LMFDB - Embedded newform 6660.2.a.v.1.3

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)

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);

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")

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,-4] 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:	5.5.10501488.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 17x^{3} - 13x^{2} + 19x - 1 $ x^5 - x^4 - 17x^3 - 13x^2 + 19*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.45735$ 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)

$f(q)$	$=$	$q+1.00000 q^{5} +0.923694 q^{7} -3.45735 q^{11} +0.823530 q^{13} +3.65972 q^{17} +2.92736 q^{19} +8.13193 q^{23} +1.00000 q^{25} +0.0875050 q^{29} -1.54485 q^{31} +0.923694 q^{35} +1.00000 q^{37} +4.51895 q^{41} -11.9422 q^{43} +8.23209 q^{47} -6.14679 q^{49} +2.73236 q^{53} -3.45735 q^{55} -10.4128 q^{59} +6.48487 q^{61} +0.823530 q^{65} +4.38104 q^{67} -9.83101 q^{71} -1.86209 q^{73} -3.19353 q^{77} -1.55751 q^{79} +8.27884 q^{83} +3.65972 q^{85} -2.49591 q^{89} +0.760690 q^{91} +2.92736 q^{95} -3.56322 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{5} - 4 q^{11} - 4 q^{13} + 10 q^{17} - 4 q^{19} - 4 q^{23} + 5 q^{25} - 4 q^{29} + 10 q^{31} + 5 q^{37} + 8 q^{41} - 18 q^{43} + 37 q^{49} + 24 q^{53} - 4 q^{55} + 14 q^{59} + 4 q^{61} - 4 q^{65}+ \cdots + 8 q^{97}+O(q^{100}) $ 5 * q + 5 * q^5 - 4 * q^11 - 4 * q^13 + 10 * q^17 - 4 * q^19 - 4 * q^23 + 5 * q^25 - 4 * q^29 + 10 * q^31 + 5 * q^37 + 8 * q^41 - 18 * q^43 + 37 * q^49 + 24 * q^53 - 4 * q^55 + 14 * q^59 + 4 * q^61 - 4 * q^65 + 4 * q^67 + 16 * q^71 - 6 * q^73 + 30 * q^77 + 2 * q^79 + 22 * q^83 + 10 * q^85 - 4 * q^89 - 18 * 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
$7$	0.923694	0.349124	0.174562	−	0.984646i	$-0.444149\pi$
$7$	0.923694	0.349124	0.174562	+	0.984646i	$0.444149\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.45735	−1.04243	−0.521215	−	0.853426i	$-0.674521\pi$
$11$	−3.45735	−1.04243	−0.521215	+	0.853426i	$0.674521\pi$
$12$	0	0
$13$	0.823530	0.228406	0.114203	−	0.993457i	$-0.463569\pi$
$13$	0.823530	0.228406	0.114203	+	0.993457i	$0.463569\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.65972	0.887612	0.443806	−	0.896123i	$-0.353628\pi$
$17$	3.65972	0.887612	0.443806	+	0.896123i	$0.353628\pi$
$18$	0	0
$19$	2.92736	0.671582	0.335791	−	0.941937i	$-0.390996\pi$
$19$	2.92736	0.671582	0.335791	+	0.941937i	$0.390996\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.13193	1.69562	0.847812	−	0.530297i	$-0.177919\pi$
$23$	8.13193	1.69562	0.847812	+	0.530297i	$0.177919\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0.0875050	0.0162493	0.00812464	−	0.999967i	$-0.497414\pi$
$29$	0.0875050	0.0162493	0.00812464	+	0.999967i	$0.497414\pi$
$30$	0	0
$31$	−1.54485	−0.277464	−0.138732	−	0.990330i	$-0.544303\pi$
$31$	−1.54485	−0.277464	−0.138732	+	0.990330i	$0.544303\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.923694	0.156133
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.51895	0.705742	0.352871	−	0.935672i	$-0.385205\pi$
$41$	4.51895	0.705742	0.352871	+	0.935672i	$0.385205\pi$
$42$	0	0
$43$	−11.9422	−1.82117	−0.910585	−	0.413321i	$-0.864369\pi$
$43$	−11.9422	−1.82117	−0.910585	+	0.413321i	$0.864369\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.23209	1.20077	0.600387	−	0.799710i	$-0.295013\pi$
$47$	8.23209	1.20077	0.600387	+	0.799710i	$0.295013\pi$
$48$	0	0
$49$	−6.14679	−0.878113
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2.73236	0.375319	0.187659	−	0.982234i	$-0.439910\pi$
$53$	2.73236	0.375319	0.187659	+	0.982234i	$0.439910\pi$
$54$	0	0
$55$	−3.45735	−0.466189
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−10.4128	−1.35563	−0.677816	−	0.735232i	$-0.737073\pi$
$59$	−10.4128	−1.35563	−0.677816	+	0.735232i	$0.737073\pi$
$60$	0	0
$61$	6.48487	0.830302	0.415151	−	0.909752i	$-0.363729\pi$
$61$	6.48487	0.830302	0.415151	+	0.909752i	$0.363729\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.823530	0.102146
$66$	0	0
$67$	4.38104	0.535230	0.267615	−	0.963526i	$-0.413765\pi$
$67$	4.38104	0.535230	0.267615	+	0.963526i	$0.413765\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−9.83101	−1.16673	−0.583363	−	0.812211i	$-0.698264\pi$
$71$	−9.83101	−1.16673	−0.583363	+	0.812211i	$0.698264\pi$
$72$	0	0
$73$	−1.86209	−0.217941	−0.108971	−	0.994045i	$-0.534755\pi$
