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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 109.16
Character \(\chi\) \(=\) 864.109
Dual form 864.2.v.a.325.16

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.0434074 + 1.41355i) q^{2} +(-1.99623 - 0.122717i) q^{4} +(2.68483 - 1.11209i) q^{5} +(1.54691 - 1.54691i) q^{7} +(0.260117 - 2.81644i) q^{8} +(1.45545 + 3.84340i) q^{10} +(0.284805 + 0.687579i) q^{11} +(-2.08570 - 0.863924i) q^{13} +(2.11949 + 2.25378i) q^{14} +(3.96988 + 0.489942i) q^{16} -6.20670i q^{17} +(-0.566840 - 0.234793i) q^{19} +(-5.49601 + 1.89052i) q^{20} +(-0.984289 + 0.372739i) q^{22} +(-4.42017 - 4.42017i) q^{23} +(2.43601 - 2.43601i) q^{25} +(1.31173 - 2.91073i) q^{26} +(-3.27783 + 2.89816i) q^{28} +(1.81939 - 4.39239i) q^{29} +7.34680 q^{31} +(-0.864879 + 5.59035i) q^{32} +(8.77346 + 0.269417i) q^{34} +(2.43288 - 5.87350i) q^{35} +(5.10608 - 2.11501i) q^{37} +(0.356496 - 0.791064i) q^{38} +(-2.43377 - 7.85093i) q^{40} +(-4.87073 - 4.87073i) q^{41} +(2.38680 + 5.76224i) q^{43} +(-0.484159 - 1.40752i) q^{44} +(6.43999 - 6.05625i) q^{46} +7.05366i q^{47} +2.21413i q^{49} +(3.33768 + 3.54916i) q^{50} +(4.05752 + 1.98054i) q^{52} +(1.34706 + 3.25209i) q^{53} +(1.52930 + 1.52930i) q^{55} +(-3.95441 - 4.75917i) q^{56} +(6.12987 + 2.76245i) q^{58} +(-6.36373 + 2.63594i) q^{59} +(3.05830 - 7.38339i) q^{61} +(-0.318905 + 10.3850i) q^{62} +(-7.86468 - 1.46521i) q^{64} -6.56050 q^{65} +(-3.58513 + 8.65526i) q^{67} +(-0.761666 + 12.3900i) q^{68} +(8.19686 + 3.69395i) q^{70} +(4.40258 - 4.40258i) q^{71} +(7.67341 + 7.67341i) q^{73} +(2.76802 + 7.30949i) q^{74} +(1.10273 + 0.538262i) q^{76} +(1.50419 + 0.623057i) q^{77} +10.9657i q^{79} +(11.2033 - 3.09946i) q^{80} +(7.09643 - 6.67358i) q^{82} +(12.4669 + 5.16396i) q^{83} +(-6.90242 - 16.6639i) q^{85} +(-8.24880 + 3.12373i) q^{86} +(2.01061 - 0.623284i) q^{88} +(9.45976 - 9.45976i) q^{89} +(-4.56281 + 1.88998i) q^{91} +(8.28126 + 9.36611i) q^{92} +(-9.97069 - 0.306181i) q^{94} -1.78298 q^{95} -13.6291 q^{97} +(-3.12977 - 0.0961094i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 128 q - 8 q^{10} - 32 q^{16} + 32 q^{22} + 64 q^{40} + 64 q^{46} + 88 q^{52} - 64 q^{55} + 64 q^{58} - 32 q^{61} - 96 q^{64} + 64 q^{67} + 48 q^{70} + 32 q^{76} + 40 q^{82} + 40 q^{88} - 48 q^{91} + 24 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.0434074 + 1.41355i −0.0306937 + 0.999529i
\(3\) 0 0
\(4\) −1.99623 − 0.122717i −0.998116 − 0.0613584i
\(5\) 2.68483 − 1.11209i 1.20069 − 0.497342i 0.309469 − 0.950910i \(-0.399849\pi\)
0.891222 + 0.453567i \(0.149849\pi\)
\(6\) 0 0
