LMFDB - Embedded newform 9300.2.a.ba.1.3

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			1.3
Root			$1.05575$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -0.649950 q^{7} +1.00000 q^{9} +5.35424 q^{11} +0.343092 q^{13} +5.13187 q^{17} -1.53116 q^{19} -0.649950 q^{21} +6.81578 q^{23} +1.00000 q^{27} +1.68424 q^{29} -1.00000 q^{31} +5.35424 q^{33} -4.23545 q^{37} +0.343092 q^{39} +4.30686 q^{41} -4.66109 q^{43} -3.25762 q^{47} -6.57757 q^{49} +5.13187 q^{51} +5.24274 q^{53} -1.53116 q^{57} +13.9245 q^{59} +4.53845 q^{61} -0.649950 q^{63} +0.342756 q^{67} +6.81578 q^{69} +3.50382 q^{71} +2.00000 q^{73} -3.47999 q^{77} +1.64961 q^{79} +1.00000 q^{81} -8.10177 q^{83} +1.68424 q^{87} -8.61186 q^{89} -0.222992 q^{91} -1.00000 q^{93} -7.92762 q^{97} +5.35424 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−0.649950	−0.245658	−0.122829	−	0.992428i	$-0.539197\pi$
$7$	−0.649950	−0.245658	−0.122829	+	0.992428i	$0.539197\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.35424	1.61436	0.807182	−	0.590303i	$-0.200992\pi$
$11$	5.35424	1.61436	0.807182	+	0.590303i	$0.200992\pi$
$12$	0	0
$13$	0.343092	0.0951565	0.0475783	−	0.998868i	$-0.484850\pi$
$13$	0.343092	0.0951565	0.0475783	+	0.998868i	$0.484850\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.13187	1.24466	0.622331	−	0.782754i	$-0.286186\pi$
$17$	5.13187	1.24466	0.622331	+	0.782754i	$0.286186\pi$
$18$	0	0
$19$	−1.53116	−0.351272	−0.175636	−	0.984455i	$-0.556198\pi$
$19$	−1.53116	−0.351272	−0.175636	+	0.984455i	$0.556198\pi$
$20$	0	0
$21$	−0.649950	−0.141831
$22$	0	0
$23$	6.81578	1.42119	0.710594	−	0.703602i	$-0.248426\pi$
$23$	6.81578	1.42119	0.710594	+	0.703602i	$0.248426\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.68424	0.312756	0.156378	−	0.987697i	$-0.450018\pi$
$29$	1.68424	0.312756	0.156378	+	0.987697i	$0.450018\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	5.35424	0.932053
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.23545	−0.696303	−0.348152	−	0.937438i	$-0.613191\pi$
$37$	−4.23545	−0.696303	−0.348152	+	0.937438i	$0.613191\pi$
$38$	0	0
$39$	0.343092	0.0549386
$40$	0	0
$41$	4.30686	0.672618	0.336309	−	0.941752i	$-0.390821\pi$
$41$	4.30686	0.672618	0.336309	+	0.941752i	$0.390821\pi$
$42$	0	0
$43$	−4.66109	−0.710810	−0.355405	−	0.934712i	$-0.615657\pi$
$43$	−4.66109	−0.710810	−0.355405	+	0.934712i	$0.615657\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.25762	−0.475173	−0.237587	−	0.971366i	$-0.576356\pi$
$47$	−3.25762	−0.475173	−0.237587	+	0.971366i	$0.576356\pi$
$48$	0	0
$49$	−6.57757	−0.939652
$50$	0	0
$51$	5.13187	0.718606
$52$	0	0
$53$	5.24274	0.720146	0.360073	−	0.932924i	$-0.382752\pi$
$53$	5.24274	0.720146	0.360073	+	0.932924i	$0.382752\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.53116	−0.202807
$58$	0	0
$59$	13.9245	1.81282	0.906408	−	0.422403i	$-0.138813\pi$
$59$	13.9245	1.81282	0.906408	+	0.422403i	$0.138813\pi$
$60$	0	0
$61$	4.53845	0.581089	0.290545	−	0.956861i	$-0.406163\pi$
$61$	4.53845	0.581089	0.290545	+	0.956861i	$0.406163\pi$
$62$	0	0
$63$	−0.649950	−0.0818860
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0.342756	0.0418743	0.0209371	−	0.999781i	$-0.493335\pi$
$67$	0.342756	0.0418743	0.0209371	+	0.999781i	$0.493335\pi$
$68$	0	0
$69$	6.81578	0.820524
$70$	0	0
$71$	3.50382	0.415827	0.207914	−	0.978147i	$-0.433333\pi$
$71$	3.50382	0.415827	0.207914	+	0.978147i	$0.433333\pi$
$72$	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
$77$	−3.47999	−0.396581
$78$	0	0
$79$	1.64961	0.185596	0.0927980	−	0.995685i	$-0.470419\pi$
$79$	1.64961	0.185596	0.0927980	+	0.995685i	$0.470419\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−8.10177	−0.889284	−0.444642	−	0.895708i	$-0.646669\pi$
$83$	−8.10177	−0.889284	−0.444642	+	0.895708i	$0.646669\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.68424	0.180570
$88$	0	0
$89$	−8.61186	−0.912855	−0.456428	−	0.889761i	$-0.650871\pi$
$89$	−8.61186	−0.912855	−0.456428	+	0.889761i	$0.650871\pi$
$90$	0	0
$91$	−0.222992	−0.0233760
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−7.92762	−0.804927	−0.402464	−	0.915436i	$-0.631846\pi$
$97$	−7.92762	−0.804927	−0.402464	+	0.915436i	$0.631846\pi$
$98$	0	0
$99$	5.35424	0.538121
$100$	0	0
$101$	18.8775	1.87838	0.939189	−	0.343401i	$-0.111579\pi$
$101$	18.8775	1.87838	0.939189	+	0.343401i	$0.111579\pi$
