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Properties

Label 9.17.d.a
Level $9$
Weight $17$
Character orbit 9.d
Analytic conductor $14.609$
Analytic rank $0$
Dimension $30$
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] = [9,17,Mod(2,9)]
 
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([1]))
 
N = Newforms(chi, 17, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9.2");
 
S:= CuspForms(chi, 17);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 17 \)
Character orbit: \([\chi]\) \(=\) 9.d (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: \(14.6092089471\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(15\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 3 q^{2} + 2049 q^{3} + 458751 q^{4} + 507588 q^{5} - 1538451 q^{6} + 220596 q^{7} + 20334015 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q - 3 q^{2} + 2049 q^{3} + 458751 q^{4} + 507588 q^{5} - 1538451 q^{6} + 220596 q^{7} + 20334015 q^{9} - 131076 q^{10} + 724073415 q^{11} - 1662191814 q^{12} + 95113194 q^{13} + 893996070 q^{14} - 3068426826 q^{15} - 12884836353 q^{16} - 7648594020 q^{18} + 3492688902 q^{19} - 25937371266 q^{20} - 52294616664 q^{21} - 26872303617 q^{22} - 103153750194 q^{23} + 40250606679 q^{24} + 243726160827 q^{25} + 362691636822 q^{27} + 66417983484 q^{28} - 795074902824 q^{29} - 2041645442958 q^{30} + 214673299890 q^{31} + 7447869772551 q^{32} - 3104707482045 q^{33} + 1235794365009 q^{34} + 23262043772511 q^{36} - 3486755684820 q^{37} - 24573009235893 q^{38} - 30273294416874 q^{39} - 1321264238994 q^{40} + 27155948705097 q^{41} - 13764931392708 q^{42} - 79735410417 q^{43} + 109086413699988 q^{45} - 53264675076192 q^{46} - 161256783274554 q^{47} - 44896377400707 q^{48} - 10559443205403 q^{49} + 217897337363187 q^{50} - 30154167970299 q^{51} - 72373532128176 q^{52} + 294017485099131 q^{54} + 40530659995296 q^{55} + 205104219592146 q^{56} - 22533136628685 q^{57} - 222257729583378 q^{58} - 390395878927419 q^{59} + 72882016957710 q^{60} + 272741853657102 q^{61} - 796502872047474 q^{63} - 231672593036166 q^{64} + 10\!\cdots\!42 q^{65}+ \cdots + 40\!\cdots\!82 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −433.587 − 250.332i −4842.16 + 4427.21i 92564.0 + 160325.i −21421.3 + 12367.6i 3.20777e6 − 707436.i −2.10083e6 + 3.63874e6i − 5.98753e7i 3.84632e6 − 4.28745e7i 1.23840e7
2.2 −370.211 − 213.741i 4659.82 − 4618.74i 58602.8 + 101503.i −173736. + 100306.i −2.71233e6 + 713912.i 4.66855e6 − 8.08617e6i − 2.20879e7i 381169. − 4.30450e7i 8.57585e7
2.3 −291.566 − 168.336i 5875.97 + 2918.84i 23905.7 + 41405.9i 584954. − 337724.i −1.22189e6 − 1.84017e6i −3.29612e6 + 5.70905e6i 5.96737e6i 2.60074e7 + 3.43021e7i −2.27404e8
2.4 −266.938 − 154.117i −4334.45 − 4925.37i 14735.8 + 25523.2i −48755.1 + 28148.7i 397947. + 1.98278e6i −1.59438e6 + 2.76155e6i 1.11162e7i −5.47183e6 + 4.26975e7i 1.73528e7
2.5 −211.776 − 122.269i 2378.87 + 6114.55i −2868.70 − 4968.74i −475482. + 274519.i 243832. − 1.58577e6i 901533. − 1.56150e6i 1.74290e7i −3.17287e7 + 2.90914e7i 1.34261e8
