LMFDB - Embedded newform 8001.2.a.u.1.14

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			1.14
Root			$1.77085$ 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)

$f(q)$	$=$	$q+1.77085 q^{2} +1.13590 q^{4} -1.38376 q^{5} +1.00000 q^{7} -1.53019 q^{8} +O(q^{10})$	\(q+1.77085 q^{2} +1.13590 q^{4} -1.38376 q^{5} +1.00000 q^{7} -1.53019 q^{8} -2.45042 q^{10} -3.51371 q^{11} +4.71315 q^{13} +1.77085 q^{14} -4.98153 q^{16} +4.68539 q^{17} -3.74479 q^{19} -1.57181 q^{20} -6.22224 q^{22} +2.00004 q^{23} -3.08521 q^{25} +8.34626 q^{26} +1.13590 q^{28} +4.37013 q^{29} -0.875947 q^{31} -5.76115 q^{32} +8.29711 q^{34} -1.38376 q^{35} -9.02904 q^{37} -6.63146 q^{38} +2.11741 q^{40} -1.57067 q^{41} -2.16122 q^{43} -3.99122 q^{44} +3.54176 q^{46} +12.9854 q^{47} +1.00000 q^{49} -5.46344 q^{50} +5.35366 q^{52} +10.7152 q^{53} +4.86212 q^{55} -1.53019 q^{56} +7.73884 q^{58} +0.801398 q^{59} +3.10892 q^{61} -1.55117 q^{62} -0.239057 q^{64} -6.52185 q^{65} +6.63235 q^{67} +5.32214 q^{68} -2.45042 q^{70} +8.91999 q^{71} +12.3019 q^{73} -15.9891 q^{74} -4.25371 q^{76} -3.51371 q^{77} +3.05893 q^{79} +6.89323 q^{80} -2.78142 q^{82} -4.04869 q^{83} -6.48345 q^{85} -3.82719 q^{86} +5.37664 q^{88} +11.2427 q^{89} +4.71315 q^{91} +2.27184 q^{92} +22.9952 q^{94} +5.18189 q^{95} -8.60399 q^{97} +1.77085 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.77085	1.25218	0.626089	−	0.779752i	$-0.284655\pi$
$2$	1.77085	1.25218	0.626089	+	0.779752i	$0.284655\pi$
$3$	0	0
$3$	0	0
$4$	1.13590	0.567950
$5$	−1.38376	−0.618835	−0.309418	−	0.950926i	$-0.600134\pi$
$5$	−1.38376	−0.618835	−0.309418	+	0.950926i	$0.600134\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.53019	−0.541004
$9$	0	0
$10$	−2.45042	−0.774892
$11$	−3.51371	−1.05942	−0.529712	−	0.848178i	$-0.677700\pi$
$11$	−3.51371	−1.05942	−0.529712	+	0.848178i	$0.677700\pi$
$12$	0	0
$13$	4.71315	1.30719	0.653596	−	0.756844i	$-0.273260\pi$
$13$	4.71315	1.30719	0.653596	+	0.756844i	$0.273260\pi$
$14$	1.77085	0.473279
$15$	0	0
$16$	−4.98153	−1.24538
$17$	4.68539	1.13637	0.568187	−	0.822899i	$-0.307645\pi$
$17$	4.68539	1.13637	0.568187	+	0.822899i	$0.307645\pi$
$18$	0	0
$19$	−3.74479	−0.859114	−0.429557	−	0.903040i	$-0.641330\pi$
$19$	−3.74479	−0.859114	−0.429557	+	0.903040i	$0.641330\pi$
$20$	−1.57181	−0.351467
$21$	0	0
$22$	−6.22224	−1.32659
$23$	2.00004	0.417036	0.208518	−	0.978018i	$-0.433136\pi$
$23$	2.00004	0.417036	0.208518	+	0.978018i	$0.433136\pi$
$24$	0	0
$25$	−3.08521	−0.617043
$26$	8.34626	1.63684
$27$	0	0
$28$	1.13590	0.214665
$29$	4.37013	0.811514	0.405757	−	0.913981i	$-0.367008\pi$
$29$	4.37013	0.811514	0.405757	+	0.913981i	$0.367008\pi$
$30$	0	0
$31$	−0.875947	−0.157325	−0.0786624	−	0.996901i	$-0.525065\pi$
$31$	−0.875947	−0.157325	−0.0786624	+	0.996901i	$0.525065\pi$
$32$	−5.76115	−1.01844
$33$	0	0
$34$	8.29711	1.42294
$35$	−1.38376	−0.233898
$36$	0	0
$37$	−9.02904	−1.48437	−0.742183	−	0.670198i	$-0.766209\pi$
$37$	−9.02904	−1.48437	−0.742183	+	0.670198i	$0.766209\pi$
$38$	−6.63146	−1.07576
$39$	0	0
$40$	2.11741	0.334792
$41$	−1.57067	−0.245298	−0.122649	−	0.992450i	$-0.539139\pi$
$41$	−1.57067	−0.245298	−0.122649	+	0.992450i	$0.539139\pi$
$42$	0	0
$43$	−2.16122	−0.329583	−0.164791	−	0.986328i	$-0.552695\pi$
$43$	−2.16122	−0.329583	−0.164791	+	0.986328i	$0.552695\pi$
$44$	−3.99122	−0.601700
$45$	0	0
$46$	3.54176	0.522204
$47$	12.9854	1.89412	0.947061	−	0.321054i	$-0.104037\pi$
$47$	12.9854	1.89412	0.947061	+	0.321054i	$0.104037\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−5.46344	−0.772648
$51$	0	0
$52$	5.35366	0.742419
$53$	10.7152	1.47184	0.735922	−	0.677066i	$-0.236749\pi$
$53$	10.7152	1.47184	0.735922	+	0.677066i	$0.236749\pi$
$54$	0	0
$55$	4.86212	0.655609
$56$	−1.53019	−0.204480
$57$	0	0
$58$	7.73884	1.01616
$59$	0.801398	0.104333	0.0521666	−	0.998638i	$-0.483387\pi$
$59$	0.801398	0.104333	0.0521666	+	0.998638i	$0.483387\pi$
$60$	0	0
$61$	3.10892	0.398056	0.199028	−	0.979994i	$-0.436222\pi$
$61$	3.10892	0.398056	0.199028	+	0.979994i	$0.436222\pi$
$62$	−1.55117	−0.196999
$63$	0	0
$64$	−0.239057	−0.0298822
$65$	−6.52185	−0.808936
$66$	0	0
$67$	6.63235	0.810271	0.405135	−	0.914257i	$-0.367224\pi$
$67$	6.63235	0.810271	0.405135	+	0.914257i	$0.367224\pi$
$68$	5.32214	0.645404
$69$	0	0
$70$	−2.45042	−0.292882
$71$	8.91999	1.05861	0.529304	−	0.848432i	$-0.322453\pi$
$71$	8.91999	1.05861	0.529304	+	0.848432i	$0.322453\pi$
$72$	0	0
$73$	12.3019	1.43983	0.719915	−	0.694062i	$-0.244181\pi$
$73$	12.3019	1.43983	0.719915	+	0.694062i	$0.244181\pi$
$74$	−15.9891	−1.85869
$75$	0	0
$76$	−4.25371	−0.487934
$77$	−3.51371	−0.400424
$78$	0	0
$79$	3.05893	0.344156	0.172078	−	0.985083i	$-0.444952\pi$
$79$	3.05893	0.344156	0.172078	+	0.985083i	$0.444952\pi$
$80$	6.89323	0.770687
$81$	0	0
$82$	−2.78142	−0.307156
