LMFDB - Embedded newform 961.4.a.d.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [961,4,Mod(1,961)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(961, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("961.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 961 = 31^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	961.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$56.7008355155$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.27702880.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 134x^{2} + 4120 $ x^4 - 134*x^2 + 4120
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-9.28490$ of defining polynomial
Character	$\chi$	$=$	961.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+3.70156 q^{2} -9.28490 q^{3} +5.70156 q^{4} -16.1047 q^{5} -34.3686 q^{6} -1.29844 q^{7} -8.50781 q^{8} +59.2094 q^{9} -59.6125 q^{10} +34.3686 q^{11} -52.9384 q^{12} +78.0222 q^{13} -4.80625 q^{14} +149.530 q^{15} -77.1047 q^{16} -38.1116 q^{17} +219.167 q^{18} -24.2828 q^{19} -91.8219 q^{20} +12.0559 q^{21} +127.218 q^{22} +69.7093 q^{23} +78.9942 q^{24} +134.361 q^{25} +288.804 q^{26} -299.061 q^{27} -7.40312 q^{28} +94.6480 q^{29} +553.496 q^{30} -217.345 q^{32} -319.109 q^{33} -141.072 q^{34} +20.9109 q^{35} +337.586 q^{36} +201.497 q^{37} -89.8844 q^{38} -724.428 q^{39} +137.016 q^{40} +93.4141 q^{41} +44.6255 q^{42} -449.418 q^{43} +195.955 q^{44} -953.548 q^{45} +258.033 q^{46} +271.822 q^{47} +715.909 q^{48} -341.314 q^{49} +497.345 q^{50} +353.862 q^{51} +444.848 q^{52} +305.430 q^{53} -1106.99 q^{54} -553.496 q^{55} +11.0469 q^{56} +225.464 q^{57} +350.345 q^{58} -577.455 q^{59} +852.557 q^{60} -637.350 q^{61} -76.8797 q^{63} -187.680 q^{64} -1256.52 q^{65} -1181.20 q^{66} +693.025 q^{67} -217.296 q^{68} -647.244 q^{69} +77.4031 q^{70} -274.711 q^{71} -503.742 q^{72} -652.177 q^{73} +745.853 q^{74} -1247.53 q^{75} -138.450 q^{76} -44.6255 q^{77} -2681.52 q^{78} +788.680 q^{79} +1241.75 q^{80} +1178.10 q^{81} +345.778 q^{82} -710.657 q^{83} +68.7373 q^{84} +613.775 q^{85} -1663.55 q^{86} -878.797 q^{87} -292.402 q^{88} +387.050 q^{89} -3529.62 q^{90} -101.307 q^{91} +397.452 q^{92} +1006.17 q^{94} +391.067 q^{95} +2018.03 q^{96} -510.795 q^{97} -1263.40 q^{98} +2034.95 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 10 q^{4} - 26 q^{5} - 18 q^{7} + 30 q^{8} + 160 q^{9} - 136 q^{10} + 32 q^{14} - 270 q^{16} + 326 q^{18} - 238 q^{19} - 188 q^{20} + 38 q^{25} - 4 q^{28} - 498 q^{32} - 380 q^{33} - 6 q^{35}+ \cdots - 994 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 + 10 * q^4 - 26 * q^5 - 18 * q^7 + 30 * q^8 + 160 * q^9 - 136 * q^10 + 32 * q^14 - 270 * q^16 + 326 * q^18 - 238 * q^19 - 188 * q^20 + 38 * q^25 - 4 * q^28 - 498 * q^32 - 380 * q^33 - 6 * q^35 + 646 * q^36 + 332 * q^38 - 1028 * q^39 + 420 * q^40 + 1078 * q^41 - 1778 * q^45 + 908 * q^47 - 1250 * q^49 + 1618 * q^50 - 736 * q^51 - 340 * q^56 - 1042 * q^59 - 474 * q^63 + 722 * q^64 - 3060 * q^66 + 928 * q^67 + 1048 * q^69 + 284 * q^70 + 630 * q^71 - 30 * q^72 - 144 * q^76 - 6500 * q^78 + 2124 * q^80 + 640 * q^81 - 1716 * q^82 - 5180 * q^87 - 7408 * q^90 + 1028 * q^94 + 194 * q^95 - 1262 * q^97 - 994 * q^98

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 $

$2$	3.70156	1.30870	0.654350	−	0.756192i	$-0.272942\pi$
$2$	3.70156	1.30870	0.654350	+	0.756192i	$0.272942\pi$
$3$	−9.28490	−1.78688	−0.893440	−	0.449183i	$-0.851715\pi$
$3$	−9.28490	−1.78688	−0.893440	+	0.449183i	$0.851715\pi$
$4$	5.70156	0.712695
$5$	−16.1047	−1.44045	−0.720223	−	0.693742i	$-0.755961\pi$
$5$	−16.1047	−1.44045	−0.720223	+	0.693742i	$0.755961\pi$
$6$	−34.3686	−2.33849
$7$	−1.29844	−0.0701091	−0.0350545	−	0.999385i	$-0.511160\pi$
$7$	−1.29844	−0.0701091	−0.0350545	+	0.999385i	$0.511160\pi$
$8$	−8.50781	−0.375996
$9$	59.2094	2.19294
$10$	−59.6125	−1.88511
$11$	34.3686	0.942048	0.471024	−	0.882120i	$-0.343884\pi$
$11$	34.3686	0.942048	0.471024	+	0.882120i	$0.343884\pi$
$12$	−52.9384	−1.27350
$13$	78.0222	1.66457	0.832287	−	0.554345i	$-0.187031\pi$
$13$	78.0222	1.66457	0.832287	+	0.554345i	$0.187031\pi$
$14$	−4.80625	−0.0917517
$15$	149.530	2.57391
$16$	−77.1047	−1.20476
$17$	−38.1116	−0.543731	−0.271865	−	0.962335i	$-0.587641\pi$
$17$	−38.1116	−0.543731	−0.271865	+	0.962335i	$0.587641\pi$
$18$	219.167	2.86990
$19$	−24.2828	−0.293203	−0.146602	−	0.989196i	$-0.546834\pi$
$19$	−24.2828	−0.293203	−0.146602	+	0.989196i	$0.546834\pi$
$20$	−91.8219	−1.02660
$21$	12.0559	0.125276
$22$	127.218	1.23286
$23$	69.7093	0.631973	0.315987	−	0.948764i	$-0.397664\pi$
$23$	69.7093	0.631973	0.315987	+	0.948764i	$0.397664\pi$
$24$	78.9942	0.671859
$25$	134.361	1.07489
$26$	288.804	2.17843
$27$	−299.061	−2.13164
$28$	−7.40312	−0.0499664
$29$	94.6480	0.606058	0.303029	−	0.952981i	$-0.402002\pi$
$29$	94.6480	0.606058	0.303029	+	0.952981i	$0.402002\pi$
$30$	553.496	3.36847
$31$	0	0
$31$	0	0
$32$	−217.345	−1.20067
$33$	−319.109	−1.68333
$34$	−141.072	−0.711580
$35$	20.9109	0.100988
$36$	337.586	1.56290
$37$	201.497	0.895294	0.447647	−	0.894210i	$-0.352262\pi$
$37$	201.497	0.895294	0.447647	+	0.894210i	$0.352262\pi$
$38$	−89.8844	−0.383715
$39$	−724.428	−2.97439
$40$	137.016	0.541602
$41$	93.4141	0.355825	0.177913	−	0.984046i	$-0.443066\pi$
$41$	93.4141	0.355825	0.177913	+	0.984046i	$0.443066\pi$
$42$	44.6255	0.163949
$43$	−449.418	−1.59385	−0.796926	−	0.604077i	$-0.793542\pi$
$43$	−449.418	−1.59385	−0.796926	+	0.604077i	$0.793542\pi$
$44$	195.955	0.671393
$45$	−953.548	−3.15881
$46$	258.033	0.827063
$47$	271.822	0.843602	0.421801	−	0.906688i	$-0.361398\pi$
$47$	271.822	0.843602	0.421801	+	0.906688i	$0.361398\pi$
$48$	715.909	2.15276
$49$	−341.314	−0.995085
$50$	497.345	1.40670
$51$	353.862	0.971581
