LMFDB - Embedded newform 552.4.a.f.1.1

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,-4] 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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ 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)

$f(q)$	$=$	$q+3.00000 q^{3} -6.24264 q^{5} -16.7279 q^{7} +9.00000 q^{9} -15.0294 q^{11} +66.9117 q^{13} -18.7279 q^{15} +86.1249 q^{17} -25.3310 q^{19} -50.1838 q^{21} +23.0000 q^{23} -86.0294 q^{25} +27.0000 q^{27} -253.161 q^{29} -242.309 q^{31} -45.0883 q^{33} +104.426 q^{35} -211.338 q^{37} +200.735 q^{39} -297.765 q^{41} -147.522 q^{43} -56.1838 q^{45} -440.191 q^{47} -63.1766 q^{49} +258.375 q^{51} +293.522 q^{53} +93.8234 q^{55} -75.9929 q^{57} -359.338 q^{59} -304.073 q^{61} -150.551 q^{63} -417.706 q^{65} -493.963 q^{67} +69.0000 q^{69} +688.528 q^{71} +1159.50 q^{73} -258.088 q^{75} +251.411 q^{77} -409.934 q^{79} +81.0000 q^{81} +230.912 q^{83} -537.647 q^{85} -759.484 q^{87} -867.991 q^{89} -1119.29 q^{91} -726.926 q^{93} +158.132 q^{95} +1716.95 q^{97} -135.265 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} - 4 q^{5} - 8 q^{7} + 18 q^{9} - 64 q^{11} + 32 q^{13} - 12 q^{15} + 28 q^{17} - 144 q^{19} - 24 q^{21} + 46 q^{23} - 206 q^{25} + 54 q^{27} - 116 q^{29} - 264 q^{31} - 192 q^{33} + 124 q^{35}+ \cdots - 576 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 - 4 * q^5 - 8 * q^7 + 18 * q^9 - 64 * q^11 + 32 * q^13 - 12 * q^15 + 28 * q^17 - 144 * q^19 - 24 * q^21 + 46 * q^23 - 206 * q^25 + 54 * q^27 - 116 * q^29 - 264 * q^31 - 192 * q^33 + 124 * q^35 - 236 * q^37 + 96 * q^39 - 324 * q^41 - 32 * q^43 - 36 * q^45 - 524 * q^47 - 330 * q^49 + 84 * q^51 + 324 * q^53 - 16 * q^55 - 432 * q^57 - 532 * q^59 - 116 * q^61 - 72 * q^63 - 496 * q^65 - 80 * q^67 + 138 * q^69 - 320 * q^71 + 588 * q^73 - 618 * q^75 - 176 * q^77 - 1032 * q^79 + 162 * q^81 + 360 * q^83 - 668 * q^85 - 348 * q^87 + 292 * q^89 - 1424 * q^91 - 792 * q^93 - 108 * q^95 + 1652 * q^97 - 576 * 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
$5$	−6.24264	−0.558359	−0.279179	−	0.960239i	$-0.590062\pi$
$5$	−6.24264	−0.558359	−0.279179	+	0.960239i	$0.590062\pi$
$6$	0	0
$7$	−16.7279	−0.903223	−0.451611	−	0.892215i	$-0.649151\pi$
$7$	−16.7279	−0.903223	−0.451611	+	0.892215i	$0.649151\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−15.0294	−0.411959	−0.205979	−	0.978556i	$-0.566038\pi$
$11$	−15.0294	−0.411959	−0.205979	+	0.978556i	$0.566038\pi$
$12$	0	0
$13$	66.9117	1.42754	0.713768	−	0.700382i	$-0.246987\pi$
$13$	66.9117	1.42754	0.713768	+	0.700382i	$0.246987\pi$
$14$	0	0
$15$	−18.7279	−0.322369
$16$	0	0
$17$	86.1249	1.22873	0.614363	−	0.789023i	$-0.289413\pi$
$17$	86.1249	1.22873	0.614363	+	0.789023i	$0.289413\pi$
$18$	0	0
$19$	−25.3310	−0.305859	−0.152929	−	0.988237i	$-0.548871\pi$
$19$	−25.3310	−0.305859	−0.152929	+	0.988237i	$0.548871\pi$
$20$	0	0
$21$	−50.1838	−0.521476
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	−86.0294	−0.688235
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−253.161	−1.62107	−0.810533	−	0.585693i	$-0.800822\pi$
$29$	−253.161	−1.62107	−0.810533	+	0.585693i	$0.800822\pi$
$30$	0	0
$31$	−242.309	−1.40387	−0.701934	−	0.712242i	$-0.747680\pi$
$31$	−242.309	−1.40387	−0.701934	+	0.712242i	$0.747680\pi$
$32$	0	0
$33$	−45.0883	−0.237844
$34$	0	0
$35$	104.426	0.504322
$36$	0	0
$37$	−211.338	−0.939021	−0.469510	−	0.882927i	$-0.655570\pi$
$37$	−211.338	−0.939021	−0.469510	+	0.882927i	$0.655570\pi$
$38$	0	0
$39$	200.735	0.824188
$40$	0	0
$41$	−297.765	−1.13422	−0.567110	−	0.823642i	$-0.691938\pi$
$41$	−297.765	−1.13422	−0.567110	+	0.823642i	$0.691938\pi$
$42$	0	0
$43$	−147.522	−0.523183	−0.261592	−	0.965179i	$-0.584247\pi$
$43$	−147.522	−0.523183	−0.261592	+	0.965179i	$0.584247\pi$
$44$	0	0
$45$	−56.1838	−0.186120
$46$	0	0
$47$	−440.191	−1.36614	−0.683069	−	0.730354i	$-0.739355\pi$
$47$	−440.191	−1.36614	−0.683069	+	0.730354i	$0.739355\pi$
$48$	0	0
$49$	−63.1766	−0.184188
$50$	0	0
$51$	258.375	0.709406
$52$	0	0
$53$	293.522	0.760723	0.380362	−	0.924838i	$-0.375800\pi$
$53$	293.522	0.760723	0.380362	+	0.924838i	$0.375800\pi$
$54$	0	0
$55$	93.8234	0.230021
$56$	0	0
$57$	−75.9929	−0.176588
$58$	0	0
$59$	−359.338	−0.792912	−0.396456	−	0.918054i	$-0.629760\pi$
$59$	−359.338	−0.792912	−0.396456	+	0.918054i	$0.629760\pi$
$60$	0	0
$61$	−304.073	−0.638239	−0.319119	−	0.947714i	$-0.603387\pi$
$61$	−304.073	−0.638239	−0.319119	+	0.947714i	$0.603387\pi$
$62$	0	0
$63$	−150.551	−0.301074
$64$	0	0
$65$	−417.706	−0.797077
$66$	0	0
$67$	−493.963	−0.900703	−0.450352	−	0.892851i	$-0.648701\pi$
$67$	−493.963	−0.900703	−0.450352	+	0.892851i	$0.648701\pi$
