LMFDB - Embedded newform 552.4.a.j.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 = [5,0,-15,0,16] 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:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 119x^{3} - 173x^{2} + 880x - 575 $ x^5 - 2x^4 - 119x^3 - 173x^2 + 880x - 575
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$12.3379$ 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)

$f(q)$	$=$	$q-3.00000 q^{3} -10.6116 q^{5} -23.9380 q^{7} +9.00000 q^{9} -56.6778 q^{11} -42.8818 q^{13} +31.8349 q^{15} -6.90498 q^{17} -6.51792 q^{19} +71.8140 q^{21} -23.0000 q^{23} -12.3931 q^{25} -27.0000 q^{27} +125.161 q^{29} -171.426 q^{31} +170.033 q^{33} +254.021 q^{35} +189.505 q^{37} +128.645 q^{39} -276.188 q^{41} -76.6612 q^{43} -95.5048 q^{45} -206.736 q^{47} +230.028 q^{49} +20.7149 q^{51} +382.960 q^{53} +601.444 q^{55} +19.5538 q^{57} +757.103 q^{59} -109.549 q^{61} -215.442 q^{63} +455.046 q^{65} +835.087 q^{67} +69.0000 q^{69} -158.799 q^{71} -619.113 q^{73} +37.1792 q^{75} +1356.75 q^{77} -996.598 q^{79} +81.0000 q^{81} -824.116 q^{83} +73.2732 q^{85} -375.484 q^{87} +1226.51 q^{89} +1026.50 q^{91} +514.279 q^{93} +69.1658 q^{95} +756.861 q^{97} -510.100 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 15 q^{3} + 16 q^{5} + 45 q^{9} + 6 q^{11} - 10 q^{13} - 48 q^{15} + 22 q^{17} + 42 q^{19} - 115 q^{23} + 291 q^{25} - 135 q^{27} + 274 q^{29} - 460 q^{31} - 18 q^{33} - 320 q^{35} + 420 q^{37} + 30 q^{39}+ \cdots + 54 q^{99}+O(q^{100}) $ 5 * q - 15 * q^3 + 16 * q^5 + 45 * q^9 + 6 * q^11 - 10 * q^13 - 48 * q^15 + 22 * q^17 + 42 * q^19 - 115 * q^23 + 291 * q^25 - 135 * q^27 + 274 * q^29 - 460 * q^31 - 18 * q^33 - 320 * q^35 + 420 * q^37 + 30 * q^39 + 362 * q^41 + 206 * q^43 + 144 * q^45 + 444 * q^47 + 1489 * q^49 - 66 * q^51 + 1240 * q^53 - 148 * q^55 - 126 * q^57 + 880 * q^59 + 584 * q^61 + 1840 * q^65 + 1554 * q^67 + 345 * q^69 + 472 * q^71 + 1350 * q^73 - 873 * q^75 + 4092 * q^77 - 224 * q^79 + 405 * q^81 + 1570 * q^83 + 2904 * q^85 - 822 * q^87 + 2074 * q^89 - 688 * q^91 + 1380 * q^93 + 2444 * q^95 + 14 * q^97 + 54 * 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
$5$	−10.6116	−0.949134	−0.474567	−	0.880219i	$-0.657395\pi$
$5$	−10.6116	−0.949134	−0.474567	+	0.880219i	$0.657395\pi$
$6$	0	0
$7$	−23.9380	−1.29253	−0.646265	−	0.763113i	$-0.723670\pi$
$7$	−23.9380	−1.29253	−0.646265	+	0.763113i	$0.723670\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−56.6778	−1.55355	−0.776773	−	0.629781i	$-0.783145\pi$
$11$	−56.6778	−1.55355	−0.776773	+	0.629781i	$0.783145\pi$
$12$	0	0
$13$	−42.8818	−0.914867	−0.457433	−	0.889244i	$-0.651231\pi$
$13$	−42.8818	−0.914867	−0.457433	+	0.889244i	$0.651231\pi$
$14$	0	0
$15$	31.8349	0.547983
$16$	0	0
$17$	−6.90498	−0.0985120	−0.0492560	−	0.998786i	$-0.515685\pi$
$17$	−6.90498	−0.0985120	−0.0492560	+	0.998786i	$0.515685\pi$
$18$	0	0
$19$	−6.51792	−0.0787007	−0.0393504	−	0.999225i	$-0.512529\pi$
$19$	−6.51792	−0.0787007	−0.0393504	+	0.999225i	$0.512529\pi$
$20$	0	0
$21$	71.8140	0.746243
$22$	0	0
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	−12.3931	−0.0991445
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	125.161	0.801444	0.400722	−	0.916200i	$-0.368759\pi$
$29$	125.161	0.801444	0.400722	+	0.916200i	$0.368759\pi$
$30$	0	0
$31$	−171.426	−0.993197	−0.496598	−	0.867980i	$-0.665418\pi$
$31$	−171.426	−0.993197	−0.496598	+	0.867980i	$0.665418\pi$
$32$	0	0
$33$	170.033	0.896940
$34$	0	0
$35$	254.021	1.22678
$36$	0	0
$37$	189.505	0.842010	0.421005	−	0.907058i	$-0.361677\pi$
$37$	189.505	0.842010	0.421005	+	0.907058i	$0.361677\pi$
$38$	0	0
$39$	128.645	0.528199
$40$	0	0
$41$	−276.188	−1.05203	−0.526015	−	0.850475i	$-0.676315\pi$
$41$	−276.188	−1.05203	−0.526015	+	0.850475i	$0.676315\pi$
$42$	0	0
$43$	−76.6612	−0.271877	−0.135939	−	0.990717i	$-0.543405\pi$
$43$	−76.6612	−0.271877	−0.135939	+	0.990717i	$0.543405\pi$
$44$	0	0
$45$	−95.5048	−0.316378
$46$	0	0
$47$	−206.736	−0.641608	−0.320804	−	0.947146i	$-0.603953\pi$
$47$	−206.736	−0.641608	−0.320804	+	0.947146i	$0.603953\pi$
$48$	0	0
$49$	230.028	0.670635
$50$	0	0
$51$	20.7149	0.0568759
$52$	0	0
$53$	382.960	0.992520	0.496260	−	0.868174i	$-0.334706\pi$
$53$	382.960	0.992520	0.496260	+	0.868174i	$0.334706\pi$
$54$	0	0
$55$	601.444	1.47452
$56$	0	0
$57$	19.5538	0.0454379
$58$	0	0
$59$	757.103	1.67062	0.835309	−	0.549781i	$-0.185289\pi$
$59$	757.103	1.67062	0.835309	+	0.549781i	$0.185289\pi$
$60$	0	0
$61$	−109.549	−0.229939	−0.114970	−	0.993369i	$-0.536677\pi$
$61$	−109.549	−0.229939	−0.114970	+	0.993369i	$0.536677\pi$
$62$	0	0
$63$	−215.442	−0.430843
$64$	0	0
$65$	455.046	0.868331
$66$	0	0
