LMFDB - Embedded newform 4900.2.a.bg.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4900,2,Mod(1,4900)] 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("4900.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4900, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4900.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0,0,-4,0,0,0,0,0,0,0,0,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

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

Self dual:	yes
Analytic conductor:	$39.1266969904$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{5})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 4 $ x^4 - 6*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.874032$ of defining polynomial
Character	$\chi$	$=$	4900.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.874032 q^{3} -2.23607 q^{9} +3.47214 q^{11} +2.28825 q^{13} -1.74806 q^{17} +0.333851 q^{19} -5.47214 q^{23} +4.57649 q^{27} -4.23607 q^{29} +1.20788 q^{31} -3.03476 q^{33} +0.236068 q^{37} -2.00000 q^{39} -1.95440 q^{41} -8.23607 q^{43} -7.73877 q^{47} +1.52786 q^{51} +1.70820 q^{53} -0.291796 q^{57} +5.11667 q^{59} +14.6035 q^{61} -3.94427 q^{67} +4.78282 q^{69} +3.29180 q^{71} +14.6035 q^{73} -2.52786 q^{79} +2.70820 q^{81} +9.02546 q^{83} +3.70246 q^{87} -15.3500 q^{89} -1.05573 q^{93} -12.1089 q^{97} -7.76393 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{11} - 4 q^{23} - 8 q^{29} - 8 q^{37} - 8 q^{39} - 24 q^{43} + 24 q^{51} - 20 q^{53} - 28 q^{57} + 20 q^{67} + 40 q^{71} - 28 q^{79} - 16 q^{81} - 40 q^{93} - 40 q^{99}+O(q^{100}) $ 4 * q - 4 * q^11 - 4 * q^23 - 8 * q^29 - 8 * q^37 - 8 * q^39 - 24 * q^43 + 24 * q^51 - 20 * q^53 - 28 * q^57 + 20 * q^67 + 40 * q^71 - 28 * q^79 - 16 * q^81 - 40 * q^93 - 40 * 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
$3$	−0.874032	−0.504623	−0.252311	−	0.967646i	$-0.581191\pi$
$3$	−0.874032	−0.504623	−0.252311	+	0.967646i	$0.581191\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.23607	−0.745356
$10$	0	0
$11$	3.47214	1.04689	0.523444	−	0.852060i	$-0.324647\pi$
$11$	3.47214	1.04689	0.523444	+	0.852060i	$0.324647\pi$
$12$	0	0
$13$	2.28825	0.634645	0.317323	−	0.948318i	$-0.397216\pi$
$13$	2.28825	0.634645	0.317323	+	0.948318i	$0.397216\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.74806	−0.423968	−0.211984	−	0.977273i	$-0.567992\pi$
$17$	−1.74806	−0.423968	−0.211984	+	0.977273i	$0.567992\pi$
$18$	0	0
$19$	0.333851	0.0765906	0.0382953	−	0.999266i	$-0.487807\pi$
$19$	0.333851	0.0765906	0.0382953	+	0.999266i	$0.487807\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.47214	−1.14102	−0.570510	−	0.821291i	$-0.693254\pi$
$23$	−5.47214	−1.14102	−0.570510	+	0.821291i	$0.693254\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.57649	0.880746
$28$	0	0
$29$	−4.23607	−0.786618	−0.393309	−	0.919406i	$-0.628670\pi$
$29$	−4.23607	−0.786618	−0.393309	+	0.919406i	$0.628670\pi$
$30$	0	0
$31$	1.20788	0.216942	0.108471	−	0.994100i	$-0.465405\pi$
$31$	1.20788	0.216942	0.108471	+	0.994100i	$0.465405\pi$
$32$	0	0
$33$	−3.03476	−0.528284
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.236068	0.0388093	0.0194047	−	0.999812i	$-0.493823\pi$
$37$	0.236068	0.0388093	0.0194047	+	0.999812i	$0.493823\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	−1.95440	−0.305225	−0.152613	−	0.988286i	$-0.548769\pi$
$41$	−1.95440	−0.305225	−0.152613	+	0.988286i	$0.548769\pi$
$42$	0	0
$43$	−8.23607	−1.25599	−0.627994	−	0.778218i	$-0.716124\pi$
$43$	−8.23607	−1.25599	−0.627994	+	0.778218i	$0.716124\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−7.73877	−1.12882	−0.564408	−	0.825496i	$-0.690895\pi$
$47$	−7.73877	−1.12882	−0.564408	+	0.825496i	$0.690895\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	1.52786	0.213944
$52$	0	0
$53$	1.70820	0.234640	0.117320	−	0.993094i	$-0.462570\pi$
$53$	1.70820	0.234640	0.117320	+	0.993094i	$0.462570\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−0.291796	−0.0386493
$58$	0	0
$59$	5.11667	0.666134	0.333067	−	0.942903i	$-0.391916\pi$
$59$	5.11667	0.666134	0.333067	+	0.942903i	$0.391916\pi$
$60$	0	0
$61$	14.6035	1.86979	0.934894	−	0.354928i	$-0.115495\pi$
$61$	14.6035	1.86979	0.934894	+	0.354928i	$0.115495\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−3.94427	−0.481870	−0.240935	−	0.970541i	$-0.577454\pi$
$67$	−3.94427	−0.481870	−0.240935	+	0.970541i	$0.577454\pi$
$68$	0	0
$69$	4.78282	0.575784
