Properties

Label 99.8.f.a
Level $99$
Weight $8$
Character orbit 99.f
Analytic conductor $30.926$
Analytic rank $0$
Dimension $24$
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] = [99,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("99.37");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 8 \)
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: \(30.9261175229\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 3 q^{2} - 505 q^{4} + 72 q^{5} + 68 q^{7} + 4545 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 3 q^{2} - 505 q^{4} + 72 q^{5} + 68 q^{7} + 4545 q^{8} + 4240 q^{10} - 5952 q^{11} - 16564 q^{13} + 29544 q^{14} + 13279 q^{16} + 65592 q^{17} - 37804 q^{19} - 120702 q^{20} + 246233 q^{22} + 26304 q^{23} - 210646 q^{25} + 336384 q^{26} + 92334 q^{28} + 231048 q^{29} + 141004 q^{31} + 260892 q^{32} + 1512162 q^{34} + 342588 q^{35} - 836112 q^{37} + 2387040 q^{38} - 1906028 q^{40} - 1930344 q^{41} + 2877456 q^{43} - 5822808 q^{44} - 4951876 q^{46} + 2462532 q^{47} - 6581826 q^{49} + 2698797 q^{50} + 1655498 q^{52} - 6157836 q^{53} + 11721612 q^{55} - 11431080 q^{56} - 15150140 q^{58} + 2480268 q^{59} - 4195044 q^{61} + 6436524 q^{62} + 11301151 q^{64} - 11117640 q^{65} + 16682528 q^{67} + 14718306 q^{68} - 16625944 q^{70} + 20942628 q^{71} - 456824 q^{73} - 8747736 q^{74} + 18091206 q^{76} - 13904988 q^{77} - 12976692 q^{79} + 11893596 q^{80} - 32706099 q^{82} + 4471404 q^{83} + 25528128 q^{85} - 20374383 q^{86} + 25246815 q^{88} + 2477316 q^{89} - 18339604 q^{91} - 1026678 q^{92} - 4961738 q^{94} + 25328772 q^{95} - 34034202 q^{97} + 51534036 q^{98}+O(q^{100}) \) 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 −14.3633 10.4355i 0 57.8489 + 178.041i −53.3657 + 38.7724i 0 −143.396 441.326i 324.804 999.643i 0 1171.11
37.2 −7.83171 5.69007i 0 −10.5954 32.6093i −9.68480 + 7.03642i 0 520.681 + 1602.49i −485.474 + 1494.14i 0 115.886
37.3 −0.987911 0.717759i 0 −39.0934 120.317i 339.439 246.617i 0 312.595 + 962.069i −96.0385 + 295.576i 0 −512.347
37.4 1.19911 + 0.871205i 0 −38.8753 119.646i −171.086 + 124.301i 0 −463.381 1426.14i 116.247 357.771i 0 −313.442
37.5 12.7113 + 9.23528i 0 36.7318 + 113.049i 268.339 194.959i 0 −72.0370 221.707i 44.3449 136.480i 0 5211.43
37.6 13.5536 + 9.84730i 0 47.1778 + 145.198i −315.392 + 229.146i 0 356.708 + 1097.83i −127.721 + 393.085i 0 −6531.19
64.1 −5.78570 + 17.8065i 0 −180.045 130.810i −151.154 465.205i 0 −544.422 395.546i 1432.12 1040.50i 0 9158.23
64.2 −5.72800 + 17.6290i 0 −174.417 126.721i 66.8497 + 205.742i 0 −327.935 238.259i 1313.52 954.328i 0 −4009.94
64.3 −2.40997 + 7.41712i 0 54.3484 + 39.4864i 139.587 + 429.605i 0 40.1674 + 29.1833i −1231.45 + 894.704i 0 −3522.83
64.4 −0.419581 + 1.29134i 0 102.063 + 74.1529i −90.0532 277.155i 0 205.463 + 149.277i −279.185 + 202.840i 0 395.685
64.5 3.78266 11.6418i 0 −17.6695 12.8376i −31.6803 97.5020i 0 601.350 + 436.906i 1051.31 763.821i 0 −1254.94
64.6 4.77944 14.7096i 0 −89.9749 65.3706i 44.2018 + 136.039i 0 −451.793 328.247i 210.025 152.592i 0 2212.34
82.1 −5.78570 17.8065i 0 −180.045 + 130.810i −151.154 + 465.205i 0 −544.422 + 395.546i 1432.12 + 1040.50i 0 9158.23
82.2 −5.72800 17.6290i 0 −174.417 + 126.721i 66.8497 205.742i 0 −327.935 + 238.259i 1313.52 + 954.328i 0 −4009.94
82.3 −2.40997 7.41712i 0 54.3484 39.4864i 139.587 429.605i 0 40.1674 29.1833i −1231.45 894.704i 0 −3522.83
82.4 −0.419581 1.29134i 0 102.063 74.1529i −90.0532 + 277.155i 0 205.463 149.277i −279.185 202.840i 0 395.685
82.5 3.78266 + 11.6418i 0 −17.6695 + 12.8376i −31.6803 + 97.5020i 0 601.350 436.906i 1051.31 + 763.821i 0 −1254.94
82.6 4.77944 + 14.7096i 0 −89.9749 + 65.3706i 44.2018 136.039i 0 −451.793 + 328.247i 210.025 + 152.592i 0 2212.34
91.1 −14.3633 + 10.4355i 0 57.8489 178.041i −53.3657 38.7724i 0 −143.396 + 441.326i 324.804 + 999.643i 0 1171.11
91.2 −7.83171 + 5.69007i 0 −10.5954 + 32.6093i −9.68480 7.03642i 0 520.681 1602.49i −485.474 1494.14i 0 115.886
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.6
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 99.8.f.a 24
3.b odd 2 1 11.8.c.a 24
11.c even 5 1 inner 99.8.f.a 24
33.f even 10 1 121.8.a.g 12
33.h odd 10 1 11.8.c.a 24
33.h odd 10 1 121.8.a.i 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.8.c.a 24 3.b odd 2 1
11.8.c.a 24 33.h odd 10 1
99.8.f.a 24 1.a even 1 1 trivial
99.8.f.a 24 11.c even 5 1 inner
121.8.a.g 12 33.f even 10 1
121.8.a.i 12 33.h odd 10 1