LMFDB - Embedded newform 522.4.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [522,4,Mod(1,522)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(522, 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("522.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 522 = 2 \cdot 3^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	522.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:	$30.7989970230$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.19816.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 42x - 54 $ x^3 - x^2 - 42*x - 54
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 58)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$7.53003$ of defining polynomial
Character	$\chi$	$=$	522.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +4.00000 q^{4} -20.9205 q^{5} +8.55839 q^{7} -8.00000 q^{8} +O(q^{10})$	\(q-2.00000 q^{2} +4.00000 q^{4} -20.9205 q^{5} +8.55839 q^{7} -8.00000 q^{8} +41.8409 q^{10} -10.8092 q^{11} +54.7046 q^{13} -17.1168 q^{14} +16.0000 q^{16} +106.127 q^{17} -113.636 q^{19} -83.6818 q^{20} +21.6184 q^{22} +112.855 q^{23} +312.666 q^{25} -109.409 q^{26} +34.2336 q^{28} -29.0000 q^{29} -102.805 q^{31} -32.0000 q^{32} -212.254 q^{34} -179.045 q^{35} -105.665 q^{37} +227.272 q^{38} +167.364 q^{40} -216.958 q^{41} -102.230 q^{43} -43.2369 q^{44} -225.711 q^{46} -455.212 q^{47} -269.754 q^{49} -625.331 q^{50} +218.819 q^{52} +593.714 q^{53} +226.134 q^{55} -68.4671 q^{56} +58.0000 q^{58} +558.141 q^{59} -473.986 q^{61} +205.610 q^{62} +64.0000 q^{64} -1144.45 q^{65} +193.132 q^{67} +424.507 q^{68} +358.091 q^{70} +2.38155 q^{71} +119.013 q^{73} +211.330 q^{74} -454.545 q^{76} -92.5096 q^{77} -964.306 q^{79} -334.727 q^{80} +433.915 q^{82} -1068.19 q^{83} -2220.22 q^{85} +204.459 q^{86} +86.4738 q^{88} -772.544 q^{89} +468.184 q^{91} +451.421 q^{92} +910.423 q^{94} +2377.32 q^{95} +1344.03 q^{97} +539.508 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 6 q^{2} + 12 q^{4} - 20 q^{5} + 24 q^{7} - 24 q^{8}+O(q^{10}) $ 3 * q - 6 * q^2 + 12 * q^4 - 20 * q^5 + 24 * q^7 - 24 * q^8	$ 3 q - 6 q^{2} + 12 q^{4} - 20 q^{5} + 24 q^{7} - 24 q^{8} + 40 q^{10} - 10 q^{11} - 4 q^{13} - 48 q^{14} + 48 q^{16} + 66 q^{17} - 164 q^{19} - 80 q^{20} + 20 q^{22} + 204 q^{23} + 79 q^{25} + 8 q^{26} + 96 q^{28} - 87 q^{29} - 86 q^{31} - 96 q^{32} - 132 q^{34} - 24 q^{35} - 42 q^{37} + 328 q^{38} + 160 q^{40} - 562 q^{41} + 18 q^{43} - 40 q^{44} - 408 q^{46} - 654 q^{47} + 539 q^{49} - 158 q^{50} - 16 q^{52} - 712 q^{53} + 142 q^{55} - 192 q^{56} + 174 q^{58} - 184 q^{59} + 322 q^{61} + 172 q^{62} + 192 q^{64} - 1494 q^{65} - 228 q^{67} + 264 q^{68} + 48 q^{70} + 52 q^{71} - 494 q^{73} + 84 q^{74} - 656 q^{76} - 872 q^{77} - 2110 q^{79} - 320 q^{80} + 1124 q^{82} + 288 q^{83} - 2704 q^{85} - 36 q^{86} + 80 q^{88} - 914 q^{89} - 2984 q^{91} + 816 q^{92} + 1308 q^{94} + 1900 q^{95} + 218 q^{97} - 1078 q^{98}+O(q^{100}) $ 3 * q - 6 * q^2 + 12 * q^4 - 20 * q^5 + 24 * q^7 - 24 * q^8 + 40 * q^10 - 10 * q^11 - 4 * q^13 - 48 * q^14 + 48 * q^16 + 66 * q^17 - 164 * q^19 - 80 * q^20 + 20 * q^22 + 204 * q^23 + 79 * q^25 + 8 * q^26 + 96 * q^28 - 87 * q^29 - 86 * q^31 - 96 * q^32 - 132 * q^34 - 24 * q^35 - 42 * q^37 + 328 * q^38 + 160 * q^40 - 562 * q^41 + 18 * q^43 - 40 * q^44 - 408 * q^46 - 654 * q^47 + 539 * q^49 - 158 * q^50 - 16 * q^52 - 712 * q^53 + 142 * q^55 - 192 * q^56 + 174 * q^58 - 184 * q^59 + 322 * q^61 + 172 * q^62 + 192 * q^64 - 1494 * q^65 - 228 * q^67 + 264 * q^68 + 48 * q^70 + 52 * q^71 - 494 * q^73 + 84 * q^74 - 656 * q^76 - 872 * q^77 - 2110 * q^79 - 320 * q^80 + 1124 * q^82 + 288 * q^83 - 2704 * q^85 - 36 * q^86 + 80 * q^88 - 914 * q^89 - 2984 * q^91 + 816 * q^92 + 1308 * q^94 + 1900 * q^95 + 218 * q^97 - 1078 * 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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	−20.9205	−1.87118	−0.935591	−	0.353085i	$-0.885133\pi$
$5$	−20.9205	−1.87118	−0.935591	+	0.353085i	$0.885133\pi$
$6$	0	0
$7$	8.55839	0.462110	0.231055	−	0.972941i	$-0.425782\pi$
$7$	8.55839	0.462110	0.231055	+	0.972941i	$0.425782\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	41.8409	1.32313
$11$	−10.8092	−0.296282	−0.148141	−	0.988966i	$-0.547329\pi$
$11$	−10.8092	−0.296282	−0.148141	+	0.988966i	$0.547329\pi$
$12$	0	0
$13$	54.7046	1.16710	0.583551	−	0.812076i	$-0.301663\pi$
$13$	54.7046	1.16710	0.583551	+	0.812076i	$0.301663\pi$
$14$	−17.1168	−0.326761
$15$	0	0
$16$	16.0000	0.250000
$17$	106.127	1.51409	0.757045	−	0.653363i	$-0.226642\pi$
$17$	106.127	1.51409	0.757045	+	0.653363i	$0.226642\pi$
$18$	0	0
$19$	−113.636	−1.37210	−0.686051	−	0.727554i	$-0.740657\pi$
$19$	−113.636	−1.37210	−0.686051	+	0.727554i	$0.740657\pi$
$20$	−83.6818	−0.935591
$21$	0	0
$22$	21.6184	0.209503
$23$	112.855	1.02313	0.511564	−	0.859245i	$-0.329066\pi$
$23$	112.855	1.02313	0.511564	+	0.859245i	$0.329066\pi$
$24$	0	0
$25$	312.666	2.50132
$26$	−109.409	−0.825266
$27$	0	0
$28$	34.2336	0.231055
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	−102.805	−0.595622	−0.297811	−	0.954625i	$-0.596257\pi$
$31$	−102.805	−0.595622	−0.297811	+	0.954625i	$0.596257\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	−212.254	−1.07062
$35$	−179.045	−0.864692
$36$	0	0
$37$	−105.665	−0.469493	−0.234746	−	0.972057i	$-0.575426\pi$
$37$	−105.665	−0.469493	−0.234746	+	0.972057i	$0.575426\pi$
