Properties

Label 9.26.c.a
Level $9$
Weight $26$
Character orbit 9.c
Analytic conductor $35.640$
Analytic rank $0$
Dimension $48$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9,26,Mod(4,9)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9.4"); S:= CuspForms(chi, 26); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 26, names="a")
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 9.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(35.6397101957\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 4095 q^{2} - 335640 q^{3} - 385875969 q^{4} + 194286528 q^{5} + 8407960119 q^{6} + 574239120 q^{7} - 333604719630 q^{8} + 1107247155912 q^{9} + 67108860 q^{10} + 15182693684712 q^{11} + 21776654328900 q^{12}+ \cdots - 20\!\cdots\!96 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −5690.87 − 9856.88i −713523. − 581527.i −4.79949e7 + 8.31295e7i 2.77799e7 − 4.81162e7i −1.67148e9 + 1.03425e10i −1.08436e10 − 1.87816e10i 7.10623e11 1.70940e11 + 8.29866e11i −6.32367e11
4.2 −4970.34 − 8608.89i 910175. − 137367.i −3.26314e7 + 5.65193e7i 4.18211e8 − 7.24362e8i −5.70646e9 − 7.15284e9i 2.08239e10 + 3.60680e10i 3.15203e11 8.09549e11 − 2.50056e11i −8.31460e12
4.3 −4616.47 − 7995.96i 703629. + 593461.i −2.58464e7 + 4.47673e7i −4.62100e8 + 8.00380e8i 1.49701e9 − 8.36588e9i −2.01951e8 − 3.49790e8i 1.67470e11 1.42898e11 + 8.35152e11i 8.53308e12
4.4 −4550.12 − 7881.03i −343156. + 854127.i −2.46299e7 + 4.26602e7i 9.17986e7 − 1.59000e8i 8.29280e9 − 1.18195e9i 2.51114e9 + 4.34942e9i 1.42923e11 −6.11777e11 − 5.86197e11i −1.67078e12
4.5 −3824.60 − 6624.40i 341634. − 854737.i −1.24779e7 + 2.16124e7i −1.31361e8 + 2.27524e8i −6.96874e9 + 1.00591e9i −1.10067e10 − 1.90642e10i −6.57720e10 −6.13861e11 − 5.84014e11i 2.00962e12
4.6 −3119.44 − 5403.03i −854398. − 342480.i −2.68459e6 + 4.64984e6i −3.95013e8 + 6.84182e8i 8.14817e8 + 5.68468e9i 3.41634e10 + 5.91727e10i −1.75844e11 6.12704e11 + 5.85228e11i 4.92887e12
4.7 −2925.10 − 5066.42i −893834. + 219887.i −335167. + 580526.i 1.06880e8 − 1.85122e8i 3.72859e9 + 3.88534e9i −1.48902e10 − 2.57905e10i −1.92378e11 7.50588e11 − 3.93084e11i −1.25054e12
4.8 −2129.61 − 3688.59i −358567. − 847772.i 7.70675e6 − 1.33485e7i 5.16859e8 − 8.95226e8i −2.36348e9 + 3.12803e9i 1.41543e10 + 2.45159e10i −2.08565e11 −5.90148e11 + 6.07967e11i −4.40283e12
4.9 −2122.04 − 3675.49i 737380. + 550962.i 7.77109e6 − 1.34599e7i 2.25558e8 − 3.90678e8i 4.60302e8 − 3.87939e9i −3.19014e10 − 5.52548e10i −2.08370e11 2.40170e11 + 8.12537e11i −1.91458e12
4.10 −1017.45 − 1762.27i 875722. − 283548.i 1.47068e7 − 2.54730e7i −1.55247e8 + 2.68896e8i −1.39069e9 − 1.25476e9i 1.15137e10 + 1.99423e10i −1.28133e11 6.86489e11 − 4.96619e11i 6.31822e11
4.11 −978.739 − 1695.23i 209312. + 896369.i 1.48614e7 − 2.57406e7i 1.47925e8 − 2.56214e8i 1.31469e9 − 1.23214e9i 2.95079e10 + 5.11091e10i −1.23864e11 −7.59666e11 + 3.75241e11i −5.79121e11
4.12 −186.752 − 323.463i −415685. + 821276.i 1.67075e7 − 2.89382e7i −4.69648e8 + 8.13454e8i 3.43283e8 − 1.89159e7i −1.53342e10 − 2.65596e10i −2.50133e10 −5.01701e11 − 6.82784e11i 3.50830e11
4.13 246.408 + 426.791i −644645. − 657055.i 1.66558e7 − 2.88487e7i −8.03289e7 + 1.39134e8i 1.21580e8 − 4.37032e8i −2.50988e10 − 4.34723e10i 3.29526e10 −1.61548e10 + 8.47135e11i −7.91747e10
4.14 893.119 + 1546.93i 23282.8 − 920188.i 1.51819e7 − 2.62958e7i −1.74761e8 + 3.02695e8i 1.44426e9 − 7.85821e8i 3.23838e9 + 5.60904e9i 1.14173e11 −8.46204e11 − 4.28490e10i −6.24329e11
4.15 1416.06 + 2452.69i −846731. + 361019.i 1.27668e7 − 2.21127e7i 2.88839e8 − 5.00284e8i −2.08449e9 − 1.56554e9i 9.28062e9 + 1.60745e10i 1.67344e11 5.86619e11 − 6.11372e11i 1.63605e12
4.16 2086.26 + 3613.51i 786249. − 478645.i 8.07223e6 − 1.39815e7i 3.88339e8 − 6.72623e8i 3.36991e9 + 1.84254e9i −1.12786e10 − 1.95351e10i 2.07370e11 3.89087e11 − 7.52668e11i 3.24071e12
4.17 2716.70 + 4705.47i 786797. + 477744.i 2.01628e6 − 3.49229e6i −2.51556e8 + 4.35707e8i −1.10518e8 + 5.00013e9i 5.60120e9 + 9.70156e9i 2.04225e11 3.90809e11 + 7.51775e11i −2.73361e12
4.18 3264.37 + 5654.05i 105903. + 914370.i −4.53496e6 + 7.85478e6i 2.30570e8 − 3.99359e8i −4.82419e9 + 3.58362e9i −1.68464e10 − 2.91789e10i 1.59853e11 −8.24858e11 + 1.93668e11i 3.01066e12
4.19 3360.08 + 5819.83i −888833. − 239300.i −5.80306e6 + 1.00512e7i −2.73191e8 + 4.73180e8i −1.59386e9 − 5.97692e9i 1.17027e10 + 2.02697e10i 1.47496e11 7.32760e11 + 4.25395e11i −3.67177e12
4.20 4042.62 + 7002.03i 49216.9 − 919166.i −1.59084e7 + 2.75541e7i 7.83410e7 − 1.35691e8i 6.63499e9 − 3.37122e9i 2.95089e10 + 5.11109e10i 1.40499e10 −8.42444e11 − 9.04770e10i 1.26681e12
See all 48 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 4.24
Significant digits:
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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 9.26.c.a ✓ 48
3.b odd 2 1 27.26.c.a 48
9.c even 3 1 inner 9.26.c.a ✓ 48
9.d odd 6 1 27.26.c.a 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.26.c.a ✓ 48 1.a even 1 1 trivial
9.26.c.a ✓ 48 9.c even 3 1 inner
27.26.c.a 48 3.b odd 2 1
27.26.c.a 48 9.d odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{26}^{\mathrm{new}}(9, [\chi])\).