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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [864,2,Mod(109,864)] 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("864.109"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(864, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 7, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 864 = 2^{5} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 864.v (of order \(8\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [128,0,0,0,0,0,0,0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.89907473464\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(32\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 109.2
Character \(\chi\) \(=\) 864.109
Dual form 864.2.v.b.325.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.39831 + 0.211463i) q^{2} +(1.91057 - 0.591382i) q^{4} +(-1.77214 + 0.734044i) q^{5} +(-0.0355868 + 0.0355868i) q^{7} +(-2.54652 + 1.23095i) q^{8} +(2.32278 - 1.40116i) q^{10} +(1.04798 + 2.53004i) q^{11} +(1.82513 + 0.755995i) q^{13} +(0.0422362 - 0.0572867i) q^{14} +(3.30053 - 2.25975i) q^{16} -2.92843i q^{17} +(-1.55044 - 0.642214i) q^{19} +(-2.95169 + 2.45045i) q^{20} +(-2.00041 - 3.31619i) q^{22} +(-0.146574 - 0.146574i) q^{23} +(-0.933880 + 0.933880i) q^{25} +(-2.71198 - 0.671172i) q^{26} +(-0.0469455 + 0.0890363i) q^{28} +(-2.17892 + 5.26037i) q^{29} -4.28852 q^{31} +(-4.13733 + 3.85778i) q^{32} +(0.619253 + 4.09486i) q^{34} +(0.0369424 - 0.0891869i) q^{35} +(-1.92396 + 0.796930i) q^{37} +(2.30381 + 0.570157i) q^{38} +(3.60921 - 4.05067i) q^{40} +(-4.22763 - 4.22763i) q^{41} +(1.29855 + 3.13498i) q^{43} +(3.49846 + 4.21406i) q^{44} +(0.235952 + 0.173962i) q^{46} +5.33727i q^{47} +6.99747i q^{49} +(1.10838 - 1.50334i) q^{50} +(3.93412 + 0.365028i) q^{52} +(-1.48773 - 3.59170i) q^{53} +(-3.71433 - 3.71433i) q^{55} +(0.0468168 - 0.134428i) q^{56} +(1.93444 - 7.81641i) q^{58} +(-1.27834 + 0.529507i) q^{59} +(-5.58246 + 13.4772i) q^{61} +(5.99670 - 0.906861i) q^{62} +(4.96952 - 6.26928i) q^{64} -3.78932 q^{65} +(1.11670 - 2.69596i) q^{67} +(-1.73182 - 5.59496i) q^{68} +(-0.0327974 + 0.132523i) q^{70} +(-3.73700 + 3.73700i) q^{71} +(-10.8405 - 10.8405i) q^{73} +(2.52178 - 1.52120i) q^{74} +(-3.34202 - 0.310089i) q^{76} +(-0.127330 - 0.0527419i) q^{77} +9.94610i q^{79} +(-4.19025 + 6.42733i) q^{80} +(6.80554 + 5.01757i) q^{82} +(11.2462 + 4.65832i) q^{83} +(2.14959 + 5.18958i) q^{85} +(-2.47871 - 4.10909i) q^{86} +(-5.78306 - 5.15279i) q^{88} +(-6.00227 + 6.00227i) q^{89} +(-0.0918540 + 0.0380472i) q^{91} +(-0.366721 - 0.193358i) q^{92} +(-1.12863 - 7.46318i) q^{94} +3.21901 q^{95} -3.69111 q^{97} +(-1.47970 - 9.78466i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 128 q + 16 q^{10} - 32 q^{16} - 16 q^{22} - 32 q^{40} - 32 q^{46} - 80 q^{52} + 32 q^{55} - 32 q^{58} + 64 q^{61} + 48 q^{64} + 64 q^{67} - 96 q^{70} + 32 q^{76} - 80 q^{82} - 80 q^{88} + 96 q^{91} - 48 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/864\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(353\) \(703\)
