LMFDB - Embedded newform 475.4.a.f.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [475,4,Mod(1,475)] 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("475.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(475, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	475.a (trivial)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$28.0259072527$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.3144.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 16x - 8 $ x^3 - x^2 - 16*x - 8
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 19)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.73549$ of defining polynomial
Character	$\chi$	$=$	475.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.89080 q^{2} -2.95388 q^{3} -4.42486 q^{4} +5.58521 q^{6} -5.94196 q^{7} +23.4930 q^{8} -18.2746 q^{9} +11.5171 q^{11} +13.0705 q^{12} -22.6093 q^{13} +11.2351 q^{14} -9.02172 q^{16} -120.560 q^{17} +34.5537 q^{18} -19.0000 q^{19} +17.5518 q^{21} -21.7766 q^{22} -63.9160 q^{23} -69.3955 q^{24} +42.7498 q^{26} +133.736 q^{27} +26.2923 q^{28} -89.7278 q^{29} -251.051 q^{31} -170.885 q^{32} -34.0201 q^{33} +227.954 q^{34} +80.8625 q^{36} -198.702 q^{37} +35.9253 q^{38} +66.7853 q^{39} +373.170 q^{41} -33.1871 q^{42} +448.586 q^{43} -50.9616 q^{44} +120.853 q^{46} +186.475 q^{47} +26.6491 q^{48} -307.693 q^{49} +356.119 q^{51} +100.043 q^{52} -364.882 q^{53} -252.868 q^{54} -139.594 q^{56} +56.1238 q^{57} +169.658 q^{58} -376.730 q^{59} +816.832 q^{61} +474.689 q^{62} +108.587 q^{63} +395.285 q^{64} +64.3254 q^{66} -220.185 q^{67} +533.459 q^{68} +188.800 q^{69} +383.466 q^{71} -429.324 q^{72} +537.689 q^{73} +375.706 q^{74} +84.0724 q^{76} -68.4341 q^{77} -126.278 q^{78} +1062.27 q^{79} +98.3741 q^{81} -705.592 q^{82} +616.932 q^{83} -77.6645 q^{84} -848.189 q^{86} +265.045 q^{87} +270.571 q^{88} -90.2686 q^{89} +134.344 q^{91} +282.819 q^{92} +741.576 q^{93} -352.589 q^{94} +504.776 q^{96} +524.636 q^{97} +581.787 q^{98} -210.470 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} - q^{3} + 21 q^{4} - 65 q^{6} + 35 q^{7} - 27 q^{8} + 48 q^{9} + 16 q^{11} + 115 q^{12} - 65 q^{13} + 37 q^{14} + 33 q^{16} - 29 q^{17} - 138 q^{18} - 57 q^{19} - 25 q^{21} - 118 q^{22} + 101 q^{23}+ \cdots + 250 q^{99}+O(q^{100}) $ 3 * q - 3 * q^2 - q^3 + 21 * q^4 - 65 * q^6 + 35 * q^7 - 27 * q^8 + 48 * q^9 + 16 * q^11 + 115 * q^12 - 65 * q^13 + 37 * q^14 + 33 * q^16 - 29 * q^17 - 138 * q^18 - 57 * q^19 - 25 * q^21 - 118 * q^22 + 101 * q^23 - 489 * q^24 + 299 * q^26 + 377 * q^27 + 493 * q^28 + 377 * q^29 - 140 * q^31 - 579 * q^32 + 130 * q^33 + 329 * q^34 + 1074 * q^36 + 290 * q^37 + 57 * q^38 - 371 * q^39 + 956 * q^41 - 1385 * q^42 + 570 * q^43 + 110 * q^44 - 239 * q^46 - 66 * q^47 + 1411 * q^48 - 98 * q^49 + 187 * q^51 - 697 * q^52 - 817 * q^53 - 947 * q^54 - 915 * q^56 + 19 * q^57 - 647 * q^58 + 265 * q^59 + 988 * q^61 + 1436 * q^62 + 1304 * q^63 - 159 * q^64 - 274 * q^66 + 207 * q^67 + 1529 * q^68 + 807 * q^69 + 846 * q^71 - 2826 * q^72 - 627 * q^73 - 770 * q^74 - 399 * q^76 - 88 * q^77 + 1817 * q^78 + 382 * q^79 - 501 * q^81 - 872 * q^82 + 766 * q^83 + 907 * q^84 + 2592 * q^86 + 1671 * q^87 - 342 * q^88 - 172 * q^89 - 457 * q^91 + 2677 * q^92 - 892 * q^93 - 96 * q^94 + 863 * q^96 + 2450 * q^97 + 2450 * q^98 + 250 * 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 $

$2$	−1.89080	−0.668500	−0.334250	−	0.942484i	$-0.608483\pi$
$2$	−1.89080	−0.668500	−0.334250	+	0.942484i	$0.608483\pi$
$3$	−2.95388	−0.568475	−0.284237	−	0.958754i	$-0.591740\pi$
$3$	−2.95388	−0.568475	−0.284237	+	0.958754i	$0.591740\pi$
$4$	−4.42486	−0.553108
$5$	0	0
$5$	0	0
$6$	5.58521	0.380025
$7$	−5.94196	−0.320836	−0.160418	−	0.987049i	$-0.551284\pi$
$7$	−5.94196	−0.320836	−0.160418	+	0.987049i	$0.551284\pi$
$8$	23.4930	1.03825
$9$	−18.2746	−0.676836
$10$	0	0
$11$	11.5171	0.315685	0.157843	−	0.987464i	$-0.449546\pi$
$11$	11.5171	0.315685	0.157843	+	0.987464i	$0.449546\pi$
$12$	13.0705	0.314428
$13$	−22.6093	−0.482362	−0.241181	−	0.970480i	$-0.577535\pi$
$13$	−22.6093	−0.482362	−0.241181	+	0.970480i	$0.577535\pi$
$14$	11.2351	0.214479
$15$	0	0
$16$	−9.02172	−0.140964
$17$	−120.560	−1.72000	−0.859999	−	0.510295i	$-0.829536\pi$
$17$	−120.560	−1.72000	−0.859999	+	0.510295i	$0.829536\pi$
$18$	34.5537	0.452465
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	0	0
$21$	17.5518	0.182387
$22$	−21.7766	−0.211035
$23$	−63.9160	−0.579452	−0.289726	−	0.957110i	$-0.593564\pi$
$23$	−63.9160	−0.579452	−0.289726	+	0.957110i	$0.593564\pi$
$24$	−69.3955	−0.590220
$25$	0	0
$26$	42.7498	0.322459
$27$	133.736	0.953239
$28$	26.2923	0.177457
$29$	−89.7278	−0.574553	−0.287276	−	0.957848i	$-0.592750\pi$
$29$	−89.7278	−0.574553	−0.287276	+	0.957848i	$0.592750\pi$
$30$	0	0
$31$	−251.051	−1.45452	−0.727260	−	0.686362i	$-0.759207\pi$
$31$	−251.051	−1.45452	−0.727260	+	0.686362i	$0.759207\pi$
$32$	−170.885	−0.944018
$33$	−34.0201	−0.179459
$34$	227.954	1.14982
$35$	0	0
$36$	80.8625	0.374363
$37$	−198.702	−0.882875	−0.441438	−	0.897292i	$-0.645531\pi$
$37$	−198.702	−0.882875	−0.441438	+	0.897292i	$0.645531\pi$
$38$	35.9253	0.153364
$39$	66.7853	0.274210
$40$	0	0
$41$	373.170	1.42145	0.710725	−	0.703470i	$-0.248367\pi$
$41$	373.170	1.42145	0.710725	+	0.703470i	$0.248367\pi$
$42$	−33.1871	−0.121926
$43$	448.586	1.59090	0.795451	−	0.606018i	$-0.207234\pi$
$43$	448.586	1.59090	0.795451	+	0.606018i	$0.207234\pi$
$44$	−50.9616	−0.174608
$45$	0	0
$46$	120.853	0.387364
$47$	186.475	0.578729	0.289364	−	0.957219i	$-0.406556\pi$
