LMFDB - Embedded newform 2151.4.a.h.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:	$59$
Twist minimal:	yes
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)

$f(q)$	$=$	$q-3.10672 q^{2} +1.65170 q^{4} -17.8784 q^{5} +0.745718 q^{7} +19.7224 q^{8} +O(q^{10})$	\(q-3.10672 q^{2} +1.65170 q^{4} -17.8784 q^{5} +0.745718 q^{7} +19.7224 q^{8} +55.5430 q^{10} -24.7007 q^{11} -48.8856 q^{13} -2.31673 q^{14} -74.4855 q^{16} +98.7517 q^{17} +29.0037 q^{19} -29.5297 q^{20} +76.7380 q^{22} +104.592 q^{23} +194.636 q^{25} +151.874 q^{26} +1.23170 q^{28} -143.025 q^{29} -9.12534 q^{31} +73.6264 q^{32} -306.794 q^{34} -13.3322 q^{35} -280.097 q^{37} -90.1064 q^{38} -352.604 q^{40} +350.333 q^{41} -161.036 q^{43} -40.7981 q^{44} -324.938 q^{46} -160.894 q^{47} -342.444 q^{49} -604.679 q^{50} -80.7443 q^{52} -316.963 q^{53} +441.607 q^{55} +14.7073 q^{56} +444.339 q^{58} +295.547 q^{59} -431.505 q^{61} +28.3499 q^{62} +367.147 q^{64} +873.994 q^{65} -547.646 q^{67} +163.108 q^{68} +41.4194 q^{70} +841.853 q^{71} +189.720 q^{73} +870.184 q^{74} +47.9054 q^{76} -18.4197 q^{77} -587.506 q^{79} +1331.68 q^{80} -1088.39 q^{82} +237.277 q^{83} -1765.52 q^{85} +500.292 q^{86} -487.156 q^{88} -1303.76 q^{89} -36.4548 q^{91} +172.754 q^{92} +499.854 q^{94} -518.539 q^{95} -364.918 q^{97} +1063.88 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−3.10672	−1.09839	−0.549195	−	0.835694i	$-0.685066\pi$
$2$	−3.10672	−1.09839	−0.549195	+	0.835694i	$0.685066\pi$
$3$	0	0
$3$	0	0
$4$	1.65170	0.206462
$5$	−17.8784	−1.59909	−0.799545	−	0.600607i	$-0.794926\pi$
$5$	−17.8784	−1.59909	−0.799545	+	0.600607i	$0.794926\pi$
$6$	0	0
$7$	0.745718	0.0402650	0.0201325	−	0.999797i	$-0.493591\pi$
$7$	0.745718	0.0402650	0.0201325	+	0.999797i	$0.493591\pi$
$8$	19.7224	0.871614
$9$	0	0
$10$	55.5430	1.75643
$11$	−24.7007	−0.677048	−0.338524	−	0.940958i	$-0.609928\pi$
$11$	−24.7007	−0.677048	−0.338524	+	0.940958i	$0.609928\pi$
$12$	0	0
$13$	−48.8856	−1.04296	−0.521478	−	0.853265i	$-0.674619\pi$
$13$	−48.8856	−1.04296	−0.521478	+	0.853265i	$0.674619\pi$
$14$	−2.31673	−0.0442267
$15$	0	0
$16$	−74.4855	−1.16384
$17$	98.7517	1.40887	0.704435	−	0.709768i	$-0.251200\pi$
$17$	98.7517	1.40887	0.704435	+	0.709768i	$0.251200\pi$
$18$	0	0
$19$	29.0037	0.350206	0.175103	−	0.984550i	$-0.443974\pi$
$19$	29.0037	0.350206	0.175103	+	0.984550i	$0.443974\pi$
$20$	−29.5297	−0.330152
$21$	0	0
$22$	76.7380	0.743664
$23$	104.592	0.948214	0.474107	−	0.880467i	$-0.342771\pi$
$23$	104.592	0.948214	0.474107	+	0.880467i	$0.342771\pi$
$24$	0	0
$25$	194.636	1.55709
$26$	151.874	1.14557
$27$	0	0
$28$	1.23170	0.00831320
$29$	−143.025	−0.915831	−0.457915	−	0.888996i	$-0.651404\pi$
$29$	−143.025	−0.915831	−0.457915	+	0.888996i	$0.651404\pi$
$30$	0	0
$31$	−9.12534	−0.0528697	−0.0264348	−	0.999651i	$-0.508415\pi$
$31$	−9.12534	−0.0528697	−0.0264348	+	0.999651i	$0.508415\pi$
$32$	73.6264	0.406732
$33$	0	0
$34$	−306.794	−1.54749
$35$	−13.3322	−0.0643873
$36$	0	0
$37$	−280.097	−1.24453	−0.622267	−	0.782805i	$-0.713788\pi$
$37$	−280.097	−1.24453	−0.622267	+	0.782805i	$0.713788\pi$
$38$	−90.1064	−0.384663
$39$	0	0
$40$	−352.604	−1.39379
$41$	350.333	1.33446	0.667230	−	0.744852i	$-0.267480\pi$
$41$	350.333	1.33446	0.667230	+	0.744852i	$0.267480\pi$
$42$	0	0
$43$	−161.036	−0.571109	−0.285555	−	0.958362i	$-0.592178\pi$
$43$	−161.036	−0.571109	−0.285555	+	0.958362i	$0.592178\pi$
$44$	−40.7981	−0.139785
$45$	0	0
$46$	−324.938	−1.04151
$47$	−160.894	−0.499337	−0.249669	−	0.968331i	$-0.580322\pi$
$47$	−160.894	−0.499337	−0.249669	+	0.968331i	$0.580322\pi$
$48$	0	0
$49$	−342.444	−0.998379
$50$	−604.679	−1.71029
$51$	0	0
$52$	−80.7443	−0.215331
$53$	−316.963	−0.821476	−0.410738	−	0.911754i	$-0.634729\pi$
$53$	−316.963	−0.821476	−0.410738	+	0.911754i	$0.634729\pi$
$54$	0	0
$55$	441.607	1.08266
$56$	14.7073	0.0350955
$57$	0	0
$58$	444.339	1.00594
$59$	295.547	0.652151	0.326076	−	0.945344i	$-0.394274\pi$
$59$	295.547	0.652151	0.326076	+	0.945344i	$0.394274\pi$
$60$	0	0
$61$	−431.505	−0.905713	−0.452857	−	0.891583i	$-0.649595\pi$
$61$	−431.505	−0.905713	−0.452857	+	0.891583i	$0.649595\pi$
$62$	28.3499	0.0580715
$63$	0	0
$64$	367.147	0.717085
$65$	873.994	1.66778
$66$	0	0
$67$	−547.646	−0.998591	−0.499296	−	0.866432i	$-0.666408\pi$
$67$	−547.646	−0.998591	−0.499296	+	0.866432i	$0.666408\pi$
$68$	163.108	0.290879
$69$	0	0
$70$	41.4194	0.0707224
$71$	841.853	1.40718	0.703588	−	0.710608i	$-0.251580\pi$
$71$	841.853	1.40718	0.703588	+	0.710608i	$0.251580\pi$
$72$	0	0
$73$	189.720	0.304178	0.152089	−	0.988367i	$-0.451400\pi$
$73$	189.720	0.304178	0.152089	+	0.988367i	$0.451400\pi$
$74$	870.184	1.36698
$75$	0	0
$76$	47.9054	0.0723043
$77$	−18.4197	−0.0272613
$78$	0	0
$79$	−587.506	−0.836703	−0.418352	−	0.908285i	$-0.637392\pi$
$79$	−587.506	−0.836703	−0.418352	+	0.908285i	$0.637392\pi$
