LMFDB - Embedded newform 2151.4.a.h.1.4

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			1.4
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)

$f(q)$	$=$	$q-5.11618 q^{2} +18.1753 q^{4} +6.15957 q^{5} +24.2123 q^{7} -52.0584 q^{8} +O(q^{10})$	\(q-5.11618 q^{2} +18.1753 q^{4} +6.15957 q^{5} +24.2123 q^{7} -52.0584 q^{8} -31.5134 q^{10} +60.5329 q^{11} -2.25761 q^{13} -123.874 q^{14} +120.938 q^{16} +114.457 q^{17} -56.9025 q^{19} +111.952 q^{20} -309.697 q^{22} +11.1459 q^{23} -87.0598 q^{25} +11.5503 q^{26} +440.064 q^{28} +177.408 q^{29} -199.533 q^{31} -202.272 q^{32} -585.582 q^{34} +149.137 q^{35} +267.387 q^{37} +291.123 q^{38} -320.657 q^{40} +475.722 q^{41} -207.667 q^{43} +1100.20 q^{44} -57.0244 q^{46} -598.232 q^{47} +243.233 q^{49} +445.413 q^{50} -41.0325 q^{52} -275.859 q^{53} +372.856 q^{55} -1260.45 q^{56} -907.651 q^{58} +25.8847 q^{59} +247.309 q^{61} +1020.85 q^{62} +67.3571 q^{64} -13.9059 q^{65} -24.8585 q^{67} +2080.29 q^{68} -763.011 q^{70} +260.252 q^{71} +1076.96 q^{73} -1368.00 q^{74} -1034.22 q^{76} +1465.64 q^{77} +644.964 q^{79} +744.925 q^{80} -2433.88 q^{82} -276.207 q^{83} +705.006 q^{85} +1062.46 q^{86} -3151.25 q^{88} +446.913 q^{89} -54.6617 q^{91} +202.580 q^{92} +3060.66 q^{94} -350.494 q^{95} +1294.37 q^{97} -1244.42 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−5.11618	−1.80884	−0.904421	−	0.426642i	$-0.859696\pi$
$2$	−5.11618	−1.80884	−0.904421	+	0.426642i	$0.859696\pi$
$3$	0	0
$3$	0	0
$4$	18.1753	2.27191
$5$	6.15957	0.550928	0.275464	−	0.961311i	$-0.411168\pi$
$5$	6.15957	0.550928	0.275464	+	0.961311i	$0.411168\pi$
$6$	0	0
$7$	24.2123	1.30734	0.653669	−	0.756780i	$-0.273229\pi$
$7$	24.2123	1.30734	0.653669	+	0.756780i	$0.273229\pi$
$8$	−52.0584	−2.30068
$9$	0	0
$10$	−31.5134	−0.996542
$11$	60.5329	1.65921	0.829607	−	0.558348i	$-0.188564\pi$
$11$	60.5329	1.65921	0.829607	+	0.558348i	$0.188564\pi$
$12$	0	0
$13$	−2.25761	−0.0481652	−0.0240826	−	0.999710i	$-0.507666\pi$
$13$	−2.25761	−0.0481652	−0.0240826	+	0.999710i	$0.507666\pi$
$14$	−123.874	−2.36477
$15$	0	0
$16$	120.938	1.88965
$17$	114.457	1.63294	0.816468	−	0.577391i	$-0.195929\pi$
$17$	114.457	1.63294	0.816468	+	0.577391i	$0.195929\pi$
$18$	0	0
$19$	−56.9025	−0.687070	−0.343535	−	0.939140i	$-0.611624\pi$
$19$	−56.9025	−0.687070	−0.343535	+	0.939140i	$0.611624\pi$
$20$	111.952	1.25166
$21$	0	0
$22$	−309.697	−3.00126
$23$	11.1459	0.101047	0.0505235	−	0.998723i	$-0.483911\pi$
$23$	11.1459	0.101047	0.0505235	+	0.998723i	$0.483911\pi$
$24$	0	0
$25$	−87.0598	−0.696478
$26$	11.5503	0.0871231
$27$	0	0
$28$	440.064	2.97015
$29$	177.408	1.13600	0.567998	−	0.823030i	$-0.307718\pi$
$29$	177.408	1.13600	0.567998	+	0.823030i	$0.307718\pi$
$30$	0	0
$31$	−199.533	−1.15604	−0.578020	−	0.816022i	$-0.696175\pi$
$31$	−199.533	−1.15604	−0.578020	+	0.816022i	$0.696175\pi$
$32$	−202.272	−1.11741
$33$	0	0
$34$	−585.582	−2.95372
$35$	149.137	0.720250
$36$	0	0
$37$	267.387	1.18806	0.594028	−	0.804444i	$-0.297537\pi$
$37$	267.387	1.18806	0.594028	+	0.804444i	$0.297537\pi$
$38$	291.123	1.24280
$39$	0	0
$40$	−320.657	−1.26751
$41$	475.722	1.81208	0.906040	−	0.423192i	$-0.139090\pi$
$41$	475.722	1.81208	0.906040	+	0.423192i	$0.139090\pi$
$42$	0	0
$43$	−207.667	−0.736487	−0.368244	−	0.929729i	$-0.620041\pi$
$43$	−207.667	−0.736487	−0.368244	+	0.929729i	$0.620041\pi$
$44$	1100.20	3.76958
$45$	0	0
$46$	−57.0244	−0.182778
$47$	−598.232	−1.85662	−0.928309	−	0.371809i	$-0.878737\pi$
$47$	−598.232	−1.85662	−0.928309	+	0.371809i	$0.878737\pi$
$48$	0	0
$49$	243.233	0.709135
$50$	445.413	1.25982
$51$	0	0
$52$	−41.0325	−0.109427
$53$	−275.859	−0.714947	−0.357473	−	0.933923i	$-0.616362\pi$
$53$	−275.859	−0.714947	−0.357473	+	0.933923i	$0.616362\pi$
$54$	0	0
$55$	372.856	0.914108
$56$	−1260.45	−3.00777
$57$	0	0
$58$	−907.651	−2.05484
$59$	25.8847	0.0571169	0.0285585	−	0.999592i	$-0.490908\pi$
$59$	25.8847	0.0571169	0.0285585	+	0.999592i	$0.490908\pi$
$60$	0	0
$61$	247.309	0.519092	0.259546	−	0.965731i	$-0.416427\pi$
$61$	247.309	0.519092	0.259546	+	0.965731i	$0.416427\pi$
$62$	1020.85	2.09109
$63$	0	0
$64$	67.3571	0.131557
$65$	−13.9059	−0.0265355
$66$	0	0
$67$	−24.8585	−0.0453276	−0.0226638	−	0.999743i	$-0.507215\pi$
$67$	−24.8585	−0.0453276	−0.0226638	+	0.999743i	$0.507215\pi$
$68$	2080.29	3.70988
$69$	0	0
$70$	−763.011	−1.30282
$71$	260.252	0.435017	0.217509	−	0.976058i	$-0.430207\pi$
$71$	260.252	0.435017	0.217509	+	0.976058i	$0.430207\pi$
$72$	0	0
$73$	1076.96	1.72670	0.863349	−	0.504607i	$-0.168363\pi$
$73$	1076.96	1.72670	0.863349	+	0.504607i	$0.168363\pi$
$74$	−1368.00	−2.14900
$75$	0	0
$76$	−1034.22	−1.56096
$77$	1465.64	2.16915
$78$	0	0
$79$	644.964	0.918533	0.459266	−	0.888299i	$-0.348112\pi$
$79$	644.964	0.918533	0.459266	+	0.888299i	$0.348112\pi$
$80$	744.925	1.04106
