LMFDB - Embedded newform 2169.4.a.e.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2169,4,Mod(1,2169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2169, 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("2169.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2169 = 3^{2} \cdot 241 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2169.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:	$127.975142802$
Analytic rank:	$1$
Dimension:	$33$
Twist minimal:	no (minimal twist has level 241)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	2169.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.52337 q^{2} -1.63263 q^{4} +2.94898 q^{5} -33.1605 q^{7} +24.3066 q^{8} +O(q^{10})$	\(q-2.52337 q^{2} -1.63263 q^{4} +2.94898 q^{5} -33.1605 q^{7} +24.3066 q^{8} -7.44136 q^{10} -63.4292 q^{11} +1.98833 q^{13} +83.6761 q^{14} -48.2735 q^{16} +58.9476 q^{17} +110.065 q^{19} -4.81460 q^{20} +160.055 q^{22} -45.7188 q^{23} -116.303 q^{25} -5.01729 q^{26} +54.1388 q^{28} -69.0210 q^{29} -142.534 q^{31} -72.6414 q^{32} -148.746 q^{34} -97.7898 q^{35} +110.799 q^{37} -277.734 q^{38} +71.6799 q^{40} +249.075 q^{41} +413.051 q^{43} +103.556 q^{44} +115.365 q^{46} -243.951 q^{47} +756.619 q^{49} +293.476 q^{50} -3.24621 q^{52} +570.739 q^{53} -187.052 q^{55} -806.021 q^{56} +174.165 q^{58} +461.556 q^{59} +741.511 q^{61} +359.665 q^{62} +569.489 q^{64} +5.86356 q^{65} +34.2081 q^{67} -96.2395 q^{68} +246.759 q^{70} -1093.72 q^{71} -261.992 q^{73} -279.585 q^{74} -179.695 q^{76} +2103.35 q^{77} -238.436 q^{79} -142.358 q^{80} -628.506 q^{82} -381.845 q^{83} +173.835 q^{85} -1042.28 q^{86} -1541.75 q^{88} +48.7251 q^{89} -65.9341 q^{91} +74.6419 q^{92} +615.577 q^{94} +324.580 q^{95} +381.832 q^{97} -1909.23 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 33 q - 9 q^{2} + 161 q^{4} - 50 q^{5} + 58 q^{7} - 111 q^{8}+O(q^{10}) $ 33 * q - 9 * q^2 + 161 * q^4 - 50 * q^5 + 58 * q^7 - 111 * q^8	$ 33 q - 9 q^{2} + 161 q^{4} - 50 q^{5} + 58 q^{7} - 111 q^{8} + q^{10} - 274 q^{11} + 24 q^{13} - 220 q^{14} + 789 q^{16} - 140 q^{17} + 164 q^{19} - 630 q^{20} + 270 q^{22} - 980 q^{23} + 1121 q^{25} - 484 q^{26} + 433 q^{28} - 772 q^{29} + 804 q^{31} - 947 q^{32} + 211 q^{34} - 558 q^{35} + 342 q^{37} - 1092 q^{38} + 564 q^{40} - 978 q^{41} + 230 q^{43} - 2788 q^{44} + 1103 q^{46} - 2312 q^{47} + 2501 q^{49} - 1787 q^{50} - 57 q^{52} - 1352 q^{53} + 1058 q^{55} - 2367 q^{56} + 2237 q^{58} - 3546 q^{59} + 156 q^{61} - 1349 q^{62} + 47 q^{64} - 900 q^{65} - 236 q^{67} + 3439 q^{68} - 7312 q^{70} - 5966 q^{71} - 2020 q^{73} - 488 q^{74} - 7552 q^{76} + 8 q^{77} - 198 q^{79} + 261 q^{80} - 4465 q^{82} - 502 q^{83} - 4778 q^{85} + 185 q^{86} - 9372 q^{88} + 192 q^{89} + 1896 q^{91} - 3214 q^{92} - 14222 q^{94} - 5610 q^{95} - 846 q^{97} + 3726 q^{98}+O(q^{100}) $ 33 * q - 9 * q^2 + 161 * q^4 - 50 * q^5 + 58 * q^7 - 111 * q^8 + q^10 - 274 * q^11 + 24 * q^13 - 220 * q^14 + 789 * q^16 - 140 * q^17 + 164 * q^19 - 630 * q^20 + 270 * q^22 - 980 * q^23 + 1121 * q^25 - 484 * q^26 + 433 * q^28 - 772 * q^29 + 804 * q^31 - 947 * q^32 + 211 * q^34 - 558 * q^35 + 342 * q^37 - 1092 * q^38 + 564 * q^40 - 978 * q^41 + 230 * q^43 - 2788 * q^44 + 1103 * q^46 - 2312 * q^47 + 2501 * q^49 - 1787 * q^50 - 57 * q^52 - 1352 * q^53 + 1058 * q^55 - 2367 * q^56 + 2237 * q^58 - 3546 * q^59 + 156 * q^61 - 1349 * q^62 + 47 * q^64 - 900 * q^65 - 236 * q^67 + 3439 * q^68 - 7312 * q^70 - 5966 * q^71 - 2020 * q^73 - 488 * q^74 - 7552 * q^76 + 8 * q^77 - 198 * q^79 + 261 * q^80 - 4465 * q^82 - 502 * q^83 - 4778 * q^85 + 185 * q^86 - 9372 * q^88 + 192 * q^89 + 1896 * q^91 - 3214 * q^92 - 14222 * q^94 - 5610 * q^95 - 846 * q^97 + 3726 * 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 $

$2$	−2.52337	−0.892144	−0.446072	−	0.894997i	$-0.647178\pi$
$2$	−2.52337	−0.892144	−0.446072	+	0.894997i	$0.647178\pi$
$3$	0	0
$3$	0	0
$4$	−1.63263	−0.204079
$5$	2.94898	0.263765	0.131883	−	0.991265i	$-0.457898\pi$
$5$	2.94898	0.263765	0.131883	+	0.991265i	$0.457898\pi$
$6$	0	0
$7$	−33.1605	−1.79050	−0.895250	−	0.445565i	$-0.853003\pi$
$7$	−33.1605	−1.79050	−0.895250	+	0.445565i	$0.853003\pi$
$8$	24.3066	1.07421
$9$	0	0
$10$	−7.44136	−0.235317
$11$	−63.4292	−1.73860	−0.869302	−	0.494282i	$-0.835431\pi$
$11$	−63.4292	−1.73860	−0.869302	+	0.494282i	$0.835431\pi$
$12$	0	0
$13$	1.98833	0.0424203	0.0212102	−	0.999775i	$-0.493248\pi$
$13$	1.98833	0.0424203	0.0212102	+	0.999775i	$0.493248\pi$
$14$	83.6761	1.59738
$15$	0	0
$16$	−48.2735	−0.754273
$17$	58.9476	0.840993	0.420497	−	0.907294i	$-0.361856\pi$
$17$	58.9476	0.840993	0.420497	+	0.907294i	$0.361856\pi$
$18$	0	0
$19$	110.065	1.32898	0.664490	−	0.747298i	$-0.268649\pi$
$19$	110.065	1.32898	0.664490	+	0.747298i	$0.268649\pi$
$20$	−4.81460	−0.0538288
$21$	0	0
$22$	160.055	1.55108
$23$	−45.7188	−0.414480	−0.207240	−	0.978290i	$-0.566448\pi$
$23$	−45.7188	−0.414480	−0.207240	+	0.978290i	$0.566448\pi$
$24$	0	0
$25$	−116.303	−0.930428
$26$	−5.01729	−0.0378451
$27$	0	0
$28$	54.1388	0.365402
$29$	−69.0210	−0.441961	−0.220981	−	0.975278i	$-0.570926\pi$
$29$	−69.0210	−0.441961	−0.220981	+	0.975278i	$0.570926\pi$
$30$	0	0
$31$	−142.534	−0.825801	−0.412901	−	0.910776i	$-0.635484\pi$
$31$	−142.534	−0.825801	−0.412901	+	0.910776i	$0.635484\pi$
$32$	−72.6414	−0.401291
$33$	0	0
$34$	−148.746	−0.750287
$35$	−97.7898	−0.472271
