LMFDB - Embedded newform 552.4.a.d.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [552,4,Mod(1,552)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(552, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("552.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 552 = 2^{3} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	552.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-6,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$32.5690543232$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{93}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 23 $ x^2 - x - 23
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$5.32183$ of defining polynomial
Character	$\chi$	$=$	552.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +10.6437 q^{5} -4.64365 q^{7} +9.00000 q^{9} +30.5746 q^{11} -12.7127 q^{13} -31.9310 q^{15} -38.5056 q^{17} +124.506 q^{19} +13.9310 q^{21} +23.0000 q^{23} -11.7127 q^{25} -27.0000 q^{27} -187.011 q^{29} +222.437 q^{31} -91.7238 q^{33} -49.4254 q^{35} +124.160 q^{37} +38.1381 q^{39} -186.873 q^{41} -32.9198 q^{43} +95.7929 q^{45} +305.425 q^{47} -321.437 q^{49} +115.517 q^{51} +603.655 q^{53} +325.425 q^{55} -373.517 q^{57} +413.702 q^{59} +353.862 q^{61} -41.7929 q^{63} -135.310 q^{65} +598.345 q^{67} -69.0000 q^{69} -412.044 q^{71} +562.276 q^{73} +35.1381 q^{75} -141.978 q^{77} -896.871 q^{79} +81.0000 q^{81} +1043.17 q^{83} -409.840 q^{85} +561.033 q^{87} +1254.55 q^{89} +59.0333 q^{91} -667.310 q^{93} +1325.19 q^{95} +517.033 q^{97} +275.171 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} + 2 q^{5} + 10 q^{7} + 18 q^{9} - 16 q^{11} - 64 q^{13} - 6 q^{15} + 58 q^{17} + 114 q^{19} - 30 q^{21} + 46 q^{23} - 62 q^{25} - 54 q^{27} - 104 q^{29} + 252 q^{31} + 48 q^{33} - 176 q^{35}+ \cdots - 144 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 + 2 * q^5 + 10 * q^7 + 18 * q^9 - 16 * q^11 - 64 * q^13 - 6 * q^15 + 58 * q^17 + 114 * q^19 - 30 * q^21 + 46 * q^23 - 62 * q^25 - 54 * q^27 - 104 * q^29 + 252 * q^31 + 48 * q^33 - 176 * q^35 - 176 * q^37 + 192 * q^39 + 12 * q^41 - 278 * q^43 + 18 * q^45 + 688 * q^47 - 450 * q^49 - 174 * q^51 + 918 * q^53 + 728 * q^55 - 342 * q^57 + 1136 * q^59 + 592 * q^61 + 90 * q^63 + 308 * q^65 + 1486 * q^67 - 138 * q^69 + 256 * q^71 + 1356 * q^73 + 186 * q^75 - 824 * q^77 + 270 * q^79 + 162 * q^81 + 1392 * q^83 - 1244 * q^85 + 312 * q^87 + 1294 * q^89 - 692 * q^91 - 756 * q^93 + 1416 * q^95 + 224 * q^97 - 144 * 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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	10.6437	0.951997	0.475999	−	0.879446i	$-0.342087\pi$
$5$	10.6437	0.951997	0.475999	+	0.879446i	$0.342087\pi$
$6$	0	0
$7$	−4.64365	−0.250734	−0.125367	−	0.992110i	$-0.540011\pi$
$7$	−4.64365	−0.250734	−0.125367	+	0.992110i	$0.540011\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	30.5746	0.838054	0.419027	−	0.907974i	$-0.362371\pi$
$11$	30.5746	0.838054	0.419027	+	0.907974i	$0.362371\pi$
$12$	0	0
$13$	−12.7127	−0.271221	−0.135610	−	0.990762i	$-0.543300\pi$
$13$	−12.7127	−0.271221	−0.135610	+	0.990762i	$0.543300\pi$
$14$	0	0
$15$	−31.9310	−0.549636
$16$	0	0
$17$	−38.5056	−0.549351	−0.274676	−	0.961537i	$-0.588570\pi$
$17$	−38.5056	−0.549351	−0.274676	+	0.961537i	$0.588570\pi$
$18$	0	0
$19$	124.506	1.50334	0.751672	−	0.659537i	$-0.229248\pi$
$19$	124.506	1.50334	0.751672	+	0.659537i	$0.229248\pi$
$20$	0	0
$21$	13.9310	0.144761
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	−11.7127	−0.0937016
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−187.011	−1.19749	−0.598743	−	0.800941i	$-0.704333\pi$
$29$	−187.011	−1.19749	−0.598743	+	0.800941i	$0.704333\pi$
$30$	0	0
$31$	222.437	1.28873	0.644367	−	0.764716i	$-0.277121\pi$
$31$	222.437	1.28873	0.644367	+	0.764716i	$0.277121\pi$
$32$	0	0
$33$	−91.7238	−0.483850
$34$	0	0
$35$	−49.4254	−0.238698
$36$	0	0
$37$	124.160	0.551671	0.275836	−	0.961205i	$-0.411045\pi$
$37$	124.160	0.551671	0.275836	+	0.961205i	$0.411045\pi$
$38$	0	0
$39$	38.1381	0.156589
$40$	0	0
$41$	−186.873	−0.711821	−0.355911	−	0.934520i	$-0.615829\pi$
$41$	−186.873	−0.711821	−0.355911	+	0.934520i	$0.615829\pi$
$42$	0	0
$43$	−32.9198	−0.116750	−0.0583748	−	0.998295i	$-0.518592\pi$
$43$	−32.9198	−0.116750	−0.0583748	+	0.998295i	$0.518592\pi$
$44$	0	0
$45$	95.7929	0.317332
$46$	0	0
$47$	305.425	0.947891	0.473945	−	0.880554i	$-0.342829\pi$
$47$	305.425	0.947891	0.473945	+	0.880554i	$0.342829\pi$
$48$	0	0
$49$	−321.437	−0.937133
$50$	0	0
$51$	115.517	0.317168
$52$	0	0
$53$	603.655	1.56450	0.782249	−	0.622966i	$-0.214073\pi$
$53$	603.655	1.56450	0.782249	+	0.622966i	$0.214073\pi$
$54$	0	0
$55$	325.425	0.797825
$56$	0	0
$57$	−373.517	−0.867956
$58$	0	0
$59$	413.702	0.912870	0.456435	−	0.889757i	$-0.349126\pi$