$68$	0	0
$69$	69.0000	0.120386
$70$	0	0
$71$	688.528	1.15089	0.575445	−	0.817840i	$-0.304829\pi$
$71$	688.528	1.15089	0.575445	+	0.817840i	$0.304829\pi$
$72$	0	0
$73$	1159.50	1.85903	0.929514	−	0.368787i	$-0.120227\pi$
$73$	1159.50	1.85903	0.929514	+	0.368787i	$0.120227\pi$
$74$	0	0
$75$	−258.088	−0.397353
$76$	0	0
$77$	251.411	0.372091
$78$	0	0
$79$	−409.934	−0.583812	−0.291906	−	0.956447i	$-0.594290\pi$
$79$	−409.934	−0.583812	−0.291906	+	0.956447i	$0.594290\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	230.912	0.305372	0.152686	−	0.988275i	$-0.451208\pi$
$83$	230.912	0.305372	0.152686	+	0.988275i	$0.451208\pi$
$84$	0	0
$85$	−537.647	−0.686070
$86$	0	0
$87$	−759.484	−0.935923
$88$	0	0
$89$	−867.991	−1.03379	−0.516893	−	0.856050i	$-0.672911\pi$
$89$	−867.991	−1.03379	−0.516893	+	0.856050i	$0.672911\pi$
$90$	0	0
$91$	−1119.29	−1.28938
$92$	0	0
$93$	−726.926	−0.810524
$94$	0	0
$95$	158.132	0.170779
$96$	0	0
$97$	1716.95	1.79722	0.898610	−	0.438748i	$-0.144578\pi$
$97$	1716.95	1.79722	0.898610	+	0.438748i	$0.144578\pi$
$98$	0	0
$99$	−135.265	−0.137320
$100$	0	0
$101$	328.177	0.323315	0.161657	−	0.986847i	$-0.448316\pi$
$101$	328.177	0.323315	0.161657	+	0.986847i	$0.448316\pi$
$102$	0	0
$103$	−303.258	−0.290106	−0.145053	−	0.989424i	$-0.546335\pi$
$103$	−303.258	−0.290106	−0.145053	+	0.989424i	$0.546335\pi$
$104$	0	0
$105$	313.279	0.291171
$106$	0	0
$107$	2091.20	1.88939	0.944693	−	0.327955i	$-0.106360\pi$
$107$	2091.20	1.88939	0.944693	+	0.327955i	$0.106360\pi$
$108$	0	0
$109$	−1644.97	−1.44550	−0.722750	−	0.691110i	$-0.757122\pi$
$109$	−1644.97	−1.44550	−0.722750	+	0.691110i	$0.757122\pi$
$110$	0	0
$111$	−634.014	−0.542144
$112$	0	0
$113$	770.860	0.641738	0.320869	−	0.947124i	$-0.396025\pi$
$113$	770.860	0.641738	0.320869	+	0.947124i	$0.396025\pi$
$114$	0	0
$115$	−143.581	−0.116426
$116$	0	0
$117$	602.205	0.475845
$118$	0	0
$119$	−1440.69	−1.10981
$120$	0	0
$121$	−1105.12	−0.830290
$122$	0	0
$123$	−893.294	−0.654842
$124$	0	0
$125$	1317.38	0.942641
$126$	0	0
$127$	−2023.51	−1.41384	−0.706921	−	0.707293i	$-0.749916\pi$
$127$	−2023.51	−1.41384	−0.706921	+	0.707293i	$0.749916\pi$
$128$	0	0
$129$	−442.566	−0.302060
$130$	0	0
$131$	−1307.79	−0.872232	−0.436116	−	0.899890i	$-0.643646\pi$
$131$	−1307.79	−0.872232	−0.436116	+	0.899890i	$0.643646\pi$
$132$	0	0
$133$	423.734	0.276259
$134$	0	0
$135$	−168.551	−0.107456
$136$	0	0
$137$	1837.17	1.14569	0.572846	−	0.819663i	$-0.305839\pi$
$137$	1837.17	1.14569	0.572846	+	0.819663i	$0.305839\pi$
$138$	0	0
$139$	−1733.91	−1.05805	−0.529024	−	0.848607i	$-0.677442\pi$
$139$	−1733.91	−1.05805	−0.529024	+	0.848607i	$0.677442\pi$
$140$	0	0
$141$	−1320.57	−0.788740
$142$	0	0
$143$	−1005.65	−0.588086
$144$	0	0
$145$	1580.40	0.905136
$146$	0	0
$147$	−189.530	−0.106341
$148$	0	0
$149$	2180.92	1.19911	0.599556	−	0.800333i	$-0.295344\pi$
$149$	2180.92	1.19911	0.599556	+	0.800333i	$0.295344\pi$
$150$	0	0
$151$	−2041.03	−1.09998	−0.549988	−	0.835172i	$-0.685368\pi$
$151$	−2041.03	−1.09998	−0.549988	+	0.835172i	$0.685368\pi$
$152$	0	0
$153$	775.124	0.409576
$154$	0	0
$155$	1512.65	0.783862
$156$	0	0
$157$	−2645.84	−1.34497	−0.672486	−	0.740110i	$-0.734774\pi$
$157$	−2645.84	−1.34497	−0.672486	+	0.740110i	$0.734774\pi$
$158$	0	0
$159$	880.566	0.439204
$160$	0	0
$161$	−384.742	−0.188335
$162$	0	0
$163$	2530.80	1.21612	0.608061	−	0.793890i	$-0.291948\pi$
$163$	2530.80	1.21612	0.608061	+	0.793890i	$0.291948\pi$
$164$	0	0
$165$	281.470	0.132803
$166$	0	0
$167$	−3501.97	−1.62270	−0.811348	−	0.584563i	$-0.801266\pi$
$167$	−3501.97	−1.62270	−0.811348	+	0.584563i	$0.801266\pi$
$168$	0	0
$169$	2280.17	1.03786
$170$	0	0
$171$	−227.979	−0.101953
$172$	0	0
$173$	−1272.19	−0.559090	−0.279545	−	0.960133i	$-0.590184\pi$
$173$	−1272.19	−0.559090	−0.279545	+	0.960133i	$0.590184\pi$
$174$	0	0
$175$	1439.09	0.621630
$176$	0	0
$177$	−1078.01	−0.457788
$178$	0	0
$179$	523.634	0.218649	0.109325	−	0.994006i	$-0.465131\pi$
$179$	523.634	0.218649	0.109325	+	0.994006i	$0.465131\pi$
$180$	0	0
$181$	3757.18	1.54292	0.771460	−	0.636277i	$-0.219527\pi$
$181$	3757.18	1.54292	0.771460	+	0.636277i	$0.219527\pi$
$182$	0	0
$183$	−912.219	−0.368487
$184$	0	0
$185$	1319.31	0.524310
$186$	0	0
$187$	−1294.41	−0.506185
$188$	0	0
$189$	−451.654	−0.173825
$190$	0	0
$191$	392.425	0.148664	0.0743321	−	0.997234i	$-0.476318\pi$
$191$	392.425	0.148664	0.0743321	+	0.997234i	$0.476318\pi$
$192$	0	0
$193$	763.532	0.284768	0.142384	−	0.989811i	$-0.454523\pi$
$193$	763.532	0.284768	0.142384	+	0.989811i	$0.454523\pi$
$194$	0	0
$195$	−1253.12	−0.460193
$196$	0	0
$197$	−1826.92	−0.660726	−0.330363	−	0.943854i	$-0.607171\pi$