$73$	−1.86209	−0.217941	−0.108971	+	0.994045i	$0.534755\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.19353	−0.363937
$78$	0	0
$79$	−1.55751	−0.175234	−0.0876169	−	0.996154i	$-0.527925\pi$
$79$	−1.55751	−0.175234	−0.0876169	+	0.996154i	$0.527925\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	8.27884	0.908720	0.454360	−	0.890818i	$-0.349868\pi$
$83$	8.27884	0.908720	0.454360	+	0.890818i	$0.349868\pi$
$84$	0	0
$85$	3.65972	0.396952
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−2.49591	−0.264566	−0.132283	−	0.991212i	$-0.542231\pi$
$89$	−2.49591	−0.264566	−0.132283	+	0.991212i	$0.542231\pi$
$90$	0	0
$91$	0.760690	0.0797419
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.92736	0.300340
$96$	0	0
$97$	−3.56322	−0.361790	−0.180895	−	0.983502i	$-0.557899\pi$
$97$	−3.56322	−0.361790	−0.180895	+	0.983502i	$0.557899\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	8.38104	0.833945	0.416972	−	0.908919i	$-0.363091\pi$
$101$	8.38104	0.833945	0.416972	+	0.908919i	$0.363091\pi$
$102$	0	0
$103$	15.7679	1.55366	0.776831	−	0.629709i	$-0.216826\pi$
$103$	15.7679	1.55366	0.776831	+	0.629709i	$0.216826\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.68056	0.452487	0.226244	−	0.974071i	$-0.427355\pi$
$107$	4.68056	0.452487	0.226244	+	0.974071i	$0.427355\pi$
$108$	0	0
$109$	8.90204	0.852660	0.426330	−	0.904568i	$-0.359806\pi$
$109$	8.90204	0.852660	0.426330	+	0.904568i	$0.359806\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	3.18217	0.299354	0.149677	−	0.988735i	$-0.452177\pi$
$113$	3.18217	0.299354	0.149677	+	0.988735i	$0.452177\pi$
$114$	0	0
$115$	8.13193	0.758306
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.38046	0.309886
$120$	0	0
$121$	0.953256	0.0866597
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−16.2268	−1.43989	−0.719946	−	0.694030i	$-0.755833\pi$
$127$	−16.2268	−1.43989	−0.719946	+	0.694030i	$0.755833\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	18.6782	1.63193	0.815963	−	0.578104i	$-0.196207\pi$
$131$	18.6782	1.63193	0.815963	+	0.578104i	$0.196207\pi$
$132$	0	0
$133$	2.70398	0.234465
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−11.2506	−0.961205	−0.480602	−	0.876939i	$-0.659582\pi$
$137$	−11.2506	−0.961205	−0.480602	+	0.876939i	$0.659582\pi$
$138$	0	0
$139$	2.78643	0.236342	0.118171	−	0.992993i	$-0.462297\pi$
$139$	2.78643	0.236342	0.118171	+	0.992993i	$0.462297\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2.84723	−0.238097
$144$	0	0
$145$	0.0875050	0.00726690
$146$	0	0
$147$	0	0
$148$	0	0
$149$	9.33576	0.764815	0.382408	−	0.923994i	$-0.375095\pi$
$149$	9.33576	0.764815	0.382408	+	0.923994i	$0.375095\pi$
$150$	0	0
$151$	−19.1468	−1.55814	−0.779072	−	0.626935i	$-0.784309\pi$
$151$	−19.1468	−1.55814	−0.779072	+	0.626935i	$0.784309\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1.54485	−0.124086
$156$	0	0
$157$	18.6005	1.48448	0.742240	−	0.670134i	$-0.233764\pi$
$157$	18.6005	1.48448	0.742240	+	0.670134i	$0.233764\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	7.51142	0.591982
$162$	0	0
$163$	6.05180	0.474014	0.237007	−	0.971508i	$-0.423834\pi$
$163$	6.05180	0.474014	0.237007	+	0.971508i	$0.423834\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	22.0762	1.70831	0.854153	−	0.520021i	$-0.174076\pi$
$167$	22.0762	1.70831	0.854153	+	0.520021i	$0.174076\pi$
$168$	0	0
$169$	−12.3218	−0.947831
$170$	0	0
$171$	0	0
$172$	0	0
$173$	19.6523	1.49414	0.747070	−	0.664745i	$-0.231460\pi$
$173$	19.6523	1.49414	0.747070	+	0.664745i	$0.231460\pi$
$174$	0	0
$175$	0.923694	0.0698247
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2.87533	0.214912	0.107456	−	0.994210i	$-0.465729\pi$
$179$	2.87533	0.214912	0.107456	+	0.994210i	$0.465729\pi$
$180$	0	0
$181$	24.0147	1.78500	0.892501	−	0.451046i	$-0.148949\pi$
$181$	24.0147	1.78500	0.892501	+	0.451046i	$0.148949\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.00000	0.0735215
$186$	0	0
$187$	−12.6529	−0.925273
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3.44265	0.249101	0.124551	−	0.992213i	$-0.460251\pi$
$191$	3.44265	0.249101	0.124551	+	0.992213i	$0.460251\pi$
$192$	0	0
$193$	14.9517	1.07625	0.538125	−	0.842865i	$-0.319133\pi$
$193$	14.9517	1.07625	0.538125	+	0.842865i	$0.319133\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−10.4662	−0.745688	−0.372844	−	0.927894i	$-0.621617\pi$
$197$	−10.4662	−0.745688	−0.372844	+	0.927894i	$0.621617\pi$
$198$	0	0
$199$	16.6419	1.17971	0.589856	−	0.807509i	$-0.299185\pi$
