LMFDB - Embedded newform 1692.2.h.a.845.7

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			845.7
Root			$4.38784 + 1.41421i$ of defining polynomial
Character	$\chi$	$=$	1692.845
Dual form			1692.2.h.a.845.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.213445 q^{5} +4.15994 q^{7} +O(q^{10})$	$q-0.213445 q^{5} +4.15994 q^{7} -4.61574 q^{11} -3.89168i q^{13} -3.63817i q^{17} -6.62565i q^{19} -4.75435 q^{23} -4.95444 q^{25} +3.65298 q^{29} +2.43212i q^{31} -0.887919 q^{35} -0.809295 q^{37} -0.962762 q^{41} -3.98968i q^{43} +(5.71711 - 3.78348i) q^{47} +10.3051 q^{49} -1.10036i q^{53} +0.985208 q^{55} -3.65909i q^{59} +5.35064 q^{61} +0.830660i q^{65} +2.32628i q^{67} +7.00664i q^{71} -4.95137i q^{73} -19.2012 q^{77} -2.15994 q^{79} +0.809741i q^{83} +0.776549i q^{85} -14.5736i q^{89} -16.1891i q^{91} +1.41421i q^{95} +2.36543 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−0.213445			−0.0954556			−0.0477278	−	0.998860i	$-0.515198\pi$
$5$	−0.213445			−0.0954556			−0.0477278	+	0.998860i	$0.515198\pi$
$6$		0			0
$7$	4.15994			1.57231			0.786154	−	0.618030i	$-0.212069\pi$
$7$	4.15994			1.57231			0.786154	+	0.618030i	$0.212069\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.61574			−1.39170			−0.695849	−	0.718188i	$-0.744972\pi$
$11$	−4.61574			−1.39170			−0.695849	+	0.718188i	$0.744972\pi$
$12$		0			0
$13$		−	3.89168i		−	1.07936i	−0.841871	−	0.539678i	$-0.818546\pi$
$13$		−	3.89168i		−	1.07936i	0.841871	−	0.539678i	$-0.181454\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	3.63817i		−	0.882385i	−0.897412	−	0.441193i	$-0.854556\pi$
$17$		−	3.63817i		−	0.882385i	0.897412	−	0.441193i	$-0.145444\pi$
$18$		0			0
$19$		−	6.62565i		−	1.52003i	−0.649906	−	0.760015i	$-0.725192\pi$
$19$		−	6.62565i		−	1.52003i	0.649906	−	0.760015i	$-0.274808\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.75435			−0.991350			−0.495675	−	0.868508i	$-0.665079\pi$
$23$	−4.75435			−0.991350			−0.495675	+	0.868508i	$0.665079\pi$
$24$		0			0
$25$	−4.95444			−0.990888
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.65298			0.678342			0.339171	−	0.940725i	$-0.389854\pi$
$29$	3.65298			0.678342			0.339171	+	0.940725i	$0.389854\pi$
$30$		0			0
$31$			2.43212i			0.436822i	0.975857	+	0.218411i	$0.0700873\pi$
$31$			2.43212i			0.436822i	−0.975857	+	0.218411i	$0.929913\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.887919			−0.150086
$36$		0			0
$37$	−0.809295			−0.133047			−0.0665237	−	0.997785i	$-0.521191\pi$
$37$	−0.809295			−0.133047			−0.0665237	+	0.997785i	$0.521191\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−0.962762			−0.150358			−0.0751791	−	0.997170i	$-0.523953\pi$
$41$	−0.962762			−0.150358			−0.0751791	+	0.997170i	$0.523953\pi$
$42$		0			0
$43$		−	3.98968i		−	0.608421i	−0.952605	−	0.304210i	$-0.901607\pi$
$43$		−	3.98968i		−	0.608421i	0.952605	−	0.304210i	$-0.0983926\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	5.71711	−	3.78348i	0.833926	−	0.551877i
$47$	5.71711	−	3.78348i	0.833926	−	0.551877i
$48$		0			0
$49$	10.3051			1.47215
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	1.10036i		−	0.151146i	−0.997140	−	0.0755728i	$-0.975921\pi$
$53$		−	1.10036i		−	0.151146i	0.997140	−	0.0755728i	$-0.0240785\pi$
$54$		0			0
$55$	0.985208			0.132845
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	3.65909i		−	0.476373i	−0.971219	−	0.238186i	$-0.923447\pi$
$59$		−	3.65909i		−	0.476373i	0.971219	−	0.238186i	$-0.0765529\pi$
$60$		0			0
$61$	5.35064			0.685080			0.342540	−	0.939503i	$-0.388713\pi$
$61$	5.35064			0.685080			0.342540	+	0.939503i	$0.388713\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			0.830660i			0.103031i
$66$		0			0
$67$			2.32628i			0.284200i	0.989852	+	0.142100i	$0.0453854\pi$
$67$			2.32628i			0.284200i	−0.989852	+	0.142100i	$0.954615\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			7.00664i			0.831535i	0.909471	+	0.415767i	$0.136487\pi$
$71$			7.00664i			0.831535i	−0.909471	+	0.415767i	$0.863513\pi$
$72$		0			0
$73$		−	4.95137i		−	0.579514i	−0.957100	−	0.289757i	$-0.906426\pi$
$73$		−	4.95137i		−	0.579514i	0.957100	−	0.289757i	$-0.0935745\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−19.2012			−2.18818
$78$		0			0
$79$	−2.15994			−0.243012			−0.121506	−	0.992591i	$-0.538772\pi$
$79$	−2.15994			−0.243012			−0.121506	+	0.992591i	$0.538772\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			0.809741i			0.0888806i	0.999012	+	0.0444403i	$0.0141504\pi$
$83$			0.809741i			0.0888806i	−0.999012	+	0.0444403i	$0.985850\pi$
$84$		0			0
$85$			0.776549i			0.0842286i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	14.5736i		−	1.54480i	−0.635138	−	0.772399i	$-0.719057\pi$
$89$		−	14.5736i		−	1.54480i	0.635138	−	0.772399i	$-0.280943\pi$
$90$		0			0
