LMFDB - Embedded newform 531.2.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,2,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, 2, names="a")

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

chi := DirichletCharacter("531.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
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:	$4.24005634733$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.254102$ 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)

$f(q)$	$=$	$q+0.254102 q^{2} -1.93543 q^{4} -1.68133 q^{5} +2.74590 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q+0.254102 q^{2} -1.93543 q^{4} -1.68133 q^{5} +2.74590 q^{7} -1.00000 q^{8} -0.427229 q^{10} -2.18953 q^{11} +4.18953 q^{13} +0.697737 q^{14} +3.61676 q^{16} +6.29809 q^{17} +4.93543 q^{19} +3.25410 q^{20} -0.556364 q^{22} -3.44364 q^{23} -2.17313 q^{25} +1.06457 q^{26} -5.31450 q^{28} +9.12497 q^{29} -0.616763 q^{31} +2.91903 q^{32} +1.60036 q^{34} -4.61676 q^{35} +1.44364 q^{37} +1.25410 q^{38} +1.68133 q^{40} -4.10856 q^{41} +8.53579 q^{43} +4.23769 q^{44} -0.875034 q^{46} +6.48763 q^{47} +0.539958 q^{49} -0.552195 q^{50} -8.10856 q^{52} +0.664924 q^{53} +3.68133 q^{55} -2.74590 q^{56} +2.31867 q^{58} -1.00000 q^{59} -10.9958 q^{61} -0.156721 q^{62} -6.49180 q^{64} -7.04399 q^{65} -9.93126 q^{67} -12.1895 q^{68} -1.17313 q^{70} -6.82687 q^{71} +13.5040 q^{73} +0.366830 q^{74} -9.55220 q^{76} -6.01224 q^{77} -1.17313 q^{79} -6.08097 q^{80} -1.04399 q^{82} +15.3421 q^{83} -10.5892 q^{85} +2.16896 q^{86} +2.18953 q^{88} -1.97942 q^{89} +11.5040 q^{91} +6.66492 q^{92} +1.64852 q^{94} -8.29809 q^{95} +3.33508 q^{97} +0.137204 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{4} + 2 q^{5} + 9 q^{7} - 3 q^{8}+O(q^{10}) $ 3 * q + 2 * q^4 + 2 * q^5 + 9 * q^7 - 3 * q^8	$ 3 q + 2 q^{4} + 2 q^{5} + 9 q^{7} - 3 q^{8} + 5 q^{10} + 2 q^{11} + 4 q^{13} - 8 q^{14} - 4 q^{16} - 3 q^{17} + 7 q^{19} + 9 q^{20} - 11 q^{22} - q^{23} - q^{25} + 11 q^{26} + 9 q^{28} + 11 q^{29} + 13 q^{31} + 4 q^{32} - 7 q^{34} + q^{35} - 5 q^{37} + 3 q^{38} - 2 q^{40} + q^{41} + 6 q^{43} + 15 q^{44} - 19 q^{46} - 11 q^{47} + 14 q^{49} + 21 q^{50} - 11 q^{52} - 2 q^{53} + 4 q^{55} - 9 q^{56} + 14 q^{58} - 3 q^{59} - q^{61} + 2 q^{62} - 21 q^{64} + 10 q^{67} - 28 q^{68} + 2 q^{70} - 26 q^{71} + 7 q^{73} + 19 q^{74} - 6 q^{76} + 17 q^{77} + 2 q^{79} - 23 q^{80} + 18 q^{82} + 3 q^{83} - 35 q^{85} - 31 q^{86} - 2 q^{88} + 23 q^{89} + q^{91} + 16 q^{92} + 4 q^{94} - 3 q^{95} + 14 q^{97} - 51 q^{98}+O(q^{100}) $ 3 * q + 2 * q^4 + 2 * q^5 + 9 * q^7 - 3 * q^8 + 5 * q^10 + 2 * q^11 + 4 * q^13 - 8 * q^14 - 4 * q^16 - 3 * q^17 + 7 * q^19 + 9 * q^20 - 11 * q^22 - q^23 - q^25 + 11 * q^26 + 9 * q^28 + 11 * q^29 + 13 * q^31 + 4 * q^32 - 7 * q^34 + q^35 - 5 * q^37 + 3 * q^38 - 2 * q^40 + q^41 + 6 * q^43 + 15 * q^44 - 19 * q^46 - 11 * q^47 + 14 * q^49 + 21 * q^50 - 11 * q^52 - 2 * q^53 + 4 * q^55 - 9 * q^56 + 14 * q^58 - 3 * q^59 - q^61 + 2 * q^62 - 21 * q^64 + 10 * q^67 - 28 * q^68 + 2 * q^70 - 26 * q^71 + 7 * q^73 + 19 * q^74 - 6 * q^76 + 17 * q^77 + 2 * q^79 - 23 * q^80 + 18 * q^82 + 3 * q^83 - 35 * q^85 - 31 * q^86 - 2 * q^88 + 23 * q^89 + q^91 + 16 * q^92 + 4 * q^94 - 3 * q^95 + 14 * q^97 - 51 * 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 $

$2$	0.254102	0.179677	0.0898385	−	0.995956i	$-0.471365\pi$
$2$	0.254102	0.179677	0.0898385	+	0.995956i	$0.471365\pi$
$3$	0	0
$3$	0	0
$4$	−1.93543	−0.967716
$5$	−1.68133	−0.751914	−0.375957	−	0.926637i	$-0.622686\pi$
$5$	−1.68133	−0.751914	−0.375957	+	0.926637i	$0.622686\pi$
$6$	0	0
$7$	2.74590	1.03785	0.518926	−	0.854819i	$-0.326332\pi$
$7$	2.74590	1.03785	0.518926	+	0.854819i	$0.326332\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−0.427229	−0.135102
$11$	−2.18953	−0.660169	−0.330085	−	0.943951i	$-0.607077\pi$
$11$	−2.18953	−0.660169	−0.330085	+	0.943951i	$0.607077\pi$
$12$	0	0
$13$	4.18953	1.16197	0.580984	−	0.813915i	$-0.302668\pi$
$13$	4.18953	1.16197	0.580984	+	0.813915i	$0.302668\pi$
$14$	0.697737	0.186478
$15$	0	0
$16$	3.61676	0.904191
$17$	6.29809	1.52751	0.763756	−	0.645505i	$-0.223353\pi$
$17$	6.29809	1.52751	0.763756	+	0.645505i	$0.223353\pi$
$18$	0	0
$19$	4.93543	1.13227	0.566133	−	0.824314i	$-0.308439\pi$
$19$	4.93543	1.13227	0.566133	+	0.824314i	$0.308439\pi$
$20$	3.25410	0.727639
$21$	0	0
$22$	−0.556364	−0.118617
$23$	−3.44364	−0.718048	−0.359024	−	0.933328i	$-0.616890\pi$
$23$	−3.44364	−0.718048	−0.359024	+	0.933328i	$0.616890\pi$
$24$	0	0
$25$	−2.17313	−0.434625
$26$	1.06457	0.208779
$27$	0	0
$28$	−5.31450	−1.00435
$29$	9.12497	1.69446	0.847232	−	0.531223i	$-0.178267\pi$
$29$	9.12497	1.69446	0.847232	+	0.531223i	$0.178267\pi$
$30$	0	0
$31$	−0.616763	−0.110774	−0.0553870	−	0.998465i	$-0.517639\pi$
$31$	−0.616763	−0.110774	−0.0553870	+	0.998465i	$0.517639\pi$
$32$	2.91903	0.516016
$33$	0	0
$34$	1.60036	0.274459
$35$	−4.61676	−0.780375
$36$	0	0
$37$	1.44364	0.237332	0.118666	−	0.992934i	$-0.462138\pi$
$37$	1.44364	0.237332	0.118666	+	0.992934i	$0.462138\pi$
$38$	1.25410	0.203442
$39$	0	0
$40$	1.68133	0.265842
$41$	−4.10856	−0.641649	−0.320825	−	0.947139i	$-0.603960\pi$
$41$	−4.10856	−0.641649	−0.320825	+	0.947139i	$0.603960\pi$
$42$	0	0
$43$	8.53579	1.30170	0.650848	−	0.759208i	$-0.274414\pi$
