LMFDB - Embedded newform 531.4.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,4,Mod(1,531)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(531, 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("531.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	531.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:	$31.3300142130$
Analytic rank:	$1$
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} - 2x^{7} - 45x^{6} + 47x^{5} + 654x^{4} - 157x^{3} - 2898x^{2} + 96x + 2432 $ x^8 - 2x^7 - 45x^6 + 47x^5 + 654x^4 - 157x^3 - 2898x^2 + 96*x + 2432
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.15242$ of defining polynomial
Character	$\chi$	$=$	531.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.15242 q^{2} +18.5474 q^{4} -1.69104 q^{5} -11.0943 q^{7} -54.3449 q^{8} +O(q^{10})$	\(q-5.15242 q^{2} +18.5474 q^{4} -1.69104 q^{5} -11.0943 q^{7} -54.3449 q^{8} +8.71294 q^{10} +10.1415 q^{11} +29.5495 q^{13} +57.1625 q^{14} +131.628 q^{16} +29.3698 q^{17} -46.9309 q^{19} -31.3645 q^{20} -52.2532 q^{22} -22.2505 q^{23} -122.140 q^{25} -152.252 q^{26} -205.771 q^{28} +103.959 q^{29} -52.5878 q^{31} -243.445 q^{32} -151.326 q^{34} +18.7609 q^{35} +5.51956 q^{37} +241.808 q^{38} +91.8993 q^{40} -201.493 q^{41} +479.341 q^{43} +188.099 q^{44} +114.644 q^{46} +133.168 q^{47} -219.917 q^{49} +629.319 q^{50} +548.068 q^{52} +484.443 q^{53} -17.1497 q^{55} +602.918 q^{56} -535.639 q^{58} -59.0000 q^{59} +578.161 q^{61} +270.954 q^{62} +201.305 q^{64} -49.9694 q^{65} +52.3009 q^{67} +544.735 q^{68} -96.6639 q^{70} -399.262 q^{71} -1045.81 q^{73} -28.4391 q^{74} -870.449 q^{76} -112.513 q^{77} +269.263 q^{79} -222.588 q^{80} +1038.18 q^{82} -174.118 q^{83} -49.6655 q^{85} -2469.77 q^{86} -551.138 q^{88} -1380.40 q^{89} -327.831 q^{91} -412.691 q^{92} -686.140 q^{94} +79.3620 q^{95} +628.773 q^{97} +1133.10 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{2} + 34 q^{4} - 42 q^{5} + 53 q^{7} - 51 q^{8}+O(q^{10}) $ 8 * q - 6 * q^2 + 34 * q^4 - 42 * q^5 + 53 * q^7 - 51 * q^8	$ 8 q - 6 q^{2} + 34 q^{4} - 42 q^{5} + 53 q^{7} - 51 q^{8} + 21 q^{10} - 67 q^{11} + 33 q^{13} - 79 q^{14} - 30 q^{16} - 139 q^{17} + 64 q^{19} - 117 q^{20} - 84 q^{22} - 226 q^{23} + 96 q^{25} - 24 q^{26} + 34 q^{28} - 456 q^{29} + 124 q^{31} - 174 q^{32} - 114 q^{34} - 556 q^{35} + 127 q^{37} - 237 q^{38} - 188 q^{40} - 425 q^{41} - 115 q^{43} - 510 q^{44} - 711 q^{46} - 420 q^{47} + 171 q^{49} + 137 q^{50} - 922 q^{52} - 98 q^{53} - 616 q^{55} + 412 q^{56} - 1548 q^{58} - 472 q^{59} - 1254 q^{61} + 766 q^{62} - 2019 q^{64} + 734 q^{65} - 1010 q^{67} + 503 q^{68} - 2956 q^{70} + 17 q^{71} - 1180 q^{73} + 1228 q^{74} - 2008 q^{76} - 441 q^{77} - 873 q^{79} + 865 q^{80} - 3645 q^{82} - 759 q^{83} - 850 q^{85} + 1226 q^{86} - 3047 q^{88} - 988 q^{89} - 2111 q^{91} + 1062 q^{92} - 2240 q^{94} - 1822 q^{95} - 668 q^{97} + 1368 q^{98}+O(q^{100}) $ 8 * q - 6 * q^2 + 34 * q^4 - 42 * q^5 + 53 * q^7 - 51 * q^8 + 21 * q^10 - 67 * q^11 + 33 * q^13 - 79 * q^14 - 30 * q^16 - 139 * q^17 + 64 * q^19 - 117 * q^20 - 84 * q^22 - 226 * q^23 + 96 * q^25 - 24 * q^26 + 34 * q^28 - 456 * q^29 + 124 * q^31 - 174 * q^32 - 114 * q^34 - 556 * q^35 + 127 * q^37 - 237 * q^38 - 188 * q^40 - 425 * q^41 - 115 * q^43 - 510 * q^44 - 711 * q^46 - 420 * q^47 + 171 * q^49 + 137 * q^50 - 922 * q^52 - 98 * q^53 - 616 * q^55 + 412 * q^56 - 1548 * q^58 - 472 * q^59 - 1254 * q^61 + 766 * q^62 - 2019 * q^64 + 734 * q^65 - 1010 * q^67 + 503 * q^68 - 2956 * q^70 + 17 * q^71 - 1180 * q^73 + 1228 * q^74 - 2008 * q^76 - 441 * q^77 - 873 * q^79 + 865 * q^80 - 3645 * q^82 - 759 * q^83 - 850 * q^85 + 1226 * q^86 - 3047 * q^88 - 988 * q^89 - 2111 * q^91 + 1062 * q^92 - 2240 * q^94 - 1822 * q^95 - 668 * q^97 + 1368 * q^98

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 $

$2$	−5.15242	−1.82166	−0.910828	−	0.412786i	$-0.864556\pi$
$2$	−5.15242	−1.82166	−0.910828	+	0.412786i	$0.864556\pi$
$3$	0	0
$3$	0	0
$4$	18.5474	2.31843
$5$	−1.69104	−0.151251	−0.0756256	−	0.997136i	$-0.524095\pi$
$5$	−1.69104	−0.151251	−0.0756256	+	0.997136i	$0.524095\pi$
$6$	0	0
$7$	−11.0943	−0.599035	−0.299518	−	0.954091i	$-0.596826\pi$
$7$	−11.0943	−0.599035	−0.299518	+	0.954091i	$0.596826\pi$
$8$	−54.3449	−2.40173
$9$	0	0
$10$	8.71294	0.275528
$11$	10.1415	0.277980	0.138990	−	0.990294i	$-0.455614\pi$
$11$	10.1415	0.277980	0.138990	+	0.990294i	$0.455614\pi$
$12$	0	0
$13$	29.5495	0.630428	0.315214	−	0.949021i	$-0.397924\pi$
$13$	29.5495	0.630428	0.315214	+	0.949021i	$0.397924\pi$
$14$	57.1625	1.09124
$15$	0	0
$16$	131.628	2.05669
$17$	29.3698	0.419013	0.209507	−	0.977807i	$-0.432814\pi$
$17$	29.3698	0.419013	0.209507	+	0.977807i	$0.432814\pi$
$18$	0	0
$19$	−46.9309	−0.566668	−0.283334	−	0.959021i	$-0.591441\pi$
$19$	−46.9309	−0.566668	−0.283334	+	0.959021i	$0.591441\pi$
$20$	−31.3645	−0.350665
$21$	0	0
$22$	−52.2532	−0.506383
$23$	−22.2505	−0.201720	−0.100860	−	0.994901i	$-0.532159\pi$
$23$	−22.2505	−0.201720	−0.100860	+	0.994901i	$0.532159\pi$
$24$	0	0
$25$	−122.140	−0.977123
$26$	−152.252	−1.14842
$27$	0	0
$28$	−205.771	−1.38882
$29$	103.959	0.665677	0.332838	−	0.942984i	$-0.391994\pi$
$29$	103.959	0.665677	0.332838	+	0.942984i	$0.391994\pi$
$30$	0	0
$31$	−52.5878	−0.304679	−0.152339	−	0.988328i	$-0.548681\pi$
$31$	−52.5878	−0.304679	−0.152339	+	0.988328i	$0.548681\pi$
