LMFDB - Embedded newform 1692.2.h.a.845.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1692,2,Mod(845,1692)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1692, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1692.845");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1692 = 2^{2} \cdot 3^{2} \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1692.h (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$13.5106880220$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 26 x^{14} - 16 x^{13} + 269 x^{12} + 288 x^{11} + 850 x^{10} + 2032 x^{9} + 6628 x^{8} + \cdots + 253609 $ x^16 - 26x^14 - 16x^13 + 269x^12 + 288x^11 + 850x^10 + 2032x^9 + 6628x^8 + 7312x^7 + 53790x^6 + 18944x^5 + 201613x^4 + 27344x^3 + 355010x^2 + 13984x + 253609
Coefficient ring:	$\Z[a_1, \ldots, a_{47}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			845.11
Root			$0.457675 + 1.41421i$ of defining polynomial
Character	$\chi$	$=$	1692.845
Dual form			1692.2.h.a.845.12

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.591052 q^{5} -1.72728 q^{7} +O(q^{10})$	$q+0.591052 q^{5} -1.72728 q^{7} -2.64263 q^{11} -2.59052i q^{13} +2.32583i q^{17} +2.39270i q^{19} +6.02643 q^{23} -4.65066 q^{25} -6.45629 q^{29} -4.14735i q^{31} -1.02091 q^{35} -8.93987 q^{37} -9.09892 q^{41} +3.53943i q^{43} +(3.07249 + 6.12860i) q^{47} -4.01650 q^{49} +12.7598i q^{53} -1.56193 q^{55} -1.29730i q^{59} -8.66715 q^{61} -1.53113i q^{65} -15.5714i q^{67} +6.57702i q^{71} -16.2942i q^{73} +4.56457 q^{77} +3.72728 q^{79} -5.15425i q^{83} +1.37469i q^{85} +5.68019i q^{89} +4.47455i q^{91} +1.41421i q^{95} -9.10522 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 16 q^{25} + 8 q^{37} + 24 q^{49} + 8 q^{55} + 40 q^{61} + 32 q^{79}+O(q^{100}) $ 16 * q + 16 * q^25 + 8 * q^37 + 24 * q^49 + 8 * q^55 + 40 * q^61 + 32 * q^79

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1692\mathbb{Z}\right)^\times$.

$n$	$757$	$847$	$1505$
$\chi(n)$	$-1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.591052			0.264327			0.132163	−	0.991228i	$-0.457808\pi$
$5$	0.591052			0.264327			0.132163	+	0.991228i	$0.457808\pi$
$6$		0			0
$7$	−1.72728			−0.652851			−0.326426	−	0.945223i	$-0.605844\pi$
$7$	−1.72728			−0.652851			−0.326426	+	0.945223i	$0.605844\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.64263			−0.796783			−0.398392	−	0.917215i	$-0.630432\pi$
$11$	−2.64263			−0.796783			−0.398392	+	0.917215i	$0.630432\pi$
$12$		0			0
$13$		−	2.59052i		−	0.718480i	−0.933245	−	0.359240i	$-0.883036\pi$
$13$		−	2.59052i		−	0.718480i	0.933245	−	0.359240i	$-0.116964\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			2.32583i			0.564096i	0.959400	+	0.282048i	$0.0910137\pi$
$17$			2.32583i			0.564096i	−0.959400	+	0.282048i	$0.908986\pi$
$18$		0			0
$19$			2.39270i			0.548924i	0.961598	+	0.274462i	$0.0884997\pi$
$19$			2.39270i			0.548924i	−0.961598	+	0.274462i	$0.911500\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	6.02643			1.25660			0.628298	−	0.777972i	$-0.283752\pi$
$23$	6.02643			1.25660			0.628298	+	0.777972i	$0.283752\pi$
$24$		0			0
$25$	−4.65066			−0.930131
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−6.45629			−1.19890			−0.599451	−	0.800411i	$-0.704614\pi$
$29$	−6.45629			−1.19890			−0.599451	+	0.800411i	$0.704614\pi$
$30$		0			0
$31$		−	4.14735i		−	0.744886i	−0.928055	−	0.372443i	$-0.878520\pi$
$31$		−	4.14735i		−	0.744886i	0.928055	−	0.372443i	$-0.121480\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.02091			−0.172566
$36$		0			0
$37$	−8.93987			−1.46971			−0.734853	−	0.678226i	$-0.762749\pi$
$37$	−8.93987			−1.46971			−0.734853	+	0.678226i	$0.762749\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−9.09892			−1.42101			−0.710506	−	0.703691i	$-0.751534\pi$
$41$	−9.09892			−1.42101			−0.710506	+	0.703691i	$0.751534\pi$
$42$		0			0
$43$			3.53943i			0.539758i	0.962894	+	0.269879i	$0.0869837\pi$
$43$			3.53943i			0.539758i	−0.962894	+	0.269879i	$0.913016\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.07249	+	6.12860i	0.448169	+	0.893949i
$47$	3.07249	+	6.12860i	0.448169	+	0.893949i
$48$		0			0
$49$	−4.01650			−0.573785
$50$		0			0
$51$		0			0
$52$		0			0
$53$			12.7598i			1.75269i	0.481680	+	0.876347i	$0.340027\pi$
$53$			12.7598i			1.75269i	−0.481680	+	0.876347i	$0.659973\pi$
$54$		0			0
$55$	−1.56193			−0.210611
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	1.29730i		−	0.168893i	−0.996428	−	0.0844467i	$-0.973088\pi$
$59$		−	1.29730i		−	0.168893i	0.996428	−	0.0844467i	$-0.0269123\pi$
$60$		0			0
$61$	−8.66715			−1.10972			−0.554858	−	0.831945i	$-0.687227\pi$
$61$	−8.66715			−1.10972			−0.554858	+	0.831945i	$0.687227\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	1.53113i		−	0.189914i
$66$		0			0
$67$		−	15.5714i		−	1.90235i	−0.308658	−	0.951173i	$-0.599880\pi$
$67$		−	15.5714i		−	1.90235i	0.308658	−	0.951173i	$-0.400120\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			6.57702i			0.780549i	0.920699	+	0.390274i	$0.127620\pi$
