LMFDB - Embedded newform 4732.2.a.r.1.6

Error: no document with id 273130214 found in table mf_hecke_traces.

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4732,2,Mod(1,4732)] 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("4732.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4732, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4732 = 2^{2} \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4732.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,6,0,0,0,6] 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:	$37.7852102365$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.2854789.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 6x^{4} + 17x^{3} + 11x^{2} - 20x - 13 $ x^6 - 3x^5 - 6x^4 + 17x^3 + 11x^2 - 20*x - 13
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.87590$ of defining polynomial
Character	$\chi$	$=$	4732.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.28803 q^{3} -3.77007 q^{5} +1.00000 q^{7} +7.81112 q^{9} -0.166194 q^{11} -12.3961 q^{15} -6.66424 q^{17} -3.14115 q^{19} +3.28803 q^{21} +8.71947 q^{23} +9.21343 q^{25} +15.8191 q^{27} +2.50377 q^{29} +2.07228 q^{31} -0.546452 q^{33} -3.77007 q^{35} +5.24067 q^{37} +6.25901 q^{41} -3.39957 q^{43} -29.4485 q^{45} +8.74103 q^{47} +1.00000 q^{49} -21.9122 q^{51} +5.35058 q^{53} +0.626565 q^{55} -10.3282 q^{57} +2.20341 q^{59} -8.97348 q^{61} +7.81112 q^{63} +3.94316 q^{67} +28.6698 q^{69} +12.2041 q^{71} -2.07244 q^{73} +30.2940 q^{75} -0.166194 q^{77} +0.710134 q^{79} +28.5802 q^{81} +8.92521 q^{83} +25.1246 q^{85} +8.23247 q^{87} -9.36913 q^{89} +6.81371 q^{93} +11.8423 q^{95} -15.9901 q^{97} -1.29816 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} + 6 q^{7} + 20 q^{9} + 4 q^{11} + 3 q^{15} - 3 q^{17} + 5 q^{19} + 6 q^{21} + 18 q^{23} + 2 q^{25} + 24 q^{27} + 17 q^{29} - 17 q^{31} + 21 q^{33} + q^{37} + 14 q^{41} + 14 q^{43} - 29 q^{45}+ \cdots + 3 q^{99}+O(q^{100}) $ 6 * q + 6 * q^3 + 6 * q^7 + 20 * q^9 + 4 * q^11 + 3 * q^15 - 3 * q^17 + 5 * q^19 + 6 * q^21 + 18 * q^23 + 2 * q^25 + 24 * q^27 + 17 * q^29 - 17 * q^31 + 21 * q^33 + q^37 + 14 * q^41 + 14 * q^43 - 29 * q^45 - 6 * q^47 + 6 * q^49 + 2 * q^51 + 3 * q^53 + 18 * q^55 - 14 * q^59 - 4 * q^61 + 20 * q^63 + 4 * q^67 + 39 * q^69 + 3 * q^71 - 31 * q^73 + 37 * q^75 + 4 * q^77 + 22 * q^79 + 6 * q^81 - 15 * q^83 + 51 * q^85 + 51 * q^87 - 20 * q^89 + 13 * q^93 + 19 * q^95 - 29 * 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
$3$	3.28803	1.89834	0.949171	−	0.314760i	$-0.101924\pi$
$3$	3.28803	1.89834	0.949171	+	0.314760i	$0.101924\pi$
$4$	0	0
$5$	−3.77007	−1.68603	−0.843013	−	0.537893i	$-0.819221\pi$
$5$	−3.77007	−1.68603	−0.843013	+	0.537893i	$0.819221\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	7.81112	2.60371
$10$	0	0
$11$	−0.166194	−0.0501095	−0.0250548	−	0.999686i	$-0.507976\pi$
$11$	−0.166194	−0.0501095	−0.0250548	+	0.999686i	$0.507976\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−12.3961	−3.20066
$16$	0	0
$17$	−6.66424	−1.61632	−0.808158	−	0.588966i	$-0.799535\pi$
$17$	−6.66424	−1.61632	−0.808158	+	0.588966i	$0.799535\pi$
$18$	0	0
$19$	−3.14115	−0.720629	−0.360314	−	0.932831i	$-0.617331\pi$
$19$	−3.14115	−0.720629	−0.360314	+	0.932831i	$0.617331\pi$
$20$	0	0
$21$	3.28803	0.717506
$22$	0	0
$23$	8.71947	1.81813	0.909067	−	0.416649i	$-0.136796\pi$
$23$	8.71947	1.81813	0.909067	+	0.416649i	$0.136796\pi$
$24$	0	0
$25$	9.21343	1.84269
$26$	0	0
$27$	15.8191	3.04438
$28$	0	0
$29$	2.50377	0.464939	0.232470	−	0.972604i	$-0.425319\pi$
$29$	2.50377	0.464939	0.232470	+	0.972604i	$0.425319\pi$
$30$	0	0
$31$	2.07228	0.372192	0.186096	−	0.982532i	$-0.440416\pi$
$31$	2.07228	0.372192	0.186096	+	0.982532i	$0.440416\pi$
$32$	0	0
$33$	−0.546452	−0.0951250
$34$	0	0
$35$	−3.77007	−0.637258
$36$	0	0
$37$	5.24067	0.861560	0.430780	−	0.902457i	$-0.358238\pi$
$37$	5.24067	0.861560	0.430780	+	0.902457i	$0.358238\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.25901	0.977493	0.488746	−	0.872426i	$-0.337454\pi$
$41$	6.25901	0.977493	0.488746	+	0.872426i	$0.337454\pi$
$42$	0	0
$43$	−3.39957	−0.518430	−0.259215	−	0.965820i	$-0.583464\pi$
$43$	−3.39957	−0.518430	−0.259215	+	0.965820i	$0.583464\pi$
$44$	0	0
$45$	−29.4485	−4.38992
$46$	0	0
$47$	8.74103	1.27501	0.637505	−	0.770446i	$-0.279967\pi$
$47$	8.74103	1.27501	0.637505	+	0.770446i	$0.279967\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−21.9122	−3.06832
$52$	0	0
$53$	5.35058	0.734959	0.367480	−	0.930032i	$-0.380221\pi$
$53$	5.35058	0.734959	0.367480	+	0.930032i	$0.380221\pi$
$54$	0	0
$55$	0.626565	0.0844859
$56$	0	0
$57$	−10.3282	−1.36800
$58$	0	0
$59$	2.20341	0.286859	0.143430	−	0.989661i	$-0.454187\pi$
