LMFDB - Embedded newform 9300.2.a.ba.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9300,2,Mod(1,9300)] 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("9300.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9300.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,6,0,0,0,4,0,6,0,-3] 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:	$74.2608738798$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 22x^{4} + 30x^{3} + 112x^{2} - 64x - 64 $ x^6 - 2x^5 - 22x^4 + 30x^3 + 112x^2 - 64*x - 64
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-3.68688$ of defining polynomial
Character	$\chi$	$=$	9300.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +3.18902 q^{7} +1.00000 q^{9} +0.0550609 q^{11} +5.87876 q^{13} +5.79383 q^{17} +5.64811 q^{19} +3.18902 q^{21} -4.12967 q^{23} +1.00000 q^{27} +4.59908 q^{29} -1.00000 q^{31} +0.0550609 q^{33} +4.40403 q^{37} +5.87876 q^{39} -5.06778 q^{41} +10.0127 q^{43} -5.56318 q^{47} +3.16986 q^{49} +5.79383 q^{51} +9.42881 q^{53} +5.64811 q^{57} -13.9476 q^{59} +10.1847 q^{61} +3.18902 q^{63} -8.64382 q^{67} -4.12967 q^{69} -12.1260 q^{71} +2.00000 q^{73} +0.175590 q^{77} -16.7116 q^{79} +1.00000 q^{81} +7.12823 q^{83} +4.59908 q^{87} -5.61824 q^{89} +18.7475 q^{91} -1.00000 q^{93} -2.01916 q^{97} +0.0550609 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} + 4 q^{7} + 6 q^{9} - 3 q^{11} + 2 q^{13} + 4 q^{17} - 3 q^{19} + 4 q^{21} + 5 q^{23} + 6 q^{27} - 6 q^{31} - 3 q^{33} + 8 q^{37} + 2 q^{39} + 18 q^{41} + 15 q^{43} + q^{47} + 14 q^{49} + 4 q^{51}+ \cdots - 3 q^{99}+O(q^{100}) $ 6 * q + 6 * q^3 + 4 * q^7 + 6 * q^9 - 3 * q^11 + 2 * q^13 + 4 * q^17 - 3 * q^19 + 4 * q^21 + 5 * q^23 + 6 * q^27 - 6 * q^31 - 3 * q^33 + 8 * q^37 + 2 * q^39 + 18 * q^41 + 15 * q^43 + q^47 + 14 * q^49 + 4 * q^51 + 5 * q^53 - 3 * q^57 - 12 * q^59 + 28 * q^61 + 4 * q^63 + 3 * q^67 + 5 * q^69 + 25 * q^71 + 12 * q^73 + 8 * q^77 + 3 * q^79 + 6 * q^81 + 19 * q^83 + 4 * q^89 + 16 * q^91 - 6 * q^93 - 2 * q^97 - 3 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.18902	1.20534	0.602668	−	0.797992i	$-0.294104\pi$
$7$	3.18902	1.20534	0.602668	+	0.797992i	$0.294104\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.0550609	0.0166015	0.00830075	−	0.999966i	$-0.497358\pi$
$11$	0.0550609	0.0166015	0.00830075	+	0.999966i	$0.497358\pi$
$12$	0	0
$13$	5.87876	1.63048	0.815238	−	0.579126i	$-0.196606\pi$
$13$	5.87876	1.63048	0.815238	+	0.579126i	$0.196606\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.79383	1.40521	0.702605	−	0.711580i	$-0.252020\pi$
$17$	5.79383	1.40521	0.702605	+	0.711580i	$0.252020\pi$
$18$	0	0
$19$	5.64811	1.29577	0.647883	−	0.761740i	$-0.275655\pi$
$19$	5.64811	1.29577	0.647883	+	0.761740i	$0.275655\pi$
$20$	0	0
$21$	3.18902	0.695902
$22$	0	0
$23$	−4.12967	−0.861095	−0.430548	−	0.902568i	$-0.641680\pi$
$23$	−4.12967	−0.861095	−0.430548	+	0.902568i	$0.641680\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	4.59908	0.854028	0.427014	−	0.904245i	$-0.359566\pi$
$29$	4.59908	0.854028	0.427014	+	0.904245i	$0.359566\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	0.0550609	0.00958487
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.40403	0.724018	0.362009	−	0.932175i	$-0.382091\pi$
$37$	4.40403	0.724018	0.362009	+	0.932175i	$0.382091\pi$
$38$	0	0
$39$	5.87876	0.941355
$40$	0	0
$41$	−5.06778	−0.791455	−0.395727	−	0.918368i	$-0.629508\pi$
$41$	−5.06778	−0.791455	−0.395727	+	0.918368i	$0.629508\pi$
$42$	0	0
$43$	10.0127	1.52693	0.763463	−	0.645852i	$-0.223498\pi$
$43$	10.0127	1.52693	0.763463	+	0.645852i	$0.223498\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.56318	−0.811473	−0.405737	−	0.913990i	$-0.632985\pi$
$47$	−5.56318	−0.811473	−0.405737	+	0.913990i	$0.632985\pi$
$48$	0	0
$49$	3.16986	0.452837
$50$	0	0
$51$	5.79383	0.811299
$52$	0	0
$53$	9.42881	1.29515	0.647573	−	0.762003i	$-0.275784\pi$
$53$	9.42881	1.29515	0.647573	+	0.762003i	$0.275784\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	5.64811	0.748111
$58$	0	0
$59$	−13.9476	−1.81583	−0.907914	−	0.419157i	$-0.862326\pi$
$59$	−13.9476	−1.81583	−0.907914	+	0.419157i	$0.862326\pi$
$60$	0	0
$61$	10.1847	1.30402	0.652010	−	0.758210i	$-0.273926\pi$
$61$	10.1847	1.30402	0.652010	+	0.758210i	$0.273926\pi$
$62$	0	0
$63$	3.18902	0.401779
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−8.64382	−1.05601	−0.528005	−	0.849241i	$-0.677060\pi$
$67$	−8.64382	−1.05601	−0.528005	+	0.849241i	$0.677060\pi$
$68$	0	0
$69$	−4.12967	−0.497154
$70$	0	0
$71$	−12.1260	−1.43909	−0.719543	−	0.694448i	$-0.755649\pi$
