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Properties

Label 9.16.c.a
Level $9$
Weight $16$
Character orbit 9.c
Analytic conductor $12.842$
Analytic rank $0$
Dimension $28$
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] = [9,16,Mod(4,9)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2]))
 
N = Newforms(chi, 16, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9.4");
 
S:= CuspForms(chi, 16);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 9.c (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: \(12.8424154590\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 129 q^{2} + 3345 q^{3} - 212993 q^{4} - 152655 q^{5} + 2208231 q^{6} + 803705 q^{7} + 20162658 q^{8} - 15596991 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 129 q^{2} + 3345 q^{3} - 212993 q^{4} - 152655 q^{5} + 2208231 q^{6} + 803705 q^{7} + 20162658 q^{8} - 15596991 q^{9} + 65532 q^{10} - 60735990 q^{11} + 182013108 q^{12} - 41509093 q^{13} + 215313024 q^{14} - 657143253 q^{15} - 2952822785 q^{16} - 2121112242 q^{17} + 7761499596 q^{18} + 5670809246 q^{19} - 1434155028 q^{20} + 18429715101 q^{21} + 7841223807 q^{22} - 49433203761 q^{23} - 22375042227 q^{24} - 59779497779 q^{25} + 28088907384 q^{26} + 162819945072 q^{27} - 103195803652 q^{28} - 152663830725 q^{29} + 588495022128 q^{30} - 13955906443 q^{31} - 779638216401 q^{32} - 752623919694 q^{33} - 15287791959 q^{34} + 2803990417206 q^{35} - 98909775117 q^{36} + 531598359836 q^{37} - 3598723579461 q^{38} + 3663944150079 q^{39} - 1083028910664 q^{40} - 1656362316264 q^{41} - 12459707404686 q^{42} + 2394872112236 q^{43} + 13185661781850 q^{44} + 4884357020085 q^{45} + 1511321018232 q^{46} - 7686086213559 q^{47} + 12130882824291 q^{48} - 7946464864203 q^{49} - 2177050903941 q^{50} + 5493352002093 q^{51} + 1969798905818 q^{52} - 20778802614756 q^{53} - 8013203434503 q^{54} - 21700105111626 q^{55} + 13789887516654 q^{56} - 15469760310585 q^{57} + 15429496587216 q^{58} + 17132922121350 q^{59} + 143327945409084 q^{60} - 17229834579991 q^{61} - 190512185539596 q^{62} + 41573408563611 q^{63} + 11322374934466 q^{64} + 92170621979115 q^{65} - 193311009227574 q^{66} - 19847122719016 q^{67} + 106742875854159 q^{68} + 270237544770987 q^{69} - 85984052928798 q^{70} + 60954771085488 q^{71} - 121608306427509 q^{72} + 206937948168362 q^{73} - 169102773629016 q^{74} - 327902975378103 q^{75} - 274786877071699 q^{76} - 121510644149499 q^{77} - 527565898825578 q^{78} + 137921083996235 q^{79} + 11\!\cdots\!92 q^{80}+ \cdots + 54\!\cdots\!45 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −168.709 − 292.213i 406.522 + 3766.12i −40541.7 + 70220.2i −118733. + 205651.i 1.03192e6 − 754170.i −1.07628e6 − 1.86418e6i 1.63025e7 −1.40184e7 + 3.06202e6i 8.01252e7
4.2 −166.126 − 287.739i −2565.89 − 2786.59i −38811.7 + 67223.8i 91736.2 − 158892.i −375549. + 1.20123e6i 1.67796e6 + 2.90631e6i 1.49033e7 −1.18130e6 + 1.43002e7i −6.09590e7
4.3 −116.313 − 201.461i 2882.05 − 2458.18i −10673.6 + 18487.2i −14612.7 + 25309.9i −830448. − 294701.i −936776. − 1.62254e6i −2.65680e6 2.26356e6 − 1.41692e7i 6.79859e6
4.4 −90.7396 − 157.166i −2860.12 + 2483.67i −83.3453 + 144.358i 114284. − 197945.i 649873. + 224146.i −926548. − 1.60483e6i −5.91646e6 2.01169e6 − 1.42072e7i −4.14803e7