$36$	0	0
$37$	110.799	0.492302	0.246151	−	0.969231i	$-0.420834\pi$
$37$	110.799	0.492302	0.246151	+	0.969231i	$0.420834\pi$
$38$	−277.734	−1.18564
$39$	0	0
$40$	71.6799	0.283340
$41$	249.075	0.948754	0.474377	−	0.880322i	$-0.342673\pi$
$41$	249.075	0.948754	0.474377	+	0.880322i	$0.342673\pi$
$42$	0	0
$43$	413.051	1.46488	0.732438	−	0.680834i	$-0.238382\pi$
$43$	413.051	1.46488	0.732438	+	0.680834i	$0.238382\pi$
$44$	103.556	0.354812
$45$	0	0
$46$	115.365	0.369776
$47$	−243.951	−0.757104	−0.378552	−	0.925580i	$-0.623578\pi$
$47$	−243.951	−0.757104	−0.378552	+	0.925580i	$0.623578\pi$
$48$	0	0
$49$	756.619	2.20589
$50$	293.476	0.830076
$51$	0	0
$52$	−3.24621	−0.00865708
$53$	570.739	1.47919	0.739594	−	0.673053i	$-0.235018\pi$
$53$	570.739	1.47919	0.739594	+	0.673053i	$0.235018\pi$
$54$	0	0
$55$	−187.052	−0.458583
$56$	−806.021	−1.92338
$57$	0	0
$58$	174.165	0.394293
$59$	461.556	1.01847	0.509233	−	0.860629i	$-0.329929\pi$
$59$	461.556	1.01847	0.509233	+	0.860629i	$0.329929\pi$
$60$	0	0
$61$	741.511	1.55641	0.778203	−	0.628013i	$-0.216131\pi$
$61$	741.511	1.55641	0.778203	+	0.628013i	$0.216131\pi$
$62$	359.665	0.736734
$63$	0	0
$64$	569.489	1.11228
$65$	5.86356	0.0111890
$66$	0	0
$67$	34.2081	0.0623759	0.0311879	−	0.999514i	$-0.490071\pi$
$67$	34.2081	0.0623759	0.0311879	+	0.999514i	$0.490071\pi$
$68$	−96.2395	−0.171629
$69$	0	0
$70$	246.759	0.421334
$71$	−1093.72	−1.82817	−0.914085	−	0.405522i	$-0.867090\pi$
$71$	−1093.72	−1.82817	−0.914085	+	0.405522i	$0.867090\pi$
$72$	0	0
$73$	−261.992	−0.420053	−0.210027	−	0.977696i	$-0.567355\pi$
$73$	−261.992	−0.420053	−0.210027	+	0.977696i	$0.567355\pi$
$74$	−279.585	−0.439205
$75$	0	0
$76$	−179.695	−0.271216
$77$	2103.35	3.11297
$78$	0	0
$79$	−238.436	−0.339572	−0.169786	−	0.985481i	$-0.554308\pi$
$79$	−238.436	−0.339572	−0.169786	+	0.985481i	$0.554308\pi$
$80$	−142.358	−0.198951
$81$	0	0
$82$	−628.506	−0.846426
$83$	−381.845	−0.504975	−0.252488	−	0.967600i	$-0.581249\pi$
$83$	−381.845	−0.504975	−0.252488	+	0.967600i	$0.581249\pi$
$84$	0	0
$85$	173.835	0.221825
$86$	−1042.28	−1.30688
$87$	0	0
$88$	−1541.75	−1.86763
$89$	48.7251	0.0580321	0.0290160	−	0.999579i	$-0.490763\pi$
$89$	48.7251	0.0580321	0.0290160	+	0.999579i	$0.490763\pi$
$90$	0	0
$91$	−65.9341	−0.0759536
$92$	74.6419	0.0845864
$93$	0	0
$94$	615.577	0.675446
$95$	324.580	0.350538
$96$	0	0
$97$	381.832	0.399683	0.199841	−	0.979828i	$-0.435957\pi$
$97$	381.832	0.399683	0.199841	+	0.979828i	$0.435957\pi$
$98$	−1909.23	−1.96797
$99$	0	0
$100$	189.880	0.189880
$101$	8.17679	0.00805565	0.00402783	−	0.999992i	$-0.498718\pi$
$101$	8.17679	0.00805565	0.00402783	+	0.999992i	$0.498718\pi$
$102$	0	0
$103$	−1436.85	−1.37453	−0.687267	−	0.726405i	$-0.741190\pi$
$103$	−1436.85	−1.37453	−0.687267	+	0.726405i	$0.741190\pi$
$104$	48.3297	0.0455684
$105$	0	0
$106$	−1440.18	−1.31965
$107$	−1939.37	−1.75221	−0.876103	−	0.482124i	$-0.839865\pi$
$107$	−1939.37	−1.75221	−0.876103	+	0.482124i	$0.839865\pi$
$108$	0	0
$109$	212.011	0.186302	0.0931512	−	0.995652i	$-0.470306\pi$
$109$	212.011	0.186302	0.0931512	+	0.995652i	$0.470306\pi$
$110$	472.000	0.409122
$111$	0	0
$112$	1600.77	1.35053
$113$	1580.56	1.31581	0.657906	−	0.753100i	$-0.271442\pi$
$113$	1580.56	1.31581	0.657906	+	0.753100i	$0.271442\pi$
$114$	0	0
$115$	−134.824	−0.109325
$116$	112.686	0.0901948
$117$	0	0
$118$	−1164.67	−0.908618
$119$	−1954.73	−1.50580
$120$	0	0
$121$	2692.27	2.02274
$122$	−1871.10	−1.38854
$123$	0	0
$124$	232.705	0.168528
$125$	−711.600	−0.509180
$126$	0	0
$127$	2180.14	1.52328	0.761638	−	0.648003i	$-0.224395\pi$
$127$	2180.14	1.52328	0.761638	+	0.648003i	$0.224395\pi$
$128$	−855.897	−0.591026
$129$	0	0
$130$	−14.7959	−0.00998221
$131$	214.565	0.143104	0.0715521	−	0.997437i	$-0.477205\pi$
$131$	214.565	0.143104	0.0715521	+	0.997437i	$0.477205\pi$
$132$	0	0
$133$	−3649.81	−2.37954
$134$	−86.3195	−0.0556483
$135$	0	0
$136$	1432.82	0.903405
$137$	1769.83	1.10370	0.551851	−	0.833943i	$-0.313922\pi$
$137$	1769.83	1.10370	0.551851	+	0.833943i	$0.313922\pi$
$138$	0	0
$139$	2916.86	1.77989	0.889946	−	0.456065i	$-0.150742\pi$
$139$	2916.86	1.77989	0.889946	+	0.456065i	$0.150742\pi$
$140$	159.654	0.0963805
$141$	0	0
$142$	2759.84	1.63099
$143$	−126.118	−0.0737521
$144$	0	0
$145$	−203.542	−0.116574
$146$	661.102	0.374748
$147$	0	0
$148$	−180.893	−0.100468
$149$	−2677.52	−1.47215	−0.736077	−	0.676898i	$-0.763324\pi$
$149$	−2677.52	−1.47215	−0.736077	+	0.676898i	$0.763324\pi$
$150$	0	0
$151$	1573.08	0.847786	0.423893	−	0.905712i	$-0.360663\pi$
$151$	1573.08	0.847786	0.423893	+	0.905712i	$0.360663\pi$
$152$	2675.31	1.42761
$153$	0	0
$154$	−5307.51	−2.77722
$155$	−420.330	−0.217818
$156$	0	0
$157$	2486.76	1.26411	0.632055	−	0.774923i	$-0.282212\pi$
$157$	2486.76	1.26411	0.632055	+	0.774923i	$0.282212\pi$
$158$	601.662	0.302947
$159$	0	0
$160$	−214.219	−0.105847
$161$	1516.06	0.742126
$162$	0	0
$163$	785.086	0.377256	0.188628	−	0.982049i	$-0.439596\pi$
$163$	785.086	0.377256	0.188628	+	0.982049i	$0.439596\pi$
$164$	−406.646	−0.193620
$165$	0	0
$166$	963.534	0.450511
$167$	−389.091	−0.180292	−0.0901460	−	0.995929i	$-0.528733\pi$
$167$	−389.091	−0.180292	−0.0901460	+	0.995929i	$0.528733\pi$
$168$	0	0
$169$	−2193.05	−0.998201
$170$	−438.650	−0.197900
$171$	0	0
$172$	−674.359	−0.298950
$173$	−909.688	−0.399782	−0.199891	−	0.979818i	$-0.564059\pi$
