LMFDB - Embedded newform 1058.4.a.o.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1058,4,Mod(1,1058)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1058.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1058, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1058 = 2 \cdot 23^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1058.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,8,0,16,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$62.4240207861$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{13})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 9x^{2} + 10x - 1 $ x^4 - 2x^3 - 9x^2 + 10*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.71699$ of defining polynomial
Character	$\chi$	$=$	1058.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000 q^{2} +3.60555 q^{3} +4.00000 q^{4} -7.79780 q^{5} +7.21110 q^{6} +1.84240 q^{7} +8.00000 q^{8} -14.0000 q^{9} -15.5956 q^{10} +22.7963 q^{11} +14.4222 q^{12} -74.8722 q^{13} +3.68481 q^{14} -28.1154 q^{15} +16.0000 q^{16} +23.6919 q^{17} -28.0000 q^{18} +87.2804 q^{19} -31.1912 q^{20} +6.64288 q^{21} +45.5926 q^{22} +28.8444 q^{24} -64.1943 q^{25} -149.744 q^{26} -147.828 q^{27} +7.36961 q^{28} -131.239 q^{29} -56.2307 q^{30} -17.2002 q^{31} +32.0000 q^{32} +82.1933 q^{33} +47.3839 q^{34} -14.3667 q^{35} -56.0000 q^{36} -43.7347 q^{37} +174.561 q^{38} -269.955 q^{39} -62.3824 q^{40} +59.9941 q^{41} +13.2858 q^{42} -146.952 q^{43} +91.1853 q^{44} +109.169 q^{45} -171.589 q^{47} +57.6888 q^{48} -339.606 q^{49} -128.389 q^{50} +85.4225 q^{51} -299.489 q^{52} -650.011 q^{53} -295.655 q^{54} -177.761 q^{55} +14.7392 q^{56} +314.694 q^{57} -262.478 q^{58} -780.538 q^{59} -112.461 q^{60} +57.9280 q^{61} -34.4003 q^{62} -25.7936 q^{63} +64.0000 q^{64} +583.838 q^{65} +164.387 q^{66} -951.957 q^{67} +94.7678 q^{68} -28.7334 q^{70} +461.328 q^{71} -112.000 q^{72} +1170.99 q^{73} -87.4695 q^{74} -231.456 q^{75} +349.122 q^{76} +42.0000 q^{77} -539.911 q^{78} +794.049 q^{79} -124.765 q^{80} -155.000 q^{81} +119.988 q^{82} -1101.20 q^{83} +26.5715 q^{84} -184.745 q^{85} -293.904 q^{86} -473.188 q^{87} +182.371 q^{88} +934.016 q^{89} +218.338 q^{90} -137.945 q^{91} -62.0160 q^{93} -343.177 q^{94} -680.595 q^{95} +115.378 q^{96} -1343.06 q^{97} -679.211 q^{98} -319.148 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{2} + 16 q^{4} + 32 q^{8} - 56 q^{9} - 112 q^{13} + 64 q^{16} - 112 q^{18} + 248 q^{25} - 224 q^{26} - 424 q^{29} - 588 q^{31} + 128 q^{32} - 144 q^{35} - 224 q^{36} - 676 q^{39} - 784 q^{41}+ \cdots - 2688 q^{98}+O(q^{100}) $ 4 * q + 8 * q^2 + 16 * q^4 + 32 * q^8 - 56 * q^9 - 112 * q^13 + 64 * q^16 - 112 * q^18 + 248 * q^25 - 224 * q^26 - 424 * q^29 - 588 * q^31 + 128 * q^32 - 144 * q^35 - 224 * q^36 - 676 * q^39 - 784 * q^41 - 196 * q^47 - 1344 * q^49 + 496 * q^50 - 448 * q^52 - 812 * q^55 - 848 * q^58 - 1204 * q^59 - 1176 * q^62 + 256 * q^64 - 288 * q^70 + 2552 * q^71 - 448 * q^72 + 1064 * q^73 - 1820 * q^75 + 168 * q^77 - 1352 * q^78 - 620 * q^81 - 1568 * q^82 - 2960 * q^85 - 364 * q^87 + 1872 * q^93 - 392 * q^94 - 5852 * q^95 - 2688 * 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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	3.60555	0.693889	0.346944	−	0.937886i	$-0.387219\pi$
$3$	3.60555	0.693889	0.346944	+	0.937886i	$0.387219\pi$
$4$	4.00000	0.500000
$5$	−7.79780	−0.697457	−0.348728	−	0.937224i	$-0.613386\pi$
$5$	−7.79780	−0.697457	−0.348728	+	0.937224i	$0.613386\pi$
$6$	7.21110	0.490653
$7$	1.84240	0.0994804	0.0497402	−	0.998762i	$-0.484161\pi$
$7$	1.84240	0.0994804	0.0497402	+	0.998762i	$0.484161\pi$
$8$	8.00000	0.353553
$9$	−14.0000	−0.518519
$10$	−15.5956	−0.493176
$11$	22.7963	0.624850	0.312425	−	0.949942i	$-0.398859\pi$
$11$	22.7963	0.624850	0.312425	+	0.949942i	$0.398859\pi$
$12$	14.4222	0.346944
$13$	−74.8722	−1.59737	−0.798685	−	0.601750i	$-0.794470\pi$
$13$	−74.8722	−1.59737	−0.798685	+	0.601750i	$0.794470\pi$
$14$	3.68481	0.0703433
$15$	−28.1154	−0.483957
$16$	16.0000	0.250000
$17$	23.6919	0.338008	0.169004	−	0.985615i	$-0.445945\pi$
$17$	23.6919	0.338008	0.169004	+	0.985615i	$0.445945\pi$
$18$	−28.0000	−0.366648
$19$	87.2804	1.05387	0.526934	−	0.849906i	$-0.323341\pi$
$19$	87.2804	1.05387	0.526934	+	0.849906i	$0.323341\pi$
$20$	−31.1912	−0.348728
$21$	6.64288	0.0690283
$22$	45.5926	0.441836
$23$	0	0
$23$	0	0
$24$	28.8444	0.245327
$25$	−64.1943	−0.513554
$26$	−149.744	−1.12951
$27$	−147.828	−1.05368
$28$	7.36961	0.0497402
$29$	−131.239	−0.840360	−0.420180	−	0.907441i	$-0.638033\pi$
$29$	−131.239	−0.840360	−0.420180	+	0.907441i	$0.638033\pi$
$30$	−56.2307	−0.342209
$31$	−17.2002	−0.0996529	−0.0498264	−	0.998758i	$-0.515867\pi$
$31$	−17.2002	−0.0996529	−0.0498264	+	0.998758i	$0.515867\pi$
$32$	32.0000	0.176777
$33$	82.1933	0.433576
$34$	47.3839	0.239008
$35$	−14.3667	−0.0693833
$36$	−56.0000	−0.259259
$37$	−43.7347	−0.194323	−0.0971615	−	0.995269i	$-0.530976\pi$
$37$	−43.7347	−0.194323	−0.0971615	+	0.995269i	$0.530976\pi$
$38$	174.561	0.745198
$39$	−269.955	−1.10840
$40$	−62.3824	−0.246588
$41$	59.9941	0.228525	0.114262	−	0.993451i	$-0.463550\pi$
$41$	59.9941	0.228525	0.114262	+	0.993451i	$0.463550\pi$
$42$	13.2858	0.0488104
$43$	−146.952	−0.521163	−0.260581	−	0.965452i	$-0.583914\pi$
$43$	−146.952	−0.521163	−0.260581	+	0.965452i	$0.583914\pi$
$44$	91.1853	0.312425
$45$	109.169	0.361644
$46$	0	0
$47$	−171.589	−0.532527	−0.266264	−	0.963900i	$-0.585789\pi$
$47$	−171.589	−0.532527	−0.266264	+	0.963900i	$0.585789\pi$
$48$	57.6888	0.173472
$49$	−339.606	−0.990104
$50$	−128.389	−0.363138
$51$	85.4225	0.234540
$52$	−299.489	−0.798685
$53$	−650.011	−1.68464	−0.842320	−	0.538978i	$-0.818811\pi$
$53$	−650.011	−1.68464	−0.842320	+	0.538978i	$0.818811\pi$
