LMFDB - Embedded newform 2151.4.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("2151.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$126.913108422$
Analytic rank:	$1$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$	$=$	$q-5.59453 q^{2} +23.2988 q^{4} -16.4914 q^{5} -10.5009 q^{7} -85.5894 q^{8} +O(q^{10})$	\(q-5.59453 q^{2} +23.2988 q^{4} -16.4914 q^{5} -10.5009 q^{7} -85.5894 q^{8} +92.2616 q^{10} -4.40880 q^{11} -64.7884 q^{13} +58.7475 q^{14} +292.442 q^{16} -7.65474 q^{17} -84.7450 q^{19} -384.229 q^{20} +24.6651 q^{22} +145.624 q^{23} +146.966 q^{25} +362.461 q^{26} -244.658 q^{28} -113.139 q^{29} -249.383 q^{31} -951.362 q^{32} +42.8247 q^{34} +173.174 q^{35} +95.9890 q^{37} +474.108 q^{38} +1411.49 q^{40} +391.208 q^{41} +200.408 q^{43} -102.720 q^{44} -814.696 q^{46} -392.357 q^{47} -232.731 q^{49} -822.206 q^{50} -1509.49 q^{52} +614.057 q^{53} +72.7072 q^{55} +898.765 q^{56} +632.959 q^{58} +104.990 q^{59} -218.526 q^{61} +1395.18 q^{62} +2982.89 q^{64} +1068.45 q^{65} +1027.44 q^{67} -178.346 q^{68} -968.828 q^{70} -345.643 q^{71} +705.393 q^{73} -537.013 q^{74} -1974.45 q^{76} +46.2963 q^{77} +813.898 q^{79} -4822.78 q^{80} -2188.63 q^{82} +985.181 q^{83} +126.237 q^{85} -1121.19 q^{86} +377.346 q^{88} -1483.05 q^{89} +680.336 q^{91} +3392.85 q^{92} +2195.05 q^{94} +1397.56 q^{95} -932.272 q^{97} +1302.02 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8}+O(q^{10}) $ 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8	$ 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8} + 52 q^{10} - 270 q^{11} + 48 q^{13} - 184 q^{14} + 775 q^{16} - 384 q^{17} + 216 q^{19} - 534 q^{20} + 437 q^{22} - 712 q^{23} + 1190 q^{25} - 436 q^{26} + 598 q^{28} - 562 q^{29} + 384 q^{31} - 1770 q^{32} + 452 q^{34} - 1026 q^{35} + 770 q^{37} - 733 q^{38} + 877 q^{40} - 1648 q^{41} + 1592 q^{43} - 1595 q^{44} + 532 q^{46} - 1540 q^{47} + 2134 q^{49} - 1646 q^{50} - 144 q^{52} - 1708 q^{53} + 1282 q^{55} - 2155 q^{56} + 1086 q^{58} - 2396 q^{59} + 364 q^{61} - 2180 q^{62} + 1663 q^{64} - 1520 q^{65} + 2728 q^{67} - 1545 q^{68} - 4609 q^{70} - 3322 q^{71} - 188 q^{73} - 1111 q^{74} - 3134 q^{76} - 556 q^{77} - 462 q^{79} - 6076 q^{80} - 7965 q^{82} - 4604 q^{83} - 852 q^{85} - 549 q^{86} - 1127 q^{88} - 6742 q^{89} + 1390 q^{91} - 1802 q^{92} - 2796 q^{94} - 448 q^{95} - 1322 q^{97} - 1000 q^{98}+O(q^{100}) $ 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8 + 52 * q^10 - 270 * q^11 + 48 * q^13 - 184 * q^14 + 775 * q^16 - 384 * q^17 + 216 * q^19 - 534 * q^20 + 437 * q^22 - 712 * q^23 + 1190 * q^25 - 436 * q^26 + 598 * q^28 - 562 * q^29 + 384 * q^31 - 1770 * q^32 + 452 * q^34 - 1026 * q^35 + 770 * q^37 - 733 * q^38 + 877 * q^40 - 1648 * q^41 + 1592 * q^43 - 1595 * q^44 + 532 * q^46 - 1540 * q^47 + 2134 * q^49 - 1646 * q^50 - 144 * q^52 - 1708 * q^53 + 1282 * q^55 - 2155 * q^56 + 1086 * q^58 - 2396 * q^59 + 364 * q^61 - 2180 * q^62 + 1663 * q^64 - 1520 * q^65 + 2728 * q^67 - 1545 * q^68 - 4609 * q^70 - 3322 * q^71 - 188 * q^73 - 1111 * q^74 - 3134 * q^76 - 556 * q^77 - 462 * q^79 - 6076 * q^80 - 7965 * q^82 - 4604 * q^83 - 852 * q^85 - 549 * q^86 - 1127 * q^88 - 6742 * q^89 + 1390 * q^91 - 1802 * q^92 - 2796 * q^94 - 448 * q^95 - 1322 * q^97 - 1000 * 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 $

$2$	−5.59453	−1.97797	−0.988983	−	0.148032i	$-0.952706\pi$
$2$	−5.59453	−1.97797	−0.988983	+	0.148032i	$0.952706\pi$
$3$	0	0
$3$	0	0
$4$	23.2988	2.91235
$5$	−16.4914	−1.47504	−0.737518	−	0.675328i	$-0.764002\pi$
$5$	−16.4914	−1.47504	−0.737518	+	0.675328i	$0.764002\pi$
$6$	0	0
$7$	−10.5009	−0.566994	−0.283497	−	0.958973i	$-0.591495\pi$
$7$	−10.5009	−0.566994	−0.283497	+	0.958973i	$0.591495\pi$
$8$	−85.5894	−3.78255
$9$	0	0
$10$	92.2616	2.91757
$11$	−4.40880	−0.120846	−0.0604228	−	0.998173i	$-0.519245\pi$
$11$	−4.40880	−0.120846	−0.0604228	+	0.998173i	$0.519245\pi$
$12$	0	0
$13$	−64.7884	−1.38224	−0.691118	−	0.722742i	$-0.742882\pi$
$13$	−64.7884	−1.38224	−0.691118	+	0.722742i	$0.742882\pi$
$14$	58.7475	1.12150
$15$	0	0
$16$	292.442	4.56941
$17$	−7.65474	−0.109209	−0.0546043	−	0.998508i	$-0.517390\pi$
$17$	−7.65474	−0.109209	−0.0546043	+	0.998508i	$0.517390\pi$
$18$	0	0
$19$	−84.7450	−1.02325	−0.511627	−	0.859208i	$-0.670957\pi$
$19$	−84.7450	−1.02325	−0.511627	+	0.859208i	$0.670957\pi$
$20$	−384.229	−4.29581
$21$	0	0
$22$	24.6651	0.239029
$23$	145.624	1.32020	0.660101	−	0.751177i	$-0.270514\pi$
$23$	145.624	1.32020	0.660101	+	0.751177i	$0.270514\pi$
$24$	0	0
$25$	146.966	1.17573
$26$	362.461	2.73402
$27$	0	0
$28$	−244.658	−1.65128
$29$	−113.139	−0.724461	−0.362231	−	0.932088i	$-0.617985\pi$
$29$	−113.139	−0.724461	−0.362231	+	0.932088i	$0.617985\pi$
$30$	0	0
$31$	−249.383	−1.44486	−0.722428	−	0.691446i	$-0.756974\pi$
$31$	−249.383	−1.44486	−0.722428	+	0.691446i	$0.756974\pi$
$32$	−951.362	−5.25558
$33$	0	0
$34$	42.8247	0.216011
$35$	173.174	0.836337
$36$	0	0
$37$	95.9890	0.426500	0.213250	−	0.976998i	$-0.431595\pi$
$37$	95.9890	0.426500	0.213250	+	0.976998i	$0.431595\pi$
$38$	474.108	2.02396
$39$	0	0
$40$	1411.49	5.57940
$41$	391.208	1.49016	0.745079	−	0.666977i	$-0.232412\pi$
$41$	391.208	1.49016	0.745079	+	0.666977i	$0.232412\pi$
$42$	0	0