$59$	413.702	0.912870	0.456435	+	0.889757i	$0.349126\pi$
$60$	0	0
$61$	353.862	0.742744	0.371372	−	0.928484i	$-0.378887\pi$
$61$	353.862	0.742744	0.371372	+	0.928484i	$0.378887\pi$
$62$	0	0
$63$	−41.7929	−0.0835779
$64$	0	0
$65$	−135.310	−0.258201
$66$	0	0
$67$	598.345	1.09104	0.545519	−	0.838099i	$-0.316333\pi$
$67$	598.345	1.09104	0.545519	+	0.838099i	$0.316333\pi$
$68$	0	0
$69$	−69.0000	−0.120386
$70$	0	0
$71$	−412.044	−0.688742	−0.344371	−	0.938834i	$-0.611908\pi$
$71$	−412.044	−0.688742	−0.344371	+	0.938834i	$0.611908\pi$
$72$	0	0
$73$	562.276	0.901499	0.450750	−	0.892650i	$-0.351157\pi$
$73$	562.276	0.901499	0.450750	+	0.892650i	$0.351157\pi$
$74$	0	0
$75$	35.1381	0.0540986
$76$	0	0
$77$	−141.978	−0.210128
$78$	0	0
$79$	−896.871	−1.27729	−0.638644	−	0.769502i	$-0.720505\pi$
$79$	−896.871	−1.27729	−0.638644	+	0.769502i	$0.720505\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	1043.17	1.37955	0.689777	−	0.724022i	$-0.257709\pi$
$83$	1043.17	1.37955	0.689777	+	0.724022i	$0.257709\pi$
$84$	0	0
$85$	−409.840	−0.522981
$86$	0	0
$87$	561.033	0.691369
$88$	0	0
$89$	1254.55	1.49418	0.747090	−	0.664723i	$-0.231450\pi$
$89$	1254.55	1.49418	0.747090	+	0.664723i	$0.231450\pi$
$90$	0	0
$91$	59.0333	0.0680041
$92$	0	0
$93$	−667.310	−0.744051
$94$	0	0
$95$	1325.19	1.43118
$96$	0	0
$97$	517.033	0.541204	0.270602	−	0.962691i	$-0.412777\pi$
$97$	517.033	0.541204	0.270602	+	0.962691i	$0.412777\pi$
$98$	0	0
$99$	275.171	0.279351
$100$	0	0
$101$	1351.56	1.33154	0.665768	−	0.746159i	$-0.268104\pi$
$101$	1351.56	1.33154	0.665768	+	0.746159i	$0.268104\pi$
$102$	0	0
$103$	1206.39	1.15407	0.577035	−	0.816720i	$-0.304210\pi$
$103$	1206.39	1.15407	0.577035	+	0.816720i	$0.304210\pi$
$104$	0	0
$105$	148.276	0.137812
$106$	0	0
$107$	572.552	0.517297	0.258648	−	0.965972i	$-0.416723\pi$
$107$	572.552	0.517297	0.258648	+	0.965972i	$0.416723\pi$
$108$	0	0
$109$	96.8508	0.0851066	0.0425533	−	0.999094i	$-0.486451\pi$
$109$	96.8508	0.0851066	0.0425533	+	0.999094i	$0.486451\pi$
$110$	0	0
$111$	−372.481	−0.318507
$112$	0	0
$113$	923.423	0.768746	0.384373	−	0.923178i	$-0.374418\pi$
$113$	923.423	0.768746	0.384373	+	0.923178i	$0.374418\pi$
$114$	0	0
$115$	244.804	0.198505
$116$	0	0
$117$	−114.414	−0.0904069
$118$	0	0
$119$	178.806	0.137741
$120$	0	0
$121$	−396.194	−0.297666
$122$	0	0
$123$	560.619	0.410970
$124$	0	0
$125$	−1455.12	−1.04120
$126$	0	0
$127$	1853.97	1.29538	0.647691	−	0.761903i	$-0.275735\pi$
$127$	1853.97	1.29538	0.647691	+	0.761903i	$0.275735\pi$
$128$	0	0
$129$	98.7595	0.0674054
$130$	0	0
$131$	−1879.86	−1.25377	−0.626885	−	0.779112i	$-0.715670\pi$
$131$	−1879.86	−1.25377	−0.626885	+	0.779112i	$0.715670\pi$
$132$	0	0
$133$	−578.160	−0.376939
$134$	0	0
$135$	−287.379	−0.183212
$136$	0	0
$137$	1192.94	0.743938	0.371969	−	0.928245i	$-0.378683\pi$
$137$	1192.94	0.743938	0.371969	+	0.928245i	$0.378683\pi$
$138$	0	0
$139$	−1986.71	−1.21230	−0.606152	−	0.795349i	$-0.707288\pi$
$139$	−1986.71	−1.21230	−0.606152	+	0.795349i	$0.707288\pi$
$140$	0	0
$141$	−916.276	−0.547265
$142$	0	0
$143$	−388.686	−0.227297
$144$	0	0
$145$	−1990.48	−1.14000
$146$	0	0
$147$	964.310	0.541054
$148$	0	0
$149$	−729.766	−0.401240	−0.200620	−	0.979669i	$-0.564296\pi$
$149$	−729.766	−0.401240	−0.200620	+	0.979669i	$0.564296\pi$
$150$	0	0
$151$	−1675.40	−0.902930	−0.451465	−	0.892289i	$-0.649098\pi$
$151$	−1675.40	−0.902930	−0.451465	+	0.892289i	$0.649098\pi$
$152$	0	0
$153$	−346.550	−0.183117
$154$	0	0
$155$	2367.54	1.22687
$156$	0	0
$157$	−1025.72	−0.521409	−0.260705	−	0.965419i	$-0.583955\pi$
$157$	−1025.72	−0.521409	−0.260705	+	0.965419i	$0.583955\pi$
$158$	0	0
$159$	−1810.96	−0.903263
$160$	0	0
$161$	−106.804	−0.0522816
$162$	0	0
$163$	989.376	0.475423	0.237711	−	0.971336i	$-0.423603\pi$
$163$	989.376	0.475423	0.237711	+	0.971336i	$0.423603\pi$
$164$	0	0
$165$	−976.276	−0.460624
$166$	0	0
$167$	1139.30	0.527916	0.263958	−	0.964534i	$-0.414972\pi$
$167$	1139.30	0.527916	0.263958	+	0.964534i	$0.414972\pi$
$168$	0	0
$169$	−2035.39	−0.926439
$170$	0	0
$171$	1120.55	0.501115
$172$	0	0
$173$	−2542.50	−1.11736	−0.558679	−	0.829384i	$-0.688692\pi$
$173$	−2542.50	−1.11736	−0.558679	+	0.829384i	$0.688692\pi$
$174$	0	0
$175$	54.3897	0.0234941
$176$	0	0
$177$	−1241.10	−0.527046
$178$	0	0
$179$	1230.25	0.513707	0.256853	−	0.966450i	$-0.417314\pi$
$179$	1230.25	0.513707	0.256853	+	0.966450i	$0.417314\pi$
$180$	0	0
$181$	−1858.28	−0.763119	−0.381560	−	0.924344i	$-0.624613\pi$
$181$	−1858.28	−0.763119	−0.381560	+	0.924344i	$0.624613\pi$
$182$	0	0
$183$	−1061.59	−0.428823
$184$	0	0
$185$	1321.52	0.525189
$186$	0	0
$187$	−1177.29	−0.460386
$188$	0	0
$189$	125.379	0.0482537
$190$	0	0
$191$	3272.27	1.23965	0.619825	−	0.784740i	$-0.287203\pi$
