LMFDB - Embedded newform 147.4.a.j.1.2

Newspace parameters

Level:	$ N $	$=$	$ 147 = 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	147.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$8.67328077084$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
Defining polynomial:	$x^{2} - 2$
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	147.1

$q$-expansion

$f(q)$	$=$	$q+2.41421 q^{2} -3.00000 q^{3} -2.17157 q^{4} +19.8995 q^{5} -7.24264 q^{6} -24.5563 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+2.41421 q^{2} -3.00000 q^{3} -2.17157 q^{4} +19.8995 q^{5} -7.24264 q^{6} -24.5563 q^{8} +9.00000 q^{9} +48.0416 q^{10} +23.9411 q^{11} +6.51472 q^{12} +87.3553 q^{13} -59.6985 q^{15} -41.9117 q^{16} -5.63961 q^{17} +21.7279 q^{18} +64.8873 q^{19} -43.2132 q^{20} +57.7990 q^{22} -25.5980 q^{23} +73.6690 q^{24} +270.990 q^{25} +210.894 q^{26} -27.0000 q^{27} +60.3188 q^{29} -144.125 q^{30} -122.711 q^{31} +95.2670 q^{32} -71.8234 q^{33} -13.6152 q^{34} -19.5442 q^{36} -56.1177 q^{37} +156.652 q^{38} -262.066 q^{39} -488.659 q^{40} -299.713 q^{41} -501.421 q^{43} -51.9899 q^{44} +179.095 q^{45} -61.7990 q^{46} +305.553 q^{47} +125.735 q^{48} +654.227 q^{50} +16.9188 q^{51} -189.698 q^{52} -375.117 q^{53} -65.1838 q^{54} +476.416 q^{55} -194.662 q^{57} +145.622 q^{58} -627.612 q^{59} +129.640 q^{60} +3.75736 q^{61} -296.250 q^{62} +565.288 q^{64} +1738.33 q^{65} -173.397 q^{66} -813.048 q^{67} +12.2468 q^{68} +76.7939 q^{69} +165.902 q^{71} -221.007 q^{72} +619.100 q^{73} -135.480 q^{74} -812.970 q^{75} -140.908 q^{76} -632.683 q^{78} -138.246 q^{79} -834.021 q^{80} +81.0000 q^{81} -723.571 q^{82} +621.137 q^{83} -112.225 q^{85} -1210.54 q^{86} -180.956 q^{87} -587.907 q^{88} +285.418 q^{89} +432.375 q^{90} +55.5879 q^{92} +368.132 q^{93} +737.671 q^{94} +1291.22 q^{95} -285.801 q^{96} -603.114 q^{97} +215.470 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2q + 2q^{2} - 6q^{3} - 10q^{4} + 20q^{5} - 6q^{6} - 18q^{8} + 18q^{9} + O(q^{10}) $	$ 2q + 2q^{2} - 6q^{3} - 10q^{4} + 20q^{5} - 6q^{6} - 18q^{8} + 18q^{9} + 48q^{10} - 20q^{11} + 30q^{12} + 104q^{13} - 60q^{15} + 18q^{16} + 116q^{17} + 18q^{18} + 192q^{19} - 44q^{20} + 76q^{22} + 28q^{23} + 54q^{24} + 146q^{25} + 204q^{26} - 54q^{27} + 296q^{29} - 144q^{30} - 104q^{31} + 18q^{32} + 60q^{33} - 64q^{34} - 90q^{36} - 248q^{37} + 104q^{38} - 312q^{39} - 488q^{40} + 20q^{41} - 720q^{43} + 292q^{44} + 180q^{45} - 84q^{46} - 96q^{47} - 54q^{48} + 706q^{50} - 348q^{51} - 320q^{52} + 268q^{53} - 54q^{54} + 472q^{55} - 576q^{57} + 48q^{58} - 616q^{59} + 132q^{60} + 16q^{61} - 304q^{62} + 118q^{64} + 1740q^{65} - 228q^{66} - 144q^{67} - 940q^{68} - 84q^{69} + 988q^{71} - 162q^{72} + 104q^{73} - 56q^{74} - 438q^{75} - 1136q^{76} - 612q^{78} - 944q^{79} - 828q^{80} + 162q^{81} - 856q^{82} + 1016q^{83} - 100q^{85} - 1120q^{86} - 888q^{87} - 876q^{88} - 388q^{89} + 432q^{90} - 364q^{92} + 312q^{93} + 904q^{94} + 1304q^{95} - 54q^{96} + 488q^{97} - 180q^{99} + O(q^{100}) $

Display 100 coefficients 10 coefficients

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 $

$2$	2.41421	0.853553	0.426777	−	0.904357i	$-0.359649\pi$
$2$	2.41421	0.853553	0.426777	+	0.904357i	$0.359649\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−2.17157	−0.271447
$5$	19.8995	1.77986	0.889932	−	0.456092i	$-0.150751\pi$
$5$	19.8995	1.77986	0.889932	+	0.456092i	$0.150751\pi$
$6$	−7.24264	−0.492799
$7$	0	0
$7$	0	0
$8$	−24.5563	−1.08525
$9$	9.00000	0.333333
$10$	48.0416	1.51921
$11$	23.9411	0.656229	0.328115	−	0.944638i	$-0.393587\pi$
$11$	23.9411	0.656229	0.328115	+	0.944638i	$0.393587\pi$
$12$	6.51472	0.156720
$13$	87.3553	1.86369	0.931847	−	0.362852i	$-0.118197\pi$
$13$	87.3553	1.86369	0.931847	+	0.362852i	$0.118197\pi$
$14$	0	0
$15$	−59.6985	−1.02761
$16$	−41.9117	−0.654870
$17$	−5.63961	−0.0804592	−0.0402296	−	0.999190i	$-0.512809\pi$
$17$	−5.63961	−0.0804592	−0.0402296	+	0.999190i	$0.512809\pi$
$18$	21.7279	0.284518
$19$	64.8873	0.783483	0.391741	−	0.920075i	$-0.371873\pi$
$19$	64.8873	0.783483	0.391741	+	0.920075i	$0.371873\pi$
$20$	−43.2132	−0.483138
$21$	0	0
$22$	57.7990	0.560127
$23$	−25.5980	−0.232067	−0.116034	−	0.993245i	$-0.537018\pi$
$23$	−25.5980	−0.232067	−0.116034	+	0.993245i	$0.537018\pi$
$24$	73.6690	0.626568
$25$	270.990	2.16792
$26$	210.894	1.59076
$27$	−27.0000	−0.192450
$28$	0	0
$29$	60.3188	0.386238	0.193119	−	0.981175i	$-0.438140\pi$
$29$	60.3188	0.386238	0.193119	+	0.981175i	$0.438140\pi$
$30$	−144.125	−0.877116
$31$	−122.711	−0.710951	−0.355476	−	0.934686i	$-0.615681\pi$
$31$	−122.711	−0.710951	−0.355476	+	0.934686i	$0.615681\pi$
$32$	95.2670	0.526281
$33$	−71.8234	−0.378874
$34$	−13.6152	−0.0686762
$35$	0	0
$36$	−19.5442	−0.0904822
$37$	−56.1177	−0.249343	−0.124672	−	0.992198i	$-0.539788\pi$
$37$	−56.1177	−0.249343	−0.124672	+	0.992198i	$0.539788\pi$
$38$	156.652	0.668744
$39$	−262.066	−1.07600
$40$	−488.659	−1.93159
$41$	−299.713	−1.14164	−0.570820	−	0.821075i	$-0.693375\pi$
$41$	−299.713	−1.14164	−0.570820	+	0.821075i	$0.693375\pi$
$42$	0	0
$43$	−501.421	−1.77828	−0.889140	−	0.457635i	$-0.848697\pi$
$43$	−501.421	−1.77828	−0.889140	+	0.457635i	$0.848697\pi$
$44$	−51.9899	−0.178131
$45$	179.095	0.593288
$46$	−61.7990	−0.198082
$47$	305.553	0.948288	0.474144	−	0.880447i	$-0.342758\pi$
$47$	305.553	0.948288	0.474144	+	0.880447i	$0.342758\pi$
$48$	125.735	0.378089
$49$	0	0
$50$	654.227	1.85043
$51$	16.9188	0.0464531
$52$	−189.698	−0.505893
$53$	−375.117	−0.972194	−0.486097	−	0.873905i	$-0.661580\pi$
$53$	−375.117	−0.972194	−0.486097	+	0.873905i	$0.661580\pi$
$54$	−65.1838	−0.164266
$55$	476.416	1.16800