$199$	16.6419	1.17971	0.589856	+	0.807509i	$0.299185\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0.0808279	0.00567301
$204$	0	0
$205$	4.51895	0.315617
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−10.1209	−0.700077
$210$	0	0
$211$	2.36004	0.162472	0.0812359	−	0.996695i	$-0.474113\pi$
$211$	2.36004	0.162472	0.0812359	+	0.996695i	$0.474113\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−11.9422	−0.814452
$216$	0	0
$217$	−1.42697	−0.0968692
$218$	0	0
$219$	0	0
$220$	0	0
$221$	3.01389	0.202736
$222$	0	0
$223$	−23.0643	−1.54450	−0.772251	−	0.635317i	$-0.780869\pi$
$223$	−23.0643	−1.54450	−0.772251	+	0.635317i	$0.780869\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	15.1741	1.00714	0.503572	−	0.863953i	$-0.332019\pi$
$227$	15.1741	1.00714	0.503572	+	0.863953i	$0.332019\pi$
$228$	0	0
$229$	13.5953	0.898400	0.449200	−	0.893431i	$-0.351709\pi$
$229$	13.5953	0.898400	0.449200	+	0.893431i	$0.351709\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	24.0391	1.57485	0.787426	−	0.616409i	$-0.211413\pi$
$233$	24.0391	1.57485	0.787426	+	0.616409i	$0.211413\pi$
$234$	0	0
$235$	8.23209	0.537003
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−22.9674	−1.48564	−0.742819	−	0.669492i	$-0.766512\pi$
$239$	−22.9674	−1.48564	−0.742819	+	0.669492i	$0.766512\pi$
$240$	0	0
$241$	−28.0331	−1.80577	−0.902885	−	0.429881i	$-0.858555\pi$
$241$	−28.0331	−1.80577	−0.902885	+	0.429881i	$0.858555\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−6.14679	−0.392704
$246$	0	0
$247$	2.41076	0.153393
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−16.0654	−1.01404	−0.507020	−	0.861934i	$-0.669253\pi$
$251$	−16.0654	−1.01404	−0.507020	+	0.861934i	$0.669253\pi$
$252$	0	0
$253$	−28.1149	−1.76757
$254$	0	0
$255$	0	0
$256$	0	0
$257$	11.8494	0.739144	0.369572	−	0.929202i	$-0.379504\pi$
$257$	11.8494	0.739144	0.369572	+	0.929202i	$0.379504\pi$
$258$	0	0
$259$	0.923694	0.0573956
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−6.08379	−0.375142	−0.187571	−	0.982251i	$-0.560062\pi$
$263$	−6.08379	−0.375142	−0.187571	+	0.982251i	$0.560062\pi$
$264$	0	0
$265$	2.73236	0.167848
$266$	0	0
$267$	0	0
$268$	0	0
$269$	30.1320	1.83718	0.918589	−	0.395214i	$-0.129329\pi$
$269$	30.1320	1.83718	0.918589	+	0.395214i	$0.129329\pi$
$270$	0	0
$271$	−24.4168	−1.48321	−0.741607	−	0.670834i	$-0.765936\pi$
$271$	−24.4168	−1.48321	−0.741607	+	0.670834i	$0.765936\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.45735	−0.208486
$276$	0	0
$277$	23.0259	1.38349	0.691747	−	0.722140i	$-0.256841\pi$
$277$	23.0259	1.38349	0.691747	+	0.722140i	$0.256841\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	26.1466	1.55978	0.779889	−	0.625918i	$-0.215276\pi$
$281$	26.1466	1.55978	0.779889	+	0.625918i	$0.215276\pi$
$282$	0	0
$283$	−16.5975	−0.986617	−0.493308	−	0.869855i	$-0.664213\pi$
$283$	−16.5975	−0.986617	−0.493308	+	0.869855i	$0.664213\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.17413	0.246391
$288$	0	0
$289$	−3.60646	−0.212145
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−12.3209	−0.719797	−0.359898	−	0.932991i	$-0.617189\pi$
$293$	−12.3209	−0.719797	−0.359898	+	0.932991i	$0.617189\pi$
$294$	0	0
$295$	−10.4128	−0.606257
$296$	0	0
$297$	0	0
$298$	0	0
$299$	6.69688	0.387291
$300$	0	0
$301$	−11.0310	−0.635814
$302$	0	0
$303$	0	0
$304$	0	0
$305$	6.48487	0.371323
$306$	0	0
$307$	−13.3920	−0.764320	−0.382160	−	0.924096i	$-0.624820\pi$
$307$	−13.3920	−0.764320	−0.382160	+	0.924096i	$0.624820\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	19.3565	1.09761	0.548803	−	0.835952i	$-0.315084\pi$
$311$	19.3565	1.09761	0.548803	+	0.835952i	$0.315084\pi$
$312$	0	0
$313$	−11.2025	−0.633204	−0.316602	−	0.948558i	$-0.602542\pi$
$313$	−11.2025	−0.633204	−0.316602	+	0.948558i	$0.602542\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	14.4051	0.809069	0.404534	−	0.914523i	$-0.367434\pi$
$317$	14.4051	0.809069	0.404534	+	0.914523i	$0.367434\pi$
$318$	0	0
$319$	−0.302535	−0.0169387
$320$	0	0
$321$	0	0
$322$	0	0
$323$	10.7133	0.596104
$324$	0	0
$325$	0.823530	0.0456812
$326$	0	0
$327$	0	0
$328$	0	0
$329$	7.60394	0.419219
$330$	0	0
$331$	−10.9274	−0.600622	−0.300311	−	0.953841i	$-0.597090\pi$
$331$	−10.9274	−0.600622	−0.300311	+	0.953841i	$0.597090\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	4.38104	0.239362
$336$	0	0
$337$	−11.8413	−0.645037	−0.322519	−	0.946563i	$-0.604530\pi$
