LMFDB - Embedded newform 1472.4.a.bk.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1472,4,Mod(1,1472)] 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("1472.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1472, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1472 = 2^{6} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1472.a (trivial)

Newform invariants

comment:select newform

sage:traces = [9,0,0,0,0,0,42] 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:	$86.8508115285$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 147x^{7} - 97x^{6} + 4561x^{5} + 7383x^{4} - 31427x^{3} - 43981x^{2} + 17596x + 12306 $ x^9 - 3x^8 - 147x^7 - 97x^6 + 4561x^5 + 7383x^4 - 31427x^3 - 43981x^2 + 17596x + 12306
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 736)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.625016$ of defining polynomial
Character	$\chi$	$=$	1472.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.66678 q^{3} -9.23864 q^{5} -3.16263 q^{7} -24.2218 q^{9} +13.0566 q^{11} +37.8002 q^{13} -15.3988 q^{15} +7.39954 q^{17} -122.752 q^{19} -5.27141 q^{21} +23.0000 q^{23} -39.6475 q^{25} -85.3757 q^{27} -91.6626 q^{29} +187.336 q^{31} +21.7625 q^{33} +29.2184 q^{35} +223.354 q^{37} +63.0048 q^{39} +413.874 q^{41} -482.585 q^{43} +223.777 q^{45} -352.450 q^{47} -332.998 q^{49} +12.3334 q^{51} +88.5192 q^{53} -120.625 q^{55} -204.601 q^{57} -253.386 q^{59} -592.897 q^{61} +76.6046 q^{63} -349.223 q^{65} +787.886 q^{67} +38.3360 q^{69} +782.621 q^{71} +117.824 q^{73} -66.0838 q^{75} -41.2931 q^{77} +1062.77 q^{79} +511.687 q^{81} +431.000 q^{83} -68.3618 q^{85} -152.782 q^{87} +247.711 q^{89} -119.548 q^{91} +312.249 q^{93} +1134.06 q^{95} +615.910 q^{97} -316.254 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 42 q^{7} + 71 q^{9} - 66 q^{11} - 48 q^{13} + 90 q^{15} - 32 q^{17} - 6 q^{21} + 207 q^{23} + 135 q^{25} + 88 q^{29} + 556 q^{31} - 230 q^{33} - 368 q^{35} + 368 q^{37} + 468 q^{39} - 216 q^{41} + 552 q^{43}+ \cdots - 552 q^{99}+O(q^{100}) $ 9 * q + 42 * q^7 + 71 * q^9 - 66 * q^11 - 48 * q^13 + 90 * q^15 - 32 * q^17 - 6 * q^21 + 207 * q^23 + 135 * q^25 + 88 * q^29 + 556 * q^31 - 230 * q^33 - 368 * q^35 + 368 * q^37 + 468 * q^39 - 216 * q^41 + 552 * q^43 + 206 * q^45 + 1300 * q^47 + 1041 * q^49 - 1658 * q^51 - 510 * q^53 + 1948 * q^55 - 804 * q^57 - 1444 * q^59 + 882 * q^61 + 3128 * q^63 + 334 * q^65 - 1186 * q^67 + 2440 * q^71 - 368 * q^73 - 1636 * q^75 - 556 * q^77 + 2576 * q^79 + 1069 * q^81 - 1710 * q^83 - 640 * q^85 + 4844 * q^87 - 474 * q^89 - 2790 * q^91 - 1178 * q^93 + 4564 * q^95 - 884 * q^97 - 552 * 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
$3$	1.66678	0.320773	0.160386	−	0.987054i	$-0.448726\pi$
$3$	1.66678	0.320773	0.160386	+	0.987054i	$0.448726\pi$
$4$	0	0
$5$	−9.23864	−0.826329	−0.413165	−	0.910656i	$-0.635577\pi$
$5$	−9.23864	−0.826329	−0.413165	+	0.910656i	$0.635577\pi$
$6$	0	0
$7$	−3.16263	−0.170766	−0.0853829	−	0.996348i	$-0.527211\pi$
$7$	−3.16263	−0.170766	−0.0853829	+	0.996348i	$0.527211\pi$
$8$	0	0
$9$	−24.2218	−0.897105
$10$	0	0
$11$	13.0566	0.357883	0.178941	−	0.983860i	$-0.442733\pi$
$11$	13.0566	0.357883	0.178941	+	0.983860i	$0.442733\pi$
$12$	0	0
$13$	37.8002	0.806453	0.403227	−	0.915100i	$-0.367889\pi$
$13$	37.8002	0.806453	0.403227	+	0.915100i	$0.367889\pi$
$14$	0	0
$15$	−15.3988	−0.265064
$16$	0	0
$17$	7.39954	0.105568	0.0527839	−	0.998606i	$-0.483191\pi$
$17$	7.39954	0.105568	0.0527839	+	0.998606i	$0.483191\pi$
$18$	0	0
$19$	−122.752	−1.48217	−0.741084	−	0.671413i	$-0.765688\pi$
$19$	−122.752	−1.48217	−0.741084	+	0.671413i	$0.765688\pi$
$20$	0	0
$21$	−5.27141	−0.0547770
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	−39.6475	−0.317180
$26$	0	0
$27$	−85.3757	−0.608540
$28$	0	0
$29$	−91.6626	−0.586942	−0.293471	−	0.955968i	$-0.594810\pi$
$29$	−91.6626	−0.586942	−0.293471	+	0.955968i	$0.594810\pi$
$30$	0	0
$31$	187.336	1.08537	0.542687	−	0.839935i	$-0.317407\pi$
$31$	187.336	1.08537	0.542687	+	0.839935i	$0.317407\pi$
$32$	0	0
$33$	21.7625	0.114799
$34$	0	0
$35$	29.2184	0.141109
$36$	0	0
$37$	223.354	0.992409	0.496204	−	0.868206i	$-0.334727\pi$
$37$	223.354	0.992409	0.496204	+	0.868206i	$0.334727\pi$
$38$	0	0
$39$	63.0048	0.258688
$40$	0	0
$41$	413.874	1.57649	0.788247	−	0.615359i	$-0.210989\pi$
$41$	413.874	1.57649	0.788247	+	0.615359i	$0.210989\pi$
$42$	0	0
$43$	−482.585	−1.71148	−0.855738	−	0.517410i	$-0.826896\pi$
$43$	−482.585	−1.71148	−0.855738	+	0.517410i	$0.826896\pi$
$44$	0	0
$45$	223.777	0.741304
$46$	0	0
$47$	−352.450	−1.09383	−0.546917	−	0.837187i	$-0.684199\pi$
$47$	−352.450	−1.09383	−0.546917	+	0.837187i	$0.684199\pi$
$48$	0	0
$49$	−332.998	−0.970839
$50$	0	0
$51$	12.3334	0.0338633
$52$	0	0