$52$	444.848	1.18633
$53$	305.430	0.791585	0.395792	−	0.918340i	$-0.370470\pi$
$53$	305.430	0.791585	0.395792	+	0.918340i	$0.370470\pi$
$54$	−1106.99	−2.78968
$55$	−553.496	−1.35697
$56$	11.0469	0.0263607
$57$	225.464	0.523919
$58$	350.345	0.793148
$59$	−577.455	−1.27421	−0.637103	−	0.770778i	$-0.719868\pi$
$59$	−577.455	−1.27421	−0.637103	+	0.770778i	$0.719868\pi$
$60$	852.557	1.83441
$61$	−637.350	−1.33778	−0.668888	−	0.743363i	$-0.733229\pi$
$61$	−637.350	−1.33778	−0.668888	+	0.743363i	$0.733229\pi$
$62$	0	0
$63$	−76.8797	−0.153745
$64$	−187.680	−0.366562
$65$	−1256.52	−2.39773
$66$	−1181.20	−2.20297
$67$	693.025	1.26368	0.631839	−	0.775099i	$-0.282300\pi$
$67$	693.025	1.26368	0.631839	+	0.775099i	$0.282300\pi$
$68$	−217.296	−0.387514
$69$	−647.244	−1.12926
$70$	77.4031	0.132163
$71$	−274.711	−0.459186	−0.229593	−	0.973287i	$-0.573739\pi$
$71$	−274.711	−0.459186	−0.229593	+	0.973287i	$0.573739\pi$
$72$	−503.742	−0.824536
$73$	−652.177	−1.04564	−0.522819	−	0.852444i	$-0.675120\pi$
$73$	−652.177	−1.04564	−0.522819	+	0.852444i	$0.675120\pi$
$74$	745.853	1.17167
$75$	−1247.53	−1.92069
$76$	−138.450	−0.208965
$77$	−44.6255	−0.0660461
$78$	−2681.52	−3.89259
$79$	788.680	1.12321	0.561604	−	0.827406i	$-0.310185\pi$
$79$	788.680	1.12321	0.561604	+	0.827406i	$0.310185\pi$
$80$	1241.75	1.73539
$81$	1178.10	1.61604
$82$	345.778	0.465668
$83$	−710.657	−0.939817	−0.469908	−	0.882715i	$-0.655713\pi$
$83$	−710.657	−0.939817	−0.469908	+	0.882715i	$0.655713\pi$
$84$	68.7373	0.0892839
$85$	613.775	0.783215
$86$	−1663.55	−2.08587
$87$	−878.797	−1.08295
$88$	−292.402	−0.354206
$89$	387.050	0.460980	0.230490	−	0.973075i	$-0.425967\pi$
$89$	387.050	0.460980	0.230490	+	0.973075i	$0.425967\pi$
$90$	−3529.62	−4.13394
$91$	−101.307	−0.116702
$92$	397.452	0.450404
$93$	0	0
$94$	1006.17	1.10402
$95$	391.067	0.422344
$96$	2018.03	2.14546
$97$	−510.795	−0.534674	−0.267337	−	0.963603i	$-0.586144\pi$
$97$	−510.795	−0.534674	−0.267337	+	0.963603i	$0.586144\pi$
$98$	−1263.40	−1.30227
$99$	2034.95	2.06586
$100$	766.067	0.766067
$101$	197.383	0.194459	0.0972293	−	0.995262i	$-0.469002\pi$
$101$	197.383	0.194459	0.0972293	+	0.995262i	$0.469002\pi$
$102$	1309.84	1.27151
$103$	−1392.70	−1.33230	−0.666149	−	0.745819i	$-0.732058\pi$
$103$	−1392.70	−1.33230	−0.666149	+	0.745819i	$0.732058\pi$
$104$	−663.798	−0.625872
$105$	−194.156	−0.180454
$106$	1130.57	1.03595
$107$	−1706.69	−1.54198	−0.770990	−	0.636847i	$-0.780238\pi$
$107$	−1706.69	−1.54198	−0.770990	+	0.636847i	$0.780238\pi$
$108$	−1705.11	−1.51921
$109$	−202.264	−0.177738	−0.0888688	−	0.996043i	$-0.528325\pi$
$109$	−202.264	−0.177738	−0.0888688	+	0.996043i	$0.528325\pi$
$110$	−2048.80	−1.77587
$111$	−1870.88	−1.59978
$112$	100.116	0.0844646
$113$	−477.770	−0.397742	−0.198871	−	0.980026i	$-0.563727\pi$
$113$	−477.770	−0.397742	−0.198871	+	0.980026i	$0.563727\pi$
$114$	834.567	0.685653
$115$	−1122.65	−0.910324
$116$	539.641	0.431935
$117$	4619.64	3.65031
$118$	−2137.48	−1.66755
$119$	49.4855	0.0381204
$120$	−1272.18	−0.967777
$121$	−149.797	−0.112545
$122$	−2359.19	−1.75075
$123$	−867.340	−0.635817
$124$	0	0
$125$	−150.755	−0.107871
$126$	−284.575	−0.201206
$127$	12.7378	0.00889997	0.00444998	−	0.999990i	$-0.498584\pi$
$127$	12.7378	0.00889997	0.00444998	+	0.999990i	$0.498584\pi$
$128$	1044.05	0.720955
$129$	4172.80	2.84802
$130$	−4651.10	−3.13791
$131$	1225.97	0.817659	0.408830	−	0.912611i	$-0.365937\pi$
$131$	1225.97	0.817659	0.408830	+	0.912611i	$0.365937\pi$
$132$	−1819.42	−1.19970
$133$	31.5297	0.0205562
$134$	2565.27	1.65378
$135$	4816.28	3.07051
$136$	324.246	0.204440
$137$	1030.38	0.642562	0.321281	−	0.946984i	$-0.395887\pi$
$137$	1030.38	0.642562	0.321281	+	0.946984i	$0.395887\pi$
$138$	−2395.81	−1.47786
$139$	923.281	0.563394	0.281697	−	0.959503i	$-0.409103\pi$
$139$	923.281	0.563394	0.281697	+	0.959503i	$0.409103\pi$
$140$	119.225	0.0719739
$141$	−2523.84	−1.50742
$142$	−1016.86	−0.600936
$143$	2681.52	1.56811
$144$	−4565.32	−2.64197
$145$	−1524.28	−0.872995
$146$	−2414.07	−1.36843
$147$	3169.07	1.77810
$148$	1148.85	0.638072
$149$	1884.01	1.03586	0.517932	−	0.855422i	$-0.326702\pi$
$149$	1884.01	1.03586	0.517932	+	0.855422i	$0.326702\pi$
$150$	−4617.80	−2.51361
$151$	1680.81	0.905846	0.452923	−	0.891550i	$-0.350381\pi$
$151$	1680.81	0.905846	0.452923	+	0.891550i	$0.350381\pi$
$152$	206.594	0.110243
$153$	−2256.56	−1.19237
$154$	−165.184	−0.0864346
$155$	0	0
$156$	−4130.37	−2.11984
$157$	−879.014	−0.446834	−0.223417	−	0.974723i	$-0.571721\pi$
$157$	−879.014	−0.446834	−0.223417	+	0.974723i	$0.571721\pi$
$158$	2919.35	1.46994
$159$	−2835.88	−1.41447
$160$	3500.28	1.72951
$161$	−90.5132	−0.0443071
$162$	4360.80	2.11492
$163$	755.879	0.363221	0.181611	−	0.983371i	$-0.441869\pi$
$163$	755.879	0.363221	0.181611	+	0.983371i	$0.441869\pi$
$164$	532.606	0.253595
$165$	5139.16	2.42474
$166$	−2630.54	−1.22994
$167$	−1116.13	−0.517179	−0.258589	−	0.965987i	$-0.583258\pi$
$167$	−1116.13	−0.517179	−0.258589	+	0.965987i	$0.583258\pi$
$168$	−102.569	−0.0471034
$169$	3890.46	1.77081
$170$	2271.93	1.02499
$171$	−1437.77	−0.642977
$172$	−2562.39	−1.13593
$173$	433.019	0.190300	0.0951498	−	0.995463i	$-0.469667\pi$
$173$	433.019	0.190300	0.0951498	+	0.995463i	$0.469667\pi$
$174$	−3252.92	−1.41726
$175$	−174.459	−0.0753593
$176$	−2649.98	−1.13494
$177$	5361.61	2.27685
$178$	1432.69	0.603284
$179$	−2424.52	−1.01239	−0.506193	−	0.862420i	$-0.668948\pi$
$179$	−2424.52	−1.01239	−0.506193	+	0.862420i	$0.668948\pi$
$180$	−5436.72	−2.25127
$181$	2583.58	1.06097	0.530486	−	0.847693i	$-0.322009\pi$
$181$	2583.58	1.06097	0.530486	+	0.847693i	$0.322009\pi$
$182$	−374.994	−0.152727
$183$	5917.73	2.39045
$184$	−593.073	−0.237619
$185$	−3245.04	−1.28962
$186$	0	0
