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Properties

Label 9.14.c.a
Level $9$
Weight $14$
Character orbit 9.c
Analytic conductor $9.651$
Analytic rank $0$
Dimension $24$
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] = [9,14,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, 14, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9.4");
 
S:= CuspForms(chi, 14);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 14 \)
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: \(9.65078360567\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 63 q^{2} - 732 q^{3} - 45057 q^{4} + 52128 q^{5} - 179433 q^{6} - 93912 q^{7} - 528750 q^{8} + 247716 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 63 q^{2} - 732 q^{3} - 45057 q^{4} + 52128 q^{5} - 179433 q^{6} - 93912 q^{7} - 528750 q^{8} + 247716 q^{9} + 16380 q^{10} + 5374116 q^{11} + 3110244 q^{12} - 5761392 q^{13} - 17168904 q^{14} + 125678088 q^{15} - 151003137 q^{16} - 255994776 q^{17} + 302719140 q^{18} - 429803880 q^{19} + 985236444 q^{20} - 1179453480 q^{21} - 319272129 q^{22} + 902651544 q^{23} + 7227167013 q^{24} - 2148489804 q^{25} - 9138616488 q^{26} - 2661880752 q^{27} + 3211509756 q^{28} + 8581074696 q^{29} - 23828670216 q^{30} - 2657941800 q^{31} + 28248279903 q^{32} + 26485796700 q^{33} + 9945286137 q^{34} - 33040041984 q^{35} - 30678986637 q^{36} + 4058904864 q^{37} + 6148446795 q^{38} - 675795120 q^{39} + 13161798144 q^{40} + 16567334748 q^{41} - 66579812622 q^{42} - 27962130372 q^{43} + 72531160218 q^{44} + 18812710728 q^{45} - 135411199560 q^{46} - 15315949440 q^{47} + 578553075555 q^{48} - 49081707324 q^{49} - 273091973517 q^{50} - 381866937084 q^{51} - 204838453950 q^{52} + 643343610672 q^{53} - 256421097927 q^{54} + 31769369280 q^{55} - 215622477810 q^{56} + 821568380412 q^{57} - 95988424752 q^{58} + 105925680252 q^{59} - 474421590252 q^{60} - 30573149712 q^{61} - 2444547938076 q^{62} - 1601282086776 q^{63} + 2988468306690 q^{64} + 1381046429664 q^{65} + 1399091420826 q^{66} - 1151032746228 q^{67} + 4190340445911 q^{68} + 1195868876544 q^{69} + 1094619621186 q^{70} - 7748309652768 q^{71} - 8843735411109 q^{72} - 1498667197128 q^{73} + 7016277435120 q^{74} + 5046580890708 q^{75} + 2833472883117 q^{76} + 6460934627664 q^{77} + 4422197495862 q^{78} - 1478663156112 q^{79} - 32580513665568 q^{80} - 8191801732644 q^{81} - 4818820561398 q^{82} + 17002530736776 q^{83} + 34491814005378 q^{84} - 2293157799504 q^{85} + 17634834056313 q^{86} - 10413041553216 q^{87} - 180382572117 q^{88} - 18332462247264 q^{89} - 48867239563092 q^{90} - 5040327944496 q^{91} + 23410418756274 q^{92} + 62456975369328 q^{93} + 2909654324928 q^{94} + 24016410647208 q^{95} - 40790028862224 q^{96} - 3550321688604 q^{97} - 100086324465186 q^{98} - 61452155792808 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 −78.9523 − 136.749i 1184.67 + 436.900i −8370.95 + 14498.9i 31418.1 − 54417.7i −33786.5 − 196497.i −132855. − 230112.i 1.35007e6 1.21256e6 + 1.03516e6i −9.92213e6
4.2 −69.1352 − 119.746i 84.8305 − 1259.81i −5463.36 + 9462.82i −17229.3 + 29842.0i −156722. + 76939.3i −2514.98 − 4356.07i 378132. −1.57993e6 − 213741.i 4.76460e6
4.3 −65.5602 − 113.554i −991.333 + 782.036i −4500.29 + 7794.73i −2944.41 + 5099.87i 153795. + 61299.1i 158277. + 274143.i 106022. 371161. − 1.55052e6i 772146.
