LMFDB - Embedded newform 2151.4.a.c.1.14

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:	$0$
Dimension:	$28$
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
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)

$f(q)$	$=$	$q+0.608170 q^{2} -7.63013 q^{4} +1.98658 q^{5} -5.67520 q^{7} -9.50578 q^{8} +O(q^{10})$	\(q+0.608170 q^{2} -7.63013 q^{4} +1.98658 q^{5} -5.67520 q^{7} -9.50578 q^{8} +1.20818 q^{10} -36.9768 q^{11} -35.5538 q^{13} -3.45149 q^{14} +55.2599 q^{16} +39.0322 q^{17} +12.3580 q^{19} -15.1578 q^{20} -22.4882 q^{22} -99.9935 q^{23} -121.054 q^{25} -21.6228 q^{26} +43.3025 q^{28} -100.245 q^{29} -140.630 q^{31} +109.654 q^{32} +23.7382 q^{34} -11.2742 q^{35} -12.0854 q^{37} +7.51577 q^{38} -18.8840 q^{40} -344.863 q^{41} +284.536 q^{43} +282.138 q^{44} -60.8130 q^{46} +361.358 q^{47} -310.792 q^{49} -73.6211 q^{50} +271.280 q^{52} +299.949 q^{53} -73.4573 q^{55} +53.9472 q^{56} -60.9660 q^{58} -197.837 q^{59} -713.414 q^{61} -85.5267 q^{62} -375.391 q^{64} -70.6304 q^{65} -875.150 q^{67} -297.821 q^{68} -6.85664 q^{70} +426.881 q^{71} -1163.06 q^{73} -7.34995 q^{74} -94.2932 q^{76} +209.851 q^{77} +985.983 q^{79} +109.778 q^{80} -209.735 q^{82} +957.334 q^{83} +77.5404 q^{85} +173.046 q^{86} +351.494 q^{88} -115.143 q^{89} +201.775 q^{91} +762.963 q^{92} +219.767 q^{94} +24.5501 q^{95} -767.299 q^{97} -189.015 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 13 q^{2} + 99 q^{4} + 74 q^{5} - 82 q^{7} + 135 q^{8}+O(q^{10}) $ 28 * q + 13 * q^2 + 99 * q^4 + 74 * q^5 - 82 * q^7 + 135 * q^8	$ 28 q + 13 q^{2} + 99 q^{4} + 74 q^{5} - 82 q^{7} + 135 q^{8} - 68 q^{10} + 258 q^{11} - 134 q^{13} + 292 q^{14} + 327 q^{16} + 364 q^{17} - 278 q^{19} + 986 q^{20} - 179 q^{22} + 668 q^{23} + 490 q^{25} + 760 q^{26} - 802 q^{28} + 714 q^{29} - 608 q^{31} + 918 q^{32} - 228 q^{34} + 934 q^{35} - 1080 q^{37} + 1395 q^{38} - 563 q^{40} + 1796 q^{41} - 1934 q^{43} + 3157 q^{44} - 940 q^{46} + 2032 q^{47} + 762 q^{49} + 1754 q^{50} - 2328 q^{52} + 1790 q^{53} - 478 q^{55} + 3557 q^{56} - 2626 q^{58} + 3622 q^{59} + 324 q^{61} + 796 q^{62} + 2023 q^{64} + 2200 q^{65} - 2444 q^{67} - 357 q^{68} + 4305 q^{70} + 1298 q^{71} - 1368 q^{73} - 813 q^{74} + 1390 q^{76} + 1408 q^{77} - 1378 q^{79} + 7684 q^{80} + 9001 q^{82} + 3524 q^{83} + 60 q^{85} + 2543 q^{86} + 1749 q^{88} + 7854 q^{89} + 850 q^{91} + 496 q^{92} + 6634 q^{94} + 3696 q^{95} - 1746 q^{97} + 4632 q^{98}+O(q^{100}) $ 28 * q + 13 * q^2 + 99 * q^4 + 74 * q^5 - 82 * q^7 + 135 * q^8 - 68 * q^10 + 258 * q^11 - 134 * q^13 + 292 * q^14 + 327 * q^16 + 364 * q^17 - 278 * q^19 + 986 * q^20 - 179 * q^22 + 668 * q^23 + 490 * q^25 + 760 * q^26 - 802 * q^28 + 714 * q^29 - 608 * q^31 + 918 * q^32 - 228 * q^34 + 934 * q^35 - 1080 * q^37 + 1395 * q^38 - 563 * q^40 + 1796 * q^41 - 1934 * q^43 + 3157 * q^44 - 940 * q^46 + 2032 * q^47 + 762 * q^49 + 1754 * q^50 - 2328 * q^52 + 1790 * q^53 - 478 * q^55 + 3557 * q^56 - 2626 * q^58 + 3622 * q^59 + 324 * q^61 + 796 * q^62 + 2023 * q^64 + 2200 * q^65 - 2444 * q^67 - 357 * q^68 + 4305 * q^70 + 1298 * q^71 - 1368 * q^73 - 813 * q^74 + 1390 * q^76 + 1408 * q^77 - 1378 * q^79 + 7684 * q^80 + 9001 * q^82 + 3524 * q^83 + 60 * q^85 + 2543 * q^86 + 1749 * q^88 + 7854 * q^89 + 850 * q^91 + 496 * q^92 + 6634 * q^94 + 3696 * q^95 - 1746 * q^97 + 4632 * 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 $

$2$	0.608170	0.215021	0.107510	−	0.994204i	$-0.465712\pi$
$2$	0.608170	0.215021	0.107510	+	0.994204i	$0.465712\pi$
$3$	0	0
$3$	0	0
$4$	−7.63013	−0.953766
$5$	1.98658	0.177685	0.0888424	−	0.996046i	$-0.471683\pi$
$5$	1.98658	0.177685	0.0888424	+	0.996046i	$0.471683\pi$
$6$	0	0
$7$	−5.67520	−0.306432	−0.153216	−	0.988193i	$-0.548963\pi$
$7$	−5.67520	−0.306432	−0.153216	+	0.988193i	$0.548963\pi$
$8$	−9.50578	−0.420100
$9$	0	0
$10$	1.20818	0.0382059
$11$	−36.9768	−1.01354	−0.506770	−	0.862081i	$-0.669161\pi$
$11$	−36.9768	−1.01354	−0.506770	+	0.862081i	$0.669161\pi$
$12$	0	0
$13$	−35.5538	−0.758528	−0.379264	−	0.925289i	$-0.623823\pi$
$13$	−35.5538	−0.758528	−0.379264	+	0.925289i	$0.623823\pi$
$14$	−3.45149	−0.0658892
$15$	0	0
$16$	55.2599	0.863436
$17$	39.0322	0.556864	0.278432	−	0.960456i	$-0.410185\pi$
$17$	39.0322	0.556864	0.278432	+	0.960456i	$0.410185\pi$
$18$	0	0
$19$	12.3580	0.149217	0.0746085	−	0.997213i	$-0.476229\pi$
$19$	12.3580	0.149217	0.0746085	+	0.997213i	$0.476229\pi$
$20$	−15.1578	−0.169470
$21$	0	0
$22$	−22.4882	−0.217932
$23$	−99.9935	−0.906525	−0.453263	−	0.891377i	$-0.649740\pi$
$23$	−99.9935	−0.906525	−0.453263	+	0.891377i	$0.649740\pi$
$24$	0	0
$25$	−121.054	−0.968428
$26$	−21.6228	−0.163099
$27$	0	0
$28$	43.3025	0.292264
$29$	−100.245	−0.641898	−0.320949	−	0.947097i	$-0.604002\pi$
$29$	−100.245	−0.641898	−0.320949	+	0.947097i	$0.604002\pi$
$30$	0	0
$31$	−140.630	−0.814768	−0.407384	−	0.913257i	$-0.633559\pi$
$31$	−140.630	−0.814768	−0.407384	+	0.913257i	$0.633559\pi$
$32$	109.654	0.605757
$33$	0	0
$34$	23.7382	0.119737
$35$	−11.2742	−0.0544483
$36$	0	0
$37$	−12.0854	−0.0536978	−0.0268489	−	0.999640i	$-0.508547\pi$
$37$	−12.0854	−0.0536978	−0.0268489	+	0.999640i	$0.508547\pi$
$38$	7.51577	0.0320847
$39$	0	0
$40$	−18.8840	−0.0746454
$41$	−344.863	−1.31362	−0.656811	−	0.754055i	$-0.728095\pi$
$41$	−344.863	−1.31362	−0.656811	+	0.754055i	$0.728095\pi$