$80$	1331.68	1.86108
$81$	0	0
$82$	−1088.39	−1.46576
$83$	237.277	0.313790	0.156895	−	0.987615i	$-0.449852\pi$
$83$	237.277	0.313790	0.156895	+	0.987615i	$0.449852\pi$
$84$	0	0
$85$	−1765.52	−2.25291
$86$	500.292	0.627301
$87$	0	0
$88$	−487.156	−0.590125
$89$	−1303.76	−1.55279	−0.776394	−	0.630248i	$-0.782953\pi$
$89$	−1303.76	−1.55279	−0.776394	+	0.630248i	$0.782953\pi$
$90$	0	0
$91$	−36.4548	−0.0419946
$92$	172.754	0.195771
$93$	0	0
$94$	499.854	0.548468
$95$	−518.539	−0.560010
$96$	0	0
$97$	−364.918	−0.381977	−0.190989	−	0.981592i	$-0.561169\pi$
$97$	−364.918	−0.381977	−0.190989	+	0.981592i	$0.561169\pi$
$98$	1063.88	1.09661
$99$	0	0
$100$	321.480	0.321480
$101$	−1684.17	−1.65922	−0.829610	−	0.558344i	$-0.811437\pi$
$101$	−1684.17	−1.65922	−0.829610	+	0.558344i	$0.811437\pi$
$102$	0	0
$103$	−29.1605	−0.0278958	−0.0139479	−	0.999903i	$-0.504440\pi$
$103$	−29.1605	−0.0278958	−0.0139479	+	0.999903i	$0.504440\pi$
$104$	−964.140	−0.909055
$105$	0	0
$106$	984.715	0.902301
$107$	−1521.44	−1.37461	−0.687304	−	0.726370i	$-0.741206\pi$
$107$	−1521.44	−1.37461	−0.687304	+	0.726370i	$0.741206\pi$
$108$	0	0
$109$	1470.92	1.29256	0.646278	−	0.763102i	$-0.276325\pi$
$109$	1470.92	1.29256	0.646278	+	0.763102i	$0.276325\pi$
$110$	−1371.95	−1.18918
$111$	0	0
$112$	−55.5451	−0.0468618
$113$	−28.8054	−0.0239804	−0.0119902	−	0.999928i	$-0.503817\pi$
$113$	−28.8054	−0.0239804	−0.0119902	+	0.999928i	$0.503817\pi$
$114$	0	0
$115$	−1869.93	−1.51628
$116$	−236.235	−0.189085
$117$	0	0
$118$	−918.181	−0.716317
$119$	73.6409	0.0567281
$120$	0	0
$121$	−720.877	−0.541606
$122$	1340.56	0.994827
$123$	0	0
$124$	−15.0723	−0.0109156
$125$	−1244.98	−0.890832
$126$	0	0
$127$	1705.66	1.19175	0.595877	−	0.803076i	$-0.296805\pi$
$127$	1705.66	1.19175	0.595877	+	0.803076i	$0.296805\pi$
$128$	−1729.63	−1.19437
$129$	0	0
$130$	−2715.25	−1.83187
$131$	920.292	0.613789	0.306894	−	0.951744i	$-0.400710\pi$
$131$	920.292	0.613789	0.306894	+	0.951744i	$0.400710\pi$
$132$	0	0
$133$	21.6286	0.0141010
$134$	1701.38	1.09684
$135$	0	0
$136$	1947.62	1.22799
$137$	−163.174	−0.101758	−0.0508791	−	0.998705i	$-0.516202\pi$
$137$	−163.174	−0.101758	−0.0508791	+	0.998705i	$0.516202\pi$
$138$	0	0
$139$	−2133.81	−1.30207	−0.651035	−	0.759048i	$-0.725665\pi$
$139$	−2133.81	−1.30207	−0.651035	+	0.759048i	$0.725665\pi$
$140$	−22.0208	−0.0132936
$141$	0	0
$142$	−2615.40	−1.54563
$143$	1207.51	0.706131
$144$	0	0
$145$	2557.06	1.46450
$146$	−589.406	−0.334107
$147$	0	0
$148$	−462.637	−0.256949
$149$	−1922.08	−1.05680	−0.528398	−	0.848997i	$-0.677207\pi$
$149$	−1922.08	−1.05680	−0.528398	+	0.848997i	$0.677207\pi$
$150$	0	0
$151$	−2674.86	−1.44157	−0.720784	−	0.693159i	$-0.756218\pi$
$151$	−2674.86	−1.44157	−0.720784	+	0.693159i	$0.756218\pi$
$152$	572.022	0.305244
$153$	0	0
$154$	57.2249	0.0299436
$155$	163.146	0.0845433
$156$	0	0
$157$	−1031.87	−0.524536	−0.262268	−	0.964995i	$-0.584470\pi$
$157$	−1031.87	−0.524536	−0.262268	+	0.964995i	$0.584470\pi$
$158$	1825.21	0.919027
$159$	0	0
$160$	−1316.32	−0.650401
$161$	77.9960	0.0381798
$162$	0	0
$163$	−1046.99	−0.503106	−0.251553	−	0.967843i	$-0.580941\pi$
$163$	−1046.99	−0.503106	−0.251553	+	0.967843i	$0.580941\pi$
$164$	578.645	0.275516
$165$	0	0
$166$	−737.153	−0.344664
$167$	−476.791	−0.220929	−0.110465	−	0.993880i	$-0.535234\pi$
$167$	−476.791	−0.220929	−0.110465	+	0.993880i	$0.535234\pi$
$168$	0	0
$169$	192.801	0.0877565
$170$	5484.97	2.47458
$171$	0	0
$172$	−265.982	−0.117913
$173$	1492.14	0.655754	0.327877	−	0.944720i	$-0.393667\pi$
$173$	1492.14	0.655754	0.327877	+	0.944720i	$0.393667\pi$
$174$	0	0
$175$	145.143	0.0626961
$176$	1839.84	0.787973
$177$	0	0
$178$	4050.41	1.70557
$179$	−1851.15	−0.772968	−0.386484	−	0.922296i	$-0.626311\pi$
$179$	−1851.15	−0.772968	−0.386484	+	0.922296i	$0.626311\pi$
$180$	0	0
$181$	−2034.86	−0.835633	−0.417817	−	0.908531i	$-0.637205\pi$
$181$	−2034.86	−0.835633	−0.417817	+	0.908531i	$0.637205\pi$
$182$	113.255	0.0461265
$183$	0	0
$184$	2062.80	0.826477
$185$	5007.68	1.99012
$186$	0	0
$187$	−2439.23	−0.953873
$188$	−265.749	−0.103094
$189$	0	0
$190$	1610.95	0.615110
$191$	3100.11	1.17443	0.587214	−	0.809431i	$-0.300225\pi$
$191$	3100.11	1.17443	0.587214	+	0.809431i	$0.300225\pi$
$192$	0	0
$193$	1287.94	0.480353	0.240177	−	0.970729i	$-0.422795\pi$
$193$	1287.94	0.480353	0.240177	+	0.970729i	$0.422795\pi$
$194$	1133.70	0.419560
$195$	0	0
$196$	−565.614	−0.206128
$197$	4343.12	1.57073	0.785367	−	0.619031i	$-0.212474\pi$
$197$	4343.12	1.57073	0.785367	+	0.619031i	$0.212474\pi$
$198$	0	0
$199$	−30.9425	−0.0110224	−0.00551120	−	0.999985i	$-0.501754\pi$
$199$	−30.9425	−0.0110224	−0.00551120	+	0.999985i	$0.501754\pi$
$200$	3838.68	1.35718
$201$	0	0
$202$	5232.24	1.82247
$203$	−106.656	−0.0368759
$204$	0	0
$205$	−6263.39	−2.13392
$206$	90.5935	0.0306405
