LMFDB - Embedded newform 8001.2.a.v.1.9

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:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 4 x^{18} - 22 x^{17} + 101 x^{16} + 178 x^{15} - 1035 x^{14} - 583 x^{13} + 5572 x^{12} + \cdots + 210 $ x^19 - 4x^18 - 22x^17 + 101x^16 + 178x^15 - 1035x^14 - 583x^13 + 5572x^12 + 21x^11 - 17032x^10 + 4985x^9 + 29792x^8 - 13249x^7 - 28600x^6 + 14000x^5 + 13725x^4 - 5723x^3 - 2913x^2 + 608x + 210
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
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			1.9
Root			$0.823573$ 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)

$f(q)$	$=$	$q-0.823573 q^{2} -1.32173 q^{4} +2.87784 q^{5} +1.00000 q^{7} +2.73569 q^{8} +O(q^{10})$	\(q-0.823573 q^{2} -1.32173 q^{4} +2.87784 q^{5} +1.00000 q^{7} +2.73569 q^{8} -2.37011 q^{10} +2.49735 q^{11} -3.14350 q^{13} -0.823573 q^{14} +0.390415 q^{16} +2.04099 q^{17} +3.80024 q^{19} -3.80372 q^{20} -2.05675 q^{22} +0.178746 q^{23} +3.28195 q^{25} +2.58890 q^{26} -1.32173 q^{28} -7.77822 q^{29} +8.24454 q^{31} -5.79291 q^{32} -1.68090 q^{34} +2.87784 q^{35} +0.0498363 q^{37} -3.12978 q^{38} +7.87286 q^{40} +2.14558 q^{41} +4.51117 q^{43} -3.30082 q^{44} -0.147210 q^{46} +0.391553 q^{47} +1.00000 q^{49} -2.70293 q^{50} +4.15485 q^{52} +8.31101 q^{53} +7.18698 q^{55} +2.73569 q^{56} +6.40593 q^{58} -14.3285 q^{59} +8.25276 q^{61} -6.78999 q^{62} +3.99005 q^{64} -9.04648 q^{65} -5.83463 q^{67} -2.69763 q^{68} -2.37011 q^{70} -8.03307 q^{71} +14.0099 q^{73} -0.0410438 q^{74} -5.02288 q^{76} +2.49735 q^{77} -5.72861 q^{79} +1.12355 q^{80} -1.76704 q^{82} +8.13150 q^{83} +5.87364 q^{85} -3.71528 q^{86} +6.83197 q^{88} +15.4810 q^{89} -3.14350 q^{91} -0.236253 q^{92} -0.322472 q^{94} +10.9365 q^{95} -1.62954 q^{97} -0.823573 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8}+O(q^{10}) $ 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8	$ 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8} + 9 q^{11} + 24 q^{13} - 4 q^{14} + 20 q^{16} - 17 q^{17} + 23 q^{19} - 5 q^{20} - 3 q^{22} + 17 q^{23} + 38 q^{25} - 28 q^{26} + 22 q^{28} - 2 q^{29} + 16 q^{31} - 17 q^{32} + 29 q^{34} - 5 q^{35} + 56 q^{37} - 2 q^{38} - 13 q^{40} + 7 q^{41} + 19 q^{43} + 29 q^{44} + 10 q^{46} - 25 q^{47} + 19 q^{49} + 9 q^{50} + 16 q^{52} - 18 q^{53} + 10 q^{55} - 9 q^{56} + 31 q^{58} - 11 q^{59} + 26 q^{61} - 26 q^{62} + 45 q^{64} - 27 q^{65} + 24 q^{67} - 14 q^{68} + 32 q^{71} + 51 q^{73} + 12 q^{76} + 9 q^{77} + 30 q^{79} + 30 q^{80} - 52 q^{82} - q^{83} + 44 q^{85} + 24 q^{86} - 30 q^{88} - 5 q^{89} + 24 q^{91} + 88 q^{92} + 7 q^{94} + 24 q^{95} + 5 q^{97} - 4 q^{98}+O(q^{100}) $ 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8 + 9 * q^11 + 24 * q^13 - 4 * q^14 + 20 * q^16 - 17 * q^17 + 23 * q^19 - 5 * q^20 - 3 * q^22 + 17 * q^23 + 38 * q^25 - 28 * q^26 + 22 * q^28 - 2 * q^29 + 16 * q^31 - 17 * q^32 + 29 * q^34 - 5 * q^35 + 56 * q^37 - 2 * q^38 - 13 * q^40 + 7 * q^41 + 19 * q^43 + 29 * q^44 + 10 * q^46 - 25 * q^47 + 19 * q^49 + 9 * q^50 + 16 * q^52 - 18 * q^53 + 10 * q^55 - 9 * q^56 + 31 * q^58 - 11 * q^59 + 26 * q^61 - 26 * q^62 + 45 * q^64 - 27 * q^65 + 24 * q^67 - 14 * q^68 + 32 * q^71 + 51 * q^73 + 12 * q^76 + 9 * q^77 + 30 * q^79 + 30 * q^80 - 52 * q^82 - q^83 + 44 * q^85 + 24 * q^86 - 30 * q^88 - 5 * q^89 + 24 * q^91 + 88 * q^92 + 7 * q^94 + 24 * q^95 + 5 * 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 $

$2$	−0.823573	−0.582354	−0.291177	−	0.956669i	$-0.594047\pi$
$2$	−0.823573	−0.582354	−0.291177	+	0.956669i	$0.594047\pi$
$3$	0	0
$3$	0	0
$4$	−1.32173	−0.660863
$5$	2.87784	1.28701	0.643504	−	0.765443i	$-0.277480\pi$
$5$	2.87784	1.28701	0.643504	+	0.765443i	$0.277480\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.73569	0.967211
$9$	0	0
$10$	−2.37011	−0.749495
$11$	2.49735	0.752980	0.376490	−	0.926421i	$-0.377131\pi$
$11$	2.49735	0.752980	0.376490	+	0.926421i	$0.377131\pi$
$12$	0	0
$13$	−3.14350	−0.871850	−0.435925	−	0.899983i	$-0.643579\pi$
$13$	−3.14350	−0.871850	−0.435925	+	0.899983i	$0.643579\pi$
$14$	−0.823573	−0.220109
$15$	0	0
$16$	0.390415	0.0976038
$17$	2.04099	0.495012	0.247506	−	0.968886i	$-0.420389\pi$
$17$	2.04099	0.495012	0.247506	+	0.968886i	$0.420389\pi$
$18$	0	0
$19$	3.80024	0.871836	0.435918	−	0.899987i	$-0.356424\pi$
$19$	3.80024	0.871836	0.435918	+	0.899987i	$0.356424\pi$
$20$	−3.80372	−0.850537
$21$	0	0
$22$	−2.05675	−0.438501
$23$	0.178746	0.0372711	0.0186355	−	0.999826i	$-0.494068\pi$
$23$	0.178746	0.0372711	0.0186355	+	0.999826i	$0.494068\pi$
$24$	0	0
$25$	3.28195	0.656391
$26$	2.58890	0.507726
$27$	0	0
$28$	−1.32173	−0.249783
$29$	−7.77822	−1.44438	−0.722189	−	0.691695i	$-0.756864\pi$
$29$	−7.77822	−1.44438	−0.722189	+	0.691695i	$0.756864\pi$
$30$	0	0
$31$	8.24454	1.48076	0.740382	−	0.672187i	$-0.234645\pi$
$31$	8.24454	1.48076	0.740382	+	0.672187i	$0.234645\pi$
$32$	−5.79291	−1.02405
$33$	0	0
$34$	−1.68090	−0.288273
$35$	2.87784	0.486444
$36$	0	0
$37$	0.0498363	0.00819303	0.00409652	−	0.999992i	$-0.498696\pi$
$37$	0.0498363	0.00819303	0.00409652	+	0.999992i	$0.498696\pi$
$38$	−3.12978	−0.507717
$39$	0	0
$40$	7.87286	1.24481
$41$	2.14558	0.335084	0.167542	−	0.985865i	$-0.446417\pi$
$41$	2.14558	0.335084	0.167542	+	0.985865i	$0.446417\pi$
$42$	0	0