$54$	−295.655	−0.745066
$55$	−177.761	−0.435806
$56$	14.7392	0.0351716
$57$	314.694	0.731267
$58$	−262.478	−0.594224
$59$	−780.538	−1.72233	−0.861165	−	0.508326i	$-0.830264\pi$
$59$	−780.538	−1.72233	−0.861165	+	0.508326i	$0.830264\pi$
$60$	−112.461	−0.241979
$61$	57.9280	0.121589	0.0607944	−	0.998150i	$-0.480637\pi$
$61$	57.9280	0.121589	0.0607944	+	0.998150i	$0.480637\pi$
$62$	−34.4003	−0.0704652
$63$	−25.7936	−0.0515824
$64$	64.0000	0.125000
$65$	583.838	1.11410
$66$	164.387	0.306585
$67$	−951.957	−1.73582	−0.867911	−	0.496719i	$-0.834538\pi$
$67$	−951.957	−1.73582	−0.867911	+	0.496719i	$0.834538\pi$
$68$	94.7678	0.169004
$69$	0	0
$70$	−28.7334	−0.0490614
$71$	461.328	0.771121	0.385560	−	0.922683i	$-0.374008\pi$
$71$	461.328	0.771121	0.385560	+	0.922683i	$0.374008\pi$
$72$	−112.000	−0.183324
$73$	1170.99	1.87746	0.938729	−	0.344657i	$-0.112005\pi$
$73$	1170.99	1.87746	0.938729	+	0.344657i	$0.112005\pi$
$74$	−87.4695	−0.137407
$75$	−231.456	−0.356350
$76$	349.122	0.526934
$77$	42.0000	0.0621603
$78$	−539.911	−0.783755
$79$	794.049	1.13085	0.565427	−	0.824798i	$-0.308711\pi$
$79$	794.049	1.13085	0.565427	+	0.824798i	$0.308711\pi$
$80$	−124.765	−0.174364
$81$	−155.000	−0.212620
$82$	119.988	0.161591
$83$	−1101.20	−1.45629	−0.728145	−	0.685423i	$-0.759617\pi$
$83$	−1101.20	−1.45629	−0.728145	+	0.685423i	$0.759617\pi$
$84$	26.5715	0.0345142
$85$	−184.745	−0.235746
$86$	−293.904	−0.368518
$87$	−473.188	−0.583116
$88$	182.371	0.220918
$89$	934.016	1.11242	0.556211	−	0.831041i	$-0.312255\pi$
$89$	934.016	1.11242	0.556211	+	0.831041i	$0.312255\pi$
$90$	218.338	0.255721
$91$	−137.945	−0.158907
$92$	0	0
$93$	−62.0160	−0.0691480
$94$	−343.177	−0.376554
$95$	−680.595	−0.735027
$96$	115.378	0.122663
$97$	−1343.06	−1.40585	−0.702923	−	0.711266i	$-0.748122\pi$
$97$	−1343.06	−1.40585	−0.702923	+	0.711266i	$0.748122\pi$
$98$	−679.211	−0.700109
$99$	−319.148	−0.323996
$100$	−256.777	−0.256777
$101$	483.583	0.476419	0.238209	−	0.971214i	$-0.423440\pi$
$101$	483.583	0.476419	0.238209	+	0.971214i	$0.423440\pi$
$102$	170.845	0.165845
$103$	1184.67	1.13329	0.566644	−	0.823962i	$-0.308241\pi$
$103$	1184.67	1.13329	0.566644	+	0.823962i	$0.308241\pi$
$104$	−598.977	−0.564755
$105$	−51.7998	−0.0481443
$106$	−1300.02	−1.19122
$107$	−252.217	−0.227876	−0.113938	−	0.993488i	$-0.536346\pi$
$107$	−252.217	−0.227876	−0.113938	+	0.993488i	$0.536346\pi$
$108$	−591.310	−0.526841
$109$	−1604.27	−1.40973	−0.704867	−	0.709340i	$-0.748993\pi$
$109$	−1604.27	−1.40973	−0.704867	+	0.709340i	$0.748993\pi$
$110$	−355.522	−0.308161
$111$	−157.688	−0.134838
$112$	29.4784	0.0248701
$113$	197.506	0.164423	0.0822115	−	0.996615i	$-0.473802\pi$
$113$	197.506	0.164423	0.0822115	+	0.996615i	$0.473802\pi$
$114$	629.388	0.517084
$115$	0	0
$116$	−524.955	−0.420180
$117$	1048.21	0.828266
$118$	−1561.08	−1.21787
$119$	43.6501	0.0336252
$120$	−224.923	−0.171105
$121$	−811.328	−0.609563
$122$	115.856	0.0859763
$123$	216.312	0.158571
$124$	−68.8006	−0.0498264
$125$	1475.30	1.05564
$126$	−51.5873	−0.0364743
$127$	−246.761	−0.172413	−0.0862067	−	0.996277i	$-0.527475\pi$
$127$	−246.761	−0.172413	−0.0862067	+	0.996277i	$0.527475\pi$
$128$	128.000	0.0883883
$129$	−529.844	−0.361629
$130$	1167.68	0.787785
$131$	−2532.08	−1.68877	−0.844386	−	0.535735i	$-0.820035\pi$
$131$	−2532.08	−1.68877	−0.844386	+	0.535735i	$0.820035\pi$
$132$	328.773	0.216788
$133$	160.806	0.104839
$134$	−1903.91	−1.22741
$135$	1152.73	0.734898
$136$	189.536	0.119504
$137$	−1700.61	−1.06053	−0.530265	−	0.847832i	$-0.677907\pi$
$137$	−1700.61	−1.06053	−0.530265	+	0.847832i	$0.677907\pi$
$138$	0	0
$139$	−638.351	−0.389527	−0.194763	−	0.980850i	$-0.562394\pi$
$139$	−638.351	−0.389527	−0.194763	+	0.980850i	$0.562394\pi$
$140$	−57.4668	−0.0346916
$141$	−618.672	−0.369515
$142$	922.656	0.545265
$143$	−1706.81	−0.998116
$144$	−224.000	−0.129630
$145$	1023.37	0.586115
$146$	2341.99	1.32756
$147$	−1224.47	−0.687022
$148$	−174.939	−0.0971615
$149$	3082.76	1.69496	0.847481	−	0.530826i	$-0.178118\pi$
$149$	3082.76	1.69496	0.847481	+	0.530826i	$0.178118\pi$
$150$	−462.912	−0.251977
$151$	2100.65	1.13211	0.566054	−	0.824368i	$-0.308469\pi$
$151$	2100.65	1.13211	0.566054	+	0.824368i	$0.308469\pi$
$152$	698.243	0.372599
$153$	−331.687	−0.175264
$154$	84.0000	0.0439540
$155$	134.123	0.0695035
$156$	−1079.82	−0.554198
$157$	3172.40	1.61264	0.806322	−	0.591477i	$-0.201455\pi$
$157$	3172.40	1.61264	0.806322	+	0.591477i	$0.201455\pi$
$158$	1588.10	0.799635
$159$	−2343.65	−1.16895
$160$	−249.530	−0.123294
$161$	0	0
$162$	−310.000	−0.150345
$163$	3351.00	1.61025	0.805125	−	0.593105i	$-0.202098\pi$
$163$	3351.00	1.61025	0.805125	+	0.593105i	$0.202098\pi$
$164$	239.977	0.114262
$165$	−640.927	−0.302401
$166$	−2202.39	−1.02975
$167$	−1770.18	−0.820244	−0.410122	−	0.912031i	$-0.634514\pi$
$167$	−1770.18	−0.820244	−0.410122	+	0.912031i	$0.634514\pi$
$168$	53.1430	0.0244052
$169$	3408.84	1.55159
$170$	−369.490	−0.166698
$171$	−1221.93	−0.546450
$172$	−587.809	−0.260581
$173$	655.433	0.288044	0.144022	−	0.989574i	$-0.453996\pi$
$173$	655.433	0.288044	0.144022	+	0.989574i	$0.453996\pi$
$174$	−946.377	−0.412326
$175$	−118.272	−0.0510886
$176$	364.741	0.156212
$177$	−2814.27	−1.19510
$178$	1868.03	0.786601
$179$	−2312.55	−0.965634	−0.482817	−	0.875721i	$-0.660386\pi$
$179$	−2312.55	−0.965634	−0.482817	+	0.875721i	$0.660386\pi$
$180$	436.677	0.180822
$181$	2086.79	0.856959	0.428480	−	0.903551i	$-0.359049\pi$
$181$	2086.79	0.856959	0.428480	+	0.903551i	$0.359049\pi$
$182$	−275.889	−0.112364
$183$	208.862	0.0843692
$184$	0	0
$185$	341.035	0.135532
$186$	−124.032	−0.0488950
$187$	540.089	0.211204
$188$	−686.355	−0.266264
$189$	−272.358	−0.104821
$190$	−1361.19	−0.519743