$53$	88.5192	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$53$	88.5192	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$54$	0	0
$55$	−120.625	−0.295729
$56$	0	0
$57$	−204.601	−0.475439
$58$	0	0
$59$	−253.386	−0.559119	−0.279559	−	0.960128i	$-0.590188\pi$
$59$	−253.386	−0.559119	−0.279559	+	0.960128i	$0.590188\pi$
$60$	0	0
$61$	−592.897	−1.24447	−0.622236	−	0.782830i	$-0.713775\pi$
$61$	−592.897	−1.24447	−0.622236	+	0.782830i	$0.713775\pi$
$62$	0	0
$63$	76.6046	0.153195
$64$	0	0
$65$	−349.223	−0.666396
$66$	0	0
$67$	787.886	1.43665	0.718325	−	0.695708i	$-0.244909\pi$
$67$	787.886	1.43665	0.718325	+	0.695708i	$0.244909\pi$
$68$	0	0
$69$	38.3360	0.0668857
$70$	0	0
$71$	782.621	1.30817	0.654085	−	0.756421i	$-0.273054\pi$
$71$	782.621	1.30817	0.654085	+	0.756421i	$0.273054\pi$
$72$	0	0
$73$	117.824	0.188908	0.0944539	−	0.995529i	$-0.469889\pi$
$73$	117.824	0.188908	0.0944539	+	0.995529i	$0.469889\pi$
$74$	0	0
$75$	−66.0838	−0.101743
$76$	0	0
$77$	−41.2931	−0.0611141
$78$	0	0
$79$	1062.77	1.51355	0.756776	−	0.653674i	$-0.226773\pi$
$79$	1062.77	1.51355	0.756776	+	0.653674i	$0.226773\pi$
$80$	0	0
$81$	511.687	0.701902
$82$	0	0
$83$	431.000	0.569981	0.284991	−	0.958530i	$-0.408009\pi$
$83$	431.000	0.569981	0.284991	+	0.958530i	$0.408009\pi$
$84$	0	0
$85$	−68.3618	−0.0872338
$86$	0	0
$87$	−152.782	−0.188275
$88$	0	0
$89$	247.711	0.295026	0.147513	−	0.989060i	$-0.452873\pi$
$89$	247.711	0.295026	0.147513	+	0.989060i	$0.452873\pi$
$90$	0	0
$91$	−119.548	−0.137715
$92$	0	0
$93$	312.249	0.348159
$94$	0	0
$95$	1134.06	1.22476
$96$	0	0
$97$	615.910	0.644703	0.322351	−	0.946620i	$-0.395527\pi$
$97$	615.910	0.644703	0.322351	+	0.946620i	$0.395527\pi$
$98$	0	0
$99$	−316.254	−0.321058
$100$	0	0
$101$	880.587	0.867542	0.433771	−	0.901023i	$-0.357183\pi$
$101$	880.587	0.867542	0.433771	+	0.901023i	$0.357183\pi$
$102$	0	0
$103$	1514.96	1.44926	0.724628	−	0.689140i	$-0.242012\pi$
$103$	1514.96	1.44926	0.724628	+	0.689140i	$0.242012\pi$
$104$	0	0
$105$	48.7007	0.0452638
$106$	0	0
$107$	−1426.94	−1.28923	−0.644614	−	0.764508i	$-0.722982\pi$
$107$	−1426.94	−1.28923	−0.644614	+	0.764508i	$0.722982\pi$
$108$	0	0
$109$	−123.283	−0.108333	−0.0541667	−	0.998532i	$-0.517250\pi$
$109$	−123.283	−0.108333	−0.0541667	+	0.998532i	$0.517250\pi$
$110$	0	0
$111$	372.282	0.318338
$112$	0	0
$113$	795.263	0.662054	0.331027	−	0.943621i	$-0.392605\pi$
$113$	795.263	0.662054	0.331027	+	0.943621i	$0.392605\pi$
$114$	0	0
$115$	−212.489	−0.172302
$116$	0	0
$117$	−915.591	−0.723473
$118$	0	0
$119$	−23.4020	−0.0180274
$120$	0	0
$121$	−1160.53	−0.871920
$122$	0	0
$123$	689.838	0.505696
$124$	0	0
$125$	1521.12	1.08842
$126$	0	0
$127$	2450.52	1.71219	0.856097	−	0.516816i	$-0.172883\pi$
$127$	2450.52	1.71219	0.856097	+	0.516816i	$0.172883\pi$
$128$	0	0
$129$	−804.364	−0.548995
$130$	0	0
$131$	910.056	0.606962	0.303481	−	0.952838i	$-0.401851\pi$
$131$	910.056	0.606962	0.303481	+	0.952838i	$0.401851\pi$
$132$	0	0
$133$	388.218	0.253103
$134$	0	0
$135$	788.756	0.502854
$136$	0	0
$137$	1215.66	0.758110	0.379055	−	0.925374i	$-0.376249\pi$
$137$	1215.66	0.758110	0.379055	+	0.925374i	$0.376249\pi$
$138$	0	0
$139$	−803.695	−0.490421	−0.245211	−	0.969470i	$-0.578857\pi$
$139$	−803.695	−0.490421	−0.245211	+	0.969470i	$0.578857\pi$
$140$	0	0
$141$	−587.459	−0.350872
$142$	0	0
$143$	493.542	0.288616
$144$	0	0
$145$	846.838	0.485007
$146$	0	0
$147$	−555.035	−0.311419
$148$	0	0
$149$	−3321.11	−1.82601	−0.913007	−	0.407944i	$-0.866246\pi$
$149$	−3321.11	−1.82601	−0.913007	+	0.407944i	$0.866246\pi$
$150$	0	0
$151$	2843.93	1.53269	0.766344	−	0.642431i	$-0.222074\pi$
$151$	2843.93	1.53269	0.766344	+	0.642431i	$0.222074\pi$
$152$	0	0
$153$	−179.231	−0.0947054
$154$	0	0
$155$	−1730.73	−0.896877
$156$	0	0
$157$	−444.465	−0.225937	−0.112969	−	0.993599i	$-0.536036\pi$
$157$	−444.465	−0.225937	−0.112969	+	0.993599i	$0.536036\pi$
$158$	0	0
$159$	147.542	0.0735904
$160$	0	0
$161$	−72.7404	−0.0356071
$162$	0	0
$163$	254.266	0.122182	0.0610908	−	0.998132i	$-0.480542\pi$
$163$	254.266	0.122182	0.0610908	+	0.998132i	$0.480542\pi$
$164$	0	0
$165$	−201.056	−0.0948618
$166$	0	0
$167$	56.0642	0.0259783	0.0129892	−	0.999916i	$-0.495865\pi$
$167$	56.0642	0.0259783	0.0129892	+	0.999916i	$0.495865\pi$
$168$	0	0
$169$	−768.143	−0.349633
$170$	0	0
$171$	2973.27	1.32966
$172$	0	0
$173$	3903.62	1.71553	0.857765	−	0.514042i	$-0.171852\pi$
$173$	3903.62	1.71553	0.857765	+	0.514042i	$0.171852\pi$
$174$	0	0
$175$	125.390	0.0541635
$176$	0	0
$177$	−422.339	−0.179350
$178$	0	0
$179$	2099.74	0.876770	0.438385	−	0.898787i	$-0.355551\pi$
$179$	2099.74	0.876770	0.438385	+	0.898787i	$0.355551\pi$
$180$	0	0
$181$	99.5753	0.0408916	0.0204458	−	0.999791i	$-0.493491\pi$
$181$	99.5753	0.0408916	0.0204458	+	0.999791i	$0.493491\pi$