$187$	−1309.84	−0.512221
$188$	1549.81	0.601231
$189$	388.312	0.149447
$190$	1447.56	0.552721
$191$	−4629.12	−1.75367	−0.876837	−	0.480788i	$-0.840351\pi$
$191$	−4629.12	−1.75367	−0.876837	+	0.480788i	$0.840351\pi$
$192$	1742.59	0.655002
$193$	2407.24	0.897807	0.448904	−	0.893580i	$-0.351815\pi$
$193$	2407.24	0.897807	0.448904	+	0.893580i	$0.351815\pi$
$194$	−1890.74	−0.699728
$195$	11666.7	4.28446
$196$	−1946.02	−0.709192
$197$	−3159.72	−1.14275	−0.571373	−	0.820690i	$-0.693589\pi$
$197$	−3159.72	−1.14275	−0.571373	+	0.820690i	$0.693589\pi$
$198$	7532.48	2.70358
$199$	−1922.12	−0.684701	−0.342350	−	0.939572i	$-0.611223\pi$
$199$	−1922.12	−0.684701	−0.342350	+	0.939572i	$0.611223\pi$
$200$	−1143.12	−0.404153
$201$	−6434.67	−2.25804
$202$	730.625	0.254488
$203$	−122.895	−0.0424902
$204$	2017.57	0.692441
$205$	−1504.40	−0.512547
$206$	−5155.16	−1.74358
$207$	4127.44	1.38588
$208$	−6015.87	−2.00541
$209$	−834.567	−0.276212
$210$	−718.680	−0.236160
$211$	−346.024	−0.112897	−0.0564484	−	0.998406i	$-0.517978\pi$
$211$	−346.024	−0.112897	−0.0564484	+	0.998406i	$0.517978\pi$
$212$	1741.43	0.564159
$213$	2550.66	0.820510
$214$	−6317.42	−2.01799
$215$	7237.74	2.29586
$216$	2544.35	0.801487
$217$	0	0
$218$	−748.693	−0.232605
$219$	6055.40	1.86843
$220$	−3155.79	−0.967107
$221$	−2973.55	−0.905080
$222$	−6925.17	−2.09364
$223$	4651.88	1.39692	0.698459	−	0.715650i	$-0.253869\pi$
$223$	4651.88	1.39692	0.698459	+	0.715650i	$0.253869\pi$
$224$	282.209	0.0841782
$225$	7955.43	2.35716
$226$	−1768.50	−0.520525
$227$	−2010.54	−0.587859	−0.293930	−	0.955827i	$-0.594963\pi$
$227$	−2010.54	−0.587859	−0.293930	+	0.955827i	$0.594963\pi$
$228$	1285.49	0.373394
$229$	−5708.95	−1.64742	−0.823708	−	0.567015i	$-0.808098\pi$
$229$	−5708.95	−1.64742	−0.823708	+	0.567015i	$0.808098\pi$
$230$	−4155.54	−1.19134
$231$	414.344	0.118017
$232$	−805.247	−0.227875
$233$	−5766.15	−1.62126	−0.810628	−	0.585561i	$-0.800874\pi$
$233$	−5766.15	−1.62126	−0.810628	+	0.585561i	$0.800874\pi$
$234$	17099.9	4.77716
$235$	−4377.61	−1.21516
$236$	−3292.39	−0.908121
$237$	−7322.81	−2.00704
$238$	183.174	0.0498882
$239$	−4524.60	−1.22457	−0.612285	−	0.790637i	$-0.709750\pi$
$239$	−4524.60	−1.22457	−0.612285	+	0.790637i	$0.709750\pi$
$240$	−11529.5	−3.10094
$241$	−5866.78	−1.56810	−0.784051	−	0.620697i	$-0.786850\pi$
$241$	−5866.78	−1.56810	−0.784051	+	0.620697i	$0.786850\pi$
$242$	−554.483	−0.147287
$243$	−2863.87	−0.756038
$244$	−3633.89	−0.953427
$245$	5496.76	1.43337
$246$	−3210.51	−0.832093
$247$	−1894.60	−0.488058
$248$	0	0
$249$	6598.38	1.67934
$250$	−558.028	−0.141171
$251$	−3074.09	−0.773046	−0.386523	−	0.922280i	$-0.626324\pi$
$251$	−3074.09	−0.773046	−0.386523	+	0.922280i	$0.626324\pi$
$252$	−438.334	−0.109573
$253$	2395.81	0.595350
$254$	47.1497	0.0116474
$255$	−5698.84	−1.39951
$256$	5366.07	1.31008
$257$	−3915.95	−0.950469	−0.475234	−	0.879859i	$-0.657637\pi$
$257$	−3915.95	−0.950469	−0.475234	+	0.879859i	$0.657637\pi$
$258$	15445.9	3.72721
$259$	−261.631	−0.0627682
$260$	−7164.14	−1.70885
$261$	5604.05	1.32905
$262$	4538.00	1.07007
$263$	1638.42	0.384142	0.192071	−	0.981381i	$-0.438480\pi$
$263$	1638.42	0.384142	0.192071	+	0.981381i	$0.438480\pi$
$264$	2714.92	0.632924
$265$	−4918.85	−1.14024
$266$	116.709	0.0269019
$267$	−3593.72	−0.823715
$268$	3951.32	0.900618
$269$	4721.44	1.07015	0.535077	−	0.844803i	$-0.320283\pi$
$269$	4721.44	1.07015	0.535077	+	0.844803i	$0.320283\pi$
$270$	17827.8	4.01838
$271$	−4324.14	−0.969272	−0.484636	−	0.874716i	$-0.661048\pi$
$271$	−4324.14	−0.969272	−0.484636	+	0.874716i	$0.661048\pi$
$272$	2938.58	0.655065
$273$	940.625	0.208532
$274$	3814.00	0.840921
$275$	4617.80	1.01260
$276$	−3690.30	−0.804819
$277$	7569.51	1.64191	0.820953	−	0.570996i	$-0.193443\pi$
$277$	7569.51	1.64191	0.820953	+	0.570996i	$0.193443\pi$
$278$	3417.58	0.737313
$279$	0	0
$280$	−177.906	−0.0379712
$281$	−4570.05	−0.970201	−0.485101	−	0.874458i	$-0.661217\pi$
$281$	−4570.05	−0.970201	−0.485101	+	0.874458i	$0.661217\pi$
$282$	−9342.15	−1.97275
$283$	1958.83	0.411450	0.205725	−	0.978610i	$-0.434045\pi$
$283$	1958.83	0.411450	0.205725	+	0.978610i	$0.434045\pi$
$284$	−1566.28	−0.327259
$285$	−3631.02	−0.754677
$286$	9925.80	2.05218
$287$	−121.292	−0.0249466
$288$	−12868.9	−2.63301
$289$	−3460.51	−0.704357
$290$	−5642.20	−1.14249
$291$	4742.68	0.955399
$292$	−3718.43	−0.745221
$293$	−7389.79	−1.47343	−0.736717	−	0.676201i	$-0.763625\pi$
$293$	−7389.79	−1.47343	−0.736717	+	0.676201i	$0.763625\pi$
$294$	11730.5	2.32700
$295$	9299.73	1.83543
$296$	−1714.30	−0.336627
$297$	−10278.3	−2.00811
$298$	6973.77	1.35564
$299$	5438.87	1.05197
$300$	−7112.86	−1.36887
$301$	583.542	0.111743
$302$	6221.64	1.18548
$303$	−1832.68	−0.347474
$304$	1872.32	0.353240
$305$	10264.3	1.92700
$306$	−8352.81	−1.56045
$307$	−5905.23	−1.09782	−0.548908	−	0.835883i	$-0.684956\pi$
$307$	−5905.23	−1.09782	−0.548908	+	0.835883i	$0.684956\pi$
$308$	−254.435	−0.0470708
$309$	12931.1	2.38066
$310$	0	0
$311$	−1671.28	−0.304725	−0.152363	−	0.988325i	$-0.548688\pi$
$311$	−1671.28	−0.304725	−0.152363	+	0.988325i	$0.548688\pi$
$312$	6163.30	1.11836
$313$	5146.05	0.929304	0.464652	−	0.885493i	$-0.346179\pi$
$313$	5146.05	0.929304	0.464652	+	0.885493i	$0.346179\pi$
$314$	−3253.72	−0.584772
$315$	1238.12	0.221461
$316$	4496.71	0.800505
$317$	−8464.03	−1.49964	−0.749822	−	0.661639i	$-0.769861\pi$
$317$	−8464.03	−1.49964	−0.749822	+	0.661639i	$0.769861\pi$
$318$	−10497.2	−1.85111
$319$	3252.92	0.570936
$320$	3022.52	0.528013
$321$	15846.4	2.75533
$322$	−335.040	−0.0579846
$323$	925.457	0.159424
$324$	6716.99	1.15175
$325$	10483.1	1.78923
$326$	2797.93	0.475347
$327$	1878.00	0.317596
$328$	−794.749	−0.133789
$329$	−352.944	−0.0591441
$330$	19022.9	3.17326
