LMFDB - Embedded newform 2736.3.o.l.721.2

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:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-19})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} - 5x + 25 $ x^4 - x^3 - 4x^2 - 5x + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 57)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			721.2
Root			$-1.63746 - 1.52274i$ of defining polynomial
Character	$\chi$	$=$	2736.721
Dual form			2736.3.o.l.721.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.27492 q^{5} -4.72508 q^{7} +O(q^{10})$	$q-1.27492 q^{5} -4.72508 q^{7} -7.27492 q^{11} +4.30136i q^{13} +20.3746 q^{17} +(7.54983 - 17.4356i) q^{19} +5.45017 q^{23} -23.3746 q^{25} +8.60271i q^{29} +20.0624i q^{31} +6.02409 q^{35} -40.6169i q^{37} -31.2920i q^{41} -65.1238 q^{43} +55.4743 q^{47} -26.6736 q^{49} -78.1149i q^{53} +9.27492 q^{55} +69.5122i q^{59} -6.17525 q^{61} -5.48387i q^{65} +123.724i q^{67} -3.11884i q^{71} +33.8248 q^{73} +34.3746 q^{77} -87.6700i q^{79} +121.698 q^{83} -25.9759 q^{85} +69.9726i q^{89} -20.3243i q^{91} +(-9.62541 + 22.2289i) q^{95} +111.081i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 10 q^{5} - 34 q^{7}+O(q^{10}) $ 4 * q + 10 * q^5 - 34 * q^7	$ 4 q + 10 q^{5} - 34 q^{7} - 14 q^{11} + 6 q^{17} + 52 q^{23} - 18 q^{25} - 142 q^{35} - 34 q^{43} + 86 q^{47} + 150 q^{49} + 22 q^{55} - 70 q^{61} + 90 q^{73} + 62 q^{77} + 64 q^{83} - 270 q^{85} - 114 q^{95}+O(q^{100}) $ 4 * q + 10 * q^5 - 34 * q^7 - 14 * q^11 + 6 * q^17 + 52 * q^23 - 18 * q^25 - 142 * q^35 - 34 * q^43 + 86 * q^47 + 150 * q^49 + 22 * q^55 - 70 * q^61 + 90 * q^73 + 62 * q^77 + 64 * q^83 - 270 * q^85 - 114 * 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
$5$	−1.27492			−0.254983			−0.127492	−	0.991840i	$-0.540693\pi$
$5$	−1.27492			−0.254983			−0.127492	+	0.991840i	$0.540693\pi$
$6$		0			0
$7$	−4.72508			−0.675012			−0.337506	−	0.941323i	$-0.609583\pi$
$7$	−4.72508			−0.675012			−0.337506	+	0.941323i	$0.609583\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−7.27492			−0.661356			−0.330678	−	0.943744i	$-0.607277\pi$
$11$	−7.27492			−0.661356			−0.330678	+	0.943744i	$0.607277\pi$
$12$		0			0
$13$			4.30136i			0.330873i	0.986220	+	0.165437i	$0.0529034\pi$
$13$			4.30136i			0.330873i	−0.986220	+	0.165437i	$0.947097\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	20.3746			1.19851			0.599253	−	0.800560i	$-0.295465\pi$
$17$	20.3746			1.19851			0.599253	+	0.800560i	$0.295465\pi$
$18$		0			0
$19$	7.54983	−	17.4356i	0.397360	−	0.917663i
$19$	7.54983	−	17.4356i	0.397360	−	0.917663i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	5.45017			0.236964			0.118482	−	0.992956i	$-0.462197\pi$
$23$	5.45017			0.236964			0.118482	+	0.992956i	$0.462197\pi$
$24$		0			0
$25$	−23.3746			−0.934983
$26$		0			0
$27$		0			0
$28$		0			0
$29$			8.60271i			0.296645i	0.988939	+	0.148323i	$0.0473874\pi$
$29$			8.60271i			0.296645i	−0.988939	+	0.148323i	$0.952613\pi$
$30$		0			0
$31$			20.0624i			0.647176i	0.946198	+	0.323588i	$0.104889\pi$
$31$			20.0624i			0.647176i	−0.946198	+	0.323588i	$0.895111\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	6.02409			0.172117
$36$		0			0
$37$		−	40.6169i		−	1.09775i	−0.835903	−	0.548877i	$-0.815056\pi$
$37$		−	40.6169i		−	1.09775i	0.835903	−	0.548877i	$-0.184944\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	31.2920i		−	0.763220i	−0.924324	−	0.381610i	$-0.875370\pi$
$41$		−	31.2920i		−	0.763220i	0.924324	−	0.381610i	$-0.124630\pi$
$42$		0			0
$43$	−65.1238			−1.51451			−0.757253	−	0.653122i	$-0.773459\pi$
$43$	−65.1238			−1.51451			−0.757253	+	0.653122i	$0.773459\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	55.4743			1.18030			0.590152	−	0.807292i	$-0.299068\pi$
$47$	55.4743			1.18030			0.590152	+	0.807292i	$0.299068\pi$
$48$		0			0
$49$	−26.6736			−0.544359
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	78.1149i		−	1.47387i	−0.675966	−	0.736933i	$-0.736273\pi$
$53$		−	78.1149i		−	1.47387i	0.675966	−	0.736933i	$-0.263727\pi$
$54$		0			0
$55$	9.27492			0.168635
$56$		0			0
$57$		0			0
$58$		0			0
$59$			69.5122i			1.17817i	0.808070	+	0.589087i	$0.200512\pi$
$59$			69.5122i			1.17817i	−0.808070	+	0.589087i	$0.799488\pi$
$60$		0			0
$61$	−6.17525			−0.101234			−0.0506168	−	0.998718i	$-0.516119\pi$
$61$	−6.17525			−0.101234			−0.0506168	+	0.998718i	$0.516119\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	5.48387i		−	0.0843673i
$66$		0			0
$67$			123.724i			1.84662i	0.384053	+	0.923311i	$0.374528\pi$
$67$			123.724i			1.84662i	−0.384053	+	0.923311i	$0.625472\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	3.11884i		−	0.0439273i	−0.999759	−	0.0219637i	$-0.993008\pi$
$71$		−	3.11884i		−	0.0439273i	0.999759	−	0.0219637i	$-0.00699181\pi$
$72$		0			0
$73$	33.8248			0.463353			0.231676	−	0.972793i	$-0.425579\pi$
$73$	33.8248			0.463353			0.231676	+	0.972793i	$0.425579\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	34.3746			0.446423
$78$		0			0