$59$	2.20341	0.286859	0.143430	+	0.989661i	$0.454187\pi$
$60$	0	0
$61$	−8.97348	−1.14894	−0.574468	−	0.818527i	$-0.694791\pi$
$61$	−8.97348	−1.14894	−0.574468	+	0.818527i	$0.694791\pi$
$62$	0	0
$63$	7.81112	0.984108
$64$	0	0
$65$	0	0
$66$	0	0
$67$	3.94316	0.481733	0.240867	−	0.970558i	$-0.422568\pi$
$67$	3.94316	0.481733	0.240867	+	0.970558i	$0.422568\pi$
$68$	0	0
$69$	28.6698	3.45144
$70$	0	0
$71$	12.2041	1.44836	0.724179	−	0.689612i	$-0.242219\pi$
$71$	12.2041	1.44836	0.724179	+	0.689612i	$0.242219\pi$
$72$	0	0
$73$	−2.07244	−0.242561	−0.121281	−	0.992618i	$-0.538700\pi$
$73$	−2.07244	−0.242561	−0.121281	+	0.992618i	$0.538700\pi$
$74$	0	0
$75$	30.2940	3.49805
$76$	0	0
$77$	−0.166194	−0.0189396
$78$	0	0
$79$	0.710134	0.0798963	0.0399481	−	0.999202i	$-0.487281\pi$
$79$	0.710134	0.0798963	0.0399481	+	0.999202i	$0.487281\pi$
$80$	0	0
$81$	28.5802	3.17558
$82$	0	0
$83$	8.92521	0.979669	0.489834	−	0.871815i	$-0.337057\pi$
$83$	8.92521	0.979669	0.489834	+	0.871815i	$0.337057\pi$
$84$	0	0
$85$	25.1246	2.72515
$86$	0	0
$87$	8.23247	0.882614
$88$	0	0
$89$	−9.36913	−0.993126	−0.496563	−	0.868001i	$-0.665405\pi$
$89$	−9.36913	−0.993126	−0.496563	+	0.868001i	$0.665405\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	6.81371	0.706548
$94$	0	0
$95$	11.8423	1.21500
$96$	0	0
$97$	−15.9901	−1.62355	−0.811775	−	0.583971i	$-0.801498\pi$
$97$	−15.9901	−1.62355	−0.811775	+	0.583971i	$0.801498\pi$
$98$	0	0
$99$	−1.29816	−0.130470
$100$	0	0
$101$	18.8716	1.87779	0.938897	−	0.344198i	$-0.111849\pi$
$101$	18.8716	1.87779	0.938897	+	0.344198i	$0.111849\pi$
$102$	0	0
$103$	1.32073	0.130136	0.0650678	−	0.997881i	$-0.479274\pi$
$103$	1.32073	0.130136	0.0650678	+	0.997881i	$0.479274\pi$
$104$	0	0
$105$	−12.3961	−1.20973
$106$	0	0
$107$	15.6517	1.51311	0.756553	−	0.653932i	$-0.226882\pi$
$107$	15.6517	1.51311	0.756553	+	0.653932i	$0.226882\pi$
$108$	0	0
$109$	16.2365	1.55518	0.777588	−	0.628775i	$-0.216443\pi$
$109$	16.2365	1.55518	0.777588	+	0.628775i	$0.216443\pi$
$110$	0	0
$111$	17.2315	1.63554
$112$	0	0
$113$	−3.47410	−0.326816	−0.163408	−	0.986559i	$-0.552249\pi$
$113$	−3.47410	−0.326816	−0.163408	+	0.986559i	$0.552249\pi$
$114$	0	0
$115$	−32.8730	−3.06542
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.66424	−0.610910
$120$	0	0
$121$	−10.9724	−0.997489
$122$	0	0
$123$	20.5798	1.85562
$124$	0	0
$125$	−15.8849	−1.42079
$126$	0	0
$127$	−14.8850	−1.32083	−0.660417	−	0.750899i	$-0.729620\pi$
$127$	−14.8850	−1.32083	−0.660417	+	0.750899i	$0.729620\pi$
$128$	0	0
$129$	−11.1779	−0.984158
$130$	0	0
$131$	−14.1906	−1.23984	−0.619920	−	0.784665i	$-0.712835\pi$
$131$	−14.1906	−1.23984	−0.619920	+	0.784665i	$0.712835\pi$
$132$	0	0
$133$	−3.14115	−0.272372
$134$	0	0
$135$	−59.6390	−5.13291
$136$	0	0
$137$	15.6016	1.33294	0.666468	−	0.745534i	$-0.267805\pi$
$137$	15.6016	1.33294	0.666468	+	0.745534i	$0.267805\pi$
$138$	0	0
$139$	−3.43170	−0.291073	−0.145537	−	0.989353i	$-0.546491\pi$
$139$	−3.43170	−0.291073	−0.145537	+	0.989353i	$0.546491\pi$
$140$	0	0
$141$	28.7407	2.42041
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−9.43940	−0.783900
$146$	0	0
$147$	3.28803	0.271192
$148$	0	0
$149$	−7.87139	−0.644850	−0.322425	−	0.946595i	$-0.604498\pi$
$149$	−7.87139	−0.644850	−0.322425	+	0.946595i	$0.604498\pi$
$150$	0	0
$151$	−18.0392	−1.46801	−0.734005	−	0.679144i	$-0.762351\pi$
$151$	−18.0392	−1.46801	−0.734005	+	0.679144i	$0.762351\pi$
$152$	0	0
$153$	−52.0551	−4.20841
$154$	0	0
$155$	−7.81264	−0.627526
$156$	0	0
$157$	11.9607	0.954568	0.477284	−	0.878749i	$-0.341621\pi$
$157$	11.9607	0.954568	0.477284	+	0.878749i	$0.341621\pi$
$158$	0	0
$159$	17.5929	1.39520
$160$	0	0
$161$	8.71947	0.687190
$162$	0	0
$163$	−5.51608	−0.432053	−0.216027	−	0.976387i	$-0.569310\pi$
$163$	−5.51608	−0.432053	−0.216027	+	0.976387i	$0.569310\pi$
$164$	0	0
$165$	2.06016	0.160383
$166$	0	0
$167$	5.35124	0.414091	0.207046	−	0.978331i	$-0.433615\pi$
$167$	5.35124	0.414091	0.207046	+	0.978331i	$0.433615\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−24.5359	−1.87631
$172$	0	0
$173$	−4.67653	−0.355550	−0.177775	−	0.984071i	$-0.556890\pi$
$173$	−4.67653	−0.355550	−0.177775	+	0.984071i	$0.556890\pi$
$174$	0	0
$175$	9.21343	0.696470
$176$	0	0
$177$	7.24486	0.544557
$178$	0	0
$179$	19.2036	1.43535	0.717673	−	0.696381i	$-0.245207\pi$
$179$	19.2036	1.43535	0.717673	+	0.696381i	$0.245207\pi$
$180$	0	0
$181$	10.7540	0.799338	0.399669	−	0.916660i	$-0.369125\pi$
$181$	10.7540	0.799338	0.399669	+	0.916660i	$0.369125\pi$
$182$	0	0
$183$	−29.5050	−2.18107
$184$	0	0
$185$	−19.7577	−1.45261
$186$	0	0
$187$	1.10756	0.0809928
$188$	0	0
$189$	15.8191	1.15067
$190$	0	0
