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Properties

Label 99.6.e.b
Level $99$
Weight $6$
Character orbit 99.e
Analytic conductor $15.878$
Analytic rank $0$
Dimension $54$
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] = [99,6,Mod(34,99)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(99, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("99.34");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 99.e (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: \(15.8779981615\)
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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 54 q - 16 q^{3} - 480 q^{4} - 93 q^{5} - 547 q^{6} - 225 q^{7} + 426 q^{8} + 412 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 54 q - 16 q^{3} - 480 q^{4} - 93 q^{5} - 547 q^{6} - 225 q^{7} + 426 q^{8} + 412 q^{9} + 1200 q^{10} - 3267 q^{11} - 1307 q^{12} - 1509 q^{13} - 1329 q^{14} - 1402 q^{15} - 9216 q^{16} + 534 q^{17} + 8005 q^{18} + 7158 q^{19} - 5565 q^{20} - 2269 q^{21} - 6330 q^{23} + 8151 q^{24} - 19548 q^{25} + 19140 q^{26} - 19111 q^{27} - 162 q^{28} - 5970 q^{29} + 6152 q^{30} - 9591 q^{31} - 1161 q^{32} + 3872 q^{33} - 15642 q^{34} + 6480 q^{35} - 39247 q^{36} + 52344 q^{37} + 20853 q^{38} + 17360 q^{39} - 31335 q^{40} + 12030 q^{41} + 32195 q^{42} - 7455 q^{43} + 116160 q^{44} - 87125 q^{45} + 62208 q^{46} - 52014 q^{47} + 113233 q^{48} - 121638 q^{49} + 66849 q^{50} + 12803 q^{51} - 75396 q^{52} - 31386 q^{53} - 171706 q^{54} + 22506 q^{55} - 16602 q^{56} + 112551 q^{57} - 101865 q^{58} + 4515 q^{59} + 29659 q^{60} - 196533 q^{61} - 30594 q^{62} - 250463 q^{63} + 575670 q^{64} + 78987 q^{65} + 22748 q^{66} - 88959 q^{67} + 168471 q^{68} + 133456 q^{69} - 285492 q^{70} - 95664 q^{71} - 422406 q^{72} + 262860 q^{73} + 30075 q^{74} + 174664 q^{75} - 286026 q^{76} - 27225 q^{77} + 20051 q^{78} - 165651 q^{79} + 261594 q^{80} - 249164 q^{81} + 978690 q^{82} + 30672 q^{83} + 84460 q^{84} - 349959 q^{85} + 13332 q^{86} + 408183 q^{87} - 25773 q^{88} + 429384 q^{89} - 117449 q^{90} + 435798 q^{91} - 71163 q^{92} - 341479 q^{93} - 511200 q^{94} - 325926 q^{95} + 291727 q^{96} - 277011 q^{97} - 1227540 q^{98} + 49852 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
34.1 −5.62923 + 9.75012i −12.9772 + 8.63676i −47.3765 − 82.0585i 5.45632 + 9.45063i −11.1580 − 175.147i −8.90498 + 15.4239i 706.503 93.8129 − 224.161i −122.860
34.2 −5.31577 + 9.20718i 8.37966 − 13.1446i −40.5148 − 70.1737i −47.7632 − 82.7283i 76.4806 + 147.027i 8.33894 − 14.4435i 521.260 −102.562 − 220.295i 1015.59
34.3 −4.55350 + 7.88689i 4.64197 + 14.8813i −25.4687 − 44.1130i 43.2098 + 74.8415i −138.504 − 31.1511i 62.2572 − 107.833i 172.462 −199.904 + 138.157i −787.022
34.4 −4.48738 + 7.77237i −7.90333 − 13.4364i −24.2731 − 42.0423i 7.78579 + 13.4854i 139.898 − 1.13333i −113.885 + 197.255i 148.499 −118.075 + 212.385i −139.751
34.5 −4.33072 + 7.50102i 9.66469 + 12.2309i −21.5102 − 37.2568i −34.3439 − 59.4854i −133.599 + 19.5267i −103.509 + 179.283i 95.4528 −56.1876 + 236.415i 594.935