$43$	8.53579	1.30170	0.650848	+	0.759208i	$0.274414\pi$
$44$	4.23769	0.638857
$45$	0	0
$46$	−0.875034	−0.129017
$47$	6.48763	0.946318	0.473159	−	0.880977i	$-0.343114\pi$
$47$	6.48763	0.946318	0.473159	+	0.880977i	$0.343114\pi$
$48$	0	0
$49$	0.539958	0.0771368
$50$	−0.552195	−0.0780922
$51$	0	0
$52$	−8.10856	−1.12445
$53$	0.664924	0.0913343	0.0456672	−	0.998957i	$-0.485459\pi$
$53$	0.664924	0.0913343	0.0456672	+	0.998957i	$0.485459\pi$
$54$	0	0
$55$	3.68133	0.496391
$56$	−2.74590	−0.366936
$57$	0	0
$58$	2.31867	0.304456
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	−10.9958	−1.40787	−0.703936	−	0.710263i	$-0.748576\pi$
$61$	−10.9958	−1.40787	−0.703936	+	0.710263i	$0.748576\pi$
$62$	−0.156721	−0.0199035
$63$	0	0
$64$	−6.49180	−0.811475
$65$	−7.04399	−0.873700
$66$	0	0
$67$	−9.93126	−1.21330	−0.606648	−	0.794970i	$-0.707486\pi$
$67$	−9.93126	−1.21330	−0.606648	+	0.794970i	$0.707486\pi$
$68$	−12.1895	−1.47820
$69$	0	0
$70$	−1.17313	−0.140216
$71$	−6.82687	−0.810201	−0.405100	−	0.914272i	$-0.632763\pi$
$71$	−6.82687	−0.810201	−0.405100	+	0.914272i	$0.632763\pi$
$72$	0	0
$73$	13.5040	1.58053	0.790264	−	0.612767i	$-0.209943\pi$
$73$	13.5040	1.58053	0.790264	+	0.612767i	$0.209943\pi$
$74$	0.366830	0.0426432
$75$	0	0
$76$	−9.55220	−1.09571
$77$	−6.01224	−0.685158
$78$	0	0
$79$	−1.17313	−0.131987	−0.0659936	−	0.997820i	$-0.521022\pi$
$79$	−1.17313	−0.131987	−0.0659936	+	0.997820i	$0.521022\pi$
$80$	−6.08097	−0.679874
$81$	0	0
$82$	−1.04399	−0.115290
$83$	15.3421	1.68401	0.842006	−	0.539468i	$-0.181374\pi$
$83$	15.3421	1.68401	0.842006	+	0.539468i	$0.181374\pi$
$84$	0	0
$85$	−10.5892	−1.14856
$86$	2.16896	0.233885
$87$	0	0
$88$	2.18953	0.233405
$89$	−1.97942	−0.209819	−0.104909	−	0.994482i	$-0.533455\pi$
$89$	−1.97942	−0.209819	−0.104909	+	0.994482i	$0.533455\pi$
$90$	0	0
$91$	11.5040	1.20595
$92$	6.66492	0.694866
$93$	0	0
$94$	1.64852	0.170032
$95$	−8.29809	−0.851366
$96$	0	0
$97$	3.33508	0.338626	0.169313	−	0.985562i	$-0.445845\pi$
$97$	3.33508	0.338626	0.169313	+	0.985562i	$0.445845\pi$
$98$	0.137204	0.0138597
$99$	0	0
$100$	4.20594	0.420594
$101$	3.49180	0.347447	0.173723	−	0.984794i	$-0.444420\pi$
$101$	3.49180	0.347447	0.173723	+	0.984794i	$0.444420\pi$
$102$	0	0
$103$	8.21712	0.809657	0.404828	−	0.914393i	$-0.367331\pi$
$103$	8.21712	0.809657	0.404828	+	0.914393i	$0.367331\pi$
$104$	−4.18953	−0.410818
$105$	0	0
$106$	0.168958	0.0164107
$107$	−17.7899	−1.71981	−0.859907	−	0.510451i	$-0.829478\pi$
$107$	−17.7899	−1.71981	−0.859907	+	0.510451i	$0.829478\pi$
$108$	0	0
$109$	−18.2981	−1.75264	−0.876320	−	0.481730i	$-0.840009\pi$
$109$	−18.2981	−1.75264	−0.876320	+	0.481730i	$0.840009\pi$
$110$	0.935432	0.0891900
$111$	0	0
$112$	9.93126	0.938416
$113$	7.71414	0.725686	0.362843	−	0.931850i	$-0.381806\pi$
$113$	7.71414	0.725686	0.362843	+	0.931850i	$0.381806\pi$
$114$	0	0
$115$	5.78989	0.539910
$116$	−17.6608	−1.63976
$117$	0	0
$118$	−0.254102	−0.0233920
$119$	17.2939	1.58533
$120$	0	0
$121$	−6.20594	−0.564176
$122$	−2.79406	−0.252962
$123$	0	0
$124$	1.19370	0.107198
$125$	12.0604	1.07871
$126$	0	0
$127$	6.18953	0.549232	0.274616	−	0.961554i	$-0.411449\pi$
$127$	6.18953	0.549232	0.274616	+	0.961554i	$0.411449\pi$
$128$	−7.48763	−0.661819
$129$	0	0
$130$	−1.78989	−0.156984
$131$	15.5798	1.36121	0.680606	−	0.732650i	$-0.261717\pi$
$131$	15.5798	1.36121	0.680606	+	0.732650i	$0.261717\pi$
$132$	0	0
$133$	13.5522	1.17512
$134$	−2.52355	−0.218002
$135$	0	0
$136$	−6.29809	−0.540057
$137$	−0.664924	−0.0568083	−0.0284041	−	0.999597i	$-0.509043\pi$
$137$	−0.664924	−0.0568083	−0.0284041	+	0.999597i	$0.509043\pi$
$138$	0	0
$139$	−18.7857	−1.59338	−0.796692	−	0.604385i	$-0.793419\pi$
$139$	−18.7857	−1.59338	−0.796692	+	0.604385i	$0.793419\pi$
$140$	8.93543	0.755182
$141$	0	0
$142$	−1.73472	−0.145574
$143$	−9.17313	−0.767095
$144$	0	0
$145$	−15.3421	−1.27409
$146$	3.43140	0.283985
$147$	0	0
$148$	−2.79406	−0.229670
$149$	8.04816	0.659331	0.329666	−	0.944098i	$-0.393064\pi$
$149$	8.04816	0.659331	0.329666	+	0.944098i	$0.393064\pi$
$150$	0	0
$151$	−17.8503	−1.45264	−0.726318	−	0.687359i	$-0.758770\pi$
$151$	−17.8503	−1.45264	−0.726318	+	0.687359i	$0.758770\pi$
$152$	−4.93543	−0.400316
$153$	0	0
$154$	−1.52772	−0.123107
$155$	1.03698	0.0832924
$156$	0	0
$157$	10.4395	0.833160	0.416580	−	0.909099i	$-0.363229\pi$
$157$	10.4395	0.833160	0.416580	+	0.909099i	$0.363229\pi$
$158$	−0.298094	−0.0237151
$159$	0	0
$160$	−4.90785	−0.387999
$161$	−9.45587	−0.745227
$162$	0	0
$163$	−3.76231	−0.294686	−0.147343	−	0.989085i	$-0.547072\pi$
$163$	−3.76231	−0.294686	−0.147343	+	0.989085i	$0.547072\pi$
$164$	7.95184	0.620934
$165$	0	0
$166$	3.89845	0.302578
$167$	−18.1086	−1.40128	−0.700641	−	0.713514i	$-0.747103\pi$
$167$	−18.1086	−1.40128	−0.700641	+	0.713514i	$0.747103\pi$
$168$	0	0
$169$	4.55220	0.350169
$170$	−2.69073	−0.206369
$171$	0	0
$172$	−16.5204	−1.25967
$173$	9.66598	0.734891	0.367446	−	0.930045i	$-0.380232\pi$
$173$	9.66598	0.734891	0.367446	+	0.930045i	$0.380232\pi$
$174$	0	0
$175$	−5.96719	−0.451077
$176$	−7.91903	−0.596919
$177$	0	0
$178$	−0.502975	−0.0376996
$179$	8.95601	0.669403	0.334702	−	0.942324i	$-0.391364\pi$
$179$	8.95601	0.669403	0.334702	+	0.942324i	$0.391364\pi$
$180$	0	0
$181$	−0.487628	−0.0362451	−0.0181225	−	0.999836i	$-0.505769\pi$