\(7\) 1.54691 − 1.54691i 0.584678 − 0.584678i −0.351507 − 0.936185i \(-0.614331\pi\)
0.936185 + 0.351507i \(0.114331\pi\)
\(8\) 0.260117 − 2.81644i 0.0919653 − 0.995762i
\(9\) 0 0
\(10\) 1.45545 + 3.84340i 0.460255 + 1.21539i
\(11\) 0.284805 + 0.687579i 0.0858719 + 0.207313i 0.960982 − 0.276611i \(-0.0892112\pi\)
−0.875110 + 0.483924i \(0.839211\pi\)
\(12\) 0 0
\(13\) −2.08570 − 0.863924i −0.578468 − 0.239609i 0.0742124 − 0.997242i \(-0.476356\pi\)
−0.652681 + 0.757633i \(0.726356\pi\)
\(14\) 2.11949 + 2.25378i 0.566456 + 0.602348i
\(15\) 0 0
\(16\) 3.96988 + 0.489942i 0.992470 + 0.122486i
\(17\) − 6.20670i − 1.50535i −0.658395 − 0.752673i \(-0.728764\pi\)
0.658395 − 0.752673i \(-0.271236\pi\)
\(18\) 0 0
\(19\) −0.566840 − 0.234793i −0.130042 − 0.0538652i 0.316713 − 0.948521i \(-0.397421\pi\)
−0.446756 + 0.894656i \(0.647421\pi\)
\(20\) −5.49601 + 1.89052i −1.22894 + 0.422733i
\(21\) 0 0
\(22\) −0.984289 + 0.372739i −0.209851 + 0.0794682i
\(23\) −4.42017 − 4.42017i −0.921669 − 0.921669i 0.0754781 − 0.997147i \(-0.475952\pi\)
−0.997147 + 0.0754781i \(0.975952\pi\)
\(24\) 0 0
\(25\) 2.43601 − 2.43601i 0.487203 − 0.487203i
\(26\) 1.31173 − 2.91073i 0.257252 − 0.570841i
\(27\) 0 0
\(28\) −3.27783 + 2.89816i −0.619451 + 0.547701i
\(29\) 1.81939 − 4.39239i 0.337852 − 0.815646i −0.660070 − 0.751204i \(-0.729473\pi\)
0.997921 − 0.0644418i \(-0.0205267\pi\)
\(30\) 0 0
\(31\) 7.34680 1.31952 0.659762 − 0.751475i \(-0.270657\pi\)
0.659762 + 0.751475i \(0.270657\pi\)
\(32\) −0.864879 + 5.59035i −0.152890 + 0.988243i
\(33\) 0 0
\(34\) 8.77346 + 0.269417i 1.50464 + 0.0462046i
\(35\) 2.43288 − 5.87350i 0.411232 − 0.992803i
\(36\) 0 0
\(37\) 5.10608 − 2.11501i 0.839434 − 0.347705i 0.0788036 − 0.996890i \(-0.474890\pi\)
0.760630 + 0.649185i \(0.224890\pi\)
\(38\) 0.356496 − 0.791064i 0.0578313 − 0.128327i
\(39\) 0 0
\(40\) −2.43377 − 7.85093i −0.384813 − 1.24134i
\(41\) −4.87073 − 4.87073i −0.760680 − 0.760680i 0.215765 − 0.976445i \(-0.430775\pi\)
−0.976445 + 0.215765i \(0.930775\pi\)
\(42\) 0 0
\(43\) 2.38680 + 5.76224i 0.363983 + 0.878733i 0.994710 + 0.102728i \(0.0327570\pi\)
−0.630726 + 0.776005i \(0.717243\pi\)
\(44\) −0.484159 − 1.40752i −0.0729897 − 0.212191i
\(45\) 0 0
\(46\) 6.43999 − 6.05625i 0.949525 − 0.892946i
\(47\) 7.05366i 1.02888i 0.857526 + 0.514441i \(0.172001\pi\)
−0.857526 + 0.514441i \(0.827999\pi\)
\(48\) 0 0
\(49\) 2.21413i 0.316304i
\(50\) 3.33768 + 3.54916i 0.472019 + 0.501927i
\(51\) 0 0
\(52\) 4.05752 + 1.98054i 0.562676 + 0.274652i
\(53\) 1.34706 + 3.25209i 0.185033 + 0.446709i 0.988991 − 0.147977i \(-0.0472763\pi\)
−0.803958 + 0.594686i \(0.797276\pi\)
\(54\) 0 0
