Show commands: Magma / Pari/GP / SageMath

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 325.15
Character \(\chi\) \(=\) 864.325
Dual form 864.2.v.b.109.15

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.274404 + 1.38734i) q^{2} +(-1.84940 - 0.761382i) q^{4} +(-3.91016 - 1.61964i) q^{5} +(-0.876571 - 0.876571i) q^{7} +(1.56378 - 2.35682i) q^{8} +(3.31995 - 4.98027i) q^{10} +(0.423068 - 1.02138i) q^{11} +(0.652663 - 0.270342i) q^{13} +(1.45663 - 0.975564i) q^{14} +(2.84059 + 2.81621i) q^{16} +4.86916i q^{17} +(2.72502 - 1.12874i) q^{19} +(5.99830 + 5.97250i) q^{20} +(1.30090 + 0.867207i) q^{22} +(-5.27134 + 5.27134i) q^{23} +(9.13059 + 9.13059i) q^{25} +(0.195961 + 0.979645i) q^{26} +(0.953728 + 2.28854i) q^{28} +(-2.12365 - 5.12695i) q^{29} -1.63933 q^{31} +(-4.68650 + 3.16808i) q^{32} +(-6.75517 - 1.33612i) q^{34} +(2.00780 + 4.84726i) q^{35} +(7.76051 + 3.21451i) q^{37} +(0.818184 + 4.09025i) q^{38} +(-9.93183 + 6.68278i) q^{40} +(-3.84249 + 3.84249i) q^{41} +(-1.66052 + 4.00885i) q^{43} +(-1.56008 + 1.56682i) q^{44} +(-5.86665 - 8.75961i) q^{46} +1.62125i q^{47} -5.46325i q^{49} +(-15.1727 + 10.1617i) q^{50} +(-1.41287 + 0.00304527i) q^{52} +(1.11677 - 2.69613i) q^{53} +(-3.30853 + 3.30853i) q^{55} +(-3.43668 + 0.695156i) q^{56} +(7.69554 - 1.53936i) q^{58} +(12.6902 + 5.25647i) q^{59} +(2.82854 + 6.82871i) q^{61} +(0.449840 - 2.27430i) q^{62} +(-3.10919 - 7.37109i) q^{64} -2.98987 q^{65} +(3.48525 + 8.41415i) q^{67} +(3.70729 - 9.00505i) q^{68} +(-7.27573 + 1.45539i) q^{70} +(6.28212 + 6.28212i) q^{71} +(-0.766637 + 0.766637i) q^{73} +(-6.58913 + 9.88437i) q^{74} +(-5.89906 + 0.0127147i) q^{76} +(-1.26616 + 0.524459i) q^{77} -10.0672i q^{79} +(-6.54593 - 15.6126i) q^{80} +(-4.27643 - 6.38523i) q^{82} +(14.5434 - 6.02409i) q^{83} +(7.88630 - 19.0392i) q^{85} +(-5.10597 - 3.40375i) q^{86} +(-1.74561 - 2.59430i) q^{88} +(3.65963 + 3.65963i) q^{89} +(-0.809078 - 0.335131i) q^{91} +(13.7624 - 5.73534i) q^{92} +(-2.24922 - 0.444878i) q^{94} -12.4834 q^{95} -14.2161 q^{97} +(7.57936 + 1.49914i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 128 q + 16 q^{10} - 32 q^{16} - 16 q^{22} - 32 q^{40} - 32 q^{46} - 80 q^{52} + 32 q^{55} - 32 q^{58} + 64 q^{61} + 48 q^{64} + 64 q^{67} - 96 q^{70} + 32 q^{76} - 80 q^{82} - 80 q^{88} + 96 q^{91} - 48 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.274404 + 1.38734i −0.194033 + 0.980995i
\(3\) 0 0
\(4\) −1.84940 − 0.761382i −0.924702 − 0.380691i
\(5\) −3.91016 − 1.61964i −1.74868 − 0.724326i −0.997970 − 0.0636914i \(-0.979713\pi\)
−0.750708 − 0.660634i \(-0.770287\pi\)
\(6\) 0 0