$43$	200.408	0.710742	0.355371	−	0.934725i	$-0.384355\pi$
$43$	200.408	0.710742	0.355371	+	0.934725i	$0.384355\pi$
$44$	−102.720	−0.351944
$45$	0	0
$46$	−814.696	−2.61131
$47$	−392.357	−1.21768	−0.608841	−	0.793292i	$-0.708365\pi$
$47$	−392.357	−1.21768	−0.608841	+	0.793292i	$0.708365\pi$
$48$	0	0
$49$	−232.731	−0.678517
$50$	−822.206	−2.32555
$51$	0	0
$52$	−1509.49	−4.02555
$53$	614.057	1.59146	0.795729	−	0.605653i	$-0.207088\pi$
$53$	614.057	1.59146	0.795729	+	0.605653i	$0.207088\pi$
$54$	0	0
$55$	72.7072	0.178252
$56$	898.765	2.14469
$57$	0	0
$58$	632.959	1.43296
$59$	104.990	0.231669	0.115835	−	0.993269i	$-0.463046\pi$
$59$	104.990	0.231669	0.115835	+	0.993269i	$0.463046\pi$
$60$	0	0
$61$	−218.526	−0.458678	−0.229339	−	0.973347i	$-0.573656\pi$
$61$	−218.526	−0.458678	−0.229339	+	0.973347i	$0.573656\pi$
$62$	1395.18	2.85788
$63$	0	0
$64$	2982.89	5.82595
$65$	1068.45	2.03885
$66$	0	0
$67$	1027.44	1.87347	0.936733	−	0.350044i	$-0.113833\pi$
$67$	1027.44	1.87347	0.936733	+	0.350044i	$0.113833\pi$
$68$	−178.346	−0.318053
$69$	0	0
$70$	−968.828	−1.65425
$71$	−345.643	−0.577750	−0.288875	−	0.957367i	$-0.593281\pi$
$71$	−345.643	−0.577750	−0.288875	+	0.957367i	$0.593281\pi$
$72$	0	0
$73$	705.393	1.13096	0.565479	−	0.824762i	$-0.308691\pi$
$73$	705.393	1.13096	0.565479	+	0.824762i	$0.308691\pi$
$74$	−537.013	−0.843602
$75$	0	0
$76$	−1974.45	−2.98007
$77$	46.2963	0.0685188
$78$	0	0
$79$	813.898	1.15912	0.579561	−	0.814929i	$-0.303224\pi$
$79$	813.898	1.15912	0.579561	+	0.814929i	$0.303224\pi$
$80$	−4822.78	−6.74004
$81$	0	0
$82$	−2188.63	−2.94748
$83$	985.181	1.30286	0.651431	−	0.758707i	$-0.274169\pi$
$83$	985.181	1.30286	0.651431	+	0.758707i	$0.274169\pi$
$84$	0	0
$85$	126.237	0.161087
$86$	−1121.19	−1.40582
$87$	0	0
$88$	377.346	0.457105
$89$	−1483.05	−1.76633	−0.883164	−	0.469065i	$-0.844591\pi$
$89$	−1483.05	−1.76633	−0.883164	+	0.469065i	$0.844591\pi$
$90$	0	0
$91$	680.336	0.783720
$92$	3392.85	3.84488
$93$	0	0
$94$	2195.05	2.40853
$95$	1397.56	1.50934
$96$	0	0
$97$	−932.272	−0.975854	−0.487927	−	0.872884i	$-0.662247\pi$
$97$	−932.272	−0.975854	−0.487927	+	0.872884i	$0.662247\pi$
$98$	1302.02	1.34208
$99$	0	0
$100$	3424.13	3.42413
$101$	479.894	0.472784	0.236392	−	0.971658i	$-0.424035\pi$
$101$	479.894	0.472784	0.236392	+	0.971658i	$0.424035\pi$
$102$	0	0
$103$	46.9757	0.0449384	0.0224692	−	0.999748i	$-0.492847\pi$
$103$	46.9757	0.0449384	0.0224692	+	0.999748i	$0.492847\pi$
$104$	5545.20	5.22838
$105$	0	0
$106$	−3435.36	−3.14785
$107$	−1334.27	−1.20550	−0.602752	−	0.797929i	$-0.705929\pi$
$107$	−1334.27	−1.20550	−0.602752	+	0.797929i	$0.705929\pi$
$108$	0	0
$109$	−73.9090	−0.0649468	−0.0324734	−	0.999473i	$-0.510338\pi$
$109$	−73.9090	−0.0649468	−0.0324734	+	0.999473i	$0.510338\pi$
$110$	−406.763	−0.352575
$111$	0	0
$112$	−3070.90	−2.59083
$113$	136.597	0.113717	0.0568584	−	0.998382i	$-0.481892\pi$
$113$	136.597	0.113717	0.0568584	+	0.998382i	$0.481892\pi$
$114$	0	0
$115$	−2401.54	−1.94734
$116$	−2636.00	−2.10988
$117$	0	0
$118$	−587.368	−0.458234
$119$	80.3815	0.0619207
$120$	0	0
$121$	−1311.56	−0.985396
$122$	1222.55	0.907249
$123$	0	0
$124$	−5810.32	−4.20792
$125$	−362.252	−0.259206
$126$	0	0
$127$	1289.26	0.900812	0.450406	−	0.892824i	$-0.351279\pi$
$127$	1289.26	0.900812	0.450406	+	0.892824i	$0.351279\pi$
$128$	−9076.95	−6.26794
$129$	0	0
$130$	−5977.48	−4.03277
$131$	−2135.16	−1.42404	−0.712022	−	0.702157i	$-0.752221\pi$
$131$	−2135.16	−1.42404	−0.712022	+	0.702157i	$0.752221\pi$
$132$	0	0
$133$	889.897	0.580180
$134$	−5748.07	−3.70565
$135$	0	0
$136$	655.165	0.413088
$137$	114.671	0.0715108	0.0357554	−	0.999361i	$-0.488616\pi$
$137$	114.671	0.0715108	0.0357554	+	0.999361i	$0.488616\pi$
$138$	0	0
$139$	1589.58	0.969976	0.484988	−	0.874521i	$-0.338824\pi$
$139$	1589.58	0.969976	0.484988	+	0.874521i	$0.338824\pi$
$140$	4034.75	2.43570
$141$	0	0
$142$	1933.71	1.14277
$143$	285.639	0.167037
$144$	0	0
$145$	1865.82	1.06861
$146$	−3946.34	−2.23700
$147$	0	0
$148$	2236.43	1.24212
$149$	62.2139	0.0342065	0.0171032	−	0.999854i	$-0.494556\pi$
$149$	62.2139	0.0342065	0.0171032	+	0.999854i	$0.494556\pi$
$150$	0	0
$151$	2298.75	1.23887	0.619436	−	0.785047i	$-0.287361\pi$
$151$	2298.75	1.23887	0.619436	+	0.785047i	$0.287361\pi$
$152$	7253.27	3.87051
$153$	0	0
$154$	−259.006	−0.135528
$155$	4112.68	2.13121
$156$	0	0
$157$	−1528.52	−0.777000	−0.388500	−	0.921449i	$-0.627007\pi$
$157$	−1528.52	−0.777000	−0.388500	+	0.921449i	$0.627007\pi$
$158$	−4553.38	−2.29270
$159$	0	0
$160$	15689.3	7.75217
$161$	−1529.18	−0.748547
$162$	0	0
$163$	171.423	0.0823736	0.0411868	−	0.999151i	$-0.486886\pi$
$163$	171.423	0.0823736	0.0411868	+	0.999151i	$0.486886\pi$
$164$	9114.67	4.33985
$165$	0	0
$166$	−5511.62	−2.57702
$167$	1147.92	0.531910	0.265955	−	0.963985i	$-0.414313\pi$
$167$	1147.92	0.531910	0.265955	+	0.963985i	$0.414313\pi$
$168$	0	0
$169$	2000.54	0.910577
$170$	−706.239	−0.318624
$171$	0	0
$172$	4669.26	2.06993
$173$	1092.88	0.480289	0.240145	−	0.970737i	$-0.422805\pi$
$173$	1092.88	0.480289	0.240145	+	0.970737i	$0.422805\pi$
$174$	0	0
$175$	−1543.27	−0.666632
$176$	−1289.32	−0.552194
$177$	0	0
$178$	8296.98	3.49373
$179$	2224.83	0.929004	0.464502	−	0.885572i	$-0.346233\pi$
$179$	2224.83	0.929004	0.464502	+	0.885572i	$0.346233\pi$
