LMFDB - Embedded newform 8001.2.a.s.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, 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("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.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:	$63.8883066572$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 18 x^{14} + 83 x^{13} + 112 x^{12} - 668 x^{11} - 235 x^{10} + 2648 x^{9} + \cdots - 20 $ x^16 - 4x^15 - 18x^14 + 83x^13 + 112x^12 - 668x^11 - 235x^10 + 2648x^9 - 298x^8 - 5422x^7 + 2075x^6 + 5385x^5 - 3163x^4 - 1882x^3 + 1614x^2 - 200*x - 20
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.50110$ of defining polynomial
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.50110 q^{2} +0.253312 q^{4} +2.96917 q^{5} -1.00000 q^{7} +2.62196 q^{8} +O(q^{10})$	\(q-1.50110 q^{2} +0.253312 q^{4} +2.96917 q^{5} -1.00000 q^{7} +2.62196 q^{8} -4.45703 q^{10} +5.03235 q^{11} +5.28468 q^{13} +1.50110 q^{14} -4.44246 q^{16} -5.69847 q^{17} +4.97405 q^{19} +0.752126 q^{20} -7.55408 q^{22} -1.70002 q^{23} +3.81597 q^{25} -7.93285 q^{26} -0.253312 q^{28} -3.58394 q^{29} +1.14042 q^{31} +1.42467 q^{32} +8.55399 q^{34} -2.96917 q^{35} +7.99996 q^{37} -7.46656 q^{38} +7.78504 q^{40} -2.27251 q^{41} -8.39125 q^{43} +1.27475 q^{44} +2.55191 q^{46} +8.01028 q^{47} +1.00000 q^{49} -5.72816 q^{50} +1.33867 q^{52} -2.49184 q^{53} +14.9419 q^{55} -2.62196 q^{56} +5.37986 q^{58} -5.98779 q^{59} +7.87074 q^{61} -1.71189 q^{62} +6.74634 q^{64} +15.6911 q^{65} +13.4773 q^{67} -1.44349 q^{68} +4.45703 q^{70} +7.07915 q^{71} +6.23904 q^{73} -12.0088 q^{74} +1.25999 q^{76} -5.03235 q^{77} +1.41290 q^{79} -13.1904 q^{80} +3.41128 q^{82} +5.37001 q^{83} -16.9197 q^{85} +12.5961 q^{86} +13.1946 q^{88} -4.94427 q^{89} -5.28468 q^{91} -0.430635 q^{92} -12.0243 q^{94} +14.7688 q^{95} +4.90024 q^{97} -1.50110 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8}+O(q^{10}) $ 16 * q - 4 * q^2 + 20 * q^4 - 5 * q^5 - 16 * q^7 - 15 * q^8	$ 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8} - 4 q^{10} - q^{11} + 20 q^{13} + 4 q^{14} + 32 q^{16} - 3 q^{17} + 13 q^{19} - 17 q^{20} + 13 q^{22} - 5 q^{23} + 17 q^{25} + 2 q^{26} - 20 q^{28} - 22 q^{29} + 26 q^{31} - 54 q^{32} - 6 q^{34} + 5 q^{35} + 30 q^{37} - 5 q^{38} + 13 q^{40} - q^{41} + 31 q^{43} - 22 q^{44} - 2 q^{46} + q^{47} + 16 q^{49} - 5 q^{50} + 31 q^{52} - 24 q^{53} + 8 q^{55} + 15 q^{56} + 13 q^{58} + 17 q^{59} + 32 q^{61} + 5 q^{62} + 61 q^{64} + 3 q^{65} + 16 q^{67} + 10 q^{68} + 4 q^{70} + 10 q^{71} + 23 q^{73} - q^{74} + 18 q^{76} + q^{77} + 48 q^{79} - 38 q^{80} + 12 q^{82} - 9 q^{83} + 22 q^{85} + 4 q^{86} + 27 q^{88} - 17 q^{89} - 20 q^{91} - 16 q^{92} + 13 q^{94} - 22 q^{95} + 17 q^{97} - 4 q^{98}+O(q^{100}) $ 16 * q - 4 * q^2 + 20 * q^4 - 5 * q^5 - 16 * q^7 - 15 * q^8 - 4 * q^10 - q^11 + 20 * q^13 + 4 * q^14 + 32 * q^16 - 3 * q^17 + 13 * q^19 - 17 * q^20 + 13 * q^22 - 5 * q^23 + 17 * q^25 + 2 * q^26 - 20 * q^28 - 22 * q^29 + 26 * q^31 - 54 * q^32 - 6 * q^34 + 5 * q^35 + 30 * q^37 - 5 * q^38 + 13 * q^40 - q^41 + 31 * q^43 - 22 * q^44 - 2 * q^46 + q^47 + 16 * q^49 - 5 * q^50 + 31 * q^52 - 24 * q^53 + 8 * q^55 + 15 * q^56 + 13 * q^58 + 17 * q^59 + 32 * q^61 + 5 * q^62 + 61 * q^64 + 3 * q^65 + 16 * q^67 + 10 * q^68 + 4 * q^70 + 10 * q^71 + 23 * q^73 - q^74 + 18 * q^76 + q^77 + 48 * q^79 - 38 * q^80 + 12 * q^82 - 9 * q^83 + 22 * q^85 + 4 * q^86 + 27 * q^88 - 17 * q^89 - 20 * q^91 - 16 * q^92 + 13 * q^94 - 22 * q^95 + 17 * q^97 - 4 * 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 $

$2$	−1.50110	−1.06144	−0.530720	−	0.847547i	$-0.678079\pi$
$2$	−1.50110	−1.06144	−0.530720	+	0.847547i	$0.678079\pi$
$3$	0	0
$3$	0	0
$4$	0.253312	0.126656
$5$	2.96917	1.32785	0.663927	−	0.747798i	$-0.268889\pi$
$5$	2.96917	1.32785	0.663927	+	0.747798i	$0.268889\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.62196	0.927003
$9$	0	0
$10$	−4.45703	−1.40944
$11$	5.03235	1.51731	0.758656	−	0.651492i	$-0.225857\pi$
$11$	5.03235	1.51731	0.758656	+	0.651492i	$0.225857\pi$
$12$	0	0
$13$	5.28468	1.46571	0.732853	−	0.680387i	$-0.238188\pi$
$13$	5.28468	1.46571	0.732853	+	0.680387i	$0.238188\pi$
$14$	1.50110	0.401187
$15$	0	0
$16$	−4.44246	−1.11061
$17$	−5.69847	−1.38208	−0.691040	−	0.722816i	$-0.742847\pi$
$17$	−5.69847	−1.38208	−0.691040	+	0.722816i	$0.742847\pi$
$18$	0	0
$19$	4.97405	1.14113	0.570563	−	0.821254i	$-0.306725\pi$
$19$	4.97405	1.14113	0.570563	+	0.821254i	$0.306725\pi$
$20$	0.752126	0.168180
$21$	0	0
$22$	−7.55408	−1.61054
$23$	−1.70002	−0.354479	−0.177239	−	0.984168i	$-0.556717\pi$
$23$	−1.70002	−0.354479	−0.177239	+	0.984168i	$0.556717\pi$
$24$	0	0
$25$	3.81597	0.763194
$26$	−7.93285	−1.55576
$27$	0	0
$28$	−0.253312	−0.0478714
$29$	−3.58394	−0.665520	−0.332760	−	0.943011i	$-0.607980\pi$
$29$	−3.58394	−0.665520	−0.332760	+	0.943011i	$0.607980\pi$
$30$	0	0
$31$	1.14042	0.204826	0.102413	−	0.994742i	$-0.467344\pi$
$31$	1.14042	0.204826	0.102413	+	0.994742i	$0.467344\pi$
$32$	1.42467	0.251848
$33$	0	0
$34$	8.55399	1.46700
$35$	−2.96917	−0.501881
$36$	0	0
$37$	7.99996	1.31519	0.657593	−	0.753374i	$-0.271575\pi$
$37$	7.99996	1.31519	0.657593	+	0.753374i	$0.271575\pi$
$38$	−7.46656	−1.21124
$39$	0	0
$40$	7.78504	1.23092
$41$	−2.27251	−0.354907	−0.177454	−	0.984129i	$-0.556786\pi$
