LMFDB - Embedded newform 2704.2.a.x.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2704,2,Mod(1,2704)] 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("2704.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2704, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2704 = 2^{4} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2704.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,-8,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$21.5915487066$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 676)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.24698$ of defining polynomial
Character	$\chi$	$=$	2704.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.04892 q^{3} -2.55496 q^{5} -3.15883 q^{7} +6.29590 q^{9} -1.04892 q^{11} -7.78986 q^{15} -2.08815 q^{17} +2.46681 q^{19} -9.63102 q^{21} -4.38404 q^{23} +1.52781 q^{25} +10.0489 q^{27} -3.15883 q^{29} +4.93900 q^{31} -3.19806 q^{33} +8.07069 q^{35} -8.56465 q^{37} -11.4940 q^{41} -12.7289 q^{43} -16.0858 q^{45} +3.54288 q^{47} +2.97823 q^{49} -6.36658 q^{51} -8.45473 q^{53} +2.67994 q^{55} +7.52111 q^{57} +12.4819 q^{59} +3.57673 q^{61} -19.8877 q^{63} +1.74094 q^{67} -13.3666 q^{69} -4.69202 q^{71} +7.34481 q^{73} +4.65817 q^{75} +3.31336 q^{77} -6.73556 q^{79} +11.7506 q^{81} +2.19806 q^{83} +5.33513 q^{85} -9.63102 q^{87} -13.1860 q^{89} +15.0586 q^{93} -6.30260 q^{95} +13.2567 q^{97} -6.60388 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 8 q^{5} - q^{7} + 5 q^{9} + 6 q^{11} - 10 q^{17} + 4 q^{19} - 14 q^{21} - 3 q^{23} + 11 q^{25} + 21 q^{27} - q^{29} + 5 q^{31} - 14 q^{33} + 12 q^{35} - 4 q^{37} - 25 q^{41} - 5 q^{43} - 11 q^{45}+ \cdots - 11 q^{99}+O(q^{100}) $ 3 * q - 8 * q^5 - q^7 + 5 * q^9 + 6 * q^11 - 10 * q^17 + 4 * q^19 - 14 * q^21 - 3 * q^23 + 11 * q^25 + 21 * q^27 - q^29 + 5 * q^31 - 14 * q^33 + 12 * q^35 - 4 * q^37 - 25 * q^41 - 5 * q^43 - 11 * q^45 - 8 * q^47 + 12 * q^49 + 7 * q^51 - 3 * q^53 - 16 * q^55 + 7 * q^57 + 9 * q^59 + 8 * q^61 - 18 * q^63 - 9 * q^67 - 14 * q^69 - 9 * q^71 - q^73 - 7 * q^75 + 12 * q^77 - 9 * q^79 - q^81 + 11 * q^83 + 15 * q^85 - 14 * q^87 - 25 * q^89 + 14 * q^93 - 27 * q^95 + 13 * q^97 - 11 * 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
$3$	3.04892	1.76029	0.880147	−	0.474702i	$-0.157444\pi$
$3$	3.04892	1.76029	0.880147	+	0.474702i	$0.157444\pi$
$4$	0	0
$5$	−2.55496	−1.14261	−0.571306	−	0.820737i	$-0.693563\pi$
$5$	−2.55496	−1.14261	−0.571306	+	0.820737i	$0.693563\pi$
$6$	0	0
$7$	−3.15883	−1.19393	−0.596963	−	0.802268i	$-0.703626\pi$
$7$	−3.15883	−1.19393	−0.596963	+	0.802268i	$0.703626\pi$
$8$	0	0
$9$	6.29590	2.09863
$10$	0	0
$11$	−1.04892	−0.316260	−0.158130	−	0.987418i	$-0.550547\pi$
$11$	−1.04892	−0.316260	−0.158130	+	0.987418i	$0.550547\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−7.78986	−2.01133
$16$	0	0
$17$	−2.08815	−0.506450	−0.253225	−	0.967407i	$-0.581491\pi$
$17$	−2.08815	−0.506450	−0.253225	+	0.967407i	$0.581491\pi$
$18$	0	0
$19$	2.46681	0.565926	0.282963	−	0.959131i	$-0.408683\pi$
$19$	2.46681	0.565926	0.282963	+	0.959131i	$0.408683\pi$
$20$	0	0
$21$	−9.63102	−2.10166
$22$	0	0
$23$	−4.38404	−0.914136	−0.457068	−	0.889432i	$-0.651100\pi$
$23$	−4.38404	−0.914136	−0.457068	+	0.889432i	$0.651100\pi$
$24$	0	0
$25$	1.52781	0.305562
$26$	0	0
$27$	10.0489	1.93392
$28$	0	0
$29$	−3.15883	−0.586581	−0.293290	−	0.956023i	$-0.594750\pi$
$29$	−3.15883	−0.586581	−0.293290	+	0.956023i	$0.594750\pi$
$30$	0	0
$31$	4.93900	0.887071	0.443535	−	0.896257i	$-0.353724\pi$
$31$	4.93900	0.887071	0.443535	+	0.896257i	$0.353724\pi$
$32$	0	0
$33$	−3.19806	−0.556711
$34$	0	0
$35$	8.07069	1.36420
$36$	0	0
$37$	−8.56465	−1.40802	−0.704010	−	0.710190i	$-0.748609\pi$
$37$	−8.56465	−1.40802	−0.704010	+	0.710190i	$0.748609\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−11.4940	−1.79505	−0.897527	−	0.440959i	$-0.854639\pi$
$41$	−11.4940	−1.79505	−0.897527	+	0.440959i	$0.854639\pi$
$42$	0	0
$43$	−12.7289	−1.94113	−0.970566	−	0.240834i	$-0.922579\pi$
$43$	−12.7289	−1.94113	−0.970566	+	0.240834i	$0.922579\pi$
$44$	0	0
$45$	−16.0858	−2.39792
$46$	0	0
$47$	3.54288	0.516782	0.258391	−	0.966040i	$-0.416808\pi$
$47$	3.54288	0.516782	0.258391	+	0.966040i	$0.416808\pi$
$48$	0	0
$49$	2.97823	0.425461
$50$	0	0
$51$	−6.36658	−0.891500
$52$	0	0
$53$	−8.45473	−1.16135	−0.580673	−	0.814137i	$-0.697211\pi$
$53$	−8.45473	−1.16135	−0.580673	+	0.814137i	$0.697211\pi$
$54$	0	0
$55$	2.67994	0.361363
$56$	0	0
$57$	7.52111	0.996195
$58$	0	0
$59$	12.4819	1.62500	0.812501	−	0.582960i	$-0.198105\pi$
$59$	12.4819	1.62500	0.812501	+	0.582960i	$0.198105\pi$
$60$	0	0
$61$	3.57673	0.457953	0.228977	−	0.973432i	$-0.426462\pi$
$61$	3.57673	0.457953	0.228977	+	0.973432i	$0.426462\pi$
$62$	0	0
$63$	−19.8877	−2.50561