$191$	3272.27	1.23965	0.619825	+	0.784740i	$0.287203\pi$
$192$	0	0
$193$	198.183	0.0739145	0.0369572	−	0.999317i	$-0.488233\pi$
$193$	198.183	0.0739145	0.0369572	+	0.999317i	$0.488233\pi$
$194$	0	0
$195$	405.929	0.149073
$196$	0	0
$197$	−38.2270	−0.0138252	−0.00691259	−	0.999976i	$-0.502200\pi$
$197$	−38.2270	−0.0138252	−0.00691259	+	0.999976i	$0.502200\pi$
$198$	0	0
$199$	−2771.81	−0.987379	−0.493690	−	0.869638i	$-0.664352\pi$
$199$	−2771.81	−0.987379	−0.493690	+	0.869638i	$0.664352\pi$
$200$	0	0
$201$	−1795.04	−0.629911
$202$	0	0
$203$	868.414	0.300250
$204$	0	0
$205$	−1989.01	−0.677652
$206$	0	0
$207$	207.000	0.0695048
$208$	0	0
$209$	3806.71	1.25988
$210$	0	0
$211$	−3597.83	−1.17386	−0.586931	−	0.809637i	$-0.699664\pi$
$211$	−3597.83	−1.17386	−0.586931	+	0.809637i	$0.699664\pi$
$212$	0	0
$213$	1236.13	0.397645
$214$	0	0
$215$	−350.387	−0.111145
$216$	0	0
$217$	−1032.92	−0.323129
$218$	0	0
$219$	−1686.83	−0.520481
$220$	0	0
$221$	489.510	0.148995
$222$	0	0
$223$	−1565.97	−0.470247	−0.235123	−	0.971966i	$-0.575549\pi$
$223$	−1565.97	−0.470247	−0.235123	+	0.971966i	$0.575549\pi$
$224$	0	0
$225$	−105.414	−0.0312339
$226$	0	0
$227$	−477.386	−0.139582	−0.0697912	−	0.997562i	$-0.522233\pi$
$227$	−477.386	−0.139582	−0.0697912	+	0.997562i	$0.522233\pi$
$228$	0	0
$229$	3910.27	1.12837	0.564187	−	0.825647i	$-0.309190\pi$
$229$	3910.27	1.12837	0.564187	+	0.825647i	$0.309190\pi$
$230$	0	0
$231$	425.933	0.121318
$232$	0	0
$233$	−3391.83	−0.953676	−0.476838	−	0.878991i	$-0.658217\pi$
$233$	−3391.83	−0.953676	−0.476838	+	0.878991i	$0.658217\pi$
$234$	0	0
$235$	3250.84	0.902389
$236$	0	0
$237$	2690.61	0.737443
$238$	0	0
$239$	1108.70	0.300065	0.150032	−	0.988681i	$-0.452062\pi$
$239$	1108.70	0.300065	0.150032	+	0.988681i	$0.452062\pi$
$240$	0	0
$241$	−3612.90	−0.965674	−0.482837	−	0.875710i	$-0.660394\pi$
$241$	−3612.90	−0.965674	−0.482837	+	0.875710i	$0.660394\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−3421.26	−0.892148
$246$	0	0
$247$	−1582.80	−0.407738
$248$	0	0
$249$	−3129.51	−0.796486
$250$	0	0
$251$	3388.00	0.851986	0.425993	−	0.904727i	$-0.359925\pi$
$251$	3388.00	0.851986	0.425993	+	0.904727i	$0.359925\pi$
$252$	0	0
$253$	703.216	0.174746
$254$	0	0
$255$	1229.52	0.301943
$256$	0	0
$257$	1201.16	0.291542	0.145771	−	0.989318i	$-0.453434\pi$
$257$	1201.16	0.291542	0.145771	+	0.989318i	$0.453434\pi$
$258$	0	0
$259$	−576.557	−0.138322
$260$	0	0
$261$	−1683.10	−0.399162
$262$	0	0
$263$	4770.85	1.11857	0.559283	−	0.828977i	$-0.311076\pi$
$263$	4770.85	1.11857	0.559283	+	0.828977i	$0.311076\pi$
$264$	0	0
$265$	6425.09	1.48940
$266$	0	0
$267$	−3763.65	−0.862665
$268$	0	0
$269$	−6389.94	−1.44833	−0.724167	−	0.689625i	$-0.757775\pi$
$269$	−6389.94	−1.44833	−0.724167	+	0.689625i	$0.757775\pi$
$270$	0	0
$271$	−951.916	−0.213376	−0.106688	−	0.994293i	$-0.534025\pi$
$271$	−951.916	−0.213376	−0.106688	+	0.994293i	$0.534025\pi$
$272$	0	0
$273$	−177.100	−0.0392622
$274$	0	0
$275$	−358.111	−0.0785270
$276$	0	0
$277$	7.01588	0.00152182	0.000760908	−	1.00000i	$-0.499758\pi$
$277$	7.01588	0.00152182	0.000760908		1.00000i	$0.499758\pi$
$278$	0	0
$279$	2001.93	0.429578
$280$	0	0
$281$	−3166.40	−0.672213	−0.336106	−	0.941824i	$-0.609110\pi$
$281$	−3166.40	−0.672213	−0.336106	+	0.941824i	$0.609110\pi$
$282$	0	0
$283$	1761.47	0.369995	0.184998	−	0.982739i	$-0.440772\pi$
$283$	1761.47	0.369995	0.184998	+	0.982739i	$0.440772\pi$
$284$	0	0
$285$	−3975.58	−0.826292
$286$	0	0
$287$	867.773	0.178477
$288$	0	0
$289$	−3430.32	−0.698213
$290$	0	0
$291$	−1551.10	−0.312464
$292$	0	0
$293$	−4875.69	−0.972154	−0.486077	−	0.873916i	$-0.661573\pi$
$293$	−4875.69	−0.972154	−0.486077	+	0.873916i	$0.661573\pi$
$294$	0	0
$295$	4403.30	0.869050
$296$	0	0
$297$	−825.514	−0.161283
$298$	0	0
$299$	−292.392	−0.0565534
$300$	0	0
$301$	152.868	0.0292730
$302$	0	0
$303$	−4054.68	−0.768763
$304$	0	0
$305$	3766.38	0.707090
$306$	0	0
$307$	−9238.57	−1.71750	−0.858751	−	0.512393i	$-0.828759\pi$
$307$	−9238.57	−1.71750	−0.858751	+	0.512393i	$0.828759\pi$
$308$	0	0
$309$	−3619.17	−0.666302
$310$	0	0
$311$	4549.75	0.829558	0.414779	−	0.909922i	$-0.363859\pi$
$311$	4549.75	0.829558	0.414779	+	0.909922i	$0.363859\pi$
$312$	0	0
$313$	9651.45	1.74291	0.871457	−	0.490471i	$-0.163175\pi$
$313$	9651.45	1.74291	0.871457	+	0.490471i	$0.163175\pi$
$314$	0	0
$315$	−444.829	−0.0795659
$316$	0	0
$317$	−2043.28	−0.362025	−0.181012	−	0.983481i	$-0.557937\pi$
$317$	−2043.28	−0.362025	−0.181012	+	0.983481i	$0.557937\pi$
$318$	0	0
$319$	−5717.79	−1.00356
$320$	0	0
$321$	−1717.66	−0.298661
$322$	0	0
$323$	−4794.16	−0.825864
$324$	0	0
$325$	148.900	0.0254138
$326$	0	0
$327$	−290.552	−0.0491363
$328$	0	0
$329$	−1418.29	−0.237668
$330$	0	0
$331$	−8914.78	−1.48036	−0.740182	−	0.672406i	$-0.765261\pi$
