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 = [1,24] 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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
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)
 
\(f(q)\) \(=\) \(q+24.0000 q^{2} +64.0000 q^{4} +144.000 q^{5} +2401.00 q^{7} -10752.0 q^{8} +3456.00 q^{10} +15030.0 q^{11} -151486. q^{13} +57624.0 q^{14} -290816. q^{16} +350448. q^{17} -691108. q^{19} +9216.00 q^{20} +360720. q^{22} -892458. q^{23} -1.93239e6 q^{25} -3.63566e6 q^{26} +153664. q^{28} -1.64852e6 q^{29} -3.73430e6 q^{31} -1.47456e6 q^{32} +8.41075e6 q^{34} +345744. q^{35} -1.14719e7 q^{37} -1.65866e7 q^{38} -1.54829e6 q^{40} -1.39857e7 q^{41} +1.67945e7 q^{43} +961920. q^{44} -2.14190e7 q^{46} +1.40121e7 q^{47} +5.76480e6 q^{49} -4.63773e7 q^{50} -9.69510e6 q^{52} +9.74399e7 q^{53} +2.16432e6 q^{55} -2.58156e7 q^{56} -3.95644e7 q^{58} -1.10798e8 q^{59} -9.38167e7 q^{61} -8.96231e7 q^{62} +1.13508e8 q^{64} -2.18140e7 q^{65} -1.22446e8 q^{67} +2.24287e7 q^{68} +8.29786e6 q^{70} -2.06197e8 q^{71} +2.50338e8 q^{73} -2.75326e8 q^{74} -4.42309e7 q^{76} +3.60870e7 q^{77} -3.83149e7 q^{79} -4.18775e7 q^{80} -3.35657e8 q^{82} +5.14087e8 q^{83} +5.04645e7 q^{85} +4.03069e8 q^{86} -1.61603e8 q^{88} +1.06129e9 q^{89} -3.63718e8 q^{91} -5.71173e7 q^{92} +3.36289e8 q^{94} -9.95196e7 q^{95} -7.38416e7 q^{97} +1.38355e8 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 24.0000 1.06066 0.530330 − 0.847791i \(-0.322068\pi\)
0.530330 + 0.847791i \(0.322068\pi\)
\(3\) 0 0
\(4\) 64.0000 0.125000
\(5\) 144.000 0.103038 0.0515190 − 0.998672i \(-0.483594\pi\)
0.0515190 + 0.998672i \(0.483594\pi\)
\(6\) 0 0
\(7\) 2401.00 0.377964
\(8\) −10752.0 −0.928078
\(9\) 0 0
\(10\) 3456.00 0.109288
\(11\) 15030.0 0.309522 0.154761 − 0.987952i \(-0.450539\pi\)
0.154761 + 0.987952i \(0.450539\pi\)
\(12\) 0 0
\(13\) −151486. −1.47105 −0.735525 − 0.677498i \(-0.763064\pi\)
−0.735525 + 0.677498i \(0.763064\pi\)
\(14\) 57624.0 0.400892
\(15\) 0 0
\(16\) −290816. −1.10938
\(17\) 350448. 1.01766 0.508831 − 0.860867i \(-0.330078\pi\)
0.508831 + 0.860867i \(0.330078\pi\)
\(18\) 0 0
\(19\) −691108. −1.21662 −0.608310 − 0.793700i \(-0.708152\pi\)
−0.608310 + 0.793700i \(0.708152\pi\)
\(20\) 9216.00 0.0128798
\(21\) 0 0
\(22\) 360720. 0.328298
\(23\) −892458. −0.664986 −0.332493 − 0.943106i \(-0.607890\pi\)
−0.332493 + 0.943106i \(0.607890\pi\)
\(24\) 0 0
\(25\) −1.93239e6 −0.989383
\(26\) −3.63566e6 −1.56028
\(27\) 0 0
\(28\) 153664. 0.0472456
\(29\) −1.64852e6 −0.432815 −0.216408 − 0.976303i \(-0.569434\pi\)
−0.216408 + 0.976303i \(0.569434\pi\)
\(30\) 0 0
\(31\) −3.73430e6 −0.726242 −0.363121 − 0.931742i \(-0.618289\pi\)
−0.363121 + 0.931742i \(0.618289\pi\)
\(32\) −1.47456e6 −0.248592
\(33\) 0 0
\(34\) 8.41075e6 1.07939
