LMFDB - Embedded newform 475.4.a.f.1.3

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			1.3
Root			$-3.20905$ 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)

$f(q)$	$=$	\(q+3.96257 q^{2} -6.71610 q^{3} +7.70200 q^{4} -26.6130 q^{6} +25.8362 q^{7} -1.18085 q^{8} +18.1060 q^{9} -8.13420 q^{11} -51.7274 q^{12} +4.56640 q^{13} +102.378 q^{14} -66.2952 q^{16} +62.5850 q^{17} +71.7464 q^{18} -19.0000 q^{19} -173.518 q^{21} -32.2324 q^{22} +52.7502 q^{23} +7.93070 q^{24} +18.0947 q^{26} +59.7330 q^{27} +198.990 q^{28} +171.620 q^{29} +168.749 q^{31} -253.253 q^{32} +54.6301 q^{33} +247.998 q^{34} +139.452 q^{36} +147.534 q^{37} -75.2889 q^{38} -30.6684 q^{39} +308.774 q^{41} -687.580 q^{42} +448.950 q^{43} -62.6496 q^{44} +209.027 q^{46} -113.335 q^{47} +445.245 q^{48} +324.509 q^{49} -420.327 q^{51} +35.1704 q^{52} -155.402 q^{53} +236.696 q^{54} -30.5086 q^{56} +127.606 q^{57} +680.059 q^{58} +182.347 q^{59} +404.080 q^{61} +668.681 q^{62} +467.790 q^{63} -473.172 q^{64} +216.476 q^{66} +106.400 q^{67} +482.030 q^{68} -354.276 q^{69} +472.079 q^{71} -21.3805 q^{72} -843.821 q^{73} +584.616 q^{74} -146.338 q^{76} -210.157 q^{77} -121.526 q^{78} -591.036 q^{79} -890.035 q^{81} +1223.54 q^{82} -290.388 q^{83} -1336.44 q^{84} +1779.00 q^{86} -1152.62 q^{87} +9.60526 q^{88} -964.896 q^{89} +117.978 q^{91} +406.282 q^{92} -1133.34 q^{93} -449.099 q^{94} +1700.87 q^{96} +219.495 q^{97} +1285.89 q^{98} -147.278 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	3.96257	1.40098	0.700491	−	0.713661i	$-0.252964\pi$
$2$	3.96257	1.40098	0.700491	+	0.713661i	$0.252964\pi$
$3$	−6.71610	−1.29251	−0.646257	−	0.763120i	$-0.723667\pi$
$3$	−6.71610	−1.29251	−0.646257	+	0.763120i	$0.723667\pi$
$4$	7.70200	0.962750
$5$	0	0
$5$	0	0
$6$	−26.6130	−1.81079
$7$	25.8362	1.39502	0.697512	−	0.716573i	$-0.254290\pi$
$7$	25.8362	1.39502	0.697512	+	0.716573i	$0.254290\pi$
$8$	−1.18085	−0.0521866
$9$	18.1060	0.670593
$10$	0	0
$11$	−8.13420	−0.222959	−0.111480	−	0.993767i	$-0.535559\pi$
$11$	−8.13420	−0.222959	−0.111480	+	0.993767i	$0.535559\pi$
$12$	−51.7274	−1.24437
$13$	4.56640	0.0974224	0.0487112	−	0.998813i	$-0.484489\pi$
$13$	4.56640	0.0974224	0.0487112	+	0.998813i	$0.484489\pi$
$14$	102.378	1.95440
$15$	0	0
$16$	−66.2952	−1.03586
$17$	62.5850	0.892888	0.446444	−	0.894812i	$-0.352690\pi$
$17$	62.5850	0.892888	0.446444	+	0.894812i	$0.352690\pi$
$18$	71.7464	0.939488
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	0	0
$21$	−173.518	−1.80309
$22$	−32.2324	−0.312362
$23$	52.7502	0.478225	0.239113	−	0.970992i	$-0.423144\pi$
$23$	52.7502	0.478225	0.239113	+	0.970992i	$0.423144\pi$
$24$	7.93070	0.0674520
$25$	0	0
$26$	18.0947	0.136487
$27$	59.7330	0.425764
$28$	198.990	1.34306
$29$	171.620	1.09894	0.549468	−	0.835515i	$-0.314831\pi$
$29$	171.620	1.09894	0.549468	+	0.835515i	$0.314831\pi$
$30$	0	0
$31$	168.749	0.977685	0.488842	−	0.872372i	$-0.337419\pi$
$31$	168.749	0.977685	0.488842	+	0.872372i	$0.337419\pi$
$32$	−253.253	−1.39904
$33$	54.6301	0.288178
$34$	247.998	1.25092
$35$	0	0
$36$	139.452	0.645613
$37$	147.534	0.655528	0.327764	−	0.944760i	$-0.393705\pi$
$37$	147.534	0.655528	0.327764	+	0.944760i	$0.393705\pi$
$38$	−75.2889	−0.321407
$39$	−30.6684	−0.125920
$40$	0	0
$41$	308.774	1.17616	0.588078	−	0.808804i	$-0.299885\pi$
$41$	308.774	1.17616	0.588078	+	0.808804i	$0.299885\pi$
$42$	−687.580	−2.52609
$43$	448.950	1.59219	0.796096	−	0.605170i	$-0.206895\pi$
$43$	448.950	1.59219	0.796096	+	0.605170i	$0.206895\pi$
$44$	−62.6496	−0.214654
$45$	0	0
$46$	209.027	0.669985
$47$	−113.335	−0.351737	−0.175868	−	0.984414i	$-0.556273\pi$
$47$	−113.335	−0.351737	−0.175868	+	0.984414i	$0.556273\pi$
$48$	445.245	1.33887
$49$	324.509	0.946091
$50$	0	0
$51$	−420.327	−1.15407
$52$	35.1704	0.0937934
$53$	−155.402	−0.402758	−0.201379	−	0.979513i	$-0.564542\pi$
$53$	−155.402	−0.402758	−0.201379	+	0.979513i	$0.564542\pi$
$54$	236.696	0.596487
$55$	0	0
$56$	−30.5086	−0.0728016
$57$	127.606	0.296523
$58$	680.059	1.53959
$59$	182.347	0.402365	0.201183	−	0.979554i	$-0.435522\pi$
$59$	182.347	0.402365	0.201183	+	0.979554i	$0.435522\pi$
$60$	0	0
$61$	404.080	0.848149	0.424075	−	0.905627i	$-0.360599\pi$
$61$	404.080	0.848149	0.424075	+	0.905627i	$0.360599\pi$
$62$	668.681	1.36972
$63$	467.790	0.935492
$64$	−473.172	−0.924164
$65$	0	0
$66$	216.476	0.403732
$67$	106.400	0.194013	0.0970064	−	0.995284i	$-0.469073\pi$
$67$	106.400	0.194013	0.0970064	+	0.995284i	$0.469073\pi$
$68$	482.030	0.859628
$69$	−354.276	−0.618113
$70$	0	0
$71$	472.079	0.789091	0.394546	−	0.918876i	$-0.370902\pi$
$71$	472.079	0.789091	0.394546	+	0.918876i	$0.370902\pi$
$72$	−21.3805	−0.0349960
$73$	−843.821	−1.35290	−0.676451	−	0.736488i	$-0.736483\pi$
$73$	−843.821	−1.35290	−0.676451	+	0.736488i	$0.736483\pi$
$74$	584.616	0.918382
$75$	0	0
$76$	−146.338	−0.220870
$77$	−210.157	−0.311034
$78$	−121.526	−0.176411
$79$	−591.036	−0.841731	−0.420866	−	0.907123i	$-0.638274\pi$
$79$	−591.036	−0.841731	−0.420866	+	0.907123i	$0.638274\pi$
$80$	0	0
$81$	−890.035	−1.22090
$82$	1223.54	1.64777
$83$	−290.388	−0.384027	−0.192013	−	0.981392i	$-0.561502\pi$
$83$	−290.388	−0.384027	−0.192013	+	0.981392i	$0.561502\pi$