$47$	186.475	0.578729	0.289364	+	0.957219i	$0.406556\pi$
$48$	26.6491	0.0801347
$49$	−307.693	−0.897065
$50$	0	0
$51$	356.119	0.977776
$52$	100.043	0.266798
$53$	−364.882	−0.945669	−0.472834	−	0.881151i	$-0.656769\pi$
$53$	−364.882	−0.945669	−0.472834	+	0.881151i	$0.656769\pi$
$54$	−252.868	−0.637241
$55$	0	0
$56$	−139.594	−0.333108
$57$	56.1238	0.130417
$58$	169.658	0.384089
$59$	−376.730	−0.831290	−0.415645	−	0.909527i	$-0.636444\pi$
$59$	−376.730	−0.831290	−0.415645	+	0.909527i	$0.636444\pi$
$60$	0	0
$61$	816.832	1.71450	0.857251	−	0.514899i	$-0.172171\pi$
$61$	816.832	1.71450	0.857251	+	0.514899i	$0.172171\pi$
$62$	474.689	0.972347
$63$	108.587	0.217153
$64$	395.285	0.772040
$65$	0	0
$66$	64.3254	0.119968
$67$	−220.185	−0.401490	−0.200745	−	0.979644i	$-0.564336\pi$
$67$	−220.185	−0.401490	−0.200745	+	0.979644i	$0.564336\pi$
$68$	533.459	0.951344
$69$	188.800	0.329404
$70$	0	0
$71$	383.466	0.640973	0.320486	−	0.947253i	$-0.396154\pi$
$71$	383.466	0.640973	0.320486	+	0.947253i	$0.396154\pi$
$72$	−429.324	−0.702727
$73$	537.689	0.862078	0.431039	−	0.902333i	$-0.358147\pi$
$73$	537.689	0.862078	0.431039	+	0.902333i	$0.358147\pi$
$74$	375.706	0.590202
$75$	0	0
$76$	84.0724	0.126892
$77$	−68.4341	−0.101283
$78$	−126.278	−0.183310
$79$	1062.27	1.51284	0.756421	−	0.654085i	$-0.226946\pi$
$79$	1062.27	1.51284	0.756421	+	0.654085i	$0.226946\pi$
$80$	0	0
$81$	98.3741	0.134944
$82$	−705.592	−0.950239
$83$	616.932	0.815869	0.407935	−	0.913011i	$-0.366249\pi$
$83$	616.932	0.815869	0.407935	+	0.913011i	$0.366249\pi$
$84$	−77.6645	−0.100880
$85$	0	0
$86$	−848.189	−1.06352
$87$	265.045	0.326619
$88$	270.571	0.327761
$89$	−90.2686	−0.107511	−0.0537554	−	0.998554i	$-0.517119\pi$
$89$	−90.2686	−0.107511	−0.0537554	+	0.998554i	$0.517119\pi$
$90$	0	0
$91$	134.344	0.154759
$92$	282.819	0.320499
$93$	741.576	0.826858
$94$	−352.589	−0.386880
$95$	0	0
$96$	504.776	0.536650
$97$	524.636	0.549162	0.274581	−	0.961564i	$-0.411461\pi$
$97$	524.636	0.549162	0.274581	+	0.961564i	$0.411461\pi$
$98$	581.787	0.599688
$99$	−210.470	−0.213667
$100$	0	0
$101$	−336.523	−0.331538	−0.165769	−	0.986165i	$-0.553011\pi$
$101$	−336.523	−0.331538	−0.165769	+	0.986165i	$0.553011\pi$
$102$	−673.350	−0.653643
$103$	718.709	0.687539	0.343770	−	0.939054i	$-0.388296\pi$
$103$	718.709	0.687539	0.343770	+	0.939054i	$0.388296\pi$
$104$	−531.161	−0.500813
$105$	0	0
$106$	689.921	0.632180
$107$	−1321.73	−1.19417	−0.597087	−	0.802177i	$-0.703675\pi$
$107$	−1321.73	−1.19417	−0.597087	+	0.802177i	$0.703675\pi$
$108$	−591.762	−0.527244
$109$	472.039	0.414799	0.207400	−	0.978256i	$-0.433500\pi$
$109$	472.039	0.414799	0.207400	+	0.978256i	$0.433500\pi$
$110$	0	0
$111$	586.942	0.501892
$112$	53.6067	0.0452264
$113$	−442.672	−0.368523	−0.184261	−	0.982877i	$-0.558989\pi$
$113$	−442.672	−0.368523	−0.184261	+	0.982877i	$0.558989\pi$
$114$	−106.119	−0.0871838
$115$	0	0
$116$	397.033	0.317790
$117$	413.176	0.326480
$118$	712.323	0.555717
$119$	716.360	0.551837
$120$	0	0
$121$	−1198.36	−0.900343
$122$	−1544.47	−1.14614
$123$	−1102.30	−0.808058
$124$	1110.87	0.804506
$125$	0	0
$126$	−205.316	−0.145167
$127$	1482.52	1.03585	0.517923	−	0.855427i	$-0.326705\pi$
$127$	1482.52	1.03585	0.517923	+	0.855427i	$0.326705\pi$
$128$	619.678	0.427909
$129$	−1325.07	−0.904387
$130$	0	0
$131$	785.710	0.524029	0.262015	−	0.965064i	$-0.415613\pi$
$131$	785.710	0.524029	0.262015	+	0.965064i	$0.415613\pi$
$132$	150.534	0.0992601
$133$	112.897	0.0736047
$134$	416.326	0.268396
$135$	0	0
$136$	−2832.30	−1.78579
$137$	1423.76	0.887886	0.443943	−	0.896055i	$-0.353579\pi$
$137$	1423.76	0.887886	0.443943	+	0.896055i	$0.353579\pi$
$138$	−356.984	−0.220207
$139$	−1416.42	−0.864311	−0.432155	−	0.901799i	$-0.642247\pi$
$139$	−1416.42	−0.864311	−0.432155	+	0.901799i	$0.642247\pi$
$140$	0	0
$141$	−550.827	−0.328993
$142$	−725.059	−0.428490
$143$	−260.394	−0.152274
$144$	164.868	0.0954098
$145$	0	0
$146$	−1016.66	−0.576299
$147$	908.889	0.509959
$148$	879.228	0.488325
$149$	3342.97	1.83803	0.919016	−	0.394221i	$-0.128986\pi$
$149$	3342.97	1.83803	0.919016	+	0.394221i	$0.128986\pi$
$150$	0	0
$151$	2233.19	1.20354	0.601769	−	0.798670i	$-0.294463\pi$
$151$	2233.19	1.20354	0.601769	+	0.798670i	$0.294463\pi$
$152$	−446.367	−0.238191
$153$	2203.17	1.16416
$154$	129.395	0.0677077
$155$	0	0
$156$	−295.516	−0.151668
$157$	−1748.03	−0.888588	−0.444294	−	0.895881i	$-0.646545\pi$
$157$	−1748.03	−0.888588	−0.444294	+	0.895881i	$0.646545\pi$
$158$	−2008.54	−1.01134
$159$	1077.82	0.537589
$160$	0	0
$161$	379.786	0.185909
$162$	−186.006	−0.0902100
$163$	−3442.43	−1.65418	−0.827092	−	0.562067i	$-0.810006\pi$
$163$	−3442.43	−1.65418	−0.827092	+	0.562067i	$0.810006\pi$
$164$	−1651.23	−0.786215
$165$	0	0
$166$	−1166.50	−0.545409
$167$	3440.98	1.59444	0.797218	−	0.603692i	$-0.206304\pi$
$167$	3440.98	1.59444	0.797218	+	0.603692i	$0.206304\pi$
$168$	412.345	0.189364
$169$	−1685.82	−0.767327
$170$	0	0
$171$	347.217	0.155277
$172$	−1984.93	−0.879940
$173$	−3159.81	−1.38865	−0.694324	−	0.719663i	$-0.744296\pi$
$173$	−3159.81	−1.38865	−0.694324	+	0.719663i	$0.744296\pi$
$174$	−501.149	−0.218345
$175$	0	0
$176$	−103.904	−0.0445003
$177$	1112.82	0.472567
$178$	170.680	0.0718709
$179$	2056.18	0.858580	0.429290	−	0.903167i	$-0.358764\pi$
$179$	2056.18	0.858580	0.429290	+	0.903167i	$0.358764\pi$
$180$	0	0
$181$	−123.527	−0.0507277	−0.0253638	−	0.999678i	$-0.508074\pi$
$181$	−123.527	−0.0507277	−0.0253638	+	0.999678i	$0.508074\pi$
$182$	−254.018	−0.103456
$183$	−2412.82	−0.974651
$184$	−1501.58	−0.601618
$185$	0	0