$182$	0	0
$183$	−988.232	−0.399192
$184$	0	0
$185$	−2063.49	−0.820057
$186$	0	0
$187$	96.6128	0.0377809
$188$	0	0
$189$	270.012	0.103918
$190$	0	0
$191$	−1733.84	−0.656837	−0.328419	−	0.944532i	$-0.606516\pi$
$191$	−1733.84	−0.656837	−0.328419	+	0.944532i	$0.606516\pi$
$192$	0	0
$193$	−1729.41	−0.645003	−0.322501	−	0.946569i	$-0.604524\pi$
$193$	−1729.41	−0.645003	−0.322501	+	0.946569i	$0.604524\pi$
$194$	0	0
$195$	−582.079	−0.213762
$196$	0	0
$197$	−1727.26	−0.624683	−0.312341	−	0.949970i	$-0.601113\pi$
$197$	−1727.26	−0.624683	−0.312341	+	0.949970i	$0.601113\pi$
$198$	0	0
$199$	1399.12	0.498395	0.249198	−	0.968453i	$-0.419833\pi$
$199$	1399.12	0.498395	0.249198	+	0.968453i	$0.419833\pi$
$200$	0	0
$201$	1313.24	0.460838
$202$	0	0
$203$	289.895	0.100230
$204$	0	0
$205$	−3823.63	−1.30270
$206$	0	0
$207$	−557.102	−0.187059
$208$	0	0
$209$	−1602.72	−0.530442
$210$	0	0
$211$	5502.20	1.79520	0.897599	−	0.440812i	$-0.145310\pi$
$211$	5502.20	1.79520	0.897599	+	0.440812i	$0.145310\pi$
$212$	0	0
$213$	1304.46	0.419625
$214$	0	0
$215$	4458.43	1.41424
$216$	0	0
$217$	−592.475	−0.185345
$218$	0	0
$219$	196.387	0.0605965
$220$	0	0
$221$	279.704	0.0851355
$222$	0	0
$223$	240.020	0.0720759	0.0360379	−	0.999350i	$-0.488526\pi$
$223$	240.020	0.0720759	0.0360379	+	0.999350i	$0.488526\pi$
$224$	0	0
$225$	960.335	0.284544
$226$	0	0
$227$	868.383	0.253906	0.126953	−	0.991909i	$-0.459480\pi$
$227$	868.383	0.253906	0.126953	+	0.991909i	$0.459480\pi$
$228$	0	0
$229$	−863.833	−0.249274	−0.124637	−	0.992202i	$-0.539777\pi$
$229$	−863.833	−0.249274	−0.124637	+	0.992202i	$0.539777\pi$
$230$	0	0
$231$	−68.8267	−0.0196037
$232$	0	0
$233$	−1453.38	−0.408645	−0.204323	−	0.978904i	$-0.565499\pi$
$233$	−1453.38	−0.408645	−0.204323	+	0.978904i	$0.565499\pi$
$234$	0	0
$235$	3256.16	0.903866
$236$	0	0
$237$	1771.40	0.485506
$238$	0	0
$239$	3867.47	1.04672	0.523360	−	0.852112i	$-0.324678\pi$
$239$	3867.47	1.04672	0.523360	+	0.852112i	$0.324678\pi$
$240$	0	0
$241$	5238.20	1.40009	0.700046	−	0.714098i	$-0.253163\pi$
$241$	5238.20	1.40009	0.700046	+	0.714098i	$0.253163\pi$
$242$	0	0
$243$	3158.02	0.833691
$244$	0	0
$245$	3076.45	0.802233
$246$	0	0
$247$	−4640.04	−1.19530
$248$	0	0
$249$	718.385	0.182834
$250$	0	0
$251$	2286.99	0.575113	0.287557	−	0.957764i	$-0.407157\pi$
$251$	2286.99	0.575113	0.287557	+	0.957764i	$0.407157\pi$
$252$	0	0
$253$	300.301	0.0746237
$254$	0	0
$255$	−113.944	−0.0279822
$256$	0	0
$257$	−2874.64	−0.697725	−0.348863	−	0.937174i	$-0.613432\pi$
$257$	−2874.64	−0.697725	−0.348863	+	0.937174i	$0.613432\pi$
$258$	0	0
$259$	−706.384	−0.169469
$260$	0	0
$261$	2220.24	0.526548
$262$	0	0
$263$	962.922	0.225765	0.112883	−	0.993608i	$-0.463992\pi$
$263$	962.922	0.225765	0.112883	+	0.993608i	$0.463992\pi$
$264$	0	0
$265$	−817.797	−0.189573
$266$	0	0
$267$	412.881	0.0946364
$268$	0	0
$269$	−3355.11	−0.760465	−0.380232	−	0.924891i	$-0.624156\pi$
$269$	−3355.11	−0.760465	−0.380232	+	0.924891i	$0.624156\pi$
$270$	0	0
$271$	3528.43	0.790911	0.395455	−	0.918485i	$-0.370587\pi$
$271$	3528.43	0.790911	0.395455	+	0.918485i	$0.370587\pi$
$272$	0	0
$273$	−199.261	−0.0441751
$274$	0	0
$275$	−517.661	−0.113513
$276$	0	0
$277$	960.706	0.208387	0.104194	−	0.994557i	$-0.466774\pi$
$277$	960.706	0.208387	0.104194	+	0.994557i	$0.466774\pi$
$278$	0	0
$279$	−4537.63	−0.973695
$280$	0	0
$281$	3860.68	0.819605	0.409803	−	0.912174i	$-0.365598\pi$
$281$	3860.68	0.819605	0.409803	+	0.912174i	$0.365598\pi$
$282$	0	0
$283$	−3950.23	−0.829742	−0.414871	−	0.909880i	$-0.636173\pi$
$283$	−3950.23	−0.829742	−0.414871	+	0.909880i	$0.636173\pi$
$284$	0	0
$285$	1890.23	0.392869
$286$	0	0
$287$	−1308.93	−0.269211
$288$	0	0
$289$	−4858.25	−0.988855
$290$	0	0
$291$	1026.59	0.206803
$292$	0	0
$293$	−4881.86	−0.973384	−0.486692	−	0.873574i	$-0.661797\pi$
$293$	−4881.86	−0.973384	−0.486692	+	0.873574i	$0.661797\pi$
$294$	0	0
$295$	2340.94	0.462016
$296$	0	0
$297$	−1114.72	−0.217786
$298$	0	0
$299$	869.405	0.168157
$300$	0	0
$301$	1526.23	0.292261
$302$	0	0
$303$	1467.75	0.278284
$304$	0	0
$305$	5477.57	1.02834
$306$	0	0
$307$	8023.71	1.49165	0.745826	−	0.666141i	$-0.232055\pi$
$307$	8023.71	1.49165	0.745826	+	0.666141i	$0.232055\pi$
$308$	0	0
$309$	2525.11	0.464882
$310$	0	0
$311$	−8162.92	−1.48835	−0.744175	−	0.667984i	$-0.767157\pi$
$311$	−8162.92	−1.48835	−0.744175	+	0.667984i	$0.767157\pi$
$312$	0	0
$313$	−8135.11	−1.46909	−0.734543	−	0.678562i	$-0.762603\pi$
$313$	−8135.11	−1.46909	−0.734543	+	0.678562i	$0.762603\pi$
$314$	0	0
$315$	−707.722	−0.126589
$316$	0	0
$317$	−5721.01	−1.01364	−0.506820	−	0.862052i	$-0.669179\pi$
$317$	−5721.01	−1.01364	−0.506820	+	0.862052i	$0.669179\pi$
$318$	0	0
$319$	−1196.80	−0.210056
$320$	0	0
$321$	−2378.40	−0.413549
$322$	0	0
$323$	−908.307	−0.156469
