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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 109.14
Character \(\chi\) \(=\) 864.109
Dual form 864.2.v.b.325.14

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.322202 - 1.37702i) q^{2} +(-1.79237 + 0.887359i) q^{4} +(1.57161 - 0.650984i) q^{5} +(3.07534 - 3.07534i) q^{7} +(1.79942 + 2.18222i) q^{8} +(-1.40280 - 1.95440i) q^{10} +(-1.10590 - 2.66989i) q^{11} +(4.29788 + 1.78024i) q^{13} +(-5.22569 - 3.24393i) q^{14} +(2.42519 - 3.18095i) q^{16} +5.78176i q^{17} +(1.92883 + 0.798948i) q^{19} +(-2.23926 + 2.56139i) q^{20} +(-3.32016 + 2.38309i) q^{22} +(-0.525810 - 0.525810i) q^{23} +(-1.48934 + 1.48934i) q^{25} +(1.06664 - 6.49187i) q^{26} +(-2.78322 + 8.24109i) q^{28} +(1.72922 - 4.17470i) q^{29} +6.80754 q^{31} +(-5.16164 - 2.31462i) q^{32} +(7.96160 - 1.86290i) q^{34} +(2.83125 - 6.83525i) q^{35} +(-1.09955 + 0.455448i) q^{37} +(0.478693 - 2.91346i) q^{38} +(4.24858 + 2.25822i) q^{40} +(-7.20426 - 7.20426i) q^{41} +(0.960366 + 2.31853i) q^{43} +(4.35133 + 3.80409i) q^{44} +(-0.554634 + 0.893469i) q^{46} -5.24257i q^{47} -11.9155i q^{49} +(2.53072 + 1.57099i) q^{50} +(-9.28311 + 0.622911i) q^{52} +(-0.00485782 - 0.0117278i) q^{53} +(-3.47611 - 3.47611i) q^{55} +(12.2449 + 1.17726i) q^{56} +(-6.30580 - 1.03607i) q^{58} +(-8.86095 + 3.67033i) q^{59} +(0.439229 - 1.06039i) q^{61} +(-2.19341 - 9.37412i) q^{62} +(-1.52419 + 7.85346i) q^{64} +7.91352 q^{65} +(-1.17849 + 2.84512i) q^{67} +(-5.13049 - 10.3631i) q^{68} +(-10.3245 - 1.69636i) q^{70} +(7.54709 - 7.54709i) q^{71} +(-9.04188 - 9.04188i) q^{73} +(0.981439 + 1.36735i) q^{74} +(-4.16613 + 0.279554i) q^{76} +(-11.6118 - 4.80978i) q^{77} -8.85007i q^{79} +(1.74071 - 6.57799i) q^{80} +(-7.59918 + 12.2416i) q^{82} +(13.0846 + 5.41983i) q^{83} +(3.76383 + 9.08669i) q^{85} +(2.88323 - 2.06948i) q^{86} +(3.83630 - 7.21756i) q^{88} +(-12.4152 + 12.4152i) q^{89} +(18.6923 - 7.74261i) q^{91} +(1.40903 + 0.475865i) q^{92} +(-7.21913 + 1.68917i) q^{94} +3.55148 q^{95} +9.47707 q^{97} +(-16.4078 + 3.83919i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 128 q + 16 q^{10} - 32 q^{16} - 16 q^{22} - 32 q^{40} - 32 q^{46} - 80 q^{52} + 32 q^{55} - 32 q^{58} + 64 q^{61} + 48 q^{64} + 64 q^{67} - 96 q^{70} + 32 q^{76} - 80 q^{82} - 80 q^{88} + 96 q^{91} - 48 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.322202 − 1.37702i −0.227832 − 0.973701i
\(3\) 0 0
\(4\) −1.79237 + 0.887359i −0.896186 + 0.443679i
\(5\) 1.57161 − 0.650984i 0.702847 − 0.291129i −0.00249383 − 0.999997i \(-0.500794\pi\)
0.705341 + 0.708868i \(0.250794\pi\)
\(6\) 0 0
