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 = [2] 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: \(2\)
Coefficient field: \(\Q(\sqrt{1287001}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 321750 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-566.730\) 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)
 
\(f(q)\) \(=\) \(q+6968.76 q^{2} +1.50092e7 q^{4} -4.41387e8 q^{5} +2.02309e10 q^{7} -1.29237e11 q^{8} -3.07592e12 q^{10} +1.57853e13 q^{11} +4.98841e12 q^{13} +1.40984e14 q^{14} -1.40425e15 q^{16} +2.34522e15 q^{17} +1.90653e16 q^{19} -6.62489e15 q^{20} +1.10004e17 q^{22} +6.78144e16 q^{23} -1.03200e17 q^{25} +3.47630e16 q^{26} +3.03650e17 q^{28} +2.11015e18 q^{29} -6.62759e18 q^{31} -5.44940e18 q^{32} +1.63433e19 q^{34} -8.92965e18 q^{35} +1.02569e18 q^{37} +1.32861e20 q^{38} +5.70436e19 q^{40} +1.74041e20 q^{41} +3.26968e20 q^{43} +2.36926e20 q^{44} +4.72583e20 q^{46} +4.36828e20 q^{47} -9.31781e20 q^{49} -7.19179e20 q^{50} +7.48722e19 q^{52} +4.14060e21 q^{53} -6.96745e21 q^{55} -2.61458e21 q^{56} +1.47051e22 q^{58} -1.51046e21 q^{59} +4.63585e21 q^{61} -4.61861e22 q^{62} +9.14317e21 q^{64} -2.20182e21 q^{65} -1.21518e23 q^{67} +3.52000e22 q^{68} -6.22286e22 q^{70} -1.87102e23 q^{71} -4.86220e22 q^{73} +7.14782e21 q^{74} +2.86155e23 q^{76} +3.19351e23 q^{77} +4.43976e23 q^{79} +6.19818e23 q^{80} +1.21285e24 q^{82} +1.52666e23 q^{83} -1.03515e24 q^{85} +2.27856e24 q^{86} -2.04005e24 q^{88} -1.97570e24 q^{89} +1.00920e23 q^{91} +1.01784e24 q^{92} +3.04415e24 q^{94} -8.41518e24 q^{95} -3.28438e24 q^{97} -6.49336e24 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 324 q^{2} + 25607696 q^{4} - 570861756 q^{5} - 29687385728 q^{7} + 23299970112 q^{8} - 2215598822136 q^{10} + 18187785618552 q^{11} - 105143636679236 q^{13} + 472678969464576 q^{14} - 27\!\cdots\!40 q^{16}+ \cdots - 14\!\cdots\!76 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 6968.76 1.20304 0.601521 − 0.798857i \(-0.294562\pi\)
0.601521 + 0.798857i \(0.294562\pi\)
\(3\) 0 0
\(4\) 1.50092e7 0.447310
\(5\) −4.41387e8 −0.808528 −0.404264 − 0.914642i \(-0.632472\pi\)
−0.404264 + 0.914642i \(0.632472\pi\)
\(6\) 0 0
\(7\) 2.02309e10 0.552445 0.276222 − 0.961094i \(-0.410917\pi\)
0.276222 + 0.961094i \(0.410917\pi\)
\(8\) −1.29237e11 −0.664909
\(9\) 0 0
\(10\) −3.07592e12 −0.972693
\(11\) 1.57853e13 1.51651 0.758255 − 0.651958i \(-0.226052\pi\)
0.758255 + 0.651958i \(0.226052\pi\)
\(12\) 0 0
\(13\) 4.98841e12 0.0593841 0.0296920 − 0.999559i \(-0.490547\pi\)
0.0296920 + 0.999559i \(0.490547\pi\)
\(14\) 1.40984e14 0.664614
\(15\) 0 0
\(16\) −1.40425e15 −1.24722
\(17\) 2.34522e15 0.976275 0.488137 − 0.872767i \(-0.337677\pi\)
0.488137 + 0.872767i \(0.337677\pi\)
\(18\) 0 0
\(19\) 1.90653e16 1.97617 0.988083 − 0.153923i \(-0.0491908\pi\)
0.988083 + 0.153923i \(0.0491908\pi\)
\(20\) −6.62489e15 −0.361663
\(21\) 0 0
\(22\) 1.10004e17 1.82442
\(23\) 6.78144e16 0.645244 0.322622 − 0.946528i \(-0.395436\pi\)
0.322622 + 0.946528i \(0.395436\pi\)
\(24\) 0 0
\(25\) −1.03200e17 −0.346283
\(26\) 3.47630e16 0.0714415