$191$	2.70445	0.195687	0.0978435	−	0.995202i	$-0.468806\pi$
$191$	2.70445	0.195687	0.0978435	+	0.995202i	$0.468806\pi$
$192$	0	0
$193$	−7.24910	−0.521802	−0.260901	−	0.965366i	$-0.584020\pi$
$193$	−7.24910	−0.521802	−0.260901	+	0.965366i	$0.584020\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	15.8259	1.12755	0.563775	−	0.825928i	$-0.309348\pi$
$197$	15.8259	1.12755	0.563775	+	0.825928i	$0.309348\pi$
$198$	0	0
$199$	1.98977	0.141051	0.0705254	−	0.997510i	$-0.477532\pi$
$199$	1.98977	0.141051	0.0705254	+	0.997510i	$0.477532\pi$
$200$	0	0
$201$	12.9652	0.914495
$202$	0	0
$203$	2.50377	0.175730
$204$	0	0
$205$	−23.5969	−1.64808
$206$	0	0
$207$	68.1088	4.73389
$208$	0	0
$209$	0.522041	0.0361104
$210$	0	0
$211$	−4.77371	−0.328636	−0.164318	−	0.986407i	$-0.552542\pi$
$211$	−4.77371	−0.328636	−0.164318	+	0.986407i	$0.552542\pi$
$212$	0	0
$213$	40.1273	2.74948
$214$	0	0
$215$	12.8166	0.874087
$216$	0	0
$217$	2.07228	0.140675
$218$	0	0
$219$	−6.81425	−0.460465
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−24.4090	−1.63455	−0.817274	−	0.576250i	$-0.804516\pi$
$223$	−24.4090	−1.63455	−0.817274	+	0.576250i	$0.804516\pi$
$224$	0	0
$225$	71.9672	4.79781
$226$	0	0
$227$	−22.1495	−1.47011	−0.735056	−	0.678006i	$-0.762844\pi$
$227$	−22.1495	−1.47011	−0.735056	+	0.678006i	$0.762844\pi$
$228$	0	0
$229$	11.0363	0.729302	0.364651	−	0.931144i	$-0.381188\pi$
$229$	11.0363	0.729302	0.364651	+	0.931144i	$0.381188\pi$
$230$	0	0
$231$	−0.546452	−0.0359539
$232$	0	0
$233$	−17.3066	−1.13380	−0.566898	−	0.823788i	$-0.691857\pi$
$233$	−17.3066	−1.13380	−0.566898	+	0.823788i	$0.691857\pi$
$234$	0	0
$235$	−32.9543	−2.14970
$236$	0	0
$237$	2.33494	0.151671
$238$	0	0
$239$	−1.90457	−0.123196	−0.0615981	−	0.998101i	$-0.519620\pi$
$239$	−1.90457	−0.123196	−0.0615981	+	0.998101i	$0.519620\pi$
$240$	0	0
$241$	2.30854	0.148706	0.0743530	−	0.997232i	$-0.476311\pi$
$241$	2.30854	0.148706	0.0743530	+	0.997232i	$0.476311\pi$
$242$	0	0
$243$	46.5152	2.98395
$244$	0	0
$245$	−3.77007	−0.240861
$246$	0	0
$247$	0	0
$248$	0	0
$249$	29.3463	1.85975
$250$	0	0
$251$	−2.97612	−0.187851	−0.0939254	−	0.995579i	$-0.529942\pi$
$251$	−2.97612	−0.187851	−0.0939254	+	0.995579i	$0.529942\pi$
$252$	0	0
$253$	−1.44913	−0.0911058
$254$	0	0
$255$	82.6105	5.17327
$256$	0	0
$257$	−10.9271	−0.681615	−0.340808	−	0.940133i	$-0.610700\pi$
$257$	−10.9271	−0.681615	−0.340808	+	0.940133i	$0.610700\pi$
$258$	0	0
$259$	5.24067	0.325639
$260$	0	0
$261$	19.5573	1.21056
$262$	0	0
$263$	−12.9704	−0.799789	−0.399895	−	0.916561i	$-0.630953\pi$
$263$	−12.9704	−0.799789	−0.399895	+	0.916561i	$0.630953\pi$
$264$	0	0
$265$	−20.1721	−1.23916
$266$	0	0
$267$	−30.8060	−1.88529
$268$	0	0
$269$	6.74714	0.411380	0.205690	−	0.978617i	$-0.434056\pi$
$269$	6.74714	0.411380	0.205690	+	0.978617i	$0.434056\pi$
$270$	0	0
$271$	−2.71741	−0.165071	−0.0825355	−	0.996588i	$-0.526302\pi$
$271$	−2.71741	−0.165071	−0.0825355	+	0.996588i	$0.526302\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.53122	−0.0923360
$276$	0	0
$277$	6.25873	0.376051	0.188025	−	0.982164i	$-0.439791\pi$
$277$	6.25873	0.376051	0.188025	+	0.982164i	$0.439791\pi$
$278$	0	0
$279$	16.1868	0.969079
$280$	0	0
$281$	29.2013	1.74200	0.871002	−	0.491279i	$-0.163471\pi$
$281$	29.2013	1.74200	0.871002	+	0.491279i	$0.163471\pi$
$282$	0	0
$283$	2.80256	0.166595	0.0832975	−	0.996525i	$-0.473455\pi$
$283$	2.80256	0.166595	0.0832975	+	0.996525i	$0.473455\pi$
$284$	0	0
$285$	38.9380	2.30649
$286$	0	0
$287$	6.25901	0.369458
$288$	0	0
$289$	27.4121	1.61248
$290$	0	0
$291$	−52.5759	−3.08205
$292$	0	0
$293$	3.46125	0.202208	0.101104	−	0.994876i	$-0.467762\pi$
$293$	3.46125	0.202208	0.101104	+	0.994876i	$0.467762\pi$
$294$	0	0
$295$	−8.30700	−0.483652
$296$	0	0
$297$	−2.62904	−0.152553
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−3.39957	−0.195948
$302$	0	0
$303$	62.0503	3.56470
$304$	0	0
$305$	33.8306	1.93714
$306$	0	0
$307$	−16.5672	−0.945542	−0.472771	−	0.881185i	$-0.656746\pi$
$307$	−16.5672	−0.945542	−0.472771	+	0.881185i	$0.656746\pi$
$308$	0	0
$309$	4.34260	0.247042
$310$	0	0
$311$	−7.46655	−0.423389	−0.211695	−	0.977336i	$-0.567898\pi$
$311$	−7.46655	−0.423389	−0.211695	+	0.977336i	$0.567898\pi$
$312$	0	0
$313$	−6.38185	−0.360723	−0.180362	−	0.983600i	$-0.557727\pi$
$313$	−6.38185	−0.360723	−0.180362	+	0.983600i	$0.557727\pi$
$314$	0	0
$315$	−29.4485	−1.65923
$316$	0	0
$317$	8.19850	0.460474	0.230237	−	0.973135i	$-0.426050\pi$
$317$	8.19850	0.460474	0.230237	+	0.973135i	$0.426050\pi$
$318$	0	0
$319$	−0.416113	−0.0232979
$320$	0	0
$321$	51.4632	2.87239
$322$	0	0
$323$	20.9334	1.16476
$324$	0	0
$325$	0	0
$326$	0	0
$327$	53.3861	2.95226
$328$	0	0
$329$	8.74103	0.481909
$330$	0	0