$43$	4.51117	0.687946	0.343973	−	0.938979i	$-0.388227\pi$
$43$	4.51117	0.687946	0.343973	+	0.938979i	$0.388227\pi$
$44$	−3.30082	−0.497617
$45$	0	0
$46$	−0.147210	−0.0217050
$47$	0.391553	0.0571138	0.0285569	−	0.999592i	$-0.490909\pi$
$47$	0.391553	0.0571138	0.0285569	+	0.999592i	$0.490909\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−2.70293	−0.382252
$51$	0	0
$52$	4.15485	0.576174
$53$	8.31101	1.14160	0.570802	−	0.821088i	$-0.306632\pi$
$53$	8.31101	1.14160	0.570802	+	0.821088i	$0.306632\pi$
$54$	0	0
$55$	7.18698	0.969092
$56$	2.73569	0.365571
$57$	0	0
$58$	6.40593	0.841140
$59$	−14.3285	−1.86542	−0.932709	−	0.360630i	$-0.882562\pi$
$59$	−14.3285	−1.86542	−0.932709	+	0.360630i	$0.882562\pi$
$60$	0	0
$61$	8.25276	1.05666	0.528329	−	0.849040i	$-0.322819\pi$
$61$	8.25276	1.05666	0.528329	+	0.849040i	$0.322819\pi$
$62$	−6.78999	−0.862329
$63$	0	0
$64$	3.99005	0.498757
$65$	−9.04648	−1.12208
$66$	0	0
$67$	−5.83463	−0.712814	−0.356407	−	0.934331i	$-0.615998\pi$
$67$	−5.83463	−0.712814	−0.356407	+	0.934331i	$0.615998\pi$
$68$	−2.69763	−0.327136
$69$	0	0
$70$	−2.37011	−0.283283
$71$	−8.03307	−0.953349	−0.476675	−	0.879080i	$-0.658158\pi$
$71$	−8.03307	−0.953349	−0.476675	+	0.879080i	$0.658158\pi$
$72$	0	0
$73$	14.0099	1.63973	0.819866	−	0.572556i	$-0.194048\pi$
$73$	14.0099	1.63973	0.819866	+	0.572556i	$0.194048\pi$
$74$	−0.0410438	−0.00477125
$75$	0	0
$76$	−5.02288	−0.576164
$77$	2.49735	0.284600
$78$	0	0
$79$	−5.72861	−0.644520	−0.322260	−	0.946651i	$-0.604442\pi$
$79$	−5.72861	−0.644520	−0.322260	+	0.946651i	$0.604442\pi$
$80$	1.12355	0.125617
$81$	0	0
$82$	−1.76704	−0.195137
$83$	8.13150	0.892548	0.446274	−	0.894896i	$-0.352751\pi$
$83$	8.13150	0.892548	0.446274	+	0.894896i	$0.352751\pi$
$84$	0	0
$85$	5.87364	0.637085
$86$	−3.71528	−0.400629
$87$	0	0
$88$	6.83197	0.728291
$89$	15.4810	1.64098	0.820490	−	0.571660i	$-0.193701\pi$
$89$	15.4810	1.64098	0.820490	+	0.571660i	$0.193701\pi$
$90$	0	0
$91$	−3.14350	−0.329528
$92$	−0.236253	−0.0246311
$93$	0	0
$94$	−0.322472	−0.0332605
$95$	10.9365	1.12206
$96$	0	0
$97$	−1.62954	−0.165455	−0.0827275	−	0.996572i	$-0.526363\pi$
$97$	−1.62954	−0.165455	−0.0827275	+	0.996572i	$0.526363\pi$
$98$	−0.823573	−0.0831935
$99$	0	0
$100$	−4.33785	−0.433785
$101$	18.5531	1.84611	0.923054	−	0.384672i	$-0.125685\pi$
$101$	18.5531	1.84611	0.923054	+	0.384672i	$0.125685\pi$
$102$	0	0
$103$	13.8022	1.35997	0.679984	−	0.733227i	$-0.261987\pi$
$103$	13.8022	1.35997	0.679984	+	0.733227i	$0.261987\pi$
$104$	−8.59963	−0.843263
$105$	0	0
$106$	−6.84472	−0.664818
$107$	−11.5898	−1.12043	−0.560216	−	0.828347i	$-0.689282\pi$
$107$	−11.5898	−1.12043	−0.560216	+	0.828347i	$0.689282\pi$
$108$	0	0
$109$	4.90568	0.469879	0.234939	−	0.972010i	$-0.424511\pi$
$109$	4.90568	0.469879	0.234939	+	0.972010i	$0.424511\pi$
$110$	−5.91900	−0.564355
$111$	0	0
$112$	0.390415	0.0368908
$113$	−11.4900	−1.08089	−0.540446	−	0.841379i	$-0.681744\pi$
$113$	−11.4900	−1.08089	−0.540446	+	0.841379i	$0.681744\pi$
$114$	0	0
$115$	0.514401	0.0479682
$116$	10.2807	0.954537
$117$	0	0
$118$	11.8006	1.08633
$119$	2.04099	0.187097
$120$	0	0
$121$	−4.76323	−0.433021
$122$	−6.79675	−0.615349
$123$	0	0
$124$	−10.8970	−0.978582
$125$	−4.94426	−0.442228
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	8.29971	0.733598
$129$	0	0
$130$	7.45044	0.653447
$131$	−4.61325	−0.403062	−0.201531	−	0.979482i	$-0.564592\pi$
$131$	−4.61325	−0.403062	−0.201531	+	0.979482i	$0.564592\pi$
$132$	0	0
$133$	3.80024	0.329523
$134$	4.80525	0.415110
$135$	0	0
$136$	5.58350	0.478781
$137$	7.76252	0.663197	0.331599	−	0.943421i	$-0.392412\pi$
$137$	7.76252	0.663197	0.331599	+	0.943421i	$0.392412\pi$
$138$	0	0
$139$	13.1628	1.11646	0.558228	−	0.829688i	$-0.311481\pi$
$139$	13.1628	1.11646	0.558228	+	0.829688i	$0.311481\pi$
$140$	−3.80372	−0.321473
$141$	0	0
$142$	6.61582	0.555187
$143$	−7.85043	−0.656486
$144$	0	0
$145$	−22.3845	−1.85893
$146$	−11.5382	−0.954905
$147$	0	0
$148$	−0.0658699	−0.00541447
$149$	−14.6464	−1.19988	−0.599940	−	0.800045i	$-0.704809\pi$
$149$	−14.6464	−1.19988	−0.599940	+	0.800045i	$0.704809\pi$
$150$	0	0
$151$	−5.84407	−0.475584	−0.237792	−	0.971316i	$-0.576424\pi$
$151$	−5.84407	−0.475584	−0.237792	+	0.971316i	$0.576424\pi$
$152$	10.3963	0.843249
$153$	0	0
$154$	−2.05675	−0.165738
$155$	23.7265	1.90576
$156$	0	0
$157$	2.25235	0.179757	0.0898786	−	0.995953i	$-0.471352\pi$
$157$	2.25235	0.179757	0.0898786	+	0.995953i	$0.471352\pi$
$158$	4.71794	0.375339
$159$	0	0
$160$	−16.6711	−1.31796
$161$	0.178746	0.0140871
$162$	0	0
$163$	17.9604	1.40677	0.703385	−	0.710809i	$-0.251671\pi$
$163$	17.9604	1.40677	0.703385	+	0.710809i	$0.251671\pi$
$164$	−2.83587	−0.221445
$165$	0	0
$166$	−6.69689	−0.519779
$167$	−6.74574	−0.522001	−0.261000	−	0.965339i	$-0.584052\pi$
$167$	−6.74574	−0.522001	−0.261000	+	0.965339i	$0.584052\pi$
$168$	0	0
$169$	−3.11841	−0.239878
$170$	−4.83737	−0.371009
$171$	0	0
$172$	−5.96253	−0.454639
$173$	−13.0095	−0.989096	−0.494548	−	0.869150i	$-0.664666\pi$
$173$	−13.0095	−0.989096	−0.494548	+	0.869150i	$0.664666\pi$
$174$	0	0
$175$	3.28195	0.248092
$176$	0.975004	0.0734937
$177$	0	0
$178$	−12.7497	−0.955632
$179$	13.0566	0.975894	0.487947	−	0.872873i	$-0.337746\pi$
$179$	13.0566	0.975894	0.487947	+	0.872873i	$0.337746\pi$
