LMFDB - Embedded newform 2151.4.a.g.1.52

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,4,Mod(1,2151)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2151.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2151.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$126.913108422$
Analytic rank:	$1$
Dimension:	$59$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.52
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.45149 q^{2} +11.8157 q^{4} -14.0297 q^{5} +23.7055 q^{7} +16.9858 q^{8} +O(q^{10})$	\(q+4.45149 q^{2} +11.8157 q^{4} -14.0297 q^{5} +23.7055 q^{7} +16.9858 q^{8} -62.4530 q^{10} -36.5037 q^{11} +56.5575 q^{13} +105.525 q^{14} -18.9141 q^{16} +7.01736 q^{17} -45.1390 q^{19} -165.771 q^{20} -162.496 q^{22} -174.255 q^{23} +71.8322 q^{25} +251.765 q^{26} +280.099 q^{28} +226.036 q^{29} +97.8926 q^{31} -220.082 q^{32} +31.2377 q^{34} -332.581 q^{35} -338.487 q^{37} -200.936 q^{38} -238.305 q^{40} +481.138 q^{41} -58.7919 q^{43} -431.318 q^{44} -775.695 q^{46} -407.573 q^{47} +218.952 q^{49} +319.760 q^{50} +668.269 q^{52} -302.026 q^{53} +512.135 q^{55} +402.656 q^{56} +1006.20 q^{58} -785.873 q^{59} -81.0979 q^{61} +435.768 q^{62} -828.379 q^{64} -793.484 q^{65} +813.974 q^{67} +82.9154 q^{68} -1480.48 q^{70} -437.780 q^{71} -631.448 q^{73} -1506.77 q^{74} -533.351 q^{76} -865.339 q^{77} -757.443 q^{79} +265.359 q^{80} +2141.78 q^{82} -650.447 q^{83} -98.4514 q^{85} -261.712 q^{86} -620.043 q^{88} -806.535 q^{89} +1340.73 q^{91} -2058.95 q^{92} -1814.31 q^{94} +633.286 q^{95} +1218.57 q^{97} +974.663 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8}+O(q^{10}) $ 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8	$ 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8} - 36 q^{10} - 132 q^{11} + 104 q^{13} - 280 q^{14} + 822 q^{16} - 408 q^{17} + 20 q^{19} - 800 q^{20} - 2 q^{22} - 276 q^{23} + 1477 q^{25} - 780 q^{26} + 224 q^{28} - 696 q^{29} - 380 q^{31} - 896 q^{32} - 72 q^{34} - 700 q^{35} + 224 q^{37} - 988 q^{38} - 258 q^{40} - 2706 q^{41} - 156 q^{43} - 1584 q^{44} + 428 q^{46} - 1316 q^{47} + 2135 q^{49} - 1400 q^{50} + 1092 q^{52} - 1484 q^{53} - 992 q^{55} - 3360 q^{56} - 120 q^{58} - 3186 q^{59} - 254 q^{61} - 1240 q^{62} + 3054 q^{64} - 5120 q^{65} + 288 q^{67} - 9420 q^{68} + 1108 q^{70} - 4468 q^{71} - 1770 q^{73} - 6214 q^{74} + 720 q^{76} - 6352 q^{77} - 746 q^{79} - 7040 q^{80} + 276 q^{82} - 5484 q^{83} + 588 q^{85} - 10152 q^{86} + 1186 q^{88} - 11570 q^{89} + 1768 q^{91} - 15366 q^{92} - 2142 q^{94} - 5736 q^{95} + 2390 q^{97} - 6912 q^{98}+O(q^{100}) $ 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8 - 36 * q^10 - 132 * q^11 + 104 * q^13 - 280 * q^14 + 822 * q^16 - 408 * q^17 + 20 * q^19 - 800 * q^20 - 2 * q^22 - 276 * q^23 + 1477 * q^25 - 780 * q^26 + 224 * q^28 - 696 * q^29 - 380 * q^31 - 896 * q^32 - 72 * q^34 - 700 * q^35 + 224 * q^37 - 988 * q^38 - 258 * q^40 - 2706 * q^41 - 156 * q^43 - 1584 * q^44 + 428 * q^46 - 1316 * q^47 + 2135 * q^49 - 1400 * q^50 + 1092 * q^52 - 1484 * q^53 - 992 * q^55 - 3360 * q^56 - 120 * q^58 - 3186 * q^59 - 254 * q^61 - 1240 * q^62 + 3054 * q^64 - 5120 * q^65 + 288 * q^67 - 9420 * q^68 + 1108 * q^70 - 4468 * q^71 - 1770 * q^73 - 6214 * q^74 + 720 * q^76 - 6352 * q^77 - 746 * q^79 - 7040 * q^80 + 276 * q^82 - 5484 * q^83 + 588 * q^85 - 10152 * q^86 + 1186 * q^88 - 11570 * q^89 + 1768 * q^91 - 15366 * q^92 - 2142 * q^94 - 5736 * q^95 + 2390 * q^97 - 6912 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	4.45149	1.57384	0.786919	−	0.617056i	$-0.211675\pi$
$2$	4.45149	1.57384	0.786919	+	0.617056i	$0.211675\pi$
$3$	0	0
$3$	0	0
$4$	11.8157	1.47697
$5$	−14.0297	−1.25485	−0.627427	−	0.778676i	$-0.715892\pi$
$5$	−14.0297	−1.25485	−0.627427	+	0.778676i	$0.715892\pi$
$6$	0	0
$7$	23.7055	1.27998	0.639989	−	0.768384i	$-0.278939\pi$
$7$	23.7055	1.27998	0.639989	+	0.768384i	$0.278939\pi$
$8$	16.9858	0.750672
$9$	0	0
$10$	−62.4530	−1.97494
$11$	−36.5037	−1.00057	−0.500285	−	0.865861i	$-0.666771\pi$
$11$	−36.5037	−1.00057	−0.500285	+	0.865861i	$0.666771\pi$
$12$	0	0
$13$	56.5575	1.20663	0.603316	−	0.797502i	$-0.293846\pi$
$13$	56.5575	1.20663	0.603316	+	0.797502i	$0.293846\pi$
$14$	105.525	2.01448
$15$	0	0
$16$	−18.9141	−0.295532
$17$	7.01736	0.100115	0.0500577	−	0.998746i	$-0.484059\pi$
$17$	7.01736	0.100115	0.0500577	+	0.998746i	$0.484059\pi$
$18$	0	0
$19$	−45.1390	−0.545031	−0.272516	−	0.962151i	$-0.587856\pi$
$19$	−45.1390	−0.545031	−0.272516	+	0.962151i	$0.587856\pi$
$20$	−165.771	−1.85338
$21$	0	0
$22$	−162.496	−1.57474
$23$	−174.255	−1.57977	−0.789885	−	0.613255i	$-0.789860\pi$
$23$	−174.255	−1.57977	−0.789885	+	0.613255i	$0.789860\pi$
$24$	0	0
$25$	71.8322	0.574657
$26$	251.765	1.89905
$27$	0	0
$28$	280.099	1.89049
$29$	226.036	1.44737	0.723686	−	0.690130i	$-0.242446\pi$
$29$	226.036	1.44737	0.723686	+	0.690130i	$0.242446\pi$
$30$	0	0
$31$	97.8926	0.567162	0.283581	−	0.958948i	$-0.408477\pi$
$31$	97.8926	0.567162	0.283581	+	0.958948i	$0.408477\pi$
$32$	−220.082	−1.21579
$33$	0	0
$34$	31.2377	0.157565
$35$	−332.581	−1.60619
$36$	0	0
$37$	−338.487	−1.50397	−0.751984	−	0.659181i	$-0.770903\pi$
$37$	−338.487	−1.50397	−0.751984	+	0.659181i	$0.770903\pi$
$38$	−200.936	−0.857791
$39$	0	0
$40$	−238.305	−0.941983