2.6 −140.294 − 80.9989i −5028.64 + 4214.20i −19646.4 − 34028.5i 262819. − 151738.i 1.04683e6 − 183914.i 2.90166e6 − 5.02582e6i 1.69820e7i 7.52768e6 − 4.23834e7i −4.91626e7
2.7 −40.7299 − 23.5154i 5833.40 − 3003.02i −31662.0 − 54840.3i −338197. + 195258.i −308211. − 14862.4i −2.69586e6 + 4.66936e6i 6.06040e6i 2.50105e7 − 3.50356e7i 1.83663e7
2.8 −10.6459 − 6.14639i 1100.09 − 6468.12i −32692.4 − 56625.0i 524567. − 302859.i −51467.0 + 62097.1i 1.24279e6 − 2.15258e6i 1.60938e6i −4.06263e7 − 1.42311e7i −7.44596e6
2.9 80.3902 + 46.4133i −6496.31 − 919.077i −28459.6 − 49293.5i −327120. + 188863.i −479582. − 375400.i −322895. + 559270.i − 1.13671e7i 4.13573e7 + 1.19412e7i −3.50629e7
2.10 152.091 + 87.8095i 5582.02 + 3447.87i −17347.0 − 30045.8i 133136. − 76865.8i 546216. + 1.01454e6i 4.70524e6 − 8.14971e6i − 1.76023e7i 1.92711e7 + 3.84921e7i 2.69982e7
2.11 156.717 + 90.4808i −787.993 + 6513.51i −16394.4 − 28396.0i 148338. − 85643.2i −712840. + 949482.i −5.16463e6 + 8.94540e6i − 1.77930e7i −4.18049e7 − 1.02652e7i 3.09963e7
2.12 276.243 + 159.489i −570.822 − 6536.12i 18105.3 + 31359.3i −240746. + 138995.i 884752. − 1.89659e6i 894995. − 1.55018e6i − 9.35413e6i −4.23950e7 + 7.46193e6i −8.86724e7
2.13 344.075 + 198.652i −6478.91 − 1034.63i 46156.9 + 79946.1i 626449. − 361681.i −2.02370e6 − 1.64304e6i −174209. + 301738.i 1.06389e7i 4.09058e7 + 1.34065e7i 2.87394e8
2.14 372.020 + 214.786i 6451.24 − 1195.09i 59497.9 + 103053.i 84073.6 − 48539.9i 2.65668e6 + 941038.i −3.07776e6 + 5.33084e6i 2.29649e7i 4.01902e7 − 1.54196e7i 4.17027e7
2.15 382.712 + 220.959i −2317.64 + 6138.02i 64877.6 + 112371.i −485087. + 280065.i −2.24324e6 + 1.83699e6i 3.22220e6 − 5.58101e6i 2.83796e7i −3.23038e7 − 2.84514e7i −2.47531e8
5.1 −433.587 + 250.332i −4842.16 − 4427.21i 92564.0 − 160325.i −21421.3 − 12367.6i 3.20777e6 + 707436.i −2.10083e6 − 3.63874e6i 5.98753e7i 3.84632e6 + 4.28745e7i 1.23840e7
5.2 −370.211 + 213.741i 4659.82 + 4618.74i 58602.8 − 101503.i −173736. − 100306.i −2.71233e6 − 713912.i 4.66855e6 + 8.08617e6i 2.20879e7i 381169. + 4.30450e7i 8.57585e7
5.3 −291.566 + 168.336i 5875.97 − 2918.84i 23905.7 − 41405.9i 584954. + 337724.i −1.22189e6 + 1.84017e6i −3.29612e6 − 5.70905e6i − 5.96737e6i 2.60074e7 − 3.43021e7i −2.27404e8
5.4 −266.938 + 154.117i −4334.45 + 4925.37i 14735.8 − 25523.2i −48755.1 − 28148.7i 397947. − 1.98278e6i −1.59438e6 − 2.76155e6i − 1.11162e7i −5.47183e6 − 4.26975e7i 1.73528e7
5.5 −211.776 + 122.269i 2378.87 − 6114.55i −2868.70 + 4968.74i −475482. − 274519.i 243832. + 1.58577e6i 901533. + 1.56150e6i − 1.74290e7i −3.17287e7 − 2.90914e7i 1.34261e8
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.15
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9.17.d.a ✓ 30
3.b odd 2 1 27.17.d.a 30
9.c even 3 1 27.17.d.a 30
9.c even 3 1 81.17.b.a 30
9.d odd 6 1 inner 9.17.d.a ✓ 30
9.d odd 6 1 81.17.b.a 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.17.d.a ✓ 30 1.a even 1 1 trivial
9.17.d.a ✓ 30 9.d odd 6 1 inner
27.17.d.a 30 3.b odd 2 1
27.17.d.a 30 9.c even 3 1
81.17.b.a 30 9.c even 3 1
81.17.b.a 30 9.d odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{17}^{\mathrm{new}}(9, [\chi])\).