$173$	−909.688	−0.399782	−0.199891	+	0.979818i	$0.564059\pi$
$174$	0	0
$175$	3856.68	1.66593
$176$	3061.95	1.31138
$177$	0	0
$178$	−122.951	−0.0517730
$179$	1025.52	0.428219	0.214109	−	0.976810i	$-0.431315\pi$
$179$	1025.52	0.428219	0.214109	+	0.976810i	$0.431315\pi$
$180$	0	0
$181$	−1561.24	−0.641137	−0.320568	−	0.947225i	$-0.603874\pi$
$181$	−1561.24	−0.641137	−0.320568	+	0.947225i	$0.603874\pi$
$182$	166.376	0.0677616
$183$	0	0
$184$	−1111.27	−0.445239
$185$	326.743	0.129852
$186$	0	0
$187$	−3739.00	−1.46215
$188$	398.281	0.154509
$189$	0	0
$190$	−819.033	−0.312731
$191$	1635.11	0.619436	0.309718	−	0.950829i	$-0.399765\pi$
$191$	1635.11	0.619436	0.309718	+	0.950829i	$0.399765\pi$
$192$	0	0
$193$	128.963	0.0480984	0.0240492	−	0.999711i	$-0.492344\pi$
$193$	128.963	0.0480984	0.0240492	+	0.999711i	$0.492344\pi$
$194$	−963.502	−0.356574
$195$	0	0
$196$	−1235.28	−0.450174
$197$	−4639.67	−1.67798	−0.838992	−	0.544143i	$-0.816855\pi$
$197$	−4639.67	−1.67798	−0.838992	+	0.544143i	$0.816855\pi$
$198$	0	0
$199$	1995.39	0.710802	0.355401	−	0.934714i	$-0.384344\pi$
$199$	1995.39	0.710802	0.355401	+	0.934714i	$0.384344\pi$
$200$	−2826.95	−0.999477
$201$	0	0
$202$	−20.6330	−0.00718681
$203$	2288.77	0.791331
$204$	0	0
$205$	734.517	0.250248
$206$	3625.69	1.22628
$207$	0	0
$208$	−95.9838	−0.0319965
$209$	−6981.33	−2.31057
$210$	0	0
$211$	−2494.42	−0.813851	−0.406926	−	0.913461i	$-0.633399\pi$
$211$	−2494.42	−0.813851	−0.406926	+	0.913461i	$0.633399\pi$
$212$	−931.804	−0.301871
$213$	0	0
$214$	4893.74	1.56322
$215$	1218.08	0.386383
$216$	0	0
$217$	4726.50	1.47860
$218$	−534.981	−0.166209
$219$	0	0
$220$	305.386	0.0935869
$221$	117.207	0.0356752
$222$	0	0
$223$	−2523.31	−0.757728	−0.378864	−	0.925452i	$-0.623685\pi$
$223$	−2523.31	−0.757728	−0.378864	+	0.925452i	$0.623685\pi$
$224$	2408.83	0.718511
$225$	0	0
$226$	−3988.34	−1.17389
$227$	2513.90	0.735038	0.367519	−	0.930016i	$-0.380207\pi$
$227$	2513.90	0.735038	0.367519	+	0.930016i	$0.380207\pi$
$228$	0	0
$229$	3902.92	1.12625	0.563127	−	0.826370i	$-0.309598\pi$
$229$	3902.92	1.12625	0.563127	+	0.826370i	$0.309598\pi$
$230$	340.211	0.0975340
$231$	0	0
$232$	−1677.67	−0.474760
$233$	−4918.86	−1.38303	−0.691513	−	0.722364i	$-0.743055\pi$
$233$	−4918.86	−1.38303	−0.691513	+	0.722364i	$0.743055\pi$
$234$	0	0
$235$	−719.407	−0.199698
$236$	−753.549	−0.207847
$237$	0	0
$238$	4932.50	1.34339
$239$	−2457.77	−0.665187	−0.332593	−	0.943070i	$-0.607924\pi$
$239$	−2457.77	−0.665187	−0.332593	+	0.943070i	$0.607924\pi$
$240$	0	0
$241$	−241.000	−0.0644157
$241$	−241.000	−0.0644157
$242$	−6793.57	−1.80458
$243$	0	0
$244$	−1210.61	−0.317629
$245$	2231.26	0.581836
$246$	0	0
$247$	218.846	0.0563758
$248$	−3464.52	−0.887085
$249$	0	0
$250$	1795.63	0.454262
$251$	3802.75	0.956284	0.478142	−	0.878283i	$-0.341310\pi$
$251$	3802.75	0.956284	0.478142	+	0.878283i	$0.341310\pi$
$252$	0	0
$253$	2899.91	0.720616
$254$	−5501.29	−1.35898
$255$	0	0
$256$	−2396.17	−0.585003
$257$	6569.71	1.59458	0.797290	−	0.603597i	$-0.206266\pi$
$257$	6569.71	1.59458	0.797290	+	0.603597i	$0.206266\pi$
$258$	0	0
$259$	−3674.14	−0.881467
$260$	−9.57302	−0.00228344
$261$	0	0
$262$	−541.426	−0.127670
$263$	−6204.46	−1.45469	−0.727345	−	0.686272i	$-0.759246\pi$
$263$	−6204.46	−1.45469	−0.727345	+	0.686272i	$0.759246\pi$
$264$	0	0
$265$	1683.10	0.390158
$266$	9209.79	2.12289
$267$	0	0
$268$	−55.8491	−0.0127296
$269$	1997.58	0.452769	0.226384	−	0.974038i	$-0.427309\pi$
$269$	1997.58	0.452769	0.226384	+	0.974038i	$0.427309\pi$
$270$	0	0
$271$	1907.51	0.427575	0.213788	−	0.976880i	$-0.431420\pi$
$271$	1907.51	0.427575	0.213788	+	0.976880i	$0.431420\pi$
$272$	−2845.61	−0.634339
$273$	0	0
$274$	−4465.94	−0.984661
$275$	7377.04	1.61764
$276$	0	0
$277$	1637.05	0.355094	0.177547	−	0.984112i	$-0.443184\pi$
$277$	1637.05	0.355094	0.177547	+	0.984112i	$0.443184\pi$
$278$	−7360.31	−1.58792
$279$	0	0
$280$	−2376.94	−0.507319
$281$	895.392	0.190088	0.0950438	−	0.995473i	$-0.469701\pi$
$281$	895.392	0.190088	0.0950438	+	0.995473i	$0.469701\pi$
$282$	0	0
$283$	−7841.37	−1.64707	−0.823536	−	0.567265i	$-0.808002\pi$
$283$	−7841.37	−1.64707	−0.823536	+	0.567265i	$0.808002\pi$
$284$	1785.63	0.373090
$285$	0	0
$286$	318.243	0.0657975
$287$	−8259.44	−1.69874
$288$	0	0
$289$	−1438.18	−0.292730
$290$	513.611	0.104001
$291$	0	0
$292$	427.736	0.0857238
$293$	−249.565	−0.0497603	−0.0248802	−	0.999690i	$-0.507920\pi$
$293$	−249.565	−0.0497603	−0.0248802	+	0.999690i	$0.507920\pi$
$294$	0	0
$295$	1361.12	0.268636
$296$	2693.14	0.528837
$297$	0	0
$298$	6756.36	1.31337
$299$	−90.9043	−0.0175824
$300$	0	0
$301$	−13697.0	−2.62286
$302$	−3969.46	−0.756347
$303$	0	0
$304$	−5313.21	−1.00241
$305$	2186.70	0.410526
$306$	0	0
$307$	−4826.01	−0.897182	−0.448591	−	0.893737i	$-0.648074\pi$
$307$	−4826.01	−0.897182	−0.448591	+	0.893737i	$0.648074\pi$
$308$	−3433.98	−0.635290
$309$	0	0
$310$	1060.65	0.194325
$311$	2025.69	0.369346	0.184673	−	0.982800i	$-0.440877\pi$
$311$	2025.69	0.369346	0.184673	+	0.982800i	$0.440877\pi$
$312$	0	0
$313$	−1293.24	−0.233542	−0.116771	−	0.993159i	$-0.537254\pi$
$313$	−1293.24	−0.233542	−0.116771	+	0.993159i	$0.537254\pi$
$314$	−6275.01	−1.12777
$315$	0	0
$316$	389.278	0.0692993
$317$	−9507.91	−1.68460	−0.842299	−	0.539011i	$-0.818798\pi$
$317$	−9507.91	−1.68460	−0.842299	+	0.539011i	$0.818798\pi$
$318$	0	0
$319$	4377.95	0.768395
