LMFDB - Embedded newform 4005.2.a.x.1.11

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			1.11
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	$-0.374920\pi$
$2$	1.08305	0.765833	0.382917	+	0.923783i	$0.374920\pi$
$3$	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	$-0.787132\pi$
$11$	−5.20445	−1.56920	−0.784601	+	0.620001i	$0.787132\pi$
$12$	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	$-0.705960\pi$
$17$	−4.97107	−1.20566	−0.602830	+	0.797869i	$0.705960\pi$
$18$	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	$-0.465528\pi$
$23$	1.03671	0.216169	0.108085	+	0.994142i	$0.465528\pi$
$24$	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	$-0.386809\pi$
$29$	3.74972	0.696305	0.348152	+	0.937438i	$0.386809\pi$
$30$	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	$-0.502304\pi$
$41$	−0.0927013	−0.0144775	−0.00723875	+	0.999974i	$0.502304\pi$
$42$	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	$-0.0469096\pi$
$47$	13.5627	1.97832	0.989161	+	0.146838i	$0.0469096\pi$
$48$	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	$-0.376528\pi$
$53$	5.50732	0.756488	0.378244	+	0.925706i	$0.376528\pi$
$54$	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	$-0.564262\pi$
$59$	−3.08041	−0.401035	−0.200517	+	0.979690i	$0.564262\pi$
$60$	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	$-0.539289\pi$
$71$	−2.07479	−0.246232	−0.123116	+	0.992392i	$0.539289\pi$
$72$	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	$-0.338603\pi$
$83$	8.84796	0.971190	0.485595	+	0.874184i	$0.338603\pi$
$84$	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	$-0.553051\pi$
$101$	−3.33440	−0.331785	−0.165893	+	0.986144i	$0.553051\pi$
$102$	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	$-0.377081\pi$
$107$	7.79189	0.753270	0.376635	+	0.926362i	$0.377081\pi$
$108$	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	$-0.703291\pi$
$113$	−12.6737	−1.19224	−0.596119	+	0.802896i	$0.703291\pi$
$114$	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	$-0.739112\pi$
$131$	−15.6234	−1.36502	−0.682510	+	0.730876i	$0.739112\pi$
$132$	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	$-0.595694\pi$
$137$	−6.93206	−0.592246	−0.296123	+	0.955150i	$0.595694\pi$
$138$	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	$-0.611926\pi$
$149$	−8.40848	−0.688849	−0.344425	+	0.938814i	$0.611926\pi$
$150$	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	$-0.504863\pi$
$167$	−0.394822	−0.0305523	−0.0152761	+	0.999883i	$0.504863\pi$
$168$	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	$-0.759414\pi$
$173$	−19.1430	−1.45541	−0.727707	+	0.685888i	$0.759414\pi$
$174$	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	$-0.675886\pi$
$179$	−14.0446	−1.04974	−0.524871	+	0.851182i	$0.675886\pi$
$180$	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	$-0.412588\pi$
$191$	7.49540	0.542348	0.271174	+	0.962530i	$0.412588\pi$
$192$	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	$-0.235971\pi$
$197$	20.7047	1.47515	0.737575	+	0.675265i	$0.235971\pi$
$198$	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	$-0.362077\pi$
$227$	12.6519	0.839735	0.419868	+	0.907585i	$0.362077\pi$
$228$	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	$-0.289185\pi$
$233$	18.7729	1.22986	0.614928	+	0.788583i	$0.289185\pi$
$234$	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	$-0.557188\pi$
$239$	−5.52517	−0.357394	−0.178697	+	0.983904i	$0.557188\pi$
$240$	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	$-0.612544\pi$
$251$	−10.9711	−0.692491	−0.346245	+	0.938144i	$0.612544\pi$
$252$	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	$-0.427770\pi$
$257$	7.21327	0.449951	0.224976	+	0.974364i	$0.427770\pi$
$258$	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	$-0.626393\pi$
$263$	−12.5431	−0.773443	−0.386722	+	0.922197i	$0.626393\pi$
$264$	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	$-0.346726\pi$
$269$	15.1918	0.926260	0.463130	+	0.886290i	$0.346726\pi$