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1.74094	0.212690	0.106345	−	0.994329i	$-0.466085\pi$
$67$	1.74094	0.212690	0.106345	+	0.994329i	$0.466085\pi$
$68$	0	0
$69$	−13.3666	−1.60915
$70$	0	0
$71$	−4.69202	−0.556841	−0.278420	−	0.960459i	$-0.589811\pi$
$71$	−4.69202	−0.556841	−0.278420	+	0.960459i	$0.589811\pi$
$72$	0	0
$73$	7.34481	0.859645	0.429823	−	0.902913i	$-0.358576\pi$
$73$	7.34481	0.859645	0.429823	+	0.902913i	$0.358576\pi$
$74$	0	0
$75$	4.65817	0.537879
$76$	0	0
$77$	3.31336	0.377592
$78$	0	0
$79$	−6.73556	−0.757810	−0.378905	−	0.925436i	$-0.623699\pi$
$79$	−6.73556	−0.757810	−0.378905	+	0.925436i	$0.623699\pi$
$80$	0	0
$81$	11.7506	1.30563
$82$	0	0
$83$	2.19806	0.241269	0.120634	−	0.992697i	$-0.461507\pi$
$83$	2.19806	0.241269	0.120634	+	0.992697i	$0.461507\pi$
$84$	0	0
$85$	5.33513	0.578676
$86$	0	0
$87$	−9.63102	−1.03255
$88$	0	0
$89$	−13.1860	−1.39771	−0.698856	−	0.715263i	$-0.746307\pi$
$89$	−13.1860	−1.39771	−0.698856	+	0.715263i	$0.746307\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	15.0586	1.56150
$94$	0	0
$95$	−6.30260	−0.646633
$96$	0	0
$97$	13.2567	1.34601	0.673005	−	0.739638i	$-0.265003\pi$
$97$	13.2567	1.34601	0.673005	+	0.739638i	$0.265003\pi$
$98$	0	0
$99$	−6.60388	−0.663714
$100$	0	0
$101$	1.50365	0.149619	0.0748093	−	0.997198i	$-0.476165\pi$
$101$	1.50365	0.149619	0.0748093	+	0.997198i	$0.476165\pi$
$102$	0	0
$103$	5.07069	0.499630	0.249815	−	0.968294i	$-0.419630\pi$
$103$	5.07069	0.499630	0.249815	+	0.968294i	$0.419630\pi$
$104$	0	0
$105$	24.6069	2.40138
$106$	0	0
$107$	4.52781	0.437720	0.218860	−	0.975756i	$-0.429766\pi$
$107$	4.52781	0.437720	0.218860	+	0.975756i	$0.429766\pi$
$108$	0	0
$109$	−12.8659	−1.23233	−0.616166	−	0.787616i	$-0.711315\pi$
$109$	−12.8659	−1.23233	−0.616166	+	0.787616i	$0.711315\pi$
$110$	0	0
$111$	−26.1129	−2.47853
$112$	0	0
$113$	−15.4601	−1.45436	−0.727182	−	0.686444i	$-0.759171\pi$
$113$	−15.4601	−1.45436	−0.727182	+	0.686444i	$0.759171\pi$
$114$	0	0
$115$	11.2010	1.04450
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.59611	0.604664
$120$	0	0
$121$	−9.89977	−0.899979
$122$	0	0
$123$	−35.0441	−3.15982
$124$	0	0
$125$	8.87130	0.793473
$126$	0	0
$127$	11.0489	0.980433	0.490216	−	0.871601i	$-0.336918\pi$
$127$	11.0489	0.980433	0.490216	+	0.871601i	$0.336918\pi$
$128$	0	0
$129$	−38.8092	−3.41696
$130$	0	0
$131$	−6.84117	−0.597715	−0.298858	−	0.954298i	$-0.596606\pi$
$131$	−6.84117	−0.597715	−0.298858	+	0.954298i	$0.596606\pi$
$132$	0	0
$133$	−7.79225	−0.675674
$134$	0	0
$135$	−25.6746	−2.20971
$136$	0	0
$137$	−5.78017	−0.493833	−0.246917	−	0.969037i	$-0.579417\pi$
$137$	−5.78017	−0.493833	−0.246917	+	0.969037i	$0.579417\pi$
$138$	0	0
$139$	0.527811	0.0447683	0.0223842	−	0.999749i	$-0.492874\pi$
$139$	0.527811	0.0447683	0.0223842	+	0.999749i	$0.492874\pi$
$140$	0	0
$141$	10.8019	0.909687
$142$	0	0
$143$	0	0
$144$	0	0
$145$	8.07069	0.670234
$146$	0	0
$147$	9.08038	0.748937
$148$	0	0
$149$	−19.9661	−1.63569	−0.817845	−	0.575438i	$-0.804832\pi$
$149$	−19.9661	−1.63569	−0.817845	+	0.575438i	$0.804832\pi$
$150$	0	0
$151$	3.41550	0.277950	0.138975	−	0.990296i	$-0.455619\pi$
$151$	3.41550	0.277950	0.138975	+	0.990296i	$0.455619\pi$
$152$	0	0
$153$	−13.1468	−1.06285
$154$	0	0
$155$	−12.6189	−1.01358
$156$	0	0
$157$	6.32304	0.504634	0.252317	−	0.967645i	$-0.418807\pi$
$157$	6.32304	0.504634	0.252317	+	0.967645i	$0.418807\pi$
$158$	0	0
$159$	−25.7778	−2.04431
$160$	0	0
$161$	13.8485	1.09141
$162$	0	0
$163$	−11.2524	−0.881353	−0.440676	−	0.897666i	$-0.645261\pi$
$163$	−11.2524	−0.881353	−0.440676	+	0.897666i	$0.645261\pi$
$164$	0	0
$165$	8.17092	0.636105
$166$	0	0
$167$	−5.32975	−0.412428	−0.206214	−	0.978507i	$-0.566114\pi$
$167$	−5.32975	−0.412428	−0.206214	+	0.978507i	$0.566114\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	15.5308	1.18767
$172$	0	0
$173$	19.2078	1.46034	0.730169	−	0.683266i	$-0.239441\pi$
$173$	19.2078	1.46034	0.730169	+	0.683266i	$0.239441\pi$
$174$	0	0
$175$	−4.82610	−0.364819
$176$	0	0
$177$	38.0562	2.86048
$178$	0	0
$179$	5.63640	0.421284	0.210642	−	0.977563i	$-0.432445\pi$
$179$	5.63640	0.421284	0.210642	+	0.977563i	$0.432445\pi$
$180$	0	0
$181$	3.96615	0.294801	0.147401	−	0.989077i	$-0.452909\pi$
$181$	3.96615	0.294801	0.147401	+	0.989077i	$0.452909\pi$
$182$	0	0
$183$	10.9051	0.806132
$184$	0	0
$185$	21.8823	1.60882
$186$	0	0
$187$	2.19029	0.160170
$188$	0	0
$189$	−31.7429	−2.30895
$190$	0	0
$191$	20.9051	1.51264	0.756322	−	0.654200i	$-0.226994\pi$
$191$	20.9051	1.51264	0.756322	+	0.654200i	$0.226994\pi$
$192$	0	0
$193$	12.8224	0.922975	0.461488	−	0.887147i	$-0.347316\pi$
$193$	12.8224	0.922975	0.461488	+	0.887147i	$0.347316\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−17.9638	−1.27986	−0.639932	−	0.768431i	$-0.721038\pi$
