LMFDB - Embedded newform 8001.2.a.u.1.8

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:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 6 x^{17} - 11 x^{16} + 123 x^{15} - 35 x^{14} - 982 x^{13} + 988 x^{12} + 3872 x^{11} + \cdots + 16 $ x^18 - 6x^17 - 11x^16 + 123x^15 - 35x^14 - 982x^13 + 988x^12 + 3872x^11 - 5421x^10 - 7882x^9 + 13376x^8 + 7948x^7 - 15795x^6 - 3858x^5 + 8199x^4 + 1453x^3 - 1610x^2 - 380*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 5 $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-0.375443$ 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)

$f(q)$	$=$	$q-0.375443 q^{2} -1.85904 q^{4} -2.49678 q^{5} +1.00000 q^{7} +1.44885 q^{8} +O(q^{10})$	\(q-0.375443 q^{2} -1.85904 q^{4} -2.49678 q^{5} +1.00000 q^{7} +1.44885 q^{8} +0.937397 q^{10} +0.0439826 q^{11} +1.23685 q^{13} -0.375443 q^{14} +3.17413 q^{16} +4.66254 q^{17} -4.55600 q^{19} +4.64162 q^{20} -0.0165129 q^{22} +5.31043 q^{23} +1.23390 q^{25} -0.464365 q^{26} -1.85904 q^{28} +5.23792 q^{29} +0.715277 q^{31} -4.08940 q^{32} -1.75052 q^{34} -2.49678 q^{35} +7.90706 q^{37} +1.71052 q^{38} -3.61746 q^{40} +4.97134 q^{41} +0.341392 q^{43} -0.0817655 q^{44} -1.99376 q^{46} -11.4407 q^{47} +1.00000 q^{49} -0.463260 q^{50} -2.29935 q^{52} +4.41842 q^{53} -0.109815 q^{55} +1.44885 q^{56} -1.96654 q^{58} -9.78565 q^{59} -9.42399 q^{61} -0.268546 q^{62} -4.81291 q^{64} -3.08813 q^{65} +0.00811780 q^{67} -8.66785 q^{68} +0.937397 q^{70} +4.91418 q^{71} -11.8395 q^{73} -2.96865 q^{74} +8.46980 q^{76} +0.0439826 q^{77} -1.44620 q^{79} -7.92509 q^{80} -1.86645 q^{82} +7.62022 q^{83} -11.6413 q^{85} -0.128173 q^{86} +0.0637241 q^{88} +11.8328 q^{89} +1.23685 q^{91} -9.87232 q^{92} +4.29534 q^{94} +11.3753 q^{95} -8.97251 q^{97} -0.375443 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8}+O(q^{10}) $ 18 * q + 6 * q^2 + 22 * q^4 + 10 * q^5 + 18 * q^7 + 21 * q^8	$ 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8} - 4 q^{10} + 9 q^{11} - 25 q^{13} + 6 q^{14} + 34 q^{16} + 17 q^{17} - 5 q^{19} + 21 q^{20} + 5 q^{22} + 14 q^{23} + 28 q^{25} + 8 q^{26} + 22 q^{28} + 17 q^{29} + 5 q^{31} + 53 q^{32} - 19 q^{34} + 10 q^{35} - 15 q^{37} + 22 q^{38} - q^{40} + 17 q^{41} + q^{43} + 33 q^{44} + 10 q^{46} + 31 q^{47} + 18 q^{49} + 35 q^{50} - 70 q^{52} + 35 q^{53} + 4 q^{55} + 21 q^{56} + 3 q^{58} + 46 q^{59} - 5 q^{61} + 10 q^{62} + 63 q^{64} + 12 q^{65} + 6 q^{67} + 56 q^{68} - 4 q^{70} + 22 q^{71} - 16 q^{73} - 18 q^{74} + 32 q^{76} + 9 q^{77} + 46 q^{79} + 30 q^{80} - 12 q^{82} + 46 q^{83} + 4 q^{85} - 18 q^{86} + 30 q^{88} + 42 q^{89} - 25 q^{91} + 48 q^{92} + 3 q^{94} + 2 q^{95} - 35 q^{97} + 6 q^{98}+O(q^{100}) $ 18 * q + 6 * q^2 + 22 * q^4 + 10 * q^5 + 18 * q^7 + 21 * q^8 - 4 * q^10 + 9 * q^11 - 25 * q^13 + 6 * q^14 + 34 * q^16 + 17 * q^17 - 5 * q^19 + 21 * q^20 + 5 * q^22 + 14 * q^23 + 28 * q^25 + 8 * q^26 + 22 * q^28 + 17 * q^29 + 5 * q^31 + 53 * q^32 - 19 * q^34 + 10 * q^35 - 15 * q^37 + 22 * q^38 - q^40 + 17 * q^41 + q^43 + 33 * q^44 + 10 * q^46 + 31 * q^47 + 18 * q^49 + 35 * q^50 - 70 * q^52 + 35 * q^53 + 4 * q^55 + 21 * q^56 + 3 * q^58 + 46 * q^59 - 5 * q^61 + 10 * q^62 + 63 * q^64 + 12 * q^65 + 6 * q^67 + 56 * q^68 - 4 * q^70 + 22 * q^71 - 16 * q^73 - 18 * q^74 + 32 * q^76 + 9 * q^77 + 46 * q^79 + 30 * q^80 - 12 * q^82 + 46 * q^83 + 4 * q^85 - 18 * q^86 + 30 * q^88 + 42 * q^89 - 25 * q^91 + 48 * q^92 + 3 * q^94 + 2 * q^95 - 35 * q^97 + 6 * 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 $

$2$	−0.375443	−0.265478	−0.132739	−	0.991151i	$-0.542377\pi$
$2$	−0.375443	−0.265478	−0.132739	+	0.991151i	$0.542377\pi$
$3$	0	0
$3$	0	0
$4$	−1.85904	−0.929521
$5$	−2.49678	−1.11659	−0.558297	−	0.829641i	$-0.688545\pi$
$5$	−2.49678	−1.11659	−0.558297	+	0.829641i	$0.688545\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.44885	0.512246
$9$	0	0
$10$	0.937397	0.296431
$11$	0.0439826	0.0132612	0.00663062	−	0.999978i	$-0.497889\pi$
$11$	0.0439826	0.0132612	0.00663062	+	0.999978i	$0.497889\pi$
$12$	0	0
$13$	1.23685	0.343039	0.171520	−	0.985181i	$-0.445132\pi$
$13$	1.23685	0.343039	0.171520	+	0.985181i	$0.445132\pi$
$14$	−0.375443	−0.100341
$15$	0	0
$16$	3.17413	0.793531
$17$	4.66254	1.13083	0.565416	−	0.824806i	$-0.308716\pi$
$17$	4.66254	1.13083	0.565416	+	0.824806i	$0.308716\pi$
$18$	0	0
$19$	−4.55600	−1.04522	−0.522609	−	0.852572i	$-0.675041\pi$
$19$	−4.55600	−1.04522	−0.522609	+	0.852572i	$0.675041\pi$
$20$	4.64162	1.03790
$21$	0	0
$22$	−0.0165129	−0.00352057
$23$	5.31043	1.10730	0.553651	−	0.832749i	$-0.313234\pi$
$23$	5.31043	1.10730	0.553651	+	0.832749i	$0.313234\pi$
$24$	0	0
$25$	1.23390	0.246780
$26$	−0.464365	−0.0910694
$27$	0	0
$28$	−1.85904	−0.351326
$29$	5.23792	0.972657	0.486329	−	0.873776i	$-0.338336\pi$
$29$	5.23792	0.972657	0.486329	+	0.873776i	$0.338336\pi$
$30$	0	0
$31$	0.715277	0.128468	0.0642338	−	0.997935i	$-0.479540\pi$
$31$	0.715277	0.128468	0.0642338	+	0.997935i	$0.479540\pi$
$32$	−4.08940	−0.722911
$33$	0	0
$34$	−1.75052	−0.300211
$35$	−2.49678	−0.422033
$36$	0	0
$37$	7.90706	1.29991	0.649956	−	0.759972i	$-0.274787\pi$
$37$	7.90706	1.29991	0.649956	+	0.759972i	$0.274787\pi$
$38$	1.71052	0.277483
$39$	0	0
$40$	−3.61746	−0.571970
$41$	4.97134	0.776393	0.388197	−	0.921577i	$-0.373098\pi$
$41$	4.97134	0.776393	0.388197	+	0.921577i	$0.373098\pi$
