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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 109.12
Character \(\chi\) \(=\) 864.109
Dual form 864.2.v.b.325.12

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.479175 + 1.33056i) q^{2} +(-1.54078 - 1.27514i) q^{4} +(-0.855212 + 0.354240i) q^{5} +(-3.04155 + 3.04155i) q^{7} +(2.43496 - 1.43909i) q^{8} +(-0.0615424 - 1.30765i) q^{10} +(1.47477 + 3.56040i) q^{11} +(-0.524396 - 0.217212i) q^{13} +(-2.58953 - 5.50440i) q^{14} +(0.748026 + 3.92943i) q^{16} -5.34789i q^{17} +(-4.09576 - 1.69652i) q^{19} +(1.76940 + 0.544709i) q^{20} +(-5.44400 + 0.256212i) q^{22} +(2.03658 + 2.03658i) q^{23} +(-2.92963 + 2.92963i) q^{25} +(0.540290 - 0.593658i) q^{26} +(8.56477 - 0.807961i) q^{28} +(3.31480 - 8.00262i) q^{29} +2.51477 q^{31} +(-5.58679 - 0.887591i) q^{32} +(7.11569 + 2.56257i) q^{34} +(1.52373 - 3.67861i) q^{35} +(-7.84910 + 3.25120i) q^{37} +(4.21990 - 4.63673i) q^{38} +(-1.57262 + 2.09329i) q^{40} +(-8.96180 - 8.96180i) q^{41} +(-4.72038 - 11.3960i) q^{43} +(2.26772 - 7.36634i) q^{44} +(-3.68566 + 1.73391i) q^{46} -3.46822i q^{47} -11.5020i q^{49} +(-2.49425 - 5.30186i) q^{50} +(0.531004 + 1.00335i) q^{52} +(3.31524 + 8.00369i) q^{53} +(-2.52248 - 2.52248i) q^{55} +(-3.02898 + 11.7831i) q^{56} +(9.05961 + 8.24519i) q^{58} +(0.0488541 - 0.0202360i) q^{59} +(-0.0218216 + 0.0526820i) q^{61} +(-1.20501 + 3.34605i) q^{62} +(3.85804 - 7.00825i) q^{64} +0.525415 q^{65} +(2.67338 - 6.45412i) q^{67} +(-6.81931 + 8.23993i) q^{68} +(4.16448 + 3.79011i) q^{70} +(-0.889622 + 0.889622i) q^{71} +(-4.73977 - 4.73977i) q^{73} +(-0.564834 - 12.0016i) q^{74} +(4.14737 + 7.83664i) q^{76} +(-15.3147 - 6.34356i) q^{77} +12.2691i q^{79} +(-2.03169 - 3.09552i) q^{80} +(16.2185 - 7.62995i) q^{82} +(2.42613 + 1.00493i) q^{83} +(1.89444 + 4.57358i) q^{85} +(17.4250 - 0.820074i) q^{86} +(8.71473 + 6.54710i) q^{88} +(-4.26786 + 4.26786i) q^{89} +(2.25563 - 0.934314i) q^{91} +(-0.540999 - 5.73485i) q^{92} +(4.61467 + 1.66188i) q^{94} +4.10372 q^{95} -6.74287 q^{97} +(15.3042 + 5.51149i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 128 q + 16 q^{10} - 32 q^{16} - 16 q^{22} - 32 q^{40} - 32 q^{46} - 80 q^{52} + 32 q^{55} - 32 q^{58} + 64 q^{61} + 48 q^{64} + 64 q^{67} - 96 q^{70} + 32 q^{76} - 80 q^{82} - 80 q^{88} + 96 q^{91} - 48 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.479175 + 1.33056i −0.338828 + 0.940848i
\(3\) 0 0
\(4\) −1.54078 − 1.27514i −0.770392 − 0.637571i
\(5\) −0.855212 + 0.354240i −0.382462 + 0.158421i −0.565627 − 0.824661i \(-0.691366\pi\)
0.183165 + 0.983082i \(0.441366\pi\)
\(6\) 0 0
