LMFDB - Embedded newform 500.4.a.d.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [500,4,Mod(1,500)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(500, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("500.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 500 = 2^{2} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	500.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$29.5009550029$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 36x^{6} + 431x^{4} - 2016x^{2} + 2896 $ x^8 - 36x^6 + 431x^4 - 2016*x^2 + 2896
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8}\cdot 5^{4} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$4.14072$ of defining polynomial
Character	$\chi$	$=$	500.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.99947 q^{3} +13.6557 q^{7} -23.0021 q^{9} +O(q^{10})$	$q-1.99947 q^{3} +13.6557 q^{7} -23.0021 q^{9} +24.7784 q^{11} +74.4947 q^{13} -137.995 q^{17} +54.7771 q^{19} -27.3042 q^{21} -170.121 q^{23} +99.9777 q^{27} +65.8946 q^{29} +295.426 q^{31} -49.5436 q^{33} +116.513 q^{37} -148.950 q^{39} +333.050 q^{41} +304.868 q^{43} -370.955 q^{47} -156.521 q^{49} +275.918 q^{51} -110.290 q^{53} -109.525 q^{57} +93.7143 q^{59} +643.906 q^{61} -314.111 q^{63} +755.760 q^{67} +340.151 q^{69} +602.116 q^{71} +572.271 q^{73} +338.367 q^{77} -404.933 q^{79} +421.155 q^{81} +554.312 q^{83} -131.754 q^{87} +886.269 q^{89} +1017.28 q^{91} -590.694 q^{93} +203.851 q^{97} -569.956 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 132 q^{9}+O(q^{10}) $ 8 * q + 132 * q^9	$ 8 q + 132 q^{9} + 40 q^{11} - 128 q^{19} + 328 q^{21} + 452 q^{29} + 88 q^{31} + 1712 q^{39} + 1748 q^{41} + 2324 q^{49} + 2456 q^{51} + 664 q^{59} + 2316 q^{61} + 3256 q^{69} + 624 q^{71} + 2136 q^{79} + 3272 q^{81} + 2628 q^{89} + 2552 q^{91} + 880 q^{99}+O(q^{100}) $ 8 * q + 132 * q^9 + 40 * q^11 - 128 * q^19 + 328 * q^21 + 452 * q^29 + 88 * q^31 + 1712 * q^39 + 1748 * q^41 + 2324 * q^49 + 2456 * q^51 + 664 * q^59 + 2316 * q^61 + 3256 * q^69 + 624 * q^71 + 2136 * q^79 + 3272 * q^81 + 2628 * q^89 + 2552 * q^91 + 880 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	0	0
$2$	0	0
$3$	−1.99947	−0.384798	−0.192399	−	0.981317i	$-0.561627\pi$
$3$	−1.99947	−0.384798	−0.192399	+	0.981317i	$0.561627\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	13.6557	0.737340	0.368670	−	0.929560i	$-0.379813\pi$
$7$	13.6557	0.737340	0.368670	+	0.929560i	$0.379813\pi$
$8$	0	0
$9$	−23.0021	−0.851931
$10$	0	0
$11$	24.7784	0.679179	0.339590	−	0.940574i	$-0.389712\pi$
$11$	24.7784	0.679179	0.339590	+	0.940574i	$0.389712\pi$
$12$	0	0
$13$	74.4947	1.58932	0.794658	−	0.607058i	$-0.207650\pi$
$13$	74.4947	1.58932	0.794658	+	0.607058i	$0.207650\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−137.995	−1.96875	−0.984377	−	0.176073i	$-0.943661\pi$
$17$	−137.995	−1.96875	−0.984377	+	0.176073i	$0.943661\pi$
$18$	0	0
$19$	54.7771	0.661407	0.330704	−	0.943735i	$-0.392714\pi$
$19$	54.7771	0.661407	0.330704	+	0.943735i	$0.392714\pi$
$20$	0	0
$21$	−27.3042	−0.283727
$22$	0	0
$23$	−170.121	−1.54229	−0.771144	−	0.636661i	$-0.780315\pi$
$23$	−170.121	−1.54229	−0.771144	+	0.636661i	$0.780315\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	99.9777	0.712619
$28$	0	0
$29$	65.8946	0.421942	0.210971	−	0.977492i	$-0.432337\pi$
$29$	65.8946	0.421942	0.210971	+	0.977492i	$0.432337\pi$
$30$	0	0
$31$	295.426	1.71161	0.855807	−	0.517295i	$-0.173061\pi$
$31$	295.426	1.71161	0.855807	+	0.517295i	$0.173061\pi$
$32$	0	0
$33$	−49.5436	−0.261347
$34$	0	0
$35$	0	0
$36$	0	0
$37$	116.513	0.517692	0.258846	−	0.965919i	$-0.416658\pi$
$37$	116.513	0.517692	0.258846	+	0.965919i	$0.416658\pi$
$38$	0	0
$39$	−148.950	−0.611565
$40$	0	0
$41$	333.050	1.26863	0.634313	−	0.773077i	$-0.281283\pi$
$41$	333.050	1.26863	0.634313	+	0.773077i	$0.281283\pi$
$42$	0	0
$43$	304.868	1.08121	0.540604	−	0.841277i	$-0.318196\pi$
$43$	304.868	1.08121	0.540604	+	0.841277i	$0.318196\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−370.955	−1.15126	−0.575631	−	0.817710i	$-0.695243\pi$
$47$	−370.955	−1.15126	−0.575631	+	0.817710i	$0.695243\pi$
$48$	0	0
$49$	−156.521	−0.456329
$50$	0	0
$51$	275.918	0.757572
$52$	0	0
$53$	−110.290	−0.285840	−0.142920	−	0.989734i	$-0.545649\pi$
$53$	−110.290	−0.285840	−0.142920	+	0.989734i	$0.545649\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−109.525	−0.254508
$58$	0	0
$59$	93.7143	0.206789	0.103395	−	0.994640i	$-0.467030\pi$
$59$	93.7143	0.206789	0.103395	+	0.994640i	$0.467030\pi$
$60$	0	0
$61$	643.906	1.35154	0.675769	−	0.737114i	$-0.263812\pi$
$61$	643.906	1.35154	0.675769	+	0.737114i	$0.263812\pi$