$324$	0	0
$325$	−1498.68	−0.255791
$326$	0	0
$327$	−205.486	−0.0347504
$328$	0	0
$329$	1114.67	0.186789
$330$	0	0
$331$	11233.2	1.86536	0.932679	−	0.360707i	$-0.117465\pi$
$331$	11233.2	1.86536	0.932679	+	0.360707i	$0.117465\pi$
$332$	0	0
$333$	−5410.04	−0.890295
$334$	0	0
$335$	−7278.99	−1.18715
$336$	0	0
$337$	11099.8	1.79420	0.897099	−	0.441830i	$-0.145671\pi$
$337$	11099.8	1.79420	0.897099	+	0.441830i	$0.145671\pi$
$338$	0	0
$339$	1325.53	0.212369
$340$	0	0
$341$	2445.97	0.388437
$342$	0	0
$343$	2137.93	0.336552
$344$	0	0
$345$	−354.173	−0.0552696
$346$	0	0
$347$	607.635	0.0940045	0.0470022	−	0.998895i	$-0.485033\pi$
$347$	607.635	0.0940045	0.0470022	+	0.998895i	$0.485033\pi$
$348$	0	0
$349$	5030.44	0.771556	0.385778	−	0.922592i	$-0.373933\pi$
$349$	5030.44	0.771556	0.385778	+	0.922592i	$0.373933\pi$
$350$	0	0
$351$	−3227.22	−0.490759
$352$	0	0
$353$	7578.90	1.14273	0.571366	−	0.820695i	$-0.306414\pi$
$353$	7578.90	1.14273	0.571366	+	0.820695i	$0.306414\pi$
$354$	0	0
$355$	−7230.36	−1.08098
$356$	0	0
$357$	−39.0061	−0.00578269
$358$	0	0
$359$	1804.47	0.265282	0.132641	−	0.991164i	$-0.457654\pi$
$359$	1804.47	0.265282	0.132641	+	0.991164i	$0.457654\pi$
$360$	0	0
$361$	8208.98	1.19682
$362$	0	0
$363$	−1934.35	−0.279688
$364$	0	0
$365$	−1088.54	−0.156100
$366$	0	0
$367$	−2356.11	−0.335117	−0.167558	−	0.985862i	$-0.553588\pi$
$367$	−2356.11	−0.335117	−0.167558	+	0.985862i	$0.553588\pi$
$368$	0	0
$369$	−10024.8	−1.41428
$370$	0	0
$371$	−279.953	−0.0391764
$372$	0	0
$373$	−7071.80	−0.981672	−0.490836	−	0.871252i	$-0.663309\pi$
$373$	−7071.80	−0.981672	−0.490836	+	0.871252i	$0.663309\pi$
$374$	0	0
$375$	2535.38	0.349137
$376$	0	0
$377$	−3464.87	−0.473341
$378$	0	0
$379$	6752.69	0.915205	0.457602	−	0.889157i	$-0.348708\pi$
$379$	6752.69	0.915205	0.457602	+	0.889157i	$0.348708\pi$
$380$	0	0
$381$	4084.49	0.549225
$382$	0	0
$383$	−6495.53	−0.866595	−0.433297	−	0.901251i	$-0.642650\pi$
$383$	−6495.53	−0.866595	−0.433297	+	0.901251i	$0.642650\pi$
$384$	0	0
$385$	381.492	0.0505004
$386$	0	0
$387$	11689.1	1.53537
$388$	0	0
$389$	4340.24	0.565704	0.282852	−	0.959164i	$-0.408719\pi$
$389$	4340.24	0.565704	0.282852	+	0.959164i	$0.408719\pi$
$390$	0	0
$391$	170.190	0.0220124
$392$	0	0
$393$	1516.87	0.194697
$394$	0	0
$395$	−9818.52	−1.25069
$396$	0	0
$397$	10555.8	1.33446	0.667231	−	0.744851i	$-0.267480\pi$
$397$	10555.8	1.33446	0.667231	+	0.744851i	$0.267480\pi$
$398$	0	0
$399$	647.075	0.0811887
$400$	0	0
$401$	−13778.3	−1.71585	−0.857925	−	0.513774i	$-0.828247\pi$
$401$	−13778.3	−1.71585	−0.857925	+	0.513774i	$0.828247\pi$
$402$	0	0
$403$	7081.36	0.875304
$404$	0	0
$405$	−4727.29	−0.580002
$406$	0	0
$407$	2916.24	0.355166
$408$	0	0
$409$	6083.54	0.735481	0.367740	−	0.929928i	$-0.380132\pi$
$409$	6083.54	0.735481	0.367740	+	0.929928i	$0.380132\pi$
$410$	0	0
$411$	2026.25	0.243181
$412$	0	0
$413$	801.364	0.0954783
$414$	0	0
$415$	−3981.86	−0.470992
$416$	0	0
$417$	−1339.59	−0.157314
$418$	0	0
$419$	4326.69	0.504469	0.252234	−	0.967666i	$-0.418835\pi$
$419$	4326.69	0.504469	0.252234	+	0.967666i	$0.418835\pi$
$420$	0	0
$421$	1577.16	0.182580	0.0912901	−	0.995824i	$-0.470901\pi$
$421$	1577.16	0.182580	0.0912901	+	0.995824i	$0.470901\pi$
$422$	0	0
$423$	8536.99	0.981283
$424$	0	0
$425$	−293.373	−0.0334840
$426$	0	0
$427$	1875.11	0.212513
$428$	0	0
$429$	822.628	0.0925800
$430$	0	0
$431$	11095.3	1.24001	0.620004	−	0.784599i	$-0.287131\pi$
$431$	11095.3	1.24001	0.620004	+	0.784599i	$0.287131\pi$
$432$	0	0
$433$	4970.59	0.551665	0.275833	−	0.961206i	$-0.411046\pi$
$433$	4970.59	0.551665	0.275833	+	0.961206i	$0.411046\pi$
$434$	0	0
$435$	1411.50	0.155577
$436$	0	0
$437$	−2823.29	−0.309053
$438$	0	0
$439$	3189.61	0.346769	0.173385	−	0.984854i	$-0.444530\pi$
$439$	3189.61	0.346769	0.173385	+	0.984854i	$0.444530\pi$
$440$	0	0
$441$	8065.82	0.870944
$442$	0	0
$443$	−4489.76	−0.481523	−0.240762	−	0.970584i	$-0.577397\pi$
$443$	−4489.76	−0.481523	−0.240762	+	0.970584i	$0.577397\pi$
$444$	0	0
$445$	−2288.52	−0.243789
$446$	0	0
$447$	−5535.58	−0.585736
$448$	0	0
$449$	10374.1	1.09038	0.545191	−	0.838312i	$-0.316457\pi$
$449$	10374.1	1.09038	0.545191	+	0.838312i	$0.316457\pi$
$450$	0	0
$451$	5403.78	0.564200
$452$	0	0
$453$	4740.22	0.491644
$454$	0	0
$455$	1104.46	0.113798
$456$	0	0
$457$	−16384.6	−1.67711	−0.838555	−	0.544818i	$-0.816599\pi$
$457$	−16384.6	−1.67711	−0.838555	+	0.544818i	$0.816599\pi$
$458$	0	0
$459$	−631.742	−0.0642422
$460$	0	0
$461$	767.780	0.0775685	0.0387843	−	0.999248i	$-0.487651\pi$
$461$	767.780	0.0775685	0.0387843	+	0.999248i	$0.487651\pi$
$462$	0	0
$463$	−14148.6	−1.42017	−0.710086	−	0.704115i	$-0.751344\pi$
$463$	−14148.6	−1.42017	−0.710086	+	0.704115i	$0.751344\pi$
$464$	0	0
$465$	−2884.76	−0.287694
$466$	0	0
