LMFDB - Embedded newform 1584.3.i.b.881.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1584,3,Mod(881,1584)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1584, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1584.881");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1584 = 2^{4} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1584.i (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:	$43.1608738747$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.65306824704.6
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 2x^{6} + 20x^{5} + x^{4} - 40x^{3} + 36x^{2} - 12x + 27 $ x^8 - 4x^7 - 2x^6 + 20x^5 + x^4 - 40x^3 + 36x^2 - 12x + 27
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 99)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.4
Root			$-1.75726 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1584.881
Dual form			1584.3.i.b.881.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.68911i q^{5} +3.30128 q^{7} +O(q^{10})$	$q-4.68911i q^{5} +3.30128 q^{7} +3.31662i q^{11} +1.00848 q^{13} +6.45594i q^{17} -25.1291 q^{19} -24.4236i q^{23} +3.01224 q^{25} -50.8695i q^{29} -48.4055 q^{31} -15.4800i q^{35} +65.8199 q^{37} +6.41379i q^{41} +48.5582 q^{43} -56.8721i q^{47} -38.1016 q^{49} +67.3272i q^{53} +15.5520 q^{55} -0.307118i q^{59} -86.8383 q^{61} -4.72888i q^{65} -29.6749 q^{67} -40.9307i q^{71} -61.0447 q^{73} +10.9491i q^{77} -85.4268 q^{79} +64.7194i q^{83} +30.2726 q^{85} -64.1065i q^{89} +3.32928 q^{91} +117.833i q^{95} -86.8082 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{7}+O(q^{10}) $ 8 * q - 16 * q^7	$ 8 q - 16 q^{7} - 8 q^{13} - 40 q^{19} - 112 q^{25} + 56 q^{31} + 136 q^{37} + 104 q^{43} - 96 q^{49} - 8 q^{61} - 112 q^{67} + 448 q^{73} - 448 q^{79} + 48 q^{85} + 544 q^{91} - 152 q^{97}+O(q^{100}) $ 8 * q - 16 * q^7 - 8 * q^13 - 40 * q^19 - 112 * q^25 + 56 * q^31 + 136 * q^37 + 104 * q^43 - 96 * q^49 - 8 * q^61 - 112 * q^67 + 448 * q^73 - 448 * q^79 + 48 * q^85 + 544 * q^91 - 152 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1584\mathbb{Z}\right)^\times$.

$n$	$145$	$353$	$991$	$1189$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$5$	− 4.68911i	− 0.937822i	−0.883245	−	0.468911i	$-0.844646\pi$
$5$	− 4.68911i	− 0.937822i	0.883245	−	0.468911i	$-0.155354\pi$
$6$	0	0
$7$	3.30128	0.471611	0.235805	−	0.971800i	$-0.424227\pi$
$7$	3.30128	0.471611	0.235805	+	0.971800i	$0.424227\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.31662i	0.301511i
$11$	3.31662i	0.301511i
$12$	0	0
$13$	1.00848	0.0775756	0.0387878	−	0.999247i	$-0.487650\pi$
$13$	1.00848	0.0775756	0.0387878	+	0.999247i	$0.487650\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.45594i	0.379761i	0.981807	+	0.189881i	$0.0608101\pi$
$17$	6.45594i	0.379761i	−0.981807	+	0.189881i	$0.939190\pi$
$18$	0	0
$19$	−25.1291	−1.32259	−0.661293	−	0.750128i	$-0.729992\pi$
$19$	−25.1291	−1.32259	−0.661293	+	0.750128i	$0.729992\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 24.4236i	− 1.06189i	−0.847405	−	0.530947i	$-0.821836\pi$
$23$	− 24.4236i	− 1.06189i	0.847405	−	0.530947i	$-0.178164\pi$
$24$	0	0
$25$	3.01224	0.120489
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 50.8695i	− 1.75412i	−0.480379	−	0.877061i	$-0.659501\pi$
$29$	− 50.8695i	− 1.75412i	0.480379	−	0.877061i	$-0.340499\pi$
$30$	0	0
$31$	−48.4055	−1.56147	−0.780734	−	0.624864i	$-0.785154\pi$
$31$	−48.4055	−1.56147	−0.780734	+	0.624864i	$0.785154\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 15.4800i	− 0.442287i
$36$	0	0
$37$	65.8199	1.77892	0.889458	−	0.457017i	$-0.151082\pi$
$37$	65.8199	1.77892	0.889458	+	0.457017i	$0.151082\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.41379i	0.156434i	0.996936	+	0.0782169i	$0.0249227\pi$
$41$	6.41379i	0.156434i	−0.996936	+	0.0782169i	$0.975077\pi$
$42$	0	0
$43$	48.5582	1.12926	0.564631	−	0.825344i	$-0.309019\pi$
$43$	48.5582	1.12926	0.564631	+	0.825344i	$0.309019\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 56.8721i	− 1.21004i	−0.796209	−	0.605022i	$-0.793164\pi$
$47$	− 56.8721i	− 1.21004i	0.796209	−	0.605022i	$-0.206836\pi$
$48$	0	0
$49$	−38.1016	−0.777583
$50$	0	0
$51$	0	0
$52$	0	0
$53$	67.3272i	1.27033i	0.772379	+	0.635163i	$0.219067\pi$
$53$	67.3272i	1.27033i	−0.772379	+	0.635163i	$0.780933\pi$
$54$	0	0
$55$	15.5520	0.282764
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 0.307118i	− 0.00520539i	−0.999997	−	0.00260270i	$-0.999172\pi$
$59$	− 0.307118i	− 0.00520539i	0.999997	−	0.00260270i	$-0.000828465\pi$
$60$	0	0
$61$	−86.8383	−1.42358	−0.711790	−	0.702393i	$-0.752115\pi$
$61$	−86.8383	−1.42358	−0.711790	+	0.702393i	$0.752115\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 4.72888i	− 0.0727521i