$81$	0	0
$82$	−2433.88	−3.27777
$83$	−276.207	−0.365274	−0.182637	−	0.983180i	$-0.558463\pi$
$83$	−276.207	−0.365274	−0.182637	+	0.983180i	$0.558463\pi$
$84$	0	0
$85$	705.006	0.899631
$86$	1062.46	1.33219
$87$	0	0
$88$	−3151.25	−3.81732
$89$	446.913	0.532278	0.266139	−	0.963935i	$-0.414252\pi$
$89$	446.913	0.532278	0.266139	+	0.963935i	$0.414252\pi$
$90$	0	0
$91$	−54.6617	−0.0629682
$92$	202.580	0.229570
$93$	0	0
$94$	3060.66	3.35833
$95$	−350.494	−0.378526
$96$	0	0
$97$	1294.37	1.35488	0.677438	−	0.735580i	$-0.263090\pi$
$97$	1294.37	1.35488	0.677438	+	0.735580i	$0.263090\pi$
$98$	−1244.42	−1.28271
$99$	0	0
$100$	−1582.33	−1.58233
$101$	973.935	0.959506	0.479753	−	0.877404i	$-0.340726\pi$
$101$	973.935	0.959506	0.479753	+	0.877404i	$0.340726\pi$
$102$	0	0
$103$	478.119	0.457383	0.228692	−	0.973499i	$-0.426555\pi$
$103$	478.119	0.457383	0.228692	+	0.973499i	$0.426555\pi$
$104$	117.527	0.110812
$105$	0	0
$106$	1411.34	1.29323
$107$	−337.772	−0.305174	−0.152587	−	0.988290i	$-0.548760\pi$
$107$	−337.772	−0.305174	−0.152587	+	0.988290i	$0.548760\pi$
$108$	0	0
$109$	−71.4379	−0.0627754	−0.0313877	−	0.999507i	$-0.509993\pi$
$109$	−71.4379	−0.0627754	−0.0313877	+	0.999507i	$0.509993\pi$
$110$	−1907.60	−1.65348
$111$	0	0
$112$	2928.18	2.47042
$113$	952.662	0.793088	0.396544	−	0.918016i	$-0.370209\pi$
$113$	952.662	0.793088	0.396544	+	0.918016i	$0.370209\pi$
$114$	0	0
$115$	68.6539	0.0556697
$116$	3224.44	2.58088
$117$	0	0
$118$	−132.431	−0.103315
$119$	2771.26	2.13480
$120$	0	0
$121$	2333.23	1.75299
$122$	−1265.27	−0.938955
$123$	0	0
$124$	−3626.57	−2.62642
$125$	−1306.20	−0.934638
$126$	0	0
$127$	−565.205	−0.394912	−0.197456	−	0.980312i	$-0.563268\pi$
$127$	−565.205	−0.394912	−0.197456	+	0.980312i	$0.563268\pi$
$128$	1273.57	0.879441
$129$	0	0
$130$	71.1449	0.0479986
$131$	1729.24	1.15331	0.576657	−	0.816986i	$-0.304357\pi$
$131$	1729.24	1.15331	0.576657	+	0.816986i	$0.304357\pi$
$132$	0	0
$133$	−1377.74	−0.898233
$134$	127.181	0.0819905
$135$	0	0
$136$	−5958.45	−3.75686
$137$	1709.34	1.06597	0.532987	−	0.846123i	$-0.321069\pi$
$137$	1709.34	1.06597	0.532987	+	0.846123i	$0.321069\pi$
$138$	0	0
$139$	−2291.12	−1.39806	−0.699030	−	0.715092i	$-0.746385\pi$
$139$	−2291.12	−1.39806	−0.699030	+	0.715092i	$0.746385\pi$
$140$	2710.60	1.63634
$141$	0	0
$142$	−1331.49	−0.786877
$143$	−136.659	−0.0799163
$144$	0	0
$145$	1092.76	0.625852
$146$	−5509.93	−3.12332
$147$	0	0
$148$	4859.82	2.69915
$149$	−2879.38	−1.58314	−0.791570	−	0.611079i	$-0.790736\pi$
$149$	−2879.38	−1.58314	−0.791570	+	0.611079i	$0.790736\pi$
$150$	0	0
$151$	−869.813	−0.468771	−0.234385	−	0.972144i	$-0.575308\pi$
$151$	−869.813	−0.468771	−0.234385	+	0.972144i	$0.575308\pi$
$152$	2962.25	1.58073
$153$	0	0
$154$	−7498.46	−3.92366
$155$	−1229.04	−0.636896
$156$	0	0
$157$	−61.0278	−0.0310226	−0.0155113	−	0.999880i	$-0.504938\pi$
$157$	−61.0278	−0.0310226	−0.0155113	+	0.999880i	$0.504938\pi$
$158$	−3299.75	−1.66148
$159$	0	0
$160$	−1245.91	−0.615611
$161$	269.868	0.132103
$162$	0	0
$163$	−566.997	−0.272458	−0.136229	−	0.990677i	$-0.543498\pi$
$163$	−566.997	−0.272458	−0.136229	+	0.990677i	$0.543498\pi$
$164$	8646.37	4.11688
$165$	0	0
$166$	1413.13	0.660722
$167$	−833.335	−0.386140	−0.193070	−	0.981185i	$-0.561844\pi$
$167$	−833.335	−0.386140	−0.193070	+	0.981185i	$0.561844\pi$
$168$	0	0
$169$	−2191.90	−0.997680
$170$	−3606.93	−1.62729
$171$	0	0
$172$	−3774.40	−1.67323
$173$	2726.65	1.19829	0.599143	−	0.800642i	$-0.295508\pi$
$173$	2726.65	1.19829	0.599143	+	0.800642i	$0.295508\pi$
$174$	0	0
$175$	−2107.91	−0.910533
$176$	7320.72	3.13534
$177$	0	0
$178$	−2286.49	−0.962806
$179$	−2310.94	−0.964958	−0.482479	−	0.875907i	$-0.660264\pi$
$179$	−2310.94	−0.964958	−0.482479	+	0.875907i	$0.660264\pi$
$180$	0	0
$181$	−258.541	−0.106173	−0.0530863	−	0.998590i	$-0.516906\pi$
$181$	−258.541	−0.106173	−0.0530863	+	0.998590i	$0.516906\pi$
$182$	279.659	0.113899
$183$	0	0
$184$	−580.238	−0.232477
$185$	1646.98	0.654534
$186$	0	0
$187$	6928.42	2.70939
$188$	−10873.0	−4.21806
$189$	0	0
$190$	1793.19	0.684694
$191$	−3180.80	−1.20500	−0.602499	−	0.798119i	$-0.705828\pi$
$191$	−3180.80	−1.20500	−0.602499	+	0.798119i	$0.705828\pi$
$192$	0	0
$193$	1964.34	0.732624	0.366312	−	0.930492i	$-0.380620\pi$
$193$	1964.34	0.732624	0.366312	+	0.930492i	$0.380620\pi$
$194$	−6622.20	−2.45075
$195$	0	0
$196$	4420.82	1.61109
$197$	−3229.80	−1.16809	−0.584045	−	0.811721i	$-0.698531\pi$
$197$	−3229.80	−1.16809	−0.584045	+	0.811721i	$0.698531\pi$
$198$	0	0
$199$	−1069.53	−0.380991	−0.190495	−	0.981688i	$-0.561009\pi$
$199$	−1069.53	−0.380991	−0.190495	+	0.981688i	$0.561009\pi$
$200$	4532.19	1.60237
$201$	0	0
$202$	−4982.82	−1.73559
$203$	4295.45	1.48513
$204$	0	0
$205$	2930.24	0.998326
$206$	−2446.14	−0.827334
$207$	0	0
$208$	−273.030	−0.0910155