$197$	−17.9638	−1.27986	−0.639932	+	0.768431i	$0.721038\pi$
$198$	0	0
$199$	13.6746	0.969364	0.484682	−	0.874690i	$-0.338935\pi$
$199$	13.6746	0.969364	0.484682	+	0.874690i	$0.338935\pi$
$200$	0	0
$201$	5.30798	0.374396
$202$	0	0
$203$	9.97823	0.700334
$204$	0	0
$205$	29.3666	2.05105
$206$	0	0
$207$	−27.6015	−1.91844
$208$	0	0
$209$	−2.58748	−0.178980
$210$	0	0
$211$	7.54288	0.519273	0.259637	−	0.965706i	$-0.416397\pi$
$211$	7.54288	0.519273	0.259637	+	0.965706i	$0.416397\pi$
$212$	0	0
$213$	−14.3056	−0.980203
$214$	0	0
$215$	32.5217	2.21796
$216$	0	0
$217$	−15.6015	−1.05910
$218$	0	0
$219$	22.3937	1.51323
$220$	0	0
$221$	0	0
$222$	0	0
$223$	15.2131	1.01875	0.509373	−	0.860546i	$-0.329877\pi$
$223$	15.2131	1.01875	0.509373	+	0.860546i	$0.329877\pi$
$224$	0	0
$225$	9.61894	0.641263
$226$	0	0
$227$	14.0043	0.929499	0.464749	−	0.885442i	$-0.346144\pi$
$227$	14.0043	0.929499	0.464749	+	0.885442i	$0.346144\pi$
$228$	0	0
$229$	−4.72886	−0.312492	−0.156246	−	0.987718i	$-0.549939\pi$
$229$	−4.72886	−0.312492	−0.156246	+	0.987718i	$0.549939\pi$
$230$	0	0
$231$	10.1021	0.664672
$232$	0	0
$233$	19.8998	1.30368	0.651839	−	0.758358i	$-0.273998\pi$
$233$	19.8998	1.30368	0.651839	+	0.758358i	$0.273998\pi$
$234$	0	0
$235$	−9.05190	−0.590481
$236$	0	0
$237$	−20.5362	−1.33397
$238$	0	0
$239$	−3.51573	−0.227414	−0.113707	−	0.993514i	$-0.536272\pi$
$239$	−3.51573	−0.227414	−0.113707	+	0.993514i	$0.536272\pi$
$240$	0	0
$241$	−7.17092	−0.461919	−0.230960	−	0.972963i	$-0.574187\pi$
$241$	−7.17092	−0.461919	−0.230960	+	0.972963i	$0.574187\pi$
$242$	0	0
$243$	5.67994	0.364368
$244$	0	0
$245$	−7.60925	−0.486137
$246$	0	0
$247$	0	0
$248$	0	0
$249$	6.70171	0.424704
$250$	0	0
$251$	−17.7995	−1.12350	−0.561749	−	0.827308i	$-0.689871\pi$
$251$	−17.7995	−1.12350	−0.561749	+	0.827308i	$0.689871\pi$
$252$	0	0
$253$	4.59850	0.289105
$254$	0	0
$255$	16.2664	1.01864
$256$	0	0
$257$	14.1226	0.880943	0.440471	−	0.897767i	$-0.354811\pi$
$257$	14.1226	0.880943	0.440471	+	0.897767i	$0.354811\pi$
$258$	0	0
$259$	27.0543	1.68107
$260$	0	0
$261$	−19.8877	−1.23102
$262$	0	0
$263$	−24.4969	−1.51055	−0.755273	−	0.655410i	$-0.772496\pi$
$263$	−24.4969	−1.51055	−0.755273	+	0.655410i	$0.772496\pi$
$264$	0	0
$265$	21.6015	1.32697
$266$	0	0
$267$	−40.2030	−2.46038
$268$	0	0
$269$	14.1890	0.865116	0.432558	−	0.901606i	$-0.357611\pi$
$269$	14.1890	0.865116	0.432558	+	0.901606i	$0.357611\pi$
$270$	0	0
$271$	−24.7832	−1.50547	−0.752735	−	0.658324i	$-0.771266\pi$
$271$	−24.7832	−1.50547	−0.752735	+	0.658324i	$0.771266\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.60255	−0.0966373
$276$	0	0
$277$	−11.6189	−0.698115	−0.349057	−	0.937101i	$-0.613498\pi$
$277$	−11.6189	−0.698115	−0.349057	+	0.937101i	$0.613498\pi$
$278$	0	0
$279$	31.0954	1.86164
$280$	0	0
$281$	−4.10023	−0.244599	−0.122300	−	0.992493i	$-0.539027\pi$
$281$	−4.10023	−0.244599	−0.122300	+	0.992493i	$0.539027\pi$
$282$	0	0
$283$	24.9095	1.48071	0.740357	−	0.672214i	$-0.234657\pi$
$283$	24.9095	1.48071	0.740357	+	0.672214i	$0.234657\pi$
$284$	0	0
$285$	−19.2161	−1.13826
$286$	0	0
$287$	36.3075	2.14316
$288$	0	0
$289$	−12.6396	−0.743509
$290$	0	0
$291$	40.4185	2.36937
$292$	0	0
$293$	22.4010	1.30868	0.654341	−	0.756200i	$-0.272946\pi$
$293$	22.4010	1.30868	0.654341	+	0.756200i	$0.272946\pi$
$294$	0	0
$295$	−31.8907	−1.85675
$296$	0	0
$297$	−10.5405	−0.611621
$298$	0	0
$299$	0	0
$300$	0	0
$301$	40.2083	2.31757
$302$	0	0
$303$	4.58450	0.263373
$304$	0	0
$305$	−9.13839	−0.523263
$306$	0	0
$307$	13.4383	0.766966	0.383483	−	0.923548i	$-0.374724\pi$
$307$	13.4383	0.766966	0.383483	+	0.923548i	$0.374724\pi$
$308$	0	0
$309$	15.4601	0.879495
$310$	0	0
$311$	14.3424	0.813284	0.406642	−	0.913588i	$-0.366700\pi$
$311$	14.3424	0.813284	0.406642	+	0.913588i	$0.366700\pi$
$312$	0	0
$313$	4.52111	0.255548	0.127774	−	0.991803i	$-0.459217\pi$
$313$	4.52111	0.255548	0.127774	+	0.991803i	$0.459217\pi$
$314$	0	0
$315$	50.8122	2.86294
$316$	0	0
$317$	22.6655	1.27302	0.636510	−	0.771269i	$-0.280378\pi$
$317$	22.6655	1.27302	0.636510	+	0.771269i	$0.280378\pi$
$318$	0	0
$319$	3.31336	0.185512
$320$	0	0
$321$	13.8049	0.770516
$322$	0	0
$323$	−5.15106	−0.286613
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−39.2271	−2.16927
$328$	0	0
$329$	−11.1914	−0.617000
$330$	0	0
$331$	−13.7603	−0.756335	−0.378168	−	0.925737i	$-0.623446\pi$
$331$	−13.7603	−0.756335	−0.378168	+	0.925737i	$0.623446\pi$
$332$	0	0
$333$	−53.9221	−2.95491
$334$	0	0
$335$	−4.44803	−0.243022
$336$	0	0
$337$	9.84548	0.536317	0.268159	−	0.963375i	$-0.413585\pi$
$337$	9.84548	0.536317	0.268159	+	0.963375i	$0.413585\pi$
$338$	0	0
$339$	−47.1366	−2.56011
$340$	0	0
$341$	−5.18060	−0.280545
$342$	0	0
$343$	12.7041	0.685957
$344$	0	0
$345$	34.1511	1.83863
$346$	0	0
$347$	2.92453	0.156997	0.0784984	−	0.996914i	$-0.474987\pi$