$70$	0	0
$71$	3.29180	0.390664	0.195332	−	0.980737i	$-0.437422\pi$
$71$	3.29180	0.390664	0.195332	+	0.980737i	$0.437422\pi$
$72$	0	0
$73$	14.6035	1.70921	0.854606	−	0.519278i	$-0.173799\pi$
$73$	14.6035	1.70921	0.854606	+	0.519278i	$0.173799\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−2.52786	−0.284407	−0.142203	−	0.989837i	$-0.545419\pi$
$79$	−2.52786	−0.284407	−0.142203	+	0.989837i	$0.545419\pi$
$80$	0	0
$81$	2.70820	0.300912
$82$	0	0
$83$	9.02546	0.990673	0.495337	−	0.868701i	$-0.335045\pi$
$83$	9.02546	0.990673	0.495337	+	0.868701i	$0.335045\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.70246	0.396945
$88$	0	0
$89$	−15.3500	−1.62710	−0.813549	−	0.581496i	$-0.802468\pi$
$89$	−15.3500	−1.62710	−0.813549	+	0.581496i	$0.802468\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−1.05573	−0.109474
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−12.1089	−1.22948	−0.614738	−	0.788732i	$-0.710738\pi$
$97$	−12.1089	−1.22948	−0.614738	+	0.788732i	$0.710738\pi$
$98$	0	0
$99$	−7.76393	−0.780305
$100$	0	0
$101$	−6.53089	−0.649847	−0.324924	−	0.945740i	$-0.605339\pi$
$101$	−6.53089	−0.649847	−0.324924	+	0.945740i	$0.605339\pi$
$102$	0	0
$103$	−18.3848	−1.81151	−0.905753	−	0.423806i	$-0.860694\pi$
$103$	−18.3848	−1.81151	−0.905753	+	0.423806i	$0.860694\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−3.52786	−0.341051	−0.170526	−	0.985353i	$-0.554547\pi$
$107$	−3.52786	−0.341051	−0.170526	+	0.985353i	$0.554547\pi$
$108$	0	0
$109$	10.4164	0.997711	0.498855	−	0.866685i	$-0.333754\pi$
$109$	10.4164	0.997711	0.498855	+	0.866685i	$0.333754\pi$
$110$	0	0
$111$	−0.206331	−0.0195841
$112$	0	0
$113$	−18.2361	−1.71550	−0.857752	−	0.514063i	$-0.828140\pi$
$113$	−18.2361	−1.71550	−0.857752	+	0.514063i	$0.828140\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−5.11667	−0.473037
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.05573	0.0959753
$122$	0	0
$123$	1.70820	0.154024
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−11.1803	−0.992095	−0.496047	−	0.868295i	$-0.665216\pi$
$127$	−11.1803	−0.992095	−0.496047	+	0.868295i	$0.665216\pi$
$128$	0	0
$129$	7.19859	0.633800
$130$	0	0
$131$	14.8098	1.29394	0.646971	−	0.762515i	$-0.276036\pi$
$131$	14.8098	1.29394	0.646971	+	0.762515i	$0.276036\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	0	0
$139$	11.7751	0.998749	0.499375	−	0.866386i	$-0.333563\pi$
$139$	11.7751	0.998749	0.499375	+	0.866386i	$0.333563\pi$
$140$	0	0
$141$	6.76393	0.569626
$142$	0	0
$143$	7.94510	0.664403
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−6.70820	−0.549557	−0.274779	−	0.961507i	$-0.588605\pi$
$149$	−6.70820	−0.549557	−0.274779	+	0.961507i	$0.588605\pi$
$150$	0	0
$151$	3.76393	0.306304	0.153152	−	0.988203i	$-0.451058\pi$
$151$	3.76393	0.306304	0.153152	+	0.988203i	$0.451058\pi$
$152$	0	0
$153$	3.90879	0.316007
$154$	0	0
$155$	0	0
$156$	0	0
$157$	12.1877	0.972688	0.486344	−	0.873768i	$-0.338330\pi$
$157$	12.1877	0.972688	0.486344	+	0.873768i	$0.338330\pi$
$158$	0	0
$159$	−1.49302	−0.118405
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−20.1803	−1.58065	−0.790323	−	0.612690i	$-0.790087\pi$
$163$	−20.1803	−1.58065	−0.790323	+	0.612690i	$0.790087\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	19.9265	1.54196	0.770980	−	0.636860i	$-0.219767\pi$
$167$	19.9265	1.54196	0.770980	+	0.636860i	$0.219767\pi$
$168$	0	0
$169$	−7.76393	−0.597226
$170$	0	0
$171$	−0.746512	−0.0570872
$172$	0	0
$173$	−18.3848	−1.39777	−0.698884	−	0.715235i	$-0.746320\pi$
$173$	−18.3848	−1.39777	−0.698884	+	0.715235i	$0.746320\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−4.47214	−0.336146
$178$	0	0
$179$	−11.7082	−0.875112	−0.437556	−	0.899191i	$-0.644156\pi$
$179$	−11.7082	−0.875112	−0.437556	+	0.899191i	$0.644156\pi$
$180$	0	0
$181$	−24.8369	−1.84611	−0.923054	−	0.384670i	$-0.874315\pi$
$181$	−24.8369	−1.84611	−0.923054	+	0.384670i	$0.874315\pi$
$182$	0	0
$183$	−12.7639	−0.943537
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−6.06952	−0.443847
$188$	0	0
$189$	0	0
$190$	0	0
$191$	10.1803	0.736624	0.368312	−	0.929702i	$-0.379936\pi$
$191$	10.1803	0.736624	0.368312	+	0.929702i	$0.379936\pi$
$192$	0	0
$193$	−15.6525	−1.12669	−0.563345	−	0.826222i	$-0.690486\pi$
$193$	−15.6525	−1.12669	−0.563345	+	0.826222i	$0.690486\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−10.5279	−0.750079	−0.375040	−	0.927009i	$-0.622371\pi$