$42$	0	0
$43$	0.341392	0.0520618	0.0260309	−	0.999661i	$-0.491713\pi$
$43$	0.341392	0.0520618	0.0260309	+	0.999661i	$0.491713\pi$
$44$	−0.0817655	−0.0123266
$45$	0	0
$46$	−1.99376	−0.293964
$47$	−11.4407	−1.66880	−0.834402	−	0.551157i	$-0.814187\pi$
$47$	−11.4407	−1.66880	−0.834402	+	0.551157i	$0.814187\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−0.463260	−0.0655148
$51$	0	0
$52$	−2.29935	−0.318862
$53$	4.41842	0.606916	0.303458	−	0.952845i	$-0.401859\pi$
$53$	4.41842	0.606916	0.303458	+	0.952845i	$0.401859\pi$
$54$	0	0
$55$	−0.109815	−0.0148074
$56$	1.44885	0.193611
$57$	0	0
$58$	−1.96654	−0.258219
$59$	−9.78565	−1.27398	−0.636991	−	0.770871i	$-0.719821\pi$
$59$	−9.78565	−1.27398	−0.636991	+	0.770871i	$0.719821\pi$
$60$	0	0
$61$	−9.42399	−1.20662	−0.603309	−	0.797508i	$-0.706151\pi$
$61$	−9.42399	−1.20662	−0.603309	+	0.797508i	$0.706151\pi$
$62$	−0.268546	−0.0341053
$63$	0	0
$64$	−4.81291	−0.601614
$65$	−3.08813	−0.383035
$66$	0	0
$67$	0.00811780	0.000991748 0	0.000495874	−	1.00000i	$-0.499842\pi$
$67$	0.00811780	0.000991748 0	0.000495874		1.00000i	$0.499842\pi$
$68$	−8.66785	−1.05113
$69$	0	0
$70$	0.937397	0.112040
$71$	4.91418	0.583206	0.291603	−	0.956539i	$-0.405811\pi$
$71$	4.91418	0.583206	0.291603	+	0.956539i	$0.405811\pi$
$72$	0	0
$73$	−11.8395	−1.38571	−0.692856	−	0.721076i	$-0.743648\pi$
$73$	−11.8395	−1.38571	−0.692856	+	0.721076i	$0.743648\pi$
$74$	−2.96865	−0.345098
$75$	0	0
$76$	8.46980	0.971553
$77$	0.0439826	0.00501228
$78$	0	0
$79$	−1.44620	−0.162710	−0.0813552	−	0.996685i	$-0.525925\pi$
$79$	−1.44620	−0.162710	−0.0813552	+	0.996685i	$0.525925\pi$
$80$	−7.92509	−0.886052
$81$	0	0
$82$	−1.86645	−0.206115
$83$	7.62022	0.836428	0.418214	−	0.908349i	$-0.362656\pi$
$83$	7.62022	0.836428	0.418214	+	0.908349i	$0.362656\pi$
$84$	0	0
$85$	−11.6413	−1.26268
$86$	−0.128173	−0.0138213
$87$	0	0
$88$	0.0637241	0.00679301
$89$	11.8328	1.25427	0.627137	−	0.778909i	$-0.284227\pi$
$89$	11.8328	1.25427	0.627137	+	0.778909i	$0.284227\pi$
$90$	0	0
$91$	1.23685	0.129657
$92$	−9.87232	−1.02926
$93$	0	0
$94$	4.29534	0.443031
$95$	11.3753	1.16708
$96$	0	0
$97$	−8.97251	−0.911020	−0.455510	−	0.890231i	$-0.650543\pi$
$97$	−8.97251	−0.911020	−0.455510	+	0.890231i	$0.650543\pi$
$98$	−0.375443	−0.0379255
$99$	0	0
$100$	−2.29388	−0.229388
$101$	−11.8979	−1.18389	−0.591943	−	0.805980i	$-0.701639\pi$
$101$	−11.8979	−1.18389	−0.591943	+	0.805980i	$0.701639\pi$
$102$	0	0
$103$	3.32884	0.328000	0.164000	−	0.986460i	$-0.447560\pi$
$103$	3.32884	0.328000	0.164000	+	0.986460i	$0.447560\pi$
$104$	1.79200	0.175720
$105$	0	0
$106$	−1.65886	−0.161123
$107$	11.7272	1.13371	0.566856	−	0.823817i	$-0.308159\pi$
$107$	11.7272	1.13371	0.566856	+	0.823817i	$0.308159\pi$
$108$	0	0
$109$	−12.3647	−1.18432	−0.592161	−	0.805820i	$-0.701725\pi$
$109$	−12.3647	−1.18432	−0.592161	+	0.805820i	$0.701725\pi$
$110$	0.0412291	0.00393104
$111$	0	0
$112$	3.17413	0.299927
$113$	11.5463	1.08618	0.543092	−	0.839673i	$-0.317253\pi$
$113$	11.5463	1.08618	0.543092	+	0.839673i	$0.317253\pi$
$114$	0	0
$115$	−13.2590	−1.23641
$116$	−9.73752	−0.904106
$117$	0	0
$118$	3.67395	0.338215
$119$	4.66254	0.427414
$120$	0	0
$121$	−10.9981	−0.999824
$122$	3.53817	0.320331
$123$	0	0
$124$	−1.32973	−0.119413
$125$	9.40311	0.841040
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	9.98578	0.882626
$129$	0	0
$130$	1.15942	0.101688
$131$	−1.63563	−0.142906	−0.0714529	−	0.997444i	$-0.522764\pi$
$131$	−1.63563	−0.142906	−0.0714529	+	0.997444i	$0.522764\pi$
$132$	0	0
$133$	−4.55600	−0.395056
$134$	−0.00304777	−0.000263287 0
$135$	0	0
$136$	6.75531	0.579263
$137$	−4.20337	−0.359118	−0.179559	−	0.983747i	$-0.557467\pi$
$137$	−4.20337	−0.359118	−0.179559	+	0.983747i	$0.557467\pi$
$138$	0	0
$139$	−1.83457	−0.155606	−0.0778030	−	0.996969i	$-0.524791\pi$
$139$	−1.83457	−0.155606	−0.0778030	+	0.996969i	$0.524791\pi$
$140$	4.64162	0.392288
$141$	0	0
$142$	−1.84499	−0.154828
$143$	0.0543996	0.00454913
$144$	0	0
$145$	−13.0779	−1.08606
$146$	4.44506	0.367876
$147$	0	0
$148$	−14.6996	−1.20830
$149$	14.6708	1.20188	0.600940	−	0.799294i	$-0.294793\pi$
$149$	14.6708	1.20188	0.600940	+	0.799294i	$0.294793\pi$
$150$	0	0
$151$	7.31356	0.595169	0.297585	−	0.954695i	$-0.403819\pi$
$151$	7.31356	0.595169	0.297585	+	0.954695i	$0.403819\pi$
$152$	−6.60096	−0.535409
$153$	0	0
$154$	−0.0165129	−0.00133065
$155$	−1.78589	−0.143446
$156$	0	0
$157$	5.65707	0.451483	0.225742	−	0.974187i	$-0.427519\pi$
$157$	5.65707	0.451483	0.225742	+	0.974187i	$0.427519\pi$
$158$	0.542966	0.0431961
$159$	0	0
$160$	10.2103	0.807198
$161$	5.31043	0.418521
$162$	0	0
$163$	16.3284	1.27894	0.639470	−	0.768816i	$-0.279154\pi$
$163$	16.3284	1.27894	0.639470	+	0.768816i	$0.279154\pi$
$164$	−9.24194	−0.721674
$165$	0	0
$166$	−2.86096	−0.222053
$167$	12.9310	1.00063	0.500314	−	0.865844i	$-0.333218\pi$
$167$	12.9310	1.00063	0.500314	+	0.865844i	$0.333218\pi$
$168$	0	0
$169$	−11.4702	−0.882324
$170$	4.37065	0.335213
$171$	0	0
$172$	−0.634663	−0.0483926
$173$	−9.17036	−0.697209	−0.348605	−	0.937270i	$-0.613344\pi$
$173$	−9.17036	−0.697209	−0.348605	+	0.937270i	$0.613344\pi$
$174$	0	0
$175$	1.23390	0.0932742
$176$	0.139606	0.0105232
$177$	0	0
$178$	−4.44254	−0.332982
$179$	4.47476	0.334459	0.167230	−	0.985918i	$-0.446518\pi$