$83$	−4.04869	−0.444401	−0.222201	−	0.975001i	$-0.571324\pi$
$83$	−4.04869	−0.444401	−0.222201	+	0.975001i	$0.571324\pi$
$84$	0	0
$85$	−6.48345	−0.703229
$86$	−3.82719	−0.412696
$87$	0	0
$88$	5.37664	0.573152
$89$	11.2427	1.19173	0.595864	−	0.803085i	$-0.296810\pi$
$89$	11.2427	1.19173	0.595864	+	0.803085i	$0.296810\pi$
$90$	0	0
$91$	4.71315	0.494072
$92$	2.27184	0.236856
$93$	0	0
$94$	22.9952	2.37178
$95$	5.18189	0.531650
$96$	0	0
$97$	−8.60399	−0.873603	−0.436802	−	0.899558i	$-0.643889\pi$
$97$	−8.60399	−0.873603	−0.436802	+	0.899558i	$0.643889\pi$
$98$	1.77085	0.178883
$99$	0	0
$100$	−3.50449	−0.350449
$101$	12.8002	1.27366	0.636832	−	0.771003i	$-0.280245\pi$
$101$	12.8002	1.27366	0.636832	+	0.771003i	$0.280245\pi$
$102$	0	0
$103$	−15.8265	−1.55943	−0.779715	−	0.626134i	$-0.784636\pi$
$103$	−15.8265	−1.55943	−0.779715	+	0.626134i	$0.784636\pi$
$104$	−7.21201	−0.707196
$105$	0	0
$106$	18.9750	1.84301
$107$	−4.31000	−0.416663	−0.208332	−	0.978058i	$-0.566803\pi$
$107$	−4.31000	−0.416663	−0.208332	+	0.978058i	$0.566803\pi$
$108$	0	0
$109$	−4.97624	−0.476638	−0.238319	−	0.971187i	$-0.576596\pi$
$109$	−4.97624	−0.476638	−0.238319	+	0.971187i	$0.576596\pi$
$110$	8.61008	0.820939
$111$	0	0
$112$	−4.98153	−0.470710
$113$	−3.01373	−0.283508	−0.141754	−	0.989902i	$-0.545274\pi$
$113$	−3.01373	−0.283508	−0.141754	+	0.989902i	$0.545274\pi$
$114$	0	0
$115$	−2.76757	−0.258077
$116$	4.96403	0.460899
$117$	0	0
$118$	1.41915	0.130644
$119$	4.68539	0.429509
$120$	0	0
$121$	1.34616	0.122378
$122$	5.50542	0.498437
$123$	0	0
$124$	−0.994988	−0.0893526
$125$	11.1880	1.00068
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	11.0990	0.981020
$129$	0	0
$130$	−11.5492	−1.01293
$131$	1.80215	0.157454	0.0787271	−	0.996896i	$-0.474914\pi$
$131$	1.80215	0.157454	0.0787271	+	0.996896i	$0.474914\pi$
$132$	0	0
$133$	−3.74479	−0.324715
$134$	11.7449	1.01460
$135$	0	0
$136$	−7.16954	−0.614783
$137$	−8.12738	−0.694369	−0.347184	−	0.937797i	$-0.612862\pi$
$137$	−8.12738	−0.694369	−0.347184	+	0.937797i	$0.612862\pi$
$138$	0	0
$139$	17.8232	1.51174	0.755871	−	0.654720i	$-0.227213\pi$
$139$	17.8232	1.51174	0.755871	+	0.654720i	$0.227213\pi$
$140$	−1.57181	−0.132842
$141$	0	0
$142$	15.7959	1.32557
$143$	−16.5606	−1.38487
$144$	0	0
$145$	−6.04721	−0.502193
$146$	21.7848	1.80292
$147$	0	0
$148$	−10.2561	−0.843045
$149$	−7.44846	−0.610201	−0.305101	−	0.952320i	$-0.598690\pi$
$149$	−7.44846	−0.610201	−0.305101	+	0.952320i	$0.598690\pi$
$150$	0	0
$151$	−5.07938	−0.413354	−0.206677	−	0.978409i	$-0.566265\pi$
$151$	−5.07938	−0.413354	−0.206677	+	0.978409i	$0.566265\pi$
$152$	5.73024	0.464784
$153$	0	0
$154$	−6.22224	−0.501403
$155$	1.21210	0.0973581
$156$	0	0
$157$	1.60558	0.128139	0.0640695	−	0.997945i	$-0.479592\pi$
$157$	1.60558	0.128139	0.0640695	+	0.997945i	$0.479592\pi$
$158$	5.41689	0.430945
$159$	0	0
$160$	7.97204	0.630245
$161$	2.00004	0.157625
$162$	0	0
$163$	16.6976	1.30786	0.653928	−	0.756557i	$-0.273120\pi$
$163$	16.6976	1.30786	0.653928	+	0.756557i	$0.273120\pi$
$164$	−1.78413	−0.139317
$165$	0	0
$166$	−7.16961	−0.556469
$167$	18.4120	1.42477	0.712383	−	0.701791i	$-0.247616\pi$
$167$	18.4120	1.42477	0.712383	+	0.701791i	$0.247616\pi$
$168$	0	0
$169$	9.21376	0.708750
$170$	−11.4812	−0.880567
$171$	0	0
$172$	−2.45493	−0.187187
$173$	−24.4792	−1.86112	−0.930560	−	0.366138i	$-0.880680\pi$
$173$	−24.4792	−1.86112	−0.930560	+	0.366138i	$0.880680\pi$
$174$	0	0
$175$	−3.08521	−0.233220
$176$	17.5037	1.31939
$177$	0	0
$178$	19.9092	1.49226
$179$	−8.54042	−0.638341	−0.319171	−	0.947697i	$-0.603404\pi$
$179$	−8.54042	−0.638341	−0.319171	+	0.947697i	$0.603404\pi$
$180$	0	0
$181$	−12.2354	−0.909453	−0.454726	−	0.890631i	$-0.650263\pi$
$181$	−12.2354	−0.909453	−0.454726	+	0.890631i	$0.650263\pi$
$182$	8.34626	0.618666
$183$	0	0
$184$	−3.06043	−0.225618
$185$	12.4940	0.918578
$186$	0	0
$187$	−16.4631	−1.20390
$188$	14.7502	1.07577
$189$	0	0
$190$	9.17633	0.665721
$191$	23.6641	1.71227	0.856137	−	0.516749i	$-0.172858\pi$
$191$	23.6641	1.71227	0.856137	+	0.516749i	$0.172858\pi$
$192$	0	0
$193$	19.3057	1.38965	0.694827	−	0.719177i	$-0.255481\pi$
$193$	19.3057	1.38965	0.694827	+	0.719177i	$0.255481\pi$
$194$	−15.2364	−1.09391
$195$	0	0
$196$	1.13590	0.0811357
$197$	15.5858	1.11044	0.555222	−	0.831702i	$-0.312633\pi$
$197$	15.5858	1.11044	0.555222	+	0.831702i	$0.312633\pi$
$198$	0	0
$199$	8.01019	0.567827	0.283914	−	0.958850i	$-0.408367\pi$
$199$	8.01019	0.567827	0.283914	+	0.958850i	$0.408367\pi$
$200$	4.72096	0.333822
$201$	0	0
$202$	22.6671	1.59485
$203$	4.37013	0.306723
$204$	0	0
$205$	2.17343	0.151799
$206$	−28.0263	−1.95269
$207$	0	0
$208$	−23.4787	−1.62795
$209$	13.1581	0.910166
$210$	0	0
$211$	28.6873	1.97492	0.987458	−	0.157881i	$-0.0504663\pi$
$211$	28.6873	1.97492	0.987458	+	0.157881i	$0.0504663\pi$
