LMFDB - Embedded newform 4005.2.a.w.1.7

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			1.7
Root			$1.08305$ 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)

$f(q)$	$=$	$q-1.08305 q^{2} -0.826999 q^{4} -1.00000 q^{5} -2.85554 q^{7} +3.06179 q^{8} +O(q^{10})$	\(q-1.08305 q^{2} -0.826999 q^{4} -1.00000 q^{5} -2.85554 q^{7} +3.06179 q^{8} +1.08305 q^{10} +5.20445 q^{11} -1.47747 q^{13} +3.09270 q^{14} -1.66208 q^{16} +4.97107 q^{17} +5.74417 q^{19} +0.826999 q^{20} -5.63669 q^{22} -1.03671 q^{23} +1.00000 q^{25} +1.60017 q^{26} +2.36153 q^{28} -3.74972 q^{29} +4.54430 q^{31} -4.32346 q^{32} -5.38392 q^{34} +2.85554 q^{35} +1.55142 q^{37} -6.22123 q^{38} -3.06179 q^{40} +0.0927013 q^{41} -4.64981 q^{43} -4.30408 q^{44} +1.12281 q^{46} -13.5627 q^{47} +1.15411 q^{49} -1.08305 q^{50} +1.22186 q^{52} -5.50732 q^{53} -5.20445 q^{55} -8.74305 q^{56} +4.06114 q^{58} +3.08041 q^{59} -0.0385230 q^{61} -4.92171 q^{62} +8.00668 q^{64} +1.47747 q^{65} +10.2334 q^{67} -4.11107 q^{68} -3.09270 q^{70} +2.07479 q^{71} +11.9314 q^{73} -1.68027 q^{74} -4.75042 q^{76} -14.8615 q^{77} -7.74274 q^{79} +1.66208 q^{80} -0.100400 q^{82} -8.84796 q^{83} -4.97107 q^{85} +5.03599 q^{86} +15.9349 q^{88} +1.00000 q^{89} +4.21896 q^{91} +0.857359 q^{92} +14.6891 q^{94} -5.74417 q^{95} -11.8228 q^{97} -1.24996 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.08305	−0.765833	−0.382917	−	0.923783i	$-0.625080\pi$
$2$	−1.08305	−0.765833	−0.382917	+	0.923783i	$0.625080\pi$
$3$	0	0
$3$	0	0
$4$	−0.826999	−0.413499
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.85554	−1.07929	−0.539646	−	0.841892i	$-0.681442\pi$
$7$	−2.85554	−1.07929	−0.539646	+	0.841892i	$0.681442\pi$
$8$	3.06179	1.08250
$9$	0	0
$10$	1.08305	0.342491
$11$	5.20445	1.56920	0.784601	−	0.620001i	$-0.212868\pi$
$11$	5.20445	1.56920	0.784601	+	0.620001i	$0.212868\pi$
$12$	0	0
$13$	−1.47747	−0.409775	−0.204888	−	0.978785i	$-0.565683\pi$
$13$	−1.47747	−0.409775	−0.204888	+	0.978785i	$0.565683\pi$
$14$	3.09270	0.826558
$15$	0	0
$16$	−1.66208	−0.415519
$17$	4.97107	1.20566	0.602830	−	0.797869i	$-0.294040\pi$
$17$	4.97107	1.20566	0.602830	+	0.797869i	$0.294040\pi$
$18$	0	0
$19$	5.74417	1.31780	0.658901	−	0.752229i	$-0.271021\pi$
$19$	5.74417	1.31780	0.658901	+	0.752229i	$0.271021\pi$
$20$	0.826999	0.184923
$21$	0	0
$22$	−5.63669	−1.20175
$23$	−1.03671	−0.216169	−0.108085	−	0.994142i	$-0.534472\pi$
$23$	−1.03671	−0.216169	−0.108085	+	0.994142i	$0.534472\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	1.60017	0.313820
$27$	0	0
$28$	2.36153	0.446287
$29$	−3.74972	−0.696305	−0.348152	−	0.937438i	$-0.613191\pi$
$29$	−3.74972	−0.696305	−0.348152	+	0.937438i	$0.613191\pi$
$30$	0	0
$31$	4.54430	0.816180	0.408090	−	0.912942i	$-0.366195\pi$
$31$	4.54430	0.816180	0.408090	+	0.912942i	$0.366195\pi$
$32$	−4.32346	−0.764287
$33$	0	0
$34$	−5.38392	−0.923335
$35$	2.85554	0.482674
$36$	0	0
$37$	1.55142	0.255051	0.127526	−	0.991835i	$-0.459296\pi$
$37$	1.55142	0.255051	0.127526	+	0.991835i	$0.459296\pi$
$38$	−6.22123	−1.00922
$39$	0	0
$40$	−3.06179	−0.484111
$41$	0.0927013	0.0144775	0.00723875	−	0.999974i	$-0.497696\pi$
$41$	0.0927013	0.0144775	0.00723875	+	0.999974i	$0.497696\pi$
$42$	0	0
$43$	−4.64981	−0.709090	−0.354545	−	0.935039i	$-0.615364\pi$
$43$	−4.64981	−0.709090	−0.354545	+	0.935039i	$0.615364\pi$
$44$	−4.30408	−0.648864
$45$	0	0
$46$	1.12281	0.165550
$47$	−13.5627	−1.97832	−0.989161	−	0.146838i	$-0.953090\pi$
$47$	−13.5627	−1.97832	−0.989161	+	0.146838i	$0.953090\pi$
$48$	0	0
$49$	1.15411	0.164872
$50$	−1.08305	−0.153167
$51$	0	0
$52$	1.22186	0.169442
$53$	−5.50732	−0.756488	−0.378244	−	0.925706i	$-0.623472\pi$
$53$	−5.50732	−0.756488	−0.378244	+	0.925706i	$0.623472\pi$
$54$	0	0
$55$	−5.20445	−0.701768
$56$	−8.74305	−1.16834
$57$	0	0
$58$	4.06114	0.533254
$59$	3.08041	0.401035	0.200517	−	0.979690i	$-0.435738\pi$
$59$	3.08041	0.401035	0.200517	+	0.979690i	$0.435738\pi$
$60$	0	0
$61$	−0.0385230	−0.00493236	−0.00246618	−	0.999997i	$-0.500785\pi$
$61$	−0.0385230	−0.00493236	−0.00246618	+	0.999997i	$0.500785\pi$
$62$	−4.92171	−0.625058
$63$	0	0
$64$	8.00668	1.00084
$65$	1.47747	0.183257
$66$	0	0
$67$	10.2334	1.25021	0.625106	−	0.780540i	$-0.285056\pi$
$67$	10.2334	1.25021	0.625106	+	0.780540i	$0.285056\pi$
$68$	−4.11107	−0.498540
$69$	0	0
$70$	−3.09270	−0.369648
$71$	2.07479	0.246232	0.123116	−	0.992392i	$-0.460711\pi$
$71$	2.07479	0.246232	0.123116	+	0.992392i	$0.460711\pi$
$72$	0	0
$73$	11.9314	1.39647	0.698234	−	0.715870i	$-0.253970\pi$
$73$	11.9314	1.39647	0.698234	+	0.715870i	$0.253970\pi$
$74$	−1.68027	−0.195327
$75$	0	0
$76$	−4.75042	−0.544911
$77$	−14.8615	−1.69363
$78$	0	0
$79$	−7.74274	−0.871126	−0.435563	−	0.900158i	$-0.643451\pi$
$79$	−7.74274	−0.871126	−0.435563	+	0.900158i	$0.643451\pi$
$80$	1.66208	0.185826
$81$	0	0