\(7\) −3.04155 + 3.04155i −1.14960 + 1.14960i −0.162966 + 0.986632i \(0.552106\pi\)
−0.986632 + 0.162966i \(0.947894\pi\)
\(8\) 2.43496 − 1.43909i 0.860888 − 0.508795i
\(9\) 0 0
\(10\) −0.0615424 − 1.30765i −0.0194614 − 0.413517i
\(11\) 1.47477 + 3.56040i 0.444659 + 1.07350i 0.974295 + 0.225276i \(0.0723284\pi\)
−0.529636 + 0.848225i \(0.677672\pi\)
\(12\) 0 0
\(13\) −0.524396 − 0.217212i −0.145441 − 0.0602437i 0.308776 − 0.951135i \(-0.400081\pi\)
−0.454217 + 0.890891i \(0.650081\pi\)
\(14\) −2.58953 − 5.50440i −0.692082 − 1.47111i
\(15\) 0 0
\(16\) 0.748026 + 3.92943i 0.187007 + 0.982359i
\(17\) − 5.34789i − 1.29705i −0.761192 − 0.648526i \(-0.775386\pi\)
0.761192 − 0.648526i \(-0.224614\pi\)
\(18\) 0 0
\(19\) −4.09576 − 1.69652i −0.939631 − 0.389208i −0.140307 − 0.990108i \(-0.544809\pi\)
−0.799324 + 0.600900i \(0.794809\pi\)
\(20\) 1.76940 + 0.544709i 0.395651 + 0.121801i
\(21\) 0 0
\(22\) −5.44400 + 0.256212i −1.16066 + 0.0546246i
\(23\) 2.03658 + 2.03658i 0.424656 + 0.424656i 0.886803 − 0.462148i \(-0.152921\pi\)
−0.462148 + 0.886803i \(0.652921\pi\)
\(24\) 0 0
\(25\) −2.92963 + 2.92963i −0.585927 + 0.585927i
\(26\) 0.540290 − 0.593658i 0.105960 − 0.116426i
\(27\) 0 0
\(28\) 8.56477 − 0.807961i 1.61859 − 0.152690i
\(29\) 3.31480 − 8.00262i 0.615542 − 1.48605i −0.241289 − 0.970453i \(-0.577570\pi\)
0.856831 − 0.515597i \(-0.172430\pi\)
\(30\) 0 0
\(31\) 2.51477 0.451665 0.225833 − 0.974166i \(-0.427490\pi\)
0.225833 + 0.974166i \(0.427490\pi\)
\(32\) −5.58679 − 0.887591i −0.987614 − 0.156905i
\(33\) 0 0
\(34\) 7.11569 + 2.56257i 1.22033 + 0.439477i
\(35\) 1.52373 − 3.67861i 0.257557 − 0.621798i
\(36\) 0 0
\(37\) −7.84910 + 3.25120i −1.29038 + 0.534495i −0.919100 − 0.394024i \(-0.871083\pi\)
−0.371285 + 0.928519i \(0.621083\pi\)
\(38\) 4.21990 − 4.63673i 0.684559 − 0.752176i
\(39\) 0 0
\(40\) −1.57262 + 2.09329i −0.248653 + 0.330978i
\(41\) −8.96180 − 8.96180i −1.39960 − 1.39960i −0.801196 − 0.598402i \(-0.795803\pi\)
−0.598402 − 0.801196i \(-0.704197\pi\)
\(42\) 0 0
\(43\) −4.72038 − 11.3960i −0.719851 − 1.73787i −0.673776 − 0.738936i \(-0.735329\pi\)
−0.0460752 − 0.998938i \(-0.514671\pi\)
\(44\) 2.26772 − 7.36634i 0.341872 − 1.11052i
\(45\) 0 0
\(46\) −3.68566 + 1.73391i −0.543422 + 0.255651i
\(47\) − 3.46822i − 0.505892i −0.967480 − 0.252946i \(-0.918601\pi\)
0.967480 − 0.252946i \(-0.0813994\pi\)
\(48\) 0 0
\(49\) − 11.5020i − 1.64315i
\(50\) −2.49425 − 5.30186i −0.352740 − 0.749796i
\(51\) 0 0
\(52\) 0.531004 + 1.00335i 0.0736370 + 0.139140i
\(53\) 3.31524 + 8.00369i 0.455383 + 1.09939i 0.970247 + 0.242119i \(0.0778425\pi\)
−0.514864 + 0.857272i \(0.672158\pi\)
\(54\) 0 0