\(55\) 1.52930 + 1.52930i 0.206211 + 0.206211i
\(56\) −3.95441 − 4.75917i −0.528430 − 0.635970i
\(57\) 0 0
\(58\) 6.12987 + 2.76245i 0.804892 + 0.362728i
\(59\) −6.36373 + 2.63594i −0.828487 + 0.343171i −0.756304 − 0.654221i \(-0.772997\pi\)
−0.0721835 + 0.997391i \(0.522997\pi\)
\(60\) 0 0
\(61\) 3.05830 − 7.38339i 0.391575 − 0.945347i −0.598022 − 0.801480i \(-0.704046\pi\)
0.989597 − 0.143867i \(-0.0459537\pi\)
\(62\) −0.318905 + 10.3850i −0.0405010 + 1.31890i
\(63\) 0 0
\(64\) −7.86468 − 1.46521i −0.983085 − 0.183151i
\(65\) −6.56050 −0.813730
\(66\) 0 0
\(67\) −3.58513 + 8.65526i −0.437993 + 1.05741i 0.538648 + 0.842531i \(0.318935\pi\)
−0.976641 + 0.214878i \(0.931065\pi\)
\(68\) −0.761666 + 12.3900i −0.0923656 + 1.50251i
\(69\) 0 0
\(70\) 8.19686 + 3.69395i 0.979713 + 0.441511i
\(71\) 4.40258 − 4.40258i 0.522490 − 0.522490i −0.395833 − 0.918323i \(-0.629544\pi\)
0.918323 + 0.395833i \(0.129544\pi\)
\(72\) 0 0
\(73\) 7.67341 + 7.67341i 0.898105 + 0.898105i 0.995268 − 0.0971633i \(-0.0309769\pi\)
−0.0971633 + 0.995268i \(0.530977\pi\)
\(74\) 2.76802 + 7.30949i 0.321776 + 0.849711i
\(75\) 0 0
\(76\) 1.10273 + 0.538262i 0.126492 + 0.0617429i
\(77\) 1.50419 + 0.623057i 0.171419 + 0.0710039i
\(78\) 0 0
\(79\) 10.9657i 1.23374i 0.787066 + 0.616869i \(0.211599\pi\)
−0.787066 + 0.616869i \(0.788401\pi\)
\(80\) 11.2033 − 3.09946i 1.25257 − 0.346530i
\(81\) 0 0
\(82\) 7.09643 − 6.67358i 0.783670 − 0.736974i
\(83\) 12.4669 + 5.16396i 1.36842 + 0.566818i 0.941360 − 0.337405i \(-0.109549\pi\)
0.427060 + 0.904223i \(0.359549\pi\)
\(84\) 0 0
\(85\) −6.90242 − 16.6639i −0.748672 − 1.80745i
\(86\) −8.24880 + 3.12373i −0.889491 + 0.336840i
\(87\) 0 0
\(88\) 2.01061 − 0.623284i 0.214332 − 0.0664423i
\(89\) 9.45976 − 9.45976i 1.00273 − 1.00273i 0.00273639 − 0.999996i \(-0.499129\pi\)
0.999996 − 0.00273639i \(-0.000871022\pi\)
\(90\) 0 0
\(91\) −4.56281 + 1.88998i −0.478312 + 0.198123i
\(92\) 8.28126 + 9.36611i 0.863381 + 0.976485i
\(93\) 0 0
\(94\) −9.97069 − 0.306181i −1.02840 − 0.0315802i
\(95\) −1.78298 −0.182930
\(96\) 0 0
\(97\) −13.6291 −1.38383 −0.691913 − 0.721981i \(-0.743232\pi\)
−0.691913 + 0.721981i \(0.743232\pi\)
\(98\) −3.12977 − 0.0961094i −0.316155 − 0.00970852i
\(99\) 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.109.16 ✓ 128
3.2 odd 2 inner 864.2.v.a.109.17 yes 128
32.5 even 8 inner 864.2.v.a.325.16 yes 128
96.5 odd 8 inner 864.2.v.a.325.17 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.v.a.109.16 ✓ 128 1.1 even 1 trivial
864.2.v.a.109.17 yes 128 3.2 odd 2 inner
864.2.v.a.325.16 yes 128 32.5 even 8 inner
864.2.v.a.325.17 yes 128 96.5 odd 8 inner