$41$	−2.27251	−0.354907	−0.177454	+	0.984129i	$0.556786\pi$
$42$	0	0
$43$	−8.39125	−1.27965	−0.639827	−	0.768519i	$-0.720994\pi$
$43$	−8.39125	−1.27965	−0.639827	+	0.768519i	$0.720994\pi$
$44$	1.27475	0.192176
$45$	0	0
$46$	2.55191	0.376258
$47$	8.01028	1.16842	0.584210	−	0.811603i	$-0.301405\pi$
$47$	8.01028	1.16842	0.584210	+	0.811603i	$0.301405\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−5.72816	−0.810085
$51$	0	0
$52$	1.33867	0.185640
$53$	−2.49184	−0.342280	−0.171140	−	0.985247i	$-0.554745\pi$
$53$	−2.49184	−0.342280	−0.171140	+	0.985247i	$0.554745\pi$
$54$	0	0
$55$	14.9419	2.01477
$56$	−2.62196	−0.350374
$57$	0	0
$58$	5.37986	0.706410
$59$	−5.98779	−0.779544	−0.389772	−	0.920911i	$-0.627446\pi$
$59$	−5.98779	−0.779544	−0.389772	+	0.920911i	$0.627446\pi$
$60$	0	0
$61$	7.87074	1.00774	0.503872	−	0.863778i	$-0.331908\pi$
$61$	7.87074	1.00774	0.503872	+	0.863778i	$0.331908\pi$
$62$	−1.71189	−0.217411
$63$	0	0
$64$	6.74634	0.843292
$65$	15.6911	1.94624
$66$	0	0
$67$	13.4773	1.64652	0.823260	−	0.567664i	$-0.192153\pi$
$67$	13.4773	1.64652	0.823260	+	0.567664i	$0.192153\pi$
$68$	−1.44349	−0.175049
$69$	0	0
$70$	4.45703	0.532717
$71$	7.07915	0.840141	0.420071	−	0.907491i	$-0.362005\pi$
$71$	7.07915	0.840141	0.420071	+	0.907491i	$0.362005\pi$
$72$	0	0
$73$	6.23904	0.730224	0.365112	−	0.930964i	$-0.381031\pi$
$73$	6.23904	0.730224	0.365112	+	0.930964i	$0.381031\pi$
$74$	−12.0088	−1.39599
$75$	0	0
$76$	1.25999	0.144530
$77$	−5.03235	−0.573490
$78$	0	0
$79$	1.41290	0.158964	0.0794819	−	0.996836i	$-0.474673\pi$
$79$	1.41290	0.158964	0.0794819	+	0.996836i	$0.474673\pi$
$80$	−13.1904	−1.47473
$81$	0	0
$82$	3.41128	0.376713
$83$	5.37001	0.589435	0.294718	−	0.955584i	$-0.404774\pi$
$83$	5.37001	0.589435	0.294718	+	0.955584i	$0.404774\pi$
$84$	0	0
$85$	−16.9197	−1.83520
$86$	12.5961	1.35828
$87$	0	0
$88$	13.1946	1.40655
$89$	−4.94427	−0.524091	−0.262046	−	0.965056i	$-0.584397\pi$
$89$	−4.94427	−0.524091	−0.262046	+	0.965056i	$0.584397\pi$
$90$	0	0
$91$	−5.28468	−0.553985
$92$	−0.430635	−0.0448968
$93$	0	0
$94$	−12.0243	−1.24021
$95$	14.7688	1.51525
$96$	0	0
$97$	4.90024	0.497544	0.248772	−	0.968562i	$-0.419973\pi$
$97$	4.90024	0.497544	0.248772	+	0.968562i	$0.419973\pi$
$98$	−1.50110	−0.151634
$99$	0	0
$100$	0.966630	0.0966630
$101$	0.367501	0.0365678	0.0182839	−	0.999833i	$-0.494180\pi$
$101$	0.367501	0.0365678	0.0182839	+	0.999833i	$0.494180\pi$
$102$	0	0
$103$	−7.94858	−0.783197	−0.391599	−	0.920136i	$-0.628078\pi$
$103$	−7.94858	−0.783197	−0.391599	+	0.920136i	$0.628078\pi$
$104$	13.8562	1.35871
$105$	0	0
$106$	3.74051	0.363310
$107$	−10.9611	−1.05965	−0.529823	−	0.848108i	$-0.677742\pi$
$107$	−10.9611	−1.05965	−0.529823	+	0.848108i	$0.677742\pi$
$108$	0	0
$109$	10.7913	1.03362	0.516809	−	0.856101i	$-0.327120\pi$
$109$	10.7913	1.03362	0.516809	+	0.856101i	$0.327120\pi$
$110$	−22.4293	−2.13855
$111$	0	0
$112$	4.44246	0.419773
$113$	−1.89200	−0.177985	−0.0889924	−	0.996032i	$-0.528365\pi$
$113$	−1.89200	−0.177985	−0.0889924	+	0.996032i	$0.528365\pi$
$114$	0	0
$115$	−5.04765	−0.470696
$116$	−0.907853	−0.0842921
$117$	0	0
$118$	8.98829	0.827440
$119$	5.69847	0.522378
$120$	0	0
$121$	14.3246	1.30223
$122$	−11.8148	−1.06966
$123$	0	0
$124$	0.288882	0.0259424
$125$	−3.51559	−0.314444
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−12.9763	−1.14695
$129$	0	0
$130$	−23.5540	−2.06582
$131$	19.7242	1.72331	0.861656	−	0.507492i	$-0.169428\pi$
$131$	19.7242	1.72331	0.861656	+	0.507492i	$0.169428\pi$
$132$	0	0
$133$	−4.97405	−0.431305
$134$	−20.2309	−1.74768
$135$	0	0
$136$	−14.9411	−1.28119
$137$	1.23699	0.105683	0.0528415	−	0.998603i	$-0.483172\pi$
$137$	1.23699	0.105683	0.0528415	+	0.998603i	$0.483172\pi$
$138$	0	0
$139$	−18.9060	−1.60359	−0.801794	−	0.597601i	$-0.796121\pi$
$139$	−18.9060	−1.60359	−0.801794	+	0.597601i	$0.796121\pi$
$140$	−0.752126	−0.0635662
$141$	0	0
$142$	−10.6265	−0.891760
$143$	26.5943	2.22393
$144$	0	0
$145$	−10.6413	−0.883713
$146$	−9.36545	−0.775090
$147$	0	0
$148$	2.02648	0.166576
$149$	18.6688	1.52940	0.764702	−	0.644384i	$-0.222886\pi$
$149$	18.6688	1.52940	0.764702	+	0.644384i	$0.222886\pi$
$150$	0	0
$151$	14.9340	1.21531	0.607656	−	0.794201i	$-0.292110\pi$
$151$	14.9340	1.21531	0.607656	+	0.794201i	$0.292110\pi$
$152$	13.0418	1.05783
$153$	0	0
$154$	7.55408	0.608725
$155$	3.38611	0.271979
$156$	0	0
$157$	11.6259	0.927850	0.463925	−	0.885874i	$-0.346441\pi$
$157$	11.6259	0.927850	0.463925	+	0.885874i	$0.346441\pi$
$158$	−2.12091	−0.168731
$159$	0	0
$160$	4.23008	0.334417
$161$	1.70002	0.133980
$162$	0	0
$163$	−23.6206	−1.85011	−0.925055	−	0.379833i	$-0.875982\pi$
$163$	−23.6206	−1.85011	−0.925055	+	0.379833i	$0.875982\pi$
$164$	−0.575655	−0.0449511
$165$	0	0
$166$	−8.06094	−0.625650
$167$	−16.6636	−1.28947	−0.644735	−	0.764406i	$-0.723032\pi$
$167$	−16.6636	−1.28947	−0.644735	+	0.764406i	$0.723032\pi$
$168$	0	0
$169$	14.9278	1.14829
$170$	25.3982	1.94796
$171$	0	0
$172$	−2.12560	−0.162076
$173$	−4.06588	−0.309123	−0.154561	−	0.987983i	$-0.549396\pi$
$173$	−4.06588	−0.309123	−0.154561	+	0.987983i	$0.549396\pi$
$174$	0	0
$175$	−3.81597	−0.288460
$176$	−22.3560	−1.68515
$177$	0	0
$178$	7.42185	0.556291
$179$	−11.2678	−0.842196	−0.421098	−	0.907015i	$-0.638355\pi$