$71$	−12.1260	−1.43909	−0.719543	+	0.694448i	$0.755649\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.175590	0.0200104
$78$	0	0
$79$	−16.7116	−1.88020	−0.940101	−	0.340896i	$-0.889270\pi$
$79$	−16.7116	−1.88020	−0.940101	+	0.340896i	$0.889270\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	7.12823	0.782425	0.391212	−	0.920300i	$-0.372056\pi$
$83$	7.12823	0.782425	0.391212	+	0.920300i	$0.372056\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	4.59908	0.493073
$88$	0	0
$89$	−5.61824	−0.595533	−0.297766	−	0.954639i	$-0.596242\pi$
$89$	−5.61824	−0.595533	−0.297766	+	0.954639i	$0.596242\pi$
$90$	0	0
$91$	18.7475	1.96527
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−2.01916	−0.205015	−0.102507	−	0.994732i	$-0.532687\pi$
$97$	−2.01916	−0.205015	−0.102507	+	0.994732i	$0.532687\pi$
$98$	0	0
$99$	0.0550609	0.00553383
$100$	0	0
$101$	1.45210	0.144489	0.0722446	−	0.997387i	$-0.476984\pi$
$101$	1.45210	0.144489	0.0722446	+	0.997387i	$0.476984\pi$
$102$	0	0
$103$	−3.12285	−0.307703	−0.153852	−	0.988094i	$-0.549168\pi$
$103$	−3.12285	−0.307703	−0.153852	+	0.988094i	$0.549168\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	11.2429	1.08689	0.543446	−	0.839444i	$-0.317119\pi$
$107$	11.2429	1.08689	0.543446	+	0.839444i	$0.317119\pi$
$108$	0	0
$109$	8.99356	0.861427	0.430713	−	0.902489i	$-0.358262\pi$
$109$	8.99356	0.861427	0.430713	+	0.902489i	$0.358262\pi$
$110$	0	0
$111$	4.40403	0.418012
$112$	0	0
$113$	−10.3929	−0.977683	−0.488842	−	0.872373i	$-0.662580\pi$
$113$	−10.3929	−0.977683	−0.488842	+	0.872373i	$0.662580\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	5.87876	0.543492
$118$	0	0
$119$	18.4767	1.69375
$120$	0	0
$121$	−10.9970	−0.999724
$122$	0	0
$123$	−5.06778	−0.456947
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−10.0276	−0.889805	−0.444903	−	0.895579i	$-0.646762\pi$
$127$	−10.0276	−0.889805	−0.444903	+	0.895579i	$0.646762\pi$
$128$	0	0
$129$	10.0127	0.881571
$130$	0	0
$131$	18.6907	1.63301	0.816507	−	0.577335i	$-0.195907\pi$
$131$	18.6907	1.63301	0.816507	+	0.577335i	$0.195907\pi$
$132$	0	0
$133$	18.0119	1.56183
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−8.69714	−0.743047	−0.371524	−	0.928424i	$-0.621164\pi$
$137$	−8.69714	−0.743047	−0.371524	+	0.928424i	$0.621164\pi$
$138$	0	0
$139$	−4.17229	−0.353889	−0.176944	−	0.984221i	$-0.556621\pi$
$139$	−4.17229	−0.353889	−0.176944	+	0.984221i	$0.556621\pi$
$140$	0	0
$141$	−5.56318	−0.468504
$142$	0	0
$143$	0.323690	0.0270683
$144$	0	0
$145$	0	0
$146$	0	0
$147$	3.16986	0.261446
$148$	0	0
$149$	−18.2276	−1.49326	−0.746632	−	0.665237i	$-0.768331\pi$
$149$	−18.2276	−1.49326	−0.746632	+	0.665237i	$0.768331\pi$
$150$	0	0
$151$	1.17057	0.0952594	0.0476297	−	0.998865i	$-0.484833\pi$
$151$	1.17057	0.0952594	0.0476297	+	0.998865i	$0.484833\pi$
$152$	0	0
$153$	5.79383	0.468404
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−6.47473	−0.516740	−0.258370	−	0.966046i	$-0.583185\pi$
$157$	−6.47473	−0.516740	−0.258370	+	0.966046i	$0.583185\pi$
$158$	0	0
$159$	9.42881	0.747753
$160$	0	0
$161$	−13.1696	−1.03791
$162$	0	0
$163$	−19.7986	−1.55074	−0.775371	−	0.631506i	$-0.782437\pi$
$163$	−19.7986	−1.55074	−0.775371	+	0.631506i	$0.782437\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−22.1121	−1.71109	−0.855543	−	0.517731i	$-0.826777\pi$
$167$	−22.1121	−1.71109	−0.855543	+	0.517731i	$0.826777\pi$
$168$	0	0
$169$	21.5599	1.65845
$170$	0	0
$171$	5.64811	0.431922
$172$	0	0
$173$	−12.6214	−0.959584	−0.479792	−	0.877382i	$-0.659288\pi$
$173$	−12.6214	−0.959584	−0.479792	+	0.877382i	$0.659288\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−13.9476	−1.04837
$178$	0	0
$179$	−9.85851	−0.736860	−0.368430	−	0.929656i	$-0.620104\pi$
$179$	−9.85851	−0.736860	−0.368430	+	0.929656i	$0.620104\pi$
$180$	0	0
$181$	10.1822	0.756837	0.378418	−	0.925635i	$-0.376468\pi$
$181$	10.1822	0.756837	0.378418	+	0.925635i	$0.376468\pi$
$182$	0	0
$183$	10.1847	0.752877
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0.319014	0.0233286
$188$	0	0
$189$	3.18902	0.231967
$190$	0	0
$191$	12.9480	0.936888	0.468444	−	0.883493i	$-0.344815\pi$
$191$	12.9480	0.936888	0.468444	+	0.883493i	$0.344815\pi$
$192$	0	0
$193$	3.80883	0.274166	0.137083	−	0.990560i	$-0.456227\pi$
$193$	3.80883	0.274166	0.137083	+	0.990560i	$0.456227\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2.70474	−0.192705	−0.0963524	−	0.995347i	$-0.530718\pi$
$197$	−2.70474	−0.192705	−0.0963524	+	0.995347i	$0.530718\pi$
$198$	0	0
$199$	−24.8750	−1.76334	−0.881669	−	0.471867i	$-0.843580\pi$