$191$	21.4048	0.00810888	0.00405444	−	0.999992i	$-0.498709\pi$
$191$	21.4048	0.00810888	0.00405444	+	0.999992i	$0.498709\pi$
$192$	230.755	0.0867361
$193$	−54.9750	−0.0205036	−0.0102518	−	0.999947i	$-0.503263\pi$
$193$	−54.9750	−0.0205036	−0.0102518	+	0.999947i	$0.503263\pi$
$194$	−2686.12	−0.994083
$195$	2105.06	0.773058
$196$	−1358.42	−0.495052
$197$	−960.230	−0.347277	−0.173638	−	0.984809i	$-0.555552\pi$
$197$	−960.230	−0.347277	−0.173638	+	0.984809i	$0.555552\pi$
$198$	−638.297	−0.229100
$199$	1942.97	0.692127	0.346063	−	0.938211i	$-0.387518\pi$
$199$	1942.97	0.692127	0.346063	+	0.938211i	$0.387518\pi$
$200$	−513.554	−0.181569
$201$	−3432.33	−1.20447
$202$	967.166	0.336879
$203$	−241.795	−0.0835994
$204$	341.690	0.117270
$205$	−467.822	−0.159386
$206$	2369.33	0.801356
$207$	0	0
$208$	−1197.95	−0.399342
$209$	1989.67	0.658510
$210$	−103.600	−0.0340431
$211$	−3741.23	−1.22065	−0.610324	−	0.792152i	$-0.708961\pi$
$211$	−3741.23	−1.22065	−0.610324	+	0.792152i	$0.708961\pi$
$212$	−2600.05	−0.842320
$213$	1663.34	0.535072
$214$	−504.433	−0.161132
$215$	1145.90	0.363488
$216$	−1182.62	−0.372533
$217$	−31.6896	−0.00991351
$218$	−3208.54	−0.996832
$219$	4222.08	1.30275
$220$	−711.045	−0.217903
$221$	−1773.87	−0.539924
$222$	−315.376	−0.0953452
$223$	−2950.52	−0.886016	−0.443008	−	0.896518i	$-0.646089\pi$
$223$	−2950.52	−0.886016	−0.443008	+	0.896518i	$0.646089\pi$
$224$	58.9569	0.0175858
$225$	898.720	0.266287
$226$	395.012	0.116265
$227$	4198.06	1.22747	0.613733	−	0.789514i	$-0.289667\pi$
$227$	4198.06	1.22747	0.613733	+	0.789514i	$0.289667\pi$
$228$	1258.78	0.365634
$229$	3707.05	1.06973	0.534866	−	0.844937i	$-0.320362\pi$
$229$	3707.05	1.06973	0.534866	+	0.844937i	$0.320362\pi$
$230$	0	0
$231$	151.433	0.0431323
$232$	−1049.91	−0.297112
$233$	−2115.45	−0.594797	−0.297398	−	0.954753i	$-0.596119\pi$
$233$	−2115.45	−0.594797	−0.297398	+	0.954753i	$0.596119\pi$
$234$	2096.42	0.585672
$235$	1338.01	0.371415
$236$	−3122.15	−0.861165
$237$	2862.99	0.784687
$238$	87.3002	0.0237766
$239$	6802.89	1.84118	0.920591	−	0.390528i	$-0.127707\pi$
$239$	6802.89	1.84118	0.920591	+	0.390528i	$0.127707\pi$
$240$	−449.846	−0.120989
$241$	−598.999	−0.160103	−0.0800517	−	0.996791i	$-0.525509\pi$
$241$	−598.999	−0.160103	−0.0800517	+	0.996791i	$0.525509\pi$
$242$	−1622.66	−0.431026
$243$	3432.48	0.906148
$244$	231.712	0.0607944
$245$	2648.18	0.690554
$246$	432.624	0.112126
$247$	−6534.87	−1.68342
$248$	−137.601	−0.0352326
$249$	−3970.42	−1.01050
$250$	2950.60	0.746449
$251$	91.8760	0.0231042	0.0115521	−	0.999933i	$-0.496323\pi$
$251$	91.8760	0.0231042	0.0115521	+	0.999933i	$0.496323\pi$
$252$	−103.175	−0.0257912
$253$	0	0
$254$	−493.522	−0.121915
$255$	−666.108	−0.163582
$256$	256.000	0.0625000
$257$	−2628.63	−0.638013	−0.319007	−	0.947752i	$-0.603349\pi$
$257$	−2628.63	−0.638013	−0.319007	+	0.947752i	$0.603349\pi$
$258$	−1059.69	−0.255710
$259$	−80.5770	−0.0193313
$260$	2335.35	0.557048
$261$	1837.34	0.435742
$262$	−5064.17	−1.19414
$263$	1857.46	0.435497	0.217749	−	0.976005i	$-0.430129\pi$
$263$	1857.46	0.435497	0.217749	+	0.976005i	$0.430129\pi$
$264$	657.546	0.153292
$265$	5068.66	1.17496
$266$	321.611	0.0741326
$267$	3367.64	0.771897
$268$	−3807.83	−0.867911
$269$	1589.64	0.360306	0.180153	−	0.983639i	$-0.442341\pi$
$269$	1589.64	0.360306	0.180153	+	0.983639i	$0.442341\pi$
$270$	2305.46	0.519651
$271$	515.280	0.115502	0.0577510	−	0.998331i	$-0.481607\pi$
$271$	515.280	0.115502	0.0577510	+	0.998331i	$0.481607\pi$
$272$	379.071	0.0845021
$273$	−497.367	−0.110264
$274$	−3401.21	−0.749908
$275$	−1463.39	−0.320894
$276$	0	0
$277$	−6186.50	−1.34192	−0.670959	−	0.741495i	$-0.734117\pi$
$277$	−6186.50	−1.34192	−0.670959	+	0.741495i	$0.734117\pi$
$278$	−1276.70	−0.275437
$279$	240.802	0.0516719
$280$	−114.934	−0.0245307
$281$	5544.51	1.17707	0.588537	−	0.808470i	$-0.299704\pi$
$281$	5544.51	1.17707	0.588537	+	0.808470i	$0.299704\pi$
$282$	−1237.34	−0.261286
$283$	−7551.96	−1.58628	−0.793140	−	0.609039i	$-0.791555\pi$
$283$	−7551.96	−1.58628	−0.793140	+	0.609039i	$0.791555\pi$
$284$	1845.31	0.385560
$285$	−2453.92	−0.510027
$286$	−3413.62	−0.705775
$287$	110.533	0.0227337
$288$	−448.000	−0.0916620
$289$	−4351.69	−0.885750
$290$	2046.75	0.414446
$291$	−4842.47	−0.975500
$292$	4683.97	0.938729
$293$	−6877.46	−1.37128	−0.685641	−	0.727940i	$-0.740478\pi$
$293$	−6877.46	−1.37128	−0.685641	+	0.727940i	$0.740478\pi$
$294$	−2448.93	−0.485798
$295$	6086.48	1.20125
$296$	−349.878	−0.0687035
$297$	−3369.92	−0.658393
$298$	6165.51	1.19852
$299$	0	0
$300$	−925.823	−0.178175
$301$	−270.745	−0.0518455
$302$	4201.29	0.800521
$303$	1743.58	0.330582
$304$	1396.49	0.263467
$305$	−451.711	−0.0848030
$306$	−663.375	−0.123930
$307$	−7077.40	−1.31573	−0.657864	−	0.753137i	$-0.728540\pi$
$307$	−7077.40	−1.31573	−0.657864	+	0.753137i	$0.728540\pi$
$308$	168.000	0.0310802
$309$	4271.38	0.786376
$310$	268.247	0.0491464
$311$	−5312.79	−0.968683	−0.484342	−	0.874879i	$-0.660941\pi$
$311$	−5312.79	−0.968683	−0.484342	+	0.874879i	$0.660941\pi$
$312$	−2159.64	−0.391877
$313$	3031.30	0.547409	0.273705	−	0.961814i	$-0.411751\pi$
$313$	3031.30	0.547409	0.273705	+	0.961814i	$0.411751\pi$
$314$	6344.80	1.14031
$315$	201.134	0.0359765
$316$	3176.20	0.565427
$317$	6807.14	1.20608	0.603040	−	0.797711i	$-0.293956\pi$
$317$	6807.14	1.20608	0.603040	+	0.797711i	$0.293956\pi$
$318$	−4687.30	−0.826574
$319$	−2991.76	−0.525099
$320$	−499.059	−0.0871821
$321$	−909.380	−0.158120
$322$	0	0
$323$	2067.84	0.356216
$324$	−620.000	−0.106310
$325$	4806.37	0.820336
$326$	6702.01	1.13862
$327$	−5784.27	−0.978198
$328$	479.953	0.0807957
$329$	−316.136	−0.0529760
$330$	−1281.85	−0.213829
$331$	6305.34	1.04705	0.523524	−	0.852011i	$-0.324617\pi$