$180$	0	0
$181$	3822.94	1.56993	0.784964	−	0.619541i	$-0.212681\pi$
$181$	3822.94	1.56993	0.784964	+	0.619541i	$0.212681\pi$
$182$	−3806.16	−1.55017
$183$	0	0
$184$	−12463.8	−4.99373
$185$	−1582.99	−0.629102
$186$	0	0
$187$	33.7482	0.0131974
$188$	−9141.42	−3.54631
$189$	0	0
$190$	−7818.71	−2.98541
$191$	1404.49	0.532070	0.266035	−	0.963963i	$-0.414286\pi$
$191$	1404.49	0.532070	0.266035	+	0.963963i	$0.414286\pi$
$192$	0	0
$193$	3208.89	1.19679	0.598397	−	0.801200i	$-0.295805\pi$
$193$	3208.89	1.19679	0.598397	+	0.801200i	$0.295805\pi$
$194$	5215.62	1.93021
$195$	0	0
$196$	−5422.36	−1.97608
$197$	−429.982	−0.155507	−0.0777537	−	0.996973i	$-0.524775\pi$
$197$	−429.982	−0.155507	−0.0777537	+	0.996973i	$0.524775\pi$
$198$	0	0
$199$	−739.825	−0.263542	−0.131771	−	0.991280i	$-0.542066\pi$
$199$	−739.825	−0.263542	−0.131771	+	0.991280i	$0.542066\pi$
$200$	−12578.7	−4.44726
$201$	0	0
$202$	−2684.78	−0.935151
$203$	1188.06	0.410766
$204$	0	0
$205$	−6451.57	−2.19803
$206$	−262.807	−0.0888865
$207$	0	0
$208$	−18946.9	−6.31601
$209$	373.623	0.123656
$210$	0	0
$211$	−516.163	−0.168408	−0.0842040	−	0.996449i	$-0.526835\pi$
$211$	−516.163	−0.168408	−0.0842040	+	0.996449i	$0.526835\pi$
$212$	14306.8	4.63487
$213$	0	0
$214$	7464.62	2.38444
$215$	−3305.00	−1.04837
$216$	0	0
$217$	2618.74	0.819225
$218$	413.486	0.128462
$219$	0	0
$220$	1693.99	0.519130
$221$	495.938	0.150952
$222$	0	0
$223$	−2817.87	−0.846183	−0.423091	−	0.906087i	$-0.639055\pi$
$223$	−2817.87	−0.846183	−0.423091	+	0.906087i	$0.639055\pi$
$224$	9990.15	2.97989
$225$	0	0
$226$	−764.198	−0.224928
$227$	102.810	0.0300607	0.0150303	−	0.999887i	$-0.495216\pi$
$227$	102.810	0.0300607	0.0150303	+	0.999887i	$0.495216\pi$
$228$	0	0
$229$	832.288	0.240171	0.120085	−	0.992764i	$-0.461683\pi$
$229$	832.288	0.240171	0.120085	+	0.992764i	$0.461683\pi$
$230$	13435.5	3.85178
$231$	0	0
$232$	9683.50	2.74031
$233$	6241.12	1.75480	0.877402	−	0.479755i	$-0.159274\pi$
$233$	6241.12	1.75480	0.877402	+	0.479755i	$0.159274\pi$
$234$	0	0
$235$	6470.51	1.79612
$236$	2446.13	0.674701
$237$	0	0
$238$	−449.697	−0.122477
$239$	−239.000	−0.0646846
$239$	−239.000	−0.0646846
$240$	0	0
$241$	−4792.89	−1.28107	−0.640534	−	0.767930i	$-0.721287\pi$
$241$	−4792.89	−1.28107	−0.640534	+	0.767930i	$0.721287\pi$
$242$	7337.58	1.94908
$243$	0	0
$244$	−5091.38	−1.33583
$245$	3838.07	1.00084
$246$	0	0
$247$	5490.49	1.41438
$248$	21344.6	5.46525
$249$	0	0
$250$	2026.63	0.512701
$251$	−6350.33	−1.59693	−0.798465	−	0.602041i	$-0.794354\pi$
$251$	−6350.33	−1.59693	−0.798465	+	0.602041i	$0.794354\pi$
$252$	0	0
$253$	−642.025	−0.159541
$254$	−7212.79	−1.78177
$255$	0	0
$256$	26918.2	6.57182
$257$	635.383	0.154218	0.0771092	−	0.997023i	$-0.475431\pi$
$257$	635.383	0.154218	0.0771092	+	0.997023i	$0.475431\pi$
$258$	0	0
$259$	−1007.97	−0.241823
$260$	24893.6	5.93783
$261$	0	0
$262$	11945.2	2.81671
$263$	7718.20	1.80960	0.904800	−	0.425837i	$-0.140020\pi$
$263$	7718.20	1.80960	0.904800	+	0.425837i	$0.140020\pi$
$264$	0	0
$265$	−10126.7	−2.34746
$266$	−4978.56	−1.14757
$267$	0	0
$268$	23938.2	5.45618
$269$	−1216.31	−0.275687	−0.137844	−	0.990454i	$-0.544017\pi$
$269$	−1216.31	−0.275687	−0.137844	+	0.990454i	$0.544017\pi$
$270$	0	0
$271$	913.818	0.204836	0.102418	−	0.994741i	$-0.467342\pi$
$271$	913.818	0.204836	0.102418	+	0.994741i	$0.467342\pi$
$272$	−2238.57	−0.499019
$273$	0	0
$274$	−641.529	−0.141446
$275$	−647.944	−0.142082
$276$	0	0
$277$	7171.71	1.55562	0.777809	−	0.628500i	$-0.216331\pi$
$277$	7171.71	1.55562	0.777809	+	0.628500i	$0.216331\pi$
$278$	−8892.97	−1.91858
$279$	0	0
$280$	−14821.9	−3.16349
$281$	3380.63	0.717692	0.358846	−	0.933397i	$-0.383170\pi$
$281$	3380.63	0.717692	0.358846	+	0.933397i	$0.383170\pi$
$282$	0	0
$283$	−6581.12	−1.38236	−0.691178	−	0.722685i	$-0.742908\pi$
$283$	−6581.12	−1.38236	−0.691178	+	0.722685i	$0.742908\pi$
$284$	−8053.04	−1.68261
$285$	0	0
$286$	−1598.02	−0.330394
$287$	−4108.03	−0.844911
$288$	0	0
$289$	−4854.40	−0.988073
$290$	−10438.4	−2.11367
$291$	0	0
$292$	16434.8	3.29374
$293$	7076.95	1.41106	0.705528	−	0.708682i	$-0.250710\pi$
$293$	7076.95	1.41106	0.705528	+	0.708682i	$0.250710\pi$
$294$	0	0
$295$	−1731.43	−0.341720
$296$	−8215.64	−1.61326
$297$	0	0
$298$	−348.058	−0.0676592
$299$	−9434.72	−1.82483
$300$	0	0
$301$	−2104.46	−0.402987
$302$	−12860.4	−2.45044
$303$	0	0
$304$	−24783.0	−4.67567
$305$	3603.79	0.676566
$306$	0	0
$307$	281.442	0.0523217	0.0261608	−	0.999658i	$-0.491672\pi$
$307$	281.442	0.0523217	0.0261608	+	0.999658i	$0.491672\pi$
$308$	1078.65	0.199551
$309$	0	0
$310$	−23008.5	−4.21547
$311$	4634.16	0.844949	0.422475	−	0.906375i	$-0.361162\pi$
$311$	4634.16	0.844949	0.422475	+	0.906375i	$0.361162\pi$
$312$	0	0
$313$	937.982	0.169386	0.0846931	−	0.996407i	$-0.473009\pi$
$313$	937.982	0.169386	0.0846931	+	0.996407i	$0.473009\pi$
$314$	8551.34	1.53688
$315$	0	0
$316$	18962.8	3.37576
$317$	10357.0	1.83503	0.917516	−	0.397700i	$-0.130191\pi$
$317$	10357.0	1.83503	0.917516	+	0.397700i	$0.130191\pi$
$318$	0	0
$319$	498.807	0.0875480
$320$	−49192.0	−8.59348
$321$	0	0
$322$	8555.03	1.48060
$323$	648.701	0.111748
$324$	0	0
$325$	−9521.70	−1.62514
$326$	−959.032	−0.162932
$327$	0	0
$328$	−33483.3	−5.63660
$329$	4120.09	0.690419