$331$	−8914.78	−1.48036	−0.740182	+	0.672406i	$0.765261\pi$
$332$	0	0
$333$	1117.44	0.183890
$334$	0	0
$335$	6368.58	1.03866
$336$	0	0
$337$	−8497.74	−1.37360	−0.686798	−	0.726849i	$-0.740984\pi$
$337$	−8497.74	−1.37360	−0.686798	+	0.726849i	$0.740984\pi$
$338$	0	0
$339$	−2770.27	−0.443836
$340$	0	0
$341$	6800.91	1.08003
$342$	0	0
$343$	3085.41	0.485704
$344$	0	0
$345$	−734.412	−0.114607
$346$	0	0
$347$	−109.781	−0.0169837	−0.00849186	−	0.999964i	$-0.502703\pi$
$347$	−109.781	−0.0169837	−0.00849186	+	0.999964i	$0.502703\pi$
$348$	0	0
$349$	3883.37	0.595621	0.297811	−	0.954625i	$-0.403744\pi$
$349$	3883.37	0.595621	0.297811	+	0.954625i	$0.403744\pi$
$350$	0	0
$351$	343.243	0.0521964
$352$	0	0
$353$	3298.47	0.497336	0.248668	−	0.968589i	$-0.420007\pi$
$353$	3298.47	0.497336	0.248668	+	0.968589i	$0.420007\pi$
$354$	0	0
$355$	−4385.66	−0.655680
$356$	0	0
$357$	−536.419	−0.0795247
$358$	0	0
$359$	7435.67	1.09315	0.546573	−	0.837411i	$-0.315932\pi$
$359$	7435.67	1.09315	0.546573	+	0.837411i	$0.315932\pi$
$360$	0	0
$361$	8642.63	1.26004
$362$	0	0
$363$	1188.58	0.171858
$364$	0	0
$365$	5984.67	0.858225
$366$	0	0
$367$	−5618.69	−0.799164	−0.399582	−	0.916698i	$-0.630845\pi$
$367$	−5618.69	−0.799164	−0.399582	+	0.916698i	$0.630845\pi$
$368$	0	0
$369$	−1681.86	−0.237274
$370$	0	0
$371$	−2803.16	−0.392272
$372$	0	0
$373$	−502.405	−0.0697414	−0.0348707	−	0.999392i	$-0.511102\pi$
$373$	−502.405	−0.0697414	−0.0348707	+	0.999392i	$0.511102\pi$
$374$	0	0
$375$	4365.37	0.601138
$376$	0	0
$377$	2377.42	0.324783
$378$	0	0
$379$	−432.190	−0.0585754	−0.0292877	−	0.999571i	$-0.509324\pi$
$379$	−432.190	−0.0585754	−0.0292877	+	0.999571i	$0.509324\pi$
$380$	0	0
$381$	−5561.92	−0.747889
$382$	0	0
$383$	−4051.40	−0.540514	−0.270257	−	0.962788i	$-0.587109\pi$
$383$	−4051.40	−0.540514	−0.270257	+	0.962788i	$0.587109\pi$
$384$	0	0
$385$	−1511.16	−0.200041
$386$	0	0
$387$	−296.279	−0.0389165
$388$	0	0
$389$	1358.95	0.177125	0.0885623	−	0.996071i	$-0.471773\pi$
$389$	1358.95	0.177125	0.0885623	+	0.996071i	$0.471773\pi$
$390$	0	0
$391$	−885.628	−0.114548
$392$	0	0
$393$	5639.57	0.723864
$394$	0	0
$395$	−9545.98	−1.21598
$396$	0	0
$397$	−8722.62	−1.10271	−0.551355	−	0.834271i	$-0.685889\pi$
$397$	−8722.62	−1.10271	−0.551355	+	0.834271i	$0.685889\pi$
$398$	0	0
$399$	1734.48	0.217626
$400$	0	0
$401$	−62.7817	−0.00781838	−0.00390919	−	0.999992i	$-0.501244\pi$
$401$	−62.7817	−0.00781838	−0.00390919	+	0.999992i	$0.501244\pi$
$402$	0	0
$403$	−2827.77	−0.349531
$404$	0	0
$405$	862.136	0.105777
$406$	0	0
$407$	3796.15	0.462330
$408$	0	0
$409$	−1025.00	−0.123919	−0.0619595	−	0.998079i	$-0.519735\pi$
$409$	−1025.00	−0.123919	−0.0619595	+	0.998079i	$0.519735\pi$
$410$	0	0
$411$	−3578.81	−0.429513
$412$	0	0
$413$	−1921.09	−0.228887
$414$	0	0
$415$	11103.2	1.31333
$416$	0	0
$417$	5960.12	0.699925
$418$	0	0
$419$	−7162.60	−0.835121	−0.417561	−	0.908649i	$-0.637115\pi$
$419$	−7162.60	−0.835121	−0.417561	+	0.908649i	$0.637115\pi$
$420$	0	0
$421$	17091.8	1.97863	0.989314	−	0.145804i	$-0.0465769\pi$
$421$	17091.8	1.97863	0.989314	+	0.145804i	$0.0465769\pi$
$422$	0	0
$423$	2748.83	0.315964
$424$	0	0
$425$	451.004	0.0514751
$426$	0	0
$427$	−1643.21	−0.186231
$428$	0	0
$429$	1166.06	0.131230
$430$	0	0
$431$	11740.0	1.31206	0.656029	−	0.754736i	$-0.272235\pi$
$431$	11740.0	1.31206	0.656029	+	0.754736i	$0.272235\pi$
$432$	0	0
$433$	5440.54	0.603824	0.301912	−	0.953336i	$-0.402375\pi$
$433$	5440.54	0.603824	0.301912	+	0.953336i	$0.402375\pi$
$434$	0	0
$435$	5971.44	0.658181
$436$	0	0
$437$	2863.63	0.313469
$438$	0	0
$439$	8576.16	0.932387	0.466193	−	0.884683i	$-0.345625\pi$
$439$	8576.16	0.932387	0.466193	+	0.884683i	$0.345625\pi$
$440$	0	0
$441$	−2892.93	−0.312378
$442$	0	0
$443$	14185.6	1.52139	0.760696	−	0.649108i	$-0.224858\pi$
$443$	14185.6	1.52139	0.760696	+	0.649108i	$0.224858\pi$
$444$	0	0
$445$	13353.0	1.42246
$446$	0	0
$447$	2189.30	0.231656
$448$	0	0
$449$	−455.203	−0.0478449	−0.0239225	−	0.999714i	$-0.507615\pi$
$449$	−455.203	−0.0478449	−0.0239225	+	0.999714i	$0.507615\pi$
$450$	0	0
$451$	−5713.57	−0.596544
$452$	0	0
$453$	5026.21	0.521307
$454$	0	0
$455$	628.330	0.0647397
$456$	0	0
$457$	−4388.32	−0.449184	−0.224592	−	0.974453i	$-0.572105\pi$
$457$	−4388.32	−0.449184	−0.224592	+	0.974453i	$0.572105\pi$
$458$	0	0
$459$	1039.65	0.105723
$460$	0	0
$461$	6335.11	0.640034	0.320017	−	0.947412i	$-0.396311\pi$
$461$	6335.11	0.640034	0.320017	+	0.947412i	$0.396311\pi$
$462$	0	0
$463$	−18364.5	−1.84335	−0.921674	−	0.387966i	$-0.873178\pi$
$463$	−18364.5	−1.84335	−0.921674	+	0.387966i	$0.873178\pi$
$464$	0	0
$465$	−7102.61	−0.708335
$466$	0	0
$467$	11326.3	1.12231	0.561153	−	0.827712i	$-0.310358\pi$
$467$	11326.3	1.12231	0.561153	+	0.827712i	$0.310358\pi$
