LMFDB - Embedded newform 1519.4.a.h.1.17

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1519,4,Mod(1,1519)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1519, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1519.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 1519 = 7^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1519.a (trivial)

Newform invariants

comment:select newform

sage:traces = [23,5,-6,91,-40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$89.6239012987$
Analytic rank:	$1$
Dimension:	$23$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Character	$\chi$	$=$	1519.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+3.43289 q^{2} -7.72707 q^{3} +3.78474 q^{4} -12.7805 q^{5} -26.5262 q^{6} -14.4705 q^{8} +32.7076 q^{9} -43.8739 q^{10} +36.3512 q^{11} -29.2450 q^{12} -18.8337 q^{13} +98.7554 q^{15} -79.9537 q^{16} +29.3324 q^{17} +112.282 q^{18} -20.5569 q^{19} -48.3707 q^{20} +124.790 q^{22} +179.957 q^{23} +111.815 q^{24} +38.3400 q^{25} -64.6539 q^{26} -44.1031 q^{27} +125.863 q^{29} +339.017 q^{30} -31.0000 q^{31} -158.708 q^{32} -280.888 q^{33} +100.695 q^{34} +123.790 q^{36} -46.7737 q^{37} -70.5697 q^{38} +145.529 q^{39} +184.940 q^{40} +378.421 q^{41} +286.604 q^{43} +137.580 q^{44} -418.018 q^{45} +617.774 q^{46} -608.901 q^{47} +617.808 q^{48} +131.617 q^{50} -226.653 q^{51} -71.2806 q^{52} -28.6235 q^{53} -151.401 q^{54} -464.585 q^{55} +158.845 q^{57} +432.075 q^{58} +26.1799 q^{59} +373.764 q^{60} +22.1075 q^{61} -106.420 q^{62} +94.8020 q^{64} +240.703 q^{65} -964.259 q^{66} -613.441 q^{67} +111.016 q^{68} -1390.54 q^{69} -486.486 q^{71} -473.296 q^{72} +356.421 q^{73} -160.569 q^{74} -296.255 q^{75} -77.8026 q^{76} +499.586 q^{78} +286.380 q^{79} +1021.84 q^{80} -542.318 q^{81} +1299.08 q^{82} +106.507 q^{83} -374.881 q^{85} +983.880 q^{86} -972.554 q^{87} -526.021 q^{88} -352.687 q^{89} -1435.01 q^{90} +681.092 q^{92} +239.539 q^{93} -2090.29 q^{94} +262.727 q^{95} +1226.35 q^{96} +631.893 q^{97} +1188.96 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q + 5 q^{2} - 6 q^{3} + 91 q^{4} - 40 q^{5} - 36 q^{6} + 39 q^{8} + 211 q^{9} - 40 q^{10} + 44 q^{11} - 414 q^{12} + 20 q^{13} + 523 q^{16} - 306 q^{17} + 51 q^{18} - 296 q^{19} - 400 q^{20} - 326 q^{22}+ \cdots - 3456 q^{99}+O(q^{100}) $ 23 * q + 5 * q^2 - 6 * q^3 + 91 * q^4 - 40 * q^5 - 36 * q^6 + 39 * q^8 + 211 * q^9 - 40 * q^10 + 44 * q^11 - 414 * q^12 + 20 * q^13 + 523 * q^16 - 306 * q^17 + 51 * q^18 - 296 * q^19 - 400 * q^20 - 326 * q^22 - 130 * q^23 - 432 * q^24 + 801 * q^25 - 474 * q^26 - 192 * q^27 - 16 * q^29 - 74 * q^30 - 713 * q^31 + 287 * q^32 - 488 * q^33 - 692 * q^34 + 623 * q^36 - 296 * q^37 + 40 * q^38 - 124 * q^39 - 48 * q^40 - 2344 * q^41 - 48 * q^43 + 1352 * q^44 - 132 * q^45 - 836 * q^46 - 436 * q^47 - 4190 * q^48 - 767 * q^50 - 872 * q^51 + 632 * q^52 - 884 * q^53 - 100 * q^54 - 3308 * q^55 + 404 * q^57 - 562 * q^58 - 400 * q^59 + 2248 * q^60 - 860 * q^61 - 155 * q^62 + 3125 * q^64 - 1676 * q^65 - 128 * q^66 - 164 * q^67 - 302 * q^68 - 2136 * q^69 + 56 * q^71 + 1613 * q^72 - 1610 * q^73 + 908 * q^74 + 910 * q^75 - 5208 * q^76 + 1424 * q^78 + 318 * q^79 - 11480 * q^80 + 231 * q^81 + 2608 * q^82 - 3166 * q^83 - 716 * q^85 - 224 * q^86 - 1248 * q^87 - 5562 * q^88 - 2854 * q^89 - 7172 * q^90 - 3558 * q^92 + 186 * q^93 - 4228 * q^94 + 984 * q^95 + 4178 * q^96 - 3332 * q^97 - 3456 * 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 $

$2$	3.43289	1.21371	0.606855	−	0.794812i	$-0.292431\pi$
$2$	3.43289	1.21371	0.606855	+	0.794812i	$0.292431\pi$
$3$	−7.72707	−1.48708	−0.743538	−	0.668694i	$-0.766854\pi$
$3$	−7.72707	−1.48708	−0.743538	+	0.668694i	$0.766854\pi$
$4$	3.78474	0.473093
$5$	−12.7805	−1.14312	−0.571559	−	0.820561i	$-0.693661\pi$
$5$	−12.7805	−1.14312	−0.571559	+	0.820561i	$0.693661\pi$
$6$	−26.5262	−1.80488
$7$	0	0
$7$	0	0
$8$	−14.4705	−0.639513
$9$	32.7076	1.21139
$10$	−43.8739	−1.38741
$11$	36.3512	0.996391	0.498195	−	0.867065i	$-0.333996\pi$
$11$	36.3512	0.996391	0.498195	+	0.867065i	$0.333996\pi$
$12$	−29.2450	−0.703524
$13$	−18.8337	−0.401809	−0.200905	−	0.979611i	$-0.564388\pi$
$13$	−18.8337	−0.401809	−0.200905	+	0.979611i	$0.564388\pi$
$14$	0	0
$15$	98.7554	1.69990
$16$	−79.9537	−1.24928
$17$	29.3324	0.418480	0.209240	−	0.977864i	$-0.432901\pi$
$17$	29.3324	0.418480	0.209240	+	0.977864i	$0.432901\pi$
$18$	112.282	1.47028
$19$	−20.5569	−0.248215	−0.124107	−	0.992269i	$-0.539607\pi$
$19$	−20.5569	−0.248215	−0.124107	+	0.992269i	$0.539607\pi$
$20$	−48.3707	−0.540801
$21$	0	0
$22$	124.790	1.20933
$23$	179.957	1.63147	0.815733	−	0.578429i	$-0.196334\pi$
$23$	179.957	1.63147	0.815733	+	0.578429i	$0.196334\pi$
$24$	111.815	0.951004
$25$	38.3400	0.306720
$26$	−64.6539	−0.487680
$27$	−44.1031	−0.314357
$28$	0	0
$29$	125.863	0.805938	0.402969	−	0.915214i	$-0.367978\pi$
$29$	125.863	0.805938	0.402969	+	0.915214i	$0.367978\pi$
$30$	339.017	2.06319
$31$	−31.0000	−0.179605
$31$	−31.0000	−0.179605
$32$	−158.708	−0.876746
$33$	−280.888	−1.48171
$34$	100.695	0.507913
$35$	0	0
$36$	123.790	0.573101
$37$	−46.7737	−0.207826	−0.103913	−	0.994586i	$-0.533136\pi$
$37$	−46.7737	−0.207826	−0.103913	+	0.994586i	$0.533136\pi$
$38$	−70.5697	−0.301261
$39$	145.529	0.597521
$40$	184.940	0.731039
$41$	378.421	1.44145	0.720725	−	0.693222i	$-0.243809\pi$
$41$	378.421	1.44145	0.720725	+	0.693222i	$0.243809\pi$
$42$	0	0
$43$	286.604	1.01644	0.508218	−	0.861229i	$-0.330305\pi$
$43$	286.604	1.01644	0.508218	+	0.861229i	$0.330305\pi$
$44$	137.580	0.471385
$45$	−418.018	−1.38477
$46$	617.774	1.98013