$331$	23.8711	1.31208	0.656039	−	0.754727i	$-0.272231\pi$
$331$	23.8711	1.31208	0.656039	+	0.754727i	$0.272231\pi$
$332$	0	0
$333$	40.9355	2.24325
$334$	0	0
$335$	−14.8660	−0.812215
$336$	0	0
$337$	−14.2185	−0.774534	−0.387267	−	0.921968i	$-0.626581\pi$
$337$	−14.2185	−0.774534	−0.387267	+	0.921968i	$0.626581\pi$
$338$	0	0
$339$	−11.4229	−0.620409
$340$	0	0
$341$	−0.344401	−0.0186504
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−108.087	−5.81922
$346$	0	0
$347$	−15.5065	−0.832432	−0.416216	−	0.909266i	$-0.636644\pi$
$347$	−15.5065	−0.832432	−0.416216	+	0.909266i	$0.636644\pi$
$348$	0	0
$349$	−13.2890	−0.711346	−0.355673	−	0.934610i	$-0.615748\pi$
$349$	−13.2890	−0.711346	−0.355673	+	0.934610i	$0.615748\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	26.1697	1.39287	0.696437	−	0.717618i	$-0.254768\pi$
$353$	26.1697	1.39287	0.696437	+	0.717618i	$0.254768\pi$
$354$	0	0
$355$	−46.0102	−2.44197
$356$	0	0
$357$	−21.9122	−1.15972
$358$	0	0
$359$	−24.0498	−1.26930	−0.634649	−	0.772800i	$-0.718855\pi$
$359$	−24.0498	−1.26930	−0.634649	+	0.772800i	$0.718855\pi$
$360$	0	0
$361$	−9.13319	−0.480694
$362$	0	0
$363$	−36.0775	−1.89358
$364$	0	0
$365$	7.81326	0.408965
$366$	0	0
$367$	5.27863	0.275542	0.137771	−	0.990464i	$-0.456006\pi$
$367$	5.27863	0.275542	0.137771	+	0.990464i	$0.456006\pi$
$368$	0	0
$369$	48.8898	2.54510
$370$	0	0
$371$	5.35058	0.277788
$372$	0	0
$373$	20.0485	1.03807	0.519036	−	0.854752i	$-0.326291\pi$
$373$	20.0485	1.03807	0.519036	+	0.854752i	$0.326291\pi$
$374$	0	0
$375$	−52.2300	−2.69715
$376$	0	0
$377$	0	0
$378$	0	0
$379$	19.2760	0.990142	0.495071	−	0.868852i	$-0.335142\pi$
$379$	19.2760	0.990142	0.495071	+	0.868852i	$0.335142\pi$
$380$	0	0
$381$	−48.9424	−2.50740
$382$	0	0
$383$	−20.0986	−1.02699	−0.513495	−	0.858093i	$-0.671650\pi$
$383$	−20.0986	−1.02699	−0.513495	+	0.858093i	$0.671650\pi$
$384$	0	0
$385$	0.626565	0.0319327
$386$	0	0
$387$	−26.5545	−1.34984
$388$	0	0
$389$	−7.32852	−0.371571	−0.185785	−	0.982590i	$-0.559483\pi$
$389$	−7.32852	−0.371571	−0.185785	+	0.982590i	$0.559483\pi$
$390$	0	0
$391$	−58.1086	−2.93868
$392$	0	0
$393$	−46.6591	−2.35364
$394$	0	0
$395$	−2.67725	−0.134707
$396$	0	0
$397$	23.3190	1.17035	0.585175	−	0.810907i	$-0.301026\pi$
$397$	23.3190	1.17035	0.585175	+	0.810907i	$0.301026\pi$
$398$	0	0
$399$	−10.3282	−0.517056
$400$	0	0
$401$	3.62784	0.181166	0.0905829	−	0.995889i	$-0.471127\pi$
$401$	3.62784	0.181166	0.0905829	+	0.995889i	$0.471127\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−107.749	−5.35411
$406$	0	0
$407$	−0.870970	−0.0431724
$408$	0	0
$409$	16.5099	0.816364	0.408182	−	0.912901i	$-0.366163\pi$
$409$	16.5099	0.816364	0.408182	+	0.912901i	$0.366163\pi$
$410$	0	0
$411$	51.2985	2.53037
$412$	0	0
$413$	2.20341	0.108423
$414$	0	0
$415$	−33.6487	−1.65175
$416$	0	0
$417$	−11.2835	−0.552556
$418$	0	0
$419$	1.39271	0.0680383	0.0340191	−	0.999421i	$-0.489169\pi$
$419$	1.39271	0.0680383	0.0340191	+	0.999421i	$0.489169\pi$
$420$	0	0
$421$	3.31700	0.161661	0.0808303	−	0.996728i	$-0.474243\pi$
$421$	3.31700	0.161661	0.0808303	+	0.996728i	$0.474243\pi$
$422$	0	0
$423$	68.2772	3.31975
$424$	0	0
$425$	−61.4005	−2.97836
$426$	0	0
$427$	−8.97348	−0.434257
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−8.72035	−0.420045	−0.210022	−	0.977697i	$-0.567354\pi$
$431$	−8.72035	−0.420045	−0.210022	+	0.977697i	$0.567354\pi$
$432$	0	0
$433$	−11.4153	−0.548585	−0.274293	−	0.961646i	$-0.588444\pi$
$433$	−11.4153	−0.548585	−0.274293	+	0.961646i	$0.588444\pi$
$434$	0	0
$435$	−31.0370	−1.48811
$436$	0	0
$437$	−27.3891	−1.31020
$438$	0	0
$439$	−15.8781	−0.757822	−0.378911	−	0.925433i	$-0.623701\pi$
$439$	−15.8781	−0.757822	−0.378911	+	0.925433i	$0.623701\pi$
$440$	0	0
$441$	7.81112	0.371958
$442$	0	0
$443$	5.51966	0.262247	0.131123	−	0.991366i	$-0.458142\pi$
$443$	5.51966	0.262247	0.131123	+	0.991366i	$0.458142\pi$
$444$	0	0
$445$	35.3223	1.67444
$446$	0	0
$447$	−25.8813	−1.22415
$448$	0	0
$449$	−1.60128	−0.0755692	−0.0377846	−	0.999286i	$-0.512030\pi$
$449$	−1.60128	−0.0755692	−0.0377846	+	0.999286i	$0.512030\pi$
$450$	0	0
$451$	−1.04021	−0.0489817
$452$	0	0
$453$	−59.3134	−2.78679
$454$	0	0
$455$	0	0
$456$	0	0
$457$	9.23757	0.432115	0.216058	−	0.976381i	$-0.430680\pi$
$457$	9.23757	0.432115	0.216058	+	0.976381i	$0.430680\pi$
$458$	0	0
$459$	−105.422	−4.92068
$460$	0	0
$461$	−14.7131	−0.685256	−0.342628	−	0.939471i	$-0.611317\pi$
$461$	−14.7131	−0.685256	−0.342628	+	0.939471i	$0.611317\pi$
$462$	0	0
$463$	−41.2688	−1.91792	−0.958961	−	0.283539i	$-0.908492\pi$
$463$	−41.2688	−1.91792	−0.958961	+	0.283539i	$0.908492\pi$
$464$	0	0
$465$	−25.6881	−1.19126
$466$	0	0