$320$	1679.41	0.293382
$321$	0	0
$322$	−3825.57	−0.662083
$323$	6488.05	1.11766
$324$	0	0
$325$	−231.250	−0.0394691
$326$	−1981.06	−0.336567
$327$	0	0
$328$	6054.17	1.01916
$329$	8089.53	1.35559
$330$	0	0
$331$	−7732.72	−1.28407	−0.642037	−	0.766673i	$-0.721911\pi$
$331$	−7732.72	−1.28407	−0.642037	+	0.766673i	$0.721911\pi$
$332$	623.411	0.103055
$333$	0	0
$334$	981.819	0.160846
$335$	100.879	0.0164526
$336$	0	0
$337$	1576.75	0.254869	0.127435	−	0.991847i	$-0.459326\pi$
$337$	1576.75	0.254869	0.127435	+	0.991847i	$0.459326\pi$
$338$	5533.86	0.890539
$339$	0	0
$340$	−283.809	−0.0452697
$341$	9040.81	1.43574
$342$	0	0
$343$	−13715.8	−2.15914
$344$	10039.9	1.57359
$345$	0	0
$346$	2295.47	0.356663
$347$	−85.1841	−0.0131785	−0.00658923	−	0.999978i	$-0.502097\pi$
$347$	−85.1841	−0.0131785	−0.00658923	+	0.999978i	$0.502097\pi$
$348$	0	0
$349$	−10959.4	−1.68092	−0.840460	−	0.541874i	$-0.817715\pi$
$349$	−10959.4	−1.68092	−0.840460	+	0.541874i	$0.817715\pi$
$350$	−9731.82	−1.48625
$351$	0	0
$352$	4607.59	0.697686
$353$	5735.09	0.864725	0.432362	−	0.901700i	$-0.357680\pi$
$353$	5735.09	0.864725	0.432362	+	0.901700i	$0.357680\pi$
$354$	0	0
$355$	−3225.35	−0.482208
$356$	−79.5500	−0.0118431
$357$	0	0
$358$	−2587.77	−0.382033
$359$	−9597.03	−1.41090	−0.705448	−	0.708761i	$-0.749254\pi$
$359$	−9597.03	−1.41090	−0.705448	+	0.708761i	$0.749254\pi$
$360$	0	0
$361$	5255.27	0.766186
$362$	3939.57	0.571987
$363$	0	0
$364$	107.646	0.0155005
$365$	−772.611	−0.110795
$366$	0	0
$367$	9653.79	1.37309	0.686545	−	0.727088i	$-0.259127\pi$
$367$	9653.79	1.37309	0.686545	+	0.727088i	$0.259127\pi$
$368$	2207.01	0.312631
$369$	0	0
$370$	−824.493	−0.115847
$371$	−18926.0	−2.64849
$372$	0	0
$373$	2630.18	0.365109	0.182555	−	0.983196i	$-0.441563\pi$
$373$	2630.18	0.365109	0.182555	+	0.983196i	$0.441563\pi$
$374$	9434.86	1.30445
$375$	0	0
$376$	−5929.62	−0.813290
$377$	−137.237	−0.0187482
$378$	0	0
$379$	−2789.59	−0.378078	−0.189039	−	0.981970i	$-0.560537\pi$
$379$	−2789.59	−0.378078	−0.189039	+	0.981970i	$0.560537\pi$
$380$	−529.918	−0.0715374
$381$	0	0
$382$	−4125.97	−0.552626
$383$	−10226.2	−1.36432	−0.682161	−	0.731202i	$-0.738960\pi$
$383$	−10226.2	−1.36432	−0.682161	+	0.731202i	$0.738960\pi$
$384$	0	0
$385$	6202.73	0.821092
$386$	−325.422	−0.0429107
$387$	0	0
$388$	−623.390	−0.0815666
$389$	−7673.90	−1.00021	−0.500106	−	0.865964i	$-0.666706\pi$
$389$	−7673.90	−1.00021	−0.500106	+	0.865964i	$0.666706\pi$
$390$	0	0
$391$	−2695.01	−0.348575
$392$	18390.9	2.36959
$393$	0	0
$394$	11707.6	1.49700
$395$	−703.145	−0.0895672
$396$	0	0
$397$	−8605.85	−1.08795	−0.543974	−	0.839102i	$-0.683081\pi$
$397$	−8605.85	−1.08795	−0.543974	+	0.839102i	$0.683081\pi$
$398$	−5035.10	−0.634138
$399$	0	0
$400$	5614.38	0.701797
$401$	−8278.61	−1.03096	−0.515479	−	0.856902i	$-0.672386\pi$
$401$	−8278.61	−1.03096	−0.515479	+	0.856902i	$0.672386\pi$
$402$	0	0
$403$	−283.405	−0.0350308
$404$	−13.3497	−0.00164399
$405$	0	0
$406$	−5775.41	−0.705982
$407$	−7027.87	−0.855918
$408$	0	0
$409$	8755.01	1.05845	0.529227	−	0.848481i	$-0.322482\pi$
$409$	8755.01	1.05845	0.529227	+	0.848481i	$0.322482\pi$
$410$	−1853.46	−0.223258
$411$	0	0
$412$	2345.84	0.280513
$413$	−15305.4	−1.82356
$414$	0	0
$415$	−1126.06	−0.133195
$416$	−144.435	−0.0170229
$417$	0	0
$418$	17616.4	2.06136
$419$	−10035.9	−1.17013	−0.585066	−	0.810985i	$-0.698932\pi$
$419$	−10035.9	−1.17013	−0.585066	+	0.810985i	$0.698932\pi$
$420$	0	0
$421$	−12774.4	−1.47882	−0.739412	−	0.673254i	$-0.764896\pi$
$421$	−12774.4	−1.47882	−0.739412	+	0.673254i	$0.764896\pi$
$422$	6294.32	0.726073
$423$	0	0
$424$	13872.7	1.58896
$425$	−6855.81	−0.782484
$426$	0	0
$427$	−24588.9	−2.78674
$428$	3166.27	0.357588
$429$	0	0
$430$	−3073.66	−0.344710
$431$	14356.2	1.60444	0.802219	−	0.597029i	$-0.203652\pi$
$431$	14356.2	1.60444	0.802219	+	0.597029i	$0.203652\pi$
$432$	0	0
$433$	1937.23	0.215006	0.107503	−	0.994205i	$-0.465715\pi$
$433$	1937.23	0.215006	0.107503	+	0.994205i	$0.465715\pi$
$434$	−11926.7	−1.31912
$435$	0	0
$436$	−346.135	−0.0380203
$437$	−5032.04	−0.550835
$438$	0	0
$439$	13668.0	1.48596	0.742981	−	0.669312i	$-0.233411\pi$
$439$	13668.0	1.48596	0.742981	+	0.669312i	$0.233411\pi$
$440$	−4546.60	−0.492615
$441$	0	0
$442$	−295.757	−0.0318274
$443$	6153.49	0.659958	0.329979	−	0.943988i	$-0.392958\pi$
$443$	6153.49	0.659958	0.329979	+	0.943988i	$0.392958\pi$
$444$	0	0
$445$	143.690	0.0153068
$446$	6367.24	0.676003
$447$	0	0
$448$	−18884.5	−1.99154
$449$	−2177.68	−0.228889	−0.114445	−	0.993430i	$-0.536509\pi$
$449$	−2177.68	−0.228889	−0.114445	+	0.993430i	$0.536509\pi$
$450$	0	0
$451$	−15798.6	−1.64951
$452$	−2580.47	−0.268529
$453$	0	0
$454$	−6343.50	−0.655760
$455$	−194.439	−0.0200339
$456$	0	0
$457$	3855.60	0.394655	0.197327	−	0.980338i	$-0.436774\pi$
$457$	3855.60	0.394655	0.197327	+	0.980338i	$0.436774\pi$
$458$	−9848.49	−1.00478
$459$	0	0
$460$	220.118	0.0223110
$461$	−6623.59	−0.669179	−0.334589	−	0.942364i	$-0.608598\pi$
$461$	−6623.59	−0.669179	−0.334589	+	0.942364i	$0.608598\pi$
$462$	0	0
$463$	−14263.7	−1.43173	−0.715863	−	0.698241i	$-0.753966\pi$
$463$	−14263.7	−1.43173	−0.715863	+	0.698241i	$0.753966\pi$
$464$	3331.89	0.333360
$465$	0	0
$466$	12412.1	1.23386
$467$	11083.5	1.09825	0.549124	−	0.835741i	$-0.314961\pi$
$467$	11083.5	1.09825	0.549124	+	0.835741i	$0.314961\pi$
$468$	0	0
$469$	−1134.36	−0.111684