$330$	0	0
$331$	−4348.43	−0.722088	−0.361044	−	0.932549i	$-0.617580\pi$
$331$	−4348.43	−0.722088	−0.361044	+	0.932549i	$0.617580\pi$
$332$	22953.5	3.79439
$333$	0	0
$334$	−6422.09	−1.05210
$335$	−16944.0	−2.76343
$336$	0	0
$337$	909.879	0.147075	0.0735375	−	0.997292i	$-0.476571\pi$
$337$	909.879	0.147075	0.0735375	+	0.997292i	$0.476571\pi$
$338$	−11192.1	−1.80109
$339$	0	0
$340$	2941.17	0.469140
$341$	1099.48	0.174605
$342$	0	0
$343$	6045.69	0.951710
$344$	−17152.8	−2.68842
$345$	0	0
$346$	−6114.14	−0.949995
$347$	−4889.20	−0.756386	−0.378193	−	0.925727i	$-0.623454\pi$
$347$	−4889.20	−0.756386	−0.378193	+	0.925727i	$0.623454\pi$
$348$	0	0
$349$	2738.75	0.420062	0.210031	−	0.977695i	$-0.432643\pi$
$349$	2738.75	0.420062	0.210031	+	0.977695i	$0.432643\pi$
$350$	8633.89	1.31857
$351$	0	0
$352$	4194.36	0.635115
$353$	−9458.10	−1.42607	−0.713037	−	0.701127i	$-0.752681\pi$
$353$	−9458.10	−1.42607	−0.713037	+	0.701127i	$0.752681\pi$
$354$	0	0
$355$	5700.13	0.852201
$356$	−34553.3	−5.14416
$357$	0	0
$358$	−12446.9	−1.83754
$359$	−8799.52	−1.29365	−0.646826	−	0.762638i	$-0.723904\pi$
$359$	−8799.52	−1.29365	−0.646826	+	0.762638i	$0.723904\pi$
$360$	0	0
$361$	322.712	0.0470495
$362$	−21387.6	−3.10526
$363$	0	0
$364$	15851.0	2.28246
$365$	−11632.9	−1.66820
$366$	0	0
$367$	6214.38	0.883891	0.441946	−	0.897042i	$-0.354288\pi$
$367$	6214.38	0.883891	0.441946	+	0.897042i	$0.354288\pi$
$368$	42586.5	6.03254
$369$	0	0
$370$	8856.10	1.24434
$371$	−6448.14	−0.902347
$372$	0	0
$373$	−2127.14	−0.295280	−0.147640	−	0.989041i	$-0.547168\pi$
$373$	−2127.14	−0.295280	−0.147640	+	0.989041i	$0.547168\pi$
$374$	−188.805	−0.0261040
$375$	0	0
$376$	33581.6	4.60595
$377$	7330.09	1.00138
$378$	0	0
$379$	−12161.3	−1.64824	−0.824121	−	0.566414i	$-0.808330\pi$
$379$	−12161.3	−1.64824	−0.824121	+	0.566414i	$0.808330\pi$
$380$	32561.5	4.39571
$381$	0	0
$382$	−7857.47	−1.05242
$383$	−4412.21	−0.588652	−0.294326	−	0.955705i	$-0.595095\pi$
$383$	−4412.21	−0.588652	−0.294326	+	0.955705i	$0.595095\pi$
$384$	0	0
$385$	−763.490	−0.101068
$386$	−17952.2	−2.36722
$387$	0	0
$388$	−21720.8	−2.84203
$389$	7047.64	0.918585	0.459292	−	0.888285i	$-0.348103\pi$
$389$	7047.64	0.918585	0.459292	+	0.888285i	$0.348103\pi$
$390$	0	0
$391$	−1114.71	−0.144177
$392$	19919.3	2.56653
$393$	0	0
$394$	2405.55	0.307588
$395$	−13422.3	−1.70975
$396$	0	0
$397$	−13089.0	−1.65471	−0.827353	−	0.561683i	$-0.810154\pi$
$397$	−13089.0	−1.65471	−0.827353	+	0.561683i	$0.810154\pi$
$398$	4138.97	0.521276
$399$	0	0
$400$	42979.1	5.37239
$401$	−13604.2	−1.69417	−0.847085	−	0.531457i	$-0.821645\pi$
$401$	−13604.2	−1.69417	−0.847085	+	0.531457i	$0.821645\pi$
$402$	0	0
$403$	16157.1	1.99713
$404$	11180.9	1.37691
$405$	0	0
$406$	−6646.63	−0.812480
$407$	−423.196	−0.0515407
$408$	0	0
$409$	5192.48	0.627755	0.313877	−	0.949464i	$-0.398372\pi$
$409$	5192.48	0.627755	0.313877	+	0.949464i	$0.398372\pi$
$410$	36093.5	4.34763
$411$	0	0
$412$	1094.48	0.130876
$413$	−1102.48	−0.131355
$414$	0	0
$415$	−16247.0	−1.92177
$416$	61637.3	7.26446
$417$	0	0
$418$	−2090.25	−0.244587
$419$	−16171.1	−1.88547	−0.942733	−	0.333548i	$-0.891754\pi$
$419$	−16171.1	−1.88547	−0.942733	+	0.333548i	$0.891754\pi$
$420$	0	0
$421$	−2991.06	−0.346260	−0.173130	−	0.984899i	$-0.555388\pi$
$421$	−2991.06	−0.346260	−0.173130	+	0.984899i	$0.555388\pi$
$422$	2887.69	0.333105
$423$	0	0
$424$	−52556.8	−6.01977
$425$	−1124.99	−0.128400
$426$	0	0
$427$	2294.71	0.260068
$428$	−31086.9	−3.51084
$429$	0	0
$430$	18489.9	2.07364
$431$	2498.72	0.279256	0.139628	−	0.990204i	$-0.455409\pi$
$431$	2498.72	0.279256	0.139628	+	0.990204i	$0.455409\pi$
$432$	0	0
$433$	8402.81	0.932595	0.466297	−	0.884628i	$-0.345588\pi$
$433$	8402.81	0.932595	0.466297	+	0.884628i	$0.345588\pi$
$434$	−14650.6	−1.62040
$435$	0	0
$436$	−1721.99	−0.189147
$437$	−12340.9	−1.35090
$438$	0	0
$439$	2876.16	0.312691	0.156346	−	0.987702i	$-0.450029\pi$
$439$	2876.16	0.312691	0.156346	+	0.987702i	$0.450029\pi$
$440$	−6222.97	−0.674246
$441$	0	0
$442$	−2774.54	−0.298578
$443$	−9100.17	−0.975986	−0.487993	−	0.872847i	$-0.662271\pi$
$443$	−9100.17	−0.975986	−0.487993	+	0.872847i	$0.662271\pi$
$444$	0	0
$445$	24457.6	2.60540
$446$	15764.7	1.67372
$447$	0	0
$448$	−31323.0	−3.30328
$449$	−13503.1	−1.41927	−0.709636	−	0.704569i	$-0.751141\pi$
$449$	−13503.1	−1.41927	−0.709636	+	0.704569i	$0.751141\pi$
$450$	0	0
$451$	−1724.76	−0.180079
$452$	3182.55	0.331183
$453$	0	0
$454$	−575.176	−0.0594589
$455$	−11219.7	−1.15602
$456$	0	0
$457$	−16456.4	−1.68446	−0.842229	−	0.539120i	$-0.818757\pi$
$457$	−16456.4	−1.68446	−0.842229	+	0.539120i	$0.818757\pi$
$458$	−4656.26	−0.475049
$459$	0	0
$460$	−55952.8	−5.67134
$461$	15740.1	1.59022	0.795110	−	0.606465i	$-0.207413\pi$
$461$	15740.1	1.59022	0.795110	+	0.606465i	$0.207413\pi$
$462$	0	0
$463$	−17654.4	−1.77208	−0.886038	−	0.463612i	$-0.846553\pi$
$463$	−17654.4	−1.77208	−0.886038	+	0.463612i	$0.846553\pi$
$464$	−33086.6	−3.31036
$465$	0	0
$466$	−34916.1	−3.47094
$467$	−8443.75	−0.836682	−0.418341	−	0.908290i	$-0.637388\pi$
$467$	−8443.75	−0.836682	−0.418341	+	0.908290i	$0.637388\pi$
$468$	0	0
$469$	−10789.1	−1.06225
$470$	−36199.4	−3.55267
$471$	0	0
$472$	−8986.00	−0.876301
$473$	−883.558	−0.0858901
$474$	0	0
