LMFDB - Embedded newform 2736.3.o.q.721.15

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:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 264 x^{18} + 28274 x^{16} - 1545308 x^{14} + 45358441 x^{12} - 637328868 x^{10} + \cdots + 194396337216 $ x^20 - 264x^18 + 28274x^16 - 1545308x^14 + 45358441x^12 - 637328868x^10 + 1825819356x^8 + 32794262368x^6 + 135580415344x^4 + 245217530816*x^2 + 194396337216
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{32} $
Twist minimal:	no (minimal twist has level 1368)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			721.15
Root			$5.08927 - 1.41421i$ of defining polynomial
Character	$\chi$	$=$	2736.721
Dual form			2736.3.o.q.721.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.08927 q^{5} +11.6968 q^{7} +O(q^{10})$	$q+5.08927 q^{5} +11.6968 q^{7} -14.6358 q^{11} -16.9834i q^{13} -8.79536 q^{17} +(11.9638 + 14.7603i) q^{19} -6.84997 q^{23} +0.900640 q^{25} +45.8127i q^{29} -50.1201i q^{31} +59.5283 q^{35} -35.1916i q^{37} -60.4216i q^{41} +75.0196 q^{43} +52.6783 q^{47} +87.8158 q^{49} -93.2242i q^{53} -74.4857 q^{55} -13.0988i q^{59} -1.36673 q^{61} -86.4331i q^{65} -69.4258i q^{67} +58.7253i q^{71} -41.4946 q^{73} -171.193 q^{77} +90.3616i q^{79} +155.789 q^{83} -44.7620 q^{85} -73.4230i q^{89} -198.652i q^{91} +(60.8869 + 75.1194i) q^{95} -36.7235i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 16 q^{7}+O(q^{10}) $ 20 * q + 16 * q^7	$ 20 q + 16 q^{7} + 8 q^{19} + 68 q^{25} + 128 q^{43} + 116 q^{49} - 144 q^{55} - 104 q^{61} - 88 q^{73} - 280 q^{85}+O(q^{100}) $ 20 * q + 16 * q^7 + 8 * q^19 + 68 * q^25 + 128 * q^43 + 116 * q^49 - 144 * q^55 - 104 * q^61 - 88 * q^73 - 280 * q^85

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
$5$	5.08927			1.01785			0.508927	−	0.860810i	$-0.330042\pi$
$5$	5.08927			1.01785			0.508927	+	0.860810i	$0.330042\pi$
$6$		0			0
$7$	11.6968			1.67098			0.835488	−	0.549509i	$-0.185185\pi$
$7$	11.6968			1.67098			0.835488	+	0.549509i	$0.185185\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−14.6358			−1.33053			−0.665266	−	0.746607i	$-0.731682\pi$
$11$	−14.6358			−1.33053			−0.665266	+	0.746607i	$0.731682\pi$
$12$		0			0
$13$		−	16.9834i		−	1.30642i	−0.757179	−	0.653208i	$-0.773423\pi$
$13$		−	16.9834i		−	1.30642i	0.757179	−	0.653208i	$-0.226577\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−8.79536			−0.517374			−0.258687	−	0.965961i	$-0.583290\pi$
$17$	−8.79536			−0.517374			−0.258687	+	0.965961i	$0.583290\pi$
$18$		0			0
$19$	11.9638	+	14.7603i	0.629673	+	0.776861i
$19$	11.9638	+	14.7603i	0.629673	+	0.776861i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−6.84997			−0.297825			−0.148912	−	0.988850i	$-0.547577\pi$
$23$	−6.84997			−0.297825			−0.148912	+	0.988850i	$0.547577\pi$
$24$		0			0
$25$	0.900640			0.0360256
$26$		0			0
$27$		0			0
$28$		0			0
$29$			45.8127i			1.57975i	0.613270	+	0.789873i	$0.289854\pi$
$29$			45.8127i			1.57975i	−0.613270	+	0.789873i	$0.710146\pi$
$30$		0			0
$31$		−	50.1201i		−	1.61678i	−0.588648	−	0.808389i	$-0.700340\pi$
$31$		−	50.1201i		−	1.61678i	0.588648	−	0.808389i	$-0.299660\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	59.5283			1.70081
$36$		0			0
$37$		−	35.1916i		−	0.951124i	−0.879682	−	0.475562i	$-0.842245\pi$
$37$		−	35.1916i		−	0.951124i	0.879682	−	0.475562i	$-0.157755\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	60.4216i		−	1.47370i	−0.676057	−	0.736849i	$-0.736313\pi$
$41$		−	60.4216i		−	1.47370i	0.676057	−	0.736849i	$-0.263687\pi$
$42$		0			0
$43$	75.0196			1.74464			0.872321	−	0.488933i	$-0.162614\pi$
$43$	75.0196			1.74464			0.872321	+	0.488933i	$0.162614\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	52.6783			1.12082			0.560408	−	0.828217i	$-0.310644\pi$
$47$	52.6783			1.12082			0.560408	+	0.828217i	$0.310644\pi$
$48$		0			0
$49$	87.8158			1.79216
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	93.2242i		−	1.75895i	−0.475947	−	0.879474i	$-0.657895\pi$
$53$		−	93.2242i		−	1.75895i	0.475947	−	0.879474i	$-0.342105\pi$
$54$		0			0
$55$	−74.4857			−1.35429
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	13.0988i		−	0.222014i	−0.993820	−	0.111007i	$-0.964592\pi$
$59$		−	13.0988i		−	0.222014i	0.993820	−	0.111007i	$-0.0354077\pi$
$60$		0			0
$61$	−1.36673			−0.0224055			−0.0112027	−	0.999937i	$-0.503566\pi$
$61$	−1.36673			−0.0224055			−0.0112027	+	0.999937i	$0.503566\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	86.4331i		−	1.32974i
$66$		0			0
$67$		−	69.4258i		−	1.03621i	−0.855318	−	0.518103i	$-0.826638\pi$
$67$		−	69.4258i		−	1.03621i	0.855318	−	0.518103i	$-0.173362\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			58.7253i			0.827116i	0.910478	+	0.413558i	$0.135714\pi$
$71$			58.7253i			0.827116i	−0.910478	+	0.413558i	$0.864286\pi$
$72$		0			0
$73$	−41.4946			−0.568419			−0.284210	−	0.958762i	$-0.591731\pi$
$73$	−41.4946			−0.568419			−0.284210	+	0.958762i	$0.591731\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−171.193			−2.22329
