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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,-8] 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 109.1
Character \(\chi\) \(=\) 864.109
Dual form 864.2.v.a.325.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.39805 - 0.213220i) q^{2} +(1.90907 + 0.596184i) q^{4} +(0.0181233 - 0.00750691i) q^{5} +(-1.64798 + 1.64798i) q^{7} +(-2.54186 - 1.24055i) q^{8} +(-0.0269378 + 0.00663076i) q^{10} +(-1.05521 - 2.54750i) q^{11} +(-0.619443 - 0.256582i) q^{13} +(2.65534 - 1.95258i) q^{14} +(3.28913 + 2.27632i) q^{16} -1.88186i q^{17} +(7.70774 + 3.19265i) q^{19} +(0.0390742 - 0.00352644i) q^{20} +(0.932055 + 3.78652i) q^{22} +(2.54144 + 2.54144i) q^{23} +(-3.53526 + 3.53526i) q^{25} +(0.811303 + 0.490791i) q^{26} +(-4.12863 + 2.16362i) q^{28} +(-2.16330 + 5.22266i) q^{29} +8.26107 q^{31} +(-4.11300 - 3.88371i) q^{32} +(-0.401250 + 2.63092i) q^{34} +(-0.0174956 + 0.0422381i) q^{35} +(-5.35925 + 2.21987i) q^{37} +(-10.0951 - 6.10693i) q^{38} +(-0.0553795 - 0.00340128i) q^{40} +(3.00044 + 3.00044i) q^{41} +(-1.48765 - 3.59150i) q^{43} +(-0.495695 - 5.49247i) q^{44} +(-3.01116 - 4.09494i) q^{46} +7.48247i q^{47} +1.56829i q^{49} +(5.69625 - 4.18868i) q^{50} +(-1.02959 - 0.859136i) q^{52} +(1.80541 + 4.35866i) q^{53} +(-0.0382477 - 0.0382477i) q^{55} +(6.23334 - 2.14454i) q^{56} +(4.13797 - 6.84027i) q^{58} +(3.62927 - 1.50329i) q^{59} +(-0.177788 + 0.429218i) q^{61} +(-11.5494 - 1.76143i) q^{62} +(4.92209 + 6.30659i) q^{64} -0.0131525 q^{65} +(0.264108 - 0.637614i) q^{67} +(1.12193 - 3.59260i) q^{68} +(0.0334657 - 0.0553205i) q^{70} +(-10.9496 + 10.9496i) q^{71} +(1.35090 + 1.35090i) q^{73} +(7.96581 - 1.96079i) q^{74} +(12.8112 + 10.6902i) q^{76} +(5.93722 + 2.45928i) q^{77} +2.39418i q^{79} +(0.0766979 + 0.0165632i) q^{80} +(-3.55500 - 4.83450i) q^{82} +(13.6807 + 5.66672i) q^{83} +(-0.0141269 - 0.0341054i) q^{85} +(1.31402 + 5.33829i) q^{86} +(-0.478101 + 7.78443i) q^{88} +(-2.85123 + 2.85123i) q^{89} +(1.44368 - 0.597990i) q^{91} +(3.33663 + 6.36696i) q^{92} +(1.59541 - 10.4608i) q^{94} +0.163656 q^{95} -12.3625 q^{97} +(0.334392 - 2.19255i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 128 q - 8 q^{10} - 32 q^{16} + 32 q^{22} + 64 q^{40} + 64 q^{46} + 88 q^{52} - 64 q^{55} + 64 q^{58} - 32 q^{61} - 96 q^{64} + 64 q^{67} + 48 q^{70} + 32 q^{76} + 40 q^{82} + 40 q^{88} - 48 q^{91} + 24 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.39805 − 0.213220i −0.988569 − 0.150769i
\(3\) 0 0
\(4\) 1.90907 + 0.596184i 0.954537 + 0.298092i
\(5\) 0.0181233 − 0.00750691i 0.00810498 − 0.00335719i −0.378627 − 0.925549i \(-0.623604\pi\)
0.386732 + 0.922192i \(0.373604\pi\)