$209$	−3444.47	−1.14000
$210$	0	0
$211$	3260.16	1.06369	0.531845	−	0.846842i	$-0.321499\pi$
$211$	3260.16	1.06369	0.531845	+	0.846842i	$0.321499\pi$
$212$	−5013.81	−1.62429
$213$	0	0
$214$	1728.10	0.552012
$215$	−1279.14	−0.405752
$216$	0	0
$217$	−4831.15	−1.51134
$218$	365.489	0.113551
$219$	0	0
$220$	6776.76	2.07677
$221$	−258.399	−0.0786506
$222$	0	0
$223$	−5181.92	−1.55609	−0.778043	−	0.628211i	$-0.783787\pi$
$223$	−5181.92	−1.55609	−0.778043	+	0.628211i	$0.783787\pi$
$224$	−4897.46	−1.46083
$225$	0	0
$226$	−4873.99	−1.43457
$227$	−3885.67	−1.13613	−0.568064	−	0.822984i	$-0.692307\pi$
$227$	−3885.67	−1.13613	−0.568064	+	0.822984i	$0.692307\pi$
$228$	0	0
$229$	−996.362	−0.287517	−0.143759	−	0.989613i	$-0.545919\pi$
$229$	−996.362	−0.287517	−0.143759	+	0.989613i	$0.545919\pi$
$230$	−351.246	−0.100698
$231$	0	0
$232$	−9235.58	−2.61356
$233$	4588.29	1.29008	0.645040	−	0.764149i	$-0.276841\pi$
$233$	4588.29	1.29008	0.645040	+	0.764149i	$0.276841\pi$
$234$	0	0
$235$	−3684.85	−1.02286
$236$	470.461	0.129764
$237$	0	0
$238$	−14178.3	−3.86151
$239$	−239.000	−0.0646846
$239$	−239.000	−0.0646846
$240$	0	0
$241$	−862.038	−0.230410	−0.115205	−	0.993342i	$-0.536752\pi$
$241$	−862.038	−0.230410	−0.115205	+	0.993342i	$0.536752\pi$
$242$	−11937.2	−3.17088
$243$	0	0
$244$	4494.90	1.17933
$245$	1498.21	0.390682
$246$	0	0
$247$	128.463	0.0330928
$248$	10387.4	2.65968
$249$	0	0
$250$	6682.73	1.69061
$251$	2970.96	0.747113	0.373557	−	0.927607i	$-0.378138\pi$
$251$	2970.96	0.747113	0.373557	+	0.927607i	$0.378138\pi$
$252$	0	0
$253$	674.694	0.167659
$254$	2891.69	0.714334
$255$	0	0
$256$	−7054.65	−1.72233
$257$	−3291.02	−0.798787	−0.399393	−	0.916780i	$-0.630779\pi$
$257$	−3291.02	−0.798787	−0.399393	+	0.916780i	$0.630779\pi$
$258$	0	0
$259$	6474.03	1.55319
$260$	−252.743	−0.0602863
$261$	0	0
$262$	−8847.09	−2.08616
$263$	5559.72	1.30353	0.651763	−	0.758423i	$-0.274030\pi$
$263$	5559.72	1.30353	0.651763	+	0.758423i	$0.274030\pi$
$264$	0	0
$265$	−1699.17	−0.393884
$266$	7048.74	1.62476
$267$	0	0
$268$	−451.810	−0.102980
$269$	−4835.44	−1.09599	−0.547996	−	0.836481i	$-0.684609\pi$
$269$	−4835.44	−1.09599	−0.547996	+	0.836481i	$0.684609\pi$
$270$	0	0
$271$	4159.85	0.932447	0.466223	−	0.884667i	$-0.345614\pi$
$271$	4159.85	0.932447	0.466223	+	0.884667i	$0.345614\pi$
$272$	13842.2	3.08568
$273$	0	0
$274$	−8745.27	−1.92818
$275$	−5269.98	−1.15561
$276$	0	0
$277$	−1547.96	−0.335768	−0.167884	−	0.985807i	$-0.553693\pi$
$277$	−1547.96	−0.335768	−0.167884	+	0.985807i	$0.553693\pi$
$278$	11721.8	2.52887
$279$	0	0
$280$	−7763.83	−1.65706
$281$	6222.21	1.32095	0.660474	−	0.750849i	$-0.270356\pi$
$281$	6222.21	1.32095	0.660474	+	0.750849i	$0.270356\pi$
$282$	0	0
$283$	−7453.16	−1.56553	−0.782764	−	0.622319i	$-0.786191\pi$
$283$	−7453.16	−1.56553	−0.782764	+	0.622319i	$0.786191\pi$
$284$	4730.15	0.988319
$285$	0	0
$286$	699.173	0.144556
$287$	11518.3	2.36900
$288$	0	0
$289$	8187.41	1.66648
$290$	−5590.74	−1.13207
$291$	0	0
$292$	19574.1	3.92290
$293$	9507.05	1.89559	0.947795	−	0.318881i	$-0.103307\pi$
$293$	9507.05	1.89559	0.947795	+	0.318881i	$0.103307\pi$
$294$	0	0
$295$	159.438	0.0314673
$296$	−13919.7	−2.73333
$297$	0	0
$298$	14731.4	2.86365
$299$	−25.1631	−0.00486695
$300$	0	0
$301$	−5028.09	−0.962838
$302$	4450.12	0.847932
$303$	0	0
$304$	−6881.66	−1.29832
$305$	1523.31	0.285982
$306$	0	0
$307$	−4201.33	−0.781051	−0.390526	−	0.920592i	$-0.627707\pi$
$307$	−4201.33	−0.781051	−0.390526	+	0.920592i	$0.627707\pi$
$308$	26638.3	4.92812
$309$	0	0
$310$	6287.98	1.15204
$311$	−1055.44	−0.192438	−0.0962190	−	0.995360i	$-0.530675\pi$
$311$	−1055.44	−0.192438	−0.0962190	+	0.995360i	$0.530675\pi$
$312$	0	0
$313$	−9121.61	−1.64723	−0.823617	−	0.567147i	$-0.808047\pi$
$313$	−9121.61	−1.64723	−0.823617	+	0.567147i	$0.808047\pi$
$314$	312.229	0.0561149
$315$	0	0
$316$	11722.4	2.08682
$317$	−5793.06	−1.02641	−0.513203	−	0.858267i	$-0.671541\pi$
$317$	−5793.06	−1.02641	−0.513203	+	0.858267i	$0.671541\pi$
$318$	0	0
$319$	10739.0	1.88486
$320$	414.891	0.0724784
$321$	0	0
$322$	−1380.69	−0.238953
$323$	−6512.89	−1.12194
$324$	0	0
$325$	196.547	0.0335460
$326$	2900.86	0.492833
$327$	0	0
$328$	−24765.3	−4.16901
$329$	−14484.5	−2.42723
$330$	0	0
$331$	−3979.44	−0.660815	−0.330408	−	0.943838i	$-0.607186\pi$
$331$	−3979.44	−0.660815	−0.330408	+	0.943838i	$0.607186\pi$
$332$	−5020.14	−0.829868
$333$	0	0
$334$	4263.49	0.698466
$335$	−153.118	−0.0249723
$336$	0	0
$337$	2938.64	0.475009	0.237504	−	0.971386i	$-0.423671\pi$
$337$	2938.64	0.475009	0.237504	+	0.971386i	$0.423671\pi$
$338$	11214.2	1.80465
$339$	0	0
$340$	12813.7	2.04388
$341$	−12078.3	−1.91812
$342$	0	0
$343$	−2415.58	−0.380260
$344$	10810.8	1.69442
$345$	0	0
$346$	−13950.0	−2.16751
$347$	−3152.37	−0.487688	−0.243844	−	0.969814i	$-0.578409\pi$
