LMFDB - Embedded newform 552.4.a.k.1.5

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 = [6,0,18,0,-6] 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:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 394x^{4} + 1617x^{3} + 37348x^{2} - 237382x + 130496 $ x^6 - 2x^5 - 394x^4 + 1617x^3 + 37348x^2 - 237382*x + 130496
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$9.55982$ 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)

$f(q)$	$=$	$q+3.00000 q^{3} +12.9270 q^{5} -4.19260 q^{7} +9.00000 q^{9} -64.0171 q^{11} +82.4586 q^{13} +38.7811 q^{15} +71.6401 q^{17} -23.7886 q^{19} -12.5778 q^{21} +23.0000 q^{23} +42.1084 q^{25} +27.0000 q^{27} +264.917 q^{29} +239.892 q^{31} -192.051 q^{33} -54.1979 q^{35} +229.073 q^{37} +247.376 q^{39} -230.641 q^{41} -46.3845 q^{43} +116.343 q^{45} -119.812 q^{47} -325.422 q^{49} +214.920 q^{51} +122.208 q^{53} -827.551 q^{55} -71.3658 q^{57} +647.813 q^{59} -835.524 q^{61} -37.7334 q^{63} +1065.95 q^{65} -24.3400 q^{67} +69.0000 q^{69} +411.045 q^{71} +961.375 q^{73} +126.325 q^{75} +268.398 q^{77} +1013.43 q^{79} +81.0000 q^{81} -1180.90 q^{83} +926.094 q^{85} +794.750 q^{87} +246.712 q^{89} -345.716 q^{91} +719.675 q^{93} -307.516 q^{95} -822.248 q^{97} -576.154 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 18 q^{3} - 6 q^{5} + 2 q^{7} + 54 q^{9} + 20 q^{11} + 116 q^{13} - 18 q^{15} + 58 q^{17} + 122 q^{19} + 6 q^{21} + 138 q^{23} + 210 q^{25} + 162 q^{27} + 120 q^{29} + 492 q^{31} + 60 q^{33} + 580 q^{35}+ \cdots + 180 q^{99}+O(q^{100}) $ 6 * q + 18 * q^3 - 6 * q^5 + 2 * q^7 + 54 * q^9 + 20 * q^11 + 116 * q^13 - 18 * q^15 + 58 * q^17 + 122 * q^19 + 6 * q^21 + 138 * q^23 + 210 * q^25 + 162 * q^27 + 120 * q^29 + 492 * q^31 + 60 * q^33 + 580 * q^35 + 420 * q^37 + 348 * q^39 + 420 * q^41 + 706 * q^43 - 54 * q^45 + 844 * q^47 + 1318 * q^49 + 174 * q^51 + 102 * q^53 + 448 * q^55 + 366 * q^57 + 692 * q^59 + 732 * q^61 + 18 * q^63 + 356 * q^65 + 734 * q^67 + 414 * q^69 - 600 * q^71 + 2092 * q^73 + 630 * q^75 + 408 * q^77 - 978 * q^79 + 486 * q^81 - 948 * q^83 + 1680 * q^85 + 360 * q^87 + 1166 * q^89 + 836 * q^91 + 1476 * q^93 - 2204 * q^95 + 3472 * q^97 + 180 * 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
$5$	12.9270	1.15623	0.578115	−	0.815955i	$-0.303789\pi$
$5$	12.9270	1.15623	0.578115	+	0.815955i	$0.303789\pi$
$6$	0	0
$7$	−4.19260	−0.226379	−0.113190	−	0.993573i	$-0.536107\pi$
$7$	−4.19260	−0.226379	−0.113190	+	0.993573i	$0.536107\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−64.0171	−1.75472	−0.877358	−	0.479837i	$-0.840696\pi$
$11$	−64.0171	−1.75472	−0.877358	+	0.479837i	$0.840696\pi$
$12$	0	0
$13$	82.4586	1.75922	0.879611	−	0.475693i	$-0.157803\pi$
$13$	82.4586	1.75922	0.879611	+	0.475693i	$0.157803\pi$
$14$	0	0
$15$	38.7811	0.667550
$16$	0	0
$17$	71.6401	1.02207	0.511037	−	0.859559i	$-0.329261\pi$
$17$	71.6401	1.02207	0.511037	+	0.859559i	$0.329261\pi$
$18$	0	0
$19$	−23.7886	−0.287236	−0.143618	−	0.989633i	$-0.545874\pi$
$19$	−23.7886	−0.287236	−0.143618	+	0.989633i	$0.545874\pi$
$20$	0	0
$21$	−12.5778	−0.130700
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	42.1084	0.336867
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	264.917	1.69634	0.848169	−	0.529725i	$-0.177705\pi$
$29$	264.917	1.69634	0.848169	+	0.529725i	$0.177705\pi$
$30$	0	0
$31$	239.892	1.38987	0.694933	−	0.719075i	$-0.255434\pi$
$31$	239.892	1.38987	0.694933	+	0.719075i	$0.255434\pi$
$32$	0	0
$33$	−192.051	−1.01309
$34$	0	0
$35$	−54.1979	−0.261746
$36$	0	0
$37$	229.073	1.01782	0.508910	−	0.860820i	$-0.330049\pi$
$37$	229.073	1.01782	0.508910	+	0.860820i	$0.330049\pi$
$38$	0	0
$39$	247.376	1.01569
$40$	0	0
$41$	−230.641	−0.878538	−0.439269	−	0.898356i	$-0.644762\pi$
$41$	−230.641	−0.878538	−0.439269	+	0.898356i	$0.644762\pi$
$42$	0	0
$43$	−46.3845	−0.164502	−0.0822508	−	0.996612i	$-0.526211\pi$
$43$	−46.3845	−0.164502	−0.0822508	+	0.996612i	$0.526211\pi$
$44$	0	0
$45$	116.343	0.385410
$46$	0	0
$47$	−119.812	−0.371839	−0.185919	−	0.982565i	$-0.559526\pi$
$47$	−119.812	−0.371839	−0.185919	+	0.982565i	$0.559526\pi$
$48$	0	0
$49$	−325.422	−0.948753
$50$	0	0
$51$	214.920	0.590095
$52$	0	0
$53$	122.208	0.316726	0.158363	−	0.987381i	$-0.449378\pi$
$53$	122.208	0.316726	0.158363	+	0.987381i	$0.449378\pi$
$54$	0	0
$55$	−827.551	−2.02885
$56$	0	0
$57$	−71.3658	−0.165836
$58$	0	0
$59$	647.813	1.42946	0.714730	−	0.699401i	$-0.246550\pi$
$59$	647.813	1.42946	0.714730	+	0.699401i	$0.246550\pi$
$60$	0	0
$61$	−835.524	−1.75374	−0.876868	−	0.480732i	$-0.840371\pi$
$61$	−835.524	−1.75374	−0.876868	+	0.480732i	$0.840371\pi$
$62$	0	0
$63$	−37.7334	−0.0754597
$64$	0	0
$65$	1065.95	2.03407