$47$	−608.901	−1.88973	−0.944866	−	0.327458i	$-0.893808\pi$
$47$	−608.901	−1.88973	−0.944866	+	0.327458i	$0.893808\pi$
$48$	617.808	1.85777
$49$	0	0
$50$	131.617	0.372269
$51$	−226.653	−0.622311
$52$	−71.2806	−0.190093
$53$	−28.6235	−0.0741838	−0.0370919	−	0.999312i	$-0.511809\pi$
$53$	−28.6235	−0.0741838	−0.0370919	+	0.999312i	$0.511809\pi$
$54$	−151.401	−0.381539
$55$	−464.585	−1.13899
$56$	0	0
$57$	158.845	0.369114
$58$	432.075	0.978175
$59$	26.1799	0.0577683	0.0288842	−	0.999583i	$-0.490805\pi$
$59$	26.1799	0.0577683	0.0288842	+	0.999583i	$0.490805\pi$
$60$	373.764	0.804212
$61$	22.1075	0.0464030	0.0232015	−	0.999731i	$-0.492614\pi$
$61$	22.1075	0.0464030	0.0232015	+	0.999731i	$0.492614\pi$
$62$	−106.420	−0.217989
$63$	0	0
$64$	94.8020	0.185160
$65$	240.703	0.459316
$66$	−964.259	−1.79836
$67$	−613.441	−1.11856	−0.559282	−	0.828978i	$-0.688923\pi$
$67$	−613.441	−1.11856	−0.559282	+	0.828978i	$0.688923\pi$
$68$	111.016	0.197980
$69$	−1390.54	−2.42611
$70$	0	0
$71$	−486.486	−0.813172	−0.406586	−	0.913612i	$-0.633281\pi$
$71$	−486.486	−0.813172	−0.406586	+	0.913612i	$0.633281\pi$
$72$	−473.296	−0.774702
$73$	356.421	0.571452	0.285726	−	0.958311i	$-0.407765\pi$
$73$	356.421	0.571452	0.285726	+	0.958311i	$0.407765\pi$
$74$	−160.569	−0.252240
$75$	−296.255	−0.456115
$76$	−77.8026	−0.117429
$77$	0	0
$78$	499.586	0.725217
$79$	286.380	0.407851	0.203926	−	0.978986i	$-0.434630\pi$
$79$	286.380	0.407851	0.203926	+	0.978986i	$0.434630\pi$
$80$	1021.84	1.42807
$81$	−542.318	−0.743920
$82$	1299.08	1.74950
$83$	106.507	0.140852	0.0704260	−	0.997517i	$-0.477564\pi$
$83$	106.507	0.140852	0.0704260	+	0.997517i	$0.477564\pi$
$84$	0	0
$85$	−374.881	−0.478372
$86$	983.880	1.23366
$87$	−972.554	−1.19849
$88$	−526.021	−0.637205
$89$	−352.687	−0.420054	−0.210027	−	0.977696i	$-0.567355\pi$
$89$	−352.687	−0.420054	−0.210027	+	0.977696i	$0.567355\pi$
$90$	−1435.01	−1.68070
$91$	0	0
$92$	681.092	0.771834
$93$	239.539	0.267087
$94$	−2090.29	−2.29359
$95$	262.727	0.283739
$96$	1226.35	1.30379
$97$	631.893	0.661433	0.330717	−	0.943730i	$-0.392710\pi$
$97$	631.893	0.661433	0.330717	+	0.943730i	$0.392710\pi$
$98$	0	0
$99$	1188.96	1.20702
$100$	145.107	0.145107
$101$	−236.439	−0.232937	−0.116468	−	0.993194i	$-0.537157\pi$
$101$	−236.439	−0.232937	−0.116468	+	0.993194i	$0.537157\pi$
$102$	−778.077	−0.755305
$103$	−1316.95	−1.25984	−0.629918	−	0.776662i	$-0.716911\pi$
$103$	−1316.95	−1.25984	−0.629918	+	0.776662i	$0.716911\pi$
$104$	272.533	0.256962
$105$	0	0
$106$	−98.2613	−0.0900376
$107$	527.486	0.476579	0.238289	−	0.971194i	$-0.423413\pi$
$107$	527.486	0.476579	0.238289	+	0.971194i	$0.423413\pi$
$108$	−166.919	−0.148720
$109$	−1094.66	−0.961921	−0.480961	−	0.876742i	$-0.659712\pi$
$109$	−1094.66	−0.961921	−0.480961	+	0.876742i	$0.659712\pi$
$110$	−1594.87	−1.38241
$111$	361.424	0.309052
$112$	0	0
$113$	−991.836	−0.825700	−0.412850	−	0.910799i	$-0.635467\pi$
$113$	−991.836	−0.825700	−0.412850	+	0.910799i	$0.635467\pi$
$114$	545.297	0.447998
$115$	−2299.94	−1.86496
$116$	476.359	0.381283
$117$	−616.004	−0.486749
$118$	89.8727	0.0701140
$119$	0	0
$120$	−1429.04	−1.08711
$121$	−9.59069	−0.00720563
$122$	75.8928	0.0563198
$123$	−2924.08	−2.14354
$124$	−117.327	−0.0849699
$125$	1107.55	0.792502
$126$	0	0
$127$	2066.58	1.44393	0.721966	−	0.691929i	$-0.243239\pi$
$127$	2066.58	1.44393	0.721966	+	0.691929i	$0.243239\pi$
$128$	1595.11	1.10148
$129$	−2214.61	−1.51152
$130$	826.307	0.557476
$131$	676.993	0.451521	0.225760	−	0.974183i	$-0.427513\pi$
$131$	676.993	0.451521	0.225760	+	0.974183i	$0.427513\pi$
$132$	−1063.09	−0.700985
$133$	0	0
$134$	−2105.88	−1.35761
$135$	563.658	0.359348
$136$	−424.455	−0.267623
$137$	−326.837	−0.203822	−0.101911	−	0.994794i	$-0.532496\pi$
$137$	−326.837	−0.203822	−0.101911	+	0.994794i	$0.532496\pi$
$138$	−4773.58	−2.94460
$139$	−952.506	−0.581227	−0.290613	−	0.956841i	$-0.593859\pi$
$139$	−952.506	−0.581227	−0.290613	+	0.956841i	$0.593859\pi$
$140$	0	0
$141$	4705.02	2.81017
$142$	−1670.05	−0.986955
$143$	−684.626	−0.400359
$144$	−2615.09	−1.51336
$145$	−1608.59	−0.921283
$146$	1223.56	0.693577
$147$	0	0
$148$	−177.026	−0.0983208
$149$	−2826.15	−1.55387	−0.776936	−	0.629579i	$-0.783227\pi$
$149$	−2826.15	−1.55387	−0.776936	+	0.629579i	$0.783227\pi$
$150$	−1017.01	−0.553592
$151$	−3518.74	−1.89636	−0.948181	−	0.317730i	$-0.897079\pi$
$151$	−3518.74	−1.89636	−0.948181	+	0.317730i	$0.897079\pi$
$152$	297.469	0.158737
$153$	959.393	0.506943
$154$	0	0
$155$	396.194	0.205310
$156$	550.790	0.282683
$157$	−3036.72	−1.54367	−0.771836	−	0.635821i	$-0.780661\pi$
$157$	−3036.72	−1.54367	−0.771836	+	0.635821i	$0.780661\pi$
$158$	983.111	0.495013
$159$	221.176	0.110317
$160$	2028.36	1.00222
$161$	0	0
$162$	−1861.72	−0.902903
$163$	−1492.82	−0.717341	−0.358670	−	0.933464i	$-0.616770\pi$
$163$	−1492.82	−0.717341	−0.358670	+	0.933464i	$0.616770\pi$
$164$	1432.22	0.681939
$165$	3589.88	1.69377
$166$	365.628	0.170953
$167$	−379.063	−0.175645	−0.0878227	−	0.996136i	$-0.527991\pi$
$167$	−379.063	−0.175645	−0.0878227	+	0.996136i	$0.527991\pi$
$168$	0	0
$169$	−1842.29	−0.838549
$170$	−1286.93	−0.580605
$171$	−672.368	−0.300686
$172$	1084.72	0.480868
$173$	3296.35	1.44865	0.724326	−	0.689458i	$-0.242151\pi$
$173$	3296.35	1.44865	0.724326	+	0.689458i	$0.242151\pi$
$174$	−3338.67	−1.45462
$175$	0	0
$176$	−2906.41	−1.24477
$177$	−202.294	−0.0859059
$178$	−1210.74	−0.509824
$179$	1553.09	0.648509	0.324254	−	0.945970i	$-0.394887\pi$
$179$	1553.09	0.648509	0.324254	+	0.945970i	$0.394887\pi$
$180$	−1582.09	−0.655122
$181$	3522.56	1.44657	0.723287	−	0.690547i	$-0.242630\pi$