$71$			6.57702i			0.780549i	−0.920699	+	0.390274i	$0.872380\pi$
$72$		0			0
$73$		−	16.2942i		−	1.90709i	−0.301246	−	0.953546i	$-0.597403\pi$
$73$		−	16.2942i		−	1.90709i	0.301246	−	0.953546i	$-0.402597\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	4.56457			0.520181
$78$		0			0
$79$	3.72728			0.419352			0.209676	−	0.977771i	$-0.432759\pi$
$79$	3.72728			0.419352			0.209676	+	0.977771i	$0.432759\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	5.15425i		−	0.565753i	−0.959156	−	0.282876i	$-0.908711\pi$
$83$		−	5.15425i		−	0.565753i	0.959156	−	0.282876i	$-0.0912886\pi$
$84$		0			0
$85$			1.37469i			0.149106i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			5.68019i			0.602098i	0.953609	+	0.301049i	$0.0973368\pi$
$89$			5.68019i			0.602098i	−0.953609	+	0.301049i	$0.902663\pi$
$90$		0			0
$91$			4.47455i			0.469061i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			1.41421i			0.145095i
$96$		0			0
$97$	−9.10522			−0.924495			−0.462248	−	0.886751i	$-0.652957\pi$
$97$	−9.10522			−0.924495			−0.462248	+	0.886751i	$0.652957\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			4.13428i			0.411376i	0.978618	+	0.205688i	$0.0659432\pi$
$101$			4.13428i			0.411376i	−0.978618	+	0.205688i	$0.934057\pi$
$102$		0			0
$103$	−7.01650			−0.691356			−0.345678	−	0.938353i	$-0.612351\pi$
$103$	−7.01650			−0.691356			−0.345678	+	0.938353i	$0.612351\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−13.3535			−1.29093			−0.645467	−	0.763788i	$-0.723337\pi$
$107$	−13.3535			−1.29093			−0.645467	+	0.763788i	$0.723337\pi$
$108$		0			0
$109$			7.60388i			0.728320i	0.931336	+	0.364160i	$0.118644\pi$
$109$			7.60388i			0.728320i	−0.931336	+	0.364160i	$0.881356\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	8.37956			0.788282			0.394141	−	0.919050i	$-0.371042\pi$
$113$	8.37956			0.788282			0.394141	+	0.919050i	$0.371042\pi$
$114$		0			0
$115$	3.56193			0.332152
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	4.01736i		−	0.368271i
$120$		0			0
$121$	−4.01650			−0.365136
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−5.70404			−0.510185
$126$		0			0
$127$			5.08179i			0.450936i	0.974251	+	0.225468i	$0.0723911\pi$
$127$			5.08179i			0.450936i	−0.974251	+	0.225468i	$0.927609\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	6.96270i		−	0.608334i	−0.952619	−	0.304167i	$-0.901622\pi$
$131$		−	6.96270i		−	0.608334i	0.952619	−	0.304167i	$-0.0983781\pi$
$132$		0			0
$133$		−	4.13287i		−	0.358366i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	8.51895			0.727823			0.363911	−	0.931434i	$-0.381441\pi$
$137$	8.51895			0.727823			0.363911	+	0.931434i	$0.381441\pi$
$138$		0			0
$139$			19.8180i			1.68094i	0.541859	+	0.840469i	$0.317720\pi$
$139$			19.8180i			1.68094i	−0.541859	+	0.840469i	$0.682280\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			6.84579i			0.572473i
$144$		0			0
$145$	−3.81600			−0.316902
$146$		0			0
$147$		0			0
$148$		0			0
$149$			3.47128i			0.284378i	0.989840	+	0.142189i	$0.0454141\pi$
$149$			3.47128i			0.284378i	−0.989840	+	0.142189i	$0.954586\pi$
$150$		0			0
$151$		−	13.8186i		−	1.12454i	−0.826953	−	0.562271i	$-0.809928\pi$
$151$		−	13.8186i		−	1.12454i	0.826953	−	0.562271i	$-0.190072\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	2.45130i		−	0.196893i
$156$		0			0
$157$	−2.85115			−0.227547			−0.113773	−	0.993507i	$-0.536294\pi$
$157$	−2.85115			−0.227547			−0.113773	+	0.993507i	$0.536294\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−10.4093			−0.820371
$162$		0			0
$163$			0.722836i			0.0566168i	0.999599	+	0.0283084i	$0.00901206\pi$
$163$			0.722836i			0.0566168i	−0.999599	+	0.0283084i	$0.990988\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−15.5978			−1.20700			−0.603498	−	0.797364i	$-0.706227\pi$
$167$	−15.5978			−1.20700			−0.603498	+	0.797364i	$0.706227\pi$
$168$		0			0
$169$	6.28922			0.483786
$170$		0			0
$171$		0			0
$172$		0			0
$173$			15.3109i			1.16407i	0.813165	+	0.582034i	$0.197743\pi$
$173$			15.3109i			1.16407i	−0.813165	+	0.582034i	$0.802257\pi$
$174$		0			0
$175$	8.03300			0.607237
$176$		0			0
$177$		0			0
$178$		0			0
$179$	12.4938			0.933830			0.466915	−	0.884302i	$-0.345365\pi$
$179$	12.4938			0.933830			0.466915	+	0.884302i	$0.345365\pi$
$180$		0			0
$181$		−	16.2942i		−	1.21114i	−0.795792	−	0.605569i	$-0.792945\pi$
$181$		−	16.2942i		−	1.21114i	0.795792	−	0.605569i	$-0.207055\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−5.28393			−0.388483
$186$		0			0
$187$		−	6.14630i		−	0.449462i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	21.8879i		−	1.58376i	−0.610680	−	0.791878i	$-0.709104\pi$
$191$		−	21.8879i		−	1.58376i	0.610680	−	0.791878i	$-0.290896\pi$
$192$		0			0
$193$			6.76801i			0.487172i	0.969879	+	0.243586i	$0.0783238\pi$
$193$			6.76801i			0.487172i	−0.969879	+	0.243586i	$0.921676\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	3.05371i		−	0.217568i	−0.994065	−	0.108784i	$-0.965304\pi$