$467$	10364.7	1.02703	0.513514	−	0.858081i	$-0.328343\pi$
$467$	10364.7	1.02703	0.513514	+	0.858081i	$0.328343\pi$
$468$	0	0
$469$	−2491.79	−0.245331
$470$	0	0
$471$	−740.827	−0.0724745
$472$	0	0
$473$	−6300.91	−0.612507
$474$	0	0
$475$	4866.80	0.470114
$476$	0	0
$477$	−2144.10	−0.205810
$478$	0	0
$479$	−6248.79	−0.596063	−0.298032	−	0.954556i	$-0.596330\pi$
$479$	−6248.79	−0.596063	−0.298032	+	0.954556i	$0.596330\pi$
$480$	0	0
$481$	8442.82	0.800332
$482$	0	0
$483$	−121.243	−0.0114218
$484$	0	0
$485$	−5690.17	−0.532737
$486$	0	0
$487$	−2405.30	−0.223808	−0.111904	−	0.993719i	$-0.535695\pi$
$487$	−2405.30	−0.223808	−0.111904	+	0.993719i	$0.535695\pi$
$488$	0	0
$489$	423.806	0.0391926
$490$	0	0
$491$	−10022.0	−0.921157	−0.460578	−	0.887619i	$-0.652358\pi$
$491$	−10022.0	−0.921157	−0.460578	+	0.887619i	$0.652358\pi$
$492$	0	0
$493$	−678.261	−0.0619622
$494$	0	0
$495$	2921.76	0.265300
$496$	0	0
$497$	−2475.14	−0.223391
$498$	0	0
$499$	−300.268	−0.0269376	−0.0134688	−	0.999909i	$-0.504287\pi$
$499$	−300.268	−0.0269376	−0.0134688	+	0.999909i	$0.504287\pi$
$500$	0	0
$501$	93.4470	0.00833314
$502$	0	0
$503$	−6790.34	−0.601921	−0.300961	−	0.953637i	$-0.597307\pi$
$503$	−6790.34	−0.601921	−0.300961	+	0.953637i	$0.597307\pi$
$504$	0	0
$505$	−8135.43	−0.716875
$506$	0	0
$507$	−1280.33	−0.112153
$508$	0	0
$509$	−15177.3	−1.32166	−0.660829	−	0.750537i	$-0.729795\pi$
$509$	−15177.3	−1.32166	−0.660829	+	0.750537i	$0.729795\pi$
$510$	0	0
$511$	−372.634	−0.0322590
$512$	0	0
$513$	10480.0	0.901957
$514$	0	0
$515$	−13996.2	−1.19756
$516$	0	0
$517$	−4601.80	−0.391464
$518$	0	0
$519$	6506.49	0.550295
$520$	0	0
$521$	−9295.72	−0.781675	−0.390838	−	0.920460i	$-0.627815\pi$
$521$	−9295.72	−0.781675	−0.390838	+	0.920460i	$0.627815\pi$
$522$	0	0
$523$	−7803.60	−0.652443	−0.326221	−	0.945293i	$-0.605775\pi$
$523$	−7803.60	−0.652443	−0.326221	+	0.945293i	$0.605775\pi$
$524$	0	0
$525$	208.998	0.0173742
$526$	0	0
$527$	1386.20	0.114581
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	6137.47	0.501588
$532$	0	0
$533$	15644.5	1.27137
$534$	0	0
$535$	13183.0	1.06533
$536$	0	0
$537$	3499.81	0.281244
$538$	0	0
$539$	−4347.81	−0.347446
$540$	0	0
$541$	22303.2	1.77244	0.886220	−	0.463265i	$-0.153322\pi$
$541$	22303.2	1.77244	0.886220	+	0.463265i	$0.153322\pi$
$542$	0	0
$543$	165.970	0.0131169
$544$	0	0
$545$	1138.96	0.0895190
$546$	0	0
$547$	−17044.9	−1.33233	−0.666166	−	0.745803i	$-0.732066\pi$
$547$	−17044.9	−1.33233	−0.666166	+	0.745803i	$0.732066\pi$
$548$	0	0
$549$	14361.1	1.11642
$550$	0	0
$551$	11251.7	0.869946
$552$	0	0
$553$	−3361.13	−0.258463
$554$	0	0
$555$	−3439.38	−0.263052
$556$	0	0
$557$	14046.8	1.06855	0.534273	−	0.845312i	$-0.320585\pi$
$557$	14046.8	1.06855	0.534273	+	0.845312i	$0.320585\pi$
$558$	0	0
$559$	−18241.8	−1.38023
$560$	0	0
$561$	161.033	0.0121191
$562$	0	0
$563$	−23296.2	−1.74390	−0.871950	−	0.489595i	$-0.837145\pi$
$563$	−23296.2	−1.74390	−0.871950	+	0.489595i	$0.837145\pi$
$564$	0	0
$565$	−7347.15	−0.547074
$566$	0	0
$567$	−1618.27	−0.119861
$568$	0	0
$569$	19503.6	1.43696	0.718481	−	0.695546i	$-0.244838\pi$
$569$	19503.6	1.43696	0.718481	+	0.695546i	$0.244838\pi$
$570$	0	0
$571$	−11030.4	−0.808420	−0.404210	−	0.914666i	$-0.632454\pi$
$571$	−11030.4	−0.808420	−0.404210	+	0.914666i	$0.632454\pi$
$572$	0	0
$573$	−2889.93	−0.210696
$574$	0	0
$575$	−911.892	−0.0661366
$576$	0	0
$577$	−8912.75	−0.643055	−0.321527	−	0.946900i	$-0.604196\pi$
$577$	−8912.75	−0.643055	−0.321527	+	0.946900i	$0.604196\pi$
$578$	0	0
$579$	−2882.55	−0.206899
$580$	0	0
$581$	−1363.09	−0.0973333
$582$	0	0
$583$	1155.76	0.0821040
$584$	0	0
$585$	8458.81	0.597827
$586$	0	0
$587$	−25244.2	−1.77502	−0.887512	−	0.460784i	$-0.847568\pi$
$587$	−25244.2	−1.77502	−0.887512	+	0.460784i	$0.847568\pi$
$588$	0	0
$589$	−22995.9	−1.60871
$590$	0	0
$591$	−2878.98	−0.200381
$592$	0	0
$593$	−1620.86	−0.112244	−0.0561222	−	0.998424i	$-0.517874\pi$
$593$	−1620.86	−0.112244	−0.0561222	+	0.998424i	$0.517874\pi$
$594$	0	0
$595$	216.203	0.0148965
$596$	0	0
$597$	2332.02	0.159872
$598$	0	0
$599$	14225.4	0.970341	0.485171	−	0.874420i	$-0.338758\pi$
$599$	14225.4	0.970341	0.485171	+	0.874420i	$0.338758\pi$
$600$	0	0
$601$	−99.7031	−0.00676702	−0.00338351	−	0.999994i	$-0.501077\pi$
$601$	−99.7031	−0.00676702	−0.00338351	+	0.999994i	$0.501077\pi$
$602$	0	0
$603$	−19084.0	−1.28883
$604$	0	0
$605$	10721.7	0.720493
$606$	0	0
$607$	−22971.3	−1.53604	−0.768019	−	0.640427i	$-0.778757\pi$
$607$	−22971.3	−1.53604	−0.768019	+	0.640427i	$0.778757\pi$
$608$	0	0
$609$	483.192	0.0321509
$610$	0	0
$611$	−13322.7	−0.882125
$612$	0	0
$613$	−5333.56	−0.351420	−0.175710	−	0.984442i	$-0.556222\pi$
$613$	−5333.56	−0.351420	−0.175710	+	0.984442i	$0.556222\pi$
$614$	0	0
$615$	−6373.17	−0.417872
$616$	0	0