$67$	835.087	1.52272	0.761359	−	0.648331i	$-0.224533\pi$
$67$	835.087	1.52272	0.761359	+	0.648331i	$0.224533\pi$
$68$	0	0
$69$	69.0000	0.120386
$70$	0	0
$71$	−158.799	−0.265437	−0.132719	−	0.991154i	$-0.542371\pi$
$71$	−158.799	−0.265437	−0.132719	+	0.991154i	$0.542371\pi$
$72$	0	0
$73$	−619.113	−0.992626	−0.496313	−	0.868144i	$-0.665313\pi$
$73$	−619.113	−0.992626	−0.496313	+	0.868144i	$0.665313\pi$
$74$	0	0
$75$	37.1792	0.0572411
$76$	0	0
$77$	1356.75	2.00800
$78$	0	0
$79$	−996.598	−1.41932	−0.709658	−	0.704546i	$-0.751151\pi$
$79$	−996.598	−1.41932	−0.709658	+	0.704546i	$0.751151\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−824.116	−1.08986	−0.544931	−	0.838481i	$-0.683444\pi$
$83$	−824.116	−1.08986	−0.544931	+	0.838481i	$0.683444\pi$
$84$	0	0
$85$	73.2732	0.0935011
$86$	0	0
$87$	−375.484	−0.462714
$88$	0	0
$89$	1226.51	1.46078	0.730390	−	0.683030i	$-0.239338\pi$
$89$	1226.51	1.46078	0.730390	+	0.683030i	$0.239338\pi$
$90$	0	0
$91$	1026.50	1.18249
$92$	0	0
$93$	514.279	0.573422
$94$	0	0
$95$	69.1658	0.0746975
$96$	0	0
$97$	756.861	0.792243	0.396121	−	0.918198i	$-0.370356\pi$
$97$	756.861	0.792243	0.396121	+	0.918198i	$0.370356\pi$
$98$	0	0
$99$	−510.100	−0.517848
$100$	0	0
$101$	190.542	0.187719	0.0938597	−	0.995585i	$-0.470079\pi$
$101$	190.542	0.187719	0.0938597	+	0.995585i	$0.470079\pi$
$102$	0	0
$103$	−738.006	−0.705999	−0.353000	−	0.935623i	$-0.614838\pi$
$103$	−738.006	−0.705999	−0.353000	+	0.935623i	$0.614838\pi$
$104$	0	0
$105$	−762.064	−0.708285
$106$	0	0
$107$	−443.423	−0.400629	−0.200315	−	0.979732i	$-0.564196\pi$
$107$	−443.423	−0.400629	−0.200315	+	0.979732i	$0.564196\pi$
$108$	0	0
$109$	−592.014	−0.520227	−0.260113	−	0.965578i	$-0.583760\pi$
$109$	−592.014	−0.520227	−0.260113	+	0.965578i	$0.583760\pi$
$110$	0	0
$111$	−568.514	−0.486135
$112$	0	0
$113$	2043.52	1.70122	0.850612	−	0.525793i	$-0.176232\pi$
$113$	2043.52	1.70122	0.850612	+	0.525793i	$0.176232\pi$
$114$	0	0
$115$	244.068	0.197908
$116$	0	0
$117$	−385.936	−0.304956
$118$	0	0
$119$	165.291	0.127330
$120$	0	0
$121$	1881.37	1.41350
$122$	0	0
$123$	828.563	0.607390
$124$	0	0
$125$	1457.97	1.04324
$126$	0	0
$127$	−423.322	−0.295777	−0.147889	−	0.989004i	$-0.547248\pi$
$127$	−423.322	−0.295777	−0.147889	+	0.989004i	$0.547248\pi$
$128$	0	0
$129$	229.984	0.156968
$130$	0	0
$131$	−2255.29	−1.50416	−0.752081	−	0.659070i	$-0.770950\pi$
$131$	−2255.29	−1.50416	−0.752081	+	0.659070i	$0.770950\pi$
$132$	0	0
$133$	156.026	0.101723
$134$	0	0
$135$	286.514	0.182661
$136$	0	0
$137$	2477.27	1.54487	0.772435	−	0.635094i	$-0.219039\pi$
$137$	2477.27	1.54487	0.772435	+	0.635094i	$0.219039\pi$
$138$	0	0
$139$	1024.14	0.624940	0.312470	−	0.949928i	$-0.398844\pi$
$139$	1024.14	0.624940	0.312470	+	0.949928i	$0.398844\pi$
$140$	0	0
$141$	620.208	0.370432
$142$	0	0
$143$	2430.44	1.42129
$144$	0	0
$145$	−1328.17	−0.760678
$146$	0	0
$147$	−690.083	−0.387191
$148$	0	0
$149$	−365.212	−0.200801	−0.100401	−	0.994947i	$-0.532012\pi$
$149$	−365.212	−0.200801	−0.100401	+	0.994947i	$0.532012\pi$
$150$	0	0
$151$	−3117.91	−1.68035	−0.840173	−	0.542319i	$-0.817547\pi$
$151$	−3117.91	−1.68035	−0.840173	+	0.542319i	$0.817547\pi$
$152$	0	0
$153$	−62.1448	−0.0328373
$154$	0	0
$155$	1819.12	0.942677
$156$	0	0
$157$	−2671.09	−1.35781	−0.678904	−	0.734227i	$-0.737545\pi$
$157$	−2671.09	−1.35781	−0.678904	+	0.734227i	$0.737545\pi$
$158$	0	0
$159$	−1148.88	−0.573031
$160$	0	0
$161$	550.574	0.269511
$162$	0	0
$163$	2046.87	0.983576	0.491788	−	0.870715i	$-0.336344\pi$
$163$	2046.87	0.983576	0.491788	+	0.870715i	$0.336344\pi$
$164$	0	0
$165$	−1804.33	−0.851316
$166$	0	0
$167$	−935.107	−0.433298	−0.216649	−	0.976250i	$-0.569513\pi$
$167$	−935.107	−0.433298	−0.216649	+	0.976250i	$0.569513\pi$
$168$	0	0
$169$	−358.152	−0.163019
$170$	0	0
$171$	−58.6613	−0.0262336
$172$	0	0
$173$	−1487.01	−0.653500	−0.326750	−	0.945111i	$-0.605954\pi$
$173$	−1487.01	−0.653500	−0.326750	+	0.945111i	$0.605954\pi$
$174$	0	0
$175$	296.665	0.128147
$176$	0	0
$177$	−2271.31	−0.964532
$178$	0	0
$179$	−2342.99	−0.978341	−0.489170	−	0.872188i	$-0.662700\pi$
$179$	−2342.99	−0.978341	−0.489170	+	0.872188i	$0.662700\pi$
$180$	0	0
$181$	595.008	0.244346	0.122173	−	0.992509i	$-0.461014\pi$
$181$	595.008	0.244346	0.122173	+	0.992509i	$0.461014\pi$
$182$	0	0
$183$	328.646	0.132755
$184$	0	0
$185$	−2010.96	−0.799181
$186$	0	0
$187$	391.359	0.153043
$188$	0	0
$189$	646.326	0.248748
$190$	0	0
$191$	−1358.46	−0.514631	−0.257316	−	0.966327i	$-0.582838\pi$
$191$	−1358.46	−0.514631	−0.257316	+	0.966327i	$0.582838\pi$
$192$	0	0
$193$	−2821.91	−1.05247	−0.526233	−	0.850341i	$-0.676396\pi$
$193$	−2821.91	−1.05247	−0.526233	+	0.850341i	$0.676396\pi$
$194$	0	0
