LMFDB - Embedded newform 2736.3.o.p.721.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,3,Mod(721,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2736.721");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2736.o (of order $2$, degree $1$, not 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:	$74.5506003290$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 34x^{6} + 345x^{4} + 1064x^{2} + 256 $ x^8 + 34x^6 + 345x^4 + 1064*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			721.4
Root			$-0.512197i$ of defining polynomial
Character	$\chi$	$=$	2736.721
Dual form			2736.3.o.p.721.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.30665 q^{5} -4.59559 q^{7} +O(q^{10})$	$q+2.30665 q^{5} -4.59559 q^{7} +13.7820 q^{11} +22.2405i q^{13} -5.43101 q^{17} +(7.35095 + 17.5204i) q^{19} +24.9908 q^{23} -19.6794 q^{25} -34.7357i q^{29} +41.2508i q^{31} -10.6004 q^{35} -20.5969i q^{37} +27.8883i q^{41} +10.8491 q^{43} -33.8609 q^{47} -27.8805 q^{49} -29.2996i q^{53} +31.7901 q^{55} -65.8291i q^{59} -107.373 q^{61} +51.3010i q^{65} +1.78720i q^{67} +95.8822i q^{71} +65.4556 q^{73} -63.3362 q^{77} -37.2168i q^{79} -32.6916 q^{83} -12.5274 q^{85} +74.3815i q^{89} -102.208i q^{91} +(16.9560 + 40.4133i) q^{95} -128.116i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 14 q^{5} + 6 q^{7}+O(q^{10}) $ 8 * q + 14 * q^5 + 6 * q^7	$ 8 q + 14 q^{5} + 6 q^{7} - 26 q^{11} + 18 q^{17} - 16 q^{19} + 12 q^{23} + 34 q^{25} - 50 q^{35} - 62 q^{43} + 22 q^{47} + 22 q^{49} - 174 q^{55} - 158 q^{61} - 170 q^{73} - 82 q^{77} - 64 q^{83} + 410 q^{85} + 222 q^{95}+O(q^{100}) $ 8 * q + 14 * q^5 + 6 * q^7 - 26 * q^11 + 18 * q^17 - 16 * q^19 + 12 * q^23 + 34 * q^25 - 50 * q^35 - 62 * q^43 + 22 * q^47 + 22 * q^49 - 174 * q^55 - 158 * q^61 - 170 * q^73 - 82 * q^77 - 64 * q^83 + 410 * q^85 + 222 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$5$	2.30665			0.461329			0.230665	−	0.973033i	$-0.425910\pi$
$5$	2.30665			0.461329			0.230665	+	0.973033i	$0.425910\pi$
$6$		0			0
$7$	−4.59559			−0.656513			−0.328257	−	0.944589i	$-0.606461\pi$
$7$	−4.59559			−0.656513			−0.328257	+	0.944589i	$0.606461\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	13.7820			1.25291			0.626453	−	0.779460i	$-0.284506\pi$
$11$	13.7820			1.25291			0.626453	+	0.779460i	$0.284506\pi$
$12$		0			0
$13$			22.2405i			1.71081i	0.517960	+	0.855405i	$0.326692\pi$
$13$			22.2405i			1.71081i	−0.517960	+	0.855405i	$0.673308\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−5.43101			−0.319471			−0.159736	−	0.987160i	$-0.551064\pi$
$17$	−5.43101			−0.319471			−0.159736	+	0.987160i	$0.551064\pi$
$18$		0			0
$19$	7.35095	+	17.5204i	0.386892	+	0.922125i
$19$	7.35095	+	17.5204i	0.386892	+	0.922125i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	24.9908			1.08656			0.543279	−	0.839552i	$-0.317183\pi$
$23$	24.9908			1.08656			0.543279	+	0.839552i	$0.317183\pi$
$24$		0			0
$25$	−19.6794			−0.787175
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	34.7357i		−	1.19778i	−0.800830	−	0.598891i	$-0.795608\pi$
$29$		−	34.7357i		−	1.19778i	0.800830	−	0.598891i	$-0.204392\pi$
$30$		0			0
$31$			41.2508i			1.33067i	0.746544	+	0.665336i	$0.231712\pi$
$31$			41.2508i			1.33067i	−0.746544	+	0.665336i	$0.768288\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−10.6004			−0.302869
$36$		0			0
$37$		−	20.5969i		−	0.556674i	−0.960483	−	0.278337i	$-0.910217\pi$
$37$		−	20.5969i		−	0.556674i	0.960483	−	0.278337i	$-0.0897832\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			27.8883i			0.680203i	0.940389	+	0.340101i	$0.110461\pi$
$41$			27.8883i			0.680203i	−0.940389	+	0.340101i	$0.889539\pi$
$42$		0			0
$43$	10.8491			0.252305			0.126153	−	0.992011i	$-0.459737\pi$
$43$	10.8491			0.252305			0.126153	+	0.992011i	$0.459737\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−33.8609			−0.720445			−0.360223	−	0.932866i	$-0.617299\pi$
$47$	−33.8609			−0.720445			−0.360223	+	0.932866i	$0.617299\pi$
$48$		0			0
$49$	−27.8805			−0.568991
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	29.2996i		−	0.552823i	−0.961039	−	0.276411i	$-0.910855\pi$
$53$		−	29.2996i		−	0.552823i	0.961039	−	0.276411i	$-0.0891452\pi$
$54$		0			0
$55$	31.7901			0.578002
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	65.8291i		−	1.11575i	−0.829926	−	0.557874i	$-0.811617\pi$
$59$		−	65.8291i		−	1.11575i	0.829926	−	0.557874i	$-0.188383\pi$
$60$		0			0
$61$	−107.373			−1.76021			−0.880103	−	0.474782i	$-0.842527\pi$
$61$	−107.373			−1.76021			−0.880103	+	0.474782i	$0.842527\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			51.3010i			0.789247i
$66$		0			0
$67$			1.78720i			0.0266746i	0.999911	+	0.0133373i	$0.00424552\pi$
$67$			1.78720i			0.0266746i	−0.999911	+	0.0133373i	$0.995754\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			95.8822i			1.35045i	0.737610	+	0.675227i	$0.235954\pi$
$71$			95.8822i			1.35045i	−0.737610	+	0.675227i	$0.764046\pi$
$72$		0			0
$73$	65.4556			0.896651			0.448326	−	0.893870i	$-0.352020\pi$