$56$	0	0
$57$	−194.662	−0.452344
$58$	145.622	0.329675
$59$	−627.612	−1.38488	−0.692442	−	0.721474i	$-0.743465\pi$
$59$	−627.612	−1.38488	−0.692442	+	0.721474i	$0.743465\pi$
$60$	129.640	0.278940
$61$	3.75736	0.00788657	0.00394328	−	0.999992i	$-0.498745\pi$
$61$	3.75736	0.00788657	0.00394328	+	0.999992i	$0.498745\pi$
$62$	−296.250	−0.606835
$63$	0	0
$64$	565.288	1.10408
$65$	1738.33	3.31712
$66$	−173.397	−0.323389
$67$	−813.048	−1.48253	−0.741266	−	0.671212i	$-0.765774\pi$
$67$	−813.048	−1.48253	−0.741266	+	0.671212i	$0.765774\pi$
$68$	12.2468	0.0218404
$69$	76.7939	0.133984
$70$	0	0
$71$	165.902	0.277310	0.138655	−	0.990341i	$-0.455722\pi$
$71$	165.902	0.277310	0.138655	+	0.990341i	$0.455722\pi$
$72$	−221.007	−0.361749
$73$	619.100	0.992605	0.496302	−	0.868150i	$-0.334691\pi$
$73$	619.100	0.992605	0.496302	+	0.868150i	$0.334691\pi$
$74$	−135.480	−0.212828
$75$	−812.970	−1.25165
$76$	−140.908	−0.212674
$77$	0	0
$78$	−632.683	−0.918427
$79$	−138.246	−0.196884	−0.0984421	−	0.995143i	$-0.531386\pi$
$79$	−138.246	−0.196884	−0.0984421	+	0.995143i	$0.531386\pi$
$80$	−834.021	−1.16558
$81$	81.0000	0.111111
$82$	−723.571	−0.974451
$83$	621.137	0.821430	0.410715	−	0.911764i	$-0.365279\pi$
$83$	621.137	0.821430	0.410715	+	0.911764i	$0.365279\pi$
$84$	0	0
$85$	−112.225	−0.143207
$86$	−1210.54	−1.51786
$87$	−180.956	−0.222995
$88$	−587.907	−0.712171
$89$	285.418	0.339936	0.169968	−	0.985450i	$-0.445634\pi$
$89$	285.418	0.339936	0.169968	+	0.985450i	$0.445634\pi$
$90$	432.375	0.506403
$91$	0	0
$92$	55.5879	0.0629939
$93$	368.132	0.410468
$94$	737.671	0.809415
$95$	1291.22	1.39449
$96$	−285.801	−0.303848
$97$	−603.114	−0.631309	−0.315654	−	0.948874i	$-0.602224\pi$
$97$	−603.114	−0.631309	−0.315654	+	0.948874i	$0.602224\pi$
$98$	0	0
$99$	215.470	0.218743
$100$	−588.474	−0.588474
$101$	457.209	0.450436	0.225218	−	0.974308i	$-0.427691\pi$
$101$	457.209	0.450436	0.225218	+	0.974308i	$0.427691\pi$
$102$	40.8457	0.0396502
$103$	786.045	0.751954	0.375977	−	0.926629i	$-0.377307\pi$
$103$	786.045	0.751954	0.375977	+	0.926629i	$0.377307\pi$
$104$	−2145.13	−2.02257
$105$	0	0
$106$	−905.612	−0.829819
$107$	−196.461	−0.177501	−0.0887504	−	0.996054i	$-0.528287\pi$
$107$	−196.461	−0.177501	−0.0887504	+	0.996054i	$0.528287\pi$
$108$	58.6325	0.0522399
$109$	−306.343	−0.269196	−0.134598	−	0.990900i	$-0.542974\pi$
$109$	−306.343	−0.269196	−0.134598	+	0.990900i	$0.542974\pi$
$110$	1150.17	0.996950
$111$	168.353	0.143958
$112$	0	0
$113$	1997.63	1.66302	0.831508	−	0.555512i	$-0.187478\pi$
$113$	1997.63	1.66302	0.831508	+	0.555512i	$0.187478\pi$
$114$	−469.955	−0.386100
$115$	−509.387	−0.413048
$116$	−130.987	−0.104843
$117$	786.198	0.621231
$118$	−1515.19	−1.18207
$119$	0	0
$120$	1465.98	1.11521
$121$	−757.823	−0.569363
$122$	9.07107	0.00673161
$123$	899.138	0.659127
$124$	266.475	0.192985
$125$	2905.13	2.07874
$126$	0	0
$127$	−2311.40	−1.61499	−0.807494	−	0.589875i	$-0.799177\pi$
$127$	−2311.40	−1.61499	−0.807494	+	0.589875i	$0.799177\pi$
$128$	602.591	0.416109
$129$	1504.26	1.02669
$130$	4196.69	2.83134
$131$	−155.018	−0.103389	−0.0516945	−	0.998663i	$-0.516462\pi$
$131$	−155.018	−0.103389	−0.0516945	+	0.998663i	$0.516462\pi$
$132$	155.970	0.102844
$133$	0	0
$134$	−1962.87	−1.26542
$135$	−537.286	−0.342535
$136$	138.488	0.0873182
$137$	516.936	0.322371	0.161186	−	0.986924i	$-0.448468\pi$
$137$	516.936	0.322371	0.161186	+	0.986924i	$0.448468\pi$
$138$	185.397	0.114363
$139$	958.067	0.584620	0.292310	−	0.956324i	$-0.405576\pi$
$139$	958.067	0.584620	0.292310	+	0.956324i	$0.405576\pi$
$140$	0	0
$141$	−916.660	−0.547494
$142$	400.524	0.236699
$143$	2091.39	1.22301
$144$	−377.205	−0.218290
$145$	1200.31	0.687452
$146$	1494.64	0.847241
$147$	0	0
$148$	121.864	0.0676834
$149$	−1770.63	−0.973526	−0.486763	−	0.873534i	$-0.661822\pi$
$149$	−1770.63	−0.973526	−0.486763	+	0.873534i	$0.661822\pi$
$150$	−1962.68	−1.06835
$151$	−2540.24	−1.36902	−0.684508	−	0.729005i	$-0.739983\pi$
$151$	−2540.24	−1.36902	−0.684508	+	0.729005i	$0.739983\pi$
$152$	−1593.40	−0.850272
$153$	−50.7565	−0.0268197
$154$	0	0
$155$	−2441.88	−1.26540
$156$	569.095	0.292078
$157$	−1083.34	−0.550702	−0.275351	−	0.961344i	$-0.588794\pi$
$157$	−1083.34	−0.550702	−0.275351	+	0.961344i	$0.588794\pi$
$158$	−333.754	−0.168051
$159$	1125.35	0.561296
$160$	1895.77	0.936709
$161$	0	0
$162$	195.551	0.0948393
$163$	2968.72	1.42655	0.713277	−	0.700882i	$-0.247210\pi$
$163$	2968.72	1.42655	0.713277	+	0.700882i	$0.247210\pi$
$164$	650.848	0.309895
$165$	−1429.25	−0.674345
$166$	1499.56	0.701134
$167$	−2091.53	−0.969149	−0.484574	−	0.874750i	$-0.661025\pi$
$167$	−2091.53	−0.969149	−0.484574	+	0.874750i	$0.661025\pi$
$168$	0	0
$169$	5433.96	2.47335
$170$	−270.936	−0.122234
$171$	583.986	0.261161
$172$	1088.87	0.482708
$173$	−470.148	−0.206617	−0.103308	−	0.994649i	$-0.532943\pi$
$173$	−470.148	−0.206617	−0.103308	+	0.994649i	$0.532943\pi$
$174$	−436.867	−0.190338
$175$	0	0
$176$	−1003.41	−0.429745
$177$	1882.84	0.799563
$178$	689.061	0.290153
$179$	1056.46	0.441137	0.220569	−	0.975371i	$-0.429209\pi$
$179$	1056.46	0.441137	0.220569	+	0.975371i	$0.429209\pi$
$180$	−388.919	−0.161046
$181$	−406.470	−0.166921	−0.0834605	−	0.996511i	$-0.526597\pi$
$181$	−406.470	−0.166921	−0.0834605	+	0.996511i	$0.526597\pi$
$182$	0	0
$183$	−11.2721	−0.00455331
$184$	628.593	0.251850
$185$	−1116.71	−0.443797
$186$	888.749	0.350356
$187$	−135.019	−0.0527997
$188$	−663.531	−0.257410
$189$	0	0
$190$	3117.29	1.19027
$191$	179.410	0.0679666	0.0339833	−	0.999422i	$-0.489181\pi$
$191$	179.410	0.0679666	0.0339833	+	0.999422i	$0.489181\pi$
$192$	−1695.87	−0.637440