\(6\) 0 0
\(7\) −1.64798 + 1.64798i −0.622880 + 0.622880i −0.946267 − 0.323387i \(-0.895178\pi\)
0.323387 + 0.946267i \(0.395178\pi\)
\(8\) −2.54186 − 1.24055i −0.898683 − 0.438599i
\(9\) 0 0
\(10\) −0.0269378 + 0.00663076i −0.00851849 + 0.00209683i
\(11\) −1.05521 − 2.54750i −0.318158 − 0.768101i −0.999352 − 0.0359982i \(-0.988539\pi\)
0.681194 − 0.732103i \(-0.261461\pi\)
\(12\) 0 0
\(13\) −0.619443 − 0.256582i −0.171803 − 0.0711630i 0.295124 − 0.955459i \(-0.404639\pi\)
−0.466927 + 0.884296i \(0.654639\pi\)
\(14\) 2.65534 − 1.95258i 0.709671 − 0.521848i
\(15\) 0 0
\(16\) 3.28913 + 2.27632i 0.822282 + 0.569080i
\(17\) − 1.88186i − 0.456417i −0.973612 − 0.228209i \(-0.926713\pi\)
0.973612 − 0.228209i \(-0.0732868\pi\)
\(18\) 0 0
\(19\) 7.70774 + 3.19265i 1.76828 + 0.732445i 0.995170 + 0.0981701i \(0.0312989\pi\)
0.773108 + 0.634274i \(0.218701\pi\)
\(20\) 0.0390742 − 0.00352644i 0.00873725 − 0.000788535i
\(21\) 0 0
\(22\) 0.932055 + 3.78652i 0.198715 + 0.807289i
\(23\) 2.54144 + 2.54144i 0.529926 + 0.529926i 0.920550 − 0.390624i \(-0.127741\pi\)
−0.390624 + 0.920550i \(0.627741\pi\)
\(24\) 0 0
\(25\) −3.53526 + 3.53526i −0.707052 + 0.707052i
\(26\) 0.811303 + 0.490791i 0.159110 + 0.0962521i
\(27\) 0 0
\(28\) −4.12863 + 2.16362i −0.780237 + 0.408886i
\(29\) −2.16330 + 5.22266i −0.401714 + 0.969823i 0.585536 + 0.810646i \(0.300884\pi\)
−0.987250 + 0.159177i \(0.949116\pi\)
\(30\) 0 0
\(31\) 8.26107 1.48373 0.741866 − 0.670548i \(-0.233941\pi\)
0.741866 + 0.670548i \(0.233941\pi\)
\(32\) −4.11300 − 3.88371i −0.727083 − 0.686550i
\(33\) 0 0
\(34\) −0.401250 + 2.63092i −0.0688137 + 0.451200i
\(35\) −0.0174956 + 0.0422381i −0.00295730 + 0.00713955i
\(36\) 0 0
\(37\) −5.35925 + 2.21987i −0.881056 + 0.364945i −0.776906 − 0.629616i \(-0.783212\pi\)
−0.104150 + 0.994562i \(0.533212\pi\)
\(38\) −10.0951 − 6.10693i −1.63763 − 0.990674i
\(39\) 0 0
\(40\) −0.0553795 − 0.00340128i −0.00875626 − 0.000537789i
\(41\) 3.00044 + 3.00044i 0.468589 + 0.468589i 0.901457 − 0.432868i \(-0.142498\pi\)
−0.432868 + 0.901457i \(0.642498\pi\)
\(42\) 0 0
\(43\) −1.48765 − 3.59150i −0.226864 − 0.547699i 0.768928 − 0.639335i \(-0.220790\pi\)
−0.995793 + 0.0916363i \(0.970790\pi\)
\(44\) −0.495695 − 5.49247i −0.0747288 − 0.828021i
\(45\) 0 0
\(46\) −3.01116 − 4.09494i −0.443972 − 0.603765i
\(47\) 7.48247i 1.09143i 0.837971 + 0.545715i \(0.183742\pi\)
−0.837971 + 0.545715i \(0.816258\pi\)
\(48\) 0 0
\(49\) 1.56829i 0.224042i
\(50\) 5.69625 − 4.18868i 0.805572 − 0.592368i
\(51\) 0 0
\(52\) −1.02959 − 0.859136i −0.142779 − 0.119141i
\(53\) 1.80541 + 4.35866i 0.247993 + 0.598707i 0.998033 − 0.0626843i \(-0.0199661\pi\)