$186$	−1402.17	−0.552755
$187$	−1388.50	−0.542978
$188$	−825.128	−0.320099
$189$	−794.652	−0.305833
$190$	0	0
$191$	−3379.82	−1.28039	−0.640197	−	0.768210i	$-0.721147\pi$
$191$	−3379.82	−1.28039	−0.640197	+	0.768210i	$0.721147\pi$
$192$	−1167.62	−0.438886
$193$	386.034	0.143976	0.0719880	−	0.997405i	$-0.477066\pi$
$193$	386.034	0.143976	0.0719880	+	0.997405i	$0.477066\pi$
$194$	−991.983	−0.367115
$195$	0	0
$196$	1361.50	0.496173
$197$	3242.43	1.17266	0.586328	−	0.810073i	$-0.300573\pi$
$197$	3242.43	1.17266	0.586328	+	0.810073i	$0.300573\pi$
$198$	397.958	0.142836
$199$	−752.507	−0.268059	−0.134030	−	0.990977i	$-0.542792\pi$
$199$	−752.507	−0.268059	−0.134030	+	0.990977i	$0.542792\pi$
$200$	0	0
$201$	650.400	0.228237
$202$	636.300	0.221633
$203$	533.159	0.184337
$204$	−1575.77	−0.540815
$205$	0	0
$206$	−1358.94	−0.459620
$207$	1168.04	0.392194
$208$	203.975	0.0679958
$209$	−218.825	−0.0724231
$210$	0	0
$211$	−3996.39	−1.30390	−0.651949	−	0.758263i	$-0.726048\pi$
$211$	−3996.39	−1.30390	−0.651949	+	0.758263i	$0.726048\pi$
$212$	1614.55	0.523057
$213$	−1132.71	−0.364377
$214$	2499.13	0.798305
$215$	0	0
$216$	3141.85	0.989703
$217$	1491.74	0.466662
$218$	−892.533	−0.277293
$219$	−1588.27	−0.490070
$220$	0	0
$221$	2725.77	0.829661
$222$	−1109.79	−0.335515
$223$	−5465.82	−1.64134	−0.820669	−	0.571404i	$-0.806399\pi$
$223$	−5465.82	−1.64134	−0.820669	+	0.571404i	$0.806399\pi$
$224$	1015.39	0.302875
$225$	0	0
$226$	837.006	0.246358
$227$	1866.81	0.545836	0.272918	−	0.962037i	$-0.412011\pi$
$227$	1866.81	0.545836	0.272918	+	0.962037i	$0.412011\pi$
$228$	−248.340	−0.0721347
$229$	456.437	0.131713	0.0658563	−	0.997829i	$-0.479022\pi$
$229$	456.437	0.131713	0.0658563	+	0.997829i	$0.479022\pi$
$230$	0	0
$231$	202.146	0.0575768
$232$	−2107.97	−0.596531
$233$	1268.44	0.356645	0.178323	−	0.983972i	$-0.442933\pi$
$233$	1268.44	0.356645	0.178323	+	0.983972i	$0.442933\pi$
$234$	−781.235	−0.218252
$235$	0	0
$236$	1666.98	0.459793
$237$	−3137.82	−0.860013
$238$	−1354.50	−0.368903
$239$	−5115.26	−1.38443	−0.692214	−	0.721692i	$-0.743365\pi$
$239$	−5115.26	−1.38443	−0.692214	+	0.721692i	$0.743365\pi$
$240$	0	0
$241$	−1509.15	−0.403372	−0.201686	−	0.979450i	$-0.564642\pi$
$241$	−1509.15	−0.403372	−0.201686	+	0.979450i	$0.564642\pi$
$242$	2265.86	0.601879
$243$	−3901.45	−1.02995
$244$	−3614.37	−0.948304
$245$	0	0
$246$	2084.24	0.540187
$247$	429.577	0.110661
$248$	−5897.94	−1.51016
$249$	−1822.35	−0.463801
$250$	0	0
$251$	−4695.51	−1.18079	−0.590394	−	0.807115i	$-0.701028\pi$
$251$	−4695.51	−1.18079	−0.590394	+	0.807115i	$0.701028\pi$
$252$	−480.481	−0.120109
$253$	−736.127	−0.182924
$254$	−2803.16	−0.692464
$255$	0	0
$256$	−4333.97	−1.05810
$257$	6383.64	1.54942	0.774710	−	0.632317i	$-0.217896\pi$
$257$	6383.64	1.54942	0.774710	+	0.632317i	$0.217896\pi$
$258$	2505.45	0.604583
$259$	1180.68	0.283258
$260$	0	0
$261$	1639.74	0.388878
$262$	−1485.62	−0.350314
$263$	3805.82	0.892308	0.446154	−	0.894956i	$-0.352793\pi$
$263$	3805.82	0.892308	0.446154	+	0.894956i	$0.352793\pi$
$264$	−799.234	−0.186324
$265$	0	0
$266$	−213.466	−0.0492048
$267$	266.643	0.0611171
$268$	974.287	0.222067
$269$	6390.72	1.44851	0.724255	−	0.689533i	$-0.242184\pi$
$269$	6390.72	1.44851	0.724255	+	0.689533i	$0.242184\pi$
$270$	0	0
$271$	−2526.33	−0.566286	−0.283143	−	0.959078i	$-0.591377\pi$
$271$	−2526.33	−0.566286	−0.283143	+	0.959078i	$0.591377\pi$
$272$	1087.65	0.242459
$273$	−396.835	−0.0879765
$274$	−2692.06	−0.593552
$275$	0	0
$276$	−835.415	−0.182196
$277$	−3156.63	−0.684706	−0.342353	−	0.939571i	$-0.611224\pi$
$277$	−3156.63	−0.684706	−0.342353	+	0.939571i	$0.611224\pi$
$278$	2678.17	0.577792
$279$	4587.86	0.984472
$280$	0	0
$281$	8144.80	1.72910	0.864552	−	0.502544i	$-0.167602\pi$
$281$	8144.80	1.72910	0.864552	+	0.502544i	$0.167602\pi$
$282$	1041.50	0.219932
$283$	3378.88	0.709729	0.354865	−	0.934918i	$-0.384527\pi$
$283$	3378.88	0.709729	0.354865	+	0.934918i	$0.384527\pi$
$284$	−1696.78	−0.354527
$285$	0	0
$286$	492.354	0.101795
$287$	−2217.36	−0.456052
$288$	3122.86	0.638946
$289$	9621.60	1.95840
$290$	0	0
$291$	−1549.71	−0.312185
$292$	−2379.20	−0.476822
$293$	−3191.30	−0.636307	−0.318154	−	0.948039i	$-0.603063\pi$
$293$	−3191.30	−0.636307	−0.318154	+	0.948039i	$0.603063\pi$
$294$	−1718.53	−0.340907
$295$	0	0
$296$	−4668.10	−0.916647
$297$	1540.25	0.300923
$298$	−6320.90	−1.22872
$299$	1445.10	0.279506
$300$	0	0
$301$	−2665.48	−0.510418
$302$	−4222.52	−0.804566
$303$	994.050	0.188471
$304$	171.413	0.0323394
$305$	0	0
$306$	−4165.77	−0.778240
$307$	−4313.64	−0.801931	−0.400965	−	0.916093i	$-0.631325\pi$
$307$	−4313.64	−0.801931	−0.400965	+	0.916093i	$0.631325\pi$
$308$	302.811	0.0560204
$309$	−2122.98	−0.390849
$310$	0	0
$311$	−296.966	−0.0541459	−0.0270730	−	0.999633i	$-0.508619\pi$
$311$	−296.966	−0.0541459	−0.0270730	+	0.999633i	$0.508619\pi$
$312$	1568.99	0.284700
$313$	74.6344	0.0134779	0.00673895	−	0.999977i	$-0.497855\pi$
$313$	74.6344	0.0134779	0.00673895	+	0.999977i	$0.497855\pi$
$314$	3305.19	0.594021
$315$	0	0
$316$	−4700.39	−0.836764
$317$	634.636	0.112444	0.0562220	−	0.998418i	$-0.482095\pi$
$317$	634.636	0.112444	0.0562220	+	0.998418i	$0.482095\pi$
$318$	−2037.94	−0.359378
$319$	−1033.40	−0.181378
$320$	0	0
$321$	3904.24	0.678858
$322$	−718.101	−0.124280
$323$	2290.63	0.394595
$324$	−435.292	−0.0746385
$325$	0	0
$326$	6508.96	1.10582
$327$	−1394.35	−0.235803
$328$	8766.88	1.47582
$329$	−1108.03	−0.185677
$330$	0	0
$331$	−1010.00	−0.167718	−0.0838588	−	0.996478i	$-0.526724\pi$
$331$	−1010.00	−0.167718	−0.0838588	+	0.996478i	$0.526724\pi$