$195$	−1365.14	−0.501331
$196$	0	0
$197$	2180.29	0.788524	0.394262	−	0.918998i	$-0.371000\pi$
$197$	2180.29	0.788524	0.394262	+	0.918998i	$0.371000\pi$
$198$	0	0
$199$	−894.363	−0.318591	−0.159296	−	0.987231i	$-0.550922\pi$
$199$	−894.363	−0.318591	−0.159296	+	0.987231i	$0.550922\pi$
$200$	0	0
$201$	−2505.26	−0.879142
$202$	0	0
$203$	−2996.11	−1.03589
$204$	0	0
$205$	2930.80	0.998518
$206$	0	0
$207$	−207.000	−0.0695048
$208$	0	0
$209$	369.421	0.122265
$210$	0	0
$211$	−4286.93	−1.39869	−0.699347	−	0.714783i	$-0.746526\pi$
$211$	−4286.93	−1.39869	−0.699347	+	0.714783i	$0.746526\pi$
$212$	0	0
$213$	476.398	0.153250
$214$	0	0
$215$	813.501	0.258048
$216$	0	0
$217$	4103.61	1.28374
$218$	0	0
$219$	1857.34	0.573093
$220$	0	0
$221$	296.098	0.0901253
$222$	0	0
$223$	−1236.58	−0.371334	−0.185667	−	0.982613i	$-0.559445\pi$
$223$	−1236.58	−0.371334	−0.185667	+	0.982613i	$0.559445\pi$
$224$	0	0
$225$	−111.538	−0.0330482
$226$	0	0
$227$	2123.19	0.620799	0.310399	−	0.950606i	$-0.399537\pi$
$227$	2123.19	0.620799	0.310399	+	0.950606i	$0.399537\pi$
$228$	0	0
$229$	−2009.27	−0.579810	−0.289905	−	0.957055i	$-0.593624\pi$
$229$	−2009.27	−0.579810	−0.289905	+	0.957055i	$0.593624\pi$
$230$	0	0
$231$	−4070.26	−1.15932
$232$	0	0
$233$	−2384.67	−0.670494	−0.335247	−	0.942130i	$-0.608820\pi$
$233$	−2384.67	−0.670494	−0.335247	+	0.942130i	$0.608820\pi$
$234$	0	0
$235$	2193.81	0.608972
$236$	0	0
$237$	2989.79	0.819443
$238$	0	0
$239$	4057.12	1.09805	0.549023	−	0.835807i	$-0.315000\pi$
$239$	4057.12	1.09805	0.549023	+	0.835807i	$0.315000\pi$
$240$	0	0
$241$	611.129	0.163345	0.0816727	−	0.996659i	$-0.473974\pi$
$241$	611.129	0.163345	0.0816727	+	0.996659i	$0.473974\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−2440.97	−0.636523
$246$	0	0
$247$	279.500	0.0720007
$248$	0	0
$249$	2472.35	0.629232
$250$	0	0
$251$	5936.57	1.49288	0.746441	−	0.665452i	$-0.231761\pi$
$251$	5936.57	1.49288	0.746441	+	0.665452i	$0.231761\pi$
$252$	0	0
$253$	1303.59	0.323937
$254$	0	0
$255$	−219.820	−0.0539829
$256$	0	0
$257$	−1969.70	−0.478080	−0.239040	−	0.971010i	$-0.576833\pi$
$257$	−1969.70	−0.478080	−0.239040	+	0.971010i	$0.576833\pi$
$258$	0	0
$259$	−4536.36	−1.08832
$260$	0	0
$261$	1126.45	0.267148
$262$	0	0
$263$	−4243.92	−0.995024	−0.497512	−	0.867457i	$-0.665753\pi$
$263$	−4243.92	−0.995024	−0.497512	+	0.867457i	$0.665753\pi$
$264$	0	0
$265$	−4063.83	−0.942034
$266$	0	0
$267$	−3679.52	−0.843382
$268$	0	0
$269$	−818.053	−0.185418	−0.0927092	−	0.995693i	$-0.529553\pi$
$269$	−818.053	−0.185418	−0.0927092	+	0.995693i	$0.529553\pi$
$270$	0	0
$271$	−8551.04	−1.91675	−0.958374	−	0.285514i	$-0.907836\pi$
$271$	−8551.04	−1.91675	−0.958374	+	0.285514i	$0.907836\pi$
$272$	0	0
$273$	−3079.51	−0.682713
$274$	0	0
$275$	702.411	0.154025
$276$	0	0
$277$	−1935.68	−0.419868	−0.209934	−	0.977716i	$-0.567325\pi$
$277$	−1935.68	−0.419868	−0.209934	+	0.977716i	$0.567325\pi$
$278$	0	0
$279$	−1542.84	−0.331066
$280$	0	0
$281$	−226.498	−0.0480844	−0.0240422	−	0.999711i	$-0.507654\pi$
$281$	−226.498	−0.0480844	−0.0240422	+	0.999711i	$0.507654\pi$
$282$	0	0
$283$	−664.631	−0.139605	−0.0698025	−	0.997561i	$-0.522237\pi$
$283$	−664.631	−0.139605	−0.0698025	+	0.997561i	$0.522237\pi$
$284$	0	0
$285$	−207.497	−0.0431266
$286$	0	0
$287$	6611.38	1.35978
$288$	0	0
$289$	−4865.32	−0.990295
$290$	0	0
$291$	−2270.58	−0.457402
$292$	0	0
$293$	6083.42	1.21296	0.606480	−	0.795099i	$-0.292581\pi$
$293$	6083.42	1.21296	0.606480	+	0.795099i	$0.292581\pi$
$294$	0	0
$295$	−8034.11	−1.58564
$296$	0	0
$297$	1530.30	0.298980
$298$	0	0
$299$	986.281	0.190763
$300$	0	0
$301$	1835.12	0.351410
$302$	0	0
$303$	−571.627	−0.108380
$304$	0	0
$305$	1162.49	0.218243
$306$	0	0
$307$	8366.57	1.55539	0.777696	−	0.628640i	$-0.216388\pi$
$307$	8366.57	1.55539	0.777696	+	0.628640i	$0.216388\pi$
$308$	0	0
$309$	2214.02	0.407609
$310$	0	0
$311$	−419.976	−0.0765745	−0.0382872	−	0.999267i	$-0.512190\pi$
$311$	−419.976	−0.0765745	−0.0382872	+	0.999267i	$0.512190\pi$
$312$	0	0
$313$	−10053.3	−1.81547	−0.907737	−	0.419539i	$-0.862192\pi$
$313$	−10053.3	−1.81547	−0.907737	+	0.419539i	$0.862192\pi$
$314$	0	0
$315$	2286.19	0.408928
$316$	0	0
$317$	796.576	0.141136	0.0705681	−	0.997507i	$-0.477519\pi$
$317$	796.576	0.141136	0.0705681	+	0.997507i	$0.477519\pi$
$318$	0	0
$319$	−7093.87	−1.24508
$320$	0	0
$321$	1330.27	0.231304
$322$	0	0
$323$	45.0061	0.00775296
$324$	0	0
$325$	531.437	0.0907040
$326$	0	0
$327$	1776.04	0.300353
$328$	0	0
$329$	4948.85	0.829297
$330$	0	0
$331$	327.269	0.0543455	0.0271727	−	0.999631i	$-0.491350\pi$
$331$	327.269	0.0543455	0.0271727	+	0.999631i	$0.491350\pi$
$332$	0	0
$333$	1705.54	0.280670
$334$	0	0
$335$	−8861.64	−1.44526