$193$	−2388.37	−0.890769	−0.445385	−	0.895339i	$-0.646933\pi$
$193$	−2388.37	−0.890769	−0.445385	+	0.895339i	$0.646933\pi$
$194$	−1456.05	−0.538856
$195$	−5214.98	−1.91514
$196$	0	0
$197$	−2665.35	−0.963952	−0.481976	−	0.876184i	$-0.660081\pi$
$197$	−2665.35	−0.963952	−0.481976	+	0.876184i	$0.660081\pi$
$198$	520.191	0.186709
$199$	−1342.31	−0.478159	−0.239079	−	0.971000i	$-0.576846\pi$
$199$	−1342.31	−0.478159	−0.239079	+	0.971000i	$0.576846\pi$
$200$	−6654.52	−2.35273
$201$	2439.14	0.855940
$202$	1103.80	0.384471
$203$	0	0
$204$	−36.7405	−0.0126095
$205$	−5964.13	−2.03197
$206$	1897.68	0.641833
$207$	−230.382	−0.0773558
$208$	−3661.21	−1.22048
$209$	1553.48	0.514144
$210$	0	0
$211$	628.442	0.205042	0.102521	−	0.994731i	$-0.467309\pi$
$211$	628.442	0.205042	0.102521	+	0.994731i	$0.467309\pi$
$212$	814.594	0.263899
$213$	−497.707	−0.160105
$214$	−474.299	−0.151506
$215$	−9978.03	−3.16510
$216$	663.021	0.208856
$217$	0	0
$218$	−739.578	−0.229773
$219$	−1857.30	−0.573081
$220$	−1034.57	−0.317049
$221$	−492.650	−0.149951
$222$	406.441	0.122876
$223$	−969.970	−0.291273	−0.145637	−	0.989338i	$-0.546523\pi$
$223$	−969.970	−0.291273	−0.145637	+	0.989338i	$0.546523\pi$
$224$	0	0
$225$	2438.91	0.722640
$226$	4822.70	1.41947
$227$	−4748.64	−1.38845	−0.694225	−	0.719758i	$-0.744253\pi$
$227$	−4748.64	−1.38845	−0.694225	+	0.719758i	$0.744253\pi$
$228$	422.723	0.122787
$229$	−4801.99	−1.38570	−0.692848	−	0.721083i	$-0.743644\pi$
$229$	−4801.99	−1.38570	−0.692848	+	0.721083i	$0.743644\pi$
$230$	−1229.77	−0.352559
$231$	0	0
$232$	−1481.21	−0.419164
$233$	−3155.29	−0.887166	−0.443583	−	0.896233i	$-0.646293\pi$
$233$	−3155.29	−0.887166	−0.443583	+	0.896233i	$0.646293\pi$
$234$	1898.05	0.530254
$235$	6080.36	1.68782
$236$	1362.91	0.375922
$237$	414.737	0.113671
$238$	0	0
$239$	4241.93	1.14806	0.574032	−	0.818833i	$-0.305378\pi$
$239$	4241.93	1.14806	0.574032	+	0.818833i	$0.305378\pi$
$240$	2502.06	0.672948
$241$	−4342.99	−1.16081	−0.580407	−	0.814326i	$-0.697107\pi$
$241$	−4342.99	−1.16081	−0.580407	+	0.814326i	$0.697107\pi$
$242$	−1829.55	−0.485982
$243$	−243.000	−0.0641500
$244$	−8.15938	−0.00214078
$245$	0	0
$246$	2170.71	0.562600
$247$	5668.25	1.46017
$248$	3013.33	0.771558
$249$	−1863.41	−0.474253
$250$	7013.59	1.77431
$251$	3003.01	0.755172	0.377586	−	0.925974i	$-0.376754\pi$
$251$	3003.01	0.755172	0.377586	+	0.925974i	$0.376754\pi$
$252$	0	0
$253$	−612.844	−0.152289
$254$	−5580.21	−1.37848
$255$	336.676	0.0826803
$256$	−3067.52	−0.748907
$257$	4468.84	1.08466	0.542332	−	0.840164i	$-0.317541\pi$
$257$	4468.84	1.08466	0.542332	+	0.840164i	$0.317541\pi$
$258$	3631.61	0.876335
$259$	0	0
$260$	−3774.90	−0.900422
$261$	542.869	0.128746
$262$	−374.246	−0.0882480
$263$	6160.13	1.44430	0.722148	−	0.691738i	$-0.243155\pi$
$263$	6160.13	1.44430	0.722148	+	0.691738i	$0.243155\pi$
$264$	1763.72	0.411172
$265$	−7464.64	−1.73037
$266$	0	0
$267$	−856.255	−0.196262
$268$	1765.59	0.402428
$269$	4988.90	1.13078	0.565388	−	0.824825i	$-0.308726\pi$
$269$	4988.90	1.13078	0.565388	+	0.824825i	$0.308726\pi$
$270$	−1297.12	−0.292372
$271$	4433.73	0.993837	0.496918	−	0.867797i	$-0.334465\pi$
$271$	4433.73	0.993837	0.496918	+	0.867797i	$0.334465\pi$
$272$	236.366	0.0526903
$273$	0	0
$274$	1247.99	0.275161
$275$	6487.80	1.42265
$276$	−166.764	−0.0363695
$277$	1112.37	0.241284	0.120642	−	0.992696i	$-0.461505\pi$
$277$	1112.37	0.241284	0.120642	+	0.992696i	$0.461505\pi$
$278$	2312.98	0.499005
$279$	−1104.40	−0.236984
$280$	0	0
$281$	2813.22	0.597233	0.298616	−	0.954373i	$-0.403475\pi$
$281$	2813.22	0.597233	0.298616	+	0.954373i	$0.403475\pi$
$282$	−2213.01	−0.467316
$283$	3147.54	0.661137	0.330569	−	0.943782i	$-0.392759\pi$
$283$	3147.54	0.661137	0.330569	+	0.943782i	$0.392759\pi$
$284$	−360.269	−0.0752748
$285$	−3873.67	−0.805111
$286$	5049.05	1.04390
$287$	0	0
$288$	857.403	0.175427
$289$	−4881.19	−0.993526
$290$	2897.81	0.586777
$291$	1809.34	0.364486
$292$	−1344.42	−0.269439
$293$	9143.04	1.82301	0.911505	−	0.411289i	$-0.134921\pi$
$293$	9143.04	1.82301	0.911505	+	0.411289i	$0.134921\pi$
$294$	0	0
$295$	−12489.2	−2.46491
$296$	1378.05	0.270599
$297$	−646.410	−0.126291
$298$	−4274.67	−0.830956
$299$	−2236.12	−0.432502
$300$	1765.42	0.339756
$301$	0	0
$302$	−6132.67	−1.16853
$303$	−1371.63	−0.260059
$304$	−2719.54	−0.513079
$305$	74.7696	0.0140370
$306$	−122.537	−0.0228921
$307$	−4648.90	−0.864257	−0.432129	−	0.901812i	$-0.642237\pi$
$307$	−4648.90	−0.864257	−0.432129	+	0.901812i	$0.642237\pi$
$308$	0	0
$309$	−2358.13	−0.434141
$310$	−5895.22	−1.08008
$311$	−6417.18	−1.17005	−0.585024	−	0.811016i	$-0.698915\pi$
$311$	−6417.18	−1.17005	−0.585024	+	0.811016i	$0.698915\pi$
$312$	6435.38	1.16773
$313$	5868.22	1.05972	0.529858	−	0.848086i	$-0.322245\pi$
$313$	5868.22	1.05972	0.529858	+	0.848086i	$0.322245\pi$
$314$	−2615.42	−0.470054
$315$	0	0
$316$	300.210	0.0534435
$317$	−3974.52	−0.704200	−0.352100	−	0.935962i	$-0.614532\pi$
$317$	−3974.52	−0.704200	−0.352100	+	0.935962i	$0.614532\pi$
$318$	2716.84	0.479096
$319$	1444.10	0.253461
$320$	11249.0	1.96511
$321$	589.383	0.102480
$322$	0	0
$323$	−365.939	−0.0630384
$324$	−175.897	−0.0301607
$325$	23672.4	4.04034
$326$	7167.13	1.21764
$327$	919.029	0.155420
$328$	7359.85	1.23896
$329$	0	0
$330$	−3450.51	−0.575589
$331$	−8912.82	−1.48004	−0.740020	−	0.672585i	$-0.765184\pi$
$331$	−8912.82	−1.48004	−0.740020	+	0.672585i	$0.765184\pi$
$332$	−1348.84	−0.222974
$333$	−505.060	−0.0831144
$334$	−5049.41	−0.827220
$335$	−16179.2	−2.63871
$336$	0	0
$337$	3977.06	0.642862	0.321431	−	0.946933i	$-0.395836\pi$
$337$	3977.06	0.642862	0.321431	+	0.946933i	$0.395836\pi$
$338$	13118.7	2.11114