$42$	0	0
$43$	284.536	1.00910	0.504551	−	0.863382i	$-0.331658\pi$
$43$	284.536	1.00910	0.504551	+	0.863382i	$0.331658\pi$
$44$	282.138	0.966680
$45$	0	0
$46$	−60.8130	−0.194922
$47$	361.358	1.12148	0.560739	−	0.827993i	$-0.310517\pi$
$47$	361.358	1.12148	0.560739	+	0.827993i	$0.310517\pi$
$48$	0	0
$49$	−310.792	−0.906099
$50$	−73.6211	−0.208232
$51$	0	0
$52$	271.280	0.723458
$53$	299.949	0.777381	0.388690	−	0.921368i	$-0.372928\pi$
$53$	299.949	0.777381	0.388690	+	0.921368i	$0.372928\pi$
$54$	0	0
$55$	−73.4573	−0.180091
$56$	53.9472	0.128732
$57$	0	0
$58$	−60.9660	−0.138021
$59$	−197.837	−0.436545	−0.218273	−	0.975888i	$-0.570042\pi$
$59$	−197.837	−0.436545	−0.218273	+	0.975888i	$0.570042\pi$
$60$	0	0
$61$	−713.414	−1.49743	−0.748715	−	0.662892i	$-0.769329\pi$
$61$	−713.414	−1.49743	−0.748715	+	0.662892i	$0.769329\pi$
$62$	−85.5267	−0.175192
$63$	0	0
$64$	−375.391	−0.733186
$65$	−70.6304	−0.134779
$66$	0	0
$67$	−875.150	−1.59577	−0.797885	−	0.602810i	$-0.794048\pi$
$67$	−875.150	−1.59577	−0.797885	+	0.602810i	$0.794048\pi$
$68$	−297.821	−0.531118
$69$	0	0
$70$	−6.85664	−0.0117075
$71$	426.881	0.713542	0.356771	−	0.934192i	$-0.383878\pi$
$71$	426.881	0.713542	0.356771	+	0.934192i	$0.383878\pi$
$72$	0	0
$73$	−1163.06	−1.86474	−0.932371	−	0.361502i	$-0.882264\pi$
$73$	−1163.06	−1.86474	−0.932371	+	0.361502i	$0.882264\pi$
$74$	−7.34995	−0.0115461
$75$	0	0
$76$	−94.2932	−0.142318
$77$	209.851	0.310581
$78$	0	0
$79$	985.983	1.40420	0.702100	−	0.712078i	$-0.252246\pi$
$79$	985.983	1.40420	0.702100	+	0.712078i	$0.252246\pi$
$80$	109.778	0.153419
$81$	0	0
$82$	−209.735	−0.282456
$83$	957.334	1.26604	0.633018	−	0.774137i	$-0.281816\pi$
$83$	957.334	1.26604	0.633018	+	0.774137i	$0.281816\pi$
$84$	0	0
$85$	77.5404	0.0989463
$86$	173.046	0.216978
$87$	0	0
$88$	351.494	0.425788
$89$	−115.143	−0.137136	−0.0685680	−	0.997646i	$-0.521843\pi$
$89$	−115.143	−0.137136	−0.0685680	+	0.997646i	$0.521843\pi$
$90$	0	0
$91$	201.775	0.232437
$92$	762.963	0.864613
$93$	0	0
$94$	219.767	0.241141
$95$	24.5501	0.0265136
$96$	0	0
$97$	−767.299	−0.803169	−0.401585	−	0.915822i	$-0.631540\pi$
$97$	−767.299	−0.803169	−0.401585	+	0.915822i	$0.631540\pi$
$98$	−189.015	−0.194830
$99$	0	0
$100$	923.654	0.923654
$101$	1244.30	1.22587	0.612933	−	0.790135i	$-0.289989\pi$
$101$	1244.30	1.22587	0.612933	+	0.790135i	$0.289989\pi$
$102$	0	0
$103$	1820.64	1.74168	0.870839	−	0.491569i	$-0.163576\pi$
$103$	1820.64	1.74168	0.870839	+	0.491569i	$0.163576\pi$
$104$	337.967	0.318658
$105$	0	0
$106$	182.420	0.167153
$107$	−43.5196	−0.0393196	−0.0196598	−	0.999807i	$-0.506258\pi$
$107$	−43.5196	−0.0393196	−0.0196598	+	0.999807i	$0.506258\pi$
$108$	0	0
$109$	−1155.09	−1.01502	−0.507511	−	0.861645i	$-0.669434\pi$
$109$	−1155.09	−1.01502	−0.507511	+	0.861645i	$0.669434\pi$
$110$	−44.6745	−0.0387232
$111$	0	0
$112$	−313.611	−0.264584
$113$	470.403	0.391609	0.195805	−	0.980643i	$-0.437268\pi$
$113$	470.403	0.391609	0.195805	+	0.980643i	$0.437268\pi$
$114$	0	0
$115$	−198.645	−0.161076
$116$	764.883	0.612220
$117$	0	0
$118$	−120.318	−0.0938662
$119$	−221.515	−0.170641
$120$	0	0
$121$	36.2861	0.0272623
$122$	−433.877	−0.321978
$123$	0	0
$124$	1073.02	0.777098
$125$	−488.804	−0.349760
$126$	0	0
$127$	1204.72	0.841745	0.420872	−	0.907120i	$-0.361724\pi$
$127$	1204.72	0.841745	0.420872	+	0.907120i	$0.361724\pi$
$128$	−1105.53	−0.763407
$129$	0	0
$130$	−42.9553	−0.0289802
$131$	−220.941	−0.147357	−0.0736784	−	0.997282i	$-0.523474\pi$
$131$	−220.941	−0.147357	−0.0736784	+	0.997282i	$0.523474\pi$
$132$	0	0
$133$	−70.1341	−0.0457248
$134$	−532.240	−0.343123
$135$	0	0
$136$	−371.031	−0.233939
$137$	2999.29	1.87041	0.935207	−	0.354102i	$-0.115214\pi$
$137$	2999.29	1.87041	0.935207	+	0.354102i	$0.115214\pi$
$138$	0	0
$139$	−1134.72	−0.692416	−0.346208	−	0.938158i	$-0.612531\pi$
$139$	−1134.72	−0.692416	−0.346208	+	0.938158i	$0.612531\pi$
$140$	86.0237	0.0519309
$141$	0	0
$142$	259.617	0.153426
$143$	1314.67	0.768798
$144$	0	0
$145$	−199.144	−0.114055
$146$	−707.340	−0.400958
$147$	0	0
$148$	92.2128	0.0512152
$149$	−916.879	−0.504119	−0.252059	−	0.967712i	$-0.581108\pi$
$149$	−916.879	−0.504119	−0.252059	+	0.967712i	$0.581108\pi$
$150$	0	0
$151$	−1253.23	−0.675407	−0.337703	−	0.941253i	$-0.609650\pi$
$151$	−1253.23	−0.675407	−0.337703	+	0.941253i	$0.609650\pi$
$152$	−117.472	−0.0626860
$153$	0	0
$154$	127.625	0.0667813
$155$	−279.371	−0.144772
$156$	0	0
$157$	2944.84	1.49697	0.748483	−	0.663153i	$-0.230782\pi$
$157$	2944.84	1.49697	0.748483	+	0.663153i	$0.230782\pi$
$158$	599.646	0.301932
$159$	0	0
$160$	217.835	0.107634
$161$	567.483	0.277788
$162$	0	0
$163$	−1133.72	−0.544785	−0.272392	−	0.962186i	$-0.587815\pi$
$163$	−1133.72	−0.544785	−0.272392	+	0.962186i	$0.587815\pi$
$164$	2631.35	1.25289
$165$	0	0
$166$	582.222	0.272224
$167$	3528.35	1.63492	0.817460	−	0.575985i	$-0.195381\pi$
$167$	3528.35	1.63492	0.817460	+	0.575985i	$0.195381\pi$
$168$	0	0
$169$	−932.924	−0.424636
$170$	47.1578	0.0212755
$171$	0	0
$172$	−2171.05	−0.962447
$173$	741.816	0.326007	0.163003	−	0.986626i	$-0.447882\pi$
$173$	741.816	0.326007	0.163003	+	0.986626i	$0.447882\pi$
$174$	0	0
$175$	687.003	0.296757
$176$	−2043.34	−0.875126
$177$	0	0
$178$	−70.0264	−0.0294871
$179$	−3844.69	−1.60539	−0.802697	−	0.596387i	$-0.796602\pi$