$212$	12.1714	0.835934
$213$	0	0
$214$	−7.63235	−0.521736
$215$	2.99060	0.203958
$216$	0	0
$217$	−0.875947	−0.0594632
$218$	−8.81217	−0.596835
$219$	0	0
$220$	5.52289	0.372353
$221$	22.0829	1.48546
$222$	0	0
$223$	20.1324	1.34817	0.674083	−	0.738656i	$-0.264539\pi$
$223$	20.1324	1.34817	0.674083	+	0.738656i	$0.264539\pi$
$224$	−5.76115	−0.384933
$225$	0	0
$226$	−5.33685	−0.355002
$227$	17.6437	1.17106	0.585528	−	0.810652i	$-0.300887\pi$
$227$	17.6437	1.17106	0.585528	+	0.810652i	$0.300887\pi$
$228$	0	0
$229$	−17.9149	−1.18385	−0.591925	−	0.805993i	$-0.701632\pi$
$229$	−17.9149	−1.18385	−0.591925	+	0.805993i	$0.701632\pi$
$230$	−4.90094	−0.323158
$231$	0	0
$232$	−6.68713	−0.439032
$233$	16.1065	1.05517	0.527585	−	0.849502i	$-0.323098\pi$
$233$	16.1065	1.05517	0.527585	+	0.849502i	$0.323098\pi$
$234$	0	0
$235$	−17.9687	−1.17215
$236$	0.910308	0.0592560
$237$	0	0
$238$	8.29711	0.537822
$239$	1.89730	0.122726	0.0613631	−	0.998116i	$-0.480455\pi$
$239$	1.89730	0.122726	0.0613631	+	0.998116i	$0.480455\pi$
$240$	0	0
$241$	2.23972	0.144273	0.0721365	−	0.997395i	$-0.477018\pi$
$241$	2.23972	0.144273	0.0721365	+	0.997395i	$0.477018\pi$
$242$	2.38385	0.153240
$243$	0	0
$244$	3.53142	0.226076
$245$	−1.38376	−0.0884050
$246$	0	0
$247$	−17.6498	−1.12303
$248$	1.34037	0.0851133
$249$	0	0
$250$	19.8122	1.25303
$251$	6.80187	0.429330	0.214665	−	0.976688i	$-0.431134\pi$
$251$	6.80187	0.429330	0.214665	+	0.976688i	$0.431134\pi$
$252$	0	0
$253$	−7.02755	−0.441818
$254$	1.77085	0.111113
$255$	0	0
$256$	20.1327	1.25829
$257$	19.0414	1.18777	0.593886	−	0.804549i	$-0.297593\pi$
$257$	19.0414	1.18777	0.593886	+	0.804549i	$0.297593\pi$
$258$	0	0
$259$	−9.02904	−0.561037
$260$	−7.40817	−0.459435
$261$	0	0
$262$	3.19132	0.197161
$263$	18.7401	1.15556	0.577782	−	0.816191i	$-0.303918\pi$
$263$	18.7401	1.15556	0.577782	+	0.816191i	$0.303918\pi$
$264$	0	0
$265$	−14.8272	−0.910829
$266$	−6.63146	−0.406601
$267$	0	0
$268$	7.53369	0.460193
$269$	8.31015	0.506679	0.253339	−	0.967377i	$-0.418471\pi$
$269$	8.31015	0.506679	0.253339	+	0.967377i	$0.418471\pi$
$270$	0	0
$271$	8.07002	0.490219	0.245110	−	0.969495i	$-0.421176\pi$
$271$	8.07002	0.490219	0.245110	+	0.969495i	$0.421176\pi$
$272$	−23.3404	−1.41522
$273$	0	0
$274$	−14.3923	−0.869473
$275$	10.8406	0.653710
$276$	0	0
$277$	1.87325	0.112553	0.0562764	−	0.998415i	$-0.482077\pi$
$277$	1.87325	0.112553	0.0562764	+	0.998415i	$0.482077\pi$
$278$	31.5621	1.89297
$279$	0	0
$280$	2.11741	0.126540
$281$	−19.1711	−1.14365	−0.571825	−	0.820376i	$-0.693764\pi$
$281$	−19.1711	−1.14365	−0.571825	+	0.820376i	$0.693764\pi$
$282$	0	0
$283$	−8.16883	−0.485587	−0.242793	−	0.970078i	$-0.578064\pi$
$283$	−8.16883	−0.485587	−0.242793	+	0.970078i	$0.578064\pi$
$284$	10.1322	0.601236
$285$	0	0
$286$	−29.3264	−1.73410
$287$	−1.57067	−0.0927138
$288$	0	0
$289$	4.95289	0.291347
$290$	−10.7087	−0.628835
$291$	0	0
$292$	13.9737	0.817751
$293$	−2.53599	−0.148154	−0.0740771	−	0.997253i	$-0.523601\pi$
$293$	−2.53599	−0.148154	−0.0740771	+	0.997253i	$0.523601\pi$
$294$	0	0
$295$	−1.10894	−0.0645650
$296$	13.8161	0.803047
$297$	0	0
$298$	−13.1901	−0.764081
$299$	9.42646	0.545146
$300$	0	0
$301$	−2.16122	−0.124571
$302$	−8.99481	−0.517593
$303$	0	0
$304$	18.6548	1.06993
$305$	−4.30199	−0.246331
$306$	0	0
$307$	12.9910	0.741436	0.370718	−	0.928745i	$-0.379112\pi$
$307$	12.9910	0.741436	0.370718	+	0.928745i	$0.379112\pi$
$308$	−3.99122	−0.227421
$309$	0	0
$310$	2.14644	0.121910
$311$	−19.1662	−1.08682	−0.543408	−	0.839469i	$-0.682866\pi$
$311$	−19.1662	−1.08682	−0.543408	+	0.839469i	$0.682866\pi$
$312$	0	0
$313$	−22.9716	−1.29843	−0.649216	−	0.760604i	$-0.724903\pi$
$313$	−22.9716	−1.29843	−0.649216	+	0.760604i	$0.724903\pi$
$314$	2.84323	0.160453
$315$	0	0
$316$	3.47464	0.195464
$317$	−6.48402	−0.364179	−0.182089	−	0.983282i	$-0.558286\pi$
$317$	−6.48402	−0.364179	−0.182089	+	0.983282i	$0.558286\pi$
$318$	0	0
$319$	−15.3554	−0.859737
$320$	0.330797	0.0184921
$321$	0	0
$322$	3.54176	0.197374
$323$	−17.5458	−0.976276
$324$	0	0
$325$	−14.5411	−0.806593
$326$	29.5689	1.63767
$327$	0	0
$328$	2.40343	0.132707
$329$	12.9854	0.715911
$330$	0	0
$331$	33.4592	1.83909	0.919543	−	0.392990i	$-0.128559\pi$
$331$	33.4592	1.83909	0.919543	+	0.392990i	$0.128559\pi$
$332$	−4.59890	−0.252398
$333$	0	0
$334$	32.6049	1.78406
$335$	−9.17757	−0.501424
$336$	0	0
$337$	−21.8026	−1.18766	−0.593832	−	0.804589i	$-0.702386\pi$
$337$	−21.8026	−1.18766	−0.593832	+	0.804589i	$0.702386\pi$
$338$	16.3162	0.887482
$339$	0	0
$340$	−7.36455	−0.399399
$341$	3.07783	0.166674
$342$	0	0
$343$	1.00000	0.0539949
$344$	3.30708	0.178306
$345$	0	0
$346$	−43.3490	−2.33045
$347$	36.6342	1.96663	0.983314	−	0.181918i	$-0.0582307\pi$
$347$	36.6342	1.96663	0.983314	+	0.181918i	$0.0582307\pi$
$348$	0	0
