LMFDB - Embedded newform 4600.2.a.w.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4600,2,Mod(1,4600)] 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("4600.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4600 = 2^{3} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4600.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-2,0,0,0,-4,0,-3,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$36.7311849298$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.445042$ of defining polynomial
Character	$\chi$	$=$	4600.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.801938 q^{3} +1.60388 q^{7} -2.35690 q^{9} -6.09783 q^{11} +1.35690 q^{13} +1.60388 q^{17} +1.78017 q^{19} +1.28621 q^{21} +1.00000 q^{23} -4.29590 q^{27} -4.82908 q^{29} +6.15883 q^{31} -4.89008 q^{33} +9.87800 q^{37} +1.08815 q^{39} -8.89977 q^{41} +11.3840 q^{43} -9.85086 q^{47} -4.42758 q^{49} +1.28621 q^{51} -9.20775 q^{53} +1.42758 q^{57} -7.27413 q^{59} +0.933624 q^{61} -3.78017 q^{63} -9.42758 q^{67} +0.801938 q^{69} +12.6189 q^{71} +2.60925 q^{73} -9.78017 q^{77} -5.16421 q^{79} +3.62565 q^{81} -15.4034 q^{83} -3.87263 q^{87} -15.7017 q^{89} +2.17629 q^{91} +4.93900 q^{93} -12.8901 q^{97} +14.3720 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{3} - 4 q^{7} - 3 q^{9} - 4 q^{17} + 4 q^{19} + 12 q^{21} + 3 q^{23} + q^{27} - 4 q^{29} + 10 q^{31} - 14 q^{33} + 10 q^{37} + 7 q^{39} - 4 q^{41} + 24 q^{43} - 16 q^{47} + 3 q^{49} + 12 q^{51}+ \cdots + 14 q^{99}+O(q^{100}) $ 3 * q - 2 * q^3 - 4 * q^7 - 3 * q^9 - 4 * q^17 + 4 * q^19 + 12 * q^21 + 3 * q^23 + q^27 - 4 * q^29 + 10 * q^31 - 14 * q^33 + 10 * q^37 + 7 * q^39 - 4 * q^41 + 24 * q^43 - 16 * q^47 + 3 * q^49 + 12 * q^51 - 10 * q^53 - 12 * q^57 - 11 * q^59 - 4 * q^61 - 10 * q^63 - 12 * q^67 - 2 * q^69 + 4 * q^71 - 4 * q^73 - 28 * q^77 - 4 * q^79 - q^81 + 8 * q^83 + 5 * q^87 - 20 * q^89 + 14 * q^91 + 5 * q^93 - 38 * q^97 + 14 * q^99

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
$3$	0.801938	0.462999	0.231499	−	0.972835i	$-0.425637\pi$
$3$	0.801938	0.462999	0.231499	+	0.972835i	$0.425637\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.60388	0.606208	0.303104	−	0.952957i	$-0.401977\pi$
$7$	1.60388	0.606208	0.303104	+	0.952957i	$0.401977\pi$
$8$	0	0
$9$	−2.35690	−0.785632
$10$	0	0
$11$	−6.09783	−1.83857	−0.919283	−	0.393597i	$-0.871231\pi$
$11$	−6.09783	−1.83857	−0.919283	+	0.393597i	$0.871231\pi$
$12$	0	0
$13$	1.35690	0.376335	0.188168	−	0.982137i	$-0.439745\pi$
$13$	1.35690	0.376335	0.188168	+	0.982137i	$0.439745\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.60388	0.388997	0.194498	−	0.980903i	$-0.437692\pi$
$17$	1.60388	0.388997	0.194498	+	0.980903i	$0.437692\pi$
$18$	0	0
$19$	1.78017	0.408398	0.204199	−	0.978929i	$-0.434541\pi$
$19$	1.78017	0.408398	0.204199	+	0.978929i	$0.434541\pi$
$20$	0	0
$21$	1.28621	0.280674
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−4.29590	−0.826746
$28$	0	0
$29$	−4.82908	−0.896739	−0.448369	−	0.893848i	$-0.647995\pi$
$29$	−4.82908	−0.896739	−0.448369	+	0.893848i	$0.647995\pi$
$30$	0	0
$31$	6.15883	1.10616	0.553080	−	0.833128i	$-0.313453\pi$
$31$	6.15883	1.10616	0.553080	+	0.833128i	$0.313453\pi$
$32$	0	0
$33$	−4.89008	−0.851254
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.87800	1.62393	0.811967	−	0.583704i	$-0.198397\pi$
$37$	9.87800	1.62393	0.811967	+	0.583704i	$0.198397\pi$
$38$	0	0
$39$	1.08815	0.174243
$40$	0	0
$41$	−8.89977	−1.38991	−0.694955	−	0.719053i	$-0.744576\pi$
$41$	−8.89977	−1.38991	−0.694955	+	0.719053i	$0.744576\pi$
$42$	0	0
$43$	11.3840	1.73605	0.868025	−	0.496520i	$-0.165389\pi$
$43$	11.3840	1.73605	0.868025	+	0.496520i	$0.165389\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.85086	−1.43689	−0.718447	−	0.695581i	$-0.755147\pi$
$47$	−9.85086	−1.43689	−0.718447	+	0.695581i	$0.755147\pi$
$48$	0	0
$49$	−4.42758	−0.632512
$50$	0	0
$51$	1.28621	0.180105
$52$	0	0
$53$	−9.20775	−1.26478	−0.632391	−	0.774649i	$-0.717926\pi$
$53$	−9.20775	−1.26478	−0.632391	+	0.774649i	$0.717926\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.42758	0.189088
$58$	0	0
$59$	−7.27413	−0.947011	−0.473505	−	0.880791i	$-0.657012\pi$
$59$	−7.27413	−0.947011	−0.473505	+	0.880791i	$0.657012\pi$
$60$	0	0
$61$	0.933624	0.119538	0.0597692	−	0.998212i	$-0.480964\pi$
$61$	0.933624	0.119538	0.0597692	+	0.998212i	$0.480964\pi$
$62$	0	0
$63$	−3.78017	−0.476256
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−9.42758	−1.15176	−0.575881	−	0.817533i	$-0.695341\pi$
$67$	−9.42758	−1.15176	−0.575881	+	0.817533i	$0.695341\pi$
$68$	0	0
$69$	0.801938	0.0965420
$70$	0	0