$41$	481.138	1.83271	0.916356	−	0.400365i	$-0.131117\pi$
$41$	481.138	1.83271	0.916356	+	0.400365i	$0.131117\pi$
$42$	0	0
$43$	−58.7919	−0.208504	−0.104252	−	0.994551i	$-0.533245\pi$
$43$	−58.7919	−0.208504	−0.104252	+	0.994551i	$0.533245\pi$
$44$	−431.318	−1.47781
$45$	0	0
$46$	−775.695	−2.48630
$47$	−407.573	−1.26491	−0.632454	−	0.774598i	$-0.717952\pi$
$47$	−407.573	−1.26491	−0.632454	+	0.774598i	$0.717952\pi$
$48$	0	0
$49$	218.952	0.638344
$50$	319.760	0.904418
$51$	0	0
$52$	668.269	1.78216
$53$	−302.026	−0.782762	−0.391381	−	0.920229i	$-0.628003\pi$
$53$	−302.026	−0.782762	−0.391381	+	0.920229i	$0.628003\pi$
$54$	0	0
$55$	512.135	1.25557
$56$	402.656	0.960844
$57$	0	0
$58$	1006.20	2.27793
$59$	−785.873	−1.73410	−0.867050	−	0.498221i	$-0.833987\pi$
$59$	−785.873	−1.73410	−0.867050	+	0.498221i	$0.833987\pi$
$60$	0	0
$61$	−81.0979	−0.170222	−0.0851108	−	0.996371i	$-0.527124\pi$
$61$	−81.0979	−0.170222	−0.0851108	+	0.996371i	$0.527124\pi$
$62$	435.768	0.892622
$63$	0	0
$64$	−828.379	−1.61793
$65$	−793.484	−1.51415
$66$	0	0
$67$	813.974	1.48422	0.742110	−	0.670278i	$-0.233825\pi$
$67$	813.974	1.48422	0.742110	+	0.670278i	$0.233825\pi$
$68$	82.9154	0.147867
$69$	0	0
$70$	−1480.48	−2.52788
$71$	−437.780	−0.731760	−0.365880	−	0.930662i	$-0.619232\pi$
$71$	−437.780	−0.731760	−0.365880	+	0.930662i	$0.619232\pi$
$72$	0	0
$73$	−631.448	−1.01240	−0.506201	−	0.862415i	$-0.668951\pi$
$73$	−631.448	−1.01240	−0.506201	+	0.862415i	$0.668951\pi$
$74$	−1506.77	−2.36700
$75$	0	0
$76$	−533.351	−0.804994
$77$	−865.339	−1.28071
$78$	0	0
$79$	−757.443	−1.07872	−0.539361	−	0.842075i	$-0.681334\pi$
$79$	−757.443	−1.07872	−0.539361	+	0.842075i	$0.681334\pi$
$80$	265.359	0.370850
$81$	0	0
$82$	2141.78	2.88439
$83$	−650.447	−0.860191	−0.430096	−	0.902783i	$-0.641520\pi$
$83$	−650.447	−0.860191	−0.430096	+	0.902783i	$0.641520\pi$
$84$	0	0
$85$	−98.4514	−0.125630
$86$	−261.712	−0.328152
$87$	0	0
$88$	−620.043	−0.751100
$89$	−806.535	−0.960590	−0.480295	−	0.877107i	$-0.659470\pi$
$89$	−806.535	−0.960590	−0.480295	+	0.877107i	$0.659470\pi$
$90$	0	0
$91$	1340.73	1.54446
$92$	−2058.95	−2.33327
$93$	0	0
$94$	−1814.31	−1.99076
$95$	633.286	0.683934
$96$	0	0
$97$	1218.57	1.27554	0.637768	−	0.770228i	$-0.279858\pi$
$97$	1218.57	1.27554	0.637768	+	0.770228i	$0.279858\pi$
$98$	974.663	1.00465
$99$	0	0
$100$	848.751	0.848751
$101$	1266.55	1.24779	0.623895	−	0.781508i	$-0.285549\pi$
$101$	1266.55	1.24779	0.623895	+	0.781508i	$0.285549\pi$
$102$	0	0
$103$	1515.00	1.44930	0.724650	−	0.689117i	$-0.242002\pi$
$103$	1515.00	1.44930	0.724650	+	0.689117i	$0.242002\pi$
$104$	960.672	0.905785
$105$	0	0
$106$	−1344.46	−1.23194
$107$	−111.078	−0.100358	−0.0501789	−	0.998740i	$-0.515979\pi$
$107$	−111.078	−0.100358	−0.0501789	+	0.998740i	$0.515979\pi$
$108$	0	0
$109$	−1871.22	−1.64431	−0.822157	−	0.569260i	$-0.807230\pi$
$109$	−1871.22	−1.64431	−0.822157	+	0.569260i	$0.807230\pi$
$110$	2279.76	1.97606
$111$	0	0
$112$	−448.368	−0.378275
$113$	−2092.28	−1.74182	−0.870908	−	0.491446i	$-0.836468\pi$
$113$	−2092.28	−1.74182	−0.870908	+	0.491446i	$0.836468\pi$
$114$	0	0
$115$	2444.74	1.98238
$116$	2670.78	2.13772
$117$	0	0
$118$	−3498.30	−2.72919
$119$	166.350	0.128145
$120$	0	0
$121$	1.51748	0.00114011
$122$	−361.006	−0.267901
$123$	0	0
$124$	1156.67	0.837681
$125$	745.928	0.533743
$126$	0	0
$127$	384.445	0.268614	0.134307	−	0.990940i	$-0.457119\pi$
$127$	384.445	0.268614	0.134307	+	0.990940i	$0.457119\pi$
$128$	−1926.87	−1.33057
$129$	0	0
$130$	−3532.19	−2.38302
$131$	−2035.31	−1.35745	−0.678724	−	0.734393i	$-0.737467\pi$
$131$	−2035.31	−1.35745	−0.678724	+	0.734393i	$0.737467\pi$
$132$	0	0
$133$	−1070.04	−0.697628
$134$	3623.40	2.33592
$135$	0	0
$136$	119.195	0.0751537
$137$	−2884.67	−1.79894	−0.899468	−	0.436987i	$-0.856046\pi$
$137$	−2884.67	−1.79894	−0.899468	+	0.436987i	$0.856046\pi$
$138$	0	0
$139$	1709.80	1.04333	0.521666	−	0.853150i	$-0.325311\pi$
$139$	1709.80	1.04333	0.521666	+	0.853150i	$0.325311\pi$
$140$	−3929.70	−2.37229
$141$	0	0
$142$	−1948.77	−1.15167
$143$	−2064.56	−1.20732
$144$	0	0
$145$	−3171.21	−1.81624
$146$	−2810.88	−1.59336
$147$	0	0
$148$	−3999.47	−2.22131
$149$	414.246	0.227761	0.113880	−	0.993494i	$-0.463672\pi$
$149$	414.246	0.227761	0.113880	+	0.993494i	$0.463672\pi$
$150$	0	0
$151$	1120.72	0.603991	0.301995	−	0.953309i	$-0.402347\pi$
$151$	1120.72	0.603991	0.301995	+	0.953309i	$0.402347\pi$
$152$	−766.720	−0.409140
$153$	0	0
$154$	−3852.05	−2.01563
$155$	−1373.40	−0.711706
$156$	0	0
$157$	−346.520	−0.176149	−0.0880743	−	0.996114i	$-0.528071\pi$
$157$	−346.520	−0.176149	−0.0880743	+	0.996114i	$0.528071\pi$
$158$	−3371.75	−1.69773
$159$	0	0
$160$	3087.68	1.52564
$161$	−4130.81	−2.02207
$162$	0	0
$163$	−3269.21	−1.57095	−0.785473	−	0.618896i	$-0.787580\pi$
$163$	−3269.21	−1.57095	−0.785473	+	0.618896i	$0.787580\pi$
$164$	5685.01	2.70686
$165$	0	0
$166$	−2895.46	−1.35380
$167$	281.159	0.130280	0.0651399	−	0.997876i	$-0.479251\pi$
$167$	281.159	0.130280	0.0651399	+	0.997876i	$0.479251\pi$
$168$	0	0
$169$	1001.75	0.455963
$170$	−438.255	−0.197721
$171$	0	0
$172$	−694.671	−0.307954
$173$	1132.71	0.497794	0.248897	−	0.968530i	$-0.419932\pi$
$173$	1132.71	0.497794	0.248897	+	0.968530i	$0.419932\pi$
$174$	0	0
$175$	1702.82	0.735549