$475$	−12454.6	−1.20307
$476$	1872.79	0.180335
$477$	0	0
$478$	1337.09	0.127944
$479$	14429.0	1.37636	0.688182	−	0.725538i	$-0.258409\pi$
$479$	14429.0	1.37636	0.688182	+	0.725538i	$0.258409\pi$
$480$	0	0
$481$	−6218.98	−0.589524
$482$	26814.0	2.53391
$483$	0	0
$484$	−30557.8	−2.86981
$485$	15374.5	1.43942
$486$	0	0
$487$	9285.21	0.863970	0.431985	−	0.901881i	$-0.357813\pi$
$487$	9285.21	0.863970	0.431985	+	0.901881i	$0.357813\pi$
$488$	18703.5	1.73497
$489$	0	0
$490$	−21472.2	−1.97962
$491$	−1349.85	−0.124069	−0.0620344	−	0.998074i	$-0.519759\pi$
$491$	−1349.85	−0.124069	−0.0620344	+	0.998074i	$0.519759\pi$
$492$	0	0
$493$	866.049	0.0791175
$494$	−30716.7	−2.79759
$495$	0	0
$496$	−72930.2	−6.60214
$497$	3629.55	0.327581
$498$	0	0
$499$	10490.1	0.941088	0.470544	−	0.882377i	$-0.344058\pi$
$499$	10490.1	0.941088	0.470544	+	0.882377i	$0.344058\pi$
$500$	−8440.02	−0.754898
$501$	0	0
$502$	35527.1	3.15867
$503$	12923.4	1.14558	0.572788	−	0.819704i	$-0.305862\pi$
$503$	12923.4	1.14558	0.572788	+	0.819704i	$0.305862\pi$
$504$	0	0
$505$	−7914.12	−0.697373
$506$	3591.83	0.315566
$507$	0	0
$508$	30038.1	2.62347
$509$	−10543.1	−0.918104	−0.459052	−	0.888409i	$-0.651811\pi$
$509$	−10543.1	−0.918104	−0.459052	+	0.888409i	$0.651811\pi$
$510$	0	0
$511$	−7407.25	−0.641247
$512$	−77978.9	−6.73089
$513$	0	0
$514$	−3554.67	−0.305038
$515$	−774.694	−0.0662857
$516$	0	0
$517$	1729.82	0.147152
$518$	5639.12	0.478318
$519$	0	0
$520$	−91448.1	−7.71205
$521$	−1006.15	−0.0846074	−0.0423037	−	0.999105i	$-0.513470\pi$
$521$	−1006.15	−0.0846074	−0.0423037	+	0.999105i	$0.513470\pi$
$522$	0	0
$523$	8828.89	0.738165	0.369082	−	0.929397i	$-0.379672\pi$
$523$	8828.89	0.738165	0.369082	+	0.929397i	$0.379672\pi$
$524$	−49746.6	−4.14731
$525$	0	0
$526$	−43179.7	−3.57933
$527$	1908.96	0.157791
$528$	0	0
$529$	9039.24	0.742931
$530$	56653.9	4.64318
$531$	0	0
$532$	20733.5	1.68968
$533$	−25345.8	−2.05975
$534$	0	0
$535$	22004.0	1.77816
$536$	−87938.3	−7.08649
$537$	0	0
$538$	6804.70	0.545300
$539$	1026.07	0.0819959
$540$	0	0
$541$	20069.4	1.59492	0.797460	−	0.603372i	$-0.206176\pi$
$541$	20069.4	1.59492	0.797460	+	0.603372i	$0.206176\pi$
$542$	−5112.38	−0.405158
$543$	0	0
$544$	7282.43	0.573955
$545$	1218.86	0.0957987
$546$	0	0
$547$	−4946.56	−0.386654	−0.193327	−	0.981134i	$-0.561928\pi$
$547$	−4946.56	−0.386654	−0.193327	+	0.981134i	$0.561928\pi$
$548$	2671.69	0.208264
$549$	0	0
$550$	3624.94	0.281033
$551$	9587.96	0.741308
$552$	0	0
$553$	−8546.65	−0.657216
$554$	−40122.3	−3.07696
$555$	0	0
$556$	37035.3	2.82491
$557$	8033.46	0.611111	0.305555	−	0.952174i	$-0.401158\pi$
$557$	8033.46	0.611111	0.305555	+	0.952174i	$0.401158\pi$
$558$	0	0
$559$	−12984.1	−0.982413
$560$	50643.5	3.82157
$561$	0	0
$562$	−18913.0	−1.41957
$563$	−2628.38	−0.196755	−0.0983774	−	0.995149i	$-0.531365\pi$
$563$	−2628.38	−0.196755	−0.0983774	+	0.995149i	$0.531365\pi$
$564$	0	0
$565$	−2252.68	−0.167736
$566$	36818.3	2.73425
$567$	0	0
$568$	29583.3	2.18537
$569$	−1517.90	−0.111835	−0.0559173	−	0.998435i	$-0.517808\pi$
$569$	−1517.90	−0.111835	−0.0559173	+	0.998435i	$0.517808\pi$
$570$	0	0
$571$	−14148.4	−1.03694	−0.518470	−	0.855096i	$-0.673498\pi$
$571$	−14148.4	−1.03694	−0.518470	+	0.855096i	$0.673498\pi$
$572$	6655.04	0.486470
$573$	0	0
$574$	22982.5	1.67120
$575$	21401.7	1.55220
$576$	0	0
$577$	1780.86	0.128489	0.0642447	−	0.997934i	$-0.479536\pi$
$577$	1780.86	0.128489	0.0642447	+	0.997934i	$0.479536\pi$
$578$	27158.1	1.95437
$579$	0	0
$580$	43471.3	3.11215
$581$	−10345.3	−0.738716
$582$	0	0
$583$	−2707.25	−0.192321
$584$	−60374.2	−4.27791
$585$	0	0
$586$	−39592.2	−2.79102
$587$	−23685.2	−1.66541	−0.832703	−	0.553720i	$-0.813208\pi$
$587$	−23685.2	−1.66541	−0.832703	+	0.553720i	$0.813208\pi$
$588$	0	0
$589$	21134.0	1.47846
$590$	9686.51	0.675911
$591$	0	0
$592$	28071.3	1.94885
$593$	3349.18	0.231930	0.115965	−	0.993253i	$-0.463004\pi$
$593$	3349.18	0.231930	0.115965	+	0.993253i	$0.463004\pi$
$594$	0	0
$595$	−1325.60	−0.0913352
$596$	1449.51	0.0996211
$597$	0	0
$598$	52782.8	3.60945
$599$	14390.2	0.981582	0.490791	−	0.871277i	$-0.336708\pi$
$599$	14390.2	0.981582	0.490791	+	0.871277i	$0.336708\pi$
$600$	0	0
$601$	12665.4	0.859618	0.429809	−	0.902920i	$-0.358581\pi$
$601$	12665.4	0.859618	0.429809	+	0.902920i	$0.358581\pi$
$602$	11773.5	0.797094
$603$	0	0
$604$	53558.0	3.60802
$605$	21629.5	1.45349
$606$	0	0
$607$	−24017.4	−1.60599	−0.802994	−	0.595987i	$-0.796761\pi$
$607$	−24017.4	−1.60599	−0.802994	+	0.595987i	$0.796761\pi$
$608$	80623.2	5.37780
$609$	0	0
$610$	−20161.5	−1.33822
$611$	25420.2	1.68313
$612$	0	0
$613$	16998.9	1.12003	0.560016	−	0.828482i	$-0.310795\pi$
$613$	16998.9	1.12003	0.560016	+	0.828482i	$0.310795\pi$
$614$	−1574.54	−0.103490
$615$	0	0
$616$	−3962.47	−0.259176
$617$	−25333.5	−1.65298	−0.826490	−	0.562951i	$-0.809666\pi$
$617$	−25333.5	−1.65298	−0.826490	+	0.562951i	$0.809666\pi$
$618$	0	0
$619$	1379.37	0.0895665	0.0447832	−	0.998997i	$-0.485740\pi$
$619$	1379.37	0.0895665	0.0447832	+	0.998997i	$0.485740\pi$
$620$	95820.3	6.20683
$621$	0	0
$622$	−25926.0	−1.67128
$623$	15573.4	1.00150
$624$	0	0
$625$	−12396.7	−0.793390
$626$	−5247.57	−0.335040
$627$	0	0
$628$	−35612.6	−2.26289
$629$	−734.771	−0.0465775
$630$	0	0