$467$	10.0035	0.462907	0.231453	−	0.972846i	$-0.425652\pi$
$467$	10.0035	0.462907	0.231453	+	0.972846i	$0.425652\pi$
$468$	0	0
$469$	3.94316	0.182078
$470$	0	0
$471$	39.3271	1.81210
$472$	0	0
$473$	0.564990	0.0259783
$474$	0	0
$475$	−28.9407	−1.32789
$476$	0	0
$477$	41.7940	1.91362
$478$	0	0
$479$	−17.1559	−0.783873	−0.391937	−	0.919992i	$-0.628195\pi$
$479$	−17.1559	−0.783873	−0.391937	+	0.919992i	$0.628195\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	28.6698	1.30452
$484$	0	0
$485$	60.2838	2.73735
$486$	0	0
$487$	8.08467	0.366351	0.183176	−	0.983080i	$-0.441362\pi$
$487$	8.08467	0.366351	0.183176	+	0.983080i	$0.441362\pi$
$488$	0	0
$489$	−18.1370	−0.820185
$490$	0	0
$491$	−5.79104	−0.261346	−0.130673	−	0.991426i	$-0.541714\pi$
$491$	−5.79104	−0.261346	−0.130673	+	0.991426i	$0.541714\pi$
$492$	0	0
$493$	−16.6857	−0.751488
$494$	0	0
$495$	4.89417	0.219977
$496$	0	0
$497$	12.2041	0.547428
$498$	0	0
$499$	−13.3966	−0.599714	−0.299857	−	0.953984i	$-0.596939\pi$
$499$	−13.3966	−0.599714	−0.299857	+	0.953984i	$0.596939\pi$
$500$	0	0
$501$	17.5950	0.786087
$502$	0	0
$503$	3.23034	0.144034	0.0720168	−	0.997403i	$-0.477056\pi$
$503$	3.23034	0.144034	0.0720168	+	0.997403i	$0.477056\pi$
$504$	0	0
$505$	−71.1472	−3.16601
$506$	0	0
$507$	0	0
$508$	0	0
$509$	21.3664	0.947049	0.473524	−	0.880781i	$-0.342982\pi$
$509$	21.3664	0.947049	0.473524	+	0.880781i	$0.342982\pi$
$510$	0	0
$511$	−2.07244	−0.0916796
$512$	0	0
$513$	−49.6901	−2.19387
$514$	0	0
$515$	−4.97925	−0.219412
$516$	0	0
$517$	−1.45271	−0.0638901
$518$	0	0
$519$	−15.3765	−0.674955
$520$	0	0
$521$	−32.2857	−1.41446	−0.707231	−	0.706983i	$-0.750056\pi$
$521$	−32.2857	−1.41446	−0.707231	+	0.706983i	$0.750056\pi$
$522$	0	0
$523$	2.46204	0.107658	0.0538288	−	0.998550i	$-0.482857\pi$
$523$	2.46204	0.107658	0.0538288	+	0.998550i	$0.482857\pi$
$524$	0	0
$525$	30.2940	1.32214
$526$	0	0
$527$	−13.8102	−0.601580
$528$	0	0
$529$	53.0291	2.30561
$530$	0	0
$531$	17.2111	0.746897
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−59.0080	−2.55114
$536$	0	0
$537$	63.1420	2.72478
$538$	0	0
$539$	−0.166194	−0.00715850
$540$	0	0
$541$	−23.9903	−1.03142	−0.515711	−	0.856762i	$-0.672472\pi$
$541$	−23.9903	−1.03142	−0.515711	+	0.856762i	$0.672472\pi$
$542$	0	0
$543$	35.3594	1.51742
$544$	0	0
$545$	−61.2128	−2.62207
$546$	0	0
$547$	36.2013	1.54786	0.773929	−	0.633273i	$-0.218289\pi$
$547$	36.2013	1.54786	0.773929	+	0.633273i	$0.218289\pi$
$548$	0	0
$549$	−70.0929	−2.99149
$550$	0	0
$551$	−7.86472	−0.335049
$552$	0	0
$553$	0.710134	0.0301980
$554$	0	0
$555$	−64.9638	−2.75756
$556$	0	0
$557$	−1.48127	−0.0627636	−0.0313818	−	0.999507i	$-0.509991\pi$
$557$	−1.48127	−0.0627636	−0.0313818	+	0.999507i	$0.509991\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	3.64168	0.153752
$562$	0	0
$563$	13.3255	0.561602	0.280801	−	0.959766i	$-0.409400\pi$
$563$	13.3255	0.561602	0.280801	+	0.959766i	$0.409400\pi$
$564$	0	0
$565$	13.0976	0.551020
$566$	0	0
$567$	28.5802	1.20026
$568$	0	0
$569$	−9.37782	−0.393139	−0.196569	−	0.980490i	$-0.562980\pi$
$569$	−9.37782	−0.393139	−0.196569	+	0.980490i	$0.562980\pi$
$570$	0	0
$571$	4.70554	0.196921	0.0984603	−	0.995141i	$-0.468608\pi$
$571$	4.70554	0.196921	0.0984603	+	0.995141i	$0.468608\pi$
$572$	0	0
$573$	8.89229	0.371481
$574$	0	0
$575$	80.3362	3.35025
$576$	0	0
$577$	41.4130	1.72405	0.862023	−	0.506870i	$-0.169197\pi$
$577$	41.4130	1.72405	0.862023	+	0.506870i	$0.169197\pi$
$578$	0	0
$579$	−23.8352	−0.990559
$580$	0	0
$581$	8.92521	0.370280
$582$	0	0
$583$	−0.889237	−0.0368284
$584$	0	0
$585$	0	0
$586$	0	0
$587$	22.7773	0.940120	0.470060	−	0.882634i	$-0.344232\pi$
$587$	22.7773	0.940120	0.470060	+	0.882634i	$0.344232\pi$
$588$	0	0
$589$	−6.50933	−0.268212
$590$	0	0
$591$	52.0360	2.14048
$592$	0	0
$593$	−30.3573	−1.24662	−0.623311	−	0.781974i	$-0.714213\pi$
$593$	−30.3573	−1.24662	−0.623311	+	0.781974i	$0.714213\pi$
$594$	0	0
$595$	25.1246	1.03001
$596$	0	0
$597$	6.54240	0.267763
$598$	0	0
$599$	14.9609	0.611287	0.305643	−	0.952146i	$-0.401128\pi$
$599$	14.9609	0.611287	0.305643	+	0.952146i	$0.401128\pi$
$600$	0	0
$601$	9.88738	0.403315	0.201657	−	0.979456i	$-0.435367\pi$
$601$	9.88738	0.403315	0.201657	+	0.979456i	$0.435367\pi$
$602$	0	0
$603$	30.8005	1.25429
$604$	0	0
$605$	41.3666	1.68179
$606$	0	0
$607$	19.0570	0.773502	0.386751	−	0.922184i	$-0.373597\pi$
$607$	19.0570	0.773502	0.386751	+	0.922184i	$0.373597\pi$
$608$	0	0
$609$	8.23247	0.333597
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−11.3334	−0.457752	−0.228876	−	0.973456i	$-0.573505\pi$
$613$	−11.3334	−0.457752	−0.228876	+	0.973456i	$0.573505\pi$
$614$	0	0
$615$	−77.5872	−3.12862
$616$	0	0
$617$	−18.4204	−0.741577	−0.370788	−	0.928717i	$-0.620913\pi$