$179$	−11.2678	−0.842196	−0.421098	+	0.907015i	$0.638355\pi$
$180$	0	0
$181$	−14.9248	−1.10935	−0.554677	−	0.832066i	$-0.687158\pi$
$181$	−14.9248	−1.10935	−0.554677	+	0.832066i	$0.687158\pi$
$182$	7.93285	0.588022
$183$	0	0
$184$	−4.45738	−0.328603
$185$	23.7532	1.74637
$186$	0	0
$187$	−28.6767	−2.09705
$188$	2.02910	0.147987
$189$	0	0
$190$	−22.1695	−1.60834
$191$	5.10449	0.369348	0.184674	−	0.982800i	$-0.440877\pi$
$191$	5.10449	0.369348	0.184674	+	0.982800i	$0.440877\pi$
$192$	0	0
$193$	20.4463	1.47176	0.735878	−	0.677114i	$-0.236770\pi$
$193$	20.4463	1.47176	0.735878	+	0.677114i	$0.236770\pi$
$194$	−7.35576	−0.528113
$195$	0	0
$196$	0.253312	0.0180937
$197$	−10.8131	−0.770405	−0.385202	−	0.922832i	$-0.625868\pi$
$197$	−10.8131	−0.770405	−0.385202	+	0.922832i	$0.625868\pi$
$198$	0	0
$199$	−14.8057	−1.04955	−0.524774	−	0.851242i	$-0.675850\pi$
$199$	−14.8057	−1.04955	−0.524774	+	0.851242i	$0.675850\pi$
$200$	10.0053	0.707483
$201$	0	0
$202$	−0.551658	−0.0388145
$203$	3.58394	0.251543
$204$	0	0
$205$	−6.74748	−0.471265
$206$	11.9316	0.831317
$207$	0	0
$208$	−23.4769	−1.62783
$209$	25.0312	1.73144
$210$	0	0
$211$	−0.296253	−0.0203949	−0.0101975	−	0.999948i	$-0.503246\pi$
$211$	−0.296253	−0.0203949	−0.0101975	+	0.999948i	$0.503246\pi$
$212$	−0.631212	−0.0433518
$213$	0	0
$214$	16.4537	1.12475
$215$	−24.9150	−1.69919
$216$	0	0
$217$	−1.14042	−0.0774169
$218$	−16.1988	−1.09712
$219$	0	0
$220$	3.78496	0.255182
$221$	−30.1145	−2.02572
$222$	0	0
$223$	14.9296	0.999762	0.499881	−	0.866094i	$-0.333377\pi$
$223$	14.9296	0.999762	0.499881	+	0.866094i	$0.333377\pi$
$224$	−1.42467	−0.0951896
$225$	0	0
$226$	2.84010	0.188920
$227$	28.6612	1.90231	0.951155	−	0.308714i	$-0.0998986\pi$
$227$	28.6612	1.90231	0.951155	+	0.308714i	$0.0998986\pi$
$228$	0	0
$229$	−6.72133	−0.444158	−0.222079	−	0.975029i	$-0.571284\pi$
$229$	−6.72133	−0.444158	−0.222079	+	0.975029i	$0.571284\pi$
$230$	7.57704	0.499615
$231$	0	0
$232$	−9.39694	−0.616939
$233$	−1.90481	−0.124788	−0.0623941	−	0.998052i	$-0.519874\pi$
$233$	−1.90481	−0.124788	−0.0623941	+	0.998052i	$0.519874\pi$
$234$	0	0
$235$	23.7839	1.55149
$236$	−1.51678	−0.0987338
$237$	0	0
$238$	−8.55399	−0.554473
$239$	2.36861	0.153213	0.0766064	−	0.997061i	$-0.475592\pi$
$239$	2.36861	0.153213	0.0766064	+	0.997061i	$0.475592\pi$
$240$	0	0
$241$	11.9592	0.770359	0.385180	−	0.922842i	$-0.374139\pi$
$241$	11.9592	0.770359	0.385180	+	0.922842i	$0.374139\pi$
$242$	−21.5026	−1.38224
$243$	0	0
$244$	1.99375	0.127637
$245$	2.96917	0.189693
$246$	0	0
$247$	26.2862	1.67255
$248$	2.99014	0.189874
$249$	0	0
$250$	5.27726	0.333763
$251$	12.9770	0.819101	0.409551	−	0.912287i	$-0.365685\pi$
$251$	12.9770	0.819101	0.409551	+	0.912287i	$0.365685\pi$
$252$	0	0
$253$	−8.55510	−0.537854
$254$	1.50110	0.0941876
$255$	0	0
$256$	5.98607	0.374130
$257$	−0.945782	−0.0589963	−0.0294981	−	0.999565i	$-0.509391\pi$
$257$	−0.945782	−0.0589963	−0.0294981	+	0.999565i	$0.509391\pi$
$258$	0	0
$259$	−7.99996	−0.497093
$260$	3.97474	0.246503
$261$	0	0
$262$	−29.6081	−1.82919
$263$	−18.5648	−1.14476	−0.572378	−	0.819990i	$-0.693979\pi$
$263$	−18.5648	−1.14476	−0.572378	+	0.819990i	$0.693979\pi$
$264$	0	0
$265$	−7.39869	−0.454498
$266$	7.46656	0.457804
$267$	0	0
$268$	3.41397	0.208541
$269$	−27.9998	−1.70718	−0.853589	−	0.520947i	$-0.825579\pi$
$269$	−27.9998	−1.70718	−0.853589	+	0.520947i	$0.825579\pi$
$270$	0	0
$271$	12.0427	0.731543	0.365772	−	0.930705i	$-0.380805\pi$
$271$	12.0427	0.731543	0.365772	+	0.930705i	$0.380805\pi$
$272$	25.3152	1.53496
$273$	0	0
$274$	−1.85685	−0.112176
$275$	19.2033	1.15800
$276$	0	0
$277$	−2.97940	−0.179015	−0.0895074	−	0.995986i	$-0.528529\pi$
$277$	−2.97940	−0.179015	−0.0895074	+	0.995986i	$0.528529\pi$
$278$	28.3799	1.70211
$279$	0	0
$280$	−7.78504	−0.465245
$281$	−18.5742	−1.10804	−0.554021	−	0.832503i	$-0.686907\pi$
$281$	−18.5742	−1.10804	−0.554021	+	0.832503i	$0.686907\pi$
$282$	0	0
$283$	−7.99311	−0.475141	−0.237571	−	0.971370i	$-0.576351\pi$
$283$	−7.99311	−0.475141	−0.237571	+	0.971370i	$0.576351\pi$
$284$	1.79323	0.106409
$285$	0	0
$286$	−39.9209	−2.36057
$287$	2.27251	0.134142
$288$	0	0
$289$	15.4725	0.910148
$290$	15.9737	0.938009
$291$	0	0
$292$	1.58042	0.0924872
$293$	−10.6556	−0.622508	−0.311254	−	0.950327i	$-0.600749\pi$
$293$	−10.6556	−0.622508	−0.311254	+	0.950327i	$0.600749\pi$
$294$	0	0
$295$	−17.7788	−1.03512
$296$	20.9756	1.21918
$297$	0	0
$298$	−28.0237	−1.62337
$299$	−8.98406	−0.519561
$300$	0	0
$301$	8.39125	0.483663
$302$	−22.4175	−1.28998
$303$	0	0
$304$	−22.0970	−1.26735
$305$	23.3696	1.33814
$306$	0	0
$307$	29.9512	1.70941	0.854703	−	0.519117i	$-0.173739\pi$
$307$	29.9512	1.70941	0.854703	+	0.519117i	$0.173739\pi$
$308$	−1.27475	−0.0726358
$309$	0	0
$310$	−5.08290	−0.288689
$311$	−10.0081	−0.567507	−0.283753	−	0.958897i	$-0.591580\pi$
$311$	−10.0081	−0.567507	−0.283753	+	0.958897i	$0.591580\pi$
$312$	0	0
$313$	−10.1282	−0.572482	−0.286241	−	0.958158i	$-0.592406\pi$
$313$	−10.1282	−0.572482	−0.286241	+	0.958158i	$0.592406\pi$
$314$	−17.4517	−0.984858
$315$	0	0
$316$	0.357905	0.0201337
$317$	23.0112	1.29244	0.646220	−	0.763151i	$-0.276349\pi$
$317$	23.0112	1.29244	0.646220	+	0.763151i	$0.276349\pi$
$318$	0	0
$319$	−18.0356	−1.00980
$320$	20.0310	1.11977