$349$	−13.6984	−0.733256	−0.366628	−	0.930368i	$-0.619488\pi$
$349$	−13.6984	−0.733256	−0.366628	+	0.930368i	$0.619488\pi$
$350$	−5.46344	−0.292033
$351$	0	0
$352$	20.2430	1.07896
$353$	28.7738	1.53148	0.765738	−	0.643153i	$-0.222374\pi$
$353$	28.7738	1.53148	0.765738	+	0.643153i	$0.222374\pi$
$354$	0	0
$355$	−12.3431	−0.655104
$356$	12.7706	0.676842
$357$	0	0
$358$	−15.1238	−0.799317
$359$	−32.9067	−1.73675	−0.868376	−	0.495907i	$-0.834836\pi$
$359$	−32.9067	−1.73675	−0.868376	+	0.495907i	$0.834836\pi$
$360$	0	0
$361$	−4.97653	−0.261922
$362$	−21.6671	−1.13880
$363$	0	0
$364$	5.35366	0.280608
$365$	−17.0229	−0.891018
$366$	0	0
$367$	3.44086	0.179611	0.0898057	−	0.995959i	$-0.471375\pi$
$367$	3.44086	0.179611	0.0898057	+	0.995959i	$0.471375\pi$
$368$	−9.96324	−0.519370
$369$	0	0
$370$	22.1250	1.15022
$371$	10.7152	0.556305
$372$	0	0
$373$	−16.1404	−0.835719	−0.417859	−	0.908512i	$-0.637220\pi$
$373$	−16.1404	−0.835719	−0.417859	+	0.908512i	$0.637220\pi$
$374$	−29.1537	−1.50750
$375$	0	0
$376$	−19.8702	−1.02473
$377$	20.5971	1.06080
$378$	0	0
$379$	−7.03351	−0.361287	−0.180644	−	0.983549i	$-0.557818\pi$
$379$	−7.03351	−0.361287	−0.180644	+	0.983549i	$0.557818\pi$
$380$	5.88610	0.301951
$381$	0	0
$382$	41.9055	2.14407
$383$	−5.54087	−0.283125	−0.141563	−	0.989929i	$-0.545213\pi$
$383$	−5.54087	−0.283125	−0.141563	+	0.989929i	$0.545213\pi$
$384$	0	0
$385$	4.86212	0.247797
$386$	34.1874	1.74009
$387$	0	0
$388$	−9.77328	−0.496163
$389$	−9.88826	−0.501354	−0.250677	−	0.968071i	$-0.580653\pi$
$389$	−9.88826	−0.501354	−0.250677	+	0.968071i	$0.580653\pi$
$390$	0	0
$391$	9.37095	0.473909
$392$	−1.53019	−0.0772862
$393$	0	0
$394$	27.6001	1.39047
$395$	−4.23282	−0.212976
$396$	0	0
$397$	−32.5678	−1.63453	−0.817265	−	0.576262i	$-0.804511\pi$
$397$	−32.5678	−1.63453	−0.817265	+	0.576262i	$0.804511\pi$
$398$	14.1848	0.711021
$399$	0	0
$400$	15.3691	0.768455
$401$	−0.767983	−0.0383512	−0.0191756	−	0.999816i	$-0.506104\pi$
$401$	−0.767983	−0.0383512	−0.0191756	+	0.999816i	$0.506104\pi$
$402$	0	0
$403$	−4.12847	−0.205654
$404$	14.5397	0.723377
$405$	0	0
$406$	7.73884	0.384072
$407$	31.7254	1.57257
$408$	0	0
$409$	13.7626	0.680517	0.340258	−	0.940332i	$-0.389485\pi$
$409$	13.7626	0.680517	0.340258	+	0.940332i	$0.389485\pi$
$410$	3.84881	0.190079
$411$	0	0
$412$	−17.9773	−0.885679
$413$	0.801398	0.0394342
$414$	0	0
$415$	5.60240	0.275011
$416$	−27.1532	−1.33129
$417$	0	0
$418$	23.3010	1.13969
$419$	−25.5079	−1.24614	−0.623072	−	0.782165i	$-0.714116\pi$
$419$	−25.5079	−1.24614	−0.623072	+	0.782165i	$0.714116\pi$
$420$	0	0
$421$	25.0885	1.22274	0.611370	−	0.791345i	$-0.290619\pi$
$421$	25.0885	1.22274	0.611370	+	0.791345i	$0.290619\pi$
$422$	50.8009	2.47295
$423$	0	0
$424$	−16.3963	−0.796273
$425$	−14.4554	−0.701192
$426$	0	0
$427$	3.10892	0.150451
$428$	−4.89572	−0.236644
$429$	0	0
$430$	5.29590	0.255391
$431$	1.42166	0.0684788	0.0342394	−	0.999414i	$-0.489099\pi$
$431$	1.42166	0.0684788	0.0342394	+	0.999414i	$0.489099\pi$
$432$	0	0
$433$	−0.288528	−0.0138658	−0.00693288	−	0.999976i	$-0.502207\pi$
$433$	−0.288528	−0.0138658	−0.00693288	+	0.999976i	$0.502207\pi$
$434$	−1.55117	−0.0744585
$435$	0	0
$436$	−5.65251	−0.270706
$437$	−7.48972	−0.358282
$438$	0	0
$439$	5.66246	0.270254	0.135127	−	0.990828i	$-0.456856\pi$
$439$	5.66246	0.270254	0.135127	+	0.990828i	$0.456856\pi$
$440$	−7.43997	−0.354687
$441$	0	0
$442$	39.1055	1.86006
$443$	35.1768	1.67130	0.835650	−	0.549263i	$-0.185091\pi$
$443$	35.1768	1.67130	0.835650	+	0.549263i	$0.185091\pi$
$444$	0	0
$445$	−15.5572	−0.737483
$446$	35.6514	1.68814
$447$	0	0
$448$	−0.239057	−0.0112944
$449$	10.2609	0.484244	0.242122	−	0.970246i	$-0.422157\pi$
$449$	10.2609	0.484244	0.242122	+	0.970246i	$0.422157\pi$
$450$	0	0
$451$	5.51889	0.259874
$452$	−3.42329	−0.161018
$453$	0	0
$454$	31.2444	1.46637
$455$	−6.52185	−0.305749
$456$	0	0
$457$	26.8801	1.25740	0.628700	−	0.777648i	$-0.283587\pi$
$457$	26.8801	1.25740	0.628700	+	0.777648i	$0.283587\pi$
$458$	−31.7246	−1.48239
$459$	0	0
$460$	−3.14368	−0.146575
$461$	−34.5198	−1.60775	−0.803873	−	0.594800i	$-0.797231\pi$
$461$	−34.5198	−1.60775	−0.803873	+	0.594800i	$0.797231\pi$
$462$	0	0
$463$	23.9679	1.11388	0.556941	−	0.830552i	$-0.311975\pi$
$463$	23.9679	1.11388	0.556941	+	0.830552i	$0.311975\pi$
$464$	−21.7700	−1.01065
$465$	0	0
$466$	28.5221	1.32126
$467$	−32.7795	−1.51686	−0.758428	−	0.651757i	$-0.774032\pi$
$467$	−32.7795	−1.51686	−0.758428	+	0.651757i	$0.774032\pi$
$468$	0	0
$469$	6.63235	0.306253
$470$	−31.8198	−1.46774
$471$	0	0
$472$	−1.22629	−0.0564446
$473$	7.59390	0.349168
$474$	0	0
$475$	11.5535	0.530110
$476$	5.32214	0.243940
$477$	0	0
$478$	3.35983	0.153675
$479$	−6.54768	−0.299171	−0.149586	−	0.988749i	$-0.547794\pi$
$479$	−6.54768	−0.299171	−0.149586	+	0.988749i	$0.547794\pi$
$480$	0	0
