LMFDB - Embedded newform 8001.2.a.w.1.4

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:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 8 x^{19} + 152 x^{17} - 274 x^{16} - 1061 x^{15} + 3125 x^{14} + 2977 x^{13} - 15474 x^{12} + \cdots + 31 $ x^20 - 8x^19 + 152x^17 - 274x^16 - 1061x^15 + 3125x^14 + 2977x^13 - 15474x^12 - 56x^11 + 39579x^10 - 17664x^9 - 52271x^8 + 35701x^7 + 32493x^6 - 25504x^5 - 8607x^4 + 6812x^3 + 609x^2 - 425x + 31
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 889)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.24329$ 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)

$f(q)$	$=$	$q-2.24329 q^{2} +3.03233 q^{4} +1.67114 q^{5} +1.00000 q^{7} -2.31581 q^{8} +O(q^{10})$	\(q-2.24329 q^{2} +3.03233 q^{4} +1.67114 q^{5} +1.00000 q^{7} -2.31581 q^{8} -3.74884 q^{10} +0.0139093 q^{11} -1.81387 q^{13} -2.24329 q^{14} -0.869643 q^{16} +6.81811 q^{17} +7.14593 q^{19} +5.06743 q^{20} -0.0312026 q^{22} +0.0905788 q^{23} -2.20730 q^{25} +4.06904 q^{26} +3.03233 q^{28} -7.70047 q^{29} -8.56257 q^{31} +6.58247 q^{32} -15.2950 q^{34} +1.67114 q^{35} +1.24400 q^{37} -16.0304 q^{38} -3.87003 q^{40} +6.73120 q^{41} -6.87624 q^{43} +0.0421777 q^{44} -0.203194 q^{46} -7.32805 q^{47} +1.00000 q^{49} +4.95161 q^{50} -5.50026 q^{52} -13.0181 q^{53} +0.0232444 q^{55} -2.31581 q^{56} +17.2743 q^{58} -9.03994 q^{59} -1.91415 q^{61} +19.2083 q^{62} -13.0271 q^{64} -3.03123 q^{65} -2.93024 q^{67} +20.6747 q^{68} -3.74884 q^{70} -5.54335 q^{71} -3.15233 q^{73} -2.79064 q^{74} +21.6688 q^{76} +0.0139093 q^{77} -0.921078 q^{79} -1.45329 q^{80} -15.1000 q^{82} +4.65022 q^{83} +11.3940 q^{85} +15.4254 q^{86} -0.0322114 q^{88} +11.4784 q^{89} -1.81387 q^{91} +0.274665 q^{92} +16.4389 q^{94} +11.9418 q^{95} -8.25542 q^{97} -2.24329 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8}+O(q^{10}) $ 20 * q - 8 * q^2 + 24 * q^4 - 3 * q^5 + 20 * q^7 - 24 * q^8	$ 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8} - 8 q^{10} - 26 q^{11} - 4 q^{13} - 8 q^{14} + 24 q^{16} - 4 q^{17} + q^{19} + 2 q^{20} + q^{22} - 31 q^{23} + 27 q^{25} - 4 q^{26} + 24 q^{28} - 16 q^{29} + 6 q^{31} - 41 q^{32} - 10 q^{34} - 3 q^{35} + 2 q^{37} - 3 q^{38} - 38 q^{40} - 25 q^{41} + 13 q^{43} - 66 q^{44} + 20 q^{46} - 19 q^{47} + 20 q^{49} + 4 q^{50} + 20 q^{52} - 24 q^{53} - 3 q^{55} - 24 q^{56} + 12 q^{58} - 23 q^{59} - 27 q^{61} - 7 