$66$	0	0
$67$	−24.3400	−0.0443822	−0.0221911	−	0.999754i	$-0.507064\pi$
$67$	−24.3400	−0.0443822	−0.0221911	+	0.999754i	$0.507064\pi$
$68$	0	0
$69$	69.0000	0.120386
$70$	0	0
$71$	411.045	0.687072	0.343536	−	0.939140i	$-0.388375\pi$
$71$	411.045	0.687072	0.343536	+	0.939140i	$0.388375\pi$
$72$	0	0
$73$	961.375	1.54138	0.770688	−	0.637213i	$-0.219913\pi$
$73$	961.375	1.54138	0.770688	+	0.637213i	$0.219913\pi$
$74$	0	0
$75$	126.325	0.194490
$76$	0	0
$77$	268.398	0.397231
$78$	0	0
$79$	1013.43	1.44328	0.721641	−	0.692267i	$-0.243388\pi$
$79$	1013.43	1.44328	0.721641	+	0.692267i	$0.243388\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−1180.90	−1.56170	−0.780848	−	0.624721i	$-0.785213\pi$
$83$	−1180.90	−1.56170	−0.780848	+	0.624721i	$0.785213\pi$
$84$	0	0
$85$	926.094	1.18175
$86$	0	0
$87$	794.750	0.979382
$88$	0	0
$89$	246.712	0.293837	0.146918	−	0.989149i	$-0.453065\pi$
$89$	246.712	0.293837	0.146918	+	0.989149i	$0.453065\pi$
$90$	0	0
$91$	−345.716	−0.398251
$92$	0	0
$93$	719.675	0.802439
$94$	0	0
$95$	−307.516	−0.332110
$96$	0	0
$97$	−822.248	−0.860687	−0.430343	−	0.902665i	$-0.641608\pi$
$97$	−822.248	−0.860687	−0.430343	+	0.902665i	$0.641608\pi$
$98$	0	0
$99$	−576.154	−0.584905
$100$	0	0
$101$	355.731	0.350461	0.175231	−	0.984527i	$-0.443933\pi$
$101$	355.731	0.350461	0.175231	+	0.984527i	$0.443933\pi$
$102$	0	0
$103$	1148.01	1.09822	0.549112	−	0.835748i	$-0.314966\pi$
$103$	1148.01	1.09822	0.549112	+	0.835748i	$0.314966\pi$
$104$	0	0
$105$	−162.594	−0.151119
$106$	0	0
$107$	−1004.83	−0.907860	−0.453930	−	0.891037i	$-0.649978\pi$
$107$	−1004.83	−0.907860	−0.453930	+	0.891037i	$0.649978\pi$
$108$	0	0
$109$	8.60720	0.00756349	0.00378174	−	0.999993i	$-0.498796\pi$
$109$	8.60720	0.00756349	0.00378174	+	0.999993i	$0.498796\pi$
$110$	0	0
$111$	687.219	0.587639
$112$	0	0
$113$	−1175.23	−0.978374	−0.489187	−	0.872179i	$-0.662706\pi$
$113$	−1175.23	−0.978374	−0.489187	+	0.872179i	$0.662706\pi$
$114$	0	0
$115$	297.322	0.241091
$116$	0	0
$117$	742.127	0.586407
$118$	0	0
$119$	−300.358	−0.231376
$120$	0	0
$121$	2767.19	2.07903
$122$	0	0
$123$	−691.922	−0.507224
$124$	0	0
$125$	−1071.54	−0.766734
$126$	0	0
$127$	1424.70	0.995448	0.497724	−	0.867335i	$-0.334169\pi$
$127$	1424.70	0.995448	0.497724	+	0.867335i	$0.334169\pi$
$128$	0	0
$129$	−139.153	−0.0949750
$130$	0	0
$131$	655.270	0.437032	0.218516	−	0.975833i	$-0.429878\pi$
$131$	655.270	0.437032	0.218516	+	0.975833i	$0.429878\pi$
$132$	0	0
$133$	99.7360	0.0650242
$134$	0	0
$135$	349.030	0.222517
$136$	0	0
$137$	−481.031	−0.299980	−0.149990	−	0.988687i	$-0.547924\pi$
$137$	−481.031	−0.299980	−0.149990	+	0.988687i	$0.547924\pi$
$138$	0	0
$139$	2459.48	1.50079	0.750396	−	0.660989i	$-0.229863\pi$
$139$	2459.48	1.50079	0.750396	+	0.660989i	$0.229863\pi$
$140$	0	0
$141$	−359.437	−0.214681
$142$	0	0
$143$	−5278.76	−3.08694
$144$	0	0
$145$	3424.59	1.96136
$146$	0	0
$147$	−976.266	−0.547763
$148$	0	0
$149$	1012.74	0.556827	0.278414	−	0.960461i	$-0.410191\pi$
$149$	1012.74	0.556827	0.278414	+	0.960461i	$0.410191\pi$
$150$	0	0
$151$	−1570.47	−0.846379	−0.423190	−	0.906041i	$-0.639090\pi$
$151$	−1570.47	−0.846379	−0.423190	+	0.906041i	$0.639090\pi$
$152$	0	0
$153$	644.761	0.340692
$154$	0	0
$155$	3101.09	1.60700
$156$	0	0
$157$	−295.345	−0.150134	−0.0750671	−	0.997178i	$-0.523917\pi$
$157$	−295.345	−0.150134	−0.0750671	+	0.997178i	$0.523917\pi$
$158$	0	0
$159$	366.623	0.182862
$160$	0	0
$161$	−96.4298	−0.0472033
$162$	0	0
$163$	−2915.11	−1.40079	−0.700395	−	0.713755i	$-0.746993\pi$
$163$	−2915.11	−1.40079	−0.700395	+	0.713755i	$0.746993\pi$
$164$	0	0
$165$	−2482.65	−1.17136
$166$	0	0
$167$	1518.11	0.703444	0.351722	−	0.936105i	$-0.385596\pi$
$167$	1518.11	0.703444	0.351722	+	0.936105i	$0.385596\pi$
$168$	0	0
$169$	4602.41	2.09486
$170$	0	0
$171$	−214.097	−0.0957452
$172$	0	0
$173$	−3205.84	−1.40887	−0.704437	−	0.709766i	$-0.748800\pi$
$173$	−3205.84	−1.40887	−0.704437	+	0.709766i	$0.748800\pi$
$174$	0	0
$175$	−176.544	−0.0762597
$176$	0	0
$177$	1943.44	0.825299
$178$	0	0
$179$	−3875.31	−1.61818	−0.809090	−	0.587685i	$-0.800040\pi$
$179$	−3875.31	−1.61818	−0.809090	+	0.587685i	$0.800040\pi$
$180$	0	0
$181$	1295.72	0.532100	0.266050	−	0.963959i	$-0.414281\pi$
$181$	1295.72	0.532100	0.266050	+	0.963959i	$0.414281\pi$
$182$	0	0
$183$	−2506.57	−1.01252
$184$	0	0
$185$	2961.23	1.17683
$186$	0	0
$187$	−4586.19	−1.79345
$188$	0	0
$189$	−113.200	−0.0435667
$190$	0	0
$191$	−4515.49	−1.71063	−0.855313	−	0.518112i	$-0.826635\pi$
$191$	−4515.49	−1.71063	−0.855313	+	0.518112i	$0.826635\pi$
$192$	0	0
$193$	−1746.88	−0.651521	−0.325760	−	0.945452i	$-0.605620\pi$
$193$	−1746.88	−0.651521	−0.325760	+	0.945452i	$0.605620\pi$
