LMFDB - Embedded newform 475.4.a.f.1.1

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			1.1
Root			$-0.526440$ 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)

$f(q)$	$=$	\(q-5.07177 q^{2} +8.66998 q^{3} +17.7229 q^{4} -43.9722 q^{6} +15.1058 q^{7} -49.3121 q^{8} +48.1686 q^{9} +12.6171 q^{11} +153.657 q^{12} -46.9571 q^{13} -76.6130 q^{14} +108.317 q^{16} +28.9745 q^{17} -244.300 q^{18} -19.0000 q^{19} +130.967 q^{21} -63.9911 q^{22} +112.166 q^{23} -427.535 q^{24} +238.155 q^{26} +183.531 q^{27} +267.717 q^{28} +295.107 q^{29} -57.6979 q^{31} -154.862 q^{32} +109.390 q^{33} -146.952 q^{34} +853.685 q^{36} +341.167 q^{37} +96.3636 q^{38} -407.117 q^{39} +274.056 q^{41} -664.233 q^{42} -327.536 q^{43} +223.611 q^{44} -568.879 q^{46} -139.140 q^{47} +939.106 q^{48} -114.816 q^{49} +251.209 q^{51} -832.214 q^{52} -296.715 q^{53} -930.828 q^{54} -744.897 q^{56} -164.730 q^{57} -1496.72 q^{58} +459.383 q^{59} -232.911 q^{61} +292.631 q^{62} +727.623 q^{63} -81.1127 q^{64} -554.801 q^{66} +320.784 q^{67} +513.511 q^{68} +972.475 q^{69} -9.54518 q^{71} -2375.30 q^{72} -320.868 q^{73} -1730.32 q^{74} -336.734 q^{76} +190.591 q^{77} +2064.80 q^{78} -89.2323 q^{79} +290.661 q^{81} -1389.95 q^{82} +439.455 q^{83} +2321.10 q^{84} +1661.19 q^{86} +2558.58 q^{87} -622.176 q^{88} +883.164 q^{89} -709.322 q^{91} +1987.90 q^{92} -500.240 q^{93} +705.688 q^{94} -1342.65 q^{96} +1705.87 q^{97} +582.321 q^{98} +607.748 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−5.07177	−1.79314	−0.896571	−	0.442900i	$-0.853950\pi$
$2$	−5.07177	−1.79314	−0.896571	+	0.442900i	$0.853950\pi$
$3$	8.66998	1.66854	0.834269	−	0.551357i	$-0.185890\pi$
$3$	8.66998	1.66854	0.834269	+	0.551357i	$0.185890\pi$
$4$	17.7229	2.21536
$5$	0	0
$5$	0	0
$6$	−43.9722	−2.99193
$7$	15.1058	0.815634	0.407817	−	0.913064i	$-0.366290\pi$
$7$	15.1058	0.815634	0.407817	+	0.913064i	$0.366290\pi$
$8$	−49.3121	−2.17931
$9$	48.1686	1.78402
$10$	0	0
$11$	12.6171	0.345836	0.172918	−	0.984936i	$-0.444680\pi$
$11$	12.6171	0.345836	0.172918	+	0.984936i	$0.444680\pi$
$12$	153.657	3.69641
$13$	−46.9571	−1.00181	−0.500906	−	0.865502i	$-0.667000\pi$
$13$	−46.9571	−1.00181	−0.500906	+	0.865502i	$0.667000\pi$
$14$	−76.6130	−1.46255
$15$	0	0
$16$	108.317	1.69245
$17$	28.9745	0.413374	0.206687	−	0.978407i	$-0.433732\pi$
$17$	28.9745	0.413374	0.206687	+	0.978407i	$0.433732\pi$
$18$	−244.300	−3.19900
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	0	0
$21$	130.967	1.36092
$22$	−63.9911	−0.620134
$23$	112.166	1.01688	0.508439	−	0.861098i	$-0.330223\pi$
$23$	112.166	1.01688	0.508439	+	0.861098i	$0.330223\pi$
$24$	−427.535	−3.63626
$25$	0	0
$26$	238.155	1.79639
$27$	183.531	1.30817
$28$	267.717	1.80692
$29$	295.107	1.88966	0.944828	−	0.327565i	$-0.106228\pi$
$29$	295.107	1.88966	0.944828	+	0.327565i	$0.106228\pi$
$30$	0	0
$31$	−57.6979	−0.334285	−0.167143	−	0.985933i	$-0.553454\pi$
$31$	−57.6979	−0.334285	−0.167143	+	0.985933i	$0.553454\pi$
$32$	−154.862	−0.855498
$33$	109.390	0.577041
$34$	−146.952	−0.741238
$35$	0	0
$36$	853.685	3.95225
$37$	341.167	1.51588	0.757940	−	0.652324i	$-0.226206\pi$
$37$	341.167	1.51588	0.757940	+	0.652324i	$0.226206\pi$
$38$	96.3636	0.411375
$39$	−407.117	−1.67156
$40$	0	0
$41$	274.056	1.04391	0.521955	−	0.852973i	$-0.325203\pi$
$41$	274.056	1.04391	0.521955	+	0.852973i	$0.325203\pi$
$42$	−664.233	−2.44032
$43$	−327.536	−1.16160	−0.580800	−	0.814046i	$-0.697260\pi$
$43$	−327.536	−1.16160	−0.580800	+	0.814046i	$0.697260\pi$
$44$	223.611	0.766151
$45$	0	0
$46$	−568.879	−1.82341
$47$	−139.140	−0.431823	−0.215912	−	0.976413i	$-0.569272\pi$
$47$	−139.140	−0.431823	−0.215912	+	0.976413i	$0.569272\pi$
$48$	939.106	2.82392
$49$	−114.816	−0.334741
$50$	0	0
$51$	251.209	0.689730
$52$	−832.214	−2.21937
$53$	−296.715	−0.769000	−0.384500	−	0.923125i	$-0.625626\pi$
$53$	−296.715	−0.769000	−0.384500	+	0.923125i	$0.625626\pi$
$54$	−930.828	−2.34574
$55$	0	0
$56$	−744.897	−1.77752
$57$	−164.730	−0.382789
$58$	−1496.72	−3.38842
$59$	459.383	1.01367	0.506836	−	0.862043i	$-0.330815\pi$
$59$	459.383	1.01367	0.506836	+	0.862043i	$0.330815\pi$
$60$	0	0
$61$	−232.911	−0.488873	−0.244436	−	0.969665i	$-0.578603\pi$
$61$	−232.911	−0.488873	−0.244436	+	0.969665i	$0.578603\pi$
$62$	292.631	0.599421
$63$	727.623	1.45511
$64$	−81.1127	−0.158423
$65$	0	0
$66$	−554.801	−1.03472
$67$	320.784	0.584926	0.292463	−	0.956277i	$-0.405525\pi$
$67$	320.784	0.584926	0.292463	+	0.956277i	$0.405525\pi$
$68$	513.511	0.915771
$69$	972.475	1.69670
$70$	0	0
$71$	−9.54518	−0.0159550	−0.00797749	−	0.999968i	$-0.502539\pi$
$71$	−9.54518	−0.0159550	−0.00797749	+	0.999968i	$0.502539\pi$
$72$	−2375.30	−3.88793
$73$	−320.868	−0.514448	−0.257224	−	0.966352i	$-0.582808\pi$
$73$	−320.868	−0.514448	−0.257224	+	0.966352i	$0.582808\pi$
$74$	−1730.32	−2.71819
$75$	0	0
$76$	−336.734	−0.508238
$77$	190.591	0.282076
$78$	2064.80	2.99735
$79$	−89.2323	−0.127081	−0.0635406	−	0.997979i	$-0.520239\pi$
$79$	−89.2323	−0.127081	−0.0635406	+	0.997979i	$0.520239\pi$
$80$	0	0
$81$	290.661	0.398712
$82$	−1389.95	−1.87188
$83$	439.455	0.581163	0.290581	−	0.956850i	$-0.406151\pi$