$38$	227.272	0.970222
$39$	0	0
$40$	167.364	0.661563
$41$	−216.958	−0.826417	−0.413209	−	0.910636i	$-0.635592\pi$
$41$	−216.958	−0.826417	−0.413209	+	0.910636i	$0.635592\pi$
$42$	0	0
$43$	−102.230	−0.362555	−0.181278	−	0.983432i	$-0.558023\pi$
$43$	−102.230	−0.362555	−0.181278	+	0.983432i	$0.558023\pi$
$44$	−43.2369	−0.148141
$45$	0	0
$46$	−225.711	−0.723461
$47$	−455.212	−1.41275	−0.706377	−	0.707836i	$-0.749672\pi$
$47$	−455.212	−1.41275	−0.706377	+	0.707836i	$0.749672\pi$
$48$	0	0
$49$	−269.754	−0.786455
$50$	−625.331	−1.76870
$51$	0	0
$52$	218.819	0.583551
$53$	593.714	1.53873	0.769367	−	0.638807i	$-0.220572\pi$
$53$	593.714	1.53873	0.769367	+	0.638807i	$0.220572\pi$
$54$	0	0
$55$	226.134	0.554398
$56$	−68.4671	−0.163380
$57$	0	0
$58$	58.0000	0.131306
$59$	558.141	1.23159	0.615794	−	0.787907i	$-0.288835\pi$
$59$	558.141	1.23159	0.615794	+	0.787907i	$0.288835\pi$
$60$	0	0
$61$	−473.986	−0.994880	−0.497440	−	0.867498i	$-0.665727\pi$
$61$	−473.986	−0.994880	−0.497440	+	0.867498i	$0.665727\pi$
$62$	205.610	0.421168
$63$	0	0
$64$	64.0000	0.125000
$65$	−1144.45	−2.18386
$66$	0	0
$67$	193.132	0.352162	0.176081	−	0.984376i	$-0.443658\pi$
$67$	193.132	0.352162	0.176081	+	0.984376i	$0.443658\pi$
$68$	424.507	0.757045
$69$	0	0
$70$	358.091	0.611429
$71$	2.38155	0.00398082	0.00199041	−	0.999998i	$-0.499366\pi$
$71$	2.38155	0.00398082	0.00199041	+	0.999998i	$0.499366\pi$
$72$	0	0
$73$	119.013	0.190814	0.0954071	−	0.995438i	$-0.469585\pi$
$73$	119.013	0.190814	0.0954071	+	0.995438i	$0.469585\pi$
$74$	211.330	0.331981
$75$	0	0
$76$	−454.545	−0.686051
$77$	−92.5096	−0.136915
$78$	0	0
$79$	−964.306	−1.37333	−0.686664	−	0.726975i	$-0.740926\pi$
$79$	−964.306	−1.37333	−0.686664	+	0.726975i	$0.740926\pi$
$80$	−334.727	−0.467796
$81$	0	0
$82$	433.915	0.584365
$83$	−1068.19	−1.41264	−0.706319	−	0.707893i	$-0.749646\pi$
$83$	−1068.19	−1.41264	−0.706319	+	0.707893i	$0.749646\pi$
$84$	0	0
$85$	−2220.22	−2.83314
$86$	204.459	0.256365
$87$	0	0
$88$	86.4738	0.104752
$89$	−772.544	−0.920106	−0.460053	−	0.887891i	$-0.652170\pi$
$89$	−772.544	−0.920106	−0.460053	+	0.887891i	$0.652170\pi$
$90$	0	0
$91$	468.184	0.539330
$92$	451.421	0.511564
$93$	0	0
$94$	910.423	0.998968
$95$	2377.32	2.56745
$96$	0	0
$97$	1344.03	1.40686	0.703431	−	0.710763i	$-0.251650\pi$
$97$	1344.03	1.40686	0.703431	+	0.710763i	$0.251650\pi$
$98$	539.508	0.556107
$99$	0	0
$100$	1250.66	1.25066
$101$	−986.733	−0.972115	−0.486057	−	0.873927i	$-0.661565\pi$
$101$	−986.733	−0.972115	−0.486057	+	0.873927i	$0.661565\pi$
$102$	0	0
$103$	548.272	0.524493	0.262247	−	0.965001i	$-0.415537\pi$
$103$	548.272	0.524493	0.262247	+	0.965001i	$0.415537\pi$
$104$	−437.637	−0.412633
$105$	0	0
$106$	−1187.43	−1.08805
$107$	−1387.51	−1.25361	−0.626803	−	0.779178i	$-0.715637\pi$
$107$	−1387.51	−1.25361	−0.626803	+	0.779178i	$0.715637\pi$
$108$	0	0
$109$	−1293.32	−1.13649	−0.568246	−	0.822859i	$-0.692378\pi$
$109$	−1293.32	−1.13649	−0.568246	+	0.822859i	$0.692378\pi$
$110$	−452.268	−0.392019
$111$	0	0
$112$	136.934	0.115527
$113$	−302.883	−0.252149	−0.126075	−	0.992021i	$-0.540238\pi$
$113$	−302.883	−0.252149	−0.126075	+	0.992021i	$0.540238\pi$
$114$	0	0
$115$	−2360.99	−1.91446
$116$	−116.000	−0.0928477
$117$	0	0
$118$	−1116.28	−0.870865
$119$	908.274	0.699676
$120$	0	0
$121$	−1214.16	−0.912217
$122$	947.972	0.703487
$123$	0	0
$124$	−411.219	−0.297811
$125$	−3926.05	−2.80925
$126$	0	0
$127$	2021.68	1.41256	0.706281	−	0.707931i	$-0.250371\pi$
$127$	2021.68	1.41256	0.706281	+	0.707931i	$0.250371\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	2288.89	1.54422
$131$	−854.726	−0.570059	−0.285030	−	0.958519i	$-0.592003\pi$
$131$	−854.726	−0.570059	−0.285030	+	0.958519i	$0.592003\pi$
$132$	0	0
$133$	−972.543	−0.634062
$134$	−386.264	−0.249016
$135$	0	0
$136$	−849.014	−0.535311
$137$	−365.024	−0.227636	−0.113818	−	0.993502i	$-0.536308\pi$
$137$	−365.024	−0.227636	−0.113818	+	0.993502i	$0.536308\pi$
$138$	0	0
$139$	−1010.10	−0.616370	−0.308185	−	0.951326i	$-0.599722\pi$
$139$	−1010.10	−0.616370	−0.308185	+	0.951326i	$0.599722\pi$
$140$	−716.182	−0.432346
$141$	0	0
$142$	−4.76310	−0.00281487
$143$	−591.315	−0.345792
$144$	0	0
$145$	606.693	0.347470
$146$	−238.026	−0.134926
$147$	0	0
$148$	−422.660	−0.234746
$149$	−819.765	−0.450723	−0.225362	−	0.974275i	$-0.572356\pi$
$149$	−819.765	−0.450723	−0.225362	+	0.974275i	$0.572356\pi$
$150$	0	0
$151$	1000.25	0.539068	0.269534	−	0.962991i	$-0.413130\pi$
$151$	1000.25	0.539068	0.269534	+	0.962991i	$0.413130\pi$
$152$	909.090	0.485111
$153$	0	0
$154$	185.019	0.0968134
$155$	2150.72	1.11452
$156$	0	0
$157$	−1702.38	−0.865382	−0.432691	−	0.901542i	$-0.642436\pi$
$157$	−1702.38	−0.865382	−0.432691	+	0.901542i	$0.642436\pi$
$158$	1928.61	0.971090
$159$	0	0
$160$	669.455	0.330781
$161$	965.860	0.472798
$162$	0	0
$163$	−3451.76	−1.65867	−0.829334	−	0.558753i	$-0.811280\pi$
$163$	−3451.76	−1.65867	−0.829334	+	0.558753i	$0.811280\pi$
$164$	−867.831	−0.413209
$165$	0	0
$166$	2136.38	0.998886
$167$	−1409.23	−0.652990	−0.326495	−	0.945199i	$-0.605868\pi$
$167$	−1409.23	−0.652990	−0.326495	+	0.945199i	$0.605868\pi$
$168$	0	0
$169$	795.598	0.362129
$170$	4440.44	2.00333
$171$	0	0
$172$	−408.918	−0.181278
$173$	−1358.39	−0.596973	−0.298487	−	0.954414i	$-0.596482\pi$
$173$	−1358.39	−0.596973	−0.298487	+	0.954414i	$0.596482\pi$
$174$	0	0
$175$	2675.91	1.15589
