + 3.45159i −0.500000 + 0.866025i −1.84638 + 3.81412i −0.119145 − 0.206366i 4.85772 −1.10740 + 2.78813i 2.44654
346.5 −1.00932 − 1.74820i −0.592849 + 1.62743i −1.03746 + 1.79694i −0.500000 + 0.866025i 3.44344 − 0.606185i −2.53464 − 4.39013i 0.151242 −2.29706 − 1.92964i 2.01864
346.6 −0.945588 − 1.63781i 0.759549 − 1.55663i −0.788275 + 1.36533i −0.500000 + 0.866025i −3.26767 + 0.227933i −1.69675 − 2.93885i −0.800818 −1.84617 − 2.36467i 1.89118
346.7 −0.876139 − 1.51752i −1.57243 − 0.726276i −0.535240 + 0.927063i −0.500000 + 0.866025i 0.275527 + 3.02250i 1.12140 + 1.94232i −1.62878 1.94505 + 2.28403i 1.75228
346.8 −0.769380 − 1.33260i 0.176139 − 1.72307i −0.183891 + 0.318508i −0.500000 + 0.866025i −2.43169 + 1.09097i 2.54505 + 4.40815i −2.51159 −2.93795 − 0.606999i 1.53876
346.9 −0.700890 − 1.21398i 1.70107 − 0.326156i 0.0175071 − 0.0303232i −0.500000 + 0.866025i −1.58821 − 1.83645i 1.04186 + 1.80455i −2.85264 2.78724 − 1.10963i 1.40178
346.10 −0.506081 − 0.876559i 1.55289 + 0.767164i 0.487763 − 0.844831i −0.500000 + 0.866025i −0.113423 − 1.74944i −1.16266 − 2.01379i −3.01172 1.82292 + 2.38264i 1.01216
346.11 −0.383041 − 0.663447i 0.404548 + 1.68414i 0.706559 − 1.22380i −0.500000 + 0.866025i 0.962381 − 0.913492i 0.651755 + 1.12887i −2.61473 −2.67268 + 1.36263i 0.766082
346.12 −0.182646 − 0.316352i −1.72848 − 0.111203i 0.933281 − 1.61649i −0.500000 + 0.866025i 0.280520 + 0.567119i −1.67673 − 2.90417i −1.41242 2.97527 + 0.384424i 0.365292
346.13 −0.0689820 − 0.119480i −1.24651 + 1.20258i 0.990483 − 1.71557i −0.500000 + 0.866025i 0.229672 + 0.0659772i 1.24168 + 2.15066i −0.549230 0.107588 − 2.99807i 0.137964
346.14 −0.0274226 − 0.0474973i −0.449828 − 1.67262i 0.998496 − 1.72945i −0.500000 + 0.866025i −0.0671094 + 0.0672331i −0.723120 − 1.25248i −0.219216 −2.59531 + 1.50478i 0.0548451
346.15 0.109407 + 0.189498i 1.53938 − 0.793923i 0.976060 − 1.69059i −0.500000 + 0.866025i 0.318865 + 0.204849i 2.14071 + 3.70782i 0.864777 1.73937 − 2.44430i −0.218813
346.16 0.331652 + 0.574438i −1.63724 − 0.565207i 0.780014 − 1.35102i −0.500000 + 0.866025i −0.218316 − 1.12794i −0.712672 − 1.23438i 2.36138 2.36108 + 1.85075i −0.663304
346.17 0.456073 + 0.789942i −1.13041 + 1.31231i 0.583994 − 1.01151i −0.500000 + 0.866025i −1.55220 − 0.294450i −0.446483 − 0.773332i 2.88967 −0.444334 − 2.96691i −0.912147
346.18 0.587940 + 1.01834i 0.500594 + 1.65813i 0.308653 − 0.534603i −0.500000 + 0.866025i −1.39423 + 1.48466i 2.08036 + 3.60329i 3.07764 −2.49881 + 1.66010i −1.17588
346.19 0.678092 + 1.17449i 1.63797 − 0.563069i 0.0803821 − 0.139226i −0.500000 + 0.866025i 1.77202 + 1.54197i −2.51204 − 4.35099i 2.93039 2.36591 − 1.84458i −1.35618
346.20 0.687266 + 1.19038i 0.564865 − 1.63735i 0.0553306 − 0.0958355i −0.500000 + 0.866025i 2.33729 − 0.452895i 1.79065 + 3.10150i 2.90117 −2.36186 − 1.84977i −1.37453
See all 50 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 346.25
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1035.2.i.f ✓ 50
9.c even 3 1 inner 1035.2.i.f ✓ 50
9.c even 3 1 9315.2.a.bn 25
9.d odd 6 1 9315.2.a.bm 25
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1035.2.i.f ✓ 50 1.a even 1 1 trivial
1035.2.i.f ✓ 50 9.c even 3 1 inner
9315.2.a.bm 25 9.d odd 6 1
9315.2.a.bn 25 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{50} + 4 T_{2}^{49} + 47 T_{2}^{48} + 150 T_{2}^{47} + 1118 T_{2}^{46} + 3122 T_{2}^{45} + \cdots + 729 \) acting on \(S_{2}^{\mathrm{new}}(1035, [\chi])\). Copy content Toggle raw display