LMFDB - Embedded newform 1452.4.a.o.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1452,4,Mod(1,1452)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1452, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1452.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 1452 = 2^{2} \cdot 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1452.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,-12,0,-6,0,-11] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$85.6707733283$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.885025.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 93x^{2} + 94x + 2189 $ x^4 - 2x^3 - 93x^2 + 94*x + 2189
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 132)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$7.04048$ of defining polynomial
Character	$\chi$	$=$	1452.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -1.66773 q^{5} +27.3211 q^{7} +9.00000 q^{9} -6.61093 q^{13} +5.00318 q^{15} -25.9424 q^{17} -25.0751 q^{19} -81.9633 q^{21} -89.6533 q^{23} -122.219 q^{25} -27.0000 q^{27} +181.874 q^{29} +62.9586 q^{31} -45.5641 q^{35} -332.007 q^{37} +19.8328 q^{39} -41.3231 q^{41} -9.61174 q^{43} -15.0095 q^{45} -360.922 q^{47} +403.443 q^{49} +77.8271 q^{51} -513.601 q^{53} +75.2252 q^{57} +211.844 q^{59} +645.477 q^{61} +245.890 q^{63} +11.0252 q^{65} +947.111 q^{67} +268.960 q^{69} +486.431 q^{71} -795.268 q^{73} +366.656 q^{75} +652.849 q^{79} +81.0000 q^{81} -1421.81 q^{83} +43.2647 q^{85} -545.621 q^{87} -754.014 q^{89} -180.618 q^{91} -188.876 q^{93} +41.8183 q^{95} +446.787 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{3} - 6 q^{5} - 11 q^{7} + 36 q^{9} + 158 q^{13} + 18 q^{15} - 117 q^{17} - 139 q^{19} + 33 q^{21} - 174 q^{23} - 122 q^{25} - 108 q^{27} - 177 q^{29} + 104 q^{31} + 285 q^{35} + 240 q^{37} - 474 q^{39}+ \cdots + 2016 q^{97}+O(q^{100}) $ 4 * q - 12 * q^3 - 6 * q^5 - 11 * q^7 + 36 * q^9 + 158 * q^13 + 18 * q^15 - 117 * q^17 - 139 * q^19 + 33 * q^21 - 174 * q^23 - 122 * q^25 - 108 * q^27 - 177 * q^29 + 104 * q^31 + 285 * q^35 + 240 * q^37 - 474 * q^39 + 534 * q^41 + 342 * q^43 - 54 * q^45 - 99 * q^47 + 1041 * q^49 + 351 * q^51 - 114 * q^53 + 417 * q^57 + 108 * q^59 + 1007 * q^61 - 99 * q^63 - 57 * q^65 + 235 * q^67 + 522 * q^69 - 273 * q^71 - 869 * q^73 + 366 * q^75 + 63 * q^79 + 324 * q^81 - 2025 * q^83 - 320 * q^85 + 531 * q^87 - 858 * q^89 - 4492 * q^91 - 312 * q^93 - 33 * q^95 + 2016 * 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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	−1.66773	−0.149166	−0.0745829	−	0.997215i	$-0.523763\pi$
$5$	−1.66773	−0.149166	−0.0745829	+	0.997215i	$0.523763\pi$
$6$	0	0
$7$	27.3211	1.47520	0.737600	−	0.675237i	$-0.235959\pi$
$7$	27.3211	1.47520	0.737600	+	0.675237i	$0.235959\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−6.61093	−0.141042	−0.0705209	−	0.997510i	$-0.522466\pi$
$13$	−6.61093	−0.141042	−0.0705209	+	0.997510i	$0.522466\pi$
$14$	0	0
$15$	5.00318	0.0861210
$16$	0	0
$17$	−25.9424	−0.370115	−0.185057	−	0.982728i	$-0.559247\pi$
$17$	−25.9424	−0.370115	−0.185057	+	0.982728i	$0.559247\pi$
$18$	0	0
$19$	−25.0751	−0.302769	−0.151385	−	0.988475i	$-0.548373\pi$
$19$	−25.0751	−0.302769	−0.151385	+	0.988475i	$0.548373\pi$
$20$	0	0
$21$	−81.9633	−0.851708
$22$	0	0
$23$	−89.6533	−0.812783	−0.406391	−	0.913699i	$-0.633213\pi$
$23$	−89.6533	−0.812783	−0.406391	+	0.913699i	$0.633213\pi$
$24$	0	0
$25$	−122.219	−0.977750
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	181.874	1.16459	0.582294	−	0.812978i	$-0.302155\pi$
$29$	181.874	1.16459	0.582294	+	0.812978i	$0.302155\pi$
$30$	0	0
$31$	62.9586	0.364765	0.182382	−	0.983228i	$-0.441619\pi$
$31$	62.9586	0.364765	0.182382	+	0.983228i	$0.441619\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−45.5641	−0.220050
$36$	0	0
$37$	−332.007	−1.47518	−0.737588	−	0.675251i	$-0.764035\pi$
$37$	−332.007	−1.47518	−0.737588	+	0.675251i	$0.764035\pi$
$38$	0	0
$39$	19.8328	0.0814305
$40$	0	0
$41$	−41.3231	−0.157405	−0.0787023	−	0.996898i	$-0.525078\pi$
$41$	−41.3231	−0.157405	−0.0787023	+	0.996898i	$0.525078\pi$
$42$	0	0
$43$	−9.61174	−0.0340878	−0.0170439	−	0.999855i	$-0.505426\pi$
$43$	−9.61174	−0.0340878	−0.0170439	+	0.999855i	$0.505426\pi$
$44$	0	0
$45$	−15.0095	−0.0497220
$46$	0	0
$47$	−360.922	−1.12013	−0.560063	−	0.828450i	$-0.689223\pi$
$47$	−360.922	−1.12013	−0.560063	+	0.828450i	$0.689223\pi$
$48$	0	0
$49$	403.443	1.17622
$50$	0	0
$51$	77.8271	0.213686
$52$	0	0
$53$	−513.601	−1.33110	−0.665552	−	0.746352i	$-0.731804\pi$
$53$	−513.601	−1.33110	−0.665552	+	0.746352i	$0.731804\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	75.2252	0.174804
$58$	0	0