$470$	1815.33	0.178159
$471$	0	0
$472$	11218.9	1.09405
$473$	−26199.5	−2.54684
$474$	0	0
$475$	−12800.9	−1.23652
$476$	3191.35	0.307301
$477$	0	0
$478$	6201.84	0.593443
$479$	−1475.21	−0.140719	−0.0703593	−	0.997522i	$-0.522415\pi$
$479$	−1475.21	−0.140719	−0.0703593	+	0.997522i	$0.522415\pi$
$480$	0	0
$481$	220.305	0.0208836
$482$	608.131	0.0574681
$483$	0	0
$484$	−4395.47	−0.412798
$485$	1126.02	0.105422
$486$	0	0
$487$	9924.78	0.923480	0.461740	−	0.887015i	$-0.347225\pi$
$487$	9924.78	0.923480	0.461740	+	0.887015i	$0.347225\pi$
$488$	18023.6	1.67191
$489$	0	0
$490$	−5630.28	−0.519082
$491$	−5082.89	−0.467185	−0.233592	−	0.972335i	$-0.575048\pi$
$491$	−5082.89	−0.467185	−0.233592	+	0.972335i	$0.575048\pi$
$492$	0	0
$493$	−4068.62	−0.371686
$494$	−552.227	−0.0502953
$495$	0	0
$496$	6880.61	0.622880
$497$	36268.2	3.27334
$498$	0	0
$499$	1117.36	0.100240	0.0501199	−	0.998743i	$-0.484040\pi$
$499$	1117.36	0.100240	0.0501199	+	0.998743i	$0.484040\pi$
$500$	1161.78	0.103913
$501$	0	0
$502$	−9595.72	−0.853143
$503$	−16122.0	−1.42912	−0.714558	−	0.699576i	$-0.753372\pi$
$503$	−16122.0	−1.42912	−0.714558	+	0.699576i	$0.753372\pi$
$504$	0	0
$505$	24.1132	0.00212480
$506$	−7317.53	−0.642893
$507$	0	0
$508$	−3559.36	−0.310868
$509$	5532.31	0.481759	0.240879	−	0.970555i	$-0.422564\pi$
$509$	5532.31	0.481759	0.240879	+	0.970555i	$0.422564\pi$
$510$	0	0
$511$	8687.80	0.752105
$512$	12893.6	1.11293
$513$	0	0
$514$	−16577.8	−1.42260
$515$	−4237.24	−0.362554
$516$	0	0
$517$	15473.6	1.31630
$518$	9271.20	0.786396
$519$	0	0
$520$	142.524	0.0120194
$521$	1750.54	0.147202	0.0736011	−	0.997288i	$-0.476551\pi$
$521$	1750.54	0.147202	0.0736011	+	0.997288i	$0.476551\pi$
$522$	0	0
$523$	−10636.0	−0.889257	−0.444629	−	0.895715i	$-0.646664\pi$
$523$	−10636.0	−0.889257	−0.444629	+	0.895715i	$0.646664\pi$
$524$	−350.305	−0.0292045
$525$	0	0
$526$	15656.1	1.29779
$527$	−8402.02	−0.694493
$528$	0	0
$529$	−10076.8	−0.828206
$530$	−4247.07	−0.348078
$531$	0	0
$532$	5958.78	0.485612
$533$	495.243	0.0402465
$534$	0	0
$535$	−5719.17	−0.462171
$536$	831.484	0.0670049
$537$	0	0
$538$	−5040.63	−0.403935
$539$	−47991.8	−3.83516
$540$	0	0
$541$	−10993.8	−0.873678	−0.436839	−	0.899540i	$-0.643902\pi$
$541$	−10993.8	−0.873678	−0.436839	+	0.899540i	$0.643902\pi$
$542$	−4813.34	−0.381459
$543$	0	0
$544$	−4282.04	−0.337483
$545$	625.217	0.0491401
$546$	0	0
$547$	24443.6	1.91066	0.955331	−	0.295537i	$-0.0954985\pi$
$547$	24443.6	1.91066	0.955331	+	0.295537i	$0.0954985\pi$
$548$	−2889.48	−0.225242
$549$	0	0
$550$	−18615.0	−1.44317
$551$	−7596.79	−0.587357
$552$	0	0
$553$	7906.67	0.608003
$554$	−4130.88	−0.316795
$555$	0	0
$556$	−4762.15	−0.363238
$557$	−15207.1	−1.15681	−0.578405	−	0.815750i	$-0.696325\pi$
$557$	−15207.1	−1.15681	−0.578405	+	0.815750i	$0.696325\pi$
$558$	0	0
$559$	821.283	0.0621406
$560$	4720.66	0.356222
$561$	0	0
$562$	−2259.40	−0.169586
$563$	−21328.6	−1.59662	−0.798308	−	0.602250i	$-0.794271\pi$
$563$	−21328.6	−1.59662	−0.798308	+	0.602250i	$0.794271\pi$
$564$	0	0
$565$	4661.05	0.347065
$566$	19786.6	1.46943
$567$	0	0
$568$	−26584.5	−1.96384
$569$	−4297.28	−0.316611	−0.158305	−	0.987390i	$-0.550603\pi$
$569$	−4297.28	−0.316611	−0.158305	+	0.987390i	$0.550603\pi$
$570$	0	0
$571$	−17590.1	−1.28918	−0.644591	−	0.764527i	$-0.722972\pi$
$571$	−17590.1	−1.28918	−0.644591	+	0.764527i	$0.722972\pi$
$572$	205.905	0.0150512
$573$	0	0
$574$	20841.6	1.51552
$575$	5317.26	0.385644
$576$	0	0
$577$	12080.4	0.871602	0.435801	−	0.900043i	$-0.356465\pi$
$577$	12080.4	0.871602	0.435801	+	0.900043i	$0.356465\pi$
$578$	3629.07	0.261158
$579$	0	0
$580$	332.308	0.0237903
$581$	12662.2	0.904158
$582$	0	0
$583$	−36201.5	−2.57172
$584$	−6368.15	−0.451226
$585$	0	0
$586$	629.745	0.0443934
$587$	−19681.0	−1.38385	−0.691927	−	0.721968i	$-0.743238\pi$
$587$	−19681.0	−1.38385	−0.691927	+	0.721968i	$0.743238\pi$
$588$	0	0
$589$	−15688.0	−1.09747
$590$	−3434.61	−0.239662
$591$	0	0
$592$	−5348.64	−0.371331
$593$	18867.2	1.30655	0.653274	−	0.757122i	$-0.273395\pi$
$593$	18867.2	1.30655	0.653274	+	0.757122i	$0.273395\pi$
$594$	0	0
$595$	−5764.47	−0.397177
$596$	4371.39	0.300435
$597$	0	0
$598$	229.385	0.0156860
$599$	13175.2	0.898705	0.449353	−	0.893355i	$-0.351655\pi$
$599$	13175.2	0.898705	0.449353	+	0.893355i	$0.351655\pi$
$600$	0	0
$601$	20483.0	1.39022	0.695108	−	0.718905i	$-0.255356\pi$
$601$	20483.0	1.39022	0.695108	+	0.718905i	$0.255356\pi$
$602$	34562.5	2.33997
$603$	0	0
$604$	−2568.26	−0.173015
$605$	7939.46	0.533529
$606$	0	0
$607$	−20433.1	−1.36631	−0.683157	−	0.730271i	$-0.739394\pi$
$607$	−20433.1	−1.36631	−0.683157	+	0.730271i	$0.739394\pi$
$608$	−7995.27	−0.533308
$609$	0	0
$610$	−5517.85	−0.366248
$611$	−485.056	−0.0321166
$612$	0	0
$613$	−4411.81	−0.290687	−0.145344	−	0.989381i	$-0.546429\pi$
$613$	−4411.81	−0.290687	−0.145344	+	0.989381i	$0.546429\pi$
$614$	12177.8	0.800415
$615$	0	0
$616$	51125.3	3.34399
$617$	−20503.6	−1.33783	−0.668916	−	0.743338i	$-0.733242\pi$
$617$	−20503.6	−1.33783	−0.668916	+	0.743338i	$0.733242\pi$
$618$	0	0
$619$	27871.0	1.80974	0.904872	−	0.425683i	$-0.139966\pi$
$619$	27871.0	1.80974	0.904872	+	0.425683i	$0.139966\pi$
$620$	686.243	0.0444519
$621$	0	0
$622$	−5111.56	−0.329510
$623$	−1615.75	−0.103906
$624$	0	0
$625$	12439.4	0.796124
$626$	3263.33	0.208353
$627$	0	0
$628$	−4059.96	−0.257978
$629$	6531.31	0.414023