$181$	−0.487628	−0.0362451	−0.0181225	+	0.999836i	$0.505769\pi$
$182$	2.92319	0.216682
$183$	0	0
$184$	3.44364	0.253868
$185$	−2.42723	−0.178453
$186$	0	0
$187$	−13.7899	−1.00842
$188$	−12.5564	−0.915767
$189$	0	0
$190$	−2.10856	−0.152971
$191$	−5.07681	−0.367345	−0.183672	−	0.982988i	$-0.558799\pi$
$191$	−5.07681	−0.367345	−0.183672	+	0.982988i	$0.558799\pi$
$192$	0	0
$193$	9.36266	0.673939	0.336970	−	0.941516i	$-0.390598\pi$
$193$	9.36266	0.673939	0.336970	+	0.941516i	$0.390598\pi$
$194$	0.847448	0.0608433
$195$	0	0
$196$	−1.04505	−0.0746465
$197$	16.2499	1.15776	0.578880	−	0.815413i	$-0.303490\pi$
$197$	16.2499	1.15776	0.578880	+	0.815413i	$0.303490\pi$
$198$	0	0
$199$	6.64958	0.471376	0.235688	−	0.971829i	$-0.424266\pi$
$199$	6.64958	0.471376	0.235688	+	0.971829i	$0.424266\pi$
$200$	2.17313	0.153663
$201$	0	0
$202$	0.887271	0.0624282
$203$	25.0562	1.75860
$204$	0	0
$205$	6.90785	0.482465
$206$	2.08798	0.145477
$207$	0	0
$208$	15.1526	1.05064
$209$	−10.8063	−0.747487
$210$	0	0
$211$	13.1526	0.905459	0.452729	−	0.891648i	$-0.350450\pi$
$211$	13.1526	0.905459	0.452729	+	0.891648i	$0.350450\pi$
$212$	−1.28692	−0.0883857
$213$	0	0
$214$	−4.52044	−0.309011
$215$	−14.3515	−0.978763
$216$	0	0
$217$	−1.69357	−0.114967
$218$	−4.64958	−0.314909
$219$	0	0
$220$	−7.12497	−0.480365
$221$	26.3861	1.77492
$222$	0	0
$223$	−9.70892	−0.650157	−0.325079	−	0.945687i	$-0.605391\pi$
$223$	−9.70892	−0.650157	−0.325079	+	0.945687i	$0.605391\pi$
$224$	8.01535	0.535548
$225$	0	0
$226$	1.96018	0.130389
$227$	−6.16896	−0.409448	−0.204724	−	0.978820i	$-0.565630\pi$
$227$	−6.16896	−0.409448	−0.204724	+	0.978820i	$0.565630\pi$
$228$	0	0
$229$	25.8175	1.70607	0.853033	−	0.521856i	$-0.174760\pi$
$229$	25.8175	1.70607	0.853033	+	0.521856i	$0.174760\pi$
$230$	1.47122	0.0970094
$231$	0	0
$232$	−9.12497	−0.599083
$233$	−12.2499	−0.802520	−0.401260	−	0.915964i	$-0.631428\pi$
$233$	−12.2499	−0.802520	−0.401260	+	0.915964i	$0.631428\pi$
$234$	0	0
$235$	−10.9078	−0.711549
$236$	1.93543	0.125986
$237$	0	0
$238$	4.39442	0.284848
$239$	22.9477	1.48436	0.742181	−	0.670200i	$-0.233792\pi$
$239$	22.9477	1.48436	0.742181	+	0.670200i	$0.233792\pi$
$240$	0	0
$241$	4.67015	0.300831	0.150415	−	0.988623i	$-0.451939\pi$
$241$	4.67015	0.300831	0.150415	+	0.988623i	$0.451939\pi$
$242$	−1.57694	−0.101370
$243$	0	0
$244$	21.2817	1.36242
$245$	−0.907847	−0.0580002
$246$	0	0
$247$	20.6772	1.31566
$248$	0.616763	0.0391645
$249$	0	0
$250$	3.06457	0.193820
$251$	−22.4067	−1.41430	−0.707148	−	0.707066i	$-0.750018\pi$
$251$	−22.4067	−1.41430	−0.707148	+	0.707066i	$0.750018\pi$
$252$	0	0
$253$	7.53996	0.474033
$254$	1.57277	0.0986844
$255$	0	0
$256$	11.0810	0.692561
$257$	−31.1372	−1.94229	−0.971143	−	0.238499i	$-0.923345\pi$
$257$	−31.1372	−1.94229	−0.971143	+	0.238499i	$0.923345\pi$
$258$	0	0
$259$	3.96408	0.246316
$260$	13.6332	0.845493
$261$	0	0
$262$	3.95885	0.244578
$263$	−24.9547	−1.53877	−0.769386	−	0.638784i	$-0.779438\pi$
$263$	−24.9547	−1.53877	−0.769386	+	0.638784i	$0.779438\pi$
$264$	0	0
$265$	−1.11796	−0.0686755
$266$	3.44364	0.211143
$267$	0	0
$268$	19.2213	1.17413
$269$	−5.99477	−0.365508	−0.182754	−	0.983159i	$-0.558501\pi$
$269$	−5.99477	−0.365508	−0.182754	+	0.983159i	$0.558501\pi$
$270$	0	0
$271$	−1.97241	−0.119816	−0.0599078	−	0.998204i	$-0.519081\pi$
$271$	−1.97241	−0.119816	−0.0599078	+	0.998204i	$0.519081\pi$
$272$	22.7787	1.38116
$273$	0	0
$274$	−0.168958	−0.0102071
$275$	4.75814	0.286926
$276$	0	0
$277$	−2.53579	−0.152361	−0.0761804	−	0.997094i	$-0.524272\pi$
$277$	−2.53579	−0.152361	−0.0761804	+	0.997094i	$0.524272\pi$
$278$	−4.77348	−0.286295
$279$	0	0
$280$	4.61676	0.275904
$281$	18.8545	1.12476	0.562381	−	0.826878i	$-0.309885\pi$
$281$	18.8545	1.12476	0.562381	+	0.826878i	$0.309885\pi$
$282$	0	0
$283$	−5.78989	−0.344173	−0.172087	−	0.985082i	$-0.555051\pi$
$283$	−5.78989	−0.344173	−0.172087	+	0.985082i	$0.555051\pi$
$284$	13.2130	0.784044
$285$	0	0
$286$	−2.33091	−0.137829
$287$	−11.2817	−0.665937
$288$	0	0
$289$	22.6660	1.33329
$290$	−3.89845	−0.228925
$291$	0	0
$292$	−26.1361	−1.52950
$293$	−7.73472	−0.451867	−0.225934	−	0.974143i	$-0.572543\pi$
$293$	−7.73472	−0.451867	−0.225934	+	0.974143i	$0.572543\pi$
$294$	0	0
$295$	1.68133	0.0978909
$296$	−1.44364	−0.0839096
$297$	0	0
$298$	2.04505	0.118467
$299$	−14.4272	−0.834348
$300$	0	0
$301$	23.4384	1.35097
$302$	−4.53579	−0.261005
$303$	0	0
$304$	17.8503	1.02378
$305$	18.4876	1.05860
$306$	0	0
$307$	16.4999	0.941697	0.470849	−	0.882214i	$-0.343948\pi$
$307$	16.4999	0.941697	0.470849	+	0.882214i	$0.343948\pi$
$308$	11.6363	0.663039
$309$	0	0
$310$	0.263499	0.0149657
$311$	−24.9354	−1.41396	−0.706979	−	0.707234i	$-0.749943\pi$
$311$	−24.9354	−1.41396	−0.706979	+	0.707234i	$0.749943\pi$
$312$	0	0
$313$	−24.1690	−1.36611	−0.683055	−	0.730367i	$-0.739349\pi$
$313$	−24.1690	−1.36611	−0.683055	+	0.730367i	$0.739349\pi$
$314$	2.65269	0.149700
$315$	0	0
$316$	2.27051	0.127726
$317$	18.1812	1.02116	0.510579	−	0.859831i	$-0.329431\pi$
$317$	18.1812	1.02116	0.510579	+	0.859831i	$0.329431\pi$
$318$	0	0
$319$	−19.9794	−1.11863
$320$	10.9149	0.610159
$321$	0	0
$322$	−2.40275	−0.133900
$323$	31.0838	1.72955
$324$	0	0
$325$	−9.10439	−0.505021
$326$	−0.956008	−0.0529484
$327$	0	0
$328$	4.10856	0.226857
$329$	17.8144	0.982138
$330$	0	0