$91$		−	16.1891i		−	1.69708i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			1.41421i			0.145095i
$96$		0			0
$97$	2.36543			0.240174			0.120087	−	0.992763i	$-0.461683\pi$
$97$	2.36543			0.240174			0.120087	+	0.992763i	$0.461683\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			12.8897i			1.28257i	0.767302	+	0.641285i	$0.221599\pi$
$101$			12.8897i			1.28257i	−0.767302	+	0.641285i	$0.778401\pi$
$102$		0			0
$103$	7.30508			0.719791			0.359896	−	0.932993i	$-0.382812\pi$
$103$	7.30508			0.719791			0.359896	+	0.932993i	$0.382812\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−6.25298			−0.604498			−0.302249	−	0.953229i	$-0.597737\pi$
$107$	−6.25298			−0.604498			−0.302249	+	0.953229i	$0.597737\pi$
$108$		0			0
$109$		−	11.2191i		−	1.07459i	−0.843394	−	0.537295i	$-0.819446\pi$
$109$		−	11.2191i		−	1.07459i	0.843394	−	0.537295i	$-0.180554\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−18.0657			−1.69948			−0.849739	−	0.527203i	$-0.823241\pi$
$113$	−18.0657			−1.69948			−0.849739	+	0.527203i	$0.823241\pi$
$114$		0			0
$115$	1.01479			0.0946298
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	15.1346i		−	1.38738i
$120$		0			0
$121$	10.3051			0.936826
$122$		0			0
$123$		0			0
$124$		0			0
$125$	2.12473			0.190041
$126$		0			0
$127$			19.6809i			1.74640i	0.487361	+	0.873201i	$0.337960\pi$
$127$			19.6809i			1.74640i	−0.487361	+	0.873201i	$0.662040\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	15.7181i		−	1.37330i	−0.726989	−	0.686649i	$-0.759081\pi$
$131$		−	15.7181i		−	1.37330i	0.726989	−	0.686649i	$-0.240919\pi$
$132$		0			0
$133$		−	27.5623i		−	2.38995i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	13.6338			1.16481			0.582406	−	0.812898i	$-0.302111\pi$
$137$	13.6338			1.16481			0.582406	+	0.812898i	$0.302111\pi$
$138$		0			0
$139$		−	16.6560i		−	1.41274i	−0.707842	−	0.706371i	$-0.750331\pi$
$139$		−	16.6560i		−	1.41274i	0.707842	−	0.706371i	$-0.249669\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			17.9630i			1.50214i
$144$		0			0
$145$	−0.779711			−0.0647515
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	13.1803i		−	1.07977i	−0.841738	−	0.539886i	$-0.818467\pi$
$149$		−	13.1803i		−	1.07977i	0.841738	−	0.539886i	$-0.181533\pi$
$150$		0			0
$151$		−	6.77978i		−	0.551730i	−0.961196	−	0.275865i	$-0.911036\pi$
$151$		−	6.77978i		−	0.551730i	0.961196	−	0.275865i	$-0.0889643\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	0.519124i		−	0.0416971i
$156$		0			0
$157$	8.13035			0.648873			0.324436	−	0.945907i	$-0.394825\pi$
$157$	8.13035			0.648873			0.324436	+	0.945907i	$0.394825\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−19.7778			−1.55871
$162$		0			0
$163$			7.27765i			0.570029i	0.958523	+	0.285015i	$0.0919984\pi$
$163$			7.27765i			0.570029i	−0.958523	+	0.285015i	$0.908002\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	20.1936			1.56263			0.781313	−	0.624139i	$-0.214550\pi$
$167$	20.1936			1.56263			0.781313	+	0.624139i	$0.214550\pi$
$168$		0			0
$169$	−2.14515			−0.165011
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	15.6304i		−	1.18836i	−0.804332	−	0.594180i	$-0.797477\pi$
$173$		−	15.6304i		−	1.18836i	0.804332	−	0.594180i	$-0.202523\pi$
$174$		0			0
$175$	−20.6102			−1.55798
$176$		0			0
$177$		0			0
$178$		0			0
$179$	4.05025			0.302730			0.151365	−	0.988478i	$-0.451633\pi$
$179$	4.05025			0.302730			0.151365	+	0.988478i	$0.451633\pi$
$180$		0			0
$181$		−	4.95137i		−	0.368032i	−0.982923	−	0.184016i	$-0.941090\pi$
$181$		−	4.95137i		−	0.368032i	0.982923	−	0.184016i	$-0.0589099\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.172740			0.0127001
$186$		0			0
$187$			16.7928i			1.22801i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			8.62380i			0.623996i	0.950083	+	0.311998i	$0.100998\pi$
$191$			8.62380i			0.623996i	−0.950083	+	0.311998i	$0.899002\pi$
$192$		0			0
$193$		−	10.9172i		−	0.785837i	−0.919573	−	0.392918i	$-0.871465\pi$
$193$		−	10.9172i		−	0.785837i	0.919573	−	0.392918i	$-0.128535\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			8.06348i			0.574499i	0.957856	+	0.287250i	$0.0927410\pi$
$197$			8.06348i			0.574499i	−0.957856	+	0.287250i	$0.907259\pi$
$198$		0			0
$199$			18.4967i			1.31120i	0.755110	+	0.655598i	$0.227583\pi$
$199$			18.4967i			1.31120i	−0.755110	+	0.655598i	$0.772417\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	15.1962			1.06656
$204$		0			0
$205$	0.205497			0.0143525
$206$		0			0
$207$		0			0
$208$		0			0
$209$			30.5823i			2.11542i
$210$		0			0
$211$			26.5587i			1.82838i	0.405288	+	0.914189i	$0.367171\pi$
$211$			26.5587i			1.82838i	−0.405288	+	0.914189i	$0.632829\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			0.851579i			0.0580772i
$216$		0			0
$217$			10.1175i			0.686818i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−14.1586			−0.952409