$630$	0	0
$631$	17180.0	1.08388	0.541938	−	0.840418i	$-0.317691\pi$
$631$	17180.0	1.08388	0.541938	+	0.840418i	$0.317691\pi$
$632$	−5795.58	−0.364772
$633$	0	0
$634$	23991.9	1.50290
$635$	6429.20	0.401787
$636$	0	0
$637$	1504.41	0.0935745
$638$	−11047.2	−0.685520
$639$	0	0
$640$	−2524.03	−0.155892
$641$	1356.40	0.0835794	0.0417897	−	0.999126i	$-0.486694\pi$
$641$	1356.40	0.0835794	0.0417897	+	0.999126i	$0.486694\pi$
$642$	0	0
$643$	−7520.54	−0.461246	−0.230623	−	0.973043i	$-0.574076\pi$
$643$	−7520.54	−0.461246	−0.230623	+	0.973043i	$0.574076\pi$
$644$	−2475.16	−0.151452
$645$	0	0
$646$	−16371.7	−0.997116
$647$	−5359.06	−0.325636	−0.162818	−	0.986656i	$-0.552058\pi$
$647$	−5359.06	−0.325636	−0.162818	+	0.986656i	$0.552058\pi$
$648$	0	0
$649$	−29276.1	−1.77071
$650$	583.528	0.0352121
$651$	0	0
$652$	−1281.75	−0.0769898
$653$	2232.60	0.133795	0.0668977	−	0.997760i	$-0.478690\pi$
$653$	2232.60	0.133795	0.0668977	+	0.997760i	$0.478690\pi$
$654$	0	0
$655$	632.749	0.0377459
$656$	−12023.7	−0.715620
$657$	0	0
$658$	−20412.8	−1.20939
$659$	−8997.69	−0.531867	−0.265934	−	0.963991i	$-0.585680\pi$
$659$	−8997.69	−0.531867	−0.265934	+	0.963991i	$0.585680\pi$
$660$	0	0
$661$	23110.3	1.35989	0.679945	−	0.733263i	$-0.262004\pi$
$661$	23110.3	1.35989	0.679945	+	0.733263i	$0.262004\pi$
$662$	19512.5	1.14558
$663$	0	0
$664$	−9281.37	−0.542450
$665$	−10763.2	−0.627639
$666$	0	0
$667$	3155.56	0.183184
$668$	635.241	0.0367937
$669$	0	0
$670$	−254.555	−0.0146781
$671$	−47033.5	−2.70597
$672$	0	0
$673$	5668.67	0.324683	0.162341	−	0.986735i	$-0.448095\pi$
$673$	5668.67	0.324683	0.162341	+	0.986735i	$0.448095\pi$
$674$	−3978.71	−0.227380
$675$	0	0
$676$	3580.43	0.203711
$677$	1026.23	0.0582591	0.0291295	−	0.999576i	$-0.490726\pi$
$677$	1026.23	0.0582591	0.0291295	+	0.999576i	$0.490726\pi$
$678$	0	0
$679$	−12661.8	−0.715631
$680$	4225.36	0.238287
$681$	0	0
$682$	−22813.3	−1.28089
$683$	10816.7	0.605989	0.302994	−	0.952992i	$-0.402014\pi$
$683$	10816.7	0.605989	0.302994	+	0.952992i	$0.402014\pi$
$684$	0	0
$685$	5219.21	0.291118
$686$	34610.0	1.92626
$687$	0	0
$688$	−19939.4	−1.10492
$689$	1134.82	0.0627477
$690$	0	0
$691$	−4149.05	−0.228419	−0.114209	−	0.993457i	$-0.536433\pi$
$691$	−4149.05	−0.228419	−0.114209	+	0.993457i	$0.536433\pi$
$692$	1485.18	0.0815869
$693$	0	0
$694$	214.951	0.0117571
$695$	8601.78	0.469474
$696$	0	0
$697$	14682.3	0.797896
$698$	27654.5	1.49962
$699$	0	0
$700$	−6296.53	−0.339981
$701$	−16684.2	−0.898937	−0.449468	−	0.893296i	$-0.648387\pi$
$701$	−16684.2	−0.898937	−0.449468	+	0.893296i	$0.648387\pi$
$702$	0	0
$703$	12195.0	0.654260
$704$	−36122.2	−1.93382
$705$	0	0
$706$	−14471.7	−0.771459
$707$	−271.147	−0.0144236
$708$	0	0
$709$	−4556.14	−0.241339	−0.120670	−	0.992693i	$-0.538504\pi$
$709$	−4556.14	−0.241339	−0.120670	+	0.992693i	$0.538504\pi$
$710$	8138.74	0.430199
$711$	0	0
$712$	1184.34	0.0623387
$713$	6516.48	0.342278
$714$	0	0
$715$	−371.921	−0.0194532
$716$	−1674.30	−0.0873903
$717$	0	0
$718$	24216.8	1.25872
$719$	19259.4	0.998966	0.499483	−	0.866324i	$-0.333523\pi$
$719$	19259.4	0.998966	0.499483	+	0.866324i	$0.333523\pi$
$720$	0	0
$721$	47646.6	2.46110
$722$	−13261.0	−0.683548
$723$	0	0
$724$	2548.92	0.130842
$725$	8027.38	0.411213
$726$	0	0
$727$	1342.92	0.0685094	0.0342547	−	0.999413i	$-0.489094\pi$
$727$	1342.92	0.0685094	0.0342547	+	0.999413i	$0.489094\pi$
$728$	−1602.64	−0.0815902
$729$	0	0
$730$	1949.58	0.0988455
$731$	24348.3	1.23195
$732$	0	0
$733$	15828.0	0.797573	0.398786	−	0.917044i	$-0.369431\pi$
$733$	15828.0	0.797573	0.398786	+	0.917044i	$0.369431\pi$
$734$	−24360.0	−1.22499
$735$	0	0
$736$	3321.08	0.166327
$737$	−2169.79	−0.108447
$738$	0	0
$739$	36440.2	1.81390	0.906950	−	0.421237i	$-0.138404\pi$
$739$	36440.2	1.81390	0.906950	+	0.421237i	$0.138404\pi$
$740$	−533.451	−0.0265001
$741$	0	0
$742$	47757.2	2.36283
$743$	1764.82	0.0871399	0.0435700	−	0.999050i	$-0.486127\pi$
$743$	1764.82	0.0871399	0.0435700	+	0.999050i	$0.486127\pi$
$744$	0	0
$745$	−7895.96	−0.388303
$746$	−6636.92	−0.325730
$747$	0	0
$748$	6104.40	0.298394
$749$	64310.5	3.13732
$750$	0	0
$751$	21906.0	1.06440	0.532199	−	0.846619i	$-0.321366\pi$
$751$	21906.0	1.06440	0.532199	+	0.846619i	$0.321366\pi$
$752$	11776.4	0.571063
$753$	0	0
$754$	346.299	0.0167261
$755$	4639.00	0.223616
$756$	0	0
$757$	36776.8	1.76576	0.882878	−	0.469603i	$-0.155603\pi$
$757$	36776.8	1.76576	0.882878	+	0.469603i	$0.155603\pi$
$758$	7039.15	0.337300
$759$	0	0
$760$	7889.44	0.376553
$761$	23531.7	1.12092	0.560462	−	0.828180i	$-0.310624\pi$
$761$	23531.7	1.12092	0.560462	+	0.828180i	$0.310624\pi$
$762$	0	0
$763$	−7030.39	−0.333574
$764$	−2669.52	−0.126414
$765$	0	0
$766$	25804.5	1.21717
$767$	917.727	0.0432037
$768$	0	0
$769$	−18469.7	−0.866106	−0.433053	−	0.901368i	$-0.642564\pi$
$769$	−18469.7	−0.866106	−0.433053	+	0.901368i	$0.642564\pi$
$770$	−15651.8	−0.732533
$771$	0	0
$772$	−210.549	−0.00981584
$773$	−2075.26	−0.0965612	−0.0482806	−	0.998834i	$-0.515374\pi$
$773$	−2075.26	−0.0965612	−0.0482806	+	0.998834i	$0.515374\pi$
$774$	0	0
$775$	16577.2	0.768349
$776$	9281.06	0.429344
$777$	0	0
$778$	19364.1	0.892333
$779$	27414.4	1.26087
$780$	0	0
$781$	69373.5	3.17846
$782$	6800.50	0.310979
$783$	0	0
$784$	−36524.7	−1.66384
$785$	7333.43	0.333428
$786$	0	0
$787$	11473.9	0.519697	0.259849	−	0.965649i	$-0.416327\pi$