$82$	−0.100400	−0.0110874
$83$	−8.84796	−0.971190	−0.485595	−	0.874184i	$-0.661397\pi$
$83$	−8.84796	−0.971190	−0.485595	+	0.874184i	$0.661397\pi$
$84$	0	0
$85$	−4.97107	−0.539188
$86$	5.03599	0.543045
$87$	0	0
$88$	15.9349	1.69867
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	4.21896	0.442268
$92$	0.857359	0.0893858
$93$	0	0
$94$	14.6891	1.51506
$95$	−5.74417	−0.589339
$96$	0	0
$97$	−11.8228	−1.20042	−0.600209	−	0.799843i	$-0.704916\pi$
$97$	−11.8228	−1.20042	−0.600209	+	0.799843i	$0.704916\pi$
$98$	−1.24996	−0.126265
$99$	0	0
$100$	−0.826999	−0.0826999
$101$	3.33440	0.331785	0.165893	−	0.986144i	$-0.446949\pi$
$101$	3.33440	0.331785	0.165893	+	0.986144i	$0.446949\pi$
$102$	0	0
$103$	13.1709	1.29777	0.648884	−	0.760887i	$-0.275236\pi$
$103$	13.1709	1.29777	0.648884	+	0.760887i	$0.275236\pi$
$104$	−4.52369	−0.443584
$105$	0	0
$106$	5.96471	0.579344
$107$	−7.79189	−0.753270	−0.376635	−	0.926362i	$-0.622919\pi$
$107$	−7.79189	−0.753270	−0.376635	+	0.926362i	$0.622919\pi$
$108$	0	0
$109$	8.28038	0.793116	0.396558	−	0.918010i	$-0.370204\pi$
$109$	8.28038	0.793116	0.396558	+	0.918010i	$0.370204\pi$
$110$	5.63669	0.537438
$111$	0	0
$112$	4.74612	0.448466
$113$	12.6737	1.19224	0.596119	−	0.802896i	$-0.296709\pi$
$113$	12.6737	1.19224	0.596119	+	0.802896i	$0.296709\pi$
$114$	0	0
$115$	1.03671	0.0966738
$116$	3.10101	0.287922
$117$	0	0
$118$	−3.33624	−0.307126
$119$	−14.1951	−1.30126
$120$	0	0
$121$	16.0863	1.46239
$122$	0.0417224	0.00377736
$123$	0	0
$124$	−3.75813	−0.337490
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	2.40123	0.213075	0.106538	−	0.994309i	$-0.466024\pi$
$127$	2.40123	0.213075	0.106538	+	0.994309i	$0.466024\pi$
$128$	−0.0247324	−0.00218606
$129$	0	0
$130$	−1.60017	−0.140344
$131$	15.6234	1.36502	0.682510	−	0.730876i	$-0.260888\pi$
$131$	15.6234	1.36502	0.682510	+	0.730876i	$0.260888\pi$
$132$	0	0
$133$	−16.4027	−1.42229
$134$	−11.0833	−0.957454
$135$	0	0
$136$	15.2203	1.30513
$137$	6.93206	0.592246	0.296123	−	0.955150i	$-0.404306\pi$
$137$	6.93206	0.592246	0.296123	+	0.955150i	$0.404306\pi$
$138$	0	0
$139$	−15.9764	−1.35510	−0.677551	−	0.735476i	$-0.736959\pi$
$139$	−15.9764	−1.35510	−0.677551	+	0.735476i	$0.736959\pi$
$140$	−2.36153	−0.199586
$141$	0	0
$142$	−2.24710	−0.188572
$143$	−7.68941	−0.643020
$144$	0	0
$145$	3.74972	0.311397
$146$	−12.9224	−1.06946
$147$	0	0
$148$	−1.28302	−0.105464
$149$	8.40848	0.688849	0.344425	−	0.938814i	$-0.388074\pi$
$149$	8.40848	0.688849	0.344425	+	0.938814i	$0.388074\pi$
$150$	0	0
$151$	6.37812	0.519044	0.259522	−	0.965737i	$-0.416435\pi$
$151$	6.37812	0.519044	0.259522	+	0.965737i	$0.416435\pi$
$152$	17.5874	1.42653
$153$	0	0
$154$	16.0958	1.29704
$155$	−4.54430	−0.365007
$156$	0	0
$157$	−0.908265	−0.0724874	−0.0362437	−	0.999343i	$-0.511539\pi$
$157$	−0.908265	−0.0724874	−0.0362437	+	0.999343i	$0.511539\pi$
$158$	8.38579	0.667138
$159$	0	0
$160$	4.32346	0.341799
$161$	2.96037	0.233310
$162$	0	0
$163$	1.59158	0.124662	0.0623309	−	0.998056i	$-0.480147\pi$
$163$	1.59158	0.124662	0.0623309	+	0.998056i	$0.480147\pi$
$164$	−0.0766638	−0.00598644
$165$	0	0
$166$	9.58280	0.743770
$167$	0.394822	0.0305523	0.0152761	−	0.999883i	$-0.495137\pi$
$167$	0.394822	0.0305523	0.0152761	+	0.999883i	$0.495137\pi$
$168$	0	0
$169$	−10.8171	−0.832084
$170$	5.38392	0.412928
$171$	0	0
$172$	3.84539	0.293208
$173$	19.1430	1.45541	0.727707	−	0.685888i	$-0.240586\pi$
$173$	19.1430	1.45541	0.727707	+	0.685888i	$0.240586\pi$
$174$	0	0
$175$	−2.85554	−0.215858
$176$	−8.65020	−0.652033
$177$	0	0
$178$	−1.08305	−0.0811782
$179$	14.0446	1.04974	0.524871	−	0.851182i	$-0.324114\pi$
$179$	14.0446	1.04974	0.524871	+	0.851182i	$0.324114\pi$
$180$	0	0
$181$	22.6255	1.68174	0.840869	−	0.541239i	$-0.182044\pi$
$181$	22.6255	1.68174	0.840869	+	0.541239i	$0.182044\pi$
$182$	−4.56936	−0.338703
$183$	0	0
$184$	−3.17419	−0.234004
$185$	−1.55142	−0.114062
$186$	0	0
$187$	25.8717	1.89192
$188$	11.2163	0.818035
$189$	0	0
$190$	6.22123	0.451336
$191$	−7.49540	−0.542348	−0.271174	−	0.962530i	$-0.587412\pi$
$191$	−7.49540	−0.542348	−0.271174	+	0.962530i	$0.587412\pi$
$192$	0	0
$193$	19.4231	1.39811	0.699053	−	0.715070i	$-0.253605\pi$
$193$	19.4231	1.39811	0.699053	+	0.715070i	$0.253605\pi$
$194$	12.8047	0.919321
$195$	0	0
$196$	−0.954444	−0.0681745
$197$	−20.7047	−1.47515	−0.737575	−	0.675265i	$-0.764029\pi$
$197$	−20.7047	−1.47515	−0.737575	+	0.675265i	$0.764029\pi$
$198$	0	0
$199$	−8.27586	−0.586660	−0.293330	−	0.956011i	$-0.594764\pi$
$199$	−8.27586	−0.586660	−0.293330	+	0.956011i	$0.594764\pi$
$200$	3.06179	0.216501
$201$	0	0
$202$	−3.61133	−0.254092
$203$	10.7075	0.751517
$204$	0	0
$205$	−0.0927013	−0.00647454
$206$	−14.2648	−0.993874
$207$	0	0
$208$	2.45566	0.170269
$209$	29.8953	2.06790
$210$	0	0
$211$	1.73015	0.119109	0.0595543	−	0.998225i	$-0.481032\pi$