q^{62} + 2 q^{64} - 26 q^{65} + 9 q^{67} + 25 q^{68} - 8 q^{70} - 63 q^{71} - 21 q^{73} - 21 q^{74} - 10 q^{76} - 26 q^{77} + 18 q^{79} + 23 q^{80} - 42 q^{82} + q^{83} - 41 q^{85} + 12 q^{86} + 57 q^{88} + 16 q^{89} - 4 q^{91} - 17 q^{92} + 7 q^{94} - 75 q^{95} - 32 q^{97} - 8 q^{98}+O(q^{100}) $ 20 * q - 8 * q^2 + 24 * q^4 - 3 * q^5 + 20 * q^7 - 24 * q^8 - 8 * q^10 - 26 * q^11 - 4 * q^13 - 8 * q^14 + 24 * q^16 - 4 * q^17 + q^19 + 2 * q^20 + q^22 - 31 * q^23 + 27 * q^25 - 4 * q^26 + 24 * q^28 - 16 * q^29 + 6 * q^31 - 41 * q^32 - 10 * q^34 - 3 * q^35 + 2 * q^37 - 3 * q^38 - 38 * q^40 - 25 * q^41 + 13 * q^43 - 66 * q^44 + 20 * q^46 - 19 * q^47 + 20 * q^49 + 4 * q^50 + 20 * q^52 - 24 * q^53 - 3 * q^55 - 24 * q^56 + 12 * q^58 - 23 * q^59 - 27 * q^61 - 7 * q^62 + 2 * q^64 - 26 * q^65 + 9 * q^67 + 25 * q^68 - 8 * q^70 - 63 * q^71 - 21 * q^73 - 21 * q^74 - 10 * q^76 - 26 * q^77 + 18 * q^79 + 23 * q^80 - 42 * q^82 + q^83 - 41 * q^85 + 12 * q^86 + 57 * q^88 + 16 * q^89 - 4 * q^91 - 17 * q^92 + 7 * q^94 - 75 * q^95 - 32 * q^97 - 8 * 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 $

$2$	−2.24329	−1.58624	−0.793121	−	0.609064i	$-0.791545\pi$
$2$	−2.24329	−1.58624	−0.793121	+	0.609064i	$0.791545\pi$
$3$	0	0
$3$	0	0
$4$	3.03233	1.51616
$5$	1.67114	0.747355	0.373677	−	0.927559i	$-0.378097\pi$
$5$	1.67114	0.747355	0.373677	+	0.927559i	$0.378097\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.31581	−0.818761
$9$	0	0
$10$	−3.74884	−1.18549
$11$	0.0139093	0.00419383	0.00209691	−	0.999998i	$-0.499333\pi$
$11$	0.0139093	0.00419383	0.00209691	+	0.999998i	$0.499333\pi$
$12$	0	0
$13$	−1.81387	−0.503078	−0.251539	−	0.967847i	$-0.580937\pi$
$13$	−1.81387	−0.503078	−0.251539	+	0.967847i	$0.580937\pi$
$14$	−2.24329	−0.599543
$15$	0	0
$16$	−0.869643	−0.217411
$17$	6.81811	1.65363	0.826817	−	0.562471i	$-0.190149\pi$
$17$	6.81811	1.65363	0.826817	+	0.562471i	$0.190149\pi$
$18$	0	0
$19$	7.14593	1.63939	0.819695	−	0.572801i	$-0.194143\pi$
$19$	7.14593	1.63939	0.819695	+	0.572801i	$0.194143\pi$
$20$	5.06743	1.13311
$21$	0	0
$22$	−0.0312026	−0.00665242
$23$	0.0905788	0.0188870	0.00944349	−	0.999955i	$-0.496994\pi$
$23$	0.0905788	0.0188870	0.00944349	+	0.999955i	$0.496994\pi$
$24$	0	0
$25$	−2.20730	−0.441461
$26$	4.06904	0.798004
$27$	0	0
$28$	3.03233	0.573056
$29$	−7.70047	−1.42994	−0.714970	−	0.699155i	$-0.753560\pi$
$29$	−7.70047	−1.42994	−0.714970	+	0.699155i	$0.753560\pi$