$102$	0	0
$103$	0.952622	0.0938646	0.0469323	−	0.998898i	$-0.485055\pi$
$103$	0.952622	0.0938646	0.0469323	+	0.998898i	$0.485055\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−0.658511	−0.0636607	−0.0318303	−	0.999493i	$-0.510134\pi$
$107$	−0.658511	−0.0636607	−0.0318303	+	0.999493i	$0.510134\pi$
$108$	0	0
$109$	−11.5887	−1.11000	−0.554999	−	0.831851i	$-0.687281\pi$
$109$	−11.5887	−1.11000	−0.554999	+	0.831851i	$0.687281\pi$
$110$	0	0
$111$	−4.23545	−0.402011
$112$	0	0
$113$	−6.81612	−0.641207	−0.320603	−	0.947214i	$-0.603886\pi$
$113$	−6.81612	−0.641207	−0.320603	+	0.947214i	$0.603886\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0.343092	0.0317188
$118$	0	0
$119$	−3.33546	−0.305761
$120$	0	0
$121$	17.6679	1.60617
$122$	0	0
$123$	4.30686	0.388336
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0.545076	0.0483677	0.0241838	−	0.999708i	$-0.492301\pi$
$127$	0.545076	0.0483677	0.0241838	+	0.999708i	$0.492301\pi$
$128$	0	0
$129$	−4.66109	−0.410387
$130$	0	0
$131$	−13.3353	−1.16511	−0.582554	−	0.812792i	$-0.697947\pi$
$131$	−13.3353	−1.16511	−0.582554	+	0.812792i	$0.697947\pi$
$132$	0	0
$133$	0.995177	0.0862928
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.74656	0.234655	0.117327	−	0.993093i	$-0.462567\pi$
$137$	2.74656	0.234655	0.117327	+	0.993093i	$0.462567\pi$
$138$	0	0
$139$	−6.85104	−0.581098	−0.290549	−	0.956860i	$-0.593838\pi$
$139$	−6.85104	−0.581098	−0.290549	+	0.956860i	$0.593838\pi$
$140$	0	0
$141$	−3.25762	−0.274341
$142$	0	0
$143$	1.83699	0.153617
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−6.57757	−0.542508
$148$	0	0
$149$	8.79360	0.720400	0.360200	−	0.932875i	$-0.382708\pi$
$149$	8.79360	0.720400	0.360200	+	0.932875i	$0.382708\pi$
$150$	0	0
$151$	5.91357	0.481239	0.240620	−	0.970620i	$-0.422649\pi$
$151$	5.91357	0.481239	0.240620	+	0.970620i	$0.422649\pi$
$152$	0	0
$153$	5.13187	0.414887
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−9.57854	−0.764451	−0.382225	−	0.924069i	$-0.624842\pi$
$157$	−9.57854	−0.764451	−0.382225	+	0.924069i	$0.624842\pi$
$158$	0	0
$159$	5.24274	0.415776
$160$	0	0
$161$	−4.42992	−0.349126
$162$	0	0
$163$	2.02886	0.158912	0.0794562	−	0.996838i	$-0.474682\pi$
$163$	2.02886	0.158912	0.0794562	+	0.996838i	$0.474682\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.35293	0.336840	0.168420	−	0.985715i	$-0.446134\pi$
$167$	4.35293	0.336840	0.168420	+	0.985715i	$0.446134\pi$
$168$	0	0
$169$	−12.8823	−0.990945
$170$	0	0
$171$	−1.53116	−0.117091
$172$	0	0
$173$	−4.06066	−0.308726	−0.154363	−	0.988014i	$-0.549333\pi$
$173$	−4.06066	−0.308726	−0.154363	+	0.988014i	$0.549333\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	13.9245	1.04663
$178$	0	0
$179$	−11.3762	−0.850298	−0.425149	−	0.905123i	$-0.639778\pi$
$179$	−11.3762	−0.850298	−0.425149	+	0.905123i	$0.639778\pi$
$180$	0	0
$181$	13.2132	0.982130	0.491065	−	0.871123i	$-0.336608\pi$
$181$	13.2132	0.982130	0.491065	+	0.871123i	$0.336608\pi$
$182$	0	0
$183$	4.53845	0.335492
$184$	0	0
$185$	0	0
$186$	0	0
$187$	27.4773	2.00934
$188$	0	0
$189$	−0.649950	−0.0472769
$190$	0	0
$191$	−3.90544	−0.282588	−0.141294	−	0.989968i	$-0.545126\pi$
$191$	−3.90544	−0.282588	−0.141294	+	0.989968i	$0.545126\pi$
$192$	0	0
$193$	−11.1272	−0.800951	−0.400475	−	0.916308i	$-0.631155\pi$
$193$	−11.1272	−0.800951	−0.400475	+	0.916308i	$0.631155\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	13.2660	0.945163	0.472582	−	0.881287i	$-0.343322\pi$
$197$	13.2660	0.945163	0.472582	+	0.881287i	$0.343322\pi$
$198$	0	0
$199$	11.6685	0.827157	0.413579	−	0.910468i	$-0.364279\pi$
$199$	11.6685	0.827157	0.413579	+	0.910468i	$0.364279\pi$
$200$	0	0
$201$	0.342756	0.0241761
$202$	0	0
$203$	−1.09467	−0.0768311
$204$	0	0
$205$	0	0
$206$	0	0
$207$	6.81578	0.473730
$208$	0	0
$209$	−8.19819	−0.567081
$210$	0	0
$211$	5.28429	0.363785	0.181893	−	0.983318i	$-0.441778\pi$
$211$	5.28429	0.363785	0.181893	+	0.983318i	$0.441778\pi$
$212$	0	0
$213$	3.50382	0.240078
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0.649950	0.0441215
$218$	0	0
$219$	2.00000	0.135147
$220$	0	0
$221$	1.76070	0.118438
$222$	0	0
$223$	17.5849	1.17757	0.588785	−	0.808290i	$-0.299607\pi$
$223$	17.5849	1.17757	0.588785	+	0.808290i	$0.299607\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−21.7285	−1.44217	−0.721086	−	0.692845i	$-0.756357\pi$