$331$	−14.6290	−0.804083	−0.402041	−	0.915622i	$-0.631699\pi$
$331$	−14.6290	−0.804083	−0.402041	+	0.915622i	$0.631699\pi$
$332$	−29.6936	−1.62965
$333$	0	0
$334$	−4.60142	−0.251778
$335$	16.6977	0.912295
$336$	0	0
$337$	−19.1526	−1.04331	−0.521653	−	0.853158i	$-0.674684\pi$
$337$	−19.1526	−1.04331	−0.521653	+	0.853158i	$0.674684\pi$
$338$	1.15672	0.0629173
$339$	0	0
$340$	20.4946	1.11148
$341$	1.35042	0.0731295
$342$	0	0
$343$	−17.7386	−0.957795
$344$	−8.53579	−0.460219
$345$	0	0
$346$	2.45614	0.132043
$347$	−29.4301	−1.57989	−0.789944	−	0.613178i	$-0.789891\pi$
$347$	−29.4301	−1.57989	−0.789944	+	0.613178i	$0.789891\pi$
$348$	0	0
$349$	28.9065	1.54733	0.773665	−	0.633595i	$-0.218421\pi$
$349$	28.9065	1.54733	0.773665	+	0.633595i	$0.218421\pi$
$350$	−1.51627	−0.0810482
$351$	0	0
$352$	−6.39131	−0.340658
$353$	−0.0481609	−0.00256335	−0.00128167	−	0.999999i	$-0.500408\pi$
$353$	−0.0481609	−0.00256335	−0.00128167	+	0.999999i	$0.500408\pi$
$354$	0	0
$355$	11.4782	0.609201
$356$	3.83104	0.203045
$357$	0	0
$358$	2.27574	0.120276
$359$	−11.7417	−0.619705	−0.309852	−	0.950785i	$-0.600280\pi$
$359$	−11.7417	−0.619705	−0.309852	+	0.950785i	$0.600280\pi$
$360$	0	0
$361$	5.35849	0.282026
$362$	−0.123907	−0.00651241
$363$	0	0
$364$	−22.2653	−1.16702
$365$	−22.7047	−1.18842
$366$	0	0
$367$	−7.04399	−0.367693	−0.183847	−	0.982955i	$-0.558855\pi$
$367$	−7.04399	−0.367693	−0.183847	+	0.982955i	$0.558855\pi$
$368$	−12.4548	−0.649252
$369$	0	0
$370$	−0.616763	−0.0320640
$371$	1.82581	0.0947915
$372$	0	0
$373$	31.5316	1.63265	0.816323	−	0.577596i	$-0.196009\pi$
$373$	31.5316	1.63265	0.816323	+	0.577596i	$0.196009\pi$
$374$	−3.50403	−0.181189
$375$	0	0
$376$	−6.48763	−0.334574
$377$	38.2294	1.96891
$378$	0	0
$379$	−10.5030	−0.539502	−0.269751	−	0.962930i	$-0.586941\pi$
$379$	−10.5030	−0.539502	−0.269751	+	0.962930i	$0.586941\pi$
$380$	16.0604	0.823881
$381$	0	0
$382$	−1.29002	−0.0660034
$383$	−23.3023	−1.19069	−0.595345	−	0.803470i	$-0.702985\pi$
$383$	−23.3023	−1.19069	−0.595345	+	0.803470i	$0.702985\pi$
$384$	0	0
$385$	10.1086	0.515180
$386$	2.37907	0.121091
$387$	0	0
$388$	−6.45481	−0.327694
$389$	33.4835	1.69768	0.848839	−	0.528651i	$-0.177302\pi$
$389$	33.4835	1.69768	0.848839	+	0.528651i	$0.177302\pi$
$390$	0	0
$391$	−21.6883	−1.09683
$392$	−0.539958	−0.0272720
$393$	0	0
$394$	4.12914	0.208023
$395$	1.97241	0.0992430
$396$	0	0
$397$	−10.6977	−0.536904	−0.268452	−	0.963293i	$-0.586512\pi$
$397$	−10.6977	−0.536904	−0.268452	+	0.963293i	$0.586512\pi$
$398$	1.68967	0.0846954
$399$	0	0
$400$	−7.85969	−0.392984
$401$	23.3145	1.16427	0.582135	−	0.813092i	$-0.302217\pi$
$401$	23.3145	1.16427	0.582135	+	0.813092i	$0.302217\pi$
$402$	0	0
$403$	−2.58395	−0.128716
$404$	−6.75814	−0.336230
$405$	0	0
$406$	6.36683	0.315980
$407$	−3.16089	−0.156679
$408$	0	0
$409$	−8.65970	−0.428194	−0.214097	−	0.976812i	$-0.568681\pi$
$409$	−8.65970	−0.428194	−0.214097	+	0.976812i	$0.568681\pi$
$410$	1.75530	0.0866879
$411$	0	0
$412$	−15.9037	−0.783518
$413$	−2.74590	−0.135117
$414$	0	0
$415$	−25.7951	−1.26623
$416$	12.2294	0.599594
$417$	0	0
$418$	−2.74590	−0.134306
$419$	−4.63734	−0.226549	−0.113274	−	0.993564i	$-0.536134\pi$
$419$	−4.63734	−0.226549	−0.113274	+	0.993564i	$0.536134\pi$
$420$	0	0
$421$	1.68133	0.0819430	0.0409715	−	0.999160i	$-0.486955\pi$
$421$	1.68133	0.0819430	0.0409715	+	0.999160i	$0.486955\pi$
$422$	3.34209	0.162690
$423$	0	0
$424$	−0.664924	−0.0322916
$425$	−13.6866	−0.663896
$426$	0	0
$427$	−30.1934	−1.46116
$428$	34.4311	1.66429
$429$	0	0
$430$	−3.64674	−0.175861
$431$	−11.5040	−0.554130	−0.277065	−	0.960851i	$-0.589362\pi$
$431$	−11.5040	−0.554130	−0.277065	+	0.960851i	$0.589362\pi$
$432$	0	0
$433$	24.5069	1.17773	0.588863	−	0.808233i	$-0.299576\pi$
$433$	24.5069	1.17773	0.588863	+	0.808233i	$0.299576\pi$
$434$	−0.430339	−0.0206569
$435$	0	0
$436$	35.4147	1.69606
$437$	−16.9958	−0.813021
$438$	0	0
$439$	−5.63840	−0.269106	−0.134553	−	0.990906i	$-0.542960\pi$
$439$	−5.63840	−0.269106	−0.134553	+	0.990906i	$0.542960\pi$
$440$	−3.68133	−0.175501
$441$	0	0
$442$	6.70475	0.318912
$443$	38.0070	1.80577	0.902884	−	0.429885i	$-0.141446\pi$
$443$	38.0070	1.80577	0.902884	+	0.429885i	$0.141446\pi$
$444$	0	0
$445$	3.32807	0.157765
$446$	−2.46705	−0.116818
$447$	0	0
$448$	−17.8258	−0.842191
$449$	11.0357	0.520805	0.260402	−	0.965500i	$-0.416145\pi$
$449$	11.0357	0.520805	0.260402	+	0.965500i	$0.416145\pi$
$450$	0	0
$451$	8.99583	0.423597
$452$	−14.9302	−0.702258
$453$	0	0
$454$	−1.56754	−0.0735684
$455$	−19.3421	−0.906771
$456$	0	0
$457$	−32.3913	−1.51520	−0.757601	−	0.652718i	$-0.773629\pi$
$457$	−32.3913	−1.51520	−0.757601	+	0.652718i	$0.773629\pi$
$458$	6.56026	0.306541
$459$	0	0
$460$	−11.2059	−0.522480
$461$	−34.9599	−1.62825	−0.814123	−	0.580693i	$-0.802782\pi$
$461$	−34.9599	−1.62825	−0.814123	+	0.580693i	$0.802782\pi$
$462$	0	0
$463$	−22.8063	−1.05990	−0.529949	−	0.848029i	$-0.677789\pi$
$463$	−22.8063	−1.05990	−0.529949	+	0.848029i	$0.677789\pi$
$464$	33.0028	1.53212
$465$	0	0
$466$	−3.11273	−0.144194
$467$	36.2416	1.67706	0.838530	−	0.544855i	$-0.183415\pi$
$467$	36.2416	1.67706	0.838530	+	0.544855i	$0.183415\pi$
$468$	0	0
$469$	−27.2702	−1.25922
$470$	−2.77170	−0.127849
$471$	0	0
$472$	1.00000	0.0460287
$473$	−18.6894	−0.859340
$474$	0	0
$475$	−10.7253	−0.492112
$476$	−33.4712	−1.53415
