LMFDB - Embedded newform 4005.2.a.p.1.2

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			1.2
Root			$2.23321$ 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)

$f(q)$	$=$	$q-2.23321 q^{2} +2.98721 q^{4} +1.00000 q^{5} +1.87039 q^{7} -2.20463 q^{8} +O(q^{10})$	\(q-2.23321 q^{2} +2.98721 q^{4} +1.00000 q^{5} +1.87039 q^{7} -2.20463 q^{8} -2.23321 q^{10} -2.02857 q^{11} +5.20418 q^{13} -4.17696 q^{14} -1.05101 q^{16} +1.73901 q^{17} -5.29579 q^{19} +2.98721 q^{20} +4.53022 q^{22} -3.33379 q^{23} +1.00000 q^{25} -11.6220 q^{26} +5.58724 q^{28} +3.53903 q^{29} -4.98395 q^{31} +6.75639 q^{32} -3.88356 q^{34} +1.87039 q^{35} -7.17561 q^{37} +11.8266 q^{38} -2.20463 q^{40} +1.08840 q^{41} -4.73941 q^{43} -6.05976 q^{44} +7.44504 q^{46} -7.45521 q^{47} -3.50165 q^{49} -2.23321 q^{50} +15.5460 q^{52} -12.1190 q^{53} -2.02857 q^{55} -4.12352 q^{56} -7.90338 q^{58} -3.66019 q^{59} +11.4032 q^{61} +11.1302 q^{62} -12.9864 q^{64} +5.20418 q^{65} -9.52615 q^{67} +5.19477 q^{68} -4.17696 q^{70} -2.81801 q^{71} +1.11516 q^{73} +16.0246 q^{74} -15.8196 q^{76} -3.79422 q^{77} -6.37686 q^{79} -1.05101 q^{80} -2.43062 q^{82} -11.6127 q^{83} +1.73901 q^{85} +10.5841 q^{86} +4.47226 q^{88} -1.00000 q^{89} +9.73384 q^{91} -9.95872 q^{92} +16.6490 q^{94} -5.29579 q^{95} +16.0880 q^{97} +7.81989 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.23321	−1.57911	−0.789557	−	0.613677i	$-0.789690\pi$
$2$	−2.23321	−1.57911	−0.789557	+	0.613677i	$0.789690\pi$
$3$	0	0
$3$	0	0
$4$	2.98721	1.49360
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.87039	0.706941	0.353470	−	0.935446i	$-0.385002\pi$
$7$	1.87039	0.706941	0.353470	+	0.935446i	$0.385002\pi$
$8$	−2.20463	−0.779456
$9$	0	0
$10$	−2.23321	−0.706202
$11$	−2.02857	−0.611637	−0.305819	−	0.952090i	$-0.598930\pi$
$11$	−2.02857	−0.611637	−0.305819	+	0.952090i	$0.598930\pi$
$12$	0	0
$13$	5.20418	1.44338	0.721690	−	0.692216i	$-0.243366\pi$
$13$	5.20418	1.44338	0.721690	+	0.692216i	$0.243366\pi$
$14$	−4.17696	−1.11634
$15$	0	0
$16$	−1.05101	−0.262753
$17$	1.73901	0.421771	0.210886	−	0.977511i	$-0.432365\pi$
$17$	1.73901	0.421771	0.210886	+	0.977511i	$0.432365\pi$
$18$	0	0
$19$	−5.29579	−1.21494	−0.607469	−	0.794343i	$-0.707815\pi$
$19$	−5.29579	−1.21494	−0.607469	+	0.794343i	$0.707815\pi$
$20$	2.98721	0.667960
$21$	0	0
$22$	4.53022	0.965845
$23$	−3.33379	−0.695143	−0.347572	−	0.937653i	$-0.612994\pi$
$23$	−3.33379	−0.695143	−0.347572	+	0.937653i	$0.612994\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−11.6220	−2.27926
$27$	0	0
$28$	5.58724	1.05589
$29$	3.53903	0.657182	0.328591	−	0.944472i	$-0.393426\pi$
$29$	3.53903	0.657182	0.328591	+	0.944472i	$0.393426\pi$
$30$	0	0
$31$	−4.98395	−0.895144	−0.447572	−	0.894248i	$-0.647711\pi$
$31$	−4.98395	−0.895144	−0.447572	+	0.894248i	$0.647711\pi$
$32$	6.75639	1.19437
$33$	0	0
$34$	−3.88356	−0.666025
$35$	1.87039	0.316153
$36$	0	0
$37$	−7.17561	−1.17966	−0.589831	−	0.807526i	$-0.700806\pi$
$37$	−7.17561	−1.17966	−0.589831	+	0.807526i	$0.700806\pi$
$38$	11.8266	1.91853
$39$	0	0
$40$	−2.20463	−0.348583
$41$	1.08840	0.169979	0.0849896	−	0.996382i	$-0.472914\pi$
$41$	1.08840	0.169979	0.0849896	+	0.996382i	$0.472914\pi$
$42$	0	0
$43$	−4.73941	−0.722753	−0.361377	−	0.932420i	$-0.617693\pi$
$43$	−4.73941	−0.722753	−0.361377	+	0.932420i	$0.617693\pi$
$44$	−6.05976	−0.913543
$45$	0	0
$46$	7.44504	1.09771
$47$	−7.45521	−1.08745	−0.543727	−	0.839262i	$-0.682987\pi$
$47$	−7.45521	−1.08745	−0.543727	+	0.839262i	$0.682987\pi$
$48$	0	0
$49$	−3.50165	−0.500235
$50$	−2.23321	−0.315823
$51$	0	0
$52$	15.5460	2.15584
$53$	−12.1190	−1.66468	−0.832340	−	0.554266i	$-0.812999\pi$
$53$	−12.1190	−1.66468	−0.832340	+	0.554266i	$0.812999\pi$
$54$	0	0
$55$	−2.02857	−0.273533
$56$	−4.12352	−0.551029
$57$	0	0
$58$	−7.90338	−1.03777
$59$	−3.66019	−0.476516	−0.238258	−	0.971202i	$-0.576576\pi$
$59$	−3.66019	−0.476516	−0.238258	+	0.971202i	$0.576576\pi$
$60$	0	0
$61$	11.4032	1.46003	0.730017	−	0.683429i	$-0.239512\pi$
$61$	11.4032	1.46003	0.730017	+	0.683429i	$0.239512\pi$
$62$	11.1302	1.41353
$63$	0	0
$64$	−12.9864	−1.62330
$65$	5.20418	0.645499
$66$	0	0
$67$	−9.52615	−1.16380	−0.581902	−	0.813259i	$-0.697691\pi$
$67$	−9.52615	−1.16380	−0.581902	+	0.813259i	$0.697691\pi$
$68$	5.19477	0.629959
$69$	0	0
$70$	−4.17696	−0.499243
$71$	−2.81801	−0.334436	−0.167218	−	0.985920i	$-0.553478\pi$
$71$	−2.81801	−0.334436	−0.167218	+	0.985920i	$0.553478\pi$
$72$	0	0
$73$	1.11516	0.130519	0.0652597	−	0.997868i	$-0.479212\pi$
$73$	1.11516	0.130519	0.0652597	+	0.997868i	$0.479212\pi$
$74$	16.0246	1.86282
$75$	0	0
$76$	−15.8196	−1.81464
$77$	−3.79422	−0.432391
$78$	0	0
$79$	−6.37686	−0.717453	−0.358727	−	0.933443i	$-0.616789\pi$
$79$	−6.37686	−0.717453	−0.358727	+	0.933443i	$0.616789\pi$
$80$	−1.05101	−0.117507
$81$	0	0