$227$	−21.7285	−1.44217	−0.721086	+	0.692845i	$0.756357\pi$
$228$	0	0
$229$	25.8445	1.70785	0.853926	−	0.520395i	$-0.174215\pi$
$229$	25.8445	1.70785	0.853926	+	0.520395i	$0.174215\pi$
$230$	0	0
$231$	−3.47999	−0.228966
$232$	0	0
$233$	11.4699	0.751419	0.375710	−	0.926737i	$-0.377399\pi$
$233$	11.4699	0.751419	0.375710	+	0.926737i	$0.377399\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.64961	0.107154
$238$	0	0
$239$	4.54264	0.293839	0.146919	−	0.989148i	$-0.453064\pi$
$239$	4.54264	0.293839	0.146919	+	0.989148i	$0.453064\pi$
$240$	0	0
$241$	−13.2216	−0.851677	−0.425839	−	0.904799i	$-0.640021\pi$
$241$	−13.2216	−0.851677	−0.425839	+	0.904799i	$0.640021\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−0.525328	−0.0334258
$248$	0	0
$249$	−8.10177	−0.513429
$250$	0	0
$251$	21.5605	1.36089	0.680443	−	0.732801i	$-0.261787\pi$
$251$	21.5605	1.36089	0.680443	+	0.732801i	$0.261787\pi$
$252$	0	0
$253$	36.4933	2.29432
$254$	0	0
$255$	0	0
$256$	0	0
$257$	11.8169	0.737116	0.368558	−	0.929605i	$-0.379852\pi$
$257$	11.8169	0.737116	0.368558	+	0.929605i	$0.379852\pi$
$258$	0	0
$259$	2.75283	0.171052
$260$	0	0
$261$	1.68424	0.104252
$262$	0	0
$263$	15.8413	0.976817	0.488409	−	0.872615i	$-0.337578\pi$
$263$	15.8413	0.976817	0.488409	+	0.872615i	$0.337578\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−8.61186	−0.527037
$268$	0	0
$269$	−19.1761	−1.16919	−0.584595	−	0.811325i	$-0.698747\pi$
$269$	−19.1761	−1.16919	−0.584595	+	0.811325i	$0.698747\pi$
$270$	0	0
$271$	−0.842540	−0.0511807	−0.0255903	−	0.999673i	$-0.508147\pi$
$271$	−0.842540	−0.0511807	−0.0255903	+	0.999673i	$0.508147\pi$
$272$	0	0
$273$	−0.222992	−0.0134961
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−31.9176	−1.91774	−0.958871	−	0.283841i	$-0.908391\pi$
$277$	−31.9176	−1.91774	−0.958871	+	0.283841i	$0.908391\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	15.5023	0.924790	0.462395	−	0.886674i	$-0.346990\pi$
$281$	15.5023	0.924790	0.462395	+	0.886674i	$0.346990\pi$
$282$	0	0
$283$	−10.9026	−0.648093	−0.324046	−	0.946041i	$-0.605043\pi$
$283$	−10.9026	−0.648093	−0.324046	+	0.946041i	$0.605043\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.79924	−0.165234
$288$	0	0
$289$	9.33613	0.549184
$290$	0	0
$291$	−7.92762	−0.464725
$292$	0	0
$293$	−7.74714	−0.452593	−0.226296	−	0.974058i	$-0.572662\pi$
$293$	−7.74714	−0.452593	−0.226296	+	0.974058i	$0.572662\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	5.35424	0.310684
$298$	0	0
$299$	2.33844	0.135235
$300$	0	0
$301$	3.02948	0.174616
$302$	0	0
$303$	18.8775	1.08448
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1.46995	0.0838943	0.0419471	−	0.999120i	$-0.486644\pi$
$307$	1.46995	0.0838943	0.0419471	+	0.999120i	$0.486644\pi$
$308$	0	0
$309$	0.952622	0.0541927
$310$	0	0
$311$	19.8469	1.12541	0.562707	−	0.826657i	$-0.309760\pi$
$311$	19.8469	1.12541	0.562707	+	0.826657i	$0.309760\pi$
$312$	0	0
$313$	1.89449	0.107083	0.0535414	−	0.998566i	$-0.482949\pi$
$313$	1.89449	0.107083	0.0535414	+	0.998566i	$0.482949\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−27.7231	−1.55709	−0.778543	−	0.627591i	$-0.784041\pi$
$317$	−27.7231	−1.55709	−0.778543	+	0.627591i	$0.784041\pi$
$318$	0	0
$319$	9.01784	0.504902
$320$	0	0
$321$	−0.658511	−0.0367545
$322$	0	0
$323$	−7.85772	−0.437215
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−11.5887	−0.640857
$328$	0	0
$329$	2.11729	0.116730
$330$	0	0
$331$	5.93845	0.326407	0.163203	−	0.986592i	$-0.447817\pi$
$331$	5.93845	0.326407	0.163203	+	0.986592i	$0.447817\pi$
$332$	0	0
$333$	−4.23545	−0.232101
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−7.17313	−0.390745	−0.195373	−	0.980729i	$-0.562592\pi$
$337$	−7.17313	−0.390745	−0.195373	+	0.980729i	$0.562592\pi$
$338$	0	0
$339$	−6.81612	−0.370201
$340$	0	0
$341$	−5.35424	−0.289948
$342$	0	0
$343$	8.82474	0.476491
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−18.3362	−0.984338	−0.492169	−	0.870500i	$-0.663796\pi$
$347$	−18.3362	−0.984338	−0.492169	+	0.870500i	$0.663796\pi$
$348$	0	0
$349$	15.7837	0.844884	0.422442	−	0.906390i	$-0.361173\pi$
$349$	15.7837	0.844884	0.422442	+	0.906390i	$0.361173\pi$
$350$	0	0
$351$	0.343092	0.0183129
$352$	0	0
$353$	6.23635	0.331928	0.165964	−	0.986132i	$-0.446926\pi$
$353$	6.23635	0.331928	0.165964	+	0.986132i	$0.446926\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−3.33546	−0.176531