$336$	0	0
$337$	−458.322	−0.0740842	−0.0370421	−	0.999314i	$-0.511794\pi$
$337$	−458.322	−0.0740842	−0.0370421	+	0.999314i	$0.511794\pi$
$338$	0	0
$339$	−6130.57	−0.982202
$340$	0	0
$341$	9716.07	1.54298
$342$	0	0
$343$	2704.33	0.425714
$344$	0	0
$345$	−732.203	−0.114262
$346$	0	0
$347$	−1135.87	−0.175725	−0.0878624	−	0.996133i	$-0.528004\pi$
$347$	−1135.87	−0.175725	−0.0878624	+	0.996133i	$0.528004\pi$
$348$	0	0
$349$	1679.79	0.257643	0.128821	−	0.991668i	$-0.458881\pi$
$349$	1679.79	0.257643	0.128821	+	0.991668i	$0.458881\pi$
$350$	0	0
$351$	1157.81	0.176066
$352$	0	0
$353$	9318.23	1.40498	0.702492	−	0.711691i	$-0.252071\pi$
$353$	9318.23	1.40498	0.702492	+	0.711691i	$0.252071\pi$
$354$	0	0
$355$	1685.12	0.251935
$356$	0	0
$357$	−495.874	−0.0735139
$358$	0	0
$359$	8757.29	1.28744	0.643721	−	0.765260i	$-0.277390\pi$
$359$	8757.29	1.28744	0.643721	+	0.765260i	$0.277390\pi$
$360$	0	0
$361$	−6816.52	−0.993806
$362$	0	0
$363$	−5644.12	−0.816086
$364$	0	0
$365$	6569.80	0.942135
$366$	0	0
$367$	−1384.02	−0.196854	−0.0984271	−	0.995144i	$-0.531381\pi$
$367$	−1384.02	−0.196854	−0.0984271	+	0.995144i	$0.531381\pi$
$368$	0	0
$369$	−2485.69	−0.350677
$370$	0	0
$371$	−9167.29	−1.28286
$372$	0	0
$373$	−5627.27	−0.781150	−0.390575	−	0.920571i	$-0.627724\pi$
$373$	−5627.27	−0.781150	−0.390575	+	0.920571i	$0.627724\pi$
$374$	0	0
$375$	−4373.90	−0.602312
$376$	0	0
$377$	−5367.14	−0.733215
$378$	0	0
$379$	13693.5	1.85590	0.927951	−	0.372702i	$-0.121569\pi$
$379$	13693.5	1.85590	0.927951	+	0.372702i	$0.121569\pi$
$380$	0	0
$381$	1269.97	0.170767
$382$	0	0
$383$	14194.7	1.89378	0.946888	−	0.321565i	$-0.104209\pi$
$383$	14194.7	1.89378	0.946888	+	0.321565i	$0.104209\pi$
$384$	0	0
$385$	−14397.4	−1.90587
$386$	0	0
$387$	−689.951	−0.0906257
$388$	0	0
$389$	6997.48	0.912047	0.456024	−	0.889968i	$-0.349273\pi$
$389$	6997.48	0.912047	0.456024	+	0.889968i	$0.349273\pi$
$390$	0	0
$391$	158.815	0.0205412
$392$	0	0
$393$	6765.86	0.868429
$394$	0	0
$395$	10575.5	1.34712
$396$	0	0
$397$	4376.03	0.553216	0.276608	−	0.960983i	$-0.410790\pi$
$397$	4376.03	0.553216	0.276608	+	0.960983i	$0.410790\pi$
$398$	0	0
$399$	−468.078	−0.0587298
$400$	0	0
$401$	12336.5	1.53630	0.768148	−	0.640273i	$-0.221179\pi$
$401$	12336.5	1.53630	0.768148	+	0.640273i	$0.221179\pi$
$402$	0	0
$403$	7351.07	0.908643
$404$	0	0
$405$	−859.543	−0.105459
$406$	0	0
$407$	−10740.7	−1.30810
$408$	0	0
$409$	−55.5800	−0.00671945	−0.00335973	−	0.999994i	$-0.501069\pi$
$409$	−55.5800	−0.00671945	−0.00335973	+	0.999994i	$0.501069\pi$
$410$	0	0
$411$	−7431.80	−0.891931
$412$	0	0
$413$	−18123.5	−2.15932
$414$	0	0
$415$	8745.22	1.03442
$416$	0	0
$417$	−3072.43	−0.360809
$418$	0	0
$419$	−4196.16	−0.489250	−0.244625	−	0.969618i	$-0.578665\pi$
$419$	−4196.16	−0.489250	−0.244625	+	0.969618i	$0.578665\pi$
$420$	0	0
$421$	6162.19	0.713365	0.356682	−	0.934226i	$-0.383908\pi$
$421$	6162.19	0.713365	0.356682	+	0.934226i	$0.383908\pi$
$422$	0	0
$423$	−1860.62	−0.213869
$424$	0	0
$425$	85.5738	0.00976692
$426$	0	0
$427$	2622.38	0.297203
$428$	0	0
$429$	−7291.33	−0.820580
$430$	0	0
$431$	−10406.2	−1.16299	−0.581493	−	0.813551i	$-0.697531\pi$
$431$	−10406.2	−1.16299	−0.581493	+	0.813551i	$0.697531\pi$
$432$	0	0
$433$	−4625.62	−0.513380	−0.256690	−	0.966494i	$-0.582632\pi$
$433$	−4625.62	−0.513380	−0.256690	+	0.966494i	$0.582632\pi$
$434$	0	0
$435$	3984.50	0.439178
$436$	0	0
$437$	149.912	0.0164102
$438$	0	0
$439$	11547.8	1.25546	0.627728	−	0.778433i	$-0.283985\pi$
$439$	11547.8	1.25546	0.627728	+	0.778433i	$0.283985\pi$
$440$	0	0
$441$	2070.25	0.223545
$442$	0	0
$443$	4495.30	0.482118	0.241059	−	0.970510i	$-0.422505\pi$
$443$	4495.30	0.482118	0.241059	+	0.970510i	$0.422505\pi$
$444$	0	0
$445$	−13015.3	−1.38648
$446$	0	0
$447$	1095.64	0.115933
$448$	0	0
$449$	−11909.9	−1.25181	−0.625903	−	0.779901i	$-0.715269\pi$
$449$	−11909.9	−1.25181	−0.625903	+	0.779901i	$0.715269\pi$
$450$	0	0
$451$	15653.7	1.63438
$452$	0	0
$453$	9353.74	0.970148
$454$	0	0
$455$	−10892.9	−1.12234
$456$	0	0
$457$	17167.0	1.75719	0.878596	−	0.477566i	$-0.158481\pi$
$457$	17167.0	1.75719	0.878596	+	0.477566i	$0.158481\pi$
$458$	0	0
$459$	186.434	0.0189586
$460$	0	0
$461$	−11601.2	−1.17206	−0.586032	−	0.810288i	$-0.699311\pi$
$461$	−11601.2	−1.17206	−0.586032	+	0.810288i	$0.699311\pi$
$462$	0	0
$463$	14155.3	1.42085	0.710424	−	0.703773i	$-0.248503\pi$
$463$	14155.3	1.42085	0.710424	+	0.703773i	$0.248503\pi$
$464$	0	0
$465$	−5457.35	−0.544255
$466$	0	0
$467$	−266.993	−0.0264560	−0.0132280	−	0.999913i	$-0.504211\pi$
$467$	−266.993	−0.0264560	−0.0132280	+	0.999913i	$0.504211\pi$
$468$	0	0
$469$	−19990.3	−1.96816
$470$	0	0
$471$	8013.26	0.783931
$472$	0	0