$197$	−1826.92	−0.660726	−0.330363	+	0.943854i	$0.607171\pi$
$198$	0	0
$199$	618.519	0.220330	0.110165	−	0.993913i	$-0.464862\pi$
$199$	618.519	0.220330	0.110165	+	0.993913i	$0.464862\pi$
$200$	0	0
$201$	−1481.89	−0.520021
$202$	0	0
$203$	4234.87	1.46418
$204$	0	0
$205$	1858.84	0.633301
$206$	0	0
$207$	207.000	0.0695048
$208$	0	0
$209$	380.710	0.126001
$210$	0	0
$211$	−2841.93	−0.927234	−0.463617	−	0.886036i	$-0.653449\pi$
$211$	−2841.93	−0.927234	−0.463617	+	0.886036i	$0.653449\pi$
$212$	0	0
$213$	2065.58	0.664467
$214$	0	0
$215$	920.926	0.292124
$216$	0	0
$217$	4053.32	1.26801
$218$	0	0
$219$	3478.50	1.07331
$220$	0	0
$221$	5762.76	1.75405
$222$	0	0
$223$	−1264.35	−0.379674	−0.189837	−	0.981816i	$-0.560796\pi$
$223$	−1264.35	−0.379674	−0.189837	+	0.981816i	$0.560796\pi$
$224$	0	0
$225$	−774.265	−0.229412
$226$	0	0
$227$	−2767.81	−0.809277	−0.404638	−	0.914477i	$-0.632603\pi$
$227$	−2767.81	−0.809277	−0.404638	+	0.914477i	$0.632603\pi$
$228$	0	0
$229$	−2155.85	−0.622107	−0.311053	−	0.950392i	$-0.600682\pi$
$229$	−2155.85	−0.622107	−0.311053	+	0.950392i	$0.600682\pi$
$230$	0	0
$231$	754.234	0.214827
$232$	0	0
$233$	3688.85	1.03719	0.518594	−	0.855021i	$-0.326456\pi$
$233$	3688.85	1.03719	0.518594	+	0.855021i	$0.326456\pi$
$234$	0	0
$235$	2747.95	0.762795
$236$	0	0
$237$	−1229.80	−0.337064
$238$	0	0
$239$	2656.67	0.719020	0.359510	−	0.933141i	$-0.382944\pi$
$239$	2656.67	0.719020	0.359510	+	0.933141i	$0.382944\pi$
$240$	0	0
$241$	−2916.54	−0.779547	−0.389774	−	0.920911i	$-0.627447\pi$
$241$	−2916.54	−0.779547	−0.389774	+	0.920911i	$0.627447\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	394.389	0.102843
$246$	0	0
$247$	−1694.94	−0.436624
$248$	0	0
$249$	692.735	0.176306
$250$	0	0
$251$	6264.90	1.57545	0.787723	−	0.616030i	$-0.211260\pi$
$251$	6264.90	1.57545	0.787723	+	0.616030i	$0.211260\pi$
$252$	0	0
$253$	−345.677	−0.0858993
$254$	0	0
$255$	−1612.94	−0.396103
$256$	0	0
$257$	−1349.94	−0.327653	−0.163827	−	0.986489i	$-0.552384\pi$
$257$	−1349.94	−0.327653	−0.163827	+	0.986489i	$0.552384\pi$
$258$	0	0
$259$	3535.25	0.848145
$260$	0	0
$261$	−2278.45	−0.540355
$262$	0	0
$263$	3890.78	0.912227	0.456114	−	0.889922i	$-0.349241\pi$
$263$	3890.78	0.912227	0.456114	+	0.889922i	$0.349241\pi$
$264$	0	0
$265$	−1832.35	−0.424756
$266$	0	0
$267$	−2603.97	−0.596856
$268$	0	0
$269$	1585.90	0.359457	0.179729	−	0.983716i	$-0.442478\pi$
$269$	1585.90	0.359457	0.179729	+	0.983716i	$0.442478\pi$
$270$	0	0
$271$	−5414.04	−1.21358	−0.606788	−	0.794863i	$-0.707543\pi$
$271$	−5414.04	−1.21358	−0.606788	+	0.794863i	$0.707543\pi$
$272$	0	0
$273$	−3357.88	−0.744426
$274$	0	0
$275$	1292.97	0.283525
$276$	0	0
$277$	6493.76	1.40856	0.704282	−	0.709921i	$-0.251269\pi$
$277$	6493.76	1.40856	0.704282	+	0.709921i	$0.251269\pi$
$278$	0	0
$279$	−2180.78	−0.467956
$280$	0	0
$281$	7344.93	1.55929	0.779647	−	0.626219i	$-0.215398\pi$
$281$	7344.93	1.55929	0.779647	+	0.626219i	$0.215398\pi$
$282$	0	0
$283$	−4398.28	−0.923853	−0.461927	−	0.886918i	$-0.652842\pi$
$283$	−4398.28	−0.923853	−0.461927	+	0.886918i	$0.652842\pi$
$284$	0	0
$285$	474.396	0.0985993
$286$	0	0
$287$	4980.98	1.02445
$288$	0	0
$289$	2504.50	0.509769
$290$	0	0
$291$	5150.86	1.03763
$292$	0	0
$293$	4812.44	0.959542	0.479771	−	0.877394i	$-0.340720\pi$
$293$	4812.44	0.959542	0.479771	+	0.877394i	$0.340720\pi$
$294$	0	0
$295$	2243.22	0.442730
$296$	0	0
$297$	−405.795	−0.0792815
$298$	0	0
$299$	1538.97	0.297662
$300$	0	0
$301$	2467.73	0.472551
$302$	0	0
$303$	984.530	0.186666
$304$	0	0
$305$	1898.22	0.356366
$306$	0	0
$307$	−5481.16	−1.01898	−0.509489	−	0.860477i	$-0.670166\pi$
$307$	−5481.16	−1.01898	−0.509489	+	0.860477i	$0.670166\pi$
$308$	0	0
$309$	−909.773	−0.167493
$310$	0	0
$311$	3224.60	0.587943	0.293972	−	0.955814i	$-0.405023\pi$
$311$	3224.60	0.587943	0.293972	+	0.955814i	$0.405023\pi$
$312$	0	0
$313$	1119.38	0.202144	0.101072	−	0.994879i	$-0.467773\pi$
$313$	1119.38	0.202144	0.101072	+	0.994879i	$0.467773\pi$
$314$	0	0
$315$	939.838	0.168107
$316$	0	0
$317$	−6647.39	−1.17778	−0.588888	−	0.808215i	$-0.700434\pi$
$317$	−6647.39	−1.17778	−0.588888	+	0.808215i	$0.700434\pi$
$318$	0	0
$319$	3804.87	0.667812
$320$	0	0
$321$	6273.61	1.09084
$322$	0	0
$323$	−2181.63	−0.375817
$324$	0	0
$325$	−5756.37	−0.982481
$326$	0	0
$327$	−4934.91	−0.834559
$328$	0	0
$329$	7363.48	1.23393
$330$	0	0
$331$	−3471.22	−0.576421	−0.288211	−	0.957567i	$-0.593060\pi$
$331$	−3471.22	−0.576421	−0.288211	+	0.957567i	$0.593060\pi$
$332$	0	0
$333$	−1902.04	−0.313007
$334$	0	0
$335$	3083.63	0.502916
$336$	0	0
$337$	−8157.68	−1.31863	−0.659313	−	0.751868i	$-0.729153\pi$