$468$	0	0
$469$	−2778.51	−0.273560
$470$	0	0
$471$	3077.16	0.301036
$472$	0	0
$473$	−1006.51	−0.0978423
$474$	0	0
$475$	−1458.30	−0.140866
$476$	0	0
$477$	5432.89	0.521499
$478$	0	0
$479$	−2283.85	−0.217853	−0.108927	−	0.994050i	$-0.534741\pi$
$479$	−2283.85	−0.217853	−0.108927	+	0.994050i	$0.534741\pi$
$480$	0	0
$481$	−1578.41	−0.149625
$482$	0	0
$483$	320.412	0.0301848
$484$	0	0
$485$	5503.12	0.515225
$486$	0	0
$487$	1845.24	0.171696	0.0858478	−	0.996308i	$-0.472640\pi$
$487$	1845.24	0.171696	0.0858478	+	0.996308i	$0.472640\pi$
$488$	0	0
$489$	−2968.13	−0.274486
$490$	0	0
$491$	−16279.4	−1.49629	−0.748147	−	0.663534i	$-0.769056\pi$
$491$	−16279.4	−1.49629	−0.748147	+	0.663534i	$0.769056\pi$
$492$	0	0
$493$	7200.97	0.657840
$494$	0	0
$495$	2928.83	0.265942
$496$	0	0
$497$	1913.39	0.172691
$498$	0	0
$499$	−12925.5	−1.15956	−0.579782	−	0.814771i	$-0.696862\pi$
$499$	−12925.5	−1.15956	−0.579782	+	0.814771i	$0.696862\pi$
$500$	0	0
$501$	−3417.91	−0.304793
$502$	0	0
$503$	12898.0	1.14333	0.571664	−	0.820488i	$-0.306298\pi$
$503$	12898.0	1.14333	0.571664	+	0.820488i	$0.306298\pi$
$504$	0	0
$505$	14385.5	1.26762
$506$	0	0
$507$	6106.16	0.534880
$508$	0	0
$509$	−20960.9	−1.82530	−0.912649	−	0.408744i	$-0.865967\pi$
$509$	−20960.9	−1.82530	−0.912649	+	0.408744i	$0.865967\pi$
$510$	0	0
$511$	−2611.01	−0.226036
$512$	0	0
$513$	−3361.65	−0.289319
$514$	0	0
$515$	12840.4	1.09867
$516$	0	0
$517$	9338.26	0.794383
$518$	0	0
$519$	7627.51	0.645107
$520$	0	0
$521$	15683.9	1.31886	0.659429	−	0.751767i	$-0.270798\pi$
$521$	15683.9	1.31886	0.659429	+	0.751767i	$0.270798\pi$
$522$	0	0
$523$	−5107.30	−0.427011	−0.213505	−	0.976942i	$-0.568488\pi$
$523$	−5107.30	−0.427011	−0.213505	+	0.976942i	$0.568488\pi$
$524$	0	0
$525$	−163.169	−0.0135643
$526$	0	0
$527$	−8565.04	−0.707968
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	3723.31	0.304290
$532$	0	0
$533$	2375.66	0.193061
$534$	0	0
$535$	6094.05	0.492465
$536$	0	0
$537$	−3690.76	−0.296589
$538$	0	0
$539$	−9827.79	−0.785367
$540$	0	0
$541$	4135.63	0.328659	0.164329	−	0.986406i	$-0.447454\pi$
$541$	4135.63	0.328659	0.164329	+	0.986406i	$0.447454\pi$
$542$	0	0
$543$	5574.83	0.440587
$544$	0	0
$545$	1030.85	0.0810213
$546$	0	0
$547$	−15256.1	−1.19251	−0.596257	−	0.802794i	$-0.703346\pi$
$547$	−15256.1	−1.19251	−0.596257	+	0.802794i	$0.703346\pi$
$548$	0	0
$549$	3184.76	0.247581
$550$	0	0
$551$	−23283.9	−1.80023
$552$	0	0
$553$	4164.75	0.320259
$554$	0	0
$555$	−3964.56	−0.303218
$556$	0	0
$557$	17303.7	1.31630	0.658152	−	0.752885i	$-0.271339\pi$
$557$	17303.7	1.31630	0.658152	+	0.752885i	$0.271339\pi$
$558$	0	0
$559$	418.500	0.0316649
$560$	0	0
$561$	3531.88	0.265804
$562$	0	0
$563$	−14821.8	−1.10953	−0.554766	−	0.832006i	$-0.687192\pi$
$563$	−14821.8	−1.10953	−0.554766	+	0.832006i	$0.687192\pi$
$564$	0	0
$565$	9828.59	0.731844
$566$	0	0
$567$	−376.136	−0.0278593
$568$	0	0
$569$	25352.3	1.86788	0.933939	−	0.357433i	$-0.116348\pi$
$569$	25352.3	1.86788	0.933939	+	0.357433i	$0.116348\pi$
$570$	0	0
$571$	−3674.00	−0.269268	−0.134634	−	0.990895i	$-0.542986\pi$
$571$	−3674.00	−0.269268	−0.134634	+	0.990895i	$0.542986\pi$
$572$	0	0
$573$	−9816.81	−0.715713
$574$	0	0
$575$	−269.392	−0.0195381
$576$	0	0
$577$	10442.8	0.753445	0.376723	−	0.926326i	$-0.377051\pi$
$577$	10442.8	0.753445	0.376723	+	0.926326i	$0.377051\pi$
$578$	0	0
$579$	−594.548	−0.0426745
$580$	0	0
$581$	−4844.12	−0.345900
$582$	0	0
$583$	18456.5	1.31113
$584$	0	0
$585$	−1217.79	−0.0860671
$586$	0	0
$587$	−12505.2	−0.879293	−0.439647	−	0.898171i	$-0.644896\pi$
$587$	−12505.2	−0.879293	−0.439647	+	0.898171i	$0.644896\pi$
$588$	0	0
$589$	27694.6	1.93741
$590$	0	0
$591$	114.681	0.00798197
$592$	0	0
$593$	−2119.35	−0.146765	−0.0733823	−	0.997304i	$-0.523379\pi$
$593$	−2119.35	−0.146765	−0.0733823	+	0.997304i	$0.523379\pi$
$594$	0	0
$595$	1903.15	0.131129
$596$	0	0
$597$	8315.43	0.570064
$598$	0	0
$599$	5667.32	0.386579	0.193289	−	0.981142i	$-0.438084\pi$
$599$	5667.32	0.386579	0.193289	+	0.981142i	$0.438084\pi$
$600$	0	0
$601$	−6485.13	−0.440156	−0.220078	−	0.975482i	$-0.570631\pi$
$601$	−6485.13	−0.440156	−0.220078	+	0.975482i	$0.570631\pi$
$602$	0	0
$603$	5385.11	0.363679
$604$	0	0
$605$	−4216.95	−0.283377
$606$	0	0
$607$	23917.8	1.59933	0.799666	−	0.600445i	$-0.205010\pi$
$607$	23917.8	1.59933	0.799666	+	0.600445i	$0.205010\pi$
$608$	0	0
$609$	−2605.24	−0.173349
$610$	0	0
$611$	−3882.78	−0.257088
$612$	0	0
$613$	−13727.9	−0.904513	−0.452256	−	0.891888i	$-0.649381\pi$
$613$	−13727.9	−0.904513	−0.452256	+	0.891888i	$0.649381\pi$
$614$	0	0
$615$	5967.03	0.391242
$616$	0	0
$617$	−6433.75	−0.419794	−0.209897	−	0.977724i	$-0.567313\pi$
$617$	−6433.75	−0.419794	−0.209897	+	0.977724i	$0.567313\pi$
