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Properties

Label 63.10.f.a
Level $63$
Weight $10$
Character orbit 63.f
Analytic conductor $32.447$
Analytic rank $0$
Dimension $54$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [63,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("63.22");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
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: \(32.4472576783\)
Analytic rank: \(0\)
Dimension: \(54\)
Relative dimension: \(27\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 54 q - 48 q^{2} + 9 q^{3} - 6912 q^{4} - 1704 q^{5} - 540 q^{6} - 64827 q^{7} + 64650 q^{8} - 13275 q^{9} - 150009 q^{11} - 6156 q^{12} - 89478 q^{13} - 115248 q^{14} - 582588 q^{15} - 1769472 q^{16} + 924474 q^{17}+ \cdots - 5802933888 q^{99}+O(q^{100}) \) 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 −21.9947 − 38.0960i −72.8107 − 119.923i −711.535 + 1232.41i −933.466 + 1616.81i −2967.14 + 5411.47i −1200.50 − 2079.33i 40077.5 −9080.19 + 17463.4i 82125.3
22.2 −20.7841 − 35.9991i 138.883 + 19.8644i −607.956 + 1053.01i 431.689 − 747.707i −2171.45 − 5412.51i −1200.50 − 2079.33i 29260.3 18893.8 + 5517.65i −35889.0
22.3 −20.1553 − 34.9100i −63.9215 + 124.888i −556.470 + 963.834i −597.295 + 1034.55i 5648.19 − 285.655i −1200.50 − 2079.33i 24224.2 −11511.1 − 15966.1i 48154.6
22.4 −18.8236 − 32.6034i 57.5087 − 127.968i −452.656 + 784.023i 875.867 − 1517.05i −5254.71 + 533.833i −1200.50 − 2079.33i 14807.1 −13068.5 − 14718.5i −65947.9
22.5 −15.6011 − 27.0219i −120.204 − 72.3459i −230.788 + 399.737i 224.203 − 388.331i −79.6031 + 4376.82i −1200.50 − 2079.33i −1573.32 9215.15 + 17392.6i −13991.2
22.6 −15.3318 − 26.5554i 94.1003 + 104.058i −214.128 + 370.880i −1208.32 + 2092.86i 1320.59 − 4094.28i −1200.50 − 2079.33i −2567.90 −1973.27 + 19583.8i 74102.6
22.7 −15.0772 − 26.1144i −130.356 + 51.8680i −198.641 + 344.056i 534.305 − 925.443i 3319.90 + 2622.15i −1200.50 − 2079.33i −3459.24 14302.4 − 13522.6i −32223.2
22.8 −13.0201 − 22.5515i 85.0977 − 111.541i −83.0468 + 143.841i −799.287 + 1384.41i −3623.40 − 466.805i −1200.50 − 2079.33i −9007.48 −5199.76 − 18983.8i 41627.2
22.9 −12.7613 − 22.1032i 60.6817 + 126.494i −69.7006 + 120.725i 1146.62 − 1986.01i 2021.54 − 2955.48i −1200.50 − 2079.33i −9509.68 −12318.5 + 15351.7i −58529.5
22.10 −6.38254 − 11.0549i −79.5122 + 115.589i 174.526 − 302.289i −535.587 + 927.664i 1785.31 + 141.247i −1200.50 − 2079.33i −10991.4 −7038.62 − 18381.5i 13673.6
22.11 −5.50373 − 9.53274i 23.6879 − 138.282i 195.418 − 338.474i 227.775 − 394.518i −1448.58 + 535.256i −1200.50 − 2079.33i −9937.93 −18560.8 − 6551.21i −5014.45
22.12 −4.52912 − 7.84467i −135.687 − 35.6664i 214.974 − 372.346i −1038.21 + 1798.24i 334.751 + 1225.96i −1200.50 − 2079.33i −8532.40 17138.8 + 9678.91i 18808.8
22.13 −3.86694 − 6.69773i −42.4923 − 133.706i 226.094 − 391.606i 1082.49 − 1874.93i −731.214 + 801.636i −1200.50 − 2079.33i −7456.90 −16071.8 + 11363.0i −16743.7
22.14 −2.49097 − 4.31449i 19.0872 + 138.992i 243.590 − 421.910i 130.494 − 226.022i 552.133 − 428.576i −1200.50 − 2079.33i −4977.86 −18954.4 + 5305.91i −1300.23
22.15 −1.70004 − 2.94456i 134.921 + 38.4618i 250.220 − 433.393i −30.2545 + 52.4023i −116.118 − 462.670i −1200.50 − 2079.33i −3442.38 16724.4 + 10378.6i 205.736
22.16 5.33869 + 9.24689i −43.8624 − 133.263i 198.997 − 344.672i −667.909 + 1156.85i 998.102 − 1117.04i −1200.50 − 2079.33i 9716.35 −15835.2 + 11690.5i −14263.1
22.17 5.54457 + 9.60347i 138.579 − 21.8801i 194.516 − 336.911i 1340.86 − 2322.44i 978.487 + 1209.53i −1200.50 − 2079.33i 9991.65 18725.5 − 6064.26i 29738.0
22.18 6.17786 + 10.7004i −138.713 − 21.0197i 179.668 − 311.194i 571.192 − 989.334i −632.028 − 1614.13i −1200.50 − 2079.33i 10766.0 18799.3 + 5831.40i 14115.0
22.19 7.62540 + 13.2076i 114.234 − 81.4470i 139.707 − 241.979i −752.794 + 1303.88i 1946.80 + 887.687i −1200.50 − 2079.33i 12069.7 6415.77 − 18608.0i −22961.4
22.20 10.8750 + 18.8360i −109.873 + 87.2406i 19.4689 − 33.7212i −1121.31 + 1942.16i −2838.14 − 1120.83i −1200.50 − 2079.33i 11982.9 4461.16 − 19170.8i −48776.8
See all 54 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 22.27
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 63.10.f.a ✓ 54
9.c even 3 1 inner 63.10.f.a ✓ 54
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.10.f.a ✓ 54 1.a even 1 1 trivial
63.10.f.a ✓ 54 9.c even 3 1 inner