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Properties

Label 99.6.e.a
Level $99$
Weight $6$
Character orbit 99.e
Analytic conductor $15.878$
Analytic rank $0$
Dimension $46$
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: \(46\)
Relative dimension: \(23\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 46 q + 15 q^{3} - 320 q^{4} - 36 q^{5} + 273 q^{6} + 167 q^{7} + 426 q^{8} - 63 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 46 q + 15 q^{3} - 320 q^{4} - 36 q^{5} + 273 q^{6} + 167 q^{7} + 426 q^{8} - 63 q^{9} - 1200 q^{10} + 2783 q^{11} - 1419 q^{12} + 1871 q^{13} - 1329 q^{14} + 3090 q^{15} - 3584 q^{16} + 534 q^{17} + 5391 q^{18} - 7282 q^{19} - 1917 q^{20} - 8319 q^{21} - 8292 q^{23} + 351 q^{24} - 10049 q^{25} + 19140 q^{26} + 19953 q^{27} + 7586 q^{28} - 5970 q^{29} - 15150 q^{30} + 9542 q^{31} - 3831 q^{32} + 3630 q^{33} + 2982 q^{34} + 6480 q^{35} + 28881 q^{36} - 32014 q^{37} + 1221 q^{38} - 17544 q^{39} + 40635 q^{40} + 12030 q^{41} - 55365 q^{42} + 25943 q^{43} - 77440 q^{44} + 70578 q^{45} - 154008 q^{46} + 9756 q^{47} - 65463 q^{48} - 6990 q^{49} + 101805 q^{50} - 42117 q^{51} + 144446 q^{52} + 107838 q^{53} - 2322 q^{54} - 8712 q^{55} - 16602 q^{56} - 100233 q^{57} + 95367 q^{58} - 20310 q^{59} - 215301 q^{60} + 100247 q^{61} - 30594 q^{62} + 28323 q^{63} - 169154 q^{64} + 20931 q^{65} + 26862 q^{66} + 84956 q^{67} + 168471 q^{68} - 30243 q^{69} + 212292 q^{70} + 72186 q^{71} + 144522 q^{72} - 346888 q^{73} + 86619 q^{74} + 103413 q^{75} + 340334 q^{76} - 20207 q^{77} - 386061 q^{78} + 123113 q^{79} - 30246 q^{80} - 73683 q^{81} - 399966 q^{82} + 30672 q^{83} - 38850 q^{84} + 268335 q^{85} - 211260 q^{86} + 180009 q^{87} + 25773 q^{88} + 65028 q^{89} + 460017 q^{90} - 656042 q^{91} - 196731 q^{92} + 232206 q^{93} + 230262 q^{94} - 325926 q^{95} - 517071 q^{96} + 357002 q^{97} - 428928 q^{98} + 7623 q^{99}+O(q^{100}) \) 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 −5.31385 + 9.20387i 15.1010 + 3.86789i −40.4741 − 70.1032i 19.4283 + 33.6508i −115.844 + 118.434i −68.8108 + 119.184i 520.207 213.079 + 116.818i −412.956
34.2 −4.76758 + 8.25768i 6.13362 − 14.3310i −29.4596 − 51.0254i 34.6730 + 60.0554i 89.0987 + 118.974i 14.5433 − 25.1897i 256.678 −167.758 − 175.802i −661.225
34.3 −4.74574 + 8.21986i −12.6839 − 9.06197i −29.0441 − 50.3058i −15.5606 − 26.9517i 134.683 − 61.2540i 58.4765 − 101.284i 247.615 78.7614 + 229.882i 295.386
34.4 −4.58663 + 7.94427i 0.424588 + 15.5827i −26.0743 − 45.1620i −26.3116 − 45.5730i −125.740 − 68.0989i 57.4649 − 99.5322i 184.828 −242.639 + 13.2324i 482.726
34.5 −3.74438 + 6.48546i −10.3943 + 11.6171i −12.0408 − 20.8552i 2.53771 + 4.39545i −36.4222 − 110.911i −85.0063 + 147.235i −59.2996 −26.9162 − 241.505i −38.0087
34.6 −3.09256 + 5.35646i 14.8684 − 4.68287i −3.12780 − 5.41751i −26.4438 − 45.8020i −20.8979 + 94.1243i −11.2115 + 19.4189i −159.232 199.142 − 139.254i 327.116