$179$	4.47476	0.334459	0.167230	+	0.985918i	$0.446518\pi$
$180$	0	0
$181$	−10.6152	−0.789023	−0.394511	−	0.918891i	$-0.629086\pi$
$181$	−10.6152	−0.789023	−0.394511	+	0.918891i	$0.629086\pi$
$182$	−0.464365	−0.0344210
$183$	0	0
$184$	7.69402	0.567210
$185$	−19.7422	−1.45147
$186$	0	0
$187$	0.205070	0.0149962
$188$	21.2688	1.55119
$189$	0	0
$190$	−4.27079	−0.309835
$191$	−10.6972	−0.774020	−0.387010	−	0.922075i	$-0.626492\pi$
$191$	−10.6972	−0.774020	−0.387010	+	0.922075i	$0.626492\pi$
$192$	0	0
$193$	3.42294	0.246388	0.123194	−	0.992383i	$-0.460686\pi$
$193$	3.42294	0.246388	0.123194	+	0.992383i	$0.460686\pi$
$194$	3.36866	0.241856
$195$	0	0
$196$	−1.85904	−0.132789
$197$	−12.6206	−0.899181	−0.449591	−	0.893235i	$-0.648430\pi$
$197$	−12.6206	−0.899181	−0.449591	+	0.893235i	$0.648430\pi$
$198$	0	0
$199$	21.2114	1.50364	0.751819	−	0.659369i	$-0.229177\pi$
$199$	21.2114	1.50364	0.751819	+	0.659369i	$0.229177\pi$
$200$	1.78774	0.126412
$201$	0	0
$202$	4.46699	0.314296
$203$	5.23792	0.367630
$204$	0	0
$205$	−12.4123	−0.866915
$206$	−1.24979	−0.0870769
$207$	0	0
$208$	3.92590	0.272212
$209$	−0.200385	−0.0138609
$210$	0	0
$211$	1.41420	0.0973575	0.0486788	−	0.998814i	$-0.484499\pi$
$211$	1.41420	0.0973575	0.0486788	+	0.998814i	$0.484499\pi$
$212$	−8.21403	−0.564142
$213$	0	0
$214$	−4.40290	−0.300976
$215$	−0.852381	−0.0581319
$216$	0	0
$217$	0.715277	0.0485562
$218$	4.64223	0.314411
$219$	0	0
$220$	0.204150	0.0137638
$221$	5.76684	0.387919
$222$	0	0
$223$	−11.6705	−0.781512	−0.390756	−	0.920494i	$-0.627786\pi$
$223$	−11.6705	−0.781512	−0.390756	+	0.920494i	$0.627786\pi$
$224$	−4.08940	−0.273235
$225$	0	0
$226$	−4.33497	−0.288358
$227$	6.98427	0.463562	0.231781	−	0.972768i	$-0.425545\pi$
$227$	6.98427	0.463562	0.231781	+	0.972768i	$0.425545\pi$
$228$	0	0
$229$	3.76335	0.248689	0.124345	−	0.992239i	$-0.460317\pi$
$229$	3.76335	0.248689	0.124345	+	0.992239i	$0.460317\pi$
$230$	4.97798	0.328239
$231$	0	0
$232$	7.58896	0.498240
$233$	6.17050	0.404243	0.202121	−	0.979360i	$-0.435216\pi$
$233$	6.17050	0.404243	0.202121	+	0.979360i	$0.435216\pi$
$234$	0	0
$235$	28.5650	1.86337
$236$	18.1919	1.18419
$237$	0	0
$238$	−1.75052	−0.113469
$239$	−5.01726	−0.324540	−0.162270	−	0.986746i	$-0.551882\pi$
$239$	−5.01726	−0.324540	−0.162270	+	0.986746i	$0.551882\pi$
$240$	0	0
$241$	−4.66693	−0.300623	−0.150312	−	0.988639i	$-0.548028\pi$
$241$	−4.66693	−0.300623	−0.150312	+	0.988639i	$0.548028\pi$
$242$	4.12914	0.265431
$243$	0	0
$244$	17.5196	1.12158
$245$	−2.49678	−0.159513
$246$	0	0
$247$	−5.63507	−0.358551
$248$	1.03633	0.0658070
$249$	0	0
$250$	−3.53033	−0.223278
$251$	14.9362	0.942765	0.471382	−	0.881929i	$-0.343755\pi$
$251$	14.9362	0.942765	0.471382	+	0.881929i	$0.343755\pi$
$252$	0	0
$253$	0.233566	0.0146842
$254$	−0.375443	−0.0235574
$255$	0	0
$256$	5.87674	0.367296
$257$	−2.50206	−0.156074	−0.0780370	−	0.996950i	$-0.524865\pi$
$257$	−2.50206	−0.156074	−0.0780370	+	0.996950i	$0.524865\pi$
$258$	0	0
$259$	7.90706	0.491321
$260$	5.74096	0.356040
$261$	0	0
$262$	0.614086	0.0379383
$263$	3.74847	0.231141	0.115570	−	0.993299i	$-0.463130\pi$
$263$	3.74847	0.231141	0.115570	+	0.993299i	$0.463130\pi$
$264$	0	0
$265$	−11.0318	−0.677679
$266$	1.71052	0.104879
$267$	0	0
$268$	−0.0150913	−0.000921851 0
$269$	−8.16292	−0.497702	−0.248851	−	0.968542i	$-0.580053\pi$
$269$	−8.16292	−0.497702	−0.248851	+	0.968542i	$0.580053\pi$
$270$	0	0
$271$	−8.16035	−0.495706	−0.247853	−	0.968798i	$-0.579725\pi$
$271$	−8.16035	−0.495706	−0.247853	+	0.968798i	$0.579725\pi$
$272$	14.7995	0.897350
$273$	0	0
$274$	1.57812	0.0953380
$275$	0.0542702	0.00327261
$276$	0	0
$277$	−19.5232	−1.17304	−0.586519	−	0.809935i	$-0.699502\pi$
$277$	−19.5232	−1.17304	−0.586519	+	0.809935i	$0.699502\pi$
$278$	0.688775	0.0413100
$279$	0	0
$280$	−3.61746	−0.216184
$281$	13.5957	0.811052	0.405526	−	0.914084i	$-0.367088\pi$
$281$	13.5957	0.811052	0.405526	+	0.914084i	$0.367088\pi$
$282$	0	0
$283$	24.0199	1.42783	0.713917	−	0.700230i	$-0.246919\pi$
$283$	24.0199	1.42783	0.713917	+	0.700230i	$0.246919\pi$
$284$	−9.13567	−0.542102
$285$	0	0
$286$	−0.0204240	−0.00120769
$287$	4.97134	0.293449
$288$	0	0
$289$	4.73924	0.278779
$290$	4.91001	0.288326
$291$	0	0
$292$	22.0102	1.28805
$293$	27.5825	1.61138	0.805692	−	0.592334i	$-0.201794\pi$
$293$	27.5825	1.61138	0.805692	+	0.592334i	$0.201794\pi$
$294$	0	0
$295$	24.4326	1.42252
$296$	11.4561	0.665875
$297$	0	0
$298$	−5.50806	−0.319073
$299$	6.56818	0.379848
$300$	0	0
$301$	0.341392	0.0196775
$302$	−2.74582	−0.158004
$303$	0	0
$304$	−14.4613	−0.829414
$305$	23.5296	1.34730
$306$	0	0
$307$	−24.8153	−1.41629	−0.708143	−	0.706069i	$-0.750467\pi$
$307$	−24.8153	−1.41629	−0.708143	+	0.706069i	$0.750467\pi$
$308$	−0.0817655	−0.00465902
$309$	0	0
$310$	0.670499	0.0380818
$311$	0.788961	0.0447379	0.0223689	−	0.999750i	$-0.492879\pi$
$311$	0.788961	0.0447379	0.0223689	+	0.999750i	$0.492879\pi$
$312$	0	0
$313$	16.4791	0.931453	0.465727	−	0.884929i	$-0.345793\pi$
$313$	16.4791	0.931453	0.465727	+	0.884929i	$0.345793\pi$
$314$	−2.12391	−0.119859
$315$	0	0
$316$	2.68855	0.151243
$317$	−15.2893	−0.858735	−0.429367	−	0.903130i	$-0.641263\pi$
$317$	−15.2893	−0.858735	−0.429367	+	0.903130i	$0.641263\pi$
$318$	0	0
$319$	0.230377	0.0128986
$320$	12.0168	0.671758
$321$	0	0
$322$	−1.99376	−0.111108