$79$		−	87.6700i		−	1.10975i	−0.831935	−	0.554873i	$-0.812767\pi$
$79$		−	87.6700i		−	1.10975i	0.831935	−	0.554873i	$-0.187233\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	121.698			1.46624			0.733119	−	0.680101i	$-0.238064\pi$
$83$	121.698			1.46624			0.733119	+	0.680101i	$0.238064\pi$
$84$		0			0
$85$	−25.9759			−0.305599
$86$		0			0
$87$		0			0
$88$		0			0
$89$			69.9726i			0.786209i	0.919494	+	0.393104i	$0.128599\pi$
$89$			69.9726i			0.786209i	−0.919494	+	0.393104i	$0.871401\pi$
$90$		0			0
$91$		−	20.3243i		−	0.223344i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−9.62541	+	22.2289i	−0.101320	+	0.233989i
$96$		0			0
$97$			111.081i			1.14517i	0.819846	+	0.572585i	$0.194059\pi$
$97$			111.081i			1.14517i	−0.819846	+	0.572585i	$0.805941\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−75.2508			−0.745058			−0.372529	−	0.928021i	$-0.621509\pi$
$101$	−75.2508			−0.745058			−0.372529	+	0.928021i	$0.621509\pi$
$102$		0			0
$103$			188.474i			1.82985i	0.403628	+	0.914923i	$0.367749\pi$
$103$			188.474i			1.82985i	−0.403628	+	0.914923i	$0.632251\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	8.14236i		−	0.0760968i	−0.999276	−	0.0380484i	$-0.987886\pi$
$107$		−	8.14236i		−	0.0760968i	0.999276	−	0.0380484i	$-0.0121141\pi$
$108$		0			0
$109$			119.192i			1.09351i	0.837294	+	0.546753i	$0.184136\pi$
$109$			119.192i			1.09351i	−0.837294	+	0.546753i	$0.815864\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			196.355i			1.73765i	0.495117	+	0.868826i	$0.335125\pi$
$113$			196.355i			1.73765i	−0.495117	+	0.868826i	$0.664875\pi$
$114$		0			0
$115$	−6.94851			−0.0604218
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−96.2716			−0.809005
$120$		0			0
$121$	−68.0756			−0.562608
$122$		0			0
$123$		0			0
$124$		0			0
$125$	61.6736			0.493389
$126$		0			0
$127$			16.0229i			0.126165i	0.998008	+	0.0630823i	$0.0200931\pi$
$127$			16.0229i			0.126165i	−0.998008	+	0.0630823i	$0.979907\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	165.921			1.26657			0.633287	−	0.773917i	$-0.281705\pi$
$131$	165.921			1.26657			0.633287	+	0.773917i	$0.281705\pi$
$132$		0			0
$133$	−35.6736	+	82.3846i	−0.268223	+	0.619433i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	199.674			1.45747			0.728736	−	0.684795i	$-0.240108\pi$
$137$	199.674			1.45747			0.728736	+	0.684795i	$0.240108\pi$
$138$		0			0
$139$	54.5739			0.392618			0.196309	−	0.980542i	$-0.437104\pi$
$139$	54.5739			0.392618			0.196309	+	0.980542i	$0.437104\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	31.2920i		−	0.218825i
$144$		0			0
$145$		−	10.9677i		−	0.0756396i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	245.323			1.64646			0.823232	−	0.567705i	$-0.192169\pi$
$149$	245.323			1.64646			0.823232	+	0.567705i	$0.192169\pi$
$150$		0			0
$151$		−	79.2974i		−	0.525149i	−0.964912	−	0.262574i	$-0.915429\pi$
$151$		−	79.2974i		−	0.525149i	0.964912	−	0.262574i	$-0.0845715\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	25.5780i		−	0.165019i
$156$		0			0
$157$	−53.6977			−0.342023			−0.171012	−	0.985269i	$-0.554704\pi$
$157$	−53.6977			−0.342023			−0.171012	+	0.985269i	$0.554704\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−25.7525			−0.159953
$162$		0			0
$163$	−78.3987			−0.480973			−0.240487	−	0.970652i	$-0.577307\pi$
$163$	−78.3987			−0.480973			−0.240487	+	0.970652i	$0.577307\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			293.350i			1.75658i	0.478124	+	0.878292i	$0.341317\pi$
$167$			293.350i			1.75658i	−0.478124	+	0.878292i	$0.658683\pi$
$168$		0			0
$169$	150.498			0.890523
$170$		0			0
$171$		0			0
$172$		0			0
$173$			10.5074i			0.0607364i	0.999539	+	0.0303682i	$0.00966798\pi$
$173$			10.5074i			0.0607364i	−0.999539	+	0.0303682i	$0.990332\pi$
$174$		0			0
$175$	110.447			0.631125
$176$		0			0
$177$		0			0
$178$		0			0
$179$			12.4121i			0.0693412i	0.999399	+	0.0346706i	$0.0110382\pi$
$179$			12.4121i			0.0693412i	−0.999399	+	0.0346706i	$0.988962\pi$
$180$		0			0
$181$		−	106.780i		−	0.589945i	−0.955506	−	0.294973i	$-0.904689\pi$
$181$		−	106.780i		−	0.589945i	0.955506	−	0.294973i	$-0.0953105\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			51.7832i			0.279909i
$186$		0			0
$187$	−148.223			−0.792639
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−19.0274			−0.0996199			−0.0498099	−	0.998759i	$-0.515862\pi$
$191$	−19.0274			−0.0996199			−0.0498099	+	0.998759i	$0.515862\pi$
$192$		0			0
$193$		−	110.391i		−	0.571974i	−0.958234	−	0.285987i	$-0.907679\pi$
$193$		−	110.391i		−	0.571974i	0.958234	−	0.285987i	$-0.0923213\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−328.749			−1.66878			−0.834389	−	0.551176i	$-0.814179\pi$
$197$	−328.749			−1.66878			−0.834389	+	0.551176i	$0.814179\pi$
$198$		0			0
$199$	266.973			1.34157			0.670785	−	0.741651i	$-0.265957\pi$
$199$	266.973			1.34157			0.670785	+	0.741651i	$0.265957\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	40.6485i		−	0.200239i
$204$		0			0