$197$	−10.5279	−0.750079	−0.375040	+	0.927009i	$0.622371\pi$
$198$	0	0
$199$	−0.461370	−0.0327057	−0.0163528	−	0.999866i	$-0.505205\pi$
$199$	−0.461370	−0.0327057	−0.0163528	+	0.999866i	$0.505205\pi$
$200$	0	0
$201$	3.44742	0.243162
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	12.2361	0.850466
$208$	0	0
$209$	1.15917	0.0801818
$210$	0	0
$211$	18.4721	1.27167	0.635837	−	0.771823i	$-0.280655\pi$
$211$	18.4721	1.27167	0.635837	+	0.771823i	$0.280655\pi$
$212$	0	0
$213$	−2.87714	−0.197138
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−12.7639	−0.862507
$220$	0	0
$221$	−4.00000	−0.269069
$222$	0	0
$223$	15.0162	1.00556	0.502778	−	0.864415i	$-0.332311\pi$
$223$	15.0162	1.00556	0.502778	+	0.864415i	$0.332311\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−19.0525	−1.26456	−0.632279	−	0.774741i	$-0.717880\pi$
$227$	−19.0525	−1.26456	−0.632279	+	0.774741i	$0.717880\pi$
$228$	0	0
$229$	−19.7202	−1.30315	−0.651573	−	0.758586i	$-0.725891\pi$
$229$	−19.7202	−1.30315	−0.651573	+	0.758586i	$0.725891\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−19.4721	−1.27566	−0.637831	−	0.770176i	$-0.720168\pi$
$233$	−19.4721	−1.27566	−0.637831	+	0.770176i	$0.720168\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	2.20943	0.143518
$238$	0	0
$239$	−22.6525	−1.46527	−0.732633	−	0.680623i	$-0.761709\pi$
$239$	−22.6525	−1.46527	−0.732633	+	0.680623i	$0.761709\pi$
$240$	0	0
$241$	1.41421	0.0910975	0.0455488	−	0.998962i	$-0.485496\pi$
$241$	1.41421	0.0910975	0.0455488	+	0.998962i	$0.485496\pi$
$242$	0	0
$243$	−16.0965	−1.03259
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0.763932	0.0486078
$248$	0	0
$249$	−7.88854	−0.499916
$250$	0	0
$251$	4.57649	0.288866	0.144433	−	0.989515i	$-0.453864\pi$
$251$	4.57649	0.288866	0.144433	+	0.989515i	$0.453864\pi$
$252$	0	0
$253$	−19.0000	−1.19452
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.20788	−0.0753456	−0.0376728	−	0.999290i	$-0.511994\pi$
$257$	−1.20788	−0.0753456	−0.0376728	+	0.999290i	$0.511994\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	9.47214	0.586310
$262$	0	0
$263$	−2.70820	−0.166995	−0.0834975	−	0.996508i	$-0.526609\pi$
$263$	−2.70820	−0.166995	−0.0834975	+	0.996508i	$0.526609\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	13.4164	0.821071
$268$	0	0
$269$	−2.54328	−0.155067	−0.0775334	−	0.996990i	$-0.524704\pi$
$269$	−2.54328	−0.155067	−0.0775334	+	0.996990i	$0.524704\pi$
$270$	0	0
$271$	−3.62365	−0.220121	−0.110060	−	0.993925i	$-0.535104\pi$
$271$	−3.62365	−0.220121	−0.110060	+	0.993925i	$0.535104\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−3.05573	−0.183601	−0.0918005	−	0.995777i	$-0.529262\pi$
$277$	−3.05573	−0.183601	−0.0918005	+	0.995777i	$0.529262\pi$
$278$	0	0
$279$	−2.70091	−0.161699
$280$	0	0
$281$	16.1246	0.961914	0.480957	−	0.876744i	$-0.340289\pi$
$281$	16.1246	0.961914	0.480957	+	0.876744i	$0.340289\pi$
$282$	0	0
$283$	−2.08191	−0.123757	−0.0618785	−	0.998084i	$-0.519709\pi$
$283$	−2.08191	−0.123757	−0.0618785	+	0.998084i	$0.519709\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.9443	−0.820251
$290$	0	0
$291$	10.5836	0.620421
$292$	0	0
$293$	21.2619	1.24213	0.621067	−	0.783757i	$-0.286699\pi$
$293$	21.2619	1.24213	0.621067	+	0.783757i	$0.286699\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	15.8902	0.922043
$298$	0	0
$299$	−12.5216	−0.724142
$300$	0	0
$301$	0	0
$302$	0	0
$303$	5.70820	0.327928
$304$	0	0
$305$	0	0
$306$	0	0
$307$	12.3941	0.707367	0.353684	−	0.935365i	$-0.384929\pi$
$307$	12.3941	0.707367	0.353684	+	0.935365i	$0.384929\pi$
$308$	0	0
$309$	16.0689	0.914127
$310$	0	0
$311$	−34.8639	−1.97695	−0.988474	−	0.151389i	$-0.951625\pi$
$311$	−34.8639	−1.97695	−0.988474	+	0.151389i	$0.951625\pi$
$312$	0	0
$313$	−16.2241	−0.917038	−0.458519	−	0.888685i	$-0.651620\pi$
$313$	−16.2241	−0.917038	−0.458519	+	0.888685i	$0.651620\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−26.4164	−1.48369	−0.741847	−	0.670570i	$-0.766050\pi$
$317$	−26.4164	−1.48369	−0.741847	+	0.670570i	$0.766050\pi$
$318$	0	0
$319$	−14.7082	−0.823501
$320$	0	0
$321$	3.08347	0.172102
$322$	0	0
$323$	−0.583592	−0.0324719
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−9.10427	−0.503468
$328$	0	0
$329$	0	0
$330$	0	0
$331$	27.3607	1.50388	0.751939	−	0.659232i	$-0.229119\pi$
$331$	27.3607	1.50388	0.751939	+	0.659232i	$0.229119\pi$
$332$	0	0
$333$	−0.527864	−0.0289268