$347$	2.92453	0.156997	0.0784984	+	0.996914i	$0.474987\pi$
$348$	0	0
$349$	−9.33752	−0.499826	−0.249913	−	0.968268i	$-0.580402\pi$
$349$	−9.33752	−0.499826	−0.249913	+	0.968268i	$0.580402\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−15.7748	−0.839607	−0.419804	−	0.907615i	$-0.637901\pi$
$353$	−15.7748	−0.839607	−0.419804	+	0.907615i	$0.637901\pi$
$354$	0	0
$355$	11.9879	0.636253
$356$	0	0
$357$	20.1110	1.06439
$358$	0	0
$359$	−21.5676	−1.13830	−0.569148	−	0.822235i	$-0.692727\pi$
$359$	−21.5676	−1.13830	−0.569148	+	0.822235i	$0.692727\pi$
$360$	0	0
$361$	−12.9148	−0.679728
$362$	0	0
$363$	−30.1836	−1.58423
$364$	0	0
$365$	−18.7657	−0.982241
$366$	0	0
$367$	13.8006	0.720386	0.360193	−	0.932878i	$-0.382711\pi$
$367$	13.8006	0.720386	0.360193	+	0.932878i	$0.382711\pi$
$368$	0	0
$369$	−72.3648	−3.76716
$370$	0	0
$371$	26.7071	1.38656
$372$	0	0
$373$	24.0780	1.24671	0.623355	−	0.781939i	$-0.285769\pi$
$373$	24.0780	1.24671	0.623355	+	0.781939i	$0.285769\pi$
$374$	0	0
$375$	27.0479	1.39675
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−16.3575	−0.840228	−0.420114	−	0.907471i	$-0.638010\pi$
$379$	−16.3575	−0.840228	−0.420114	+	0.907471i	$0.638010\pi$
$380$	0	0
$381$	33.6872	1.72585
$382$	0	0
$383$	6.98792	0.357066	0.178533	−	0.983934i	$-0.442865\pi$
$383$	6.98792	0.357066	0.178533	+	0.983934i	$0.442865\pi$
$384$	0	0
$385$	−8.46548	−0.431441
$386$	0	0
$387$	−80.1396	−4.07372
$388$	0	0
$389$	9.57971	0.485711	0.242855	−	0.970063i	$-0.421916\pi$
$389$	9.57971	0.485711	0.242855	+	0.970063i	$0.421916\pi$
$390$	0	0
$391$	9.15452	0.462964
$392$	0	0
$393$	−20.8582	−1.05215
$394$	0	0
$395$	17.2091	0.865883
$396$	0	0
$397$	16.9554	0.850967	0.425483	−	0.904966i	$-0.360104\pi$
$397$	16.9554	0.850967	0.425483	+	0.904966i	$0.360104\pi$
$398$	0	0
$399$	−23.7579	−1.18938
$400$	0	0
$401$	19.2640	0.961996	0.480998	−	0.876722i	$-0.340274\pi$
$401$	19.2640	0.961996	0.480998	+	0.876722i	$0.340274\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−30.0224	−1.49182
$406$	0	0
$407$	8.98361	0.445301
$408$	0	0
$409$	−21.3424	−1.05531	−0.527657	−	0.849457i	$-0.676930\pi$
$409$	−21.3424	−1.05531	−0.527657	+	0.849457i	$0.676930\pi$
$410$	0	0
$411$	−17.6233	−0.869291
$412$	0	0
$413$	−39.4282	−1.94013
$414$	0	0
$415$	−5.61596	−0.275676
$416$	0	0
$417$	1.60925	0.0788054
$418$	0	0
$419$	−13.8019	−0.674269	−0.337134	−	0.941457i	$-0.609458\pi$
$419$	−13.8019	−0.674269	−0.337134	+	0.941457i	$0.609458\pi$
$420$	0	0
$421$	16.9836	0.827730	0.413865	−	0.910338i	$-0.364178\pi$
$421$	16.9836	0.827730	0.413865	+	0.910338i	$0.364178\pi$
$422$	0	0
$423$	22.3056	1.08453
$424$	0	0
$425$	−3.19029	−0.154752
$426$	0	0
$427$	−11.2983	−0.546763
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−16.6910	−0.803975	−0.401988	−	0.915645i	$-0.631680\pi$
$431$	−16.6910	−0.803975	−0.401988	+	0.915645i	$0.631680\pi$
$432$	0	0
$433$	−2.54420	−0.122267	−0.0611333	−	0.998130i	$-0.519471\pi$
$433$	−2.54420	−0.122267	−0.0611333	+	0.998130i	$0.519471\pi$
$434$	0	0
$435$	24.6069	1.17981
$436$	0	0
$437$	−10.8146	−0.517333
$438$	0	0
$439$	6.10752	0.291496	0.145748	−	0.989322i	$-0.453441\pi$
$439$	6.10752	0.291496	0.145748	+	0.989322i	$0.453441\pi$
$440$	0	0
$441$	18.7506	0.892887
$442$	0	0
$443$	29.2030	1.38747	0.693737	−	0.720228i	$-0.255963\pi$
$443$	29.2030	1.38747	0.693737	+	0.720228i	$0.255963\pi$
$444$	0	0
$445$	33.6896	1.59704
$446$	0	0
$447$	−60.8751	−2.87930
$448$	0	0
$449$	−32.6698	−1.54178	−0.770891	−	0.636967i	$-0.780189\pi$
$449$	−32.6698	−1.54178	−0.770891	+	0.636967i	$0.780189\pi$
$450$	0	0
$451$	12.0562	0.567705
$452$	0	0
$453$	10.4136	0.489273
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−3.22952	−0.151071	−0.0755353	−	0.997143i	$-0.524067\pi$
$457$	−3.22952	−0.151071	−0.0755353	+	0.997143i	$0.524067\pi$
$458$	0	0
$459$	−20.9836	−0.979431
$460$	0	0
$461$	−10.4354	−0.486023	−0.243011	−	0.970023i	$-0.578135\pi$
$461$	−10.4354	−0.486023	−0.243011	+	0.970023i	$0.578135\pi$
$462$	0	0
$463$	−6.17928	−0.287175	−0.143588	−	0.989638i	$-0.545864\pi$
$463$	−6.17928	−0.287175	−0.143588	+	0.989638i	$0.545864\pi$
$464$	0	0
$465$	−38.4741	−1.78419
$466$	0	0
$467$	16.4601	0.761683	0.380841	−	0.924640i	$-0.375634\pi$
$467$	16.4601	0.761683	0.380841	+	0.924640i	$0.375634\pi$
$468$	0	0
$469$	−5.49934	−0.253936
$470$	0	0
$471$	19.2784	0.888304
$472$	0	0
$473$	13.3515	0.613904
$474$	0	0
$475$	3.76882	0.172925
$476$	0	0
$477$	−53.2301	−2.43724
$478$	0	0
$479$	−38.2597	−1.74813	−0.874064	−	0.485811i	$-0.838524\pi$
$479$	−38.2597	−1.74813	−0.874064	+	0.485811i	$0.838524\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	42.2228	1.92120
$484$	0	0
$485$	−33.8702	−1.53797
$486$	0	0
$487$	29.5937	1.34102	0.670510	−	0.741901i	$-0.266076\pi$