$71$	12.6189	1.49759	0.748796	−	0.662800i	$-0.230632\pi$
$71$	12.6189	1.49759	0.748796	+	0.662800i	$0.230632\pi$
$72$	0	0
$73$	2.60925	0.305390	0.152695	−	0.988273i	$-0.451205\pi$
$73$	2.60925	0.305390	0.152695	+	0.988273i	$0.451205\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−9.78017	−1.11455
$78$	0	0
$79$	−5.16421	−0.581019	−0.290510	−	0.956872i	$-0.593825\pi$
$79$	−5.16421	−0.581019	−0.290510	+	0.956872i	$0.593825\pi$
$80$	0	0
$81$	3.62565	0.402850
$82$	0	0
$83$	−15.4034	−1.69075	−0.845373	−	0.534177i	$-0.820621\pi$
$83$	−15.4034	−1.69075	−0.845373	+	0.534177i	$0.820621\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−3.87263	−0.415189
$88$	0	0
$89$	−15.7017	−1.66438	−0.832189	−	0.554492i	$-0.812913\pi$
$89$	−15.7017	−1.66438	−0.832189	+	0.554492i	$0.812913\pi$
$90$	0	0
$91$	2.17629	0.228137
$92$	0	0
$93$	4.93900	0.512151
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−12.8901	−1.30879	−0.654395	−	0.756153i	$-0.727077\pi$
$97$	−12.8901	−1.30879	−0.654395	+	0.756153i	$0.727077\pi$
$98$	0	0
$99$	14.3720	1.44444
$100$	0	0
$101$	6.56033	0.652778	0.326389	−	0.945236i	$-0.394168\pi$
$101$	6.56033	0.652778	0.326389	+	0.945236i	$0.394168\pi$
$102$	0	0
$103$	−9.52542	−0.938567	−0.469284	−	0.883047i	$-0.655488\pi$
$103$	−9.52542	−0.938567	−0.469284	+	0.883047i	$0.655488\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−13.3840	−1.29388	−0.646942	−	0.762539i	$-0.723952\pi$
$107$	−13.3840	−1.29388	−0.646942	+	0.762539i	$0.723952\pi$
$108$	0	0
$109$	−15.4819	−1.48289	−0.741447	−	0.671011i	$-0.765860\pi$
$109$	−15.4819	−1.48289	−0.741447	+	0.671011i	$0.765860\pi$
$110$	0	0
$111$	7.92154	0.751880
$112$	0	0
$113$	−12.9879	−1.22180	−0.610900	−	0.791708i	$-0.709192\pi$
$113$	−12.9879	−1.22180	−0.610900	+	0.791708i	$0.709192\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−3.19806	−0.295661
$118$	0	0
$119$	2.57242	0.235813
$120$	0	0
$121$	26.1836	2.38033
$122$	0	0
$123$	−7.13706	−0.643527
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−0.0609989	−0.00541278	−0.00270639	−	0.999996i	$-0.500861\pi$
$127$	−0.0609989	−0.00541278	−0.00270639	+	0.999996i	$0.500861\pi$
$128$	0	0
$129$	9.12929	0.803789
$130$	0	0
$131$	−17.9855	−1.57140	−0.785701	−	0.618606i	$-0.787698\pi$
$131$	−17.9855	−1.57140	−0.785701	+	0.618606i	$0.787698\pi$
$132$	0	0
$133$	2.85517	0.247574
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.10992	0.436570	0.218285	−	0.975885i	$-0.429954\pi$
$137$	5.10992	0.436570	0.218285	+	0.975885i	$0.429954\pi$
$138$	0	0
$139$	−17.8509	−1.51409	−0.757045	−	0.653363i	$-0.773358\pi$
$139$	−17.8509	−1.51409	−0.757045	+	0.653363i	$0.773358\pi$
$140$	0	0
$141$	−7.89977	−0.665281
$142$	0	0
$143$	−8.27413	−0.691917
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−3.55065	−0.292852
$148$	0	0
$149$	13.3599	1.09448	0.547242	−	0.836974i	$-0.315678\pi$
$149$	13.3599	1.09448	0.547242	+	0.836974i	$0.315678\pi$
$150$	0	0
$151$	−4.91723	−0.400159	−0.200079	−	0.979780i	$-0.564120\pi$
$151$	−4.91723	−0.400159	−0.200079	+	0.979780i	$0.564120\pi$
$152$	0	0
$153$	−3.78017	−0.305608
$154$	0	0
$155$	0	0
$156$	0	0
$157$	5.92154	0.472591	0.236295	−	0.971681i	$-0.424067\pi$
$157$	5.92154	0.472591	0.236295	+	0.971681i	$0.424067\pi$
$158$	0	0
$159$	−7.38404	−0.585593
$160$	0	0
$161$	1.60388	0.126403
$162$	0	0
$163$	5.71917	0.447960	0.223980	−	0.974594i	$-0.428095\pi$
$163$	5.71917	0.447960	0.223980	+	0.974594i	$0.428095\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.6775	1.29055	0.645274	−	0.763952i	$-0.276743\pi$
$167$	16.6775	1.29055	0.645274	+	0.763952i	$0.276743\pi$
$168$	0	0
$169$	−11.1588	−0.858372
$170$	0	0
$171$	−4.19567	−0.320851
$172$	0	0
$173$	−1.00000	−0.0760286	−0.0380143	−	0.999277i	$-0.512103\pi$
$173$	−1.00000	−0.0760286	−0.0380143	+	0.999277i	$0.512103\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−5.83340	−0.438465
$178$	0	0
$179$	13.0489	0.975322	0.487661	−	0.873033i	$-0.337850\pi$
$179$	13.0489	0.975322	0.487661	+	0.873033i	$0.337850\pi$
$180$	0	0
$181$	−17.3840	−1.29215	−0.646073	−	0.763276i	$-0.723590\pi$
$181$	−17.3840	−1.29215	−0.646073	+	0.763276i	$0.723590\pi$
$182$	0	0
$183$	0.748709	0.0553461
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−9.78017	−0.715197
$188$	0	0
$189$	−6.89008	−0.501180
$190$	0	0
$191$	23.6233	1.70932	0.854659	−	0.519189i	$-0.173766\pi$
$191$	23.6233	1.70932	0.854659	+	0.519189i	$0.173766\pi$
$192$	0	0
$193$	2.06398	0.148569	0.0742844	−	0.997237i	$-0.476333\pi$
$193$	2.06398	0.148569	0.0742844	+	0.997237i	$0.476333\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−24.1075	−1.71759	−0.858795	−	0.512319i	$-0.828786\pi$