$617$	3633.60	0.237088	0.118544	−	0.992949i	$-0.462177\pi$
$617$	3633.60	0.237088	0.118544	+	0.992949i	$0.462177\pi$
$618$	0	0
$619$	−9984.55	−0.648325	−0.324162	−	0.946001i	$-0.605082\pi$
$619$	−9984.55	−0.648325	−0.324162	+	0.946001i	$0.605082\pi$
$620$	0	0
$621$	−1963.64	−0.126889
$622$	0	0
$623$	−783.418	−0.0503804
$624$	0	0
$625$	−9097.14	−0.582217
$626$	0	0
$627$	−2671.39	−0.170151
$628$	0	0
$629$	1652.72	0.104766
$630$	0	0
$631$	19104.9	1.20532	0.602659	−	0.797999i	$-0.294108\pi$
$631$	19104.9	1.20532	0.602659	+	0.797999i	$0.294108\pi$
$632$	0	0
$633$	9170.97	0.575851
$634$	0	0
$635$	−22639.5	−1.41484
$636$	0	0
$637$	−12587.4	−0.782936
$638$	0	0
$639$	−18956.5	−1.17357
$640$	0	0
$641$	−19282.8	−1.18818	−0.594092	−	0.804397i	$-0.702488\pi$
$641$	−19282.8	−1.18818	−0.594092	+	0.804397i	$0.702488\pi$
$642$	0	0
$643$	3074.46	0.188562	0.0942808	−	0.995546i	$-0.469945\pi$
$643$	3074.46	0.188562	0.0942808	+	0.995546i	$0.469945\pi$
$644$	0	0
$645$	7431.23	0.453650
$646$	0	0
$647$	24653.6	1.49804	0.749021	−	0.662546i	$-0.230524\pi$
$647$	24653.6	1.49804	0.749021	+	0.662546i	$0.230524\pi$
$648$	0	0
$649$	−3308.35	−0.200099
$650$	0	0
$651$	−987.528	−0.0594536
$652$	0	0
$653$	−8394.63	−0.503074	−0.251537	−	0.967848i	$-0.580936\pi$
$653$	−8394.63	−0.503074	−0.251537	+	0.967848i	$0.580936\pi$
$654$	0	0
$655$	−8407.68	−0.501550
$656$	0	0
$657$	−2853.92	−0.169470
$658$	0	0
$659$	−16885.4	−0.998121	−0.499060	−	0.866567i	$-0.666321\pi$
$659$	−16885.4	−0.998121	−0.499060	+	0.866567i	$0.666321\pi$
$660$	0	0
$661$	−6002.36	−0.353200	−0.176600	−	0.984283i	$-0.556510\pi$
$661$	−6002.36	−0.353200	−0.176600	+	0.984283i	$0.556510\pi$
$662$	0	0
$663$	466.207	0.0273092
$664$	0	0
$665$	−3586.61	−0.209147
$666$	0	0
$667$	−2108.24	−0.122386
$668$	0	0
$669$	400.061	0.0231200
$670$	0	0
$671$	−7741.22	−0.445375
$672$	0	0
$673$	−23919.0	−1.37000	−0.684998	−	0.728545i	$-0.740197\pi$
$673$	−23919.0	−1.37000	−0.684998	+	0.728545i	$0.740197\pi$
$674$	0	0
$675$	3384.93	0.193016
$676$	0	0
$677$	−9402.50	−0.533778	−0.266889	−	0.963727i	$-0.585996\pi$
$677$	−9402.50	−0.533778	−0.266889	+	0.963727i	$0.585996\pi$
$678$	0	0
$679$	−1947.89	−0.110093
$680$	0	0
$681$	1447.41	0.0814460
$682$	0	0
$683$	12019.5	0.673371	0.336685	−	0.941617i	$-0.390694\pi$
$683$	12019.5	0.673371	0.336685	+	0.941617i	$0.390694\pi$
$684$	0	0
$685$	−11231.1	−0.626448
$686$	0	0
$687$	−1439.82	−0.0799602
$688$	0	0
$689$	3346.05	0.185013
$690$	0	0
$691$	9350.34	0.514767	0.257383	−	0.966309i	$-0.417140\pi$
$691$	9350.34	0.514767	0.257383	+	0.966309i	$0.417140\pi$
$692$	0	0
$693$	1000.19	0.0548258
$694$	0	0
$695$	7425.05	0.405249
$696$	0	0
$697$	3062.48	0.166427
$698$	0	0
$699$	−2422.48	−0.131082
$700$	0	0
$701$	−21765.1	−1.17269	−0.586346	−	0.810061i	$-0.699434\pi$
$701$	−21765.1	−1.17269	−0.586346	+	0.810061i	$0.699434\pi$
$702$	0	0
$703$	−27417.1	−1.47092
$704$	0	0
$705$	5427.32	0.289936
$706$	0	0
$707$	−2784.97	−0.148146
$708$	0	0
$709$	426.079	0.0225694	0.0112847	−	0.999936i	$-0.496408\pi$
$709$	426.079	0.0225694	0.0112847	+	0.999936i	$0.496408\pi$
$710$	0	0
$711$	−25742.2	−1.35781
$712$	0	0
$713$	4308.74	0.226316
$714$	0	0
$715$	−4559.66	−0.238492
$716$	0	0
$717$	6446.24	0.335759
$718$	0	0
$719$	−13266.1	−0.688097	−0.344049	−	0.938952i	$-0.611799\pi$
$719$	−13266.1	−0.688097	−0.344049	+	0.938952i	$0.611799\pi$
$720$	0	0
$721$	−4791.25	−0.247483
$722$	0	0
$723$	8730.95	0.449111
$724$	0	0
$725$	3634.19	0.186166
$726$	0	0
$727$	−6451.40	−0.329119	−0.164559	−	0.986367i	$-0.552620\pi$
$727$	−6451.40	−0.329119	−0.164559	+	0.986367i	$0.552620\pi$
$728$	0	0
$729$	−8551.81	−0.434477
$730$	0	0
$731$	−3570.91	−0.180677
$732$	0	0
$733$	8411.80	0.423870	0.211935	−	0.977284i	$-0.432023\pi$
$733$	8411.80	0.423870	0.211935	+	0.977284i	$0.432023\pi$
$734$	0	0
$735$	5127.77	0.257334
$736$	0	0
$737$	10287.1	0.514152
$738$	0	0
$739$	22558.5	1.12291	0.561453	−	0.827509i	$-0.310243\pi$
$739$	22558.5	1.12291	0.561453	+	0.827509i	$0.310243\pi$
$740$	0	0
$741$	−7733.95	−0.383419
$742$	0	0
$743$	26995.7	1.33294	0.666472	−	0.745530i	$-0.267803\pi$
$743$	26995.7	1.33294	0.666472	+	0.745530i	$0.267803\pi$
$744$	0	0
$745$	30682.6	1.50889
$746$	0	0
$747$	−10439.6	−0.511333
$748$	0	0
$749$	4512.87	0.220156
$750$	0	0
$751$	35427.7	1.72140	0.860702	−	0.509109i	$-0.170025\pi$
$751$	35427.7	1.72140	0.860702	+	0.509109i	$0.170025\pi$
$752$	0	0
$753$	3811.92	0.184481
$754$	0	0
$755$	−26274.1	−1.26650
$756$	0	0
$757$	22067.2	1.05951	0.529753	−	0.848152i	$-0.322284\pi$
$757$	22067.2	1.05951	0.529753	+	0.848152i	$0.322284\pi$
$758$	0	0
$759$	500.538	0.0239372
$760$	0	0
$761$	6125.32	0.291777	0.145889	−	0.989301i	$-0.453396\pi$
$761$	6125.32	0.291777	0.145889	+	0.989301i	$0.453396\pi$
$762$	0	0
$763$	389.897	0.0184996
