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 = [4,0] 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: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 479x^{2} + 46396 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(11.6099\) 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)
 
\(f(q)\) \(=\) \(q+23.2198 q^{2} +27.1587 q^{4} +1279.53 q^{5} -2401.00 q^{7} -11257.9 q^{8} +29710.4 q^{10} -50160.8 q^{11} -70449.0 q^{13} -55750.7 q^{14} -275312. q^{16} -26127.5 q^{17} +925085. q^{19} +34750.4 q^{20} -1.16472e6 q^{22} -1.41558e6 q^{23} -315927. q^{25} -1.63581e6 q^{26} -65208.1 q^{28} -5.98102e6 q^{29} -8.14364e6 q^{31} -628628. q^{32} -606675. q^{34} -3.07215e6 q^{35} +3.35651e6 q^{37} +2.14803e7 q^{38} -1.44048e7 q^{40} +1.70063e6 q^{41} -3.72496e7 q^{43} -1.36230e6 q^{44} -3.28695e7 q^{46} +3.99838e7 q^{47} +5.76480e6 q^{49} -7.33576e6 q^{50} -1.91330e6 q^{52} +9.71630e6 q^{53} -6.41822e7 q^{55} +2.70302e7 q^{56} -1.38878e8 q^{58} +1.08146e8 q^{59} -4.86405e7 q^{61} -1.89094e8 q^{62} +1.26363e8 q^{64} -9.01416e7 q^{65} -3.45149e7 q^{67} -709589. q^{68} -7.13347e7 q^{70} -6.49760e7 q^{71} -1.34365e8 q^{73} +7.79374e7 q^{74} +2.51242e7 q^{76} +1.20436e8 q^{77} +4.22502e8 q^{79} -3.52270e8 q^{80} +3.94883e7 q^{82} +1.11555e8 q^{83} -3.34309e7 q^{85} -8.64928e8 q^{86} +5.64706e8 q^{88} -2.57565e8 q^{89} +1.69148e8 q^{91} -3.84453e7 q^{92} +9.28416e8 q^{94} +1.18367e9 q^{95} +2.63076e8 q^{97} +1.33857e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 1784 q^{4} - 9604 q^{7} - 77176 q^{10} - 291848 q^{13} - 464608 q^{16} + 527200 q^{19} + 660392 q^{22} + 2237804 q^{25} - 4283384 q^{28} - 10657440 q^{31} - 42084264 q^{34} - 26229864 q^{37} - 77006688 q^{40}+ \cdots - 1678668264 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 23.2198 1.02618 0.513090 − 0.858335i \(-0.328501\pi\)
0.513090 + 0.858335i \(0.328501\pi\)
\(3\) 0 0
\(4\) 27.1587 0.0530444
\(5\) 1279.53 0.915557 0.457779 − 0.889066i \(-0.348645\pi\)
0.457779 + 0.889066i \(0.348645\pi\)
\(6\) 0 0
\(7\) −2401.00 −0.377964
\(8\) −11257.9 −0.971746
\(9\) 0 0
\(10\) 29710.4 0.939526
\(11\) −50160.8 −1.03299 −0.516496 − 0.856289i \(-0.672764\pi\)
−0.516496 + 0.856289i \(0.672764\pi\)
\(12\) 0 0
\(13\) −70449.0 −0.684116 −0.342058 − 0.939679i \(-0.611124\pi\)
−0.342058 + 0.939679i \(0.611124\pi\)
\(14\) −55750.7 −0.387859
\(15\) 0 0
\(16\) −275312. −1.05023
\(17\) −26127.5 −0.0758713 −0.0379356 − 0.999280i \(-0.512078\pi\)
−0.0379356 + 0.999280i \(0.512078\pi\)
\(18\) 0 0
\(19\) 925085. 1.62851 0.814255 − 0.580507i \(-0.197146\pi\)
0.814255 + 0.580507i \(0.197146\pi\)
\(20\) 34750.4 0.0485652
\(21\) 0 0
\(22\) −1.16472e6 −1.06004
\(23\) −1.41558e6 −1.05477 −0.527386 − 0.849626i \(-0.676828\pi\)
−0.527386 + 0.849626i \(0.676828\pi\)
\(24\) 0 0
\(25\) −315927. −0.161755
\(26\) −1.63581e6 −0.702025
\(27\) 0 0
\(28\) −65208.1 −0.0200489
\(29\) −5.98102e6 −1.57031 −0.785154 − 0.619301i \(-0.787416\pi\)