$197$	−24.1075	−1.71759	−0.858795	+	0.512319i	$0.828786\pi$
$198$	0	0
$199$	0.439665	0.0311670	0.0155835	−	0.999879i	$-0.495039\pi$
$199$	0.439665	0.0311670	0.0155835	+	0.999879i	$0.495039\pi$
$200$	0	0
$201$	−7.56033	−0.533265
$202$	0	0
$203$	−7.74525	−0.543610
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−2.35690	−0.163816
$208$	0	0
$209$	−10.8552	−0.750868
$210$	0	0
$211$	14.0664	0.968369	0.484185	−	0.874966i	$-0.339116\pi$
$211$	14.0664	0.968369	0.484185	+	0.874966i	$0.339116\pi$
$212$	0	0
$213$	10.1196	0.693384
$214$	0	0
$215$	0	0
$216$	0	0
$217$	9.87800	0.670562
$218$	0	0
$219$	2.09246	0.141395
$220$	0	0
$221$	2.17629	0.146393
$222$	0	0
$223$	16.9215	1.13315	0.566575	−	0.824010i	$-0.308268\pi$
$223$	16.9215	1.13315	0.566575	+	0.824010i	$0.308268\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3.80433	−0.252502	−0.126251	−	0.991998i	$-0.540295\pi$
$227$	−3.80433	−0.252502	−0.126251	+	0.991998i	$0.540295\pi$
$228$	0	0
$229$	3.04221	0.201035	0.100518	−	0.994935i	$-0.467950\pi$
$229$	3.04221	0.201035	0.100518	+	0.994935i	$0.467950\pi$
$230$	0	0
$231$	−7.84309	−0.516037
$232$	0	0
$233$	4.12498	0.270237	0.135118	−	0.990829i	$-0.456859\pi$
$233$	4.12498	0.270237	0.135118	+	0.990829i	$0.456859\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−4.14138	−0.269011
$238$	0	0
$239$	16.7342	1.08245	0.541224	−	0.840879i	$-0.317961\pi$
$239$	16.7342	1.08245	0.541224	+	0.840879i	$0.317961\pi$
$240$	0	0
$241$	26.5676	1.71137	0.855686	−	0.517496i	$-0.173136\pi$
$241$	26.5676	1.71137	0.855686	+	0.517496i	$0.173136\pi$
$242$	0	0
$243$	15.7952	1.01326
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.41550	0.153695
$248$	0	0
$249$	−12.3526	−0.782813
$250$	0	0
$251$	20.2392	1.27749	0.638744	−	0.769419i	$-0.279454\pi$
$251$	20.2392	1.27749	0.638744	+	0.769419i	$0.279454\pi$
$252$	0	0
$253$	−6.09783	−0.383368
$254$	0	0
$255$	0	0
$256$	0	0
$257$	14.2403	0.888284	0.444142	−	0.895956i	$-0.353509\pi$
$257$	14.2403	0.888284	0.444142	+	0.895956i	$0.353509\pi$
$258$	0	0
$259$	15.8431	0.984441
$260$	0	0
$261$	11.3817	0.704506
$262$	0	0
$263$	1.82371	0.112455	0.0562273	−	0.998418i	$-0.482093\pi$
$263$	1.82371	0.112455	0.0562273	+	0.998418i	$0.482093\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−12.5918	−0.770605
$268$	0	0
$269$	−6.20344	−0.378230	−0.189115	−	0.981955i	$-0.560562\pi$
$269$	−6.20344	−0.378230	−0.189115	+	0.981955i	$0.560562\pi$
$270$	0	0
$271$	−1.49396	−0.0907516	−0.0453758	−	0.998970i	$-0.514449\pi$
$271$	−1.49396	−0.0907516	−0.0453758	+	0.998970i	$0.514449\pi$
$272$	0	0
$273$	1.74525	0.105627
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−7.47219	−0.448960	−0.224480	−	0.974479i	$-0.572068\pi$
$277$	−7.47219	−0.448960	−0.224480	+	0.974479i	$0.572068\pi$
$278$	0	0
$279$	−14.5157	−0.869034
$280$	0	0
$281$	−6.29350	−0.375439	−0.187719	−	0.982223i	$-0.560110\pi$
$281$	−6.29350	−0.375439	−0.187719	+	0.982223i	$0.560110\pi$
$282$	0	0
$283$	−3.73663	−0.222119	−0.111060	−	0.993814i	$-0.535424\pi$
$283$	−3.73663	−0.222119	−0.111060	+	0.993814i	$0.535424\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−14.2741	−0.842575
$288$	0	0
$289$	−14.4276	−0.848681
$290$	0	0
$291$	−10.3370	−0.605968
$292$	0	0
$293$	5.00730	0.292529	0.146265	−	0.989245i	$-0.453275\pi$
$293$	5.00730	0.292529	0.146265	+	0.989245i	$0.453275\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	26.1957	1.52003
$298$	0	0
$299$	1.35690	0.0784713
$300$	0	0
$301$	18.2586	1.05241
$302$	0	0
$303$	5.26098	0.302235
$304$	0	0
$305$	0	0
$306$	0	0
$307$	5.78017	0.329892	0.164946	−	0.986303i	$-0.447255\pi$
$307$	5.78017	0.329892	0.164946	+	0.986303i	$0.447255\pi$
$308$	0	0
$309$	−7.63879	−0.434556
$310$	0	0
$311$	33.7579	1.91424	0.957118	−	0.289698i	$-0.0935549\pi$
$311$	33.7579	1.91424	0.957118	+	0.289698i	$0.0935549\pi$
$312$	0	0
$313$	−27.6775	−1.56443	−0.782214	−	0.623010i	$-0.785910\pi$
$313$	−27.6775	−1.56443	−0.782214	+	0.623010i	$0.785910\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−31.3793	−1.76243	−0.881217	−	0.472711i	$-0.843275\pi$
$317$	−31.3793	−1.76243	−0.881217	+	0.472711i	$0.843275\pi$
$318$	0	0
$319$	29.4470	1.64871
$320$	0	0
$321$	−10.7332	−0.599067
$322$	0	0
$323$	2.85517	0.158866
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−12.4155	−0.686579
$328$	0	0
$329$	−15.7995	−0.871057
$330$	0	0
$331$	−10.1618	−0.558544	−0.279272	−	0.960212i	$-0.590093\pi$
$331$	−10.1618	−0.558544	−0.279272	+	0.960212i	$0.590093\pi$
$332$	0	0
$333$	−23.2814	−1.27581
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−16.4892	−0.898223	−0.449111	−	0.893476i	$-0.648259\pi$