$358$	0	0
$359$	7.03911	0.371510	0.185755	−	0.982596i	$-0.440527\pi$
$359$	7.03911	0.371510	0.185755	+	0.982596i	$0.440527\pi$
$360$	0	0
$361$	−16.6556	−0.876608
$362$	0	0
$363$	17.6679	0.927322
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−4.29037	−0.223956	−0.111978	−	0.993711i	$-0.535719\pi$
$367$	−4.29037	−0.223956	−0.111978	+	0.993711i	$0.535719\pi$
$368$	0	0
$369$	4.30686	0.224206
$370$	0	0
$371$	−3.40752	−0.176910
$372$	0	0
$373$	33.1822	1.71811	0.859056	−	0.511882i	$-0.171052\pi$
$373$	33.1822	1.71811	0.859056	+	0.511882i	$0.171052\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.577850	0.0297608
$378$	0	0
$379$	0.194003	0.00996524	0.00498262	−	0.999988i	$-0.498414\pi$
$379$	0.194003	0.00996524	0.00498262	+	0.999988i	$0.498414\pi$
$380$	0	0
$381$	0.545076	0.0279251
$382$	0	0
$383$	0.304031	0.0155353	0.00776763	−	0.999970i	$-0.497527\pi$
$383$	0.304031	0.0155353	0.00776763	+	0.999970i	$0.497527\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4.66109	−0.236937
$388$	0	0
$389$	10.0759	0.510870	0.255435	−	0.966826i	$-0.417781\pi$
$389$	10.0759	0.510870	0.255435	+	0.966826i	$0.417781\pi$
$390$	0	0
$391$	34.9777	1.76890
$392$	0	0
$393$	−13.3353	−0.672676
$394$	0	0
$395$	0	0
$396$	0	0
$397$	6.12089	0.307199	0.153599	−	0.988133i	$-0.450913\pi$
$397$	6.12089	0.307199	0.153599	+	0.988133i	$0.450913\pi$
$398$	0	0
$399$	0.995177	0.0498212
$400$	0	0
$401$	32.6625	1.63109	0.815544	−	0.578695i	$-0.196438\pi$
$401$	32.6625	1.63109	0.815544	+	0.578695i	$0.196438\pi$
$402$	0	0
$403$	−0.343092	−0.0170906
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−22.6776	−1.12409
$408$	0	0
$409$	36.4199	1.80085	0.900423	−	0.435016i	$-0.143257\pi$
$409$	36.4199	1.80085	0.900423	+	0.435016i	$0.143257\pi$
$410$	0	0
$411$	2.74656	0.135478
$412$	0	0
$413$	−9.05023	−0.445333
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−6.85104	−0.335497
$418$	0	0
$419$	−3.50449	−0.171206	−0.0856029	−	0.996329i	$-0.527282\pi$
$419$	−3.50449	−0.171206	−0.0856029	+	0.996329i	$0.527282\pi$
$420$	0	0
$421$	30.8953	1.50574	0.752872	−	0.658167i	$-0.228668\pi$
$421$	30.8953	1.50574	0.752872	+	0.658167i	$0.228668\pi$
$422$	0	0
$423$	−3.25762	−0.158391
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−2.94977	−0.142749
$428$	0	0
$429$	1.83699	0.0886909
$430$	0	0
$431$	7.44637	0.358679	0.179339	−	0.983787i	$-0.442604\pi$
$431$	7.44637	0.358679	0.179339	+	0.983787i	$0.442604\pi$
$432$	0	0
$433$	−20.7559	−0.997465	−0.498733	−	0.866756i	$-0.666201\pi$
$433$	−20.7559	−0.997465	−0.498733	+	0.866756i	$0.666201\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−10.4360	−0.499224
$438$	0	0
$439$	2.48163	0.118442	0.0592209	−	0.998245i	$-0.481138\pi$
$439$	2.48163	0.118442	0.0592209	+	0.998245i	$0.481138\pi$
$440$	0	0
$441$	−6.57757	−0.313217
$442$	0	0
$443$	0.757538	0.0359917	0.0179959	−	0.999838i	$-0.494271\pi$
$443$	0.757538	0.0359917	0.0179959	+	0.999838i	$0.494271\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	8.79360	0.415923
$448$	0	0
$449$	21.0354	0.992723	0.496362	−	0.868116i	$-0.334669\pi$
$449$	21.0354	0.992723	0.496362	+	0.868116i	$0.334669\pi$
$450$	0	0
$451$	23.0599	1.08585
$452$	0	0
$453$	5.91357	0.277844
$454$	0	0
$455$	0	0
$456$	0	0
$457$	35.0835	1.64114	0.820568	−	0.571549i	$-0.193657\pi$
$457$	35.0835	1.64114	0.820568	+	0.571549i	$0.193657\pi$
$458$	0	0
$459$	5.13187	0.239535
$460$	0	0
$461$	41.0666	1.91266	0.956332	−	0.292282i	$-0.0944147\pi$
$461$	41.0666	1.91266	0.956332	+	0.292282i	$0.0944147\pi$
$462$	0	0
$463$	17.3619	0.806874	0.403437	−	0.915007i	$-0.367815\pi$
$463$	17.3619	0.806874	0.403437	+	0.915007i	$0.367815\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−17.6429	−0.816418	−0.408209	−	0.912889i	$-0.633846\pi$
$467$	−17.6429	−0.816418	−0.408209	+	0.912889i	$0.633846\pi$
$468$	0	0
$469$	−0.222774	−0.0102867
$470$	0	0
$471$	−9.57854	−0.441356
$472$	0	0
$473$	−24.9566	−1.14751
$474$	0	0
$475$	0	0
$476$	0	0
$477$	5.24274	0.240049
$478$	0	0
$479$	−3.00292	−0.137207	−0.0686035	−	0.997644i	$-0.521854\pi$
$479$	−3.00292	−0.137207	−0.0686035	+	0.997644i	$0.521854\pi$
$480$	0	0
$481$	−1.45315	−0.0662578
$482$	0	0
$483$	−4.42992	−0.201568
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−15.3888	−0.697334	−0.348667	−	0.937247i	$-0.613366\pi$
$487$	−15.3888	−0.697334	−0.348667	+	0.937247i	$0.613366\pi$