$618$	0	0
$619$	−16614.0	−1.07879	−0.539396	−	0.842053i	$-0.681347\pi$
$619$	−16614.0	−1.07879	−0.539396	+	0.842053i	$0.681347\pi$
$620$	0	0
$621$	−621.000	−0.0401286
$622$	0	0
$623$	−5825.69	−0.374641
$624$	0	0
$625$	−14023.7	−0.897518
$626$	0	0
$627$	−11420.1	−0.727394
$628$	0	0
$629$	−4780.86	−0.303061
$630$	0	0
$631$	13976.7	0.881781	0.440890	−	0.897561i	$-0.354663\pi$
$631$	13976.7	0.881781	0.440890	+	0.897561i	$0.354663\pi$
$632$	0	0
$633$	10793.5	0.677729
$634$	0	0
$635$	19733.0	1.23320
$636$	0	0
$637$	4086.33	0.254170
$638$	0	0
$639$	−3708.40	−0.229581
$640$	0	0
$641$	−19465.5	−1.19944	−0.599719	−	0.800211i	$-0.704721\pi$
$641$	−19465.5	−1.19944	−0.599719	+	0.800211i	$0.704721\pi$
$642$	0	0
$643$	6166.13	0.378178	0.189089	−	0.981960i	$-0.439446\pi$
$643$	6166.13	0.378178	0.189089	+	0.981960i	$0.439446\pi$
$644$	0	0
$645$	1051.16	0.0641697
$646$	0	0
$647$	−21911.9	−1.33145	−0.665723	−	0.746199i	$-0.731877\pi$
$647$	−21911.9	−1.33145	−0.665723	+	0.746199i	$0.731877\pi$
$648$	0	0
$649$	12648.8	0.765034
$650$	0	0
$651$	3098.75	0.186559
$652$	0	0
$653$	−14627.8	−0.876615	−0.438307	−	0.898825i	$-0.644422\pi$
$653$	−14627.8	−0.876615	−0.438307	+	0.898825i	$0.644422\pi$
$654$	0	0
$655$	−20008.5	−1.19359
$656$	0	0
$657$	5060.49	0.300500
$658$	0	0
$659$	2846.84	0.168281	0.0841404	−	0.996454i	$-0.473186\pi$
$659$	2846.84	0.168281	0.0841404	+	0.996454i	$0.473186\pi$
$660$	0	0
$661$	15798.8	0.929657	0.464829	−	0.885401i	$-0.346116\pi$
$661$	15798.8	0.929657	0.464829	+	0.885401i	$0.346116\pi$
$662$	0	0
$663$	−1468.53	−0.0860225
$664$	0	0
$665$	−6153.74	−0.358845
$666$	0	0
$667$	−4301.26	−0.249693
$668$	0	0
$669$	4697.90	0.271497
$670$	0	0
$671$	10819.2	0.622459
$672$	0	0
$673$	−10461.9	−0.599222	−0.299611	−	0.954061i	$-0.596857\pi$
$673$	−10461.9	−0.599222	−0.299611	+	0.954061i	$0.596857\pi$
$674$	0	0
$675$	316.243	0.0180329
$676$	0	0
$677$	−3689.41	−0.209447	−0.104723	−	0.994501i	$-0.533396\pi$
$677$	−3689.41	−0.209447	−0.104723	+	0.994501i	$0.533396\pi$
$678$	0	0
$679$	−2400.92	−0.135698
$680$	0	0
$681$	1432.16	0.0805879
$682$	0	0
$683$	−14454.8	−0.809808	−0.404904	−	0.914359i	$-0.632695\pi$
$683$	−14454.8	−0.809808	−0.404904	+	0.914359i	$0.632695\pi$
$684$	0	0
$685$	12697.2	0.708227
$686$	0	0
$687$	−11730.8	−0.651467
$688$	0	0
$689$	−7674.08	−0.424324
$690$	0	0
$691$	−34363.1	−1.89180	−0.945899	−	0.324460i	$-0.894817\pi$
$691$	−34363.1	−1.89180	−0.945899	+	0.324460i	$0.894817\pi$
$692$	0	0
$693$	−1277.80	−0.0700427
$694$	0	0
$695$	−21145.8	−1.15411
$696$	0	0
$697$	7195.65	0.391040
$698$	0	0
$699$	10175.5	0.550605
$700$	0	0
$701$	−31456.5	−1.69486	−0.847428	−	0.530911i	$-0.821850\pi$
$701$	−31456.5	−1.69486	−0.847428	+	0.530911i	$0.821850\pi$
$702$	0	0
$703$	15458.6	0.829351
$704$	0	0
$705$	−9752.52	−0.520995
$706$	0	0
$707$	−6276.17	−0.333861
$708$	0	0
$709$	−5721.57	−0.303072	−0.151536	−	0.988452i	$-0.548422\pi$
$709$	−5721.57	−0.303072	−0.151536	+	0.988452i	$0.548422\pi$
$710$	0	0
$711$	−8071.84	−0.425763
$712$	0	0
$713$	5116.04	0.268720
$714$	0	0
$715$	−4137.03	−0.216386
$716$	0	0
$717$	−3326.09	−0.173243
$718$	0	0
$719$	405.540	0.0210349	0.0105174	−	0.999945i	$-0.496652\pi$
$719$	405.540	0.0210349	0.0105174	+	0.999945i	$0.496652\pi$
$720$	0	0
$721$	−5602.05	−0.289364
$722$	0	0
$723$	10838.7	0.557532
$724$	0	0
$725$	2190.40	0.112206
$726$	0	0
$727$	−13075.6	−0.667051	−0.333525	−	0.942741i	$-0.608238\pi$
$727$	−13075.6	−0.667051	−0.333525	+	0.942741i	$0.608238\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	1267.60	0.0641365
$732$	0	0
$733$	10812.4	0.544834	0.272417	−	0.962179i	$-0.412177\pi$
$733$	10812.4	0.544834	0.272417	+	0.962179i	$0.412177\pi$
$734$	0	0
$735$	10263.8	0.515082
$736$	0	0
$737$	18294.2	0.914348
$738$	0	0
$739$	37507.4	1.86702	0.933512	−	0.358545i	$-0.116727\pi$
$739$	37507.4	1.86702	0.933512	+	0.358545i	$0.116727\pi$
$740$	0	0
$741$	4748.40	0.235408
$742$	0	0
$743$	16188.5	0.799327	0.399663	−	0.916662i	$-0.369127\pi$
$743$	16188.5	0.799327	0.399663	+	0.916662i	$0.369127\pi$
$744$	0	0
$745$	−7767.37	−0.381979
$746$	0	0
$747$	9388.54	0.459851
$748$	0	0
$749$	−2658.73	−0.129704
$750$	0	0
$751$	17911.2	0.870292	0.435146	−	0.900360i	$-0.356697\pi$
$751$	17911.2	0.870292	0.435146	+	0.900360i	$0.356697\pi$
$752$	0	0
$753$	−10164.0	−0.491894
$754$	0	0
$755$	−17832.4	−0.859586
$756$	0	0
$757$	−6837.02	−0.328264	−0.164132	−	0.986438i	$-0.552482\pi$
$757$	−6837.02	−0.328264	−0.164132	+	0.986438i	$0.552482\pi$
$758$	0	0
$759$	−2109.65	−0.100890
$760$	0	0
$761$	−31986.3	−1.52366	−0.761829	−	0.647779i	$-0.775698\pi$
$761$	−31986.3	−1.52366	−0.761829	+	0.647779i	$0.775698\pi$
$762$	0	0
$763$	−449.741	−0.0213391
$764$	0	0
$765$	−3688.56	−0.174327
$766$	0	0
$767$	−5259.26	−0.247589
$768$	0	0