$473$	4344.99	0.422374
$474$	0	0
$475$	80.7770	0.00780274
$476$	0	0
$477$	3446.64	0.330840
$478$	0	0
$479$	1512.77	0.144301	0.0721504	−	0.997394i	$-0.477014\pi$
$479$	1512.77	0.144301	0.0721504	+	0.997394i	$0.477014\pi$
$480$	0	0
$481$	−8126.30	−0.770327
$482$	0	0
$483$	−1651.72	−0.155602
$484$	0	0
$485$	−8031.53	−0.751945
$486$	0	0
$487$	19045.5	1.77215	0.886073	−	0.463545i	$-0.153423\pi$
$487$	19045.5	1.77215	0.886073	+	0.463545i	$0.153423\pi$
$488$	0	0
$489$	−6140.60	−0.567868
$490$	0	0
$491$	17593.5	1.61708	0.808539	−	0.588442i	$-0.200258\pi$
$491$	17593.5	1.61708	0.808539	+	0.588442i	$0.200258\pi$
$492$	0	0
$493$	−864.237	−0.0789519
$494$	0	0
$495$	5413.00	0.491508
$496$	0	0
$497$	3801.34	0.343085
$498$	0	0
$499$	−487.931	−0.0437731	−0.0218866	−	0.999760i	$-0.506967\pi$
$499$	−487.931	−0.0437731	−0.0218866	+	0.999760i	$0.506967\pi$
$500$	0	0
$501$	2805.32	0.250165
$502$	0	0
$503$	−5519.58	−0.489276	−0.244638	−	0.969614i	$-0.578669\pi$
$503$	−5519.58	−0.489276	−0.244638	+	0.969614i	$0.578669\pi$
$504$	0	0
$505$	−2021.97	−0.178171
$506$	0	0
$507$	1074.46	0.0941190
$508$	0	0
$509$	−15942.0	−1.38824	−0.694122	−	0.719858i	$-0.744207\pi$
$509$	−15942.0	−1.38824	−0.694122	+	0.719858i	$0.744207\pi$
$510$	0	0
$511$	14820.3	1.28300
$512$	0	0
$513$	175.984	0.0151460
$514$	0	0
$515$	7831.46	0.670088
$516$	0	0
$517$	11717.3	0.996766
$518$	0	0
$519$	4461.04	0.377299
$520$	0	0
$521$	10119.4	0.850943	0.425471	−	0.904972i	$-0.360108\pi$
$521$	10119.4	0.850943	0.425471	+	0.904972i	$0.360108\pi$
$522$	0	0
$523$	19967.4	1.66943	0.834715	−	0.550683i	$-0.185632\pi$
$523$	19967.4	1.66943	0.834715	+	0.550683i	$0.185632\pi$
$524$	0	0
$525$	−889.995	−0.0739859
$526$	0	0
$527$	1183.70	0.0978418
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	6813.93	0.556873
$532$	0	0
$533$	11843.4	0.962468
$534$	0	0
$535$	4705.45	0.380251
$536$	0	0
$537$	7028.96	0.564845
$538$	0	0
$539$	−13037.5	−1.04186
$540$	0	0
$541$	−9132.13	−0.725732	−0.362866	−	0.931841i	$-0.618202\pi$
$541$	−9132.13	−0.725732	−0.362866	+	0.931841i	$0.618202\pi$
$542$	0	0
$543$	−1785.02	−0.141073
$544$	0	0
$545$	6282.25	0.493765
$546$	0	0
$547$	16329.9	1.27645	0.638224	−	0.769851i	$-0.279670\pi$
$547$	16329.9	1.27645	0.638224	+	0.769851i	$0.279670\pi$
$548$	0	0
$549$	−985.939	−0.0766463
$550$	0	0
$551$	−815.792	−0.0630742
$552$	0	0
$553$	23856.6	1.83451
$554$	0	0
$555$	6032.87	0.461407
$556$	0	0
$557$	−17494.1	−1.33079	−0.665393	−	0.746494i	$-0.731736\pi$
$557$	−17494.1	−1.33079	−0.665393	+	0.746494i	$0.731736\pi$
$558$	0	0
$559$	3287.37	0.248731
$560$	0	0
$561$	−1174.08	−0.0883593
$562$	0	0
$563$	15509.2	1.16099	0.580494	−	0.814265i	$-0.302860\pi$
$563$	15509.2	1.16099	0.580494	+	0.814265i	$0.302860\pi$
$564$	0	0
$565$	−21685.1	−1.61469
$566$	0	0
$567$	−1938.98	−0.143614
$568$	0	0
$569$	13154.8	0.969203	0.484601	−	0.874735i	$-0.338965\pi$
$569$	13154.8	0.969203	0.484601	+	0.874735i	$0.338965\pi$
$570$	0	0
$571$	−5633.40	−0.412873	−0.206436	−	0.978460i	$-0.566187\pi$
$571$	−5633.40	−0.412873	−0.206436	+	0.978460i	$0.566187\pi$
$572$	0	0
$573$	4075.37	0.297122
$574$	0	0
$575$	285.040	0.0206731
$576$	0	0
$577$	24998.5	1.80364	0.901821	−	0.432109i	$-0.142231\pi$
$577$	24998.5	1.80364	0.901821	+	0.432109i	$0.142231\pi$
$578$	0	0
$579$	8465.74	0.607641
$580$	0	0
$581$	19727.7	1.40868
$582$	0	0
$583$	−21705.3	−1.54192
$584$	0	0
$585$	4095.42	0.289444
$586$	0	0
$587$	15578.6	1.09540	0.547698	−	0.836676i	$-0.315504\pi$
$587$	15578.6	1.09540	0.547698	+	0.836676i	$0.315504\pi$
$588$	0	0
$589$	1117.34	0.0781653
$590$	0	0
$591$	−6540.87	−0.455255
$592$	0	0
$593$	9044.52	0.626331	0.313165	−	0.949699i	$-0.398611\pi$
$593$	9044.52	0.626331	0.313165	+	0.949699i	$0.398611\pi$
$594$	0	0
$595$	−1754.01	−0.120853
$596$	0	0
$597$	2683.09	0.183939
$598$	0	0
$599$	−5390.40	−0.367689	−0.183845	−	0.982955i	$-0.558854\pi$
$599$	−5390.40	−0.367689	−0.183845	+	0.982955i	$0.558854\pi$
$600$	0	0
$601$	1663.90	0.112932	0.0564658	−	0.998405i	$-0.482017\pi$
$601$	1663.90	0.112932	0.0564658	+	0.998405i	$0.482017\pi$
$602$	0	0
$603$	7515.78	0.507573
$604$	0	0
$605$	−19964.4	−1.34160
$606$	0	0
$607$	−23418.1	−1.56592	−0.782959	−	0.622074i	$-0.786290\pi$
$607$	−23418.1	−1.56592	−0.782959	+	0.622074i	$0.786290\pi$
$608$	0	0
$609$	8988.34	0.598072
$610$	0	0
$611$	8865.21	0.586985
$612$	0	0
$613$	10609.6	0.699049	0.349525	−	0.936927i	$-0.386343\pi$
$613$	10609.6	0.699049	0.349525	+	0.936927i	$0.386343\pi$
$614$	0	0
$615$	−8792.41	−0.576495
$616$	0	0
$617$	−8462.32	−0.552156	−0.276078	−	0.961135i	$-0.589035\pi$
$617$	−8462.32	−0.552156	−0.276078	+	0.961135i	$0.589035\pi$
$618$	0	0
$619$	−29421.9	−1.91045	−0.955224	−	0.295883i	$-0.904386\pi$