$331$	2152.98	0.357518	0.178759	−	0.983893i	$-0.442792\pi$
$331$	2152.98	0.357518	0.178759	+	0.983893i	$0.442792\pi$
$332$	−4051.86	−0.669803
$333$	11930.5	1.96333
$334$	−4131.43	−0.676832
$335$	−11160.9	−1.82026
$336$	−929.564	−0.150928
$337$	−3973.05	−0.642213	−0.321107	−	0.947043i	$-0.604055\pi$
$337$	−3973.05	−0.642213	−0.321107	+	0.947043i	$0.604055\pi$
$338$	14400.8	2.31745
$339$	4436.05	0.710717
$340$	3499.48	0.558194
$341$	0	0
$342$	−5322.00	−0.841464
$343$	888.539	0.139874
$344$	3823.56	0.599281
$345$	10423.7	1.62664
$346$	1602.85	0.249045
$347$	−9040.79	−1.39866	−0.699330	−	0.714799i	$-0.746518\pi$
$347$	−9040.79	−1.39866	−0.699330	+	0.714799i	$0.746518\pi$
$348$	−5010.52	−0.771816
$349$	3333.06	0.511216	0.255608	−	0.966780i	$-0.417724\pi$
$349$	3333.06	0.511216	0.255608	+	0.966780i	$0.417724\pi$
$350$	−645.772	−0.0986228
$351$	−23333.4	−3.54827
$352$	−7469.86	−1.13109
$353$	−5968.19	−0.899871	−0.449936	−	0.893061i	$-0.648553\pi$
$353$	−5968.19	−0.899871	−0.449936	+	0.893061i	$0.648553\pi$
$354$	19846.3	2.97972
$355$	4424.13	0.661433
$356$	2206.79	0.328538
$357$	−459.468	−0.0681167
$358$	−8974.51	−1.32491
$359$	1481.94	0.217865	0.108933	−	0.994049i	$-0.465257\pi$
$359$	1481.94	0.217865	0.108933	+	0.994049i	$0.465257\pi$
$360$	8112.61	1.18770
$361$	−6269.34	−0.914032
$362$	9563.29	1.38850
$363$	1390.85	0.201104
$364$	−577.608	−0.0831727
$365$	10503.1	1.50619
$366$	21904.9	3.12838
$367$	5962.73	0.848098	0.424049	−	0.905639i	$-0.360608\pi$
$367$	5962.73	0.848098	0.424049	+	0.905639i	$0.360608\pi$
$368$	−5374.91	−0.761377
$369$	5530.99	0.780303
$370$	−12011.7	−1.68773
$371$	−396.582	−0.0554973
$372$	0	0
$373$	11549.8	1.60328	0.801640	−	0.597807i	$-0.203961\pi$
$373$	11549.8	1.60328	0.801640	+	0.597807i	$0.203961\pi$
$374$	−4848.47	−0.670343
$375$	1399.74	0.192753
$376$	−2312.61	−0.317191
$377$	7384.64	1.00883
$378$	1437.36	0.195582
$379$	−1000.65	−0.135620	−0.0678099	−	0.997698i	$-0.521601\pi$
$379$	−1000.65	−0.135620	−0.0678099	+	0.997698i	$0.521601\pi$
$380$	2229.69	0.301002
$381$	−118.269	−0.0159032
$382$	−17135.0	−2.29503
$383$	−6639.34	−0.885782	−0.442891	−	0.896575i	$-0.646047\pi$
$383$	−6639.34	−0.885782	−0.442891	+	0.896575i	$0.646047\pi$
$384$	−9693.94	−1.28826
$385$	718.680	0.0951359
$386$	8910.54	1.17496
$387$	−26609.8	−3.49522
$388$	−2912.33	−0.381060
$389$	1496.59	0.195065	0.0975326	−	0.995232i	$-0.468905\pi$
$389$	1496.59	0.195065	0.0975326	+	0.995232i	$0.468905\pi$
$390$	43185.0	5.60707
$391$	−2656.73	−0.343623
$392$	2903.84	0.374148
$393$	−11383.0	−1.46106
$394$	−11695.9	−1.49551
$395$	−12701.4	−1.61792
$396$	11602.4	1.47233
$397$	14816.9	1.87314	0.936572	−	0.350475i	$-0.113980\pi$
$397$	14816.9	1.87314	0.936572	+	0.350475i	$0.113980\pi$
$398$	−7114.84	−0.896068
$399$	−292.750	−0.0367315
$400$	−10359.9	−1.29498
$401$	−14142.0	−1.76114	−0.880572	−	0.473912i	$-0.842841\pi$
$401$	−14142.0	−1.76114	−0.880572	+	0.473912i	$0.842841\pi$
$402$	−23818.3	−2.95510
$403$	0	0
$404$	1125.39	0.138590
$405$	−18972.9	−2.32783
$406$	−454.902	−0.0556069
$407$	6925.17	0.843410
$408$	−3010.59	−0.365310
$409$	−1156.80	−0.139853	−0.0699266	−	0.997552i	$-0.522277\pi$
$409$	−1156.80	−0.139853	−0.0699266	+	0.997552i	$0.522277\pi$
$410$	−5568.65	−0.670770
$411$	−9566.95	−1.14818
$412$	−7940.56	−0.949522
$413$	749.789	0.0893334
$414$	15278.0	1.81370
$415$	11444.9	1.35376
$416$	−16957.8	−1.99861
$417$	−8572.58	−1.00672
$418$	−3089.20	−0.361478
$419$	−15791.8	−1.84124	−0.920618	−	0.390463i	$-0.872315\pi$
$419$	−15791.8	−1.84124	−0.920618	+	0.390463i	$0.872315\pi$
$420$	−1106.99	−0.128609
$421$	−4921.06	−0.569686	−0.284843	−	0.958574i	$-0.591941\pi$
$421$	−4921.06	−0.569686	−0.284843	+	0.958574i	$0.591941\pi$
$422$	−1280.83	−0.147748
$423$	16094.4	1.84997
$424$	−2598.54	−0.297632
$425$	−5120.71	−0.584449
$426$	9441.44	1.07380
$427$	827.560	0.0937902
$428$	−9730.79	−1.09896
$429$	−24897.6	−2.80202
$430$	26790.9	3.00459
$431$	4918.80	0.549722	0.274861	−	0.961484i	$-0.411368\pi$
$431$	4918.80	0.549722	0.274861	+	0.961484i	$0.411368\pi$
$432$	23059.0	2.56812
$433$	5726.51	0.635562	0.317781	−	0.948164i	$-0.397062\pi$
$433$	5726.51	0.635562	0.317781	+	0.948164i	$0.397062\pi$
$434$	0	0
$435$	14152.7	1.55994
$436$	−1153.22	−0.126673
$437$	−1692.74	−0.185297
$438$	22414.4	2.44521
$439$	−2184.41	−0.237485	−0.118743	−	0.992925i	$-0.537886\pi$
$439$	−2184.41	−0.237485	−0.118743	+	0.992925i	$0.537886\pi$
$440$	4709.04	0.510215
$441$	−20209.0	−2.18216
$442$	−11006.8	−1.18448
$443$	6698.91	0.718453	0.359227	−	0.933250i	$-0.383040\pi$
$443$	6698.91	0.718453	0.359227	+	0.933250i	$0.383040\pi$
$444$	−10666.9	−1.14016
$445$	−6233.32	−0.664017
$446$	17219.2	1.82815
$447$	−17492.8	−1.85097
$448$	243.690	0.0256993
$449$	13246.5	1.39229	0.696147	−	0.717899i	$-0.254896\pi$
$449$	13246.5	1.39229	0.696147	+	0.717899i	$0.254896\pi$
$450$	29447.5	3.08482
$451$	3210.51	0.335204
$452$	−2724.04	−0.283469
$453$	−15606.2	−1.61864
$454$	−7442.13	−0.769331
$455$	1631.52	0.168103
$456$	−1918.20	−0.196991
$457$	−1004.60	−0.102829	−0.0514147	−	0.998677i	$-0.516373\pi$
$457$	−1004.60	−0.102829	−0.0514147	+	0.998677i	$0.516373\pi$
$458$	−21132.0	−2.15597
$459$	11397.7	1.15904
$460$	−6400.84	−0.648784
$461$	6851.60	0.692215	0.346107	−	0.938195i	$-0.387503\pi$
$461$	6851.60	0.692215	0.346107	+	0.938195i	$0.387503\pi$
$462$	1533.72	0.154448
$463$	−16120.9	−1.61815	−0.809074	−	0.587707i	$-0.800031\pi$
$463$	−16120.9	−1.61815	−0.809074	+	0.587707i	$0.800031\pi$
$464$	−7297.80	−0.730155
$465$	0	0
$466$	−21343.7	−2.12174
$467$	−15545.3	−1.54037	−0.770184	−	0.637821i	$-0.779836\pi$
$467$	−15545.3	−1.54037	−0.770184	+	0.637821i	$0.779836\pi$
$468$	26339.2	2.60156
$469$	−899.850	−0.0885953