$337$	−16.4892	−0.898223	−0.449111	+	0.893476i	$0.648259\pi$
$338$	0	0
$339$	−10.4155	−0.565692
$340$	0	0
$341$	−37.5555	−2.03375
$342$	0	0
$343$	−18.3284	−0.989642
$344$	0	0
$345$	0	0
$346$	0	0
$347$	26.1051	1.40140	0.700698	−	0.713458i	$-0.252872\pi$
$347$	26.1051	1.40140	0.700698	+	0.713458i	$0.252872\pi$
$348$	0	0
$349$	−12.8629	−0.688537	−0.344269	−	0.938871i	$-0.611873\pi$
$349$	−12.8629	−0.688537	−0.344269	+	0.938871i	$0.611873\pi$
$350$	0	0
$351$	−5.82908	−0.311134
$352$	0	0
$353$	1.51573	0.0806741	0.0403371	−	0.999186i	$-0.487157\pi$
$353$	1.51573	0.0806741	0.0403371	+	0.999186i	$0.487157\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	2.06292	0.109181
$358$	0	0
$359$	−5.82371	−0.307364	−0.153682	−	0.988120i	$-0.549113\pi$
$359$	−5.82371	−0.307364	−0.153682	+	0.988120i	$0.549113\pi$
$360$	0	0
$361$	−15.8310	−0.833211
$362$	0	0
$363$	20.9976	1.10209
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−35.4142	−1.84860	−0.924302	−	0.381661i	$-0.875352\pi$
$367$	−35.4142	−1.84860	−0.924302	+	0.381661i	$0.875352\pi$
$368$	0	0
$369$	20.9758	1.09196
$370$	0	0
$371$	−14.7681	−0.766721
$372$	0	0
$373$	27.0616	1.40120	0.700598	−	0.713556i	$-0.252917\pi$
$373$	27.0616	1.40120	0.700598	+	0.713556i	$0.252917\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6.55257	−0.337474
$378$	0	0
$379$	7.12929	0.366207	0.183104	−	0.983094i	$-0.441386\pi$
$379$	7.12929	0.366207	0.183104	+	0.983094i	$0.441386\pi$
$380$	0	0
$381$	−0.0489173	−0.00250611
$382$	0	0
$383$	19.9866	1.02127	0.510634	−	0.859798i	$-0.329411\pi$
$383$	19.9866	1.02127	0.510634	+	0.859798i	$0.329411\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−26.8310	−1.36390
$388$	0	0
$389$	−16.1957	−0.821153	−0.410577	−	0.911826i	$-0.634673\pi$
$389$	−16.1957	−0.821153	−0.410577	+	0.911826i	$0.634673\pi$
$390$	0	0
$391$	1.60388	0.0811115
$392$	0	0
$393$	−14.4233	−0.727558
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−27.3317	−1.37174	−0.685869	−	0.727725i	$-0.740578\pi$
$397$	−27.3317	−1.37174	−0.685869	+	0.727725i	$0.740578\pi$
$398$	0	0
$399$	2.28967	0.114627
$400$	0	0
$401$	12.7681	0.637608	0.318804	−	0.947821i	$-0.396719\pi$
$401$	12.7681	0.637608	0.318804	+	0.947821i	$0.396719\pi$
$402$	0	0
$403$	8.35690	0.416287
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−60.2344	−2.98571
$408$	0	0
$409$	−28.4795	−1.40822	−0.704110	−	0.710091i	$-0.748654\pi$
$409$	−28.4795	−1.40822	−0.704110	+	0.710091i	$0.748654\pi$
$410$	0	0
$411$	4.09783	0.202131
$412$	0	0
$413$	−11.6668	−0.574085
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−14.3153	−0.701022
$418$	0	0
$419$	25.4905	1.24529	0.622646	−	0.782503i	$-0.286058\pi$
$419$	25.4905	1.24529	0.622646	+	0.782503i	$0.286058\pi$
$420$	0	0
$421$	32.4892	1.58343	0.791713	−	0.610894i	$-0.209190\pi$
$421$	32.4892	1.58343	0.791713	+	0.610894i	$0.209190\pi$
$422$	0	0
$423$	23.2174	1.12887
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.49742	0.0724651
$428$	0	0
$429$	−6.63533	−0.320357
$430$	0	0
$431$	6.23059	0.300117	0.150058	−	0.988677i	$-0.452054\pi$
$431$	6.23059	0.300117	0.150058	+	0.988677i	$0.452054\pi$
$432$	0	0
$433$	13.2271	0.635655	0.317828	−	0.948149i	$-0.397047\pi$
$433$	13.2271	0.635655	0.317828	+	0.948149i	$0.397047\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.78017	0.0851570
$438$	0	0
$439$	18.1424	0.865891	0.432946	−	0.901420i	$-0.357474\pi$
$439$	18.1424	0.865891	0.432946	+	0.901420i	$0.357474\pi$
$440$	0	0
$441$	10.4354	0.496922
$442$	0	0
$443$	−9.76377	−0.463891	−0.231945	−	0.972729i	$-0.574509\pi$
$443$	−9.76377	−0.463891	−0.231945	+	0.972729i	$0.574509\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	10.7138	0.506745
$448$	0	0
$449$	33.6112	1.58621	0.793105	−	0.609085i	$-0.208463\pi$
$449$	33.6112	1.58621	0.793105	+	0.609085i	$0.208463\pi$
$450$	0	0
$451$	54.2693	2.55544
$452$	0	0
$453$	−3.94331	−0.185273
$454$	0	0
$455$	0	0
$456$	0	0
$457$	2.14138	0.100169	0.0500847	−	0.998745i	$-0.484051\pi$
$457$	2.14138	0.100169	0.0500847	+	0.998745i	$0.484051\pi$
$458$	0	0
$459$	−6.89008	−0.321602
$460$	0	0
$461$	−39.6310	−1.84580	−0.922900	−	0.385039i	$-0.874188\pi$
$461$	−39.6310	−1.84580	−0.922900	+	0.385039i	$0.874188\pi$
$462$	0	0
$463$	−1.16554	−0.0541672	−0.0270836	−	0.999633i	$-0.508622\pi$
$463$	−1.16554	−0.0541672	−0.0270836	+	0.999633i	$0.508622\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	33.3986	1.54550	0.772752	−	0.634708i	$-0.218880\pi$
$467$	33.3986	1.54550	0.772752	+	0.634708i	$0.218880\pi$
$468$	0	0
$469$	−15.1207	−0.698208
$470$	0	0
$471$	4.74871	0.218809
$472$	0	0