$78$		0			0
$79$			90.3616i			1.14382i	0.820317	+	0.571909i	$0.193797\pi$
$79$			90.3616i			1.14382i	−0.820317	+	0.571909i	$0.806203\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	155.789			1.87698			0.938491	−	0.345304i	$-0.112224\pi$
$83$	155.789			1.87698			0.938491	+	0.345304i	$0.112224\pi$
$84$		0			0
$85$	−44.7620			−0.526611
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	73.4230i		−	0.824978i	−0.910963	−	0.412489i	$-0.864660\pi$
$89$		−	73.4230i		−	0.824978i	0.910963	−	0.412489i	$-0.135340\pi$
$90$		0			0
$91$		−	198.652i		−	2.18299i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	60.8869	+	75.1194i	0.640915	+	0.790730i
$96$		0			0
$97$		−	36.7235i		−	0.378592i	−0.981920	−	0.189296i	$-0.939379\pi$
$97$		−	36.7235i		−	0.378592i	0.981920	−	0.189296i	$-0.0606206\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	102.594			1.01578			0.507892	−	0.861421i	$-0.330425\pi$
$101$	102.594			1.01578			0.507892	+	0.861421i	$0.330425\pi$
$102$		0			0
$103$			64.2479i			0.623766i	0.950121	+	0.311883i	$0.100960\pi$
$103$			64.2479i			0.623766i	−0.950121	+	0.311883i	$0.899040\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	19.9788i		−	0.186718i	−0.995633	−	0.0933588i	$-0.970240\pi$
$107$		−	19.9788i		−	0.186718i	0.995633	−	0.0933588i	$-0.0297604\pi$
$108$		0			0
$109$		−	144.748i		−	1.32796i	−0.747749	−	0.663981i	$-0.768866\pi$
$109$		−	144.748i		−	1.32796i	0.747749	−	0.663981i	$-0.231134\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			127.440i			1.12779i	0.825846	+	0.563895i	$0.190698\pi$
$113$			127.440i			1.12779i	−0.825846	+	0.563895i	$0.809302\pi$
$114$		0			0
$115$	−34.8613			−0.303142
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−102.878			−0.864520
$120$		0			0
$121$	93.2079			0.770313
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−122.648			−0.981185
$126$		0			0
$127$		−	34.1643i		−	0.269010i	−0.990913	−	0.134505i	$-0.957056\pi$
$127$		−	34.1643i		−	0.269010i	0.990913	−	0.134505i	$-0.0429445\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	33.9589			0.259228			0.129614	−	0.991565i	$-0.458626\pi$
$131$	33.9589			0.259228			0.129614	+	0.991565i	$0.458626\pi$
$132$		0			0
$133$	139.938	+	172.649i	1.05217	+	1.29811i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	110.031			0.803148			0.401574	−	0.915827i	$-0.368463\pi$
$137$	110.031			0.803148			0.401574	+	0.915827i	$0.368463\pi$
$138$		0			0
$139$	16.3672			0.117750			0.0588748	−	0.998265i	$-0.481249\pi$
$139$	16.3672			0.117750			0.0588748	+	0.998265i	$0.481249\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			248.566i			1.73823i
$144$		0			0
$145$			233.153i			1.60795i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−249.958			−1.67757			−0.838784	−	0.544465i	$-0.816733\pi$
$149$	−249.958			−1.67757			−0.838784	+	0.544465i	$0.816733\pi$
$150$		0			0
$151$			153.037i			1.01349i	0.862097	+	0.506743i	$0.169151\pi$
$151$			153.037i			1.01349i	−0.862097	+	0.506743i	$0.830849\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	255.075i		−	1.64564i
$156$		0			0
$157$	80.1194			0.510315			0.255157	−	0.966900i	$-0.417873\pi$
$157$	80.1194			0.510315			0.255157	+	0.966900i	$0.417873\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−80.1229			−0.497658
$162$		0			0
$163$	209.684			1.28640			0.643202	−	0.765696i	$-0.277605\pi$
$163$	209.684			1.28640			0.643202	+	0.765696i	$0.277605\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	26.1320i		−	0.156479i	−0.996935	−	0.0782395i	$-0.975070\pi$
$167$		−	26.1320i		−	0.156479i	0.996935	−	0.0782395i	$-0.0249299\pi$
$168$		0			0
$169$	−119.436			−0.706722
$170$		0			0
$171$		0			0
$172$		0			0
$173$			148.817i			0.860216i	0.902777	+	0.430108i	$0.141525\pi$
$173$			148.817i			0.860216i	−0.902777	+	0.430108i	$0.858475\pi$
$174$		0			0
$175$	10.5346			0.0601979
$176$		0			0
$177$		0			0
$178$		0			0
$179$			346.022i			1.93308i	0.256512	+	0.966541i	$0.417427\pi$
$179$			346.022i			1.93308i	−0.256512	+	0.966541i	$0.582573\pi$
$180$		0			0
$181$		−	283.289i		−	1.56514i	−0.622566	−	0.782568i	$-0.713910\pi$
$181$		−	283.289i		−	1.56514i	0.622566	−	0.782568i	$-0.286090\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	179.099i		−	0.968105i
$186$		0			0
$187$	128.728			0.688383
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−315.646			−1.65260			−0.826300	−	0.563231i	$-0.809558\pi$
$191$	−315.646			−1.65260			−0.826300	+	0.563231i	$0.809558\pi$
$192$		0			0
$193$		−	47.4748i		−	0.245983i	−0.992408	−	0.122992i	$-0.960751\pi$
$193$		−	47.4748i		−	0.245983i	0.992408	−	0.122992i	$-0.0392489\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−197.715			−1.00363			−0.501816	−	0.864974i	$-0.667335\pi$
$197$	−197.715			−1.00363			−0.501816	+	0.864974i	$0.667335\pi$
$198$		0			0
$199$	107.198			0.538684			0.269342	−	0.963045i	$-0.413194\pi$
$199$	107.198			0.538684			0.269342	+	0.963045i	$0.413194\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			535.863i			2.63972i
