LMFDB - Embedded newform 531.4.a.c.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,4,Mod(1,531)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(531, 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("531.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	531.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:	$31.3300142130$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 41x^{5} - 7x^{4} + 484x^{3} + 63x^{2} - 1736x - 44 $ x^7 - 41x^5 - 7x^4 + 484x^3 + 63x^2 - 1736*x - 44
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$5.07078$ of defining polynomial
Character	$\chi$	$=$	531.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.07078 q^{2} +17.7128 q^{4} -5.77165 q^{5} -31.1296 q^{7} +49.2517 q^{8} +O(q^{10})$	\(q+5.07078 q^{2} +17.7128 q^{4} -5.77165 q^{5} -31.1296 q^{7} +49.2517 q^{8} -29.2668 q^{10} -52.2494 q^{11} -16.2393 q^{13} -157.851 q^{14} +108.042 q^{16} -102.687 q^{17} +46.2494 q^{19} -102.232 q^{20} -264.945 q^{22} +99.1068 q^{23} -91.6880 q^{25} -82.3459 q^{26} -551.394 q^{28} +119.005 q^{29} +20.2372 q^{31} +153.844 q^{32} -520.703 q^{34} +179.669 q^{35} +117.880 q^{37} +234.521 q^{38} -284.264 q^{40} +278.005 q^{41} -484.571 q^{43} -925.485 q^{44} +502.549 q^{46} +347.541 q^{47} +626.051 q^{49} -464.930 q^{50} -287.644 q^{52} -161.039 q^{53} +301.565 q^{55} -1533.19 q^{56} +603.449 q^{58} -59.0000 q^{59} -845.543 q^{61} +102.618 q^{62} -84.2264 q^{64} +93.7275 q^{65} -740.885 q^{67} -1818.88 q^{68} +911.063 q^{70} +738.136 q^{71} +539.769 q^{73} +597.743 q^{74} +819.208 q^{76} +1626.50 q^{77} -412.312 q^{79} -623.581 q^{80} +1409.70 q^{82} +75.5904 q^{83} +592.673 q^{85} -2457.15 q^{86} -2573.37 q^{88} -163.421 q^{89} +505.523 q^{91} +1755.46 q^{92} +1762.31 q^{94} -266.935 q^{95} +857.136 q^{97} +3174.57 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 26 q^{4} + 2 q^{5} - 59 q^{7} + 21 q^{8}+O(q^{10}) $ 7 * q + 26 * q^4 + 2 * q^5 - 59 * q^7 + 21 * q^8	$ 7 q + 26 q^{4} + 2 q^{5} - 59 q^{7} + 21 q^{8} - 71 q^{10} + 5 q^{11} - 67 q^{13} + 65 q^{14} - 94 q^{16} + 23 q^{17} - 176 q^{19} + 207 q^{20} - 704 q^{22} + 218 q^{23} - 183 q^{25} - 58 q^{26} - 938 q^{28} - 168 q^{29} - 604 q^{31} + 448 q^{32} - 610 q^{34} + 336 q^{35} - 505 q^{37} + 453 q^{38} - 1080 q^{40} + 265 q^{41} - 493 q^{43} - 504 q^{44} + 381 q^{46} + 244 q^{47} + 770 q^{49} - 1639 q^{50} + 160 q^{52} - 686 q^{53} - 116 q^{55} - 2190 q^{56} + 1584 q^{58} - 413 q^{59} - 838 q^{61} - 286 q^{62} + 205 q^{64} - 490 q^{65} - 1504 q^{67} - 3047 q^{68} + 1530 q^{70} + 1267 q^{71} - 666 q^{73} - 528 q^{74} - 64 q^{76} - 1109 q^{77} - 2741 q^{79} - 1213 q^{80} + 953 q^{82} + 2025 q^{83} - 1274 q^{85} - 4394 q^{86} - 1639 q^{88} - 616 q^{89} - 2415 q^{91} - 218 q^{92} + 900 q^{94} - 2554 q^{95} - 1298 q^{97} + 172 q^{98}+O(q^{100}) $ 7 * q + 26 * q^4 + 2 * q^5 - 59 * q^7 + 21 * q^8 - 71 * q^10 + 5 * q^11 - 67 * q^13 + 65 * q^14 - 94 * q^16 + 23 * q^17 - 176 * q^19 + 207 * q^20 - 704 * q^22 + 218 * q^23 - 183 * q^25 - 58 * q^26 - 938 * q^28 - 168 * q^29 - 604 * q^31 + 448 * q^32 - 610 * q^34 + 336 * q^35 - 505 * q^37 + 453 * q^38 - 1080 * q^40 + 265 * q^41 - 493 * q^43 - 504 * q^44 + 381 * q^46 + 244 * q^47 + 770 * q^49 - 1639 * q^50 + 160 * q^52 - 686 * q^53 - 116 * q^55 - 2190 * q^56 + 1584 * q^58 - 413 * q^59 - 838 * q^61 - 286 * q^62 + 205 * q^64 - 490 * q^65 - 1504 * q^67 - 3047 * q^68 + 1530 * q^70 + 1267 * q^71 - 666 * q^73 - 528 * q^74 - 64 * q^76 - 1109 * q^77 - 2741 * q^79 - 1213 * q^80 + 953 * q^82 + 2025 * q^83 - 1274 * q^85 - 4394 * q^86 - 1639 * q^88 - 616 * q^89 - 2415 * q^91 - 218 * q^92 + 900 * q^94 - 2554 * q^95 - 1298 * q^97 + 172 * 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 $

$2$	5.07078	1.79279	0.896396	−	0.443253i	$-0.146176\pi$
$2$	5.07078	1.79279	0.896396	+	0.443253i	$0.146176\pi$
$3$	0	0
$3$	0	0
$4$	17.7128	2.21411
$5$	−5.77165	−0.516232	−0.258116	−	0.966114i	$-0.583102\pi$
$5$	−5.77165	−0.516232	−0.258116	+	0.966114i	$0.583102\pi$
$6$	0	0
$7$	−31.1296	−1.68084	−0.840420	−	0.541936i	$-0.817692\pi$
$7$	−31.1296	−1.68084	−0.840420	+	0.541936i	$0.817692\pi$
$8$	49.2517	2.17664
$9$	0	0
$10$	−29.2668	−0.925497
$11$	−52.2494	−1.43216	−0.716081	−	0.698018i	$-0.754066\pi$
$11$	−52.2494	−1.43216	−0.716081	+	0.698018i	$0.754066\pi$
$12$	0	0
$13$	−16.2393	−0.346459	−0.173230	−	0.984881i	$-0.555420\pi$
$13$	−16.2393	−0.346459	−0.173230	+	0.984881i	$0.555420\pi$
$14$	−157.851	−3.01340
$15$	0	0
$16$	108.042	1.68816
$17$	−102.687	−1.46501	−0.732507	−	0.680760i	$-0.761650\pi$
$17$	−102.687	−1.46501	−0.732507	+	0.680760i	$0.761650\pi$
$18$	0	0
$19$	46.2494	0.558439	0.279219	−	0.960227i	$-0.409924\pi$
$19$	46.2494	0.558439	0.279219	+	0.960227i	$0.409924\pi$
$20$	−102.232	−1.14299
$21$	0	0
$22$	−264.945	−2.56757
$23$	99.1068	0.898487	0.449243	−	0.893409i	$-0.351694\pi$
$23$	99.1068	0.898487	0.449243	+	0.893409i	$0.351694\pi$
$24$	0	0
$25$	−91.6880	−0.733504
$26$	−82.3459	−0.621129
$27$	0	0
$28$	−551.394	−3.72156
$29$	119.005	0.762024	0.381012	−	0.924570i	$-0.375576\pi$
$29$	119.005	0.762024	0.381012	+	0.924570i	$0.375576\pi$
$30$	0	0
$31$	20.2372	0.117249	0.0586243	−	0.998280i	$-0.481329\pi$
$31$	20.2372	0.117249	0.0586243	+	0.998280i	$0.481329\pi$
$32$	153.844	0.849877
$33$	0	0
$34$	−520.703	−2.62647