$194$	0	0
$195$	3197.84	1.17437
$196$	0	0
$197$	−1425.11	−0.515407	−0.257703	−	0.966224i	$-0.582966\pi$
$197$	−1425.11	−0.515407	−0.257703	+	0.966224i	$0.582966\pi$
$198$	0	0
$199$	−2743.12	−0.977158	−0.488579	−	0.872520i	$-0.662485\pi$
$199$	−2743.12	−0.977158	−0.488579	+	0.872520i	$0.662485\pi$
$200$	0	0
$201$	−73.0201	−0.0256241
$202$	0	0
$203$	−1110.69	−0.384016
$204$	0	0
$205$	−2981.50	−1.01579
$206$	0	0
$207$	207.000	0.0695048
$208$	0	0
$209$	1522.88	0.504017
$210$	0	0
$211$	−2902.19	−0.946895	−0.473447	−	0.880822i	$-0.656991\pi$
$211$	−2902.19	−0.946895	−0.473447	+	0.880822i	$0.656991\pi$
$212$	0	0
$213$	1233.14	0.396681
$214$	0	0
$215$	−599.614	−0.190202
$216$	0	0
$217$	−1005.77	−0.314637
$218$	0	0
$219$	2884.12	0.889913
$220$	0	0
$221$	5907.34	1.79806
$222$	0	0
$223$	−17.3636	−0.00521414	−0.00260707	−	0.999997i	$-0.500830\pi$
$223$	−17.3636	−0.00521414	−0.00260707	+	0.999997i	$0.500830\pi$
$224$	0	0
$225$	378.976	0.112289
$226$	0	0
$227$	1358.90	0.397328	0.198664	−	0.980068i	$-0.436340\pi$
$227$	1358.90	0.397328	0.198664	+	0.980068i	$0.436340\pi$
$228$	0	0
$229$	5976.65	1.72466	0.862332	−	0.506343i	$-0.169003\pi$
$229$	5976.65	1.72466	0.862332	+	0.506343i	$0.169003\pi$
$230$	0	0
$231$	805.194	0.229341
$232$	0	0
$233$	−5556.79	−1.56239	−0.781196	−	0.624286i	$-0.785390\pi$
$233$	−5556.79	−1.56239	−0.781196	+	0.624286i	$0.785390\pi$
$234$	0	0
$235$	−1548.82	−0.429931
$236$	0	0
$237$	3040.28	0.833280
$238$	0	0
$239$	−3441.37	−0.931395	−0.465698	−	0.884944i	$-0.654197\pi$
$239$	−3441.37	−0.931395	−0.465698	+	0.884944i	$0.654197\pi$
$240$	0	0
$241$	2410.17	0.644202	0.322101	−	0.946705i	$-0.395611\pi$
$241$	2410.17	0.644202	0.322101	+	0.946705i	$0.395611\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	−4206.75	−1.09698
$246$	0	0
$247$	−1961.57	−0.505312
$248$	0	0
$249$	−3542.70	−0.901646
$250$	0	0
$251$	1450.26	0.364699	0.182350	−	0.983234i	$-0.441630\pi$
$251$	1450.26	0.364699	0.182350	+	0.983234i	$0.441630\pi$
$252$	0	0
$253$	−1472.39	−0.365884
$254$	0	0
$255$	2778.28	0.682285
$256$	0	0
$257$	4278.10	1.03837	0.519184	−	0.854663i	$-0.326236\pi$
$257$	4278.10	1.03837	0.519184	+	0.854663i	$0.326236\pi$
$258$	0	0
$259$	−960.411	−0.230413
$260$	0	0
$261$	2384.25	0.565446
$262$	0	0
$263$	−5204.33	−1.22020	−0.610101	−	0.792324i	$-0.708871\pi$
$263$	−5204.33	−1.22020	−0.610101	+	0.792324i	$0.708871\pi$
$264$	0	0
$265$	1579.78	0.366208
$266$	0	0
$267$	740.137	0.169647
$268$	0	0
$269$	−8287.54	−1.87844	−0.939220	−	0.343316i	$-0.888450\pi$
$269$	−8287.54	−1.87844	−0.939220	+	0.343316i	$0.888450\pi$
$270$	0	0
$271$	261.850	0.0586947	0.0293473	−	0.999569i	$-0.490657\pi$
$271$	261.850	0.0586947	0.0293473	+	0.999569i	$0.490657\pi$
$272$	0	0
$273$	−1037.15	−0.229930
$274$	0	0
$275$	−2695.66	−0.591106
$276$	0	0
$277$	−4663.05	−1.01146	−0.505732	−	0.862691i	$-0.668777\pi$
$277$	−4663.05	−1.01146	−0.505732	+	0.862691i	$0.668777\pi$
$278$	0	0
$279$	2159.03	0.463289
$280$	0	0
$281$	−6903.29	−1.46554	−0.732768	−	0.680479i	$-0.761772\pi$
$281$	−6903.29	−1.46554	−0.732768	+	0.680479i	$0.761772\pi$
$282$	0	0
$283$	5843.36	1.22739	0.613695	−	0.789543i	$-0.289682\pi$
$283$	5843.36	1.22739	0.613695	+	0.789543i	$0.289682\pi$
$284$	0	0
$285$	−922.549	−0.191744
$286$	0	0
$287$	966.984	0.198883
$288$	0	0
$289$	219.298	0.0446363
$290$	0	0
$291$	−2466.74	−0.496918
$292$	0	0
$293$	−7169.89	−1.42959	−0.714794	−	0.699335i	$-0.753479\pi$
$293$	−7169.89	−1.42959	−0.714794	+	0.699335i	$0.753479\pi$
$294$	0	0
$295$	8374.31	1.65278
$296$	0	0
$297$	−1728.46	−0.337695
$298$	0	0
$299$	1896.55	0.366823
$300$	0	0
$301$	194.472	0.0372397
$302$	0	0
$303$	1067.19	0.202339
$304$	0	0
$305$	−10800.9	−2.02772
$306$	0	0
$307$	−6115.90	−1.13698	−0.568490	−	0.822690i	$-0.692472\pi$
$307$	−6115.90	−1.13698	−0.568490	+	0.822690i	$0.692472\pi$
$308$	0	0
$309$	3444.04	0.634060
$310$	0	0
$311$	1186.53	0.216341	0.108170	−	0.994132i	$-0.465501\pi$
$311$	1186.53	0.216341	0.108170	+	0.994132i	$0.465501\pi$
$312$	0	0
$313$	−6725.70	−1.21457	−0.607283	−	0.794485i	$-0.707741\pi$
$313$	−6725.70	−1.21457	−0.607283	+	0.794485i	$0.707741\pi$
$314$	0	0
$315$	−487.781	−0.0872487
$316$	0	0
$317$	−4545.00	−0.805277	−0.402638	−	0.915359i	$-0.631907\pi$
$317$	−4545.00	−0.805277	−0.402638	+	0.915359i	$0.631907\pi$
$318$	0	0
$319$	−16959.2	−2.97659
$320$	0	0
$321$	−3014.50	−0.524153
$322$	0	0
$323$	−1704.22	−0.293576
$324$	0	0
$325$	3472.20	0.592624
$326$	0	0
$327$	25.8216	0.00436678
$328$	0	0
$329$	502.325	0.0841766
$330$	0	0
$331$	−2039.67	−0.338702	−0.169351	−	0.985556i	$-0.554167\pi$
$331$	−2039.67	−0.338702	−0.169351	+	0.985556i	$0.554167\pi$
$332$	0	0
$333$	2061.66	0.339273
$334$	0	0