$59$	211.844	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$59$	211.844	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$60$	0	0
$61$	645.477	1.35483	0.677417	−	0.735599i	$-0.263099\pi$
$61$	645.477	1.35483	0.677417	+	0.735599i	$0.263099\pi$
$62$	0	0
$63$	245.890	0.491734
$64$	0	0
$65$	11.0252	0.0210386
$66$	0	0
$67$	947.111	1.72699	0.863493	−	0.504361i	$-0.168272\pi$
$67$	947.111	1.72699	0.863493	+	0.504361i	$0.168272\pi$
$68$	0	0
$69$	268.960	0.469260
$70$	0	0
$71$	486.431	0.813081	0.406540	−	0.913633i	$-0.366735\pi$
$71$	486.431	0.813081	0.406540	+	0.913633i	$0.366735\pi$
$72$	0	0
$73$	−795.268	−1.27506	−0.637528	−	0.770427i	$-0.720043\pi$
$73$	−795.268	−1.27506	−0.637528	+	0.770427i	$0.720043\pi$
$74$	0	0
$75$	366.656	0.564504
$76$	0	0
$77$	0	0
$78$	0	0
$79$	652.849	0.929762	0.464881	−	0.885373i	$-0.346097\pi$
$79$	652.849	0.929762	0.464881	+	0.885373i	$0.346097\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−1421.81	−1.88028	−0.940141	−	0.340785i	$-0.889307\pi$
$83$	−1421.81	−1.88028	−0.940141	+	0.340785i	$0.889307\pi$
$84$	0	0
$85$	43.2647	0.0552085
$86$	0	0
$87$	−545.621	−0.672376
$88$	0	0
$89$	−754.014	−0.898038	−0.449019	−	0.893522i	$-0.648226\pi$
$89$	−754.014	−0.898038	−0.449019	+	0.893522i	$0.648226\pi$
$90$	0	0
$91$	−180.618	−0.208065
$92$	0	0
$93$	−188.876	−0.210597
$94$	0	0
$95$	41.8183	0.0451628
$96$	0	0
$97$	446.787	0.467674	0.233837	−	0.972276i	$-0.424872\pi$
$97$	446.787	0.467674	0.233837	+	0.972276i	$0.424872\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1254.57	1.23598	0.617990	−	0.786186i	$-0.287947\pi$
$101$	1254.57	1.23598	0.617990	+	0.786186i	$0.287947\pi$
$102$	0	0
$103$	582.126	0.556880	0.278440	−	0.960454i	$-0.410183\pi$
$103$	582.126	0.556880	0.278440	+	0.960454i	$0.410183\pi$
$104$	0	0
$105$	136.692	0.127046
$106$	0	0
$107$	−561.090	−0.506940	−0.253470	−	0.967343i	$-0.581572\pi$
$107$	−561.090	−0.506940	−0.253470	+	0.967343i	$0.581572\pi$
$108$	0	0
$109$	−1339.20	−1.17681	−0.588405	−	0.808566i	$-0.700244\pi$
$109$	−1339.20	−1.17681	−0.588405	+	0.808566i	$0.700244\pi$
$110$	0	0
$111$	996.020	0.851694
$112$	0	0
$113$	−863.293	−0.718688	−0.359344	−	0.933205i	$-0.617000\pi$
$113$	−863.293	−0.718688	−0.359344	+	0.933205i	$0.617000\pi$
$114$	0	0
$115$	149.517	0.121239
$116$	0	0
$117$	−59.4984	−0.0470139
$118$	0	0
$119$	−708.774	−0.545993
$120$	0	0
$121$	0	0
$122$	0	0
$123$	123.969	0.0908776
$124$	0	0
$125$	412.293	0.295013
$126$	0	0
$127$	−2328.51	−1.62694	−0.813472	−	0.581604i	$-0.802425\pi$
$127$	−2328.51	−1.62694	−0.813472	+	0.581604i	$0.802425\pi$
$128$	0	0
$129$	28.8352	0.0196806
$130$	0	0
$131$	−762.449	−0.508515	−0.254257	−	0.967137i	$-0.581831\pi$
$131$	−762.449	−0.508515	−0.254257	+	0.967137i	$0.581831\pi$
$132$	0	0
$133$	−685.078	−0.446645
$134$	0	0
$135$	45.0286	0.0287070
$136$	0	0
$137$	879.336	0.548370	0.274185	−	0.961677i	$-0.411592\pi$
$137$	879.336	0.548370	0.274185	+	0.961677i	$0.411592\pi$
$138$	0	0
$139$	513.121	0.313111	0.156555	−	0.987669i	$-0.449961\pi$
$139$	513.121	0.313111	0.156555	+	0.987669i	$0.449961\pi$
$140$	0	0
$141$	1082.77	0.646705
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−303.315	−0.173717
$146$	0	0
$147$	−1210.33	−0.679089
$148$	0	0
$149$	−944.681	−0.519405	−0.259702	−	0.965689i	$-0.583624\pi$
$149$	−944.681	−0.519405	−0.259702	+	0.965689i	$0.583624\pi$
$150$	0	0
$151$	1725.70	0.930036	0.465018	−	0.885301i	$-0.346048\pi$
$151$	1725.70	0.930036	0.465018	+	0.885301i	$0.346048\pi$
$152$	0	0
$153$	−233.481	−0.123372
$154$	0	0
$155$	−104.998	−0.0544104
$156$	0	0
$157$	−2339.29	−1.18914	−0.594571	−	0.804043i	$-0.702678\pi$
$157$	−2339.29	−1.18914	−0.594571	+	0.804043i	$0.702678\pi$
$158$	0	0
$159$	1540.80	0.768513
$160$	0	0
$161$	−2449.43	−1.19902
$162$	0	0
$163$	2350.19	1.12933	0.564665	−	0.825320i	$-0.309005\pi$
$163$	2350.19	1.12933	0.564665	+	0.825320i	$0.309005\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3373.46	−1.56315	−0.781576	−	0.623810i	$-0.785584\pi$
$167$	−3373.46	−1.56315	−0.781576	+	0.623810i	$0.785584\pi$
$168$	0	0
$169$	−2153.30	−0.980107
$170$	0	0
$171$	−225.675	−0.100923
$172$	0	0
$173$	−1789.32	−0.786356	−0.393178	−	0.919462i	$-0.628624\pi$
$173$	−1789.32	−0.786356	−0.393178	+	0.919462i	$0.628624\pi$
$174$	0	0
$175$	−3339.15	−1.44238
$176$	0	0
$177$	−635.533	−0.269885
$178$	0	0
$179$	−3344.50	−1.39653	−0.698267	−	0.715837i	$-0.746045\pi$
$179$	−3344.50	−1.39653	−0.698267	+	0.715837i	$0.746045\pi$
$180$	0	0
$181$	1778.16	0.730219	0.365110	−	0.930965i	$-0.381031\pi$
$181$	1778.16	0.730219	0.365110	+	0.930965i	$0.381031\pi$
$182$	0	0
$183$	−1936.43	−0.782214
$184$	0	0
$185$	553.696	0.220046
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−737.670	−0.283903
$190$	0	0