$332$	−2729.84	−0.451263
$333$	3631.19	0.597562
$334$	−6506.21	−1.06588
$335$	0	0
$336$	−158.348	−0.0257101
$337$	3136.03	0.506916	0.253458	−	0.967346i	$-0.418432\pi$
$337$	3136.03	0.506916	0.253458	+	0.967346i	$0.418432\pi$
$338$	3187.55	0.512958
$339$	1307.60	0.209496
$340$	0	0
$341$	−2891.38	−0.459170
$342$	−656.519	−0.103803
$343$	3866.39	0.608646
$344$	10538.6	1.65176
$345$	0	0
$346$	5974.58	0.928311
$347$	11450.8	1.77150	0.885749	−	0.464164i	$-0.153645\pi$
$347$	11450.8	1.77150	0.885749	+	0.464164i	$0.153645\pi$
$348$	−1172.79	−0.180655
$349$	4403.78	0.675442	0.337721	−	0.941246i	$-0.390344\pi$
$349$	4403.78	0.675442	0.337721	+	0.941246i	$0.390344\pi$
$350$	0	0
$351$	−3023.68	−0.459806
$352$	−1968.10	−0.298012
$353$	−601.300	−0.0906629	−0.0453314	−	0.998972i	$-0.514434\pi$
$353$	−601.300	−0.0906629	−0.0453314	+	0.998972i	$0.514434\pi$
$354$	−2104.12	−0.315911
$355$	0	0
$356$	399.426	0.0594650
$357$	−2116.04	−0.313705
$358$	−3887.83	−0.573961
$359$	7924.45	1.16500	0.582502	−	0.812829i	$-0.302074\pi$
$359$	7924.45	1.16500	0.582502	+	0.812829i	$0.302074\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	233.566	0.0339115
$363$	3539.80	0.511822
$364$	−594.452	−0.0855982
$365$	0	0
$366$	4562.18	0.651554
$367$	2232.61	0.317552	0.158776	−	0.987315i	$-0.449245\pi$
$367$	2232.61	0.317552	0.158776	+	0.987315i	$0.449245\pi$
$368$	576.632	0.0816821
$369$	−6819.53	−0.962089
$370$	0	0
$371$	2168.12	0.303404
$372$	−3281.37	−0.457341
$373$	9994.88	1.38744	0.693720	−	0.720245i	$-0.255970\pi$
$373$	9994.88	1.38744	0.693720	+	0.720245i	$0.255970\pi$
$374$	2625.37	0.362981
$375$	0	0
$376$	4380.86	0.600867
$377$	2028.69	0.277142
$378$	1502.53	0.204449
$379$	13486.3	1.82782	0.913912	−	0.405913i	$-0.133046\pi$
$379$	13486.3	1.82782	0.913912	+	0.405913i	$0.133046\pi$
$380$	0	0
$381$	−4379.19	−0.588853
$382$	6390.58	0.855944
$383$	−5181.58	−0.691296	−0.345648	−	0.938364i	$-0.612341\pi$
$383$	−5181.58	−0.691296	−0.345648	+	0.938364i	$0.612341\pi$
$384$	−1830.46	−0.243255
$385$	0	0
$386$	−729.915	−0.0962480
$387$	−8197.73	−1.07678
$388$	−2321.44	−0.303745
$389$	−371.727	−0.0484507	−0.0242253	−	0.999707i	$-0.507712\pi$
$389$	−371.727	−0.0484507	−0.0242253	+	0.999707i	$0.507712\pi$
$390$	0	0
$391$	7705.68	0.996657
$392$	−7228.63	−0.931380
$393$	−2320.90	−0.297897
$394$	−6130.79	−0.783921
$395$	0	0
$396$	931.301	0.118181
$397$	4571.57	0.577936	0.288968	−	0.957339i	$-0.406688\pi$
$397$	4571.57	0.577936	0.288968	+	0.957339i	$0.406688\pi$
$398$	1422.84	0.179198
$399$	−333.485	−0.0418424
$400$	0	0
$401$	13606.8	1.69449	0.847245	−	0.531203i	$-0.178260\pi$
$401$	13606.8	1.69449	0.847245	+	0.531203i	$0.178260\pi$
$402$	−1229.78	−0.152577
$403$	5676.10	0.701605
$404$	1489.07	0.183376
$405$	0	0
$406$	−1008.10	−0.123229
$407$	−2288.47	−0.278710
$408$	8366.28	1.01518
$409$	−3182.34	−0.384735	−0.192368	−	0.981323i	$-0.561617\pi$
$409$	−3182.34	−0.384735	−0.192368	+	0.981323i	$0.561617\pi$
$410$	0	0
$411$	−4205.63	−0.504741
$412$	−3180.19	−0.380283
$413$	2238.51	0.266707
$414$	−2208.53	−0.262182
$415$	0	0
$416$	3863.61	0.455358
$417$	4183.94	0.491339
$418$	413.755	0.0484149
$419$	7452.18	0.868885	0.434442	−	0.900700i	$-0.356945\pi$
$419$	7452.18	0.868885	0.434442	+	0.900700i	$0.356945\pi$
$420$	0	0
$421$	−9082.58	−1.05144	−0.525722	−	0.850656i	$-0.676205\pi$
$421$	−9082.58	−1.05144	−0.525722	+	0.850656i	$0.676205\pi$
$422$	7556.38	0.871656
$423$	−3407.76	−0.391705
$424$	−8572.17	−0.981843
$425$	0	0
$426$	2141.74	0.243586
$427$	−4853.58	−0.550073
$428$	5848.48	0.660507
$429$	769.173	0.0865641
$430$	0	0
$431$	−10010.1	−1.11872	−0.559362	−	0.828923i	$-0.688954\pi$
$431$	−10010.1	−1.11872	−0.559362	+	0.828923i	$0.688954\pi$
$432$	−1206.53	−0.134373
$433$	8137.16	0.903111	0.451555	−	0.892243i	$-0.350869\pi$
$433$	8137.16	0.903111	0.451555	+	0.892243i	$0.350869\pi$
$434$	−2820.58	−0.311963
$435$	0	0
$436$	−2088.71	−0.229429
$437$	1214.40	0.132935
$438$	3003.10	0.327612
$439$	−5582.62	−0.606933	−0.303467	−	0.952842i	$-0.598144\pi$
$439$	−5582.62	−0.606933	−0.303467	+	0.952842i	$0.598144\pi$
$440$	0	0
$441$	5622.96	0.607166
$442$	−5153.90	−0.554629
$443$	−5823.52	−0.624568	−0.312284	−	0.949989i	$-0.601094\pi$
$443$	−5823.52	−0.624568	−0.312284	+	0.949989i	$0.601094\pi$
$444$	−2597.14	−0.277600
$445$	0	0
$446$	10334.8	1.09723
$447$	−9874.73	−1.04487
$448$	−2348.77	−0.247698
$449$	3999.42	0.420366	0.210183	−	0.977662i	$-0.432594\pi$
$449$	3999.42	0.420366	0.210183	+	0.977662i	$0.432594\pi$
$450$	0	0
$451$	4297.84	0.448730
$452$	1958.76	0.203833
$453$	−6596.58	−0.684182
$454$	−3529.78	−0.364892
$455$	0	0
$456$	1318.51	0.135406
$457$	−10800.0	−1.10548	−0.552739	−	0.833354i	$-0.686417\pi$
$457$	−10800.0	−1.10548	−0.552739	+	0.833354i	$0.686417\pi$
$458$	−863.033	−0.0880499
$459$	−16123.1	−1.63957
$460$	0	0
$461$	11879.6	1.20019	0.600096	−	0.799928i	$-0.295129\pi$
$461$	11879.6	1.20019	0.600096	+	0.799928i	$0.295129\pi$
$462$	−382.219	−0.0384901
$463$	−10252.9	−1.02914	−0.514569	−	0.857449i	$-0.672048\pi$
$463$	−10252.9	−1.02914	−0.514569	+	0.857449i	$0.672048\pi$
$464$	809.499	0.0809915
$465$	0	0
$466$	−2398.37	−0.238417
$467$	4190.63	0.415245	0.207622	−	0.978209i	$-0.433427\pi$
$467$	4190.63	0.415245	0.207622	+	0.978209i	$0.433427\pi$
$468$	−1828.25	−0.180579
$469$	1308.33	0.128812
$470$	0	0
$471$	5163.48	0.505140
$472$	−8850.51	−0.863089
$473$	5166.41	0.502224
$474$	5932.99	0.574919
$475$	0	0
$476$	−3169.79	−0.305225
$477$	6668.07	0.640063
$478$	9671.94	0.925490
$479$	−11777.5	−1.12344	−0.561721	−	0.827327i	$-0.689861\pi$