$207$	0	0
$208$	3641.27	1.21383
$209$	−716.411	−0.237106
$210$	0	0
$211$	3748.26	1.22294	0.611471	−	0.791267i	$-0.290578\pi$
$211$	3748.26	1.22294	0.611471	+	0.791267i	$0.290578\pi$
$212$	−523.527	−0.169604
$213$	0	0
$214$	4726.68	1.50986
$215$	2879.05	0.913255
$216$	0	0
$217$	−6.80493	−0.00212879
$218$	−4569.73	−1.41973
$219$	0	0
$220$	729.403	0.223529
$221$	−4827.53	−1.46939
$222$	0	0
$223$	−2771.21	−0.832171	−0.416085	−	0.909326i	$-0.636598\pi$
$223$	−2771.21	−0.832171	−0.416085	+	0.909326i	$0.636598\pi$
$224$	54.9045	0.0163771
$225$	0	0
$226$	89.4902	0.0263398
$227$	236.430	0.0691296	0.0345648	−	0.999402i	$-0.488995\pi$
$227$	236.430	0.0691296	0.0345648	+	0.999402i	$0.488995\pi$
$228$	0	0
$229$	322.526	0.0930705	0.0465352	−	0.998917i	$-0.485182\pi$
$229$	322.526	0.0930705	0.0465352	+	0.998917i	$0.485182\pi$
$230$	5809.35	1.66547
$231$	0	0
$232$	−2820.80	−0.798251
$233$	1999.92	0.562312	0.281156	−	0.959662i	$-0.409282\pi$
$233$	1999.92	0.562312	0.281156	+	0.959662i	$0.409282\pi$
$234$	0	0
$235$	2876.53	0.798485
$236$	488.154	0.134645
$237$	0	0
$238$	−228.781	−0.0623097
$239$	−239.000	−0.0646846
$239$	−239.000	−0.0646846
$240$	0	0
$241$	−5112.71	−1.36655	−0.683275	−	0.730161i	$-0.739445\pi$
$241$	−5112.71	−1.36655	−0.683275	+	0.730161i	$0.739445\pi$
$242$	2239.56	0.594895
$243$	0	0
$244$	−712.716	−0.186996
$245$	6122.34	1.59650
$246$	0	0
$247$	−1417.86	−0.365249
$248$	−179.973	−0.0460820
$249$	0	0
$250$	3867.79	0.978482
$251$	−2554.30	−0.642334	−0.321167	−	0.947023i	$-0.604075\pi$
$251$	−2554.30	−0.642334	−0.321167	+	0.947023i	$0.604075\pi$
$252$	0	0
$253$	−2583.49	−0.641987
$254$	−5299.00	−1.30901
$255$	0	0
$256$	2436.31	0.594802
$257$	4075.19	0.989119	0.494559	−	0.869144i	$-0.335329\pi$
$257$	4075.19	0.989119	0.494559	+	0.869144i	$0.335329\pi$
$258$	0	0
$259$	−208.874	−0.0501111
$260$	1443.58	0.344334
$261$	0	0
$262$	−2859.09	−0.674180
$263$	737.047	0.172807	0.0864035	−	0.996260i	$-0.472463\pi$
$263$	737.047	0.172807	0.0864035	+	0.996260i	$0.472463\pi$
$264$	0	0
$265$	5666.78	1.31361
$266$	−67.1939	−0.0154884
$267$	0	0
$268$	−904.547	−0.206172
$269$	3909.75	0.886178	0.443089	−	0.896478i	$-0.353883\pi$
$269$	3909.75	0.886178	0.443089	+	0.896478i	$0.353883\pi$
$270$	0	0
$271$	−843.576	−0.189091	−0.0945454	−	0.995521i	$-0.530140\pi$
$271$	−843.576	−0.189091	−0.0945454	+	0.995521i	$0.530140\pi$
$272$	−7355.57	−1.63969
$273$	0	0
$274$	506.935	0.111770
$275$	−4807.64	−1.05422
$276$	0	0
$277$	1297.21	0.281379	0.140690	−	0.990054i	$-0.455068\pi$
$277$	1297.21	0.281379	0.140690	+	0.990054i	$0.455068\pi$
$278$	6629.16	1.43018
$279$	0	0
$280$	−262.943	−0.0561209
$281$	−6877.66	−1.46010	−0.730048	−	0.683396i	$-0.760502\pi$
$281$	−6877.66	−1.46010	−0.730048	+	0.683396i	$0.760502\pi$
$282$	0	0
$283$	177.597	0.0373041	0.0186520	−	0.999826i	$-0.494063\pi$
$283$	177.597	0.0373041	0.0186520	+	0.999826i	$0.494063\pi$
$284$	1390.49	0.290529
$285$	0	0
$286$	−3751.38	−0.775608
$287$	261.250	0.0537320
$288$	0	0
$289$	4838.90	0.984917
$290$	−7944.05	−1.60859
$291$	0	0
$292$	313.360	0.0628014
$293$	−231.891	−0.0462362	−0.0231181	−	0.999733i	$-0.507359\pi$
$293$	−231.891	−0.0462362	−0.0231181	+	0.999733i	$0.507359\pi$
$294$	0	0
$295$	−5283.89	−1.04285
$296$	−5524.19	−1.08475
$297$	0	0
$298$	5971.35	1.16078
$299$	−5113.04	−0.988945
$300$	0	0
$301$	−120.087	−0.0229957
$302$	8310.03	1.58341
$303$	0	0
$304$	−2160.36	−0.407582
$305$	7714.60	1.44832
$306$	0	0
$307$	1985.03	0.369029	0.184514	−	0.982830i	$-0.440929\pi$
$307$	1985.03	0.369029	0.184514	+	0.982830i	$0.440929\pi$
$308$	−30.4238	−0.00562844
$309$	0	0
$310$	−506.849	−0.0928616
$311$	3843.49	0.700786	0.350393	−	0.936603i	$-0.386048\pi$
$311$	3843.49	0.700786	0.350393	+	0.936603i	$0.386048\pi$
$312$	0	0
$313$	3979.76	0.718688	0.359344	−	0.933205i	$-0.383001\pi$
$313$	3979.76	0.718688	0.359344	+	0.933205i	$0.383001\pi$
$314$	3205.73	0.576145
$315$	0	0
$316$	−970.383	−0.172748
$317$	−8025.86	−1.42201	−0.711005	−	0.703187i	$-0.751760\pi$
$317$	−8025.86	−1.42201	−0.711005	+	0.703187i	$0.751760\pi$
$318$	0	0
$319$	3532.82	0.620062
$320$	−6564.00	−1.14668
$321$	0	0
$322$	−242.312	−0.0419364
$323$	2864.17	0.493395
$324$	0	0
$325$	−9514.89	−1.62397
$326$	3252.69	0.552607
$327$	0	0
$328$	6909.41	1.16313
$329$	−119.982	−0.0201058
$330$	0	0
$331$	2201.00	0.365493	0.182746	−	0.983160i	$-0.441501\pi$
$331$	2201.00	0.365493	0.182746	+	0.983160i	$0.441501\pi$
$332$	391.910	0.0647858
$333$	0	0
$334$	1481.26	0.242667
$335$	9791.02	1.59684
$336$	0	0
$337$	9699.26	1.56781	0.783905	−	0.620880i	$-0.213225\pi$
$337$	9699.26	1.56781	0.783905	+	0.620880i	$0.213225\pi$
$338$	−598.978	−0.0963909
$339$	0	0
$340$	−2916.11	−0.465141
$341$	225.402	0.0357953
$342$	0	0
$343$	−511.148	−0.0804646
$344$	−3176.01	−0.497787
$345$	0	0
$346$	−4635.67	−0.720274