$222$		0			0
$223$			28.1739i			1.88666i	0.331855	+	0.943331i	$0.392326\pi$
$223$			28.1739i			1.88666i	−0.331855	+	0.943331i	$0.607674\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	3.37893			0.224268			0.112134	−	0.993693i	$-0.464231\pi$
$227$	3.37893			0.224268			0.112134	+	0.993693i	$0.464231\pi$
$228$		0			0
$229$			0.0901690i			0.00595854i	0.999996	+	0.00297927i	$0.000948332\pi$
$229$			0.0901690i			0.00595854i	−0.999996	+	0.00297927i	$0.999052\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−6.50056			−0.425866			−0.212933	−	0.977067i	$-0.568302\pi$
$233$	−6.50056			−0.425866			−0.212933	+	0.977067i	$0.568302\pi$
$234$		0			0
$235$	−1.22029	+	0.807565i	−0.0796029	+	0.0526797i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			28.0445i			1.81405i	0.421079	+	0.907024i	$0.361652\pi$
$239$			28.0445i			1.81405i	−0.421079	+	0.907024i	$0.638348\pi$
$240$		0			0
$241$	3.42907			0.220886			0.110443	−	0.993882i	$-0.464773\pi$
$241$	3.42907			0.220886			0.110443	+	0.993882i	$0.464773\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2.19957			−0.140525
$246$		0			0
$247$	−25.7849			−1.64065
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	15.6304i		−	0.986584i	−0.869864	−	0.493292i	$-0.835793\pi$
$251$		−	15.6304i		−	0.986584i	0.869864	−	0.493292i	$-0.164207\pi$
$252$		0			0
$253$	21.9448			1.37966
$254$		0			0
$255$		0			0
$256$		0			0
$257$	21.9663			1.37022			0.685109	−	0.728441i	$-0.259755\pi$
$257$	21.9663			1.37022			0.685109	+	0.728441i	$0.259755\pi$
$258$		0			0
$259$	−3.36662			−0.209191
$260$		0			0
$261$		0			0
$262$		0			0
$263$			20.5234i			1.26553i	0.774346	+	0.632763i	$0.218079\pi$
$263$			20.5234i			1.26553i	−0.774346	+	0.632763i	$0.781921\pi$
$264$		0			0
$265$			0.234866i			0.0144277i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	32.0384i		−	1.95341i	−0.214579	−	0.976707i	$-0.568838\pi$
$269$		−	32.0384i		−	1.95341i	0.214579	−	0.976707i	$-0.431162\pi$
$270$		0			0
$271$	−8.98521			−0.545813			−0.272906	−	0.962041i	$-0.587985\pi$
$271$	−8.98521			−0.545813			−0.272906	+	0.962041i	$0.587985\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	22.8684			1.37902
$276$		0			0
$277$	−6.99882			−0.420518			−0.210259	−	0.977646i	$-0.567431\pi$
$277$	−6.99882			−0.420518			−0.210259	+	0.977646i	$0.567431\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	22.3778			1.33495			0.667473	−	0.744634i	$-0.267376\pi$
$281$	22.3778			1.33495			0.667473	+	0.744634i	$0.267376\pi$
$282$		0			0
$283$	−9.30508			−0.553130			−0.276565	−	0.960995i	$-0.589196\pi$
$283$	−9.30508			−0.553130			−0.276565	+	0.960995i	$0.589196\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−4.00503			−0.236409
$288$		0			0
$289$	3.76374			0.221396
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−23.7421			−1.38703			−0.693514	−	0.720443i	$-0.743939\pi$
$293$	−23.7421			−1.38703			−0.693514	+	0.720443i	$0.743939\pi$
$294$		0			0
$295$			0.781014i			0.0454724i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			18.5024i			1.07002i
$300$		0			0
$301$		−	16.5968i		−	0.956626i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.14207			−0.0653947
$306$		0			0
$307$	−27.5010			−1.56956			−0.784782	−	0.619772i	$-0.787225\pi$
$307$	−27.5010			−1.56956			−0.784782	+	0.619772i	$0.787225\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	28.7210			1.62862			0.814309	−	0.580432i	$-0.197116\pi$
$311$	28.7210			1.62862			0.814309	+	0.580432i	$0.197116\pi$
$312$		0			0
$313$			12.9076i			0.729578i	0.931090	+	0.364789i	$0.118859\pi$
$313$			12.9076i			0.729578i	−0.931090	+	0.364789i	$0.881141\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	8.66148			0.486477			0.243239	−	0.969967i	$-0.421790\pi$
$317$	8.66148			0.486477			0.243239	+	0.969967i	$0.421790\pi$
$318$		0			0
$319$	−16.8612			−0.944047
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−24.1052			−1.34125
$324$		0			0
$325$			19.2811i			1.06952i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	23.7828	−	15.7390i	1.31119	−	0.867721i
$330$		0			0
$331$	25.4035			1.39630			0.698151	−	0.715951i	$-0.254006\pi$
$331$	25.4035			1.39630			0.698151	+	0.715951i	$0.254006\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	0.496532i		−	0.0271285i
$336$		0			0
$337$	−19.6250			−1.06904			−0.534520	−	0.845156i	$-0.679508\pi$
$337$	−19.6250			−1.06904			−0.534520	+	0.845156i	$0.679508\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	11.2260i		−	0.607924i
$342$		0			0
$343$	13.7489			0.742373
$344$		0			0
$345$		0			0
$346$		0			0
$347$			30.2040i			1.62144i	0.585436	+	0.810719i	$0.300923\pi$
$347$			30.2040i			1.62144i	−0.585436	+	0.810719i	$0.699077\pi$
$348$		0			0
$349$		−	18.0673i		−	0.967119i	−0.875312	−	0.483559i	$-0.839344\pi$