$66$	0	0
$67$	−29.6749	−0.442909	−0.221455	−	0.975171i	$-0.571080\pi$
$67$	−29.6749	−0.442909	−0.221455	+	0.975171i	$0.571080\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 40.9307i	− 0.576489i	−0.957557	−	0.288244i	$-0.906928\pi$
$71$	− 40.9307i	− 0.576489i	0.957557	−	0.288244i	$-0.0930716\pi$
$72$	0	0
$73$	−61.0447	−0.836229	−0.418115	−	0.908394i	$-0.637309\pi$
$73$	−61.0447	−0.836229	−0.418115	+	0.908394i	$0.637309\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	10.9491i	0.142196i
$78$	0	0
$79$	−85.4268	−1.08135	−0.540676	−	0.841231i	$-0.681831\pi$
$79$	−85.4268	−1.08135	−0.540676	+	0.841231i	$0.681831\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	64.7194i	0.779751i	0.920867	+	0.389876i	$0.127482\pi$
$83$	64.7194i	0.779751i	−0.920867	+	0.389876i	$0.872518\pi$
$84$	0	0
$85$	30.2726	0.356149
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 64.1065i	− 0.720297i	−0.932895	−	0.360149i	$-0.882726\pi$
$89$	− 64.1065i	− 0.720297i	0.932895	−	0.360149i	$-0.117274\pi$
$90$	0	0
$91$	3.32928	0.0365855
$92$	0	0
$93$	0	0
$94$	0	0
$95$	117.833i	1.24035i
$96$	0	0
$97$	−86.8082	−0.894930	−0.447465	−	0.894302i	$-0.647673\pi$
$97$	−86.8082	−0.894930	−0.447465	+	0.894302i	$0.647673\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 114.951i	− 1.13813i	−0.822292	−	0.569065i	$-0.807305\pi$
$101$	− 114.951i	− 1.13813i	0.822292	−	0.569065i	$-0.192695\pi$
$102$	0	0
$103$	−28.6367	−0.278026	−0.139013	−	0.990291i	$-0.544393\pi$
$103$	−28.6367	−0.278026	−0.139013	+	0.990291i	$0.544393\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 119.479i	− 1.11662i	−0.829631	−	0.558312i	$-0.811449\pi$
$107$	− 119.479i	− 1.11662i	0.829631	−	0.558312i	$-0.188551\pi$
$108$	0	0
$109$	−76.7577	−0.704199	−0.352100	−	0.935962i	$-0.614532\pi$
$109$	−76.7577	−0.704199	−0.352100	+	0.935962i	$0.614532\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	84.4614i	0.747446i	0.927540	+	0.373723i	$0.121919\pi$
$113$	84.4614i	0.747446i	−0.927540	+	0.373723i	$0.878081\pi$
$114$	0	0
$115$	−114.525	−0.995867
$116$	0	0
$117$	0	0
$118$	0	0
$119$	21.3129i	0.179100i
$120$	0	0
$121$	−11.0000	−0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 131.352i	− 1.05082i
$126$	0	0
$127$	99.8661	0.786347	0.393174	−	0.919464i	$-0.371377\pi$
$127$	99.8661	0.786347	0.393174	+	0.919464i	$0.371377\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 165.140i	− 1.26061i	−0.776347	−	0.630305i	$-0.782930\pi$
$131$	− 165.140i	− 1.26061i	0.776347	−	0.630305i	$-0.217070\pi$
$132$	0	0
$133$	−82.9582	−0.623746
$134$	0	0
$135$	0	0
$136$	0	0
$137$	23.5961i	0.172234i	0.996285	+	0.0861170i	$0.0274459\pi$
$137$	23.5961i	0.172234i	−0.996285	+	0.0861170i	$0.972554\pi$
$138$	0	0
$139$	−272.764	−1.96233	−0.981167	−	0.193162i	$-0.938126\pi$
$139$	−272.764	−1.96233	−0.981167	+	0.193162i	$0.938126\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3.34476i	0.0233899i
$144$	0	0
$145$	−238.533	−1.64505
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 34.7567i	− 0.233266i	−0.993175	−	0.116633i	$-0.962790\pi$
$149$	− 34.7567i	− 0.233266i	0.993175	−	0.116633i	$-0.0372102\pi$
$150$	0	0
$151$	−123.990	−0.821124	−0.410562	−	0.911833i	$-0.634667\pi$
$151$	−123.990	−0.821124	−0.410562	+	0.911833i	$0.634667\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	226.979i	1.46438i
$156$	0	0
$157$	157.296	1.00188	0.500942	−	0.865481i	$-0.332987\pi$
$157$	157.296	1.00188	0.500942	+	0.865481i	$0.332987\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 80.6289i	− 0.500800i
$162$	0	0
$163$	300.686	1.84470	0.922350	−	0.386356i	$-0.126266\pi$
$163$	300.686	1.84470	0.922350	+	0.386356i	$0.126266\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	90.7043i	0.543139i	0.962419	+	0.271570i	$0.0875427\pi$
$167$	90.7043i	0.543139i	−0.962419	+	0.271570i	$0.912457\pi$
$168$	0	0
$169$	−167.983	−0.993982
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 194.822i	− 1.12614i	−0.826410	−	0.563069i	$-0.809620\pi$
$173$	− 194.822i	− 1.12614i	0.826410	−	0.563069i	$-0.190380\pi$
$174$	0	0
$175$	9.94422	0.0568241
$176$	0	0
$177$	0	0
$178$	0	0
$179$	27.8164i	0.155399i	0.996977	+	0.0776994i	$0.0247574\pi$
$179$	27.8164i	0.155399i	−0.996977	+	0.0776994i	$0.975243\pi$
$180$	0	0
$181$	194.094	1.07234	0.536171	−	0.844109i	$-0.319870\pi$
$181$	194.094	1.07234	0.536171	+	0.844109i	$0.319870\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 308.637i	− 1.66831i
$186$	0	0
$187$	−21.4119	−0.114502
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 297.244i	− 1.55625i	−0.628108	−	0.778126i	$-0.716171\pi$
$191$	− 297.244i	− 1.55625i	0.628108	−	0.778126i	$-0.283829\pi$
$192$	0	0
