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Properties

Label 63.6.o.a
Level $63$
Weight $6$
Character orbit 63.o
Analytic conductor $10.104$
Analytic rank $0$
Dimension $76$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [63,6,Mod(20,63)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(63, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 3]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("63.20");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 63.o (of order \(6\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.1041806482\)
Analytic rank: \(0\)
Dimension: \(76\)
Relative dimension: \(38\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 76 q - 6 q^{2} + 574 q^{4} - 30 q^{7} + 156 q^{9} - 576 q^{11} + 1392 q^{14} - 666 q^{15} - 8130 q^{16} + 3798 q^{18} + 4194 q^{21} - 130 q^{22} - 8112 q^{23} - 18752 q^{25} - 1732 q^{28} - 35826 q^{29}+ \cdots + 325080 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
20.1 −9.18933 − 5.30546i −14.8309 + 4.80033i 40.2959 + 69.7945i 32.1784 + 55.7346i 161.754 + 34.5732i −128.300 + 18.6049i − 515.603i 196.914 − 142.387i − 682.885i
20.2 −9.18933 − 5.30546i 14.8309 − 4.80033i 40.2959 + 69.7945i −32.1784 − 55.7346i −161.754 − 34.5732i 48.0376 + 120.413i − 515.603i 196.914 − 142.387i 682.885i
20.3 −8.40544 − 4.85288i −6.76084 + 14.0460i 31.1009 + 53.8683i −32.8857 − 56.9596i 124.992 − 85.2535i 113.770 − 62.1566i − 293.131i −151.582 − 189.926i 638.361i
20.4 −8.40544 − 4.85288i 6.76084 − 14.0460i 31.1009 + 53.8683i 32.8857 + 56.9596i −124.992 + 85.2535i −3.05564 − 129.606i − 293.131i −151.582 − 189.926i − 638.361i
20.5 −8.10965 − 4.68211i −9.16477 − 12.6098i 27.8443 + 48.2277i −41.6759 − 72.1848i 15.2827 + 145.171i −128.575 + 16.6010i − 221.824i −75.0139 + 231.132i 780.524i
20.6 −8.10965 − 4.68211i 9.16477 + 12.6098i 27.8443 + 48.2277i 41.6759 + 72.1848i −15.2827 − 145.171i 49.9104 + 119.649i − 221.824i −75.0139 + 231.132i − 780.524i
20.7 −6.90679 − 3.98764i −13.5113 − 7.77467i 15.8025 + 27.3707i 0.726757 + 1.25878i 62.3170 + 107.576i 119.993 − 49.0780i 3.15069i 122.109 + 210.091i − 11.5922i
20.8 −6.90679 − 3.98764i 13.5113 + 7.77467i 15.8025 + 27.3707i −0.726757 − 1.25878i −62.3170 − 107.576i −17.4938 − 128.456i 3.15069i 122.109 + 210.091i 11.5922i
20.9 −5.56054 − 3.21038i −2.07465 + 15.4498i 4.61309 + 7.99011i −9.29267 − 16.0954i 61.1359 − 79.2488i −120.355 + 48.1831i 146.225i −234.392 − 64.1057i 119.332i
20.10 −5.56054 − 3.21038i 2.07465 − 15.4498i 4.61309 + 7.99011i 9.29267 + 16.0954i −61.1359 + 79.2488i 18.4499 + 128.322i 146.225i −234.392 − 64.1057i − 119.332i
20.11 −4.66729 − 2.69466i −15.1594 + 3.63216i −1.47762 − 2.55931i 13.2780 + 22.9982i 80.5407 + 23.8971i 58.2095 + 115.839i 188.385i 216.615 − 110.123i − 143.119i
20.12 −4.66729 − 2.69466i 15.1594 − 3.63216i −1.47762 − 2.55931i −13.2780 − 22.9982i −80.5407 − 23.8971i −129.424 + 7.50861i 188.385i 216.615 − 110.123i 143.119i
20.13 −2.93016 − 1.69173i −3.58345 + 15.1710i −10.2761 − 17.7988i 47.2965 + 81.9200i 36.1652 − 38.3911i 59.9583 − 114.943i 177.808i −217.318 − 108.729i − 320.051i
20.14 −2.93016 − 1.69173i 3.58345 − 15.1710i −10.2761 − 17.7988i −47.2965 − 81.9200i −36.1652 + 38.3911i 69.5648 − 109.397i 177.808i −217.318 − 108.729i 320.051i
20.15 −2.69116 − 1.55374i −10.2344 − 11.7583i −11.1718 − 19.3501i 48.7868 + 84.5013i 9.27294 + 47.5451i −124.613 − 35.7563i 168.872i −33.5157 + 240.678i − 303.209i
20.16 −2.69116 − 1.55374i 10.2344 + 11.7583i −11.1718 − 19.3501i −48.7868 − 84.5013i −9.27294 − 47.5451i 93.2725 + 90.0402i 168.872i −33.5157 + 240.678i 303.209i
20.17 −2.26766 − 1.30923i −15.1407 + 3.70948i −12.5718 − 21.7750i −26.1836 − 45.3513i 39.1904 + 11.4108i −66.1554 − 111.492i 149.629i 215.479 − 112.328i 137.122i
20.18 −2.26766 − 1.30923i 15.1407 − 3.70948i −12.5718 − 21.7750i 26.1836 + 45.3513i −39.1904 − 11.4108i 129.633 + 1.54629i 149.629i 215.479 − 112.328i − 137.122i
20.19 0.868434 + 0.501391i −8.37086 + 13.1502i −15.4972 − 26.8420i −10.2165 − 17.6955i −13.8630 + 7.22305i 101.350 + 80.8400i − 63.1697i −102.857 − 220.158i − 20.4898i
20.20 0.868434 + 0.501391i 8.37086 − 13.1502i −15.4972 − 26.8420i 10.2165 + 17.6955i 13.8630 − 7.22305i −120.685 − 47.3520i − 63.1697i −102.857 − 220.158i 20.4898i
See all 76 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 20.38
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
9.d odd 6 1 inner
63.o even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.6.o.a ✓ 76
3.b odd 2 1 189.6.o.a 76
7.b odd 2 1 inner 63.6.o.a ✓ 76
9.c even 3 1 189.6.o.a 76
9.d odd 6 1 inner 63.6.o.a ✓ 76
21.c even 2 1 189.6.o.a 76
63.l odd 6 1 189.6.o.a 76
63.o even 6 1 inner 63.6.o.a ✓ 76
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.6.o.a ✓ 76 1.a even 1 1 trivial
63.6.o.a ✓ 76 7.b odd 2 1 inner
63.6.o.a ✓ 76 9.d odd 6 1 inner
63.6.o.a ✓ 76 63.o even 6 1 inner
189.6.o.a 76 3.b odd 2 1
189.6.o.a 76 9.c even 3 1
189.6.o.a 76 21.c even 2 1
189.6.o.a 76 63.l odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(63, [\chi])\).