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−23.1246	−1.25968	−0.629839	−	0.776726i	$-0.716879\pi$
$337$	−23.1246	−1.25968	−0.629839	+	0.776726i	$0.716879\pi$
$338$	0	0
$339$	15.9389	0.865683
$340$	0	0
$341$	4.19393	0.227114
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	19.3607	1.03934	0.519668	−	0.854368i	$-0.326056\pi$
$347$	19.3607	1.03934	0.519668	+	0.854368i	$0.326056\pi$
$348$	0	0
$349$	15.9690	0.854802	0.427401	−	0.904062i	$-0.359429\pi$
$349$	15.9690	0.854802	0.427401	+	0.904062i	$0.359429\pi$
$350$	0	0
$351$	10.4721	0.558961
$352$	0	0
$353$	10.3609	0.551453	0.275727	−	0.961236i	$-0.411082\pi$
$353$	10.3609	0.551453	0.275727	+	0.961236i	$0.411082\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	29.0689	1.53420	0.767099	−	0.641529i	$-0.221700\pi$
$359$	29.0689	1.53420	0.767099	+	0.641529i	$0.221700\pi$
$360$	0	0
$361$	−18.8885	−0.994134
$362$	0	0
$363$	−0.922740	−0.0484313
$364$	0	0
$365$	0	0
$366$	0	0
$367$	10.0270	0.523406	0.261703	−	0.965148i	$-0.415716\pi$
$367$	10.0270	0.523406	0.261703	+	0.965148i	$0.415716\pi$
$368$	0	0
$369$	4.37016	0.227501
$370$	0	0
$371$	0	0
$372$	0	0
$373$	10.1246	0.524233	0.262116	−	0.965036i	$-0.415580\pi$
$373$	10.1246	0.524233	0.262116	+	0.965036i	$0.415580\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−9.69316	−0.499223
$378$	0	0
$379$	−30.1246	−1.54740	−0.773699	−	0.633554i	$-0.781596\pi$
$379$	−30.1246	−1.54740	−0.773699	+	0.633554i	$0.781596\pi$
$380$	0	0
$381$	9.77198	0.500633
$382$	0	0
$383$	−25.8384	−1.32028	−0.660140	−	0.751142i	$-0.729503\pi$
$383$	−25.8384	−1.32028	−0.660140	+	0.751142i	$0.729503\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	18.4164	0.936159
$388$	0	0
$389$	19.1803	0.972482	0.486241	−	0.873825i	$-0.338368\pi$
$389$	19.1803	0.972482	0.486241	+	0.873825i	$0.338368\pi$
$390$	0	0
$391$	9.56564	0.483755
$392$	0	0
$393$	−12.9443	−0.652952
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−4.91034	−0.246443	−0.123221	−	0.992379i	$-0.539323\pi$
$397$	−4.91034	−0.246443	−0.123221	+	0.992379i	$0.539323\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−12.7082	−0.634617	−0.317309	−	0.948322i	$-0.602779\pi$
$401$	−12.7082	−0.634617	−0.317309	+	0.948322i	$0.602779\pi$
$402$	0	0
$403$	2.76393	0.137681
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.819660	0.0406290
$408$	0	0
$409$	−9.89949	−0.489499	−0.244749	−	0.969586i	$-0.578706\pi$
$409$	−9.89949	−0.489499	−0.244749	+	0.969586i	$0.578706\pi$
$410$	0	0
$411$	8.74032	0.431128
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−10.2918	−0.503991
$418$	0	0
$419$	−12.9343	−0.631880	−0.315940	−	0.948779i	$-0.602320\pi$
$419$	−12.9343	−0.631880	−0.315940	+	0.948779i	$0.602320\pi$
$420$	0	0
$421$	−18.5967	−0.906350	−0.453175	−	0.891422i	$-0.649709\pi$
$421$	−18.5967	−0.906350	−0.453175	+	0.891422i	$0.649709\pi$
$422$	0	0
$423$	17.3044	0.841369
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−6.94427	−0.335273
$430$	0	0
$431$	−16.6525	−0.802122	−0.401061	−	0.916051i	$-0.631358\pi$
$431$	−16.6525	−0.802122	−0.401061	+	0.916051i	$0.631358\pi$
$432$	0	0
$433$	−16.5579	−0.795722	−0.397861	−	0.917446i	$-0.630247\pi$
$433$	−16.5579	−0.795722	−0.397861	+	0.917446i	$0.630247\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.82688	−0.0873913
$438$	0	0
$439$	−11.0286	−0.526365	−0.263182	−	0.964746i	$-0.584772\pi$
$439$	−11.0286	−0.526365	−0.263182	+	0.964746i	$0.584772\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.12461	0.0534319	0.0267160	−	0.999643i	$-0.491495\pi$
$443$	1.12461	0.0534319	0.0267160	+	0.999643i	$0.491495\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	5.86319	0.277319
$448$	0	0
$449$	14.2361	0.671842	0.335921	−	0.941890i	$-0.390953\pi$
$449$	14.2361	0.671842	0.335921	+	0.941890i	$0.390953\pi$
$450$	0	0
$451$	−6.78593	−0.319537
$452$	0	0
$453$	−3.28980	−0.154568
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−5.65248	−0.264412	−0.132206	−	0.991222i	$-0.542206\pi$
$457$	−5.65248	−0.264412	−0.132206	+	0.991222i	$0.542206\pi$
$458$	0	0
$459$	−8.00000	−0.373408
$460$	0	0
$461$	−24.0903	−1.12200	−0.560999	−	0.827816i	$-0.689583\pi$
$461$	−24.0903	−1.12200	−0.560999	+	0.827816i	$0.689583\pi$
$462$	0	0
$463$	13.4164	0.623513	0.311757	−	0.950162i	$-0.399083\pi$
$463$	13.4164	0.623513	0.311757	+	0.950162i	$0.399083\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	20.3879	0.943439	0.471719	−	0.881749i	$-0.343634\pi$