$211$	1.73015	0.119109	0.0595543	+	0.998225i	$0.481032\pi$
$212$	4.55454	0.312807
$213$	0	0
$214$	8.43902	0.576880
$215$	4.64981	0.317115
$216$	0	0
$217$	−12.9764	−0.880897
$218$	−8.96808	−0.607395
$219$	0	0
$220$	4.30408	0.290181
$221$	−7.34458	−0.494050
$222$	0	0
$223$	5.23136	0.350318	0.175159	−	0.984540i	$-0.443956\pi$
$223$	5.23136	0.350318	0.175159	+	0.984540i	$0.443956\pi$
$224$	12.3458	0.824889
$225$	0	0
$226$	−13.7262	−0.913056
$227$	−12.6519	−0.839735	−0.419868	−	0.907585i	$-0.637923\pi$
$227$	−12.6519	−0.839735	−0.419868	+	0.907585i	$0.637923\pi$
$228$	0	0
$229$	−0.812245	−0.0536747	−0.0268373	−	0.999640i	$-0.508544\pi$
$229$	−0.812245	−0.0536747	−0.0268373	+	0.999640i	$0.508544\pi$
$230$	−1.12281	−0.0740360
$231$	0	0
$232$	−11.4808	−0.753754
$233$	−18.7729	−1.22986	−0.614928	−	0.788583i	$-0.710815\pi$
$233$	−18.7729	−1.22986	−0.614928	+	0.788583i	$0.710815\pi$
$234$	0	0
$235$	13.5627	0.884732
$236$	−2.54749	−0.165828
$237$	0	0
$238$	15.3740	0.996548
$239$	5.52517	0.357394	0.178697	−	0.983904i	$-0.442812\pi$
$239$	5.52517	0.357394	0.178697	+	0.983904i	$0.442812\pi$
$240$	0	0
$241$	13.2878	0.855941	0.427970	−	0.903793i	$-0.359229\pi$
$241$	13.2878	0.855941	0.427970	+	0.903793i	$0.359229\pi$
$242$	−17.4223	−1.11995
$243$	0	0
$244$	0.0318584	0.00203953
$245$	−1.15411	−0.0737331
$246$	0	0
$247$	−8.48682	−0.540003
$248$	13.9137	0.883519
$249$	0	0
$250$	1.08305	0.0684982
$251$	10.9711	0.692491	0.346245	−	0.938144i	$-0.387456\pi$
$251$	10.9711	0.692491	0.346245	+	0.938144i	$0.387456\pi$
$252$	0	0
$253$	−5.39551	−0.339213
$254$	−2.60066	−0.163180
$255$	0	0
$256$	−15.9866	−0.999161
$257$	−7.21327	−0.449951	−0.224976	−	0.974364i	$-0.572230\pi$
$257$	−7.21327	−0.449951	−0.224976	+	0.974364i	$0.572230\pi$
$258$	0	0
$259$	−4.43013	−0.275275
$260$	−1.22186	−0.0757767
$261$	0	0
$262$	−16.9209	−1.04538
$263$	12.5431	0.773443	0.386722	−	0.922197i	$-0.373607\pi$
$263$	12.5431	0.773443	0.386722	+	0.922197i	$0.373607\pi$
$264$	0	0
$265$	5.50732	0.338312
$266$	17.7650	1.08924
$267$	0	0
$268$	−8.46303	−0.516962
$269$	−15.1918	−0.926260	−0.463130	−	0.886290i	$-0.653274\pi$
$269$	−15.1918	−0.926260	−0.463130	+	0.886290i	$0.653274\pi$
$270$	0	0
$271$	21.6796	1.31694	0.658471	−	0.752606i	$-0.271204\pi$
$271$	21.6796	1.31694	0.658471	+	0.752606i	$0.271204\pi$
$272$	−8.26229	−0.500975
$273$	0	0
$274$	−7.50778	−0.453561
$275$	5.20445	0.313840
$276$	0	0
$277$	5.83291	0.350465	0.175233	−	0.984527i	$-0.443932\pi$
$277$	5.83291	0.350465	0.175233	+	0.984527i	$0.443932\pi$
$278$	17.3033	1.03778
$279$	0	0
$280$	8.74305	0.522497
$281$	−5.68182	−0.338949	−0.169474	−	0.985535i	$-0.554207\pi$
$281$	−5.68182	−0.338949	−0.169474	+	0.985535i	$0.554207\pi$
$282$	0	0
$283$	12.8969	0.766640	0.383320	−	0.923616i	$-0.374781\pi$
$283$	12.8969	0.766640	0.383320	+	0.923616i	$0.374781\pi$
$284$	−1.71585	−0.101817
$285$	0	0
$286$	8.32802	0.492446
$287$	−0.264712	−0.0156255
$288$	0	0
$289$	7.71149	0.453617
$290$	−4.06114	−0.238478
$291$	0	0
$292$	−9.86727	−0.577439
$293$	1.11682	0.0652452	0.0326226	−	0.999468i	$-0.489614\pi$
$293$	1.11682	0.0652452	0.0326226	+	0.999468i	$0.489614\pi$
$294$	0	0
$295$	−3.08041	−0.179348
$296$	4.75011	0.276094
$297$	0	0
$298$	−9.10682	−0.527544
$299$	1.53171	0.0885808
$300$	0	0
$301$	13.2777	0.765316
$302$	−6.90783	−0.397501
$303$	0	0
$304$	−9.54724	−0.547572
$305$	0.0385230	0.00220582
$306$	0	0
$307$	6.74996	0.385241	0.192620	−	0.981273i	$-0.438301\pi$
$307$	6.74996	0.385241	0.192620	+	0.981273i	$0.438301\pi$
$308$	12.2905	0.700314
$309$	0	0
$310$	4.92171	0.279534
$311$	−27.0904	−1.53615	−0.768077	−	0.640357i	$-0.778786\pi$
$311$	−27.0904	−1.53615	−0.768077	+	0.640357i	$0.778786\pi$
$312$	0	0
$313$	−14.6657	−0.828953	−0.414476	−	0.910060i	$-0.636035\pi$
$313$	−14.6657	−0.828953	−0.414476	+	0.910060i	$0.636035\pi$
$314$	0.983698	0.0555133
$315$	0	0
$316$	6.40324	0.360210
$317$	3.87134	0.217436	0.108718	−	0.994073i	$-0.465325\pi$
$317$	3.87134	0.217436	0.108718	+	0.994073i	$0.465325\pi$
$318$	0	0
$319$	−19.5152	−1.09264
$320$	−8.00668	−0.447587
$321$	0	0
$322$	−3.20623	−0.178676
$323$	28.5546	1.58882
$324$	0	0
$325$	−1.47747	−0.0819551
$326$	−1.72376	−0.0954701
$327$	0	0
$328$	0.283831	0.0156720
$329$	38.7288	2.13519
$330$	0	0
$331$	11.8126	0.649280	0.324640	−	0.945838i	$-0.394757\pi$
$331$	11.8126	0.649280	0.324640	+	0.945838i	$0.394757\pi$
$332$	7.31726	0.401587
$333$	0	0
$334$	−0.427613	−0.0233979
$335$	−10.2334	−0.559112
$336$	0	0
$337$	20.9510	1.14127	0.570636	−	0.821203i	$-0.306697\pi$
$337$	20.9510	1.14127	0.570636	+	0.821203i	$0.306697\pi$
$338$	11.7155	0.637238
$339$	0	0
$340$	4.11107	0.222954
$341$	23.6506	1.28075
$342$	0	0
$343$	16.6932	0.901347
$344$	−14.2367	−0.767593
$345$	0	0
$346$	−20.7328	−1.11460
$347$	−29.5442	−1.58601	−0.793007	−	0.609213i	$-0.791486\pi$
