LMFDB - Embedded newform 1725.4.a.bf.1.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1725,4,Mod(1,1725)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1725.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1725, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1725 = 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1725.a (trivial)

Newform invariants

comment:select newform

sage:traces = [11,1,-33,39,0,-3,-19] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$101.778294760$
Analytic rank:	$1$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - x^{10} - 63 x^{9} + 64 x^{8} + 1341 x^{7} - 1301 x^{6} - 11425 x^{5} + 10222 x^{4} + \cdots + 37920 $ x^11 - x^10 - 63x^9 + 64x^8 + 1341x^7 - 1301x^6 - 11425x^5 + 10222x^4 + 38938x^3 - 34600x^2 - 42408*x + 37920
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$1.86984$ of defining polynomial
Character	$\chi$	$=$	1725.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.86984 q^{2} -3.00000 q^{3} -4.50369 q^{4} -5.60953 q^{6} -34.2037 q^{7} -23.3799 q^{8} +9.00000 q^{9} -22.6245 q^{11} +13.5111 q^{12} +83.1897 q^{13} -63.9556 q^{14} -7.68732 q^{16} +40.1809 q^{17} +16.8286 q^{18} +46.4457 q^{19} +102.611 q^{21} -42.3043 q^{22} -23.0000 q^{23} +70.1398 q^{24} +155.552 q^{26} -27.0000 q^{27} +154.043 q^{28} +119.225 q^{29} -35.3890 q^{31} +172.665 q^{32} +67.8736 q^{33} +75.1320 q^{34} -40.5332 q^{36} +117.335 q^{37} +86.8462 q^{38} -249.569 q^{39} +290.561 q^{41} +191.867 q^{42} -324.311 q^{43} +101.894 q^{44} -43.0064 q^{46} +83.8133 q^{47} +23.0620 q^{48} +826.893 q^{49} -120.543 q^{51} -374.660 q^{52} -131.722 q^{53} -50.4858 q^{54} +799.680 q^{56} -139.337 q^{57} +222.933 q^{58} -96.7145 q^{59} -656.006 q^{61} -66.1718 q^{62} -307.833 q^{63} +384.356 q^{64} +126.913 q^{66} -548.538 q^{67} -180.962 q^{68} +69.0000 q^{69} +114.598 q^{71} -210.419 q^{72} -25.6346 q^{73} +219.398 q^{74} -209.177 q^{76} +773.843 q^{77} -466.655 q^{78} +291.728 q^{79} +81.0000 q^{81} +543.303 q^{82} -1490.11 q^{83} -462.128 q^{84} -606.411 q^{86} -357.676 q^{87} +528.960 q^{88} +1308.37 q^{89} -2845.40 q^{91} +103.585 q^{92} +106.167 q^{93} +156.718 q^{94} -517.996 q^{96} +1166.79 q^{97} +1546.16 q^{98} -203.621 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + q^{2} - 33 q^{3} + 39 q^{4} - 3 q^{6} - 19 q^{7} - 18 q^{8} + 99 q^{9} - 41 q^{11} - 117 q^{12} - 57 q^{13} + 8 q^{14} + 227 q^{16} - 66 q^{17} + 9 q^{18} - 215 q^{19} + 57 q^{21} + 342 q^{22} - 253 q^{23}+ \cdots - 369 q^{99}+O(q^{100}) $ 11 * q + q^2 - 33 * q^3 + 39 * q^4 - 3 * q^6 - 19 * q^7 - 18 * q^8 + 99 * q^9 - 41 * q^11 - 117 * q^12 - 57 * q^13 + 8 * q^14 + 227 * q^16 - 66 * q^17 + 9 * q^18 - 215 * q^19 + 57 * q^21 + 342 * q^22 - 253 * q^23 + 54 * q^24 + 364 * q^26 - 297 * q^27 - 657 * q^28 + 513 * q^29 + 340 * q^31 - 263 * q^32 + 123 * q^33 - 125 * q^34 + 351 * q^36 - 20 * q^37 - 124 * q^38 + 171 * q^39 + 515 * q^41 - 24 * q^42 - 1401 * q^43 - 879 * q^44 - 23 * q^46 + 152 * q^47 - 681 * q^48 + 1108 * q^49 + 198 * q^51 - 697 * q^52 - 794 * q^53 - 27 * q^54 + 1113 * q^56 + 645 * q^57 - 3058 * q^58 + 1332 * q^59 + 148 * q^61 - 984 * q^62 - 171 * q^63 + 1422 * q^64 - 1026 * q^66 - 56 * q^67 - 2277 * q^68 + 759 * q^69 - 198 * q^71 - 162 * q^72 + 205 * q^73 + 2901 * q^74 - 4522 * q^76 + 851 * q^77 - 1092 * q^78 - 645 * q^79 + 891 * q^81 - 37 * q^82 - 2591 * q^83 + 1971 * q^84 - 567 * q^86 - 1539 * q^87 + 2247 * q^88 + 4130 * q^89 - 1623 * q^91 - 897 * q^92 - 1020 * q^93 + 49 * q^94 + 789 * q^96 - 2348 * q^97 - 2804 * q^98 - 369 * q^99

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 $

$2$	1.86984	0.661089	0.330545	−	0.943790i	$-0.392768\pi$
$2$	1.86984	0.661089	0.330545	+	0.943790i	$0.392768\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−4.50369	−0.562961
$5$	0	0
$5$	0	0
$6$	−5.60953	−0.381680
$7$	−34.2037	−1.84683	−0.923413	−	0.383808i	$-0.874613\pi$
$7$	−34.2037	−1.84683	−0.923413	+	0.383808i	$0.874613\pi$
$8$	−23.3799	−1.03326
$9$	9.00000	0.333333
$10$	0	0
$11$	−22.6245	−0.620141	−0.310071	−	0.950714i	$-0.600353\pi$
$11$	−22.6245	−0.620141	−0.310071	+	0.950714i	$0.600353\pi$
$12$	13.5111	0.325026
$13$	83.1897	1.77482	0.887411	−	0.460979i	$-0.152502\pi$
$13$	83.1897	1.77482	0.887411	+	0.460979i	$0.152502\pi$
$14$	−63.9556	−1.22092
$15$	0	0
$16$	−7.68732	−0.120114
$17$	40.1809	0.573253	0.286627	−	0.958042i	$-0.407466\pi$
$17$	40.1809	0.573253	0.286627	+	0.958042i	$0.407466\pi$
$18$	16.8286	0.220363
$19$	46.4457	0.560810	0.280405	−	0.959882i	$-0.409531\pi$
$19$	46.4457	0.560810	0.280405	+	0.959882i	$0.409531\pi$
$20$	0	0
$21$	102.611	1.06627
$22$	−42.3043	−0.409969
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	70.1398	0.596551
$25$	0	0
$26$	155.552	1.17332
$27$	−27.0000	−0.192450
$28$	154.043	1.03969
$29$	119.225	0.763433	0.381717	−	0.924279i	$-0.375333\pi$
$29$	119.225	0.763433	0.381717	+	0.924279i	$0.375333\pi$
$30$	0	0
$31$	−35.3890	−0.205034	−0.102517	−	0.994731i	$-0.532690\pi$
$31$	−35.3890	−0.205034	−0.102517	+	0.994731i	$0.532690\pi$
$32$	172.665	0.953850
$33$	67.8736	0.358039
$34$	75.1320	0.378972
$35$	0	0
$36$	−40.5332	−0.187654
$37$	117.335	0.521345	0.260672	−	0.965427i	$-0.416056\pi$
$37$	117.335	0.521345	0.260672	+	0.965427i	$0.416056\pi$
$38$	86.8462	0.370745
$39$	−249.569	−1.02469
$40$	0	0
$41$	290.561	1.10678	0.553390	−	0.832922i	$-0.313334\pi$
$41$	290.561	1.10678	0.553390	+	0.832922i	$0.313334\pi$
$42$	191.867	0.704897
$43$	−324.311	−1.15016	−0.575081	−	0.818096i	$-0.695030\pi$