$193$	−197.890	−1.02533	−0.512667	−	0.858587i	$-0.671343\pi$
$193$	−197.890	−1.02533	−0.512667	+	0.858587i	$0.671343\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	249.339i	1.26568i	0.774282	+	0.632841i	$0.218111\pi$
$197$	249.339i	1.26568i	−0.774282	+	0.632841i	$0.781889\pi$
$198$	0	0
$199$	−101.803	−0.511571	−0.255786	−	0.966733i	$-0.582334\pi$
$199$	−101.803	−0.511571	−0.255786	+	0.966733i	$0.582334\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 167.934i	− 0.827263i
$204$	0	0
$205$	30.0750	0.146707
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 83.3439i	− 0.398775i
$210$	0	0
$211$	37.9452	0.179835	0.0899175	−	0.995949i	$-0.471340\pi$
$211$	37.9452	0.179835	0.0899175	+	0.995949i	$0.471340\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 227.695i	− 1.05905i
$216$	0	0
$217$	−159.800	−0.736405
$218$	0	0
$219$	0	0
$220$	0	0
$221$	6.51071i	0.0294602i
$222$	0	0
$223$	−119.504	−0.535893	−0.267946	−	0.963434i	$-0.586345\pi$
$223$	−119.504	−0.535893	−0.267946	+	0.963434i	$0.586345\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 233.484i	− 1.02856i	−0.857621	−	0.514282i	$-0.828059\pi$
$227$	− 233.484i	− 1.02856i	0.857621	−	0.514282i	$-0.171941\pi$
$228$	0	0
$229$	−170.303	−0.743683	−0.371841	−	0.928296i	$-0.621274\pi$
$229$	−170.303	−0.743683	−0.371841	+	0.928296i	$0.621274\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	149.406i	0.641228i	0.947210	+	0.320614i	$0.103889\pi$
$233$	149.406i	0.641228i	−0.947210	+	0.320614i	$0.896111\pi$
$234$	0	0
$235$	−266.680	−1.13481
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 267.637i	− 1.11982i	−0.828554	−	0.559909i	$-0.810836\pi$
$239$	− 267.637i	− 1.11982i	0.828554	−	0.559909i	$-0.189164\pi$
$240$	0	0
$241$	202.338	0.839578	0.419789	−	0.907622i	$-0.362104\pi$
$241$	202.338	0.839578	0.419789	+	0.907622i	$0.362104\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	178.663i	0.729235i
$246$	0	0
$247$	−25.3423	−0.102600
$248$	0	0
$249$	0	0
$250$	0	0
$251$	169.400i	0.674901i	0.941343	+	0.337450i	$0.109565\pi$
$251$	169.400i	0.674901i	−0.941343	+	0.337450i	$0.890435\pi$
$252$	0	0
$253$	81.0038	0.320173
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 208.654i	− 0.811885i	−0.913899	−	0.405942i	$-0.866943\pi$
$257$	− 208.654i	− 0.811885i	0.913899	−	0.405942i	$-0.133057\pi$
$258$	0	0
$259$	217.290	0.838956
$260$	0	0
$261$	0	0
$262$	0	0
$263$	117.552i	0.446967i	0.974708	+	0.223483i	$0.0717429\pi$
$263$	117.552i	0.446967i	−0.974708	+	0.223483i	$0.928257\pi$
$264$	0	0
$265$	315.705	1.19134
$266$	0	0
$267$	0	0
$268$	0	0
$269$	467.823i	1.73912i	0.493830	+	0.869559i	$0.335597\pi$
$269$	467.823i	1.73912i	−0.493830	+	0.869559i	$0.664403\pi$
$270$	0	0
$271$	116.293	0.429126	0.214563	−	0.976710i	$-0.431167\pi$
$271$	116.293	0.429126	0.214563	+	0.976710i	$0.431167\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	9.99046i	0.0363289i
$276$	0	0
$277$	290.493	1.04871	0.524356	−	0.851499i	$-0.324306\pi$
$277$	290.493	1.04871	0.524356	+	0.851499i	$0.324306\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 26.7812i	− 0.0953066i	−0.998864	−	0.0476533i	$-0.984826\pi$
$281$	− 26.7812i	− 0.0953066i	0.998864	−	0.0476533i	$-0.0151743\pi$
$282$	0	0
$283$	−23.7154	−0.0838001	−0.0419001	−	0.999122i	$-0.513341\pi$
$283$	−23.7154	−0.0838001	−0.0419001	+	0.999122i	$0.513341\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	21.1737i	0.0737759i
$288$	0	0
$289$	247.321	0.855781
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 379.831i	− 1.29635i	−0.761491	−	0.648175i	$-0.775532\pi$
$293$	− 379.831i	− 1.29635i	0.761491	−	0.648175i	$-0.224468\pi$
$294$	0	0
$295$	−1.44011	−0.00488173
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 24.6307i	− 0.0823770i
$300$	0	0
$301$	160.304	0.532572
$302$	0	0
$303$	0	0
$304$	0	0
$305$	407.195i	1.33506i
$306$	0	0
$307$	−41.2092	−0.134232	−0.0671160	−	0.997745i	$-0.521380\pi$
$307$	−41.2092	−0.134232	−0.0671160	+	0.997745i	$0.521380\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	520.132i	1.67245i	0.548387	+	0.836224i	$0.315242\pi$
$311$	520.132i	1.67245i	−0.548387	+	0.836224i	$0.684758\pi$
$312$	0	0
$313$	−558.820	−1.78537	−0.892684	−	0.450683i	$-0.851181\pi$
$313$	−558.820	−1.78537	−0.892684	+	0.450683i	$0.851181\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	434.973i	1.37215i	0.727529	+	0.686077i	$0.240669\pi$
$317$	434.973i	1.37215i	−0.727529	+	0.686077i	$0.759331\pi$
$318$	0	0
$319$	168.715	0.528888
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 162.232i	− 0.502267i
$324$	0	0
$325$	3.03779	0.00934704
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 187.750i	− 0.570670i
$330$	0	0
