Properties

Label 99.10.f.a
Level $99$
Weight $10$
Character orbit 99.f
Analytic conductor $50.989$
Analytic rank $0$
Dimension $32$
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] = [99,10,Mod(37,99)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(99, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("99.37");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 99.f (of order \(5\), degree \(4\), 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: \(50.9885477802\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{5})\)
Twist minimal: no (minimal twist has level 11)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 21 q^{2} - 1625 q^{4} - 225 q^{5} - 10675 q^{7} - 22863 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 21 q^{2} - 1625 q^{4} - 225 q^{5} - 10675 q^{7} - 22863 q^{8} - 16272 q^{10} + 44637 q^{11} + 164099 q^{13} + 125832 q^{14} - 1428705 q^{16} - 1430145 q^{17} + 2175116 q^{19} + 79194 q^{20} - 4280751 q^{22} + 4045452 q^{23} - 1591199 q^{25} + 11064048 q^{26} + 3790710 q^{28} - 2621889 q^{29} + 3571591 q^{31} + 56548860 q^{32} + 8515154 q^{34} - 3015333 q^{35} + 20945529 q^{37} - 55333848 q^{38} - 21310356 q^{40} + 53829591 q^{41} - 4383434 q^{43} + 21997776 q^{44} - 2025868 q^{46} - 78754497 q^{47} + 227397397 q^{49} + 74287941 q^{50} - 157487230 q^{52} + 30528081 q^{53} - 248082383 q^{55} + 327187608 q^{56} + 279888572 q^{58} - 254767908 q^{59} - 10620465 q^{61} - 686572332 q^{62} - 1338445825 q^{64} + 869006394 q^{65} - 946203354 q^{67} - 2053775934 q^{68} - 10909448 q^{70} - 1396168719 q^{71} + 1566720997 q^{73} + 604530504 q^{74} - 4233309866 q^{76} + 3256040955 q^{77} + 2890358835 q^{79} - 5092570836 q^{80} + 4014664309 q^{82} + 480997020 q^{83} - 3775782881 q^{85} + 3721184697 q^{86} - 8695160705 q^{88} + 1559422158 q^{89} + 5038455823 q^{91} - 6839439726 q^{92} + 1219591670 q^{94} - 2592967899 q^{95} - 3010541778 q^{97} + 7371061380 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 −35.8136 26.0201i 0 447.352 + 1376.81i −376.543 + 273.575i 0 −442.773 1362.72i 12799.4 39392.6i 0 20603.8
37.2 −24.5942 17.8687i 0 127.366 + 391.991i 601.873 437.286i 0 430.583 + 1325.20i −937.875 + 2886.48i 0 −22616.3
37.3 −12.1558 8.83167i 0 −88.4528 272.230i −1436.78 + 1043.88i 0 741.893 + 2283.31i −3706.29 + 11406.8i 0 26684.4
37.4 −7.79135 5.66074i 0 −129.556 398.731i 1985.48 1442.54i 0 −2299.51 7077.16i −2771.43 + 8529.59i 0 −23635.4
37.5 7.91186 + 5.74830i 0 −128.662 395.981i −690.317 + 501.544i 0 −2608.39 8027.81i 2805.56 8634.62i 0 −8344.72
37.6 14.4842 + 10.5234i 0 −59.1658 182.094i 591.288 429.596i 0 2103.13 + 6472.78i 3891.91 11978.1i 0 13085.2
37.7 21.4033 + 15.5504i 0 58.0694 + 178.719i −1489.97 + 1082.53i 0 −720.731 2218.18i 2649.49 8154.29i 0 −48724.0
37.8 33.4202 + 24.2812i 0 369.117 + 1136.03i 314.303 228.354i 0 −1491.31 4589.77i −8712.23 + 26813.5i 0 16048.8
64.1 −11.8250 + 36.3935i 0 −770.441 559.758i 276.050 + 849.593i 0 42.1653 + 30.6349i 13631.4 9903.80i 0 −34184.0
64.2 −6.35502 + 19.5587i 0 72.0591 + 52.3540i −411.773 1267.31i 0 2149.58 + 1561.76i −10000.4 + 7265.71i 0 27403.7
64.3 −4.40341 + 13.5523i 0 249.942 + 181.594i −63.1636 194.398i 0 −1584.15 1150.95i −9464.08 + 6876.06i 0 2912.67
64.4 1.04500 3.21619i 0 404.965 + 294.224i 656.025 + 2019.04i 0 −6768.61 4917.68i 2770.23 2012.69i 0 7179.16
64.5 4.28817 13.1976i 0 258.428 + 187.759i 237.559 + 731.131i 0 9125.19 + 6629.84i 9334.16 6781.66i 0 10667.9
64.6 6.86703 21.1346i 0 14.7033 + 10.6826i −597.948 1840.30i 0 −7706.91 5599.40i 9531.54 6925.07i 0 −43000.0
64.7 10.8459 33.3802i 0 −582.388 423.130i −311.221 957.840i 0 7268.43 + 5280.82i −5902.49 + 4288.41i 0 −35348.4
64.8 13.1726 40.5409i 0 −1055.84 767.109i 602.641 + 1854.74i 0 −3576.09 2598.18i −27350.5 + 19871.3i 0 83131.1
82.1 −11.8250 36.3935i 0 −770.441 + 559.758i 276.050 849.593i 0 42.1653 30.6349i 13631.4 + 9903.80i 0 −34184.0
82.2 −6.35502 19.5587i 0 72.0591 52.3540i −411.773 + 1267.31i 0 2149.58 1561.76i −10000.4 7265.71i 0 27403.7
82.3 −4.40341 13.5523i 0 249.942 181.594i −63.1636 + 194.398i 0 −1584.15 + 1150.95i −9464.08 6876.06i 0 2912.67
82.4 1.04500 + 3.21619i 0 404.965 294.224i 656.025 2019.04i 0 −6768.61 + 4917.68i 2770.23 + 2012.69i 0 7179.16
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.10.f.a 32
3.b odd 2 1 11.10.c.a 32
11.c even 5 1 inner 99.10.f.a 32
33.f even 10 1 121.10.a.i 16
33.h odd 10 1 11.10.c.a 32
33.h odd 10 1 121.10.a.h 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.10.c.a 32 3.b odd 2 1
11.10.c.a 32 33.h odd 10 1
99.10.f.a 32 1.a even 1 1 trivial
99.10.f.a 32 11.c even 5 1 inner
121.10.a.h 16 33.h odd 10 1
121.10.a.i 16 33.f even 10 1