Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [63,10,Mod(1,63)] 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("63.1"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(63, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 63.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,6] 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: \(32.4472576783\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{193}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 7)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(7.44622\) of defining polynomial
Character \(\chi\) \(=\) 63.1

$q$-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)
 
\(f(q)\) \(=\) \(q+16.8924 q^{2} -226.645 q^{4} +2438.78 q^{5} -2401.00 q^{7} -12477.5 q^{8} +41197.0 q^{10} +28548.3 q^{11} +138149. q^{13} -40558.8 q^{14} -94733.5 q^{16} +101010. q^{17} -488928. q^{19} -552739. q^{20} +482250. q^{22} +140071. q^{23} +3.99453e6 q^{25} +2.33367e6 q^{26} +544175. q^{28} +6.31716e6 q^{29} -1.00903e6 q^{31} +4.78821e6 q^{32} +1.70630e6 q^{34} -5.85552e6 q^{35} +1.19206e7 q^{37} -8.25919e6 q^{38} -3.04300e7 q^{40} +2.15106e7 q^{41} +1.65957e7 q^{43} -6.47033e6 q^{44} +2.36615e6 q^{46} +2.67441e7 q^{47} +5.76480e6 q^{49} +6.74774e7 q^{50} -3.13108e7 q^{52} -3.74991e7 q^{53} +6.96230e7 q^{55} +2.99585e7 q^{56} +1.06712e8 q^{58} -1.81907e7 q^{59} -2.50111e7 q^{61} -1.70449e7 q^{62} +1.29388e8 q^{64} +3.36915e8 q^{65} -2.18572e8 q^{67} -2.28934e7 q^{68} -9.89140e7 q^{70} -3.12688e8 q^{71} -2.89038e8 q^{73} +2.01369e8 q^{74} +1.10813e8 q^{76} -6.85444e7 q^{77} +4.68685e8 q^{79} -2.31034e8 q^{80} +3.63366e8 q^{82} +7.75407e7 q^{83} +2.46341e8 q^{85} +2.80342e8 q^{86} -3.56212e8 q^{88} -3.37680e8 q^{89} -3.31695e8 q^{91} -3.17465e7 q^{92} +4.51773e8 q^{94} -1.19239e9 q^{95} -7.36733e8 q^{97} +9.73816e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{2} - 620 q^{4} + 2238 q^{5} - 4802 q^{7} - 2616 q^{8} + 43384 q^{10} - 35316 q^{11} - 26530 q^{13} - 14406 q^{14} - 752 q^{16} + 463920 q^{17} - 925426 q^{19} - 473760 q^{20} + 1177888 q^{22}+ \cdots + 34588806 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 16.8924 0.746548 0.373274 − 0.927721i \(-0.378235\pi\)
0.373274 + 0.927721i \(0.378235\pi\)
\(3\) 0 0
\(4\) −226.645 −0.442667
\(5\) 2438.78 1.74505 0.872525 − 0.488569i \(-0.162481\pi\)
0.872525 + 0.488569i \(0.162481\pi\)
\(6\) 0 0
\(7\) −2401.00 −0.377964
\(8\) −12477.5 −1.07702
\(9\) 0 0
\(10\) 41197.0 1.30276
\(11\) 28548.3 0.587912 0.293956 − 0.955819i \(-0.405028\pi\)
0.293956 + 0.955819i \(0.405028\pi\)
\(12\) 0 0
\(13\) 138149. 1.34153 0.670767 − 0.741668i \(-0.265965\pi\)
0.670767 + 0.741668i \(0.265965\pi\)
\(14\) −40558.8 −0.282168
\(15\) 0 0
\(16\) −94733.5 −0.361380
\(17\) 101010. 0.293321 0.146661 − 0.989187i \(-0.453147\pi\)
0.146661 + 0.989187i \(0.453147\pi\)
\(18\) 0 0
\(19\) −488928. −0.860704 −0.430352 − 0.902661i \(-0.641611\pi\)
−0.430352 + 0.902661i \(0.641611\pi\)
\(20\) −552739. −0.772476
\(21\) 0 0
\(22\) 482250. 0.438905
\(23\) 140071. 0.104370 0.0521848 − 0.998637i \(-0.483382\pi\)
0.0521848 + 0.998637i \(0.483382\pi\)
\(24\) 0 0
\(25\) 3.99453e6 2.04520
\(26\) 2.33367e6 1.00152
\(27\) 0 0
\(28\) 544175. 0.167312
\(29\) 6.31716e6 1.65856 0.829279 − 0.558835i \(-0.188751\pi\)
0.829279 + 0.558835i \(0.188751\pi\)
\(30\) 0 0