$619$	−29421.9	−1.91045	−0.955224	+	0.295883i	$0.904386\pi$
$620$	0	0
$621$	621.000	0.0401286
$622$	0	0
$623$	−29360.1	−1.88810
$624$	0	0
$625$	−13922.3	−0.891026
$626$	0	0
$627$	−1108.26	−0.0705898
$628$	0	0
$629$	−1308.53	−0.0829481
$630$	0	0
$631$	−568.767	−0.0358832	−0.0179416	−	0.999839i	$-0.505711\pi$
$631$	−568.767	−0.0358832	−0.0179416	+	0.999839i	$0.505711\pi$
$632$	0	0
$633$	12860.8	0.807536
$634$	0	0
$635$	4492.14	0.280732
$636$	0	0
$637$	−9864.00	−0.613542
$638$	0	0
$639$	−1429.20	−0.0884790
$640$	0	0
$641$	−3575.12	−0.220294	−0.110147	−	0.993915i	$-0.535132\pi$
$641$	−3575.12	−0.220294	−0.110147	+	0.993915i	$0.535132\pi$
$642$	0	0
$643$	18881.2	1.15801	0.579006	−	0.815323i	$-0.303441\pi$
$643$	18881.2	1.15801	0.579006	+	0.815323i	$0.303441\pi$
$644$	0	0
$645$	−2440.50	−0.148984
$646$	0	0
$647$	−24841.6	−1.50947	−0.754734	−	0.656031i	$-0.772234\pi$
$647$	−24841.6	−1.50947	−0.754734	+	0.656031i	$0.772234\pi$
$648$	0	0
$649$	−42910.9	−2.59538
$650$	0	0
$651$	−12310.8	−0.741166
$652$	0	0
$653$	22151.2	1.32748	0.663739	−	0.747964i	$-0.268969\pi$
$653$	22151.2	1.32748	0.663739	+	0.747964i	$0.268969\pi$
$654$	0	0
$655$	23932.3	1.42765
$656$	0	0
$657$	−5572.02	−0.330875
$658$	0	0
$659$	−24528.7	−1.44993	−0.724964	−	0.688786i	$-0.758144\pi$
$659$	−24528.7	−1.44993	−0.724964	+	0.688786i	$0.758144\pi$
$660$	0	0
$661$	−4798.52	−0.282361	−0.141181	−	0.989984i	$-0.545090\pi$
$661$	−4798.52	−0.282361	−0.141181	+	0.989984i	$0.545090\pi$
$662$	0	0
$663$	−888.294	−0.0520339
$664$	0	0
$665$	−1655.69	−0.0965488
$666$	0	0
$667$	−2878.71	−0.167113
$668$	0	0
$669$	3709.74	0.214390
$670$	0	0
$671$	6208.98	0.357221
$672$	0	0
$673$	25372.8	1.45327	0.726635	−	0.687023i	$-0.241083\pi$
$673$	25372.8	1.45327	0.726635	+	0.687023i	$0.241083\pi$
$674$	0	0
$675$	334.613	0.0190804
$676$	0	0
$677$	−21369.7	−1.21315	−0.606576	−	0.795025i	$-0.707458\pi$
$677$	−21369.7	−1.21315	−0.606576	+	0.795025i	$0.707458\pi$
$678$	0	0
$679$	−18117.7	−1.02400
$680$	0	0
$681$	−6369.58	−0.358418
$682$	0	0
$683$	−1784.74	−0.0999873	−0.0499936	−	0.998750i	$-0.515920\pi$
$683$	−1784.74	−0.0999873	−0.0499936	+	0.998750i	$0.515920\pi$
$684$	0	0
$685$	−26287.9	−1.46629
$686$	0	0
$687$	6027.82	0.334753
$688$	0	0
$689$	−16422.0	−0.908023
$690$	0	0
$691$	−1019.61	−0.0561328	−0.0280664	−	0.999606i	$-0.508935\pi$
$691$	−1019.61	−0.0561328	−0.0280664	+	0.999606i	$0.508935\pi$
$692$	0	0
$693$	12210.8	0.669335
$694$	0	0
$695$	−10867.8	−0.593152
$696$	0	0
$697$	1907.07	0.103638
$698$	0	0
$699$	7154.02	0.387110
$700$	0	0
$701$	5491.06	0.295855	0.147928	−	0.988998i	$-0.452740\pi$
$701$	5491.06	0.295855	0.147928	+	0.988998i	$0.452740\pi$
$702$	0	0
$703$	−1235.18	−0.0662668
$704$	0	0
$705$	−6581.43	−0.351590
$706$	0	0
$707$	−4561.20	−0.242633
$708$	0	0
$709$	−3653.09	−0.193504	−0.0967522	−	0.995309i	$-0.530845\pi$
$709$	−3653.09	−0.193504	−0.0967522	+	0.995309i	$0.530845\pi$
$710$	0	0
$711$	−8969.38	−0.473105
$712$	0	0
$713$	3942.81	0.207096
$714$	0	0
$715$	−25791.0	−1.34899
$716$	0	0
$717$	−12171.3	−0.633957
$718$	0	0
$719$	23252.3	1.20607	0.603035	−	0.797715i	$-0.293958\pi$
$719$	23252.3	1.20607	0.603035	+	0.797715i	$0.293958\pi$
$720$	0	0
$721$	17666.4	0.912526
$722$	0	0
$723$	−1833.39	−0.0943076
$724$	0	0
$725$	−1551.13	−0.0794588
$726$	0	0
$727$	−30710.6	−1.56670	−0.783352	−	0.621579i	$-0.786492\pi$
$727$	−30710.6	−1.56670	−0.783352	+	0.621579i	$0.786492\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	529.344	0.0267832
$732$	0	0
$733$	6775.47	0.341416	0.170708	−	0.985322i	$-0.445395\pi$
$733$	6775.47	0.341416	0.170708	+	0.985322i	$0.445395\pi$
$734$	0	0
$735$	7322.92	0.367496
$736$	0	0
$737$	−47330.9	−2.36561
$738$	0	0
$739$	24106.3	1.19995	0.599976	−	0.800018i	$-0.295177\pi$
$739$	24106.3	1.19995	0.599976	+	0.800018i	$0.295177\pi$
$740$	0	0
$741$	−838.500	−0.0415696
$742$	0	0
$743$	−27021.5	−1.33421	−0.667107	−	0.744962i	$-0.732468\pi$
$743$	−27021.5	−1.33421	−0.667107	+	0.744962i	$0.732468\pi$
$744$	0	0
$745$	3875.50	0.190587
$746$	0	0
$747$	−7417.04	−0.363287
$748$	0	0
$749$	10614.7	0.517826
$750$	0	0
$751$	12789.3	0.621420	0.310710	−	0.950505i	$-0.399433\pi$
$751$	12789.3	0.621420	0.310710	+	0.950505i	$0.399433\pi$
$752$	0	0
$753$	−17809.7	−0.861915
$754$	0	0
$755$	33086.2	1.59487
$756$	0	0
$757$	10581.1	0.508029	0.254014	−	0.967200i	$-0.418249\pi$
$757$	10581.1	0.508029	0.254014	+	0.967200i	$0.418249\pi$
$758$	0	0
$759$	−3910.77	−0.187025
$760$	0	0
$761$	−14840.7	−0.706933	−0.353466	−	0.935447i	$-0.614997\pi$
$761$	−14840.7	−0.706933	−0.353466	+	0.935447i	$0.614997\pi$
$762$	0	0
$763$	14171.6	0.672409
$764$	0	0
$765$	659.459	0.0311670
$766$	0	0
$767$	−32465.9	−1.52839