$349$		−	18.0673i		−	0.967119i	0.875312	−	0.483559i	$-0.160656\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	6.53167i		−	0.347646i	−0.984777	−	0.173823i	$-0.944388\pi$
$353$		−	6.53167i		−	0.347646i	0.984777	−	0.173823i	$-0.0556120\pi$
$354$		0			0
$355$		−	1.49553i		−	0.0793746i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−18.3585			−0.968925			−0.484462	−	0.874812i	$-0.660985\pi$
$359$	−18.3585			−0.968925			−0.484462	+	0.874812i	$0.660985\pi$
$360$		0			0
$361$	−24.8993			−1.31049
$362$		0			0
$363$		0			0
$364$		0			0
$365$			1.05685i			0.0553178i
$366$		0			0
$367$			12.2290i			0.638349i	0.947696	+	0.319175i	$0.103406\pi$
$367$			12.2290i			0.638349i	−0.947696	+	0.319175i	$0.896594\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	4.57742i		−	0.237648i
$372$		0			0
$373$		−	15.9261i		−	0.824624i	−0.911043	−	0.412312i	$-0.864721\pi$
$373$		−	15.9261i		−	0.824624i	0.911043	−	0.412312i	$-0.135279\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	14.2162i		−	0.732173i
$378$		0			0
$379$	−6.22147			−0.319576			−0.159788	−	0.987151i	$-0.551081\pi$
$379$	−6.22147			−0.319576			−0.159788	+	0.987151i	$0.551081\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			7.50020i			0.383242i	0.981469	+	0.191621i	$0.0613745\pi$
$383$			7.50020i			0.383242i	−0.981469	+	0.191621i	$0.938626\pi$
$384$		0			0
$385$	4.09840			0.208874
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−12.4917			−0.633355			−0.316678	−	0.948533i	$-0.602567\pi$
$389$	−12.4917			−0.633355			−0.316678	+	0.948533i	$0.602567\pi$
$390$		0			0
$391$			17.2971i			0.874752i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	0.461028			0.0231969
$396$		0			0
$397$	32.9596			1.65420			0.827098	−	0.562058i	$-0.189990\pi$
$397$	32.9596			1.65420			0.827098	+	0.562058i	$0.189990\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			5.94979i			0.297118i	0.988904	+	0.148559i	$0.0474636\pi$
$401$			5.94979i			0.297118i	−0.988904	+	0.148559i	$0.952536\pi$
$402$		0			0
$403$	9.46502			0.471486
$404$		0			0
$405$		0			0
$406$		0			0
$407$	3.73550			0.185162
$408$		0			0
$409$			27.3895i			1.35432i	0.735835	+	0.677161i	$0.236790\pi$
$409$			27.3895i			1.35432i	−0.735835	+	0.677161i	$0.763210\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	15.2216i		−	0.749005i
$414$		0			0
$415$		−	0.172835i		−	0.00848415i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	33.1264			1.61833			0.809166	−	0.587580i	$-0.199919\pi$
$419$	33.1264			1.61833			0.809166	+	0.587580i	$0.199919\pi$
$420$		0			0
$421$			31.8164i			1.55064i	0.631570	+	0.775319i	$0.282411\pi$
$421$			31.8164i			1.55064i	−0.631570	+	0.775319i	$0.717589\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			18.0251i			0.874345i
$426$		0			0
$427$	22.2583			1.07716
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	36.3777i		−	1.75225i	−0.482083	−	0.876126i	$-0.660120\pi$
$431$		−	36.3777i		−	1.75225i	0.482083	−	0.876126i	$-0.339880\pi$
$432$		0			0
$433$		−	8.06203i		−	0.387437i	−0.981057	−	0.193718i	$-0.937945\pi$
$433$		−	8.06203i		−	0.387437i	0.981057	−	0.193718i	$-0.0620548\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			31.5006i			1.50688i
$438$		0			0
$439$	−30.2351			−1.44304			−0.721522	−	0.692392i	$-0.756557\pi$
$439$	−30.2351			−1.44304			−0.721522	+	0.692392i	$0.756557\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	10.2921			0.488995			0.244497	−	0.969650i	$-0.421377\pi$
$443$	10.2921			0.488995			0.244497	+	0.969650i	$0.421377\pi$
$444$		0			0
$445$			3.11066i			0.147460i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−14.2488			−0.672442			−0.336221	−	0.941783i	$-0.609149\pi$
$449$	−14.2488			−0.672442			−0.336221	+	0.941783i	$0.609149\pi$
$450$		0			0
$451$	4.44386			0.209253
$452$		0			0
$453$		0			0
$454$		0			0
$455$			3.45549i			0.161996i
$456$		0			0
$457$	−39.7689			−1.86031			−0.930156	−	0.367165i	$-0.880328\pi$
$457$	−39.7689			−1.86031			−0.930156	+	0.367165i	$0.880328\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	15.4438			0.719288			0.359644	−	0.933090i	$-0.382898\pi$
$461$	15.4438			0.719288			0.359644	+	0.933090i	$0.382898\pi$
$462$		0			0
$463$		−	14.0683i		−	0.653809i	−0.945057	−	0.326905i	$-0.893994\pi$
$463$		−	14.0683i		−	0.653809i	0.945057	−	0.326905i	$-0.106006\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−12.7490			−0.589954			−0.294977	−	0.955504i	$-0.595312\pi$
$467$	−12.7490			−0.589954			−0.294977	+	0.955504i	$0.595312\pi$
$468$		0			0
$469$			9.67716i			0.446850i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			18.4154i			0.846739i
$474$		0			0
$475$			32.8264i			1.50618i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	2.62548i		−	0.119961i	−0.998200	−	0.0599806i	$-0.980896\pi$
$479$		−	2.62548i		−	0.119961i	0.998200	−	0.0599806i	$-0.0191039\pi$
$480$		0			0