$176$	−172.948	−0.0740705
$177$	0	0
$178$	1545.09	0.650614
$179$	1702.56	0.710924	0.355462	−	0.934691i	$-0.384323\pi$
$179$	1702.56	0.710924	0.355462	+	0.934691i	$0.384323\pi$
$180$	0	0
$181$	−7.33698	−0.00301300	−0.00150650	−	0.999999i	$-0.500480\pi$
$181$	−7.33698	−0.00301300	−0.00150650	+	0.999999i	$0.500480\pi$
$182$	−936.368	−0.381364
$183$	0	0
$184$	−902.843	−0.361731
$185$	2210.56	0.878507
$186$	0	0
$187$	−1147.15	−0.448598
$188$	−1820.85	−0.706377
$189$	0	0
$190$	−4754.64	−1.81546
$191$	−1324.78	−0.501873	−0.250936	−	0.968004i	$-0.580738\pi$
$191$	−1324.78	−0.501873	−0.250936	+	0.968004i	$0.580738\pi$
$192$	0	0
$193$	1834.47	0.684187	0.342094	−	0.939666i	$-0.388864\pi$
$193$	1834.47	0.684187	0.342094	+	0.939666i	$0.388864\pi$
$194$	−2688.06	−0.994802
$195$	0	0
$196$	−1079.02	−0.393227
$197$	4949.99	1.79021	0.895107	−	0.445851i	$-0.147099\pi$
$197$	4949.99	1.79021	0.895107	+	0.445851i	$0.147099\pi$
$198$	0	0
$199$	3554.04	1.26603	0.633014	−	0.774140i	$-0.281818\pi$
$199$	3554.04	1.26603	0.633014	+	0.774140i	$0.281818\pi$
$200$	−2501.32	−0.884352
$201$	0	0
$202$	1973.47	0.687389
$203$	−248.193	−0.0858116
$204$	0	0
$205$	4538.85	1.54638
$206$	−1096.54	−0.370873
$207$	0	0
$208$	875.274	0.291776
$209$	1228.32	0.406529
$210$	0	0
$211$	2475.23	0.807590	0.403795	−	0.914849i	$-0.367691\pi$
$211$	2475.23	0.807590	0.403795	+	0.914849i	$0.367691\pi$
$212$	2374.86	0.769367
$213$	0	0
$214$	2775.03	0.886434
$215$	2138.69	0.678407
$216$	0	0
$217$	−879.844	−0.275243
$218$	2586.64	0.803622
$219$	0	0
$220$	904.536	0.277199
$221$	5805.63	1.76710
$222$	0	0
$223$	2381.72	0.715211	0.357605	−	0.933873i	$-0.383593\pi$
$223$	2381.72	0.715211	0.357605	+	0.933873i	$0.383593\pi$
$224$	−273.869	−0.0816902
$225$	0	0
$226$	605.767	0.178297
$227$	−5452.74	−1.59432	−0.797160	−	0.603768i	$-0.793666\pi$
$227$	−5452.74	−1.59432	−0.797160	+	0.603768i	$0.793666\pi$
$228$	0	0
$229$	596.232	0.172053	0.0860264	−	0.996293i	$-0.472583\pi$
$229$	596.232	0.172053	0.0860264	+	0.996293i	$0.472583\pi$
$230$	4721.97	1.35373
$231$	0	0
$232$	232.000	0.0656532
$233$	5623.04	1.58102	0.790509	−	0.612450i	$-0.209816\pi$
$233$	5623.04	1.58102	0.790509	+	0.612450i	$0.209816\pi$
$234$	0	0
$235$	9523.24	2.64352
$236$	2232.56	0.615794
$237$	0	0
$238$	−1816.55	−0.494745
$239$	−1564.27	−0.423366	−0.211683	−	0.977338i	$-0.567894\pi$
$239$	−1564.27	−0.423366	−0.211683	+	0.977338i	$0.567894\pi$
$240$	0	0
$241$	−730.326	−0.195205	−0.0976026	−	0.995225i	$-0.531117\pi$
$241$	−730.326	−0.195205	−0.0976026	+	0.995225i	$0.531117\pi$
$242$	2428.32	0.645035
$243$	0	0
$244$	−1895.94	−0.497440
$245$	5643.38	1.47160
$246$	0	0
$247$	−6216.43	−1.60138
$248$	822.438	0.210584
$249$	0	0
$250$	7852.10	1.98644
$251$	4244.39	1.06735	0.533673	−	0.845691i	$-0.320811\pi$
$251$	4244.39	1.06735	0.533673	+	0.845691i	$0.320811\pi$
$252$	0	0
$253$	−1219.88	−0.303135
$254$	−4043.37	−0.998833
$255$	0	0
$256$	256.000	0.0625000
$257$	235.374	0.0571292	0.0285646	−	0.999592i	$-0.490906\pi$
$257$	235.374	0.0571292	0.0285646	+	0.999592i	$0.490906\pi$
$258$	0	0
$259$	−904.323	−0.216957
$260$	−4577.78	−1.09193
$261$	0	0
$262$	1709.45	0.403093
$263$	−3453.63	−0.809734	−0.404867	−	0.914376i	$-0.632682\pi$
$263$	−3453.63	−0.809734	−0.404867	+	0.914376i	$0.632682\pi$
$264$	0	0
$265$	−12420.8	−2.87925
$266$	1945.09	0.448349
$267$	0	0
$268$	772.528	0.176081
$269$	1921.31	0.435481	0.217741	−	0.976007i	$-0.430131\pi$
$269$	1921.31	0.435481	0.217741	+	0.976007i	$0.430131\pi$
$270$	0	0
$271$	−2480.09	−0.555921	−0.277961	−	0.960592i	$-0.589658\pi$
$271$	−2480.09	−0.555921	−0.277961	+	0.960592i	$0.589658\pi$
$272$	1698.03	0.378522
$273$	0	0
$274$	730.048	0.160963
$275$	−3379.67	−0.741098
$276$	0	0
$277$	−4766.90	−1.03399	−0.516995	−	0.855989i	$-0.672949\pi$
$277$	−4766.90	−1.03399	−0.516995	+	0.855989i	$0.672949\pi$
$278$	2020.20	0.435840
$279$	0	0
$280$	1432.36	0.305715
$281$	−194.329	−0.0412552	−0.0206276	−	0.999787i	$-0.506566\pi$
$281$	−194.329	−0.0412552	−0.0206276	+	0.999787i	$0.506566\pi$
$282$	0	0
$283$	−2817.42	−0.591797	−0.295898	−	0.955219i	$-0.595619\pi$
$283$	−2817.42	−0.591797	−0.295898	+	0.955219i	$0.595619\pi$
$284$	9.52621	0.00199041
$285$	0	0
$286$	1182.63	0.244512
$287$	−1856.81	−0.381895
$288$	0	0
$289$	6349.89	1.29247
$290$	−1213.39	−0.245698
$291$	0	0
$292$	476.053	0.0954071
$293$	−4059.12	−0.809339	−0.404670	−	0.914463i	$-0.632614\pi$
$293$	−4059.12	−0.809339	−0.404670	+	0.914463i	$0.632614\pi$
$294$	0	0
$295$	−11676.6	−2.30453
$296$	845.320	0.165991
$297$	0	0
$298$	1639.53	0.318709
$299$	6173.71	1.19410
$300$	0	0
$301$	−874.921	−0.167540
$302$	−2000.50	−0.381178
$303$	0	0
$304$	−1818.18	−0.343025
$305$	9916.01	1.86160
$306$	0	0
$307$	−131.572	−0.0244600	−0.0122300	−	0.999925i	$-0.503893\pi$
$307$	−131.572	−0.0244600	−0.0122300	+	0.999925i	$0.503893\pi$
$308$	−370.038	−0.0684574
$309$	0	0
$310$	−4301.45	−0.788083
$311$	−1517.84	−0.276748	−0.138374	−	0.990380i	$-0.544188\pi$
$311$	−1517.84	−0.276748	−0.138374	+	0.990380i	$0.544188\pi$
$312$	0	0
$313$	4244.17	0.766436	0.383218	−	0.923658i	$-0.374816\pi$
$313$	4244.17	0.766436	0.383218	+	0.923658i	$0.374816\pi$
$314$	3404.77	0.611917
$315$	0	0
$316$	−3857.23	−0.686664
$317$	−5596.56	−0.991590	−0.495795	−	0.868439i	$-0.665123\pi$
$317$	−5596.56	−0.991590	−0.495795	+	0.868439i	$0.665123\pi$
$318$	0	0
$319$	313.467	0.0550182
$320$	−1338.91	−0.233898