$270$	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	$-0.445793\pi$
$281$	5.68182	0.338949	0.169474	+	0.985535i	$0.445793\pi$
$282$	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	$-0.510386\pi$
$293$	−1.11682	−0.0652452	−0.0326226	+	0.999468i	$0.510386\pi$
$294$	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	$-0.221214\pi$
$311$	27.0904	1.53615	0.768077	+	0.640357i	$0.221214\pi$
$312$	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	$-0.534675\pi$
$317$	−3.87134	−0.217436	−0.108718	+	0.994073i	$0.534675\pi$
$318$	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	$-0.208514\pi$
$347$	29.5442	1.58601	0.793007	+	0.609213i	$0.208514\pi$
$348$	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	$-0.592524\pi$
$353$	−10.7693	−0.573191	−0.286596	+	0.958052i	$0.592524\pi$
$354$	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	$-0.596789\pi$
$359$	−11.3459	−0.598816	−0.299408	+	0.954125i	$0.596789\pi$
$360$	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	$-0.250520\pi$
$383$	27.6314	1.41190	0.705950	+	0.708261i	$0.250520\pi$
$384$	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	$-0.462116\pi$
$389$	4.68360	0.237468	0.118734	+	0.992926i	$0.462116\pi$
$390$	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	$-0.684319\pi$
$401$	−21.9167	−1.09447	−0.547234	+	0.836980i	$0.684319\pi$
$402$	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	$-0.285701\pi$
$419$	25.5263	1.24704	0.623522	+	0.781806i	$0.285701\pi$
$420$	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	$-0.263066\pi$
$431$	28.1303	1.35499	0.677494	+	0.735528i	$0.263066\pi$
$432$	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	$-0.756100\pi$
$443$	−30.3306	−1.44105	−0.720526	+	0.693428i	$0.756100\pi$
$444$	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	$-0.836545\pi$
$449$	−36.9134	−1.74205	−0.871026	+	0.491237i	$0.836545\pi$
$450$	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	$-0.566866\pi$
$461$	−8.95443	−0.417049	−0.208525	+	0.978017i	$0.566866\pi$
$462$	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	$-0.641711\pi$
$467$	−18.6122	−0.861271	−0.430636	+	0.902526i	$0.641711\pi$
$468$	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	$-0.698104\pi$
$479$	−25.5172	−1.16591	−0.582956	+	0.812504i	$0.698104\pi$
$480$	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	$-0.265591\pi$
$491$	29.7650	1.34328	0.671639	+	0.740879i	$0.265591\pi$
$492$	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	$-0.346460\pi$
$503$	20.8071	0.927742	0.463871	+	0.885903i	$0.346460\pi$
$504$	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	$-0.629835\pi$
$509$	−17.8987	−0.793344	−0.396672	+	0.917960i	$0.629835\pi$
$510$	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	$-0.843048\pi$
$521$	−40.2129	−1.76176	−0.880880	+	0.473340i	$0.843048\pi$
$522$	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	$-0.427070\pi$
$557$	10.7203	0.454234	0.227117	+	0.973867i	$0.427070\pi$
$558$	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	$-0.489154\pi$
$563$	1.61674	0.0681374	0.0340687	+	0.999419i	$0.489154\pi$
$564$	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	$-0.425521\pi$
$569$	11.0612	0.463709	0.231854	+	0.972751i	$0.425521\pi$
$570$	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	$-0.690403\pi$
$587$	−27.2872	−1.12626	−0.563131	+	0.826368i	$0.690403\pi$
$588$	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	$-0.408593\pi$
$593$	13.7944	0.566468	0.283234	+	0.959051i	$0.408593\pi$
$594$	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	$-0.643418\pi$
$599$	−21.3158	−0.870940	−0.435470	+	0.900203i	$0.643418\pi$
$600$	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	$-0.617784\pi$
$617$	−17.9660	−0.723286	−0.361643	+	0.932317i	$0.617784\pi$
$618$	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	$-0.582450\pi$
$641$	−12.9698	−0.512275	−0.256137	+	0.966640i	$0.582450\pi$
$642$	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	$-0.648266\pi$