$339$	−5992.88	−0.960143
$340$	243.706	0.0388729
$341$	−2937.83	−0.466547
$342$	1409.87	0.222915
$343$	0	0
$344$	12313.1	1.92987
$345$	1528.16	0.238474
$346$	−1135.04	−0.176358
$347$	6826.43	1.05609	0.528043	−	0.849218i	$-0.322926\pi$
$347$	6826.43	1.05609	0.528043	+	0.849218i	$0.322926\pi$
$348$	392.960	0.0605312
$349$	−807.342	−0.123828	−0.0619141	−	0.998081i	$-0.519720\pi$
$349$	−807.342	−0.123828	−0.0619141	+	0.998081i	$0.519720\pi$
$350$	0	0
$351$	−2358.59	−0.358668
$352$	2280.80	0.345361
$353$	−7919.20	−1.19404	−0.597020	−	0.802226i	$-0.703649\pi$
$353$	−7919.20	−1.19404	−0.597020	+	0.802226i	$0.703649\pi$
$354$	4545.57	0.682470
$355$	3301.38	0.493574
$356$	−619.807	−0.0922744
$357$	0	0
$358$	2550.52	0.376534
$359$	−8819.21	−1.29655	−0.648273	−	0.761408i	$-0.724509\pi$
$359$	−8819.21	−1.29655	−0.648273	+	0.761408i	$0.724509\pi$
$360$	−4397.93	−0.643865
$361$	−2648.64	−0.386155
$362$	−981.307	−0.142476
$363$	2273.47	0.328722
$364$	0	0
$365$	12319.8	1.76670
$366$	−27.2132	−0.00388649
$367$	−11161.8	−1.58758	−0.793788	−	0.608194i	$-0.791894\pi$
$367$	−11161.8	−1.58758	−0.793788	+	0.608194i	$0.791894\pi$
$368$	1072.85	0.151974
$369$	−2697.41	−0.380547
$370$	−2695.99	−0.378805
$371$	0	0
$372$	−799.426	−0.111420
$373$	2727.86	0.378668	0.189334	−	0.981913i	$-0.439367\pi$
$373$	2727.86	0.378668	0.189334	+	0.981913i	$0.439367\pi$
$374$	−325.964	−0.0450673
$375$	−8715.38	−1.20016
$376$	−7503.28	−1.02913
$377$	5269.17	0.719830
$378$	0	0
$379$	4086.49	0.553849	0.276924	−	0.960892i	$-0.410685\pi$
$379$	4086.49	0.553849	0.276924	+	0.960892i	$0.410685\pi$
$380$	−2803.99	−0.378530
$381$	6934.20	0.932414
$382$	433.133	0.0580131
$383$	13032.9	1.73878	0.869389	−	0.494129i	$-0.164513\pi$
$383$	13032.9	1.73878	0.869389	+	0.494129i	$0.164513\pi$
$384$	−1807.77	−0.240241
$385$	0	0
$386$	−5766.03	−0.760319
$387$	−4512.79	−0.592760
$388$	1309.71	0.171367
$389$	−194.991	−0.0254151	−0.0127075	−	0.999919i	$-0.504045\pi$
$389$	−194.991	−0.0254151	−0.0127075	+	0.999919i	$0.504045\pi$
$390$	−12590.1	−1.63468
$391$	144.363	0.0186719
$392$	0	0
$393$	465.053	0.0596916
$394$	−6434.73	−0.822785
$395$	−2751.02	−0.350427
$396$	−467.909	−0.0593771
$397$	−14183.0	−1.79300	−0.896501	−	0.443042i	$-0.853899\pi$
$397$	−14183.0	−1.79300	−0.896501	+	0.443042i	$0.853899\pi$
$398$	−3240.61	−0.408134
$399$	0	0
$400$	−11357.6	−1.41971
$401$	10005.0	1.24596	0.622978	−	0.782239i	$-0.285923\pi$
$401$	10005.0	1.24596	0.622978	+	0.782239i	$0.285923\pi$
$402$	5888.61	0.730590
$403$	−10719.4	−1.32500
$404$	−992.863	−0.122269
$405$	1611.86	0.197763
$406$	0	0
$407$	−1343.52	−0.163626
$408$	−415.465	−0.0504132
$409$	4634.93	0.560349	0.280174	−	0.959949i	$-0.409608\pi$
$409$	4634.93	0.560349	0.280174	+	0.959949i	$0.409608\pi$
$410$	−14398.7	−1.73439
$411$	−1550.81	−0.186121
$412$	−1706.95	−0.204115
$413$	0	0
$414$	−556.191	−0.0660273
$415$	12360.3	1.46203
$416$	8322.08	0.980826
$417$	−2874.20	−0.337531
$418$	3750.42	0.438849
$419$	4998.31	0.582777	0.291388	−	0.956605i	$-0.405883\pi$
$419$	4998.31	0.582777	0.291388	+	0.956605i	$0.405883\pi$
$420$	0	0
$421$	−704.160	−0.0815170	−0.0407585	−	0.999169i	$-0.512977\pi$
$421$	−704.160	−0.0815170	−0.0407585	+	0.999169i	$0.512977\pi$
$422$	1517.19	0.175014
$423$	2749.98	0.316096
$424$	9211.50	1.05507
$425$	−1528.28	−0.174429
$426$	−1201.57	−0.136658
$427$	0	0
$428$	426.629	0.0481820
$429$	−6274.16	−0.706105
$430$	−24089.1	−2.70158
$431$	−10332.8	−1.15479	−0.577393	−	0.816466i	$-0.695930\pi$
$431$	−10332.8	−1.15479	−0.577393	+	0.816466i	$0.695930\pi$
$432$	1131.62	0.126030
$433$	11106.8	1.23270	0.616348	−	0.787474i	$-0.288611\pi$
$433$	11106.8	1.23270	0.616348	+	0.787474i	$0.288611\pi$
$434$	0	0
$435$	−3600.94	−0.396901
$436$	665.246	0.0730723
$437$	−1660.98	−0.181821
$438$	−4483.92	−0.489155
$439$	−7299.28	−0.793566	−0.396783	−	0.917912i	$-0.629874\pi$
$439$	−7299.28	−0.793566	−0.396783	+	0.917912i	$0.629874\pi$
$440$	−11699.0	−1.26757
$441$	0	0
$442$	−1189.36	−0.127991
$443$	16089.7	1.72560	0.862802	−	0.505542i	$-0.168707\pi$
$443$	16089.7	1.72560	0.862802	+	0.505542i	$0.168707\pi$
$444$	−365.591	−0.0390770
$445$	5679.68	0.605040
$446$	−2341.71	−0.248617
$447$	5311.88	0.562065
$448$	0	0
$449$	13561.7	1.42543	0.712715	−	0.701454i	$-0.247466\pi$
$449$	13561.7	1.42543	0.712715	+	0.701454i	$0.247466\pi$
$450$	5888.05	0.616812
$451$	−7175.46	−0.749178
$452$	−4337.99	−0.451420
$453$	7620.71	0.790402
$454$	−11464.2	−1.18512
$455$	0	0
$456$	4780.19	0.490905
$457$	10848.6	1.11045	0.555224	−	0.831701i	$-0.312633\pi$
$457$	10848.6	1.11045	0.555224	+	0.831701i	$0.312633\pi$
$458$	−11593.0	−1.18277
$459$	152.269	0.0154844
$460$	1106.17	0.112121
$461$	−1758.69	−0.177679	−0.0888397	−	0.996046i	$-0.528316\pi$
$461$	−1758.69	−0.177679	−0.0888397	+	0.996046i	$0.528316\pi$
$462$	0	0
$463$	−5411.95	−0.543228	−0.271614	−	0.962406i	$-0.587557\pi$
$463$	−5411.95	−0.543228	−0.271614	+	0.962406i	$0.587557\pi$
$464$	−2528.06	−0.252936
$465$	7325.64	0.730577
$466$	−7617.54	−0.757244
$467$	8111.34	0.803744	0.401872	−	0.915696i	$-0.368360\pi$
$467$	8111.34	0.803744	0.401872	+	0.915696i	$0.368360\pi$
$468$	−1707.29	−0.168631
$469$	0	0
$470$	14679.3	1.44065
$471$	3250.03	0.317948
$472$	15411.9	1.50294
$473$	−12004.6	−1.16696
$474$	1001.26	0.0970244
$475$	17583.8	1.69853
$476$	0	0
$477$	−3376.05	−0.324065
$478$	10240.9	0.979935
$479$	2095.76	0.199912	0.0999559	−	0.994992i	$-0.468130\pi$
$479$	2095.76	0.199912	0.0999559	+	0.994992i	$0.468130\pi$
$480$	−5687.30	−0.540809
$481$	−4902.18	−0.464699
$482$	−10484.9	−0.990817
$483$	0	0
$484$	1645.67	0.154552
$485$	−12001.7	−1.12364
$486$	−586.654	−0.0547555