$180$	0	0
$181$	−8.28417	−0.615758	−0.307879	−	0.951426i	$-0.599619\pi$
$181$	−8.28417	−0.615758	−0.307879	+	0.951426i	$0.599619\pi$
$182$	2.58890	0.191902
$183$	0	0
$184$	0.488992	0.0360490
$185$	0.143421	0.0105445
$186$	0	0
$187$	5.09707	0.372734
$188$	−0.517526	−0.0377444
$189$	0	0
$190$	−9.00700	−0.653436
$191$	−1.47363	−0.106628	−0.0533140	−	0.998578i	$-0.516978\pi$
$191$	−1.47363	−0.106628	−0.0533140	+	0.998578i	$0.516978\pi$
$192$	0	0
$193$	−3.64272	−0.262209	−0.131104	−	0.991369i	$-0.541852\pi$
$193$	−3.64272	−0.262209	−0.131104	+	0.991369i	$0.541852\pi$
$194$	1.34205	0.0963534
$195$	0	0
$196$	−1.32173	−0.0944091
$197$	−1.55636	−0.110886	−0.0554429	−	0.998462i	$-0.517657\pi$
$197$	−1.55636	−0.110886	−0.0554429	+	0.998462i	$0.517657\pi$
$198$	0	0
$199$	−4.82683	−0.342165	−0.171083	−	0.985257i	$-0.554726\pi$
$199$	−4.82683	−0.342165	−0.171083	+	0.985257i	$0.554726\pi$
$200$	8.97840	0.634869
$201$	0	0
$202$	−15.2799	−1.07509
$203$	−7.77822	−0.545924
$204$	0	0
$205$	6.17464	0.431256
$206$	−11.3671	−0.791984
$207$	0	0
$208$	−1.22727	−0.0850959
$209$	9.49055	0.656475
$210$	0	0
$211$	13.2751	0.913893	0.456947	−	0.889494i	$-0.348943\pi$
$211$	13.2751	0.913893	0.456947	+	0.889494i	$0.348943\pi$
$212$	−10.9849	−0.754445
$213$	0	0
$214$	9.54508	0.652488
$215$	12.9824	0.885393
$216$	0	0
$217$	8.24454	0.559676
$218$	−4.04019	−0.273636
$219$	0	0
$220$	−9.49922	−0.640437
$221$	−6.41585	−0.431577
$222$	0	0
$223$	17.5530	1.17544	0.587718	−	0.809066i	$-0.300027\pi$
$223$	17.5530	1.17544	0.587718	+	0.809066i	$0.300027\pi$
$224$	−5.79291	−0.387055
$225$	0	0
$226$	9.46288	0.629462
$227$	−18.5774	−1.23303	−0.616514	−	0.787344i	$-0.711455\pi$
$227$	−18.5774	−1.23303	−0.616514	+	0.787344i	$0.711455\pi$
$228$	0	0
$229$	22.9178	1.51445	0.757226	−	0.653153i	$-0.226554\pi$
$229$	22.9178	1.51445	0.757226	+	0.653153i	$0.226554\pi$
$230$	−0.423647	−0.0279345
$231$	0	0
$232$	−21.2788	−1.39702
$233$	4.19204	0.274630	0.137315	−	0.990527i	$-0.456153\pi$
$233$	4.19204	0.274630	0.137315	+	0.990527i	$0.456153\pi$
$234$	0	0
$235$	1.12683	0.0735060
$236$	18.9384	1.23279
$237$	0	0
$238$	−1.68090	−0.108957
$239$	16.4112	1.06155	0.530776	−	0.847512i	$-0.321901\pi$
$239$	16.4112	1.06155	0.530776	+	0.847512i	$0.321901\pi$
$240$	0	0
$241$	−19.4130	−1.25050	−0.625252	−	0.780423i	$-0.715004\pi$
$241$	−19.4130	−1.25050	−0.625252	+	0.780423i	$0.715004\pi$
$242$	3.92287	0.252172
$243$	0	0
$244$	−10.9079	−0.698306
$245$	2.87784	0.183858
$246$	0	0
$247$	−11.9461	−0.760110
$248$	22.5545	1.43221
$249$	0	0
$250$	4.07196	0.257533
$251$	−2.54869	−0.160872	−0.0804359	−	0.996760i	$-0.525631\pi$
$251$	−2.54869	−0.160872	−0.0804359	+	0.996760i	$0.525631\pi$
$252$	0	0
$253$	0.446391	0.0280644
$254$	−0.823573	−0.0516756
$255$	0	0
$256$	−14.8155	−0.925971
$257$	−9.24122	−0.576451	−0.288226	−	0.957563i	$-0.593065\pi$
$257$	−9.24122	−0.576451	−0.288226	+	0.957563i	$0.593065\pi$
$258$	0	0
$259$	0.0498363	0.00309668
$260$	11.9570	0.741540
$261$	0	0
$262$	3.79935	0.234725
$263$	−9.94843	−0.613447	−0.306723	−	0.951799i	$-0.599233\pi$
$263$	−9.94843	−0.613447	−0.306723	+	0.951799i	$0.599233\pi$
$264$	0	0
$265$	23.9177	1.46925
$266$	−3.12978	−0.191899
$267$	0	0
$268$	7.71179	0.471072
$269$	−3.68337	−0.224579	−0.112290	−	0.993676i	$-0.535818\pi$
$269$	−3.68337	−0.224579	−0.112290	+	0.993676i	$0.535818\pi$
$270$	0	0
$271$	8.76862	0.532656	0.266328	−	0.963883i	$-0.414190\pi$
$271$	8.76862	0.532656	0.266328	+	0.963883i	$0.414190\pi$
$272$	0.796833	0.0483151
$273$	0	0
$274$	−6.39301	−0.386216
$275$	8.19620	0.494249
$276$	0	0
$277$	6.04083	0.362958	0.181479	−	0.983395i	$-0.441912\pi$
$277$	6.04083	0.362958	0.181479	+	0.983395i	$0.441912\pi$
$278$	−10.8406	−0.650173
$279$	0	0
$280$	7.87286	0.470494
$281$	29.8481	1.78059	0.890293	−	0.455388i	$-0.150499\pi$
$281$	29.8481	1.78059	0.890293	+	0.455388i	$0.150499\pi$
$282$	0	0
$283$	−0.479453	−0.0285005	−0.0142502	−	0.999898i	$-0.504536\pi$
$283$	−0.479453	−0.0285005	−0.0142502	+	0.999898i	$0.504536\pi$
$284$	10.6175	0.630034
$285$	0	0
$286$	6.46540	0.382307
$287$	2.14558	0.126650
$288$	0	0
$289$	−12.8344	−0.754963
$290$	18.4352	1.08255
$291$	0	0
$292$	−18.5172	−1.08364
$293$	20.9011	1.22106	0.610528	−	0.791994i	$-0.290957\pi$
$293$	20.9011	1.22106	0.610528	+	0.791994i	$0.290957\pi$
$294$	0	0
$295$	−41.2352	−2.40081
$296$	0.136336	0.00792439
$297$	0	0
$298$	12.0624	0.698755
$299$	−0.561887	−0.0324948
$300$	0	0
$301$	4.51117	0.260019
$302$	4.81302	0.276958
$303$	0	0
$304$	1.48367	0.0850944
$305$	23.7501	1.35993
$306$	0	0
$307$	−23.1940	−1.32375	−0.661876	−	0.749613i	$-0.730240\pi$
$307$	−23.1940	−1.32375	−0.661876	+	0.749613i	$0.730240\pi$
$308$	−3.30082	−0.188082
$309$	0	0
$310$	−19.5405	−1.10982
$311$	−25.8909	−1.46814	−0.734069	−	0.679075i	$-0.762381\pi$
$311$	−25.8909	−1.46814	−0.734069	+	0.679075i	$0.762381\pi$
$312$	0	0
$313$	18.2780	1.03313	0.516566	−	0.856247i	$-0.327210\pi$
$313$	18.2780	1.03313	0.516566	+	0.856247i	$0.327210\pi$
$314$	−1.85498	−0.104682
$315$	0	0
$316$	7.57166	0.425939
$317$	−25.2643	−1.41898	−0.709491	−	0.704714i	$-0.751075\pi$
$317$	−25.2643	−1.41898	−0.709491	+	0.704714i	$0.751075\pi$
$318$	0	0
$319$	−19.4249	−1.08759
$320$	11.4827	0.641904
$321$	0	0
$322$	−0.147210	−0.00820370
$323$	7.75625	0.431569