$35$	179.669	0.867704
$36$	0	0
$37$	117.880	0.523766	0.261883	−	0.965100i	$-0.415657\pi$
$37$	117.880	0.523766	0.261883	+	0.965100i	$0.415657\pi$
$38$	234.521	1.00117
$39$	0	0
$40$	−284.264	−1.12365
$41$	278.005	1.05895	0.529476	−	0.848325i	$-0.322389\pi$
$41$	278.005	1.05895	0.529476	+	0.848325i	$0.322389\pi$
$42$	0	0
$43$	−484.571	−1.71852	−0.859260	−	0.511539i	$-0.829076\pi$
$43$	−484.571	−1.71852	−0.859260	+	0.511539i	$0.829076\pi$
$44$	−925.485	−3.17096
$45$	0	0
$46$	502.549	1.61080
$47$	347.541	1.07860	0.539299	−	0.842114i	$-0.318689\pi$
$47$	347.541	1.07860	0.539299	+	0.842114i	$0.318689\pi$
$48$	0	0
$49$	626.051	1.82522
$50$	−464.930	−1.31502
$51$	0	0
$52$	−287.644	−0.767097
$53$	−161.039	−0.417367	−0.208684	−	0.977983i	$-0.566918\pi$
$53$	−161.039	−0.417367	−0.208684	+	0.977983i	$0.566918\pi$
$54$	0	0
$55$	301.565	0.739328
$56$	−1533.19	−3.65858
$57$	0	0
$58$	603.449	1.36615
$59$	−59.0000	−0.130189
$59$	−59.0000	−0.130189
$60$	0	0
$61$	−845.543	−1.77477	−0.887383	−	0.461033i	$-0.847479\pi$
$61$	−845.543	−1.77477	−0.887383	+	0.461033i	$0.847479\pi$
$62$	102.618	0.210203
$63$	0	0
$64$	−84.2264	−0.164505
$65$	93.7275	0.178853
$66$	0	0
$67$	−740.885	−1.35095	−0.675474	−	0.737384i	$-0.736061\pi$
$67$	−740.885	−1.35095	−0.675474	+	0.737384i	$0.736061\pi$
$68$	−1818.88	−3.24369
$69$	0	0
$70$	911.063	1.55561
$71$	738.136	1.23381	0.616906	−	0.787037i	$-0.288386\pi$
$71$	738.136	1.23381	0.616906	+	0.787037i	$0.288386\pi$
$72$	0	0
$73$	539.769	0.865413	0.432706	−	0.901535i	$-0.357559\pi$
$73$	539.769	0.865413	0.432706	+	0.901535i	$0.357559\pi$
$74$	597.743	0.939003
$75$	0	0
$76$	819.208	1.23644
$77$	1626.50	2.40723
$78$	0	0
$79$	−412.312	−0.587199	−0.293599	−	0.955929i	$-0.594853\pi$
$79$	−412.312	−0.587199	−0.293599	+	0.955929i	$0.594853\pi$
$80$	−623.581	−0.871481
$81$	0	0
$82$	1409.70	1.89848
$83$	75.5904	0.0999654	0.0499827	−	0.998750i	$-0.484083\pi$
$83$	75.5904	0.0999654	0.0499827	+	0.998750i	$0.484083\pi$
$84$	0	0
$85$	592.673	0.756287
$86$	−2457.15	−3.08095
$87$	0	0
$88$	−2573.37	−3.11730
$89$	−163.421	−0.194635	−0.0973177	−	0.995253i	$-0.531026\pi$
$89$	−163.421	−0.194635	−0.0973177	+	0.995253i	$0.531026\pi$
$90$	0	0
$91$	505.523	0.582342
$92$	1755.46	1.98934
$93$	0	0
$94$	1762.31	1.93370
$95$	−266.935	−0.288284
$96$	0	0
$97$	857.136	0.897205	0.448603	−	0.893731i	$-0.351922\pi$
$97$	857.136	0.897205	0.448603	+	0.893731i	$0.351922\pi$
$98$	3174.57	3.27225
$99$	0	0
$100$	−1624.06	−1.62406
$101$	−1635.06	−1.61083	−0.805417	−	0.592708i	$-0.798059\pi$
$101$	−1635.06	−1.61083	−0.805417	+	0.592708i	$0.798059\pi$
$102$	0	0
$103$	−127.175	−0.121659	−0.0608296	−	0.998148i	$-0.519375\pi$
$103$	−127.175	−0.121659	−0.0608296	+	0.998148i	$0.519375\pi$
$104$	−799.813	−0.754117
$105$	0	0
$106$	−816.596	−0.748253
$107$	1769.35	1.59859	0.799296	−	0.600938i	$-0.205206\pi$
$107$	1769.35	1.59859	0.799296	+	0.600938i	$0.205206\pi$
$108$	0	0
$109$	−1862.08	−1.63628	−0.818140	−	0.575019i	$-0.804995\pi$
$109$	−1862.08	−1.63628	−0.818140	+	0.575019i	$0.804995\pi$
$110$	1529.17	1.32546
$111$	0	0
$112$	−3363.31	−2.83752
$113$	−2135.01	−1.77739	−0.888694	−	0.458501i	$-0.848386\pi$
$113$	−2135.01	−1.77739	−0.888694	+	0.458501i	$0.848386\pi$
$114$	0	0
$115$	−572.010	−0.463828
$116$	2107.92	1.68720
$117$	0	0
$118$	−299.176	−0.233402
$119$	3196.60	2.46245
$120$	0	0
$121$	1399.00	1.05109
$122$	−4287.57	−3.18179
$123$	0	0
$124$	358.458	0.259601
$125$	1250.65	0.894891
$126$	0	0
$127$	487.180	0.340396	0.170198	−	0.985410i	$-0.445559\pi$
$127$	487.180	0.340396	0.170198	+	0.985410i	$0.445559\pi$
$128$	−1657.85	−1.14480
$129$	0	0
$130$	475.272	0.320647
$131$	−1162.04	−0.775022	−0.387511	−	0.921865i	$-0.626665\pi$
$131$	−1162.04	−0.775022	−0.387511	+	0.921865i	$0.626665\pi$
$132$	0	0
$133$	−1439.72	−0.938647
$134$	−3756.87	−2.42197
$135$	0	0
$136$	−5057.51	−3.18881
$137$	−47.9021	−0.0298727	−0.0149363	−	0.999888i	$-0.504755\pi$
$137$	−47.9021	−0.0298727	−0.0149363	+	0.999888i	$0.504755\pi$
$138$	0	0
$139$	−2217.52	−1.35315	−0.676573	−	0.736375i	$-0.736536\pi$
$139$	−2217.52	−1.35315	−0.676573	+	0.736375i	$0.736536\pi$
$140$	3182.45	1.92119
$141$	0	0
$142$	3742.93	2.21197
$143$	848.493	0.496185
$144$	0	0
$145$	−686.856	−0.393382
$146$	2737.05	1.55151
$147$	0	0
$148$	2087.99	1.15967
$149$	830.176	0.456448	0.228224	−	0.973609i	$-0.426708\pi$
$149$	830.176	0.456448	0.228224	+	0.973609i	$0.426708\pi$
$150$	0	0
$151$	−2874.89	−1.54937	−0.774685	−	0.632347i	$-0.782092\pi$
$151$	−2874.89	−1.54937	−0.774685	+	0.632347i	$0.782092\pi$
$152$	2277.86	1.21552
$153$	0	0
$154$	8247.64	4.31567
$155$	−116.802	−0.0605275
$156$	0	0
$157$	−341.385	−0.173538	−0.0867691	−	0.996228i	$-0.527654\pi$
$157$	−341.385	−0.173538	−0.0867691	+	0.996228i	$0.527654\pi$
$158$	−2090.74	−1.05273
$159$	0	0
$160$	−887.935	−0.438734
$161$	−3085.15	−1.51021
$162$	0	0
$163$	3435.09	1.65065	0.825327	−	0.564655i	$-0.190991\pi$
$163$	3435.09	1.65065	0.825327	+	0.564655i	$0.190991\pi$
$164$	4924.25	2.34463
$165$	0	0
$166$	383.303	0.179217
$167$	2320.44	1.07522	0.537608	−	0.843195i	$-0.319328\pi$
$167$	2320.44	1.07522	0.537608	+	0.843195i	$0.319328\pi$
$168$	0	0
$169$	−1933.29	−0.879966
$170$	3005.32	1.35587
$171$	0	0
$172$	−8583.13	−3.80498
$173$	2665.42	1.17138	0.585689	−	0.810536i	$-0.300824\pi$