$205$			39.8947i			0.194608i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−54.9244	+	126.843i	−0.262796	+	0.606902i
$210$		0			0
$211$			70.4329i			0.333805i	0.985973	+	0.166903i	$0.0533766\pi$
$211$			70.4329i			0.333805i	−0.985973	+	0.166903i	$0.946623\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	83.0274			0.386174
$216$		0			0
$217$		−	94.7967i		−	0.436851i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			87.6383i			0.396554i
$222$		0			0
$223$		−	237.432i		−	1.06472i	−0.846519	−	0.532359i	$-0.821306\pi$
$223$		−	237.432i		−	1.06472i	0.846519	−	0.532359i	$-0.178694\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			269.740i			1.18828i	0.804362	+	0.594140i	$0.202508\pi$
$227$			269.740i			1.18828i	−0.804362	+	0.594140i	$0.797492\pi$
$228$		0			0
$229$	259.419			1.13284			0.566418	−	0.824118i	$-0.308329\pi$
$229$	259.419			1.13284			0.566418	+	0.824118i	$0.308329\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	52.6769			0.226081			0.113041	−	0.993590i	$-0.463941\pi$
$233$	52.6769			0.226081			0.113041	+	0.993590i	$0.463941\pi$
$234$		0			0
$235$	−70.7251			−0.300958
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−346.120			−1.44820			−0.724101	−	0.689694i	$-0.757745\pi$
$239$	−346.120			−1.44820			−0.724101	+	0.689694i	$0.757745\pi$
$240$		0			0
$241$			300.278i			1.24597i	0.782235	+	0.622983i	$0.214079\pi$
$241$			300.278i			1.24597i	−0.782235	+	0.622983i	$0.785921\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	34.0066			0.138803
$246$		0			0
$247$	74.9967	+	32.4745i	0.303630	+	0.131476i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	190.223			0.757862			0.378931	−	0.925425i	$-0.376292\pi$
$251$	190.223			0.757862			0.378931	+	0.925425i	$0.376292\pi$
$252$		0			0
$253$	−39.6495			−0.156717
$254$		0			0
$255$		0			0
$256$		0			0
$257$			25.3478i			0.0986295i	0.998783	+	0.0493147i	$0.0157037\pi$
$257$			25.3478i			0.0986295i	−0.998783	+	0.0493147i	$0.984296\pi$
$258$		0			0
$259$			191.918i			0.740997i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−450.120			−1.71148			−0.855742	−	0.517402i	$-0.826899\pi$
$263$	−450.120			−1.71148			−0.855742	+	0.517402i	$0.826899\pi$
$264$		0			0
$265$			99.5901i			0.375812i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			485.141i			1.80350i	0.432259	+	0.901749i	$0.357716\pi$
$269$			485.141i			1.80350i	−0.432259	+	0.901749i	$0.642284\pi$
$270$		0			0
$271$	−156.997			−0.579324			−0.289662	−	0.957129i	$-0.593543\pi$
$271$	−156.997			−0.579324			−0.289662	+	0.957129i	$0.593543\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	170.048			0.618357
$276$		0			0
$277$	156.629			0.565447			0.282723	−	0.959202i	$-0.408762\pi$
$277$	156.629			0.565447			0.282723	+	0.959202i	$0.408762\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			457.658i			1.62868i	0.580390	+	0.814339i	$0.302900\pi$
$281$			457.658i			1.62868i	−0.580390	+	0.814339i	$0.697100\pi$
$282$		0			0
$283$	46.5739			0.164572			0.0822861	−	0.996609i	$-0.473778\pi$
$283$	46.5739			0.164572			0.0822861	+	0.996609i	$0.473778\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			147.857i			0.515182i
$288$		0			0
$289$	126.124			0.436414
$290$		0			0
$291$		0			0
$292$		0			0
$293$			222.163i			0.758235i	0.925349	+	0.379117i	$0.123772\pi$
$293$			222.163i			0.758235i	−0.925349	+	0.379117i	$0.876228\pi$
$294$		0			0
$295$		−	88.6223i		−	0.300415i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			23.4431i			0.0784050i
$300$		0			0
$301$	307.715			1.02231
$302$		0			0
$303$		0			0
$304$		0			0
$305$	7.87293			0.0258129
$306$		0			0
$307$			492.069i			1.60283i	0.598108	+	0.801416i	$0.295920\pi$
$307$			492.069i			1.60283i	−0.598108	+	0.801416i	$0.704080\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	162.773			0.523387			0.261693	−	0.965151i	$-0.415719\pi$
$311$	162.773			0.523387			0.261693	+	0.965151i	$0.415719\pi$
$312$		0			0
$313$	414.894			1.32554			0.662770	−	0.748823i	$-0.269381\pi$
$313$	414.894			1.32554			0.662770	+	0.748823i	$0.269381\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	365.750i		−	1.15379i	−0.816819	−	0.576893i	$-0.804265\pi$
$317$		−	365.750i		−	1.15379i	0.816819	−	0.576893i	$-0.195735\pi$
$318$		0			0
$319$		−	62.5840i		−	0.196188i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	153.825	−	355.243i	0.476238	−	1.09982i
$324$		0			0
$325$		−	100.542i		−	0.309361i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−262.120			−0.796719
$330$		0			0
$331$		−	220.258i		−	0.665433i	−0.943027	−	0.332716i	$-0.892035\pi$
$331$		−	220.258i		−	0.665433i	0.943027	−	0.332716i	$-0.107965\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	157.737i		−	0.470858i
$336$		0			0
$337$			240.980i			0.715073i	0.933899	+	0.357536i	$0.116383\pi$
$337$			240.980i			0.715073i	−0.933899	+	0.357536i	$0.883617\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	145.953i		−	0.428014i
$342$		0			0
$343$	357.564			1.04246