$83$	439.455	0.581163	0.290581	+	0.956850i	$0.406151\pi$
$84$	2321.10	3.01492
$85$	0	0
$86$	1661.19	2.08292
$87$	2558.58	3.15297
$88$	−622.176	−0.753684
$89$	883.164	1.05186	0.525928	−	0.850529i	$-0.323718\pi$
$89$	883.164	1.05186	0.525928	+	0.850529i	$0.323718\pi$
$90$	0	0
$91$	−709.322	−0.817112
$92$	1987.90	2.25275
$93$	−500.240	−0.557768
$94$	705.688	0.774321
$95$	0	0
$96$	−1342.65	−1.42743
$97$	1705.87	1.78562	0.892808	−	0.450437i	$-0.148732\pi$
$97$	1705.87	1.78562	0.892808	+	0.450437i	$0.148732\pi$
$98$	582.321	0.600237
$99$	607.748	0.616979
$100$	0	0
$101$	−961.422	−0.947178	−0.473589	−	0.880746i	$-0.657042\pi$
$101$	−961.422	−0.947178	−0.473589	+	0.880746i	$0.657042\pi$
$102$	−1274.07	−1.23678
$103$	−173.142	−0.165633	−0.0828166	−	0.996565i	$-0.526392\pi$
$103$	−173.142	−0.165633	−0.0828166	+	0.996565i	$0.526392\pi$
$104$	2315.55	2.18326
$105$	0	0
$106$	1504.87	1.37893
$107$	−921.087	−0.832194	−0.416097	−	0.909320i	$-0.636602\pi$
$107$	−921.087	−0.832194	−0.416097	+	0.909320i	$0.636602\pi$
$108$	3252.70	2.89807
$109$	552.454	0.485463	0.242731	−	0.970094i	$-0.421957\pi$
$109$	552.454	0.485463	0.242731	+	0.970094i	$0.421957\pi$
$110$	0	0
$111$	2957.91	2.52930
$112$	1636.21	1.38042
$113$	1395.26	1.16155	0.580774	−	0.814064i	$-0.302750\pi$
$113$	1395.26	1.16155	0.580774	+	0.814064i	$0.302750\pi$
$114$	835.471	0.686395
$115$	0	0
$116$	5230.15	4.18627
$117$	−2261.86	−1.78725
$118$	−2329.89	−1.81766
$119$	437.682	0.337162
$120$	0	0
$121$	−1171.81	−0.880397
$122$	1181.27	0.876618
$123$	2376.06	1.74180
$124$	−1022.57	−0.740562
$125$	0	0
$126$	−3690.34	−2.60922
$127$	−793.013	−0.554083	−0.277041	−	0.960858i	$-0.589354\pi$
$127$	−793.013	−0.554083	−0.277041	+	0.960858i	$0.589354\pi$
$128$	1650.28	1.13957
$129$	−2839.73	−1.93818
$130$	0	0
$131$	25.0544	0.0167100	0.00835501	−	0.999965i	$-0.497340\pi$
$131$	25.0544	0.0167100	0.00835501	+	0.999965i	$0.497340\pi$
$132$	1938.70	1.27835
$133$	−287.009	−0.187119
$134$	−1626.95	−1.04886
$135$	0	0
$136$	−1428.80	−0.900869
$137$	716.056	0.446546	0.223273	−	0.974756i	$-0.428326\pi$
$137$	716.056	0.446546	0.223273	+	0.974756i	$0.428326\pi$
$138$	−4932.17	−3.04242
$139$	666.566	0.406744	0.203372	−	0.979102i	$-0.434810\pi$
$139$	666.566	0.406744	0.203372	+	0.979102i	$0.434810\pi$
$140$	0	0
$141$	−1206.34	−0.720514
$142$	48.4109	0.0286095
$143$	−592.462	−0.346463
$144$	5217.47	3.01937
$145$	0	0
$146$	1627.37	0.922479
$147$	−995.453	−0.558528
$148$	6046.46	3.35822
$149$	−268.063	−0.147386	−0.0736931	−	0.997281i	$-0.523479\pi$
$149$	−268.063	−0.147386	−0.0736931	+	0.997281i	$0.523479\pi$
$150$	0	0
$151$	2809.46	1.51411	0.757055	−	0.653352i	$-0.226638\pi$
$151$	2809.46	1.51411	0.757055	+	0.653352i	$0.226638\pi$
$152$	936.930	0.499968
$153$	1395.66	0.737468
$154$	−966.633	−0.505802
$155$	0	0
$156$	−7215.28	−3.70311
$157$	1999.77	1.01656	0.508278	−	0.861193i	$-0.330282\pi$
$157$	1999.77	1.01656	0.508278	+	0.861193i	$0.330282\pi$
$158$	452.566	0.227875
$159$	−2572.52	−1.28311
$160$	0	0
$161$	1694.35	0.829400
$162$	−1474.16	−0.714946
$163$	−1642.08	−0.789064	−0.394532	−	0.918882i	$-0.629093\pi$
$163$	−1642.08	−0.789064	−0.394532	+	0.918882i	$0.629093\pi$
$164$	4857.05	2.31263
$165$	0	0
$166$	−2228.82	−1.04211
$167$	−2965.92	−1.37431	−0.687155	−	0.726511i	$-0.741141\pi$
$167$	−2965.92	−1.37431	−0.687155	+	0.726511i	$0.741141\pi$
$168$	−6458.24	−2.96586
$169$	7.96603	0.00362587
$170$	0	0
$171$	−915.203	−0.409283
$172$	−5804.88	−2.57336
$173$	−92.7872	−0.0407773	−0.0203887	−	0.999792i	$-0.506490\pi$
$173$	−92.7872	−0.0407773	−0.0203887	+	0.999792i	$0.506490\pi$
$174$	−12976.5	−5.65372
$175$	0	0
$176$	1366.65	0.585311
$177$	3982.84	1.69135
$178$	−4479.21	−1.88613
$179$	−3294.07	−1.37548	−0.687738	−	0.725959i	$-0.741396\pi$
$179$	−3294.07	−1.37548	−0.687738	+	0.725959i	$0.741396\pi$
$180$	0	0
$181$	3590.17	1.47434	0.737168	−	0.675709i	$-0.236162\pi$
$181$	3590.17	1.47434	0.737168	+	0.675709i	$0.236162\pi$
$182$	3597.52	1.46520
$183$	−2019.34	−0.815703
$184$	−5531.13	−2.21609
$185$	0	0
$186$	2537.10	1.00016
$187$	365.574	0.142960
$188$	−2465.96	−0.956643
$189$	2772.38	1.06699
$190$	0	0
$191$	480.480	0.182023	0.0910114	−	0.995850i	$-0.470990\pi$
$191$	480.480	0.182023	0.0910114	+	0.995850i	$0.470990\pi$
$192$	−703.246	−0.264335
$193$	−3504.07	−1.30688	−0.653442	−	0.756977i	$-0.726675\pi$
$193$	−3504.07	−1.30688	−0.653442	+	0.756977i	$0.726675\pi$
$194$	−8651.78	−3.20186
$195$	0	0
$196$	−2034.87	−0.741570
$197$	−603.790	−0.218367	−0.109183	−	0.994022i	$-0.534824\pi$
$197$	−603.790	−0.218367	−0.109183	+	0.994022i	$0.534824\pi$
$198$	−3082.36	−1.10633
$199$	3063.80	1.09139	0.545695	−	0.837984i	$-0.316266\pi$
$199$	3063.80	1.09139	0.545695	+	0.837984i	$0.316266\pi$
$200$	0	0
$201$	2781.20	0.975972
$202$	4876.11	1.69843
$203$	4457.82	1.54127
$204$	4452.13	1.52800
$205$	0	0
$206$	878.139	0.297004
$207$	5402.87	1.81413
$208$	−5086.24	−1.69552
$209$	−239.725	−0.0793403
$210$	0	0
$211$	−2772.16	−0.904470	−0.452235	−	0.891899i	$-0.649373\pi$
$211$	−2772.16	−0.904470	−0.452235	+	0.891899i	$0.649373\pi$
$212$	−5258.65	−1.70361
$213$	−82.7565	−0.0266215
$214$	4671.54	1.49224
$215$	0	0
$216$	−9050.32	−2.85091