$479$	−11777.5	−1.12344	−0.561721	+	0.827327i	$0.689861\pi$
$480$	0	0
$481$	4492.52	0.425865
$482$	2853.50	0.269654
$483$	−1121.84	−0.105685
$484$	5302.56	0.497987
$485$	0	0
$486$	7376.88	0.688523
$487$	−17288.2	−1.60863	−0.804315	−	0.594203i	$-0.797468\pi$
$487$	−17288.2	−1.60863	−0.804315	+	0.594203i	$0.797468\pi$
$488$	19189.8	1.78009
$489$	10168.5	0.940361
$490$	0	0
$491$	−4426.69	−0.406872	−0.203436	−	0.979088i	$-0.565211\pi$
$491$	−4426.69	−0.406872	−0.203436	+	0.979088i	$0.565211\pi$
$492$	4877.53	0.446943
$493$	10817.5	0.988230
$494$	−812.246	−0.0739771
$495$	0	0
$496$	2264.91	0.205036
$497$	−2278.54	−0.205647
$498$	3445.70	0.310051
$499$	12876.9	1.15521	0.577606	−	0.816316i	$-0.303987\pi$
$499$	12876.9	1.15521	0.577606	+	0.816316i	$0.303987\pi$
$500$	0	0
$501$	−10164.2	−0.906396
$502$	8878.28	0.789357
$503$	3852.90	0.341535	0.170768	−	0.985311i	$-0.445375\pi$
$503$	3852.90	0.341535	0.170768	+	0.985311i	$0.445375\pi$
$504$	2551.03	0.225460
$505$	0	0
$506$	1391.87	0.122285
$507$	4979.71	0.436206
$508$	−6559.95	−0.572935
$509$	−20087.1	−1.74920	−0.874601	−	0.484844i	$-0.838876\pi$
$509$	−20087.1	−1.74920	−0.874601	+	0.484844i	$0.838876\pi$
$510$	0	0
$511$	−3194.92	−0.276585
$512$	3237.26	0.279430
$513$	−2540.98	−0.218688
$514$	−12070.2	−1.03579
$515$	0	0
$516$	5863.25	0.500224
$517$	2147.66	0.182696
$518$	−2232.43	−0.189358
$519$	9333.71	0.789411
$520$	0	0
$521$	21715.8	1.82608	0.913038	−	0.407874i	$-0.133730\pi$
$521$	21715.8	1.82608	0.913038	+	0.407874i	$0.133730\pi$
$522$	−3100.42	−0.259965
$523$	−3451.54	−0.288576	−0.144288	−	0.989536i	$-0.546089\pi$
$523$	−3451.54	−0.288576	−0.144288	+	0.989536i	$0.546089\pi$
$524$	−3476.66	−0.289845
$525$	0	0
$526$	−7196.06	−0.596508
$527$	30266.6	2.50177
$528$	306.920	0.0252973
$529$	−8081.75	−0.664235
$530$	0	0
$531$	6884.59	0.562647
$532$	−499.554	−0.0407113
$533$	−8437.13	−0.685653
$534$	−504.169	−0.0408568
$535$	0	0
$536$	−5172.79	−0.416848
$537$	−6073.70	−0.488081
$538$	−12083.6	−0.968329
$539$	−3543.73	−0.283190
$540$	0	0
$541$	−5540.30	−0.440288	−0.220144	−	0.975467i	$-0.570653\pi$
$541$	−5540.30	−0.440288	−0.220144	+	0.975467i	$0.570653\pi$
$542$	4776.79	0.378562
$543$	364.885	0.0288374
$544$	20601.9	1.62371
$545$	0	0
$546$	750.338	0.0588123
$547$	463.137	0.0362016	0.0181008	−	0.999836i	$-0.494238\pi$
$547$	463.137	0.0362016	0.0181008	+	0.999836i	$0.494238\pi$
$548$	−6299.96	−0.491096
$549$	−14927.3	−1.16044
$550$	0	0
$551$	1704.83	0.131811
$552$	4435.48	0.342005
$553$	−6311.95	−0.485374
$554$	5968.57	0.457726
$555$	0	0
$556$	6267.46	0.478057
$557$	−2283.84	−0.173733	−0.0868667	−	0.996220i	$-0.527685\pi$
$557$	−2283.84	−0.173733	−0.0868667	+	0.996220i	$0.527685\pi$
$558$	−8674.73	−0.658120
$559$	−10142.2	−0.767390
$560$	0	0
$561$	4101.45	0.308669
$562$	−15400.2	−1.15591
$563$	−12065.7	−0.903211	−0.451606	−	0.892218i	$-0.649149\pi$
$563$	−12065.7	−0.903211	−0.451606	+	0.892218i	$0.649149\pi$
$564$	2437.33	0.181968
$565$	0	0
$566$	−6388.79	−0.474454
$567$	−584.535	−0.0432948
$568$	9008.76	0.665492
$569$	4496.79	0.331309	0.165655	−	0.986184i	$-0.447026\pi$
$569$	4496.79	0.331309	0.165655	+	0.986184i	$0.447026\pi$
$570$	0	0
$571$	−8884.67	−0.651159	−0.325580	−	0.945515i	$-0.605559\pi$
$571$	−8884.67	−0.651159	−0.325580	+	0.945515i	$0.605559\pi$
$572$	1152.21	0.0842241
$573$	9983.60	0.727872
$574$	4192.60	0.304871
$575$	0	0
$576$	−7223.66	−0.522545
$577$	−8760.77	−0.632089	−0.316045	−	0.948744i	$-0.602355\pi$
$577$	−8760.77	−0.632089	−0.316045	+	0.948744i	$0.602355\pi$
$578$	−18192.6	−1.30919
$579$	−1140.30	−0.0818467
$580$	0	0
$581$	−3665.79	−0.261760
$582$	2930.20	0.208695
$583$	−4202.39	−0.298533
$584$	12631.9	0.895055
$585$	0	0
$586$	6034.13	0.425371
$587$	−4691.92	−0.329909	−0.164954	−	0.986301i	$-0.552748\pi$
$587$	−4691.92	−0.329909	−0.164954	+	0.986301i	$0.552748\pi$
$588$	−4021.71	−0.282062
$589$	4769.97	0.333690
$590$	0	0
$591$	−9577.75	−0.666626
$592$	1792.63	0.124454
$593$	−9399.23	−0.650894	−0.325447	−	0.945560i	$-0.605515\pi$
$593$	−9399.23	−0.650894	−0.325447	+	0.945560i	$0.605515\pi$
$594$	−2912.31	−0.201167
$595$	0	0
$596$	−14792.2	−1.01663
$597$	2222.82	0.152385
$598$	−2732.40	−0.186850
$599$	15981.9	1.09016	0.545078	−	0.838386i	$-0.316500\pi$
$599$	15981.9	1.09016	0.545078	+	0.838386i	$0.316500\pi$
$600$	0	0
$601$	−24969.1	−1.69470	−0.847348	−	0.531038i	$-0.821802\pi$
$601$	−24969.1	−1.69470	−0.847348	+	0.531038i	$0.821802\pi$
$602$	5039.90	0.341214
$603$	4023.78	0.271743
$604$	−9881.55	−0.665687
$605$	0	0
$606$	−1879.55	−0.125993
$607$	12651.2	0.845956	0.422978	−	0.906140i	$-0.360985\pi$
$607$	12651.2	0.845956	0.422978	+	0.906140i	$0.360985\pi$
$608$	3246.82	0.216573
$609$	−1574.89	−0.104791
$610$	0	0
$611$	−4216.09	−0.279156
$612$	−9748.74	−0.643904
$613$	−11851.0	−0.780847	−0.390424	−	0.920635i	$-0.627671\pi$
$613$	−11851.0	−0.780847	−0.390424	+	0.920635i	$0.627671\pi$
$614$	8156.26	0.536091
$615$	0	0
$616$	−1607.72	−0.105157
$617$	−19473.2	−1.27060	−0.635301	−	0.772265i	$-0.719124\pi$
$617$	−19473.2	−1.27060	−0.635301	+	0.772265i	$0.719124\pi$
$618$	4014.14	0.261282
$619$	−25664.7	−1.66648	−0.833240	−	0.552912i	$-0.813517\pi$
$619$	−25664.7	−1.66648	−0.833240	+	0.552912i	$0.813517\pi$
$620$	0	0
$621$	−8547.85	−0.552357
$622$	561.504	0.0361966
$623$	536.372	0.0344933
$624$	−602.518	−0.0386539
$625$	0	0
$626$	−141.119	−0.00900998
$627$	646.383	0.0411707
$628$	7734.80	0.491485
$629$	23955.4	1.51854
$630$	0	0
$631$	−19618.1	−1.23769	−0.618846	−	0.785512i	$-0.712399\pi$