$331$	6305.34	1.04705	0.523524	+	0.852011i	$0.324617\pi$
$332$	−4404.79	−0.728145
$333$	612.286	0.100760
$334$	−3540.36	−0.580000
$335$	7423.17	1.21066
$336$	106.286	0.0172571
$337$	−6797.91	−1.09883	−0.549415	−	0.835550i	$-0.685149\pi$
$337$	−6797.91	−1.09883	−0.549415	+	0.835550i	$0.685149\pi$
$338$	6817.68	1.09714
$339$	712.118	0.114091
$340$	−738.980	−0.117873
$341$	−392.100	−0.0622681
$342$	−2443.85	−0.386399
$343$	−1257.63	−0.197976
$344$	−1175.62	−0.184259
$345$	0	0
$346$	1310.87	0.203678
$347$	−4127.74	−0.638584	−0.319292	−	0.947656i	$-0.603445\pi$
$347$	−4127.74	−0.638584	−0.319292	+	0.947656i	$0.603445\pi$
$348$	−1892.75	−0.291558
$349$	−4092.11	−0.627638	−0.313819	−	0.949483i	$-0.601609\pi$
$349$	−4092.11	−0.627638	−0.313819	+	0.949483i	$0.601609\pi$
$350$	−236.544	−0.0361251
$351$	11068.2	1.68312
$352$	729.482	0.110459
$353$	−1238.49	−0.186737	−0.0933685	−	0.995632i	$-0.529763\pi$
$353$	−1238.49	−0.186737	−0.0933685	+	0.995632i	$0.529763\pi$
$354$	−5628.54	−0.845067
$355$	−3597.34	−0.537823
$356$	3736.06	0.556211
$357$	157.383	0.0233321
$358$	−4625.11	−0.682806
$359$	−3965.35	−0.582962	−0.291481	−	0.956577i	$-0.594148\pi$
$359$	−3965.35	−0.582962	−0.291481	+	0.956577i	$0.594148\pi$
$360$	873.354	0.127861
$361$	758.872	0.110639
$362$	4173.57	0.605962
$363$	−2925.28	−0.422969
$364$	−551.779	−0.0794535
$365$	−9131.17	−1.30945
$366$	417.725	0.0596580
$367$	−10025.5	−1.42596	−0.712978	−	0.701187i	$-0.752654\pi$
$367$	−10025.5	−1.42596	−0.712978	+	0.701187i	$0.752654\pi$
$368$	0	0
$369$	−839.918	−0.118494
$370$	682.070	0.0958354
$371$	−1197.58	−0.167589
$372$	−248.064	−0.0345740
$373$	9141.61	1.26899	0.634496	−	0.772926i	$-0.281208\pi$
$373$	9141.61	1.26899	0.634496	+	0.772926i	$0.281208\pi$
$374$	1080.18	0.149344
$375$	5319.27	0.732496
$376$	−1372.71	−0.188277
$377$	9826.14	1.34237
$378$	−544.716	−0.0741195
$379$	−6949.04	−0.941815	−0.470908	−	0.882182i	$-0.656074\pi$
$379$	−6949.04	−0.941815	−0.470908	+	0.882182i	$0.656074\pi$
$380$	−2722.38	−0.367514
$381$	−889.710	−0.119636
$382$	42.8096	0.00573384
$383$	10433.9	1.39203	0.696017	−	0.718025i	$-0.254954\pi$
$383$	10433.9	1.39203	0.696017	+	0.718025i	$0.254954\pi$
$384$	461.511	0.0613317
$385$	−327.508	−0.0433541
$386$	−109.950	−0.0144982
$387$	2057.33	0.270232
$388$	−5372.24	−0.702923
$389$	11114.2	1.44862	0.724310	−	0.689475i	$-0.242159\pi$
$389$	11114.2	1.44862	0.724310	+	0.689475i	$0.242159\pi$
$390$	4210.12	0.546635
$391$	0	0
$392$	−2716.84	−0.350055
$393$	−9129.56	−1.17182
$394$	−1920.46	−0.245562
$395$	−6191.84	−0.788722
$396$	−1276.59	−0.161998
$397$	−2607.12	−0.329591	−0.164795	−	0.986328i	$-0.552696\pi$
$397$	−2607.12	−0.329591	−0.164795	+	0.986328i	$0.552696\pi$
$398$	3885.93	0.489407
$399$	579.793	0.0727468
$400$	−1027.11	−0.128389
$401$	11431.1	1.42354	0.711772	−	0.702411i	$-0.247893\pi$
$401$	11431.1	1.42354	0.711772	+	0.702411i	$0.247893\pi$
$402$	−6864.66	−0.851687
$403$	1287.81	0.159182
$404$	1934.33	0.238209
$405$	1208.66	0.148293
$406$	−483.590	−0.0591137
$407$	−996.991	−0.121423
$408$	683.380	0.0829225
$409$	7625.67	0.921920	0.460960	−	0.887421i	$-0.347505\pi$
$409$	7625.67	0.921920	0.460960	+	0.887421i	$0.347505\pi$
$410$	−935.645	−0.112703
$411$	−6131.62	−0.735889
$412$	4738.67	0.566644
$413$	−1438.07	−0.171338
$414$	0	0
$415$	8586.91	1.01570
$416$	−2395.91	−0.282378
$417$	−2301.61	−0.270288
$418$	3979.34	0.465637
$419$	12552.4	1.46354	0.731772	−	0.681550i	$-0.238694\pi$
$419$	12552.4	1.46354	0.731772	+	0.681550i	$0.238694\pi$
$420$	−207.199	−0.0240721
$421$	−5286.23	−0.611960	−0.305980	−	0.952038i	$-0.598984\pi$
$421$	−5286.23	−0.611960	−0.305980	+	0.952038i	$0.598984\pi$
$422$	−7482.45	−0.863128
$423$	2402.24	0.276125
$424$	−5200.09	−0.595610
$425$	−1520.89	−0.173586
$426$	3326.68	0.378353
$427$	106.727	0.0120957
$428$	−1008.87	−0.113938
$429$	−6153.99	−0.692581
$430$	2291.81	0.257025
$431$	14386.0	1.60777	0.803885	−	0.594784i	$-0.202762\pi$
$431$	14386.0	1.60777	0.803885	+	0.594784i	$0.202762\pi$
$432$	−2365.24	−0.263421
$433$	7015.47	0.778619	0.389309	−	0.921107i	$-0.372714\pi$
$433$	7015.47	0.778619	0.389309	+	0.921107i	$0.372714\pi$
$434$	−63.3792	−0.00700991
$435$	3689.83	0.406698
$436$	−6417.07	−0.704867
$437$	0	0
$438$	8444.15	0.921181
$439$	−1499.70	−0.163045	−0.0815224	−	0.996672i	$-0.525978\pi$
$439$	−1499.70	−0.163045	−0.0815224	+	0.996672i	$0.525978\pi$
$440$	−1422.09	−0.154081
$441$	4754.48	0.513387
$442$	−3547.73	−0.381784
$443$	13925.8	1.49353	0.746765	−	0.665088i	$-0.231606\pi$
$443$	13925.8	1.49353	0.746765	+	0.665088i	$0.231606\pi$
$444$	−630.751	−0.0674192
$445$	−7283.27	−0.775866
$446$	−5901.04	−0.626508
$447$	11115.0	1.17611
$448$	117.914	0.0124351
$449$	10754.9	1.13041	0.565205	−	0.824951i	$-0.308797\pi$
$449$	10754.9	1.13041	0.565205	+	0.824951i	$0.308797\pi$
$450$	1797.44	0.188294
$451$	1367.65	0.142794
$452$	790.024	0.0822115
$453$	7573.99	0.785557
$454$	8396.11	0.867949
$455$	1075.67	0.110831
$456$	2517.55	0.258542
$457$	−14754.2	−1.51023	−0.755114	−	0.655593i	$-0.772419\pi$
$457$	−14754.2	−1.51023	−0.755114	+	0.655593i	$0.772419\pi$
$458$	7414.09	0.756414
$459$	−3502.32	−0.356154
$460$	0	0
$461$	14234.2	1.43808	0.719040	−	0.694969i	$-0.244582\pi$
$461$	14234.2	1.43808	0.719040	+	0.694969i	$0.244582\pi$
$462$	302.866	0.0304992
$463$	12778.0	1.28260	0.641301	−	0.767289i	$-0.278395\pi$
$463$	12778.0	1.28260	0.641301	+	0.767289i	$0.278395\pi$
$464$	−2099.82	−0.210090
$465$	483.589	0.0482277
$466$	−4230.90	−0.420585
$467$	20.5016	0.00203148	0.00101574	−	0.999999i	$-0.499677\pi$
$467$	20.5016	0.00203148	0.00101574	+	0.999999i	$0.499677\pi$
$468$	4192.84	0.414133
$469$	−1753.89	−0.172680
$470$	2676.03	0.262630
$471$	11438.3	1.11900
$472$	−6244.31	−0.608935