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−234.876			−0.676877			−0.338438	−	0.940989i	$-0.609899\pi$
$347$	−234.876			−0.676877			−0.338438	+	0.940989i	$0.609899\pi$
$348$		0			0
$349$	−440.677			−1.26268			−0.631342	−	0.775504i	$-0.717496\pi$
$349$	−440.677			−1.26268			−0.631342	+	0.775504i	$0.717496\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	543.292			1.53907			0.769536	−	0.638603i	$-0.220488\pi$
$353$	543.292			1.53907			0.769536	+	0.638603i	$0.220488\pi$
$354$		0			0
$355$			3.97626i			0.0112007i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−38.5191			−0.107296			−0.0536478	−	0.998560i	$-0.517085\pi$
$359$	−38.5191			−0.107296			−0.0536478	+	0.998560i	$0.517085\pi$
$360$		0			0
$361$	−247.000	−	263.272i	−0.684211	−	0.729285i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−43.1238			−0.118147
$366$		0			0
$367$	568.997			1.55040			0.775200	−	0.631716i	$-0.217649\pi$
$367$	568.997			1.55040			0.775200	+	0.631716i	$0.217649\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			369.099i			0.994877i
$372$		0			0
$373$			61.8618i			0.165849i	0.996556	+	0.0829247i	$0.0264261\pi$
$373$			61.8618i			0.165849i	−0.996556	+	0.0829247i	$0.973574\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−37.0033			−0.0981520
$378$		0			0
$379$		−	342.601i		−	0.903960i	−0.892028	−	0.451980i	$-0.850718\pi$
$379$		−	342.601i		−	0.903960i	0.892028	−	0.451980i	$-0.149282\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			77.1309i			0.201386i	0.994918	+	0.100693i	$0.0321060\pi$
$383$			77.1309i			0.201386i	−0.994918	+	0.100693i	$0.967894\pi$
$384$		0			0
$385$	−43.8248			−0.113831
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−433.770			−1.11509			−0.557545	−	0.830147i	$-0.688257\pi$
$389$	−433.770			−1.11509			−0.557545	+	0.830147i	$0.688257\pi$
$390$		0			0
$391$	111.045			0.284002
$392$		0			0
$393$		0			0
$394$		0			0
$395$			111.772i			0.282967i
$396$		0			0
$397$	−283.069			−0.713020			−0.356510	−	0.934291i	$-0.616033\pi$
$397$	−283.069			−0.713020			−0.356510	+	0.934291i	$0.616033\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	323.491i		−	0.806710i	−0.915044	−	0.403355i	$-0.867844\pi$
$401$		−	323.491i		−	0.806710i	0.915044	−	0.403355i	$-0.132156\pi$
$402$		0			0
$403$	−86.2957			−0.214133
$404$		0			0
$405$		0			0
$406$		0			0
$407$			295.484i			0.726006i
$408$		0			0
$409$			546.741i			1.33678i	0.743813	+	0.668388i	$0.233015\pi$
$409$			546.741i			1.33678i	−0.743813	+	0.668388i	$0.766985\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	328.451i		−	0.795281i
$414$		0			0
$415$	−155.154			−0.373866
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−352.997			−0.842474			−0.421237	−	0.906951i	$-0.638404\pi$
$419$	−352.997			−0.842474			−0.421237	+	0.906951i	$0.638404\pi$
$420$		0			0
$421$		−	578.525i		−	1.37417i	−0.726578	−	0.687084i	$-0.758890\pi$
$421$		−	578.525i		−	1.37417i	0.726578	−	0.687084i	$-0.241110\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−476.248			−1.12058
$426$		0			0
$427$	29.1786			0.0683339
$428$		0			0
$429$		0			0
$430$		0			0
$431$			628.896i			1.45915i	0.683898	+	0.729577i	$0.260283\pi$
$431$			628.896i			1.45915i	−0.683898	+	0.729577i	$0.739717\pi$
$432$		0			0
$433$			122.803i			0.283610i	0.989895	+	0.141805i	$0.0452905\pi$
$433$			122.803i			0.283610i	−0.989895	+	0.141805i	$0.954709\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	41.1478	−	95.0269i	0.0941598	−	0.217453i
$438$		0			0
$439$		−	63.3694i		−	0.144350i	−0.997392	−	0.0721748i	$-0.977006\pi$
$439$		−	63.3694i		−	0.144350i	0.997392	−	0.0721748i	$-0.0229939\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	610.320			1.37770			0.688849	−	0.724905i	$-0.258117\pi$
$443$	610.320			1.37770			0.688849	+	0.724905i	$0.258117\pi$
$444$		0			0
$445$		−	89.2092i		−	0.200470i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			410.772i			0.914860i	0.889246	+	0.457430i	$0.151230\pi$
$449$			410.772i			0.914860i	−0.889246	+	0.457430i	$0.848770\pi$
$450$		0			0
$451$			227.647i			0.504760i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			25.9117i			0.0569489i
$456$		0			0
$457$	251.722			0.550814			0.275407	−	0.961328i	$-0.411187\pi$
$457$	251.722			0.550814			0.275407	+	0.961328i	$0.411187\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	224.622			0.487250			0.243625	−	0.969870i	$-0.421663\pi$
$461$	224.622			0.487250			0.243625	+	0.969870i	$0.421663\pi$
$462$		0			0
$463$	196.375			0.424135			0.212068	−	0.977255i	$-0.431980\pi$
$463$	196.375			0.424135			0.212068	+	0.977255i	$0.431980\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−106.381			−0.227797			−0.113899	−	0.993492i	$-0.536334\pi$
$467$	−106.381			−0.227797			−0.113899	+	0.993492i	$0.536334\pi$
$468$		0			0
$469$		−	584.605i		−	1.24649i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	473.770			1.00163
$474$		0			0
$475$	−176.474	+	407.550i	−0.371525	+	0.858000i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	463.154			0.966920			0.483460	−	0.875367i	$-0.339380\pi$