34.6 −4.15741 + 7.20084i 15.5883 − 0.0753899i −18.5681 − 32.1609i 3.31007 + 5.73321i −64.2640 + 112.562i 77.1936 − 133.703i 42.7064 242.989 − 2.35040i −55.0452
34.7 −3.25108 + 5.63104i −15.0337 + 4.12175i −5.13906 − 8.90111i −47.5771 − 82.4060i 25.6659 − 98.0553i 15.8192 − 27.3997i −141.239 209.022 − 123.930i 618.708
34.8 −2.86900 + 4.96925i 0.826774 − 15.5665i −0.462311 − 0.800746i 6.42997 + 11.1370i 74.9819 + 48.7688i 110.934 − 192.144i −178.310 −241.633 − 25.7400i −73.7904
34.9 −2.36142 + 4.09010i 12.0863 − 9.84482i 4.84738 + 8.39590i 30.9574 + 53.6199i 11.7254 + 72.6821i −86.0470 + 149.038i −196.918 49.1590 − 237.976i −292.414
34.10 −2.23074 + 3.86376i −2.99004 + 15.2990i 6.04758 + 10.4747i 23.0417 + 39.9094i −52.4417 − 45.6809i −25.4492 + 44.0793i −196.730 −225.119 − 91.4892i −205.600
34.11 −1.81122 + 3.13712i −11.7970 − 10.1898i 9.43900 + 16.3488i −7.28842 − 12.6239i 53.3334 − 18.5525i 13.2437 − 22.9388i −184.302 35.3361 + 240.417i 52.8036
34.12 −0.793357 + 1.37413i −1.91348 + 15.4706i 14.7412 + 25.5325i −21.9115 − 37.9519i −19.7406 − 14.9031i −24.9219 + 43.1659i −97.5548 −235.677 − 59.2054i 69.5346
34.13 0.187970 − 0.325573i 6.22879 − 14.2899i 15.9293 + 27.5904i −30.4550 − 52.7497i −3.48160 − 4.71401i −1.63348 + 2.82927i 24.0070 −165.404 − 178.018i −22.8985
34.14 0.201317 − 0.348691i 15.5707 + 0.744389i 15.9189 + 27.5724i −39.4247 − 68.2856i 3.39420 − 5.27950i −99.3099 + 172.010i 25.7033 241.892 + 23.1813i −31.7475
34.15 0.488949 − 0.846885i −15.3435 + 2.75284i 15.5219 + 26.8846i 10.2980 + 17.8366i −5.17084 + 14.3402i 58.5363 − 101.388i 61.6504 227.844 − 84.4762i 20.1407
34.16 0.637654 − 1.10445i 15.2838 + 3.06663i 15.1868 + 26.3043i 31.8478 + 55.1620i 13.1327 − 14.9248i 34.7413 − 60.1737i 79.5455 224.192 + 93.7398i 81.2315
34.17 1.18195 − 2.04720i −12.6585 + 9.09740i 13.2060 + 22.8735i 51.3087 + 88.8693i 3.66246 + 36.6671i −105.742 + 183.151i 138.080 77.4748 − 230.319i 242.577
34.18 2.07791 − 3.59904i −9.30817 − 12.5043i 7.36460 + 12.7559i 10.4989 + 18.1846i −64.3451 + 7.51769i 7.60589 − 13.1738i 194.198 −69.7158 + 232.785i 87.2628
34.19 2.83718 − 4.91414i 6.51437 + 14.1620i −0.0992010 − 0.171821i −21.7995 − 37.7579i 88.0767 + 8.16767i 98.3413 − 170.332i 180.454 −158.126 + 184.513i −247.397
34.20 3.35706 − 5.81460i −9.46176 + 12.3885i −6.53972 − 11.3271i −16.5826 − 28.7220i 40.2705 + 96.6053i 10.0991 − 17.4922i 127.035 −63.9500 − 234.434i −222.676
See all 54 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 34.27
Significant digits:
Format:

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 99.6.e.b ✓ 54
3.b odd 2 1 297.6.e.b 54
9.c even 3 1 inner 99.6.e.b ✓ 54
9.c even 3 1 891.6.a.j 27
9.d odd 6 1 297.6.e.b 54
9.d odd 6 1 891.6.a.i 27
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.6.e.b ✓ 54 1.a even 1 1 trivial
99.6.e.b ✓ 54 9.c even 3 1 inner
297.6.e.b 54 3.b odd 2 1
297.6.e.b 54 9.d odd 6 1
891.6.a.i 27 9.d odd 6 1
891.6.a.j 27 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{54} + 672 T_{2}^{52} - 142 T_{2}^{51} + 253440 T_{2}^{50} - 89739 T_{2}^{49} + \cdots + 68\!\cdots\!00 \) acting on \(S_{6}^{\mathrm{new}}(99, [\chi])\). Copy content Toggle raw display