$473$	−3349.97	−0.325648
$474$	5725.97	0.554858
$475$	−5602.90	−0.541219
$476$	174.600	0.0168126
$477$	9100.16	0.873517
$478$	13605.8	1.30191
$479$	−19549.7	−1.86482	−0.932408	−	0.361408i	$-0.882296\pi$
$479$	−19549.7	−1.86482	−0.932408	+	0.361408i	$0.882296\pi$
$480$	−899.692	−0.0855524
$481$	3274.52	0.310405
$482$	−1198.00	−0.113210
$483$	0	0
$484$	−3245.31	−0.304781
$485$	10472.9	0.980516
$486$	6864.97	0.640744
$487$	8252.59	0.767886	0.383943	−	0.923357i	$-0.374566\pi$
$487$	8252.59	0.767886	0.383943	+	0.923357i	$0.374566\pi$
$488$	463.424	0.0429882
$489$	12082.2	1.11733
$490$	5296.35	0.488296
$491$	−3487.38	−0.320536	−0.160268	−	0.987074i	$-0.551236\pi$
$491$	−3487.38	−0.320536	−0.160268	+	0.987074i	$0.551236\pi$
$492$	865.248	0.0792853
$493$	−3109.30	−0.284049
$494$	−13069.7	−1.19036
$495$	2488.66	0.225973
$496$	−275.202	−0.0249132
$497$	849.952	0.0767114
$498$	−7940.84	−0.714533
$499$	−4702.81	−0.421897	−0.210949	−	0.977497i	$-0.567655\pi$
$499$	−4702.81	−0.421897	−0.210949	+	0.977497i	$0.567655\pi$
$500$	5901.20	0.527819
$501$	−6382.48	−0.569158
$502$	183.752	0.0163372
$503$	−2273.98	−0.201574	−0.100787	−	0.994908i	$-0.532136\pi$
$503$	−2273.98	−0.201574	−0.100787	+	0.994908i	$0.532136\pi$
$504$	−206.349	−0.0182371
$505$	−3770.88	−0.332281
$506$	0	0
$507$	12290.8	1.07663
$508$	−987.045	−0.0862067
$509$	−2827.62	−0.246232	−0.123116	−	0.992392i	$-0.539289\pi$
$509$	−2827.62	−0.246232	−0.123116	+	0.992392i	$0.539289\pi$
$510$	−1332.22	−0.115670
$511$	2157.44	0.186770
$512$	512.000	0.0441942
$513$	−12902.5	−1.11044
$514$	−5257.26	−0.451144
$515$	−9237.80	−0.790420
$516$	−2119.37	−0.180814
$517$	−3911.59	−0.332750
$518$	−161.154	−0.0136693
$519$	2363.20	0.199871
$520$	4670.71	0.393892
$521$	−5631.21	−0.473527	−0.236764	−	0.971567i	$-0.576087\pi$
$521$	−5631.21	−0.473527	−0.236764	+	0.971567i	$0.576087\pi$
$522$	3674.69	0.308116
$523$	17389.5	1.45390	0.726949	−	0.686691i	$-0.240937\pi$
$523$	17389.5	1.45390	0.726949	+	0.686691i	$0.240937\pi$
$524$	−10128.3	−0.844386
$525$	−426.435	−0.0354498
$526$	3714.91	0.307943
$527$	−407.505	−0.0336835
$528$	1315.09	0.108394
$529$	0	0
$530$	10137.3	0.830825
$531$	10927.5	0.893060
$532$	643.223	0.0524196
$533$	−4491.89	−0.365038
$534$	6735.29	0.545813
$535$	1966.73	0.158933
$536$	−7615.66	−0.613706
$537$	−8338.03	−0.670042
$538$	3179.29	0.254775
$539$	−7741.76	−0.618666
$540$	4610.92	0.367449
$541$	2457.61	0.195306	0.0976532	−	0.995221i	$-0.468866\pi$
$541$	2457.61	0.195306	0.0976532	+	0.995221i	$0.468866\pi$
$542$	1030.56	0.0816722
$543$	7524.02	0.594634
$544$	758.142	0.0597520
$545$	12509.8	0.983228
$546$	−994.733	−0.0779682
$547$	−1492.89	−0.116693	−0.0583467	−	0.998296i	$-0.518583\pi$
$547$	−1492.89	−0.116693	−0.0583467	+	0.998296i	$0.518583\pi$
$548$	−6802.42	−0.530265
$549$	−810.992	−0.0630461
$550$	−2926.79	−0.226907
$551$	−11454.6	−0.885629
$552$	0	0
$553$	1462.96	0.112498
$554$	−12373.0	−0.948879
$555$	1229.62	0.0940440
$556$	−2553.40	−0.194763
$557$	12138.1	0.923356	0.461678	−	0.887048i	$-0.347248\pi$
$557$	12138.1	0.923356	0.461678	+	0.887048i	$0.347248\pi$
$558$	481.604	0.0365375
$559$	11002.6	0.832489
$560$	−229.867	−0.0173458
$561$	1947.32	0.146552
$562$	11089.0	0.832317
$563$	−24269.4	−1.81675	−0.908377	−	0.418152i	$-0.862678\pi$
$563$	−24269.4	−1.81675	−0.908377	+	0.418152i	$0.862678\pi$
$564$	−2474.69	−0.184757
$565$	−1540.11	−0.114678
$566$	−15103.9	−1.12167
$567$	−285.572	−0.0211515
$568$	3690.62	0.272632
$569$	−12781.0	−0.941665	−0.470832	−	0.882223i	$-0.656046\pi$
$569$	−12781.0	−0.941665	−0.470832	+	0.882223i	$0.656046\pi$
$570$	−4907.84	−0.360644
$571$	2211.63	0.162091	0.0810455	−	0.996710i	$-0.474174\pi$
$571$	2211.63	0.162091	0.0810455	+	0.996710i	$0.474174\pi$
$572$	−6827.24	−0.499058
$573$	77.1760	0.00562666
$574$	221.067	0.0160752
$575$	0	0
$576$	−896.000	−0.0648148
$577$	−8713.28	−0.628663	−0.314331	−	0.949313i	$-0.601780\pi$
$577$	−8713.28	−0.628663	−0.314331	+	0.949313i	$0.601780\pi$
$578$	−8703.38	−0.626320
$579$	−198.215	−0.0142272
$580$	4093.50	0.293057
$581$	−2028.85	−0.144872
$582$	−9684.94	−0.689783
$583$	−14817.9	−1.05265
$584$	9367.95	0.663782
$585$	−8173.74	−0.577679
$586$	−13754.9	−0.969643
$587$	1622.69	0.114098	0.0570491	−	0.998371i	$-0.481831\pi$
$587$	1622.69	0.114098	0.0570491	+	0.998371i	$0.481831\pi$
$588$	−4897.86	−0.343511
$589$	−1501.24	−0.105021
$590$	12173.0	0.849412
$591$	−3462.16	−0.240972
$592$	−699.756	−0.0485807
$593$	2048.67	0.141870	0.0709350	−	0.997481i	$-0.477402\pi$
$593$	2048.67	0.141870	0.0709350	+	0.997481i	$0.477402\pi$
$594$	−6739.85	−0.465554
$595$	−340.375	−0.0234521
$596$	12331.0	0.847481
$597$	7005.46	0.480259
$598$	0	0
$599$	−13615.9	−0.928767	−0.464383	−	0.885634i	$-0.653724\pi$
$599$	−13615.9	−0.928767	−0.464383	+	0.885634i	$0.653724\pi$
$600$	−1851.65	−0.125989
$601$	11927.4	0.809529	0.404765	−	0.914421i	$-0.367353\pi$
$601$	11927.4	0.809529	0.404765	+	0.914421i	$0.367353\pi$
$602$	−541.490	−0.0366603
$603$	13327.4	0.900056
$604$	8402.59	0.566054
$605$	6326.57	0.425144
$606$	3487.17	0.233756
$607$	−12420.7	−0.830545	−0.415272	−	0.909697i	$-0.636314\pi$
$607$	−12420.7	−0.830545	−0.415272	+	0.909697i	$0.636314\pi$
$608$	2792.97	0.186299
$609$	−871.804	−0.0580087
$610$	−903.422	−0.0599648
$611$	12847.2	0.850643
$612$	−1326.75	−0.0876318
$613$	−1484.18	−0.0977905	−0.0488953	−	0.998804i	$-0.515570\pi$
$613$	−1484.18	−0.0977905	−0.0488953	+	0.998804i	$0.515570\pi$
$614$	−14154.8	−0.930360
$615$	−1686.76	−0.110596
$616$	336.000	0.0219770
$617$	6434.71	0.419857	0.209929	−	0.977717i	$-0.432677\pi$
$617$	6434.71	0.419857	0.209929	+	0.977717i	$0.432677\pi$
$618$	8542.76	0.556052
$619$	8881.08	0.576674	0.288337	−	0.957529i	$-0.406898\pi$