$481$	−42.5552	−1.94035
$482$	3.96620	0.180656
$483$	0	0
$484$	1.52911	0.0695048
$485$	11.9058	0.540617
$486$	0	0
$487$	−18.2161	−0.825451	−0.412726	−	0.910855i	$-0.635423\pi$
$487$	−18.2161	−0.825451	−0.412726	+	0.910855i	$0.635423\pi$
$488$	−4.75723	−0.215350
$489$	0	0
$490$	−2.45042	−0.110699
$491$	−43.7839	−1.97594	−0.987969	−	0.154650i	$-0.950575\pi$
$491$	−43.7839	−1.97594	−0.987969	+	0.154650i	$0.950575\pi$
$492$	0	0
$493$	20.4758	0.922183
$494$	−31.2550	−1.40623
$495$	0	0
$496$	4.36356	0.195930
$497$	8.91999	0.400116
$498$	0	0
$499$	11.9415	0.534574	0.267287	−	0.963617i	$-0.413873\pi$
$499$	11.9415	0.534574	0.267287	+	0.963617i	$0.413873\pi$
$500$	12.7084	0.568338
$501$	0	0
$502$	12.0451	0.537598
$503$	−12.6088	−0.562197	−0.281098	−	0.959679i	$-0.590699\pi$
$503$	−12.6088	−0.562197	−0.281098	+	0.959679i	$0.590699\pi$
$504$	0	0
$505$	−17.7123	−0.788188
$506$	−12.4447	−0.553235
$507$	0	0
$508$	1.13590	0.0503974
$509$	−7.41939	−0.328859	−0.164429	−	0.986389i	$-0.552578\pi$
$509$	−7.41939	−0.328859	−0.164429	+	0.986389i	$0.552578\pi$
$510$	0	0
$511$	12.3019	0.544205
$512$	13.4540	0.594588
$513$	0	0
$514$	33.7195	1.48730
$515$	21.9000	0.965031
$516$	0	0
$517$	−45.6271	−2.00668
$518$	−15.9891	−0.702519
$519$	0	0
$520$	9.97967	0.437638
$521$	−15.8190	−0.693044	−0.346522	−	0.938042i	$-0.612637\pi$
$521$	−15.8190	−0.693044	−0.346522	+	0.938042i	$0.612637\pi$
$522$	0	0
$523$	−17.9670	−0.785643	−0.392821	−	0.919615i	$-0.628501\pi$
$523$	−17.9670	−0.785643	−0.392821	+	0.919615i	$0.628501\pi$
$524$	2.04706	0.0894261
$525$	0	0
$526$	33.1858	1.44697
$527$	−4.10416	−0.178780
$528$	0	0
$529$	−18.9999	−0.826081
$530$	−26.2567	−1.14052
$531$	0	0
$532$	−4.25371	−0.184422
$533$	−7.40281	−0.320651
$534$	0	0
$535$	5.96399	0.257846
$536$	−10.1488	−0.438359
$537$	0	0
$538$	14.7160	0.634452
$539$	−3.51371	−0.151346
$540$	0	0
$541$	−23.0200	−0.989706	−0.494853	−	0.868977i	$-0.664778\pi$
$541$	−23.0200	−0.989706	−0.494853	+	0.868977i	$0.664778\pi$
$542$	14.2908	0.613842
$543$	0	0
$544$	−26.9933	−1.15733
$545$	6.88591	0.294960
$546$	0	0
$547$	29.1769	1.24752	0.623758	−	0.781617i	$-0.285605\pi$
$547$	29.1769	1.24752	0.623758	+	0.781617i	$0.285605\pi$
$548$	−9.23188	−0.394367
$549$	0	0
$550$	19.1970	0.818561
$551$	−16.3652	−0.697183
$552$	0	0
$553$	3.05893	0.130079
$554$	3.31724	0.140936
$555$	0	0
$556$	20.2454	0.858594
$557$	−35.5947	−1.50819	−0.754097	−	0.656763i	$-0.771925\pi$
$557$	−35.5947	−1.50819	−0.754097	+	0.656763i	$0.771925\pi$
$558$	0	0
$559$	−10.1861	−0.430828
$560$	6.89323	0.291292
$561$	0	0
$562$	−33.9490	−1.43205
$563$	16.3319	0.688307	0.344154	−	0.938913i	$-0.388166\pi$
$563$	16.3319	0.688307	0.344154	+	0.938913i	$0.388166\pi$
$564$	0	0
$565$	4.17027	0.175445
$566$	−14.4658	−0.608041
$567$	0	0
$568$	−13.6493	−0.572711
$569$	−39.0238	−1.63596	−0.817981	−	0.575245i	$-0.804907\pi$
$569$	−39.0238	−1.63596	−0.817981	+	0.575245i	$0.804907\pi$
$570$	0	0
$571$	−13.5080	−0.565291	−0.282646	−	0.959224i	$-0.591212\pi$
$571$	−13.5080	−0.565291	−0.282646	+	0.959224i	$0.591212\pi$
$572$	−18.8112	−0.786537
$573$	0	0
$574$	−2.78142	−0.116094
$575$	−6.17054	−0.257329
$576$	0	0
$577$	23.3145	0.970595	0.485298	−	0.874349i	$-0.338711\pi$
$577$	23.3145	0.970595	0.485298	+	0.874349i	$0.338711\pi$
$578$	8.77082	0.364818
$579$	0	0
$580$	−6.86902	−0.285221
$581$	−4.04869	−0.167968
$582$	0	0
$583$	−37.6501	−1.55931
$584$	−18.8243	−0.778953
$585$	0	0
$586$	−4.49085	−0.185515
$587$	−31.1722	−1.28661	−0.643307	−	0.765609i	$-0.722438\pi$
$587$	−31.1722	−1.28661	−0.643307	+	0.765609i	$0.722438\pi$
$588$	0	0
$589$	3.28024	0.135160
$590$	−1.96376	−0.0808469
$591$	0	0
$592$	44.9785	1.84860
$593$	4.36484	0.179242	0.0896211	−	0.995976i	$-0.471434\pi$
$593$	4.36484	0.179242	0.0896211	+	0.995976i	$0.471434\pi$
$594$	0	0
$595$	−6.48345	−0.265795
$596$	−8.46070	−0.346564
$597$	0	0
$598$	16.6928	0.682620
$599$	11.6431	0.475724	0.237862	−	0.971299i	$-0.423553\pi$
$599$	11.6431	0.475724	0.237862	+	0.971299i	$0.423553\pi$
$600$	0	0
$601$	−45.2104	−1.84417	−0.922086	−	0.386986i	$-0.873516\pi$
$601$	−45.2104	−1.84417	−0.922086	+	0.386986i	$0.873516\pi$
$602$	−3.82719	−0.155985
$603$	0	0
$604$	−5.76967	−0.234765
$605$	−1.86276	−0.0757321
$606$	0	0
$607$	−17.6142	−0.714938	−0.357469	−	0.933925i	$-0.616360\pi$
$607$	−17.6142	−0.714938	−0.357469	+	0.933925i	$0.616360\pi$
$608$	21.5743	0.874954
$609$	0	0
$610$	−7.61817	−0.308451
$611$	61.2023	2.47598
$612$	0	0
$613$	−3.47398	−0.140313	−0.0701563	−	0.997536i	$-0.522350\pi$
$613$	−3.47398	−0.140313	−0.0701563	+	0.997536i	$0.522350\pi$
$614$	23.0051	0.928410
$615$	0	0
$616$	5.37664	0.216631
$617$	12.7145	0.511867	0.255934	−	0.966694i	$-0.417617\pi$
$617$	12.7145	0.511867	0.255934	+	0.966694i	$0.417617\pi$
$618$	0	0
$619$	−17.1760	−0.690360	−0.345180	−	0.938536i	$-0.612182\pi$