$347$	−3152.37	−0.487688	−0.243844	+	0.969814i	$0.578409\pi$
$348$	0	0
$349$	−4348.01	−0.666887	−0.333444	−	0.942770i	$-0.608211\pi$
$349$	−4348.01	−0.666887	−0.333444	+	0.942770i	$0.608211\pi$
$350$	10784.5	1.64701
$351$	0	0
$352$	−12244.1	−1.85402
$353$	10646.0	1.60518	0.802589	−	0.596532i	$-0.203455\pi$
$353$	10646.0	1.60518	0.802589	+	0.596532i	$0.203455\pi$
$354$	0	0
$355$	1603.04	0.239663
$356$	8122.76	1.20929
$357$	0	0
$358$	11823.2	1.74546
$359$	7666.22	1.12704	0.563520	−	0.826102i	$-0.309447\pi$
$359$	7666.22	1.12704	0.563520	+	0.826102i	$0.309447\pi$
$360$	0	0
$361$	−3621.11	−0.527935
$362$	1322.74	0.192049
$363$	0	0
$364$	−993.490	−0.143058
$365$	6633.62	0.951287
$366$	0	0
$367$	−9305.91	−1.32361	−0.661805	−	0.749676i	$-0.730209\pi$
$367$	−9305.91	−1.32361	−0.661805	+	0.749676i	$0.730209\pi$
$368$	1347.96	0.190944
$369$	0	0
$370$	−8426.26	−1.18395
$371$	−6679.17	−0.934678
$372$	0	0
$373$	13038.1	1.80989	0.904943	−	0.425533i	$-0.139913\pi$
$373$	13038.1	1.80989	0.904943	+	0.425533i	$0.139913\pi$
$374$	−35447.0	−4.90086
$375$	0	0
$376$	31143.0	4.27148
$377$	−400.518	−0.0547154
$378$	0	0
$379$	7405.15	1.00363	0.501816	−	0.864974i	$-0.332665\pi$
$379$	7405.15	1.00363	0.501816	+	0.864974i	$0.332665\pi$
$380$	−6370.33	−0.859976
$381$	0	0
$382$	16273.5	2.17965
$383$	−8586.05	−1.14550	−0.572750	−	0.819730i	$-0.694123\pi$
$383$	−8586.05	−1.14550	−0.572750	+	0.819730i	$0.694123\pi$
$384$	0	0
$385$	9027.69	1.19505
$386$	−10049.9	−1.32520
$387$	0	0
$388$	23525.4	3.07815
$389$	−6051.43	−0.788740	−0.394370	−	0.918952i	$-0.629037\pi$
$389$	−6051.43	−0.788740	−0.394370	+	0.918952i	$0.629037\pi$
$390$	0	0
$391$	1275.73	0.165003
$392$	−12662.3	−1.63149
$393$	0	0
$394$	16524.2	2.11289
$395$	3972.70	0.506046
$396$	0	0
$397$	−5248.00	−0.663450	−0.331725	−	0.943376i	$-0.607631\pi$
$397$	−5248.00	−0.663450	−0.331725	+	0.943376i	$0.607631\pi$
$398$	5471.91	0.689151
$399$	0	0
$400$	−10528.8	−1.31610
$401$	−10290.3	−1.28147	−0.640737	−	0.767760i	$-0.721371\pi$
$401$	−10290.3	−1.28147	−0.640737	+	0.767760i	$0.721371\pi$
$402$	0	0
$403$	450.468	0.0556809
$404$	17701.5	2.17991
$405$	0	0
$406$	−21976.3	−2.68637
$407$	16185.7	1.97124
$408$	0	0
$409$	8720.00	1.05422	0.527111	−	0.849797i	$-0.323275\pi$
$409$	8720.00	1.05422	0.527111	+	0.849797i	$0.323275\pi$
$410$	−14991.6	−1.80581
$411$	0	0
$412$	8689.94	1.03913
$413$	626.726	0.0746712
$414$	0	0
$415$	−1701.32	−0.201240
$416$	456.651	0.0538201
$417$	0	0
$418$	17622.5	2.06207
$419$	7078.88	0.825361	0.412680	−	0.910876i	$-0.364593\pi$
$419$	7078.88	0.825361	0.412680	+	0.910876i	$0.364593\pi$
$420$	0	0
$421$	14011.5	1.62203	0.811017	−	0.585022i	$-0.198914\pi$
$421$	14011.5	1.62203	0.811017	+	0.585022i	$0.198914\pi$
$422$	−16679.5	−1.92405
$423$	0	0
$424$	14360.8	1.64486
$425$	−9964.60	−1.13730
$426$	0	0
$427$	5987.90	0.678629
$428$	−6139.09	−0.693327
$429$	0	0
$430$	6544.30	0.733940
$431$	15783.4	1.76394	0.881970	−	0.471305i	$-0.156217\pi$
$431$	15783.4	1.76394	0.881970	+	0.471305i	$0.156217\pi$
$432$	0	0
$433$	11461.2	1.27203	0.636014	−	0.771677i	$-0.280582\pi$
$433$	11461.2	1.27203	0.636014	+	0.771677i	$0.280582\pi$
$434$	24717.0	2.73377
$435$	0	0
$436$	−1298.40	−0.142620
$437$	−634.230	−0.0694264
$438$	0	0
$439$	6354.81	0.690885	0.345442	−	0.938440i	$-0.387729\pi$
$439$	6354.81	0.690885	0.345442	+	0.938440i	$0.387729\pi$
$440$	−19410.3	−2.10307
$441$	0	0
$442$	1322.01	0.142266
$443$	1129.37	0.121124	0.0605619	−	0.998164i	$-0.480711\pi$
$443$	1129.37	0.121124	0.0605619	+	0.998164i	$0.480711\pi$
$444$	0	0
$445$	2752.79	0.293247
$446$	26511.6	2.81471
$447$	0	0
$448$	1630.87	0.171989
$449$	−5699.02	−0.599006	−0.299503	−	0.954095i	$-0.596821\pi$
$449$	−5699.02	−0.599006	−0.299503	+	0.954095i	$0.596821\pi$
$450$	0	0
$451$	28796.8	3.00663
$452$	17314.9	1.80182
$453$	0	0
$454$	19879.8	2.05508
$455$	−336.692	−0.0346909
$456$	0	0
$457$	−9585.31	−0.981142	−0.490571	−	0.871401i	$-0.663212\pi$
$457$	−9585.31	−0.981142	−0.490571	+	0.871401i	$0.663212\pi$
$458$	5097.57	0.520073
$459$	0	0
$460$	1247.80	0.126476
$461$	15284.2	1.54416	0.772078	−	0.635527i	$-0.219217\pi$
$461$	15284.2	1.54416	0.772078	+	0.635527i	$0.219217\pi$
$462$	0	0
$463$	3329.90	0.334241	0.167121	−	0.985936i	$-0.446553\pi$
$463$	3329.90	0.334241	0.167121	+	0.985936i	$0.446553\pi$
$464$	21455.4	2.14664
$465$	0	0
$466$	−23474.5	−2.33355
$467$	−459.551	−0.0455364	−0.0227682	−	0.999741i	$-0.507248\pi$
$467$	−459.551	−0.0455364	−0.0227682	+	0.999741i	$0.507248\pi$
$468$	0	0
$469$	−601.881	−0.0592586
$470$	18852.3	1.85020
$471$	0	0
$472$	−1347.51	−0.131408
$473$	−12570.7	−1.22199
$474$	0	0
$475$	4953.92	0.478529
$476$	50368.4	4.85007
$477$	0	0
$478$	1222.77	0.117004
$479$	−4162.34	−0.397040	−0.198520	−	0.980097i	$-0.563613\pi$
$479$	−4162.34	−0.397040	−0.198520	+	0.980097i	$0.563613\pi$