$477$	0	0
$478$	5.83104	0.266706
$479$	−17.0562	−0.779319	−0.389660	−	0.920959i	$-0.627407\pi$
$479$	−17.0562	−0.779319	−0.389660	+	0.920959i	$0.627407\pi$
$480$	0	0
$481$	6.04816	0.275772
$482$	1.18669	0.0540524
$483$	0	0
$484$	12.0112	0.545963
$485$	−5.60737	−0.254617
$486$	0	0
$487$	15.3473	0.695453	0.347727	−	0.937596i	$-0.386954\pi$
$487$	15.3473	0.695453	0.347727	+	0.937596i	$0.386954\pi$
$488$	10.9958	0.497758
$489$	0	0
$490$	−0.230685	−0.0104213
$491$	−18.0122	−0.812881	−0.406440	−	0.913677i	$-0.633230\pi$
$491$	−18.0122	−0.812881	−0.406440	+	0.913677i	$0.633230\pi$
$492$	0	0
$493$	57.4699	2.58831
$494$	5.25410	0.236393
$495$	0	0
$496$	−2.23069	−0.100161
$497$	−18.7459	−0.840868
$498$	0	0
$499$	−21.6074	−0.967279	−0.483639	−	0.875267i	$-0.660685\pi$
$499$	−21.6074	−0.967279	−0.483639	+	0.875267i	$0.660685\pi$
$500$	−23.3421	−1.04389
$501$	0	0
$502$	−5.69357	−0.254116
$503$	−7.76231	−0.346104	−0.173052	−	0.984913i	$-0.555363\pi$
$503$	−7.76231	−0.346104	−0.173052	+	0.984913i	$0.555363\pi$
$504$	0	0
$505$	−5.87086	−0.261250
$506$	1.91592	0.0851729
$507$	0	0
$508$	−11.9794	−0.531501
$509$	−40.7704	−1.80712	−0.903558	−	0.428467i	$-0.859054\pi$
$509$	−40.7704	−1.80712	−0.903558	+	0.428467i	$0.859054\pi$
$510$	0	0
$511$	37.0807	1.64035
$512$	17.7909	0.786256
$513$	0	0
$514$	−7.91202	−0.348984
$515$	−13.8157	−0.608792
$516$	0	0
$517$	−14.2049	−0.624730
$518$	1.00728	0.0442573
$519$	0	0
$520$	7.04399	0.308899
$521$	−41.0427	−1.79811	−0.899056	−	0.437834i	$-0.855746\pi$
$521$	−41.0427	−1.79811	−0.899056	+	0.437834i	$0.855746\pi$
$522$	0	0
$523$	−22.5686	−0.986856	−0.493428	−	0.869787i	$-0.664256\pi$
$523$	−22.5686	−0.986856	−0.493428	+	0.869787i	$0.664256\pi$
$524$	−30.1536	−1.31727
$525$	0	0
$526$	−6.34103	−0.276482
$527$	−3.88443	−0.169208
$528$	0	0
$529$	−11.1414	−0.484408
$530$	−0.284075	−0.0123394
$531$	0	0
$532$	−26.2294	−1.13719
$533$	−17.2130	−0.745576
$534$	0	0
$535$	29.9107	1.29315
$536$	9.93126	0.428965
$537$	0	0
$538$	−1.52328	−0.0656733
$539$	−1.18226	−0.0509233
$540$	0	0
$541$	35.0357	1.50630	0.753150	−	0.657849i	$-0.228533\pi$
$541$	35.0357	1.50630	0.753150	+	0.657849i	$0.228533\pi$
$542$	−0.501194	−0.0215281
$543$	0	0
$544$	18.3843	0.788220
$545$	30.7651	1.31783
$546$	0	0
$547$	3.70191	0.158282	0.0791410	−	0.996863i	$-0.474782\pi$
$547$	3.70191	0.158282	0.0791410	+	0.996863i	$0.474782\pi$
$548$	1.28692	0.0549743
$549$	0	0
$550$	1.20905	0.0515541
$551$	45.0357	1.91858
$552$	0	0
$553$	−3.22129	−0.136983
$554$	−0.644348	−0.0273757
$555$	0	0
$556$	36.3585	1.54194
$557$	16.7253	0.708675	0.354337	−	0.935118i	$-0.384706\pi$
$557$	16.7253	0.708675	0.354337	+	0.935118i	$0.384706\pi$
$558$	0	0
$559$	35.7610	1.51253
$560$	−16.6977	−0.705608
$561$	0	0
$562$	4.79095	0.202094
$563$	−24.9149	−1.05004	−0.525018	−	0.851091i	$-0.675941\pi$
$563$	−24.9149	−1.05004	−0.525018	+	0.851091i	$0.675941\pi$
$564$	0	0
$565$	−12.9700	−0.545653
$566$	−1.47122	−0.0618400
$567$	0	0
$568$	6.82687	0.286449
$569$	−42.4548	−1.77980	−0.889899	−	0.456157i	$-0.849225\pi$
$569$	−42.4548	−1.77980	−0.889899	+	0.456157i	$0.849225\pi$
$570$	0	0
$571$	−32.6977	−1.36836	−0.684179	−	0.729314i	$-0.739839\pi$
$571$	−32.6977	−1.36836	−0.684179	+	0.729314i	$0.739839\pi$
$572$	17.7540	0.742331
$573$	0	0
$574$	−2.86670	−0.119654
$575$	7.48346	0.312082
$576$	0	0
$577$	−13.2716	−0.552503	−0.276251	−	0.961085i	$-0.589092\pi$
$577$	−13.2716	−0.552503	−0.276251	+	0.961085i	$0.589092\pi$
$578$	5.75946	0.239562
$579$	0	0
$580$	29.6936	1.23296
$581$	42.1278	1.74776
$582$	0	0
$583$	−1.45587	−0.0602961
$584$	−13.5040	−0.558801
$585$	0	0
$586$	−1.96541	−0.0811901
$587$	14.4067	0.594626	0.297313	−	0.954780i	$-0.403909\pi$
$587$	14.4067	0.594626	0.297313	+	0.954780i	$0.403909\pi$
$588$	0	0
$589$	−3.04399	−0.125426
$590$	0.427229	0.0175887
$591$	0	0
$592$	5.22129	0.214594
$593$	1.85863	0.0763247	0.0381623	−	0.999272i	$-0.487850\pi$
$593$	1.85863	0.0763247	0.0381623	+	0.999272i	$0.487850\pi$
$594$	0	0
$595$	−29.0768	−1.19203
$596$	−15.5767	−0.638045
$597$	0	0
$598$	−3.66598	−0.149913
$599$	20.2171	0.826049	0.413025	−	0.910720i	$-0.364472\pi$
$599$	20.2171	0.826049	0.413025	+	0.910720i	$0.364472\pi$
$600$	0	0
$601$	−29.6608	−1.20989	−0.604944	−	0.796268i	$-0.706804\pi$
$601$	−29.6608	−1.20989	−0.604944	+	0.796268i	$0.706804\pi$
$602$	5.95574	0.242738
$603$	0	0
$604$	34.5480	1.40574
$605$	10.4342	0.424212
$606$	0	0
$607$	8.06040	0.327161	0.163581	−	0.986530i	$-0.447696\pi$
$607$	8.06040	0.327161	0.163581	+	0.986530i	$0.447696\pi$
$608$	14.4067	0.584267
$609$	0	0
$610$	4.69774	0.190206
$611$	27.1801	1.09959
$612$	0	0
$613$	−35.7886	−1.44549	−0.722743	−	0.691117i	$-0.757119\pi$
$613$	−35.7886	−1.44549	−0.722743	+	0.691117i	$0.757119\pi$
$614$	4.19264	0.169201
$615$	0	0
$616$	6.01224	0.242240
$617$	−21.7159	−0.874250	−0.437125	−	0.899401i	$-0.644003\pi$
$617$	−21.7159	−0.874250	−0.437125	+	0.899401i	$0.644003\pi$
$618$	0	0
$619$	−15.3679	−0.617688	−0.308844	−	0.951113i	$-0.599942\pi$
$619$	−15.3679	−0.617688	−0.308844	+	0.951113i	$0.599942\pi$
$620$	−2.00701	−0.0806034
$621$	0	0
$622$	−6.33614	−0.254056
$623$	−5.43530	−0.217761
$624$	0	0
$625$	−9.41188	−0.376475
$626$	−6.14137	−0.245459
$627$	0	0
$628$	−20.2049	−0.806263
$629$	9.09215	0.362528
$630$	0	0
