LMFDB - Embedded newform 1980.3.b.a.901.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1980,3,Mod(901,1980)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1980.901"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1980, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1980 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1980.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$53.9510923433$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 67x^{6} + 1356x^{4} + 9065x^{2} + 17275 $ x^8 + 67x^6 + 1356x^4 + 9065*x^2 + 17275
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5}\cdot 5 $
Twist minimal:	no (minimal twist has level 220)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			901.5
Root			$-2.68549i$ of defining polynomial
Character	$\chi$	$=$	1980.901
Dual form			1980.3.b.a.901.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23607 q^{5} -13.1585i q^{7} +(3.18499 - 10.5288i) q^{11} +14.6396i q^{13} -9.90801i q^{17} +11.5333i q^{19} +14.6044 q^{23} +5.00000 q^{25} -8.28275i q^{29} +53.2285 q^{31} -29.4234i q^{35} +41.1146 q^{37} -74.6449i q^{41} -30.4377i q^{43} -0.697244 q^{47} -124.147 q^{49} -17.0571 q^{53} +(7.12186 - 23.5431i) q^{55} +6.29242 q^{59} +50.6864i q^{61} +32.7351i q^{65} -81.8378 q^{67} +62.4365 q^{71} +11.5551i q^{73} +(-138.544 - 41.9098i) q^{77} +79.2397i q^{79} -81.8105i q^{83} -22.1550i q^{85} -136.935 q^{89} +192.636 q^{91} +25.7892i q^{95} -80.2686 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 24 q^{11} - 56 q^{23} + 40 q^{25} + 20 q^{31} + 72 q^{37} + 184 q^{47} - 244 q^{49} - 136 q^{53} + 60 q^{55} + 16 q^{59} - 264 q^{67} + 220 q^{71} - 208 q^{77} - 220 q^{89} - 72 q^{91} - 8 q^{97}+O(q^{100}) $ 8 * q - 24 * q^11 - 56 * q^23 + 40 * q^25 + 20 * q^31 + 72 * q^37 + 184 * q^47 - 244 * q^49 - 136 * q^53 + 60 * q^55 + 16 * q^59 - 264 * q^67 + 220 * q^71 - 208 * q^77 - 220 * q^89 - 72 * q^91 - 8 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1980\mathbb{Z}\right)^\times$.

$n$	$397$	$541$	$991$	$1541$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	2.23607			0.447214
$5$	2.23607			0.447214
$6$		0			0
$7$		−	13.1585i		−	1.87979i	−0.341462	−	0.939896i	$-0.610922\pi$
$7$		−	13.1585i		−	1.87979i	0.341462	−	0.939896i	$-0.389078\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.18499	−	10.5288i	0.289545	−	0.957165i
$11$	3.18499	−	10.5288i	0.289545	−	0.957165i
$12$		0			0
$13$			14.6396i			1.12612i	0.826415	+	0.563061i	$0.190376\pi$
$13$			14.6396i			1.12612i	−0.826415	+	0.563061i	$0.809624\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	9.90801i		−	0.582824i	−0.956598	−	0.291412i	$-0.905875\pi$
$17$		−	9.90801i		−	0.582824i	0.956598	−	0.291412i	$-0.0941251\pi$
$18$		0			0
$19$			11.5333i			0.607015i	0.952829	+	0.303507i	$0.0981577\pi$
$19$			11.5333i			0.607015i	−0.952829	+	0.303507i	$0.901842\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	14.6044			0.634974			0.317487	−	0.948263i	$-0.397161\pi$
$23$	14.6044			0.634974			0.317487	+	0.948263i	$0.397161\pi$
$24$		0			0
$25$	5.00000			0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	8.28275i		−	0.285612i	−0.989751	−	0.142806i	$-0.954387\pi$
$29$		−	8.28275i		−	0.285612i	0.989751	−	0.142806i	$-0.0456125\pi$
$30$		0			0
$31$	53.2285			1.71705			0.858524	−	0.512773i	$-0.171382\pi$
$31$	53.2285			1.71705			0.858524	+	0.512773i	$0.171382\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	29.4234i		−	0.840668i
$36$		0			0
$37$	41.1146			1.11121			0.555603	−	0.831448i	$-0.312488\pi$
$37$	41.1146			1.11121			0.555603	+	0.831448i	$0.312488\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	74.6449i		−	1.82061i	−0.413940	−	0.910304i	$-0.635848\pi$
$41$		−	74.6449i		−	1.82061i	0.413940	−	0.910304i	$-0.364152\pi$
$42$		0			0
$43$		−	30.4377i		−	0.707855i	−0.935273	−	0.353927i	$-0.884846\pi$
$43$		−	30.4377i		−	0.707855i	0.935273	−	0.353927i	$-0.115154\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−0.697244			−0.0148350			−0.00741749	−	0.999972i	$-0.502361\pi$
$47$	−0.697244			−0.0148350			−0.00741749	+	0.999972i	$0.502361\pi$
$48$		0			0
$49$	−124.147			−2.53361
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−17.0571			−0.321831			−0.160916	−	0.986968i	$-0.551445\pi$
$53$	−17.0571			−0.321831			−0.160916	+	0.986968i	$0.551445\pi$
$54$		0			0
$55$	7.12186	−	23.5431i	0.129488	−	0.428057i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	6.29242			0.106651			0.0533256	−	0.998577i	$-0.483018\pi$
$59$	6.29242			0.106651			0.0533256	+	0.998577i	$0.483018\pi$
$60$		0			0
$61$			50.6864i			0.830925i	0.909610	+	0.415462i	$0.136380\pi$
$61$			50.6864i			0.830925i	−0.909610	+	0.415462i	$0.863620\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			32.7351i			0.503617i
$66$		0			0
$67$	−81.8378			−1.22146			−0.610730	−	0.791839i	$-0.709124\pi$
$67$	−81.8378			−1.22146			−0.610730	+	0.791839i	$0.709124\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	62.4365			0.879387			0.439694	−	0.898148i	$-0.355087\pi$
$71$	62.4365			0.879387			0.439694	+	0.898148i	$0.355087\pi$
$72$		0			0