$487$	9610.08	0.894197	0.447099	−	0.894485i	$-0.352457\pi$
$487$	9610.08	0.894197	0.447099	+	0.894485i	$0.352457\pi$
$488$	−92.2670	−0.00855888
$489$	−8906.17	−0.823622
$490$	0	0
$491$	−11717.3	−1.07698	−0.538488	−	0.842633i	$-0.681004\pi$
$491$	−11717.3	−1.07698	−0.538488	+	0.842633i	$0.681004\pi$
$492$	−1952.54	−0.178918
$493$	−340.174	−0.0310764
$494$	13684.4	1.24633
$495$	4287.75	0.389333
$496$	5143.01	0.465581
$497$	0	0
$498$	−4498.67	−0.404800
$499$	−9195.19	−0.824916	−0.412458	−	0.910977i	$-0.635330\pi$
$499$	−9195.19	−0.824916	−0.412458	+	0.910977i	$0.635330\pi$
$500$	−6308.69	−0.564266
$501$	6274.60	0.559538
$502$	7249.90	0.644580
$503$	16118.8	1.42883	0.714414	−	0.699724i	$-0.246693\pi$
$503$	16118.8	1.42883	0.714414	+	0.699724i	$0.246693\pi$
$504$	0	0
$505$	9098.23	0.801715
$506$	−1479.54	−0.129987
$507$	−16301.9	−1.42799
$508$	5019.37	0.438383
$509$	4918.78	0.428333	0.214166	−	0.976797i	$-0.431297\pi$
$509$	4918.78	0.428333	0.214166	+	0.976797i	$0.431297\pi$
$510$	812.808	0.0705721
$511$	0	0
$512$	−12226.4	−1.05534
$513$	−1751.96	−0.150781
$514$	10788.7	0.925819
$515$	15641.9	1.33838
$516$	−3266.62	−0.278692
$517$	7315.29	0.622294
$518$	0	0
$519$	1410.44	0.119290
$520$	−42687.0	−3.59990
$521$	13963.4	1.17418	0.587089	−	0.809522i	$-0.300274\pi$
$521$	13963.4	1.17418	0.587089	+	0.809522i	$0.300274\pi$
$522$	1310.60	0.109892
$523$	−13755.3	−1.15005	−0.575024	−	0.818136i	$-0.695007\pi$
$523$	−13755.3	−1.15005	−0.575024	+	0.818136i	$0.695007\pi$
$524$	336.632	0.0280646
$525$	0	0
$526$	14871.9	1.23278
$527$	692.040	0.0572026
$528$	3010.24	0.248113
$529$	−11511.7	−0.946145
$530$	−18021.2	−1.47697
$531$	−5648.51	−0.461628
$532$	0	0
$533$	−26181.5	−2.12767
$534$	−2067.18	−0.167520
$535$	−3909.47	−0.315928
$536$	19965.5	1.60891
$537$	−3169.38	−0.254691
$538$	12044.3	0.965178
$539$	0	0
$540$	1166.76	0.0929800
$541$	14462.7	1.14935	0.574676	−	0.818381i	$-0.305128\pi$
$541$	14462.7	1.14935	0.574676	+	0.818381i	$0.305128\pi$
$542$	10704.0	0.848293
$543$	1219.41	0.0963719
$544$	−537.269	−0.0423441
$545$	−6096.07	−0.479132
$546$	0	0
$547$	13682.5	1.06951	0.534755	−	0.845007i	$-0.320404\pi$
$547$	13682.5	1.06951	0.534755	+	0.845007i	$0.320404\pi$
$548$	−1122.56	−0.0875065
$549$	33.8162	0.00262886
$550$	15662.9	1.21431
$551$	3913.92	0.302611
$552$	−1885.78	−0.145406
$553$	0	0
$554$	2685.49	0.205949
$555$	3350.14	0.256227
$556$	−2080.51	−0.158693
$557$	−7663.13	−0.582939	−0.291470	−	0.956580i	$-0.594144\pi$
$557$	−7663.13	−0.582939	−0.291470	+	0.956580i	$0.594144\pi$
$558$	−2666.25	−0.202278
$559$	−43801.8	−3.31417
$560$	0	0
$561$	405.056	0.0304839
$562$	6791.71	0.509770
$563$	17470.5	1.30780	0.653902	−	0.756580i	$-0.273131\pi$
$563$	17470.5	1.30780	0.653902	+	0.756580i	$0.273131\pi$
$564$	1990.59	0.148616
$565$	39751.8	2.95995
$566$	7598.83	0.564316
$567$	0	0
$568$	−4073.96	−0.300950
$569$	−13873.9	−1.02219	−0.511094	−	0.859525i	$-0.670760\pi$
$569$	−13873.9	−1.02219	−0.511094	+	0.859525i	$0.670760\pi$
$570$	−9351.88	−0.687205
$571$	−3777.52	−0.276855	−0.138428	−	0.990373i	$-0.544205\pi$
$571$	−3777.52	−0.276855	−0.138428	+	0.990373i	$0.544205\pi$
$572$	−4541.60	−0.331982
$573$	−538.229	−0.0392405
$574$	0	0
$575$	−6936.79	−0.503103
$576$	5087.60	0.368026
$577$	13880.5	1.00148	0.500738	−	0.865599i	$-0.333062\pi$
$577$	13880.5	1.00148	0.500738	+	0.865599i	$0.333062\pi$
$578$	−11784.2	−0.848028
$579$	7165.10	0.514286
$580$	−2606.57	−0.186607
$581$	0	0
$582$	4368.14	0.311108
$583$	−8980.72	−0.637982
$584$	−15202.8	−1.07722
$585$	15644.9	1.10571
$586$	22073.2	1.55604
$587$	−2395.61	−0.168445	−0.0842227	−	0.996447i	$-0.526841\pi$
$587$	−2395.61	−0.168445	−0.0842227	+	0.996447i	$0.526841\pi$
$588$	0	0
$589$	−7962.36	−0.557018
$590$	−30151.5	−2.10393
$591$	7996.06	0.556538
$592$	2351.99	0.163287
$593$	6603.50	0.457290	0.228645	−	0.973510i	$-0.426570\pi$
$593$	6603.50	0.457290	0.228645	+	0.973510i	$0.426570\pi$
$594$	−1560.57	−0.107796
$595$	0	0
$596$	3845.04	0.264260
$597$	4026.92	0.276065
$598$	−5398.47	−0.369164
$599$	17252.1	1.17680	0.588399	−	0.808571i	$-0.299759\pi$
$599$	17252.1	1.17680	0.588399	+	0.808571i	$0.299759\pi$
$600$	19963.6	1.35835
$601$	12833.1	0.871005	0.435503	−	0.900187i	$-0.356571\pi$
$601$	12833.1	0.871005	0.435503	+	0.900187i	$0.356571\pi$
$602$	0	0
$603$	−7317.43	−0.494177
$604$	5516.31	0.371615
$605$	−15080.3	−1.01339
$606$	−3311.40	−0.221974
$607$	8620.16	0.576411	0.288206	−	0.957569i	$-0.406941\pi$
$607$	8620.16	0.576411	0.288206	+	0.957569i	$0.406941\pi$
$608$	6181.62	0.412332
$609$	0	0
$610$	180.510	0.0119813
$611$	26691.7	1.76732
$612$	110.221	0.00728013
$613$	5136.73	0.338451	0.169226	−	0.985577i	$-0.445873\pi$
$613$	5136.73	0.338451	0.169226	+	0.985577i	$0.445873\pi$
$614$	−11223.4	−0.737690
$615$	17892.4	1.17316
$616$	0	0
$617$	−1759.82	−0.114826	−0.0574131	−	0.998351i	$-0.518285\pi$
$617$	−1759.82	−0.114826	−0.0574131	+	0.998351i	$0.518285\pi$
$618$	−5693.04	−0.370562
$619$	−3560.24	−0.231176	−0.115588	−	0.993297i	$-0.536875\pi$
$619$	−3560.24	−0.231176	−0.115588	+	0.993297i	$0.536875\pi$
$620$	5302.72	0.343488
$621$	691.145	0.0446614
$622$	−15492.4	−0.998698
$623$	0	0
$624$	10983.6	0.704643
$625$	23936.8	1.53195
$626$	14167.1	0.904524
$627$	−4660.43	−0.296841
$628$	2352.56	0.149486
$629$	316.482	0.0200620
$630$	0	0
$631$	−27321.4	−1.72369	−0.861845	−	0.507172i	$-0.830691\pi$
$631$	−27321.4	−1.72369	−0.861845	+	0.507172i	$0.830691\pi$
$632$	3394.81	0.213668
$633$	−1885.33	−0.118381
$634$	−9595.34	−0.601072
$635$	−45995.7	−2.87446
$636$	−2443.78	−0.152362
$637$	0	0
$638$	3486.36	0.216342
$639$	1493.12	0.0924366
$640$	11991.3	0.740619
