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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 109.8
Character \(\chi\) \(=\) 864.109
Dual form 864.2.v.b.325.8

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.11341 + 0.871966i) q^{2} +(0.479352 - 1.94171i) q^{4} +(-2.70909 + 1.12214i) q^{5} +(0.103960 - 0.103960i) q^{7} +(1.15939 + 2.57989i) q^{8} +(2.03785 - 3.61163i) q^{10} +(-2.45541 - 5.92789i) q^{11} +(5.82208 + 2.41158i) q^{13} +(-0.0251002 + 0.206399i) q^{14} +(-3.54044 - 1.86152i) q^{16} +2.21295i q^{17} +(-0.136914 - 0.0567118i) q^{19} +(0.880262 + 5.79815i) q^{20} +(7.90280 + 4.45912i) q^{22} +(4.61190 + 4.61190i) q^{23} +(2.54442 - 2.54442i) q^{25} +(-8.58516 + 2.39158i) q^{26} +(-0.152026 - 0.251692i) q^{28} +(0.234725 - 0.566675i) q^{29} -8.85403 q^{31} +(5.56514 - 1.01452i) q^{32} +(-1.92962 - 2.46392i) q^{34} +(-0.164979 + 0.398293i) q^{35} +(-4.52845 + 1.87575i) q^{37} +(0.201892 - 0.0562413i) q^{38} +(-6.03588 - 5.68815i) q^{40} +(3.66902 + 3.66902i) q^{41} +(1.55993 + 3.76601i) q^{43} +(-12.6872 + 1.92615i) q^{44} +(-9.15633 - 1.11350i) q^{46} +6.58283i q^{47} +6.97838i q^{49} +(-0.614329 + 5.05163i) q^{50} +(7.47341 - 10.1488i) q^{52} +(3.68433 + 8.89477i) q^{53} +(13.3039 + 13.3039i) q^{55} +(0.388734 + 0.147675i) q^{56} +(0.232777 + 0.835612i) q^{58} +(4.19204 - 1.73640i) q^{59} +(2.57150 - 6.20815i) q^{61} +(9.85814 - 7.72041i) q^{62} +(-5.31164 + 5.98218i) q^{64} -18.4787 q^{65} +(-2.81795 + 6.80313i) q^{67} +(4.29691 + 1.06078i) q^{68} +(-0.163610 - 0.587319i) q^{70} +(2.74210 - 2.74210i) q^{71} +(4.87325 + 4.87325i) q^{73} +(3.40642 - 6.03712i) q^{74} +(-0.175748 + 0.238663i) q^{76} +(-0.871527 - 0.360998i) q^{77} +10.4247i q^{79} +(11.6803 + 1.07014i) q^{80} +(-7.28437 - 0.885853i) q^{82} +(-6.12008 - 2.53502i) q^{83} +(-2.48325 - 5.99509i) q^{85} +(-5.02067 - 2.83289i) q^{86} +(12.4465 - 13.2074i) q^{88} +(-7.98297 + 7.98297i) q^{89} +(0.855969 - 0.354554i) q^{91} +(11.1657 - 6.74423i) q^{92} +(-5.74000 - 7.32937i) q^{94} +0.434552 q^{95} -16.1885 q^{97} +(-6.08491 - 7.76978i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 128 q + 16 q^{10} - 32 q^{16} - 16 q^{22} - 32 q^{40} - 32 q^{46} - 80 q^{52} + 32 q^{55} - 32 q^{58} + 64 q^{61} + 48 q^{64} + 64 q^{67} - 96 q^{70} + 32 q^{76} - 80 q^{82} - 80 q^{88} + 96 q^{91} - 48 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.11341 + 0.871966i −0.787298 + 0.616573i
\(3\) 0 0
\(4\) 0.479352 − 1.94171i 0.239676 − 0.970853i
\(5\) −2.70909 + 1.12214i −1.21154 + 0.501837i −0.894711 − 0.446646i \(-0.852618\pi\)
−0.316830 + 0.948482i \(0.602618\pi\)
\(6\) 0 0