$337$	−11.8413	−0.645037	−0.322519	+	0.946563i	$0.604530\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	5.34110	0.289237
$342$	0	0
$343$	−12.1436	−0.655693
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3.67238	−0.197144	−0.0985718	−	0.995130i	$-0.531427\pi$
$347$	−3.67238	−0.197144	−0.0985718	+	0.995130i	$0.531427\pi$
$348$	0	0
$349$	−14.5150	−0.776971	−0.388485	−	0.921455i	$-0.627002\pi$
$349$	−14.5150	−0.776971	−0.388485	+	0.921455i	$0.627002\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−5.39957	−0.287390	−0.143695	−	0.989622i	$-0.545898\pi$
$353$	−5.39957	−0.287390	−0.143695	+	0.989622i	$0.545898\pi$
$354$	0	0
$355$	−9.83101	−0.521776
$356$	0	0
$357$	0	0
$358$	0	0
$359$	16.4001	0.865567	0.432783	−	0.901498i	$-0.357532\pi$
$359$	16.4001	0.865567	0.432783	+	0.901498i	$0.357532\pi$
$360$	0	0
$361$	−10.4306	−0.548978
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−1.86209	−0.0974662
$366$	0	0
$367$	33.7970	1.76419	0.882094	−	0.471074i	$-0.156134\pi$
$367$	33.7970	1.76419	0.882094	+	0.471074i	$0.156134\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.52387	0.131033
$372$	0	0
$373$	−20.7902	−1.07648	−0.538238	−	0.842793i	$-0.680910\pi$
$373$	−20.7902	−1.07648	−0.538238	+	0.842793i	$0.680910\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.0720630	0.00371143
$378$	0	0
$379$	21.8080	1.12020	0.560100	−	0.828425i	$-0.310763\pi$
$379$	21.8080	1.12020	0.560100	+	0.828425i	$0.310763\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−38.3834	−1.96130	−0.980651	−	0.195763i	$-0.937282\pi$
$383$	−38.3834	−1.96130	−0.980651	+	0.195763i	$0.937282\pi$
$384$	0	0
$385$	−3.19353	−0.162757
$386$	0	0
$387$	0	0
$388$	0	0
$389$	3.96620	0.201094	0.100547	−	0.994932i	$-0.467941\pi$
$389$	3.96620	0.201094	0.100547	+	0.994932i	$0.467941\pi$
$390$	0	0
$391$	29.7606	1.50506
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1.55751	−0.0783670
$396$	0	0
$397$	0.960739	0.0482181	0.0241091	−	0.999709i	$-0.492325\pi$
$397$	0.960739	0.0482181	0.0241091	+	0.999709i	$0.492325\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5.42025	0.270674	0.135337	−	0.990800i	$-0.456788\pi$
$401$	5.42025	0.270674	0.135337	+	0.990800i	$0.456788\pi$
$402$	0	0
$403$	−1.27223	−0.0633744
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−3.45735	−0.171374
$408$	0	0
$409$	−4.42913	−0.219006	−0.109503	−	0.993986i	$-0.534926\pi$
$409$	−4.42913	−0.219006	−0.109503	+	0.993986i	$0.534926\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−9.61825	−0.473283
$414$	0	0
$415$	8.27884	0.406392
$416$	0	0
$417$	0	0
$418$	0	0
$419$	21.7286	1.06151	0.530755	−	0.847525i	$-0.321908\pi$
$419$	21.7286	1.06151	0.530755	+	0.847525i	$0.321908\pi$
$420$	0	0
$421$	28.2135	1.37504	0.687520	−	0.726166i	$-0.258699\pi$
$421$	28.2135	1.37504	0.687520	+	0.726166i	$0.258699\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.65972	0.177522
$426$	0	0
$427$	5.99004	0.289878
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−11.4573	−0.551878	−0.275939	−	0.961175i	$-0.588989\pi$
$431$	−11.4573	−0.551878	−0.275939	+	0.961175i	$0.588989\pi$
$432$	0	0
$433$	11.4516	0.550328	0.275164	−	0.961397i	$-0.411268\pi$
$433$	11.4516	0.550328	0.275164	+	0.961397i	$0.411268\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	23.8050	1.13875
$438$	0	0
$439$	35.7950	1.70840	0.854202	−	0.519942i	$-0.174046\pi$
$439$	35.7950	1.70840	0.854202	+	0.519942i	$0.174046\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−12.5894	−0.598142	−0.299071	−	0.954231i	$-0.596677\pi$
$443$	−12.5894	−0.598142	−0.299071	+	0.954231i	$0.596677\pi$
$444$	0	0
$445$	−2.49591	−0.118317
$446$	0	0
$447$	0	0
$448$	0	0
$449$	10.1016	0.476722	0.238361	−	0.971177i	$-0.423390\pi$
$449$	10.1016	0.476722	0.238361	+	0.971177i	$0.423390\pi$
$450$	0	0
$451$	−15.6236	−0.735686
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0.760690	0.0356617
$456$	0	0
$457$	−16.8894	−0.790052	−0.395026	−	0.918670i	$-0.629264\pi$
$457$	−16.8894	−0.790052	−0.395026	+	0.918670i	$0.629264\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	7.60801	0.354340	0.177170	−	0.984180i	$-0.443306\pi$
$461$	7.60801	0.354340	0.177170	+	0.984180i	$0.443306\pi$
$462$	0	0
$463$	−13.0673	−0.607289	−0.303645	−	0.952785i	$-0.598204\pi$
$463$	−13.0673	−0.607289	−0.303645	+	0.952785i	$0.598204\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−13.9887	−0.647321	−0.323661	−	0.946173i	$-0.604914\pi$