$337$	−8157.68	−1.31863	−0.659313	+	0.751868i	$0.729153\pi$
$338$	0	0
$339$	2312.58	0.370508
$340$	0	0
$341$	3641.76	0.578336
$342$	0	0
$343$	6794.49	1.06959
$344$	0	0
$345$	−430.742	−0.0672185
$346$	0	0
$347$	10809.4	1.67227	0.836134	−	0.548525i	$-0.184810\pi$
$347$	10809.4	1.67227	0.836134	+	0.548525i	$0.184810\pi$
$348$	0	0
$349$	−3674.47	−0.563581	−0.281791	−	0.959476i	$-0.590928\pi$
$349$	−3674.47	−0.563581	−0.281791	+	0.959476i	$0.590928\pi$
$350$	0	0
$351$	1806.62	0.274729
$352$	0	0
$353$	7055.50	1.06381	0.531907	−	0.846803i	$-0.321475\pi$
$353$	7055.50	1.06381	0.531907	+	0.846803i	$0.321475\pi$
$354$	0	0
$355$	−4298.23	−0.642610
$356$	0	0
$357$	−4322.07	−0.640751
$358$	0	0
$359$	−12759.1	−1.87576	−0.937882	−	0.346955i	$-0.887216\pi$
$359$	−12759.1	−1.87576	−0.937882	+	0.346955i	$0.887216\pi$
$360$	0	0
$361$	−6217.34	−0.906450
$362$	0	0
$363$	−3315.35	−0.479368
$364$	0	0
$365$	−7238.33	−1.03800
$366$	0	0
$367$	−9532.35	−1.35582	−0.677908	−	0.735147i	$-0.737113\pi$
$367$	−9532.35	−1.35582	−0.677908	+	0.735147i	$0.737113\pi$
$368$	0	0
$369$	−2679.88	−0.378073
$370$	0	0
$371$	−4910.01	−0.687103
$372$	0	0
$373$	7953.59	1.10408	0.552039	−	0.833818i	$-0.313850\pi$
$373$	7953.59	1.10408	0.552039	+	0.833818i	$0.313850\pi$
$374$	0	0
$375$	3952.14	0.544234
$376$	0	0
$377$	−16939.5	−2.31413
$378$	0	0
$379$	3599.35	0.487827	0.243913	−	0.969797i	$-0.421569\pi$
$379$	3599.35	0.487827	0.243913	+	0.969797i	$0.421569\pi$
$380$	0	0
$381$	−6070.54	−0.816282
$382$	0	0
$383$	−9334.89	−1.24541	−0.622703	−	0.782458i	$-0.713965\pi$
$383$	−9334.89	−1.24541	−0.622703	+	0.782458i	$0.713965\pi$
$384$	0	0
$385$	−1569.47	−0.207760
$386$	0	0
$387$	−1327.70	−0.174394
$388$	0	0
$389$	1355.94	0.176733	0.0883663	−	0.996088i	$-0.471835\pi$
$389$	1355.94	0.176733	0.0883663	+	0.996088i	$0.471835\pi$
$390$	0	0
$391$	1980.87	0.256207
$392$	0	0
$393$	−3923.38	−0.503583
$394$	0	0
$395$	2559.07	0.325977
$396$	0	0
$397$	11441.2	1.44639	0.723194	−	0.690645i	$-0.242673\pi$
$397$	11441.2	1.44639	0.723194	+	0.690645i	$0.242673\pi$
$398$	0	0
$399$	1271.20	0.159498
$400$	0	0
$401$	−14903.0	−1.85591	−0.927953	−	0.372698i	$-0.878433\pi$
$401$	−14903.0	−1.85591	−0.927953	+	0.372698i	$0.878433\pi$
$402$	0	0
$403$	−16213.3	−2.00407
$404$	0	0
$405$	−505.654	−0.0620399
$406$	0	0
$407$	3176.29	0.386838
$408$	0	0
$409$	2793.90	0.337774	0.168887	−	0.985635i	$-0.445983\pi$
$409$	2793.90	0.337774	0.168887	+	0.985635i	$0.445983\pi$
$410$	0	0
$411$	5511.51	0.661466
$412$	0	0
$413$	6010.98	0.716177
$414$	0	0
$415$	−1441.50	−0.170507
$416$	0	0
$417$	−5201.74	−0.610864
$418$	0	0
$419$	5099.89	0.594620	0.297310	−	0.954781i	$-0.403911\pi$
$419$	5099.89	0.594620	0.297310	+	0.954781i	$0.403911\pi$
$420$	0	0
$421$	−9540.93	−1.10450	−0.552252	−	0.833677i	$-0.686232\pi$
$421$	−9540.93	−1.10450	−0.552252	+	0.833677i	$0.686232\pi$
$422$	0	0
$423$	−3961.72	−0.455379
$424$	0	0
$425$	−7409.28	−0.845653
$426$	0	0
$427$	5086.51	0.576472
$428$	0	0
$429$	−3016.94	−0.339531
$430$	0	0
$431$	6837.04	0.764104	0.382052	−	0.924141i	$-0.375218\pi$
$431$	6837.04	0.764104	0.382052	+	0.924141i	$0.375218\pi$
$432$	0	0
$433$	8428.43	0.935438	0.467719	−	0.883877i	$-0.345076\pi$
$433$	8428.43	0.935438	0.467719	+	0.883877i	$0.345076\pi$
$434$	0	0
$435$	4741.19	0.522581
$436$	0	0
$437$	−582.612	−0.0637760
$438$	0	0
$439$	6386.90	0.694374	0.347187	−	0.937796i	$-0.387137\pi$
$439$	6386.90	0.694374	0.347187	+	0.937796i	$0.387137\pi$
$440$	0	0
$441$	−568.590	−0.0613961
$442$	0	0
$443$	11042.2	1.18427	0.592134	−	0.805839i	$-0.298286\pi$
$443$	11042.2	1.18427	0.592134	+	0.805839i	$0.298286\pi$
$444$	0	0
$445$	5418.56	0.577223
$446$	0	0
$447$	6542.75	0.692307
$448$	0	0
$449$	−9301.96	−0.977699	−0.488849	−	0.872368i	$-0.662583\pi$
$449$	−9301.96	−0.977699	−0.488849	+	0.872368i	$0.662583\pi$
$450$	0	0
$451$	4475.23	0.467252
$452$	0	0
$453$	−6123.08	−0.635072
$454$	0	0
$455$	6987.35	0.719938
$456$	0	0
$457$	2481.48	0.254002	0.127001	−	0.991903i	$-0.459465\pi$
$457$	2481.48	0.254002	0.127001	+	0.991903i	$0.459465\pi$
$458$	0	0
$459$	2325.37	0.236469
$460$	0	0
$461$	−5204.22	−0.525780	−0.262890	−	0.964826i	$-0.584676\pi$
$461$	−5204.22	−0.525780	−0.262890	+	0.964826i	$0.584676\pi$
$462$	0	0
$463$	−2965.33	−0.297647	−0.148824	−	0.988864i	$-0.547549\pi$
$463$	−2965.33	−0.297647	−0.148824	+	0.988864i	$0.547549\pi$
$464$	0	0
$465$	4537.94	0.452563
$466$	0	0
$467$	10281.5	1.01878	0.509392	−	0.860534i	$-0.329870\pi$
$467$	10281.5	1.01878	0.509392	+	0.860534i	$0.329870\pi$
$468$	0	0
$469$	8262.97	0.813536
$470$	0	0
$471$	−7937.51	−0.776520
$472$	0	0
$473$	2217.17	0.215530
$474$	0	0
