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Properties

Label 31.5.f.a
Level $31$
Weight $5$
Character orbit 31.f
Analytic conductor $3.204$
Analytic rank $0$
Dimension $36$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [31,5,Mod(15,31)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(31, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([7])) N = Newforms(chi, 5, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("31.15"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Level: \( N \) \(=\) \( 31 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 31.f (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.20446885560\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(9\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 3 q^{2} - 5 q^{3} - 79 q^{4} - 2 q^{5} - 73 q^{7} + 173 q^{8} + 62 q^{9} + 84 q^{10} + 100 q^{11} + 150 q^{12} - 260 q^{13} + 488 q^{14} - 1075 q^{15} + 777 q^{16} - 725 q^{17} - 786 q^{18} - 1581 q^{19}+ \cdots + 74456 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1 −5.57132 + 4.04780i 3.06627 − 4.22036i 9.71064 − 29.8863i 23.0787 35.9246i −5.48427 + 16.8789i 32.8237 + 101.021i 16.6209 + 51.1540i −128.579 + 93.4180i
15.2 −3.99299 + 2.90108i −6.95155 + 9.56799i 2.58345 − 7.95104i −0.0618873 − 58.3719i 5.65199 − 17.3950i −11.6521 − 35.8615i −18.1920 − 55.9892i 0.247115 − 0.179540i
15.3 −3.66939 + 2.66597i 7.06689 − 9.72674i 1.41276 − 4.34804i −44.3313 54.5313i 17.8399 − 54.9055i −16.0176 − 49.2970i −19.6382 − 60.4400i 162.669 − 118.186i
15.4 −1.19136 + 0.865573i 0.644517 − 0.887101i −4.27415 + 13.1545i −1.23246 1.61473i −23.4764 + 72.2530i −13.5751 − 41.7797i 24.6588 + 75.8921i 1.46831 − 1.06679i
15.5 0.129011 − 0.0937321i 4.21329 − 5.79909i −4.93641 + 15.1927i 33.7795 − 1.14307i 24.9042 − 76.6471i 1.57564 + 4.84932i 9.15268 + 28.1691i 4.35793 − 3.16622i
15.6 1.65757 − 1.20430i −5.47296 + 7.53288i −3.64705 + 11.2245i −37.3945 19.0774i 8.17809 − 25.1696i 17.6025 + 54.1751i −1.76066 − 5.41876i −61.9842 + 45.0341i
15.7 3.01837 − 2.19297i 9.74278 − 13.4098i −0.642851 + 1.97849i −9.66586 − 61.8414i −17.5449 + 53.9977i 20.8451 + 64.1545i −59.8702 − 184.262i −29.1751 + 21.1970i
15.8 3.11734 − 2.26488i −8.18387 + 11.2641i −0.356154 + 1.09613i 40.0060 53.6495i −8.49904 + 26.1574i 20.4238 + 62.8580i −34.8745 − 107.333i 124.712 − 90.6087i
15.9 5.19375 − 3.77348i 0.773818 − 1.06507i 7.79160 − 23.9801i −1.32406 − 8.45169i 3.10021 − 9.54147i −18.2794 − 56.2581i 24.4948 + 75.3872i −6.87683 + 4.99631i
23.1 −2.30154 + 7.08340i 0.135338 − 0.0439738i −31.9333 − 23.2009i −18.1577 1.05986i 10.5920 + 7.69552i 141.429 − 102.754i −65.5140 + 47.5987i 41.7905 − 128.618i
23.2 −1.60665 + 4.94476i −15.6281 + 5.07789i −8.92504 − 6.48442i 30.2396 − 85.4357i −52.6343 − 38.2411i −20.8969 + 15.1825i 152.923 − 111.105i −48.5844 + 149.527i
23.3 −1.33487 + 4.10831i 7.83134 − 2.54456i −2.15208 − 1.56358i 21.7607 35.5702i 7.98302 + 5.80000i −46.6194 + 33.8710i −10.6753 + 7.75606i −29.0478 + 89.3999i
23.4 −0.697431 + 2.14647i −5.17987 + 1.68304i 8.82334 + 6.41053i −35.2158 − 12.2923i −13.1009 − 9.51835i −49.1281 + 35.6936i −41.5319 + 30.1747i 24.5606 − 75.5898i
23.5 0.300296 − 0.924216i 13.0181 − 4.22985i 12.1803 + 8.84949i −23.3923 − 13.3018i 2.85368 + 2.07332i 24.4155 − 17.7389i 86.0501 − 62.5190i −7.02461 + 21.6195i
23.6 0.416650 − 1.28232i −7.95500 + 2.58474i 11.4735 + 8.33601i 20.3736 11.2778i 51.5849 + 37.4786i 32.9227 − 23.9197i −8.92916 + 6.48742i 8.48864 − 26.1253i
23.7 1.17020 − 3.60149i 4.81539 − 1.56461i 1.34288 + 0.975660i 26.3902 − 19.1735i −57.3808 − 41.6896i 54.1031 − 39.3082i −44.7904 + 32.5422i 30.8817 − 95.0441i
23.8 1.73579 − 5.34220i −11.2256 + 3.64741i −12.5819 − 9.14128i −23.2172 66.3004i −34.4697 − 25.0437i 2.03552 − 1.47889i 47.1794 − 34.2779i −40.3002 + 124.031i
23.9 2.12658 − 6.54494i 6.78918 − 2.20594i −25.3696 − 18.4321i −2.63517 − 49.1258i 43.4024 + 31.5337i −85.5079 + 62.1251i −24.3036 + 17.6576i −5.60390 + 17.2470i
27.1 −2.30154 − 7.08340i 0.135338 + 0.0439738i −31.9333 + 23.2009i −18.1577 − 1.05986i 10.5920 − 7.69552i 141.429 + 102.754i −65.5140 − 47.5987i 41.7905 + 128.618i
27.2 −1.60665 − 4.94476i −15.6281 − 5.07789i −8.92504 + 6.48442i 30.2396 85.4357i −52.6343 + 38.2411i −20.8969 − 15.1825i 152.923 + 111.105i −48.5844 − 149.527i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 15.9
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.f odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 31.5.f.a ✓ 36
31.f odd 10 1 inner 31.5.f.a ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.5.f.a ✓ 36 1.a even 1 1 trivial
31.5.f.a ✓ 36 31.f odd 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(31, [\chi])\).