$73$			11.5551i			0.158289i	0.996863	+	0.0791445i	$0.0252188\pi$
$73$			11.5551i			0.158289i	−0.996863	+	0.0791445i	$0.974781\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−138.544	−	41.9098i	−1.79927	−	0.544283i
$78$		0			0
$79$			79.2397i			1.00303i	0.865148	+	0.501517i	$0.167224\pi$
$79$			79.2397i			1.00303i	−0.865148	+	0.501517i	$0.832776\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	81.8105i		−	0.985669i	−0.870123	−	0.492834i	$-0.835961\pi$
$83$		−	81.8105i		−	0.985669i	0.870123	−	0.492834i	$-0.164039\pi$
$84$		0			0
$85$		−	22.1550i		−	0.260647i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−136.935			−1.53860			−0.769300	−	0.638888i	$-0.779395\pi$
$89$	−136.935			−1.53860			−0.769300	+	0.638888i	$0.779395\pi$
$90$		0			0
$91$	192.636			2.11687
$92$		0			0
$93$		0			0
$94$		0			0
$95$			25.7892i			0.271465i
$96$		0			0
$97$	−80.2686			−0.827511			−0.413756	−	0.910388i	$-0.635783\pi$
$97$	−80.2686			−0.827511			−0.413756	+	0.910388i	$0.635783\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			19.3173i			0.191261i	0.995417	+	0.0956303i	$0.0304867\pi$
$101$			19.3173i			0.191261i	−0.995417	+	0.0956303i	$0.969513\pi$
$102$		0			0
$103$	−71.4905			−0.694082			−0.347041	−	0.937850i	$-0.612814\pi$
$103$	−71.4905			−0.694082			−0.347041	+	0.937850i	$0.612814\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	148.254i		−	1.38555i	−0.721154	−	0.692774i	$-0.756388\pi$
$107$		−	148.254i		−	1.38555i	0.721154	−	0.692774i	$-0.243612\pi$
$108$		0			0
$109$		−	201.904i		−	1.85233i	−0.377116	−	0.926166i	$-0.623084\pi$
$109$		−	201.904i		−	1.85233i	0.377116	−	0.926166i	$-0.376916\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−179.659			−1.58990			−0.794951	−	0.606673i	$-0.792504\pi$
$113$	−179.659			−1.58990			−0.794951	+	0.606673i	$0.792504\pi$
$114$		0			0
$115$	32.6564			0.283969
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−130.375			−1.09559
$120$		0			0
$121$	−100.712	−	67.0683i	−0.832328	−	0.554284i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	11.1803			0.0894427
$126$		0			0
$127$			173.681i			1.36757i	0.729685	+	0.683784i	$0.239667\pi$
$127$			173.681i			1.36757i	−0.729685	+	0.683784i	$0.760333\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	99.4932i		−	0.759490i	−0.925091	−	0.379745i	$-0.876012\pi$
$131$		−	99.4932i		−	0.759490i	0.925091	−	0.379745i	$-0.123988\pi$
$132$		0			0
$133$	151.761			1.14106
$134$		0			0
$135$		0			0
$136$		0			0
$137$	155.034			1.13163			0.565816	−	0.824531i	$-0.308561\pi$
$137$	155.034			1.13163			0.565816	+	0.824531i	$0.308561\pi$
$138$		0			0
$139$			62.7620i			0.451525i	0.974182	+	0.225763i	$0.0724874\pi$
$139$			62.7620i			0.451525i	−0.974182	+	0.225763i	$0.927513\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	154.137	+	46.6269i	1.07788	+	0.326063i
$144$		0			0
$145$		−	18.5208i		−	0.127730i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			137.649i			0.923818i	0.886927	+	0.461909i	$0.152835\pi$
$149$			137.649i			0.923818i	−0.886927	+	0.461909i	$0.847165\pi$
$150$		0			0
$151$		−	229.174i		−	1.51771i	−0.651259	−	0.758856i	$-0.725759\pi$
$151$		−	229.174i		−	1.51771i	0.651259	−	0.758856i	$-0.274241\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	119.023			0.767887
$156$		0			0
$157$	−137.708			−0.877118			−0.438559	−	0.898702i	$-0.644511\pi$
$157$	−137.708			−0.877118			−0.438559	+	0.898702i	$0.644511\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	192.172i		−	1.19362i
$162$		0			0
$163$	−38.9324			−0.238849			−0.119425	−	0.992843i	$-0.538105\pi$
$163$	−38.9324			−0.238849			−0.119425	+	0.992843i	$0.538105\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	165.244i		−	0.989488i	−0.869039	−	0.494744i	$-0.835262\pi$
$167$		−	165.244i		−	0.989488i	0.869039	−	0.494744i	$-0.164738\pi$
$168$		0			0
$169$	−45.3175			−0.268151
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	297.612i		−	1.72030i	−0.510042	−	0.860149i	$-0.670370\pi$
$173$		−	297.612i		−	1.72030i	0.510042	−	0.860149i	$-0.329630\pi$
$174$		0			0
$175$		−	65.7927i		−	0.375958i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	208.567			1.16518			0.582588	−	0.812767i	$-0.302040\pi$
$179$	208.567			1.16518			0.582588	+	0.812767i	$0.302040\pi$
$180$		0			0
$181$	234.341			1.29470			0.647351	−	0.762192i	$-0.275877\pi$
$181$	234.341			1.29470			0.647351	+	0.762192i	$0.275877\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	91.9350			0.496946
$186$		0			0
$187$	−104.320	−	31.5569i	−0.557859	−	0.168754i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−186.250			−0.975128			−0.487564	−	0.873087i	$-0.662115\pi$
$191$	−186.250			−0.975128			−0.487564	+	0.873087i	$0.662115\pi$
$192$		0			0
$193$			144.553i			0.748982i	0.927231	+	0.374491i	$0.122182\pi$
$193$			144.553i			0.748982i	−0.927231	+	0.374491i	$0.877818\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	159.129i		−	0.807762i	−0.914812	−	0.403881i	$-0.867661\pi$
$197$		−	159.129i		−	0.807762i	0.914812	−	0.403881i	$-0.132339\pi$
$198$		0			0
$199$	208.591			1.04820			0.524098	−	0.851658i	$-0.324402\pi$