$619$	−17.1760	−0.690360	−0.345180	+	0.938536i	$0.612182\pi$
$620$	1.37682	0.0552945
$621$	0	0
$622$	−33.9404	−1.36089
$623$	11.2427	0.450431
$624$	0	0
$625$	−0.0553790	−0.00221516
$626$	−40.6792	−1.62587
$627$	0	0
$628$	1.82378	0.0727766
$629$	−42.3046	−1.68679
$630$	0	0
$631$	15.7815	0.628252	0.314126	−	0.949381i	$-0.398289\pi$
$631$	15.7815	0.628252	0.314126	+	0.949381i	$0.398289\pi$
$632$	−4.68074	−0.186190
$633$	0	0
$634$	−11.4822	−0.456017
$635$	−1.38376	−0.0549128
$636$	0	0
$637$	4.71315	0.186742
$638$	−27.1920	−1.07654
$639$	0	0
$640$	−15.3583	−0.607090
$641$	−0.538037	−0.0212512	−0.0106256	−	0.999944i	$-0.503382\pi$
$641$	−0.538037	−0.0212512	−0.0106256	+	0.999944i	$0.503382\pi$
$642$	0	0
$643$	−24.7771	−0.977112	−0.488556	−	0.872532i	$-0.662476\pi$
$643$	−24.7771	−0.977112	−0.488556	+	0.872532i	$0.662476\pi$
$644$	2.27184	0.0895231
$645$	0	0
$646$	−31.0710	−1.22247
$647$	14.8933	0.585517	0.292758	−	0.956186i	$-0.405427\pi$
$647$	14.8933	0.585517	0.292758	+	0.956186i	$0.405427\pi$
$648$	0	0
$649$	−2.81588	−0.110533
$650$	−25.7500	−1.01000
$651$	0	0
$652$	18.9668	0.742796
$653$	5.43582	0.212720	0.106360	−	0.994328i	$-0.466080\pi$
$653$	5.43582	0.212720	0.106360	+	0.994328i	$0.466080\pi$
$654$	0	0
$655$	−2.49373	−0.0974382
$656$	7.82435	0.305490
$657$	0	0
$658$	22.9952	0.896448
$659$	−11.5047	−0.448158	−0.224079	−	0.974571i	$-0.571937\pi$
$659$	−11.5047	−0.448158	−0.224079	+	0.974571i	$0.571937\pi$
$660$	0	0
$661$	3.46362	0.134719	0.0673596	−	0.997729i	$-0.478543\pi$
$661$	3.46362	0.134719	0.0673596	+	0.997729i	$0.478543\pi$
$662$	59.2512	2.30286
$663$	0	0
$664$	6.19526	0.240423
$665$	5.18189	0.200945
$666$	0	0
$667$	8.74043	0.338431
$668$	20.9142	0.809195
$669$	0	0
$670$	−16.2521	−0.627872
$671$	−10.9238	−0.421710
$672$	0	0
$673$	−25.4769	−0.982063	−0.491032	−	0.871142i	$-0.663380\pi$
$673$	−25.4769	−0.982063	−0.491032	+	0.871142i	$0.663380\pi$
$674$	−38.6091	−1.48717
$675$	0	0
$676$	10.4659	0.402535
$677$	13.2125	0.507799	0.253900	−	0.967231i	$-0.418287\pi$
$677$	13.2125	0.507799	0.253900	+	0.967231i	$0.418287\pi$
$678$	0	0
$679$	−8.60399	−0.330191
$680$	9.92090	0.380449
$681$	0	0
$682$	5.45036	0.208705
$683$	31.7115	1.21341	0.606704	−	0.794928i	$-0.292491\pi$
$683$	31.7115	1.21341	0.606704	+	0.794928i	$0.292491\pi$
$684$	0	0
$685$	11.2463	0.429700
$686$	1.77085	0.0676113
$687$	0	0
$688$	10.7662	0.410457
$689$	50.5023	1.92398
$690$	0	0
$691$	−31.5039	−1.19847	−0.599233	−	0.800575i	$-0.704528\pi$
$691$	−31.5039	−1.19847	−0.599233	+	0.800575i	$0.704528\pi$
$692$	−27.8059	−1.05702
$693$	0	0
$694$	64.8736	2.46257
$695$	−24.6630	−0.935520
$696$	0	0
$697$	−7.35921	−0.278750
$698$	−24.2577	−0.918168
$699$	0	0
$700$	−3.50449	−0.132457
$701$	−0.903370	−0.0341198	−0.0170599	−	0.999854i	$-0.505431\pi$
$701$	−0.903370	−0.0341198	−0.0170599	+	0.999854i	$0.505431\pi$
$702$	0	0
$703$	33.8119	1.27524
$704$	0.839978	0.0316579
$705$	0	0
$706$	50.9540	1.91768
$707$	12.8002	0.481399
$708$	0	0
$709$	46.7863	1.75710	0.878549	−	0.477653i	$-0.158512\pi$
$709$	46.7863	1.75710	0.878549	+	0.477653i	$0.158512\pi$
$710$	−21.8578	−0.820307
$711$	0	0
$712$	−17.2035	−0.644729
$713$	−1.75193	−0.0656101
$714$	0	0
$715$	22.9159	0.857006
$716$	−9.70106	−0.362546
$717$	0	0
$718$	−58.2728	−2.17472
$719$	−5.39158	−0.201072	−0.100536	−	0.994933i	$-0.532056\pi$
$719$	−5.39158	−0.201072	−0.100536	+	0.994933i	$0.532056\pi$
$720$	0	0
$721$	−15.8265	−0.589409
$722$	−8.81267	−0.327974
$723$	0	0
$724$	−13.8982	−0.516524
$725$	−13.4828	−0.500739
$726$	0	0
$727$	−2.00859	−0.0744946	−0.0372473	−	0.999306i	$-0.511859\pi$
$727$	−2.00859	−0.0744946	−0.0372473	+	0.999306i	$0.511859\pi$
$728$	−7.21201	−0.267295
$729$	0	0
$730$	−30.1449	−1.11571
$731$	−10.1262	−0.374530
$732$	0	0
$733$	−51.9571	−1.91908	−0.959540	−	0.281571i	$-0.909144\pi$
$733$	−51.9571	−1.91908	−0.959540	+	0.281571i	$0.909144\pi$
$734$	6.09324	0.224906
$735$	0	0
$736$	−11.5225	−0.424725
$737$	−23.3042	−0.858420
$738$	0	0
$739$	49.8689	1.83446	0.917229	−	0.398360i	$-0.130421\pi$
$739$	49.8689	1.83446	0.917229	+	0.398360i	$0.130421\pi$
$740$	14.1919	0.521706
$741$	0	0
$742$	18.9750	0.696593
$743$	37.0545	1.35940	0.679699	−	0.733491i	$-0.262110\pi$
$743$	37.0545	1.35940	0.679699	+	0.733491i	$0.262110\pi$
$744$	0	0
$745$	10.3069	0.377614
$746$	−28.5822	−1.04647
$747$	0	0
$748$	−18.7004	−0.683756
$749$	−4.31000	−0.157484
$750$	0	0
$751$	17.8645	0.651886	0.325943	−	0.945389i	$-0.394318\pi$
$751$	17.8645	0.651886	0.325943	+	0.945389i	$0.394318\pi$
$752$	−64.6874	−2.35891
$753$	0	0
$754$	36.4743	1.32832
$755$	7.02864	0.255798
$756$	0	0
$757$	18.7291	0.680722	0.340361	−	0.940295i	$-0.389451\pi$
$757$	18.7291	0.680722	0.340361	+	0.940295i	$0.389451\pi$
$758$	−12.4553	−0.452396
$759$	0	0
$760$	−7.92927	−0.287625
$761$	36.5839	1.32616	0.663082	−	0.748546i	$-0.269248\pi$
