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 1.1
Root \(-6.44622\) 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)
 
\(f(q)\) \(=\) \(q-10.8924 q^{2} -393.355 q^{4} -200.782 q^{5} -2401.00 q^{7} +9861.52 q^{8} +2187.01 q^{10} -63864.3 q^{11} -164679. q^{13} +26152.8 q^{14} +93981.5 q^{16} +362910. q^{17} -436498. q^{19} +78978.6 q^{20} +695638. q^{22} -918199. q^{23} -1.91281e6 q^{25} +1.79375e6 q^{26} +944445. q^{28} +3.68643e6 q^{29} +3.47629e6 q^{31} -6.07279e6 q^{32} -3.95298e6 q^{34} +482078. q^{35} +1.88149e7 q^{37} +4.75453e6 q^{38} -1.98002e6 q^{40} -2.40714e6 q^{41} -1.25306e7 q^{43} +2.51213e7 q^{44} +1.00014e7 q^{46} +5.54509e7 q^{47} +5.76480e6 q^{49} +2.08352e7 q^{50} +6.47772e7 q^{52} +9.26889e7 q^{53} +1.28228e7 q^{55} -2.36775e7 q^{56} -4.01542e7 q^{58} +2.52600e7 q^{59} +6.93275e7 q^{61} -3.78653e7 q^{62} +1.80290e7 q^{64} +3.30646e7 q^{65} -2.33494e7 q^{67} -1.42752e8 q^{68} -5.25101e6 q^{70} +1.06194e8 q^{71} -2.10115e8 q^{73} -2.04940e8 q^{74} +1.71699e8 q^{76} +1.53338e8 q^{77} -149606. q^{79} -1.88698e7 q^{80} +2.62197e7 q^{82} -5.21565e8 q^{83} -7.28659e7 q^{85} +1.36489e8 q^{86} -6.29799e8 q^{88} -2.98587e8 q^{89} +3.95394e8 q^{91} +3.61178e8 q^{92} -6.03996e8 q^{94} +8.76410e7 q^{95} -8.95983e8 q^{97} -6.27928e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{2} - 620 q^{4} + 2238 q^{5} - 4802 q^{7} - 2616 q^{8} + 43384 q^{10} - 35316 q^{11} - 26530 q^{13} - 14406 q^{14} - 752 q^{16} + 463920 q^{17} - 925426 q^{19} - 473760 q^{20} + 1177888 q^{22}+ \cdots + 34588806 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −10.8924 −0.481383 −0.240691 − 0.970602i \(-0.577374\pi\)
−0.240691 + 0.970602i \(0.577374\pi\)
\(3\) 0 0
\(4\) −393.355 −0.768271
\(5\) −200.782 −0.143668 −0.0718340 − 0.997417i \(-0.522885\pi\)
−0.0718340 + 0.997417i \(0.522885\pi\)
\(6\) 0 0
\(7\) −2401.00 −0.377964
\(8\) 9861.52 0.851215
\(9\) 0 0
\(10\) 2187.01 0.0691593
\(11\) −63864.3 −1.31520 −0.657599 − 0.753369i \(-0.728428\pi\)
−0.657599 + 0.753369i \(0.728428\pi\)
\(12\) 0 0
\(13\) −164679. −1.59916 −0.799581 − 0.600558i \(-0.794945\pi\)
−0.799581 + 0.600558i \(0.794945\pi\)
\(14\) 26152.8 0.181946
\(15\) 0 0
\(16\) 93981.5 0.358511
\(17\) 362910. 1.05385 0.526925 − 0.849912i \(-0.323345\pi\)
0.526925 + 0.849912i \(0.323345\pi\)
\(18\) 0 0
\(19\) −436498. −0.768406 −0.384203 − 0.923249i \(-0.625524\pi\)
−0.384203 + 0.923249i \(0.625524\pi\)
\(20\) 78978.6 0.110376
\(21\) 0 0
\(22\) 695638. 0.633113
\(23\) −918199. −0.684166 −0.342083 − 0.939670i \(-0.611132\pi\)
−0.342083 + 0.939670i \(0.611132\pi\)
\(24\) 0 0
\(25\) −1.91281e6 −0.979359
\(26\) 1.79375e6 0.769809
\(27\) 0 0
\(28\) 944445. 0.290379
\(29\) 3.68643e6 0.967865 0.483932 − 0.875105i \(-0.339208\pi\)
0.483932 + 0.875105i \(0.339208\pi\)
\(30\) 0 0
\(31\) 3.47629e6 0.676064 0.338032 − 0.941135i \(-0.390239\pi\)
0.338032 + 0.941135i \(0.390239\pi\)
\(32\) −6.07279e6 −1.02380
\(33\) 0 0
\(34\) −3.95298e6 −0.507305
\(35\) 482078. 0.0543014
\(36\) 0 0
\(37\) 1.88149e7 1.65042 0.825210 − 0.564826i \(-0.191057\pi\)
0.825210 + 0.564826i \(0.191057\pi\)