$176$	690.433	0.295701
$177$	0	0
$178$	−3590.28	−1.51181
$179$	2743.00	1.14537	0.572685	−	0.819775i	$-0.305902\pi$
$179$	2743.00	1.14537	0.572685	+	0.819775i	$0.305902\pi$
$180$	0	0
$181$	−3516.42	−1.44405	−0.722025	−	0.691867i	$-0.756789\pi$
$181$	−3516.42	−1.44405	−0.722025	+	0.691867i	$0.756789\pi$
$182$	5968.22	2.43074
$183$	0	0
$184$	−2959.86	−1.18589
$185$	4748.86	1.88726
$186$	0	0
$187$	−256.159	−0.100172
$188$	−4815.78	−1.86823
$189$	0	0
$190$	2819.07	1.07640
$191$	817.974	0.309877	0.154939	−	0.987924i	$-0.450482\pi$
$191$	817.974	0.309877	0.154939	+	0.987924i	$0.450482\pi$
$192$	0	0
$193$	−1107.75	−0.413148	−0.206574	−	0.978431i	$-0.566231\pi$
$193$	−1107.75	−0.413148	−0.206574	+	0.978431i	$0.566231\pi$
$194$	5424.45	2.00749
$195$	0	0
$196$	2587.08	0.942815
$197$	−3084.42	−1.11551	−0.557756	−	0.830005i	$-0.688338\pi$
$197$	−3084.42	−1.11551	−0.557756	+	0.830005i	$0.688338\pi$
$198$	0	0
$199$	−1855.10	−0.660828	−0.330414	−	0.943836i	$-0.607188\pi$
$199$	−1855.10	−0.660828	−0.330414	+	0.943836i	$0.607188\pi$
$200$	1220.12	0.431379
$201$	0	0
$202$	5638.05	1.96382
$203$	5358.30	1.85260
$204$	0	0
$205$	−6750.22	−2.29978
$206$	6744.03	2.28096
$207$	0	0
$208$	−1069.73	−0.356599
$209$	1647.74	0.545342
$210$	0	0
$211$	5350.57	1.74573	0.872864	−	0.487964i	$-0.162260\pi$
$211$	5350.57	1.74573	0.872864	+	0.487964i	$0.162260\pi$
$212$	−3568.66	−1.15612
$213$	0	0
$214$	−494.461	−0.157947
$215$	824.833	0.261642
$216$	0	0
$217$	2320.60	0.725955
$218$	−8329.71	−2.58789
$219$	0	0
$220$	6051.26	1.85444
$221$	396.885	0.120802
$222$	0	0
$223$	2047.76	0.614923	0.307462	−	0.951560i	$-0.400520\pi$
$223$	2047.76	0.614923	0.307462	+	0.951560i	$0.400520\pi$
$224$	−5217.16	−1.55619
$225$	0	0
$226$	−9313.76	−2.74134
$227$	1577.39	0.461211	0.230605	−	0.973047i	$-0.425929\pi$
$227$	1577.39	0.461211	0.230605	+	0.973047i	$0.425929\pi$
$228$	0	0
$229$	−3083.95	−0.889926	−0.444963	−	0.895549i	$-0.646783\pi$
$229$	−3083.95	−0.889926	−0.444963	+	0.895549i	$0.646783\pi$
$230$	10882.8	3.11995
$231$	0	0
$232$	3839.39	1.08650
$233$	6317.07	1.77616	0.888080	−	0.459689i	$-0.152039\pi$
$233$	6317.07	1.77616	0.888080	+	0.459689i	$0.152039\pi$
$234$	0	0
$235$	5718.13	1.58727
$236$	−9285.68	−2.56121
$237$	0	0
$238$	740.507	0.201680
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	−3832.63	−1.02440	−0.512202	−	0.858865i	$-0.671170\pi$
$241$	−3832.63	−1.02440	−0.512202	+	0.858865i	$0.671170\pi$
$242$	6.75506	0.00179434
$243$	0	0
$244$	−958.232	−0.251412
$245$	−3071.83	−0.801029
$246$	0	0
$247$	−2552.95	−0.657653
$248$	1662.78	0.425753
$249$	0	0
$250$	3320.49	0.840025
$251$	−5227.23	−1.31450	−0.657251	−	0.753672i	$-0.728281\pi$
$251$	−5227.23	−1.31450	−0.657251	+	0.753672i	$0.728281\pi$
$252$	0	0
$253$	6360.95	1.58067
$254$	1711.35	0.422755
$255$	0	0
$256$	−1950.39	−0.476169
$257$	−7083.43	−1.71927	−0.859635	−	0.510909i	$-0.829309\pi$
$257$	−7083.43	−1.71927	−0.859635	+	0.510909i	$0.829309\pi$
$258$	0	0
$259$	−8024.00	−1.92505
$260$	−9375.61	−2.23635
$261$	0	0
$262$	−9060.16	−2.13641
$263$	1996.26	0.468040	0.234020	−	0.972232i	$-0.424812\pi$
$263$	1996.26	0.468040	0.234020	+	0.972232i	$0.424812\pi$
$264$	0	0
$265$	4237.33	0.982252
$266$	−4763.29	−1.09795
$267$	0	0
$268$	9617.71	2.19215
$269$	7347.36	1.66534	0.832670	−	0.553770i	$-0.186811\pi$
$269$	7347.36	1.66534	0.832670	+	0.553770i	$0.186811\pi$
$270$	0	0
$271$	−289.105	−0.0648041	−0.0324020	−	0.999475i	$-0.510316\pi$
$271$	−289.105	−0.0648041	−0.0324020	+	0.999475i	$0.510316\pi$
$272$	−132.727	−0.0295873
$273$	0	0
$274$	−12841.1	−2.83123
$275$	−2622.14	−0.574985
$276$	0	0
$277$	−74.8453	−0.0162347	−0.00811736	−	0.999967i	$-0.502584\pi$
$277$	−74.8453	−0.0162347	−0.00811736	+	0.999967i	$0.502584\pi$
$278$	7611.15	1.64204
$279$	0	0
$280$	−5649.15	−1.20572
$281$	−5156.24	−1.09464	−0.547322	−	0.836922i	$-0.684353\pi$
$281$	−5156.24	−1.09464	−0.547322	+	0.836922i	$0.684353\pi$
$282$	0	0
$283$	5977.40	1.25555	0.627773	−	0.778397i	$-0.283967\pi$
$283$	5977.40	1.25555	0.627773	+	0.778397i	$0.283967\pi$
$284$	−5172.70	−1.08079
$285$	0	0
$286$	−9190.35	−1.90013
$287$	11405.6	2.34583
$288$	0	0
$289$	−4863.76	−0.989977
$290$	−14116.6	−2.85847
$291$	0	0
$292$	−7461.03	−1.49529
$293$	507.037	0.101097	0.0505485	−	0.998722i	$-0.483903\pi$
$293$	507.037	0.101097	0.0505485	+	0.998722i	$0.483903\pi$
$294$	0	0
$295$	11025.6	2.17604
$296$	−5749.45	−1.12899
$297$	0	0
$298$	1844.01	0.358459
$299$	−9855.43	−1.90620
$300$	0	0
$301$	−1393.69	−0.266881
$302$	4988.85	0.950584
$303$	0	0
$304$	853.762	0.161074
$305$	1137.78	0.213603
$306$	0	0
$307$	−6594.14	−1.22589	−0.612944	−	0.790126i	$-0.710015\pi$
$307$	−6594.14	−1.22589	−0.612944	+	0.790126i	$0.710015\pi$
$308$	−10224.6	−1.89157
$309$	0	0
$310$	−6113.69	−1.12011
$311$	−2390.42	−0.435847	−0.217924	−	0.975966i	$-0.569928\pi$
$311$	−2390.42	−0.435847	−0.217924	+	0.975966i	$0.569928\pi$
$312$	0	0
$313$	3916.45	0.707256	0.353628	−	0.935386i	$-0.384948\pi$
$313$	3916.45	0.707256	0.353628	+	0.935386i	$0.384948\pi$
$314$	−1542.53	−0.277229
$315$	0	0
$316$	−8949.76	−1.59324
$317$	−8075.22	−1.43076	−0.715378	−	0.698738i	$-0.753745\pi$
$317$	−8075.22	−1.43076	−0.715378	+	0.698738i	$0.753745\pi$
$318$	0	0
$319$	−8251.13	−1.44820
$320$	11621.9	2.03026
$321$	0	0
