LMFDB - Embedded newform 4005.2.a.w.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4005,2,Mod(1,4005)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4005, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4005 = 3^{2} \cdot 5 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4005.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:	$31.9800860095$
Analytic rank:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - 5 x^{16} - 15 x^{15} + 105 x^{14} + 45 x^{13} - 849 x^{12} + 320 x^{11} + 3371 x^{10} + \cdots + 24 $ x^17 - 5x^16 - 15x^15 + 105x^14 + 45x^13 - 849x^12 + 320x^11 + 3371x^10 - 2456x^9 - 7002x^8 + 6279x^7 + 7299x^6 - 7119x^5 - 3066x^4 + 3184x^3 + 99x^2 - 231x + 24
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Root			$-2.53341$ of defining polynomial
Character	$\chi$	$=$	4005.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.53341 q^{2} +4.41816 q^{4} -1.00000 q^{5} +3.58669 q^{7} +6.12619 q^{8} +O(q^{10})$	\(q+2.53341 q^{2} +4.41816 q^{4} -1.00000 q^{5} +3.58669 q^{7} +6.12619 q^{8} -2.53341 q^{10} -1.67318 q^{11} +4.71471 q^{13} +9.08656 q^{14} +6.68383 q^{16} -2.27235 q^{17} +2.67631 q^{19} -4.41816 q^{20} -4.23884 q^{22} +1.58933 q^{23} +1.00000 q^{25} +11.9443 q^{26} +15.8466 q^{28} -1.55619 q^{29} +3.18374 q^{31} +4.68049 q^{32} -5.75680 q^{34} -3.58669 q^{35} +1.24043 q^{37} +6.78018 q^{38} -6.12619 q^{40} -3.11783 q^{41} -2.21379 q^{43} -7.39237 q^{44} +4.02643 q^{46} -1.60661 q^{47} +5.86436 q^{49} +2.53341 q^{50} +20.8304 q^{52} +3.96549 q^{53} +1.67318 q^{55} +21.9728 q^{56} -3.94247 q^{58} -5.94080 q^{59} -0.872940 q^{61} +8.06572 q^{62} -1.51006 q^{64} -4.71471 q^{65} +12.8368 q^{67} -10.0396 q^{68} -9.08656 q^{70} +9.63119 q^{71} -1.34058 q^{73} +3.14253 q^{74} +11.8244 q^{76} -6.00117 q^{77} -12.1486 q^{79} -6.68383 q^{80} -7.89874 q^{82} -11.4279 q^{83} +2.27235 q^{85} -5.60844 q^{86} -10.2502 q^{88} +1.00000 q^{89} +16.9102 q^{91} +7.02192 q^{92} -4.07021 q^{94} -2.67631 q^{95} +16.3266 q^{97} +14.8568 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q - 5 q^{2} + 21 q^{4} - 17 q^{5} + 12 q^{7} - 15 q^{8}+O(q^{10}) $ 17 * q - 5 * q^2 + 21 * q^4 - 17 * q^5 + 12 * q^7 - 15 * q^8	$ 17 q - 5 q^{2} + 21 q^{4} - 17 q^{5} + 12 q^{7} - 15 q^{8} + 5 q^{10} - 2 q^{11} + 8 q^{13} - 4 q^{14} + 33 q^{16} - 10 q^{17} + 32 q^{19} - 21 q^{20} + 8 q^{22} - 15 q^{23} + 17 q^{25} + 15 q^{26} + 24 q^{28} - q^{29} + 18 q^{31} - 25 q^{32} + 14 q^{34} - 12 q^{35} + 12 q^{37} - 22 q^{38} + 15 q^{40} + 7 q^{41} + 28 q^{43} + 14 q^{44} + 4 q^{46} - 26 q^{47} + 41 q^{49} - 5 q^{50} + 10 q^{52} - 12 q^{53} + 2 q^{55} - 13 q^{56} + 16 q^{58} + 23 q^{59} + 26 q^{61} - 10 q^{62} + 59 q^{64} - 8 q^{65} + 31 q^{67} + q^{68} + 4 q^{70} + 2 q^{71} + 33 q^{73} + 10 q^{74} + 66 q^{76} - 12 q^{77} + 33 q^{79} - 33 q^{80} + 30 q^{82} - 13 q^{83} + 10 q^{85} + 20 q^{86} + 12 q^{88} + 17 q^{89} + 40 q^{91} - 16 q^{92} + 38 q^{94} - 32 q^{95} + 45 q^{97} - 2 q^{98}+O(q^{100}) $ 17 * q - 5 * q^2 + 21 * q^4 - 17 * q^5 + 12 * q^7 - 15 * q^8 + 5 * q^10 - 2 * q^11 + 8 * q^13 - 4 * q^14 + 33 * q^16 - 10 * q^17 + 32 * q^19 - 21 * q^20 + 8 * q^22 - 15 * q^23 + 17 * q^25 + 15 * q^26 + 24 * q^28 - q^29 + 18 * q^31 - 25 * q^32 + 14 * q^34 - 12 * q^35 + 12 * q^37 - 22 * q^38 + 15 * q^40 + 7 * q^41 + 28 * q^43 + 14 * q^44 + 4 * q^46 - 26 * q^47 + 41 * q^49 - 5 * q^50 + 10 * q^52 - 12 * q^53 + 2 * q^55 - 13 * q^56 + 16 * q^58 + 23 * q^59 + 26 * q^61 - 10 * q^62 + 59 * q^64 - 8 * q^65 + 31 * q^67 + q^68 + 4 * q^70 + 2 * q^71 + 33 * q^73 + 10 * q^74 + 66 * q^76 - 12 * q^77 + 33 * q^79 - 33 * q^80 + 30 * q^82 - 13 * q^83 + 10 * q^85 + 20 * q^86 + 12 * q^88 + 17 * q^89 + 40 * q^91 - 16 * q^92 + 38 * q^94 - 32 * q^95 + 45 * q^97 - 2 * 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 $

$2$	2.53341	1.79139	0.895695	−	0.444668i	$-0.146678\pi$
$2$	2.53341	1.79139	0.895695	+	0.444668i	$0.146678\pi$
$3$	0	0
$3$	0	0
$4$	4.41816	2.20908
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.58669	1.35564	0.677821	−	0.735227i	$-0.262924\pi$
$7$	3.58669	1.35564	0.677821	+	0.735227i	$0.262924\pi$
$8$	6.12619	2.16594
$9$	0	0
$10$	−2.53341	−0.801134
$11$	−1.67318	−0.504482	−0.252241	−	0.967664i	$-0.581168\pi$
$11$	−1.67318	−0.504482	−0.252241	+	0.967664i	$0.581168\pi$
$12$	0	0
$13$	4.71471	1.30763	0.653813	−	0.756656i	$-0.273168\pi$
$13$	4.71471	1.30763	0.653813	+	0.756656i	$0.273168\pi$
$14$	9.08656	2.42849
$15$	0	0
$16$	6.68383	1.67096
$17$	−2.27235	−0.551127	−0.275563	−	0.961283i	$-0.588864\pi$
$17$	−2.27235	−0.551127	−0.275563	+	0.961283i	$0.588864\pi$
$18$	0	0
$19$	2.67631	0.613987	0.306993	−	0.951712i	$-0.400677\pi$
$19$	2.67631	0.613987	0.306993	+	0.951712i	$0.400677\pi$
$20$	−4.41816	−0.987931
$21$	0	0
$22$	−4.23884	−0.903724
$23$	1.58933	0.331398	0.165699	−	0.986176i	$-0.447012\pi$
$23$	1.58933	0.331398	0.165699	+	0.986176i	$0.447012\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	11.9443	2.34247
$27$	0	0
$28$	15.8466	2.99472
$29$	−1.55619	−0.288977	−0.144489	−	0.989506i	$-0.546154\pi$
$29$	−1.55619	−0.288977	−0.144489	+	0.989506i	$0.546154\pi$
$30$	0	0
$31$	3.18374	0.571817	0.285908	−	0.958257i	$-0.407705\pi$
$31$	3.18374	0.571817	0.285908	+	0.958257i	$0.407705\pi$
$32$	4.68049	0.827402
$33$	0	0
$34$	−5.75680	−0.987284
$35$	−3.58669	−0.606262
