LMFDB - Embedded newform 4005.2.a.p.1.3

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:	$1$
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} - x^{7} - 11x^{6} + 9x^{5} + 34x^{4} - 19x^{3} - 27x^{2} + 11x - 1 $ x^8 - x^7 - 11x^6 + 9x^5 + 34x^4 - 19x^3 - 27x^2 + 11x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 445)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.16343$ 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)

$f(q)$	$=$	$q-1.16343 q^{2} -0.646427 q^{4} +1.00000 q^{5} +2.23155 q^{7} +3.07894 q^{8} +O(q^{10})$	\(q-1.16343 q^{2} -0.646427 q^{4} +1.00000 q^{5} +2.23155 q^{7} +3.07894 q^{8} -1.16343 q^{10} -6.24237 q^{11} -4.63493 q^{13} -2.59625 q^{14} -2.28928 q^{16} +2.75732 q^{17} +3.29630 q^{19} -0.646427 q^{20} +7.26257 q^{22} +6.86648 q^{23} +1.00000 q^{25} +5.39242 q^{26} -1.44253 q^{28} -1.22407 q^{29} -5.69744 q^{31} -3.49446 q^{32} -3.20795 q^{34} +2.23155 q^{35} +6.87730 q^{37} -3.83501 q^{38} +3.07894 q^{40} -0.954991 q^{41} -12.5627 q^{43} +4.03524 q^{44} -7.98867 q^{46} +3.00176 q^{47} -2.02020 q^{49} -1.16343 q^{50} +2.99615 q^{52} +10.1149 q^{53} -6.24237 q^{55} +6.87079 q^{56} +1.42412 q^{58} -7.06511 q^{59} -2.99922 q^{61} +6.62858 q^{62} +8.64411 q^{64} -4.63493 q^{65} -10.0152 q^{67} -1.78241 q^{68} -2.59625 q^{70} -9.04184 q^{71} +11.2191 q^{73} -8.00126 q^{74} -2.13082 q^{76} -13.9301 q^{77} +0.814296 q^{79} -2.28928 q^{80} +1.11107 q^{82} -2.94088 q^{83} +2.75732 q^{85} +14.6158 q^{86} -19.2199 q^{88} -1.00000 q^{89} -10.3431 q^{91} -4.43868 q^{92} -3.49234 q^{94} +3.29630 q^{95} +11.5840 q^{97} +2.35036 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{2} + 7 q^{4} + 8 q^{5} - 6 q^{7} - 3 q^{8}+O(q^{10}) $ 8 * q - q^2 + 7 * q^4 + 8 * q^5 - 6 * q^7 - 3 * q^8	$ 8 q - q^{2} + 7 q^{4} + 8 q^{5} - 6 q^{7} - 3 q^{8} - q^{10} - 14 q^{11} - 7 q^{13} - 15 q^{14} + 9 q^{16} - 17 q^{17} + 17 q^{19} + 7 q^{20} + 2 q^{22} + q^{23} + 8 q^{25} - 3 q^{26} - 29 q^{28} - 10 q^{29} + q^{31} - 2 q^{32} - 16 q^{34} - 6 q^{35} - 11 q^{37} + 30 q^{38} - 3 q^{40} - 15 q^{41} - 5 q^{43} - 7 q^{44} - 12 q^{46} - 12 q^{47} + 4 q^{49} - q^{50} - 14 q^{52} + q^{53} - 14 q^{55} - 3 q^{56} - 37 q^{58} - 26 q^{59} + 13 q^{61} - 22 q^{62} - 15 q^{64} - 7 q^{65} - 25 q^{67} - 23 q^{68} - 15 q^{70} - 28 q^{71} - 17 q^{73} + 5 q^{74} + 8 q^{76} - 7 q^{79} + 9 q^{80} + 5 q^{82} - 44 q^{83} - 17 q^{85} + 13 q^{86} - 66 q^{88} - 8 q^{89} + 27 q^{91} - 15 q^{92} - 27 q^{94} + 17 q^{95} + q^{97} + 34 q^{98}+O(q^{100}) $ 8 * q - q^2 + 7 * q^4 + 8 * q^5 - 6 * q^7 - 3 * q^8 - q^10 - 14 * q^11 - 7 * q^13 - 15 * q^14 + 9 * q^16 - 17 * q^17 + 17 * q^19 + 7 * q^20 + 2 * q^22 + q^23 + 8 * q^25 - 3 * q^26 - 29 * q^28 - 10 * q^29 + q^31 - 2 * q^32 - 16 * q^34 - 6 * q^35 - 11 * q^37 + 30 * q^38 - 3 * q^40 - 15 * q^41 - 5 * q^43 - 7 * q^44 - 12 * q^46 - 12 * q^47 + 4 * q^49 - q^50 - 14 * q^52 + q^53 - 14 * q^55 - 3 * q^56 - 37 * q^58 - 26 * q^59 + 13 * q^61 - 22 * q^62 - 15 * q^64 - 7 * q^65 - 25 * q^67 - 23 * q^68 - 15 * q^70 - 28 * q^71 - 17 * q^73 + 5 * q^74 + 8 * q^76 - 7 * q^79 + 9 * q^80 + 5 * q^82 - 44 * q^83 - 17 * q^85 + 13 * q^86 - 66 * q^88 - 8 * q^89 + 27 * q^91 - 15 * q^92 - 27 * q^94 + 17 * q^95 + q^97 + 34 * 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 $

$2$	−1.16343	−0.822670	−0.411335	−	0.911484i	$-0.634937\pi$
$2$	−1.16343	−0.822670	−0.411335	+	0.911484i	$0.634937\pi$
$3$	0	0
$3$	0	0
$4$	−0.646427	−0.323214
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.23155	0.843445	0.421723	−	0.906725i	$-0.361426\pi$
$7$	2.23155	0.843445	0.421723	+	0.906725i	$0.361426\pi$
$8$	3.07894	1.08857
$9$	0	0
$10$	−1.16343	−0.367909
$11$	−6.24237	−1.88214	−0.941072	−	0.338205i	$-0.890180\pi$
$11$	−6.24237	−1.88214	−0.941072	+	0.338205i	$0.890180\pi$
$12$	0	0
$13$	−4.63493	−1.28550	−0.642749	−	0.766077i	$-0.722206\pi$
$13$	−4.63493	−1.28550	−0.642749	+	0.766077i	$0.722206\pi$
$14$	−2.59625	−0.693877
$15$	0	0
$16$	−2.28928	−0.572319
$17$	2.75732	0.668748	0.334374	−	0.942441i	$-0.391475\pi$
$17$	2.75732	0.668748	0.334374	+	0.942441i	$0.391475\pi$
$18$	0	0
$19$	3.29630	0.756222	0.378111	−	0.925760i	$-0.376574\pi$
$19$	3.29630	0.756222	0.378111	+	0.925760i	$0.376574\pi$
$20$	−0.646427	−0.144546
$21$	0	0
$22$	7.26257	1.54838
$23$	6.86648	1.43176	0.715880	−	0.698224i	$-0.246026\pi$
$23$	6.86648	1.43176	0.715880	+	0.698224i	$0.246026\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	5.39242	1.05754
$27$	0	0
$28$	−1.44253	−0.272613
$29$	−1.22407	−0.227304	−0.113652	−	0.993521i	$-0.536255\pi$
$29$	−1.22407	−0.227304	−0.113652	+	0.993521i	$0.536255\pi$
$30$	0	0
$31$	−5.69744	−1.02329	−0.511645	−	0.859197i	$-0.670964\pi$
$31$	−5.69744	−1.02329	−0.511645	+	0.859197i	$0.670964\pi$
$32$	−3.49446	−0.617738
$33$	0	0
$34$	−3.20795	−0.550159
$35$	2.23155	0.377200
$36$	0	0
$37$	6.87730	1.13062	0.565310	−	0.824878i	$-0.308756\pi$
$37$	6.87730	1.13062	0.565310	+	0.824878i	$0.308756\pi$
$38$	−3.83501	−0.622121
$39$	0	0
$40$	3.07894	0.486823
$41$	−0.954991	−0.149144	−0.0745722	−	0.997216i	$-0.523759\pi$
$41$	−0.954991	−0.149144	−0.0745722	+	0.997216i	$0.523759\pi$
$42$	0	0
$43$	−12.5627	−1.91579	−0.957894	−	0.287121i	$-0.907302\pi$
$43$	−12.5627	−1.91579	−0.957894	+	0.287121i	$0.907302\pi$