$321$	0	0
$322$	−1931.72	−0.334319
$323$	−12059.8	−2.07748
$324$	0	0
$325$	17104.3	2.91930
$326$	6903.53	1.17286
$327$	0	0
$328$	1735.66	0.292183
$329$	−3895.88	−0.652848
$330$	0	0
$331$	−8383.85	−1.39220	−0.696100	−	0.717945i	$-0.745083\pi$
$331$	−8383.85	−1.39220	−0.696100	+	0.717945i	$0.745083\pi$
$332$	−4272.76	−0.706319
$333$	0	0
$334$	2818.45	0.461733
$335$	−4040.41	−0.658959
$336$	0	0
$337$	9887.12	1.59818	0.799089	−	0.601213i	$-0.205316\pi$
$337$	9887.12	1.59818	0.799089	+	0.601213i	$0.205316\pi$
$338$	−1591.20	−0.256064
$339$	0	0
$340$	−8880.88	−1.41657
$341$	1111.24	0.176472
$342$	0	0
$343$	−5244.19	−0.825538
$344$	817.837	0.128183
$345$	0	0
$346$	2716.78	0.422124
$347$	−10678.8	−1.65206	−0.826032	−	0.563623i	$-0.809407\pi$
$347$	−10678.8	−1.65206	−0.826032	+	0.563623i	$0.809407\pi$
$348$	0	0
$349$	−1457.88	−0.223605	−0.111803	−	0.993730i	$-0.535662\pi$
$349$	−1457.88	−0.223605	−0.111803	+	0.993730i	$0.535662\pi$
$350$	−5351.83	−0.817335
$351$	0	0
$352$	345.895	0.0523758
$353$	−6737.20	−1.01582	−0.507911	−	0.861410i	$-0.669582\pi$
$353$	−6737.20	−1.01582	−0.507911	+	0.861410i	$0.669582\pi$
$354$	0	0
$355$	−49.8232	−0.00744884
$356$	−3090.18	−0.460053
$357$	0	0
$358$	−3405.12	−0.502699
$359$	3539.85	0.520407	0.260204	−	0.965554i	$-0.416210\pi$
$359$	3539.85	0.520407	0.260204	+	0.965554i	$0.416210\pi$
$360$	0	0
$361$	6054.19	0.882663
$362$	14.6740	0.00213051
$363$	0	0
$364$	1872.74	0.269665
$365$	−2489.81	−0.357048
$366$	0	0
$367$	−4917.53	−0.699436	−0.349718	−	0.936855i	$-0.613723\pi$
$367$	−4917.53	−0.699436	−0.349718	+	0.936855i	$0.613723\pi$
$368$	1805.69	0.255782
$369$	0	0
$370$	−4421.12	−0.621198
$371$	5081.24	0.711064
$372$	0	0
$373$	2032.31	0.282115	0.141057	−	0.990001i	$-0.454950\pi$
$373$	2032.31	0.282115	0.141057	+	0.990001i	$0.454950\pi$
$374$	2294.30	0.317206
$375$	0	0
$376$	3641.69	0.499484
$377$	−1586.43	−0.216726
$378$	0	0
$379$	−7051.47	−0.955699	−0.477849	−	0.878442i	$-0.658584\pi$
$379$	−7051.47	−0.955699	−0.477849	+	0.878442i	$0.658584\pi$
$380$	9509.29	1.28373
$381$	0	0
$382$	2649.56	0.354878
$383$	9334.72	1.24538	0.622691	−	0.782467i	$-0.286039\pi$
$383$	9334.72	1.24538	0.622691	+	0.782467i	$0.286039\pi$
$384$	0	0
$385$	1935.34	0.256193
$386$	−3668.94	−0.483793
$387$	0	0
$388$	5376.12	0.703431
$389$	1901.32	0.247816	0.123908	−	0.992294i	$-0.460457\pi$
$389$	1901.32	0.247816	0.123908	+	0.992294i	$0.460457\pi$
$390$	0	0
$391$	11977.0	1.54911
$392$	2158.03	0.278054
$393$	0	0
$394$	−9899.98	−1.26587
$395$	20173.7	2.56975
$396$	0	0
$397$	1995.81	0.252309	0.126155	−	0.992011i	$-0.459736\pi$
$397$	1995.81	0.252309	0.126155	+	0.992011i	$0.459736\pi$
$398$	−7108.09	−0.895217
$399$	0	0
$400$	5002.65	0.625331
$401$	−12920.6	−1.60904	−0.804520	−	0.593925i	$-0.797578\pi$
$401$	−12920.6	−1.60904	−0.804520	+	0.593925i	$0.797578\pi$
$402$	0	0
$403$	−5623.90	−0.695152
$404$	−3946.93	−0.486057
$405$	0	0
$406$	496.387	0.0606780
$407$	1142.16	0.139102
$408$	0	0
$409$	−9713.54	−1.17434	−0.587168	−	0.809465i	$-0.699757\pi$
$409$	−9713.54	−1.17434	−0.587168	+	0.809465i	$0.699757\pi$
$410$	−9077.71	−1.09345
$411$	0	0
$412$	2193.09	0.262247
$413$	4776.79	0.569129
$414$	0	0
$415$	22347.0	2.64330
$416$	−1750.55	−0.206317
$417$	0	0
$418$	−2456.64	−0.287460
$419$	−15925.4	−1.85682	−0.928411	−	0.371555i	$-0.878825\pi$
$419$	−15925.4	−1.85682	−0.928411	+	0.371555i	$0.878825\pi$
$420$	0	0
$421$	10849.9	1.25604	0.628019	−	0.778198i	$-0.283866\pi$
$421$	10849.9	1.25604	0.628019	+	0.778198i	$0.283866\pi$
$422$	−4950.45	−0.571053
$423$	0	0
$424$	−4749.71	−0.544025
$425$	33182.2	3.78723
$426$	0	0
$427$	−4056.56	−0.459744
$428$	−5550.05	−0.626803
$429$	0	0
$430$	−4277.38	−0.479706
$431$	532.335	0.0594934	0.0297467	−	0.999557i	$-0.490530\pi$
$431$	532.335	0.0594934	0.0297467	+	0.999557i	$0.490530\pi$
$432$	0	0
$433$	−6995.94	−0.776451	−0.388225	−	0.921564i	$-0.626912\pi$
$433$	−6995.94	−0.776451	−0.388225	+	0.921564i	$0.626912\pi$
$434$	1759.69	0.194626
$435$	0	0
$436$	−5173.28	−0.568246
$437$	−12824.5	−1.40384
$438$	0	0
$439$	3272.22	0.355750	0.177875	−	0.984053i	$-0.443078\pi$
$439$	3272.22	0.355750	0.177875	+	0.984053i	$0.443078\pi$
$440$	−1809.07	−0.196009
$441$	0	0
$442$	−11611.3	−1.24953
$443$	3818.14	0.409493	0.204746	−	0.978815i	$-0.434363\pi$
$443$	3818.14	0.409493	0.204746	+	0.978815i	$0.434363\pi$
$444$	0	0
$445$	16162.0	1.72169
$446$	−4763.45	−0.505731
$447$	0	0
$448$	547.737	0.0577637
$449$	4323.19	0.454396	0.227198	−	0.973849i	$-0.427043\pi$
$449$	4323.19	0.454396	0.227198	+	0.973849i	$0.427043\pi$
$450$	0	0
$451$	2345.14	0.244853
$452$	−1211.53	−0.126075
$453$	0	0
$454$	10905.5	1.12736
$455$	−9794.62	−1.00918
$456$	0	0
$457$	−8367.43	−0.856481	−0.428240	−	0.903665i	$-0.640866\pi$
$457$	−8367.43	−0.856481	−0.428240	+	0.903665i	$0.640866\pi$
$458$	−1192.46	−0.121660
$459$	0	0
$460$	−9443.94	−0.957231
$461$	17249.0	1.74266	0.871328	−	0.490701i	$-0.163259\pi$
$461$	17249.0	1.74266	0.871328	+	0.490701i	$0.163259\pi$
$462$	0	0
$463$	−16774.3	−1.68373	−0.841864	−	0.539690i	$-0.818542\pi$
$463$	−16774.3	−1.68373	−0.841864	+	0.539690i	$0.818542\pi$
$464$	−464.000	−0.0464238
$465$	0	0
$466$	−11246.1	−1.11795
$467$	−7701.05	−0.763088	−0.381544	−	0.924351i	$-0.624608\pi$
$467$	−7701.05	−0.763088	−0.381544	+	0.924351i	$0.624608\pi$
$468$	0	0
$469$	1652.90	0.162737
$470$	−19046.5	−1.86925
$471$	0	0
$472$	−4465.13	−0.435432