$199$	208.591			1.04820			0.524098	+	0.851658i	$0.324402\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−108.989			−0.536891
$204$		0			0
$205$		−	166.911i		−	0.814201i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	121.432	+	36.7334i	0.581013	+	0.175758i
$210$		0			0
$211$			249.692i			1.18337i	0.806168	+	0.591687i	$0.201538\pi$
$211$			249.692i			1.18337i	−0.806168	+	0.591687i	$0.798462\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	68.0609i		−	0.316562i
$216$		0			0
$217$		−	700.409i		−	3.22769i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	145.049			0.656331
$222$		0			0
$223$	−94.8922			−0.425526			−0.212763	−	0.977104i	$-0.568246\pi$
$223$	−94.8922			−0.425526			−0.212763	+	0.977104i	$0.568246\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			365.237i			1.60897i	0.593970	+	0.804487i	$0.297560\pi$
$227$			365.237i			1.60897i	−0.593970	+	0.804487i	$0.702440\pi$
$228$		0			0
$229$	−181.047			−0.790600			−0.395300	−	0.918552i	$-0.629359\pi$
$229$	−181.047			−0.790600			−0.395300	+	0.918552i	$0.629359\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			291.907i			1.25282i	0.779494	+	0.626410i	$0.215476\pi$
$233$			291.907i			1.25282i	−0.779494	+	0.626410i	$0.784524\pi$
$234$		0			0
$235$	−1.55909			−0.00663441
$236$		0			0
$237$		0			0
$238$		0			0
$239$			43.0882i			0.180285i	0.995929	+	0.0901427i	$0.0287323\pi$
$239$			43.0882i			0.180285i	−0.995929	+	0.0901427i	$0.971268\pi$
$240$		0			0
$241$		−	77.0546i		−	0.319729i	−0.987139	−	0.159864i	$-0.948894\pi$
$241$		−	77.0546i		−	0.319729i	0.987139	−	0.159864i	$-0.0511057\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−277.601			−1.13307
$246$		0			0
$247$	−168.842			−0.683572
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−7.50457			−0.0298987			−0.0149494	−	0.999888i	$-0.504759\pi$
$251$	−7.50457			−0.0298987			−0.0149494	+	0.999888i	$0.504759\pi$
$252$		0			0
$253$	46.5149	−	153.767i	0.183853	−	0.607774i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	86.1842			0.335347			0.167674	−	0.985843i	$-0.446375\pi$
$257$	86.1842			0.335347			0.167674	+	0.985843i	$0.446375\pi$
$258$		0			0
$259$		−	541.008i		−	2.08883i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	328.758i		−	1.25003i	−0.780612	−	0.625016i	$-0.785093\pi$
$263$		−	328.758i		−	1.25003i	0.780612	−	0.625016i	$-0.214907\pi$
$264$		0			0
$265$	−38.1408			−0.143927
$266$		0			0
$267$		0			0
$268$		0			0
$269$	358.896			1.33418			0.667092	−	0.744975i	$-0.267539\pi$
$269$	358.896			1.33418			0.667092	+	0.744975i	$0.267539\pi$
$270$		0			0
$271$			223.030i			0.822989i	0.911412	+	0.411494i	$0.134993\pi$
$271$			223.030i			0.822989i	−0.911412	+	0.411494i	$0.865007\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	15.9250	−	52.6440i	0.0579089	−	0.191433i
$276$		0			0
$277$		−	81.1615i		−	0.293002i	−0.989211	−	0.146501i	$-0.953199\pi$
$277$		−	81.1615i		−	0.293002i	0.989211	−	0.146501i	$-0.0468011\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			91.9342i			0.327168i	0.986529	+	0.163584i	$0.0523054\pi$
$281$			91.9342i			0.327168i	−0.986529	+	0.163584i	$0.947695\pi$
$282$		0			0
$283$		−	1.55872i		−	0.00550784i	−0.999996	−	0.00275392i	$-0.999123\pi$
$283$		−	1.55872i		−	0.00550784i	0.999996	−	0.00275392i	$-0.000876601\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−982.218			−3.42236
$288$		0			0
$289$	190.831			0.660316
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	361.504i		−	1.23380i	−0.787041	−	0.616900i	$-0.788388\pi$
$293$		−	361.504i		−	1.23380i	0.787041	−	0.616900i	$-0.211612\pi$
$294$		0			0
$295$	14.0703			0.0476958
$296$		0			0
$297$		0			0
$298$		0			0
$299$			213.802i			0.715058i
$300$		0			0
$301$	−400.516			−1.33062
$302$		0			0
$303$		0			0
$304$		0			0
$305$			113.338i			0.371601i
$306$		0			0
$307$			51.3776i			0.167354i	0.996493	+	0.0836769i	$0.0266664\pi$
$307$			51.3776i			0.167354i	−0.996493	+	0.0836769i	$0.973334\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−198.075			−0.636898			−0.318449	−	0.947940i	$-0.603162\pi$
$311$	−198.075			−0.636898			−0.318449	+	0.947940i	$0.603162\pi$
$312$		0			0
$313$	−22.2970			−0.0712365			−0.0356183	−	0.999365i	$-0.511340\pi$
$313$	−22.2970			−0.0712365			−0.0356183	+	0.999365i	$0.511340\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	102.256			0.322574			0.161287	−	0.986908i	$-0.448436\pi$
$317$	102.256			0.322574			0.161287	+	0.986908i	$0.448436\pi$
$318$		0			0
$319$	−87.2075	−	26.3805i	−0.273378	−	0.0826975i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	114.272			0.353783
$324$		0			0
$325$			73.1979i			0.225224i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			9.17471i			0.0278867i
$330$		0			0
$331$	−353.647			−1.06842			−0.534211	−	0.845351i	$-0.679391\pi$
$331$	−353.647			−1.06842			−0.534211	+	0.845351i	$0.679391\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−182.995			−0.546253
$336$		0			0
$337$		−	237.983i		−	0.706182i	−0.935589	−	0.353091i	$-0.885131\pi$
$337$		−	237.983i		−	0.706182i	0.935589	−	0.353091i	$-0.114869\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	169.532	−	560.433i	0.497162	−	1.64350i