$647$	−22.8483	−0.898258	−0.449129	+	0.893467i	$0.648266\pi$
$648$	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	$-0.498855\pi$
$653$	0.183855	0.00719480	0.00359740	+	0.999994i	$0.498855\pi$
$654$	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	$-0.605033\pi$
$659$	−16.6355	−0.648028	−0.324014	+	0.946052i	$0.605033\pi$
$660$	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	$-0.384275\pi$
$677$	18.5052	0.711211	0.355606	+	0.934636i	$0.384275\pi$
$678$	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	$-0.379489\pi$
$683$	19.3194	0.739236	0.369618	+	0.929184i	$0.379489\pi$
$684$	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	$-0.600140\pi$
$701$	−16.3854	−0.618869	−0.309434	+	0.950921i	$0.600140\pi$
$702$	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	$-0.562230\pi$
$719$	−10.4178	−0.388518	−0.194259	+	0.980950i	$0.562230\pi$
$720$	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	$-0.543469\pi$
$743$	−7.42164	−0.272274	−0.136137	+	0.990690i	$0.543469\pi$
$744$	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	$-0.370483\pi$
$761$	21.8348	0.791512	0.395756	+	0.918356i	$0.370483\pi$
$762$	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	$-0.767152\pi$
$773$	−41.3798	−1.48833	−0.744163	+	0.667998i	$0.767152\pi$
$774$	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	$-0.333661\pi$
$797$	28.1809	0.998218	0.499109	+	0.866539i	$0.333661\pi$
$798$	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	$-0.540166\pi$
$809$	−7.15905	−0.251699	−0.125849	+	0.992049i	$0.540166\pi$
$810$	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	$-0.341825\pi$
$821$	27.3191	0.953442	0.476721	+	0.879055i	$0.341825\pi$
$822$	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	$-0.332189\pi$
$827$	28.9366	1.00622	0.503111	+	0.864222i	$0.332189\pi$
$828$	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	$-0.799817\pi$
$839$	−46.8475	−1.61736	−0.808678	+	0.588251i	$0.799817\pi$
$840$	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	$-0.674289\pi$
$857$	−30.4803	−1.04119	−0.520593	+	0.853805i	$0.674289\pi$
$858$	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	$-0.532772\pi$
$863$	−6.03830	−0.205546	−0.102773	+	0.994705i	$0.532772\pi$
$864$	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	$-0.308382\pi$
$881$	33.6163	1.13256	0.566281	+	0.824212i	$0.308382\pi$
$882$	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	$-0.318401\pi$
$887$	32.1688	1.08012	0.540062	+	0.841625i	$0.318401\pi$
$888$	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	$-0.579697\pi$
$911$	−14.9566	−0.495534	−0.247767	+	0.968820i	$0.579697\pi$
$912$	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	$-0.793032\pi$
$929$	−48.5207	−1.59191	−0.795957	+	0.605354i	$0.793032\pi$
$930$	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	$-0.564336\pi$
$941$	−12.3159	−0.401486	−0.200743	+	0.979644i	$0.564336\pi$
$942$	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	$-0.516504\pi$
$947$	−3.18967	−0.103650	−0.0518251	+	0.998656i	$0.516504\pi$
$948$	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	$-0.313556\pi$
$953$	34.1312	1.10562	0.552809	+	0.833308i	$0.313556\pi$
$954$	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	$-0.696850\pi$
$971$	−36.1310	−1.15950	−0.579750	+	0.814794i	$0.696850\pi$
$972$	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	$-0.216419\pi$
$977$	48.6131	1.55527	0.777636	+	0.628715i	$0.216419\pi$
$978$	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	$-0.379505\pi$
$983$	23.1742	0.739142	0.369571	+	0.929202i	$0.379505\pi$
$984$	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.x.1.11	yes	17
3.2	odd	2		4005.2.a.w.1.7	✓	17

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

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

L-functions

Modular forms

Varieties

Fields

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Groups

Database

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