$217$	−871.571	−0.272655
$218$	−2801.92	−0.870504
$219$	−2781.92	−0.858377
$220$	0	0
$221$	−1360.56	−0.414122
$222$	−15001.9	−4.53540
$223$	5653.47	1.69769	0.848844	−	0.528644i	$-0.177299\pi$
$223$	5653.47	1.69769	0.848844	+	0.528644i	$0.177299\pi$
$224$	−2339.30	−0.697773
$225$	0	0
$226$	−7076.44	−2.08282
$227$	5083.00	1.48621	0.743106	−	0.669173i	$-0.233352\pi$
$227$	5083.00	1.48621	0.743106	+	0.669173i	$0.233352\pi$
$228$	−2919.48	−0.848015
$229$	501.966	0.144851	0.0724254	−	0.997374i	$-0.476926\pi$
$229$	501.966	0.144851	0.0724254	+	0.997374i	$0.476926\pi$
$230$	0	0
$231$	1652.42	0.470655
$232$	−14552.4	−4.11815
$233$	−2809.38	−0.789909	−0.394955	−	0.918701i	$-0.629240\pi$
$233$	−2809.38	−0.789909	−0.394955	+	0.918701i	$0.629240\pi$
$234$	11471.6	3.20480
$235$	0	0
$236$	8141.58	2.24564
$237$	−773.642	−0.212040
$238$	−2219.82	−0.604579
$239$	2239.56	0.606130	0.303065	−	0.952970i	$-0.401990\pi$
$239$	2239.56	0.606130	0.303065	+	0.952970i	$0.401990\pi$
$240$	0	0
$241$	−7214.25	−1.92826	−0.964130	−	0.265431i	$-0.914486\pi$
$241$	−7214.25	−1.92826	−0.964130	+	0.265431i	$0.914486\pi$
$242$	5943.15	1.57868
$243$	−2435.32	−0.642905
$244$	−4127.85	−1.08303
$245$	0	0
$246$	−12050.8	−3.12330
$247$	892.184	0.229831
$248$	2845.21	0.728511
$249$	3810.07	0.969693
$250$	0	0
$251$	6212.82	1.56235	0.781174	−	0.624313i	$-0.214621\pi$
$251$	6212.82	1.56235	0.781174	+	0.624313i	$0.214621\pi$
$252$	12895.6	3.22359
$253$	1415.21	0.351673
$254$	4021.98	0.993549
$255$	0	0
$256$	−7720.93	−1.88499
$257$	2796.37	0.678727	0.339364	−	0.940655i	$-0.389788\pi$
$257$	2796.37	0.678727	0.339364	+	0.940655i	$0.389788\pi$
$258$	14402.5	3.47542
$259$	5153.59	1.23640
$260$	0	0
$261$	14214.9	3.37119
$262$	−127.070	−0.0299634
$263$	−2976.00	−0.697748	−0.348874	−	0.937170i	$-0.613436\pi$
$263$	−2976.00	−0.697748	−0.348874	+	0.937170i	$0.613436\pi$
$264$	−5394.26	−1.25755
$265$	0	0
$266$	1455.65	0.335532
$267$	7657.02	1.75506
$268$	5685.22	1.29582
$269$	−25.4734	−0.00577376	−0.00288688	−	0.999996i	$-0.500919\pi$
$269$	−25.4734	−0.00577376	−0.00288688	+	0.999996i	$0.500919\pi$
$270$	0	0
$271$	−4338.19	−0.972421	−0.486211	−	0.873842i	$-0.661621\pi$
$271$	−4338.19	−0.972421	−0.486211	+	0.873842i	$0.661621\pi$
$272$	3138.43	0.699615
$273$	−6149.81	−1.36338
$274$	−3631.67	−0.800720
$275$	0	0
$276$	17235.0	3.75880
$277$	−4276.02	−0.927514	−0.463757	−	0.885962i	$-0.653499\pi$
$277$	−4276.02	−0.927514	−0.463757	+	0.885962i	$0.653499\pi$
$278$	−3380.67	−0.729349
$279$	−2779.23	−0.596373
$280$	0	0
$281$	−3716.43	−0.788980	−0.394490	−	0.918900i	$-0.629079\pi$
$281$	−3716.43	−0.788980	−0.394490	+	0.918900i	$0.629079\pi$
$282$	6118.30	1.29198
$283$	−3748.01	−0.787265	−0.393633	−	0.919268i	$-0.628782\pi$
$283$	−3748.01	−0.787265	−0.393633	+	0.919268i	$0.628782\pi$
$284$	−169.168	−0.0353460
$285$	0	0
$286$	3004.83	0.621257
$287$	4139.82	0.851449
$288$	−7459.46	−1.52623
$289$	−4073.48	−0.829122
$290$	0	0
$291$	14789.9	2.97937
$292$	−5686.69	−1.13969
$293$	−8210.10	−1.63699	−0.818497	−	0.574510i	$-0.805192\pi$
$293$	−8210.10	−1.63699	−0.818497	+	0.574510i	$0.805192\pi$
$294$	5048.71	1.00152
$295$	0	0
$296$	−16823.7	−3.30357
$297$	2315.63	0.452413
$298$	1359.55	0.264284
$299$	−5266.98	−1.01872
$300$	0	0
$301$	−4947.69	−0.947442
$302$	−14248.9	−2.71501
$303$	−8335.51	−1.58040
$304$	−2058.02	−0.388275
$305$	0	0
$306$	−7078.48	−1.32238
$307$	2814.79	0.523285	0.261642	−	0.965165i	$-0.415736\pi$
$307$	2814.79	0.523285	0.261642	+	0.965165i	$0.415736\pi$
$308$	3377.82	0.624899
$309$	−1501.14	−0.276366
$310$	0	0
$311$	−8650.75	−1.57730	−0.788648	−	0.614845i	$-0.789218\pi$
$311$	−8650.75	−1.57730	−0.788648	+	0.614845i	$0.789218\pi$
$312$	20075.8	3.64285
$313$	−1623.48	−0.293178	−0.146589	−	0.989197i	$-0.546829\pi$
$313$	−1623.48	−0.293178	−0.146589	+	0.989197i	$0.546829\pi$
$314$	−10142.4	−1.82283
$315$	0	0
$316$	−1581.45	−0.281530
$317$	−9372.31	−1.66057	−0.830286	−	0.557337i	$-0.811823\pi$
$317$	−9372.31	−1.66057	−0.830286	+	0.557337i	$0.811823\pi$
$318$	13047.2	2.30079
$319$	3723.40	0.653512
$320$	0	0
$321$	−7985.80	−1.38855
$322$	−8593.35	−1.48723
$323$	−550.516	−0.0948344
$324$	5151.34	0.883289
$325$	0	0
$326$	8328.24	1.41490
$327$	4789.76	0.810014
$328$	−13514.3	−2.27500
$329$	−2101.82	−0.352210
$330$	0	0
$331$	−1765.55	−0.293182	−0.146591	−	0.989197i	$-0.546830\pi$
$331$	−1765.55	−0.293182	−0.146591	+	0.989197i	$0.546830\pi$
$332$	7788.41	1.28748
$333$	16433.5	2.70436
$334$	15042.5	2.46433
$335$	0	0
$336$	14185.9	2.30329
$337$	−10189.8	−1.64711	−0.823555	−	0.567237i	$-0.808012\pi$
$337$	−10189.8	−1.64711	−0.823555	+	0.567237i	$0.808012\pi$
$338$	−40.4019	−0.00650169
$339$	12096.9	1.93809
$340$	0	0
$341$	−727.980	−0.115608
$342$	4641.70	0.733902
$343$	−6915.66	−1.08866
$344$	16151.5	2.53149
$345$	0	0
$346$	470.595	0.0731195
$347$	11350.2	1.75594	0.877971	−	0.478714i	$-0.158897\pi$
$347$	11350.2	1.75594	0.877971	+	0.478714i	$0.158897\pi$
$348$	45345.3	6.98495
$349$	511.619	0.0784709	0.0392354	−	0.999230i	$-0.487508\pi$
$349$	511.619	0.0784709	0.0392354	+	0.999230i	$0.487508\pi$
$350$	0	0
$351$	−8618.09	−1.31054
$352$	−1953.90	−0.295862
