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Properties

Label 63.8.f.b
Level $63$
Weight $8$
Character orbit 63.f
Analytic conductor $19.680$
Analytic rank $0$
Dimension $42$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [63,8,Mod(22,63)]
 
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([2, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("63.22");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 63.f (of order \(3\), 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: \(19.6802566055\)
Analytic rank: \(0\)
Dimension: \(42\)
Relative dimension: \(21\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 42 q + 24 q^{2} - 108 q^{3} - 1344 q^{4} + 429 q^{5} - 414 q^{6} - 7203 q^{7} - 11814 q^{8} - 4590 q^{9} + 12306 q^{11} + 24264 q^{12} + 6177 q^{13} + 8232 q^{14} - 3789 q^{15} - 86016 q^{16} - 96324 q^{17}+ \cdots + 47404458 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1 −10.5562 − 18.2838i −42.0824 + 20.3978i −158.865 + 275.163i 9.81890 − 17.0068i 817.178 + 554.105i −171.500 − 297.047i 4005.65 1354.86 − 1716.77i −414.600
22.2 −8.75497 − 15.1641i −30.3123 − 35.6113i −89.2990 + 154.670i 242.339 − 419.744i −274.629 + 771.433i −171.500 − 297.047i 885.966 −349.334 + 2158.92i −8486.69
22.3 −8.72830 − 15.1179i 46.4096 + 5.75724i −88.3664 + 153.055i −37.5173 + 64.9818i −318.040 − 751.865i −171.500 − 297.047i 850.710 2120.71 + 534.383i 1309.85
22.4 −7.85131 − 13.5989i −6.13219 − 46.3616i −59.2862 + 102.687i −213.909 + 370.502i −582.319 + 447.390i −171.500 − 297.047i −148.039 −2111.79 + 568.596i 6717.87
22.5 −7.72419 − 13.3787i 11.6262 + 45.2971i −55.3263 + 95.8280i 15.6459 − 27.0996i 516.213 − 505.428i −171.500 − 297.047i −267.989 −1916.66 + 1053.27i −483.409
22.6 −4.30291 − 7.45286i 36.5560 − 29.1661i 26.9699 − 46.7132i 143.860 − 249.173i −374.668 − 146.948i −171.500 − 297.047i −1565.74 485.680 − 2132.39i −2476.07
22.7 −4.18468 − 7.24807i −46.5293 + 4.69264i 28.9770 − 50.1896i 47.2438 − 81.8286i 228.723 + 317.611i −171.500 − 297.047i −1556.31 2142.96 − 436.691i −790.800
22.8 −3.17864 − 5.50556i −36.0198 − 29.8257i 43.7925 − 75.8509i −73.9619 + 128.106i −49.7131 + 293.114i −171.500 − 297.047i −1370.53 407.857 + 2148.63i 940.392
22.9 −1.42494 − 2.46806i −23.9426 + 40.1715i 59.9391 − 103.818i 267.089 − 462.612i 133.263 + 1.84996i −171.500 − 297.047i −706.422 −1040.50 − 1923.62i −1522.34
22.10 −0.724752 − 1.25531i 43.0642 + 18.2339i 62.9495 − 109.032i −154.597 + 267.769i −8.32176 − 67.2738i −171.500 − 297.047i −368.027 1522.05 + 1570.45i 448.177
22.11 0.128401 + 0.222396i 28.5208 − 37.0616i 63.9670 − 110.794i −170.587 + 295.465i 11.9045 + 1.58419i −171.500 − 297.047i 65.7241 −560.127 − 2114.05i −87.6138
22.12 1.07109 + 1.85518i −5.81249 + 46.4027i 61.7055 − 106.877i −93.9498 + 162.726i −92.3112 + 38.9182i −171.500 − 297.047i 538.567 −2119.43 − 539.431i −402.514
22.13 4.20811 + 7.28866i 33.7867 + 32.3336i 28.5836 − 49.5082i 126.650 − 219.365i −93.4907 + 382.323i −171.500 − 297.047i 1558.41 96.0764 + 2184.89i 2131.84
22.14 4.46158 + 7.72768i 8.34683 − 46.0145i 24.1887 − 41.8960i 166.337 − 288.105i 392.825 − 140.795i −171.500 − 297.047i 1573.84 −2047.66 − 768.150i 2968.51
22.15 4.57691 + 7.92743i −45.7230 + 9.81857i 22.1039 − 38.2850i −22.0792 + 38.2424i −287.106 − 317.528i −171.500 − 297.047i 1576.36 1994.19 − 897.870i −404.218
22.16 6.99699 + 12.1191i −31.2198 − 34.8184i −33.9158 + 58.7439i −123.337 + 213.626i 203.525 − 621.982i −171.500 − 297.047i 841.995 −237.649 + 2174.05i −3451.96
22.17 8.42248 + 14.5882i −7.74956 + 46.1188i −77.8762 + 134.886i −239.015 + 413.987i −738.059 + 275.383i −171.500 − 297.047i −467.488 −2066.89 − 714.801i −8052.40
22.18 8.53322 + 14.7800i 46.7551 + 0.978140i −81.6317 + 141.390i 117.163 − 202.932i 384.515 + 699.386i −171.500 − 297.047i −601.820 2185.09 + 91.4662i 3999.11
22.19 10.0790 + 17.4574i −2.43782 + 46.7018i −139.173 + 241.055i 102.888 − 178.208i −839.861 + 428.150i −171.500 − 297.047i −3030.68 −2175.11 − 227.701i 4148.05
22.20 10.1579 + 17.5940i −44.8233 − 13.3369i −142.366 + 246.585i 235.375 − 407.681i −220.662 − 924.096i −171.500 − 297.047i −3184.14 1831.26 + 1195.60i 9563.65
See all 42 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 22.21
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.8.f.b ✓ 42
3.b odd 2 1 189.8.f.a 42
9.c even 3 1 inner 63.8.f.b ✓ 42
9.c even 3 1 567.8.a.f 21
9.d odd 6 1 189.8.f.a 42
9.d odd 6 1 567.8.a.i 21
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.8.f.b ✓ 42 1.a even 1 1 trivial
63.8.f.b ✓ 42 9.c even 3 1 inner
189.8.f.a 42 3.b odd 2 1
189.8.f.a 42 9.d odd 6 1
567.8.a.f 21 9.c even 3 1
567.8.a.i 21 9.d odd 6 1