$480$	0	0
$481$	−603.653	−0.0572229
$482$	4410.34	0.416775
$483$	0	0
$484$	42407.1	3.98263
$485$	7972.73	0.746439
$486$	0	0
$487$	−6565.21	−0.610879	−0.305439	−	0.952211i	$-0.598803\pi$
$487$	−6565.21	−0.610879	−0.305439	+	0.952211i	$0.598803\pi$
$488$	−12874.5	−1.19426
$489$	0	0
$490$	−7665.11	−0.706682
$491$	6302.45	0.579278	0.289639	−	0.957136i	$-0.406465\pi$
$491$	6302.45	0.579278	0.289639	+	0.957136i	$0.406465\pi$
$492$	0	0
$493$	20305.6	1.85501
$494$	−657.241	−0.0598596
$495$	0	0
$496$	−24131.1	−2.18452
$497$	6301.29	0.568715
$498$	0	0
$499$	10128.2	0.908620	0.454310	−	0.890844i	$-0.349886\pi$
$499$	10128.2	0.908620	0.454310	+	0.890844i	$0.349886\pi$
$500$	−23740.4	−2.12341
$501$	0	0
$502$	−15200.0	−1.35141
$503$	10440.7	0.925499	0.462750	−	0.886489i	$-0.346863\pi$
$503$	10440.7	0.925499	0.462750	+	0.886489i	$0.346863\pi$
$504$	0	0
$505$	5999.01	0.528619
$506$	−3451.85	−0.303268
$507$	0	0
$508$	−10272.8	−0.897204
$509$	7114.47	0.619535	0.309767	−	0.950812i	$-0.399749\pi$
$509$	7114.47	0.619535	0.309767	+	0.950812i	$0.399749\pi$
$510$	0	0
$511$	26075.7	2.25738
$512$	25904.3	2.23597
$513$	0	0
$514$	16837.4	1.44488
$515$	2945.01	0.251985
$516$	0	0
$517$	−36212.7	−3.08053
$518$	−33122.3	−2.80948
$519$	0	0
$520$	723.917	0.0610497
$521$	17164.2	1.44333	0.721665	−	0.692242i	$-0.243377\pi$
$521$	17164.2	1.44333	0.721665	+	0.692242i	$0.243377\pi$
$522$	0	0
$523$	7814.01	0.653313	0.326657	−	0.945143i	$-0.394078\pi$
$523$	7814.01	0.653313	0.326657	+	0.945143i	$0.394078\pi$
$524$	31429.3	2.62022
$525$	0	0
$526$	−28444.5	−2.35787
$527$	−22838.0	−1.88774
$528$	0	0
$529$	−12042.8	−0.989789
$530$	8693.27	0.712474
$531$	0	0
$532$	−25040.7	−2.04070
$533$	−1073.99	−0.0872791
$534$	0	0
$535$	−2080.53	−0.168129
$536$	1294.09	0.104284
$537$	0	0
$538$	24739.0	1.98248
$539$	14723.6	1.17661
$540$	0	0
$541$	−5367.07	−0.426522	−0.213261	−	0.976995i	$-0.568409\pi$
$541$	−5367.07	−0.426522	−0.213261	+	0.976995i	$0.568409\pi$
$542$	−21282.5	−1.68665
$543$	0	0
$544$	−23151.5	−1.82465
$545$	−440.027	−0.0345847
$546$	0	0
$547$	21500.9	1.68064	0.840322	−	0.542088i	$-0.182366\pi$
$547$	21500.9	1.68064	0.840322	+	0.542088i	$0.182366\pi$
$548$	31067.6	2.42180
$549$	0	0
$550$	26962.1	2.09031
$551$	−10095.0	−0.780508
$552$	0	0
$553$	15616.0	1.20083
$554$	7919.61	0.607350
$555$	0	0
$556$	−41641.7	−3.17626
$557$	−17048.6	−1.29690	−0.648449	−	0.761258i	$-0.724582\pi$
$557$	−17048.6	−1.29690	−0.648449	+	0.761258i	$0.724582\pi$
$558$	0	0
$559$	468.831	0.0354730
$560$	18036.3	1.36102
$561$	0	0
$562$	−31833.9	−2.38938
$563$	−613.122	−0.0458970	−0.0229485	−	0.999737i	$-0.507305\pi$
$563$	−613.122	−0.0458970	−0.0229485	+	0.999737i	$0.507305\pi$
$564$	0	0
$565$	5867.99	0.436935
$566$	38131.7	2.83179
$567$	0	0
$568$	−13548.3	−1.00083
$569$	−16212.2	−1.19446	−0.597232	−	0.802069i	$-0.703733\pi$
$569$	−16212.2	−1.19446	−0.597232	+	0.802069i	$0.703733\pi$
$570$	0	0
$571$	−16513.8	−1.21030	−0.605150	−	0.796112i	$-0.706887\pi$
$571$	−16513.8	−1.21030	−0.605150	+	0.796112i	$0.706887\pi$
$572$	−2483.82	−0.181562
$573$	0	0
$574$	−58929.7	−4.28515
$575$	−970.360	−0.0703771
$576$	0	0
$577$	−1929.46	−0.139210	−0.0696051	−	0.997575i	$-0.522174\pi$
$577$	−1929.46	−0.139210	−0.0696051	+	0.997575i	$0.522174\pi$
$578$	−41888.2	−3.01440
$579$	0	0
$580$	19861.1	1.42188
$581$	−6687.60	−0.477536
$582$	0	0
$583$	−16698.6	−1.18625
$584$	−56065.0	−3.97258
$585$	0	0
$586$	−48639.7	−3.42882
$587$	−7835.41	−0.550941	−0.275470	−	0.961310i	$-0.588834\pi$
$587$	−7835.41	−0.550941	−0.275470	+	0.961310i	$0.588834\pi$
$588$	0	0
$589$	11353.9	0.794281
$590$	−815.715	−0.0569194
$591$	0	0
$592$	32337.2	2.24501
$593$	−11655.4	−0.807131	−0.403566	−	0.914951i	$-0.632229\pi$
$593$	−11655.4	−0.807131	−0.403566	+	0.914951i	$0.632229\pi$
$594$	0	0
$595$	17069.8	1.17612
$596$	−52333.4	−3.59675
$597$	0	0
$598$	128.739	0.00880354
$599$	8154.09	0.556205	0.278103	−	0.960551i	$-0.410294\pi$
$599$	8154.09	0.556205	0.278103	+	0.960551i	$0.410294\pi$
$600$	0	0
$601$	−3889.65	−0.263997	−0.131999	−	0.991250i	$-0.542139\pi$
$601$	−3889.65	−0.263997	−0.131999	+	0.991250i	$0.542139\pi$
$602$	25724.6	1.74162
$603$	0	0
$604$	−15809.1	−1.06500
$605$	14371.7	0.965773
$606$	0	0
$607$	−15961.4	−1.06730	−0.533651	−	0.845705i	$-0.679180\pi$
$607$	−15961.4	−1.06730	−0.533651	+	0.845705i	$0.679180\pi$
$608$	11509.8	0.767736
$609$	0	0
$610$	−7793.54	−0.517297
$611$	1350.57	0.0894243
$612$	0	0
$613$	−8958.59	−0.590268	−0.295134	−	0.955456i	$-0.595364\pi$
$613$	−8958.59	−0.590268	−0.295134	+	0.955456i	$0.595364\pi$
$614$	21494.8	1.41280
$615$	0	0
$616$	−76298.7	−4.99053
$617$	−25935.8	−1.69228	−0.846139	−	0.532962i	$-0.821079\pi$
$617$	−25935.8	−1.69228	−0.846139	+	0.532962i	$0.821079\pi$
$618$	0	0
$619$	28793.1	1.86962	0.934810	−	0.355149i	$-0.115570\pi$