$84$	−1336.44	−1.73592
$85$	0	0
$86$	1779.00	2.23063
$87$	−1152.62	−1.42039
$88$	9.60526	0.0116355
$89$	−964.896	−1.14920	−0.574600	−	0.818435i	$-0.694842\pi$
$89$	−964.896	−1.14920	−0.574600	+	0.818435i	$0.694842\pi$
$90$	0	0
$91$	117.978	0.135907
$92$	406.282	0.460411
$93$	−1133.34	−1.26367
$94$	−449.099	−0.492777
$95$	0	0
$96$	1700.87	1.80828
$97$	219.495	0.229756	0.114878	−	0.993380i	$-0.463352\pi$
$97$	219.495	0.229756	0.114878	+	0.993380i	$0.463352\pi$
$98$	1285.89	1.32546
$99$	−147.278	−0.149515
$100$	0	0
$101$	1447.94	1.42649	0.713247	−	0.700913i	$-0.247224\pi$
$101$	1447.94	1.42649	0.713247	+	0.700913i	$0.247224\pi$
$102$	−1665.58	−1.61683
$103$	−883.567	−0.845247	−0.422623	−	0.906305i	$-0.638891\pi$
$103$	−883.567	−0.845247	−0.422623	+	0.906305i	$0.638891\pi$
$104$	−5.39223	−0.00508415
$105$	0	0
$106$	−615.793	−0.564256
$107$	1307.82	1.18160	0.590801	−	0.806817i	$-0.298812\pi$
$107$	1307.82	1.18160	0.590801	+	0.806817i	$0.298812\pi$
$108$	460.064	0.409904
$109$	870.507	0.764949	0.382475	−	0.923966i	$-0.375072\pi$
$109$	870.507	0.764949	0.382475	+	0.923966i	$0.375072\pi$
$110$	0	0
$111$	−990.856	−0.847279
$112$	−1712.82	−1.44505
$113$	1181.41	0.983521	0.491761	−	0.870730i	$-0.336354\pi$
$113$	1181.41	0.983521	0.491761	+	0.870730i	$0.336354\pi$
$114$	505.648	0.415423
$115$	0	0
$116$	1321.82	1.05800
$117$	82.6792	0.0653307
$118$	722.564	0.563707
$119$	1616.96	1.24560
$120$	0	0
$121$	−1264.83	−0.950289
$122$	1601.20	1.18824
$123$	−2073.76	−1.52020
$124$	1299.71	0.941266
$125$	0	0
$126$	1853.65	1.31061
$127$	−887.509	−0.620108	−0.310054	−	0.950719i	$-0.600347\pi$
$127$	−887.509	−0.620108	−0.310054	+	0.950719i	$0.600347\pi$
$128$	151.044	0.104301
$129$	−3015.19	−2.05793
$130$	0	0
$131$	−2344.76	−1.56384	−0.781920	−	0.623379i	$-0.785759\pi$
$131$	−2344.76	−1.56384	−0.781920	+	0.623379i	$0.785759\pi$
$132$	420.761	0.277444
$133$	−490.888	−0.320040
$134$	421.619	0.271809
$135$	0	0
$136$	−73.9034	−0.0465968
$137$	−2244.82	−1.39991	−0.699956	−	0.714186i	$-0.746797\pi$
$137$	−2244.82	−1.39991	−0.699956	+	0.714186i	$0.746797\pi$
$138$	−1403.84	−0.865964
$139$	−296.146	−0.180711	−0.0903554	−	0.995910i	$-0.528800\pi$
$139$	−296.146	−0.180711	−0.0903554	+	0.995910i	$0.528800\pi$
$140$	0	0
$141$	761.170	0.454625
$142$	1870.65	1.10550
$143$	−37.1440	−0.0217212
$144$	−1200.34	−0.694642
$145$	0	0
$146$	−3343.70	−1.89539
$147$	−2179.44	−1.22284
$148$	1136.31	0.631109
$149$	1791.09	0.984780	0.492390	−	0.870375i	$-0.336123\pi$
$149$	1791.09	0.984780	0.492390	+	0.870375i	$0.336123\pi$
$150$	0	0
$151$	−2352.65	−1.26792	−0.633960	−	0.773366i	$-0.718571\pi$
$151$	−2352.65	−1.26792	−0.633960	+	0.773366i	$0.718571\pi$
$152$	22.4361	0.0119724
$153$	1133.16	0.598764
$154$	−832.762	−0.435752
$155$	0	0
$156$	−236.208	−0.121229
$157$	1438.26	0.731118	0.365559	−	0.930788i	$-0.380878\pi$
$157$	1438.26	0.731118	0.365559	+	0.930788i	$0.380878\pi$
$158$	−2342.02	−1.17925
$159$	1043.70	0.520570
$160$	0	0
$161$	1362.86	0.667135
$162$	−3526.83	−1.71046
$163$	−127.493	−0.0612640	−0.0306320	−	0.999531i	$-0.509752\pi$
$163$	−127.493	−0.0612640	−0.0306320	+	0.999531i	$0.509752\pi$
$164$	2378.18	1.13234
$165$	0	0
$166$	−1150.68	−0.538014
$167$	−3419.05	−1.58428	−0.792139	−	0.610341i	$-0.791033\pi$
$167$	−3419.05	−1.58428	−0.792139	+	0.610341i	$0.791033\pi$
$168$	204.899	0.0940971
$169$	−2176.15	−0.990509
$170$	0	0
$171$	−344.014	−0.153844
$172$	3457.81	1.53288
$173$	362.598	0.159352	0.0796758	−	0.996821i	$-0.474611\pi$
$173$	362.598	0.159352	0.0796758	+	0.996821i	$0.474611\pi$
$174$	−4567.34	−1.98994
$175$	0	0
$176$	539.258	0.230955
$177$	−1224.66	−0.520063
$178$	−3823.47	−1.61001
$179$	2417.89	1.00962	0.504809	−	0.863231i	$-0.331563\pi$
$179$	2417.89	1.00962	0.504809	+	0.863231i	$0.331563\pi$
$180$	0	0
$181$	−2444.64	−1.00391	−0.501957	−	0.864892i	$-0.667387\pi$
$181$	−2444.64	−1.00391	−0.501957	+	0.864892i	$0.667387\pi$
$182$	467.498	0.190403
$183$	−2713.84	−1.09624
$184$	−62.2900	−0.0249570
$185$	0	0
$186$	−4490.93	−1.77038
$187$	−509.079	−0.199078
$188$	−872.908	−0.338635
$189$	1543.27	0.593951
$190$	0	0
$191$	−1387.66	−0.525693	−0.262846	−	0.964838i	$-0.584661\pi$
$191$	−1387.66	−0.525693	−0.262846	+	0.964838i	$0.584661\pi$
$192$	3177.87	1.19449
$193$	3208.03	1.19647	0.598237	−	0.801319i	$-0.295868\pi$
$193$	3208.03	1.19647	0.598237	+	0.801319i	$0.295868\pi$
$194$	869.764	0.321884
$195$	0	0
$196$	2499.37	0.910849
$197$	3445.36	1.24605	0.623025	−	0.782202i	$-0.285903\pi$
$197$	3445.36	1.24605	0.623025	+	0.782202i	$0.285903\pi$
$198$	−583.599	−0.209468
$199$	2025.71	0.721602	0.360801	−	0.932643i	$-0.382503\pi$
$199$	2025.71	0.721602	0.360801	+	0.932643i	$0.382503\pi$
$200$	0	0
$201$	−714.595	−0.250764
$202$	5737.59	1.99849
$203$	4434.02	1.53304
$204$	−3237.36	−1.11108
$205$	0	0
$206$	−3501.20	−1.18418
$207$	955.095	0.320694
$208$	−302.730	−0.100916
$209$	154.550	0.0511504
$210$	0	0
$211$	4309.54	1.40607	0.703036	−	0.711155i	$-0.251827\pi$
$211$	4309.54	1.40607	0.703036	+	0.711155i	$0.251827\pi$
$212$	−1196.91	−0.387755
$213$	−3170.53	−1.01991
$214$	5182.33	1.65540
$215$	0	0
$216$	−70.5357	−0.0222192
$217$	4359.84	1.36389
