LMFDB - Embedded newform 6660.2.a.u.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6660,2,Mod(1,6660)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6660, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6660.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 6660 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6660.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,0,0,-5,0,0,0,0,0,4,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$53.1803677462$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.2240944.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 11x^{3} + 17x^{2} - x - 3 $ x^5 - x^4 - 11x^3 + 17x^2 - x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.85539$ of defining polynomial
Character	$\chi$	$=$	6660.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +2.46535 q^{7} +3.32875 q^{11} +6.26208 q^{13} +5.40669 q^{17} -2.30409 q^{19} +4.01665 q^{23} +1.00000 q^{25} +7.59393 q^{29} +0.640850 q^{31} -2.46535 q^{35} -1.00000 q^{37} +6.34803 q^{41} -7.47662 q^{43} -2.88269 q^{47} -0.922059 q^{49} +12.3436 q^{53} -3.32875 q^{55} -3.57418 q^{59} +4.62483 q^{61} -6.26208 q^{65} -9.89539 q^{67} +3.79410 q^{71} -12.9222 q^{73} +8.20653 q^{77} +6.14883 q^{79} -7.49686 q^{83} -5.40669 q^{85} -3.75519 q^{89} +15.4382 q^{91} +2.30409 q^{95} -13.3187 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{5} + 4 q^{11} - q^{13} + 8 q^{17} - 4 q^{19} + 2 q^{23} + 5 q^{25} + 13 q^{29} - 4 q^{31} - 5 q^{37} + 6 q^{41} + q^{43} - q^{47} - 11 q^{49} + 15 q^{53} - 4 q^{55} + 9 q^{59} - 10 q^{61}+ \cdots + 8 q^{97}+O(q^{100}) $ 5 * q - 5 * q^5 + 4 * q^11 - q^13 + 8 * q^17 - 4 * q^19 + 2 * q^23 + 5 * q^25 + 13 * q^29 - 4 * q^31 - 5 * q^37 + 6 * q^41 + q^43 - q^47 - 11 * q^49 + 15 * q^53 - 4 * q^55 + 9 * q^59 - 10 * q^61 + q^65 - 12 * q^67 - 6 * q^71 - 12 * q^73 + 22 * q^77 - 8 * q^79 + 27 * q^83 - 8 * q^85 + 11 * q^89 + 6 * q^91 + 4 * q^95 + 8 * q^97

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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.46535	0.931814	0.465907	−	0.884834i	$-0.345728\pi$
$7$	2.46535	0.931814	0.465907	+	0.884834i	$0.345728\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.32875	1.00366	0.501828	−	0.864967i	$-0.332661\pi$
$11$	3.32875	1.00366	0.501828	+	0.864967i	$0.332661\pi$
$12$	0	0
$13$	6.26208	1.73679	0.868394	−	0.495874i	$-0.165152\pi$
$13$	6.26208	1.73679	0.868394	+	0.495874i	$0.165152\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.40669	1.31132	0.655658	−	0.755058i	$-0.272392\pi$
$17$	5.40669	1.31132	0.655658	+	0.755058i	$0.272392\pi$
$18$	0	0
$19$	−2.30409	−0.528594	−0.264297	−	0.964441i	$-0.585140\pi$
$19$	−2.30409	−0.528594	−0.264297	+	0.964441i	$0.585140\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.01665	0.837529	0.418765	−	0.908095i	$-0.362463\pi$
$23$	4.01665	0.837529	0.418765	+	0.908095i	$0.362463\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.59393	1.41016	0.705079	−	0.709129i	$-0.250911\pi$
$29$	7.59393	1.41016	0.705079	+	0.709129i	$0.250911\pi$
$30$	0	0
$31$	0.640850	0.115100	0.0575500	−	0.998343i	$-0.481671\pi$
$31$	0.640850	0.115100	0.0575500	+	0.998343i	$0.481671\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.46535	−0.416720
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.34803	0.991396	0.495698	−	0.868495i	$-0.334912\pi$
$41$	6.34803	0.991396	0.495698	+	0.868495i	$0.334912\pi$
$42$	0	0
$43$	−7.47662	−1.14017	−0.570087	−	0.821584i	$-0.693090\pi$
$43$	−7.47662	−1.14017	−0.570087	+	0.821584i	$0.693090\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.88269	−0.420483	−0.210241	−	0.977649i	$-0.567425\pi$
$47$	−2.88269	−0.420483	−0.210241	+	0.977649i	$0.567425\pi$
$48$	0	0
$49$	−0.922059	−0.131723
$50$	0	0
$51$	0	0
$52$	0	0
$53$	12.3436	1.69553	0.847763	−	0.530375i	$-0.177949\pi$
$53$	12.3436	1.69553	0.847763	+	0.530375i	$0.177949\pi$
$54$	0	0
$55$	−3.32875	−0.448849
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−3.57418	−0.465319	−0.232659	−	0.972558i	$-0.574743\pi$
$59$	−3.57418	−0.465319	−0.232659	+	0.972558i	$0.574743\pi$
$60$	0	0
$61$	4.62483	0.592148	0.296074	−	0.955165i	$-0.404322\pi$
$61$	4.62483	0.592148	0.296074	+	0.955165i	$0.404322\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−6.26208	−0.776715
$66$	0	0
$67$	−9.89539	−1.20891	−0.604457	−	0.796638i	$-0.706610\pi$
$67$	−9.89539	−1.20891	−0.604457	+	0.796638i	$0.706610\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3.79410	0.450277	0.225138	−	0.974327i	$-0.427717\pi$