$173$	2665.42	1.17138	0.585689	+	0.810536i	$0.300824\pi$
$174$	0	0
$175$	2854.21	1.23290
$176$	−5645.13	−2.41771
$177$	0	0
$178$	−828.670	−0.348941
$179$	−771.425	−0.322117	−0.161059	−	0.986945i	$-0.551491\pi$
$179$	−771.425	−0.322117	−0.161059	+	0.986945i	$0.551491\pi$
$180$	0	0
$181$	1057.15	0.434129	0.217065	−	0.976157i	$-0.430352\pi$
$181$	1057.15	0.434129	0.217065	+	0.976157i	$0.430352\pi$
$182$	2563.40	1.04402
$183$	0	0
$184$	4881.18	1.95568
$185$	−680.362	−0.270385
$186$	0	0
$187$	5365.32	2.09814
$188$	6155.95	2.38813
$189$	0	0
$190$	−1353.57	−0.516834
$191$	−2555.11	−0.967966	−0.483983	−	0.875078i	$-0.660810\pi$
$191$	−2555.11	−0.967966	−0.483983	+	0.875078i	$0.660810\pi$
$192$	0	0
$193$	−2038.27	−0.760197	−0.380098	−	0.924946i	$-0.624110\pi$
$193$	−2038.27	−0.760197	−0.380098	+	0.924946i	$0.624110\pi$
$194$	4346.35	1.60850
$195$	0	0
$196$	11089.2	4.04124
$197$	−1204.93	−0.435776	−0.217888	−	0.975974i	$-0.569917\pi$
$197$	−1204.93	−0.435776	−0.217888	+	0.975974i	$0.569917\pi$
$198$	0	0
$199$	3829.73	1.36423	0.682116	−	0.731244i	$-0.261060\pi$
$199$	3829.73	1.36423	0.682116	+	0.731244i	$0.261060\pi$
$200$	−4515.79	−1.59657
$201$	0	0
$202$	−8291.02	−2.88789
$203$	−3704.58	−1.28084
$204$	0	0
$205$	−1604.55	−0.546665
$206$	−644.876	−0.218110
$207$	0	0
$208$	−1754.53	−0.584878
$209$	−2416.50	−0.799775
$210$	0	0
$211$	3285.19	1.07186	0.535929	−	0.844263i	$-0.319961\pi$
$211$	3285.19	1.07186	0.535929	+	0.844263i	$0.319961\pi$
$212$	−2852.46	−0.924095
$213$	0	0
$214$	8971.98	2.86594
$215$	2796.77	0.887155
$216$	0	0
$217$	−629.976	−0.197076
$218$	−9442.19	−2.93351
$219$	0	0
$220$	5341.58	1.63695
$221$	1667.56	0.507567
$222$	0	0
$223$	426.083	0.127949	0.0639745	−	0.997952i	$-0.479622\pi$
$223$	426.083	0.127949	0.0639745	+	0.997952i	$0.479622\pi$
$224$	−4789.10	−1.42851
$225$	0	0
$226$	−10826.2	−3.18649
$227$	4529.34	1.32433	0.662164	−	0.749359i	$-0.269638\pi$
$227$	4529.34	1.32433	0.662164	+	0.749359i	$0.269638\pi$
$228$	0	0
$229$	2137.81	0.616900	0.308450	−	0.951241i	$-0.400190\pi$
$229$	2137.81	0.616900	0.308450	+	0.951241i	$0.400190\pi$
$230$	−2900.54	−0.831547
$231$	0	0
$232$	5861.21	1.65865
$233$	−5634.00	−1.58410	−0.792051	−	0.610455i	$-0.790987\pi$
$233$	−5634.00	−1.58410	−0.792051	+	0.610455i	$0.790987\pi$
$234$	0	0
$235$	−2005.89	−0.556807
$236$	−1045.06	−0.288252
$237$	0	0
$238$	16209.3	4.41467
$239$	4143.54	1.12144	0.560718	−	0.828007i	$-0.310525\pi$
$239$	4143.54	1.12144	0.560718	+	0.828007i	$0.310525\pi$
$240$	0	0
$241$	−5880.72	−1.57183	−0.785914	−	0.618335i	$-0.787807\pi$
$241$	−5880.72	−1.57183	−0.785914	+	0.618335i	$0.787807\pi$
$242$	7094.00	1.88438
$243$	0	0
$244$	−14977.0	−3.92952
$245$	−3613.35	−0.942239
$246$	0	0
$247$	−751.057	−0.193476
$248$	996.717	0.255208
$249$	0	0
$250$	6341.76	1.60435
$251$	2345.45	0.589814	0.294907	−	0.955526i	$-0.404711\pi$
$251$	2345.45	0.589814	0.294907	+	0.955526i	$0.404711\pi$
$252$	0	0
$253$	−5178.27	−1.28678
$254$	2470.38	0.610259
$255$	0	0
$256$	−7732.77	−1.88788
$257$	4401.18	1.06824	0.534120	−	0.845409i	$-0.320643\pi$
$257$	4401.18	1.06824	0.534120	+	0.845409i	$0.320643\pi$
$258$	0	0
$259$	−3669.55	−0.880366
$260$	1660.18	0.396000
$261$	0	0
$262$	−5892.45	−1.38945
$263$	−6751.11	−1.58286	−0.791428	−	0.611262i	$-0.790662\pi$
$263$	−6751.11	−1.58286	−0.791428	+	0.611262i	$0.790662\pi$
$264$	0	0
$265$	929.463	0.215458
$266$	−7300.53	−1.68280
$267$	0	0
$268$	−13123.2	−2.99114
$269$	−2097.51	−0.475419	−0.237709	−	0.971336i	$-0.576397\pi$
$269$	−2097.51	−0.475419	−0.237709	+	0.971336i	$0.576397\pi$
$270$	0	0
$271$	−5765.27	−1.29231	−0.646153	−	0.763208i	$-0.723623\pi$
$271$	−5765.27	−1.29231	−0.646153	+	0.763208i	$0.723623\pi$
$272$	−11094.5	−2.47317
$273$	0	0
$274$	−242.901	−0.0535555
$275$	4790.64	1.05050
$276$	0	0
$277$	−3383.35	−0.733884	−0.366942	−	0.930244i	$-0.619595\pi$
$277$	−3383.35	−0.733884	−0.366942	+	0.930244i	$0.619595\pi$
$278$	−11244.5	−2.42591
$279$	0	0
$280$	8849.02	1.88868
$281$	2239.18	0.475368	0.237684	−	0.971343i	$-0.423612\pi$
$281$	2239.18	0.475368	0.237684	+	0.971343i	$0.423612\pi$
$282$	0	0
$283$	−4303.21	−0.903884	−0.451942	−	0.892047i	$-0.649269\pi$
$283$	−4303.21	−0.903884	−0.451942	+	0.892047i	$0.649269\pi$
$284$	13074.5	2.73179
$285$	0	0
$286$	4302.52	0.889558
$287$	−8654.17	−1.77993
$288$	0	0
$289$	5631.60	1.14626
$290$	−3482.90	−0.705252
$291$	0	0
$292$	9560.83	1.91611
$293$	3649.23	0.727612	0.363806	−	0.931475i	$-0.381477\pi$
$293$	3649.23	0.727612	0.363806	+	0.931475i	$0.381477\pi$
$294$	0	0
$295$	340.527	0.0672077
$296$	5805.79	1.14005
$297$	0	0
$298$	4209.64	0.818316
$299$	−1609.42	−0.311289
$300$	0	0
$301$	15084.5	2.88856
$302$	−14577.9	−2.77770
$303$	0	0
$304$	4996.88	0.942733
$305$	4880.18	0.916192
$306$	0	0
$307$	−4242.25	−0.788658	−0.394329	−	0.918969i	$-0.629023\pi$
$307$	−4242.25	−0.788658	−0.394329	+	0.918969i	$0.629023\pi$
$308$	28810.0	5.32987
$309$	0	0
$310$	−592.278	−0.108513
$311$	292.249	0.0532859	0.0266430	−	0.999645i	$-0.491518\pi$
$311$	292.249	0.0532859	0.0266430	+	0.999645i	$0.491518\pi$
$312$	0	0
$313$	5053.05	0.912509	0.456254	−	0.889849i	$-0.349191\pi$
$313$	5053.05	0.912509	0.456254	+	0.889849i	$0.349191\pi$
$314$	−1731.09	−0.311118
$315$	0	0
$316$	−7303.21	−1.30012
$317$	7858.38	1.39234	0.696168	−	0.717879i	$-0.254887\pi$