$43$	−324.311	−1.15016	−0.575081	+	0.818096i	$0.695030\pi$
$44$	101.894	0.349115
$45$	0	0
$46$	−43.0064	−0.137847
$47$	83.8133	0.260115	0.130058	−	0.991506i	$-0.458484\pi$
$47$	83.8133	0.260115	0.130058	+	0.991506i	$0.458484\pi$
$48$	23.0620	0.0693481
$49$	826.893	2.41077
$50$	0	0
$51$	−120.543	−0.330968
$52$	−374.660	−0.999155
$53$	−131.722	−0.341385	−0.170693	−	0.985324i	$-0.554600\pi$
$53$	−131.722	−0.341385	−0.170693	+	0.985324i	$0.554600\pi$
$54$	−50.4858	−0.127227
$55$	0	0
$56$	799.680	1.90825
$57$	−139.337	−0.323784
$58$	222.933	0.504698
$59$	−96.7145	−0.213409	−0.106705	−	0.994291i	$-0.534030\pi$
$59$	−96.7145	−0.213409	−0.106705	+	0.994291i	$0.534030\pi$
$60$	0	0
$61$	−656.006	−1.37693	−0.688467	−	0.725268i	$-0.741716\pi$
$61$	−656.006	−1.37693	−0.688467	+	0.725268i	$0.741716\pi$
$62$	−66.1718	−0.135546
$63$	−307.833	−0.615609
$64$	384.356	0.750695
$65$	0	0
$66$	126.913	0.236696
$67$	−548.538	−1.00022	−0.500109	−	0.865963i	$-0.666707\pi$
$67$	−548.538	−1.00022	−0.500109	+	0.865963i	$0.666707\pi$
$68$	−180.962	−0.322719
$69$	69.0000	0.120386
$70$	0	0
$71$	114.598	0.191554	0.0957768	−	0.995403i	$-0.469467\pi$
$71$	114.598	0.191554	0.0957768	+	0.995403i	$0.469467\pi$
$72$	−210.419	−0.344419
$73$	−25.6346	−0.0411001	−0.0205500	−	0.999789i	$-0.506542\pi$
$73$	−25.6346	−0.0411001	−0.0205500	+	0.999789i	$0.506542\pi$
$74$	219.398	0.344655
$75$	0	0
$76$	−209.177	−0.315714
$77$	773.843	1.14529
$78$	−466.655	−0.677414
$79$	291.728	0.415468	0.207734	−	0.978185i	$-0.433391\pi$
$79$	291.728	0.415468	0.207734	+	0.978185i	$0.433391\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	543.303	0.731681
$83$	−1490.11	−1.97061	−0.985307	−	0.170791i	$-0.945368\pi$
$83$	−1490.11	−1.97061	−0.985307	+	0.170791i	$0.945368\pi$
$84$	−462.128	−0.600266
$85$	0	0
$86$	−606.411	−0.760360
$87$	−357.676	−0.440769
$88$	528.960	0.640765
$89$	1308.37	1.55828	0.779140	−	0.626850i	$-0.215656\pi$
$89$	1308.37	1.55828	0.779140	+	0.626850i	$0.215656\pi$
$90$	0	0
$91$	−2845.40	−3.27779
$92$	103.585	0.117385
$93$	106.167	0.118376
$94$	156.718	0.171960
$95$	0	0
$96$	−517.996	−0.550706
$97$	1166.79	1.22134	0.610669	−	0.791886i	$-0.290901\pi$
$97$	1166.79	1.22134	0.610669	+	0.791886i	$0.290901\pi$
$98$	1546.16	1.59373
$99$	−203.621	−0.206714
$100$	0	0
$101$	−1310.48	−1.29107	−0.645533	−	0.763732i	$-0.723365\pi$
$101$	−1310.48	−1.29107	−0.645533	+	0.763732i	$0.723365\pi$
$102$	−225.396	−0.218799
$103$	379.399	0.362945	0.181472	−	0.983396i	$-0.441914\pi$
$103$	379.399	0.362945	0.181472	+	0.983396i	$0.441914\pi$
$104$	−1944.97	−1.83385
$105$	0	0
$106$	−246.300	−0.225686
$107$	4.58258	0.00414033	0.00207016	−	0.999998i	$-0.499341\pi$
$107$	4.58258	0.00414033	0.00207016	+	0.999998i	$0.499341\pi$
$108$	121.600	0.108342
$109$	−801.637	−0.704430	−0.352215	−	0.935919i	$-0.614571\pi$
$109$	−801.637	−0.704430	−0.352215	+	0.935919i	$0.614571\pi$
$110$	0	0
$111$	−352.005	−0.300999
$112$	262.935	0.221830
$113$	−43.1499	−0.0359222	−0.0179611	−	0.999839i	$-0.505717\pi$
$113$	−43.1499	−0.0359222	−0.0179611	+	0.999839i	$0.505717\pi$
$114$	−260.539	−0.214050
$115$	0	0
$116$	−536.953	−0.429783
$117$	748.708	0.591607
$118$	−180.841	−0.141083
$119$	−1374.34	−1.05870
$120$	0	0
$121$	−819.131	−0.615425
$122$	−1226.63	−0.910277
$123$	−871.683	−0.639000
$124$	159.381	0.115426
$125$	0	0
$126$	−575.600	−0.406972
$127$	426.223	0.297805	0.148902	−	0.988852i	$-0.452426\pi$
$127$	426.223	0.297805	0.148902	+	0.988852i	$0.452426\pi$
$128$	−662.638	−0.457574
$129$	972.933	0.664047
$130$	0	0
$131$	−561.989	−0.374819	−0.187409	−	0.982282i	$-0.560009\pi$
$131$	−561.989	−0.374819	−0.187409	+	0.982282i	$0.560009\pi$
$132$	−305.681	−0.201562
$133$	−1588.62	−1.03572
$134$	−1025.68	−0.661233
$135$	0	0
$136$	−939.427	−0.592318
$137$	−1067.60	−0.665776	−0.332888	−	0.942966i	$-0.608023\pi$
$137$	−1067.60	−0.665776	−0.332888	+	0.942966i	$0.608023\pi$
$138$	129.019	0.0795858
$139$	2013.90	1.22890	0.614448	−	0.788957i	$-0.289379\pi$
$139$	2013.90	1.22890	0.614448	+	0.788957i	$0.289379\pi$
$140$	0	0
$141$	−251.440	−0.150178
$142$	214.281	0.126634
$143$	−1882.13	−1.10064
$144$	−69.1859	−0.0400381
$145$	0	0
$146$	−47.9327	−0.0271708
$147$	−2480.68	−1.39186
$148$	−528.440	−0.293497
$149$	−674.511	−0.370860	−0.185430	−	0.982658i	$-0.559368\pi$
$149$	−674.511	−0.370860	−0.185430	+	0.982658i	$0.559368\pi$
$150$	0	0
$151$	701.714	0.378176	0.189088	−	0.981960i	$-0.439447\pi$
$151$	701.714	0.378176	0.189088	+	0.981960i	$0.439447\pi$
$152$	−1085.90	−0.579460
$153$	361.628	0.191084
$154$	1446.96	0.757141
$155$	0	0
$156$	1123.98	0.576862
$157$	185.518	0.0943053	0.0471526	−	0.998888i	$-0.484985\pi$
$157$	185.518	0.0943053	0.0471526	+	0.998888i	$0.484985\pi$
$158$	545.486	0.274662
$159$	395.166	0.197099
$160$	0	0
$161$	786.685	0.385090
$162$	151.457	0.0734544
$163$	−2431.90	−1.16860	−0.584298	−	0.811539i	$-0.698630\pi$
$163$	−2431.90	−1.16860	−0.584298	+	0.811539i	$0.698630\pi$
$164$	−1308.60	−0.623074
$165$	0	0
$166$	−2786.28	−1.30275
$167$	1173.63	0.543820	0.271910	−	0.962323i	$-0.412345\pi$
$167$	1173.63	0.543820	0.271910	+	0.962323i	$0.412345\pi$
$168$	−2399.04	−1.10173
$169$	4723.53	2.14999
$170$	0	0
$171$	418.012	0.186937
$172$	1460.60	0.647496
$173$	−3438.01	−1.51091	−0.755453	−	0.655202i	$-0.772583\pi$
$173$	−3438.01	−1.51091	−0.755453	+	0.655202i	$0.772583\pi$
$174$	−668.798	−0.291387
$175$	0	0
$176$	173.922	0.0744879
$177$	290.144	0.123212
$178$	2446.45	1.03016
$179$	2815.45	1.17562	0.587811	−	0.808998i	$-0.299990\pi$