$321$	0	0
$322$	−2.55191	−0.142212
$323$	−28.3445	−1.57713
$324$	0	0
$325$	20.1662	1.11862
$326$	35.4570	1.96378
$327$	0	0
$328$	−5.95844	−0.329000
$329$	−8.01028	−0.441621
$330$	0	0
$331$	19.9378	1.09588	0.547941	−	0.836517i	$-0.315412\pi$
$331$	19.9378	1.09588	0.547941	+	0.836517i	$0.315412\pi$
$332$	1.36029	0.0746554
$333$	0	0
$334$	25.0138	1.36870
$335$	40.0165	2.18634
$336$	0	0
$337$	−19.4694	−1.06057	−0.530283	−	0.847821i	$-0.677914\pi$
$337$	−19.4694	−1.06057	−0.530283	+	0.847821i	$0.677914\pi$
$338$	−22.4082	−1.21884
$339$	0	0
$340$	−4.28596	−0.232439
$341$	5.73901	0.310785
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−22.0015	−1.18624
$345$	0	0
$346$	6.10330	0.328116
$347$	10.8161	0.580639	0.290320	−	0.956930i	$-0.406238\pi$
$347$	10.8161	0.580639	0.290320	+	0.956930i	$0.406238\pi$
$348$	0	0
$349$	16.6549	0.891518	0.445759	−	0.895153i	$-0.352934\pi$
$349$	16.6549	0.891518	0.445759	+	0.895153i	$0.352934\pi$
$350$	5.72816	0.306183
$351$	0	0
$352$	7.16943	0.382132
$353$	34.3191	1.82662	0.913312	−	0.407262i	$-0.133516\pi$
$353$	34.3191	1.82662	0.913312	+	0.407262i	$0.133516\pi$
$354$	0	0
$355$	21.0192	1.11558
$356$	−1.25244	−0.0663792
$357$	0	0
$358$	16.9141	0.893941
$359$	−13.5371	−0.714463	−0.357232	−	0.934016i	$-0.616279\pi$
$359$	−13.5371	−0.714463	−0.357232	+	0.934016i	$0.616279\pi$
$360$	0	0
$361$	5.74118	0.302167
$362$	22.4037	1.17751
$363$	0	0
$364$	−1.33867	−0.0701654
$365$	18.5248	0.969631
$366$	0	0
$367$	2.36464	0.123433	0.0617165	−	0.998094i	$-0.480343\pi$
$367$	2.36464	0.123433	0.0617165	+	0.998094i	$0.480343\pi$
$368$	7.55227	0.393689
$369$	0	0
$370$	−35.6561	−1.85367
$371$	2.49184	0.129370
$372$	0	0
$373$	−10.6616	−0.552035	−0.276018	−	0.961153i	$-0.589015\pi$
$373$	−10.6616	−0.552035	−0.276018	+	0.961153i	$0.589015\pi$
$374$	43.0467	2.22589
$375$	0	0
$376$	21.0026	1.08313
$377$	−18.9399	−0.975457
$378$	0	0
$379$	26.1609	1.34379	0.671896	−	0.740645i	$-0.265480\pi$
$379$	26.1609	1.34379	0.671896	+	0.740645i	$0.265480\pi$
$380$	3.74111	0.191915
$381$	0	0
$382$	−7.66237	−0.392041
$383$	7.78799	0.397948	0.198974	−	0.980005i	$-0.436239\pi$
$383$	7.78799	0.397948	0.198974	+	0.980005i	$0.436239\pi$
$384$	0	0
$385$	−14.9419	−0.761510
$386$	−30.6920	−1.56218
$387$	0	0
$388$	1.24129	0.0630168
$389$	−2.64982	−0.134351	−0.0671757	−	0.997741i	$-0.521399\pi$
$389$	−2.64982	−0.134351	−0.0671757	+	0.997741i	$0.521399\pi$
$390$	0	0
$391$	9.68751	0.489918
$392$	2.62196	0.132429
$393$	0	0
$394$	16.2317	0.817739
$395$	4.19514	0.211081
$396$	0	0
$397$	−29.0810	−1.45953	−0.729767	−	0.683696i	$-0.760371\pi$
$397$	−29.0810	−1.45953	−0.729767	+	0.683696i	$0.760371\pi$
$398$	22.2249	1.11403
$399$	0	0
$400$	−16.9523	−0.847614
$401$	−6.88884	−0.344012	−0.172006	−	0.985096i	$-0.555025\pi$
$401$	−6.88884	−0.344012	−0.172006	+	0.985096i	$0.555025\pi$
$402$	0	0
$403$	6.02676	0.300214
$404$	0.0930924	0.00463152
$405$	0	0
$406$	−5.37986	−0.266998
$407$	40.2586	1.99554
$408$	0	0
$409$	32.9194	1.62776	0.813880	−	0.581032i	$-0.197351\pi$
$409$	32.9194	1.62776	0.813880	+	0.581032i	$0.197351\pi$
$410$	10.1287	0.500219
$411$	0	0
$412$	−2.01347	−0.0991965
$413$	5.98779	0.294640
$414$	0	0
$415$	15.9445	0.782683
$416$	7.52891	0.369135
$417$	0	0
$418$	−37.5744	−1.83782
$419$	9.10685	0.444899	0.222449	−	0.974944i	$-0.428595\pi$
$419$	9.10685	0.444899	0.222449	+	0.974944i	$0.428595\pi$
$420$	0	0
$421$	−18.5215	−0.902685	−0.451343	−	0.892351i	$-0.649055\pi$
$421$	−18.5215	−0.902685	−0.451343	+	0.892351i	$0.649055\pi$
$422$	0.444707	0.0216480
$423$	0	0
$424$	−6.53350	−0.317295
$425$	−21.7452	−1.05480
$426$	0	0
$427$	−7.87074	−0.380892
$428$	−2.77657	−0.134211
$429$	0	0
$430$	37.4001	1.80359
$431$	27.2836	1.31421	0.657103	−	0.753801i	$-0.271782\pi$
$431$	27.2836	1.31421	0.657103	+	0.753801i	$0.271782\pi$
$432$	0	0
$433$	−25.5517	−1.22794	−0.613968	−	0.789331i	$-0.710427\pi$
$433$	−25.5517	−1.22794	−0.613968	+	0.789331i	$0.710427\pi$
$434$	1.71189	0.0821735
$435$	0	0
$436$	2.73356	0.130914
$437$	−8.45599	−0.404505
$438$	0	0
$439$	−21.2340	−1.01344	−0.506721	−	0.862110i	$-0.669143\pi$
$439$	−21.2340	−1.01344	−0.506721	+	0.862110i	$0.669143\pi$
$440$	39.1771	1.86769
$441$	0	0
$442$	45.2051	2.15019
$443$	−12.1933	−0.579322	−0.289661	−	0.957129i	$-0.593543\pi$
$443$	−12.1933	−0.579322	−0.289661	+	0.957129i	$0.593543\pi$
$444$	0	0
$445$	−14.6804	−0.695916
$446$	−22.4109	−1.06119
$447$	0	0
$448$	−6.74634	−0.318735
$449$	−19.6586	−0.927748	−0.463874	−	0.885901i	$-0.653541\pi$
$449$	−19.6586	−0.927748	−0.463874	+	0.885901i	$0.653541\pi$
$450$	0	0
$451$	−11.4361	−0.538505
$452$	−0.479267	−0.0225428
$453$	0	0
$454$	−43.0234	−2.01919
$455$	−15.6911	−0.735610
$456$	0	0
$457$	−25.9369	−1.21328	−0.606639	−	0.794977i	$-0.707483\pi$
$457$	−25.9369	−1.21328	−0.606639	+	0.794977i	$0.707483\pi$
$458$	10.0894	0.471447
$459$	0	0
$460$	−1.27863	−0.0596164
$461$	−16.5556	−0.771073	−0.385537	−	0.922693i	$-0.625984\pi$
$461$	−16.5556	−0.771073	−0.385537	+	0.922693i	$0.625984\pi$
$462$	0	0
$463$	21.7070	1.00881	0.504404	−	0.863468i	$-0.331712\pi$
$463$	21.7070	1.00881	0.504404	+	0.863468i	$0.331712\pi$
$464$	15.9215	0.739136
$465$	0	0
$466$	2.85932	0.132455
$467$	−22.5624	−1.04406	−0.522032	−	0.852926i	$-0.674826\pi$