$353$	−816.097	−0.123049	−0.0615247	−	0.998106i	$-0.519596\pi$
$353$	−816.097	−0.123049	−0.0615247	+	0.998106i	$0.519596\pi$
$354$	−20200.1	−3.03283
$355$	0	0
$356$	15652.2	2.33024
$357$	3794.70	0.562567
$358$	16706.8	2.46642
$359$	3998.35	0.587813	0.293906	−	0.955834i	$-0.405045\pi$
$359$	3998.35	0.587813	0.293906	+	0.955834i	$0.405045\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	−18208.5	−2.64369
$363$	−10159.6	−1.46898
$364$	−12571.2	−1.81019
$365$	0	0
$366$	10241.6	1.46267
$367$	3123.19	0.444221	0.222111	−	0.975021i	$-0.428705\pi$
$367$	3123.19	0.444221	0.222111	+	0.975021i	$0.428705\pi$
$368$	12149.5	1.72102
$369$	13200.9	1.86236
$370$	0	0
$371$	−4482.11	−0.627223
$372$	−8865.68	−1.23566
$373$	7026.28	0.975354	0.487677	−	0.873024i	$-0.337844\pi$
$373$	7026.28	0.975354	0.487677	+	0.873024i	$0.337844\pi$
$374$	−1854.11	−0.256347
$375$	0	0
$376$	6861.30	0.941076
$377$	−13857.4	−1.89308
$378$	−14060.9	−1.91326
$379$	−11595.9	−1.57162	−0.785808	−	0.618470i	$-0.787753\pi$
$379$	−11595.9	−1.57162	−0.785808	+	0.618470i	$0.787753\pi$
$380$	0	0
$381$	−6875.41	−0.924508
$382$	−2436.89	−0.326393
$383$	10962.4	1.46254	0.731268	−	0.682091i	$-0.238929\pi$
$383$	10962.4	1.46254	0.731268	+	0.682091i	$0.238929\pi$
$384$	14307.9	1.90142
$385$	0	0
$386$	17771.8	2.34343
$387$	−15777.0	−2.07232
$388$	30232.9	3.95578
$389$	7126.21	0.928825	0.464413	−	0.885619i	$-0.346265\pi$
$389$	7126.21	0.928825	0.464413	+	0.885619i	$0.346265\pi$
$390$	0	0
$391$	3249.95	0.420350
$392$	5661.82	0.729503
$393$	217.221	0.0278813
$394$	3062.29	0.391563
$395$	0	0
$396$	10771.0	1.36683
$397$	−4895.59	−0.618899	−0.309449	−	0.950916i	$-0.600145\pi$
$397$	−4895.59	−0.618899	−0.309449	+	0.950916i	$0.600145\pi$
$398$	−15538.9	−1.95702
$399$	−2488.37	−0.312216
$400$	0	0
$401$	−6148.05	−0.765634	−0.382817	−	0.923824i	$-0.625046\pi$
$401$	−6148.05	−0.765634	−0.382817	+	0.923824i	$0.625046\pi$
$402$	−14105.6	−1.75006
$403$	2709.32	0.334891
$404$	−17039.1	−2.09834
$405$	0	0
$406$	−22609.0	−2.76371
$407$	4304.54	0.524246
$408$	−12387.6	−1.50313
$409$	2968.47	0.358878	0.179439	−	0.983769i	$-0.442572\pi$
$409$	2968.47	0.358878	0.179439	+	0.983769i	$0.442572\pi$
$410$	0	0
$411$	6208.19	0.745079
$412$	−3068.58	−0.366937
$413$	6939.33	0.826785
$414$	−27402.1	−3.25300
$415$	0	0
$416$	7271.85	0.857047
$417$	5779.12	0.678668
$418$	1215.83	0.142268
$419$	−11101.1	−1.29432	−0.647162	−	0.762352i	$-0.724044\pi$
$419$	−11101.1	−1.29432	−0.647162	+	0.762352i	$0.724044\pi$
$420$	0	0
$421$	3241.73	0.375278	0.187639	−	0.982238i	$-0.439916\pi$
$421$	3241.73	0.375278	0.187639	+	0.982238i	$0.439916\pi$
$422$	14059.7	1.62184
$423$	−6702.19	−0.770382
$424$	14631.7	1.67589
$425$	0	0
$426$	419.722	0.0477361
$427$	−3518.30	−0.398741
$428$	−16324.3	−1.84361
$429$	−5136.64	−0.578086
$430$	0	0
$431$	290.271	0.0324405	0.0162202	−	0.999868i	$-0.494837\pi$
$431$	290.271	0.0324405	0.0162202	+	0.999868i	$0.494837\pi$
$432$	19879.5	2.21402
$433$	−2104.16	−0.233533	−0.116766	−	0.993159i	$-0.537253\pi$
$433$	−2104.16	−0.233533	−0.116766	+	0.993159i	$0.537253\pi$
$434$	4420.41	0.488909
$435$	0	0
$436$	9791.06	1.07547
$437$	−2131.15	−0.233288
$438$	14109.2	1.53919
$439$	4013.82	0.436376	0.218188	−	0.975907i	$-0.429985\pi$
$439$	4013.82	0.436376	0.218188	+	0.975907i	$0.429985\pi$
$440$	0	0
$441$	−5530.53	−0.597185
$442$	6900.44	0.742580
$443$	−12013.8	−1.28847	−0.644236	−	0.764827i	$-0.722824\pi$
$443$	−12013.8	−1.28847	−0.644236	+	0.764827i	$0.722824\pi$
$444$	52422.7	5.60331
$445$	0	0
$446$	−28673.1	−3.04419
$447$	−2324.10	−0.245920
$448$	−1225.27	−0.129215
$449$	4281.55	0.450020	0.225010	−	0.974356i	$-0.427759\pi$
$449$	4281.55	0.450020	0.225010	+	0.974356i	$0.427759\pi$
$450$	0	0
$451$	3457.79	0.361022
$452$	24728.0	2.57325
$453$	24358.0	2.52635
$454$	−25779.8	−2.66499
$455$	0	0
$456$	8123.17	0.834215
$457$	−293.330	−0.0300249	−0.0150125	−	0.999887i	$-0.504779\pi$
$457$	−293.330	−0.0300249	−0.0150125	+	0.999887i	$0.504779\pi$
$458$	−2545.85	−0.259738
$459$	5317.73	0.540763
$460$	0	0
$461$	−5267.87	−0.532210	−0.266105	−	0.963944i	$-0.585737\pi$
$461$	−5267.87	−0.532210	−0.266105	+	0.963944i	$0.585737\pi$
$462$	−8380.69	−0.843951
$463$	−8076.98	−0.810733	−0.405366	−	0.914154i	$-0.632856\pi$
$463$	−8076.98	−0.810733	−0.405366	+	0.914154i	$0.632856\pi$
$464$	31965.1	3.19815
$465$	0	0
$466$	14248.5	1.41642
$467$	3160.43	0.313163	0.156582	−	0.987665i	$-0.449953\pi$
$467$	3160.43	0.313163	0.156582	+	0.987665i	$0.449953\pi$
$468$	−40086.5	−3.95940
$469$	4845.69	0.477086
$470$	0	0
$471$	17338.0	1.69616
$472$	−22653.2	−2.20910
$473$	−4132.56	−0.401724
$474$	3923.74	0.380218
$475$	0	0
$476$	7756.98	0.746934
$477$	−14292.4	−1.37191
$478$	−11358.5	−1.08688
$479$	−249.277	−0.0237782	−0.0118891	−	0.999929i	$-0.503785\pi$
$479$	−249.277	−0.0237782	−0.0118891	+	0.999929i	$0.503785\pi$
$480$	0	0
$481$	−16020.2	−1.51863
$482$	36589.0	3.45764
$483$	14690.0	1.38389
$484$	−20767.8	−1.95039
$485$	0	0
$486$	12351.4	1.15282
$487$	−7265.71	−0.676059	−0.338029	−	0.941136i	$-0.609760\pi$
$487$	−7265.71	−0.676059	−0.338029	+	0.941136i	$0.609760\pi$
$488$	11485.4	1.06540
$489$	−14236.8	−1.31658
$490$	0	0
