Properties

Label 9.19.d.a
Level $9$
Weight $19$
Character orbit 9.d
Analytic conductor $18.485$
Analytic rank $0$
Dimension $34$
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] = [9,19,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, 19, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9.2");
 
S:= CuspForms(chi, 19);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 19 \)
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: \(18.4847523939\)
Analytic rank: \(0\)
Dimension: \(34\)
Relative dimension: \(17\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 34 q - 3 q^{2} + 3804 q^{3} + 2097151 q^{4} + 2192181 q^{5} + 3257181 q^{6} + 4302359 q^{7} + 397977264 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 34 q - 3 q^{2} + 3804 q^{3} + 2097151 q^{4} + 2192181 q^{5} + 3257181 q^{6} + 4302359 q^{7} + 397977264 q^{9} - 524292 q^{10} + 1022992575 q^{11} + 32417762010 q^{12} + 3839449319 q^{13} + 129938709678 q^{14} - 124119415395 q^{15} - 240517906433 q^{16} - 285581999484 q^{18} - 608222566264 q^{19} + 1471193748174 q^{20} - 1381493495229 q^{21} + 851254438911 q^{22} + 375441179325 q^{23} + 305135298351 q^{24} + 12156817117552 q^{25} + 11936276558406 q^{27} + 4648789868540 q^{28} - 31101186224271 q^{29} + 62775784981818 q^{30} - 20376334153801 q^{31} - 113411230023465 q^{32} - 20701744605945 q^{33} - 99737362554111 q^{34} - 472069890962721 q^{36} + 56970364072604 q^{37} + 12\!\cdots\!23 q^{38}+ \cdots - 52\!\cdots\!15 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −826.948 477.438i −4479.77 19166.4i 324823. + 562610.i 2.90361e6 1.67640e6i −5.44626e6 + 1.79885e7i −4.86570e6 + 8.42763e6i 3.70017e8i −3.47284e8 + 1.71722e8i −3.20151e9
2.2 −778.562 449.503i 19565.6 + 2146.79i 273034. + 472909.i −2.42540e6 + 1.40030e6i −1.42680e7 1.04662e7i −3.53698e7 + 6.12622e7i 2.55250e8i 3.78203e8 + 8.40066e7i 2.51777e9
2.3 −645.469 372.662i −1755.59 + 19604.6i 146682. + 254060.i 690555. 398692.i 8.43904e6 1.19999e7i 1.15930e7 2.00797e7i 2.32684e7i −3.81256e8 6.88350e7i −5.94309e8
2.4 −633.468 365.733i −19383.5 3420.35i 136449. + 236336.i −2.15900e6 + 1.24650e6i 1.10279e7 + 9.25588e6i 1.48825e7 2.57772e7i 7.86595e6i 3.64023e8 + 1.32597e8i 1.82355e9
2.5 −462.218 266.862i 18990.6 5174.86i 11358.3 + 19673.1i 1.29042e6 745022.i −1.01587e7 2.67594e6i 2.69368e7 4.66559e7i 1.27788e8i 3.33862e8 1.96547e8i −7.95271e8
2.6 −307.242 177.386i 3077.23 19441.0i −68140.4 118023.i −996976. + 575604.i −4.39401e6 + 5.42722e6i −8.05977e6 + 1.39599e7i 1.41350e8i −3.68482e8 1.19649e8i 4.08417e8
2.7 −301.302 173.957i −18971.4 + 5244.55i −70549.9 122196.i 2.15861e6 1.24627e6i 6.62846e6 + 1.72002e6i −3.75558e7 + 6.50485e7i 1.40294e8i 3.32410e8 1.98993e8i −8.67191e8