$467$	20.3879	0.943439	0.471719	+	0.881749i	$0.343634\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−10.6525	−0.490840
$472$	0	0
$473$	−28.5967	−1.31488
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−3.81966	−0.174890
$478$	0	0
$479$	30.5725	1.39689	0.698447	−	0.715662i	$-0.253875\pi$
$479$	30.5725	1.39689	0.698447	+	0.715662i	$0.253875\pi$
$480$	0	0
$481$	0.540182	0.0246302
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	22.5279	1.02083	0.510417	−	0.859927i	$-0.329491\pi$
$487$	22.5279	1.02083	0.510417	+	0.859927i	$0.329491\pi$
$488$	0	0
$489$	17.6383	0.797630
$490$	0	0
$491$	6.23607	0.281430	0.140715	−	0.990050i	$-0.455060\pi$
$491$	6.23607	0.281430	0.140715	+	0.990050i	$0.455060\pi$
$492$	0	0
$493$	7.40492	0.333501
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−9.05573	−0.405390	−0.202695	−	0.979242i	$-0.564970\pi$
$499$	−9.05573	−0.405390	−0.202695	+	0.979242i	$0.564970\pi$
$500$	0	0
$501$	−17.4164	−0.778108
$502$	0	0
$503$	−6.65841	−0.296884	−0.148442	−	0.988921i	$-0.547426\pi$
$503$	−6.65841	−0.296884	−0.148442	+	0.988921i	$0.547426\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	6.78593	0.301374
$508$	0	0
$509$	−11.8539	−0.525414	−0.262707	−	0.964876i	$-0.584615\pi$
$509$	−11.8539	−0.525414	−0.262707	+	0.964876i	$0.584615\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	1.52786	0.0674568
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−26.8701	−1.18174
$518$	0	0
$519$	16.0689	0.705346
$520$	0	0
$521$	3.41732	0.149715	0.0748577	−	0.997194i	$-0.476150\pi$
$521$	3.41732	0.149715	0.0748577	+	0.997194i	$0.476150\pi$
$522$	0	0
$523$	10.2333	0.447473	0.223736	−	0.974650i	$-0.428175\pi$
$523$	10.2333	0.447473	0.223736	+	0.974650i	$0.428175\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.11146	−0.0919765
$528$	0	0
$529$	6.94427	0.301925
$530$	0	0
$531$	−11.4412	−0.496507
$532$	0	0
$533$	−4.47214	−0.193710
$534$	0	0
$535$	0	0
$536$	0	0
$537$	10.2333	0.441601
$538$	0	0
$539$	0	0
$540$	0	0
$541$	2.70820	0.116435	0.0582174	−	0.998304i	$-0.481458\pi$
$541$	2.70820	0.116435	0.0582174	+	0.998304i	$0.481458\pi$
$542$	0	0
$543$	21.7082	0.931588
$544$	0	0
$545$	0	0
$546$	0	0
$547$	16.7082	0.714391	0.357196	−	0.934030i	$-0.383733\pi$
$547$	16.7082	0.714391	0.357196	+	0.934030i	$0.383733\pi$
$548$	0	0
$549$	−32.6544	−1.39366
$550$	0	0
$551$	−1.41421	−0.0602475
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−45.6525	−1.93436	−0.967179	−	0.254097i	$-0.918222\pi$
$557$	−45.6525	−1.93436	−0.967179	+	0.254097i	$0.918222\pi$
$558$	0	0
$559$	−18.8461	−0.797107
$560$	0	0
$561$	5.30495	0.223975
$562$	0	0
$563$	−29.0308	−1.22350	−0.611751	−	0.791051i	$-0.709534\pi$
$563$	−29.0308	−1.22350	−0.611751	+	0.791051i	$0.709534\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	6.05573	0.253869	0.126935	−	0.991911i	$-0.459486\pi$
$569$	6.05573	0.253869	0.126935	+	0.991911i	$0.459486\pi$
$570$	0	0
$571$	35.6525	1.49201	0.746005	−	0.665941i	$-0.231970\pi$
$571$	35.6525	1.49201	0.746005	+	0.665941i	$0.231970\pi$
$572$	0	0
$573$	−8.89794	−0.371717
$574$	0	0
$575$	0	0
$576$	0	0
$577$	14.2209	0.592026	0.296013	−	0.955184i	$-0.404343\pi$
$577$	14.2209	0.592026	0.296013	+	0.955184i	$0.404343\pi$
$578$	0	0
$579$	13.6808	0.568553
$580$	0	0
$581$	0	0
$582$	0	0
$583$	5.93112	0.245642
$584$	0	0
$585$	0	0
$586$	0	0
$587$	21.0857	0.870299	0.435150	−	0.900358i	$-0.356695\pi$
$587$	21.0857	0.870299	0.435150	+	0.900358i	$0.356695\pi$
$588$	0	0
$589$	0.403252	0.0166157
$590$	0	0
$591$	9.20169	0.378507
$592$	0	0
$593$	39.0277	1.60268	0.801338	−	0.598212i	$-0.204122\pi$
$593$	39.0277	1.60268	0.801338	+	0.598212i	$0.204122\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0.403252	0.0165040
$598$	0	0
$599$	17.4721	0.713892	0.356946	−	0.934125i	$-0.383818\pi$
$599$	17.4721	0.713892	0.356946	+	0.934125i	$0.383818\pi$
$600$	0	0
$601$	−7.14988	−0.291650	−0.145825	−	0.989310i	$-0.546584\pi$
$601$	−7.14988	−0.291650	−0.145825	+	0.989310i	$0.546584\pi$
$602$	0	0
$603$	8.81966	0.359164
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−23.4226	−0.950696	−0.475348	−	0.879798i	$-0.657678\pi$
$607$	−23.4226	−0.950696	−0.475348	+	0.879798i	$0.657678\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−17.7082	−0.716397
$612$	0	0
$613$	8.88854	0.359005	0.179502	−	0.983758i	$-0.442551\pi$
$613$	8.88854	0.359005	0.179502	+	0.983758i	$0.442551\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−29.3607	−1.18202	−0.591008	−	0.806666i	$-0.701270\pi$