$769$	−32722.9	−1.53448	−0.767241	−	0.641359i	$-0.778371\pi$
$769$	−32722.9	−1.53448	−0.767241	+	0.641359i	$0.778371\pi$
$770$	0	0
$771$	−3603.49	−0.168322
$772$	0	0
$773$	−18354.2	−0.854018	−0.427009	−	0.904247i	$-0.640433\pi$
$773$	−18354.2	−0.854018	−0.427009	+	0.904247i	$0.640433\pi$
$774$	0	0
$775$	−2605.33	−0.120756
$776$	0	0
$777$	1729.67	0.0798605
$778$	0	0
$779$	−23266.7	−1.07011
$780$	0	0
$781$	−12598.1	−0.577203
$782$	0	0
$783$	5049.30	0.230456
$784$	0	0
$785$	−10917.4	−0.496380
$786$	0	0
$787$	−7921.56	−0.358797	−0.179398	−	0.983776i	$-0.557415\pi$
$787$	−7921.56	−0.358797	−0.179398	+	0.983776i	$0.557415\pi$
$788$	0	0
$789$	−14312.5	−0.645805
$790$	0	0
$791$	−4288.05	−0.192751
$792$	0	0
$793$	−4498.54	−0.201447
$794$	0	0
$795$	−19275.3	−0.859904
$796$	0	0
$797$	−26374.7	−1.17219	−0.586097	−	0.810241i	$-0.699336\pi$
$797$	−26374.7	−1.17219	−0.586097	+	0.810241i	$0.699336\pi$
$798$	0	0
$799$	−11760.6	−0.520725
$800$	0	0
$801$	11290.9	0.498060
$802$	0	0
$803$	17191.4	0.755505
$804$	0	0
$805$	−1136.78	−0.0497719
$806$	0	0
$807$	19169.8	0.836195
$808$	0	0
$809$	−24337.2	−1.05766	−0.528832	−	0.848726i	$-0.677370\pi$
$809$	−24337.2	−1.05766	−0.528832	+	0.848726i	$0.677370\pi$
$810$	0	0
$811$	−6501.11	−0.281486	−0.140743	−	0.990046i	$-0.544949\pi$
$811$	−6501.11	−0.281486	−0.140743	+	0.990046i	$0.544949\pi$
$812$	0	0
$813$	2855.75	0.123192
$814$	0	0
$815$	10530.6	0.452601
$816$	0	0
$817$	−4098.70	−0.175515
$818$	0	0
$819$	531.300	0.0226680
$820$	0	0
$821$	6448.21	0.274110	0.137055	−	0.990563i	$-0.456236\pi$
$821$	6448.21	0.274110	0.137055	+	0.990563i	$0.456236\pi$
$822$	0	0
$823$	−20647.9	−0.874534	−0.437267	−	0.899332i	$-0.644054\pi$
$823$	−20647.9	−0.874534	−0.437267	+	0.899332i	$0.644054\pi$
$824$	0	0
$825$	1074.33	0.0453376
$826$	0	0
$827$	−4547.07	−0.191194	−0.0955968	−	0.995420i	$-0.530476\pi$
$827$	−4547.07	−0.191194	−0.0955968	+	0.995420i	$0.530476\pi$
$828$	0	0
$829$	24013.9	1.00608	0.503038	−	0.864264i	$-0.332216\pi$
$829$	24013.9	1.00608	0.503038	+	0.864264i	$0.332216\pi$
$830$	0	0
$831$	−21.0476	−0.000878621 0
$832$	0	0
$833$	12377.1	0.514815
$834$	0	0
$835$	12126.4	0.502575
$836$	0	0
$837$	−6005.79	−0.248017
$838$	0	0
$839$	906.392	0.0372969	0.0186485	−	0.999826i	$-0.494064\pi$
$839$	906.392	0.0372969	0.0186485	+	0.999826i	$0.494064\pi$
$840$	0	0
$841$	10584.2	0.433973
$842$	0	0
$843$	9499.21	0.388102
$844$	0	0
$845$	−21664.0	−0.881968
$846$	0	0
$847$	1839.78	0.0746349
$848$	0	0
$849$	−5284.42	−0.213617
$850$	0	0
$851$	2855.69	0.115031
$852$	0	0
$853$	18820.4	0.755450	0.377725	−	0.925918i	$-0.376707\pi$
$853$	18820.4	0.755450	0.377725	+	0.925918i	$0.376707\pi$
$854$	0	0
$855$	11926.7	0.477060
$856$	0	0
$857$	−39314.2	−1.56703	−0.783516	−	0.621371i	$-0.786576\pi$
$857$	−39314.2	−1.56703	−0.783516	+	0.621371i	$0.786576\pi$
$858$	0	0
$859$	−6387.76	−0.253722	−0.126861	−	0.991920i	$-0.540490\pi$
$859$	−6387.76	−0.253722	−0.126861	+	0.991920i	$0.540490\pi$
$860$	0	0
$861$	−2603.32	−0.103044
$862$	0	0
$863$	23555.1	0.929113	0.464556	−	0.885544i	$-0.346214\pi$
$863$	23555.1	0.929113	0.464556	+	0.885544i	$0.346214\pi$
$864$	0	0
$865$	−27061.5	−1.06372
$866$	0	0
$867$	10291.0	0.403114
$868$	0	0
$869$	−27421.5	−1.07044
$870$	0	0
$871$	−7606.58	−0.295912
$872$	0	0
$873$	4653.30	0.180401
$874$	0	0
$875$	6757.08	0.261064
$876$	0	0
$877$	−8537.95	−0.328741	−0.164371	−	0.986399i	$-0.552559\pi$
$877$	−8537.95	−0.328741	−0.164371	+	0.986399i	$0.552559\pi$
$878$	0	0
$879$	14627.1	0.561273
$880$	0	0
$881$	−23574.5	−0.901528	−0.450764	−	0.892643i	$-0.648848\pi$
$881$	−23574.5	−0.901528	−0.450764	+	0.892643i	$0.648848\pi$
$882$	0	0
$883$	40392.6	1.53943	0.769717	−	0.638385i	$-0.220397\pi$
$883$	40392.6	1.53943	0.769717	+	0.638385i	$0.220397\pi$
$884$	0	0
$885$	−13209.9	−0.501746
$886$	0	0
$887$	26612.4	1.00739	0.503696	−	0.863881i	$-0.331973\pi$
$887$	26612.4	1.00739	0.503696	+	0.863881i	$0.331973\pi$
$888$	0	0
$889$	−8609.20	−0.324796
$890$	0	0
$891$	2476.54	0.0931171
$892$	0	0
$893$	38027.2	1.42501
$894$	0	0
$895$	13094.4	0.489047
$896$	0	0
$897$	877.176	0.0326511
$898$	0	0
$899$	−41598.1	−1.54324
$900$	0	0
$901$	−23244.1	−0.859458
$902$	0	0
$903$	−458.605	−0.0169008
$904$	0	0
$905$	−19778.8	−0.726487
$906$	0	0
$907$	25937.5	0.949547	0.474774	−	0.880108i	$-0.342530\pi$
$907$	25937.5	0.949547	0.474774	+	0.880108i	$0.342530\pi$
$908$	0	0
$909$	12164.0	0.443845
$910$	0	0
$911$	3170.66	0.115311	0.0576557	−	0.998337i	$-0.481637\pi$
$911$	3170.66	0.115311	0.0576557	+	0.998337i	$0.481637\pi$
$912$	0	0
$913$	31894.6	1.15614
$914$	0	0
$915$	−11299.1	−0.408239
$916$	0	0
$917$	8729.40	0.314362
$918$	0	0
$919$	−41315.7	−1.48300	−0.741502	−	0.670951i	$-0.765886\pi$
$919$	−41315.7	−1.48300	−0.741502	+	0.670951i	$0.765886\pi$