$331$	−185.292	−0.559795	−0.279897	−	0.960030i	$-0.590300\pi$
$331$	−185.292	−0.559795	−0.279897	+	0.960030i	$0.590300\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	139.149i	0.415370i
$336$	0	0
$337$	−366.174	−1.08657	−0.543285	−	0.839549i	$-0.682820\pi$
$337$	−366.174	−1.08657	−0.543285	+	0.839549i	$0.682820\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 160.543i	− 0.470800i
$342$	0	0
$343$	−287.546	−0.838327
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 232.794i	− 0.670876i	−0.942062	−	0.335438i	$-0.891116\pi$
$347$	− 232.794i	− 0.670876i	0.942062	−	0.335438i	$-0.108884\pi$
$348$	0	0
$349$	−190.183	−0.544938	−0.272469	−	0.962165i	$-0.587840\pi$
$349$	−190.183	−0.544938	−0.272469	+	0.962165i	$0.587840\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 650.001i	− 1.84136i	−0.390316	−	0.920681i	$-0.627634\pi$
$353$	− 650.001i	− 1.84136i	0.390316	−	0.920681i	$-0.372366\pi$
$354$	0	0
$355$	−191.929	−0.540644
$356$	0	0
$357$	0	0
$358$	0	0
$359$	318.134i	0.886167i	0.896480	+	0.443083i	$0.146115\pi$
$359$	318.134i	0.886167i	−0.896480	+	0.443083i	$0.853885\pi$
$360$	0	0
$361$	270.473	0.749233
$362$	0	0
$363$	0	0
$364$	0	0
$365$	286.246i	0.784234i
$366$	0	0
$367$	10.0024	0.0272546	0.0136273	−	0.999907i	$-0.495662\pi$
$367$	10.0024	0.0272546	0.0136273	+	0.999907i	$0.495662\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	222.266i	0.599099i
$372$	0	0
$373$	373.434	1.00116	0.500582	−	0.865689i	$-0.333119\pi$
$373$	373.434	1.00116	0.500582	+	0.865689i	$0.333119\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 51.3010i	− 0.136077i
$378$	0	0
$379$	273.614	0.721937	0.360969	−	0.932578i	$-0.382446\pi$
$379$	273.614	0.721937	0.360969	+	0.932578i	$0.382446\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	129.063i	0.336978i	0.985703	+	0.168489i	$0.0538888\pi$
$383$	129.063i	0.336978i	−0.985703	+	0.168489i	$0.946111\pi$
$384$	0	0
$385$	51.3415	0.133355
$386$	0	0
$387$	0	0
$388$	0	0
$389$	311.684i	0.801244i	0.916243	+	0.400622i	$0.131206\pi$
$389$	311.684i	0.801244i	−0.916243	+	0.400622i	$0.868794\pi$
$390$	0	0
$391$	157.677	0.403266
$392$	0	0
$393$	0	0
$394$	0	0
$395$	400.576i	1.01412i
$396$	0	0
$397$	449.573	1.13243	0.566213	−	0.824259i	$-0.308408\pi$
$397$	449.573	1.13243	0.566213	+	0.824259i	$0.308408\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 348.583i	− 0.869284i	−0.900603	−	0.434642i	$-0.856875\pi$
$401$	− 348.583i	− 0.869284i	0.900603	−	0.434642i	$-0.143125\pi$
$402$	0	0
$403$	−48.8161	−0.121132
$404$	0	0
$405$	0	0
$406$	0	0
$407$	218.300i	0.536363i
$408$	0	0
$409$	−532.578	−1.30215	−0.651073	−	0.759015i	$-0.725681\pi$
$409$	−532.578	−1.30215	−0.651073	+	0.759015i	$0.725681\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 1.01388i	− 0.00245492i
$414$	0	0
$415$	303.476	0.731268
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 267.800i	− 0.639141i	−0.947563	−	0.319570i	$-0.896461\pi$
$419$	− 267.800i	− 0.639141i	0.947563	−	0.319570i	$-0.103539\pi$
$420$	0	0
$421$	371.455	0.882315	0.441158	−	0.897430i	$-0.354568\pi$
$421$	371.455	0.882315	0.441158	+	0.897430i	$0.354568\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	19.4468i	0.0457572i
$426$	0	0
$427$	−286.677	−0.671375
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 285.915i	− 0.663376i	−0.943389	−	0.331688i	$-0.892382\pi$
$431$	− 285.915i	− 0.663376i	0.943389	−	0.331688i	$-0.107618\pi$
$432$	0	0
$433$	228.990	0.528845	0.264423	−	0.964407i	$-0.414819\pi$
$433$	228.990	0.528845	0.264423	+	0.964407i	$0.414819\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	613.743i	1.40445i
$438$	0	0
$439$	815.871	1.85848	0.929238	−	0.369481i	$-0.120464\pi$
$439$	815.871	1.85848	0.929238	+	0.369481i	$0.120464\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	434.432i	0.980660i	0.871537	+	0.490330i	$0.163124\pi$
$443$	434.432i	0.980660i	−0.871537	+	0.490330i	$0.836876\pi$
$444$	0	0
$445$	−300.602	−0.675511
$446$	0	0
$447$	0	0
$448$	0	0
$449$	73.1022i	0.162811i	0.996681	+	0.0814056i	$0.0259409\pi$
$449$	73.1022i	0.162811i	−0.996681	+	0.0814056i	$0.974059\pi$
$450$	0	0
$451$	−21.2721	−0.0471666
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 15.6114i	− 0.0343107i
$456$	0	0
$457$	−49.1411	−0.107530	−0.0537649	−	0.998554i	$-0.517122\pi$
$457$	−49.1411	−0.107530	−0.0537649	+	0.998554i	$0.517122\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	657.645i	1.42656i	0.700878	+	0.713281i	$0.252792\pi$
$461$	657.645i	1.42656i	−0.700878	+	0.713281i	$0.747208\pi$
$462$	0	0
$463$	195.703	0.422685	0.211343	−	0.977412i	$-0.432216\pi$
$463$	195.703	0.422685	0.211343	+	0.977412i	$0.432216\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	13.5937i	0.0291086i	0.999894	+	0.0145543i	$0.00463295\pi$
