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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [32,9,Mod(31,32)] 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("32.31"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(32, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 9, names="a")
 
Level: \( N \) \(=\) \( 32 = 2^{5} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 32.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,-728] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.0361155220\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{39})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 19x^{2} + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{18} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 31.1
Root \(-3.12250 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 32.31
Dual form 32.9.c.a.31.4

$q$-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)
 
\(f(q)\) \(=\) \(q-135.920i q^{3} -581.680 q^{5} -2670.24i q^{7} -11913.2 q^{9} +14148.7i q^{11} +42805.2 q^{13} +79061.9i q^{15} -140364. q^{17} +129717. i q^{19} -362939. q^{21} -196909. i q^{23} -52273.5 q^{25} +727476. i q^{27} -355830. q^{29} +3357.86i q^{31} +1.92309e6 q^{33} +1.55322e6i q^{35} +907325. q^{37} -5.81808e6i q^{39} -3.06821e6 q^{41} -5.01099e6i q^{43} +6.92969e6 q^{45} -3.25359e6i q^{47} -1.36538e6 q^{49} +1.90783e7i q^{51} +234649. q^{53} -8.23002e6i q^{55} +1.76312e7 q^{57} -7.78979e6i q^{59} -2.42160e7 q^{61} +3.18112e7i q^{63} -2.48989e7 q^{65} -6.87944e6i q^{67} -2.67639e7 q^{69} -5.76593e6i q^{71} -1.19654e7 q^{73} +7.10502e6i q^{75} +3.77805e7 q^{77} +3.55920e7i q^{79} +2.07158e7 q^{81} -2.03804e7i q^{83} +8.16470e7 q^{85} +4.83644e7i q^{87} -1.19259e7 q^{89} -1.14300e8i q^{91} +456400. q^{93} -7.54539e7i q^{95} -3.19199e7 q^{97} -1.68557e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 728 q^{5} - 18876 q^{9} - 12632 q^{13} - 391992 q^{17} - 946560 q^{21} - 791028 q^{25} - 705496 q^{29} + 3084864 q^{33} + 4443048 q^{37} + 2953352 q^{41} + 14937000 q^{45} + 2839044 q^{49} - 4501848 q^{53}+ \cdots + 56444872 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/32\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(31\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 135.920i − 1.67802i −0.544113 − 0.839012i \(-0.683134\pi\)
0.544113 − 0.839012i \(-0.316866\pi\)
\(4\) 0 0
\(5\) −581.680 −0.930688 −0.465344 − 0.885130i \(-0.654069\pi\)
−0.465344 + 0.885130i \(0.654069\pi\)
\(6\) 0 0
\(7\) − 2670.24i − 1.11214i −0.831137 − 0.556068i \(-0.812309\pi\)
0.831137 − 0.556068i \(-0.187691\pi\)
\(8\) 0 0
\(9\) −11913.2 −1.81577
\(10\) 0 0
\(11\) 14148.7i 0.966377i 0.875516 + 0.483188i \(0.160521\pi\)
−0.875516 + 0.483188i \(0.839479\pi\)
\(12\) 0 0
\(13\) 42805.2 1.49873 0.749364 − 0.662158i \(-0.230359\pi\)
0.749364 + 0.662158i \(0.230359\pi\)
\(14\) 0 0
\(15\) 79061.9i 1.56172i
\(16\) 0 0
\(17\) −140364. −1.68058 −0.840292 − 0.542134i \(-0.817617\pi\)
−0.840292 + 0.542134i \(0.817617\pi\)
\(18\) 0 0
\(19\) 129717.i 0.995367i 0.867359 + 0.497683i \(0.165816\pi\)
−0.867359 + 0.497683i \(0.834184\pi\)
\(20\) 0 0
\(21\) −362939. −1.86619
\(22\) 0 0
\(23\) − 196909.i − 0.703647i −0.936066 − 0.351823i \(-0.885562\pi\)
0.936066 − 0.351823i \(-0.114438\pi\)
\(24\) 0 0
\(25\) −52273.5 −0.133820
\(26\) 0 0
\(27\) 727476.i 1.36887i
\(28\) 0 0