$487$	29.5937	1.34102	0.670510	+	0.741901i	$0.266076\pi$
$488$	0	0
$489$	−34.3075	−1.55144
$490$	0	0
$491$	−37.6993	−1.70135	−0.850673	−	0.525695i	$-0.823805\pi$
$491$	−37.6993	−1.70135	−0.850673	+	0.525695i	$0.823805\pi$
$492$	0	0
$493$	6.59611	0.297074
$494$	0	0
$495$	16.8726	0.758368
$496$	0	0
$497$	14.8213	0.664827
$498$	0	0
$499$	−13.0774	−0.585424	−0.292712	−	0.956201i	$-0.594558\pi$
$499$	−13.0774	−0.585424	−0.292712	+	0.956201i	$0.594558\pi$
$500$	0	0
$501$	−16.2500	−0.725995
$502$	0	0
$503$	2.82610	0.126010	0.0630048	−	0.998013i	$-0.479932\pi$
$503$	2.82610	0.126010	0.0630048	+	0.998013i	$0.479932\pi$
$504$	0	0
$505$	−3.84176	−0.170956
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−37.1336	−1.64592	−0.822959	−	0.568101i	$-0.807678\pi$
$509$	−37.1336	−1.64592	−0.822959	+	0.568101i	$0.807678\pi$
$510$	0	0
$511$	−23.2010	−1.02635
$512$	0	0
$513$	24.7888	1.09445
$514$	0	0
$515$	−12.9554	−0.570883
$516$	0	0
$517$	−3.71618	−0.163438
$518$	0	0
$519$	58.5628	2.57062
$520$	0	0
$521$	−29.5375	−1.29406	−0.647031	−	0.762464i	$-0.723989\pi$
$521$	−29.5375	−1.29406	−0.647031	+	0.762464i	$0.723989\pi$
$522$	0	0
$523$	−31.1637	−1.36270	−0.681348	−	0.731960i	$-0.738606\pi$
$523$	−31.1637	−1.36270	−0.681348	+	0.731960i	$0.738606\pi$
$524$	0	0
$525$	−14.7144	−0.642188
$526$	0	0
$527$	−10.3134	−0.449257
$528$	0	0
$529$	−3.78017	−0.164355
$530$	0	0
$531$	78.5846	3.41028
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−11.5684	−0.500144
$536$	0	0
$537$	17.1849	0.741584
$538$	0	0
$539$	−3.12392	−0.134557
$540$	0	0
$541$	40.0810	1.72322	0.861608	−	0.507575i	$-0.169458\pi$
$541$	40.0810	1.72322	0.861608	+	0.507575i	$0.169458\pi$
$542$	0	0
$543$	12.0925	0.518937
$544$	0	0
$545$	32.8719	1.40808
$546$	0	0
$547$	−16.4534	−0.703497	−0.351748	−	0.936095i	$-0.614413\pi$
$547$	−16.4534	−0.703497	−0.351748	+	0.936095i	$0.614413\pi$
$548$	0	0
$549$	22.5187	0.961075
$550$	0	0
$551$	−7.79225	−0.331961
$552$	0	0
$553$	21.2765	0.904770
$554$	0	0
$555$	66.7174	2.83199
$556$	0	0
$557$	−12.6799	−0.537266	−0.268633	−	0.963243i	$-0.586572\pi$
$557$	−12.6799	−0.537266	−0.268633	+	0.963243i	$0.586572\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	6.67802	0.281946
$562$	0	0
$563$	−22.1588	−0.933883	−0.466942	−	0.884288i	$-0.654644\pi$
$563$	−22.1588	−0.933883	−0.466942	+	0.884288i	$0.654644\pi$
$564$	0	0
$565$	39.4999	1.66177
$566$	0	0
$567$	−37.1183	−1.55882
$568$	0	0
$569$	3.31468	0.138959	0.0694794	−	0.997583i	$-0.477866\pi$
$569$	3.31468	0.138959	0.0694794	+	0.997583i	$0.477866\pi$
$570$	0	0
$571$	15.7429	0.658818	0.329409	−	0.944187i	$-0.393150\pi$
$571$	15.7429	0.658818	0.329409	+	0.944187i	$0.393150\pi$
$572$	0	0
$573$	63.7381	2.66270
$574$	0	0
$575$	−6.69799	−0.279325
$576$	0	0
$577$	2.05131	0.0853972	0.0426986	−	0.999088i	$-0.486404\pi$
$577$	2.05131	0.0853972	0.0426986	+	0.999088i	$0.486404\pi$
$578$	0	0
$579$	39.0944	1.62471
$580$	0	0
$581$	−6.94331	−0.288057
$582$	0	0
$583$	8.86831	0.367288
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.79763	0.0741960	0.0370980	−	0.999312i	$-0.488189\pi$
$587$	1.79763	0.0741960	0.0370980	+	0.999312i	$0.488189\pi$
$588$	0	0
$589$	12.1836	0.502016
$590$	0	0
$591$	−54.7700	−2.25294
$592$	0	0
$593$	−28.7332	−1.17993	−0.589965	−	0.807429i	$-0.700858\pi$
$593$	−28.7332	−1.17993	−0.589965	+	0.807429i	$0.700858\pi$
$594$	0	0
$595$	−16.8528	−0.690896
$596$	0	0
$597$	41.6926	1.70637
$598$	0	0
$599$	−44.5840	−1.82165	−0.910827	−	0.412788i	$-0.864555\pi$
$599$	−44.5840	−1.82165	−0.910827	+	0.412788i	$0.864555\pi$
$600$	0	0
$601$	−9.72455	−0.396672	−0.198336	−	0.980134i	$-0.563554\pi$
$601$	−9.72455	−0.396672	−0.198336	+	0.980134i	$0.563554\pi$
$602$	0	0
$603$	10.9608	0.446357
$604$	0	0
$605$	25.2935	1.02833
$606$	0	0
$607$	−45.8649	−1.86160	−0.930799	−	0.365533i	$-0.880887\pi$
$607$	−45.8649	−1.86160	−0.930799	+	0.365533i	$0.880887\pi$
$608$	0	0
$609$	30.4228	1.23279
$610$	0	0
$611$	0	0
$612$	0	0
$613$	10.0392	0.405481	0.202740	−	0.979233i	$-0.435015\pi$
$613$	10.0392	0.405481	0.202740	+	0.979233i	$0.435015\pi$
$614$	0	0
$615$	89.5363	3.61045
$616$	0	0
$617$	11.7922	0.474738	0.237369	−	0.971420i	$-0.423715\pi$
$617$	11.7922	0.474738	0.237369	+	0.971420i	$0.423715\pi$
$618$	0	0
$619$	21.2711	0.854959	0.427480	−	0.904025i	$-0.359402\pi$
$619$	21.2711	0.854959	0.427480	+	0.904025i	$0.359402\pi$
$620$	0	0
$621$	−44.0549	−1.76786
$622$	0	0
$623$	41.6523	1.66876
$624$	0	0
$625$	−30.3048	−1.21219
$626$	0	0
$627$	−7.88902	−0.315057
$628$	0	0
$629$	17.8842	0.713091
$630$	0	0
$631$	−3.82610	−0.152315	−0.0761573	−	0.997096i	$-0.524265\pi$
$631$	−3.82610	−0.152315	−0.0761573	+	0.997096i	$0.524265\pi$
$632$	0	0
$633$	22.9976	0.914073
$634$	0	0
$635$	−28.2295	−1.12025
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−29.5405	−1.16860
$640$	0	0