$317$	7858.38	1.39234	0.696168	+	0.717879i	$0.254887\pi$
$318$	0	0
$319$	−6217.94	−1.09134
$320$	486.126	0.0849226
$321$	0	0
$322$	−15644.2	−2.70750
$323$	−4749.21	−0.818121
$324$	0	0
$325$	1488.95	0.254129
$326$	17418.6	2.95928
$327$	0	0
$328$	13692.2	2.30496
$329$	−10818.8	−1.81295
$330$	0	0
$331$	6703.97	1.11324	0.556622	−	0.830766i	$-0.312097\pi$
$331$	6703.97	1.11324	0.556622	+	0.830766i	$0.312097\pi$
$332$	1338.92	0.221334
$333$	0	0
$334$	11766.4	1.92764
$335$	4276.13	0.697403
$336$	0	0
$337$	−1034.27	−0.167182	−0.0835910	−	0.996500i	$-0.526639\pi$
$337$	−1034.27	−0.167182	−0.0835910	+	0.996500i	$0.526639\pi$
$338$	−9803.27	−1.57760
$339$	0	0
$340$	10497.9	1.67450
$341$	−1057.38	−0.167919
$342$	0	0
$343$	−8811.28	−1.38707
$344$	−23865.9	−3.74060
$345$	0	0
$346$	13515.8	2.10004
$347$	2414.72	0.373571	0.186785	−	0.982401i	$-0.440193\pi$
$347$	2414.72	0.373571	0.186785	+	0.982401i	$0.440193\pi$
$348$	0	0
$349$	1429.79	0.219298	0.109649	−	0.993970i	$-0.465027\pi$
$349$	1429.79	0.219298	0.109649	+	0.993970i	$0.465027\pi$
$350$	14473.1	2.21034
$351$	0	0
$352$	−8038.26	−1.21716
$353$	−9657.60	−1.45615	−0.728077	−	0.685496i	$-0.759585\pi$
$353$	−9657.60	−1.45615	−0.728077	+	0.685496i	$0.759585\pi$
$354$	0	0
$355$	−4260.27	−0.636934
$356$	−2894.64	−0.430943
$357$	0	0
$358$	−3911.73	−0.577490
$359$	−1728.23	−0.254074	−0.127037	−	0.991898i	$-0.540547\pi$
$359$	−1728.23	−0.254074	−0.127037	+	0.991898i	$0.540547\pi$
$360$	0	0
$361$	−4719.99	−0.688146
$362$	5360.58	0.778304
$363$	0	0
$364$	8954.24	1.28937
$365$	−3115.36	−0.446754
$366$	0	0
$367$	−2452.53	−0.348831	−0.174416	−	0.984672i	$-0.555804\pi$
$367$	−2452.53	−0.348831	−0.174416	+	0.984672i	$0.555804\pi$
$368$	10707.7	1.51679
$369$	0	0
$370$	−3449.97	−0.484744
$371$	5013.09	0.701527
$372$	0	0
$373$	8584.89	1.19171	0.595856	−	0.803091i	$-0.296813\pi$
$373$	8584.89	1.19171	0.595856	+	0.803091i	$0.296813\pi$
$374$	27206.4	3.76152
$375$	0	0
$376$	17117.0	2.34772
$377$	−1932.56	−0.264010
$378$	0	0
$379$	−6441.28	−0.872998	−0.436499	−	0.899705i	$-0.643782\pi$
$379$	−6441.28	−0.872998	−0.436499	+	0.899705i	$0.643782\pi$
$380$	−4728.19	−0.638292
$381$	0	0
$382$	−12956.4	−1.73536
$383$	10167.8	1.35653	0.678264	−	0.734818i	$-0.262732\pi$
$383$	10167.8	1.35653	0.678264	+	0.734818i	$0.262732\pi$
$384$	0	0
$385$	−9387.60	−1.24269
$386$	−10335.6	−1.36288
$387$	0	0
$388$	15182.3	1.98651
$389$	−3426.40	−0.446595	−0.223297	−	0.974750i	$-0.571682\pi$
$389$	−3426.40	−0.446595	−0.223297	+	0.974750i	$0.571682\pi$
$390$	0	0
$391$	−10177.0	−1.31630
$392$	30834.1	3.97285
$393$	0	0
$394$	−6109.95	−0.781256
$395$	2379.72	0.303131
$396$	0	0
$397$	−8161.03	−1.03171	−0.515857	−	0.856675i	$-0.672526\pi$
$397$	−8161.03	−1.03171	−0.515857	+	0.856675i	$0.672526\pi$
$398$	19419.7	2.44578
$399$	0	0
$400$	−9906.16	−1.23827
$401$	−8303.40	−1.03404	−0.517022	−	0.855972i	$-0.672960\pi$
$401$	−8303.40	−1.03404	−0.517022	+	0.855972i	$0.672960\pi$
$402$	0	0
$403$	−328.638	−0.0406219
$404$	−28961.5	−3.56656
$405$	0	0
$406$	−18785.1	−2.29628
$407$	−6159.15	−0.750117
$408$	0	0
$409$	−1611.03	−0.194769	−0.0973844	−	0.995247i	$-0.531048\pi$
$409$	−1611.03	−0.194769	−0.0973844	+	0.995247i	$0.531048\pi$
$410$	−8136.31	−0.980057
$411$	0	0
$412$	−2252.63	−0.269366
$413$	1836.65	0.218827
$414$	0	0
$415$	−436.282	−0.0516053
$416$	−2498.32	−0.294448
$417$	0	0
$418$	−12253.6	−1.43383
$419$	−12278.0	−1.43155	−0.715776	−	0.698330i	$-0.753927\pi$
$419$	−12278.0	−1.43155	−0.715776	+	0.698330i	$0.753927\pi$
$420$	0	0
$421$	−2402.19	−0.278089	−0.139045	−	0.990286i	$-0.544403\pi$
$421$	−2402.19	−0.278089	−0.139045	+	0.990286i	$0.544403\pi$
$422$	16658.5	1.92162
$423$	0	0
$424$	−7931.46	−0.908457
$425$	9415.16	1.07459
$426$	0	0
$427$	26321.4	2.98310
$428$	31340.2	3.53945
$429$	0	0
$430$	14181.8	1.59049
$431$	−7294.64	−0.815245	−0.407622	−	0.913151i	$-0.633642\pi$
$431$	−7294.64	−0.815245	−0.407622	+	0.913151i	$0.633642\pi$
$432$	0	0
$433$	13340.5	1.48060	0.740302	−	0.672275i	$-0.234683\pi$
$433$	13340.5	1.48060	0.740302	+	0.672275i	$0.234683\pi$
$434$	−3194.47	−0.353317
$435$	0	0
$436$	−32982.7	−3.62290
$437$	4583.63	0.501750
$438$	0	0
$439$	7920.79	0.861136	0.430568	−	0.902558i	$-0.358313\pi$
$439$	7920.79	0.861136	0.430568	+	0.902558i	$0.358313\pi$
$440$	14852.6	1.60925
$441$	0	0
$442$	8455.85	0.909963
$443$	−776.340	−0.0832619	−0.0416310	−	0.999133i	$-0.513255\pi$
$443$	−776.340	−0.0832619	−0.0416310	+	0.999133i	$0.513255\pi$
$444$	0	0
$445$	943.207	0.100477
$446$	2160.57	0.229386
$447$	0	0
$448$	2621.93	0.276506
$449$	−2796.54	−0.293935	−0.146968	−	0.989141i	$-0.546951\pi$
$449$	−2796.54	−0.293935	−0.146968	+	0.989141i	$0.546951\pi$
$450$	0	0
$451$	−14525.6	−1.51659
$452$	−37817.1	−3.93532
$453$	0	0
$454$	22967.3	2.37425
$455$	−2917.70	−0.300624
$456$	0	0
$457$	−3189.75	−0.326499	−0.163250	−	0.986585i	$-0.552198\pi$
$457$	−3189.75	−0.326499	−0.163250	+	0.986585i	$0.552198\pi$
$458$	10840.3	1.10597
$459$	0	0
$460$	−10131.9	−1.02696
$461$	−11072.8	−1.11868	−0.559339	−	0.828939i	$-0.688945\pi$
$461$	−11072.8	−1.11868	−0.559339	+	0.828939i	$0.688945\pi$
$462$	0	0
$463$	15208.6	1.52658	0.763288	−	0.646059i	$-0.223584\pi$
$463$	15208.6	1.52658	0.763288	+	0.646059i	$0.223584\pi$
$464$	12857.6	1.28642
$465$	0	0
$466$	−28568.8	−2.83997
$467$	2666.62	0.264233	0.132116	−	0.991234i	$-0.457823\pi$