$631$	25706.3	1.62179	0.810897	−	0.585188i	$-0.198979\pi$
$631$	25706.3	1.62179	0.810897	+	0.585188i	$0.198979\pi$
$632$	−69661.0	−4.38444
$633$	0	0
$634$	−57942.3	−3.62963
$635$	−21261.6	−1.32873
$636$	0	0
$637$	15078.3	0.937871
$638$	−2790.59	−0.173167
$639$	0	0
$640$	149692.	9.24544
$641$	−19384.4	−1.19444	−0.597222	−	0.802076i	$-0.703729\pi$
$641$	−19384.4	−1.19444	−0.597222	+	0.802076i	$0.703729\pi$
$642$	0	0
$643$	−6363.87	−0.390306	−0.195153	−	0.980773i	$-0.562520\pi$
$643$	−6363.87	−0.390306	−0.195153	+	0.980773i	$0.562520\pi$
$644$	−35627.9	−2.18003
$645$	0	0
$646$	−3629.18	−0.221034
$647$	−31505.5	−1.91439	−0.957194	−	0.289448i	$-0.906528\pi$
$647$	−31505.5	−1.91439	−0.957194	+	0.289448i	$0.906528\pi$
$648$	0	0
$649$	−462.878	−0.0279962
$650$	53269.4	3.21446
$651$	0	0
$652$	3993.95	0.239900
$653$	−7940.72	−0.475872	−0.237936	−	0.971281i	$-0.576471\pi$
$653$	−7940.72	−0.475872	−0.237936	+	0.971281i	$0.576471\pi$
$654$	0	0
$655$	35211.8	2.10052
$656$	114406.	6.80914
$657$	0	0
$658$	−23050.0	−1.36563
$659$	16195.5	0.957340	0.478670	−	0.877995i	$-0.341119\pi$
$659$	16195.5	0.957340	0.478670	+	0.877995i	$0.341119\pi$
$660$	0	0
$661$	−25863.9	−1.52192	−0.760961	−	0.648798i	$-0.775272\pi$
$661$	−25863.9	−1.52192	−0.760961	+	0.648798i	$0.775272\pi$
$662$	24327.4	1.42826
$663$	0	0
$664$	−84321.0	−4.92815
$665$	−14675.6	−0.855785
$666$	0	0
$667$	−16475.7	−0.956435
$668$	26745.2	1.54911
$669$	0	0
$670$	94793.6	5.46597
$671$	963.435	0.0554292
$672$	0	0
$673$	−5680.25	−0.325346	−0.162673	−	0.986680i	$-0.552011\pi$
$673$	−5680.25	−0.325346	−0.162673	+	0.986680i	$0.552011\pi$
$674$	−5090.34	−0.290909
$675$	0	0
$676$	46610.1	2.65192
$677$	−13257.1	−0.752604	−0.376302	−	0.926497i	$-0.622804\pi$
$677$	−13257.1	−0.752604	−0.376302	+	0.926497i	$0.622804\pi$
$678$	0	0
$679$	9789.68	0.553304
$680$	−10804.6	−0.609319
$681$	0	0
$682$	−6151.07	−0.345362
$683$	11979.6	0.671138	0.335569	−	0.942016i	$-0.391071\pi$
$683$	11979.6	0.671138	0.335569	+	0.942016i	$0.391071\pi$
$684$	0	0
$685$	−1891.08	−0.105481
$686$	−33822.8	−1.88245
$687$	0	0
$688$	58607.7	3.24767
$689$	−39783.8	−2.19977
$690$	0	0
$691$	2558.83	0.140872	0.0704361	−	0.997516i	$-0.477561\pi$
$691$	2558.83	0.140872	0.0704361	+	0.997516i	$0.477561\pi$
$692$	25462.7	1.39877
$693$	0	0
$694$	27352.8	1.49610
$695$	−26214.4	−1.43075
$696$	0	0
$697$	−2994.60	−0.162738
$698$	−15322.0	−0.830868
$699$	0	0
$700$	−35956.4	−1.94146
$701$	−20570.0	−1.10830	−0.554150	−	0.832417i	$-0.686956\pi$
$701$	−20570.0	−1.10830	−0.554150	+	0.832417i	$0.686956\pi$
$702$	0	0
$703$	−8134.59	−0.436418
$704$	−13150.9	−0.704041
$705$	0	0
$706$	52913.6	2.82072
$707$	−5039.31	−0.268066
$708$	0	0
$709$	15591.8	0.825901	0.412951	−	0.910753i	$-0.364498\pi$
$709$	15591.8	0.825901	0.412951	+	0.910753i	$0.364498\pi$
$710$	−31889.5	−1.68562
$711$	0	0
$712$	126934.	6.68123
$713$	−36316.1	−1.90750
$714$	0	0
$715$	−4710.58	−0.246386
$716$	51835.8	2.70558
$717$	0	0
$718$	49229.2	2.55880
$719$	−6558.63	−0.340189	−0.170094	−	0.985428i	$-0.554407\pi$
$719$	−6558.63	−0.340189	−0.170094	+	0.985428i	$0.554407\pi$
$720$	0	0
$721$	−493.286	−0.0254798
$722$	−1805.42	−0.0930622
$723$	0	0
$724$	89069.8	4.57218
$725$	−16627.6	−0.851770
$726$	0	0
$727$	38714.0	1.97500	0.987499	−	0.157626i	$-0.0503842\pi$
$727$	38714.0	1.97500	0.987499	+	0.157626i	$0.0503842\pi$
$728$	−58229.5	−2.96446
$729$	0	0
$730$	65080.7	3.29965
$731$	−1534.07	−0.0776192
$732$	0	0
$733$	590.331	0.0297467	0.0148734	−	0.999889i	$-0.495265\pi$
$733$	590.331	0.0297467	0.0148734	+	0.999889i	$0.495265\pi$
$734$	−34766.5	−1.74831
$735$	0	0
$736$	−138541.	−6.93843
$737$	−4529.79	−0.226400
$738$	0	0
$739$	−35859.1	−1.78498	−0.892488	−	0.451071i	$-0.851042\pi$
$739$	−35859.1	−1.78498	−0.892488	+	0.451071i	$0.851042\pi$
$740$	−36881.8	−1.83216
$741$	0	0
$742$	36074.3	1.78481
$743$	−24748.4	−1.22198	−0.610991	−	0.791638i	$-0.709229\pi$
$743$	−24748.4	−1.22198	−0.610991	+	0.791638i	$0.709229\pi$
$744$	0	0
$745$	−1025.99	−0.0504558
$746$	11900.4	0.584053
$747$	0	0
$748$	786.291	0.0384354
$749$	14011.0	0.683514
$750$	0	0
$751$	−15649.6	−0.760403	−0.380202	−	0.924904i	$-0.624145\pi$
$751$	−15649.6	−0.760403	−0.380202	+	0.924904i	$0.624145\pi$
$752$	−114742.	−5.56409
$753$	0	0
$754$	−41008.4	−1.98069
$755$	−37909.6	−1.82738
$756$	0	0
$757$	30325.5	1.45601	0.728003	−	0.685573i	$-0.240448\pi$
$757$	30325.5	1.45601	0.728003	+	0.685573i	$0.240448\pi$
$758$	68036.7	3.26016
$759$	0	0
$760$	−119617.	−5.70914
$761$	35662.2	1.69876	0.849378	−	0.527785i	$-0.176977\pi$
$761$	35662.2	1.69876	0.849378	+	0.527785i	$0.176977\pi$
$762$	0	0
$763$	776.110	0.0368244
$764$	32722.9	1.54957
$765$	0	0
$766$	24684.3	1.16433
$767$	−6802.11	−0.320222
$768$	0	0
$769$	−31970.0	−1.49918	−0.749589	−	0.661904i	$-0.769749\pi$
$769$	−31970.0	−1.49918	−0.749589	+	0.661904i	$0.769749\pi$
$770$	4271.37	0.199908
$771$	0	0
$772$	74763.2	3.48548
$773$	−2448.41	−0.113924	−0.0569619	−	0.998376i	$-0.518141\pi$
$773$	−2448.41	−0.113924	−0.0569619	+	0.998376i	$0.518141\pi$
$774$	0	0
$775$	−36650.9	−1.69876
$776$	79792.6	3.69122
$777$	0	0
$778$	−39428.2	−1.81693
$779$	−33152.9	−1.52481
$780$	0	0
$781$	1523.87	0.0698185
$782$	6236.28	0.285178
$783$	0	0
$784$	−68060.5	−3.10043
$785$	25207.4	1.14610