$342$		0			0
$343$			988.826i			2.88288i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	142.124i		−	0.409580i	−0.978806	−	0.204790i	$-0.934349\pi$
$347$		−	142.124i		−	0.409580i	0.978806	−	0.204790i	$-0.0656512\pi$
$348$		0			0
$349$			25.4867i			0.0730278i	0.999333	+	0.0365139i	$0.0116253\pi$
$349$			25.4867i			0.0730278i	−0.999333	+	0.0365139i	$0.988375\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−155.144			−0.439502			−0.219751	−	0.975556i	$-0.570524\pi$
$353$	−155.144			−0.439502			−0.219751	+	0.975556i	$0.570524\pi$
$354$		0			0
$355$	139.612			0.393274
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	6.51482i		−	0.0181471i	−0.999959	−	0.00907357i	$-0.997112\pi$
$359$		−	6.51482i		−	0.0181471i	0.999959	−	0.00907357i	$-0.00288825\pi$
$360$		0			0
$361$	227.984			0.631533
$362$		0			0
$363$		0			0
$364$		0			0
$365$			25.8380i			0.0707890i
$366$		0			0
$367$	102.889			0.280353			0.140176	−	0.990127i	$-0.455233\pi$
$367$	102.889			0.280353			0.140176	+	0.990127i	$0.455233\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			224.446i			0.604976i
$372$		0			0
$373$		−	119.790i		−	0.321152i	−0.987024	−	0.160576i	$-0.948665\pi$
$373$		−	119.790i		−	0.321152i	0.987024	−	0.160576i	$-0.0513351\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	121.256			0.321634
$378$		0			0
$379$	175.693			0.463570			0.231785	−	0.972767i	$-0.425543\pi$
$379$	175.693			0.463570			0.231785	+	0.972767i	$0.425543\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−225.422			−0.588570			−0.294285	−	0.955718i	$-0.595082\pi$
$383$	−225.422			−0.588570			−0.294285	+	0.955718i	$0.595082\pi$
$384$		0			0
$385$	−309.793	−	93.7132i	−0.804658	−	0.243411i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−91.2920			−0.234684			−0.117342	−	0.993092i	$-0.537437\pi$
$389$	−91.2920			−0.234684			−0.117342	+	0.993092i	$0.537437\pi$
$390$		0			0
$391$		−	144.701i		−	0.370078i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			177.185i			0.448570i
$396$		0			0
$397$	602.789			1.51836			0.759181	−	0.650880i	$-0.225600\pi$
$397$	602.789			1.51836			0.759181	+	0.650880i	$0.225600\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−277.075			−0.690961			−0.345480	−	0.938426i	$-0.612284\pi$
$401$	−277.075			−0.690961			−0.345480	+	0.938426i	$0.612284\pi$
$402$		0			0
$403$			779.243i			1.93361i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	130.950	−	432.888i	0.321744	−	1.06361i
$408$		0			0
$409$		−	165.156i		−	0.403804i	−0.979406	−	0.201902i	$-0.935288\pi$
$409$		−	165.156i		−	0.403804i	0.979406	−	0.201902i	$-0.0647122\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	82.7990i		−	0.200482i
$414$		0			0
$415$		−	182.934i		−	0.440805i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	619.744			1.47910			0.739551	−	0.673100i	$-0.235038\pi$
$419$	619.744			1.47910			0.739551	+	0.673100i	$0.235038\pi$
$420$		0			0
$421$	232.190			0.551520			0.275760	−	0.961226i	$-0.411070\pi$
$421$	232.190			0.551520			0.275760	+	0.961226i	$0.411070\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	49.5401i		−	0.116565i
$426$		0			0
$427$	666.959			1.56197
$428$		0			0
$429$		0			0
$430$		0			0
$431$			606.608i			1.40744i	0.710476	+	0.703721i	$0.248480\pi$
$431$			606.608i			1.40744i	−0.710476	+	0.703721i	$0.751520\pi$
$432$		0			0
$433$	85.1841			0.196730			0.0983650	−	0.995150i	$-0.468639\pi$
$433$	85.1841			0.196730			0.0983650	+	0.995150i	$0.468639\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			168.436i			0.385438i
$438$		0			0
$439$			559.497i			1.27448i	0.770665	+	0.637240i	$0.219924\pi$
$439$			559.497i			1.27448i	−0.770665	+	0.637240i	$0.780076\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	171.266			0.386606			0.193303	−	0.981139i	$-0.438080\pi$
$443$	171.266			0.386606			0.193303	+	0.981139i	$0.438080\pi$
$444$		0			0
$445$	−306.197			−0.688083
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−701.180			−1.56165			−0.780824	−	0.624751i	$-0.785200\pi$
$449$	−701.180			−1.56165			−0.780824	+	0.624751i	$0.785200\pi$
$450$		0			0
$451$	−785.922	−	237.743i	−1.74262	−	0.527147i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	430.746			0.946695
$456$		0			0
$457$			94.6651i			0.207145i	0.994622	+	0.103572i	$0.0330273\pi$
$457$			94.6651i			0.207145i	−0.994622	+	0.103572i	$0.966973\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			386.694i			0.838815i	0.907798	+	0.419408i	$0.137762\pi$
$461$			386.694i			0.838815i	−0.907798	+	0.419408i	$0.862238\pi$
$462$		0			0
$463$	364.997			0.788330			0.394165	−	0.919040i	$-0.371034\pi$
$463$	364.997			0.788330			0.394165	+	0.919040i	$0.371034\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	719.317			1.54029			0.770147	−	0.637867i	$-0.220183\pi$
$467$	719.317			1.54029			0.770147	+	0.637867i	$0.220183\pi$
$468$		0			0
$469$			1076.87i			2.29609i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−320.473	−	96.9440i	−0.677533	−	0.204955i
$474$		0			0