$631$	−24.5358	−0.976754	−0.488377	−	0.872633i	$-0.662411\pi$
$631$	−24.5358	−0.976754	−0.488377	+	0.872633i	$0.662411\pi$
$632$	1.17313	0.0466645
$633$	0	0
$634$	4.61987	0.183479
$635$	−10.4067	−0.412975
$636$	0	0
$637$	2.26217	0.0896305
$638$	−5.07681	−0.200993
$639$	0	0
$640$	12.5892	0.497631
$641$	16.0521	0.634018	0.317009	−	0.948422i	$-0.397321\pi$
$641$	16.0521	0.634018	0.317009	+	0.948422i	$0.397321\pi$
$642$	0	0
$643$	−9.47645	−0.373715	−0.186857	−	0.982387i	$-0.559830\pi$
$643$	−9.47645	−0.373715	−0.186857	+	0.982387i	$0.559830\pi$
$644$	18.3012	0.721168
$645$	0	0
$646$	7.89845	0.310760
$647$	−41.1924	−1.61944	−0.809720	−	0.586817i	$-0.800381\pi$
$647$	−41.1924	−1.61944	−0.809720	+	0.586817i	$0.800381\pi$
$648$	0	0
$649$	2.18953	0.0859467
$650$	−2.31344	−0.0907406
$651$	0	0
$652$	7.28169	0.285173
$653$	−14.0757	−0.550827	−0.275413	−	0.961326i	$-0.588815\pi$
$653$	−14.0757	−0.550827	−0.275413	+	0.961326i	$0.588815\pi$
$654$	0	0
$655$	−26.1948	−1.02351
$656$	−14.8597	−0.580173
$657$	0	0
$658$	4.52666	0.176468
$659$	20.6220	0.803319	0.401659	−	0.915789i	$-0.368434\pi$
$659$	20.6220	0.803319	0.401659	+	0.915789i	$0.368434\pi$
$660$	0	0
$661$	−30.1414	−1.17236	−0.586182	−	0.810180i	$-0.699370\pi$
$661$	−30.1414	−1.17236	−0.586182	+	0.810180i	$0.699370\pi$
$662$	−3.71725	−0.144475
$663$	0	0
$664$	−15.3421	−0.595388
$665$	−22.7857	−0.883592
$666$	0	0
$667$	−31.4231	−1.21671
$668$	35.0479	1.35604
$669$	0	0
$670$	4.24292	0.163918
$671$	24.0757	0.929434
$672$	0	0
$673$	3.27468	0.126230	0.0631148	−	0.998006i	$-0.479897\pi$
$673$	3.27468	0.126230	0.0631148	+	0.998006i	$0.479897\pi$
$674$	−4.86670	−0.187458
$675$	0	0
$676$	−8.81047	−0.338864
$677$	−19.6126	−0.753773	−0.376887	−	0.926259i	$-0.623005\pi$
$677$	−19.6126	−0.753773	−0.376887	+	0.926259i	$0.623005\pi$
$678$	0	0
$679$	9.15778	0.351443
$680$	10.5892	0.406076
$681$	0	0
$682$	0.343145	0.0131397
$683$	−12.1812	−0.466101	−0.233050	−	0.972465i	$-0.574871\pi$
$683$	−12.1812	−0.466101	−0.233050	+	0.972465i	$0.574871\pi$
$684$	0	0
$685$	1.11796	0.0427149
$686$	−4.50741	−0.172094
$687$	0	0
$688$	30.8719	1.17698
$689$	2.78572	0.106128
$690$	0	0
$691$	−13.8052	−0.525176	−0.262588	−	0.964908i	$-0.584576\pi$
$691$	−13.8052	−0.525176	−0.262588	+	0.964908i	$0.584576\pi$
$692$	−18.7079	−0.711166
$693$	0	0
$694$	−7.47823	−0.283870
$695$	31.5850	1.19809
$696$	0	0
$697$	−25.8761	−0.980127
$698$	7.34520	0.278020
$699$	0	0
$700$	11.5491	0.436514
$701$	9.06980	0.342561	0.171281	−	0.985222i	$-0.445209\pi$
$701$	9.06980	0.342561	0.171281	+	0.985222i	$0.445209\pi$
$702$	0	0
$703$	7.12497	0.268723
$704$	14.2140	0.535711
$705$	0	0
$706$	−0.0122378	−0.000460575 0
$707$	9.58812	0.360598
$708$	0	0
$709$	26.2171	0.984605	0.492302	−	0.870424i	$-0.336155\pi$
$709$	26.2171	0.984605	0.492302	+	0.870424i	$0.336155\pi$
$710$	2.91664	0.109459
$711$	0	0
$712$	1.97942	0.0741821
$713$	2.12391	0.0795409
$714$	0	0
$715$	15.4231	0.576790
$716$	−17.3337	−0.647793
$717$	0	0
$718$	−2.98359	−0.111347
$719$	37.5972	1.40214	0.701070	−	0.713092i	$-0.252706\pi$
$719$	37.5972	1.40214	0.701070	+	0.713092i	$0.252706\pi$
$720$	0	0
$721$	22.5634	0.840304
$722$	1.36160	0.0506736
$723$	0	0
$724$	0.943770	0.0350749
$725$	−19.8297	−0.736457
$726$	0	0
$727$	42.3463	1.57054	0.785268	−	0.619156i	$-0.212525\pi$
$727$	42.3463	1.57054	0.785268	+	0.619156i	$0.212525\pi$
$728$	−11.5040	−0.426368
$729$	0	0
$730$	−5.76931	−0.213532
$731$	53.7592	1.98836
$732$	0	0
$733$	−5.96408	−0.220288	−0.110144	−	0.993916i	$-0.535131\pi$
$733$	−5.96408	−0.220288	−0.110144	+	0.993916i	$0.535131\pi$
$734$	−1.78989	−0.0660661
$735$	0	0
$736$	−10.0521	−0.370524
$737$	21.7448	0.800981
$738$	0	0
$739$	39.9435	1.46935	0.734673	−	0.678422i	$-0.237336\pi$
$739$	39.9435	1.46935	0.734673	+	0.678422i	$0.237336\pi$
$740$	4.69774	0.172692
$741$	0	0
$742$	0.463942	0.0170319
$743$	−22.7407	−0.834274	−0.417137	−	0.908844i	$-0.636967\pi$
$743$	−22.7407	−0.834274	−0.417137	+	0.908844i	$0.636967\pi$
$744$	0	0
$745$	−13.5316	−0.495760
$746$	8.01224	0.293349
$747$	0	0
$748$	26.6894	0.975861
$749$	−48.8492	−1.78491
$750$	0	0
$751$	17.8052	0.649722	0.324861	−	0.945762i	$-0.394682\pi$
$751$	17.8052	0.649722	0.324861	+	0.945762i	$0.394682\pi$
$752$	23.4642	0.855652
$753$	0	0
$754$	9.71414	0.353768
$755$	30.0122	1.09226
$756$	0	0
$757$	27.4353	0.997153	0.498576	−	0.866846i	$-0.333856\pi$
$757$	27.4353	0.997153	0.498576	+	0.866846i	$0.333856\pi$
$758$	−2.66882	−0.0969360
$759$	0	0
$760$	8.29809	0.301003
$761$	−10.1812	−0.369068	−0.184534	−	0.982826i	$-0.559078\pi$
$761$	−10.1812	−0.369068	−0.184534	+	0.982826i	$0.559078\pi$
$762$	0	0
$763$	−50.2447	−1.81898
$764$	9.82581	0.355485
$765$	0	0
$766$	−5.92114	−0.213940
$767$	−4.18953	−0.151275
$768$	0	0
$769$	38.0398	1.37175	0.685876	−	0.727719i	$-0.259419\pi$
$769$	38.0398	1.37175	0.685876	+	0.727719i	$0.259419\pi$
$770$	2.56860	0.0925660
$771$	0	0
$772$	−18.1208	−0.652182
$773$	44.1676	1.58860	0.794300	−	0.607526i	$-0.207838\pi$
$773$	44.1676	1.58860	0.794300	+	0.607526i	$0.207838\pi$
$774$	0	0
$775$	1.34030	0.0481452
$776$	−3.33508	−0.119722
$777$	0	0
$778$	8.50820	0.305034
$779$	−20.2775	−0.726517
$780$	0	0
$781$	14.9477	0.534870
$782$	−5.51104	−0.197075
$783$	0	0
$784$	1.95290	0.0697464
$785$	−17.5522	−0.626465
$786$	0	0
$787$	−16.3379	−0.582384	−0.291192	−	0.956665i	$-0.594052\pi$