$199$	−24.8750	−1.76334	−0.881669	+	0.471867i	$0.843580\pi$
$200$	0	0
$201$	−8.64382	−0.609688
$202$	0	0
$203$	14.6666	1.02939
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−4.12967	−0.287032
$208$	0	0
$209$	0.310990	0.0215116
$210$	0	0
$211$	−13.0041	−0.895242	−0.447621	−	0.894223i	$-0.647729\pi$
$211$	−13.0041	−0.895242	−0.447621	+	0.894223i	$0.647729\pi$
$212$	0	0
$213$	−12.1260	−0.830857
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−3.18902	−0.216485
$218$	0	0
$219$	2.00000	0.135147
$220$	0	0
$221$	34.0606	2.29116
$222$	0	0
$223$	20.6630	1.38370	0.691848	−	0.722043i	$-0.256797\pi$
$223$	20.6630	1.38370	0.691848	+	0.722043i	$0.256797\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−6.75512	−0.448353	−0.224177	−	0.974549i	$-0.571969\pi$
$227$	−6.75512	−0.448353	−0.224177	+	0.974549i	$0.571969\pi$
$228$	0	0
$229$	23.3234	1.54126	0.770628	−	0.637285i	$-0.219943\pi$
$229$	23.3234	1.54126	0.770628	+	0.637285i	$0.219943\pi$
$230$	0	0
$231$	0.175590	0.0115530
$232$	0	0
$233$	−12.4527	−0.815800	−0.407900	−	0.913027i	$-0.633739\pi$
$233$	−12.4527	−0.815800	−0.407900	+	0.913027i	$0.633739\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−16.7116	−1.08554
$238$	0	0
$239$	1.05077	0.0679686	0.0339843	−	0.999422i	$-0.489180\pi$
$239$	1.05077	0.0679686	0.0339843	+	0.999422i	$0.489180\pi$
$240$	0	0
$241$	8.08572	0.520847	0.260424	−	0.965494i	$-0.416138\pi$
$241$	8.08572	0.520847	0.260424	+	0.965494i	$0.416138\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	33.2039	2.11271
$248$	0	0
$249$	7.12823	0.451733
$250$	0	0
$251$	14.9270	0.942184	0.471092	−	0.882084i	$-0.343860\pi$
$251$	14.9270	0.942184	0.471092	+	0.882084i	$0.343860\pi$
$252$	0	0
$253$	−0.227383	−0.0142955
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−30.2304	−1.88572	−0.942861	−	0.333186i	$-0.891876\pi$
$257$	−30.2304	−1.88572	−0.942861	+	0.333186i	$0.891876\pi$
$258$	0	0
$259$	14.0445	0.872685
$260$	0	0
$261$	4.59908	0.284676
$262$	0	0
$263$	7.41781	0.457402	0.228701	−	0.973497i	$-0.426552\pi$
$263$	7.41781	0.457402	0.228701	+	0.973497i	$0.426552\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−5.61824	−0.343831
$268$	0	0
$269$	−1.94988	−0.118886	−0.0594431	−	0.998232i	$-0.518932\pi$
$269$	−1.94988	−0.118886	−0.0594431	+	0.998232i	$0.518932\pi$
$270$	0	0
$271$	10.3273	0.627337	0.313669	−	0.949533i	$-0.398442\pi$
$271$	10.3273	0.627337	0.313669	+	0.949533i	$0.398442\pi$
$272$	0	0
$273$	18.7475	1.13465
$274$	0	0
$275$	0	0
$276$	0	0
$277$	10.3191	0.620013	0.310007	−	0.950734i	$-0.399669\pi$
$277$	10.3191	0.620013	0.310007	+	0.950734i	$0.399669\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	30.1504	1.79862	0.899312	−	0.437307i	$-0.144068\pi$
$281$	30.1504	1.79862	0.899312	+	0.437307i	$0.144068\pi$
$282$	0	0
$283$	−3.16117	−0.187912	−0.0939560	−	0.995576i	$-0.529951\pi$
$283$	−3.16117	−0.187912	−0.0939560	+	0.995576i	$0.529951\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−16.1613	−0.953970
$288$	0	0
$289$	16.5685	0.974618
$290$	0	0
$291$	−2.01916	−0.118365
$292$	0	0
$293$	−4.97831	−0.290836	−0.145418	−	0.989370i	$-0.546453\pi$
$293$	−4.97831	−0.290836	−0.145418	+	0.989370i	$0.546453\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0.0550609	0.00319496
$298$	0	0
$299$	−24.2773	−1.40399
$300$	0	0
$301$	31.9308	1.84046
$302$	0	0
$303$	1.45210	0.0834208
$304$	0	0
$305$	0	0
$306$	0	0
$307$	17.5208	0.999964	0.499982	−	0.866036i	$-0.333340\pi$
$307$	17.5208	0.999964	0.499982	+	0.866036i	$0.333340\pi$
$308$	0	0
$309$	−3.12285	−0.177652
$310$	0	0
$311$	−30.2206	−1.71365	−0.856827	−	0.515604i	$-0.827568\pi$
$311$	−30.2206	−1.71365	−0.856827	+	0.515604i	$0.827568\pi$
$312$	0	0
$313$	3.03942	0.171798	0.0858990	−	0.996304i	$-0.472624\pi$
$313$	3.03942	0.171798	0.0858990	+	0.996304i	$0.472624\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	31.8927	1.79127	0.895636	−	0.444787i	$-0.146721\pi$
$317$	31.8927	1.79127	0.895636	+	0.444787i	$0.146721\pi$
$318$	0	0
$319$	0.253230	0.0141781
$320$	0	0
$321$	11.2429	0.627517
$322$	0	0
$323$	32.7242	1.82082
$324$	0	0
$325$	0	0
$326$	0	0
$327$	8.99356	0.497345
$328$	0	0
$329$	−17.7411	−0.978099
$330$	0	0
$331$	14.6463	0.805034	0.402517	−	0.915412i	$-0.368135\pi$
$331$	14.6463	0.805034	0.402517	+	0.915412i	$0.368135\pi$
$332$	0	0
$333$	4.40403	0.241339
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−12.8922	−0.702283	−0.351141	−	0.936322i	$-0.614206\pi$
$337$	−12.8922	−0.702283	−0.351141	+	0.936322i	$0.614206\pi$
$338$	0	0
$339$	−10.3929	−0.564466
$340$	0	0
