↑

Properties

Label 9.66.a.b.1.1
Level $9$
Weight $66$
Character 9.1
Self dual yes
Analytic conductor $240.815$
Analytic rank $0$
Dimension $5$
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] = [9,66,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, 66); 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, 66, names="a")
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 66 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,3959709648] 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: \(240.815120825\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} + \cdots - 10\!\cdots\!36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{40}\cdot 3^{20}\cdot 5^{4}\cdot 7^{2}\cdot 11\cdot 13^{2} \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.84762e8\) 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)
 
\(f(q)\) \(=\) \(q-8.07662e9 q^{2} +2.83384e19 q^{4} +4.30990e22 q^{5} -4.31488e27 q^{7} +6.90965e28 q^{8} -3.48095e32 q^{10} +3.09454e33 q^{11} -1.30257e36 q^{13} +3.48497e37 q^{14} -1.60357e39 q^{16} -1.01238e40 q^{17} +6.77371e41 q^{19} +1.22136e42 q^{20} -2.49934e43 q^{22} -2.44678e44 q^{23} -8.52980e44 q^{25} +1.05204e46 q^{26} -1.22277e47 q^{28} -6.24422e46 q^{29} -4.23499e48 q^{31} +1.04022e49 q^{32} +8.17660e49 q^{34} -1.85967e50 q^{35} -6.58889e50 q^{37} -5.47087e51 q^{38} +2.97799e51 q^{40} +3.22294e51 q^{41} -7.33033e52 q^{43} +8.76942e52 q^{44} +1.97617e54 q^{46} -4.96764e53 q^{47} +1.00799e55 q^{49} +6.88920e54 q^{50} -3.69126e55 q^{52} +1.14911e56 q^{53} +1.33372e56 q^{55} -2.98143e56 q^{56} +5.04323e56 q^{58} +7.39266e56 q^{59} +1.27879e58 q^{61} +3.42044e58 q^{62} -2.48535e58 q^{64} -5.61394e58 q^{65} -1.92730e59 q^{67} -2.86892e59 q^{68} +1.50199e60 q^{70} +1.96706e60 q^{71} -4.77567e60 q^{73} +5.32160e60 q^{74} +1.91956e61 q^{76} -1.33526e61 q^{77} +1.98178e61 q^{79} -6.91122e61 q^{80} -2.60304e61 q^{82} -3.39503e62 q^{83} -4.36325e62 q^{85} +5.92043e62 q^{86} +2.13822e62 q^{88} -8.38014e62 q^{89} +5.62042e63 q^{91} -6.93378e63 q^{92} +4.01217e63 q^{94} +2.91940e64 q^{95} +1.21487e64 q^{97} -8.14112e64 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 3959709648 q^{2} + 11\!\cdots\!60 q^{4} - 26\!\cdots\!50 q^{5} - 69\!\cdots\!08 q^{7} + 44\!\cdots\!20 q^{8} - 46\!\cdots\!00 q^{10} + 54\!\cdots\!40 q^{11} - 28\!\cdots\!14 q^{13} - 41\!\cdots\!20 q^{14}+ \cdots + 46\!\cdots\!36 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −8.07662e9 −1.32970 −0.664852 − 0.746975i \(-0.731505\pi\)
−0.664852 + 0.746975i \(0.731505\pi\)
\(3\) 0 0
\(4\) 2.83384e19 0.768113
\(5\) 4.30990e22 0.827832 0.413916 − 0.910315i \(-0.364161\pi\)
0.413916 + 0.910315i \(0.364161\pi\)
\(6\) 0 0
\(7\) −4.31488e27 −1.47667 −0.738333 − 0.674436i \(-0.764387\pi\)
−0.738333 + 0.674436i \(0.764387\pi\)
\(8\) 6.90965e28 0.308341
\(9\) 0 0
\(10\) −3.48095e32 −1.10077
\(11\) 3.09454e33 0.441910 0.220955 − 0.975284i \(-0.429083\pi\)
0.220955 + 0.975284i \(0.429083\pi\)
\(12\) 0 0
\(13\) −1.30257e36 −0.815908 −0.407954 − 0.913002i \(-0.633758\pi\)
−0.407954 + 0.913002i \(0.633758\pi\)
\(14\) 3.48497e37 1.96353
\(15\) 0 0
\(16\) −1.60357e39 −1.17812
\(17\) −1.01238e40 −1.03694 −0.518469 − 0.855096i \(-0.673498\pi\)
−0.518469 + 0.855096i \(0.673498\pi\)
\(18\) 0 0
\(19\) 6.77371e41 1.86782 0.933909 − 0.357512i \(-0.116375\pi\)
