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Properties

Label 99.6.g.b
Level $99$
Weight $6$
Character orbit 99.g
Analytic conductor $15.878$
Analytic rank $0$
Dimension $112$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [99,6,Mod(32,99)]
 
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([5, 3]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("99.32");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 99.g (of order \(6\), 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: \(112\)
Relative dimension: \(56\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 112 q + 48 q^{3} - 962 q^{4} - 90 q^{5} + 876 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 112 q + 48 q^{3} - 962 q^{4} - 90 q^{5} + 876 q^{9} - 120 q^{11} - 2622 q^{12} - 6 q^{14} + 3534 q^{15} - 11330 q^{16} - 10854 q^{20} - 885 q^{22} - 204 q^{23} + 26346 q^{25} - 18558 q^{27} - 9994 q^{31} - 18042 q^{33} - 7722 q^{34} + 61296 q^{36} + 7472 q^{37} - 68532 q^{38} - 107166 q^{42} + 13248 q^{45} + 105810 q^{47} - 100488 q^{48} + 144058 q^{49} - 54880 q^{55} - 243876 q^{56} - 9906 q^{58} - 157698 q^{59} + 267198 q^{60} + 495736 q^{64} + 166020 q^{66} - 96730 q^{67} + 81102 q^{69} - 32280 q^{70} - 154290 q^{75} - 144522 q^{77} + 61818 q^{78} - 221892 q^{81} + 111528 q^{82} + 393138 q^{86} - 166029 q^{88} - 131916 q^{91} + 794142 q^{92} + 282522 q^{93} - 37750 q^{97} + 32826 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
32.1 −5.63724 − 9.76398i 13.1034 − 8.44393i −47.5569 + 82.3709i 30.2534 + 17.4668i −156.314 − 80.3412i 80.8973 − 46.7061i 711.574 100.400 − 221.289i − 393.858i
32.2 −5.20322 − 9.01224i 0.290633 + 15.5857i −38.1470 + 66.0725i −76.6595 − 44.2594i 138.950 − 83.7153i −184.826 + 106.709i 460.942 −242.831 + 9.05946i 921.165i
32.3 −5.18732 − 8.98471i −15.4751 − 1.87672i −37.8166 + 65.5003i −28.6969 − 16.5682i 63.4125 + 148.774i 63.9232 − 36.9061i 452.680 235.956 + 58.0847i 343.778i
32.4 −5.03327 − 8.71787i −9.65118 − 12.2415i −34.6675 + 60.0459i 68.4163 + 39.5002i −58.1430 + 145.753i −186.989 + 107.958i 375.834 −56.7095 + 236.290i − 795.259i
32.5 −4.89349 − 8.47578i 4.63637 + 14.8830i −31.8925 + 55.2395i 75.0033 + 43.3032i 103.457 − 112.127i −15.7562 + 9.09687i 311.080 −200.008 + 138.006i − 847.615i
32.6 −4.85489 − 8.40892i −10.5454 + 11.4802i −31.1400 + 53.9360i 15.9898 + 9.23172i 147.733 + 32.9406i 77.7833 − 44.9082i 294.012 −20.5883 − 242.126i − 179.276i
32.7 −4.68704 − 8.11820i 13.3287 + 8.08362i −27.9368 + 48.3879i −60.1141 − 34.7069i 3.15205 − 146.094i 183.997 − 106.231i 223.793 112.310 + 215.489i 650.691i
32.8 −4.46292 − 7.73001i −1.84633 − 15.4787i −23.8353 + 41.2840i −12.7170 − 7.34215i −111.411 + 83.3525i 38.7521 − 22.3735i 139.874 −236.182 + 57.1577i 131.070i
32.9 −4.43557 − 7.68263i 10.3330 − 11.6717i −23.3486 + 40.4409i −57.4717 − 33.1813i −135.502 − 27.6135i −130.818 + 75.5275i 130.380 −29.4595 − 241.208i 588.712i
32.10 −4.26332 − 7.38429i 14.2739 + 6.26537i −20.3519 + 35.2504i 22.3597 + 12.9094i −14.5891 − 132.114i −52.8640 + 30.5211i 74.2135 164.490 + 178.863i − 220.147i
32.11 −3.48907 − 6.04324i −1.08782 − 15.5505i −8.34717 + 14.4577i 36.0041 + 20.7870i −90.1797 + 60.8305i 186.787 − 107.841i −106.805 −240.633 + 33.8322i − 290.108i
32.12 −3.46070 − 5.99411i −13.4011 − 7.96314i −7.95289 + 13.7748i −63.2921 − 36.5417i −1.35488 + 107.885i −23.2305 + 13.4122i −111.395 116.177 + 213.429i 505.840i
32.13 −3.38781 − 5.86786i −15.2257 + 3.34341i −6.95454 + 12.0456i 75.5844 + 43.6387i 71.2004 + 78.0154i 37.6538 − 21.7394i −122.577 220.643 − 101.811i − 591.358i
32.14 −3.35193 − 5.80571i −13.2811 + 8.16164i −6.47084 + 11.2078i 7.90517 + 4.56405i 91.9014 + 49.7490i −200.686 + 115.866i −127.764 109.775 − 216.791i − 61.1935i
32.15 −3.32030 − 5.75092i 13.3180 − 8.10134i −6.04874 + 10.4767i 83.9056 + 48.4429i −90.8098 − 49.6917i −3.38055 + 1.95176i −132.165 111.737 − 215.787i − 643.380i
32.16 −3.17739 − 5.50339i 15.5215 + 1.44373i −4.19157 + 7.26001i −13.1586 − 7.59710i −41.3723 − 90.0080i −70.0268 + 40.4300i −150.080 238.831 + 44.8175i 96.5557i
32.17 −3.12059 − 5.40502i −0.487448 + 15.5808i −3.47618 + 6.02093i 5.76255 + 3.32701i 85.7359 − 45.9868i 140.859 − 81.3248i −156.327 −242.525 − 15.1897i − 41.5289i
32.18 −2.74522 − 4.75486i −8.77752 + 12.8824i 0.927533 − 1.60653i −76.8352 − 44.3608i 85.3500 + 6.37100i 66.5283 − 38.4101i −185.879 −88.9102 − 226.150i 487.121i
32.19 −2.14538 − 3.71591i 5.00165 + 14.7643i 6.79469 − 11.7688i 13.7378 + 7.93153i 44.1322 − 50.2606i −90.7518 + 52.3956i −195.613 −192.967 + 147.691i − 68.0646i
32.20 −2.08560 − 3.61236i 11.7874 − 10.2009i 7.30056 − 12.6449i −69.0112 − 39.8436i −61.4329 − 21.3054i 165.542 − 95.5759i −194.382 34.8852 − 240.483i 332.391i
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 32.56
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
11.b odd 2 1 inner
99.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.6.g.b ✓ 112
9.d odd 6 1 inner 99.6.g.b ✓ 112
11.b odd 2 1 inner 99.6.g.b ✓ 112
99.g even 6 1 inner 99.6.g.b ✓ 112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.6.g.b ✓ 112 1.a even 1 1 trivial
99.6.g.b ✓ 112 9.d odd 6 1 inner
99.6.g.b ✓ 112 11.b odd 2 1 inner
99.6.g.b ✓ 112 99.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{112} + 1377 T_{2}^{110} + 1002657 T_{2}^{108} + 503634978 T_{2}^{106} + 194074401180 T_{2}^{104} + \cdots + 11\!\cdots\!96 \) acting on \(S_{6}^{\mathrm{new}}(99, [\chi])\). Copy content Toggle raw display