$179$	−3844.69	−1.60539	−0.802697	+	0.596387i	$0.796602\pi$
$180$	0	0
$181$	1594.70	0.654881	0.327441	−	0.944872i	$-0.393814\pi$
$181$	1594.70	0.654881	0.327441	+	0.944872i	$0.393814\pi$
$182$	122.714	0.0499788
$183$	0	0
$184$	950.516	0.380831
$185$	−24.0085	−0.00954129
$186$	0	0
$187$	−1443.29	−0.564404
$188$	−2757.21	−1.06963
$189$	0	0
$190$	14.9307	0.00570097
$191$	−2542.08	−0.963030	−0.481515	−	0.876438i	$-0.659913\pi$
$191$	−2542.08	−0.963030	−0.481515	+	0.876438i	$0.659913\pi$
$192$	0	0
$193$	−2826.29	−1.05410	−0.527049	−	0.849835i	$-0.676701\pi$
$193$	−2826.29	−1.05410	−0.527049	+	0.849835i	$0.676701\pi$
$194$	−466.648	−0.172698
$195$	0	0
$196$	2371.38	0.864207
$197$	4299.11	1.55482	0.777409	−	0.628995i	$-0.216534\pi$
$197$	4299.11	1.55482	0.777409	+	0.628995i	$0.216534\pi$
$198$	0	0
$199$	1325.88	0.472307	0.236154	−	0.971716i	$-0.424113\pi$
$199$	1325.88	0.472307	0.236154	+	0.971716i	$0.424113\pi$
$200$	1150.71	0.406837
$201$	0	0
$202$	756.746	0.263587
$203$	568.911	0.196698
$204$	0	0
$205$	−685.096	−0.233411
$206$	1107.26	0.374497
$207$	0	0
$208$	−1964.70	−0.654940
$209$	−456.960	−0.151237
$210$	0	0
$211$	1486.60	0.485032	0.242516	−	0.970147i	$-0.422027\pi$
$211$	1486.60	0.485032	0.242516	+	0.970147i	$0.422027\pi$
$212$	−2288.65	−0.741440
$213$	0	0
$214$	−26.4673	−0.00845452
$215$	565.253	0.179302
$216$	0	0
$217$	798.101	0.249671
$218$	−702.491	−0.218251
$219$	0	0
$220$	560.489	0.171764
$221$	−1387.74	−0.422397
$222$	0	0
$223$	1462.38	0.439139	0.219570	−	0.975597i	$-0.429535\pi$
$223$	1462.38	0.439139	0.219570	+	0.975597i	$0.429535\pi$
$224$	−622.306	−0.185623
$225$	0	0
$226$	286.085	0.0842040
$227$	−715.294	−0.209144	−0.104572	−	0.994517i	$-0.533347\pi$
$227$	−715.294	−0.209144	−0.104572	+	0.994517i	$0.533347\pi$
$228$	0	0
$229$	6460.40	1.86426	0.932130	−	0.362124i	$-0.117948\pi$
$229$	6460.40	1.86426	0.932130	+	0.362124i	$0.117948\pi$
$230$	−120.810	−0.0346346
$231$	0	0
$232$	952.907	0.269661
$233$	5737.87	1.61331	0.806653	−	0.591026i	$-0.201277\pi$
$233$	5737.87	1.61331	0.806653	+	0.591026i	$0.201277\pi$
$234$	0	0
$235$	717.865	0.199270
$236$	1509.52	0.416362
$237$	0	0
$238$	−134.719	−0.0366913
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	19.4713	0.00520439	0.00260219	−	0.999997i	$-0.499172\pi$
$241$	19.4713	0.00520439	0.00260219	+	0.999997i	$0.499172\pi$
$242$	22.0681	0.00586196
$243$	0	0
$244$	5443.44	1.42820
$245$	−617.412	−0.161000
$246$	0	0
$247$	−439.375	−0.113185
$248$	1336.79	0.342284
$249$	0	0
$250$	−297.276	−0.0752056
$251$	1529.01	0.384504	0.192252	−	0.981346i	$-0.438421\pi$
$251$	1529.01	0.384504	0.192252	+	0.981346i	$0.438421\pi$
$252$	0	0
$253$	3697.44	0.918799
$254$	732.675	0.180993
$255$	0	0
$256$	2330.78	0.569038
$257$	4447.54	1.07949	0.539747	−	0.841827i	$-0.318520\pi$
$257$	4447.54	1.07949	0.539747	+	0.841827i	$0.318520\pi$
$258$	0	0
$259$	68.5868	0.0164547
$260$	538.919	0.128548
$261$	0	0
$262$	−134.370	−0.0316847
$263$	5700.34	1.33649	0.668247	−	0.743940i	$-0.267045\pi$
$263$	5700.34	1.33649	0.668247	+	0.743940i	$0.267045\pi$
$264$	0	0
$265$	595.872	0.138129
$266$	−42.6535	−0.00983178
$267$	0	0
$268$	6677.50	1.52199
$269$	390.139	0.0884282	0.0442141	−	0.999022i	$-0.485922\pi$
$269$	390.139	0.0884282	0.0442141	+	0.999022i	$0.485922\pi$
$270$	0	0
$271$	−1577.04	−0.353500	−0.176750	−	0.984256i	$-0.556558\pi$
$271$	−1577.04	−0.353500	−0.176750	+	0.984256i	$0.556558\pi$
$272$	2156.91	0.480817
$273$	0	0
$274$	1824.08	0.402177
$275$	4476.18	0.981540
$276$	0	0
$277$	1582.72	0.343308	0.171654	−	0.985157i	$-0.445089\pi$
$277$	1582.72	0.343308	0.171654	+	0.985157i	$0.445089\pi$
$278$	−690.104	−0.148884
$279$	0	0
$280$	107.170	0.0228737
$281$	−2728.51	−0.579250	−0.289625	−	0.957140i	$-0.593531\pi$
$281$	−2728.51	−0.579250	−0.289625	+	0.957140i	$0.593531\pi$
$282$	0	0
$283$	−5681.42	−1.19338	−0.596688	−	0.802474i	$-0.703517\pi$
$283$	−5681.42	−1.19338	−0.596688	+	0.802474i	$0.703517\pi$
$284$	−3257.16	−0.680553
$285$	0	0
$286$	799.542	0.165307
$287$	1957.16	0.402536
$288$	0	0
$289$	−3389.49	−0.689902
$290$	−121.114	−0.0245243
$291$	0	0
$292$	8874.32	1.77853
$293$	−7733.72	−1.54201	−0.771005	−	0.636830i	$-0.780245\pi$
$293$	−7733.72	−1.54201	−0.771005	+	0.636830i	$0.780245\pi$
$294$	0	0
$295$	−393.018	−0.0775674
$296$	114.881	0.0225585
$297$	0	0
$298$	−557.619	−0.108396
$299$	3555.15	0.687625
$300$	0	0
$301$	−1614.80	−0.309221
$302$	−762.177	−0.145226
$303$	0	0
$304$	682.902	0.128839
$305$	−1417.25	−0.266071
$306$	0	0
$307$	9144.78	1.70007	0.850033	−	0.526730i	$-0.176582\pi$
$307$	9144.78	1.70007	0.850033	+	0.526730i	$0.176582\pi$
$308$	−1601.19	−0.296222
$309$	0	0
$310$	−169.905	−0.0311290
$311$	−5530.38	−1.00836	−0.504179	−	0.863599i	$-0.668205\pi$
$311$	−5530.38	−1.00836	−0.504179	+	0.863599i	$0.668205\pi$
$312$	0	0
$313$	8252.27	1.49024	0.745121	−	0.666929i	$-0.232392\pi$
$313$	8252.27	1.49024	0.745121	+	0.666929i	$0.232392\pi$
$314$	1790.96	0.321879
$315$	0	0
$316$	−7523.18	−1.33928
$317$	7445.86	1.31925	0.659623	−	0.751597i	$-0.270716\pi$
$317$	7445.86	1.31925	0.659623	+	0.751597i	$0.270716\pi$
$318$	0	0
$319$	3706.74	0.650589
$320$	−745.743	−0.130276
$321$	0	0
$322$	345.126	0.0597302
$323$	482.360	0.0830936
$324$	0	0
$325$	4303.92	0.734580
$326$	−689.496	−0.117140
$327$	0	0
$328$	3278.19	0.551853