$322$	−18388.3	−3.18241
$323$	−316.757	−0.0545660
$324$	0	0
$325$	4062.65	0.693400
$326$	−14552.8	−2.47242
$327$	0	0
$328$	8172.50	1.37576
$329$	−9661.74	−1.61905
$330$	0	0
$331$	68.3404	0.0113484	0.00567421	−	0.999984i	$-0.498194\pi$
$331$	68.3404	0.0113484	0.00567421	+	0.999984i	$0.498194\pi$
$332$	−7685.52	−1.27048
$333$	0	0
$334$	1251.57	0.205039
$335$	−11419.8	−1.86248
$336$	0	0
$337$	10721.2	1.73300	0.866500	−	0.499177i	$-0.166364\pi$
$337$	10721.2	1.73300	0.866500	+	0.499177i	$0.166364\pi$
$338$	4459.28	0.717612
$339$	0	0
$340$	−1163.28	−0.185552
$341$	−3573.44	−0.567485
$342$	0	0
$343$	−2940.62	−0.462911
$344$	−998.626	−0.156518
$345$	0	0
$346$	5042.25	0.783448
$347$	11225.7	1.73668	0.868338	−	0.495973i	$-0.165189\pi$
$347$	11225.7	1.73668	0.868338	+	0.495973i	$0.165189\pi$
$348$	0	0
$349$	2207.14	0.338526	0.169263	−	0.985571i	$-0.445861\pi$
$349$	2207.14	0.338526	0.169263	+	0.985571i	$0.445861\pi$
$350$	7580.08	1.15764
$351$	0	0
$352$	8033.79	1.21648
$353$	1614.73	0.243466	0.121733	−	0.992563i	$-0.461155\pi$
$353$	1614.73	0.243466	0.121733	+	0.992563i	$0.461155\pi$
$354$	0	0
$355$	6141.92	0.918252
$356$	−9529.81	−1.41876
$357$	0	0
$358$	12210.4	1.80263
$359$	6709.88	0.986445	0.493223	−	0.869903i	$-0.335819\pi$
$359$	6709.88	0.986445	0.493223	+	0.869903i	$0.335819\pi$
$360$	0	0
$361$	−4821.47	−0.702941
$362$	−15653.3	−2.27270
$363$	0	0
$364$	15841.7	2.28112
$365$	8859.02	1.27042
$366$	0	0
$367$	−4429.43	−0.630012	−0.315006	−	0.949090i	$-0.602006\pi$
$367$	−4429.43	−0.630012	−0.315006	+	0.949090i	$0.602006\pi$
$368$	3295.87	0.466873
$369$	0	0
$370$	21139.5	2.97024
$371$	−7159.68	−1.00192
$372$	0	0
$373$	7732.75	1.07342	0.536711	−	0.843766i	$-0.319667\pi$
$373$	7732.75	1.07342	0.536711	+	0.843766i	$0.319667\pi$
$374$	−1140.29	−0.157655
$375$	0	0
$376$	−6922.94	−0.949531
$377$	12784.0	1.74645
$378$	0	0
$379$	13844.7	1.87640	0.938198	−	0.346100i	$-0.112494\pi$
$379$	13844.7	1.87640	0.938198	+	0.346100i	$0.112494\pi$
$380$	7482.75	1.01015
$381$	0	0
$382$	3641.20	0.487697
$383$	−4062.56	−0.542004	−0.271002	−	0.962579i	$-0.587355\pi$
$383$	−4062.56	−0.542004	−0.271002	+	0.962579i	$0.587355\pi$
$384$	0	0
$385$	12140.4	1.60710
$386$	−4931.14	−0.650229
$387$	0	0
$388$	14398.3	1.88393
$389$	5590.09	0.728609	0.364305	−	0.931280i	$-0.381307\pi$
$389$	5590.09	0.728609	0.364305	+	0.931280i	$0.381307\pi$
$390$	0	0
$391$	−1222.81	−0.158159
$392$	3719.07	0.479187
$393$	0	0
$394$	−13730.3	−1.75564
$395$	10626.7	1.35364
$396$	0	0
$397$	12532.4	1.58434	0.792170	−	0.610301i	$-0.208951\pi$
$397$	12532.4	1.58434	0.792170	+	0.610301i	$0.208951\pi$
$398$	−8257.97	−1.04004
$399$	0	0
$400$	−1358.64	−0.169830
$401$	−6998.95	−0.871598	−0.435799	−	0.900044i	$-0.643534\pi$
$401$	−6998.95	−0.871598	−0.435799	+	0.900044i	$0.643534\pi$
$402$	0	0
$403$	5536.56	0.684357
$404$	14965.3	1.84295
$405$	0	0
$406$	23852.4	2.91570
$407$	12356.0	1.50483
$408$	0	0
$409$	7220.74	0.872964	0.436482	−	0.899713i	$-0.356224\pi$
$409$	7220.74	0.872964	0.436482	+	0.899713i	$0.356224\pi$
$410$	−30048.5	−3.61949
$411$	0	0
$412$	17900.9	2.14057
$413$	−18629.5	−2.21961
$414$	0	0
$415$	9125.58	1.07941
$416$	−12447.3	−1.46701
$417$	0	0
$418$	7334.89	0.858280
$419$	12911.8	1.50545	0.752724	−	0.658336i	$-0.228740\pi$
$419$	12911.8	1.50545	0.752724	+	0.658336i	$0.228740\pi$
$420$	0	0
$421$	650.097	0.0752584	0.0376292	−	0.999292i	$-0.488019\pi$
$421$	650.097	0.0752584	0.0376292	+	0.999292i	$0.488019\pi$
$422$	23818.0	2.74749
$423$	0	0
$424$	−5130.14	−0.587598
$425$	504.072	0.0575320
$426$	0	0
$427$	−1922.47	−0.217880
$428$	−1312.46	−0.148225
$429$	0	0
$430$	3671.73	0.411783
$431$	−6911.08	−0.772378	−0.386189	−	0.922420i	$-0.626209\pi$
$431$	−6911.08	−0.772378	−0.386189	+	0.922420i	$0.626209\pi$
$432$	0	0
$433$	6163.77	0.684092	0.342046	−	0.939683i	$-0.388880\pi$
$433$	6163.77	0.684092	0.342046	+	0.939683i	$0.388880\pi$
$434$	10330.1	1.14254
$435$	0	0
$436$	−22109.9	−2.42860
$437$	7865.70	0.861024
$438$	0	0
$439$	−2441.71	−0.265458	−0.132729	−	0.991152i	$-0.542374\pi$
$439$	−2441.71	−0.265458	−0.132729	+	0.991152i	$0.542374\pi$
$440$	8699.00	0.942520
$441$	0	0
$442$	1766.73	0.190124
$443$	1068.49	0.114594	0.0572972	−	0.998357i	$-0.481752\pi$
$443$	1068.49	0.114594	0.0572972	+	0.998357i	$0.481752\pi$
$444$	0	0
$445$	11315.4	1.20540
$446$	9115.56	0.967790
$447$	0	0
$448$	−19637.2	−2.07091
$449$	−15299.1	−1.60804	−0.804018	−	0.594605i	$-0.797309\pi$
$449$	−15299.1	−1.60804	−0.804018	+	0.594605i	$0.797309\pi$
$450$	0	0
$451$	−17563.3	−1.83376
$452$	−24721.9	−2.57261
$453$	0	0
$454$	7021.72	0.725871
$455$	−18810.0	−1.93808
$456$	0	0
$457$	−1516.60	−0.155237	−0.0776186	−	0.996983i	$-0.524732\pi$
$457$	−1516.60	−0.155237	−0.0776186	+	0.996983i	$0.524732\pi$
$458$	−13728.2	−1.40060
$459$	0	0
$460$	28886.5	2.92791
$461$	17802.2	1.79855	0.899275	−	0.437384i	$-0.144095\pi$
$461$	17802.2	1.79855	0.899275	+	0.437384i	$0.144095\pi$
$462$	0	0
$463$	−4493.23	−0.451011	−0.225506	−	0.974242i	$-0.572403\pi$
$463$	−4493.23	−0.451011	−0.225506	+	0.974242i	$0.572403\pi$
$464$	−4275.26	−0.427745
$465$	0	0
$466$	28120.4	2.79539
$467$	1115.34	0.110518	0.0552590	−	0.998472i	$-0.482402\pi$
$467$	1115.34	0.110518	0.0552590	+	0.998472i	$0.482402\pi$
$468$	0	0
$469$	19295.7	1.89977