$764$	0	0
$765$	1655.85	0.0782579
$766$	0	0
$767$	−9578.03	−0.450903
$768$	0	0
$769$	3552.49	0.166588	0.0832939	−	0.996525i	$-0.473456\pi$
$769$	3552.49	0.166588	0.0832939	+	0.996525i	$0.473456\pi$
$770$	0	0
$771$	−4791.41	−0.223811
$772$	0	0
$773$	1348.20	0.0627313	0.0313656	−	0.999508i	$-0.490014\pi$
$773$	1348.20	0.0627313	0.0313656	+	0.999508i	$0.490014\pi$
$774$	0	0
$775$	−7427.42	−0.344259
$776$	0	0
$777$	−1177.39	−0.0543612
$778$	0	0
$779$	−50803.7	−2.33663
$780$	0	0
$781$	10218.4	0.468171
$782$	0	0
$783$	7825.76	0.357177
$784$	0	0
$785$	4106.25	0.186699
$786$	0	0
$787$	34885.5	1.58009	0.790047	−	0.613047i	$-0.210056\pi$
$787$	34885.5	1.58009	0.790047	+	0.613047i	$0.210056\pi$
$788$	0	0
$789$	1604.98	0.0724194
$790$	0	0
$791$	−2515.12	−0.113056
$792$	0	0
$793$	−22411.7	−1.00361
$794$	0	0
$795$	−1363.09	−0.0608099
$796$	0	0
$797$	11000.3	0.488897	0.244449	−	0.969662i	$-0.421393\pi$
$797$	11000.3	0.488897	0.244449	+	0.969662i	$0.421393\pi$
$798$	0	0
$799$	−2607.97	−0.115474
$800$	0	0
$801$	−6000.02	−0.264670
$802$	0	0
$803$	1538.38	0.0676068
$804$	0	0
$805$	672.023	0.0294232
$806$	0	0
$807$	−5592.25	−0.243936
$808$	0	0
$809$	−12167.3	−0.528775	−0.264388	−	0.964416i	$-0.585170\pi$
$809$	−12167.3	−0.528775	−0.264388	+	0.964416i	$0.585170\pi$
$810$	0	0
$811$	28930.7	1.25264	0.626322	−	0.779565i	$-0.284560\pi$
$811$	28930.7	1.25264	0.626322	+	0.779565i	$0.284560\pi$
$812$	0	0
$813$	5881.13	0.253703
$814$	0	0
$815$	−2349.07	−0.100962
$816$	0	0
$817$	59238.1	2.53669
$818$	0	0
$819$	2895.67	0.123544
$820$	0	0
$821$	−46160.7	−1.96227	−0.981133	−	0.193332i	$-0.938071\pi$
$821$	−46160.7	−1.96227	−0.981133	+	0.193332i	$0.938071\pi$
$822$	0	0
$823$	13199.6	0.559064	0.279532	−	0.960136i	$-0.409821\pi$
$823$	13199.6	0.559064	0.279532	+	0.960136i	$0.409821\pi$
$824$	0	0
$825$	−862.829	−0.0364119
$826$	0	0
$827$	−18213.5	−0.765833	−0.382917	−	0.923783i	$-0.625080\pi$
$827$	−18213.5	−0.765833	−0.382917	+	0.923783i	$0.625080\pi$
$828$	0	0
$829$	−14514.5	−0.608094	−0.304047	−	0.952657i	$-0.598338\pi$
$829$	−14514.5	−0.608094	−0.304047	+	0.952657i	$0.598338\pi$
$830$	0	0
$831$	1601.29	0.0668449
$832$	0	0
$833$	−2464.03	−0.102489
$834$	0	0
$835$	−517.957	−0.0214666
$836$	0	0
$837$	−15994.0	−0.660494
$838$	0	0
$839$	17152.8	0.705816	0.352908	−	0.935658i	$-0.385193\pi$
$839$	17152.8	0.705816	0.352908	+	0.935658i	$0.385193\pi$
$840$	0	0
$841$	−15987.0	−0.655499
$842$	0	0
$843$	6434.93	0.262907
$844$	0	0
$845$	7096.60	0.288912
$846$	0	0
$847$	3670.31	0.148894
$848$	0	0
$849$	−6584.18	−0.266158
$850$	0	0
$851$	5137.14	0.206932
$852$	0	0
$853$	5172.11	0.207608	0.103804	−	0.994598i	$-0.466899\pi$
$853$	5172.11	0.207608	0.103804	+	0.994598i	$0.466899\pi$
$854$	0	0
$855$	−27469.0	−1.09874
$856$	0	0
$857$	−10403.6	−0.414679	−0.207340	−	0.978269i	$-0.566481\pi$
$857$	−10403.6	−0.414679	−0.207340	+	0.978269i	$0.566481\pi$
$858$	0	0
$859$	−6774.47	−0.269083	−0.134541	−	0.990908i	$-0.542956\pi$
$859$	−6774.47	−0.269083	−0.134541	+	0.990908i	$0.542956\pi$
$860$	0	0
$861$	−2181.70	−0.0863556
$862$	0	0
$863$	16651.9	0.656822	0.328411	−	0.944535i	$-0.393487\pi$
$863$	16651.9	0.656822	0.328411	+	0.944535i	$0.393487\pi$
$864$	0	0
$865$	−36064.2	−1.41759
$866$	0	0
$867$	−8097.65	−0.317198
$868$	0	0
$869$	13876.1	0.541674
$870$	0	0
$871$	29782.2	1.15859
$872$	0	0
$873$	−14918.5	−0.578366
$874$	0	0
$875$	−4810.73	−0.185866
$876$	0	0
$877$	19710.8	0.758935	0.379467	−	0.925205i	$-0.376107\pi$
$877$	19710.8	0.758935	0.379467	+	0.925205i	$0.376107\pi$
$878$	0	0
$879$	−8137.01	−0.312235
$880$	0	0
$881$	−4425.01	−0.169219	−0.0846097	−	0.996414i	$-0.526964\pi$
$881$	−4425.01	−0.169219	−0.0846097	+	0.996414i	$0.526964\pi$
$882$	0	0
$883$	−2282.94	−0.0870069	−0.0435035	−	0.999053i	$-0.513852\pi$
$883$	−2282.94	−0.0870069	−0.0435035	+	0.999053i	$0.513852\pi$
$884$	0	0
$885$	3901.84	0.148202
$886$	0	0
$887$	−9323.50	−0.352934	−0.176467	−	0.984307i	$-0.556467\pi$
$887$	−9323.50	−0.352934	−0.176467	+	0.984307i	$0.556467\pi$
$888$	0	0
$889$	−7750.08	−0.292384
$890$	0	0
$891$	6680.88	0.251199
$892$	0	0
$893$	43263.9	1.62124
$894$	0	0
$895$	−19398.7	−0.724501
$896$	0	0
$897$	1449.11	0.0539402
$898$	0	0
$899$	−17171.7	−0.637052
$900$	0	0
$901$	655.002	0.0242190
$902$	0	0
$903$	2543.90	0.0937495
$904$	0	0
$905$	−919.940	−0.0337899
$906$	0	0
$907$	−18256.8	−0.668365	−0.334182	−	0.942508i	$-0.608460\pi$
$907$	−18256.8	−0.668365	−0.334182	+	0.942508i	$0.608460\pi$
$908$	0	0
$909$	−21329.4	−0.778276
$910$	0	0
$911$	6264.20	0.227818	0.113909	−	0.993491i	$-0.463663\pi$
$911$	6264.20	0.227818	0.113909	+	0.993491i	$0.463663\pi$
$912$	0	0
$913$	5627.39	0.203986
$914$	0	0
$915$	9129.92	0.329864
$916$	0	0
$917$	−2878.17	−0.103648
$918$	0	0
$919$	1446.18	0.0519098	0.0259549	−	0.999663i	$-0.491737\pi$