$329$	−2050.78	−0.343657
$330$	0	0
$331$	−1983.40	−0.329358	−0.164679	−	0.986347i	$-0.552659\pi$
$331$	−1983.40	−0.329358	−0.164679	+	0.986347i	$0.552659\pi$
$332$	−7304.58	−1.20750
$333$	0	0
$334$	2145.84	0.351542
$335$	−1738.55	−0.283544
$336$	0	0
$337$	8095.45	1.30857	0.654284	−	0.756249i	$-0.272970\pi$
$337$	8095.45	1.30857	0.654284	+	0.756249i	$0.272970\pi$
$338$	−567.377	−0.0913054
$339$	0	0
$340$	−591.643	−0.0943716
$341$	5200.04	0.825800
$342$	0	0
$343$	3710.40	0.584090
$344$	−2704.74	−0.423924
$345$	0	0
$346$	451.150	0.0700982
$347$	−7749.65	−1.19891	−0.599457	−	0.800407i	$-0.704617\pi$
$347$	−7749.65	−1.19891	−0.599457	+	0.800407i	$0.704617\pi$
$348$	0	0
$349$	10689.2	1.63948	0.819741	−	0.572734i	$-0.194117\pi$
$349$	10689.2	1.63948	0.819741	+	0.572734i	$0.194117\pi$
$350$	417.815	0.0638089
$351$	0	0
$352$	−4054.64	−0.613958
$353$	−11039.3	−1.66449	−0.832245	−	0.554408i	$-0.812945\pi$
$353$	−11039.3	−1.66449	−0.832245	+	0.554408i	$0.812945\pi$
$354$	0	0
$355$	848.033	0.126786
$356$	878.554	0.130796
$357$	0	0
$358$	−2338.22	−0.345193
$359$	−7625.24	−1.12102	−0.560508	−	0.828149i	$-0.689394\pi$
$359$	−7625.24	−1.12102	−0.560508	+	0.828149i	$0.689394\pi$
$360$	0	0
$361$	−6706.28	−0.977734
$362$	969.852	0.140813
$363$	0	0
$364$	−1539.57	−0.221691
$365$	−2310.51	−0.331336
$366$	0	0
$367$	4302.91	0.612016	0.306008	−	0.952029i	$-0.401007\pi$
$367$	4302.91	0.612016	0.306008	+	0.952029i	$0.401007\pi$
$368$	−5525.63	−0.782726
$369$	0	0
$370$	−14.6012	−0.00205157
$371$	−1702.27	−0.238214
$372$	0	0
$373$	−7178.53	−0.996489	−0.498245	−	0.867037i	$-0.666022\pi$
$373$	−7178.53	−0.996489	−0.498245	+	0.867037i	$0.666022\pi$
$374$	−877.764	−0.121359
$375$	0	0
$376$	−3434.99	−0.471133
$377$	3564.10	0.486897
$378$	0	0
$379$	7392.00	1.00185	0.500926	−	0.865490i	$-0.332993\pi$
$379$	7392.00	1.00185	0.500926	+	0.865490i	$0.332993\pi$
$380$	−187.321	−0.0252877
$381$	0	0
$382$	−1546.02	−0.207071
$383$	745.908	0.0995146	0.0497573	−	0.998761i	$-0.484155\pi$
$383$	745.908	0.0995146	0.0497573	+	0.998761i	$0.484155\pi$
$384$	0	0
$385$	416.885	0.0551855
$386$	−1718.87	−0.226653
$387$	0	0
$388$	5854.59	0.766036
$389$	11990.8	1.56288	0.781439	−	0.623982i	$-0.214486\pi$
$389$	11990.8	1.56288	0.781439	+	0.623982i	$0.214486\pi$
$390$	0	0
$391$	−3902.96	−0.504811
$392$	2954.32	0.380652
$393$	0	0
$394$	2614.59	0.334318
$395$	1958.73	0.249505
$396$	0	0
$397$	3957.34	0.500286	0.250143	−	0.968209i	$-0.419522\pi$
$397$	3957.34	0.500286	0.250143	+	0.968209i	$0.419522\pi$
$398$	806.361	0.101556
$399$	0	0
$400$	−6689.41	−0.836176
$401$	8232.21	1.02518	0.512590	−	0.858634i	$-0.328686\pi$
$401$	8232.21	1.02518	0.512590	+	0.858634i	$0.328686\pi$
$402$	0	0
$403$	4999.92	0.618024
$404$	−9494.17	−1.16919
$405$	0	0
$406$	345.994	0.0422941
$407$	446.878	0.0544249
$408$	0	0
$409$	9624.73	1.16360	0.581800	−	0.813332i	$-0.302349\pi$
$409$	9624.73	1.16360	0.581800	+	0.813332i	$0.302349\pi$
$410$	−416.655	−0.0501881
$411$	0	0
$412$	−13891.7	−1.66115
$413$	1122.76	0.133771
$414$	0	0
$415$	1901.82	0.224955
$416$	−3898.61	−0.459483
$417$	0	0
$418$	−277.909	−0.0325191
$419$	−7422.63	−0.865440	−0.432720	−	0.901528i	$-0.642446\pi$
$419$	−7422.63	−0.865440	−0.432720	+	0.901528i	$0.642446\pi$
$420$	0	0
$421$	−3616.95	−0.418716	−0.209358	−	0.977839i	$-0.567137\pi$
$421$	−3616.95	−0.418716	−0.209358	+	0.977839i	$0.567137\pi$
$422$	904.106	0.104292
$423$	0	0
$424$	−2851.25	−0.326578
$425$	−4724.98	−0.539283
$426$	0	0
$427$	4048.76	0.458861
$428$	332.060	0.0375017
$429$	0	0
$430$	343.770	0.0385536
$431$	4814.33	0.538047	0.269023	−	0.963134i	$-0.413299\pi$
$431$	4814.33	0.538047	0.269023	+	0.963134i	$0.413299\pi$
$432$	0	0
$433$	8152.90	0.904858	0.452429	−	0.891800i	$-0.350558\pi$
$433$	8152.90	0.904858	0.452429	+	0.891800i	$0.350558\pi$
$434$	485.381	0.0536844
$435$	0	0
$436$	8813.48	0.968094
$437$	−1235.72	−0.135269
$438$	0	0
$439$	−14610.9	−1.58847	−0.794236	−	0.607609i	$-0.792129\pi$
$439$	−14610.9	−1.58847	−0.794236	+	0.607609i	$0.792129\pi$
$440$	698.269	0.0756560
$441$	0	0
$442$	−843.984	−0.0908241
$443$	13071.6	1.40192	0.700961	−	0.713199i	$-0.252754\pi$
$443$	13071.6	1.40192	0.700961	+	0.713199i	$0.252754\pi$
$444$	0	0
$445$	−228.740	−0.0243670
$446$	889.374	0.0944240
$447$	0	0
$448$	2130.42	0.224672
$449$	13794.2	1.44986	0.724931	−	0.688821i	$-0.241872\pi$
$449$	13794.2	1.44986	0.724931	+	0.688821i	$0.241872\pi$
$450$	0	0
$451$	12751.9	1.33141
$452$	−3589.24	−0.373503
$453$	0	0
$454$	−435.021	−0.0449703
$455$	400.842	0.0413005
$456$	0	0
$457$	−11156.6	−1.14197	−0.570987	−	0.820959i	$-0.693439\pi$
$457$	−11156.6	−1.14197	−0.570987	+	0.820959i	$0.693439\pi$
$458$	3929.02	0.400854
$459$	0	0
$460$	1515.68	0.153629
$461$	−647.133	−0.0653796	−0.0326898	−	0.999466i	$-0.510407\pi$
$461$	−647.133	−0.0653796	−0.0326898	+	0.999466i	$0.510407\pi$
$462$	0	0
$463$	9975.48	1.00130	0.500648	−	0.865651i	$-0.333095\pi$
$463$	9975.48	1.00130	0.500648	+	0.865651i	$0.333095\pi$
$464$	−5539.53	−0.554238
$465$	0	0
$466$	3489.60	0.346894
$467$	−16875.6	−1.67219	−0.836093	−	0.548587i	$-0.815166\pi$
$467$	−16875.6	−1.67219	−0.836093	+	0.548587i	$0.815166\pi$
$468$	0	0
$469$	4966.65	0.488995
$470$	436.584	0.0428471
$471$	0	0
$472$	1880.59	0.183393
$473$	−10521.2	−1.02276
$474$	0	0
$475$	−1495.98	−0.144506