$323$	−21.2425	−1.18197
$324$	0	0
$325$	1.52615	0.0846554
$326$	−6.13039	−0.339531
$327$	0	0
$328$	7.20273	0.397704
$329$	−11.4407	−0.630748
$330$	0	0
$331$	−8.79412	−0.483369	−0.241684	−	0.970355i	$-0.577700\pi$
$331$	−8.79412	−0.483369	−0.241684	+	0.970355i	$0.577700\pi$
$332$	−14.1663	−0.777477
$333$	0	0
$334$	−4.85484	−0.265645
$335$	−0.0202684	−0.00110738
$336$	0	0
$337$	10.3907	0.566015	0.283008	−	0.959118i	$-0.408668\pi$
$337$	10.3907	0.566015	0.283008	+	0.959118i	$0.408668\pi$
$338$	4.30641	0.234238
$339$	0	0
$340$	21.6417	1.17369
$341$	0.0314597	0.00170364
$342$	0	0
$343$	1.00000	0.0539949
$344$	0.494626	0.0266684
$345$	0	0
$346$	3.44294	0.185094
$347$	3.73798	0.200665	0.100333	−	0.994954i	$-0.468009\pi$
$347$	3.73798	0.200665	0.100333	+	0.994954i	$0.468009\pi$
$348$	0	0
$349$	13.8895	0.743487	0.371744	−	0.928335i	$-0.378760\pi$
$349$	13.8895	0.743487	0.371744	+	0.928335i	$0.378760\pi$
$350$	−0.463260	−0.0247623
$351$	0	0
$352$	−0.179862	−0.00958670
$353$	29.3769	1.56357	0.781787	−	0.623546i	$-0.214309\pi$
$353$	29.3769	1.56357	0.781787	+	0.623546i	$0.214309\pi$
$354$	0	0
$355$	−12.2696	−0.651204
$356$	−21.9977	−1.16587
$357$	0	0
$358$	−1.68002	−0.0887917
$359$	3.99547	0.210873	0.105436	−	0.994426i	$-0.466376\pi$
$359$	3.99547	0.210873	0.105436	+	0.994426i	$0.466376\pi$
$360$	0	0
$361$	1.75716	0.0924823
$362$	3.98541	0.209468
$363$	0	0
$364$	−2.29935	−0.120519
$365$	29.5607	1.54728
$366$	0	0
$367$	−15.8689	−0.828348	−0.414174	−	0.910198i	$-0.635929\pi$
$367$	−15.8689	−0.828348	−0.414174	+	0.910198i	$0.635929\pi$
$368$	16.8560	0.878678
$369$	0	0
$370$	7.41206	0.385335
$371$	4.41842	0.229393
$372$	0	0
$373$	−1.79342	−0.0928595	−0.0464298	−	0.998922i	$-0.514784\pi$
$373$	−1.79342	−0.0928595	−0.0464298	+	0.998922i	$0.514784\pi$
$374$	−0.0769922	−0.00398117
$375$	0	0
$376$	−16.5759	−0.854837
$377$	6.47850	0.333660
$378$	0	0
$379$	30.6924	1.57656	0.788280	−	0.615316i	$-0.210972\pi$
$379$	30.6924	1.57656	0.788280	+	0.615316i	$0.210972\pi$
$380$	−21.1472	−1.08483
$381$	0	0
$382$	4.01618	0.205485
$383$	13.9575	0.713194	0.356597	−	0.934258i	$-0.383937\pi$
$383$	13.9575	0.713194	0.356597	+	0.934258i	$0.383937\pi$
$384$	0	0
$385$	−0.109815	−0.00559668
$386$	−1.28512	−0.0654107
$387$	0	0
$388$	16.6803	0.846813
$389$	−31.4166	−1.59288	−0.796442	−	0.604715i	$-0.793287\pi$
$389$	−31.4166	−1.59288	−0.796442	+	0.604715i	$0.793287\pi$
$390$	0	0
$391$	24.7601	1.25217
$392$	1.44885	0.0731780
$393$	0	0
$394$	4.73832	0.238713
$395$	3.61085	0.181681
$396$	0	0
$397$	4.14664	0.208114	0.104057	−	0.994571i	$-0.466818\pi$
$397$	4.14664	0.208114	0.104057	+	0.994571i	$0.466818\pi$
$398$	−7.96368	−0.399183
$399$	0	0
$400$	3.91656	0.195828
$401$	−23.0458	−1.15085	−0.575427	−	0.817853i	$-0.695164\pi$
$401$	−23.0458	−1.15085	−0.575427	+	0.817853i	$0.695164\pi$
$402$	0	0
$403$	0.884688	0.0440694
$404$	22.1187	1.10045
$405$	0	0
$406$	−1.96654	−0.0975977
$407$	0.347773	0.0172385
$408$	0	0
$409$	10.6304	0.525639	0.262819	−	0.964845i	$-0.415348\pi$
$409$	10.6304	0.525639	0.262819	+	0.964845i	$0.415348\pi$
$410$	4.66012	0.230147
$411$	0	0
$412$	−6.18845	−0.304883
$413$	−9.78565	−0.481520
$414$	0	0
$415$	−19.0260	−0.933950
$416$	−5.05796	−0.247987
$417$	0	0
$418$	0.0752330	0.00367977
$419$	−0.439830	−0.0214871	−0.0107436	−	0.999942i	$-0.503420\pi$
$419$	−0.439830	−0.0214871	−0.0107436	+	0.999942i	$0.503420\pi$
$420$	0	0
$421$	30.3492	1.47913	0.739564	−	0.673087i	$-0.235032\pi$
$421$	30.3492	1.47913	0.739564	+	0.673087i	$0.235032\pi$
$422$	−0.530951	−0.0258463
$423$	0	0
$424$	6.40162	0.310890
$425$	5.75311	0.279067
$426$	0	0
$427$	−9.42399	−0.456059
$428$	−21.8014	−1.05381
$429$	0	0
$430$	0.320020	0.0154327
$431$	−8.63673	−0.416017	−0.208008	−	0.978127i	$-0.566698\pi$
$431$	−8.63673	−0.416017	−0.208008	+	0.978127i	$0.566698\pi$
$432$	0	0
$433$	9.34002	0.448853	0.224426	−	0.974491i	$-0.427949\pi$
$433$	9.34002	0.448853	0.224426	+	0.974491i	$0.427949\pi$
$434$	−0.268546	−0.0128906
$435$	0	0
$436$	22.9865	1.10085
$437$	−24.1943	−1.15737
$438$	0	0
$439$	4.79533	0.228869	0.114434	−	0.993431i	$-0.463494\pi$
$439$	4.79533	0.228869	0.114434	+	0.993431i	$0.463494\pi$
$440$	−0.159105	−0.00758503
$441$	0	0
$442$	−2.16512	−0.102984
$443$	9.57969	0.455145	0.227572	−	0.973761i	$-0.426921\pi$
$443$	9.57969	0.455145	0.227572	+	0.973761i	$0.426921\pi$
$444$	0	0
$445$	−29.5439	−1.40051
$446$	4.38159	0.207474
$447$	0	0
$448$	−4.81291	−0.227389
$449$	22.2178	1.04852	0.524262	−	0.851557i	$-0.324341\pi$
$449$	22.2178	1.04852	0.524262	+	0.851557i	$0.324341\pi$
$450$	0	0
$451$	0.218652	0.0102959
$452$	−21.4651	−1.00963
$453$	0	0
$454$	−2.62219	−0.123066
$455$	−3.08813	−0.144774
$456$	0	0
$457$	32.8310	1.53577	0.767884	−	0.640589i	$-0.221310\pi$
$457$	32.8310	1.53577	0.767884	+	0.640589i	$0.221310\pi$
$458$	−1.41292	−0.0660216
$459$	0	0
$460$	24.6490	1.14927
$461$	7.65926	0.356727	0.178364	−	0.983965i	$-0.442920\pi$
$461$	7.65926	0.356727	0.178364	+	0.983965i	$0.442920\pi$
$462$	0	0
$463$	38.3105	1.78044	0.890220	−	0.455530i	$-0.150550\pi$
$463$	38.3105	1.78044	0.890220	+	0.455530i	$0.150550\pi$
$464$	16.6258	0.771834
$465$	0	0
$466$	−2.31667	−0.107318
$467$	35.9613	1.66409	0.832046	−	0.554707i	$-0.187170\pi$
$467$	35.9613	1.66409	0.832046	+	0.554707i	$0.187170\pi$
$468$	0	0
$469$	0.00811780	0.000374845 0
