Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9,22,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, 22); 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, 22, names="a")
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,288] 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: \(25.1529609858\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
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)
 
\(f(q)\) \(=\) \(q+288.000 q^{2} -2.01421e6 q^{4} -2.16410e7 q^{5} -7.68079e8 q^{7} -1.18407e9 q^{8} -6.23259e9 q^{10} +9.47249e10 q^{11} -8.06218e10 q^{13} -2.21207e11 q^{14} +3.88309e12 q^{16} -3.05228e12 q^{17} -7.92079e12 q^{19} +4.35894e13 q^{20} +2.72808e13 q^{22} +7.38454e13 q^{23} -8.50644e12 q^{25} -2.32191e13 q^{26} +1.54707e15 q^{28} +4.25303e15 q^{29} +1.90054e15 q^{31} +3.60151e15 q^{32} -8.79057e14 q^{34} +1.66220e16 q^{35} +2.21914e16 q^{37} -2.28119e15 q^{38} +2.56244e16 q^{40} +2.06228e16 q^{41} -1.93606e17 q^{43} -1.90796e17 q^{44} +2.12675e16 q^{46} -1.46961e17 q^{47} +3.13992e16 q^{49} -2.44986e15 q^{50} +1.62389e17 q^{52} -2.03827e18 q^{53} -2.04994e18 q^{55} +9.09460e17 q^{56} +1.22487e18 q^{58} +5.97588e18 q^{59} +6.19062e18 q^{61} +5.47356e17 q^{62} -7.10619e18 q^{64} +1.74473e18 q^{65} +1.69613e19 q^{67} +6.14793e18 q^{68} +4.78712e18 q^{70} +5.63276e18 q^{71} -4.32848e19 q^{73} +6.39113e18 q^{74} +1.59541e19 q^{76} -7.27562e19 q^{77} -5.12649e19 q^{79} -8.40337e19 q^{80} +5.93937e18 q^{82} -4.89119e19 q^{83} +6.60543e19 q^{85} -5.57585e19 q^{86} -1.12161e20 q^{88} +5.04303e20 q^{89} +6.19239e19 q^{91} -1.48740e20 q^{92} -4.23246e19 q^{94} +1.71413e20 q^{95} +8.08275e20 q^{97} +9.04297e18 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 288.000 0.198874 0.0994369 − 0.995044i \(-0.468296\pi\)
0.0994369 + 0.995044i \(0.468296\pi\)
\(3\) 0 0
\(4\) −2.01421e6 −0.960449
\(5\) −2.16410e7 −0.991040 −0.495520 − 0.868596i \(-0.665023\pi\)
−0.495520 + 0.868596i \(0.665023\pi\)
\(6\) 0 0
\(7\) −7.68079e8 −1.02772 −0.513862 − 0.857873i \(-0.671786\pi\)
−0.513862 + 0.857873i \(0.671786\pi\)
\(8\) −1.18407e9 −0.389882
\(9\) 0 0
\(10\) −6.23259e9 −0.197092
\(11\) 9.47249e10 1.10114 0.550568 − 0.834790i \(-0.314411\pi\)
0.550568 + 0.834790i \(0.314411\pi\)
\(12\) 0 0
\(13\) −8.06218e10 −0.162199 −0.0810993 − 0.996706i \(-0.525843\pi\)
−0.0810993 + 0.996706i \(0.525843\pi\)
\(14\) −2.21207e11 −0.204387
\(15\) 0 0
\(16\) 3.88309e12 0.882912
\(17\) −3.05228e12 −0.367207 −0.183604 − 0.983000i \(-0.558776\pi\)
−0.183604 + 0.983000i \(0.558776\pi\)
\(18\) 0 0
\(19\) −7.92079e12 −0.296385 −0.148192 − 0.988959i \(-0.547345\pi\)
−0.148192 + 0.988959i \(0.547345\pi\)
\(20\) 4.35894e13 0.951844
\(21\) 0 0
\(22\) 2.72808e13 0.218987
\(23\) 7.38454e13 0.371690 0.185845 − 0.982579i \(-0.440498\pi\)
0.185845 + 0.982579i \(0.440498\pi\)
\(24\) 0 0
\(25\) −8.50644e12 −0.0178393
\(26\) −2.32191e13 −0.0322571
\(27\) 0 0
\(28\) 1.54707e15 0.987076
\(29\) 4.25303e15 1.87724 0.938620 − 0.344954i \(-0.112105\pi\)
0.938620 + 0.344954i \(0.112105\pi\)
\(30\) 0 0
\(31\) 1.90054e15 0.416466 0.208233 − 0.978079i \(-0.433229\pi\)
0.208233 + 0.978079i \(0.433229\pi\)
\(32\) 3.60151e15 0.565470