$768$	0	0
$769$	−22803.1	−1.06931	−0.534655	−	0.845070i	$-0.679559\pi$
$769$	−22803.1	−1.06931	−0.534655	+	0.845070i	$0.679559\pi$
$770$	0	0
$771$	5909.11	0.276020
$772$	0	0
$773$	−40457.8	−1.88249	−0.941245	−	0.337725i	$-0.890342\pi$
$773$	−40457.8	−1.88249	−0.941245	+	0.337725i	$0.890342\pi$
$774$	0	0
$775$	2124.50	0.0984700
$776$	0	0
$777$	13609.1	0.628344
$778$	0	0
$779$	1800.17	0.0827955
$780$	0	0
$781$	9000.40	0.412368
$782$	0	0
$783$	−3379.36	−0.154238
$784$	0	0
$785$	28344.6	1.28874
$786$	0	0
$787$	27887.6	1.26313	0.631566	−	0.775322i	$-0.282413\pi$
$787$	27887.6	1.26313	0.631566	+	0.775322i	$0.282413\pi$
$788$	0	0
$789$	12731.8	0.574477
$790$	0	0
$791$	−48917.8	−2.19888
$792$	0	0
$793$	4697.65	0.210364
$794$	0	0
$795$	12191.5	0.543884
$796$	0	0
$797$	−21037.5	−0.934991	−0.467496	−	0.883995i	$-0.654844\pi$
$797$	−21037.5	−0.934991	−0.467496	+	0.883995i	$0.654844\pi$
$798$	0	0
$799$	1427.51	0.0632060
$800$	0	0
$801$	11038.6	0.486927
$802$	0	0
$803$	35090.0	1.54209
$804$	0	0
$805$	−5842.49	−0.255802
$806$	0	0
$807$	2454.16	0.107051
$808$	0	0
$809$	21600.7	0.938741	0.469371	−	0.883001i	$-0.344481\pi$
$809$	21600.7	0.938741	0.469371	+	0.883001i	$0.344481\pi$
$810$	0	0
$811$	−6101.67	−0.264191	−0.132095	−	0.991237i	$-0.542171\pi$
$811$	−6101.67	−0.264191	−0.132095	+	0.991237i	$0.542171\pi$
$812$	0	0
$813$	25653.1	1.10664
$814$	0	0
$815$	−21720.6	−0.933545
$816$	0	0
$817$	499.671	0.0213969
$818$	0	0
$819$	9238.54	0.394164
$820$	0	0
$821$	5358.32	0.227779	0.113889	−	0.993493i	$-0.463669\pi$
$821$	5358.32	0.227779	0.113889	+	0.993493i	$0.463669\pi$
$822$	0	0
$823$	21845.2	0.925246	0.462623	−	0.886555i	$-0.346908\pi$
$823$	21845.2	0.925246	0.462623	+	0.886555i	$0.346908\pi$
$824$	0	0
$825$	−2107.23	−0.0889266
$826$	0	0
$827$	−32998.5	−1.38751	−0.693755	−	0.720211i	$-0.744045\pi$
$827$	−32998.5	−1.38751	−0.693755	+	0.720211i	$0.744045\pi$
$828$	0	0
$829$	−25560.5	−1.07087	−0.535435	−	0.844576i	$-0.679852\pi$
$829$	−25560.5	−1.07087	−0.535435	+	0.844576i	$0.679852\pi$
$830$	0	0
$831$	5807.03	0.242411
$832$	0	0
$833$	−1588.34	−0.0660656
$834$	0	0
$835$	9923.02	0.411258
$836$	0	0
$837$	4628.51	0.191141
$838$	0	0
$839$	−19753.0	−0.812813	−0.406407	−	0.913692i	$-0.633218\pi$
$839$	−19753.0	−0.812813	−0.406407	+	0.913692i	$0.633218\pi$
$840$	0	0
$841$	−8723.63	−0.357687
$842$	0	0
$843$	679.493	0.0277615
$844$	0	0
$845$	3800.59	0.154727
$846$	0	0
$847$	−45036.3	−1.82700
$848$	0	0
$849$	1993.89	0.0806010
$850$	0	0
$851$	−4358.61	−0.175571
$852$	0	0
$853$	−17608.4	−0.706800	−0.353400	−	0.935472i	$-0.614975\pi$
$853$	−17608.4	−0.706800	−0.353400	+	0.935472i	$0.614975\pi$
$854$	0	0
$855$	622.492	0.0248992
$856$	0	0
$857$	−29487.6	−1.17535	−0.587676	−	0.809096i	$-0.699957\pi$
$857$	−29487.6	−1.17535	−0.587676	+	0.809096i	$0.699957\pi$
$858$	0	0
$859$	18762.8	0.745259	0.372629	−	0.927980i	$-0.378456\pi$
$859$	18762.8	0.745259	0.372629	+	0.927980i	$0.378456\pi$
$860$	0	0
$861$	−19834.1	−0.785070
$862$	0	0
$863$	−3744.37	−0.147694	−0.0738470	−	0.997270i	$-0.523528\pi$
$863$	−3744.37	−0.147694	−0.0738470	+	0.997270i	$0.523528\pi$
$864$	0	0
$865$	15779.7	0.620259
$866$	0	0
$867$	14596.0	0.571747
$868$	0	0
$869$	56485.0	2.20497
$870$	0	0
$871$	−35810.0	−1.39308
$872$	0	0
$873$	6811.74	0.264081
$874$	0	0
$875$	−34900.8	−1.34841
$876$	0	0
$877$	43793.8	1.68621	0.843107	−	0.537745i	$-0.180724\pi$
$877$	43793.8	1.68621	0.843107	+	0.537745i	$0.180724\pi$
$878$	0	0
$879$	−18250.3	−0.700303
$880$	0	0
$881$	−3679.93	−0.140726	−0.0703632	−	0.997521i	$-0.522416\pi$
$881$	−3679.93	−0.140726	−0.0703632	+	0.997521i	$0.522416\pi$
$882$	0	0
$883$	−20809.5	−0.793085	−0.396542	−	0.918016i	$-0.629790\pi$
$883$	−20809.5	−0.793085	−0.396542	+	0.918016i	$0.629790\pi$
$884$	0	0
$885$	24102.3	0.915470
$886$	0	0
$887$	5960.34	0.225624	0.112812	−	0.993616i	$-0.464014\pi$
$887$	5960.34	0.225624	0.112812	+	0.993616i	$0.464014\pi$
$888$	0	0
$889$	10133.5	0.382301
$890$	0	0
$891$	−4590.90	−0.172616
$892$	0	0
$893$	1347.49	0.0504950
$894$	0	0
$895$	24862.9	0.928577
$896$	0	0
$897$	−2958.84	−0.110137
$898$	0	0
$899$	−21456.0	−0.795992
$900$	0	0
$901$	−2644.33	−0.0977751
$902$	0	0
$903$	−5505.35	−0.202886
$904$	0	0
$905$	−6314.01	−0.231917
$906$	0	0
$907$	−24021.9	−0.879421	−0.439710	−	0.898140i	$-0.644919\pi$
$907$	−24021.9	−0.879421	−0.439710	+	0.898140i	$0.644919\pi$
$908$	0	0
$909$	1714.88	0.0625731
$910$	0	0
$911$	42723.9	1.55379	0.776897	−	0.629627i	$-0.216792\pi$
$911$	42723.9	1.55379	0.776897	+	0.629627i	$0.216792\pi$
$912$	0	0
$913$	46709.1	1.69315
$914$	0	0
$915$	−3487.48	−0.126003
$916$	0	0
$917$	53987.1	1.94418
$918$	0	0
$919$	42416.4	1.52251	0.761257	−	0.648451i	$-0.224583\pi$