$467$	−22.5624	−1.04406	−0.522032	+	0.852926i	$0.674826\pi$
$468$	0	0
$469$	−13.4773	−0.622326
$470$	−35.7021	−1.64681
$471$	0	0
$472$	−15.6997	−0.722639
$473$	−42.2277	−1.94163
$474$	0	0
$475$	18.9808	0.870900
$476$	1.44349	0.0661622
$477$	0	0
$478$	−3.55553	−0.162626
$479$	−27.0168	−1.23443	−0.617214	−	0.786795i	$-0.711739\pi$
$479$	−27.0168	−1.23443	−0.617214	+	0.786795i	$0.711739\pi$
$480$	0	0
$481$	42.2772	1.92767
$482$	−17.9520	−0.817691
$483$	0	0
$484$	3.62858	0.164935
$485$	14.5496	0.660665
$486$	0	0
$487$	−18.5223	−0.839325	−0.419662	−	0.907680i	$-0.637851\pi$
$487$	−18.5223	−0.839325	−0.419662	+	0.907680i	$0.637851\pi$
$488$	20.6368	0.934182
$489$	0	0
$490$	−4.45703	−0.201348
$491$	15.1509	0.683750	0.341875	−	0.939745i	$-0.388938\pi$
$491$	15.1509	0.683750	0.341875	+	0.939745i	$0.388938\pi$
$492$	0	0
$493$	20.4229	0.919803
$494$	−39.4584	−1.77532
$495$	0	0
$496$	−5.06628	−0.227483
$497$	−7.07915	−0.317543
$498$	0	0
$499$	44.4265	1.98880	0.994401	−	0.105676i	$-0.0337008\pi$
$499$	44.4265	1.98880	0.994401	+	0.105676i	$0.0337008\pi$
$500$	−0.890540	−0.0398262
$501$	0	0
$502$	−19.4798	−0.869427
$503$	−36.4849	−1.62678	−0.813390	−	0.581719i	$-0.802380\pi$
$503$	−36.4849	−1.62678	−0.813390	+	0.581719i	$0.802380\pi$
$504$	0	0
$505$	1.09117	0.0485566
$506$	12.8421	0.570900
$507$	0	0
$508$	−0.253312	−0.0112389
$509$	−20.1520	−0.893222	−0.446611	−	0.894728i	$-0.647369\pi$
$509$	−20.1520	−0.893222	−0.446611	+	0.894728i	$0.647369\pi$
$510$	0	0
$511$	−6.23904	−0.275999
$512$	16.9669	0.749836
$513$	0	0
$514$	1.41972	0.0626210
$515$	−23.6007	−1.03997
$516$	0	0
$517$	40.3105	1.77286
$518$	12.0088	0.527635
$519$	0	0
$520$	41.1414	1.80417
$521$	18.5745	0.813763	0.406882	−	0.913481i	$-0.366616\pi$
$521$	18.5745	0.813763	0.406882	+	0.913481i	$0.366616\pi$
$522$	0	0
$523$	−10.8272	−0.473440	−0.236720	−	0.971578i	$-0.576072\pi$
$523$	−10.8272	−0.473440	−0.236720	+	0.971578i	$0.576072\pi$
$524$	4.99637	0.218268
$525$	0	0
$526$	27.8677	1.21509
$527$	−6.49866	−0.283086
$528$	0	0
$529$	−20.1099	−0.874345
$530$	11.1062	0.482423
$531$	0	0
$532$	−1.25999	−0.0546273
$533$	−12.0095	−0.520189
$534$	0	0
$535$	−32.5453	−1.40706
$536$	35.3371	1.52633
$537$	0	0
$538$	42.0306	1.81207
$539$	5.03235	0.216759
$540$	0	0
$541$	20.8734	0.897417	0.448709	−	0.893678i	$-0.351884\pi$
$541$	20.8734	0.897417	0.448709	+	0.893678i	$0.351884\pi$
$542$	−18.0774	−0.776490
$543$	0	0
$544$	−8.11842	−0.348074
$545$	32.0412	1.37249
$546$	0	0
$547$	13.2787	0.567754	0.283877	−	0.958861i	$-0.408379\pi$
$547$	13.2787	0.567754	0.283877	+	0.958861i	$0.408379\pi$
$548$	0.313343	0.0133854
$549$	0	0
$550$	−28.8261	−1.22915
$551$	−17.8267	−0.759442
$552$	0	0
$553$	−1.41290	−0.0600827
$554$	4.47239	0.190014
$555$	0	0
$556$	−4.78912	−0.203104
$557$	−30.6231	−1.29754	−0.648772	−	0.760983i	$-0.724717\pi$
$557$	−30.6231	−1.29754	−0.648772	+	0.760983i	$0.724717\pi$
$558$	0	0
$559$	−44.3450	−1.87559
$560$	13.1904	0.557396
$561$	0	0
$562$	27.8817	1.17612
$563$	−44.0774	−1.85764	−0.928820	−	0.370531i	$-0.879176\pi$
$563$	−44.0774	−1.85764	−0.928820	+	0.370531i	$0.879176\pi$
$564$	0	0
$565$	−5.61768	−0.236338
$566$	11.9985	0.504334
$567$	0	0
$568$	18.5613	0.778813
$569$	13.4726	0.564803	0.282401	−	0.959296i	$-0.408869\pi$
$569$	13.4726	0.564803	0.282401	+	0.959296i	$0.408869\pi$
$570$	0	0
$571$	−10.8076	−0.452282	−0.226141	−	0.974095i	$-0.572611\pi$
$571$	−10.8076	−0.452282	−0.226141	+	0.974095i	$0.572611\pi$
$572$	6.73666	0.281674
$573$	0	0
$574$	−3.41128	−0.142384
$575$	−6.48722	−0.270536
$576$	0	0
$577$	−2.47813	−0.103166	−0.0515830	−	0.998669i	$-0.516427\pi$
$577$	−2.47813	−0.103166	−0.0515830	+	0.998669i	$0.516427\pi$
$578$	−23.2258	−0.966068
$579$	0	0
$580$	−2.69557	−0.111927
$581$	−5.37001	−0.222786
$582$	0	0
$583$	−12.5398	−0.519346
$584$	16.3585	0.676920
$585$	0	0
$586$	15.9952	0.660755
$587$	22.1676	0.914957	0.457478	−	0.889221i	$-0.348753\pi$
$587$	22.1676	0.914957	0.457478	+	0.889221i	$0.348753\pi$
$588$	0	0
$589$	5.67252	0.233732
$590$	26.6878	1.09872
$591$	0	0
$592$	−35.5395	−1.46066
$593$	38.4169	1.57759	0.788797	−	0.614654i	$-0.210704\pi$
$593$	38.4169	1.57759	0.788797	+	0.614654i	$0.210704\pi$
$594$	0	0
$595$	16.9197	0.693641
$596$	4.72902	0.193708
$597$	0	0
$598$	13.4860	0.551483
$599$	−13.4735	−0.550513	−0.275256	−	0.961371i	$-0.588763\pi$
$599$	−13.4735	−0.550513	−0.275256	+	0.961371i	$0.588763\pi$
$600$	0	0
$601$	25.1750	1.02691	0.513454	−	0.858117i	$-0.328366\pi$
$601$	25.1750	1.02691	0.513454	+	0.858117i	$0.328366\pi$
$602$	−12.5961	−0.513380
$603$	0	0
$604$	3.78296	0.153926
$605$	42.5321	1.72917
$606$	0	0
$607$	29.5557	1.19963	0.599814	−	0.800140i	$-0.295241\pi$
$607$	29.5557	1.19963	0.599814	+	0.800140i	$0.295241\pi$
$608$	7.08637	0.287390
$609$	0	0
$610$	−35.0801	−1.42035
$611$	42.3317	1.71256
$612$	0	0
$613$	11.7295	0.473752	0.236876	−	0.971540i	$-0.423877\pi$
$613$	11.7295	0.473752	0.236876	+	0.971540i	$0.423877\pi$
$614$	−44.9599	−1.81443
$615$	0	0
$616$	−13.1946	−0.531627
$617$	14.0770	0.566717	0.283359	−	0.959014i	$-0.408551\pi$
$617$	14.0770	0.566717	0.283359	+	0.959014i	$0.408551\pi$
$618$	0	0
$619$	22.4709	0.903183	0.451591	−	0.892225i	$-0.350856\pi$
$619$	22.4709	0.903183	0.451591	+	0.892225i	$0.350856\pi$
$620$	0.857741	0.0344477
$621$	0	0