$641$	−21927.0	−1.35112	−0.675558	−	0.737307i	$-0.736097\pi$
$641$	−21927.0	−1.35112	−0.675558	+	0.737307i	$0.736097\pi$
$642$	1422.90	0.0874723
$643$	−5826.04	−0.357320	−0.178660	−	0.983911i	$-0.557176\pi$
$643$	−5826.04	−0.357320	−0.178660	+	0.983911i	$0.557176\pi$
$644$	0	0
$645$	29934.1	1.82737
$646$	−883.455	−0.0538066
$647$	24210.7	1.47113	0.735565	−	0.677454i	$-0.236917\pi$
$647$	24210.7	1.47113	0.735565	+	0.677454i	$0.236917\pi$
$648$	−1989.06	−0.120583
$649$	−15025.7	−0.908801
$650$	57150.3	3.44864
$651$	0	0
$652$	−6446.80	−0.387233
$653$	−25623.3	−1.53556	−0.767778	−	0.640716i	$-0.778637\pi$
$653$	−25623.3	−1.53556	−0.767778	+	0.640716i	$0.778637\pi$
$654$	2218.73	0.132660
$655$	−3084.77	−0.184018
$656$	12561.5	0.747626
$657$	5571.90	0.330868
$658$	0	0
$659$	−23273.7	−1.37574	−0.687871	−	0.725833i	$-0.741455\pi$
$659$	−23273.7	−1.37574	−0.687871	+	0.725833i	$0.741455\pi$
$660$	3103.72	0.183049
$661$	−20036.5	−1.17902	−0.589508	−	0.807763i	$-0.700678\pi$
$661$	−20036.5	−1.17902	−0.589508	+	0.807763i	$0.700678\pi$
$662$	−21517.5	−1.26329
$663$	1477.95	0.0865744
$664$	−15252.9	−0.891454
$665$	0	0
$666$	−1219.32	−0.0709426
$667$	−1544.04	−0.0896333
$668$	4541.92	0.263072
$669$	2909.91	0.168167
$670$	−39060.1	−2.25228
$671$	89.9554	0.00517540
$672$	0	0
$673$	−18127.8	−1.03830	−0.519149	−	0.854684i	$-0.673751\pi$
$673$	−18127.8	−1.03830	−0.519149	+	0.854684i	$0.673751\pi$
$674$	9601.48	0.548717
$675$	−7316.73	−0.417216
$676$	−11800.2	−0.671383
$677$	13815.5	0.784301	0.392150	−	0.919901i	$-0.371731\pi$
$677$	13815.5	0.784301	0.392150	+	0.919901i	$0.371731\pi$
$678$	−14468.1	−0.819533
$679$	0	0
$680$	2755.85	0.155415
$681$	14245.9	0.801622
$682$	−7092.55	−0.398223
$683$	−4604.23	−0.257944	−0.128972	−	0.991648i	$-0.541168\pi$
$683$	−4604.23	−0.257944	−0.128972	+	0.991648i	$0.541168\pi$
$684$	−1268.17	−0.0708912
$685$	10286.8	0.573777
$686$	0	0
$687$	14406.0	0.800032
$688$	21015.4	1.16454
$689$	−32768.5	−1.81187
$690$	3689.31	0.203550
$691$	17913.7	0.986205	0.493103	−	0.869971i	$-0.335863\pi$
$691$	17913.7	0.986205	0.493103	+	0.869971i	$0.335863\pi$
$692$	1020.96	0.0560854
$693$	0	0
$694$	16480.5	0.901426
$695$	19065.1	1.04054
$696$	4443.63	0.242005
$697$	1690.26	0.0918555
$698$	−1949.10	−0.105694
$699$	9465.86	0.512206
$700$	0	0
$701$	11303.7	0.609035	0.304518	−	0.952507i	$-0.401505\pi$
$701$	11303.7	0.609035	0.304518	+	0.952507i	$0.401505\pi$
$702$	−5694.15	−0.306142
$703$	−3641.33	−0.195356
$704$	13533.6	0.724529
$705$	−18241.1	−0.974466
$706$	−19118.6	−1.01918
$707$	0	0
$708$	−4088.72	−0.217039
$709$	−16046.3	−0.849973	−0.424987	−	0.905200i	$-0.639721\pi$
$709$	−16046.3	−0.849973	−0.424987	+	0.905200i	$0.639721\pi$
$710$	7970.22	0.421292
$711$	−1244.21	−0.0656280
$712$	−7008.83	−0.368915
$713$	3141.15	0.164989
$714$	0	0
$715$	41617.5	2.17679
$716$	−2294.18	−0.119745
$717$	−12725.8	−0.662836
$718$	−21291.5	−1.10667
$719$	−25190.5	−1.30660	−0.653300	−	0.757099i	$-0.726616\pi$
$719$	−25190.5	−1.30660	−0.653300	+	0.757099i	$0.726616\pi$
$720$	−7506.19	−0.388527
$721$	0	0
$722$	−6394.38	−0.329604
$723$	13029.0	0.670197
$724$	882.680	0.0453102
$725$	16345.8	0.837334
$726$	5488.64	0.280582
$727$	11277.2	0.575307	0.287653	−	0.957735i	$-0.407125\pi$
$727$	11277.2	0.575307	0.287653	+	0.957735i	$0.407125\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	29742.6	1.50797
$731$	2827.82	0.143079
$732$	24.4781	0.00123598
$733$	23720.0	1.19525	0.597626	−	0.801775i	$-0.296111\pi$
$733$	23720.0	1.19525	0.597626	+	0.801775i	$0.296111\pi$
$734$	−26946.9	−1.35508
$735$	0	0
$736$	−2438.64	−0.122133
$737$	−19465.3	−0.972880
$738$	−6512.14	−0.324817
$739$	8124.72	0.404428	0.202214	−	0.979341i	$-0.435186\pi$
$739$	8124.72	0.404428	0.202214	+	0.979341i	$0.435186\pi$
$740$	2425.03	0.120467
$741$	−17004.8	−0.843030
$742$	0	0
$743$	20955.3	1.03469	0.517346	−	0.855777i	$-0.326920\pi$
$743$	20955.3	1.03469	0.517346	+	0.855777i	$0.326920\pi$
$744$	−9039.98	−0.445459
$745$	−35234.6	−1.73274
$746$	6585.63	0.323213
$747$	5590.23	0.273810
$748$	293.203	0.0143323
$749$	0	0
$750$	−21040.8	−1.02440
$751$	−38208.1	−1.85650	−0.928251	−	0.371954i	$-0.878688\pi$
$751$	−38208.1	−1.85650	−0.928251	+	0.371954i	$0.878688\pi$
$752$	−12806.3	−0.621006
$753$	−9009.02	−0.435999
$754$	12720.9	0.614413
$755$	−50549.4	−2.43666
$756$	0	0
$757$	30958.1	1.48638	0.743191	−	0.669079i	$-0.233311\pi$
$757$	30958.1	1.48638	0.743191	+	0.669079i	$0.233311\pi$
$758$	9865.65	0.472740
$759$	1838.53	0.0879243
$760$	−31707.8	−1.51337
$761$	−40049.4	−1.90774	−0.953871	−	0.300218i	$-0.902941\pi$
$761$	−40049.4	−1.90774	−0.953871	+	0.300218i	$0.902941\pi$
$762$	16740.6	0.795865
$763$	0	0
$764$	−389.601	−0.0184493
$765$	−1010.03	−0.0477355
$766$	31464.3	1.48414
$767$	−54825.3	−2.58100
$768$	9202.57	0.432382
$769$	−8002.01	−0.375240	−0.187620	−	0.982242i	$-0.560077\pi$
$769$	−8002.01	−0.375240	−0.187620	+	0.982242i	$0.560077\pi$
$770$	0	0
$771$	−13406.5	−0.626231
$772$	5186.52	0.241796
$773$	−13933.4	−0.648316	−0.324158	−	0.946003i	$-0.605081\pi$
$773$	−13933.4	−0.648316	−0.324158	+	0.946003i	$0.605081\pi$
$774$	−10894.8	−0.505952
$775$	−33253.4	−1.54128
$776$	14810.3	0.685126
$777$	0	0
$778$	−470.751	−0.0216931
$779$	−19447.6	−0.894456
$780$	11324.7	0.519859
$781$	3971.89	0.181979
$782$	348.522	0.0159375
$783$	−1628.61	−0.0743316
$784$	0	0
$785$	−21558.0	−0.980175
$786$	1122.74	0.0509500
$787$	37581.7	1.70222	0.851108	−	0.524991i	$-0.175931\pi$
$787$	37581.7	1.70222	0.851108	+	0.524991i	$0.175931\pi$
$788$	5788.01	0.261662
$789$	−18480.4	−0.833865
$790$	−6641.54	−0.299108
$791$	0	0
$792$	−5291.16	−0.237390
$793$	328.225	0.0146981