$71$	3.79410	0.450277	0.225138	+	0.974327i	$0.427717\pi$
$72$	0	0
$73$	−12.9222	−1.51243	−0.756215	−	0.654323i	$-0.772954\pi$
$73$	−12.9222	−1.51243	−0.756215	+	0.654323i	$0.772954\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	8.20653	0.935221
$78$	0	0
$79$	6.14883	0.691797	0.345899	−	0.938272i	$-0.387574\pi$
$79$	6.14883	0.691797	0.345899	+	0.938272i	$0.387574\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−7.49686	−0.822888	−0.411444	−	0.911435i	$-0.634975\pi$
$83$	−7.49686	−0.822888	−0.411444	+	0.911435i	$0.634975\pi$
$84$	0	0
$85$	−5.40669	−0.586438
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−3.75519	−0.398050	−0.199025	−	0.979994i	$-0.563777\pi$
$89$	−3.75519	−0.398050	−0.199025	+	0.979994i	$0.563777\pi$
$90$	0	0
$91$	15.4382	1.61836
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.30409	0.236394
$96$	0	0
$97$	−13.3187	−1.35231	−0.676156	−	0.736759i	$-0.736355\pi$
$97$	−13.3187	−1.35231	−0.676156	+	0.736759i	$0.736355\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	4.70752	0.468416	0.234208	−	0.972187i	$-0.424750\pi$
$101$	4.70752	0.468416	0.234208	+	0.972187i	$0.424750\pi$
$102$	0	0
$103$	−1.98888	−0.195971	−0.0979853	−	0.995188i	$-0.531240\pi$
$103$	−1.98888	−0.195971	−0.0979853	+	0.995188i	$0.531240\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.68085	0.742536	0.371268	−	0.928526i	$-0.378923\pi$
$107$	7.68085	0.742536	0.371268	+	0.928526i	$0.378923\pi$
$108$	0	0
$109$	−3.10020	−0.296945	−0.148472	−	0.988917i	$-0.547436\pi$
$109$	−3.10020	−0.296945	−0.148472	+	0.988917i	$0.547436\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−5.40732	−0.508677	−0.254339	−	0.967115i	$-0.581858\pi$
$113$	−5.40732	−0.508677	−0.254339	+	0.967115i	$0.581858\pi$
$114$	0	0
$115$	−4.01665	−0.374554
$116$	0	0
$117$	0	0
$118$	0	0
$119$	13.3294	1.22190
$120$	0	0
$121$	0.0805750	0.00732500
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	14.1261	1.25349	0.626745	−	0.779225i	$-0.284387\pi$
$127$	14.1261	1.25349	0.626745	+	0.779225i	$0.284387\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−21.4191	−1.87139	−0.935697	−	0.352805i	$-0.885228\pi$
$131$	−21.4191	−1.87139	−0.935697	+	0.352805i	$0.885228\pi$
$132$	0	0
$133$	−5.68038	−0.492551
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−13.1297	−1.12175	−0.560873	−	0.827902i	$-0.689534\pi$
$137$	−13.1297	−1.12175	−0.560873	+	0.827902i	$0.689534\pi$
$138$	0	0
$139$	1.88005	0.159464	0.0797319	−	0.996816i	$-0.474594\pi$
$139$	1.88005	0.159464	0.0797319	+	0.996816i	$0.474594\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	20.8449	1.74314
$144$	0	0
$145$	−7.59393	−0.630642
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.36933	0.194103	0.0970514	−	0.995279i	$-0.469059\pi$
$149$	2.36933	0.194103	0.0970514	+	0.995279i	$0.469059\pi$
$150$	0	0
$151$	1.99150	0.162066	0.0810328	−	0.996711i	$-0.474178\pi$
$151$	1.99150	0.162066	0.0810328	+	0.996711i	$0.474178\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−0.640850	−0.0514743
$156$	0	0
$157$	−15.3405	−1.22431	−0.612153	−	0.790739i	$-0.709696\pi$
$157$	−15.3405	−1.22431	−0.612153	+	0.790739i	$0.709696\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	9.90244	0.780422
$162$	0	0
$163$	−1.20916	−0.0947088	−0.0473544	−	0.998878i	$-0.515079\pi$
$163$	−1.20916	−0.0947088	−0.0473544	+	0.998878i	$0.515079\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−8.10370	−0.627083	−0.313541	−	0.949575i	$-0.601515\pi$
$167$	−8.10370	−0.627083	−0.313541	+	0.949575i	$0.601515\pi$
$168$	0	0
$169$	26.2136	2.01643
$170$	0	0
$171$	0	0
$172$	0	0
$173$	7.60291	0.578038	0.289019	−	0.957323i	$-0.406671\pi$
$173$	7.60291	0.578038	0.289019	+	0.957323i	$0.406671\pi$
$174$	0	0
$175$	2.46535	0.186363
$176$	0	0
$177$	0	0
$178$	0	0
$179$	7.25358	0.542158	0.271079	−	0.962557i	$-0.412620\pi$
$179$	7.25358	0.542158	0.271079	+	0.962557i	$0.412620\pi$
$180$	0	0
$181$	4.13709	0.307507	0.153754	−	0.988109i	$-0.450864\pi$
$181$	4.13709	0.307507	0.153754	+	0.988109i	$0.450864\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.00000	0.0735215
$186$	0	0
$187$	17.9975	1.31611
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4.04049	−0.292360	−0.146180	−	0.989258i	$-0.546698\pi$
$191$	−4.04049	−0.292360	−0.146180	+	0.989258i	$0.546698\pi$
$192$	0	0
$193$	2.98384	0.214782	0.107391	−	0.994217i	$-0.465750\pi$
$193$	2.98384	0.214782	0.107391	+	0.994217i	$0.465750\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−16.0792	−1.14559	−0.572797	−	0.819697i	$-0.694142\pi$