$179$	2815.45	1.17562	0.587811	+	0.808998i	$0.299990\pi$
$180$	0	0
$181$	4802.08	1.97202	0.986011	−	0.166680i	$-0.0533045\pi$
$181$	4802.08	1.97202	0.986011	+	0.166680i	$0.0533045\pi$
$182$	−5320.45	−2.16691
$183$	1968.02	0.794973
$184$	537.738	0.215449
$185$	0	0
$186$	198.516	0.0782573
$187$	−909.075	−0.355498
$188$	−377.469	−0.146435
$189$	923.500	0.355422
$190$	0	0
$191$	−3070.94	−1.16338	−0.581689	−	0.813411i	$-0.697608\pi$
$191$	−3070.94	−1.16338	−0.581689	+	0.813411i	$0.697608\pi$
$192$	−1153.07	−0.433414
$193$	−2299.27	−0.857541	−0.428770	−	0.903413i	$-0.641053\pi$
$193$	−2299.27	−0.857541	−0.428770	+	0.903413i	$0.641053\pi$
$194$	2181.72	0.807414
$195$	0	0
$196$	−3724.07	−1.35717
$197$	1233.41	0.446077	0.223038	−	0.974810i	$-0.428402\pi$
$197$	1233.41	0.446077	0.223038	+	0.974810i	$0.428402\pi$
$198$	−380.739	−0.136656
$199$	−1732.24	−0.617062	−0.308531	−	0.951214i	$-0.599837\pi$
$199$	−1732.24	−0.617062	−0.308531	+	0.951214i	$0.599837\pi$
$200$	0	0
$201$	1645.61	0.577476
$202$	−2450.39	−0.853510
$203$	−4077.94	−1.40993
$204$	542.887	0.186322
$205$	0	0
$206$	709.417	0.239939
$207$	−207.000	−0.0695048
$208$	−639.506	−0.213182
$209$	−1050.81	−0.347781
$210$	0	0
$211$	5018.43	1.63736	0.818680	−	0.574250i	$-0.194706\pi$
$211$	5018.43	1.63736	0.818680	+	0.574250i	$0.194706\pi$
$212$	593.235	0.192186
$213$	−343.795	−0.110593
$214$	8.56871	0.00273713
$215$	0	0
$216$	631.258	0.198850
$217$	1210.43	0.378662
$218$	−1498.94	−0.465691
$219$	76.9039	0.0237291
$220$	0	0
$221$	3342.64	1.01742
$222$	−658.194	−0.198987
$223$	4157.93	1.24859	0.624296	−	0.781188i	$-0.285386\pi$
$223$	4157.93	1.24859	0.624296	+	0.781188i	$0.285386\pi$
$224$	−5905.79	−1.76160
$225$	0	0
$226$	−80.6836	−0.0237478
$227$	−6431.87	−1.88061	−0.940305	−	0.340333i	$-0.889460\pi$
$227$	−6431.87	−1.88061	−0.940305	+	0.340333i	$0.889460\pi$
$228$	627.531	0.182277
$229$	−2441.64	−0.704577	−0.352289	−	0.935891i	$-0.614596\pi$
$229$	−2441.64	−0.704577	−0.352289	+	0.935891i	$0.614596\pi$
$230$	0	0
$231$	−2321.53	−0.661235
$232$	−2787.48	−0.788823
$233$	3913.37	1.10032	0.550158	−	0.835061i	$-0.314567\pi$
$233$	3913.37	1.10032	0.550158	+	0.835061i	$0.314567\pi$
$234$	1399.97	0.391105
$235$	0	0
$236$	435.572	0.120141
$237$	−875.185	−0.239871
$238$	−2569.79	−0.699895
$239$	5348.03	1.44743	0.723714	−	0.690100i	$-0.242433\pi$
$239$	5348.03	1.44743	0.723714	+	0.690100i	$0.242433\pi$
$240$	0	0
$241$	1285.90	0.343701	0.171851	−	0.985123i	$-0.445025\pi$
$241$	1285.90	0.343701	0.171851	+	0.985123i	$0.445025\pi$
$242$	−1531.65	−0.406851
$243$	−243.000	−0.0641500
$244$	2954.45	0.775160
$245$	0	0
$246$	−1629.91	−0.422436
$247$	3863.81	0.995337
$248$	827.392	0.211853
$249$	4470.33	1.13773
$250$	0	0
$251$	−4889.96	−1.22969	−0.614844	−	0.788649i	$-0.710781\pi$
$251$	−4889.96	−1.22969	−0.614844	+	0.788649i	$0.710781\pi$
$252$	1386.38	0.346564
$253$	520.364	0.129308
$254$	796.971	0.196876
$255$	0	0
$256$	−4313.88	−1.05319
$257$	−4742.25	−1.15103	−0.575513	−	0.817793i	$-0.695197\pi$
$257$	−4742.25	−1.15103	−0.575513	+	0.817793i	$0.695197\pi$
$258$	1819.23	0.438994
$259$	−4013.29	−0.962833
$260$	0	0
$261$	1073.03	0.254478
$262$	−1050.83	−0.247789
$263$	3232.34	0.757850	0.378925	−	0.925427i	$-0.376294\pi$
$263$	3232.34	0.757850	0.378925	+	0.925427i	$0.376294\pi$
$264$	−1586.88	−0.369946
$265$	0	0
$266$	−2970.46	−0.684702
$267$	−3925.11	−0.899673
$268$	2470.44	0.563083
$269$	7397.49	1.67670	0.838351	−	0.545130i	$-0.183520\pi$
$269$	7397.49	1.67670	0.838351	+	0.545130i	$0.183520\pi$
$270$	0	0
$271$	−8543.99	−1.91517	−0.957584	−	0.288155i	$-0.906958\pi$
$271$	−8543.99	−1.91517	−0.957584	+	0.288155i	$0.906958\pi$
$272$	−308.884	−0.0688560
$273$	8536.19	1.89243
$274$	−1996.25	−0.440137
$275$	0	0
$276$	−310.754	−0.0677725
$277$	−6487.13	−1.40713	−0.703563	−	0.710633i	$-0.748409\pi$
$277$	−6487.13	−1.40713	−0.703563	+	0.710633i	$0.748409\pi$
$278$	3765.68	0.812411
$279$	−318.501	−0.0683446
$280$	0	0
$281$	4227.82	0.897546	0.448773	−	0.893646i	$-0.351861\pi$
$281$	4227.82	0.897546	0.448773	+	0.893646i	$0.351861\pi$
$282$	−470.153	−0.0992809
$283$	38.5911	0.00810602	0.00405301	−	0.999992i	$-0.498710\pi$
$283$	38.5911	0.00810602	0.00405301	+	0.999992i	$0.498710\pi$
$284$	−516.114	−0.107837
$285$	0	0
$286$	−3519.29	−0.727621
$287$	−9938.26	−2.04403
$288$	1553.99	0.317950
$289$	−3298.49	−0.671381
$290$	0	0
$291$	−3500.38	−0.705140
$292$	115.450	0.0231377
$293$	−6722.74	−1.34043	−0.670216	−	0.742166i	$-0.733799\pi$
$293$	−6722.74	−1.34043	−0.670216	+	0.742166i	$0.733799\pi$
$294$	−4638.48	−0.920142
$295$	0	0
$296$	−2743.28	−0.538683
$297$	610.862	0.119346
$298$	−1261.23	−0.245171
$299$	−1913.36	−0.370076
$300$	0	0
$301$	11092.6	2.12415
$302$	1312.09	0.250008
$303$	3931.44	0.745398
$304$	−357.043	−0.0673613
$305$	0	0
$306$	676.188	0.126324
$307$	1901.68	0.353532	0.176766	−	0.984253i	$-0.443436\pi$
$307$	1901.68	0.353532	0.176766	+	0.984253i	$0.443436\pi$
$308$	−3485.14	−0.644755
$309$	−1138.20	−0.209546
$310$	0	0
$311$	4658.79	0.849439	0.424720	−	0.905325i	$-0.360373\pi$
$311$	4658.79	0.849439	0.424720	+	0.905325i	$0.360373\pi$
$312$	5834.91	1.05877
$313$	−1962.03	−0.354314	−0.177157	−	0.984183i	$-0.556690\pi$
$313$	−1962.03	−0.354314	−0.177157	+	0.984183i	$0.556690\pi$
$314$	346.889	0.0623442
$315$	0	0
$316$	−1313.85	−0.233892
$317$	−6420.06	−1.13750	−0.568749	−	0.822511i	$-0.692572\pi$
$317$	−6420.06	−1.13750	−0.568749	+	0.822511i	$0.692572\pi$
$318$	738.899	0.130300
$319$	−2697.42	−0.473437
$320$	0	0