$181$	3522.56	1.44657	0.723287	+	0.690547i	$0.242630\pi$
$182$	0	0
$183$	−170.827	−0.0690047
$184$	−2604.08	−1.04334
$185$	597.789	0.237569
$186$	822.312	0.324166
$187$	1066.27	0.416969
$188$	−2304.53	−0.894018
$189$	0	0
$190$	901.912	0.344377
$191$	43.2443	0.0163824	0.00819122	−	0.999966i	$-0.497393\pi$
$191$	43.2443	0.0163824	0.00819122	+	0.999966i	$0.497393\pi$
$192$	−732.542	−0.275347
$193$	−206.333	−0.0769541	−0.0384771	−	0.999259i	$-0.512251\pi$
$193$	−206.333	−0.0769541	−0.0384771	+	0.999259i	$0.512251\pi$
$194$	2169.22	0.802788
$195$	−1859.93	−0.683037
$196$	0	0
$197$	4425.94	1.60069	0.800344	−	0.599541i	$-0.204650\pi$
$197$	4425.94	1.60069	0.800344	+	0.599541i	$0.204650\pi$
$198$	4081.57	1.46497
$199$	−1658.93	−0.590947	−0.295473	−	0.955351i	$-0.595477\pi$
$199$	−1658.93	−0.590947	−0.295473	+	0.955351i	$0.595477\pi$
$200$	−554.799	−0.196151
$201$	4740.10	1.66339
$202$	−811.671	−0.282718
$203$	0	0
$204$	−857.825	−0.294410
$205$	−4836.39	−1.64775
$206$	−4520.95	−1.52908
$207$	5885.97	1.97635
$208$	1505.82	0.501971
$209$	−747.269	−0.247319
$210$	0	0
$211$	−4319.63	−1.40936	−0.704681	−	0.709524i	$-0.748910\pi$
$211$	−4319.63	−1.40936	−0.704681	+	0.709524i	$0.748910\pi$
$212$	−108.333	−0.0350958
$213$	3759.11	1.20925
$214$	1810.80	0.578429
$215$	−3662.93	−1.16191
$216$	638.195	0.201036
$217$	0	0
$218$	−3757.85	−1.16749
$219$	−2754.09	−0.849792
$220$	−1758.33	−0.538849
$221$	−552.437	−0.168149
$222$	1240.73	0.375100
$223$	−1449.39	−0.435240	−0.217620	−	0.976034i	$-0.569829\pi$
$223$	−1449.39	−0.435240	−0.217620	+	0.976034i	$0.569829\pi$
$224$	0	0
$225$	1254.01	0.371558
$226$	−3404.87	−1.00216
$227$	−3298.69	−0.964500	−0.482250	−	0.876034i	$-0.660180\pi$
$227$	−3298.69	−0.964500	−0.482250	+	0.876034i	$0.660180\pi$
$228$	601.186	0.174625
$229$	5290.18	1.52657	0.763285	−	0.646062i	$-0.223585\pi$
$229$	5290.18	1.52657	0.763285	+	0.646062i	$0.223585\pi$
$230$	−7895.43	−2.26352
$231$	0	0
$232$	−1821.31	−0.515408
$233$	−4213.70	−1.18476	−0.592379	−	0.805659i	$-0.701811\pi$
$233$	−4213.70	−1.18476	−0.592379	+	0.805659i	$0.701811\pi$
$234$	−2114.68	−0.590772
$235$	7782.03	2.16019
$236$	99.0841	0.0273298
$237$	−2212.88	−0.606506
$238$	0	0
$239$	−1366.82	−0.369927	−0.184963	−	0.982745i	$-0.559217\pi$
$239$	−1366.82	−0.369927	−0.184963	+	0.982745i	$0.559217\pi$
$240$	−7895.86	−2.12365
$241$	−2678.22	−0.715847	−0.357923	−	0.933751i	$-0.616515\pi$
$241$	−2678.22	−0.715847	−0.357923	+	0.933751i	$0.616515\pi$
$242$	−32.9238	−0.00874555
$243$	5381.31	1.42062
$244$	83.6713	0.0219529
$245$	0	0
$246$	−10038.1	−2.60164
$247$	387.162	0.0997350
$248$	448.586	0.114860
$249$	−822.991	−0.209457
$250$	3802.11	0.961867
$251$	6136.68	1.54320	0.771601	−	0.636107i	$-0.219456\pi$
$251$	6136.68	1.54320	0.771601	+	0.636107i	$0.219456\pi$
$252$	0	0
$253$	6541.66	1.62558
$254$	7094.34	1.75251
$255$	2896.73	0.711375
$256$	4717.42	1.15171
$257$	4865.51	1.18094	0.590472	−	0.807058i	$-0.298942\pi$
$257$	4865.51	1.18094	0.590472	+	0.807058i	$0.298942\pi$
$258$	−7602.51	−1.83454
$259$	0	0
$260$	910.998	0.217299
$261$	4116.68	0.976308
$262$	2324.04	0.548015
$263$	6961.98	1.63230	0.816149	−	0.577842i	$-0.196105\pi$
$263$	6961.98	1.63230	0.816149	+	0.577842i	$0.196105\pi$
$264$	4064.60	0.947571
$265$	365.821	0.0848008
$266$	0	0
$267$	2725.24	0.624652
$268$	−2321.71	−0.529184
$269$	1232.38	0.279329	0.139664	−	0.990199i	$-0.455398\pi$
$269$	1232.38	0.279329	0.139664	+	0.990199i	$0.455398\pi$
$270$	1934.98	0.436144
$271$	−6481.64	−1.45289	−0.726443	−	0.687227i	$-0.758828\pi$
$271$	−6481.64	−1.45289	−0.726443	+	0.687227i	$0.758828\pi$
$272$	−2345.23	−0.522796
$273$	0	0
$274$	−1122.00	−0.247380
$275$	1393.70	0.305613
$276$	−5262.84	−1.14778
$277$	6000.65	1.30160	0.650802	−	0.759247i	$-0.274433\pi$
$277$	6000.65	1.30160	0.650802	+	0.759247i	$0.274433\pi$
$278$	−3269.85	−0.705441
$279$	−1013.94	−0.217573
$280$	0	0
$281$	1126.57	0.239166	0.119583	−	0.992824i	$-0.461844\pi$
$281$	1126.57	0.239166	0.119583	+	0.992824i	$0.461844\pi$
$282$	16151.8	3.41074
$283$	2585.20	0.543019	0.271510	−	0.962436i	$-0.412477\pi$
$283$	2585.20	0.543019	0.271510	+	0.962436i	$0.412477\pi$
$284$	−1841.22	−0.384706
$285$	−2030.11	−0.421941
$286$	−2350.25	−0.485920
$287$	0	0
$288$	−5190.96	−1.06208
$289$	−4052.61	−0.824875
$290$	−5522.11	−1.11817
$291$	−4882.68	−0.983601
$292$	1348.96	0.270350
$293$	1414.69	0.282073	0.141036	−	0.990004i	$-0.454957\pi$
$293$	1414.69	0.282073	0.141036	+	0.990004i	$0.454957\pi$
$294$	0	0
$295$	−334.591	−0.0660361
$296$	676.840	0.132907
$297$	−1603.20	−0.313223
$298$	−9701.86	−1.88595
$299$	−3389.26	−0.655538
$300$	−1121.25	−0.215785
$301$	0	0
$302$	−12079.4	−2.30163
$303$	1826.98	0.346394
$304$	1643.60	0.310089
$305$	−282.544	−0.0530441
$306$	3293.49	0.615282
$307$	−4177.55	−0.776630	−0.388315	−	0.921527i	$-0.626943\pi$
$307$	−4177.55	−0.776630	−0.388315	+	0.921527i	$0.626943\pi$
$308$	0	0
$309$	10176.2	1.87347
$310$	1360.09	0.249187
$311$	6143.55	1.12016	0.560079	−	0.828439i	$-0.310771\pi$
$311$	6143.55	1.12016	0.560079	+	0.828439i	$0.310771\pi$
$312$	−2105.88	−0.382122
$313$	−4527.90	−0.817675	−0.408837	−	0.912607i	$-0.634066\pi$
$313$	−4527.90	−0.817675	−0.408837	+	0.912607i	$0.634066\pi$
$314$	−10424.7	−1.87357
$315$	0	0
$316$	1083.87	0.192951
$317$	1801.20	0.319134	0.159567	−	0.987187i	$-0.448990\pi$
$317$	1801.20	0.319134	0.159567	+	0.987187i	$0.448990\pi$
$318$	759.272	0.133893
$319$	4575.28	0.803029
$320$	−1211.61	−0.211660
$321$	−4075.92	−0.708709
$322$	0	0
$323$	−602.984	−0.103873
$324$	−2052.53	−0.351943
$325$	−722.082	−0.123243
$326$	−5124.68	−0.870644
$327$	8458.51	1.43045