$619$	28793.1	1.86962	0.934810	+	0.355149i	$0.115570\pi$
$620$	−22338.1	−1.44697
$621$	0	0
$622$	5399.79	0.348090
$623$	10820.8	0.695867
$624$	0	0
$625$	2836.87	0.181560
$626$	46667.8	2.97958
$627$	0	0
$628$	−1109.20	−0.0704804
$629$	30604.3	1.94002
$630$	0	0
$631$	−10699.7	−0.675034	−0.337517	−	0.941319i	$-0.609587\pi$
$631$	−10699.7	−0.675034	−0.337517	+	0.941319i	$0.609587\pi$
$632$	−33575.8	−2.11325
$633$	0	0
$634$	29638.3	1.85661
$635$	−3481.42	−0.217568
$636$	0	0
$637$	−549.124	−0.0341556
$638$	−54942.8	−3.40941
$639$	0	0
$640$	7844.61	0.484509
$641$	−18742.7	−1.15490	−0.577450	−	0.816426i	$-0.695952\pi$
$641$	−18742.7	−1.15490	−0.577450	+	0.816426i	$0.695952\pi$
$642$	0	0
$643$	−1338.90	−0.0821166	−0.0410583	−	0.999157i	$-0.513073\pi$
$643$	−1338.90	−0.0821166	−0.0410583	+	0.999157i	$0.513073\pi$
$644$	4904.91	0.300125
$645$	0	0
$646$	33321.1	2.02941
$647$	−14253.8	−0.866112	−0.433056	−	0.901367i	$-0.642565\pi$
$647$	−14253.8	−0.866112	−0.433056	+	0.901367i	$0.642565\pi$
$648$	0	0
$649$	1566.87	0.0947692
$650$	−1005.57	−0.0606793
$651$	0	0
$652$	−10305.3	−0.618999
$653$	7106.59	0.425884	0.212942	−	0.977065i	$-0.431695\pi$
$653$	7106.59	0.425884	0.212942	+	0.977065i	$0.431695\pi$
$654$	0	0
$655$	10651.4	0.635394
$656$	57532.8	3.42420
$657$	0	0
$658$	74105.4	4.39047
$659$	32587.0	1.92627	0.963133	−	0.269024i	$-0.0867011\pi$
$659$	32587.0	1.92627	0.963133	+	0.269024i	$0.0867011\pi$
$660$	0	0
$661$	−29315.3	−1.72501	−0.862506	−	0.506046i	$-0.831106\pi$
$661$	−29315.3	−1.72501	−0.862506	+	0.506046i	$0.831106\pi$
$662$	20359.5	1.19531
$663$	0	0
$664$	14378.9	0.840377
$665$	−8486.26	−0.494862
$666$	0	0
$667$	1977.37	0.114789
$668$	−15146.1	−0.877275
$669$	0	0
$670$	783.377	0.0451709
$671$	14970.3	0.861285
$672$	0	0
$673$	4257.99	0.243884	0.121942	−	0.992537i	$-0.461088\pi$
$673$	4257.99	0.243884	0.121942	+	0.992537i	$0.461088\pi$
$674$	−15034.6	−0.859215
$675$	0	0
$676$	−39838.4	−2.26664
$677$	−10449.7	−0.593227	−0.296613	−	0.954998i	$-0.595857\pi$
$677$	−10449.7	−0.593227	−0.296613	+	0.954998i	$0.595857\pi$
$678$	0	0
$679$	31339.5	1.77128
$680$	−36701.5	−2.06976
$681$	0	0
$682$	61794.9	3.46957
$683$	20866.7	1.16902	0.584510	−	0.811386i	$-0.301287\pi$
$683$	20866.7	1.16902	0.584510	+	0.811386i	$0.301287\pi$
$684$	0	0
$685$	10528.8	0.587276
$686$	12358.5	0.687829
$687$	0	0
$688$	−25114.8	−1.39171
$689$	622.781	0.0344355
$690$	0	0
$691$	31753.3	1.74812	0.874061	−	0.485815i	$-0.161477\pi$
$691$	31753.3	1.74812	0.874061	+	0.485815i	$0.161477\pi$
$692$	49557.6	2.72239
$693$	0	0
$694$	16128.1	0.882151
$695$	−14112.3	−0.770231
$696$	0	0
$697$	54449.7	2.95901
$698$	22245.2	1.20629
$699$	0	0
$700$	−38311.9	−2.06865
$701$	33086.2	1.78266	0.891332	−	0.453350i	$-0.149771\pi$
$701$	33086.2	1.78266	0.891332	+	0.453350i	$0.149771\pi$
$702$	0	0
$703$	−15215.0	−0.816277
$704$	4077.32	0.218281
$705$	0	0
$706$	−54466.7	−2.90351
$707$	23581.2	1.25440
$708$	0	0
$709$	13001.5	0.688689	0.344345	−	0.938843i	$-0.388101\pi$
$709$	13001.5	0.688689	0.344345	+	0.938843i	$0.388101\pi$
$710$	−8201.43	−0.433513
$711$	0	0
$712$	−23265.6	−1.22460
$713$	−2223.98	−0.116815
$714$	0	0
$715$	−841.762	−0.0440281
$716$	−42001.9	−2.19230
$717$	0	0
$718$	−39221.7	−2.03864
$719$	−22832.0	−1.18427	−0.592136	−	0.805838i	$-0.701715\pi$
$719$	−22832.0	−1.18427	−0.592136	+	0.805838i	$0.701715\pi$
$720$	0	0
$721$	11576.3	0.597955
$722$	18526.2	0.954951
$723$	0	0
$724$	−4699.05	−0.241214
$725$	−15445.1	−0.791196
$726$	0	0
$727$	−12675.4	−0.646634	−0.323317	−	0.946291i	$-0.604798\pi$
$727$	−12675.4	−0.646634	−0.323317	+	0.946291i	$0.604798\pi$
$728$	2845.60	0.144869
$729$	0	0
$730$	−33938.8	−1.72073
$731$	−23769.0	−1.20264
$732$	0	0
$733$	12531.2	0.631448	0.315724	−	0.948851i	$-0.397753\pi$
$733$	12531.2	0.631448	0.315724	+	0.948851i	$0.397753\pi$
$734$	47610.7	2.39420
$735$	0	0
$736$	−2254.51	−0.112911
$737$	−1504.76	−0.0752082
$738$	0	0
$739$	28474.9	1.41741	0.708706	−	0.705504i	$-0.249279\pi$
$739$	28474.9	1.41741	0.708706	+	0.705504i	$0.249279\pi$
$740$	29934.4	1.48704
$741$	0	0
$742$	34171.8	1.69068
$743$	−33823.3	−1.67006	−0.835032	−	0.550202i	$-0.814551\pi$
$743$	−33823.3	−1.67006	−0.835032	+	0.550202i	$0.814551\pi$
$744$	0	0
$745$	−17735.7	−0.872196
$746$	−66705.2	−3.27380
$747$	0	0
$748$	125926.	6.15548
$749$	−8178.22	−0.398966
$750$	0	0
$751$	−32411.1	−1.57483	−0.787414	−	0.616424i	$-0.788581\pi$
$751$	−32411.1	−1.57483	−0.787414	+	0.616424i	$0.788581\pi$
$752$	−72348.8	−3.50837
$753$	0	0
$754$	2049.12	0.0989715
$755$	−5357.67	−0.258259
$756$	0	0
$757$	−19616.4	−0.941834	−0.470917	−	0.882178i	$-0.656077\pi$
$757$	−19616.4	−0.941834	−0.470917	+	0.882178i	$0.656077\pi$
$758$	−37886.0	−1.81541
$759$	0	0
$760$	18246.2	0.870866
$761$	11000.9	0.524025	0.262012	−	0.965065i	$-0.415614\pi$