$321$	−13.7477	−0.00239042
$322$	1470.98	0.254579
$323$	1866.23	0.321486
$324$	−364.799	−0.0625512
$325$	0	0
$326$	−4547.28	−0.772547
$327$	2404.91	0.406703
$328$	−6793.29	−1.14359
$329$	−2866.72	−0.480388
$330$	0	0
$331$	−5517.64	−0.916244	−0.458122	−	0.888889i	$-0.651478\pi$
$331$	−5517.64	−0.916244	−0.458122	+	0.888889i	$0.651478\pi$
$332$	6710.99	1.10938
$333$	1056.01	0.173782
$334$	2194.50	0.359513
$335$	0	0
$336$	−788.804	−0.128074
$337$	−10066.6	−1.62719	−0.813594	−	0.581433i	$-0.802492\pi$
$337$	−10066.6	−1.62719	−0.813594	+	0.581433i	$0.802492\pi$
$338$	8832.26	1.42134
$339$	129.450	0.0207397
$340$	0	0
$341$	800.659	0.127150
$342$	781.616	0.123582
$343$	−16550.9	−2.60544
$344$	7582.37	1.18841
$345$	0	0
$346$	−6428.54	−0.998845
$347$	−3636.99	−0.562663	−0.281331	−	0.959611i	$-0.590776\pi$
$347$	−3636.99	−0.562663	−0.281331	+	0.959611i	$0.590776\pi$
$348$	1610.86	0.248135
$349$	−5540.05	−0.849720	−0.424860	−	0.905259i	$-0.639677\pi$
$349$	−5540.05	−0.849720	−0.424860	+	0.905259i	$0.639677\pi$
$350$	0	0
$351$	−2246.12	−0.341565
$352$	−3906.47	−0.591522
$353$	−11826.6	−1.78319	−0.891596	−	0.452832i	$-0.850414\pi$
$353$	−11826.6	−1.78319	−0.891596	+	0.452832i	$0.850414\pi$
$354$	542.523	0.0814542
$355$	0	0
$356$	−5892.49	−0.877250
$357$	4123.01	0.611240
$358$	5264.44	0.777191
$359$	−8918.82	−1.31119	−0.655595	−	0.755113i	$-0.727582\pi$
$359$	−8918.82	−1.31119	−0.655595	+	0.755113i	$0.727582\pi$
$360$	0	0
$361$	−4701.79	−0.685493
$362$	8979.14	1.30368
$363$	2457.39	0.355316
$364$	12814.8	1.84527
$365$	0	0
$366$	3679.89	0.525548
$367$	−4072.54	−0.579250	−0.289625	−	0.957140i	$-0.593531\pi$
$367$	−4072.54	−0.579250	−0.289625	+	0.957140i	$0.593531\pi$
$368$	176.808	0.0250456
$369$	2615.05	0.368927
$370$	0	0
$371$	4505.38	0.630479
$372$	−478.143	−0.0666412
$373$	351.539	0.0487989	0.0243995	−	0.999702i	$-0.492233\pi$
$373$	351.539	0.0487989	0.0243995	+	0.999702i	$0.492233\pi$
$374$	−1699.83	−0.235016
$375$	0	0
$376$	−1959.55	−0.268766
$377$	9918.32	1.35496
$378$	1726.80	0.234966
$379$	−1677.01	−0.227288	−0.113644	−	0.993522i	$-0.536252\pi$
$379$	−1677.01	−0.227288	−0.113644	+	0.993522i	$0.536252\pi$
$380$	0	0
$381$	−1278.67	−0.171938
$382$	−5742.17	−0.769097
$383$	1410.06	0.188122	0.0940610	−	0.995566i	$-0.470015\pi$
$383$	1410.06	0.188122	0.0940610	+	0.995566i	$0.470015\pi$
$384$	1987.91	0.264180
$385$	0	0
$386$	−4299.28	−0.566911
$387$	−2918.80	−0.383387
$388$	−5254.86	−0.687565
$389$	6765.59	0.881823	0.440912	−	0.897551i	$-0.354655\pi$
$389$	6765.59	0.881823	0.440912	+	0.897551i	$0.354655\pi$
$390$	0	0
$391$	−924.161	−0.119532
$392$	−19332.7	−2.49094
$393$	1685.97	0.216402
$394$	2306.29	0.294897
$395$	0	0
$396$	917.044	0.116372
$397$	201.450	0.0254672	0.0127336	−	0.999919i	$-0.495947\pi$
$397$	201.450	0.0254672	0.0127336	+	0.999919i	$0.495947\pi$
$398$	−3239.02	−0.407933
$399$	4765.85	0.597972
$400$	0	0
$401$	15001.7	1.86821	0.934104	−	0.357002i	$-0.116201\pi$
$401$	15001.7	1.86821	0.934104	+	0.357002i	$0.116201\pi$
$402$	3077.04	0.381763
$403$	−2944.00	−0.363898
$404$	5901.99	0.726820
$405$	0	0
$406$	−7625.12	−0.932089
$407$	−2654.65	−0.323307
$408$	2818.28	0.341975
$409$	8299.23	1.00335	0.501675	−	0.865056i	$-0.332717\pi$
$409$	8299.23	1.00335	0.501675	+	0.865056i	$0.332717\pi$
$410$	0	0
$411$	3202.80	0.384386
$412$	−1708.70	−0.204324
$413$	3307.99	0.394130
$414$	−387.058	−0.0459489
$415$	0	0
$416$	14364.0	1.69291
$417$	−6041.70	−0.709504
$418$	−1964.86	−0.229914
$419$	3016.52	0.351711	0.175855	−	0.984416i	$-0.443731\pi$
$419$	3016.52	0.351711	0.175855	+	0.984416i	$0.443731\pi$
$420$	0	0
$421$	−7055.89	−0.816824	−0.408412	−	0.912798i	$-0.633917\pi$
$421$	−7055.89	−0.816824	−0.408412	+	0.912798i	$0.633917\pi$
$422$	9383.68	1.08244
$423$	754.320	0.0867052
$424$	3079.65	0.352738
$425$	0	0
$426$	−642.842	−0.0731122
$427$	22437.8	2.54296
$428$	−20.6385	−0.00233084
$429$	5646.39	0.635455
$430$	0	0
$431$	−1934.19	−0.216164	−0.108082	−	0.994142i	$-0.534471\pi$
$431$	−1934.19	−0.216164	−0.108082	+	0.994142i	$0.534471\pi$
$432$	207.558	0.0231160
$433$	−3651.12	−0.405223	−0.202611	−	0.979259i	$-0.564943\pi$
$433$	−3651.12	−0.405223	−0.202611	+	0.979259i	$0.564943\pi$
$434$	2263.32	0.250329
$435$	0	0
$436$	3610.32	0.396567
$437$	−1068.25	−0.116937
$438$	143.798	0.0156871
$439$	−9406.77	−1.02269	−0.511345	−	0.859376i	$-0.670852\pi$
$439$	−9406.77	−1.02269	−0.511345	+	0.859376i	$0.670852\pi$
$440$	0	0
$441$	7442.04	0.803589
$442$	6250.21	0.672607
$443$	3789.41	0.406412	0.203206	−	0.979136i	$-0.434864\pi$
$443$	3789.41	0.406412	0.203206	+	0.979136i	$0.434864\pi$
$444$	1585.32	0.169450
$445$	0	0
$446$	7774.68	0.825430
$447$	2023.53	0.214116
$448$	−13146.4	−1.38640
$449$	8810.30	0.926021	0.463011	−	0.886353i	$-0.346769\pi$
$449$	8810.30	0.926021	0.463011	+	0.886353i	$0.346769\pi$
$450$	0	0
$451$	−6573.80	−0.686360
$452$	194.334	0.0202228
$453$	−2105.14	−0.218340
$454$	−12026.6	−1.24325
$455$	0	0
$456$	3257.69	0.334552
$457$	−10516.3	−1.07644	−0.538221	−	0.842804i	$-0.680903\pi$
$457$	−10516.3	−1.07644	−0.538221	+	0.842804i	$0.680903\pi$
$458$	−4565.49	−0.465789
$459$	−1084.89	−0.110323
$460$	0	0
$461$	−16605.5	−1.67764	−0.838822	−	0.544405i	$-0.816755\pi$
$461$	−16605.5	−1.67764	−0.838822	+	0.544405i	$0.816755\pi$
$462$	−4340.89	−0.437136
$463$	14904.6	1.49606	0.748028	−	0.663667i	$-0.231001\pi$
$463$	14904.6	1.49606	0.748028	+	0.663667i	$0.231001\pi$
$464$	−916.523	−0.0916994
$465$	0	0
$466$	7317.40	0.727407
$467$	−15749.0	−1.56055	−0.780273	−	0.625439i	$-0.784920\pi$