$467$	2666.62	0.264233	0.132116	+	0.991234i	$0.457823\pi$
$468$	0	0
$469$	23063.4	2.27073
$470$	−10171.4	−0.998240
$471$	0	0
$472$	−2905.85	−0.283374
$473$	25318.5	2.46120
$474$	0	0
$475$	−4240.52	−0.409617
$476$	56620.9	5.45213
$477$	0	0
$478$	21011.0	2.01050
$479$	1470.72	0.140290	0.0701451	−	0.997537i	$-0.477654\pi$
$479$	1470.72	0.140290	0.0701451	+	0.997537i	$0.477654\pi$
$480$	0	0
$481$	−1914.29	−0.181463
$482$	−29819.9	−2.81796
$483$	0	0
$484$	24780.2	2.32722
$485$	−4947.09	−0.463166
$486$	0	0
$487$	−10466.9	−0.973921	−0.486960	−	0.873424i	$-0.661894\pi$
$487$	−10466.9	−0.973921	−0.486960	+	0.873424i	$0.661894\pi$
$488$	−41644.5	−3.86303
$489$	0	0
$490$	−18322.5	−1.68924
$491$	−8125.93	−0.746880	−0.373440	−	0.927654i	$-0.621822\pi$
$491$	−8125.93	−0.746880	−0.373440	+	0.927654i	$0.621822\pi$
$492$	0	0
$493$	−12220.3	−1.11638
$494$	−3808.45	−0.346863
$495$	0	0
$496$	2186.47	0.197934
$497$	−22977.9	−2.07384
$498$	0	0
$499$	−14989.0	−1.34469	−0.672344	−	0.740239i	$-0.734712\pi$
$499$	−14989.0	−1.34469	−0.672344	+	0.740239i	$0.734712\pi$
$500$	22152.5	1.98138
$501$	0	0
$502$	11893.2	1.05741
$503$	5526.27	0.489869	0.244935	−	0.969540i	$-0.421234\pi$
$503$	5526.27	0.489869	0.244935	+	0.969540i	$0.421234\pi$
$504$	0	0
$505$	9436.98	0.831565
$506$	−26257.9	−2.30693
$507$	0	0
$508$	8629.34	0.753672
$509$	−16053.1	−1.39792	−0.698962	−	0.715159i	$-0.746354\pi$
$509$	−16053.1	−1.39792	−0.698962	+	0.715159i	$0.746354\pi$
$510$	0	0
$511$	−16802.8	−1.45462
$512$	−25948.4	−2.23978
$513$	0	0
$514$	22317.4	1.91513
$515$	734.008	0.0628044
$516$	0	0
$517$	−18158.8	−1.54473
$518$	−18607.5	−1.57831
$519$	0	0
$520$	4616.24	0.389299
$521$	16059.8	1.35047	0.675234	−	0.737604i	$-0.264043\pi$
$521$	16059.8	1.35047	0.675234	+	0.737604i	$0.264043\pi$
$522$	0	0
$523$	−18602.9	−1.55535	−0.777675	−	0.628666i	$-0.783601\pi$
$523$	−18602.9	−1.55535	−0.777675	+	0.628666i	$0.783601\pi$
$524$	−20583.0	−1.71598
$525$	0	0
$526$	−34233.4	−2.83773
$527$	−2078.10	−0.171771
$528$	0	0
$529$	−2344.84	−0.192721
$530$	4713.11	0.386272
$531$	0	0
$532$	−25501.6	−2.07826
$533$	−4514.60	−0.366884
$534$	0	0
$535$	−10212.1	−0.825245
$536$	−36489.9	−2.94053
$537$	0	0
$538$	−10636.0	−0.852327
$539$	−32710.8	−2.61401
$540$	0	0
$541$	−21283.9	−1.69144	−0.845718	−	0.533630i	$-0.820827\pi$
$541$	−21283.9	−1.69144	−0.845718	+	0.533630i	$0.820827\pi$
$542$	−29234.4	−2.31684
$543$	0	0
$544$	−15797.8	−1.24508
$545$	10747.3	0.844701
$546$	0	0
$547$	17983.2	1.40568	0.702839	−	0.711349i	$-0.251915\pi$
$547$	17983.2	1.40568	0.702839	+	0.711349i	$0.251915\pi$
$548$	−848.483	−0.0661412
$549$	0	0
$550$	24292.3	1.88332
$551$	5503.92	0.425544
$552$	0	0
$553$	12835.1	0.986987
$554$	−17156.2	−1.31570
$555$	0	0
$556$	−39278.5	−2.99601
$557$	−22854.0	−1.73852	−0.869261	−	0.494354i	$-0.835405\pi$
$557$	−22854.0	−1.73852	−0.869261	+	0.494354i	$0.835405\pi$
$558$	0	0
$559$	7869.09	0.595397
$560$	19411.8	1.46482
$561$	0	0
$562$	11354.4	0.852236
$563$	9314.59	0.697271	0.348635	−	0.937258i	$-0.386645\pi$
$563$	9314.59	0.697271	0.348635	+	0.937258i	$0.386645\pi$
$564$	0	0
$565$	12322.5	0.917545
$566$	−21820.6	−1.62048
$567$	0	0
$568$	36354.5	2.68556
$569$	−13990.2	−1.03075	−0.515376	−	0.856964i	$-0.672348\pi$
$569$	−13990.2	−1.03075	−0.515376	+	0.856964i	$0.672348\pi$
$570$	0	0
$571$	19287.1	1.41355	0.706777	−	0.707437i	$-0.250149\pi$
$571$	19287.1	1.41355	0.706777	+	0.707437i	$0.250149\pi$
$572$	15029.2	1.09861
$573$	0	0
$574$	−43883.4	−3.19104
$575$	−9086.91	−0.659044
$576$	0	0
$577$	−23173.6	−1.67197	−0.835986	−	0.548750i	$-0.815104\pi$
$577$	−23173.6	−1.67197	−0.835986	+	0.548750i	$0.815104\pi$
$578$	28556.6	2.05501
$579$	0	0
$580$	−12166.2	−0.870988
$581$	−2353.10	−0.168026
$582$	0	0
$583$	8414.20	0.597737
$584$	26584.5	1.88369
$585$	0	0
$586$	18504.5	1.30446
$587$	−26048.4	−1.83157	−0.915787	−	0.401663i	$-0.868432\pi$
$587$	−26048.4	−1.83157	−0.915787	+	0.401663i	$0.868432\pi$
$588$	0	0
$589$	935.958	0.0654762
$590$	1726.74	0.120489
$591$	0	0
$592$	12736.0	0.884199
$593$	54.2397	0.00375609	0.00187804	−	0.999998i	$-0.499402\pi$
$593$	54.2397	0.00375609	0.00187804	+	0.999998i	$0.499402\pi$
$594$	0	0
$595$	−18449.7	−1.27120
$596$	14704.8	1.01062
$597$	0	0
$598$	−8161.04	−0.558077
$599$	7953.75	0.542540	0.271270	−	0.962503i	$-0.412556\pi$
$599$	7953.75	0.542540	0.271270	+	0.962503i	$0.412556\pi$
$600$	0	0
$601$	10456.8	0.709720	0.354860	−	0.934919i	$-0.384528\pi$
$601$	10456.8	0.709720	0.354860	+	0.934919i	$0.384528\pi$
$602$	76490.2	5.17858
$603$	0	0
$604$	−50922.4	−3.43047
$605$	−8074.52	−0.542605
$606$	0	0
$607$	6233.80	0.416840	0.208420	−	0.978039i	$-0.433168\pi$
$607$	6233.80	0.416840	0.208420	+	0.978039i	$0.433168\pi$
$608$	7115.20	0.474604
$609$	0	0
$610$	24746.3	1.64254
$611$	−5643.83	−0.373690
$612$	0	0
$613$	−12563.6	−0.827794	−0.413897	−	0.910324i	$-0.635833\pi$
$613$	−12563.6	−0.827794	−0.413897	+	0.910324i	$0.635833\pi$
$614$	−21511.5	−1.41390
$615$	0	0
$616$	80108.0	5.23968
$617$	14795.6	0.965396	0.482698	−	0.875787i	$-0.339657\pi$
$617$	14795.6	0.965396	0.482698	+	0.875787i	$0.339657\pi$
$618$	0	0
$619$	−1136.52	−0.0737976	−0.0368988	−	0.999319i	$-0.511748\pi$
$619$	−1136.52	−0.0737976	−0.0368988	+	0.999319i	$0.511748\pi$
$620$	−2068.90	−0.134014
$621$	0	0
$622$	1481.93	0.0955306