$30$	0	0
$31$	−8.56257	−1.53788	−0.768942	−	0.639319i	$-0.779216\pi$
$31$	−8.56257	−1.53788	−0.768942	+	0.639319i	$0.779216\pi$
$32$	6.58247	1.16363
$33$	0	0
$34$	−15.2950	−2.62306
$35$	1.67114	0.282474
$36$	0	0
$37$	1.24400	0.204512	0.102256	−	0.994758i	$-0.467394\pi$
$37$	1.24400	0.204512	0.102256	+	0.994758i	$0.467394\pi$
$38$	−16.0304	−2.60047
$39$	0	0
$40$	−3.87003	−0.611905
$41$	6.73120	1.05124	0.525618	−	0.850721i	$-0.323834\pi$
$41$	6.73120	1.05124	0.525618	+	0.850721i	$0.323834\pi$
$42$	0	0
$43$	−6.87624	−1.04862	−0.524308	−	0.851529i	$-0.675676\pi$
$43$	−6.87624	−1.04862	−0.524308	+	0.851529i	$0.675676\pi$
$44$	0.0421777	0.00635853
$45$	0	0
$46$	−0.203194	−0.0299593
$47$	−7.32805	−1.06891	−0.534453	−	0.845198i	$-0.679482\pi$
$47$	−7.32805	−1.06891	−0.534453	+	0.845198i	$0.679482\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	4.95161	0.700263
$51$	0	0
$52$	−5.50026	−0.762749
$53$	−13.0181	−1.78817	−0.894087	−	0.447894i	$-0.852174\pi$
$53$	−13.0181	−1.78817	−0.894087	+	0.447894i	$0.852174\pi$
$54$	0	0
$55$	0.0232444	0.00313428
$56$	−2.31581	−0.309463
$57$	0	0
$58$	17.2743	2.26823
$59$	−9.03994	−1.17690	−0.588450	−	0.808534i	$-0.700262\pi$
$59$	−9.03994	−1.17690	−0.588450	+	0.808534i	$0.700262\pi$
$60$	0	0
$61$	−1.91415	−0.245082	−0.122541	−	0.992463i	$-0.539104\pi$
$61$	−1.91415	−0.245082	−0.122541	+	0.992463i	$0.539104\pi$
$62$	19.2083	2.43946
$63$	0	0
$64$	−13.0271	−1.62838
$65$	−3.03123	−0.375978
$66$	0	0
$67$	−2.93024	−0.357985	−0.178993	−	0.983850i	$-0.557284\pi$
$67$	−2.93024	−0.357985	−0.178993	+	0.983850i	$0.557284\pi$
$68$	20.6747	2.50718
$69$	0	0
$70$	−3.74884	−0.448072
$71$	−5.54335	−0.657875	−0.328937	−	0.944352i	$-0.606690\pi$
$71$	−5.54335	−0.657875	−0.328937	+	0.944352i	$0.606690\pi$
$72$	0	0
$73$	−3.15233	−0.368952	−0.184476	−	0.982837i	$-0.559059\pi$
$73$	−3.15233	−0.368952	−0.184476	+	0.982837i	$0.559059\pi$
$74$	−2.79064	−0.324405
$75$	0	0
$76$	21.6688	2.48558
$77$	0.0139093	0.00158512
$78$	0	0
$79$	−0.921078	−0.103629	−0.0518147	−	0.998657i	$-0.516501\pi$
$79$	−0.921078	−0.103629	−0.0518147	+	0.998657i	$0.516501\pi$
$80$	−1.45329	−0.162483
$81$	0	0
$82$	−15.1000	−1.66751
$83$	4.65022	0.510428	0.255214	−	0.966885i	$-0.417854\pi$
$83$	4.65022	0.510428	0.255214	+	0.966885i	$0.417854\pi$
$84$	0	0
$85$	11.3940	1.23585
$86$	15.4254	1.66336
$87$	0	0
$88$	−0.0322114	−0.00343374