\(\chi(n)\) \(e\left(\frac{7}{8}\right)\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.39831 + 0.211463i −0.988758 + 0.149527i
\(3\) 0 0
\(4\) 1.91057 − 0.591382i 0.955284 − 0.295691i
\(5\) −1.77214 + 0.734044i −0.792524 + 0.328274i −0.741958 − 0.670446i \(-0.766103\pi\)
−0.0505664 + 0.998721i \(0.516103\pi\)
\(6\) 0 0
\(7\) −0.0355868 + 0.0355868i −0.0134505 + 0.0134505i −0.713800 − 0.700350i \(-0.753027\pi\)
0.700350 + 0.713800i \(0.253027\pi\)
\(8\) −2.54652 + 1.23095i −0.900330 + 0.435207i
\(9\) 0 0
\(10\) 2.32278 − 1.40116i 0.734529 − 0.443087i
\(11\) 1.04798 + 2.53004i 0.315977 + 0.762837i 0.999460 + 0.0328698i \(0.0104647\pi\)
−0.683482 + 0.729967i \(0.739535\pi\)
\(12\) 0 0
\(13\) 1.82513 + 0.755995i 0.506201 + 0.209675i 0.621144 − 0.783697i \(-0.286668\pi\)
−0.114942 + 0.993372i \(0.536668\pi\)
\(14\) 0.0422362 − 0.0572867i 0.0112881 − 0.0153105i
\(15\) 0 0
\(16\) 3.30053 − 2.25975i 0.825134 − 0.564938i
\(17\) − 2.92843i − 0.710248i −0.934819 − 0.355124i \(-0.884439\pi\)
0.934819 − 0.355124i \(-0.115561\pi\)
\(18\) 0 0
\(19\) −1.55044 − 0.642214i −0.355696 − 0.147334i 0.197678 − 0.980267i \(-0.436660\pi\)
−0.553374 + 0.832933i \(0.686660\pi\)
\(20\) −2.95169 + 2.45045i −0.660018 + 0.547937i
\(21\) 0 0
\(22\) −2.00041 − 3.31619i −0.426490 − 0.707014i
\(23\) −0.146574 − 0.146574i −0.0305628 − 0.0305628i 0.691660 − 0.722223i \(-0.256880\pi\)
−0.722223 + 0.691660i \(0.756880\pi\)
\(24\) 0 0
\(25\) −0.933880 + 0.933880i −0.186776 + 0.186776i
\(26\) −2.71198 − 0.671172i −0.531862 − 0.131628i
\(27\) 0 0
\(28\) −0.0469455 + 0.0890363i −0.00887187 + 0.0168263i
\(29\) −2.17892 + 5.26037i −0.404614 + 0.976826i 0.581916 + 0.813249i \(0.302303\pi\)
−0.986531 + 0.163577i \(0.947697\pi\)
\(30\) 0 0
\(31\) −4.28852 −0.770241 −0.385120 − 0.922866i \(-0.625840\pi\)
−0.385120 + 0.922866i \(0.625840\pi\)
\(32\) −4.13733 + 3.85778i −0.731384 + 0.681966i
\(33\) 0 0
\(34\) 0.619253 + 4.09486i 0.106201 + 0.702263i
\(35\) 0.0369424 − 0.0891869i 0.00624441 − 0.0150753i
\(36\) 0 0
\(37\) −1.92396 + 0.796930i −0.316297 + 0.131014i −0.535183 − 0.844736i \(-0.679758\pi\)
0.218887 + 0.975750i \(0.429758\pi\)
\(38\) 2.30381 + 0.570157i 0.373727 + 0.0924916i
\(39\) 0 0
\(40\) 3.60921 − 4.05067i 0.570666 − 0.640468i
\(41\) −4.22763 − 4.22763i −0.660244 − 0.660244i 0.295193 − 0.955438i \(-0.404616\pi\)
−0.955438 + 0.295193i \(0.904616\pi\)
\(42\) 0 0
\(43\) 1.29855 + 3.13498i 0.198027 + 0.478080i 0.991433 − 0.130613i \(-0.0416944\pi\)
−0.793406 + 0.608692i \(0.791694\pi\)
\(44\) 3.49846 + 4.21406i 0.527412 + 0.635294i
\(45\) 0 0
\(46\) 0.235952 + 0.173962i 0.0347892 + 0.0256493i
\(47\) 5.33727i 0.778520i 0.921128 + 0.389260i \(0.127269\pi\)
−0.921128 + 0.389260i \(0.872731\pi\)
\(48\) 0 0
\(49\) 6.99747i 0.999638i