$787$	11473.9	0.519697	0.259849	+	0.965649i	$0.416327\pi$
$788$	7574.86	0.342441
$789$	0	0
$790$	1774.29	0.0799069
$791$	−52412.2	−2.35596
$792$	0	0
$793$	1474.37	0.0660233
$794$	21715.7	0.970607
$795$	0	0
$796$	−3257.73	−0.145059
$797$	−23163.7	−1.02949	−0.514743	−	0.857345i	$-0.672113\pi$
$797$	−23163.7	−1.02949	−0.514743	+	0.857345i	$0.672113\pi$
$798$	0	0
$799$	−14380.3	−0.636719
$800$	8448.45	0.373372
$801$	0	0
$802$	20890.0	0.919763
$803$	16618.0	0.730306
$804$	0	0
$805$	4470.84	0.195747
$806$	715.134	0.0312525
$807$	0	0
$808$	198.750	0.00865348
$809$	−21358.0	−0.928194	−0.464097	−	0.885785i	$-0.653621\pi$
$809$	−21358.0	−0.928194	−0.464097	+	0.885785i	$0.653621\pi$
$810$	0	0
$811$	−23430.2	−1.01448	−0.507240	−	0.861805i	$-0.669334\pi$
$811$	−23430.2	−1.01448	−0.507240	+	0.861805i	$0.669334\pi$
$812$	−3736.71	−0.161494
$813$	0	0
$814$	17733.9	0.763603
$815$	2315.21	0.0995069
$816$	0	0
$817$	45462.4	1.94679
$818$	−22092.1	−0.944293
$819$	0	0
$820$	−1199.19	−0.0510703
$821$	−3446.33	−0.146502	−0.0732508	−	0.997314i	$-0.523337\pi$
$821$	−3446.33	−0.146502	−0.0732508	+	0.997314i	$0.523337\pi$
$822$	0	0
$823$	−26554.1	−1.12469	−0.562345	−	0.826903i	$-0.690101\pi$
$823$	−26554.1	−1.12469	−0.562345	+	0.826903i	$0.690101\pi$
$824$	−34925.0	−1.47654
$825$	0	0
$826$	38621.2	1.62688
$827$	−5217.79	−0.219396	−0.109698	−	0.993965i	$-0.534988\pi$
$827$	−5217.79	−0.219396	−0.109698	+	0.993965i	$0.534988\pi$
$828$	0	0
$829$	−18254.2	−0.764771	−0.382386	−	0.924003i	$-0.624897\pi$
$829$	−18254.2	−0.764771	−0.382386	+	0.924003i	$0.624897\pi$
$830$	2841.45	0.118829
$831$	0	0
$832$	1132.33	0.0471834
$833$	44600.9	1.85514
$834$	0	0
$835$	−1147.42	−0.0475548
$836$	11397.9	0.471537
$837$	0	0
$838$	25324.2	1.04393
$839$	−40927.2	−1.68410	−0.842052	−	0.539397i	$-0.818652\pi$
$839$	−40927.2	−1.68410	−0.842052	+	0.539397i	$0.818652\pi$
$840$	0	0
$841$	−19625.1	−0.804670
$842$	32234.4	1.31932
$843$	0	0
$844$	4072.45	0.166090
$845$	−6467.26	−0.263291
$846$	0	0
$847$	−89277.0	−3.62172
$848$	−27551.5	−1.11571
$849$	0	0
$850$	17299.7	0.698088
$851$	−5065.59	−0.204049
$852$	0	0
$853$	−34434.4	−1.38219	−0.691097	−	0.722762i	$-0.742872\pi$
$853$	−34434.4	−1.38219	−0.691097	+	0.722762i	$0.742872\pi$
$854$	62046.7	2.48618
$855$	0	0
$856$	−47139.6	−1.88224
$857$	5759.74	0.229579	0.114789	−	0.993390i	$-0.463381\pi$
$857$	5759.74	0.229579	0.114789	+	0.993390i	$0.463381\pi$
$858$	0	0
$859$	−24439.7	−0.970747	−0.485373	−	0.874307i	$-0.661316\pi$
$859$	−24439.7	−0.970747	−0.485373	+	0.874307i	$0.661316\pi$
$860$	−1988.67	−0.0788526
$861$	0	0
$862$	−36225.9	−1.43139
$863$	4977.39	0.196330	0.0981648	−	0.995170i	$-0.468703\pi$
$863$	4977.39	0.196330	0.0981648	+	0.995170i	$0.468703\pi$
$864$	0	0
$865$	−2682.66	−0.105449
$866$	−4888.35	−0.191816
$867$	0	0
$868$	−7716.61	−0.301750
$869$	15123.8	0.590381
$870$	0	0
$871$	68.0171	0.00264601
$872$	5153.27	0.200128
$873$	0	0
$874$	12697.7	0.491424
$875$	23597.0	0.911686
$876$	0	0
$877$	−6648.16	−0.255978	−0.127989	−	0.991776i	$-0.540852\pi$
$877$	−6648.16	−0.255978	−0.127989	+	0.991776i	$0.540852\pi$
$878$	−34489.3	−1.32569
$879$	0	0
$880$	9029.65	0.345897
$881$	12545.0	0.479742	0.239871	−	0.970805i	$-0.422895\pi$
$881$	12545.0	0.479742	0.239871	+	0.970805i	$0.422895\pi$
$882$	0	0
$883$	39011.6	1.48680	0.743400	−	0.668847i	$-0.233212\pi$
$883$	39011.6	1.48680	0.743400	+	0.668847i	$0.233212\pi$
$884$	−191.356	−0.00728055
$885$	0	0
$886$	−15527.5	−0.588778
$887$	−7985.71	−0.302293	−0.151147	−	0.988511i	$-0.548297\pi$
$887$	−7985.71	−0.302293	−0.151147	+	0.988511i	$0.548297\pi$
$888$	0	0
$889$	−72294.5	−2.72742
$890$	−362.581	−0.0136559
$891$	0	0
$892$	4119.63	0.154636
$893$	−26850.4	−1.00618
$894$	0	0
$895$	3024.25	0.112949
$896$	28382.0	1.05823
$897$	0	0
$898$	5495.09	0.204202
$899$	9837.83	0.364972
$900$	0	0
$901$	33643.6	1.24399
$902$	39865.7	1.47160
$903$	0	0
$904$	38418.1	1.41346
$905$	−4604.06	−0.169110
$906$	0	0
$907$	26238.5	0.960568	0.480284	−	0.877113i	$-0.340534\pi$
$907$	26238.5	0.960568	0.480284	+	0.877113i	$0.340534\pi$
$908$	−4104.27	−0.150006
$909$	0	0
$910$	490.640	0.0178731
$911$	−14570.2	−0.529894	−0.264947	−	0.964263i	$-0.585355\pi$
$911$	−14570.2	−0.529894	−0.264947	+	0.964263i	$0.585355\pi$
$912$	0	0
$913$	24220.1	0.877951
$914$	−9729.08	−0.352089
$915$	0	0
$916$	−6372.01	−0.229844
$917$	−7115.09	−0.256228
$918$	0	0
$919$	−5591.79	−0.200714	−0.100357	−	0.994952i	$-0.531998\pi$
$919$	−5591.79	−0.200714	−0.100357	+	0.994952i	$0.531998\pi$
$920$	−3277.12	−0.117439
$921$	0	0
$922$	16713.7	0.597004
$923$	−2174.67	−0.0775516
$924$	0	0
$925$	−12886.3	−0.458052
$926$	35992.4	1.27731
$927$	0	0
$928$	5013.79	0.177355
$929$	−12439.5	−0.439318	−0.219659	−	0.975577i	$-0.570494\pi$
$929$	−12439.5	−0.439318	−0.219659	+	0.975577i	$0.570494\pi$
$930$	0	0
$931$	83277.2	2.93158
$932$	8030.66	0.282246
$933$	0	0
$934$	−27967.7	−0.979796
$935$	−11026.2	−0.385665
$936$	0	0
$937$	−35372.8	−1.23327	−0.616637	−	0.787247i	$-0.711506\pi$
$937$	−35372.8	−1.23327	−0.616637	+	0.787247i	$0.711506\pi$
$938$	2862.40	0.0996382
$939$	0	0
$940$	1174.52	0.0407540
$941$	−42606.0	−1.47600	−0.738000	−	0.674801i	$-0.764229\pi$
$941$	−42606.0	−1.47600	−0.738000	+	0.674801i	$0.764229\pi$
$942$	0	0
$943$	−11387.4	−0.393240
$944$	−22280.9	−0.768202
$945$	0	0
$946$	66110.9	2.27215
$947$	−43877.3	−1.50562	−0.752810	−	0.658238i	$-0.771302\pi$