$761$	11000.9	0.524025	0.262012	+	0.965065i	$0.415614\pi$
$762$	0	0
$763$	−1729.67	−0.0820687
$764$	−57811.9	−2.73764
$765$	0	0
$766$	43927.7	2.07203
$767$	−58.4374	−0.00275104
$768$	0	0
$769$	23498.7	1.10193	0.550966	−	0.834528i	$-0.314260\pi$
$769$	23498.7	1.10193	0.550966	+	0.834528i	$0.314260\pi$
$770$	−46187.3	−2.16165
$771$	0	0
$772$	35702.4	1.66445
$773$	−20105.2	−0.935489	−0.467745	−	0.883864i	$-0.654933\pi$
$773$	−20105.2	−0.935489	−0.467745	+	0.883864i	$0.654933\pi$
$774$	0	0
$775$	17371.3	0.805157
$776$	−67382.6	−3.11713
$777$	0	0
$778$	30960.2	1.42670
$779$	−27069.8	−1.24503
$780$	0	0
$781$	15753.8	0.721787
$782$	−6526.85	−0.298465
$783$	0	0
$784$	29416.1	1.34002
$785$	−375.904	−0.0170912
$786$	0	0
$787$	−34093.2	−1.54421	−0.772103	−	0.635497i	$-0.780795\pi$
$787$	−34093.2	−1.54421	−0.772103	+	0.635497i	$0.780795\pi$
$788$	−58702.5	−2.65379
$789$	0	0
$790$	−20325.0	−0.915356
$791$	23066.1	1.03683
$792$	0	0
$793$	−558.325	−0.0250021
$794$	26849.7	1.20008
$795$	0	0
$796$	−19439.0	−0.865575
$797$	−9128.51	−0.405707	−0.202853	−	0.979209i	$-0.565021\pi$
$797$	−9128.51	−0.405707	−0.202853	+	0.979209i	$0.565021\pi$
$798$	0	0
$799$	−68471.8	−3.03174
$800$	17609.8	0.778249
$801$	0	0
$802$	52646.8	2.31798
$803$	65191.7	2.86496
$804$	0	0
$805$	1662.27	0.0727791
$806$	−2304.67	−0.100718
$807$	0	0
$808$	−50701.5	−2.20751
$809$	−8581.63	−0.372947	−0.186474	−	0.982460i	$-0.559706\pi$
$809$	−8581.63	−0.372947	−0.186474	+	0.982460i	$0.559706\pi$
$810$	0	0
$811$	−7419.25	−0.321239	−0.160620	−	0.987016i	$-0.551349\pi$
$811$	−7419.25	−0.321239	−0.160620	+	0.987016i	$0.551349\pi$
$812$	78070.9	3.37408
$813$	0	0
$814$	−82808.8	−3.56566
$815$	−3492.46	−0.150105
$816$	0	0
$817$	11816.8	0.506018
$818$	−44613.1	−1.90692
$819$	0	0
$820$	53257.9	2.26810
$821$	−395.175	−0.0167987	−0.00839933	−	0.999965i	$-0.502674\pi$
$821$	−395.175	−0.0167987	−0.00839933	+	0.999965i	$0.502674\pi$
$822$	0	0
$823$	−13815.4	−0.585145	−0.292572	−	0.956243i	$-0.594511\pi$
$823$	−13815.4	−0.585145	−0.292572	+	0.956243i	$0.594511\pi$
$824$	−24890.1	−1.05229
$825$	0	0
$826$	−3206.44	−0.135068
$827$	−4972.77	−0.209093	−0.104547	−	0.994520i	$-0.533339\pi$
$827$	−4972.77	−0.209093	−0.104547	+	0.994520i	$0.533339\pi$
$828$	0	0
$829$	−2223.76	−0.0931656	−0.0465828	−	0.998914i	$-0.514833\pi$
$829$	−2223.76	−0.0931656	−0.0465828	+	0.998914i	$0.514833\pi$
$830$	8704.24	0.364010
$831$	0	0
$832$	−152.066	−0.00633646
$833$	27839.7	1.15797
$834$	0	0
$835$	−5132.98	−0.212736
$836$	−62604.1	−2.58996
$837$	0	0
$838$	−36216.8	−1.49295
$839$	6945.72	0.285808	0.142904	−	0.989737i	$-0.454356\pi$
$839$	6945.72	0.285808	0.142904	+	0.989737i	$0.454356\pi$
$840$	0	0
$841$	7084.65	0.290486
$842$	−71685.1	−2.93400
$843$	0	0
$844$	59254.2	2.41660
$845$	−13501.2	−0.549650
$846$	0	0
$847$	56492.8	2.29175
$848$	−33361.8	−1.35100
$849$	0	0
$850$	50980.7	2.05720
$851$	2980.27	0.120050
$852$	0	0
$853$	−21002.3	−0.843029	−0.421515	−	0.906822i	$-0.638501\pi$
$853$	−21002.3	−0.843029	−0.421515	+	0.906822i	$0.638501\pi$
$854$	−30635.1	−1.22753
$855$	0	0
$856$	17583.9	0.702107
$857$	17224.3	0.686545	0.343273	−	0.939236i	$-0.388465\pi$
$857$	17224.3	0.686545	0.343273	+	0.939236i	$0.388465\pi$
$858$	0	0
$859$	−39760.0	−1.57927	−0.789635	−	0.613576i	$-0.789730\pi$
$859$	−39760.0	−1.57927	−0.789635	+	0.613576i	$0.789730\pi$
$860$	−23248.7	−0.921830
$861$	0	0
$862$	−80750.5	−3.19069
$863$	32018.9	1.26296	0.631480	−	0.775392i	$-0.282448\pi$
$863$	32018.9	1.26296	0.631480	+	0.775392i	$0.282448\pi$
$864$	0	0
$865$	16795.0	0.660170
$866$	−58637.3	−2.30090
$867$	0	0
$868$	−87807.5	−3.43362
$869$	39041.5	1.52404
$870$	0	0
$871$	56.1207	0.00218321
$872$	3718.94	0.144426
$873$	0	0
$874$	3244.83	0.125581
$875$	−31625.9	−1.22189
$876$	0	0
$877$	24682.8	0.950376	0.475188	−	0.879884i	$-0.342380\pi$
$877$	24682.8	0.950376	0.475188	+	0.879884i	$0.342380\pi$
$878$	−32512.3	−1.24970
$879$	0	0
$880$	45092.4	1.72735
$881$	11864.7	0.453726	0.226863	−	0.973927i	$-0.427153\pi$
$881$	11864.7	0.453726	0.226863	+	0.973927i	$0.427153\pi$
$882$	0	0
$883$	16157.7	0.615797	0.307899	−	0.951419i	$-0.400374\pi$
$883$	16157.7	0.615797	0.307899	+	0.951419i	$0.400374\pi$
$884$	−4696.46	−0.178687
$885$	0	0
$886$	−5778.04	−0.219094
$887$	26782.5	1.01383	0.506916	−	0.861995i	$-0.330785\pi$
$887$	26782.5	1.01383	0.506916	+	0.861995i	$0.330785\pi$
$888$	0	0
$889$	−13684.9	−0.516284
$890$	−14083.8	−0.530437
$891$	0	0
$892$	−94182.7	−3.53528
$893$	34040.9	1.27563
$894$	0	0
$895$	−14234.4	−0.531623
$896$	30835.9	1.14973
$897$	0	0
$898$	29157.2	1.08351
$899$	−35398.9	−1.31326
$900$	0	0
$901$	−31574.0	−1.16746
$902$	−147330.	−5.43852
$903$	0	0
$904$	−49594.1	−1.82464
$905$	−1592.50	−0.0584934
$906$	0	0
$907$	−16780.3	−0.614312	−0.307156	−	0.951659i	$-0.599377\pi$