$473$	−69.4180	−3.19184
$474$	0	0
$475$	0	0
$476$	0	0
$477$	21.7017	0.993653
$478$	0	0
$479$	−16.9638	−0.775094	−0.387547	−	0.921850i	$-0.626677\pi$
$479$	−16.9638	−0.775094	−0.387547	+	0.921850i	$0.626677\pi$
$480$	0	0
$481$	13.4034	0.611143
$482$	0	0
$483$	1.28621	0.0585245
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−26.7362	−1.21153	−0.605765	−	0.795643i	$-0.707133\pi$
$487$	−26.7362	−1.21153	−0.605765	+	0.795643i	$0.707133\pi$
$488$	0	0
$489$	4.58642	0.207405
$490$	0	0
$491$	−26.2905	−1.18647	−0.593237	−	0.805028i	$-0.702150\pi$
$491$	−26.2905	−1.18647	−0.593237	+	0.805028i	$0.702150\pi$
$492$	0	0
$493$	−7.74525	−0.348829
$494$	0	0
$495$	0	0
$496$	0	0
$497$	20.2392	0.907853
$498$	0	0
$499$	3.83340	0.171606	0.0858032	−	0.996312i	$-0.472654\pi$
$499$	3.83340	0.171606	0.0858032	+	0.996312i	$0.472654\pi$
$500$	0	0
$501$	13.3744	0.597522
$502$	0	0
$503$	−15.3056	−0.682442	−0.341221	−	0.939983i	$-0.610841\pi$
$503$	−15.3056	−0.682442	−0.341221	+	0.939983i	$0.610841\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−8.94869	−0.397425
$508$	0	0
$509$	−29.9071	−1.32561	−0.662804	−	0.748793i	$-0.730634\pi$
$509$	−29.9071	−1.32561	−0.662804	+	0.748793i	$0.730634\pi$
$510$	0	0
$511$	4.18492	0.185130
$512$	0	0
$513$	−7.64742	−0.337642
$514$	0	0
$515$	0	0
$516$	0	0
$517$	60.0689	2.64183
$518$	0	0
$519$	−0.801938	−0.0352012
$520$	0	0
$521$	−13.6146	−0.596468	−0.298234	−	0.954493i	$-0.596398\pi$
$521$	−13.6146	−0.596468	−0.298234	+	0.954493i	$0.596398\pi$
$522$	0	0
$523$	20.5133	0.896986	0.448493	−	0.893786i	$-0.351961\pi$
$523$	20.5133	0.896986	0.448493	+	0.893786i	$0.351961\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9.87800	0.430293
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	17.1444	0.744002
$532$	0	0
$533$	−12.0761	−0.523072
$534$	0	0
$535$	0	0
$536$	0	0
$537$	10.4644	0.451573
$538$	0	0
$539$	26.9987	1.16292
$540$	0	0
$541$	−17.5157	−0.753060	−0.376530	−	0.926404i	$-0.622883\pi$
$541$	−17.5157	−0.753060	−0.376530	+	0.926404i	$0.622883\pi$
$542$	0	0
$543$	−13.9409	−0.598262
$544$	0	0
$545$	0	0
$546$	0	0
$547$	19.4741	0.832653	0.416326	−	0.909215i	$-0.363317\pi$
$547$	19.4741	0.832653	0.416326	+	0.909215i	$0.363317\pi$
$548$	0	0
$549$	−2.20046	−0.0939131
$550$	0	0
$551$	−8.59658	−0.366227
$552$	0	0
$553$	−8.28275	−0.352218
$554$	0	0
$555$	0	0
$556$	0	0
$557$	41.5555	1.76077	0.880383	−	0.474264i	$-0.157286\pi$
$557$	41.5555	1.76077	0.880383	+	0.474264i	$0.157286\pi$
$558$	0	0
$559$	15.4470	0.653337
$560$	0	0
$561$	−7.84309	−0.331135
$562$	0	0
$563$	1.87800	0.0791484	0.0395742	−	0.999217i	$-0.487400\pi$
$563$	1.87800	0.0791484	0.0395742	+	0.999217i	$0.487400\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	5.81508	0.244211
$568$	0	0
$569$	6.64609	0.278619	0.139309	−	0.990249i	$-0.455512\pi$
$569$	6.64609	0.278619	0.139309	+	0.990249i	$0.455512\pi$
$570$	0	0
$571$	−15.2948	−0.640069	−0.320034	−	0.947406i	$-0.603694\pi$
$571$	−15.2948	−0.640069	−0.320034	+	0.947406i	$0.603694\pi$
$572$	0	0
$573$	18.9444	0.791413
$574$	0	0
$575$	0	0
$576$	0	0
$577$	11.1873	0.465734	0.232867	−	0.972509i	$-0.425189\pi$
$577$	11.1873	0.465734	0.232867	+	0.972509i	$0.425189\pi$
$578$	0	0
$579$	1.65519	0.0687872
$580$	0	0
$581$	−24.7052	−1.02494
$582$	0	0
$583$	56.1473	2.32539
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−14.6431	−0.604386	−0.302193	−	0.953247i	$-0.597719\pi$
$587$	−14.6431	−0.604386	−0.302193	+	0.953247i	$0.597719\pi$
$588$	0	0
$589$	10.9638	0.451754
$590$	0	0
$591$	−19.3327	−0.795242
$592$	0	0
$593$	12.5241	0.514303	0.257151	−	0.966371i	$-0.417216\pi$
$593$	12.5241	0.514303	0.257151	+	0.966371i	$0.417216\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0.352584	0.0144303
$598$	0	0
$599$	0.882788	0.0360697	0.0180349	−	0.999837i	$-0.494259\pi$
$599$	0.882788	0.0360697	0.0180349	+	0.999837i	$0.494259\pi$
$600$	0	0
$601$	−16.2325	−0.662138	−0.331069	−	0.943607i	$-0.607409\pi$
$601$	−16.2325	−0.662138	−0.331069	+	0.943607i	$0.607409\pi$
$602$	0	0
$603$	22.2198	0.904862
$604$	0	0
$605$	0	0
$606$	0	0
$607$	31.1594	1.26472	0.632361	−	0.774674i	$-0.282086\pi$
$607$	31.1594	1.26472	0.632361	+	0.774674i	$0.282086\pi$
$608$	0	0
$609$	−6.21121	−0.251691
$610$	0	0
$611$	−13.3666	−0.540754
$612$	0	0
$613$	26.0086	1.05048	0.525239	−	0.850955i	$-0.323976\pi$
$613$	26.0086	1.05048	0.525239	+	0.850955i	$0.323976\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	12.8310	0.516557	0.258278	−	0.966071i	$-0.416845\pi$
$617$	12.8310	0.516557	0.258278	+	0.966071i	$0.416845\pi$
$618$	0	0
$619$	35.8103	1.43934	0.719669	−	0.694318i	$-0.244294\pi$