$335$	−314.645	−0.0513160
$336$	0	0
$337$	1940.10	0.313603	0.156802	−	0.987630i	$-0.449882\pi$
$337$	1940.10	0.313603	0.156802	+	0.987630i	$0.449882\pi$
$338$	0	0
$339$	−3525.69	−0.564864
$340$	0	0
$341$	−15357.2	−2.43882
$342$	0	0
$343$	2802.43	0.441157
$344$	0	0
$345$	891.966	0.139194
$346$	0	0
$347$	6391.31	0.988772	0.494386	−	0.869243i	$-0.335393\pi$
$347$	6391.31	0.988772	0.494386	+	0.869243i	$0.335393\pi$
$348$	0	0
$349$	9878.96	1.51521	0.757605	−	0.652713i	$-0.226369\pi$
$349$	9878.96	1.51521	0.757605	+	0.652713i	$0.226369\pi$
$350$	0	0
$351$	2226.38	0.338562
$352$	0	0
$353$	8331.06	1.25614	0.628070	−	0.778157i	$-0.283845\pi$
$353$	8331.06	1.25614	0.628070	+	0.778157i	$0.283845\pi$
$354$	0	0
$355$	5313.60	0.794413
$356$	0	0
$357$	−901.074	−0.133585
$358$	0	0
$359$	11191.5	1.64531	0.822654	−	0.568543i	$-0.192493\pi$
$359$	11191.5	1.64531	0.822654	+	0.568543i	$0.192493\pi$
$360$	0	0
$361$	−6293.10	−0.917496
$362$	0	0
$363$	8301.56	1.20033
$364$	0	0
$365$	12427.7	1.78218
$366$	0	0
$367$	−3201.82	−0.455405	−0.227703	−	0.973731i	$-0.573121\pi$
$367$	−3201.82	−0.455405	−0.227703	+	0.973731i	$0.573121\pi$
$368$	0	0
$369$	−2075.77	−0.292846
$370$	0	0
$371$	−512.367	−0.0717002
$372$	0	0
$373$	6828.79	0.947939	0.473970	−	0.880541i	$-0.342821\pi$
$373$	6828.79	0.947939	0.473970	+	0.880541i	$0.342821\pi$
$374$	0	0
$375$	−3214.63	−0.442674
$376$	0	0
$377$	21844.7	2.98424
$378$	0	0
$379$	3177.07	0.430594	0.215297	−	0.976549i	$-0.430928\pi$
$379$	3177.07	0.430594	0.215297	+	0.976549i	$0.430928\pi$
$380$	0	0
$381$	4274.11	0.574722
$382$	0	0
$383$	1184.55	0.158035	0.0790176	−	0.996873i	$-0.474822\pi$
$383$	1184.55	0.158035	0.0790176	+	0.996873i	$0.474822\pi$
$384$	0	0
$385$	3469.59	0.459290
$386$	0	0
$387$	−417.460	−0.0548339
$388$	0	0
$389$	−8707.33	−1.13491	−0.567454	−	0.823405i	$-0.692072\pi$
$389$	−8707.33	−1.13491	−0.567454	+	0.823405i	$0.692072\pi$
$390$	0	0
$391$	1647.72	0.213117
$392$	0	0
$393$	1965.81	0.252321
$394$	0	0
$395$	13100.6	1.66877
$396$	0	0
$397$	9174.40	1.15982	0.579912	−	0.814679i	$-0.303087\pi$
$397$	9174.40	1.15982	0.579912	+	0.814679i	$0.303087\pi$
$398$	0	0
$399$	299.208	0.0375417
$400$	0	0
$401$	−12258.1	−1.52653	−0.763267	−	0.646083i	$-0.776406\pi$
$401$	−12258.1	−1.52653	−0.763267	+	0.646083i	$0.776406\pi$
$402$	0	0
$403$	19781.1	2.44508
$404$	0	0
$405$	1047.09	0.128470
$406$	0	0
$407$	−14664.6	−1.78599
$408$	0	0
$409$	−13802.4	−1.66866	−0.834332	−	0.551262i	$-0.814146\pi$
$409$	−13802.4	−1.66866	−0.834332	+	0.551262i	$0.814146\pi$
$410$	0	0
$411$	−1443.09	−0.173194
$412$	0	0
$413$	−2716.02	−0.323600
$414$	0	0
$415$	−15265.6	−1.80568
$416$	0	0
$417$	7378.43	0.866482
$418$	0	0
$419$	10173.9	1.18622	0.593110	−	0.805121i	$-0.297900\pi$
$419$	10173.9	1.18622	0.593110	+	0.805121i	$0.297900\pi$
$420$	0	0
$421$	9880.41	1.14381	0.571903	−	0.820322i	$-0.306206\pi$
$421$	9880.41	1.14381	0.571903	+	0.820322i	$0.306206\pi$
$422$	0	0
$423$	−1078.31	−0.123946
$424$	0	0
$425$	3016.65	0.344303
$426$	0	0
$427$	3503.02	0.397009
$428$	0	0
$429$	−15836.3	−1.78224
$430$	0	0
$431$	3803.87	0.425119	0.212559	−	0.977148i	$-0.431820\pi$
$431$	3803.87	0.425119	0.212559	+	0.977148i	$0.431820\pi$
$432$	0	0
$433$	−9433.35	−1.04697	−0.523485	−	0.852035i	$-0.675368\pi$
$433$	−9433.35	−1.04697	−0.523485	+	0.852035i	$0.675368\pi$
$434$	0	0
$435$	10273.8	1.13239
$436$	0	0
$437$	−547.138	−0.0598928
$438$	0	0
$439$	729.216	0.0792793	0.0396396	−	0.999214i	$-0.487379\pi$
$439$	729.216	0.0792793	0.0396396	+	0.999214i	$0.487379\pi$
$440$	0	0
$441$	−2928.80	−0.316251
$442$	0	0
$443$	5268.44	0.565036	0.282518	−	0.959262i	$-0.408830\pi$
$443$	5268.44	0.565036	0.282518	+	0.959262i	$0.408830\pi$
$444$	0	0
$445$	3189.26	0.339743
$446$	0	0
$447$	3038.23	0.321484
$448$	0	0
$449$	4134.67	0.434582	0.217291	−	0.976107i	$-0.430278\pi$
$449$	4134.67	0.434582	0.217291	+	0.976107i	$0.430278\pi$
$450$	0	0
$451$	14764.9	1.54158
$452$	0	0
$453$	−4711.42	−0.488657
$454$	0	0
$455$	−4469.08	−0.460470
$456$	0	0
$457$	−3919.06	−0.401151	−0.200575	−	0.979678i	$-0.564281\pi$
$457$	−3919.06	−0.401151	−0.200575	+	0.979678i	$0.564281\pi$
$458$	0	0
$459$	1934.28	0.196698
$460$	0	0
$461$	−8254.74	−0.833973	−0.416987	−	0.908913i	$-0.636914\pi$
$461$	−8254.74	−0.833973	−0.416987	+	0.908913i	$0.636914\pi$
$462$	0	0
$463$	1329.71	0.133471	0.0667353	−	0.997771i	$-0.478742\pi$
$463$	1329.71	0.133471	0.0667353	+	0.997771i	$0.478742\pi$
$464$	0	0
$465$	9303.27	0.927804
$466$	0	0
$467$	2900.60	0.287417	0.143709	−	0.989620i	$-0.454097\pi$
$467$	2900.60	0.287417	0.143709	+	0.989620i	$0.454097\pi$
$468$	0	0
$469$	102.048	0.0100472
$470$	0	0
$471$	−886.034	−0.0866800
$472$	0	0
$473$	2969.40	0.288654
