Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9,26,Mod(1,9)] 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("9.1"); S:= CuspForms(chi, 26); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 26, names="a")
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(35.6397101957\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 783420x + 148321440 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{6}\cdot 3^{5}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(199.394\) of defining polynomial
Character \(\chi\) \(=\) 9.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+10558.1 q^{2} +7.79199e7 q^{4} +8.66156e8 q^{5} -1.75156e10 q^{7} +4.68416e11 q^{8} +9.14500e12 q^{10} +8.03769e12 q^{11} -2.74406e12 q^{13} -1.84932e14 q^{14} +2.33105e15 q^{16} -3.84314e15 q^{17} +5.80352e15 q^{19} +6.74908e16 q^{20} +8.48630e16 q^{22} -1.28434e16 q^{23} +4.52204e17 q^{25} -2.89722e16 q^{26} -1.36481e18 q^{28} -1.23594e18 q^{29} +3.64474e18 q^{31} +8.89410e18 q^{32} -4.05764e19 q^{34} -1.51712e19 q^{35} +3.54993e19 q^{37} +6.12744e19 q^{38} +4.05722e20 q^{40} +2.62561e20 q^{41} -7.82469e19 q^{43} +6.26295e20 q^{44} -1.35602e20 q^{46} -1.42679e21 q^{47} -1.03427e21 q^{49} +4.77443e21 q^{50} -2.13817e20 q^{52} -3.15289e21 q^{53} +6.96189e21 q^{55} -8.20457e21 q^{56} -1.30493e22 q^{58} -5.95554e21 q^{59} +1.46494e22 q^{61} +3.84817e22 q^{62} +1.56881e22 q^{64} -2.37679e21 q^{65} -6.68064e22 q^{67} -2.99457e23 q^{68} -1.60180e23 q^{70} -1.43174e22 q^{71} -6.25968e22 q^{73} +3.74807e23 q^{74} +4.52210e23 q^{76} -1.40785e23 q^{77} -3.24882e23 q^{79} +2.01905e24 q^{80} +2.77216e24 q^{82} -1.01939e24 q^{83} -3.32876e24 q^{85} -8.26142e23 q^{86} +3.76498e24 q^{88} +8.94156e23 q^{89} +4.80638e22 q^{91} -1.00075e24 q^{92} -1.50643e25 q^{94} +5.02676e24 q^{95} -8.14572e24 q^{97} -1.09200e25 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3678 q^{2} + 90747588 q^{4} + 163152750 q^{5} - 9622572744 q^{7} + 268012830648 q^{8} + 8363188874700 q^{10} + 5946998130780 q^{11} + 248137774407690 q^{13} - 754361641264848 q^{14} + 23\!\cdots\!16 q^{16}+ \cdots - 24\!\cdots\!50 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 10558.1 1.82269 0.911344 − 0.411645i \(-0.135046\pi\)
0.911344 + 0.411645i \(0.135046\pi\)
\(3\) 0 0
\(4\) 7.79199e7 2.32219
\(5\) 8.66156e8 1.58661 0.793307 − 0.608822i \(-0.208358\pi\)
0.793307 + 0.608822i \(0.208358\pi\)
\(6\) 0 0
\(7\) −1.75156e10 −0.478298 −0.239149 − 0.970983i \(-0.576868\pi\)
−0.239149 + 0.970983i \(0.576868\pi\)
\(8\) 4.68416e11 2.40995
\(9\) 0 0
\(10\) 9.14500e12 2.89190
\(11\) 8.03769e12 0.772187 0.386093 − 0.922460i \(-0.373824\pi\)
0.386093 + 0.922460i \(0.373824\pi\)
\(12\) 0 0
\(13\) −2.74406e12 −0.0326664 −0.0163332 − 0.999867i \(-0.505199\pi\)
−0.0163332 + 0.999867i \(0.505199\pi\)
\(14\) −1.84932e14 −0.871788
\(15\) 0 0
\(16\) 2.33105e15 2.07039
\(17\) −3.84314e15 −1.59983 −0.799917 − 0.600110i \(-0.795123\pi\)
−0.799917 + 0.600110i \(0.795123\pi\)
\(18\) 0 0
\(19\) 5.80352e15 0.601550 0.300775 − 0.953695i \(-0.402755\pi\)
0.300775 + 0.953695i \(0.402755\pi\)
\(20\) 6.74908e16 3.68442
\(21\) 0 0
\(22\) 8.48630e16 1.40746
\(23\) −1.28434e16 −0.122203 −0.0611013 − 0.998132i \(-0.519461\pi\)