$470$	25454.2	2.49811
$471$	0	0
$472$	−13348.6	−1.30174
$473$	2146.12	0.208623
$474$	0	0
$475$	−3242.43	−0.313206
$476$	1965.55	0.189267
$477$	0	0
$478$	1063.91	0.101803
$479$	850.433	0.0811217	0.0405608	−	0.999177i	$-0.487086\pi$
$479$	850.433	0.0811217	0.0405608	+	0.999177i	$0.487086\pi$
$480$	0	0
$481$	−19144.0	−1.81474
$482$	−17060.9	−1.61225
$483$	0	0
$484$	17.9302	0.00168390
$485$	−17096.2	−1.60061
$486$	0	0
$487$	3249.57	0.302366	0.151183	−	0.988506i	$-0.451692\pi$
$487$	3249.57	0.302366	0.151183	+	0.988506i	$0.451692\pi$
$488$	−1377.51	−0.127781
$489$	0	0
$490$	−13674.2	−1.26069
$491$	21221.2	1.95051	0.975256	−	0.221080i	$-0.0709583\pi$
$491$	21221.2	1.95051	0.975256	+	0.221080i	$0.0709583\pi$
$492$	0	0
$493$	1586.17	0.144904
$494$	−11364.4	−1.03504
$495$	0	0
$496$	−1851.55	−0.167615
$497$	−10377.8	−0.936637
$498$	0	0
$499$	1277.06	0.114567	0.0572836	−	0.998358i	$-0.481756\pi$
$499$	1277.06	0.114567	0.0572836	+	0.998358i	$0.481756\pi$
$500$	8813.70	0.788321
$501$	0	0
$502$	−23268.9	−2.06881
$503$	7789.84	0.690521	0.345260	−	0.938507i	$-0.387791\pi$
$503$	7789.84	0.690521	0.345260	+	0.938507i	$0.387791\pi$
$504$	0	0
$505$	−17769.4	−1.56579
$506$	28315.7	2.48772
$507$	0	0
$508$	4542.51	0.396735
$509$	−714.739	−0.0622401	−0.0311201	−	0.999516i	$-0.509907\pi$
$509$	−714.739	−0.0622401	−0.0311201	+	0.999516i	$0.509907\pi$
$510$	0	0
$511$	−14968.8	−1.29585
$512$	6732.80	0.581154
$513$	0	0
$514$	−31531.8	−2.70585
$515$	−21255.0	−1.81866
$516$	0	0
$517$	14877.9	1.26563
$518$	−35718.8	−3.02971
$519$	0	0
$520$	−13477.9	−1.13663
$521$	−16998.2	−1.42938	−0.714688	−	0.699443i	$-0.753431\pi$
$521$	−16998.2	−1.42938	−0.714688	+	0.699443i	$0.753431\pi$
$522$	0	0
$523$	20805.5	1.73951	0.869754	−	0.493485i	$-0.164277\pi$
$523$	20805.5	1.73951	0.869754	+	0.493485i	$0.164277\pi$
$524$	−24048.7	−2.00491
$525$	0	0
$526$	8886.32	0.736620
$527$	686.948	0.0567816
$528$	0	0
$529$	18197.8	1.49567
$530$	18862.4	1.54591
$531$	0	0
$532$	−12643.4	−1.03038
$533$	27212.0	2.21141
$534$	0	0
$535$	1558.38	0.125934
$536$	13826.0	1.11416
$537$	0	0
$538$	32706.7	2.62098
$539$	−7992.56	−0.638708
$540$	0	0
$541$	−10482.0	−0.833010	−0.416505	−	0.909133i	$-0.636745\pi$
$541$	−10482.0	−0.833010	−0.416505	+	0.909133i	$0.636745\pi$
$542$	−1286.95	−0.101991
$543$	0	0
$544$	−1544.39	−0.121719
$545$	26252.6	2.06337
$546$	0	0
$547$	4223.09	0.330103	0.165051	−	0.986285i	$-0.447221\pi$
$547$	4223.09	0.330103	0.165051	+	0.986285i	$0.447221\pi$
$548$	−34084.6	−2.65697
$549$	0	0
$550$	−11672.4	−0.904933
$551$	−10203.0	−0.788863
$552$	0	0
$553$	−17955.6	−1.38074
$554$	−333.173	−0.0255508
$555$	0	0
$556$	20202.6	1.54097
$557$	−6773.17	−0.515240	−0.257620	−	0.966246i	$-0.582938\pi$
$557$	−6773.17	−0.515240	−0.257620	+	0.966246i	$0.582938\pi$
$558$	0	0
$559$	−3325.13	−0.251588
$560$	6290.46	0.474680
$561$	0	0
$562$	−22952.9	−1.72279
$563$	3722.21	0.278637	0.139318	−	0.990248i	$-0.455509\pi$
$563$	3722.21	0.278637	0.139318	+	0.990248i	$0.455509\pi$
$564$	0	0
$565$	29354.0	2.18572
$566$	26608.3	1.97603
$567$	0	0
$568$	−7436.03	−0.549312
$569$	18264.4	1.34566	0.672831	−	0.739796i	$-0.265078\pi$
$569$	18264.4	1.34566	0.672831	+	0.739796i	$0.265078\pi$
$570$	0	0
$571$	14134.6	1.03593	0.517963	−	0.855403i	$-0.326690\pi$
$571$	14134.6	1.03593	0.517963	+	0.855403i	$0.326690\pi$
$572$	−24394.3	−1.78317
$573$	0	0
$574$	50772.1	3.69196
$575$	−12517.1	−0.907826
$576$	0	0
$577$	9177.26	0.662139	0.331069	−	0.943606i	$-0.392591\pi$
$577$	9177.26	0.662139	0.331069	+	0.943606i	$0.392591\pi$
$578$	−21651.0	−1.55806
$579$	0	0
$580$	−37470.2	−2.68253
$581$	−15419.2	−1.10103
$582$	0	0
$583$	11025.0	0.783209
$584$	−10725.6	−0.759982
$585$	0	0
$586$	2257.07	0.159110
$587$	9434.81	0.663401	0.331701	−	0.943385i	$-0.392378\pi$
$587$	9434.81	0.663401	0.331701	+	0.943385i	$0.392378\pi$
$588$	0	0
$589$	−4418.77	−0.309121
$590$	49080.1	3.42474
$591$	0	0
$592$	6402.16	0.444471
$593$	20277.0	1.40418	0.702090	−	0.712089i	$-0.252251\pi$
$593$	20277.0	1.40418	0.702090	+	0.712089i	$0.252251\pi$
$594$	0	0
$595$	−2333.84	−0.160804
$596$	4894.63	0.336396
$597$	0	0
$598$	−43871.3	−3.00005
$599$	2802.75	0.191181	0.0955904	−	0.995421i	$-0.469526\pi$
$599$	2802.75	0.191181	0.0955904	+	0.995421i	$0.469526\pi$
$600$	0	0
$601$	−4087.51	−0.277426	−0.138713	−	0.990333i	$-0.544297\pi$
$601$	−4087.51	−0.277426	−0.138713	+	0.990333i	$0.544297\pi$
$602$	−6204.01	−0.420028
$603$	0	0
$604$	13242.1	0.892075
$605$	−21.2898	−0.00143067
$606$	0	0
$607$	−4522.40	−0.302403	−0.151201	−	0.988503i	$-0.548314\pi$
$607$	−4522.40	−0.302403	−0.151201	+	0.988503i	$0.548314\pi$
$608$	9934.27	0.662645
$609$	0	0
$610$	5064.81	0.336177
$611$	−23051.3	−1.52628
$612$	0	0
$613$	22808.1	1.50279	0.751397	−	0.659851i	$-0.229381\pi$
$613$	22808.1	1.50279	0.751397	+	0.659851i	$0.229381\pi$
$614$	−29353.7	−1.92935
$615$	0	0
$616$	−14698.4	−0.961391
$617$	−22347.1	−1.45812	−0.729062	−	0.684448i	$-0.760043\pi$
$617$	−22347.1	−1.45812	−0.729062	+	0.684448i	$0.760043\pi$
$618$	0	0
$619$	−13531.6	−0.878644	−0.439322	−	0.898330i	$-0.644781\pi$
$619$	−13531.6	−0.878644	−0.439322	+	0.898330i	$0.644781\pi$
$620$	−16227.8	−1.05117
$621$	0	0
$622$	−10640.9	−0.685954
$623$	−19119.3	−1.22953
$624$	0	0
$625$	−19444.2	−1.24443
$626$	17434.0	1.11311