$341$	−0.0550609	−0.00298172
$342$	0	0
$343$	−12.2144	−0.659516
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−9.50733	−0.510380	−0.255190	−	0.966891i	$-0.582138\pi$
$347$	−9.50733	−0.510380	−0.255190	+	0.966891i	$0.582138\pi$
$348$	0	0
$349$	−19.2102	−1.02830	−0.514149	−	0.857701i	$-0.671892\pi$
$349$	−19.2102	−1.02830	−0.514149	+	0.857701i	$0.671892\pi$
$350$	0	0
$351$	5.87876	0.313785
$352$	0	0
$353$	15.6633	0.833671	0.416836	−	0.908982i	$-0.363139\pi$
$353$	15.6633	0.833671	0.416836	+	0.908982i	$0.363139\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	18.4767	0.977889
$358$	0	0
$359$	−8.35459	−0.440938	−0.220469	−	0.975394i	$-0.570759\pi$
$359$	−8.35459	−0.440938	−0.220469	+	0.975394i	$0.570759\pi$
$360$	0	0
$361$	12.9012	0.679009
$362$	0	0
$363$	−10.9970	−0.577191
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−3.16198	−0.165054	−0.0825271	−	0.996589i	$-0.526299\pi$
$367$	−3.16198	−0.165054	−0.0825271	+	0.996589i	$0.526299\pi$
$368$	0	0
$369$	−5.06778	−0.263818
$370$	0	0
$371$	30.0687	1.56109
$372$	0	0
$373$	−24.9991	−1.29440	−0.647201	−	0.762319i	$-0.724061\pi$
$373$	−24.9991	−1.29440	−0.647201	+	0.762319i	$0.724061\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	27.0369	1.39247
$378$	0	0
$379$	0.822971	0.0422732	0.0211366	−	0.999777i	$-0.493272\pi$
$379$	0.822971	0.0422732	0.0211366	+	0.999777i	$0.493272\pi$
$380$	0	0
$381$	−10.0276	−0.513729
$382$	0	0
$383$	−0.450828	−0.0230362	−0.0115181	−	0.999934i	$-0.503666\pi$
$383$	−0.450828	−0.0230362	−0.0115181	+	0.999934i	$0.503666\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	10.0127	0.508975
$388$	0	0
$389$	14.7249	0.746580	0.373290	−	0.927715i	$-0.378230\pi$
$389$	14.7249	0.746580	0.373290	+	0.927715i	$0.378230\pi$
$390$	0	0
$391$	−23.9266	−1.21002
$392$	0	0
$393$	18.6907	0.942821
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−36.6813	−1.84098	−0.920490	−	0.390767i	$-0.872210\pi$
$397$	−36.6813	−1.84098	−0.920490	+	0.390767i	$0.872210\pi$
$398$	0	0
$399$	18.0119	0.901725
$400$	0	0
$401$	1.40910	0.0703669	0.0351835	−	0.999381i	$-0.488798\pi$
$401$	1.40910	0.0703669	0.0351835	+	0.999381i	$0.488798\pi$
$402$	0	0
$403$	−5.87876	−0.292842
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.242490	0.0120198
$408$	0	0
$409$	24.8925	1.23086	0.615428	−	0.788193i	$-0.288983\pi$
$409$	24.8925	1.23086	0.615428	+	0.788193i	$0.288983\pi$
$410$	0	0
$411$	−8.69714	−0.428998
$412$	0	0
$413$	−44.4793	−2.18868
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−4.17229	−0.204318
$418$	0	0
$419$	−16.9192	−0.826557	−0.413279	−	0.910605i	$-0.635616\pi$
$419$	−16.9192	−0.826557	−0.413279	+	0.910605i	$0.635616\pi$
$420$	0	0
$421$	−35.2681	−1.71886	−0.859431	−	0.511251i	$-0.829182\pi$
$421$	−35.2681	−1.71886	−0.859431	+	0.511251i	$0.829182\pi$
$422$	0	0
$423$	−5.56318	−0.270491
$424$	0	0
$425$	0	0
$426$	0	0
$427$	32.4793	1.57178
$428$	0	0
$429$	0.323690	0.0156279
$430$	0	0
$431$	31.2122	1.50344	0.751720	−	0.659483i	$-0.229225\pi$
$431$	31.2122	1.50344	0.751720	+	0.659483i	$0.229225\pi$
$432$	0	0
$433$	7.45121	0.358082	0.179041	−	0.983842i	$-0.442700\pi$
$433$	7.45121	0.358082	0.179041	+	0.983842i	$0.442700\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−23.3248	−1.11578
$438$	0	0
$439$	34.5142	1.64727	0.823636	−	0.567119i	$-0.191942\pi$
$439$	34.5142	1.64727	0.823636	+	0.567119i	$0.191942\pi$
$440$	0	0
$441$	3.16986	0.150946
$442$	0	0
$443$	27.6472	1.31356	0.656779	−	0.754083i	$-0.271918\pi$
$443$	27.6472	1.31356	0.656779	+	0.754083i	$0.271918\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−18.2276	−0.862137
$448$	0	0
$449$	−24.9968	−1.17967	−0.589837	−	0.807522i	$-0.700808\pi$
$449$	−24.9968	−1.17967	−0.589837	+	0.807522i	$0.700808\pi$
$450$	0	0
$451$	−0.279037	−0.0131393
$452$	0	0
$453$	1.17057	0.0549980
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−35.9439	−1.68138	−0.840692	−	0.541513i	$-0.817852\pi$
$457$	−35.9439	−1.68138	−0.840692	+	0.541513i	$0.817852\pi$
$458$	0	0
$459$	5.79383	0.270433
$460$	0	0
$461$	25.4209	1.18397	0.591985	−	0.805949i	$-0.298345\pi$
$461$	25.4209	1.18397	0.591985	+	0.805949i	$0.298345\pi$
$462$	0	0
$463$	39.4105	1.83156	0.915780	−	0.401680i	$-0.131573\pi$
$463$	39.4105	1.83156	0.915780	+	0.401680i	$0.131573\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−32.0542	−1.48329	−0.741645	−	0.670793i	$-0.765954\pi$
$467$	−32.0542	−1.48329	−0.741645	+	0.670793i	$0.765954\pi$
$468$	0	0
$469$	−27.5653	−1.27285
$470$	0	0
$471$	−6.47473	−0.298340
$472$	0	0
$473$	0.551310	0.0253492
$474$	0	0
$475$	0	0
$476$	0	0