$197$		−	3.05371i		−	0.217568i	0.994065	−	0.108784i	$-0.0346957\pi$
$198$		0			0
$199$		−	6.88105i		−	0.487785i	−0.969802	−	0.243892i	$-0.921576\pi$
$199$		−	6.88105i		−	0.487785i	0.969802	−	0.243892i	$-0.0784243\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	11.1518			0.782705
$204$		0			0
$205$	−5.37794			−0.375612
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	6.32304i		−	0.437374i
$210$		0			0
$211$			12.7704i			0.879154i	0.898205	+	0.439577i	$0.144872\pi$
$211$			12.7704i			0.879154i	−0.898205	+	0.439577i	$0.855128\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			2.09199i			0.142673i
$216$		0			0
$217$			7.16364i			0.486299i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	6.02510			0.405292
$222$		0			0
$223$			20.0151i			1.34031i	0.742221	+	0.670155i	$0.233772\pi$
$223$			20.0151i			1.34031i	−0.742221	+	0.670155i	$0.766228\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	9.36759			0.621749			0.310874	−	0.950451i	$-0.399378\pi$
$227$	9.36759			0.621749			0.310874	+	0.950451i	$0.399378\pi$
$228$		0			0
$229$		−	23.6839i		−	1.56508i	−0.622602	−	0.782539i	$-0.713924\pi$
$229$		−	23.6839i		−	1.56508i	0.622602	−	0.782539i	$-0.286076\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−16.1476			−1.05786			−0.528932	−	0.848664i	$-0.677407\pi$
$233$	−16.1476			−1.05786			−0.528932	+	0.848664i	$0.677407\pi$
$234$		0			0
$235$	1.81600	+	3.62233i	0.118463	+	0.236295i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	16.7570i		−	1.08392i	−0.840404	−	0.541961i	$-0.817682\pi$
$239$		−	16.7570i		−	1.08392i	0.840404	−	0.541961i	$-0.182318\pi$
$240$		0			0
$241$	20.4832			1.31944			0.659718	−	0.751513i	$-0.270676\pi$
$241$	20.4832			1.31944			0.659718	+	0.751513i	$0.270676\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2.37396			−0.151667
$246$		0			0
$247$	6.19834			0.394391
$248$		0			0
$249$		0			0
$250$		0			0
$251$			15.3109i			0.966417i	0.875505	+	0.483208i	$0.160529\pi$
$251$			15.3109i			0.966417i	−0.875505	+	0.483208i	$0.839471\pi$
$252$		0			0
$253$	−15.9256			−1.00124
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−12.0418			−0.751145			−0.375573	−	0.926793i	$-0.622554\pi$
$257$	−12.0418			−0.751145			−0.375573	+	0.926793i	$0.622554\pi$
$258$		0			0
$259$	15.4417			0.959499
$260$		0			0
$261$		0			0
$262$		0			0
$263$			10.5276i			0.649158i	0.945859	+	0.324579i	$0.105223\pi$
$263$			10.5276i			0.649158i	−0.945859	+	0.324579i	$0.894777\pi$
$264$		0			0
$265$			7.54172i			0.463284i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	2.33742i		−	0.142515i	−0.997458	−	0.0712576i	$-0.977299\pi$
$269$		−	2.33742i		−	0.142515i	0.997458	−	0.0712576i	$-0.0227012\pi$
$270$		0			0
$271$	−6.43807			−0.391085			−0.195542	−	0.980695i	$-0.562647\pi$
$271$	−6.43807			−0.391085			−0.195542	+	0.980695i	$0.562647\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	12.2900			0.741113
$276$		0			0
$277$	−14.3365			−0.861394			−0.430697	−	0.902497i	$-0.641732\pi$
$277$	−14.3365			−0.861394			−0.430697	+	0.902497i	$0.641732\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	30.2191			1.80272			0.901361	−	0.433068i	$-0.142569\pi$
$281$	30.2191			1.80272			0.901361	+	0.433068i	$0.142569\pi$
$282$		0			0
$283$	5.01650			0.298200			0.149100	−	0.988822i	$-0.452362\pi$
$283$	5.01650			0.298200			0.149100	+	0.988822i	$0.452362\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	15.7164			0.927710
$288$		0			0
$289$	11.5905			0.681796
$290$		0			0
$291$		0			0
$292$		0			0
$293$	9.99994			0.584203			0.292101	−	0.956387i	$-0.405646\pi$
$293$	9.99994			0.584203			0.292101	+	0.956387i	$0.405646\pi$
$294$		0			0
$295$		−	0.766769i		−	0.0446430i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	15.6116i		−	0.902840i
$300$		0			0
$301$		−	6.11360i		−	0.352382i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−5.12274			−0.293327
$306$		0			0
$307$	29.9707			1.71052			0.855260	−	0.518200i	$-0.173398\pi$
$307$	29.9707			1.71052			0.855260	+	0.518200i	$0.173398\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	8.20765			0.465413			0.232706	−	0.972547i	$-0.425242\pi$
$311$	8.20765			0.465413			0.232706	+	0.972547i	$0.425242\pi$
$312$		0			0
$313$			24.5217i			1.38605i	0.720915	+	0.693024i	$0.243722\pi$
$313$			24.5217i			1.38605i	−0.720915	+	0.693024i	$0.756278\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−8.38089			−0.470717			−0.235359	−	0.971909i	$-0.575626\pi$
$317$	−8.38089			−0.470717			−0.235359	+	0.971909i	$0.575626\pi$
$318$		0			0
$319$	17.0616			0.955266
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−5.56501			−0.309646
$324$		0			0
$325$			12.0476i			0.668281i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−5.30706	−	10.5858i	−0.292588	−	0.583615i
$330$		0			0
$331$	9.68140			0.532138			0.266069	−	0.963954i	$-0.414275\pi$
$331$	9.68140			0.532138			0.266069	+	0.963954i	$0.414275\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	9.20350i		−	0.502841i
$336$		0			0