$786$	0	0
$787$	−1909.26	−0.0864774	−0.0432387	−	0.999065i	$-0.513768\pi$
$787$	−1909.26	−0.0864774	−0.0432387	+	0.999065i	$0.513768\pi$
$788$	−10018.1	−0.452891
$789$	0	0
$790$	75091.5	3.38182
$791$	−1434.39	−0.0644768
$792$	0	0
$793$	14157.9	0.634001
$794$	73226.8	3.27295
$795$	0	0
$796$	−17237.0	−0.767525
$797$	29068.3	1.29191	0.645955	−	0.763376i	$-0.276459\pi$
$797$	29068.3	1.29191	0.645955	+	0.763376i	$0.276459\pi$
$798$	0	0
$799$	3003.39	0.132981
$800$	−139818.	−6.17914
$801$	0	0
$802$	76109.2	3.35101
$803$	−3109.93	−0.136671
$804$	0	0
$805$	25218.3	1.10413
$806$	−90391.6	−3.95026
$807$	0	0
$808$	−41073.8	−1.78833
$809$	22381.3	0.972663	0.486331	−	0.873775i	$-0.338335\pi$
$809$	22381.3	0.972663	0.486331	+	0.873775i	$0.338335\pi$
$810$	0	0
$811$	14697.0	0.636351	0.318176	−	0.948032i	$-0.396930\pi$
$811$	14697.0	0.636351	0.318176	+	0.948032i	$0.396930\pi$
$812$	27680.3	1.19629
$813$	0	0
$814$	2367.58	0.101946
$815$	−2827.01	−0.121504
$816$	0	0
$817$	−16983.6	−0.727270
$818$	−29049.5	−1.24168
$819$	0	0
$820$	−150314.	−6.40144
$821$	16897.3	0.718297	0.359148	−	0.933280i	$-0.383067\pi$
$821$	16897.3	0.718297	0.359148	+	0.933280i	$0.383067\pi$
$822$	0	0
$823$	−14004.3	−0.593145	−0.296572	−	0.955010i	$-0.595844\pi$
$823$	−14004.3	−0.593145	−0.296572	+	0.955010i	$0.595844\pi$
$824$	−4020.62	−0.169982
$825$	0	0
$826$	6167.88	0.259816
$827$	12562.9	0.528239	0.264120	−	0.964490i	$-0.414919\pi$
$827$	12562.9	0.528239	0.264120	+	0.964490i	$0.414919\pi$
$828$	0	0
$829$	33257.2	1.39333	0.696666	−	0.717396i	$-0.254666\pi$
$829$	33257.2	1.39333	0.696666	+	0.717396i	$0.254666\pi$
$830$	90894.3	3.80119
$831$	0	0
$832$	−193256.	−8.05284
$833$	1781.50	0.0741000
$834$	0	0
$835$	−18930.9	−0.784586
$836$	8704.97	0.360129
$837$	0	0
$838$	90469.8	3.72939
$839$	−27823.7	−1.14491	−0.572457	−	0.819935i	$-0.694010\pi$
$839$	−27823.7	−1.14491	−0.572457	+	0.819935i	$0.694010\pi$
$840$	0	0
$841$	−11588.6	−0.475156
$842$	16733.6	0.684891
$843$	0	0
$844$	−12026.0	−0.490463
$845$	−32991.7	−1.34313
$846$	0	0
$847$	13772.6	0.558714
$848$	179576.	7.27202
$849$	0	0
$850$	6293.78	0.253970
$851$	13978.3	0.563066
$852$	0	0
$853$	−13485.9	−0.541322	−0.270661	−	0.962675i	$-0.587242\pi$
$853$	−13485.9	−0.541322	−0.270661	+	0.962675i	$0.587242\pi$
$854$	−12837.8	−0.514405
$855$	0	0
$856$	114199.	4.55988
$857$	−13390.1	−0.533718	−0.266859	−	0.963736i	$-0.585986\pi$
$857$	−13390.1	−0.533718	−0.266859	+	0.963736i	$0.585986\pi$
$858$	0	0
$859$	−36846.7	−1.46355	−0.731777	−	0.681544i	$-0.761309\pi$
$859$	−36846.7	−1.46355	−0.731777	+	0.681544i	$0.761309\pi$
$860$	−77002.5	−3.05321
$861$	0	0
$862$	−13979.2	−0.552358
$863$	−22566.7	−0.890126	−0.445063	−	0.895499i	$-0.646819\pi$
$863$	−22566.7	−0.890126	−0.445063	+	0.895499i	$0.646819\pi$
$864$	0	0
$865$	−18023.1	−0.708443
$866$	−47009.8	−1.84464
$867$	0	0
$868$	61013.5	2.38587
$869$	−3588.31	−0.140075
$870$	0	0
$871$	−66566.5	−2.58957
$872$	6325.83	0.245665
$873$	0	0
$874$	69041.4	2.67204
$875$	3803.96	0.146969
$876$	0	0
$877$	−30291.8	−1.16634	−0.583170	−	0.812350i	$-0.698188\pi$
$877$	−30291.8	−1.16634	−0.583170	+	0.812350i	$0.698188\pi$
$878$	−16090.7	−0.618492
$879$	0	0
$880$	21262.7	0.814505
$881$	34510.4	1.31973	0.659867	−	0.751382i	$-0.270613\pi$
$881$	34510.4	1.31973	0.659867	+	0.751382i	$0.270613\pi$
$882$	0	0
$883$	27430.5	1.04543	0.522713	−	0.852509i	$-0.324920\pi$
$883$	27430.5	1.04543	0.522713	+	0.852509i	$0.324920\pi$
$884$	11554.8	0.439625
$885$	0	0
$886$	50911.1	1.93047
$887$	−20649.9	−0.781686	−0.390843	−	0.920457i	$-0.627816\pi$
$887$	−20649.9	−0.781686	−0.390843	+	0.920457i	$0.627816\pi$
$888$	0	0
$889$	−13538.3	−0.510755
$890$	−136829.	−5.15338
$891$	0	0
$892$	−65653.0	−2.46438
$893$	33250.3	1.24600
$894$	0	0
$895$	−36690.6	−1.37031
$896$	95316.0	3.55389
$897$	0	0
$898$	75543.8	2.80727
$899$	28215.0	1.04674
$900$	0	0
$901$	−4700.45	−0.173801
$902$	9649.21	0.356190
$903$	0	0
$904$	−11691.3	−0.430140
$905$	−63045.7	−2.31570
$906$	0	0
$907$	31290.5	1.14552	0.572758	−	0.819724i	$-0.305873\pi$
$907$	31290.5	1.14552	0.572758	+	0.819724i	$0.305873\pi$
$908$	2395.36	0.0875470
$909$	0	0
$910$	62768.9	2.28656
$911$	−45514.1	−1.65527	−0.827635	−	0.561267i	$-0.810314\pi$
$911$	−45514.1	−1.65527	−0.827635	+	0.561267i	$0.810314\pi$
$912$	0	0
$913$	−4343.46	−0.157445
$914$	92065.7	3.33180
$915$	0	0
$916$	19391.3	0.699460
$917$	22421.1	0.807425
$918$	0	0
$919$	−7152.00	−0.256717	−0.128358	−	0.991728i	$-0.540971\pi$
$919$	−7152.00	−0.256717	−0.128358	+	0.991728i	$0.540971\pi$
$920$	205546.	7.36593
$921$	0	0
$922$	−88058.7	−3.14540
$923$	22393.6	0.798586
$924$	0	0
$925$	14107.1	0.501448
$926$	98768.3	3.50511
$927$	0	0
$928$	107636.	3.80747
$929$	1004.21	0.0354651	0.0177325	−	0.999843i	$-0.494355\pi$
$929$	1004.21	0.0354651	0.0177325	+	0.999843i	$0.494355\pi$
$930$	0	0
$931$	19722.8	0.694296
$932$	145410.	5.11060
$933$	0	0
$934$	47238.8	1.65493
$935$	−556.555	−0.0194666
$936$	0	0
$937$	42201.2	1.47135	0.735674	−	0.677336i	$-0.236866\pi$
$937$	42201.2	1.47135	0.735674	+	0.677336i	$0.236866\pi$
$938$	60359.8	2.10108
$939$	0	0
$940$	150755.	5.23094
$941$	−1345.69	−0.0466187	−0.0233093	−	0.999728i	$-0.507420\pi$
$941$	−1345.69	−0.0466187	−0.0233093	+	0.999728i	$0.507420\pi$