$488$	0	0
$489$	2.02886	0.0917481
$490$	0	0
$491$	16.5958	0.748956	0.374478	−	0.927236i	$-0.377822\pi$
$491$	16.5958	0.748956	0.374478	+	0.927236i	$0.377822\pi$
$492$	0	0
$493$	8.64333	0.389276
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.27731	−0.102151
$498$	0	0
$499$	−12.2579	−0.548740	−0.274370	−	0.961624i	$-0.588469\pi$
$499$	−12.2579	−0.548740	−0.274370	+	0.961624i	$0.588469\pi$
$500$	0	0
$501$	4.35293	0.194474
$502$	0	0
$503$	−2.64670	−0.118011	−0.0590053	−	0.998258i	$-0.518793\pi$
$503$	−2.64670	−0.118011	−0.0590053	+	0.998258i	$0.518793\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−12.8823	−0.572122
$508$	0	0
$509$	21.6152	0.958075	0.479038	−	0.877794i	$-0.340986\pi$
$509$	21.6152	0.958075	0.479038	+	0.877794i	$0.340986\pi$
$510$	0	0
$511$	−1.29990	−0.0575042
$512$	0	0
$513$	−1.53116	−0.0676023
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−17.4421	−0.767102
$518$	0	0
$519$	−4.06066	−0.178243
$520$	0	0
$521$	−6.59610	−0.288980	−0.144490	−	0.989506i	$-0.546154\pi$
$521$	−6.59610	−0.288980	−0.144490	+	0.989506i	$0.546154\pi$
$522$	0	0
$523$	42.9488	1.87802	0.939011	−	0.343888i	$-0.111744\pi$
$523$	42.9488	1.87802	0.939011	+	0.343888i	$0.111744\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5.13187	−0.223548
$528$	0	0
$529$	23.4549	1.01978
$530$	0	0
$531$	13.9245	0.604272
$532$	0	0
$533$	1.47765	0.0640040
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−11.3762	−0.490920
$538$	0	0
$539$	−35.2178	−1.51694
$540$	0	0
$541$	−14.0430	−0.603754	−0.301877	−	0.953347i	$-0.597613\pi$
$541$	−14.0430	−0.603754	−0.301877	+	0.953347i	$0.597613\pi$
$542$	0	0
$543$	13.2132	0.567033
$544$	0	0
$545$	0	0
$546$	0	0
$547$	29.5256	1.26242	0.631212	−	0.775611i	$-0.282558\pi$
$547$	29.5256	1.26242	0.631212	+	0.775611i	$0.282558\pi$
$548$	0	0
$549$	4.53845	0.193696
$550$	0	0
$551$	−2.57885	−0.109863
$552$	0	0
$553$	−1.07217	−0.0455932
$554$	0	0
$555$	0	0
$556$	0	0
$557$	19.1673	0.812145	0.406072	−	0.913841i	$-0.366898\pi$
$557$	19.1673	0.812145	0.406072	+	0.913841i	$0.366898\pi$
$558$	0	0
$559$	−1.59918	−0.0676382
$560$	0	0
$561$	27.4773	1.16009
$562$	0	0
$563$	−11.2021	−0.472114	−0.236057	−	0.971739i	$-0.575855\pi$
$563$	−11.2021	−0.472114	−0.236057	+	0.971739i	$0.575855\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−0.649950	−0.0272953
$568$	0	0
$569$	−6.30788	−0.264440	−0.132220	−	0.991220i	$-0.542211\pi$
$569$	−6.30788	−0.264440	−0.132220	+	0.991220i	$0.542211\pi$
$570$	0	0
$571$	−32.4125	−1.35642	−0.678211	−	0.734868i	$-0.737244\pi$
$571$	−32.4125	−1.35642	−0.678211	+	0.734868i	$0.737244\pi$
$572$	0	0
$573$	−3.90544	−0.163152
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−34.7319	−1.44591	−0.722953	−	0.690897i	$-0.757216\pi$
$577$	−34.7319	−1.44591	−0.722953	+	0.690897i	$0.757216\pi$
$578$	0	0
$579$	−11.1272	−0.462429
$580$	0	0
$581$	5.26574	0.218460
$582$	0	0
$583$	28.0709	1.16258
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−43.2265	−1.78415	−0.892073	−	0.451890i	$-0.850750\pi$
$587$	−43.2265	−1.78415	−0.892073	+	0.451890i	$0.850750\pi$
$588$	0	0
$589$	1.53116	0.0630903
$590$	0	0
$591$	13.2660	0.545690
$592$	0	0
$593$	−45.1582	−1.85442	−0.927212	−	0.374536i	$-0.877802\pi$
$593$	−45.1582	−1.85442	−0.927212	+	0.374536i	$0.877802\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	11.6685	0.477559
$598$	0	0
$599$	−8.13772	−0.332498	−0.166249	−	0.986084i	$-0.553166\pi$
$599$	−8.13772	−0.332498	−0.166249	+	0.986084i	$0.553166\pi$
$600$	0	0
$601$	−2.08104	−0.0848875	−0.0424438	−	0.999099i	$-0.513514\pi$
$601$	−2.08104	−0.0848875	−0.0424438	+	0.999099i	$0.513514\pi$
$602$	0	0
$603$	0.342756	0.0139581
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−3.68238	−0.149463	−0.0747317	−	0.997204i	$-0.523810\pi$
$607$	−3.68238	−0.149463	−0.0747317	+	0.997204i	$0.523810\pi$
$608$	0	0
$609$	−1.09467	−0.0443585
$610$	0	0
$611$	−1.11766	−0.0452158
$612$	0	0
$613$	−37.6568	−1.52094	−0.760472	−	0.649370i	$-0.775032\pi$
$613$	−37.6568	−1.52094	−0.760472	+	0.649370i	$0.775032\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−34.0450	−1.37060	−0.685300	−	0.728260i	$-0.740329\pi$
$617$	−34.0450	−1.37060	−0.685300	+	0.728260i	$0.740329\pi$
$618$	0	0
$619$	−2.97114	−0.119420	−0.0597102	−	0.998216i	$-0.519018\pi$
$619$	−2.97114	−0.119420	−0.0597102	+	0.998216i	$0.519018\pi$
$620$	0	0