$328$	−5475.95	−0.921825
$329$	0	0
$330$	12323.7	2.05574
$331$	−2982.43	−0.495254	−0.247627	−	0.968855i	$-0.579651\pi$
$331$	−2982.43	−0.495254	−0.247627	+	0.968855i	$0.579651\pi$
$332$	403.103	0.0666360
$333$	−1529.86	−0.251759
$334$	−1301.28	−0.213183
$335$	7840.05	1.27865
$336$	0	0
$337$	−2479.66	−0.400818	−0.200409	−	0.979712i	$-0.564227\pi$
$337$	−2479.66	−0.400818	−0.200409	+	0.979712i	$0.564227\pi$
$338$	−6324.39	−1.01776
$339$	7663.99	1.22788
$340$	−1418.83	−0.226314
$341$	−1126.89	−0.178957
$342$	−2308.17	−0.364945
$343$	0	0
$344$	−4147.31	−0.650023
$345$	17771.8	2.77333
$346$	11316.0	1.75824
$347$	−4408.14	−0.681964	−0.340982	−	0.940070i	$-0.610760\pi$
$347$	−4408.14	−0.681964	−0.340982	+	0.940070i	$0.610760\pi$
$348$	−3680.86	−0.566997
$349$	111.860	0.0171569	0.00857843	−	0.999963i	$-0.497269\pi$
$349$	111.860	0.0171569	0.00857843	+	0.999963i	$0.497269\pi$
$350$	0	0
$351$	830.624	0.126312
$352$	−5769.23	−0.873582
$353$	−3996.59	−0.602598	−0.301299	−	0.953530i	$-0.597420\pi$
$353$	−3996.59	−0.602598	−0.301299	+	0.953530i	$0.597420\pi$
$354$	−694.453	−0.104265
$355$	6217.51	0.929552
$356$	−1334.83	−0.198724
$357$	0	0
$358$	5331.58	0.787102
$359$	−616.601	−0.0906490	−0.0453245	−	0.998972i	$-0.514432\pi$
$359$	−616.601	−0.0906490	−0.0453245	+	0.998972i	$0.514432\pi$
$360$	6048.94	0.885576
$361$	−6436.41	−0.938389
$362$	12092.6	1.75572
$363$	74.1080	0.0107153
$364$	0	0
$365$	−4555.23	−0.653237
$366$	−586.429	−0.0837517
$367$	−5179.64	−0.736717	−0.368359	−	0.929684i	$-0.620080\pi$
$367$	−5179.64	−0.736717	−0.368359	+	0.929684i	$0.620080\pi$
$368$	−14388.2	−2.03815
$369$	12377.2	1.74616
$370$	2052.15	0.288340
$371$	0	0
$372$	906.594	0.126357
$373$	−9348.00	−1.29764	−0.648822	−	0.760940i	$-0.724738\pi$
$373$	−9348.00	−1.29764	−0.648822	+	0.760940i	$0.724738\pi$
$374$	3660.38	0.506080
$375$	−8558.15	−1.17851
$376$	8811.12	1.20851
$377$	−2370.47	−0.323833
$378$	0	0
$379$	−11968.2	−1.62208	−0.811039	−	0.584992i	$-0.801097\pi$
$379$	−11968.2	−1.62208	−0.811039	+	0.584992i	$0.801097\pi$
$380$	994.352	0.134235
$381$	−15968.6	−2.14723
$382$	148.453	0.0198835
$383$	−12318.1	−1.64341	−0.821706	−	0.569911i	$-0.806978\pi$
$383$	−12318.1	−1.64341	−0.821706	+	0.569911i	$0.806978\pi$
$384$	−12325.5	−1.63798
$385$	0	0
$386$	−708.318	−0.0934000
$387$	9374.13	1.23130
$388$	2391.55	0.312919
$389$	−1073.05	−0.139860	−0.0699301	−	0.997552i	$-0.522278\pi$
$389$	−1073.05	−0.139860	−0.0699301	+	0.997552i	$0.522278\pi$
$390$	−6384.93	−0.829009
$391$	5278.58	0.682735
$392$	0	0
$393$	−5231.18	−0.671445
$394$	15193.8	1.94277
$395$	−3660.07	−0.466222
$396$	4499.91	0.571033
$397$	4565.61	0.577182	0.288591	−	0.957452i	$-0.406813\pi$
$397$	4565.61	0.577182	0.288591	+	0.957452i	$0.406813\pi$
$398$	−5694.92	−0.717238
$399$	0	0
$400$	−3065.42	−0.383177
$401$	−1932.00	−0.240598	−0.120299	−	0.992738i	$-0.538385\pi$
$401$	−1932.00	−0.240598	−0.120299	+	0.992738i	$0.538385\pi$
$402$	16272.2	2.01887
$403$	583.844	0.0721671
$404$	−894.862	−0.110201
$405$	6931.06	0.850389
$406$	0	0
$407$	−1700.28	−0.207076
$408$	3279.80	0.397976
$409$	−6119.00	−0.739768	−0.369884	−	0.929078i	$-0.620603\pi$
$409$	−6119.00	−0.739768	−0.369884	+	0.929078i	$0.620603\pi$
$410$	−16602.8	−1.99989
$411$	2525.49	0.303098
$412$	−4984.32	−0.596019
$413$	0	0
$414$	20205.9	2.39871
$415$	−1361.21	−0.161010
$416$	2989.05	0.352285
$417$	7360.08	0.864328
$418$	−2565.29	−0.300173
$419$	8203.40	0.956473	0.478236	−	0.878231i	$-0.341276\pi$
$419$	8203.40	0.956473	0.478236	+	0.878231i	$0.341276\pi$
$420$	0	0
$421$	9371.53	1.08489	0.542447	−	0.840090i	$-0.317498\pi$
$421$	9371.53	1.08489	0.542447	+	0.840090i	$0.317498\pi$
$422$	−14828.8	−1.71056
$423$	−19915.7	−2.28921
$424$	414.197	0.0474415
$425$	1124.60	0.128356
$426$	12904.6	1.46768
$427$	0	0
$428$	1996.40	0.225466
$429$	5290.16	0.595364
$430$	−12574.4	−1.41022
$431$	13294.8	1.48581	0.742907	−	0.669395i	$-0.233447\pi$
$431$	13294.8	1.48581	0.742907	+	0.669395i	$0.233447\pi$
$432$	3526.21	0.392719
$433$	3573.48	0.396606	0.198303	−	0.980141i	$-0.436457\pi$
$433$	3573.48	0.396606	0.198303	+	0.980141i	$0.436457\pi$
$434$	0	0
$435$	12429.7	1.37002
$436$	−4143.00	−0.455078
$437$	−3699.37	−0.404954
$438$	−9454.50	−1.03140
$439$	5381.51	0.585070	0.292535	−	0.956255i	$-0.405501\pi$
$439$	5381.51	0.585070	0.292535	+	0.956255i	$0.405501\pi$
$440$	6722.78	0.728400
$441$	0	0
$442$	−1896.46	−0.204084
$443$	−11113.7	−1.19193	−0.595966	−	0.803009i	$-0.703231\pi$
$443$	−11113.7	−1.19193	−0.595966	+	0.803009i	$0.703231\pi$
$444$	1367.90	0.146210
$445$	4507.51	0.480171
$446$	−4975.60	−0.528255
$447$	21837.8	2.31073
$448$	0	0
$449$	−8324.31	−0.874941	−0.437471	−	0.899233i	$-0.644126\pi$
$449$	−8324.31	−0.874941	−0.437471	+	0.899233i	$0.644126\pi$
$450$	4304.87	0.450964
$451$	13756.1	1.43625
$452$	−3753.84	−0.390633
$453$	27189.5	2.82003
$454$	−11324.0	−1.17062
$455$	0	0
$456$	−2298.57	−0.236053
$457$	−8388.31	−0.858618	−0.429309	−	0.903158i	$-0.641243\pi$
$457$	−8388.31	−0.858618	−0.429309	+	0.903158i	$0.641243\pi$
$458$	18160.6	1.85281
$459$	−1293.65	−0.131552
$460$	−8704.66	−0.882298
$461$	−8725.26	−0.881510	−0.440755	−	0.897628i	$-0.645289\pi$
$461$	−8725.26	−0.881510	−0.440755	+	0.897628i	$0.645289\pi$
$462$	0	0
$463$	15540.8	1.55992	0.779961	−	0.625828i	$-0.215239\pi$
$463$	15540.8	1.55992	0.779961	+	0.625828i	$0.215239\pi$
$464$	−10063.2	−1.00684
$465$	−3061.42	−0.305312
$466$	−14465.2	−1.43795
$467$	−12674.3	−1.25588	−0.627941	−	0.778261i	$-0.716102\pi$
$467$	−12674.3	−1.25588	−0.627941	+	0.778261i	$0.716102\pi$
$468$	−2331.42	−0.230277
$469$	0	0
$470$	26714.9	2.62184