$347$	9863.87	1.52599	0.762997	−	0.646402i	$-0.223727\pi$
$347$	9863.87	1.52599	0.762997	+	0.646402i	$0.223727\pi$
$348$	0	0
$349$	−11050.6	−1.69491	−0.847454	−	0.530869i	$-0.821866\pi$
$349$	−11050.6	−1.69491	−0.847454	+	0.530869i	$0.821866\pi$
$350$	−450.920	−0.0688648
$351$	0	0
$352$	−1818.62	−0.275377
$353$	−511.811	−0.0771698	−0.0385849	−	0.999255i	$-0.512285\pi$
$353$	−511.811	−0.0771698	−0.0385849	+	0.999255i	$0.512285\pi$
$354$	0	0
$355$	−15051.0	−2.25020
$356$	−2153.42	−0.320592
$357$	0	0
$358$	5750.99	0.849021
$359$	12746.3	1.87389	0.936943	−	0.349481i	$-0.113642\pi$
$359$	12746.3	1.87389	0.936943	+	0.349481i	$0.113642\pi$
$360$	0	0
$361$	−6017.78	−0.877356
$362$	6321.72	0.917852
$363$	0	0
$364$	−60.2124	−0.00867030
$365$	−3391.88	−0.486409
$366$	0	0
$367$	7419.01	1.05523	0.527615	−	0.849484i	$-0.323086\pi$
$367$	7419.01	1.05523	0.527615	+	0.849484i	$0.323086\pi$
$368$	−7790.58	−1.10357
$369$	0	0
$370$	−15557.5	−2.18593
$371$	−236.365	−0.0330767
$372$	0	0
$373$	−1108.86	−0.153927	−0.0769633	−	0.997034i	$-0.524522\pi$
$373$	−1108.86	−0.153927	−0.0769633	+	0.997034i	$0.524522\pi$
$374$	7578.01	1.04773
$375$	0	0
$376$	−3173.22	−0.435230
$377$	6991.87	0.955171
$378$	0	0
$379$	10686.4	1.44835	0.724173	−	0.689618i	$-0.242222\pi$
$379$	10686.4	1.44835	0.724173	+	0.689618i	$0.242222\pi$
$380$	−856.471	−0.115621
$381$	0	0
$382$	−9631.16	−1.28998
$383$	−12048.2	−1.60740	−0.803702	−	0.595031i	$-0.797140\pi$
$383$	−12048.2	−1.60740	−0.803702	+	0.595031i	$0.797140\pi$
$384$	0	0
$385$	329.314	0.0435933
$386$	−4001.28	−0.527615
$387$	0	0
$388$	−602.734	−0.0788639
$389$	−1473.17	−0.192011	−0.0960057	−	0.995381i	$-0.530607\pi$
$389$	−1473.17	−0.192011	−0.0960057	+	0.995381i	$0.530607\pi$
$390$	0	0
$391$	10328.6	1.33591
$392$	−6753.81	−0.870201
$393$	0	0
$394$	−13492.9	−1.72528
$395$	10503.6	1.33796
$396$	0	0
$397$	5818.69	0.735596	0.367798	−	0.929906i	$-0.380112\pi$
$397$	5818.69	0.735596	0.367798	+	0.929906i	$0.380112\pi$
$398$	96.1297	0.0121069
$399$	0	0
$400$	−14497.5	−1.81219
$401$	−3009.09	−0.374730	−0.187365	−	0.982290i	$-0.559995\pi$
$401$	−3009.09	−0.374730	−0.187365	+	0.982290i	$0.559995\pi$
$402$	0	0
$403$	446.098	0.0551407
$404$	−2781.74	−0.342566
$405$	0	0
$406$	331.351	0.0405042
$407$	6918.59	0.842609
$408$	0	0
$409$	−12146.2	−1.46844	−0.734218	−	0.678914i	$-0.762451\pi$
$409$	−12146.2	−1.46844	−0.734218	+	0.678914i	$0.762451\pi$
$410$	19458.6	2.34388
$411$	0	0
$412$	−48.1644	−0.00575944
$413$	220.394	0.0262588
$414$	0	0
$415$	−4242.13	−0.501778
$416$	−3599.27	−0.424203
$417$	0	0
$418$	2225.69	0.260435
$419$	7078.38	0.825302	0.412651	−	0.910889i	$-0.364603\pi$
$419$	7078.38	0.825302	0.412651	+	0.910889i	$0.364603\pi$
$420$	0	0
$421$	−1350.43	−0.156332	−0.0781662	−	0.996940i	$-0.524906\pi$
$421$	−1350.43	−0.156332	−0.0781662	+	0.996940i	$0.524906\pi$
$422$	−11644.8	−1.34327
$423$	0	0
$424$	−6251.26	−0.716010
$425$	19220.6	2.19373
$426$	0	0
$427$	−321.781	−0.0364685
$428$	−2512.96	−0.283805
$429$	0	0
$430$	−8944.41	−1.00311
$431$	−6841.23	−0.764572	−0.382286	−	0.924044i	$-0.624863\pi$
$431$	−6841.23	−0.764572	−0.382286	+	0.924044i	$0.624863\pi$
$432$	0	0
$433$	14287.5	1.58572	0.792859	−	0.609406i	$-0.208592\pi$
$433$	14287.5	1.58572	0.792859	+	0.609406i	$0.208592\pi$
$434$	21.1410	0.00233825
$435$	0	0
$436$	2429.52	0.266864
$437$	3033.56	0.332070
$438$	0	0
$439$	1490.40	0.162034	0.0810171	−	0.996713i	$-0.474183\pi$
$439$	1490.40	0.162034	0.0810171	+	0.996713i	$0.474183\pi$
$440$	8709.55	0.943663
$441$	0	0
$442$	14997.8	1.61396
$443$	11204.7	1.20169	0.600847	−	0.799364i	$-0.294830\pi$
$443$	11204.7	1.20169	0.600847	+	0.799364i	$0.294830\pi$
$444$	0	0
$445$	23309.1	2.48305
$446$	8609.37	0.914049
$447$	0	0
$448$	273.788	0.0288734
$449$	15726.5	1.65296	0.826480	−	0.562966i	$-0.190340\pi$
$449$	15726.5	1.65296	0.826480	+	0.562966i	$0.190340\pi$
$450$	0	0
$451$	−8653.47	−0.903494
$452$	−47.5778	−0.00495104
$453$	0	0
$454$	−734.522	−0.0759313
$455$	651.753	0.0671531
$456$	0	0
$457$	9442.99	0.966574	0.483287	−	0.875462i	$-0.339443\pi$
$457$	9442.99	0.966574	0.483287	+	0.875462i	$0.339443\pi$
$458$	−1002.00	−0.102228
$459$	0	0
$460$	−3088.57	−0.313055
$461$	4345.33	0.439007	0.219504	−	0.975612i	$-0.429556\pi$
$461$	4345.33	0.439007	0.219504	+	0.975612i	$0.429556\pi$
$462$	0	0
$463$	11702.1	1.17461	0.587303	−	0.809367i	$-0.300190\pi$
$463$	11702.1	1.17461	0.587303	+	0.809367i	$0.300190\pi$
$464$	10653.3	1.06588
$465$	0	0
$466$	−6213.17	−0.617639
$467$	−10980.5	−1.08805	−0.544023	−	0.839070i	$-0.683100\pi$
$467$	−10980.5	−1.08805	−0.544023	+	0.839070i	$0.683100\pi$
$468$	0	0
$469$	−408.389	−0.0402082
$470$	−8936.56	−0.877049
$471$	0	0
$472$	5828.89	0.568424
$473$	3977.69	0.386669
$474$	0	0
$475$	5645.16	0.545301
$476$	121.633	0.0117122
$477$	0	0
$478$	742.506	0.0710490
$479$	16368.3	1.56135	0.780673	−	0.624939i	$-0.214876\pi$