$324$	0	0
$325$	−10.3168	−0.572274
$326$	−14.7917	−0.819239
$327$	0	0
$328$	5.86964	0.324097
$329$	0.391553	0.0215870
$330$	0	0
$331$	−9.71108	−0.533769	−0.266885	−	0.963728i	$-0.585994\pi$
$331$	−9.71108	−0.533769	−0.266885	+	0.963728i	$0.585994\pi$
$332$	−10.7476	−0.589852
$333$	0	0
$334$	5.55561	0.303989
$335$	−16.7911	−0.917397
$336$	0	0
$337$	−0.338072	−0.0184159	−0.00920797	−	0.999958i	$-0.502931\pi$
$337$	−0.338072	−0.0184159	−0.00920797	+	0.999958i	$0.502931\pi$
$338$	2.56824	0.139694
$339$	0	0
$340$	−7.76334	−0.421026
$341$	20.5895	1.11499
$342$	0	0
$343$	1.00000	0.0539949
$344$	12.3411	0.665389
$345$	0	0
$346$	10.7143	0.576005
$347$	−10.8081	−0.580211	−0.290106	−	0.956995i	$-0.593690\pi$
$347$	−10.8081	−0.580211	−0.290106	+	0.956995i	$0.593690\pi$
$348$	0	0
$349$	−30.2039	−1.61678	−0.808388	−	0.588650i	$-0.799660\pi$
$349$	−30.2039	−1.61678	−0.808388	+	0.588650i	$0.799660\pi$
$350$	−2.70293	−0.144478
$351$	0	0
$352$	−14.4669	−0.771090
$353$	24.0686	1.28104	0.640521	−	0.767941i	$-0.278719\pi$
$353$	24.0686	1.28104	0.640521	+	0.767941i	$0.278719\pi$
$354$	0	0
$355$	−23.1179	−1.22697
$356$	−20.4616	−1.08446
$357$	0	0
$358$	−10.7530	−0.568316
$359$	−6.18593	−0.326481	−0.163240	−	0.986586i	$-0.552195\pi$
$359$	−6.18593	−0.326481	−0.163240	+	0.986586i	$0.552195\pi$
$360$	0	0
$361$	−4.55815	−0.239903
$362$	6.82262	0.358589
$363$	0	0
$364$	4.15485	0.217773
$365$	40.3182	2.11035
$366$	0	0
$367$	−11.5359	−0.602170	−0.301085	−	0.953597i	$-0.597349\pi$
$367$	−11.5359	−0.602170	−0.301085	+	0.953597i	$0.597349\pi$
$368$	0.0697850	0.00363780
$369$	0	0
$370$	−0.118118	−0.00614064
$371$	8.31101	0.431486
$372$	0	0
$373$	−19.8284	−1.02668	−0.513338	−	0.858186i	$-0.671591\pi$
$373$	−19.8284	−1.02668	−0.513338	+	0.858186i	$0.671591\pi$
$374$	−4.19781	−0.217064
$375$	0	0
$376$	1.07116	0.0552411
$377$	24.4508	1.25928
$378$	0	0
$379$	−9.80003	−0.503394	−0.251697	−	0.967806i	$-0.580989\pi$
$379$	−9.80003	−0.503394	−0.251697	+	0.967806i	$0.580989\pi$
$380$	−14.4550	−0.741528
$381$	0	0
$382$	1.21364	0.0620953
$383$	−8.47434	−0.433018	−0.216509	−	0.976281i	$-0.569467\pi$
$383$	−8.47434	−0.433018	−0.216509	+	0.976281i	$0.569467\pi$
$384$	0	0
$385$	7.18698	0.366282
$386$	3.00005	0.152698
$387$	0	0
$388$	2.15381	0.109343
$389$	38.7291	1.96365	0.981823	−	0.189800i	$-0.0607840\pi$
$389$	38.7291	1.96365	0.981823	+	0.189800i	$0.0607840\pi$
$390$	0	0
$391$	0.364818	0.0184496
$392$	2.73569	0.138173
$393$	0	0
$394$	1.28177	0.0645749
$395$	−16.4860	−0.829502
$396$	0	0
$397$	15.0600	0.755838	0.377919	−	0.925839i	$-0.376640\pi$
$397$	15.0600	0.755838	0.377919	+	0.925839i	$0.376640\pi$
$398$	3.97525	0.199261
$399$	0	0
$400$	1.28132	0.0640662
$401$	−7.21253	−0.360177	−0.180088	−	0.983650i	$-0.557638\pi$
$401$	−7.21253	−0.360177	−0.180088	+	0.983650i	$0.557638\pi$
$402$	0	0
$403$	−25.9167	−1.29100
$404$	−24.5222	−1.22002
$405$	0	0
$406$	6.40593	0.317921
$407$	0.124459	0.00616919
$408$	0	0
$409$	22.4241	1.10880	0.554400	−	0.832250i	$-0.312948\pi$
$409$	22.4241	1.10880	0.554400	+	0.832250i	$0.312948\pi$
$410$	−5.08527	−0.251144
$411$	0	0
$412$	−18.2427	−0.898753
$413$	−14.3285	−0.705062
$414$	0	0
$415$	23.4011	1.14872
$416$	18.2100	0.892819
$417$	0	0
$418$	−7.81616	−0.382301
$419$	35.0231	1.71099	0.855496	−	0.517809i	$-0.173252\pi$
$419$	35.0231	1.71099	0.855496	+	0.517809i	$0.173252\pi$
$420$	0	0
$421$	4.53382	0.220965	0.110483	−	0.993878i	$-0.464760\pi$
$421$	4.53382	0.220965	0.110483	+	0.993878i	$0.464760\pi$
$422$	−10.9330	−0.532210
$423$	0	0
$424$	22.7363	1.10417
$425$	6.69843	0.324922
$426$	0	0
$427$	8.25276	0.399379
$428$	15.3186	0.740452
$429$	0	0
$430$	−10.6920	−0.515612
$431$	7.51951	0.362202	0.181101	−	0.983464i	$-0.442034\pi$
$431$	7.51951	0.362202	0.181101	+	0.983464i	$0.442034\pi$
$432$	0	0
$433$	28.4018	1.36490	0.682452	−	0.730930i	$-0.260914\pi$
$433$	28.4018	1.36490	0.682452	+	0.730930i	$0.260914\pi$
$434$	−6.78999	−0.325930
$435$	0	0
$436$	−6.48397	−0.310526
$437$	0.679277	0.0324942
$438$	0	0
$439$	21.3607	1.01949	0.509745	−	0.860325i	$-0.329740\pi$
$439$	21.3607	1.01949	0.509745	+	0.860325i	$0.329740\pi$
$440$	19.6613	0.937316
$441$	0	0
$442$	5.28392	0.251330
$443$	−14.5312	−0.690400	−0.345200	−	0.938529i	$-0.612189\pi$
$443$	−14.5312	−0.690400	−0.345200	+	0.938529i	$0.612189\pi$
$444$	0	0
$445$	44.5518	2.11196
$446$	−14.4562	−0.684520
$447$	0	0
$448$	3.99005	0.188512
$449$	1.55717	0.0734876	0.0367438	−	0.999325i	$-0.488301\pi$
$449$	1.55717	0.0734876	0.0367438	+	0.999325i	$0.488301\pi$
$450$	0	0
$451$	5.35827	0.252311
$452$	15.1867	0.714321
$453$	0	0
$454$	15.2999	0.718059
$455$	−9.04648	−0.424106
$456$	0	0
$457$	−1.20252	−0.0562515	−0.0281258	−	0.999604i	$-0.508954\pi$
$457$	−1.20252	−0.0562515	−0.0281258	+	0.999604i	$0.508954\pi$
$458$	−18.8745	−0.881948
$459$	0	0
$460$	−0.679898	−0.0317004
$461$	−32.4395	−1.51086	−0.755429	−	0.655231i	$-0.772571\pi$
$461$	−32.4395	−1.51086	−0.755429	+	0.655231i	$0.772571\pi$
$462$	0	0
$463$	24.1154	1.12074	0.560368	−	0.828244i	$-0.310660\pi$
$463$	24.1154	1.12074	0.560368	+	0.828244i	$0.310660\pi$
$464$	−3.03673	−0.140977
$465$	0	0
$466$	−3.45245	−0.159932
$467$	−27.5828	−1.27638	−0.638191	−	0.769878i	$-0.720317\pi$
$467$	−27.5828	−1.27638	−0.638191	+	0.769878i	$0.720317\pi$