$475$	2179.21	0.210503
$476$	0	0
$477$	2641.70	0.253574
$478$	0	0
$479$	−8370.34	−0.798435	−0.399218	−	0.916856i	$-0.630718\pi$
$479$	−8370.34	−0.798435	−0.399218	+	0.916856i	$0.630718\pi$
$480$	0	0
$481$	−14141.0	−1.34049
$482$	0	0
$483$	−1154.23	−0.108735
$484$	0	0
$485$	−10718.3	−1.00349
$486$	0	0
$487$	−15786.6	−1.46891	−0.734455	−	0.678657i	$-0.762562\pi$
$487$	−15786.6	−1.46891	−0.734455	+	0.678657i	$0.762562\pi$
$488$	0	0
$489$	7592.41	0.702129
$490$	0	0
$491$	1223.74	0.112478	0.0562390	−	0.998417i	$-0.482089\pi$
$491$	1223.74	0.112478	0.0562390	+	0.998417i	$0.482089\pi$
$492$	0	0
$493$	−21803.5	−1.99185
$494$	0	0
$495$	844.410	0.0766736
$496$	0	0
$497$	−11517.6	−1.03951
$498$	0	0
$499$	14900.5	1.33675	0.668375	−	0.743825i	$-0.266990\pi$
$499$	14900.5	1.33675	0.668375	+	0.743825i	$0.266990\pi$
$500$	0	0
$501$	−10505.9	−0.936864
$502$	0	0
$503$	−4105.12	−0.363893	−0.181946	−	0.983308i	$-0.558240\pi$
$503$	−4105.12	−0.363893	−0.181946	+	0.983308i	$0.558240\pi$
$504$	0	0
$505$	−2048.69	−0.180526
$506$	0	0
$507$	6840.52	0.599208
$508$	0	0
$509$	−9815.68	−0.854759	−0.427380	−	0.904072i	$-0.640563\pi$
$509$	−9815.68	−0.854759	−0.427380	+	0.904072i	$0.640563\pi$
$510$	0	0
$511$	−19396.0	−1.67912
$512$	0	0
$513$	−683.936	−0.0588626
$514$	0	0
$515$	1893.13	0.161983
$516$	0	0
$517$	6615.82	0.562792
$518$	0	0
$519$	−3816.56	−0.322791
$520$	0	0
$521$	12019.0	1.01067	0.505337	−	0.862922i	$-0.331368\pi$
$521$	12019.0	1.01067	0.505337	+	0.862922i	$0.331368\pi$
$522$	0	0
$523$	−10319.3	−0.862773	−0.431386	−	0.902167i	$-0.641975\pi$
$523$	−10319.3	−0.862773	−0.431386	+	0.902167i	$0.641975\pi$
$524$	0	0
$525$	4317.28	0.358898
$526$	0	0
$527$	−20868.8	−1.72497
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	−3234.04	−0.264304
$532$	0	0
$533$	−19923.9	−1.61914
$534$	0	0
$535$	−13054.6	−1.05496
$536$	0	0
$537$	1570.90	0.126237
$538$	0	0
$539$	949.509	0.0758780
$540$	0	0
$541$	18497.4	1.46999	0.734994	−	0.678074i	$-0.237185\pi$
$541$	18497.4	1.46999	0.734994	+	0.678074i	$0.237185\pi$
$542$	0	0
$543$	11271.5	0.890806
$544$	0	0
$545$	10268.9	0.807107
$546$	0	0
$547$	16432.9	1.28449	0.642247	−	0.766498i	$-0.278002\pi$
$547$	16432.9	1.28449	0.642247	+	0.766498i	$0.278002\pi$
$548$	0	0
$549$	−2736.66	−0.212746
$550$	0	0
$551$	6412.82	0.495817
$552$	0	0
$553$	6857.34	0.527313
$554$	0	0
$555$	3957.92	0.302711
$556$	0	0
$557$	−2545.03	−0.193602	−0.0968009	−	0.995304i	$-0.530861\pi$
$557$	−2545.03	−0.193602	−0.0968009	+	0.995304i	$0.530861\pi$
$558$	0	0
$559$	−9870.94	−0.746863
$560$	0	0
$561$	−3883.23	−0.292246
$562$	0	0
$563$	1771.93	0.132643	0.0663213	−	0.997798i	$-0.478874\pi$
$563$	1771.93	0.132643	0.0663213	+	0.997798i	$0.478874\pi$
$564$	0	0
$565$	−4812.20	−0.358320
$566$	0	0
$567$	−1354.96	−0.100358
$568$	0	0
$569$	−17820.3	−1.31294	−0.656472	−	0.754351i	$-0.727952\pi$
$569$	−17820.3	−1.31294	−0.656472	+	0.754351i	$0.727952\pi$
$570$	0	0
$571$	3208.29	0.235136	0.117568	−	0.993065i	$-0.462490\pi$
$571$	3208.29	0.235136	0.117568	+	0.993065i	$0.462490\pi$
$572$	0	0
$573$	1177.27	0.0858313
$574$	0	0
$575$	−1978.68	−0.143507
$576$	0	0
$577$	24532.6	1.77003	0.885015	−	0.465563i	$-0.154148\pi$
$577$	24532.6	1.77003	0.885015	+	0.465563i	$0.154148\pi$
$578$	0	0
$579$	2290.60	0.164411
$580$	0	0
$581$	−3862.67	−0.275819
$582$	0	0
$583$	−4411.47	−0.313387
$584$	0	0
$585$	−3759.35	−0.265692
$586$	0	0
$587$	2684.90	0.188786	0.0943932	−	0.995535i	$-0.469909\pi$
$587$	2684.90	0.188786	0.0943932	+	0.995535i	$0.469909\pi$
$588$	0	0
$589$	6137.91	0.429386
$590$	0	0
$591$	−5480.77	−0.381470
$592$	0	0
$593$	−4003.67	−0.277253	−0.138627	−	0.990345i	$-0.544269\pi$
$593$	−4003.67	−0.277253	−0.138627	+	0.990345i	$0.544269\pi$
$594$	0	0
$595$	8993.71	0.619674
$596$	0	0
$597$	1855.56	0.127208
$598$	0	0
$599$	−5702.89	−0.389005	−0.194502	−	0.980902i	$-0.562309\pi$
$599$	−5702.89	−0.389005	−0.194502	+	0.980902i	$0.562309\pi$
$600$	0	0
$601$	1015.27	0.0689082	0.0344541	−	0.999406i	$-0.489031\pi$
$601$	1015.27	0.0689082	0.0344541	+	0.999406i	$0.489031\pi$
$602$	0	0
$603$	−4445.66	−0.300234
$604$	0	0
$605$	6898.84	0.463600
$606$	0	0
$607$	7260.55	0.485497	0.242748	−	0.970089i	$-0.421951\pi$
$607$	7260.55	0.485497	0.242748	+	0.970089i	$0.421951\pi$
$608$	0	0
$609$	12704.6	0.845347
$610$	0	0
$611$	−29453.9	−1.95021
$612$	0	0
$613$	−8696.58	−0.573004	−0.286502	−	0.958080i	$-0.592493\pi$
$613$	−8696.58	−0.573004	−0.286502	+	0.958080i	$0.592493\pi$
$614$	0	0
$615$	5576.51	0.365637
$616$	0	0
$617$	−12353.0	−0.806018	−0.403009	−	0.915196i	$-0.632036\pi$
$617$	−12353.0	−0.806018	−0.403009	+	0.915196i	$0.632036\pi$
$618$	0	0