$191$	−3908.30	−1.48060	−0.740300	−	0.672277i	$-0.765316\pi$
$191$	−3908.30	−1.48060	−0.740300	+	0.672277i	$0.765316\pi$
$192$	0	0
$193$	−359.764	−0.134178	−0.0670892	−	0.997747i	$-0.521371\pi$
$193$	−359.764	−0.134178	−0.0670892	+	0.997747i	$0.521371\pi$
$194$	0	0
$195$	−33.0757	−0.0121467
$196$	0	0
$197$	−4139.93	−1.49725	−0.748624	−	0.662995i	$-0.769285\pi$
$197$	−4139.93	−1.49725	−0.748624	+	0.662995i	$0.769285\pi$
$198$	0	0
$199$	−3664.80	−1.30548	−0.652740	−	0.757582i	$-0.726381\pi$
$199$	−3664.80	−1.30548	−0.652740	+	0.757582i	$0.726381\pi$
$200$	0	0
$201$	−2841.33	−0.997076
$202$	0	0
$203$	4968.99	1.71800
$204$	0	0
$205$	68.9156	0.0234794
$206$	0	0
$207$	−806.879	−0.270928
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−1653.43	−0.539465	−0.269733	−	0.962935i	$-0.586935\pi$
$211$	−1653.43	−0.539465	−0.269733	+	0.962935i	$0.586935\pi$
$212$	0	0
$213$	−1459.29	−0.469432
$214$	0	0
$215$	16.0297	0.00508474
$216$	0	0
$217$	1720.10	0.538101
$218$	0	0
$219$	2385.80	0.736154
$220$	0	0
$221$	171.503	0.0522016
$222$	0	0
$223$	−74.3016	−0.0223121	−0.0111561	−	0.999938i	$-0.503551\pi$
$223$	−74.3016	−0.0223121	−0.0111561	+	0.999938i	$0.503551\pi$
$224$	0	0
$225$	−1099.97	−0.325917
$226$	0	0
$227$	−5850.02	−1.71048	−0.855241	−	0.518231i	$-0.826591\pi$
$227$	−5850.02	−1.71048	−0.855241	+	0.518231i	$0.826591\pi$
$228$	0	0
$229$	2051.85	0.592097	0.296048	−	0.955173i	$-0.404331\pi$
$229$	2051.85	0.592097	0.296048	+	0.955173i	$0.404331\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4407.45	1.23924	0.619618	−	0.784904i	$-0.287288\pi$
$233$	4407.45	1.23924	0.619618	+	0.784904i	$0.287288\pi$
$234$	0	0
$235$	601.919	0.167084
$236$	0	0
$237$	−1958.55	−0.536799
$238$	0	0
$239$	1411.70	0.382072	0.191036	−	0.981583i	$-0.438815\pi$
$239$	1411.70	0.382072	0.191036	+	0.981583i	$0.438815\pi$
$240$	0	0
$241$	−1511.63	−0.404035	−0.202018	−	0.979382i	$-0.564750\pi$
$241$	−1511.63	−0.404035	−0.202018	+	0.979382i	$0.564750\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−672.831	−0.175451
$246$	0	0
$247$	165.770	0.0427031
$248$	0	0
$249$	4265.42	1.08558
$250$	0	0
$251$	4759.95	1.19699	0.598497	−	0.801125i	$-0.295765\pi$
$251$	4759.95	1.19699	0.598497	+	0.801125i	$0.295765\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−129.794	−0.0318746
$256$	0	0
$257$	6414.55	1.55692	0.778461	−	0.627693i	$-0.216001\pi$
$257$	6414.55	1.55692	0.778461	+	0.627693i	$0.216001\pi$
$258$	0	0
$259$	−9070.78	−2.17618
$260$	0	0
$261$	1636.86	0.388196
$262$	0	0
$263$	−6345.85	−1.48784	−0.743920	−	0.668269i	$-0.767036\pi$
$263$	−6345.85	−1.48784	−0.743920	+	0.668269i	$0.767036\pi$
$264$	0	0
$265$	856.545	0.198555
$266$	0	0
$267$	2262.04	0.518482
$268$	0	0
$269$	977.706	0.221605	0.110803	−	0.993842i	$-0.464658\pi$
$269$	977.706	0.221605	0.110803	+	0.993842i	$0.464658\pi$
$270$	0	0
$271$	−3415.84	−0.765673	−0.382837	−	0.923816i	$-0.625053\pi$
$271$	−3415.84	−0.765673	−0.382837	+	0.923816i	$0.625053\pi$
$272$	0	0
$273$	541.854	0.120126
$274$	0	0
$275$	0	0
$276$	0	0
$277$	3537.59	0.767340	0.383670	−	0.923470i	$-0.374660\pi$
$277$	3537.59	0.767340	0.383670	+	0.923470i	$0.374660\pi$
$278$	0	0
$279$	566.628	0.121588
$280$	0	0
$281$	−4012.49	−0.851833	−0.425917	−	0.904762i	$-0.640048\pi$
$281$	−4012.49	−0.851833	−0.425917	+	0.904762i	$0.640048\pi$
$282$	0	0
$283$	−2856.26	−0.599954	−0.299977	−	0.953946i	$-0.596979\pi$
$283$	−2856.26	−0.599954	−0.299977	+	0.953946i	$0.596979\pi$
$284$	0	0
$285$	−125.455	−0.0260748
$286$	0	0
$287$	−1128.99	−0.232203
$288$	0	0
$289$	−4239.99	−0.863015
$290$	0	0
$291$	−1340.36	−0.270012
$292$	0	0
$293$	−6744.61	−1.34479	−0.672397	−	0.740191i	$-0.734735\pi$
$293$	−6744.61	−1.34479	−0.672397	+	0.740191i	$0.734735\pi$
$294$	0	0
$295$	−353.298	−0.0697282
$296$	0	0
$297$	0	0
$298$	0	0
$299$	592.692	0.114636
$300$	0	0
$301$	−262.603	−0.0502864
$302$	0	0
$303$	−3763.70	−0.713593
$304$	0	0
$305$	−1076.48	−0.202095
$306$	0	0
$307$	3900.45	0.725116	0.362558	−	0.931961i	$-0.381904\pi$
$307$	3900.45	0.725116	0.362558	+	0.931961i	$0.381904\pi$
$308$	0	0
$309$	−1746.38	−0.321515
$310$	0	0
$311$	5428.47	0.989775	0.494888	−	0.868957i	$-0.335209\pi$
$311$	5428.47	0.989775	0.494888	+	0.868957i	$0.335209\pi$
$312$	0	0
$313$	5143.49	0.928841	0.464421	−	0.885615i	$-0.346263\pi$
$313$	5143.49	0.928841	0.464421	+	0.885615i	$0.346263\pi$
$314$	0	0
$315$	−410.077	−0.0733499
$316$	0	0
$317$	2628.40	0.465697	0.232848	−	0.972513i	$-0.425195\pi$
$317$	2628.40	0.465697	0.232848	+	0.972513i	$0.425195\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	1683.27	0.292682
$322$	0	0
$323$	650.506	0.112059
$324$	0	0
$325$	807.980	0.137904
$326$	0	0
$327$	4017.61	0.679432
$328$	0	0
$329$	−9860.79	−1.65241