$471$	23464.9	2.29556
$472$	−378.837	−0.0369436
$473$	10418.4	1.01277
$474$	−7596.57	−0.736122
$475$	−788.151	−0.0761323
$476$	0	0
$477$	−936.206	−0.0898657
$478$	−4692.16	−0.448984
$479$	3648.70	0.348045	0.174022	−	0.984742i	$-0.444323\pi$
$479$	3648.70	0.348045	0.174022	+	0.984742i	$0.444323\pi$
$480$	−15673.3	−1.49038
$481$	880.921	0.0835063
$482$	−9194.02	−0.868830
$483$	0	0
$484$	−36.2983	−0.00340893
$485$	−8075.88	−0.756097
$486$	18473.5	1.72422
$487$	11578.4	1.07735	0.538674	−	0.842514i	$-0.318925\pi$
$487$	11578.4	1.07735	0.538674	+	0.842514i	$0.318925\pi$
$488$	−319.908	−0.0296753
$489$	11535.1	1.06674
$490$	0	0
$491$	−2138.19	−0.196527	−0.0982637	−	0.995160i	$-0.531329\pi$
$491$	−2138.19	−0.196527	−0.0982637	+	0.995160i	$0.531329\pi$
$492$	−11066.9	−1.01409
$493$	3691.87	0.337269
$494$	1329.09	0.121049
$495$	−15195.5	−1.37977
$496$	2478.56	0.224377
$497$	0	0
$498$	−2825.24	−0.254221
$499$	15360.9	1.37805	0.689025	−	0.724737i	$-0.258039\pi$
$499$	15360.9	1.37805	0.689025	+	0.724737i	$0.258039\pi$
$500$	4191.81	0.374927
$501$	2929.05	0.261198
$502$	21066.5	1.87300
$503$	−15109.5	−1.33937	−0.669683	−	0.742647i	$-0.733570\pi$
$503$	−15109.5	−1.33937	−0.669683	+	0.742647i	$0.733570\pi$
$504$	0	0
$505$	3021.80	0.266274
$506$	22456.8	1.97298
$507$	14235.5	1.24699
$508$	7821.47	0.683113
$509$	449.913	0.0391788	0.0195894	−	0.999808i	$-0.493764\pi$
$509$	449.913	0.0391788	0.0195894	+	0.999808i	$0.493764\pi$
$510$	9944.17	0.863403
$511$	0	0
$512$	3433.51	0.296370
$513$	906.624	0.0780281
$514$	16702.8	1.43332
$515$	16831.2	1.44014
$516$	−8381.72	−0.715087
$517$	−22134.3	−1.88291
$518$	0	0
$519$	−25471.1	−2.15425
$520$	−3483.10	−0.293738
$521$	4086.94	0.343670	0.171835	−	0.985126i	$-0.445030\pi$
$521$	4086.94	0.343670	0.171835	+	0.985126i	$0.445030\pi$
$522$	14132.1	1.18495
$523$	−10729.8	−0.897097	−0.448549	−	0.893758i	$-0.648059\pi$
$523$	−10729.8	−0.897097	−0.448549	+	0.893758i	$0.648059\pi$
$524$	2562.24	0.213611
$525$	0	0
$526$	23899.7	1.98114
$527$	−909.304	−0.0751611
$528$	22458.0	1.85106
$529$	20217.6	1.66168
$530$	1255.82	0.102924
$531$	856.282	0.0699802
$532$	0	0
$533$	−7127.05	−0.579188
$534$	9355.45	0.758146
$535$	−6741.50	−0.544786
$536$	8876.81	0.715336
$537$	−12000.8	−0.964382
$538$	4230.62	0.339024
$539$	0	0
$540$	2133.30	0.170005
$541$	−950.986	−0.0755750	−0.0377875	−	0.999286i	$-0.512031\pi$
$541$	−950.986	−0.0755750	−0.0377875	+	0.999286i	$0.512031\pi$
$542$	−22250.8	−1.76338
$543$	−27219.1	−2.15117
$544$	−4655.29	−0.366900
$545$	13990.2	1.09959
$546$	0	0
$547$	18719.6	1.46324	0.731622	−	0.681711i	$-0.238764\pi$
$547$	18719.6	1.46324	0.731622	+	0.681711i	$0.238764\pi$
$548$	−1236.99	−0.0964265
$549$	723.085	0.0562122
$550$	4784.43	0.370925
$551$	−2587.36	−0.200046
$552$	20121.9	1.55153
$553$	0	0
$554$	20599.6	1.57977
$555$	−4619.16	−0.353284
$556$	−3604.99	−0.274974
$557$	−18162.8	−1.38166	−0.690829	−	0.723018i	$-0.742754\pi$
$557$	−18162.8	−1.38166	−0.690829	+	0.723018i	$0.742754\pi$
$558$	−3480.73	−0.264070
$559$	−5397.81	−0.408413
$560$	0	0
$561$	−8239.13	−0.620064
$562$	3867.41	0.290279
$563$	15511.6	1.16117	0.580584	−	0.814200i	$-0.302824\pi$
$563$	15511.6	1.16117	0.580584	+	0.814200i	$0.302824\pi$
$564$	17807.3	1.32947
$565$	12676.1	0.943873
$566$	8874.72	0.659068
$567$	0	0
$568$	7039.70	0.520034
$569$	−17928.9	−1.32095	−0.660473	−	0.750850i	$-0.729644\pi$
$569$	−17928.9	−1.32095	−0.660473	+	0.750850i	$0.729644\pi$
$570$	−6969.14	−0.512114
$571$	−910.568	−0.0667357	−0.0333679	−	0.999443i	$-0.510623\pi$
$571$	−910.568	−0.0667357	−0.0333679	+	0.999443i	$0.510623\pi$
$572$	−2591.13	−0.189407
$573$	−334.151	−0.0243619
$574$	0	0
$575$	6899.56	0.500402
$576$	3100.75	0.224302
$577$	−26496.1	−1.91170	−0.955848	−	0.293862i	$-0.905059\pi$
$577$	−26496.1	−1.91170	−0.955848	+	0.293862i	$0.905059\pi$
$578$	−13912.2	−1.00116
$579$	1594.35	0.114437
$580$	−6088.09	−0.435852
$581$	0	0
$582$	−16761.7	−1.19381
$583$	−1040.50	−0.0739160
$584$	−5157.61	−0.365451
$585$	7872.82	0.556412
$586$	4856.49	0.342355
$587$	25543.4	1.79606	0.898031	−	0.439932i	$-0.144997\pi$
$587$	25543.4	1.79606	0.898031	+	0.439932i	$0.144997\pi$
$588$	0	0
$589$	637.264	0.0445807
$590$	−1148.61	−0.0801486
$591$	−34199.6	−2.38034
$592$	3739.73	0.259632
$593$	−21320.4	−1.47643	−0.738215	−	0.674565i	$-0.764331\pi$
$593$	−21320.4	−1.47643	−0.738215	+	0.674565i	$0.764331\pi$
$594$	−5503.61	−0.380162
$595$	0	0
$596$	−10696.2	−0.735126
$597$	12818.7	0.878782
$598$	−11635.0	−0.795633
$599$	−19402.4	−1.32347	−0.661737	−	0.749736i	$-0.730180\pi$
$599$	−19402.4	−1.32347	−0.661737	+	0.749736i	$0.730180\pi$
$600$	4286.97	0.291692
$601$	−16408.5	−1.11367	−0.556837	−	0.830622i	$-0.687985\pi$
$601$	−16408.5	−1.11367	−0.556837	+	0.830622i	$0.687985\pi$
$602$	0	0
$603$	−20064.2	−1.35502
$604$	−13317.5	−0.897155
$605$	122.573	0.00823689
$606$	6271.84	0.420422
$607$	18288.3	1.22290	0.611448	−	0.791284i	$-0.290587\pi$
$607$	18288.3	1.22290	0.611448	+	0.791284i	$0.290587\pi$
$608$	3262.55	0.217621
$609$	0	0
$610$	−969.944	−0.0643801
$611$	11467.8	0.759312
$612$	3631.05	0.239831
$613$	575.666	0.0379298	0.0189649	−	0.999820i	$-0.493963\pi$
$613$	575.666	0.0379298	0.0189649	+	0.999820i	$0.493963\pi$
$614$	−14341.1	−0.942603
$615$	37371.1	2.45032
$616$	0	0
$617$	17703.9	1.15516	0.577580	−	0.816334i	$-0.303997\pi$
$617$	17703.9	1.15516	0.577580	+	0.816334i	$0.303997\pi$
$618$	34933.7	2.27385
$619$	−9375.12	−0.608753	−0.304376	−	0.952552i	$-0.598448\pi$
$619$	−9375.12	−0.608753	−0.304376	+	0.952552i	$0.598448\pi$
$620$	1499.49	0.0971307
$621$	−7936.68	−0.512863
$622$	21090.2	1.35955