$479$	16368.3	1.56135	0.780673	+	0.624939i	$0.214876\pi$
$480$	0	0
$481$	13692.7	1.29799
$482$	15883.8	1.50101
$483$	0	0
$484$	−1190.67	−0.111821
$485$	6524.13	0.610815
$486$	0	0
$487$	−5105.40	−0.475047	−0.237523	−	0.971382i	$-0.576336\pi$
$487$	−5105.40	−0.475047	−0.237523	+	0.971382i	$0.576336\pi$
$488$	−8510.30	−0.789433
$489$	0	0
$490$	−19020.4	−1.75358
$491$	1265.56	0.116322	0.0581608	−	0.998307i	$-0.481476\pi$
$491$	1265.56	0.116322	0.0581608	+	0.998307i	$0.481476\pi$
$492$	0	0
$493$	−14124.0	−1.29029
$494$	4404.90	0.401186
$495$	0	0
$496$	679.705	0.0615316
$497$	627.784	0.0566599
$498$	0	0
$499$	4373.23	0.392330	0.196165	−	0.980571i	$-0.437151\pi$
$499$	4373.23	0.392330	0.196165	+	0.980571i	$0.437151\pi$
$500$	−2056.33	−0.183923
$501$	0	0
$502$	7935.48	0.705534
$503$	−19926.2	−1.76633	−0.883164	−	0.469064i	$-0.844591\pi$
$503$	−19926.2	−1.76633	−0.883164	+	0.469064i	$0.844591\pi$
$504$	0	0
$505$	30110.2	2.65324
$506$	8026.18	0.705152
$507$	0	0
$508$	2817.23	0.246052
$509$	−15679.4	−1.36538	−0.682691	−	0.730708i	$-0.739190\pi$
$509$	−15679.4	−1.36538	−0.682691	+	0.730708i	$0.739190\pi$
$510$	0	0
$511$	141.477	0.0122477
$512$	6268.15	0.541047
$513$	0	0
$514$	−12660.5	−1.08644
$515$	521.343	0.0446080
$516$	0	0
$517$	3974.20	0.338075
$518$	648.911	0.0550416
$519$	0	0
$520$	17237.3	1.45366
$521$	16954.5	1.42570	0.712850	−	0.701316i	$-0.247404\pi$
$521$	16954.5	1.42570	0.712850	+	0.701316i	$0.247404\pi$
$522$	0	0
$523$	20850.4	1.74326	0.871628	−	0.490167i	$-0.163064\pi$
$523$	20850.4	1.74326	0.871628	+	0.490167i	$0.163064\pi$
$524$	1520.05	0.126724
$525$	0	0
$526$	−2289.80	−0.189810
$527$	−901.143	−0.0744865
$528$	0	0
$529$	−1227.53	−0.100890
$530$	−17605.1	−1.44286
$531$	0	0
$532$	35.7239	0.00291133
$533$	−17126.3	−1.39178
$534$	0	0
$535$	27200.8	2.19812
$536$	−10800.9	−0.870387
$537$	0	0
$538$	−12146.5	−0.973370
$539$	8458.59	0.675951
$540$	0	0
$541$	3440.60	0.273425	0.136712	−	0.990611i	$-0.456346\pi$
$541$	3440.60	0.273425	0.136712	+	0.990611i	$0.456346\pi$
$542$	2620.75	0.207696
$543$	0	0
$544$	7270.73	0.573033
$545$	−26297.6	−2.06691
$546$	0	0
$547$	11784.0	0.921113	0.460556	−	0.887630i	$-0.347650\pi$
$547$	11784.0	0.921113	0.460556	+	0.887630i	$0.347650\pi$
$548$	−269.514	−0.0210092
$549$	0	0
$550$	14936.0	1.15795
$551$	−4148.26	−0.320729
$552$	0	0
$553$	−438.113	−0.0336898
$554$	−4030.08	−0.309064
$555$	0	0
$556$	−3524.42	−0.268829
$557$	−620.891	−0.0472316	−0.0236158	−	0.999721i	$-0.507518\pi$
$557$	−620.891	−0.0472316	−0.0236158	+	0.999721i	$0.507518\pi$
$558$	0	0
$559$	7872.32	0.595642
$560$	993.056	0.0749362
$561$	0	0
$562$	21366.9	1.60376
$563$	26002.2	1.94647	0.973233	−	0.229821i	$-0.0738140\pi$
$563$	26002.2	1.94647	0.973233	+	0.229821i	$0.0738140\pi$
$564$	0	0
$565$	514.993	0.0383467
$566$	−551.744	−0.0409745
$567$	0	0
$568$	16603.3	1.22652
$569$	−10004.8	−0.737120	−0.368560	−	0.929604i	$-0.620149\pi$
$569$	−10004.8	−0.737120	−0.368560	+	0.929604i	$0.620149\pi$
$570$	0	0
$571$	−9283.06	−0.680357	−0.340179	−	0.940361i	$-0.610488\pi$
$571$	−9283.06	−0.680357	−0.340179	+	0.940361i	$0.610488\pi$
$572$	1994.44	0.145790
$573$	0	0
$574$	−811.629	−0.0590187
$575$	20357.3	1.47645
$576$	0	0
$577$	−4371.52	−0.315405	−0.157703	−	0.987487i	$-0.550409\pi$
$577$	−4371.52	−0.315405	−0.157703	+	0.987487i	$0.550409\pi$
$578$	−15033.1	−1.08182
$579$	0	0
$580$	4223.49	0.302363
$581$	176.942	0.0126347
$582$	0	0
$583$	7829.19	0.556179
$584$	3741.73	0.265126
$585$	0	0
$586$	720.419	0.0507854
$587$	5076.25	0.356932	0.178466	−	0.983946i	$-0.442886\pi$
$587$	5076.25	0.356932	0.178466	+	0.983946i	$0.442886\pi$
$588$	0	0
$589$	−264.669	−0.0185153
$590$	16415.6	1.14545
$591$	0	0
$592$	20863.2	1.44843
$593$	−14397.4	−0.997019	−0.498509	−	0.866884i	$-0.666119\pi$
$593$	−14397.4	−0.997019	−0.498509	+	0.866884i	$0.666119\pi$
$594$	0	0
$595$	−1316.58	−0.0907134
$596$	−3174.69	−0.218189
$597$	0	0
$598$	15884.8	1.08625
$599$	−13497.3	−0.920676	−0.460338	−	0.887744i	$-0.652272\pi$
$599$	−13497.3	−0.920676	−0.460338	+	0.887744i	$0.652272\pi$
$600$	0	0
$601$	7012.74	0.475966	0.237983	−	0.971269i	$-0.423514\pi$
$601$	7012.74	0.475966	0.237983	+	0.971269i	$0.423514\pi$
$602$	373.077	0.0252583
$603$	0	0
$604$	−4418.06	−0.297630
$605$	12888.1	0.866076
$606$	0	0
$607$	−22822.2	−1.52607	−0.763035	−	0.646357i	$-0.776292\pi$
$607$	−22822.2	−1.52607	−0.763035	+	0.646357i	$0.776292\pi$
$608$	2135.44	0.142440
$609$	0	0
$610$	−23967.1	−1.59082
$611$	7865.42	0.520787
$612$	0	0
$613$	−12441.8	−0.819771	−0.409885	−	0.912137i	$-0.634431\pi$
$613$	−12441.8	−0.819771	−0.409885	+	0.912137i	$0.634431\pi$
$614$	−6166.94	−0.405338
$615$	0	0
$616$	−363.281	−0.0237614
$617$	8903.80	0.580962	0.290481	−	0.956881i	$-0.406185\pi$
$617$	8903.80	0.580962	0.290481	+	0.956881i	$0.406185\pi$
$618$	0	0