$82$	−2.43062	−0.268417
$83$	−11.6127	−1.27466	−0.637328	−	0.770593i	$-0.719960\pi$
$83$	−11.6127	−1.27466	−0.637328	+	0.770593i	$0.719960\pi$
$84$	0	0
$85$	1.73901	0.188622
$86$	10.5841	1.14131
$87$	0	0
$88$	4.47226	0.476744
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	9.73384	1.02038
$92$	−9.95872	−1.03827
$93$	0	0
$94$	16.6490	1.71721
$95$	−5.29579	−0.543337
$96$	0	0
$97$	16.0880	1.63349	0.816745	−	0.576999i	$-0.195776\pi$
$97$	16.0880	1.63349	0.816745	+	0.576999i	$0.195776\pi$
$98$	7.81989	0.789928
$99$	0	0
$100$	2.98721	0.298721
$101$	−4.93597	−0.491147	−0.245574	−	0.969378i	$-0.578976\pi$
$101$	−4.93597	−0.491147	−0.245574	+	0.969378i	$0.578976\pi$
$102$	0	0
$103$	−17.7749	−1.75141	−0.875706	−	0.482844i	$-0.839604\pi$
$103$	−17.7749	−1.75141	−0.875706	+	0.482844i	$0.839604\pi$
$104$	−11.4733	−1.12505
$105$	0	0
$106$	27.0643	2.62872
$107$	−12.1644	−1.17597	−0.587987	−	0.808871i	$-0.700079\pi$
$107$	−12.1644	−1.17597	−0.587987	+	0.808871i	$0.700079\pi$
$108$	0	0
$109$	2.87280	0.275164	0.137582	−	0.990490i	$-0.456067\pi$
$109$	2.87280	0.275164	0.137582	+	0.990490i	$0.456067\pi$
$110$	4.53022	0.431939
$111$	0	0
$112$	−1.96580	−0.185751
$113$	1.44516	0.135949	0.0679744	−	0.997687i	$-0.478346\pi$
$113$	1.44516	0.135949	0.0679744	+	0.997687i	$0.478346\pi$
$114$	0	0
$115$	−3.33379	−0.310878
$116$	10.5718	0.981568
$117$	0	0
$118$	8.17395	0.752473
$119$	3.25262	0.298167
$120$	0	0
$121$	−6.88490	−0.625900
$122$	−25.4657	−2.30556
$123$	0	0
$124$	−14.8881	−1.33699
$125$	1.00000	0.0894427
$126$	0	0
$127$	12.5720	1.11559	0.557794	−	0.829979i	$-0.311648\pi$
$127$	12.5720	1.11559	0.557794	+	0.829979i	$0.311648\pi$
$128$	15.4885	1.36900
$129$	0	0
$130$	−11.6220	−1.01932
$131$	−0.192392	−0.0168093	−0.00840467	−	0.999965i	$-0.502675\pi$
$131$	−0.192392	−0.0168093	−0.00840467	+	0.999965i	$0.502675\pi$
$132$	0	0
$133$	−9.90520	−0.858889
$134$	21.2738	1.83778
$135$	0	0
$136$	−3.83388	−0.328752
$137$	−12.8576	−1.09850	−0.549249	−	0.835659i	$-0.685086\pi$
$137$	−12.8576	−1.09850	−0.549249	+	0.835659i	$0.685086\pi$
$138$	0	0
$139$	8.48167	0.719406	0.359703	−	0.933067i	$-0.382878\pi$
$139$	8.48167	0.719406	0.359703	+	0.933067i	$0.382878\pi$
$140$	5.58724	0.472208
$141$	0	0
$142$	6.29319	0.528113
$143$	−10.5571	−0.882825
$144$	0	0
$145$	3.53903	0.293901
$146$	−2.49038	−0.206105
$147$	0	0
$148$	−21.4350	−1.76195
$149$	17.0248	1.39473	0.697363	−	0.716718i	$-0.254357\pi$
$149$	17.0248	1.39473	0.697363	+	0.716718i	$0.254357\pi$
$150$	0	0
$151$	14.8408	1.20772	0.603862	−	0.797089i	$-0.293628\pi$
$151$	14.8408	1.20772	0.603862	+	0.797089i	$0.293628\pi$
$152$	11.6753	0.946991
$153$	0	0
$154$	8.47327	0.682795
$155$	−4.98395	−0.400320
$156$	0	0
$157$	−9.36339	−0.747280	−0.373640	−	0.927574i	$-0.621890\pi$
$157$	−9.36339	−0.747280	−0.373640	+	0.927574i	$0.621890\pi$
$158$	14.2408	1.13294
$159$	0	0
$160$	6.75639	0.534140
$161$	−6.23549	−0.491425
$162$	0	0
$163$	−10.4206	−0.816203	−0.408101	−	0.912937i	$-0.633809\pi$
$163$	−10.4206	−0.816203	−0.408101	+	0.912937i	$0.633809\pi$
$164$	3.25127	0.253882
$165$	0	0
$166$	25.9335	2.01283
$167$	11.1930	0.866143	0.433072	−	0.901359i	$-0.357430\pi$
$167$	11.1930	0.866143	0.433072	+	0.901359i	$0.357430\pi$
$168$	0	0
$169$	14.0835	1.08335
$170$	−3.88356	−0.297856
$171$	0	0
$172$	−14.1576	−1.07951
$173$	9.43068	0.717001	0.358501	−	0.933530i	$-0.383288\pi$
$173$	9.43068	0.717001	0.358501	+	0.933530i	$0.383288\pi$
$174$	0	0
$175$	1.87039	0.141388
$176$	2.13205	0.160709
$177$	0	0
$178$	2.23321	0.167386
$179$	−2.26640	−0.169399	−0.0846993	−	0.996407i	$-0.526993\pi$
$179$	−2.26640	−0.169399	−0.0846993	+	0.996407i	$0.526993\pi$
$180$	0	0
$181$	−4.33424	−0.322161	−0.161081	−	0.986941i	$-0.551498\pi$
$181$	−4.33424	−0.322161	−0.161081	+	0.986941i	$0.551498\pi$
$182$	−21.7377	−1.61130
$183$	0	0
$184$	7.34979	0.541834
$185$	−7.17561	−0.527561
$186$	0	0
$187$	−3.52770	−0.257971
$188$	−22.2702	−1.62422
$189$	0	0
$190$	11.8266	0.857991
$191$	−12.6806	−0.917534	−0.458767	−	0.888557i	$-0.651709\pi$
$191$	−12.6806	−0.917534	−0.458767	+	0.888557i	$0.651709\pi$
$192$	0	0
$193$	−20.3722	−1.46643	−0.733213	−	0.679999i	$-0.761980\pi$
$193$	−20.3722	−1.46643	−0.733213	+	0.679999i	$0.761980\pi$
$194$	−35.9278	−2.57947
$195$	0	0
$196$	−10.4601	−0.747153
$197$	19.8739	1.41595	0.707977	−	0.706236i	$-0.249608\pi$
$197$	19.8739	1.41595	0.707977	+	0.706236i	$0.249608\pi$
$198$	0	0
$199$	11.7504	0.832966	0.416483	−	0.909144i	$-0.363263\pi$
$199$	11.7504	0.832966	0.416483	+	0.909144i	$0.363263\pi$
$200$	−2.20463	−0.155891
$201$	0	0
$202$	11.0230	0.775578
$203$	6.61937	0.464588
$204$	0	0
$205$	1.08840	0.0760170
$206$	39.6950	2.76568
$207$	0	0
$208$	−5.46965	−0.379252
$209$	10.7429	0.743102
$210$	0	0
$211$	14.8159	1.01997	0.509983	−	0.860184i	$-0.329652\pi$