\(7\) 0.103960 − 0.103960i 0.0392931 − 0.0392931i −0.687187 − 0.726480i \(-0.741155\pi\)
0.726480 + 0.687187i \(0.241155\pi\)
\(8\) 1.15939 + 2.57989i 0.409905 + 0.912128i
\(9\) 0 0
\(10\) 2.03785 − 3.61163i 0.644425 − 1.14210i
\(11\) −2.45541 − 5.92789i −0.740335 − 1.78733i −0.604514 − 0.796594i \(-0.706633\pi\)
−0.135821 − 0.990733i \(-0.543367\pi\)
\(12\) 0 0
\(13\) 5.82208 + 2.41158i 1.61475 + 0.668853i 0.993403 − 0.114680i \(-0.0365842\pi\)
0.621351 + 0.783532i \(0.286584\pi\)
\(14\) −0.0251002 + 0.206399i −0.00670831 + 0.0551624i
\(15\) 0 0
\(16\) −3.54044 − 1.86152i −0.885111 − 0.465380i
\(17\) 2.21295i 0.536720i 0.963319 + 0.268360i \(0.0864817\pi\)
−0.963319 + 0.268360i \(0.913518\pi\)
\(18\) 0 0
\(19\) −0.136914 − 0.0567118i −0.0314103 − 0.0130106i 0.366923 − 0.930251i \(-0.380411\pi\)
−0.398333 + 0.917241i \(0.630411\pi\)
\(20\) 0.880262 + 5.79815i 0.196833 + 1.29651i
\(21\) 0 0
\(22\) 7.90280 + 4.45912i 1.68488 + 0.950688i
\(23\) 4.61190 + 4.61190i 0.961647 + 0.961647i 0.999291 − 0.0376445i \(-0.0119854\pi\)
−0.0376445 + 0.999291i \(0.511985\pi\)
\(24\) 0 0
\(25\) 2.54442 − 2.54442i 0.508884 − 0.508884i
\(26\) −8.58516 + 2.39158i −1.68369 + 0.469027i
\(27\) 0 0
\(28\) −0.152026 − 0.251692i −0.0287302 − 0.0475654i
\(29\) 0.234725 − 0.566675i 0.0435873 − 0.105229i −0.900586 − 0.434677i \(-0.856863\pi\)
0.944174 + 0.329448i \(0.106863\pi\)
\(30\) 0 0
\(31\) −8.85403 −1.59023 −0.795115 − 0.606458i \(-0.792590\pi\)
−0.795115 + 0.606458i \(0.792590\pi\)
\(32\) 5.56514 − 1.01452i 0.983787 − 0.179343i
\(33\) 0 0
\(34\) −1.92962 − 2.46392i −0.330927 − 0.422559i
\(35\) −0.164979 + 0.398293i −0.0278865 + 0.0673239i
\(36\) 0 0
\(37\) −4.52845 + 1.87575i −0.744473 + 0.308371i −0.722484 − 0.691387i \(-0.757000\pi\)
−0.0219887 + 0.999758i \(0.507000\pi\)
\(38\) 0.201892 − 0.0562413i 0.0327513 − 0.00912355i
\(39\) 0 0
\(40\) −6.03588 − 5.68815i −0.954356 − 0.899375i
\(41\) 3.66902 + 3.66902i 0.573004 + 0.573004i 0.932967 − 0.359963i \(-0.117211\pi\)
−0.359963 + 0.932967i \(0.617211\pi\)
\(42\) 0 0
\(43\) 1.55993 + 3.76601i 0.237887 + 0.574311i 0.997064 − 0.0765741i \(-0.0243982\pi\)
−0.759177 + 0.650885i \(0.774398\pi\)
\(44\) −12.6872 + 1.92615i −1.91267 + 0.290378i
\(45\) 0 0
\(46\) −9.15633 − 1.11350i −1.35003 − 0.164177i
\(47\) 6.58283i 0.960204i 0.877213 + 0.480102i \(0.159400\pi\)
−0.877213 + 0.480102i \(0.840600\pi\)
\(48\) 0 0
\(49\) 6.97838i 0.996912i
\(50\) −0.614329 + 5.05163i −0.0868793 + 0.714408i
\(51\) 0 0
\(52\) 7.47341 − 10.1488i 1.03637 − 1.40738i
\(53\) 3.68433 + 8.89477i 0.506082 + 1.22179i 0.946121 + 0.323812i \(0.104965\pi\)
−0.440039 + 0.897979i \(0.645035\pi\)
\(54\) 0 0