$36$	0	0
$37$	1.24043	0.203926	0.101963	−	0.994788i	$-0.467488\pi$
$37$	1.24043	0.203926	0.101963	+	0.994788i	$0.467488\pi$
$38$	6.78018	1.09989
$39$	0	0
$40$	−6.12619	−0.968636
$41$	−3.11783	−0.486924	−0.243462	−	0.969910i	$-0.578283\pi$
$41$	−3.11783	−0.486924	−0.243462	+	0.969910i	$0.578283\pi$
$42$	0	0
$43$	−2.21379	−0.337600	−0.168800	−	0.985650i	$-0.553989\pi$
$43$	−2.21379	−0.337600	−0.168800	+	0.985650i	$0.553989\pi$
$44$	−7.39237	−1.11444
$45$	0	0
$46$	4.02643	0.593664
$47$	−1.60661	−0.234348	−0.117174	−	0.993111i	$-0.537384\pi$
$47$	−1.60661	−0.234348	−0.117174	+	0.993111i	$0.537384\pi$
$48$	0	0
$49$	5.86436	0.837766
$50$	2.53341	0.358278
$51$	0	0
$52$	20.8304	2.88865
$53$	3.96549	0.544702	0.272351	−	0.962198i	$-0.412199\pi$
$53$	3.96549	0.544702	0.272351	+	0.962198i	$0.412199\pi$
$54$	0	0
$55$	1.67318	0.225611
$56$	21.9728	2.93624
$57$	0	0
$58$	−3.94247	−0.517671
$59$	−5.94080	−0.773427	−0.386713	−	0.922200i	$-0.626390\pi$
$59$	−5.94080	−0.773427	−0.386713	+	0.922200i	$0.626390\pi$
$60$	0	0
$61$	−0.872940	−0.111769	−0.0558843	−	0.998437i	$-0.517798\pi$
$61$	−0.872940	−0.111769	−0.0558843	+	0.998437i	$0.517798\pi$
$62$	8.06572	1.02435
$63$	0	0
$64$	−1.51006	−0.188758
$65$	−4.71471	−0.584788
$66$	0	0
$67$	12.8368	1.56826	0.784132	−	0.620594i	$-0.213108\pi$
$67$	12.8368	1.56826	0.784132	+	0.620594i	$0.213108\pi$
$68$	−10.0396	−1.21748
$69$	0	0
$70$	−9.08656	−1.08605
$71$	9.63119	1.14301	0.571506	−	0.820598i	$-0.306359\pi$
$71$	9.63119	1.14301	0.571506	+	0.820598i	$0.306359\pi$
$72$	0	0
$73$	−1.34058	−0.156904	−0.0784518	−	0.996918i	$-0.524998\pi$
$73$	−1.34058	−0.156904	−0.0784518	+	0.996918i	$0.524998\pi$
$74$	3.14253	0.365312
$75$	0	0
$76$	11.8244	1.35635
$77$	−6.00117	−0.683897
$78$	0	0
$79$	−12.1486	−1.36683	−0.683413	−	0.730032i	$-0.739505\pi$
$79$	−12.1486	−1.36683	−0.683413	+	0.730032i	$0.739505\pi$
$80$	−6.68383	−0.747275
$81$	0	0
$82$	−7.89874	−0.872270
$83$	−11.4279	−1.25438	−0.627188	−	0.778868i	$-0.715794\pi$
$83$	−11.4279	−1.25438	−0.627188	+	0.778868i	$0.715794\pi$
$84$	0	0
$85$	2.27235	0.246471
$86$	−5.60844	−0.604774
$87$	0	0
$88$	−10.2502	−1.09268
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	16.9102	1.77267
$92$	7.02192	0.732086
$93$	0	0
$94$	−4.07021	−0.419810
$95$	−2.67631	−0.274583
$96$	0	0
$97$	16.3266	1.65772	0.828858	−	0.559460i	$-0.188991\pi$
$97$	16.3266	1.65772	0.828858	+	0.559460i	$0.188991\pi$
$98$	14.8568	1.50077
$99$	0	0
$100$	4.41816	0.441816
$101$	7.08170	0.704655	0.352328	−	0.935877i	$-0.385390\pi$
$101$	7.08170	0.704655	0.352328	+	0.935877i	$0.385390\pi$
$102$	0	0
$103$	11.5091	1.13402	0.567011	−	0.823711i	$-0.308100\pi$
$103$	11.5091	1.13402	0.567011	+	0.823711i	$0.308100\pi$
$104$	28.8832	2.83223
$105$	0	0
$106$	10.0462	0.975774
$107$	1.39105	0.134478	0.0672389	−	0.997737i	$-0.478581\pi$
$107$	1.39105	0.134478	0.0672389	+	0.997737i	$0.478581\pi$
$108$	0	0
$109$	1.40895	0.134952	0.0674762	−	0.997721i	$-0.478505\pi$
$109$	1.40895	0.134952	0.0674762	+	0.997721i	$0.478505\pi$
$110$	4.23884	0.404158
$111$	0	0
$112$	23.9728	2.26522
$113$	−1.94872	−0.183320	−0.0916601	−	0.995790i	$-0.529217\pi$
$113$	−1.94872	−0.183320	−0.0916601	+	0.995790i	$0.529217\pi$
$114$	0	0
$115$	−1.58933	−0.148206
$116$	−6.87550	−0.638374
$117$	0	0
$118$	−15.0505	−1.38551
$119$	−8.15024	−0.747131
$120$	0	0
$121$	−8.20048	−0.745498
$122$	−2.21151	−0.200221
$123$	0	0
$124$	14.0663	1.26319
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−12.5324	−1.11207	−0.556033	−	0.831160i	$-0.687677\pi$
$127$	−12.5324	−1.11207	−0.556033	+	0.831160i	$0.687677\pi$
$128$	−13.1866	−1.16554
$129$	0	0
$130$	−11.9443	−1.04758
$131$	−2.81601	−0.246036	−0.123018	−	0.992404i	$-0.539257\pi$
$131$	−2.81601	−0.246036	−0.123018	+	0.992404i	$0.539257\pi$
$132$	0	0
$133$	9.59909	0.832346
$134$	32.5209	2.80938
$135$	0	0
$136$	−13.9209	−1.19371
$137$	−10.8759	−0.929187	−0.464593	−	0.885524i	$-0.653800\pi$
$137$	−10.8759	−0.929187	−0.464593	+	0.885524i	$0.653800\pi$
$138$	0	0
$139$	−12.0783	−1.02447	−0.512234	−	0.858846i	$-0.671182\pi$
$139$	−12.0783	−1.02447	−0.512234	+	0.858846i	$0.671182\pi$
$140$	−15.8466	−1.33928
$141$	0	0
$142$	24.3997	2.04758
$143$	−7.88855	−0.659673
$144$	0	0
$145$	1.55619	0.129235
$146$	−3.39625	−0.281076
$147$	0	0
$148$	5.48044	0.450490
$149$	13.6104	1.11501	0.557505	−	0.830174i	$-0.311759\pi$
$149$	13.6104	1.11501	0.557505	+	0.830174i	$0.311759\pi$
$150$	0	0
$151$	−9.71627	−0.790699	−0.395350	−	0.918531i	$-0.629377\pi$
$151$	−9.71627	−0.790699	−0.395350	+	0.918531i	$0.629377\pi$
$152$	16.3956	1.32986
$153$	0	0
$154$	−15.2034	−1.22513
$155$	−3.18374	−0.255724
$156$	0	0
$157$	−16.5891	−1.32396	−0.661978	−	0.749524i	$-0.730283\pi$
$157$	−16.5891	−1.32396	−0.661978	+	0.749524i	$0.730283\pi$
$158$	−30.7774	−2.44852
$159$	0	0
$160$	−4.68049	−0.370025
$161$	5.70044	0.449258
$162$	0	0
$163$	9.41503	0.737442	0.368721	−	0.929540i	$-0.379796\pi$
$163$	9.41503	0.737442	0.368721	+	0.929540i	$0.379796\pi$
$164$	−13.7751	−1.07565
$165$	0	0
$166$	−28.9516	−2.24708
$167$	6.61479	0.511868	0.255934	−	0.966694i	$-0.417617\pi$
$167$	6.61479	0.511868	0.255934	+	0.966694i	$0.417617\pi$
$168$	0	0
$169$	9.22850	0.709885
$170$	5.75680	0.441527
$171$	0	0
$172$	−9.78089	−0.745786
$173$	11.1540	0.848023	0.424011	−	0.905657i	$-0.360622\pi$