$32$	−243.445	−1.34486
$33$	0	0
$34$	−151.326	−0.763298
$35$	18.7609	0.0906048
$36$	0	0
$37$	5.51956	0.0245246	0.0122623	−	0.999925i	$-0.496097\pi$
$37$	5.51956	0.0245246	0.0122623	+	0.999925i	$0.496097\pi$
$38$	241.808	1.03227
$39$	0	0
$40$	91.8993	0.363264
$41$	−201.493	−0.767512	−0.383756	−	0.923435i	$-0.625370\pi$
$41$	−201.493	−0.767512	−0.383756	+	0.923435i	$0.625370\pi$
$42$	0	0
$43$	479.341	1.69997	0.849987	−	0.526804i	$-0.176610\pi$
$43$	479.341	1.69997	0.849987	+	0.526804i	$0.176610\pi$
$44$	188.099	0.644476
$45$	0	0
$46$	114.644	0.367464
$47$	133.168	0.413290	0.206645	−	0.978416i	$-0.433746\pi$
$47$	133.168	0.413290	0.206645	+	0.978416i	$0.433746\pi$
$48$	0	0
$49$	−219.917	−0.641157
$50$	629.319	1.77998
$51$	0	0
$52$	548.068	1.46160
$53$	484.443	1.25553	0.627767	−	0.778401i	$-0.283969\pi$
$53$	484.443	1.25553	0.627767	+	0.778401i	$0.283969\pi$
$54$	0	0
$55$	−17.1497	−0.0420447
$56$	602.918	1.43872
$57$	0	0
$58$	−535.639	−1.21263
$59$	−59.0000	−0.130189
$59$	−59.0000	−0.130189
$60$	0	0
$61$	578.161	1.21354	0.606770	−	0.794877i	$-0.292465\pi$
$61$	578.161	1.21354	0.606770	+	0.794877i	$0.292465\pi$
$62$	270.954	0.555020
$63$	0	0
$64$	201.305	0.393174
$65$	−49.9694	−0.0953529
$66$	0	0
$67$	52.3009	0.0953668	0.0476834	−	0.998862i	$-0.484816\pi$
$67$	52.3009	0.0953668	0.0476834	+	0.998862i	$0.484816\pi$
$68$	544.735	0.971453
$69$	0	0
$70$	−96.6639	−0.165051
$71$	−399.262	−0.667375	−0.333687	−	0.942684i	$-0.608293\pi$
$71$	−399.262	−0.667375	−0.333687	+	0.942684i	$0.608293\pi$
$72$	0	0
$73$	−1045.81	−1.67675	−0.838374	−	0.545096i	$-0.816493\pi$
$73$	−1045.81	−1.67675	−0.838374	+	0.545096i	$0.816493\pi$
$74$	−28.4391	−0.0446754
$75$	0	0
$76$	−870.449	−1.31378
$77$	−112.513	−0.166520
$78$	0	0
$79$	269.263	0.383474	0.191737	−	0.981446i	$-0.438588\pi$
$79$	269.263	0.383474	0.191737	+	0.981446i	$0.438588\pi$
$80$	−222.588	−0.311077
$81$	0	0
$82$	1038.18	1.39814
$83$	−174.118	−0.230265	−0.115132	−	0.993350i	$-0.536729\pi$
$83$	−174.118	−0.230265	−0.115132	+	0.993350i	$0.536729\pi$
$84$	0	0
$85$	−49.6655	−0.0633762
$86$	−2469.77	−3.09677
$87$	0	0
$88$	−551.138	−0.667631
$89$	−1380.40	−1.64407	−0.822036	−	0.569435i	$-0.807162\pi$
$89$	−1380.40	−1.64407	−0.822036	+	0.569435i	$0.807162\pi$
$90$	0	0
$91$	−327.831	−0.377648
$92$	−412.691	−0.467674
$93$	0	0
$94$	−686.140	−0.752871
$95$	79.3620	0.0857092
$96$	0	0
$97$	628.773	0.658167	0.329084	−	0.944301i	$-0.393260\pi$
$97$	628.773	0.658167	0.329084	+	0.944301i	$0.393260\pi$
$98$	1133.10	1.16797
$99$	0	0
$100$	−2265.39	−2.26539
$101$	−553.980	−0.545773	−0.272887	−	0.962046i	$-0.587978\pi$
$101$	−553.980	−0.545773	−0.272887	+	0.962046i	$0.587978\pi$
$102$	0	0
$103$	401.263	0.383861	0.191930	−	0.981409i	$-0.438525\pi$
$103$	401.263	0.383861	0.191930	+	0.981409i	$0.438525\pi$
$104$	−1605.86	−1.51412
$105$	0	0
$106$	−2496.05	−2.28715
$107$	129.942	0.117401	0.0587007	−	0.998276i	$-0.481304\pi$
$107$	129.942	0.117401	0.0587007	+	0.998276i	$0.481304\pi$
$108$	0	0
$109$	−1248.43	−1.09705	−0.548524	−	0.836135i	$-0.684810\pi$
$109$	−1248.43	−1.09705	−0.548524	+	0.836135i	$0.684810\pi$
$110$	88.3623	0.0765910
$111$	0	0
$112$	−1460.32	−1.23203
$113$	−1175.90	−0.978933	−0.489466	−	0.872022i	$-0.662808\pi$
$113$	−1175.90	−0.978933	−0.489466	+	0.872022i	$0.662808\pi$
$114$	0	0
$115$	37.6265	0.0305104
$116$	1928.17	1.54333
$117$	0	0
$118$	303.993	0.237159
$119$	−325.837	−0.251004
$120$	0	0
$121$	−1228.15	−0.922727
$122$	−2978.93	−2.21065
$123$	0	0
$124$	−975.369	−0.706377
$125$	417.924	0.299042
$126$	0	0
$127$	−1993.40	−1.39280	−0.696399	−	0.717654i	$-0.745216\pi$
$127$	−1993.40	−1.39280	−0.696399	+	0.717654i	$0.745216\pi$
$128$	910.351	0.628629
$129$	0	0
$130$	257.463	0.173700
$131$	−1519.30	−1.01330	−0.506650	−	0.862152i	$-0.669116\pi$
$131$	−1519.30	−1.01330	−0.506650	+	0.862152i	$0.669116\pi$
$132$	0	0
$133$	520.665	0.339454
$134$	−269.476	−0.173726
$135$	0	0
$136$	−1596.10	−1.00636
$137$	−1596.83	−0.995812	−0.497906	−	0.867231i	$-0.665898\pi$
$137$	−1596.83	−0.995812	−0.497906	+	0.867231i	$0.665898\pi$
$138$	0	0
$139$	666.080	0.406447	0.203224	−	0.979132i	$-0.434858\pi$
$139$	666.080	0.406447	0.203224	+	0.979132i	$0.434858\pi$
$140$	347.966	0.210061
$141$	0	0
$142$	2057.16	1.21573
$143$	299.676	0.175246
$144$	0	0
$145$	−175.798	−0.100684
$146$	5388.44	3.05446
$147$	0	0
$148$	102.374	0.0568586
$149$	−1767.90	−0.972029	−0.486014	−	0.873951i	$-0.661550\pi$
$149$	−1767.90	−0.972029	−0.486014	+	0.873951i	$0.661550\pi$
$150$	0	0
$151$	−1053.25	−0.567633	−0.283817	−	0.958879i	$-0.591601\pi$
$151$	−1053.25	−0.567633	−0.283817	+	0.958879i	$0.591601\pi$
$152$	2550.46	1.36098
$153$	0	0
$154$	579.713	0.303341
$155$	88.9280	0.0460830
$156$	0	0
$157$	−47.5287	−0.0241605	−0.0120803	−	0.999927i	$-0.503845\pi$
$157$	−47.5287	−0.0241605	−0.0120803	+	0.999927i	$0.503845\pi$
$158$	−1387.36	−0.698558
$159$	0	0
$160$	411.675	0.203411
$161$	246.854	0.120837
$162$	0	0
$163$	3084.59	1.48223	0.741115	−	0.671378i	$-0.234297\pi$
$163$	3084.59	1.48223	0.741115	+	0.671378i	$0.234297\pi$
$164$	−3737.19	−1.77942
$165$	0	0
$166$	897.132	0.419463
$167$	−3685.43	−1.70771	−0.853854	−	0.520513i	$-0.825741\pi$
$167$	−3685.43	−1.70771	−0.853854	+	0.520513i	$0.825741\pi$
$168$	0	0
$169$	−1323.83	−0.602561
$170$	255.898	0.115450
$171$	0	0
$172$	8890.56	3.94127