$920$	0	0
$921$	27715.7	0.991600
$922$	0	0
$923$	5238.20	0.186801
$924$	0	0
$925$	−1454.25	−0.0516925
$926$	0	0
$927$	10857.5	0.384690
$928$	0	0
$929$	−6554.73	−0.231489	−0.115745	−	0.993279i	$-0.536925\pi$
$929$	−6554.73	−0.231489	−0.115745	+	0.993279i	$0.536925\pi$
$930$	0	0
$931$	−40020.6	−1.40883
$932$	0	0
$933$	−13649.2	−0.478945
$934$	0	0
$935$	−12530.7	−0.438286
$936$	0	0
$937$	−26127.5	−0.910939	−0.455469	−	0.890251i	$-0.650529\pi$
$937$	−26127.5	−0.910939	−0.455469	+	0.890251i	$0.650529\pi$
$938$	0	0
$939$	−28954.4	−1.00627
$940$	0	0
$941$	15701.7	0.543954	0.271977	−	0.962304i	$-0.412323\pi$
$941$	15701.7	0.543954	0.271977	+	0.962304i	$0.412323\pi$
$942$	0	0
$943$	−4298.08	−0.148425
$944$	0	0
$945$	1334.49	0.0459374
$946$	0	0
$947$	−45373.2	−1.55695	−0.778474	−	0.627676i	$-0.784006\pi$
$947$	−45373.2	−1.55695	−0.778474	+	0.627676i	$0.784006\pi$
$948$	0	0
$949$	−7148.05	−0.244505
$950$	0	0
$951$	6129.83	0.209015
$952$	0	0
$953$	47332.6	1.60887	0.804435	−	0.594041i	$-0.202468\pi$
$953$	47332.6	1.60887	0.804435	+	0.594041i	$0.202468\pi$
$954$	0	0
$955$	34828.9	1.18014
$956$	0	0
$957$	17153.4	0.579404
$958$	0	0
$959$	−5539.58	−0.186530
$960$	0	0
$961$	19687.0	0.660837
$962$	0	0
$963$	5152.97	0.172432
$964$	0	0
$965$	2109.39	0.0703664
$966$	0	0
$967$	49811.5	1.65649	0.828247	−	0.560363i	$-0.189339\pi$
$967$	49811.5	1.65649	0.828247	+	0.560363i	$0.189339\pi$
$968$	0	0
$969$	14382.5	0.476813
$970$	0	0
$971$	31020.6	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$971$	31020.6	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$972$	0	0
$973$	9225.58	0.303966
$974$	0	0
$975$	−446.700	−0.0146727
$976$	0	0
$977$	−39196.4	−1.28353	−0.641763	−	0.766903i	$-0.721797\pi$
$977$	−39196.4	−1.28353	−0.641763	+	0.766903i	$0.721797\pi$
$978$	0	0
$979$	38357.4	1.25220
$980$	0	0
$981$	871.657	0.0283689
$982$	0	0
$983$	10579.5	0.343268	0.171634	−	0.985161i	$-0.445095\pi$
$983$	10579.5	0.343268	0.171634	+	0.985161i	$0.445095\pi$
$984$	0	0
$985$	−406.875	−0.0131615
$986$	0	0
$987$	4254.87	0.137218
$988$	0	0
$989$	−757.156	−0.0243440
$990$	0	0
$991$	37854.1	1.21340	0.606698	−	0.794932i	$-0.292494\pi$
$991$	37854.1	1.21340	0.606698	+	0.794932i	$0.292494\pi$
$992$	0	0
$993$	26744.3	0.854689
$994$	0	0
$995$	−29502.2	−0.939982
$996$	0	0
$997$	−37876.2	−1.20316	−0.601580	−	0.798813i	$-0.705462\pi$
$997$	−37876.2	−1.20316	−0.601580	+	0.798813i	$0.705462\pi$
$998$	0	0
$999$	−3352.33	−0.106169

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.4.a.d.1.2	✓	2
3.2	odd	2		1656.4.a.e.1.1		2
4.3	odd	2		1104.4.a.o.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.4.a.d.1.2	✓	2	1.1	even	1	trivial
1104.4.a.o.1.2		2	4.3	odd	2
1656.4.a.e.1.1		2	3.2	odd	2

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:	\( N \)	\(=\)	\( 552 = 2^{3} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	552.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +10.6437 q^{5} -4.64365 q^{7} +9.00000 q^{9} +30.5746 q^{11} -12.7127 q^{13} -31.9310 q^{15} -38.5056 q^{17} +124.506 q^{19} +13.9310 q^{21} +23.0000 q^{23} -11.7127 q^{25} -27.0000 q^{27} -187.011 q^{29} +222.437 q^{31} -91.7238 q^{33} -49.4254 q^{35} +124.160 q^{37} +38.1381 q^{39} -186.873 q^{41} -32.9198 q^{43} +95.7929 q^{45} +305.425 q^{47} -321.437 q^{49} +115.517 q^{51} +603.655 q^{53} +325.425 q^{55} -373.517 q^{57} +413.702 q^{59} +353.862 q^{61} -41.7929 q^{63} -135.310 q^{65} +598.345 q^{67} -69.0000 q^{69} -412.044 q^{71} +562.276 q^{73} +35.1381 q^{75} -141.978 q^{77} -896.871 q^{79} +81.0000 q^{81} +1043.17 q^{83} -409.840 q^{85} +561.033 q^{87} +1254.55 q^{89} +59.0333 q^{91} -667.310 q^{93} +1325.19 q^{95} +517.033 q^{97} +275.171 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} + 2 q^{5} + 10 q^{7} + 18 q^{9} - 16 q^{11} - 64 q^{13} - 6 q^{15} + 58 q^{17} + 114 q^{19} - 30 q^{21} + 46 q^{23} - 62 q^{25} - 54 q^{27} - 104 q^{29} + 252 q^{31} + 48 q^{33} - 176 q^{35}+ \cdots - 144 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 + 2 * q^5 + 10 * q^7 + 18 * q^9 - 16 * q^11 - 64 * q^13 - 6 * q^15 + 58 * q^17 + 114 * q^19 - 30 * q^21 + 46 * q^23 - 62 * q^25 - 54 * q^27 - 104 * q^29 + 252 * q^31 + 48 * q^33 - 176 * q^35 - 176 * q^37 + 192 * q^39 + 12 * q^41 - 278 * q^43 + 18 * q^45 + 688 * q^47 - 450 * q^49 - 174 * q^51 + 918 * q^53 + 728 * q^55 - 342 * q^57 + 1136 * q^59 + 592 * q^61 + 90 * q^63 + 308 * q^65 + 1486 * q^67 - 138 * q^69 + 256 * q^71 + 1356 * q^73 + 186 * q^75 - 824 * q^77 + 270 * q^79 + 162 * q^81 + 1392 * q^83 - 1244 * q^85 + 312 * q^87 + 1294 * q^89 - 692 * q^91 - 756 * q^93 + 1416 * q^95 + 224 * q^97 - 144 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

Properties

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