\(7\) −0.876571 − 0.876571i −0.331313 − 0.331313i 0.521772 − 0.853085i \(-0.325271\pi\)
−0.853085 + 0.521772i \(0.825271\pi\)
\(8\) 1.56378 − 2.35682i 0.552879 − 0.833261i
\(9\) 0 0
\(10\) 3.31995 − 4.98027i 1.04986 − 1.57490i
\(11\) 0.423068 − 1.02138i 0.127560 − 0.307956i −0.847178 − 0.531309i \(-0.821700\pi\)
0.974738 + 0.223353i \(0.0717002\pi\)
\(12\) 0 0
\(13\) 0.652663 − 0.270342i 0.181016 − 0.0749793i −0.290335 − 0.956925i \(-0.593767\pi\)
0.471351 + 0.881946i \(0.343767\pi\)
\(14\) 1.45663 − 0.975564i 0.389302 − 0.260730i
\(15\) 0 0
\(16\) 2.84059 + 2.81621i 0.710148 + 0.704052i
\(17\) 4.86916i 1.18095i 0.807058 + 0.590473i \(0.201059\pi\)
−0.807058 + 0.590473i \(0.798941\pi\)
\(18\) 0 0
\(19\) 2.72502 − 1.12874i 0.625162 − 0.258951i −0.0475340 − 0.998870i \(-0.515136\pi\)
0.672696 + 0.739919i \(0.265136\pi\)
\(20\) 5.99830 + 5.97250i 1.34126 + 1.33549i
\(21\) 0 0
\(22\) 1.30090 + 0.867207i 0.277353 + 0.184889i
\(23\) −5.27134 + 5.27134i −1.09915 + 1.09915i −0.104641 + 0.994510i \(0.533369\pi\)
−0.994510 + 0.104641i \(0.966631\pi\)
\(24\) 0 0
\(25\) 9.13059 + 9.13059i 1.82612 + 1.82612i
\(26\) 0.195961 + 0.979645i 0.0384312 + 0.192124i
\(27\) 0 0
\(28\) 0.953728 + 2.28854i 0.180238 + 0.432493i
\(29\) −2.12365 − 5.12695i −0.394352 − 0.952051i −0.988980 − 0.148049i \(-0.952701\pi\)
0.594628 − 0.804001i \(-0.297299\pi\)
\(30\) 0 0
\(31\) −1.63933 −0.294433 −0.147216 − 0.989104i \(-0.547031\pi\)
−0.147216 + 0.989104i \(0.547031\pi\)
\(32\) −4.68650 + 3.16808i −0.828464 + 0.560042i
\(33\) 0 0
\(34\) −6.75517 − 1.33612i −1.15850 − 0.229143i
\(35\) 2.00780 + 4.84726i 0.339381 + 0.819337i
\(36\) 0 0
\(37\) 7.76051 + 3.21451i 1.27582 + 0.528462i 0.914729 − 0.404068i \(-0.132404\pi\)
0.361092 + 0.932530i \(0.382404\pi\)
\(38\) 0.818184 + 4.09025i 0.132727 + 0.663526i
\(39\) 0 0
\(40\) −9.93183 + 6.68278i −1.57036 + 1.05664i
\(41\) −3.84249 + 3.84249i −0.600096 + 0.600096i −0.940338 − 0.340242i \(-0.889491\pi\)
0.340242 + 0.940338i \(0.389491\pi\)
\(42\) 0 0
\(43\) −1.66052 + 4.00885i −0.253227 + 0.611344i −0.998461 − 0.0554592i \(-0.982338\pi\)
0.745234 + 0.666803i \(0.232338\pi\)
\(44\) −1.56008 + 1.56682i −0.235191 + 0.236207i
\(45\) 0 0
\(46\) −5.86665 − 8.75961i −0.864990 − 1.29153i
\(47\) 1.62125i 0.236484i 0.992985 + 0.118242i \(0.0377258\pi\)
−0.992985 + 0.118242i \(0.962274\pi\)
\(48\) 0 0
\(49\) − 5.46325i − 0.780464i
\(50\) −15.1727 + 10.1617i −2.14574 + 1.43708i
\(51\) 0 0
\(52\) −1.41287 + 0.00304527i −0.195930 + 0.000422303i
\(53\) 1.11677 − 2.69613i 0.153400 − 0.370341i −0.828433 − 0.560089i \(-0.810767\pi\)
0.981833 + 0.189748i \(0.0607670\pi\)
\(54\) 0 0