$468$	0	0
$469$	−5.83463	−0.269418
$470$	−0.928023	−0.0428065
$471$	0	0
$472$	−39.1984	−1.80425
$473$	11.2660	0.518010
$474$	0	0
$475$	12.4722	0.572265
$476$	−2.69763	−0.123646
$477$	0	0
$478$	−13.5158	−0.618199
$479$	26.6971	1.21982	0.609910	−	0.792471i	$-0.291206\pi$
$479$	26.6971	1.21982	0.609910	+	0.792471i	$0.291206\pi$
$480$	0	0
$481$	−0.156660	−0.00714309
$482$	15.9881	0.728237
$483$	0	0
$484$	6.29569	0.286168
$485$	−4.68956	−0.212942
$486$	0	0
$487$	20.3423	0.921798	0.460899	−	0.887453i	$-0.347527\pi$
$487$	20.3423	0.921798	0.460899	+	0.887453i	$0.347527\pi$
$488$	22.5770	1.02201
$489$	0	0
$490$	−2.37011	−0.107071
$491$	10.3444	0.466838	0.233419	−	0.972376i	$-0.425009\pi$
$491$	10.3444	0.466838	0.233419	+	0.972376i	$0.425009\pi$
$492$	0	0
$493$	−15.8753	−0.714985
$494$	9.83846	0.442653
$495$	0	0
$496$	3.21879	0.144528
$497$	−8.03307	−0.360332
$498$	0	0
$499$	8.63127	0.386389	0.193194	−	0.981161i	$-0.438115\pi$
$499$	8.63127	0.386389	0.193194	+	0.981161i	$0.438115\pi$
$500$	6.53496	0.292252
$501$	0	0
$502$	2.09903	0.0936844
$503$	−3.53121	−0.157449	−0.0787245	−	0.996896i	$-0.525085\pi$
$503$	−3.53121	−0.157449	−0.0787245	+	0.996896i	$0.525085\pi$
$504$	0	0
$505$	53.3930	2.37596
$506$	−0.367636	−0.0163434
$507$	0	0
$508$	−1.32173	−0.0586421
$509$	14.8864	0.659830	0.329915	−	0.944011i	$-0.392980\pi$
$509$	14.8864	0.659830	0.329915	+	0.944011i	$0.392980\pi$
$510$	0	0
$511$	14.0099	0.619760
$512$	−4.39775	−0.194355
$513$	0	0
$514$	7.61082	0.335699
$515$	39.7204	1.75029
$516$	0	0
$517$	0.977845	0.0430056
$518$	−0.0410438	−0.00180336
$519$	0	0
$520$	−24.7483	−1.08529
$521$	35.0975	1.53765	0.768826	−	0.639458i	$-0.220841\pi$
$521$	35.0975	1.53765	0.768826	+	0.639458i	$0.220841\pi$
$522$	0	0
$523$	31.8044	1.39071	0.695355	−	0.718667i	$-0.255247\pi$
$523$	31.8044	1.39071	0.695355	+	0.718667i	$0.255247\pi$
$524$	6.09746	0.266369
$525$	0	0
$526$	8.19326	0.357243
$527$	16.8270	0.732996
$528$	0	0
$529$	−22.9680	−0.998611
$530$	−19.6980	−0.855627
$531$	0	0
$532$	−5.02288	−0.217770
$533$	−6.74464	−0.292143
$534$	0	0
$535$	−33.3537	−1.44201
$536$	−15.9617	−0.689441
$537$	0	0
$538$	3.03353	0.130785
$539$	2.49735	0.107569
$540$	0	0
$541$	2.82729	0.121555	0.0607774	−	0.998151i	$-0.480642\pi$
$541$	2.82729	0.121555	0.0607774	+	0.998151i	$0.480642\pi$
$542$	−7.22160	−0.310194
$543$	0	0
$544$	−11.8233	−0.506918
$545$	14.1178	0.604738
$546$	0	0
$547$	30.1788	1.29035	0.645176	−	0.764034i	$-0.276784\pi$
$547$	30.1788	1.29035	0.645176	+	0.764034i	$0.276784\pi$
$548$	−10.2599	−0.438283
$549$	0	0
$550$	−6.75017	−0.287828
$551$	−29.5591	−1.25926
$552$	0	0
$553$	−5.72861	−0.243605
$554$	−4.97507	−0.211370
$555$	0	0
$556$	−17.3977	−0.737825
$557$	31.9667	1.35447	0.677236	−	0.735766i	$-0.263178\pi$
$557$	31.9667	1.35447	0.677236	+	0.735766i	$0.263178\pi$
$558$	0	0
$559$	−14.1809	−0.599786
$560$	1.12355	0.0474787
$561$	0	0
$562$	−24.5821	−1.03693
$563$	−45.6546	−1.92411	−0.962057	−	0.272849i	$-0.912034\pi$
$563$	−45.6546	−1.92411	−0.962057	+	0.272849i	$0.912034\pi$
$564$	0	0
$565$	−33.0665	−1.39112
$566$	0.394864	0.0165974
$567$	0	0
$568$	−21.9759	−0.922090
$569$	18.7255	0.785012	0.392506	−	0.919749i	$-0.371608\pi$
$569$	18.7255	0.785012	0.392506	+	0.919749i	$0.371608\pi$
$570$	0	0
$571$	−4.52907	−0.189536	−0.0947678	−	0.995499i	$-0.530211\pi$
$571$	−4.52907	−0.189536	−0.0947678	+	0.995499i	$0.530211\pi$
$572$	10.3761	0.433847
$573$	0	0
$574$	−1.76704	−0.0737550
$575$	0.586635	0.0244644
$576$	0	0
$577$	−8.47904	−0.352987	−0.176494	−	0.984302i	$-0.556475\pi$
$577$	−8.47904	−0.352987	−0.176494	+	0.984302i	$0.556475\pi$
$578$	10.5700	0.439656
$579$	0	0
$580$	29.5861	1.22850
$581$	8.13150	0.337352
$582$	0	0
$583$	20.7555	0.859605
$584$	38.3266	1.58597
$585$	0	0
$586$	−17.2136	−0.711088
$587$	0.755169	0.0311691	0.0155846	−	0.999879i	$-0.495039\pi$
$587$	0.755169	0.0311691	0.0155846	+	0.999879i	$0.495039\pi$
$588$	0	0
$589$	31.3313	1.29098
$590$	33.9603	1.39812
$591$	0	0
$592$	0.0194568	0.000799671 0
$593$	−29.2749	−1.20218	−0.601088	−	0.799183i	$-0.705266\pi$
$593$	−29.2749	−1.20218	−0.601088	+	0.799183i	$0.705266\pi$
$594$	0	0
$595$	5.87364	0.240796
$596$	19.3585	0.792957
$597$	0	0
$598$	0.462755	0.0189235
$599$	1.60224	0.0654658	0.0327329	−	0.999464i	$-0.489579\pi$
$599$	1.60224	0.0654658	0.0327329	+	0.999464i	$0.489579\pi$
$600$	0	0
$601$	28.8736	1.17778	0.588889	−	0.808214i	$-0.299566\pi$
$601$	28.8736	1.17778	0.588889	+	0.808214i	$0.299566\pi$
$602$	−3.71528	−0.151423
$603$	0	0
$604$	7.72427	0.314296
$605$	−13.7078	−0.557302
$606$	0	0
$607$	45.4619	1.84524	0.922621	−	0.385707i	$-0.126043\pi$
$607$	45.4619	1.84524	0.922621	+	0.385707i	$0.126043\pi$
$608$	−22.0145	−0.892804
$609$	0	0
$610$	−19.5600	−0.791959
$611$	−1.23085	−0.0497947
$612$	0	0
$613$	−2.01541	−0.0814015	−0.0407008	−	0.999171i	$-0.512959\pi$
$613$	−2.01541	−0.0814015	−0.0407008	+	0.999171i	$0.512959\pi$
$614$	19.1020	0.770893
$615$	0	0
$616$	6.83197	0.275268
$617$	17.0285	0.685541	0.342771	−	0.939419i	$-0.388635\pi$
$617$	17.0285	0.685541	0.342771	+	0.939419i	$0.388635\pi$
$618$	0	0
$619$	34.6985	1.39465	0.697325	−	0.716755i	$-0.254374\pi$
$619$	34.6985	1.39465	0.697325	+	0.716755i	$0.254374\pi$
$620$	−31.3599	−1.25944
$621$	0	0
$622$	21.3230	0.854976
$623$	15.4810	0.620232