$470$	−10.7245	−0.494685
$471$	0	0
$472$	−14.1779	−0.652592
$473$	0.0150153	0.000690404 0
$474$	0	0
$475$	−5.62166	−0.257939
$476$	−8.66785	−0.397290
$477$	0	0
$478$	1.88370	0.0861582
$479$	−9.37274	−0.428251	−0.214126	−	0.976806i	$-0.568690\pi$
$479$	−9.37274	−0.428251	−0.214126	+	0.976806i	$0.568690\pi$
$480$	0	0
$481$	9.77981	0.445921
$482$	1.75217	0.0798089
$483$	0	0
$484$	20.4459	0.929358
$485$	22.4024	1.01724
$486$	0	0
$487$	41.8227	1.89517	0.947584	−	0.319507i	$-0.103517\pi$
$487$	41.8227	1.89517	0.947584	+	0.319507i	$0.103517\pi$
$488$	−13.6539	−0.618085
$489$	0	0
$490$	0.937397	0.0423473
$491$	−15.2407	−0.687802	−0.343901	−	0.939006i	$-0.611748\pi$
$491$	−15.2407	−0.687802	−0.343901	+	0.939006i	$0.611748\pi$
$492$	0	0
$493$	24.4220	1.09991
$494$	2.11565	0.0951875
$495$	0	0
$496$	2.27038	0.101943
$497$	4.91418	0.220431
$498$	0	0
$499$	18.6214	0.833608	0.416804	−	0.908996i	$-0.363150\pi$
$499$	18.6214	0.833608	0.416804	+	0.908996i	$0.363150\pi$
$500$	−17.4808	−0.781765
$501$	0	0
$502$	−5.60769	−0.250283
$503$	40.6766	1.81368	0.906841	−	0.421474i	$-0.138487\pi$
$503$	40.6766	1.81368	0.906841	+	0.421474i	$0.138487\pi$
$504$	0	0
$505$	29.7064	1.32192
$506$	−0.0876908	−0.00389833
$507$	0	0
$508$	−1.85904	−0.0824817
$509$	−33.9922	−1.50668	−0.753340	−	0.657631i	$-0.771558\pi$
$509$	−33.9922	−1.50668	−0.753340	+	0.657631i	$0.771558\pi$
$510$	0	0
$511$	−11.8395	−0.523750
$512$	−22.1779	−0.980136
$513$	0	0
$514$	0.939379	0.0414342
$515$	−8.31137	−0.366243
$516$	0	0
$517$	−0.503193	−0.0221304
$518$	−2.96865	−0.130435
$519$	0	0
$520$	−4.47424	−0.196208
$521$	1.31867	0.0577722	0.0288861	−	0.999583i	$-0.490804\pi$
$521$	1.31867	0.0577722	0.0288861	+	0.999583i	$0.490804\pi$
$522$	0	0
$523$	−30.7412	−1.34422	−0.672109	−	0.740452i	$-0.734611\pi$
$523$	−30.7412	−1.34422	−0.672109	+	0.740452i	$0.734611\pi$
$524$	3.04071	0.132834
$525$	0	0
$526$	−1.40734	−0.0613628
$527$	3.33501	0.145275
$528$	0	0
$529$	5.20068	0.226116
$530$	4.14181	0.179909
$531$	0	0
$532$	8.46980	0.367213
$533$	6.14878	0.266333
$534$	0	0
$535$	−29.2803	−1.26590
$536$	0.0117615	0.000508019 0
$537$	0	0
$538$	3.06471	0.132129
$539$	0.0439826	0.00189446
$540$	0	0
$541$	16.8366	0.723860	0.361930	−	0.932205i	$-0.382118\pi$
$541$	16.8366	0.723860	0.361930	+	0.932205i	$0.382118\pi$
$542$	3.06374	0.131599
$543$	0	0
$544$	−19.0670	−0.817490
$545$	30.8719	1.32241
$546$	0	0
$547$	16.8752	0.721531	0.360766	−	0.932656i	$-0.382515\pi$
$547$	16.8752	0.721531	0.360766	+	0.932656i	$0.382515\pi$
$548$	7.81424	0.333808
$549$	0	0
$550$	−0.0203753	−0.000868807 0
$551$	−23.8640	−1.01664
$552$	0	0
$553$	−1.44620	−0.0614988
$554$	7.32986	0.311416
$555$	0	0
$556$	3.41054	0.144639
$557$	31.2255	1.32307	0.661534	−	0.749915i	$-0.269906\pi$
$557$	31.2255	1.32307	0.661534	+	0.749915i	$0.269906\pi$
$558$	0	0
$559$	0.422249	0.0178592
$560$	−7.92509	−0.334896
$561$	0	0
$562$	−5.10441	−0.215317
$563$	12.2339	0.515595	0.257798	−	0.966199i	$-0.417003\pi$
$563$	12.2339	0.515595	0.257798	+	0.966199i	$0.417003\pi$
$564$	0	0
$565$	−28.8285	−1.21283
$566$	−9.01810	−0.379059
$567$	0	0
$568$	7.11991	0.298745
$569$	16.3199	0.684164	0.342082	−	0.939670i	$-0.388868\pi$
$569$	16.3199	0.684164	0.342082	+	0.939670i	$0.388868\pi$
$570$	0	0
$571$	0.774915	0.0324292	0.0162146	−	0.999869i	$-0.494839\pi$
$571$	0.774915	0.0324292	0.0162146	+	0.999869i	$0.494839\pi$
$572$	−0.101131	−0.00422851
$573$	0	0
$574$	−1.86645	−0.0779043
$575$	6.55255	0.273260
$576$	0	0
$577$	13.3038	0.553845	0.276922	−	0.960892i	$-0.410685\pi$
$577$	13.3038	0.553845	0.276922	+	0.960892i	$0.410685\pi$
$578$	−1.77931	−0.0740097
$579$	0	0
$580$	24.3124	1.00952
$581$	7.62022	0.316140
$582$	0	0
$583$	0.194333	0.00804846
$584$	−17.1537	−0.709825
$585$	0	0
$586$	−10.3556	−0.427787
$587$	−1.02046	−0.0421187	−0.0210594	−	0.999778i	$-0.506704\pi$
$587$	−1.02046	−0.0421187	−0.0210594	+	0.999778i	$0.506704\pi$
$588$	0	0
$589$	−3.25881	−0.134277
$590$	−9.17304	−0.377648
$591$	0	0
$592$	25.0980	1.03152
$593$	22.7116	0.932655	0.466327	−	0.884612i	$-0.345577\pi$
$593$	22.7116	0.932655	0.466327	+	0.884612i	$0.345577\pi$
$594$	0	0
$595$	−11.6413	−0.477248
$596$	−27.2737	−1.11717
$597$	0	0
$598$	−2.46598	−0.100841
$599$	−25.5601	−1.04436	−0.522180	−	0.852835i	$-0.674881\pi$
$599$	−25.5601	−1.04436	−0.522180	+	0.852835i	$0.674881\pi$
$600$	0	0
$601$	−26.9352	−1.09871	−0.549355	−	0.835589i	$-0.685127\pi$
$601$	−26.9352	−1.09871	−0.549355	+	0.835589i	$0.685127\pi$
$602$	−0.128173	−0.00522395
$603$	0	0
$604$	−13.5962	−0.553222
$605$	27.4597	1.11640
$606$	0	0
$607$	37.3334	1.51532	0.757659	−	0.652651i	$-0.226343\pi$
$607$	37.3334	1.51532	0.757659	+	0.652651i	$0.226343\pi$
$608$	18.6313	0.755600
$609$	0	0
$610$	−8.83402	−0.357679
$611$	−14.1504	−0.572465
$612$	0	0
$613$	24.7374	0.999135	0.499568	−	0.866275i	$-0.333492\pi$
$613$	24.7374	0.999135	0.499568	+	0.866275i	$0.333492\pi$
$614$	9.31674	0.375993
$615$	0	0
$616$	0.0637241	0.00256752
$617$	−1.99796	−0.0804347	−0.0402173	−	0.999191i	$-0.512805\pi$
$617$	−1.99796	−0.0804347	−0.0402173	+	0.999191i	$0.512805\pi$
$618$	0	0
$619$	−12.8044	−0.514654	−0.257327	−	0.966324i	$-0.582842\pi$
$619$	−12.8044	−0.514654	−0.257327	+	0.966324i	$0.582842\pi$
$620$	3.32004	0.133336
$621$	0	0
$622$	−0.296210	−0.0118769
$623$	11.8328	0.474071
$624$	0	0
