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Properties

Label 110.3.l.a
Level $110$
Weight $3$
Character orbit 110.l
Analytic conductor $2.997$
Analytic rank $0$
Dimension $48$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [110,3,Mod(3,110)] 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("110.3"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(110, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([15, 16])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 110 = 2 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 110.l (of order \(20\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48,-12] 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: \(2.99728290796\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(6\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 12 q^{2} + 2 q^{3} + 4 q^{5} + 4 q^{6} + 38 q^{7} + 24 q^{8} + 36 q^{10} + 80 q^{11} - 16 q^{12} - 16 q^{13} + 38 q^{15} + 48 q^{16} - 18 q^{17} - 44 q^{18} + 4 q^{20} + 24 q^{21} + 64 q^{23} + 70 q^{25}+ \cdots - 440 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −0.221232 + 1.39680i −2.02588 + 3.97601i −1.90211 − 0.618034i 2.39781 + 4.38754i −5.10551 − 3.70937i 0.699150 + 1.37216i 1.28408 − 2.52015i −6.41440 − 8.82867i −6.65900 + 2.37861i
3.2 −0.221232 + 1.39680i −1.45699 + 2.85950i −1.90211 − 0.618034i −4.92092 − 0.885733i −3.67182 − 2.66774i −4.56016 − 8.94982i 1.28408 − 2.52015i −0.763859 − 1.05136i 2.32586 − 6.67760i
3.3 −0.221232 + 1.39680i −0.190659 + 0.374189i −1.90211 − 0.618034i 4.53254 − 2.11095i −0.480488 − 0.349095i 3.05982 + 6.00524i 1.28408 − 2.52015i 5.18640 + 7.13847i 1.94584 + 6.79807i
3.4 −0.221232 + 1.39680i 0.736983 − 1.44641i −1.90211 − 0.618034i 2.22216 − 4.47906i 1.85731 + 1.34941i −5.87986 − 11.5399i 1.28408 − 2.52015i 3.74111 + 5.14919i 5.76475 + 4.09483i
3.5 −0.221232 + 1.39680i 1.09534 − 2.14972i −1.90211 − 0.618034i 1.48006 + 4.77592i 2.76042 + 2.00556i 1.47655 + 2.89789i 1.28408 − 2.52015i 1.86852 + 2.57180i −6.99845 + 1.01077i
3.6 −0.221232 + 1.39680i 2.48324 − 4.87364i −1.90211 − 0.618034i −4.17484 − 2.75149i 6.25814 + 4.54680i 1.14280 + 2.24288i 1.28408 − 2.52015i −12.2958 − 16.9237i 4.76690 − 5.22271i
27.1 −0.642040 + 1.26007i −5.30750 + 0.840626i −1.17557 − 1.61803i −4.94213 + 0.758547i 2.34838 − 7.22756i 8.10082 + 1.28304i 2.79360 − 0.442463i 18.9034 − 6.14210i 2.21721 − 6.71446i
27.2 −0.642040 + 1.26007i −1.97586 + 0.312945i −1.17557 − 1.61803i 3.86028 − 3.17777i 0.874244 − 2.69065i 8.65808 + 1.37131i 2.79360 − 0.442463i −4.75344 + 1.54449i 1.52577 + 6.90449i
27.3 −0.642040 + 1.26007i −0.0723941 + 0.0114661i −1.17557 − 1.61803i −3.09837 + 3.92429i 0.0320317 − 0.0985835i −1.99053 − 0.315269i 2.79360 − 0.442463i −8.55440 + 2.77949i −2.95562 − 6.42373i