$473$	1105.02	0.107419
$474$	0	0
$475$	−35530.1	−3.43207
$476$	3633.10	0.349838
$477$	0	0
$478$	3128.55	0.299365
$479$	−5988.77	−0.571261	−0.285630	−	0.958340i	$-0.592203\pi$
$479$	−5988.77	−0.571261	−0.285630	+	0.958340i	$0.592203\pi$
$480$	0	0
$481$	−5780.37	−0.547946
$482$	1460.65	0.138031
$483$	0	0
$484$	−4856.64	−0.456108
$485$	−28117.7	−2.63250
$486$	0	0
$487$	5790.34	0.538779	0.269390	−	0.963031i	$-0.413178\pi$
$487$	5790.34	0.538779	0.269390	+	0.963031i	$0.413178\pi$
$488$	3791.89	0.351743
$489$	0	0
$490$	−11286.8	−1.04058
$491$	−19228.8	−1.76738	−0.883692	−	0.468069i	$-0.844950\pi$
$491$	−19228.8	−1.76738	−0.883692	+	0.468069i	$0.844950\pi$
$492$	0	0
$493$	−3077.68	−0.281159
$494$	12432.9	1.13235
$495$	0	0
$496$	−1644.88	−0.148905
$497$	20.3823	0.00183958
$498$	0	0
$499$	−7081.65	−0.635307	−0.317653	−	0.948207i	$-0.602895\pi$
$499$	−7081.65	−0.635307	−0.317653	+	0.948207i	$0.602895\pi$
$500$	−15704.2	−1.40463
$501$	0	0
$502$	−8488.79	−0.754727
$503$	5312.64	0.470932	0.235466	−	0.971883i	$-0.424338\pi$
$503$	5312.64	0.470932	0.235466	+	0.971883i	$0.424338\pi$
$504$	0	0
$505$	20642.9	1.81900
$506$	2439.76	0.214349
$507$	0	0
$508$	8086.74	0.706281
$509$	13862.7	1.20717	0.603587	−	0.797297i	$-0.293738\pi$
$509$	13862.7	1.20717	0.603587	+	0.797297i	$0.293738\pi$
$510$	0	0
$511$	1018.56	0.0881771
$512$	−512.000	−0.0441942
$513$	0	0
$514$	−470.747	−0.0403965
$515$	−11470.1	−0.981423
$516$	0	0
$517$	4920.49	0.418574
$518$	1808.65	0.153412
$519$	0	0
$520$	9155.57	0.772112
$521$	−15447.5	−1.29897	−0.649487	−	0.760373i	$-0.725016\pi$
$521$	−15447.5	−1.29897	−0.649487	+	0.760373i	$0.725016\pi$
$522$	0	0
$523$	4349.54	0.363656	0.181828	−	0.983330i	$-0.441799\pi$
$523$	4349.54	0.363656	0.181828	+	0.983330i	$0.441799\pi$
$524$	−3418.90	−0.285030
$525$	0	0
$526$	6907.26	0.572568
$527$	−10910.3	−0.901825
$528$	0	0
$529$	569.329	0.0467928
$530$	24841.5	2.03594
$531$	0	0
$532$	−3890.17	−0.317031
$533$	−11868.6	−0.964514
$534$	0	0
$535$	29027.4	2.34573
$536$	−1545.06	−0.124508
$537$	0	0
$538$	−3842.62	−0.307932
$539$	2915.83	0.233012
$540$	0	0
$541$	3206.84	0.254848	0.127424	−	0.991848i	$-0.459329\pi$
$541$	3206.84	0.254848	0.127424	+	0.991848i	$0.459329\pi$
$542$	4960.18	0.393096
$543$	0	0
$544$	−3396.06	−0.267656
$545$	27056.9	2.12658
$546$	0	0
$547$	−3289.81	−0.257152	−0.128576	−	0.991700i	$-0.541041\pi$
$547$	−3289.81	−0.257152	−0.128576	+	0.991700i	$0.541041\pi$
$548$	−1460.10	−0.113818
$549$	0	0
$550$	6759.34	0.524035
$551$	3295.45	0.254793
$552$	0	0
$553$	−8252.91	−0.634628
$554$	9533.79	0.731141
$555$	0	0
$556$	−4040.40	−0.308185
$557$	313.140	0.0238207	0.0119104	−	0.999929i	$-0.496209\pi$
$557$	313.140	0.0238207	0.0119104	+	0.999929i	$0.496209\pi$
$558$	0	0
$559$	−5592.43	−0.423139
$560$	−2864.73	−0.216173
$561$	0	0
$562$	388.659	0.0291718
$563$	3425.84	0.256451	0.128225	−	0.991745i	$-0.459072\pi$
$563$	3425.84	0.256451	0.128225	+	0.991745i	$0.459072\pi$
$564$	0	0
$565$	6336.46	0.471818
$566$	5634.85	0.418463
$567$	0	0
$568$	−19.0524	−0.00140743
$569$	−17763.8	−1.30878	−0.654390	−	0.756158i	$-0.727074\pi$
$569$	−17763.8	−1.30878	−0.654390	+	0.756158i	$0.727074\pi$
$570$	0	0
$571$	5741.80	0.420818	0.210409	−	0.977613i	$-0.432520\pi$
$571$	5741.80	0.420818	0.210409	+	0.977613i	$0.432520\pi$
$572$	−2365.26	−0.172896
$573$	0	0
$574$	3713.62	0.270041
$575$	35286.0	2.55918
$576$	0	0
$577$	14477.5	1.04455	0.522275	−	0.852777i	$-0.325083\pi$
$577$	14477.5	1.04455	0.522275	+	0.852777i	$0.325083\pi$
$578$	−12699.8	−0.913912
$579$	0	0
$580$	2426.77	0.173735
$581$	−9141.98	−0.652794
$582$	0	0
$583$	−6417.59	−0.455899
$584$	−952.105	−0.0674630
$585$	0	0
$586$	8118.24	0.572289
$587$	18082.8	1.27148	0.635738	−	0.771905i	$-0.280696\pi$
$587$	18082.8	1.27148	0.635738	+	0.771905i	$0.280696\pi$
$588$	0	0
$589$	11682.3	0.817254
$590$	23353.1	1.62955
$591$	0	0
$592$	−1690.64	−0.117373
$593$	−1731.57	−0.119911	−0.0599553	−	0.998201i	$-0.519096\pi$
$593$	−1731.57	−0.119911	−0.0599553	+	0.998201i	$0.519096\pi$
$594$	0	0
$595$	−19001.5	−1.30922
$596$	−3279.06	−0.225362
$597$	0	0
$598$	−12347.4	−0.844354
$599$	11480.5	0.783105	0.391552	−	0.920156i	$-0.371938\pi$
$599$	11480.5	0.783105	0.391552	+	0.920156i	$0.371938\pi$
$600$	0	0
$601$	2924.56	0.198495	0.0992474	−	0.995063i	$-0.468356\pi$
$601$	2924.56	0.198495	0.0992474	+	0.995063i	$0.468356\pi$
$602$	1749.84	0.118469
$603$	0	0
$604$	4001.00	0.269534
$605$	25400.8	1.70692
$606$	0	0
$607$	3586.09	0.239794	0.119897	−	0.992786i	$-0.461744\pi$
$607$	3586.09	0.239794	0.119897	+	0.992786i	$0.461744\pi$
$608$	3636.36	0.242556
$609$	0	0
$610$	−19832.0	−1.31635
$611$	−24902.2	−1.64883
$612$	0	0
$613$	−11679.8	−0.769561	−0.384781	−	0.923008i	$-0.625723\pi$
$613$	−11679.8	−0.769561	−0.384781	+	0.923008i	$0.625723\pi$
$614$	263.145	0.0172958
$615$	0	0
$616$	740.077	0.0484067
$617$	−11063.0	−0.721847	−0.360924	−	0.932595i	$-0.617539\pi$
$617$	−11063.0	−0.721847	−0.360924	+	0.932595i	$0.617539\pi$
$618$	0	0
$619$	−2463.60	−0.159969	−0.0799843	−	0.996796i	$-0.525487\pi$
$619$	−2463.60	−0.159969	−0.0799843	+	0.996796i	$0.525487\pi$
$620$	8602.89	0.557259
$621$	0	0
$622$	3035.67	0.195690
$623$	−6611.73	−0.425190
$624$	0	0
$625$	43051.5	2.75530
$626$	−8488.34	−0.541952
$627$	0	0
$628$	−6809.53	−0.432691
$629$	−11213.9	−0.710854
$630$	0	0
$631$	1032.43	0.0651352	0.0325676	−	0.999470i	$-0.489632\pi$