\(55\) −2.52248 − 2.52248i −0.340131 − 0.340131i
\(56\) −3.02898 + 11.7831i −0.404765 + 1.57458i
\(57\) 0 0
\(58\) 9.05961 + 8.24519i 1.18959 + 1.08265i
\(59\) 0.0488541 − 0.0202360i 0.00636027 − 0.00263451i −0.379501 − 0.925191i \(-0.623904\pi\)
0.385861 + 0.922557i \(0.373904\pi\)
\(60\) 0 0
\(61\) −0.0218216 + 0.0526820i −0.00279397 + 0.00674523i −0.925270 − 0.379308i \(-0.876162\pi\)
0.922476 + 0.386053i \(0.126162\pi\)
\(62\) −1.20501 + 3.34605i −0.153037 + 0.424949i
\(63\) 0 0
\(64\) 3.85804 − 7.00825i 0.482255 − 0.876031i
\(65\) 0.525415 0.0651696
\(66\) 0 0
\(67\) 2.67338 − 6.45412i 0.326606 − 0.788496i −0.672234 − 0.740339i \(-0.734665\pi\)
0.998840 − 0.0481572i \(-0.0153348\pi\)
\(68\) −6.81931 + 8.23993i −0.826963 + 0.999239i
\(69\) 0 0
\(70\) 4.16448 + 3.79011i 0.497750 + 0.453005i
\(71\) −0.889622 + 0.889622i −0.105579 + 0.105579i −0.757923 − 0.652344i \(-0.773786\pi\)
0.652344 + 0.757923i \(0.273786\pi\)
\(72\) 0 0
\(73\) −4.73977 − 4.73977i −0.554748 − 0.554748i 0.373060 − 0.927807i \(-0.378309\pi\)
−0.927807 + 0.373060i \(0.878309\pi\)
\(74\) −0.564834 − 12.0016i −0.0656606 − 1.39516i
\(75\) 0 0
\(76\) 4.14737 + 7.83664i 0.475736 + 0.898924i
\(77\) −15.3147 − 6.34356i −1.74527 − 0.722916i
\(78\) 0 0
\(79\) 12.2691i 1.38038i 0.723630 + 0.690189i \(0.242472\pi\)
−0.723630 + 0.690189i \(0.757528\pi\)
\(80\) −2.03169 − 3.09552i −0.227149 − 0.346090i
\(81\) 0 0
\(82\) 16.2185 − 7.62995i 1.79103 − 0.842587i
\(83\) 2.42613 + 1.00493i 0.266302 + 0.110306i 0.511839 − 0.859081i \(-0.328964\pi\)
−0.245537 + 0.969387i \(0.578964\pi\)
\(84\) 0 0
\(85\) 1.89444 + 4.57358i 0.205481 + 0.496074i
\(86\) 17.4250 − 0.820074i 1.87898 − 0.0884309i
\(87\) 0 0
\(88\) 8.71473 + 6.54710i 0.928993 + 0.697924i
\(89\) −4.26786 + 4.26786i −0.452392 + 0.452392i −0.896148 − 0.443756i \(-0.853646\pi\)
0.443756 + 0.896148i \(0.353646\pi\)
\(90\) 0 0
\(91\) 2.25563 − 0.934314i 0.236455 − 0.0979428i
\(92\) −0.540999 − 5.73485i −0.0564030 − 0.597899i
\(93\) 0 0
\(94\) 4.61467 + 1.66188i 0.475967 + 0.171410i
\(95\) 4.10372 0.421032
\(96\) 0 0
\(97\) −6.74287 −0.684635 −0.342317 − 0.939584i \(-0.611212\pi\)
−0.342317 + 0.939584i \(0.611212\pi\)
\(98\) 15.3042 + 5.51149i 1.54595 + 0.556744i
\(99\) 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.109.12 ✓ 128
3.2 odd 2 inner 864.2.v.b.109.21 yes 128
32.5 even 8 inner 864.2.v.b.325.12 yes 128
96.5 odd 8 inner 864.2.v.b.325.21 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.v.b.109.12 ✓ 128 1.1 even 1 trivial
864.2.v.b.109.21 yes 128 3.2 odd 2 inner
864.2.v.b.325.12 yes 128 32.5 even 8 inner
864.2.v.b.325.21 yes 128 96.5 odd 8 inner