$919$	42416.4	1.52251	0.761257	+	0.648451i	$0.224583\pi$
$920$	0	0
$921$	−25099.7	−0.898006
$922$	0	0
$923$	6809.61	0.242840
$924$	0	0
$925$	−2348.54	−0.0834807
$926$	0	0
$927$	−6642.06	−0.235333
$928$	0	0
$929$	−25754.7	−0.909564	−0.454782	−	0.890603i	$-0.650283\pi$
$929$	−25754.7	−0.909564	−0.454782	+	0.890603i	$0.650283\pi$
$930$	0	0
$931$	−1499.30	−0.0527795
$932$	0	0
$933$	1259.93	0.0442103
$934$	0	0
$935$	−4152.96	−0.145258
$936$	0	0
$937$	9759.21	0.340255	0.170128	−	0.985422i	$-0.445582\pi$
$937$	9759.21	0.340255	0.170128	+	0.985422i	$0.445582\pi$
$938$	0	0
$939$	30159.8	1.04816
$940$	0	0
$941$	−32013.0	−1.10903	−0.554514	−	0.832174i	$-0.687096\pi$
$941$	−32013.0	−1.10903	−0.554514	+	0.832174i	$0.687096\pi$
$942$	0	0
$943$	6352.31	0.219364
$944$	0	0
$945$	−6858.58	−0.236095
$946$	0	0
$947$	−30699.1	−1.05342	−0.526709	−	0.850046i	$-0.676574\pi$
$947$	−30699.1	−1.05342	−0.526709	+	0.850046i	$0.676574\pi$
$948$	0	0
$949$	26548.7	0.908120
$950$	0	0
$951$	−2389.73	−0.0814850
$952$	0	0
$953$	−40451.8	−1.37499	−0.687493	−	0.726191i	$-0.741289\pi$
$953$	−40451.8	−1.37499	−0.687493	+	0.726191i	$0.741289\pi$
$954$	0	0
$955$	14415.5	0.488454
$956$	0	0
$957$	21281.6	0.718847
$958$	0	0
$959$	−59300.8	−1.99679
$960$	0	0
$961$	−403.972	−0.0135602
$962$	0	0
$963$	−3990.81	−0.133543
$964$	0	0
$965$	29945.1	0.998931
$966$	0	0
$967$	−4775.46	−0.158809	−0.0794046	−	0.996842i	$-0.525302\pi$
$967$	−4775.46	−0.158809	−0.0794046	+	0.996842i	$0.525302\pi$
$968$	0	0
$969$	−135.018	−0.00447617
$970$	0	0
$971$	26994.0	0.892151	0.446076	−	0.894995i	$-0.352821\pi$
$971$	26994.0	0.892151	0.446076	+	0.894995i	$0.352821\pi$
$972$	0	0
$973$	−24515.9	−0.807754
$974$	0	0
$975$	−1594.31	−0.0523680
$976$	0	0
$977$	−8376.58	−0.274299	−0.137150	−	0.990550i	$-0.543794\pi$
$977$	−8376.58	−0.274299	−0.137150	+	0.990550i	$0.543794\pi$
$978$	0	0
$979$	−69515.7	−2.26939
$980$	0	0
$981$	−5328.13	−0.173409
$982$	0	0
$983$	4759.06	0.154416	0.0772078	−	0.997015i	$-0.475400\pi$
$983$	4759.06	0.154416	0.0772078	+	0.997015i	$0.475400\pi$
$984$	0	0
$985$	−23136.5	−0.748415
$986$	0	0
$987$	−14846.5	−0.478795
$988$	0	0
$989$	1763.21	0.0566903
$990$	0	0
$991$	−54632.9	−1.75123	−0.875616	−	0.483008i	$-0.839544\pi$
$991$	−54632.9	−1.75123	−0.875616	+	0.483008i	$0.839544\pi$
$992$	0	0
$993$	−981.808	−0.0313764
$994$	0	0
$995$	9490.66	0.302386
$996$	0	0
$997$	−55666.7	−1.76829	−0.884144	−	0.467215i	$-0.845257\pi$
$997$	−55666.7	−1.76829	−0.884144	+	0.467215i	$0.845257\pi$
$998$	0	0
$999$	−5116.63	−0.162045

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.4.a.j.1.2	✓	5
3.2	odd	2		1656.4.a.o.1.4		5
4.3	odd	2		1104.4.a.y.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.4.a.j.1.2	✓	5	1.1	even	1	trivial
1104.4.a.y.1.2		5	4.3	odd	2
1656.4.a.o.1.4		5	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)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -10.6116 q^{5} -23.9380 q^{7} +9.00000 q^{9} -56.6778 q^{11} -42.8818 q^{13} +31.8349 q^{15} -6.90498 q^{17} -6.51792 q^{19} +71.8140 q^{21} -23.0000 q^{23} -12.3931 q^{25} -27.0000 q^{27} +125.161 q^{29} -171.426 q^{31} +170.033 q^{33} +254.021 q^{35} +189.505 q^{37} +128.645 q^{39} -276.188 q^{41} -76.6612 q^{43} -95.5048 q^{45} -206.736 q^{47} +230.028 q^{49} +20.7149 q^{51} +382.960 q^{53} +601.444 q^{55} +19.5538 q^{57} +757.103 q^{59} -109.549 q^{61} -215.442 q^{63} +455.046 q^{65} +835.087 q^{67} +69.0000 q^{69} -158.799 q^{71} -619.113 q^{73} +37.1792 q^{75} +1356.75 q^{77} -996.598 q^{79} +81.0000 q^{81} -824.116 q^{83} +73.2732 q^{85} -375.484 q^{87} +1226.51 q^{89} +1026.50 q^{91} +514.279 q^{93} +69.1658 q^{95} +756.861 q^{97} -510.100 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 15 q^{3} + 16 q^{5} + 45 q^{9} + 6 q^{11} - 10 q^{13} - 48 q^{15} + 22 q^{17} + 42 q^{19} - 115 q^{23} + 291 q^{25} - 135 q^{27} + 274 q^{29} - 460 q^{31} - 18 q^{33} - 320 q^{35} + 420 q^{37} + 30 q^{39}+ \cdots + 54 q^{99}+O(q^{100}) \) 5 * q - 15 * q^3 + 16 * q^5 + 45 * q^9 + 6 * q^11 - 10 * q^13 - 48 * q^15 + 22 * q^17 + 42 * q^19 - 115 * q^23 + 291 * q^25 - 135 * q^27 + 274 * q^29 - 460 * q^31 - 18 * q^33 - 320 * q^35 + 420 * q^37 + 30 * q^39 + 362 * q^41 + 206 * q^43 + 144 * q^45 + 444 * q^47 + 1489 * q^49 - 66 * q^51 + 1240 * q^53 - 148 * q^55 - 126 * q^57 + 880 * q^59 + 584 * q^61 + 1840 * q^65 + 1554 * q^67 + 345 * q^69 + 472 * q^71 + 1350 * q^73 - 873 * q^75 + 4092 * q^77 - 224 * q^79 + 405 * q^81 + 1570 * q^83 + 2904 * q^85 - 822 * q^87 + 2074 * q^89 - 688 * q^91 + 1380 * q^93 + 2444 * q^95 + 14 * q^97 + 54 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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