$481$			3.14951i			0.143606i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−0.504891			−0.0229259
$486$		0			0
$487$	30.5191			1.38295			0.691475	−	0.722400i	$-0.256961\pi$
$487$	30.5191			1.38295			0.691475	+	0.722400i	$0.256961\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			30.9815i			1.39818i	0.715035	+	0.699088i	$0.246411\pi$
$491$			30.9815i			1.39818i	−0.715035	+	0.699088i	$0.753589\pi$
$492$		0			0
$493$		−	13.2902i		−	0.598559i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			29.1472i			1.30743i
$498$		0			0
$499$			31.5101i			1.41059i	0.708916	+	0.705293i	$0.249184\pi$
$499$			31.5101i			1.41059i	−0.708916	+	0.705293i	$0.750816\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	31.7601			1.41611			0.708056	−	0.706156i	$-0.249572\pi$
$503$	31.7601			1.41611			0.708056	+	0.706156i	$0.249572\pi$
$504$		0			0
$505$		−	2.75124i		−	0.122429i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	31.6477			1.40276			0.701380	−	0.712788i	$-0.252568\pi$
$509$	31.6477			1.40276			0.701380	+	0.712788i	$0.252568\pi$
$510$		0			0
$511$		−	20.5974i		−	0.911175i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1.55923			−0.0687081
$516$		0			0
$517$	−26.3887	+	17.4636i	−1.16057	+	0.768046i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			27.4842i			1.20410i	0.798457	+	0.602052i	$0.205650\pi$
$521$			27.4842i			1.20410i	−0.798457	+	0.602052i	$0.794350\pi$
$522$		0			0
$523$	−26.2976			−1.14991			−0.574956	−	0.818184i	$-0.694981\pi$
$523$	−26.2976			−1.14991			−0.574956	+	0.818184i	$0.694981\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	8.84846			0.385445
$528$		0			0
$529$	−0.396202			−0.0172262
$530$		0			0
$531$		0			0
$532$		0			0
$533$			3.74676i			0.162290i
$534$		0			0
$535$	1.33467			0.0577027
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−47.5656			−2.04880
$540$		0			0
$541$	−32.1672			−1.38298			−0.691489	−	0.722387i	$-0.743045\pi$
$541$	−32.1672			−1.38298			−0.691489	+	0.722387i	$0.743045\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			2.39465i			0.102576i
$546$		0			0
$547$			11.8149i			0.505170i	0.967575	+	0.252585i	$0.0812807\pi$
$547$			11.8149i			0.505170i	−0.967575	+	0.252585i	$0.918719\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	24.2034i		−	1.03110i
$552$		0			0
$553$	−8.98521			−0.382090
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−27.8814			−1.18137			−0.590687	−	0.806901i	$-0.701143\pi$
$557$	−27.8814			−1.18137			−0.590687	+	0.806901i	$0.701143\pi$
$558$		0			0
$559$	−15.5266			−0.656703
$560$		0			0
$561$		0			0
$562$		0			0
$563$	10.0059			0.421699			0.210849	−	0.977519i	$-0.432377\pi$
$563$	10.0059			0.421699			0.210849	+	0.977519i	$0.432377\pi$
$564$		0			0
$565$	3.85604			0.162225
$566$		0			0
$567$		0			0
$568$		0			0
$569$	21.9398			0.919764			0.459882	−	0.887980i	$-0.347892\pi$
$569$	21.9398			0.919764			0.459882	+	0.887980i	$0.347892\pi$
$570$		0			0
$571$	2.28701			0.0957084			0.0478542	−	0.998854i	$-0.484762\pi$
$571$	2.28701			0.0957084			0.0478542	+	0.998854i	$0.484762\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	23.5551			0.982317
$576$		0			0
$577$		−	12.7146i		−	0.529315i	−0.964342	−	0.264658i	$-0.914741\pi$
$577$		−	12.7146i		−	0.529315i	0.964342	−	0.264658i	$-0.0852589\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			3.36847i			0.139748i
$582$		0			0
$583$			5.07896i			0.210349i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	20.9101			0.863053			0.431527	−	0.902100i	$-0.357975\pi$
$587$	20.9101			0.863053			0.431527	+	0.902100i	$0.357975\pi$
$588$		0			0
$589$	16.1144			0.663981
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−48.2401			−1.98098			−0.990492	−	0.137573i	$-0.956070\pi$
$593$	−48.2401			−1.98098			−0.990492	+	0.137573i	$0.956070\pi$
$594$		0			0
$595$			3.23040i			0.132433i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−1.25986			−0.0514766			−0.0257383	−	0.999669i	$-0.508194\pi$
$599$	−1.25986			−0.0514766			−0.0257383	+	0.999669i	$0.508194\pi$
$600$		0			0
$601$	10.5646			0.430939			0.215470	−	0.976511i	$-0.430872\pi$
$601$	10.5646			0.430939			0.215470	+	0.976511i	$0.430872\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−2.19957			−0.0894252
$606$		0			0
$607$		−	26.5587i		−	1.07799i	−0.842310	−	0.538993i	$-0.818805\pi$
$607$		−	26.5587i		−	1.07799i	0.842310	−	0.538993i	$-0.181195\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−14.7241	−	22.2491i	−0.595672	−	0.900103i
$612$		0			0
$613$	14.9523			0.603919			0.301960	−	0.953321i	$-0.402359\pi$
$613$	14.9523			0.603919			0.301960	+	0.953321i	$0.402359\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	24.7943i		−	0.998180i	−0.866550	−	0.499090i	$-0.833668\pi$
$617$		−	24.7943i		−	0.998180i	0.866550	−	0.499090i	$-0.166332\pi$
$618$		0			0
$619$	−25.5306			−1.02616			−0.513080	−	0.858341i	$-0.671496\pi$