−0.0611013 + 0.998132i \(0.519461\pi\)
\(24\) 0 0
\(25\) 4.52204e17 1.51734
\(26\) −2.89722e16 −0.0595408
\(27\) 0 0
\(28\) −1.36481e18 −1.11070
\(29\) −1.23594e18 −0.648670 −0.324335 − 0.945942i \(-0.605141\pi\)
−0.324335 + 0.945942i \(0.605141\pi\)
\(30\) 0 0
\(31\) 3.64474e18 0.831085 0.415542 − 0.909574i \(-0.363592\pi\)
0.415542 + 0.909574i \(0.363592\pi\)
\(32\) 8.89410e18 1.36373
\(33\) 0 0
\(34\) −4.05764e19 −2.91600
\(35\) −1.51712e19 −0.758874
\(36\) 0 0
\(37\) 3.54993e19 0.886541 0.443270 − 0.896388i \(-0.353818\pi\)
0.443270 + 0.896388i \(0.353818\pi\)
\(38\) 6.12744e19 1.09644
\(39\) 0 0
\(40\) 4.05722e20 3.82365
\(41\) 2.62561e20 1.81732 0.908661 − 0.417534i \(-0.137106\pi\)
0.908661 + 0.417534i \(0.137106\pi\)
\(42\) 0 0
\(43\) −7.82469e19 −0.298615 −0.149307 − 0.988791i \(-0.547704\pi\)
−0.149307 + 0.988791i \(0.547704\pi\)
\(44\) 6.26295e20 1.79317
\(45\) 0 0
\(46\) −1.35602e20 −0.222737
\(47\) −1.42679e21 −1.79117 −0.895587 − 0.444886i \(-0.853244\pi\)
−0.895587 + 0.444886i \(0.853244\pi\)
\(48\) 0 0
\(49\) −1.03427e21 −0.771231
\(50\) 4.77443e21 2.76564
\(51\) 0 0
\(52\) −2.13817e20 −0.0758578
\(53\) −3.15289e21 −0.881578 −0.440789 − 0.897611i \(-0.645301\pi\)
−0.440789 + 0.897611i \(0.645301\pi\)
\(54\) 0 0
\(55\) 6.96189e21 1.22516
\(56\) −8.20457e21 −1.15267
\(57\) 0 0
\(58\) −1.30493e22 −1.18232
\(59\) −5.95554e21 −0.435784 −0.217892 − 0.975973i \(-0.569918\pi\)
−0.217892 + 0.975973i \(0.569918\pi\)
\(60\) 0 0
\(61\) 1.46494e22 0.706637 0.353318 − 0.935503i \(-0.385053\pi\)
0.353318 + 0.935503i \(0.385053\pi\)
\(62\) 3.84817e22 1.51481
\(63\) 0 0
\(64\) 1.56881e22 0.415259
\(65\) −2.37679e21 −0.0518290
\(66\) 0 0
\(67\) −6.68064e22 −0.997431 −0.498716 − 0.866766i \(-0.666195\pi\)
−0.498716 + 0.866766i \(0.666195\pi\)
\(68\) −2.99457e23 −3.71512
\(69\) 0 0
\(70\) −1.60180e23 −1.38319
\(71\) −1.43174e22 −0.103546 −0.0517732 − 0.998659i \(-0.516487\pi\)
−0.0517732 + 0.998659i \(0.516487\pi\)
\(72\) 0 0
\(73\) −6.25968e22 −0.319901 −0.159951 − 0.987125i \(-0.551134\pi\)
−0.159951 + 0.987125i \(0.551134\pi\)
\(74\) 3.74807e23 1.61589
\(75\) 0 0
\(76\) 4.52210e23 1.39691
\(77\) −1.40785e23 −0.369335
\(78\) 0 0
\(79\) −3.24882e23 −0.618567 −0.309284 − 0.950970i \(-0.600089\pi\)
−0.309284 + 0.950970i \(0.600089\pi\)
\(80\) 2.01905e24 3.28491
\(81\) 0 0
\(82\) 2.77216e24 3.31241
\(83\) −1.01939e24 −1.04681 −0.523403 − 0.852085i \(-0.675338\pi\)
−0.523403 + 0.852085i \(0.675338\pi\)
\(84\) 0 0
\(85\) −3.32876e24 −2.53832
\(86\) −8.26142e23 −0.544282
\(87\) 0 0
\(88\) 3.76498e24 1.86093
\(89\) 8.94156e23 0.383741 0.191871 − 0.981420i \(-0.438545\pi\)
0.191871 + 0.981420i \(0.438545\pi\)
\(90\) 0 0
\(91\) 4.80638e22 0.0156243
\(92\) −1.00075e24 −0.283778
\(93\) 0 0
\(94\) −1.50643e25 −3.26475
\(95\) 5.02676e24 0.954427
\(96\) 0 0
\(97\) −8.14572e24 −1.19202 −0.596009 − 0.802978i \(-0.703248\pi\)
−0.596009 + 0.802978i \(0.703248\pi\)
\(98\) −1.09200e25 −1.40571
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 9.26.a.c.1.3 3
3.2 odd 2 3.26.a.b.1.1 ✓ 3
12.11 even 2 48.26.a.i.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.26.a.b.1.1 ✓ 3 3.2 odd 2
9.26.a.c.1.3 3 1.1 even 1 trivial
48.26.a.i.1.1 3 12.11 even 2