$44$	4.03524	0.608335
$45$	0	0
$46$	−7.98867	−1.17787
$47$	3.00176	0.437852	0.218926	−	0.975741i	$-0.429745\pi$
$47$	3.00176	0.437852	0.218926	+	0.975741i	$0.429745\pi$
$48$	0	0
$49$	−2.02020	−0.288600
$50$	−1.16343	−0.164534
$51$	0	0
$52$	2.99615	0.415491
$53$	10.1149	1.38939	0.694695	−	0.719304i	$-0.255539\pi$
$53$	10.1149	1.38939	0.694695	+	0.719304i	$0.255539\pi$
$54$	0	0
$55$	−6.24237	−0.841721
$56$	6.87079	0.918148
$57$	0	0
$58$	1.42412	0.186996
$59$	−7.06511	−0.919799	−0.459900	−	0.887971i	$-0.652115\pi$
$59$	−7.06511	−0.919799	−0.459900	+	0.887971i	$0.652115\pi$
$60$	0	0
$61$	−2.99922	−0.384010	−0.192005	−	0.981394i	$-0.561499\pi$
$61$	−2.99922	−0.384010	−0.192005	+	0.981394i	$0.561499\pi$
$62$	6.62858	0.841831
$63$	0	0
$64$	8.64411	1.08051
$65$	−4.63493	−0.574892
$66$	0	0
$67$	−10.0152	−1.22355	−0.611776	−	0.791031i	$-0.709545\pi$
$67$	−10.0152	−1.22355	−0.611776	+	0.791031i	$0.709545\pi$
$68$	−1.78241	−0.216148
$69$	0	0
$70$	−2.59625	−0.310311
$71$	−9.04184	−1.07307	−0.536535	−	0.843878i	$-0.680267\pi$
$71$	−9.04184	−1.07307	−0.536535	+	0.843878i	$0.680267\pi$
$72$	0	0
$73$	11.2191	1.31310	0.656548	−	0.754284i	$-0.272016\pi$
$73$	11.2191	1.31310	0.656548	+	0.754284i	$0.272016\pi$
$74$	−8.00126	−0.930128
$75$	0	0
$76$	−2.13082	−0.244421
$77$	−13.9301	−1.58749
$78$	0	0
$79$	0.814296	0.0916155	0.0458077	−	0.998950i	$-0.485414\pi$
$79$	0.814296	0.0916155	0.0458077	+	0.998950i	$0.485414\pi$
$80$	−2.28928	−0.255949
$81$	0	0
$82$	1.11107	0.122697
$83$	−2.94088	−0.322804	−0.161402	−	0.986889i	$-0.551601\pi$
$83$	−2.94088	−0.322804	−0.161402	+	0.986889i	$0.551601\pi$
$84$	0	0
$85$	2.75732	0.299073
$86$	14.6158	1.57606
$87$	0	0
$88$	−19.2199	−2.04884
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	−10.3431	−1.08425
$92$	−4.43868	−0.462764
$93$	0	0
$94$	−3.49234	−0.360208
$95$	3.29630	0.338193
$96$	0	0
$97$	11.5840	1.17617	0.588086	−	0.808798i	$-0.299882\pi$
$97$	11.5840	1.17617	0.588086	+	0.808798i	$0.299882\pi$
$98$	2.35036	0.237423
$99$	0	0
$100$	−0.646427	−0.0646427
$101$	−10.8997	−1.08456	−0.542279	−	0.840199i	$-0.682438\pi$
$101$	−10.8997	−1.08456	−0.542279	+	0.840199i	$0.682438\pi$
$102$	0	0
$103$	13.2170	1.30231	0.651153	−	0.758947i	$-0.274286\pi$
$103$	13.2170	1.30231	0.651153	+	0.758947i	$0.274286\pi$
$104$	−14.2707	−1.39935
$105$	0	0
$106$	−11.7680	−1.14301
$107$	2.56843	0.248299	0.124150	−	0.992264i	$-0.460380\pi$
$107$	2.56843	0.248299	0.124150	+	0.992264i	$0.460380\pi$
$108$	0	0
$109$	−20.8218	−1.99437	−0.997185	−	0.0749844i	$-0.976109\pi$
$109$	−20.8218	−1.99437	−0.997185	+	0.0749844i	$0.976109\pi$
$110$	7.26257	0.692459
$111$	0	0
$112$	−5.10863	−0.482720
$113$	−10.2342	−0.962756	−0.481378	−	0.876513i	$-0.659864\pi$
$113$	−10.2342	−0.962756	−0.481378	+	0.876513i	$0.659864\pi$
$114$	0	0
$115$	6.86648	0.640302
$116$	0.791271	0.0734677
$117$	0	0
$118$	8.21977	0.756691
$119$	6.15308	0.564052
$120$	0	0
$121$	27.9672	2.54247
$122$	3.48938	0.315914
$123$	0	0
$124$	3.68298	0.330742
$125$	1.00000	0.0894427
$126$	0	0
$127$	−18.9903	−1.68512	−0.842559	−	0.538605i	$-0.818952\pi$
$127$	−18.9903	−1.68512	−0.842559	+	0.538605i	$0.818952\pi$
$128$	−3.06792	−0.271169
$129$	0	0
$130$	5.39242	0.472947
$131$	−5.28918	−0.462118	−0.231059	−	0.972940i	$-0.574219\pi$
$131$	−5.28918	−0.462118	−0.231059	+	0.972940i	$0.574219\pi$
$132$	0	0
$133$	7.35584	0.637832
$134$	11.6520	1.00658
$135$	0	0
$136$	8.48961	0.727978
$137$	1.36108	0.116285	0.0581425	−	0.998308i	$-0.481482\pi$
$137$	1.36108	0.116285	0.0581425	+	0.998308i	$0.481482\pi$
$138$	0	0
$139$	−1.45928	−0.123775	−0.0618874	−	0.998083i	$-0.519712\pi$
$139$	−1.45928	−0.123775	−0.0618874	+	0.998083i	$0.519712\pi$
$140$	−1.44253	−0.121916
$141$	0	0
$142$	10.5196	0.882782
$143$	28.9329	2.41949
$144$	0	0
$145$	−1.22407	−0.101653
$146$	−13.0526	−1.08024
$147$	0	0
$148$	−4.44567	−0.365432
$149$	−10.1991	−0.835547	−0.417773	−	0.908551i	$-0.637189\pi$
$149$	−10.1991	−0.835547	−0.417773	+	0.908551i	$0.637189\pi$
$150$	0	0
$151$	19.6293	1.59741	0.798703	−	0.601725i	$-0.205520\pi$
$151$	19.6293	1.59741	0.798703	+	0.601725i	$0.205520\pi$
$152$	10.1491	0.823200
$153$	0	0
$154$	16.2068	1.30598
$155$	−5.69744	−0.457630
$156$	0	0
$157$	−2.79087	−0.222735	−0.111368	−	0.993779i	$-0.535523\pi$
$157$	−2.79087	−0.222735	−0.111368	+	0.993779i	$0.535523\pi$
$158$	−0.947378	−0.0753693
$159$	0	0
$160$	−3.49446	−0.276261
$161$	15.3229	1.20761
$162$	0	0
$163$	2.22536	0.174303	0.0871517	−	0.996195i	$-0.472224\pi$
$163$	2.22536	0.174303	0.0871517	+	0.996195i	$0.472224\pi$
$164$	0.617332	0.0482055
$165$	0	0
$166$	3.42151	0.265561
$167$	−6.89641	−0.533660	−0.266830	−	0.963744i	$-0.585976\pi$
$167$	−6.89641	−0.533660	−0.266830	+	0.963744i	$0.585976\pi$
$168$	0	0
$169$	8.48258	0.652506
$170$	−3.20795	−0.246039
$171$	0	0
$172$	8.12085	0.619209
$173$	−6.88629	−0.523555	−0.261777	−	0.965128i	$-0.584309\pi$
$173$	−6.88629	−0.523555	−0.261777	+	0.965128i	$0.584309\pi$
$174$	0	0
$175$	2.23155	0.168689
$176$	14.2905	1.07719
$177$	0	0
$178$	1.16343	0.0872029
$179$	18.4452	1.37866	0.689331	−	0.724446i	$-0.257904\pi$
$179$	18.4452	1.37866	0.689331	+	0.724446i	$0.257904\pi$
$180$	0	0
$181$	6.69862	0.497905	0.248952	−	0.968516i	$-0.419914\pi$
$181$	6.69862	0.497905	0.248952	+	0.968516i	$0.419914\pi$