4.5 −87.7622 − 152.009i 3223.10 + 1990.12i 979.600 − 1696.72i 31725.2 − 54949.7i 19648.6 − 664595.i 1.89991e6 + 3.29074e6i −6.09547e6 6.42779e6 + 1.28287e7i −1.11371e7
4.6 −71.0488 − 123.060i −3420.09 − 1628.46i 6288.14 − 10891.4i −134844. + 233556.i 42594.5 + 536577.i −263672. − 456694.i −6.44331e6 9.04513e6 + 1.11390e7i 3.83220e7
4.7 5.85162 + 10.1353i −1216.56 + 3587.32i 16315.5 − 28259.3i −75278.1 + 130386.i −43477.5 + 8661.42i 1.21608e6 + 2.10632e6i 765381. −1.13889e7 − 8.72841e6i −1.76200e6
4.8 7.65532 + 13.2594i −2.84524 − 3787.99i 16266.8 − 28174.9i 62137.0 − 107624.i 50204.8 − 29036.0i 214112. + 370852.i 999809. −1.43489e7 + 21555.5i 1.90271e6
4.9 30.1976 + 52.3037i 2829.55 + 2518.45i 14560.2 − 25219.0i 62931.9 − 109001.i −46278.7 + 224047.i −2.04281e6 − 3.53826e6i 3.73776e6 1.66376e6 + 1.42521e7i 7.60156e6
4.10 64.3019 + 111.374i 3592.76 − 1200.40i 8114.53 − 14054.8i −162694. + 281794.i 364715. + 322953.i 160110. + 277318.i 6.30121e6 1.14670e7 − 8.62549e6i −4.18462e7
4.11 90.1139 + 156.082i −3787.95 − 17.8502i 142.961 − 247.616i 43232.1 − 74880.2i −338561. − 592839.i 142580. + 246956.i 5.95724e6 1.43483e7 + 135231.i 1.55833e7
4.12 137.732 + 238.559i 3667.27 − 948.716i −21556.3 + 37336.6i 132801. − 230018.i 731425. + 744191.i 1.52495e6 + 2.64129e6i −2.84956e6 1.25488e7 − 6.95839e6i 7.31638e7
4.13 146.377 + 253.533i −62.1065 + 3787.49i −26468.7 + 45845.2i −28285.5 + 48991.9i −969344. + 538657.i −125194. − 216842.i −5.90471e6 −1.43412e7 − 470455.i −1.65614e7
4.14 153.969 + 266.682i −1013.18 − 3649.98i −31029.0 + 53743.8i −80727.8 + 139825.i 817389. − 832181.i −1.06256e6 − 1.84041e6i −9.01952e6 −1.22959e7 + 7.39615e6i −4.97183e7
7.1 −168.709 + 292.213i 406.522 − 3766.12i −40541.7 − 70220.2i −118733. − 205651.i 1.03192e6 + 754170.i −1.07628e6 + 1.86418e6i 1.63025e7 −1.40184e7 − 3.06202e6i 8.01252e7
7.2 −166.126 + 287.739i −2565.89 + 2786.59i −38811.7 − 67223.8i 91736.2 + 158892.i −375549. − 1.20123e6i 1.67796e6 − 2.90631e6i 1.49033e7 −1.18130e6 − 1.43002e7i −6.09590e7
7.3 −116.313 + 201.461i 2882.05 + 2458.18i −10673.6 − 18487.2i −14612.7 − 25309.9i −830448. + 294701.i −936776. + 1.62254e6i −2.65680e6 2.26356e6 + 1.41692e7i 6.79859e6
7.4 −90.7396 + 157.166i −2860.12 − 2483.67i −83.3453 − 144.358i 114284. + 197945.i 649873. − 224146.i −926548. + 1.60483e6i −5.91646e6 2.01169e6 + 1.42072e7i −4.14803e7
7.5 −87.7622 + 152.009i 3223.10 − 1990.12i 979.600 + 1696.72i 31725.2 + 54949.7i 19648.6 + 664595.i 1.89991e6 − 3.29074e6i −6.09547e6 6.42779e6 − 1.28287e7i −1.11371e7
7.6 −71.0488 + 123.060i −3420.09 + 1628.46i 6288.14 + 10891.4i −134844. − 233556.i 42594.5 − 536577.i −263672. + 456694.i −6.44331e6 9.04513e6 − 1.11390e7i 3.83220e7
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.14
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 9.16.c.a ✓ 28
3.b odd 2 1 27.16.c.a 28
9.c even 3 1 inner 9.16.c.a ✓ 28
9.c even 3 1 81.16.a.e 14
9.d odd 6 1 27.16.c.a 28
9.d odd 6 1 81.16.a.c 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.16.c.a ✓ 28 1.a even 1 1 trivial
9.16.c.a ✓ 28 9.c even 3 1 inner
27.16.c.a 28 3.b odd 2 1
27.16.c.a 28 9.d odd 6 1
81.16.a.c 14 9.d odd 6 1
81.16.a.e 14 9.c even 3 1

Hecke kernels

This newform subspace is the entire newspace \(S_{16}^{\mathrm{new}}(9, [\chi])\).