$474$	0	0
$475$	−1001.70	−0.0967603
$476$	0	0
$477$	1099.87	0.105575
$478$	0	0
$479$	−13010.2	−1.24102	−0.620512	−	0.784197i	$-0.713075\pi$
$479$	−13010.2	−1.24102	−0.620512	+	0.784197i	$0.713075\pi$
$480$	0	0
$481$	18889.0	1.79057
$482$	0	0
$483$	−289.289	−0.0272528
$484$	0	0
$485$	−10629.2	−0.995152
$486$	0	0
$487$	−9048.93	−0.841984	−0.420992	−	0.907064i	$-0.638318\pi$
$487$	−9048.93	−0.841984	−0.420992	+	0.907064i	$0.638318\pi$
$488$	0	0
$489$	−8745.33	−0.808747
$490$	0	0
$491$	−9859.56	−0.906223	−0.453112	−	0.891454i	$-0.649686\pi$
$491$	−9859.56	−0.906223	−0.453112	+	0.891454i	$0.649686\pi$
$492$	0	0
$493$	18978.7	1.73378
$494$	0	0
$495$	−7447.96	−0.676285
$496$	0	0
$497$	−1723.35	−0.155539
$498$	0	0
$499$	−8920.80	−0.800301	−0.400150	−	0.916450i	$-0.631042\pi$
$499$	−8920.80	−0.800301	−0.400150	+	0.916450i	$0.631042\pi$
$500$	0	0
$501$	4554.34	0.406133
$502$	0	0
$503$	18943.0	1.67917	0.839587	−	0.543225i	$-0.182797\pi$
$503$	18943.0	1.67917	0.839587	+	0.543225i	$0.182797\pi$
$504$	0	0
$505$	4598.55	0.405214
$506$	0	0
$507$	13807.2	1.20947
$508$	0	0
$509$	4830.63	0.420656	0.210328	−	0.977631i	$-0.432547\pi$
$509$	4830.63	0.420656	0.210328	+	0.977631i	$0.432547\pi$
$510$	0	0
$511$	−4030.66	−0.348935
$512$	0	0
$513$	−642.292	−0.0552785
$514$	0	0
$515$	14840.4	1.26980
$516$	0	0
$517$	7670.04	0.652472
$518$	0	0
$519$	−9617.51	−0.813414
$520$	0	0
$521$	12036.9	1.01218	0.506091	−	0.862480i	$-0.331090\pi$
$521$	12036.9	1.01218	0.506091	+	0.862480i	$0.331090\pi$
$522$	0	0
$523$	6133.13	0.512778	0.256389	−	0.966574i	$-0.417467\pi$
$523$	6133.13	0.512778	0.256389	+	0.966574i	$0.417467\pi$
$524$	0	0
$525$	−529.631	−0.0440285
$526$	0	0
$527$	17185.9	1.42055
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	5830.32	0.476486
$532$	0	0
$533$	−19018.3	−1.54554
$534$	0	0
$535$	−12989.5	−1.04969
$536$	0	0
$537$	−11625.9	−0.934257
$538$	0	0
$539$	20832.6	1.66479
$540$	0	0
$541$	22882.9	1.81851	0.909253	−	0.416243i	$-0.136653\pi$
$541$	22882.9	1.81851	0.909253	+	0.416243i	$0.136653\pi$
$542$	0	0
$543$	3887.16	0.307208
$544$	0	0
$545$	111.266	0.00874513
$546$	0	0
$547$	4377.18	0.342148	0.171074	−	0.985258i	$-0.445276\pi$
$547$	4377.18	0.342148	0.171074	+	0.985258i	$0.445276\pi$
$548$	0	0
$549$	−7519.72	−0.584579
$550$	0	0
$551$	−6302.00	−0.487249
$552$	0	0
$553$	−4248.89	−0.326729
$554$	0	0
$555$	8883.70	0.679445
$556$	0	0
$557$	−9386.41	−0.714030	−0.357015	−	0.934099i	$-0.616206\pi$
$557$	−9386.41	−0.714030	−0.357015	+	0.934099i	$0.616206\pi$
$558$	0	0
$559$	−3824.80	−0.289395
$560$	0	0
$561$	−13758.6	−1.03545
$562$	0	0
$563$	−16508.3	−1.23578	−0.617889	−	0.786265i	$-0.712012\pi$
$563$	−16508.3	−1.23578	−0.617889	+	0.786265i	$0.712012\pi$
$564$	0	0
$565$	−15192.2	−1.13122
$566$	0	0
$567$	−339.600	−0.0251532
$568$	0	0
$569$	−16177.9	−1.19194	−0.595968	−	0.803008i	$-0.703232\pi$
$569$	−16177.9	−1.19194	−0.595968	+	0.803008i	$0.703232\pi$
$570$	0	0
$571$	−10865.0	−0.796299	−0.398150	−	0.917321i	$-0.630347\pi$
$571$	−10865.0	−0.796299	−0.398150	+	0.917321i	$0.630347\pi$
$572$	0	0
$573$	−13546.5	−0.987630
$574$	0	0
$575$	968.493	0.0702417
$576$	0	0
$577$	8829.62	0.637057	0.318529	−	0.947913i	$-0.396811\pi$
$577$	8829.62	0.637057	0.318529	+	0.947913i	$0.396811\pi$
$578$	0	0
$579$	−5240.65	−0.376156
$580$	0	0
$581$	4951.05	0.353535
$582$	0	0
$583$	−7823.37	−0.555765
$584$	0	0
$585$	9593.51	0.678022
$586$	0	0
$587$	−2733.60	−0.192211	−0.0961054	−	0.995371i	$-0.530639\pi$
$587$	−2733.60	−0.192211	−0.0961054	+	0.995371i	$0.530639\pi$
$588$	0	0
$589$	−5706.69	−0.399219
$590$	0	0
$591$	−4275.34	−0.297570
$592$	0	0
$593$	18752.6	1.29861	0.649307	−	0.760527i	$-0.275059\pi$
$593$	18752.6	1.29861	0.649307	+	0.760527i	$0.275059\pi$
$594$	0	0
$595$	−3882.74	−0.267524
$596$	0	0
$597$	−8229.35	−0.564163
$598$	0	0
$599$	−4868.99	−0.332123	−0.166061	−	0.986115i	$-0.553105\pi$
$599$	−4868.99	−0.332123	−0.166061	+	0.986115i	$0.553105\pi$
$600$	0	0
$601$	12147.7	0.824482	0.412241	−	0.911075i	$-0.364746\pi$
$601$	12147.7	0.824482	0.412241	+	0.911075i	$0.364746\pi$
$602$	0	0
$603$	−219.060	−0.0147941
$604$	0	0
$605$	35771.5	2.40383
$606$	0	0
$607$	−10976.8	−0.733993	−0.366997	−	0.930222i	$-0.619614\pi$
$607$	−10976.8	−0.733993	−0.366997	+	0.930222i	$0.619614\pi$
$608$	0	0
$609$	−3332.07	−0.221711
$610$	0	0
$611$	−9879.56	−0.654147
$612$	0	0
$613$	−13319.6	−0.877605	−0.438802	−	0.898584i	$-0.644597\pi$
$613$	−13319.6	−0.877605	−0.438802	+	0.898584i	$0.644597\pi$
$614$	0	0
$615$	−8944.51	−0.586467
$616$	0	0
$617$	−6758.54	−0.440986	−0.220493	−	0.975389i	$-0.570767\pi$
$617$	−6758.54	−0.440986	−0.220493	+	0.975389i	$0.570767\pi$
$618$	0	0