0.933909 + 0.357512i \(0.116375\pi\)
\(20\) 1.22136e42 0.635869
\(21\) 0 0
\(22\) −2.49934e43 −0.587610
\(23\) −2.44678e44 −1.35656 −0.678282 − 0.734801i \(-0.737275\pi\)
−0.678282 + 0.734801i \(0.737275\pi\)
\(24\) 0 0
\(25\) −8.52980e44 −0.314694
\(26\) 1.05204e46 1.08492
\(27\) 0 0
\(28\) −1.22277e47 −1.13425
\(29\) −6.24422e46 −0.185158 −0.0925792 − 0.995705i \(-0.529511\pi\)
−0.0925792 + 0.995705i \(0.529511\pi\)
\(30\) 0 0
\(31\) −4.23499e48 −1.43746 −0.718732 − 0.695287i \(-0.755277\pi\)
−0.718732 + 0.695287i \(0.755277\pi\)
\(32\) 1.04022e49 1.25820
\(33\) 0 0
\(34\) 8.17660e49 1.37882
\(35\) −1.85967e50 −1.22243
\(36\) 0 0
\(37\) −6.58889e50 −0.711633 −0.355817 − 0.934556i \(-0.615797\pi\)
−0.355817 + 0.934556i \(0.615797\pi\)
\(38\) −5.47087e51 −2.48364
\(39\) 0 0
\(40\) 2.97799e51 0.255255
\(41\) 3.22294e51 0.123816 0.0619079 − 0.998082i \(-0.480281\pi\)
0.0619079 + 0.998082i \(0.480281\pi\)
\(42\) 0 0
\(43\) −7.33033e52 −0.598961 −0.299481 − 0.954102i \(-0.596813\pi\)
−0.299481 + 0.954102i \(0.596813\pi\)
\(44\) 8.76942e52 0.339437
\(45\) 0 0
\(46\) 1.97617e54 1.80383
\(47\) −4.96764e53 −0.225408 −0.112704 − 0.993629i \(-0.535951\pi\)
−0.112704 + 0.993629i \(0.535951\pi\)
\(48\) 0 0
\(49\) 1.00799e55 1.18054
\(50\) 6.88920e54 0.418450
\(51\) 0 0
\(52\) −3.69126e55 −0.626710
\(53\) 1.14911e56 1.05050 0.525249 − 0.850948i \(-0.323972\pi\)
0.525249 + 0.850948i \(0.323972\pi\)
\(54\) 0 0
\(55\) 1.33372e56 0.365827
\(56\) −2.98143e56 −0.455317
\(57\) 0 0
\(58\) 5.04323e56 0.246206
\(59\) 7.39266e56 0.207067 0.103533 − 0.994626i \(-0.466985\pi\)
0.103533 + 0.994626i \(0.466985\pi\)
\(60\) 0 0
\(61\) 1.27879e58 1.21222 0.606109 − 0.795381i \(-0.292729\pi\)
0.606109 + 0.795381i \(0.292729\pi\)
\(62\) 3.42044e58 1.91140
\(63\) 0 0
\(64\) −2.48535e58 −0.494923
\(65\) −5.61394e58 −0.675435
\(66\) 0 0
\(67\) −1.92730e59 −0.866000 −0.433000 − 0.901394i \(-0.642545\pi\)
−0.433000 + 0.901394i \(0.642545\pi\)
\(68\) −2.86892e59 −0.796486
\(69\) 0 0
\(70\) 1.50199e60 1.62547
\(71\) 1.96706e60 1.34252 0.671259 − 0.741223i \(-0.265754\pi\)
0.671259 + 0.741223i \(0.265754\pi\)
\(72\) 0 0
\(73\) −4.77567e60 −1.32142 −0.660711 − 0.750640i \(-0.729745\pi\)
−0.660711 + 0.750640i \(0.729745\pi\)
\(74\) 5.32160e60 0.946262
\(75\) 0 0
\(76\) 1.91956e61 1.43469
\(77\) −1.33526e61 −0.652554
\(78\) 0 0
\(79\) 1.98178e61 0.420896 0.210448 − 0.977605i \(-0.432508\pi\)
0.210448 + 0.977605i \(0.432508\pi\)
\(80\) −6.91122e61 −0.975282
\(81\) 0 0
\(82\) −2.60304e61 −0.164638
\(83\) −3.39503e62 −1.44812 −0.724062 − 0.689735i \(-0.757727\pi\)
−0.724062 + 0.689735i \(0.757727\pi\)
\(84\) 0 0
\(85\) −4.36325e62 −0.858411
\(86\) 5.92043e62 0.796441
\(87\) 0 0
\(88\) 2.13822e62 0.136259
\(89\) −8.38014e62 −0.369893 −0.184947 − 0.982749i \(-0.559211\pi\)
−0.184947 + 0.982749i \(0.559211\pi\)
\(90\) 0 0
\(91\) 5.62042e63 1.20482
\(92\) −6.93378e63 −1.04199
\(93\) 0 0
\(94\) 4.01217e63 0.299726
\(95\) 2.91940e64 1.54624
\(96\) 0 0
\(97\) 1.21487e64 0.326925 0.163462 − 0.986550i \(-0.447734\pi\)
0.163462 + 0.986550i \(0.447734\pi\)
\(98\) −8.14112e64 −1.56977
\(99\) 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 9.66.a.b.1.1 5
3.2 odd 2 1.66.a.a.1.5 ✓ 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.66.a.a.1.5 ✓ 5 3.2 odd 2
9.66.a.b.1.1 5 1.1 even 1 trivial