$204$		0			0
$205$		−	307.502i		−	1.50001i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−175.100	−	216.030i	−0.837799	−	1.03364i
$210$		0			0
$211$		−	377.831i		−	1.79067i	−0.445394	−	0.895335i	$-0.646936\pi$
$211$		−	377.831i		−	1.79067i	0.445394	−	0.895335i	$-0.353064\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	381.795			1.77579
$216$		0			0
$217$		−	586.247i		−	2.70160i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			149.375i			0.675906i
$222$		0			0
$223$		−	29.5498i		−	0.132510i	−0.997803	−	0.0662551i	$-0.978895\pi$
$223$		−	29.5498i		−	0.132510i	0.997803	−	0.0662551i	$-0.0211051\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	234.224i		−	1.03182i	−0.856642	−	0.515911i	$-0.827454\pi$
$227$		−	234.224i		−	1.03182i	0.856642	−	0.515911i	$-0.172546\pi$
$228$		0			0
$229$	68.7496			0.300216			0.150108	−	0.988670i	$-0.452038\pi$
$229$	68.7496			0.300216			0.150108	+	0.988670i	$0.452038\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−190.805			−0.818904			−0.409452	−	0.912332i	$-0.634280\pi$
$233$	−190.805			−0.818904			−0.409452	+	0.912332i	$0.634280\pi$
$234$		0			0
$235$	268.094			1.14083
$236$		0			0
$237$		0			0
$238$		0			0
$239$	413.869			1.73167			0.865835	−	0.500330i	$-0.166788\pi$
$239$	413.869			1.73167			0.865835	+	0.500330i	$0.166788\pi$
$240$		0			0
$241$			271.817i			1.12787i	0.825819	+	0.563935i	$0.190713\pi$
$241$			271.817i			1.12787i	−0.825819	+	0.563935i	$0.809287\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	446.918			1.82416
$246$		0			0
$247$	250.681	−	203.186i	1.01490	−	0.822614i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−218.692			−0.871283			−0.435641	−	0.900120i	$-0.643478\pi$
$251$	−218.692			−0.871283			−0.435641	+	0.900120i	$0.643478\pi$
$252$		0			0
$253$	100.255			0.396265
$254$		0			0
$255$		0			0
$256$		0			0
$257$			115.480i			0.449339i	0.974435	+	0.224669i	$0.0721302\pi$
$257$			115.480i			0.449339i	−0.974435	+	0.224669i	$0.927870\pi$
$258$		0			0
$259$		−	411.630i		−	1.58931i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	101.290			0.385134			0.192567	−	0.981284i	$-0.438319\pi$
$263$	101.290			0.385134			0.192567	+	0.981284i	$0.438319\pi$
$264$		0			0
$265$		−	474.443i		−	1.79035i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			223.290i			0.830075i	0.909804	+	0.415037i	$0.136232\pi$
$269$			223.290i			0.830075i	−0.909804	+	0.415037i	$0.863768\pi$
$270$		0			0
$271$	−486.033			−1.79348			−0.896741	−	0.442556i	$-0.854072\pi$
$271$	−486.033			−1.79348			−0.896741	+	0.442556i	$0.854072\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−13.1816			−0.0479332
$276$		0			0
$277$	16.0729			0.0580250			0.0290125	−	0.999579i	$-0.490764\pi$
$277$	16.0729			0.0580250			0.0290125	+	0.999579i	$0.490764\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	165.956i		−	0.590589i	−0.955406	−	0.295295i	$-0.904582\pi$
$281$		−	165.956i		−	0.590589i	0.955406	−	0.295295i	$-0.0954178\pi$
$282$		0			0
$283$	154.300			0.545230			0.272615	−	0.962123i	$-0.412111\pi$
$283$	154.300			0.545230			0.272615	+	0.962123i	$0.412111\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	706.741i		−	2.46251i
$288$		0			0
$289$	−211.642			−0.732324
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	448.580i		−	1.53099i	−0.643441	−	0.765496i	$-0.722494\pi$
$293$		−	448.580i		−	1.53099i	0.643441	−	0.765496i	$-0.277506\pi$
$294$		0			0
$295$		−	66.6635i		−	0.225978i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			116.336i			0.389083i
$300$		0			0
$301$	877.492			2.91525
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−6.95568			−0.0228055
$306$		0			0
$307$		−	164.603i		−	0.536166i	−0.963396	−	0.268083i	$-0.913610\pi$
$307$		−	164.603i		−	0.536166i	0.963396	−	0.268083i	$-0.0863901\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−290.496			−0.934072			−0.467036	−	0.884238i	$-0.654678\pi$
$311$	−290.496			−0.934072			−0.467036	+	0.884238i	$0.654678\pi$
$312$		0			0
$313$	568.470			1.81620			0.908100	−	0.418754i	$-0.137533\pi$
$313$	568.470			1.81620			0.908100	+	0.418754i	$0.137533\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	150.231i		−	0.473915i	−0.971520	−	0.236957i	$-0.923850\pi$
$317$		−	150.231i		−	0.473915i	0.971520	−	0.236957i	$-0.0761502\pi$
$318$		0			0
$319$		−	670.507i		−	2.10190i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−105.226	−	129.823i	−0.325777	−	0.401928i
$324$		0			0
$325$		−	15.2959i		−	0.0470644i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	616.169			1.87285
$330$		0			0
$331$			13.4836i			0.0407360i	0.999793	+	0.0203680i	$0.00648378\pi$
$331$			13.4836i			0.0407360i	−0.999793	+	0.0203680i	$0.993516\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	353.327i		−	1.05471i
$336$		0			0
$337$			209.994i			0.623129i	0.950225	+	0.311565i	$0.100853\pi$
$337$			209.994i			0.623129i	−0.950225	+	0.311565i	$0.899147\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			733.550i			2.15117i