$476$	1690.19	0.162752
$477$	0	0
$478$	145.353	0.0139085
$479$	−792.249	−0.0755715	−0.0377858	−	0.999286i	$-0.512030\pi$
$479$	−792.249	−0.0755715	−0.0377858	+	0.999286i	$0.512030\pi$
$480$	0	0
$481$	429.681	0.0407313
$482$	11.8419	0.00111905
$483$	0	0
$484$	−276.868	−0.0260019
$485$	−1524.30	−0.142711
$486$	0	0
$487$	−10793.5	−1.00432	−0.502158	−	0.864776i	$-0.667460\pi$
$487$	−10793.5	−1.00432	−0.502158	+	0.864776i	$0.667460\pi$
$488$	6781.55	0.629071
$489$	0	0
$490$	−375.492	−0.0346183
$491$	11341.9	1.04247	0.521233	−	0.853415i	$-0.325472\pi$
$491$	11341.9	1.04247	0.521233	+	0.853415i	$0.325472\pi$
$492$	0	0
$493$	−3912.78	−0.357450
$494$	−267.215	−0.0243371
$495$	0	0
$496$	−7771.18	−0.703500
$497$	−2422.64	−0.218652
$498$	0	0
$499$	−6330.78	−0.567945	−0.283973	−	0.958832i	$-0.591653\pi$
$499$	−6330.78	−0.567945	−0.283973	+	0.958832i	$0.591653\pi$
$500$	3729.64	0.333589
$501$	0	0
$502$	929.901	0.0826764
$503$	−6617.34	−0.586586	−0.293293	−	0.956023i	$-0.594751\pi$
$503$	−6617.34	−0.586586	−0.293293	+	0.956023i	$0.594751\pi$
$504$	0	0
$505$	2471.90	0.217818
$506$	2248.67	0.197561
$507$	0	0
$508$	−9192.17	−0.802828
$509$	22275.5	1.93977	0.969886	−	0.243557i	$-0.0783144\pi$
$509$	22275.5	1.93977	0.969886	+	0.243557i	$0.0783144\pi$
$510$	0	0
$511$	6600.61	0.571417
$512$	10261.8	0.885761
$513$	0	0
$514$	2704.86	0.232113
$515$	3616.84	0.309470
$516$	0	0
$517$	−13361.9	−1.13666
$518$	41.7124	0.00353811
$519$	0	0
$520$	671.397	0.0566206
$521$	3734.54	0.314037	0.157019	−	0.987596i	$-0.449812\pi$
$521$	3734.54	0.314037	0.157019	+	0.987596i	$0.449812\pi$
$522$	0	0
$523$	−15286.9	−1.27810	−0.639052	−	0.769164i	$-0.720673\pi$
$523$	−15286.9	−1.27810	−0.639052	+	0.769164i	$0.720673\pi$
$524$	1685.81	0.140544
$525$	0	0
$526$	3466.77	0.287374
$527$	−5489.08	−0.453715
$528$	0	0
$529$	−2168.31	−0.178212
$530$	362.392	0.0297005
$531$	0	0
$532$	535.133	0.0436108
$533$	12261.2	0.996419
$534$	0	0
$535$	−86.4549	−0.00698649
$536$	8318.98	0.670383
$537$	0	0
$538$	237.271	0.0190139
$539$	11492.1	0.918368
$540$	0	0
$541$	5802.24	0.461105	0.230552	−	0.973060i	$-0.425947\pi$
$541$	5802.24	0.461105	0.230552	+	0.973060i	$0.425947\pi$
$542$	−959.109	−0.0760097
$543$	0	0
$544$	4280.02	0.337324
$545$	−2294.67	−0.180354
$546$	0	0
$547$	−15245.1	−1.19165	−0.595827	−	0.803113i	$-0.703176\pi$
$547$	−15245.1	−1.19165	−0.595827	+	0.803113i	$0.703176\pi$
$548$	−22885.0	−1.78394
$549$	0	0
$550$	2722.28	0.211051
$551$	−1238.83	−0.0957820
$552$	0	0
$553$	−5595.65	−0.430292
$554$	962.563	0.0738184
$555$	0	0
$556$	8658.07	0.660403
$557$	10931.2	0.831544	0.415772	−	0.909469i	$-0.363512\pi$
$557$	10931.2	0.831544	0.415772	+	0.909469i	$0.363512\pi$
$558$	0	0
$559$	−10116.4	−0.765432
$560$	−623.012	−0.0470126
$561$	0	0
$562$	−1659.40	−0.124551
$563$	−9953.45	−0.745094	−0.372547	−	0.928013i	$-0.621515\pi$
$563$	−9953.45	−0.745094	−0.372547	+	0.928013i	$0.621515\pi$
$564$	0	0
$565$	934.492	0.0695830
$566$	−3455.27	−0.256600
$567$	0	0
$568$	−4057.84	−0.299759
$569$	13811.5	1.01759	0.508794	−	0.860888i	$-0.330092\pi$
$569$	13811.5	1.01759	0.508794	+	0.860888i	$0.330092\pi$
$570$	0	0
$571$	−1299.73	−0.0952575	−0.0476287	−	0.998865i	$-0.515166\pi$
$571$	−1299.73	−0.0952575	−0.0476287	+	0.998865i	$0.515166\pi$
$572$	−10031.1	−0.733253
$573$	0	0
$574$	1190.29	0.0865535
$575$	12104.6	0.877904
$576$	0	0
$577$	−21138.2	−1.52512	−0.762561	−	0.646916i	$-0.776059\pi$
$577$	−21138.2	−1.52512	−0.762561	+	0.646916i	$0.776059\pi$
$578$	−2061.39	−0.148343
$579$	0	0
$580$	1519.50	0.108782
$581$	−5433.06	−0.387954
$582$	0	0
$583$	−11091.2	−0.787906
$584$	11055.8	0.783378
$585$	0	0
$586$	−4703.42	−0.331564
$587$	14655.0	1.03045	0.515226	−	0.857054i	$-0.327708\pi$
$587$	14655.0	1.03045	0.515226	+	0.857054i	$0.327708\pi$
$588$	0	0
$589$	−1737.90	−0.121577
$590$	−239.022	−0.0166786
$591$	0	0
$592$	−667.835	−0.0463646
$593$	−10063.5	−0.696897	−0.348449	−	0.937328i	$-0.613291\pi$
$593$	−10063.5	−0.696897	−0.348449	+	0.937328i	$0.613291\pi$
$594$	0	0
$595$	−440.057	−0.0303203
$596$	6995.91	0.480811
$597$	0	0
$598$	2162.14	0.147853
$599$	−7162.48	−0.488566	−0.244283	−	0.969704i	$-0.578553\pi$
$599$	−7162.48	−0.488566	−0.244283	+	0.969704i	$0.578553\pi$
$600$	0	0
$601$	−24187.2	−1.64162	−0.820812	−	0.571198i	$-0.806479\pi$
$601$	−24187.2	−1.64162	−0.820812	+	0.571198i	$0.806479\pi$
$602$	−982.073	−0.0664889
$603$	0	0
$604$	9562.31	0.644180
$605$	72.0852	0.00484410
$606$	0	0
$607$	−5955.29	−0.398217	−0.199108	−	0.979977i	$-0.563805\pi$
$607$	−5955.29	−0.398217	−0.199108	+	0.979977i	$0.563805\pi$
$608$	1355.10	0.0903891
$609$	0	0
$610$	−861.930	−0.0572107
$611$	−12847.7	−0.850672
$612$	0	0
$613$	19084.3	1.25743	0.628716	−	0.777635i	$-0.283581\pi$
$613$	19084.3	1.25743	0.628716	+	0.777635i	$0.283581\pi$
$614$	5561.58	0.365549
$615$	0	0
$616$	−1994.80	−0.130475
$617$	−419.064	−0.0273434	−0.0136717	−	0.999907i	$-0.504352\pi$
$617$	−419.064	−0.0273434	−0.0136717	+	0.999907i	$0.504352\pi$
$618$	0	0
$619$	5004.93	0.324984	0.162492	−	0.986710i	$-0.448047\pi$
$619$	5004.93	0.324984	0.162492	+	0.986710i	$0.448047\pi$
$620$	2131.64	0.138079
$621$	0	0
$622$	−3363.41	−0.216818
$623$	653.458	0.0420228
$624$	0	0
$625$	14160.6	0.906281
$626$	5018.78	0.320433
$627$	0	0
$628$	−22469.5	−1.42776
$629$	−471.718	−0.0299024