4.4 −30.8920 − 53.5065i 605.559 + 1107.98i 2187.37 − 3788.63i −17641.4 + 30555.7i 40577.2 − 66629.1i −85490.7 − 148074.i −776423. −860919. + 1.34190e6i 2.17991e6
4.5 −23.4366 − 40.5935i −1025.21 − 737.064i 2997.45 − 5191.73i 16523.3 − 28619.2i −5892.48 + 58891.2i −92965.5 − 161021.i −664986. 507795. + 1.51129e6i −1.54901e6
4.6 −9.39684 − 16.2758i 1134.35 − 554.598i 3919.40 − 6788.60i 286.316 − 495.914i −19685.8 − 13250.9i 76714.6 + 132874.i −301278. 979165. − 1.25821e6i −10761.9
4.7 24.8071 + 42.9671i −116.728 + 1257.26i 2865.22 − 4962.71i 32928.9 − 57034.6i −56916.4 + 26173.4i 180578. + 312771.i 690749. −1.56707e6 − 293515.i 3.26748e6
4.8 26.8486 + 46.5032i −1097.86 + 623.726i 2654.30 − 4597.39i −18329.9 + 31748.3i −58481.2 − 34307.7i −206963. − 358471.i 724946. 816255. − 1.36952e6i −1.96853e6
4.9 43.9238 + 76.0782i −564.624 − 1129.39i 237.402 − 411.192i −16723.9 + 28966.6i 61121.6 − 92562.7i 275772. + 477652.i 761358. −956723. + 1.27536e6i −2.93830e6
4.10 59.2148 + 102.563i 628.723 − 1095.00i −2916.77 + 5052.00i 17776.0 − 30788.9i 149536. − 356.639i −289092. − 500721.i 279310. −803738. − 1.37691e6i 4.21040e6
4.11 64.8956 + 112.402i 1030.55 + 729.580i −4326.88 + 7494.37i −12375.9 + 21435.6i −15128.3 + 163183.i 16248.7 + 28143.5i −59931.6 529750. + 1.50374e6i −3.21256e6
4.12 89.1835 + 154.470i −1238.93 + 243.687i −11811.4 + 20457.9i 12376.0 − 21435.9i −148134. − 169645.i 55335.1 + 95843.2i −2.75234e6 1.47556e6 − 603822.i 4.41495e6
7.1 −78.9523 + 136.749i 1184.67 − 436.900i −8370.95 − 14498.9i 31418.1 + 54417.7i −33786.5 + 196497.i −132855. + 230112.i 1.35007e6 1.21256e6 − 1.03516e6i −9.92213e6
7.2 −69.1352 + 119.746i 84.8305 + 1259.81i −5463.36 − 9462.82i −17229.3 − 29842.0i −156722. − 76939.3i −2514.98 + 4356.07i 378132. −1.57993e6 + 213741.i 4.76460e6
7.3 −65.5602 + 113.554i −991.333 − 782.036i −4500.29 − 7794.73i −2944.41 − 5099.87i 153795. − 61299.1i 158277. − 274143.i 106022. 371161. + 1.55052e6i 772146.
7.4 −30.8920 + 53.5065i 605.559 − 1107.98i 2187.37 + 3788.63i −17641.4 − 30555.7i 40577.2 + 66629.1i −85490.7 + 148074.i −776423. −860919. − 1.34190e6i 2.17991e6
7.5 −23.4366 + 40.5935i −1025.21 + 737.064i 2997.45 + 5191.73i 16523.3 + 28619.2i −5892.48 − 58891.2i −92965.5 + 161021.i −664986. 507795. − 1.51129e6i −1.54901e6
7.6 −9.39684 + 16.2758i 1134.35 + 554.598i 3919.40 + 6788.60i 286.316 + 495.914i −19685.8 + 13250.9i 76714.6 − 132874.i −301278. 979165. + 1.25821e6i −10761.9
7.7 24.8071 − 42.9671i −116.728 − 1257.26i 2865.22 + 4962.71i 32928.9 + 57034.6i −56916.4 − 26173.4i 180578. − 312771.i 690749. −1.56707e6 + 293515.i 3.26748e6
7.8 26.8486 − 46.5032i −1097.86 − 623.726i 2654.30 + 4597.39i −18329.9 − 31748.3i −58481.2 + 34307.7i −206963. + 358471.i 724946. 816255. + 1.36952e6i −1.96853e6
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.12
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 9.14.c.a ✓ 24
3.b odd 2 1 27.14.c.a 24
9.c even 3 1 inner 9.14.c.a ✓ 24
9.c even 3 1 81.14.a.c 12
9.d odd 6 1 27.14.c.a 24
9.d odd 6 1 81.14.a.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.14.c.a ✓ 24 1.a even 1 1 trivial
9.14.c.a ✓ 24 9.c even 3 1 inner
27.14.c.a 24 3.b odd 2 1
27.14.c.a 24 9.d odd 6 1
81.14.a.c 12 9.c even 3 1
81.14.a.d 12 9.d odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{14}^{\mathrm{new}}(9, [\chi])\).