$330$	0	0
$331$	2053.63	0.341020	0.170510	−	0.985356i	$-0.445459\pi$
$331$	2053.63	0.341020	0.170510	+	0.985356i	$0.445459\pi$
$332$	0	0
$333$	−2988.06	−0.491726
$334$	0	0
$335$	−1579.52	−0.257607
$336$	0	0
$337$	191.631	0.0309757	0.0154878	−	0.999880i	$-0.495070\pi$
$337$	191.631	0.0309757	0.0154878	+	0.999880i	$0.495070\pi$
$338$	0	0
$339$	2589.88	0.414935
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1651.36	0.259956
$344$	0	0
$345$	−448.551	−0.0699976
$346$	0	0
$347$	7135.84	1.10395	0.551977	−	0.833859i	$-0.313874\pi$
$347$	7135.84	1.10395	0.551977	+	0.833859i	$0.313874\pi$
$348$	0	0
$349$	4635.13	0.710924	0.355462	−	0.934691i	$-0.384323\pi$
$349$	4635.13	0.710924	0.355462	+	0.934691i	$0.384323\pi$
$350$	0	0
$351$	178.495	0.0271435
$352$	0	0
$353$	8016.13	1.20866	0.604328	−	0.796736i	$-0.293442\pi$
$353$	8016.13	1.20866	0.604328	+	0.796736i	$0.293442\pi$
$354$	0	0
$355$	−811.233	−0.121284
$356$	0	0
$357$	2126.32	0.315229
$358$	0	0
$359$	−1255.97	−0.184645	−0.0923223	−	0.995729i	$-0.529429\pi$
$359$	−1255.97	−0.184645	−0.0923223	+	0.995729i	$0.529429\pi$
$360$	0	0
$361$	−6230.24	−0.908331
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1326.29	0.190195
$366$	0	0
$367$	−13377.7	−1.90275	−0.951375	−	0.308036i	$-0.900328\pi$
$367$	−13377.7	−1.90275	−0.951375	+	0.308036i	$0.900328\pi$
$368$	0	0
$369$	−371.908	−0.0524682
$370$	0	0
$371$	−14032.1	−1.96364
$372$	0	0
$373$	9231.57	1.28148	0.640741	−	0.767757i	$-0.278627\pi$
$373$	9231.57	1.28148	0.640741	+	0.767757i	$0.278627\pi$
$374$	0	0
$375$	−1236.88	−0.170326
$376$	0	0
$377$	−1202.35	−0.164256
$378$	0	0
$379$	−2186.92	−0.296397	−0.148198	−	0.988958i	$-0.547347\pi$
$379$	−2186.92	−0.296397	−0.148198	+	0.988958i	$0.547347\pi$
$380$	0	0
$381$	6985.53	0.939317
$382$	0	0
$383$	−14022.7	−1.87082	−0.935411	−	0.353562i	$-0.884970\pi$
$383$	−14022.7	−1.87082	−0.935411	+	0.353562i	$0.884970\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−86.5056	−0.0113626
$388$	0	0
$389$	9801.96	1.27758	0.638790	−	0.769381i	$-0.279435\pi$
$389$	9801.96	1.27758	0.638790	+	0.769381i	$0.279435\pi$
$390$	0	0
$391$	2325.82	0.300823
$392$	0	0
$393$	2287.35	0.293591
$394$	0	0
$395$	−1088.77	−0.138689
$396$	0	0
$397$	10353.5	1.30889	0.654445	−	0.756109i	$-0.272902\pi$
$397$	10353.5	1.30889	0.654445	+	0.756109i	$0.272902\pi$
$398$	0	0
$399$	2055.23	0.257871
$400$	0	0
$401$	10636.7	1.32461	0.662307	−	0.749232i	$-0.269577\pi$
$401$	10636.7	1.32461	0.662307	+	0.749232i	$0.269577\pi$
$402$	0	0
$403$	−416.215	−0.0514471
$404$	0	0
$405$	−135.086	−0.0165740
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−10369.1	−1.25359	−0.626794	−	0.779185i	$-0.715633\pi$
$409$	−10369.1	−1.25359	−0.626794	+	0.779185i	$0.715633\pi$
$410$	0	0
$411$	−2638.01	−0.316602
$412$	0	0
$413$	5787.82	0.689588
$414$	0	0
$415$	2371.18	0.280474
$416$	0	0
$417$	−1539.36	−0.180775
$418$	0	0
$419$	−16756.8	−1.95375	−0.976876	−	0.213809i	$-0.931413\pi$
$419$	−16756.8	−1.95375	−0.976876	+	0.213809i	$0.931413\pi$
$420$	0	0
$421$	−6222.69	−0.720369	−0.360185	−	0.932881i	$-0.617286\pi$
$421$	−6222.69	−0.720369	−0.360185	+	0.932881i	$0.617286\pi$
$422$	0	0
$423$	−3248.30	−0.373375
$424$	0	0
$425$	3170.64	0.361879
$426$	0	0
$427$	17635.1	1.99865
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−3917.50	−0.437817	−0.218909	−	0.975745i	$-0.570250\pi$
$431$	−3917.50	−0.437817	−0.218909	+	0.975745i	$0.570250\pi$
$432$	0	0
$433$	−8133.11	−0.902661	−0.451330	−	0.892357i	$-0.649050\pi$
$433$	−8133.11	−0.902661	−0.451330	+	0.892357i	$0.649050\pi$
$434$	0	0
$435$	909.945	0.100295
$436$	0	0
$437$	2248.06	0.246085
$438$	0	0
$439$	−7855.20	−0.854006	−0.427003	−	0.904250i	$-0.640431\pi$
$439$	−7855.20	−0.854006	−0.427003	+	0.904250i	$0.640431\pi$
$440$	0	0
$441$	3630.98	0.392072
$442$	0	0
$443$	−15877.0	−1.70279	−0.851397	−	0.524522i	$-0.824244\pi$
$443$	−15877.0	−1.70279	−0.851397	+	0.524522i	$0.824244\pi$
$444$	0	0
$445$	1257.49	0.133957
$446$	0	0
$447$	2834.04	0.299878
$448$	0	0
$449$	−17223.2	−1.81027	−0.905137	−	0.425121i	$-0.860232\pi$
$449$	−17223.2	−1.81027	−0.905137	+	0.425121i	$0.860232\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−5177.10	−0.536957
$454$	0	0
$455$	301.221	0.0310362
$456$	0	0
$457$	10859.3	1.11154	0.555771	−	0.831335i	$-0.312423\pi$
$457$	10859.3	1.11154	0.555771	+	0.831335i	$0.312423\pi$
$458$	0	0
$459$	700.444	0.0712286
$460$	0	0
$461$	6639.83	0.670820	0.335410	−	0.942072i	$-0.391125\pi$
$461$	6639.83	0.670820	0.335410	+	0.942072i	$0.391125\pi$
$462$	0	0
$463$	−1039.92	−0.104383	−0.0521913	−	0.998637i	$-0.516621\pi$
$463$	−1039.92	−0.104383	−0.0521913	+	0.998637i	$0.516621\pi$
$464$	0	0
$465$	314.993	0.0314139
$466$	0	0