$62$	0	0
$63$	−314.111	−0.628163
$64$	0	0
$65$	0	0
$66$	0	0
$67$	755.760	1.37807	0.689036	−	0.724727i	$-0.258034\pi$
$67$	755.760	1.37807	0.689036	+	0.724727i	$0.258034\pi$
$68$	0	0
$69$	340.151	0.593469
$70$	0	0
$71$	602.116	1.00645	0.503225	−	0.864155i	$-0.332147\pi$
$71$	602.116	1.00645	0.503225	+	0.864155i	$0.332147\pi$
$72$	0	0
$73$	572.271	0.917524	0.458762	−	0.888559i	$-0.348293\pi$
$73$	572.271	0.917524	0.458762	+	0.888559i	$0.348293\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	338.367	0.500786
$78$	0	0
$79$	−404.933	−0.576690	−0.288345	−	0.957527i	$-0.593105\pi$
$79$	−404.933	−0.576690	−0.288345	+	0.957527i	$0.593105\pi$
$80$	0	0
$81$	421.155	0.577716
$82$	0	0
$83$	554.312	0.733056	0.366528	−	0.930407i	$-0.380546\pi$
$83$	554.312	0.733056	0.366528	+	0.930407i	$0.380546\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−131.754	−0.162362
$88$	0	0
$89$	886.269	1.05555	0.527777	−	0.849383i	$-0.323026\pi$
$89$	886.269	1.05555	0.527777	+	0.849383i	$0.323026\pi$
$90$	0	0
$91$	1017.28	1.17187
$92$	0	0
$93$	−590.694	−0.658625
$94$	0	0
$95$	0	0
$96$	0	0
$97$	203.851	0.213381	0.106690	−	0.994292i	$-0.465975\pi$
$97$	203.851	0.213381	0.106690	+	0.994292i	$0.465975\pi$
$98$	0	0
$99$	−569.956	−0.578613
$100$	0	0
$101$	1784.80	1.75836	0.879178	−	0.476493i	$-0.158092\pi$
$101$	1784.80	1.75836	0.879178	+	0.476493i	$0.158092\pi$
$102$	0	0
$103$	−1354.67	−1.29592	−0.647958	−	0.761676i	$-0.724377\pi$
$103$	−1354.67	−1.29592	−0.647958	+	0.761676i	$0.724377\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	246.651	0.222847	0.111424	−	0.993773i	$-0.464459\pi$
$107$	246.651	0.222847	0.111424	+	0.993773i	$0.464459\pi$
$108$	0	0
$109$	529.178	0.465010	0.232505	−	0.972595i	$-0.425308\pi$
$109$	529.178	0.465010	0.232505	+	0.972595i	$0.425308\pi$
$110$	0	0
$111$	−232.964	−0.199207
$112$	0	0
$113$	196.929	0.163943	0.0819713	−	0.996635i	$-0.473878\pi$
$113$	196.929	0.163943	0.0819713	+	0.996635i	$0.473878\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1713.54	−1.35399
$118$	0	0
$119$	−1884.43	−1.45164
$120$	0	0
$121$	−717.031	−0.538716
$122$	0	0
$123$	−665.922	−0.488164
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−1409.42	−0.984770	−0.492385	−	0.870378i	$-0.663875\pi$
$127$	−1409.42	−0.984770	−0.492385	+	0.870378i	$0.663875\pi$
$128$	0	0
$129$	−609.574	−0.416047
$130$	0	0
$131$	−658.838	−0.439412	−0.219706	−	0.975566i	$-0.570510\pi$
$131$	−658.838	−0.439412	−0.219706	+	0.975566i	$0.570510\pi$
$132$	0	0
$133$	748.022	0.487682
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1665.87	1.03887	0.519434	−	0.854511i	$-0.326143\pi$
$137$	1665.87	1.03887	0.519434	+	0.854511i	$0.326143\pi$
$138$	0	0
$139$	1783.10	1.08806	0.544031	−	0.839065i	$-0.316897\pi$
$139$	1783.10	1.08806	0.544031	+	0.839065i	$0.316897\pi$
$140$	0	0
$141$	741.712	0.443003
$142$	0	0
$143$	1845.86	1.07943
$144$	0	0
$145$	0	0
$146$	0	0
$147$	312.959	0.175595
$148$	0	0
$149$	105.787	0.0581640	0.0290820	−	0.999577i	$-0.490742\pi$
$149$	105.787	0.0581640	0.0290820	+	0.999577i	$0.490742\pi$
$150$	0	0
$151$	−264.773	−0.142695	−0.0713473	−	0.997452i	$-0.522730\pi$
$151$	−264.773	−0.142695	−0.0713473	+	0.997452i	$0.522730\pi$
$152$	0	0
$153$	3174.19	1.67724
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1613.18	0.820035	0.410018	−	0.912078i	$-0.365523\pi$
$157$	1613.18	0.820035	0.410018	+	0.912078i	$0.365523\pi$
$158$	0	0
$159$	220.522	0.109991
$160$	0	0
$161$	−2323.12	−1.13719
$162$	0	0
$163$	−2164.84	−1.04027	−0.520134	−	0.854085i	$-0.674118\pi$
$163$	−2164.84	−1.04027	−0.520134	+	0.854085i	$0.674118\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1540.11	−0.713637	−0.356818	−	0.934174i	$-0.616138\pi$
$167$	−1540.11	−0.713637	−0.356818	+	0.934174i	$0.616138\pi$
$168$	0	0
$169$	3352.45	1.52592
$170$	0	0
$171$	−1259.99	−0.563473
$172$	0	0
$173$	−2127.54	−0.934993	−0.467496	−	0.883995i	$-0.654844\pi$
$173$	−2127.54	−0.934993	−0.467496	+	0.883995i	$0.654844\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−187.379	−0.0795721
$178$	0	0
$179$	−3198.36	−1.33551	−0.667756	−	0.744381i	$-0.732745\pi$
$179$	−3198.36	−1.33551	−0.667756	+	0.744381i	$0.732745\pi$
$180$	0	0
$181$	−1515.70	−0.622435	−0.311218	−	0.950339i	$-0.600737\pi$
$181$	−1515.70	−0.622435	−0.311218	+	0.950339i	$0.600737\pi$
$182$	0	0
$183$	−1287.47	−0.520069
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−3419.31	−1.33714
$188$	0	0
$189$	1365.27	0.525443
$190$	0	0
$191$	788.293	0.298633	0.149316	−	0.988789i	$-0.452293\pi$
$191$	788.293	0.298633	0.149316	+	0.988789i	$0.452293\pi$
$192$	0	0