\(27\) 0 0
\(28\) 3.03650e17 0.247114
\(29\) 2.11015e18 1.10748 0.553742 − 0.832688i \(-0.313199\pi\)
0.553742 + 0.832688i \(0.313199\pi\)
\(30\) 0 0
\(31\) −6.62759e18 −1.51124 −0.755621 − 0.655009i \(-0.772665\pi\)
−0.755621 + 0.655009i \(0.772665\pi\)
\(32\) −5.44940e18 −0.835554
\(33\) 0 0
\(34\) 1.63433e19 1.17450
\(35\) −8.92965e18 −0.446667
\(36\) 0 0
\(37\) 1.02569e18 0.0256151 0.0128075 − 0.999918i \(-0.495923\pi\)
0.0128075 + 0.999918i \(0.495923\pi\)
\(38\) 1.32861e20 2.37741
\(39\) 0 0
\(40\) 5.70436e19 0.537597
\(41\) 1.74041e20 1.20463 0.602314 − 0.798259i \(-0.294246\pi\)
0.602314 + 0.798259i \(0.294246\pi\)
\(42\) 0 0
\(43\) 3.26968e20 1.24781 0.623907 − 0.781499i \(-0.285545\pi\)
0.623907 + 0.781499i \(0.285545\pi\)
\(44\) 2.36926e20 0.678350
\(45\) 0 0
\(46\) 4.72583e20 0.776256
\(47\) 4.36828e20 0.548387 0.274194 − 0.961674i \(-0.411589\pi\)
0.274194 + 0.961674i \(0.411589\pi\)
\(48\) 0 0
\(49\) −9.31781e20 −0.694805
\(50\) −7.19179e20 −0.416593
\(51\) 0 0
\(52\) 7.48722e19 0.0265631
\(53\) 4.14060e21 1.15775 0.578875 − 0.815416i \(-0.303492\pi\)
0.578875 + 0.815416i \(0.303492\pi\)
\(54\) 0 0
\(55\) −6.96745e21 −1.22614
\(56\) −2.61458e21 −0.367325
\(57\) 0 0
\(58\) 1.47051e22 1.33235
\(59\) −1.51046e21 −0.110524 −0.0552622 − 0.998472i \(-0.517599\pi\)
−0.0552622 + 0.998472i \(0.517599\pi\)
\(60\) 0 0
\(61\) 4.63585e21 0.223618 0.111809 − 0.993730i \(-0.464336\pi\)
0.111809 + 0.993730i \(0.464336\pi\)
\(62\) −4.61861e22 −1.81809
\(63\) 0 0
\(64\) 9.14317e21 0.242018
\(65\) −2.20182e21 −0.0480137
\(66\) 0 0
\(67\) −1.21518e23 −1.81429 −0.907144 − 0.420821i \(-0.861742\pi\)
−0.907144 + 0.420821i \(0.861742\pi\)
\(68\) 3.52000e22 0.436698
\(69\) 0 0
\(70\) −6.22286e22 −0.537359
\(71\) −1.87102e23 −1.35316 −0.676581 − 0.736368i \(-0.736539\pi\)
−0.676581 + 0.736368i \(0.736539\pi\)
\(72\) 0 0
\(73\) −4.86220e22 −0.248483 −0.124242 − 0.992252i \(-0.539650\pi\)
−0.124242 + 0.992252i \(0.539650\pi\)
\(74\) 7.14782e21 0.0308160
\(75\) 0 0
\(76\) 2.86155e23 0.883959
\(77\) 3.19351e23 0.837787
\(78\) 0 0
\(79\) 4.43976e23 0.845319 0.422660 − 0.906288i \(-0.361097\pi\)
0.422660 + 0.906288i \(0.361097\pi\)
\(80\) 6.19818e23 1.00842
\(81\) 0 0
\(82\) 1.21285e24 1.44922
\(83\) 1.52666e23 0.156771 0.0783857 − 0.996923i \(-0.475023\pi\)
0.0783857 + 0.996923i \(0.475023\pi\)
\(84\) 0 0
\(85\) −1.03515e24 −0.789345
\(86\) 2.27856e24 1.50117
\(87\) 0 0
\(88\) −2.04005e24 −1.00834
\(89\) −1.97570e24 −0.847901 −0.423951 − 0.905685i \(-0.639357\pi\)
−0.423951 + 0.905685i \(0.639357\pi\)
\(90\) 0 0
\(91\) 1.00920e23 0.0328064
\(92\) 1.01784e24 0.288624
\(93\) 0 0
\(94\) 3.04415e24 0.659733
\(95\) −8.41518e24 −1.59778
\(96\) 0 0
\(97\) −3.28438e24 −0.480626 −0.240313 − 0.970695i \(-0.577250\pi\)
−0.240313 + 0.970695i \(0.577250\pi\)
\(98\) −6.49336e24 −0.835880
\(99\) 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.b.1.2 2
3.2 odd 2 3.26.a.a.1.1 ✓ 2
12.11 even 2 48.26.a.f.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.26.a.a.1.1 ✓ 2 3.2 odd 2
9.26.a.b.1.2 2 1.1 even 1 trivial
48.26.a.f.1.2 2 12.11 even 2