$623$	0	0
$624$	−11635.6	−0.746468
$625$	−18947.5	−1.21264
$626$	−15543.8	−0.992420
$627$	5774.20	0.367782
$628$	−11493.2	−0.730300
$629$	−1371.99	−0.0869708
$630$	0	0
$631$	−25659.2	−1.61882	−0.809410	−	0.587244i	$-0.800213\pi$
$631$	−25659.2	−1.61882	−0.809410	+	0.587244i	$0.800213\pi$
$632$	−4144.07	−0.260826
$633$	33378.1	2.09583
$634$	6183.32	0.387336
$635$	−26411.8	−1.65058
$636$	837.093	0.0521901
$637$	0	0
$638$	15706.4	0.974645
$639$	−15911.8	−0.985071
$640$	−20386.2	−1.25912
$641$	17.1131	0.00105449	0.000527243	−	1.00000i	$-0.499832\pi$
$641$	17.1131	0.00105449	0.000527243		1.00000i	$0.499832\pi$
$642$	−13992.2	−0.860167
$643$	−8808.39	−0.540232	−0.270116	−	0.962828i	$-0.587062\pi$
$643$	−8808.39	−0.540232	−0.270116	+	0.962828i	$0.587062\pi$
$644$	0	0
$645$	28303.7	1.72784
$646$	−2069.98	−0.126071
$647$	68.0956	0.00413773	0.00206887	−	0.999998i	$-0.499341\pi$
$647$	68.0956	0.00413773	0.00206887	+	0.999998i	$0.499341\pi$
$648$	7847.62	0.475746
$649$	951.671	0.0575598
$650$	−2478.83	−0.149581
$651$	0	0
$652$	−5649.93	−0.339369
$653$	−24699.4	−1.48018	−0.740092	−	0.672505i	$-0.765218\pi$
$653$	−24699.4	−1.48018	−0.740092	+	0.672505i	$0.765218\pi$
$654$	29037.2	1.73615
$655$	−8652.28	−0.516141
$656$	−30256.1	−1.80077
$657$	11657.7	0.692253
$658$	0	0
$659$	−30523.5	−1.80429	−0.902144	−	0.431434i	$-0.858008\pi$
$659$	−30523.5	−1.80429	−0.902144	+	0.431434i	$0.858008\pi$
$660$	13586.8	0.801309
$661$	−24621.0	−1.44878	−0.724392	−	0.689389i	$-0.757879\pi$
$661$	−24621.0	−1.44878	−0.724392	+	0.689389i	$0.757879\pi$
$662$	−10238.3	−0.601095
$663$	4268.72	0.250050
$664$	−1541.22	−0.0900766
$665$	0	0
$666$	−5251.83	−0.305562
$667$	22650.0	1.31486
$668$	−1434.65	−0.0830965
$669$	11199.5	0.647234
$670$	26914.0	1.55191
$671$	803.636	0.0462355
$672$	0	0
$673$	32140.8	1.84092	0.920458	−	0.390842i	$-0.127816\pi$
$673$	32140.8	1.84092	0.920458	+	0.390842i	$0.127816\pi$
$674$	−8512.40	−0.486477
$675$	−1690.91	−0.0964196
$676$	−6972.60	−0.396711
$677$	−16017.0	−0.909284	−0.454642	−	0.890674i	$-0.650233\pi$
$677$	−16017.0	−0.909284	−0.454642	+	0.890674i	$0.650233\pi$
$678$	26309.6	1.49029
$679$	0	0
$680$	5424.73	0.305925
$681$	25489.2	1.43428
$682$	−3868.48	−0.217202
$683$	−18001.7	−1.00852	−0.504259	−	0.863552i	$-0.668234\pi$
$683$	−18001.7	−1.00852	−0.504259	+	0.863552i	$0.668234\pi$
$684$	−2544.74	−0.142252
$685$	4177.12	0.232992
$686$	0	0
$687$	−40877.6	−2.27013
$688$	−22915.0	−1.26981
$689$	539.086	0.0298077
$690$	61008.5	3.36602
$691$	19740.8	1.08680	0.543398	−	0.839475i	$-0.317137\pi$
$691$	19740.8	1.08680	0.543398	+	0.839475i	$0.317137\pi$
$692$	12475.8	0.685346
$693$	0	0
$694$	−15132.7	−0.827707
$695$	12173.5	0.664411
$696$	14073.4	0.766450
$697$	11100.0	0.603217
$698$	384.004	0.0208235
$699$	32559.6	1.76183
$700$	0	0
$701$	−28044.2	−1.51101	−0.755503	−	0.655145i	$-0.772607\pi$
$701$	−28044.2	−1.51101	−0.755503	+	0.655145i	$0.772607\pi$
$702$	2851.44	0.153306
$703$	961.523	0.0515854
$704$	3446.17	0.184492
$705$	−60132.3	−3.21236
$706$	−13719.9	−0.731379
$707$	0	0
$708$	−765.630	−0.0406414
$709$	−18840.1	−0.997962	−0.498981	−	0.866613i	$-0.666292\pi$
$709$	−18840.1	−0.997962	−0.498981	+	0.866613i	$0.666292\pi$
$710$	21344.0	1.12821
$711$	9366.80	0.494068
$712$	5103.57	0.268630
$713$	−5578.68	−0.293020
$714$	0	0
$715$	8749.84	0.457658
$716$	5878.03	0.306805
$717$	10561.5	0.550109
$718$	−2116.73	−0.110022
$719$	400.627	0.0207800	0.0103900	−	0.999946i	$-0.496693\pi$
$719$	400.627	0.0207800	0.0103900	+	0.999946i	$0.496693\pi$
$720$	33422.1	1.72995
$721$	0	0
$722$	−22095.5	−1.13893
$723$	20694.8	1.06452
$724$	13332.0	0.684364
$725$	4825.59	0.247197
$726$	254.405	0.0130053
$727$	30143.9	1.53779	0.768897	−	0.639372i	$-0.220806\pi$
$727$	30143.9	1.53779	0.768897	+	0.639372i	$0.220806\pi$
$728$	0	0
$729$	−26939.2	−1.36865
$730$	−15637.6	−0.792840
$731$	8406.78	0.425357
$732$	−646.534	−0.0326456
$733$	−36814.5	−1.85508	−0.927540	−	0.373724i	$-0.878081\pi$
$733$	−36814.5	−1.85508	−0.927540	+	0.373724i	$0.878081\pi$
$734$	−17781.2	−0.894161
$735$	0	0
$736$	−28560.7	−1.43038
$737$	−22299.3	−1.11453
$738$	42489.7	2.11933
$739$	−5909.07	−0.294139	−0.147069	−	0.989126i	$-0.546984\pi$
$739$	−5909.07	−0.294139	−0.147069	+	0.989126i	$0.546984\pi$
$740$	2262.48	0.112392
$741$	−2991.63	−0.148313
$742$	0	0
$743$	−19203.6	−0.948197	−0.474099	−	0.880472i	$-0.657226\pi$
$743$	−19203.6	−0.948197	−0.474099	+	0.880472i	$0.657226\pi$
$744$	−3466.26	−0.170805
$745$	36119.4	1.77626
$746$	−32090.7	−1.57496
$747$	3483.60	0.170627
$748$	4035.55	0.197265
$749$	0	0
$750$	−29379.2	−1.43037
$751$	28140.3	1.36731	0.683657	−	0.729803i	$-0.260388\pi$
$751$	28140.3	1.36731	0.683657	+	0.729803i	$0.260388\pi$
$752$	48683.9	2.36080
$753$	−47418.5	−2.29486
$754$	−8137.55	−0.393040
$755$	44971.0	2.16777
$756$	0	0
$757$	−34796.3	−1.67067	−0.835333	−	0.549744i	$-0.814725\pi$
$757$	−34796.3	−1.67067	−0.835333	+	0.549744i	$0.814725\pi$
$758$	−41085.7	−1.96873
$759$	−50547.9	−2.41735
$760$	−3801.79	−0.181455
$761$	−19782.6	−0.942340	−0.471170	−	0.882043i	$-0.656168\pi$
$761$	−19782.6	−0.942340	−0.471170	+	0.882043i	$0.656168\pi$
$762$	−54818.5	−2.60612
$763$	0	0
$764$	163.668	0.00775041
$765$	−12261.5	−0.579496
$766$	−42286.8	−1.99463
$767$	−493.064	−0.0232119
$768$	−36451.8	−1.71268
$769$	25017.3	1.17314	0.586570	−	0.809898i	$-0.300478\pi$
$769$	25017.3	1.17314	0.586570	+	0.809898i	$0.300478\pi$
$770$	0	0
$771$	−37596.2	−1.75615
$772$	−780.916	−0.0364064
$773$	9380.91	0.436491	0.218246	−	0.975894i	$-0.429967\pi$
$773$	9380.91	0.436491	0.218246	+	0.975894i	$0.429967\pi$