$218$	3449.45	1.07168
$219$	5667.19	1.74864
$220$	0	0
$221$	285.788	0.0869873
$222$	−3926.34	−1.18702
$223$	−825.648	−0.247935	−0.123968	−	0.992286i	$-0.539562\pi$
$223$	−825.648	−0.247935	−0.123968	+	0.992286i	$0.539562\pi$
$224$	−6543.09	−1.95169
$225$	0	0
$226$	4681.43	1.37790
$227$	1501.19	0.438931	0.219466	−	0.975620i	$-0.429569\pi$
$227$	1501.19	0.438931	0.219466	+	0.975620i	$0.429569\pi$
$228$	982.821	0.285478
$229$	−5250.40	−1.51509	−0.757547	−	0.652781i	$-0.773602\pi$
$229$	−5250.40	−1.51509	−0.757547	+	0.652781i	$0.773602\pi$
$230$	0	0
$231$	1411.43	0.402015
$232$	−202.658	−0.0573497
$233$	−2139.06	−0.601435	−0.300717	−	0.953713i	$-0.597226\pi$
$233$	−2139.06	−0.601435	−0.300717	+	0.953713i	$0.597226\pi$
$234$	327.623	0.0915272
$235$	0	0
$236$	1404.44	0.387377
$237$	3969.46	1.08795
$238$	6407.32	1.74506
$239$	3772.70	1.02107	0.510534	−	0.859857i	$-0.329448\pi$
$239$	3772.70	1.02107	0.510534	+	0.859857i	$0.329448\pi$
$240$	0	0
$241$	6415.39	1.71474	0.857369	−	0.514702i	$-0.172097\pi$
$241$	6415.39	1.71474	0.857369	+	0.514702i	$0.172097\pi$
$242$	−5012.00	−1.33134
$243$	4364.77	1.15226
$244$	3112.22	0.816555
$245$	0	0
$246$	−8217.41	−2.12977
$247$	−86.7616	−0.0223502
$248$	−199.267	−0.0510221
$249$	1950.27	0.496360
$250$	0	0
$251$	−6277.31	−1.57857	−0.789283	−	0.614029i	$-0.789548\pi$
$251$	−6277.31	−1.57857	−0.789283	+	0.614029i	$0.789548\pi$
$252$	3602.92	0.900645
$253$	−429.081	−0.106625
$254$	−3516.82	−0.868760
$255$	0	0
$256$	4383.90	1.07029
$257$	3183.98	0.772807	0.386404	−	0.922330i	$-0.373717\pi$
$257$	3183.98	0.772807	0.386404	+	0.922330i	$0.373717\pi$
$258$	−11947.9	−2.88312
$259$	3811.73	0.914476
$260$	0	0
$261$	3107.36	0.736938
$262$	−9291.31	−2.19091
$263$	2624.18	0.615261	0.307630	−	0.951506i	$-0.400464\pi$
$263$	2624.18	0.615261	0.307630	+	0.951506i	$0.400464\pi$
$264$	−64.5099	−0.0150391
$265$	0	0
$266$	−1945.18	−0.448371
$267$	6480.34	1.48536
$268$	819.495	0.186786
$269$	7444.76	1.68742	0.843708	−	0.536803i	$-0.180368\pi$
$269$	7444.76	1.68742	0.843708	+	0.536803i	$0.180368\pi$
$270$	0	0
$271$	−4004.49	−0.897621	−0.448810	−	0.893627i	$-0.648152\pi$
$271$	−4004.49	−0.897621	−0.448810	+	0.893627i	$0.648152\pi$
$272$	−4149.08	−0.924909
$273$	−792.355	−0.175661
$274$	−8895.27	−1.96125
$275$	0	0
$276$	−2728.63	−0.595088
$277$	5830.66	1.26473	0.632365	−	0.774671i	$-0.282084\pi$
$277$	5830.66	1.26473	0.632365	+	0.774671i	$0.282084\pi$
$278$	−1173.50	−0.253172
$279$	3055.37	0.655628
$280$	0	0
$281$	−7504.37	−1.59314	−0.796572	−	0.604544i	$-0.793355\pi$
$281$	−7504.37	−1.59314	−0.796572	+	0.604544i	$0.793355\pi$
$282$	3016.19	0.636921
$283$	−5910.87	−1.24157	−0.620785	−	0.783980i	$-0.713186\pi$
$283$	−5910.87	−1.24157	−0.620785	+	0.783980i	$0.713186\pi$
$284$	3635.95	0.759697
$285$	0	0
$286$	−147.186	−0.0304311
$287$	7977.54	1.64076
$288$	−4585.40	−0.938184
$289$	−996.118	−0.202752
$290$	0	0
$291$	−1474.15	−0.296962
$292$	−6499.11	−1.30251
$293$	−3245.59	−0.647131	−0.323566	−	0.946206i	$-0.604882\pi$
$293$	−3245.59	−0.647131	−0.323566	+	0.946206i	$0.604882\pi$
$294$	−8636.18	−1.71317
$295$	0	0
$296$	−174.216	−0.0342098
$297$	−485.880	−0.0949280
$298$	7097.35	1.37966
$299$	240.878	0.0465898
$300$	0	0
$301$	11599.2	2.22115
$302$	−9322.54	−1.77633
$303$	−9724.54	−1.84376
$304$	1259.61	0.237643
$305$	0	0
$306$	4490.25	0.838857
$307$	−7489.14	−1.39227	−0.696137	−	0.717909i	$-0.745099\pi$
$307$	−7489.14	−1.39227	−0.696137	+	0.717909i	$0.745099\pi$
$308$	−1618.63	−0.299448
$309$	5934.12	1.09249
$310$	0	0
$311$	2136.71	0.389588	0.194794	−	0.980844i	$-0.437596\pi$
$311$	2136.71	0.389588	0.194794	+	0.980844i	$0.437596\pi$
$312$	36.2147	0.00657133
$313$	−2212.15	−0.399483	−0.199742	−	0.979849i	$-0.564010\pi$
$313$	−2212.15	−0.399483	−0.199742	+	0.979849i	$0.564010\pi$
$314$	5699.21	1.02428
$315$	0	0
$316$	−4552.16	−0.810377
$317$	−429.326	−0.0760674	−0.0380337	−	0.999276i	$-0.512109\pi$
$317$	−429.326	−0.0760674	−0.0380337	+	0.999276i	$0.512109\pi$
$318$	4135.73	0.729309
$319$	−1396.00	−0.245018
$320$	0	0
$321$	−8783.43	−1.52724
$322$	5400.45	0.934644
$323$	−1189.11	−0.204842
$324$	−6855.05	−1.17542
$325$	0	0
$326$	−505.201	−0.0858297
$327$	−5846.42	−0.988708
$328$	−364.615	−0.0613796
$329$	−2928.15	−0.490681
$330$	0	0
$331$	−765.454	−0.127109	−0.0635546	−	0.997978i	$-0.520244\pi$
$331$	−765.454	−0.127109	−0.0635546	+	0.997978i	$0.520244\pi$
$332$	−2236.57	−0.369722
$333$	2671.26	0.439592
$334$	−13548.3	−2.21954
$335$	0	0
$336$	11503.4	1.86775
$337$	3049.81	0.492978	0.246489	−	0.969146i	$-0.420723\pi$
$337$	3049.81	0.492978	0.246489	+	0.969146i	$0.420723\pi$
$338$	−8623.15	−1.38768
$339$	−7934.48	−1.27121
$340$	0	0
$341$	−1372.64	−0.217984
$342$	−1363.18	−0.215533
$343$	−477.732	−0.0752045
$344$	−530.142	−0.0830912
$345$	0	0
$346$	1436.82	0.223249
$347$	5907.00	0.913845	0.456922	−	0.889507i	$-0.348952\pi$
$347$	5907.00	0.913845	0.456922	+	0.889507i	$0.348952\pi$
$348$	−8877.48	−1.36748
$349$	−12107.4	−1.85700	−0.928502	−	0.371327i	$-0.878903\pi$
$349$	−12107.4	−1.85700	−0.928502	+	0.371327i	$0.878903\pi$
$350$	0	0
$351$	272.765	0.0414789
$352$	2060.01	0.311929
$353$	2420.40	0.364943	0.182471	−	0.983211i	$-0.441590\pi$