$619$	8881.08	0.576674	0.288337	+	0.957529i	$0.406898\pi$
$620$	536.494	0.0347518
$621$	0	0
$622$	−10625.6	−0.684962
$623$	1720.83	0.110664
$624$	−4319.29	−0.277099
$625$	−3479.81	−0.222708
$626$	6062.60	0.387077
$627$	7173.86	0.456932
$628$	12689.6	0.806322
$629$	−1036.16	−0.0656828
$630$	402.267	0.0254392
$631$	4645.22	0.293064	0.146532	−	0.989206i	$-0.453189\pi$
$631$	4645.22	0.293064	0.146532	+	0.989206i	$0.453189\pi$
$632$	6352.40	0.399818
$633$	−13489.2	−0.846994
$634$	13614.3	0.852827
$635$	1924.19	0.120251
$636$	−9374.60	−0.584476
$637$	25427.0	1.58156
$638$	−5983.52	−0.371301
$639$	−6458.59	−0.399840
$640$	−998.119	−0.0616470
$641$	5349.41	0.329624	0.164812	−	0.986325i	$-0.447298\pi$
$641$	5349.41	0.329624	0.164812	+	0.986325i	$0.447298\pi$
$642$	−1818.76	−0.111808
$643$	−28328.5	−1.73743	−0.868714	−	0.495314i	$-0.835053\pi$
$643$	−28328.5	−1.73743	−0.868714	+	0.495314i	$0.835053\pi$
$644$	0	0
$645$	4131.61	0.252220
$646$	4135.69	0.251883
$647$	4873.99	0.296161	0.148081	−	0.988975i	$-0.452690\pi$
$647$	4873.99	0.296161	0.148081	+	0.988975i	$0.452690\pi$
$648$	−1240.00	−0.0751725
$649$	−17793.4	−1.07620
$650$	9612.73	0.580065
$651$	−114.259	−0.00687887
$652$	13404.0	0.805125
$653$	24436.4	1.46442	0.732212	−	0.681077i	$-0.238488\pi$
$653$	24436.4	1.46442	0.732212	+	0.681077i	$0.238488\pi$
$654$	−11568.5	−0.691691
$655$	19744.7	1.17785
$656$	959.906	0.0571312
$657$	−16393.9	−0.973497
$658$	−632.271	−0.0374597
$659$	−10301.4	−0.608929	−0.304464	−	0.952524i	$-0.598477\pi$
$659$	−10301.4	−0.608929	−0.304464	+	0.952524i	$0.598477\pi$
$660$	−2563.71	−0.151200
$661$	−6099.80	−0.358933	−0.179466	−	0.983764i	$-0.557437\pi$
$661$	−6099.80	−0.358933	−0.179466	+	0.983764i	$0.557437\pi$
$662$	12610.7	0.740374
$663$	−6395.77	−0.374647
$664$	−8809.57	−0.514876
$665$	−1253.93	−0.0731208
$666$	1224.57	0.0712481
$667$	0	0
$668$	−7080.73	−0.410122
$669$	−10638.3	−0.614796
$670$	14846.3	0.856066
$671$	1320.55	0.0759748
$672$	212.572	0.0122026
$673$	−19926.5	−1.14132	−0.570662	−	0.821185i	$-0.693313\pi$
$673$	−19926.5	−1.14132	−0.570662	+	0.821185i	$0.693313\pi$
$674$	−13595.8	−0.776990
$675$	9489.69	0.541123
$676$	13635.4	0.775795
$677$	−24188.7	−1.37319	−0.686594	−	0.727041i	$-0.740895\pi$
$677$	−24188.7	−1.37319	−0.686594	+	0.727041i	$0.740895\pi$
$678$	1424.24	0.0806747
$679$	−2474.46	−0.139854
$680$	−1477.96	−0.0833488
$681$	15136.3	0.851725
$682$	−784.200	−0.0440302
$683$	−3461.63	−0.193932	−0.0969660	−	0.995288i	$-0.530914\pi$
$683$	−3461.63	−0.193932	−0.0969660	+	0.995288i	$0.530914\pi$
$684$	−4887.70	−0.273225
$685$	13261.0	0.739673
$686$	−2515.27	−0.139990
$687$	13365.9	0.742274
$688$	−2351.23	−0.130291
$689$	48667.8	2.69099
$690$	0	0
$691$	−12662.9	−0.697134	−0.348567	−	0.937284i	$-0.613332\pi$
$691$	−12662.9	−0.697134	−0.348567	+	0.937284i	$0.613332\pi$
$692$	2621.73	0.144022
$693$	−588.000	−0.0322313
$694$	−8255.48	−0.451547
$695$	4977.73	0.271678
$696$	−3785.51	−0.206163
$697$	1421.38	0.0772432
$698$	−8184.22	−0.443807
$699$	−7627.36	−0.412723
$700$	−473.087	−0.0255443
$701$	4878.72	0.262863	0.131431	−	0.991325i	$-0.458043\pi$
$701$	4878.72	0.262863	0.131431	+	0.991325i	$0.458043\pi$
$702$	22136.3	1.19015
$703$	−3817.19	−0.204791
$704$	1458.96	0.0781062
$705$	4824.28	0.257720
$706$	−2476.98	−0.132043
$707$	890.955	0.0473943
$708$	−11257.1	−0.597552
$709$	30366.6	1.60852	0.804260	−	0.594277i	$-0.202562\pi$
$709$	30366.6	1.60852	0.804260	+	0.594277i	$0.202562\pi$
$710$	−7194.69	−0.380298
$711$	−11116.7	−0.586369
$712$	7472.13	0.393300
$713$	0	0
$714$	314.765	0.0164983
$715$	13309.4	0.696143
$716$	−9250.22	−0.482817
$717$	24528.2	1.27758
$718$	−7930.70	−0.412216
$719$	−14861.5	−0.770848	−0.385424	−	0.922740i	$-0.625945\pi$
$719$	−14861.5	−0.770848	−0.385424	+	0.922740i	$0.625945\pi$
$720$	1746.71	0.0904110
$721$	2182.63	0.112740
$722$	1517.74	0.0782335
$723$	−2159.72	−0.111094
$724$	8347.15	0.428480
$725$	8424.79	0.431571
$726$	−5850.57	−0.299084
$727$	12094.0	0.616974	0.308487	−	0.951229i	$-0.400177\pi$
$727$	12094.0	0.616974	0.308487	+	0.951229i	$0.400177\pi$
$728$	−1103.56	−0.0561821
$729$	16561.0	0.841386
$730$	−18262.3	−0.925918
$731$	−3481.58	−0.176157
$732$	835.450	0.0421846
$733$	−7794.17	−0.392748	−0.196374	−	0.980529i	$-0.562917\pi$
$733$	−7794.17	−0.392748	−0.196374	+	0.980529i	$0.562917\pi$
$734$	−20051.0	−1.00830
$735$	9548.14	0.479168
$736$	0	0
$737$	−21701.1	−1.08463
$738$	−1679.84	−0.0837881
$739$	15351.1	0.764140	0.382070	−	0.924133i	$-0.375211\pi$
$739$	15351.1	0.764140	0.382070	+	0.924133i	$0.375211\pi$
$740$	1364.14	0.0677659
$741$	−23561.8	−1.16810
$742$	−2395.17	−0.118503
$743$	−1899.77	−0.0938032	−0.0469016	−	0.998900i	$-0.514935\pi$
$743$	−1899.77	−0.0938032	−0.0469016	+	0.998900i	$0.514935\pi$
$744$	−496.128	−0.0244475
$745$	−24038.7	−1.18216
$746$	18283.2	0.897314
$747$	15416.8	0.755113
$748$	2160.36	0.105602
$749$	−464.685	−0.0226692
$750$	10638.5	0.517953
$751$	9853.32	0.478765	0.239383	−	0.970925i	$-0.423055\pi$
$751$	9853.32	0.478765	0.239383	+	0.970925i	$0.423055\pi$
$752$	−2745.42	−0.133132
$753$	331.264	0.0160318
$754$	19652.3	0.949196
$755$	−16380.4	−0.789596
$756$	−1089.43	−0.0524104
$757$	−40064.3	−1.92360	−0.961798	−	0.273761i	$-0.911732\pi$
$757$	−40064.3	−1.92360	−0.961798	+	0.273761i	$0.911732\pi$
$758$	−13898.1	−0.665964
$759$	0	0
$760$	−5444.76	−0.259871
$761$	−23242.8	−1.10716	−0.553582	−	0.832794i	$-0.686740\pi$
$761$	−23242.8	−1.10716	−0.553582	+	0.832794i	$0.686740\pi$
$762$	−1779.42	−0.0845953
$763$	−2955.71	−0.140241
$764$	85.6191	0.00405444
$765$	2586.43	0.122239
$766$	20867.9	0.984317
$767$	58440.6	2.75120
$768$	923.021	0.0433680
$769$	8815.52	0.413389	0.206694	−	0.978406i	$-0.433729\pi$