$211$	14.8159	1.01997	0.509983	+	0.860184i	$0.329652\pi$
$212$	−36.2021	−2.48637
$213$	0	0
$214$	27.1655	1.85700
$215$	−4.73941	−0.323225
$216$	0	0
$217$	−9.32192	−0.632813
$218$	−6.41555	−0.434516
$219$	0	0
$220$	−6.05976	−0.408549
$221$	9.05011	0.608776
$222$	0	0
$223$	11.4234	0.764965	0.382482	−	0.923963i	$-0.375069\pi$
$223$	11.4234	0.764965	0.382482	+	0.923963i	$0.375069\pi$
$224$	12.6371	0.844351
$225$	0	0
$226$	−3.22733	−0.214679
$227$	−4.98795	−0.331062	−0.165531	−	0.986205i	$-0.552934\pi$
$227$	−4.98795	−0.331062	−0.165531	+	0.986205i	$0.552934\pi$
$228$	0	0
$229$	10.1895	0.673340	0.336670	−	0.941623i	$-0.390699\pi$
$229$	10.1895	0.673340	0.336670	+	0.941623i	$0.390699\pi$
$230$	7.44504	0.490911
$231$	0	0
$232$	−7.80227	−0.512244
$233$	12.1002	0.792710	0.396355	−	0.918097i	$-0.370275\pi$
$233$	12.1002	0.792710	0.396355	+	0.918097i	$0.370275\pi$
$234$	0	0
$235$	−7.45521	−0.486324
$236$	−10.9337	−0.711725
$237$	0	0
$238$	−7.26377	−0.470840
$239$	−18.0630	−1.16840	−0.584200	−	0.811610i	$-0.698592\pi$
$239$	−18.0630	−1.16840	−0.584200	+	0.811610i	$0.698592\pi$
$240$	0	0
$241$	24.6768	1.58957	0.794785	−	0.606892i	$-0.207584\pi$
$241$	24.6768	1.58957	0.794785	+	0.606892i	$0.207584\pi$
$242$	15.3754	0.988368
$243$	0	0
$244$	34.0638	2.18071
$245$	−3.50165	−0.223712
$246$	0	0
$247$	−27.5603	−1.75362
$248$	10.9878	0.697725
$249$	0	0
$250$	−2.23321	−0.141240
$251$	10.2474	0.646809	0.323404	−	0.946261i	$-0.395173\pi$
$251$	10.2474	0.646809	0.323404	+	0.946261i	$0.395173\pi$
$252$	0	0
$253$	6.76283	0.425176
$254$	−28.0760	−1.76164
$255$	0	0
$256$	−8.61620	−0.538512
$257$	−1.31880	−0.0822644	−0.0411322	−	0.999154i	$-0.513096\pi$
$257$	−1.31880	−0.0822644	−0.0411322	+	0.999154i	$0.513096\pi$
$258$	0	0
$259$	−13.4212	−0.833952
$260$	15.5460	0.964119
$261$	0	0
$262$	0.429650	0.0265439
$263$	19.4751	1.20089	0.600444	−	0.799667i	$-0.294990\pi$
$263$	19.4751	1.20089	0.600444	+	0.799667i	$0.294990\pi$
$264$	0	0
$265$	−12.1190	−0.744467
$266$	22.1203	1.35628
$267$	0	0
$268$	−28.4566	−1.73826
$269$	−24.0346	−1.46541	−0.732707	−	0.680544i	$-0.761743\pi$
$269$	−24.0346	−1.46541	−0.732707	+	0.680544i	$0.761743\pi$
$270$	0	0
$271$	10.4215	0.633061	0.316530	−	0.948582i	$-0.397482\pi$
$271$	10.4215	0.633061	0.316530	+	0.948582i	$0.397482\pi$
$272$	−1.82772	−0.110822
$273$	0	0
$274$	28.7136	1.73465
$275$	−2.02857	−0.122327
$276$	0	0
$277$	−22.0943	−1.32752	−0.663760	−	0.747945i	$-0.731041\pi$
$277$	−22.0943	−1.32752	−0.663760	+	0.747945i	$0.731041\pi$
$278$	−18.9413	−1.13602
$279$	0	0
$280$	−4.12352	−0.246428
$281$	20.7756	1.23937	0.619685	−	0.784851i	$-0.287260\pi$
$281$	20.7756	1.23937	0.619685	+	0.784851i	$0.287260\pi$
$282$	0	0
$283$	30.0530	1.78646	0.893232	−	0.449597i	$-0.148432\pi$
$283$	30.0530	1.78646	0.893232	+	0.449597i	$0.148432\pi$
$284$	−8.41798	−0.499515
$285$	0	0
$286$	23.5761	1.39408
$287$	2.03573	0.120165
$288$	0	0
$289$	−13.9759	−0.822109
$290$	−7.90338	−0.464103
$291$	0	0
$292$	3.33121	0.194944
$293$	−21.0943	−1.23234	−0.616172	−	0.787612i	$-0.711317\pi$
$293$	−21.0943	−1.23234	−0.616172	+	0.787612i	$0.711317\pi$
$294$	0	0
$295$	−3.66019	−0.213104
$296$	15.8196	0.919495
$297$	0	0
$298$	−38.0199	−2.20243
$299$	−17.3496	−1.00336
$300$	0	0
$301$	−8.86454	−0.510944
$302$	−33.1425	−1.90713
$303$	0	0
$304$	5.56594	0.319229
$305$	11.4032	0.652947
$306$	0	0
$307$	−15.2829	−0.872242	−0.436121	−	0.899888i	$-0.643648\pi$
$307$	−15.2829	−0.872242	−0.436121	+	0.899888i	$0.643648\pi$
$308$	−11.3341	−0.645821
$309$	0	0
$310$	11.1302	0.632152
$311$	−4.45836	−0.252810	−0.126405	−	0.991979i	$-0.540344\pi$
$311$	−4.45836	−0.252810	−0.126405	+	0.991979i	$0.540344\pi$
$312$	0	0
$313$	8.03862	0.454369	0.227185	−	0.973852i	$-0.427048\pi$
$313$	8.03862	0.454369	0.227185	+	0.973852i	$0.427048\pi$
$314$	20.9104	1.18004
$315$	0	0
$316$	−19.0490	−1.07159
$317$	34.1258	1.91670	0.958349	−	0.285600i	$-0.0921928\pi$
$317$	34.1258	1.91670	0.958349	+	0.285600i	$0.0921928\pi$
$318$	0	0
$319$	−7.17918	−0.401957
$320$	−12.9864	−0.725961
$321$	0	0
$322$	13.9251	0.776017
$323$	−9.20943	−0.512426
$324$	0	0
$325$	5.20418	0.288676
$326$	23.2713	1.28888
$327$	0	0
$328$	−2.39952	−0.132491
$329$	−13.9441	−0.768765
$330$	0	0
$331$	−4.76450	−0.261881	−0.130940	−	0.991390i	$-0.541800\pi$
$331$	−4.76450	−0.261881	−0.130940	+	0.991390i	$0.541800\pi$
$332$	−34.6894	−1.90383
$333$	0	0
$334$	−24.9964	−1.36774
$335$	−9.52615	−0.520469
$336$	0	0
$337$	−23.4733	−1.27867	−0.639335	−	0.768928i	$-0.720790\pi$
$337$	−23.4733	−1.27867	−0.639335	+	0.768928i	$0.720790\pi$
$338$	−31.4513	−1.71073
$339$	0	0
$340$	5.19477	0.281726
$341$	10.1103	0.547503
$342$	0	0
$343$	−19.6422	−1.06058
$344$	10.4487	0.563354
$345$	0	0
$346$	−21.0606	−1.13223
$347$	−25.5270	−1.37036	−0.685181	−	0.728373i	$-0.740277\pi$
$347$	−25.5270	−1.37036	−0.685181	+	0.728373i	$0.740277\pi$