$353$	2420.40	0.364943	0.182471	+	0.983211i	$0.441590\pi$
$354$	−4852.81	−0.728599
$355$	0	0
$356$	−7431.63	−1.10639
$357$	−10859.7	−1.60995
$358$	9581.07	1.41446
$359$	−1455.80	−0.214023	−0.107011	−	0.994258i	$-0.534128\pi$
$359$	−1455.80	−0.214023	−0.107011	+	0.994258i	$0.534128\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	−9687.06	−1.40647
$363$	8494.76	1.22826
$364$	908.670	0.130844
$365$	0	0
$366$	−10753.8	−1.53582
$367$	−8783.80	−1.24935	−0.624674	−	0.780886i	$-0.714768\pi$
$367$	−8783.80	−1.24935	−0.624674	+	0.780886i	$0.714768\pi$
$368$	−3497.08	−0.495375
$369$	5590.66	0.788721
$370$	0	0
$371$	−4015.00	−0.561856
$372$	−8728.95	−1.21660
$373$	9199.84	1.27708	0.638538	−	0.769590i	$-0.279539\pi$
$373$	9199.84	1.27708	0.638538	+	0.769590i	$0.279539\pi$
$374$	−2017.26	−0.278904
$375$	0	0
$376$	133.832	0.0183560
$377$	783.688	0.107061
$378$	6115.34	0.832114
$379$	−6161.38	−0.835063	−0.417531	−	0.908662i	$-0.637105\pi$
$379$	−6161.38	−0.835063	−0.417531	+	0.908662i	$0.637105\pi$
$380$	0	0
$381$	5960.60	0.801498
$382$	−5498.70	−0.736486
$383$	−2630.79	−0.350985	−0.175492	−	0.984481i	$-0.556152\pi$
$383$	−2630.79	−0.350985	−0.175492	+	0.984481i	$0.556152\pi$
$384$	−1014.43	−0.134810
$385$	0	0
$386$	12712.1	1.67624
$387$	8128.69	1.06771
$388$	1690.55	0.221197
$389$	−5866.48	−0.764633	−0.382317	−	0.924031i	$-0.624874\pi$
$389$	−5866.48	−0.764633	−0.382317	+	0.924031i	$0.624874\pi$
$390$	0	0
$391$	3301.37	0.427001
$392$	−383.196	−0.0493733
$393$	15747.7	2.02129
$394$	13652.5	1.74569
$395$	0	0
$396$	−1134.33	−0.143945
$397$	14254.0	1.80199	0.900993	−	0.433833i	$-0.142839\pi$
$397$	14254.0	1.80199	0.900993	+	0.433833i	$0.142839\pi$
$398$	8027.03	1.01095
$399$	3296.85	0.413657
$400$	0	0
$401$	9909.27	1.23403	0.617014	−	0.786952i	$-0.288342\pi$
$401$	9909.27	1.23403	0.617014	+	0.786952i	$0.288342\pi$
$402$	−2831.64	−0.351316
$403$	770.576	0.0952484
$404$	11152.1	1.37336
$405$	0	0
$406$	17570.1	2.14776
$407$	−1200.07	−0.146156
$408$	496.343	0.0602270
$409$	5805.87	0.701912	0.350956	−	0.936392i	$-0.385857\pi$
$409$	5805.87	0.701912	0.350956	+	0.936392i	$0.385857\pi$
$410$	0	0
$411$	15076.4	1.80941
$412$	−6805.23	−0.813761
$413$	4711.15	0.561309
$414$	3784.64	0.449287
$415$	0	0
$416$	−1156.45	−0.136298
$417$	1988.95	0.233571
$418$	612.415	0.0716608
$419$	12260.9	1.42955	0.714777	−	0.699353i	$-0.246528\pi$
$419$	12260.9	1.42955	0.714777	+	0.699353i	$0.246528\pi$
$420$	0	0
$421$	5837.85	0.675818	0.337909	−	0.941179i	$-0.390280\pi$
$421$	5837.85	0.675818	0.337909	+	0.941179i	$0.390280\pi$
$422$	17076.9	1.96988
$423$	−2052.05	−0.235872
$424$	183.507	0.0210186
$425$	0	0
$426$	−12563.5	−1.42888
$427$	10439.9	1.18319
$428$	10072.8	1.13759
$429$	249.463	0.0280750
$430$	0	0
$431$	−2770.16	−0.309591	−0.154796	−	0.987946i	$-0.549472\pi$
$431$	−2770.16	−0.309591	−0.154796	+	0.987946i	$0.549472\pi$
$432$	−3960.01	−0.441033
$433$	−5663.00	−0.628513	−0.314257	−	0.949338i	$-0.601755\pi$
$433$	−5663.00	−0.628513	−0.314257	+	0.949338i	$0.601755\pi$
$434$	17276.2	1.91079
$435$	0	0
$436$	6704.65	0.736455
$437$	−1002.25	−0.109712
$438$	22456.6	2.44982
$439$	−8399.20	−0.913148	−0.456574	−	0.889685i	$-0.650924\pi$
$439$	−8399.20	−0.913148	−0.456574	+	0.889685i	$0.650924\pi$
$440$	0	0
$441$	5875.56	0.634442
$442$	1132.46	0.121868
$443$	−6154.68	−0.660085	−0.330043	−	0.943966i	$-0.607063\pi$
$443$	−6154.68	−0.660085	−0.330043	+	0.943966i	$0.607063\pi$
$444$	−7631.58	−0.815717
$445$	0	0
$446$	−3271.69	−0.347352
$447$	−12029.2	−1.27284
$448$	−12225.0	−1.28923
$449$	3445.03	0.362095	0.181048	−	0.983474i	$-0.442051\pi$
$449$	3445.03	0.362095	0.181048	+	0.983474i	$0.442051\pi$
$450$	0	0
$451$	−2511.63	−0.262235
$452$	9099.24	0.946885
$453$	15800.6	1.63880
$454$	5948.57	0.614935
$455$	0	0
$456$	−150.683	−0.0154745
$457$	502.346	0.0514196	0.0257098	−	0.999669i	$-0.491815\pi$
$457$	502.346	0.0514196	0.0257098	+	0.999669i	$0.491815\pi$
$458$	−20805.1	−2.12262
$459$	3738.39	0.380159
$460$	0	0
$461$	546.259	0.0551883	0.0275942	−	0.999619i	$-0.491215\pi$
$461$	546.259	0.0551883	0.0275942	+	0.999619i	$0.491215\pi$
$462$	5592.91	0.563216
$463$	−18540.2	−1.86098	−0.930490	−	0.366316i	$-0.880619\pi$
$463$	−18540.2	−1.86098	−0.930490	+	0.366316i	$0.880619\pi$
$464$	−11377.6	−1.13835
$465$	0	0
$466$	−8476.18	−0.842599
$467$	−12475.1	−1.23614	−0.618070	−	0.786123i	$-0.712085\pi$
$467$	−12475.1	−1.23614	−0.618070	+	0.786123i	$0.712085\pi$
$468$	636.795	0.0628972
$469$	2748.98	0.270653
$470$	0	0
$471$	−9659.49	−0.944981
$472$	−215.324	−0.0209981
$473$	−3651.85	−0.354994
$474$	15729.3	1.52420
$475$	0	0
$476$	12453.8	1.19920
$477$	−2813.71	−0.270086
$478$	14949.6	1.43050
$479$	−10569.2	−1.00818	−0.504091	−	0.863651i	$-0.668172\pi$
$479$	−10569.2	−1.00818	−0.504091	+	0.863651i	$0.668172\pi$
$480$	0	0
$481$	673.701	0.0638631
$482$	25421.5	2.40232
$483$	−9153.13	−0.862282
$484$	−9741.76	−0.914891
$485$	0	0
$486$	17295.7	1.61430
$487$	11227.9	1.04473	0.522366	−	0.852721i	$-0.325049\pi$
$487$	11227.9	1.04473	0.522366	+	0.852721i	$0.325049\pi$
$488$	−477.157	−0.0442621
$489$	856.256	0.0791846
$490$	0	0
$491$	−536.840	−0.0493427	−0.0246713	−	0.999696i	$-0.507854\pi$
