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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 109.9
Character \(\chi\) \(=\) 864.109
Dual form 864.2.v.b.325.9

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.933793 - 1.06209i) q^{2} +(-0.256062 + 1.98354i) q^{4} +(1.30670 - 0.541253i) q^{5} +(-1.80429 + 1.80429i) q^{7} +(2.34580 - 1.58026i) q^{8} +(-1.79505 - 0.882413i) q^{10} +(0.711373 + 1.71741i) q^{11} +(-1.64790 - 0.682581i) q^{13} +(3.60115 + 0.231482i) q^{14} +(-3.86886 - 1.01582i) q^{16} -0.517642i q^{17} +(2.48448 + 1.02911i) q^{19} +(0.739002 + 2.73049i) q^{20} +(1.15976 - 2.35924i) q^{22} +(4.76398 + 4.76398i) q^{23} +(-2.12102 + 2.12102i) q^{25} +(0.813833 + 2.38760i) q^{26} +(-3.11688 - 4.04090i) q^{28} +(-0.367694 + 0.887691i) q^{29} -4.87060 q^{31} +(2.53383 + 5.05764i) q^{32} +(-0.549781 + 0.483371i) q^{34} +(-1.38109 + 3.33425i) q^{35} +(-0.997414 + 0.413142i) q^{37} +(-1.22699 - 3.59971i) q^{38} +(2.20994 - 3.33460i) q^{40} +(-1.37996 - 1.37996i) q^{41} +(4.74827 + 11.4633i) q^{43} +(-3.58870 + 0.971275i) q^{44} +(0.611195 - 9.50833i) q^{46} +10.8254i q^{47} +0.489051i q^{49} +(4.23331 + 0.272117i) q^{50} +(1.77589 - 3.09389i) q^{52} +(-1.87709 - 4.53169i) q^{53} +(1.85910 + 1.85910i) q^{55} +(-1.38127 + 7.08376i) q^{56} +(1.28616 - 0.438396i) q^{58} +(12.7983 - 5.30122i) q^{59} +(-3.76200 + 9.08227i) q^{61} +(4.54813 + 5.17300i) q^{62} +(3.00558 - 7.41394i) q^{64} -2.52276 q^{65} +(-1.27556 + 3.07947i) q^{67} +(1.02676 + 0.132548i) q^{68} +(4.83092 - 1.64666i) q^{70} +(4.55636 - 4.55636i) q^{71} +(5.71852 + 5.71852i) q^{73} +(1.37017 + 0.673552i) q^{74} +(-2.67746 + 4.66456i) q^{76} +(-4.38223 - 1.81518i) q^{77} -5.84936i q^{79} +(-5.60526 + 0.766666i) q^{80} +(-0.177042 + 2.75423i) q^{82} +(-3.04294 - 1.26043i) q^{83} +(-0.280176 - 0.676404i) q^{85} +(7.74117 - 15.7474i) q^{86} +(4.38268 + 2.90455i) q^{88} +(-3.05260 + 3.05260i) q^{89} +(4.20486 - 1.74171i) q^{91} +(-10.6694 + 8.22967i) q^{92} +(11.4975 - 10.1087i) q^{94} +3.80349 q^{95} +7.01433 q^{97} +(0.519415 - 0.456673i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 128 q + 16 q^{10} - 32 q^{16} - 16 q^{22} - 32 q^{40} - 32 q^{46} - 80 q^{52} + 32 q^{55} - 32 q^{58} + 64 q^{61} + 48 q^{64} + 64 q^{67} - 96 q^{70} + 32 q^{76} - 80 q^{82} - 80 q^{88} + 96 q^{91} - 48 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.933793 − 1.06209i −0.660291 − 0.751010i
\(3\) 0 0
\(4\) −0.256062 + 1.98354i −0.128031 + 0.991770i
\(5\) 1.30670 − 0.541253i 0.584375 − 0.242056i −0.0708538 − 0.997487i \(-0.522572\pi\)
0.655228 + 0.755431i \(0.272572\pi\)
\(6\) 0 0