$182$	12.0334	0.891978
$183$	0	0
$184$	21.1414	1.55857
$185$	6.87730	0.505629
$186$	0	0
$187$	−17.2122	−1.25868
$188$	−1.94042	−0.141520
$189$	0	0
$190$	−3.83501	−0.278221
$191$	5.40267	0.390923	0.195462	−	0.980711i	$-0.437379\pi$
$191$	5.40267	0.390923	0.195462	+	0.980711i	$0.437379\pi$
$192$	0	0
$193$	−1.28529	−0.0925172	−0.0462586	−	0.998929i	$-0.514730\pi$
$193$	−1.28529	−0.0925172	−0.0462586	+	0.998929i	$0.514730\pi$
$194$	−13.4771	−0.967602
$195$	0	0
$196$	1.30591	0.0932794
$197$	−11.7541	−0.837443	−0.418722	−	0.908115i	$-0.637522\pi$
$197$	−11.7541	−0.837443	−0.418722	+	0.908115i	$0.637522\pi$
$198$	0	0
$199$	−14.5646	−1.03246	−0.516230	−	0.856450i	$-0.672665\pi$
$199$	−14.5646	−1.03246	−0.516230	+	0.856450i	$0.672665\pi$
$200$	3.07894	0.217714
$201$	0	0
$202$	12.6810	0.892233
$203$	−2.73157	−0.191718
$204$	0	0
$205$	−0.954991	−0.0666994
$206$	−15.3770	−1.07137
$207$	0	0
$208$	10.6106	0.735715
$209$	−20.5767	−1.42332
$210$	0	0
$211$	2.66223	0.183276	0.0916379	−	0.995792i	$-0.470790\pi$
$211$	2.66223	0.183276	0.0916379	+	0.995792i	$0.470790\pi$
$212$	−6.53856	−0.449070
$213$	0	0
$214$	−2.98819	−0.204268
$215$	−12.5627	−0.856767
$216$	0	0
$217$	−12.7141	−0.863090
$218$	24.2248	1.64071
$219$	0	0
$220$	4.03524	0.272056
$221$	−12.7800	−0.859674
$222$	0	0
$223$	3.03312	0.203113	0.101556	−	0.994830i	$-0.467618\pi$
$223$	3.03312	0.203113	0.101556	+	0.994830i	$0.467618\pi$
$224$	−7.79804	−0.521029
$225$	0	0
$226$	11.9068	0.792031
$227$	−14.8971	−0.988757	−0.494378	−	0.869247i	$-0.664604\pi$
$227$	−14.8971	−0.988757	−0.494378	+	0.869247i	$0.664604\pi$
$228$	0	0
$229$	11.8049	0.780090	0.390045	−	0.920796i	$-0.372459\pi$
$229$	11.8049	0.780090	0.390045	+	0.920796i	$0.372459\pi$
$230$	−7.98867	−0.526758
$231$	0	0
$232$	−3.76883	−0.247436
$233$	−15.3878	−1.00809	−0.504043	−	0.863679i	$-0.668155\pi$
$233$	−15.3878	−1.00809	−0.504043	+	0.863679i	$0.668155\pi$
$234$	0	0
$235$	3.00176	0.195813
$236$	4.56708	0.297292
$237$	0	0
$238$	−7.15869	−0.464029
$239$	−18.9114	−1.22328	−0.611638	−	0.791138i	$-0.709489\pi$
$239$	−18.9114	−1.22328	−0.611638	+	0.791138i	$0.709489\pi$
$240$	0	0
$241$	−18.0672	−1.16381	−0.581905	−	0.813257i	$-0.697693\pi$
$241$	−18.0672	−1.16381	−0.581905	+	0.813257i	$0.697693\pi$
$242$	−32.5379	−2.09161
$243$	0	0
$244$	1.93877	0.124117
$245$	−2.02020	−0.129066
$246$	0	0
$247$	−15.2781	−0.972122
$248$	−17.5421	−1.11392
$249$	0	0
$250$	−1.16343	−0.0735819
$251$	−1.89277	−0.119471	−0.0597353	−	0.998214i	$-0.519026\pi$
$251$	−1.89277	−0.119471	−0.0597353	+	0.998214i	$0.519026\pi$
$252$	0	0
$253$	−42.8631	−2.69478
$254$	22.0939	1.38630
$255$	0	0
$256$	−13.7189	−0.857432
$257$	6.30045	0.393011	0.196506	−	0.980503i	$-0.437041\pi$
$257$	6.30045	0.393011	0.196506	+	0.980503i	$0.437041\pi$
$258$	0	0
$259$	15.3470	0.953617
$260$	2.99615	0.185813
$261$	0	0
$262$	6.15360	0.380171
$263$	−16.7289	−1.03155	−0.515774	−	0.856725i	$-0.672496\pi$
$263$	−16.7289	−1.03155	−0.515774	+	0.856725i	$0.672496\pi$
$264$	0	0
$265$	10.1149	0.621354
$266$	−8.55801	−0.524725
$267$	0	0
$268$	6.47410	0.395469
$269$	−16.5062	−1.00640	−0.503200	−	0.864170i	$-0.667844\pi$
$269$	−16.5062	−1.00640	−0.503200	+	0.864170i	$0.667844\pi$
$270$	0	0
$271$	4.84164	0.294108	0.147054	−	0.989128i	$-0.453021\pi$
$271$	4.84164	0.294108	0.147054	+	0.989128i	$0.453021\pi$
$272$	−6.31226	−0.382737
$273$	0	0
$274$	−1.58352	−0.0956642
$275$	−6.24237	−0.376429
$276$	0	0
$277$	−24.8548	−1.49338	−0.746690	−	0.665172i	$-0.768358\pi$
$277$	−24.8548	−1.49338	−0.746690	+	0.665172i	$0.768358\pi$
$278$	1.69778	0.101826
$279$	0	0
$280$	6.87079	0.410608
$281$	−4.50358	−0.268661	−0.134330	−	0.990937i	$-0.542888\pi$
$281$	−4.50358	−0.268661	−0.134330	+	0.990937i	$0.542888\pi$
$282$	0	0
$283$	22.3918	1.33106	0.665528	−	0.746373i	$-0.268206\pi$
$283$	22.3918	1.33106	0.665528	+	0.746373i	$0.268206\pi$
$284$	5.84489	0.346831
$285$	0	0
$286$	−33.6615	−1.99045
$287$	−2.13111	−0.125795
$288$	0	0
$289$	−9.39720	−0.552776
$290$	1.42412	0.0836272
$291$	0	0
$292$	−7.25233	−0.424410
$293$	11.3874	0.665260	0.332630	−	0.943057i	$-0.392064\pi$
$293$	11.3874	0.665260	0.332630	+	0.943057i	$0.392064\pi$
$294$	0	0
$295$	−7.06511	−0.411347
$296$	21.1748	1.23076
$297$	0	0
$298$	11.8660	0.687379
$299$	−31.8256	−1.84052
$300$	0	0
$301$	−28.0342	−1.61586
$302$	−22.8373	−1.31414
$303$	0	0
$304$	−7.54613	−0.432801
$305$	−2.99922	−0.171735
$306$	0	0
$307$	−25.3517	−1.44690	−0.723450	−	0.690377i	$-0.757445\pi$
$307$	−25.3517	−1.44690	−0.723450	+	0.690377i	$0.757445\pi$
$308$	9.00482	0.513097
$309$	0	0
$310$	6.62858	0.376478
$311$	−10.7151	−0.607600	−0.303800	−	0.952736i	$-0.598255\pi$
$311$	−10.7151	−0.607600	−0.303800	+	0.952736i	$0.598255\pi$
$312$	0	0
$313$	−25.6198	−1.44811	−0.724057	−	0.689740i	$-0.757725\pi$
$313$	−25.6198	−1.44811	−0.724057	+	0.689740i	$0.757725\pi$
$314$	3.24698	0.183238
$315$	0	0
$316$	−0.526383	−0.0296114
$317$	−34.3428	−1.92889	−0.964443	−	0.264292i	$-0.914862\pi$
$317$	−34.3428	−1.92889	−0.964443	+	0.264292i	$0.914862\pi$
$318$	0	0
$319$	7.64109	0.427819
$320$	8.64411	0.483221
$321$	0	0
$322$	−17.8271	−0.993466
$323$	9.08893	0.505722
$324$	0	0
$325$	−4.63493	−0.257100
$326$	−2.58905	−0.143394
$327$	0	0
$328$	−2.94036	−0.162354
$329$	6.69857	0.369304
$330$	0	0
