Properties

Label 100.9.k.a
Level $100$
Weight $9$
Character orbit 100.k
Analytic conductor $40.738$
Analytic rank $0$
Dimension $160$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [100,9,Mod(13,100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(100, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([0, 19]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("100.13");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 100.k (of order \(20\), degree \(8\), 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: \(40.7378610061\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(20\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 160 q + 70 q^{3} - 894 q^{5} + 2030 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 160 q + 70 q^{3} - 894 q^{5} + 2030 q^{7} - 33180 q^{13} + 48478 q^{15} - 414270 q^{17} + 718900 q^{19} - 207030 q^{23} + 1528574 q^{25} + 720310 q^{27} - 1956900 q^{29} + 6217120 q^{33} - 3418728 q^{35} - 14424480 q^{37} + 22510400 q^{39} + 4374720 q^{41} - 4033290 q^{43} + 13482846 q^{45} + 7278150 q^{47} - 15887760 q^{53} - 11288940 q^{55} + 39717680 q^{57} + 119809950 q^{59} + 18369120 q^{61} - 19887500 q^{63} + 136188762 q^{65} - 58954530 q^{67} - 128418850 q^{69} + 60703860 q^{71} + 33399920 q^{73} + 54011742 q^{75} + 184811940 q^{77} + 138506200 q^{79} + 196903220 q^{81} - 176936400 q^{83} - 186109196 q^{85} - 418946050 q^{87} + 442507050 q^{89} - 205667130 q^{93} + 256003896 q^{95} + 310913960 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 0 −66.0807 + 129.691i 0 555.433 286.564i 0 −667.736 + 667.736i 0 −8596.54 11832.1i 0
13.2 0 −63.9528 + 125.514i 0 −579.045 235.228i 0 −1129.62 + 1129.62i 0 −7807.47 10746.1i 0
13.3 0 −52.9510 + 103.922i 0 219.117 + 585.331i 0 1577.92 1577.92i 0 −4139.56 5697.62i 0
13.4 0 −46.2266 + 90.7248i 0 133.820 + 610.506i 0 −3112.48 + 3112.48i 0 −2237.63 3079.84i 0
13.5 0 −45.5201 + 89.3382i 0 284.067 556.714i 0 3352.66 3352.66i 0 −2052.77 2825.40i 0
13.6 0 −34.3799 + 67.4744i 0 −511.671 + 358.912i 0 1064.76 1064.76i 0 485.646 + 668.435i 0
13.7 0 −28.4093 + 55.7564i 0 −156.403 605.114i 0 −283.795 + 283.795i 0 1554.77 + 2139.95i 0
13.8 0 −19.2787 + 37.8365i 0 612.390 + 124.915i 0 51.5911 51.5911i 0 2796.53 + 3849.09i 0
13.9 0 −12.8247 + 25.1698i 0 63.1427 621.802i 0 −2466.61 + 2466.61i 0 3387.41 + 4662.37i 0
13.10 0 −10.0486 + 19.7216i 0 −609.311 139.160i 0 1210.11 1210.11i 0 3568.49 + 4911.61i 0
13.11 0 0.979552 1.92248i 0 606.877 149.415i 0 −1169.07 + 1169.07i 0 3853.72 + 5304.19i 0
13.12 0 11.7856 23.1305i 0 −326.301 + 533.060i 0 −2075.47 + 2075.47i 0 3460.34 + 4762.75i 0
13.13 0 24.2582 47.6095i 0 482.915 + 396.760i 0 2135.79 2135.79i 0 2178.26 + 2998.12i 0
13.14 0 24.5038 48.0914i 0 −95.2880 + 617.693i 0 1342.85 1342.85i 0 2144.11 + 2951.12i 0
13.15 0 26.2979 51.6125i 0 −137.231 609.748i 0 2691.16 2691.16i 0 1884.19 + 2593.37i 0
13.16 0 30.4495 59.7605i 0 −601.113 171.139i 0 −261.477 + 261.477i 0 1212.31 + 1668.60i 0
13.17 0 49.5425 97.2326i 0 −507.916 364.206i 0 −2319.68 + 2319.68i 0 −3143.26 4326.33i 0
13.18 0 52.2840 102.613i 0 405.825 475.322i 0 210.976 210.976i 0 −3939.37 5422.08i 0
13.19 0 58.4622 114.739i 0 420.580 + 462.317i 0 −1981.02 + 1981.02i 0 −5890.65 8107.78i 0
13.20 0 69.7211 136.835i 0 −513.580 + 356.175i 0 2020.02 2020.02i 0 −10006.4 13772.7i 0
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.f odd 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 100.9.k.a 160
25.f odd 20 1 inner 100.9.k.a 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.9.k.a 160 1.a even 1 1 trivial
100.9.k.a 160 25.f odd 20 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(100, [\chi])\).