$627$	0	0
$628$	−4094.40	−0.260166
$629$	−2375.28	−0.150570
$630$	0	0
$631$	19524.9	1.23182	0.615908	−	0.787818i	$-0.288789\pi$
$631$	19524.9	1.23182	0.615908	+	0.787818i	$0.288789\pi$
$632$	−12865.8	−0.809766
$633$	0	0
$634$	−35946.8	−2.25178
$635$	−5393.65	−0.337071
$636$	0	0
$637$	12383.4	0.770248
$638$	−36729.8	−2.27923
$639$	0	0
$640$	27033.3	1.66967
$641$	4541.75	0.279857	0.139929	−	0.990162i	$-0.455313\pi$
$641$	4541.75	0.279857	0.139929	+	0.990162i	$0.455313\pi$
$642$	0	0
$643$	−27511.4	−1.68732	−0.843658	−	0.536880i	$-0.819603\pi$
$643$	−27511.4	−1.68732	−0.843658	+	0.536880i	$0.819603\pi$
$644$	−48808.6	−2.98654
$645$	0	0
$646$	−1410.04	−0.0858781
$647$	−22096.3	−1.34265	−0.671327	−	0.741162i	$-0.734275\pi$
$647$	−22096.3	−1.34265	−0.671327	+	0.741162i	$0.734275\pi$
$648$	0	0
$649$	28687.2	1.73509
$650$	18084.8	1.09130
$651$	0	0
$652$	−38628.1	−2.32024
$653$	−10058.9	−0.602812	−0.301406	−	0.953496i	$-0.597456\pi$
$653$	−10058.9	−0.602812	−0.301406	+	0.953496i	$0.597456\pi$
$654$	0	0
$655$	28554.8	1.70340
$656$	−9100.28	−0.541626
$657$	0	0
$658$	−43009.1	−2.54813
$659$	377.712	0.0223271	0.0111636	−	0.999938i	$-0.496446\pi$
$659$	377.712	0.0223271	0.0111636	+	0.999938i	$0.496446\pi$
$660$	0	0
$661$	−31831.9	−1.87310	−0.936549	−	0.350536i	$-0.885999\pi$
$661$	−31831.9	−1.87310	−0.936549	+	0.350536i	$0.885999\pi$
$662$	304.217	0.0178606
$663$	0	0
$664$	−11048.3	−0.645721
$665$	15012.4	0.875421
$666$	0	0
$667$	−39387.9	−2.28651
$668$	3322.10	0.192419
$669$	0	0
$670$	−50835.1	−2.93124
$671$	2960.37	0.170319
$672$	0	0
$673$	4977.25	0.285080	0.142540	−	0.989789i	$-0.454473\pi$
$673$	4977.25	0.285080	0.142540	+	0.989789i	$0.454473\pi$
$674$	47725.3	2.72746
$675$	0	0
$676$	11836.4	0.673443
$677$	17528.8	0.995106	0.497553	−	0.867433i	$-0.334232\pi$
$677$	17528.8	0.995106	0.497553	+	0.867433i	$0.334232\pi$
$678$	0	0
$679$	28886.9	1.63266
$680$	−1672.27	−0.0943069
$681$	0	0
$682$	−15907.1	−0.893131
$683$	4754.01	0.266335	0.133168	−	0.991094i	$-0.457485\pi$
$683$	4754.01	0.266335	0.133168	+	0.991094i	$0.457485\pi$
$684$	0	0
$685$	40471.1	2.25740
$686$	−13090.1	−0.728548
$687$	0	0
$688$	1111.99	0.0616198
$689$	−17081.8	−0.944507
$690$	0	0
$691$	−7006.36	−0.385723	−0.192861	−	0.981226i	$-0.561777\pi$
$691$	−7006.36	−0.385723	−0.192861	+	0.981226i	$0.561777\pi$
$692$	13383.8	0.735226
$693$	0	0
$694$	49971.0	2.73325
$695$	−23987.9	−1.30923
$696$	0	0
$697$	3376.32	0.183483
$698$	9825.06	0.532785
$699$	0	0
$700$	20120.1	1.08638
$701$	11572.3	0.623511	0.311756	−	0.950162i	$-0.399083\pi$
$701$	11572.3	0.623511	0.311756	+	0.950162i	$0.399083\pi$
$702$	0	0
$703$	15278.9	0.819710
$704$	30238.9	1.61885
$705$	0	0
$706$	7187.97	0.383177
$707$	30024.3	1.59715
$708$	0	0
$709$	−25780.5	−1.36559	−0.682796	−	0.730609i	$-0.739236\pi$
$709$	−25780.5	−1.36559	−0.682796	+	0.730609i	$0.739236\pi$
$710$	27340.7	1.44518
$711$	0	0
$712$	−13699.6	−0.721088
$713$	−17058.3	−0.895986
$714$	0	0
$715$	28965.1	1.51501
$716$	32410.6	1.69168
$717$	0	0
$718$	29868.9	1.55251
$719$	−5465.62	−0.283495	−0.141748	−	0.989903i	$-0.545272\pi$
$719$	−5465.62	−0.283495	−0.141748	+	0.989903i	$0.545272\pi$
$720$	0	0
$721$	35914.0	1.85507
$722$	−21462.7	−1.10632
$723$	0	0
$724$	−41549.1	−2.13282
$725$	16236.6	0.831743
$726$	0	0
$727$	−18761.8	−0.957134	−0.478567	−	0.878051i	$-0.658844\pi$
$727$	−18761.8	−0.957134	−0.478567	+	0.878051i	$0.658844\pi$
$728$	22773.2	1.15939
$729$	0	0
$730$	39435.8	1.99943
$731$	−412.564	−0.0208745
$732$	0	0
$733$	17761.5	0.895000	0.447500	−	0.894284i	$-0.352314\pi$
$733$	17761.5	0.895000	0.447500	+	0.894284i	$0.352314\pi$
$734$	−19717.5	−0.991537
$735$	0	0
$736$	38350.4	1.92067
$737$	−29713.0	−1.48507
$738$	0	0
$739$	14899.8	0.741676	0.370838	−	0.928698i	$-0.379071\pi$
$739$	14899.8	0.741676	0.370838	+	0.928698i	$0.379071\pi$
$740$	56111.3	2.78742
$741$	0	0
$742$	−31871.2	−1.57686
$743$	1841.01	0.0909019	0.0454509	−	0.998967i	$-0.485528\pi$
$743$	1841.01	0.0909019	0.0454509	+	0.998967i	$0.485528\pi$
$744$	0	0
$745$	−5811.74	−0.285806
$746$	34422.2	1.68939
$747$	0	0
$748$	−3026.72	−0.147951
$749$	−2633.15	−0.128456
$750$	0	0
$751$	−401.276	−0.0194977	−0.00974885	−	0.999952i	$-0.503103\pi$
$751$	−401.276	−0.0194977	−0.00974885	+	0.999952i	$0.503103\pi$
$752$	7708.87	0.373821
$753$	0	0
$754$	56907.9	2.74863
$755$	−15723.3	−0.757920
$756$	0	0
$757$	−40970.6	−1.96711	−0.983554	−	0.180612i	$-0.942192\pi$
$757$	−40970.6	−1.96711	−0.983554	+	0.180612i	$0.942192\pi$
$758$	61629.5	2.95314
$759$	0	0
$760$	10756.8	0.513410
$761$	−26698.7	−1.27178	−0.635892	−	0.771778i	$-0.719368\pi$
$761$	−26698.7	−1.27178	−0.635892	+	0.771778i	$0.719368\pi$
$762$	0	0
$763$	−44358.2	−2.10469
$764$	9664.98	0.457679
$765$	0	0
$766$	−18084.5	−0.853026
$767$	−44447.0	−2.09242
$768$	0	0
$769$	−32725.1	−1.53459	−0.767294	−	0.641296i	$-0.778397\pi$
$769$	−32725.1	−1.53459	−0.767294	+	0.641296i	$0.778397\pi$
$770$	54043.0	2.52932
$771$	0	0
$772$	−13088.9	−0.610207
$773$	21560.3	1.00319	0.501597	−	0.865101i	$-0.332746\pi$
$773$	21560.3	1.00319	0.501597	+	0.865101i	$0.332746\pi$
$774$	0	0
$775$	7031.84	0.325924
$776$	20698.3	0.957509
$777$	0	0
$778$	24884.2	1.14671
$779$	−21718.1	−0.998885
$780$	0	0
$781$	15980.6	0.732177
$782$	−5443.33	−0.248917
$783$	0	0