$348$	0	0
$349$	−1.62004	−0.0867185	−0.0433593	−	0.999060i	$-0.513806\pi$
$349$	−1.62004	−0.0867185	−0.0433593	+	0.999060i	$0.513806\pi$
$350$	−4.17696	−0.223268
$351$	0	0
$352$	−13.7058	−0.730523
$353$	−6.69225	−0.356192	−0.178096	−	0.984013i	$-0.556994\pi$
$353$	−6.69225	−0.356192	−0.178096	+	0.984013i	$0.556994\pi$
$354$	0	0
$355$	−2.81801	−0.149564
$356$	−2.98721	−0.158322
$357$	0	0
$358$	5.06133	0.267500
$359$	−30.0892	−1.58805	−0.794025	−	0.607886i	$-0.792018\pi$
$359$	−30.0892	−1.58805	−0.794025	+	0.607886i	$0.792018\pi$
$360$	0	0
$361$	9.04544	0.476076
$362$	9.67924	0.508729
$363$	0	0
$364$	29.0770	1.52405
$365$	1.11516	0.0583700
$366$	0	0
$367$	−2.37916	−0.124191	−0.0620955	−	0.998070i	$-0.519778\pi$
$367$	−2.37916	−0.124191	−0.0620955	+	0.998070i	$0.519778\pi$
$368$	3.50385	0.182651
$369$	0	0
$370$	16.0246	0.833080
$371$	−22.6673	−1.17683
$372$	0	0
$373$	−27.0739	−1.40183	−0.700917	−	0.713242i	$-0.747226\pi$
$373$	−27.0739	−1.40183	−0.700917	+	0.713242i	$0.747226\pi$
$374$	7.87808	0.407366
$375$	0	0
$376$	16.4360	0.847622
$377$	18.4178	0.948563
$378$	0	0
$379$	−3.29019	−0.169006	−0.0845029	−	0.996423i	$-0.526930\pi$
$379$	−3.29019	−0.169006	−0.0845029	+	0.996423i	$0.526930\pi$
$380$	−15.8196	−0.811530
$381$	0	0
$382$	28.3183	1.44889
$383$	10.8612	0.554981	0.277491	−	0.960728i	$-0.410497\pi$
$383$	10.8612	0.554981	0.277491	+	0.960728i	$0.410497\pi$
$384$	0	0
$385$	−3.79422	−0.193371
$386$	45.4954	2.31566
$387$	0	0
$388$	48.0582	2.43979
$389$	−14.3915	−0.729678	−0.364839	−	0.931071i	$-0.618876\pi$
$389$	−14.3915	−0.729678	−0.364839	+	0.931071i	$0.618876\pi$
$390$	0	0
$391$	−5.79749	−0.293192
$392$	7.71985	0.389911
$393$	0	0
$394$	−44.3824	−2.23595
$395$	−6.37686	−0.320855
$396$	0	0
$397$	6.69988	0.336258	0.168129	−	0.985765i	$-0.446228\pi$
$397$	6.69988	0.336258	0.168129	+	0.985765i	$0.446228\pi$
$398$	−26.2411	−1.31535
$399$	0	0
$400$	−1.05101	−0.0525506
$401$	4.20979	0.210227	0.105113	−	0.994460i	$-0.466479\pi$
$401$	4.20979	0.210227	0.105113	+	0.994460i	$0.466479\pi$
$402$	0	0
$403$	−25.9374	−1.29203
$404$	−14.7448	−0.733579
$405$	0	0
$406$	−14.7824	−0.733638
$407$	14.5562	0.721526
$408$	0	0
$409$	−38.7813	−1.91761	−0.958805	−	0.284065i	$-0.908317\pi$
$409$	−38.7813	−1.91761	−0.958805	+	0.284065i	$0.908317\pi$
$410$	−2.43062	−0.120040
$411$	0	0
$412$	−53.0973	−2.61592
$413$	−6.84597	−0.336868
$414$	0	0
$415$	−11.6127	−0.570044
$416$	35.1615	1.72393
$417$	0	0
$418$	−23.9911	−1.17344
$419$	−12.4519	−0.608313	−0.304156	−	0.952622i	$-0.598375\pi$
$419$	−12.4519	−0.608313	−0.304156	+	0.952622i	$0.598375\pi$
$420$	0	0
$421$	−39.4273	−1.92157	−0.960785	−	0.277294i	$-0.910562\pi$
$421$	−39.4273	−1.92157	−0.960785	+	0.277294i	$0.910562\pi$
$422$	−33.0869	−1.61064
$423$	0	0
$424$	26.7181	1.29754
$425$	1.73901	0.0843543
$426$	0	0
$427$	21.3285	1.03216
$428$	−36.3375	−1.75644
$429$	0	0
$430$	10.5841	0.510410
$431$	31.3424	1.50971	0.754855	−	0.655892i	$-0.227707\pi$
$431$	31.3424	1.50971	0.754855	+	0.655892i	$0.227707\pi$
$432$	0	0
$433$	33.1203	1.59166	0.795830	−	0.605520i	$-0.207035\pi$
$433$	33.1203	1.59166	0.795830	+	0.605520i	$0.207035\pi$
$434$	20.8178	0.999285
$435$	0	0
$436$	8.58164	0.410986
$437$	17.6551	0.844557
$438$	0	0
$439$	−31.6765	−1.51184	−0.755918	−	0.654666i	$-0.772809\pi$
$439$	−31.6765	−1.51184	−0.755918	+	0.654666i	$0.772809\pi$
$440$	4.47226	0.213207
$441$	0	0
$442$	−20.2107	−0.961327
$443$	−16.2715	−0.773083	−0.386542	−	0.922272i	$-0.626330\pi$
$443$	−16.2715	−0.773083	−0.386542	+	0.922272i	$0.626330\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	−25.5107	−1.20797
$447$	0	0
$448$	−24.2896	−1.14758
$449$	1.33672	0.0630839	0.0315420	−	0.999502i	$-0.489958\pi$
$449$	1.33672	0.0630839	0.0315420	+	0.999502i	$0.489958\pi$
$450$	0	0
$451$	−2.20789	−0.103966
$452$	4.31698	0.203054
$453$	0	0
$454$	11.1391	0.522785
$455$	9.73384	0.456330
$456$	0	0
$457$	10.8622	0.508114	0.254057	−	0.967189i	$-0.418235\pi$
$457$	10.8622	0.508114	0.254057	+	0.967189i	$0.418235\pi$
$458$	−22.7552	−1.06328
$459$	0	0
$460$	−9.95872	−0.464328
$461$	21.5031	1.00150	0.500750	−	0.865592i	$-0.333058\pi$
$461$	21.5031	1.00150	0.500750	+	0.865592i	$0.333058\pi$
$462$	0	0
$463$	−27.7658	−1.29038	−0.645192	−	0.764020i	$-0.723223\pi$
$463$	−27.7658	−1.29038	−0.645192	+	0.764020i	$0.723223\pi$
$464$	−3.71956	−0.172676
$465$	0	0
$466$	−27.0222	−1.25178
$467$	7.57551	0.350553	0.175276	−	0.984519i	$-0.443918\pi$
$467$	7.57551	0.350553	0.175276	+	0.984519i	$0.443918\pi$
$468$	0	0
$469$	−17.8176	−0.822740
$470$	16.6490	0.767962
$471$	0	0
$472$	8.06937	0.371423
$473$	9.61423	0.442063
$474$	0	0
$475$	−5.29579	−0.242988
$476$	9.71625	0.445343
$477$	0	0
$478$	40.3385	1.84504
$479$	11.9206	0.544668	0.272334	−	0.962203i	$-0.412204\pi$