$491$	−536.840	−0.0493427	−0.0246713	+	0.999696i	$0.507854\pi$
$492$	−15972.1	−1.46357
$493$	10740.9	0.981226
$494$	−343.799	−0.0313123
$495$	0	0
$496$	−11187.3	−1.01275
$497$	12196.7	1.10080
$498$	7728.11	0.695391
$499$	1319.91	0.118412	0.0592058	−	0.998246i	$-0.481143\pi$
$499$	1319.91	0.118412	0.0592058	+	0.998246i	$0.481143\pi$
$500$	0	0
$501$	22962.7	2.04770
$502$	−24874.3	−2.21154
$503$	−1749.27	−0.155062	−0.0775310	−	0.996990i	$-0.524704\pi$
$503$	−1749.27	−0.155062	−0.0775310	+	0.996990i	$0.524704\pi$
$504$	−552.390	−0.0488202
$505$	0	0
$506$	−1700.26	−0.149379
$507$	14615.2	1.28025
$508$	−6835.59	−0.597009
$509$	1882.19	0.163903	0.0819516	−	0.996636i	$-0.473885\pi$
$509$	1882.19	0.163903	0.0819516	+	0.996636i	$0.473885\pi$
$510$	0	0
$511$	−21801.1	−1.88733
$512$	16163.2	1.39515
$513$	−1134.93	−0.0976769
$514$	12616.8	1.08269
$515$	0	0
$516$	−23223.0	−1.98127
$517$	921.891	0.0784231
$518$	15104.3	1.28116
$519$	−2435.25	−0.205964
$520$	0	0
$521$	−3238.50	−0.272325	−0.136163	−	0.990686i	$-0.543477\pi$
$521$	−3238.50	−0.272325	−0.136163	+	0.990686i	$0.543477\pi$
$522$	12313.1	1.03244
$523$	−99.0144	−0.00827839	−0.00413919	−	0.999991i	$-0.501318\pi$
$523$	−99.0144	−0.00827839	−0.00413919	+	0.999991i	$0.501318\pi$
$524$	−18059.4	−1.50559
$525$	0	0
$526$	10398.5	0.861969
$527$	10561.2	0.872963
$528$	−3621.71	−0.298513
$529$	−9384.42	−0.771301
$530$	0	0
$531$	3301.57	0.269823
$532$	−3780.82	−0.308119
$533$	1409.98	0.114584
$534$	25678.8	2.08096
$535$	0	0
$536$	−125.643	−0.0101249
$537$	−16238.8	−1.30495
$538$	29500.4	2.36404
$539$	−2639.62	−0.210940
$540$	0	0
$541$	17183.7	1.36559	0.682794	−	0.730611i	$-0.260765\pi$
$541$	17183.7	1.36559	0.682794	+	0.730611i	$0.260765\pi$
$542$	−15868.1	−1.25755
$543$	16418.4	1.29757
$544$	−15849.8	−1.24918
$545$	0	0
$546$	−3139.77	−0.246098
$547$	1965.86	0.153664	0.0768319	−	0.997044i	$-0.475520\pi$
$547$	1965.86	0.153664	0.0768319	+	0.997044i	$0.475520\pi$
$548$	−17289.6	−1.34777
$549$	7316.27	0.568762
$550$	0	0
$551$	−3260.79	−0.252113
$552$	418.346	0.0322572
$553$	−15270.1	−1.17423
$554$	23104.4	1.77186
$555$	0	0
$556$	−2280.92	−0.173979
$557$	−6039.93	−0.459461	−0.229731	−	0.973254i	$-0.573785\pi$
$557$	−6039.93	−0.459461	−0.229731	+	0.973254i	$0.573785\pi$
$558$	12107.1	0.918523
$559$	2050.09	0.155115
$560$	0	0
$561$	3419.02	0.257311
$562$	−29736.6	−2.23197
$563$	−5260.06	−0.393757	−0.196878	−	0.980428i	$-0.563080\pi$
$563$	−5260.06	−0.393757	−0.196878	+	0.980428i	$0.563080\pi$
$564$	5862.53	0.437690
$565$	0	0
$566$	−23422.3	−1.73942
$567$	−22995.1	−1.70318
$568$	−557.454	−0.0411800
$569$	20567.4	1.51534	0.757672	−	0.652635i	$-0.226337\pi$
$569$	20567.4	1.51534	0.757672	+	0.652635i	$0.226337\pi$
$570$	0	0
$571$	11462.4	0.840080	0.420040	−	0.907506i	$-0.362016\pi$
$571$	11462.4	0.840080	0.420040	+	0.907506i	$0.362016\pi$
$572$	−286.083	−0.0209121
$573$	9319.64	0.679466
$574$	31611.6	2.29868
$575$	0	0
$576$	−8567.25	−0.619737
$577$	27029.6	1.95019	0.975094	−	0.221790i	$-0.0711899\pi$
$577$	27029.6	1.95019	0.975094	+	0.221790i	$0.0711899\pi$
$578$	−3947.19	−0.284051
$579$	−21545.5	−1.54646
$580$	0	0
$581$	−7502.52	−0.535726
$582$	−5841.42	−0.416039
$583$	1264.07	0.0897986
$584$	996.425	0.0706034
$585$	0	0
$586$	−12860.9	−0.906619
$587$	−15200.4	−1.06881	−0.534403	−	0.845230i	$-0.679464\pi$
$587$	−15200.4	−1.06881	−0.534403	+	0.845230i	$0.679464\pi$
$588$	−16786.0	−1.17729
$589$	−3206.23	−0.224296
$590$	0	0
$591$	−23139.4	−1.61054
$592$	−9780.83	−0.679036
$593$	19026.7	1.31759	0.658796	−	0.752322i	$-0.271066\pi$
$593$	19026.7	1.31759	0.658796	+	0.752322i	$0.271066\pi$
$594$	−1925.34	−0.132992
$595$	0	0
$596$	13795.0	0.948096
$597$	−13604.9	−0.932681
$598$	954.499	0.0652715
$599$	3927.31	0.267889	0.133945	−	0.990989i	$-0.457236\pi$
$599$	3927.31	0.267889	0.133945	+	0.990989i	$0.457236\pi$
$600$	0	0
$601$	13718.1	0.931069	0.465534	−	0.885030i	$-0.345862\pi$
$601$	13718.1	0.931069	0.465534	+	0.885030i	$0.345862\pi$
$602$	45962.6	3.11178
$603$	1926.48	0.130104
$604$	−18120.1	−1.22069
$605$	0	0
$606$	−38534.2	−2.58308
$607$	−26461.5	−1.76942	−0.884712	−	0.466138i	$-0.845645\pi$
$607$	−26461.5	−1.76942	−0.884712	+	0.466138i	$0.845645\pi$
$608$	4811.81	0.320961
$609$	−29779.3	−1.98148
$610$	0	0
$611$	−517.534	−0.0342671
$612$	8727.63	0.576460
$613$	233.384	0.0153773	0.00768865	−	0.999970i	$-0.497553\pi$
$613$	233.384	0.0153773	0.00768865	+	0.999970i	$0.497553\pi$
$614$	−29676.3	−1.95055
$615$	0	0
$616$	248.163	0.0162318
$617$	−4202.77	−0.274225	−0.137113	−	0.990555i	$-0.543782\pi$
$617$	−4202.77	−0.274225	−0.137113	+	0.990555i	$0.543782\pi$
$618$	23514.4	1.53056
$619$	−23009.4	−1.49407	−0.747033	−	0.664787i	$-0.768522\pi$
$619$	−23009.4	−1.49407	−0.747033	+	0.664787i	$0.768522\pi$
$620$	0	0
$621$	3150.93	0.203611
$622$	8466.89	0.545806
$623$	−24929.2	−1.60316
$624$	2033.17	0.130436
$625$	0	0
$626$	−8765.82	−0.559668
$627$	−1037.97	−0.0661126
$628$	11077.5	0.703884
$629$	9233.45	0.585313
$630$	0	0
$631$	18819.3	1.18730	0.593650	−	0.804723i	$-0.297686\pi$
$631$	18819.3	1.18730	0.593650	+	0.804723i	$0.297686\pi$
$632$	697.924	0.0439271
$633$	−28943.3	−1.81737
$634$	−1701.24	−0.106569
$635$	0	0