$467$	13.5937i	0.0291086i	−0.999894	+	0.0145543i	$0.995367\pi$
$468$	0	0
$469$	−97.9651	−0.208881
$470$	0	0
$471$	0	0
$472$	0	0
$473$	161.049i	0.340485i
$474$	0	0
$475$	−75.6949	−0.159358
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 369.614i	− 0.771637i	−0.922575	−	0.385818i	$-0.873919\pi$
$479$	− 369.614i	− 0.771637i	0.922575	−	0.385818i	$-0.126081\pi$
$480$	0	0
$481$	66.3782	0.138000
$482$	0	0
$483$	0	0
$484$	0	0
$485$	407.053i	0.839285i
$486$	0	0
$487$	485.893	0.997727	0.498863	−	0.866681i	$-0.333751\pi$
$487$	485.893	0.997727	0.498863	+	0.866681i	$0.333751\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	538.550i	1.09684i	0.836202	+	0.548421i	$0.184771\pi$
$491$	538.550i	1.09684i	−0.836202	+	0.548421i	$0.815229\pi$
$492$	0	0
$493$	328.411	0.666148
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 135.124i	− 0.271878i
$498$	0	0
$499$	949.252	1.90231	0.951154	−	0.308716i	$-0.0998993\pi$
$499$	949.252	1.90231	0.951154	+	0.308716i	$0.0998993\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 207.815i	− 0.413151i	−0.978431	−	0.206575i	$-0.933768\pi$
$503$	− 207.815i	− 0.413151i	0.978431	−	0.206575i	$-0.0662319\pi$
$504$	0	0
$505$	−539.019	−1.06736
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 473.501i	− 0.930258i	−0.885243	−	0.465129i	$-0.846008\pi$
$509$	− 473.501i	− 0.930258i	0.885243	−	0.465129i	$-0.153992\pi$
$510$	0	0
$511$	−201.526	−0.394375
$512$	0	0
$513$	0	0
$514$	0	0
$515$	134.281i	0.260739i
$516$	0	0
$517$	188.623	0.364842
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 756.599i	− 1.45221i	−0.687586	−	0.726103i	$-0.741330\pi$
$521$	− 756.599i	− 1.45221i	0.687586	−	0.726103i	$-0.258670\pi$
$522$	0	0
$523$	782.107	1.49543	0.747713	−	0.664022i	$-0.231152\pi$
$523$	782.107	1.49543	0.747713	+	0.664022i	$0.231152\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 312.503i	− 0.592985i
$528$	0	0
$529$	−67.5098	−0.127618
$530$	0	0
$531$	0	0
$532$	0	0
$533$	6.46819i	0.0121354i
$534$	0	0
$535$	−560.249	−1.04719
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 126.369i	− 0.234450i
$540$	0	0
$541$	−55.2149	−0.102061	−0.0510304	−	0.998697i	$-0.516251\pi$
$541$	−55.2149	−0.102061	−0.0510304	+	0.998697i	$0.516251\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	359.926i	0.660414i
$546$	0	0
$547$	358.392	0.655195	0.327597	−	0.944817i	$-0.393761\pi$
$547$	358.392	0.655195	0.327597	+	0.944817i	$0.393761\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1278.31i	2.31998i
$552$	0	0
$553$	−282.017	−0.509977
$554$	0	0
$555$	0	0
$556$	0	0
$557$	644.969i	1.15793i	0.815351	+	0.578967i	$0.196544\pi$
$557$	644.969i	1.15793i	−0.815351	+	0.578967i	$0.803456\pi$
$558$	0	0
$559$	48.9701	0.0876031
$560$	0	0
$561$	0	0
$562$	0	0
$563$	730.584i	1.29766i	0.760932	+	0.648831i	$0.224742\pi$
$563$	730.584i	1.29766i	−0.760932	+	0.648831i	$0.775258\pi$
$564$	0	0
$565$	396.049	0.700972
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 1037.85i	− 1.82399i	−0.410200	−	0.911996i	$-0.634541\pi$
$569$	− 1037.85i	− 1.82399i	0.410200	−	0.911996i	$-0.365459\pi$
$570$	0	0
$571$	161.245	0.282391	0.141196	−	0.989982i	$-0.454905\pi$
$571$	161.245	0.282391	0.141196	+	0.989982i	$0.454905\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 73.5695i	− 0.127947i
$576$	0	0
$577$	988.690	1.71350	0.856750	−	0.515732i	$-0.172480\pi$
$577$	988.690	1.71350	0.856750	+	0.515732i	$0.172480\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	213.656i	0.367739i
$582$	0	0
$583$	−223.299	−0.383017
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 517.799i	− 0.882110i	−0.897480	−	0.441055i	$-0.854604\pi$
$587$	− 517.799i	− 0.882110i	0.897480	−	0.441055i	$-0.145396\pi$
$588$	0	0
$589$	1216.39	2.06517
$590$	0	0
$591$	0	0
$592$	0	0
$593$	396.829i	0.669188i	0.942362	+	0.334594i	$0.108599\pi$
$593$	396.829i	0.669188i	−0.942362	+	0.334594i	$0.891401\pi$
$594$	0	0
$595$	99.9383	0.167964
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 865.942i	− 1.44565i	−0.691034	−	0.722823i	$-0.742844\pi$
$599$	− 865.942i	− 1.44565i	0.691034	−	0.722823i	$-0.257156\pi$
$600$	0	0
$601$	412.367	0.686134	0.343067	−	0.939311i	$-0.388534\pi$
$601$	412.367	0.686134	0.343067	+	0.939311i	$0.388534\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	51.5802i	0.0852566i
$606$	0	0
$607$	248.291	0.409045	0.204523	−	0.978862i	$-0.434436\pi$
$607$	248.291	0.409045	0.204523	+	0.978862i	$0.434436\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 57.3545i	− 0.0938699i
$612$	0	0
$613$	59.7914	0.0975390	0.0487695	−	0.998810i	$-0.484470\pi$
$613$	59.7914	0.0975390	0.0487695	+	0.998810i	$0.484470\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 615.297i	− 0.997240i	−0.866821	−	0.498620i	$-0.833840\pi$
