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Properties

Label 31.3.h.a
Level $31$
Weight $3$
Character orbit 31.h
Analytic conductor $0.845$
Analytic rank $0$
Dimension $32$
Inner twists $2$

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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] = [31,3,Mod(3,31)] 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("31.3"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(31, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 31 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 31.h (of order \(30\), degree \(8\), 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: \(0.844688819517\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(4\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 6 q^{2} - 10 q^{3} - 18 q^{4} - 7 q^{5} - 9 q^{6} - 22 q^{7} + 43 q^{8} - 2 q^{9} + 18 q^{10} - 5 q^{11} - 87 q^{12} - 49 q^{13} + 12 q^{14} + 70 q^{15} + 102 q^{16} - 62 q^{17} - 69 q^{18} - 132 q^{19}+ \cdots - 606 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −2.11749 + 1.53844i 3.04553 + 0.320098i 0.880871 − 2.71104i −3.92081 + 6.79105i −6.94133 + 4.00758i 8.43740 − 1.79342i −0.929678 − 2.86126i 0.369475 + 0.0785343i −2.14538 − 20.4119i
3.2 −1.18905 + 0.863898i −5.38491 − 0.565977i −0.568540 + 1.74979i −1.12075 + 1.94120i 6.89190 − 3.97904i 0.108231 − 0.0230052i −2.65232 − 8.16301i 19.8736 + 4.22427i −0.344364 − 3.27641i
3.3 −0.0969862 + 0.0704646i 2.02977 + 0.213337i −1.23163 + 3.79056i 3.06856 − 5.31490i −0.211892 + 0.122336i −3.98697 + 0.847457i −0.295831 − 0.910475i −4.72888 − 1.00516i 0.0769045 + 0.731698i
3.4 2.09451 − 1.52175i −1.49940 − 0.157594i 0.835177 − 2.57041i −1.26763 + 2.19560i −3.38033 + 1.95163i −0.0160743 + 0.00341670i 1.03789 + 3.19430i −6.57995 − 1.39861i 0.686092 + 6.52773i
11.1 −1.15629 − 3.55870i 2.16404 − 1.94851i −8.09128 + 5.87866i 3.11288 + 5.39167i −9.43644 − 5.44813i −0.873658 − 8.31230i 18.1674 + 13.1994i −0.0543773 + 0.517365i 15.5879 − 17.3122i
11.2 −0.331437 − 1.02006i 0.293175 − 0.263976i 2.30540 − 1.67497i −0.793103 − 1.37369i −0.366440 − 0.211564i 0.503493 + 4.79042i −5.94352 − 4.31822i −0.924488 + 8.79591i −1.13839 + 1.26431i
11.3 0.374220 + 1.15173i −3.80923 + 3.42985i 2.04962 − 1.48914i 2.70856 + 4.69136i −5.37576 − 3.10370i −0.963561 − 9.16768i 6.40098 + 4.65059i 1.80564 − 17.1795i −4.38959 + 4.87513i
11.4 0.922526 + 2.83924i 0.661033 − 0.595197i −3.97418 + 2.88741i −2.59389 − 4.49275i 2.29973 + 1.32775i −0.323932 − 3.08201i −2.20353 − 1.60096i −0.858051 + 8.16381i 10.3631 − 11.5094i
12.1 −2.74183 − 1.99206i −1.97791 + 4.44246i 2.31329 + 7.11957i 0.372590 − 0.645345i 14.2727 − 8.24037i −8.14446 + 9.04534i 3.65080 − 11.2360i −9.80113 − 10.8853i −2.30715 + 1.02721i
12.2 −1.30250 − 0.946319i 0.680823 − 1.52915i −0.435091 − 1.33907i 0.969949 − 1.68000i −2.33384 + 1.34744i 2.69654 − 2.99481i −2.69052 + 8.28058i 4.14738 + 4.60614i −2.85317 + 1.27031i
12.3 1.32183 + 0.960366i 1.06787 − 2.39847i −0.411134 − 1.26534i −2.97289 + 5.14920i 3.71495 − 2.14482i −7.48191 + 8.30951i 2.69132 − 8.28303i 1.40987 + 1.56582i −8.87478 + 3.95131i
12.4 1.41348 + 1.02696i −1.57980 + 3.54828i −0.292772 − 0.901060i 1.44394 − 2.50098i −5.87694 + 3.39305i 2.88725 − 3.20662i 2.67113 − 8.22089i −4.07237 − 4.52282i 4.60937 − 2.05223i
13.1 −2.74183 + 1.99206i −1.97791 − 4.44246i 2.31329 − 7.11957i 0.372590 + 0.645345i 14.2727 + 8.24037i −8.14446 − 9.04534i 3.65080 + 11.2360i −9.80113 + 10.8853i −2.30715 − 1.02721i
13.2 −1.30250 + 0.946319i 0.680823 + 1.52915i −0.435091 + 1.33907i 0.969949 + 1.68000i −2.33384 − 1.34744i 2.69654 + 2.99481i −2.69052 − 8.28058i 4.14738 − 4.60614i −2.85317 − 1.27031i
13.3 1.32183 − 0.960366i 1.06787 + 2.39847i −0.411134 + 1.26534i −2.97289 − 5.14920i 3.71495 + 2.14482i −7.48191 − 8.30951i 2.69132 + 8.28303i 1.40987 − 1.56582i −8.87478 − 3.95131i
13.4 1.41348 − 1.02696i −1.57980 − 3.54828i −0.292772 + 0.901060i 1.44394 + 2.50098i −5.87694 − 3.39305i 2.88725 + 3.20662i 2.67113 + 8.22089i −4.07237 + 4.52282i 4.60937 + 2.05223i
17.1 −1.15629 + 3.55870i 2.16404 + 1.94851i −8.09128 − 5.87866i 3.11288 − 5.39167i −9.43644 + 5.44813i −0.873658 + 8.31230i 18.1674 − 13.1994i −0.0543773 − 0.517365i 15.5879 + 17.3122i
17.2 −0.331437 + 1.02006i 0.293175 + 0.263976i 2.30540 + 1.67497i −0.793103 + 1.37369i −0.366440 + 0.211564i 0.503493 − 4.79042i −5.94352 + 4.31822i −0.924488 − 8.79591i −1.13839 − 1.26431i
17.3 0.374220 − 1.15173i −3.80923 − 3.42985i 2.04962 + 1.48914i 2.70856 − 4.69136i −5.37576 + 3.10370i −0.963561 + 9.16768i 6.40098 − 4.65059i 1.80564 + 17.1795i −4.38959 − 4.87513i
17.4 0.922526 − 2.83924i 0.661033 + 0.595197i −3.97418 − 2.88741i −2.59389 + 4.49275i 2.29973 − 1.32775i −0.323932 + 3.08201i −2.20353 + 1.60096i −0.858051 − 8.16381i 10.3631 + 11.5094i
See all 32 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 3.4
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.h odd 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 31.3.h.a ✓ 32
3.b odd 2 1 279.3.bc.b 32
31.h odd 30 1 inner 31.3.h.a ✓ 32
93.p even 30 1 279.3.bc.b 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.3.h.a ✓ 32 1.a even 1 1 trivial
31.3.h.a ✓ 32 31.h odd 30 1 inner
279.3.bc.b 32 3.b odd 2 1
279.3.bc.b 32 93.p even 30 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(31, [\chi])\).