$619$	8311.83	0.539710	0.269855	−	0.962901i	$-0.413024\pi$
$619$	8311.83	0.539710	0.269855	+	0.962901i	$0.413024\pi$
$620$	0	0
$621$	621.000	0.0401286
$622$	0	0
$623$	−1034.37	−0.0665184
$624$	0	0
$625$	−19115.4	−1.22339
$626$	0	0
$627$	4568.63	0.290994
$628$	0	0
$629$	16410.8	1.04029
$630$	0	0
$631$	22464.1	1.41724	0.708622	−	0.705588i	$-0.249317\pi$
$631$	22464.1	1.41724	0.708622	+	0.705588i	$0.249317\pi$
$632$	0	0
$633$	−8706.56	−0.546690
$634$	0	0
$635$	18417.2	1.15097
$636$	0	0
$637$	−26833.8	−1.66907
$638$	0	0
$639$	3699.41	0.229024
$640$	0	0
$641$	9424.40	0.580720	0.290360	−	0.956917i	$-0.406225\pi$
$641$	9424.40	0.580720	0.290360	+	0.956917i	$0.406225\pi$
$642$	0	0
$643$	25518.4	1.56508	0.782541	−	0.622599i	$-0.213923\pi$
$643$	25518.4	1.56508	0.782541	+	0.622599i	$0.213923\pi$
$644$	0	0
$645$	−1798.84	−0.109813
$646$	0	0
$647$	−24543.2	−1.49133	−0.745667	−	0.666318i	$-0.767869\pi$
$647$	−24543.2	−1.49133	−0.745667	+	0.666318i	$0.767869\pi$
$648$	0	0
$649$	−41471.1	−2.50830
$650$	0	0
$651$	−3017.31	−0.181655
$652$	0	0
$653$	10669.6	0.639407	0.319704	−	0.947518i	$-0.396417\pi$
$653$	10669.6	0.639407	0.319704	+	0.947518i	$0.396417\pi$
$654$	0	0
$655$	8470.70	0.505309
$656$	0	0
$657$	8652.37	0.513792
$658$	0	0
$659$	8547.77	0.505272	0.252636	−	0.967561i	$-0.418703\pi$
$659$	8547.77	0.505272	0.252636	+	0.967561i	$0.418703\pi$
$660$	0	0
$661$	−4136.55	−0.243409	−0.121704	−	0.992566i	$-0.538836\pi$
$661$	−4136.55	−0.243409	−0.121704	+	0.992566i	$0.538836\pi$
$662$	0	0
$663$	17722.0	1.03811
$664$	0	0
$665$	1289.29	0.0751829
$666$	0	0
$667$	6093.09	0.353711
$668$	0	0
$669$	−52.0908	−0.00301038
$670$	0	0
$671$	53487.8	3.07731
$672$	0	0
$673$	11868.3	0.679775	0.339887	−	0.940466i	$-0.389611\pi$
$673$	11868.3	0.679775	0.339887	+	0.940466i	$0.389611\pi$
$674$	0	0
$675$	1136.93	0.0648301
$676$	0	0
$677$	−15683.8	−0.890367	−0.445183	−	0.895439i	$-0.646861\pi$
$677$	−15683.8	−0.890367	−0.445183	+	0.895439i	$0.646861\pi$
$678$	0	0
$679$	3447.36	0.194841
$680$	0	0
$681$	4076.70	0.229397
$682$	0	0
$683$	11729.8	0.657141	0.328570	−	0.944479i	$-0.393433\pi$
$683$	11729.8	0.657141	0.328570	+	0.944479i	$0.393433\pi$
$684$	0	0
$685$	−6218.31	−0.346846
$686$	0	0
$687$	17930.0	0.995736
$688$	0	0
$689$	10077.1	0.557192
$690$	0	0
$691$	15929.9	0.876991	0.438496	−	0.898733i	$-0.355511\pi$
$691$	15929.9	0.876991	0.438496	+	0.898733i	$0.355511\pi$
$692$	0	0
$693$	2415.58	0.132410
$694$	0	0
$695$	31793.7	1.73526
$696$	0	0
$697$	−16523.1	−0.897931
$698$	0	0
$699$	−16670.4	−0.902047
$700$	0	0
$701$	3804.66	0.204993	0.102496	−	0.994733i	$-0.467317\pi$
$701$	3804.66	0.204993	0.102496	+	0.994733i	$0.467317\pi$
$702$	0	0
$703$	−5449.32	−0.292354
$704$	0	0
$705$	−4646.46	−0.248221
$706$	0	0
$707$	−1491.44	−0.0793371
$708$	0	0
$709$	16162.5	0.856130	0.428065	−	0.903748i	$-0.359195\pi$
$709$	16162.5	0.856130	0.428065	+	0.903748i	$0.359195\pi$
$710$	0	0
$711$	9120.83	0.481094
$712$	0	0
$713$	5517.51	0.289807
$714$	0	0
$715$	−68238.7	−3.56921
$716$	0	0
$717$	−10324.1	−0.537741
$718$	0	0
$719$	−37375.9	−1.93864	−0.969321	−	0.245797i	$-0.920950\pi$
$719$	−37375.9	−1.93864	−0.969321	+	0.245797i	$0.920950\pi$
$720$	0	0
$721$	−4813.16	−0.248615
$722$	0	0
$723$	7230.51	0.371930
$724$	0	0
$725$	11155.2	0.571441
$726$	0	0
$727$	−22250.1	−1.13509	−0.567546	−	0.823342i	$-0.692107\pi$
$727$	−22250.1	−1.13509	−0.567546	+	0.823342i	$0.692107\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−3322.99	−0.168133
$732$	0	0
$733$	−4449.29	−0.224200	−0.112100	−	0.993697i	$-0.535758\pi$
$733$	−4449.29	−0.224200	−0.112100	+	0.993697i	$0.535758\pi$
$734$	0	0
$735$	−12620.2	−0.633339
$736$	0	0
$737$	1558.18	0.0778782
$738$	0	0
$739$	5449.60	0.271268	0.135634	−	0.990759i	$-0.456693\pi$
$739$	5449.60	0.271268	0.135634	+	0.990759i	$0.456693\pi$
$740$	0	0
$741$	−5884.72	−0.291742
$742$	0	0
$743$	−30768.5	−1.51923	−0.759615	−	0.650373i	$-0.774613\pi$
$743$	−30768.5	−1.51923	−0.759615	+	0.650373i	$0.774613\pi$
$744$	0	0
$745$	13091.8	0.643820
$746$	0	0
$747$	−10628.1	−0.520565
$748$	0	0
$749$	4212.87	0.205520
$750$	0	0
$751$	−36087.1	−1.75345	−0.876723	−	0.480995i	$-0.840275\pi$
$751$	−36087.1	−1.75345	−0.876723	+	0.480995i	$0.840275\pi$
$752$	0	0
$753$	4350.78	0.210559
$754$	0	0
$755$	−20301.6	−0.978609
$756$	0	0
$757$	9434.21	0.452962	0.226481	−	0.974016i	$-0.427278\pi$
$757$	9434.21	0.452962	0.226481	+	0.974016i	$0.427278\pi$
$758$	0	0
$759$	−4417.18	−0.211243
$760$	0	0
$761$	−11744.3	−0.559433	−0.279717	−	0.960083i	$-0.590241\pi$
$761$	−11744.3	−0.559433	−0.279717	+	0.960083i	$0.590241\pi$
$762$	0	0
$763$	−36.0865	−0.00171222
$764$	0	0
$765$	8334.85	0.393918
$766$	0	0
$767$	53417.8	2.51474
$768$	0	0