$617$	− 615.297i	− 0.997240i	0.866821	−	0.498620i	$-0.166160\pi$
$618$	0	0
$619$	308.597	0.498541	0.249270	−	0.968434i	$-0.419809\pi$
$619$	308.597	0.498541	0.249270	+	0.968434i	$0.419809\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 211.633i	− 0.339700i
$624$	0	0
$625$	−540.621	−0.864993
$626$	0	0
$627$	0	0
$628$	0	0
$629$	424.930i	0.675564i
$630$	0	0
$631$	473.277	0.750042	0.375021	−	0.927016i	$-0.377635\pi$
$631$	473.277	0.750042	0.375021	+	0.927016i	$0.377635\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 468.283i	− 0.737454i
$636$	0	0
$637$	−38.4248	−0.0603214
$638$	0	0
$639$	0	0
$640$	0	0
$641$	280.129i	0.437019i	0.975835	+	0.218509i	$0.0701194\pi$
$641$	280.129i	0.437019i	−0.975835	+	0.218509i	$0.929881\pi$
$642$	0	0
$643$	1239.37	1.92748	0.963740	−	0.266843i	$-0.0859805\pi$
$643$	1239.37	1.92748	0.963740	+	0.266843i	$0.0859805\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 19.9618i	− 0.0308529i	−0.999881	−	0.0154264i	$-0.995089\pi$
$647$	− 19.9618i	− 0.0308529i	0.999881	−	0.0154264i	$-0.00491059\pi$
$648$	0	0
$649$	1.01860	0.00156949
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 420.634i	− 0.644156i	−0.946713	−	0.322078i	$-0.895619\pi$
$653$	− 420.634i	− 0.644156i	0.946713	−	0.322078i	$-0.104381\pi$
$654$	0	0
$655$	−774.360	−1.18223
$656$	0	0
$657$	0	0
$658$	0	0
$659$	380.713i	0.577714i	0.957372	+	0.288857i	$0.0932751\pi$
$659$	380.713i	0.577714i	−0.957372	+	0.288857i	$0.906725\pi$
$660$	0	0
$661$	339.630	0.513812	0.256906	−	0.966436i	$-0.417297\pi$
$661$	339.630	0.513812	0.256906	+	0.966436i	$0.417297\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	389.000i	0.584963i
$666$	0	0
$667$	−1242.41	−1.86269
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 288.010i	− 0.429225i
$672$	0	0
$673$	−219.920	−0.326775	−0.163388	−	0.986562i	$-0.552242\pi$
$673$	−219.920	−0.326775	−0.163388	+	0.986562i	$0.552242\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	336.004i	0.496313i	0.968720	+	0.248157i	$0.0798248\pi$
$677$	336.004i	0.496313i	−0.968720	+	0.248157i	$0.920175\pi$
$678$	0	0
$679$	−286.578	−0.422058
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 231.915i	− 0.339553i	−0.985483	−	0.169777i	$-0.945695\pi$
$683$	− 231.915i	− 0.339553i	0.985483	−	0.169777i	$-0.0543046\pi$
$684$	0	0
$685$	110.645	0.161525
$686$	0	0
$687$	0	0
$688$	0	0
$689$	67.8983i	0.0985462i
$690$	0	0
$691$	−404.654	−0.585606	−0.292803	−	0.956173i	$-0.594588\pi$
$691$	−404.654	−0.585606	−0.292803	+	0.956173i	$0.594588\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1279.02i	1.84032i
$696$	0	0
$697$	−41.4071	−0.0594075
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 144.543i	− 0.206196i	−0.994671	−	0.103098i	$-0.967125\pi$
$701$	− 144.543i	− 0.206196i	0.994671	−	0.103098i	$-0.0328755\pi$
$702$	0	0
$703$	−1654.00	−2.35277
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 379.485i	− 0.536755i
$708$	0	0
$709$	1033.34	1.45746	0.728730	−	0.684802i	$-0.240111\pi$
$709$	1033.34	1.45746	0.728730	+	0.684802i	$0.240111\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1182.23i	1.65811i
$714$	0	0
$715$	15.6839	0.0219356
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1286.83i	1.78974i	0.446324	+	0.894872i	$0.352733\pi$
$719$	1286.83i	1.78974i	−0.446324	+	0.894872i	$0.647267\pi$
$720$	0	0
$721$	−94.5376	−0.131120
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 153.231i	− 0.211353i
$726$	0	0
$727$	835.384	1.14908	0.574542	−	0.818475i	$-0.305180\pi$
$727$	835.384	1.14908	0.574542	+	0.818475i	$0.305180\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	313.489i	0.428850i
$732$	0	0
$733$	1300.03	1.77358	0.886790	−	0.462173i	$-0.152930\pi$
$733$	1300.03	1.77358	0.886790	+	0.462173i	$0.152930\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 98.4206i	− 0.133542i
$738$	0	0
$739$	−993.426	−1.34428	−0.672142	−	0.740422i	$-0.734626\pi$
$739$	−993.426	−1.34428	−0.672142	+	0.740422i	$0.734626\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 762.477i	− 1.02621i	−0.858325	−	0.513107i	$-0.828494\pi$
$743$	− 762.477i	− 1.02621i	0.858325	−	0.513107i	$-0.171506\pi$
$744$	0	0
$745$	−162.978	−0.218762
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 394.432i	− 0.526612i
$750$	0	0
$751$	647.654	0.862389	0.431194	−	0.902259i	$-0.358092\pi$
$751$	647.654	0.862389	0.431194	+	0.902259i	$0.358092\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	581.401i	0.770068i
$756$	0	0
$757$	−43.6130	−0.0576130	−0.0288065	−	0.999585i	$-0.509171\pi$
$757$	−43.6130	−0.0576130	−0.0288065	+	0.999585i	$0.509171\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	130.629i	0.171655i	0.996310	+	0.0858274i	$0.0273534\pi$