$173$	11.1540	0.848023	0.424011	+	0.905657i	$0.360622\pi$
$174$	0	0
$175$	3.58669	0.271128
$176$	−11.1832	−0.842968
$177$	0	0
$178$	2.53341	0.189887
$179$	−1.81904	−0.135962	−0.0679808	−	0.997687i	$-0.521656\pi$
$179$	−1.81904	−0.135962	−0.0679808	+	0.997687i	$0.521656\pi$
$180$	0	0
$181$	−22.4676	−1.67001	−0.835003	−	0.550246i	$-0.814534\pi$
$181$	−22.4676	−1.67001	−0.835003	+	0.550246i	$0.814534\pi$
$182$	42.8405	3.17555
$183$	0	0
$184$	9.73655	0.717788
$185$	−1.24043	−0.0911986
$186$	0	0
$187$	3.80205	0.278034
$188$	−7.09827	−0.517695
$189$	0	0
$190$	−6.78018	−0.491886
$191$	10.6757	0.772463	0.386231	−	0.922402i	$-0.373777\pi$
$191$	10.6757	0.772463	0.386231	+	0.922402i	$0.373777\pi$
$192$	0	0
$193$	2.41737	0.174006	0.0870032	−	0.996208i	$-0.472271\pi$
$193$	2.41737	0.174006	0.0870032	+	0.996208i	$0.472271\pi$
$194$	41.3620	2.96962
$195$	0	0
$196$	25.9097	1.85069
$197$	6.09355	0.434147	0.217074	−	0.976155i	$-0.430349\pi$
$197$	6.09355	0.434147	0.217074	+	0.976155i	$0.430349\pi$
$198$	0	0
$199$	3.85403	0.273205	0.136602	−	0.990626i	$-0.456382\pi$
$199$	3.85403	0.273205	0.136602	+	0.990626i	$0.456382\pi$
$200$	6.12619	0.433187
$201$	0	0
$202$	17.9408	1.26231
$203$	−5.58158	−0.391750
$204$	0	0
$205$	3.11783	0.217759
$206$	29.1571	2.03147
$207$	0	0
$208$	31.5123	2.18499
$209$	−4.47793	−0.309745
$210$	0	0
$211$	−27.6440	−1.90309	−0.951545	−	0.307508i	$-0.900505\pi$
$211$	−27.6440	−1.90309	−0.951545	+	0.307508i	$0.900505\pi$
$212$	17.5202	1.20329
$213$	0	0
$214$	3.52410	0.240902
$215$	2.21379	0.150979
$216$	0	0
$217$	11.4191	0.775179
$218$	3.56943	0.241753
$219$	0	0
$220$	7.39237	0.498393
$221$	−10.7135	−0.720668
$222$	0	0
$223$	−5.25590	−0.351961	−0.175981	−	0.984394i	$-0.556310\pi$
$223$	−5.25590	−0.351961	−0.175981	+	0.984394i	$0.556310\pi$
$224$	16.7875	1.12166
$225$	0	0
$226$	−4.93691	−0.328398
$227$	19.6218	1.30235	0.651173	−	0.758930i	$-0.274277\pi$
$227$	19.6218	1.30235	0.651173	+	0.758930i	$0.274277\pi$
$228$	0	0
$229$	−21.9609	−1.45122	−0.725610	−	0.688107i	$-0.758442\pi$
$229$	−21.9609	−1.45122	−0.725610	+	0.688107i	$0.758442\pi$
$230$	−4.02643	−0.265495
$231$	0	0
$232$	−9.53352	−0.625906
$233$	−27.1489	−1.77858	−0.889292	−	0.457341i	$-0.848802\pi$
$233$	−27.1489	−1.77858	−0.889292	+	0.457341i	$0.848802\pi$
$234$	0	0
$235$	1.60661	0.104804
$236$	−26.2474	−1.70856
$237$	0	0
$238$	−20.6479	−1.33840
$239$	−3.96350	−0.256378	−0.128189	−	0.991750i	$-0.540916\pi$
$239$	−3.96350	−0.256378	−0.128189	+	0.991750i	$0.540916\pi$
$240$	0	0
$241$	−12.3272	−0.794065	−0.397033	−	0.917805i	$-0.629960\pi$
$241$	−12.3272	−0.794065	−0.397033	+	0.917805i	$0.629960\pi$
$242$	−20.7752	−1.33548
$243$	0	0
$244$	−3.85679	−0.246906
$245$	−5.86436	−0.374660
$246$	0	0
$247$	12.6180	0.802865
$248$	19.5042	1.23852
$249$	0	0
$250$	−2.53341	−0.160227
$251$	−6.40360	−0.404192	−0.202096	−	0.979366i	$-0.564775\pi$
$251$	−6.40360	−0.404192	−0.202096	+	0.979366i	$0.564775\pi$
$252$	0	0
$253$	−2.65923	−0.167185
$254$	−31.7496	−1.99215
$255$	0	0
$256$	−30.3869	−1.89918
$257$	16.9722	1.05870	0.529349	−	0.848404i	$-0.322436\pi$
$257$	16.9722	1.05870	0.529349	+	0.848404i	$0.322436\pi$
$258$	0	0
$259$	4.44906	0.276451
$260$	−20.8304	−1.29184
$261$	0	0
$262$	−7.13409	−0.440746
$263$	−15.1119	−0.931837	−0.465919	−	0.884828i	$-0.654276\pi$
$263$	−15.1119	−0.931837	−0.465919	+	0.884828i	$0.654276\pi$
$264$	0	0
$265$	−3.96549	−0.243598
$266$	24.3184	1.49106
$267$	0	0
$268$	56.7151	3.46442
$269$	8.70992	0.531053	0.265526	−	0.964104i	$-0.414454\pi$
$269$	8.70992	0.531053	0.265526	+	0.964104i	$0.414454\pi$
$270$	0	0
$271$	13.7022	0.832349	0.416175	−	0.909285i	$-0.363370\pi$
$271$	13.7022	0.832349	0.416175	+	0.909285i	$0.363370\pi$
$272$	−15.1880	−0.920910
$273$	0	0
$274$	−27.5530	−1.66454
$275$	−1.67318	−0.100896
$276$	0	0
$277$	24.5620	1.47579	0.737894	−	0.674917i	$-0.235820\pi$
$277$	24.5620	1.47579	0.737894	+	0.674917i	$0.235820\pi$
$278$	−30.5993	−1.83522
$279$	0	0
$280$	−21.9728	−1.31312
$281$	−23.1248	−1.37951	−0.689756	−	0.724042i	$-0.742282\pi$
$281$	−23.1248	−1.37951	−0.689756	+	0.724042i	$0.742282\pi$
$282$	0	0
$283$	−13.3546	−0.793849	−0.396925	−	0.917851i	$-0.629923\pi$
$283$	−13.3546	−0.793849	−0.396925	+	0.917851i	$0.629923\pi$
$284$	42.5521	2.52501
$285$	0	0
$286$	−19.9849	−1.18173
$287$	−11.1827	−0.660094
$288$	0	0
$289$	−11.8364	−0.696259
$290$	3.94247	0.231510
$291$	0	0
$292$	−5.92292	−0.346613
$293$	6.69864	0.391339	0.195669	−	0.980670i	$-0.437312\pi$
$293$	6.69864	0.391339	0.195669	+	0.980670i	$0.437312\pi$
$294$	0	0
$295$	5.94080	0.345887
$296$	7.59914	0.441691
$297$	0	0
$298$	34.4808	1.99742
$299$	7.49324	0.433345
$300$	0	0
$301$	−7.94019	−0.457665
$302$	−24.6153	−1.41645
$303$	0	0
$304$	17.8880	1.02595
$305$	0.872940	0.0499844
$306$	0	0
$307$	−3.34098	−0.190680	−0.0953398	−	0.995445i	$-0.530394\pi$
$307$	−3.34098	−0.190680	−0.0953398	+	0.995445i	$0.530394\pi$
$308$	−26.5141	−1.51078
$309$	0	0
$310$	−8.06572	−0.458102
$311$	10.9861	0.622966	0.311483	−	0.950252i	$-0.399174\pi$
$311$	10.9861	0.622966	0.311483	+	0.950252i	$0.399174\pi$
$312$	0	0
$313$	−9.76267	−0.551819	−0.275909	−	0.961184i	$-0.588979\pi$
$313$	−9.76267	−0.551819	−0.275909	+	0.961184i	$0.588979\pi$
$314$	−42.0270	−2.37172
$315$	0	0
$316$	−53.6746	−3.01943
$317$	11.8262	0.664224	0.332112	−	0.943240i	$-0.392239\pi$