$342$		0			0
$343$	454.022			1.32368
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−195.718			−0.564028			−0.282014	−	0.959410i	$-0.591002\pi$
$347$	−195.718			−0.564028			−0.282014	+	0.959410i	$0.591002\pi$
$348$		0			0
$349$	283.953			0.813618			0.406809	−	0.913513i	$-0.366641\pi$
$349$	283.953			0.813618			0.406809	+	0.913513i	$0.366641\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	395.888			1.12150			0.560748	−	0.827986i	$-0.310514\pi$
$353$	395.888			1.12150			0.560748	+	0.827986i	$0.310514\pi$
$354$		0			0
$355$			298.869i			0.841883i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−442.196			−1.23174			−0.615872	−	0.787846i	$-0.711196\pi$
$359$	−442.196			−1.23174			−0.615872	+	0.787846i	$0.711196\pi$
$360$		0			0
$361$	−74.7358	+	353.179i	−0.207025	+	0.978336i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−211.177			−0.578567
$366$		0			0
$367$	−3.85524			−0.0105048			−0.00525238	−	0.999986i	$-0.501672\pi$
$367$	−3.85524			−0.0105048			−0.00525238	+	0.999986i	$0.501672\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	1090.43i		−	2.93916i
$372$		0			0
$373$		−	547.618i		−	1.46814i	−0.679072	−	0.734072i	$-0.737618\pi$
$373$		−	547.618i		−	1.46814i	0.679072	−	0.734072i	$-0.262382\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	778.055			2.06381
$378$		0			0
$379$			495.686i			1.30788i	0.756548	+	0.653939i	$0.226885\pi$
$379$			495.686i			1.30788i	−0.756548	+	0.653939i	$0.773115\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			334.651i			0.873763i	0.899519	+	0.436881i	$0.143917\pi$
$383$			334.651i			0.873763i	−0.899519	+	0.436881i	$0.856083\pi$
$384$		0			0
$385$	−871.247			−2.26298
$386$		0			0
$387$		0			0
$388$		0			0
$389$	348.223			0.895175			0.447588	−	0.894240i	$-0.352283\pi$
$389$	348.223			0.895175			0.447588	+	0.894240i	$0.352283\pi$
$390$		0			0
$391$	60.2480			0.154087
$392$		0			0
$393$		0			0
$394$		0			0
$395$			459.874i			1.16424i
$396$		0			0
$397$	55.7961			0.140544			0.0702721	−	0.997528i	$-0.477613\pi$
$397$	55.7961			0.140544			0.0702721	+	0.997528i	$0.477613\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			422.248i			1.05299i	0.850179	+	0.526494i	$0.176494\pi$
$401$			422.248i			1.05299i	−0.850179	+	0.526494i	$0.823506\pi$
$402$		0			0
$403$	−851.210			−2.11218
$404$		0			0
$405$		0			0
$406$		0			0
$407$			515.059i			1.26550i
$408$		0			0
$409$		−	3.23852i		−	0.00791814i	−0.999992	−	0.00395907i	$-0.998740\pi$
$409$		−	3.23852i		−	0.00791814i	0.999992	−	0.00395907i	$-0.00126021\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	153.215i		−	0.370981i
$414$		0			0
$415$	792.854			1.91049
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−427.843			−1.02110			−0.510552	−	0.859847i	$-0.670559\pi$
$419$	−427.843			−1.02110			−0.510552	+	0.859847i	$0.670559\pi$
$420$		0			0
$421$			592.172i			1.40658i	0.710901	+	0.703292i	$0.248287\pi$
$421$			592.172i			1.40658i	−0.710901	+	0.703292i	$0.751713\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−7.92146			−0.0186387
$426$		0			0
$427$	−15.9865			−0.0374390
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	411.855i		−	0.955579i	−0.878474	−	0.477790i	$-0.841438\pi$
$431$		−	411.855i		−	0.955579i	0.878474	−	0.477790i	$-0.158562\pi$
$432$		0			0
$433$			607.670i			1.40340i	0.712475	+	0.701698i	$0.247574\pi$
$433$			607.670i			1.40340i	−0.712475	+	0.701698i	$0.752426\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−81.9515	−	101.108i	−0.187532	−	0.231368i
$438$		0			0
$439$			6.58398i			0.0149977i	0.999972	+	0.00749884i	$0.00238698\pi$
$439$			6.58398i			0.0149977i	−0.999972	+	0.00749884i	$0.997613\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	270.742			0.611156			0.305578	−	0.952167i	$-0.401150\pi$
$443$	270.742			0.611156			0.305578	+	0.952167i	$0.401150\pi$
$444$		0			0
$445$		−	373.669i		−	0.839706i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			229.268i			0.510620i	0.966859	+	0.255310i	$0.0821775\pi$
$449$			229.268i			0.510620i	−0.966859	+	0.255310i	$0.917822\pi$
$450$		0			0
$451$			884.321i			1.96080i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	1010.99i		−	2.22196i
$456$		0			0
$457$	25.6628			0.0561548			0.0280774	−	0.999606i	$-0.491062\pi$
$457$	25.6628			0.0561548			0.0280774	+	0.999606i	$0.491062\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	546.064			1.18452			0.592260	−	0.805747i	$-0.298236\pi$
$461$	546.064			1.18452			0.592260	+	0.805747i	$0.298236\pi$
$462$		0			0
$463$	218.764			0.472492			0.236246	−	0.971693i	$-0.424083\pi$
$463$	218.764			0.472492			0.236246	+	0.971693i	$0.424083\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−19.8980			−0.0426082			−0.0213041	−	0.999773i	$-0.506782\pi$
$467$	−19.8980			−0.0426082			−0.0213041	+	0.999773i	$0.506782\pi$
$468$		0			0
$469$		−	812.062i		−	1.73148i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−1097.98			−2.32130
$474$		0			0