$636$	8038.56	0.501178
$637$	1481.84	0.0921705
$638$	−5531.74	−0.343266
$639$	8547.46	0.529159
$640$	0	0
$641$	25253.7	1.55610	0.778052	−	0.628200i	$-0.216208\pi$
$641$	25253.7	1.55610	0.778052	+	0.628200i	$0.216208\pi$
$642$	−34805.0	−2.13963
$643$	−11712.1	−0.718324	−0.359162	−	0.933275i	$-0.616937\pi$
$643$	−11712.1	−0.718324	−0.359162	+	0.933275i	$0.616937\pi$
$644$	10496.8	0.642284
$645$	0	0
$646$	−4711.96	−0.286981
$647$	26533.3	1.61226	0.806131	−	0.591737i	$-0.201558\pi$
$647$	26533.3	1.61226	0.806131	+	0.591737i	$0.201558\pi$
$648$	1051.00	0.0637146
$649$	−1483.25	−0.0897111
$650$	0	0
$651$	−29281.1	−1.76285
$652$	−981.952	−0.0589819
$653$	−27898.9	−1.67193	−0.835964	−	0.548785i	$-0.815091\pi$
$653$	−27898.9	−1.67193	−0.835964	+	0.548785i	$0.815091\pi$
$654$	−23166.9	−1.38516
$655$	0	0
$656$	−20470.2	−1.21834
$657$	−15278.2	−0.907245
$658$	−11603.0	−0.687436
$659$	−1274.66	−0.0753468	−0.0376734	−	0.999290i	$-0.511995\pi$
$659$	−1274.66	−0.0753468	−0.0376734	+	0.999290i	$0.511995\pi$
$660$	0	0
$661$	−5049.52	−0.297131	−0.148565	−	0.988903i	$-0.547466\pi$
$661$	−5049.52	−0.297131	−0.148565	+	0.988903i	$0.547466\pi$
$662$	−3033.17	−0.178078
$663$	−1919.38	−0.112432
$664$	342.904	0.0200411
$665$	0	0
$666$	10585.1	0.615860
$667$	9053.01	0.525538
$668$	−26333.6	−1.52526
$669$	5545.14	0.320460
$670$	0	0
$671$	−3286.86	−0.189103
$672$	43944.1	2.52259
$673$	−8398.64	−0.481045	−0.240523	−	0.970644i	$-0.577319\pi$
$673$	−8398.64	−0.481045	−0.240523	+	0.970644i	$0.577319\pi$
$674$	12085.1	0.690653
$675$	0	0
$676$	−16760.7	−0.953612
$677$	−9875.31	−0.560619	−0.280309	−	0.959910i	$-0.590437\pi$
$677$	−9875.31	−0.560619	−0.280309	+	0.959910i	$0.590437\pi$
$678$	−31441.0	−1.78095
$679$	5670.91	0.320515
$680$	0	0
$681$	−10082.1	−0.567325
$682$	−5439.18	−0.305392
$683$	8653.78	0.484814	0.242407	−	0.970175i	$-0.422063\pi$
$683$	8653.78	0.484814	0.242407	+	0.970175i	$0.422063\pi$
$684$	−2649.60	−0.148114
$685$	0	0
$686$	−1893.05	−0.105360
$687$	35262.2	1.95828
$688$	−29763.2	−1.64929
$689$	−709.629	−0.0392376
$690$	0	0
$691$	2916.50	0.160563	0.0802813	−	0.996772i	$-0.474418\pi$
$691$	2916.50	0.160563	0.0802813	+	0.996772i	$0.474418\pi$
$692$	2792.73	0.153416
$693$	−3805.10	−0.208577
$694$	23406.9	1.28028
$695$	0	0
$696$	1361.07	0.0741254
$697$	19324.6	1.05017
$698$	−47976.5	−2.60163
$699$	14366.1	0.777363
$700$	0	0
$701$	−9070.78	−0.488729	−0.244364	−	0.969683i	$-0.578579\pi$
$701$	−9070.78	−0.488729	−0.244364	+	0.969683i	$0.578579\pi$
$702$	1080.85	0.0581112
$703$	−2803.16	−0.150388
$704$	3848.88	0.206051
$705$	0	0
$706$	9591.00	0.511278
$707$	37409.4	1.98999
$708$	−9432.34	−0.500690
$709$	−5957.11	−0.315549	−0.157774	−	0.987475i	$-0.550432\pi$
$709$	−5957.11	−0.315549	−0.157774	+	0.987475i	$0.550432\pi$
$710$	0	0
$711$	−10701.3	−0.564459
$712$	1139.40	0.0599729
$713$	8901.55	0.467553
$714$	−43032.2	−2.25552
$715$	0	0
$716$	18622.6	0.972010
$717$	−25337.8	−1.31975
$718$	−5768.71	−0.299842
$719$	−31140.8	−1.61524	−0.807620	−	0.589703i	$-0.799245\pi$
$719$	−31140.8	−1.61524	−0.807620	+	0.589703i	$0.799245\pi$
$720$	0	0
$721$	−22828.0	−1.17914
$722$	1430.49	0.0737359
$723$	−43086.4	−2.21632
$724$	−18828.6	−0.966519
$725$	0	0
$726$	33661.1	1.72077
$727$	−14969.7	−0.763682	−0.381841	−	0.924228i	$-0.624710\pi$
$727$	−14969.7	−0.763682	−0.381841	+	0.924228i	$0.624710\pi$
$728$	−139.315	−0.00709251
$729$	−5283.30	−0.268420
$730$	0	0
$731$	28097.5	1.42165
$732$	−20902.0	−1.05541
$733$	12414.1	0.625545	0.312772	−	0.949828i	$-0.398742\pi$
$733$	12414.1	0.625545	0.312772	+	0.949828i	$0.398742\pi$
$734$	−34806.5	−1.75031
$735$	0	0
$736$	−13359.1	−0.669055
$737$	−865.481	−0.0432570
$738$	22153.4	1.10498
$739$	1324.11	0.0659111	0.0329555	−	0.999457i	$-0.489508\pi$
$739$	1324.11	0.0659111	0.0329555	+	0.999457i	$0.489508\pi$
$740$	0	0
$741$	582.700	0.0288880
$742$	−15909.8	−0.787150
$743$	4391.55	0.216838	0.108419	−	0.994105i	$-0.465421\pi$
$743$	4391.55	0.216838	0.108419	+	0.994105i	$0.465421\pi$
$744$	1338.30	0.0659468
$745$	0	0
$746$	36455.0	1.78916
$747$	−5257.76	−0.257525
$748$	−3920.92	−0.191662
$749$	33789.0	1.64836
$750$	0	0
$751$	−31947.5	−1.55230	−0.776152	−	0.630546i	$-0.782831\pi$
$751$	−31947.5	−1.55230	−0.776152	+	0.630546i	$0.782831\pi$
$752$	7513.58	0.364351
$753$	42159.0	2.04032
$754$	3105.42	0.149990
$755$	0	0
$756$	11886.3	0.571826
$757$	−18569.8	−0.891585	−0.445793	−	0.895136i	$-0.647078\pi$
$757$	−18569.8	−0.891585	−0.445793	+	0.895136i	$0.647078\pi$
$758$	−24414.9	−1.16991
$759$	2881.75	0.137814
$760$	0	0
$761$	5507.32	0.262339	0.131170	−	0.991360i	$-0.458127\pi$
$761$	5507.32	0.262339	0.131170	+	0.991360i	$0.458127\pi$
$762$	23619.3	1.12288
$763$	22490.6	1.06712
$764$	−10687.7	−0.506111
$765$	0	0
$766$	−10424.7	−0.491723
$767$	832.669	0.0391994
$768$	−29442.7	−1.38336
$769$	−14977.9	−0.702362	−0.351181	−	0.936308i	$-0.614220\pi$
$769$	−14977.9	−0.702362	−0.351181	+	0.936308i	$0.614220\pi$
$770$	0	0
$771$	−21384.0	−0.998864
$772$	24708.3	1.15190
$773$	−19545.6	−0.909450	−0.454725	−	0.890632i	$-0.650262\pi$
$773$	−19545.6	−0.909450	−0.454725	+	0.890632i	$0.650262\pi$
$774$	32210.5	1.49585
$775$	0	0
$776$	−259.190	−0.0119902