$625$	−29.6470	−1.18588
$626$	−6.18696	−0.247280
$627$	0	0
$628$	−10.5167	−0.419663
$629$	36.8670	1.46998
$630$	0	0
$631$	48.1211	1.91567	0.957836	−	0.287315i	$-0.0927625\pi$
$631$	48.1211	1.91567	0.957836	+	0.287315i	$0.0927625\pi$
$632$	−2.09533	−0.0833477
$633$	0	0
$634$	5.74027	0.227975
$635$	−2.49678	−0.0990816
$636$	0	0
$637$	1.23685	0.0490056
$638$	−0.0864934	−0.00342431
$639$	0	0
$640$	−24.9323	−0.985535
$641$	27.7591	1.09642	0.548209	−	0.836341i	$-0.315310\pi$
$641$	27.7591	1.09642	0.548209	+	0.836341i	$0.315310\pi$
$642$	0	0
$643$	10.3622	0.408647	0.204323	−	0.978903i	$-0.434501\pi$
$643$	10.3622	0.408647	0.204323	+	0.978903i	$0.434501\pi$
$644$	−9.87232	−0.389024
$645$	0	0
$646$	7.97535	0.313786
$647$	−27.9465	−1.09869	−0.549345	−	0.835596i	$-0.685123\pi$
$647$	−27.9465	−1.09869	−0.549345	+	0.835596i	$0.685123\pi$
$648$	0	0
$649$	−0.430398	−0.0168946
$650$	−0.572981	−0.0224741
$651$	0	0
$652$	−30.3552	−1.18880
$653$	−28.5640	−1.11780	−0.558898	−	0.829237i	$-0.688775\pi$
$653$	−28.5640	−1.11780	−0.558898	+	0.829237i	$0.688775\pi$
$654$	0	0
$655$	4.08381	0.159568
$656$	15.7797	0.616092
$657$	0	0
$658$	4.29534	0.167450
$659$	21.4088	0.833967	0.416983	−	0.908914i	$-0.363087\pi$
$659$	21.4088	0.833967	0.416983	+	0.908914i	$0.363087\pi$
$660$	0	0
$661$	−9.97679	−0.388052	−0.194026	−	0.980996i	$-0.562155\pi$
$661$	−9.97679	−0.388052	−0.194026	+	0.980996i	$0.562155\pi$
$662$	3.30169	0.128324
$663$	0	0
$664$	11.0406	0.428457
$665$	11.3753	0.441116
$666$	0	0
$667$	27.8156	1.07702
$668$	−24.0392	−0.930105
$669$	0	0
$670$	0.00760961	0.000293985 0
$671$	−0.414491	−0.0160012
$672$	0	0
$673$	13.9270	0.536846	0.268423	−	0.963301i	$-0.413498\pi$
$673$	13.9270	0.536846	0.268423	+	0.963301i	$0.413498\pi$
$674$	−3.90110	−0.150265
$675$	0	0
$676$	21.3236	0.820139
$677$	−36.3413	−1.39671	−0.698355	−	0.715752i	$-0.746084\pi$
$677$	−36.3413	−1.39671	−0.698355	+	0.715752i	$0.746084\pi$
$678$	0	0
$679$	−8.97251	−0.344333
$680$	−16.8665	−0.646802
$681$	0	0
$682$	−0.0118113	−0.000452279 0
$683$	−29.4074	−1.12524	−0.562621	−	0.826715i	$-0.690207\pi$
$683$	−29.4074	−1.12524	−0.562621	+	0.826715i	$0.690207\pi$
$684$	0	0
$685$	10.4949	0.400989
$686$	−0.375443	−0.0143345
$687$	0	0
$688$	1.08362	0.0413127
$689$	5.46490	0.208196
$690$	0	0
$691$	−30.7917	−1.17137	−0.585685	−	0.810539i	$-0.699175\pi$
$691$	−30.7917	−1.17137	−0.585685	+	0.810539i	$0.699175\pi$
$692$	17.0481	0.648071
$693$	0	0
$694$	−1.40340	−0.0532722
$695$	4.58051	0.173749
$696$	0	0
$697$	23.1791	0.877970
$698$	−5.21471	−0.197380
$699$	0	0
$700$	−2.29388	−0.0867004
$701$	−18.0737	−0.682636	−0.341318	−	0.939948i	$-0.610873\pi$
$701$	−18.0737	−0.682636	−0.341318	+	0.939948i	$0.610873\pi$
$702$	0	0
$703$	−36.0246	−1.35869
$704$	−0.211684	−0.00797815
$705$	0	0
$706$	−11.0293	−0.415095
$707$	−11.8979	−0.447467
$708$	0	0
$709$	−16.5066	−0.619919	−0.309960	−	0.950750i	$-0.600316\pi$
$709$	−16.5066	−0.619919	−0.309960	+	0.950750i	$0.600316\pi$
$710$	4.60654	0.172880
$711$	0	0
$712$	17.1439	0.642496
$713$	3.79843	0.142252
$714$	0	0
$715$	−0.135824	−0.00507952
$716$	−8.31878	−0.310887
$717$	0	0
$718$	−1.50007	−0.0559821
$719$	28.8143	1.07459	0.537296	−	0.843394i	$-0.319446\pi$
$719$	28.8143	1.07459	0.537296	+	0.843394i	$0.319446\pi$
$720$	0	0
$721$	3.32884	0.123972
$722$	−0.659715	−0.0245520
$723$	0	0
$724$	19.7341	0.733413
$725$	6.46308	0.240033
$726$	0	0
$727$	37.2045	1.37984	0.689920	−	0.723886i	$-0.257646\pi$
$727$	37.2045	1.37984	0.689920	+	0.723886i	$0.257646\pi$
$728$	1.79200	0.0664161
$729$	0	0
$730$	−11.0983	−0.410768
$731$	1.59175	0.0588731
$732$	0	0
$733$	17.7329	0.654980	0.327490	−	0.944855i	$-0.393797\pi$
$733$	17.7329	0.654980	0.327490	+	0.944855i	$0.393797\pi$
$734$	5.95785	0.219908
$735$	0	0
$736$	−21.7165	−0.800480
$737$	0.000357042 0	1.31518e−5 0
$738$	0	0
$739$	−15.1053	−0.555658	−0.277829	−	0.960631i	$-0.589615\pi$
$739$	−15.1053	−0.555658	−0.277829	+	0.960631i	$0.589615\pi$
$740$	36.7015	1.34918
$741$	0	0
$742$	−1.65886	−0.0608988
$743$	47.6088	1.74660	0.873299	−	0.487184i	$-0.161976\pi$
$743$	47.6088	1.74660	0.873299	+	0.487184i	$0.161976\pi$
$744$	0	0
$745$	−36.6298	−1.34201
$746$	0.673325	0.0246522
$747$	0	0
$748$	−0.381234	−0.0139393
$749$	11.7272	0.428503
$750$	0	0
$751$	7.36875	0.268890	0.134445	−	0.990921i	$-0.457075\pi$
$751$	7.36875	0.268890	0.134445	+	0.990921i	$0.457075\pi$
$752$	−36.3143	−1.32425
$753$	0	0
$754$	−2.43231	−0.0885793
$755$	−18.2603	−0.664562
$756$	0	0
$757$	−2.31274	−0.0840581	−0.0420291	−	0.999116i	$-0.513382\pi$
$757$	−2.31274	−0.0840581	−0.0420291	+	0.999116i	$0.513382\pi$
$758$	−11.5232	−0.418543
$759$	0	0
$760$	16.4811	0.597834
$761$	34.4910	1.25030	0.625149	−	0.780506i	$-0.285038\pi$
$761$	34.4910	1.25030	0.625149	+	0.780506i	$0.285038\pi$
$762$	0	0
$763$	−12.3647	−0.447631
$764$	19.8865	0.719468
$765$	0	0
$766$	−5.24024	−0.189337
$767$	−12.1033	−0.437026
$768$	0	0
$769$	−42.3387	−1.52677	−0.763387	−	0.645941i	$-0.776465\pi$
$769$	−42.3387	−1.52677	−0.763387	+	0.645941i	$0.776465\pi$
$770$	0.0412291	0.00148580
$771$	0	0
$772$	−6.36339	−0.229023
$773$	35.6992	1.28401	0.642006	−	0.766700i	$-0.278102\pi$
$773$	35.6992	1.28401	0.642006	+	0.766700i	$0.278102\pi$
$774$	0	0
$775$	0.882582	0.0317033
$776$	−12.9998	−0.466666
$777$	0	0
$778$	11.7951	0.422876