\(55\) 13.3039 + 13.3039i 1.79389 + 1.79389i
\(56\) 0.388734 + 0.147675i 0.0519468 + 0.0197339i
\(57\) 0 0
\(58\) 0.232777 + 0.835612i 0.0305652 + 0.109721i
\(59\) 4.19204 − 1.73640i 0.545757 − 0.226060i −0.0927318 − 0.995691i \(-0.529560\pi\)
0.638489 + 0.769631i \(0.279560\pi\)
\(60\) 0 0
\(61\) 2.57150 − 6.20815i 0.329247 − 0.794872i −0.669402 − 0.742901i \(-0.733449\pi\)
0.998649 − 0.0519716i \(-0.0165505\pi\)
\(62\) 9.85814 − 7.72041i 1.25198 − 0.980493i
\(63\) 0 0
\(64\) −5.31164 + 5.98218i −0.663955 + 0.747772i
\(65\) −18.4787 −2.29199
\(66\) 0 0
\(67\) −2.81795 + 6.80313i −0.344267 + 0.831135i 0.653007 + 0.757352i \(0.273507\pi\)
−0.997274 + 0.0737829i \(0.976493\pi\)
\(68\) 4.29691 + 1.06078i 0.521076 + 0.128639i
\(69\) 0 0
\(70\) −0.163610 − 0.587319i −0.0195551 − 0.0701980i
\(71\) 2.74210 − 2.74210i 0.325427 − 0.325427i −0.525418 − 0.850844i \(-0.676091\pi\)
0.850844 + 0.525418i \(0.176091\pi\)
\(72\) 0 0
\(73\) 4.87325 + 4.87325i 0.570371 + 0.570371i 0.932232 − 0.361861i \(-0.117859\pi\)
−0.361861 + 0.932232i \(0.617859\pi\)
\(74\) 3.40642 − 6.03712i 0.395989 − 0.701801i
\(75\) 0 0
\(76\) −0.175748 + 0.238663i −0.0201597 + 0.0273765i
\(77\) −0.871527 − 0.360998i −0.0993197 − 0.0411396i
\(78\) 0 0
\(79\) 10.4247i 1.17287i 0.809995 + 0.586437i \(0.199470\pi\)
−0.809995 + 0.586437i \(0.800530\pi\)
\(80\) 11.6803 + 1.07014i 1.30589 + 0.119646i
\(81\) 0 0
\(82\) −7.28437 − 0.885853i −0.804424 − 0.0978261i
\(83\) −6.12008 − 2.53502i −0.671766 − 0.278255i 0.0206141 − 0.999788i \(-0.493438\pi\)
−0.692380 + 0.721533i \(0.743438\pi\)
\(84\) 0 0
\(85\) −2.48325 − 5.99509i −0.269346 − 0.650258i
\(86\) −5.02067 − 2.83289i −0.541392 − 0.305479i
\(87\) 0 0
\(88\) 12.4465 − 13.2074i 1.32680 − 1.40792i
\(89\) −7.98297 + 7.98297i −0.846193 + 0.846193i −0.989656 − 0.143463i \(-0.954176\pi\)
0.143463 + 0.989656i \(0.454176\pi\)
\(90\) 0 0
\(91\) 0.855969 − 0.354554i 0.0897299 − 0.0371674i
\(92\) 11.1657 − 6.74423i 1.16410 − 0.703134i
\(93\) 0 0
\(94\) −5.74000 − 7.32937i −0.592036 − 0.755967i
\(95\) 0.434552 0.0445841
\(96\) 0 0
\(97\) −16.1885 −1.64369 −0.821845 − 0.569711i \(-0.807055\pi\)
−0.821845 + 0.569711i \(0.807055\pi\)
\(98\) −6.08491 − 7.76978i −0.614669 − 0.784867i
\(99\) 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.109.8 ✓ 128
3.2 odd 2 inner 864.2.v.b.109.25 yes 128
32.5 even 8 inner 864.2.v.b.325.8 yes 128
96.5 odd 8 inner 864.2.v.b.325.25 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.v.b.109.8 ✓ 128 1.1 even 1 trivial
864.2.v.b.109.25 yes 128 3.2 odd 2 inner
864.2.v.b.325.8 yes 128 32.5 even 8 inner
864.2.v.b.325.25 yes 128 96.5 odd 8 inner