$624$	0	0
$625$	−30.6385	−1.22554
$626$	−15.0533	−0.601649
$627$	0	0
$628$	−2.97699	−0.118795
$629$	0.101715	0.00405565
$630$	0	0
$631$	45.6586	1.81764	0.908820	−	0.417187i	$-0.136984\pi$
$631$	45.6586	1.81764	0.908820	+	0.417187i	$0.136984\pi$
$632$	−15.6717	−0.623386
$633$	0	0
$634$	20.8070	0.826350
$635$	2.87784	0.114204
$636$	0	0
$637$	−3.14350	−0.124550
$638$	15.9979	0.633362
$639$	0	0
$640$	23.8852	0.944147
$641$	−29.6589	−1.17146	−0.585728	−	0.810508i	$-0.699191\pi$
$641$	−29.6589	−1.17146	−0.585728	+	0.810508i	$0.699191\pi$
$642$	0	0
$643$	38.6046	1.52242	0.761209	−	0.648507i	$-0.224606\pi$
$643$	38.6046	1.52242	0.761209	+	0.648507i	$0.224606\pi$
$644$	−0.236253	−0.00930967
$645$	0	0
$646$	−6.38784	−0.251326
$647$	43.7708	1.72081	0.860404	−	0.509613i	$-0.170211\pi$
$647$	43.7708	1.72081	0.860404	+	0.509613i	$0.170211\pi$
$648$	0	0
$649$	−35.7834	−1.40462
$650$	8.49666	0.333267
$651$	0	0
$652$	−23.7388	−0.929683
$653$	−29.4964	−1.15428	−0.577142	−	0.816644i	$-0.695832\pi$
$653$	−29.4964	−1.15428	−0.577142	+	0.816644i	$0.695832\pi$
$654$	0	0
$655$	−13.2762	−0.518744
$656$	0.837668	0.0327054
$657$	0	0
$658$	−0.322472	−0.0125713
$659$	19.0408	0.741724	0.370862	−	0.928688i	$-0.379062\pi$
$659$	19.0408	0.741724	0.370862	+	0.928688i	$0.379062\pi$
$660$	0	0
$661$	23.6329	0.919213	0.459607	−	0.888123i	$-0.347990\pi$
$661$	23.6329	0.919213	0.459607	+	0.888123i	$0.347990\pi$
$662$	7.99779	0.310843
$663$	0	0
$664$	22.2452	0.863282
$665$	10.9365	0.424099
$666$	0	0
$667$	−1.39032	−0.0538335
$668$	8.91602	0.344971
$669$	0	0
$670$	13.8287	0.534250
$671$	20.6100	0.795642
$672$	0	0
$673$	−10.4509	−0.402853	−0.201426	−	0.979504i	$-0.564558\pi$
$673$	−10.4509	−0.402853	−0.201426	+	0.979504i	$0.564558\pi$
$674$	0.278427	0.0107246
$675$	0	0
$676$	4.12169	0.158526
$677$	−26.1735	−1.00593	−0.502965	−	0.864307i	$-0.667758\pi$
$677$	−26.1735	−1.00593	−0.502965	+	0.864307i	$0.667758\pi$
$678$	0	0
$679$	−1.62954	−0.0625361
$680$	16.0684	0.616196
$681$	0	0
$682$	−16.9570	−0.649317
$683$	−40.9830	−1.56817	−0.784085	−	0.620653i	$-0.786867\pi$
$683$	−40.9830	−1.56817	−0.784085	+	0.620653i	$0.786867\pi$
$684$	0	0
$685$	22.3393	0.853540
$686$	−0.823573	−0.0314442
$687$	0	0
$688$	1.76123	0.0671462
$689$	−26.1256	−0.995308
$690$	0	0
$691$	−26.1490	−0.994754	−0.497377	−	0.867534i	$-0.665703\pi$
$691$	−26.1490	−0.994754	−0.497377	+	0.867534i	$0.665703\pi$
$692$	17.1950	0.653658
$693$	0	0
$694$	8.90130	0.337889
$695$	37.8805	1.43689
$696$	0	0
$697$	4.37911	0.165871
$698$	24.8751	0.941537
$699$	0	0
$700$	−4.33785	−0.163955
$701$	−4.34759	−0.164206	−0.0821030	−	0.996624i	$-0.526164\pi$
$701$	−4.34759	−0.164206	−0.0821030	+	0.996624i	$0.526164\pi$
$702$	0	0
$703$	0.189390	0.00714298
$704$	9.96457	0.375554
$705$	0	0
$706$	−19.8222	−0.746020
$707$	18.5531	0.697763
$708$	0	0
$709$	32.4246	1.21773	0.608865	−	0.793274i	$-0.291625\pi$
$709$	32.4246	1.21773	0.608865	+	0.793274i	$0.291625\pi$
$710$	19.0393	0.714531
$711$	0	0
$712$	42.3511	1.58717
$713$	1.47368	0.0551896
$714$	0	0
$715$	−22.5923	−0.844903
$716$	−17.2572	−0.644933
$717$	0	0
$718$	5.09457	0.190128
$719$	8.83585	0.329522	0.164761	−	0.986334i	$-0.447315\pi$
$719$	8.83585	0.329522	0.164761	+	0.986334i	$0.447315\pi$
$720$	0	0
$721$	13.8022	0.514020
$722$	3.75397	0.139708
$723$	0	0
$724$	10.9494	0.406932
$725$	−25.5278	−0.948077
$726$	0	0
$727$	22.5666	0.836950	0.418475	−	0.908228i	$-0.362565\pi$
$727$	22.5666	0.836950	0.418475	+	0.908228i	$0.362565\pi$
$728$	−8.59963	−0.318723
$729$	0	0
$730$	−33.2050	−1.22897
$731$	9.20724	0.340542
$732$	0	0
$733$	18.1667	0.671001	0.335501	−	0.942040i	$-0.391095\pi$
$733$	18.1667	0.671001	0.335501	+	0.942040i	$0.391095\pi$
$734$	9.50068	0.350677
$735$	0	0
$736$	−1.03546	−0.0381675
$737$	−14.5711	−0.536734
$738$	0	0
$739$	15.3064	0.563055	0.281527	−	0.959553i	$-0.409159\pi$
$739$	15.3064	0.563055	0.281527	+	0.959553i	$0.409159\pi$
$740$	−0.189563	−0.00696848
$741$	0	0
$742$	−6.84472	−0.251278
$743$	12.1332	0.445125	0.222562	−	0.974918i	$-0.428558\pi$
$743$	12.1332	0.445125	0.222562	+	0.974918i	$0.428558\pi$
$744$	0	0
$745$	−42.1500	−1.54426
$746$	16.3302	0.597889
$747$	0	0
$748$	−6.73693	−0.246327
$749$	−11.5898	−0.423483
$750$	0	0
$751$	−41.2242	−1.50429	−0.752147	−	0.658996i	$-0.770981\pi$
$751$	−41.2242	−1.50429	−0.752147	+	0.658996i	$0.770981\pi$
$752$	0.152868	0.00557452
$753$	0	0
$754$	−20.1371	−0.733348
$755$	−16.8183	−0.612081
$756$	0	0
$757$	−41.3673	−1.50352	−0.751759	−	0.659437i	$-0.770795\pi$
$757$	−41.3673	−1.50352	−0.751759	+	0.659437i	$0.770795\pi$
$758$	8.07105	0.293154
$759$	0	0
$760$	29.9188	1.08527
$761$	1.38604	0.0502440	0.0251220	−	0.999684i	$-0.492003\pi$
$761$	1.38604	0.0502440	0.0251220	+	0.999684i	$0.492003\pi$
$762$	0	0
$763$	4.90568	0.177598
$764$	1.94773	0.0704666
$765$	0	0
$766$	6.97924	0.252170
$767$	45.0418	1.62636
$768$	0	0
$769$	−30.4676	−1.09869	−0.549344	−	0.835596i	$-0.685122\pi$
$769$	−30.4676	−1.09869	−0.549344	+	0.835596i	$0.685122\pi$
$770$	−5.91900	−0.213306
$771$	0	0
$772$	4.81468	0.173284
$773$	40.1159	1.44287	0.721435	−	0.692482i	$-0.243483\pi$
$773$	40.1159	1.44287	0.721435	+	0.692482i	$0.243483\pi$
$774$	0	0
$775$	27.0582	0.971960
$776$	−4.45792	−0.160030
$777$	0	0
$778$	−31.8963	−1.14354
$779$	8.15373	0.292138