$784$	−4141.28	−0.188651
$785$	4861.57	0.221041
$786$	0	0
$787$	−4776.67	−0.216353	−0.108177	−	0.994132i	$-0.534501\pi$
$787$	−4776.67	−0.216353	−0.108177	+	0.994132i	$0.534501\pi$
$788$	−36444.8	−1.64758
$789$	0	0
$790$	47304.6	2.13041
$791$	−49598.6	−2.22949
$792$	0	0
$793$	−4586.69	−0.205395
$794$	55787.8	2.49349
$795$	0	0
$796$	−21919.4	−0.976022
$797$	−39883.7	−1.77259	−0.886294	−	0.463123i	$-0.846729\pi$
$797$	−39883.7	−1.77259	−0.886294	+	0.463123i	$0.846729\pi$
$798$	0	0
$799$	−2860.09	−0.126637
$800$	−15809.0	−0.698664
$801$	0	0
$802$	−31155.7	−1.37175
$803$	23050.2	1.01298
$804$	0	0
$805$	57954.0	2.53740
$806$	24645.9	1.07707
$807$	0	0
$808$	21513.4	0.936681
$809$	19395.9	0.842921	0.421460	−	0.906847i	$-0.361518\pi$
$809$	19395.9	0.842921	0.421460	+	0.906847i	$0.361518\pi$
$810$	0	0
$811$	34882.1	1.51033	0.755163	−	0.655537i	$-0.227558\pi$
$811$	34882.1	1.51033	0.755163	+	0.655537i	$0.227558\pi$
$812$	63312.3	2.73624
$813$	0	0
$814$	55002.6	2.36835
$815$	45866.0	1.97131
$816$	0	0
$817$	2653.81	0.113641
$818$	32143.0	1.37390
$819$	0	0
$820$	−79758.9	−3.39671
$821$	15825.4	0.672729	0.336365	−	0.941732i	$-0.390803\pi$
$821$	15825.4	0.672729	0.336365	+	0.941732i	$0.390803\pi$
$822$	0	0
$823$	−24669.0	−1.04485	−0.522423	−	0.852687i	$-0.674972\pi$
$823$	−24669.0	−1.04485	−0.522423	+	0.852687i	$0.674972\pi$
$824$	25733.5	1.08795
$825$	0	0
$826$	−82929.2	−3.49331
$827$	22234.8	0.934921	0.467461	−	0.884014i	$-0.345169\pi$
$827$	22234.8	0.934921	0.467461	+	0.884014i	$0.345169\pi$
$828$	0	0
$829$	−14531.7	−0.608813	−0.304407	−	0.952542i	$-0.598458\pi$
$829$	−14531.7	−0.608813	−0.304407	+	0.952542i	$0.598458\pi$
$830$	40622.4	1.69882
$831$	0	0
$832$	−46851.1	−1.95225
$833$	1536.47	0.0639081
$834$	0	0
$835$	−3944.57	−0.163482
$836$	19469.3	0.805453
$837$	0	0
$838$	57476.7	2.36933
$839$	−10721.4	−0.441171	−0.220586	−	0.975368i	$-0.570797\pi$
$839$	−10721.4	−0.441171	−0.220586	+	0.975368i	$0.570797\pi$
$840$	0	0
$841$	26703.1	1.09488
$842$	2893.90	0.118445
$843$	0	0
$844$	63221.0	2.57838
$845$	−14054.3	−0.572167
$846$	0	0
$847$	35.9727	0.00145931
$848$	5712.53	0.231332
$849$	0	0
$850$	2243.87	0.0905461
$851$	58983.0	2.37592
$852$	0	0
$853$	−42748.0	−1.71590	−0.857951	−	0.513731i	$-0.828263\pi$
$853$	−42748.0	−1.71590	−0.857951	+	0.513731i	$0.828263\pi$
$854$	−8557.84	−0.342908
$855$	0	0
$856$	−1886.74	−0.0753357
$857$	29280.0	1.16708	0.583539	−	0.812085i	$-0.301668\pi$
$857$	29280.0	1.16708	0.583539	+	0.812085i	$0.301668\pi$
$858$	0	0
$859$	−21086.2	−0.837545	−0.418773	−	0.908091i	$-0.637540\pi$
$859$	−21086.2	−0.837545	−0.418773	+	0.908091i	$0.637540\pi$
$860$	9746.02	0.386438
$861$	0	0
$862$	−30764.6	−1.21560
$863$	10275.9	0.405326	0.202663	−	0.979249i	$-0.435040\pi$
$863$	10275.9	0.405326	0.202663	+	0.979249i	$0.435040\pi$
$864$	0	0
$865$	−15891.6	−0.624659
$866$	27438.0	1.07665
$867$	0	0
$868$	27419.6	1.07221
$869$	27649.5	1.07934
$870$	0	0
$871$	46036.3	1.79091
$872$	−31784.1	−1.23434
$873$	0	0
$874$	35014.1	1.35511
$875$	17682.6	0.683179
$876$	0	0
$877$	−42597.5	−1.64015	−0.820077	−	0.572253i	$-0.806070\pi$
$877$	−42597.5	−1.64015	−0.820077	+	0.572253i	$0.806070\pi$
$878$	−10869.2	−0.417789
$879$	0	0
$880$	−9686.56	−0.371061
$881$	−5871.79	−0.224547	−0.112273	−	0.993677i	$-0.535813\pi$
$881$	−5871.79	−0.224547	−0.112273	+	0.993677i	$0.535813\pi$
$882$	0	0
$883$	16601.8	0.632725	0.316362	−	0.948638i	$-0.397538\pi$
$883$	16601.8	0.632725	0.316362	+	0.948638i	$0.397538\pi$
$884$	4689.49	0.178421
$885$	0	0
$886$	4756.36	0.180353
$887$	−7463.42	−0.282522	−0.141261	−	0.989972i	$-0.545116\pi$
$887$	−7463.42	−0.282522	−0.141261	+	0.989972i	$0.545116\pi$
$888$	0	0
$889$	9113.48	0.343820
$890$	50370.5	1.89710
$891$	0	0
$892$	24195.8	0.908222
$893$	18397.4	0.689415
$894$	0	0
$895$	−38483.4	−1.43727
$896$	−45677.4	−1.70310
$897$	0	0
$898$	−68103.7	−2.53079
$899$	22127.2	0.820895
$900$	0	0
$901$	−2119.42	−0.0783665
$902$	−78182.9	−2.88604
$903$	0	0
$904$	−35539.0	−1.30753
$905$	49334.2	1.81207
$906$	0	0
$907$	14908.7	0.545795	0.272897	−	0.962043i	$-0.412018\pi$
$907$	14908.7	0.545795	0.272897	+	0.962043i	$0.412018\pi$
$908$	18638.0	0.681194
$909$	0	0
$910$	−83732.3	−3.05022
$911$	17426.7	0.633778	0.316889	−	0.948463i	$-0.397362\pi$
$911$	17426.7	0.633778	0.316889	+	0.948463i	$0.397362\pi$
$912$	0	0
$913$	23743.7	0.860682
$914$	−6751.11	−0.244318
$915$	0	0
$916$	−36439.1	−1.31439
$917$	−48248.1	−1.73751
$918$	0	0
$919$	4025.33	0.144487	0.0722434	−	0.997387i	$-0.476984\pi$
$919$	4025.33	0.144487	0.0722434	+	0.997387i	$0.476984\pi$
$920$	41525.9	1.48812
$921$	0	0
$922$	79246.3	2.83063
$923$	−24759.8	−0.882966
$924$	0	0
$925$	−24314.2	−0.864267
$926$	−20001.6	−0.709819
$927$	0	0
$928$	−49746.4	−1.75970
$929$	14049.4	0.496173	0.248086	−	0.968738i	$-0.420198\pi$
$929$	14049.4	0.496173	0.248086	+	0.968738i	$0.420198\pi$
$930$	0	0
$931$	−9883.28	−0.347918
$932$	74641.0	2.62333
$933$	0	0
$934$	4964.94	0.173938
$935$	3593.84	0.125702
$936$	0	0
$937$	10784.7	0.376010	0.188005	−	0.982168i	$-0.439798\pi$
$937$	10784.7	0.376010	0.188005	+	0.982168i	$0.439798\pi$
$938$	85894.5	2.98993
$939$	0	0
$940$	67563.9	2.34435
$941$	807.886	0.0279876	0.0139938	−	0.999902i	$-0.495545\pi$
$941$	807.886	0.0279876	0.0139938	+	0.999902i	$0.495545\pi$