$619$	15797.8	1.02579	0.512897	−	0.858450i	$-0.328572\pi$
$619$	15797.8	1.02579	0.512897	+	0.858450i	$0.328572\pi$
$620$	0	0
$621$	621.000	0.0401286
$622$	0	0
$623$	14519.7	0.933738
$624$	0	0
$625$	2529.74	0.161904
$626$	0	0
$627$	1142.13	0.0727468
$628$	0	0
$629$	−18201.5	−1.15380
$630$	0	0
$631$	−12275.8	−0.774475	−0.387237	−	0.921980i	$-0.626571\pi$
$631$	−12275.8	−0.774475	−0.387237	+	0.921980i	$0.626571\pi$
$632$	0	0
$633$	−8525.78	−0.535339
$634$	0	0
$635$	12632.1	0.789431
$636$	0	0
$637$	−4227.25	−0.262936
$638$	0	0
$639$	6196.75	0.383630
$640$	0	0
$641$	−3829.83	−0.235989	−0.117995	−	0.993014i	$-0.537647\pi$
$641$	−3829.83	−0.235989	−0.117995	+	0.993014i	$0.537647\pi$
$642$	0	0
$643$	2859.19	0.175358	0.0876791	−	0.996149i	$-0.472055\pi$
$643$	2859.19	0.175358	0.0876791	+	0.996149i	$0.472055\pi$
$644$	0	0
$645$	2762.78	0.168658
$646$	0	0
$647$	−13101.5	−0.796095	−0.398048	−	0.917365i	$-0.630312\pi$
$647$	−13101.5	−0.796095	−0.398048	+	0.917365i	$0.630312\pi$
$648$	0	0
$649$	5400.65	0.326647
$650$	0	0
$651$	12160.0	0.732084
$652$	0	0
$653$	7193.19	0.431074	0.215537	−	0.976496i	$-0.430850\pi$
$653$	7193.19	0.431074	0.215537	+	0.976496i	$0.430850\pi$
$654$	0	0
$655$	8164.08	0.487018
$656$	0	0
$657$	10435.5	0.619676
$658$	0	0
$659$	21910.6	1.29517	0.647585	−	0.761993i	$-0.275779\pi$
$659$	21910.6	1.29517	0.647585	+	0.761993i	$0.275779\pi$
$660$	0	0
$661$	28372.2	1.66952	0.834758	−	0.550616i	$-0.185607\pi$
$661$	28372.2	1.66952	0.834758	+	0.550616i	$0.185607\pi$
$662$	0	0
$663$	17288.3	1.01270
$664$	0	0
$665$	−2645.22	−0.154251
$666$	0	0
$667$	−5822.71	−0.338016
$668$	0	0
$669$	−3793.06	−0.219205
$670$	0	0
$671$	4570.05	0.262928
$672$	0	0
$673$	−17915.4	−1.02613	−0.513066	−	0.858349i	$-0.671490\pi$
$673$	−17915.4	−1.02613	−0.513066	+	0.858349i	$0.671490\pi$
$674$	0	0
$675$	−2322.79	−0.132451
$676$	0	0
$677$	−16473.8	−0.935216	−0.467608	−	0.883936i	$-0.654884\pi$
$677$	−16473.8	−0.935216	−0.467608	+	0.883936i	$0.654884\pi$
$678$	0	0
$679$	−28721.1	−1.62329
$680$	0	0
$681$	−8303.42	−0.467236
$682$	0	0
$683$	−15021.3	−0.841543	−0.420771	−	0.907167i	$-0.638241\pi$
$683$	−15021.3	−0.841543	−0.420771	+	0.907167i	$0.638241\pi$
$684$	0	0
$685$	−11468.8	−0.639708
$686$	0	0
$687$	−6467.55	−0.359174
$688$	0	0
$689$	19640.0	1.08596
$690$	0	0
$691$	−19095.2	−1.05125	−0.525626	−	0.850716i	$-0.676169\pi$
$691$	−19095.2	−1.05125	−0.525626	+	0.850716i	$0.676169\pi$
$692$	0	0
$693$	2262.70	0.124030
$694$	0	0
$695$	10824.2	0.590770
$696$	0	0
$697$	−25644.9	−1.39365
$698$	0	0
$699$	11066.6	0.598821
$700$	0	0
$701$	33274.0	1.79279	0.896393	−	0.443261i	$-0.146178\pi$
$701$	33274.0	1.79279	0.896393	+	0.443261i	$0.146178\pi$
$702$	0	0
$703$	5353.40	0.287208
$704$	0	0
$705$	8243.86	0.440400
$706$	0	0
$707$	−5489.71	−0.292025
$708$	0	0
$709$	29929.3	1.58536	0.792678	−	0.609640i	$-0.208686\pi$
$709$	29929.3	1.58536	0.792678	+	0.609640i	$0.208686\pi$
$710$	0	0
$711$	−3689.41	−0.194604
$712$	0	0
$713$	−5573.10	−0.292727
$714$	0	0
$715$	6277.88	0.328363
$716$	0	0
$717$	7970.01	0.415126
$718$	0	0
$719$	16408.9	0.851110	0.425555	−	0.904933i	$-0.360079\pi$
$719$	16408.9	0.851110	0.425555	+	0.904933i	$0.360079\pi$
$720$	0	0
$721$	5072.87	0.262030
$722$	0	0
$723$	−8749.62	−0.450072
$724$	0	0
$725$	21779.3	1.11567
$726$	0	0
$727$	19328.4	0.986037	0.493019	−	0.870019i	$-0.335893\pi$
$727$	19328.4	0.986037	0.493019	+	0.870019i	$0.335893\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−12705.3	−0.642849
$732$	0	0
$733$	4889.32	0.246373	0.123186	−	0.992384i	$-0.460689\pi$
$733$	4889.32	0.246373	0.123186	+	0.992384i	$0.460689\pi$
$734$	0	0
$735$	1183.17	0.0593766
$736$	0	0
$737$	7423.98	0.371053
$738$	0	0
$739$	9774.43	0.486547	0.243273	−	0.969958i	$-0.421779\pi$
$739$	9774.43	0.486547	0.243273	+	0.969958i	$0.421779\pi$
$740$	0	0
$741$	−5084.81	−0.252085
$742$	0	0
$743$	17144.9	0.846546	0.423273	−	0.906002i	$-0.360881\pi$
$743$	17144.9	0.846546	0.423273	+	0.906002i	$0.360881\pi$
$744$	0	0
$745$	−13614.7	−0.669534
$746$	0	0
$747$	2078.21	0.101791
$748$	0	0
$749$	−34981.5	−1.70654
$750$	0	0
$751$	1879.13	0.0913055	0.0456527	−	0.998957i	$-0.485463\pi$
$751$	1879.13	0.0913055	0.0456527	+	0.998957i	$0.485463\pi$
$752$	0	0
$753$	18794.7	0.909584
$754$	0	0
$755$	12741.4	0.614182
$756$	0	0
$757$	36180.8	1.73714	0.868569	−	0.495568i	$-0.165040\pi$
$757$	36180.8	1.73714	0.868569	+	0.495568i	$0.165040\pi$
$758$	0	0
$759$	−1037.03	−0.0495940
$760$	0	0
$761$	−18211.9	−0.867518	−0.433759	−	0.901029i	$-0.642813\pi$
$761$	−18211.9	−0.867518	−0.433759	+	0.901029i	$0.642813\pi$
$762$	0	0
$763$	27516.9	1.30561
$764$	0	0
$765$	−4838.82	−0.228690
$766$	0	0
$767$	−24043.9	−1.13191