$193$	−1107.93	−0.413216	−0.206608	−	0.978424i	$-0.566243\pi$
$193$	−1107.93	−0.413216	−0.206608	+	0.978424i	$0.566243\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3498.46	−1.26525	−0.632627	−	0.774457i	$-0.718023\pi$
$197$	−3498.46	−1.26525	−0.632627	+	0.774457i	$0.718023\pi$
$198$	0	0
$199$	−4402.46	−1.56825	−0.784126	−	0.620601i	$-0.786889\pi$
$199$	−4402.46	−1.56825	−0.784126	+	0.620601i	$0.786889\pi$
$200$	0	0
$201$	−1511.12	−0.530279
$202$	0	0
$203$	899.839	0.311115
$204$	0	0
$205$	0	0
$206$	0	0
$207$	3913.14	1.31392
$208$	0	0
$209$	1357.29	0.449214
$210$	0	0
$211$	−5002.19	−1.63206	−0.816031	−	0.578008i	$-0.803830\pi$
$211$	−5002.19	−1.63206	−0.816031	+	0.578008i	$0.803830\pi$
$212$	0	0
$213$	−1203.91	−0.387280
$214$	0	0
$215$	0	0
$216$	0	0
$217$	4034.26	1.26204
$218$	0	0
$219$	−1144.24	−0.353061
$220$	0	0
$221$	−10279.9	−3.12897
$222$	0	0
$223$	6500.45	1.95203	0.976014	−	0.217708i	$-0.0698581\pi$
$223$	6500.45	1.95203	0.976014	+	0.217708i	$0.0698581\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4123.83	1.20576	0.602882	−	0.797830i	$-0.294019\pi$
$227$	4123.83	1.20576	0.602882	+	0.797830i	$0.294019\pi$
$228$	0	0
$229$	1878.04	0.541940	0.270970	−	0.962588i	$-0.412656\pi$
$229$	1878.04	0.541940	0.270970	+	0.962588i	$0.412656\pi$
$230$	0	0
$231$	−676.555	−0.192701
$232$	0	0
$233$	−1298.63	−0.365132	−0.182566	−	0.983194i	$-0.558440\pi$
$233$	−1298.63	−0.365132	−0.182566	+	0.983194i	$0.558440\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	809.651	0.221909
$238$	0	0
$239$	−140.854	−0.0381218	−0.0190609	−	0.999818i	$-0.506068\pi$
$239$	−140.854	−0.0381218	−0.0190609	+	0.999818i	$0.506068\pi$
$240$	0	0
$241$	2807.16	0.750312	0.375156	−	0.926962i	$-0.377589\pi$
$241$	2807.16	0.750312	0.375156	+	0.926962i	$0.377589\pi$
$242$	0	0
$243$	−3541.48	−0.934923
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4080.60	1.05118
$248$	0	0
$249$	−1108.33	−0.282079
$250$	0	0
$251$	751.276	0.188925	0.0944624	−	0.995528i	$-0.469887\pi$
$251$	751.276	0.188925	0.0944624	+	0.995528i	$0.469887\pi$
$252$	0	0
$253$	−4215.32	−1.04749
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−7093.64	−1.72175	−0.860873	−	0.508819i	$-0.830082\pi$
$257$	−7093.64	−1.72175	−0.860873	+	0.508819i	$0.830082\pi$
$258$	0	0
$259$	1591.07	0.381715
$260$	0	0
$261$	−1515.72	−0.359465
$262$	0	0
$263$	3072.15	0.720292	0.360146	−	0.932896i	$-0.382727\pi$
$263$	3072.15	0.720292	0.360146	+	0.932896i	$0.382727\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−1772.07	−0.406175
$268$	0	0
$269$	3908.06	0.885794	0.442897	−	0.896572i	$-0.353951\pi$
$269$	3908.06	0.885794	0.442897	+	0.896572i	$0.353951\pi$
$270$	0	0
$271$	−7337.41	−1.64471	−0.822354	−	0.568976i	$-0.807340\pi$
$271$	−7337.41	−1.64471	−0.822354	+	0.568976i	$0.807340\pi$
$272$	0	0
$273$	−2034.02	−0.450932
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−4708.63	−1.02135	−0.510675	−	0.859774i	$-0.670605\pi$
$277$	−4708.63	−1.02135	−0.510675	+	0.859774i	$0.670605\pi$
$278$	0	0
$279$	−6795.42	−1.45818
$280$	0	0
$281$	43.0163	0.00913216	0.00456608	−	0.999990i	$-0.498547\pi$
$281$	43.0163	0.00913216	0.00456608	+	0.999990i	$0.498547\pi$
$282$	0	0
$283$	5634.99	1.18362	0.591811	−	0.806076i	$-0.298413\pi$
$283$	5634.99	1.18362	0.591811	+	0.806076i	$0.298413\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4548.04	0.935408
$288$	0	0
$289$	14129.8	2.87599
$290$	0	0
$291$	−407.593	−0.0821084
$292$	0	0
$293$	7824.15	1.56004	0.780020	−	0.625755i	$-0.215209\pi$
$293$	7824.15	1.56004	0.780020	+	0.625755i	$0.215209\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2477.29	0.483996
$298$	0	0
$299$	−12673.1	−2.45118
$300$	0	0
$301$	4163.20	0.797218
$302$	0	0
$303$	−3568.65	−0.676612
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−7484.42	−1.39140	−0.695698	−	0.718334i	$-0.744905\pi$
$307$	−7484.42	−1.39140	−0.695698	+	0.718334i	$0.744905\pi$
$308$	0	0
$309$	2708.62	0.498666
$310$	0	0
$311$	61.5677	0.0112257	0.00561284	−	0.999984i	$-0.498213\pi$
$311$	61.5677	0.0112257	0.00561284	+	0.999984i	$0.498213\pi$
$312$	0	0
$313$	5777.49	1.04333	0.521666	−	0.853150i	$-0.325311\pi$
$313$	5777.49	1.04333	0.521666	+	0.853150i	$0.325311\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6217.74	1.10165	0.550825	−	0.834621i	$-0.314313\pi$
$317$	6217.74	1.10165	0.550825	+	0.834621i	$0.314313\pi$
$318$	0	0
$319$	1632.76	0.286574
$320$	0	0
$321$	−493.171	−0.0857511
$322$	0	0
$323$	−7559.00	−1.30215
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−1058.08	−0.178935
$328$	0	0
$329$	−5065.66	−0.848871
$330$	0	0
$331$	−5133.82	−0.852508	−0.426254	−	0.904604i	$-0.640167\pi$