$347$	−29.5442	−1.58601	−0.793007	+	0.609213i	$0.791486\pi$
$348$	0	0
$349$	−16.6412	−0.890781	−0.445390	−	0.895336i	$-0.646935\pi$
$349$	−16.6412	−0.890781	−0.445390	+	0.895336i	$0.646935\pi$
$350$	3.09270	0.165312
$351$	0	0
$352$	−22.5012	−1.19932
$353$	10.7693	0.573191	0.286596	−	0.958052i	$-0.407476\pi$
$353$	10.7693	0.573191	0.286596	+	0.958052i	$0.407476\pi$
$354$	0	0
$355$	−2.07479	−0.110118
$356$	−0.826999	−0.0438308
$357$	0	0
$358$	−15.2110	−0.803927
$359$	11.3459	0.598816	0.299408	−	0.954125i	$-0.403211\pi$
$359$	11.3459	0.598816	0.299408	+	0.954125i	$0.403211\pi$
$360$	0	0
$361$	13.9955	0.736604
$362$	−24.5046	−1.28793
$363$	0	0
$364$	−3.48908	−0.182877
$365$	−11.9314	−0.624519
$366$	0	0
$367$	5.24392	0.273731	0.136865	−	0.990590i	$-0.456297\pi$
$367$	5.24392	0.273731	0.136865	+	0.990590i	$0.456297\pi$
$368$	1.72309	0.0898224
$369$	0	0
$370$	1.68027	0.0873528
$371$	15.7264	0.816472
$372$	0	0
$373$	14.1375	0.732012	0.366006	−	0.930613i	$-0.380725\pi$
$373$	14.1375	0.732012	0.366006	+	0.930613i	$0.380725\pi$
$374$	−28.0204	−1.44890
$375$	0	0
$376$	−41.5260	−2.14154
$377$	5.54008	0.285329
$378$	0	0
$379$	38.5279	1.97904	0.989522	−	0.144381i	$-0.0461189\pi$
$379$	38.5279	1.97904	0.989522	+	0.144381i	$0.0461189\pi$
$380$	4.75042	0.243691
$381$	0	0
$382$	8.11790	0.415348
$383$	−27.6314	−1.41190	−0.705950	−	0.708261i	$-0.749480\pi$
$383$	−27.6314	−1.41190	−0.705950	+	0.708261i	$0.749480\pi$
$384$	0	0
$385$	14.8615	0.757413
$386$	−21.0362	−1.07072
$387$	0	0
$388$	9.77740	0.496372
$389$	−4.68360	−0.237468	−0.118734	−	0.992926i	$-0.537884\pi$
$389$	−4.68360	−0.237468	−0.118734	+	0.992926i	$0.537884\pi$
$390$	0	0
$391$	−5.15356	−0.260627
$392$	3.53362	0.178475
$393$	0	0
$394$	22.4243	1.12972
$395$	7.74274	0.389580
$396$	0	0
$397$	−23.9284	−1.20093	−0.600466	−	0.799650i	$-0.705018\pi$
$397$	−23.9284	−1.20093	−0.600466	+	0.799650i	$0.705018\pi$
$398$	8.96319	0.449284
$399$	0	0
$400$	−1.66208	−0.0831038
$401$	21.9167	1.09447	0.547234	−	0.836980i	$-0.315681\pi$
$401$	21.9167	1.09447	0.547234	+	0.836980i	$0.315681\pi$
$402$	0	0
$403$	−6.71405	−0.334451
$404$	−2.75755	−0.137193
$405$	0	0
$406$	−11.5967	−0.575536
$407$	8.07428	0.400227
$408$	0	0
$409$	−14.3658	−0.710341	−0.355170	−	0.934802i	$-0.615577\pi$
$409$	−14.3658	−0.710341	−0.355170	+	0.934802i	$0.615577\pi$
$410$	0.100400	0.00495842
$411$	0	0
$412$	−10.8923	−0.536626
$413$	−8.79622	−0.432834
$414$	0	0
$415$	8.84796	0.434330
$416$	6.38776	0.313186
$417$	0	0
$418$	−32.3781	−1.58367
$419$	−25.5263	−1.24704	−0.623522	−	0.781806i	$-0.714299\pi$
$419$	−25.5263	−1.24704	−0.623522	+	0.781806i	$0.714299\pi$
$420$	0	0
$421$	25.9863	1.26649	0.633247	−	0.773950i	$-0.281722\pi$
$421$	25.9863	1.26649	0.633247	+	0.773950i	$0.281722\pi$
$422$	−1.87384	−0.0912173
$423$	0	0
$424$	−16.8622	−0.818902
$425$	4.97107	0.241132
$426$	0	0
$427$	0.110004	0.00532346
$428$	6.44388	0.311477
$429$	0	0
$430$	−5.03599	−0.242857
$431$	−28.1303	−1.35499	−0.677494	−	0.735528i	$-0.736934\pi$
$431$	−28.1303	−1.35499	−0.677494	+	0.735528i	$0.736934\pi$
$432$	0	0
$433$	−9.81800	−0.471823	−0.235911	−	0.971775i	$-0.575808\pi$
$433$	−9.81800	−0.471823	−0.235911	+	0.971775i	$0.575808\pi$
$434$	14.0541	0.674620
$435$	0	0
$436$	−6.84786	−0.327953
$437$	−5.95504	−0.284868
$438$	0	0
$439$	26.1376	1.24748	0.623740	−	0.781632i	$-0.285612\pi$
$439$	26.1376	1.24748	0.623740	+	0.781632i	$0.285612\pi$
$440$	−15.9349	−0.759668
$441$	0	0
$442$	7.95456	0.378360
$443$	30.3306	1.44105	0.720526	−	0.693428i	$-0.243900\pi$
$443$	30.3306	1.44105	0.720526	+	0.693428i	$0.243900\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	−5.66583	−0.268285
$447$	0	0
$448$	−22.8634	−1.08019
$449$	36.9134	1.74205	0.871026	−	0.491237i	$-0.163455\pi$
$449$	36.9134	1.74205	0.871026	+	0.491237i	$0.163455\pi$
$450$	0	0
$451$	0.482459	0.0227181
$452$	−10.4811	−0.492990
$453$	0	0
$454$	13.7026	0.643097
$455$	−4.21896	−0.197788
$456$	0	0
$457$	38.9450	1.82177	0.910884	−	0.412662i	$-0.135401\pi$
$457$	38.9450	1.82177	0.910884	+	0.412662i	$0.135401\pi$
$458$	0.879703	0.0411058
$459$	0	0
$460$	−0.857359	−0.0399746
$461$	8.95443	0.417049	0.208525	−	0.978017i	$-0.433134\pi$
$461$	8.95443	0.417049	0.208525	+	0.978017i	$0.433134\pi$
$462$	0	0
$463$	16.6168	0.772248	0.386124	−	0.922447i	$-0.373814\pi$
$463$	16.6168	0.772248	0.386124	+	0.922447i	$0.373814\pi$
$464$	6.23231	0.289328
$465$	0	0
$466$	20.3321	0.941865
$467$	18.6122	0.861271	0.430636	−	0.902526i	$-0.358289\pi$
$467$	18.6122	0.861271	0.430636	+	0.902526i	$0.358289\pi$
$468$	0	0
$469$	−29.2220	−1.34934
$470$	−14.6891	−0.677557
$471$	0	0
$472$	9.43154	0.434122
$473$	−24.1997	−1.11271
$474$	0	0
$475$	5.74417	0.263561
$476$	11.7393	0.538070
$477$	0	0
$478$	−5.98405	−0.273704
$479$	25.5172	1.16591	0.582956	−	0.812504i	$-0.301896\pi$