$337$	6.47106			0.352501			0.176251	−	0.984345i	$-0.443603\pi$
$337$	6.47106			0.352501			0.176251	+	0.984345i	$0.443603\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			10.9599i			0.593513i
$342$		0			0
$343$	19.0286			1.02745
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	20.9911i		−	1.12686i	−0.826163	−	0.563431i	$-0.809481\pi$
$347$		−	20.9911i		−	1.12686i	0.826163	−	0.563431i	$-0.190519\pi$
$348$		0			0
$349$			17.8674i			0.956420i	0.878246	+	0.478210i	$0.158714\pi$
$349$			17.8674i			0.956420i	−0.878246	+	0.478210i	$0.841286\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	29.5284i		−	1.57164i	−0.618456	−	0.785820i	$-0.712241\pi$
$353$		−	29.5284i		−	1.57164i	0.618456	−	0.785820i	$-0.287759\pi$
$354$		0			0
$355$			3.88737i			0.206320i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−22.6247			−1.19409			−0.597043	−	0.802209i	$-0.703658\pi$
$359$	−22.6247			−1.19409			−0.597043	+	0.802209i	$0.703658\pi$
$360$		0			0
$361$	13.2750			0.698683
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	9.63073i		−	0.504096i
$366$		0			0
$367$			17.0170i			0.888282i	0.895957	+	0.444141i	$0.146491\pi$
$367$			17.0170i			0.888282i	−0.895957	+	0.444141i	$0.853509\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	22.0398i		−	1.14425i
$372$		0			0
$373$		−	16.1629i		−	0.836885i	−0.908243	−	0.418443i	$-0.862576\pi$
$373$		−	16.1629i		−	0.836885i	0.908243	−	0.418443i	$-0.137424\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			16.7251i			0.861388i
$378$		0			0
$379$	4.15246			0.213298			0.106649	−	0.994297i	$-0.465988\pi$
$379$	4.15246			0.213298			0.106649	+	0.994297i	$0.465988\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	30.9077i		−	1.57931i	−0.613551	−	0.789655i	$-0.710260\pi$
$383$		−	30.9077i		−	1.57931i	0.613551	−	0.789655i	$-0.289740\pi$
$384$		0			0
$385$	2.69790			0.137498
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−3.39620			−0.172194			−0.0860972	−	0.996287i	$-0.527440\pi$
$389$	−3.39620			−0.172194			−0.0860972	+	0.996287i	$0.527440\pi$
$390$		0			0
$391$			14.0164i			0.708841i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	2.20302			0.110846
$396$		0			0
$397$	−2.36369			−0.118630			−0.0593151	−	0.998239i	$-0.518892\pi$
$397$	−2.36369			−0.118630			−0.0593151	+	0.998239i	$0.518892\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			16.2078i			0.809377i	0.914455	+	0.404688i	$0.132620\pi$
$401$			16.2078i			0.809377i	−0.914455	+	0.404688i	$0.867380\pi$
$402$		0			0
$403$	−10.7438			−0.535186
$404$		0			0
$405$		0			0
$406$		0			0
$407$	23.6248			1.17104
$408$		0			0
$409$			1.08644i			0.0537210i	0.999639	+	0.0268605i	$0.00855099\pi$
$409$			1.08644i			0.0537210i	−0.999639	+	0.0268605i	$0.991449\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			2.24079i			0.110262i
$414$		0			0
$415$		−	3.04643i		−	0.149544i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	9.92709			0.484970			0.242485	−	0.970155i	$-0.422037\pi$
$419$	9.92709			0.484970			0.242485	+	0.970155i	$0.422037\pi$
$420$		0			0
$421$		−	35.7486i		−	1.74228i	−0.491036	−	0.871139i	$-0.663382\pi$
$421$		−	35.7486i		−	1.74228i	0.491036	−	0.871139i	$-0.336618\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	10.8166i		−	0.524683i
$426$		0			0
$427$	14.9706			0.724479
$428$		0			0
$429$		0			0
$430$		0			0
$431$			31.0394i			1.49512i	0.664197	+	0.747558i	$0.268774\pi$
$431$			31.0394i			1.49512i	−0.664197	+	0.747558i	$0.731226\pi$
$432$		0			0
$433$		−	19.6515i		−	0.944391i	−0.881494	−	0.472195i	$-0.843462\pi$
$433$		−	19.6515i		−	0.944391i	0.881494	−	0.472195i	$-0.156538\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			14.4195i			0.689776i
$438$		0			0
$439$	24.5041			1.16951			0.584757	−	0.811208i	$-0.301190\pi$
$439$	24.5041			1.16951			0.584757	+	0.811208i	$0.301190\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	1.02224			0.0485683			0.0242841	−	0.999705i	$-0.492269\pi$
$443$	1.02224			0.0485683			0.0242841	+	0.999705i	$0.492269\pi$
$444$		0			0
$445$			3.35729i			0.159151i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	41.3901			1.95332			0.976660	−	0.214791i	$-0.0689071\pi$
$449$	41.3901			1.95332			0.976660	+	0.214791i	$0.0689071\pi$
$450$		0			0
$451$	24.0451			1.13224
$452$		0			0
$453$		0			0
$454$		0			0
$455$			2.64470i			0.123985i
$456$		0			0
$457$	−12.5762			−0.588289			−0.294144	−	0.955761i	$-0.595035\pi$
$457$	−12.5762			−0.588289			−0.294144	+	0.955761i	$0.595035\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	13.9459			0.649525			0.324763	−	0.945796i	$-0.394716\pi$
$461$	13.9459			0.649525			0.324763	+	0.945796i	$0.394716\pi$
$462$		0			0
$463$			16.1774i			0.751828i	0.926655	+	0.375914i	$0.122671\pi$
$463$			16.1774i			0.751828i	−0.926655	+	0.375914i	$0.877329\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−5.98380			−0.276897			−0.138449	−	0.990370i	$-0.544212\pi$
$467$	−5.98380			−0.276897			−0.138449	+	0.990370i	$0.544212\pi$
$468$		0			0