$331$	−5133.82	−0.852508	−0.426254	+	0.904604i	$0.640167\pi$
$332$	0	0
$333$	−2680.04	−0.441038
$334$	0	0
$335$	0	0
$336$	0	0
$337$	128.796	0.0208189	0.0104094	−	0.999946i	$-0.496687\pi$
$337$	128.796	0.0208189	0.0104094	+	0.999946i	$0.496687\pi$
$338$	0	0
$339$	−393.753	−0.0630848
$340$	0	0
$341$	7320.18	1.16249
$342$	0	0
$343$	−6821.32	−1.07381
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1599.35	−0.247428	−0.123714	−	0.992318i	$-0.539480\pi$
$347$	−1599.35	−0.247428	−0.123714	+	0.992318i	$0.539480\pi$
$348$	0	0
$349$	−763.597	−0.117119	−0.0585593	−	0.998284i	$-0.518651\pi$
$349$	−763.597	−0.117119	−0.0585593	+	0.998284i	$0.518651\pi$
$350$	0	0
$351$	7447.80	1.13258
$352$	0	0
$353$	−4732.04	−0.713487	−0.356743	−	0.934202i	$-0.616113\pi$
$353$	−4732.04	−0.713487	−0.356743	+	0.934202i	$0.616113\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	3767.86	0.558589
$358$	0	0
$359$	3796.67	0.558164	0.279082	−	0.960267i	$-0.409970\pi$
$359$	3796.67	0.558164	0.279082	+	0.960267i	$0.409970\pi$
$360$	0	0
$361$	−3858.47	−0.562541
$362$	0	0
$363$	1433.68	0.207297
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−8295.93	−1.17996	−0.589978	−	0.807419i	$-0.700864\pi$
$367$	−8295.93	−1.17996	−0.589978	+	0.807419i	$0.700864\pi$
$368$	0	0
$369$	−7660.85	−1.08078
$370$	0	0
$371$	−1506.09	−0.210762
$372$	0	0
$373$	−48.0927	−0.00667600	−0.00333800	−	0.999994i	$-0.501063\pi$
$373$	−48.0927	−0.00667600	−0.00333800	+	0.999994i	$0.501063\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4908.80	0.670599
$378$	0	0
$379$	−5302.29	−0.718629	−0.359315	−	0.933216i	$-0.616990\pi$
$379$	−5302.29	−0.718629	−0.359315	+	0.933216i	$0.616990\pi$
$380$	0	0
$381$	2818.09	0.378937
$382$	0	0
$383$	−4446.99	−0.593292	−0.296646	−	0.954988i	$-0.595868\pi$
$383$	−4446.99	−0.593292	−0.296646	+	0.954988i	$0.595868\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−7012.62	−0.921114
$388$	0	0
$389$	11997.4	1.56373	0.781865	−	0.623448i	$-0.214269\pi$
$389$	11997.4	1.56373	0.781865	+	0.623448i	$0.214269\pi$
$390$	0	0
$391$	23475.9	3.03639
$392$	0	0
$393$	1317.33	0.169085
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−4758.66	−0.601587	−0.300794	−	0.953689i	$-0.597252\pi$
$397$	−4758.66	−0.601587	−0.300794	+	0.953689i	$0.597252\pi$
$398$	0	0
$399$	−1495.65	−0.187659
$400$	0	0
$401$	13281.8	1.65401	0.827007	−	0.562192i	$-0.190042\pi$
$401$	13281.8	1.65401	0.827007	+	0.562192i	$0.190042\pi$
$402$	0	0
$403$	22007.6	2.72029
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2887.00	0.351606
$408$	0	0
$409$	12483.6	1.50922	0.754612	−	0.656171i	$-0.227825\pi$
$409$	12483.6	1.50922	0.754612	+	0.656171i	$0.227825\pi$
$410$	0	0
$411$	−3330.86	−0.399754
$412$	0	0
$413$	1279.74	0.152474
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−3565.25	−0.418684
$418$	0	0
$419$	716.066	0.0834895	0.0417447	−	0.999128i	$-0.486708\pi$
$419$	716.066	0.0834895	0.0417447	+	0.999128i	$0.486708\pi$
$420$	0	0
$421$	1491.65	0.172681	0.0863403	−	0.996266i	$-0.472483\pi$
$421$	1491.65	0.172681	0.0863403	+	0.996266i	$0.472483\pi$
$422$	0	0
$423$	8532.74	0.980795
$424$	0	0
$425$	0	0
$426$	0	0
$427$	8793.01	0.996543
$428$	0	0
$429$	−3690.74	−0.415362
$430$	0	0
$431$	7554.08	0.844239	0.422120	−	0.906540i	$-0.361286\pi$
$431$	7554.08	0.844239	0.422120	+	0.906540i	$0.361286\pi$
$432$	0	0
$433$	−15182.5	−1.68505	−0.842525	−	0.538658i	$-0.818932\pi$
$433$	−15182.5	−1.68505	−0.842525	+	0.538658i	$0.818932\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−9318.72	−1.02008
$438$	0	0
$439$	7814.45	0.849575	0.424787	−	0.905293i	$-0.360349\pi$
$439$	7814.45	0.849575	0.424787	+	0.905293i	$0.360349\pi$
$440$	0	0
$441$	3600.32	0.388761
$442$	0	0
$443$	818.772	0.0878127	0.0439063	−	0.999036i	$-0.486020\pi$
$443$	818.772	0.0878127	0.0439063	+	0.999036i	$0.486020\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−211.518	−0.0223814
$448$	0	0
$449$	−7560.87	−0.794698	−0.397349	−	0.917668i	$-0.630070\pi$
$449$	−7560.87	−0.794698	−0.397349	+	0.917668i	$0.630070\pi$
$450$	0	0
$451$	8252.44	0.861624
$452$	0	0
$453$	529.405	0.0549086
$454$	0	0
$455$	0	0
$456$	0	0
$457$	8712.71	0.891823	0.445912	−	0.895077i	$-0.352880\pi$
$457$	8712.71	0.891823	0.445912	+	0.895077i	$0.352880\pi$
$458$	0	0
$459$	−13796.5	−1.40297
$460$	0	0
$461$	−3164.97	−0.319756	−0.159878	−	0.987137i	$-0.551110\pi$
$461$	−3164.97	−0.319756	−0.159878	+	0.987137i	$0.551110\pi$
$462$	0	0
$463$	−11205.8	−1.12479	−0.562395	−	0.826868i	$-0.690120\pi$
$463$	−11205.8	−1.12479	−0.562395	+	0.826868i	$0.690120\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	18094.8	1.79299	0.896494	−	0.443056i	$-0.146106\pi$