$467$	−13.9887	−0.647321	−0.323661	+	0.946173i	$0.604914\pi$
$468$	0	0
$469$	4.04674	0.186861
$470$	0	0
$471$	0	0
$472$	0	0
$473$	41.2884	1.89844
$474$	0	0
$475$	2.92736	0.134316
$476$	0	0
$477$	0	0
$478$	0	0
$479$	13.0947	0.598313	0.299156	−	0.954204i	$-0.403295\pi$
$479$	13.0947	0.598313	0.299156	+	0.954204i	$0.403295\pi$
$480$	0	0
$481$	0.823530	0.0375497
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3.56322	−0.161797
$486$	0	0
$487$	−6.48993	−0.294087	−0.147043	−	0.989130i	$-0.546976\pi$
$487$	−6.48993	−0.294087	−0.147043	+	0.989130i	$0.546976\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	12.4181	0.560423	0.280211	−	0.959938i	$-0.409595\pi$
$491$	12.4181	0.560423	0.280211	+	0.959938i	$0.409595\pi$
$492$	0	0
$493$	0.320244	0.0144231
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−9.08085	−0.407332
$498$	0	0
$499$	24.5618	1.09954	0.549768	−	0.835318i	$-0.314716\pi$
$499$	24.5618	1.09954	0.549768	+	0.835318i	$0.314716\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2.19709	−0.0979633	−0.0489817	−	0.998800i	$-0.515598\pi$
$503$	−2.19709	−0.0979633	−0.0489817	+	0.998800i	$0.515598\pi$
$504$	0	0
$505$	8.38104	0.372951
$506$	0	0
$507$	0	0
$508$	0	0
$509$	16.9624	0.751846	0.375923	−	0.926651i	$-0.377326\pi$
$509$	16.9624	0.751846	0.375923	+	0.926651i	$0.377326\pi$
$510$	0	0
$511$	−1.72000	−0.0760884
$512$	0	0
$513$	0	0
$514$	0	0
$515$	15.7679	0.694819
$516$	0	0
$517$	−28.4612	−1.25172
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−18.2031	−0.797493	−0.398746	−	0.917061i	$-0.630555\pi$
$521$	−18.2031	−0.797493	−0.398746	+	0.917061i	$0.630555\pi$
$522$	0	0
$523$	15.9547	0.697648	0.348824	−	0.937188i	$-0.386581\pi$
$523$	15.9547	0.697648	0.348824	+	0.937188i	$0.386581\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5.65373	−0.246280
$528$	0	0
$529$	43.1283	1.87514
$530$	0	0
$531$	0	0
$532$	0	0
$533$	3.72149	0.161196
$534$	0	0
$535$	4.68056	0.202358
$536$	0	0
$537$	0	0
$538$	0	0
$539$	21.2516	0.915371
$540$	0	0
$541$	0.179326	0.00770982	0.00385491	−	0.999993i	$-0.498773\pi$
$541$	0.179326	0.00770982	0.00385491	+	0.999993i	$0.498773\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	8.90204	0.381321
$546$	0	0
$547$	28.0414	1.19897	0.599483	−	0.800388i	$-0.295373\pi$
$547$	28.0414	1.19897	0.599483	+	0.800388i	$0.295373\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0.256158	0.0109127
$552$	0	0
$553$	−1.43867	−0.0611783
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−13.8096	−0.585133	−0.292567	−	0.956245i	$-0.594509\pi$
$557$	−13.8096	−0.585133	−0.292567	+	0.956245i	$0.594509\pi$
$558$	0	0
$559$	−9.83477	−0.415966
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−12.7832	−0.538749	−0.269374	−	0.963036i	$-0.586817\pi$
$563$	−12.7832	−0.538749	−0.269374	+	0.963036i	$0.586817\pi$
$564$	0	0
$565$	3.18217	0.133875
$566$	0	0
$567$	0	0
$568$	0	0
$569$	12.0193	0.503877	0.251938	−	0.967743i	$-0.418932\pi$
$569$	12.0193	0.503877	0.251938	+	0.967743i	$0.418932\pi$
$570$	0	0
$571$	−13.1097	−0.548626	−0.274313	−	0.961641i	$-0.588450\pi$
$571$	−13.1097	−0.548626	−0.274313	+	0.961641i	$0.588450\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	8.13193	0.339125
$576$	0	0
$577$	1.09129	0.0454308	0.0227154	−	0.999742i	$-0.492769\pi$
$577$	1.09129	0.0454308	0.0227154	+	0.999742i	$0.492769\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	7.64711	0.317256
$582$	0	0
$583$	−9.44673	−0.391244
$584$	0	0
$585$	0	0
$586$	0	0
$587$	24.4252	1.00814	0.504068	−	0.863664i	$-0.331836\pi$
$587$	24.4252	1.00814	0.504068	+	0.863664i	$0.331836\pi$
$588$	0	0
$589$	−4.52234	−0.186340
$590$	0	0
$591$	0	0
$592$	0	0
$593$	21.9077	0.899642	0.449821	−	0.893119i	$-0.351488\pi$
$593$	21.9077	0.899642	0.449821	+	0.893119i	$0.351488\pi$
$594$	0	0
$595$	3.38046	0.138585
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−26.0565	−1.06464	−0.532319	−	0.846544i	$-0.678679\pi$
$599$	−26.0565	−1.06464	−0.532319	+	0.846544i	$0.678679\pi$
$600$	0	0
$601$	−31.7852	−1.29655	−0.648273	−	0.761408i	$-0.724509\pi$
$601$	−31.7852	−1.29655	−0.648273	+	0.761408i	$0.724509\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0.953256	0.0387554
$606$	0	0
$607$	36.8353	1.49510	0.747548	−	0.664207i	$-0.231231\pi$
$607$	36.8353	1.49510	0.747548	+	0.664207i	$0.231231\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	6.77937	0.274264
$612$	0	0
$613$	−13.4797	−0.544440	−0.272220	−	0.962235i	$-0.587758\pi$