\(50\) 1.10838 − 1.50334i 0.156748 − 0.212604i
\(51\) 0 0
\(52\) 3.93412 + 0.365028i 0.545565 + 0.0506203i
\(53\) −1.48773 − 3.59170i −0.204355 − 0.493357i 0.788161 − 0.615469i \(-0.211033\pi\)
−0.992516 + 0.122112i \(0.961033\pi\)
\(54\) 0 0
\(55\) −3.71433 − 3.71433i −0.500840 − 0.500840i
\(56\) 0.0468168 − 0.134428i 0.00625615 − 0.0179637i
\(57\) 0 0
\(58\) 1.93444 − 7.81641i 0.254004 − 1.02634i
\(59\) −1.27834 + 0.529507i −0.166426 + 0.0689359i −0.464341 − 0.885657i \(-0.653709\pi\)
0.297915 + 0.954592i \(0.403709\pi\)
\(60\) 0 0
\(61\) −5.58246 + 13.4772i −0.714761 + 1.72558i −0.0270124 + 0.999635i \(0.508599\pi\)
−0.687748 + 0.725949i \(0.741401\pi\)
\(62\) 5.99670 − 0.906861i 0.761581 − 0.115171i
\(63\) 0 0
\(64\) 4.96952 − 6.26928i 0.621190 − 0.783660i
\(65\) −3.78932 −0.470008
\(66\) 0 0
\(67\) 1.11670 − 2.69596i 0.136427 − 0.329364i −0.840870 − 0.541237i \(-0.817956\pi\)
0.977297 + 0.211873i \(0.0679562\pi\)
\(68\) −1.73182 − 5.59496i −0.210014 − 0.678488i
\(69\) 0 0
\(70\) −0.0327974 + 0.132523i −0.00392004 + 0.0158396i
\(71\) −3.73700 + 3.73700i −0.443501 + 0.443501i −0.893187 − 0.449686i \(-0.851536\pi\)
0.449686 + 0.893187i \(0.351536\pi\)
\(72\) 0 0
\(73\) −10.8405 − 10.8405i −1.26878 − 1.26878i −0.946718 − 0.322065i \(-0.895623\pi\)
−0.322065 − 0.946718i \(-0.604377\pi\)
\(74\) 2.52178 − 1.52120i 0.293151 − 0.176836i
\(75\) 0 0
\(76\) −3.34202 − 0.310089i −0.383355 − 0.0355697i
\(77\) −0.127330 − 0.0527419i −0.0145106 − 0.00601050i
\(78\) 0 0
\(79\) 9.94610i 1.11902i 0.828822 + 0.559512i \(0.189011\pi\)
−0.828822 + 0.559512i \(0.810989\pi\)
\(80\) −4.19025 + 6.42733i −0.468484 + 0.718597i
\(81\) 0 0
\(82\) 6.80554 + 5.01757i 0.751546 + 0.554098i
\(83\) 11.2462 + 4.65832i 1.23443 + 0.511318i 0.901970 − 0.431800i \(-0.142121\pi\)
0.332460 + 0.943117i \(0.392121\pi\)
\(84\) 0 0
\(85\) 2.14959 + 5.18958i 0.233156 + 0.562889i
\(86\) −2.47871 − 4.10909i −0.267286 − 0.443095i
\(87\) 0 0
\(88\) −5.78306 − 5.15279i −0.616476 − 0.549290i
\(89\) −6.00227 + 6.00227i −0.636239 + 0.636239i −0.949626 − 0.313386i \(-0.898537\pi\)
0.313386 + 0.949626i \(0.398537\pi\)
\(90\) 0 0
\(91\) −0.0918540 + 0.0380472i −0.00962892 + 0.00398843i
\(92\) −0.366721 − 0.193358i −0.0382333 − 0.0201590i
\(93\) 0 0
\(94\) −1.12863 − 7.46318i −0.116409 − 0.769768i
\(95\) 3.21901 0.330263
\(96\) 0 0
\(97\) −3.69111 −0.374776 −0.187388 − 0.982286i \(-0.560002\pi\)
−0.187388 + 0.982286i \(0.560002\pi\)
\(98\) −1.47970 − 9.78466i −0.149472 − 0.988400i
\(99\) 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 864.2.v.b.109.2 ✓ 128
3.2 odd 2 inner 864.2.v.b.109.31 yes 128
32.5 even 8 inner 864.2.v.b.325.2 yes 128
96.5 odd 8 inner 864.2.v.b.325.31 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.v.b.109.2 ✓ 128 1.1 even 1 trivial
864.2.v.b.109.31 yes 128 3.2 odd 2 inner
864.2.v.b.325.2 yes 128 32.5 even 8 inner
864.2.v.b.325.31 yes 128 96.5 odd 8 inner