$617$	−18.4204	−0.741577	−0.370788	+	0.928717i	$0.620913\pi$
$618$	0	0
$619$	−27.3881	−1.10082	−0.550411	−	0.834894i	$-0.685529\pi$
$619$	−27.3881	−1.10082	−0.550411	+	0.834894i	$0.685529\pi$
$620$	0	0
$621$	137.934	5.53510
$622$	0	0
$623$	−9.36913	−0.375366
$624$	0	0
$625$	13.8201	0.552804
$626$	0	0
$627$	1.71649	0.0685498
$628$	0	0
$629$	−34.9251	−1.39255
$630$	0	0
$631$	−41.8121	−1.66451	−0.832257	−	0.554389i	$-0.812952\pi$
$631$	−41.8121	−1.66451	−0.832257	+	0.554389i	$0.812952\pi$
$632$	0	0
$633$	−15.6961	−0.623864
$634$	0	0
$635$	56.1177	2.22696
$636$	0	0
$637$	0	0
$638$	0	0
$639$	95.3275	3.77110
$640$	0	0
$641$	−34.0360	−1.34434	−0.672170	−	0.740397i	$-0.734638\pi$
$641$	−34.0360	−1.34434	−0.672170	+	0.740397i	$0.734638\pi$
$642$	0	0
$643$	−3.51097	−0.138459	−0.0692295	−	0.997601i	$-0.522054\pi$
$643$	−3.51097	−0.138459	−0.0692295	+	0.997601i	$0.522054\pi$
$644$	0	0
$645$	42.1414	1.65932
$646$	0	0
$647$	38.4112	1.51010	0.755049	−	0.655668i	$-0.227613\pi$
$647$	38.4112	1.51010	0.755049	+	0.655668i	$0.227613\pi$
$648$	0	0
$649$	−0.366194	−0.0143744
$650$	0	0
$651$	6.81371	0.267050
$652$	0	0
$653$	−32.3895	−1.26750	−0.633749	−	0.773539i	$-0.718485\pi$
$653$	−32.3895	−1.26750	−0.633749	+	0.773539i	$0.718485\pi$
$654$	0	0
$655$	53.4996	2.09040
$656$	0	0
$657$	−16.1881	−0.631558
$658$	0	0
$659$	−47.3291	−1.84368	−0.921840	−	0.387570i	$-0.873315\pi$
$659$	−47.3291	−1.84368	−0.921840	+	0.387570i	$0.873315\pi$
$660$	0	0
$661$	17.6770	0.687554	0.343777	−	0.939051i	$-0.388294\pi$
$661$	17.6770	0.687554	0.343777	+	0.939051i	$0.388294\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	11.8423	0.459227
$666$	0	0
$667$	21.8316	0.845322
$668$	0	0
$669$	−80.2574	−3.10293
$670$	0	0
$671$	1.49134	0.0575726
$672$	0	0
$673$	−36.2939	−1.39903	−0.699513	−	0.714620i	$-0.746600\pi$
$673$	−36.2939	−1.39903	−0.699513	+	0.714620i	$0.746600\pi$
$674$	0	0
$675$	145.748	5.60984
$676$	0	0
$677$	30.7764	1.18283	0.591416	−	0.806367i	$-0.298569\pi$
$677$	30.7764	1.18283	0.591416	+	0.806367i	$0.298569\pi$
$678$	0	0
$679$	−15.9901	−0.613644
$680$	0	0
$681$	−72.8281	−2.79078
$682$	0	0
$683$	−10.9755	−0.419966	−0.209983	−	0.977705i	$-0.567341\pi$
$683$	−10.9755	−0.419966	−0.209983	+	0.977705i	$0.567341\pi$
$684$	0	0
$685$	−58.8192	−2.24736
$686$	0	0
$687$	36.2878	1.38446
$688$	0	0
$689$	0	0
$690$	0	0
$691$	45.3865	1.72659	0.863293	−	0.504704i	$-0.168398\pi$
$691$	45.3865	1.72659	0.863293	+	0.504704i	$0.168398\pi$
$692$	0	0
$693$	−1.29816	−0.0493132
$694$	0	0
$695$	12.9378	0.490757
$696$	0	0
$697$	−41.7115	−1.57994
$698$	0	0
$699$	−56.9047	−2.15233
$700$	0	0
$701$	−23.8023	−0.898999	−0.449500	−	0.893281i	$-0.648398\pi$
$701$	−23.8023	−0.898999	−0.449500	+	0.893281i	$0.648398\pi$
$702$	0	0
$703$	−16.4617	−0.620865
$704$	0	0
$705$	−108.355	−4.08087
$706$	0	0
$707$	18.8716	0.709739
$708$	0	0
$709$	15.8067	0.593631	0.296816	−	0.954935i	$-0.404075\pi$
$709$	15.8067	0.593631	0.296816	+	0.954935i	$0.404075\pi$
$710$	0	0
$711$	5.54694	0.208026
$712$	0	0
$713$	18.0692	0.676696
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−6.26227	−0.233869
$718$	0	0
$719$	26.6935	0.995499	0.497749	−	0.867321i	$-0.334160\pi$
$719$	26.6935	0.995499	0.497749	+	0.867321i	$0.334160\pi$
$720$	0	0
$721$	1.32073	0.0491866
$722$	0	0
$723$	7.59053	0.282295
$724$	0	0
$725$	23.0683	0.856737
$726$	0	0
$727$	−17.8836	−0.663267	−0.331634	−	0.943408i	$-0.607600\pi$
$727$	−17.8836	−0.663267	−0.331634	+	0.943408i	$0.607600\pi$
$728$	0	0
$729$	67.2026	2.48898
$730$	0	0
$731$	22.6556	0.837947
$732$	0	0
$733$	−27.2625	−1.00696	−0.503481	−	0.864006i	$-0.667948\pi$
$733$	−27.2625	−1.00696	−0.503481	+	0.864006i	$0.667948\pi$
$734$	0	0
$735$	−12.3961	−0.457237
$736$	0	0
$737$	−0.655331	−0.0241394
$738$	0	0
$739$	37.7631	1.38914	0.694569	−	0.719426i	$-0.255595\pi$
$739$	37.7631	1.38914	0.694569	+	0.719426i	$0.255595\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−38.9179	−1.42776	−0.713880	−	0.700268i	$-0.753064\pi$
$743$	−38.9179	−1.42776	−0.713880	+	0.700268i	$0.753064\pi$
$744$	0	0
$745$	29.6757	1.08723
$746$	0	0
$747$	69.7158	2.55077
$748$	0	0
$749$	15.6517	0.571900
$750$	0	0
$751$	−11.3301	−0.413442	−0.206721	−	0.978400i	$-0.566279\pi$
$751$	−11.3301	−0.413442	−0.206721	+	0.978400i	$0.566279\pi$
$752$	0	0
$753$	−9.78555	−0.356605
$754$	0	0
$755$	68.0091	2.47510
$756$	0	0
$757$	−32.9459	−1.19744	−0.598720	−	0.800958i	$-0.704324\pi$
$757$	−32.9459	−1.19744	−0.598720	+	0.800958i	$0.704324\pi$
$758$	0	0
$759$	−4.76477	−0.172950
$760$	0	0
$761$	33.7013	1.22167	0.610835	−	0.791758i	$-0.290834\pi$
$761$	33.7013	1.22167	0.610835	+	0.791758i	$0.290834\pi$
$762$	0	0