$619$	−25.5306			−1.02616			−0.513080	+	0.858341i	$0.671496\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	60.6252i		−	2.42890i
$624$		0			0
$625$	24.3187			0.972748
$626$		0			0
$627$		0			0
$628$		0			0
$629$			2.94435i			0.117399i
$630$		0			0
$631$		−	22.6420i		−	0.901362i	−0.892685	−	0.450681i	$-0.851181\pi$
$631$		−	22.6420i		−	0.901362i	0.892685	−	0.450681i	$-0.148819\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	4.20080i		−	0.166704i
$636$		0			0
$637$		−	40.1041i		−	1.58898i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	42.8409			1.69211			0.846057	−	0.533093i	$-0.178970\pi$
$641$	42.8409			1.69211			0.846057	+	0.533093i	$0.178970\pi$
$642$		0			0
$643$	29.0244			1.14461			0.572306	−	0.820040i	$-0.306049\pi$
$643$	29.0244			1.14461			0.572306	+	0.820040i	$0.306049\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	37.4916i		−	1.47395i	−0.675921	−	0.736974i	$-0.736254\pi$
$647$		−	37.4916i		−	1.47395i	0.675921	−	0.736974i	$-0.263746\pi$
$648$		0			0
$649$			16.8894i			0.662967i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	39.6053i		−	1.54988i	−0.632037	−	0.774938i	$-0.717781\pi$
$653$		−	39.6053i		−	1.54988i	0.632037	−	0.774938i	$-0.282219\pi$
$654$		0			0
$655$			3.35495i			0.131089i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			38.1283i			1.48527i	0.669697	+	0.742635i	$0.266424\pi$
$659$			38.1283i			1.48527i	−0.669697	+	0.742635i	$0.733576\pi$
$660$		0			0
$661$	37.3643			1.45330			0.726651	−	0.687006i	$-0.241076\pi$
$661$	37.3643			1.45330			0.726651	+	0.687006i	$0.241076\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			5.88304i			0.228135i
$666$		0			0
$667$	−17.3675			−0.672474
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−24.6972			−0.953425
$672$		0			0
$673$			9.97277i			0.384422i	0.981354	+	0.192211i	$0.0615658\pi$
$673$			9.97277i			0.384422i	−0.981354	+	0.192211i	$0.938434\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	9.92134			0.381308			0.190654	−	0.981657i	$-0.438939\pi$
$677$	9.92134			0.381308			0.190654	+	0.981657i	$0.438939\pi$
$678$		0			0
$679$	9.84006			0.377627
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	31.8856i		−	1.22007i	−0.792375	−	0.610035i	$-0.791156\pi$
$683$		−	31.8856i		−	1.22007i	0.792375	−	0.610035i	$-0.208844\pi$
$684$		0			0
$685$	−2.91007			−0.111188
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−4.28223			−0.163140
$690$		0			0
$691$		−	21.2228i		−	0.807355i	−0.914901	−	0.403677i	$-0.867732\pi$
$691$		−	21.2228i		−	0.807355i	0.914901	−	0.403677i	$-0.132268\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			3.55514i			0.134854i
$696$		0			0
$697$			3.50269i			0.132674i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−42.1103			−1.59049			−0.795243	−	0.606291i	$-0.792657\pi$
$701$	−42.1103			−1.59049			−0.795243	+	0.606291i	$0.792657\pi$
$702$		0			0
$703$			5.36211i			0.202236i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			53.6203i			2.01660i
$708$		0			0
$709$	38.0000			1.42712			0.713561	−	0.700593i	$-0.247081\pi$
$709$	38.0000			1.42712			0.713561	+	0.700593i	$0.247081\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	11.5631i		−	0.433043i
$714$		0			0
$715$		−	3.83411i		−	0.143388i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			36.3710i			1.35641i	0.734873	+	0.678205i	$0.237242\pi$
$719$			36.3710i			1.35641i	−0.734873	+	0.678205i	$0.762758\pi$
$720$		0			0
$721$	30.3887			1.13173
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−18.0985			−0.672161
$726$		0			0
$727$		−	33.5779i		−	1.24533i	−0.782487	−	0.622667i	$-0.786049\pi$
$727$		−	33.5779i		−	1.24533i	0.782487	−	0.622667i	$-0.213951\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−14.5151			−0.536862
$732$		0			0
$733$	24.0975			0.890061			0.445031	−	0.895515i	$-0.353193\pi$
$733$	24.0975			0.890061			0.445031	+	0.895515i	$0.353193\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	10.7375i		−	0.395521i
$738$		0			0
$739$	31.8241			1.17067			0.585335	−	0.810792i	$-0.300963\pi$
$739$	31.8241			1.17067			0.585335	+	0.810792i	$0.300963\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−3.71244			−0.136196			−0.0680982	−	0.997679i	$-0.521693\pi$
$743$	−3.71244			−0.136196			−0.0680982	+	0.997679i	$0.521693\pi$
$744$		0			0
$745$			2.81327i			0.103070i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−26.0120			−0.950458
$750$		0			0
$751$		−	11.5351i		−	0.420923i	−0.977602	−	0.210461i	$-0.932503\pi$
$751$		−	11.5351i		−	0.420923i	0.977602	−	0.210461i	$-0.0674967\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			1.44711i			0.0526657i
$756$		0			0
$757$		−	46.9724i		−	1.70724i	−0.520895	−	0.853621i	$-0.674402\pi$