\(29\) −355830. −0.503096 −0.251548 − 0.967845i \(-0.580940\pi\)
−0.251548 + 0.967845i \(0.580940\pi\)
\(30\) 0 0
\(31\) 3357.86i 0.00363593i 0.999998 + 0.00181797i \(0.000578677\pi\)
−0.999998 + 0.00181797i \(0.999421\pi\)
\(32\) 0 0
\(33\) 1.92309e6 1.62160
\(34\) 0 0
\(35\) 1.55322e6i 1.03505i
\(36\) 0 0
\(37\) 907325. 0.484123 0.242062 − 0.970261i \(-0.422176\pi\)
0.242062 + 0.970261i \(0.422176\pi\)
\(38\) 0 0
\(39\) − 5.81808e6i − 2.51490i
\(40\) 0 0
\(41\) −3.06821e6 −1.08580 −0.542900 − 0.839797i \(-0.682674\pi\)
−0.542900 + 0.839797i \(0.682674\pi\)
\(42\) 0 0
\(43\) − 5.01099e6i − 1.46572i −0.680382 − 0.732858i \(-0.738186\pi\)
0.680382 − 0.732858i \(-0.261814\pi\)
\(44\) 0 0
\(45\) 6.92969e6 1.68991
\(46\) 0 0
\(47\) − 3.25359e6i − 0.666764i −0.942792 − 0.333382i \(-0.891810\pi\)
0.942792 − 0.333382i \(-0.108190\pi\)
\(48\) 0 0
\(49\) −1.36538e6 −0.236847
\(50\) 0 0
\(51\) 1.90783e7i 2.82006i
\(52\) 0 0
\(53\) 234649. 0.0297382 0.0148691 − 0.999889i \(-0.495267\pi\)
0.0148691 + 0.999889i \(0.495267\pi\)
\(54\) 0 0
\(55\) − 8.23002e6i − 0.899395i
\(56\) 0 0
\(57\) 1.76312e7 1.67025
\(58\) 0 0
\(59\) − 7.78979e6i − 0.642862i −0.946933 − 0.321431i \(-0.895836\pi\)
0.946933 − 0.321431i \(-0.104164\pi\)
\(60\) 0 0
\(61\) −2.42160e7 −1.74897 −0.874487 − 0.485049i \(-0.838802\pi\)
−0.874487 + 0.485049i \(0.838802\pi\)
\(62\) 0 0
\(63\) 3.18112e7i 2.01938i
\(64\) 0 0
\(65\) −2.48989e7 −1.39485
\(66\) 0 0
\(67\) − 6.87944e6i − 0.341392i −0.985324 − 0.170696i \(-0.945398\pi\)
0.985324 − 0.170696i \(-0.0546017\pi\)
\(68\) 0 0
\(69\) −2.67639e7 −1.18074
\(70\) 0 0
\(71\) − 5.76593e6i − 0.226901i −0.993544 − 0.113450i \(-0.963810\pi\)
0.993544 − 0.113450i \(-0.0361903\pi\)
\(72\) 0 0
\(73\) −1.19654e7 −0.421343 −0.210672 − 0.977557i \(-0.567565\pi\)
−0.210672 + 0.977557i \(0.567565\pi\)
\(74\) 0 0
\(75\) 7.10502e6i 0.224554i
\(76\) 0 0
\(77\) 3.77805e7 1.07474
\(78\) 0 0
\(79\) 3.55920e7i 0.913784i 0.889522 + 0.456892i \(0.151037\pi\)
−0.889522 + 0.456892i \(0.848963\pi\)
\(80\) 0 0
\(81\) 2.07158e7 0.481239
\(82\) 0 0
\(83\) − 2.03804e7i − 0.429438i −0.976676 − 0.214719i \(-0.931116\pi\)
0.976676 − 0.214719i \(-0.0688836\pi\)
\(84\) 0 0
\(85\) 8.16470e7 1.56410
\(86\) 0 0
\(87\) 4.83644e7i 0.844207i
\(88\) 0 0
\(89\) −1.19259e7 −0.190078 −0.0950390 − 0.995474i \(-0.530298\pi\)
−0.0950390 + 0.995474i \(0.530298\pi\)
\(90\) 0 0
\(91\) − 1.14300e8i − 1.66679i
\(92\) 0 0
\(93\) 456400. 0.00610119
\(94\) 0 0
\(95\) − 7.54539e7i − 0.926376i
\(96\) 0 0
\(97\) −3.19199e7 −0.360558 −0.180279 − 0.983616i \(-0.557700\pi\)
−0.180279 + 0.983616i \(0.557700\pi\)
\(98\) 0 0
\(99\) − 1.68557e8i − 1.75471i
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 32.9.c.a.31.1 ✓ 4
3.2 odd 2 288.9.g.b.127.3 4
4.3 odd 2 inner 32.9.c.a.31.4 yes 4
8.3 odd 2 64.9.c.f.63.1 4
8.5 even 2 64.9.c.f.63.4 4
12.11 even 2 288.9.g.b.127.4 4
16.3 odd 4 256.9.d.b.127.2 4
16.5 even 4 256.9.d.b.127.1 4
16.11 odd 4 256.9.d.h.127.3 4
16.13 even 4 256.9.d.h.127.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.9.c.a.31.1 ✓ 4 1.1 even 1 trivial
32.9.c.a.31.4 yes 4 4.3 odd 2 inner
64.9.c.f.63.1 4 8.3 odd 2
64.9.c.f.63.4 4 8.5 even 2
256.9.d.b.127.1 4 16.5 even 4
256.9.d.b.127.2 4 16.3 odd 4
256.9.d.h.127.3 4 16.11 odd 4
256.9.d.h.127.4 4 16.13 even 4
288.9.g.b.127.3 4 3.2 odd 2
288.9.g.b.127.4 4 12.11 even 2