$331$	−20.2047	−1.11055	−0.555276	−	0.831666i	$-0.687387\pi$
$331$	−20.2047	−1.11055	−0.555276	+	0.831666i	$0.687387\pi$
$332$	1.90107	0.104335
$333$	0	0
$334$	8.02350	0.439026
$335$	−10.0152	−0.547189
$336$	0	0
$337$	−13.4799	−0.734299	−0.367149	−	0.930162i	$-0.619666\pi$
$337$	−13.4799	−0.734299	−0.367149	+	0.930162i	$0.619666\pi$
$338$	−9.86890	−0.536797
$339$	0	0
$340$	−1.78241	−0.0966645
$341$	35.5655	1.92598
$342$	0	0
$343$	−20.1290	−1.08686
$344$	−38.6797	−2.08547
$345$	0	0
$346$	8.01172	0.430713
$347$	−11.8450	−0.635875	−0.317938	−	0.948112i	$-0.602990\pi$
$347$	−11.8450	−0.635875	−0.317938	+	0.948112i	$0.602990\pi$
$348$	0	0
$349$	5.01356	0.268370	0.134185	−	0.990956i	$-0.457158\pi$
$349$	5.01356	0.268370	0.134185	+	0.990956i	$0.457158\pi$
$350$	−2.59625	−0.138775
$351$	0	0
$352$	21.8137	1.16267
$353$	4.72417	0.251442	0.125721	−	0.992066i	$-0.459876\pi$
$353$	4.72417	0.251442	0.125721	+	0.992066i	$0.459876\pi$
$354$	0	0
$355$	−9.04184	−0.479891
$356$	0.646427	0.0342606
$357$	0	0
$358$	−21.4598	−1.13418
$359$	21.2719	1.12269	0.561345	−	0.827582i	$-0.310284\pi$
$359$	21.2719	1.12269	0.561345	+	0.827582i	$0.310284\pi$
$360$	0	0
$361$	−8.13443	−0.428128
$362$	−7.79339	−0.409611
$363$	0	0
$364$	6.68604	0.350444
$365$	11.2191	0.587234
$366$	0	0
$367$	17.4395	0.910335	0.455167	−	0.890406i	$-0.349579\pi$
$367$	17.4395	0.910335	0.455167	+	0.890406i	$0.349579\pi$
$368$	−15.7193	−0.819424
$369$	0	0
$370$	−8.00126	−0.415966
$371$	22.5719	1.17187
$372$	0	0
$373$	22.9561	1.18862	0.594312	−	0.804235i	$-0.297424\pi$
$373$	22.9561	1.18862	0.594312	+	0.804235i	$0.297424\pi$
$374$	20.0252	1.03548
$375$	0	0
$376$	9.24223	0.476632
$377$	5.67347	0.292199
$378$	0	0
$379$	7.92479	0.407069	0.203535	−	0.979068i	$-0.434757\pi$
$379$	7.92479	0.407069	0.203535	+	0.979068i	$0.434757\pi$
$380$	−2.13082	−0.109309
$381$	0	0
$382$	−6.28563	−0.321601
$383$	−1.13829	−0.0581641	−0.0290821	−	0.999577i	$-0.509258\pi$
$383$	−1.13829	−0.0581641	−0.0290821	+	0.999577i	$0.509258\pi$
$384$	0	0
$385$	−13.9301	−0.709945
$386$	1.49535	0.0761111
$387$	0	0
$388$	−7.48818	−0.380155
$389$	9.12407	0.462609	0.231304	−	0.972881i	$-0.425701\pi$
$389$	9.12407	0.462609	0.231304	+	0.972881i	$0.425701\pi$
$390$	0	0
$391$	18.9331	0.957486
$392$	−6.22007	−0.314161
$393$	0	0
$394$	13.6751	0.688940
$395$	0.814296	0.0409717
$396$	0	0
$397$	−36.7768	−1.84577	−0.922887	−	0.385072i	$-0.874177\pi$
$397$	−36.7768	−1.84577	−0.922887	+	0.385072i	$0.874177\pi$
$398$	16.9450	0.849374
$399$	0	0
$400$	−2.28928	−0.114464
$401$	16.6783	0.832875	0.416437	−	0.909164i	$-0.363279\pi$
$401$	16.6783	0.832875	0.416437	+	0.909164i	$0.363279\pi$
$402$	0	0
$403$	26.4072	1.31544
$404$	7.04584	0.350544
$405$	0	0
$406$	3.17799	0.157721
$407$	−42.9306	−2.12799
$408$	0	0
$409$	−4.71669	−0.233225	−0.116613	−	0.993177i	$-0.537204\pi$
$409$	−4.71669	−0.233225	−0.116613	+	0.993177i	$0.537204\pi$
$410$	1.11107	0.0548716
$411$	0	0
$412$	−8.54380	−0.420923
$413$	−15.7661	−0.775800
$414$	0	0
$415$	−2.94088	−0.144362
$416$	16.1966	0.794102
$417$	0	0
$418$	23.9396	1.17092
$419$	0.0448056	0.00218890	0.00109445	−	0.999999i	$-0.499652\pi$
$419$	0.0448056	0.00218890	0.00109445	+	0.999999i	$0.499652\pi$
$420$	0	0
$421$	15.1761	0.739639	0.369820	−	0.929104i	$-0.379420\pi$
$421$	15.1761	0.739639	0.369820	+	0.929104i	$0.379420\pi$
$422$	−3.09733	−0.150776
$423$	0	0
$424$	31.1432	1.51245
$425$	2.75732	0.133750
$426$	0	0
$427$	−6.69289	−0.323892
$428$	−1.66030	−0.0802537
$429$	0	0
$430$	14.6158	0.704836
$431$	25.6222	1.23418	0.617089	−	0.786893i	$-0.288312\pi$
$431$	25.6222	1.23418	0.617089	+	0.786893i	$0.288312\pi$
$432$	0	0
$433$	−0.162207	−0.00779517	−0.00389759	−	0.999992i	$-0.501241\pi$
$433$	−0.162207	−0.00779517	−0.00389759	+	0.999992i	$0.501241\pi$
$434$	14.7920	0.710038
$435$	0	0
$436$	13.4598	0.644607
$437$	22.6339	1.08273
$438$	0	0
$439$	−23.2447	−1.10941	−0.554704	−	0.832048i	$-0.687168\pi$
$439$	−23.2447	−1.10941	−0.554704	+	0.832048i	$0.687168\pi$
$440$	−19.2199	−0.916271
$441$	0	0
$442$	14.8686	0.707228
$443$	40.0330	1.90202	0.951012	−	0.309153i	$-0.100045\pi$
$443$	40.0330	1.90202	0.951012	+	0.309153i	$0.100045\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	−3.52883	−0.167095
$447$	0	0
$448$	19.2897	0.911355
$449$	−4.85303	−0.229029	−0.114514	−	0.993422i	$-0.536531\pi$
$449$	−4.85303	−0.229029	−0.114514	+	0.993422i	$0.536531\pi$
$450$	0	0
$451$	5.96140	0.280711
$452$	6.61569	0.311176
$453$	0	0
$454$	17.3318	0.813421
$455$	−10.3431	−0.484890
$456$	0	0
$457$	19.5616	0.915054	0.457527	−	0.889196i	$-0.348735\pi$
$457$	19.5616	0.915054	0.457527	+	0.889196i	$0.348735\pi$
$458$	−13.7342	−0.641757
$459$	0	0
$460$	−4.43868	−0.206954
$461$	16.8856	0.786440	0.393220	−	0.919444i	$-0.371361\pi$
$461$	16.8856	0.786440	0.393220	+	0.919444i	$0.371361\pi$
$462$	0	0
$463$	−23.2862	−1.08220	−0.541102	−	0.840957i	$-0.681993\pi$
$463$	−23.2862	−1.08220	−0.541102	+	0.840957i	$0.681993\pi$
$464$	2.80223	0.130090
$465$	0	0
$466$	17.9026	0.829322
$467$	−38.4332	−1.77848	−0.889239	−	0.457443i	$-0.848765\pi$
$467$	−38.4332	−1.77848	−0.889239	+	0.457443i	$0.848765\pi$
$468$	0	0
$469$	−22.3494	−1.03200
$470$	−3.49234	−0.161090
$471$	0	0
$472$	−21.7530	−1.00126
$473$	78.4208	3.60579
$474$	0	0
$475$	3.29630	0.151244