$491$	−8456.47	−0.777261	−0.388630	−	0.921394i	$-0.627052\pi$
$491$	−8456.47	−0.777261	−0.388630	+	0.921394i	$0.627052\pi$
$492$	42110.5	3.85872
$493$	8550.59	0.781134
$494$	−4524.95	−0.412120
$495$	0	0
$496$	−6249.66	−0.565762
$497$	−144.187	−0.0130134
$498$	−19323.8	−1.73880
$499$	1905.15	0.170914	0.0854571	−	0.996342i	$-0.472765\pi$
$499$	1905.15	0.170914	0.0854571	+	0.996342i	$0.472765\pi$
$500$	0	0
$501$	−25714.5	−2.29309
$502$	−31510.0	−2.80151
$503$	−6082.63	−0.539187	−0.269593	−	0.962974i	$-0.586889\pi$
$503$	−6082.63	−0.539187	−0.269593	+	0.962974i	$0.586889\pi$
$504$	−35880.6	−3.17113
$505$	0	0
$506$	−7177.61	−0.630600
$507$	69.0653	0.00604990
$508$	−14054.5	−1.22749
$509$	13858.9	1.20684	0.603422	−	0.797422i	$-0.293803\pi$
$509$	13858.9	1.20684	0.603422	+	0.797422i	$0.293803\pi$
$510$	0	0
$511$	−4846.95	−0.419602
$512$	25956.6	2.24049
$513$	−3487.09	−0.300115
$514$	−14182.6	−1.21705
$515$	0	0
$516$	−50328.2	−4.29375
$517$	−1755.55	−0.149340
$518$	−26137.8	−2.21705
$519$	−804.463	−0.0680385
$520$	0	0
$521$	4086.72	0.343651	0.171826	−	0.985127i	$-0.445033\pi$
$521$	4086.72	0.343651	0.171826	+	0.985127i	$0.445033\pi$
$522$	−72094.7	−6.04502
$523$	−20188.4	−1.68791	−0.843957	−	0.536411i	$-0.819780\pi$
$523$	−20188.4	−1.68791	−0.843957	+	0.536411i	$0.819780\pi$
$524$	444.036	0.0370187
$525$	0	0
$526$	15093.6	1.25116
$527$	−1671.77	−0.138185
$528$	11848.8	0.976615
$529$	414.164	0.0340399
$530$	0	0
$531$	22127.8	1.80841
$532$	−5086.63	−0.414536
$533$	−12868.9	−1.04580
$534$	−38834.7	−3.14708
$535$	0	0
$536$	−15818.6	−1.27473
$537$	−28559.5	−2.29503
$538$	129.195	0.0103532
$539$	−1448.65	−0.115765
$540$	0	0
$541$	16356.6	1.29986	0.649932	−	0.759992i	$-0.274797\pi$
$541$	16356.6	1.29986	0.649932	+	0.759992i	$0.274797\pi$
$542$	22002.3	1.74369
$543$	31126.7	2.45999
$544$	−4487.04	−0.353640
$545$	0	0
$546$	31190.4	2.44474
$547$	12751.0	0.996697	0.498349	−	0.866977i	$-0.333940\pi$
$547$	12751.0	0.996697	0.498349	+	0.866977i	$0.333940\pi$
$548$	12690.6	0.989259
$549$	−11219.0	−0.872159
$550$	0	0
$551$	−5607.04	−0.433517
$552$	−47954.8	−3.69763
$553$	−1347.92	−0.103652
$554$	21687.0	1.66316
$555$	0	0
$556$	11813.5	0.901083
$557$	7353.77	0.559406	0.279703	−	0.960087i	$-0.409764\pi$
$557$	7353.77	0.559406	0.279703	+	0.960087i	$0.409764\pi$
$558$	14095.6	1.06938
$559$	15380.1	1.16370
$560$	0	0
$561$	3169.52	0.238534
$562$	18848.9	1.41475
$563$	5597.74	0.419035	0.209517	−	0.977805i	$-0.432811\pi$
$563$	5597.74	0.419035	0.209517	+	0.977805i	$0.432811\pi$
$564$	−21379.9	−1.59620
$565$	0	0
$566$	19009.0	1.41168
$567$	4390.65	0.325203
$568$	470.693	0.0347708
$569$	−19640.2	−1.44703	−0.723515	−	0.690309i	$-0.757475\pi$
$569$	−19640.2	−1.44703	−0.723515	+	0.690309i	$0.757475\pi$
$570$	0	0
$571$	−9749.71	−0.714558	−0.357279	−	0.933998i	$-0.616295\pi$
$571$	−9749.71	−0.714558	−0.357279	+	0.933998i	$0.616295\pi$
$572$	−10500.1	−0.767539
$573$	4165.76	0.303712
$574$	−20996.2	−1.52677
$575$	0	0
$576$	−3907.09	−0.282631
$577$	−11197.9	−0.807927	−0.403964	−	0.914775i	$-0.632368\pi$
$577$	−11197.9	−0.807927	−0.403964	+	0.914775i	$0.632368\pi$
$578$	20659.7	1.48673
$579$	−30380.2	−2.18059
$580$	0	0
$581$	6638.31	0.474016
$582$	−75010.8	−5.34243
$583$	−3743.69	−0.265948
$584$	15822.7	1.12114
$585$	0	0
$586$	41639.8	2.93536
$587$	18654.4	1.31167	0.655833	−	0.754906i	$-0.272318\pi$
$587$	18654.4	1.31167	0.655833	+	0.754906i	$0.272318\pi$
$588$	−17642.3	−1.23734
$589$	1096.26	0.0766903
$590$	0	0
$591$	−5234.85	−0.364354
$592$	36954.2	2.56555
$593$	−21513.4	−1.48980	−0.744900	−	0.667176i	$-0.767503\pi$
$593$	−21513.4	−1.48980	−0.744900	+	0.667176i	$0.767503\pi$
$594$	−11744.4	−0.811240
$595$	0	0
$596$	−4750.84	−0.326513
$597$	26563.1	1.82103
$598$	26712.9	1.82671
$599$	22762.8	1.55269	0.776346	−	0.630308i	$-0.217071\pi$
$599$	22762.8	1.55269	0.776346	+	0.630308i	$0.217071\pi$
$600$	0	0
$601$	−13941.0	−0.946195	−0.473098	−	0.881010i	$-0.656864\pi$
$601$	−13941.0	−0.946195	−0.473098	+	0.881010i	$0.656864\pi$
$602$	25093.5	1.69890
$603$	15451.7	1.04352
$604$	49791.6	3.35429
$605$	0	0
$606$	42275.8	2.83389
$607$	−8031.64	−0.537058	−0.268529	−	0.963272i	$-0.586538\pi$
$607$	−8031.64	−0.537058	−0.268529	+	0.963272i	$0.586538\pi$
$608$	2942.37	0.196265
$609$	38649.2	2.57167
$610$	0	0
$611$	6533.62	0.432606
$612$	24735.1	1.63375
$613$	19429.7	1.28019	0.640095	−	0.768296i	$-0.278895\pi$
$613$	19429.7	1.28019	0.640095	+	0.768296i	$0.278895\pi$
$614$	−14276.0	−0.938324
$615$	0	0
$616$	−9398.44	−0.614731
$617$	11264.0	0.734959	0.367480	−	0.930032i	$-0.380221\pi$
$617$	11264.0	0.734959	0.367480	+	0.930032i	$0.380221\pi$
$618$	7613.45	0.495563
$619$	−22183.9	−1.44046	−0.720231	−	0.693734i	$-0.755964\pi$
$619$	−22183.9	−1.44046	−0.720231	+	0.693734i	$0.755964\pi$
$620$	0	0
$621$	20585.9	1.33025
$622$	43874.6	2.82831
$623$	13340.9	0.857930
$624$	−44097.6	−2.82904
$625$	0	0
$626$	8233.93	0.525710
$627$	−2078.41	−0.132382
$628$	35441.7	2.25204
$629$	9885.16	0.626625
$630$	0	0
$631$	−23139.3	−1.45984	−0.729920	−	0.683532i	$-0.760443\pi$
$631$	−23139.3	−1.45984	−0.729920	+	0.683532i	$0.760443\pi$
$632$	4400.23	0.276949
$633$	−24034.5	−1.50914
$634$	47534.2	2.97764
$635$	0	0