$89$	11.4784	1.21671	0.608353	−	0.793667i	$-0.291830\pi$
$89$	11.4784	1.21671	0.608353	+	0.793667i	$0.291830\pi$
$90$	0	0
$91$	−1.81387	−0.190146
$92$	0.274665	0.0286358
$93$	0	0
$94$	16.4389	1.69554
$95$	11.9418	1.22521
$96$	0	0
$97$	−8.25542	−0.838211	−0.419106	−	0.907937i	$-0.637656\pi$
$97$	−8.25542	−0.838211	−0.419106	+	0.907937i	$0.637656\pi$
$98$	−2.24329	−0.226606
$99$	0	0
$100$	−6.69327	−0.669327
$101$	−13.6032	−1.35357	−0.676785	−	0.736181i	$-0.736627\pi$
$101$	−13.6032	−1.35357	−0.676785	+	0.736181i	$0.736627\pi$
$102$	0	0
$103$	−11.2022	−1.10378	−0.551891	−	0.833916i	$-0.686094\pi$
$103$	−11.2022	−1.10378	−0.551891	+	0.833916i	$0.686094\pi$
$104$	4.20058	0.411901
$105$	0	0
$106$	29.2033	2.83648
$107$	−0.369436	−0.0357147	−0.0178574	−	0.999841i	$-0.505684\pi$
$107$	−0.369436	−0.0357147	−0.0178574	+	0.999841i	$0.505684\pi$
$108$	0	0
$109$	18.7281	1.79382	0.896912	−	0.442210i	$-0.145805\pi$
$109$	18.7281	1.79382	0.896912	+	0.442210i	$0.145805\pi$
$110$	−0.0521439	−0.00497172
$111$	0	0
$112$	−0.869643	−0.0821735
$113$	−6.92572	−0.651517	−0.325758	−	0.945453i	$-0.605620\pi$
$113$	−6.92572	−0.651517	−0.325758	+	0.945453i	$0.605620\pi$
$114$	0	0
$115$	0.151369	0.0141153
$116$	−23.3503	−2.16803
$117$	0	0
$118$	20.2792	1.86685
$119$	6.81811	0.625015
$120$	0	0
$121$	−10.9998	−0.999982
$122$	4.29398	0.388759
$123$	0	0
$124$	−25.9645	−2.33168
$125$	−12.0444	−1.07728
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	16.0585	1.41938
$129$	0	0
$130$	6.79992	0.596392
$131$	−12.1147	−1.05847	−0.529233	−	0.848476i	$-0.677520\pi$
$131$	−12.1147	−1.05847	−0.529233	+	0.848476i	$0.677520\pi$
$132$	0	0
$133$	7.14593	0.619631
$134$	6.57335	0.567851
$135$	0	0
$136$	−15.7894	−1.35393
$137$	−4.69402	−0.401037	−0.200519	−	0.979690i	$-0.564263\pi$
$137$	−4.69402	−0.401037	−0.200519	+	0.979690i	$0.564263\pi$
$138$	0	0
$139$	−17.9782	−1.52489	−0.762445	−	0.647053i	$-0.776001\pi$
$139$	−17.9782	−1.52489	−0.762445	+	0.647053i	$0.776001\pi$
$140$	5.06743	0.428276
$141$	0	0
$142$	12.4353	1.04355
$143$	−0.0252298	−0.00210982
$144$	0	0
$145$	−12.8685	−1.06867
$146$	7.07157	0.585247
$147$	0	0
$148$	3.77221	0.310073
$149$	0.549953	0.0450539	0.0225270	−	0.999746i	$-0.492829\pi$
$149$	0.549953	0.0450539	0.0225270	+	0.999746i	$0.492829\pi$
$150$	0	0
$151$	10.3675	0.843698	0.421849	−	0.906666i	$-0.361381\pi$
$151$	10.3675	0.843698	0.421849	+	0.906666i	$0.361381\pi$