$479$	11.9206	0.544668	0.272334	+	0.962203i	$0.412204\pi$
$480$	0	0
$481$	−37.3432	−1.70270
$482$	−55.1083	−2.51011
$483$	0	0
$484$	−20.5666	−0.934846
$485$	16.0880	0.730519
$486$	0	0
$487$	−24.0047	−1.08776	−0.543879	−	0.839164i	$-0.683045\pi$
$487$	−24.0047	−1.08776	−0.543879	+	0.839164i	$0.683045\pi$
$488$	−25.1399	−1.13803
$489$	0	0
$490$	7.81989	0.353267
$491$	−7.03441	−0.317459	−0.158729	−	0.987322i	$-0.550740\pi$
$491$	−7.03441	−0.317459	−0.158729	+	0.987322i	$0.550740\pi$
$492$	0	0
$493$	6.15440	0.277180
$494$	61.5477	2.76916
$495$	0	0
$496$	5.23819	0.235202
$497$	−5.27077	−0.236427
$498$	0	0
$499$	14.3771	0.643606	0.321803	−	0.946807i	$-0.395711\pi$
$499$	14.3771	0.643606	0.321803	+	0.946807i	$0.395711\pi$
$500$	2.98721	0.133592
$501$	0	0
$502$	−22.8845	−1.02139
$503$	23.9794	1.06919	0.534595	−	0.845108i	$-0.320464\pi$
$503$	23.9794	1.06919	0.534595	+	0.845108i	$0.320464\pi$
$504$	0	0
$505$	−4.93597	−0.219648
$506$	−15.1028	−0.671401
$507$	0	0
$508$	37.5553	1.66625
$509$	−5.59380	−0.247941	−0.123970	−	0.992286i	$-0.539563\pi$
$509$	−5.59380	−0.247941	−0.123970	+	0.992286i	$0.539563\pi$
$510$	0	0
$511$	2.08578	0.0922694
$512$	−11.7352	−0.518629
$513$	0	0
$514$	2.94515	0.129905
$515$	−17.7749	−0.783255
$516$	0	0
$517$	15.1234	0.665127
$518$	29.9722	1.31691
$519$	0	0
$520$	−11.4733	−0.503138
$521$	−9.47103	−0.414933	−0.207467	−	0.978242i	$-0.566522\pi$
$521$	−9.47103	−0.414933	−0.207467	+	0.978242i	$0.566522\pi$
$522$	0	0
$523$	−1.98058	−0.0866048	−0.0433024	−	0.999062i	$-0.513788\pi$
$523$	−1.98058	−0.0866048	−0.0433024	+	0.999062i	$0.513788\pi$
$524$	−0.574714	−0.0251065
$525$	0	0
$526$	−43.4920	−1.89634
$527$	−8.66712	−0.377546
$528$	0	0
$529$	−11.8858	−0.516776
$530$	27.0643	1.17560
$531$	0	0
$532$	−29.5889	−1.28284
$533$	5.66422	0.245345
$534$	0	0
$535$	−12.1644	−0.525911
$536$	21.0017	0.907134
$537$	0	0
$538$	53.6742	2.31406
$539$	7.10334	0.305962
$540$	0	0
$541$	1.70941	0.0734931	0.0367466	−	0.999325i	$-0.488301\pi$
$541$	1.70941	0.0734931	0.0367466	+	0.999325i	$0.488301\pi$
$542$	−23.2733	−0.999675
$543$	0	0
$544$	11.7494	0.503752
$545$	2.87280	0.123057
$546$	0	0
$547$	9.08547	0.388467	0.194233	−	0.980955i	$-0.437778\pi$
$547$	9.08547	0.388467	0.194233	+	0.980955i	$0.437778\pi$
$548$	−38.4083	−1.64072
$549$	0	0
$550$	4.53022	0.193169
$551$	−18.7420	−0.798435
$552$	0	0
$553$	−11.9272	−0.507197
$554$	49.3412	2.09631
$555$	0	0
$556$	25.3365	1.07451
$557$	−17.6171	−0.746462	−0.373231	−	0.927739i	$-0.621750\pi$
$557$	−17.6171	−0.746462	−0.373231	+	0.927739i	$0.621750\pi$
$558$	0	0
$559$	−24.6647	−1.04321
$560$	−1.96580	−0.0830702
$561$	0	0
$562$	−46.3962	−1.95711
$563$	9.92488	0.418284	0.209142	−	0.977885i	$-0.432933\pi$
$563$	9.92488	0.418284	0.209142	+	0.977885i	$0.432933\pi$
$564$	0	0
$565$	1.44516	0.0607982
$566$	−67.1144	−2.82103
$567$	0	0
$568$	6.21268	0.260678
$569$	−15.0033	−0.628970	−0.314485	−	0.949262i	$-0.601832\pi$
$569$	−15.0033	−0.628970	−0.314485	+	0.949262i	$0.601832\pi$
$570$	0	0
$571$	23.1676	0.969532	0.484766	−	0.874644i	$-0.338905\pi$
$571$	23.1676	0.969532	0.484766	+	0.874644i	$0.338905\pi$
$572$	−31.5361	−1.31859
$573$	0	0
$574$	−4.54620	−0.189755
$575$	−3.33379	−0.139029
$576$	0	0
$577$	14.4982	0.603566	0.301783	−	0.953377i	$-0.402418\pi$
$577$	14.4982	0.603566	0.301783	+	0.953377i	$0.402418\pi$
$578$	31.2109	1.29820
$579$	0	0
$580$	10.5718	0.438971
$581$	−21.7202	−0.901106
$582$	0	0
$583$	24.5844	1.01818
$584$	−2.45852	−0.101734
$585$	0	0
$586$	47.1080	1.94601
$587$	−14.6158	−0.603259	−0.301629	−	0.953425i	$-0.597531\pi$
$587$	−14.6158	−0.603259	−0.301629	+	0.953425i	$0.597531\pi$
$588$	0	0
$589$	26.3940	1.08754
$590$	8.17395	0.336516
$591$	0	0
$592$	7.54165	0.309960
$593$	42.3122	1.73755	0.868776	−	0.495205i	$-0.164907\pi$
$593$	42.3122	1.73755	0.868776	+	0.495205i	$0.164907\pi$
$594$	0	0
$595$	3.25262	0.133344
$596$	50.8566	2.08317
$597$	0	0
$598$	38.7453	1.58441
$599$	9.93515	0.405939	0.202970	−	0.979185i	$-0.434941\pi$
$599$	9.93515	0.405939	0.202970	+	0.979185i	$0.434941\pi$
$600$	0	0
$601$	−37.0451	−1.51110	−0.755551	−	0.655090i	$-0.772631\pi$
$601$	−37.0451	−1.51110	−0.755551	+	0.655090i	$0.772631\pi$
$602$	19.7963	0.806839
$603$	0	0
$604$	44.3324	1.80386
$605$	−6.88490	−0.279911
$606$	0	0
$607$	−43.6916	−1.77339	−0.886693	−	0.462359i	$-0.847003\pi$
$607$	−43.6916	−1.77339	−0.886693	+	0.462359i	$0.847003\pi$
$608$	−35.7805	−1.45109
$609$	0	0
$610$	−25.4657	−1.03108
$611$	−38.7982	−1.56961
$612$	0	0
$613$	−33.8920	−1.36889	−0.684443	−	0.729066i	$-0.739955\pi$
$613$	−33.8920	−1.36889	−0.684443	+	0.729066i	$0.739955\pi$
$614$	34.1299	1.37737
$615$	0	0
$616$	8.36486	0.337030
$617$	−28.5621	−1.14987	−0.574933	−	0.818201i	$-0.694972\pi$
$617$	−28.5621	−1.14987	−0.574933	+	0.818201i	$0.694972\pi$
$618$	0	0
$619$	15.7926	0.634759	0.317380	−	0.948299i	$-0.397197\pi$