$479$	463.154			0.966920			0.483460	+	0.875367i	$0.339380\pi$
$480$		0			0
$481$	174.708			0.363217
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	141.620i		−	0.291999i
$486$		0			0
$487$			248.463i			0.510191i	0.966916	+	0.255095i	$0.0821069\pi$
$487$			248.463i			0.510191i	−0.966916	+	0.255095i	$0.917893\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	305.588			0.622379			0.311189	−	0.950348i	$-0.399273\pi$
$491$	305.588			0.622379			0.311189	+	0.950348i	$0.399273\pi$
$492$		0			0
$493$			175.277i			0.355531i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			14.7368i			0.0296514i
$498$		0			0
$499$	−268.918			−0.538913			−0.269457	−	0.963013i	$-0.586844\pi$
$499$	−268.918			−0.538913			−0.269457	+	0.963013i	$0.586844\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−507.154			−1.00826			−0.504130	−	0.863628i	$-0.668187\pi$
$503$	−507.154			−1.00826			−0.504130	+	0.863628i	$0.668187\pi$
$504$		0			0
$505$	95.9386			0.189977
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	407.256i		−	0.800111i	−0.916491	−	0.400055i	$-0.868991\pi$
$509$		−	407.256i		−	0.800111i	0.916491	−	0.400055i	$-0.131009\pi$
$510$		0			0
$511$	−159.825			−0.312769
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	240.289i		−	0.466581i
$516$		0			0
$517$	−403.571			−0.780601
$518$		0			0
$519$		0			0
$520$		0			0
$521$			607.190i			1.16543i	0.812676	+	0.582716i	$0.198010\pi$
$521$			607.190i			1.16543i	−0.812676	+	0.582716i	$0.801990\pi$
$522$		0			0
$523$			209.227i			0.400052i	0.979791	+	0.200026i	$0.0641026\pi$
$523$			209.227i			0.400052i	−0.979791	+	0.200026i	$0.935897\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			408.764i			0.775643i
$528$		0			0
$529$	−499.296			−0.943848
$530$		0			0
$531$		0			0
$532$		0			0
$533$	134.598			0.252529
$534$		0			0
$535$			10.3808i			0.0194034i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	194.048			0.360015
$540$		0			0
$541$	−44.6637			−0.0825576			−0.0412788	−	0.999148i	$-0.513143\pi$
$541$	−44.6637			−0.0825576			−0.0412788	+	0.999148i	$0.513143\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	151.960i		−	0.278826i
$546$		0			0
$547$		−	595.405i		−	1.08849i	−0.838925	−	0.544246i	$-0.816816\pi$
$547$		−	595.405i		−	1.08849i	0.838925	−	0.544246i	$-0.183184\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	149.993	+	64.9490i	0.272220	+	0.117875i
$552$		0			0
$553$			414.248i			0.749092i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	78.4228			0.140795			0.0703975	−	0.997519i	$-0.477573\pi$
$557$	78.4228			0.140795			0.0703975	+	0.997519i	$0.477573\pi$
$558$		0			0
$559$		−	280.120i		−	0.501110i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	457.825i		−	0.813189i	−0.913609	−	0.406594i	$-0.866716\pi$
$563$		−	457.825i		−	0.813189i	0.913609	−	0.406594i	$-0.133284\pi$
$564$		0			0
$565$		−	250.336i		−	0.443073i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			307.833i			0.541007i	0.962719	+	0.270504i	$0.0871902\pi$
$569$			307.833i			0.541007i	−0.962719	+	0.270504i	$0.912810\pi$
$570$		0			0
$571$	−589.492			−1.03238			−0.516192	−	0.856473i	$-0.672651\pi$
$571$	−589.492			−1.03238			−0.516192	+	0.856473i	$0.672651\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−127.395			−0.221557
$576$		0			0
$577$	−363.262			−0.629570			−0.314785	−	0.949163i	$-0.601932\pi$
$577$	−363.262			−0.629570			−0.314785	+	0.949163i	$0.601932\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−575.032			−0.989727
$582$		0			0
$583$			568.280i			0.974751i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	391.625			0.667164			0.333582	−	0.942721i	$-0.391743\pi$
$587$	391.625			0.667164			0.333582	+	0.942721i	$0.391743\pi$
$588$		0			0
$589$	349.801	+	151.468i	0.593889	+	0.257162i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	212.900			0.359022			0.179511	−	0.983756i	$-0.442548\pi$
$593$	212.900			0.359022			0.179511	+	0.983756i	$0.442548\pi$
$594$		0			0
$595$	122.738			0.206283
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	916.888i		−	1.53070i	−0.643616	−	0.765349i	$-0.722566\pi$
$599$		−	916.888i		−	1.53070i	0.643616	−	0.765349i	$-0.277434\pi$
$600$		0			0
$601$			485.078i			0.807118i	0.914954	+	0.403559i	$0.132227\pi$
$601$			485.078i			0.807118i	−0.914954	+	0.403559i	$0.867773\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	86.7907			0.143456
$606$		0			0
$607$			193.101i			0.318123i	0.987269	+	0.159061i	$0.0508468\pi$
$607$			193.101i			0.318123i	−0.987269	+	0.159061i	$0.949153\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			238.614i			0.390531i
$612$		0			0
$613$	−142.368			−0.232248			−0.116124	−	0.993235i	$-0.537047\pi$
$613$	−142.368			−0.232248			−0.116124	+	0.993235i	$0.537047\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	547.770			0.887796			0.443898	−	0.896077i	$-0.353595\pi$
$617$	547.770			0.887796			0.443898	+	0.896077i	$0.353595\pi$
$618$		0			0
$619$	−814.482			−1.31580			−0.657901	−	0.753104i	$-0.728556\pi$