$477$	9.42881	0.431716
$478$	0	0
$479$	−14.7528	−0.674071	−0.337035	−	0.941492i	$-0.609424\pi$
$479$	−14.7528	−0.674071	−0.337035	+	0.941492i	$0.609424\pi$
$480$	0	0
$481$	25.8902	1.18049
$482$	0	0
$483$	−13.1696	−0.599238
$484$	0	0
$485$	0	0
$486$	0	0
$487$	10.6881	0.484326	0.242163	−	0.970236i	$-0.422143\pi$
$487$	10.6881	0.484326	0.242163	+	0.970236i	$0.422143\pi$
$488$	0	0
$489$	−19.7986	−0.895321
$490$	0	0
$491$	18.2290	0.822661	0.411331	−	0.911486i	$-0.365064\pi$
$491$	18.2290	0.822661	0.411331	+	0.911486i	$0.365064\pi$
$492$	0	0
$493$	26.6463	1.20009
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−38.6699	−1.73458
$498$	0	0
$499$	12.9681	0.580533	0.290267	−	0.956946i	$-0.406256\pi$
$499$	12.9681	0.580533	0.290267	+	0.956946i	$0.406256\pi$
$500$	0	0
$501$	−22.1121	−0.987896
$502$	0	0
$503$	30.5146	1.36058	0.680290	−	0.732943i	$-0.261854\pi$
$503$	30.5146	1.36058	0.680290	+	0.732943i	$0.261854\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	21.5599	0.957507
$508$	0	0
$509$	31.8827	1.41318	0.706588	−	0.707626i	$-0.250234\pi$
$509$	31.8827	1.41318	0.706588	+	0.707626i	$0.250234\pi$
$510$	0	0
$511$	6.37804	0.282148
$512$	0	0
$513$	5.64811	0.249370
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−0.306314	−0.0134717
$518$	0	0
$519$	−12.6214	−0.554016
$520$	0	0
$521$	35.8558	1.57087	0.785435	−	0.618944i	$-0.212439\pi$
$521$	35.8558	1.57087	0.785435	+	0.618944i	$0.212439\pi$
$522$	0	0
$523$	27.5561	1.20494	0.602472	−	0.798140i	$-0.294182\pi$
$523$	27.5561	1.20494	0.602472	+	0.798140i	$0.294182\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5.79383	−0.252383
$528$	0	0
$529$	−5.94584	−0.258515
$530$	0	0
$531$	−13.9476	−0.605276
$532$	0	0
$533$	−29.7923	−1.29045
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−9.85851	−0.425426
$538$	0	0
$539$	0.174535	0.00751777
$540$	0	0
$541$	25.0111	1.07531	0.537656	−	0.843164i	$-0.319310\pi$
$541$	25.0111	1.07531	0.537656	+	0.843164i	$0.319310\pi$
$542$	0	0
$543$	10.1822	0.436960
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−32.5100	−1.39003	−0.695013	−	0.718997i	$-0.744602\pi$
$547$	−32.5100	−1.39003	−0.695013	+	0.718997i	$0.744602\pi$
$548$	0	0
$549$	10.1847	0.434674
$550$	0	0
$551$	25.9761	1.10662
$552$	0	0
$553$	−53.2937	−2.26628
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−21.0723	−0.892861	−0.446430	−	0.894818i	$-0.647305\pi$
$557$	−21.0723	−0.892861	−0.446430	+	0.894818i	$0.647305\pi$
$558$	0	0
$559$	58.8624	2.48962
$560$	0	0
$561$	0.319014	0.0134688
$562$	0	0
$563$	0.597819	0.0251950	0.0125975	−	0.999921i	$-0.495990\pi$
$563$	0.597819	0.0251950	0.0125975	+	0.999921i	$0.495990\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	3.18902	0.133926
$568$	0	0
$569$	18.1074	0.759100	0.379550	−	0.925171i	$-0.376079\pi$
$569$	18.1074	0.759100	0.379550	+	0.925171i	$0.376079\pi$
$570$	0	0
$571$	−29.7439	−1.24474	−0.622371	−	0.782722i	$-0.713831\pi$
$571$	−29.7439	−1.24474	−0.622371	+	0.782722i	$0.713831\pi$
$572$	0	0
$573$	12.9480	0.540912
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−11.3354	−0.471898	−0.235949	−	0.971765i	$-0.575820\pi$
$577$	−11.3354	−0.471898	−0.235949	+	0.971765i	$0.575820\pi$
$578$	0	0
$579$	3.80883	0.158290
$580$	0	0
$581$	22.7321	0.943085
$582$	0	0
$583$	0.519159	0.0215014
$584$	0	0
$585$	0	0
$586$	0	0
$587$	16.7819	0.692661	0.346331	−	0.938113i	$-0.387428\pi$
$587$	16.7819	0.692661	0.346331	+	0.938113i	$0.387428\pi$
$588$	0	0
$589$	−5.64811	−0.232726
$590$	0	0
$591$	−2.70474	−0.111258
$592$	0	0
$593$	−44.7719	−1.83856	−0.919280	−	0.393604i	$-0.871228\pi$
$593$	−44.7719	−1.83856	−0.919280	+	0.393604i	$0.871228\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−24.8750	−1.01806
$598$	0	0
$599$	23.2577	0.950284	0.475142	−	0.879909i	$-0.342397\pi$
$599$	23.2577	0.950284	0.475142	+	0.879909i	$0.342397\pi$
$600$	0	0
$601$	−25.9197	−1.05729	−0.528643	−	0.848844i	$-0.677299\pi$
$601$	−25.9197	−1.05729	−0.528643	+	0.848844i	$0.677299\pi$
$602$	0	0
$603$	−8.64382	−0.352004
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−16.7299	−0.679045	−0.339523	−	0.940598i	$-0.610266\pi$
$607$	−16.7299	−0.679045	−0.339523	+	0.940598i	$0.610266\pi$
$608$	0	0
$609$	14.6666	0.594319
$610$	0	0
$611$	−32.7046	−1.32309
$612$	0	0
$613$	12.2956	0.496615	0.248308	−	0.968681i	$-0.420126\pi$
$613$	12.2956	0.496615	0.248308	+	0.968681i	$0.420126\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	32.5176	1.30911	0.654554	−	0.756015i	$-0.272856\pi$
$617$	32.5176	1.30911	0.654554	+	0.756015i	$0.272856\pi$
$618$	0	0