$317$	11.8262	0.664224	0.332112	+	0.943240i	$0.392239\pi$
$318$	0	0
$319$	2.60378	0.145784
$320$	1.51006	0.0844150
$321$	0	0
$322$	14.4416	0.804796
$323$	−6.08152	−0.338385
$324$	0	0
$325$	4.71471	0.261525
$326$	23.8521	1.32105
$327$	0	0
$328$	−19.1004	−1.05465
$329$	−5.76242	−0.317693
$330$	0	0
$331$	−21.3348	−1.17267	−0.586334	−	0.810069i	$-0.699429\pi$
$331$	−21.3348	−1.17267	−0.586334	+	0.810069i	$0.699429\pi$
$332$	−50.4904	−2.77102
$333$	0	0
$334$	16.7580	0.916955
$335$	−12.8368	−0.701349
$336$	0	0
$337$	9.49160	0.517041	0.258520	−	0.966006i	$-0.416765\pi$
$337$	9.49160	0.517041	0.258520	+	0.966006i	$0.416765\pi$
$338$	23.3796	1.27168
$339$	0	0
$340$	10.0396	0.544475
$341$	−5.32696	−0.288471
$342$	0	0
$343$	−4.07319	−0.219931
$344$	−13.5621	−0.731220
$345$	0	0
$346$	28.2576	1.51914
$347$	−6.69050	−0.359165	−0.179582	−	0.983743i	$-0.557475\pi$
$347$	−6.69050	−0.359165	−0.179582	+	0.983743i	$0.557475\pi$
$348$	0	0
$349$	15.7829	0.844838	0.422419	−	0.906401i	$-0.361181\pi$
$349$	15.7829	0.844838	0.422419	+	0.906401i	$0.361181\pi$
$350$	9.08656	0.485697
$351$	0	0
$352$	−7.83129	−0.417409
$353$	4.26421	0.226961	0.113480	−	0.993540i	$-0.463800\pi$
$353$	4.26421	0.226961	0.113480	+	0.993540i	$0.463800\pi$
$354$	0	0
$355$	−9.63119	−0.511170
$356$	4.41816	0.234162
$357$	0	0
$358$	−4.60838	−0.243560
$359$	−33.4199	−1.76383	−0.881917	−	0.471404i	$-0.843747\pi$
$359$	−33.4199	−1.76383	−0.881917	+	0.471404i	$0.843747\pi$
$360$	0	0
$361$	−11.8374	−0.623020
$362$	−56.9197	−2.99163
$363$	0	0
$364$	74.7121	3.91598
$365$	1.34058	0.0701694
$366$	0	0
$367$	16.4570	0.859050	0.429525	−	0.903055i	$-0.358681\pi$
$367$	16.4570	0.859050	0.429525	+	0.903055i	$0.358681\pi$
$368$	10.6228	0.553753
$369$	0	0
$370$	−3.14253	−0.163372
$371$	14.2230	0.738421
$372$	0	0
$373$	6.83377	0.353839	0.176920	−	0.984225i	$-0.443387\pi$
$373$	6.83377	0.353839	0.176920	+	0.984225i	$0.443387\pi$
$374$	9.63215	0.498067
$375$	0	0
$376$	−9.84242	−0.507584
$377$	−7.33699	−0.377874
$378$	0	0
$379$	11.7684	0.604504	0.302252	−	0.953228i	$-0.402262\pi$
$379$	11.7684	0.604504	0.302252	+	0.953228i	$0.402262\pi$
$380$	−11.8244	−0.606576
$381$	0	0
$382$	27.0458	1.38378
$383$	18.1281	0.926301	0.463150	−	0.886280i	$-0.346719\pi$
$383$	18.1281	0.926301	0.463150	+	0.886280i	$0.346719\pi$
$384$	0	0
$385$	6.00117	0.305848
$386$	6.12420	0.311713
$387$	0	0
$388$	72.1336	3.66203
$389$	−31.4645	−1.59531	−0.797657	−	0.603112i	$-0.793927\pi$
$389$	−31.4645	−1.59531	−0.797657	+	0.603112i	$0.793927\pi$
$390$	0	0
$391$	−3.61152	−0.182643
$392$	35.9262	1.81455
$393$	0	0
$394$	15.4375	0.777728
$395$	12.1486	0.611264
$396$	0	0
$397$	−6.24567	−0.313461	−0.156731	−	0.987641i	$-0.550095\pi$
$397$	−6.24567	−0.313461	−0.156731	+	0.987641i	$0.550095\pi$
$398$	9.76383	0.489416
$399$	0	0
$400$	6.68383	0.334192
$401$	−9.19844	−0.459348	−0.229674	−	0.973268i	$-0.573766\pi$
$401$	−9.19844	−0.459348	−0.229674	+	0.973268i	$0.573766\pi$
$402$	0	0
$403$	15.0104	0.747722
$404$	31.2881	1.55664
$405$	0	0
$406$	−14.1404	−0.701777
$407$	−2.07547	−0.102877
$408$	0	0
$409$	27.0437	1.33723	0.668613	−	0.743611i	$-0.266888\pi$
$409$	27.0437	1.33723	0.668613	+	0.743611i	$0.266888\pi$
$410$	7.89874	0.390091
$411$	0	0
$412$	50.8489	2.50514
$413$	−21.3078	−1.04849
$414$	0	0
$415$	11.4279	0.560974
$416$	22.0672	1.08193
$417$	0	0
$418$	−11.3444	−0.554875
$419$	−27.4100	−1.33907	−0.669533	−	0.742782i	$-0.733506\pi$
$419$	−27.4100	−1.33907	−0.669533	+	0.742782i	$0.733506\pi$
$420$	0	0
$421$	32.7588	1.59657	0.798284	−	0.602281i	$-0.205741\pi$
$421$	32.7588	1.59657	0.798284	+	0.602281i	$0.205741\pi$
$422$	−70.0335	−3.40918
$423$	0	0
$424$	24.2934	1.17979
$425$	−2.27235	−0.110225
$426$	0	0
$427$	−3.13097	−0.151518
$428$	6.14588	0.297072
$429$	0	0
$430$	5.60844	0.270463
$431$	6.00642	0.289319	0.144660	−	0.989481i	$-0.453791\pi$
$431$	6.00642	0.289319	0.144660	+	0.989481i	$0.453791\pi$
$432$	0	0
$433$	−10.2725	−0.493666	−0.246833	−	0.969058i	$-0.579390\pi$
$433$	−10.2725	−0.493666	−0.246833	+	0.969058i	$0.579390\pi$
$434$	28.9292	1.38865
$435$	0	0
$436$	6.22495	0.298121
$437$	4.25354	0.203474
$438$	0	0
$439$	16.0109	0.764159	0.382079	−	0.924129i	$-0.375208\pi$
$439$	16.0109	0.764159	0.382079	+	0.924129i	$0.375208\pi$
$440$	10.2502	0.488659
$441$	0	0
$442$	−27.1417	−1.29100
$443$	−37.6765	−1.79006	−0.895032	−	0.446002i	$-0.852848\pi$
$443$	−37.6765	−1.79006	−0.895032	+	0.446002i	$0.852848\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	−13.3153	−0.630500
$447$	0	0
$448$	−5.41613	−0.255888
$449$	−1.30190	−0.0614405	−0.0307203	−	0.999528i	$-0.509780\pi$
$449$	−1.30190	−0.0614405	−0.0307203	+	0.999528i	$0.509780\pi$
$450$	0	0
$451$	5.21669	0.245644
$452$	−8.60976	−0.404969
$453$	0	0
$454$	49.7101	2.33301
$455$	−16.9102	−0.792763
$456$	0	0
$457$	−22.3853	−1.04714	−0.523570	−	0.851983i	$-0.675400\pi$
$457$	−22.3853	−1.04714	−0.523570	+	0.851983i	$0.675400\pi$
$458$	−55.6360	−2.59970
$459$	0	0
$460$	−7.02192	−0.327399
$461$	41.6600	1.94030	0.970150	−	0.242507i	$-0.0779698\pi$
$461$	41.6600	1.94030	0.970150	+	0.242507i	$0.0779698\pi$
$462$	0	0
$463$	−19.7433	−0.917547	−0.458773	−	0.888553i	$-0.651711\pi$
$463$	−19.7433	−0.917547	−0.458773	+	0.888553i	$0.651711\pi$
$464$	−10.4013	−0.482869
$465$	0	0
$466$	−68.7793	−3.18614
$467$	7.99380	0.369909	0.184955	−	0.982747i	$-0.440786\pi$