$621$	6.81578	0.273508
$622$	0	0
$623$	5.59728	0.224250
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−8.19819	−0.327404
$628$	0	0
$629$	−21.7358	−0.866662
$630$	0	0
$631$	5.40274	0.215080	0.107540	−	0.994201i	$-0.465703\pi$
$631$	5.40274	0.215080	0.107540	+	0.994201i	$0.465703\pi$
$632$	0	0
$633$	5.28429	0.210032
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−2.25671	−0.0894140
$638$	0	0
$639$	3.50382	0.138609
$640$	0	0
$641$	30.8883	1.22002	0.610008	−	0.792395i	$-0.291166\pi$
$641$	30.8883	1.22002	0.610008	+	0.792395i	$0.291166\pi$
$642$	0	0
$643$	36.5110	1.43985	0.719926	−	0.694050i	$-0.244176\pi$
$643$	36.5110	1.43985	0.719926	+	0.694050i	$0.244176\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	3.19555	0.125630	0.0628150	−	0.998025i	$-0.479992\pi$
$647$	3.19555	0.125630	0.0628150	+	0.998025i	$0.479992\pi$
$648$	0	0
$649$	74.5551	2.92654
$650$	0	0
$651$	0.649950	0.0254735
$652$	0	0
$653$	−0.885526	−0.0346533	−0.0173267	−	0.999850i	$-0.505516\pi$
$653$	−0.885526	−0.0346533	−0.0173267	+	0.999850i	$0.505516\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2.00000	0.0780274
$658$	0	0
$659$	−36.7476	−1.43148	−0.715742	−	0.698364i	$-0.753912\pi$
$659$	−36.7476	−1.43148	−0.715742	+	0.698364i	$0.753912\pi$
$660$	0	0
$661$	−37.4321	−1.45594	−0.727969	−	0.685610i	$-0.759536\pi$
$661$	−37.4321	−1.45594	−0.727969	+	0.685610i	$0.759536\pi$
$662$	0	0
$663$	1.76070	0.0683801
$664$	0	0
$665$	0	0
$666$	0	0
$667$	11.4794	0.444486
$668$	0	0
$669$	17.5849	0.679870
$670$	0	0
$671$	24.3000	0.938089
$672$	0	0
$673$	14.3238	0.552140	0.276070	−	0.961137i	$-0.410968\pi$
$673$	14.3238	0.552140	0.276070	+	0.961137i	$0.410968\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−12.5132	−0.480921	−0.240461	−	0.970659i	$-0.577298\pi$
$677$	−12.5132	−0.480921	−0.240461	+	0.970659i	$0.577298\pi$
$678$	0	0
$679$	5.15255	0.197737
$680$	0	0
$681$	−21.7285	−0.832639
$682$	0	0
$683$	−17.2965	−0.661834	−0.330917	−	0.943660i	$-0.607358\pi$
$683$	−17.2965	−0.661834	−0.330917	+	0.943660i	$0.607358\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	25.8445	0.986028
$688$	0	0
$689$	1.79874	0.0685266
$690$	0	0
$691$	−45.5203	−1.73167	−0.865837	−	0.500326i	$-0.833214\pi$
$691$	−45.5203	−1.73167	−0.865837	+	0.500326i	$0.833214\pi$
$692$	0	0
$693$	−3.47999	−0.132194
$694$	0	0
$695$	0	0
$696$	0	0
$697$	22.1023	0.837183
$698$	0	0
$699$	11.4699	0.433832
$700$	0	0
$701$	39.6022	1.49576	0.747878	−	0.663836i	$-0.231073\pi$
$701$	39.6022	1.49576	0.747878	+	0.663836i	$0.231073\pi$
$702$	0	0
$703$	6.48514	0.244592
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−12.2694	−0.461439
$708$	0	0
$709$	−8.12306	−0.305068	−0.152534	−	0.988298i	$-0.548743\pi$
$709$	−8.12306	−0.305068	−0.152534	+	0.988298i	$0.548743\pi$
$710$	0	0
$711$	1.64961	0.0618654
$712$	0	0
$713$	−6.81578	−0.255253
$714$	0	0
$715$	0	0
$716$	0	0
$717$	4.54264	0.169648
$718$	0	0
$719$	4.93397	0.184006	0.0920030	−	0.995759i	$-0.470673\pi$
$719$	4.93397	0.184006	0.0920030	+	0.995759i	$0.470673\pi$
$720$	0	0
$721$	−0.619156	−0.0230586
$722$	0	0
$723$	−13.2216	−0.491716
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−6.66944	−0.247356	−0.123678	−	0.992322i	$-0.539469\pi$
$727$	−6.66944	−0.247356	−0.123678	+	0.992322i	$0.539469\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−23.9202	−0.884719
$732$	0	0
$733$	−20.2657	−0.748533	−0.374266	−	0.927321i	$-0.622105\pi$
$733$	−20.2657	−0.748533	−0.374266	+	0.927321i	$0.622105\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.83519	0.0676003
$738$	0	0
$739$	−18.9049	−0.695429	−0.347714	−	0.937601i	$-0.613042\pi$
$739$	−18.9049	−0.695429	−0.347714	+	0.937601i	$0.613042\pi$
$740$	0	0
$741$	−0.525328	−0.0192984
$742$	0	0
$743$	−49.8029	−1.82709	−0.913547	−	0.406734i	$-0.866667\pi$
$743$	−49.8029	−1.82709	−0.913547	+	0.406734i	$0.866667\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−8.10177	−0.296428
$748$	0	0
$749$	0.427999	0.0156387
$750$	0	0
$751$	−30.7190	−1.12095	−0.560476	−	0.828171i	$-0.689382\pi$
$751$	−30.7190	−1.12095	−0.560476	+	0.828171i	$0.689382\pi$
$752$	0	0
$753$	21.5605	0.785708
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−29.9090	−1.08706	−0.543531	−	0.839389i	$-0.682913\pi$
$757$	−29.9090	−1.08706	−0.543531	+	0.839389i	$0.682913\pi$
$758$	0	0
$759$	36.4933	1.32462
$760$	0	0