$467$	18094.8	1.79299	0.896494	+	0.443056i	$0.146106\pi$
$468$	0	0
$469$	10320.5	1.01611
$470$	0	0
$471$	−3225.50	−0.315548
$472$	0	0
$473$	7554.15	0.734334
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2536.91	0.243516
$478$	0	0
$479$	−18448.8	−1.75981	−0.879903	−	0.475153i	$-0.842393\pi$
$479$	−18448.8	−1.75981	−0.879903	+	0.475153i	$0.842393\pi$
$480$	0	0
$481$	8679.59	0.822776
$482$	0	0
$483$	4645.01	0.437589
$484$	0	0
$485$	0	0
$486$	0	0
$487$	9787.53	0.910709	0.455354	−	0.890310i	$-0.349513\pi$
$487$	9787.53	0.910709	0.455354	+	0.890310i	$0.349513\pi$
$488$	0	0
$489$	4328.53	0.400293
$490$	0	0
$491$	10795.8	0.992278	0.496139	−	0.868243i	$-0.334751\pi$
$491$	10795.8	0.992278	0.496139	+	0.868243i	$0.334751\pi$
$492$	0	0
$493$	−9093.16	−0.830700
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8222.33	0.742097
$498$	0	0
$499$	2268.16	0.203481	0.101740	−	0.994811i	$-0.467559\pi$
$499$	2268.16	0.203481	0.101740	+	0.994811i	$0.467559\pi$
$500$	0	0
$501$	3079.40	0.274606
$502$	0	0
$503$	−6490.71	−0.575360	−0.287680	−	0.957727i	$-0.592884\pi$
$503$	−6490.71	−0.575360	−0.287680	+	0.957727i	$0.592884\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−6703.13	−0.587172
$508$	0	0
$509$	−5879.39	−0.511983	−0.255992	−	0.966679i	$-0.582402\pi$
$509$	−5879.39	−0.511983	−0.255992	+	0.966679i	$0.582402\pi$
$510$	0	0
$511$	7814.78	0.676527
$512$	0	0
$513$	5476.49	0.471331
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−9191.66	−0.781913
$518$	0	0
$519$	4253.95	0.359783
$520$	0	0
$521$	−6665.77	−0.560523	−0.280262	−	0.959924i	$-0.590421\pi$
$521$	−6665.77	−0.560523	−0.280262	+	0.959924i	$0.590421\pi$
$522$	0	0
$523$	−12790.3	−1.06937	−0.534687	−	0.845050i	$-0.679570\pi$
$523$	−12790.3	−1.06937	−0.534687	+	0.845050i	$0.679570\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−40767.4	−3.36975
$528$	0	0
$529$	16774.1	1.37865
$530$	0	0
$531$	−2155.63	−0.176170
$532$	0	0
$533$	24810.4	2.01625
$534$	0	0
$535$	0	0
$536$	0	0
$537$	6395.02	0.513902
$538$	0	0
$539$	−3878.34	−0.309929
$540$	0	0
$541$	2372.59	0.188550	0.0942749	−	0.995546i	$-0.469947\pi$
$541$	2372.59	0.188550	0.0942749	+	0.995546i	$0.469947\pi$
$542$	0	0
$543$	3030.59	0.239512
$544$	0	0
$545$	0	0
$546$	0	0
$547$	14144.2	1.10560	0.552799	−	0.833315i	$-0.313560\pi$
$547$	14144.2	1.10560	0.552799	+	0.833315i	$0.313560\pi$
$548$	0	0
$549$	−14811.2	−1.15142
$550$	0	0
$551$	3609.52	0.279076
$552$	0	0
$553$	−5529.66	−0.425217
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−13003.0	−0.989144	−0.494572	−	0.869137i	$-0.664675\pi$
$557$	−13003.0	−0.989144	−0.494572	+	0.869137i	$0.664675\pi$
$558$	0	0
$559$	22711.0	1.71838
$560$	0	0
$561$	6836.80	0.514527
$562$	0	0
$563$	−11524.7	−0.862716	−0.431358	−	0.902181i	$-0.641965\pi$
$563$	−11524.7	−0.862716	−0.431358	+	0.902181i	$0.641965\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	5751.18	0.425974
$568$	0	0
$569$	−12205.6	−0.899269	−0.449635	−	0.893213i	$-0.648446\pi$
$569$	−12205.6	−0.899269	−0.449635	+	0.893213i	$0.648446\pi$
$570$	0	0
$571$	−15921.2	−1.16687	−0.583436	−	0.812159i	$-0.698292\pi$
$571$	−15921.2	−1.16687	−0.583436	+	0.812159i	$0.698292\pi$
$572$	0	0
$573$	−1576.17	−0.114913
$574$	0	0
$575$	0	0
$576$	0	0
$577$	12931.9	0.933035	0.466518	−	0.884512i	$-0.345508\pi$
$577$	12931.9	0.933035	0.466518	+	0.884512i	$0.345508\pi$
$578$	0	0
$579$	2215.28	0.159005
$580$	0	0
$581$	7569.54	0.540512
$582$	0	0
$583$	−2732.82	−0.194137
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2295.98	0.161440	0.0807201	−	0.996737i	$-0.474278\pi$
$587$	2295.98	0.161440	0.0807201	+	0.996737i	$0.474278\pi$
$588$	0	0
$589$	16182.6	1.13207
$590$	0	0
$591$	6995.06	0.486867
$592$	0	0
$593$	−5360.96	−0.371245	−0.185623	−	0.982621i	$-0.559430\pi$
$593$	−5360.96	−0.371245	−0.185623	+	0.982621i	$0.559430\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	8802.58	0.603460
$598$	0	0
$599$	−13909.7	−0.948809	−0.474405	−	0.880307i	$-0.657337\pi$
$599$	−13909.7	−0.948809	−0.474405	+	0.880307i	$0.657337\pi$
$600$	0	0
$601$	−10274.1	−0.697320	−0.348660	−	0.937249i	$-0.613363\pi$
$601$	−10274.1	−0.697320	−0.348660	+	0.937249i	$0.613363\pi$
$602$	0	0
$603$	−17384.1	−1.17402
$604$	0	0
$605$	0	0
$606$	0	0
$607$	16143.4	1.07948	0.539738	−	0.841833i	$-0.318523\pi$
$607$	16143.4	1.07948	0.539738	+	0.841833i	$0.318523\pi$
$608$	0	0
$609$	−1799.20	−0.119716
$610$	0	0
$611$	−27634.1	−1.82972
$612$	0	0
$613$	10766.5	0.709389	0.354695	−	0.934982i	$-0.384585\pi$
$613$	10766.5	0.709389	0.354695	+	0.934982i	$0.384585\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−9169.31	−0.598286	−0.299143	−	0.954208i	$-0.596701\pi$