$619$	−24.7986	−0.996738	−0.498369	−	0.866965i	$-0.666067\pi$
$619$	−24.7986	−0.996738	−0.498369	+	0.866965i	$0.666067\pi$
$620$	0	0
$621$	−4.12967	−0.165718
$622$	0	0
$623$	−17.9167	−0.717817
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0.310990	0.0124198
$628$	0	0
$629$	25.5162	1.01740
$630$	0	0
$631$	−24.0676	−0.958117	−0.479059	−	0.877783i	$-0.659022\pi$
$631$	−24.0676	−0.958117	−0.479059	+	0.877783i	$0.659022\pi$
$632$	0	0
$633$	−13.0041	−0.516868
$634$	0	0
$635$	0	0
$636$	0	0
$637$	18.6348	0.738340
$638$	0	0
$639$	−12.1260	−0.479695
$640$	0	0
$641$	−33.5784	−1.32627	−0.663133	−	0.748502i	$-0.730774\pi$
$641$	−33.5784	−1.32627	−0.663133	+	0.748502i	$0.730774\pi$
$642$	0	0
$643$	7.43167	0.293076	0.146538	−	0.989205i	$-0.453187\pi$
$643$	7.43167	0.293076	0.146538	+	0.989205i	$0.453187\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−10.5271	−0.413864	−0.206932	−	0.978355i	$-0.566348\pi$
$647$	−10.5271	−0.413864	−0.206932	+	0.978355i	$0.566348\pi$
$648$	0	0
$649$	−0.767970	−0.0301454
$650$	0	0
$651$	−3.18902	−0.124988
$652$	0	0
$653$	−15.1744	−0.593821	−0.296911	−	0.954905i	$-0.595956\pi$
$653$	−15.1744	−0.593821	−0.296911	+	0.954905i	$0.595956\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2.00000	0.0780274
$658$	0	0
$659$	5.30564	0.206678	0.103339	−	0.994646i	$-0.467047\pi$
$659$	5.30564	0.206678	0.103339	+	0.994646i	$0.467047\pi$
$660$	0	0
$661$	16.0319	0.623570	0.311785	−	0.950153i	$-0.399073\pi$
$661$	16.0319	0.623570	0.311785	+	0.950153i	$0.399073\pi$
$662$	0	0
$663$	34.0606	1.32280
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−18.9927	−0.735399
$668$	0	0
$669$	20.6630	0.798877
$670$	0	0
$671$	0.560781	0.0216487
$672$	0	0
$673$	35.1373	1.35444	0.677221	−	0.735779i	$-0.263184\pi$
$673$	35.1373	1.35444	0.677221	+	0.735779i	$0.263184\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−25.2242	−0.969443	−0.484721	−	0.874669i	$-0.661079\pi$
$677$	−25.2242	−0.969443	−0.484721	+	0.874669i	$0.661079\pi$
$678$	0	0
$679$	−6.43915	−0.247112
$680$	0	0
$681$	−6.75512	−0.258857
$682$	0	0
$683$	6.49771	0.248628	0.124314	−	0.992243i	$-0.460327\pi$
$683$	6.49771	0.248628	0.124314	+	0.992243i	$0.460327\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	23.3234	0.889844
$688$	0	0
$689$	55.4297	2.11171
$690$	0	0
$691$	−3.22014	−0.122500	−0.0612499	−	0.998122i	$-0.519509\pi$
$691$	−3.22014	−0.122500	−0.0612499	+	0.998122i	$0.519509\pi$
$692$	0	0
$693$	0.175590	0.00667013
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−29.3619	−1.11216
$698$	0	0
$699$	−12.4527	−0.471003
$700$	0	0
$701$	−1.75085	−0.0661287	−0.0330643	−	0.999453i	$-0.510527\pi$
$701$	−1.75085	−0.0661287	−0.0330643	+	0.999453i	$0.510527\pi$
$702$	0	0
$703$	24.8744	0.938157
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.63077	0.174158
$708$	0	0
$709$	−26.2374	−0.985366	−0.492683	−	0.870209i	$-0.663984\pi$
$709$	−26.2374	−0.985366	−0.492683	+	0.870209i	$0.663984\pi$
$710$	0	0
$711$	−16.7116	−0.626734
$712$	0	0
$713$	4.12967	0.154657
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1.05077	0.0392417
$718$	0	0
$719$	50.8038	1.89466	0.947332	−	0.320254i	$-0.103768\pi$
$719$	50.8038	1.89466	0.947332	+	0.320254i	$0.103768\pi$
$720$	0	0
$721$	−9.95882	−0.370886
$722$	0	0
$723$	8.08572	0.300711
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−35.3770	−1.31206	−0.656030	−	0.754735i	$-0.727765\pi$
$727$	−35.3770	−1.31206	−0.656030	+	0.754735i	$0.727765\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	58.0121	2.14565
$732$	0	0
$733$	−18.8157	−0.694973	−0.347486	−	0.937685i	$-0.612965\pi$
$733$	−18.8157	−0.694973	−0.347486	+	0.937685i	$0.612965\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−0.475937	−0.0175313
$738$	0	0
$739$	22.6847	0.834469	0.417235	−	0.908799i	$-0.362999\pi$
$739$	22.6847	0.834469	0.417235	+	0.908799i	$0.362999\pi$
$740$	0	0
$741$	33.2039	1.21978
$742$	0	0
$743$	−16.7721	−0.615307	−0.307654	−	0.951498i	$-0.599544\pi$
$743$	−16.7721	−0.615307	−0.307654	+	0.951498i	$0.599544\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	7.12823	0.260808
$748$	0	0
$749$	35.8538	1.31007
$750$	0	0
$751$	35.8199	1.30709	0.653544	−	0.756889i	$-0.273282\pi$
$751$	35.8199	1.30709	0.653544	+	0.756889i	$0.273282\pi$
$752$	0	0
$753$	14.9270	0.543970
$754$	0	0
$755$	0	0
$756$	0	0
$757$	6.34566	0.230637	0.115318	−	0.993329i	$-0.463211\pi$
$757$	6.34566	0.230637	0.115318	+	0.993329i	$0.463211\pi$
$758$	0	0
$759$	−0.227383	−0.00825349
$760$	0	0
$761$	−11.5352	−0.418151	−0.209076	−	0.977899i	$-0.567046\pi$