$630$	0	0
$631$	−21560.3	−1.36022	−0.680112	−	0.733108i	$-0.738069\pi$
$631$	−21560.3	−1.36022	−0.680112	+	0.733108i	$0.738069\pi$
$632$	−9372.54	−0.589904
$633$	0	0
$634$	4528.35	0.283665
$635$	2393.27	0.149565
$636$	0	0
$637$	11049.9	0.687302
$638$	2254.33	0.139890
$639$	0	0
$640$	−2196.22	−0.135646
$641$	10167.3	0.626495	0.313248	−	0.949671i	$-0.398583\pi$
$641$	10167.3	0.626495	0.313248	+	0.949671i	$0.398583\pi$
$642$	0	0
$643$	15152.7	0.929339	0.464670	−	0.885484i	$-0.346173\pi$
$643$	15152.7	0.929339	0.464670	+	0.885484i	$0.346173\pi$
$644$	−4329.97	−0.264945
$645$	0	0
$646$	293.357	0.0178668
$647$	−9289.92	−0.564489	−0.282245	−	0.959343i	$-0.591079\pi$
$647$	−9289.92	−0.564489	−0.282245	+	0.959343i	$0.591079\pi$
$648$	0	0
$649$	7315.38	0.442456
$650$	2617.51	0.157950
$651$	0	0
$652$	8650.44	0.519597
$653$	−14599.6	−0.874926	−0.437463	−	0.899236i	$-0.644123\pi$
$653$	−14599.6	−0.874926	−0.437463	+	0.899236i	$0.644123\pi$
$654$	0	0
$655$	−438.917	−0.0261831
$656$	−19057.1	−1.13423
$657$	0	0
$658$	−1247.22	−0.0738933
$659$	16884.1	0.998042	0.499021	−	0.866590i	$-0.333693\pi$
$659$	16884.1	0.998042	0.499021	+	0.866590i	$0.333693\pi$
$660$	0	0
$661$	−17939.0	−1.05559	−0.527795	−	0.849372i	$-0.676981\pi$
$661$	−17939.0	−1.05559	−0.527795	+	0.849372i	$0.676981\pi$
$662$	−1206.24	−0.0708187
$663$	0	0
$664$	−9100.20	−0.531862
$665$	−139.327	−0.00812461
$666$	0	0
$667$	10023.8	0.581896
$668$	−26921.8	−1.55933
$669$	0	0
$670$	−1057.34	−0.0609678
$671$	26379.8	1.51771
$672$	0	0
$673$	5198.48	0.297752	0.148876	−	0.988856i	$-0.452435\pi$
$673$	5198.48	0.297752	0.148876	+	0.988856i	$0.452435\pi$
$674$	4923.41	0.281369
$675$	0	0
$676$	7118.33	0.405003
$677$	−27249.4	−1.54694	−0.773471	−	0.633831i	$-0.781481\pi$
$677$	−27249.4	−1.54694	−0.773471	+	0.633831i	$0.781481\pi$
$678$	0	0
$679$	4354.58	0.246117
$680$	−737.082	−0.0415674
$681$	0	0
$682$	3162.51	0.177564
$683$	−33970.8	−1.90316	−0.951578	−	0.307408i	$-0.900538\pi$
$683$	−33970.8	−1.90316	−0.951578	+	0.307408i	$0.900538\pi$
$684$	0	0
$685$	5958.32	0.332344
$686$	2256.55	0.125591
$687$	0	0
$688$	15723.4	0.871295
$689$	−10664.3	−0.589665
$690$	0	0
$691$	−4857.27	−0.267408	−0.133704	−	0.991021i	$-0.542687\pi$
$691$	−4857.27	−0.267408	−0.133704	+	0.991021i	$0.542687\pi$
$692$	−5660.15	−0.310934
$693$	0	0
$694$	−4713.11	−0.257791
$695$	−2254.21	−0.123032
$696$	0	0
$697$	−13460.7	−0.731509
$698$	6500.85	0.352523
$699$	0	0
$700$	−5241.92	−0.283037
$701$	−35151.2	−1.89393	−0.946963	−	0.321341i	$-0.895866\pi$
$701$	−35151.2	−1.89393	−0.946963	+	0.321341i	$0.895866\pi$
$702$	0	0
$703$	−149.351	−0.00801262
$704$	13880.8	0.743113
$705$	0	0
$706$	−6713.80	−0.357900
$707$	−7061.65	−0.375645
$708$	0	0
$709$	201.512	0.0106741	0.00533705	−	0.999986i	$-0.498301\pi$
$709$	201.512	0.0106741	0.00533705	+	0.999986i	$0.498301\pi$
$710$	515.748	0.0272615
$711$	0	0
$712$	1094.52	0.0576108
$713$	14062.0	0.738608
$714$	0	0
$715$	2611.69	0.136604
$716$	29335.5	1.53117
$717$	0	0
$718$	−4637.45	−0.241042
$719$	−29058.3	−1.50722	−0.753610	−	0.657321i	$-0.771689\pi$
$719$	−29058.3	−1.50722	−0.753610	+	0.657321i	$0.771689\pi$
$720$	0	0
$721$	−10332.5	−0.533706
$722$	−4078.56	−0.210233
$723$	0	0
$724$	−12167.8	−0.624603
$725$	12135.0	0.621632
$726$	0	0
$727$	22064.5	1.12562	0.562812	−	0.826585i	$-0.309720\pi$
$727$	22064.5	1.12562	0.562812	+	0.826585i	$0.309720\pi$
$728$	−1918.03	−0.0976469
$729$	0	0
$730$	−1405.19	−0.0712442
$731$	11106.1	0.561933
$732$	0	0
$733$	−10137.1	−0.510806	−0.255403	−	0.966835i	$-0.582208\pi$
$733$	−10137.1	−0.510806	−0.255403	+	0.966835i	$0.582208\pi$
$734$	2616.90	0.131596
$735$	0	0
$736$	−10964.6	−0.549134
$737$	32360.3	1.61738
$738$	0	0
$739$	12920.4	0.643146	0.321573	−	0.946885i	$-0.395788\pi$
$739$	12920.4	0.643146	0.321573	+	0.946885i	$0.395788\pi$
$740$	183.188	0.00910016
$741$	0	0
$742$	−1035.27	−0.0512210
$743$	33715.8	1.66476	0.832378	−	0.554208i	$-0.186979\pi$
$743$	33715.8	1.66476	0.832378	+	0.554208i	$0.186979\pi$
$744$	0	0
$745$	−1821.45	−0.0895742
$746$	−4365.77	−0.214266
$747$	0	0
$748$	11012.5	0.538309
$749$	246.982	0.0120488
$750$	0	0
$751$	−12877.8	−0.625721	−0.312860	−	0.949799i	$-0.601287\pi$
$751$	−12877.8	−0.625721	−0.312860	+	0.949799i	$0.601287\pi$
$752$	19968.6	0.968324
$753$	0	0
$754$	2167.58	0.104693
$755$	−2489.64	−0.120010
$756$	0	0
$757$	1195.99	0.0574228	0.0287114	−	0.999588i	$-0.490860\pi$
$757$	1195.99	0.0574228	0.0287114	+	0.999588i	$0.490860\pi$
$758$	4495.60	0.215419
$759$	0	0
$760$	−233.368	−0.0111384
$761$	892.481	0.0425130	0.0212565	−	0.999774i	$-0.493233\pi$
$761$	892.481	0.0425130	0.0212565	+	0.999774i	$0.493233\pi$
$762$	0	0
$763$	6555.36	0.311035
$764$	19396.4	0.918505
$765$	0	0
$766$	453.639	0.0213977
$767$	7033.86	0.331132
$768$	0	0
$769$	8029.88	0.376547	0.188274	−	0.982117i	$-0.439711\pi$
$769$	8029.88	0.376547	0.188274	+	0.982117i	$0.439711\pi$
$770$	253.537	0.0118660
$771$	0	0
$772$	21565.0	1.00536
$773$	11854.1	0.551569	0.275784	−	0.961220i	$-0.411062\pi$
$773$	11854.1	0.551569	0.275784	+	0.961220i	$0.411062\pi$
$774$	0	0
$775$	17023.7	0.789045
$776$	7293.78	0.337411
$777$	0	0
$778$	7292.47	0.336051
$779$	−4261.82	−0.196015
$780$	0	0
$781$	−15784.7	−0.723203
$782$	−2373.67	−0.108545
$783$	0	0
$784$	−17174.3	−0.782359
$785$	5850.15	0.265988
$786$	0	0