$467$	−15749.0	−1.56055	−0.780273	+	0.625439i	$0.784920\pi$
$468$	−3371.94	−0.333052
$469$	18762.0	1.84723
$470$	0	0
$471$	−556.553	−0.0544472
$472$	2261.18	0.220507
$473$	7337.39	0.713263
$474$	−1636.46	−0.158576
$475$	0	0
$476$	6189.58	0.596006
$477$	−1185.50	−0.113795
$478$	9999.98	0.956880
$479$	10098.0	0.963236	0.481618	−	0.876381i	$-0.340049\pi$
$479$	10098.0	0.963236	0.481618	+	0.876381i	$0.340049\pi$
$480$	0	0
$481$	9761.07	0.925294
$482$	2404.43	0.227217
$483$	−2360.06	−0.222332
$484$	3689.11	0.346460
$485$	0	0
$486$	−454.372	−0.0424089
$487$	9446.10	0.878940	0.439470	−	0.898257i	$-0.355166\pi$
$487$	9446.10	0.878940	0.439470	+	0.898257i	$0.355166\pi$
$488$	15337.4	1.42273
$489$	7295.71	0.674690
$490$	0	0
$491$	7549.51	0.693900	0.346950	−	0.937884i	$-0.387217\pi$
$491$	7549.51	0.693900	0.346950	+	0.937884i	$0.387217\pi$
$492$	3925.79	0.359732
$493$	4790.58	0.437641
$494$	7224.72	0.658007
$495$	0	0
$496$	272.046	0.0246275
$497$	−3919.68	−0.353766
$498$	8358.83	0.752144
$499$	8631.11	0.774311	0.387156	−	0.922014i	$-0.373458\pi$
$499$	8631.11	0.774311	0.387156	+	0.922014i	$0.373458\pi$
$500$	0	0
$501$	−3520.88	−0.313974
$502$	−9143.46	−0.812934
$503$	−7678.91	−0.680687	−0.340344	−	0.940301i	$-0.610543\pi$
$503$	−7678.91	−0.680687	−0.340344	+	0.940301i	$0.610543\pi$
$504$	7197.12	0.636082
$505$	0	0
$506$	973.000	0.0854844
$507$	−14170.6	−1.24130
$508$	−1919.58	−0.167652
$509$	−9718.97	−0.846338	−0.423169	−	0.906051i	$-0.639082\pi$
$509$	−9718.97	−0.846338	−0.423169	+	0.906051i	$0.639082\pi$
$510$	0	0
$511$	876.799	0.0759047
$512$	−2765.17	−0.238680
$513$	−1254.03	−0.107928
$514$	−8867.26	−0.760930
$515$	0	0
$516$	−4381.79	−0.373832
$517$	−1896.24	−0.161308
$518$	−7504.23	−0.636519
$519$	10314.0	0.872323
$520$	0	0
$521$	−1228.16	−0.103276	−0.0516379	−	0.998666i	$-0.516444\pi$
$521$	−1228.16	−0.103276	−0.0516379	+	0.998666i	$0.516444\pi$
$522$	2006.39	0.168233
$523$	5354.94	0.447716	0.223858	−	0.974622i	$-0.428135\pi$
$523$	5354.94	0.447716	0.223858	+	0.974622i	$0.428135\pi$
$524$	2531.02	0.211008
$525$	0	0
$526$	6043.97	0.501007
$527$	−1421.96	−0.117536
$528$	−521.766	−0.0430056
$529$	529.000	0.0434783
$530$	0	0
$531$	−870.431	−0.0711365
$532$	7154.63	0.583068
$533$	24171.7	1.96434
$534$	−7339.34	−0.594765
$535$	0	0
$536$	12824.8	1.03348
$537$	−8446.34	−0.678745
$538$	13832.1	1.10845
$539$	−18708.1	−1.49502
$540$	0	0
$541$	−2880.33	−0.228900	−0.114450	−	0.993429i	$-0.536511\pi$
$541$	−2880.33	−0.228900	−0.114450	+	0.993429i	$0.536511\pi$
$542$	−15975.9	−1.26610
$543$	−14406.2	−1.13855
$544$	6937.86	0.546798
$545$	0	0
$546$	15961.3	1.25107
$547$	10379.5	0.811324	0.405662	−	0.914023i	$-0.367041\pi$
$547$	10379.5	0.811324	0.405662	+	0.914023i	$0.367041\pi$
$548$	4808.14	0.374806
$549$	−5904.06	−0.458978
$550$	0	0
$551$	5537.50	0.428141
$552$	−1613.22	−0.124390
$553$	−9978.18	−0.767298
$554$	−12129.9	−0.930237
$555$	0	0
$556$	−9069.97	−0.691821
$557$	17030.5	1.29552	0.647760	−	0.761845i	$-0.275706\pi$
$557$	17030.5	1.29552	0.647760	+	0.761845i	$0.275706\pi$
$558$	−595.547	−0.0451819
$559$	−26979.4	−2.04133
$560$	0	0
$561$	2727.22	0.205247
$562$	7905.36	0.593358
$563$	−21904.9	−1.63976	−0.819878	−	0.572538i	$-0.805959\pi$
$563$	−21904.9	−1.63976	−0.819878	+	0.572538i	$0.805959\pi$
$564$	1132.41	0.0845442
$565$	0	0
$566$	72.1594	0.00535881
$567$	−2770.50	−0.205203
$568$	−2679.30	−0.197924
$569$	−19242.5	−1.41773	−0.708864	−	0.705346i	$-0.750792\pi$
$569$	−19242.5	−1.41773	−0.708864	+	0.705346i	$0.750792\pi$
$570$	0	0
$571$	2942.65	0.215667	0.107834	−	0.994169i	$-0.465609\pi$
$571$	2942.65	0.215667	0.107834	+	0.994169i	$0.465609\pi$
$572$	8476.52	0.619617
$573$	9212.81	0.671677
$574$	−18583.0	−1.35129
$575$	0	0
$576$	3459.20	0.250232
$577$	20129.5	1.45234	0.726172	−	0.687513i	$-0.241298\pi$
$577$	20129.5	1.45234	0.726172	+	0.687513i	$0.241298\pi$
$578$	−6167.66	−0.443843
$579$	6897.82	0.495101
$580$	0	0
$581$	50967.3	3.63938
$582$	−6545.15	−0.466160
$583$	2980.15	0.211707
$584$	599.336	0.0424670
$585$	0	0
$586$	−12570.5	−0.886145
$587$	−4374.44	−0.307585	−0.153793	−	0.988103i	$-0.549149\pi$
$587$	−4374.44	−0.307585	−0.153793	+	0.988103i	$0.549149\pi$
$588$	11172.2	0.783561
$589$	−1643.67	−0.114985
$590$	0	0
$591$	−3700.24	−0.257543
$592$	−901.992	−0.0626210
$593$	1916.97	0.132750	0.0663750	−	0.997795i	$-0.478857\pi$
$593$	1916.97	0.132750	0.0663750	+	0.997795i	$0.478857\pi$
$594$	1142.22	0.0788985
$595$	0	0
$596$	3037.79	0.208779
$597$	5196.73	0.356261
$598$	−3577.69	−0.244653
$599$	4506.84	0.307420	0.153710	−	0.988116i	$-0.450878\pi$
$599$	4506.84	0.307420	0.153710	+	0.988116i	$0.450878\pi$
$600$	0	0
$601$	−24178.8	−1.64106	−0.820528	−	0.571606i	$-0.806321\pi$
$601$	−24178.8	−1.64106	−0.820528	+	0.571606i	$0.806321\pi$
$602$	20741.5	1.40425
$603$	−4936.84	−0.333406
$604$	−3160.30	−0.212899
$605$	0	0
$606$	7351.18	0.492774
$607$	−19267.1	−1.28835	−0.644173	−	0.764880i	$-0.722798\pi$
$607$	−19267.1	−1.28835	−0.644173	+	0.764880i	$0.722798\pi$
$608$	8019.57	0.534928
$609$	12233.8	0.814023
$610$	0	0
$611$	6972.41	0.461659
$612$	−1628.66	−0.107573
$613$	21699.4	1.42974	0.714869	−	0.699259i	$-0.246486\pi$
$613$	21699.4	1.42974	0.714869	+	0.699259i	$0.246486\pi$
$614$	3555.84	0.233716
$615$	0	0
$616$	−18092.4	−1.18338
$617$	−16571.0	−1.08124	−0.540619	−	0.841268i	$-0.681810\pi$
$617$	−16571.0	−1.08124	−0.540619	+	0.841268i	$0.681810\pi$
$618$	−2128.25	−0.138529
$619$	−9776.97	−0.634846	−0.317423	−	0.948284i	$-0.602817\pi$