$470$	−16204.0	−1.59028
$471$	8161.56	0.798439
$472$	4912.87	0.479096
$473$	−15445.9	−1.50149
$474$	−27105.8	−2.62661
$475$	−3262.66	−0.315160
$476$	282.145	0.0271683
$477$	18084.3	1.73590
$478$	−16748.1	−1.60259
$479$	−9950.26	−0.949142	−0.474571	−	0.880217i	$-0.657397\pi$
$479$	−9950.26	−0.949142	−0.474571	+	0.880217i	$0.657397\pi$
$480$	−32499.7	−3.09042
$481$	15721.2	1.49028
$482$	−21716.2	−2.05217
$483$	840.406	0.0791714
$484$	−854.076	−0.0802100
$485$	8226.20	0.770170
$486$	−10600.8	−0.989427
$487$	15535.8	1.44557	0.722785	−	0.691073i	$-0.242862\pi$
$487$	15535.8	1.44557	0.722785	+	0.691073i	$0.242862\pi$
$488$	5422.46	0.502998
$489$	−7018.27	−0.649033
$490$	20346.6	1.87585
$491$	−10686.8	−0.982259	−0.491130	−	0.871086i	$-0.663416\pi$
$491$	−10686.8	−0.982259	−0.491130	+	0.871086i	$0.663416\pi$
$492$	−4945.20	−0.453144
$493$	−3607.19	−0.329532
$494$	−7012.97	−0.638722
$495$	−32772.2	−2.97576
$496$	0	0
$497$	356.695	0.0321931
$498$	24424.3	2.19775
$499$	−515.153	−0.0462153	−0.0231076	−	0.999733i	$-0.507356\pi$
$499$	−515.153	−0.0462153	−0.0231076	+	0.999733i	$0.507356\pi$
$500$	−859.537	−0.0768794
$501$	10363.2	0.924137
$502$	−11378.9	−1.01169
$503$	−16316.5	−1.44636	−0.723178	−	0.690662i	$-0.757319\pi$
$503$	−16316.5	−1.44636	−0.723178	+	0.690662i	$0.757319\pi$
$504$	654.078	0.0578074
$505$	−3178.79	−0.280107
$506$	8868.25	0.779134
$507$	−36122.5	−3.16422
$508$	72.6253	0.00634296
$509$	13161.2	1.14609	0.573044	−	0.819525i	$-0.305762\pi$
$509$	13161.2	1.14609	0.573044	+	0.819525i	$0.305762\pi$
$510$	−21094.6	−1.83154
$511$	846.811	0.0733087
$512$	11510.4	0.993541
$513$	7262.04	0.625004
$514$	−14495.1	−1.24388
$515$	22429.0	1.91910
$516$	23791.5	2.02977
$517$	9342.15	0.794714
$518$	−968.444	−0.0821448
$519$	−4020.54	−0.340042
$520$	10690.3	0.901536
$521$	−7039.94	−0.591987	−0.295994	−	0.955190i	$-0.595651\pi$
$521$	−7039.94	−0.591987	−0.295994	+	0.955190i	$0.595651\pi$
$522$	20743.7	1.73933
$523$	2313.01	0.193386	0.0966932	−	0.995314i	$-0.469173\pi$
$523$	2313.01	0.193386	0.0966932	+	0.995314i	$0.469173\pi$
$524$	6989.94	0.582742
$525$	1619.84	0.134658
$526$	6064.72	0.502727
$527$	0	0
$528$	24604.8	2.02801
$529$	−7307.62	−0.600610
$530$	−18207.4	−1.49223
$531$	−34190.7	−2.79426
$532$	179.769	0.0146503
$533$	7288.37	0.592297
$534$	−13302.4	−1.07800
$535$	27485.7	2.22114
$536$	−5896.12	−0.475138
$537$	22511.4	1.80901
$538$	17476.7	1.40051
$539$	−11730.5	−0.937418
$540$	27460.3	2.18834
$541$	21561.2	1.71347	0.856736	−	0.515755i	$-0.172488\pi$
$541$	21561.2	1.71347	0.856736	+	0.515755i	$0.172488\pi$
$542$	−16006.1	−1.26849
$543$	−23988.3	−1.89583
$544$	8283.38	0.652844
$545$	3257.40	0.256022
$546$	3481.78	0.272906
$547$	14356.0	1.12216	0.561078	−	0.827763i	$-0.310387\pi$
$547$	14356.0	1.12216	0.561078	+	0.827763i	$0.310387\pi$
$548$	5874.76	0.457951
$549$	−37737.1	−2.93366
$550$	17093.1	1.32518
$551$	−2298.32	−0.177698
$552$	5506.63	0.424597
$553$	−1024.05	−0.0787470
$554$	28019.0	2.14876
$555$	30129.9	2.30440
$556$	5264.15	0.401528
$557$	14741.8	1.12142	0.560708	−	0.828014i	$-0.310529\pi$
$557$	14741.8	1.12142	0.560708	+	0.828014i	$0.310529\pi$
$558$	0	0
$559$	−35064.6	−2.65308
$560$	−1612.33	−0.121667
$561$	12161.8	0.915277
$562$	−16916.3	−1.26970
$563$	2748.64	0.205758	0.102879	−	0.994694i	$-0.467195\pi$
$563$	2748.64	0.205758	0.102879	+	0.994694i	$0.467195\pi$
$564$	−14389.8	−1.07433
$565$	7694.34	0.572926
$566$	7250.74	0.538465
$567$	−1529.69	−0.113299
$568$	2337.19	0.172652
$569$	26634.8	1.96237	0.981185	−	0.193072i	$-0.0618452\pi$
$569$	26634.8	1.96237	0.981185	+	0.193072i	$0.0618452\pi$
$570$	−13440.4	−0.987646
$571$	−31.9766	−0.00234357	−0.00117179	−	0.999999i	$-0.500373\pi$
$571$	−31.9766	−0.00234357	−0.00117179	+	0.999999i	$0.500373\pi$
$572$	15288.8	1.11758
$573$	42980.9	3.13360
$574$	−448.971	−0.0326476
$575$	9366.20	0.679300
$576$	−11112.4	−0.803848
$577$	−20700.9	−1.49357	−0.746784	−	0.665067i	$-0.768403\pi$
$577$	−20700.9	−1.49357	−0.746784	+	0.665067i	$0.768403\pi$
$578$	−12809.3	−0.921792
$579$	−22351.0	−1.60427
$580$	−8690.75	−0.622179
$581$	922.745	0.0658897
$582$	17555.3	1.25033
$583$	10497.2	0.745711
$584$	5548.60	0.393155
$585$	−74397.9	−5.25808
$586$	−27353.8	−1.92828
$587$	−21647.9	−1.52215	−0.761077	−	0.648661i	$-0.775329\pi$
$587$	−21647.9	−1.52215	−0.761077	+	0.648661i	$0.775329\pi$
$588$	18068.6	1.26724
$589$	0	0
$590$	34423.5	2.40202
$591$	29337.7	2.04195
$592$	−15536.4	−1.07861
$593$	−2483.54	−0.171984	−0.0859921	−	0.996296i	$-0.527406\pi$
$593$	−2483.54	−0.171984	−0.0859921	+	0.996296i	$0.527406\pi$
$594$	−38045.8	−2.62801
$595$	−796.949	−0.0549105
$596$	10741.8	0.738256
$597$	17846.7	1.22348
$598$	20132.3	1.37671
$599$	−9572.52	−0.652959	−0.326480	−	0.945204i	$-0.605862\pi$
$599$	−9572.52	−0.652959	−0.326480	+	0.945204i	$0.605862\pi$
$600$	10613.7	0.722173
$601$	−24741.3	−1.67923	−0.839615	−	0.543183i	$-0.817219\pi$
$601$	−24741.3	−1.67923	−0.839615	+	0.543183i	$0.817219\pi$
$602$	2160.02	0.146239
$603$	41033.6	2.77117
$604$	9583.26	0.645592
$605$	2412.43	0.162115
$606$	−6783.78	−0.454740
$607$	7302.60	0.488309	0.244154	−	0.969736i	$-0.421490\pi$
$607$	7302.60	0.488309	0.244154	+	0.969736i	$0.421490\pi$
$608$	5277.76	0.352042
$609$	1141.06	0.0759248
$610$	37994.0	2.52186
$611$	21208.1	1.40424
$612$	−12865.9	−0.849795
$613$	22393.1	1.47545	0.737724	−	0.675102i	$-0.235900\pi$
$613$	22393.1	1.47545	0.737724	+	0.675102i	$0.235900\pi$
$614$	−21858.6	−1.43671
$615$	13968.2	0.915860
$616$	379.666	0.0248331
$617$	4557.77	0.297389	0.148695	−	0.988883i	$-0.452493\pi$
$617$	4557.77	0.297389	0.148695	+	0.988883i	$0.452493\pi$
$618$	47865.1	3.11556
$619$	7282.58	0.472878	0.236439	−	0.971646i	$-0.424020\pi$
$619$	7282.58	0.472878	0.236439	+	0.971646i	$0.424020\pi$