$197$	−16.0792	−1.14559	−0.572797	+	0.819697i	$0.694142\pi$
$198$	0	0
$199$	−27.1205	−1.92252	−0.961261	−	0.275641i	$-0.911110\pi$
$199$	−27.1205	−1.92252	−0.961261	+	0.275641i	$0.911110\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	18.7217	1.31401
$204$	0	0
$205$	−6.34803	−0.443366
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−7.66973	−0.530526
$210$	0	0
$211$	27.2005	1.87256	0.936278	−	0.351259i	$-0.114246\pi$
$211$	27.2005	1.87256	0.936278	+	0.351259i	$0.114246\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	7.47662	0.509901
$216$	0	0
$217$	1.57992	0.107252
$218$	0	0
$219$	0	0
$220$	0	0
$221$	33.8571	2.27748
$222$	0	0
$223$	17.4506	1.16858	0.584291	−	0.811545i	$-0.301373\pi$
$223$	17.4506	1.16858	0.584291	+	0.811545i	$0.301373\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3.26568	−0.216750	−0.108375	−	0.994110i	$-0.534565\pi$
$227$	−3.26568	−0.216750	−0.108375	+	0.994110i	$0.534565\pi$
$228$	0	0
$229$	−18.4449	−1.21887	−0.609437	−	0.792835i	$-0.708604\pi$
$229$	−18.4449	−1.21887	−0.609437	+	0.792835i	$0.708604\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	10.5070	0.688338	0.344169	−	0.938908i	$-0.388161\pi$
$233$	10.5070	0.688338	0.344169	+	0.938908i	$0.388161\pi$
$234$	0	0
$235$	2.88269	0.188046
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−15.0594	−0.974114	−0.487057	−	0.873370i	$-0.661930\pi$
$239$	−15.0594	−0.974114	−0.487057	+	0.873370i	$0.661930\pi$
$240$	0	0
$241$	11.6903	0.753040	0.376520	−	0.926408i	$-0.377121\pi$
$241$	11.6903	0.753040	0.376520	+	0.926408i	$0.377121\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0.922059	0.0589082
$246$	0	0
$247$	−14.4284	−0.918056
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4.55311	0.287390	0.143695	−	0.989622i	$-0.454102\pi$
$251$	4.55311	0.287390	0.143695	+	0.989622i	$0.454102\pi$
$252$	0	0
$253$	13.3704	0.840591
$254$	0	0
$255$	0	0
$256$	0	0
$257$	20.0482	1.25057	0.625285	−	0.780396i	$-0.284983\pi$
$257$	20.0482	1.25057	0.625285	+	0.780396i	$0.284983\pi$
$258$	0	0
$259$	−2.46535	−0.153189
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−10.5277	−0.649168	−0.324584	−	0.945857i	$-0.605224\pi$
$263$	−10.5277	−0.649168	−0.324584	+	0.945857i	$0.605224\pi$
$264$	0	0
$265$	−12.3436	−0.758262
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−5.21695	−0.318083	−0.159041	−	0.987272i	$-0.550840\pi$
$269$	−5.21695	−0.318083	−0.159041	+	0.987272i	$0.550840\pi$
$270$	0	0
$271$	−26.2488	−1.59450	−0.797252	−	0.603647i	$-0.793714\pi$
$271$	−26.2488	−1.59450	−0.797252	+	0.603647i	$0.793714\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.32875	0.200731
$276$	0	0
$277$	9.00850	0.541269	0.270634	−	0.962682i	$-0.412767\pi$
$277$	9.00850	0.541269	0.270634	+	0.962682i	$0.412767\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	21.5331	1.28456	0.642280	−	0.766470i	$-0.277989\pi$
$281$	21.5331	1.28456	0.642280	+	0.766470i	$0.277989\pi$
$282$	0	0
$283$	4.40489	0.261843	0.130922	−	0.991393i	$-0.458206\pi$
$283$	4.40489	0.261843	0.130922	+	0.991393i	$0.458206\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	15.6501	0.923797
$288$	0	0
$289$	12.2323	0.719547
$290$	0	0
$291$	0	0
$292$	0	0
$293$	6.34395	0.370617	0.185309	−	0.982680i	$-0.440672\pi$
$293$	6.34395	0.370617	0.185309	+	0.982680i	$0.440672\pi$
$294$	0	0
$295$	3.57418	0.208097
$296$	0	0
$297$	0	0
$298$	0	0
$299$	25.1526	1.45461
$300$	0	0
$301$	−18.4325	−1.06243
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−4.62483	−0.264817
$306$	0	0
$307$	19.1121	1.09078	0.545392	−	0.838181i	$-0.316381\pi$
$307$	19.1121	1.09078	0.545392	+	0.838181i	$0.316381\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	4.23251	0.240004	0.120002	−	0.992774i	$-0.461710\pi$
$311$	4.23251	0.240004	0.120002	+	0.992774i	$0.461710\pi$
$312$	0	0
$313$	31.8396	1.79968	0.899841	−	0.436218i	$-0.143682\pi$
$313$	31.8396	1.79968	0.899841	+	0.436218i	$0.143682\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	17.7740	0.998288	0.499144	−	0.866519i	$-0.333648\pi$
$317$	17.7740	0.998288	0.499144	+	0.866519i	$0.333648\pi$
$318$	0	0
$319$	25.2783	1.41531
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−12.4575	−0.693153
$324$	0	0
$325$	6.26208	0.347358
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−7.10682	−0.391812
$330$	0	0
$331$	−3.81974	−0.209952	−0.104976	−	0.994475i	$-0.533477\pi$
$331$	−3.81974	−0.209952	−0.104976	+	0.994475i	$0.533477\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	9.89539	0.540643
