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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9,18,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, 18); 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, 18, names="a")
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-594] 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: \(16.4899878610\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{14569}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3642 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(60.8511\) 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)
 
\(f(q)\) \(=\) \(q-659.106 q^{2} +303349. q^{4} +1.08318e6 q^{5} +1.59855e6 q^{7} -1.13549e8 q^{8} -7.13934e8 q^{10} +4.47381e8 q^{11} -2.48486e9 q^{13} -1.05361e9 q^{14} +3.50803e10 q^{16} +2.48745e10 q^{17} +8.23042e10 q^{19} +3.28583e11 q^{20} -2.94872e11 q^{22} -6.43083e11 q^{23} +4.10349e11 q^{25} +1.63779e12 q^{26} +4.84918e11 q^{28} +9.82213e11 q^{29} +3.28632e12 q^{31} -8.23853e12 q^{32} -1.63949e13 q^{34} +1.73152e12 q^{35} +2.63492e13 q^{37} -5.42472e13 q^{38} -1.22994e14 q^{40} +3.33007e13 q^{41} +9.83107e13 q^{43} +1.35713e14 q^{44} +4.23860e14 q^{46} +1.62068e14 q^{47} -2.30075e14 q^{49} -2.70463e14 q^{50} -7.53780e14 q^{52} +1.40921e14 q^{53} +4.84596e14 q^{55} -1.81514e14 q^{56} -6.47383e14 q^{58} +9.80930e13 q^{59} +1.37376e15 q^{61} -2.16603e15 q^{62} +8.32030e14 q^{64} -2.69156e15 q^{65} -1.85816e15 q^{67} +7.54565e15 q^{68} -1.14126e15 q^{70} +6.17500e15 q^{71} -1.30214e16 q^{73} -1.73670e16 q^{74} +2.49669e16 q^{76} +7.15160e14 q^{77} +1.27538e16 q^{79} +3.79984e16 q^{80} -2.19487e16 q^{82} +1.42886e16 q^{83} +2.69436e16 q^{85} -6.47972e16 q^{86} -5.07996e16 q^{88} +3.77818e16 q^{89} -3.97217e15 q^{91} -1.95079e17 q^{92} -1.06820e17 q^{94} +8.91506e16 q^{95} +1.09404e17 q^{97} +1.51644e17 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 594 q^{2} + 176516 q^{4} - 382860 q^{5} + 24471568 q^{7} - 130340232 q^{8} - 809382420 q^{10} + 987553512 q^{11} - 2519398244 q^{13} + 435565296 q^{14} + 50611323920 q^{16} + 34313126364 q^{17} + 80053542184 q^{19}+ \cdots + 17\!\cdots\!26 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −659.106 −1.82054 −0.910271 − 0.414014i \(-0.864127\pi\)
−0.910271 + 0.414014i \(0.864127\pi\)
\(3\) 0 0
\(4\) 303349. 2.31437
\(5\) 1.08318e6 1.24010 0.620051 − 0.784562i \(-0.287112\pi\)
0.620051 + 0.784562i \(0.287112\pi\)
\(6\) 0 0
\(7\) 1.59855e6 0.104808 0.0524038 − 0.998626i \(-0.483312\pi\)
0.0524038 + 0.998626i \(0.483312\pi\)
\(8\) −1.13549e8 −2.39287
\(9\) 0 0
\(10\) −7.13934e8 −2.25766
\(11\) 4.47381e8 0.629274 0.314637 − 0.949212i \(-0.398117\pi\)
0.314637 + 0.949212i \(0.398117\pi\)
\(12\) 0 0
\(13\) −2.48486e9 −0.844857 −0.422428 − 0.906396i \(-0.638822\pi\)
−0.422428 + 0.906396i \(0.638822\pi\)
\(14\) −1.05361e9 −0.190806
\(15\) 0 0
\(16\) 3.50803e10 2.04194
\(17\) 2.48745e10 0.864844 0.432422 − 0.901671i \(-0.357659\pi\)
0.432422 + 0.901671i \(0.357659\pi\)
\(18\) 0 0
\(19\) 8.23042e10 1.11177 0.555887 − 0.831258i \(-0.312379\pi\)
0.555887 + 0.831258i \(0.312379\pi\)
\(20\) 3.28583e11 2.87005
\(21\) 0 0
\(22\) −2.94872e11 −1.14562
\(23\) −6.43083e11 −1.71230 −0.856151 − 0.516726i \(-0.827150\pi\)
−0.856151 + 0.516726i \(0.827150\pi\)
\(24\) 0 0