$794$	−34240.7	−1.53042
$795$	22393.9	0.999032
$796$	2914.92	0.129795
$797$	−9458.78	−0.420385	−0.210193	−	0.977660i	$-0.567409\pi$
$797$	−9458.78	−0.420385	−0.210193	+	0.977660i	$0.567409\pi$
$798$	0	0
$799$	−1723.20	−0.0762985
$800$	25816.4	1.14093
$801$	2568.77	0.113312
$802$	24154.3	1.06349
$803$	14821.9	0.651376
$804$	−5296.78	−0.232342
$805$	0	0
$806$	−25879.0	−1.13095
$807$	−14966.7	−0.652854
$808$	−11227.4	−0.488834
$809$	−1909.94	−0.0830034	−0.0415017	−	0.999138i	$-0.513214\pi$
$809$	−1909.94	−0.0830034	−0.0415017	+	0.999138i	$0.513214\pi$
$810$	3891.37	0.168801
$811$	43110.6	1.86661	0.933303	−	0.359091i	$-0.116913\pi$
$811$	43110.6	1.86661	0.933303	+	0.359091i	$0.116913\pi$
$812$	0	0
$813$	−13301.2	−0.573792
$814$	−3243.55	−0.139664
$815$	59076.1	2.53907
$816$	−709.097	−0.0304208
$817$	−32535.9	−1.39325
$818$	11189.7	0.478287
$819$	0	0
$820$	12951.5	0.551570
$821$	4026.56	0.171167	0.0855834	−	0.996331i	$-0.472725\pi$
$821$	4026.56	0.171167	0.0855834	+	0.996331i	$0.472725\pi$
$822$	−3743.98	−0.158864
$823$	39668.1	1.68012	0.840062	−	0.542491i	$-0.182519\pi$
$823$	39668.1	1.68012	0.840062	+	0.542491i	$0.182519\pi$
$824$	−19302.4	−0.816056
$825$	−19463.4	−0.821368
$826$	0	0
$827$	30137.7	1.26722	0.633611	−	0.773652i	$-0.281572\pi$
$827$	30137.7	1.26722	0.633611	+	0.773652i	$0.281572\pi$
$828$	500.291	0.0209980
$829$	23278.0	0.975245	0.487622	−	0.873055i	$-0.337864\pi$
$829$	23278.0	0.975245	0.487622	+	0.873055i	$0.337864\pi$
$830$	29840.4	1.24792
$831$	−3337.10	−0.139306
$832$	49381.0	2.05766
$833$	0	0
$834$	−6938.94	−0.288100
$835$	−41620.5	−1.72495
$836$	−3373.48	−0.139563
$837$	3313.19	0.136823
$838$	12067.0	0.497431
$839$	−9494.43	−0.390684	−0.195342	−	0.980735i	$-0.562582\pi$
$839$	−9494.43	−0.390684	−0.195342	+	0.980735i	$0.562582\pi$
$840$	0	0
$841$	−20750.6	−0.850820
$842$	−1699.99	−0.0695791
$843$	−8439.65	−0.344813
$844$	−1364.71	−0.0556578
$845$	108133.	4.40223
$846$	6639.04	0.269805
$847$	0	0
$848$	15721.8	0.636661
$849$	−9442.62	−0.381708
$850$	−3689.59	−0.148885
$851$	1436.50	0.0578644
$852$	1080.81	0.0434599
$853$	−12692.4	−0.509471	−0.254736	−	0.967011i	$-0.581988\pi$
$853$	−12692.4	−0.509471	−0.254736	+	0.967011i	$0.581988\pi$
$854$	0	0
$855$	11621.0	0.464831
$856$	4824.36	0.192632
$857$	22206.0	0.885114	0.442557	−	0.896740i	$-0.354071\pi$
$857$	22206.0	0.885114	0.442557	+	0.896740i	$0.354071\pi$
$858$	−15147.2	−0.602698
$859$	19820.5	0.787271	0.393636	−	0.919267i	$-0.371217\pi$
$859$	19820.5	0.787271	0.393636	+	0.919267i	$0.371217\pi$
$860$	21668.0	0.859155
$861$	0	0
$862$	−24945.5	−0.985671
$863$	36413.8	1.43631	0.718157	−	0.695881i	$-0.244986\pi$
$863$	36413.8	1.43631	0.718157	+	0.695881i	$0.244986\pi$
$864$	−2572.21	−0.101283
$865$	−9355.70	−0.367749
$866$	26814.1	1.05217
$867$	14643.6	0.573613
$868$	0	0
$869$	−3309.76	−0.129201
$870$	−8693.43	−0.338776
$871$	−71024.1	−2.76298
$872$	7522.67	0.292144
$873$	−5428.03	−0.210436
$874$	−4009.97	−0.155194
$875$	0	0
$876$	4033.26	0.155561
$877$	19442.6	0.748609	0.374305	−	0.927306i	$-0.377881\pi$
$877$	19442.6	0.748609	0.374305	+	0.927306i	$0.377881\pi$
$878$	−17622.0	−0.677351
$879$	−27429.1	−1.05252
$880$	−19967.4	−0.764888
$881$	−25184.2	−0.963082	−0.481541	−	0.876423i	$-0.659923\pi$
$881$	−25184.2	−0.963082	−0.481541	+	0.876423i	$0.659923\pi$
$882$	0	0
$883$	−4050.03	−0.154354	−0.0771769	−	0.997017i	$-0.524591\pi$
$883$	−4050.03	−0.154354	−0.0771769	+	0.997017i	$0.524591\pi$
$884$	1069.83	0.0407038
$885$	37467.5	1.42311
$886$	38843.9	1.47290
$887$	41604.2	1.57489	0.787447	−	0.616383i	$-0.211402\pi$
$887$	41604.2	1.57489	0.787447	+	0.616383i	$0.211402\pi$
$888$	−4134.14	−0.156231
$889$	0	0
$890$	13712.0	0.516434
$891$	1939.23	0.0729144
$892$	2106.36	0.0790652
$893$	19826.5	0.742967
$894$	12824.0	0.479753
$895$	21023.0	0.785165
$896$	0	0
$897$	6708.36	0.249705
$898$	32740.9	1.21668
$899$	−7401.76	−0.274597
$900$	−5296.27	−0.196158
$901$	2115.51	0.0782219
$902$	−17323.1	−0.639463
$903$	0	0
$904$	−49054.4	−1.80478
$905$	−8088.56	−0.297097
$906$	18398.0	0.674650
$907$	−3572.60	−0.130790	−0.0653949	−	0.997859i	$-0.520831\pi$
$907$	−3572.60	−0.130790	−0.0653949	+	0.997859i	$0.520831\pi$
$908$	10312.0	0.376890
$909$	4114.88	0.150145
$910$	0	0
$911$	29457.7	1.07133	0.535663	−	0.844432i	$-0.320062\pi$
$911$	29457.7	1.07133	0.535663	+	0.844432i	$0.320062\pi$
$912$	8158.61	0.296226
$913$	14870.7	0.539046
$914$	26190.8	0.947827
$915$	−224.309	−0.00810428
$916$	10427.9	0.376143
$917$	0	0
$918$	367.611	0.0132167
$919$	−3310.65	−0.118834	−0.0594168	−	0.998233i	$-0.518924\pi$
$919$	−3310.65	−0.118834	−0.0594168	+	0.998233i	$0.518924\pi$
$920$	12508.7	0.448260
$921$	13946.7	0.498979
$922$	−4245.85	−0.151659
$923$	14492.5	0.516820
$924$	0	0
$925$	−15207.3	−0.540556
$926$	−13065.6	−0.463674
$927$	7074.40	0.250651
$928$	5746.39	0.203270
$929$	−31467.5	−1.11132	−0.555660	−	0.831410i	$-0.687534\pi$
$929$	−31467.5	−1.11132	−0.555660	+	0.831410i	$0.687534\pi$
$930$	17685.7	0.623587
$931$	0	0
$932$	6851.94	0.240818
$933$	19251.5	0.675527
$934$	19582.5	0.686038
$935$	−2686.80	−0.0939763
$936$	−19306.2	−0.674190
$937$	−17363.4	−0.605375	−0.302688	−	0.953090i	$-0.597884\pi$
$937$	−17363.4	−0.605375	−0.302688	+	0.953090i	$0.597884\pi$
$938$	0	0
$939$	−17604.6	−0.611828
$940$	−13203.9	−0.458154
$941$	−5547.77	−0.192192	−0.0960958	−	0.995372i	$-0.530636\pi$
$941$	−5547.77	−0.192192	−0.0960958	+	0.995372i	$0.530636\pi$
$942$	7846.27	0.271386
$943$	7672.04	0.264937
$944$	26304.3	0.906919
$945$	0	0
$946$	−28981.6	−0.996062
$947$	−37960.3	−1.30258	−0.651290	−	0.758829i	$-0.725772\pi$