$173$	−504.194	−0.221579	−0.110790	−	0.993844i	$-0.535338\pi$
$173$	−504.194	−0.221579	−0.110790	+	0.993844i	$0.535338\pi$
$174$	0	0
$175$	1355.06	0.585331
$176$	1334.91	0.571718
$177$	0	0
$178$	7112.42	2.99494
$179$	−2931.88	−1.22424	−0.612120	−	0.790765i	$-0.709683\pi$
$179$	−2931.88	−1.22424	−0.612120	+	0.790765i	$0.709683\pi$
$180$	0	0
$181$	4335.05	1.78023	0.890116	−	0.455734i	$-0.150623\pi$
$181$	4335.05	1.78023	0.890116	+	0.455734i	$0.150623\pi$
$182$	1689.12	0.687946
$183$	0	0
$184$	1209.20	0.484476
$185$	−9.33380	−0.00370937
$186$	0	0
$187$	297.854	0.116477
$188$	2469.93	0.958183
$189$	0	0
$190$	−408.907	−0.156133
$191$	−2286.00	−0.866016	−0.433008	−	0.901390i	$-0.642548\pi$
$191$	−2286.00	−0.866016	−0.433008	+	0.901390i	$0.642548\pi$
$192$	0	0
$193$	−1685.13	−0.628487	−0.314244	−	0.949342i	$-0.601751\pi$
$193$	−1685.13	−0.628487	−0.314244	+	0.949342i	$0.601751\pi$
$194$	−3239.70	−1.19895
$195$	0	0
$196$	−4078.89	−1.48648
$197$	−690.913	−0.249876	−0.124938	−	0.992165i	$-0.539873\pi$
$197$	−690.913	−0.249876	−0.124938	+	0.992165i	$0.539873\pi$
$198$	0	0
$199$	−1448.24	−0.515894	−0.257947	−	0.966159i	$-0.583046\pi$
$199$	−1448.24	−0.515894	−0.257947	+	0.966159i	$0.583046\pi$
$200$	6637.71	2.34678
$201$	0	0
$202$	2854.34	0.994211
$203$	−1153.35	−0.398764
$204$	0	0
$205$	340.733	0.116087
$206$	−2067.48	−0.699262
$207$	0	0
$208$	3889.55	1.29659
$209$	−475.950	−0.157522
$210$	0	0
$211$	−5007.79	−1.63389	−0.816944	−	0.576717i	$-0.804333\pi$
$211$	−5007.79	−1.63389	−0.816944	+	0.576717i	$0.804333\pi$
$212$	8985.18	2.91087
$213$	0	0
$214$	−669.515	−0.213865
$215$	−810.585	−0.257123
$216$	0	0
$217$	583.424	0.182513
$218$	6432.46	1.99844
$219$	0	0
$220$	−318.082	−0.0974778
$221$	867.864	0.264158
$222$	0	0
$223$	1275.24	0.382943	0.191471	−	0.981498i	$-0.438674\pi$
$223$	1275.24	0.382943	0.191471	+	0.981498i	$0.438674\pi$
$224$	2700.85	0.805616
$225$	0	0
$226$	6058.73	1.78328
$227$	1302.24	0.380760	0.190380	−	0.981711i	$-0.439028\pi$
$227$	1302.24	0.380760	0.190380	+	0.981711i	$0.439028\pi$
$228$	0	0
$229$	−3359.40	−0.969413	−0.484706	−	0.874677i	$-0.661073\pi$
$229$	−3359.40	−0.969413	−0.484706	+	0.874677i	$0.661073\pi$
$230$	−193.868	−0.0555794
$231$	0	0
$232$	−5649.62	−1.59877
$233$	2610.02	0.733855	0.366927	−	0.930250i	$-0.380410\pi$
$233$	2610.02	0.733855	0.366927	+	0.930250i	$0.380410\pi$
$234$	0	0
$235$	−225.193	−0.0625105
$236$	−1094.30	−0.301834
$237$	0	0
$238$	1678.85	0.457243
$239$	4236.99	1.14673	0.573364	−	0.819301i	$-0.305638\pi$
$239$	4236.99	1.14673	0.573364	+	0.819301i	$0.305638\pi$
$240$	0	0
$241$	−3164.38	−0.845792	−0.422896	−	0.906178i	$-0.638986\pi$
$241$	−3164.38	−0.845792	−0.422896	+	0.906178i	$0.638986\pi$
$242$	6327.95	1.68089
$243$	0	0
$244$	10723.4	2.81351
$245$	371.888	0.0969756
$246$	0	0
$247$	−1386.79	−0.357243
$248$	2857.88	0.731756
$249$	0	0
$250$	−2153.32	−0.544752
$251$	6495.46	1.63343	0.816713	−	0.577044i	$-0.195794\pi$
$251$	6495.46	1.63343	0.816713	+	0.577044i	$0.195794\pi$
$252$	0	0
$253$	−225.654	−0.0560740
$254$	10270.8	2.53720
$255$	0	0
$256$	−6300.95	−1.53832
$257$	4426.51	1.07439	0.537195	−	0.843458i	$-0.319484\pi$
$257$	4426.51	1.07439	0.537195	+	0.843458i	$0.319484\pi$
$258$	0	0
$259$	−61.2356	−0.0146911
$260$	−926.804	−0.221069
$261$	0	0
$262$	7828.10	1.84588
$263$	−6798.01	−1.59385	−0.796927	−	0.604076i	$-0.793542\pi$
$263$	−6798.01	−1.59385	−0.796927	+	0.604076i	$0.793542\pi$
$264$	0	0
$265$	−819.212	−0.189901
$266$	−2682.69	−0.618369
$267$	0	0
$268$	970.049	0.221101
$269$	−6894.51	−1.56270	−0.781349	−	0.624095i	$-0.785468\pi$
$269$	−6894.51	−1.56270	−0.781349	+	0.624095i	$0.785468\pi$
$270$	0	0
$271$	2936.35	0.658195	0.329097	−	0.944296i	$-0.393256\pi$
$271$	2936.35	0.658195	0.329097	+	0.944296i	$0.393256\pi$
$272$	3865.90	0.861781
$273$	0	0
$274$	8227.53	1.81403
$275$	−1238.69	−0.271620
$276$	0	0
$277$	−7457.80	−1.61767	−0.808837	−	0.588033i	$-0.799902\pi$
$277$	−7457.80	−1.61767	−0.808837	+	0.588033i	$0.799902\pi$
$278$	−3431.92	−0.740407
$279$	0	0
$280$	−1019.56	−0.217608
$281$	5450.35	1.15708	0.578542	−	0.815652i	$-0.303622\pi$
$281$	5450.35	1.15708	0.578542	+	0.815652i	$0.303622\pi$
$282$	0	0
$283$	2984.42	0.626873	0.313437	−	0.949609i	$-0.398520\pi$
$283$	2984.42	0.626873	0.313437	+	0.949609i	$0.398520\pi$
$284$	−7405.28	−1.54726
$285$	0	0
$286$	−1544.06	−0.319238
$287$	2235.43	0.459767
$288$	0	0
$289$	−4050.41	−0.824428
$290$	905.786	0.183412
$291$	0	0
$292$	−19397.1	−3.88742
$293$	−2189.77	−0.436613	−0.218307	−	0.975880i	$-0.570053\pi$
$293$	−2189.77	−0.436613	−0.218307	+	0.975880i	$0.570053\pi$
$294$	0	0
$295$	99.7713	0.0196912
$296$	−299.960	−0.0589014
$297$	0	0
$298$	9108.98	1.77070
$299$	−657.493	−0.127170
$300$	0	0
$301$	−5317.95	−1.01834
$302$	5426.81	1.03403
$303$	0	0
$304$	−6177.43	−1.16546
$305$	−977.693	−0.183549
$306$	0	0
$307$	−8606.40	−1.59998	−0.799989	−	0.600014i	$-0.795162\pi$
$307$	−8606.40	−1.59998	−0.799989	+	0.600014i	$0.795162\pi$
$308$	−2086.82	−0.386064
$309$	0	0
$310$	−458.195	−0.0839474
$311$	−757.395	−0.138096	−0.0690482	−	0.997613i	$-0.521996\pi$
$311$	−757.395	−0.138096	−0.0690482	+	0.997613i	$0.521996\pi$
$312$	0	0
$313$	−2172.72	−0.392363	−0.196182	−	0.980568i	$-0.562854\pi$
$313$	−2172.72	−0.392363	−0.196182	+	0.980568i	$0.562854\pi$
$314$	244.888	0.0440122
$315$	0	0
$316$	4994.14	0.889058