$620$	0	0
$621$	−20847.3	−1.34714
$622$	−6186.35	−0.398794
$623$	−502.560	−0.0323189
$624$	55856.8	3.58343
$625$	−14367.3	−0.919504
$626$	19048.4	1.21618
$627$	7748.87	0.493557
$628$	−5011.75	−0.318457
$629$	−7679.37	−0.486799
$630$	4582.99	0.289827
$631$	−19578.5	−1.23520	−0.617598	−	0.786494i	$-0.711894\pi$
$631$	−19578.5	−1.23520	−0.617598	+	0.786494i	$0.711894\pi$
$632$	−6709.94	−0.422321
$633$	3212.79	0.201733
$634$	−31330.1	−1.96258
$635$	−205.138	−0.0128199
$636$	−16169.0	−1.00808
$637$	−26630.1	−1.65639
$638$	12040.9	0.747184
$639$	−16265.5	−1.00697
$640$	−16814.2	−1.03850
$641$	−19476.4	−1.20011	−0.600055	−	0.799959i	$-0.704855\pi$
$641$	−19476.4	−1.20011	−0.600055	+	0.799959i	$0.704855\pi$
$642$	58656.6	3.60590
$643$	−19003.6	−1.16552	−0.582759	−	0.812645i	$-0.698027\pi$
$643$	−19003.6	−1.16552	−0.582759	+	0.812645i	$0.698027\pi$
$644$	−516.066	−0.0315774
$645$	−67201.7	−4.10242
$646$	3425.64	0.208638
$647$	445.008	0.0270403	0.0135202	−	0.999909i	$-0.495696\pi$
$647$	445.008	0.0270403	0.0135202	+	0.999909i	$0.495696\pi$
$648$	−10023.0	−0.607626
$649$	−19846.3	−1.20036
$650$	38804.0	2.34156
$651$	0	0
$652$	4309.69	0.258866
$653$	21404.0	1.28270	0.641349	−	0.767250i	$-0.278375\pi$
$653$	21404.0	1.28270	0.641349	+	0.767250i	$0.278375\pi$
$654$	6951.54	0.415637
$655$	−19743.8	−1.17780
$656$	−7202.66	−0.428684
$657$	−38615.0	−2.29302
$658$	−1306.44	−0.0774019
$659$	−8508.17	−0.502931	−0.251465	−	0.967866i	$-0.580912\pi$
$659$	−8508.17	−0.502931	−0.251465	+	0.967866i	$0.580912\pi$
$660$	29301.2	1.72810
$661$	−18003.2	−1.05937	−0.529686	−	0.848194i	$-0.677690\pi$
$661$	−18003.2	−1.05937	−0.529686	+	0.848194i	$0.677690\pi$
$662$	7969.39	0.467884
$663$	27609.1	1.61727
$664$	6046.14	0.353367
$665$	−507.776	−0.0296101
$666$	44161.5	2.56940
$667$	6597.84	0.383013
$668$	−6363.70	−0.368591
$669$	−43192.2	−2.49613
$670$	−41312.9	−2.38218
$671$	−21904.9	−1.26025
$672$	−2620.29	−0.150416
$673$	29209.0	1.67299	0.836497	−	0.547971i	$-0.184600\pi$
$673$	29209.0	1.67299	0.836497	+	0.547971i	$0.184600\pi$
$674$	−14706.5	−0.840465
$675$	−40182.1	−2.29127
$676$	22181.7	1.26204
$677$	15318.0	0.869598	0.434799	−	0.900528i	$-0.356819\pi$
$677$	15318.0	0.869598	0.434799	+	0.900528i	$0.356819\pi$
$678$	16420.3	0.930116
$679$	663.236	0.0374855
$680$	−5221.89	−0.294486
$681$	18667.6	1.05043
$682$	0	0
$683$	24503.3	1.37276	0.686379	−	0.727244i	$-0.259199\pi$
$683$	24503.3	1.37276	0.686379	+	0.727244i	$0.259199\pi$
$684$	−8197.54	−0.458247
$685$	−16593.9	−0.925577
$686$	3288.98	0.183052
$687$	53007.0	2.94373
$688$	34652.2	1.92021
$689$	23830.3	1.31765
$690$	38583.8	2.12878
$691$	4356.01	0.239812	0.119906	−	0.992785i	$-0.461741\pi$
$691$	4356.01	0.239812	0.119906	+	0.992785i	$0.461741\pi$
$692$	2468.88	0.135626
$693$	−2642.25	−0.144835
$694$	−33465.1	−1.83043
$695$	−14869.2	−0.811539
$696$	7476.64	0.407186
$697$	−3560.16	−0.193473
$698$	12337.5	0.669028
$699$	53538.1	2.89699
$700$	−994.691	−0.0537082
$701$	11272.1	0.607335	0.303667	−	0.952778i	$-0.401789\pi$
$701$	11272.1	0.607335	0.303667	+	0.952778i	$0.401789\pi$
$702$	−86369.9	−4.64362
$703$	−4892.91	−0.262503
$704$	−6450.29	−0.345319
$705$	40645.6	2.17135
$706$	−22091.6	−1.17766
$707$	−256.289	−0.0136333
$708$	30569.5	1.62270
$709$	10200.0	0.540293	0.270147	−	0.962819i	$-0.412928\pi$
$709$	10200.0	0.540293	0.270147	+	0.962819i	$0.412928\pi$
$710$	16376.2	0.865617
$711$	46697.2	2.46313
$712$	−3292.95	−0.173326
$713$	0	0
$714$	−1700.75	−0.0891443
$715$	−43185.0	−2.25878
$716$	−13823.5	−0.721523
$717$	42010.5	2.18816
$718$	5485.48	0.285120
$719$	7893.86	0.409445	0.204723	−	0.978820i	$-0.434371\pi$
$719$	7893.86	0.409445	0.204723	+	0.978820i	$0.434371\pi$
$720$	73523.0	3.80561
$721$	1808.33	0.0934061
$722$	−23206.4	−1.19619
$723$	54472.5	2.80201
$724$	14730.4	0.756150
$725$	12717.0	0.651444
$726$	5148.32	0.263184
$727$	15170.9	0.773946	0.386973	−	0.922091i	$-0.373521\pi$
$727$	15170.9	0.773946	0.386973	+	0.922091i	$0.373521\pi$
$728$	861.900	0.0438793
$729$	−5217.88	−0.265096
$730$	38877.9	1.97114
$731$	17128.0	0.866626
$732$	33740.3	1.70366
$733$	9028.93	0.454967	0.227484	−	0.973782i	$-0.426950\pi$
$733$	9028.93	0.454967	0.227484	+	0.973782i	$0.426950\pi$
$734$	22071.4	1.10991
$735$	−51036.8	−2.56125
$736$	−15151.0	−0.758794
$737$	23818.3	1.19045
$738$	20473.3	1.02118
$739$	−35356.1	−1.75994	−0.879970	−	0.475029i	$-0.842437\pi$
$739$	−35356.1	−1.75994	−0.879970	+	0.475029i	$0.842437\pi$
$740$	−18501.8	−0.919109
$741$	17591.2	0.872101
$742$	−1467.97	−0.0726293
$743$	3611.87	0.178340	0.0891700	−	0.996016i	$-0.471579\pi$
$743$	3611.87	0.178340	0.0891700	+	0.996016i	$0.471579\pi$
$744$	0	0
$745$	−30341.3	−1.49211
$746$	42752.1	2.09821
$747$	−42077.6	−2.06096
$748$	−7468.16	−0.365057
$749$	2216.03	0.108107
$750$	5181.23	0.252256
$751$	−1039.20	−0.0504938	−0.0252469	−	0.999681i	$-0.508037\pi$
$751$	−1039.20	−0.0504938	−0.0252469	+	0.999681i	$0.508037\pi$
$752$	−20958.7	−1.01634
$753$	28542.6	1.38134
$754$	27334.7	1.32025
$755$	−27069.0	−1.30482
$756$	2213.98	0.106510
$757$	7435.20	0.356984	0.178492	−	0.983941i	$-0.442878\pi$
$757$	7435.20	0.356984	0.178492	+	0.983941i	$0.442878\pi$
$758$	−3703.97	−0.177486
$759$	−22244.9	−1.06382
$760$	−3327.13	−0.158799
$761$	8641.89	0.411653	0.205827	−	0.978588i	$-0.434012\pi$
$761$	8641.89	0.411653	0.205827	+	0.978588i	$0.434012\pi$
$762$	−437.780	−0.0208125
$763$	262.627	0.0124610
$764$	−26393.2	−1.24983
$765$	36341.3	1.71754
$766$	−24575.9	−1.15922
$767$	−45054.3	−2.12101
$768$	−49823.4	−2.34095
$769$	−168.328	−0.00789343	−0.00394672	−	0.999992i	$-0.501256\pi$
$769$	−168.328	−0.00789343	−0.00394672	+	0.999992i	$0.501256\pi$
$770$	2660.24	0.124504
$771$	36359.2	1.69837