$757$		−	46.9724i		−	1.70724i	0.520895	−	0.853621i	$-0.325598\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	10.0516i		−	0.364370i	−0.983264	−	0.182185i	$-0.941683\pi$
$761$		−	10.0516i		−	0.364370i	0.983264	−	0.182185i	$-0.0583170\pi$
$762$		0			0
$763$		−	46.6706i		−	1.68959i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−14.2400			−0.514176
$768$		0			0
$769$	−31.1060			−1.12171			−0.560855	−	0.827914i	$-0.689527\pi$
$769$	−31.1060			−1.12171			−0.560855	+	0.827914i	$0.689527\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			23.4146i			0.842163i	0.907023	+	0.421082i	$0.138349\pi$
$773$			23.4146i			0.842163i	−0.907023	+	0.421082i	$0.861651\pi$
$774$		0			0
$775$		−	12.0498i		−	0.432841i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			6.37892i			0.228549i
$780$		0			0
$781$		−	32.3408i		−	1.15725i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1.73538			−0.0619385
$786$		0			0
$787$			27.8750i			0.993638i	0.867854	+	0.496819i	$0.165499\pi$
$787$			27.8750i			0.993638i	−0.867854	+	0.496819i	$0.834501\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−75.1522			−2.67210
$792$		0			0
$793$		−	20.8230i		−	0.739445i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−14.0837			−0.498871			−0.249436	−	0.968391i	$-0.580245\pi$
$797$	−14.0837			−0.498871			−0.249436	+	0.968391i	$0.580245\pi$
$798$		0			0
$799$	−13.7649	−	20.7998i	−0.486968	−	0.735844i
$800$		0			0
$801$		0			0
$802$		0			0
$803$			22.8543i			0.806509i
$804$		0			0
$805$	4.22147			0.148787
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−17.6465			−0.620417			−0.310209	−	0.950668i	$-0.600399\pi$
$809$	−17.6465			−0.620417			−0.310209	+	0.950668i	$0.600399\pi$
$810$		0			0
$811$	−34.0073			−1.19416			−0.597079	−	0.802183i	$-0.703672\pi$
$811$	−34.0073			−1.19416			−0.597079	+	0.802183i	$0.703672\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	1.55338i		−	0.0544125i
$816$		0			0
$817$	−26.4343			−0.924818
$818$		0			0
$819$		0			0
$820$		0			0
$821$	29.4671			1.02841			0.514205	−	0.857667i	$-0.328087\pi$
$821$	29.4671			1.02841			0.514205	+	0.857667i	$0.328087\pi$
$822$		0			0
$823$	46.9068			1.63507			0.817535	−	0.575879i	$-0.195340\pi$
$823$	46.9068			1.63507			0.817535	+	0.575879i	$0.195340\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	25.3360i		−	0.881019i	−0.897748	−	0.440509i	$-0.854798\pi$
$827$		−	25.3360i		−	0.881019i	0.897748	−	0.440509i	$-0.145202\pi$
$828$		0			0
$829$		−	5.06664i		−	0.175972i	−0.996122	−	0.0879858i	$-0.971957\pi$
$829$		−	5.06664i		−	0.175972i	0.996122	−	0.0879858i	$-0.0280430\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	37.4916i		−	1.29901i
$834$		0			0
$835$	−4.31022			−0.149161
$836$		0			0
$837$		0			0
$838$		0			0
$839$	13.4753			0.465219			0.232609	−	0.972570i	$-0.425274\pi$
$839$	13.4753			0.465219			0.232609	+	0.972570i	$0.425274\pi$
$840$		0			0
$841$	−15.6557			−0.539853
$842$		0			0
$843$		0			0
$844$		0			0
$845$	0.457871			0.0157512
$846$		0			0
$847$	42.8685			1.47298
$848$		0			0
$849$		0			0
$850$		0			0
$851$	3.84767			0.131896
$852$		0			0
$853$	23.0507			0.789242			0.394621	−	0.918844i	$-0.370876\pi$
$853$	23.0507			0.789242			0.394621	+	0.918844i	$0.370876\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	24.8842			0.850027			0.425014	−	0.905187i	$-0.360269\pi$
$857$	24.8842			0.850027			0.425014	+	0.905187i	$0.360269\pi$
$858$		0			0
$859$			7.81294i			0.266574i	0.991077	+	0.133287i	$0.0425532\pi$
$859$			7.81294i			0.266574i	−0.991077	+	0.133287i	$0.957447\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			12.9757i			0.441697i	0.975308	+	0.220849i	$0.0708827\pi$
$863$			12.9757i			0.441697i	−0.975308	+	0.220849i	$0.929117\pi$
$864$		0			0
$865$			3.33624i			0.113436i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	9.96972			0.338200
$870$		0			0
$871$	9.05311			0.306753
$872$		0			0
$873$		0			0
$874$		0			0
$875$	8.83873			0.298804
$876$		0			0
$877$		−	38.6085i		−	1.30372i	−0.758341	−	0.651859i	$-0.773990\pi$
$877$		−	38.6085i		−	1.30372i	0.758341	−	0.651859i	$-0.226010\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−15.6310			−0.526622			−0.263311	−	0.964711i	$-0.584814\pi$
$881$	−15.6310			−0.526622			−0.263311	+	0.964711i	$0.584814\pi$
$882$		0			0
$883$	−52.7774			−1.77610			−0.888050	−	0.459746i	$-0.847940\pi$
$883$	−52.7774			−1.77610			−0.888050	+	0.459746i	$0.847940\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−24.7300			−0.830351			−0.415176	−	0.909741i	$-0.636280\pi$
$887$	−24.7300			−0.830351			−0.415176	+	0.909741i	$0.636280\pi$
$888$		0			0
$889$			81.8715i			2.74588i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−25.0680	−	37.8796i	−0.838869	−	1.26759i
$894$		0			0
$895$	−0.864506			−0.0288973
$896$		0			0
$897$		0			0
$898$		0			0
$899$			8.88449i			0.296314i
$900$		0			0
$901$	−4.00328			−0.133369
$902$		0			0
$903$		0			0