$479$	25.5172	1.16591	0.582956	+	0.812504i	$0.301896\pi$
$480$	0	0
$481$	−2.29217	−0.104514
$482$	−14.3913	−0.655508
$483$	0	0
$484$	−13.3034	−0.604699
$485$	11.8228	0.536844
$486$	0	0
$487$	31.8036	1.44116	0.720580	−	0.693372i	$-0.243876\pi$
$487$	31.8036	1.44116	0.720580	+	0.693372i	$0.243876\pi$
$488$	−0.117949	−0.00533930
$489$	0	0
$490$	1.24996	0.0564672
$491$	−29.7650	−1.34328	−0.671639	−	0.740879i	$-0.734409\pi$
$491$	−29.7650	−1.34328	−0.671639	+	0.740879i	$0.734409\pi$
$492$	0	0
$493$	−18.6401	−0.839507
$494$	9.19166	0.413552
$495$	0	0
$496$	−7.55297	−0.339138
$497$	−5.92463	−0.265756
$498$	0	0
$499$	−21.8976	−0.980273	−0.490137	−	0.871646i	$-0.663053\pi$
$499$	−21.8976	−0.980273	−0.490137	+	0.871646i	$0.663053\pi$
$500$	0.826999	0.0369845
$501$	0	0
$502$	−11.8823	−0.530332
$503$	−20.8071	−0.927742	−0.463871	−	0.885903i	$-0.653540\pi$
$503$	−20.8071	−0.927742	−0.463871	+	0.885903i	$0.653540\pi$
$504$	0	0
$505$	−3.33440	−0.148379
$506$	5.84362	0.259781
$507$	0	0
$508$	−1.98582	−0.0881064
$509$	17.8987	0.793344	0.396672	−	0.917960i	$-0.370165\pi$
$509$	17.8987	0.793344	0.396672	+	0.917960i	$0.370165\pi$
$510$	0	0
$511$	−34.0707	−1.50720
$512$	17.3638	0.767377
$513$	0	0
$514$	7.81234	0.344588
$515$	−13.1709	−0.580380
$516$	0	0
$517$	−70.5864	−3.10439
$518$	4.79806	0.210815
$519$	0	0
$520$	4.52369	0.198377
$521$	40.2129	1.76176	0.880880	−	0.473340i	$-0.156952\pi$
$521$	40.2129	1.76176	0.880880	+	0.473340i	$0.156952\pi$
$522$	0	0
$523$	−19.4054	−0.848540	−0.424270	−	0.905536i	$-0.639469\pi$
$523$	−19.4054	−0.848540	−0.424270	+	0.905536i	$0.639469\pi$
$524$	−12.9205	−0.564435
$525$	0	0
$526$	−13.5849	−0.592328
$527$	22.5900	0.984036
$528$	0	0
$529$	−21.9252	−0.953271
$530$	−5.96471	−0.259090
$531$	0	0
$532$	13.5650	0.588118
$533$	−0.136963	−0.00593253
$534$	0	0
$535$	7.79189	0.336873
$536$	31.3326	1.35336
$537$	0	0
$538$	16.4535	0.709361
$539$	6.00649	0.258718
$540$	0	0
$541$	2.83057	0.121696	0.0608478	−	0.998147i	$-0.480620\pi$
$541$	2.83057	0.121696	0.0608478	+	0.998147i	$0.480620\pi$
$542$	−23.4801	−1.00856
$543$	0	0
$544$	−21.4922	−0.921470
$545$	−8.28038	−0.354692
$546$	0	0
$547$	21.6811	0.927017	0.463508	−	0.886093i	$-0.346590\pi$
$547$	21.6811	0.927017	0.463508	+	0.886093i	$0.346590\pi$
$548$	−5.73280	−0.244893
$549$	0	0
$550$	−5.63669	−0.240349
$551$	−21.5390	−0.917593
$552$	0	0
$553$	22.1097	0.940200
$554$	−6.31734	−0.268398
$555$	0	0
$556$	13.2125	0.560334
$557$	−10.7203	−0.454234	−0.227117	−	0.973867i	$-0.572930\pi$
$557$	−10.7203	−0.454234	−0.227117	+	0.973867i	$0.572930\pi$
$558$	0	0
$559$	6.86994	0.290568
$560$	−4.74612	−0.200560
$561$	0	0
$562$	6.15370	0.259578
$563$	−1.61674	−0.0681374	−0.0340687	−	0.999419i	$-0.510846\pi$
$563$	−1.61674	−0.0681374	−0.0340687	+	0.999419i	$0.510846\pi$
$564$	0	0
$565$	−12.6737	−0.533185
$566$	−13.9680	−0.587118
$567$	0	0
$568$	6.35255	0.266547
$569$	−11.0612	−0.463709	−0.231854	−	0.972751i	$-0.574479\pi$
$569$	−11.0612	−0.463709	−0.231854	+	0.972751i	$0.574479\pi$
$570$	0	0
$571$	22.9928	0.962221	0.481110	−	0.876660i	$-0.340234\pi$
$571$	22.9928	0.962221	0.481110	+	0.876660i	$0.340234\pi$
$572$	6.35913	0.265889
$573$	0	0
$574$	0.286697	0.0119665
$575$	−1.03671	−0.0432338
$576$	0	0
$577$	−39.0157	−1.62425	−0.812123	−	0.583486i	$-0.801688\pi$
$577$	−39.0157	−1.62425	−0.812123	+	0.583486i	$0.801688\pi$
$578$	−8.35195	−0.347395
$579$	0	0
$580$	−3.10101	−0.128762
$581$	25.2657	1.04820
$582$	0	0
$583$	−28.6626	−1.18708
$584$	36.5315	1.51168
$585$	0	0
$586$	−1.20957	−0.0499669
$587$	27.2872	1.12626	0.563131	−	0.826368i	$-0.309597\pi$
$587$	27.2872	1.12626	0.563131	+	0.826368i	$0.309597\pi$
$588$	0	0
$589$	26.1032	1.07556
$590$	3.33624	0.137351
$591$	0	0
$592$	−2.57857	−0.105979
$593$	−13.7944	−0.566468	−0.283234	−	0.959051i	$-0.591407\pi$
$593$	−13.7944	−0.566468	−0.283234	+	0.959051i	$0.591407\pi$
$594$	0	0
$595$	14.1951	0.581941
$596$	−6.95380	−0.284839
$597$	0	0
$598$	−1.65892	−0.0678381
$599$	21.3158	0.870940	0.435470	−	0.900203i	$-0.356582\pi$
$599$	21.3158	0.870940	0.435470	+	0.900203i	$0.356582\pi$
$600$	0	0
$601$	30.5203	1.24495	0.622475	−	0.782640i	$-0.286127\pi$
$601$	30.5203	1.24495	0.622475	+	0.782640i	$0.286127\pi$
$602$	−14.3805	−0.586104
$603$	0	0
$604$	−5.27470	−0.214624
$605$	−16.0863	−0.654003
$606$	0	0
$607$	46.0194	1.86787	0.933934	−	0.357445i	$-0.116352\pi$
$607$	46.0194	1.86787	0.933934	+	0.357445i	$0.116352\pi$
$608$	−24.8347	−1.00718
$609$	0	0
$610$	−0.0417224	−0.00168929
$611$	20.0384	0.810667
$612$	0	0
$613$	−23.5013	−0.949208	−0.474604	−	0.880199i	$-0.657409\pi$
$613$	−23.5013	−0.949208	−0.474604	+	0.880199i	$0.657409\pi$
$614$	−7.31056	−0.295030
$615$	0	0
$616$	−45.5028	−1.83336
$617$	17.9660	0.723286	0.361643	−	0.932317i	$-0.382216\pi$
$617$	17.9660	0.723286	0.361643	+	0.932317i	$0.382216\pi$
$618$	0	0