$761$	130.629i	0.171655i	−0.996310	+	0.0858274i	$0.972647\pi$
$762$	0	0
$763$	−253.398	−0.332108
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 0.309723i	0 0.000403811i
$768$	0	0
$769$	−1261.92	−1.64099	−0.820495	−	0.571654i	$-0.806302\pi$
$769$	−1261.92	−1.64099	−0.820495	+	0.571654i	$0.806302\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	436.493i	0.564674i	0.959315	+	0.282337i	$0.0911097\pi$
$773$	436.493i	0.564674i	−0.959315	+	0.282337i	$0.908890\pi$
$774$	0	0
$775$	−145.809	−0.188140
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 161.173i	− 0.206897i
$780$	0	0
$781$	135.752	0.173818
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 737.578i	− 0.939590i
$786$	0	0
$787$	−38.8895	−0.0494148	−0.0247074	−	0.999695i	$-0.507865\pi$
$787$	−38.8895	−0.0494148	−0.0247074	+	0.999695i	$0.507865\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	278.830i	0.352504i
$792$	0	0
$793$	−87.5749	−0.110435
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 864.722i	− 1.08497i	−0.840065	−	0.542486i	$-0.817483\pi$
$797$	− 864.722i	− 1.08497i	0.840065	−	0.542486i	$-0.182517\pi$
$798$	0	0
$799$	367.163	0.459528
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 202.462i	− 0.252133i
$804$	0	0
$805$	−378.078	−0.469662
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 267.814i	− 0.331043i	−0.986206	−	0.165521i	$-0.947069\pi$
$809$	− 267.814i	− 0.331043i	0.986206	−	0.165521i	$-0.0529307\pi$
$810$	0	0
$811$	−62.9645	−0.0776381	−0.0388190	−	0.999246i	$-0.512360\pi$
$811$	−62.9645	−0.0776381	−0.0388190	+	0.999246i	$0.512360\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 1409.95i	− 1.73000i
$816$	0	0
$817$	−1220.23	−1.49354
$818$	0	0
$819$	0	0
$820$	0	0
$821$	205.138i	0.249863i	0.992165	+	0.124932i	$0.0398712\pi$
$821$	205.138i	0.249863i	−0.992165	+	0.124932i	$0.960129\pi$
$822$	0	0
$823$	−92.1457	−0.111963	−0.0559816	−	0.998432i	$-0.517829\pi$
$823$	−92.1457	−0.111963	−0.0559816	+	0.998432i	$0.517829\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 152.560i	− 0.184475i	−0.995737	−	0.0922373i	$-0.970598\pi$
$827$	− 152.560i	− 0.184475i	0.995737	−	0.0922373i	$-0.0294018\pi$
$828$	0	0
$829$	−527.763	−0.636626	−0.318313	−	0.947986i	$-0.603116\pi$
$829$	−527.763	−0.636626	−0.318313	+	0.947986i	$0.603116\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 245.982i	− 0.295296i
$834$	0	0
$835$	425.322	0.509368
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 548.065i	− 0.653237i	−0.945156	−	0.326618i	$-0.894091\pi$
$839$	− 548.065i	− 0.653237i	0.945156	−	0.326618i	$-0.105909\pi$
$840$	0	0
$841$	−1746.71	−2.07694
$842$	0	0
$843$	0	0
$844$	0	0
$845$	787.691i	0.932178i
$846$	0	0
$847$	−36.3140	−0.0428737
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 1607.56i	− 1.88902i
$852$	0	0
$853$	−88.0263	−0.103196	−0.0515980	−	0.998668i	$-0.516431\pi$
$853$	−88.0263	−0.103196	−0.0515980	+	0.998668i	$0.516431\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1341.27i	1.56508i	0.622601	+	0.782539i	$0.286076\pi$
$857$	1341.27i	1.56508i	−0.622601	+	0.782539i	$0.713924\pi$
$858$	0	0
$859$	−1230.21	−1.43214	−0.716069	−	0.698030i	$-0.754060\pi$
$859$	−1230.21	−1.43214	−0.716069	+	0.698030i	$0.754060\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	218.670i	0.253383i	0.991942	+	0.126692i	$0.0404358\pi$
$863$	218.670i	0.253383i	−0.991942	+	0.126692i	$0.959564\pi$
$864$	0	0
$865$	−913.542	−1.05612
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 283.329i	− 0.326040i
$870$	0	0
$871$	−29.9266	−0.0343589
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 433.631i	− 0.495578i
$876$	0	0
$877$	547.663	0.624474	0.312237	−	0.950004i	$-0.398922\pi$
$877$	547.663	0.624474	0.312237	+	0.950004i	$0.398922\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	235.271i	0.267050i	0.991045	+	0.133525i	$0.0426297\pi$
$881$	235.271i	0.267050i	−0.991045	+	0.133525i	$0.957370\pi$
$882$	0	0
$883$	246.902	0.279617	0.139808	−	0.990179i	$-0.455351\pi$
$883$	246.902	0.279617	0.139808	+	0.990179i	$0.455351\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	667.037i	0.752014i	0.926617	+	0.376007i	$0.122703\pi$
$887$	667.037i	0.752014i	−0.926617	+	0.376007i	$0.877297\pi$
$888$	0	0
$889$	329.686	0.370850
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1429.15i	1.60039i
$894$	0	0
$895$	130.434	0.145736
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2462.36i	2.73900i
$900$	0	0
$901$	−434.661	−0.482420
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 910.128i	− 1.00567i
$906$	0	0
$907$	115.024	0.126819	0.0634093	−	0.997988i	$-0.479803\pi$