$947$	−37960.3	−1.30258	−0.651290	+	0.758829i	$0.725772\pi$
$948$	−900.631	−0.0308556
$949$	54081.7	1.84991
$950$	42451.1	1.44978
$951$	11923.6	0.406570
$952$	0	0
$953$	−10019.3	−0.340563	−0.170282	−	0.985395i	$-0.554468\pi$
$953$	−10019.3	−0.340563	−0.170282	+	0.985395i	$0.554468\pi$
$954$	−8150.51	−0.276606
$955$	3570.16	0.120971
$956$	−9211.65	−0.311638
$957$	−4332.30	−0.146336
$958$	5059.61	0.170635
$959$	0	0
$960$	−33746.9	−1.13456
$961$	−14733.1	−0.494548
$962$	−11834.9	−0.396646
$963$	−1768.15	−0.0591670
$964$	9431.11	0.315099
$965$	−47527.3	−1.58545
$966$	0	0
$967$	27834.4	0.925641	0.462820	−	0.886452i	$-0.346837\pi$
$967$	27834.4	0.925641	0.462820	+	0.886452i	$0.346837\pi$
$968$	18609.4	0.617900
$969$	1097.82	0.0363952
$970$	−28974.6	−0.959090
$971$	−18275.3	−0.604000	−0.302000	−	0.953308i	$-0.597654\pi$
$971$	−18275.3	−0.604000	−0.302000	+	0.953308i	$0.597654\pi$
$972$	527.692	0.0174133
$973$	0	0
$974$	23200.8	0.763245
$975$	−71017.2	−2.33269
$976$	−157.477	−0.00516468
$977$	−43028.9	−1.40903	−0.704513	−	0.709691i	$-0.748834\pi$
$977$	−43028.9	−1.40903	−0.704513	+	0.709691i	$0.748834\pi$
$978$	−21501.4	−0.703005
$979$	6833.24	0.223076
$980$	0	0
$981$	−2757.09	−0.0897320
$982$	−28288.1	−0.919256
$983$	30559.2	0.991544	0.495772	−	0.868453i	$-0.334885\pi$
$983$	30559.2	0.991544	0.495772	+	0.868453i	$0.334885\pi$
$984$	−22079.6	−0.715316
$985$	−53039.2	−1.71571
$986$	−821.253	−0.0265254
$987$	0	0
$988$	−12309.0	−0.396358
$989$	12835.4	0.412681
$990$	10351.5	0.332317
$991$	−44945.9	−1.44072	−0.720360	−	0.693600i	$-0.756024\pi$
$991$	−44945.9	−1.44072	−0.720360	+	0.693600i	$0.756024\pi$
$992$	−11690.3	−0.374160
$993$	26738.5	0.854501
$994$	0	0
$995$	−26711.2	−0.851058
$996$	4046.53	0.128734
$997$	29006.1	0.921397	0.460698	−	0.887557i	$-0.347599\pi$
$997$	29006.1	0.921397	0.460698	+	0.887557i	$0.347599\pi$
$998$	−22199.1	−0.704110
$999$	1515.18	0.0479861

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.4.a.j.1.2	✓	2
3.2	odd	2		441.4.a.n.1.1		2
4.3	odd	2		2352.4.a.cf.1.2		2
7.2	even	3		147.4.e.k.67.1		4
7.3	odd	6		147.4.e.j.79.1		4
7.4	even	3		147.4.e.k.79.1		4
7.5	odd	6		147.4.e.j.67.1		4
7.6	odd	2		147.4.a.k.1.2	yes	2
21.2	odd	6		441.4.e.v.361.2		4
21.5	even	6		441.4.e.u.361.2		4
21.11	odd	6		441.4.e.v.226.2		4
21.17	even	6		441.4.e.u.226.2		4
21.20	even	2		441.4.a.o.1.1		2
28.27	even	2		2352.4.a.bl.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.4.a.j.1.2	✓	2	1.1	even	1	trivial
147.4.a.k.1.2	yes	2	7.6	odd	2
147.4.e.j.67.1		4	7.5	odd	6
147.4.e.j.79.1		4	7.3	odd	6
147.4.e.k.67.1		4	7.2	even	3
147.4.e.k.79.1		4	7.4	even	3
441.4.a.n.1.1		2	3.2	odd	2
441.4.a.o.1.1		2	21.20	even	2
441.4.e.u.226.2		4	21.17	even	6
441.4.e.u.361.2		4	21.5	even	6
441.4.e.v.226.2		4	21.11	odd	6
441.4.e.v.361.2		4	21.2	odd	6
2352.4.a.bl.1.1		2	28.27	even	2
2352.4.a.cf.1.2		2	4.3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	147.a (trivial)

\(f(q)\)	\(=\)	\(q+2.41421 q^{2} -3.00000 q^{3} -2.17157 q^{4} +19.8995 q^{5} -7.24264 q^{6} -24.5563 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+2.41421 q^{2} -3.00000 q^{3} -2.17157 q^{4} +19.8995 q^{5} -7.24264 q^{6} -24.5563 q^{8} +9.00000 q^{9} +48.0416 q^{10} +23.9411 q^{11} +6.51472 q^{12} +87.3553 q^{13} -59.6985 q^{15} -41.9117 q^{16} -5.63961 q^{17} +21.7279 q^{18} +64.8873 q^{19} -43.2132 q^{20} +57.7990 q^{22} -25.5980 q^{23} +73.6690 q^{24} +270.990 q^{25} +210.894 q^{26} -27.0000 q^{27} +60.3188 q^{29} -144.125 q^{30} -122.711 q^{31} +95.2670 q^{32} -71.8234 q^{33} -13.6152 q^{34} -19.5442 q^{36} -56.1177 q^{37} +156.652 q^{38} -262.066 q^{39} -488.659 q^{40} -299.713 q^{41} -501.421 q^{43} -51.9899 q^{44} +179.095 q^{45} -61.7990 q^{46} +305.553 q^{47} +125.735 q^{48} +654.227 q^{50} +16.9188 q^{51} -189.698 q^{52} -375.117 q^{53} -65.1838 q^{54} +476.416 q^{55} -194.662 q^{57} +145.622 q^{58} -627.612 q^{59} +129.640 q^{60} +3.75736 q^{61} -296.250 q^{62} +565.288 q^{64} +1738.33 q^{65} -173.397 q^{66} -813.048 q^{67} +12.2468 q^{68} +76.7939 q^{69} +165.902 q^{71} -221.007 q^{72} +619.100 q^{73} -135.480 q^{74} -812.970 q^{75} -140.908 q^{76} -632.683 q^{78} -138.246 q^{79} -834.021 q^{80} +81.0000 q^{81} -723.571 q^{82} +621.137 q^{83} -112.225 q^{85} -1210.54 q^{86} -180.956 q^{87} -587.907 q^{88} +285.418 q^{89} +432.375 q^{90} +55.5879 q^{92} +368.132 q^{93} +737.671 q^{94} +1291.22 q^{95} -285.801 q^{96} -603.114 q^{97} +215.470 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2q + 2q^{2} - 6q^{3} - 10q^{4} + 20q^{5} - 6q^{6} - 18q^{8} + 18q^{9} + O(q^{10}) \)	\( 2q + 2q^{2} - 6q^{3} - 10q^{4} + 20q^{5} - 6q^{6} - 18q^{8} + 18q^{9} + 48q^{10} - 20q^{11} + 30q^{12} + 104q^{13} - 60q^{15} + 18q^{16} + 116q^{17} + 18q^{18} + 192q^{19} - 44q^{20} + 76q^{22} + 28q^{23} + 54q^{24} + 146q^{25} + 204q^{26} - 54q^{27} + 296q^{29} - 144q^{30} - 104q^{31} + 18q^{32} + 60q^{33} - 64q^{34} - 90q^{36} - 248q^{37} + 104q^{38} - 312q^{39} - 488q^{40} + 20q^{41} - 720q^{43} + 292q^{44} + 180q^{45} - 84q^{46} - 96q^{47} - 54q^{48} + 706q^{50} - 348q^{51} - 320q^{52} + 268q^{53} - 54q^{54} + 472q^{55} - 576q^{57} + 48q^{58} - 616q^{59} + 132q^{60} + 16q^{61} - 304q^{62} + 118q^{64} + 1740q^{65} - 228q^{66} - 144q^{67} - 940q^{68} - 84q^{69} + 988q^{71} - 162q^{72} + 104q^{73} - 56q^{74} - 438q^{75} - 1136q^{76} - 612q^{78} - 944q^{79} - 828q^{80} + 162q^{81} - 856q^{82} + 1016q^{83} - 100q^{85} - 1120q^{86} - 888q^{87} - 876q^{88} - 388q^{89} + 432q^{90} - 364q^{92} + 312q^{93} + 904q^{94} + 1304q^{95} - 54q^{96} + 488q^{97} - 180q^{99} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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