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Properties

Label 231.4.u.a
Level $231$
Weight $4$
Character orbit 231.u
Analytic conductor $13.629$
Analytic rank $0$
Dimension $368$
Inner twists $8$

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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] = [231,4,Mod(20,231)] 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("231.20"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(231, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([5, 5, 6])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 231 = 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 231.u (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: \(13.6294412113\)
Analytic rank: \(0\)
Dimension: \(368\)
Relative dimension: \(92\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 368 q + 340 q^{4} - 18 q^{7} + 42 q^{9} + 42 q^{15} - 1116 q^{16} - 262 q^{18} - 222 q^{21} + 696 q^{22} - 1516 q^{25} + 38 q^{28} + 624 q^{30} + 1430 q^{36} + 36 q^{37} + 390 q^{39} + 2130 q^{42} - 1376 q^{43}+ \cdots - 6218 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
20.1 −5.23865 − 1.70214i −5.19044 − 0.243508i 18.0740 + 13.1315i −3.51328 − 10.8128i 26.7764 + 10.1105i 18.1267 − 3.79766i −46.4303 − 63.9058i 26.8814 + 2.52783i 62.6244i
20.2 −5.23865 − 1.70214i 5.19044 + 0.243508i 18.0740 + 13.1315i 3.51328 + 10.8128i −26.7764 − 10.1105i 1.98968 + 18.4131i −46.4303 − 63.9058i 26.8814 + 2.52783i − 62.6244i
20.3 −5.16622 − 1.67861i −4.69469 + 2.22707i 17.4000 + 12.6418i 5.52182 + 16.9944i 27.9922 − 3.62501i −16.9124 − 7.54789i −43.1284 − 59.3611i 17.0803 − 20.9109i − 97.0658i
20.4 −5.16622 − 1.67861i 4.69469 − 2.22707i 17.4000 + 12.6418i −5.52182 − 16.9944i −27.9922 + 3.62501i −12.4047 − 13.7522i −43.1284 − 59.3611i 17.0803 − 20.9109i 97.0658i
20.5 −4.80477 − 1.56117i −2.74997 − 4.40882i 14.1765 + 10.2998i −1.47694 − 4.54554i 6.33009 + 25.4765i −10.9453 + 14.9399i −28.2789 − 38.9226i −11.8753 + 24.2482i 24.1460i
20.6 −4.80477 − 1.56117i 2.74997 + 4.40882i 14.1765 + 10.2998i 1.47694 + 4.54554i −6.33009 − 25.4765i 10.8264 − 15.0263i −28.2789 − 38.9226i −11.8753 + 24.2482i − 24.1460i
20.7 −4.68308 − 1.52162i −0.649279 − 5.15543i 13.1437 + 9.54947i −0.436121 − 1.34224i −4.80400 + 25.1312i 16.1366 − 9.08906i −23.8679 − 32.8514i −26.1569 + 6.69463i 6.94944i
20.8 −4.68308 − 1.52162i 0.649279 + 5.15543i 13.1437 + 9.54947i 0.436121 + 1.34224i 4.80400 − 25.1312i −3.65774 + 18.1555i −23.8679 − 32.8514i −26.1569 + 6.69463i − 6.94944i
20.9 −4.60266 − 1.49550i −1.25395 + 5.04258i 12.4759 + 9.06425i −3.57575 − 11.0050i 13.3126 − 21.3340i −13.9159 − 12.2208i −21.1099 − 29.0552i −23.8552 − 12.6462i 55.9999i
20.10 −4.60266 − 1.49550i 1.25395 − 5.04258i 12.4759 + 9.06425i 3.57575 + 11.0050i −13.3126 + 21.3340i −15.9229 − 9.45841i −21.1099 − 29.0552i −23.8552 − 12.6462i − 55.9999i