$761$	−13.2632	−0.480790	−0.240395	−	0.970675i	$-0.577277\pi$
$761$	−13.2632	−0.480790	−0.240395	+	0.970675i	$0.577277\pi$
$762$	0	0
$763$	7.53208	0.272680
$764$	0	0
$765$	0	0
$766$	0	0
$767$	4.77738	0.172501
$768$	0	0
$769$	−17.7608	−0.640470	−0.320235	−	0.947338i	$-0.603762\pi$
$769$	−17.7608	−0.640470	−0.320235	+	0.947338i	$0.603762\pi$
$770$	0	0
$771$	11.8169	0.425574
$772$	0	0
$773$	29.6130	1.06511	0.532554	−	0.846396i	$-0.321232\pi$
$773$	29.6130	1.06511	0.532554	+	0.846396i	$0.321232\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	2.75283	0.0987571
$778$	0	0
$779$	−6.59449	−0.236272
$780$	0	0
$781$	18.7603	0.671296
$782$	0	0
$783$	1.68424	0.0601900
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.41312	0.0503724	0.0251862	−	0.999683i	$-0.491982\pi$
$787$	1.41312	0.0503724	0.0251862	+	0.999683i	$0.491982\pi$
$788$	0	0
$789$	15.8413	0.563966
$790$	0	0
$791$	4.43014	0.157518
$792$	0	0
$793$	1.55711	0.0552944
$794$	0	0
$795$	0	0
$796$	0	0
$797$	23.2353	0.823037	0.411519	−	0.911401i	$-0.364998\pi$
$797$	23.2353	0.823037	0.411519	+	0.911401i	$0.364998\pi$
$798$	0	0
$799$	−16.7177	−0.591430
$800$	0	0
$801$	−8.61186	−0.304285
$802$	0	0
$803$	10.7085	0.377894
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−19.1761	−0.675032
$808$	0	0
$809$	−28.7950	−1.01238	−0.506190	−	0.862422i	$-0.668946\pi$
$809$	−28.7950	−1.01238	−0.506190	+	0.862422i	$0.668946\pi$
$810$	0	0
$811$	−39.2861	−1.37952	−0.689761	−	0.724037i	$-0.742284\pi$
$811$	−39.2861	−1.37952	−0.689761	+	0.724037i	$0.742284\pi$
$812$	0	0
$813$	−0.842540	−0.0295492
$814$	0	0
$815$	0	0
$816$	0	0
$817$	7.13688	0.249688
$818$	0	0
$819$	−0.222992	−0.00779199
$820$	0	0
$821$	35.6506	1.24421	0.622107	−	0.782932i	$-0.286277\pi$
$821$	35.6506	1.24421	0.622107	+	0.782932i	$0.286277\pi$
$822$	0	0
$823$	12.7389	0.444051	0.222025	−	0.975041i	$-0.428733\pi$
$823$	12.7389	0.444051	0.222025	+	0.975041i	$0.428733\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	47.7979	1.66210	0.831048	−	0.556201i	$-0.187742\pi$
$827$	47.7979	1.66210	0.831048	+	0.556201i	$0.187742\pi$
$828$	0	0
$829$	23.0923	0.802029	0.401015	−	0.916072i	$-0.368658\pi$
$829$	23.0923	0.802029	0.401015	+	0.916072i	$0.368658\pi$
$830$	0	0
$831$	−31.9176	−1.10721
$832$	0	0
$833$	−33.7552	−1.16955
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	14.5270	0.501527	0.250763	−	0.968048i	$-0.419318\pi$
$839$	14.5270	0.501527	0.250763	+	0.968048i	$0.419318\pi$
$840$	0	0
$841$	−26.1633	−0.902183
$842$	0	0
$843$	15.5023	0.533928
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−11.4832	−0.394568
$848$	0	0
$849$	−10.9026	−0.374177
$850$	0	0
$851$	−28.8679	−0.989578
$852$	0	0
$853$	−45.1017	−1.54425	−0.772126	−	0.635470i	$-0.780807\pi$
$853$	−45.1017	−1.54425	−0.772126	+	0.635470i	$0.780807\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	5.22869	0.178609	0.0893043	−	0.996004i	$-0.471536\pi$
$857$	5.22869	0.178609	0.0893043	+	0.996004i	$0.471536\pi$
$858$	0	0
$859$	17.4749	0.596235	0.298118	−	0.954529i	$-0.403641\pi$
$859$	17.4749	0.596235	0.298118	+	0.954529i	$0.403641\pi$
$860$	0	0
$861$	−2.79924	−0.0953979
$862$	0	0
$863$	16.6956	0.568324	0.284162	−	0.958776i	$-0.408285\pi$
$863$	16.6956	0.568324	0.284162	+	0.958776i	$0.408285\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	9.33613	0.317072
$868$	0	0
$869$	8.83242	0.299619
$870$	0	0
$871$	0.117597	0.00398461
$872$	0	0
$873$	−7.92762	−0.268309
$874$	0	0
$875$	0	0
$876$	0	0
$877$	13.7028	0.462711	0.231356	−	0.972869i	$-0.425684\pi$
$877$	13.7028	0.462711	0.231356	+	0.972869i	$0.425684\pi$
$878$	0	0
$879$	−7.74714	−0.261305
$880$	0	0
$881$	−8.95675	−0.301761	−0.150880	−	0.988552i	$-0.548211\pi$
$881$	−8.95675	−0.301761	−0.150880	+	0.988552i	$0.548211\pi$
$882$	0	0
$883$	40.8511	1.37475	0.687375	−	0.726303i	$-0.258763\pi$
$883$	40.8511	1.37475	0.687375	+	0.726303i	$0.258763\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−46.2905	−1.55428	−0.777141	−	0.629326i	$-0.783331\pi$
$887$	−46.2905	−1.55428	−0.777141	+	0.629326i	$0.783331\pi$
$888$	0	0
$889$	−0.354272	−0.0118819
$890$	0	0
$891$	5.35424	0.179374
$892$	0	0
$893$	4.98794	0.166915
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2.33844	0.0780782
$898$	0	0
$899$	−1.68424	−0.0561727
$900$	0	0
$901$	26.9051	0.896338
$902$	0	0
$903$	3.02948	0.100815
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−29.9467	−0.994365	−0.497182	−	0.867646i	$-0.665632\pi$