$777$	−25600.0	−1.18197
$778$	−23246.4	−1.07124
$779$	−5866.70	−0.269829
$780$	0	0
$781$	−3839.98	−0.175935
$782$	13081.9	0.598221
$783$	10251.4	0.467887
$784$	−21513.4	−0.980020
$785$	0	0
$786$	62401.3	2.83178
$787$	4274.62	0.193613	0.0968067	−	0.995303i	$-0.469137\pi$
$787$	4274.62	0.193613	0.0968067	+	0.995303i	$0.469137\pi$
$788$	26536.2	1.19963
$789$	−17624.2	−0.795233
$790$	0	0
$791$	30523.2	1.37204
$792$	173.913	0.00780268
$793$	1845.19	0.0826287
$794$	56482.6	2.52455
$795$	0	0
$796$	15602.0	0.694722
$797$	25450.6	1.13112	0.565562	−	0.824706i	$-0.308659\pi$
$797$	25450.6	1.13112	0.565562	+	0.824706i	$0.308659\pi$
$798$	13064.0	0.579525
$799$	−7093.08	−0.314062
$800$	0	0
$801$	−17470.4	−0.770645
$802$	39266.2	1.72885
$803$	6863.81	0.301642
$804$	−5503.81	−0.241423
$805$	0	0
$806$	3053.46	0.133441
$807$	−49999.7	−2.18101
$808$	−1709.80	−0.0744439
$809$	4002.04	0.173924	0.0869619	−	0.996212i	$-0.472284\pi$
$809$	4002.04	0.173924	0.0869619	+	0.996212i	$0.472284\pi$
$810$	0	0
$811$	−37915.1	−1.64165	−0.820826	−	0.571179i	$-0.806486\pi$
$811$	−37915.1	−1.64165	−0.820826	+	0.571179i	$0.806486\pi$
$812$	34150.8	1.47593
$813$	26894.5	1.16019
$814$	−4755.39	−0.204762
$815$	0	0
$816$	27865.7	1.19546
$817$	−8530.05	−0.365274
$818$	23006.2	0.983366
$819$	2136.12	0.0911379
$820$	0	0
$821$	4739.43	0.201470	0.100735	−	0.994913i	$-0.467881\pi$
$821$	4739.43	0.201470	0.100735	+	0.994913i	$0.467881\pi$
$822$	59741.5	2.53494
$823$	−20752.2	−0.878952	−0.439476	−	0.898254i	$-0.644836\pi$
$823$	−20752.2	−0.878952	−0.439476	+	0.898254i	$0.644836\pi$
$824$	1043.36	0.0441106
$825$	0	0
$826$	18668.3	0.786384
$827$	−34264.8	−1.44075	−0.720377	−	0.693583i	$-0.756031\pi$
$827$	−34264.8	−1.44075	−0.720377	+	0.693583i	$0.756031\pi$
$828$	7356.14	0.308748
$829$	−39707.5	−1.66357	−0.831784	−	0.555100i	$-0.812680\pi$
$829$	−39707.5	−1.66357	−0.831784	+	0.555100i	$0.812680\pi$
$830$	0	0
$831$	−39159.3	−1.63468
$832$	−2160.69	−0.0900343
$833$	20309.4	0.844753
$834$	7881.35	0.327229
$835$	0	0
$836$	1190.34	0.0492450
$837$	10079.9	0.416263
$838$	48584.6	2.00278
$839$	−4524.04	−0.186159	−0.0930794	−	0.995659i	$-0.529671\pi$
$839$	−4524.04	−0.186159	−0.0930794	+	0.995659i	$0.529671\pi$
$840$	0	0
$841$	5064.59	0.207659
$842$	23132.9	0.946809
$843$	50400.1	2.05916
$844$	33192.1	1.35370
$845$	0	0
$846$	−8131.39	−0.330453
$847$	−32678.5	−1.32568
$848$	10302.4	0.417201
$849$	39698.0	1.60475
$850$	0	0
$851$	7782.47	0.313490
$852$	−24419.4	−0.981920
$853$	7595.54	0.304884	0.152442	−	0.988312i	$-0.451286\pi$
$853$	7595.54	0.304884	0.152442	+	0.988312i	$0.451286\pi$
$854$	41368.8	1.65762
$855$	0	0
$856$	−1544.34	−0.0616639
$857$	19528.9	0.778405	0.389203	−	0.921152i	$-0.372751\pi$
$857$	19528.9	0.778405	0.389203	+	0.921152i	$0.372751\pi$
$858$	988.515	0.0393326
$859$	25980.8	1.03196	0.515979	−	0.856601i	$-0.327428\pi$
$859$	25980.8	1.03196	0.515979	+	0.856601i	$0.327428\pi$
$860$	0	0
$861$	−53578.0	−2.12071
$862$	−10977.0	−0.433732
$863$	48294.6	1.90494	0.952472	−	0.304625i	$-0.0985311\pi$
$863$	48294.6	1.90494	0.952472	+	0.304625i	$0.0985311\pi$
$864$	−15127.6	−0.595660
$865$	0	0
$866$	−22440.1	−0.880536
$867$	6690.03	0.262059
$868$	33579.4	1.31309
$869$	4807.61	0.187672
$870$	0	0
$871$	485.866	0.0189012
$872$	−1027.94	−0.0399201
$873$	3974.17	0.154072
$874$	−3971.51	−0.153705
$875$	0	0
$876$	43648.7	1.68351
$877$	−44377.5	−1.70869	−0.854346	−	0.519705i	$-0.826042\pi$
$877$	−44377.5	−1.70869	−0.854346	+	0.519705i	$0.826042\pi$
$878$	−33282.5	−1.27930
$879$	21797.7	0.836426
$880$	0	0
$881$	−12139.4	−0.464231	−0.232116	−	0.972688i	$-0.574565\pi$
$881$	−12139.4	−0.464231	−0.232116	+	0.972688i	$0.574565\pi$
$882$	23282.4	0.888841
$883$	−5048.07	−0.192391	−0.0961954	−	0.995362i	$-0.530667\pi$
$883$	−5048.07	−0.192391	−0.0961954	+	0.995362i	$0.530667\pi$
$884$	2201.14	0.0837470
$885$	0	0
$886$	−24388.4	−0.924767
$887$	−20373.4	−0.771221	−0.385610	−	0.922662i	$-0.626009\pi$
$887$	−20373.4	−0.771221	−0.385610	+	0.922662i	$0.626009\pi$
$888$	1170.05	0.0442166
$889$	−22929.9	−0.865065
$890$	0	0
$891$	7239.72	0.272211
$892$	−6359.14	−0.238699
$893$	2153.37	0.0806940
$894$	−47666.5	−1.78323
$895$	0	0
$896$	3902.40	0.145502
$897$	−1617.76	−0.0602180
$898$	13651.2	0.507289
$899$	28960.8	1.07441
$900$	0	0
$901$	−9725.85	−0.359617
$902$	−9952.52	−0.367386
$903$	−77901.2	−2.87086
$904$	−1395.07	−0.0513267
$905$	0	0
$906$	62611.1	2.29593
$907$	7456.13	0.272962	0.136481	−	0.990643i	$-0.456421\pi$
$907$	7456.13	0.272962	0.136481	+	0.990643i	$0.456421\pi$
$908$	11562.2	0.422581
$909$	26216.5	0.956596
$910$	0	0
$911$	−10653.2	−0.387440	−0.193720	−	0.981057i	$-0.562055\pi$
$911$	−10653.2	−0.387440	−0.193720	+	0.981057i	$0.562055\pi$
$912$	−8459.66	−0.307157
$913$	2362.07	0.0856224
$914$	1990.59	0.0720380
$915$	0	0
$916$	−40438.6	−1.45866
$917$	−60579.8	−2.18159
$918$	14813.6	0.532596
$919$	−12569.7	−0.451183	−0.225591	−	0.974222i	$-0.572431\pi$
$919$	−12569.7	−0.451183	−0.225591	+	0.974222i	$0.572431\pi$
$920$	0	0
$921$	50297.8	1.79953
$922$	2164.59	0.0773179
$923$	2155.70	0.0768752
$924$	10870.9	0.387040
$925$	0	0
$926$	−73466.8	−2.60720
$927$	−15997.9	−0.566816