$619$	35.8103	1.43934	0.719669	+	0.694318i	$0.244294\pi$
$620$	0	0
$621$	−4.29590	−0.172388
$622$	0	0
$623$	−25.1836	−1.00896
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−8.70517	−0.347651
$628$	0	0
$629$	15.8431	0.631705
$630$	0	0
$631$	−31.0810	−1.23731	−0.618657	−	0.785661i	$-0.712323\pi$
$631$	−31.0810	−1.23731	−0.618657	+	0.785661i	$0.712323\pi$
$632$	0	0
$633$	11.2804	0.448354
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−6.00777	−0.238037
$638$	0	0
$639$	−29.7415	−1.17656
$640$	0	0
$641$	26.0978	1.03080	0.515401	−	0.856949i	$-0.327643\pi$
$641$	26.0978	1.03080	0.515401	+	0.856949i	$0.327643\pi$
$642$	0	0
$643$	−48.9783	−1.93152	−0.965759	−	0.259442i	$-0.916461\pi$
$643$	−48.9783	−1.93152	−0.965759	+	0.259442i	$0.916461\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	20.8920	0.821349	0.410675	−	0.911782i	$-0.365293\pi$
$647$	20.8920	0.821349	0.410675	+	0.911782i	$0.365293\pi$
$648$	0	0
$649$	44.3564	1.74114
$650$	0	0
$651$	7.92154	0.310470
$652$	0	0
$653$	−18.8049	−0.735893	−0.367947	−	0.929847i	$-0.619939\pi$
$653$	−18.8049	−0.735893	−0.367947	+	0.929847i	$0.619939\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−6.14974	−0.239924
$658$	0	0
$659$	2.49875	0.0973373	0.0486686	−	0.998815i	$-0.484502\pi$
$659$	2.49875	0.0973373	0.0486686	+	0.998815i	$0.484502\pi$
$660$	0	0
$661$	−11.9892	−0.466328	−0.233164	−	0.972437i	$-0.574908\pi$
$661$	−11.9892	−0.466328	−0.233164	+	0.972437i	$0.574908\pi$
$662$	0	0
$663$	1.74525	0.0677799
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−4.82908	−0.186983
$668$	0	0
$669$	13.5700	0.524647
$670$	0	0
$671$	−5.69309	−0.219779
$672$	0	0
$673$	−26.8079	−1.03337	−0.516684	−	0.856176i	$-0.672834\pi$
$673$	−26.8079	−1.03337	−0.516684	+	0.856176i	$0.672834\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−27.2379	−1.04684	−0.523418	−	0.852076i	$-0.675344\pi$
$677$	−27.2379	−1.04684	−0.523418	+	0.852076i	$0.675344\pi$
$678$	0	0
$679$	−20.6741	−0.793399
$680$	0	0
$681$	−3.05084	−0.116908
$682$	0	0
$683$	−0.0609989	−0.00233406	−0.00116703	−	0.999999i	$-0.500371\pi$
$683$	−0.0609989	−0.00233406	−0.00116703	+	0.999999i	$0.500371\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	2.43967	0.0930790
$688$	0	0
$689$	−12.4940	−0.475982
$690$	0	0
$691$	19.0086	0.723122	0.361561	−	0.932348i	$-0.382244\pi$
$691$	19.0086	0.723122	0.361561	+	0.932348i	$0.382244\pi$
$692$	0	0
$693$	23.0508	0.875629
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−14.2741	−0.540671
$698$	0	0
$699$	3.30798	0.125119
$700$	0	0
$701$	16.4590	0.621649	0.310825	−	0.950467i	$-0.399395\pi$
$701$	16.4590	0.621649	0.310825	+	0.950467i	$0.399395\pi$
$702$	0	0
$703$	17.5845	0.663212
$704$	0	0
$705$	0	0
$706$	0	0
$707$	10.5220	0.395719
$708$	0	0
$709$	−10.1306	−0.380463	−0.190232	−	0.981739i	$-0.560924\pi$
$709$	−10.1306	−0.380463	−0.190232	+	0.981739i	$0.560924\pi$
$710$	0	0
$711$	12.1715	0.456467
$712$	0	0
$713$	6.15883	0.230650
$714$	0	0
$715$	0	0
$716$	0	0
$717$	13.4198	0.501172
$718$	0	0
$719$	19.2078	0.716328	0.358164	−	0.933659i	$-0.383403\pi$
$719$	19.2078	0.716328	0.358164	+	0.933659i	$0.383403\pi$
$720$	0	0
$721$	−15.2776	−0.568967
$722$	0	0
$723$	21.3056	0.792363
$724$	0	0
$725$	0	0
$726$	0	0
$727$	9.90217	0.367251	0.183625	−	0.982996i	$-0.441217\pi$
$727$	9.90217	0.367251	0.183625	+	0.982996i	$0.441217\pi$
$728$	0	0
$729$	1.78986	0.0662910
$730$	0	0
$731$	18.2586	0.675318
$732$	0	0
$733$	−3.38404	−0.124992	−0.0624962	−	0.998045i	$-0.519906\pi$
$733$	−3.38404	−0.124992	−0.0624962	+	0.998045i	$0.519906\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	57.4878	2.11759
$738$	0	0
$739$	−6.95108	−0.255700	−0.127850	−	0.991794i	$-0.540808\pi$
$739$	−6.95108	−0.255700	−0.127850	+	0.991794i	$0.540808\pi$
$740$	0	0
$741$	1.93708	0.0711605
$742$	0	0
$743$	29.8974	1.09683	0.548414	−	0.836207i	$-0.315232\pi$
$743$	29.8974	1.09683	0.548414	+	0.836207i	$0.315232\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	36.3043	1.32830
$748$	0	0
$749$	−21.4663	−0.784363
$750$	0	0
$751$	26.1763	0.955186	0.477593	−	0.878581i	$-0.341509\pi$
$751$	26.1763	0.955186	0.477593	+	0.878581i	$0.341509\pi$
$752$	0	0
$753$	16.2306	0.591475
$754$	0	0
$755$	0	0
$756$	0	0
$757$	10.3526	0.376271	0.188136	−	0.982143i	$-0.439756\pi$
$757$	10.3526	0.376271	0.188136	+	0.982143i	$0.439756\pi$
$758$	0	0
$759$	−4.89008	−0.177499
$760$	0	0
$761$	50.8786	1.84435	0.922174	−	0.386776i	$-0.126411\pi$
$761$	50.8786	1.84435	0.922174	+	0.386776i	$0.126411\pi$
$762$	0	0
$763$	−24.8310	−0.898943
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−9.87023	−0.356393