$769$	8815.52	0.413389	0.206694	+	0.978406i	$0.433729\pi$
$770$	−655.015	−0.0306560
$771$	−9477.66	−0.442710
$772$	−219.900	−0.0102518
$773$	−15898.8	−0.739769	−0.369884	−	0.929078i	$-0.620603\pi$
$773$	−15898.8	−0.739769	−0.369884	+	0.929078i	$0.620603\pi$
$774$	4114.66	0.191083
$775$	1104.15	0.0511772
$776$	−10744.5	−0.497041
$777$	−290.525	−0.0134138
$778$	22228.4	1.02433
$779$	5236.31	0.240835
$780$	8420.24	0.386529
$781$	10516.6	0.481835
$782$	0	0
$783$	19400.7	0.885473
$784$	−5433.69	−0.247526
$785$	−24737.8	−1.12475
$786$	−18259.1	−0.828602
$787$	−30097.5	−1.36323	−0.681614	−	0.731712i	$-0.738721\pi$
$787$	−30097.5	−1.36323	−0.681614	+	0.731712i	$0.738721\pi$
$788$	−3840.92	−0.173638
$789$	6697.16	0.302186
$790$	−12383.7	−0.557711
$791$	363.885	0.0163569
$792$	−2553.19	−0.114550
$793$	−4337.20	−0.194222
$794$	−5214.24	−0.233056
$795$	18275.3	0.815294
$796$	7771.86	0.346063
$797$	26253.9	1.16683	0.583413	−	0.812175i	$-0.301717\pi$
$797$	26253.9	1.16683	0.583413	+	0.812175i	$0.301717\pi$
$798$	1159.59	0.0514397
$799$	−4065.27	−0.179999
$800$	−2054.22	−0.0907844
$801$	−13076.2	−0.576811
$802$	22862.2	1.00660
$803$	26694.3	1.17313
$804$	−13729.3	−0.602234
$805$	0	0
$806$	2575.63	0.112559
$807$	5731.55	0.250012
$808$	3868.66	0.168439
$809$	−26332.8	−1.14439	−0.572195	−	0.820117i	$-0.693908\pi$
$809$	−26332.8	−1.14439	−0.572195	+	0.820117i	$0.693908\pi$
$810$	2417.32	0.104859
$811$	−11565.4	−0.500758	−0.250379	−	0.968148i	$-0.580555\pi$
$811$	−11565.4	−0.500758	−0.250379	+	0.968148i	$0.580555\pi$
$812$	−967.179	−0.0417997
$813$	1857.87	0.0801455
$814$	−1993.98	−0.0858588
$815$	−26130.5	−1.12308
$816$	1366.76	0.0586350
$817$	−12826.0	−0.549237
$818$	15251.3	0.651896
$819$	1931.23	0.0823962
$820$	−1871.29	−0.0796930
$821$	−42248.8	−1.79597	−0.897985	−	0.440025i	$-0.854970\pi$
$821$	−42248.8	−1.79597	−0.897985	+	0.440025i	$0.854970\pi$
$822$	−12263.2	−0.520352
$823$	−9432.30	−0.399501	−0.199751	−	0.979847i	$-0.564013\pi$
$823$	−9432.30	−0.399501	−0.199751	+	0.979847i	$0.564013\pi$
$824$	9477.34	0.400678
$825$	−5276.34	−0.222665
$826$	−2876.13	−0.121154
$827$	41067.2	1.72678	0.863389	−	0.504538i	$-0.168337\pi$
$827$	41067.2	1.72678	0.863389	+	0.504538i	$0.168337\pi$
$828$	0	0
$829$	−32113.6	−1.34542	−0.672710	−	0.739907i	$-0.734870\pi$
$829$	−32113.6	−1.34542	−0.672710	+	0.739907i	$0.734870\pi$
$830$	17173.8	0.718207
$831$	−22305.8	−0.931141
$832$	−4791.82	−0.199671
$833$	−8045.92	−0.334663
$834$	−4603.21	−0.191123
$835$	13803.5	0.572085
$836$	7958.69	0.329255
$837$	2542.66	0.105003
$838$	25104.8	1.03488
$839$	−36694.6	−1.50994	−0.754968	−	0.655761i	$-0.772348\pi$
$839$	−36694.6	−1.50994	−0.754968	+	0.655761i	$0.772348\pi$
$840$	−414.399	−0.0170216
$841$	−7165.36	−0.293795
$842$	−10572.5	−0.432721
$843$	19991.0	0.816758
$844$	−14964.9	−0.610324
$845$	−26581.5	−1.08217
$846$	4804.48	0.195250
$847$	−1494.79	−0.0606395
$848$	−10400.2	−0.421160
$849$	−27229.0	−1.10070
$850$	−3041.78	−0.122744
$851$	0	0
$852$	6653.37	0.267536
$853$	−18306.3	−0.734815	−0.367407	−	0.930060i	$-0.619755\pi$
$853$	−18306.3	−0.734815	−0.367407	+	0.930060i	$0.619755\pi$
$854$	213.454	0.00855296
$855$	9528.34	0.381125
$856$	−2017.73	−0.0805662
$857$	2051.88	0.0817862	0.0408931	−	0.999164i	$-0.486980\pi$
$857$	2051.88	0.0817862	0.0408931	+	0.999164i	$0.486980\pi$
$858$	−12308.0	−0.489729
$859$	9292.08	0.369082	0.184541	−	0.982825i	$-0.440920\pi$
$859$	9292.08	0.369082	0.184541	+	0.982825i	$0.440920\pi$
$860$	4583.62	0.181744
$861$	398.534	0.0157747
$862$	28772.0	1.13687
$863$	−10654.1	−0.420242	−0.210121	−	0.977675i	$-0.567386\pi$
$863$	−10654.1	−0.420242	−0.210121	+	0.977675i	$0.567386\pi$
$864$	−4730.48	−0.186267
$865$	−5110.94	−0.200898
$866$	14030.9	0.550567
$867$	−15690.2	−0.614612
$868$	−126.758	−0.00495675
$869$	18101.4	0.706614
$870$	7379.66	0.287579
$871$	71275.1	2.77275
$872$	−12834.1	−0.498416
$873$	18802.8	0.728957
$874$	0	0
$875$	2718.10	0.105015
$876$	16888.3	0.651373
$877$	6310.78	0.242987	0.121494	−	0.992592i	$-0.461232\pi$
$877$	6310.78	0.242987	0.121494	+	0.992592i	$0.461232\pi$
$878$	−2999.40	−0.115290
$879$	−24797.1	−0.951517
$880$	−2844.18	−0.108951
$881$	−2992.09	−0.114422	−0.0572111	−	0.998362i	$-0.518221\pi$
$881$	−2992.09	−0.114422	−0.0572111	+	0.998362i	$0.518221\pi$
$882$	9508.96	0.363019
$883$	−28610.9	−1.09041	−0.545205	−	0.838303i	$-0.683548\pi$
$883$	−28610.9	−1.09041	−0.545205	+	0.838303i	$0.683548\pi$
$884$	−7095.47	−0.269962
$885$	21945.1	0.833534
$886$	27851.6	1.05609
$887$	−20621.7	−0.780618	−0.390309	−	0.920684i	$-0.627632\pi$
$887$	−20621.7	−0.780618	−0.390309	+	0.920684i	$0.627632\pi$
$888$	−1261.50	−0.0476726
$889$	−454.633	−0.0171518
$890$	−14566.5	−0.548620
$891$	−3533.43	−0.132856
$892$	−11802.1	−0.443008
$893$	−14976.3	−0.561214
$894$	22230.1	0.831638
$895$	18032.8	0.673488
$896$	235.828	0.00879291
$897$	0	0
$898$	21509.7	0.799320
$899$	2257.33	0.0837443
$900$	3594.88	0.133144
$901$	−15400.0	−0.569422
$902$	2735.29	0.100970
$903$	−976.185	−0.0359750
$904$	1580.05	0.0581323
$905$	−16272.3	−0.597692
$906$	15148.0	0.555472
$907$	−12562.7	−0.459908	−0.229954	−	0.973201i	$-0.573858\pi$
$907$	−12562.7	−0.459908	−0.229954	+	0.973201i	$0.573858\pi$
$908$	16792.2	0.613733
$909$	−6770.16	−0.247032
$910$	2151.33	0.0783691
$911$	−41139.8	−1.49618	−0.748091	−	0.663596i	$-0.769029\pi$
$911$	−41139.8	−1.49618	−0.748091	+	0.663596i	$0.769029\pi$
$912$	5035.10	0.182817
$913$	−25103.2	−0.909962
$914$	−29508.5	−1.06789
$915$	−1628.67	−0.0588438
$916$	14828.2	0.534866
$917$	−4665.12	−0.168000
$918$	−7004.65	−0.251839
$919$	−43420.9	−1.55857	−0.779284	−	0.626671i	$-0.784417\pi$
$919$	−43420.9	−1.55857	−0.779284	+	0.626671i	$0.784417\pi$
$920$	0	0
$921$	−25517.9	−0.912969