$623$	5087.22	0.327151
$624$	0	0
$625$	4242.70	0.271533
$626$	25622.9	1.63594
$627$	0	0
$628$	−6046.90	−0.384232
$629$	−12104.7	−0.767324
$630$	0	0
$631$	−8230.74	−0.519272	−0.259636	−	0.965707i	$-0.583603\pi$
$631$	−8230.74	−0.519272	−0.259636	+	0.965707i	$0.583603\pi$
$632$	−20307.1	−1.27812
$633$	0	0
$634$	39848.1	2.49617
$635$	−2811.83	−0.175723
$636$	0	0
$637$	−10166.6	−0.632365
$638$	−31529.8	−1.95655
$639$	0	0
$640$	9568.51	0.590982
$641$	20600.4	1.26937	0.634685	−	0.772771i	$-0.281130\pi$
$641$	20600.4	1.26937	0.634685	+	0.772771i	$0.281130\pi$
$642$	0	0
$643$	8955.01	0.549224	0.274612	−	0.961555i	$-0.411451\pi$
$643$	8955.01	0.549224	0.274612	+	0.961555i	$0.411451\pi$
$644$	−54646.9	−3.34377
$645$	0	0
$646$	−24082.2	−1.46672
$647$	19489.8	1.18427	0.592136	−	0.805838i	$-0.298285\pi$
$647$	19489.8	1.18427	0.592136	+	0.805838i	$0.298285\pi$
$648$	0	0
$649$	3082.71	0.186452
$650$	7550.14	0.455601
$651$	0	0
$652$	60845.1	3.65472
$653$	−24922.7	−1.49357	−0.746784	−	0.665067i	$-0.768403\pi$
$653$	−24922.7	−1.49357	−0.746784	+	0.665067i	$0.768403\pi$
$654$	0	0
$655$	6706.89	0.400092
$656$	30036.2	1.78768
$657$	0	0
$658$	−54859.9	−3.25025
$659$	29532.0	1.74568	0.872839	−	0.488008i	$-0.162276\pi$
$659$	29532.0	1.74568	0.872839	+	0.488008i	$0.162276\pi$
$660$	0	0
$661$	23688.8	1.39393	0.696965	−	0.717105i	$-0.254533\pi$
$661$	23688.8	1.39393	0.696965	+	0.717105i	$0.254533\pi$
$662$	33994.4	1.99581
$663$	0	0
$664$	3722.96	0.217589
$665$	8309.59	0.484560
$666$	0	0
$667$	11794.2	0.684669
$668$	41101.6	2.38064
$669$	0	0
$670$	21683.3	1.25030
$671$	44179.1	2.54175
$672$	0	0
$673$	11534.1	0.660636	0.330318	−	0.943870i	$-0.392844\pi$
$673$	11534.1	0.660636	0.330318	+	0.943870i	$0.392844\pi$
$674$	−5244.56	−0.299723
$675$	0	0
$676$	−34244.0	−1.94834
$677$	−3710.82	−0.210662	−0.105331	−	0.994437i	$-0.533590\pi$
$677$	−3710.82	−0.210662	−0.105331	+	0.994437i	$0.533590\pi$
$678$	0	0
$679$	−26682.3	−1.50806
$680$	29190.2	1.64616
$681$	0	0
$682$	−5361.75	−0.301044
$683$	8278.65	0.463797	0.231899	−	0.972740i	$-0.425506\pi$
$683$	8278.65	0.463797	0.231899	+	0.972740i	$0.425506\pi$
$684$	0	0
$685$	276.474	0.0154212
$686$	−44680.1	−2.48672
$687$	0	0
$688$	−52354.0	−2.90113
$689$	2615.16	0.144601
$690$	0	0
$691$	19296.8	1.06235	0.531176	−	0.847261i	$-0.321750\pi$
$691$	19296.8	1.06235	0.531176	+	0.847261i	$0.321750\pi$
$692$	47212.2	2.59355
$693$	0	0
$694$	12244.5	0.669735
$695$	12798.7	0.698538
$696$	0	0
$697$	−28547.4	−1.55138
$698$	7250.17	0.393156
$699$	0	0
$700$	50556.2	2.72978
$701$	−1826.74	−0.0984237	−0.0492119	−	0.998788i	$-0.515671\pi$
$701$	−1826.74	−0.0984237	−0.0492119	+	0.998788i	$0.515671\pi$
$702$	0	0
$703$	5451.87	0.292491
$704$	4400.78	0.235597
$705$	0	0
$706$	−48971.6	−2.61058
$707$	50898.7	2.70756
$708$	0	0
$709$	−10531.6	−0.557862	−0.278931	−	0.960311i	$-0.589980\pi$
$709$	−10531.6	−0.557862	−0.278931	+	0.960311i	$0.589980\pi$
$710$	−21602.9	−1.14189
$711$	0	0
$712$	−8048.75	−0.423651
$713$	2005.64	0.105346
$714$	0	0
$715$	−4897.20	−0.256147
$716$	−13664.1	−0.713202
$717$	0	0
$718$	−8763.47	−0.455501
$719$	−20954.3	−1.08688	−0.543438	−	0.839449i	$-0.682878\pi$
$719$	−20954.3	−1.08688	−0.543438	+	0.839449i	$0.682878\pi$
$720$	0	0
$721$	3958.90	0.204490
$722$	−23934.1	−1.23370
$723$	0	0
$724$	18725.2	0.961208
$725$	−10911.4	−0.558948
$726$	0	0
$727$	12108.1	0.617697	0.308849	−	0.951111i	$-0.400056\pi$
$727$	12108.1	0.617697	0.308849	+	0.951111i	$0.400056\pi$
$728$	24897.9	1.26755
$729$	0	0
$730$	−15797.3	−0.800937
$731$	49759.1	2.51765
$732$	0	0
$733$	11182.5	0.563486	0.281743	−	0.959490i	$-0.409087\pi$
$733$	11182.5	0.563486	0.281743	+	0.959490i	$0.409087\pi$
$734$	−12436.3	−0.625382
$735$	0	0
$736$	15247.0	0.763603
$737$	38710.8	1.93477
$738$	0	0
$739$	−9024.78	−0.449231	−0.224616	−	0.974447i	$-0.572113\pi$
$739$	−9024.78	−0.449231	−0.224616	+	0.974447i	$0.572113\pi$
$740$	−12051.1	−0.598660
$741$	0	0
$742$	25420.3	1.25769
$743$	−17214.3	−0.849974	−0.424987	−	0.905200i	$-0.639721\pi$
$743$	−17214.3	−0.849974	−0.424987	+	0.905200i	$0.639721\pi$
$744$	0	0
$745$	−4791.49	−0.235633
$746$	43532.1	2.13649
$747$	0	0
$748$	95035.1	4.64549
$749$	−55079.1	−2.68698
$750$	0	0
$751$	−10115.7	−0.491517	−0.245758	−	0.969331i	$-0.579037\pi$
$751$	−10115.7	−0.491517	−0.245758	+	0.969331i	$0.579037\pi$
$752$	37549.1	1.82084
$753$	0	0
$754$	−9799.59	−0.473316
$755$	16592.8	0.799835
$756$	0	0
$757$	21416.7	1.02827	0.514137	−	0.857708i	$-0.328112\pi$
$757$	21416.7	1.02827	0.514137	+	0.857708i	$0.328112\pi$
$758$	−32662.3	−1.56510
$759$	0	0
$760$	−13147.0	−0.627491
$761$	26539.8	1.26421	0.632107	−	0.774881i	$-0.282190\pi$
$761$	26539.8	1.26421	0.632107	+	0.774881i	$0.282190\pi$
$762$	0	0
$763$	57965.7	2.75033
$764$	−45258.3	−2.14318
$765$	0	0
$766$	51558.7	2.43197
$767$	958.118	0.0451051
$768$	0	0
$769$	−8581.69	−0.402423	−0.201212	−	0.979548i	$-0.564488\pi$
$769$	−8581.69	−0.402423	−0.201212	+	0.979548i	$0.564488\pi$
$770$	−47602.5	−2.22789
$771$	0	0
$772$	−36103.6	−1.68316
$773$	9126.70	0.424663	0.212331	−	0.977198i	$-0.431894\pi$
$773$	9126.70	0.424663	0.212331	+	0.977198i	$0.431894\pi$
$774$	0	0
$775$	−1855.51	−0.0860024
$776$	42215.4	1.95289
$777$	0	0
$778$	−17374.5	−0.800652
$779$	12857.5	0.591360
$780$	0	0
$781$	−38567.2	−1.76702