$467$	−1886.23	−0.186904	−0.0934519	−	0.995624i	$-0.529790\pi$
$467$	−1886.23	−0.186904	−0.0934519	+	0.995624i	$0.529790\pi$
$468$	0	0
$469$	25876.1	2.54765
$470$	0	0
$471$	7017.86	0.686552
$472$	0	0
$473$	0	0
$474$	0	0
$475$	3064.64	0.296032
$476$	0	0
$477$	−4622.41	−0.443701
$478$	0	0
$479$	−7128.06	−0.679936	−0.339968	−	0.940437i	$-0.610416\pi$
$479$	−7128.06	−0.679936	−0.339968	+	0.940437i	$0.610416\pi$
$480$	0	0
$481$	2194.87	0.208062
$482$	0	0
$483$	7348.28	0.692253
$484$	0	0
$485$	−745.118	−0.0697610
$486$	0	0
$487$	11726.1	1.09109	0.545543	−	0.838083i	$-0.316324\pi$
$487$	11726.1	1.09109	0.545543	+	0.838083i	$0.316324\pi$
$488$	0	0
$489$	−7050.56	−0.652020
$490$	0	0
$491$	514.617	0.0473001	0.0236501	−	0.999720i	$-0.492471\pi$
$491$	514.617	0.0473001	0.0236501	+	0.999720i	$0.492471\pi$
$492$	0	0
$493$	−4718.23	−0.431031
$494$	0	0
$495$	0	0
$496$	0	0
$497$	13289.8	1.19946
$498$	0	0
$499$	17802.4	1.59708	0.798542	−	0.601939i	$-0.205605\pi$
$499$	17802.4	1.59708	0.798542	+	0.601939i	$0.205605\pi$
$500$	0	0
$501$	10120.4	0.902487
$502$	0	0
$503$	−16949.1	−1.50243	−0.751217	−	0.660055i	$-0.770533\pi$
$503$	−16949.1	−1.50243	−0.751217	+	0.660055i	$0.770533\pi$
$504$	0	0
$505$	−2092.27	−0.184366
$506$	0	0
$507$	6459.89	0.565865
$508$	0	0
$509$	5216.69	0.454274	0.227137	−	0.973863i	$-0.427063\pi$
$509$	5216.69	0.454274	0.227137	+	0.973863i	$0.427063\pi$
$510$	0	0
$511$	−21727.6	−1.88096
$512$	0	0
$513$	677.026	0.0582679
$514$	0	0
$515$	−970.827	−0.0830675
$516$	0	0
$517$	0	0
$518$	0	0
$519$	5367.97	0.454003
$520$	0	0
$521$	13965.1	1.17432	0.587162	−	0.809470i	$-0.300245\pi$
$521$	13965.1	1.17432	0.587162	+	0.809470i	$0.300245\pi$
$522$	0	0
$523$	938.690	0.0784819	0.0392410	−	0.999230i	$-0.487506\pi$
$523$	938.690	0.0784819	0.0392410	+	0.999230i	$0.487506\pi$
$524$	0	0
$525$	10017.4	0.832757
$526$	0	0
$527$	−1633.30	−0.135005
$528$	0	0
$529$	−4129.29	−0.339384
$530$	0	0
$531$	1906.60	0.155818
$532$	0	0
$533$	273.184	0.0222006
$534$	0	0
$535$	935.743	0.0756182
$536$	0	0
$537$	10033.5	0.806289
$538$	0	0
$539$	0	0
$540$	0	0
$541$	15255.6	1.21237	0.606183	−	0.795325i	$-0.292700\pi$
$541$	15255.6	1.21237	0.606183	+	0.795325i	$0.292700\pi$
$542$	0	0
$543$	−5334.48	−0.421592
$544$	0	0
$545$	2233.42	0.175540
$546$	0	0
$547$	9060.49	0.708224	0.354112	−	0.935203i	$-0.384783\pi$
$547$	9060.49	0.708224	0.354112	+	0.935203i	$0.384783\pi$
$548$	0	0
$549$	5809.30	0.451611
$550$	0	0
$551$	−4560.49	−0.352601
$552$	0	0
$553$	17836.5	1.37159
$554$	0	0
$555$	−1661.09	−0.127044
$556$	0	0
$557$	−12659.4	−0.963011	−0.481506	−	0.876443i	$-0.659910\pi$
$557$	−12659.4	−0.963011	−0.481506	+	0.876443i	$0.659910\pi$
$558$	0	0
$559$	63.5426	0.00480781
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−17572.5	−1.31544	−0.657722	−	0.753261i	$-0.728480\pi$
$563$	−17572.5	−1.31544	−0.657722	+	0.753261i	$0.728480\pi$
$564$	0	0
$565$	1439.74	0.107204
$566$	0	0
$567$	2213.01	0.163911
$568$	0	0
$569$	−23691.6	−1.74553	−0.872764	−	0.488143i	$-0.837675\pi$
$569$	−23691.6	−1.74553	−0.872764	+	0.488143i	$0.837675\pi$
$570$	0	0
$571$	25916.4	1.89942	0.949709	−	0.313134i	$-0.101379\pi$
$571$	25916.4	1.89942	0.949709	+	0.313134i	$0.101379\pi$
$572$	0	0
$573$	11724.9	0.854825
$574$	0	0
$575$	10957.3	0.794698
$576$	0	0
$577$	−13650.5	−0.984880	−0.492440	−	0.870346i	$-0.663895\pi$
$577$	−13650.5	−0.984880	−0.492440	+	0.870346i	$0.663895\pi$
$578$	0	0
$579$	1079.29	0.0774679
$580$	0	0
$581$	−38845.3	−2.77379
$582$	0	0
$583$	0	0
$584$	0	0
$585$	99.2270	0.00701287
$586$	0	0
$587$	18316.7	1.28792	0.643961	−	0.765058i	$-0.277290\pi$
$587$	18316.7	1.28792	0.643961	+	0.765058i	$0.277290\pi$
$588$	0	0
$589$	−1578.69	−0.110439
$590$	0	0
$591$	12419.8	0.864436
$592$	0	0
$593$	17082.6	1.18296	0.591481	−	0.806319i	$-0.298543\pi$
$593$	17082.6	1.18296	0.591481	+	0.806319i	$0.298543\pi$
$594$	0	0
$595$	1182.04	0.0814436
$596$	0	0
$597$	10994.4	0.753719
$598$	0	0
$599$	−6204.85	−0.423244	−0.211622	−	0.977352i	$-0.567875\pi$
$599$	−6204.85	−0.423244	−0.211622	+	0.977352i	$0.567875\pi$
$600$	0	0
$601$	11452.0	0.777263	0.388631	−	0.921393i	$-0.372948\pi$
$601$	11452.0	0.777263	0.388631	+	0.921393i	$0.372948\pi$
$602$	0	0
$603$	8524.00	0.575662
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−20401.6	−1.36421	−0.682106	−	0.731253i	$-0.738936\pi$
$607$	−20401.6	−1.36421	−0.682106	+	0.731253i	$0.738936\pi$
$608$	0	0
$609$	−14907.0	−0.991889
$610$	0	0
$611$	2386.03	0.157985
$612$	0	0
$613$	8446.77	0.556545	0.278272	−	0.960502i	$-0.410238\pi$
$613$	8446.77	0.556545	0.278272	+	0.960502i	$0.410238\pi$
$614$	0	0
$615$	−206.747	−0.0135558
$616$	0	0
$617$	4392.68	0.286617	0.143309	−	0.989678i	$-0.454226\pi$
$617$	4392.68	0.286617	0.143309	+	0.989678i	$0.454226\pi$