$475$	10.7751	+	13.2938i	0.0226843	+	0.0279869i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	95.8635			0.200133			0.100066	−	0.994981i	$-0.468095\pi$
$479$	95.8635			0.200133			0.100066	+	0.994981i	$0.468095\pi$
$480$		0			0
$481$	−597.673			−1.24256
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	186.895i		−	0.385352i
$486$		0			0
$487$		−	305.504i		−	0.627318i	−0.949536	−	0.313659i	$-0.898445\pi$
$487$		−	305.504i		−	0.627318i	0.949536	−	0.313659i	$-0.101555\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	379.267			0.772438			0.386219	−	0.922407i	$-0.373781\pi$
$491$	379.267			0.772438			0.386219	+	0.922407i	$0.373781\pi$
$492$		0			0
$493$		−	402.939i		−	0.817321i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			686.899i			1.38209i
$498$		0			0
$499$	−189.530			−0.379819			−0.189910	−	0.981802i	$-0.560820\pi$
$499$	−189.530			−0.379819			−0.189910	+	0.981802i	$0.560820\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	30.3009			0.0602403			0.0301201	−	0.999546i	$-0.490411\pi$
$503$	30.3009			0.0602403			0.0301201	+	0.999546i	$0.490411\pi$
$504$		0			0
$505$	522.129			1.03392
$506$		0			0
$507$		0			0
$508$		0			0
$509$			481.358i			0.945693i	0.881145	+	0.472846i	$0.156773\pi$
$509$			481.358i			0.945693i	−0.881145	+	0.472846i	$0.843227\pi$
$510$		0			0
$511$	−485.355			−0.949814
$512$		0			0
$513$		0			0
$514$		0			0
$515$			326.975i			0.634902i
$516$		0			0
$517$	−770.992			−1.49128
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	311.850i		−	0.598561i	−0.954165	−	0.299281i	$-0.903253\pi$
$521$		−	311.850i		−	0.598561i	0.954165	−	0.299281i	$-0.0967467\pi$
$522$		0			0
$523$			22.6368i			0.0432826i	0.999766	+	0.0216413i	$0.00688918\pi$
$523$			22.6368i			0.0432826i	−0.999766	+	0.0216413i	$0.993111\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			440.825i			0.836480i
$528$		0			0
$529$	−482.078			−0.911300
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1026.16			−1.92526
$534$		0			0
$535$		−	101.677i		−	0.190051i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−1285.26			−2.38452
$540$		0			0
$541$	−767.716			−1.41907			−0.709534	−	0.704671i	$-0.751095\pi$
$541$	−767.716			−1.41907			−0.709534	+	0.704671i	$0.751095\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	736.661i		−	1.35167i
$546$		0			0
$547$			237.682i			0.434519i	0.976114	+	0.217260i	$0.0697118\pi$
$547$			237.682i			0.434519i	−0.976114	+	0.217260i	$0.930288\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−676.211	+	548.093i	−1.22724	+	0.994724i
$552$		0			0
$553$			1056.94i			1.91129i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	227.023			0.407581			0.203790	−	0.979015i	$-0.434674\pi$
$557$	227.023			0.407581			0.203790	+	0.979015i	$0.434674\pi$
$558$		0			0
$559$		−	1274.09i		−	2.27923i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			99.7946i			0.177255i	0.996065	+	0.0886275i	$0.0282481\pi$
$563$			99.7946i			0.177255i	−0.996065	+	0.0886275i	$0.971752\pi$
$564$		0			0
$565$			648.578i			1.14793i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			635.343i			1.11660i	0.829641	+	0.558298i	$0.188545\pi$
$569$			635.343i			1.11660i	−0.829641	+	0.558298i	$0.811455\pi$
$570$		0			0
$571$	−587.396			−1.02871			−0.514357	−	0.857576i	$-0.671969\pi$
$571$	−587.396			−1.02871			−0.514357	+	0.857576i	$0.671969\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−6.16936			−0.0107293
$576$		0			0
$577$	889.609			1.54178			0.770892	−	0.636966i	$-0.219811\pi$
$577$	889.609			1.54178			0.770892	+	0.636966i	$0.219811\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	1822.24			3.13639
$582$		0			0
$583$			1364.41i			2.34033i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	400.577			0.682414			0.341207	−	0.939988i	$-0.389164\pi$
$587$	400.577			0.682414			0.341207	+	0.939988i	$0.389164\pi$
$588$		0			0
$589$	739.791	−	599.626i	1.25601	−	1.01804i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	402.176			0.678205			0.339103	−	0.940749i	$-0.389877\pi$
$593$	402.176			0.678205			0.339103	+	0.940749i	$0.389877\pi$
$594$		0			0
$595$	−523.573			−0.879955
$596$		0			0
$597$		0			0
$598$		0			0
$599$			479.823i			0.801039i	0.916288	+	0.400520i	$0.131170\pi$
$599$			479.823i			0.801039i	−0.916288	+	0.400520i	$0.868830\pi$
$600$		0			0
$601$			207.462i			0.345195i	0.984992	+	0.172597i	$0.0552160\pi$
$601$			207.462i			0.345195i	−0.984992	+	0.172597i	$0.944784\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	474.360			0.784066
$606$		0			0
$607$			96.0000i			0.158155i	0.996868	+	0.0790774i	$0.0251974\pi$
$607$			96.0000i			0.158155i	−0.996868	+	0.0790774i	$0.974803\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	894.657i		−	1.46425i
$612$		0			0
$613$	208.034			0.339370			0.169685	−	0.985498i	$-0.445725\pi$
$613$	208.034			0.339370			0.169685	+	0.985498i	$0.445725\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−67.9068			−0.110060			−0.0550298	−	0.998485i	$-0.517525\pi$