$636$	−45592.4	−2.84254
$637$	5391.42	0.335347
$638$	−18884.2	−1.17184
$639$	−459.778	−0.0284640
$640$	0	0
$641$	−4988.45	−0.307382	−0.153691	−	0.988119i	$-0.549116\pi$
$641$	−4988.45	−0.307382	−0.153691	+	0.988119i	$0.549116\pi$
$642$	40502.2	2.48986
$643$	11115.2	0.681712	0.340856	−	0.940115i	$-0.389283\pi$
$643$	11115.2	0.681712	0.340856	+	0.940115i	$0.389283\pi$
$644$	30028.7	1.83742
$645$	0	0
$646$	2792.09	0.170052
$647$	−11916.2	−0.724071	−0.362036	−	0.932164i	$-0.617918\pi$
$647$	−11916.2	−0.724071	−0.362036	+	0.932164i	$0.617918\pi$
$648$	−14333.1	−0.868915
$649$	5796.09	0.350564
$650$	0	0
$651$	−7556.50	−0.454935
$652$	−29102.3	−1.74806
$653$	18100.5	1.08473	0.542363	−	0.840144i	$-0.317530\pi$
$653$	18100.5	1.08473	0.542363	+	0.840144i	$0.317530\pi$
$654$	−24292.6	−1.45247
$655$	0	0
$656$	29684.9	1.76677
$657$	−15455.7	−0.917787
$658$	10659.9	0.631562
$659$	331.740	0.0196096	0.00980481	−	0.999952i	$-0.496879\pi$
$659$	331.740	0.0196096	0.00980481	+	0.999952i	$0.496879\pi$
$660$	0	0
$661$	−30555.8	−1.79801	−0.899004	−	0.437939i	$-0.855708\pi$
$661$	−30555.8	−1.79801	−0.899004	+	0.437939i	$0.855708\pi$
$662$	8954.46	0.525717
$663$	−11796.0	−0.690979
$664$	−21670.5	−1.26653
$665$	0	0
$666$	−83347.2	−4.84931
$667$	33100.9	1.92155
$668$	−52564.6	−3.04459
$669$	49015.5	2.83266
$670$	0	0
$671$	−2938.67	−0.169070
$672$	−20281.7	−1.16426
$673$	−3261.14	−0.186787	−0.0933936	−	0.995629i	$-0.529772\pi$
$673$	−3261.14	−0.186787	−0.0933936	+	0.995629i	$0.529772\pi$
$674$	51680.5	2.95350
$675$	0	0
$676$	141.181	0.00803259
$677$	−11556.6	−0.656063	−0.328031	−	0.944667i	$-0.606385\pi$
$677$	−11556.6	−0.656063	−0.328031	+	0.944667i	$0.606385\pi$
$678$	−61352.6	−3.47527
$679$	25768.5	1.45641
$680$	0	0
$681$	44069.5	2.47980
$682$	3692.15	0.207302
$683$	−19704.1	−1.10389	−0.551944	−	0.833881i	$-0.686114\pi$
$683$	−19704.1	−1.10389	−0.551944	+	0.833881i	$0.686114\pi$
$684$	−16220.0	−0.906707
$685$	0	0
$686$	35074.6	1.95212
$687$	4352.03	0.241689
$688$	−35477.7	−1.96595
$689$	13932.9	0.770393
$690$	0	0
$691$	−2956.51	−0.162765	−0.0813827	−	0.996683i	$-0.525934\pi$
$691$	−2956.51	−0.162765	−0.0813827	+	0.996683i	$0.525934\pi$
$692$	−1644.45	−0.0903364
$693$	9180.49	0.503230
$694$	−57565.7	−3.14865
$695$	0	0
$696$	−126169.	−6.87129
$697$	7940.63	0.431525
$698$	−2594.81	−0.140709
$699$	−24357.3	−1.31799
$700$	0	0
$701$	29022.1	1.56370	0.781848	−	0.623469i	$-0.214277\pi$
$701$	29022.1	1.56370	0.781848	+	0.623469i	$0.214277\pi$
$702$	43709.0	2.34998
$703$	−6482.18	−0.347767
$704$	−1023.41	−0.0547885
$705$	0	0
$706$	4139.05	0.220645
$707$	−14523.0	−0.772551
$708$	70587.4	3.74694
$709$	−5110.64	−0.270711	−0.135356	−	0.990797i	$-0.543218\pi$
$709$	−5110.64	−0.270711	−0.135356	+	0.990797i	$0.543218\pi$
$710$	0	0
$711$	−4298.19	−0.226716
$712$	−43550.7	−2.29232
$713$	−6471.73	−0.339927
$714$	−19245.8	−1.00876
$715$	0	0
$716$	−58380.3	−3.04717
$717$	19416.9	1.01135
$718$	−20278.7	−1.05403
$719$	−25225.9	−1.30844	−0.654219	−	0.756305i	$-0.727002\pi$
$719$	−25225.9	−1.30844	−0.654219	+	0.756305i	$0.727002\pi$
$720$	0	0
$721$	−2615.45	−0.135096
$722$	−1830.91	−0.0943759
$723$	−62547.4	−3.21738
$724$	63628.0	3.26618
$725$	0	0
$726$	51527.0	2.63408
$727$	12817.7	0.653893	0.326947	−	0.945043i	$-0.393980\pi$
$727$	12817.7	0.653893	0.326947	+	0.945043i	$0.393980\pi$
$728$	34978.2	1.78074
$729$	−28962.0	−1.47142
$730$	0	0
$731$	−9490.21	−0.480175
$732$	−35788.4	−1.80707
$733$	15307.5	0.771342	0.385671	−	0.922636i	$-0.373970\pi$
$733$	15307.5	0.771342	0.385671	+	0.922636i	$0.373970\pi$
$734$	−15840.1	−0.796551
$735$	0	0
$736$	−17370.2	−0.869936
$737$	4047.37	0.202289
$738$	−66951.8	−3.33947
$739$	34340.1	1.70936	0.854682	−	0.519152i	$-0.173752\pi$
$739$	34340.1	1.70936	0.854682	+	0.519152i	$0.173752\pi$
$740$	0	0
$741$	7735.22	0.383482
$742$	22732.2	1.12470
$743$	1883.40	0.0929948	0.0464974	−	0.998918i	$-0.485194\pi$
$743$	1883.40	0.0929948	0.0464974	+	0.998918i	$0.485194\pi$
$744$	24667.9	1.21555
$745$	0	0
$746$	−35635.7	−1.74895
$747$	21167.9	1.03681
$748$	6479.03	0.316707
$749$	−13913.7	−0.678766
$750$	0	0
$751$	−9431.98	−0.458292	−0.229146	−	0.973392i	$-0.573593\pi$
$751$	−9431.98	−0.458292	−0.229146	+	0.973392i	$0.573593\pi$
$752$	−15071.2	−0.730840
$753$	53865.0	2.60684
$754$	70281.4	3.39456
$755$	0	0
$756$	49134.5	2.36376
$757$	12355.0	0.593196	0.296598	−	0.955002i	$-0.404148\pi$
$757$	12355.0	0.593196	0.296598	+	0.955002i	$0.404148\pi$
$758$	58811.9	2.81813
$759$	12269.8	0.586780
$760$	0	0
$761$	−27257.6	−1.29841	−0.649204	−	0.760614i	$-0.724898\pi$
$761$	−27257.6	−1.29841	−0.649204	+	0.760614i	$0.724898\pi$
$762$	34870.5	1.65777
$763$	8345.23	0.395960
$764$	8515.49	0.403246
$765$	0	0
$766$	−55598.6	−2.62253
$767$	−21571.3	−1.01551
$768$	−66940.3	−3.14518
$769$	−1191.72	−0.0558835	−0.0279418	−	0.999610i	$-0.508895\pi$
$769$	−1191.72	−0.0558835	−0.0279418	+	0.999610i	$0.508895\pi$
$770$	0	0
$771$	24244.5	1.13248
$772$	−62102.1	−2.89521
$773$	7481.03	0.348091	0.174045	−	0.984738i	$-0.444316\pi$
$773$	7481.03	0.348091	0.174045	+	0.984738i	$0.444316\pi$
$774$	80017.2	3.71597
$775$	0	0
$776$	−84120.1	−3.89141
$777$	44681.5	2.06299