\(7\) −1.80429 + 1.80429i −0.681959 + 0.681959i −0.960441 − 0.278483i \(-0.910169\pi\)
0.278483 + 0.960441i \(0.410169\pi\)
\(8\) 2.34580 − 1.58026i 0.829367 − 0.558705i
\(9\) 0 0
\(10\) −1.79505 − 0.882413i −0.567644 − 0.279044i
\(11\) 0.711373 + 1.71741i 0.214487 + 0.517818i 0.994103 − 0.108440i \(-0.0345857\pi\)
−0.779616 + 0.626258i \(0.784586\pi\)
\(12\) 0 0
\(13\) −1.64790 − 0.682581i −0.457044 − 0.189314i 0.142270 − 0.989828i \(-0.454560\pi\)
−0.599314 + 0.800514i \(0.704560\pi\)
\(14\) 3.60115 + 0.231482i 0.962449 + 0.0618662i
\(15\) 0 0
\(16\) −3.86886 − 1.01582i −0.967216 − 0.253954i
\(17\) − 0.517642i − 0.125547i −0.998028 − 0.0627733i \(-0.980005\pi\)
0.998028 − 0.0627733i \(-0.0199945\pi\)
\(18\) 0 0
\(19\) 2.48448 + 1.02911i 0.569980 + 0.236093i 0.649011 − 0.760779i \(-0.275183\pi\)
−0.0790315 + 0.996872i \(0.525183\pi\)
\(20\) 0.739002 + 2.73049i 0.165246 + 0.610556i
\(21\) 0 0
\(22\) 1.15976 − 2.35924i 0.247262 − 0.502992i
\(23\) 4.76398 + 4.76398i 0.993358 + 0.993358i 0.999978 − 0.00661976i \(-0.00210715\pi\)
−0.00661976 + 0.999978i \(0.502107\pi\)
\(24\) 0 0
\(25\) −2.12102 + 2.12102i −0.424204 + 0.424204i
\(26\) 0.813833 + 2.38760i 0.159606 + 0.468247i
\(27\) 0 0
\(28\) −3.11688 − 4.04090i −0.589035 − 0.763658i
\(29\) −0.367694 + 0.887691i −0.0682790 + 0.164840i −0.954335 − 0.298738i \(-0.903434\pi\)
0.886056 + 0.463578i \(0.153434\pi\)
\(30\) 0 0
\(31\) −4.87060 −0.874785 −0.437393 − 0.899271i \(-0.644098\pi\)
−0.437393 + 0.899271i \(0.644098\pi\)
\(32\) 2.53383 + 5.05764i 0.447922 + 0.894073i
\(33\) 0 0
\(34\) −0.549781 + 0.483371i −0.0942867 + 0.0828974i
\(35\) −1.38109 + 3.33425i −0.233447 + 0.563591i
\(36\) 0 0
\(37\) −0.997414 + 0.413142i −0.163974 + 0.0679202i −0.463161 − 0.886274i \(-0.653285\pi\)
0.299187 + 0.954195i \(0.403285\pi\)
\(38\) −1.22699 − 3.59971i −0.199044 − 0.583951i
\(39\) 0 0
\(40\) 2.20994 − 3.33460i 0.349423 − 0.527246i
\(41\) −1.37996 − 1.37996i −0.215513 − 0.215513i 0.591092 − 0.806604i \(-0.298697\pi\)
−0.806604 + 0.591092i \(0.798697\pi\)
\(42\) 0 0
\(43\) 4.74827 + 11.4633i 0.724104 + 1.74814i 0.661308 + 0.750115i \(0.270002\pi\)
0.0627958 + 0.998026i \(0.479998\pi\)
\(44\) −3.58870 + 0.971275i −0.541017 + 0.146425i
\(45\) 0 0
\(46\) 0.611195 − 9.50833i 0.0901158 − 1.40193i
\(47\) 10.8254i 1.57905i 0.613719 + 0.789525i \(0.289673\pi\)
−0.613719 + 0.789525i \(0.710327\pi\)
\(48\) 0 0
\(49\) 0.489051i 0.0698645i
\(50\) 4.23331 + 0.272117i 0.598680 + 0.0384831i
\(51\) 0 0
\(52\) 1.77589 − 3.09389i 0.246272 − 0.429045i
\(53\) −1.87709 − 4.53169i −0.257838 − 0.622475i 0.740957 − 0.671552i \(-0.234372\pi\)
−0.998795 + 0.0490768i \(0.984372\pi\)