$469$			26.8961i			1.24195i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	9.35341i		−	0.430070i
$474$		0			0
$475$		−	11.1276i		−	0.510571i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	32.7076i		−	1.49445i	−0.664572	−	0.747224i	$-0.731386\pi$
$479$		−	32.7076i		−	1.49445i	0.664572	−	0.747224i	$-0.268614\pi$
$480$		0			0
$481$			23.1589i			1.05596i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−5.38166			−0.244369
$486$		0			0
$487$	1.26832			0.0574730			0.0287365	−	0.999587i	$-0.490852\pi$
$487$	1.26832			0.0574730			0.0287365	+	0.999587i	$0.490852\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			11.9682i			0.540115i	0.962844	+	0.270058i	$0.0870427\pi$
$491$			11.9682i			0.540115i	−0.962844	+	0.270058i	$0.912957\pi$
$492$		0			0
$493$		−	15.0162i		−	0.676296i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	11.3604i		−	0.509582i
$498$		0			0
$499$			29.0647i			1.30111i	0.759458	+	0.650556i	$0.225464\pi$
$499$			29.0647i			1.30111i	−0.759458	+	0.650556i	$0.774536\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−15.0934			−0.672983			−0.336491	−	0.941687i	$-0.609240\pi$
$503$	−15.0934			−0.672983			−0.336491	+	0.941687i	$0.609240\pi$
$504$		0			0
$505$			2.44357i			0.108738i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−29.3786			−1.30218			−0.651091	−	0.759000i	$-0.725688\pi$
$509$	−29.3786			−1.30218			−0.651091	+	0.759000i	$0.725688\pi$
$510$		0			0
$511$			28.1447i			1.24505i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−4.14712			−0.182744
$516$		0			0
$517$	−8.11947	−	16.1956i	−0.357094	−	0.712284i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			2.07721i			0.0910044i	0.998964	+	0.0455022i	$0.0144888\pi$
$521$			2.07721i			0.0910044i	−0.998964	+	0.0455022i	$0.985511\pi$
$522$		0			0
$523$	−7.42078			−0.324488			−0.162244	−	0.986751i	$-0.551873\pi$
$523$	−7.42078			−0.324488			−0.162244	+	0.986751i	$0.551873\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	9.64601			0.420187
$528$		0			0
$529$	13.3178			0.579035
$530$		0			0
$531$		0			0
$532$		0			0
$533$			23.5709i			1.02097i
$534$		0			0
$535$	−7.89263			−0.341228
$536$		0			0
$537$		0			0
$538$		0			0
$539$	10.6141			0.457183
$540$		0			0
$541$	−24.2719			−1.04353			−0.521766	−	0.853089i	$-0.674727\pi$
$541$	−24.2719			−1.04353			−0.521766	+	0.853089i	$0.674727\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			4.49429i			0.192514i
$546$		0			0
$547$		−	26.8296i		−	1.14715i	−0.819153	−	0.573576i	$-0.805556\pi$
$547$		−	26.8296i		−	1.14715i	0.819153	−	0.573576i	$-0.194444\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	15.4480i		−	0.658106i
$552$		0			0
$553$	−6.43807			−0.273774
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−33.8827			−1.43566			−0.717828	−	0.696221i	$-0.754863\pi$
$557$	−33.8827			−1.43566			−0.717828	+	0.696221i	$0.754863\pi$
$558$		0			0
$559$	9.16896			0.387806
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−28.6743			−1.20848			−0.604238	−	0.796804i	$-0.706522\pi$
$563$	−28.6743			−1.20848			−0.604238	+	0.796804i	$0.706522\pi$
$564$		0			0
$565$	4.95276			0.208364
$566$		0			0
$567$		0			0
$568$		0			0
$569$	6.57618			0.275688			0.137844	−	0.990454i	$-0.455983\pi$
$569$	6.57618			0.275688			0.137844	+	0.990454i	$0.455983\pi$
$570$		0			0
$571$	−40.2555			−1.68464			−0.842321	−	0.538976i	$-0.818811\pi$
$571$	−40.2555			−1.68464			−0.842321	+	0.538976i	$0.818811\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−28.0268			−1.16880
$576$		0			0
$577$			11.4912i			0.478387i	0.970972	+	0.239193i	$0.0768830\pi$
$577$			11.4912i			0.478387i	−0.970972	+	0.239193i	$0.923117\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			8.90285i			0.369352i
$582$		0			0
$583$		−	33.7195i		−	1.39652i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	15.7386			0.649600			0.324800	−	0.945783i	$-0.394703\pi$
$587$	15.7386			0.649600			0.324800	+	0.945783i	$0.394703\pi$
$588$		0			0
$589$	9.92338			0.408886
$590$		0			0
$591$		0			0
$592$		0			0
$593$	9.00216			0.369674			0.184837	−	0.982769i	$-0.440824\pi$
$593$	9.00216			0.369674			0.184837	+	0.982769i	$0.440824\pi$
$594$		0			0
$595$		−	2.37447i		−	0.0973438i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	28.1649			1.15079			0.575393	−	0.817877i	$-0.304849\pi$
$599$	28.1649			1.15079			0.575393	+	0.817877i	$0.304849\pi$
$600$		0			0
$601$	−18.3823			−0.749831			−0.374916	−	0.927059i	$-0.622328\pi$
$601$	−18.3823			−0.749831			−0.374916	+	0.927059i	$0.622328\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−2.37396			−0.0965152
$606$		0			0
$607$		−	12.7704i		−	0.518336i	−0.965832	−	0.259168i	$-0.916552\pi$
$607$		−	12.7704i		−	0.518336i	0.965832	−	0.259168i	$-0.0834484\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	15.8763	−	7.95935i	0.642285	−	0.322001i
$612$		0			0
$613$	−18.3629			−0.741671			−0.370835	−	0.928699i	$-0.620929\pi$