$467$	7.99380	0.369909	0.184955	+	0.982747i	$0.440786\pi$
$468$	0	0
$469$	46.0417	2.12601
$470$	4.07021	0.187745
$471$	0	0
$472$	−36.3945	−1.67519
$473$	3.70407	0.170313
$474$	0	0
$475$	2.67631	0.122797
$476$	−36.0091	−1.65047
$477$	0	0
$478$	−10.0412	−0.459273
$479$	−32.7638	−1.49702	−0.748509	−	0.663125i	$-0.769230\pi$
$479$	−32.7638	−1.49702	−0.748509	+	0.663125i	$0.769230\pi$
$480$	0	0
$481$	5.84829	0.266659
$482$	−31.2299	−1.42248
$483$	0	0
$484$	−36.2310	−1.64687
$485$	−16.3266	−0.741353
$486$	0	0
$487$	1.32900	0.0602228	0.0301114	−	0.999547i	$-0.490414\pi$
$487$	1.32900	0.0602228	0.0301114	+	0.999547i	$0.490414\pi$
$488$	−5.34780	−0.242083
$489$	0	0
$490$	−14.8568	−0.671163
$491$	−13.7450	−0.620303	−0.310151	−	0.950687i	$-0.600380\pi$
$491$	−13.7450	−0.620303	−0.310151	+	0.950687i	$0.600380\pi$
$492$	0	0
$493$	3.53622	0.159263
$494$	31.9666	1.43824
$495$	0	0
$496$	21.2796	0.955481
$497$	34.5441	1.54951
$498$	0	0
$499$	15.1965	0.680288	0.340144	−	0.940373i	$-0.389524\pi$
$499$	15.1965	0.680288	0.340144	+	0.940373i	$0.389524\pi$
$500$	−4.41816	−0.197586
$501$	0	0
$502$	−16.2229	−0.724065
$503$	35.6824	1.59100	0.795500	−	0.605954i	$-0.207209\pi$
$503$	35.6824	1.59100	0.795500	+	0.605954i	$0.207209\pi$
$504$	0	0
$505$	−7.08170	−0.315131
$506$	−6.73692	−0.299493
$507$	0	0
$508$	−55.3700	−2.45664
$509$	2.02268	0.0896538	0.0448269	−	0.998995i	$-0.485726\pi$
$509$	2.02268	0.0896538	0.0448269	+	0.998995i	$0.485726\pi$
$510$	0	0
$511$	−4.80826	−0.212705
$512$	−50.6093	−2.23663
$513$	0	0
$514$	42.9976	1.89654
$515$	−11.5091	−0.507150
$516$	0	0
$517$	2.68815	0.118225
$518$	11.2713	0.495232
$519$	0	0
$520$	−28.8832	−1.26661
$521$	−19.0641	−0.835215	−0.417607	−	0.908628i	$-0.637131\pi$
$521$	−19.0641	−0.835215	−0.417607	+	0.908628i	$0.637131\pi$
$522$	0	0
$523$	−24.7160	−1.08076	−0.540378	−	0.841423i	$-0.681719\pi$
$523$	−24.7160	−1.08076	−0.540378	+	0.841423i	$0.681719\pi$
$524$	−12.4416	−0.543512
$525$	0	0
$526$	−38.2845	−1.66928
$527$	−7.23459	−0.315144
$528$	0	0
$529$	−20.4740	−0.890175
$530$	−10.0462	−0.436379
$531$	0	0
$532$	42.4103	1.83872
$533$	−14.6997	−0.636714
$534$	0	0
$535$	−1.39105	−0.0601403
$536$	78.6407	3.39676
$537$	0	0
$538$	22.0658	0.951323
$539$	−9.81211	−0.422638
$540$	0	0
$541$	9.93121	0.426976	0.213488	−	0.976946i	$-0.431518\pi$
$541$	9.93121	0.426976	0.213488	+	0.976946i	$0.431518\pi$
$542$	34.7133	1.49106
$543$	0	0
$544$	−10.6357	−0.456003
$545$	−1.40895	−0.0603526
$546$	0	0
$547$	13.1436	0.561981	0.280990	−	0.959711i	$-0.409337\pi$
$547$	13.1436	0.561981	0.280990	+	0.959711i	$0.409337\pi$
$548$	−48.0513	−2.05265
$549$	0	0
$550$	−4.23884	−0.180745
$551$	−4.16484	−0.177428
$552$	0	0
$553$	−43.5734	−1.85293
$554$	62.2256	2.64371
$555$	0	0
$556$	−53.3639	−2.26313
$557$	−11.0830	−0.469600	−0.234800	−	0.972044i	$-0.575443\pi$
$557$	−11.0830	−0.469600	−0.234800	+	0.972044i	$0.575443\pi$
$558$	0	0
$559$	−10.4374	−0.441455
$560$	−23.9728	−1.01304
$561$	0	0
$562$	−58.5847	−2.47124
$563$	19.3774	0.816659	0.408329	−	0.912835i	$-0.366112\pi$
$563$	19.3774	0.816659	0.408329	+	0.912835i	$0.366112\pi$
$564$	0	0
$565$	1.94872	0.0819833
$566$	−33.8327	−1.42209
$567$	0	0
$568$	59.0025	2.47569
$569$	8.53303	0.357723	0.178862	−	0.983874i	$-0.442759\pi$
$569$	8.53303	0.357723	0.178862	+	0.983874i	$0.442759\pi$
$570$	0	0
$571$	−3.91245	−0.163731	−0.0818654	−	0.996643i	$-0.526088\pi$
$571$	−3.91245	−0.163731	−0.0818654	+	0.996643i	$0.526088\pi$
$572$	−34.8529	−1.45727
$573$	0	0
$574$	−28.3304	−1.18249
$575$	1.58933	0.0662797
$576$	0	0
$577$	3.45758	0.143941	0.0719704	−	0.997407i	$-0.477071\pi$
$577$	3.45758	0.143941	0.0719704	+	0.997407i	$0.477071\pi$
$578$	−29.9865	−1.24727
$579$	0	0
$580$	6.87550	0.285490
$581$	−40.9884	−1.70049
$582$	0	0
$583$	−6.63497	−0.274792
$584$	−8.21268	−0.339843
$585$	0	0
$586$	16.9704	0.701040
$587$	−25.3992	−1.04834	−0.524169	−	0.851614i	$-0.675624\pi$
$587$	−25.3992	−1.04834	−0.524169	+	0.851614i	$0.675624\pi$
$588$	0	0
$589$	8.52066	0.351088
$590$	15.0505	0.619619
$591$	0	0
$592$	8.29086	0.340752
$593$	−39.9795	−1.64176	−0.820880	−	0.571100i	$-0.806517\pi$
$593$	−39.9795	−1.64176	−0.820880	+	0.571100i	$0.806517\pi$
$594$	0	0
$595$	8.15024	0.334127
$596$	60.1331	2.46315
$597$	0	0
$598$	18.9834	0.776290
$599$	0.481735	0.0196832	0.00984158	−	0.999952i	$-0.496867\pi$
$599$	0.481735	0.0196832	0.00984158	+	0.999952i	$0.496867\pi$
$600$	0	0
$601$	5.80819	0.236921	0.118460	−	0.992959i	$-0.462204\pi$
$601$	5.80819	0.236921	0.118460	+	0.992959i	$0.462204\pi$
$602$	−20.1158	−0.819857
$603$	0	0
$604$	−42.9281	−1.74672
$605$	8.20048	0.333397
$606$	0	0
$607$	−0.0895567	−0.00363499	−0.00181750	−	0.999998i	$-0.500579\pi$
$607$	−0.0895567	−0.00363499	−0.00181750	+	0.999998i	$0.500579\pi$
$608$	12.5264	0.508014
$609$	0	0
$610$	2.21151	0.0895416
$611$	−7.57471	−0.306440
$612$	0	0
$613$	2.64606	0.106874	0.0534368	−	0.998571i	$-0.482982\pi$
$613$	2.64606	0.106874	0.0534368	+	0.998571i	$0.482982\pi$
$614$	−8.46406	−0.341582
$615$	0	0
$616$	−36.7643	−1.48128
$617$	−4.79628	−0.193091	−0.0965455	−	0.995329i	$-0.530779\pi$
$617$	−4.79628	−0.193091	−0.0965455	+	0.995329i	$0.530779\pi$
$618$	0	0
$619$	20.6938	0.831755	0.415877	−	0.909421i	$-0.363475\pi$
$619$	20.6938	0.831755	0.415877	+	0.909421i	$0.363475\pi$
$620$	−14.0663	−0.564915
$621$	0	0
$622$	27.8323	1.11597