−0.785154 + 0.619301i \(0.787416\pi\)
\(30\) 0 0
\(31\) −8.14364e6 −1.58377 −0.791883 − 0.610673i \(-0.790899\pi\)
−0.791883 + 0.610673i \(0.790899\pi\)
\(32\) −628628. −0.105979
\(33\) 0 0
\(34\) −606675. −0.0778575
\(35\) −3.07215e6 −0.346048
\(36\) 0 0
\(37\) 3.35651e6 0.294428 0.147214 − 0.989105i \(-0.452969\pi\)
0.147214 + 0.989105i \(0.452969\pi\)
\(38\) 2.14803e7 1.67114
\(39\) 0 0
\(40\) −1.44048e7 −0.889690
\(41\) 1.70063e6 0.0939901 0.0469951 − 0.998895i \(-0.485035\pi\)
0.0469951 + 0.998895i \(0.485035\pi\)
\(42\) 0 0
\(43\) −3.72496e7 −1.66155 −0.830775 − 0.556608i \(-0.812103\pi\)
−0.830775 + 0.556608i \(0.812103\pi\)
\(44\) −1.36230e6 −0.0547945
\(45\) 0 0
\(46\) −3.28695e7 −1.08239
\(47\) 3.99838e7 1.19521 0.597605 − 0.801791i \(-0.296119\pi\)
0.597605 + 0.801791i \(0.296119\pi\)
\(48\) 0 0
\(49\) 5.76480e6 0.142857
\(50\) −7.33576e6 −0.165989
\(51\) 0 0
\(52\) −1.91330e6 −0.0362885
\(53\) 9.71630e6 0.169145 0.0845725 − 0.996417i \(-0.473048\pi\)
0.0845725 + 0.996417i \(0.473048\pi\)
\(54\) 0 0
\(55\) −6.41822e7 −0.945764
\(56\) 2.70302e7 0.367286
\(57\) 0 0
\(58\) −1.38878e8 −1.61142
\(59\) 1.08146e8 1.16192 0.580959 − 0.813933i \(-0.302678\pi\)
0.580959 + 0.813933i \(0.302678\pi\)
\(60\) 0 0
\(61\) −4.86405e7 −0.449794 −0.224897 − 0.974383i \(-0.572205\pi\)
−0.224897 + 0.974383i \(0.572205\pi\)
\(62\) −1.89094e8 −1.62523
\(63\) 0 0
\(64\) 1.26363e8 0.941477
\(65\) −9.01416e7 −0.626347
\(66\) 0 0
\(67\) −3.45149e7 −0.209252 −0.104626 − 0.994512i \(-0.533365\pi\)
−0.104626 + 0.994512i \(0.533365\pi\)
\(68\) −709589. −0.00402455
\(69\) 0 0
\(70\) −7.13347e7 −0.355108
\(71\) −6.49760e7 −0.303452 −0.151726 − 0.988423i \(-0.548483\pi\)
−0.151726 + 0.988423i \(0.548483\pi\)
\(72\) 0 0
\(73\) −1.34365e8 −0.553775 −0.276888 − 0.960902i \(-0.589303\pi\)
−0.276888 + 0.960902i \(0.589303\pi\)
\(74\) 7.79374e7 0.302136
\(75\) 0 0
\(76\) 2.51242e7 0.0863834
\(77\) 1.20436e8 0.390435
\(78\) 0 0
\(79\) 4.22502e8 1.22041 0.610207 − 0.792242i \(-0.291086\pi\)
0.610207 + 0.792242i \(0.291086\pi\)
\(80\) −3.52270e8 −0.961546
\(81\) 0 0
\(82\) 3.94883e7 0.0964508
\(83\) 1.11555e8 0.258011 0.129006 − 0.991644i \(-0.458821\pi\)
0.129006 + 0.991644i \(0.458821\pi\)
\(84\) 0 0
\(85\) −3.34309e7 −0.0694645
\(86\) −8.64928e8 −1.70505
\(87\) 0 0
\(88\) 5.64706e8 1.00381
\(89\) −2.57565e8 −0.435143 −0.217572 − 0.976044i \(-0.569814\pi\)
−0.217572 + 0.976044i \(0.569814\pi\)
\(90\) 0 0
\(91\) 1.69148e8 0.258571
\(92\) −3.84453e7 −0.0559498
\(93\) 0 0
\(94\) 9.28416e8 1.22650
\(95\) 1.18367e9 1.49099
\(96\) 0 0
\(97\) 2.63076e8 0.301723 0.150862 − 0.988555i \(-0.451795\pi\)
0.150862 + 0.988555i \(0.451795\pi\)
\(98\) 1.33857e8 0.146597
\(99\) 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.g.1.3 yes 4
3.2 odd 2 inner 63.10.a.g.1.2 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.10.a.g.1.2 ✓ 4 3.2 odd 2 inner
63.10.a.g.1.3 yes 4 1.1 even 1 trivial