\(38\) 4.75453e6 0.369897
\(39\) 0 0
\(40\) −1.98002e6 −0.122292
\(41\) −2.40714e6 −0.133038 −0.0665188 − 0.997785i \(-0.521189\pi\)
−0.0665188 + 0.997785i \(0.521189\pi\)
\(42\) 0 0
\(43\) −1.25306e7 −0.558938 −0.279469 − 0.960155i \(-0.590158\pi\)
−0.279469 + 0.960155i \(0.590158\pi\)
\(44\) 2.51213e7 1.01043
\(45\) 0 0
\(46\) 1.00014e7 0.329346
\(47\) 5.54509e7 1.65756 0.828779 − 0.559577i \(-0.189036\pi\)
0.828779 + 0.559577i \(0.189036\pi\)
\(48\) 0 0
\(49\) 5.76480e6 0.142857
\(50\) 2.08352e7 0.471447
\(51\) 0 0
\(52\) 6.47772e7 1.22859
\(53\) 9.26889e7 1.61356 0.806782 − 0.590849i \(-0.201207\pi\)
0.806782 + 0.590849i \(0.201207\pi\)
\(54\) 0 0
\(55\) 1.28228e7 0.188952
\(56\) −2.36775e7 −0.321729
\(57\) 0 0
\(58\) −4.01542e7 −0.465913
\(59\) 2.52600e7 0.271393 0.135696 − 0.990750i \(-0.456673\pi\)
0.135696 + 0.990750i \(0.456673\pi\)
\(60\) 0 0
\(61\) 6.93275e7 0.641093 0.320547 − 0.947233i \(-0.396133\pi\)
0.320547 + 0.947233i \(0.396133\pi\)
\(62\) −3.78653e7 −0.325446
\(63\) 0 0
\(64\) 1.80290e7 0.134326
\(65\) 3.30646e7 0.229748
\(66\) 0 0
\(67\) −2.33494e7 −0.141559 −0.0707796 − 0.997492i \(-0.522549\pi\)
−0.0707796 + 0.997492i \(0.522549\pi\)
\(68\) −1.42752e8 −0.809643
\(69\) 0 0
\(70\) −5.25101e6 −0.0261398
\(71\) 1.06194e8 0.495950 0.247975 − 0.968766i \(-0.420235\pi\)
0.247975 + 0.968766i \(0.420235\pi\)
\(72\) 0 0
\(73\) −2.10115e8 −0.865974 −0.432987 − 0.901400i \(-0.642540\pi\)
−0.432987 + 0.901400i \(0.642540\pi\)
\(74\) −2.04940e8 −0.794483
\(75\) 0 0
\(76\) 1.71699e8 0.590344
\(77\) 1.53338e8 0.497098
\(78\) 0 0
\(79\) −149606. −0.000432144 0 −0.000216072 − 1.00000i \(-0.500069\pi\)
−0.000216072 1.00000i \(0.500069\pi\)
\(80\) −1.88698e7 −0.0515066
\(81\) 0 0
\(82\) 2.62197e7 0.0640420
\(83\) −5.21565e8 −1.20630 −0.603152 − 0.797626i \(-0.706089\pi\)
−0.603152 + 0.797626i \(0.706089\pi\)
\(84\) 0 0
\(85\) −7.28659e7 −0.151405
\(86\) 1.36489e8 0.269063
\(87\) 0 0
\(88\) −6.29799e8 −1.11952
\(89\) −2.98587e8 −0.504448 −0.252224 − 0.967669i \(-0.581162\pi\)
−0.252224 + 0.967669i \(0.581162\pi\)
\(90\) 0 0
\(91\) 3.95394e8 0.604426
\(92\) 3.61178e8 0.525625
\(93\) 0 0
\(94\) −6.03996e8 −0.797919
\(95\) 8.76410e7 0.110395
\(96\) 0 0
\(97\) −8.95983e8 −1.02761 −0.513803 − 0.857908i \(-0.671764\pi\)
−0.513803 + 0.857908i \(0.671764\pi\)
\(98\) −6.27928e7 −0.0687689
\(99\) 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.1.1 2
3.2 odd 2 7.10.a.a.1.2 ✓ 2
12.11 even 2 112.10.a.e.1.2 2
15.2 even 4 175.10.b.b.99.3 4
15.8 even 4 175.10.b.b.99.2 4
15.14 odd 2 175.10.a.b.1.1 2
21.2 odd 6 49.10.c.c.18.1 4
21.5 even 6 49.10.c.b.18.1 4
21.11 odd 6 49.10.c.c.30.1 4
21.17 even 6 49.10.c.b.30.1 4
21.20 even 2 49.10.a.b.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.10.a.a.1.2 ✓ 2 3.2 odd 2
49.10.a.b.1.2 2 21.20 even 2
49.10.c.b.18.1 4 21.5 even 6
49.10.c.b.30.1 4 21.17 even 6
49.10.c.c.18.1 4 21.2 odd 6
49.10.c.c.30.1 4 21.11 odd 6
63.10.a.d.1.1 2 1.1 even 1 trivial
112.10.a.e.1.2 2 12.11 even 2
175.10.a.b.1.1 2 15.14 odd 2
175.10.b.b.99.2 4 15.8 even 4
175.10.b.b.99.3 4 15.2 even 4