$618$	0	0
$619$	26484.1	1.71968	0.859842	−	0.510560i	$-0.170562\pi$
$619$	26484.1	1.71968	0.859842	+	0.510560i	$0.170562\pi$
$620$	0	0
$621$	2420.64	0.156420
$622$	0	0
$623$	−20600.5	−1.32479
$624$	0	0
$625$	14589.7	0.933744
$626$	0	0
$627$	0	0
$628$	0	0
$629$	8613.04	0.545985
$630$	0	0
$631$	−3349.81	−0.211337	−0.105669	−	0.994401i	$-0.533698\pi$
$631$	−3349.81	−0.211337	−0.105669	+	0.994401i	$0.533698\pi$
$632$	0	0
$633$	4960.30	0.311460
$634$	0	0
$635$	3883.32	0.242684
$636$	0	0
$637$	−2667.13	−0.165896
$638$	0	0
$639$	4377.88	0.271027
$640$	0	0
$641$	1260.72	0.0776842	0.0388421	−	0.999245i	$-0.487633\pi$
$641$	1260.72	0.0776842	0.0388421	+	0.999245i	$0.487633\pi$
$642$	0	0
$643$	13766.3	0.844308	0.422154	−	0.906524i	$-0.361274\pi$
$643$	13766.3	0.844308	0.422154	+	0.906524i	$0.361274\pi$
$644$	0	0
$645$	−48.0892	−0.00293568
$646$	0	0
$647$	−9404.81	−0.571470	−0.285735	−	0.958309i	$-0.592238\pi$
$647$	−9404.81	−0.571470	−0.285735	+	0.958309i	$0.592238\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−5160.30	−0.310673
$652$	0	0
$653$	2810.10	0.168404	0.0842020	−	0.996449i	$-0.473166\pi$
$653$	2810.10	0.168404	0.0842020	+	0.996449i	$0.473166\pi$
$654$	0	0
$655$	1271.55	0.0758530
$656$	0	0
$657$	−7157.41	−0.425019
$658$	0	0
$659$	28829.5	1.70416	0.852078	−	0.523415i	$-0.175342\pi$
$659$	28829.5	1.70416	0.852078	+	0.523415i	$0.175342\pi$
$660$	0	0
$661$	1946.42	0.114534	0.0572671	−	0.998359i	$-0.481761\pi$
$661$	1946.42	0.114534	0.0572671	+	0.998359i	$0.481761\pi$
$662$	0	0
$663$	−514.510	−0.0301386
$664$	0	0
$665$	1142.52	0.0666242
$666$	0	0
$667$	−16305.6	−0.946558
$668$	0	0
$669$	222.905	0.0128819
$670$	0	0
$671$	0	0
$672$	0	0
$673$	10075.9	0.577113	0.288557	−	0.957463i	$-0.406825\pi$
$673$	10075.9	0.577113	0.288557	+	0.957463i	$0.406825\pi$
$674$	0	0
$675$	3299.90	0.188168
$676$	0	0
$677$	28273.7	1.60509	0.802546	−	0.596590i	$-0.203478\pi$
$677$	28273.7	1.60509	0.802546	+	0.596590i	$0.203478\pi$
$678$	0	0
$679$	12206.7	0.689913
$680$	0	0
$681$	17550.1	0.987547
$682$	0	0
$683$	−18285.8	−1.02443	−0.512215	−	0.858857i	$-0.671175\pi$
$683$	−18285.8	−1.02443	−0.512215	+	0.858857i	$0.671175\pi$
$684$	0	0
$685$	−1466.49	−0.0817982
$686$	0	0
$687$	−6155.56	−0.341847
$688$	0	0
$689$	3395.38	0.187741
$690$	0	0
$691$	−7815.29	−0.430257	−0.215128	−	0.976586i	$-0.569017\pi$
$691$	−7815.29	−0.430257	−0.215128	+	0.976586i	$0.569017\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−855.746	−0.0467054
$696$	0	0
$697$	1072.02	0.0582578
$698$	0	0
$699$	−13222.4	−0.715473
$700$	0	0
$701$	−34661.4	−1.86753	−0.933767	−	0.357881i	$-0.883499\pi$
$701$	−34661.4	−1.86753	−0.933767	+	0.357881i	$0.883499\pi$
$702$	0	0
$703$	8325.08	0.446638
$704$	0	0
$705$	−1805.76	−0.0964663
$706$	0	0
$707$	34276.1	1.82332
$708$	0	0
$709$	−37748.5	−1.99954	−0.999769	−	0.0214801i	$-0.993162\pi$
$709$	−37748.5	−1.99954	−0.999769	+	0.0214801i	$0.993162\pi$
$710$	0	0
$711$	5875.64	0.309921
$712$	0	0
$713$	−5644.45	−0.296474
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−4235.10	−0.220589
$718$	0	0
$719$	20484.7	1.06252	0.531258	−	0.847210i	$-0.321720\pi$
$719$	20484.7	1.06252	0.531258	+	0.847210i	$0.321720\pi$
$720$	0	0
$721$	15904.3	0.821509
$722$	0	0
$723$	4534.88	0.233270
$724$	0	0
$725$	−22228.4	−1.13868
$726$	0	0
$727$	14451.2	0.737231	0.368615	−	0.929582i	$-0.379832\pi$
$727$	14451.2	0.737231	0.368615	+	0.929582i	$0.379832\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	249.351	0.0126164
$732$	0	0
$733$	29331.6	1.47802	0.739010	−	0.673694i	$-0.235293\pi$
$733$	29331.6	1.47802	0.739010	+	0.673694i	$0.235293\pi$
$734$	0	0
$735$	2018.49	0.101297
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−25755.1	−1.28203	−0.641013	−	0.767530i	$-0.721486\pi$
$739$	−25755.1	−1.28203	−0.641013	+	0.767530i	$0.721486\pi$
$740$	0	0
$741$	−497.309	−0.0246546
$742$	0	0
$743$	7709.43	0.380662	0.190331	−	0.981720i	$-0.439044\pi$
$743$	7709.43	0.380662	0.190331	+	0.981720i	$0.439044\pi$
$744$	0	0
$745$	1575.47	0.0774774
$746$	0	0
$747$	−12796.3	−0.626761
$748$	0	0
$749$	−15329.6	−0.747838
$750$	0	0
$751$	−17089.5	−0.830364	−0.415182	−	0.909738i	$-0.636282\pi$
$751$	−17089.5	−0.830364	−0.415182	+	0.909738i	$0.636282\pi$
$752$	0	0
$753$	−14279.9	−0.691085
$754$	0	0
$755$	−2877.99	−0.138730
$756$	0	0
$757$	20347.2	0.976926	0.488463	−	0.872585i	$-0.337558\pi$
$757$	20347.2	0.976926	0.488463	+	0.872585i	$0.337558\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−10417.7	−0.496244	−0.248122	−	0.968729i	$-0.579813\pi$
$761$	−10417.7	−0.496244	−0.248122	+	0.968729i	$0.579813\pi$
$762$	0	0
$763$	−36588.5	−1.73603
$764$	0	0
$765$	389.383	0.0184028
$766$	0	0
$767$	−1400.49	−0.0659305