$641$	−28.0417	−1.10758	−0.553791	−	0.832656i	$-0.686819\pi$
$641$	−28.0417	−1.10758	−0.553791	+	0.832656i	$0.686819\pi$
$642$	0	0
$643$	11.3593	0.447967	0.223983	−	0.974593i	$-0.428094\pi$
$643$	11.3593	0.447967	0.223983	+	0.974593i	$0.428094\pi$
$644$	0	0
$645$	99.1560	3.90426
$646$	0	0
$647$	18.1328	0.712872	0.356436	−	0.934320i	$-0.383992\pi$
$647$	18.1328	0.712872	0.356436	+	0.934320i	$0.383992\pi$
$648$	0	0
$649$	−13.0925	−0.513924
$650$	0	0
$651$	−47.5676	−1.86432
$652$	0	0
$653$	−8.91915	−0.349033	−0.174517	−	0.984654i	$-0.555836\pi$
$653$	−8.91915	−0.349033	−0.174517	+	0.984654i	$0.555836\pi$
$654$	0	0
$655$	17.4789	0.682957
$656$	0	0
$657$	46.2422	1.80408
$658$	0	0
$659$	−18.0575	−0.703422	−0.351711	−	0.936109i	$-0.614400\pi$
$659$	−18.0575	−0.703422	−0.351711	+	0.936109i	$0.614400\pi$
$660$	0	0
$661$	39.3860	1.53194	0.765968	−	0.642878i	$-0.222260\pi$
$661$	39.3860	1.53194	0.765968	+	0.642878i	$0.222260\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	19.9089	0.772033
$666$	0	0
$667$	13.8485	0.536215
$668$	0	0
$669$	46.3836	1.79329
$670$	0	0
$671$	−3.75169	−0.144832
$672$	0	0
$673$	23.6203	0.910494	0.455247	−	0.890365i	$-0.349551\pi$
$673$	23.6203	0.910494	0.455247	+	0.890365i	$0.349551\pi$
$674$	0	0
$675$	15.3528	0.590931
$676$	0	0
$677$	46.7741	1.79767	0.898836	−	0.438284i	$-0.144414\pi$
$677$	46.7741	1.79767	0.898836	+	0.438284i	$0.144414\pi$
$678$	0	0
$679$	−41.8756	−1.60704
$680$	0	0
$681$	42.6980	1.63619
$682$	0	0
$683$	6.74094	0.257935	0.128967	−	0.991649i	$-0.458834\pi$
$683$	6.74094	0.257935	0.128967	+	0.991649i	$0.458834\pi$
$684$	0	0
$685$	14.7681	0.564260
$686$	0	0
$687$	−14.4179	−0.550077
$688$	0	0
$689$	0	0
$690$	0	0
$691$	28.1879	1.07232	0.536159	−	0.844117i	$-0.319875\pi$
$691$	28.1879	1.07232	0.536159	+	0.844117i	$0.319875\pi$
$692$	0	0
$693$	20.8605	0.792427
$694$	0	0
$695$	−1.34854	−0.0511529
$696$	0	0
$697$	24.0011	0.909105
$698$	0	0
$699$	60.6728	2.29485
$700$	0	0
$701$	−18.2078	−0.687697	−0.343849	−	0.939025i	$-0.611731\pi$
$701$	−18.2078	−0.687697	−0.343849	+	0.939025i	$0.611731\pi$
$702$	0	0
$703$	−21.1274	−0.796834
$704$	0	0
$705$	−27.5985	−1.03942
$706$	0	0
$707$	−4.74977	−0.178634
$708$	0	0
$709$	24.9269	0.936150	0.468075	−	0.883689i	$-0.344948\pi$
$709$	24.9269	0.936150	0.468075	+	0.883689i	$0.344948\pi$
$710$	0	0
$711$	−42.4064	−1.59036
$712$	0	0
$713$	−21.6528	−0.810903
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−10.7192	−0.400315
$718$	0	0
$719$	−33.1963	−1.23801	−0.619006	−	0.785386i	$-0.712464\pi$
$719$	−33.1963	−1.23801	−0.619006	+	0.785386i	$0.712464\pi$
$720$	0	0
$721$	−16.0175	−0.596521
$722$	0	0
$723$	−21.8635	−0.813113
$724$	0	0
$725$	−4.82610	−0.179237
$726$	0	0
$727$	−36.0006	−1.33519	−0.667594	−	0.744526i	$-0.732676\pi$
$727$	−36.0006	−1.33519	−0.667594	+	0.744526i	$0.732676\pi$
$728$	0	0
$729$	−17.9342	−0.664230
$730$	0	0
$731$	26.5797	0.983086
$732$	0	0
$733$	−7.47757	−0.276190	−0.138095	−	0.990419i	$-0.544098\pi$
$733$	−7.47757	−0.276190	−0.138095	+	0.990419i	$0.544098\pi$
$734$	0	0
$735$	−23.2000	−0.855744
$736$	0	0
$737$	−1.82610	−0.0672653
$738$	0	0
$739$	39.0858	1.43779	0.718896	−	0.695117i	$-0.244648\pi$
$739$	39.0858	1.43779	0.718896	+	0.695117i	$0.244648\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−32.5295	−1.19339	−0.596695	−	0.802468i	$-0.703520\pi$
$743$	−32.5295	−1.19339	−0.596695	+	0.802468i	$0.703520\pi$
$744$	0	0
$745$	51.0127	1.86896
$746$	0	0
$747$	13.8388	0.506334
$748$	0	0
$749$	−14.3026	−0.522606
$750$	0	0
$751$	18.3263	0.668736	0.334368	−	0.942443i	$-0.391477\pi$
$751$	18.3263	0.668736	0.334368	+	0.942443i	$0.391477\pi$
$752$	0	0
$753$	−54.2693	−1.97768
$754$	0	0
$755$	−8.72646	−0.317589
$756$	0	0
$757$	−23.1172	−0.840209	−0.420105	−	0.907476i	$-0.638007\pi$
$757$	−23.1172	−0.840209	−0.420105	+	0.907476i	$0.638007\pi$
$758$	0	0
$759$	14.0204	0.508910
$760$	0	0
$761$	−13.6987	−0.496578	−0.248289	−	0.968686i	$-0.579868\pi$
$761$	−13.6987	−0.496578	−0.248289	+	0.968686i	$0.579868\pi$
$762$	0	0
$763$	40.6413	1.47131
$764$	0	0
$765$	33.5894	1.21443
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−24.4480	−0.881618	−0.440809	−	0.897601i	$-0.645308\pi$
$769$	−24.4480	−0.881618	−0.440809	+	0.897601i	$0.645308\pi$
$770$	0	0
$771$	43.0586	1.55072
$772$	0	0
$773$	34.0694	1.22539	0.612695	−	0.790320i	$-0.290085\pi$
$773$	34.0694	1.22539	0.612695	+	0.790320i	$0.290085\pi$
$774$	0	0
$775$	7.54586	0.271055
$776$	0	0
$777$	82.4863	2.95918
$778$	0	0
$779$	−28.3534	−1.01587
$780$	0	0
$781$	4.92154	0.176107
$782$	0	0
$783$	−31.7429	−1.13440
$784$	0	0
$785$	−16.1551	−0.576601
$786$	0	0
$787$	−2.07739	−0.0740510	−0.0370255	−	0.999314i	$-0.511788\pi$
$787$	−2.07739	−0.0740510	−0.0370255	+	0.999314i	$0.511788\pi$
$788$	0	0
$789$	−74.6892	−2.65900
$790$	0	0