$619$	−9776.97	−0.634846	−0.317423	+	0.948284i	$0.602817\pi$
$620$	0	0
$621$	621.000	0.0401286
$622$	8711.20	0.561555
$623$	−44751.1	−2.87787
$624$	1918.52	0.123080
$625$	0	0
$626$	−3668.68	−0.234233
$627$	3152.44	0.200792
$628$	−835.514	−0.0530902
$629$	4714.63	0.298863
$630$	0	0
$631$	−11930.8	−0.752709	−0.376354	−	0.926476i	$-0.622822\pi$
$631$	−11930.8	−0.752709	−0.376354	+	0.926476i	$0.622822\pi$
$632$	−6820.59	−0.429285
$633$	−15055.3	−0.945330
$634$	−12004.5	−0.751987
$635$	0	0
$636$	−1779.70	−0.110959
$637$	68789.0	4.27868
$638$	−5043.74	−0.312984
$639$	1031.38	0.0638512
$640$	0	0
$641$	1975.80	0.121746	0.0608730	−	0.998146i	$-0.480612\pi$
$641$	1975.80	0.121746	0.0608730	+	0.998146i	$0.480612\pi$
$642$	−25.7061	−0.00158028
$643$	825.292	0.0506164	0.0253082	−	0.999680i	$-0.491943\pi$
$643$	825.292	0.0506164	0.0253082	+	0.999680i	$0.491943\pi$
$644$	−3542.98	−0.216790
$645$	0	0
$646$	3489.56	0.212531
$647$	22005.7	1.33714	0.668572	−	0.743648i	$-0.266906\pi$
$647$	22005.7	1.33714	0.668572	+	0.743648i	$0.266906\pi$
$648$	−1893.77	−0.114806
$649$	2188.12	0.132344
$650$	0	0
$651$	−3631.30	−0.218620
$652$	10952.5	0.657874
$653$	−17408.5	−1.04326	−0.521628	−	0.853173i	$-0.674675\pi$
$653$	−17408.5	−1.04326	−0.521628	+	0.853173i	$0.674675\pi$
$654$	4496.81	0.268867
$655$	0	0
$656$	−2233.63	−0.132940
$657$	−230.712	−0.0137000
$658$	−5360.33	−0.317579
$659$	−865.634	−0.0511689	−0.0255845	−	0.999673i	$-0.508145\pi$
$659$	−865.634	−0.0511689	−0.0255845	+	0.999673i	$0.508145\pi$
$660$	0	0
$661$	29141.5	1.71478	0.857392	−	0.514664i	$-0.172083\pi$
$661$	29141.5	1.71478	0.857392	+	0.514664i	$0.172083\pi$
$662$	−10317.1	−0.605719
$663$	−10027.9	−0.587409
$664$	34838.7	2.03615
$665$	0	0
$666$	1974.58	0.114885
$667$	−2742.18	−0.159187
$668$	−5285.64	−0.306149
$669$	−12473.8	−0.720875
$670$	0	0
$671$	14841.8	0.853894
$672$	17717.4	1.01706
$673$	13481.1	0.772150	0.386075	−	0.922467i	$-0.373831\pi$
$673$	13481.1	0.772150	0.386075	+	0.922467i	$0.373831\pi$
$674$	−18823.0	−1.07572
$675$	0	0
$676$	−21273.3	−1.21036
$677$	−1986.51	−0.112774	−0.0563868	−	0.998409i	$-0.517958\pi$
$677$	−1986.51	−0.112774	−0.0563868	+	0.998409i	$0.517958\pi$
$678$	242.051	0.0137108
$679$	−39908.6	−2.25560
$680$	0	0
$681$	19295.6	1.08577
$682$	1497.11	0.0840575
$683$	27558.4	1.54391	0.771957	−	0.635675i	$-0.219278\pi$
$683$	27558.4	1.54391	0.771957	+	0.635675i	$0.219278\pi$
$684$	−1882.59	−0.105238
$685$	0	0
$686$	−30947.6	−1.72243
$687$	7324.92	0.406788
$688$	2493.08	0.138151
$689$	−10957.9	−0.605898
$690$	0	0
$691$	−11705.2	−0.644410	−0.322205	−	0.946670i	$-0.604424\pi$
$691$	−11705.2	−0.644410	−0.322205	+	0.946670i	$0.604424\pi$
$692$	15483.7	0.850581
$693$	6964.58	0.381764
$694$	−6800.61	−0.371970
$695$	0	0
$696$	8362.43	0.455427
$697$	11675.0	0.634465
$698$	−10359.0	−0.561741
$699$	−11740.1	−0.635268
$700$	0	0
$701$	32540.8	1.75328	0.876641	−	0.481146i	$-0.159779\pi$
$701$	32540.8	1.75328	0.876641	+	0.481146i	$0.159779\pi$
$702$	−4199.90	−0.225805
$703$	5449.71	0.292375
$704$	−8695.87	−0.465537
$705$	0	0
$706$	−22113.9	−1.17885
$707$	44823.3	2.38438
$708$	−1306.72	−0.0693635
$709$	14326.7	0.758887	0.379444	−	0.925215i	$-0.376115\pi$
$709$	14326.7	0.758887	0.379444	+	0.925215i	$0.376115\pi$
$710$	0	0
$711$	2625.55	0.138489
$712$	−30589.6	−1.61010
$713$	813.947	0.0427525
$714$	7709.38	0.404085
$715$	0	0
$716$	−12679.9	−0.661829
$717$	−16044.1	−0.835673
$718$	−16676.8	−0.866814
$719$	−20081.1	−1.04159	−0.520793	−	0.853683i	$-0.674363\pi$
$719$	−20081.1	−1.04159	−0.520793	+	0.853683i	$0.674363\pi$
$720$	0	0
$721$	−12976.9	−0.670296
$722$	−8791.62	−0.453172
$723$	−3857.69	−0.198436
$724$	−21627.1	−1.11017
$725$	0	0
$726$	4594.94	0.234895
$727$	19631.1	1.00148	0.500741	−	0.865597i	$-0.333061\pi$
$727$	19631.1	1.00148	0.500741	+	0.865597i	$0.333061\pi$
$728$	66525.2	3.38680
$729$	729.000	0.0370370
$730$	0	0
$731$	−13031.1	−0.659334
$732$	−8863.34	−0.447539
$733$	−26544.6	−1.33758	−0.668789	−	0.743452i	$-0.733187\pi$
$733$	−26544.6	−1.33758	−0.668789	+	0.743452i	$0.733187\pi$
$734$	−7615.01	−0.382936
$735$	0	0
$736$	−3971.30	−0.198892
$737$	12410.4	0.620276
$738$	4889.73	0.243894
$739$	16697.6	0.831167	0.415583	−	0.909555i	$-0.363577\pi$
$739$	16697.6	0.831167	0.415583	+	0.909555i	$0.363577\pi$
$740$	0	0
$741$	−11591.4	−0.574658
$742$	8424.35	0.416803
$743$	−34375.7	−1.69734	−0.848669	−	0.528924i	$-0.822596\pi$
$743$	−34375.7	−1.69734	−0.848669	+	0.528924i	$0.822596\pi$
$744$	−2482.18	−0.122313
$745$	0	0
$746$	657.323	0.0322605
$747$	−13411.0	−0.656871
$748$	4094.19	0.200131
$749$	−156.741	−0.00764646
$750$	0	0
$751$	−29075.8	−1.41277	−0.706386	−	0.707827i	$-0.749676\pi$
$751$	−29075.8	−1.41277	−0.706386	+	0.707827i	$0.749676\pi$
$752$	−644.300	−0.0312436
$753$	14669.9	0.709961
$754$	18545.7	0.895749
$755$	0	0
$756$	−4159.15	−0.200089
$757$	−27305.2	−1.31100	−0.655499	−	0.755196i	$-0.727542\pi$
$757$	−27305.2	−1.31100	−0.655499	+	0.755196i	$0.727542\pi$
$758$	−3135.74	−0.150258
$759$	−1561.09	−0.0746562
$760$	0	0
$761$	−13307.1	−0.633878	−0.316939	−	0.948446i	$-0.602655\pi$
$761$	−13307.1	−0.633878	−0.316939	+	0.948446i	$0.602655\pi$
$762$	−2390.91	−0.113666
$763$	27419.0	1.30096
$764$	13830.5	0.654936
$765$	0	0
$766$	2636.59	0.124366
$767$	−8045.66	−0.378764
$768$	12941.6	0.608061
$769$	−25341.7	−1.18835	−0.594177	−	0.804334i	$-0.702522\pi$
$769$	−25341.7	−1.18835	−0.594177	+	0.804334i	$0.702522\pi$
$770$	0	0
$771$	14226.7	0.664545
$772$	10355.2	0.482762
