↑

Properties

Label 9.96.a.c.1.3
Level $9$
Weight $96$
Character 9.1
Self dual yes
Analytic conductor $514.382$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9,96,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, 96); 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, 96, names="a")
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 96 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,5835659138280] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(514.382317934\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: multiple of \( 2^{104}\cdot 3^{57}\cdot 5^{12}\cdot 7^{7}\cdot 11\cdot 13\cdot 19^{3} \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(6.86315e12\) 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)
 
\(f(q)\) \(=\) \(q-1.63986e14 q^{2} -1.27226e28 q^{4} -1.20020e33 q^{5} -2.14893e40 q^{7} +8.58249e42 q^{8} +1.96817e47 q^{10} -1.70307e49 q^{11} +1.01181e53 q^{13} +3.52395e54 q^{14} -9.03414e56 q^{16} -2.27059e58 q^{17} +5.33207e60 q^{19} +1.52698e61 q^{20} +2.79280e63 q^{22} -5.74222e62 q^{23} -1.08387e66 q^{25} -1.65923e67 q^{26} +2.73401e68 q^{28} -2.72768e69 q^{29} +8.24705e70 q^{31} -1.91840e71 q^{32} +3.72345e72 q^{34} +2.57916e73 q^{35} -5.13786e74 q^{37} -8.74385e74 q^{38} -1.03007e76 q^{40} -7.35670e76 q^{41} -6.17805e76 q^{43} +2.16676e77 q^{44} +9.41643e76 q^{46} -3.75012e78 q^{47} +2.69343e80 q^{49} +1.77739e80 q^{50} -1.28729e81 q^{52} -2.90261e81 q^{53} +2.04403e82 q^{55} -1.84432e83 q^{56} +4.47301e83 q^{58} -1.68978e84 q^{59} +2.77082e84 q^{61} -1.35240e85 q^{62} +6.72471e85 q^{64} -1.21438e86 q^{65} +3.14536e85 q^{67} +2.88879e86 q^{68} -4.22946e87 q^{70} +4.28767e87 q^{71} -4.92275e88 q^{73} +8.42538e88 q^{74} -6.78380e88 q^{76} +3.65979e89 q^{77} +2.67794e90 q^{79} +1.08428e90 q^{80} +1.20640e91 q^{82} +1.35258e91 q^{83} +2.72517e91 q^{85} +1.01311e91 q^{86} -1.46166e92 q^{88} -3.19257e91 q^{89} -2.17432e93 q^{91} +7.30562e90 q^{92} +6.14967e92 q^{94} -6.39957e93 q^{95} +2.36851e94 q^{97} -4.41685e94 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 5835659138280 q^{2} + 20\!\cdots\!84 q^{4} - 19\!\cdots\!60 q^{5} + 31\!\cdots\!00 q^{7} + 14\!\cdots\!60 q^{8} - 35\!\cdots\!40 q^{10} - 53\!\cdots\!16 q^{11} + 11\!\cdots\!40 q^{13} - 88\!\cdots\!08 q^{14}+ \cdots - 14\!\cdots\!60 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.63986e14 −0.823915 −0.411957 − 0.911203i \(-0.635155\pi\)
−0.411957 + 0.911203i \(0.635155\pi\)
\(3\) 0 0
\(4\) −1.27226e28 −0.321165
\(5\) −1.20020e33 −0.755405 −0.377702 − 0.925927i \(-0.623286\pi\)
−0.377702 + 0.925927i \(0.623286\pi\)
\(6\) 0 0
\(7\) −2.14893e40 −1.54905 −0.774526 − 0.632542i \(-0.782012\pi\)
−0.774526 + 0.632542i \(0.782012\pi\)
\(8\) 8.58249e42 1.08853
\(9\) 0 0
\(10\) 1.96817e47 0.622389
\(11\) −1.70307e49 −0.582211 −0.291106 − 0.956691i \(-0.594023\pi\)
−0.291106 + 0.956691i \(0.594023\pi\)
\(12\) 0 0
\(13\) 1.01181e53 1.23820 0.619101 − 0.785312i \(-0.287497\pi\)
0.619101 + 0.785312i \(0.287497\pi\)
\(14\) 3.52395e54 1.27629
\(15\) 0 0
\(16\) −9.03414e56 −0.575688
\(17\) −2.27059e58 −0.812484 −0.406242 − 0.913765i \(-0.633161\pi\)
−0.406242 + 0.913765i \(0.633161\pi\)
\(18\) 0 0
\(19\) 5.33207e60 0.968500 0.484250 − 0.874930i \(-0.339093\pi\)
0.484250 + 0.874930i \(0.339093\pi\)