$778$	−36142.5	−1.66552
$779$	−5207.06	−0.239489
$780$	0	0
$781$	−120.432	−0.00551781
$782$	−16483.0	−0.753748
$783$	54161.4	2.47199
$784$	−12436.5	−0.566532
$785$	0	0
$786$	−1101.70	−0.0499952
$787$	−27403.6	−1.24121	−0.620604	−	0.784124i	$-0.713113\pi$
$787$	−27403.6	−1.24121	−0.620604	+	0.784124i	$0.713113\pi$
$788$	−10700.9	−0.483761
$789$	−25801.8	−1.16422
$790$	0	0
$791$	21076.5	0.947399
$792$	−29969.3	−1.34459
$793$	10936.8	0.489758
$794$	24829.3	1.10977
$795$	0	0
$796$	54299.2	2.41782
$797$	−30558.5	−1.35814	−0.679070	−	0.734073i	$-0.737617\pi$
$797$	−30558.5	−1.35814	−0.679070	+	0.734073i	$0.737617\pi$
$798$	12620.4	0.559847
$799$	−4031.52	−0.178504
$800$	0	0
$801$	42540.8	1.87654
$802$	31181.5	1.37289
$803$	−4048.42	−0.177915
$804$	49290.7	2.16213
$805$	0	0
$806$	−13741.1	−0.600507
$807$	−220.854	−0.00963374
$808$	47409.7	2.06419
$809$	−28018.7	−1.21766	−0.608829	−	0.793302i	$-0.708360\pi$
$809$	−28018.7	−1.21766	−0.608829	+	0.793302i	$0.708360\pi$
$810$	0	0
$811$	2520.45	0.109131	0.0545654	−	0.998510i	$-0.482623\pi$
$811$	2520.45	0.109131	0.0545654	+	0.998510i	$0.482623\pi$
$812$	79005.3	3.41446
$813$	−37612.0	−1.62252
$814$	−21831.7	−0.940048
$815$	0	0
$816$	27210.1	1.16733
$817$	6223.19	0.266490
$818$	−15055.4	−0.643520
$819$	−34167.0	−1.45774
$820$	0	0
$821$	21887.7	0.930434	0.465217	−	0.885197i	$-0.345976\pi$
$821$	21887.7	0.930434	0.465217	+	0.885197i	$0.345976\pi$
$822$	−31486.5	−1.33603
$823$	−8149.95	−0.345188	−0.172594	−	0.984993i	$-0.555215\pi$
$823$	−8149.95	−0.345188	−0.172594	+	0.984993i	$0.555215\pi$
$824$	8538.02	0.360966
$825$	0	0
$826$	−35194.7	−1.48254
$827$	−22007.0	−0.925345	−0.462672	−	0.886529i	$-0.653109\pi$
$827$	−22007.0	−0.925345	−0.462672	+	0.886529i	$0.653109\pi$
$828$	95754.3	4.01895
$829$	8082.69	0.338629	0.169314	−	0.985562i	$-0.445845\pi$
$829$	8082.69	0.338629	0.169314	+	0.985562i	$0.445845\pi$
$830$	0	0
$831$	−37073.0	−1.54759
$832$	3808.82	0.158710
$833$	−3326.74	−0.138373
$834$	−29310.4	−1.21695
$835$	0	0
$836$	−4248.61	−0.175767
$837$	−10589.4	−0.437302
$838$	56302.0	2.32091
$839$	−1144.88	−0.0471105	−0.0235552	−	0.999723i	$-0.507499\pi$
$839$	−1144.88	−0.0471105	−0.0235552	+	0.999723i	$0.507499\pi$
$840$	0	0
$841$	62699.3	2.57080
$842$	−16441.3	−0.672927
$843$	−32221.3	−1.31644
$844$	−49130.5	−2.00372
$845$	0	0
$846$	33992.0	1.38140
$847$	−17701.1	−0.718082
$848$	−32139.3	−1.30150
$849$	−32495.2	−1.31358
$850$	0	0
$851$	38267.3	1.54146
$852$	−1466.68	−0.0589762
$853$	12853.7	0.515948	0.257974	−	0.966152i	$-0.416945\pi$
$853$	12853.7	0.515948	0.257974	+	0.966152i	$0.416945\pi$
$854$	17844.0	0.715000
$855$	0	0
$856$	45420.7	1.81361
$857$	−24528.1	−0.977669	−0.488834	−	0.872377i	$-0.662578\pi$
$857$	−24528.1	−0.977669	−0.488834	+	0.872377i	$0.662578\pi$
$858$	26051.8	1.03659
$859$	−37536.3	−1.49095	−0.745473	−	0.666536i	$-0.767776\pi$
$859$	−37536.3	−1.49095	−0.745473	+	0.666536i	$0.767776\pi$
$860$	0	0
$861$	35892.2	1.42068
$862$	−1472.19	−0.0581704
$863$	2098.30	0.0827659	0.0413830	−	0.999143i	$-0.486824\pi$
$863$	2098.30	0.0827659	0.0413830	+	0.999143i	$0.486824\pi$
$864$	−28421.9	−1.11914
$865$	0	0
$866$	10671.8	0.418757
$867$	−35317.0	−1.38342
$868$	−15446.7	−0.604028
$869$	−1125.85	−0.0439493
$870$	0	0
$871$	−15063.1	−0.585986
$872$	−27242.7	−1.05797
$873$	82169.3	3.18558
$874$	10808.7	0.418318
$875$	0	0
$876$	−49303.5	−1.90161
$877$	9857.12	0.379534	0.189767	−	0.981829i	$-0.439227\pi$
$877$	9857.12	0.379534	0.189767	+	0.981829i	$0.439227\pi$
$878$	−20357.2	−0.782484
$879$	−71181.5	−2.73139
$880$	0	0
$881$	2301.91	0.0880289	0.0440144	−	0.999031i	$-0.485985\pi$
$881$	2301.91	0.0880289	0.0440144	+	0.999031i	$0.485985\pi$
$882$	28049.6	1.07084
$883$	−25401.4	−0.968093	−0.484047	−	0.875042i	$-0.660834\pi$
$883$	−25401.4	−0.968093	−0.484047	+	0.875042i	$0.660834\pi$
$884$	−24113.0	−0.917429
$885$	0	0
$886$	60931.2	2.31041
$887$	11451.6	0.433493	0.216746	−	0.976228i	$-0.430456\pi$
$887$	11451.6	0.433493	0.216746	+	0.976228i	$0.430456\pi$
$888$	−145861.	−5.51214
$889$	−11979.1	−0.451929
$890$	0	0
$891$	3667.30	0.137889
$892$	100196.	3.76098
$893$	2643.67	0.0990671
$894$	11787.3	0.440969
$895$	0	0
$896$	24928.7	0.929475
$897$	−45664.6	−1.69977
$898$	−21715.0	−0.806949
$899$	−17027.1	−0.631685
$900$	0	0
$901$	−8597.18	−0.317884
$902$	−17537.1	−0.647364
$903$	−42896.4	−1.58084
$904$	−68803.2	−2.53137
$905$	0	0
$906$	−123538.	−4.53010
$907$	−53074.7	−1.94302	−0.971508	−	0.237005i	$-0.923834\pi$
$907$	−53074.7	−1.94302	−0.971508	+	0.237005i	$0.923834\pi$
$908$	90085.2	3.29249
$909$	−46310.3	−1.68979
$910$	0	0
$911$	−20284.7	−0.737718	−0.368859	−	0.929485i	$-0.620251\pi$
$911$	−20284.7	−0.737718	−0.368859	+	0.929485i	$0.620251\pi$
$912$	−17843.0	−0.647852
$913$	5544.65	0.200987
$914$	1487.70	0.0538389
$915$	0	0
$916$	8896.27	0.320896
$917$	378.466	0.0136293
$918$	−26970.3	−0.969665
$919$	34782.0	1.24848	0.624240	−	0.781233i	$-0.285409\pi$
$919$	34782.0	1.24848	0.624240	+	0.781233i	$0.285409\pi$
$920$	0	0
$921$	24404.2	0.873121
$922$	26717.4	0.954329
$923$	448.213	0.0159839
$924$	29285.6	1.04267
$925$	0	0
$926$	40964.6	1.45376
$927$	−8340.02	−0.295493