$73$	65.4556			0.896651			0.448326	+	0.893870i	$0.352020\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−63.3362			−0.822549
$78$		0			0
$79$		−	37.2168i		−	0.471098i	−0.971862	−	0.235549i	$-0.924311\pi$
$79$		−	37.2168i		−	0.471098i	0.971862	−	0.235549i	$-0.0756888\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−32.6916			−0.393874			−0.196937	−	0.980416i	$-0.563099\pi$
$83$	−32.6916			−0.393874			−0.196937	+	0.980416i	$0.563099\pi$
$84$		0			0
$85$	−12.5274			−0.147381
$86$		0			0
$87$		0			0
$88$		0			0
$89$			74.3815i			0.835747i	0.908505	+	0.417874i	$0.137225\pi$
$89$			74.3815i			0.835747i	−0.908505	+	0.417874i	$0.862775\pi$
$90$		0			0
$91$		−	102.208i		−	1.12317i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	16.9560	+	40.4133i	0.178485	+	0.425403i
$96$		0			0
$97$		−	128.116i		−	1.32078i	−0.750921	−	0.660392i	$-0.770390\pi$
$97$		−	128.116i		−	1.32078i	0.750921	−	0.660392i	$-0.229610\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−66.1257			−0.654710			−0.327355	−	0.944902i	$-0.606157\pi$
$101$	−66.1257			−0.654710			−0.327355	+	0.944902i	$0.606157\pi$
$102$		0			0
$103$			68.6107i			0.666123i	0.942905	+	0.333061i	$0.108082\pi$
$103$			68.6107i			0.666123i	−0.942905	+	0.333061i	$0.891918\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			163.182i			1.52507i	0.646949	+	0.762534i	$0.276045\pi$
$107$			163.182i			1.52507i	−0.646949	+	0.762534i	$0.723955\pi$
$108$		0			0
$109$			98.8345i			0.906738i	0.891323	+	0.453369i	$0.149778\pi$
$109$			98.8345i			0.906738i	−0.891323	+	0.453369i	$0.850222\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			94.2226i			0.833829i	0.908946	+	0.416914i	$0.136888\pi$
$113$			94.2226i			0.833829i	−0.908946	+	0.416914i	$0.863112\pi$
$114$		0			0
$115$	57.6451			0.501261
$116$		0			0
$117$		0			0
$118$		0			0
$119$	24.9587			0.209737
$120$		0			0
$121$	68.9423			0.569771
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−103.060			−0.824477
$126$		0			0
$127$			119.017i			0.937146i	0.883425	+	0.468573i	$0.155232\pi$
$127$			119.017i			0.937146i	−0.883425	+	0.468573i	$0.844768\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	126.661			0.966878			0.483439	−	0.875378i	$-0.339387\pi$
$131$	126.661			0.966878			0.483439	+	0.875378i	$0.339387\pi$
$132$		0			0
$133$	−33.7820	−	80.5165i	−0.254000	−	0.605387i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−48.2645			−0.352296			−0.176148	−	0.984364i	$-0.556364\pi$
$137$	−48.2645			−0.352296			−0.176148	+	0.984364i	$0.556364\pi$
$138$		0			0
$139$	243.120			1.74907			0.874534	−	0.484965i	$-0.161168\pi$
$139$	243.120			1.74907			0.874534	+	0.484965i	$0.161168\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			306.518i			2.14348i
$144$		0			0
$145$		−	80.1230i		−	0.552572i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	131.025			0.879361			0.439681	−	0.898154i	$-0.355092\pi$
$149$	131.025			0.879361			0.439681	+	0.898154i	$0.355092\pi$
$150$		0			0
$151$			173.521i			1.14915i	0.818453	+	0.574574i	$0.194832\pi$
$151$			173.521i			1.14915i	−0.818453	+	0.574574i	$0.805168\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			95.1511i			0.613878i
$156$		0			0
$157$	−259.490			−1.65280			−0.826402	−	0.563081i	$-0.809616\pi$
$157$	−259.490			−1.65280			−0.826402	+	0.563081i	$0.809616\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−114.848			−0.713340
$162$		0			0
$163$	−62.7264			−0.384825			−0.192412	−	0.981314i	$-0.561631\pi$
$163$	−62.7264			−0.384825			−0.192412	+	0.981314i	$0.561631\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	134.688i		−	0.806518i	−0.915086	−	0.403259i	$-0.867877\pi$
$167$		−	134.688i		−	0.806518i	0.915086	−	0.403259i	$-0.132123\pi$
$168$		0			0
$169$	−325.641			−1.92687
$170$		0			0
$171$		0			0
$172$		0			0
$173$			90.4432i			0.522793i	0.965232	+	0.261397i	$0.0841830\pi$
$173$			90.4432i			0.522793i	−0.965232	+	0.261397i	$0.915817\pi$
$174$		0			0
$175$	90.4384			0.516791
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	322.972i		−	1.80431i	−0.431409	−	0.902156i	$-0.641984\pi$
$179$		−	322.972i		−	1.80431i	0.431409	−	0.902156i	$-0.358016\pi$
$180$		0			0
$181$			241.679i			1.33524i	0.744502	+	0.667621i	$0.232687\pi$
$181$			241.679i			1.33524i	−0.744502	+	0.667621i	$0.767313\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	47.5099i		−	0.256810i
$186$		0			0
$187$	−74.8499			−0.400267
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−115.733			−0.605931			−0.302966	−	0.953001i	$-0.597977\pi$
$191$	−115.733			−0.605931			−0.302966	+	0.953001i	$0.597977\pi$
$192$		0			0
$193$			277.929i			1.44005i	0.693951	+	0.720023i	$0.255869\pi$
$193$			277.929i			1.44005i	−0.693951	+	0.720023i	$0.744131\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	78.7308			0.399649			0.199824	−	0.979832i	$-0.435963\pi$
$197$	78.7308			0.399649			0.199824	+	0.979832i	$0.435963\pi$
$198$		0			0
$199$	93.6642			0.470674			0.235337	−	0.971914i	$-0.424381\pi$
$199$	93.6642			0.470674			0.235337	+	0.971914i	$0.424381\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			159.631i			0.786360i