$782$	−51605.2	−2.35984
$783$	0	0
$784$	67639.9	3.08126
$785$	1970.36	0.0895860
$786$	0	0
$787$	29121.3	1.31901	0.659507	−	0.751699i	$-0.270765\pi$
$787$	29121.3	1.31901	0.659507	+	0.751699i	$0.270765\pi$
$788$	−21342.8	−0.964854
$789$	0	0
$790$	12067.0	0.543451
$791$	66462.0	2.98750
$792$	0	0
$793$	13731.0	0.614884
$794$	−41382.8	−1.84965
$795$	0	0
$796$	67835.4	3.02055
$797$	29979.5	1.33241	0.666204	−	0.745770i	$-0.267918\pi$
$797$	29979.5	1.33241	0.666204	+	0.745770i	$0.267918\pi$
$798$	0	0
$799$	−35687.9	−1.58016
$800$	−14105.7	−0.623388
$801$	0	0
$802$	−42104.7	−1.85383
$803$	−28202.6	−1.23941
$804$	0	0
$805$	17806.4	0.779621
$806$	−1666.45	−0.0728266
$807$	0	0
$808$	−80529.4	−3.50621
$809$	−14674.6	−0.637739	−0.318870	−	0.947799i	$-0.603303\pi$
$809$	−14674.6	−0.637739	−0.318870	+	0.947799i	$0.603303\pi$
$810$	0	0
$811$	23037.9	0.997496	0.498748	−	0.866747i	$-0.333793\pi$
$811$	23037.9	0.997496	0.498748	+	0.866747i	$0.333793\pi$
$812$	−65618.7	−2.83592
$813$	0	0
$814$	−31231.7	−1.34480
$815$	−19826.1	−0.852121
$816$	0	0
$817$	−22411.1	−0.959689
$818$	−8169.20	−0.349180
$819$	0	0
$820$	−28421.1	−1.21037
$821$	−22136.0	−0.940987	−0.470494	−	0.882403i	$-0.655924\pi$
$821$	−22136.0	−0.940987	−0.470494	+	0.882403i	$0.655924\pi$
$822$	0	0
$823$	−31318.9	−1.32650	−0.663249	−	0.748399i	$-0.730823\pi$
$823$	−31318.9	−1.32650	−0.663249	+	0.748399i	$0.730823\pi$
$824$	−6263.58	−0.264808
$825$	0	0
$826$	9313.23	0.392311
$827$	−17055.7	−0.717151	−0.358576	−	0.933501i	$-0.616738\pi$
$827$	−17055.7	−0.717151	−0.358576	+	0.933501i	$0.616738\pi$
$828$	0	0
$829$	−23830.8	−0.998406	−0.499203	−	0.866485i	$-0.666374\pi$
$829$	−23830.8	−0.998406	−0.499203	+	0.866485i	$0.666374\pi$
$830$	−2212.29	−0.0925177
$831$	0	0
$832$	1367.78	0.0569942
$833$	−64287.3	−2.67398
$834$	0	0
$835$	−13392.8	−0.555061
$836$	−42803.1	−1.77079
$837$	0	0
$838$	−62259.2	−2.56648
$839$	6153.41	0.253205	0.126603	−	0.991954i	$-0.459593\pi$
$839$	6153.41	0.253205	0.126603	+	0.991954i	$0.459593\pi$
$840$	0	0
$841$	−10226.8	−0.419319
$842$	−12181.0	−0.498556
$843$	0	0
$844$	58190.1	2.37321
$845$	11158.3	0.454267
$846$	0	0
$847$	−43550.2	−1.76671
$848$	−17399.0	−0.704581
$849$	0	0
$850$	47742.2	1.92652
$851$	11682.7	0.470597
$852$	0	0
$853$	23127.0	0.928318	0.464159	−	0.885752i	$-0.346357\pi$
$853$	23127.0	0.928318	0.464159	+	0.885752i	$0.346357\pi$
$854$	133470.	5.34808
$855$	0	0
$856$	87143.4	3.47956
$857$	−9069.34	−0.361497	−0.180748	−	0.983529i	$-0.557852\pi$
$857$	−9069.34	−0.361497	−0.180748	+	0.983529i	$0.557852\pi$
$858$	0	0
$859$	5101.91	0.202649	0.101324	−	0.994853i	$-0.467692\pi$
$859$	5101.91	0.202649	0.101324	+	0.994853i	$0.467692\pi$
$860$	49538.8	1.96426
$861$	0	0
$862$	−36989.6	−1.46157
$863$	28547.6	1.12604	0.563020	−	0.826444i	$-0.309640\pi$
$863$	28547.6	1.12604	0.563020	+	0.826444i	$0.309640\pi$
$864$	0	0
$865$	−15383.9	−0.604703
$866$	67646.5	2.65441
$867$	0	0
$868$	−11158.7	−0.436348
$869$	21543.0	0.840963
$870$	0	0
$871$	12031.4	0.468048
$872$	−91710.5	−3.56159
$873$	0	0
$874$	23242.6	0.899534
$875$	−38932.2	−1.50417
$876$	0	0
$877$	−810.103	−0.0311918	−0.0155959	−	0.999878i	$-0.504965\pi$
$877$	−810.103	−0.0311918	−0.0155959	+	0.999878i	$0.504965\pi$
$878$	40164.6	1.54384
$879$	0	0
$880$	32581.7	1.24810
$881$	26009.1	0.994630	0.497315	−	0.867570i	$-0.334319\pi$
$881$	26009.1	0.994630	0.497315	+	0.867570i	$0.334319\pi$
$882$	0	0
$883$	19343.3	0.737208	0.368604	−	0.929586i	$-0.379836\pi$
$883$	19343.3	0.737208	0.368604	+	0.929586i	$0.379836\pi$
$884$	29537.3	1.12381
$885$	0	0
$886$	−3936.65	−0.149271
$887$	−10442.1	−0.395279	−0.197639	−	0.980275i	$-0.563327\pi$
$887$	−10442.1	−0.395279	−0.197639	+	0.980275i	$0.563327\pi$
$888$	0	0
$889$	−15165.7	−0.572150
$890$	4782.80	0.180135
$891$	0	0
$892$	7547.14	0.283293
$893$	16073.6	0.602331
$894$	0	0
$895$	4452.40	0.166287
$896$	51608.1	1.92422
$897$	0	0
$898$	−14180.6	−0.526965
$899$	2408.33	0.0893463
$900$	0	0
$901$	16536.6	0.611448
$902$	−73656.0	−2.71893
$903$	0	0
$904$	−105153.	−3.86873
$905$	−6101.51	−0.224112
$906$	0	0
$907$	31550.6	1.15504	0.577519	−	0.816377i	$-0.304021\pi$
$907$	31550.6	1.15504	0.577519	+	0.816377i	$0.304021\pi$
$908$	80227.4	2.93220
$909$	0	0
$910$	−14795.0	−0.538956
$911$	−7389.17	−0.268731	−0.134366	−	0.990932i	$-0.542900\pi$
$911$	−7389.17	−0.268731	−0.134366	+	0.990932i	$0.542900\pi$
$912$	0	0
$913$	−3949.55	−0.143167
$914$	−16174.5	−0.585345
$915$	0	0
$916$	37866.6	1.36588
$917$	36173.8	1.30269
$918$	0	0
$919$	25793.5	0.925842	0.462921	−	0.886399i	$-0.346801\pi$
$919$	25793.5	0.925842	0.462921	+	0.886399i	$0.346801\pi$
$920$	−28172.5	−1.00959
$921$	0	0
$922$	−56147.6	−2.00556
$923$	−11986.8	−0.427466
$924$	0	0
$925$	−10808.2	−0.384184
$926$	77119.6	2.73683
$927$	0	0
$928$	18308.2	0.647627
$929$	29660.4	1.04750	0.523749	−	0.851872i	$-0.324533\pi$
$929$	29660.4	1.04750	0.523749	+	0.851872i	$0.324533\pi$
$930$	0	0
$931$	28954.5	1.01928
$932$	−99794.2	−3.50737
$933$	0	0
$934$	13521.9	0.473714
$935$	−30966.8	−1.08313
$936$	0	0
$937$	29585.3	1.03149	0.515747	−	0.856741i	$-0.327514\pi$
$937$	29585.3	1.03149	0.515747	+	0.856741i	$0.327514\pi$
$938$	116950.	4.07094
$939$	0	0
$940$	−35530.0	−1.23283
$941$	−26146.0	−0.905778	−0.452889	−	0.891567i	$-0.649607\pi$
$941$	−26146.0	−0.905778	−0.452889	+	0.891567i	$0.649607\pi$