$631$	−19618.1	−1.23769	−0.618846	+	0.785512i	$0.712399\pi$
$632$	24955.8	1.57071
$633$	11804.8	0.741233
$634$	−1199.97	−0.0751688
$635$	0	0
$636$	−4769.20	−0.297344
$637$	6956.74	0.432709
$638$	1953.96	0.121251
$639$	−7007.68	−0.433834
$640$	0	0
$641$	−683.277	−0.0421027	−0.0210513	−	0.999778i	$-0.506701\pi$
$641$	−683.277	−0.0421027	−0.0210513	+	0.999778i	$0.506701\pi$
$642$	−7382.15	−0.453816
$643$	−7107.06	−0.435886	−0.217943	−	0.975961i	$-0.569935\pi$
$643$	−7107.06	−0.435886	−0.217943	+	0.975961i	$0.569935\pi$
$644$	−1680.50	−0.102828
$645$	0	0
$646$	−4331.13	−0.263787
$647$	−23626.1	−1.43561	−0.717805	−	0.696244i	$-0.754853\pi$
$647$	−23626.1	−1.43561	−0.717805	+	0.696244i	$0.754853\pi$
$648$	2311.10	0.140106
$649$	−4338.84	−0.262426
$650$	0	0
$651$	−4406.41	−0.265285
$652$	15232.3	0.914941
$653$	−1155.57	−0.0692508	−0.0346254	−	0.999400i	$-0.511024\pi$
$653$	−1155.57	−0.0692508	−0.0346254	+	0.999400i	$0.511024\pi$
$654$	2636.44	0.157634
$655$	0	0
$656$	−3366.64	−0.200374
$657$	−9826.04	−0.583486
$658$	2095.07	0.124125
$659$	10881.9	0.643246	0.321623	−	0.946868i	$-0.395772\pi$
$659$	10881.9	0.643246	0.321623	+	0.946868i	$0.395772\pi$
$660$	0	0
$661$	18028.3	1.06085	0.530425	−	0.847732i	$-0.322032\pi$
$661$	18028.3	1.06085	0.530425	+	0.847732i	$0.322032\pi$
$662$	1909.71	0.112119
$663$	−8051.60	−0.471642
$664$	14493.6	0.847078
$665$	0	0
$666$	−6865.87	−0.399470
$667$	5735.04	0.332926
$668$	−15225.8	−0.881894
$669$	16145.4	0.933059
$670$	0	0
$671$	9407.53	0.541242
$672$	−2999.36	−0.172177
$673$	8571.78	0.490963	0.245481	−	0.969401i	$-0.421054\pi$
$673$	8571.78	0.490963	0.245481	+	0.969401i	$0.421054\pi$
$674$	−5929.62	−0.338873
$675$	0	0
$676$	7459.51	0.424415
$677$	19800.9	1.12409	0.562045	−	0.827107i	$-0.310015\pi$
$677$	19800.9	1.12409	0.562045	+	0.827107i	$0.310015\pi$
$678$	−2472.42	−0.140048
$679$	−3117.36	−0.176191
$680$	0	0
$681$	−5514.35	−0.310294
$682$	5467.03	0.306955
$683$	−11953.7	−0.669686	−0.334843	−	0.942274i	$-0.608683\pi$
$683$	−11953.7	−0.669686	−0.334843	+	0.942274i	$0.608683\pi$
$684$	−1536.39	−0.0858848
$685$	0	0
$686$	−7310.59	−0.406880
$687$	−1348.26	−0.0748753
$688$	−4047.02	−0.224260
$689$	8249.75	0.456154
$690$	0	0
$691$	15664.0	0.862355	0.431177	−	0.902267i	$-0.358098\pi$
$691$	15664.0	0.862355	0.431177	+	0.902267i	$0.358098\pi$
$692$	13981.7	0.768071
$693$	1250.60	0.0685520
$694$	−21651.2	−1.18425
$695$	0	0
$696$	6226.70	0.339113
$697$	−44989.2	−2.44489
$698$	−8326.69	−0.451533
$699$	−3746.82	−0.202744
$700$	0	0
$701$	27240.6	1.46771	0.733855	−	0.679306i	$-0.237719\pi$
$701$	27240.6	1.46771	0.733855	+	0.679306i	$0.237719\pi$
$702$	5717.18	0.307380
$703$	3775.33	0.202545
$704$	4552.53	0.243722
$705$	0	0
$706$	1136.94	0.0606081
$707$	1999.61	0.106369
$708$	−4924.06	−0.261381
$709$	−30706.2	−1.62651	−0.813256	−	0.581906i	$-0.802307\pi$
$709$	−30706.2	−1.62651	−0.813256	+	0.581906i	$0.802307\pi$
$710$	0	0
$711$	−19412.5	−1.02395
$712$	−2120.68	−0.111623
$713$	16046.2	0.842825
$714$	4001.02	0.209712
$715$	0	0
$716$	−9098.30	−0.474887
$717$	15109.9	0.787013
$718$	−14983.6	−0.778806
$719$	−10948.3	−0.567874	−0.283937	−	0.958843i	$-0.591641\pi$
$719$	−10948.3	−0.567874	−0.283937	+	0.958843i	$0.591641\pi$
$720$	0	0
$721$	−4270.54	−0.220587
$722$	−682.580	−0.0351842
$723$	4457.84	0.229307
$724$	546.591	0.0280579
$725$	0	0
$726$	−6693.07	−0.342153
$727$	32291.1	1.64733	0.823666	−	0.567076i	$-0.191925\pi$
$727$	32291.1	1.64733	0.823666	+	0.567076i	$0.191925\pi$
$728$	3156.13	0.160679
$729$	8868.32	0.450558
$730$	0	0
$731$	−54081.3	−2.73635
$732$	10676.4	0.539087
$733$	12538.5	0.631813	0.315907	−	0.948790i	$-0.397691\pi$
$733$	12538.5	0.631813	0.315907	+	0.948790i	$0.397691\pi$
$734$	−4221.43	−0.212283
$735$	0	0
$736$	10922.3	0.547013
$737$	−2535.89	−0.126744
$738$	12894.4	0.643156
$739$	7341.82	0.365458	0.182729	−	0.983163i	$-0.441507\pi$
$739$	7341.82	0.365458	0.182729	+	0.983163i	$0.441507\pi$
$740$	0	0
$741$	−1268.92	−0.0629082
$742$	−4099.48	−0.202826
$743$	−10142.9	−0.500819	−0.250410	−	0.968140i	$-0.580565\pi$
$743$	−10142.9	−0.500819	−0.250410	+	0.968140i	$0.580565\pi$
$744$	17421.8	0.858488
$745$	0	0
$746$	−18898.4	−0.927504
$747$	−11274.2	−0.552210
$748$	6143.90	0.300325
$749$	7853.67	0.383133
$750$	0	0
$751$	17843.5	0.867000	0.433500	−	0.901153i	$-0.357278\pi$
$751$	17843.5	0.867000	0.433500	+	0.901153i	$0.357278\pi$
$752$	−1682.33	−0.0815801
$753$	13870.0	0.671248
$754$	−3835.85	−0.185270
$755$	0	0
$756$	3516.23	0.169159
$757$	8510.80	0.408627	0.204313	−	0.978906i	$-0.434504\pi$
$757$	8510.80	0.408627	0.204313	+	0.978906i	$0.434504\pi$
$758$	−25500.0	−1.22190
$759$	2174.43	0.103988
$760$	0	0
$761$	11227.3	0.534809	0.267404	−	0.963584i	$-0.413834\pi$
$761$	11227.3	0.534809	0.267404	+	0.963584i	$0.413834\pi$
$762$	8280.20	0.393648
$763$	−2804.84	−0.133082
$764$	14955.2	0.708196
$765$	0	0
$766$	9797.35	0.462131
$767$	8517.62	0.400982
$768$	12802.0	0.601502
$769$	7372.60	0.345725	0.172863	−	0.984946i	$-0.444698\pi$
$769$	7372.60	0.345725	0.172863	+	0.984946i	$0.444698\pi$
$770$	0	0
$771$	−18856.5	−0.880806
$772$	−1708.15	−0.0796342
$773$	−2922.48	−0.135982	−0.0679910	−	0.997686i	$-0.521659\pi$
$773$	−2922.48	−0.135982	−0.0679910	+	0.997686i	$0.521659\pi$
$774$	15500.3	0.719827
$775$	0	0
$776$	12325.3	0.570169
$777$	−3487.58	−0.161025
$778$	702.863	0.0323893
$779$	−7090.24	−0.326103
$780$	0	0
$781$	4416.42	0.202345
$782$	−14569.9	−0.666265
$783$	−11999.8	−0.547686
$784$	2775.92	0.126454
$785$	0	0
$786$	4388.36	0.199144
$787$	7067.94	0.320133	0.160067	−	0.987106i	$-0.448829\pi$