$336$	0	0
$337$	−16.5372	−0.900839	−0.450420	−	0.892817i	$-0.648726\pi$
$337$	−16.5372	−0.900839	−0.450420	+	0.892817i	$0.648726\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2.13323	0.115521
$342$	0	0
$343$	−19.5306	−1.05456
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7.09779	0.381029	0.190515	−	0.981684i	$-0.438984\pi$
$347$	7.09779	0.381029	0.190515	+	0.981684i	$0.438984\pi$
$348$	0	0
$349$	−4.64505	−0.248644	−0.124322	−	0.992242i	$-0.539676\pi$
$349$	−4.64505	−0.248644	−0.124322	+	0.992242i	$0.539676\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−9.88331	−0.526036	−0.263018	−	0.964791i	$-0.584718\pi$
$353$	−9.88331	−0.526036	−0.263018	+	0.964791i	$0.584718\pi$
$354$	0	0
$355$	−3.79410	−0.201370
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2.86113	−0.151005	−0.0755023	−	0.997146i	$-0.524056\pi$
$359$	−2.86113	−0.151005	−0.0755023	+	0.997146i	$0.524056\pi$
$360$	0	0
$361$	−13.6912	−0.720588
$362$	0	0
$363$	0	0
$364$	0	0
$365$	12.9222	0.676380
$366$	0	0
$367$	−15.7173	−0.820435	−0.410217	−	0.911988i	$-0.634547\pi$
$367$	−15.7173	−0.820435	−0.410217	+	0.911988i	$0.634547\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	30.4313	1.57991
$372$	0	0
$373$	27.9584	1.44763	0.723816	−	0.689993i	$-0.242386\pi$
$373$	27.9584	1.44763	0.723816	+	0.689993i	$0.242386\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	47.5538	2.44915
$378$	0	0
$379$	−2.61344	−0.134243	−0.0671216	−	0.997745i	$-0.521382\pi$
$379$	−2.61344	−0.134243	−0.0671216	+	0.997745i	$0.521382\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	34.9795	1.78737	0.893684	−	0.448697i	$-0.148112\pi$
$383$	34.9795	1.78737	0.893684	+	0.448697i	$0.148112\pi$
$384$	0	0
$385$	−8.20653	−0.418243
$386$	0	0
$387$	0	0
$388$	0	0
$389$	9.83767	0.498790	0.249395	−	0.968402i	$-0.419768\pi$
$389$	9.83767	0.498790	0.249395	+	0.968402i	$0.419768\pi$
$390$	0	0
$391$	21.7168	1.09826
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−6.14883	−0.309381
$396$	0	0
$397$	−4.35522	−0.218582	−0.109291	−	0.994010i	$-0.534858\pi$
$397$	−4.35522	−0.218582	−0.109291	+	0.994010i	$0.534858\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−34.4785	−1.72177	−0.860887	−	0.508796i	$-0.830091\pi$
$401$	−34.4785	−1.72177	−0.860887	+	0.508796i	$0.830091\pi$
$402$	0	0
$403$	4.01305	0.199904
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−3.32875	−0.165000
$408$	0	0
$409$	−1.38942	−0.0687022	−0.0343511	−	0.999410i	$-0.510936\pi$
$409$	−1.38942	−0.0687022	−0.0343511	+	0.999410i	$0.510936\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−8.81160	−0.433590
$414$	0	0
$415$	7.49686	0.368007
$416$	0	0
$417$	0	0
$418$	0	0
$419$	21.8454	1.06722	0.533608	−	0.845732i	$-0.320836\pi$
$419$	21.8454	1.06722	0.533608	+	0.845732i	$0.320836\pi$
$420$	0	0
$421$	4.64085	0.226181	0.113091	−	0.993585i	$-0.463925\pi$
$421$	4.64085	0.226181	0.113091	+	0.993585i	$0.463925\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	5.40669	0.262263
$426$	0	0
$427$	11.4018	0.551772
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−2.75754	−0.132826	−0.0664130	−	0.997792i	$-0.521155\pi$
$431$	−2.75754	−0.132826	−0.0664130	+	0.997792i	$0.521155\pi$
$432$	0	0
$433$	38.3587	1.84340	0.921702	−	0.387899i	$-0.126799\pi$
$433$	38.3587	1.84340	0.921702	+	0.387899i	$0.126799\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−9.25471	−0.442713
$438$	0	0
$439$	−30.2457	−1.44355	−0.721774	−	0.692129i	$-0.756673\pi$
$439$	−30.2457	−1.44355	−0.721774	+	0.692129i	$0.756673\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−23.7964	−1.13060	−0.565301	−	0.824885i	$-0.691240\pi$
$443$	−23.7964	−1.13060	−0.565301	+	0.824885i	$0.691240\pi$
$444$	0	0
$445$	3.75519	0.178013
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−36.1295	−1.70506	−0.852529	−	0.522680i	$-0.824932\pi$
$449$	−36.1295	−1.70506	−0.852529	+	0.522680i	$0.824932\pi$
$450$	0	0
$451$	21.1310	0.995021
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−15.4382	−0.723754
$456$	0	0
$457$	−34.1955	−1.59960	−0.799799	−	0.600268i	$-0.795061\pi$
$457$	−34.1955	−1.59960	−0.799799	+	0.600268i	$0.795061\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	9.51405	0.443113	0.221557	−	0.975147i	$-0.428886\pi$
$461$	9.51405	0.443113	0.221557	+	0.975147i	$0.428886\pi$
$462$	0	0
$463$	−0.406339	−0.0188842	−0.00944208	−	0.999955i	$-0.503006\pi$
$463$	−0.406339	−0.0188842	−0.00944208	+	0.999955i	$0.503006\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3.13769	0.145195	0.0725974	−	0.997361i	$-0.476871\pi$