$780$	0	0
$781$	−20.0614	−0.717853
$782$	−0.300454	−0.0107442
$783$	0	0
$784$	0.390415	0.0139434
$785$	6.48190	0.231349
$786$	0	0
$787$	−31.5702	−1.12536	−0.562678	−	0.826676i	$-0.690229\pi$
$787$	−31.5702	−1.12536	−0.562678	+	0.826676i	$0.690229\pi$
$788$	2.05708	0.0732804
$789$	0	0
$790$	13.5775	0.483064
$791$	−11.4900	−0.408538
$792$	0	0
$793$	−25.9425	−0.921247
$794$	−12.4030	−0.440166
$795$	0	0
$796$	6.37976	0.226124
$797$	−46.5952	−1.65049	−0.825244	−	0.564777i	$-0.808962\pi$
$797$	−46.5952	−1.65049	−0.825244	+	0.564777i	$0.808962\pi$
$798$	0	0
$799$	0.799154	0.0282720
$800$	−19.0121	−0.672178
$801$	0	0
$802$	5.94005	0.209750
$803$	34.9876	1.23469
$804$	0	0
$805$	0.514401	0.0181303
$806$	21.3443	0.751822
$807$	0	0
$808$	50.7556	1.78558
$809$	1.54408	0.0542868	0.0271434	−	0.999632i	$-0.491359\pi$
$809$	1.54408	0.0542868	0.0271434	+	0.999632i	$0.491359\pi$
$810$	0	0
$811$	20.5329	0.721009	0.360505	−	0.932757i	$-0.382604\pi$
$811$	20.5329	0.721009	0.360505	+	0.932757i	$0.382604\pi$
$812$	10.2807	0.360781
$813$	0	0
$814$	−0.102501	−0.00359265
$815$	51.6873	1.81053
$816$	0	0
$817$	17.1435	0.599776
$818$	−18.4679	−0.645715
$819$	0	0
$820$	−8.16119	−0.285001
$821$	−15.0246	−0.524362	−0.262181	−	0.965019i	$-0.584442\pi$
$821$	−15.0246	−0.524362	−0.262181	+	0.965019i	$0.584442\pi$
$822$	0	0
$823$	−19.6086	−0.683513	−0.341756	−	0.939789i	$-0.611022\pi$
$823$	−19.6086	−0.683513	−0.341756	+	0.939789i	$0.611022\pi$
$824$	37.7584	1.31538
$825$	0	0
$826$	11.8006	0.410596
$827$	21.4027	0.744246	0.372123	−	0.928183i	$-0.378630\pi$
$827$	21.4027	0.744246	0.372123	+	0.928183i	$0.378630\pi$
$828$	0	0
$829$	−7.52819	−0.261465	−0.130732	−	0.991418i	$-0.541733\pi$
$829$	−7.52819	−0.261465	−0.130732	+	0.991418i	$0.541733\pi$
$830$	−19.2726	−0.668960
$831$	0	0
$832$	−12.5427	−0.434841
$833$	2.04099	0.0707161
$834$	0	0
$835$	−19.4131	−0.671819
$836$	−12.5439	−0.433840
$837$	0	0
$838$	−28.8441	−0.996404
$839$	−1.62392	−0.0560640	−0.0280320	−	0.999607i	$-0.508924\pi$
$839$	−1.62392	−0.0560640	−0.0280320	+	0.999607i	$0.508924\pi$
$840$	0	0
$841$	31.5007	1.08623
$842$	−3.73394	−0.128680
$843$	0	0
$844$	−17.5460	−0.603959
$845$	−8.97428	−0.308725
$846$	0	0
$847$	−4.76323	−0.163667
$848$	3.24474	0.111425
$849$	0	0
$850$	−5.51665	−0.189220
$851$	0.00890802	0.000305363 0
$852$	0	0
$853$	50.2026	1.71891	0.859453	−	0.511215i	$-0.170804\pi$
$853$	50.2026	1.71891	0.859453	+	0.511215i	$0.170804\pi$
$854$	−6.79675	−0.232580
$855$	0	0
$856$	−31.7062	−1.08369
$857$	14.6499	0.500430	0.250215	−	0.968190i	$-0.419499\pi$
$857$	14.6499	0.500430	0.250215	+	0.968190i	$0.419499\pi$
$858$	0	0
$859$	15.5012	0.528893	0.264447	−	0.964400i	$-0.414811\pi$
$859$	15.5012	0.528893	0.264447	+	0.964400i	$0.414811\pi$
$860$	−17.1592	−0.585124
$861$	0	0
$862$	−6.19287	−0.210930
$863$	4.15587	0.141468	0.0707338	−	0.997495i	$-0.477466\pi$
$863$	4.15587	0.141468	0.0707338	+	0.997495i	$0.477466\pi$
$864$	0	0
$865$	−37.4393	−1.27298
$866$	−23.3910	−0.794858
$867$	0	0
$868$	−10.8970	−0.369869
$869$	−14.3064	−0.485310
$870$	0	0
$871$	18.3412	0.621466
$872$	13.4204	0.454472
$873$	0	0
$874$	−0.559435	−0.0189232
$875$	−4.94426	−0.167146
$876$	0	0
$877$	−29.3465	−0.990963	−0.495481	−	0.868619i	$-0.665008\pi$
$877$	−29.3465	−0.990963	−0.495481	+	0.868619i	$0.665008\pi$
$878$	−17.5921	−0.593705
$879$	0	0
$880$	2.80590	0.0945870
$881$	43.6729	1.47138	0.735688	−	0.677320i	$-0.236859\pi$
$881$	43.6729	1.47138	0.735688	+	0.677320i	$0.236859\pi$
$882$	0	0
$883$	−45.3693	−1.52680	−0.763399	−	0.645927i	$-0.776471\pi$
$883$	−45.3693	−1.52680	−0.763399	+	0.645927i	$0.776471\pi$
$884$	8.48000	0.285213
$885$	0	0
$886$	11.9675	0.402057
$887$	29.4945	0.990330	0.495165	−	0.868799i	$-0.335108\pi$
$887$	29.4945	0.990330	0.495165	+	0.868799i	$0.335108\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−36.6916	−1.22991
$891$	0	0
$892$	−23.2003	−0.776802
$893$	1.48799	0.0497939
$894$	0	0
$895$	37.5747	1.25598
$896$	8.29971	0.277274
$897$	0	0
$898$	−1.28245	−0.0427958
$899$	−64.1278	−2.13878
$900$	0	0
$901$	16.9627	0.565108
$902$	−4.41293	−0.146935
$903$	0	0
$904$	−31.4331	−1.04545
$905$	−23.8405	−0.792485
$906$	0	0
$907$	−39.5099	−1.31190	−0.655952	−	0.754802i	$-0.727733\pi$
$907$	−39.5099	−1.31190	−0.655952	+	0.754802i	$0.727733\pi$
$908$	24.5543	0.814863
$909$	0	0
$910$	7.45044	0.246980
$911$	−14.6770	−0.486271	−0.243135	−	0.969992i	$-0.578176\pi$
$911$	−14.6770	−0.486271	−0.243135	+	0.969992i	$0.578176\pi$
$912$	0	0
$913$	20.3072	0.672071
$914$	0.990364	0.0327583
$915$	0	0
$916$	−30.2911	−1.00085
$917$	−4.61325	−0.152343
$918$	0	0
$919$	15.1107	0.498457	0.249229	−	0.968445i	$-0.419823\pi$
$919$	15.1107	0.498457	0.249229	+	0.968445i	$0.419823\pi$
$920$	1.40724	0.0463953
$921$	0	0
$922$	26.7163	0.879854
$923$	25.2519	0.831178
$924$	0	0
$925$	0.163560	0.00537783
$926$	−19.8608	−0.652665
$927$	0	0
$928$	45.0585	1.47912
$929$	27.4154	0.899469	0.449735	−	0.893162i	$-0.351519\pi$
$929$	27.4154	0.899469	0.449735	+	0.893162i	$0.351519\pi$
$930$	0	0
$931$	3.80024	0.124548
$932$	−5.54073	−0.181493
$933$	0	0
$934$	22.7165	0.743306
$935$	14.6685	0.479712
$936$	0	0
$937$	2.63753	0.0861644	0.0430822	−	0.999072i	$-0.486282\pi$
$937$	2.63753	0.0861644	0.0430822	+	0.999072i	$0.486282\pi$