$773$	−29885.2	−1.39055	−0.695276	−	0.718743i	$-0.744718\pi$
$773$	−29885.2	−1.39055	−0.695276	+	0.718743i	$0.744718\pi$
$774$	−5457.70	−0.253453
$775$	0	0
$776$	−27279.5	−1.26196
$777$	12039.9	0.555892
$778$	12650.6	0.582964
$779$	13495.3	0.620693
$780$	0	0
$781$	−2592.73	−0.118790
$782$	−1728.04	−0.0790211
$783$	−3219.08	−0.146923
$784$	−6356.59	−0.289568
$785$	0	0
$786$	3152.50	0.143061
$787$	−5402.56	−0.244702	−0.122351	−	0.992487i	$-0.539043\pi$
$787$	−5402.56	−0.244702	−0.122351	+	0.992487i	$0.539043\pi$
$788$	−5554.91	−0.251124
$789$	−9697.02	−0.437545
$790$	0	0
$791$	1475.89	0.0663420
$792$	4760.64	0.213588
$793$	−54573.0	−2.44381
$794$	376.680	0.0168361
$795$	0	0
$796$	7801.47	0.347382
$797$	4791.85	0.212969	0.106484	−	0.994314i	$-0.466041\pi$
$797$	4791.85	0.212969	0.106484	+	0.994314i	$0.466041\pi$
$798$	8911.39	0.395313
$799$	3367.70	0.149112
$800$	0	0
$801$	11775.3	0.519427
$802$	28050.9	1.23505
$803$	579.972	0.0254879
$804$	−7411.33	−0.325096
$805$	0	0
$806$	−5504.82	−0.240569
$807$	−22192.5	−0.968045
$808$	30639.0	1.33400
$809$	−54.1806	−0.00235462	−0.00117731	−	0.999999i	$-0.500375\pi$
$809$	−54.1806	−0.00235462	−0.00117731	+	0.999999i	$0.500375\pi$
$810$	0	0
$811$	−12145.0	−0.525857	−0.262929	−	0.964815i	$-0.584688\pi$
$811$	−12145.0	−0.525857	−0.262929	+	0.964815i	$0.584688\pi$
$812$	18365.8	0.793735
$813$	25632.0	1.10572
$814$	−4963.78	−0.213735
$815$	0	0
$816$	926.651	0.0397540
$817$	−15062.9	−0.645022
$818$	15518.3	0.663305
$819$	−25608.6	−1.09260
$820$	0	0
$821$	3092.61	0.131465	0.0657325	−	0.997837i	$-0.479062\pi$
$821$	3092.61	0.131465	0.0657325	+	0.997837i	$0.479062\pi$
$822$	5988.74	0.254113
$823$	−40471.2	−1.71414	−0.857070	−	0.515199i	$-0.827718\pi$
$823$	−40471.2	−1.71414	−0.857070	+	0.515199i	$0.827718\pi$
$824$	−8870.33	−0.375015
$825$	0	0
$826$	6185.43	0.260555
$827$	33386.5	1.40383	0.701913	−	0.712263i	$-0.252330\pi$
$827$	33386.5	1.40383	0.701913	+	0.712263i	$0.252330\pi$
$828$	932.263	0.0391285
$829$	5605.72	0.234855	0.117427	−	0.993081i	$-0.462535\pi$
$829$	5605.72	0.234855	0.117427	+	0.993081i	$0.462535\pi$
$830$	0	0
$831$	19461.4	0.812405
$832$	31974.5	1.33235
$833$	33225.3	1.38198
$834$	−11297.0	−0.469046
$835$	0	0
$836$	4732.53	0.195787
$837$	955.502	0.0394588
$838$	5640.43	0.232512
$839$	9214.29	0.379157	0.189578	−	0.981866i	$-0.439288\pi$
$839$	9214.29	0.379157	0.189578	+	0.981866i	$0.439288\pi$
$840$	0	0
$841$	−10174.3	−0.417169
$842$	−13193.4	−0.539994
$843$	−12683.5	−0.518199
$844$	−22601.4	−0.921769
$845$	0	0
$846$	1410.46	0.0573199
$847$	28017.3	1.13658
$848$	1012.59	0.0410053
$849$	−115.773	−0.00468002
$850$	0	0
$851$	−2698.70	−0.108708
$852$	1548.34	0.0622598
$853$	−36178.7	−1.45221	−0.726105	−	0.687584i	$-0.758671\pi$
$853$	−36178.7	−1.45221	−0.726105	+	0.687584i	$0.758671\pi$
$854$	41955.2	1.68112
$855$	0	0
$856$	−107.140	−0.00427802
$857$	−49397.3	−1.96894	−0.984469	−	0.175556i	$-0.943828\pi$
$857$	−49397.3	−1.96894	−0.984469	+	0.175556i	$0.943828\pi$
$858$	10557.9	0.420092
$859$	−36610.6	−1.45418	−0.727089	−	0.686544i	$-0.759127\pi$
$859$	−36610.6	−1.45418	−0.727089	+	0.686544i	$0.759127\pi$
$860$	0	0
$861$	29814.8	1.18012
$862$	−3616.64	−0.142904
$863$	23158.2	0.913459	0.456729	−	0.889606i	$-0.349021\pi$
$863$	23158.2	0.913459	0.456729	+	0.889606i	$0.349021\pi$
$864$	−4661.97	−0.183569
$865$	0	0
$866$	−6827.02	−0.267889
$867$	9895.48	0.387622
$868$	−5451.41	−0.213172
$869$	−6600.21	−0.257649
$870$	0	0
$871$	−45632.7	−1.77521
$872$	18742.2	0.727857
$873$	10501.1	0.407113
$874$	−1997.46	−0.0773057
$875$	0	0
$876$	−346.351	−0.0133586
$877$	−17503.2	−0.673935	−0.336968	−	0.941516i	$-0.609401\pi$
$877$	−17503.2	−0.673935	−0.336968	+	0.941516i	$0.609401\pi$
$878$	−17589.2	−0.676089
$879$	20168.2	0.773899
$880$	0	0
$881$	6121.17	0.234083	0.117042	−	0.993127i	$-0.462659\pi$
$881$	6121.17	0.234083	0.117042	+	0.993127i	$0.462659\pi$
$882$	13915.4	0.531244
$883$	−11325.5	−0.431636	−0.215818	−	0.976434i	$-0.569242\pi$
$883$	−11325.5	−0.431636	−0.215818	+	0.976434i	$0.569242\pi$
$884$	−15054.2	−0.572769
$885$	0	0
$886$	7085.61	0.268674
$887$	42981.5	1.62703	0.813516	−	0.581542i	$-0.197551\pi$
$887$	42981.5	1.62703	0.813516	+	0.581542i	$0.197551\pi$
$888$	8229.85	0.311009
$889$	−14578.4	−0.549994
$890$	0	0
$891$	−1832.59	−0.0689046
$892$	−18726.0	−0.702908
$893$	3892.77	0.145875
$894$	3783.69	0.141550
$895$	0	0
$896$	22664.7	0.845060
$897$	5740.09	0.213663
$898$	16473.9	0.612183
$899$	−4219.26	−0.156530
$900$	0	0
$901$	−5292.71	−0.195700
$902$	−12292.0	−0.453745
$903$	−33277.9	−1.22638
$904$	1008.84	0.0371168
$905$	0	0
$906$	−3936.28	−0.144342
$907$	35034.3	1.28258	0.641288	−	0.767301i	$-0.278401\pi$
$907$	35034.3	1.28258	0.641288	+	0.767301i	$0.278401\pi$
$908$	28967.1	1.05871
$909$	−11794.3	−0.430355
$910$	0	0
$911$	37328.6	1.35758	0.678789	−	0.734333i	$-0.262505\pi$
$911$	37328.6	1.35758	0.678789	+	0.734333i	$0.262505\pi$
$912$	1071.13	0.0388911
$913$	33713.1	1.22206
$914$	−19663.9	−0.711624
$915$	0	0
$916$	10996.4	0.396649
$917$	19222.1	0.692225
$918$	−2028.57	−0.0729331
$919$	39014.0	1.40038	0.700192	−	0.713955i	$-0.253098\pi$
$919$	39014.0	1.40038	0.700192	+	0.713955i	$0.253098\pi$
$920$	0	0
$921$	−5705.03	−0.204112
$922$	−31049.6	−1.10907
$923$	9533.39	0.339973
$924$	10455.4	0.372249
$925$	0	0
$926$	27869.2	0.989027
$927$	3414.59	0.120982
$928$	20586.1	0.728201
$929$	31773.1	1.12211	0.561055	−	0.827778i	$-0.310396\pi$
$929$	31773.1	1.12211	0.561055	+	0.827778i	$0.310396\pi$
$930$	0	0
$931$	38405.7	1.35198