$317$	6786.82	1.20248	0.601239	−	0.799069i	$-0.294674\pi$
$317$	6786.82	1.20248	0.601239	+	0.799069i	$0.294674\pi$
$318$	0	0
$319$	1054.30	0.185045
$320$	−340.414	−0.0594679
$321$	0	0
$322$	−1271.90	−0.220124
$323$	−1378.35	−0.237441
$324$	0	0
$325$	−3609.19	−0.616005
$326$	−15893.1	−2.70011
$327$	0	0
$328$	10950.1	1.84335
$329$	−1477.41	−0.247575
$330$	0	0
$331$	−2842.29	−0.471984	−0.235992	−	0.971755i	$-0.575834\pi$
$331$	−2842.29	−0.471984	−0.235992	+	0.971755i	$0.575834\pi$
$332$	−3229.45	−0.533853
$333$	0	0
$334$	18988.9	3.11086
$335$	−88.4429	−0.0144243
$336$	0	0
$337$	−6547.58	−1.05837	−0.529183	−	0.848508i	$-0.677501\pi$
$337$	−6547.58	−1.05837	−0.529183	+	0.848508i	$0.677501\pi$
$338$	6820.91	1.09766
$339$	0	0
$340$	−921.168	−0.146933
$341$	−533.319	−0.0846945
$342$	0	0
$343$	6245.16	0.983111
$344$	−26049.7	−4.08287
$345$	0	0
$346$	2597.82	0.403641
$347$	35.2935	0.00546010	0.00273005	−	0.999996i	$-0.499131\pi$
$347$	35.2935	0.00546010	0.00273005	+	0.999996i	$0.499131\pi$
$348$	0	0
$349$	10692.0	1.63991	0.819956	−	0.572427i	$-0.193998\pi$
$349$	10692.0	1.63991	0.819956	+	0.572427i	$0.193998\pi$
$350$	−6981.85	−1.06627
$351$	0	0
$352$	−2468.89	−0.373842
$353$	8661.89	1.30602	0.653011	−	0.757348i	$-0.273505\pi$
$353$	8661.89	1.30602	0.653011	+	0.757348i	$0.273505\pi$
$354$	0	0
$355$	675.167	0.100941
$356$	−25603.0	−3.81167
$357$	0	0
$358$	15106.3	2.23014
$359$	5422.56	0.797192	0.398596	−	0.917127i	$-0.369498\pi$
$359$	5422.56	0.797192	0.398596	+	0.917127i	$0.369498\pi$
$360$	0	0
$361$	−4656.49	−0.678887
$362$	−22336.0	−3.24297
$363$	0	0
$364$	−6080.43	−0.875552
$365$	1768.50	0.253610
$366$	0	0
$367$	−1346.75	−0.191552	−0.0957762	−	0.995403i	$-0.530533\pi$
$367$	−1346.75	−0.191552	−0.0957762	+	0.995403i	$0.530533\pi$
$368$	−2928.80	−0.414876
$369$	0	0
$370$	48.0916	0.00675720
$371$	−5374.55	−0.752110
$372$	0	0
$373$	−4622.11	−0.641618	−0.320809	−	0.947144i	$-0.603955\pi$
$373$	−4622.11	−0.641618	−0.320809	+	0.947144i	$0.603955\pi$
$374$	−1534.67	−0.212181
$375$	0	0
$376$	−7237.02	−0.992609
$377$	3071.93	0.419661
$378$	0	0
$379$	8654.75	1.17299	0.586497	−	0.809952i	$-0.300507\pi$
$379$	8654.75	1.17299	0.586497	+	0.809952i	$0.300507\pi$
$380$	1471.96	0.198711
$381$	0	0
$382$	11778.4	1.57758
$383$	2461.00	0.328333	0.164166	−	0.986433i	$-0.447507\pi$
$383$	2461.00	0.328333	0.164166	+	0.986433i	$0.447507\pi$
$384$	0	0
$385$	190.263	0.0251863
$386$	8682.48	1.14489
$387$	0	0
$388$	11662.1	1.52592
$389$	8258.56	1.07642	0.538208	−	0.842812i	$-0.319102\pi$
$389$	8258.56	1.07642	0.538208	+	0.842812i	$0.319102\pi$
$390$	0	0
$391$	−653.494	−0.0845234
$392$	11951.3	1.53988
$393$	0	0
$394$	3559.87	0.455187
$395$	−455.334	−0.0580009
$396$	0	0
$397$	−4331.00	−0.547523	−0.273761	−	0.961798i	$-0.588268\pi$
$397$	−4331.00	−0.547523	−0.273761	+	0.961798i	$0.588268\pi$
$398$	7461.94	0.939781
$399$	0	0
$400$	−16077.1	−2.00964
$401$	2292.26	0.285461	0.142731	−	0.989762i	$-0.454412\pi$
$401$	2292.26	0.285461	0.142731	+	0.989762i	$0.454412\pi$
$402$	0	0
$403$	−1553.94	−0.192078
$404$	−10274.9	−1.26534
$405$	0	0
$406$	5942.53	0.726411
$407$	55.9766	0.00681734
$408$	0	0
$409$	−13032.9	−1.57563	−0.787817	−	0.615909i	$-0.788789\pi$
$409$	−13032.9	−1.57563	−0.787817	+	0.615909i	$0.788789\pi$
$410$	−1755.60	−0.211471
$411$	0	0
$412$	7442.41	0.889954
$413$	654.563	0.0779878
$414$	0	0
$415$	294.441	0.0348278
$416$	−7193.68	−0.847834
$417$	0	0
$418$	2452.29	0.286951
$419$	3919.78	0.457026	0.228513	−	0.973541i	$-0.426614\pi$
$419$	3919.78	0.457026	0.228513	+	0.973541i	$0.426614\pi$
$420$	0	0
$421$	4905.30	0.567862	0.283931	−	0.958845i	$-0.408361\pi$
$421$	4905.30	0.567862	0.283931	+	0.958845i	$0.408361\pi$
$422$	25802.2	2.97638
$423$	0	0
$424$	−26327.0	−3.01545
$425$	−3587.24	−0.409428
$426$	0	0
$427$	−6414.29	−0.726954
$428$	2410.09	0.272187
$429$	0	0
$430$	4176.47	0.468389
$431$	−13528.1	−1.51189	−0.755947	−	0.654633i	$-0.772823\pi$
$431$	−13528.1	−1.51189	−0.755947	+	0.654633i	$0.772823\pi$
$432$	0	0
$433$	15378.7	1.70683	0.853413	−	0.521236i	$-0.174529\pi$
$433$	15378.7	1.70683	0.853413	+	0.521236i	$0.174529\pi$
$434$	−3006.05	−0.332477
$435$	0	0
$436$	−23155.3	−2.54343
$437$	1044.24	0.114308
$438$	0	0
$439$	2929.73	0.318515	0.159258	−	0.987237i	$-0.449090\pi$
$439$	2929.73	0.318515	0.159258	+	0.987237i	$0.449090\pi$
$440$	931.996	0.100980
$441$	0	0
$442$	−4471.60	−0.481204
$443$	5825.07	0.624734	0.312367	−	0.949961i	$-0.398878\pi$
$443$	5825.07	0.624734	0.312367	+	0.949961i	$0.398878\pi$
$444$	0	0
$445$	2334.32	0.248668
$446$	−6570.56	−0.697590
$447$	0	0
$448$	−2233.33	−0.235525
$449$	9997.96	1.05085	0.525426	−	0.850839i	$-0.323906\pi$
$449$	9997.96	1.05085	0.525426	+	0.850839i	$0.323906\pi$
$450$	0	0
$451$	−2043.44	−0.213353
$452$	−21810.0	−2.26959
$453$	0	0
$454$	−6709.67	−0.693613
$455$	554.375	0.0571198
$456$	0	0
$457$	−11063.7	−1.13246	−0.566232	−	0.824246i	$-0.691599\pi$
$457$	−11063.7	−1.13246	−0.566232	+	0.824246i	$0.691599\pi$
$458$	17309.0	1.76594
$459$	0	0
$460$	697.876	0.0707362
$461$	−9657.22	−0.975665	−0.487832	−	0.872937i	$-0.662212\pi$
$461$	−9657.22	−0.975665	−0.487832	+	0.872937i	$0.662212\pi$
$462$	0	0
$463$	2678.74	0.268880	0.134440	−	0.990922i	$-0.457076\pi$
$463$	2678.74	0.268880	0.134440	+	0.990922i	$0.457076\pi$
$464$	13683.9	1.36909
$465$	0	0
$466$	−13447.9	−1.33683