$475$			57.6664i			0.121403i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			373.191i			0.779104i	0.921004	+	0.389552i	$0.127370\pi$
$479$			373.191i			0.779104i	−0.921004	+	0.389552i	$0.872630\pi$
$480$		0			0
$481$			601.901i			1.25135i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−179.486			−0.370074
$486$		0			0
$487$	−480.297			−0.986237			−0.493118	−	0.869962i	$-0.664143\pi$
$487$	−480.297			−0.986237			−0.493118	+	0.869962i	$0.664143\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	107.055i		−	0.218035i	−0.994040	−	0.109017i	$-0.965230\pi$
$491$		−	107.055i		−	0.218035i	0.994040	−	0.109017i	$-0.0347704\pi$
$492$		0			0
$493$	−82.0656			−0.166462
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	821.573i		−	1.65306i
$498$		0			0
$499$	296.804			0.594798			0.297399	−	0.954753i	$-0.403881\pi$
$499$	296.804			0.594798			0.297399	+	0.954753i	$0.403881\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	538.023i		−	1.06963i	−0.844969	−	0.534815i	$-0.820381\pi$
$503$		−	538.023i		−	1.06963i	0.844969	−	0.534815i	$-0.179619\pi$
$504$		0			0
$505$			43.1948i			0.0855344i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	383.073			0.752599			0.376299	−	0.926498i	$-0.377196\pi$
$509$	383.073			0.752599			0.376299	+	0.926498i	$0.377196\pi$
$510$		0			0
$511$	152.048			0.297550
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−159.858			−0.310403
$516$		0			0
$517$	−2.22072	+	7.34115i	−0.00429539	+	0.0141995i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	484.670			0.930270			0.465135	−	0.885240i	$-0.346006\pi$
$521$	484.670			0.930270			0.465135	+	0.885240i	$0.346006\pi$
$522$		0			0
$523$		−	481.382i		−	0.920424i	−0.887809	−	0.460212i	$-0.847773\pi$
$523$		−	481.382i		−	0.920424i	0.887809	−	0.460212i	$-0.152227\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	527.389i		−	1.00074i
$528$		0			0
$529$	−315.712			−0.596809
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1092.77			2.05023
$534$		0			0
$535$		−	331.505i		−	0.619636i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−395.407	+	1307.12i	−0.733595	+	2.42509i
$540$		0			0
$541$			534.638i			0.988241i	0.869393	+	0.494121i	$0.164510\pi$
$541$			534.638i			0.988241i	−0.869393	+	0.494121i	$0.835490\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	451.472i		−	0.828388i
$546$		0			0
$547$		−	601.686i		−	1.09997i	−0.835173	−	0.549987i	$-0.814633\pi$
$547$		−	601.686i		−	1.09997i	0.835173	−	0.549987i	$-0.185367\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	95.5273			0.173371
$552$		0			0
$553$	1042.68			1.88549
$554$		0			0
$555$		0			0
$556$		0			0
$557$			802.187i			1.44019i	0.693874	+	0.720096i	$0.255902\pi$
$557$			802.187i			1.44019i	−0.693874	+	0.720096i	$0.744098\pi$
$558$		0			0
$559$	445.596			0.797131
$560$		0			0
$561$		0			0
$562$		0			0
$563$			729.072i			1.29498i	0.762075	+	0.647489i	$0.224181\pi$
$563$			729.072i			1.29498i	−0.762075	+	0.647489i	$0.775819\pi$
$564$		0			0
$565$	−401.730			−0.711026
$566$		0			0
$567$		0			0
$568$		0			0
$569$			629.473i			1.10628i	0.833089	+	0.553140i	$0.186570\pi$
$569$			629.473i			1.10628i	−0.833089	+	0.553140i	$0.813430\pi$
$570$		0			0
$571$			615.487i			1.07791i	0.842335	+	0.538955i	$0.181181\pi$
$571$			615.487i			1.07791i	−0.842335	+	0.538955i	$0.818819\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	73.0220			0.126995
$576$		0			0
$577$	−162.104			−0.280942			−0.140471	−	0.990085i	$-0.544862\pi$
$577$	−162.104			−0.280942			−0.140471	+	0.990085i	$0.544862\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−1076.51			−1.85285
$582$		0			0
$583$	−54.3266	+	179.591i	−0.0931845	+	0.308046i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	661.750			1.12734			0.563671	−	0.825999i	$-0.309389\pi$
$587$	661.750			1.12734			0.563671	+	0.825999i	$0.309389\pi$
$588$		0			0
$589$			613.899i			1.04227i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	77.0827i		−	0.129988i	−0.997886	−	0.0649939i	$-0.979297\pi$
$593$		−	77.0827i		−	0.129988i	0.997886	−	0.0649939i	$-0.0207028\pi$
$594$		0			0
$595$	−291.527			−0.489962
$596$		0			0
$597$		0			0
$598$		0			0
$599$	414.443			0.691891			0.345945	−	0.938255i	$-0.387558\pi$
$599$	414.443			0.691891			0.345945	+	0.938255i	$0.387558\pi$
$600$		0			0
$601$		−	258.961i		−	0.430883i	−0.976517	−	0.215441i	$-0.930881\pi$
$601$		−	258.961i		−	0.430883i	0.976517	−	0.215441i	$-0.0691191\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−225.198	−	149.969i	−0.372228	−	0.247883i
$606$		0			0
$607$		−	447.958i		−	0.737986i	−0.929432	−	0.368993i	$-0.879703\pi$
$607$		−	447.958i		−	0.737986i	0.929432	−	0.368993i	$-0.120297\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	10.2074i		−	0.0167060i
$612$		0			0
$613$			762.804i			1.24438i	0.782867	+	0.622190i	$0.213757\pi$
$613$			762.804i			1.24438i	−0.782867	+	0.622190i	$0.786243\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	531.407			0.861276			0.430638	−	0.902525i	$-0.358289\pi$
$617$	531.407			0.861276			0.430638	+	0.902525i	$0.358289\pi$
$618$		0			0
$619$	648.349			1.04741			0.523707	−	0.851898i	$-0.324549\pi$