$617$	−67.9068			−0.110060			−0.0550298	+	0.998485i	$0.517525\pi$
$618$		0			0
$619$	485.579			0.784458			0.392229	−	0.919868i	$-0.371704\pi$
$619$	485.579			0.784458			0.392229	+	0.919868i	$0.371704\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	858.816i		−	1.37852i
$624$		0			0
$625$	−646.705			−1.03473
$626$		0			0
$627$		0			0
$628$		0			0
$629$			309.523i			0.492087i
$630$		0			0
$631$	−378.427			−0.599726			−0.299863	−	0.953982i	$-0.596941\pi$
$631$	−378.427			−0.599726			−0.299863	+	0.953982i	$0.596941\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	173.871i		−	0.273813i
$636$		0			0
$637$		−	1491.41i		−	2.34130i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	152.435i		−	0.237808i	−0.992906	−	0.118904i	$-0.962062\pi$
$641$		−	152.435i		−	0.237808i	0.992906	−	0.118904i	$-0.0379380\pi$
$642$		0			0
$643$	214.750			0.333981			0.166991	−	0.985958i	$-0.446595\pi$
$643$	214.750			0.333981			0.166991	+	0.985958i	$0.446595\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−93.1807			−0.144020			−0.0720098	−	0.997404i	$-0.522941\pi$
$647$	−93.1807			−0.144020			−0.0720098	+	0.997404i	$0.522941\pi$
$648$		0			0
$649$			191.713i			0.295397i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−995.847			−1.52503			−0.762517	−	0.646968i	$-0.776037\pi$
$653$	−995.847			−1.52503			−0.762517	+	0.646968i	$0.776037\pi$
$654$		0			0
$655$	172.826			0.263857
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	26.3579i		−	0.0399968i	−0.999800	−	0.0199984i	$-0.993634\pi$
$659$		−	26.3579i		−	0.0399968i	0.999800	−	0.0199984i	$-0.00636612\pi$
$660$		0			0
$661$			808.835i			1.22365i	0.790992	+	0.611827i	$0.209565\pi$
$661$			808.835i			1.22365i	−0.790992	+	0.611827i	$0.790435\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	712.183	+	878.658i	1.07095	+	1.32129i
$666$		0			0
$667$		−	313.815i		−	0.470488i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	20.0033			0.0298112
$672$		0			0
$673$			450.410i			0.669257i	0.942350	+	0.334629i	$0.108611\pi$
$673$			450.410i			0.669257i	−0.942350	+	0.334629i	$0.891389\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	743.973i		−	1.09893i	−0.835518	−	0.549463i	$-0.814832\pi$
$677$		−	743.973i		−	1.09893i	0.835518	−	0.549463i	$-0.185168\pi$
$678$		0			0
$679$		−	429.548i		−	0.632619i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	158.596i		−	0.232205i	−0.993237	−	0.116102i	$-0.962960\pi$
$683$		−	158.596i		−	0.232205i	0.993237	−	0.116102i	$-0.0370400\pi$
$684$		0			0
$685$	559.979			0.817487
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1583.26			−2.29792
$690$		0			0
$691$	1120.91			1.62216			0.811080	−	0.584936i	$-0.198880\pi$
$691$	1120.91			1.62216			0.811080	+	0.584936i	$0.198880\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	83.2971			0.119852
$696$		0			0
$697$			531.430i			0.762454i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	272.473			0.388691			0.194346	−	0.980933i	$-0.437742\pi$
$701$	272.473			0.388691			0.194346	+	0.980933i	$0.437742\pi$
$702$		0			0
$703$	519.440	−	421.025i	0.738891	−	0.598897i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1200.03			1.69735
$708$		0			0
$709$	−4.34415			−0.00612716			−0.00306358	−	0.999995i	$-0.500975\pi$
$709$	−4.34415			−0.00612716			−0.00306358	+	0.999995i	$0.500975\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			343.321i			0.481517i
$714$		0			0
$715$			1265.02i			1.76926i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−969.937			−1.34901			−0.674504	−	0.738271i	$-0.735642\pi$
$719$	−969.937			−1.34901			−0.674504	+	0.738271i	$0.735642\pi$
$720$		0			0
$721$			751.497i			1.04230i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			41.2607i			0.0569113i
$726$		0			0
$727$	−213.362			−0.293482			−0.146741	−	0.989175i	$-0.546878\pi$
$727$	−213.362			−0.293482			−0.146741	+	0.989175i	$0.546878\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−659.825			−0.902633
$732$		0			0
$733$	1281.86			1.74879			0.874393	−	0.485219i	$-0.161260\pi$
$733$	1281.86			1.74879			0.874393	+	0.485219i	$0.161260\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			1016.11i			1.37870i
$738$		0			0
$739$	−1339.79			−1.81298			−0.906491	−	0.422226i	$-0.861249\pi$
$739$	−1339.79			−1.81298			−0.906491	+	0.422226i	$0.861249\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			650.113i			0.874984i	0.899222	+	0.437492i	$0.144133\pi$
$743$			650.113i			0.874984i	−0.899222	+	0.437492i	$0.855867\pi$
$744$		0			0
$745$	−1272.10			−1.70752
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	233.688i		−	0.312000i
$750$		0			0
$751$			1408.26i			1.87518i	0.347745	+	0.937589i	$0.386948\pi$
$751$			1408.26i			1.87518i	−0.347745	+	0.937589i	$0.613052\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			778.844i			1.03158i
$756$		0			0
$757$	881.249			1.16413			0.582067	−	0.813141i	$-0.302244\pi$
$757$	881.249			1.16413			0.582067	+	0.813141i	$0.302244\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−218.184			−0.286707			−0.143353	−	0.989672i	$-0.545789\pi$