$617$	−29.3607	−1.18202	−0.591008	+	0.806666i	$0.701270\pi$
$618$	0	0
$619$	−27.5865	−1.10879	−0.554397	−	0.832252i	$-0.687051\pi$
$619$	−27.5865	−1.10879	−0.554397	+	0.832252i	$0.687051\pi$
$620$	0	0
$621$	−25.0432	−1.00495
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−1.01316	−0.0404615
$628$	0	0
$629$	−0.412662	−0.0164539
$630$	0	0
$631$	−1.00000	−0.0398094	−0.0199047	−	0.999802i	$-0.506336\pi$
$631$	−1.00000	−0.0398094	−0.0199047	+	0.999802i	$0.506336\pi$
$632$	0	0
$633$	−16.1452	−0.641716
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−7.36068	−0.291184
$640$	0	0
$641$	5.36068	0.211734	0.105867	−	0.994380i	$-0.466238\pi$
$641$	5.36068	0.211734	0.105867	+	0.994380i	$0.466238\pi$
$642$	0	0
$643$	17.8933	0.705643	0.352821	−	0.935691i	$-0.385222\pi$
$643$	17.8933	0.705643	0.352821	+	0.935691i	$0.385222\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−44.0957	−1.73358	−0.866790	−	0.498674i	$-0.833821\pi$
$647$	−44.0957	−1.73358	−0.866790	+	0.498674i	$0.833821\pi$
$648$	0	0
$649$	17.7658	0.697368
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−27.3050	−1.06853	−0.534263	−	0.845319i	$-0.679411\pi$
$653$	−27.3050	−1.06853	−0.534263	+	0.845319i	$0.679411\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−32.6544	−1.27397
$658$	0	0
$659$	7.52786	0.293244	0.146622	−	0.989193i	$-0.453160\pi$
$659$	7.52786	0.293244	0.146622	+	0.989193i	$0.453160\pi$
$660$	0	0
$661$	12.4729	0.485139	0.242569	−	0.970134i	$-0.422010\pi$
$661$	12.4729	0.485139	0.242569	+	0.970134i	$0.422010\pi$
$662$	0	0
$663$	3.49613	0.135778
$664$	0	0
$665$	0	0
$666$	0	0
$667$	23.1803	0.897546
$668$	0	0
$669$	−13.1246	−0.507427
$670$	0	0
$671$	50.7054	1.95746
$672$	0	0
$673$	21.7082	0.836790	0.418395	−	0.908265i	$-0.362593\pi$
$673$	21.7082	0.836790	0.418395	+	0.908265i	$0.362593\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	37.1034	1.42600	0.713000	−	0.701164i	$-0.247336\pi$
$677$	37.1034	1.42600	0.713000	+	0.701164i	$0.247336\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	16.6525	0.638124
$682$	0	0
$683$	−0.527864	−0.0201982	−0.0100991	−	0.999949i	$-0.503215\pi$
$683$	−0.527864	−0.0201982	−0.0100991	+	0.999949i	$0.503215\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	17.2361	0.657597
$688$	0	0
$689$	3.90879	0.148913
$690$	0	0
$691$	−36.5632	−1.39093	−0.695465	−	0.718560i	$-0.744802\pi$
$691$	−36.5632	−1.39093	−0.695465	+	0.718560i	$0.744802\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	3.41641	0.129406
$698$	0	0
$699$	17.0193	0.643728
$700$	0	0
$701$	−46.2492	−1.74681	−0.873405	−	0.486995i	$-0.838093\pi$
$701$	−46.2492	−1.74681	−0.873405	+	0.486995i	$0.838093\pi$
$702$	0	0
$703$	0.0788114	0.00297243
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	19.8885	0.746930	0.373465	−	0.927644i	$-0.378170\pi$
$709$	19.8885	0.746930	0.373465	+	0.927644i	$0.378170\pi$
$710$	0	0
$711$	5.65248	0.211984
$712$	0	0
$713$	−6.60970	−0.247535
$714$	0	0
$715$	0	0
$716$	0	0
$717$	19.7990	0.739407
$718$	0	0
$719$	36.4056	1.35770	0.678850	−	0.734277i	$-0.262479\pi$
$719$	36.4056	1.35770	0.678850	+	0.734277i	$0.262479\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−1.23607	−0.0459699
$724$	0	0
$725$	0	0
$726$	0	0
$727$	37.9473	1.40739	0.703694	−	0.710503i	$-0.251532\pi$
$727$	37.9473	1.40739	0.703694	+	0.710503i	$0.251532\pi$
$728$	0	0
$729$	5.94427	0.220158
$730$	0	0
$731$	14.3972	0.532499
$732$	0	0
$733$	31.7804	1.17384	0.586918	−	0.809646i	$-0.300341\pi$
$733$	31.7804	1.17384	0.586918	+	0.809646i	$0.300341\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−13.6950	−0.504464
$738$	0	0
$739$	−35.9443	−1.32223	−0.661116	−	0.750284i	$-0.729917\pi$
$739$	−35.9443	−1.32223	−0.661116	+	0.750284i	$0.729917\pi$
$740$	0	0
$741$	−0.667701	−0.0245286
$742$	0	0
$743$	23.8197	0.873859	0.436929	−	0.899496i	$-0.356066\pi$
$743$	23.8197	0.873859	0.436929	+	0.899496i	$0.356066\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−20.1815	−0.738404
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−12.9443	−0.472343	−0.236172	−	0.971711i	$-0.575893\pi$
$751$	−12.9443	−0.472343	−0.236172	+	0.971711i	$0.575893\pi$
$752$	0	0
$753$	−4.00000	−0.145768
$754$	0	0
$755$	0	0
$756$	0	0
$757$	39.7214	1.44370	0.721849	−	0.692051i	$-0.243293\pi$
$757$	39.7214	1.44370	0.721849	+	0.692051i	$0.243293\pi$
$758$	0	0
$759$	16.6066	0.602782
$760$	0	0