$204$		0			0
$205$			64.3285i			0.313798i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	101.310	+	241.465i	0.484739	+	1.15534i
$210$		0			0
$211$			362.818i			1.71952i	0.510701	+	0.859758i	$0.329386\pi$
$211$			362.818i			1.71952i	−0.510701	+	0.859758i	$0.670614\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	25.0251			0.116396
$216$		0			0
$217$		−	189.572i		−	0.873603i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	120.788i		−	0.546554i
$222$		0			0
$223$		−	221.055i		−	0.991277i	−0.868529	−	0.495639i	$-0.834934\pi$
$223$		−	221.055i		−	0.991277i	0.868529	−	0.495639i	$-0.165066\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			46.2298i			0.203656i	0.994802	+	0.101828i	$0.0324691\pi$
$227$			46.2298i			0.203656i	−0.994802	+	0.101828i	$0.967531\pi$
$228$		0			0
$229$	44.5210			0.194415			0.0972075	−	0.995264i	$-0.469009\pi$
$229$	44.5210			0.194415			0.0972075	+	0.995264i	$0.469009\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−276.616			−1.18719			−0.593596	−	0.804763i	$-0.702292\pi$
$233$	−276.616			−1.18719			−0.593596	+	0.804763i	$0.702292\pi$
$234$		0			0
$235$	−78.1052			−0.332363
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−431.105			−1.80379			−0.901893	−	0.431959i	$-0.857823\pi$
$239$	−431.105			−1.80379			−0.901893	+	0.431959i	$0.857823\pi$
$240$		0			0
$241$			90.5900i			0.375892i	0.982179	+	0.187946i	$0.0601830\pi$
$241$			90.5900i			0.375892i	−0.982179	+	0.187946i	$0.939817\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−64.3106			−0.262492
$246$		0			0
$247$	−389.662	+	163.489i	−1.57758	+	0.661899i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	165.585			0.659702			0.329851	−	0.944033i	$-0.393001\pi$
$251$	165.585			0.659702			0.329851	+	0.944033i	$0.393001\pi$
$252$		0			0
$253$	344.423			1.36135
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	500.700i		−	1.94825i	−0.226011	−	0.974125i	$-0.572569\pi$
$257$		−	500.700i		−	1.94825i	0.226011	−	0.974125i	$-0.427431\pi$
$258$		0			0
$259$			94.6551i			0.365464i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	370.994			1.41062			0.705312	−	0.708897i	$-0.250807\pi$
$263$	370.994			1.41062			0.705312	+	0.708897i	$0.250807\pi$
$264$		0			0
$265$		−	67.5838i		−	0.255033i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			273.479i			1.01665i	0.861165	+	0.508326i	$0.169735\pi$
$269$			273.479i			1.01665i	−0.861165	+	0.508326i	$0.830265\pi$
$270$		0			0
$271$	−82.1566			−0.303161			−0.151580	−	0.988445i	$-0.548436\pi$
$271$	−82.1566			−0.303161			−0.151580	+	0.988445i	$0.548436\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−271.220			−0.986256
$276$		0			0
$277$	228.921			0.826430			0.413215	−	0.910633i	$-0.364406\pi$
$277$	228.921			0.826430			0.413215	+	0.910633i	$0.364406\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	170.230i		−	0.605800i	−0.953022	−	0.302900i	$-0.902045\pi$
$281$		−	170.230i		−	0.605800i	0.953022	−	0.302900i	$-0.0979549\pi$
$282$		0			0
$283$	−37.0280			−0.130841			−0.0654205	−	0.997858i	$-0.520839\pi$
$283$	−37.0280			−0.130841			−0.0654205	+	0.997858i	$0.520839\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	128.163i		−	0.446562i
$288$		0			0
$289$	−259.504			−0.897938
$290$		0			0
$291$		0			0
$292$		0			0
$293$			379.390i			1.29485i	0.762130	+	0.647424i	$0.224153\pi$
$293$			379.390i			1.29485i	−0.762130	+	0.647424i	$0.775847\pi$
$294$		0			0
$295$		−	151.844i		−	0.514727i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			555.809i			1.85889i
$300$		0			0
$301$	−49.8582			−0.165642
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−247.671			−0.812035
$306$		0			0
$307$		−	207.485i		−	0.675846i	−0.941174	−	0.337923i	$-0.890276\pi$
$307$		−	207.485i		−	0.675846i	0.941174	−	0.337923i	$-0.109724\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	250.191			0.804474			0.402237	−	0.915536i	$-0.368233\pi$
$311$	250.191			0.804474			0.402237	+	0.915536i	$0.368233\pi$
$312$		0			0
$313$	490.013			1.56554			0.782768	−	0.622313i	$-0.213807\pi$
$313$	490.013			1.56554			0.782768	+	0.622313i	$0.213807\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			193.572i			0.610637i	0.952250	+	0.305319i	$0.0987630\pi$
$317$			193.572i			0.610637i	−0.952250	+	0.305319i	$0.901237\pi$
$318$		0			0
$319$		−	478.726i		−	1.50071i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−39.9231	−	95.1533i	−0.123601	−	0.294592i
$324$		0			0
$325$		−	437.680i		−	1.34671i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	155.611			0.472982
$330$		0			0
$331$		−	77.3740i		−	0.233758i	−0.993146	−	0.116879i	$-0.962711\pi$
$331$		−	77.3740i		−	0.233758i	0.993146	−	0.116879i	$-0.0372890\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			4.12244i			0.0123058i
$336$		0			0
$337$			307.055i			0.911143i	0.890199	+	0.455571i	$0.150565\pi$
$337$			307.055i			0.911143i	−0.890199	+	0.455571i	$0.849435\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			568.517i			1.66720i
$342$		0			0
$343$	353.312			1.03006