$619$	−814.482			−1.31580			−0.657901	+	0.753104i	$0.728556\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	330.626i		−	0.530700i
$624$		0			0
$625$	505.736			0.809177
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	827.552i		−	1.31566i
$630$		0			0
$631$	−641.509			−1.01665			−0.508327	−	0.861164i	$-0.669736\pi$
$631$	−641.509			−1.01665			−0.508327	+	0.861164i	$0.669736\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	20.4279i		−	0.0321699i
$636$		0			0
$637$		−	114.733i		−	0.180114i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			352.878i			0.550512i	0.961371	+	0.275256i	$0.0887626\pi$
$641$			352.878i			0.550512i	−0.961371	+	0.275256i	$0.911237\pi$
$642$		0			0
$643$	455.757			0.708797			0.354399	−	0.935094i	$-0.384686\pi$
$643$	455.757			0.708797			0.354399	+	0.935094i	$0.384686\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	484.368			0.748637			0.374318	−	0.927300i	$-0.377877\pi$
$647$	484.368			0.748637			0.374318	+	0.927300i	$0.377877\pi$
$648$		0			0
$649$		−	505.696i		−	0.779192i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−280.780			−0.429985			−0.214992	−	0.976616i	$-0.568973\pi$
$653$	−280.780			−0.429985			−0.214992	+	0.976616i	$0.568973\pi$
$654$		0			0
$655$	−211.536			−0.322955
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	800.259i		−	1.21435i	−0.794567	−	0.607177i	$-0.792302\pi$
$659$		−	800.259i		−	1.21435i	0.794567	−	0.607177i	$-0.207698\pi$
$660$		0			0
$661$			812.996i			1.22995i	0.788547	+	0.614975i	$0.210834\pi$
$661$			812.996i			1.22995i	−0.788547	+	0.614975i	$0.789166\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	45.4809	−	105.034i	0.0683923	−	0.157945i
$666$		0			0
$667$			46.8862i			0.0702941i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	44.9244			0.0669514
$672$		0			0
$673$			79.8930i			0.118712i	0.998237	+	0.0593559i	$0.0189047\pi$
$673$			79.8930i			0.118712i	−0.998237	+	0.0593559i	$0.981095\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	1153.64i		−	1.70404i	−0.523506	−	0.852022i	$-0.675376\pi$
$677$		−	1153.64i		−	1.70404i	0.523506	−	0.852022i	$-0.324624\pi$
$678$		0			0
$679$		−	524.869i		−	0.773003i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	382.766i		−	0.560419i	−0.959939	−	0.280209i	$-0.909596\pi$
$683$		−	382.766i		−	0.560419i	0.959939	−	0.280209i	$-0.0904039\pi$
$684$		0			0
$685$	−254.567			−0.371631
$686$		0			0
$687$		0			0
$688$		0			0
$689$	336.000			0.487663
$690$		0			0
$691$	−133.330			−0.192952			−0.0964759	−	0.995335i	$-0.530757\pi$
$691$	−133.330			−0.192952			−0.0964759	+	0.995335i	$0.530757\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−69.5772			−0.100111
$696$		0			0
$697$		−	637.562i		−	0.914723i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	934.042			1.33244			0.666221	−	0.745755i	$-0.267911\pi$
$701$	934.042			1.33244			0.666221	+	0.745755i	$0.267911\pi$
$702$		0			0
$703$	−708.179	−	306.651i	−1.00737	−	0.436203i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	355.566			0.502923
$708$		0			0
$709$	287.993			0.406197			0.203098	−	0.979158i	$-0.434899\pi$
$709$	287.993			0.406197			0.203098	+	0.979158i	$0.434899\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			109.344i			0.153357i
$714$		0			0
$715$			39.8947i			0.0557968i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	78.0723			0.108585			0.0542923	−	0.998525i	$-0.482710\pi$
$719$	78.0723			0.108585			0.0542923	+	0.998525i	$0.482710\pi$
$720$		0			0
$721$		−	890.556i		−	1.23517i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	201.085i		−	0.277358i
$726$		0			0
$727$	320.169			0.440397			0.220199	−	0.975455i	$-0.429329\pi$
$727$	320.169			0.440397			0.220199	+	0.975455i	$0.429329\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1326.87			−1.81514
$732$		0			0
$733$	−565.781			−0.771870			−0.385935	−	0.922526i	$-0.626121\pi$
$733$	−565.781			−0.771870			−0.385935	+	0.922526i	$0.626121\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	900.080i		−	1.22127i
$738$		0			0
$739$	869.557			1.17667			0.588334	−	0.808618i	$-0.299784\pi$
$739$	869.557			1.17667			0.588334	+	0.808618i	$0.299784\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			7.74529i			0.0104243i	0.999986	+	0.00521217i	$0.00165909\pi$
$743$			7.74529i			0.0104243i	−0.999986	+	0.00521217i	$0.998341\pi$
$744$		0			0
$745$	−312.767			−0.419821
$746$		0			0
$747$		0			0
$748$		0			0
$749$			38.4733i			0.0513663i
$750$		0			0
$751$			1281.23i			1.70603i	0.521890	+	0.853013i	$0.325227\pi$
$751$			1281.23i			1.70603i	−0.521890	+	0.853013i	$0.674773\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			101.098i			0.133904i
$756$		0			0
$757$	−441.172			−0.582790			−0.291395	−	0.956603i	$-0.594119\pi$
$757$	−441.172			−0.582790			−0.291395	+	0.956603i	$0.594119\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−476.532			−0.626192			−0.313096	−	0.949721i	$-0.601366\pi$
$761$	−476.532			−0.626192			−0.313096	+	0.949721i	$0.601366\pi$