$622$	15.0232	0.602375
$623$	4.94427	0.198088
$624$	0	0
$625$	−29.5182	−1.18073
$626$	15.2035	0.607655
$627$	0	0
$628$	2.94499	0.117518
$629$	−45.5875	−1.81769
$630$	0	0
$631$	23.2901	0.927163	0.463581	−	0.886054i	$-0.346564\pi$
$631$	23.2901	0.927163	0.463581	+	0.886054i	$0.346564\pi$
$632$	3.70457	0.147360
$633$	0	0
$634$	−34.5423	−1.37185
$635$	−2.96917	−0.117828
$636$	0	0
$637$	5.28468	0.209387
$638$	27.0734	1.07184
$639$	0	0
$640$	−38.5288	−1.52298
$641$	−16.9438	−0.669239	−0.334620	−	0.942353i	$-0.608608\pi$
$641$	−16.9438	−0.669239	−0.334620	+	0.942353i	$0.608608\pi$
$642$	0	0
$643$	−41.7811	−1.64768	−0.823842	−	0.566819i	$-0.808174\pi$
$643$	−41.7811	−1.64768	−0.823842	+	0.566819i	$0.808174\pi$
$644$	0.430635	0.0169694
$645$	0	0
$646$	42.5480	1.67403
$647$	19.5797	0.769759	0.384879	−	0.922967i	$-0.374243\pi$
$647$	19.5797	0.769759	0.384879	+	0.922967i	$0.374243\pi$
$648$	0	0
$649$	−30.1327	−1.18281
$650$	−30.2715	−1.18735
$651$	0	0
$652$	−5.98338	−0.234327
$653$	2.54800	0.0997108	0.0498554	−	0.998756i	$-0.484124\pi$
$653$	2.54800	0.0997108	0.0498554	+	0.998756i	$0.484124\pi$
$654$	0	0
$655$	58.5645	2.28831
$656$	10.0955	0.394165
$657$	0	0
$658$	12.0243	0.468754
$659$	5.36661	0.209053	0.104527	−	0.994522i	$-0.466667\pi$
$659$	5.36661	0.209053	0.104527	+	0.994522i	$0.466667\pi$
$660$	0	0
$661$	41.8201	1.62661	0.813306	−	0.581836i	$-0.197666\pi$
$661$	41.8201	1.62661	0.813306	+	0.581836i	$0.197666\pi$
$662$	−29.9287	−1.16321
$663$	0	0
$664$	14.0800	0.546408
$665$	−14.7688	−0.572710
$666$	0	0
$667$	6.09277	0.235913
$668$	−4.22109	−0.163319
$669$	0	0
$670$	−60.0690	−2.32067
$671$	39.6083	1.52906
$672$	0	0
$673$	−5.98792	−0.230818	−0.115409	−	0.993318i	$-0.536818\pi$
$673$	−5.98792	−0.230818	−0.115409	+	0.993318i	$0.536818\pi$
$674$	29.2256	1.12573
$675$	0	0
$676$	3.78139	0.145438
$677$	28.6767	1.10214	0.551068	−	0.834460i	$-0.314221\pi$
$677$	28.6767	1.10214	0.551068	+	0.834460i	$0.314221\pi$
$678$	0	0
$679$	−4.90024	−0.188054
$680$	−44.3628	−1.70124
$681$	0	0
$682$	−8.61484	−0.329879
$683$	50.5969	1.93604	0.968018	−	0.250880i	$-0.0807200\pi$
$683$	50.5969	1.93604	0.968018	+	0.250880i	$0.0807200\pi$
$684$	0	0
$685$	3.67283	0.140331
$686$	1.50110	0.0573124
$687$	0	0
$688$	37.2778	1.42120
$689$	−13.1686	−0.501682
$690$	0	0
$691$	43.4389	1.65249	0.826247	−	0.563309i	$-0.190472\pi$
$691$	43.4389	1.65249	0.826247	+	0.563309i	$0.190472\pi$
$692$	−1.02993	−0.0391522
$693$	0	0
$694$	−16.2361	−0.616314
$695$	−56.1352	−2.12933
$696$	0	0
$697$	12.9498	0.490510
$698$	−25.0008	−0.946293
$699$	0	0
$700$	−0.966630	−0.0365352
$701$	47.2200	1.78347	0.891737	−	0.452554i	$-0.149487\pi$
$701$	47.2200	1.78347	0.891737	+	0.452554i	$0.149487\pi$
$702$	0	0
$703$	39.7922	1.50079
$704$	33.9500	1.27954
$705$	0	0
$706$	−51.5166	−1.93885
$707$	−0.367501	−0.0138213
$708$	0	0
$709$	7.33102	0.275322	0.137661	−	0.990479i	$-0.456041\pi$
$709$	7.33102	0.275322	0.137661	+	0.990479i	$0.456041\pi$
$710$	−31.5520	−1.18413
$711$	0	0
$712$	−12.9637	−0.485834
$713$	−1.93874	−0.0726064
$714$	0	0
$715$	78.9631	2.95305
$716$	−2.85427	−0.106669
$717$	0	0
$718$	20.3207	0.758360
$719$	45.2821	1.68874	0.844368	−	0.535763i	$-0.179976\pi$
$719$	45.2821	1.68874	0.844368	+	0.535763i	$0.179976\pi$
$720$	0	0
$721$	7.94858	0.296021
$722$	−8.61811	−0.320733
$723$	0	0
$724$	−3.78064	−0.140506
$725$	−13.6762	−0.507921
$726$	0	0
$727$	−28.5675	−1.05951	−0.529754	−	0.848151i	$-0.677716\pi$
$727$	−28.5675	−1.05951	−0.529754	+	0.848151i	$0.677716\pi$
$728$	−13.8562	−0.513545
$729$	0	0
$730$	−27.8076	−1.02921
$731$	47.8172	1.76858
$732$	0	0
$733$	−17.8036	−0.657590	−0.328795	−	0.944401i	$-0.606643\pi$
$733$	−17.8036	−0.657590	−0.328795	+	0.944401i	$0.606643\pi$
$734$	−3.54956	−0.131017
$735$	0	0
$736$	−2.42196	−0.0892748
$737$	67.8228	2.49828
$738$	0	0
$739$	12.0671	0.443897	0.221948	−	0.975058i	$-0.428758\pi$
$739$	12.0671	0.443897	0.221948	+	0.975058i	$0.428758\pi$
$740$	6.01697	0.221188
$741$	0	0
$742$	−3.74051	−0.137318
$743$	−40.7391	−1.49457	−0.747286	−	0.664503i	$-0.768644\pi$
$743$	−40.7391	−1.49457	−0.747286	+	0.664503i	$0.768644\pi$
$744$	0	0
$745$	55.4307	2.03082
$746$	16.0041	0.585952
$747$	0	0
$748$	−7.26414	−0.265603
$749$	10.9611	0.400509
$750$	0	0
$751$	−15.2733	−0.557331	−0.278666	−	0.960388i	$-0.589892\pi$
$751$	−15.2733	−0.557331	−0.278666	+	0.960388i	$0.589892\pi$
$752$	−35.5853	−1.29766
$753$	0	0
$754$	28.4308	1.03539
$755$	44.3416	1.61376
$756$	0	0
$757$	−10.0211	−0.364222	−0.182111	−	0.983278i	$-0.558293\pi$
$757$	−10.0211	−0.364222	−0.182111	+	0.983278i	$0.558293\pi$
$758$	−39.2702	−1.42636
$759$	0	0
$760$	38.7232	1.40464
$761$	−24.6930	−0.895122	−0.447561	−	0.894254i	$-0.647707\pi$
$761$	−24.6930	−0.895122	−0.447561	+	0.894254i	$0.647707\pi$
$762$	0	0
$763$	−10.7913	−0.390671
$764$	1.29303	0.0467801
$765$	0	0
$766$	−11.6906	−0.422398
$767$	−31.6435	−1.14258
$768$	0	0
$769$	−47.6588	−1.71862	−0.859310	−	0.511455i	$-0.829107\pi$
$769$	−47.6588	−1.71862	−0.859310	+	0.511455i	$0.829107\pi$
$770$	22.4293	0.808298
$771$	0	0
$772$	5.17928	0.186407
$773$	−27.7526	−0.998193	−0.499097	−	0.866546i	$-0.666335\pi$
$773$	−27.7526	−0.998193	−0.499097	+	0.866546i	$0.666335\pi$
$774$	0	0
$775$	4.35182	0.156322
$776$	12.8482	0.461224
$777$	0	0