$904$		0			0
$905$			1.05685i			0.0351307i
$906$		0			0
$907$	25.2426			0.838168			0.419084	−	0.907947i	$-0.362351\pi$
$907$	25.2426			0.838168			0.419084	+	0.907947i	$0.362351\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	13.5167i		−	0.447830i	−0.974609	−	0.223915i	$-0.928116\pi$
$911$		−	13.5167i		−	0.447830i	0.974609	−	0.223915i	$-0.0718838\pi$
$912$		0			0
$913$		−	3.73755i		−	0.123695i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	65.3863i		−	2.15925i
$918$		0			0
$919$			34.4415i			1.13612i	0.822987	+	0.568061i	$0.192306\pi$
$919$			34.4415i			1.13612i	−0.822987	+	0.568061i	$0.807694\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	27.2676			0.897523
$924$		0			0
$925$	4.00961			0.131835
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	1.02692i		−	0.0336921i	−0.999858	−	0.0168461i	$-0.994637\pi$
$929$		−	1.02692i		−	0.0336921i	0.999858	−	0.0168461i	$-0.00536252\pi$
$930$		0			0
$931$		−	68.2779i		−	2.23772i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	3.58435i		−	0.117221i
$936$		0			0
$937$			43.2147i			1.41176i	0.708330	+	0.705882i	$0.249449\pi$
$937$			43.2147i			1.41176i	−0.708330	+	0.705882i	$0.750551\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	6.78946i		−	0.221330i	−0.993858	−	0.110665i	$-0.964702\pi$
$941$		−	6.78946i		−	0.221330i	0.993858	−	0.110665i	$-0.0352981\pi$
$942$		0			0
$943$	4.57730			0.149057
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	7.63370i		−	0.248062i	−0.992278	−	0.124031i	$-0.960418\pi$
$947$		−	7.63370i		−	0.248062i	0.992278	−	0.124031i	$-0.0395822\pi$
$948$		0			0
$949$	−19.2691			−0.625502
$950$		0			0
$951$		0			0
$952$		0			0
$953$	53.4478			1.73134			0.865672	−	0.500612i	$-0.166892\pi$
$953$	53.4478			1.73134			0.865672	+	0.500612i	$0.166892\pi$
$954$		0			0
$955$		−	1.84071i		−	0.0595639i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	56.7157			1.83145
$960$		0			0
$961$	25.0848			0.809187
$962$		0			0
$963$		0			0
$964$		0			0
$965$			2.33022i			0.0750125i
$966$		0			0
$967$	20.0688			0.645370			0.322685	−	0.946506i	$-0.395415\pi$
$967$	20.0688			0.645370			0.322685	+	0.946506i	$0.395415\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−14.1640			−0.454544			−0.227272	−	0.973831i	$-0.572981\pi$
$971$	−14.1640			−0.454544			−0.227272	+	0.973831i	$0.572981\pi$
$972$		0			0
$973$		−	69.2879i		−	2.22127i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	21.3009i		−	0.681475i	−0.940158	−	0.340738i	$-0.889323\pi$
$977$		−	21.3009i		−	0.681475i	0.940158	−	0.340738i	$-0.110677\pi$
$978$		0			0
$979$			67.2679i			2.14989i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−21.7264			−0.692963			−0.346482	−	0.938057i	$-0.612624\pi$
$983$	−21.7264			−0.692963			−0.346482	+	0.938057i	$0.612624\pi$
$984$		0			0
$985$		−	1.72111i		−	0.0548392i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			18.9683i			0.603158i
$990$		0			0
$991$	13.7776			0.437660			0.218830	−	0.975763i	$-0.429776\pi$
$991$	13.7776			0.437660			0.218830	+	0.975763i	$0.429776\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	3.94803i		−	0.125161i
$996$		0			0
$997$		−	30.6992i		−	0.972253i	−0.873888	−	0.486126i	$-0.838409\pi$
$997$		−	30.6992i		−	0.972253i	0.873888	−	0.486126i	$-0.161591\pi$
$998$		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.845.7	✓	16
3.2	odd	2	inner	1692.2.h.a.845.9	yes	16
4.3	odd	2		6768.2.o.d.5921.7		16
12.11	even	2		6768.2.o.d.5921.9		16
47.46	odd	2	inner	1692.2.h.a.845.10	yes	16
141.140	even	2	inner	1692.2.h.a.845.8	yes	16
188.187	even	2		6768.2.o.d.5921.10		16
564.563	odd	2		6768.2.o.d.5921.8		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1692.2.h.a.845.7	✓	16	1.1	even	1	trivial
1692.2.h.a.845.8	yes	16	141.140	even	2	inner
1692.2.h.a.845.9	yes	16	3.2	odd	2	inner
1692.2.h.a.845.10	yes	16	47.46	odd	2	inner
6768.2.o.d.5921.7		16	4.3	odd	2
6768.2.o.d.5921.8		16	564.563	odd	2
6768.2.o.d.5921.9		16	12.11	even	2
6768.2.o.d.5921.10		16	188.187	even	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)

\(f(q)\)	\(=\)	\(q-0.213445 q^{5} +4.15994 q^{7} +O(q^{10})\)	\(q-0.213445 q^{5} +4.15994 q^{7} -4.61574 q^{11} -3.89168i q^{13} -3.63817i q^{17} -6.62565i q^{19} -4.75435 q^{23} -4.95444 q^{25} +3.65298 q^{29} +2.43212i q^{31} -0.887919 q^{35} -0.809295 q^{37} -0.962762 q^{41} -3.98968i q^{43} +(5.71711 - 3.78348i) q^{47} +10.3051 q^{49} -1.10036i q^{53} +0.985208 q^{55} -3.65909i q^{59} +5.35064 q^{61} +0.830660i q^{65} +2.32628i q^{67} +7.00664i q^{71} -4.95137i q^{73} -19.2012 q^{77} -2.15994 q^{79} +0.809741i q^{83} +0.776549i q^{85} -14.5736i q^{89} -16.1891i q^{91} +1.41421i q^{95} +2.36543 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 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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