$467$	1146.00	0.113556	0.0567781	−	0.998387i	$-0.481917\pi$
$467$	1146.00	0.113556	0.0567781	+	0.998387i	$0.481917\pi$
$468$	0	0
$469$	−580.242	−0.0571281
$470$	1160.29	0.113873
$471$	0	0
$472$	3206.35	0.312678
$473$	4861.24	0.472558
$474$	0	0
$475$	5732.16	0.553704
$476$	−6043.45	−0.581935
$477$	0	0
$478$	−21830.7	−2.08894
$479$	−18182.6	−1.73441	−0.867207	−	0.497948i	$-0.834087\pi$
$479$	−18182.6	−1.73441	−0.867207	+	0.497948i	$0.834087\pi$
$480$	0	0
$481$	163.100	0.0154610
$482$	16304.2	1.54074
$483$	0	0
$484$	−22779.0	−2.13928
$485$	−1063.28	−0.0995485
$486$	0	0
$487$	3231.33	0.300668	0.150334	−	0.988635i	$-0.451965\pi$
$487$	3231.33	0.300668	0.150334	+	0.988635i	$0.451965\pi$
$488$	−31420.1	−2.91459
$489$	0	0
$490$	−1916.12	−0.176656
$491$	−6518.84	−0.599168	−0.299584	−	0.954070i	$-0.596848\pi$
$491$	−6518.84	−0.599168	−0.299584	+	0.954070i	$0.596848\pi$
$492$	0	0
$493$	3053.25	0.278927
$494$	7145.31	0.650774
$495$	0	0
$496$	−6922.04	−0.626630
$497$	4429.52	0.399781
$498$	0	0
$499$	14603.3	1.31009	0.655045	−	0.755590i	$-0.272650\pi$
$499$	14603.3	1.31009	0.655045	+	0.755590i	$0.272650\pi$
$500$	7751.42	0.693308
$501$	0	0
$502$	−33467.4	−2.97554
$503$	1522.39	0.134950	0.0674751	−	0.997721i	$-0.478506\pi$
$503$	1522.39	0.134950	0.0674751	+	0.997721i	$0.478506\pi$
$504$	0	0
$505$	936.802	0.0825488
$506$	1162.66	0.102148
$507$	0	0
$508$	−36972.4	−3.22911
$509$	−33.4888	−0.00291624	−0.00145812	−	0.999999i	$-0.500464\pi$
$509$	−33.4888	−0.00291624	−0.00145812	+	0.999999i	$0.500464\pi$
$510$	0	0
$511$	11602.5	1.00443
$512$	25182.4	2.17366
$513$	0	0
$514$	−22807.3	−1.95717
$515$	−678.552	−0.0580593
$516$	0	0
$517$	1350.53	0.114886
$518$	315.512	0.0267622
$519$	0	0
$520$	2715.58	0.229012
$521$	−12883.7	−1.08339	−0.541694	−	0.840576i	$-0.682217\pi$
$521$	−12883.7	−1.08339	−0.541694	+	0.840576i	$0.682217\pi$
$522$	0	0
$523$	14819.1	1.23899	0.619497	−	0.784999i	$-0.287337\pi$
$523$	14819.1	1.23899	0.619497	+	0.784999i	$0.287337\pi$
$524$	−28179.2	−2.34927
$525$	0	0
$526$	35026.2	2.90345
$527$	−1544.49	−0.127665
$528$	0	0
$529$	−11671.9	−0.959309
$530$	4220.92	0.345934
$531$	0	0
$532$	9657.01	0.787001
$533$	−5954.03	−0.483861
$534$	0	0
$535$	−219.737	−0.0177571
$536$	−2842.29	−0.229045
$537$	0	0
$538$	35523.4	2.84670
$539$	−2230.28	−0.178228
$540$	0	0
$541$	−14334.6	−1.13917	−0.569585	−	0.821932i	$-0.692896\pi$
$541$	−14334.6	−1.13917	−0.569585	+	0.821932i	$0.692896\pi$
$542$	−15129.3	−1.19900
$543$	0	0
$544$	−7149.93	−0.563512
$545$	2111.15	0.165930
$546$	0	0
$547$	23516.6	1.83820	0.919102	−	0.394019i	$-0.128915\pi$
$547$	23516.6	1.83820	0.919102	+	0.394019i	$0.128915\pi$
$548$	−29617.1	−2.30872
$549$	0	0
$550$	6382.23	0.494799
$551$	−4878.87	−0.377218
$552$	0	0
$553$	−2987.28	−0.229715
$554$	38425.7	2.94685
$555$	0	0
$556$	12354.1	0.942320
$557$	2834.56	0.215627	0.107813	−	0.994171i	$-0.465615\pi$
$557$	2834.56	0.215627	0.107813	+	0.994171i	$0.465615\pi$
$558$	0	0
$559$	14164.3	1.07171
$560$	2469.46	0.186346
$561$	0	0
$562$	−28082.5	−2.10781
$563$	15215.1	1.13897	0.569484	−	0.822002i	$-0.307143\pi$
$563$	15215.1	1.13897	0.569484	+	0.822002i	$0.307143\pi$
$564$	0	0
$565$	1988.49	0.148065
$566$	−15377.0	−1.14195
$567$	0	0
$568$	21697.8	1.60285
$569$	18473.2	1.36105	0.680526	−	0.732724i	$-0.261751\pi$
$569$	18473.2	1.36105	0.680526	+	0.732724i	$0.261751\pi$
$570$	0	0
$571$	2472.87	0.181237	0.0906185	−	0.995886i	$-0.471116\pi$
$571$	2472.87	0.181237	0.0906185	+	0.995886i	$0.471116\pi$
$572$	5558.23	0.406296
$573$	0	0
$574$	−11517.9	−0.837537
$575$	2717.69	0.197105
$576$	0	0
$577$	1987.70	0.143413	0.0717064	−	0.997426i	$-0.477156\pi$
$577$	1987.70	0.143413	0.0717064	+	0.997426i	$0.477156\pi$
$578$	20869.4	1.50182
$579$	0	0
$580$	−3260.60	−0.233430
$581$	1931.72	0.137937
$582$	0	0
$583$	4912.97	0.349013
$584$	56834.3	4.02709
$585$	0	0
$586$	11282.6	0.795359
$587$	−1325.75	−0.0932192	−0.0466096	−	0.998913i	$-0.514842\pi$
$587$	−1325.75	−0.0932192	−0.0466096	+	0.998913i	$0.514842\pi$
$588$	0	0
$589$	2467.99	0.172652
$590$	−514.064	−0.0358706
$591$	0	0
$592$	726.530	0.0504395
$593$	−23945.6	−1.65822	−0.829112	−	0.559083i	$-0.811153\pi$
$593$	−23945.6	−1.65822	−0.829112	+	0.559083i	$0.811153\pi$
$594$	0	0
$595$	551.004	0.0379646
$596$	−32790.1	−2.25358
$597$	0	0
$598$	3387.68	0.231660
$599$	21924.8	1.49553	0.747766	−	0.663962i	$-0.231126\pi$
$599$	21924.8	1.49553	0.747766	+	0.663962i	$0.231126\pi$
$600$	0	0
$601$	−6361.22	−0.431746	−0.215873	−	0.976421i	$-0.569260\pi$
$601$	−6361.22	−0.431746	−0.215873	+	0.976421i	$0.569260\pi$
$602$	27400.3	1.85507
$603$	0	0
$604$	−19535.2	−1.31602
$605$	2076.85	0.139564
$606$	0	0
$607$	−4019.17	−0.268753	−0.134376	−	0.990930i	$-0.542903\pi$
$607$	−4019.17	−0.268753	−0.134376	+	0.990930i	$0.542903\pi$
$608$	11425.1	0.762087
$609$	0	0
$610$	5037.49	0.334364
$611$	3935.06	0.260549
$612$	0	0
$613$	7840.19	0.516578	0.258289	−	0.966068i	$-0.416841\pi$
$613$	7840.19	0.516578	0.258289	+	0.966068i	$0.416841\pi$
$614$	44343.8	2.91461
$615$	0	0
$616$	6114.49	0.399935
$617$	−1831.09	−0.119476	−0.0597382	−	0.998214i	$-0.519027\pi$
$617$	−1831.09	−0.119476	−0.0597382	+	0.998214i	$0.519027\pi$
$618$	0	0
$619$	7868.38	0.510916	0.255458	−	0.966820i	$-0.417774\pi$
$619$	7868.38	0.510916	0.255458	+	0.966820i	$0.417774\pi$
$620$	1649.39	0.106840
$621$	0	0
$622$	3902.42	0.251564
$623$	15314.6	0.984858