$778$	3.97766	0.142606
$779$	−11.3036	−0.404994
$780$	0	0
$781$	35.6248	1.27476
$782$	−14.5420	−0.520019
$783$	0	0
$784$	−4.44246	−0.158659
$785$	34.5194	1.23205
$786$	0	0
$787$	−45.3775	−1.61753	−0.808767	−	0.588129i	$-0.799865\pi$
$787$	−45.3775	−1.61753	−0.808767	+	0.588129i	$0.799865\pi$
$788$	−2.73910	−0.0975763
$789$	0	0
$790$	−6.29735	−0.224050
$791$	1.89200	0.0672720
$792$	0	0
$793$	41.5943	1.47706
$794$	43.6536	1.54921
$795$	0	0
$796$	−3.75045	−0.132931
$797$	−0.707980	−0.0250779	−0.0125390	−	0.999921i	$-0.503991\pi$
$797$	−0.707980	−0.0250779	−0.0125390	+	0.999921i	$0.503991\pi$
$798$	0	0
$799$	−45.6463	−1.61485
$800$	5.43649	0.192209
$801$	0	0
$802$	10.3409	0.365148
$803$	31.3970	1.10798
$804$	0	0
$805$	5.04765	0.177906
$806$	−9.04680	−0.318660
$807$	0	0
$808$	0.963574	0.0338984
$809$	33.5412	1.17924	0.589622	−	0.807679i	$-0.299277\pi$
$809$	33.5412	1.17924	0.589622	+	0.807679i	$0.299277\pi$
$810$	0	0
$811$	53.7878	1.88875	0.944373	−	0.328877i	$-0.106670\pi$
$811$	53.7878	1.88875	0.944373	+	0.328877i	$0.106670\pi$
$812$	0.907853	0.0318594
$813$	0	0
$814$	−60.4323	−2.11815
$815$	−70.1337	−2.45667
$816$	0	0
$817$	−41.7385	−1.46025
$818$	−49.4155	−1.72777
$819$	0	0
$820$	−1.70922	−0.0596884
$821$	56.4836	1.97129	0.985646	−	0.168825i	$-0.0539973\pi$
$821$	56.4836	1.97129	0.985646	+	0.168825i	$0.0539973\pi$
$822$	0	0
$823$	18.1949	0.634233	0.317116	−	0.948387i	$-0.397285\pi$
$823$	18.1949	0.634233	0.317116	+	0.948387i	$0.397285\pi$
$824$	−20.8409	−0.726026
$825$	0	0
$826$	−8.98829	−0.312743
$827$	−5.17250	−0.179865	−0.0899327	−	0.995948i	$-0.528665\pi$
$827$	−5.17250	−0.179865	−0.0899327	+	0.995948i	$0.528665\pi$
$828$	0	0
$829$	10.2834	0.357157	0.178579	−	0.983926i	$-0.442850\pi$
$829$	10.2834	0.357157	0.178579	+	0.983926i	$0.442850\pi$
$830$	−23.9343	−0.830772
$831$	0	0
$832$	35.6522	1.23602
$833$	−5.69847	−0.197440
$834$	0	0
$835$	−49.4771	−1.71223
$836$	6.34069	0.219297
$837$	0	0
$838$	−13.6703	−0.472234
$839$	15.1684	0.523670	0.261835	−	0.965113i	$-0.415672\pi$
$839$	15.1684	0.523670	0.261835	+	0.965113i	$0.415672\pi$
$840$	0	0
$841$	−16.1554	−0.557083
$842$	27.8028	0.958146
$843$	0	0
$844$	−0.0750445	−0.00258314
$845$	44.3232	1.52476
$846$	0	0
$847$	−14.3246	−0.492198
$848$	11.0699	0.380142
$849$	0	0
$850$	32.6417	1.11960
$851$	−13.6001	−0.466205
$852$	0	0
$853$	−2.92663	−0.100206	−0.0501029	−	0.998744i	$-0.515955\pi$
$853$	−2.92663	−0.100206	−0.0501029	+	0.998744i	$0.515955\pi$
$854$	11.8148	0.404294
$855$	0	0
$856$	−28.7395	−0.982296
$857$	39.1115	1.33602	0.668012	−	0.744151i	$-0.267146\pi$
$857$	39.1115	1.33602	0.668012	+	0.744151i	$0.267146\pi$
$858$	0	0
$859$	−47.4971	−1.62058	−0.810290	−	0.586030i	$-0.800690\pi$
$859$	−47.4971	−1.62058	−0.810290	+	0.586030i	$0.800690\pi$
$860$	−6.31127	−0.215213
$861$	0	0
$862$	−40.9555	−1.39495
$863$	−20.1050	−0.684382	−0.342191	−	0.939630i	$-0.611169\pi$
$863$	−20.1050	−0.684382	−0.342191	+	0.939630i	$0.611169\pi$
$864$	0	0
$865$	−12.0723	−0.410470
$866$	38.3557	1.30338
$867$	0	0
$868$	−0.288882	−0.00980531
$869$	7.11022	0.241198
$870$	0	0
$871$	71.2234	2.41331
$872$	28.2943	0.958167
$873$	0	0
$874$	12.6933	0.429358
$875$	3.51559	0.118849
$876$	0	0
$877$	−46.8020	−1.58039	−0.790195	−	0.612855i	$-0.790021\pi$
$877$	−46.8020	−1.58039	−0.790195	+	0.612855i	$0.790021\pi$
$878$	31.8744	1.07571
$879$	0	0
$880$	−66.3788	−2.23763
$881$	7.29777	0.245868	0.122934	−	0.992415i	$-0.460770\pi$
$881$	7.29777	0.245868	0.122934	+	0.992415i	$0.460770\pi$
$882$	0	0
$883$	51.5206	1.73381	0.866904	−	0.498475i	$-0.166107\pi$
$883$	51.5206	1.73381	0.866904	+	0.498475i	$0.166107\pi$
$884$	−7.62837	−0.256570
$885$	0	0
$886$	18.3034	0.614916
$887$	50.8793	1.70836	0.854180	−	0.519977i	$-0.174059\pi$
$887$	50.8793	1.70836	0.854180	+	0.519977i	$0.174059\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	22.0367	0.738673
$891$	0	0
$892$	3.78185	0.126626
$893$	39.8435	1.33331
$894$	0	0
$895$	−33.4560	−1.11831
$896$	12.9763	0.433507
$897$	0	0
$898$	29.5096	0.984749
$899$	−4.08720	−0.136316
$900$	0	0
$901$	14.1997	0.473059
$902$	17.1668	0.571591
$903$	0	0
$904$	−4.96076	−0.164992
$905$	−44.3144	−1.47306
$906$	0	0
$907$	31.3616	1.04134	0.520672	−	0.853757i	$-0.325682\pi$
$907$	31.3616	1.04134	0.520672	+	0.853757i	$0.325682\pi$
$908$	7.26022	0.240939
$909$	0	0
$910$	23.5540	0.780806
$911$	10.5764	0.350411	0.175206	−	0.984532i	$-0.443941\pi$
$911$	10.5764	0.350411	0.175206	+	0.984532i	$0.443941\pi$
$912$	0	0
$913$	27.0238	0.894357
$914$	38.9340	1.28782
$915$	0	0
$916$	−1.70259	−0.0562552
$917$	−19.7242	−0.651351
$918$	0	0
$919$	18.5312	0.611289	0.305644	−	0.952146i	$-0.401128\pi$
$919$	18.5312	0.611289	0.305644	+	0.952146i	$0.401128\pi$
$920$	−13.2347	−0.436336
$921$	0	0
$922$	24.8517	0.818448
$923$	37.4110	1.23140
$924$	0	0
$925$	30.5276	1.00374
$926$	−32.5844	−1.07079
$927$	0	0
$928$	−5.10592	−0.167610
$929$	8.09475	0.265580	0.132790	−	0.991144i	$-0.457606\pi$
$929$	8.09475	0.265580	0.132790	+	0.991144i	$0.457606\pi$
$930$	0	0
$931$	4.97405	0.163018
$932$	−0.482511	−0.0158052
$933$	0	0
$934$	33.8685	1.10821
$935$	−85.1459	−2.78457
$936$	0	0
$937$	−49.2094	−1.60760	−0.803800	−	0.594900i	$-0.797192\pi$
$937$	−49.2094	−1.60760	−0.803800	+	0.594900i	$0.797192\pi$
$938$	20.2309	0.660562