$774$	32180.4	1.49444
$775$	−1188.54	−0.0550885
$776$	−9143.83	−0.422995
$777$	0	0
$778$	−3683.65	−0.169750
$779$	−7779.17	−0.357789
$780$	−7039.34	−0.323140
$781$	−17684.3	−0.810237
$782$	18120.8	0.828642
$783$	−5550.96	−0.253353
$784$	0	0
$785$	38810.6	1.76460
$786$	−17958.1	−0.814940
$787$	25327.8	1.14719	0.573596	−	0.819138i	$-0.305548\pi$
$787$	25327.8	1.14719	0.573596	+	0.819138i	$0.305548\pi$
$788$	16751.0	0.757273
$789$	−53795.7	−2.42735
$790$	−12564.6	−0.565859
$791$	0	0
$792$	−17204.9	−0.771905
$793$	−416.366	−0.0186451
$794$	15673.2	0.700532
$795$	−2826.73	−0.126105
$796$	−6278.62	−0.279572
$797$	38975.6	1.73223	0.866114	−	0.499847i	$-0.166610\pi$
$797$	38975.6	1.73223	0.866114	+	0.499847i	$0.166610\pi$
$798$	0	0
$799$	−17860.5	−0.790814
$800$	−6084.86	−0.268915
$801$	−11535.6	−0.508850
$802$	−6632.36	−0.292016
$803$	12956.3	0.569389
$804$	17940.1	0.786936
$805$	0	0
$806$	2004.27	0.0875899
$807$	−9522.68	−0.415383
$808$	3421.40	0.148966
$809$	1297.61	0.0563925	0.0281962	−	0.999602i	$-0.491024\pi$
$809$	1297.61	0.0563925	0.0281962	+	0.999602i	$0.491024\pi$
$810$	23793.6	1.03213
$811$	−37846.6	−1.63869	−0.819343	−	0.573304i	$-0.805662\pi$
$811$	−37846.6	−1.63869	−0.819343	+	0.573304i	$0.805662\pi$
$812$	0	0
$813$	50084.1	2.16055
$814$	−5836.88	−0.251330
$815$	19078.9	0.820005
$816$	18121.8	0.777438
$817$	−5891.70	−0.252294
$818$	−21005.9	−0.897864
$819$	0	0
$820$	−18304.5	−0.779537
$821$	−4428.01	−0.188232	−0.0941160	−	0.995561i	$-0.530002\pi$
$821$	−4428.01	−0.188232	−0.0941160	+	0.995561i	$0.530002\pi$
$822$	8669.74	0.367873
$823$	−5773.25	−0.244524	−0.122262	−	0.992498i	$-0.539015\pi$
$823$	−5773.25	−0.244524	−0.122262	+	0.992498i	$0.539015\pi$
$824$	19057.0	0.805681
$825$	−10769.2	−0.454469
$826$	0	0
$827$	24517.5	1.03090	0.515451	−	0.856919i	$-0.327624\pi$
$827$	24517.5	1.03090	0.515451	+	0.856919i	$0.327624\pi$
$828$	22276.9	0.934994
$829$	−26528.1	−1.11141	−0.555705	−	0.831380i	$-0.687552\pi$
$829$	−26528.1	−1.11141	−0.555705	+	0.831380i	$0.687552\pi$
$830$	−4672.90	−0.195420
$831$	−46367.5	−1.93558
$832$	−1785.47	−0.0743991
$833$	0	0
$834$	25266.4	1.04904
$835$	4844.60	0.200783
$836$	−2828.22	−0.117005
$837$	1367.20	0.0564602
$838$	28161.4	1.16088
$839$	−2805.47	−0.115442	−0.0577208	−	0.998333i	$-0.518383\pi$
$839$	−2805.47	−0.115442	−0.0577208	+	0.998333i	$0.518383\pi$
$840$	0	0
$841$	−8547.46	−0.350464
$842$	32171.4	1.31675
$843$	−8705.12	−0.355658
$844$	−16348.7	−0.666759
$845$	23545.3	0.958561
$846$	−68368.4	−2.77843
$847$	0	0
$848$	2288.55	0.0926760
$849$	−19976.1	−0.807510
$850$	3860.64	0.155787
$851$	−8417.27	−0.339060
$852$	14227.3	0.572086
$853$	6907.55	0.277268	0.138634	−	0.990344i	$-0.455729\pi$
$853$	6907.55	0.277268	0.138634	+	0.990344i	$0.455729\pi$
$854$	0	0
$855$	8593.16	0.343719
$856$	−7632.99	−0.304778
$857$	−39021.0	−1.55535	−0.777674	−	0.628668i	$-0.783601\pi$
$857$	−39021.0	−1.55535	−0.777674	+	0.628668i	$0.783601\pi$
$858$	18160.5	0.722599
$859$	−9486.04	−0.376786	−0.188393	−	0.982094i	$-0.560328\pi$
$859$	−9486.04	−0.376786	−0.188393	+	0.982094i	$0.560328\pi$
$860$	−13863.2	−0.549689
$861$	0	0
$862$	45639.4	1.80335
$863$	−38698.9	−1.52645	−0.763225	−	0.646133i	$-0.776385\pi$
$863$	−38698.9	−1.52645	−0.763225	+	0.646133i	$0.776385\pi$
$864$	6999.52	0.275612
$865$	−42128.8	−1.65598
$866$	12267.4	0.481365
$867$	31314.8	1.22665
$868$	0	0
$869$	10410.3	0.406379
$870$	42669.7	1.66280
$871$	11553.3	0.449449
$872$	15840.3	0.615161
$873$	20667.7	0.801256
$874$	−12699.5	−0.491496
$875$	0	0
$876$	−10423.5	−0.402030
$877$	−2116.94	−0.0815098	−0.0407549	−	0.999169i	$-0.512976\pi$
$877$	−2116.94	−0.0815098	−0.0407549	+	0.999169i	$0.512976\pi$
$878$	18474.2	0.710105
$879$	−10931.4	−0.419464
$880$	37145.2	1.42292
$881$	−25926.2	−0.991458	−0.495729	−	0.868477i	$-0.665099\pi$
$881$	−25926.2	−0.991458	−0.495729	+	0.868477i	$0.665099\pi$
$882$	0	0
$883$	−8378.15	−0.319306	−0.159653	−	0.987173i	$-0.551038\pi$
$883$	−8378.15	−0.319306	−0.159653	+	0.987173i	$0.551038\pi$
$884$	−2090.83	−0.0795500
$885$	2585.41	0.0982006
$886$	−38152.0	−1.44666
$887$	42851.8	1.62212	0.811061	−	0.584962i	$-0.198890\pi$
$887$	42851.8	1.62212	0.811061	+	0.584962i	$0.198890\pi$
$888$	−5229.99	−0.197643
$889$	0	0
$890$	15473.8	0.582789
$891$	−19713.9	−0.741235
$892$	−5485.57	−0.205909
$893$	12517.1	0.469059
$894$	74966.9	2.80455
$895$	−19849.1	−0.741323
$896$	0	0
$897$	26189.0	0.974834
$898$	−28576.5	−1.06193
$899$	−3901.76	−0.144751
$900$	4746.10	0.175781
$901$	−839.596	−0.0310444
$902$	47223.0	1.74319
$903$	0	0
$904$	14352.4	0.528046
$905$	−45020.0	−1.65361
$906$	93338.7	3.42270
$907$	6476.75	0.237108	0.118554	−	0.992948i	$-0.462174\pi$
$907$	6476.75	0.237108	0.118554	+	0.992948i	$0.462174\pi$
$908$	−12484.7	−0.456298
$909$	−7733.37	−0.282178
$910$	0	0
$911$	8590.89	0.312436	0.156218	−	0.987723i	$-0.450070\pi$
$911$	8590.89	0.312436	0.156218	+	0.987723i	$0.450070\pi$
$912$	−12700.2	−0.461125
$913$	3871.67	0.140344
$914$	−28796.2	−1.04211
$915$	2183.24	0.0788805
$916$	20021.9	0.722209
$917$	0	0
$918$	−4440.96	−0.159666
$919$	32029.4	1.14968	0.574838	−	0.818267i	$-0.305065\pi$
$919$	32029.4	1.14968	0.574838	+	0.818267i	$0.305065\pi$
$920$	33281.3	1.19266
$921$	32280.2	1.15491
$922$	−29952.9	−1.06990
$923$	9162.31	0.326740
$924$	0	0
$925$	−1793.30	−0.0637442
$926$	53350.0	1.89329
$927$	−43074.3	−1.52616
$928$	−19975.5	−0.706603
$929$	−37416.2	−1.32140	−0.660702	−	0.750648i	$-0.729741\pi$
$929$	−37416.2	−1.32140	−0.660702	+	0.750648i	$0.729741\pi$
$930$	−10509.5	−0.370560
$931$	0	0
$932$	−15947.8	−0.560500
$933$	−47471.7	−1.66576