$624$	0	0
$625$	14560.8	0.931893
$626$	11194.8	0.714751
$627$	0	0
$628$	−881.536	−0.0560145
$629$	162.109	0.0102761
$630$	0	0
$631$	−4830.13	−0.304730	−0.152365	−	0.988324i	$-0.548689\pi$
$631$	−4830.13	−0.304730	−0.152365	+	0.988324i	$0.548689\pi$
$632$	−14633.1	−0.921000
$633$	0	0
$634$	−34968.5	−2.19050
$635$	3370.91	0.210662
$636$	0	0
$637$	−6498.43	−0.404203
$638$	−5432.17	−0.337088
$639$	0	0
$640$	−1539.44	−0.0950808
$641$	−5637.67	−0.347386	−0.173693	−	0.984800i	$-0.555570\pi$
$641$	−5637.67	−0.347386	−0.173693	+	0.984800i	$0.555570\pi$
$642$	0	0
$643$	−16485.8	−1.01110	−0.505548	−	0.862798i	$-0.668710\pi$
$643$	−16485.8	−1.01110	−0.505548	+	0.862798i	$0.668710\pi$
$644$	4578.51	0.280153
$645$	0	0
$646$	7101.85	0.432537
$647$	−9709.68	−0.589995	−0.294998	−	0.955498i	$-0.595319\pi$
$647$	−9709.68	−0.589995	−0.294998	+	0.955498i	$0.595319\pi$
$648$	0	0
$649$	−598.348	−0.0361899
$650$	18596.1	1.12215
$651$	0	0
$652$	57211.2	3.43645
$653$	21099.7	1.26446	0.632231	−	0.774780i	$-0.282139\pi$
$653$	21099.7	1.26446	0.632231	+	0.774780i	$0.282139\pi$
$654$	0	0
$655$	2569.20	0.153263
$656$	−26522.2	−1.57853
$657$	0	0
$658$	7612.23	0.450997
$659$	−18800.6	−1.11133	−0.555665	−	0.831407i	$-0.687536\pi$
$659$	−18800.6	−1.11133	−0.555665	+	0.831407i	$0.687536\pi$
$660$	0	0
$661$	−11341.7	−0.667384	−0.333692	−	0.942682i	$-0.608295\pi$
$661$	−11341.7	−0.667384	−0.333692	+	0.942682i	$0.608295\pi$
$662$	14644.7	0.859792
$663$	0	0
$664$	9462.45	0.553033
$665$	−880.465	−0.0513428
$666$	0	0
$667$	−2313.14	−0.134280
$668$	−68355.3	−3.95920
$669$	0	0
$670$	455.695	0.0262762
$671$	5863.42	0.337339
$672$	0	0
$673$	29906.1	1.71292	0.856460	−	0.516214i	$-0.172659\pi$
$673$	29906.1	1.71292	0.856460	+	0.516214i	$0.172659\pi$
$674$	33735.9	1.92798
$675$	0	0
$676$	−24553.6	−1.39700
$677$	−7445.33	−0.422670	−0.211335	−	0.977414i	$-0.567781\pi$
$677$	−7445.33	−0.422670	−0.211335	+	0.977414i	$0.567781\pi$
$678$	0	0
$679$	−6975.79	−0.394266
$680$	2699.07	0.152212
$681$	0	0
$682$	2747.88	0.154284
$683$	6445.33	0.361089	0.180544	−	0.983567i	$-0.442214\pi$
$683$	6445.33	0.361089	0.180544	+	0.983567i	$0.442214\pi$
$684$	0	0
$685$	2700.30	0.150618
$686$	−32177.7	−1.79089
$687$	0	0
$688$	63094.8	3.49632
$689$	14315.1	0.791524
$690$	0	0
$691$	−3347.76	−0.184305	−0.0921525	−	0.995745i	$-0.529375\pi$
$691$	−3347.76	−0.184305	−0.0921525	+	0.995745i	$0.529375\pi$
$692$	−9351.52	−0.513716
$693$	0	0
$694$	−181.847	−0.00994643
$695$	−1126.37	−0.0614756
$696$	0	0
$697$	−5917.82	−0.321598
$698$	−55089.6	−2.98735
$699$	0	0
$700$	25132.9	1.35705
$701$	4893.37	0.263652	0.131826	−	0.991273i	$-0.457916\pi$
$701$	4893.37	0.263652	0.131826	+	0.991273i	$0.457916\pi$
$702$	0	0
$703$	−259.038	−0.0138973
$704$	2041.53	0.109294
$705$	0	0
$706$	−44629.7	−2.37912
$707$	6146.02	0.326938
$708$	0	0
$709$	11964.9	0.633782	0.316891	−	0.948462i	$-0.397361\pi$
$709$	11964.9	0.633782	0.316891	+	0.948462i	$0.397361\pi$
$710$	−3478.74	−0.183880
$711$	0	0
$712$	75017.9	3.94861
$713$	1170.11	0.0614598
$714$	0	0
$715$	−506.764	−0.0265061
$716$	−54378.9	−2.83831
$717$	0	0
$718$	−27939.3	−1.45221
$719$	−25587.9	−1.32722	−0.663608	−	0.748081i	$-0.730976\pi$
$719$	−25587.9	−1.32722	−0.663608	+	0.748081i	$0.730976\pi$
$720$	0	0
$721$	−4451.73	−0.229946
$722$	23992.2	1.23670
$723$	0	0
$724$	80404.2	4.12734
$725$	−12697.5	−0.650448
$726$	0	0
$727$	−18321.4	−0.934667	−0.467334	−	0.884081i	$-0.654785\pi$
$727$	−18321.4	−0.934667	−0.467334	+	0.884081i	$0.654785\pi$
$728$	17815.9	0.907009
$729$	0	0
$730$	−9112.07	−0.461990
$731$	14078.2	0.712312
$732$	0	0
$733$	−11139.1	−0.561298	−0.280649	−	0.959810i	$-0.590550\pi$
$733$	−11139.1	−0.561298	−0.280649	+	0.959810i	$0.590550\pi$
$734$	6939.02	0.348943
$735$	0	0
$736$	5416.78	0.271284
$737$	530.410	0.0265100
$738$	0	0
$739$	−16786.9	−0.835610	−0.417805	−	0.908537i	$-0.637200\pi$
$739$	−16786.9	−0.835610	−0.417805	+	0.908537i	$0.637200\pi$
$740$	−173.118	−0.00859993
$741$	0	0
$742$	27692.0	1.37009
$743$	−35364.4	−1.74615	−0.873077	−	0.487582i	$-0.837879\pi$
$743$	−35364.4	−1.74615	−0.873077	+	0.487582i	$0.837879\pi$
$744$	0	0
$745$	2989.59	0.147020
$746$	23815.0	1.16881
$747$	0	0
$748$	5524.43	0.270044
$749$	−1441.61	−0.0703276
$750$	0	0
$751$	−14920.5	−0.724977	−0.362488	−	0.931988i	$-0.618073\pi$
$751$	−14920.5	−0.724977	−0.362488	+	0.931988i	$0.618073\pi$
$752$	17528.7	0.850009
$753$	0	0
$754$	−15827.9	−0.764478
$755$	1781.09	0.0858551
$756$	0	0
$757$	−17934.5	−0.861083	−0.430541	−	0.902571i	$-0.641677\pi$
$757$	−17934.5	−0.861083	−0.430541	+	0.902571i	$0.641677\pi$
$758$	−44592.9	−2.13679
$759$	0	0
$760$	−4312.92	−0.205850
$761$	11900.1	0.566858	0.283429	−	0.958993i	$-0.408528\pi$
$761$	11900.1	0.566858	0.283429	+	0.958993i	$0.408528\pi$
$762$	0	0
$763$	13850.5	0.657171
$764$	−42399.4	−2.00780
$765$	0	0
$766$	−12680.1	−0.598109
$767$	−1743.42	−0.0820747
$768$	0	0
$769$	−3012.80	−0.141280	−0.0706400	−	0.997502i	$-0.522504\pi$
$769$	−3012.80	−0.141280	−0.0706400	+	0.997502i	$0.522504\pi$
$770$	−980.317	−0.0458807
$771$	0	0
$772$	−31254.8	−1.45710
$773$	−24647.7	−1.14685	−0.573427	−	0.819257i	$-0.694386\pi$
$773$	−24647.7	−1.14685	−0.573427	+	0.819257i	$0.694386\pi$
$774$	0	0
$775$	6423.09	0.297709
$776$	−34170.6	−1.58074
$777$	0	0
$778$	−42551.6	−1.96086
$779$	9456.27	0.434924
$780$	0	0
$781$	−4049.11	−0.185517