$938$	4.80525	0.156897
$939$	0	0
$940$	−1.48935	−0.0485774
$941$	−47.8351	−1.55938	−0.779690	−	0.626166i	$-0.784623\pi$
$941$	−47.8351	−1.55938	−0.779690	+	0.626166i	$0.784623\pi$
$942$	0	0
$943$	0.383514	0.0124889
$944$	−5.59408	−0.182072
$945$	0	0
$946$	−9.27836	−0.301665
$947$	59.9898	1.94941	0.974704	−	0.223500i	$-0.0717483\pi$
$947$	59.9898	1.94941	0.974704	+	0.223500i	$0.0717483\pi$
$948$	0	0
$949$	−44.0400	−1.42960
$950$	−10.2718	−0.333261
$951$	0	0
$952$	5.58350	0.180962
$953$	10.9525	0.354785	0.177393	−	0.984140i	$-0.443234\pi$
$953$	10.9525	0.354785	0.177393	+	0.984140i	$0.443234\pi$
$954$	0	0
$955$	−4.24087	−0.137231
$956$	−21.6911	−0.701541
$957$	0	0
$958$	−21.9870	−0.710367
$959$	7.76252	0.250665
$960$	0	0
$961$	36.9725	1.19266
$962$	0.129021	0.00415981
$963$	0	0
$964$	25.6587	0.826412
$965$	−10.4832	−0.337465
$966$	0	0
$967$	5.81937	0.187138	0.0935692	−	0.995613i	$-0.470172\pi$
$967$	5.81937	0.187138	0.0935692	+	0.995613i	$0.470172\pi$
$968$	−13.0307	−0.418823
$969$	0	0
$970$	3.86220	0.124008
$971$	−27.0322	−0.867505	−0.433752	−	0.901032i	$-0.642811\pi$
$971$	−27.0322	−0.867505	−0.433752	+	0.901032i	$0.642811\pi$
$972$	0	0
$973$	13.1628	0.421981
$974$	−16.7534	−0.536813
$975$	0	0
$976$	3.22200	0.103134
$977$	−18.5562	−0.593666	−0.296833	−	0.954929i	$-0.595930\pi$
$977$	−18.5562	−0.593666	−0.296833	+	0.954929i	$0.595930\pi$
$978$	0	0
$979$	38.6615	1.23563
$980$	−3.80372	−0.121505
$981$	0	0
$982$	−8.51941	−0.271865
$983$	−0.886444	−0.0282732	−0.0141366	−	0.999900i	$-0.504500\pi$
$983$	−0.886444	−0.0282732	−0.0141366	+	0.999900i	$0.504500\pi$
$984$	0	0
$985$	−4.47895	−0.142711
$986$	13.0744	0.416375
$987$	0	0
$988$	15.7894	0.502329
$989$	0.806352	0.0256405
$990$	0	0
$991$	3.64119	0.115666	0.0578331	−	0.998326i	$-0.481581\pi$
$991$	3.64119	0.115666	0.0578331	+	0.998326i	$0.481581\pi$
$992$	−47.7599	−1.51638
$993$	0	0
$994$	6.61582	0.209841
$995$	−13.8908	−0.440370
$996$	0	0
$997$	−34.8191	−1.10273	−0.551367	−	0.834263i	$-0.685893\pi$
$997$	−34.8191	−1.10273	−0.551367	+	0.834263i	$0.685893\pi$
$998$	−7.10848	−0.225015
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.v.1.9		19
3.2	odd	2		2667.2.a.q.1.11	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.q.1.11	✓	19	3.2	odd	2
8001.2.a.v.1.9		19	1.1	even	1	trivial

Overview	Random
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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)

\(f(q)\)	\(=\)	\(q-0.823573 q^{2} -1.32173 q^{4} +2.87784 q^{5} +1.00000 q^{7} +2.73569 q^{8} +O(q^{10})\)	\(q-0.823573 q^{2} -1.32173 q^{4} +2.87784 q^{5} +1.00000 q^{7} +2.73569 q^{8} -2.37011 q^{10} +2.49735 q^{11} -3.14350 q^{13} -0.823573 q^{14} +0.390415 q^{16} +2.04099 q^{17} +3.80024 q^{19} -3.80372 q^{20} -2.05675 q^{22} +0.178746 q^{23} +3.28195 q^{25} +2.58890 q^{26} -1.32173 q^{28} -7.77822 q^{29} +8.24454 q^{31} -5.79291 q^{32} -1.68090 q^{34} +2.87784 q^{35} +0.0498363 q^{37} -3.12978 q^{38} +7.87286 q^{40} +2.14558 q^{41} +4.51117 q^{43} -3.30082 q^{44} -0.147210 q^{46} +0.391553 q^{47} +1.00000 q^{49} -2.70293 q^{50} +4.15485 q^{52} +8.31101 q^{53} +7.18698 q^{55} +2.73569 q^{56} +6.40593 q^{58} -14.3285 q^{59} +8.25276 q^{61} -6.78999 q^{62} +3.99005 q^{64} -9.04648 q^{65} -5.83463 q^{67} -2.69763 q^{68} -2.37011 q^{70} -8.03307 q^{71} +14.0099 q^{73} -0.0410438 q^{74} -5.02288 q^{76} +2.49735 q^{77} -5.72861 q^{79} +1.12355 q^{80} -1.76704 q^{82} +8.13150 q^{83} +5.87364 q^{85} -3.71528 q^{86} +6.83197 q^{88} +15.4810 q^{89} -3.14350 q^{91} -0.236253 q^{92} -0.322472 q^{94} +10.9365 q^{95} -1.62954 q^{97} -0.823573 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8}+O(q^{10}) \) 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8	\( 19 q - 4 q^{2} + 22 q^{4} - 5 q^{5} + 19 q^{7} - 9 q^{8} + 9 q^{11} + 24 q^{13} - 4 q^{14} + 20 q^{16} - 17 q^{17} + 23 q^{19} - 5 q^{20} - 3 q^{22} + 17 q^{23} + 38 q^{25} - 28 q^{26} + 22 q^{28} - 2 q^{29} + 16 q^{31} - 17 q^{32} + 29 q^{34} - 5 q^{35} + 56 q^{37} - 2 q^{38} - 13 q^{40} + 7 q^{41} + 19 q^{43} + 29 q^{44} + 10 q^{46} - 25 q^{47} + 19 q^{49} + 9 q^{50} + 16 q^{52} - 18 q^{53} + 10 q^{55} - 9 q^{56} + 31 q^{58} - 11 q^{59} + 26 q^{61} - 26 q^{62} + 45 q^{64} - 27 q^{65} + 24 q^{67} - 14 q^{68} + 32 q^{71} + 51 q^{73} + 12 q^{76} + 9 q^{77} + 30 q^{79} + 30 q^{80} - 52 q^{82} - q^{83} + 44 q^{85} + 24 q^{86} - 30 q^{88} - 5 q^{89} + 24 q^{91} + 88 q^{92} + 7 q^{94} + 24 q^{95} + 5 q^{97} - 4 q^{98}+O(q^{100}) \) 19 * q - 4 * q^2 + 22 * q^4 - 5 * q^5 + 19 * q^7 - 9 * q^8 + 9 * q^11 + 24 * q^13 - 4 * q^14 + 20 * q^16 - 17 * q^17 + 23 * q^19 - 5 * q^20 - 3 * q^22 + 17 * q^23 + 38 * q^25 - 28 * q^26 + 22 * q^28 - 2 * q^29 + 16 * q^31 - 17 * q^32 + 29 * q^34 - 5 * q^35 + 56 * q^37 - 2 * q^38 - 13 * q^40 + 7 * q^41 + 19 * q^43 + 29 * q^44 + 10 * q^46 - 25 * q^47 + 19 * q^49 + 9 * q^50 + 16 * q^52 - 18 * q^53 + 10 * q^55 - 9 * q^56 + 31 * q^58 - 11 * q^59 + 26 * q^61 - 26 * q^62 + 45 * q^64 - 27 * q^65 + 24 * q^67 - 14 * q^68 + 32 * q^71 + 51 * q^73 + 12 * q^76 + 9 * q^77 + 30 * q^79 + 30 * q^80 - 52 * q^82 - q^83 + 44 * q^85 + 24 * q^86 - 30 * q^88 - 5 * q^89 + 24 * q^91 + 88 * q^92 + 7 * q^94 + 24 * q^95 + 5 * q^97 - 4 * q^98

Introduction

L-functions

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