$768$	0	0
$769$	−17563.2	−0.823598	−0.411799	−	0.911275i	$-0.635099\pi$
$769$	−17563.2	−0.823598	−0.411799	+	0.911275i	$0.635099\pi$
$770$	0	0
$771$	−4049.82	−0.189171
$772$	0	0
$773$	8310.20	0.386672	0.193336	−	0.981133i	$-0.438069\pi$
$773$	8310.20	0.386672	0.193336	+	0.981133i	$0.438069\pi$
$774$	0	0
$775$	20845.7	0.966192
$776$	0	0
$777$	10605.7	0.489677
$778$	0	0
$779$	7542.66	0.346911
$780$	0	0
$781$	−10348.2	−0.474120
$782$	0	0
$783$	−6835.36	−0.311974
$784$	0	0
$785$	16517.0	0.750977
$786$	0	0
$787$	19931.3	0.902763	0.451381	−	0.892331i	$-0.350931\pi$
$787$	19931.3	0.902763	0.451381	+	0.892331i	$0.350931\pi$
$788$	0	0
$789$	11672.3	0.526675
$790$	0	0
$791$	−12894.9	−0.579633
$792$	0	0
$793$	−20346.0	−0.911109
$794$	0	0
$795$	−5497.05	−0.245233
$796$	0	0
$797$	227.752	0.0101222	0.00506110	−	0.999987i	$-0.498389\pi$
$797$	227.752	0.0101222	0.00506110	+	0.999987i	$0.498389\pi$
$798$	0	0
$799$	−37911.4	−1.67861
$800$	0	0
$801$	−7811.92	−0.344595
$802$	0	0
$803$	−17426.6	−0.765843
$804$	0	0
$805$	2401.81	0.105158
$806$	0	0
$807$	4757.70	0.207533
$808$	0	0
$809$	17759.9	0.771824	0.385912	−	0.922536i	$-0.373887\pi$
$809$	17759.9	0.771824	0.385912	+	0.922536i	$0.373887\pi$
$810$	0	0
$811$	15077.3	0.652818	0.326409	−	0.945229i	$-0.394161\pi$
$811$	15077.3	0.652818	0.326409	+	0.945229i	$0.394161\pi$
$812$	0	0
$813$	−16242.1	−0.700659
$814$	0	0
$815$	−15798.9	−0.679033
$816$	0	0
$817$	3736.87	0.160020
$818$	0	0
$819$	−10073.6	−0.429794
$820$	0	0
$821$	11686.6	0.496792	0.248396	−	0.968659i	$-0.420096\pi$
$821$	11686.6	0.496792	0.248396	+	0.968659i	$0.420096\pi$
$822$	0	0
$823$	29233.5	1.23817	0.619087	−	0.785322i	$-0.287503\pi$
$823$	29233.5	1.23817	0.619087	+	0.785322i	$0.287503\pi$
$824$	0	0
$825$	3878.92	0.163693
$826$	0	0
$827$	3637.65	0.152955	0.0764773	−	0.997071i	$-0.475633\pi$
$827$	3637.65	0.152955	0.0764773	+	0.997071i	$0.475633\pi$
$828$	0	0
$829$	−41264.3	−1.72879	−0.864395	−	0.502813i	$-0.832298\pi$
$829$	−41264.3	−1.72879	−0.864395	+	0.502813i	$0.832298\pi$
$830$	0	0
$831$	19481.3	0.813234
$832$	0	0
$833$	−5441.08	−0.226317
$834$	0	0
$835$	21861.5	0.906047
$836$	0	0
$837$	−6542.33	−0.270175
$838$	0	0
$839$	−19245.1	−0.791913	−0.395956	−	0.918269i	$-0.629587\pi$
$839$	−19245.1	−0.791913	−0.395956	+	0.918269i	$0.629587\pi$
$840$	0	0
$841$	39701.7	1.62785
$842$	0	0
$843$	22034.8	0.900259
$844$	0	0
$845$	−14234.3	−0.579497
$846$	0	0
$847$	18486.3	0.749937
$848$	0	0
$849$	−13194.8	−0.533387
$850$	0	0
$851$	−4860.78	−0.195799
$852$	0	0
$853$	16254.5	0.652455	0.326227	−	0.945291i	$-0.394223\pi$
$853$	16254.5	0.652455	0.326227	+	0.945291i	$0.394223\pi$
$854$	0	0
$855$	1423.19	0.0569263
$856$	0	0
$857$	−2228.95	−0.0888441	−0.0444221	−	0.999013i	$-0.514145\pi$
$857$	−2228.95	−0.0888441	−0.0444221	+	0.999013i	$0.514145\pi$
$858$	0	0
$859$	−4415.65	−0.175390	−0.0876950	−	0.996147i	$-0.527950\pi$
$859$	−4415.65	−0.175390	−0.0876950	+	0.996147i	$0.527950\pi$
$860$	0	0
$861$	14942.9	0.591468
$862$	0	0
$863$	−50409.9	−1.98838	−0.994191	−	0.107628i	$-0.965674\pi$
$863$	−50409.9	−1.98838	−0.994191	+	0.107628i	$0.965674\pi$
$864$	0	0
$865$	7941.81	0.312173
$866$	0	0
$867$	7513.49	0.294315
$868$	0	0
$869$	6161.08	0.240507
$870$	0	0
$871$	−33051.9	−1.28579
$872$	0	0
$873$	15452.6	0.599073
$874$	0	0
$875$	−22037.0	−0.851415
$876$	0	0
$877$	8893.98	0.342450	0.171225	−	0.985232i	$-0.445228\pi$
$877$	8893.98	0.342450	0.171225	+	0.985232i	$0.445228\pi$
$878$	0	0
$879$	14437.3	0.553992
$880$	0	0
$881$	−40971.8	−1.56683	−0.783414	−	0.621500i	$-0.786524\pi$
$881$	−40971.8	−1.56683	−0.783414	+	0.621500i	$0.786524\pi$
$882$	0	0
$883$	−9426.35	−0.359255	−0.179627	−	0.983735i	$-0.557489\pi$
$883$	−9426.35	−0.359255	−0.179627	+	0.983735i	$0.557489\pi$
$884$	0	0
$885$	6729.66	0.255610
$886$	0	0
$887$	9819.37	0.371705	0.185852	−	0.982578i	$-0.440495\pi$
$887$	9819.37	0.371705	0.185852	+	0.982578i	$0.440495\pi$
$888$	0	0
$889$	33849.2	1.27701
$890$	0	0
$891$	−1217.38	−0.0457732
$892$	0	0
$893$	11150.5	0.417845
$894$	0	0
$895$	−3268.86	−0.122085
$896$	0	0
$897$	4616.91	0.171855
$898$	0	0
$899$	61343.2	2.27576
$900$	0	0
$901$	25279.5	0.934721
$902$	0	0
$903$	7403.20	0.272827
$904$	0	0
$905$	−23454.7	−0.861503
$906$	0	0
$907$	17605.9	0.644535	0.322268	−	0.946649i	$-0.395555\pi$
$907$	17605.9	0.644535	0.322268	+	0.946649i	$0.395555\pi$
$908$	0	0
$909$	2953.59	0.107772
$910$	0	0
$911$	−11290.4	−0.410613	−0.205306	−	0.978698i	$-0.565819\pi$
$911$	−11290.4	−0.410613	−0.205306	+	0.978698i	$0.565819\pi$
$912$	0	0
$913$	−3470.47	−0.125801
$914$	0	0
$915$	5694.66	0.205748
$916$	0	0
$917$	21876.7	0.787820
$918$	0	0
