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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 109.19
Character \(\chi\) \(=\) 864.109
Dual form 864.2.v.b.325.19

$q$-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 −0.705341 − 0.708868i \(-0.749206\pi\)
0.00249383 + 0.999997i \(0.499206\pi\)
\(6\) 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 \(0.0184861\pi\)
−0.664872 + 0.746957i \(0.731514\pi\)
\(12\) 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 \(-0.752674\pi\)
0.713022 − 0.701141i \(-0.247326\pi\)
\(18\) 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 0.759798 − 0.650159i \(-0.225298\pi\)
−0.650159 + 0.759798i \(0.725298\pi\)
\(24\) 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 \(0.262811\pi\)
−0.999190 + 0.0402362i \(0.987189\pi\)
\(30\) 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 \(0.0428322\pi\)
0.134156 + 0.990960i \(0.457168\pi\)
\(42\) 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 \(0.124886\pi\)
−0.924016 + 0.382354i \(0.875114\pi\)
\(48\) 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 0.924213 − 0.381878i \(-0.124722\pi\)
−0.923546 + 0.383489i \(0.874722\pi\)
\(54\) 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 0.277859 − 0.960622i \(-0.410375\pi\)
0.875738 + 0.482786i \(0.160375\pi\)
\(60\) 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 −0.995050 − 0.0993754i \(-0.968316\pi\)
0.0993754 + 0.995050i \(0.468316\pi\)
\(72\) 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 −0.477344 − 0.878717i \(-0.658400\pi\)
−0.958880 + 0.283813i \(0.908400\pi\)
\(84\) 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 \(-0.369320\pi\)
0.916904 − 0.399108i \(-0.130680\pi\)
\(90\) 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.109.19 yes 128
3.2 odd 2 inner 864.2.v.b.109.14 ✓ 128
32.5 even 8 inner 864.2.v.b.325.19 yes 128
96.5 odd 8 inner 864.2.v.b.325.14 yes 128
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.v.b.109.14 ✓ 128 3.2 odd 2 inner
864.2.v.b.109.19 yes 128 1.1 even 1 trivial
864.2.v.b.325.14 yes 128 96.5 odd 8 inner
864.2.v.b.325.19 yes 128 32.5 even 8 inner