$922$	28468.5	1.01688
$923$	−34540.6	−1.23176
$924$	605.733	0.0215662
$925$	2807.52	0.0997954
$926$	25556.0	0.906937
$927$	−16585.3	−0.587631
$928$	−4199.64	−0.148556
$929$	−7268.51	−0.256697	−0.128349	−	0.991729i	$-0.540968\pi$
$929$	−7268.51	−0.256697	−0.128349	+	0.991729i	$0.540968\pi$
$930$	967.177	0.0341021
$931$	−29640.9	−1.04344
$932$	−8461.80	−0.297398
$933$	−19155.5	−0.672158
$934$	41.0032	0.00143647
$935$	−4211.51	−0.147306
$936$	8385.68	0.292836
$937$	25991.6	0.906198	0.453099	−	0.891460i	$-0.350318\pi$
$937$	25991.6	0.906198	0.453099	+	0.891460i	$0.350318\pi$
$938$	−3507.78	−0.122103
$939$	10929.5	0.379841
$940$	5352.06	0.185707
$941$	−18872.1	−0.653786	−0.326893	−	0.945061i	$-0.606002\pi$
$941$	−18872.1	−0.653786	−0.326893	+	0.945061i	$0.606002\pi$
$942$	22876.5	0.791249
$943$	0	0
$944$	−12488.6	−0.430582
$945$	2123.79	0.0731080
$946$	−6699.94	−0.230268
$947$	51374.6	1.76288	0.881441	−	0.472294i	$-0.156574\pi$
$947$	51374.6	1.76288	0.881441	+	0.472294i	$0.156574\pi$
$948$	11451.9	0.392344
$949$	−87674.8	−2.99899
$950$	−11205.8	−0.382699
$951$	24543.5	0.836885
$952$	349.201	0.0118883
$953$	18673.5	0.634727	0.317364	−	0.948304i	$-0.397202\pi$
$953$	18673.5	0.634727	0.317364	+	0.948304i	$0.397202\pi$
$954$	18200.3	0.617670
$955$	−166.910	−0.00565559
$956$	27211.6	0.920591
$957$	−10787.0	−0.364360
$958$	−39099.3	−1.31862
$959$	−3133.20	−0.105502
$960$	−1799.38	−0.0604946
$961$	−29495.2	−0.990069
$962$	6549.03	0.219490
$963$	3531.03	0.118158
$964$	−2396.00	−0.0800517
$965$	428.684	0.0143003
$966$	0	0
$967$	23675.0	0.787319	0.393659	−	0.919256i	$-0.371209\pi$
$967$	23675.0	0.787319	0.393659	+	0.919256i	$0.371209\pi$
$968$	−6490.62	−0.215513
$969$	7455.71	0.247174
$970$	20945.8	0.693330
$971$	−47302.8	−1.56336	−0.781678	−	0.623682i	$-0.785636\pi$
$971$	−47302.8	−1.56336	−0.781678	+	0.623682i	$0.785636\pi$
$972$	13729.9	0.453074
$973$	−1176.10	−0.0387503
$974$	16505.2	0.542978
$975$	17329.6	0.569222
$976$	926.848	0.0303972
$977$	37093.1	1.21465	0.607326	−	0.794453i	$-0.292242\pi$
$977$	37093.1	1.21465	0.607326	+	0.794453i	$0.292242\pi$
$978$	24164.4	0.790075
$979$	21292.1	0.695096
$980$	10592.7	0.345277
$981$	22459.7	0.730973
$982$	−6974.76	−0.226653
$983$	42409.4	1.37604	0.688021	−	0.725691i	$-0.258480\pi$
$983$	42409.4	1.37604	0.688021	+	0.725691i	$0.258480\pi$
$984$	1730.50	0.0560632
$985$	7487.68	0.242211
$986$	−6218.61	−0.200853
$987$	−1139.84	−0.0367595
$988$	−26139.5	−0.841709
$989$	0	0
$990$	4977.31	0.159787
$991$	57549.3	1.84472	0.922359	−	0.386335i	$-0.126259\pi$
$991$	57549.3	1.84472	0.922359	+	0.386335i	$0.126259\pi$
$992$	−550.405	−0.0176163
$993$	22734.2	0.726534
$994$	1699.90	0.0542431
$995$	−15150.9	−0.482728
$996$	−15881.7	−0.505251
$997$	−11761.0	−0.373596	−0.186798	−	0.982398i	$-0.559811\pi$
$997$	−11761.0	−0.373596	−0.186798	+	0.982398i	$0.559811\pi$
$998$	−9405.62	−0.298326
$999$	6465.20	0.204755

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1058.4.a.o.1.3	✓	4
23.22	odd	2	inner	1058.4.a.o.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1058.4.a.o.1.3	✓	4	1.1	even	1	trivial
1058.4.a.o.1.4	yes	4	23.22	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:	\( N \)	\(=\)	\( 1058 = 2 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1058.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +3.60555 q^{3} +4.00000 q^{4} -7.79780 q^{5} +7.21110 q^{6} +1.84240 q^{7} +8.00000 q^{8} -14.0000 q^{9} -15.5956 q^{10} +22.7963 q^{11} +14.4222 q^{12} -74.8722 q^{13} +3.68481 q^{14} -28.1154 q^{15} +16.0000 q^{16} +23.6919 q^{17} -28.0000 q^{18} +87.2804 q^{19} -31.1912 q^{20} +6.64288 q^{21} +45.5926 q^{22} +28.8444 q^{24} -64.1943 q^{25} -149.744 q^{26} -147.828 q^{27} +7.36961 q^{28} -131.239 q^{29} -56.2307 q^{30} -17.2002 q^{31} +32.0000 q^{32} +82.1933 q^{33} +47.3839 q^{34} -14.3667 q^{35} -56.0000 q^{36} -43.7347 q^{37} +174.561 q^{38} -269.955 q^{39} -62.3824 q^{40} +59.9941 q^{41} +13.2858 q^{42} -146.952 q^{43} +91.1853 q^{44} +109.169 q^{45} -171.589 q^{47} +57.6888 q^{48} -339.606 q^{49} -128.389 q^{50} +85.4225 q^{51} -299.489 q^{52} -650.011 q^{53} -295.655 q^{54} -177.761 q^{55} +14.7392 q^{56} +314.694 q^{57} -262.478 q^{58} -780.538 q^{59} -112.461 q^{60} +57.9280 q^{61} -34.4003 q^{62} -25.7936 q^{63} +64.0000 q^{64} +583.838 q^{65} +164.387 q^{66} -951.957 q^{67} +94.7678 q^{68} -28.7334 q^{70} +461.328 q^{71} -112.000 q^{72} +1170.99 q^{73} -87.4695 q^{74} -231.456 q^{75} +349.122 q^{76} +42.0000 q^{77} -539.911 q^{78} +794.049 q^{79} -124.765 q^{80} -155.000 q^{81} +119.988 q^{82} -1101.20 q^{83} +26.5715 q^{84} -184.745 q^{85} -293.904 q^{86} -473.188 q^{87} +182.371 q^{88} +934.016 q^{89} +218.338 q^{90} -137.945 q^{91} -62.0160 q^{93} -343.177 q^{94} -680.595 q^{95} +115.378 q^{96} -1343.06 q^{97} -679.211 q^{98} -319.148 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{2} + 16 q^{4} + 32 q^{8} - 56 q^{9} - 112 q^{13} + 64 q^{16} - 112 q^{18} + 248 q^{25} - 224 q^{26} - 424 q^{29} - 588 q^{31} + 128 q^{32} - 144 q^{35} - 224 q^{36} - 676 q^{39} - 784 q^{41}+ \cdots - 2688 q^{98}+O(q^{100}) \) 4 * q + 8 * q^2 + 16 * q^4 + 32 * q^8 - 56 * q^9 - 112 * q^13 + 64 * q^16 - 112 * q^18 + 248 * q^25 - 224 * q^26 - 424 * q^29 - 588 * q^31 + 128 * q^32 - 144 * q^35 - 224 * q^36 - 676 * q^39 - 784 * q^41 - 196 * q^47 - 1344 * q^49 + 496 * q^50 - 448 * q^52 - 812 * q^55 - 848 * q^58 - 1204 * q^59 - 1176 * q^62 + 256 * q^64 - 288 * q^70 + 2552 * q^71 - 448 * q^72 + 1064 * q^73 - 1820 * q^75 + 168 * q^77 - 1352 * q^78 - 620 * q^81 - 1568 * q^82 - 2960 * q^85 - 364 * q^87 + 1872 * q^93 - 392 * q^94 - 5852 * q^95 - 2688 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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