\(54\) 0 0
\(55\) 1.85910 + 1.85910i 0.250682 + 0.250682i
\(56\) −1.38127 + 7.08376i −0.184580 + 0.946607i
\(57\) 0 0
\(58\) 1.28616 − 0.438396i 0.168880 − 0.0575643i
\(59\) 12.7983 − 5.30122i 1.66619 − 0.690160i 0.667669 − 0.744458i \(-0.267292\pi\)
0.998525 + 0.0542984i \(0.0172922\pi\)
\(60\) 0 0
\(61\) −3.76200 + 9.08227i −0.481675 + 1.16287i 0.477139 + 0.878828i \(0.341674\pi\)
−0.958813 + 0.284037i \(0.908326\pi\)
\(62\) 4.54813 + 5.17300i 0.577613 + 0.656972i
\(63\) 0 0
\(64\) 3.00558 − 7.41394i 0.375698 − 0.926742i
\(65\) −2.52276 −0.312910
\(66\) 0 0
\(67\) −1.27556 + 3.07947i −0.155834 + 0.376217i −0.982444 − 0.186559i \(-0.940266\pi\)
0.826610 + 0.562776i \(0.190266\pi\)
\(68\) 1.02676 + 0.132548i 0.124513 + 0.0160738i
\(69\) 0 0
\(70\) 4.83092 − 1.64666i 0.577406 − 0.196813i
\(71\) 4.55636 − 4.55636i 0.540740 − 0.540740i −0.383006 − 0.923746i \(-0.625111\pi\)
0.923746 + 0.383006i \(0.125111\pi\)
\(72\) 0 0
\(73\) 5.71852 + 5.71852i 0.669302 + 0.669302i 0.957555 − 0.288252i \(-0.0930741\pi\)
−0.288252 + 0.957555i \(0.593074\pi\)
\(74\) 1.37017 + 0.673552i 0.159279 + 0.0782988i
\(75\) 0 0
\(76\) −2.67746 + 4.66456i −0.307125 + 0.535062i
\(77\) −4.38223 − 1.81518i −0.499402 − 0.206859i
\(78\) 0 0
\(79\) − 5.84936i − 0.658104i −0.944312 − 0.329052i \(-0.893271\pi\)
0.944312 − 0.329052i \(-0.106729\pi\)
\(80\) −5.60526 + 0.766666i −0.626688 + 0.0857158i
\(81\) 0 0
\(82\) −0.177042 + 2.75423i −0.0195510 + 0.304154i
\(83\) −3.04294 − 1.26043i −0.334006 − 0.138350i 0.209376 − 0.977835i \(-0.432857\pi\)
−0.543383 + 0.839485i \(0.682857\pi\)
\(84\) 0 0
\(85\) −0.280176 − 0.676404i −0.0303893 − 0.0733663i
\(86\) 7.74117 − 15.7474i 0.834751 − 1.69809i
\(87\) 0 0
\(88\) 4.38268 + 2.90455i 0.467195 + 0.309626i
\(89\) −3.05260 + 3.05260i −0.323575 + 0.323575i −0.850137 − 0.526562i \(-0.823481\pi\)
0.526562 + 0.850137i \(0.323481\pi\)
\(90\) 0 0
\(91\) 4.20486 − 1.74171i 0.440790 − 0.182581i
\(92\) −10.6694 + 8.22967i −1.11236 + 0.858003i
\(93\) 0 0
\(94\) 11.4975 − 10.1087i 1.18588 − 1.04263i
\(95\) 3.80349 0.390229
\(96\) 0 0
\(97\) 7.01433 0.712197 0.356099 − 0.934448i \(-0.384107\pi\)
0.356099 + 0.934448i \(0.384107\pi\)
\(98\) 0.519415 − 0.456673i 0.0524689 − 0.0461309i
\(99\) 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.109.9 ✓ 128
3.2 odd 2 inner 864.2.v.b.109.24 yes 128
32.5 even 8 inner 864.2.v.b.325.9 yes 128
96.5 odd 8 inner 864.2.v.b.325.24 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.v.b.109.9 ✓ 128 1.1 even 1 trivial
864.2.v.b.109.24 yes 128 3.2 odd 2 inner
864.2.v.b.325.9 yes 128 32.5 even 8 inner
864.2.v.b.325.24 yes 128 96.5 odd 8 inner