$619$	15.7926	0.634759	0.317380	+	0.948299i	$0.397197\pi$
$620$	−14.8881	−0.597920
$621$	0	0
$622$	9.95642	0.399216
$623$	−1.87039	−0.0749356
$624$	0	0
$625$	1.00000	0.0400000
$626$	−17.9519	−0.717502
$627$	0	0
$628$	−27.9704	−1.11614
$629$	−12.4784	−0.497548
$630$	0	0
$631$	30.6532	1.22029	0.610143	−	0.792291i	$-0.291112\pi$
$631$	30.6532	1.22029	0.610143	+	0.792291i	$0.291112\pi$
$632$	14.0587	0.559223
$633$	0	0
$634$	−76.2100	−3.02669
$635$	12.5720	0.498906
$636$	0	0
$637$	−18.2232	−0.722029
$638$	16.0326	0.634736
$639$	0	0
$640$	15.4885	0.612236
$641$	36.2170	1.43049	0.715243	−	0.698876i	$-0.246316\pi$
$641$	36.2170	1.43049	0.715243	+	0.698876i	$0.246316\pi$
$642$	0	0
$643$	2.66217	0.104986	0.0524929	−	0.998621i	$-0.483283\pi$
$643$	2.66217	0.104986	0.0524929	+	0.998621i	$0.483283\pi$
$644$	−18.6267	−0.733994
$645$	0	0
$646$	20.5665	0.809180
$647$	−17.0540	−0.670463	−0.335232	−	0.942136i	$-0.608815\pi$
$647$	−17.0540	−0.670463	−0.335232	+	0.942136i	$0.608815\pi$
$648$	0	0
$649$	7.42495	0.291455
$650$	−11.6220	−0.455852
$651$	0	0
$652$	−31.1284	−1.21908
$653$	−35.8249	−1.40194	−0.700969	−	0.713192i	$-0.747249\pi$
$653$	−35.8249	−1.40194	−0.700969	+	0.713192i	$0.747249\pi$
$654$	0	0
$655$	−0.192392	−0.00751737
$656$	−1.14392	−0.0446625
$657$	0	0
$658$	31.1401	1.21397
$659$	28.2716	1.10131	0.550653	−	0.834735i	$-0.314379\pi$
$659$	28.2716	1.10131	0.550653	+	0.834735i	$0.314379\pi$
$660$	0	0
$661$	−9.72688	−0.378332	−0.189166	−	0.981945i	$-0.560578\pi$
$661$	−9.72688	−0.378332	−0.189166	+	0.981945i	$0.560578\pi$
$662$	10.6401	0.413540
$663$	0	0
$664$	25.6017	0.993538
$665$	−9.90520	−0.384107
$666$	0	0
$667$	−11.7984	−0.456836
$668$	33.4359	1.29367
$669$	0	0
$670$	21.2738	0.821880
$671$	−23.1323	−0.893011
$672$	0	0
$673$	−16.4032	−0.632298	−0.316149	−	0.948710i	$-0.602390\pi$
$673$	−16.4032	−0.632298	−0.316149	+	0.948710i	$0.602390\pi$
$674$	52.4206	2.01917
$675$	0	0
$676$	42.0703	1.61809
$677$	−11.8450	−0.455242	−0.227621	−	0.973750i	$-0.573095\pi$
$677$	−11.8450	−0.455242	−0.227621	+	0.973750i	$0.573095\pi$
$678$	0	0
$679$	30.0908	1.15478
$680$	−3.83388	−0.147022
$681$	0	0
$682$	−22.5784	−0.864570
$683$	34.4819	1.31941	0.659706	−	0.751524i	$-0.270681\pi$
$683$	34.4819	1.31941	0.659706	+	0.751524i	$0.270681\pi$
$684$	0	0
$685$	−12.8576	−0.491263
$686$	43.8650	1.67477
$687$	0	0
$688$	4.98118	0.189906
$689$	−63.0697	−2.40276
$690$	0	0
$691$	46.7850	1.77979	0.889893	−	0.456169i	$-0.150779\pi$
$691$	46.7850	1.77979	0.889893	+	0.456169i	$0.150779\pi$
$692$	28.1714	1.07092
$693$	0	0
$694$	57.0070	2.16396
$695$	8.48167	0.321728
$696$	0	0
$697$	1.89273	0.0716924
$698$	3.61787	0.136939
$699$	0	0
$700$	5.58724	0.211178
$701$	31.5835	1.19289	0.596447	−	0.802653i	$-0.296579\pi$
$701$	31.5835	1.19289	0.596447	+	0.802653i	$0.296579\pi$
$702$	0	0
$703$	38.0005	1.43322
$704$	26.3438	0.992870
$705$	0	0
$706$	14.9452	0.562469
$707$	−9.23218	−0.347212
$708$	0	0
$709$	−26.1926	−0.983682	−0.491841	−	0.870685i	$-0.663676\pi$
$709$	−26.1926	−0.983682	−0.491841	+	0.870685i	$0.663676\pi$
$710$	6.29319	0.236179
$711$	0	0
$712$	2.20463	0.0826222
$713$	16.6154	0.622253
$714$	0	0
$715$	−10.5571	−0.394811
$716$	−6.77020	−0.253014
$717$	0	0
$718$	67.1954	2.50771
$719$	−21.3275	−0.795382	−0.397691	−	0.917519i	$-0.630188\pi$
$719$	−21.3275	−0.795382	−0.397691	+	0.917519i	$0.630188\pi$
$720$	0	0
$721$	−33.2460	−1.23814
$722$	−20.2003	−0.751778
$723$	0	0
$724$	−12.9473	−0.481181
$725$	3.53903	0.131436
$726$	0	0
$727$	−13.9598	−0.517739	−0.258869	−	0.965912i	$-0.583350\pi$
$727$	−13.9598	−0.517739	−0.258869	+	0.965912i	$0.583350\pi$
$728$	−21.4596	−0.795344
$729$	0	0
$730$	−2.49038	−0.0921730
$731$	−8.24187	−0.304837
$732$	0	0
$733$	46.7619	1.72719	0.863595	−	0.504186i	$-0.168207\pi$
$733$	46.7619	1.72719	0.863595	+	0.504186i	$0.168207\pi$
$734$	5.31315	0.196112
$735$	0	0
$736$	−22.5244	−0.830261
$737$	19.3245	0.711826
$738$	0	0
$739$	41.6242	1.53117	0.765586	−	0.643334i	$-0.222449\pi$
$739$	41.6242	1.53117	0.765586	+	0.643334i	$0.222449\pi$
$740$	−21.4350	−0.787967
$741$	0	0
$742$	50.6208	1.85835
$743$	−24.3902	−0.894790	−0.447395	−	0.894336i	$-0.647648\pi$
$743$	−24.3902	−0.894790	−0.447395	+	0.894336i	$0.647648\pi$
$744$	0	0
$745$	17.0248	0.623740
$746$	60.4616	2.21366
$747$	0	0
$748$	−10.5380	−0.385306
$749$	−22.7521	−0.831344
$750$	0	0
$751$	44.6465	1.62917	0.814587	−	0.580041i	$-0.196963\pi$
$751$	44.6465	1.62917	0.814587	+	0.580041i	$0.196963\pi$
$752$	7.83551	0.285732
$753$	0	0
$754$	−41.1306	−1.49789
$755$	14.8408	0.540110
$756$	0	0
$757$	−5.48985	−0.199532	−0.0997659	−	0.995011i	$-0.531809\pi$
$757$	−5.48985	−0.199532	−0.0997659	+	0.995011i	$0.531809\pi$
$758$	7.34768	0.266880
$759$	0	0
$760$	11.6753	0.423507
$761$	13.9520	0.505761	0.252880	−	0.967498i	$-0.418622\pi$