$619$	648.349			1.04741			0.523707	+	0.851898i	$0.324549\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			1801.87i			2.89225i
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	407.364i		−	0.647638i
$630$		0			0
$631$	−500.262			−0.792808			−0.396404	−	0.918076i	$-0.629742\pi$
$631$	−500.262			−0.792808			−0.396404	+	0.918076i	$0.629742\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			388.363i			0.611595i
$636$		0			0
$637$		−	1817.46i		−	2.85316i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−195.726			−0.305345			−0.152673	−	0.988277i	$-0.548788\pi$
$641$	−195.726			−0.305345			−0.152673	+	0.988277i	$0.548788\pi$
$642$		0			0
$643$	1016.58			1.58100			0.790498	−	0.612465i	$-0.209822\pi$
$643$	1016.58			1.58100			0.790498	+	0.612465i	$0.209822\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−489.683			−0.756851			−0.378426	−	0.925632i	$-0.623534\pi$
$647$	−489.683			−0.756851			−0.378426	+	0.925632i	$0.623534\pi$
$648$		0			0
$649$	20.0413	−	66.2517i	0.0308803	−	0.102083i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−293.173			−0.448964			−0.224482	−	0.974478i	$-0.572069\pi$
$653$	−293.173			−0.448964			−0.224482	+	0.974478i	$0.572069\pi$
$654$		0			0
$655$		−	222.474i		−	0.339654i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			590.485i			0.896032i	0.894026	+	0.448016i	$0.147869\pi$
$659$			590.485i			0.896032i	−0.894026	+	0.448016i	$0.852131\pi$
$660$		0			0
$661$	43.8951			0.0664072			0.0332036	−	0.999449i	$-0.489429\pi$
$661$	43.8951			0.0664072			0.0332036	+	0.999449i	$0.489429\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	339.348			0.510298
$666$		0			0
$667$		−	120.965i		−	0.181356i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	533.668	+	161.436i	0.795332	+	0.240590i
$672$		0			0
$673$		−	869.115i		−	1.29140i	−0.763590	−	0.645702i	$-0.776565\pi$
$673$		−	869.115i		−	1.29140i	0.763590	−	0.645702i	$-0.223435\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	480.512i		−	0.709766i	−0.934911	−	0.354883i	$-0.884521\pi$
$677$		−	480.512i		−	0.709766i	0.934911	−	0.354883i	$-0.115479\pi$
$678$		0			0
$679$			1056.22i			1.55555i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1111.42			−1.62726			−0.813632	−	0.581380i	$-0.802513\pi$
$683$	−1111.42			−1.62726			−0.813632	+	0.581380i	$0.802513\pi$
$684$		0			0
$685$	346.666			0.506081
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	249.708i		−	0.362421i
$690$		0			0
$691$	373.681			0.540783			0.270391	−	0.962750i	$-0.412847\pi$
$691$	373.681			0.540783			0.270391	+	0.962750i	$0.412847\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			140.340i			0.201928i
$696$		0			0
$697$	−739.583			−1.06109
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	732.819i		−	1.04539i	−0.852520	−	0.522695i	$-0.824927\pi$
$701$		−	732.819i		−	1.04539i	0.852520	−	0.522695i	$-0.175073\pi$
$702$		0			0
$703$			474.186i			0.674518i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	254.188			0.359530
$708$		0			0
$709$	−799.109			−1.12709			−0.563547	−	0.826084i	$-0.690563\pi$
$709$	−799.109			−1.12709			−0.563547	+	0.826084i	$0.690563\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	777.370			1.09028
$714$		0			0
$715$	344.662	+	104.261i	0.482044	+	0.145820i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	375.329			0.522015			0.261008	−	0.965337i	$-0.415945\pi$
$719$	375.329			0.522015			0.261008	+	0.965337i	$0.415945\pi$
$720$		0			0
$721$			940.710i			1.30473i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	41.4138i		−	0.0571224i
$726$		0			0
$727$	−21.7939			−0.0299779			−0.0149889	−	0.999888i	$-0.504771\pi$
$727$	−21.7939			−0.0299779			−0.0149889	+	0.999888i	$0.504771\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−301.578			−0.412555
$732$		0			0
$733$		−	34.7783i		−	0.0474466i	−0.999719	−	0.0237233i	$-0.992448\pi$
$733$		−	34.7783i		−	0.0474466i	0.999719	−	0.0237233i	$-0.00755206\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−260.653	+	861.654i	−0.353667	+	1.16914i
$738$		0			0
$739$		−	332.608i		−	0.450079i	−0.974350	−	0.225039i	$-0.927749\pi$
$739$		−	332.608i		−	0.450079i	0.974350	−	0.225039i	$-0.0722511\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			999.033i			1.34459i	0.740282	+	0.672297i	$0.234692\pi$
$743$			999.033i			1.34459i	−0.740282	+	0.672297i	$0.765308\pi$
$744$		0			0
$745$			307.792i			0.413144i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1950.80			−2.60454
$750$		0			0
$751$	−218.656			−0.291153			−0.145576	−	0.989347i	$-0.546504\pi$
$751$	−218.656			−0.291153			−0.145576	+	0.989347i	$0.546504\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	512.450i		−	0.678741i
$756$		0			0
$757$	−158.061			−0.208800			−0.104400	−	0.994535i	$-0.533292\pi$
$757$	−158.061			−0.208800			−0.104400	+	0.994535i	$0.533292\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			389.588i			0.511942i	0.966684	+	0.255971i	$0.0823952\pi$
$761$			389.588i			0.511942i	−0.966684	+	0.255971i	$0.917605\pi$
$762$		0			0
$763$	−2656.76			−3.48200
$764$		0			0
$765$		0			0
$766$		0			0
