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Properties

Label 230.4.g.c
Level $230$
Weight $4$
Character orbit 230.g
Analytic conductor $13.570$
Analytic rank $0$
Dimension $60$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [230,4,Mod(31,230)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(230, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([0, 6])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("230.31"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 230.g (of order \(11\), degree \(10\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [60,12,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.5704393013\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(6\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q + 12 q^{2} + 5 q^{3} - 24 q^{4} + 30 q^{5} - 10 q^{6} - 3 q^{7} + 48 q^{8} - 107 q^{9} - 60 q^{10} + 19 q^{11} - 156 q^{12} - 258 q^{13} + 28 q^{14} - 25 q^{15} - 96 q^{16} - 446 q^{17} + 214 q^{18}+ \cdots + 16327 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 0.284630 − 1.97964i −3.24201 − 7.09901i −3.83797 − 1.12693i 3.27430 + 3.77875i −14.9763 + 4.39743i 0.612440 + 0.393591i −3.32332 + 7.27706i −22.2041 + 25.6249i 8.41254 − 5.40641i
31.2 0.284630 − 1.97964i −2.17613 − 4.76506i −3.83797 − 1.12693i 3.27430 + 3.77875i −10.0525 + 2.95168i 25.2044 + 16.1979i −3.32332 + 7.27706i −0.288990 + 0.333512i 8.41254 − 5.40641i
31.3 0.284630 − 1.97964i −0.0256086 − 0.0560750i −3.83797 − 1.12693i 3.27430 + 3.77875i −0.118297 + 0.0347352i −17.3463 − 11.1478i −3.32332 + 7.27706i 17.6788 − 20.4024i 8.41254 − 5.40641i
31.4 0.284630 − 1.97964i 1.06754 + 2.33758i −3.83797 − 1.12693i 3.27430 + 3.77875i 4.93143 − 1.44800i −7.25082 − 4.65982i −3.32332 + 7.27706i 13.3566 − 15.4143i 8.41254 − 5.40641i
31.5 0.284630 − 1.97964i 1.94712 + 4.26361i −3.83797 − 1.12693i 3.27430 + 3.77875i 8.99462 − 2.64106i 26.0282 + 16.7273i −3.32332 + 7.27706i 3.29420 − 3.80171i 8.41254 − 5.40641i
31.6 0.284630 − 1.97964i 4.14829 + 9.08348i −3.83797 − 1.12693i 3.27430 + 3.77875i 19.1628 − 5.62670i −29.1912 − 18.7601i −3.32332 + 7.27706i −47.6202 + 54.9566i 8.41254 − 5.40641i
41.1 1.91899 + 0.563465i −6.55594 + 7.56596i 3.36501 + 2.16256i 0.711574 − 4.94911i −16.8439 + 10.8249i −4.49226 − 9.83668i 5.23889 + 6.04600i −10.4209 − 72.4788i 4.15415 − 9.09632i
41.2 1.91899 + 0.563465i −4.07580 + 4.70373i 3.36501 + 2.16256i 0.711574 − 4.94911i −10.4718 + 6.72982i 7.45631 + 16.3271i 5.23889 + 6.04600i −1.67038 − 11.6178i 4.15415 − 9.09632i
41.3 1.91899 + 0.563465i −2.32777 + 2.68639i 3.36501 + 2.16256i 0.711574 − 4.94911i −5.98066 + 3.84353i −3.30940 − 7.24658i 5.23889 + 6.04600i 2.04432 + 14.2185i 4.15415 − 9.09632i
41.4 1.91899 + 0.563465i −1.99911 + 2.30709i 3.36501 + 2.16256i 0.711574 − 4.94911i −5.13622 + 3.30085i −9.87496 − 21.6231i 5.23889 + 6.04600i 2.51625 + 17.5009i 4.15415 − 9.09632i
41.5 1.91899 + 0.563465i 2.35383 − 2.71646i 3.36501 + 2.16256i 0.711574 − 4.94911i 6.04760 − 3.88656i 8.39133 + 18.3744i 5.23889 + 6.04600i 2.00384 + 13.9370i 4.15415 − 9.09632i
41.6 1.91899 + 0.563465i 5.61354 − 6.47837i 3.36501 + 2.16256i 0.711574 − 4.94911i 14.4226 − 9.26887i −4.23694 − 9.27760i 5.23889 + 6.04600i −6.61495 − 46.0080i 4.15415 − 9.09632i
71.1 −1.68251 − 1.08128i −1.23804 − 8.61075i 1.66166 + 3.63853i 4.79746 + 1.40866i −7.22764 + 15.8263i 11.7975 − 13.6150i 1.13852 − 7.91857i −46.7059 + 13.7141i −6.54861 − 7.55750i
71.2 −1.68251 − 1.08128i −0.573098 − 3.98599i 1.66166 + 3.63853i 4.79746 + 1.40866i −3.34573 + 7.32613i −4.78089 + 5.51744i 1.13852 − 7.91857i 10.3467 − 3.03806i −6.54861 − 7.55750i
71.3 −1.68251 − 1.08128i −0.426338 − 2.96525i 1.66166 + 3.63853i 4.79746 + 1.40866i −2.48895 + 5.45004i −17.2356 + 19.8909i 1.13852 − 7.91857i 17.2954 − 5.07839i −6.54861 − 7.55750i
71.4 −1.68251 − 1.08128i 0.642467 + 4.46846i 1.66166 + 3.63853i 4.79746 + 1.40866i 3.75071 − 8.21290i −10.3344 + 11.9265i 1.13852 − 7.91857i 6.35196 − 1.86510i −6.54861 − 7.55750i
71.5 −1.68251 − 1.08128i 0.662883 + 4.61045i 1.66166 + 3.63853i 4.79746 + 1.40866i 3.86989 − 8.47388i 15.5985 − 18.0016i 1.13852 − 7.91857i 5.08944 − 1.49439i −6.54861 − 7.55750i
71.6 −1.68251 − 1.08128i 1.46296 + 10.1751i 1.66166 + 3.63853i 4.79746 + 1.40866i 8.54074 − 18.7016i −0.590347 + 0.681297i 1.13852 − 7.91857i −75.4867 + 22.1649i −6.54861 − 7.55750i
81.1 −1.68251 + 1.08128i −1.23804 + 8.61075i 1.66166 − 3.63853i 4.79746 − 1.40866i −7.22764 − 15.8263i 11.7975 + 13.6150i 1.13852 + 7.91857i −46.7059 − 13.7141i −6.54861 + 7.55750i
81.2 −1.68251 + 1.08128i −0.573098 + 3.98599i 1.66166 − 3.63853i 4.79746 − 1.40866i −3.34573 − 7.32613i −4.78089 − 5.51744i 1.13852 + 7.91857i 10.3467 + 3.03806i −6.54861 + 7.55750i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.6
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 230.4.g.c ✓ 60
23.c even 11 1 inner 230.4.g.c ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.4.g.c ✓ 60 1.a even 1 1 trivial
230.4.g.c ✓ 60 23.c even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{60} - 5 T_{3}^{59} + 147 T_{3}^{58} - 487 T_{3}^{57} + 12168 T_{3}^{56} - 50910 T_{3}^{55} + \cdots + 92\!\cdots\!76 \) acting on \(S_{4}^{\mathrm{new}}(230, [\chi])\). Copy content Toggle raw display