$763$	16.2365	0.587801
$764$	0	0
$765$	196.252	7.09549
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−46.6993	−1.68402	−0.842010	−	0.539461i	$-0.818628\pi$
$769$	−46.6993	−1.68402	−0.842010	+	0.539461i	$0.818628\pi$
$770$	0	0
$771$	−35.9287	−1.29394
$772$	0	0
$773$	10.3745	0.373144	0.186572	−	0.982441i	$-0.440262\pi$
$773$	10.3745	0.373144	0.186572	+	0.982441i	$0.440262\pi$
$774$	0	0
$775$	19.0928	0.685833
$776$	0	0
$777$	17.2315	0.618175
$778$	0	0
$779$	−19.6605	−0.704409
$780$	0	0
$781$	−2.02825	−0.0725765
$782$	0	0
$783$	39.6074	1.41545
$784$	0	0
$785$	−45.0927	−1.60943
$786$	0	0
$787$	−16.2732	−0.580077	−0.290039	−	0.957015i	$-0.593668\pi$
$787$	−16.2732	−0.580077	−0.290039	+	0.957015i	$0.593668\pi$
$788$	0	0
$789$	−42.6470	−1.51827
$790$	0	0
$791$	−3.47410	−0.123525
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−66.3263	−2.35235
$796$	0	0
$797$	−15.8959	−0.563063	−0.281531	−	0.959552i	$-0.590842\pi$
$797$	−15.8959	−0.563063	−0.281531	+	0.959552i	$0.590842\pi$
$798$	0	0
$799$	−58.2523	−2.06082
$800$	0	0
$801$	−73.1834	−2.58581
$802$	0	0
$803$	0.344429	0.0121546
$804$	0	0
$805$	−32.8730	−1.15862
$806$	0	0
$807$	22.1848	0.780941
$808$	0	0
$809$	27.2970	0.959711	0.479856	−	0.877347i	$-0.340689\pi$
$809$	27.2970	0.959711	0.479856	+	0.877347i	$0.340689\pi$
$810$	0	0
$811$	−40.9306	−1.43727	−0.718635	−	0.695388i	$-0.755233\pi$
$811$	−40.9306	−1.43727	−0.718635	+	0.695388i	$0.755233\pi$
$812$	0	0
$813$	−8.93492	−0.313361
$814$	0	0
$815$	20.7960	0.728453
$816$	0	0
$817$	10.6786	0.373596
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−19.4546	−0.678970	−0.339485	−	0.940611i	$-0.610253\pi$
$821$	−19.4546	−0.678970	−0.339485	+	0.940611i	$0.610253\pi$
$822$	0	0
$823$	31.4806	1.09734	0.548672	−	0.836038i	$-0.315134\pi$
$823$	31.4806	1.09734	0.548672	+	0.836038i	$0.315134\pi$
$824$	0	0
$825$	−5.03469	−0.175285
$826$	0	0
$827$	−26.4656	−0.920299	−0.460150	−	0.887841i	$-0.652204\pi$
$827$	−26.4656	−0.920299	−0.460150	+	0.887841i	$0.652204\pi$
$828$	0	0
$829$	−15.7739	−0.547849	−0.273925	−	0.961751i	$-0.588322\pi$
$829$	−15.7739	−0.547849	−0.273925	+	0.961751i	$0.588322\pi$
$830$	0	0
$831$	20.5789	0.713873
$832$	0	0
$833$	−6.66424	−0.230902
$834$	0	0
$835$	−20.1745	−0.698169
$836$	0	0
$837$	32.7815	1.13310
$838$	0	0
$839$	41.3642	1.42805	0.714025	−	0.700120i	$-0.246870\pi$
$839$	41.3642	1.42805	0.714025	+	0.700120i	$0.246870\pi$
$840$	0	0
$841$	−22.7311	−0.783832
$842$	0	0
$843$	96.0147	3.30692
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−10.9724	−0.377015
$848$	0	0
$849$	9.21490	0.316254
$850$	0	0
$851$	45.6958	1.56643
$852$	0	0
$853$	8.48032	0.290361	0.145180	−	0.989405i	$-0.453624\pi$
$853$	8.48032	0.290361	0.145180	+	0.989405i	$0.453624\pi$
$854$	0	0
$855$	92.5020	3.16350
$856$	0	0
$857$	37.9916	1.29777	0.648884	−	0.760888i	$-0.275236\pi$
$857$	37.9916	1.29777	0.648884	+	0.760888i	$0.275236\pi$
$858$	0	0
$859$	−42.3745	−1.44580	−0.722900	−	0.690953i	$-0.757191\pi$
$859$	−42.3745	−1.44580	−0.722900	+	0.690953i	$0.757191\pi$
$860$	0	0
$861$	20.5798	0.701357
$862$	0	0
$863$	−15.0922	−0.513743	−0.256871	−	0.966446i	$-0.582692\pi$
$863$	−15.0922	−0.513743	−0.256871	+	0.966446i	$0.582692\pi$
$864$	0	0
$865$	17.6308	0.599466
$866$	0	0
$867$	90.1316	3.06103
$868$	0	0
$869$	−0.118020	−0.00400356
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−124.901	−4.22724
$874$	0	0
$875$	−15.8849	−0.537008
$876$	0	0
$877$	54.3132	1.83403	0.917014	−	0.398856i	$-0.130593\pi$
$877$	54.3132	1.83403	0.917014	+	0.398856i	$0.130593\pi$
$878$	0	0
$879$	11.3807	0.383861
$880$	0	0
$881$	16.1167	0.542987	0.271493	−	0.962440i	$-0.412483\pi$
$881$	16.1167	0.542987	0.271493	+	0.962440i	$0.412483\pi$
$882$	0	0
$883$	22.6462	0.762104	0.381052	−	0.924554i	$-0.375562\pi$
$883$	22.6462	0.762104	0.381052	+	0.924554i	$0.375562\pi$
$884$	0	0
$885$	−27.3136	−0.918138
$886$	0	0
$887$	21.1922	0.711565	0.355782	−	0.934569i	$-0.384214\pi$
$887$	21.1922	0.711565	0.355782	+	0.934569i	$0.384214\pi$
$888$	0	0
$889$	−14.8850	−0.499228
$890$	0	0
$891$	−4.74987	−0.159127
$892$	0	0
$893$	−27.4569	−0.918809
$894$	0	0
$895$	−72.3989	−2.42003
$896$	0	0
$897$	0	0
$898$	0	0
$899$	5.18852	0.173047
$900$	0	0
$901$	−35.6576	−1.18793
$902$	0	0
$903$	−11.1779	−0.371977
$904$	0	0
$905$	−40.5433	−1.34771
$906$	0	0
$907$	−14.7802	−0.490769	−0.245385	−	0.969426i	$-0.578914\pi$
$907$	−14.7802	−0.490769	−0.245385	+	0.969426i	$0.578914\pi$
$908$	0	0
$909$	147.408	4.88922
$910$	0	0
$911$	−32.6578	−1.08200	−0.541000	−	0.841022i	$-0.681954\pi$
$911$	−32.6578	−1.08200	−0.541000	+	0.841022i	$0.681954\pi$
$912$	0	0
$913$	−1.48332	−0.0490907
$914$	0	0
$915$	111.236	3.67735
$916$	0	0