$782$	3367.08	0.153973
$783$	0	0
$784$	−28947.2	−1.31866
$785$	80.3729	0.00365431
$786$	0	0
$787$	16431.2	0.744230	0.372115	−	0.928187i	$-0.378633\pi$
$787$	16431.2	0.744230	0.372115	+	0.928187i	$0.378633\pi$
$788$	−12814.7	−0.579319
$789$	0	0
$790$	2346.07	0.105658
$791$	13045.8	0.586415
$792$	0	0
$793$	17084.4	0.765050
$794$	22315.1	0.997398
$795$	0	0
$796$	−26861.1	−1.19606
$797$	2015.52	0.0895776	0.0447888	−	0.998996i	$-0.485739\pi$
$797$	2015.52	0.0895776	0.0447888	+	0.998996i	$0.485739\pi$
$798$	0	0
$799$	3911.13	0.173174
$800$	29734.4	1.31409
$801$	0	0
$802$	−11810.7	−0.520012
$803$	−10606.1	−0.466102
$804$	0	0
$805$	−417.440	−0.0182768
$806$	8006.57	0.349900
$807$	0	0
$808$	30106.0	1.31080
$809$	−5483.10	−0.238289	−0.119144	−	0.992877i	$-0.538015\pi$
$809$	−5483.10	−0.238289	−0.119144	+	0.992877i	$0.538015\pi$
$810$	0	0
$811$	−12029.9	−0.520870	−0.260435	−	0.965491i	$-0.583866\pi$
$811$	−12029.9	−0.520870	−0.260435	+	0.965491i	$0.583866\pi$
$812$	−21391.6	−0.924507
$813$	0	0
$814$	−288.415	−0.0124188
$815$	−5216.15	−0.224189
$816$	0	0
$817$	−22495.9	−0.963321
$818$	67150.9	2.87026
$819$	0	0
$820$	6319.73	0.269140
$821$	−13978.8	−0.594230	−0.297115	−	0.954842i	$-0.596025\pi$
$821$	−13978.8	−0.594230	−0.297115	+	0.954842i	$0.596025\pi$
$822$	0	0
$823$	−7223.95	−0.305967	−0.152984	−	0.988229i	$-0.548888\pi$
$823$	−7223.95	−0.305967	−0.152984	+	0.988229i	$0.548888\pi$
$824$	−21806.6	−0.921928
$825$	0	0
$826$	−3372.59	−0.142067
$827$	34864.4	1.46597	0.732983	−	0.680247i	$-0.238128\pi$
$827$	34864.4	1.46597	0.732983	+	0.680247i	$0.238128\pi$
$828$	0	0
$829$	−14737.1	−0.617420	−0.308710	−	0.951156i	$-0.599897\pi$
$829$	−14737.1	−0.617420	−0.308710	+	0.951156i	$0.599897\pi$
$830$	−1517.08	−0.0634443
$831$	0	0
$832$	5948.46	0.247867
$833$	−6458.91	−0.268653
$834$	0	0
$835$	6232.20	0.258293
$836$	−8827.65	−0.365204
$837$	0	0
$838$	−20196.4	−0.832544
$839$	33895.2	1.39475	0.697374	−	0.716708i	$-0.254352\pi$
$839$	33895.2	1.39475	0.697374	+	0.716708i	$0.254352\pi$
$840$	0	0
$841$	−13581.6	−0.556874
$842$	−25274.2	−1.03445
$843$	0	0
$844$	−92881.7	−3.78806
$845$	2238.64	0.0911380
$846$	0	0
$847$	13625.5	0.552746
$848$	63766.4	2.58225
$849$	0	0
$850$	18483.0	0.745836
$851$	−122.813	−0.00494710
$852$	0	0
$853$	−36174.1	−1.45202	−0.726012	−	0.687682i	$-0.758629\pi$
$853$	−36174.1	−1.45202	−0.726012	+	0.687682i	$0.758629\pi$
$854$	33049.1	1.32426
$855$	0	0
$856$	−7061.67	−0.281966
$857$	−20528.3	−0.818241	−0.409120	−	0.912480i	$-0.634164\pi$
$857$	−20528.3	−0.818241	−0.409120	+	0.912480i	$0.634164\pi$
$858$	0	0
$859$	1704.21	0.0676912	0.0338456	−	0.999427i	$-0.489225\pi$
$859$	1704.21	0.0676912	0.0338456	+	0.999427i	$0.489225\pi$
$860$	−15034.3	−0.596122
$861$	0	0
$862$	69702.6	2.75415
$863$	−39538.0	−1.55955	−0.779773	−	0.626063i	$-0.784665\pi$
$863$	−39538.0	−1.55955	−0.779773	+	0.626063i	$0.784665\pi$
$864$	0	0
$865$	852.612	0.0335141
$866$	−79237.8	−3.10925
$867$	0	0
$868$	10821.0	0.423145
$869$	2730.73	0.106598
$870$	0	0
$871$	1545.47	0.0601219
$872$	67846.0	2.63481
$873$	0	0
$874$	−5380.36	−0.208230
$875$	−4636.57	−0.179137
$876$	0	0
$877$	25216.9	0.970938	0.485469	−	0.874254i	$-0.338649\pi$
$877$	25216.9	0.970938	0.485469	+	0.874254i	$0.338649\pi$
$878$	−15095.2	−0.580226
$879$	0	0
$880$	−2257.38	−0.0864730
$881$	35122.3	1.34313	0.671567	−	0.740944i	$-0.265622\pi$
$881$	35122.3	1.34313	0.671567	+	0.740944i	$0.265622\pi$
$882$	0	0
$883$	20139.2	0.767539	0.383770	−	0.923429i	$-0.374626\pi$
$883$	20139.2	0.767539	0.383770	+	0.923429i	$0.374626\pi$
$884$	16096.7	0.612431
$885$	0	0
$886$	−30013.2	−1.13805
$887$	−35541.7	−1.34540	−0.672702	−	0.739914i	$-0.734866\pi$
$887$	−35541.7	−1.34540	−0.672702	+	0.739914i	$0.734866\pi$
$888$	0	0
$889$	22115.3	0.834336
$890$	−12027.4	−0.452987
$891$	0	0
$892$	23652.4	0.887826
$893$	−6249.72	−0.234198
$894$	0	0
$895$	4957.92	0.185168
$896$	−10099.7	−0.376571
$897$	0	0
$898$	−51513.7	−1.91429
$899$	−5466.95	−0.202818
$900$	0	0
$901$	14228.0	0.526086
$902$	10528.7	0.388655
$903$	0	0
$904$	63904.2	2.35113
$905$	−7330.74	−0.269262
$906$	0	0
$907$	−5257.49	−0.192472	−0.0962360	−	0.995359i	$-0.530680\pi$
$907$	−5257.49	−0.192472	−0.0962360	+	0.995359i	$0.530680\pi$
$908$	24153.1	0.882765
$909$	0	0
$910$	−2856.37	−0.104053
$911$	25638.3	0.932419	0.466210	−	0.884674i	$-0.345619\pi$
$911$	25638.3	0.932419	0.466210	+	0.884674i	$0.345619\pi$
$912$	0	0
$913$	−1765.82	−0.0640089
$914$	57004.6	2.06296
$915$	0	0
$916$	−62308.3	−2.24752
$917$	16855.6	0.607002
$918$	0	0
$919$	40415.4	1.45069	0.725344	−	0.688387i	$-0.241681\pi$
$919$	40415.4	1.45069	0.725344	+	0.688387i	$0.241681\pi$
$920$	−2044.81	−0.0732776
$921$	0	0
$922$	49758.1	1.77733
$923$	−11798.0	−0.420732
$924$	0	0
$925$	−674.162	−0.0239636
$926$	−13802.0	−0.489807
$927$	0	0
$928$	−25308.2	−0.895239
$929$	−22855.5	−0.807176	−0.403588	−	0.914941i	$-0.632237\pi$
$929$	−22855.5	−0.807176	−0.403588	+	0.914941i	$0.632237\pi$
$930$	0	0
$931$	10320.9	0.363323
$932$	48409.2	1.70139
$933$	0	0
$934$	−5904.70	−0.206860
$935$	−503.682	−0.0176173
$936$	0	0
$937$	−17288.6	−0.602770	−0.301385	−	0.953503i	$-0.597449\pi$
$937$	−17288.6	−0.602770	−0.301385	+	0.953503i	$0.597449\pi$
$938$	2989.65	0.104068
$939$	0	0
$940$	−4176.75	−0.144926
$941$	−31998.0	−1.10851	−0.554253	−	0.832348i	$-0.686996\pi$