$613$	−18.3629			−0.741671			−0.370835	+	0.928699i	$0.620929\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			31.1244i			1.25302i	0.779412	+	0.626511i	$0.215518\pi$
$617$			31.1244i			1.25302i	−0.779412	+	0.626511i	$0.784482\pi$
$618$		0			0
$619$	26.8469			1.07907			0.539533	−	0.841964i	$-0.318601\pi$
$619$	26.8469			1.07907			0.539533	+	0.841964i	$0.318601\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	9.81128i		−	0.393081i
$624$		0			0
$625$	19.8819			0.795276
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	20.7926i		−	0.829055i
$630$		0			0
$631$			9.91369i			0.394658i	0.980337	+	0.197329i	$0.0632267\pi$
$631$			9.91369i			0.394658i	−0.980337	+	0.197329i	$0.936773\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			3.00360i			0.119194i
$636$		0			0
$637$			10.4048i			0.412254i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−24.7199			−0.976377			−0.488188	−	0.872738i	$-0.662342\pi$
$641$	−24.7199			−0.976377			−0.488188	+	0.872738i	$0.662342\pi$
$642$		0			0
$643$	14.8882			0.587134			0.293567	−	0.955938i	$-0.405158\pi$
$643$	14.8882			0.587134			0.293567	+	0.955938i	$0.405158\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	9.34168i		−	0.367259i	−0.982996	−	0.183630i	$-0.941215\pi$
$647$		−	9.34168i		−	0.367259i	0.982996	−	0.183630i	$-0.0587847\pi$
$648$		0			0
$649$			3.42827i			0.134571i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			9.91979i			0.388191i	0.980983	+	0.194096i	$0.0621772\pi$
$653$			9.91979i			0.388191i	−0.980983	+	0.194096i	$0.937823\pi$
$654$		0			0
$655$		−	4.11532i		−	0.160799i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	22.2034i		−	0.864920i	−0.901653	−	0.432460i	$-0.857646\pi$
$659$		−	22.2034i		−	0.864920i	0.901653	−	0.432460i	$-0.142354\pi$
$660$		0			0
$661$	−21.0187			−0.817535			−0.408767	−	0.912639i	$-0.634041\pi$
$661$	−21.0187			−0.817535			−0.408767	+	0.912639i	$0.634041\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	2.44275i		−	0.0947256i
$666$		0			0
$667$	−38.9083			−1.50654
$668$		0			0
$669$		0			0
$670$		0			0
$671$	22.9041			0.884203
$672$		0			0
$673$		−	24.0620i		−	0.927522i	−0.885960	−	0.463761i	$-0.846500\pi$
$673$		−	24.0620i		−	0.927522i	0.885960	−	0.463761i	$-0.153500\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−36.5458			−1.40457			−0.702284	−	0.711897i	$-0.747836\pi$
$677$	−36.5458			−1.40457			−0.702284	+	0.711897i	$0.747836\pi$
$678$		0			0
$679$	15.7273			0.603558
$680$		0			0
$681$		0			0
$682$		0			0
$683$			48.9620i			1.87348i	0.350027	+	0.936740i	$0.386173\pi$
$683$			48.9620i			1.87348i	−0.350027	+	0.936740i	$0.613827\pi$
$684$		0			0
$685$	5.03514			0.192383
$686$		0			0
$687$		0			0
$688$		0			0
$689$	33.0545			1.25928
$690$		0			0
$691$			29.4182i			1.11912i	0.828789	+	0.559561i	$0.189030\pi$
$691$			29.4182i			1.11912i	−0.828789	+	0.559561i	$0.810970\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			11.7135i			0.444317i
$696$		0			0
$697$		−	21.1625i		−	0.801587i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−12.4183			−0.469032			−0.234516	−	0.972112i	$-0.575351\pi$
$701$	−12.4183			−0.469032			−0.234516	+	0.972112i	$0.575351\pi$
$702$		0			0
$703$		−	21.3905i		−	0.806757i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	7.14106i		−	0.268567i
$708$		0			0
$709$	−16.2500			−0.610281			−0.305141	−	0.952307i	$-0.598703\pi$
$709$	−16.2500			−0.610281			−0.305141	+	0.952307i	$0.598703\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	24.9937i		−	0.936021i
$714$		0			0
$715$			4.04622i			0.151320i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			47.0101i			1.75318i	0.481236	+	0.876591i	$0.340188\pi$
$719$			47.0101i			1.75318i	−0.481236	+	0.876591i	$0.659812\pi$
$720$		0			0
$721$	12.1195			0.451353
$722$		0			0
$723$		0			0
$724$		0			0
$725$	30.0260			1.11514
$726$		0			0
$727$			29.8466i			1.10695i	0.832866	+	0.553474i	$0.186698\pi$
$727$			29.8466i			1.10695i	−0.832866	+	0.553474i	$0.813302\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−8.23210			−0.304475
$732$		0			0
$733$	−17.6521			−0.651996			−0.325998	−	0.945370i	$-0.605700\pi$
$733$	−17.6521			−0.651996			−0.325998	+	0.945370i	$0.605700\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			41.1494i			1.51576i
$738$		0			0
$739$	−11.7482			−0.432164			−0.216082	−	0.976375i	$-0.569328\pi$
$739$	−11.7482			−0.432164			−0.216082	+	0.976375i	$0.569328\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−45.0647			−1.65326			−0.826632	−	0.562742i	$-0.809746\pi$
$743$	−45.0647			−1.65326			−0.826632	+	0.562742i	$0.809746\pi$
$744$		0			0
$745$			2.05171i			0.0751687i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	23.0653			0.842787
$750$		0			0
$751$		−	42.3727i		−	1.54620i	−0.634283	−	0.773101i	$-0.718704\pi$
$751$		−	42.3727i		−	1.54620i	0.634283	−	0.773101i	$-0.281296\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	8.16752i		−	0.297247i
$756$		0			0
$757$		−	12.2982i		−	0.446985i	−0.974706	−	0.223492i	$-0.928254\pi$