$932$	−17624.6	−0.619435
$933$	−13976.4	−0.490424
$934$	−29448.1	−1.03166
$935$	0	0
$936$	−17504.7	−0.611282
$937$	28602.9	0.997241	0.498620	−	0.866820i	$-0.333840\pi$
$937$	28602.9	0.997241	0.498620	+	0.866820i	$0.333840\pi$
$938$	35082.1	1.22118
$939$	5886.08	0.204563
$940$	0	0
$941$	9518.49	0.329749	0.164874	−	0.986315i	$-0.447278\pi$
$941$	9518.49	0.329749	0.164874	+	0.986315i	$0.447278\pi$
$942$	−1040.67	−0.0359945
$943$	−6682.90	−0.230780
$944$	743.476	0.0256335
$945$	0	0
$946$	13719.8	0.471531
$947$	−54519.0	−1.87078	−0.935390	−	0.353619i	$-0.884951\pi$
$947$	−54519.0	−1.87078	−0.935390	+	0.353619i	$0.884951\pi$
$948$	3941.56	0.135038
$949$	−2132.54	−0.0729453
$950$	0	0
$951$	19260.2	0.656734
$952$	32131.9	1.09391
$953$	13333.1	0.453201	0.226601	−	0.973988i	$-0.427239\pi$
$953$	13333.1	0.453201	0.226601	+	0.973988i	$0.427239\pi$
$954$	−2216.70	−0.0752287
$955$	0	0
$956$	−24085.9	−0.814846
$957$	8092.25	0.273339
$958$	18881.7	0.636785
$959$	36515.9	1.22957
$960$	0	0
$961$	−28538.6	−0.957961
$962$	18251.7	0.611702
$963$	41.2432	0.00138011
$964$	−5791.28	−0.193490
$965$	0	0
$966$	−4412.93	−0.146981
$967$	−9694.24	−0.322385	−0.161192	−	0.986923i	$-0.551534\pi$
$967$	−9694.24	−0.322385	−0.161192	+	0.986923i	$0.551534\pi$
$968$	19151.2	0.635892
$969$	−5598.70	−0.185610
$970$	0	0
$971$	58374.6	1.92928	0.964640	−	0.263573i	$-0.0849009\pi$
$971$	58374.6	1.92928	0.964640	+	0.263573i	$0.0849009\pi$
$972$	1094.40	0.0361140
$973$	−68882.8	−2.26956
$974$	17662.7	0.581058
$975$	0	0
$976$	5042.93	0.165390
$977$	−14542.0	−0.476193	−0.238097	−	0.971242i	$-0.576523\pi$
$977$	−14542.0	−0.476193	−0.238097	+	0.971242i	$0.576523\pi$
$978$	13641.8	0.446030
$979$	−29601.3	−0.966354
$980$	0	0
$981$	−7214.73	−0.234810
$982$	14116.4	0.458730
$983$	12611.9	0.409214	0.204607	−	0.978844i	$-0.434408\pi$
$983$	12611.9	0.409214	0.204607	+	0.978844i	$0.434408\pi$
$984$	20379.9	0.660251
$985$	0	0
$986$	8957.64	0.289320
$987$	8600.17	0.277352
$988$	−17401.4	−0.560336
$989$	7459.16	0.239825
$990$	0	0
$991$	29148.6	0.934345	0.467172	−	0.884166i	$-0.345273\pi$
$991$	29148.6	0.934345	0.467172	+	0.884166i	$0.345273\pi$
$992$	−6110.45	−0.195572
$993$	16552.9	0.528994
$994$	−7329.19	−0.233871
$995$	0	0
$996$	−20133.0	−0.640500
$997$	1143.97	0.0363388	0.0181694	−	0.999835i	$-0.494216\pi$
$997$	1143.97	0.0363388	0.0181694	+	0.999835i	$0.494216\pi$
$998$	16138.8	0.511889
$999$	−3168.04	−0.100333

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1725.4.a.bf.1.8	yes	11
5.4	even	2		1725.4.a.be.1.4	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1725.4.a.be.1.4	✓	11	5.4	even	2
1725.4.a.bf.1.8	yes	11	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1725 = 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1725.a (trivial)

\(f(q)\)	\(=\)	\(q+1.86984 q^{2} -3.00000 q^{3} -4.50369 q^{4} -5.60953 q^{6} -34.2037 q^{7} -23.3799 q^{8} +9.00000 q^{9} -22.6245 q^{11} +13.5111 q^{12} +83.1897 q^{13} -63.9556 q^{14} -7.68732 q^{16} +40.1809 q^{17} +16.8286 q^{18} +46.4457 q^{19} +102.611 q^{21} -42.3043 q^{22} -23.0000 q^{23} +70.1398 q^{24} +155.552 q^{26} -27.0000 q^{27} +154.043 q^{28} +119.225 q^{29} -35.3890 q^{31} +172.665 q^{32} +67.8736 q^{33} +75.1320 q^{34} -40.5332 q^{36} +117.335 q^{37} +86.8462 q^{38} -249.569 q^{39} +290.561 q^{41} +191.867 q^{42} -324.311 q^{43} +101.894 q^{44} -43.0064 q^{46} +83.8133 q^{47} +23.0620 q^{48} +826.893 q^{49} -120.543 q^{51} -374.660 q^{52} -131.722 q^{53} -50.4858 q^{54} +799.680 q^{56} -139.337 q^{57} +222.933 q^{58} -96.7145 q^{59} -656.006 q^{61} -66.1718 q^{62} -307.833 q^{63} +384.356 q^{64} +126.913 q^{66} -548.538 q^{67} -180.962 q^{68} +69.0000 q^{69} +114.598 q^{71} -210.419 q^{72} -25.6346 q^{73} +219.398 q^{74} -209.177 q^{76} +773.843 q^{77} -466.655 q^{78} +291.728 q^{79} +81.0000 q^{81} +543.303 q^{82} -1490.11 q^{83} -462.128 q^{84} -606.411 q^{86} -357.676 q^{87} +528.960 q^{88} +1308.37 q^{89} -2845.40 q^{91} +103.585 q^{92} +106.167 q^{93} +156.718 q^{94} -517.996 q^{96} +1166.79 q^{97} +1546.16 q^{98} -203.621 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + q^{2} - 33 q^{3} + 39 q^{4} - 3 q^{6} - 19 q^{7} - 18 q^{8} + 99 q^{9} - 41 q^{11} - 117 q^{12} - 57 q^{13} + 8 q^{14} + 227 q^{16} - 66 q^{17} + 9 q^{18} - 215 q^{19} + 57 q^{21} + 342 q^{22} - 253 q^{23}+ \cdots - 369 q^{99}+O(q^{100}) \) 11 * q + q^2 - 33 * q^3 + 39 * q^4 - 3 * q^6 - 19 * q^7 - 18 * q^8 + 99 * q^9 - 41 * q^11 - 117 * q^12 - 57 * q^13 + 8 * q^14 + 227 * q^16 - 66 * q^17 + 9 * q^18 - 215 * q^19 + 57 * q^21 + 342 * q^22 - 253 * q^23 + 54 * q^24 + 364 * q^26 - 297 * q^27 - 657 * q^28 + 513 * q^29 + 340 * q^31 - 263 * q^32 + 123 * q^33 - 125 * q^34 + 351 * q^36 - 20 * q^37 - 124 * q^38 + 171 * q^39 + 515 * q^41 - 24 * q^42 - 1401 * q^43 - 879 * q^44 - 23 * q^46 + 152 * q^47 - 681 * q^48 + 1108 * q^49 + 198 * q^51 - 697 * q^52 - 794 * q^53 - 27 * q^54 + 1113 * q^56 + 645 * q^57 - 3058 * q^58 + 1332 * q^59 + 148 * q^61 - 984 * q^62 - 171 * q^63 + 1422 * q^64 - 1026 * q^66 - 56 * q^67 - 2277 * q^68 + 759 * q^69 - 198 * q^71 - 162 * q^72 + 205 * q^73 + 2901 * q^74 - 4522 * q^76 + 851 * q^77 - 1092 * q^78 - 645 * q^79 + 891 * q^81 - 37 * q^82 - 2591 * q^83 + 1971 * q^84 - 567 * q^86 - 1539 * q^87 + 2247 * q^88 + 4130 * q^89 - 1623 * q^91 - 897 * q^92 - 1020 * q^93 + 49 * q^94 + 789 * q^96 - 2348 * q^97 - 2804 * q^98 - 369 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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