$787$	−33354.2	−1.51073	−0.755367	−	0.655301i	$-0.772542\pi$
$787$	−33354.2	−1.51073	−0.755367	+	0.655301i	$0.772542\pi$
$788$	−32802.8	−1.48293
$789$	0	0
$790$	1191.24	0.0536487
$791$	−2669.63	−0.120002
$792$	0	0
$793$	25364.6	1.13584
$794$	2406.74	0.107572
$795$	0	0
$796$	−10116.6	−0.450471
$797$	−16099.7	−0.715533	−0.357767	−	0.933811i	$-0.616462\pi$
$797$	−16099.7	−0.715533	−0.357767	+	0.933811i	$0.616462\pi$
$798$	0	0
$799$	14104.6	0.624511
$800$	−13274.0	−0.586632
$801$	0	0
$802$	5006.59	0.220435
$803$	43006.4	1.88999
$804$	0	0
$805$	1127.35	0.0493587
$806$	3040.80	0.132888
$807$	0	0
$808$	−11828.0	−0.514986
$809$	−22730.2	−0.987824	−0.493912	−	0.869512i	$-0.664434\pi$
$809$	−22730.2	−0.987824	−0.493912	+	0.869512i	$0.664434\pi$
$810$	0	0
$811$	−10739.2	−0.464988	−0.232494	−	0.972598i	$-0.574689\pi$
$811$	−10739.2	−0.464988	−0.232494	+	0.972598i	$0.574689\pi$
$812$	−4340.86	−0.187604
$813$	0	0
$814$	271.778	0.0117025
$815$	−2252.22	−0.0968000
$816$	0	0
$817$	3516.30	0.150575
$818$	5853.47	0.250198
$819$	0	0
$820$	5227.37	0.222619
$821$	−13956.0	−0.593263	−0.296631	−	0.954992i	$-0.595863\pi$
$821$	−13956.0	−0.593263	−0.296631	+	0.954992i	$0.595863\pi$
$822$	0	0
$823$	−44979.4	−1.90508	−0.952541	−	0.304412i	$-0.901540\pi$
$823$	−44979.4	−1.90508	−0.952541	+	0.304412i	$0.901540\pi$
$824$	−17306.6	−0.731679
$825$	0	0
$826$	682.831	0.0287636
$827$	40831.2	1.71686	0.858428	−	0.512934i	$-0.171441\pi$
$827$	40831.2	1.71686	0.858428	+	0.512934i	$0.171441\pi$
$828$	0	0
$829$	32065.9	1.34342	0.671710	−	0.740815i	$-0.265560\pi$
$829$	32065.9	1.34342	0.671710	+	0.740815i	$0.265560\pi$
$830$	1156.63	0.0483701
$831$	0	0
$832$	13346.6	0.556142
$833$	−12130.9	−0.504574
$834$	0	0
$835$	7009.33	0.290501
$836$	3486.66	0.144245
$837$	0	0
$838$	−4514.22	−0.186087
$839$	−41649.1	−1.71381	−0.856906	−	0.515473i	$-0.827616\pi$
$839$	−41649.1	−1.71381	−0.856906	+	0.515473i	$0.827616\pi$
$840$	0	0
$841$	−14339.9	−0.587967
$842$	−2199.72	−0.0900325
$843$	0	0
$844$	−11342.9	−0.462607
$845$	−1853.33	−0.0754513
$846$	0	0
$847$	−205.931	−0.00835404
$848$	16575.2	0.671219
$849$	0	0
$850$	−2873.59	−0.115957
$851$	1208.46	0.0486784
$852$	0	0
$853$	40151.0	1.61166	0.805830	−	0.592147i	$-0.201720\pi$
$853$	40151.0	1.61166	0.805830	+	0.592147i	$0.201720\pi$
$854$	2462.34	0.0986645
$855$	0	0
$856$	413.687	0.0165182
$857$	6166.74	0.245801	0.122901	−	0.992419i	$-0.460780\pi$
$857$	6166.74	0.245801	0.122901	+	0.992419i	$0.460780\pi$
$858$	0	0
$859$	17412.5	0.691627	0.345813	−	0.938303i	$-0.387603\pi$
$859$	17412.5	0.691627	0.345813	+	0.938303i	$0.387603\pi$
$860$	−4312.95	−0.171012
$861$	0	0
$862$	2927.93	0.115691
$863$	−28194.1	−1.11210	−0.556048	−	0.831150i	$-0.687683\pi$
$863$	−28194.1	−1.11210	−0.556048	+	0.831150i	$0.687683\pi$
$864$	0	0
$865$	1473.67	0.0579265
$866$	4958.35	0.194563
$867$	0	0
$868$	−6089.61	−0.238128
$869$	−36458.5	−1.42321
$870$	0	0
$871$	31114.9	1.21044
$872$	10980.0	0.426411
$873$	0	0
$874$	−751.528	−0.0290856
$875$	2774.06	0.107178
$876$	0	0
$877$	−26350.8	−1.01460	−0.507300	−	0.861770i	$-0.669356\pi$
$877$	−26350.8	−1.01460	−0.507300	+	0.861770i	$0.669356\pi$
$878$	−8885.91	−0.341554
$879$	0	0
$880$	−4059.24	−0.155497
$881$	18214.9	0.696566	0.348283	−	0.937390i	$-0.386765\pi$
$881$	18214.9	0.696566	0.348283	+	0.937390i	$0.386765\pi$
$882$	0	0
$883$	−36170.1	−1.37850	−0.689252	−	0.724521i	$-0.742061\pi$
$883$	−36170.1	−1.37850	−0.689252	+	0.724521i	$0.742061\pi$
$884$	10588.7	0.402868
$885$	0	0
$886$	7949.78	0.301442
$887$	−15910.3	−0.602271	−0.301136	−	0.953581i	$-0.597366\pi$
$887$	−15910.3	−0.602271	−0.301136	+	0.953581i	$0.597366\pi$
$888$	0	0
$889$	−6837.02	−0.257938
$890$	−139.113	−0.00523940
$891$	0	0
$892$	−11158.1	−0.418836
$893$	4465.66	0.167343
$894$	0	0
$895$	−7637.77	−0.285254
$896$	6274.11	0.233932
$897$	0	0
$898$	8389.22	0.311750
$899$	14097.4	0.522998
$900$	0	0
$901$	11707.7	0.432896
$902$	7755.34	0.286280
$903$	0	0
$904$	−4471.55	−0.164515
$905$	3168.00	0.116362
$906$	0	0
$907$	−17925.1	−0.656221	−0.328111	−	0.944639i	$-0.606412\pi$
$907$	−17925.1	−0.656221	−0.328111	+	0.944639i	$0.606412\pi$
$908$	5457.79	0.199475
$909$	0	0
$910$	243.780	0.00888047
$911$	8730.85	0.317526	0.158763	−	0.987317i	$-0.449249\pi$
$911$	8730.85	0.317526	0.158763	+	0.987317i	$0.449249\pi$
$912$	0	0
$913$	−35399.2	−1.28318
$914$	−6785.09	−0.245548
$915$	0	0
$916$	−49293.7	−1.77807
$917$	1253.89	0.0451548
$918$	0	0
$919$	17649.9	0.633534	0.316767	−	0.948503i	$-0.397403\pi$
$919$	17649.9	0.633534	0.316767	+	0.948503i	$0.397403\pi$
$920$	1888.27	0.0676679
$921$	0	0
$922$	−393.567	−0.0140580
$923$	−15177.3	−0.541242
$924$	0	0
$925$	1462.97	0.0520025
$926$	6066.79	0.215299
$927$	0	0
$928$	−10992.2	−0.388834
$929$	25256.4	0.891967	0.445983	−	0.895041i	$-0.352854\pi$
$929$	25256.4	0.891967	0.445983	+	0.895041i	$0.352854\pi$
$930$	0	0
$931$	−3840.77	−0.135205
$932$	−43780.7	−1.53872
$933$	0	0
$934$	−10263.3	−0.359555
$935$	−2867.20	−0.100286
$936$	0	0
$937$	28695.1	1.00046	0.500229	−	0.865893i	$-0.333249\pi$
$937$	28695.1	1.00046	0.500229	+	0.865893i	$0.333249\pi$
$938$	3020.57	0.105144
$939$	0	0
$940$	−5477.40	−0.190057
$941$	1790.57	0.0620306	0.0310153	−	0.999519i	$-0.490126\pi$
$941$	1790.57	0.0620306	0.0310153	+	0.999519i	$0.490126\pi$
$942$	0	0