$761$	36.5839	1.32616	0.663082	+	0.748546i	$0.269248\pi$
$762$	0	0
$763$	−4.97624	−0.180152
$764$	26.8800	0.972486
$765$	0	0
$766$	−9.81204	−0.354523
$767$	3.77711	0.136383
$768$	0	0
$769$	24.9186	0.898589	0.449295	−	0.893384i	$-0.351675\pi$
$769$	24.9186	0.898589	0.449295	+	0.893384i	$0.351675\pi$
$770$	8.61008	0.310286
$771$	0	0
$772$	21.9293	0.789254
$773$	−30.9862	−1.11450	−0.557249	−	0.830346i	$-0.688143\pi$
$773$	−30.9862	−1.11450	−0.557249	+	0.830346i	$0.688143\pi$
$774$	0	0
$775$	2.70249	0.0970761
$776$	13.1657	0.472623
$777$	0	0
$778$	−17.5106	−0.627785
$779$	5.88184	0.210739
$780$	0	0
$781$	−31.3423	−1.12151
$782$	16.5945	0.593419
$783$	0	0
$784$	−4.98153	−0.177912
$785$	−2.22173	−0.0792970
$786$	0	0
$787$	21.0310	0.749675	0.374837	−	0.927091i	$-0.377699\pi$
$787$	21.0310	0.749675	0.374837	+	0.927091i	$0.377699\pi$
$788$	17.7039	0.630676
$789$	0	0
$790$	−7.49567	−0.266684
$791$	−3.01373	−0.107156
$792$	0	0
$793$	14.6528	0.520336
$794$	−57.6726	−2.04672
$795$	0	0
$796$	9.09877	0.322497
$797$	37.2731	1.32028	0.660141	−	0.751142i	$-0.270496\pi$
$797$	37.2731	1.32028	0.660141	+	0.751142i	$0.270496\pi$
$798$	0	0
$799$	60.8419	2.15243
$800$	17.7744	0.628420
$801$	0	0
$802$	−1.35998	−0.0480226
$803$	−43.2254	−1.52539
$804$	0	0
$805$	−2.76757	−0.0975439
$806$	−7.31089	−0.257515
$807$	0	0
$808$	−19.5867	−0.689056
$809$	16.9337	0.595357	0.297678	−	0.954666i	$-0.403788\pi$
$809$	16.9337	0.595357	0.297678	+	0.954666i	$0.403788\pi$
$810$	0	0
$811$	20.6275	0.724329	0.362165	−	0.932114i	$-0.382038\pi$
$811$	20.6275	0.724329	0.362165	+	0.932114i	$0.382038\pi$
$812$	4.96403	0.174203
$813$	0	0
$814$	56.1809	1.96914
$815$	−23.1054	−0.809347
$816$	0	0
$817$	8.09332	0.283149
$818$	24.3715	0.852128
$819$	0	0
$820$	2.46880	0.0862142
$821$	−16.7244	−0.583685	−0.291842	−	0.956466i	$-0.594268\pi$
$821$	−16.7244	−0.583685	−0.291842	+	0.956466i	$0.594268\pi$
$822$	0	0
$823$	−4.86071	−0.169434	−0.0847168	−	0.996405i	$-0.526999\pi$
$823$	−4.86071	−0.169434	−0.0847168	+	0.996405i	$0.526999\pi$
$824$	24.2175	0.843658
$825$	0	0
$826$	1.41915	0.0493787
$827$	37.6413	1.30892	0.654458	−	0.756098i	$-0.272897\pi$
$827$	37.6413	1.30892	0.654458	+	0.756098i	$0.272897\pi$
$828$	0	0
$829$	−20.1585	−0.700133	−0.350067	−	0.936725i	$-0.613841\pi$
$829$	−20.1585	−0.700133	−0.350067	+	0.936725i	$0.613841\pi$
$830$	9.92100	0.344363
$831$	0	0
$832$	−1.12671	−0.0390617
$833$	4.68539	0.162339
$834$	0	0
$835$	−25.4778	−0.881695
$836$	14.9463	0.516929
$837$	0	0
$838$	−45.1706	−1.56039
$839$	−29.9945	−1.03553	−0.517763	−	0.855524i	$-0.673235\pi$
$839$	−29.9945	−1.03553	−0.517763	+	0.855524i	$0.673235\pi$
$840$	0	0
$841$	−9.90193	−0.341446
$842$	44.4279	1.53109
$843$	0	0
$844$	32.5859	1.12165
$845$	−12.7496	−0.438600
$846$	0	0
$847$	1.34616	0.0462547
$848$	−53.3780	−1.83301
$849$	0	0
$850$	−25.5984	−0.878017
$851$	−18.0584	−0.619034
$852$	0	0
$853$	−17.6458	−0.604182	−0.302091	−	0.953279i	$-0.597685\pi$
$853$	−17.6458	−0.604182	−0.302091	+	0.953279i	$0.597685\pi$
$854$	5.50542	0.188392
$855$	0	0
$856$	6.59511	0.225416
$857$	33.4988	1.14430	0.572149	−	0.820149i	$-0.306110\pi$
$857$	33.4988	1.14430	0.572149	+	0.820149i	$0.306110\pi$
$858$	0	0
$859$	3.95415	0.134914	0.0674569	−	0.997722i	$-0.478511\pi$
$859$	3.95415	0.134914	0.0674569	+	0.997722i	$0.478511\pi$
$860$	3.39703	0.115838
$861$	0	0
$862$	2.51754	0.0857476
$863$	28.8938	0.983557	0.491778	−	0.870720i	$-0.336347\pi$
$863$	28.8938	0.983557	0.491778	+	0.870720i	$0.336347\pi$
$864$	0	0
$865$	33.8733	1.15173
$866$	−0.510938	−0.0173624
$867$	0	0
$868$	−0.994988	−0.0337721
$869$	−10.7482	−0.364607
$870$	0	0
$871$	31.2592	1.05918
$872$	7.61459	0.257863
$873$	0	0
$874$	−13.2632	−0.448633
$875$	11.1880	0.378223
$876$	0	0
$877$	−2.53706	−0.0856703	−0.0428352	−	0.999082i	$-0.513639\pi$
$877$	−2.53706	−0.0856703	−0.0428352	+	0.999082i	$0.513639\pi$
$878$	10.0273	0.338407
$879$	0	0
$880$	−24.2208	−0.816484
$881$	−7.31984	−0.246612	−0.123306	−	0.992369i	$-0.539350\pi$
$881$	−7.31984	−0.246612	−0.123306	+	0.992369i	$0.539350\pi$
$882$	0	0
$883$	−38.0071	−1.27904	−0.639521	−	0.768774i	$-0.720867\pi$
$883$	−38.0071	−1.27904	−0.639521	+	0.768774i	$0.720867\pi$
$884$	25.0840	0.843666
$885$	0	0
$886$	62.2927	2.09276
$887$	20.9072	0.701993	0.350997	−	0.936377i	$-0.385843\pi$
$887$	20.9072	0.701993	0.350997	+	0.936377i	$0.385843\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−27.5495	−0.923460
$891$	0	0
$892$	22.8684	0.765691
$893$	−48.6278	−1.62727
$894$	0	0
$895$	11.8179	0.395028
$896$	11.0990	0.370791
$897$	0	0
$898$	18.1706	0.606359
$899$	−3.82801	−0.127671
$900$	0	0
$901$	50.2048	1.67257
$902$	9.77311	0.325409
$903$	0	0
$904$	4.61158	0.153379
$905$	16.9309	0.562801
$906$	0	0
$907$	26.8915	0.892918	0.446459	−	0.894804i	$-0.352685\pi$