$791$	48.8359	1.73640
$792$	0	0
$793$	0	0
$794$	0	0
$795$	65.8611	2.33585
$796$	0	0
$797$	−27.1299	−0.960990	−0.480495	−	0.876998i	$-0.659543\pi$
$797$	−27.1299	−0.960990	−0.480495	+	0.876998i	$0.659543\pi$
$798$	0	0
$799$	−7.39804	−0.261724
$800$	0	0
$801$	−83.0176	−2.93328
$802$	0	0
$803$	−7.70410	−0.271872
$804$	0	0
$805$	−35.3822	−1.24706
$806$	0	0
$807$	43.2610	1.52286
$808$	0	0
$809$	−12.1870	−0.428474	−0.214237	−	0.976782i	$-0.568726\pi$
$809$	−12.1870	−0.428474	−0.214237	+	0.976782i	$0.568726\pi$
$810$	0	0
$811$	17.7178	0.622158	0.311079	−	0.950384i	$-0.399310\pi$
$811$	17.7178	0.622158	0.311079	+	0.950384i	$0.399310\pi$
$812$	0	0
$813$	−75.5618	−2.65007
$814$	0	0
$815$	28.7493	1.00704
$816$	0	0
$817$	−31.3997	−1.09854
$818$	0	0
$819$	0	0
$820$	0	0
$821$	28.9560	1.01057	0.505285	−	0.862952i	$-0.331387\pi$
$821$	28.9560	1.01057	0.505285	+	0.862952i	$0.331387\pi$
$822$	0	0
$823$	40.6883	1.41830	0.709152	−	0.705055i	$-0.249078\pi$
$823$	40.6883	1.41830	0.709152	+	0.705055i	$0.249078\pi$
$824$	0	0
$825$	−4.88603	−0.170110
$826$	0	0
$827$	−20.5244	−0.713702	−0.356851	−	0.934161i	$-0.616150\pi$
$827$	−20.5244	−0.713702	−0.356851	+	0.934161i	$0.616150\pi$
$828$	0	0
$829$	−31.6155	−1.09805	−0.549026	−	0.835806i	$-0.685001\pi$
$829$	−31.6155	−1.09805	−0.549026	+	0.835806i	$0.685001\pi$
$830$	0	0
$831$	−35.4252	−1.22889
$832$	0	0
$833$	−6.21898	−0.215475
$834$	0	0
$835$	13.6173	0.471246
$836$	0	0
$837$	49.6316	1.71552
$838$	0	0
$839$	−10.2306	−0.353199	−0.176600	−	0.984283i	$-0.556510\pi$
$839$	−10.2306	−0.353199	−0.176600	+	0.984283i	$0.556510\pi$
$840$	0	0
$841$	−19.0218	−0.655923
$842$	0	0
$843$	−12.5013	−0.430566
$844$	0	0
$845$	0	0
$846$	0	0
$847$	31.2717	1.07451
$848$	0	0
$849$	75.9469	2.60649
$850$	0	0
$851$	37.5478	1.28712
$852$	0	0
$853$	21.6233	0.740366	0.370183	−	0.928959i	$-0.379295\pi$
$853$	21.6233	0.740366	0.370183	+	0.928959i	$0.379295\pi$
$854$	0	0
$855$	−39.6805	−1.35705
$856$	0	0
$857$	43.6082	1.48963	0.744814	−	0.667273i	$-0.232538\pi$
$857$	43.6082	1.48963	0.744814	+	0.667273i	$0.232538\pi$
$858$	0	0
$859$	45.5991	1.55582	0.777910	−	0.628375i	$-0.216280\pi$
$859$	45.5991	1.55582	0.777910	+	0.628375i	$0.216280\pi$
$860$	0	0
$861$	110.699	3.77260
$862$	0	0
$863$	−16.3104	−0.555212	−0.277606	−	0.960695i	$-0.589541\pi$
$863$	−16.3104	−0.555212	−0.277606	+	0.960695i	$0.589541\pi$
$864$	0	0
$865$	−49.0750	−1.66860
$866$	0	0
$867$	−38.5372	−1.30879
$868$	0	0
$869$	7.06505	0.239665
$870$	0	0
$871$	0	0
$872$	0	0
$873$	83.4626	2.82478
$874$	0	0
$875$	−28.0230	−0.947349
$876$	0	0
$877$	−12.7579	−0.430804	−0.215402	−	0.976525i	$-0.569106\pi$
$877$	−12.7579	−0.430804	−0.215402	+	0.976525i	$0.569106\pi$
$878$	0	0
$879$	68.2989	2.30366
$880$	0	0
$881$	−9.85815	−0.332130	−0.166065	−	0.986115i	$-0.553106\pi$
$881$	−9.85815	−0.332130	−0.166065	+	0.986115i	$0.553106\pi$
$882$	0	0
$883$	−15.8649	−0.533895	−0.266947	−	0.963711i	$-0.586015\pi$
$883$	−15.8649	−0.533895	−0.266947	+	0.963711i	$0.586015\pi$
$884$	0	0
$885$	−97.2320	−3.26842
$886$	0	0
$887$	2.95407	0.0991878	0.0495939	−	0.998769i	$-0.484207\pi$
$887$	2.95407	0.0991878	0.0495939	+	0.998769i	$0.484207\pi$
$888$	0	0
$889$	−34.9017	−1.17057
$890$	0	0
$891$	−12.3254	−0.412918
$892$	0	0
$893$	8.73961	0.292460
$894$	0	0
$895$	−14.4008	−0.481364
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−15.6015	−0.520339
$900$	0	0
$901$	17.6547	0.588164
$902$	0	0
$903$	122.592	4.07960
$904$	0	0
$905$	−10.1333	−0.336844
$906$	0	0
$907$	10.9933	0.365026	0.182513	−	0.983203i	$-0.441577\pi$
$907$	10.9933	0.365026	0.182513	+	0.983203i	$0.441577\pi$
$908$	0	0
$909$	9.46681	0.313994
$910$	0	0
$911$	−11.5104	−0.381355	−0.190677	−	0.981653i	$-0.561068\pi$
$911$	−11.5104	−0.381355	−0.190677	+	0.981653i	$0.561068\pi$
$912$	0	0
$913$	−2.30559	−0.0763037
$914$	0	0
$915$	−27.8622	−0.921096
$916$	0	0
$917$	21.6101	0.713629
$918$	0	0
$919$	25.0814	0.827360	0.413680	−	0.910422i	$-0.364243\pi$
$919$	25.0814	0.827360	0.413680	+	0.910422i	$0.364243\pi$
$920$	0	0
$921$	40.9724	1.35009
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−13.0852	−0.430237
$926$	0	0
$927$	31.9245	1.04854
$928$	0	0
$929$	9.28514	0.304636	0.152318	−	0.988332i	$-0.451326\pi$
$929$	9.28514	0.304636	0.152318	+	0.988332i	$0.451326\pi$
$930$	0	0
$931$	7.34673	0.240779
$932$	0	0
$933$	43.7289	1.43162
$934$	0	0
$935$	−5.59611	−0.183012
$936$	0	0
$937$	−34.3830	−1.12324	−0.561621	−	0.827394i	$-0.689822\pi$
$937$	−34.3830	−1.12324	−0.561621	+	0.827394i	$0.689822\pi$
$938$	0	0
$939$	13.7845	0.449839
$940$	0	0
$941$	−37.3284	−1.21687	−0.608436	−	0.793603i	$-0.708203\pi$
$941$	−37.3284	−1.21687	−0.608436	+	0.793603i	$0.708203\pi$
$942$	0	0
$943$	50.3900	1.64092
$944$	0	0
$945$	81.1017	2.63824
$946$	0	0
$947$	29.2887	0.951755	0.475878	−	0.879511i	$-0.342130\pi$