$619$	−1315.39	−0.0854120	−0.0427060	−	0.999088i	$-0.513598\pi$
$619$	−1315.39	−0.0854120	−0.0427060	+	0.999088i	$0.513598\pi$
$620$	269.468	0.0174550
$621$	0	0
$622$	−11940.7	−0.769737
$623$	−972.236	−0.0625230
$624$	0	0
$625$	−2071.36	−0.132567
$626$	−12364.0	−0.789400
$627$	0	0
$628$	−1704.34	−0.108297
$629$	−27660.1	−1.75339
$630$	0	0
$631$	27108.3	1.71024	0.855121	−	0.518429i	$-0.173483\pi$
$631$	27108.3	1.71024	0.855121	+	0.518429i	$0.173483\pi$
$632$	−11587.0	−0.729282
$633$	0	0
$634$	24934.1	1.56192
$635$	−30494.4	−1.90572
$636$	0	0
$637$	16740.6	1.04126
$638$	−10975.5	−0.681070
$639$	0	0
$640$	30923.0	1.90991
$641$	11334.4	0.698409	0.349204	−	0.937047i	$-0.386452\pi$
$641$	11334.4	0.698409	0.349204	+	0.937047i	$0.386452\pi$
$642$	0	0
$643$	1526.23	0.0936063	0.0468031	−	0.998904i	$-0.485097\pi$
$643$	1526.23	0.0936063	0.0468031	+	0.998904i	$0.485097\pi$
$644$	128.826	0.00788270
$645$	0	0
$646$	−8898.16	−0.541940
$647$	−1836.25	−0.111577	−0.0557886	−	0.998443i	$-0.517767\pi$
$647$	−1836.25	−0.111577	−0.0557886	+	0.998443i	$0.517767\pi$
$648$	0	0
$649$	−7300.20	−0.441538
$650$	29560.1	1.78376
$651$	0	0
$652$	−1729.31	−0.103873
$653$	−18006.4	−1.07909	−0.539543	−	0.841958i	$-0.681403\pi$
$653$	−18006.4	−1.07909	−0.539543	+	0.841958i	$0.681403\pi$
$654$	0	0
$655$	−16453.3	−0.981503
$656$	−26094.7	−1.55309
$657$	0	0
$658$	372.750	0.0220840
$659$	13881.1	0.820531	0.410266	−	0.911966i	$-0.365436\pi$
$659$	13881.1	0.820531	0.410266	+	0.911966i	$0.365436\pi$
$660$	0	0
$661$	−13229.9	−0.778495	−0.389247	−	0.921133i	$-0.627265\pi$
$661$	−13229.9	−0.778495	−0.389247	+	0.921133i	$0.627265\pi$
$662$	−6837.90	−0.401454
$663$	0	0
$664$	4679.67	0.273504
$665$	−386.684	−0.0225488
$666$	0	0
$667$	−14959.3	−0.868404
$668$	−787.515	−0.0456136
$669$	0	0
$670$	−30417.9	−1.75395
$671$	10658.4	0.613211
$672$	0	0
$673$	6521.08	0.373505	0.186753	−	0.982407i	$-0.440204\pi$
$673$	6521.08	0.373505	0.186753	+	0.982407i	$0.440204\pi$
$674$	−30132.9	−1.72207
$675$	0	0
$676$	318.449	0.0181184
$677$	−14707.0	−0.834914	−0.417457	−	0.908697i	$-0.637079\pi$
$677$	−14707.0	−0.834914	−0.417457	+	0.908697i	$0.637079\pi$
$678$	0	0
$679$	−272.125	−0.0153803
$680$	−34820.2	−1.96367
$681$	0	0
$682$	−700.260	−0.0393172
$683$	−30336.2	−1.69954	−0.849768	−	0.527157i	$-0.823258\pi$
$683$	−30336.2	−1.69954	−0.849768	+	0.527157i	$0.823258\pi$
$684$	0	0
$685$	2917.28	0.162720
$686$	1587.99	0.0883816
$687$	0	0
$688$	11994.8	0.664677
$689$	15494.9	0.856763
$690$	0	0
$691$	22793.2	1.25484	0.627421	−	0.778680i	$-0.284110\pi$
$691$	22793.2	1.25484	0.627421	+	0.778680i	$0.284110\pi$
$692$	2464.57	0.135389
$693$	0	0
$694$	−30644.3	−1.67614
$695$	38149.1	2.08213
$696$	0	0
$697$	34596.0	1.88008
$698$	34331.0	1.86167
$699$	0	0
$700$	239.733	0.0129444
$701$	4226.10	0.227700	0.113850	−	0.993498i	$-0.463682\pi$
$701$	4226.10	0.227700	0.113850	+	0.993498i	$0.463682\pi$
$702$	0	0
$703$	−8123.87	−0.435843
$704$	−9068.79	−0.485501
$705$	0	0
$706$	1590.05	0.0847626
$707$	−1255.91	−0.0668084
$708$	0	0
$709$	−29662.6	−1.57123	−0.785615	−	0.618715i	$-0.787653\pi$
$709$	−29662.6	−1.57123	−0.785615	+	0.618715i	$0.787653\pi$
$710$	46759.1	2.47160
$711$	0	0
$712$	−25713.2	−1.35343
$713$	−954.437	−0.0501318
$714$	0	0
$715$	−21588.2	−1.12917
$716$	−3057.54	−0.159589
$717$	0	0
$718$	−39599.2	−2.05826
$719$	−28079.9	−1.45647	−0.728237	−	0.685325i	$-0.759660\pi$
$719$	−28079.9	−1.45647	−0.728237	+	0.685325i	$0.759660\pi$
$720$	0	0
$721$	−21.7455	−0.00112323
$722$	18695.6	0.963680
$723$	0	0
$724$	−3360.97	−0.172527
$725$	−27837.8	−1.42603
$726$	0	0
$727$	6488.86	0.331030	0.165515	−	0.986207i	$-0.447071\pi$
$727$	6488.86	0.331030	0.165515	+	0.986207i	$0.447071\pi$
$728$	−718.976	−0.0366031
$729$	0	0
$730$	10537.6	0.534267
$731$	−15902.5	−0.804619
$732$	0	0
$733$	5082.45	0.256104	0.128052	−	0.991767i	$-0.459127\pi$
$733$	5082.45	0.256104	0.128052	+	0.991767i	$0.459127\pi$
$734$	−23048.8	−1.15905
$735$	0	0
$736$	7700.72	0.385669
$737$	13527.2	0.676094
$738$	0	0
$739$	−2586.05	−0.128727	−0.0643635	−	0.997927i	$-0.520502\pi$
$739$	−2586.05	−0.128727	−0.0643635	+	0.997927i	$0.520502\pi$
$740$	8271.19	0.410885
$741$	0	0
$742$	734.319	0.0363311
$743$	−13473.8	−0.665283	−0.332641	−	0.943053i	$-0.607940\pi$
$743$	−13473.8	−0.665283	−0.332641	+	0.943053i	$0.607940\pi$
$744$	0	0
$745$	34363.6	1.68991
$746$	3444.92	0.169072
$747$	0	0
$748$	−4028.88	−0.196939
$749$	−1134.56	−0.0553486
$750$	0	0
$751$	17858.4	0.867724	0.433862	−	0.900979i	$-0.357150\pi$
$751$	17858.4	0.867724	0.433862	+	0.900979i	$0.357150\pi$
$752$	11984.3	0.581147
$753$	0	0
$754$	−21721.8	−1.04915
$755$	47822.1	2.30520
$756$	0	0
$757$	−3622.31	−0.173917	−0.0869583	−	0.996212i	$-0.527715\pi$
$757$	−3622.31	−0.173917	−0.0869583	+	0.996212i	$0.527715\pi$
$758$	−33199.6	−1.59085
$759$	0	0
$760$	−10226.8	−0.488113