$631$	1032.43	0.0651352	0.0325676	+	0.999470i	$0.489632\pi$
$632$	7714.45	0.485545
$633$	0	0
$634$	11193.1	0.701160
$635$	−42294.6	−2.64316
$636$	0	0
$637$	−14756.8	−0.917873
$638$	−626.935	−0.0389038
$639$	0	0
$640$	2677.82	0.165391
$641$	−12841.6	−0.791282	−0.395641	−	0.918405i	$-0.629478\pi$
$641$	−12841.6	−0.791282	−0.395641	+	0.918405i	$0.629478\pi$
$642$	0	0
$643$	8449.14	0.518198	0.259099	−	0.965851i	$-0.416574\pi$
$643$	8449.14	0.518198	0.259099	+	0.965851i	$0.416574\pi$
$644$	3863.44	0.236399
$645$	0	0
$646$	24119.7	1.46900
$647$	27036.8	1.64285	0.821426	−	0.570315i	$-0.193179\pi$
$647$	27036.8	1.64285	0.821426	+	0.570315i	$0.193179\pi$
$648$	0	0
$649$	−6033.07	−0.364898
$650$	−34208.5	−2.06426
$651$	0	0
$652$	−13807.1	−0.829334
$653$	27105.2	1.62436	0.812181	−	0.583405i	$-0.198280\pi$
$653$	27105.2	1.62436	0.812181	+	0.583405i	$0.198280\pi$
$654$	0	0
$655$	17881.3	1.06668
$656$	−3471.32	−0.206604
$657$	0	0
$658$	7791.76	0.461633
$659$	22622.4	1.33724	0.668621	−	0.743603i	$-0.266885\pi$
$659$	22622.4	1.33724	0.668621	+	0.743603i	$0.266885\pi$
$660$	0	0
$661$	−21000.2	−1.23572	−0.617862	−	0.786287i	$-0.712001\pi$
$661$	−21000.2	−1.23572	−0.617862	+	0.786287i	$0.712001\pi$
$662$	16767.7	0.984434
$663$	0	0
$664$	8545.51	0.499443
$665$	20346.0	1.18645
$666$	0	0
$667$	−3272.80	−0.189990
$668$	−5636.91	−0.326495
$669$	0	0
$670$	8080.82	0.465954
$671$	5123.42	0.294765
$672$	0	0
$673$	−13691.2	−0.784186	−0.392093	−	0.919926i	$-0.628249\pi$
$673$	−13691.2	−0.784186	−0.392093	+	0.919926i	$0.628249\pi$
$674$	−19774.2	−1.13008
$675$	0	0
$676$	3182.39	0.181065
$677$	−9694.60	−0.550360	−0.275180	−	0.961393i	$-0.588737\pi$
$677$	−9694.60	−0.550360	−0.275180	+	0.961393i	$0.588737\pi$
$678$	0	0
$679$	11502.7	0.650125
$680$	17761.8	1.00167
$681$	0	0
$682$	−2222.48	−0.124785
$683$	−6012.25	−0.336826	−0.168413	−	0.985717i	$-0.553864\pi$
$683$	−6012.25	−0.336826	−0.168413	+	0.985717i	$0.553864\pi$
$684$	0	0
$685$	7636.47	0.425948
$686$	10488.4	0.583744
$687$	0	0
$688$	−1635.67	−0.0906388
$689$	32478.9	1.79586
$690$	0	0
$691$	−10304.9	−0.567317	−0.283659	−	0.958925i	$-0.591548\pi$
$691$	−10304.9	−0.567317	−0.283659	+	0.958925i	$0.591548\pi$
$692$	−5433.55	−0.298487
$693$	0	0
$694$	21357.5	1.16819
$695$	21131.7	1.15334
$696$	0	0
$697$	−23025.0	−1.25127
$698$	2915.75	0.158113
$699$	0	0
$700$	10703.7	0.577943
$701$	−11785.8	−0.635011	−0.317506	−	0.948256i	$-0.602845\pi$
$701$	−11785.8	−0.635011	−0.317506	+	0.948256i	$0.602845\pi$
$702$	0	0
$703$	12007.4	0.644192
$704$	−691.790	−0.0370353
$705$	0	0
$706$	13474.4	0.718294
$707$	−8444.85	−0.449224
$708$	0	0
$709$	29226.1	1.54811	0.774054	−	0.633120i	$-0.218226\pi$
$709$	29226.1	1.54811	0.774054	+	0.633120i	$0.218226\pi$
$710$	99.6463	0.00526713
$711$	0	0
$712$	6180.35	0.325307
$713$	−11602.1	−0.609398
$714$	0	0
$715$	12370.6	0.647040
$716$	6810.25	0.355462
$717$	0	0
$718$	−7079.70	−0.367983
$719$	−14706.5	−0.762809	−0.381404	−	0.924408i	$-0.624559\pi$
$719$	−14706.5	−0.762809	−0.381404	+	0.924408i	$0.624559\pi$
$720$	0	0
$721$	4692.32	0.242374
$722$	−12108.4	−0.624137
$723$	0	0
$724$	−29.3479	−0.00150650
$725$	−9067.30	−0.464484
$726$	0	0
$727$	−35975.1	−1.83527	−0.917637	−	0.397419i	$-0.869906\pi$
$727$	−35975.1	−1.83527	−0.917637	+	0.397419i	$0.869906\pi$
$728$	−3745.47	−0.190682
$729$	0	0
$730$	4979.62	0.252471
$731$	−10849.3	−0.548941
$732$	0	0
$733$	10872.8	0.547879	0.273939	−	0.961747i	$-0.411673\pi$
$733$	10872.8	0.547879	0.273939	+	0.961747i	$0.411673\pi$
$734$	9835.06	0.494576
$735$	0	0
$736$	−3611.37	−0.180865
$737$	−2087.61	−0.104339
$738$	0	0
$739$	−1078.25	−0.0536727	−0.0268363	−	0.999640i	$-0.508543\pi$
$739$	−1078.25	−0.0536727	−0.0268363	+	0.999640i	$0.508543\pi$
$740$	8842.25	0.439253
$741$	0	0
$742$	−10162.5	−0.502798
$743$	23176.4	1.14436	0.572180	−	0.820128i	$-0.306098\pi$
$743$	23176.4	1.14436	0.572180	+	0.820128i	$0.306098\pi$
$744$	0	0
$745$	17149.9	0.843385
$746$	−4064.61	−0.199485
$747$	0	0
$748$	−4588.59	−0.224299
$749$	−11874.9	−0.579304
$750$	0	0
$751$	8738.85	0.424614	0.212307	−	0.977203i	$-0.431902\pi$
$751$	8738.85	0.424614	0.212307	+	0.977203i	$0.431902\pi$
$752$	−7283.39	−0.353189
$753$	0	0
$754$	3172.87	0.153248
$755$	−20925.7	−1.00869
$756$	0	0
$757$	4121.15	0.197868	0.0989338	−	0.995094i	$-0.468457\pi$
$757$	4121.15	0.197868	0.0989338	+	0.995094i	$0.468457\pi$
$758$	14102.9	0.675781
$759$	0	0
$760$	−19018.6	−0.907732
$761$	−8706.54	−0.414733	−0.207367	−	0.978263i	$-0.566489\pi$
$761$	−8706.54	−0.414733	−0.207367	+	0.978263i	$0.566489\pi$
$762$	0	0
$763$	−11068.7	−0.525184
$764$	−5299.12	−0.250936
$765$	0	0
$766$	−18669.4	−0.880619
$767$	30532.9	1.43739
$768$	0	0
$769$	−32258.8	−1.51272	−0.756361	−	0.654154i	$-0.773025\pi$
$769$	−32258.8	−1.51272	−0.756361	+	0.654154i	$0.773025\pi$
$770$	−3870.69	−0.181156
$771$	0	0
$772$	7337.88	0.342094
$773$	−24794.1	−1.15366	−0.576832	−	0.816863i	$-0.695711\pi$
$773$	−24794.1	−1.15366	−0.576832	+	0.816863i	$0.695711\pi$
$774$	0	0
$775$	−32143.5	−1.48984
$776$	−10752.2	−0.497401
$777$	0	0
$778$	−3802.63	−0.175233
$779$	24654.2	1.13393
$780$	0	0
$781$	−25.7427	−0.00117945
$782$	−23953.9	−1.09539
$783$	0	0
$784$	−4316.06	−0.196614
$785$	35614.6	1.61929
$786$	0	0
$787$	−11114.7	−0.503427	−0.251714	−	0.967802i	$-0.580994\pi$
$787$	−11114.7	−0.503427	−0.251714	+	0.967802i	$0.580994\pi$
$788$	19800.0	0.895107