$772$	13725.0	0.639863
$773$	7869.07	0.366146	0.183073	−	0.983099i	$-0.441396\pi$
$773$	7869.07	0.366146	0.183073	+	0.983099i	$0.441396\pi$
$774$	−98497.7	−4.57420
$775$	0	0
$776$	4345.75	0.201035
$777$	2429.22	0.112159
$778$	5539.74	0.255282
$779$	−2268.36	−0.104329
$780$	66518.3	3.05351
$781$	−9441.44	−0.432575
$782$	−9834.06	−0.449700
$783$	−28305.5	−1.29190
$784$	26316.9	1.19884
$785$	14156.2	0.643641
$786$	−42134.9	−1.91209
$787$	18356.6	0.831440	0.415720	−	0.909493i	$-0.363530\pi$
$787$	18356.6	0.831440	0.415720	+	0.909493i	$0.363530\pi$
$788$	−18015.4	−0.814430
$789$	−15212.6	−0.686416
$790$	−47015.2	−2.11737
$791$	620.355	0.0278853
$792$	−17312.9	−0.776753
$793$	−49727.5	−2.22683
$794$	54845.6	2.45138
$795$	45671.0	2.03746
$796$	−10959.1	−0.487983
$797$	9837.74	0.437228	0.218614	−	0.975811i	$-0.429846\pi$
$797$	9837.74	0.437228	0.218614	+	0.975811i	$0.429846\pi$
$798$	−1083.63	−0.0480704
$799$	−10359.6	−0.458692
$800$	−29202.7	−1.29059
$801$	22917.0	1.01090
$802$	−52347.6	−2.30481
$803$	−22414.4	−0.985041
$804$	−36687.7	−1.60930
$805$	1457.69	0.0638220
$806$	0	0
$807$	−43838.1	−1.91224
$808$	−1679.30	−0.0731156
$809$	29925.6	1.30053	0.650264	−	0.759708i	$-0.274658\pi$
$809$	29925.6	1.30053	0.650264	+	0.759708i	$0.274658\pi$
$810$	−70229.3	−3.04643
$811$	−21548.5	−0.933008	−0.466504	−	0.884519i	$-0.654487\pi$
$811$	−21548.5	−0.933008	−0.466504	+	0.884519i	$0.654487\pi$
$812$	−700.691	−0.0302825
$813$	40149.2	1.73197
$814$	25634.0	1.10377
$815$	−12173.2	−0.523201
$816$	−27284.5	−1.17052
$817$	10913.1	0.467322
$818$	−4281.96	−0.183026
$819$	−5998.32	−0.255920
$820$	−8577.46	−0.365290
$821$	−17005.5	−0.722893	−0.361446	−	0.932393i	$-0.617717\pi$
$821$	−17005.5	−0.722893	−0.361446	+	0.932393i	$0.617717\pi$
$822$	−35412.7	−1.50263
$823$	14757.6	0.625051	0.312525	−	0.949909i	$-0.398825\pi$
$823$	14757.6	0.625051	0.312525	+	0.949909i	$0.398825\pi$
$824$	11848.8	0.500938
$825$	−42875.8	−1.80939
$826$	2775.39	0.116911
$827$	2221.47	0.0934075	0.0467038	−	0.998909i	$-0.485128\pi$
$827$	2221.47	0.0934075	0.0467038	+	0.998909i	$0.485128\pi$
$828$	23532.9	0.987710
$829$	−31879.9	−1.33563	−0.667813	−	0.744329i	$-0.732769\pi$
$829$	−31879.9	−1.33563	−0.667813	+	0.744329i	$0.732769\pi$
$830$	42364.1	1.77166
$831$	−70282.2	−2.93389
$832$	−14643.2	−0.610169
$833$	13008.0	0.541058
$834$	−31731.9	−1.31749
$835$	17975.0	0.744969
$836$	−4758.34	−0.196855
$837$	0	0
$838$	−58454.2	−2.40963
$839$	2744.17	0.112919	0.0564595	−	0.998405i	$-0.482019\pi$
$839$	2744.17	0.112919	0.0564595	+	0.998405i	$0.482019\pi$
$840$	1651.84	0.0678500
$841$	−15430.8	−0.632694
$842$	−18215.6	−0.745548
$843$	42432.5	1.73363
$844$	−1972.87	−0.0804611
$845$	−62654.6	−2.55075
$846$	59574.4	2.42105
$847$	194.502	0.00789040
$848$	−23550.1	−0.953670
$849$	−18187.6	−0.735212
$850$	−18954.6	−0.764869
$851$	14046.2	0.565802
$852$	14542.8	0.584773
$853$	28692.3	1.15170	0.575852	−	0.817554i	$-0.304670\pi$
$853$	28692.3	1.15170	0.575852	+	0.817554i	$0.304670\pi$
$854$	3063.26	0.122743
$855$	23154.8	0.926174
$856$	14520.2	0.579778
$857$	−17445.3	−0.695356	−0.347678	−	0.937614i	$-0.613030\pi$
$857$	−17445.3	−0.695356	−0.347678	+	0.937614i	$0.613030\pi$
$858$	−92160.0	−3.66701
$859$	−31398.6	−1.24715	−0.623577	−	0.781762i	$-0.714321\pi$
$859$	−31398.6	−1.24715	−0.623577	+	0.781762i	$0.714321\pi$
$860$	41266.4	1.63625
$861$	1126.19	0.0445765
$862$	18207.2	0.719421
$863$	−35281.8	−1.39167	−0.695833	−	0.718204i	$-0.744964\pi$
$863$	−35281.8	−1.39167	−0.695833	+	0.718204i	$0.744964\pi$
$864$	64999.5	2.55941
$865$	−6973.63	−0.274116
$866$	21197.0	0.831760
$867$	32130.5	1.25860
$868$	0	0
$869$	27105.8	1.05812
$870$	52387.3	2.04149
$871$	54071.3	2.10349
$872$	1720.83	0.0668286
$873$	−30243.9	−1.17251
$874$	−6265.77	−0.242498
$875$	195.746	0.00756275
$876$	34525.2	1.33162
$877$	16236.2	0.625151	0.312575	−	0.949893i	$-0.398808\pi$
$877$	16236.2	0.625151	0.312575	+	0.949893i	$0.398808\pi$
$878$	−8085.72	−0.310797
$879$	68613.5	2.63285
$880$	42677.1	1.63483
$881$	−14528.0	−0.555574	−0.277787	−	0.960643i	$-0.589601\pi$
$881$	−14528.0	−0.555574	−0.277787	+	0.960643i	$0.589601\pi$
$882$	−74804.8	−2.85579
$883$	−3540.90	−0.134950	−0.0674750	−	0.997721i	$-0.521494\pi$
$883$	−3540.90	−0.134950	−0.0674750	+	0.997721i	$0.521494\pi$
$884$	−16953.9	−0.645046
$885$	−86347.0	−3.27969
$886$	24796.4	0.940240
$887$	−12314.5	−0.466158	−0.233079	−	0.972458i	$-0.574880\pi$
$887$	−12314.5	−0.466158	−0.233079	+	0.972458i	$0.574880\pi$
$888$	15917.1	0.601511
$889$	−16.5392	−0.000623968 0
$890$	−23073.0	−0.868999
$891$	40489.6	1.52239
$892$	26523.0	0.995578
$893$	−6600.60	−0.247347
$894$	−64750.7	−2.42236
$895$	39046.1	1.45829
$896$	−1355.64	−0.0505455
$897$	−50499.4	−1.87974
$898$	49032.7	1.82209
$899$	0	0
$900$	45358.4	1.67994
$901$	−11640.4	−0.430409
$902$	11883.9	0.438682
$903$	−5418.13	−0.199672
$904$	4064.78	0.149549
$905$	−41607.8	−1.52828
$906$	−57767.3	−2.11831
$907$	25600.5	0.937210	0.468605	−	0.883408i	$-0.344757\pi$
$907$	25600.5	0.937210	0.468605	+	0.883408i	$0.344757\pi$
$908$	−11463.2	−0.418965
$909$	11686.9	0.426436
$910$	6039.16	0.219996
$911$	636.381	0.0231440	0.0115720	−	0.999933i	$-0.496316\pi$
$911$	636.381	0.0231440	0.0115720	+	0.999933i	$0.496316\pi$
$912$	−17384.3	−0.631197
$913$	−24424.3	−0.885353
$914$	−3718.58	−0.134573
$915$	−95303.2	−3.44331
$916$	−32549.9	−1.17411
$917$	−1591.84	−0.0573253
$918$	42189.2	1.51683
$919$	−16746.4	−0.601100	−0.300550	−	0.953766i	$-0.597170\pi$
$919$	−16746.4	−0.601100	−0.300550	+	0.953766i	$0.597170\pi$
$920$	9551.26	0.342278
$921$	54829.5	1.96166
$922$	25361.6	0.905901
$923$	−21433.5	−0.764348
$924$	2362.41	0.0841098
$925$	27073.3	0.962340
$926$	−59672.5	−2.11767
$927$	−82460.8	−2.92165