$767$			92.1184i			0.120102i
$768$		0			0
$769$			687.354i			0.893828i	0.894577	+	0.446914i	$0.147477\pi$
$769$			687.354i			0.893828i	−0.894577	+	0.446914i	$0.852523\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	934.266			1.20862			0.604312	−	0.796748i	$-0.293448\pi$
$773$	934.266			1.20862			0.604312	+	0.796748i	$0.293448\pi$
$774$		0			0
$775$	266.142			0.343410
$776$		0			0
$777$		0			0
$778$		0			0
$779$	860.901			1.10514
$780$		0			0
$781$	198.860	−	657.382i	0.254622	−	0.841718i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−307.924			−0.392259
$786$		0			0
$787$		−	793.478i		−	1.00823i	−0.863636	−	0.504116i	$-0.831819\pi$
$787$		−	793.478i		−	1.00823i	0.863636	−	0.504116i	$-0.168181\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			2364.05i			2.98869i
$792$		0			0
$793$	−742.028			−0.935723
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1015.43			−1.27407			−0.637033	−	0.770837i	$-0.719838\pi$
$797$	−1015.43			−1.27407			−0.637033	+	0.770837i	$0.719838\pi$
$798$		0			0
$799$			6.90831i			0.00864619i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	121.661	+	36.8029i	0.151509	+	0.0458317i
$804$		0			0
$805$		−	429.711i		−	0.533802i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			759.906i			0.939316i	0.882849	+	0.469658i	$0.155623\pi$
$809$			759.906i			0.939316i	−0.882849	+	0.469658i	$0.844377\pi$
$810$		0			0
$811$			626.134i			0.772052i	0.922488	+	0.386026i	$0.126152\pi$
$811$			626.134i			0.772052i	−0.922488	+	0.386026i	$0.873848\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−87.0555			−0.106817
$816$		0			0
$817$	351.047			0.429678
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	677.557i		−	0.825283i	−0.910894	−	0.412641i	$-0.864606\pi$
$821$		−	677.557i		−	0.825283i	0.910894	−	0.412641i	$-0.135394\pi$
$822$		0			0
$823$	55.0463			0.0668850			0.0334425	−	0.999441i	$-0.489353\pi$
$823$	55.0463			0.0668850			0.0334425	+	0.999441i	$0.489353\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	198.508i		−	0.240034i	−0.992772	−	0.120017i	$-0.961705\pi$
$827$		−	198.508i		−	0.240034i	0.992772	−	0.120017i	$-0.0382950\pi$
$828$		0			0
$829$	−1191.24			−1.43696			−0.718481	−	0.695547i	$-0.755162\pi$
$829$	−1191.24			−1.43696			−0.718481	+	0.695547i	$0.755162\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			1230.05i			1.47665i
$834$		0			0
$835$		−	369.498i		−	0.442512i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−159.788			−0.190450			−0.0952252	−	0.995456i	$-0.530357\pi$
$839$	−159.788			−0.190450			−0.0952252	+	0.995456i	$0.530357\pi$
$840$		0			0
$841$	772.396			0.918426
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−101.333			−0.119921
$846$		0			0
$847$	−882.521	+	1325.22i	−1.04194	+	1.56460i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	600.454			0.705586
$852$		0			0
$853$			214.501i			0.251467i	0.992064	+	0.125733i	$0.0401284\pi$
$853$			214.501i			0.251467i	−0.992064	+	0.125733i	$0.959872\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	773.417i		−	0.902471i	−0.892405	−	0.451235i	$-0.850983\pi$
$857$		−	773.417i		−	0.902471i	0.892405	−	0.451235i	$-0.149017\pi$
$858$		0			0
$859$	497.526			0.579193			0.289596	−	0.957149i	$-0.406479\pi$
$859$	497.526			0.579193			0.289596	+	0.957149i	$0.406479\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−476.209			−0.551807			−0.275903	−	0.961185i	$-0.588977\pi$
$863$	−476.209			−0.551807			−0.275903	+	0.961185i	$0.588977\pi$
$864$		0			0
$865$		−	665.480i		−	0.769341i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	834.299	+	252.378i	0.960068	+	0.290423i
$870$		0			0
$871$		−	1198.07i		−	1.37551i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	147.117i		−	0.168134i
$876$		0			0
$877$			329.549i			0.375768i	0.982191	+	0.187884i	$0.0601629\pi$
$877$			329.549i			0.375768i	−0.982191	+	0.187884i	$0.939837\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−4.25551			−0.00483031			−0.00241516	−	0.999997i	$-0.500769\pi$
$881$	−4.25551			−0.00483031			−0.00241516	+	0.999997i	$0.500769\pi$
$882$		0			0
$883$	288.392			0.326605			0.163302	−	0.986576i	$-0.447785\pi$
$883$	288.392			0.326605			0.163302	+	0.986576i	$0.447785\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			437.564i			0.493308i	0.969104	+	0.246654i	$0.0793312\pi$
$887$			437.564i			0.493308i	−0.969104	+	0.246654i	$0.920669\pi$
$888$		0			0
$889$	2285.39			2.57074
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	8.04151i		−	0.00900505i
$894$		0			0
$895$	466.369			0.521083
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	440.878i		−	0.490410i
$900$		0			0
$901$			169.002i			0.187571i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	524.003			0.579009
$906$		0			0
$907$	1071.31			1.18115			0.590576	−	0.806982i	$-0.298900\pi$
$907$	1071.31			1.18115			0.590576	+	0.806982i	$0.298900\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	503.288			0.552457			0.276228	−	0.961092i	$-0.410915\pi$
$911$	503.288			0.552457			0.276228	+	0.961092i	$0.410915\pi$
$912$		0			0