27.4 −0.642040 + 1.26007i −0.0541671 + 0.00857923i −1.17557 − 1.61803i −1.39544 − 4.80133i 0.0239670 − 0.0737628i −9.88997 − 1.56642i 2.79360 − 0.442463i −8.55665 + 2.78022i 6.94595 + 1.32429i
27.5 −0.642040 + 1.26007i 3.86350 − 0.611918i −1.17557 − 1.61803i 3.87433 + 3.16063i −1.70946 + 5.26117i −2.25703 − 0.357478i 2.79360 − 0.442463i 5.99265 − 1.94713i −6.47010 + 2.85268i
27.6 −0.642040 + 1.26007i 4.94323 − 0.782930i −1.17557 − 1.61803i −3.72952 − 3.33027i −2.18720 + 6.73150i 10.4055 + 1.64808i 2.79360 − 0.442463i 15.2630 − 4.95925i 6.59088 − 2.56131i
37.1 −0.221232 − 1.39680i −2.02588 − 3.97601i −1.90211 + 0.618034i 2.39781 − 4.38754i −5.10551 + 3.70937i 0.699150 − 1.37216i 1.28408 + 2.52015i −6.41440 + 8.82867i −6.65900 − 2.37861i
37.2 −0.221232 − 1.39680i −1.45699 − 2.85950i −1.90211 + 0.618034i −4.92092 + 0.885733i −3.67182 + 2.66774i −4.56016 + 8.94982i 1.28408 + 2.52015i −0.763859 + 1.05136i 2.32586 + 6.67760i
37.3 −0.221232 − 1.39680i −0.190659 − 0.374189i −1.90211 + 0.618034i 4.53254 + 2.11095i −0.480488 + 0.349095i 3.05982 − 6.00524i 1.28408 + 2.52015i 5.18640 − 7.13847i 1.94584 − 6.79807i
37.4 −0.221232 − 1.39680i 0.736983 + 1.44641i −1.90211 + 0.618034i 2.22216 + 4.47906i 1.85731 − 1.34941i −5.87986 + 11.5399i 1.28408 + 2.52015i 3.74111 − 5.14919i 5.76475 − 4.09483i
37.5 −0.221232 − 1.39680i 1.09534 + 2.14972i −1.90211 + 0.618034i 1.48006 − 4.77592i 2.76042 − 2.00556i 1.47655 − 2.89789i 1.28408 + 2.52015i 1.86852 − 2.57180i −6.99845 − 1.01077i
37.6 −0.221232 − 1.39680i 2.48324 + 4.87364i −1.90211 + 0.618034i −4.17484 + 2.75149i 6.25814 − 4.54680i 1.14280 − 2.24288i 1.28408 + 2.52015i −12.2958 + 16.9237i 4.76690 + 5.22271i
47.1 −1.39680 − 0.221232i −4.87364 − 2.48324i 1.90211 + 0.618034i −3.90692 − 3.12025i 6.25814 + 4.54680i −2.24288 + 1.14280i −2.52015 − 1.28408i 12.2958 + 16.9237i 4.76690 + 5.22271i
47.2 −1.39680 − 0.221232i −2.14972 − 1.09534i 1.90211 + 0.618034i 4.99953 − 0.0682179i 2.76042 + 2.00556i −2.89789 + 1.47655i −2.52015 − 1.28408i −1.86852 − 2.57180i −6.99845 − 1.01077i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.6
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
11.c even 5 1 inner
55.k odd 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 110.3.l.a ✓ 48
5.c odd 4 1 inner 110.3.l.a ✓ 48
11.c even 5 1 inner 110.3.l.a ✓ 48
55.k odd 20 1 inner 110.3.l.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.3.l.a ✓ 48 1.a even 1 1 trivial
110.3.l.a ✓ 48 5.c odd 4 1 inner
110.3.l.a ✓ 48 11.c even 5 1 inner
110.3.l.a ✓ 48 55.k odd 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{48} - 2 T_{3}^{47} + 2 T_{3}^{46} + 38 T_{3}^{45} - 1018 T_{3}^{44} + 588 T_{3}^{43} + \cdots + 141158161 \) acting on \(S_{3}^{\mathrm{new}}(110, [\chi])\). Copy content Toggle raw display