$779$	−22.6495	−0.811501
$780$	0	0
$781$	0.216138	0.00773404
$782$	−9.29599	−0.332424
$783$	0	0
$784$	3.17413	0.113362
$785$	−14.1244	−0.504123
$786$	0	0
$787$	−43.8015	−1.56135	−0.780677	−	0.624935i	$-0.785125\pi$
$787$	−43.8015	−1.56135	−0.780677	+	0.624935i	$0.785125\pi$
$788$	23.4623	0.835808
$789$	0	0
$790$	−1.35567	−0.0482324
$791$	11.5463	0.410539
$792$	0	0
$793$	−11.6560	−0.413917
$794$	−1.55683	−0.0552497
$795$	0	0
$796$	−39.4329	−1.39766
$797$	31.2751	1.10782	0.553910	−	0.832577i	$-0.313135\pi$
$797$	31.2751	1.10782	0.553910	+	0.832577i	$0.313135\pi$
$798$	0	0
$799$	−53.3428	−1.88713
$800$	−5.04592	−0.178400
$801$	0	0
$802$	8.65240	0.305527
$803$	−0.520733	−0.0183763
$804$	0	0
$805$	−13.2590	−0.467317
$806$	−0.332150	−0.0116995
$807$	0	0
$808$	−17.2383	−0.606441
$809$	−15.2691	−0.536834	−0.268417	−	0.963303i	$-0.586501\pi$
$809$	−15.2691	−0.536834	−0.268417	+	0.963303i	$0.586501\pi$
$810$	0	0
$811$	12.6537	0.444332	0.222166	−	0.975009i	$-0.428687\pi$
$811$	12.6537	0.444332	0.222166	+	0.975009i	$0.428687\pi$
$812$	−9.73752	−0.341720
$813$	0	0
$814$	−0.130569	−0.00457643
$815$	−40.7684	−1.42806
$816$	0	0
$817$	−1.55538	−0.0544160
$818$	−3.99110	−0.139546
$819$	0	0
$820$	23.0751	0.805816
$821$	−34.4322	−1.20169	−0.600846	−	0.799365i	$-0.705169\pi$
$821$	−34.4322	−1.20169	−0.600846	+	0.799365i	$0.705169\pi$
$822$	0	0
$823$	−37.0847	−1.29269	−0.646346	−	0.763044i	$-0.723704\pi$
$823$	−37.0847	−1.29269	−0.646346	+	0.763044i	$0.723704\pi$
$824$	4.82299	0.168017
$825$	0	0
$826$	3.67395	0.127833
$827$	−40.9578	−1.42424	−0.712120	−	0.702057i	$-0.752265\pi$
$827$	−40.9578	−1.42424	−0.712120	+	0.702057i	$0.752265\pi$
$828$	0	0
$829$	38.7169	1.34469	0.672346	−	0.740237i	$-0.265287\pi$
$829$	38.7169	1.34469	0.672346	+	0.740237i	$0.265287\pi$
$830$	7.14317	0.247943
$831$	0	0
$832$	−5.95283	−0.206377
$833$	4.66254	0.161547
$834$	0	0
$835$	−32.2857	−1.11729
$836$	0.372524	0.0128840
$837$	0	0
$838$	0.165131	0.00570436
$839$	−41.1323	−1.42005	−0.710023	−	0.704179i	$-0.751315\pi$
$839$	−41.1323	−1.42005	−0.710023	+	0.704179i	$0.751315\pi$
$840$	0	0
$841$	−1.56420	−0.0539380
$842$	−11.3944	−0.392676
$843$	0	0
$844$	−2.62906	−0.0904959
$845$	28.6386	0.985197
$846$	0	0
$847$	−10.9981	−0.377898
$848$	14.0246	0.481607
$849$	0	0
$850$	−2.15996	−0.0740862
$851$	41.9899	1.43940
$852$	0	0
$853$	−17.8259	−0.610348	−0.305174	−	0.952297i	$-0.598715\pi$
$853$	−17.8259	−0.610348	−0.305174	+	0.952297i	$0.598715\pi$
$854$	3.53817	0.121074
$855$	0	0
$856$	16.9910	0.580739
$857$	52.7623	1.80233	0.901164	−	0.433479i	$-0.142714\pi$
$857$	52.7623	1.80233	0.901164	+	0.433479i	$0.142714\pi$
$858$	0	0
$859$	−52.2180	−1.78166	−0.890828	−	0.454341i	$-0.849875\pi$
$859$	−52.2180	−1.78166	−0.890828	+	0.454341i	$0.849875\pi$
$860$	1.58461	0.0540348
$861$	0	0
$862$	3.24260	0.110443
$863$	16.7285	0.569446	0.284723	−	0.958610i	$-0.408098\pi$
$863$	16.7285	0.569446	0.284723	+	0.958610i	$0.408098\pi$
$864$	0	0
$865$	22.8963	0.778499
$866$	−3.50664	−0.119161
$867$	0	0
$868$	−1.32973	−0.0451340
$869$	−0.0636077	−0.00215774
$870$	0	0
$871$	0.0100405	0.000340208 0
$872$	−17.9146	−0.606664
$873$	0	0
$874$	9.08359	0.307257
$875$	9.40311	0.317883
$876$	0	0
$877$	−3.86764	−0.130601	−0.0653005	−	0.997866i	$-0.520801\pi$
$877$	−3.86764	−0.130601	−0.0653005	+	0.997866i	$0.520801\pi$
$878$	−1.80037	−0.0607596
$879$	0	0
$880$	−0.348566	−0.0117501
$881$	−22.2193	−0.748586	−0.374293	−	0.927311i	$-0.622115\pi$
$881$	−22.2193	−0.748586	−0.374293	+	0.927311i	$0.622115\pi$
$882$	0	0
$883$	−3.75514	−0.126370	−0.0631852	−	0.998002i	$-0.520126\pi$
$883$	−3.75514	−0.126370	−0.0631852	+	0.998002i	$0.520126\pi$
$884$	−10.7208	−0.360579
$885$	0	0
$886$	−3.59662	−0.120831
$887$	−54.5326	−1.83102	−0.915512	−	0.402291i	$-0.868214\pi$
$887$	−54.5326	−1.83102	−0.915512	+	0.402291i	$0.868214\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	11.0920	0.371806
$891$	0	0
$892$	21.6959	0.726432
$893$	52.1240	1.74426
$894$	0	0
$895$	−11.1725	−0.373455
$896$	9.98578	0.333601
$897$	0	0
$898$	−8.34152	−0.278360
$899$	3.74657	0.124955
$900$	0	0
$901$	20.6010	0.686320
$902$	−0.0820915	−0.00273335
$903$	0	0
$904$	16.7288	0.556393
$905$	26.5038	0.881017
$906$	0	0
$907$	42.1002	1.39791	0.698957	−	0.715163i	$-0.253648\pi$
$907$	42.1002	1.39791	0.698957	+	0.715163i	$0.253648\pi$
$908$	−12.9841	−0.430891
$909$	0	0
$910$	1.15942	0.0384343
$911$	−25.1777	−0.834173	−0.417086	−	0.908867i	$-0.636949\pi$
$911$	−25.1777	−0.834173	−0.417086	+	0.908867i	$0.636949\pi$
$912$	0	0
$913$	0.335157	0.0110921
$914$	−12.3261	−0.407713
$915$	0	0
$916$	−6.99624	−0.231162
$917$	−1.63563	−0.0540133
$918$	0	0
$919$	3.16307	0.104340	0.0521699	−	0.998638i	$-0.483386\pi$
$919$	3.16307	0.104340	0.0521699	+	0.998638i	$0.483386\pi$
$920$	−19.2103	−0.633343
$921$	0	0
$922$	−2.87561	−0.0947033
$923$	6.07808	0.200063
$924$	0	0
$925$	9.75654	0.320793
$926$	−14.3834	−0.472668
$927$	0	0
$928$	−21.4200	−0.703145
$929$	−14.3412	−0.470519	−0.235260	−	0.971933i	$-0.575594\pi$
$929$	−14.3412	−0.470519	−0.235260	+	0.971933i	$0.575594\pi$
$930$	0	0
$931$	−4.55600	−0.149317
$932$	−11.4712	−0.375752
$933$	0	0
$934$	−13.5014	−0.441780
$935$	−0.512015	−0.0167447
$936$	0	0
$937$	42.7371	1.39616	0.698080	−	0.716020i	$-0.254038\pi$
$937$	42.7371	1.39616	0.698080	+	0.716020i	$0.254038\pi$