$762$		0			0
$763$		−	563.193i		−	0.738129i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−298.997			−0.389826
$768$		0			0
$769$	−624.567			−0.812181			−0.406091	−	0.913833i	$-0.633108\pi$
$769$	−624.567			−0.812181			−0.406091	+	0.913833i	$0.633108\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	334.038i		−	0.432133i	−0.976379	−	0.216066i	$-0.930677\pi$
$773$		−	334.038i		−	0.432133i	0.976379	−	0.216066i	$-0.0693227\pi$
$774$		0			0
$775$		−	468.951i		−	0.605098i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−545.595	−	236.249i	−0.700378	−	0.303273i
$780$		0			0
$781$			22.6893i			0.0290516i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	68.4601			0.0872103
$786$		0			0
$787$		−	479.323i		−	0.609051i	−0.952504	−	0.304526i	$-0.901502\pi$
$787$		−	479.323i		−	0.609051i	0.952504	−	0.304526i	$-0.0984980\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	927.792i		−	1.17294i
$792$		0			0
$793$		−	26.5619i		−	0.0334955i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	126.025i		−	0.158125i	−0.996870	−	0.0790624i	$-0.974807\pi$
$797$		−	126.025i		−	0.158125i	0.996870	−	0.0790624i	$-0.0251926\pi$
$798$		0			0
$799$	1130.26			1.41460
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−246.072			−0.306441
$804$		0			0
$805$	32.8323			0.0407854
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−901.825			−1.11474			−0.557370	−	0.830264i	$-0.688190\pi$
$809$	−901.825			−1.11474			−0.557370	+	0.830264i	$0.688190\pi$
$810$		0			0
$811$		−	183.776i		−	0.226604i	−0.993561	−	0.113302i	$-0.963857\pi$
$811$		−	183.776i		−	0.226604i	0.993561	−	0.113302i	$-0.0361427\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	99.9518			0.122640
$816$		0			0
$817$	−491.674	+	1135.47i	−0.601804	+	1.38981i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1034.35			−1.25986			−0.629932	−	0.776650i	$-0.716917\pi$
$821$	−1034.35			−1.25986			−0.629932	+	0.776650i	$0.716917\pi$
$822$		0			0
$823$	−952.107			−1.15687			−0.578437	−	0.815727i	$-0.696337\pi$
$823$	−952.107			−1.15687			−0.578437	+	0.815727i	$0.696337\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	970.806i		−	1.17389i	−0.809627	−	0.586944i	$-0.800331\pi$
$827$		−	970.806i		−	1.17389i	0.809627	−	0.586944i	$-0.199669\pi$
$828$		0			0
$829$			374.908i			0.452242i	0.974099	+	0.226121i	$0.0726044\pi$
$829$			374.908i			0.452242i	−0.974099	+	0.226121i	$0.927396\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−543.463			−0.652417
$834$		0			0
$835$		−	373.996i		−	0.447900i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			1282.87i			1.52904i	0.644597	+	0.764522i	$0.277025\pi$
$839$			1282.87i			1.52904i	−0.644597	+	0.764522i	$0.722975\pi$
$840$		0			0
$841$	766.993			0.912002
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−191.873			−0.227069
$846$		0			0
$847$	321.663			0.379767
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	221.369i		−	0.260128i
$852$		0			0
$853$	827.595			0.970217			0.485108	−	0.874454i	$-0.338780\pi$
$853$	827.595			0.970217			0.485108	+	0.874454i	$0.338780\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			986.671i			1.15131i	0.817694	+	0.575654i	$0.195252\pi$
$857$			986.671i			1.15131i	−0.817694	+	0.575654i	$0.804748\pi$
$858$		0			0
$859$	1382.27			1.60916			0.804582	−	0.593842i	$-0.202389\pi$
$859$	1382.27			1.60916			0.804582	+	0.593842i	$0.202389\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	485.475i		−	0.562543i	−0.959628	−	0.281272i	$-0.909244\pi$
$863$		−	485.475i		−	0.562543i	0.959628	−	0.281272i	$-0.0907562\pi$
$864$		0			0
$865$		−	13.3961i		−	0.0154868i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			637.792i			0.733938i
$870$		0			0
$871$	−532.179			−0.610998
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−291.413			−0.333043
$876$		0			0
$877$		−	266.756i		−	0.304169i	−0.988368	−	0.152084i	$-0.951401\pi$
$877$		−	266.756i		−	0.304169i	0.988368	−	0.152084i	$-0.0485985\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	979.248			1.11152			0.555760	−	0.831343i	$-0.312427\pi$
$881$	979.248			1.11152			0.555760	+	0.831343i	$0.312427\pi$
$882$		0			0
$883$	−1551.91			−1.75755			−0.878774	−	0.477239i	$-0.841638\pi$
$883$	−1551.91			−1.75755			−0.878774	+	0.477239i	$0.841638\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	700.543i		−	0.789789i	−0.918727	−	0.394894i	$-0.870781\pi$
$887$		−	700.543i		−	0.789789i	0.918727	−	0.394894i	$-0.129219\pi$
$888$		0			0
$889$		−	75.7095i		−	0.0851626i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	418.821	−	967.227i	0.469005	−	1.08312i
$894$		0			0
$895$		−	15.8244i		−	0.0176809i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−172.591			−0.191982
$900$		0			0
$901$		−	1591.56i		−	1.76644i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			136.136i			0.150426i
$906$		0			0
$907$		−	1228.07i		−	1.35399i	−0.735987	−	0.676996i	$-0.763282\pi$