$761$	38.6150	1.39979	0.699897	−	0.714244i	$-0.253229\pi$
$761$	38.6150	1.39979	0.699897	+	0.714244i	$0.253229\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	11.7082	0.422759
$768$	0	0
$769$	−18.0996	−0.652689	−0.326345	−	0.945251i	$-0.605817\pi$
$769$	−18.0996	−0.652689	−0.326345	+	0.945251i	$0.605817\pi$
$770$	0	0
$771$	1.05573	0.0380211
$772$	0	0
$773$	−22.7549	−0.818438	−0.409219	−	0.912436i	$-0.634199\pi$
$773$	−22.7549	−0.818438	−0.409219	+	0.912436i	$0.634199\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−0.652476	−0.0233774
$780$	0	0
$781$	11.4296	0.408982
$782$	0	0
$783$	−19.3863	−0.692811
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−34.2449	−1.22070	−0.610349	−	0.792133i	$-0.708971\pi$
$787$	−34.2449	−1.22070	−0.610349	+	0.792133i	$0.708971\pi$
$788$	0	0
$789$	2.36706	0.0842695
$790$	0	0
$791$	0	0
$792$	0	0
$793$	33.4164	1.18665
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−17.2256	−0.610162	−0.305081	−	0.952326i	$-0.598684\pi$
$797$	−17.2256	−0.610162	−0.305081	+	0.952326i	$0.598684\pi$
$798$	0	0
$799$	13.5279	0.478581
$800$	0	0
$801$	34.3237	1.21277
$802$	0	0
$803$	50.7054	1.78935
$804$	0	0
$805$	0	0
$806$	0	0
$807$	2.22291	0.0782502
$808$	0	0
$809$	52.1246	1.83260	0.916302	−	0.400488i	$-0.131159\pi$
$809$	52.1246	1.83260	0.916302	+	0.400488i	$0.131159\pi$
$810$	0	0
$811$	21.4682	0.753852	0.376926	−	0.926243i	$-0.376981\pi$
$811$	21.4682	0.753852	0.376926	+	0.926243i	$0.376981\pi$
$812$	0	0
$813$	3.16718	0.111078
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−2.74962	−0.0961969
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−28.4721	−0.993684	−0.496842	−	0.867841i	$-0.665507\pi$
$821$	−28.4721	−0.993684	−0.496842	+	0.867841i	$0.665507\pi$
$822$	0	0
$823$	33.9443	1.18322	0.591611	−	0.806223i	$-0.298492\pi$
$823$	33.9443	1.18322	0.591611	+	0.806223i	$0.298492\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	16.3475	0.568459	0.284230	−	0.958756i	$-0.408262\pi$
$827$	16.3475	0.568459	0.284230	+	0.958756i	$0.408262\pi$
$828$	0	0
$829$	0.922740	0.0320481	0.0160240	−	0.999872i	$-0.494899\pi$
$829$	0.922740	0.0320481	0.0160240	+	0.999872i	$0.494899\pi$
$830$	0	0
$831$	2.67080	0.0926492
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	5.52786	0.191071
$838$	0	0
$839$	−45.7162	−1.57830	−0.789149	−	0.614201i	$-0.789478\pi$
$839$	−45.7162	−1.57830	−0.789149	+	0.614201i	$0.789478\pi$
$840$	0	0
$841$	−11.0557	−0.381232
$842$	0	0
$843$	−14.0934	−0.485403
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	1.81966	0.0624506
$850$	0	0
$851$	−1.29180	−0.0442822
$852$	0	0
$853$	−31.3677	−1.07401	−0.537005	−	0.843579i	$-0.680445\pi$
$853$	−31.3677	−1.07401	−0.537005	+	0.843579i	$0.680445\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−41.3948	−1.41402	−0.707009	−	0.707205i	$-0.749956\pi$
$857$	−41.3948	−1.41402	−0.707009	+	0.707205i	$0.749956\pi$
$858$	0	0
$859$	52.8360	1.80274	0.901370	−	0.433049i	$-0.142562\pi$
$859$	52.8360	1.80274	0.901370	+	0.433049i	$0.142562\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	50.1246	1.70626	0.853131	−	0.521697i	$-0.174701\pi$
$863$	50.1246	1.70626	0.853131	+	0.521697i	$0.174701\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	12.1877	0.413917
$868$	0	0
$869$	−8.77709	−0.297742
$870$	0	0
$871$	−9.02546	−0.305816
$872$	0	0
$873$	27.0764	0.916397
$874$	0	0
$875$	0	0
$876$	0	0
$877$	58.4296	1.97303	0.986513	−	0.163682i	$-0.0523372\pi$
$877$	58.4296	1.97303	0.986513	+	0.163682i	$0.0523372\pi$
$878$	0	0
$879$	−18.5836	−0.626809
$880$	0	0
$881$	−6.60970	−0.222686	−0.111343	−	0.993782i	$-0.535515\pi$
$881$	−6.60970	−0.222686	−0.111343	+	0.993782i	$0.535515\pi$
$882$	0	0
$883$	23.9443	0.805789	0.402894	−	0.915246i	$-0.368004\pi$
$883$	23.9443	0.805789	0.402894	+	0.915246i	$0.368004\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.03165	0.0346396	0.0173198	−	0.999850i	$-0.494487\pi$
$887$	1.03165	0.0346396	0.0173198	+	0.999850i	$0.494487\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	9.40325	0.315021
$892$	0	0
$893$	−2.58359	−0.0864566
$894$	0	0
$895$	0	0
$896$	0	0
$897$	10.9443	0.365419
$898$	0	0
$899$	−5.11667	−0.170651
$900$	0	0
$901$	−2.98605	−0.0994797
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	43.7771	1.45359	0.726797	−	0.686852i	$-0.241008\pi$
$907$	43.7771	1.45359	0.726797	+	0.686852i	$0.241008\pi$
$908$	0	0
$909$	14.6035	0.484368
$910$	0	0
$911$	−42.8885	−1.42096	−0.710480	−	0.703717i	$-0.751522\pi$