$613$	−13.4797	−0.544440	−0.272220	+	0.962235i	$0.587758\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−13.1224	−0.528290	−0.264145	−	0.964483i	$-0.585090\pi$
$617$	−13.1224	−0.528290	−0.264145	+	0.964483i	$0.585090\pi$
$618$	0	0
$619$	−38.0592	−1.52973	−0.764864	−	0.644191i	$-0.777194\pi$
$619$	−38.0592	−1.52973	−0.764864	+	0.644191i	$0.777194\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−2.30546	−0.0923661
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	3.65972	0.145923
$630$	0	0
$631$	21.0119	0.836470	0.418235	−	0.908339i	$-0.362649\pi$
$631$	21.0119	0.836470	0.418235	+	0.908339i	$0.362649\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−16.2268	−0.643939
$636$	0	0
$637$	−5.06206	−0.200566
$638$	0	0
$639$	0	0
$640$	0	0
$641$	30.7799	1.21573	0.607866	−	0.794039i	$-0.292026\pi$
$641$	30.7799	1.21573	0.607866	+	0.794039i	$0.292026\pi$
$642$	0	0
$643$	7.70284	0.303770	0.151885	−	0.988398i	$-0.451466\pi$
$643$	7.70284	0.303770	0.151885	+	0.988398i	$0.451466\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−24.2642	−0.953923	−0.476962	−	0.878924i	$-0.658262\pi$
$647$	−24.2642	−0.953923	−0.476962	+	0.878924i	$0.658262\pi$
$648$	0	0
$649$	36.0007	1.41315
$650$	0	0
$651$	0	0
$652$	0	0
$653$	33.3568	1.30535	0.652676	−	0.757637i	$-0.273646\pi$
$653$	33.3568	1.30535	0.652676	+	0.757637i	$0.273646\pi$
$654$	0	0
$655$	18.6782	0.729819
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−15.9913	−0.622933	−0.311467	−	0.950257i	$-0.600820\pi$
$659$	−15.9913	−0.622933	−0.311467	+	0.950257i	$0.600820\pi$
$660$	0	0
$661$	−4.53064	−0.176221	−0.0881107	−	0.996111i	$-0.528083\pi$
$661$	−4.53064	−0.176221	−0.0881107	+	0.996111i	$0.528083\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.70398	0.104856
$666$	0	0
$667$	0.711585	0.0275527
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−22.4204	−0.865532
$672$	0	0
$673$	−24.4449	−0.942281	−0.471141	−	0.882058i	$-0.656158\pi$
$673$	−24.4449	−0.942281	−0.471141	+	0.882058i	$0.656158\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−27.4851	−1.05634	−0.528169	−	0.849140i	$-0.677121\pi$
$677$	−27.4851	−1.05634	−0.528169	+	0.849140i	$0.677121\pi$
$678$	0	0
$679$	−3.29132	−0.126309
$680$	0	0
$681$	0	0
$682$	0	0
$683$	3.12351	0.119518	0.0597589	−	0.998213i	$-0.480967\pi$
$683$	3.12351	0.119518	0.0597589	+	0.998213i	$0.480967\pi$
$684$	0	0
$685$	−11.2506	−0.429864
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.25018	0.0857251
$690$	0	0
$691$	41.3469	1.57291	0.786455	−	0.617648i	$-0.211914\pi$
$691$	41.3469	1.57291	0.786455	+	0.617648i	$0.211914\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.78643	0.105695
$696$	0	0
$697$	16.5381	0.626425
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−31.2898	−1.18180	−0.590900	−	0.806745i	$-0.701227\pi$
$701$	−31.2898	−1.18180	−0.590900	+	0.806745i	$0.701227\pi$
$702$	0	0
$703$	2.92736	0.110407
$704$	0	0
$705$	0	0
$706$	0	0
$707$	7.74152	0.291150
$708$	0	0
$709$	16.4866	0.619167	0.309583	−	0.950872i	$-0.399810\pi$
$709$	16.4866	0.619167	0.309583	+	0.950872i	$0.399810\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−12.5626	−0.470474
$714$	0	0
$715$	−2.84723	−0.106480
$716$	0	0
$717$	0	0
$718$	0	0
$719$	11.5673	0.431387	0.215694	−	0.976461i	$-0.430799\pi$
$719$	11.5673	0.431387	0.215694	+	0.976461i	$0.430799\pi$
$720$	0	0
$721$	14.5648	0.542420
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0.0875050	0.00324986
$726$	0	0
$727$	−32.3590	−1.20013	−0.600065	−	0.799952i	$-0.704858\pi$
$727$	−32.3590	−1.20013	−0.600065	+	0.799952i	$0.704858\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−43.7052	−1.61649
$732$	0	0
$733$	−6.41490	−0.236940	−0.118470	−	0.992958i	$-0.537799\pi$
$733$	−6.41490	−0.236940	−0.118470	+	0.992958i	$0.537799\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−15.1468	−0.557939
$738$	0	0
$739$	−12.6411	−0.465012	−0.232506	−	0.972595i	$-0.574693\pi$
$739$	−12.6411	−0.465012	−0.232506	+	0.972595i	$0.574693\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	15.7659	0.578393	0.289197	−	0.957270i	$-0.406612\pi$
$743$	15.7659	0.578393	0.289197	+	0.957270i	$0.406612\pi$
$744$	0	0
$745$	9.33576	0.342036
$746$	0	0
$747$	0	0
$748$	0	0
$749$	4.32341	0.157974
$750$	0	0
$751$	−36.4672	−1.33071	−0.665354	−	0.746528i	$-0.731719\pi$
$751$	−36.4672	−1.33071	−0.665354	+	0.746528i	$0.731719\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−19.1468	−0.696823