$768$	0	0
$769$	30.5133	1.10034	0.550170	−	0.835053i	$-0.314563\pi$
$769$	30.5133	1.10034	0.550170	+	0.835053i	$0.314563\pi$
$770$	0	0
$771$	11.4198	0.411275
$772$	0	0
$773$	10.6025	0.381347	0.190674	−	0.981653i	$-0.438933\pi$
$773$	10.6025	0.381347	0.190674	+	0.981653i	$0.438933\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	12.7052	0.455795
$778$	0	0
$779$	−15.8431	−0.567637
$780$	0	0
$781$	−76.9482	−2.75342
$782$	0	0
$783$	20.7453	0.741375
$784$	0	0
$785$	0	0
$786$	0	0
$787$	21.3056	0.759462	0.379731	−	0.925097i	$-0.376017\pi$
$787$	21.3056	0.759462	0.379731	+	0.925097i	$0.376017\pi$
$788$	0	0
$789$	1.46250	0.0520664
$790$	0	0
$791$	−20.8310	−0.740665
$792$	0	0
$793$	1.26683	0.0449865
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−0.611171	−0.0216488	−0.0108244	−	0.999941i	$-0.503446\pi$
$797$	−0.611171	−0.0216488	−0.0108244	+	0.999941i	$0.503446\pi$
$798$	0	0
$799$	−15.7995	−0.558948
$800$	0	0
$801$	37.0073	1.30759
$802$	0	0
$803$	−15.9108	−0.561480
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−4.97477	−0.175120
$808$	0	0
$809$	15.0242	0.528221	0.264111	−	0.964492i	$-0.414922\pi$
$809$	15.0242	0.528221	0.264111	+	0.964492i	$0.414922\pi$
$810$	0	0
$811$	−28.6517	−1.00610	−0.503049	−	0.864258i	$-0.667789\pi$
$811$	−28.6517	−1.00610	−0.503049	+	0.864258i	$0.667789\pi$
$812$	0	0
$813$	−1.19806	−0.0420179
$814$	0	0
$815$	0	0
$816$	0	0
$817$	20.2655	0.709000
$818$	0	0
$819$	−5.12929	−0.179232
$820$	0	0
$821$	−51.4155	−1.79441	−0.897207	−	0.441611i	$-0.854407\pi$
$821$	−51.4155	−1.79441	−0.897207	+	0.441611i	$0.854407\pi$
$822$	0	0
$823$	−24.9299	−0.869002	−0.434501	−	0.900671i	$-0.643075\pi$
$823$	−24.9299	−0.869002	−0.434501	+	0.900671i	$0.643075\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.59525	0.0554723	0.0277362	−	0.999615i	$-0.491170\pi$
$827$	1.59525	0.0554723	0.0277362	+	0.999615i	$0.491170\pi$
$828$	0	0
$829$	−36.5362	−1.26895	−0.634477	−	0.772942i	$-0.718784\pi$
$829$	−36.5362	−1.26895	−0.634477	+	0.772942i	$0.718784\pi$
$830$	0	0
$831$	−5.99223	−0.207868
$832$	0	0
$833$	−7.10129	−0.246045
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−26.4577	−0.914512
$838$	0	0
$839$	39.5883	1.36674	0.683371	−	0.730072i	$-0.260513\pi$
$839$	39.5883	1.36674	0.683371	+	0.730072i	$0.260513\pi$
$840$	0	0
$841$	−5.67994	−0.195860
$842$	0	0
$843$	−5.04700	−0.173828
$844$	0	0
$845$	0	0
$846$	0	0
$847$	41.9952	1.44297
$848$	0	0
$849$	−2.99654	−0.102841
$850$	0	0
$851$	9.87800	0.338614
$852$	0	0
$853$	−20.0146	−0.685287	−0.342643	−	0.939466i	$-0.611322\pi$
$853$	−20.0146	−0.685287	−0.342643	+	0.939466i	$0.611322\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	3.61058	0.123335	0.0616675	−	0.998097i	$-0.480358\pi$
$857$	3.61058	0.123335	0.0616675	+	0.998097i	$0.480358\pi$
$858$	0	0
$859$	−43.4161	−1.48134	−0.740669	−	0.671870i	$-0.765491\pi$
$859$	−43.4161	−1.48134	−0.740669	+	0.671870i	$0.765491\pi$
$860$	0	0
$861$	−11.4470	−0.390111
$862$	0	0
$863$	14.6886	0.500005	0.250002	−	0.968245i	$-0.419569\pi$
$863$	14.6886	0.500005	0.250002	+	0.968245i	$0.419569\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−11.5700	−0.392939
$868$	0	0
$869$	31.4905	1.06824
$870$	0	0
$871$	−12.7922	−0.433449
$872$	0	0
$873$	30.3806	1.02823
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−13.7439	−0.464099	−0.232050	−	0.972704i	$-0.574543\pi$
$877$	−13.7439	−0.464099	−0.232050	+	0.972704i	$0.574543\pi$
$878$	0	0
$879$	4.01554	0.135441
$880$	0	0
$881$	18.8224	0.634142	0.317071	−	0.948402i	$-0.397301\pi$
$881$	18.8224	0.634142	0.317071	+	0.948402i	$0.397301\pi$
$882$	0	0
$883$	−22.0422	−0.741780	−0.370890	−	0.928677i	$-0.620947\pi$
$883$	−22.0422	−0.741780	−0.370890	+	0.928677i	$0.620947\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−18.9928	−0.637717	−0.318858	−	0.947802i	$-0.603299\pi$
$887$	−18.9928	−0.637717	−0.318858	+	0.947802i	$0.603299\pi$
$888$	0	0
$889$	−0.0978347	−0.00328127
$890$	0	0
$891$	−22.1086	−0.740666
$892$	0	0
$893$	−17.5362	−0.586826
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1.08815	0.0363321
$898$	0	0
$899$	−29.7415	−0.991936
$900$	0	0
$901$	−14.7681	−0.491996
$902$	0	0
$903$	14.6423	0.487264
$904$	0	0
$905$	0	0
$906$	0	0
$907$	23.6560	0.785486	0.392743	−	0.919648i	$-0.371526\pi$
$907$	23.6560	0.785486	0.392743	+	0.919648i	$0.371526\pi$
$908$	0	0
$909$	−15.4620	−0.512843
$910$	0	0
$911$	−3.44312	−0.114076	−0.0570379	−	0.998372i	$-0.518166\pi$
$911$	−3.44312	−0.114076	−0.0570379	+	0.998372i	$0.518166\pi$
$912$	0	0
$913$	93.9275	3.10855
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−28.8465	−0.952597
$918$	0	0