$938$	−0.00304777	−9.95133e−5 0
$939$	0	0
$940$	−53.1035	−1.73205
$941$	4.82197	0.157192	0.0785958	−	0.996907i	$-0.474956\pi$
$941$	4.82197	0.157192	0.0785958	+	0.996907i	$0.474956\pi$
$942$	0	0
$943$	26.4000	0.859701
$944$	−31.0609	−1.01095
$945$	0	0
$946$	−0.00563739	−0.000183287 0
$947$	−14.0659	−0.457080	−0.228540	−	0.973535i	$-0.573395\pi$
$947$	−14.0659	−0.457080	−0.228540	+	0.973535i	$0.573395\pi$
$948$	0	0
$949$	−14.6437	−0.475354
$950$	2.11061	0.0684773
$951$	0	0
$952$	6.75531	0.218941
$953$	−37.8805	−1.22707	−0.613534	−	0.789668i	$-0.710253\pi$
$953$	−37.8805	−1.22707	−0.613534	+	0.789668i	$0.710253\pi$
$954$	0	0
$955$	26.7085	0.864266
$956$	9.32731	0.301667
$957$	0	0
$958$	3.51893	0.113691
$959$	−4.20337	−0.135734
$960$	0	0
$961$	−30.4884	−0.983496
$962$	−3.67176	−0.118382
$963$	0	0
$964$	8.67602	0.279436
$965$	−8.54631	−0.275116
$966$	0	0
$967$	−4.19261	−0.134825	−0.0674126	−	0.997725i	$-0.521474\pi$
$967$	−4.19261	−0.134825	−0.0674126	+	0.997725i	$0.521474\pi$
$968$	−15.9345	−0.512156
$969$	0	0
$970$	−8.41081	−0.270055
$971$	−48.6618	−1.56163	−0.780815	−	0.624762i	$-0.785196\pi$
$971$	−48.6618	−1.56163	−0.780815	+	0.624762i	$0.785196\pi$
$972$	0	0
$973$	−1.83457	−0.0588135
$974$	−15.7020	−0.503126
$975$	0	0
$976$	−29.9129	−0.957489
$977$	9.44891	0.302297	0.151149	−	0.988511i	$-0.451703\pi$
$977$	9.44891	0.302297	0.151149	+	0.988511i	$0.451703\pi$
$978$	0	0
$979$	0.520437	0.0166332
$980$	4.64162	0.148271
$981$	0	0
$982$	5.72200	0.182596
$983$	41.1626	1.31288	0.656441	−	0.754377i	$-0.272061\pi$
$983$	41.1626	1.31288	0.656441	+	0.754377i	$0.272061\pi$
$984$	0	0
$985$	31.5109	1.00402
$986$	−9.16906	−0.292002
$987$	0	0
$988$	10.4758	0.333281
$989$	1.81294	0.0576481
$990$	0	0
$991$	−24.3889	−0.774740	−0.387370	−	0.921924i	$-0.626616\pi$
$991$	−24.3889	−0.774740	−0.387370	+	0.921924i	$0.626616\pi$
$992$	−2.92506	−0.0928707
$993$	0	0
$994$	−1.84499	−0.0585197
$995$	−52.9602	−1.67895
$996$	0	0
$997$	2.58240	0.0817855	0.0408927	−	0.999164i	$-0.486980\pi$
$997$	2.58240	0.0817855	0.0408927	+	0.999164i	$0.486980\pi$
$998$	−6.99127	−0.221305
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.u.1.8		18
3.2	odd	2		2667.2.a.p.1.11	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.p.1.11	✓	18	3.2	odd	2
8001.2.a.u.1.8		18	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)

\(f(q)\)	\(=\)	\(q-0.375443 q^{2} -1.85904 q^{4} -2.49678 q^{5} +1.00000 q^{7} +1.44885 q^{8} +O(q^{10})\)	\(q-0.375443 q^{2} -1.85904 q^{4} -2.49678 q^{5} +1.00000 q^{7} +1.44885 q^{8} +0.937397 q^{10} +0.0439826 q^{11} +1.23685 q^{13} -0.375443 q^{14} +3.17413 q^{16} +4.66254 q^{17} -4.55600 q^{19} +4.64162 q^{20} -0.0165129 q^{22} +5.31043 q^{23} +1.23390 q^{25} -0.464365 q^{26} -1.85904 q^{28} +5.23792 q^{29} +0.715277 q^{31} -4.08940 q^{32} -1.75052 q^{34} -2.49678 q^{35} +7.90706 q^{37} +1.71052 q^{38} -3.61746 q^{40} +4.97134 q^{41} +0.341392 q^{43} -0.0817655 q^{44} -1.99376 q^{46} -11.4407 q^{47} +1.00000 q^{49} -0.463260 q^{50} -2.29935 q^{52} +4.41842 q^{53} -0.109815 q^{55} +1.44885 q^{56} -1.96654 q^{58} -9.78565 q^{59} -9.42399 q^{61} -0.268546 q^{62} -4.81291 q^{64} -3.08813 q^{65} +0.00811780 q^{67} -8.66785 q^{68} +0.937397 q^{70} +4.91418 q^{71} -11.8395 q^{73} -2.96865 q^{74} +8.46980 q^{76} +0.0439826 q^{77} -1.44620 q^{79} -7.92509 q^{80} -1.86645 q^{82} +7.62022 q^{83} -11.6413 q^{85} -0.128173 q^{86} +0.0637241 q^{88} +11.8328 q^{89} +1.23685 q^{91} -9.87232 q^{92} +4.29534 q^{94} +11.3753 q^{95} -8.97251 q^{97} -0.375443 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8}+O(q^{10}) \) 18 * q + 6 * q^2 + 22 * q^4 + 10 * q^5 + 18 * q^7 + 21 * q^8	\( 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8} - 4 q^{10} + 9 q^{11} - 25 q^{13} + 6 q^{14} + 34 q^{16} + 17 q^{17} - 5 q^{19} + 21 q^{20} + 5 q^{22} + 14 q^{23} + 28 q^{25} + 8 q^{26} + 22 q^{28} + 17 q^{29} + 5 q^{31} + 53 q^{32} - 19 q^{34} + 10 q^{35} - 15 q^{37} + 22 q^{38} - q^{40} + 17 q^{41} + q^{43} + 33 q^{44} + 10 q^{46} + 31 q^{47} + 18 q^{49} + 35 q^{50} - 70 q^{52} + 35 q^{53} + 4 q^{55} + 21 q^{56} + 3 q^{58} + 46 q^{59} - 5 q^{61} + 10 q^{62} + 63 q^{64} + 12 q^{65} + 6 q^{67} + 56 q^{68} - 4 q^{70} + 22 q^{71} - 16 q^{73} - 18 q^{74} + 32 q^{76} + 9 q^{77} + 46 q^{79} + 30 q^{80} - 12 q^{82} + 46 q^{83} + 4 q^{85} - 18 q^{86} + 30 q^{88} + 42 q^{89} - 25 q^{91} + 48 q^{92} + 3 q^{94} + 2 q^{95} - 35 q^{97} + 6 q^{98}+O(q^{100}) \) 18 * q + 6 * q^2 + 22 * q^4 + 10 * q^5 + 18 * q^7 + 21 * q^8 - 4 * q^10 + 9 * q^11 - 25 * q^13 + 6 * q^14 + 34 * q^16 + 17 * q^17 - 5 * q^19 + 21 * q^20 + 5 * q^22 + 14 * q^23 + 28 * q^25 + 8 * q^26 + 22 * q^28 + 17 * q^29 + 5 * q^31 + 53 * q^32 - 19 * q^34 + 10 * q^35 - 15 * q^37 + 22 * q^38 - q^40 + 17 * q^41 + q^43 + 33 * q^44 + 10 * q^46 + 31 * q^47 + 18 * q^49 + 35 * q^50 - 70 * q^52 + 35 * q^53 + 4 * q^55 + 21 * q^56 + 3 * q^58 + 46 * q^59 - 5 * q^61 + 10 * q^62 + 63 * q^64 + 12 * q^65 + 6 * q^67 + 56 * q^68 - 4 * q^70 + 22 * q^71 - 16 * q^73 - 18 * q^74 + 32 * q^76 + 9 * q^77 + 46 * q^79 + 30 * q^80 - 12 * q^82 + 46 * q^83 + 4 * q^85 - 18 * q^86 + 30 * q^88 + 42 * q^89 - 25 * q^91 + 48 * q^92 + 3 * q^94 + 2 * q^95 - 35 * q^97 + 6 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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