$623$	3.58669	0.143698
$624$	0	0
$625$	1.00000	0.0400000
$626$	−24.7328	−0.988523
$627$	0	0
$628$	−73.2934	−2.92472
$629$	−2.81871	−0.112389
$630$	0	0
$631$	17.1609	0.683166	0.341583	−	0.939852i	$-0.389037\pi$
$631$	17.1609	0.683166	0.341583	+	0.939852i	$0.389037\pi$
$632$	−74.4248	−2.96046
$633$	0	0
$634$	29.9605	1.18988
$635$	12.5324	0.497331
$636$	0	0
$637$	27.6488	1.09548
$638$	6.59645	0.261156
$639$	0	0
$640$	13.1866	0.521246
$641$	−30.0870	−1.18836	−0.594182	−	0.804331i	$-0.702524\pi$
$641$	−30.0870	−1.18836	−0.594182	+	0.804331i	$0.702524\pi$
$642$	0	0
$643$	−5.36263	−0.211482	−0.105741	−	0.994394i	$-0.533721\pi$
$643$	−5.36263	−0.211482	−0.105741	+	0.994394i	$0.533721\pi$
$644$	25.1855	0.992447
$645$	0	0
$646$	−15.4070	−0.606179
$647$	−34.0319	−1.33793	−0.668966	−	0.743293i	$-0.733263\pi$
$647$	−34.0319	−1.33793	−0.668966	+	0.743293i	$0.733263\pi$
$648$	0	0
$649$	9.94002	0.390180
$650$	11.9443	0.468494
$651$	0	0
$652$	41.5971	1.62907
$653$	−31.2152	−1.22155	−0.610773	−	0.791806i	$-0.709141\pi$
$653$	−31.2152	−1.22155	−0.610773	+	0.791806i	$0.709141\pi$
$654$	0	0
$655$	2.81601	0.110030
$656$	−20.8391	−0.813629
$657$	0	0
$658$	−14.5986	−0.569112
$659$	−3.33398	−0.129874	−0.0649368	−	0.997889i	$-0.520685\pi$
$659$	−3.33398	−0.129874	−0.0649368	+	0.997889i	$0.520685\pi$
$660$	0	0
$661$	−18.6309	−0.724658	−0.362329	−	0.932050i	$-0.618018\pi$
$661$	−18.6309	−0.724658	−0.362329	+	0.932050i	$0.618018\pi$
$662$	−54.0499	−2.10071
$663$	0	0
$664$	−70.0096	−2.71690
$665$	−9.59909	−0.372237
$666$	0	0
$667$	−2.47330	−0.0957666
$668$	29.2252	1.13076
$669$	0	0
$670$	−32.5209	−1.25639
$671$	1.46058	0.0563852
$672$	0	0
$673$	−36.6892	−1.41427	−0.707133	−	0.707080i	$-0.750012\pi$
$673$	−36.6892	−1.41427	−0.707133	+	0.707080i	$0.750012\pi$
$674$	24.0461	0.926222
$675$	0	0
$676$	40.7730	1.56819
$677$	0.513509	0.0197358	0.00986788	−	0.999951i	$-0.496859\pi$
$677$	0.513509	0.0197358	0.00986788	+	0.999951i	$0.496859\pi$
$678$	0	0
$679$	58.5585	2.24727
$680$	13.9209	0.533842
$681$	0	0
$682$	−13.4954	−0.516764
$683$	27.2469	1.04257	0.521286	−	0.853382i	$-0.325452\pi$
$683$	27.2469	1.04257	0.521286	+	0.853382i	$0.325452\pi$
$684$	0	0
$685$	10.8759	0.415545
$686$	−10.3190	−0.393983
$687$	0	0
$688$	−14.7966	−0.564116
$689$	18.6961	0.712266
$690$	0	0
$691$	−6.42411	−0.244384	−0.122192	−	0.992506i	$-0.538992\pi$
$691$	−6.42411	−0.244384	−0.122192	+	0.992506i	$0.538992\pi$
$692$	49.2802	1.87335
$693$	0	0
$694$	−16.9498	−0.643405
$695$	12.0783	0.458156
$696$	0	0
$697$	7.08482	0.268357
$698$	39.9845	1.51344
$699$	0	0
$700$	15.8466	0.598945
$701$	47.8619	1.80772	0.903859	−	0.427831i	$-0.140722\pi$
$701$	47.8619	1.80772	0.903859	+	0.427831i	$0.140722\pi$
$702$	0	0
$703$	3.31978	0.125208
$704$	2.52660	0.0952248
$705$	0	0
$706$	10.8030	0.406576
$707$	25.3999	0.955260
$708$	0	0
$709$	21.4103	0.804082	0.402041	−	0.915622i	$-0.368301\pi$
$709$	21.4103	0.804082	0.402041	+	0.915622i	$0.368301\pi$
$710$	−24.3997	−0.915706
$711$	0	0
$712$	6.12619	0.229589
$713$	5.06002	0.189499
$714$	0	0
$715$	7.88855	0.295015
$716$	−8.03683	−0.300350
$717$	0	0
$718$	−84.6663	−3.15972
$719$	22.8373	0.851688	0.425844	−	0.904797i	$-0.359977\pi$
$719$	22.8373	0.851688	0.425844	+	0.904797i	$0.359977\pi$
$720$	0	0
$721$	41.2794	1.53733
$722$	−29.9889	−1.11607
$723$	0	0
$724$	−99.2656	−3.68918
$725$	−1.55619	−0.0577955
$726$	0	0
$727$	25.4307	0.943172	0.471586	−	0.881820i	$-0.343682\pi$
$727$	25.4307	0.943172	0.471586	+	0.881820i	$0.343682\pi$
$728$	103.595	3.83950
$729$	0	0
$730$	3.39625	0.125701
$731$	5.03052	0.186061
$732$	0	0
$733$	−11.4253	−0.422003	−0.211002	−	0.977486i	$-0.567673\pi$
$733$	−11.4253	−0.422003	−0.211002	+	0.977486i	$0.567673\pi$
$734$	41.6924	1.53889
$735$	0	0
$736$	7.43885	0.274200
$737$	−21.4782	−0.791161
$738$	0	0
$739$	27.2320	1.00174	0.500872	−	0.865521i	$-0.333013\pi$
$739$	27.2320	1.00174	0.500872	+	0.865521i	$0.333013\pi$
$740$	−5.48044	−0.201465
$741$	0	0
$742$	36.0326	1.32280
$743$	44.0819	1.61721	0.808603	−	0.588354i	$-0.200224\pi$
$743$	44.0819	1.61721	0.808603	+	0.588354i	$0.200224\pi$
$744$	0	0
$745$	−13.6104	−0.498647
$746$	17.3127	0.633865
$747$	0	0
$748$	16.7981	0.614199
$749$	4.98926	0.182304
$750$	0	0
$751$	−39.9861	−1.45911	−0.729557	−	0.683920i	$-0.760274\pi$
$751$	−39.9861	−1.45911	−0.729557	+	0.683920i	$0.760274\pi$
$752$	−10.7383	−0.391586
$753$	0	0
$754$	−18.5876	−0.676920
$755$	9.71627	0.353611
$756$	0	0
$757$	28.9162	1.05098	0.525488	−	0.850801i	$-0.323883\pi$
$757$	28.9162	1.05098	0.525488	+	0.850801i	$0.323883\pi$
$758$	29.8142	1.08290
$759$	0	0
$760$	−16.3956	−0.594730
$761$	28.2266	1.02321	0.511607	−	0.859219i	$-0.329050\pi$
$761$	28.2266	1.02321	0.511607	+	0.859219i	$0.329050\pi$
$762$	0	0
$763$	5.05345	0.182947
$764$	47.1668	1.70643
$765$	0	0
$766$	45.9258	1.65937
$767$	−28.0092	−1.01135
$768$	0	0
$769$	28.4286	1.02516	0.512581	−	0.858639i	$-0.328689\pi$
$769$	28.4286	1.02516	0.512581	+	0.858639i	$0.328689\pi$
$770$	15.2034	0.547893
$771$	0	0
$772$	10.6803	0.384394
$773$	4.03218	0.145027	0.0725137	−	0.997367i	$-0.476898\pi$
$773$	4.03218	0.145027	0.0725137	+	0.997367i	$0.476898\pi$
$774$	0	0
$775$	3.18374	0.114363
$776$	100.020	3.59051
$777$	0	0
$778$	−79.7124	−2.85783
$779$	−8.34427	−0.298965
$780$	0	0
$781$	−16.1147	−0.576629
$782$	−9.14947	−0.327184