$941$	−31998.0	−1.10851	−0.554253	+	0.832348i	$0.686996\pi$
$942$	0	0
$943$	4483.34	0.154822
$944$	−7766.06	−0.267758
$945$	0	0
$946$	−25047.1	−0.860838
$947$	13177.3	0.452170	0.226085	−	0.974108i	$-0.427407\pi$
$947$	13177.3	0.452170	0.226085	+	0.974108i	$0.427407\pi$
$948$	0	0
$949$	−30903.1	−1.05707
$950$	−29534.5	−1.00866
$951$	0	0
$952$	17707.6	0.602843
$953$	−35095.6	−1.19292	−0.596462	−	0.802641i	$-0.703427\pi$
$953$	−35095.6	−1.19292	−0.596462	+	0.802641i	$0.703427\pi$
$954$	0	0
$955$	3865.71	0.130986
$956$	78585.3	2.65861
$957$	0	0
$958$	93684.4	3.15951
$959$	17715.7	0.596527
$960$	0	0
$961$	−27025.5	−0.907171
$962$	−840.362	−0.0281646
$963$	0	0
$964$	−58691.2	−1.96091
$965$	2849.61	0.0950594
$966$	0	0
$967$	−556.001	−0.0184900	−0.00924498	−	0.999957i	$-0.502943\pi$
$967$	−556.001	−0.0184900	−0.00924498	+	0.999957i	$0.502943\pi$
$968$	66743.7	2.21614
$969$	0	0
$970$	5478.47	0.181343
$971$	46251.0	1.52859	0.764297	−	0.644865i	$-0.223086\pi$
$971$	46251.0	1.52859	0.764297	+	0.644865i	$0.223086\pi$
$972$	0	0
$973$	−7389.68	−0.243476
$974$	−16649.2	−0.547714
$975$	0	0
$976$	76102.3	2.49588
$977$	−29081.6	−0.952306	−0.476153	−	0.879362i	$-0.657969\pi$
$977$	−29081.6	−0.952306	−0.476153	+	0.879362i	$0.657969\pi$
$978$	0	0
$979$	−13999.4	−0.457019
$980$	6897.57	0.224831
$981$	0	0
$982$	33587.8	1.09148
$983$	43019.6	1.39584	0.697920	−	0.716175i	$-0.254109\pi$
$983$	43019.6	1.39584	0.697920	+	0.716175i	$0.254109\pi$
$984$	0	0
$985$	1168.36	0.0377940
$986$	−15731.6	−0.508110
$987$	0	0
$988$	−25721.3	−0.828244
$989$	−10665.6	−0.342919
$990$	0	0
$991$	−19398.1	−0.621798	−0.310899	−	0.950443i	$-0.600630\pi$
$991$	−19398.1	−0.621798	−0.310899	+	0.950443i	$0.600630\pi$
$992$	12802.2	0.409749
$993$	0	0
$994$	−22822.8	−0.728264
$995$	2449.03	0.0780295
$996$	0	0
$997$	48069.8	1.52697	0.763484	−	0.645827i	$-0.223487\pi$
$997$	48069.8	1.52697	0.763484	+	0.645827i	$0.223487\pi$
$998$	−75242.5	−2.38653
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.4.a.e.1.1		8
3.2	odd	2		177.4.a.d.1.8	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.a.d.1.8	✓	8	3.2	odd	2
531.4.a.e.1.1		8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	531.a (trivial)

\(f(q)\)	\(=\)	\(q-5.15242 q^{2} +18.5474 q^{4} -1.69104 q^{5} -11.0943 q^{7} -54.3449 q^{8} +O(q^{10})\)	\(q-5.15242 q^{2} +18.5474 q^{4} -1.69104 q^{5} -11.0943 q^{7} -54.3449 q^{8} +8.71294 q^{10} +10.1415 q^{11} +29.5495 q^{13} +57.1625 q^{14} +131.628 q^{16} +29.3698 q^{17} -46.9309 q^{19} -31.3645 q^{20} -52.2532 q^{22} -22.2505 q^{23} -122.140 q^{25} -152.252 q^{26} -205.771 q^{28} +103.959 q^{29} -52.5878 q^{31} -243.445 q^{32} -151.326 q^{34} +18.7609 q^{35} +5.51956 q^{37} +241.808 q^{38} +91.8993 q^{40} -201.493 q^{41} +479.341 q^{43} +188.099 q^{44} +114.644 q^{46} +133.168 q^{47} -219.917 q^{49} +629.319 q^{50} +548.068 q^{52} +484.443 q^{53} -17.1497 q^{55} +602.918 q^{56} -535.639 q^{58} -59.0000 q^{59} +578.161 q^{61} +270.954 q^{62} +201.305 q^{64} -49.9694 q^{65} +52.3009 q^{67} +544.735 q^{68} -96.6639 q^{70} -399.262 q^{71} -1045.81 q^{73} -28.4391 q^{74} -870.449 q^{76} -112.513 q^{77} +269.263 q^{79} -222.588 q^{80} +1038.18 q^{82} -174.118 q^{83} -49.6655 q^{85} -2469.77 q^{86} -551.138 q^{88} -1380.40 q^{89} -327.831 q^{91} -412.691 q^{92} -686.140 q^{94} +79.3620 q^{95} +628.773 q^{97} +1133.10 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{2} + 34 q^{4} - 42 q^{5} + 53 q^{7} - 51 q^{8}+O(q^{10}) \) 8 * q - 6 * q^2 + 34 * q^4 - 42 * q^5 + 53 * q^7 - 51 * q^8	\( 8 q - 6 q^{2} + 34 q^{4} - 42 q^{5} + 53 q^{7} - 51 q^{8} + 21 q^{10} - 67 q^{11} + 33 q^{13} - 79 q^{14} - 30 q^{16} - 139 q^{17} + 64 q^{19} - 117 q^{20} - 84 q^{22} - 226 q^{23} + 96 q^{25} - 24 q^{26} + 34 q^{28} - 456 q^{29} + 124 q^{31} - 174 q^{32} - 114 q^{34} - 556 q^{35} + 127 q^{37} - 237 q^{38} - 188 q^{40} - 425 q^{41} - 115 q^{43} - 510 q^{44} - 711 q^{46} - 420 q^{47} + 171 q^{49} + 137 q^{50} - 922 q^{52} - 98 q^{53} - 616 q^{55} + 412 q^{56} - 1548 q^{58} - 472 q^{59} - 1254 q^{61} + 766 q^{62} - 2019 q^{64} + 734 q^{65} - 1010 q^{67} + 503 q^{68} - 2956 q^{70} + 17 q^{71} - 1180 q^{73} + 1228 q^{74} - 2008 q^{76} - 441 q^{77} - 873 q^{79} + 865 q^{80} - 3645 q^{82} - 759 q^{83} - 850 q^{85} + 1226 q^{86} - 3047 q^{88} - 988 q^{89} - 2111 q^{91} + 1062 q^{92} - 2240 q^{94} - 1822 q^{95} - 668 q^{97} + 1368 q^{98}+O(q^{100}) \) 8 * q - 6 * q^2 + 34 * q^4 - 42 * q^5 + 53 * q^7 - 51 * q^8 + 21 * q^10 - 67 * q^11 + 33 * q^13 - 79 * q^14 - 30 * q^16 - 139 * q^17 + 64 * q^19 - 117 * q^20 - 84 * q^22 - 226 * q^23 + 96 * q^25 - 24 * q^26 + 34 * q^28 - 456 * q^29 + 124 * q^31 - 174 * q^32 - 114 * q^34 - 556 * q^35 + 127 * q^37 - 237 * q^38 - 188 * q^40 - 425 * q^41 - 115 * q^43 - 510 * q^44 - 711 * q^46 - 420 * q^47 + 171 * q^49 + 137 * q^50 - 922 * q^52 - 98 * q^53 - 616 * q^55 + 412 * q^56 - 1548 * q^58 - 472 * q^59 - 1254 * q^61 + 766 * q^62 - 2019 * q^64 + 734 * q^65 - 1010 * q^67 + 503 * q^68 - 2956 * q^70 + 17 * q^71 - 1180 * q^73 + 1228 * q^74 - 2008 * q^76 - 441 * q^77 - 873 * q^79 + 865 * q^80 - 3645 * q^82 - 759 * q^83 - 850 * q^85 + 1226 * q^86 - 3047 * q^88 - 988 * q^89 - 2111 * q^91 + 1062 * q^92 - 2240 * q^94 - 1822 * q^95 - 668 * q^97 + 1368 * q^98

Introduction

L-functions

Modular forms

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