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−343.072			−0.988679			−0.494340	−	0.869269i	$-0.664590\pi$
$347$	−343.072			−0.988679			−0.494340	+	0.869269i	$0.664590\pi$
$348$		0			0
$349$	−135.313			−0.387716			−0.193858	−	0.981030i	$-0.562100\pi$
$349$	−135.313			−0.387716			−0.193858	+	0.981030i	$0.562100\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−95.3649			−0.270156			−0.135078	−	0.990835i	$-0.543128\pi$
$353$	−95.3649			−0.270156			−0.135078	+	0.990835i	$0.543128\pi$
$354$		0			0
$355$			221.167i			0.623004i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−339.918			−0.946846			−0.473423	−	0.880835i	$-0.656982\pi$
$359$	−339.918			−0.946846			−0.473423	+	0.880835i	$0.656982\pi$
$360$		0			0
$361$	−252.927	+	257.583i	−0.700629	+	0.713526i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	150.983			0.413652
$366$		0			0
$367$	207.153			0.564449			0.282225	−	0.959348i	$-0.408928\pi$
$367$	207.153			0.564449			0.282225	+	0.959348i	$0.408928\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			134.649i			0.362935i
$372$		0			0
$373$		−	221.266i		−	0.593207i	−0.955001	−	0.296604i	$-0.904146\pi$
$373$		−	221.266i		−	0.593207i	0.955001	−	0.296604i	$-0.0958540\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	772.540			2.04918
$378$		0			0
$379$			251.511i			0.663619i	0.943346	+	0.331809i	$0.107659\pi$
$379$			251.511i			0.663619i	−0.943346	+	0.331809i	$0.892341\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			367.906i			0.960590i	0.877107	+	0.480295i	$0.159470\pi$
$383$			367.906i			0.960590i	−0.877107	+	0.480295i	$0.840530\pi$
$384$		0			0
$385$	−146.094			−0.379466
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−480.911			−1.23628			−0.618138	−	0.786070i	$-0.712113\pi$
$389$	−480.911			−1.23628			−0.618138	+	0.786070i	$0.712113\pi$
$390$		0			0
$391$	−135.725			−0.347124
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	85.8459i		−	0.217331i
$396$		0			0
$397$	432.964			1.09059			0.545294	−	0.838245i	$-0.316418\pi$
$397$	432.964			1.09059			0.545294	+	0.838245i	$0.316418\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			314.930i			0.785361i	0.919675	+	0.392680i	$0.128452\pi$
$401$			314.930i			0.785361i	−0.919675	+	0.392680i	$0.871548\pi$
$402$		0			0
$403$	−917.440			−2.27653
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	283.866i		−	0.697460i
$408$		0			0
$409$		−	13.2424i		−	0.0323775i	−0.999869	−	0.0161888i	$-0.994847\pi$
$409$		−	13.2424i		−	0.0323775i	0.999869	−	0.0161888i	$-0.00515327\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			302.524i			0.732503i
$414$		0			0
$415$	−75.4079			−0.181706
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−667.426			−1.59290			−0.796451	−	0.604703i	$-0.793292\pi$
$419$	−667.426			−1.59290			−0.796451	+	0.604703i	$0.793292\pi$
$420$		0			0
$421$			437.582i			1.03939i	0.854352	+	0.519694i	$0.173954\pi$
$421$			437.582i			1.03939i	−0.854352	+	0.519694i	$0.826046\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	106.879			0.251480
$426$		0			0
$427$	493.441			1.15560
$428$		0			0
$429$		0			0
$430$		0			0
$431$			680.592i			1.57910i	0.613686	+	0.789550i	$0.289686\pi$
$431$			680.592i			1.57910i	−0.613686	+	0.789550i	$0.710314\pi$
$432$		0			0
$433$			391.348i			0.903807i	0.892067	+	0.451904i	$0.149255\pi$
$433$			391.348i			0.903807i	−0.892067	+	0.451904i	$0.850745\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	183.706	+	437.849i	0.420381	+	1.00194i
$438$		0			0
$439$			433.093i			0.986544i	0.869875	+	0.493272i	$0.164199\pi$
$439$			433.093i			0.986544i	−0.869875	+	0.493272i	$0.835801\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−372.634			−0.841160			−0.420580	−	0.907256i	$-0.638173\pi$
$443$	−372.634			−0.841160			−0.420580	+	0.907256i	$0.638173\pi$
$444$		0			0
$445$			171.572i			0.385555i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	96.4188i		−	0.214741i	−0.994219	−	0.107371i	$-0.965757\pi$
$449$		−	96.4188i		−	0.214741i	0.994219	−	0.107371i	$-0.0342431\pi$
$450$		0			0
$451$			384.355i			0.852229i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	235.759i		−	0.518151i
$456$		0			0
$457$	741.777			1.62314			0.811572	−	0.584253i	$-0.198612\pi$
$457$	741.777			1.62314			0.811572	+	0.584253i	$0.198612\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−40.2016			−0.0872053			−0.0436026	−	0.999049i	$-0.513884\pi$
$461$	−40.2016			−0.0872053			−0.0436026	+	0.999049i	$0.513884\pi$
$462$		0			0
$463$	−534.775			−1.15502			−0.577510	−	0.816383i	$-0.695976\pi$
$463$	−534.775			−1.15502			−0.577510	+	0.816383i	$0.695976\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	390.715			0.836649			0.418325	−	0.908298i	$-0.362617\pi$
$467$	390.715			0.836649			0.418325	+	0.908298i	$0.362617\pi$
$468$		0			0
$469$		−	8.21324i		−	0.0175122i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	149.522			0.316115
$474$		0			0
$475$	−144.662	−	344.790i	−0.304552	−	0.725874i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−4.79318			−0.0100066			−0.00500332	−	0.999987i	$-0.501593\pi$
