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Properties

Label 820.2.j.c
Level $820$
Weight $2$
Character orbit 820.j
Analytic conductor $6.548$
Analytic rank $0$
Dimension $240$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [820,2,Mod(483,820)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(820, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 3, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("820.483"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 820 = 2^{2} \cdot 5 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 820.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [240,0] 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: \(6.54773296574\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(120\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q - 4 q^{6} + 12 q^{8} - 240 q^{9} - 20 q^{10} - 32 q^{13} - 8 q^{14} + 8 q^{16} + 32 q^{17} - 12 q^{18} + 16 q^{20} - 28 q^{22} + 12 q^{24} - 16 q^{25} - 8 q^{28} - 10 q^{30} + 24 q^{33} - 20 q^{34}+ \cdots - 12 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
483.1 −1.41355 + 0.0431962i − 1.95083i 1.99627 − 0.122120i 0.661743 + 2.13591i 0.0842684 + 2.75760i 3.68722 −2.81656 + 0.258855i −0.805738 −1.02767 − 2.99063i
483.2 −1.41330 − 0.0508885i 2.76323i 1.99482 + 0.143841i 0.630427 − 2.14536i 0.140616 − 3.90526i −4.56297 −2.81196 − 0.304804i −4.63543 −1.00015 + 2.99995i
483.3 −1.41313 − 0.0554718i − 1.54369i 1.99385 + 0.156777i −1.38671 + 1.75415i −0.0856310 + 2.18142i −2.96451 −2.80886 − 0.332148i 0.617035 2.05690 − 2.40191i
483.4 −1.41116 − 0.0928905i 0.528253i 1.98274 + 0.262167i −2.08490 − 0.808196i 0.0490697 − 0.745449i −2.69881 −2.77361 − 0.554137i 2.72095 2.86706 + 1.33416i
483.5 −1.40801 − 0.132345i 2.00302i 1.96497 + 0.372687i 2.05738 + 0.875886i 0.265090 − 2.82026i 0.526563 −2.71737 − 0.784800i −1.01208 −2.78089 − 1.50554i
483.6 −1.39510 − 0.231725i 0.177636i 1.89261 + 0.646558i 2.02814 − 0.941627i 0.0411626 − 0.247820i 1.24733 −2.49055 − 1.34058i 2.96845 −3.04765 + 0.843694i
483.7 −1.38088 − 0.305216i 2.75443i 1.81369 + 0.842936i 1.14803 + 1.91886i 0.840694 − 3.80355i 2.02593 −2.24722 − 1.71756i −4.58686 −0.999638 − 3.00012i
483.8 −1.37740 + 0.320588i − 2.73986i 1.79445 − 0.883155i 1.68639 − 1.46836i 0.878367 + 3.77387i 2.27751 −2.18854 + 1.79173i −4.50682 −1.85210 + 2.56315i
483.9 −1.36734 + 0.361076i − 0.837081i 1.73925 − 0.987428i 2.12147 + 0.706651i 0.302250 + 1.14458i −1.14908 −2.02161 + 1.97815i 2.29930 −3.15593 − 0.200221i
483.10 −1.36551 + 0.367958i 0.984353i 1.72921 − 1.00490i 0.127778 + 2.23241i −0.362201 − 1.34414i −2.37783 −1.99149 + 2.00847i 2.03105 −0.995917 − 3.00136i
483.11 −1.35630 + 0.400571i 0.278394i 1.67909 − 1.08659i −0.759482 − 2.10314i −0.111517 − 0.377586i 3.39519 −1.84208 + 2.14633i 2.92250 1.87254 + 2.54825i
483.12 −1.35590 + 0.401898i 2.73899i 1.67696 − 1.08987i −2.07460 + 0.834293i −1.10079 − 3.71381i 0.506296 −1.83578 + 2.15173i −4.50206 2.47766 − 1.96500i
483.13 −1.34514 − 0.436587i − 1.46343i 1.61878 + 1.17454i −0.431484 − 2.19404i −0.638915 + 1.96851i 1.92859 −1.66469 − 2.28666i 0.858371 −0.377487 + 3.13967i
483.14 −1.34442 − 0.438801i − 2.44417i 1.61491 + 1.17986i 1.52940 − 1.63124i −1.07250 + 3.28598i −4.45018 −1.65338 − 2.29485i −2.97396 −2.77193 + 1.52197i
483.15 −1.34160 + 0.447349i − 2.33166i 1.59976 − 1.20032i −1.06811 − 1.96447i 1.04307 + 3.12815i −2.60871 −1.60926 + 2.32600i −2.43664 2.31178 + 2.15771i
483.16 −1.33221 − 0.474575i 0.567590i 1.54956 + 1.26447i −1.84871 + 1.25788i 0.269364 − 0.756148i 4.85551 −1.46425 − 2.41991i 2.67784 3.05983 − 0.798412i
483.17 −1.26315 − 0.635967i 3.24721i 1.19109 + 1.60664i −1.25619 − 1.84986i 2.06512 − 4.10171i 3.82568 −0.482756 − 2.78692i −7.54436 0.410298 + 3.13555i
483.18 −1.23863 + 0.682492i − 1.46723i 1.06841 − 1.69071i −2.23604 + 0.0119321i 1.00137 + 1.81736i 1.28612 −0.169471 + 2.82335i 0.847235 2.76148 − 1.54086i
483.19 −1.23475 + 0.689491i 2.13292i 1.04921 − 1.70269i 1.66560 − 1.49190i −1.47063 − 2.63362i 3.79559 −0.121513 + 2.82582i −1.54935 −1.02794 + 2.99054i
483.20 −1.19471 − 0.756742i − 2.38102i 0.854683 + 1.80818i −0.119205 + 2.23289i −1.80181 + 2.84463i −1.01521 0.347225 − 2.80703i −2.66924 1.83214 − 2.57746i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 483.120
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
205.i odd 4 1 inner
820.j even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 820.2.j.c ✓ 240
4.b odd 2 1 inner 820.2.j.c ✓ 240
5.c odd 4 1 820.2.s.c yes 240
20.e even 4 1 820.2.s.c yes 240
41.c even 4 1 820.2.s.c yes 240
164.e odd 4 1 820.2.s.c yes 240
205.i odd 4 1 inner 820.2.j.c ✓ 240
820.j even 4 1 inner 820.2.j.c ✓ 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
820.2.j.c ✓ 240 1.a even 1 1 trivial
820.2.j.c ✓ 240 4.b odd 2 1 inner
820.2.j.c ✓ 240 205.i odd 4 1 inner
820.2.j.c ✓ 240 820.j even 4 1 inner
820.2.s.c yes 240 5.c odd 4 1
820.2.s.c yes 240 20.e even 4 1
820.2.s.c yes 240 41.c even 4 1
820.2.s.c yes 240 164.e odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(820, [\chi])\):

\( T_{3}^{120} + 240 T_{3}^{118} + 27984 T_{3}^{116} + 2112528 T_{3}^{114} + 116091792 T_{3}^{112} + \cdots + 40\!\cdots\!72 \) Copy content Toggle raw display
\( T_{13}^{60} + 8 T_{13}^{59} - 370 T_{13}^{58} - 3024 T_{13}^{57} + 63754 T_{13}^{56} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display