$787$	−16.3379	−0.582384	−0.291192	+	0.956665i	$0.594052\pi$
$788$	−31.4506	−1.12038
$789$	0	0
$790$	0.501194	0.0178317
$791$	21.1823	0.753154
$792$	0	0
$793$	−46.0674	−1.63590
$794$	−2.71831	−0.0964693
$795$	0	0
$796$	−12.8698	−0.456158
$797$	32.9424	1.16688	0.583441	−	0.812156i	$-0.301706\pi$
$797$	32.9424	1.16688	0.583441	+	0.812156i	$0.301706\pi$
$798$	0	0
$799$	40.8597	1.44551
$800$	−6.34341	−0.224274
$801$	0	0
$802$	5.92425	0.209193
$803$	−29.5675	−1.04342
$804$	0	0
$805$	15.8984	0.560347
$806$	−0.656586	−0.0231273
$807$	0	0
$808$	−3.49180	−0.122841
$809$	51.9659	1.82702	0.913511	−	0.406814i	$-0.133360\pi$
$809$	51.9659	1.82702	0.913511	+	0.406814i	$0.133360\pi$
$810$	0	0
$811$	26.9424	0.946077	0.473039	−	0.881042i	$-0.343157\pi$
$811$	26.9424	0.946077	0.473039	+	0.881042i	$0.343157\pi$
$812$	−48.4946	−1.70183
$813$	0	0
$814$	−0.803187	−0.0281517
$815$	6.32568	0.221579
$816$	0	0
$817$	42.1278	1.47387
$818$	−2.20044	−0.0769367
$819$	0	0
$820$	−13.3697	−0.466889
$821$	27.8914	0.973418	0.486709	−	0.873564i	$-0.338197\pi$
$821$	27.8914	0.973418	0.486709	+	0.873564i	$0.338197\pi$
$822$	0	0
$823$	18.3103	0.638258	0.319129	−	0.947711i	$-0.396610\pi$
$823$	18.3103	0.638258	0.319129	+	0.947711i	$0.396610\pi$
$824$	−8.21712	−0.286257
$825$	0	0
$826$	−0.697737	−0.0242774
$827$	4.37384	0.152093	0.0760467	−	0.997104i	$-0.475770\pi$
$827$	4.37384	0.152093	0.0760467	+	0.997104i	$0.475770\pi$
$828$	0	0
$829$	30.5204	1.06002	0.530009	−	0.847992i	$-0.322188\pi$
$829$	30.5204	1.06002	0.530009	+	0.847992i	$0.322188\pi$
$830$	−6.55458	−0.227513
$831$	0	0
$832$	−27.1976	−0.942907
$833$	3.40070	0.117827
$834$	0	0
$835$	30.4465	1.05364
$836$	20.9149	0.723355
$837$	0	0
$838$	−1.17836	−0.0407056
$839$	0.859686	0.0296797	0.0148398	−	0.999890i	$-0.495276\pi$
$839$	0.859686	0.0296797	0.0148398	+	0.999890i	$0.495276\pi$
$840$	0	0
$841$	54.2650	1.87121
$842$	0.427229	0.0147233
$843$	0	0
$844$	−25.4559	−0.876227
$845$	−7.65375	−0.263297
$846$	0	0
$847$	−17.0409	−0.585532
$848$	2.40487	0.0825836
$849$	0	0
$850$	−3.47778	−0.119287
$851$	−4.97136	−0.170416
$852$	0	0
$853$	50.4741	1.72820	0.864099	−	0.503321i	$-0.167889\pi$
$853$	50.4741	1.72820	0.864099	+	0.503321i	$0.167889\pi$
$854$	−7.67220	−0.262537
$855$	0	0
$856$	17.7899	0.608046
$857$	−38.4946	−1.31495	−0.657476	−	0.753476i	$-0.728376\pi$
$857$	−38.4946	−1.31495	−0.657476	+	0.753476i	$0.728376\pi$
$858$	0	0
$859$	−27.1044	−0.924790	−0.462395	−	0.886674i	$-0.653010\pi$
$859$	−27.1044	−0.924790	−0.462395	+	0.886674i	$0.653010\pi$
$860$	27.7763	0.947165
$861$	0	0
$862$	−2.92319	−0.0995644
$863$	−10.6443	−0.362338	−0.181169	−	0.983452i	$-0.557988\pi$
$863$	−10.6443	−0.362338	−0.181169	+	0.983452i	$0.557988\pi$
$864$	0	0
$865$	−16.2517	−0.552575
$866$	6.22724	0.211610
$867$	0	0
$868$	3.27779	0.111255
$869$	2.56860	0.0871339
$870$	0	0
$871$	−41.6074	−1.40981
$872$	18.2981	0.619652
$873$	0	0
$874$	−4.31867	−0.146081
$875$	33.1166	1.11955
$876$	0	0
$877$	40.0328	1.35181	0.675906	−	0.736988i	$-0.263753\pi$
$877$	40.0328	1.35181	0.675906	+	0.736988i	$0.263753\pi$
$878$	−1.43273	−0.0483522
$879$	0	0
$880$	13.3145	0.448832
$881$	−29.9588	−1.00934	−0.504670	−	0.863313i	$-0.668386\pi$
$881$	−29.9588	−1.00934	−0.504670	+	0.863313i	$0.668386\pi$
$882$	0	0
$883$	−12.3239	−0.414732	−0.207366	−	0.978263i	$-0.566489\pi$
$883$	−12.3239	−0.414732	−0.207366	+	0.978263i	$0.566489\pi$
$884$	−51.0685	−1.71762
$885$	0	0
$886$	9.65765	0.324455
$887$	−25.4559	−0.854725	−0.427362	−	0.904080i	$-0.640557\pi$
$887$	−25.4559	−0.854725	−0.427362	+	0.904080i	$0.640557\pi$
$888$	0	0
$889$	16.9958	0.570022
$890$	0.845667	0.0283468
$891$	0	0
$892$	18.7909	0.629168
$893$	32.0192	1.07148
$894$	0	0
$895$	−15.0580	−0.503334
$896$	−20.5603	−0.686870
$897$	0	0
$898$	2.80418	0.0935766
$899$	−5.62794	−0.187702
$900$	0	0
$901$	4.18775	0.139514
$902$	2.28586	0.0761107
$903$	0	0
$904$	−7.71414	−0.256569
$905$	0.819863	0.0272532
$906$	0	0
$907$	38.4863	1.27792	0.638958	−	0.769241i	$-0.279366\pi$
$907$	38.4863	1.27792	0.638958	+	0.769241i	$0.279366\pi$
$908$	11.9396	0.396230
$909$	0	0
$910$	−4.91486	−0.162926
$911$	−21.2234	−0.703163	−0.351581	−	0.936157i	$-0.614356\pi$
$911$	−21.2234	−0.703163	−0.351581	+	0.936157i	$0.614356\pi$
$912$	0	0
$913$	−33.5920	−1.11173
$914$	−8.23069	−0.272247
$915$	0	0
$916$	−49.9680	−1.65099
$917$	42.7805	1.41274
$918$	0	0
$919$	46.3051	1.52746	0.763732	−	0.645533i	$-0.223365\pi$
$919$	46.3051	1.52746	0.763732	+	0.645533i	$0.223365\pi$
$920$	−5.78989	−0.190887
$921$	0	0
$922$	−8.88337	−0.292558
$923$	−28.6014	−0.941427
$924$	0	0
$925$	−3.13720	−0.103151
$926$	−5.79512	−0.190439
$927$	0	0
$928$	26.6360	0.874370
$929$	−31.9313	−1.04763	−0.523815	−	0.851832i	$-0.675492\pi$
$929$	−31.9313	−1.04763	−0.523815	+	0.851832i	$0.675492\pi$
$930$	0	0
$931$	2.66492	0.0873394
$932$	23.7089	0.776611
$933$	0	0
$934$	9.20905	0.301329
$935$	23.1854	0.758243
$936$	0	0
$937$	−33.6813	−1.10032	−0.550161	−	0.835059i	$-0.685433\pi$
$937$	−33.6813	−1.10032	−0.550161	+	0.835059i	$0.685433\pi$
$938$	−6.92941	−0.226253
$939$	0	0
$940$	21.1114	0.688578
$941$	22.6977	0.739925	0.369963	−	0.929047i	$-0.379371\pi$
$941$	22.6977	0.739925	0.369963	+	0.929047i	$0.379371\pi$
$942$	0	0
$943$	14.1484	0.460735
$944$	−3.61676	−0.117716
$945$	0	0