$942$	0	0
$943$	56969.1	1.96731
$944$	30703.4	1.05859
$945$	0	0
$946$	4943.09	0.169888
$947$	5871.87	0.201489	0.100745	−	0.994912i	$-0.467878\pi$
$947$	5871.87	0.201489	0.100745	+	0.994912i	$0.467878\pi$
$948$	0	0
$949$	−45701.3	−1.56325
$950$	69677.9	2.37963
$951$	0	0
$952$	−6879.81	−0.234218
$953$	−42193.5	−1.43419	−0.717094	−	0.696977i	$-0.754528\pi$
$953$	−42193.5	−1.43419	−0.717094	+	0.696977i	$0.754528\pi$
$954$	0	0
$955$	−23162.0	−0.784822
$956$	−5568.41	−0.188384
$957$	0	0
$958$	−80723.6	−2.72240
$959$	−1204.14	−0.0405462
$960$	0	0
$961$	32401.0	1.08761
$962$	34792.2	1.16606
$963$	0	0
$964$	−111668.	−3.73091
$965$	−52919.1	−1.76531
$966$	0	0
$967$	−18692.9	−0.621637	−0.310819	−	0.950469i	$-0.600603\pi$
$967$	−18692.9	−0.621637	−0.310819	+	0.950469i	$0.600603\pi$
$968$	112256.	3.72731
$969$	0	0
$970$	−86012.9	−2.84712
$971$	−5054.69	−0.167057	−0.0835287	−	0.996505i	$-0.526619\pi$
$971$	−5054.69	−0.167057	−0.0835287	+	0.996505i	$0.526619\pi$
$972$	0	0
$973$	−16692.0	−0.549971
$974$	−51946.4	−1.70890
$975$	0	0
$976$	−63906.2	−2.09589
$977$	−48228.8	−1.57930	−0.789650	−	0.613557i	$-0.789738\pi$
$977$	−48228.8	−1.57930	−0.789650	+	0.613557i	$0.789738\pi$
$978$	0	0
$979$	6538.47	0.213453
$980$	89422.2	2.91478
$981$	0	0
$982$	7551.76	0.245404
$983$	9488.25	0.307862	0.153931	−	0.988082i	$-0.450807\pi$
$983$	9488.25	0.307862	0.153931	+	0.988082i	$0.450807\pi$
$984$	0	0
$985$	7091.01	0.229379
$986$	−4845.14	−0.156492
$987$	0	0
$988$	127922.	4.11916
$989$	29184.1	0.938322
$990$	0	0
$991$	−5840.06	−0.187201	−0.0936003	−	0.995610i	$-0.529838\pi$
$991$	−5840.06	−0.187201	−0.0936003	+	0.995610i	$0.529838\pi$
$992$	237254.	7.59356
$993$	0	0
$994$	−20305.6	−0.647943
$995$	12200.7	0.388733
$996$	0	0
$997$	−25652.7	−0.814875	−0.407437	−	0.913233i	$-0.633578\pi$
$997$	−25652.7	−0.814875	−0.407437	+	0.913233i	$0.633578\pi$
$998$	−58687.4	−1.86144
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.d.1.1		32
3.2	odd	2		717.4.a.d.1.32	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.4.a.d.1.32	✓	32	3.2	odd	2
2151.4.a.d.1.1		32	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)

\(f(q)\)	\(=\)	\(q-5.59453 q^{2} +23.2988 q^{4} -16.4914 q^{5} -10.5009 q^{7} -85.5894 q^{8} +O(q^{10})\)	\(q-5.59453 q^{2} +23.2988 q^{4} -16.4914 q^{5} -10.5009 q^{7} -85.5894 q^{8} +92.2616 q^{10} -4.40880 q^{11} -64.7884 q^{13} +58.7475 q^{14} +292.442 q^{16} -7.65474 q^{17} -84.7450 q^{19} -384.229 q^{20} +24.6651 q^{22} +145.624 q^{23} +146.966 q^{25} +362.461 q^{26} -244.658 q^{28} -113.139 q^{29} -249.383 q^{31} -951.362 q^{32} +42.8247 q^{34} +173.174 q^{35} +95.9890 q^{37} +474.108 q^{38} +1411.49 q^{40} +391.208 q^{41} +200.408 q^{43} -102.720 q^{44} -814.696 q^{46} -392.357 q^{47} -232.731 q^{49} -822.206 q^{50} -1509.49 q^{52} +614.057 q^{53} +72.7072 q^{55} +898.765 q^{56} +632.959 q^{58} +104.990 q^{59} -218.526 q^{61} +1395.18 q^{62} +2982.89 q^{64} +1068.45 q^{65} +1027.44 q^{67} -178.346 q^{68} -968.828 q^{70} -345.643 q^{71} +705.393 q^{73} -537.013 q^{74} -1974.45 q^{76} +46.2963 q^{77} +813.898 q^{79} -4822.78 q^{80} -2188.63 q^{82} +985.181 q^{83} +126.237 q^{85} -1121.19 q^{86} +377.346 q^{88} -1483.05 q^{89} +680.336 q^{91} +3392.85 q^{92} +2195.05 q^{94} +1397.56 q^{95} -932.272 q^{97} +1302.02 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8}+O(q^{10}) \) 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8	\( 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8} + 52 q^{10} - 270 q^{11} + 48 q^{13} - 184 q^{14} + 775 q^{16} - 384 q^{17} + 216 q^{19} - 534 q^{20} + 437 q^{22} - 712 q^{23} + 1190 q^{25} - 436 q^{26} + 598 q^{28} - 562 q^{29} + 384 q^{31} - 1770 q^{32} + 452 q^{34} - 1026 q^{35} + 770 q^{37} - 733 q^{38} + 877 q^{40} - 1648 q^{41} + 1592 q^{43} - 1595 q^{44} + 532 q^{46} - 1540 q^{47} + 2134 q^{49} - 1646 q^{50} - 144 q^{52} - 1708 q^{53} + 1282 q^{55} - 2155 q^{56} + 1086 q^{58} - 2396 q^{59} + 364 q^{61} - 2180 q^{62} + 1663 q^{64} - 1520 q^{65} + 2728 q^{67} - 1545 q^{68} - 4609 q^{70} - 3322 q^{71} - 188 q^{73} - 1111 q^{74} - 3134 q^{76} - 556 q^{77} - 462 q^{79} - 6076 q^{80} - 7965 q^{82} - 4604 q^{83} - 852 q^{85} - 549 q^{86} - 1127 q^{88} - 6742 q^{89} + 1390 q^{91} - 1802 q^{92} - 2796 q^{94} - 448 q^{95} - 1322 q^{97} - 1000 q^{98}+O(q^{100}) \) 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8 + 52 * q^10 - 270 * q^11 + 48 * q^13 - 184 * q^14 + 775 * q^16 - 384 * q^17 + 216 * q^19 - 534 * q^20 + 437 * q^22 - 712 * q^23 + 1190 * q^25 - 436 * q^26 + 598 * q^28 - 562 * q^29 + 384 * q^31 - 1770 * q^32 + 452 * q^34 - 1026 * q^35 + 770 * q^37 - 733 * q^38 + 877 * q^40 - 1648 * q^41 + 1592 * q^43 - 1595 * q^44 + 532 * q^46 - 1540 * q^47 + 2134 * q^49 - 1646 * q^50 - 144 * q^52 - 1708 * q^53 + 1282 * q^55 - 2155 * q^56 + 1086 * q^58 - 2396 * q^59 + 364 * q^61 - 2180 * q^62 + 1663 * q^64 - 1520 * q^65 + 2728 * q^67 - 1545 * q^68 - 4609 * q^70 - 3322 * q^71 - 188 * q^73 - 1111 * q^74 - 3134 * q^76 - 556 * q^77 - 462 * q^79 - 6076 * q^80 - 7965 * q^82 - 4604 * q^83 - 852 * q^85 - 549 * q^86 - 1127 * q^88 - 6742 * q^89 + 1390 * q^91 - 1802 * q^92 - 2796 * q^94 - 448 * q^95 - 1322 * q^97 - 1000 * q^98

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