\(7\) 3.07534 − 3.07534i 1.16237 − 1.16237i 0.178415 − 0.983955i \(-0.442903\pi\)
0.983955 − 0.178415i \(-0.0570969\pi\)
\(8\) 1.79942 + 2.18222i 0.636190 + 0.771532i
\(9\) 0 0
\(10\) −1.40280 − 1.95440i −0.443603 − 0.618035i
\(11\) −1.10590 − 2.66989i −0.333442 − 0.805001i −0.998314 − 0.0580432i \(-0.981514\pi\)
0.664872 − 0.746957i \(-0.268486\pi\)
\(12\) 0 0
\(13\) 4.29788 + 1.78024i 1.19202 + 0.493750i 0.888412 − 0.459047i \(-0.151809\pi\)
0.303606 + 0.952798i \(0.401809\pi\)
\(14\) −5.22569 − 3.24393i −1.39663 − 0.866976i
\(15\) 0 0
\(16\) 2.42519 − 3.18095i 0.606297 − 0.795238i
\(17\) 5.78176i 1.40228i 0.713022 + 0.701141i \(0.247326\pi\)
−0.713022 + 0.701141i \(0.752674\pi\)
\(18\) 0 0
\(19\) 1.92883 + 0.798948i 0.442504 + 0.183291i 0.592800 − 0.805350i \(-0.298023\pi\)
−0.150296 + 0.988641i \(0.548023\pi\)
\(20\) −2.23926 + 2.56139i −0.500714 + 0.572744i
\(21\) 0 0
\(22\) −3.32016 + 2.38309i −0.707861 + 0.508077i
\(23\) −0.525810 − 0.525810i −0.109639 − 0.109639i 0.650159 − 0.759798i \(-0.274702\pi\)
−0.759798 + 0.650159i \(0.774702\pi\)
\(24\) 0 0
\(25\) −1.48934 + 1.48934i −0.297868 + 0.297868i
\(26\) 1.06664 − 6.49187i 0.209185 − 1.27316i
\(27\) 0 0
\(28\) −2.78322 + 8.24109i −0.525980 + 1.55742i
\(29\) 1.72922 − 4.17470i 0.321107 − 0.775222i −0.678083 − 0.734985i \(-0.737189\pi\)
0.999190 − 0.0402362i \(-0.0128110\pi\)
\(30\) 0 0
\(31\) 6.80754 1.22267 0.611335 − 0.791372i \(-0.290633\pi\)
0.611335 + 0.791372i \(0.290633\pi\)
\(32\) −5.16164 − 2.31462i −0.912457 − 0.409172i
\(33\) 0 0
\(34\) 7.96160 − 1.86290i 1.36540 − 0.319484i
\(35\) 2.83125 − 6.83525i 0.478569 − 1.15537i
\(36\) 0 0
\(37\) −1.09955 + 0.455448i −0.180765 + 0.0748752i −0.471230 − 0.882010i \(-0.656190\pi\)
0.290465 + 0.956885i \(0.406190\pi\)
\(38\) 0.478693 − 2.91346i 0.0776543 − 0.472626i
\(39\) 0 0
\(40\) 4.24858 + 2.25822i 0.671760 + 0.357056i
\(41\) −7.20426 − 7.20426i −1.12512 − 1.12512i −0.990960 − 0.134156i \(-0.957168\pi\)
−0.134156 − 0.990960i \(-0.542832\pi\)
\(42\) 0 0
\(43\) 0.960366 + 2.31853i 0.146455 + 0.353572i 0.980035 − 0.198826i \(-0.0637128\pi\)
−0.833580 + 0.552398i \(0.813713\pi\)
\(44\) 4.35133 + 3.80409i 0.655988 + 0.573489i
\(45\) 0 0
\(46\) −0.554634 + 0.893469i −0.0817764 + 0.131735i
\(47\) − 5.24257i − 0.764708i −0.924016 − 0.382354i \(-0.875114\pi\)
0.924016 − 0.382354i \(-0.124886\pi\)
\(48\) 0 0
\(49\) − 11.9155i − 1.70221i
\(50\) 2.53072 + 1.57099i 0.357898 + 0.222171i
\(51\) 0 0
\(52\) −9.28311 + 0.622911i −1.28734 + 0.0863822i
\(53\) −0.00485782 − 0.0117278i −0.000667273 − 0.00161094i 0.923546 − 0.383489i \(-0.125278\pi\)
−0.924213 + 0.381878i \(0.875278\pi\)