$928$	−45700.8	−1.61660
$929$	−15586.2	−0.550448	−0.275224	−	0.961380i	$-0.588752\pi$
$929$	−15586.2	−0.550448	−0.275224	+	0.961380i	$0.588752\pi$
$930$	0	0
$931$	2181.50	0.0767948
$932$	−49790.3	−1.74993
$933$	−75001.8	−2.63178
$934$	−16029.0	−0.561546
$935$	0	0
$936$	111537.	3.89498
$937$	15194.9	0.529770	0.264885	−	0.964280i	$-0.414666\pi$
$937$	15194.9	0.529770	0.264885	+	0.964280i	$0.414666\pi$
$938$	−24576.2	−0.855483
$939$	−14075.6	−0.489179
$940$	0	0
$941$	−48650.1	−1.68538	−0.842692	−	0.538396i	$-0.819031\pi$
$941$	−48650.1	−1.68538	−0.842692	+	0.538396i	$0.819031\pi$
$942$	−87934.4	−3.04146
$943$	30739.7	1.06153
$944$	49759.0	1.71559
$945$	0	0
$946$	20959.4	0.720348
$947$	−3258.29	−0.111806	−0.0559030	−	0.998436i	$-0.517804\pi$
$947$	−3258.29	−0.111806	−0.0559030	+	0.998436i	$0.517804\pi$
$948$	−13711.2	−0.469744
$949$	15067.0	0.515380
$950$	0	0
$951$	−81257.8	−2.77073
$952$	−21583.0	−0.734780
$953$	44488.8	1.51221	0.756104	−	0.654451i	$-0.227100\pi$
$953$	44488.8	1.51221	0.756104	+	0.654451i	$0.227100\pi$
$954$	72487.6	2.46003
$955$	0	0
$956$	39691.4	1.34279
$957$	32281.8	1.09041
$958$	1264.27	0.0426376
$959$	10816.6	0.364218
$960$	0	0
$961$	−26462.0	−0.888253
$962$	81250.9	2.72311
$963$	−44367.4	−1.48465
$964$	−127857.	−4.27179
$965$	0	0
$966$	−74504.2	−2.48150
$967$	5791.53	0.192599	0.0962994	−	0.995352i	$-0.469299\pi$
$967$	5791.53	0.192599	0.0962994	+	0.995352i	$0.469299\pi$
$968$	57784.4	1.91866
$969$	−4772.96	−0.158235
$970$	0	0
$971$	−17829.5	−0.589265	−0.294632	−	0.955611i	$-0.595197\pi$
$971$	−17829.5	−0.589265	−0.294632	+	0.955611i	$0.595197\pi$
$972$	−43160.8	−1.42426
$973$	10069.0	0.331754
$974$	36850.0	1.21227
$975$	0	0
$976$	−25228.2	−0.827394
$977$	2645.66	0.0866349	0.0433174	−	0.999061i	$-0.486207\pi$
$977$	2645.66	0.0866349	0.0433174	+	0.999061i	$0.486207\pi$
$978$	72205.7	2.36082
$979$	11143.0	0.363770
$980$	0	0
$981$	26610.9	0.866076
$982$	42889.3	1.39374
$983$	−3880.50	−0.125909	−0.0629545	−	0.998016i	$-0.520052\pi$
$983$	−3880.50	−0.125909	−0.0629545	+	0.998016i	$0.520052\pi$
$984$	−117168.	−3.79593
$985$	0	0
$986$	−43366.6	−1.40068
$987$	−18222.7	−0.587676
$988$	15812.1	0.509158
$989$	−36738.4	−1.18121
$990$	0	0
$991$	57787.1	1.85234	0.926170	−	0.377106i	$-0.123081\pi$
$991$	57787.1	1.85234	0.926170	+	0.377106i	$0.123081\pi$
$992$	8935.19	0.285980
$993$	−15307.3	−0.489186
$994$	731.284	0.0233349
$995$	0	0
$996$	67525.4	2.14822
$997$	−30649.0	−0.973585	−0.486793	−	0.873518i	$-0.661833\pi$
$997$	−30649.0	−0.973585	−0.486793	+	0.873518i	$0.661833\pi$
$998$	−9662.48	−0.306473
$999$	62614.9	1.98303

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.4.a.f.1.1		3
5.2	odd	4		475.4.b.f.324.1		6
5.3	odd	4		475.4.b.f.324.6		6
5.4	even	2		19.4.a.b.1.3	✓	3
15.14	odd	2		171.4.a.f.1.1		3
20.19	odd	2		304.4.a.i.1.3		3
35.34	odd	2		931.4.a.c.1.3		3
40.19	odd	2		1216.4.a.u.1.1		3
40.29	even	2		1216.4.a.s.1.3		3
55.54	odd	2		2299.4.a.h.1.1		3
95.94	odd	2		361.4.a.i.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.4.a.b.1.3	✓	3	5.4	even	2
171.4.a.f.1.1		3	15.14	odd	2
304.4.a.i.1.3		3	20.19	odd	2
361.4.a.i.1.1		3	95.94	odd	2
475.4.a.f.1.1		3	1.1	even	1	trivial
475.4.b.f.324.1		6	5.2	odd	4
475.4.b.f.324.6		6	5.3	odd	4
931.4.a.c.1.3		3	35.34	odd	2
1216.4.a.s.1.3		3	40.29	even	2
1216.4.a.u.1.1		3	40.19	odd	2
2299.4.a.h.1.1		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)

\(f(q)\)	\(=\)	\(q-5.07177 q^{2} +8.66998 q^{3} +17.7229 q^{4} -43.9722 q^{6} +15.1058 q^{7} -49.3121 q^{8} +48.1686 q^{9} +12.6171 q^{11} +153.657 q^{12} -46.9571 q^{13} -76.6130 q^{14} +108.317 q^{16} +28.9745 q^{17} -244.300 q^{18} -19.0000 q^{19} +130.967 q^{21} -63.9911 q^{22} +112.166 q^{23} -427.535 q^{24} +238.155 q^{26} +183.531 q^{27} +267.717 q^{28} +295.107 q^{29} -57.6979 q^{31} -154.862 q^{32} +109.390 q^{33} -146.952 q^{34} +853.685 q^{36} +341.167 q^{37} +96.3636 q^{38} -407.117 q^{39} +274.056 q^{41} -664.233 q^{42} -327.536 q^{43} +223.611 q^{44} -568.879 q^{46} -139.140 q^{47} +939.106 q^{48} -114.816 q^{49} +251.209 q^{51} -832.214 q^{52} -296.715 q^{53} -930.828 q^{54} -744.897 q^{56} -164.730 q^{57} -1496.72 q^{58} +459.383 q^{59} -232.911 q^{61} +292.631 q^{62} +727.623 q^{63} -81.1127 q^{64} -554.801 q^{66} +320.784 q^{67} +513.511 q^{68} +972.475 q^{69} -9.54518 q^{71} -2375.30 q^{72} -320.868 q^{73} -1730.32 q^{74} -336.734 q^{76} +190.591 q^{77} +2064.80 q^{78} -89.2323 q^{79} +290.661 q^{81} -1389.95 q^{82} +439.455 q^{83} +2321.10 q^{84} +1661.19 q^{86} +2558.58 q^{87} -622.176 q^{88} +883.164 q^{89} -709.322 q^{91} +1987.90 q^{92} -500.240 q^{93} +705.688 q^{94} -1342.65 q^{96} +1705.87 q^{97} +582.321 q^{98} +607.748 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} - q^{3} + 21 q^{4} - 65 q^{6} + 35 q^{7} - 27 q^{8} + 48 q^{9} + 16 q^{11} + 115 q^{12} - 65 q^{13} + 37 q^{14} + 33 q^{16} - 29 q^{17} - 138 q^{18} - 57 q^{19} - 25 q^{21} - 118 q^{22} + 101 q^{23}+ \cdots + 250 q^{99}+O(q^{100}) \) 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

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Database

Properties

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