$467$	3.13769	0.145195	0.0725974	+	0.997361i	$0.476871\pi$
$468$	0	0
$469$	−24.3956	−1.12648
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−24.8878	−1.14434
$474$	0	0
$475$	−2.30409	−0.105719
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−2.55657	−0.116813	−0.0584063	−	0.998293i	$-0.518602\pi$
$479$	−2.55657	−0.116813	−0.0584063	+	0.998293i	$0.518602\pi$
$480$	0	0
$481$	−6.26208	−0.285526
$482$	0	0
$483$	0	0
$484$	0	0
$485$	13.3187	0.604772
$486$	0	0
$487$	−2.50984	−0.113732	−0.0568658	−	0.998382i	$-0.518111\pi$
$487$	−2.50984	−0.113732	−0.0568658	+	0.998382i	$0.518111\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−42.7681	−1.93010	−0.965049	−	0.262068i	$-0.915596\pi$
$491$	−42.7681	−1.93010	−0.965049	+	0.262068i	$0.915596\pi$
$492$	0	0
$493$	41.0581	1.84916
$494$	0	0
$495$	0	0
$496$	0	0
$497$	9.35377	0.419574
$498$	0	0
$499$	−0.835615	−0.0374072	−0.0187036	−	0.999825i	$-0.505954\pi$
$499$	−0.835615	−0.0374072	−0.0187036	+	0.999825i	$0.505954\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	30.5245	1.36102	0.680509	−	0.732739i	$-0.261759\pi$
$503$	30.5245	1.36102	0.680509	+	0.732739i	$0.261759\pi$
$504$	0	0
$505$	−4.70752	−0.209482
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−4.20785	−0.186510	−0.0932548	−	0.995642i	$-0.529727\pi$
$509$	−4.20785	−0.186510	−0.0932548	+	0.995642i	$0.529727\pi$
$510$	0	0
$511$	−31.8578	−1.40930
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.98888	0.0876407
$516$	0	0
$517$	−9.59574	−0.422020
$518$	0	0
$519$	0	0
$520$	0	0
$521$	7.66432	0.335780	0.167890	−	0.985806i	$-0.446305\pi$
$521$	7.66432	0.335780	0.167890	+	0.985806i	$0.446305\pi$
$522$	0	0
$523$	−32.7827	−1.43349	−0.716743	−	0.697337i	$-0.754368\pi$
$523$	−32.7827	−1.43349	−0.716743	+	0.697337i	$0.754368\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.46488	0.150932
$528$	0	0
$529$	−6.86653	−0.298545
$530$	0	0
$531$	0	0
$532$	0	0
$533$	39.7519	1.72185
$534$	0	0
$535$	−7.68085	−0.332072
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−3.06930	−0.132204
$540$	0	0
$541$	−23.6901	−1.01852	−0.509259	−	0.860613i	$-0.670080\pi$
$541$	−23.6901	−1.01852	−0.509259	+	0.860613i	$0.670080\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	3.10020	0.132798
$546$	0	0
$547$	15.0638	0.644083	0.322042	−	0.946726i	$-0.395631\pi$
$547$	15.0638	0.644083	0.322042	+	0.946726i	$0.395631\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−17.4971	−0.745401
$552$	0	0
$553$	15.1590	0.644627
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2.88410	0.122203	0.0611015	−	0.998132i	$-0.480539\pi$
$557$	2.88410	0.122203	0.0611015	+	0.998132i	$0.480539\pi$
$558$	0	0
$559$	−46.8192	−1.98024
$560$	0	0
$561$	0	0
$562$	0	0
$563$	18.0920	0.762487	0.381243	−	0.924475i	$-0.375496\pi$
$563$	18.0920	0.762487	0.381243	+	0.924475i	$0.375496\pi$
$564$	0	0
$565$	5.40732	0.227487
$566$	0	0
$567$	0	0
$568$	0	0
$569$	36.5457	1.53208	0.766038	−	0.642795i	$-0.222225\pi$
$569$	36.5457	1.53208	0.766038	+	0.642795i	$0.222225\pi$
$570$	0	0
$571$	−12.6639	−0.529967	−0.264984	−	0.964253i	$-0.585367\pi$
$571$	−12.6639	−0.529967	−0.264984	+	0.964253i	$0.585367\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	4.01665	0.167506
$576$	0	0
$577$	10.9415	0.455501	0.227750	−	0.973720i	$-0.426863\pi$
$577$	10.9415	0.455501	0.227750	+	0.973720i	$0.426863\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−18.4824	−0.766778
$582$	0	0
$583$	41.0888	1.70172
$584$	0	0
$585$	0	0
$586$	0	0
$587$	19.9238	0.822343	0.411172	−	0.911558i	$-0.365120\pi$
$587$	19.9238	0.822343	0.411172	+	0.911558i	$0.365120\pi$
$588$	0	0
$589$	−1.47657	−0.0608412
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−24.5748	−1.00917	−0.504583	−	0.863363i	$-0.668354\pi$
$593$	−24.5748	−1.00917	−0.504583	+	0.863363i	$0.668354\pi$
$594$	0	0
$595$	−13.3294	−0.546451
$596$	0	0
$597$	0	0
$598$	0	0
$599$	37.7153	1.54101	0.770503	−	0.637437i	$-0.220005\pi$
$599$	37.7153	1.54101	0.770503	+	0.637437i	$0.220005\pi$
$600$	0	0
$601$	−7.77495	−0.317147	−0.158573	−	0.987347i	$-0.550689\pi$
$601$	−7.77495	−0.317147	−0.158573	+	0.987347i	$0.550689\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−0.0805750	−0.00327584
$606$	0	0
$607$	20.7491	0.842181	0.421090	−	0.907019i	$-0.361648\pi$
$607$	20.7491	0.842181	0.421090	+	0.907019i	$0.361648\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−18.0516	−0.730290
$612$	0	0
$613$	−13.7685	−0.556106	−0.278053	−	0.960566i	$-0.589689\pi$