$476$	−3.97752	−0.182309
$477$	0	0
$478$	22.0021	1.00635
$479$	−34.0667	−1.55655	−0.778273	−	0.627926i	$-0.783904\pi$
$479$	−34.0667	−1.55655	−0.778273	+	0.627926i	$0.783904\pi$
$480$	0	0
$481$	−31.8758	−1.45341
$482$	21.0200	0.957432
$483$	0	0
$484$	−18.0787	−0.821761
$485$	11.5840	0.526000
$486$	0	0
$487$	−32.0563	−1.45261	−0.726304	−	0.687374i	$-0.758764\pi$
$487$	−32.0563	−1.45261	−0.726304	+	0.687374i	$0.758764\pi$
$488$	−9.23439	−0.418021
$489$	0	0
$490$	2.35036	0.106179
$491$	−35.1106	−1.58452	−0.792260	−	0.610183i	$-0.791096\pi$
$491$	−35.1106	−1.58452	−0.792260	+	0.610183i	$0.791096\pi$
$492$	0	0
$493$	−3.37515	−0.152009
$494$	17.7750	0.799736
$495$	0	0
$496$	13.0430	0.585649
$497$	−20.1773	−0.905075
$498$	0	0
$499$	−17.9968	−0.805648	−0.402824	−	0.915277i	$-0.631971\pi$
$499$	−17.9968	−0.805648	−0.402824	+	0.915277i	$0.631971\pi$
$500$	−0.646427	−0.0289091
$501$	0	0
$502$	2.20211	0.0982849
$503$	−5.23480	−0.233408	−0.116704	−	0.993167i	$-0.537233\pi$
$503$	−5.23480	−0.233408	−0.116704	+	0.993167i	$0.537233\pi$
$504$	0	0
$505$	−10.8997	−0.485029
$506$	49.8682	2.21691
$507$	0	0
$508$	12.2759	0.544653
$509$	0.165304	0.00732699	0.00366350	−	0.999993i	$-0.498834\pi$
$509$	0.165304	0.00732699	0.00366350	+	0.999993i	$0.498834\pi$
$510$	0	0
$511$	25.0359	1.10752
$512$	22.0969	0.976552
$513$	0	0
$514$	−7.33014	−0.323319
$515$	13.2170	0.582409
$516$	0	0
$517$	−18.7381	−0.824101
$518$	−17.8552	−0.784512
$519$	0	0
$520$	−14.2707	−0.625810
$521$	−4.79905	−0.210250	−0.105125	−	0.994459i	$-0.533524\pi$
$521$	−4.79905	−0.210250	−0.105125	+	0.994459i	$0.533524\pi$
$522$	0	0
$523$	20.0143	0.875164	0.437582	−	0.899179i	$-0.355835\pi$
$523$	20.0143	0.875164	0.437582	+	0.899179i	$0.355835\pi$
$524$	3.41907	0.149363
$525$	0	0
$526$	19.4629	0.848624
$527$	−15.7097	−0.684323
$528$	0	0
$529$	24.1485	1.04993
$530$	−11.7680	−0.511170
$531$	0	0
$532$	−4.75501	−0.206156
$533$	4.42631	0.191725
$534$	0	0
$535$	2.56843	0.111043
$536$	−30.8362	−1.33192
$537$	0	0
$538$	19.2038	0.827936
$539$	12.6108	0.543187
$540$	0	0
$541$	34.1351	1.46758	0.733792	−	0.679374i	$-0.237749\pi$
$541$	34.1351	1.46758	0.733792	+	0.679374i	$0.237749\pi$
$542$	−5.63291	−0.241954
$543$	0	0
$544$	−9.63533	−0.413111
$545$	−20.8218	−0.891909
$546$	0	0
$547$	7.41855	0.317194	0.158597	−	0.987343i	$-0.449303\pi$
$547$	7.41855	0.317194	0.158597	+	0.987343i	$0.449303\pi$
$548$	−0.879840	−0.0375849
$549$	0	0
$550$	7.26257	0.309677
$551$	−4.03489	−0.171892
$552$	0	0
$553$	1.81714	0.0772726
$554$	28.9168	1.22856
$555$	0	0
$556$	0.943320	0.0400057
$557$	0.699946	0.0296577	0.0148288	−	0.999890i	$-0.495280\pi$
$557$	0.699946	0.0296577	0.0148288	+	0.999890i	$0.495280\pi$
$558$	0	0
$559$	58.2271	2.46274
$560$	−5.10863	−0.215879
$561$	0	0
$562$	5.23960	0.221019
$563$	−37.4498	−1.57832	−0.789161	−	0.614186i	$-0.789484\pi$
$563$	−37.4498	−1.57832	−0.789161	+	0.614186i	$0.789484\pi$
$564$	0	0
$565$	−10.2342	−0.430558
$566$	−26.0514	−1.09502
$567$	0	0
$568$	−27.8393	−1.16811
$569$	16.5455	0.693625	0.346813	−	0.937934i	$-0.387264\pi$
$569$	16.5455	0.693625	0.346813	+	0.937934i	$0.387264\pi$
$570$	0	0
$571$	−26.1728	−1.09530	−0.547650	−	0.836708i	$-0.684477\pi$
$571$	−26.1728	−1.09530	−0.547650	+	0.836708i	$0.684477\pi$
$572$	−18.7030	−0.782013
$573$	0	0
$574$	2.47940	0.103488
$575$	6.86648	0.286352
$576$	0	0
$577$	38.9896	1.62316	0.811580	−	0.584241i	$-0.198608\pi$
$577$	38.9896	1.62316	0.811580	+	0.584241i	$0.198608\pi$
$578$	10.9330	0.454753
$579$	0	0
$580$	0.791271	0.0328558
$581$	−6.56271	−0.272267
$582$	0	0
$583$	−63.1410	−2.61503
$584$	34.5429	1.42939
$585$	0	0
$586$	−13.2485	−0.547289
$587$	26.5530	1.09596	0.547979	−	0.836492i	$-0.315397\pi$
$587$	26.5530	1.09596	0.547979	+	0.836492i	$0.315397\pi$
$588$	0	0
$589$	−18.7805	−0.773835
$590$	8.21977	0.338403
$591$	0	0
$592$	−15.7440	−0.647076
$593$	17.3600	0.712891	0.356445	−	0.934316i	$-0.383989\pi$
$593$	17.3600	0.712891	0.356445	+	0.934316i	$0.383989\pi$
$594$	0	0
$595$	6.15308	0.252252
$596$	6.59301	0.270060
$597$	0	0
$598$	37.0269	1.51414
$599$	−4.39556	−0.179598	−0.0897989	−	0.995960i	$-0.528622\pi$
$599$	−4.39556	−0.179598	−0.0897989	+	0.995960i	$0.528622\pi$
$600$	0	0
$601$	−29.7115	−1.21196	−0.605978	−	0.795481i	$-0.707218\pi$
$601$	−29.7115	−1.21196	−0.605978	+	0.795481i	$0.707218\pi$
$602$	32.6158	1.32932
$603$	0	0
$604$	−12.6889	−0.516303
$605$	27.9672	1.13703
$606$	0	0
$607$	−40.6274	−1.64902	−0.824508	−	0.565851i	$-0.808548\pi$
$607$	−40.6274	−1.64902	−0.824508	+	0.565851i	$0.808548\pi$
$608$	−11.5188	−0.467147
$609$	0	0
$610$	3.48938	0.141281
$611$	−13.9130	−0.562858
$612$	0	0
$613$	−13.8543	−0.559571	−0.279785	−	0.960063i	$-0.590263\pi$
$613$	−13.8543	−0.559571	−0.279785	+	0.960063i	$0.590263\pi$
$614$	29.4950	1.19032
$615$	0	0
$616$	−42.8900	−1.72809
$617$	30.0491	1.20973	0.604866	−	0.796327i	$-0.293227\pi$
$617$	30.0491	1.20973	0.604866	+	0.796327i	$0.293227\pi$
$618$	0	0
$619$	−16.3623	−0.657654	−0.328827	−	0.944390i	$-0.606653\pi$
$619$	−16.3623	−0.657654	−0.328827	+	0.944390i	$0.606653\pi$
$620$	3.68298	0.147912
$621$	0	0
$622$	12.4663	0.499854
$623$	−2.23155	−0.0894050
$624$	0	0
$625$	1.00000	0.0400000
$626$	29.8068	1.19132
$627$	0	0
$628$	1.80409	0.0719911
$629$	18.9629	0.756100
$630$	0	0