$757$		−	12.2982i		−	0.446985i	0.974706	−	0.223492i	$-0.0717458\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	32.6672i		−	1.18418i	−0.805870	−	0.592092i	$-0.798302\pi$
$761$		−	32.6672i		−	1.18418i	0.805870	−	0.592092i	$-0.201698\pi$
$762$		0			0
$763$		−	13.1341i		−	0.475485i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−3.36067			−0.121347
$768$		0			0
$769$	19.9894			0.720835			0.360417	−	0.932791i	$-0.382634\pi$
$769$	19.9894			0.720835			0.360417	+	0.932791i	$0.382634\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			24.2254i			0.871326i	0.900110	+	0.435663i	$0.143486\pi$
$773$			24.2254i			0.871326i	−0.900110	+	0.435663i	$0.856514\pi$
$774$		0			0
$775$			19.2879i			0.692842i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	21.7710i		−	0.780028i
$780$		0			0
$781$		−	17.3806i		−	0.621928i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1.68518			−0.0601466
$786$		0			0
$787$		−	27.4219i		−	0.977484i	−0.872429	−	0.488742i	$-0.837456\pi$
$787$		−	27.4219i		−	0.977484i	0.872429	−	0.488742i	$-0.162544\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−14.4739			−0.514631
$792$		0			0
$793$			22.4524i			0.797309i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	14.1031			0.499557			0.249779	−	0.968303i	$-0.419642\pi$
$797$	14.1031			0.499557			0.249779	+	0.968303i	$0.419642\pi$
$798$		0			0
$799$	−14.2541	+	7.14609i	−0.504273	+	0.252810i
$800$		0			0
$801$		0			0
$802$		0			0
$803$			43.0596i			1.51954i
$804$		0			0
$805$	−6.15246			−0.216846
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−14.8056			−0.520538			−0.260269	−	0.965536i	$-0.583811\pi$
$809$	−14.8056			−0.520538			−0.260269	+	0.965536i	$0.583811\pi$
$810$		0			0
$811$	−31.9992			−1.12364			−0.561822	−	0.827258i	$-0.689899\pi$
$811$	−31.9992			−1.12364			−0.561822	+	0.827258i	$0.689899\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			0.427234i			0.0149653i
$816$		0			0
$817$	−8.46881			−0.296286
$818$		0			0
$819$		0			0
$820$		0			0
$821$	19.4119			0.677478			0.338739	−	0.940880i	$-0.390000\pi$
$821$	19.4119			0.677478			0.338739	+	0.940880i	$0.390000\pi$
$822$		0			0
$823$	−40.9622			−1.42785			−0.713927	−	0.700220i	$-0.753085\pi$
$823$	−40.9622			−1.42785			−0.713927	+	0.700220i	$0.753085\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			35.4253i			1.23186i	0.787801	+	0.615930i	$0.211219\pi$
$827$			35.4253i			1.23186i	−0.787801	+	0.615930i	$0.788781\pi$
$828$		0			0
$829$		−	12.7039i		−	0.441225i	−0.975362	−	0.220612i	$-0.929194\pi$
$829$		−	12.7039i		−	0.441225i	0.975362	−	0.220612i	$-0.0708056\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	9.34168i		−	0.323670i
$834$		0			0
$835$	−9.21914			−0.319042
$836$		0			0
$837$		0			0
$838$		0			0
$839$	37.1137			1.28131			0.640654	−	0.767830i	$-0.278663\pi$
$839$	37.1137			1.28131			0.640654	+	0.767830i	$0.278663\pi$
$840$		0			0
$841$	12.6837			0.437367
$842$		0			0
$843$		0			0
$844$		0			0
$845$	3.71726			0.127878
$846$		0			0
$847$	6.93762			0.238380
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−53.8755			−1.84683
$852$		0			0
$853$	−11.6650			−0.399402			−0.199701	−	0.979857i	$-0.563997\pi$
$853$	−11.6650			−0.399402			−0.199701	+	0.979857i	$0.563997\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−4.87720			−0.166602			−0.0833010	−	0.996524i	$-0.526546\pi$
$857$	−4.87720			−0.166602			−0.0833010	+	0.996524i	$0.526546\pi$
$858$		0			0
$859$		−	38.7027i		−	1.32052i	−0.751038	−	0.660259i	$-0.770446\pi$
$859$		−	38.7027i		−	1.32052i	0.751038	−	0.660259i	$-0.229554\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			36.7832i			1.25212i	0.779777	+	0.626058i	$0.215333\pi$
$863$			36.7832i			1.25212i	−0.779777	+	0.626058i	$0.784667\pi$
$864$		0			0
$865$			9.04956i			0.307694i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−9.84983			−0.334133
$870$		0			0
$871$	−40.3379			−1.36680
$872$		0			0
$873$		0			0
$874$		0			0
$875$	9.85249			0.333075
$876$		0			0
$877$			6.51744i			0.220078i	0.993927	+	0.110039i	$0.0350977\pi$
$877$			6.51744i			0.220078i	−0.993927	+	0.110039i	$0.964902\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−36.3090			−1.22328			−0.611641	−	0.791135i	$-0.709490\pi$
$881$	−36.3090			−1.22328			−0.611641	+	0.791135i	$0.709490\pi$
$882$		0			0
$883$	−16.2389			−0.546484			−0.273242	−	0.961945i	$-0.588096\pi$
$883$	−16.2389			−0.546484			−0.273242	+	0.961945i	$0.588096\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	44.5505			1.49586			0.747930	−	0.663778i	$-0.231048\pi$
$887$	44.5505			1.49586			0.747930	+	0.663778i	$0.231048\pi$
$888$		0			0
$889$		−	8.77768i		−	0.294394i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−14.6639	+	7.35157i	−0.490710	+	0.246011i
$894$		0			0
$895$	7.38449			0.246836
$896$		0			0
$897$		0			0
$898$		0			0
$899$			26.7765i			0.893045i
$900$		0			0
$901$	−29.6771			−0.988688
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	9.63073i		−	0.320136i
$906$		0			0
$907$	−24.9413			−0.828163			−0.414082	−	0.910240i	$-0.635897\pi$