$613$	−13.7685	−0.556106	−0.278053	+	0.960566i	$0.589689\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	3.45493	0.139090	0.0695451	−	0.997579i	$-0.477845\pi$
$617$	3.45493	0.139090	0.0695451	+	0.997579i	$0.477845\pi$
$618$	0	0
$619$	4.19182	0.168484	0.0842418	−	0.996445i	$-0.473153\pi$
$619$	4.19182	0.168484	0.0842418	+	0.996445i	$0.473153\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−9.25786	−0.370908
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−5.40669	−0.215579
$630$	0	0
$631$	10.6334	0.423311	0.211655	−	0.977344i	$-0.432115\pi$
$631$	10.6334	0.423311	0.211655	+	0.977344i	$0.432115\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−14.1261	−0.560577
$636$	0	0
$637$	−5.77401	−0.228775
$638$	0	0
$639$	0	0
$640$	0	0
$641$	30.9055	1.22069	0.610347	−	0.792134i	$-0.291030\pi$
$641$	30.9055	1.22069	0.610347	+	0.792134i	$0.291030\pi$
$642$	0	0
$643$	23.6594	0.933037	0.466519	−	0.884511i	$-0.345508\pi$
$643$	23.6594	0.933037	0.466519	+	0.884511i	$0.345508\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−41.4513	−1.62962	−0.814809	−	0.579729i	$-0.803158\pi$
$647$	−41.4513	−1.62962	−0.814809	+	0.579729i	$0.803158\pi$
$648$	0	0
$649$	−11.8976	−0.467020
$650$	0	0
$651$	0	0
$652$	0	0
$653$	7.41504	0.290173	0.145086	−	0.989419i	$-0.453654\pi$
$653$	7.41504	0.290173	0.145086	+	0.989419i	$0.453654\pi$
$654$	0	0
$655$	21.4191	0.836913
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−23.8408	−0.928704	−0.464352	−	0.885651i	$-0.653713\pi$
$659$	−23.8408	−0.928704	−0.464352	+	0.885651i	$0.653713\pi$
$660$	0	0
$661$	−33.3022	−1.29531	−0.647653	−	0.761935i	$-0.724249\pi$
$661$	−33.3022	−1.29531	−0.647653	+	0.761935i	$0.724249\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	5.68038	0.220276
$666$	0	0
$667$	30.5022	1.18105
$668$	0	0
$669$	0	0
$670$	0	0
$671$	15.3949	0.594313
$672$	0	0
$673$	22.2744	0.858616	0.429308	−	0.903158i	$-0.358758\pi$
$673$	22.2744	0.858616	0.429308	+	0.903158i	$0.358758\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	25.0143	0.961378	0.480689	−	0.876891i	$-0.340387\pi$
$677$	25.0143	0.961378	0.480689	+	0.876891i	$0.340387\pi$
$678$	0	0
$679$	−32.8353	−1.26010
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−40.3633	−1.54446	−0.772230	−	0.635343i	$-0.780859\pi$
$683$	−40.3633	−1.54446	−0.772230	+	0.635343i	$0.780859\pi$
$684$	0	0
$685$	13.1297	0.501660
$686$	0	0
$687$	0	0
$688$	0	0
$689$	77.2967	2.94477
$690$	0	0
$691$	11.5915	0.440961	0.220480	−	0.975391i	$-0.429238\pi$
$691$	11.5915	0.440961	0.220480	+	0.975391i	$0.429238\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.88005	−0.0713144
$696$	0	0
$697$	34.3219	1.30003
$698$	0	0
$699$	0	0
$700$	0	0
$701$	18.1224	0.684473	0.342237	−	0.939614i	$-0.388816\pi$
$701$	18.1224	0.684473	0.342237	+	0.939614i	$0.388816\pi$
$702$	0	0
$703$	2.30409	0.0869003
$704$	0	0
$705$	0	0
$706$	0	0
$707$	11.6057	0.436476
$708$	0	0
$709$	−16.9915	−0.638129	−0.319065	−	0.947733i	$-0.603369\pi$
$709$	−16.9915	−0.638129	−0.319065	+	0.947733i	$0.603369\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.57407	0.0963997
$714$	0	0
$715$	−20.8449	−0.779555
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−26.4675	−0.987073	−0.493536	−	0.869725i	$-0.664296\pi$
$719$	−26.4675	−0.987073	−0.493536	+	0.869725i	$0.664296\pi$
$720$	0	0
$721$	−4.90329	−0.182608
$722$	0	0
$723$	0	0
$724$	0	0
$725$	7.59393	0.282032
$726$	0	0
$727$	−39.0725	−1.44912	−0.724560	−	0.689212i	$-0.757957\pi$
$727$	−39.0725	−1.44912	−0.724560	+	0.689212i	$0.757957\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−40.4238	−1.49513
$732$	0	0
$733$	−3.50617	−0.129503	−0.0647516	−	0.997901i	$-0.520626\pi$
$733$	−3.50617	−0.129503	−0.0647516	+	0.997901i	$0.520626\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−32.9393	−1.21333
$738$	0	0
$739$	−41.6153	−1.53084	−0.765422	−	0.643529i	$-0.777470\pi$
$739$	−41.6153	−1.53084	−0.765422	+	0.643529i	$0.777470\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−0.0947006	−0.00347423	−0.00173711	−	0.999998i	$-0.500553\pi$
$743$	−0.0947006	−0.00347423	−0.00173711	+	0.999998i	$0.500553\pi$
$744$	0	0
$745$	−2.36933	−0.0868054
$746$	0	0
$747$	0	0
$748$	0	0
$749$	18.9360	0.691905
$750$	0	0
$751$	36.1790	1.32019	0.660096	−	0.751182i	$-0.270516\pi$
$751$	36.1790	1.32019	0.660096	+	0.751182i	$0.270516\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−1.99150	−0.0724780
$756$	0	0
$757$	−1.20080	−0.0436437	−0.0218218	−	0.999762i	$-0.506947\pi$