$928$	−43463.4	−1.53745
$929$	4920.06	0.173759	0.0868795	−	0.996219i	$-0.472311\pi$
$929$	4920.06	0.173759	0.0868795	+	0.996219i	$0.472311\pi$
$930$	0	0
$931$	−6165.67	−0.217048
$932$	−16475.0	−0.579031
$933$	−14350.4	−0.503548
$934$	−49433.4	−1.73181
$935$	0	0
$936$	−97.6317	−0.00340939
$937$	−1991.87	−0.0694465	−0.0347233	−	0.999397i	$-0.511055\pi$
$937$	−1991.87	−0.0694465	−0.0347233	+	0.999397i	$0.511055\pi$
$938$	10893.0	0.379179
$939$	14857.0	0.516337
$940$	0	0
$941$	−7640.33	−0.264684	−0.132342	−	0.991204i	$-0.542250\pi$
$941$	−7640.33	−0.264684	−0.132342	+	0.991204i	$0.542250\pi$
$942$	−38276.5	−1.32390
$943$	16287.9	0.562467
$944$	−12088.7	−0.416795
$945$	0	0
$946$	−14470.7	−0.497340
$947$	6521.15	0.223769	0.111884	−	0.993721i	$-0.464311\pi$
$947$	6521.15	0.223769	0.111884	+	0.993721i	$0.464311\pi$
$948$	30572.8	1.04742
$949$	−3853.22	−0.131803
$950$	0	0
$951$	2883.40	0.0983182
$952$	−1909.38	−0.0650037
$953$	35757.9	1.21544	0.607719	−	0.794152i	$-0.292085\pi$
$953$	35757.9	1.21544	0.607719	+	0.794152i	$0.292085\pi$
$954$	−11149.6	−0.378386
$955$	0	0
$956$	29057.3	0.983034
$957$	9375.64	0.316689
$958$	−41881.3	−1.41244
$959$	−57997.6	−1.95291
$960$	0	0
$961$	−1314.75	−0.0441323
$962$	2669.59	0.0894710
$963$	23679.3	0.792374
$964$	49411.4	1.65086
$965$	0	0
$966$	−36270.0	−1.20804
$967$	6342.30	0.210915	0.105457	−	0.994424i	$-0.466369\pi$
$967$	6342.30	0.210915	0.105457	+	0.994424i	$0.466369\pi$
$968$	1493.58	0.0495924
$969$	7986.21	0.264762
$970$	0	0
$971$	30351.2	1.00311	0.501553	−	0.865127i	$-0.332762\pi$
$971$	30351.2	1.00311	0.501553	+	0.865127i	$0.332762\pi$
$972$	33617.5	1.10934
$973$	−7651.29	−0.252096
$974$	44491.4	1.46365
$975$	0	0
$976$	−26788.5	−0.878566
$977$	−39843.6	−1.30472	−0.652359	−	0.757910i	$-0.726220\pi$
$977$	−39843.6	−1.30472	−0.652359	+	0.757910i	$0.726220\pi$
$978$	3392.98	0.110936
$979$	7848.66	0.256225
$980$	0	0
$981$	15761.4	0.512969
$982$	−2127.27	−0.0691282
$983$	−24068.7	−0.780949	−0.390475	−	0.920614i	$-0.627689\pi$
$983$	−24068.7	−0.780949	−0.390475	+	0.920614i	$0.627689\pi$
$984$	2448.79	0.0793340
$985$	0	0
$986$	42561.5	1.37468
$987$	19665.8	0.634213
$988$	−668.238	−0.0215177
$989$	23682.2	0.761426
$990$	0	0
$991$	3235.83	0.103723	0.0518615	−	0.998654i	$-0.483485\pi$
$991$	3235.83	0.103723	0.0518615	+	0.998654i	$0.483485\pi$
$992$	−42736.2	−1.36782
$993$	5140.86	0.164290
$994$	48330.4	1.54220
$995$	0	0
$996$	15021.0	0.477871
$997$	−19444.5	−0.617665	−0.308833	−	0.951116i	$-0.599938\pi$
$997$	−19444.5	−0.617665	−0.308833	+	0.951116i	$0.599938\pi$
$998$	5230.25	0.165892
$999$	8812.68	0.279100

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.4.a.f.1.3		3
5.2	odd	4		475.4.b.f.324.5		6
5.3	odd	4		475.4.b.f.324.2		6
5.4	even	2		19.4.a.b.1.1	✓	3
15.14	odd	2		171.4.a.f.1.3		3
20.19	odd	2		304.4.a.i.1.1		3
35.34	odd	2		931.4.a.c.1.1		3
40.19	odd	2		1216.4.a.u.1.3		3
40.29	even	2		1216.4.a.s.1.1		3
55.54	odd	2		2299.4.a.h.1.3		3
95.94	odd	2		361.4.a.i.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.4.a.b.1.1	✓	3	5.4	even	2
171.4.a.f.1.3		3	15.14	odd	2
304.4.a.i.1.1		3	20.19	odd	2
361.4.a.i.1.3		3	95.94	odd	2
475.4.a.f.1.3		3	1.1	even	1	trivial
475.4.b.f.324.2		6	5.3	odd	4
475.4.b.f.324.5		6	5.2	odd	4
931.4.a.c.1.1		3	35.34	odd	2
1216.4.a.s.1.1		3	40.29	even	2
1216.4.a.u.1.3		3	40.19	odd	2
2299.4.a.h.1.3		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)

\(f(q)\)	\(=\)	\(q+3.96257 q^{2} -6.71610 q^{3} +7.70200 q^{4} -26.6130 q^{6} +25.8362 q^{7} -1.18085 q^{8} +18.1060 q^{9} -8.13420 q^{11} -51.7274 q^{12} +4.56640 q^{13} +102.378 q^{14} -66.2952 q^{16} +62.5850 q^{17} +71.7464 q^{18} -19.0000 q^{19} -173.518 q^{21} -32.2324 q^{22} +52.7502 q^{23} +7.93070 q^{24} +18.0947 q^{26} +59.7330 q^{27} +198.990 q^{28} +171.620 q^{29} +168.749 q^{31} -253.253 q^{32} +54.6301 q^{33} +247.998 q^{34} +139.452 q^{36} +147.534 q^{37} -75.2889 q^{38} -30.6684 q^{39} +308.774 q^{41} -687.580 q^{42} +448.950 q^{43} -62.6496 q^{44} +209.027 q^{46} -113.335 q^{47} +445.245 q^{48} +324.509 q^{49} -420.327 q^{51} +35.1704 q^{52} -155.402 q^{53} +236.696 q^{54} -30.5086 q^{56} +127.606 q^{57} +680.059 q^{58} +182.347 q^{59} +404.080 q^{61} +668.681 q^{62} +467.790 q^{63} -473.172 q^{64} +216.476 q^{66} +106.400 q^{67} +482.030 q^{68} -354.276 q^{69} +472.079 q^{71} -21.3805 q^{72} -843.821 q^{73} +584.616 q^{74} -146.338 q^{76} -210.157 q^{77} -121.526 q^{78} -591.036 q^{79} -890.035 q^{81} +1223.54 q^{82} -290.388 q^{83} -1336.44 q^{84} +1779.00 q^{86} -1152.62 q^{87} +9.60526 q^{88} -964.896 q^{89} +117.978 q^{91} +406.282 q^{92} -1133.34 q^{93} -449.099 q^{94} +1700.87 q^{96} +219.495 q^{97} +1285.89 q^{98} -147.278 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} - q^{3} + 21 q^{4} - 65 q^{6} + 35 q^{7} - 27 q^{8} + 48 q^{9} + 16 q^{11} + 115 q^{12} - 65 q^{13} + 37 q^{14} + 33 q^{16} - 29 q^{17} - 138 q^{18} - 57 q^{19} - 25 q^{21} - 118 q^{22} + 101 q^{23}+ \cdots + 250 q^{99}+O(q^{100}) \) 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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Representations

Groups

Database

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