$631$	27.8270	1.10778	0.553888	−	0.832591i	$-0.313144\pi$
$631$	27.8270	1.10778	0.553888	+	0.832591i	$0.313144\pi$
$632$	2.50717	0.0997297
$633$	0	0
$634$	39.9555	1.58684
$635$	−18.9903	−0.753607
$636$	0	0
$637$	9.36348	0.370995
$638$	−8.88988	−0.351954
$639$	0	0
$640$	−3.06792	−0.121270
$641$	22.6216	0.893498	0.446749	−	0.894659i	$-0.352582\pi$
$641$	22.6216	0.893498	0.446749	+	0.894659i	$0.352582\pi$
$642$	0	0
$643$	−9.16994	−0.361627	−0.180814	−	0.983517i	$-0.557873\pi$
$643$	−9.16994	−0.361627	−0.180814	+	0.983517i	$0.557873\pi$
$644$	−9.90512	−0.390316
$645$	0	0
$646$	−10.5744	−0.416042
$647$	−10.6302	−0.417918	−0.208959	−	0.977924i	$-0.567007\pi$
$647$	−10.6302	−0.417918	−0.208959	+	0.977924i	$0.567007\pi$
$648$	0	0
$649$	44.1030	1.73120
$650$	5.39242	0.211508
$651$	0	0
$652$	−1.43853	−0.0563372
$653$	6.46716	0.253079	0.126540	−	0.991962i	$-0.459613\pi$
$653$	6.46716	0.253079	0.126540	+	0.991962i	$0.459613\pi$
$654$	0	0
$655$	−5.28918	−0.206665
$656$	2.18624	0.0853583
$657$	0	0
$658$	−7.79333	−0.303816
$659$	29.7579	1.15920	0.579601	−	0.814900i	$-0.303208\pi$
$659$	29.7579	1.15920	0.579601	+	0.814900i	$0.303208\pi$
$660$	0	0
$661$	34.3396	1.33566	0.667829	−	0.744315i	$-0.267224\pi$
$661$	34.3396	1.33566	0.667829	+	0.744315i	$0.267224\pi$
$662$	23.5068	0.913618
$663$	0	0
$664$	−9.05478	−0.351394
$665$	7.35584	0.285247
$666$	0	0
$667$	−8.40504	−0.325444
$668$	4.45803	0.172486
$669$	0	0
$670$	11.6520	0.450156
$671$	18.7222	0.722763
$672$	0	0
$673$	21.0833	0.812700	0.406350	−	0.913717i	$-0.366801\pi$
$673$	21.0833	0.812700	0.406350	+	0.913717i	$0.366801\pi$
$674$	15.6830	0.604086
$675$	0	0
$676$	−5.48337	−0.210899
$677$	−12.0279	−0.462270	−0.231135	−	0.972922i	$-0.574244\pi$
$677$	−12.0279	−0.462270	−0.231135	+	0.972922i	$0.574244\pi$
$678$	0	0
$679$	25.8501	0.992037
$680$	8.48961	0.325562
$681$	0	0
$682$	−41.3781	−1.58445
$683$	−13.8096	−0.528409	−0.264204	−	0.964467i	$-0.585109\pi$
$683$	−13.8096	−0.528409	−0.264204	+	0.964467i	$0.585109\pi$
$684$	0	0
$685$	1.36108	0.0520042
$686$	23.4187	0.894130
$687$	0	0
$688$	28.7594	1.09644
$689$	−46.8819	−1.78606
$690$	0	0
$691$	23.0764	0.877869	0.438935	−	0.898519i	$-0.355356\pi$
$691$	23.0764	0.877869	0.438935	+	0.898519i	$0.355356\pi$
$692$	4.45148	0.169220
$693$	0	0
$694$	13.7809	0.523116
$695$	−1.45928	−0.0553538
$696$	0	0
$697$	−2.63321	−0.0997400
$698$	−5.83293	−0.220780
$699$	0	0
$700$	−1.44253	−0.0545226
$701$	−2.34166	−0.0884431	−0.0442216	−	0.999022i	$-0.514081\pi$
$701$	−2.34166	−0.0884431	−0.0442216	+	0.999022i	$0.514081\pi$
$702$	0	0
$703$	22.6696	0.855000
$704$	−53.9597	−2.03368
$705$	0	0
$706$	−5.49624	−0.206854
$707$	−24.3231	−0.914765
$708$	0	0
$709$	−14.8131	−0.556317	−0.278159	−	0.960535i	$-0.589724\pi$
$709$	−14.8131	−0.556317	−0.278159	+	0.960535i	$0.589724\pi$
$710$	10.5196	0.394792
$711$	0	0
$712$	−3.07894	−0.115388
$713$	−39.1214	−1.46511
$714$	0	0
$715$	28.9329	1.08203
$716$	−11.9235	−0.445603
$717$	0	0
$718$	−24.7484	−0.923603
$719$	4.73254	0.176494	0.0882469	−	0.996099i	$-0.471874\pi$
$719$	4.73254	0.176494	0.0882469	+	0.996099i	$0.471874\pi$
$720$	0	0
$721$	29.4943	1.09842
$722$	9.46386	0.352208
$723$	0	0
$724$	−4.33017	−0.160930
$725$	−1.22407	−0.0454608
$726$	0	0
$727$	0.313014	0.0116090	0.00580452	−	0.999983i	$-0.498152\pi$
$727$	0.313014	0.0116090	0.00580452	+	0.999983i	$0.498152\pi$
$728$	−31.8456	−1.18028
$729$	0	0
$730$	−13.0526	−0.483100
$731$	−34.6393	−1.28118
$732$	0	0
$733$	17.4239	0.643566	0.321783	−	0.946814i	$-0.395718\pi$
$733$	17.4239	0.643566	0.321783	+	0.946814i	$0.395718\pi$
$734$	−20.2897	−0.748905
$735$	0	0
$736$	−23.9946	−0.884453
$737$	62.5186	2.30290
$738$	0	0
$739$	2.15336	0.0792126	0.0396063	−	0.999215i	$-0.487390\pi$
$739$	2.15336	0.0792126	0.0396063	+	0.999215i	$0.487390\pi$
$740$	−4.44567	−0.163426
$741$	0	0
$742$	−26.2609	−0.964067
$743$	37.1193	1.36177	0.680887	−	0.732388i	$-0.261594\pi$
$743$	37.1193	1.36177	0.680887	+	0.732388i	$0.261594\pi$
$744$	0	0
$745$	−10.1991	−0.373668
$746$	−26.7079	−0.977846
$747$	0	0
$748$	11.1264	0.406823
$749$	5.73157	0.209427
$750$	0	0
$751$	−3.42199	−0.124870	−0.0624351	−	0.998049i	$-0.519887\pi$
$751$	−3.42199	−0.124870	−0.0624351	+	0.998049i	$0.519887\pi$
$752$	−6.87186	−0.250591
$753$	0	0
$754$	−6.60070	−0.240383
$755$	19.6293	0.714382
$756$	0	0
$757$	−42.7111	−1.55236	−0.776181	−	0.630510i	$-0.782846\pi$
$757$	−42.7111	−1.55236	−0.776181	+	0.630510i	$0.782846\pi$
$758$	−9.21995	−0.334884
$759$	0	0
$760$	10.1491	0.368146
$761$	−49.0963	−1.77974	−0.889871	−	0.456213i	$-0.849206\pi$
$761$	−49.0963	−1.77974	−0.889871	+	0.456213i	$0.849206\pi$
$762$	0	0
$763$	−46.4649	−1.68214
$764$	−3.49243	−0.126352
$765$	0	0
$766$	1.32433	0.0478499
$767$	32.7463	1.18240
$768$	0	0
$769$	33.6377	1.21301	0.606503	−	0.795081i	$-0.292572\pi$
$769$	33.6377	1.21301	0.606503	+	0.795081i	$0.292572\pi$
$770$	16.2068	0.584051
$771$	0	0
$772$	0.830846	0.0299028
$773$	−42.5005	−1.52864	−0.764319	−	0.644838i	$-0.776925\pi$
$773$	−42.5005	−1.52864	−0.764319	+	0.644838i	$0.776925\pi$
$774$	0	0
$775$	−5.69744	−0.204658
$776$	35.6663	1.28034
$777$	0	0
$778$	−10.6152	−0.380575
$779$	−3.14793	−0.112786
$780$	0	0
$781$	56.4425	2.01967
$782$	−22.0273	−0.787695
$783$	0	0
$784$	4.62480	0.165171
$785$	−2.79087	−0.0996103