$947$	−43877.3	−1.50562	−0.752810	+	0.658238i	$0.771302\pi$
$948$	0	0
$949$	−520.928	−0.0178188
$950$	32301.4	1.10315
$951$	0	0
$952$	−47512.9	−1.61755
$953$	41703.0	1.41752	0.708758	−	0.705452i	$-0.249256\pi$
$953$	41703.0	1.41752	0.708758	+	0.705452i	$0.249256\pi$
$954$	0	0
$955$	4821.91	0.163386
$956$	4012.62	0.135750
$957$	0	0
$958$	3722.50	0.125541
$959$	−58688.6	−1.97618
$960$	0	0
$961$	−9475.10	−0.318052
$962$	−555.909	−0.0186312
$963$	0	0
$964$	393.463	0.0131459
$965$	380.311	0.0126867
$966$	0	0
$967$	−12991.5	−0.432034	−0.216017	−	0.976390i	$-0.569307\pi$
$967$	−12991.5	−0.432034	−0.216017	+	0.976390i	$0.569307\pi$
$968$	65440.0	2.17285
$969$	0	0
$970$	−2841.35	−0.0940519
$971$	−14607.3	−0.482770	−0.241385	−	0.970429i	$-0.577602\pi$
$971$	−14607.3	−0.482770	−0.241385	+	0.970429i	$0.577602\pi$
$972$	0	0
$973$	−96724.7	−3.18690
$974$	−25043.8	−0.823877
$975$	0	0
$976$	−35795.3	−1.17396
$977$	44709.9	1.46407	0.732035	−	0.681267i	$-0.238571\pi$
$977$	44709.9	1.46407	0.732035	+	0.681267i	$0.238571\pi$
$978$	0	0
$979$	−3090.60	−0.100895
$980$	−3642.82	−0.118740
$981$	0	0
$982$	12826.0	0.416796
$983$	−44616.7	−1.44766	−0.723831	−	0.689977i	$-0.757621\pi$
$983$	−44616.7	−1.44766	−0.723831	+	0.689977i	$0.757621\pi$
$984$	0	0
$985$	−13682.3	−0.442594
$986$	10266.6	0.331598
$987$	0	0
$988$	−357.294	−0.0115051
$989$	−18884.2	−0.607162
$990$	0	0
$991$	19015.9	0.609545	0.304772	−	0.952425i	$-0.401420\pi$
$991$	19015.9	0.609545	0.304772	+	0.952425i	$0.401420\pi$
$992$	10353.9	0.331387
$993$	0	0
$994$	−91517.8	−2.92029
$995$	5884.38	0.187485
$996$	0	0
$997$	−10518.0	−0.334112	−0.167056	−	0.985947i	$-0.553426\pi$
$997$	−10518.0	−0.334112	−0.167056	+	0.985947i	$0.553426\pi$
$998$	−2819.50	−0.0894284
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2169.4.a.e.1.11		33
3.2	odd	2		241.4.a.b.1.23	✓	33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
241.4.a.b.1.23	✓	33	3.2	odd	2
2169.4.a.e.1.11		33	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:	\( N \)	\(=\)	\( 2169 = 3^{2} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2169.a (trivial)

\(f(q)\)	\(=\)	\(q-2.52337 q^{2} -1.63263 q^{4} +2.94898 q^{5} -33.1605 q^{7} +24.3066 q^{8} +O(q^{10})\)	\(q-2.52337 q^{2} -1.63263 q^{4} +2.94898 q^{5} -33.1605 q^{7} +24.3066 q^{8} -7.44136 q^{10} -63.4292 q^{11} +1.98833 q^{13} +83.6761 q^{14} -48.2735 q^{16} +58.9476 q^{17} +110.065 q^{19} -4.81460 q^{20} +160.055 q^{22} -45.7188 q^{23} -116.303 q^{25} -5.01729 q^{26} +54.1388 q^{28} -69.0210 q^{29} -142.534 q^{31} -72.6414 q^{32} -148.746 q^{34} -97.7898 q^{35} +110.799 q^{37} -277.734 q^{38} +71.6799 q^{40} +249.075 q^{41} +413.051 q^{43} +103.556 q^{44} +115.365 q^{46} -243.951 q^{47} +756.619 q^{49} +293.476 q^{50} -3.24621 q^{52} +570.739 q^{53} -187.052 q^{55} -806.021 q^{56} +174.165 q^{58} +461.556 q^{59} +741.511 q^{61} +359.665 q^{62} +569.489 q^{64} +5.86356 q^{65} +34.2081 q^{67} -96.2395 q^{68} +246.759 q^{70} -1093.72 q^{71} -261.992 q^{73} -279.585 q^{74} -179.695 q^{76} +2103.35 q^{77} -238.436 q^{79} -142.358 q^{80} -628.506 q^{82} -381.845 q^{83} +173.835 q^{85} -1042.28 q^{86} -1541.75 q^{88} +48.7251 q^{89} -65.9341 q^{91} +74.6419 q^{92} +615.577 q^{94} +324.580 q^{95} +381.832 q^{97} -1909.23 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q - 9 q^{2} + 161 q^{4} - 50 q^{5} + 58 q^{7} - 111 q^{8}+O(q^{10}) \) 33 * q - 9 * q^2 + 161 * q^4 - 50 * q^5 + 58 * q^7 - 111 * q^8	\( 33 q - 9 q^{2} + 161 q^{4} - 50 q^{5} + 58 q^{7} - 111 q^{8} + q^{10} - 274 q^{11} + 24 q^{13} - 220 q^{14} + 789 q^{16} - 140 q^{17} + 164 q^{19} - 630 q^{20} + 270 q^{22} - 980 q^{23} + 1121 q^{25} - 484 q^{26} + 433 q^{28} - 772 q^{29} + 804 q^{31} - 947 q^{32} + 211 q^{34} - 558 q^{35} + 342 q^{37} - 1092 q^{38} + 564 q^{40} - 978 q^{41} + 230 q^{43} - 2788 q^{44} + 1103 q^{46} - 2312 q^{47} + 2501 q^{49} - 1787 q^{50} - 57 q^{52} - 1352 q^{53} + 1058 q^{55} - 2367 q^{56} + 2237 q^{58} - 3546 q^{59} + 156 q^{61} - 1349 q^{62} + 47 q^{64} - 900 q^{65} - 236 q^{67} + 3439 q^{68} - 7312 q^{70} - 5966 q^{71} - 2020 q^{73} - 488 q^{74} - 7552 q^{76} + 8 q^{77} - 198 q^{79} + 261 q^{80} - 4465 q^{82} - 502 q^{83} - 4778 q^{85} + 185 q^{86} - 9372 q^{88} + 192 q^{89} + 1896 q^{91} - 3214 q^{92} - 14222 q^{94} - 5610 q^{95} - 846 q^{97} + 3726 q^{98}+O(q^{100}) \) 33 * q - 9 * q^2 + 161 * q^4 - 50 * q^5 + 58 * q^7 - 111 * q^8 + q^10 - 274 * q^11 + 24 * q^13 - 220 * q^14 + 789 * q^16 - 140 * q^17 + 164 * q^19 - 630 * q^20 + 270 * q^22 - 980 * q^23 + 1121 * q^25 - 484 * q^26 + 433 * q^28 - 772 * q^29 + 804 * q^31 - 947 * q^32 + 211 * q^34 - 558 * q^35 + 342 * q^37 - 1092 * q^38 + 564 * q^40 - 978 * q^41 + 230 * q^43 - 2788 * q^44 + 1103 * q^46 - 2312 * q^47 + 2501 * q^49 - 1787 * q^50 - 57 * q^52 - 1352 * q^53 + 1058 * q^55 - 2367 * q^56 + 2237 * q^58 - 3546 * q^59 + 156 * q^61 - 1349 * q^62 + 47 * q^64 - 900 * q^65 - 236 * q^67 + 3439 * q^68 - 7312 * q^70 - 5966 * q^71 - 2020 * q^73 - 488 * q^74 - 7552 * q^76 + 8 * q^77 - 198 * q^79 + 261 * q^80 - 4465 * q^82 - 502 * q^83 - 4778 * q^85 + 185 * q^86 - 9372 * q^88 + 192 * q^89 + 1896 * q^91 - 3214 * q^92 - 14222 * q^94 - 5610 * q^95 - 846 * q^97 + 3726 * q^98

Introduction

L-functions

Modular forms

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Database

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