$911$	−42.8885	−1.42096	−0.710480	+	0.703717i	$0.751522\pi$
$912$	0	0
$913$	31.3376	1.03712
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	5.29180	0.174560	0.0872801	−	0.996184i	$-0.472182\pi$
$919$	5.29180	0.174560	0.0872801	+	0.996184i	$0.472182\pi$
$920$	0	0
$921$	−10.8328	−0.356953
$922$	0	0
$923$	7.53244	0.247933
$924$	0	0
$925$	0	0
$926$	0	0
$927$	41.1096	1.35022
$928$	0	0
$929$	−6.78593	−0.222639	−0.111319	−	0.993785i	$-0.535508\pi$
$929$	−6.78593	−0.222639	−0.111319	+	0.993785i	$0.535508\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	30.4721	0.997613
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−33.4798	−1.09374	−0.546868	−	0.837219i	$-0.684180\pi$
$937$	−33.4798	−1.09374	−0.546868	+	0.837219i	$0.684180\pi$
$938$	0	0
$939$	14.1803	0.462758
$940$	0	0
$941$	−27.9991	−0.912746	−0.456373	−	0.889789i	$-0.650852\pi$
$941$	−27.9991	−0.912746	−0.456373	+	0.889789i	$0.650852\pi$
$942$	0	0
$943$	10.6947	0.348268
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−55.7771	−1.81251	−0.906256	−	0.422730i	$-0.861072\pi$
$947$	−55.7771	−1.81251	−0.906256	+	0.422730i	$0.861072\pi$
$948$	0	0
$949$	33.4164	1.08474
$950$	0	0
$951$	23.0888	0.748705
$952$	0	0
$953$	55.9017	1.81083	0.905417	−	0.424524i	$-0.139558\pi$
$953$	55.9017	1.81083	0.905417	+	0.424524i	$0.139558\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	12.8554	0.415557
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29.5410	−0.952936
$962$	0	0
$963$	7.88854	0.254205
$964$	0	0
$965$	0	0
$966$	0	0
$967$	31.4164	1.01028	0.505142	−	0.863036i	$-0.331440\pi$
$967$	31.4164	1.01028	0.505142	+	0.863036i	$0.331440\pi$
$968$	0	0
$969$	0.510078	0.0163861
$970$	0	0
$971$	−44.0957	−1.41510	−0.707549	−	0.706665i	$-0.750199\pi$
$971$	−44.0957	−1.41510	−0.707549	+	0.706665i	$0.750199\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−42.5967	−1.36279	−0.681395	−	0.731916i	$-0.738627\pi$
$977$	−42.5967	−1.36279	−0.681395	+	0.731916i	$0.738627\pi$
$978$	0	0
$979$	−53.2974	−1.70339
$980$	0	0
$981$	−23.2918	−0.743650
$982$	0	0
$983$	−1.36551	−0.0435529	−0.0217764	−	0.999763i	$-0.506932\pi$
$983$	−1.36551	−0.0435529	−0.0217764	+	0.999763i	$0.506932\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	45.0689	1.43311
$990$	0	0
$991$	35.6525	1.13254	0.566269	−	0.824220i	$-0.308386\pi$
$991$	35.6525	1.13254	0.566269	+	0.824220i	$0.308386\pi$
$992$	0	0
$993$	−23.9141	−0.758891
$994$	0	0
$995$	0	0
$996$	0	0
$997$	21.8809	0.692975	0.346488	−	0.938055i	$-0.387374\pi$
$997$	21.8809	0.692975	0.346488	+	0.938055i	$0.387374\pi$
$998$	0	0
$999$	1.08036	0.0341812

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4900.2.a.bg.1.2	✓	4
5.2	odd	4		4900.2.e.u.2549.5		8
5.3	odd	4		4900.2.e.u.2549.3		8
5.4	even	2		4900.2.a.bi.1.3	yes	4
7.6	odd	2	inner	4900.2.a.bg.1.3	yes	4
35.13	even	4		4900.2.e.u.2549.6		8
35.27	even	4		4900.2.e.u.2549.4		8
35.34	odd	2		4900.2.a.bi.1.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4900.2.a.bg.1.2	✓	4	1.1	even	1	trivial
4900.2.a.bg.1.3	yes	4	7.6	odd	2	inner
4900.2.a.bi.1.2	yes	4	35.34	odd	2
4900.2.a.bi.1.3	yes	4	5.4	even	2
4900.2.e.u.2549.3		8	5.3	odd	4
4900.2.e.u.2549.4		8	35.27	even	4
4900.2.e.u.2549.5		8	5.2	odd	4
4900.2.e.u.2549.6		8	35.13	even	4

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:	\( N \)	\(=\)	\( 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4900.a (trivial)

\(f(q)\)	\(=\)	\(q-0.874032 q^{3} -2.23607 q^{9} +3.47214 q^{11} +2.28825 q^{13} -1.74806 q^{17} +0.333851 q^{19} -5.47214 q^{23} +4.57649 q^{27} -4.23607 q^{29} +1.20788 q^{31} -3.03476 q^{33} +0.236068 q^{37} -2.00000 q^{39} -1.95440 q^{41} -8.23607 q^{43} -7.73877 q^{47} +1.52786 q^{51} +1.70820 q^{53} -0.291796 q^{57} +5.11667 q^{59} +14.6035 q^{61} -3.94427 q^{67} +4.78282 q^{69} +3.29180 q^{71} +14.6035 q^{73} -2.52786 q^{79} +2.70820 q^{81} +9.02546 q^{83} +3.70246 q^{87} -15.3500 q^{89} -1.05573 q^{93} -12.1089 q^{97} -7.76393 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{11} - 4 q^{23} - 8 q^{29} - 8 q^{37} - 8 q^{39} - 24 q^{43} + 24 q^{51} - 20 q^{53} - 28 q^{57} + 20 q^{67} + 40 q^{71} - 28 q^{79} - 16 q^{81} - 40 q^{93} - 40 q^{99}+O(q^{100}) \) 4 * q - 4 * q^11 - 4 * q^23 - 8 * q^29 - 8 * q^37 - 8 * q^39 - 24 * q^43 + 24 * q^51 - 20 * q^53 - 28 * q^57 + 20 * q^67 + 40 * q^71 - 28 * q^79 - 16 * q^81 - 40 * q^93 - 40 * q^99

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