$761$	13.9520	0.505761	0.252880	+	0.967498i	$0.418622\pi$
$762$	0	0
$763$	5.37325	0.194525
$764$	−37.8795	−1.37043
$765$	0	0
$766$	−24.2553	−0.876379
$767$	−19.0483	−0.687793
$768$	0	0
$769$	12.3923	0.446876	0.223438	−	0.974718i	$-0.428272\pi$
$769$	12.3923	0.446876	0.223438	+	0.974718i	$0.428272\pi$
$770$	8.47327	0.305355
$771$	0	0
$772$	−60.8561	−2.19026
$773$	−49.2281	−1.77061	−0.885306	−	0.465008i	$-0.846051\pi$
$773$	−49.2281	−1.77061	−0.885306	+	0.465008i	$0.846051\pi$
$774$	0	0
$775$	−4.98395	−0.179029
$776$	−35.4682	−1.27323
$777$	0	0
$778$	32.1392	1.15225
$779$	−5.76393	−0.206514
$780$	0	0
$781$	5.71653	0.204554
$782$	12.9470	0.462983
$783$	0	0
$784$	3.68027	0.131438
$785$	−9.36339	−0.334194
$786$	0	0
$787$	14.2794	0.509006	0.254503	−	0.967072i	$-0.418088\pi$
$787$	14.2794	0.509006	0.254503	+	0.967072i	$0.418088\pi$
$788$	59.3673	2.11487
$789$	0	0
$790$	14.2408	0.506667
$791$	2.70300	0.0961077
$792$	0	0
$793$	59.3444	2.10738
$794$	−14.9622	−0.530989
$795$	0	0
$796$	35.1010	1.24412
$797$	−28.9742	−1.02632	−0.513160	−	0.858293i	$-0.671525\pi$
$797$	−28.9742	−1.02632	−0.513160	+	0.858293i	$0.671525\pi$
$798$	0	0
$799$	−12.9647	−0.458657
$800$	6.75639	0.238875
$801$	0	0
$802$	−9.40132	−0.331972
$803$	−2.26218	−0.0798305
$804$	0	0
$805$	−6.23549	−0.219772
$806$	57.9235	2.04027
$807$	0	0
$808$	10.8820	0.382828
$809$	−42.1439	−1.48170	−0.740850	−	0.671671i	$-0.765577\pi$
$809$	−42.1439	−1.48170	−0.740850	+	0.671671i	$0.765577\pi$
$810$	0	0
$811$	36.3440	1.27621	0.638104	−	0.769950i	$-0.279719\pi$
$811$	36.3440	1.27621	0.638104	+	0.769950i	$0.279719\pi$
$812$	19.7734	0.693911
$813$	0	0
$814$	−32.5071	−1.13937
$815$	−10.4206	−0.365017
$816$	0	0
$817$	25.0989	0.878101
$818$	86.6065	3.02813
$819$	0	0
$820$	3.25127	0.113539
$821$	−9.36717	−0.326917	−0.163458	−	0.986550i	$-0.552265\pi$
$821$	−9.36717	−0.326917	−0.163458	+	0.986550i	$0.552265\pi$
$822$	0	0
$823$	−4.37609	−0.152541	−0.0762705	−	0.997087i	$-0.524301\pi$
$823$	−4.37609	−0.152541	−0.0762705	+	0.997087i	$0.524301\pi$
$824$	39.1871	1.36515
$825$	0	0
$826$	15.2885	0.531954
$827$	−17.4196	−0.605739	−0.302869	−	0.953032i	$-0.597945\pi$
$827$	−17.4196	−0.605739	−0.302869	+	0.953032i	$0.597945\pi$
$828$	0	0
$829$	−21.0368	−0.730640	−0.365320	−	0.930882i	$-0.619040\pi$
$829$	−21.0368	−0.730640	−0.365320	+	0.930882i	$0.619040\pi$
$830$	25.9335	0.900164
$831$	0	0
$832$	−67.5835	−2.34304
$833$	−6.08939	−0.210985
$834$	0	0
$835$	11.1930	0.387351
$836$	32.0912	1.10990
$837$	0	0
$838$	27.8075	0.960596
$839$	−41.2797	−1.42513	−0.712567	−	0.701604i	$-0.752467\pi$
$839$	−41.2797	−1.42513	−0.712567	+	0.701604i	$0.752467\pi$
$840$	0	0
$841$	−16.4753	−0.568112
$842$	88.0493	3.03438
$843$	0	0
$844$	44.2581	1.52343
$845$	14.0835	0.484487
$846$	0	0
$847$	−12.8774	−0.442474
$848$	12.7373	0.437399
$849$	0	0
$850$	−3.88356	−0.133205
$851$	23.9220	0.820035
$852$	0	0
$853$	1.62313	0.0555749	0.0277875	−	0.999614i	$-0.491154\pi$
$853$	1.62313	0.0555749	0.0277875	+	0.999614i	$0.491154\pi$
$854$	−47.6309	−1.62989
$855$	0	0
$856$	26.8180	0.916620
$857$	−21.4347	−0.732196	−0.366098	−	0.930576i	$-0.619306\pi$
$857$	−21.4347	−0.732196	−0.366098	+	0.930576i	$0.619306\pi$
$858$	0	0
$859$	1.93092	0.0658822	0.0329411	−	0.999457i	$-0.489513\pi$
$859$	1.93092	0.0658822	0.0329411	+	0.999457i	$0.489513\pi$
$860$	−14.1576	−0.482770
$861$	0	0
$862$	−69.9940	−2.38400
$863$	35.4312	1.20609	0.603046	−	0.797707i	$-0.293954\pi$
$863$	35.4312	1.20609	0.603046	+	0.797707i	$0.293954\pi$
$864$	0	0
$865$	9.43068	0.320653
$866$	−73.9644	−2.51341
$867$	0	0
$868$	−27.8465	−0.945172
$869$	12.9359	0.438821
$870$	0	0
$871$	−49.5758	−1.67981
$872$	−6.33347	−0.214478
$873$	0	0
$874$	−39.4274	−1.33365
$875$	1.87039	0.0632307
$876$	0	0
$877$	22.3502	0.754713	0.377356	−	0.926068i	$-0.376833\pi$
$877$	22.3502	0.754713	0.377356	+	0.926068i	$0.376833\pi$
$878$	70.7401	2.38736
$879$	0	0
$880$	2.13205	0.0718715
$881$	30.2395	1.01880	0.509398	−	0.860531i	$-0.329868\pi$
$881$	30.2395	1.01880	0.509398	+	0.860531i	$0.329868\pi$
$882$	0	0
$883$	−36.2381	−1.21951	−0.609755	−	0.792590i	$-0.708732\pi$
$883$	−36.2381	−1.21951	−0.609755	+	0.792590i	$0.708732\pi$
$884$	27.0345	0.909270
$885$	0	0
$886$	36.3376	1.22079
$887$	53.5776	1.79896	0.899479	−	0.436963i	$-0.143946\pi$
$887$	53.5776	1.79896	0.899479	+	0.436963i	$0.143946\pi$
$888$	0	0
$889$	23.5146	0.788655
$890$	2.23321	0.0748572
$891$	0	0
$892$	34.1239	1.14255
$893$	39.4813	1.32119
$894$	0	0
$895$	−2.26640	−0.0757574
$896$	28.9695	0.967803
$897$	0	0
$898$	−2.98518	−0.0996168
$899$	−17.6384	−0.588272
$900$	0	0
$901$	−21.0751	−0.702114
$902$	4.93068	0.164174
$903$	0	0
$904$	−3.18604	−0.105966
$905$	−4.33424	−0.144075
$906$	0	0
$907$	50.5208	1.67751	0.838757	−	0.544506i	$-0.183283\pi$
$907$	50.5208	1.67751	0.838757	+	0.544506i	$0.183283\pi$