\(54\) 0 0
\(55\) −3.47611 − 3.47611i −0.468718 − 0.468718i
\(56\) 12.2449 + 1.17726i 1.63629 + 0.157318i
\(57\) 0 0
\(58\) −6.30580 − 1.03607i −0.827992 − 0.136042i
\(59\) −8.86095 + 3.67033i −1.15360 + 0.477836i −0.875738 − 0.482786i \(-0.839625\pi\)
−0.277859 + 0.960622i \(0.589625\pi\)
\(60\) 0 0
\(61\) 0.439229 − 1.06039i 0.0562375 − 0.135769i −0.893263 − 0.449534i \(-0.851590\pi\)
0.949501 + 0.313764i \(0.101590\pi\)
\(62\) −2.19341 − 9.37412i −0.278563 − 1.19051i
\(63\) 0 0
\(64\) −1.52419 + 7.85346i −0.190524 + 0.981683i
\(65\) 7.91352 0.981552
\(66\) 0 0
\(67\) −1.17849 + 2.84512i −0.143975 + 0.347587i −0.979374 − 0.202058i \(-0.935237\pi\)
0.835398 + 0.549645i \(0.185237\pi\)
\(68\) −5.13049 − 10.3631i −0.622164 − 1.25671i
\(69\) 0 0
\(70\) −10.3245 − 1.69636i −1.23402 − 0.202754i
\(71\) 7.54709 − 7.54709i 0.895675 − 0.895675i −0.0993754 − 0.995050i \(-0.531684\pi\)
0.995050 + 0.0993754i \(0.0316845\pi\)
\(72\) 0 0
\(73\) −9.04188 − 9.04188i −1.05827 − 1.05827i −0.998194 − 0.0600782i \(-0.980865\pi\)
−0.0600782 − 0.998194i \(-0.519135\pi\)
\(74\) 0.981439 + 1.36735i 0.114090 + 0.158952i
\(75\) 0 0
\(76\) −4.16613 + 0.279554i −0.477888 + 0.0320670i
\(77\) −11.6118 − 4.80978i −1.32329 − 0.548126i
\(78\) 0 0
\(79\) − 8.85007i − 0.995711i −0.867260 − 0.497855i \(-0.834121\pi\)
0.867260 − 0.497855i \(-0.165879\pi\)
\(80\) 1.74071 − 6.57799i 0.194618 − 0.735442i
\(81\) 0 0
\(82\) −7.59918 + 12.2416i −0.839189 + 1.35186i
\(83\) 13.0846 + 5.41983i 1.43622 + 0.594903i 0.958880 − 0.283813i \(-0.0915997\pi\)
0.477344 + 0.878717i \(0.341600\pi\)
\(84\) 0 0
\(85\) 3.76383 + 9.08669i 0.408245 + 0.985590i
\(86\) 2.88323 − 2.06948i 0.310907 − 0.223158i
\(87\) 0 0
\(88\) 3.83630 − 7.21756i 0.408951 − 0.769395i
\(89\) −12.4152 + 12.4152i −1.31601 + 1.31601i −0.399108 + 0.916904i \(0.630680\pi\)
−0.916904 + 0.399108i \(0.869320\pi\)
\(90\) 0 0
\(91\) 18.6923 − 7.74261i 1.95949 − 0.811646i
\(92\) 1.40903 + 0.475865i 0.146901 + 0.0496123i
\(93\) 0 0
\(94\) −7.21913 + 1.68917i −0.744597 + 0.174225i
\(95\) 3.55148 0.364374
\(96\) 0 0
\(97\) 9.47707 0.962250 0.481125 − 0.876652i \(-0.340228\pi\)
0.481125 + 0.876652i \(0.340228\pi\)
\(98\) −16.4078 + 3.83919i −1.65744 + 0.387817i
\(99\) 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.109.14 ✓ 128
3.2 odd 2 inner 864.2.v.b.109.19 yes 128
32.5 even 8 inner 864.2.v.b.325.14 yes 128
96.5 odd 8 inner 864.2.v.b.325.19 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.v.b.109.14 ✓ 128 1.1 even 1 trivial
864.2.v.b.109.19 yes 128 3.2 odd 2 inner
864.2.v.b.325.14 yes 128 32.5 even 8 inner
864.2.v.b.325.19 yes 128 96.5 odd 8 inner