$942$	0	0
$943$	27552.2	0.951454
$944$	−6374.48	−0.219779
$945$	0	0
$946$	128385.	4.41242
$947$	22905.8	0.785998	0.392999	−	0.919539i	$-0.371438\pi$
$947$	22905.8	0.785998	0.392999	+	0.919539i	$0.371438\pi$
$948$	0	0
$949$	−8765.46	−0.299830
$950$	−21502.7	−0.734359
$951$	0	0
$952$	157438.	5.35987
$953$	−2307.17	−0.0784225	−0.0392113	−	0.999231i	$-0.512485\pi$
$953$	−2307.17	−0.0784225	−0.0392113	+	0.999231i	$0.512485\pi$
$954$	0	0
$955$	14747.2	0.499695
$956$	73393.9	2.48298
$957$	0	0
$958$	7457.71	0.251511
$959$	1491.17	0.0502112
$960$	0	0
$961$	−29381.5	−0.986253
$962$	−9706.93	−0.325326
$963$	0	0
$964$	−104164.	−3.48019
$965$	11764.2	0.392438
$966$	0	0
$967$	13468.7	0.447906	0.223953	−	0.974600i	$-0.428104\pi$
$967$	13468.7	0.447906	0.223953	+	0.974600i	$0.428104\pi$
$968$	68902.9	2.28784
$969$	0	0
$970$	−25085.6	−0.830361
$971$	−18009.0	−0.595197	−0.297598	−	0.954691i	$-0.596186\pi$
$971$	−18009.0	−0.595197	−0.297598	+	0.954691i	$0.596186\pi$
$972$	0	0
$973$	69030.4	2.27442
$974$	−53075.3	−1.74604
$975$	0	0
$976$	−91354.3	−2.99608
$977$	−39514.9	−1.29395	−0.646977	−	0.762510i	$-0.723967\pi$
$977$	−39514.9	−1.29395	−0.646977	+	0.762510i	$0.723967\pi$
$978$	0	0
$979$	8538.62	0.278749
$980$	−64002.7	−2.08622
$981$	0	0
$982$	−41204.8	−1.33900
$983$	2941.64	0.0954464	0.0477232	−	0.998861i	$-0.484803\pi$
$983$	2941.64	0.0954464	0.0477232	+	0.998861i	$0.484803\pi$
$984$	0	0
$985$	6954.45	0.224962
$986$	−61966.3	−2.00143
$987$	0	0
$988$	−13303.4	−0.428377
$989$	−48024.3	−1.54407
$990$	0	0
$991$	22197.7	0.711537	0.355769	−	0.934574i	$-0.384219\pi$
$991$	22197.7	0.711537	0.355769	+	0.934574i	$0.384219\pi$
$992$	3113.37	0.0996469
$993$	0	0
$994$	−116516.	−3.71797
$995$	−22103.9	−0.704261
$996$	0	0
$997$	2312.86	0.0734695	0.0367347	−	0.999325i	$-0.488304\pi$
$997$	2312.86	0.0734695	0.0367347	+	0.999325i	$0.488304\pi$
$998$	−76005.9	−2.41074
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.4.a.c.1.7		7
3.2	odd	2		177.4.a.b.1.1	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.a.b.1.1	✓	7	3.2	odd	2
531.4.a.c.1.7		7	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:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	531.a (trivial)

\(f(q)\)	\(=\)	\(q+5.07078 q^{2} +17.7128 q^{4} -5.77165 q^{5} -31.1296 q^{7} +49.2517 q^{8} +O(q^{10})\)	\(q+5.07078 q^{2} +17.7128 q^{4} -5.77165 q^{5} -31.1296 q^{7} +49.2517 q^{8} -29.2668 q^{10} -52.2494 q^{11} -16.2393 q^{13} -157.851 q^{14} +108.042 q^{16} -102.687 q^{17} +46.2494 q^{19} -102.232 q^{20} -264.945 q^{22} +99.1068 q^{23} -91.6880 q^{25} -82.3459 q^{26} -551.394 q^{28} +119.005 q^{29} +20.2372 q^{31} +153.844 q^{32} -520.703 q^{34} +179.669 q^{35} +117.880 q^{37} +234.521 q^{38} -284.264 q^{40} +278.005 q^{41} -484.571 q^{43} -925.485 q^{44} +502.549 q^{46} +347.541 q^{47} +626.051 q^{49} -464.930 q^{50} -287.644 q^{52} -161.039 q^{53} +301.565 q^{55} -1533.19 q^{56} +603.449 q^{58} -59.0000 q^{59} -845.543 q^{61} +102.618 q^{62} -84.2264 q^{64} +93.7275 q^{65} -740.885 q^{67} -1818.88 q^{68} +911.063 q^{70} +738.136 q^{71} +539.769 q^{73} +597.743 q^{74} +819.208 q^{76} +1626.50 q^{77} -412.312 q^{79} -623.581 q^{80} +1409.70 q^{82} +75.5904 q^{83} +592.673 q^{85} -2457.15 q^{86} -2573.37 q^{88} -163.421 q^{89} +505.523 q^{91} +1755.46 q^{92} +1762.31 q^{94} -266.935 q^{95} +857.136 q^{97} +3174.57 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 26 q^{4} + 2 q^{5} - 59 q^{7} + 21 q^{8}+O(q^{10}) \) 7 * q + 26 * q^4 + 2 * q^5 - 59 * q^7 + 21 * q^8	\( 7 q + 26 q^{4} + 2 q^{5} - 59 q^{7} + 21 q^{8} - 71 q^{10} + 5 q^{11} - 67 q^{13} + 65 q^{14} - 94 q^{16} + 23 q^{17} - 176 q^{19} + 207 q^{20} - 704 q^{22} + 218 q^{23} - 183 q^{25} - 58 q^{26} - 938 q^{28} - 168 q^{29} - 604 q^{31} + 448 q^{32} - 610 q^{34} + 336 q^{35} - 505 q^{37} + 453 q^{38} - 1080 q^{40} + 265 q^{41} - 493 q^{43} - 504 q^{44} + 381 q^{46} + 244 q^{47} + 770 q^{49} - 1639 q^{50} + 160 q^{52} - 686 q^{53} - 116 q^{55} - 2190 q^{56} + 1584 q^{58} - 413 q^{59} - 838 q^{61} - 286 q^{62} + 205 q^{64} - 490 q^{65} - 1504 q^{67} - 3047 q^{68} + 1530 q^{70} + 1267 q^{71} - 666 q^{73} - 528 q^{74} - 64 q^{76} - 1109 q^{77} - 2741 q^{79} - 1213 q^{80} + 953 q^{82} + 2025 q^{83} - 1274 q^{85} - 4394 q^{86} - 1639 q^{88} - 616 q^{89} - 2415 q^{91} - 218 q^{92} + 900 q^{94} - 2554 q^{95} - 1298 q^{97} + 172 q^{98}+O(q^{100}) \) 7 * q + 26 * q^4 + 2 * q^5 - 59 * q^7 + 21 * q^8 - 71 * q^10 + 5 * q^11 - 67 * q^13 + 65 * q^14 - 94 * q^16 + 23 * q^17 - 176 * q^19 + 207 * q^20 - 704 * q^22 + 218 * q^23 - 183 * q^25 - 58 * q^26 - 938 * q^28 - 168 * q^29 - 604 * q^31 + 448 * q^32 - 610 * q^34 + 336 * q^35 - 505 * q^37 + 453 * q^38 - 1080 * q^40 + 265 * q^41 - 493 * q^43 - 504 * q^44 + 381 * q^46 + 244 * q^47 + 770 * q^49 - 1639 * q^50 + 160 * q^52 - 686 * q^53 - 116 * q^55 - 2190 * q^56 + 1584 * q^58 - 413 * q^59 - 838 * q^61 - 286 * q^62 + 205 * q^64 - 490 * q^65 - 1504 * q^67 - 3047 * q^68 + 1530 * q^70 + 1267 * q^71 - 666 * q^73 - 528 * q^74 - 64 * q^76 - 1109 * q^77 - 2741 * q^79 - 1213 * q^80 + 953 * q^82 + 2025 * q^83 - 1274 * q^85 - 4394 * q^86 - 1639 * q^88 - 616 * q^89 - 2415 * q^91 - 218 * q^92 + 900 * q^94 - 2554 * q^95 - 1298 * q^97 + 172 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more