$619$	35.1840	1.41416	0.707082	−	0.707131i	$-0.250011\pi$
$619$	35.1840	1.41416	0.707082	+	0.707131i	$0.250011\pi$
$620$	3.75813	0.150930
$621$	0	0
$622$	29.3403	1.17644
$623$	−2.85554	−0.114405
$624$	0	0
$625$	1.00000	0.0400000
$626$	15.8837	0.634840
$627$	0	0
$628$	0.751134	0.0299735
$629$	7.71220	0.307505
$630$	0	0
$631$	45.1855	1.79881	0.899404	−	0.437119i	$-0.144001\pi$
$631$	45.1855	1.79881	0.899404	+	0.437119i	$0.144001\pi$
$632$	−23.7066	−0.942999
$633$	0	0
$634$	−4.19286	−0.166520
$635$	−2.40123	−0.0952901
$636$	0	0
$637$	−1.70515	−0.0675606
$638$	21.1360	0.836782
$639$	0	0
$640$	0.0247324	0.000977636 0
$641$	12.9698	0.512275	0.256137	−	0.966640i	$-0.417550\pi$
$641$	12.9698	0.512275	0.256137	+	0.966640i	$0.417550\pi$
$642$	0	0
$643$	23.4811	0.926004	0.463002	−	0.886357i	$-0.346772\pi$
$643$	23.4811	0.926004	0.463002	+	0.886357i	$0.346772\pi$
$644$	−2.44822	−0.0964734
$645$	0	0
$646$	−30.9262	−1.21677
$647$	22.8483	0.898258	0.449129	−	0.893467i	$-0.351734\pi$
$647$	22.8483	0.898258	0.449129	+	0.893467i	$0.351734\pi$
$648$	0	0
$649$	16.0318	0.629304
$650$	1.60017	0.0627639
$651$	0	0
$652$	−1.31623	−0.0515476
$653$	−0.183855	−0.00719480	−0.00359740	−	0.999994i	$-0.501145\pi$
$653$	−0.183855	−0.00719480	−0.00359740	+	0.999994i	$0.501145\pi$
$654$	0	0
$655$	−15.6234	−0.610455
$656$	−0.154076	−0.00601568
$657$	0	0
$658$	−41.9453	−1.63520
$659$	16.6355	0.648028	0.324014	−	0.946052i	$-0.394967\pi$
$659$	16.6355	0.648028	0.324014	+	0.946052i	$0.394967\pi$
$660$	0	0
$661$	34.5370	1.34333	0.671666	−	0.740854i	$-0.265579\pi$
$661$	34.5370	1.34333	0.671666	+	0.740854i	$0.265579\pi$
$662$	−12.7937	−0.497241
$663$	0	0
$664$	−27.0906	−1.05132
$665$	16.4027	0.636070
$666$	0	0
$667$	3.88737	0.150520
$668$	−0.326518	−0.0126333
$669$	0	0
$670$	11.0833	0.428186
$671$	−0.200491	−0.00773987
$672$	0	0
$673$	−44.7089	−1.72340	−0.861701	−	0.507416i	$-0.830601\pi$
$673$	−44.7089	−1.72340	−0.861701	+	0.507416i	$0.830601\pi$
$674$	−22.6910	−0.874023
$675$	0	0
$676$	8.94572	0.344066
$677$	−18.5052	−0.711211	−0.355606	−	0.934636i	$-0.615725\pi$
$677$	−18.5052	−0.711211	−0.355606	+	0.934636i	$0.615725\pi$
$678$	0	0
$679$	33.7603	1.29560
$680$	−15.2203	−0.583673
$681$	0	0
$682$	−25.6148	−0.980842
$683$	−19.3194	−0.739236	−0.369618	−	0.929184i	$-0.620511\pi$
$683$	−19.3194	−0.739236	−0.369618	+	0.929184i	$0.620511\pi$
$684$	0	0
$685$	−6.93206	−0.264860
$686$	−18.0796	−0.690282
$687$	0	0
$688$	7.72834	0.294640
$689$	8.13687	0.309990
$690$	0	0
$691$	−39.9625	−1.52025	−0.760123	−	0.649780i	$-0.774861\pi$
$691$	−39.9625	−1.52025	−0.760123	+	0.649780i	$0.774861\pi$
$692$	−15.8312	−0.601813
$693$	0	0
$694$	31.9979	1.21462
$695$	15.9764	0.606020
$696$	0	0
$697$	0.460824	0.0174550
$698$	18.0232	0.682190
$699$	0	0
$700$	2.36153	0.0892574
$701$	16.3854	0.618869	0.309434	−	0.950921i	$-0.399860\pi$
$701$	16.3854	0.618869	0.309434	+	0.950921i	$0.399860\pi$
$702$	0	0
$703$	8.91160	0.336108
$704$	41.6704	1.57051
$705$	0	0
$706$	−11.6637	−0.438969
$707$	−9.52152	−0.358093
$708$	0	0
$709$	−4.55272	−0.170981	−0.0854905	−	0.996339i	$-0.527246\pi$
$709$	−4.55272	−0.170981	−0.0854905	+	0.996339i	$0.527246\pi$
$710$	2.24710	0.0843322
$711$	0	0
$712$	3.06179	0.114745
$713$	−4.71113	−0.176433
$714$	0	0
$715$	7.68941	0.287567
$716$	−11.6149	−0.434068
$717$	0	0
$718$	−12.2882	−0.458593
$719$	10.4178	0.388518	0.194259	−	0.980950i	$-0.437770\pi$
$719$	10.4178	0.388518	0.194259	+	0.980950i	$0.437770\pi$
$720$	0	0
$721$	−37.6101	−1.40067
$722$	−15.1578	−0.564116
$723$	0	0
$724$	−18.7112	−0.695398
$725$	−3.74972	−0.139261
$726$	0	0
$727$	−8.31044	−0.308217	−0.154109	−	0.988054i	$-0.549251\pi$
$727$	−8.31044	−0.308217	−0.154109	+	0.988054i	$0.549251\pi$
$728$	12.9176	0.478757
$729$	0	0
$730$	12.9224	0.478278
$731$	−23.1145	−0.854922
$732$	0	0
$733$	−8.81501	−0.325590	−0.162795	−	0.986660i	$-0.552051\pi$
$733$	−8.81501	−0.325590	−0.162795	+	0.986660i	$0.552051\pi$
$734$	−5.67944	−0.209632
$735$	0	0
$736$	4.48218	0.165215
$737$	53.2594	1.96184
$738$	0	0
$739$	38.8084	1.42759	0.713795	−	0.700355i	$-0.246975\pi$
$739$	38.8084	1.42759	0.713795	+	0.700355i	$0.246975\pi$
$740$	1.28302	0.0471648
$741$	0	0
$742$	−17.0325	−0.625281
$743$	7.42164	0.272274	0.136137	−	0.990690i	$-0.456531\pi$
$743$	7.42164	0.272274	0.136137	+	0.990690i	$0.456531\pi$
$744$	0	0
$745$	−8.40848	−0.308063
$746$	−15.3116	−0.560599
$747$	0	0
$748$	−21.3959	−0.782310
$749$	22.2500	0.812999
$750$	0	0
$751$	−18.0850	−0.659930	−0.329965	−	0.943993i	$-0.607037\pi$
$751$	−18.0850	−0.659930	−0.329965	+	0.943993i	$0.607037\pi$
$752$	22.5422	0.822030
$753$	0	0
$754$	−6.00019	−0.218514
$755$	−6.37812	−0.232124
$756$	0	0
$757$	−53.9124	−1.95948	−0.979739	−	0.200277i	$-0.935816\pi$
$757$	−53.9124	−1.95948	−0.979739	+	0.200277i	$0.935816\pi$
$758$	−41.7277	−1.51562
$759$	0	0