$761$	−11.5352	−0.418151	−0.209076	+	0.977899i	$0.567046\pi$
$762$	0	0
$763$	28.6807	1.03831
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−81.9949	−2.96066
$768$	0	0
$769$	43.5243	1.56953	0.784764	−	0.619795i	$-0.212784\pi$
$769$	43.5243	1.56953	0.784764	+	0.619795i	$0.212784\pi$
$770$	0	0
$771$	−30.2304	−1.08872
$772$	0	0
$773$	−42.2279	−1.51883	−0.759416	−	0.650606i	$-0.774515\pi$
$773$	−42.2279	−1.51883	−0.759416	+	0.650606i	$0.774515\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	14.0445	0.503845
$778$	0	0
$779$	−28.6234	−1.02554
$780$	0	0
$781$	−0.667666	−0.0238910
$782$	0	0
$783$	4.59908	0.164358
$784$	0	0
$785$	0	0
$786$	0	0
$787$	43.8502	1.56309	0.781546	−	0.623848i	$-0.214432\pi$
$787$	43.8502	1.56309	0.781546	+	0.623848i	$0.214432\pi$
$788$	0	0
$789$	7.41781	0.264081
$790$	0	0
$791$	−33.1432	−1.17844
$792$	0	0
$793$	59.8736	2.12617
$794$	0	0
$795$	0	0
$796$	0	0
$797$	26.2137	0.928538	0.464269	−	0.885694i	$-0.346317\pi$
$797$	26.2137	0.928538	0.464269	+	0.885694i	$0.346317\pi$
$798$	0	0
$799$	−32.2321	−1.14029
$800$	0	0
$801$	−5.61824	−0.198511
$802$	0	0
$803$	0.110122	0.00388611
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−1.94988	−0.0686389
$808$	0	0
$809$	9.67506	0.340157	0.170078	−	0.985431i	$-0.445598\pi$
$809$	9.67506	0.340157	0.170078	+	0.985431i	$0.445598\pi$
$810$	0	0
$811$	−42.8525	−1.50476	−0.752378	−	0.658732i	$-0.771093\pi$
$811$	−42.8525	−1.50476	−0.752378	+	0.658732i	$0.771093\pi$
$812$	0	0
$813$	10.3273	0.362193
$814$	0	0
$815$	0	0
$816$	0	0
$817$	56.5530	1.97854
$818$	0	0
$819$	18.7475	0.655091
$820$	0	0
$821$	5.01659	0.175080	0.0875401	−	0.996161i	$-0.472099\pi$
$821$	5.01659	0.175080	0.0875401	+	0.996161i	$0.472099\pi$
$822$	0	0
$823$	27.4240	0.955941	0.477971	−	0.878376i	$-0.341372\pi$
$823$	27.4240	0.955941	0.477971	+	0.878376i	$0.341372\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−41.9731	−1.45955	−0.729773	−	0.683689i	$-0.760374\pi$
$827$	−41.9731	−1.45955	−0.729773	+	0.683689i	$0.760374\pi$
$828$	0	0
$829$	38.3241	1.33105	0.665525	−	0.746376i	$-0.268208\pi$
$829$	38.3241	1.33105	0.665525	+	0.746376i	$0.268208\pi$
$830$	0	0
$831$	10.3191	0.357965
$832$	0	0
$833$	18.3656	0.636332
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	16.9781	0.586149	0.293075	−	0.956090i	$-0.405322\pi$
$839$	16.9781	0.586149	0.293075	+	0.956090i	$0.405322\pi$
$840$	0	0
$841$	−7.84846	−0.270637
$842$	0	0
$843$	30.1504	1.03844
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−35.0696	−1.20500
$848$	0	0
$849$	−3.16117	−0.108491
$850$	0	0
$851$	−18.1872	−0.623448
$852$	0	0
$853$	14.5451	0.498014	0.249007	−	0.968502i	$-0.419896\pi$
$853$	14.5451	0.498014	0.249007	+	0.968502i	$0.419896\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	10.5802	0.361413	0.180707	−	0.983537i	$-0.442162\pi$
$857$	10.5802	0.361413	0.180707	+	0.983537i	$0.442162\pi$
$858$	0	0
$859$	−21.2365	−0.724581	−0.362290	−	0.932065i	$-0.618005\pi$
$859$	−21.2365	−0.724581	−0.362290	+	0.932065i	$0.618005\pi$
$860$	0	0
$861$	−16.1613	−0.550775
$862$	0	0
$863$	0.0730209	0.00248566	0.00124283	−	0.999999i	$-0.499604\pi$
$863$	0.0730209	0.00248566	0.00124283	+	0.999999i	$0.499604\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	16.5685	0.562696
$868$	0	0
$869$	−0.920156	−0.0312142
$870$	0	0
$871$	−50.8150	−1.72180
$872$	0	0
$873$	−2.01916	−0.0683383
$874$	0	0
$875$	0	0
$876$	0	0
$877$	25.9670	0.876845	0.438422	−	0.898769i	$-0.355537\pi$
$877$	25.9670	0.876845	0.438422	+	0.898769i	$0.355537\pi$
$878$	0	0
$879$	−4.97831	−0.167914
$880$	0	0
$881$	−12.2966	−0.414284	−0.207142	−	0.978311i	$-0.566416\pi$
$881$	−12.2966	−0.414284	−0.207142	+	0.978311i	$0.566416\pi$
$882$	0	0
$883$	37.5451	1.26349	0.631747	−	0.775174i	$-0.282338\pi$
$883$	37.5451	1.26349	0.631747	+	0.775174i	$0.282338\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−47.8823	−1.60773	−0.803866	−	0.594811i	$-0.797227\pi$
$887$	−47.8823	−1.60773	−0.803866	+	0.594811i	$0.797227\pi$
$888$	0	0
$889$	−31.9782	−1.07251
$890$	0	0
$891$	0.0550609	0.00184461
$892$	0	0
$893$	−31.4215	−1.05148
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−24.2773	−0.810597
$898$	0	0
$899$	−4.59908	−0.153388
$900$	0	0
$901$	54.6290	1.81995
$902$	0	0
$903$	31.9308	1.06259
$904$	0	0
$905$	0	0
$906$	0	0
$907$	29.6615	0.984893	0.492446	−	0.870343i	$-0.336103\pi$
$907$	29.6615	0.984893	0.492446	+	0.870343i	$0.336103\pi$
$908$	0	0
$909$	1.45210	0.0481630
$910$	0	0
$911$	−3.30593	−0.109530	−0.0547652	−	0.998499i	$-0.517441\pi$
$911$	−3.30593	−0.109530	−0.0547652	+	0.998499i	$0.517441\pi$