$907$	26.8915	0.892918	0.446459	+	0.894804i	$0.352685\pi$
$908$	20.0415	0.665101
$909$	0	0
$910$	−11.5492	−0.382852
$911$	−39.7733	−1.31775	−0.658875	−	0.752253i	$-0.728967\pi$
$911$	−39.7733	−1.31775	−0.658875	+	0.752253i	$0.728967\pi$
$912$	0	0
$913$	14.2259	0.470809
$914$	47.6006	1.57449
$915$	0	0
$916$	−20.3495	−0.672368
$917$	1.80215	0.0595121
$918$	0	0
$919$	−13.4951	−0.445161	−0.222580	−	0.974914i	$-0.571448\pi$
$919$	−13.4951	−0.445161	−0.222580	+	0.974914i	$0.571448\pi$
$920$	4.23490	0.139620
$921$	0	0
$922$	−61.1293	−2.01319
$923$	42.0412	1.38380
$924$	0	0
$925$	27.8565	0.915917
$926$	42.4435	1.39478
$927$	0	0
$928$	−25.1770	−0.826476
$929$	−19.7078	−0.646592	−0.323296	−	0.946298i	$-0.604791\pi$
$929$	−19.7078	−0.646592	−0.323296	+	0.946298i	$0.604791\pi$
$930$	0	0
$931$	−3.74479	−0.122731
$932$	18.2953	0.599284
$933$	0	0
$934$	−58.0475	−1.89937
$935$	22.7810	0.745017
$936$	0	0
$937$	−2.82515	−0.0922935	−0.0461467	−	0.998935i	$-0.514694\pi$
$937$	−2.82515	−0.0922935	−0.0461467	+	0.998935i	$0.514694\pi$
$938$	11.7449	0.383484
$939$	0	0
$940$	−20.4107	−0.665722
$941$	−7.68921	−0.250661	−0.125331	−	0.992115i	$-0.539999\pi$
$941$	−7.68921	−0.250661	−0.125331	+	0.992115i	$0.539999\pi$
$942$	0	0
$943$	−3.14140	−0.102298
$944$	−3.99219	−0.129935
$945$	0	0
$946$	13.4476	0.437220
$947$	−38.9226	−1.26482	−0.632408	−	0.774636i	$-0.717933\pi$
$947$	−38.9226	−1.26482	−0.632408	+	0.774636i	$0.717933\pi$
$948$	0	0
$949$	57.9807	1.88213
$950$	20.4595	0.663793
$951$	0	0
$952$	−7.16954	−0.232366
$953$	1.05657	0.0342257	0.0171129	−	0.999854i	$-0.494553\pi$
$953$	1.05657	0.0342257	0.0171129	+	0.999854i	$0.494553\pi$
$954$	0	0
$955$	−32.7454	−1.05962
$956$	2.15514	0.0697024
$957$	0	0
$958$	−11.5949	−0.374616
$959$	−8.12738	−0.262447
$960$	0	0
$961$	−30.2327	−0.975249
$962$	−75.3588	−2.42966
$963$	0	0
$964$	2.54410	0.0819399
$965$	−26.7144	−0.859967
$966$	0	0
$967$	−17.3565	−0.558148	−0.279074	−	0.960270i	$-0.590027\pi$
$967$	−17.3565	−0.558148	−0.279074	+	0.960270i	$0.590027\pi$
$968$	−2.05988	−0.0662071
$969$	0	0
$970$	21.0834	0.676948
$971$	−0.389374	−0.0124956	−0.00624780	−	0.999980i	$-0.501989\pi$
$971$	−0.389374	−0.0124956	−0.00624780	+	0.999980i	$0.501989\pi$
$972$	0	0
$973$	17.8232	0.571385
$974$	−32.2580	−1.03361
$975$	0	0
$976$	−15.4872	−0.495732
$977$	44.1149	1.41136	0.705680	−	0.708531i	$-0.250642\pi$
$977$	44.1149	1.41136	0.705680	+	0.708531i	$0.250642\pi$
$978$	0	0
$979$	−39.5037	−1.26254
$980$	−1.57181	−0.0502096
$981$	0	0
$982$	−77.5346	−2.47423
$983$	30.6578	0.977833	0.488917	−	0.872331i	$-0.337392\pi$
$983$	30.6578	0.977833	0.488917	+	0.872331i	$0.337392\pi$
$984$	0	0
$985$	−21.5670	−0.687182
$986$	36.2595	1.15474
$987$	0	0
$988$	−20.0484	−0.637823
$989$	−4.32252	−0.137448
$990$	0	0
$991$	13.2935	0.422281	0.211140	−	0.977456i	$-0.432282\pi$
$991$	13.2935	0.422281	0.211140	+	0.977456i	$0.432282\pi$
$992$	5.04647	0.160225
$993$	0	0
$994$	15.7959	0.501017
$995$	−11.0842	−0.351392
$996$	0	0
$997$	−41.6510	−1.31910	−0.659551	−	0.751660i	$-0.729254\pi$
$997$	−41.6510	−1.31910	−0.659551	+	0.751660i	$0.729254\pi$
$998$	21.1465	0.669382
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.u.1.14		18
3.2	odd	2		2667.2.a.p.1.5	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.p.1.5	✓	18	3.2	odd	2
8001.2.a.u.1.14		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)

\(f(q)\)	\(=\)	\(q+1.77085 q^{2} +1.13590 q^{4} -1.38376 q^{5} +1.00000 q^{7} -1.53019 q^{8} +O(q^{10})\)	\(q+1.77085 q^{2} +1.13590 q^{4} -1.38376 q^{5} +1.00000 q^{7} -1.53019 q^{8} -2.45042 q^{10} -3.51371 q^{11} +4.71315 q^{13} +1.77085 q^{14} -4.98153 q^{16} +4.68539 q^{17} -3.74479 q^{19} -1.57181 q^{20} -6.22224 q^{22} +2.00004 q^{23} -3.08521 q^{25} +8.34626 q^{26} +1.13590 q^{28} +4.37013 q^{29} -0.875947 q^{31} -5.76115 q^{32} +8.29711 q^{34} -1.38376 q^{35} -9.02904 q^{37} -6.63146 q^{38} +2.11741 q^{40} -1.57067 q^{41} -2.16122 q^{43} -3.99122 q^{44} +3.54176 q^{46} +12.9854 q^{47} +1.00000 q^{49} -5.46344 q^{50} +5.35366 q^{52} +10.7152 q^{53} +4.86212 q^{55} -1.53019 q^{56} +7.73884 q^{58} +0.801398 q^{59} +3.10892 q^{61} -1.55117 q^{62} -0.239057 q^{64} -6.52185 q^{65} +6.63235 q^{67} +5.32214 q^{68} -2.45042 q^{70} +8.91999 q^{71} +12.3019 q^{73} -15.9891 q^{74} -4.25371 q^{76} -3.51371 q^{77} +3.05893 q^{79} +6.89323 q^{80} -2.78142 q^{82} -4.04869 q^{83} -6.48345 q^{85} -3.82719 q^{86} +5.37664 q^{88} +11.2427 q^{89} +4.71315 q^{91} +2.27184 q^{92} +22.9952 q^{94} +5.18189 q^{95} -8.60399 q^{97} +1.77085 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 6 q^{2} + 22 q^{4} + 10 q^{5} + 18 q^{7} + 21 q^{8}+O(q^{10}) \) 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

Fields

Representations

Groups

Database

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