$760$	−17.5874	−0.637963
$761$	−21.8348	−0.791512	−0.395756	−	0.918356i	$-0.629517\pi$
$761$	−21.8348	−0.791512	−0.395756	+	0.918356i	$0.629517\pi$
$762$	0	0
$763$	−23.6449	−0.856005
$764$	6.19869	0.224261
$765$	0	0
$766$	29.9263	1.08128
$767$	−4.55120	−0.164334
$768$	0	0
$769$	17.3001	0.623858	0.311929	−	0.950105i	$-0.399025\pi$
$769$	17.3001	0.623858	0.311929	+	0.950105i	$0.399025\pi$
$770$	−16.0958	−0.580052
$771$	0	0
$772$	−16.0629	−0.578116
$773$	41.3798	1.48833	0.744163	−	0.667998i	$-0.232848\pi$
$773$	41.3798	1.48833	0.744163	+	0.667998i	$0.232848\pi$
$774$	0	0
$775$	4.54430	0.163236
$776$	−36.1987	−1.29946
$777$	0	0
$778$	5.07259	0.181861
$779$	0.532492	0.0190785
$780$	0	0
$781$	10.7981	0.386387
$782$	5.58157	0.199597
$783$	0	0
$784$	−1.91821	−0.0685075
$785$	0.908265	0.0324174
$786$	0	0
$787$	−37.3224	−1.33040	−0.665199	−	0.746666i	$-0.731653\pi$
$787$	−37.3224	−1.33040	−0.665199	+	0.746666i	$0.731653\pi$
$788$	17.1228	0.609974
$789$	0	0
$790$	−8.38579	−0.298353
$791$	−36.1902	−1.28677
$792$	0	0
$793$	0.0569164	0.00202116
$794$	25.9157	0.919713
$795$	0	0
$796$	6.84413	0.242584
$797$	−28.1809	−0.998218	−0.499109	−	0.866539i	$-0.666339\pi$
$797$	−28.1809	−0.998218	−0.499109	+	0.866539i	$0.666339\pi$
$798$	0	0
$799$	−67.4210	−2.38518
$800$	−4.32346	−0.152857
$801$	0	0
$802$	−23.7369	−0.838180
$803$	62.0966	2.19134
$804$	0	0
$805$	−2.96037	−0.104339
$806$	7.27166	0.256133
$807$	0	0
$808$	10.2092	0.359159
$809$	7.15905	0.251699	0.125849	−	0.992049i	$-0.459834\pi$
$809$	7.15905	0.251699	0.125849	+	0.992049i	$0.459834\pi$
$810$	0	0
$811$	40.5860	1.42517	0.712584	−	0.701587i	$-0.247525\pi$
$811$	40.5860	1.42517	0.712584	+	0.701587i	$0.247525\pi$
$812$	−8.85506	−0.310752
$813$	0	0
$814$	−8.74486	−0.306507
$815$	−1.59158	−0.0557504
$816$	0	0
$817$	−26.7093	−0.934441
$818$	15.5589	0.544003
$819$	0	0
$820$	0.0766638	0.00267722
$821$	−27.3191	−0.953442	−0.476721	−	0.879055i	$-0.658175\pi$
$821$	−27.3191	−0.953442	−0.476721	+	0.879055i	$0.658175\pi$
$822$	0	0
$823$	49.2387	1.71635	0.858177	−	0.513353i	$-0.171597\pi$
$823$	49.2387	1.71635	0.858177	+	0.513353i	$0.171597\pi$
$824$	40.3265	1.40484
$825$	0	0
$826$	9.52676	0.331478
$827$	−28.9366	−1.00622	−0.503111	−	0.864222i	$-0.667811\pi$
$827$	−28.9366	−1.00622	−0.503111	+	0.864222i	$0.667811\pi$
$828$	0	0
$829$	−41.5138	−1.44184	−0.720918	−	0.693021i	$-0.756279\pi$
$829$	−41.5138	−1.44184	−0.720918	+	0.693021i	$0.756279\pi$
$830$	−9.58280	−0.332624
$831$	0	0
$832$	−11.8296	−0.410118
$833$	5.73713	0.198780
$834$	0	0
$835$	−0.394822	−0.0136634
$836$	−24.7234	−0.855075
$837$	0	0
$838$	27.6463	0.955027
$839$	46.8475	1.61736	0.808678	−	0.588251i	$-0.200183\pi$
$839$	46.8475	1.61736	0.808678	+	0.588251i	$0.200183\pi$
$840$	0	0
$841$	−14.9396	−0.515159
$842$	−28.1445	−0.969923
$843$	0	0
$844$	−1.43083	−0.0492513
$845$	10.8171	0.372119
$846$	0	0
$847$	−45.9352	−1.57835
$848$	9.15357	0.314335
$849$	0	0
$850$	−5.38392	−0.184667
$851$	−1.60837	−0.0551343
$852$	0	0
$853$	−4.04849	−0.138618	−0.0693088	−	0.997595i	$-0.522079\pi$
$853$	−4.04849	−0.138618	−0.0693088	+	0.997595i	$0.522079\pi$
$854$	−0.119140	−0.00407688
$855$	0	0
$856$	−23.8571	−0.815419
$857$	30.4803	1.04119	0.520593	−	0.853805i	$-0.325711\pi$
$857$	30.4803	1.04119	0.520593	+	0.853805i	$0.325711\pi$
$858$	0	0
$859$	19.5867	0.668290	0.334145	−	0.942522i	$-0.391553\pi$
$859$	19.5867	0.668290	0.334145	+	0.942522i	$0.391553\pi$
$860$	−3.84539	−0.131127
$861$	0	0
$862$	30.4666	1.03770
$863$	6.03830	0.205546	0.102773	−	0.994705i	$-0.467228\pi$
$863$	6.03830	0.205546	0.102773	+	0.994705i	$0.467228\pi$
$864$	0	0
$865$	−19.1430	−0.650881
$866$	10.6334	0.361338
$867$	0	0
$868$	10.7315	0.364250
$869$	−40.2967	−1.36697
$870$	0	0
$871$	−15.1195	−0.512306
$872$	25.3527	0.858552
$873$	0	0
$874$	6.44962	0.218162
$875$	2.85554	0.0965349
$876$	0	0
$877$	−14.0408	−0.474123	−0.237061	−	0.971495i	$-0.576184\pi$
$877$	−14.0408	−0.474123	−0.237061	+	0.971495i	$0.576184\pi$
$878$	−28.3084	−0.955362
$879$	0	0
$880$	8.65020	0.291598
$881$	−33.6163	−1.13256	−0.566281	−	0.824212i	$-0.691618\pi$
$881$	−33.6163	−1.13256	−0.566281	+	0.824212i	$0.691618\pi$
$882$	0	0
$883$	−45.9916	−1.54774	−0.773870	−	0.633345i	$-0.781682\pi$
$883$	−45.9916	−1.54774	−0.773870	+	0.633345i	$0.781682\pi$
$884$	6.07396	0.204289
$885$	0	0
$886$	−32.8496	−1.10361
$887$	−32.1688	−1.08012	−0.540062	−	0.841625i	$-0.681599\pi$
$887$	−32.1688	−1.08012	−0.540062	+	0.841625i	$0.681599\pi$
$888$	0	0
$889$	−6.85682	−0.229970
$890$	1.08305	0.0363040
$891$	0	0
$892$	−4.32633	−0.144856
$893$	−77.9064	−2.60704
$894$	0	0
$895$	−14.0446	−0.469459
$896$	0.0706245	0.00235940
$897$	0	0
$898$	−39.9791	−1.33412
$899$	−17.0398	−0.568310
$900$	0	0
$901$	−27.3772	−0.912068
$902$	−0.522529	−0.0173983
$903$	0	0
$904$	38.8041	1.29060
$905$	−22.6255	−0.752096
$906$	0	0