$617$	−9169.31	−0.598286	−0.299143	+	0.954208i	$0.596701\pi$
$618$	0	0
$619$	21762.1	1.41308	0.706538	−	0.707675i	$-0.250256\pi$
$619$	21762.1	1.41308	0.706538	+	0.707675i	$0.250256\pi$
$620$	0	0
$621$	−17008.3	−1.09906
$622$	0	0
$623$	12102.6	0.778302
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−2713.86	−0.172857
$628$	0	0
$629$	−16078.3	−1.01921
$630$	0	0
$631$	−17399.9	−1.09775	−0.548876	−	0.835904i	$-0.684944\pi$
$631$	−17399.9	−1.09775	−0.548876	+	0.835904i	$0.684944\pi$
$632$	0	0
$633$	10001.7	0.628014
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−11660.0	−0.725251
$638$	0	0
$639$	−13849.9	−0.857426
$640$	0	0
$641$	−15912.7	−0.980520	−0.490260	−	0.871576i	$-0.663098\pi$
$641$	−15912.7	−0.980520	−0.490260	+	0.871576i	$0.663098\pi$
$642$	0	0
$643$	15068.3	0.924163	0.462082	−	0.886837i	$-0.347103\pi$
$643$	15068.3	0.924163	0.462082	+	0.886837i	$0.347103\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−16150.6	−0.981368	−0.490684	−	0.871338i	$-0.663253\pi$
$647$	−16150.6	−0.981368	−0.490684	+	0.871338i	$0.663253\pi$
$648$	0	0
$649$	2322.09	0.140447
$650$	0	0
$651$	−8066.37	−0.485631
$652$	0	0
$653$	16059.6	0.962419	0.481210	−	0.876606i	$-0.340198\pi$
$653$	16059.6	0.962419	0.481210	+	0.876606i	$0.340198\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−13163.4	−0.781667
$658$	0	0
$659$	9999.13	0.591064	0.295532	−	0.955333i	$-0.404503\pi$
$659$	9999.13	0.591064	0.295532	+	0.955333i	$0.404503\pi$
$660$	0	0
$661$	5228.54	0.307665	0.153833	−	0.988097i	$-0.450838\pi$
$661$	5228.54	0.307665	0.153833	+	0.988097i	$0.450838\pi$
$662$	0	0
$663$	20554.4	1.20402
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−11210.0	−0.650756
$668$	0	0
$669$	−12997.4	−0.751136
$670$	0	0
$671$	15955.0	0.917936
$672$	0	0
$673$	3300.82	0.189060	0.0945300	−	0.995522i	$-0.469865\pi$
$673$	3300.82	0.189060	0.0945300	+	0.995522i	$0.469865\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	16914.1	0.960208	0.480104	−	0.877212i	$-0.340599\pi$
$677$	16914.1	0.960208	0.480104	+	0.877212i	$0.340599\pi$
$678$	0	0
$679$	2783.73	0.157334
$680$	0	0
$681$	−8245.47	−0.463975
$682$	0	0
$683$	−29550.6	−1.65552	−0.827760	−	0.561082i	$-0.810385\pi$
$683$	−29550.6	−1.65552	−0.827760	+	0.561082i	$0.810385\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−3755.08	−0.208538
$688$	0	0
$689$	−8216.04	−0.454291
$690$	0	0
$691$	−28389.0	−1.56291	−0.781454	−	0.623963i	$-0.785522\pi$
$691$	−28389.0	−1.56291	−0.781454	+	0.623963i	$0.785522\pi$
$692$	0	0
$693$	−7783.17	−0.426635
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−45959.4	−2.49761
$698$	0	0
$699$	2596.56	0.140502
$700$	0	0
$701$	−6061.93	−0.326613	−0.163307	−	0.986575i	$-0.552216\pi$
$701$	−6061.93	−0.326613	−0.163307	+	0.986575i	$0.552216\pi$
$702$	0	0
$703$	6382.24	0.342405
$704$	0	0
$705$	0	0
$706$	0	0
$707$	24372.7	1.29651
$708$	0	0
$709$	−16861.2	−0.893137	−0.446568	−	0.894749i	$-0.647354\pi$
$709$	−16861.2	−0.893137	−0.446568	+	0.894749i	$0.647354\pi$
$710$	0	0
$711$	9314.32	0.491300
$712$	0	0
$713$	−50258.0	−2.63980
$714$	0	0
$715$	0	0
$716$	0	0
$717$	281.634	0.0146692
$718$	0	0
$719$	6175.67	0.320325	0.160162	−	0.987091i	$-0.448798\pi$
$719$	6175.67	0.320325	0.160162	+	0.987091i	$0.448798\pi$
$720$	0	0
$721$	−18499.0	−0.955532
$722$	0	0
$723$	−5612.83	−0.288718
$724$	0	0
$725$	0	0
$726$	0	0
$727$	33933.9	1.73114	0.865570	−	0.500787i	$-0.166956\pi$
$727$	33933.9	1.73114	0.865570	+	0.500787i	$0.166956\pi$
$728$	0	0
$729$	−4290.11	−0.217960
$730$	0	0
$731$	−42070.4	−2.12863
$732$	0	0
$733$	−3322.35	−0.167413	−0.0837066	−	0.996490i	$-0.526676\pi$
$733$	−3322.35	−0.167413	−0.0837066	+	0.996490i	$0.526676\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	18726.5	0.935957
$738$	0	0
$739$	23055.9	1.14767	0.573833	−	0.818972i	$-0.305456\pi$
$739$	23055.9	1.14767	0.573833	+	0.818972i	$0.305456\pi$
$740$	0	0
$741$	−8159.04	−0.404494
$742$	0	0
$743$	1783.49	0.0880620	0.0440310	−	0.999030i	$-0.485980\pi$
$743$	1783.49	0.0880620	0.0440310	+	0.999030i	$0.485980\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−12750.4	−0.624513
$748$	0	0
$749$	3368.20	0.164314
$750$	0	0
$751$	−3008.84	−0.146197	−0.0730986	−	0.997325i	$-0.523289\pi$
$751$	−3008.84	−0.146197	−0.0730986	+	0.997325i	$0.523289\pi$
$752$	0	0
$753$	−1502.15	−0.0726978
$754$	0	0
$755$	0	0
$756$	0	0
$757$	7214.40	0.346383	0.173192	−	0.984888i	$-0.444592\pi$
$757$	7214.40	0.346383	0.173192	+	0.984888i	$0.444592\pi$
$758$	0	0
$759$	8428.40	0.403072
$760$	0	0
$761$	−8405.66	−0.400401	−0.200200	−	0.979755i	$-0.564159\pi$