$769$	18646.4	0.874389	0.437194	−	0.899367i	$-0.355972\pi$
$769$	18646.4	0.874389	0.437194	+	0.899367i	$0.355972\pi$
$770$	0	0
$771$	12834.3	0.599502
$772$	0	0
$773$	846.985	0.0394100	0.0197050	−	0.999806i	$-0.493727\pi$
$773$	846.985	0.0394100	0.0197050	+	0.999806i	$0.493727\pi$
$774$	0	0
$775$	10101.5	0.468200
$776$	0	0
$777$	−2881.23	−0.133029
$778$	0	0
$779$	5486.62	0.252347
$780$	0	0
$781$	−26313.9	−1.20562
$782$	0	0
$783$	7152.75	0.326461
$784$	0	0
$785$	−3817.93	−0.173590
$786$	0	0
$787$	−18877.1	−0.855015	−0.427507	−	0.904012i	$-0.640608\pi$
$787$	−18877.1	−0.855015	−0.427507	+	0.904012i	$0.640608\pi$
$788$	0	0
$789$	−15613.0	−0.704484
$790$	0	0
$791$	4927.26	0.221483
$792$	0	0
$793$	−68896.1	−3.08521
$794$	0	0
$795$	4739.34	0.211430
$796$	0	0
$797$	−13161.7	−0.584959	−0.292480	−	0.956272i	$-0.594480\pi$
$797$	−13161.7	−0.584959	−0.292480	+	0.956272i	$0.594480\pi$
$798$	0	0
$799$	−8583.37	−0.380047
$800$	0	0
$801$	2220.41	0.0979455
$802$	0	0
$803$	−61544.4	−2.70468
$804$	0	0
$805$	−1246.55	−0.0545779
$806$	0	0
$807$	−24862.6	−1.08452
$808$	0	0
$809$	−38301.7	−1.66454	−0.832271	−	0.554369i	$-0.812960\pi$
$809$	−38301.7	−1.66454	−0.832271	+	0.554369i	$0.812960\pi$
$810$	0	0
$811$	−32892.0	−1.42416	−0.712081	−	0.702097i	$-0.752247\pi$
$811$	−32892.0	−1.42416	−0.712081	+	0.702097i	$0.752247\pi$
$812$	0	0
$813$	785.550	0.0338874
$814$	0	0
$815$	−37683.7	−1.61964
$816$	0	0
$817$	1103.42	0.0472507
$818$	0	0
$819$	−3111.44	−0.132750
$820$	0	0
$821$	45489.5	1.93373	0.966867	−	0.255280i	$-0.0821676\pi$
$821$	45489.5	1.93373	0.966867	+	0.255280i	$0.0821676\pi$
$822$	0	0
$823$	12461.5	0.527800	0.263900	−	0.964550i	$-0.414991\pi$
$823$	12461.5	0.527800	0.263900	+	0.964550i	$0.414991\pi$
$824$	0	0
$825$	−8086.97	−0.341275
$826$	0	0
$827$	19488.9	0.819462	0.409731	−	0.912206i	$-0.365623\pi$
$827$	19488.9	0.819462	0.409731	+	0.912206i	$0.365623\pi$
$828$	0	0
$829$	29887.3	1.25215	0.626073	−	0.779764i	$-0.284661\pi$
$829$	29887.3	1.25215	0.626073	+	0.779764i	$0.284661\pi$
$830$	0	0
$831$	−13989.1	−0.583969
$832$	0	0
$833$	−23313.3	−0.969696
$834$	0	0
$835$	19624.7	0.813343
$836$	0	0
$837$	6477.08	0.267480
$838$	0	0
$839$	−2261.66	−0.0930646	−0.0465323	−	0.998917i	$-0.514817\pi$
$839$	−2261.66	−0.0930646	−0.0465323	+	0.998917i	$0.514817\pi$
$840$	0	0
$841$	45791.9	1.87756
$842$	0	0
$843$	−20709.9	−0.846127
$844$	0	0
$845$	59495.6	2.42214
$846$	0	0
$847$	−11601.7	−0.470648
$848$	0	0
$849$	17530.1	0.708634
$850$	0	0
$851$	5268.68	0.212230
$852$	0	0
$853$	10189.5	0.409007	0.204504	−	0.978866i	$-0.434442\pi$
$853$	10189.5	0.409007	0.204504	+	0.978866i	$0.434442\pi$
$854$	0	0
$855$	−2767.65	−0.110703
$856$	0	0
$857$	−23968.1	−0.955349	−0.477674	−	0.878537i	$-0.658520\pi$
$857$	−23968.1	−0.955349	−0.477674	+	0.878537i	$0.658520\pi$
$858$	0	0
$859$	21834.1	0.867254	0.433627	−	0.901092i	$-0.357234\pi$
$859$	21834.1	0.867254	0.433627	+	0.901092i	$0.357234\pi$
$860$	0	0
$861$	2900.95	0.114825
$862$	0	0
$863$	17619.2	0.694976	0.347488	−	0.937684i	$-0.387035\pi$
$863$	17619.2	0.694976	0.347488	+	0.937684i	$0.387035\pi$
$864$	0	0
$865$	−41442.0	−1.62898
$866$	0	0
$867$	657.895	0.0257708
$868$	0	0
$869$	−64876.6	−2.53255
$870$	0	0
$871$	−2007.04	−0.0780782
$872$	0	0
$873$	−7400.23	−0.286896
$874$	0	0
$875$	4492.55	0.173572
$876$	0	0
$877$	−26798.1	−1.03182	−0.515911	−	0.856642i	$-0.672546\pi$
$877$	−26798.1	−1.03182	−0.515911	+	0.856642i	$0.672546\pi$
$878$	0	0
$879$	−21509.7	−0.825373
$880$	0	0
$881$	−17680.2	−0.676121	−0.338060	−	0.941124i	$-0.609771\pi$
$881$	−17680.2	−0.676121	−0.338060	+	0.941124i	$0.609771\pi$
$882$	0	0
$883$	−37842.7	−1.44225	−0.721126	−	0.692804i	$-0.756375\pi$
$883$	−37842.7	−1.44225	−0.721126	+	0.692804i	$0.756375\pi$
$884$	0	0
$885$	25122.9	0.954235
$886$	0	0
$887$	38642.7	1.46279	0.731395	−	0.681954i	$-0.238870\pi$
$887$	38642.7	1.46279	0.731395	+	0.681954i	$0.238870\pi$
$888$	0	0
$889$	−5973.21	−0.225349
$890$	0	0
$891$	−5185.38	−0.194968
$892$	0	0
$893$	2850.17	0.106805
$894$	0	0
$895$	−50096.3	−1.87099
$896$	0	0
$897$	5689.64	0.211785
$898$	0	0
$899$	63551.4	2.35768
$900$	0	0
$901$	8754.95	0.323718
$902$	0	0
$903$	583.415	0.0215004
$904$	0	0
$905$	16749.8	0.615230
$906$	0	0
$907$	13665.0	0.500263	0.250132	−	0.968212i	$-0.419526\pi$
$907$	13665.0	0.500263	0.250132	+	0.968212i	$0.419526\pi$
$908$	0	0
$909$	3201.58	0.116820
$910$	0	0
$911$	−15935.3	−0.579540	−0.289770	−	0.957096i	$-0.593579\pi$
$911$	−15935.3	−0.579540	−0.289770	+	0.957096i	$0.593579\pi$
$912$	0	0
$913$	75597.8	2.74033
$914$	0	0
$915$	−32402.6	−1.17071
$916$	0	0
$917$	−2747.28	−0.0989349
$918$	0	0
$919$	−32966.8	−1.18332	−0.591661	−	0.806187i	$-0.701528\pi$