$939$	0	0
$940$	6.02474	0.196505
$941$	−7.19836	−0.234660	−0.117330	−	0.993093i	$-0.537433\pi$
$941$	−7.19836	−0.234660	−0.117330	+	0.993093i	$0.537433\pi$
$942$	0	0
$943$	3.86332	0.125807
$944$	26.6005	0.865773
$945$	0	0
$946$	63.3882	2.06093
$947$	−28.4504	−0.924514	−0.462257	−	0.886746i	$-0.652960\pi$
$947$	−28.4504	−0.924514	−0.462257	+	0.886746i	$0.652960\pi$
$948$	0	0
$949$	32.9713	1.07029
$950$	−28.4922	−0.924408
$951$	0	0
$952$	14.9411	0.484245
$953$	35.2326	1.14129	0.570647	−	0.821195i	$-0.306692\pi$
$953$	35.2326	1.14129	0.570647	+	0.821195i	$0.306692\pi$
$954$	0	0
$955$	15.1561	0.490440
$956$	0.599997	0.0194053
$957$	0	0
$958$	40.5550	1.31027
$959$	−1.23699	−0.0399444
$960$	0	0
$961$	−29.6994	−0.958046
$962$	−63.4624	−2.04611
$963$	0	0
$964$	3.02940	0.0975705
$965$	60.7085	1.95428
$966$	0	0
$967$	−17.8331	−0.573475	−0.286737	−	0.958009i	$-0.592571\pi$
$967$	−17.8331	−0.573475	−0.286737	+	0.958009i	$0.592571\pi$
$968$	37.5584	1.20717
$969$	0	0
$970$	−21.8405	−0.701256
$971$	5.66021	0.181645	0.0908224	−	0.995867i	$-0.471050\pi$
$971$	5.66021	0.181645	0.0908224	+	0.995867i	$0.471050\pi$
$972$	0	0
$973$	18.9060	0.606099
$974$	27.8039	0.890893
$975$	0	0
$976$	−34.9654	−1.11922
$977$	3.13669	0.100352	0.0501758	−	0.998740i	$-0.484022\pi$
$977$	3.13669	0.100352	0.0501758	+	0.998740i	$0.484022\pi$
$978$	0	0
$979$	−24.8813	−0.795209
$980$	0.752126	0.0240258
$981$	0	0
$982$	−22.7431	−0.725760
$983$	32.6268	1.04063	0.520317	−	0.853973i	$-0.325814\pi$
$983$	32.6268	1.04063	0.520317	+	0.853973i	$0.325814\pi$
$984$	0	0
$985$	−32.1061	−1.02298
$986$	−30.6570	−0.976316
$987$	0	0
$988$	6.65862	0.211839
$989$	14.2653	0.453610
$990$	0	0
$991$	32.4762	1.03164	0.515821	−	0.856696i	$-0.327487\pi$
$991$	32.4762	1.03164	0.515821	+	0.856696i	$0.327487\pi$
$992$	1.62472	0.0515850
$993$	0	0
$994$	10.6265	0.337053
$995$	−43.9606	−1.39364
$996$	0	0
$997$	4.71956	0.149470	0.0747349	−	0.997203i	$-0.476189\pi$
$997$	4.71956	0.149470	0.0747349	+	0.997203i	$0.476189\pi$
$998$	−66.6887	−2.11099
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.s.1.5		16
3.2	odd	2		2667.2.a.n.1.12	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.n.1.12	✓	16	3.2	odd	2
8001.2.a.s.1.5		16	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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q-1.50110 q^{2} +0.253312 q^{4} +2.96917 q^{5} -1.00000 q^{7} +2.62196 q^{8} +O(q^{10})\)	\(q-1.50110 q^{2} +0.253312 q^{4} +2.96917 q^{5} -1.00000 q^{7} +2.62196 q^{8} -4.45703 q^{10} +5.03235 q^{11} +5.28468 q^{13} +1.50110 q^{14} -4.44246 q^{16} -5.69847 q^{17} +4.97405 q^{19} +0.752126 q^{20} -7.55408 q^{22} -1.70002 q^{23} +3.81597 q^{25} -7.93285 q^{26} -0.253312 q^{28} -3.58394 q^{29} +1.14042 q^{31} +1.42467 q^{32} +8.55399 q^{34} -2.96917 q^{35} +7.99996 q^{37} -7.46656 q^{38} +7.78504 q^{40} -2.27251 q^{41} -8.39125 q^{43} +1.27475 q^{44} +2.55191 q^{46} +8.01028 q^{47} +1.00000 q^{49} -5.72816 q^{50} +1.33867 q^{52} -2.49184 q^{53} +14.9419 q^{55} -2.62196 q^{56} +5.37986 q^{58} -5.98779 q^{59} +7.87074 q^{61} -1.71189 q^{62} +6.74634 q^{64} +15.6911 q^{65} +13.4773 q^{67} -1.44349 q^{68} +4.45703 q^{70} +7.07915 q^{71} +6.23904 q^{73} -12.0088 q^{74} +1.25999 q^{76} -5.03235 q^{77} +1.41290 q^{79} -13.1904 q^{80} +3.41128 q^{82} +5.37001 q^{83} -16.9197 q^{85} +12.5961 q^{86} +13.1946 q^{88} -4.94427 q^{89} -5.28468 q^{91} -0.430635 q^{92} -12.0243 q^{94} +14.7688 q^{95} +4.90024 q^{97} -1.50110 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8}+O(q^{10}) \) 16 * q - 4 * q^2 + 20 * q^4 - 5 * q^5 - 16 * q^7 - 15 * q^8	\( 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8} - 4 q^{10} - q^{11} + 20 q^{13} + 4 q^{14} + 32 q^{16} - 3 q^{17} + 13 q^{19} - 17 q^{20} + 13 q^{22} - 5 q^{23} + 17 q^{25} + 2 q^{26} - 20 q^{28} - 22 q^{29} + 26 q^{31} - 54 q^{32} - 6 q^{34} + 5 q^{35} + 30 q^{37} - 5 q^{38} + 13 q^{40} - q^{41} + 31 q^{43} - 22 q^{44} - 2 q^{46} + q^{47} + 16 q^{49} - 5 q^{50} + 31 q^{52} - 24 q^{53} + 8 q^{55} + 15 q^{56} + 13 q^{58} + 17 q^{59} + 32 q^{61} + 5 q^{62} + 61 q^{64} + 3 q^{65} + 16 q^{67} + 10 q^{68} + 4 q^{70} + 10 q^{71} + 23 q^{73} - q^{74} + 18 q^{76} + q^{77} + 48 q^{79} - 38 q^{80} + 12 q^{82} - 9 q^{83} + 22 q^{85} + 4 q^{86} + 27 q^{88} - 17 q^{89} - 20 q^{91} - 16 q^{92} + 13 q^{94} - 22 q^{95} + 17 q^{97} - 4 q^{98}+O(q^{100}) \) 16 * q - 4 * q^2 + 20 * q^4 - 5 * q^5 - 16 * q^7 - 15 * q^8 - 4 * q^10 - q^11 + 20 * q^13 + 4 * q^14 + 32 * q^16 - 3 * q^17 + 13 * q^19 - 17 * q^20 + 13 * q^22 - 5 * q^23 + 17 * q^25 + 2 * q^26 - 20 * q^28 - 22 * q^29 + 26 * q^31 - 54 * q^32 - 6 * q^34 + 5 * q^35 + 30 * q^37 - 5 * q^38 + 13 * q^40 - q^41 + 31 * q^43 - 22 * q^44 - 2 * q^46 + q^47 + 16 * q^49 - 5 * q^50 + 31 * q^52 - 24 * q^53 + 8 * q^55 + 15 * q^56 + 13 * q^58 + 17 * q^59 + 32 * q^61 + 5 * q^62 + 61 * q^64 + 3 * q^65 + 16 * q^67 + 10 * q^68 + 4 * q^70 + 10 * q^71 + 23 * q^73 - q^74 + 18 * q^76 + q^77 + 48 * q^79 - 38 * q^80 + 12 * q^82 - 9 * q^83 + 22 * q^85 + 4 * q^86 + 27 * q^88 - 17 * q^89 - 20 * q^91 - 16 * q^92 + 13 * q^94 - 22 * q^95 + 17 * q^97 - 4 * q^98

Introduction

L-functions

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Database

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