$908$	−14.9000	−0.494475
$909$	0	0
$910$	−21.7377	−0.720597
$911$	1.36013	0.0450630	0.0225315	−	0.999746i	$-0.492827\pi$
$911$	1.36013	0.0450630	0.0225315	+	0.999746i	$0.492827\pi$
$912$	0	0
$913$	23.5571	0.779627
$914$	−24.2576	−0.802371
$915$	0	0
$916$	30.4381	1.00570
$917$	−0.359847	−0.0118832
$918$	0	0
$919$	−32.2226	−1.06293	−0.531464	−	0.847081i	$-0.678358\pi$
$919$	−32.2226	−1.06293	−0.531464	+	0.847081i	$0.678358\pi$
$920$	7.34979	0.242315
$921$	0	0
$922$	−48.0209	−1.58148
$923$	−14.6654	−0.482719
$924$	0	0
$925$	−7.17561	−0.235933
$926$	62.0067	2.03767
$927$	0	0
$928$	23.9111	0.784920
$929$	60.6939	1.99130	0.995652	−	0.0931538i	$-0.0296948\pi$
$929$	60.6939	1.99130	0.995652	+	0.0931538i	$0.0296948\pi$
$930$	0	0
$931$	18.5440	0.607755
$932$	36.1458	1.18399
$933$	0	0
$934$	−16.9177	−0.553563
$935$	−3.52770	−0.115368
$936$	0	0
$937$	−40.5239	−1.32386	−0.661929	−	0.749566i	$-0.730262\pi$
$937$	−40.5239	−1.32386	−0.661929	+	0.749566i	$0.730262\pi$
$938$	39.7904	1.29920
$939$	0	0
$940$	−22.2702	−0.726375
$941$	−32.9515	−1.07419	−0.537093	−	0.843523i	$-0.680478\pi$
$941$	−32.9515	−1.07419	−0.537093	+	0.843523i	$0.680478\pi$
$942$	0	0
$943$	−3.62849	−0.118160
$944$	3.84690	0.125206
$945$	0	0
$946$	−21.4706	−0.698068
$947$	−59.2296	−1.92470	−0.962352	−	0.271808i	$-0.912379\pi$
$947$	−59.2296	−1.92470	−0.962352	+	0.271808i	$0.912379\pi$
$948$	0	0
$949$	5.80348	0.188389
$950$	11.8266	0.383705
$951$	0	0
$952$	−7.17084	−0.232408
$953$	25.0221	0.810545	0.405273	−	0.914196i	$-0.367177\pi$
$953$	25.0221	0.810545	0.405273	+	0.914196i	$0.367177\pi$
$954$	0	0
$955$	−12.6806	−0.410334
$956$	−53.9580	−1.74513
$957$	0	0
$958$	−26.6213	−0.860094
$959$	−24.0487	−0.776573
$960$	0	0
$961$	−6.16025	−0.198718
$962$	83.3949	2.68876
$963$	0	0
$964$	73.7145	2.37419
$965$	−20.3722	−0.655806
$966$	0	0
$967$	−36.4358	−1.17170	−0.585848	−	0.810421i	$-0.699239\pi$
$967$	−36.4358	−1.17170	−0.585848	+	0.810421i	$0.699239\pi$
$968$	15.1787	0.487861
$969$	0	0
$970$	−35.9278	−1.15357
$971$	57.6118	1.84885	0.924425	−	0.381363i	$-0.124545\pi$
$971$	57.6118	1.84885	0.924425	+	0.381363i	$0.124545\pi$
$972$	0	0
$973$	15.8640	0.508577
$974$	53.6075	1.71770
$975$	0	0
$976$	−11.9849	−0.383628
$977$	−25.0812	−0.802420	−0.401210	−	0.915986i	$-0.631410\pi$
$977$	−25.0812	−0.802420	−0.401210	+	0.915986i	$0.631410\pi$
$978$	0	0
$979$	2.02857	0.0648334
$980$	−10.4601	−0.334137
$981$	0	0
$982$	15.7093	0.501304
$983$	−5.65906	−0.180496	−0.0902479	−	0.995919i	$-0.528766\pi$
$983$	−5.65906	−0.180496	−0.0902479	+	0.995919i	$0.528766\pi$
$984$	0	0
$985$	19.8739	0.633234
$986$	−13.7440	−0.437700
$987$	0	0
$988$	−82.3282	−2.61921
$989$	15.8002	0.502417
$990$	0	0
$991$	29.1538	0.926100	0.463050	−	0.886332i	$-0.346755\pi$
$991$	29.1538	0.926100	0.463050	+	0.886332i	$0.346755\pi$
$992$	−33.6735	−1.06914
$993$	0	0
$994$	11.7707	0.373345
$995$	11.7504	0.372514
$996$	0	0
$997$	8.91059	0.282201	0.141101	−	0.989995i	$-0.454936\pi$
$997$	8.91059	0.282201	0.141101	+	0.989995i	$0.454936\pi$
$998$	−32.1070	−1.01633
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.p.1.2		8
3.2	odd	2		445.2.a.g.1.7	✓	8
12.11	even	2		7120.2.a.bk.1.5		8
15.14	odd	2		2225.2.a.l.1.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
445.2.a.g.1.7	✓	8	3.2	odd	2
2225.2.a.l.1.2		8	15.14	odd	2
4005.2.a.p.1.2		8	1.1	even	1	trivial
7120.2.a.bk.1.5		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)

\(f(q)\)	\(=\)	\(q-2.23321 q^{2} +2.98721 q^{4} +1.00000 q^{5} +1.87039 q^{7} -2.20463 q^{8} +O(q^{10})\)	\(q-2.23321 q^{2} +2.98721 q^{4} +1.00000 q^{5} +1.87039 q^{7} -2.20463 q^{8} -2.23321 q^{10} -2.02857 q^{11} +5.20418 q^{13} -4.17696 q^{14} -1.05101 q^{16} +1.73901 q^{17} -5.29579 q^{19} +2.98721 q^{20} +4.53022 q^{22} -3.33379 q^{23} +1.00000 q^{25} -11.6220 q^{26} +5.58724 q^{28} +3.53903 q^{29} -4.98395 q^{31} +6.75639 q^{32} -3.88356 q^{34} +1.87039 q^{35} -7.17561 q^{37} +11.8266 q^{38} -2.20463 q^{40} +1.08840 q^{41} -4.73941 q^{43} -6.05976 q^{44} +7.44504 q^{46} -7.45521 q^{47} -3.50165 q^{49} -2.23321 q^{50} +15.5460 q^{52} -12.1190 q^{53} -2.02857 q^{55} -4.12352 q^{56} -7.90338 q^{58} -3.66019 q^{59} +11.4032 q^{61} +11.1302 q^{62} -12.9864 q^{64} +5.20418 q^{65} -9.52615 q^{67} +5.19477 q^{68} -4.17696 q^{70} -2.81801 q^{71} +1.11516 q^{73} +16.0246 q^{74} -15.8196 q^{76} -3.79422 q^{77} -6.37686 q^{79} -1.05101 q^{80} -2.43062 q^{82} -11.6127 q^{83} +1.73901 q^{85} +10.5841 q^{86} +4.47226 q^{88} -1.00000 q^{89} +9.73384 q^{91} -9.95872 q^{92} +16.6490 q^{94} -5.29579 q^{95} +16.0880 q^{97} +7.81989 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} + 7 q^{4} + 8 q^{5} - 6 q^{7} - 3 q^{8}+O(q^{10}) \) 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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