$934$	−43509.5	−1.52428
$935$	−13627.4	−0.476645
$936$	8913.91	0.311282
$937$	−12407.0	−0.432570	−0.216285	−	0.976330i	$-0.569394\pi$
$937$	−12407.0	−0.432570	−0.216285	+	0.976330i	$0.569394\pi$
$938$	0	0
$939$	34987.4	1.21594
$940$	29453.0	1.02197
$941$	11985.1	0.415200	0.207600	−	0.978214i	$-0.433435\pi$
$941$	11985.1	0.415200	0.207600	+	0.978214i	$0.433435\pi$
$942$	80552.6	2.78614
$943$	68099.6	2.35167
$944$	−2093.18	−0.0721686
$945$	0	0
$946$	35765.2	1.22920
$947$	47379.7	1.62580	0.812901	−	0.582402i	$-0.197887\pi$
$947$	47379.7	1.62580	0.812901	+	0.582402i	$0.197887\pi$
$948$	−8375.17	−0.286933
$949$	−6712.73	−0.229615
$950$	−2705.64	−0.0924026
$951$	−13918.0	−0.474576
$952$	0	0
$953$	39722.5	1.35020	0.675099	−	0.737728i	$-0.264101\pi$
$953$	39722.5	1.35020	0.675099	+	0.737728i	$0.264101\pi$
$954$	−3213.89	−0.109071
$955$	−552.681	−0.0187271
$956$	−5173.07	−0.175010
$957$	−35353.5	−1.19417
$958$	12525.6	0.422426
$959$	0	0
$960$	9362.22	0.314754
$961$	961.000	0.0322581
$962$	3024.10	0.101352
$963$	17252.8	0.577324
$964$	−10136.4	−0.338662
$965$	2637.02	0.0879677
$966$	0	0
$967$	44883.6	1.49262	0.746308	−	0.665601i	$-0.231825\pi$
$967$	44883.6	1.49262	0.746308	+	0.665601i	$0.231825\pi$
$968$	138.782	0.00460809
$969$	4659.30	0.154467
$970$	−27723.6	−0.917682
$971$	−35719.0	−1.18051	−0.590256	−	0.807216i	$-0.700973\pi$
$971$	−35719.0	−1.18051	−0.590256	+	0.807216i	$0.700973\pi$
$972$	20366.9	0.672086
$973$	0	0
$974$	39747.5	1.30759
$975$	5579.58	0.183271
$976$	−1767.58	−0.0579701
$977$	−46492.5	−1.52244	−0.761222	−	0.648491i	$-0.775400\pi$
$977$	−46492.5	−1.52244	−0.761222	+	0.648491i	$0.775400\pi$
$978$	39598.8	1.29471
$979$	−12820.6	−0.418538
$980$	0	0
$981$	−35803.7	−1.16526
$982$	−7340.16	−0.238527
$983$	−58800.7	−1.90788	−0.953942	−	0.299992i	$-0.903016\pi$
$983$	−58800.7	−1.90788	−0.953942	+	0.299992i	$0.903016\pi$
$984$	42313.0	1.37082
$985$	−56565.6	−1.82978
$986$	12673.8	0.409346
$987$	0	0
$988$	1465.31	0.0471839
$989$	51576.5	1.65828
$990$	−52164.3	−1.67464
$991$	20054.4	0.642835	0.321418	−	0.946938i	$-0.395841\pi$
$991$	20054.4	0.642835	0.321418	+	0.946938i	$0.395841\pi$
$992$	4919.95	0.157468
$993$	23045.4	0.736480
$994$	0	0
$995$	21201.9	0.675522
$996$	−3114.81	−0.0990928
$997$	56811.1	1.80464	0.902320	−	0.431067i	$-0.141863\pi$
$997$	56811.1	1.80464	0.902320	+	0.431067i	$0.141863\pi$
$998$	52732.2	1.67255
$999$	2062.87	0.0653315

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1519.4.a.h.1.17	✓	23
7.6	odd	2		1519.4.a.i.1.17	yes	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1519.4.a.h.1.17	✓	23	1.1	even	1	trivial
1519.4.a.i.1.17	yes	23	7.6	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:	\( N \)	\(=\)	\( 1519 = 7^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1519.a (trivial)

\(f(q)\)	\(=\)	\(q+3.43289 q^{2} -7.72707 q^{3} +3.78474 q^{4} -12.7805 q^{5} -26.5262 q^{6} -14.4705 q^{8} +32.7076 q^{9} -43.8739 q^{10} +36.3512 q^{11} -29.2450 q^{12} -18.8337 q^{13} +98.7554 q^{15} -79.9537 q^{16} +29.3324 q^{17} +112.282 q^{18} -20.5569 q^{19} -48.3707 q^{20} +124.790 q^{22} +179.957 q^{23} +111.815 q^{24} +38.3400 q^{25} -64.6539 q^{26} -44.1031 q^{27} +125.863 q^{29} +339.017 q^{30} -31.0000 q^{31} -158.708 q^{32} -280.888 q^{33} +100.695 q^{34} +123.790 q^{36} -46.7737 q^{37} -70.5697 q^{38} +145.529 q^{39} +184.940 q^{40} +378.421 q^{41} +286.604 q^{43} +137.580 q^{44} -418.018 q^{45} +617.774 q^{46} -608.901 q^{47} +617.808 q^{48} +131.617 q^{50} -226.653 q^{51} -71.2806 q^{52} -28.6235 q^{53} -151.401 q^{54} -464.585 q^{55} +158.845 q^{57} +432.075 q^{58} +26.1799 q^{59} +373.764 q^{60} +22.1075 q^{61} -106.420 q^{62} +94.8020 q^{64} +240.703 q^{65} -964.259 q^{66} -613.441 q^{67} +111.016 q^{68} -1390.54 q^{69} -486.486 q^{71} -473.296 q^{72} +356.421 q^{73} -160.569 q^{74} -296.255 q^{75} -77.8026 q^{76} +499.586 q^{78} +286.380 q^{79} +1021.84 q^{80} -542.318 q^{81} +1299.08 q^{82} +106.507 q^{83} -374.881 q^{85} +983.880 q^{86} -972.554 q^{87} -526.021 q^{88} -352.687 q^{89} -1435.01 q^{90} +681.092 q^{92} +239.539 q^{93} -2090.29 q^{94} +262.727 q^{95} +1226.35 q^{96} +631.893 q^{97} +1188.96 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q + 5 q^{2} - 6 q^{3} + 91 q^{4} - 40 q^{5} - 36 q^{6} + 39 q^{8} + 211 q^{9} - 40 q^{10} + 44 q^{11} - 414 q^{12} + 20 q^{13} + 523 q^{16} - 306 q^{17} + 51 q^{18} - 296 q^{19} - 400 q^{20} - 326 q^{22}+ \cdots - 3456 q^{99}+O(q^{100}) \) 23 * q + 5 * q^2 - 6 * q^3 + 91 * q^4 - 40 * q^5 - 36 * q^6 + 39 * q^8 + 211 * q^9 - 40 * q^10 + 44 * q^11 - 414 * q^12 + 20 * q^13 + 523 * q^16 - 306 * q^17 + 51 * q^18 - 296 * q^19 - 400 * q^20 - 326 * q^22 - 130 * q^23 - 432 * q^24 + 801 * q^25 - 474 * q^26 - 192 * q^27 - 16 * q^29 - 74 * q^30 - 713 * q^31 + 287 * q^32 - 488 * q^33 - 692 * q^34 + 623 * q^36 - 296 * q^37 + 40 * q^38 - 124 * q^39 - 48 * q^40 - 2344 * q^41 - 48 * q^43 + 1352 * q^44 - 132 * q^45 - 836 * q^46 - 436 * q^47 - 4190 * q^48 - 767 * q^50 - 872 * q^51 + 632 * q^52 - 884 * q^53 - 100 * q^54 - 3308 * q^55 + 404 * q^57 - 562 * q^58 - 400 * q^59 + 2248 * q^60 - 860 * q^61 - 155 * q^62 + 3125 * q^64 - 1676 * q^65 - 128 * q^66 - 164 * q^67 - 302 * q^68 - 2136 * q^69 + 56 * q^71 + 1613 * q^72 - 1610 * q^73 + 908 * q^74 + 910 * q^75 - 5208 * q^76 + 1424 * q^78 + 318 * q^79 - 11480 * q^80 + 231 * q^81 + 2608 * q^82 - 3166 * q^83 - 716 * q^85 - 224 * q^86 - 1248 * q^87 - 5562 * q^88 - 2854 * q^89 - 7172 * q^90 - 3558 * q^92 + 186 * q^93 - 4228 * q^94 + 984 * q^95 + 4178 * q^96 - 3332 * q^97 - 3456 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more