20.11 −4.11007 − 1.33544i −3.27517 + 4.03401i 8.63715 + 6.27526i 3.35478 + 10.3250i 18.8484 − 12.2063i 17.4564 + 6.18659i −6.79775 − 9.35630i −5.54653 − 26.4242i − 46.9164i
20.12 −4.11007 − 1.33544i 3.27517 − 4.03401i 8.63715 + 6.27526i −3.35478 − 10.3250i −18.8484 + 12.2063i 11.2781 + 14.6903i −6.79775 − 9.35630i −5.54653 − 26.4242i 46.9164i
20.13 −4.04850 − 1.31544i −4.30346 + 2.91208i 8.18784 + 5.94881i −4.42707 − 13.6251i 21.2532 − 6.12864i −2.32894 + 18.3732i −5.30628 − 7.30347i 10.0395 − 25.0641i 60.9848i
20.14 −4.04850 − 1.31544i 4.30346 − 2.91208i 8.18784 + 5.94881i 4.42707 + 13.6251i −21.2532 + 6.12864i 16.7543 − 7.89260i −5.30628 − 7.30347i 10.0395 − 25.0641i − 60.9848i
20.15 −3.90714 − 1.26951i −4.58324 − 2.44825i 7.18195 + 5.21799i 4.81631 + 14.8231i 14.7993 + 15.3841i 9.66179 + 15.8003i −2.11865 − 2.91608i 15.0121 + 22.4418i − 64.0302i
20.16 −3.90714 − 1.26951i 4.58324 + 2.44825i 7.18195 + 5.21799i −4.81631 − 14.8231i −14.7993 − 15.3841i 18.0126 + 4.30634i −2.11865 − 2.91608i 15.0121 + 22.4418i 64.0302i
20.17 −3.82785 − 1.24374i −4.34301 − 2.85276i 6.63341 + 4.81945i 1.82332 + 5.61160i 13.0763 + 16.3215i −3.71540 − 18.1438i −0.471589 − 0.649086i 10.7235 + 24.7791i − 23.7481i
20.18 −3.82785 − 1.24374i 4.34301 + 2.85276i 6.63341 + 4.81945i −1.82332 − 5.61160i −13.0763 − 16.3215i −18.4039 + 2.07318i −0.471589 − 0.649086i 10.7235 + 24.7791i 23.7481i
20.19 −3.45308 − 1.12197i −5.17340 + 0.485754i 4.19280 + 3.04625i −2.95174 − 9.08454i 18.4092 + 4.12707i 1.71814 − 18.4404i 6.01270 + 8.27577i 26.5281 − 5.02599i 34.6814i
20.20 −3.45308 − 1.12197i 5.17340 − 0.485754i 4.19280 + 3.04625i 2.95174 + 9.08454i −18.4092 − 4.12707i −17.0069 + 7.33244i 6.01270 + 8.27577i 26.5281 − 5.02599i − 34.6814i
See next 80 embeddings (of 368 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 20.92
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
11.c even 5 1 inner
21.c even 2 1 inner
33.h odd 10 1 inner
77.j odd 10 1 inner
231.u even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 231.4.u.a ✓ 368
3.b odd 2 1 inner 231.4.u.a ✓ 368
7.b odd 2 1 inner 231.4.u.a ✓ 368
11.c even 5 1 inner 231.4.u.a ✓ 368
21.c even 2 1 inner 231.4.u.a ✓ 368
33.h odd 10 1 inner 231.4.u.a ✓ 368
77.j odd 10 1 inner 231.4.u.a ✓ 368
231.u even 10 1 inner 231.4.u.a ✓ 368
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.4.u.a ✓ 368 1.a even 1 1 trivial
231.4.u.a ✓ 368 3.b odd 2 1 inner
231.4.u.a ✓ 368 7.b odd 2 1 inner
231.4.u.a ✓ 368 11.c even 5 1 inner
231.4.u.a ✓ 368 21.c even 2 1 inner
231.4.u.a ✓ 368 33.h odd 10 1 inner
231.4.u.a ✓ 368 77.j odd 10 1 inner
231.4.u.a ✓ 368 231.u even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(231, [\chi])\).