\(35\) 345744. 0.0389447
\(36\) 0 0
\(37\) −1.14719e7 −1.00630 −0.503150 − 0.864199i \(-0.667826\pi\)
−0.503150 + 0.864199i \(0.667826\pi\)
\(38\) −1.65866e7 −1.29042
\(39\) 0 0
\(40\) −1.54829e6 −0.0956273
\(41\) −1.39857e7 −0.772961 −0.386481 − 0.922298i \(-0.626309\pi\)
−0.386481 + 0.922298i \(0.626309\pi\)
\(42\) 0 0
\(43\) 1.67945e7 0.749134 0.374567 − 0.927200i \(-0.377791\pi\)
0.374567 + 0.927200i \(0.377791\pi\)
\(44\) 961920. 0.0386903
\(45\) 0 0
\(46\) −2.14190e7 −0.705324
\(47\) 1.40121e7 0.418853 0.209426 − 0.977824i \(-0.432840\pi\)
0.209426 + 0.977824i \(0.432840\pi\)
\(48\) 0 0
\(49\) 5.76480e6 0.142857
\(50\) −4.63773e7 −1.04940
\(51\) 0 0
\(52\) −9.69510e6 −0.183881
\(53\) 9.74399e7 1.69627 0.848136 − 0.529779i \(-0.177725\pi\)
0.848136 + 0.529779i \(0.177725\pi\)
\(54\) 0 0
\(55\) 2.16432e6 0.0318926
\(56\) −2.58156e7 −0.350780
\(57\) 0 0
\(58\) −3.95644e7 −0.459070
\(59\) −1.10798e8 −1.19042 −0.595208 − 0.803571i \(-0.702931\pi\)
−0.595208 + 0.803571i \(0.702931\pi\)
\(60\) 0 0
\(61\) −9.38167e7 −0.867553 −0.433776 − 0.901021i \(-0.642819\pi\)
−0.433776 + 0.901021i \(0.642819\pi\)
\(62\) −8.96231e7 −0.770296
\(63\) 0 0
\(64\) 1.13508e8 0.845703
\(65\) −2.18140e7 −0.151574
\(66\) 0 0
\(67\) −1.22446e8 −0.742352 −0.371176 − 0.928563i \(-0.621045\pi\)
−0.371176 + 0.928563i \(0.621045\pi\)
\(68\) 2.24287e7 0.127208
\(69\) 0 0
\(70\) 8.29786e6 0.0413071
\(71\) −2.06197e8 −0.962987 −0.481494 − 0.876450i \(-0.659906\pi\)
−0.481494 + 0.876450i \(0.659906\pi\)
\(72\) 0 0
\(73\) 2.50338e8 1.03175 0.515873 − 0.856665i \(-0.327467\pi\)
0.515873 + 0.856665i \(0.327467\pi\)
\(74\) −2.75326e8 −1.06734
\(75\) 0 0
\(76\) −4.42309e7 −0.152077
\(77\) 3.60870e7 0.116988
\(78\) 0 0
\(79\) −3.83149e7 −0.110674 −0.0553370 − 0.998468i \(-0.517623\pi\)
−0.0553370 + 0.998468i \(0.517623\pi\)
\(80\) −4.18775e7 −0.114308
\(81\) 0 0
\(82\) −3.35657e8 −0.819849
\(83\) 5.14087e8 1.18901 0.594504 − 0.804092i \(-0.297348\pi\)
0.594504 + 0.804092i \(0.297348\pi\)
\(84\) 0 0
\(85\) 5.04645e7 0.104858
\(86\) 4.03069e8 0.794577
\(87\) 0 0
\(88\) −1.61603e8 −0.287261
\(89\) 1.06129e9 1.79300 0.896502 − 0.443041i \(-0.146100\pi\)
0.896502 + 0.443041i \(0.146100\pi\)
\(90\) 0 0
\(91\) −3.63718e8 −0.556005
\(92\) −5.71173e7 −0.0831233
\(93\) 0 0
\(94\) 3.36289e8 0.444260
\(95\) −9.95196e7 −0.125358
\(96\) 0 0
\(97\) −7.38416e7 −0.0846892 −0.0423446 − 0.999103i \(-0.513483\pi\)
−0.0423446 + 0.999103i \(0.513483\pi\)
\(98\) 1.38355e8 0.151523
\(99\) 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.a.1.1 1
3.2 odd 2 21.10.a.a.1.1 ✓ 1
12.11 even 2 336.10.a.d.1.1 1
21.20 even 2 147.10.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.10.a.a.1.1 ✓ 1 3.2 odd 2
63.10.a.a.1.1 1 1.1 even 1 trivial
147.10.a.b.1.1 1 21.20 even 2
336.10.a.d.1.1 1 12.11 even 2