$479$	−4.79318			−0.0100066			−0.00500332	+	0.999987i	$0.501593\pi$
$480$		0			0
$481$	458.087			0.952364
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	295.518i		−	0.609316i
$486$		0			0
$487$		−	173.322i		−	0.355898i	−0.984040	−	0.177949i	$-0.943054\pi$
$487$		−	173.322i		−	0.355898i	0.984040	−	0.177949i	$-0.0569462\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	522.057			1.06325			0.531626	−	0.846979i	$-0.321581\pi$
$491$	522.057			1.06325			0.531626	+	0.846979i	$0.321581\pi$
$492$		0			0
$493$			188.650i			0.382657i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	440.636i		−	0.886591i
$498$		0			0
$499$	−806.635			−1.61650			−0.808251	−	0.588838i	$-0.799586\pi$
$499$	−806.635			−1.61650			−0.808251	+	0.588838i	$0.799586\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	148.884			0.295992			0.147996	−	0.988988i	$-0.452718\pi$
$503$	148.884			0.295992			0.147996	+	0.988988i	$0.452718\pi$
$504$		0			0
$505$	−152.529			−0.302037
$506$		0			0
$507$		0			0
$508$		0			0
$509$			469.470i			0.922339i	0.887312	+	0.461169i	$0.152570\pi$
$509$			469.470i			0.922339i	−0.887312	+	0.461169i	$0.847430\pi$
$510$		0			0
$511$	−300.807			−0.588663
$512$		0			0
$513$		0			0
$514$		0			0
$515$			158.261i			0.307302i
$516$		0			0
$517$	−466.670			−0.902650
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	952.247i		−	1.82773i	−0.406020	−	0.913864i	$-0.633084\pi$
$521$		−	952.247i		−	1.82773i	0.406020	−	0.913864i	$-0.366916\pi$
$522$		0			0
$523$		−	883.789i		−	1.68985i	−0.534889	−	0.844923i	$-0.679646\pi$
$523$		−	883.789i		−	1.68985i	0.534889	−	0.844923i	$-0.320354\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	224.033i		−	0.425111i
$528$		0			0
$529$	95.5422			0.180609
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−620.251			−1.16370
$534$		0			0
$535$			376.404i			0.703558i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−384.248			−0.712891
$540$		0			0
$541$	370.690			0.685193			0.342597	−	0.939483i	$-0.388694\pi$
$541$	370.690			0.685193			0.342597	+	0.939483i	$0.388694\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			227.976i			0.418305i
$546$		0			0
$547$			361.146i			0.660230i	0.943941	+	0.330115i	$0.107088\pi$
$547$			361.146i			0.660230i	−0.943941	+	0.330115i	$0.892912\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	608.582	−	255.340i	1.10451	−	0.463413i
$552$		0			0
$553$			171.033i			0.309282i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−935.225			−1.67904			−0.839519	−	0.543330i	$-0.817163\pi$
$557$	−935.225			−1.67904			−0.839519	+	0.543330i	$0.817163\pi$
$558$		0			0
$559$			241.290i			0.431646i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	288.854i		−	0.513062i	−0.966536	−	0.256531i	$-0.917420\pi$
$563$		−	288.854i		−	0.513062i	0.966536	−	0.256531i	$-0.0825796\pi$
$564$		0			0
$565$			217.338i			0.384670i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	1109.10i		−	1.94921i	−0.223936	−	0.974604i	$-0.571891\pi$
$569$		−	1109.10i		−	1.94921i	0.223936	−	0.974604i	$-0.428109\pi$
$570$		0			0
$571$	−81.8698			−0.143380			−0.0716898	−	0.997427i	$-0.522839\pi$
$571$	−81.8698			−0.143380			−0.0716898	+	0.997427i	$0.522839\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−491.804			−0.855312
$576$		0			0
$577$	−282.577			−0.489735			−0.244867	−	0.969557i	$-0.578744\pi$
$577$	−282.577			−0.489735			−0.244867	+	0.969557i	$0.578744\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	150.237			0.258584
$582$		0			0
$583$		−	403.806i		−	0.692634i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	732.036			1.24708			0.623540	−	0.781792i	$-0.285694\pi$
$587$	732.036			1.24708			0.623540	+	0.781792i	$0.285694\pi$
$588$		0			0
$589$	−722.730	+	303.233i	−1.22705	+	0.514826i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	669.125			1.12837			0.564187	−	0.825647i	$-0.309190\pi$
$593$	669.125			1.12837			0.564187	+	0.825647i	$0.309190\pi$
$594$		0			0
$595$	57.5709			0.0967578
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	852.366i		−	1.42298i	−0.702695	−	0.711491i	$-0.748020\pi$
$599$		−	852.366i		−	1.42298i	0.702695	−	0.711491i	$-0.251980\pi$
$600$		0			0
$601$		−	265.429i		−	0.441646i	−0.975314	−	0.220823i	$-0.929126\pi$
$601$		−	265.429i		−	0.441646i	0.975314	−	0.220823i	$-0.0708743\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	159.026			0.262852
$606$		0			0
$607$		−	751.473i		−	1.23801i	−0.785387	−	0.619005i	$-0.787536\pi$
$607$		−	751.473i		−	1.23801i	0.785387	−	0.619005i	$-0.212464\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	753.085i		−	1.23254i
$612$		0			0
$613$	606.487			0.989376			0.494688	−	0.869071i	$-0.335282\pi$
$613$	606.487			0.989376			0.494688	+	0.869071i	$0.335282\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1020.39			1.65380			0.826900	−	0.562349i	$-0.190102\pi$
$617$	1020.39			1.65380			0.826900	+	0.562349i	$0.190102\pi$
$618$		0			0
$619$	886.720			1.43250			0.716252	−	0.697842i	$-0.245856\pi$
$619$	886.720			1.43250			0.716252	+	0.697842i	$0.245856\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	341.827i		−	0.548679i