$787$	7067.94	0.320133	0.160067	+	0.987106i	$0.448829\pi$
$788$	−14347.3	−0.648605
$789$	−11241.9	−0.507255
$790$	0	0
$791$	2630.34	0.118235
$792$	−4944.57	−0.221840
$793$	−18468.0	−0.827010
$794$	−8643.94	−0.386350
$795$	0	0
$796$	3329.74	0.148266
$797$	41550.9	1.84669	0.923343	−	0.383975i	$-0.125445\pi$
$797$	41550.9	1.84669	0.923343	+	0.383975i	$0.125445\pi$
$798$	630.555	0.0279717
$799$	−22481.4	−0.995412
$800$	0	0
$801$	1649.62	0.0727672
$802$	−25727.8	−1.13277
$803$	6192.61	0.272145
$804$	−2877.93	−0.126240
$805$	0	0
$806$	−10732.4	−0.469023
$807$	−18877.4	−0.823441
$808$	−7905.94	−0.344220
$809$	−15664.4	−0.680753	−0.340377	−	0.940289i	$-0.610555\pi$
$809$	−15664.4	−0.680753	−0.340377	+	0.940289i	$0.610555\pi$
$810$	0	0
$811$	16913.6	0.732329	0.366164	−	0.930550i	$-0.380671\pi$
$811$	16913.6	0.732329	0.366164	+	0.930550i	$0.380671\pi$
$812$	−2359.15	−0.101958
$813$	7462.47	0.321919
$814$	4327.04	0.186318
$815$	0	0
$816$	−3212.80	−0.137832
$817$	−8523.14	−0.364978
$818$	6017.18	0.257195
$819$	−2455.08	−0.104746
$820$	0	0
$821$	17022.9	0.723632	0.361816	−	0.932249i	$-0.382157\pi$
$821$	17022.9	0.723632	0.361816	+	0.932249i	$0.382157\pi$
$822$	7952.02	0.337419
$823$	−22976.8	−0.973173	−0.486586	−	0.873632i	$-0.661758\pi$
$823$	−22976.8	−0.973173	−0.486586	+	0.873632i	$0.661758\pi$
$824$	16884.6	0.713839
$825$	0	0
$826$	−4232.59	−0.178294
$827$	−597.155	−0.0251090	−0.0125545	−	0.999921i	$-0.503996\pi$
$827$	−597.155	−0.0251090	−0.0125545	+	0.999921i	$0.503996\pi$
$828$	−5168.41	−0.216926
$829$	−24800.2	−1.03902	−0.519510	−	0.854464i	$-0.673885\pi$
$829$	−24800.2	−1.03902	−0.519510	+	0.854464i	$0.673885\pi$
$830$	0	0
$831$	9324.32	0.389238
$832$	−8937.12	−0.372403
$833$	37095.3	1.54295
$834$	−7911.00	−0.328460
$835$	0	0
$836$	968.269	0.0400578
$837$	−33574.5	−1.38651
$838$	−14090.6	−0.580850
$839$	−8479.08	−0.348904	−0.174452	−	0.984666i	$-0.555815\pi$
$839$	−8479.08	−0.348904	−0.174452	+	0.984666i	$0.555815\pi$
$840$	0	0
$841$	−16337.9	−0.669889
$842$	17173.4	0.702890
$843$	−24058.8	−0.982952
$844$	17683.4	0.721196
$845$	0	0
$846$	6443.41	0.261855
$847$	7120.58	0.288862
$848$	3291.87	0.133306
$849$	−9980.80	−0.403463
$850$	0	0
$851$	12700.2	0.511584
$852$	5012.10	0.201540
$853$	26344.7	1.05748	0.528738	−	0.848785i	$-0.322666\pi$
$853$	26344.7	1.05748	0.528738	+	0.848785i	$0.322666\pi$
$854$	9177.17	0.367724
$855$	0	0
$856$	−31051.4	−1.23985
$857$	22575.2	0.899829	0.449915	−	0.893072i	$-0.351454\pi$
$857$	22575.2	0.899829	0.449915	+	0.893072i	$0.351454\pi$
$858$	−1454.35	−0.0578681
$859$	−6824.44	−0.271067	−0.135534	−	0.990773i	$-0.543275\pi$
$859$	−6824.44	−0.271067	−0.135534	+	0.990773i	$0.543275\pi$
$860$	0	0
$861$	6549.83	0.259254
$862$	18927.2	0.747867
$863$	−11178.9	−0.440944	−0.220472	−	0.975393i	$-0.570760\pi$
$863$	−11178.9	−0.440944	−0.220472	+	0.975393i	$0.570760\pi$
$864$	−22853.5	−0.899875
$865$	0	0
$866$	−15385.8	−0.603730
$867$	−28421.1	−1.11330
$868$	−6600.72	−0.258114
$869$	12234.2	0.477582
$870$	0	0
$871$	4978.23	0.193663
$872$	11089.6	0.430667
$873$	−9587.50	−0.371693
$874$	−2296.20	−0.0888674
$875$	0	0
$876$	7027.87	0.271061
$877$	−29614.6	−1.14027	−0.570133	−	0.821552i	$-0.693108\pi$
$877$	−29614.6	−1.14027	−0.570133	+	0.821552i	$0.693108\pi$
$878$	10555.6	0.405735
$879$	9426.74	0.361725
$880$	0	0
$881$	−44074.5	−1.68548	−0.842739	−	0.538322i	$-0.819059\pi$
$881$	−44074.5	−1.68548	−0.842739	+	0.538322i	$0.819059\pi$
$882$	−10631.9	−0.405890
$883$	11301.5	0.430720	0.215360	−	0.976535i	$-0.430907\pi$
$883$	11301.5	0.430720	0.215360	+	0.976535i	$0.430907\pi$
$884$	−12061.2	−0.458892
$885$	0	0
$886$	11011.1	0.417524
$887$	11839.8	0.448187	0.224093	−	0.974568i	$-0.428058\pi$
$887$	11839.8	0.448187	0.224093	+	0.974568i	$0.428058\pi$
$888$	13789.0	0.521091
$889$	−8809.08	−0.332336
$890$	0	0
$891$	1132.98	0.0425998
$892$	24185.5	0.907837
$893$	−3543.03	−0.132769
$894$	18671.2	0.698499
$895$	0	0
$896$	−3682.10	−0.137288
$897$	−4268.65	−0.158892
$898$	−7562.13	−0.281015
$899$	22526.3	0.835699
$900$	0	0
$901$	43990.0	1.62655
$902$	−8126.37	−0.299976
$903$	7873.51	0.290160
$904$	−10399.7	−0.382620
$905$	0	0
$906$	12472.8	0.457375
$907$	30147.6	1.10368	0.551838	−	0.833952i	$-0.313927\pi$
$907$	30147.6	1.10368	0.551838	+	0.833952i	$0.313927\pi$
$908$	−8260.40	−0.301906
$909$	6149.82	0.224397
$910$	0	0
$911$	42005.9	1.52768	0.763841	−	0.645405i	$-0.223311\pi$
$911$	42005.9	1.52768	0.763841	+	0.645405i	$0.223311\pi$
$912$	−506.333	−0.0183842
$913$	7105.27	0.257558
$914$	20420.7	0.739012
$915$	0	0
$916$	−2019.67	−0.0728513
$917$	−4668.66	−0.168127
$918$	30485.7	1.09605
$919$	41588.7	1.49280	0.746401	−	0.665497i	$-0.231780\pi$
$919$	41588.7	1.49280	0.746401	+	0.665497i	$0.231780\pi$
$920$	0	0
$921$	12742.0	0.455877
$922$	−22462.0	−0.802328
$923$	−8669.91	−0.309181
$924$	−894.469	−0.0318462
$925$	0	0
$926$	19386.2	0.687979
$927$	−13134.1	−0.465352
$928$	15333.2	0.542388
$929$	−32440.9	−1.14570	−0.572848	−	0.819662i	$-0.694161\pi$
$929$	−32440.9	−1.14570	−0.572848	+	0.819662i	$0.694161\pi$
$930$	0	0
$931$	5846.17	0.205801
$932$	−5612.67	−0.197263
$933$	877.202	0.0307806
$934$	−7923.66	−0.277591
$935$	0	0
$936$	9706.74	0.338969
$937$	−3765.99	−0.131302	−0.0656508	−	0.997843i	$-0.520912\pi$
$937$	−3765.99	−0.131302	−0.0656508	+	0.997843i	$0.520912\pi$
$938$	−2473.79	−0.0861111
$939$	−220.461	−0.00766185
$940$	0	0
$941$	6311.38	0.218645	0.109323	−	0.994006i	$-0.465132\pi$
$941$	6311.38	0.218645	0.109323	+	0.994006i	$0.465132\pi$
$942$	−9763.14	−0.337686