$943$	34484.0	1.19083
$944$	−10932.4	−0.376929
$945$	0	0
$946$	−6398.71	−0.219915
$947$	−13284.8	−0.455860	−0.227930	−	0.973678i	$-0.573196\pi$
$947$	−13284.8	−0.455860	−0.227930	+	0.973678i	$0.573196\pi$
$948$	0	0
$949$	41351.4	1.41446
$950$	−909.811	−0.0310717
$951$	0	0
$952$	2105.68	0.0716863
$953$	44511.2	1.51297	0.756485	−	0.654011i	$-0.226915\pi$
$953$	44511.2	1.51297	0.756485	+	0.654011i	$0.226915\pi$
$954$	0	0
$955$	−5050.04	−0.171116
$956$	−1823.60	−0.0616940
$957$	0	0
$958$	−481.822	−0.0162494
$959$	−17021.6	−0.573154
$960$	0	0
$961$	−10014.3	−0.336153
$962$	261.319	0.00875807
$963$	0	0
$964$	−148.569	−0.00496377
$965$	−5614.64	−0.187297
$966$	0	0
$967$	37025.4	1.23129	0.615644	−	0.788025i	$-0.288896\pi$
$967$	37025.4	1.23129	0.615644	+	0.788025i	$0.288896\pi$
$968$	−344.928	−0.0114529
$969$	0	0
$970$	−927.033	−0.0306858
$971$	43175.4	1.42695	0.713473	−	0.700682i	$-0.247121\pi$
$971$	43175.4	1.42695	0.713473	+	0.700682i	$0.247121\pi$
$972$	0	0
$973$	6439.77	0.212178
$974$	−6564.30	−0.215948
$975$	0	0
$976$	−39423.2	−1.29294
$977$	−10364.6	−0.339398	−0.169699	−	0.985496i	$-0.554280\pi$
$977$	−10364.6	−0.339398	−0.169699	+	0.985496i	$0.554280\pi$
$978$	0	0
$979$	4257.61	0.138993
$980$	4710.94	0.153556
$981$	0	0
$982$	6897.78	0.224152
$983$	6223.47	0.201931	0.100965	−	0.994890i	$-0.467807\pi$
$983$	6223.47	0.201931	0.100965	+	0.994890i	$0.467807\pi$
$984$	0	0
$985$	8540.52	0.276268
$986$	−2379.64	−0.0768591
$987$	0	0
$988$	3352.49	0.107952
$989$	−28451.8	−0.914776
$990$	0	0
$991$	−17421.5	−0.558439	−0.279219	−	0.960227i	$-0.590076\pi$
$991$	−17421.5	−0.558439	−0.279219	+	0.960227i	$0.590076\pi$
$992$	−15420.5	−0.493551
$993$	0	0
$994$	−1473.38	−0.0470147
$995$	2633.96	0.0839219
$996$	0	0
$997$	16970.0	0.539062	0.269531	−	0.962992i	$-0.413131\pi$
$997$	16970.0	0.539062	0.269531	+	0.962992i	$0.413131\pi$
$998$	−3850.19	−0.122120
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.c.1.14		28
3.2	odd	2		717.4.a.a.1.15	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.4.a.a.1.15	✓	28	3.2	odd	2
2151.4.a.c.1.14		28	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+0.608170 q^{2} -7.63013 q^{4} +1.98658 q^{5} -5.67520 q^{7} -9.50578 q^{8} +O(q^{10})\)	\(q+0.608170 q^{2} -7.63013 q^{4} +1.98658 q^{5} -5.67520 q^{7} -9.50578 q^{8} +1.20818 q^{10} -36.9768 q^{11} -35.5538 q^{13} -3.45149 q^{14} +55.2599 q^{16} +39.0322 q^{17} +12.3580 q^{19} -15.1578 q^{20} -22.4882 q^{22} -99.9935 q^{23} -121.054 q^{25} -21.6228 q^{26} +43.3025 q^{28} -100.245 q^{29} -140.630 q^{31} +109.654 q^{32} +23.7382 q^{34} -11.2742 q^{35} -12.0854 q^{37} +7.51577 q^{38} -18.8840 q^{40} -344.863 q^{41} +284.536 q^{43} +282.138 q^{44} -60.8130 q^{46} +361.358 q^{47} -310.792 q^{49} -73.6211 q^{50} +271.280 q^{52} +299.949 q^{53} -73.4573 q^{55} +53.9472 q^{56} -60.9660 q^{58} -197.837 q^{59} -713.414 q^{61} -85.5267 q^{62} -375.391 q^{64} -70.6304 q^{65} -875.150 q^{67} -297.821 q^{68} -6.85664 q^{70} +426.881 q^{71} -1163.06 q^{73} -7.34995 q^{74} -94.2932 q^{76} +209.851 q^{77} +985.983 q^{79} +109.778 q^{80} -209.735 q^{82} +957.334 q^{83} +77.5404 q^{85} +173.046 q^{86} +351.494 q^{88} -115.143 q^{89} +201.775 q^{91} +762.963 q^{92} +219.767 q^{94} +24.5501 q^{95} -767.299 q^{97} -189.015 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 13 q^{2} + 99 q^{4} + 74 q^{5} - 82 q^{7} + 135 q^{8}+O(q^{10}) \) 28 * q + 13 * q^2 + 99 * q^4 + 74 * q^5 - 82 * q^7 + 135 * q^8	\( 28 q + 13 q^{2} + 99 q^{4} + 74 q^{5} - 82 q^{7} + 135 q^{8} - 68 q^{10} + 258 q^{11} - 134 q^{13} + 292 q^{14} + 327 q^{16} + 364 q^{17} - 278 q^{19} + 986 q^{20} - 179 q^{22} + 668 q^{23} + 490 q^{25} + 760 q^{26} - 802 q^{28} + 714 q^{29} - 608 q^{31} + 918 q^{32} - 228 q^{34} + 934 q^{35} - 1080 q^{37} + 1395 q^{38} - 563 q^{40} + 1796 q^{41} - 1934 q^{43} + 3157 q^{44} - 940 q^{46} + 2032 q^{47} + 762 q^{49} + 1754 q^{50} - 2328 q^{52} + 1790 q^{53} - 478 q^{55} + 3557 q^{56} - 2626 q^{58} + 3622 q^{59} + 324 q^{61} + 796 q^{62} + 2023 q^{64} + 2200 q^{65} - 2444 q^{67} - 357 q^{68} + 4305 q^{70} + 1298 q^{71} - 1368 q^{73} - 813 q^{74} + 1390 q^{76} + 1408 q^{77} - 1378 q^{79} + 7684 q^{80} + 9001 q^{82} + 3524 q^{83} + 60 q^{85} + 2543 q^{86} + 1749 q^{88} + 7854 q^{89} + 850 q^{91} + 496 q^{92} + 6634 q^{94} + 3696 q^{95} - 1746 q^{97} + 4632 q^{98}+O(q^{100}) \) 28 * q + 13 * q^2 + 99 * q^4 + 74 * q^5 - 82 * q^7 + 135 * q^8 - 68 * q^10 + 258 * q^11 - 134 * q^13 + 292 * q^14 + 327 * q^16 + 364 * q^17 - 278 * q^19 + 986 * q^20 - 179 * q^22 + 668 * q^23 + 490 * q^25 + 760 * q^26 - 802 * q^28 + 714 * q^29 - 608 * q^31 + 918 * q^32 - 228 * q^34 + 934 * q^35 - 1080 * q^37 + 1395 * q^38 - 563 * q^40 + 1796 * q^41 - 1934 * q^43 + 3157 * q^44 - 940 * q^46 + 2032 * q^47 + 762 * q^49 + 1754 * q^50 - 2328 * q^52 + 1790 * q^53 - 478 * q^55 + 3557 * q^56 - 2626 * q^58 + 3622 * q^59 + 324 * q^61 + 796 * q^62 + 2023 * q^64 + 2200 * q^65 - 2444 * q^67 - 357 * q^68 + 4305 * q^70 + 1298 * q^71 - 1368 * q^73 - 813 * q^74 + 1390 * q^76 + 1408 * q^77 - 1378 * q^79 + 7684 * q^80 + 9001 * q^82 + 3524 * q^83 + 60 * q^85 + 2543 * q^86 + 1749 * q^88 + 7854 * q^89 + 850 * q^91 + 496 * q^92 + 6634 * q^94 + 3696 * q^95 - 1746 * q^97 + 4632 * q^98

Introduction

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