$757$	−1.20080	−0.0436437	−0.0218218	+	0.999762i	$0.506947\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	25.5134	0.924861	0.462430	−	0.886656i	$-0.346977\pi$
$761$	25.5134	0.924861	0.462430	+	0.886656i	$0.346977\pi$
$762$	0	0
$763$	−7.64306	−0.276697
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−22.3818	−0.808160
$768$	0	0
$769$	−9.98251	−0.359978	−0.179989	−	0.983669i	$-0.557606\pi$
$769$	−9.98251	−0.359978	−0.179989	+	0.983669i	$0.557606\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−8.34119	−0.300012	−0.150006	−	0.988685i	$-0.547929\pi$
$773$	−8.34119	−0.300012	−0.150006	+	0.988685i	$0.547929\pi$
$774$	0	0
$775$	0.640850	0.0230200
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−14.6264	−0.524046
$780$	0	0
$781$	12.6296	0.451923
$782$	0	0
$783$	0	0
$784$	0	0
$785$	15.3405	0.547526
$786$	0	0
$787$	−52.5454	−1.87304	−0.936521	−	0.350612i	$-0.885974\pi$
$787$	−52.5454	−1.87304	−0.936521	+	0.350612i	$0.885974\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−13.3309	−0.473993
$792$	0	0
$793$	28.9610	1.02844
$794$	0	0
$795$	0	0
$796$	0	0
$797$	36.7571	1.30200	0.651002	−	0.759076i	$-0.274349\pi$
$797$	36.7571	1.30200	0.651002	+	0.759076i	$0.274349\pi$
$798$	0	0
$799$	−15.5858	−0.551386
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−43.0148	−1.51796
$804$	0	0
$805$	−9.90244	−0.349015
$806$	0	0
$807$	0	0
$808$	0	0
$809$	50.3613	1.77061	0.885305	−	0.465011i	$-0.153950\pi$
$809$	50.3613	1.77061	0.885305	+	0.465011i	$0.153950\pi$
$810$	0	0
$811$	32.8563	1.15374	0.576870	−	0.816836i	$-0.304274\pi$
$811$	32.8563	1.15374	0.576870	+	0.816836i	$0.304274\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.20916	0.0423551
$816$	0	0
$817$	17.2268	0.602689
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3.07615	0.107358	0.0536792	−	0.998558i	$-0.482905\pi$
$821$	3.07615	0.107358	0.0536792	+	0.998558i	$0.482905\pi$
$822$	0	0
$823$	6.23896	0.217476	0.108738	−	0.994070i	$-0.465319\pi$
$823$	6.23896	0.217476	0.108738	+	0.994070i	$0.465319\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	57.4407	1.99741	0.998705	−	0.0508854i	$-0.0162043\pi$
$827$	57.4407	1.99741	0.998705	+	0.0508854i	$0.0162043\pi$
$828$	0	0
$829$	10.2742	0.356837	0.178418	−	0.983955i	$-0.442902\pi$
$829$	10.2742	0.356837	0.178418	+	0.983955i	$0.442902\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−4.98529	−0.172730
$834$	0	0
$835$	8.10370	0.280440
$836$	0	0
$837$	0	0
$838$	0	0
$839$	14.7882	0.510545	0.255272	−	0.966869i	$-0.417835\pi$
$839$	14.7882	0.510545	0.255272	+	0.966869i	$0.417835\pi$
$840$	0	0
$841$	28.6678	0.988546
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−26.2136	−0.901777
$846$	0	0
$847$	0.198645	0.00682554
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−4.01665	−0.137689
$852$	0	0
$853$	−13.4822	−0.461620	−0.230810	−	0.972999i	$-0.574138\pi$
$853$	−13.4822	−0.461620	−0.230810	+	0.972999i	$0.574138\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−20.3081	−0.693712	−0.346856	−	0.937918i	$-0.612751\pi$
$857$	−20.3081	−0.693712	−0.346856	+	0.937918i	$0.612751\pi$
$858$	0	0
$859$	−14.9879	−0.511382	−0.255691	−	0.966759i	$-0.582303\pi$
$859$	−14.9879	−0.511382	−0.255691	+	0.966759i	$0.582303\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−47.8357	−1.62835	−0.814173	−	0.580622i	$-0.802810\pi$
$863$	−47.8357	−1.62835	−0.814173	+	0.580622i	$0.802810\pi$
$864$	0	0
$865$	−7.60291	−0.258507
$866$	0	0
$867$	0	0
$868$	0	0
$869$	20.4679	0.694327
$870$	0	0
$871$	−61.9657	−2.09963
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−2.46535	−0.0833440
$876$	0	0
$877$	−28.9278	−0.976822	−0.488411	−	0.872614i	$-0.662423\pi$
$877$	−28.9278	−0.976822	−0.488411	+	0.872614i	$0.662423\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−23.0071	−0.775130	−0.387565	−	0.921842i	$-0.626684\pi$
$881$	−23.0071	−0.775130	−0.387565	+	0.921842i	$0.626684\pi$
$882$	0	0
$883$	1.56358	0.0526188	0.0263094	−	0.999654i	$-0.491624\pi$
$883$	1.56358	0.0526188	0.0263094	+	0.999654i	$0.491624\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−39.1397	−1.31418	−0.657091	−	0.753812i	$-0.728213\pi$
$887$	−39.1397	−1.31418	−0.657091	+	0.753812i	$0.728213\pi$
$888$	0	0
$889$	34.8258	1.16802
$890$	0	0
$891$	0	0
$892$	0	0
$893$	6.64196	0.222265
$894$	0	0
$895$	−7.25358	−0.242460
$896$	0	0
$897$	0	0
$898$	0	0
$899$	4.86657	0.162309
$900$	0	0
$901$	66.7381	2.22337
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−4.13709	−0.137522
$906$	0	0