$756$	0	0
$757$	−35.9379	−1.30619	−0.653093	−	0.757278i	$-0.726529\pi$
$757$	−35.9379	−1.30619	−0.653093	+	0.757278i	$0.726529\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−46.3794	−1.68125	−0.840627	−	0.541615i	$-0.817813\pi$
$761$	−46.3794	−1.68125	−0.840627	+	0.541615i	$0.817813\pi$
$762$	0	0
$763$	8.22276	0.297684
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8.57525	−0.309634
$768$	0	0
$769$	32.0497	1.15574	0.577871	−	0.816128i	$-0.303884\pi$
$769$	32.0497	1.15574	0.577871	+	0.816128i	$0.303884\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	10.7858	0.387937	0.193969	−	0.981008i	$-0.437864\pi$
$773$	10.7858	0.387937	0.193969	+	0.981008i	$0.437864\pi$
$774$	0	0
$775$	−1.54485	−0.0554928
$776$	0	0
$777$	0	0
$778$	0	0
$779$	13.2286	0.473963
$780$	0	0
$781$	33.9892	1.21623
$782$	0	0
$783$	0	0
$784$	0	0
$785$	18.6005	0.663879
$786$	0	0
$787$	−10.2712	−0.366130	−0.183065	−	0.983101i	$-0.558602\pi$
$787$	−10.2712	−0.366130	−0.183065	+	0.983101i	$0.558602\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2.93936	0.104511
$792$	0	0
$793$	5.34048	0.189646
$794$	0	0
$795$	0	0
$796$	0	0
$797$	27.4822	0.973471	0.486736	−	0.873549i	$-0.338188\pi$
$797$	27.4822	0.973471	0.486736	+	0.873549i	$0.338188\pi$
$798$	0	0
$799$	30.1271	1.06582
$800$	0	0
$801$	0	0
$802$	0	0
$803$	6.43789	0.227188
$804$	0	0
$805$	7.51142	0.264743
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−10.6653	−0.374970	−0.187485	−	0.982267i	$-0.560034\pi$
$809$	−10.6653	−0.374970	−0.187485	+	0.982267i	$0.560034\pi$
$810$	0	0
$811$	−51.1571	−1.79637	−0.898185	−	0.439618i	$-0.855114\pi$
$811$	−51.1571	−1.79637	−0.898185	+	0.439618i	$0.855114\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	6.05180	0.211985
$816$	0	0
$817$	−34.9591	−1.22306
$818$	0	0
$819$	0	0
$820$	0	0
$821$	43.4963	1.51803	0.759016	−	0.651072i	$-0.225680\pi$
$821$	43.4963	1.51803	0.759016	+	0.651072i	$0.225680\pi$
$822$	0	0
$823$	−9.14151	−0.318653	−0.159326	−	0.987226i	$-0.550932\pi$
$823$	−9.14151	−0.318653	−0.159326	+	0.987226i	$0.550932\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−9.49176	−0.330061	−0.165030	−	0.986288i	$-0.552772\pi$
$827$	−9.49176	−0.330061	−0.165030	+	0.986288i	$0.552772\pi$
$828$	0	0
$829$	56.1495	1.95015	0.975076	−	0.221873i	$-0.0712171\pi$
$829$	56.1495	1.95015	0.975076	+	0.221873i	$0.0712171\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−22.4955	−0.779424
$834$	0	0
$835$	22.0762	0.763978
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−22.0864	−0.762509	−0.381254	−	0.924470i	$-0.624508\pi$
$839$	−22.0864	−0.762509	−0.381254	+	0.924470i	$0.624508\pi$
$840$	0	0
$841$	−28.9923	−0.999736
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−12.3218	−0.423883
$846$	0	0
$847$	0.880517	0.0302549
$848$	0	0
$849$	0	0
$850$	0	0
$851$	8.13193	0.278759
$852$	0	0
$853$	−45.0263	−1.54167	−0.770835	−	0.637035i	$-0.780161\pi$
$853$	−45.0263	−1.54167	−0.770835	+	0.637035i	$0.780161\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−11.9896	−0.409556	−0.204778	−	0.978808i	$-0.565647\pi$
$857$	−11.9896	−0.409556	−0.204778	+	0.978808i	$0.565647\pi$
$858$	0	0
$859$	30.0104	1.02394	0.511971	−	0.859002i	$-0.328915\pi$
$859$	30.0104	1.02394	0.511971	+	0.859002i	$0.328915\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−50.7888	−1.72887	−0.864435	−	0.502744i	$-0.832324\pi$
$863$	−50.7888	−1.72887	−0.864435	+	0.502744i	$0.832324\pi$
$864$	0	0
$865$	19.6523	0.668200
$866$	0	0
$867$	0	0
$868$	0	0
$869$	5.38486	0.182669
$870$	0	0
$871$	3.60792	0.122250
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0.923694	0.0312266
$876$	0	0
$877$	−41.6117	−1.40513	−0.702563	−	0.711621i	$-0.747961\pi$
$877$	−41.6117	−1.40513	−0.702563	+	0.711621i	$0.747961\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−20.3810	−0.686655	−0.343327	−	0.939216i	$-0.611554\pi$
$881$	−20.3810	−0.686655	−0.343327	+	0.939216i	$0.611554\pi$
$882$	0	0
$883$	−18.0761	−0.608309	−0.304154	−	0.952623i	$-0.598374\pi$
$883$	−18.0761	−0.608309	−0.304154	+	0.952623i	$0.598374\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	41.2666	1.38560	0.692799	−	0.721131i	$-0.256377\pi$
$887$	41.2666	1.38560	0.692799	+	0.721131i	$0.256377\pi$
$888$	0	0
$889$	−14.9886	−0.502700
$890$	0	0
$891$	0	0
$892$	0	0
$893$	24.0983	0.806418
$894$	0	0
$895$	2.87533	0.0961117
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−0.135182	−0.00450859