$768$	0	0
$769$	41894.3	1.96456	0.982281	−	0.187417i	$-0.0600115\pi$
$769$	41894.3	1.96456	0.982281	+	0.187417i	$0.0600115\pi$
$770$	0	0
$771$	−19243.7	−0.898889
$772$	0	0
$773$	−12669.5	−0.589509	−0.294754	−	0.955573i	$-0.595238\pi$
$773$	−12669.5	−0.589509	−0.294754	+	0.955573i	$0.595238\pi$
$774$	0	0
$775$	−7694.72	−0.356648
$776$	0	0
$777$	27212.4	1.25642
$778$	0	0
$779$	1036.18	0.0476572
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−4910.59	−0.224125
$784$	0	0
$785$	3901.29	0.177379
$786$	0	0
$787$	23132.4	1.04775	0.523875	−	0.851795i	$-0.324486\pi$
$787$	23132.4	1.04775	0.523875	+	0.851795i	$0.324486\pi$
$788$	0	0
$789$	19037.5	0.859005
$790$	0	0
$791$	−23586.1	−1.06021
$792$	0	0
$793$	−4267.21	−0.191088
$794$	0	0
$795$	−2569.63	−0.114636
$796$	0	0
$797$	−8157.15	−0.362536	−0.181268	−	0.983434i	$-0.558020\pi$
$797$	−8157.15	−0.362536	−0.181268	+	0.983434i	$0.558020\pi$
$798$	0	0
$799$	9363.18	0.414575
$800$	0	0
$801$	−6786.13	−0.299346
$802$	0	0
$803$	0	0
$804$	0	0
$805$	4084.97	0.178852
$806$	0	0
$807$	−2933.12	−0.127944
$808$	0	0
$809$	−9629.79	−0.418498	−0.209249	−	0.977862i	$-0.567102\pi$
$809$	−9629.79	−0.418498	−0.209249	+	0.977862i	$0.567102\pi$
$810$	0	0
$811$	20546.5	0.889626	0.444813	−	0.895624i	$-0.353270\pi$
$811$	20546.5	0.889626	0.444813	+	0.895624i	$0.353270\pi$
$812$	0	0
$813$	10247.5	0.442062
$814$	0	0
$815$	−3919.47	−0.168458
$816$	0	0
$817$	241.015	0.0103207
$818$	0	0
$819$	−1625.56	−0.0693550
$820$	0	0
$821$	7383.11	0.313852	0.156926	−	0.987610i	$-0.449842\pi$
$821$	7383.11	0.313852	0.156926	+	0.987610i	$0.449842\pi$
$822$	0	0
$823$	12862.2	0.544774	0.272387	−	0.962188i	$-0.412187\pi$
$823$	12862.2	0.544774	0.272387	+	0.962188i	$0.412187\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−11661.2	−0.490325	−0.245162	−	0.969482i	$-0.578841\pi$
$827$	−11661.2	−0.490325	−0.245162	+	0.969482i	$0.578841\pi$
$828$	0	0
$829$	8656.55	0.362671	0.181336	−	0.983421i	$-0.441958\pi$
$829$	8656.55	0.362671	0.181336	+	0.983421i	$0.441958\pi$
$830$	0	0
$831$	−10612.8	−0.443024
$832$	0	0
$833$	−10466.3	−0.435335
$834$	0	0
$835$	5626.01	0.233169
$836$	0	0
$837$	−1699.88	−0.0701990
$838$	0	0
$839$	−7047.79	−0.290008	−0.145004	−	0.989431i	$-0.546320\pi$
$839$	−7047.79	−0.290008	−0.145004	+	0.989431i	$0.546320\pi$
$840$	0	0
$841$	8689.00	0.356267
$842$	0	0
$843$	12037.5	0.491806
$844$	0	0
$845$	3591.10	0.146199
$846$	0	0
$847$	0	0
$848$	0	0
$849$	8568.77	0.346383
$850$	0	0
$851$	29765.5	1.19900
$852$	0	0
$853$	−13842.8	−0.555651	−0.277825	−	0.960632i	$-0.589614\pi$
$853$	−13842.8	−0.555651	−0.277825	+	0.960632i	$0.589614\pi$
$854$	0	0
$855$	376.365	0.0150543
$856$	0	0
$857$	−46128.3	−1.83864	−0.919319	−	0.393512i	$-0.871260\pi$
$857$	−46128.3	−1.83864	−0.919319	+	0.393512i	$0.871260\pi$
$858$	0	0
$859$	−30332.6	−1.20481	−0.602406	−	0.798190i	$-0.705791\pi$
$859$	−30332.6	−1.20481	−0.602406	+	0.798190i	$0.705791\pi$
$860$	0	0
$861$	3386.98	0.134063
$862$	0	0
$863$	−10578.3	−0.417255	−0.208627	−	0.977995i	$-0.566900\pi$
$863$	−10578.3	−0.417255	−0.208627	+	0.977995i	$0.566900\pi$
$864$	0	0
$865$	2984.10	0.117297
$866$	0	0
$867$	12720.0	0.498262
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−6261.29	−0.243577
$872$	0	0
$873$	4021.09	0.155891
$874$	0	0
$875$	11264.3	0.435203
$876$	0	0
$877$	−41740.9	−1.60717	−0.803585	−	0.595190i	$-0.797077\pi$
$877$	−41740.9	−1.60717	−0.803585	+	0.595190i	$0.797077\pi$
$878$	0	0
$879$	20233.8	0.776417
$880$	0	0
$881$	40155.8	1.53562	0.767811	−	0.640677i	$-0.221346\pi$
$881$	40155.8	1.53562	0.767811	+	0.640677i	$0.221346\pi$
$882$	0	0
$883$	−17631.5	−0.671967	−0.335983	−	0.941868i	$-0.609069\pi$
$883$	−17631.5	−0.671967	−0.335983	+	0.941868i	$0.609069\pi$
$884$	0	0
$885$	1059.89	0.0402576
$886$	0	0
$887$	−30610.5	−1.15874	−0.579368	−	0.815066i	$-0.696701\pi$
$887$	−30610.5	−1.15874	−0.579368	+	0.815066i	$0.696701\pi$
$888$	0	0
$889$	−63617.5	−2.40007
$890$	0	0
$891$	0	0
$892$	0	0
$893$	9050.14	0.339139
$894$	0	0
$895$	5577.71	0.208315
$896$	0	0
$897$	−1778.08	−0.0661853
$898$	0	0
$899$	11450.5	0.424801
$900$	0	0
$901$	13324.0	0.492661
$902$	0	0
$903$	787.810	0.0290329
$904$	0	0
$905$	−2965.48	−0.108924
$906$	0	0
$907$	14509.2	0.531170	0.265585	−	0.964087i	$-0.414435\pi$
$907$	14509.2	0.531170	0.265585	+	0.964087i	$0.414435\pi$
$908$	0	0
$909$	11291.1	0.411993
$910$	0	0
$911$	16513.0	0.600550	0.300275	−	0.953853i	$-0.402922\pi$
$911$	16513.0	0.600550	0.300275	+	0.953853i	$0.402922\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	3229.44	0.116680
$916$	0	0
$917$	−20830.9	−0.750161
$918$	0	0
$919$	−12429.2	−0.446138	−0.223069	−	0.974803i	$-0.571608\pi$
$919$	−12429.2	−0.446138	−0.223069	+	0.974803i	$0.571608\pi$