$789$	0	0
$790$	−40347.5	−1.81709
$791$	−2592.20	−0.116521
$792$	0	0
$793$	−25929.2	−1.16113
$794$	−3991.62	−0.178410
$795$	0	0
$796$	14216.2	0.633014
$797$	12329.8	0.547986	0.273993	−	0.961732i	$-0.411656\pi$
$797$	12329.8	0.547986	0.273993	+	0.961732i	$0.411656\pi$
$798$	0	0
$799$	−48310.1	−2.13904
$800$	−10005.3	−0.442176
$801$	0	0
$802$	25841.3	1.13776
$803$	−1286.44	−0.0565348
$804$	0	0
$805$	−20206.2	−0.884691
$806$	11247.8	0.491547
$807$	0	0
$808$	7893.86	0.343695
$809$	36175.5	1.57214	0.786070	−	0.618138i	$-0.212113\pi$
$809$	36175.5	1.57214	0.786070	+	0.618138i	$0.212113\pi$
$810$	0	0
$811$	−26581.0	−1.15091	−0.575453	−	0.817835i	$-0.695174\pi$
$811$	−26581.0	−1.15091	−0.575453	+	0.817835i	$0.695174\pi$
$812$	−992.774	−0.0429058
$813$	0	0
$814$	−2284.31	−0.0983602
$815$	72212.5	3.10367
$816$	0	0
$817$	11617.0	0.497462
$818$	19427.1	0.830381
$819$	0	0
$820$	18155.4	0.773189
$821$	−10811.1	−0.459574	−0.229787	−	0.973241i	$-0.573803\pi$
$821$	−10811.1	−0.459574	−0.229787	+	0.973241i	$0.573803\pi$
$822$	0	0
$823$	28625.1	1.21240	0.606202	−	0.795311i	$-0.292692\pi$
$823$	28625.1	1.21240	0.606202	+	0.795311i	$0.292692\pi$
$824$	−4386.17	−0.185436
$825$	0	0
$826$	−9553.58	−0.402435
$827$	9970.68	0.419244	0.209622	−	0.977783i	$-0.432777\pi$
$827$	9970.68	0.419244	0.209622	+	0.977783i	$0.432777\pi$
$828$	0	0
$829$	20201.0	0.846334	0.423167	−	0.906052i	$-0.360918\pi$
$829$	20201.0	0.846334	0.423167	+	0.906052i	$0.360918\pi$
$830$	−44694.0	−1.86910
$831$	0	0
$832$	3501.10	0.145888
$833$	−28628.1	−1.19076
$834$	0	0
$835$	29481.7	1.22186
$836$	4913.28	0.203265
$837$	0	0
$838$	31850.9	1.31297
$839$	−28023.4	−1.15313	−0.576564	−	0.817052i	$-0.695607\pi$
$839$	−28023.4	−1.15313	−0.576564	+	0.817052i	$0.695607\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	−21699.8	−0.888153
$843$	0	0
$844$	9900.90	0.403795
$845$	−16644.3	−0.677610
$846$	0	0
$847$	−10391.3	−0.421544
$848$	9499.42	0.384683
$849$	0	0
$850$	−66364.4	−2.67797
$851$	−11924.9	−0.480352
$852$	0	0
$853$	−26333.8	−1.05704	−0.528518	−	0.848922i	$-0.677252\pi$
$853$	−26333.8	−1.05704	−0.528518	+	0.848922i	$0.677252\pi$
$854$	8113.12	0.325088
$855$	0	0
$856$	11100.1	0.443217
$857$	−36388.6	−1.45042	−0.725211	−	0.688527i	$-0.758258\pi$
$857$	−36388.6	−1.45042	−0.725211	+	0.688527i	$0.758258\pi$
$858$	0	0
$859$	14974.0	0.594770	0.297385	−	0.954758i	$-0.403885\pi$
$859$	14974.0	0.594770	0.297385	+	0.954758i	$0.403885\pi$
$860$	8554.76	0.339203
$861$	0	0
$862$	−1064.67	−0.0420682
$863$	40168.2	1.58440	0.792202	−	0.610259i	$-0.208935\pi$
$863$	40168.2	1.58440	0.792202	+	0.610259i	$0.208935\pi$
$864$	0	0
$865$	28418.1	1.11705
$866$	13991.9	0.549034
$867$	0	0
$868$	−3519.37	−0.137621
$869$	10423.4	0.406893
$870$	0	0
$871$	10565.2	0.411009
$872$	10346.6	0.401811
$873$	0	0
$874$	25648.9	0.992663
$875$	−33600.7	−1.29818
$876$	0	0
$877$	32161.7	1.23834	0.619169	−	0.785258i	$-0.287469\pi$
$877$	32161.7	1.23834	0.619169	+	0.785258i	$0.287469\pi$
$878$	−6544.43	−0.251553
$879$	0	0
$880$	3618.14	0.138599
$881$	8870.49	0.339222	0.169611	−	0.985511i	$-0.445749\pi$
$881$	8870.49	0.339222	0.169611	+	0.985511i	$0.445749\pi$
$882$	0	0
$883$	−24647.3	−0.939351	−0.469675	−	0.882839i	$-0.655629\pi$
$883$	−24647.3	−0.939351	−0.469675	+	0.882839i	$0.655629\pi$
$884$	23222.5	0.883549
$885$	0	0
$886$	−7636.28	−0.289555
$887$	−32371.7	−1.22541	−0.612703	−	0.790313i	$-0.709918\pi$
$887$	−32371.7	−1.22541	−0.612703	+	0.790313i	$0.709918\pi$
$888$	0	0
$889$	17302.4	0.652759
$890$	−32323.9	−1.21742
$891$	0	0
$892$	9526.90	0.357605
$893$	51728.5	1.93844
$894$	0	0
$895$	−35618.4	−1.33027
$896$	−1095.47	−0.0408451
$897$	0	0
$898$	−8646.38	−0.321307
$899$	2981.34	0.110604
$900$	0	0
$901$	63008.9	2.32978
$902$	−4690.29	−0.173137
$903$	0	0
$904$	2423.07	0.0891483
$905$	153.493	0.00563787
$906$	0	0
$907$	−43728.0	−1.60084	−0.800422	−	0.599437i	$-0.795391\pi$
$907$	−43728.0	−1.60084	−0.800422	+	0.599437i	$0.795391\pi$
$908$	−21810.9	−0.797160
$909$	0	0
$910$	19589.2	0.713601
$911$	−16033.6	−0.583115	−0.291557	−	0.956553i	$-0.594173\pi$
$911$	−16033.6	−0.583115	−0.291557	+	0.956553i	$0.594173\pi$
$912$	0	0
$913$	11546.3	0.418540
$914$	16734.9	0.605623
$915$	0	0
$916$	2384.93	0.0860264
$917$	−7315.08	−0.263430
$918$	0	0
$919$	29331.1	1.05282	0.526411	−	0.850230i	$-0.323537\pi$
$919$	29331.1	1.05282	0.526411	+	0.850230i	$0.323537\pi$
$920$	18887.9	0.676864
$921$	0	0
$922$	−34497.9	−1.23224
$923$	130.282	0.00464603
$924$	0	0
$925$	−33037.8	−1.17435
$926$	33548.5	1.19058
$927$	0	0
$928$	928.000	0.0328266
$929$	10024.6	0.354034	0.177017	−	0.984208i	$-0.443355\pi$
$929$	10024.6	0.354034	0.177017	+	0.984208i	$0.443355\pi$
$930$	0	0
$931$	30653.8	1.07910
$932$	22492.1	0.790509
$933$	0	0
$934$	15402.1	0.539585
$935$	23998.9	0.839408
$936$	0	0
$937$	21241.3	0.740578	0.370289	−	0.928917i	$-0.379259\pi$
$937$	21241.3	0.740578	0.370289	+	0.928917i	$0.379259\pi$
$938$	−3305.80	−0.115073
$939$	0	0
$940$	38092.9	1.32176
$941$	−11953.8	−0.414115	−0.207057	−	0.978329i	$-0.566389\pi$
$941$	−11953.8	−0.414115	−0.207057	+	0.978329i	$0.566389\pi$
$942$	0	0
$943$	−24484.8	−0.845531
$944$	8930.25	0.307897
$945$	0	0
$946$	−2210.05	−0.0759564
$947$	31650.6	1.08607	0.543034	−	0.839711i	$-0.317275\pi$
$947$	31650.6	1.08607	0.543034	+	0.839711i	$0.317275\pi$
$948$	0	0
$949$	6510.57	0.222700
$950$	71060.2	2.42684
$951$	0	0