$761$	−218.184			−0.286707			−0.143353	+	0.989672i	$0.545789\pi$
$762$		0			0
$763$		−	1693.09i		−	2.21899i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−222.463			−0.290043
$768$		0			0
$769$	−624.448			−0.812026			−0.406013	−	0.913867i	$-0.633081\pi$
$769$	−624.448			−0.812026			−0.406013	+	0.913867i	$0.633081\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	425.151i		−	0.550002i	−0.961444	−	0.275001i	$-0.911322\pi$
$773$		−	425.151i		−	0.550002i	0.961444	−	0.275001i	$-0.0886781\pi$
$774$		0			0
$775$		−	45.1402i		−	0.0582454i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	891.844	−	722.871i	1.14486	−	0.927947i
$780$		0			0
$781$		−	859.494i		−	1.10050i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	407.749			0.519426
$786$		0			0
$787$			757.949i			0.963086i	0.876422	+	0.481543i	$0.159924\pi$
$787$			757.949i			0.963086i	−0.876422	+	0.481543i	$0.840076\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			1490.65i			1.88451i
$792$		0			0
$793$			23.2118i			0.0292709i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	1466.54i		−	1.84008i	−0.391828	−	0.920038i	$-0.628157\pi$
$797$		−	1466.54i		−	1.84008i	0.391828	−	0.920038i	$-0.371843\pi$
$798$		0			0
$799$	−463.325			−0.579881
$800$		0			0
$801$		0			0
$802$		0			0
$803$	607.308			0.756299
$804$		0			0
$805$	−407.767			−0.506543
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1576.10			−1.94821			−0.974104	−	0.226098i	$-0.927403\pi$
$809$	−1576.10			−1.94821			−0.974104	+	0.226098i	$0.927403\pi$
$810$		0			0
$811$		−	163.777i		−	0.201945i	−0.994889	−	0.100973i	$-0.967805\pi$
$811$		−	163.777i		−	0.201945i	0.994889	−	0.100973i	$-0.0321954\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1067.14			1.30937
$816$		0			0
$817$	897.518	+	1107.32i	1.09855	+	1.35534i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−603.584			−0.735181			−0.367591	−	0.929988i	$-0.619817\pi$
$821$	−603.584			−0.735181			−0.367591	+	0.929988i	$0.619817\pi$
$822$		0			0
$823$	−790.775			−0.960844			−0.480422	−	0.877037i	$-0.659516\pi$
$823$	−790.775			−0.960844			−0.480422	+	0.877037i	$0.659516\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			290.501i			0.351271i	0.984455	+	0.175635i	$0.0561980\pi$
$827$			290.501i			0.351271i	−0.984455	+	0.175635i	$0.943802\pi$
$828$		0			0
$829$		−	915.557i		−	1.10441i	−0.833708	−	0.552206i	$-0.813786\pi$
$829$		−	915.557i		−	1.10441i	0.833708	−	0.552206i	$-0.186214\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−772.372			−0.927217
$834$		0			0
$835$		−	132.993i		−	0.159273i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			818.866i			0.976002i	0.872843	+	0.488001i	$0.162274\pi$
$839$			818.866i			0.976002i	−0.872843	+	0.488001i	$0.837726\pi$
$840$		0			0
$841$	−1257.80			−1.49560
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−607.841			−0.719339
$846$		0			0
$847$	1090.24			1.28717
$848$		0			0
$849$		0			0
$850$		0			0
$851$			241.061i			0.283268i
$852$		0			0
$853$	−1063.70			−1.24701			−0.623505	−	0.781819i	$-0.714292\pi$
$853$	−1063.70			−1.24701			−0.623505	+	0.781819i	$0.714292\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	1028.97i		−	1.20067i	−0.799749	−	0.600335i	$-0.795034\pi$
$857$		−	1028.97i		−	1.20067i	0.799749	−	0.600335i	$-0.204966\pi$
$858$		0			0
$859$	−481.326			−0.560333			−0.280166	−	0.959951i	$-0.590390\pi$
$859$	−481.326			−0.560333			−0.280166	+	0.959951i	$0.590390\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			693.908i			0.804065i	0.915625	+	0.402033i	$0.131696\pi$
$863$			693.908i			0.804065i	−0.915625	+	0.402033i	$0.868304\pi$
$864$		0			0
$865$			757.371i			0.875574i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	1322.52i		−	1.52189i
$870$		0			0
$871$	−1179.09			−1.35372
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1434.59			−1.63954
$876$		0			0
$877$			942.448i			1.07463i	0.843383	+	0.537313i	$0.180561\pi$
$877$			942.448i			1.07463i	−0.843383	+	0.537313i	$0.819439\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−84.6900			−0.0961294			−0.0480647	−	0.998844i	$-0.515305\pi$
$881$	−84.6900			−0.0961294			−0.0480647	+	0.998844i	$0.515305\pi$
$882$		0			0
$883$	−784.001			−0.887883			−0.443941	−	0.896056i	$-0.646420\pi$
$883$	−784.001			−0.887883			−0.443941	+	0.896056i	$0.646420\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			239.164i			0.269632i	0.990871	+	0.134816i	$0.0430444\pi$
$887$			239.164i			0.269632i	−0.990871	+	0.134816i	$0.956956\pi$
$888$		0			0
$889$		−	399.614i		−	0.449509i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	630.232	+	777.550i	0.705747	+	0.870717i
$894$		0			0
$895$			1761.00i			1.96759i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	2296.14			2.55410
$900$		0			0
$901$			819.941i			0.910034i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	1441.74i		−	1.59308i
$906$		0			0