$920$	0	0
$921$	−11701.4	−0.418646
$922$	0	0
$923$	−3215.76	−0.114678
$924$	0	0
$925$	40577.4	1.44235
$926$	0	0
$927$	5239.14	0.185627
$928$	0	0
$929$	23589.5	0.833097	0.416548	−	0.909114i	$-0.363240\pi$
$929$	23589.5	0.833097	0.416548	+	0.909114i	$0.363240\pi$
$930$	0	0
$931$	−10116.3	−0.356122
$932$	0	0
$933$	−16285.4	−0.571447
$934$	0	0
$935$	0	0
$936$	0	0
$937$	20009.7	0.697640	0.348820	−	0.937190i	$-0.386582\pi$
$937$	20009.7	0.697640	0.348820	+	0.937190i	$0.386582\pi$
$938$	0	0
$939$	−15430.5	−0.536267
$940$	0	0
$941$	40823.9	1.41426	0.707132	−	0.707082i	$-0.249989\pi$
$941$	40823.9	1.41426	0.707132	+	0.707082i	$0.249989\pi$
$942$	0	0
$943$	3704.75	0.127936
$944$	0	0
$945$	1230.23	0.0423486
$946$	0	0
$947$	18558.6	0.636827	0.318413	−	0.947952i	$-0.396850\pi$
$947$	18558.6	0.636827	0.318413	+	0.947952i	$0.396850\pi$
$948$	0	0
$949$	5257.47	0.179836
$950$	0	0
$951$	−7885.21	−0.268870
$952$	0	0
$953$	−181.955	−0.00618480	−0.00309240	−	0.999995i	$-0.500984\pi$
$953$	−181.955	−0.00618480	−0.00309240	+	0.999995i	$0.500984\pi$
$954$	0	0
$955$	6517.97	0.220855
$956$	0	0
$957$	0	0
$958$	0	0
$959$	24024.4	0.808957
$960$	0	0
$961$	−25827.2	−0.866947
$962$	0	0
$963$	−5049.81	−0.168980
$964$	0	0
$965$	599.988	0.0200148
$966$	0	0
$967$	23043.6	0.766320	0.383160	−	0.923682i	$-0.374836\pi$
$967$	23043.6	0.766320	0.383160	+	0.923682i	$0.374836\pi$
$968$	0	0
$969$	−1951.52	−0.0646974
$970$	0	0
$971$	56960.6	1.88255	0.941273	−	0.337645i	$-0.109630\pi$
$971$	56960.6	1.88255	0.941273	+	0.337645i	$0.109630\pi$
$972$	0	0
$973$	14019.0	0.461901
$974$	0	0
$975$	−2423.94	−0.0796186
$976$	0	0
$977$	−12105.4	−0.396402	−0.198201	−	0.980161i	$-0.563510\pi$
$977$	−12105.4	−0.396402	−0.198201	+	0.980161i	$0.563510\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−12052.8	−0.392270
$982$	0	0
$983$	22002.2	0.713897	0.356949	−	0.934124i	$-0.383817\pi$
$983$	22002.2	0.713897	0.356949	+	0.934124i	$0.383817\pi$
$984$	0	0
$985$	6904.26	0.223338
$986$	0	0
$987$	29582.4	0.954019
$988$	0	0
$989$	861.724	0.0277060
$990$	0	0
$991$	−11666.1	−0.373953	−0.186976	−	0.982364i	$-0.559869\pi$
$991$	−11666.1	−0.373953	−0.186976	+	0.982364i	$0.559869\pi$
$992$	0	0
$993$	−6160.88	−0.196888
$994$	0	0
$995$	6111.87	0.194733
$996$	0	0
$997$	3982.56	0.126508	0.0632542	−	0.997997i	$-0.479852\pi$
$997$	3982.56	0.126508	0.0632542	+	0.997997i	$0.479852\pi$
$998$	0	0
$999$	8964.18	0.283898

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1452.4.a.o.1.3		4
11.3	even	5		132.4.i.b.97.2	yes	8
11.4	even	5		132.4.i.b.49.2	✓	8
11.10	odd	2		1452.4.a.p.1.3		4
33.14	odd	10		396.4.j.b.361.1		8
33.26	odd	10		396.4.j.b.181.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
132.4.i.b.49.2	✓	8	11.4	even	5
132.4.i.b.97.2	yes	8	11.3	even	5
396.4.j.b.181.1		8	33.26	odd	10
396.4.j.b.361.1		8	33.14	odd	10
1452.4.a.o.1.3		4	1.1	even	1	trivial
1452.4.a.p.1.3		4	11.10	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:	\( N \)	\(=\)	\( 1452 = 2^{2} \cdot 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1452.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -1.66773 q^{5} +27.3211 q^{7} +9.00000 q^{9} -6.61093 q^{13} +5.00318 q^{15} -25.9424 q^{17} -25.0751 q^{19} -81.9633 q^{21} -89.6533 q^{23} -122.219 q^{25} -27.0000 q^{27} +181.874 q^{29} +62.9586 q^{31} -45.5641 q^{35} -332.007 q^{37} +19.8328 q^{39} -41.3231 q^{41} -9.61174 q^{43} -15.0095 q^{45} -360.922 q^{47} +403.443 q^{49} +77.8271 q^{51} -513.601 q^{53} +75.2252 q^{57} +211.844 q^{59} +645.477 q^{61} +245.890 q^{63} +11.0252 q^{65} +947.111 q^{67} +268.960 q^{69} +486.431 q^{71} -795.268 q^{73} +366.656 q^{75} +652.849 q^{79} +81.0000 q^{81} -1421.81 q^{83} +43.2647 q^{85} -545.621 q^{87} -754.014 q^{89} -180.618 q^{91} -188.876 q^{93} +41.8183 q^{95} +446.787 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{3} - 6 q^{5} - 11 q^{7} + 36 q^{9} + 158 q^{13} + 18 q^{15} - 117 q^{17} - 139 q^{19} + 33 q^{21} - 174 q^{23} - 122 q^{25} - 108 q^{27} - 177 q^{29} + 104 q^{31} + 285 q^{35} + 240 q^{37} - 474 q^{39}+ \cdots + 2016 q^{97}+O(q^{100}) \) 4 * q - 12 * q^3 - 6 * q^5 - 11 * q^7 + 36 * q^9 + 158 * q^13 + 18 * q^15 - 117 * q^17 - 139 * q^19 + 33 * q^21 - 174 * q^23 - 122 * q^25 - 108 * q^27 - 177 * q^29 + 104 * q^31 + 285 * q^35 + 240 * q^37 - 474 * q^39 + 534 * q^41 + 342 * q^43 - 54 * q^45 - 99 * q^47 + 1041 * q^49 + 351 * q^51 - 114 * q^53 + 417 * q^57 + 108 * q^59 + 1007 * q^61 - 99 * q^63 - 57 * q^65 + 235 * q^67 + 522 * q^69 - 273 * q^71 - 869 * q^73 + 366 * q^75 + 63 * q^79 + 324 * q^81 - 2025 * q^83 - 320 * q^85 + 531 * q^87 - 858 * q^89 - 4492 * q^91 - 312 * q^93 - 33 * q^95 + 2016 * q^97

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