2.8 −77.3719 44.6707i 14059.8 + 13774.7i −127081. 220111.i 10314.3 5954.97i −472508. 1.69384e6i −5.95096e6 + 1.03074e7i 4.61275e7i 7.93549e6 + 3.87339e8i −1.06405e6
2.9 56.6635 + 32.7147i −11098.7 + 16255.4i −128931. 223316.i −2.62400e6 + 1.51497e6i −1.16068e6 + 557998.i −1.47703e6 + 2.55830e6i 3.40238e7i −1.41057e8 3.60829e8i −1.98247e8
2.10 82.9990 + 47.9195i −16416.3 10859.3i −126479. 219069.i 1.05847e6 611105.i −842168. 1.68797e6i 3.06063e7 5.30117e7i 4.93669e7i 1.51572e8 + 3.56540e8i 1.17135e8
2.11 336.967 + 194.548i 16767.8 10308.3i −55374.0 95910.6i 2.58095e6 1.49011e6i 7.65567e6 211404.i −1.88244e7 + 3.26048e7i 1.45091e8i 1.74899e8 3.45695e8i 1.15959e9
2.12 406.376 + 234.621i 18030.6 7894.22i −20977.9 36334.9i −2.94106e6 + 1.69802e6i 9.17934e6 + 1.02233e6i 2.46134e7 4.26317e7i 1.42696e8i 2.62783e8 2.84675e8i −1.59357e9
2.13 437.639 + 252.671i −7960.89 18001.2i −3386.81 5866.13i −693411. + 400341.i 1.06440e6 9.88953e6i −2.35552e7 + 4.07987e7i 1.35895e8i −2.60669e8 + 2.86612e8i −4.04618e8
2.14 451.478 + 260.661i −6615.47 + 18538.0i 4816.35 + 8342.15i 2.21110e6 1.27658e6i −7.81886e6 + 6.64509e6i 2.07433e7 3.59285e7i 1.31640e8i −2.99892e8 2.45275e8i 1.33102e9
2.15 689.327 + 397.983i −19476.2 + 2845.44i 185709. + 321657.i −768821. + 443879.i −1.45579e7 5.78978e6i −1.10203e7 + 1.90877e7i 8.69784e7i 3.71227e8 1.10837e8i −7.06626e8
2.16 709.707 + 409.750i 13445.3 + 14375.1i 204718. + 354581.i −432139. + 249496.i 3.65207e6 + 1.57113e7i −1.14976e7 + 1.99144e7i 1.20705e8i −2.58659e7 + 3.86556e8i −4.08923e8
2.17 859.924 + 496.477i 4123.08 19246.3i 361908. + 626842.i 1.23289e6 711809.i 1.31009e7 1.45034e7i 3.09522e7 5.36108e7i 4.58419e8i −3.53421e8 1.58708e8i 1.41359e9
5.1 −826.948 + 477.438i −4479.77 + 19166.4i 324823. 562610.i 2.90361e6 + 1.67640e6i −5.44626e6 1.79885e7i −4.86570e6 8.42763e6i 3.70017e8i −3.47284e8 1.71722e8i −3.20151e9
5.2 −778.562 + 449.503i 19565.6 2146.79i 273034. 472909.i −2.42540e6 1.40030e6i −1.42680e7 + 1.04662e7i −3.53698e7 6.12622e7i 2.55250e8i 3.78203e8 8.40066e7i 2.51777e9
5.3 −645.469 + 372.662i −1755.59 19604.6i 146682. 254060.i 690555. + 398692.i 8.43904e6 + 1.19999e7i 1.15930e7 + 2.00797e7i 2.32684e7i −3.81256e8 + 6.88350e7i −5.94309e8
See all 34 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.17
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 9.19.d.a 34
3.b odd 2 1 27.19.d.a 34
9.c even 3 1 27.19.d.a 34
9.c even 3 1 81.19.b.a 34
9.d odd 6 1 inner 9.19.d.a 34
9.d odd 6 1 81.19.b.a 34
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.19.d.a 34 1.a even 1 1 trivial
9.19.d.a 34 9.d odd 6 1 inner
27.19.d.a 34 3.b odd 2 1
27.19.d.a 34 9.c even 3 1
81.19.b.a 34 9.c even 3 1
81.19.b.a 34 9.d odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{19}^{\mathrm{new}}(9, [\chi])\).