$624$		0			0
$625$	254.262			0.406820
$626$		0			0
$627$		0			0
$628$		0			0
$629$			111.862i			0.177841i
$630$		0			0
$631$	−737.311			−1.16848			−0.584240	−	0.811581i	$-0.698607\pi$
$631$	−737.311			−1.16848			−0.584240	+	0.811581i	$0.698607\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			274.531i			0.432333i
$636$		0			0
$637$		−	620.078i		−	0.973435i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	267.570i		−	0.417426i	−0.977977	−	0.208713i	$-0.933073\pi$
$641$		−	267.570i		−	0.417426i	0.977977	−	0.208713i	$-0.0669274\pi$
$642$		0			0
$643$	567.693			0.882881			0.441441	−	0.897290i	$-0.354468\pi$
$643$	567.693			0.882881			0.441441	+	0.897290i	$0.354468\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−543.577			−0.840150			−0.420075	−	0.907489i	$-0.637996\pi$
$647$	−543.577			−0.840150			−0.420075	+	0.907489i	$0.637996\pi$
$648$		0			0
$649$		−	907.254i		−	1.39793i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	39.9308			0.0611498			0.0305749	−	0.999532i	$-0.490266\pi$
$653$	39.9308			0.0611498			0.0305749	+	0.999532i	$0.490266\pi$
$654$		0			0
$655$	292.162			0.446049
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	779.029i		−	1.18214i	−0.806621	−	0.591069i	$-0.798706\pi$
$659$		−	779.029i		−	1.18214i	0.806621	−	0.591069i	$-0.201294\pi$
$660$		0			0
$661$			419.496i			0.634638i	0.948319	+	0.317319i	$0.102783\pi$
$661$			419.496i			0.634638i	−0.948319	+	0.317319i	$0.897217\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−77.9231	−	185.723i	−0.117178	−	0.279283i
$666$		0			0
$667$		−	868.074i		−	1.30146i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−1479.80			−2.20537
$672$		0			0
$673$		−	997.533i		−	1.48222i	−0.671385	−	0.741109i	$-0.734300\pi$
$673$		−	997.533i		−	1.48222i	0.671385	−	0.741109i	$-0.265700\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	441.407i		−	0.652005i	−0.945369	−	0.326003i	$-0.894298\pi$
$677$		−	441.407i		−	0.652005i	0.945369	−	0.326003i	$-0.105702\pi$
$678$		0			0
$679$			588.769i			0.867111i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			907.324i			1.32844i	0.747537	+	0.664220i	$0.231236\pi$
$683$			907.324i			1.32844i	−0.747537	+	0.664220i	$0.768764\pi$
$684$		0			0
$685$	−111.329			−0.162525
$686$		0			0
$687$		0			0
$688$		0			0
$689$	651.638			0.945774
$690$		0			0
$691$	928.930			1.34433			0.672164	−	0.740403i	$-0.265365\pi$
$691$	928.930			1.34433			0.672164	+	0.740403i	$0.265365\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	560.793			0.806896
$696$		0			0
$697$		−	151.462i		−	0.217305i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	1104.52			1.57563			0.787814	−	0.615913i	$-0.211213\pi$
$701$	1104.52			1.57563			0.787814	+	0.615913i	$0.211213\pi$
$702$		0			0
$703$	360.866	−	151.407i	0.513323	−	0.215373i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	303.886			0.429825
$708$		0			0
$709$	−382.002			−0.538790			−0.269395	−	0.963030i	$-0.586824\pi$
$709$	−382.002			−0.538790			−0.269395	+	0.963030i	$0.586824\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			1030.89i			1.44585i
$714$		0			0
$715$			707.029i			0.988851i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	784.961			1.09174			0.545870	−	0.837870i	$-0.316199\pi$
$719$	784.961			1.09174			0.545870	+	0.837870i	$0.316199\pi$
$720$		0			0
$721$		−	315.307i		−	0.437318i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			683.577i			0.942865i
$726$		0			0
$727$	776.412			1.06797			0.533983	−	0.845495i	$-0.320694\pi$
$727$	776.412			1.06797			0.533983	+	0.845495i	$0.320694\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−58.9217			−0.0806042
$732$		0			0
$733$	263.990			0.360150			0.180075	−	0.983653i	$-0.442366\pi$
$733$	263.990			0.360150			0.180075	+	0.983653i	$0.442366\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			24.6311i			0.0334208i
$738$		0			0
$739$	−696.550			−0.942558			−0.471279	−	0.881984i	$-0.656208\pi$
$739$	−696.550			−0.942558			−0.471279	+	0.881984i	$0.656208\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			544.317i			0.732593i	0.930498	+	0.366297i	$0.119374\pi$
$743$			544.317i			0.732593i	−0.930498	+	0.366297i	$0.880626\pi$
$744$		0			0
$745$	302.228			0.405675
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	749.919i		−	1.00123i
$750$		0			0
$751$			648.270i			0.863209i	0.902063	+	0.431605i	$0.142052\pi$
$751$			648.270i			0.863209i	−0.902063	+	0.431605i	$0.857948\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			400.252i			0.530136i
$756$		0			0
$757$	−726.694			−0.959965			−0.479983	−	0.877278i	$-0.659357\pi$
$757$	−726.694			−0.959965			−0.479983	+	0.877278i	$0.659357\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−425.877			−0.559629			−0.279814	−	0.960054i	$-0.590273\pi$
$761$	−425.877			−0.559629			−0.279814	+	0.960054i	$0.590273\pi$
$762$		0			0
$763$		−	454.203i		−	0.595286i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1464.07			1.90883
$768$		0			0
$769$	1060.37			1.37890			0.689448	−	0.724335i	$-0.257853\pi$