$928$	−20571.3	−0.727679
$929$	−39807.4	−1.40585	−0.702926	−	0.711263i	$-0.748124\pi$
$929$	−39807.4	−1.40585	−0.702926	+	0.711263i	$0.748124\pi$
$930$	0	0
$931$	8288.07	0.291762
$932$	−32876.0	−1.15546
$933$	15517.7	0.544508
$934$	−57542.0	−2.01588
$935$	21094.6	0.737827
$936$	−39303.1	−1.37250
$937$	−50379.6	−1.75649	−0.878244	−	0.478212i	$-0.841285\pi$
$937$	−50379.6	−1.75649	−0.878244	+	0.478212i	$0.841285\pi$
$938$	−3330.85	−0.115945
$939$	−47780.6	−1.66055
$940$	−24959.2	−0.866042
$941$	14045.5	0.486580	0.243290	−	0.969954i	$-0.421773\pi$
$941$	14045.5	0.486580	0.243290	+	0.969954i	$0.421773\pi$
$942$	30210.5	1.04492
$943$	6511.83	0.224872
$944$	44524.5	1.53511
$945$	−6253.64	−0.215271
$946$	−57173.9	−1.96499
$947$	−28709.9	−0.985159	−0.492580	−	0.870267i	$-0.663946\pi$
$947$	−28709.9	−0.985159	−0.492580	+	0.870267i	$0.663946\pi$
$948$	−41751.5	−1.43041
$949$	−50884.3	−1.74054
$950$	−12076.9	−0.412450
$951$	78587.7	2.67968
$952$	−421.014	−0.0143331
$953$	42034.4	1.42878	0.714391	−	0.699747i	$-0.246704\pi$
$953$	42034.4	1.42878	0.714391	+	0.699747i	$0.246704\pi$
$954$	66940.2	2.27177
$955$	74550.6	2.52607
$956$	−25797.3	−0.872745
$957$	−30203.1	−1.02019
$958$	−36831.5	−1.24214
$959$	−1337.88	−0.0450494
$960$	−28063.8	−0.943496
$961$	0	0
$962$	58193.1	1.95033
$963$	−101052.	−3.38147
$964$	−33449.8	−1.11758
$965$	−38767.8	−1.29324
$966$	3110.81	0.103612
$967$	−4751.37	−0.158008	−0.0790040	−	0.996874i	$-0.525174\pi$
$967$	−4751.37	−0.158008	−0.0790040	+	0.996874i	$0.525174\pi$
$968$	1274.44	0.0423163
$969$	−8592.78	−0.284871
$970$	30449.8	1.00792
$971$	8345.51	0.275819	0.137909	−	0.990445i	$-0.455962\pi$
$971$	8345.51	0.275819	0.137909	+	0.990445i	$0.455962\pi$
$972$	−16328.5	−0.538825
$973$	−1198.82	−0.0394990
$974$	57506.6	1.89182
$975$	−97334.8	−3.19714
$976$	49142.7	1.61170
$977$	8023.64	0.262742	0.131371	−	0.991333i	$-0.458062\pi$
$977$	8023.64	0.262742	0.131371	+	0.991333i	$0.458062\pi$
$978$	−25978.5	−0.849389
$979$	13302.4	0.434265
$980$	31340.1	1.02155
$981$	−11975.9	−0.389768
$982$	−39557.9	−1.28548
$983$	15707.9	0.509668	0.254834	−	0.966985i	$-0.417979\pi$
$983$	15707.9	0.509668	0.254834	+	0.966985i	$0.417979\pi$
$984$	7379.17	0.239064
$985$	50886.4	1.64607
$986$	−13352.2	−0.431259
$987$	3277.05	0.105683
$988$	−10802.2	−0.347837
$989$	−31328.6	−1.00727
$990$	−121308.	−3.89437
$991$	−55931.3	−1.79285	−0.896426	−	0.443193i	$-0.853846\pi$
$991$	−55931.3	−1.79285	−0.896426	+	0.443193i	$0.853846\pi$
$992$	0	0
$993$	−19990.2	−0.638842
$994$	1320.33	0.0421311
$995$	30955.1	0.986275
$996$	37621.1	1.19686
$997$	2267.37	0.0720245	0.0360122	−	0.999351i	$-0.488534\pi$
$997$	2267.37	0.0720245	0.0360122	+	0.999351i	$0.488534\pi$
$998$	−1906.87	−0.0604819
$999$	−60259.8	−1.90844

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	961.4.a.d.1.3	✓	4
31.30	odd	2	inner	961.4.a.d.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
961.4.a.d.1.3	✓	4	1.1	even	1	trivial
961.4.a.d.1.4	yes	4	31.30	odd	2	inner

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:	\( N \)	\(=\)	\( 961 = 31^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	961.a (trivial)

\(f(q)\)	\(=\)	\(q+3.70156 q^{2} -9.28490 q^{3} +5.70156 q^{4} -16.1047 q^{5} -34.3686 q^{6} -1.29844 q^{7} -8.50781 q^{8} +59.2094 q^{9} -59.6125 q^{10} +34.3686 q^{11} -52.9384 q^{12} +78.0222 q^{13} -4.80625 q^{14} +149.530 q^{15} -77.1047 q^{16} -38.1116 q^{17} +219.167 q^{18} -24.2828 q^{19} -91.8219 q^{20} +12.0559 q^{21} +127.218 q^{22} +69.7093 q^{23} +78.9942 q^{24} +134.361 q^{25} +288.804 q^{26} -299.061 q^{27} -7.40312 q^{28} +94.6480 q^{29} +553.496 q^{30} -217.345 q^{32} -319.109 q^{33} -141.072 q^{34} +20.9109 q^{35} +337.586 q^{36} +201.497 q^{37} -89.8844 q^{38} -724.428 q^{39} +137.016 q^{40} +93.4141 q^{41} +44.6255 q^{42} -449.418 q^{43} +195.955 q^{44} -953.548 q^{45} +258.033 q^{46} +271.822 q^{47} +715.909 q^{48} -341.314 q^{49} +497.345 q^{50} +353.862 q^{51} +444.848 q^{52} +305.430 q^{53} -1106.99 q^{54} -553.496 q^{55} +11.0469 q^{56} +225.464 q^{57} +350.345 q^{58} -577.455 q^{59} +852.557 q^{60} -637.350 q^{61} -76.8797 q^{63} -187.680 q^{64} -1256.52 q^{65} -1181.20 q^{66} +693.025 q^{67} -217.296 q^{68} -647.244 q^{69} +77.4031 q^{70} -274.711 q^{71} -503.742 q^{72} -652.177 q^{73} +745.853 q^{74} -1247.53 q^{75} -138.450 q^{76} -44.6255 q^{77} -2681.52 q^{78} +788.680 q^{79} +1241.75 q^{80} +1178.10 q^{81} +345.778 q^{82} -710.657 q^{83} +68.7373 q^{84} +613.775 q^{85} -1663.55 q^{86} -878.797 q^{87} -292.402 q^{88} +387.050 q^{89} -3529.62 q^{90} -101.307 q^{91} +397.452 q^{92} +1006.17 q^{94} +391.067 q^{95} +2018.03 q^{96} -510.795 q^{97} -1263.40 q^{98} +2034.95 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 10 q^{4} - 26 q^{5} - 18 q^{7} + 30 q^{8} + 160 q^{9} - 136 q^{10} + 32 q^{14} - 270 q^{16} + 326 q^{18} - 238 q^{19} - 188 q^{20} + 38 q^{25} - 4 q^{28} - 498 q^{32} - 380 q^{33} - 6 q^{35}+ \cdots - 994 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 + 10 * q^4 - 26 * q^5 - 18 * q^7 + 30 * q^8 + 160 * q^9 - 136 * q^10 + 32 * q^14 - 270 * q^16 + 326 * q^18 - 238 * q^19 - 188 * q^20 + 38 * q^25 - 4 * q^28 - 498 * q^32 - 380 * q^33 - 6 * q^35 + 646 * q^36 + 332 * q^38 - 1028 * q^39 + 420 * q^40 + 1078 * q^41 - 1778 * q^45 + 908 * q^47 - 1250 * q^49 + 1618 * q^50 - 736 * q^51 - 340 * q^56 - 1042 * q^59 - 474 * q^63 + 722 * q^64 - 3060 * q^66 + 928 * q^67 + 1048 * q^69 + 284 * q^70 + 630 * q^71 - 30 * q^72 - 144 * q^76 - 6500 * q^78 + 2124 * q^80 + 640 * q^81 - 1716 * q^82 - 5180 * q^87 - 7408 * q^90 + 1028 * q^94 + 194 * q^95 - 1262 * q^97 - 994 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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