$907$	6.17792	0.205134	0.102567	−	0.994726i	$-0.467294\pi$
$907$	6.17792	0.205134	0.102567	+	0.994726i	$0.467294\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	24.4709	0.810757	0.405379	−	0.914149i	$-0.367140\pi$
$911$	24.4709	0.810757	0.405379	+	0.914149i	$0.367140\pi$
$912$	0	0
$913$	−24.9552	−0.825896
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−52.8055	−1.74379
$918$	0	0
$919$	0.399655	0.0131834	0.00659170	−	0.999978i	$-0.497902\pi$
$919$	0.399655	0.0131834	0.00659170	+	0.999978i	$0.497902\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	23.7589	0.782035
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−28.8369	−0.946107	−0.473053	−	0.881034i	$-0.656848\pi$
$929$	−28.8369	−0.946107	−0.473053	+	0.881034i	$0.656848\pi$
$930$	0	0
$931$	2.12451	0.0696278
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−17.9975	−0.588582
$936$	0	0
$937$	51.4159	1.67968	0.839842	−	0.542831i	$-0.182647\pi$
$937$	51.4159	1.67968	0.839842	+	0.542831i	$0.182647\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	42.7839	1.39472	0.697358	−	0.716723i	$-0.254359\pi$
$941$	42.7839	1.39472	0.697358	+	0.716723i	$0.254359\pi$
$942$	0	0
$943$	25.4978	0.830323
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−21.6500	−0.703530	−0.351765	−	0.936088i	$-0.614418\pi$
$947$	−21.6500	−0.703530	−0.351765	+	0.936088i	$0.614418\pi$
$948$	0	0
$949$	−80.9199	−2.62677
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−43.7911	−1.41853	−0.709266	−	0.704941i	$-0.750974\pi$
$953$	−43.7911	−1.41853	−0.709266	+	0.704941i	$0.750974\pi$
$954$	0	0
$955$	4.04049	0.130747
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−32.3693	−1.04526
$960$	0	0
$961$	−30.5893	−0.986752
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−2.98384	−0.0960533
$966$	0	0
$967$	39.7337	1.27775	0.638875	−	0.769310i	$-0.279400\pi$
$967$	39.7337	1.27775	0.638875	+	0.769310i	$0.279400\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−54.3362	−1.74373	−0.871865	−	0.489746i	$-0.837090\pi$
$971$	−54.3362	−1.74373	−0.871865	+	0.489746i	$0.837090\pi$
$972$	0	0
$973$	4.63498	0.148591
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−49.7888	−1.59289	−0.796443	−	0.604713i	$-0.793288\pi$
$977$	−49.7888	−1.59289	−0.796443	+	0.604713i	$0.793288\pi$
$978$	0	0
$979$	−12.5001	−0.399505
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−10.1540	−0.323862	−0.161931	−	0.986802i	$-0.551772\pi$
$983$	−10.1540	−0.323862	−0.161931	+	0.986802i	$0.551772\pi$
$984$	0	0
$985$	16.0792	0.512326
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−30.0310	−0.954929
$990$	0	0
$991$	49.8384	1.58317	0.791584	−	0.611061i	$-0.209257\pi$
$991$	49.8384	1.58317	0.791584	+	0.611061i	$0.209257\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	27.1205	0.859778
$996$	0	0
$997$	48.7030	1.54244	0.771220	−	0.636569i	$-0.219647\pi$
$997$	48.7030	1.54244	0.771220	+	0.636569i	$0.219647\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6660.2.a.u.1.5	✓	5
3.2	odd	2		6660.2.a.w.1.5	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6660.2.a.u.1.5	✓	5	1.1	even	1	trivial
6660.2.a.w.1.5	yes	5	3.2	odd	2

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:	\( N \)	\(=\)	\( 6660 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6660.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +2.46535 q^{7} +3.32875 q^{11} +6.26208 q^{13} +5.40669 q^{17} -2.30409 q^{19} +4.01665 q^{23} +1.00000 q^{25} +7.59393 q^{29} +0.640850 q^{31} -2.46535 q^{35} -1.00000 q^{37} +6.34803 q^{41} -7.47662 q^{43} -2.88269 q^{47} -0.922059 q^{49} +12.3436 q^{53} -3.32875 q^{55} -3.57418 q^{59} +4.62483 q^{61} -6.26208 q^{65} -9.89539 q^{67} +3.79410 q^{71} -12.9222 q^{73} +8.20653 q^{77} +6.14883 q^{79} -7.49686 q^{83} -5.40669 q^{85} -3.75519 q^{89} +15.4382 q^{91} +2.30409 q^{95} -13.3187 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{5} + 4 q^{11} - q^{13} + 8 q^{17} - 4 q^{19} + 2 q^{23} + 5 q^{25} + 13 q^{29} - 4 q^{31} - 5 q^{37} + 6 q^{41} + q^{43} - q^{47} - 11 q^{49} + 15 q^{53} - 4 q^{55} + 9 q^{59} - 10 q^{61}+ \cdots + 8 q^{97}+O(q^{100}) \) 5 * q - 5 * q^5 + 4 * q^11 - q^13 + 8 * q^17 - 4 * q^19 + 2 * q^23 + 5 * q^25 + 13 * q^29 - 4 * q^31 - 5 * q^37 + 6 * q^41 + q^43 - q^47 - 11 * q^49 + 15 * q^53 - 4 * q^55 + 9 * q^59 - 10 * q^61 + q^65 - 12 * q^67 - 6 * q^71 - 12 * q^73 + 22 * q^77 - 8 * q^79 + 27 * q^83 - 8 * q^85 + 11 * q^89 + 6 * q^91 + 4 * q^95 + 8 * q^97

Introduction

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