$900$	0	0
$901$	9.99968	0.333138
$902$	0	0
$903$	0	0
$904$	0	0
$905$	24.0147	0.798277
$906$	0	0
$907$	−38.7872	−1.28791	−0.643953	−	0.765065i	$-0.722707\pi$
$907$	−38.7872	−1.28791	−0.643953	+	0.765065i	$0.722707\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−47.4805	−1.57310	−0.786549	−	0.617528i	$-0.788134\pi$
$911$	−47.4805	−1.57310	−0.786549	+	0.617528i	$0.788134\pi$
$912$	0	0
$913$	−28.6228	−0.947277
$914$	0	0
$915$	0	0
$916$	0	0
$917$	17.2530	0.569744
$918$	0	0
$919$	27.9927	0.923393	0.461697	−	0.887038i	$-0.347241\pi$
$919$	27.9927	0.923393	0.461697	+	0.887038i	$0.347241\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−8.09613	−0.266487
$924$	0	0
$925$	1.00000	0.0328798
$926$	0	0
$927$	0	0
$928$	0	0
$929$	14.5280	0.476647	0.238323	−	0.971186i	$-0.423402\pi$
$929$	14.5280	0.476647	0.238323	+	0.971186i	$0.423402\pi$
$930$	0	0
$931$	−17.9938	−0.589724
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−12.6529	−0.413795
$936$	0	0
$937$	60.9338	1.99062	0.995311	−	0.0967285i	$-0.0308379\pi$
$937$	60.9338	1.99062	0.995311	+	0.0967285i	$0.0308379\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	30.7754	1.00325	0.501624	−	0.865086i	$-0.332736\pi$
$941$	30.7754	1.00325	0.501624	+	0.865086i	$0.332736\pi$
$942$	0	0
$943$	36.7478	1.19667
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−50.6092	−1.64458	−0.822290	−	0.569069i	$-0.807304\pi$
$947$	−50.6092	−1.64458	−0.822290	+	0.569069i	$0.807304\pi$
$948$	0	0
$949$	−1.53349	−0.0497791
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−15.7362	−0.509746	−0.254873	−	0.966974i	$-0.582034\pi$
$953$	−15.7362	−0.509746	−0.254873	+	0.966974i	$0.582034\pi$
$954$	0	0
$955$	3.44265	0.111401
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−10.3921	−0.335579
$960$	0	0
$961$	−28.6134	−0.923014
$962$	0	0
$963$	0	0
$964$	0	0
$965$	14.9517	0.481314
$966$	0	0
$967$	11.4163	0.367125	0.183562	−	0.983008i	$-0.441237\pi$
$967$	11.4163	0.367125	0.183562	+	0.983008i	$0.441237\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−46.2916	−1.48557	−0.742784	−	0.669531i	$-0.766495\pi$
$971$	−46.2916	−1.48557	−0.742784	+	0.669531i	$0.766495\pi$
$972$	0	0
$973$	2.57381	0.0825125
$974$	0	0
$975$	0	0
$976$	0	0
$977$	42.0661	1.34581	0.672906	−	0.739728i	$-0.265046\pi$
$977$	42.0661	1.34581	0.672906	+	0.739728i	$0.265046\pi$
$978$	0	0
$979$	8.62922	0.275791
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.45387	0.0463711	0.0231855	−	0.999731i	$-0.492619\pi$
$983$	1.45387	0.0463711	0.0231855	+	0.999731i	$0.492619\pi$
$984$	0	0
$985$	−10.4662	−0.333482
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−97.1132	−3.08802
$990$	0	0
$991$	−20.2131	−0.642090	−0.321045	−	0.947064i	$-0.604034\pi$
$991$	−20.2131	−0.642090	−0.321045	+	0.947064i	$0.604034\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	16.6419	0.527583
$996$	0	0
$997$	5.22390	0.165443	0.0827213	−	0.996573i	$-0.473639\pi$
$997$	5.22390	0.165443	0.0827213	+	0.996573i	$0.473639\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6660.2.a.v.1.3	yes	5
3.2	odd	2		6660.2.a.t.1.3	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6660.2.a.t.1.3	✓	5	3.2	odd	2
6660.2.a.v.1.3	yes	5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +0.923694 q^{7} -3.45735 q^{11} +0.823530 q^{13} +3.65972 q^{17} +2.92736 q^{19} +8.13193 q^{23} +1.00000 q^{25} +0.0875050 q^{29} -1.54485 q^{31} +0.923694 q^{35} +1.00000 q^{37} +4.51895 q^{41} -11.9422 q^{43} +8.23209 q^{47} -6.14679 q^{49} +2.73236 q^{53} -3.45735 q^{55} -10.4128 q^{59} +6.48487 q^{61} +0.823530 q^{65} +4.38104 q^{67} -9.83101 q^{71} -1.86209 q^{73} -3.19353 q^{77} -1.55751 q^{79} +8.27884 q^{83} +3.65972 q^{85} -2.49591 q^{89} +0.760690 q^{91} +2.92736 q^{95} -3.56322 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{5} - 4 q^{11} - 4 q^{13} + 10 q^{17} - 4 q^{19} - 4 q^{23} + 5 q^{25} - 4 q^{29} + 10 q^{31} + 5 q^{37} + 8 q^{41} - 18 q^{43} + 37 q^{49} + 24 q^{53} - 4 q^{55} + 14 q^{59} + 4 q^{61} - 4 q^{65}+ \cdots + 8 q^{97}+O(q^{100}) \) 5 * q + 5 * q^5 - 4 * q^11 - 4 * q^13 + 10 * q^17 - 4 * q^19 - 4 * q^23 + 5 * q^25 - 4 * q^29 + 10 * q^31 + 5 * q^37 + 8 * q^41 - 18 * q^43 + 37 * q^49 + 24 * q^53 - 4 * q^55 + 14 * q^59 + 4 * q^61 - 4 * q^65 + 4 * q^67 + 16 * q^71 - 6 * q^73 + 30 * q^77 + 2 * q^79 + 22 * q^83 + 10 * q^85 - 4 * q^89 - 18 * q^91 - 4 * q^95 + 8 * q^97

Introduction

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