\(33\) 0 0
\(34\) −8.79057e14 −0.0730279
\(35\) 1.66220e16 1.01852
\(36\) 0 0
\(37\) 2.21914e16 0.758695 0.379347 − 0.925254i \(-0.376149\pi\)
0.379347 + 0.925254i \(0.376149\pi\)
\(38\) −2.28119e15 −0.0589431
\(39\) 0 0
\(40\) 2.56244e16 0.386389
\(41\) 2.06228e16 0.239948 0.119974 − 0.992777i \(-0.461719\pi\)
0.119974 + 0.992777i \(0.461719\pi\)
\(42\) 0 0
\(43\) −1.93606e17 −1.36615 −0.683077 − 0.730346i \(-0.739359\pi\)
−0.683077 + 0.730346i \(0.739359\pi\)
\(44\) −1.90796e17 −1.05759
\(45\) 0 0
\(46\) 2.12675e16 0.0739195
\(47\) −1.46961e17 −0.407543 −0.203771 − 0.979019i \(-0.565320\pi\)
−0.203771 + 0.979019i \(0.565320\pi\)
\(48\) 0 0
\(49\) 3.13992e16 0.0562160
\(50\) −2.44986e15 −0.00354777
\(51\) 0 0
\(52\) 1.62389e17 0.155784
\(53\) −2.03827e18 −1.60090 −0.800450 − 0.599399i \(-0.795406\pi\)
−0.800450 + 0.599399i \(0.795406\pi\)
\(54\) 0 0
\(55\) −2.04994e18 −1.09127
\(56\) 9.09460e17 0.400691
\(57\) 0 0
\(58\) 1.22487e18 0.373334
\(59\) 5.97588e18 1.52214 0.761072 − 0.648667i \(-0.224673\pi\)
0.761072 + 0.648667i \(0.224673\pi\)
\(60\) 0 0
\(61\) 6.19062e18 1.11114 0.555572 − 0.831468i \(-0.312499\pi\)
0.555572 + 0.831468i \(0.312499\pi\)
\(62\) 5.47356e17 0.0828242
\(63\) 0 0
\(64\) −7.10619e18 −0.770455
\(65\) 1.74473e18 0.160745
\(66\) 0 0
\(67\) 1.69613e19 1.13677 0.568387 − 0.822761i \(-0.307568\pi\)
0.568387 + 0.822761i \(0.307568\pi\)
\(68\) 6.14793e18 0.352684
\(69\) 0 0
\(70\) 4.78712e18 0.202556
\(71\) 5.63276e18 0.205357 0.102678 − 0.994715i \(-0.467259\pi\)
0.102678 + 0.994715i \(0.467259\pi\)
\(72\) 0 0
\(73\) −4.32848e19 −1.17881 −0.589407 − 0.807837i \(-0.700638\pi\)
−0.589407 + 0.807837i \(0.700638\pi\)
\(74\) 6.39113e18 0.150885
\(75\) 0 0
\(76\) 1.59541e19 0.284662
\(77\) −7.27562e19 −1.13166
\(78\) 0 0
\(79\) −5.12649e19 −0.609166 −0.304583 − 0.952486i \(-0.598517\pi\)
−0.304583 + 0.952486i \(0.598517\pi\)
\(80\) −8.40337e19 −0.875001
\(81\) 0 0
\(82\) 5.93937e18 0.0477194
\(83\) −4.89119e19 −0.346014 −0.173007 − 0.984921i \(-0.555348\pi\)
−0.173007 + 0.984921i \(0.555348\pi\)
\(84\) 0 0
\(85\) 6.60543e19 0.363917
\(86\) −5.57585e19 −0.271692
\(87\) 0 0
\(88\) −1.12161e20 −0.429313
\(89\) 5.04303e20 1.71434 0.857170 − 0.515034i \(-0.172221\pi\)
0.857170 + 0.515034i \(0.172221\pi\)
\(90\) 0 0
\(91\) 6.19239e19 0.166695
\(92\) −1.48740e20 −0.356990
\(93\) 0 0
\(94\) −4.23246e19 −0.0810496
\(95\) 1.71413e20 0.293729
\(96\) 0 0
\(97\) 8.08275e20 1.11290 0.556450 − 0.830881i \(-0.312163\pi\)
0.556450 + 0.830881i \(0.312163\pi\)
\(98\) 9.04297e18 0.0111799
\(99\) 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 9.22.a.c.1.1 1
3.2 odd 2 1.22.a.a.1.1 ✓ 1
12.11 even 2 16.22.a.c.1.1 1
15.2 even 4 25.22.b.a.24.1 2
15.8 even 4 25.22.b.a.24.2 2
15.14 odd 2 25.22.a.a.1.1 1
21.20 even 2 49.22.a.a.1.1 1
24.5 odd 2 64.22.a.g.1.1 1
24.11 even 2 64.22.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.22.a.a.1.1 ✓ 1 3.2 odd 2
9.22.a.c.1.1 1 1.1 even 1 trivial
16.22.a.c.1.1 1 12.11 even 2
25.22.a.a.1.1 1 15.14 odd 2
25.22.b.a.24.1 2 15.2 even 4
25.22.b.a.24.2 2 15.8 even 4
49.22.a.a.1.1 1 21.20 even 2
64.22.a.a.1.1 1 24.11 even 2
64.22.a.g.1.1 1 24.5 odd 2