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Properties

Label 63.8.h.a
Level $63$
Weight $8$
Character orbit 63.h
Analytic conductor $19.680$
Analytic rank $0$
Dimension $108$
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(25,63)]
 
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([4, 4]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("63.25");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 63.h (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: \(108\)
Relative dimension: \(54\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 108 q - 2 q^{2} - q^{3} + 6654 q^{4} - 499 q^{5} + 796 q^{6} - 84 q^{7} - 264 q^{8} - 2083 q^{9} - 258 q^{10} - 2953 q^{11} - 4262 q^{12} + 1845 q^{13} - 6064 q^{14} - 4756 q^{15} + 392958 q^{16} - 25695 q^{17}+ \cdots - 27043330 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 −22.0028 34.7176 − 31.3319i 356.122 145.681 − 252.326i −763.883 + 689.390i −303.887 − 855.100i −5019.33 223.620 − 2175.54i −3205.38 + 5551.88i
25.2 −21.7221 −10.9736 − 45.4597i 343.851 −233.757 + 404.878i 238.369 + 987.481i 739.781 + 525.612i −4688.75 −1946.16 + 997.708i 5077.69 − 8794.82i
25.3 −21.1764 −31.9133 + 34.1839i 320.441 −39.8456 + 69.0147i 675.811 − 723.893i −857.480 − 297.104i −4075.22 −150.078 − 2181.84i 843.789 − 1461.48i
25.4 −20.0358 33.2746 + 32.8603i 273.432 65.6746 − 113.752i −666.683 − 658.381i −332.989 + 844.193i −2913.84 27.4046 + 2186.83i −1315.84 + 2279.10i
25.5 −19.3904 −46.7353 − 1.67546i 247.987 143.053 − 247.775i 906.216 + 32.4878i 900.856 + 109.551i −2326.58 2181.39 + 156.606i −2773.85 + 4804.45i
25.6 −18.5497 25.9770 + 38.8870i 216.092 −264.606 + 458.310i −481.866 − 721.342i 165.435 − 892.286i −1634.08 −837.390 + 2020.33i 4908.36 − 8501.53i
25.7 −17.8618 −41.8243 − 20.9220i 191.043 −69.5880 + 120.530i 747.056 + 373.704i −819.973 − 388.829i −1126.07 1311.54 + 1750.09i 1242.97 − 2152.88i
25.8 −17.1725 46.4864 − 5.10084i 166.895 −16.8936 + 29.2606i −798.287 + 87.5943i 897.313 − 135.542i −667.927 2134.96 − 474.239i 290.106 − 502.478i
25.9 −16.4519 −25.8895 + 38.9453i 142.666 −81.2779 + 140.777i 425.933 − 640.725i 401.495 + 813.846i −241.291 −846.466 − 2016.55i 1337.18 − 2316.06i
25.10 −16.2500 0.187335 − 46.7650i 136.064 168.642 − 292.097i −3.04419 + 759.933i −289.417 + 860.105i −131.033 −2186.93 − 17.5214i −2740.44 + 4746.58i
25.11 −15.9040 1.05606 + 46.7534i 124.937 223.705 − 387.469i −16.7956 − 743.566i 357.985 − 833.900i 48.7143 −2184.77 + 98.7492i −3557.80 + 6162.30i
25.12 −14.9058 37.0045 − 28.5948i 94.1825 −151.448 + 262.315i −551.581 + 426.229i −856.219 + 300.719i 504.077 551.669 − 2116.28i 2257.44 − 3910.01i
25.13 −14.1340 −20.1996 − 42.1779i 71.7697 50.1511 − 86.8643i 285.502 + 596.142i 77.2576 − 904.198i 794.758 −1370.95 + 1703.96i −708.836 + 1227.74i
25.14 −10.9059 −38.2693 + 26.8786i −9.06177 −185.898 + 321.986i 417.360 − 293.135i 490.087 − 763.778i 1494.78 742.080 − 2057.25i 2027.39 − 3511.54i
25.15 −10.1682 −44.8504 − 13.2455i −24.6071 −179.394 + 310.719i 456.049 + 134.683i −260.605 + 869.269i 1551.74 1836.12 + 1188.13i 1824.12 − 3159.46i
25.16 −9.72250 43.2655 + 17.7509i −33.4730 12.5884 − 21.8038i −420.649 − 172.583i −714.928 − 558.946i 1569.92 1556.81 + 1536.01i −122.391 + 211.987i
25.17 −9.49162 −45.1496 + 12.1865i −37.9092 243.873 − 422.401i 428.543 − 115.669i −858.815 + 293.222i 1574.75 1889.98 − 1100.43i −2314.75 + 4009.27i
25.18 −9.47009 16.1710 − 43.8805i −38.3174 −94.8517 + 164.288i −153.141 + 415.552i 878.923 − 225.914i 1575.04 −1664.00 − 1419.18i 898.254 − 1555.82i
25.19 −8.81268 −4.01164 + 46.5930i −50.3367 33.1374 − 57.3957i 35.3533 − 410.609i −844.012 + 333.447i 1571.62 −2154.81 − 373.828i −292.029 + 505.810i
25.20 −8.75411 46.7562 − 0.924996i −51.3655 237.436 − 411.251i −409.309 + 8.09752i 482.651 + 768.499i 1570.19 2185.29 − 86.4986i −2078.54 + 3600.13i
See next 80 embeddings (of 108 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.54
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h 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.h.a yes 108
7.c even 3 1 63.8.g.a ✓ 108
9.c even 3 1 63.8.g.a ✓ 108
63.h even 3 1 inner 63.8.h.a yes 108
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.8.g.a ✓ 108 7.c even 3 1
63.8.g.a ✓ 108 9.c even 3 1
63.8.h.a yes 108 1.a even 1 1 trivial
63.8.h.a yes 108 63.h even 3 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(63, [\chi])\).