$783$	0	0
$784$	39.1964	1.39987
$785$	16.5891	0.592091
$786$	0	0
$787$	45.9979	1.63965	0.819823	−	0.572617i	$-0.194072\pi$
$787$	45.9979	1.63965	0.819823	+	0.572617i	$0.194072\pi$
$788$	26.9223	0.959067
$789$	0	0
$790$	30.7774	1.09501
$791$	−6.98946	−0.248517
$792$	0	0
$793$	−4.11566	−0.146151
$794$	−15.8228	−0.561532
$795$	0	0
$796$	17.0277	0.603531
$797$	33.9974	1.20425	0.602125	−	0.798402i	$-0.294321\pi$
$797$	33.9974	1.20425	0.602125	+	0.798402i	$0.294321\pi$
$798$	0	0
$799$	3.65079	0.129156
$800$	4.68049	0.165480
$801$	0	0
$802$	−23.3034	−0.822872
$803$	2.24304	0.0791550
$804$	0	0
$805$	−5.70044	−0.200914
$806$	38.0275	1.33946
$807$	0	0
$808$	43.3838	1.52624
$809$	15.8990	0.558977	0.279489	−	0.960149i	$-0.409835\pi$
$809$	15.8990	0.558977	0.279489	+	0.960149i	$0.409835\pi$
$810$	0	0
$811$	−18.2217	−0.639850	−0.319925	−	0.947443i	$-0.603658\pi$
$811$	−18.2217	−0.639850	−0.319925	+	0.947443i	$0.603658\pi$
$812$	−24.6603	−0.865407
$813$	0	0
$814$	−5.25801	−0.184293
$815$	−9.41503	−0.329794
$816$	0	0
$817$	−5.92478	−0.207282
$818$	68.5128	2.39549
$819$	0	0
$820$	13.7751	0.481047
$821$	−9.81954	−0.342704	−0.171352	−	0.985210i	$-0.554814\pi$
$821$	−9.81954	−0.342704	−0.171352	+	0.985210i	$0.554814\pi$
$822$	0	0
$823$	2.81056	0.0979699	0.0489850	−	0.998800i	$-0.484401\pi$
$823$	2.81056	0.0979699	0.0489850	+	0.998800i	$0.484401\pi$
$824$	70.5067	2.45622
$825$	0	0
$826$	−53.9815	−1.87826
$827$	−43.9896	−1.52967	−0.764835	−	0.644227i	$-0.777179\pi$
$827$	−43.9896	−1.52967	−0.764835	+	0.644227i	$0.777179\pi$
$828$	0	0
$829$	−12.4461	−0.432271	−0.216136	−	0.976363i	$-0.569345\pi$
$829$	−12.4461	−0.432271	−0.216136	+	0.976363i	$0.569345\pi$
$830$	28.9516	1.00492
$831$	0	0
$832$	−7.11950	−0.246824
$833$	−13.3259	−0.461715
$834$	0	0
$835$	−6.61479	−0.228914
$836$	−19.7842	−0.684252
$837$	0	0
$838$	−69.4408	−2.39879
$839$	38.4526	1.32753	0.663766	−	0.747941i	$-0.268957\pi$
$839$	38.4526	1.32753	0.663766	+	0.747941i	$0.268957\pi$
$840$	0	0
$841$	−26.5783	−0.916492
$842$	82.9915	2.86008
$843$	0	0
$844$	−122.136	−4.20408
$845$	−9.22850	−0.317470
$846$	0	0
$847$	−29.4126	−1.01063
$848$	26.5047	0.910174
$849$	0	0
$850$	−5.75680	−0.197457
$851$	1.97146	0.0675808
$852$	0	0
$853$	31.6967	1.08528	0.542638	−	0.839967i	$-0.317426\pi$
$853$	31.6967	1.08528	0.542638	+	0.839967i	$0.317426\pi$
$854$	−7.93202	−0.271428
$855$	0	0
$856$	8.52183	0.291270
$857$	27.3283	0.933516	0.466758	−	0.884385i	$-0.345422\pi$
$857$	27.3283	0.933516	0.466758	+	0.884385i	$0.345422\pi$
$858$	0	0
$859$	29.0204	0.990165	0.495082	−	0.868846i	$-0.335138\pi$
$859$	29.0204	0.990165	0.495082	+	0.868846i	$0.335138\pi$
$860$	9.78089	0.333526
$861$	0	0
$862$	15.2167	0.518284
$863$	−56.7522	−1.93187	−0.965934	−	0.258790i	$-0.916676\pi$
$863$	−56.7522	−1.93187	−0.965934	+	0.258790i	$0.916676\pi$
$864$	0	0
$865$	−11.1540	−0.379247
$866$	−26.0245	−0.884348
$867$	0	0
$868$	50.4514	1.71243
$869$	20.3268	0.689539
$870$	0	0
$871$	60.5218	2.05070
$872$	8.63147	0.292298
$873$	0	0
$874$	10.7759	0.364502
$875$	−3.58669	−0.121252
$876$	0	0
$877$	−3.47323	−0.117283	−0.0586414	−	0.998279i	$-0.518677\pi$
$877$	−3.47323	−0.117283	−0.0586414	+	0.998279i	$0.518677\pi$
$878$	40.5622	1.36891
$879$	0	0
$880$	11.1832	0.376987
$881$	−28.5649	−0.962374	−0.481187	−	0.876618i	$-0.659794\pi$
$881$	−28.5649	−0.962374	−0.481187	+	0.876618i	$0.659794\pi$
$882$	0	0
$883$	−37.1481	−1.25013	−0.625066	−	0.780572i	$-0.714928\pi$
$883$	−37.1481	−1.25013	−0.625066	+	0.780572i	$0.714928\pi$
$884$	−47.3340	−1.59201
$885$	0	0
$886$	−95.4500	−3.20670
$887$	−16.5944	−0.557185	−0.278592	−	0.960409i	$-0.589868\pi$
$887$	−16.5944	−0.557185	−0.278592	+	0.960409i	$0.589868\pi$
$888$	0	0
$889$	−44.9497	−1.50756
$890$	−2.53341	−0.0849201
$891$	0	0
$892$	−23.2214	−0.777511
$893$	−4.29979	−0.143887
$894$	0	0
$895$	1.81904	0.0608039
$896$	−47.2962	−1.58006
$897$	0	0
$898$	−3.29825	−0.110064
$899$	−4.95450	−0.165242
$900$	0	0
$901$	−9.01100	−0.300200
$902$	13.2160	0.440045
$903$	0	0
$904$	−11.9382	−0.397060
$905$	22.4676	0.746849
$906$	0	0
$907$	−7.98862	−0.265258	−0.132629	−	0.991166i	$-0.542342\pi$
$907$	−7.98862	−0.265258	−0.132629	+	0.991166i	$0.542342\pi$
$908$	86.6924	2.87699
$909$	0	0
$910$	−42.8405	−1.42015
$911$	50.7004	1.67978	0.839889	−	0.542759i	$-0.182620\pi$
$911$	50.7004	1.67978	0.839889	+	0.542759i	$0.182620\pi$
$912$	0	0
$913$	19.1209	0.632810
$914$	−56.7111	−1.87584
$915$	0	0
$916$	−97.0270	−3.20586
$917$	−10.1001	−0.333536
$918$	0	0
$919$	10.6847	0.352455	0.176228	−	0.984349i	$-0.443611\pi$
$919$	10.6847	0.352455	0.176228	+	0.984349i	$0.443611\pi$
$920$	−9.73655	−0.321005
$921$	0	0
$922$	105.542	3.47583
$923$	45.4083	1.49463
$924$	0	0
$925$	1.24043	0.0407852
$926$	−50.0177	−1.64369
$927$	0	0
$928$	−7.28374	−0.239100
$929$	0.222188	0.00728975	0.00364487	−	0.999993i	$-0.498840\pi$
$929$	0.222188	0.00728975	0.00364487	+	0.999993i	$0.498840\pi$
$930$	0	0
$931$	15.6948	0.514377
$932$	−119.948	−3.92903
$933$	0	0
$934$	20.2516	0.662652
$935$	−3.80205	−0.124340
$936$	0	0
$937$	−7.16614	−0.234108	−0.117054	−	0.993126i	$-0.537345\pi$
$937$	−7.16614	−0.234108	−0.117054	+	0.993126i	$0.537345\pi$
$938$	116.642	3.80851
$939$	0	0
$940$	7.09827	0.231520
$941$	21.0564	0.686420	0.343210	−	0.939259i	$-0.388486\pi$
$941$	21.0564	0.686420	0.343210	+	0.939259i	$0.388486\pi$