\(20\) 1.52698e61 0.242609
\(21\) 0 0
\(22\) 2.79280e63 0.479692
\(23\) −5.74222e62 −0.0119401 −0.00597004 − 0.999982i \(-0.501900\pi\)
−0.00597004 + 0.999982i \(0.501900\pi\)
\(24\) 0 0
\(25\) −1.08387e66 −0.429364
\(26\) −1.65923e67 −1.02017
\(27\) 0 0
\(28\) 2.73401e68 0.497501
\(29\) −2.72768e69 −0.937320 −0.468660 − 0.883379i \(-0.655263\pi\)
−0.468660 + 0.883379i \(0.655263\pi\)
\(30\) 0 0
\(31\) 8.24705e70 1.19294 0.596469 − 0.802636i \(-0.296570\pi\)
0.596469 + 0.802636i \(0.296570\pi\)
\(32\) −1.91840e71 −0.614209
\(33\) 0 0
\(34\) 3.72345e72 0.669418
\(35\) 2.57916e73 1.17016
\(36\) 0 0
\(37\) −5.13786e74 −1.66418 −0.832091 − 0.554639i \(-0.812856\pi\)
−0.832091 + 0.554639i \(0.812856\pi\)
\(38\) −8.74385e74 −0.797961
\(39\) 0 0
\(40\) −1.03007e76 −0.822278
\(41\) −7.35670e76 −1.81740 −0.908699 − 0.417452i \(-0.862923\pi\)
−0.908699 + 0.417452i \(0.862923\pi\)
\(42\) 0 0
\(43\) −6.17805e76 −0.158891 −0.0794456 − 0.996839i \(-0.525315\pi\)
−0.0794456 + 0.996839i \(0.525315\pi\)
\(44\) 2.16676e77 0.186986
\(45\) 0 0
\(46\) 9.41643e76 0.00983761
\(47\) −3.75012e78 −0.141058 −0.0705288 − 0.997510i \(-0.522469\pi\)
−0.0705288 + 0.997510i \(0.522469\pi\)
\(48\) 0 0
\(49\) 2.69343e80 1.39956
\(50\) 1.77739e80 0.353759
\(51\) 0 0
\(52\) −1.28729e81 −0.397667
\(53\) −2.90261e81 −0.362814 −0.181407 − 0.983408i \(-0.558065\pi\)
−0.181407 + 0.983408i \(0.558065\pi\)
\(54\) 0 0
\(55\) 2.04403e82 0.439805
\(56\) −1.84432e83 −1.68618
\(57\) 0 0
\(58\) 4.47301e83 0.772272
\(59\) −1.68978e84 −1.29527 −0.647635 − 0.761950i \(-0.724242\pi\)
−0.647635 + 0.761950i \(0.724242\pi\)
\(60\) 0 0
\(61\) 2.77082e84 0.435954 0.217977 − 0.975954i \(-0.430054\pi\)
0.217977 + 0.975954i \(0.430054\pi\)
\(62\) −1.35240e85 −0.982879
\(63\) 0 0
\(64\) 6.72471e85 1.08174
\(65\) −1.21438e86 −0.935343
\(66\) 0 0
\(67\) 3.14536e85 0.0574271 0.0287135 − 0.999588i \(-0.490859\pi\)
0.0287135 + 0.999588i \(0.490859\pi\)
\(68\) 2.88879e86 0.260941
\(69\) 0 0
\(70\) −4.22946e87 −0.964113
\(71\) 4.28767e87 0.498252 0.249126 − 0.968471i \(-0.419857\pi\)
0.249126 + 0.968471i \(0.419857\pi\)
\(72\) 0 0
\(73\) −4.92275e88 −1.52888 −0.764438 − 0.644697i \(-0.776983\pi\)
−0.764438 + 0.644697i \(0.776983\pi\)
\(74\) 8.42538e88 1.37114
\(75\) 0 0
\(76\) −6.78380e88 −0.311048
\(77\) 3.65979e89 0.901876
\(78\) 0 0
\(79\) 2.67794e90 1.95216 0.976078 − 0.217419i \(-0.0697637\pi\)
0.976078 + 0.217419i \(0.0697637\pi\)
\(80\) 1.08428e90 0.434878
\(81\) 0 0
\(82\) 1.20640e91 1.49738
\(83\) 1.35258e91 0.943960 0.471980 − 0.881609i \(-0.343539\pi\)
0.471980 + 0.881609i \(0.343539\pi\)
\(84\) 0 0
\(85\) 2.72517e91 0.613755
\(86\) 1.01311e91 0.130913
\(87\) 0 0
\(88\) −1.46166e92 −0.633753
\(89\) −3.19257e91 −0.0809310 −0.0404655 − 0.999181i \(-0.512884\pi\)
−0.0404655 + 0.999181i \(0.512884\pi\)
\(90\) 0 0
\(91\) −2.17432e93 −1.91804
\(92\) 7.30562e90 0.00383473
\(93\) 0 0
\(94\) 6.14967e92 0.116219
\(95\) −6.39957e93 −0.731609
\(96\) 0 0
\(97\) 2.36851e94 1.00651 0.503254 − 0.864139i \(-0.332136\pi\)
0.503254 + 0.864139i \(0.332136\pi\)
\(98\) −4.41685e94 −1.15312
\(99\) 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 9.96.a.c.1.3 8
3.2 odd 2 1.96.a.a.1.6 ✓ 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.96.a.a.1.6 ✓ 8 3.2 odd 2
9.96.a.c.1.3 8 1.1 even 1 trivial