$769$	1060.37			1.37890			0.689448	+	0.724335i	$0.257853\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			5.03627i			0.00651523i	0.999995	+	0.00325761i	$0.00103693\pi$
$773$			5.03627i			0.00651523i	−0.999995	+	0.00325761i	$0.998963\pi$
$774$		0			0
$775$		−	811.790i		−	1.04747i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−488.614	+	205.006i	−0.627232	+	0.263165i
$780$		0			0
$781$			1321.44i			1.69199i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−598.552			−0.762487
$786$		0			0
$787$		−	284.035i		−	0.360909i	−0.983583	−	0.180454i	$-0.942243\pi$
$787$		−	284.035i		−	0.360909i	0.983583	−	0.180454i	$-0.0577568\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	433.009i		−	0.547419i
$792$		0			0
$793$		−	2388.02i		−	3.01138i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	262.594i		−	0.329479i	−0.986337	−	0.164739i	$-0.947322\pi$
$797$		−	262.594i		−	0.329479i	0.986337	−	0.164739i	$-0.0526783\pi$
$798$		0			0
$799$	183.899			0.230161
$800$		0			0
$801$		0			0
$802$		0			0
$803$	902.106			1.12342
$804$		0			0
$805$	−264.913			−0.329085
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1302.72			−1.61028			−0.805142	−	0.593082i	$-0.797911\pi$
$809$	−1302.72			−1.61028			−0.805142	+	0.593082i	$0.797911\pi$
$810$		0			0
$811$		−	603.977i		−	0.744731i	−0.928086	−	0.372365i	$-0.878547\pi$
$811$		−	603.977i		−	0.744731i	0.928086	−	0.372365i	$-0.121453\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−144.688			−0.177531
$816$		0			0
$817$	79.7514	+	190.081i	0.0976149	+	0.232657i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−137.526			−0.167510			−0.0837550	−	0.996486i	$-0.526691\pi$
$821$	−137.526			−0.167510			−0.0837550	+	0.996486i	$0.526691\pi$
$822$		0			0
$823$	−18.1953			−0.0221085			−0.0110543	−	0.999939i	$-0.503519\pi$
$823$	−18.1953			−0.0221085			−0.0110543	+	0.999939i	$0.503519\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	1257.34i		−	1.52036i	−0.649710	−	0.760182i	$-0.725110\pi$
$827$		−	1257.34i		−	1.52036i	0.649710	−	0.760182i	$-0.274890\pi$
$828$		0			0
$829$			103.574i			0.124939i	0.998047	+	0.0624695i	$0.0198976\pi$
$829$			103.574i			0.124939i	−0.998047	+	0.0624695i	$0.980102\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	151.419			0.181776
$834$		0			0
$835$		−	310.679i		−	0.372070i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			343.417i			0.409317i	0.978833	+	0.204659i	$0.0656084\pi$
$839$			343.417i			0.409317i	−0.978833	+	0.204659i	$0.934392\pi$
$840$		0			0
$841$	−365.568			−0.434683
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−751.139			−0.888922
$846$		0			0
$847$	−316.831			−0.374062
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	514.735i		−	0.604859i
$852$		0			0
$853$	−260.020			−0.304830			−0.152415	−	0.988317i	$-0.548705\pi$
$853$	−260.020			−0.304830			−0.152415	+	0.988317i	$0.548705\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	1420.47i		−	1.65749i	−0.559623	−	0.828747i	$-0.689054\pi$
$857$		−	1420.47i		−	1.65749i	0.559623	−	0.828747i	$-0.310946\pi$
$858$		0			0
$859$	−856.333			−0.996895			−0.498447	−	0.866920i	$-0.666096\pi$
$859$	−856.333			−0.996895			−0.498447	+	0.866920i	$0.666096\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			1477.83i			1.71243i	0.516616	+	0.856217i	$0.327192\pi$
$863$			1477.83i			1.71243i	−0.516616	+	0.856217i	$0.672808\pi$
$864$		0			0
$865$			208.621i			0.241180i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	512.920i		−	0.590241i
$870$		0			0
$871$	−39.7483			−0.0456352
$872$		0			0
$873$		0			0
$874$		0			0
$875$	473.620			0.541280
$876$		0			0
$877$		−	160.901i		−	0.183467i	−0.995784	−	0.0917337i	$-0.970759\pi$
$877$		−	160.901i		−	0.183467i	0.995784	−	0.0917337i	$-0.0292408\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	711.737			0.807874			0.403937	−	0.914787i	$-0.367641\pi$
$881$	711.737			0.807874			0.403937	+	0.914787i	$0.367641\pi$
$882$		0			0
$883$	1690.91			1.91496			0.957481	−	0.288496i	$-0.0931551\pi$
$883$	1690.91			1.91496			0.957481	+	0.288496i	$0.0931551\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	52.1528i		−	0.0587968i	−0.999568	−	0.0293984i	$-0.990641\pi$
$887$		−	52.1528i		−	0.0587968i	0.999568	−	0.0293984i	$-0.00935916\pi$
$888$		0			0
$889$		−	546.956i		−	0.615248i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−248.910	−	593.256i	−0.278735	−	0.664341i
$894$		0			0
$895$		−	744.982i		−	0.832383i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1432.88			1.59385
$900$		0			0
$901$			159.126i			0.176611i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			557.467i			0.615986i
$906$		0			0
$907$			1447.89i			1.59635i	0.602425	+	0.798176i	$0.294201\pi$
$907$			1447.89i			1.59635i	−0.602425	+	0.798176i	$0.705799\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	946.717i		−	1.03921i	−0.854408	−	0.519603i	$-0.826080\pi$
$911$		−	946.717i		−	1.03921i	0.854408	−	0.519603i	$-0.173920\pi$
$912$		0			0
$913$	−450.554			−0.493487
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−582.082			−0.634768