$952$	−7266.20	−0.247373
$953$	17373.5	0.590539	0.295269	−	0.955414i	$-0.404591\pi$
$953$	17373.5	0.590539	0.295269	+	0.955414i	$0.404591\pi$
$954$	0	0
$955$	27715.0	0.939095
$956$	−6257.09	−0.211683
$957$	0	0
$958$	11977.5	0.403942
$959$	−3124.02	−0.105193
$960$	0	0
$961$	−19222.2	−0.645234
$962$	11560.7	0.387457
$963$	0	0
$964$	−2921.30	−0.0976026
$965$	−38378.0	−1.28024
$966$	0	0
$967$	−11017.6	−0.366394	−0.183197	−	0.983076i	$-0.558645\pi$
$967$	−11017.6	−0.366394	−0.183197	+	0.983076i	$0.558645\pi$
$968$	9713.29	0.322517
$969$	0	0
$970$	56235.5	1.86146
$971$	−17365.0	−0.573912	−0.286956	−	0.957944i	$-0.592643\pi$
$971$	−17365.0	−0.573912	−0.286956	+	0.957944i	$0.592643\pi$
$972$	0	0
$973$	−8644.82	−0.284831
$974$	−11580.7	−0.380975
$975$	0	0
$976$	−7583.78	−0.248720
$977$	29193.4	0.955967	0.477983	−	0.878369i	$-0.341368\pi$
$977$	29193.4	0.955967	0.477983	+	0.878369i	$0.341368\pi$
$978$	0	0
$979$	8350.60	0.272611
$980$	22573.5	0.735800
$981$	0	0
$982$	38457.7	1.24973
$983$	−50359.8	−1.63401	−0.817003	−	0.576634i	$-0.804366\pi$
$983$	−50359.8	−1.63401	−0.817003	+	0.576634i	$0.804366\pi$
$984$	0	0
$985$	−103556.	−3.34982
$986$	6155.35	0.198810
$987$	0	0
$988$	−24865.7	−0.800692
$989$	−11537.2	−0.370941
$990$	0	0
$991$	−20823.5	−0.667488	−0.333744	−	0.942664i	$-0.608312\pi$
$991$	−20823.5	−0.667488	−0.333744	+	0.942664i	$0.608312\pi$
$992$	3289.75	0.105292
$993$	0	0
$994$	−40.7645	−0.00130078
$995$	−74352.2	−2.36897
$996$	0	0
$997$	−27651.2	−0.878356	−0.439178	−	0.898400i	$-0.644730\pi$
$997$	−27651.2	−0.878356	−0.439178	+	0.898400i	$0.644730\pi$
$998$	14163.3	0.449230
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	522.4.a.k.1.1		3
3.2	odd	2		58.4.a.d.1.1	✓	3
12.11	even	2		464.4.a.i.1.3		3
15.14	odd	2		1450.4.a.h.1.3		3
24.5	odd	2		1856.4.a.r.1.3		3
24.11	even	2		1856.4.a.s.1.1		3
87.86	odd	2		1682.4.a.d.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
58.4.a.d.1.1	✓	3	3.2	odd	2
464.4.a.i.1.3		3	12.11	even	2
522.4.a.k.1.1		3	1.1	even	1	trivial
1450.4.a.h.1.3		3	15.14	odd	2
1682.4.a.d.1.3		3	87.86	odd	2
1856.4.a.r.1.3		3	24.5	odd	2
1856.4.a.s.1.1		3	24.11	even	2

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:	\( N \)	\(=\)	\( 522 = 2 \cdot 3^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	522.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} -20.9205 q^{5} +8.55839 q^{7} -8.00000 q^{8} +O(q^{10})\)	\(q-2.00000 q^{2} +4.00000 q^{4} -20.9205 q^{5} +8.55839 q^{7} -8.00000 q^{8} +41.8409 q^{10} -10.8092 q^{11} +54.7046 q^{13} -17.1168 q^{14} +16.0000 q^{16} +106.127 q^{17} -113.636 q^{19} -83.6818 q^{20} +21.6184 q^{22} +112.855 q^{23} +312.666 q^{25} -109.409 q^{26} +34.2336 q^{28} -29.0000 q^{29} -102.805 q^{31} -32.0000 q^{32} -212.254 q^{34} -179.045 q^{35} -105.665 q^{37} +227.272 q^{38} +167.364 q^{40} -216.958 q^{41} -102.230 q^{43} -43.2369 q^{44} -225.711 q^{46} -455.212 q^{47} -269.754 q^{49} -625.331 q^{50} +218.819 q^{52} +593.714 q^{53} +226.134 q^{55} -68.4671 q^{56} +58.0000 q^{58} +558.141 q^{59} -473.986 q^{61} +205.610 q^{62} +64.0000 q^{64} -1144.45 q^{65} +193.132 q^{67} +424.507 q^{68} +358.091 q^{70} +2.38155 q^{71} +119.013 q^{73} +211.330 q^{74} -454.545 q^{76} -92.5096 q^{77} -964.306 q^{79} -334.727 q^{80} +433.915 q^{82} -1068.19 q^{83} -2220.22 q^{85} +204.459 q^{86} +86.4738 q^{88} -772.544 q^{89} +468.184 q^{91} +451.421 q^{92} +910.423 q^{94} +2377.32 q^{95} +1344.03 q^{97} +539.508 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 6 q^{2} + 12 q^{4} - 20 q^{5} + 24 q^{7} - 24 q^{8}+O(q^{10}) \) 3 * q - 6 * q^2 + 12 * q^4 - 20 * q^5 + 24 * q^7 - 24 * q^8	\( 3 q - 6 q^{2} + 12 q^{4} - 20 q^{5} + 24 q^{7} - 24 q^{8} + 40 q^{10} - 10 q^{11} - 4 q^{13} - 48 q^{14} + 48 q^{16} + 66 q^{17} - 164 q^{19} - 80 q^{20} + 20 q^{22} + 204 q^{23} + 79 q^{25} + 8 q^{26} + 96 q^{28} - 87 q^{29} - 86 q^{31} - 96 q^{32} - 132 q^{34} - 24 q^{35} - 42 q^{37} + 328 q^{38} + 160 q^{40} - 562 q^{41} + 18 q^{43} - 40 q^{44} - 408 q^{46} - 654 q^{47} + 539 q^{49} - 158 q^{50} - 16 q^{52} - 712 q^{53} + 142 q^{55} - 192 q^{56} + 174 q^{58} - 184 q^{59} + 322 q^{61} + 172 q^{62} + 192 q^{64} - 1494 q^{65} - 228 q^{67} + 264 q^{68} + 48 q^{70} + 52 q^{71} - 494 q^{73} + 84 q^{74} - 656 q^{76} - 872 q^{77} - 2110 q^{79} - 320 q^{80} + 1124 q^{82} + 288 q^{83} - 2704 q^{85} - 36 q^{86} + 80 q^{88} - 914 q^{89} - 2984 q^{91} + 816 q^{92} + 1308 q^{94} + 1900 q^{95} + 218 q^{97} - 1078 q^{98}+O(q^{100}) \) 3 * q - 6 * q^2 + 12 * q^4 - 20 * q^5 + 24 * q^7 - 24 * q^8 + 40 * q^10 - 10 * q^11 - 4 * q^13 - 48 * q^14 + 48 * q^16 + 66 * q^17 - 164 * q^19 - 80 * q^20 + 20 * q^22 + 204 * q^23 + 79 * q^25 + 8 * q^26 + 96 * q^28 - 87 * q^29 - 86 * q^31 - 96 * q^32 - 132 * q^34 - 24 * q^35 - 42 * q^37 + 328 * q^38 + 160 * q^40 - 562 * q^41 + 18 * q^43 - 40 * q^44 - 408 * q^46 - 654 * q^47 + 539 * q^49 - 158 * q^50 - 16 * q^52 - 712 * q^53 + 142 * q^55 - 192 * q^56 + 174 * q^58 - 184 * q^59 + 322 * q^61 + 172 * q^62 + 192 * q^64 - 1494 * q^65 - 228 * q^67 + 264 * q^68 + 48 * q^70 + 52 * q^71 - 494 * q^73 + 84 * q^74 - 656 * q^76 - 872 * q^77 - 2110 * q^79 - 320 * q^80 + 1124 * q^82 + 288 * q^83 - 2704 * q^85 - 36 * q^86 + 80 * q^88 - 914 * q^89 - 2984 * q^91 + 816 * q^92 + 1308 * q^94 + 1900 * q^95 + 218 * q^97 - 1078 * q^98

Introduction

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