$786$	0	0
$787$	−16.9945	−0.605790	−0.302895	−	0.953024i	$-0.597953\pi$
$787$	−16.9945	−0.605790	−0.302895	+	0.953024i	$0.597953\pi$
$788$	7.59816	0.270673
$789$	0	0
$790$	−0.947378	−0.0337062
$791$	−22.8382	−0.812032
$792$	0	0
$793$	13.9012	0.493644
$794$	42.7873	1.51846
$795$	0	0
$796$	9.41498	0.333705
$797$	18.2917	0.647926	0.323963	−	0.946070i	$-0.394985\pi$
$797$	18.2917	0.647926	0.323963	+	0.946070i	$0.394985\pi$
$798$	0	0
$799$	8.27681	0.292812
$800$	−3.49446	−0.123548
$801$	0	0
$802$	−19.4041	−0.685181
$803$	−70.0337	−2.47144
$804$	0	0
$805$	15.3229	0.540060
$806$	−30.7230	−1.08217
$807$	0	0
$808$	−33.5594	−1.18062
$809$	−10.0297	−0.352626	−0.176313	−	0.984334i	$-0.556417\pi$
$809$	−10.0297	−0.352626	−0.176313	+	0.984334i	$0.556417\pi$
$810$	0	0
$811$	−4.44600	−0.156120	−0.0780601	−	0.996949i	$-0.524873\pi$
$811$	−4.44600	−0.156120	−0.0780601	+	0.996949i	$0.524873\pi$
$812$	1.76576	0.0619660
$813$	0	0
$814$	49.9468	1.75064
$815$	2.22536	0.0779508
$816$	0	0
$817$	−41.4103	−1.44876
$818$	5.48754	0.191867
$819$	0	0
$820$	0.617332	0.0215582
$821$	37.9696	1.32515	0.662575	−	0.748996i	$-0.269464\pi$
$821$	37.9696	1.32515	0.662575	+	0.748996i	$0.269464\pi$
$822$	0	0
$823$	−48.4143	−1.68761	−0.843807	−	0.536646i	$-0.819691\pi$
$823$	−48.4143	−1.68761	−0.843807	+	0.536646i	$0.819691\pi$
$824$	40.6942	1.41765
$825$	0	0
$826$	18.3428	0.638228
$827$	−10.3587	−0.360206	−0.180103	−	0.983648i	$-0.557643\pi$
$827$	−10.3587	−0.360206	−0.180103	+	0.983648i	$0.557643\pi$
$828$	0	0
$829$	6.47084	0.224742	0.112371	−	0.993666i	$-0.464156\pi$
$829$	6.47084	0.224742	0.112371	+	0.993666i	$0.464156\pi$
$830$	3.42151	0.118762
$831$	0	0
$832$	−40.0649	−1.38900
$833$	−5.57033	−0.193001
$834$	0	0
$835$	−6.89641	−0.238660
$836$	13.3013	0.460036
$837$	0	0
$838$	−0.0521283	−0.00180074
$839$	0.873216	0.0301468	0.0150734	−	0.999886i	$-0.495202\pi$
$839$	0.873216	0.0301468	0.0150734	+	0.999886i	$0.495202\pi$
$840$	0	0
$841$	−27.5017	−0.948333
$842$	−17.6564	−0.608479
$843$	0	0
$844$	−1.72094	−0.0592372
$845$	8.48258	0.291809
$846$	0	0
$847$	62.4100	2.14443
$848$	−23.1558	−0.795175
$849$	0	0
$850$	−3.20795	−0.110032
$851$	47.2228	1.61878
$852$	0	0
$853$	−13.2634	−0.454131	−0.227065	−	0.973880i	$-0.572913\pi$
$853$	−13.2634	−0.454131	−0.227065	+	0.973880i	$0.572913\pi$
$854$	7.78672	0.266456
$855$	0	0
$856$	7.90803	0.270291
$857$	46.9796	1.60479	0.802397	−	0.596791i	$-0.203558\pi$
$857$	46.9796	1.60479	0.802397	+	0.596791i	$0.203558\pi$
$858$	0	0
$859$	3.10197	0.105838	0.0529189	−	0.998599i	$-0.483148\pi$
$859$	3.10197	0.105838	0.0529189	+	0.998599i	$0.483148\pi$
$860$	8.12085	0.276919
$861$	0	0
$862$	−29.8097	−1.01532
$863$	15.4875	0.527201	0.263601	−	0.964632i	$-0.415090\pi$
$863$	15.4875	0.527201	0.263601	+	0.964632i	$0.415090\pi$
$864$	0	0
$865$	−6.88629	−0.234141
$866$	0.188717	0.00641286
$867$	0	0
$868$	8.21875	0.278962
$869$	−5.08314	−0.172434
$870$	0	0
$871$	46.4198	1.57287
$872$	−64.1091	−2.17101
$873$	0	0
$874$	−26.3330	−0.890728
$875$	2.23155	0.0754400
$876$	0	0
$877$	−18.5712	−0.627103	−0.313552	−	0.949571i	$-0.601519\pi$
$877$	−18.5712	−0.627103	−0.313552	+	0.949571i	$0.601519\pi$
$878$	27.0436	0.912677
$879$	0	0
$880$	14.2905	0.481733
$881$	57.3001	1.93049	0.965244	−	0.261350i	$-0.0841677\pi$
$881$	57.3001	1.93049	0.965244	+	0.261350i	$0.0841677\pi$
$882$	0	0
$883$	21.3909	0.719863	0.359931	−	0.932979i	$-0.382800\pi$
$883$	21.3909	0.719863	0.359931	+	0.932979i	$0.382800\pi$
$884$	8.26132	0.277858
$885$	0	0
$886$	−46.5756	−1.56474
$887$	−13.1731	−0.442309	−0.221155	−	0.975239i	$-0.570983\pi$
$887$	−13.1731	−0.442309	−0.221155	+	0.975239i	$0.570983\pi$
$888$	0	0
$889$	−42.3778	−1.42130
$890$	1.16343	0.0389983
$891$	0	0
$892$	−1.96069	−0.0656488
$893$	9.89469	0.331113
$894$	0	0
$895$	18.4452	0.616557
$896$	−6.84621	−0.228716
$897$	0	0
$898$	5.64616	0.188415
$899$	6.97406	0.232598
$900$	0	0
$901$	27.8900	0.929152
$902$	−6.93568	−0.230933
$903$	0	0
$904$	−31.5106	−1.04803
$905$	6.69862	0.222670
$906$	0	0
$907$	31.0645	1.03148	0.515740	−	0.856745i	$-0.327517\pi$
$907$	31.0645	1.03148	0.515740	+	0.856745i	$0.327517\pi$
$908$	9.62991	0.319580
$909$	0	0
$910$	12.0334	0.398905
$911$	−49.6807	−1.64600	−0.822998	−	0.568044i	$-0.807700\pi$
$911$	−49.6807	−1.64600	−0.822998	+	0.568044i	$0.807700\pi$
$912$	0	0
$913$	18.3581	0.607563
$914$	−22.7586	−0.752788
$915$	0	0
$916$	−7.63101	−0.252136
$917$	−11.8031	−0.389771
$918$	0	0
$919$	−10.1234	−0.333939	−0.166970	−	0.985962i	$-0.553398\pi$
$919$	−10.1234	−0.333939	−0.166970	+	0.985962i	$0.553398\pi$
$920$	21.1414	0.697013
$921$	0	0
$922$	−19.6452	−0.646981
$923$	41.9083	1.37943
$924$	0	0
$925$	6.87730	0.226124
$926$	27.0919	0.890297
$927$	0	0
$928$	4.27745	0.140414
$929$	−28.9742	−0.950613	−0.475307	−	0.879820i	$-0.657663\pi$
$929$	−28.9742	−0.950613	−0.475307	+	0.879820i	$0.657663\pi$
$930$	0	0
$931$	−6.65917	−0.218246
$932$	9.94707	0.325827
$933$	0	0
$934$	44.7144	1.46310
$935$	−17.2122	−0.562899
$936$	0	0
$937$	42.6921	1.39469	0.697345	−	0.716736i	$-0.254365\pi$
$937$	42.6921	1.39469	0.697345	+	0.716736i	$0.254365\pi$
$938$	26.0020	0.848995
$939$	0	0
$940$	−1.94042	−0.0632895
$941$	50.1099	1.63354	0.816768	−	0.576967i	$-0.195764\pi$
$941$	50.1099	1.63354	0.816768	+	0.576967i	$0.195764\pi$