$946$	−4.74901	−0.154404
$947$	−22.5533	−0.732882	−0.366441	−	0.930441i	$-0.619424\pi$
$947$	−22.5533	−0.732882	−0.366441	+	0.930441i	$0.619424\pi$
$948$	0	0
$949$	56.5756	1.83652
$950$	−2.72532	−0.0884211
$951$	0	0
$952$	−17.2939	−0.560499
$953$	10.3791	0.336211	0.168105	−	0.985769i	$-0.446235\pi$
$953$	10.3791	0.336211	0.168105	+	0.985769i	$0.446235\pi$
$954$	0	0
$955$	8.53579	0.276212
$956$	−44.4137	−1.43644
$957$	0	0
$958$	−4.33402	−0.140026
$959$	−1.82581	−0.0589586
$960$	0	0
$961$	−30.6196	−0.987729
$962$	1.53685	0.0495500
$963$	0	0
$964$	−9.03876	−0.291119
$965$	−15.7417	−0.506744
$966$	0	0
$967$	32.1913	1.03520	0.517601	−	0.855622i	$-0.326825\pi$
$967$	32.1913	1.03520	0.517601	+	0.855622i	$0.326825\pi$
$968$	6.20594	0.199466
$969$	0	0
$970$	−1.42484	−0.0457489
$971$	33.1114	1.06260	0.531298	−	0.847185i	$-0.321705\pi$
$971$	33.1114	1.06260	0.531298	+	0.847185i	$0.321705\pi$
$972$	0	0
$973$	−51.5837	−1.65370
$974$	3.89978	0.124957
$975$	0	0
$976$	−39.7693	−1.27298
$977$	−12.2307	−0.391294	−0.195647	−	0.980674i	$-0.562681\pi$
$977$	−12.2307	−0.391294	−0.195647	+	0.980674i	$0.562681\pi$
$978$	0	0
$979$	4.33402	0.138516
$980$	1.75708	0.0561278
$981$	0	0
$982$	−4.57694	−0.146056
$983$	7.26111	0.231593	0.115797	−	0.993273i	$-0.463058\pi$
$983$	7.26111	0.231593	0.115797	+	0.993273i	$0.463058\pi$
$984$	0	0
$985$	−27.3215	−0.870536
$986$	14.6032	0.465061
$987$	0	0
$988$	−40.0192	−1.27318
$989$	−29.3941	−0.934679
$990$	0	0
$991$	−20.1054	−0.638671	−0.319335	−	0.947642i	$-0.603460\pi$
$991$	−20.1054	−0.638671	−0.319335	+	0.947642i	$0.603460\pi$
$992$	−1.80035	−0.0571611
$993$	0	0
$994$	−4.76336	−0.151085
$995$	−11.1801	−0.354434
$996$	0	0
$997$	−42.3173	−1.34020	−0.670102	−	0.742269i	$-0.733750\pi$
$997$	−42.3173	−1.34020	−0.670102	+	0.742269i	$0.733750\pi$
$998$	−5.49047	−0.173798
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.2.a.d.1.2		3
3.2	odd	2		177.2.a.d.1.2	✓	3
4.3	odd	2		8496.2.a.bl.1.1		3
12.11	even	2		2832.2.a.t.1.3		3
15.14	odd	2		4425.2.a.w.1.2		3
21.20	even	2		8673.2.a.s.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.a.d.1.2	✓	3	3.2	odd	2
531.2.a.d.1.2		3	1.1	even	1	trivial
2832.2.a.t.1.3		3	12.11	even	2
4425.2.a.w.1.2		3	15.14	odd	2
8496.2.a.bl.1.1		3	4.3	odd	2
8673.2.a.s.1.2		3	21.20	even	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:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	531.a (trivial)

\(f(q)\)	\(=\)	\(q+0.254102 q^{2} -1.93543 q^{4} -1.68133 q^{5} +2.74590 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q+0.254102 q^{2} -1.93543 q^{4} -1.68133 q^{5} +2.74590 q^{7} -1.00000 q^{8} -0.427229 q^{10} -2.18953 q^{11} +4.18953 q^{13} +0.697737 q^{14} +3.61676 q^{16} +6.29809 q^{17} +4.93543 q^{19} +3.25410 q^{20} -0.556364 q^{22} -3.44364 q^{23} -2.17313 q^{25} +1.06457 q^{26} -5.31450 q^{28} +9.12497 q^{29} -0.616763 q^{31} +2.91903 q^{32} +1.60036 q^{34} -4.61676 q^{35} +1.44364 q^{37} +1.25410 q^{38} +1.68133 q^{40} -4.10856 q^{41} +8.53579 q^{43} +4.23769 q^{44} -0.875034 q^{46} +6.48763 q^{47} +0.539958 q^{49} -0.552195 q^{50} -8.10856 q^{52} +0.664924 q^{53} +3.68133 q^{55} -2.74590 q^{56} +2.31867 q^{58} -1.00000 q^{59} -10.9958 q^{61} -0.156721 q^{62} -6.49180 q^{64} -7.04399 q^{65} -9.93126 q^{67} -12.1895 q^{68} -1.17313 q^{70} -6.82687 q^{71} +13.5040 q^{73} +0.366830 q^{74} -9.55220 q^{76} -6.01224 q^{77} -1.17313 q^{79} -6.08097 q^{80} -1.04399 q^{82} +15.3421 q^{83} -10.5892 q^{85} +2.16896 q^{86} +2.18953 q^{88} -1.97942 q^{89} +11.5040 q^{91} +6.66492 q^{92} +1.64852 q^{94} -8.29809 q^{95} +3.33508 q^{97} +0.137204 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{4} + 2 q^{5} + 9 q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q + 2 * q^4 + 2 * q^5 + 9 * q^7 - 3 * q^8	\( 3 q + 2 q^{4} + 2 q^{5} + 9 q^{7} - 3 q^{8} + 5 q^{10} + 2 q^{11} + 4 q^{13} - 8 q^{14} - 4 q^{16} - 3 q^{17} + 7 q^{19} + 9 q^{20} - 11 q^{22} - q^{23} - q^{25} + 11 q^{26} + 9 q^{28} + 11 q^{29} + 13 q^{31} + 4 q^{32} - 7 q^{34} + q^{35} - 5 q^{37} + 3 q^{38} - 2 q^{40} + q^{41} + 6 q^{43} + 15 q^{44} - 19 q^{46} - 11 q^{47} + 14 q^{49} + 21 q^{50} - 11 q^{52} - 2 q^{53} + 4 q^{55} - 9 q^{56} + 14 q^{58} - 3 q^{59} - q^{61} + 2 q^{62} - 21 q^{64} + 10 q^{67} - 28 q^{68} + 2 q^{70} - 26 q^{71} + 7 q^{73} + 19 q^{74} - 6 q^{76} + 17 q^{77} + 2 q^{79} - 23 q^{80} + 18 q^{82} + 3 q^{83} - 35 q^{85} - 31 q^{86} - 2 q^{88} + 23 q^{89} + q^{91} + 16 q^{92} + 4 q^{94} - 3 q^{95} + 14 q^{97} - 51 q^{98}+O(q^{100}) \) 3 * q + 2 * q^4 + 2 * q^5 + 9 * q^7 - 3 * q^8 + 5 * q^10 + 2 * q^11 + 4 * q^13 - 8 * q^14 - 4 * q^16 - 3 * q^17 + 7 * q^19 + 9 * q^20 - 11 * q^22 - q^23 - q^25 + 11 * q^26 + 9 * q^28 + 11 * q^29 + 13 * q^31 + 4 * q^32 - 7 * q^34 + q^35 - 5 * q^37 + 3 * q^38 - 2 * q^40 + q^41 + 6 * q^43 + 15 * q^44 - 19 * q^46 - 11 * q^47 + 14 * q^49 + 21 * q^50 - 11 * q^52 - 2 * q^53 + 4 * q^55 - 9 * q^56 + 14 * q^58 - 3 * q^59 - q^61 + 2 * q^62 - 21 * q^64 + 10 * q^67 - 28 * q^68 + 2 * q^70 - 26 * q^71 + 7 * q^73 + 19 * q^74 - 6 * q^76 + 17 * q^77 + 2 * q^79 - 23 * q^80 + 18 * q^82 + 3 * q^83 - 35 * q^85 - 31 * q^86 - 2 * q^88 + 23 * q^89 + q^91 + 16 * q^92 + 4 * q^94 - 3 * q^95 + 14 * q^97 - 51 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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