$942$	0	0
$943$	−4.95527	−0.161366
$944$	−39.7073	−1.29236
$945$	0	0
$946$	9.38391	0.305097
$947$	50.4902	1.64071	0.820356	−	0.571853i	$-0.193775\pi$
$947$	50.4902	1.64071	0.820356	+	0.571853i	$0.193775\pi$
$948$	0	0
$949$	−6.32047	−0.205171
$950$	6.78018	0.219978
$951$	0	0
$952$	−49.9299	−1.61824
$953$	−4.68806	−0.151861	−0.0759305	−	0.997113i	$-0.524193\pi$
$953$	−4.68806	−0.151861	−0.0759305	+	0.997113i	$0.524193\pi$
$954$	0	0
$955$	−10.6757	−0.345456
$956$	−17.5114	−0.566359
$957$	0	0
$958$	−83.0042	−2.68174
$959$	−39.0083	−1.25965
$960$	0	0
$961$	−20.8638	−0.673026
$962$	14.8161	0.477691
$963$	0	0
$964$	−54.4636	−1.75415
$965$	−2.41737	−0.0778180
$966$	0	0
$967$	−41.6134	−1.33820	−0.669098	−	0.743174i	$-0.733320\pi$
$967$	−41.6134	−1.33820	−0.669098	+	0.743174i	$0.733320\pi$
$968$	−50.2377	−1.61470
$969$	0	0
$970$	−41.3620	−1.32805
$971$	−37.9596	−1.21818	−0.609090	−	0.793101i	$-0.708465\pi$
$971$	−37.9596	−1.21818	−0.609090	+	0.793101i	$0.708465\pi$
$972$	0	0
$973$	−43.3211	−1.38881
$974$	3.36690	0.107883
$975$	0	0
$976$	−5.83458	−0.186760
$977$	27.8547	0.891150	0.445575	−	0.895245i	$-0.352999\pi$
$977$	27.8547	0.891150	0.445575	+	0.895245i	$0.352999\pi$
$978$	0	0
$979$	−1.67318	−0.0534750
$980$	−25.9097	−0.827655
$981$	0	0
$982$	−34.8217	−1.11120
$983$	10.8610	0.346411	0.173205	−	0.984886i	$-0.444588\pi$
$983$	10.8610	0.346411	0.173205	+	0.984886i	$0.444588\pi$
$984$	0	0
$985$	−6.09355	−0.194157
$986$	8.95868	0.285303
$987$	0	0
$988$	55.7484	1.77359
$989$	−3.51845	−0.111880
$990$	0	0
$991$	−17.3737	−0.551895	−0.275947	−	0.961173i	$-0.588992\pi$
$991$	−17.3737	−0.551895	−0.275947	+	0.961173i	$0.588992\pi$
$992$	14.9015	0.473122
$993$	0	0
$994$	87.5144	2.77579
$995$	−3.85403	−0.122181
$996$	0	0
$997$	15.7502	0.498814	0.249407	−	0.968399i	$-0.419764\pi$
$997$	15.7502	0.498814	0.249407	+	0.968399i	$0.419764\pi$
$998$	38.4989	1.21866
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.w.1.16	✓	17
3.2	odd	2		4005.2.a.x.1.2	yes	17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4005.2.a.w.1.16	✓	17	1.1	even	1	trivial
4005.2.a.x.1.2	yes	17	3.2	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:	\( N \)	\(=\)	\( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4005.a (trivial)

\(f(q)\)	\(=\)	\(q+2.53341 q^{2} +4.41816 q^{4} -1.00000 q^{5} +3.58669 q^{7} +6.12619 q^{8} +O(q^{10})\)	\(q+2.53341 q^{2} +4.41816 q^{4} -1.00000 q^{5} +3.58669 q^{7} +6.12619 q^{8} -2.53341 q^{10} -1.67318 q^{11} +4.71471 q^{13} +9.08656 q^{14} +6.68383 q^{16} -2.27235 q^{17} +2.67631 q^{19} -4.41816 q^{20} -4.23884 q^{22} +1.58933 q^{23} +1.00000 q^{25} +11.9443 q^{26} +15.8466 q^{28} -1.55619 q^{29} +3.18374 q^{31} +4.68049 q^{32} -5.75680 q^{34} -3.58669 q^{35} +1.24043 q^{37} +6.78018 q^{38} -6.12619 q^{40} -3.11783 q^{41} -2.21379 q^{43} -7.39237 q^{44} +4.02643 q^{46} -1.60661 q^{47} +5.86436 q^{49} +2.53341 q^{50} +20.8304 q^{52} +3.96549 q^{53} +1.67318 q^{55} +21.9728 q^{56} -3.94247 q^{58} -5.94080 q^{59} -0.872940 q^{61} +8.06572 q^{62} -1.51006 q^{64} -4.71471 q^{65} +12.8368 q^{67} -10.0396 q^{68} -9.08656 q^{70} +9.63119 q^{71} -1.34058 q^{73} +3.14253 q^{74} +11.8244 q^{76} -6.00117 q^{77} -12.1486 q^{79} -6.68383 q^{80} -7.89874 q^{82} -11.4279 q^{83} +2.27235 q^{85} -5.60844 q^{86} -10.2502 q^{88} +1.00000 q^{89} +16.9102 q^{91} +7.02192 q^{92} -4.07021 q^{94} -2.67631 q^{95} +16.3266 q^{97} +14.8568 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q - 5 q^{2} + 21 q^{4} - 17 q^{5} + 12 q^{7} - 15 q^{8}+O(q^{10}) \) 17 * q - 5 * q^2 + 21 * q^4 - 17 * q^5 + 12 * q^7 - 15 * q^8	\( 17 q - 5 q^{2} + 21 q^{4} - 17 q^{5} + 12 q^{7} - 15 q^{8} + 5 q^{10} - 2 q^{11} + 8 q^{13} - 4 q^{14} + 33 q^{16} - 10 q^{17} + 32 q^{19} - 21 q^{20} + 8 q^{22} - 15 q^{23} + 17 q^{25} + 15 q^{26} + 24 q^{28} - q^{29} + 18 q^{31} - 25 q^{32} + 14 q^{34} - 12 q^{35} + 12 q^{37} - 22 q^{38} + 15 q^{40} + 7 q^{41} + 28 q^{43} + 14 q^{44} + 4 q^{46} - 26 q^{47} + 41 q^{49} - 5 q^{50} + 10 q^{52} - 12 q^{53} + 2 q^{55} - 13 q^{56} + 16 q^{58} + 23 q^{59} + 26 q^{61} - 10 q^{62} + 59 q^{64} - 8 q^{65} + 31 q^{67} + q^{68} + 4 q^{70} + 2 q^{71} + 33 q^{73} + 10 q^{74} + 66 q^{76} - 12 q^{77} + 33 q^{79} - 33 q^{80} + 30 q^{82} - 13 q^{83} + 10 q^{85} + 20 q^{86} + 12 q^{88} + 17 q^{89} + 40 q^{91} - 16 q^{92} + 38 q^{94} - 32 q^{95} + 45 q^{97} - 2 q^{98}+O(q^{100}) \) 17 * q - 5 * q^2 + 21 * q^4 - 17 * q^5 + 12 * q^7 - 15 * q^8 + 5 * q^10 - 2 * q^11 + 8 * q^13 - 4 * q^14 + 33 * q^16 - 10 * q^17 + 32 * q^19 - 21 * q^20 + 8 * q^22 - 15 * q^23 + 17 * q^25 + 15 * q^26 + 24 * q^28 - q^29 + 18 * q^31 - 25 * q^32 + 14 * q^34 - 12 * q^35 + 12 * q^37 - 22 * q^38 + 15 * q^40 + 7 * q^41 + 28 * q^43 + 14 * q^44 + 4 * q^46 - 26 * q^47 + 41 * q^49 - 5 * q^50 + 10 * q^52 - 12 * q^53 + 2 * q^55 - 13 * q^56 + 16 * q^58 + 23 * q^59 + 26 * q^61 - 10 * q^62 + 59 * q^64 - 8 * q^65 + 31 * q^67 + q^68 + 4 * q^70 + 2 * q^71 + 33 * q^73 + 10 * q^74 + 66 * q^76 - 12 * q^77 + 33 * q^79 - 33 * q^80 + 30 * q^82 - 13 * q^83 + 10 * q^85 + 20 * q^86 + 12 * q^88 + 17 * q^89 + 40 * q^91 - 16 * q^92 + 38 * q^94 - 32 * q^95 + 45 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

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Database

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