−0.750041 + 0.661392i \(0.769966\pi\)
\(54\) 0 0
\(55\) −0.0382477 − 0.0382477i −0.00515732 − 0.00515732i
\(56\) 6.23334 − 2.14454i 0.832966 − 0.286576i
\(57\) 0 0
\(58\) 4.13797 − 6.84027i 0.543342 − 0.898171i
\(59\) 3.62927 − 1.50329i 0.472491 − 0.195712i −0.133715 − 0.991020i \(-0.542691\pi\)
0.606206 + 0.795308i \(0.292691\pi\)
\(60\) 0 0
\(61\) −0.177788 + 0.429218i −0.0227634 + 0.0549557i −0.934853 − 0.355036i \(-0.884469\pi\)
0.912089 + 0.409992i \(0.134469\pi\)
\(62\) −11.5494 − 1.76143i −1.46677 − 0.223701i
\(63\) 0 0
\(64\) 4.92209 + 6.30659i 0.615261 + 0.788323i
\(65\) −0.0131525 −0.00163136
\(66\) 0 0
\(67\) 0.264108 − 0.637614i 0.0322660 − 0.0778969i −0.906925 − 0.421292i \(-0.861577\pi\)
0.939191 + 0.343395i \(0.111577\pi\)
\(68\) 1.12193 − 3.59260i 0.136054 − 0.435667i
\(69\) 0 0
\(70\) 0.0334657 − 0.0553205i 0.00399992 − 0.00661207i
\(71\) −10.9496 + 10.9496i −1.29948 + 1.29948i −0.370740 + 0.928737i \(0.620896\pi\)
−0.928737 + 0.370740i \(0.879104\pi\)
\(72\) 0 0
\(73\) 1.35090 + 1.35090i 0.158111 + 0.158111i 0.781729 − 0.623618i \(-0.214338\pi\)
−0.623618 + 0.781729i \(0.714338\pi\)
\(74\) 7.96581 − 1.96079i 0.926007 − 0.227937i
\(75\) 0 0
\(76\) 12.8112 + 10.6902i 1.46955 + 1.22625i
\(77\) 5.93722 + 2.45928i 0.676609 + 0.280260i
\(78\) 0 0
\(79\) 2.39418i 0.269367i 0.990889 + 0.134683i \(0.0430017\pi\)
−0.990889 + 0.134683i \(0.956998\pi\)
\(80\) 0.0766979 + 0.0165632i 0.00857509 + 0.00185182i
\(81\) 0 0
\(82\) −3.55500 − 4.83450i −0.392584 − 0.533882i
\(83\) 13.6807 + 5.66672i 1.50165 + 0.622004i 0.973815 − 0.227343i \(-0.0730039\pi\)
0.527835 + 0.849347i \(0.323004\pi\)
\(84\) 0 0
\(85\) −0.0141269 − 0.0341054i −0.00153228 − 0.00369925i
\(86\) 1.31402 + 5.33829i 0.141695 + 0.575642i
\(87\) 0 0
\(88\) −0.478101 + 7.78443i −0.0509658 + 0.829823i
\(89\) −2.85123 + 2.85123i −0.302230 + 0.302230i −0.841886 − 0.539656i \(-0.818554\pi\)
0.539656 + 0.841886i \(0.318554\pi\)
\(90\) 0 0
\(91\) 1.44368 − 0.597990i 0.151338 − 0.0626864i
\(92\) 3.33663 + 6.36696i 0.347868 + 0.663801i
\(93\) 0 0
\(94\) 1.59541 − 10.4608i 0.164554 − 1.07895i
\(95\) 0.163656 0.0167908
\(96\) 0 0
\(97\) −12.3625 −1.25522 −0.627609 − 0.778528i \(-0.715967\pi\)
−0.627609 + 0.778528i \(0.715967\pi\)
\(98\) 0.334392 − 2.19255i 0.0337787 − 0.221481i
\(99\) 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.a.109.1 ✓ 128
3.2 odd 2 inner 864.2.v.a.109.32 yes 128
32.5 even 8 inner 864.2.v.a.325.1 yes 128
96.5 odd 8 inner 864.2.v.a.325.32 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.v.a.109.1 ✓ 128 1.1 even 1 trivial
864.2.v.a.109.32 yes 128 3.2 odd 2 inner
864.2.v.a.325.1 yes 128 32.5 even 8 inner
864.2.v.a.325.32 yes 128 96.5 odd 8 inner