\(31\) −1.00903e6 −0.196234 −0.0981172 − 0.995175i \(-0.531282\pi\)
−0.0981172 + 0.995175i \(0.531282\pi\)
\(32\) 4.78821e6 0.807232
\(33\) 0 0
\(34\) 1.70630e6 0.218978
\(35\) −5.85552e6 −0.659567
\(36\) 0 0
\(37\) 1.19206e7 1.04566 0.522832 − 0.852436i \(-0.324876\pi\)
0.522832 + 0.852436i \(0.324876\pi\)
\(38\) −8.25919e6 −0.642556
\(39\) 0 0
\(40\) −3.04300e7 −1.87945
\(41\) 2.15106e7 1.18884 0.594422 − 0.804153i \(-0.297381\pi\)
0.594422 + 0.804153i \(0.297381\pi\)
\(42\) 0 0
\(43\) 1.65957e7 0.740265 0.370133 − 0.928979i \(-0.379312\pi\)
0.370133 + 0.928979i \(0.379312\pi\)
\(44\) −6.47033e6 −0.260249
\(45\) 0 0
\(46\) 2.36615e6 0.0779169
\(47\) 2.67441e7 0.799443 0.399721 − 0.916637i \(-0.369107\pi\)
0.399721 + 0.916637i \(0.369107\pi\)
\(48\) 0 0
\(49\) 5.76480e6 0.142857
\(50\) 6.74774e7 1.52684
\(51\) 0 0
\(52\) −3.13108e7 −0.593853
\(53\) −3.74991e7 −0.652799 −0.326399 − 0.945232i \(-0.605835\pi\)
−0.326399 + 0.945232i \(0.605835\pi\)
\(54\) 0 0
\(55\) 6.96230e7 1.02594
\(56\) 2.99585e7 0.407075
\(57\) 0 0
\(58\) 1.06712e8 1.23819
\(59\) −1.81907e7 −0.195441 −0.0977207 − 0.995214i \(-0.531155\pi\)
−0.0977207 + 0.995214i \(0.531155\pi\)
\(60\) 0 0
\(61\) −2.50111e7 −0.231285 −0.115643 − 0.993291i \(-0.536893\pi\)
−0.115643 + 0.993291i \(0.536893\pi\)
\(62\) −1.70449e7 −0.146498
\(63\) 0 0
\(64\) 1.29388e8 0.964017
\(65\) 3.36915e8 2.34105
\(66\) 0 0
\(67\) −2.18572e8 −1.32513 −0.662564 − 0.749005i \(-0.730532\pi\)
−0.662564 + 0.749005i \(0.730532\pi\)
\(68\) −2.28934e7 −0.129844
\(69\) 0 0
\(70\) −9.89140e7 −0.492398
\(71\) −3.12688e8 −1.46032 −0.730161 − 0.683275i \(-0.760555\pi\)
−0.730161 + 0.683275i \(0.760555\pi\)
\(72\) 0 0
\(73\) −2.89038e8 −1.19125 −0.595624 − 0.803264i \(-0.703095\pi\)
−0.595624 + 0.803264i \(0.703095\pi\)
\(74\) 2.01369e8 0.780638
\(75\) 0 0
\(76\) 1.10813e8 0.381005
\(77\) −6.85444e7 −0.222210
\(78\) 0 0
\(79\) 4.68685e8 1.35381 0.676907 − 0.736069i \(-0.263320\pi\)
0.676907 + 0.736069i \(0.263320\pi\)
\(80\) −2.31034e8 −0.630626
\(81\) 0 0
\(82\) 3.63366e8 0.887529
\(83\) 7.75407e7 0.179341 0.0896703 − 0.995972i \(-0.471419\pi\)
0.0896703 + 0.995972i \(0.471419\pi\)
\(84\) 0 0
\(85\) 2.46341e8 0.511860
\(86\) 2.80342e8 0.552643
\(87\) 0 0
\(88\) −3.56212e8 −0.633193
\(89\) −3.37680e8 −0.570493 −0.285246 − 0.958454i \(-0.592075\pi\)
−0.285246 + 0.958454i \(0.592075\pi\)
\(90\) 0 0
\(91\) −3.31695e8 −0.507052
\(92\) −3.17465e7 −0.0462010
\(93\) 0 0
\(94\) 4.51773e8 0.596822
\(95\) −1.19239e9 −1.50197
\(96\) 0 0
\(97\) −7.36733e8 −0.844962 −0.422481 − 0.906372i \(-0.638841\pi\)
−0.422481 + 0.906372i \(0.638841\pi\)
\(98\) 9.73816e7 0.106650
\(99\) 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 63.10.a.d.1.2 2
3.2 odd 2 7.10.a.a.1.1 ✓ 2
12.11 even 2 112.10.a.e.1.1 2
15.2 even 4 175.10.b.b.99.1 4
15.8 even 4 175.10.b.b.99.4 4
15.14 odd 2 175.10.a.b.1.2 2
21.2 odd 6 49.10.c.c.18.2 4
21.5 even 6 49.10.c.b.18.2 4
21.11 odd 6 49.10.c.c.30.2 4
21.17 even 6 49.10.c.b.30.2 4
21.20 even 2 49.10.a.b.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.10.a.a.1.1 ✓ 2 3.2 odd 2
49.10.a.b.1.1 2 21.20 even 2
49.10.c.b.18.2 4 21.5 even 6
49.10.c.b.30.2 4 21.17 even 6
49.10.c.c.18.2 4 21.2 odd 6
49.10.c.c.30.2 4 21.11 odd 6
63.10.a.d.1.2 2 1.1 even 1 trivial
112.10.a.e.1.1 2 12.11 even 2
175.10.a.b.1.2 2 15.14 odd 2
175.10.b.b.99.1 4 15.2 even 4
175.10.b.b.99.4 4 15.8 even 4