$761$	−2386.44	−0.113677	−0.0568386	−	0.998383i	$-0.518102\pi$
$761$	−2386.44	−0.113677	−0.0568386	+	0.998383i	$0.518102\pi$
$762$	0	0
$763$	1096.89	0.0520447
$764$	5120.45	0.242475
$765$	0	0
$766$	37430.5	1.76556
$767$	−14448.0	−0.680165
$768$	0	0
$769$	4038.92	0.189398	0.0946990	−	0.995506i	$-0.469811\pi$
$769$	4038.92	0.189398	0.0946990	+	0.995506i	$0.469811\pi$
$770$	−1023.09	−0.0478825
$771$	0	0
$772$	2127.29	0.0991749
$773$	−17780.2	−0.827306	−0.413653	−	0.910435i	$-0.635747\pi$
$773$	−17780.2	−0.827306	−0.413653	+	0.910435i	$0.635747\pi$
$774$	0	0
$775$	−1776.12	−0.0823227
$776$	−7197.04	−0.332937
$777$	0	0
$778$	4576.71	0.210904
$779$	10161.0	0.467336
$780$	0	0
$781$	−20794.3	−0.952727
$782$	−32088.1	−1.46735
$783$	0	0
$784$	25507.1	1.16195
$785$	18448.1	0.838779
$786$	0	0
$787$	28216.2	1.27802	0.639009	−	0.769199i	$-0.279345\pi$
$787$	28216.2	1.27802	0.639009	+	0.769199i	$0.279345\pi$
$788$	7173.53	0.324298
$789$	0	0
$790$	−32631.8	−1.46961
$791$	−21.4807	−0.000965568 0
$792$	0	0
$793$	21094.4	0.944619
$794$	−18077.0	−0.807972
$795$	0	0
$796$	−51.1077	−0.00227571
$797$	8810.39	0.391568	0.195784	−	0.980647i	$-0.437275\pi$
$797$	8810.39	0.391568	0.195784	+	0.980647i	$0.437275\pi$
$798$	0	0
$799$	−15888.6	−0.703502
$800$	14330.3	0.633317
$801$	0	0
$802$	9348.38	0.411600
$803$	−4686.21	−0.205943
$804$	0	0
$805$	−1394.44	−0.0610529
$806$	−1385.90	−0.0605660
$807$	0	0
$808$	−33215.8	−1.44620
$809$	−5436.82	−0.236277	−0.118139	−	0.992997i	$-0.537693\pi$
$809$	−5436.82	−0.236277	−0.118139	+	0.992997i	$0.537693\pi$
$810$	0	0
$811$	11563.5	0.500678	0.250339	−	0.968158i	$-0.419458\pi$
$811$	11563.5	0.500678	0.250339	+	0.968158i	$0.419458\pi$
$812$	−176.164	−0.00761349
$813$	0	0
$814$	−21494.1	−0.925514
$815$	18718.4	0.804512
$816$	0	0
$817$	−4670.63	−0.200006
$818$	37734.8	1.61292
$819$	0	0
$820$	−10345.2	−0.440575
$821$	25869.9	1.09972	0.549858	−	0.835258i	$-0.314682\pi$
$821$	25869.9	1.09972	0.549858	+	0.835258i	$0.314682\pi$
$822$	0	0
$823$	−6191.18	−0.262225	−0.131112	−	0.991368i	$-0.541855\pi$
$823$	−6191.18	−0.262225	−0.131112	+	0.991368i	$0.541855\pi$
$824$	−575.115	−0.0243144
$825$	0	0
$826$	−684.703	−0.0288425
$827$	−33775.3	−1.42017	−0.710086	−	0.704115i	$-0.751344\pi$
$827$	−33775.3	−1.42017	−0.710086	+	0.704115i	$0.751344\pi$
$828$	0	0
$829$	−9016.25	−0.377741	−0.188871	−	0.982002i	$-0.560483\pi$
$829$	−9016.25	−0.377741	−0.188871	+	0.982002i	$0.560483\pi$
$830$	13179.1	0.551148
$831$	0	0
$832$	−17948.2	−0.747888
$833$	−33816.9	−1.40659
$834$	0	0
$835$	8524.24	0.353286
$836$	−1183.30	−0.0489535
$837$	0	0
$838$	−21990.5	−0.906504
$839$	45095.2	1.85561	0.927806	−	0.373062i	$-0.121692\pi$
$839$	45095.2	1.85561	0.927806	+	0.373062i	$0.121692\pi$
$840$	0	0
$841$	−3932.81	−0.161254
$842$	4195.41	0.171714
$843$	0	0
$844$	6191.00	0.252492
$845$	−3446.97	−0.140330
$846$	0	0
$847$	−537.571	−0.0218077
$848$	23609.1	0.956063
$849$	0	0
$850$	−59713.1	−2.40958
$851$	−29295.9	−1.18008
$852$	0	0
$853$	−49211.7	−1.97535	−0.987676	−	0.156513i	$-0.949975\pi$
$853$	−49211.7	−1.97535	−0.987676	+	0.156513i	$0.949975\pi$
$854$	999.682	0.0400567
$855$	0	0
$856$	−30006.4	−1.19813
$857$	−23038.6	−0.918300	−0.459150	−	0.888359i	$-0.651846\pi$
$857$	−23038.6	−0.918300	−0.459150	+	0.888359i	$0.651846\pi$
$858$	0	0
$859$	−6529.39	−0.259348	−0.129674	−	0.991557i	$-0.541393\pi$
$859$	−6529.39	−0.259348	−0.129674	+	0.991557i	$0.541393\pi$
$860$	4755.33	0.188553
$861$	0	0
$862$	21253.8	0.839799
$863$	−18172.1	−0.716786	−0.358393	−	0.933571i	$-0.616675\pi$
$863$	−18172.1	−0.716786	−0.358393	+	0.933571i	$0.616675\pi$
$864$	0	0
$865$	−26677.1	−1.04861
$866$	−44387.4	−1.74174
$867$	0	0
$868$	−11.2397	−0.000439516 0
$869$	14511.8	0.566488
$870$	0	0
$871$	26772.0	1.04149
$872$	29010.0	1.12661
$873$	0	0
$874$	−9424.40	−0.364743
$875$	−928.400	−0.0358693
$876$	0	0
$877$	9910.10	0.381574	0.190787	−	0.981631i	$-0.438896\pi$
$877$	9910.10	0.381574	0.190787	+	0.981631i	$0.438896\pi$
$878$	−4630.26	−0.177977
$879$	0	0
$880$	−32893.3	−1.26004
$881$	3153.32	0.120588	0.0602940	−	0.998181i	$-0.480796\pi$
$881$	3153.32	0.120588	0.0602940	+	0.998181i	$0.480796\pi$
$882$	0	0
$883$	−11764.5	−0.448366	−0.224183	−	0.974547i	$-0.571971\pi$
$883$	−11764.5	−0.448366	−0.224183	+	0.974547i	$0.571971\pi$
$884$	−7973.64	−0.303374
$885$	0	0
$886$	−34809.8	−1.31993
$887$	−35545.6	−1.34555	−0.672777	−	0.739846i	$-0.734899\pi$
$887$	−35545.6	−1.34555	−0.672777	+	0.739846i	$0.734899\pi$
$888$	0	0
$889$	1271.94	0.0479859
$890$	−72414.7	−2.72736
$891$	0	0
$892$	−4577.21	−0.171812
$893$	−4666.54	−0.174871
$894$	0	0
$895$	33095.5	1.23604
$896$	−1289.82	−0.0480913
$897$	0	0
$898$	−48857.8	−1.81560
$899$	1305.15	0.0484197
$900$	0	0
$901$	−31300.6	−1.15735
$902$	26883.9	0.992390
$903$	0	0
$904$	−568.110	−0.0209016
$905$	36379.9	1.33625