$907$	115.024	0.126819	0.0634093	+	0.997988i	$0.479803\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	545.185i	0.598447i	0.954183	+	0.299223i	$0.0967276\pi$
$911$	545.185i	0.598447i	−0.954183	+	0.299223i	$0.903272\pi$
$912$	0	0
$913$	−214.650	−0.235104
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 545.173i	− 0.594518i
$918$	0	0
$919$	116.321	0.126573	0.0632866	−	0.997995i	$-0.479842\pi$
$919$	116.321	0.126573	0.0632866	+	0.997995i	$0.479842\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 41.2779i	− 0.0447214i
$924$	0	0
$925$	198.265	0.214341
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 25.3182i	− 0.0272531i	−0.999907	−	0.0136266i	$-0.995662\pi$
$929$	− 25.3182i	− 0.0272531i	0.999907	−	0.0136266i	$-0.00433760\pi$
$930$	0	0
$931$	957.459	1.02842
$932$	0	0
$933$	0	0
$934$	0	0
$935$	100.403i	0.107383i
$936$	0	0
$937$	1360.70	1.45219	0.726093	−	0.687597i	$-0.241334\pi$
$937$	1360.70	1.45219	0.726093	+	0.687597i	$0.241334\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 908.864i	− 0.965850i	−0.875662	−	0.482925i	$-0.839574\pi$
$941$	− 908.864i	− 0.965850i	0.875662	−	0.482925i	$-0.160426\pi$
$942$	0	0
$943$	156.647	0.166116
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1593.80i	− 1.68300i	−0.540256	−	0.841501i	$-0.681673\pi$
$947$	− 1593.80i	− 1.68300i	0.540256	−	0.841501i	$-0.318327\pi$
$948$	0	0
$949$	−61.5625	−0.0648709
$950$	0	0
$951$	0	0
$952$	0	0
$953$	163.175i	0.171222i	0.996329	+	0.0856110i	$0.0272842\pi$
$953$	163.175i	0.171222i	−0.996329	+	0.0856110i	$0.972716\pi$
$954$	0	0
$955$	−1393.81	−1.45949
$956$	0	0
$957$	0	0
$958$	0	0
$959$	77.8971i	0.0812274i
$960$	0	0
$961$	1382.09	1.43818
$962$	0	0
$963$	0	0
$964$	0	0
$965$	927.926i	0.961582i
$966$	0	0
$967$	−751.225	−0.776862	−0.388431	−	0.921478i	$-0.626983\pi$
$967$	−751.225	−0.776862	−0.388431	+	0.921478i	$0.626983\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	134.041i	0.138044i	0.997615	+	0.0690221i	$0.0219879\pi$
$971$	134.041i	0.138044i	−0.997615	+	0.0690221i	$0.978012\pi$
$972$	0	0
$973$	−900.470	−0.925458
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 1081.18i	− 1.10664i	−0.832970	−	0.553319i	$-0.813361\pi$
$977$	− 1081.18i	− 1.10664i	0.832970	−	0.553319i	$-0.186639\pi$
$978$	0	0
$979$	212.617	0.217178
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1114.11i	− 1.13337i	−0.823933	−	0.566687i	$-0.808225\pi$
$983$	− 1114.11i	− 1.13337i	0.823933	−	0.566687i	$-0.191775\pi$
$984$	0	0
$985$	1169.18	1.18698
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 1185.96i	− 1.19916i
$990$	0	0
$991$	1199.41	1.21031	0.605153	−	0.796109i	$-0.293112\pi$
$991$	1199.41	1.21031	0.605153	+	0.796109i	$0.293112\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	477.364i	0.479763i
$996$	0	0
$997$	−995.763	−0.998760	−0.499380	−	0.866383i	$-0.666439\pi$
$997$	−995.763	−0.998760	−0.499380	+	0.866383i	$0.666439\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1584.3.i.b.881.4		8
3.2	odd	2	inner	1584.3.i.b.881.5		8
4.3	odd	2		99.3.b.a.89.2	✓	8
12.11	even	2		99.3.b.a.89.7	yes	8
44.43	even	2		1089.3.b.g.485.7		8
132.131	odd	2		1089.3.b.g.485.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.3.b.a.89.2	✓	8	4.3	odd	2
99.3.b.a.89.7	yes	8	12.11	even	2
1089.3.b.g.485.2		8	132.131	odd	2
1089.3.b.g.485.7		8	44.43	even	2
1584.3.i.b.881.4		8	1.1	even	1	trivial
1584.3.i.b.881.5		8	3.2	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:	\( N \)	\(=\)	\( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1584.i (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.68911i q^{5} +3.30128 q^{7} +O(q^{10})\)	\(q-4.68911i q^{5} +3.30128 q^{7} +3.31662i q^{11} +1.00848 q^{13} +6.45594i q^{17} -25.1291 q^{19} -24.4236i q^{23} +3.01224 q^{25} -50.8695i q^{29} -48.4055 q^{31} -15.4800i q^{35} +65.8199 q^{37} +6.41379i q^{41} +48.5582 q^{43} -56.8721i q^{47} -38.1016 q^{49} +67.3272i q^{53} +15.5520 q^{55} -0.307118i q^{59} -86.8383 q^{61} -4.72888i q^{65} -29.6749 q^{67} -40.9307i q^{71} -61.0447 q^{73} +10.9491i q^{77} -85.4268 q^{79} +64.7194i q^{83} +30.2726 q^{85} -64.1065i q^{89} +3.32928 q^{91} +117.833i q^{95} -86.8082 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{7}+O(q^{10}) \) 8 * q - 16 * q^7	\( 8 q - 16 q^{7} - 8 q^{13} - 40 q^{19} - 112 q^{25} + 56 q^{31} + 136 q^{37} + 104 q^{43} - 96 q^{49} - 8 q^{61} - 112 q^{67} + 448 q^{73} - 448 q^{79} + 48 q^{85} + 544 q^{91} - 152 q^{97}+O(q^{100}) \) 8 * q - 16 * q^7 - 8 * q^13 - 40 * q^19 - 112 * q^25 + 56 * q^31 + 136 * q^37 + 104 * q^43 - 96 * q^49 - 8 * q^61 - 112 * q^67 + 448 * q^73 - 448 * q^79 + 48 * q^85 + 544 * q^91 - 152 * q^97

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