$761$	−8405.66	−0.400401	−0.200200	+	0.979755i	$0.564159\pi$
$762$	0	0
$763$	7226.32	0.342871
$764$	0	0
$765$	0	0
$766$	0	0
$767$	6981.22	0.328653
$768$	0	0
$769$	−8630.77	−0.404725	−0.202362	−	0.979311i	$-0.564862\pi$
$769$	−8630.77	−0.404725	−0.202362	+	0.979311i	$0.564862\pi$
$770$	0	0
$771$	14183.5	0.662524
$772$	0	0
$773$	−21391.0	−0.995319	−0.497660	−	0.867372i	$-0.665807\pi$
$773$	−21391.0	−0.995319	−0.497660	+	0.867372i	$0.665807\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−3181.29	−0.146883
$778$	0	0
$779$	18243.5	0.839078
$780$	0	0
$781$	14919.5	0.683560
$782$	0	0
$783$	6587.99	0.300684
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−14746.3	−0.667913	−0.333957	−	0.942588i	$-0.608384\pi$
$787$	−14746.3	−0.667913	−0.333957	+	0.942588i	$0.608384\pi$
$788$	0	0
$789$	−6142.66	−0.277167
$790$	0	0
$791$	2689.21	0.120882
$792$	0	0
$793$	47967.6	2.14802
$794$	0	0
$795$	0	0
$796$	0	0
$797$	29601.1	1.31559	0.657794	−	0.753197i	$-0.271490\pi$
$797$	29601.1	1.31559	0.657794	+	0.753197i	$0.271490\pi$
$798$	0	0
$799$	51190.1	2.26655
$800$	0	0
$801$	−20386.1	−0.899259
$802$	0	0
$803$	14180.0	0.623163
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−7814.04	−0.340852
$808$	0	0
$809$	−37891.7	−1.64673	−0.823364	−	0.567514i	$-0.807905\pi$
$809$	−37891.7	−1.64673	−0.823364	+	0.567514i	$0.807905\pi$
$810$	0	0
$811$	−37448.1	−1.62143	−0.810715	−	0.585441i	$-0.800922\pi$
$811$	−37448.1	−1.62143	−0.810715	+	0.585441i	$0.800922\pi$
$812$	0	0
$813$	14670.9	0.632880
$814$	0	0
$815$	0	0
$816$	0	0
$817$	16699.8	0.715119
$818$	0	0
$819$	−23399.6	−0.998349
$820$	0	0
$821$	−8278.04	−0.351895	−0.175947	−	0.984400i	$-0.556299\pi$
$821$	−8278.04	−0.351895	−0.175947	+	0.984400i	$0.556299\pi$
$822$	0	0
$823$	−18736.6	−0.793582	−0.396791	−	0.917909i	$-0.629876\pi$
$823$	−18736.6	−0.793582	−0.396791	+	0.917909i	$0.629876\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−14930.3	−0.627785	−0.313893	−	0.949458i	$-0.601633\pi$
$827$	−14930.3	−0.627785	−0.313893	+	0.949458i	$0.601633\pi$
$828$	0	0
$829$	21173.7	0.887085	0.443543	−	0.896253i	$-0.353721\pi$
$829$	21173.7	0.887085	0.443543	+	0.896253i	$0.353721\pi$
$830$	0	0
$831$	9414.75	0.393013
$832$	0	0
$833$	21599.2	0.898400
$834$	0	0
$835$	0	0
$836$	0	0
$837$	29536.0	1.21973
$838$	0	0
$839$	−32109.1	−1.32125	−0.660626	−	0.750715i	$-0.729709\pi$
$839$	−32109.1	−1.32125	−0.660626	+	0.750715i	$0.729709\pi$
$840$	0	0
$841$	−20046.9	−0.821965
$842$	0	0
$843$	−86.0097	−0.00351404
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−9791.58	−0.397217
$848$	0	0
$849$	−11267.0	−0.455456
$850$	0	0
$851$	−19821.3	−0.798430
$852$	0	0
$853$	−45916.5	−1.84308	−0.921542	−	0.388278i	$-0.873070\pi$
$853$	−45916.5	−1.84308	−0.921542	+	0.388278i	$0.873070\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1570.43	0.0625963	0.0312981	−	0.999510i	$-0.490036\pi$
$857$	1570.43	0.0625963	0.0312981	+	0.999510i	$0.490036\pi$
$858$	0	0
$859$	28861.9	1.14640	0.573198	−	0.819417i	$-0.305702\pi$
$859$	28861.9	1.14640	0.573198	+	0.819417i	$0.305702\pi$
$860$	0	0
$861$	−9093.66	−0.359943
$862$	0	0
$863$	10254.2	0.404469	0.202235	−	0.979337i	$-0.435180\pi$
$863$	10254.2	0.404469	0.202235	+	0.979337i	$0.435180\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−28252.0	−1.10668
$868$	0	0
$869$	−10033.6	−0.391676
$870$	0	0
$871$	56300.1	2.19019
$872$	0	0
$873$	−4689.00	−0.181785
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−34259.5	−1.31911	−0.659556	−	0.751655i	$-0.729256\pi$
$877$	−34259.5	−1.31911	−0.659556	+	0.751655i	$0.729256\pi$
$878$	0	0
$879$	−15644.1	−0.600300
$880$	0	0
$881$	30777.9	1.17700	0.588498	−	0.808499i	$-0.299720\pi$
$881$	30777.9	1.17700	0.588498	+	0.808499i	$0.299720\pi$
$882$	0	0
$883$	14509.4	0.552980	0.276490	−	0.961017i	$-0.410829\pi$
$883$	14509.4	0.552980	0.276490	+	0.961017i	$0.410829\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	838.430	0.0317381	0.0158691	−	0.999874i	$-0.494949\pi$
$887$	838.430	0.0317381	0.0158691	+	0.999874i	$0.494949\pi$
$888$	0	0
$889$	−19246.7	−0.726111
$890$	0	0
$891$	10435.6	0.392373
$892$	0	0
$893$	−20319.8	−0.761453
$894$	0	0
$895$	0	0
$896$	0	0
$897$	25339.4	0.943209
$898$	0	0
$899$	19467.0	0.722202
$900$	0	0
$901$	15219.6	0.562749
$902$	0	0
$903$	−8324.18	−0.306768
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−19231.8	−0.704059	−0.352030	−	0.935989i	$-0.614508\pi$
$907$	−19231.8	−0.704059	−0.352030	+	0.935989i	$0.614508\pi$
$908$	0	0
$909$	−41054.1	−1.49800
$910$	0	0
$911$	46235.9	1.68152	0.840760	−	0.541408i	$-0.182108\pi$