\(55\) −3.30853 + 3.30853i −0.446122 + 0.446122i
\(56\) −3.43668 + 0.695156i −0.459246 + 0.0928942i
\(57\) 0 0
\(58\) 7.69554 − 1.53936i 1.01047 − 0.202128i
\(59\) 12.6902 + 5.25647i 1.65213 + 0.684334i 0.997436 − 0.0715615i \(-0.0227982\pi\)
0.654692 + 0.755895i \(0.272798\pi\)
\(60\) 0 0
\(61\) 2.82854 + 6.82871i 0.362158 + 0.874327i 0.994984 + 0.100033i \(0.0318948\pi\)
−0.632826 + 0.774294i \(0.718105\pi\)
\(62\) 0.449840 − 2.27430i 0.0571297 − 0.288837i
\(63\) 0 0
\(64\) −3.10919 − 7.37109i −0.388649 − 0.921386i
\(65\) −2.98987 −0.370848
\(66\) 0 0
\(67\) 3.48525 + 8.41415i 0.425792 + 1.02795i 0.980608 + 0.195979i \(0.0627885\pi\)
−0.554816 + 0.831973i \(0.687212\pi\)
\(68\) 3.70729 − 9.00505i 0.449575 − 1.09202i
\(69\) 0 0
\(70\) −7.27573 + 1.45539i −0.869617 + 0.173952i
\(71\) 6.28212 + 6.28212i 0.745551 + 0.745551i 0.973640 − 0.228089i \(-0.0732479\pi\)
−0.228089 + 0.973640i \(0.573248\pi\)
\(72\) 0 0
\(73\) −0.766637 + 0.766637i −0.0897281 + 0.0897281i −0.750546 − 0.660818i \(-0.770209\pi\)
0.660818 + 0.750546i \(0.270209\pi\)
\(74\) −6.58913 + 9.88437i −0.765970 + 1.14903i
\(75\) 0 0
\(76\) −5.89906 + 0.0127147i −0.676669 + 0.00145848i
\(77\) −1.26616 + 0.524459i −0.144292 + 0.0597677i
\(78\) 0 0
\(79\) − 10.0672i − 1.13265i −0.824181 − 0.566327i \(-0.808364\pi\)
0.824181 − 0.566327i \(-0.191636\pi\)
\(80\) −6.54593 − 15.6126i −0.731857 − 1.74554i
\(81\) 0 0
\(82\) −4.27643 − 6.38523i −0.472253 − 0.705130i
\(83\) 14.5434 − 6.02409i 1.59635 − 0.661230i 0.605456 − 0.795878i \(-0.292991\pi\)
0.990893 + 0.134649i \(0.0429906\pi\)
\(84\) 0 0
\(85\) 7.88630 − 19.0392i 0.855389 − 2.06509i
\(86\) −5.10597 − 3.40375i −0.550591 − 0.367035i
\(87\) 0 0
\(88\) −1.74561 − 2.59430i −0.186083 − 0.276553i
\(89\) 3.65963 + 3.65963i 0.387920 + 0.387920i 0.873945 − 0.486025i \(-0.161554\pi\)
−0.486025 + 0.873945i \(0.661554\pi\)
\(90\) 0 0
\(91\) −0.809078 − 0.335131i −0.0848145 − 0.0351313i
\(92\) 13.7624 − 5.73534i 1.43482 − 0.597950i
\(93\) 0 0
\(94\) −2.24922 − 0.444878i −0.231989 − 0.0458857i
\(95\) −12.4834 −1.28077
\(96\) 0 0
\(97\) −14.2161 −1.44343 −0.721714 − 0.692191i \(-0.756645\pi\)
−0.721714 + 0.692191i \(0.756645\pi\)
\(98\) 7.57936 + 1.49914i 0.765631 + 0.151436i
\(99\) 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.325.15 yes 128
3.2 odd 2 inner 864.2.v.b.325.18 yes 128
32.13 even 8 inner 864.2.v.b.109.15 ✓ 128
96.77 odd 8 inner 864.2.v.b.109.18 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.v.b.109.15 ✓ 128 32.13 even 8 inner
864.2.v.b.109.18 yes 128 96.77 odd 8 inner
864.2.v.b.325.15 yes 128 1.1 even 1 trivial
864.2.v.b.325.18 yes 128 3.2 odd 2 inner