$942$	0	0
$943$	−6.55742	−0.213539
$944$	16.1740	0.526419
$945$	0	0
$946$	−91.2372	−2.96638
$947$	20.5148	0.666640	0.333320	−	0.942814i	$-0.391831\pi$
$947$	20.5148	0.666640	0.333320	+	0.942814i	$0.391831\pi$
$948$	0	0
$949$	−51.9997	−1.68798
$950$	−3.83501	−0.124424
$951$	0	0
$952$	18.9450	0.614009
$953$	14.5669	0.471869	0.235934	−	0.971769i	$-0.424185\pi$
$953$	14.5669	0.471869	0.235934	+	0.971769i	$0.424185\pi$
$954$	0	0
$955$	5.40267	0.174826
$956$	12.2248	0.395379
$957$	0	0
$958$	39.6343	1.28052
$959$	3.03732	0.0980800
$960$	0	0
$961$	1.46085	0.0471242
$962$	37.0853	1.19568
$963$	0	0
$964$	11.6791	0.376160
$965$	−1.28529	−0.0413749
$966$	0	0
$967$	−24.8597	−0.799433	−0.399717	−	0.916639i	$-0.630891\pi$
$967$	−24.8597	−0.799433	−0.399717	+	0.916639i	$0.630891\pi$
$968$	86.1091	2.76765
$969$	0	0
$970$	−13.4771	−0.432725
$971$	33.6301	1.07924	0.539621	−	0.841908i	$-0.318568\pi$
$971$	33.6301	1.07924	0.539621	+	0.841908i	$0.318568\pi$
$972$	0	0
$973$	−3.25646	−0.104397
$974$	37.2953	1.19502
$975$	0	0
$976$	6.86603	0.219776
$977$	10.3812	0.332125	0.166063	−	0.986115i	$-0.446895\pi$
$977$	10.3812	0.332125	0.166063	+	0.986115i	$0.446895\pi$
$978$	0	0
$979$	6.24237	0.199507
$980$	1.30591	0.0417158
$981$	0	0
$982$	40.8488	1.30354
$983$	47.5089	1.51530	0.757649	−	0.652663i	$-0.226348\pi$
$983$	47.5089	1.51530	0.757649	+	0.652663i	$0.226348\pi$
$984$	0	0
$985$	−11.7541	−0.374516
$986$	3.92675	0.125053
$987$	0	0
$988$	9.87618	0.314203
$989$	−86.2613	−2.74295
$990$	0	0
$991$	1.77873	0.0565033	0.0282517	−	0.999601i	$-0.491006\pi$
$991$	1.77873	0.0565033	0.0282517	+	0.999601i	$0.491006\pi$
$992$	19.9095	0.632126
$993$	0	0
$994$	23.4749	0.744579
$995$	−14.5646	−0.461730
$996$	0	0
$997$	38.7428	1.22700	0.613498	−	0.789696i	$-0.289762\pi$
$997$	38.7428	1.22700	0.613498	+	0.789696i	$0.289762\pi$
$998$	20.9381	0.662783
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.p.1.3		8
3.2	odd	2		445.2.a.g.1.6	✓	8
12.11	even	2		7120.2.a.bk.1.2		8
15.14	odd	2		2225.2.a.l.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
445.2.a.g.1.6	✓	8	3.2	odd	2
2225.2.a.l.1.3		8	15.14	odd	2
4005.2.a.p.1.3		8	1.1	even	1	trivial
7120.2.a.bk.1.2		8	12.11	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:	\( N \)	\(=\)	\( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4005.a (trivial)

\(f(q)\)	\(=\)	\(q-1.16343 q^{2} -0.646427 q^{4} +1.00000 q^{5} +2.23155 q^{7} +3.07894 q^{8} +O(q^{10})\)	\(q-1.16343 q^{2} -0.646427 q^{4} +1.00000 q^{5} +2.23155 q^{7} +3.07894 q^{8} -1.16343 q^{10} -6.24237 q^{11} -4.63493 q^{13} -2.59625 q^{14} -2.28928 q^{16} +2.75732 q^{17} +3.29630 q^{19} -0.646427 q^{20} +7.26257 q^{22} +6.86648 q^{23} +1.00000 q^{25} +5.39242 q^{26} -1.44253 q^{28} -1.22407 q^{29} -5.69744 q^{31} -3.49446 q^{32} -3.20795 q^{34} +2.23155 q^{35} +6.87730 q^{37} -3.83501 q^{38} +3.07894 q^{40} -0.954991 q^{41} -12.5627 q^{43} +4.03524 q^{44} -7.98867 q^{46} +3.00176 q^{47} -2.02020 q^{49} -1.16343 q^{50} +2.99615 q^{52} +10.1149 q^{53} -6.24237 q^{55} +6.87079 q^{56} +1.42412 q^{58} -7.06511 q^{59} -2.99922 q^{61} +6.62858 q^{62} +8.64411 q^{64} -4.63493 q^{65} -10.0152 q^{67} -1.78241 q^{68} -2.59625 q^{70} -9.04184 q^{71} +11.2191 q^{73} -8.00126 q^{74} -2.13082 q^{76} -13.9301 q^{77} +0.814296 q^{79} -2.28928 q^{80} +1.11107 q^{82} -2.94088 q^{83} +2.75732 q^{85} +14.6158 q^{86} -19.2199 q^{88} -1.00000 q^{89} -10.3431 q^{91} -4.43868 q^{92} -3.49234 q^{94} +3.29630 q^{95} +11.5840 q^{97} +2.35036 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} + 7 q^{4} + 8 q^{5} - 6 q^{7} - 3 q^{8}+O(q^{10}) \) 8 * q - q^2 + 7 * q^4 + 8 * q^5 - 6 * q^7 - 3 * q^8	\( 8 q - q^{2} + 7 q^{4} + 8 q^{5} - 6 q^{7} - 3 q^{8} - q^{10} - 14 q^{11} - 7 q^{13} - 15 q^{14} + 9 q^{16} - 17 q^{17} + 17 q^{19} + 7 q^{20} + 2 q^{22} + q^{23} + 8 q^{25} - 3 q^{26} - 29 q^{28} - 10 q^{29} + q^{31} - 2 q^{32} - 16 q^{34} - 6 q^{35} - 11 q^{37} + 30 q^{38} - 3 q^{40} - 15 q^{41} - 5 q^{43} - 7 q^{44} - 12 q^{46} - 12 q^{47} + 4 q^{49} - q^{50} - 14 q^{52} + q^{53} - 14 q^{55} - 3 q^{56} - 37 q^{58} - 26 q^{59} + 13 q^{61} - 22 q^{62} - 15 q^{64} - 7 q^{65} - 25 q^{67} - 23 q^{68} - 15 q^{70} - 28 q^{71} - 17 q^{73} + 5 q^{74} + 8 q^{76} - 7 q^{79} + 9 q^{80} + 5 q^{82} - 44 q^{83} - 17 q^{85} + 13 q^{86} - 66 q^{88} - 8 q^{89} + 27 q^{91} - 15 q^{92} - 27 q^{94} + 17 q^{95} + q^{97} + 34 q^{98}+O(q^{100}) \) 8 * q - q^2 + 7 * q^4 + 8 * q^5 - 6 * q^7 - 3 * q^8 - q^10 - 14 * q^11 - 7 * q^13 - 15 * q^14 + 9 * q^16 - 17 * q^17 + 17 * q^19 + 7 * q^20 + 2 * q^22 + q^23 + 8 * q^25 - 3 * q^26 - 29 * q^28 - 10 * q^29 + q^31 - 2 * q^32 - 16 * q^34 - 6 * q^35 - 11 * q^37 + 30 * q^38 - 3 * q^40 - 15 * q^41 - 5 * q^43 - 7 * q^44 - 12 * q^46 - 12 * q^47 + 4 * q^49 - q^50 - 14 * q^52 + q^53 - 14 * q^55 - 3 * q^56 - 37 * q^58 - 26 * q^59 + 13 * q^61 - 22 * q^62 - 15 * q^64 - 7 * q^65 - 25 * q^67 - 23 * q^68 - 15 * q^70 - 28 * q^71 - 17 * q^73 + 5 * q^74 + 8 * q^76 - 7 * q^79 + 9 * q^80 + 5 * q^82 - 44 * q^83 - 17 * q^85 + 13 * q^86 - 66 * q^88 - 8 * q^89 + 27 * q^91 - 15 * q^92 - 27 * q^94 + 17 * q^95 + q^97 + 34 * q^98

Introduction

L-functions

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Database

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