$911$	46235.9	1.68152	0.840760	+	0.541408i	$0.182108\pi$
$912$	0	0
$913$	13735.0	0.497877
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−8996.92	−0.323996
$918$	0	0
$919$	9102.63	0.326733	0.163367	−	0.986565i	$-0.447765\pi$
$919$	9102.63	0.326733	0.163367	+	0.986565i	$0.447765\pi$
$920$	0	0
$921$	14964.9	0.535406
$922$	0	0
$923$	44854.4	1.59957
$924$	0	0
$925$	0	0
$926$	0	0
$927$	31160.3	1.10403
$928$	0	0
$929$	19920.6	0.703523	0.351762	−	0.936090i	$-0.385583\pi$
$929$	19920.6	0.703523	0.351762	+	0.936090i	$0.385583\pi$
$930$	0	0
$931$	−8573.77	−0.301819
$932$	0	0
$933$	−123.103	−0.00431962
$934$	0	0
$935$	0	0
$936$	0	0
$937$	5005.42	0.174514	0.0872572	−	0.996186i	$-0.472190\pi$
$937$	5005.42	0.174514	0.0872572	+	0.996186i	$0.472190\pi$
$938$	0	0
$939$	−11551.9	−0.401472
$940$	0	0
$941$	−41316.9	−1.43134	−0.715671	−	0.698437i	$-0.753879\pi$
$941$	−41316.9	−1.43134	−0.715671	+	0.698437i	$0.753879\pi$
$942$	0	0
$943$	−56658.6	−1.95658
$944$	0	0
$945$	0	0
$946$	0	0
$947$	17455.7	0.598981	0.299490	−	0.954099i	$-0.403183\pi$
$947$	17455.7	0.598981	0.299490	+	0.954099i	$0.403183\pi$
$948$	0	0
$949$	42631.1	1.45823
$950$	0	0
$951$	−12432.2	−0.423912
$952$	0	0
$953$	7123.45	0.242131	0.121066	−	0.992645i	$-0.461369\pi$
$953$	7123.45	0.242131	0.121066	+	0.992645i	$0.461369\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−3264.66	−0.110273
$958$	0	0
$959$	22748.7	0.765999
$960$	0	0
$961$	57485.4	1.92962
$962$	0	0
$963$	−5673.49	−0.189850
$964$	0	0
$965$	0	0
$966$	0	0
$967$	19497.4	0.648390	0.324195	−	0.945990i	$-0.394907\pi$
$967$	19497.4	0.648390	0.324195	+	0.945990i	$0.394907\pi$
$968$	0	0
$969$	15114.0	0.501064
$970$	0	0
$971$	37108.6	1.22644	0.613219	−	0.789913i	$-0.289874\pi$
$971$	37108.6	1.22644	0.613219	+	0.789913i	$0.289874\pi$
$972$	0	0
$973$	24349.6	0.802272
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−13954.7	−0.456961	−0.228481	−	0.973548i	$-0.573376\pi$
$977$	−13954.7	−0.456961	−0.228481	+	0.973548i	$0.573376\pi$
$978$	0	0
$979$	21960.3	0.716910
$980$	0	0
$981$	−12172.2	−0.396156
$982$	0	0
$983$	16201.7	0.525691	0.262846	−	0.964838i	$-0.415339\pi$
$983$	16201.7	0.525691	0.262846	+	0.964838i	$0.415339\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	10128.6	0.326644
$988$	0	0
$989$	−51864.4	−1.66753
$990$	0	0
$991$	17644.7	0.565592	0.282796	−	0.959180i	$-0.408738\pi$
$991$	17644.7	0.565592	0.282796	+	0.959180i	$0.408738\pi$
$992$	0	0
$993$	10264.9	0.328043
$994$	0	0
$995$	0	0
$996$	0	0
$997$	3826.21	0.121542	0.0607709	−	0.998152i	$-0.480644\pi$
$997$	3826.21	0.121542	0.0607709	+	0.998152i	$0.480644\pi$
$998$	0	0
$999$	11648.7	0.368917

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	500.4.a.d.1.4	✓	8
4.3	odd	2		2000.4.a.r.1.5		8
5.2	odd	4		500.4.c.b.249.5		8
5.3	odd	4		500.4.c.b.249.4		8
5.4	even	2	inner	500.4.a.d.1.5	yes	8
20.19	odd	2		2000.4.a.r.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
500.4.a.d.1.4	✓	8	1.1	even	1	trivial
500.4.a.d.1.5	yes	8	5.4	even	2	inner
500.4.c.b.249.4		8	5.3	odd	4
500.4.c.b.249.5		8	5.2	odd	4
2000.4.a.r.1.4		8	20.19	odd	2
2000.4.a.r.1.5		8	4.3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 500 = 2^{2} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	500.a (trivial)

\(f(q)\)	\(=\)	\(q-1.99947 q^{3} +13.6557 q^{7} -23.0021 q^{9} +O(q^{10})\)	\(q-1.99947 q^{3} +13.6557 q^{7} -23.0021 q^{9} +24.7784 q^{11} +74.4947 q^{13} -137.995 q^{17} +54.7771 q^{19} -27.3042 q^{21} -170.121 q^{23} +99.9777 q^{27} +65.8946 q^{29} +295.426 q^{31} -49.5436 q^{33} +116.513 q^{37} -148.950 q^{39} +333.050 q^{41} +304.868 q^{43} -370.955 q^{47} -156.521 q^{49} +275.918 q^{51} -110.290 q^{53} -109.525 q^{57} +93.7143 q^{59} +643.906 q^{61} -314.111 q^{63} +755.760 q^{67} +340.151 q^{69} +602.116 q^{71} +572.271 q^{73} +338.367 q^{77} -404.933 q^{79} +421.155 q^{81} +554.312 q^{83} -131.754 q^{87} +886.269 q^{89} +1017.28 q^{91} -590.694 q^{93} +203.851 q^{97} -569.956 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 132 q^{9}+O(q^{10}) \) 8 * q + 132 * q^9	\( 8 q + 132 q^{9} + 40 q^{11} - 128 q^{19} + 328 q^{21} + 452 q^{29} + 88 q^{31} + 1712 q^{39} + 1748 q^{41} + 2324 q^{49} + 2456 q^{51} + 664 q^{59} + 2316 q^{61} + 3256 q^{69} + 624 q^{71} + 2136 q^{79} + 3272 q^{81} + 2628 q^{89} + 2552 q^{91} + 880 q^{99}+O(q^{100}) \) 8 * q + 132 * q^9 + 40 * q^11 - 128 * q^19 + 328 * q^21 + 452 * q^29 + 88 * q^31 + 1712 * q^39 + 1748 * q^41 + 2324 * q^49 + 2456 * q^51 + 664 * q^59 + 2316 * q^61 + 3256 * q^69 + 624 * q^71 + 2136 * q^79 + 3272 * q^81 + 2628 * q^89 + 2552 * q^91 + 880 * q^99

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