\(25\) 4.10349e11 0.537852
\(26\) 1.63779e12 1.53810
\(27\) 0 0
\(28\) 4.84918e11 0.242563
\(29\) 9.82213e11 0.364605 0.182302 − 0.983243i \(-0.441645\pi\)
0.182302 + 0.983243i \(0.441645\pi\)
\(30\) 0 0
\(31\) 3.28632e12 0.692046 0.346023 − 0.938226i \(-0.387532\pi\)
0.346023 + 0.938226i \(0.387532\pi\)
\(32\) −8.23853e12 −1.32457
\(33\) 0 0
\(34\) −1.63949e13 −1.57448
\(35\) 1.73152e12 0.129972
\(36\) 0 0
\(37\) 2.63492e13 1.23326 0.616628 − 0.787254i \(-0.288498\pi\)
0.616628 + 0.787254i \(0.288498\pi\)
\(38\) −5.42472e13 −2.02403
\(39\) 0 0
\(40\) −1.22994e14 −2.96740
\(41\) 3.33007e13 0.651314 0.325657 − 0.945488i \(-0.394415\pi\)
0.325657 + 0.945488i \(0.394415\pi\)
\(42\) 0 0
\(43\) 9.83107e13 1.28268 0.641341 − 0.767256i \(-0.278378\pi\)
0.641341 + 0.767256i \(0.278378\pi\)
\(44\) 1.35713e14 1.45637
\(45\) 0 0
\(46\) 4.23860e14 3.11732
\(47\) 1.62068e14 0.992811 0.496406 − 0.868091i \(-0.334653\pi\)
0.496406 + 0.868091i \(0.334653\pi\)
\(48\) 0 0
\(49\) −2.30075e14 −0.989015
\(50\) −2.70463e14 −0.979182
\(51\) 0 0
\(52\) −7.53780e14 −1.95531
\(53\) 1.40921e14 0.310907 0.155453 − 0.987843i \(-0.450316\pi\)
0.155453 + 0.987843i \(0.450316\pi\)
\(54\) 0 0
\(55\) 4.84596e14 0.780363
\(56\) −1.81514e14 −0.250790
\(57\) 0 0
\(58\) −6.47383e14 −0.663778
\(59\) 9.80930e13 0.0869753 0.0434876 − 0.999054i \(-0.486153\pi\)
0.0434876 + 0.999054i \(0.486153\pi\)
\(60\) 0 0
\(61\) 1.37376e15 0.917503 0.458751 − 0.888565i \(-0.348297\pi\)
0.458751 + 0.888565i \(0.348297\pi\)
\(62\) −2.16603e15 −1.25990
\(63\) 0 0
\(64\) 8.32030e14 0.369495
\(65\) −2.69156e15 −1.04771
\(66\) 0 0
\(67\) −1.85816e15 −0.559046 −0.279523 − 0.960139i \(-0.590176\pi\)
−0.279523 + 0.960139i \(0.590176\pi\)
\(68\) 7.54565e15 2.00157
\(69\) 0 0
\(70\) −1.14126e15 −0.236619
\(71\) 6.17500e15 1.13486 0.567428 − 0.823423i \(-0.307939\pi\)
0.567428 + 0.823423i \(0.307939\pi\)
\(72\) 0 0
\(73\) −1.30214e16 −1.88979 −0.944896 − 0.327371i \(-0.893837\pi\)
−0.944896 + 0.327371i \(0.893837\pi\)
\(74\) −1.73670e16 −2.24519
\(75\) 0 0
\(76\) 2.49669e16 2.57305
\(77\) 7.15160e14 0.0659526
\(78\) 0 0
\(79\) 1.27538e16 0.945824 0.472912 − 0.881110i \(-0.343203\pi\)
0.472912 + 0.881110i \(0.343203\pi\)
\(80\) 3.79984e16 2.53221
\(81\) 0 0
\(82\) −2.19487e16 −1.18574
\(83\) 1.42886e16 0.696348 0.348174 − 0.937430i \(-0.386802\pi\)
0.348174 + 0.937430i \(0.386802\pi\)
\(84\) 0 0
\(85\) 2.69436e16 1.07250
\(86\) −6.47972e16 −2.33517
\(87\) 0 0
\(88\) −5.07996e16 −1.50577
\(89\) 3.77818e16 1.01734 0.508672 − 0.860961i \(-0.330137\pi\)
0.508672 + 0.860961i \(0.330137\pi\)
\(90\) 0 0
\(91\) −3.97217e15 −0.0885473
\(92\) −1.95079e17 −3.96290
\(93\) 0 0
\(94\) −1.06820e17 −1.80745
\(95\) 8.91506e16 1.37871
\(96\) 0 0
\(97\) 1.09404e17 1.41733 0.708667 − 0.705543i \(-0.249297\pi\)
0.708667 + 0.705543i \(0.249297\pi\)
\(98\) 1.51644e17 1.80054
\(99\) 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 9.18.a.c.1.1 2
3.2 odd 2 3.18.a.b.1.2 ✓ 2
12.11 even 2 48.18.a.h.1.1 2
15.2 even 4 75.18.b.c.49.4 4
15.8 even 4 75.18.b.c.49.1 4
15.14 odd 2 75.18.a.b.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.18.a.b.1.2 ✓ 2 3.2 odd 2
9.18.a.c.1.1 2 1.1 even 1 trivial
48.18.a.h.1.1 2 12.11 even 2
75.18.a.b.1.1 2 15.14 odd 2
75.18.b.c.49.1 4 15.8 even 4
75.18.b.c.49.4 4 15.2 even 4