$913$	−861.367	−	260.566i	−0.943447	−	0.285395i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1309.19			−1.42768
$918$		0			0
$919$			1781.27i			1.93827i	0.246534	+	0.969134i	$0.420708\pi$
$919$			1781.27i			1.93827i	−0.246534	+	0.969134i	$0.579292\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			914.045i			0.990297i
$924$		0			0
$925$	205.573			0.222241
$926$		0			0
$927$		0			0
$928$		0			0
$929$	879.394			0.946603			0.473301	−	0.880901i	$-0.343062\pi$
$929$	879.394			0.946603			0.473301	+	0.880901i	$0.343062\pi$
$930$		0			0
$931$		−	1431.82i		−	1.53794i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−233.266	−	70.5635i	−0.249482	−	0.0754689i
$936$		0			0
$937$		−	446.198i		−	0.476198i	−0.971241	−	0.238099i	$-0.923476\pi$
$937$		−	446.198i		−	0.476198i	0.971241	−	0.238099i	$-0.0765243\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			245.275i			0.260653i	0.991471	+	0.130327i	$0.0416026\pi$
$941$			245.275i			0.260653i	−0.991471	+	0.130327i	$0.958397\pi$
$942$		0			0
$943$		−	1090.14i		−	1.15604i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	236.400			0.249631			0.124815	−	0.992180i	$-0.460166\pi$
$947$	236.400			0.249631			0.124815	+	0.992180i	$0.460166\pi$
$948$		0			0
$949$	−169.162			−0.178253
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	1049.09i		−	1.10083i	−0.834892	−	0.550414i	$-0.814470\pi$
$953$		−	1049.09i		−	1.10083i	0.834892	−	0.550414i	$-0.185530\pi$
$954$		0			0
$955$	−416.467			−0.436091
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	2040.02i		−	2.12723i
$960$		0			0
$961$	1872.27			1.94825
$962$		0			0
$963$		0			0
$964$		0			0
$965$			323.231i			0.334955i
$966$		0			0
$967$		−	523.118i		−	0.540970i	−0.962724	−	0.270485i	$-0.912816\pi$
$967$		−	523.118i		−	0.540970i	0.962724	−	0.270485i	$-0.0871840\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1692.15			1.74268			0.871341	−	0.490677i	$-0.163251\pi$
$971$	1692.15			1.74268			0.871341	+	0.490677i	$0.163251\pi$
$972$		0			0
$973$	825.857			0.848773
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−998.995			−1.02251			−0.511257	−	0.859428i	$-0.670820\pi$
$977$	−998.995			−1.02251			−0.511257	+	0.859428i	$0.670820\pi$
$978$		0			0
$979$	−436.138	+	1441.77i	−0.445493	+	1.47269i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1768.79			1.79938			0.899691	−	0.436528i	$-0.143792\pi$
$983$	1768.79			1.79938			0.899691	+	0.436528i	$0.143792\pi$
$984$		0			0
$985$		−	355.824i		−	0.361242i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	444.525i		−	0.449469i
$990$		0			0
$991$	854.135			0.861892			0.430946	−	0.902378i	$-0.358180\pi$
$991$	854.135			0.861892			0.430946	+	0.902378i	$0.358180\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	466.424			0.468768
$996$		0			0
$997$		−	1362.72i		−	1.36682i	−0.730036	−	0.683408i	$-0.760497\pi$
$997$		−	1362.72i		−	1.36682i	0.730036	−	0.683408i	$-0.239503\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1980.3.b.a.901.5		8
3.2	odd	2		220.3.f.a.21.1	✓	8
11.10	odd	2	inner	1980.3.b.a.901.8		8
12.11	even	2		880.3.j.b.241.8		8
15.2	even	4		1100.3.e.b.549.16		16
15.8	even	4		1100.3.e.b.549.1		16
15.14	odd	2		1100.3.f.f.901.8		8
33.32	even	2		220.3.f.a.21.2	yes	8
132.131	odd	2		880.3.j.b.241.7		8
165.32	odd	4		1100.3.e.b.549.15		16
165.98	odd	4		1100.3.e.b.549.2		16
165.164	even	2		1100.3.f.f.901.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
220.3.f.a.21.1	✓	8	3.2	odd	2
220.3.f.a.21.2	yes	8	33.32	even	2
880.3.j.b.241.7		8	132.131	odd	2
880.3.j.b.241.8		8	12.11	even	2
1100.3.e.b.549.1		16	15.8	even	4
1100.3.e.b.549.2		16	165.98	odd	4
1100.3.e.b.549.15		16	165.32	odd	4
1100.3.e.b.549.16		16	15.2	even	4
1100.3.f.f.901.7		8	165.164	even	2
1100.3.f.f.901.8		8	15.14	odd	2
1980.3.b.a.901.5		8	1.1	even	1	trivial
1980.3.b.a.901.8		8	11.10	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1980 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1980.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.23607 q^{5} -13.1585i q^{7} +(3.18499 - 10.5288i) q^{11} +14.6396i q^{13} -9.90801i q^{17} +11.5333i q^{19} +14.6044 q^{23} +5.00000 q^{25} -8.28275i q^{29} +53.2285 q^{31} -29.4234i q^{35} +41.1146 q^{37} -74.6449i q^{41} -30.4377i q^{43} -0.697244 q^{47} -124.147 q^{49} -17.0571 q^{53} +(7.12186 - 23.5431i) q^{55} +6.29242 q^{59} +50.6864i q^{61} +32.7351i q^{65} -81.8378 q^{67} +62.4365 q^{71} +11.5551i q^{73} +(-138.544 - 41.9098i) q^{77} +79.2397i q^{79} -81.8105i q^{83} -22.1550i q^{85} -136.935 q^{89} +192.636 q^{91} +25.7892i q^{95} -80.2686 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 24 q^{11} - 56 q^{23} + 40 q^{25} + 20 q^{31} + 72 q^{37} + 184 q^{47} - 244 q^{49} - 136 q^{53} + 60 q^{55} + 16 q^{59} - 264 q^{67} + 220 q^{71} - 208 q^{77} - 220 q^{89} - 72 q^{91} - 8 q^{97}+O(q^{100}) \) 8 * q - 24 * q^11 - 56 * q^23 + 40 * q^25 + 20 * q^31 + 72 * q^37 + 184 * q^47 - 244 * q^49 - 136 * q^53 + 60 * q^55 + 16 * q^59 - 264 * q^67 + 220 * q^71 - 208 * q^77 - 220 * q^89 - 72 * q^91 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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