$907$	−29.9467	−0.994365	−0.497182	+	0.867646i	$0.665632\pi$
$908$	0	0
$909$	18.8775	0.626126
$910$	0	0
$911$	−47.8940	−1.58680	−0.793399	−	0.608702i	$-0.791690\pi$
$911$	−47.8940	−1.58680	−0.793399	+	0.608702i	$0.791690\pi$
$912$	0	0
$913$	−43.3788	−1.43563
$914$	0	0
$915$	0	0
$916$	0	0
$917$	8.66726	0.286218
$918$	0	0
$919$	−47.0420	−1.55177	−0.775886	−	0.630873i	$-0.782697\pi$
$919$	−47.0420	−1.55177	−0.775886	+	0.630873i	$0.782697\pi$
$920$	0	0
$921$	1.46995	0.0484364
$922$	0	0
$923$	1.20213	0.0395687
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0.952622	0.0312882
$928$	0	0
$929$	2.38689	0.0783114	0.0391557	−	0.999233i	$-0.487533\pi$
$929$	2.38689	0.0783114	0.0391557	+	0.999233i	$0.487533\pi$
$930$	0	0
$931$	10.0713	0.330074
$932$	0	0
$933$	19.8469	0.649758
$934$	0	0
$935$	0	0
$936$	0	0
$937$	19.7649	0.645691	0.322846	−	0.946452i	$-0.395361\pi$
$937$	19.7649	0.645691	0.322846	+	0.946452i	$0.395361\pi$
$938$	0	0
$939$	1.89449	0.0618243
$940$	0	0
$941$	1.81125	0.0590450	0.0295225	−	0.999564i	$-0.490601\pi$
$941$	1.81125	0.0590450	0.0295225	+	0.999564i	$0.490601\pi$
$942$	0	0
$943$	29.3546	0.955918
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−24.6290	−0.800335	−0.400167	−	0.916442i	$-0.631048\pi$
$947$	−24.6290	−0.800335	−0.400167	+	0.916442i	$0.631048\pi$
$948$	0	0
$949$	0.686183	0.0222745
$950$	0	0
$951$	−27.7231	−0.898984
$952$	0	0
$953$	−32.1879	−1.04267	−0.521335	−	0.853352i	$-0.674566\pi$
$953$	−32.1879	−1.04267	−0.521335	+	0.853352i	$0.674566\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	9.01784	0.291506
$958$	0	0
$959$	−1.78513	−0.0576448
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−0.658511	−0.0212202
$964$	0	0
$965$	0	0
$966$	0	0
$967$	19.3094	0.620948	0.310474	−	0.950582i	$-0.399512\pi$
$967$	19.3094	0.620948	0.310474	+	0.950582i	$0.399512\pi$
$968$	0	0
$969$	−7.85772	−0.252426
$970$	0	0
$971$	44.4190	1.42547	0.712736	−	0.701432i	$-0.247456\pi$
$971$	44.4190	1.42547	0.712736	+	0.701432i	$0.247456\pi$
$972$	0	0
$973$	4.45284	0.142751
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−15.4034	−0.492800	−0.246400	−	0.969168i	$-0.579248\pi$
$977$	−15.4034	−0.492800	−0.246400	+	0.969168i	$0.579248\pi$
$978$	0	0
$979$	−46.1099	−1.47368
$980$	0	0
$981$	−11.5887	−0.369999
$982$	0	0
$983$	25.9914	0.828997	0.414498	−	0.910050i	$-0.363957\pi$
$983$	25.9914	0.828997	0.414498	+	0.910050i	$0.363957\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	2.11729	0.0673941
$988$	0	0
$989$	−31.7690	−1.01020
$990$	0	0
$991$	−7.40420	−0.235202	−0.117601	−	0.993061i	$-0.537520\pi$
$991$	−7.40420	−0.235202	−0.117601	+	0.993061i	$0.537520\pi$
$992$	0	0
$993$	5.93845	0.188451
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−2.57638	−0.0815949	−0.0407974	−	0.999167i	$-0.512990\pi$
$997$	−2.57638	−0.0815949	−0.0407974	+	0.999167i	$0.512990\pi$
$998$	0	0
$999$	−4.23545	−0.134004

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9300.2.a.ba.1.3	yes	6
5.2	odd	4		9300.2.g.t.3349.3		12
5.3	odd	4		9300.2.g.t.3349.10		12
5.4	even	2		9300.2.a.y.1.4	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9300.2.a.y.1.4	✓	6	5.4	even	2
9300.2.a.ba.1.3	yes	6	1.1	even	1	trivial
9300.2.g.t.3349.3		12	5.2	odd	4
9300.2.g.t.3349.10		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -0.649950 q^{7} +1.00000 q^{9} +5.35424 q^{11} +0.343092 q^{13} +5.13187 q^{17} -1.53116 q^{19} -0.649950 q^{21} +6.81578 q^{23} +1.00000 q^{27} +1.68424 q^{29} -1.00000 q^{31} +5.35424 q^{33} -4.23545 q^{37} +0.343092 q^{39} +4.30686 q^{41} -4.66109 q^{43} -3.25762 q^{47} -6.57757 q^{49} +5.13187 q^{51} +5.24274 q^{53} -1.53116 q^{57} +13.9245 q^{59} +4.53845 q^{61} -0.649950 q^{63} +0.342756 q^{67} +6.81578 q^{69} +3.50382 q^{71} +2.00000 q^{73} -3.47999 q^{77} +1.64961 q^{79} +1.00000 q^{81} -8.10177 q^{83} +1.68424 q^{87} -8.61186 q^{89} -0.222992 q^{91} -1.00000 q^{93} -7.92762 q^{97} +5.35424 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} + 4 q^{7} + 6 q^{9} - 3 q^{11} + 2 q^{13} + 4 q^{17} - 3 q^{19} + 4 q^{21} + 5 q^{23} + 6 q^{27} - 6 q^{31} - 3 q^{33} + 8 q^{37} + 2 q^{39} + 18 q^{41} + 15 q^{43} + q^{47} + 14 q^{49} + 4 q^{51}+ \cdots - 3 q^{99}+O(q^{100}) \) 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

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