$919$	1446.18	0.0519098	0.0259549	+	0.999663i	$0.491737\pi$
$920$	0	0
$921$	13373.8	0.478481
$922$	0	0
$923$	29583.2	1.05498
$924$	0	0
$925$	−8855.41	−0.314772
$926$	0	0
$927$	−36695.1	−1.30013
$928$	0	0
$929$	14182.6	0.500877	0.250438	−	0.968133i	$-0.419425\pi$
$929$	14182.6	0.500877	0.250438	+	0.968133i	$0.419425\pi$
$930$	0	0
$931$	40876.1	1.43895
$932$	0	0
$933$	−13605.8	−0.477422
$934$	0	0
$935$	−892.571	−0.0312195
$936$	0	0
$937$	35308.5	1.23103	0.615517	−	0.788124i	$-0.288947\pi$
$937$	35308.5	1.23103	0.615517	+	0.788124i	$0.288947\pi$
$938$	0	0
$939$	−13559.5	−0.471243
$940$	0	0
$941$	−33968.4	−1.17677	−0.588384	−	0.808581i	$-0.700236\pi$
$941$	−33968.4	−1.17677	−0.588384	+	0.808581i	$0.700236\pi$
$942$	0	0
$943$	9519.10	0.328722
$944$	0	0
$945$	−2494.54	−0.0858703
$946$	0	0
$947$	49044.4	1.68292	0.841461	−	0.540317i	$-0.181696\pi$
$947$	49044.4	1.68292	0.841461	+	0.540317i	$0.181696\pi$
$948$	0	0
$949$	4453.78	0.152345
$950$	0	0
$951$	−9535.69	−0.325148
$952$	0	0
$953$	−22796.9	−0.774882	−0.387441	−	0.921894i	$-0.626641\pi$
$953$	−22796.9	−0.774882	−0.387441	+	0.921894i	$0.626641\pi$
$954$	0	0
$955$	16018.3	0.542764
$956$	0	0
$957$	−1994.81	−0.0673803
$958$	0	0
$959$	−3844.68	−0.129459
$960$	0	0
$961$	5303.95	0.178039
$962$	0	0
$963$	34563.1	1.15657
$964$	0	0
$965$	15977.4	0.532985
$966$	0	0
$967$	−32512.1	−1.08120	−0.540599	−	0.841280i	$-0.681803\pi$
$967$	−32512.1	−1.08120	−0.540599	+	0.841280i	$0.681803\pi$
$968$	0	0
$969$	−1513.95	−0.0501910
$970$	0	0
$971$	−58023.6	−1.91768	−0.958839	−	0.283951i	$-0.908355\pi$
$971$	−58023.6	−1.91768	−0.958839	+	0.283951i	$0.908355\pi$
$972$	0	0
$973$	2541.79	0.0837471
$974$	0	0
$975$	−2497.98	−0.0820507
$976$	0	0
$977$	−14305.7	−0.468453	−0.234226	−	0.972182i	$-0.575256\pi$
$977$	−14305.7	−0.468453	−0.234226	+	0.972182i	$0.575256\pi$
$978$	0	0
$979$	3234.26	0.105585
$980$	0	0
$981$	2986.13	0.0971864
$982$	0	0
$983$	15294.9	0.496267	0.248134	−	0.968726i	$-0.420183\pi$
$983$	15294.9	0.496267	0.248134	+	0.968726i	$0.420183\pi$
$984$	0	0
$985$	15957.6	0.516194
$986$	0	0
$987$	1857.91	0.0599169
$988$	0	0
$989$	−11099.4	−0.356867
$990$	0	0
$991$	4874.24	0.156241	0.0781207	−	0.996944i	$-0.475108\pi$
$991$	4874.24	0.156241	0.0781207	+	0.996944i	$0.475108\pi$
$992$	0	0
$993$	18723.4	0.598356
$994$	0	0
$995$	−12925.9	−0.411839
$996$	0	0
$997$	53025.2	1.68438	0.842189	−	0.539183i	$-0.181267\pi$
$997$	53025.2	1.68438	0.842189	+	0.539183i	$0.181267\pi$
$998$	0	0
$999$	−19069.0	−0.603920

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1472.4.a.bk.1.5		9
4.3	odd	2		1472.4.a.bj.1.5		9
8.3	odd	2		736.4.a.h.1.5	✓	9
8.5	even	2		736.4.a.i.1.5	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
736.4.a.h.1.5	✓	9	8.3	odd	2
736.4.a.i.1.5	yes	9	8.5	even	2
1472.4.a.bj.1.5		9	4.3	odd	2
1472.4.a.bk.1.5		9	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1472 = 2^{6} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1472.a (trivial)

\(f(q)\)	\(=\)	\(q+1.66678 q^{3} -9.23864 q^{5} -3.16263 q^{7} -24.2218 q^{9} +13.0566 q^{11} +37.8002 q^{13} -15.3988 q^{15} +7.39954 q^{17} -122.752 q^{19} -5.27141 q^{21} +23.0000 q^{23} -39.6475 q^{25} -85.3757 q^{27} -91.6626 q^{29} +187.336 q^{31} +21.7625 q^{33} +29.2184 q^{35} +223.354 q^{37} +63.0048 q^{39} +413.874 q^{41} -482.585 q^{43} +223.777 q^{45} -352.450 q^{47} -332.998 q^{49} +12.3334 q^{51} +88.5192 q^{53} -120.625 q^{55} -204.601 q^{57} -253.386 q^{59} -592.897 q^{61} +76.6046 q^{63} -349.223 q^{65} +787.886 q^{67} +38.3360 q^{69} +782.621 q^{71} +117.824 q^{73} -66.0838 q^{75} -41.2931 q^{77} +1062.77 q^{79} +511.687 q^{81} +431.000 q^{83} -68.3618 q^{85} -152.782 q^{87} +247.711 q^{89} -119.548 q^{91} +312.249 q^{93} +1134.06 q^{95} +615.910 q^{97} -316.254 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 42 q^{7} + 71 q^{9} - 66 q^{11} - 48 q^{13} + 90 q^{15} - 32 q^{17} - 6 q^{21} + 207 q^{23} + 135 q^{25} + 88 q^{29} + 556 q^{31} - 230 q^{33} - 368 q^{35} + 368 q^{37} + 468 q^{39} - 216 q^{41} + 552 q^{43}+ \cdots - 552 q^{99}+O(q^{100}) \) 9 * q + 42 * q^7 + 71 * q^9 - 66 * q^11 - 48 * q^13 + 90 * q^15 - 32 * q^17 - 6 * q^21 + 207 * q^23 + 135 * q^25 + 88 * q^29 + 556 * q^31 - 230 * q^33 - 368 * q^35 + 368 * q^37 + 468 * q^39 - 216 * q^41 + 552 * q^43 + 206 * q^45 + 1300 * q^47 + 1041 * q^49 - 1658 * q^51 - 510 * q^53 + 1948 * q^55 - 804 * q^57 - 1444 * q^59 + 882 * q^61 + 3128 * q^63 + 334 * q^65 - 1186 * q^67 + 2440 * q^71 - 368 * q^73 - 1636 * q^75 - 556 * q^77 + 2576 * q^79 + 1069 * q^81 - 1710 * q^83 - 640 * q^85 + 4844 * q^87 - 474 * q^89 - 2790 * q^91 - 1178 * q^93 + 4564 * q^95 - 884 * q^97 - 552 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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