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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 109.17
Character \(\chi\) \(=\) 864.109
Dual form 864.2.v.a.325.17

$q$-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 −0.891222 − 0.453567i \(-0.850151\pi\)
−0.309469 + 0.950910i \(0.600151\pi\)
\(6\) 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 0.875110 − 0.483924i \(-0.160789\pi\)
−0.960982 + 0.276611i \(0.910789\pi\)
\(12\) 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 \(0.271236\pi\)
−0.658395 + 0.752673i \(0.728764\pi\)
\(18\) 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 0.997147 − 0.0754781i \(-0.0240483\pi\)
−0.0754781 + 0.997147i \(0.524048\pi\)
\(24\) 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 \(0.270527\pi\)
−0.997921 + 0.0644418i \(0.979473\pi\)
\(30\) 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 0.976445 − 0.215765i \(-0.0692245\pi\)
−0.215765 + 0.976445i \(0.569225\pi\)
\(42\) 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 \(-0.827999\pi\)
0.857526 − 0.514441i \(-0.172001\pi\)
\(48\) 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 0.803958 − 0.594686i \(-0.202724\pi\)
−0.988991 + 0.147977i \(0.952724\pi\)
\(54\) 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 0.0721835 − 0.997391i \(-0.477003\pi\)
0.756304 + 0.654221i \(0.227003\pi\)
\(60\) 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 −0.918323 − 0.395833i \(-0.870456\pi\)
0.395833 + 0.918323i \(0.370456\pi\)
\(72\) 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 −0.427060 − 0.904223i \(-0.640451\pi\)
−0.941360 + 0.337405i \(0.890451\pi\)
\(84\) 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 \(0.500871\pi\)
−0.999996 + 0.00273639i \(0.999129\pi\)
\(90\) 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.109.17 yes 128
3.2 odd 2 inner 864.2.v.a.109.16 ✓ 128
32.5 even 8 inner 864.2.v.a.325.17 yes 128
96.5 odd 8 inner 864.2.v.a.325.16 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.v.a.109.16 ✓ 128 3.2 odd 2 inner
864.2.v.a.109.17 yes 128 1.1 even 1 trivial
864.2.v.a.325.16 yes 128 96.5 odd 8 inner
864.2.v.a.325.17 yes 128 32.5 even 8 inner