$919$	−17552.0	−0.630017	−0.315009	−	0.949089i	$-0.602007\pi$
$919$	−17552.0	−0.630017	−0.315009	+	0.949089i	$0.602007\pi$
$920$	0	0
$921$	−16443.5	−0.588307
$922$	0	0
$923$	46070.6	1.64294
$924$	0	0
$925$	18181.3	0.646267
$926$	0	0
$927$	−2729.32	−0.0967019
$928$	0	0
$929$	16159.9	0.570709	0.285354	−	0.958422i	$-0.407889\pi$
$929$	16159.9	0.570709	0.285354	+	0.958422i	$0.407889\pi$
$930$	0	0
$931$	1600.32	0.0563357
$932$	0	0
$933$	9673.80	0.339449
$934$	0	0
$935$	8080.53	0.282633
$936$	0	0
$937$	11990.1	0.418036	0.209018	−	0.977912i	$-0.432973\pi$
$937$	11990.1	0.418036	0.209018	+	0.977912i	$0.432973\pi$
$938$	0	0
$939$	3358.13	0.116708
$940$	0	0
$941$	−267.117	−0.00925374	−0.00462687	−	0.999989i	$-0.501473\pi$
$941$	−267.117	−0.00925374	−0.00462687	+	0.999989i	$0.501473\pi$
$942$	0	0
$943$	−6848.58	−0.236501
$944$	0	0
$945$	2819.51	0.0970569
$946$	0	0
$947$	27046.2	0.928072	0.464036	−	0.885816i	$-0.346401\pi$
$947$	27046.2	0.928072	0.464036	+	0.885816i	$0.346401\pi$
$948$	0	0
$949$	77584.0	2.65383
$950$	0	0
$951$	−19942.2	−0.679989
$952$	0	0
$953$	−26262.7	−0.892689	−0.446345	−	0.894861i	$-0.647274\pi$
$953$	−26262.7	−0.892689	−0.446345	+	0.894861i	$0.647274\pi$
$954$	0	0
$955$	−2449.77	−0.0830079
$956$	0	0
$957$	11414.6	0.385562
$958$	0	0
$959$	−30732.0	−1.03482
$960$	0	0
$961$	28922.5	0.970846
$962$	0	0
$963$	18820.8	0.629795
$964$	0	0
$965$	−4766.46	−0.159003
$966$	0	0
$967$	−24422.1	−0.812163	−0.406081	−	0.913837i	$-0.633105\pi$
$967$	−24422.1	−0.812163	−0.406081	+	0.913837i	$0.633105\pi$
$968$	0	0
$969$	−6544.88	−0.216978
$970$	0	0
$971$	−25253.2	−0.834617	−0.417308	−	0.908765i	$-0.637026\pi$
$971$	−25253.2	−0.834617	−0.417308	+	0.908765i	$0.637026\pi$
$972$	0	0
$973$	29004.8	0.955652
$974$	0	0
$975$	−17269.1	−0.567235
$976$	0	0
$977$	−22675.5	−0.742531	−0.371265	−	0.928527i	$-0.621076\pi$
$977$	−22675.5	−0.742531	−0.371265	+	0.928527i	$0.621076\pi$
$978$	0	0
$979$	13045.4	0.425877
$980$	0	0
$981$	−14804.7	−0.481833
$982$	0	0
$983$	−11015.2	−0.357407	−0.178703	−	0.983903i	$-0.557190\pi$
$983$	−11015.2	−0.357407	−0.178703	+	0.983903i	$0.557190\pi$
$984$	0	0
$985$	11404.8	0.368922
$986$	0	0
$987$	22090.4	0.712408
$988$	0	0
$989$	−3393.00	−0.109091
$990$	0	0
$991$	−43162.3	−1.38355	−0.691774	−	0.722114i	$-0.743170\pi$
$991$	−43162.3	−1.38355	−0.691774	+	0.722114i	$0.743170\pi$
$992$	0	0
$993$	−10413.7	−0.332797
$994$	0	0
$995$	−3861.19	−0.123023
$996$	0	0
$997$	−9251.64	−0.293884	−0.146942	−	0.989145i	$-0.546943\pi$
$997$	−9251.64	−0.293884	−0.146942	+	0.989145i	$0.546943\pi$
$998$	0	0
$999$	−5706.13	−0.180715

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.4.a.f.1.1	✓	2	1.1	even	1	trivial
1104.4.a.j.1.1		2	4.3	odd	2
1656.4.a.f.1.2		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)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -6.24264 q^{5} -16.7279 q^{7} +9.00000 q^{9} -15.0294 q^{11} +66.9117 q^{13} -18.7279 q^{15} +86.1249 q^{17} -25.3310 q^{19} -50.1838 q^{21} +23.0000 q^{23} -86.0294 q^{25} +27.0000 q^{27} -253.161 q^{29} -242.309 q^{31} -45.0883 q^{33} +104.426 q^{35} -211.338 q^{37} +200.735 q^{39} -297.765 q^{41} -147.522 q^{43} -56.1838 q^{45} -440.191 q^{47} -63.1766 q^{49} +258.375 q^{51} +293.522 q^{53} +93.8234 q^{55} -75.9929 q^{57} -359.338 q^{59} -304.073 q^{61} -150.551 q^{63} -417.706 q^{65} -493.963 q^{67} +69.0000 q^{69} +688.528 q^{71} +1159.50 q^{73} -258.088 q^{75} +251.411 q^{77} -409.934 q^{79} +81.0000 q^{81} +230.912 q^{83} -537.647 q^{85} -759.484 q^{87} -867.991 q^{89} -1119.29 q^{91} -726.926 q^{93} +158.132 q^{95} +1716.95 q^{97} -135.265 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} - 4 q^{5} - 8 q^{7} + 18 q^{9} - 64 q^{11} + 32 q^{13} - 12 q^{15} + 28 q^{17} - 144 q^{19} - 24 q^{21} + 46 q^{23} - 206 q^{25} + 54 q^{27} - 116 q^{29} - 264 q^{31} - 192 q^{33} + 124 q^{35}+ \cdots - 576 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 - 4 * q^5 - 8 * q^7 + 18 * q^9 - 64 * q^11 + 32 * q^13 - 12 * q^15 + 28 * q^17 - 144 * q^19 - 24 * q^21 + 46 * q^23 - 206 * q^25 + 54 * q^27 - 116 * q^29 - 264 * q^31 - 192 * q^33 + 124 * q^35 - 236 * q^37 + 96 * q^39 - 324 * q^41 - 32 * q^43 - 36 * q^45 - 524 * q^47 - 330 * q^49 + 84 * q^51 + 324 * q^53 - 16 * q^55 - 432 * q^57 - 532 * q^59 - 116 * q^61 - 72 * q^63 - 496 * q^65 - 80 * q^67 + 138 * q^69 - 320 * q^71 + 588 * q^73 - 618 * q^75 - 176 * q^77 - 1032 * q^79 + 162 * q^81 + 360 * q^83 - 668 * q^85 - 348 * q^87 + 292 * q^89 - 1424 * q^91 - 792 * q^93 - 108 * q^95 + 1652 * q^97 - 576 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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