34.7 −2.57522 + 4.46041i −0.0736683 − 15.5883i 2.73651 + 4.73978i −36.1342 − 62.5863i 69.7198 + 39.8146i −46.8775 + 81.1943i −193.002 −242.989 + 2.29672i 372.214
34.8 −2.16807 + 3.75521i 11.8040 + 10.1817i 6.59895 + 11.4297i 29.6963 + 51.4354i −63.8260 + 22.2518i −30.4063 + 52.6652i −195.984 35.6678 + 240.368i −257.534
34.9 −1.69570 + 2.93704i −10.7528 + 11.2862i 10.2492 + 17.7521i 12.5191 + 21.6836i −14.9145 − 50.7193i 124.918 − 216.365i −178.043 −11.7558 − 242.715i −84.9142
34.10 −0.909131 + 1.57466i −7.01311 − 13.9218i 14.3470 + 24.8497i 48.3226 + 83.6972i 28.2979 + 1.61345i 3.10482 − 5.37770i −110.357 −144.632 + 195.270i −175.726
34.11 −0.769578 + 1.33295i −15.4100 − 2.35231i 14.8155 + 25.6612i −8.82917 − 15.2926i 14.9947 − 18.7304i −82.7694 + 143.361i −94.8598 231.933 + 72.4978i 27.1789
34.12 0.102117 − 0.176872i 11.7568 − 10.2361i 15.9791 + 27.6767i 9.22797 + 15.9833i −0.609918 − 3.12473i 66.9497 − 115.960i 13.0625 33.4432 − 240.688i 3.76934
34.13 1.19190 − 2.06443i −10.2213 + 11.7697i 13.1587 + 22.7916i −40.5317 − 70.2029i 12.1150 + 35.1294i −2.78933 + 4.83126i 139.017 −34.0513 − 240.602i −193.239
34.14 1.20052 − 2.07936i −9.90567 − 12.0365i 13.1175 + 22.7202i −52.8015 − 91.4549i −36.9202 + 6.14738i 104.403 − 180.830i 139.825 −46.7555 + 238.459i −253.557
34.15 1.29852 − 2.24911i 5.90891 + 14.4251i 12.6277 + 21.8718i −0.711324 − 1.23205i 40.1166 + 5.44162i −80.0518 + 138.654i 148.695 −173.170 + 170.474i −3.69468
34.16 1.93633 − 3.35383i −0.407119 + 15.5831i 8.50122 + 14.7245i 43.6184 + 75.5493i 51.4749 + 31.5396i 70.2256 − 121.634i 189.770 −242.669 − 12.6884i 337.840
34.17 2.92960 − 5.07422i 15.5881 + 0.109722i −1.16511 − 2.01802i −26.8186 − 46.4512i 46.2236 − 78.7758i 12.7329 − 22.0541i 173.841 242.976 + 3.42071i −314.271
34.18 3.31865 − 5.74807i −15.3128 − 2.91842i −6.02689 − 10.4389i 17.7351 + 30.7182i −67.5932 + 78.3340i 50.9494 − 88.2470i 132.389 225.966 + 89.3786i 235.427
34.19 3.34130 − 5.78730i 13.3973 − 7.96943i −6.32853 − 10.9613i 39.9952 + 69.2736i −1.35706 − 104.162i −78.3863 + 135.769i 129.261 115.976 − 213.538i 534.543
34.20 4.21753 − 7.30497i −13.1923 + 8.30442i −19.5751 − 33.9050i 4.92097 + 8.52338i 5.02465 + 131.393i −50.4728 + 87.4215i −60.3116 105.073 − 219.109i 83.0173
See all 46 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 34.23
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.a ✓ 46
3.b odd 2 1 297.6.e.a 46
9.c even 3 1 inner 99.6.e.a ✓ 46
9.c even 3 1 891.6.a.f 23
9.d odd 6 1 297.6.e.a 46
9.d odd 6 1 891.6.a.e 23
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.6.e.a ✓ 46 1.a even 1 1 trivial
99.6.e.a ✓ 46 9.c even 3 1 inner
297.6.e.a 46 3.b odd 2 1
297.6.e.a 46 9.d odd 6 1
891.6.a.e 23 9.d odd 6 1
891.6.a.f 23 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{46} + 528 T_{2}^{44} - 142 T_{2}^{43} + 159744 T_{2}^{42} - 68757 T_{2}^{41} + \cdots + 46\!\cdots\!00 \) acting on \(S_{6}^{\mathrm{new}}(99, [\chi])\). Copy content Toggle raw display