$919$	−32966.8	−1.18332	−0.591661	+	0.806187i	$0.701528\pi$
$920$	0	0
$921$	−18347.7	−0.656436
$922$	0	0
$923$	33894.2	1.20871
$924$	0	0
$925$	9645.89	0.342870
$926$	0	0
$927$	10332.1	0.366075
$928$	0	0
$929$	−975.097	−0.0344369	−0.0172184	−	0.999852i	$-0.505481\pi$
$929$	−975.097	−0.0344369	−0.0172184	+	0.999852i	$0.505481\pi$
$930$	0	0
$931$	7741.34	0.272516
$932$	0	0
$933$	3559.59	0.124904
$934$	0	0
$935$	−59285.8	−2.07364
$936$	0	0
$937$	23182.3	0.808251	0.404126	−	0.914703i	$-0.367576\pi$
$937$	23182.3	0.808251	0.404126	+	0.914703i	$0.367576\pi$
$938$	0	0
$939$	−20177.1	−0.701230
$940$	0	0
$941$	−7785.63	−0.269718	−0.134859	−	0.990865i	$-0.543058\pi$
$941$	−7785.63	−0.269718	−0.134859	+	0.990865i	$0.543058\pi$
$942$	0	0
$943$	−5304.74	−0.183188
$944$	0	0
$945$	−1463.34	−0.0503731
$946$	0	0
$947$	43868.4	1.50531	0.752656	−	0.658413i	$-0.228772\pi$
$947$	43868.4	1.50531	0.752656	+	0.658413i	$0.228772\pi$
$948$	0	0
$949$	79273.6	2.71162
$950$	0	0
$951$	−13635.0	−0.464927
$952$	0	0
$953$	37311.9	1.26826	0.634129	−	0.773227i	$-0.281359\pi$
$953$	37311.9	1.26826	0.634129	+	0.773227i	$0.281359\pi$
$954$	0	0
$955$	−58372.0	−1.97788
$956$	0	0
$957$	−50877.6	−1.71854
$958$	0	0
$959$	2016.77	0.0679092
$960$	0	0
$961$	27757.1	0.931727
$962$	0	0
$963$	−9043.51	−0.302620
$964$	0	0
$965$	−22582.0	−0.753307
$966$	0	0
$967$	21234.7	0.706166	0.353083	−	0.935592i	$-0.385133\pi$
$967$	21234.7	0.706166	0.353083	+	0.935592i	$0.385133\pi$
$968$	0	0
$969$	−5112.65	−0.169496
$970$	0	0
$971$	−27275.5	−0.901454	−0.450727	−	0.892662i	$-0.648835\pi$
$971$	−27275.5	−0.901454	−0.450727	+	0.892662i	$0.648835\pi$
$972$	0	0
$973$	−10311.6	−0.339748
$974$	0	0
$975$	10416.6	0.342152
$976$	0	0
$977$	23999.9	0.785900	0.392950	−	0.919560i	$-0.371455\pi$
$977$	23999.9	0.785900	0.392950	+	0.919560i	$0.371455\pi$
$978$	0	0
$979$	−15793.8	−0.515600
$980$	0	0
$981$	77.4648	0.00252116
$982$	0	0
$983$	−16482.7	−0.534807	−0.267403	−	0.963585i	$-0.586166\pi$
$983$	−16482.7	−0.534807	−0.267403	+	0.963585i	$0.586166\pi$
$984$	0	0
$985$	−18422.5	−0.595928
$986$	0	0
$987$	1506.98	0.0485994
$988$	0	0
$989$	−1066.84	−0.0343009
$990$	0	0
$991$	24348.4	0.780477	0.390238	−	0.920714i	$-0.372393\pi$
$991$	24348.4	0.780477	0.390238	+	0.920714i	$0.372393\pi$
$992$	0	0
$993$	−6119.02	−0.195550
$994$	0	0
$995$	−35460.4	−1.12982
$996$	0	0
$997$	−16941.8	−0.538168	−0.269084	−	0.963117i	$-0.586721\pi$
$997$	−16941.8	−0.538168	−0.269084	+	0.963117i	$0.586721\pi$
$998$	0	0
$999$	6184.97	0.195880

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.4.a.k.1.5	✓	6
3.2	odd	2		1656.4.a.q.1.2		6
4.3	odd	2		1104.4.a.z.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.4.a.k.1.5	✓	6	1.1	even	1	trivial
1104.4.a.z.1.5		6	4.3	odd	2
1656.4.a.q.1.2		6	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)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +12.9270 q^{5} -4.19260 q^{7} +9.00000 q^{9} -64.0171 q^{11} +82.4586 q^{13} +38.7811 q^{15} +71.6401 q^{17} -23.7886 q^{19} -12.5778 q^{21} +23.0000 q^{23} +42.1084 q^{25} +27.0000 q^{27} +264.917 q^{29} +239.892 q^{31} -192.051 q^{33} -54.1979 q^{35} +229.073 q^{37} +247.376 q^{39} -230.641 q^{41} -46.3845 q^{43} +116.343 q^{45} -119.812 q^{47} -325.422 q^{49} +214.920 q^{51} +122.208 q^{53} -827.551 q^{55} -71.3658 q^{57} +647.813 q^{59} -835.524 q^{61} -37.7334 q^{63} +1065.95 q^{65} -24.3400 q^{67} +69.0000 q^{69} +411.045 q^{71} +961.375 q^{73} +126.325 q^{75} +268.398 q^{77} +1013.43 q^{79} +81.0000 q^{81} -1180.90 q^{83} +926.094 q^{85} +794.750 q^{87} +246.712 q^{89} -345.716 q^{91} +719.675 q^{93} -307.516 q^{95} -822.248 q^{97} -576.154 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 18 q^{3} - 6 q^{5} + 2 q^{7} + 54 q^{9} + 20 q^{11} + 116 q^{13} - 18 q^{15} + 58 q^{17} + 122 q^{19} + 6 q^{21} + 138 q^{23} + 210 q^{25} + 162 q^{27} + 120 q^{29} + 492 q^{31} + 60 q^{33} + 580 q^{35}+ \cdots + 180 q^{99}+O(q^{100}) \) 6 * q + 18 * q^3 - 6 * q^5 + 2 * q^7 + 54 * q^9 + 20 * q^11 + 116 * q^13 - 18 * q^15 + 58 * q^17 + 122 * q^19 + 6 * q^21 + 138 * q^23 + 210 * q^25 + 162 * q^27 + 120 * q^29 + 492 * q^31 + 60 * q^33 + 580 * q^35 + 420 * q^37 + 348 * q^39 + 420 * q^41 + 706 * q^43 - 54 * q^45 + 844 * q^47 + 1318 * q^49 + 174 * q^51 + 102 * q^53 + 448 * q^55 + 366 * q^57 + 692 * q^59 + 732 * q^61 + 18 * q^63 + 356 * q^65 + 734 * q^67 + 414 * q^69 - 600 * q^71 + 2092 * q^73 + 630 * q^75 + 408 * q^77 - 978 * q^79 + 486 * q^81 - 948 * q^83 + 1680 * q^85 + 360 * q^87 + 1166 * q^89 + 836 * q^91 + 1476 * q^93 - 2204 * q^95 + 3472 * q^97 + 180 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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