$919$	−22.3913	−0.738622	−0.369311	−	0.929306i	$-0.620406\pi$
$919$	−22.3913	−0.738622	−0.369311	+	0.929306i	$0.620406\pi$
$920$	0	0
$921$	4.63533	0.152739
$922$	0	0
$923$	17.1226	0.563597
$924$	0	0
$925$	0	0
$926$	0	0
$927$	22.4504	0.737368
$928$	0	0
$929$	30.3437	0.995546	0.497773	−	0.867307i	$-0.334151\pi$
$929$	30.3437	0.995546	0.497773	+	0.867307i	$0.334151\pi$
$930$	0	0
$931$	−7.88184	−0.258317
$932$	0	0
$933$	27.0718	0.886289
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−8.37196	−0.273500	−0.136750	−	0.990606i	$-0.543666\pi$
$937$	−8.37196	−0.273500	−0.136750	+	0.990606i	$0.543666\pi$
$938$	0	0
$939$	−22.1957	−0.724328
$940$	0	0
$941$	−21.2078	−0.691353	−0.345676	−	0.938354i	$-0.612351\pi$
$941$	−21.2078	−0.691353	−0.345676	+	0.938354i	$0.612351\pi$
$942$	0	0
$943$	−8.89977	−0.289816
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4.13467	−0.134359	−0.0671794	−	0.997741i	$-0.521400\pi$
$947$	−4.13467	−0.134359	−0.0671794	+	0.997741i	$0.521400\pi$
$948$	0	0
$949$	3.54048	0.114929
$950$	0	0
$951$	−25.1642	−0.816005
$952$	0	0
$953$	−50.3806	−1.63199	−0.815994	−	0.578061i	$-0.803810\pi$
$953$	−50.3806	−1.63199	−0.815994	+	0.578061i	$0.803810\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	23.6146	0.763353
$958$	0	0
$959$	8.19567	0.264652
$960$	0	0
$961$	6.93123	0.223588
$962$	0	0
$963$	31.5448	1.01652
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−4.03790	−0.129850	−0.0649251	−	0.997890i	$-0.520681\pi$
$967$	−4.03790	−0.129850	−0.0649251	+	0.997890i	$0.520681\pi$
$968$	0	0
$969$	2.28967	0.0735547
$970$	0	0
$971$	12.9987	0.417147	0.208574	−	0.978007i	$-0.433118\pi$
$971$	12.9987	0.417147	0.208574	+	0.978007i	$0.433118\pi$
$972$	0	0
$973$	−28.6305	−0.917853
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−16.1823	−0.517716	−0.258858	−	0.965915i	$-0.583346\pi$
$977$	−16.1823	−0.517716	−0.258858	+	0.965915i	$0.583346\pi$
$978$	0	0
$979$	95.7464	3.06007
$980$	0	0
$981$	36.4892	1.16501
$982$	0	0
$983$	−1.48666	−0.0474172	−0.0237086	−	0.999719i	$-0.507547\pi$
$983$	−1.48666	−0.0474172	−0.0237086	+	0.999719i	$0.507547\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−12.6703	−0.403299
$988$	0	0
$989$	11.3840	0.361992
$990$	0	0
$991$	17.3854	0.552265	0.276132	−	0.961120i	$-0.410947\pi$
$991$	17.3854	0.552265	0.276132	+	0.961120i	$0.410947\pi$
$992$	0	0
$993$	−8.14914	−0.258605
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−20.6474	−0.653910	−0.326955	−	0.945040i	$-0.606023\pi$
$997$	−20.6474	−0.653910	−0.326955	+	0.945040i	$0.606023\pi$
$998$	0	0
$999$	−42.4349	−1.34258

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.a.w.1.3	✓	3
4.3	odd	2		9200.2.a.ch.1.1		3
5.2	odd	4		4600.2.e.s.4049.2		6
5.3	odd	4		4600.2.e.s.4049.5		6
5.4	even	2		4600.2.a.z.1.1	yes	3
20.19	odd	2		9200.2.a.cb.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4600.2.a.w.1.3	✓	3	1.1	even	1	trivial
4600.2.a.z.1.1	yes	3	5.4	even	2
4600.2.e.s.4049.2		6	5.2	odd	4
4600.2.e.s.4049.5		6	5.3	odd	4
9200.2.a.cb.1.3		3	20.19	odd	2
9200.2.a.ch.1.1		3	4.3	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:	\( N \)	\(=\)	\( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4600.a (trivial)

\(f(q)\)	\(=\)	\(q+0.801938 q^{3} +1.60388 q^{7} -2.35690 q^{9} -6.09783 q^{11} +1.35690 q^{13} +1.60388 q^{17} +1.78017 q^{19} +1.28621 q^{21} +1.00000 q^{23} -4.29590 q^{27} -4.82908 q^{29} +6.15883 q^{31} -4.89008 q^{33} +9.87800 q^{37} +1.08815 q^{39} -8.89977 q^{41} +11.3840 q^{43} -9.85086 q^{47} -4.42758 q^{49} +1.28621 q^{51} -9.20775 q^{53} +1.42758 q^{57} -7.27413 q^{59} +0.933624 q^{61} -3.78017 q^{63} -9.42758 q^{67} +0.801938 q^{69} +12.6189 q^{71} +2.60925 q^{73} -9.78017 q^{77} -5.16421 q^{79} +3.62565 q^{81} -15.4034 q^{83} -3.87263 q^{87} -15.7017 q^{89} +2.17629 q^{91} +4.93900 q^{93} -12.8901 q^{97} +14.3720 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{3} - 4 q^{7} - 3 q^{9} - 4 q^{17} + 4 q^{19} + 12 q^{21} + 3 q^{23} + q^{27} - 4 q^{29} + 10 q^{31} - 14 q^{33} + 10 q^{37} + 7 q^{39} - 4 q^{41} + 24 q^{43} - 16 q^{47} + 3 q^{49} + 12 q^{51}+ \cdots + 14 q^{99}+O(q^{100}) \) 3 * q - 2 * q^3 - 4 * q^7 - 3 * q^9 - 4 * q^17 + 4 * q^19 + 12 * q^21 + 3 * q^23 + q^27 - 4 * q^29 + 10 * q^31 - 14 * q^33 + 10 * q^37 + 7 * q^39 - 4 * q^41 + 24 * q^43 - 16 * q^47 + 3 * q^49 + 12 * q^51 - 10 * q^53 - 12 * q^57 - 11 * q^59 - 4 * q^61 - 10 * q^63 - 12 * q^67 - 2 * q^69 + 4 * q^71 - 4 * q^73 - 28 * q^77 - 4 * q^79 - q^81 + 8 * q^83 + 5 * q^87 - 20 * q^89 + 14 * q^91 + 5 * q^93 - 38 * q^97 + 14 * q^99

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