$907$		−	1228.07i		−	1.35399i	0.735987	−	0.676996i	$-0.236718\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	1077.26i		−	1.18250i	−0.806487	−	0.591252i	$-0.798634\pi$
$911$		−	1077.26i		−	1.18250i	0.806487	−	0.591252i	$-0.201366\pi$
$912$		0			0
$913$	−885.341			−0.969705
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−783.991			−0.854952
$918$		0			0
$919$	348.488			0.379204			0.189602	−	0.981861i	$-0.439280\pi$
$919$	348.488			0.379204			0.189602	+	0.981861i	$0.439280\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	13.4152			0.0145344
$924$		0			0
$925$			949.403i			1.02638i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1506.48			1.62162			0.810808	−	0.585312i	$-0.199028\pi$
$929$	1506.48			1.62162			0.810808	+	0.585312i	$0.199028\pi$
$930$		0			0
$931$	−201.381	+	465.070i	−0.216306	+	0.499538i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	188.973			0.202110
$936$		0			0
$937$	−145.777			−0.155578			−0.0777890	−	0.996970i	$-0.524786\pi$
$937$	−145.777			−0.155578			−0.0777890	+	0.996970i	$0.524786\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	1501.75i		−	1.59590i	−0.602721	−	0.797952i	$-0.705917\pi$
$941$		−	1501.75i		−	1.59590i	0.602721	−	0.797952i	$-0.294083\pi$
$942$		0			0
$943$		−	170.547i		−	0.180855i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	765.794			0.808653			0.404326	−	0.914615i	$-0.367506\pi$
$947$	765.794			0.808653			0.404326	+	0.914615i	$0.367506\pi$
$948$		0			0
$949$			145.492i			0.153311i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			1308.88i			1.37344i	0.726924	+	0.686718i	$0.240949\pi$
$953$			1308.88i			1.37344i	−0.726924	+	0.686718i	$0.759051\pi$
$954$		0			0
$955$	24.2584			0.0254014
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−943.474			−0.983810
$960$		0			0
$961$	558.498			0.581164
$962$		0			0
$963$		0			0
$964$		0			0
$965$			140.739i			0.145844i
$966$		0			0
$967$	−587.704			−0.607760			−0.303880	−	0.952710i	$-0.598282\pi$
$967$	−587.704			−0.607760			−0.303880	+	0.952710i	$0.598282\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			653.823i			0.673351i	0.941621	+	0.336675i	$0.109302\pi$
$971$			653.823i			0.673351i	−0.941621	+	0.336675i	$0.890698\pi$
$972$		0			0
$973$	−257.866			−0.265022
$974$		0			0
$975$		0			0
$976$		0			0
$977$			101.034i			0.103413i	0.998662	+	0.0517064i	$0.0164660\pi$
$977$			101.034i			0.103413i	−0.998662	+	0.0517064i	$0.983534\pi$
$978$		0			0
$979$		−	509.045i		−	0.519964i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	888.065i		−	0.903423i	−0.892164	−	0.451711i	$-0.850814\pi$
$983$		−	888.065i		−	0.903423i	0.892164	−	0.451711i	$-0.149186\pi$
$984$		0			0
$985$	419.128			0.425511
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−354.935			−0.358883
$990$		0			0
$991$		−	1782.19i		−	1.79838i	−0.437562	−	0.899188i	$-0.644158\pi$
$991$		−	1782.19i		−	1.79838i	0.437562	−	0.899188i	$-0.355842\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−340.368			−0.342078
$996$		0			0
$997$	891.200			0.893882			0.446941	−	0.894563i	$-0.352513\pi$
$997$	891.200			0.893882			0.446941	+	0.894563i	$0.352513\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.3.o.l.721.2		4
3.2	odd	2		912.3.o.b.721.2		4
4.3	odd	2		171.3.c.f.37.3		4
12.11	even	2		57.3.c.b.37.2	✓	4
19.18	odd	2	inner	2736.3.o.l.721.1		4
57.56	even	2		912.3.o.b.721.4		4
76.75	even	2		171.3.c.f.37.2		4
228.227	odd	2		57.3.c.b.37.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.3.c.b.37.2	✓	4	12.11	even	2
57.3.c.b.37.3	yes	4	228.227	odd	2
171.3.c.f.37.2		4	76.75	even	2
171.3.c.f.37.3		4	4.3	odd	2
912.3.o.b.721.2		4	3.2	odd	2
912.3.o.b.721.4		4	57.56	even	2
2736.3.o.l.721.1		4	19.18	odd	2	inner
2736.3.o.l.721.2		4	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)

\(f(q)\)	\(=\)	\(q-1.27492 q^{5} -4.72508 q^{7} +O(q^{10})\)	\(q-1.27492 q^{5} -4.72508 q^{7} -7.27492 q^{11} +4.30136i q^{13} +20.3746 q^{17} +(7.54983 - 17.4356i) q^{19} +5.45017 q^{23} -23.3746 q^{25} +8.60271i q^{29} +20.0624i q^{31} +6.02409 q^{35} -40.6169i q^{37} -31.2920i q^{41} -65.1238 q^{43} +55.4743 q^{47} -26.6736 q^{49} -78.1149i q^{53} +9.27492 q^{55} +69.5122i q^{59} -6.17525 q^{61} -5.48387i q^{65} +123.724i q^{67} -3.11884i q^{71} +33.8248 q^{73} +34.3746 q^{77} -87.6700i q^{79} +121.698 q^{83} -25.9759 q^{85} +69.9726i q^{89} -20.3243i q^{91} +(-9.62541 + 22.2289i) q^{95} +111.081i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 10 q^{5} - 34 q^{7}+O(q^{10}) \) 4 * q + 10 * q^5 - 34 * q^7	\( 4 q + 10 q^{5} - 34 q^{7} - 14 q^{11} + 6 q^{17} + 52 q^{23} - 18 q^{25} - 142 q^{35} - 34 q^{43} + 86 q^{47} + 150 q^{49} + 22 q^{55} - 70 q^{61} + 90 q^{73} + 62 q^{77} + 64 q^{83} - 270 q^{85} - 114 q^{95}+O(q^{100}) \) 4 * q + 10 * q^5 - 34 * q^7 - 14 * q^11 + 6 * q^17 + 52 * q^23 - 18 * q^25 - 142 * q^35 - 34 * q^43 + 86 * q^47 + 150 * q^49 + 22 * q^55 - 70 * q^61 + 90 * q^73 + 62 * q^77 + 64 * q^83 - 270 * q^85 - 114 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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