$947$	29.2887	0.951755	0.475878	+	0.879511i	$0.342130\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	69.1051	2.24089
$952$	0	0
$953$	7.34588	0.237956	0.118978	−	0.992897i	$-0.462038\pi$
$953$	7.34588	0.237956	0.118978	+	0.992897i	$0.462038\pi$
$954$	0	0
$955$	−53.4118	−1.72836
$956$	0	0
$957$	10.1021	0.326556
$958$	0	0
$959$	18.2586	0.589601
$960$	0	0
$961$	−6.60627	−0.213105
$962$	0	0
$963$	28.5066	0.918613
$964$	0	0
$965$	−32.7606	−1.05460
$966$	0	0
$967$	−40.6859	−1.30837	−0.654185	−	0.756334i	$-0.726988\pi$
$967$	−40.6859	−1.30837	−0.654185	+	0.756334i	$0.726988\pi$
$968$	0	0
$969$	−15.7052	−0.504523
$970$	0	0
$971$	−30.5206	−0.979454	−0.489727	−	0.871876i	$-0.662903\pi$
$971$	−30.5206	−0.979454	−0.489727	+	0.871876i	$0.662903\pi$
$972$	0	0
$973$	−1.66727	−0.0534501
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−36.2529	−1.15983	−0.579917	−	0.814675i	$-0.696915\pi$
$977$	−36.2529	−1.15983	−0.579917	+	0.814675i	$0.696915\pi$
$978$	0	0
$979$	13.8310	0.442041
$980$	0	0
$981$	−81.0025	−2.58621
$982$	0	0
$983$	−13.9632	−0.445356	−0.222678	−	0.974892i	$-0.571480\pi$
$983$	−13.9632	−0.445356	−0.222678	+	0.974892i	$0.571480\pi$
$984$	0	0
$985$	45.8966	1.46239
$986$	0	0
$987$	−34.1215	−1.08610
$988$	0	0
$989$	55.8039	1.77446
$990$	0	0
$991$	40.7542	1.29460	0.647300	−	0.762235i	$-0.275898\pi$
$991$	40.7542	1.29460	0.647300	+	0.762235i	$0.275898\pi$
$992$	0	0
$993$	−41.9541	−1.33137
$994$	0	0
$995$	−34.9379	−1.10761
$996$	0	0
$997$	−49.7109	−1.57436	−0.787180	−	0.616723i	$-0.788460\pi$
$997$	−49.7109	−1.57436	−0.787180	+	0.616723i	$0.788460\pi$
$998$	0	0
$999$	−86.0654	−2.72299

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2704.2.a.x.1.3		3
4.3	odd	2		676.2.a.g.1.1	✓	3
12.11	even	2		6084.2.a.bc.1.2		3
13.5	odd	4		2704.2.f.n.337.6		6
13.8	odd	4		2704.2.f.n.337.5		6
13.12	even	2		2704.2.a.y.1.3		3
52.3	odd	6		676.2.e.f.529.3		6
52.7	even	12		676.2.h.e.361.5		12
52.11	even	12		676.2.h.e.485.5		12
52.15	even	12		676.2.h.e.485.6		12
52.19	even	12		676.2.h.e.361.6		12
52.23	odd	6		676.2.e.g.529.3		6
52.31	even	4		676.2.d.e.337.2		6
52.35	odd	6		676.2.e.f.653.3		6
52.43	odd	6		676.2.e.g.653.3		6
52.47	even	4		676.2.d.e.337.1		6
52.51	odd	2		676.2.a.h.1.1	yes	3
156.47	odd	4		6084.2.b.p.4393.5		6
156.83	odd	4		6084.2.b.p.4393.2		6
156.155	even	2		6084.2.a.x.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
676.2.a.g.1.1	✓	3	4.3	odd	2
676.2.a.h.1.1	yes	3	52.51	odd	2
676.2.d.e.337.1		6	52.47	even	4
676.2.d.e.337.2		6	52.31	even	4
676.2.e.f.529.3		6	52.3	odd	6
676.2.e.f.653.3		6	52.35	odd	6
676.2.e.g.529.3		6	52.23	odd	6
676.2.e.g.653.3		6	52.43	odd	6
676.2.h.e.361.5		12	52.7	even	12
676.2.h.e.361.6		12	52.19	even	12
676.2.h.e.485.5		12	52.11	even	12
676.2.h.e.485.6		12	52.15	even	12
2704.2.a.x.1.3		3	1.1	even	1	trivial
2704.2.a.y.1.3		3	13.12	even	2
2704.2.f.n.337.5		6	13.8	odd	4
2704.2.f.n.337.6		6	13.5	odd	4
6084.2.a.x.1.2		3	156.155	even	2
6084.2.a.bc.1.2		3	12.11	even	2
6084.2.b.p.4393.2		6	156.83	odd	4
6084.2.b.p.4393.5		6	156.47	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:	\( N \)	\(=\)	\( 2704 = 2^{4} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2704.a (trivial)

\(f(q)\)	\(=\)	\(q+3.04892 q^{3} -2.55496 q^{5} -3.15883 q^{7} +6.29590 q^{9} -1.04892 q^{11} -7.78986 q^{15} -2.08815 q^{17} +2.46681 q^{19} -9.63102 q^{21} -4.38404 q^{23} +1.52781 q^{25} +10.0489 q^{27} -3.15883 q^{29} +4.93900 q^{31} -3.19806 q^{33} +8.07069 q^{35} -8.56465 q^{37} -11.4940 q^{41} -12.7289 q^{43} -16.0858 q^{45} +3.54288 q^{47} +2.97823 q^{49} -6.36658 q^{51} -8.45473 q^{53} +2.67994 q^{55} +7.52111 q^{57} +12.4819 q^{59} +3.57673 q^{61} -19.8877 q^{63} +1.74094 q^{67} -13.3666 q^{69} -4.69202 q^{71} +7.34481 q^{73} +4.65817 q^{75} +3.31336 q^{77} -6.73556 q^{79} +11.7506 q^{81} +2.19806 q^{83} +5.33513 q^{85} -9.63102 q^{87} -13.1860 q^{89} +15.0586 q^{93} -6.30260 q^{95} +13.2567 q^{97} -6.60388 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 8 q^{5} - q^{7} + 5 q^{9} + 6 q^{11} - 10 q^{17} + 4 q^{19} - 14 q^{21} - 3 q^{23} + 11 q^{25} + 21 q^{27} - q^{29} + 5 q^{31} - 14 q^{33} + 12 q^{35} - 4 q^{37} - 25 q^{41} - 5 q^{43} - 11 q^{45}+ \cdots - 11 q^{99}+O(q^{100}) \) 3 * q - 8 * q^5 - q^7 + 5 * q^9 + 6 * q^11 - 10 * q^17 + 4 * q^19 - 14 * q^21 - 3 * q^23 + 11 * q^25 + 21 * q^27 - q^29 + 5 * q^31 - 14 * q^33 + 12 * q^35 - 4 * q^37 - 25 * q^41 - 5 * q^43 - 11 * q^45 - 8 * q^47 + 12 * q^49 + 7 * q^51 - 3 * q^53 - 16 * q^55 + 7 * q^57 + 9 * q^59 + 8 * q^61 - 18 * q^63 - 9 * q^67 - 14 * q^69 - 9 * q^71 - q^73 - 7 * q^75 + 12 * q^77 - 9 * q^79 - q^81 + 11 * q^83 + 15 * q^85 - 14 * q^87 - 25 * q^89 + 14 * q^93 - 27 * q^95 + 13 * q^97 - 11 * q^99

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