$912$	0	0
$913$	0.392487	0.0129894
$914$	0	0
$915$	0	0
$916$	0	0
$917$	59.6051	1.96833
$918$	0	0
$919$	−44.0855	−1.45425	−0.727123	−	0.686507i	$-0.759143\pi$
$919$	−44.0855	−1.45425	−0.727123	+	0.686507i	$0.759143\pi$
$920$	0	0
$921$	17.5208	0.577329
$922$	0	0
$923$	−71.2856	−2.34639
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−3.12285	−0.102568
$928$	0	0
$929$	−26.7159	−0.876519	−0.438259	−	0.898849i	$-0.644405\pi$
$929$	−26.7159	−0.876519	−0.438259	+	0.898849i	$0.644405\pi$
$930$	0	0
$931$	17.9037	0.586771
$932$	0	0
$933$	−30.2206	−0.989379
$934$	0	0
$935$	0	0
$936$	0	0
$937$	24.0102	0.784379	0.392190	−	0.919884i	$-0.371718\pi$
$937$	24.0102	0.784379	0.392190	+	0.919884i	$0.371718\pi$
$938$	0	0
$939$	3.03942	0.0991876
$940$	0	0
$941$	43.5968	1.42121	0.710607	−	0.703589i	$-0.248420\pi$
$941$	43.5968	1.42121	0.710607	+	0.703589i	$0.248420\pi$
$942$	0	0
$943$	20.9283	0.681518
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−7.59095	−0.246673	−0.123336	−	0.992365i	$-0.539359\pi$
$947$	−7.59095	−0.246673	−0.123336	+	0.992365i	$0.539359\pi$
$948$	0	0
$949$	11.7575	0.381665
$950$	0	0
$951$	31.8927	1.03419
$952$	0	0
$953$	3.25955	0.105587	0.0527936	−	0.998605i	$-0.483187\pi$
$953$	3.25955	0.105587	0.0527936	+	0.998605i	$0.483187\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0.253230	0.00818575
$958$	0	0
$959$	−27.7354	−0.895622
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	11.2429	0.362297
$964$	0	0
$965$	0	0
$966$	0	0
$967$	18.0928	0.581826	0.290913	−	0.956749i	$-0.406041\pi$
$967$	18.0928	0.581826	0.290913	+	0.956749i	$0.406041\pi$
$968$	0	0
$969$	32.7242	1.05125
$970$	0	0
$971$	−46.3401	−1.48712	−0.743562	−	0.668667i	$-0.766865\pi$
$971$	−46.3401	−1.48712	−0.743562	+	0.668667i	$0.766865\pi$
$972$	0	0
$973$	−13.3055	−0.426555
$974$	0	0
$975$	0	0
$976$	0	0
$977$	46.6801	1.49343	0.746714	−	0.665145i	$-0.231630\pi$
$977$	46.6801	1.49343	0.746714	+	0.665145i	$0.231630\pi$
$978$	0	0
$979$	−0.309346	−0.00988673
$980$	0	0
$981$	8.99356	0.287142
$982$	0	0
$983$	−32.5394	−1.03785	−0.518923	−	0.854821i	$-0.673667\pi$
$983$	−32.5394	−1.03785	−0.518923	+	0.854821i	$0.673667\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−17.7411	−0.564706
$988$	0	0
$989$	−41.3492	−1.31483
$990$	0	0
$991$	−2.91582	−0.0926241	−0.0463120	−	0.998927i	$-0.514747\pi$
$991$	−2.91582	−0.0926241	−0.0463120	+	0.998927i	$0.514747\pi$
$992$	0	0
$993$	14.6463	0.464787
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−37.1369	−1.17614	−0.588068	−	0.808812i	$-0.700111\pi$
$997$	−37.1369	−1.17614	−0.588068	+	0.808812i	$0.700111\pi$
$998$	0	0
$999$	4.40403	0.139337

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9300.2.a.ba.1.5	yes	6
5.2	odd	4		9300.2.g.t.3349.5		12
5.3	odd	4		9300.2.g.t.3349.8		12
5.4	even	2		9300.2.a.y.1.2	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9300.2.a.y.1.2	✓	6	5.4	even	2
9300.2.a.ba.1.5	yes	6	1.1	even	1	trivial
9300.2.g.t.3349.5		12	5.2	odd	4
9300.2.g.t.3349.8		12	5.3	odd	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:	\( N \)	\(=\)	\( 9300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9300.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.18902 q^{7} +1.00000 q^{9} +0.0550609 q^{11} +5.87876 q^{13} +5.79383 q^{17} +5.64811 q^{19} +3.18902 q^{21} -4.12967 q^{23} +1.00000 q^{27} +4.59908 q^{29} -1.00000 q^{31} +0.0550609 q^{33} +4.40403 q^{37} +5.87876 q^{39} -5.06778 q^{41} +10.0127 q^{43} -5.56318 q^{47} +3.16986 q^{49} +5.79383 q^{51} +9.42881 q^{53} +5.64811 q^{57} -13.9476 q^{59} +10.1847 q^{61} +3.18902 q^{63} -8.64382 q^{67} -4.12967 q^{69} -12.1260 q^{71} +2.00000 q^{73} +0.175590 q^{77} -16.7116 q^{79} +1.00000 q^{81} +7.12823 q^{83} +4.59908 q^{87} -5.61824 q^{89} +18.7475 q^{91} -1.00000 q^{93} -2.01916 q^{97} +0.0550609 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} + 4 q^{7} + 6 q^{9} - 3 q^{11} + 2 q^{13} + 4 q^{17} - 3 q^{19} + 4 q^{21} + 5 q^{23} + 6 q^{27} - 6 q^{31} - 3 q^{33} + 8 q^{37} + 2 q^{39} + 18 q^{41} + 15 q^{43} + q^{47} + 14 q^{49} + 4 q^{51}+ \cdots - 3 q^{99}+O(q^{100}) \) 6 * q + 6 * q^3 + 4 * q^7 + 6 * q^9 - 3 * q^11 + 2 * q^13 + 4 * q^17 - 3 * q^19 + 4 * q^21 + 5 * q^23 + 6 * q^27 - 6 * q^31 - 3 * q^33 + 8 * q^37 + 2 * q^39 + 18 * q^41 + 15 * q^43 + q^47 + 14 * q^49 + 4 * q^51 + 5 * q^53 - 3 * q^57 - 12 * q^59 + 28 * q^61 + 4 * q^63 + 3 * q^67 + 5 * q^69 + 25 * q^71 + 12 * q^73 + 8 * q^77 + 3 * q^79 + 6 * q^81 + 19 * q^83 + 4 * q^89 + 16 * q^91 - 6 * q^93 - 2 * q^97 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more