$907$		−	699.827i		−	0.771584i	−0.922586	−	0.385792i	$-0.873928\pi$
$907$		−	699.827i		−	0.771584i	0.922586	−	0.385792i	$-0.126072\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	539.505i		−	0.592212i	−0.955155	−	0.296106i	$-0.904312\pi$
$911$		−	539.505i		−	0.592212i	0.955155	−	0.296106i	$-0.0956881\pi$
$912$		0			0
$913$	−2280.11			−2.49738
$914$		0			0
$915$		0			0
$916$		0			0
$917$	397.212			0.433164
$918$		0			0
$919$	−337.584			−0.367338			−0.183669	−	0.982988i	$-0.558798\pi$
$919$	−337.584			−0.367338			−0.183669	+	0.982988i	$0.558798\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	997.355			1.08056
$924$		0			0
$925$		−	31.6950i		−	0.0342648i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−998.673			−1.07500			−0.537499	−	0.843264i	$-0.680631\pi$
$929$	−998.673			−1.07500			−0.537499	+	0.843264i	$0.680631\pi$
$930$		0			0
$931$	1050.61	+	1296.19i	1.12847	+	1.39226i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	655.129			0.700673
$936$		0			0
$937$	5.98538			0.00638781			0.00319390	−	0.999995i	$-0.498983\pi$
$937$	5.98538			0.00638781			0.00319390	+	0.999995i	$0.498983\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			1196.81i			1.27185i	0.771752	+	0.635924i	$0.219381\pi$
$941$			1196.81i			1.27185i	−0.771752	+	0.635924i	$0.780619\pi$
$942$		0			0
$943$			413.886i			0.438904i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−343.433			−0.362654			−0.181327	−	0.983423i	$-0.558039\pi$
$947$	−343.433			−0.362654			−0.181327	+	0.983423i	$0.558039\pi$
$948$		0			0
$949$			704.719i			0.742591i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	1342.45i		−	1.40865i	−0.709876	−	0.704327i	$-0.751249\pi$
$953$		−	1342.45i		−	1.40865i	0.709876	−	0.704327i	$-0.248751\pi$
$954$		0			0
$955$	−1606.41			−1.68210
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1287.02			1.34204
$960$		0			0
$961$	−1551.03			−1.61397
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	241.612i		−	0.250375i
$966$		0			0
$967$	−334.092			−0.345494			−0.172747	−	0.984966i	$-0.555264\pi$
$967$	−334.092			−0.345494			−0.172747	+	0.984966i	$0.555264\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			1197.90i			1.23367i	0.787091	+	0.616836i	$0.211586\pi$
$971$			1197.90i			1.23367i	−0.787091	+	0.616836i	$0.788414\pi$
$972$		0			0
$973$	191.444			0.196757
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	798.084i		−	0.816872i	−0.912787	−	0.408436i	$-0.866074\pi$
$977$		−	798.084i		−	0.816872i	0.912787	−	0.408436i	$-0.133926\pi$
$978$		0			0
$979$			1074.61i			1.09766i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	142.172i		−	0.144631i	−0.997382	−	0.0723154i	$-0.976961\pi$
$983$		−	142.172i		−	0.144631i	0.997382	−	0.0723154i	$-0.0230388\pi$
$984$		0			0
$985$	−1006.23			−1.02155
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−513.882			−0.519598
$990$		0			0
$991$			982.532i			0.991455i	0.868478	+	0.495727i	$0.165098\pi$
$991$			982.532i			0.991455i	−0.868478	+	0.495727i	$0.834902\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	545.560			0.548302
$996$		0			0
$997$	808.860			0.811293			0.405647	−	0.914030i	$-0.367046\pi$
$997$	808.860			0.811293			0.405647	+	0.914030i	$0.367046\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.3.o.q.721.15		20
3.2	odd	2	inner	2736.3.o.q.721.5		20
4.3	odd	2		1368.3.o.d.721.15	yes	20
12.11	even	2		1368.3.o.d.721.5	✓	20
19.18	odd	2	inner	2736.3.o.q.721.16		20
57.56	even	2	inner	2736.3.o.q.721.6		20
76.75	even	2		1368.3.o.d.721.16	yes	20
228.227	odd	2		1368.3.o.d.721.6	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1368.3.o.d.721.5	✓	20	12.11	even	2
1368.3.o.d.721.6	yes	20	228.227	odd	2
1368.3.o.d.721.15	yes	20	4.3	odd	2
1368.3.o.d.721.16	yes	20	76.75	even	2
2736.3.o.q.721.5		20	3.2	odd	2	inner
2736.3.o.q.721.6		20	57.56	even	2	inner
2736.3.o.q.721.15		20	1.1	even	1	trivial
2736.3.o.q.721.16		20	19.18	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q+5.08927 q^{5} +11.6968 q^{7} +O(q^{10})\)	\(q+5.08927 q^{5} +11.6968 q^{7} -14.6358 q^{11} -16.9834i q^{13} -8.79536 q^{17} +(11.9638 + 14.7603i) q^{19} -6.84997 q^{23} +0.900640 q^{25} +45.8127i q^{29} -50.1201i q^{31} +59.5283 q^{35} -35.1916i q^{37} -60.4216i q^{41} +75.0196 q^{43} +52.6783 q^{47} +87.8158 q^{49} -93.2242i q^{53} -74.4857 q^{55} -13.0988i q^{59} -1.36673 q^{61} -86.4331i q^{65} -69.4258i q^{67} +58.7253i q^{71} -41.4946 q^{73} -171.193 q^{77} +90.3616i q^{79} +155.789 q^{83} -44.7620 q^{85} -73.4230i q^{89} -198.652i q^{91} +(60.8869 + 75.1194i) q^{95} -36.7235i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 16 q^{7}+O(q^{10}) \) 20 * q + 16 * q^7	\( 20 q + 16 q^{7} + 8 q^{19} + 68 q^{25} + 128 q^{43} + 116 q^{49} - 144 q^{55} - 104 q^{61} - 88 q^{73} - 280 q^{85}+O(q^{100}) \) 20 * q + 16 * q^7 + 8 * q^19 + 68 * q^25 + 128 * q^43 + 116 * q^49 - 144 * q^55 - 104 * q^61 - 88 * q^73 - 280 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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