$918$		0			0
$919$	236.916			0.257797			0.128899	−	0.991658i	$-0.458856\pi$
$919$	236.916			0.257797			0.128899	+	0.991658i	$0.458856\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−2132.47			−2.31037
$924$		0			0
$925$			405.335i			0.438200i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1000.34			−1.07680			−0.538398	−	0.842691i	$-0.680970\pi$
$929$	−1000.34			−1.07680			−0.538398	+	0.842691i	$0.680970\pi$
$930$		0			0
$931$	−204.948	−	488.478i	−0.220138	−	0.524681i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−172.652			−0.184655
$936$		0			0
$937$	−440.075			−0.469664			−0.234832	−	0.972036i	$-0.575454\pi$
$937$	−440.075			−0.469664			−0.234832	+	0.972036i	$0.575454\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			1176.64i			1.25042i	0.780457	+	0.625209i	$0.214986\pi$
$941$			1176.64i			1.25042i	−0.780457	+	0.625209i	$0.785014\pi$
$942$		0			0
$943$			696.952i			0.739080i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	225.120			0.237719			0.118860	−	0.992911i	$-0.462076\pi$
$947$	225.120			0.237719			0.118860	+	0.992911i	$0.462076\pi$
$948$		0			0
$949$			1455.77i			1.53400i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	589.549i		−	0.618624i	−0.950961	−	0.309312i	$-0.899901\pi$
$953$		−	589.549i		−	0.618624i	0.950961	−	0.309312i	$-0.100099\pi$
$954$		0			0
$955$	−266.955			−0.279534
$956$		0			0
$957$		0			0
$958$		0			0
$959$	221.804			0.231287
$960$		0			0
$961$	−740.629			−0.770686
$962$		0			0
$963$		0			0
$964$		0			0
$965$			641.084i			0.664335i
$966$		0			0
$967$	1641.09			1.69709			0.848545	−	0.529124i	$-0.177479\pi$
$967$	1641.09			1.69709			0.848545	+	0.529124i	$0.177479\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	721.099i		−	0.742636i	−0.928506	−	0.371318i	$-0.878906\pi$
$971$		−	721.099i		−	0.742636i	0.928506	−	0.371318i	$-0.121094\pi$
$972$		0			0
$973$	−1117.28			−1.14829
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	1222.67i		−	1.25145i	−0.780043	−	0.625725i	$-0.784803\pi$
$977$		−	1222.67i		−	1.25145i	0.780043	−	0.625725i	$-0.215197\pi$
$978$		0			0
$979$			1025.12i			1.04711i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	1493.88i		−	1.51971i	−0.650090	−	0.759857i	$-0.725269\pi$
$983$		−	1493.88i		−	1.51971i	0.650090	−	0.759857i	$-0.274731\pi$
$984$		0			0
$985$	181.604			0.184370
$986$		0			0
$987$		0			0
$988$		0			0
$989$	271.129			0.274144
$990$		0			0
$991$			1461.21i			1.47448i	0.675631	+	0.737240i	$0.263871\pi$
$991$			1461.21i			1.47448i	−0.675631	+	0.737240i	$0.736129\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	216.050			0.217136
$996$		0			0
$997$	−963.569			−0.966469			−0.483234	−	0.875491i	$-0.660538\pi$
$997$	−963.569			−0.966469			−0.483234	+	0.875491i	$0.660538\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.3.o.p.721.4		8
3.2	odd	2		304.3.e.g.113.4		8
4.3	odd	2		1368.3.o.b.721.4		8
12.11	even	2		152.3.e.b.113.5	yes	8
19.18	odd	2	inner	2736.3.o.p.721.3		8
24.5	odd	2		1216.3.e.n.1025.5		8
24.11	even	2		1216.3.e.m.1025.4		8
57.56	even	2		304.3.e.g.113.5		8
76.75	even	2		1368.3.o.b.721.3		8
228.227	odd	2		152.3.e.b.113.4	✓	8
456.227	odd	2		1216.3.e.m.1025.5		8
456.341	even	2		1216.3.e.n.1025.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.3.e.b.113.4	✓	8	228.227	odd	2
152.3.e.b.113.5	yes	8	12.11	even	2
304.3.e.g.113.4		8	3.2	odd	2
304.3.e.g.113.5		8	57.56	even	2
1216.3.e.m.1025.4		8	24.11	even	2
1216.3.e.m.1025.5		8	456.227	odd	2
1216.3.e.n.1025.4		8	456.341	even	2
1216.3.e.n.1025.5		8	24.5	odd	2
1368.3.o.b.721.3		8	76.75	even	2
1368.3.o.b.721.4		8	4.3	odd	2
2736.3.o.p.721.3		8	19.18	odd	2	inner
2736.3.o.p.721.4		8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2736.o (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.30665 q^{5} -4.59559 q^{7} +O(q^{10})\)	\(q+2.30665 q^{5} -4.59559 q^{7} +13.7820 q^{11} +22.2405i q^{13} -5.43101 q^{17} +(7.35095 + 17.5204i) q^{19} +24.9908 q^{23} -19.6794 q^{25} -34.7357i q^{29} +41.2508i q^{31} -10.6004 q^{35} -20.5969i q^{37} +27.8883i q^{41} +10.8491 q^{43} -33.8609 q^{47} -27.8805 q^{49} -29.2996i q^{53} +31.7901 q^{55} -65.8291i q^{59} -107.373 q^{61} +51.3010i q^{65} +1.78720i q^{67} +95.8822i q^{71} +65.4556 q^{73} -63.3362 q^{77} -37.2168i q^{79} -32.6916 q^{83} -12.5274 q^{85} +74.3815i q^{89} -102.208i q^{91} +(16.9560 + 40.4133i) q^{95} -128.116i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 14 q^{5} + 6 q^{7}+O(q^{10}) \) 8 * q + 14 * q^5 + 6 * q^7	\( 8 q + 14 q^{5} + 6 q^{7} - 26 q^{11} + 18 q^{17} - 16 q^{19} + 12 q^{23} + 34 q^{25} - 50 q^{35} - 62 q^{43} + 22 q^{47} + 22 q^{49} - 174 q^{55} - 158 q^{61} - 170 q^{73} - 82 q^{77} - 64 q^{83} + 410 q^{85} + 222 q^{95}+O(q^{100}) \) 8 * q + 14 * q^5 + 6 * q^7 - 26 * q^11 + 18 * q^17 - 16 * q^19 + 12 * q^23 + 34 * q^25 - 50 * q^35 - 62 * q^43 + 22 * q^47 + 22 * q^49 - 174 * q^55 - 158 * q^61 - 170 * q^73 - 82 * q^77 - 64 * q^83 + 410 * q^85 + 222 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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