$943$	−23851.6	−0.823662
$944$	3398.75	0.117182
$945$	0	0
$946$	−9768.67	−0.335737
$947$	−29630.9	−1.01676	−0.508381	−	0.861132i	$-0.669756\pi$
$947$	−29630.9	−1.01676	−0.508381	+	0.861132i	$0.669756\pi$
$948$	13884.4	0.475680
$949$	−12156.8	−0.415833
$950$	0	0
$951$	−1874.64	−0.0639216
$952$	16829.4	0.572946
$953$	23155.3	0.787065	0.393532	−	0.919311i	$-0.371253\pi$
$953$	23155.3	0.787065	0.393532	+	0.919311i	$0.371253\pi$
$954$	−12608.0	−0.427882
$955$	0	0
$956$	22634.3	0.765738
$957$	3052.55	0.103109
$958$	22269.0	0.751021
$959$	−8459.95	−0.284865
$960$	0	0
$961$	33235.7	1.11563
$962$	−8494.47	−0.284691
$963$	24154.1	0.808260
$964$	6677.76	0.223108
$965$	0	0
$966$	2121.19	0.0706501
$967$	−16357.8	−0.543984	−0.271992	−	0.962300i	$-0.587682\pi$
$967$	−16357.8	−0.543984	−0.271992	+	0.962300i	$0.587682\pi$
$968$	−28153.0	−0.934783
$969$	−6766.25	−0.224317
$970$	0	0
$971$	−32741.7	−1.08211	−0.541055	−	0.840987i	$-0.681975\pi$
$971$	−32741.7	−1.08211	−0.541055	+	0.840987i	$0.681975\pi$
$972$	17263.4	0.569674
$973$	8416.31	0.277302
$974$	32688.6	1.07537
$975$	0	0
$976$	−7369.23	−0.241684
$977$	8985.92	0.294253	0.147126	−	0.989118i	$-0.452998\pi$
$977$	8985.92	0.294253	0.147126	+	0.989118i	$0.452998\pi$
$978$	−19226.7	−0.628632
$979$	−1039.63	−0.0339395
$980$	0	0
$981$	−8626.31	−0.280751
$982$	8370.01	0.271994
$983$	10051.2	0.326128	0.163064	−	0.986615i	$-0.447862\pi$
$983$	10051.2	0.326128	0.163064	+	0.986615i	$0.447862\pi$
$984$	−25896.3	−0.838969
$985$	0	0
$986$	−20453.8	−0.660632
$987$	3272.99	0.105553
$988$	−1900.82	−0.0612076
$989$	−28671.8	−0.921852
$990$	0	0
$991$	−33527.0	−1.07469	−0.537346	−	0.843362i	$-0.680573\pi$
$991$	−33527.0	−1.07469	−0.537346	+	0.843362i	$0.680573\pi$
$992$	42901.0	1.37309
$993$	2983.41	0.0953432
$994$	4308.27	0.137475
$995$	0	0
$996$	8063.63	0.256532
$997$	−38204.5	−1.21359	−0.606795	−	0.794858i	$-0.707545\pi$
$997$	−38204.5	−1.21359	−0.606795	+	0.794858i	$0.707545\pi$
$998$	−24347.8	−0.772260
$999$	−26573.5	−0.841591

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.4.a.f.1.2		3
5.2	odd	4		475.4.b.f.324.3		6
5.3	odd	4		475.4.b.f.324.4		6
5.4	even	2		19.4.a.b.1.2	✓	3
15.14	odd	2		171.4.a.f.1.2		3
20.19	odd	2		304.4.a.i.1.2		3
35.34	odd	2		931.4.a.c.1.2		3
40.19	odd	2		1216.4.a.u.1.2		3
40.29	even	2		1216.4.a.s.1.2		3
55.54	odd	2		2299.4.a.h.1.2		3
95.94	odd	2		361.4.a.i.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.4.a.b.1.2	✓	3	5.4	even	2
171.4.a.f.1.2		3	15.14	odd	2
304.4.a.i.1.2		3	20.19	odd	2
361.4.a.i.1.2		3	95.94	odd	2
475.4.a.f.1.2		3	1.1	even	1	trivial
475.4.b.f.324.3		6	5.2	odd	4
475.4.b.f.324.4		6	5.3	odd	4
931.4.a.c.1.2		3	35.34	odd	2
1216.4.a.s.1.2		3	40.29	even	2
1216.4.a.u.1.2		3	40.19	odd	2
2299.4.a.h.1.2		3	55.54	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	475.a (trivial)

\(f(q)\)	\(=\)	\(q-1.89080 q^{2} -2.95388 q^{3} -4.42486 q^{4} +5.58521 q^{6} -5.94196 q^{7} +23.4930 q^{8} -18.2746 q^{9} +11.5171 q^{11} +13.0705 q^{12} -22.6093 q^{13} +11.2351 q^{14} -9.02172 q^{16} -120.560 q^{17} +34.5537 q^{18} -19.0000 q^{19} +17.5518 q^{21} -21.7766 q^{22} -63.9160 q^{23} -69.3955 q^{24} +42.7498 q^{26} +133.736 q^{27} +26.2923 q^{28} -89.7278 q^{29} -251.051 q^{31} -170.885 q^{32} -34.0201 q^{33} +227.954 q^{34} +80.8625 q^{36} -198.702 q^{37} +35.9253 q^{38} +66.7853 q^{39} +373.170 q^{41} -33.1871 q^{42} +448.586 q^{43} -50.9616 q^{44} +120.853 q^{46} +186.475 q^{47} +26.6491 q^{48} -307.693 q^{49} +356.119 q^{51} +100.043 q^{52} -364.882 q^{53} -252.868 q^{54} -139.594 q^{56} +56.1238 q^{57} +169.658 q^{58} -376.730 q^{59} +816.832 q^{61} +474.689 q^{62} +108.587 q^{63} +395.285 q^{64} +64.3254 q^{66} -220.185 q^{67} +533.459 q^{68} +188.800 q^{69} +383.466 q^{71} -429.324 q^{72} +537.689 q^{73} +375.706 q^{74} +84.0724 q^{76} -68.4341 q^{77} -126.278 q^{78} +1062.27 q^{79} +98.3741 q^{81} -705.592 q^{82} +616.932 q^{83} -77.6645 q^{84} -848.189 q^{86} +265.045 q^{87} +270.571 q^{88} -90.2686 q^{89} +134.344 q^{91} +282.819 q^{92} +741.576 q^{93} -352.589 q^{94} +504.776 q^{96} +524.636 q^{97} +581.787 q^{98} -210.470 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} - q^{3} + 21 q^{4} - 65 q^{6} + 35 q^{7} - 27 q^{8} + 48 q^{9} + 16 q^{11} + 115 q^{12} - 65 q^{13} + 37 q^{14} + 33 q^{16} - 29 q^{17} - 138 q^{18} - 57 q^{19} - 25 q^{21} - 118 q^{22} + 101 q^{23}+ \cdots + 250 q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 - q^3 + 21 * q^4 - 65 * q^6 + 35 * q^7 - 27 * q^8 + 48 * q^9 + 16 * q^11 + 115 * q^12 - 65 * q^13 + 37 * q^14 + 33 * q^16 - 29 * q^17 - 138 * q^18 - 57 * q^19 - 25 * q^21 - 118 * q^22 + 101 * q^23 - 489 * q^24 + 299 * q^26 + 377 * q^27 + 493 * q^28 + 377 * q^29 - 140 * q^31 - 579 * q^32 + 130 * q^33 + 329 * q^34 + 1074 * q^36 + 290 * q^37 + 57 * q^38 - 371 * q^39 + 956 * q^41 - 1385 * q^42 + 570 * q^43 + 110 * q^44 - 239 * q^46 - 66 * q^47 + 1411 * q^48 - 98 * q^49 + 187 * q^51 - 697 * q^52 - 817 * q^53 - 947 * q^54 - 915 * q^56 + 19 * q^57 - 647 * q^58 + 265 * q^59 + 988 * q^61 + 1436 * q^62 + 1304 * q^63 - 159 * q^64 - 274 * q^66 + 207 * q^67 + 1529 * q^68 + 807 * q^69 + 846 * q^71 - 2826 * q^72 - 627 * q^73 - 770 * q^74 - 399 * q^76 - 88 * q^77 + 1817 * q^78 + 382 * q^79 - 501 * q^81 - 872 * q^82 + 766 * q^83 + 907 * q^84 + 2592 * q^86 + 1671 * q^87 - 342 * q^88 - 172 * q^89 - 457 * q^91 + 2677 * q^92 - 892 * q^93 - 96 * q^94 + 863 * q^96 + 2450 * q^97 + 2450 * q^98 + 250 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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