$907$	−16780.3	−0.614312	−0.307156	+	0.951659i	$0.599377\pi$
$908$	−70623.1	−2.58118
$909$	0	0
$910$	1722.58	0.0627504
$911$	47243.2	1.71815	0.859076	−	0.511848i	$-0.171039\pi$
$911$	47243.2	1.71815	0.859076	+	0.511848i	$0.171039\pi$
$912$	0	0
$913$	−16719.6	−0.606067
$914$	49040.1	1.77473
$915$	0	0
$916$	−18109.1	−0.653213
$917$	41868.8	1.50777
$918$	0	0
$919$	16832.5	0.604191	0.302095	−	0.953278i	$-0.402314\pi$
$919$	16832.5	0.604191	0.302095	+	0.953278i	$0.402314\pi$
$920$	−3574.01	−0.128078
$921$	0	0
$922$	−78196.7	−2.79313
$923$	−587.546	−0.0209527
$924$	0	0
$925$	−23278.6	−0.827455
$926$	−17036.4	−0.604590
$927$	0	0
$928$	−35884.7	−1.26937
$929$	44392.4	1.56778	0.783890	−	0.620900i	$-0.213233\pi$
$929$	44392.4	1.56778	0.783890	+	0.620900i	$0.213233\pi$
$930$	0	0
$931$	−13840.6	−0.487225
$932$	83393.3	2.93094
$933$	0	0
$934$	2351.14	0.0823681
$935$	42676.0	1.49268
$936$	0	0
$937$	26352.9	0.918797	0.459399	−	0.888230i	$-0.348065\pi$
$937$	26352.9	0.918797	0.459399	+	0.888230i	$0.348065\pi$
$938$	3079.33	0.107189
$939$	0	0
$940$	−66973.0	−2.32385
$941$	−12263.8	−0.424853	−0.212427	−	0.977177i	$-0.568137\pi$
$941$	−12263.8	−0.424853	−0.212427	+	0.977177i	$0.568137\pi$
$942$	0	0
$943$	5302.35	0.183105
$944$	3130.44	0.107931
$945$	0	0
$946$	64313.9	2.21039
$947$	19599.5	0.672542	0.336271	−	0.941765i	$-0.390834\pi$
$947$	19599.5	0.672542	0.336271	+	0.941765i	$0.390834\pi$
$948$	0	0
$949$	−2431.36	−0.0831667
$950$	−25345.1	−0.865583
$951$	0	0
$952$	−144267.	−4.91149
$953$	26505.2	0.900932	0.450466	−	0.892794i	$-0.351258\pi$
$953$	26505.2	0.900932	0.450466	+	0.892794i	$0.351258\pi$
$954$	0	0
$955$	−19592.4	−0.663868
$956$	−4343.89	−0.146957
$957$	0	0
$958$	21295.3	0.718183
$959$	41386.9	1.39359
$960$	0	0
$961$	10022.6	0.336430
$962$	3088.40	0.103507
$963$	0	0
$964$	−15667.8	−0.523470
$965$	12099.5	0.403623
$966$	0	0
$967$	2628.01	0.0873951	0.0436976	−	0.999045i	$-0.486086\pi$
$967$	2628.01	0.0873951	0.0436976	+	0.999045i	$0.486086\pi$
$968$	−121464.	−4.03307
$969$	0	0
$970$	−40789.9	−1.35019
$971$	−4466.94	−0.147632	−0.0738161	−	0.997272i	$-0.523518\pi$
$971$	−4466.94	−0.147632	−0.0738161	+	0.997272i	$0.523518\pi$
$972$	0	0
$973$	−55473.2	−1.82774
$974$	33588.8	1.10498
$975$	0	0
$976$	29909.0	0.980904
$977$	4528.27	0.148283	0.0741413	−	0.997248i	$-0.476378\pi$
$977$	4528.27	0.148283	0.0741413	+	0.997248i	$0.476378\pi$
$978$	0	0
$979$	27053.0	0.883163
$980$	27230.4	0.887594
$981$	0	0
$982$	−32244.4	−1.04782
$983$	−37570.4	−1.21903	−0.609517	−	0.792773i	$-0.708637\pi$
$983$	−37570.4	−1.21903	−0.609517	+	0.792773i	$0.708637\pi$
$984$	0	0
$985$	−19894.2	−0.643534
$986$	−103887.	−3.35541
$987$	0	0
$988$	2334.85	0.0751838
$989$	−2314.64	−0.0744199
$990$	0	0
$991$	30386.3	0.974018	0.487009	−	0.873397i	$-0.338088\pi$
$991$	30386.3	0.974018	0.487009	+	0.873397i	$0.338088\pi$
$992$	40360.1	1.29177
$993$	0	0
$994$	−32238.5	−1.02871
$995$	−6587.85	−0.209898
$996$	0	0
$997$	−30865.5	−0.980462	−0.490231	−	0.871593i	$-0.663088\pi$
$997$	−30865.5	−0.980462	−0.490231	+	0.871593i	$0.663088\pi$
$998$	−51817.8	−1.64355
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.h.1.4	yes	59
3.2	odd	2		2151.4.a.g.1.56	✓	59

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.4.a.g.1.56	✓	59	3.2	odd	2
2151.4.a.h.1.4	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)

\(f(q)\)	\(=\)	\(q-5.11618 q^{2} +18.1753 q^{4} +6.15957 q^{5} +24.2123 q^{7} -52.0584 q^{8} +O(q^{10})\)	\(q-5.11618 q^{2} +18.1753 q^{4} +6.15957 q^{5} +24.2123 q^{7} -52.0584 q^{8} -31.5134 q^{10} +60.5329 q^{11} -2.25761 q^{13} -123.874 q^{14} +120.938 q^{16} +114.457 q^{17} -56.9025 q^{19} +111.952 q^{20} -309.697 q^{22} +11.1459 q^{23} -87.0598 q^{25} +11.5503 q^{26} +440.064 q^{28} +177.408 q^{29} -199.533 q^{31} -202.272 q^{32} -585.582 q^{34} +149.137 q^{35} +267.387 q^{37} +291.123 q^{38} -320.657 q^{40} +475.722 q^{41} -207.667 q^{43} +1100.20 q^{44} -57.0244 q^{46} -598.232 q^{47} +243.233 q^{49} +445.413 q^{50} -41.0325 q^{52} -275.859 q^{53} +372.856 q^{55} -1260.45 q^{56} -907.651 q^{58} +25.8847 q^{59} +247.309 q^{61} +1020.85 q^{62} +67.3571 q^{64} -13.9059 q^{65} -24.8585 q^{67} +2080.29 q^{68} -763.011 q^{70} +260.252 q^{71} +1076.96 q^{73} -1368.00 q^{74} -1034.22 q^{76} +1465.64 q^{77} +644.964 q^{79} +744.925 q^{80} -2433.88 q^{82} -276.207 q^{83} +705.006 q^{85} +1062.46 q^{86} -3151.25 q^{88} +446.913 q^{89} -54.6617 q^{91} +202.580 q^{92} +3060.66 q^{94} -350.494 q^{95} +1294.37 q^{97} -1244.42 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 59 q + 8 q^{2} + 238 q^{4} + 80 q^{5} - 10 q^{7} + 96 q^{8}+O(q^{10}) \) 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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Database

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