$917$	−14.1906	−0.468616
$918$	0	0
$919$	21.8819	0.721817	0.360908	−	0.932601i	$-0.382467\pi$
$919$	21.8819	0.721817	0.360908	+	0.932601i	$0.382467\pi$
$920$	0	0
$921$	−54.4735	−1.79496
$922$	0	0
$923$	0	0
$924$	0	0
$925$	48.2845	1.58758
$926$	0	0
$927$	10.3164	0.338835
$928$	0	0
$929$	−60.1004	−1.97183	−0.985915	−	0.167249i	$-0.946511\pi$
$929$	−60.1004	−1.97183	−0.985915	+	0.167249i	$0.946511\pi$
$930$	0	0
$931$	−3.14115	−0.102947
$932$	0	0
$933$	−24.5502	−0.803738
$934$	0	0
$935$	−4.17558	−0.136556
$936$	0	0
$937$	57.1562	1.86721	0.933606	−	0.358300i	$-0.116644\pi$
$937$	57.1562	1.86721	0.933606	+	0.358300i	$0.116644\pi$
$938$	0	0
$939$	−20.9837	−0.684777
$940$	0	0
$941$	12.2849	0.400476	0.200238	−	0.979747i	$-0.435829\pi$
$941$	12.2849	0.400476	0.200238	+	0.979747i	$0.435829\pi$
$942$	0	0
$943$	54.5752	1.77721
$944$	0	0
$945$	−59.6390	−1.94006
$946$	0	0
$947$	−8.33208	−0.270756	−0.135378	−	0.990794i	$-0.543225\pi$
$947$	−8.33208	−0.270756	−0.135378	+	0.990794i	$0.543225\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	26.9569	0.874137
$952$	0	0
$953$	−47.2691	−1.53120	−0.765599	−	0.643318i	$-0.777557\pi$
$953$	−47.2691	−1.53120	−0.765599	+	0.643318i	$0.777557\pi$
$954$	0	0
$955$	−10.1960	−0.329933
$956$	0	0
$957$	−1.36819	−0.0442273
$958$	0	0
$959$	15.6016	0.503802
$960$	0	0
$961$	−26.7057	−0.861473
$962$	0	0
$963$	122.257	3.93968
$964$	0	0
$965$	27.3296	0.879772
$966$	0	0
$967$	−2.60091	−0.0836397	−0.0418198	−	0.999125i	$-0.513316\pi$
$967$	−2.60091	−0.0836397	−0.0418198	+	0.999125i	$0.513316\pi$
$968$	0	0
$969$	68.8294	2.21112
$970$	0	0
$971$	−28.6175	−0.918379	−0.459189	−	0.888338i	$-0.651860\pi$
$971$	−28.6175	−0.918379	−0.459189	+	0.888338i	$0.651860\pi$
$972$	0	0
$973$	−3.43170	−0.110015
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−24.9100	−0.796942	−0.398471	−	0.917181i	$-0.630459\pi$
$977$	−24.9100	−0.796942	−0.398471	+	0.917181i	$0.630459\pi$
$978$	0	0
$979$	1.55710	0.0497651
$980$	0	0
$981$	126.825	4.04922
$982$	0	0
$983$	−29.2114	−0.931698	−0.465849	−	0.884864i	$-0.654251\pi$
$983$	−29.2114	−0.931698	−0.465849	+	0.884864i	$0.654251\pi$
$984$	0	0
$985$	−59.6648	−1.90108
$986$	0	0
$987$	28.7407	0.914828
$988$	0	0
$989$	−29.6425	−0.942576
$990$	0	0
$991$	−17.5542	−0.557627	−0.278814	−	0.960345i	$-0.589941\pi$
$991$	−17.5542	−0.557627	−0.278814	+	0.960345i	$0.589941\pi$
$992$	0	0
$993$	78.4890	2.49077
$994$	0	0
$995$	−7.50156	−0.237815
$996$	0	0
$997$	−53.5986	−1.69749	−0.848743	−	0.528805i	$-0.822640\pi$
$997$	−53.5986	−1.69749	−0.848743	+	0.528805i	$0.822640\pi$
$998$	0	0
$999$	82.9025	2.62292

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4732.2.a.r.1.6	yes	6
13.5	odd	4		4732.2.g.j.337.12		12
13.8	odd	4		4732.2.g.j.337.11		12
13.12	even	2		4732.2.a.q.1.6	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4732.2.a.q.1.6	✓	6	13.12	even	2
4732.2.a.r.1.6	yes	6	1.1	even	1	trivial
4732.2.g.j.337.11		12	13.8	odd	4
4732.2.g.j.337.12		12	13.5	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:	\( N \)	\(=\)	\( 4732 = 2^{2} \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4732.a (trivial)

\(f(q)\)	\(=\)	\(q+3.28803 q^{3} -3.77007 q^{5} +1.00000 q^{7} +7.81112 q^{9} -0.166194 q^{11} -12.3961 q^{15} -6.66424 q^{17} -3.14115 q^{19} +3.28803 q^{21} +8.71947 q^{23} +9.21343 q^{25} +15.8191 q^{27} +2.50377 q^{29} +2.07228 q^{31} -0.546452 q^{33} -3.77007 q^{35} +5.24067 q^{37} +6.25901 q^{41} -3.39957 q^{43} -29.4485 q^{45} +8.74103 q^{47} +1.00000 q^{49} -21.9122 q^{51} +5.35058 q^{53} +0.626565 q^{55} -10.3282 q^{57} +2.20341 q^{59} -8.97348 q^{61} +7.81112 q^{63} +3.94316 q^{67} +28.6698 q^{69} +12.2041 q^{71} -2.07244 q^{73} +30.2940 q^{75} -0.166194 q^{77} +0.710134 q^{79} +28.5802 q^{81} +8.92521 q^{83} +25.1246 q^{85} +8.23247 q^{87} -9.36913 q^{89} +6.81371 q^{93} +11.8423 q^{95} -15.9901 q^{97} -1.29816 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} + 6 q^{7} + 20 q^{9} + 4 q^{11} + 3 q^{15} - 3 q^{17} + 5 q^{19} + 6 q^{21} + 18 q^{23} + 2 q^{25} + 24 q^{27} + 17 q^{29} - 17 q^{31} + 21 q^{33} + q^{37} + 14 q^{41} + 14 q^{43} - 29 q^{45}+ \cdots + 3 q^{99}+O(q^{100}) \) 6 * q + 6 * q^3 + 6 * q^7 + 20 * q^9 + 4 * q^11 + 3 * q^15 - 3 * q^17 + 5 * q^19 + 6 * q^21 + 18 * q^23 + 2 * q^25 + 24 * q^27 + 17 * q^29 - 17 * q^31 + 21 * q^33 + q^37 + 14 * q^41 + 14 * q^43 - 29 * q^45 - 6 * q^47 + 6 * q^49 + 2 * q^51 + 3 * q^53 + 18 * q^55 - 14 * q^59 - 4 * q^61 + 20 * q^63 + 4 * q^67 + 39 * q^69 + 3 * q^71 - 31 * q^73 + 37 * q^75 + 4 * q^77 + 22 * q^79 + 6 * q^81 - 15 * q^83 + 51 * q^85 + 51 * q^87 - 20 * q^89 + 13 * q^93 + 19 * q^95 - 29 * q^97 + 3 * q^99

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