$907$	5.04940	0.167662	0.0838312	−	0.996480i	$-0.473284\pi$
$907$	5.04940	0.167662	0.0838312	+	0.996480i	$0.473284\pi$
$908$	10.4631	0.347230
$909$	0	0
$910$	4.56936	0.151473
$911$	14.9566	0.495534	0.247767	−	0.968820i	$-0.420303\pi$
$911$	14.9566	0.495534	0.247767	+	0.968820i	$0.420303\pi$
$912$	0	0
$913$	−46.0488	−1.52399
$914$	−42.1794	−1.39517
$915$	0	0
$916$	0.671726	0.0221944
$917$	−44.6131	−1.47326
$918$	0	0
$919$	−48.7638	−1.60857	−0.804285	−	0.594244i	$-0.797451\pi$
$919$	−48.7638	−1.60857	−0.804285	+	0.594244i	$0.797451\pi$
$920$	3.17419	0.104650
$921$	0	0
$922$	−9.69811	−0.319390
$923$	−3.06543	−0.100900
$924$	0	0
$925$	1.55142	0.0510103
$926$	−17.9969	−0.591414
$927$	0	0
$928$	16.2117	0.532177
$929$	48.5207	1.59191	0.795957	−	0.605354i	$-0.206968\pi$
$929$	48.5207	1.59191	0.795957	+	0.605354i	$0.206968\pi$
$930$	0	0
$931$	6.62938	0.217269
$932$	15.5252	0.508545
$933$	0	0
$934$	−20.1580	−0.659590
$935$	−25.8717	−0.846094
$936$	0	0
$937$	−6.86837	−0.224380	−0.112190	−	0.993687i	$-0.535786\pi$
$937$	−6.86837	−0.224380	−0.112190	+	0.993687i	$0.535786\pi$
$938$	31.6489	1.03337
$939$	0	0
$940$	−11.2163	−0.365836
$941$	12.3159	0.401486	0.200743	−	0.979644i	$-0.435664\pi$
$941$	12.3159	0.401486	0.200743	+	0.979644i	$0.435664\pi$
$942$	0	0
$943$	−0.0961044	−0.00312959
$944$	−5.11987	−0.166637
$945$	0	0
$946$	26.2096	0.852147
$947$	3.18967	0.103650	0.0518251	−	0.998656i	$-0.483496\pi$
$947$	3.18967	0.103650	0.0518251	+	0.998656i	$0.483496\pi$
$948$	0	0
$949$	−17.6283	−0.572238
$950$	−6.22123	−0.201843
$951$	0	0
$952$	−43.4623	−1.40862
$953$	−34.1312	−1.10562	−0.552809	−	0.833308i	$-0.686444\pi$
$953$	−34.1312	−1.10562	−0.552809	+	0.833308i	$0.686444\pi$
$954$	0	0
$955$	7.49540	0.242545
$956$	−4.56931	−0.147782
$957$	0	0
$958$	−27.6365	−0.892894
$959$	−19.7948	−0.639206
$960$	0	0
$961$	−10.3493	−0.333850
$962$	2.48254	0.0800401
$963$	0	0
$964$	−10.9890	−0.353931
$965$	−19.4231	−0.625252
$966$	0	0
$967$	38.3712	1.23393	0.616967	−	0.786989i	$-0.288361\pi$
$967$	38.3712	1.23393	0.616967	+	0.786989i	$0.288361\pi$
$968$	49.2529	1.58305
$969$	0	0
$970$	−12.8047	−0.411133
$971$	36.1310	1.15950	0.579750	−	0.814794i	$-0.303150\pi$
$971$	36.1310	1.15950	0.579750	+	0.814794i	$0.303150\pi$
$972$	0	0
$973$	45.6213	1.46255
$974$	−34.4450	−1.10369
$975$	0	0
$976$	0.0640281	0.00204949
$977$	−48.6131	−1.55527	−0.777636	−	0.628715i	$-0.783581\pi$
$977$	−48.6131	−1.55527	−0.777636	+	0.628715i	$0.783581\pi$
$978$	0	0
$979$	5.20445	0.166335
$980$	0.954444	0.0304886
$981$	0	0
$982$	32.2371	1.02873
$983$	−23.1742	−0.739142	−0.369571	−	0.929202i	$-0.620495\pi$
$983$	−23.1742	−0.739142	−0.369571	+	0.929202i	$0.620495\pi$
$984$	0	0
$985$	20.7047	0.659707
$986$	20.1882	0.642923
$987$	0	0
$988$	7.01859	0.223291
$989$	4.82051	0.153283
$990$	0	0
$991$	14.3848	0.456947	0.228474	−	0.973550i	$-0.426627\pi$
$991$	14.3848	0.456947	0.228474	+	0.973550i	$0.426627\pi$
$992$	−19.6471	−0.623796
$993$	0	0
$994$	6.41668	0.203525
$995$	8.27586	0.262363
$996$	0	0
$997$	11.6237	0.368127	0.184064	−	0.982914i	$-0.441075\pi$
$997$	11.6237	0.368127	0.184064	+	0.982914i	$0.441075\pi$
$998$	23.7163	0.750726
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.w.1.7	✓	17
3.2	odd	2		4005.2.a.x.1.11	yes	17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4005.2.a.w.1.7	✓	17	1.1	even	1	trivial
4005.2.a.x.1.11	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)

\(f(q)\)	\(=\)	\(q-1.08305 q^{2} -0.826999 q^{4} -1.00000 q^{5} -2.85554 q^{7} +3.06179 q^{8} +O(q^{10})\)	\(q-1.08305 q^{2} -0.826999 q^{4} -1.00000 q^{5} -2.85554 q^{7} +3.06179 q^{8} +1.08305 q^{10} +5.20445 q^{11} -1.47747 q^{13} +3.09270 q^{14} -1.66208 q^{16} +4.97107 q^{17} +5.74417 q^{19} +0.826999 q^{20} -5.63669 q^{22} -1.03671 q^{23} +1.00000 q^{25} +1.60017 q^{26} +2.36153 q^{28} -3.74972 q^{29} +4.54430 q^{31} -4.32346 q^{32} -5.38392 q^{34} +2.85554 q^{35} +1.55142 q^{37} -6.22123 q^{38} -3.06179 q^{40} +0.0927013 q^{41} -4.64981 q^{43} -4.30408 q^{44} +1.12281 q^{46} -13.5627 q^{47} +1.15411 q^{49} -1.08305 q^{50} +1.22186 q^{52} -5.50732 q^{53} -5.20445 q^{55} -8.74305 q^{56} +4.06114 q^{58} +3.08041 q^{59} -0.0385230 q^{61} -4.92171 q^{62} +8.00668 q^{64} +1.47747 q^{65} +10.2334 q^{67} -4.11107 q^{68} -3.09270 q^{70} +2.07479 q^{71} +11.9314 q^{73} -1.68027 q^{74} -4.75042 q^{76} -14.8615 q^{77} -7.74274 q^{79} +1.66208 q^{80} -0.100400 q^{82} -8.84796 q^{83} -4.97107 q^{85} +5.03599 q^{86} +15.9349 q^{88} +1.00000 q^{89} +4.21896 q^{91} +0.857359 q^{92} +14.6891 q^{94} -5.74417 q^{95} -11.8228 q^{97} -1.24996 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q - 5 q^{2} + 21 q^{4} - 17 q^{5} + 12 q^{7} - 15 q^{8}+O(q^{10}) \) 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

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