$942$	0	0
$943$	−83840.8	−2.89526
$944$	14864.1	0.512483
$945$	0	0
$946$	9553.43	0.328339
$947$	−40932.1	−1.40456	−0.702278	−	0.711902i	$-0.747834\pi$
$947$	−40932.1	−1.40456	−0.702278	+	0.711902i	$0.747834\pi$
$948$	0	0
$949$	−35713.1	−1.22160
$950$	−14433.6	−0.492936
$951$	0	0
$952$	2825.59	0.0961952
$953$	1797.29	0.0610912	0.0305456	−	0.999533i	$-0.490276\pi$
$953$	1797.29	0.0610912	0.0305456	+	0.999533i	$0.490276\pi$
$954$	0	0
$955$	−11475.9	−0.388850
$956$	2823.96	0.0955372
$957$	0	0
$958$	3785.69	0.127672
$959$	−68382.7	−2.30260
$960$	0	0
$961$	−20208.0	−0.678327
$962$	−85219.1	−2.85611
$963$	0	0
$964$	−45285.4	−1.51301
$965$	15541.4	0.518440
$966$	0	0
$967$	−12648.8	−0.420640	−0.210320	−	0.977633i	$-0.567451\pi$
$967$	−12648.8	−0.420640	−0.210320	+	0.977633i	$0.567451\pi$
$968$	25.7756	0.000855846 0
$969$	0	0
$970$	−76103.4	−2.51910
$971$	−54297.9	−1.79454	−0.897272	−	0.441477i	$-0.854455\pi$
$971$	−54297.9	−1.79454	−0.897272	+	0.441477i	$0.854455\pi$
$972$	0	0
$973$	40531.7	1.33544
$974$	14465.4	0.475875
$975$	0	0
$976$	1533.89	0.0503060
$977$	38106.3	1.24783	0.623914	−	0.781493i	$-0.285542\pi$
$977$	38106.3	1.24783	0.623914	+	0.781493i	$0.285542\pi$
$978$	0	0
$979$	29441.5	0.961137
$980$	−36296.0	−1.18309
$981$	0	0
$982$	94466.1	3.06979
$983$	35577.7	1.15438	0.577189	−	0.816611i	$-0.304150\pi$
$983$	35577.7	1.15438	0.577189	+	0.816611i	$0.304150\pi$
$984$	0	0
$985$	43273.5	1.39981
$986$	7060.84	0.228056
$987$	0	0
$988$	−30165.0	−0.971332
$989$	10244.8	0.329389
$990$	0	0
$991$	26708.2	0.856120	0.428060	−	0.903750i	$-0.359197\pi$
$991$	26708.2	0.856120	0.428060	+	0.903750i	$0.359197\pi$
$992$	−21544.4	−0.689551
$993$	0	0
$994$	−46196.7	−1.47412
$995$	26026.5	0.829242
$996$	0	0
$997$	18318.2	0.581888	0.290944	−	0.956740i	$-0.406031\pi$
$997$	18318.2	0.581888	0.290944	+	0.956740i	$0.406031\pi$
$998$	5684.81	0.180310
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.g.1.52	✓	59
3.2	odd	2		2151.4.a.h.1.8	yes	59

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.4.a.g.1.52	✓	59	1.1	even	1	trivial
2151.4.a.h.1.8	yes	59	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q+4.45149 q^{2} +11.8157 q^{4} -14.0297 q^{5} +23.7055 q^{7} +16.9858 q^{8} +O(q^{10})\)	\(q+4.45149 q^{2} +11.8157 q^{4} -14.0297 q^{5} +23.7055 q^{7} +16.9858 q^{8} -62.4530 q^{10} -36.5037 q^{11} +56.5575 q^{13} +105.525 q^{14} -18.9141 q^{16} +7.01736 q^{17} -45.1390 q^{19} -165.771 q^{20} -162.496 q^{22} -174.255 q^{23} +71.8322 q^{25} +251.765 q^{26} +280.099 q^{28} +226.036 q^{29} +97.8926 q^{31} -220.082 q^{32} +31.2377 q^{34} -332.581 q^{35} -338.487 q^{37} -200.936 q^{38} -238.305 q^{40} +481.138 q^{41} -58.7919 q^{43} -431.318 q^{44} -775.695 q^{46} -407.573 q^{47} +218.952 q^{49} +319.760 q^{50} +668.269 q^{52} -302.026 q^{53} +512.135 q^{55} +402.656 q^{56} +1006.20 q^{58} -785.873 q^{59} -81.0979 q^{61} +435.768 q^{62} -828.379 q^{64} -793.484 q^{65} +813.974 q^{67} +82.9154 q^{68} -1480.48 q^{70} -437.780 q^{71} -631.448 q^{73} -1506.77 q^{74} -533.351 q^{76} -865.339 q^{77} -757.443 q^{79} +265.359 q^{80} +2141.78 q^{82} -650.447 q^{83} -98.4514 q^{85} -261.712 q^{86} -620.043 q^{88} -806.535 q^{89} +1340.73 q^{91} -2058.95 q^{92} -1814.31 q^{94} +633.286 q^{95} +1218.57 q^{97} +974.663 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8}+O(q^{10}) \) 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8	\( 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8} - 36 q^{10} - 132 q^{11} + 104 q^{13} - 280 q^{14} + 822 q^{16} - 408 q^{17} + 20 q^{19} - 800 q^{20} - 2 q^{22} - 276 q^{23} + 1477 q^{25} - 780 q^{26} + 224 q^{28} - 696 q^{29} - 380 q^{31} - 896 q^{32} - 72 q^{34} - 700 q^{35} + 224 q^{37} - 988 q^{38} - 258 q^{40} - 2706 q^{41} - 156 q^{43} - 1584 q^{44} + 428 q^{46} - 1316 q^{47} + 2135 q^{49} - 1400 q^{50} + 1092 q^{52} - 1484 q^{53} - 992 q^{55} - 3360 q^{56} - 120 q^{58} - 3186 q^{59} - 254 q^{61} - 1240 q^{62} + 3054 q^{64} - 5120 q^{65} + 288 q^{67} - 9420 q^{68} + 1108 q^{70} - 4468 q^{71} - 1770 q^{73} - 6214 q^{74} + 720 q^{76} - 6352 q^{77} - 746 q^{79} - 7040 q^{80} + 276 q^{82} - 5484 q^{83} + 588 q^{85} - 10152 q^{86} + 1186 q^{88} - 11570 q^{89} + 1768 q^{91} - 15366 q^{92} - 2142 q^{94} - 5736 q^{95} + 2390 q^{97} - 6912 q^{98}+O(q^{100}) \) 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8 - 36 * q^10 - 132 * q^11 + 104 * q^13 - 280 * q^14 + 822 * q^16 - 408 * q^17 + 20 * q^19 - 800 * q^20 - 2 * q^22 - 276 * q^23 + 1477 * q^25 - 780 * q^26 + 224 * q^28 - 696 * q^29 - 380 * q^31 - 896 * q^32 - 72 * q^34 - 700 * q^35 + 224 * q^37 - 988 * q^38 - 258 * q^40 - 2706 * q^41 - 156 * q^43 - 1584 * q^44 + 428 * q^46 - 1316 * q^47 + 2135 * q^49 - 1400 * q^50 + 1092 * q^52 - 1484 * q^53 - 992 * q^55 - 3360 * q^56 - 120 * q^58 - 3186 * q^59 - 254 * q^61 - 1240 * q^62 + 3054 * q^64 - 5120 * q^65 + 288 * q^67 - 9420 * q^68 + 1108 * q^70 - 4468 * q^71 - 1770 * q^73 - 6214 * q^74 + 720 * q^76 - 6352 * q^77 - 746 * q^79 - 7040 * q^80 + 276 * q^82 - 5484 * q^83 + 588 * q^85 - 10152 * q^86 + 1186 * q^88 - 11570 * q^89 + 1768 * q^91 - 15366 * q^92 - 2142 * q^94 - 5736 * q^95 + 2390 * q^97 - 6912 * q^98

Introduction

L-functions

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