$907$	−24.9413			−0.828163			−0.414082	+	0.910240i	$0.635897\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	3.95055i		−	0.130887i	−0.997856	−	0.0654437i	$-0.979154\pi$
$911$		−	3.95055i		−	0.130887i	0.997856	−	0.0654437i	$-0.0208463\pi$
$912$		0			0
$913$			13.6208i			0.450783i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			12.0266i			0.397152i
$918$		0			0
$919$		−	3.88299i		−	0.128088i	−0.997947	−	0.0640441i	$-0.979600\pi$
$919$		−	3.88299i		−	0.128088i	0.997947	−	0.0640441i	$-0.0203998\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	17.0379			0.560809
$924$		0			0
$925$	41.5763			1.36702
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	46.6392i		−	1.53018i	−0.643921	−	0.765092i	$-0.722694\pi$
$929$		−	46.6392i		−	1.53018i	0.643921	−	0.765092i	$-0.277306\pi$
$930$		0			0
$931$		−	9.61029i		−	0.314965i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	3.63279i		−	0.118805i
$936$		0			0
$937$		−	7.04752i		−	0.230233i	−0.993352	−	0.115116i	$-0.963276\pi$
$937$		−	7.04752i		−	0.230233i	0.993352	−	0.115116i	$-0.0367241\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			45.2165i			1.47401i	0.675885	+	0.737007i	$0.263762\pi$
$941$			45.2165i			1.47401i	−0.675885	+	0.737007i	$0.736238\pi$
$942$		0			0
$943$	−54.8340			−1.78564
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	6.39329i		−	0.207754i	−0.994590	−	0.103877i	$-0.966875\pi$
$947$		−	6.39329i		−	0.207754i	0.994590	−	0.103877i	$-0.0331248\pi$
$948$		0			0
$949$	−42.2104			−1.37021
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−34.8286			−1.12821			−0.564105	−	0.825703i	$-0.690779\pi$
$953$	−34.8286			−1.12821			−0.564105	+	0.825703i	$0.690779\pi$
$954$		0			0
$955$		−	12.9369i		−	0.418629i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−14.7146			−0.475160
$960$		0			0
$961$	13.7995			0.445145
$962$		0			0
$963$		0			0
$964$		0			0
$965$			4.00025i			0.128773i
$966$		0			0
$967$	13.5740			0.436511			0.218256	−	0.975892i	$-0.429963\pi$
$967$	13.5740			0.436511			0.218256	+	0.975892i	$0.429963\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−62.0365			−1.99085			−0.995423	−	0.0955651i	$-0.969534\pi$
$971$	−62.0365			−1.99085			−0.995423	+	0.0955651i	$0.969534\pi$
$972$		0			0
$973$		−	34.2312i		−	1.09740i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	43.4868i		−	1.39127i	−0.718397	−	0.695633i	$-0.755124\pi$
$977$		−	43.4868i		−	1.39127i	0.718397	−	0.695633i	$-0.244876\pi$
$978$		0			0
$979$		−	15.0106i		−	0.479742i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−7.16723			−0.228599			−0.114300	−	0.993446i	$-0.536462\pi$
$983$	−7.16723			−0.228599			−0.114300	+	0.993446i	$0.536462\pi$
$984$		0			0
$985$		−	1.80490i		−	0.0575090i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			21.3301i			0.678258i
$990$		0			0
$991$	−16.1976			−0.514532			−0.257266	−	0.966341i	$-0.582822\pi$
$991$	−16.1976			−0.514532			−0.257266	+	0.966341i	$0.582822\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	4.06706i		−	0.128934i
$996$		0			0
$997$			56.0890i			1.77636i	0.459499	+	0.888178i	$0.348029\pi$
$997$			56.0890i			1.77636i	−0.459499	+	0.888178i	$0.651971\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1692.2.h.a.845.11	yes	16
3.2	odd	2	inner	1692.2.h.a.845.5	✓	16
4.3	odd	2		6768.2.o.d.5921.11		16
12.11	even	2		6768.2.o.d.5921.5		16
47.46	odd	2	inner	1692.2.h.a.845.6	yes	16
141.140	even	2	inner	1692.2.h.a.845.12	yes	16
188.187	even	2		6768.2.o.d.5921.6		16
564.563	odd	2		6768.2.o.d.5921.12		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1692.2.h.a.845.5	✓	16	3.2	odd	2	inner
1692.2.h.a.845.6	yes	16	47.46	odd	2	inner
1692.2.h.a.845.11	yes	16	1.1	even	1	trivial
1692.2.h.a.845.12	yes	16	141.140	even	2	inner
6768.2.o.d.5921.5		16	12.11	even	2
6768.2.o.d.5921.6		16	188.187	even	2
6768.2.o.d.5921.11		16	4.3	odd	2
6768.2.o.d.5921.12		16	564.563	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1692 = 2^{2} \cdot 3^{2} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1692.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.591052 q^{5} -1.72728 q^{7} +O(q^{10})\)	\(q+0.591052 q^{5} -1.72728 q^{7} -2.64263 q^{11} -2.59052i q^{13} +2.32583i q^{17} +2.39270i q^{19} +6.02643 q^{23} -4.65066 q^{25} -6.45629 q^{29} -4.14735i q^{31} -1.02091 q^{35} -8.93987 q^{37} -9.09892 q^{41} +3.53943i q^{43} +(3.07249 + 6.12860i) q^{47} -4.01650 q^{49} +12.7598i q^{53} -1.56193 q^{55} -1.29730i q^{59} -8.66715 q^{61} -1.53113i q^{65} -15.5714i q^{67} +6.57702i q^{71} -16.2942i q^{73} +4.56457 q^{77} +3.72728 q^{79} -5.15425i q^{83} +1.37469i q^{85} +5.68019i q^{89} +4.47455i q^{91} +1.41421i q^{95} -9.10522 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 16 q^{25} + 8 q^{37} + 24 q^{49} + 8 q^{55} + 40 q^{61} + 32 q^{79}+O(q^{100}) \) 16 * q + 16 * q^25 + 8 * q^37 + 24 * q^49 + 8 * q^55 + 40 * q^61 + 32 * q^79

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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