$906$	0	0
$907$	7842.99	0.287125	0.143562	−	0.989641i	$-0.454144\pi$
$907$	7842.99	0.287125	0.143562	+	0.989641i	$0.454144\pi$
$908$	390.511	0.0142727
$909$	0	0
$910$	−2024.81	−0.0737603
$911$	−17676.5	−0.642862	−0.321431	−	0.946933i	$-0.604164\pi$
$911$	−17676.5	−0.642862	−0.321431	+	0.946933i	$0.604164\pi$
$912$	0	0
$913$	−5860.90	−0.212451
$914$	−29336.7	−1.06168
$915$	0	0
$916$	532.716	0.0192156
$917$	686.278	0.0247142
$918$	0	0
$919$	40448.5	1.45187	0.725937	−	0.687761i	$-0.241406\pi$
$919$	40448.5	1.45187	0.725937	+	0.687761i	$0.241406\pi$
$920$	−36879.5	−1.32161
$921$	0	0
$922$	−13499.7	−0.482202
$923$	−41154.5	−1.46762
$924$	0	0
$925$	−54517.0	−1.93785
$926$	−36355.1	−1.29018
$927$	0	0
$928$	−10530.4	−0.372498
$929$	18479.7	0.652635	0.326317	−	0.945260i	$-0.394192\pi$
$929$	18479.7	0.652635	0.326317	+	0.945260i	$0.394192\pi$
$930$	0	0
$931$	−9932.15	−0.349638
$932$	3303.26	0.116096
$933$	0	0
$934$	34113.4	1.19510
$935$	43609.5	1.52533
$936$	0	0
$937$	−12551.8	−0.437620	−0.218810	−	0.975767i	$-0.570218\pi$
$937$	−12551.8	−0.437620	−0.218810	+	0.975767i	$0.570218\pi$
$938$	1268.75	0.0441644
$939$	0	0
$940$	4751.16	0.164857
$941$	42332.5	1.46652	0.733262	−	0.679946i	$-0.237997\pi$
$941$	42332.5	1.46652	0.733262	+	0.679946i	$0.237997\pi$
$942$	0	0
$943$	36642.0	1.26535
$944$	−22013.9	−0.758997
$945$	0	0
$946$	−12357.6	−0.424713
$947$	12726.8	0.436710	0.218355	−	0.975869i	$-0.429931\pi$
$947$	12726.8	0.436710	0.218355	+	0.975869i	$0.429931\pi$
$948$	0	0
$949$	−9274.57	−0.317245
$950$	−17537.9	−0.598953
$951$	0	0
$952$	1452.37	0.0494451
$953$	−14939.7	−0.507812	−0.253906	−	0.967229i	$-0.581715\pi$
$953$	−14939.7	−0.507812	−0.253906	+	0.967229i	$0.581715\pi$
$954$	0	0
$955$	−55424.8	−1.87802
$956$	−394.756	−0.0133549
$957$	0	0
$958$	−50851.6	−1.71497
$959$	−121.682	−0.00409729
$960$	0	0
$961$	−29707.7	−0.997205
$962$	−42539.4	−1.42570
$963$	0	0
$964$	−8444.66	−0.282141
$965$	−23026.3	−0.768128
$966$	0	0
$967$	3191.74	0.106142	0.0530710	−	0.998591i	$-0.483099\pi$
$967$	3191.74	0.106142	0.0530710	+	0.998591i	$0.483099\pi$
$968$	−14217.4	−0.472071
$969$	0	0
$970$	−20268.6	−0.670914
$971$	−43380.7	−1.43373	−0.716866	−	0.697211i	$-0.754424\pi$
$971$	−43380.7	−1.43373	−0.716866	+	0.697211i	$0.754424\pi$
$972$	0	0
$973$	−1591.22	−0.0524278
$974$	15861.0	0.521787
$975$	0	0
$976$	32140.8	1.05410
$977$	57824.4	1.89352	0.946759	−	0.321945i	$-0.104336\pi$
$977$	57824.4	1.89352	0.946759	+	0.321945i	$0.104336\pi$
$978$	0	0
$979$	32203.7	1.05131
$980$	10112.3	0.329617
$981$	0	0
$982$	−3931.74	−0.127767
$983$	36789.7	1.19370	0.596851	−	0.802352i	$-0.296418\pi$
$983$	36789.7	1.19370	0.596851	+	0.802352i	$0.296418\pi$
$984$	0	0
$985$	−77647.9	−2.51174
$986$	43879.2	1.41724
$987$	0	0
$988$	−2341.88	−0.0754102
$989$	−16843.0	−0.541534
$990$	0	0
$991$	59886.2	1.91963	0.959813	−	0.280641i	$-0.0905470\pi$
$991$	59886.2	1.91963	0.959813	+	0.280641i	$0.0905470\pi$
$992$	−671.866	−0.0215038
$993$	0	0
$994$	−1950.35	−0.0622347
$995$	553.202	0.0176258
$996$	0	0
$997$	39148.2	1.24357	0.621784	−	0.783189i	$-0.286408\pi$
$997$	39148.2	1.24357	0.621784	+	0.783189i	$0.286408\pi$
$998$	−13586.4	−0.430932
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.h.1.14	yes	59
3.2	odd	2		2151.4.a.g.1.46	✓	59

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.4.a.g.1.46	✓	59	3.2	odd	2
2151.4.a.h.1.14	yes	59	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)

\(f(q)\)	\(=\)	\(q-3.10672 q^{2} +1.65170 q^{4} -17.8784 q^{5} +0.745718 q^{7} +19.7224 q^{8} +O(q^{10})\)	\(q-3.10672 q^{2} +1.65170 q^{4} -17.8784 q^{5} +0.745718 q^{7} +19.7224 q^{8} +55.5430 q^{10} -24.7007 q^{11} -48.8856 q^{13} -2.31673 q^{14} -74.4855 q^{16} +98.7517 q^{17} +29.0037 q^{19} -29.5297 q^{20} +76.7380 q^{22} +104.592 q^{23} +194.636 q^{25} +151.874 q^{26} +1.23170 q^{28} -143.025 q^{29} -9.12534 q^{31} +73.6264 q^{32} -306.794 q^{34} -13.3322 q^{35} -280.097 q^{37} -90.1064 q^{38} -352.604 q^{40} +350.333 q^{41} -161.036 q^{43} -40.7981 q^{44} -324.938 q^{46} -160.894 q^{47} -342.444 q^{49} -604.679 q^{50} -80.7443 q^{52} -316.963 q^{53} +441.607 q^{55} +14.7073 q^{56} +444.339 q^{58} +295.547 q^{59} -431.505 q^{61} +28.3499 q^{62} +367.147 q^{64} +873.994 q^{65} -547.646 q^{67} +163.108 q^{68} +41.4194 q^{70} +841.853 q^{71} +189.720 q^{73} +870.184 q^{74} +47.9054 q^{76} -18.4197 q^{77} -587.506 q^{79} +1331.68 q^{80} -1088.39 q^{82} +237.277 q^{83} -1765.52 q^{85} +500.292 q^{86} -487.156 q^{88} -1303.76 q^{89} -36.4548 q^{91} +172.754 q^{92} +499.854 q^{94} -518.539 q^{95} -364.918 q^{97} +1063.88 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 59 q + 8 q^{2} + 238 q^{4} + 80 q^{5} - 10 q^{7} + 96 q^{8}+O(q^{10}) \) 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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