$152$	−16.5486	−1.34227
$153$	0	0
$154$	−0.0312026	−0.00251438
$155$	−14.3092	−1.14934
$156$	0	0
$157$	0.165635	0.0132191	0.00660956	−	0.999978i	$-0.497896\pi$
$157$	0.165635	0.0132191	0.00660956	+	0.999978i	$0.497896\pi$
$158$	2.06624	0.164381
$159$	0	0
$160$	11.0002	0.869642
$161$	0.0905788	0.00713861
$162$	0	0
$163$	1.07614	0.0842900	0.0421450	−	0.999112i	$-0.486581\pi$
$163$	1.07614	0.0842900	0.0421450	+	0.999112i	$0.486581\pi$
$164$	20.4112	1.59385
$165$	0	0
$166$	−10.4318	−0.809663
$167$	7.35985	0.569522	0.284761	−	0.958599i	$-0.408086\pi$
$167$	7.35985	0.569522	0.284761	+	0.958599i	$0.408086\pi$
$168$	0	0
$169$	−9.70986	−0.746912
$170$	−25.5600	−1.96036
$171$	0	0
$172$	−20.8510	−1.58987
$173$	−24.2304	−1.84221	−0.921103	−	0.389318i	$-0.872711\pi$
$173$	−24.2304	−1.84221	−0.921103	+	0.389318i	$0.872711\pi$
$174$	0	0
$175$	−2.20730	−0.166856
$176$	−0.0120962	−0.000911783 0
$177$	0	0
$178$	−25.7493	−1.92999
$179$	−13.9864	−1.04539	−0.522697	−	0.852519i	$-0.675074\pi$
$179$	−13.9864	−1.04539	−0.522697	+	0.852519i	$0.675074\pi$
$180$	0	0
$181$	8.66121	0.643783	0.321891	−	0.946777i	$-0.395681\pi$
$181$	8.66121	0.643783	0.321891	+	0.946777i	$0.395681\pi$
$182$	4.06904	0.301617
$183$	0	0
$184$	−0.209763	−0.0154639
$185$	2.07889	0.152843
$186$	0	0
$187$	0.0948354	0.00693505
$188$	−22.2211	−1.62064
$189$	0	0
$190$	−26.7889	−1.94347
$191$	−5.87506	−0.425104	−0.212552	−	0.977150i	$-0.568178\pi$
$191$	−5.87506	−0.425104	−0.212552	+	0.977150i	$0.568178\pi$
$192$	0	0
$193$	17.4207	1.25397	0.626985	−	0.779031i	$-0.284289\pi$
$193$	17.4207	1.25397	0.626985	+	0.779031i	$0.284289\pi$
$194$	18.5193	1.32961
$195$	0	0
$196$	3.03233	0.216595
$197$	−0.137384	−0.00978819	−0.00489410	−	0.999988i	$-0.501558\pi$
$197$	−0.137384	−0.00978819	−0.00489410	+	0.999988i	$0.501558\pi$
$198$	0	0
$199$	−10.7655	−0.763146	−0.381573	−	0.924339i	$-0.624618\pi$
$199$	−10.7655	−0.763146	−0.381573	+	0.924339i	$0.624618\pi$
$200$	5.11169	0.361451
$201$	0	0
$202$	30.5159	2.14709
$203$	−7.70047	−0.540467
$204$	0	0
$205$	11.2487	0.785646
$206$	25.1296	1.75087
$207$	0	0
$208$	1.57742	0.109375
$209$	0.0993953	0.00687531
$210$	0	0
$211$	25.6149	1.76340	0.881700	−	0.471811i	$-0.156400\pi$
$211$	25.6149	1.76340	0.881700	+	0.471811i	$0.156400\pi$
$212$	−39.4752	−2.71116
$213$	0	0
$214$	0.828750	0.0566522
$215$	−11.4911	−0.783689
$216$	0	0
$217$	−8.56257	−0.581265
$218$	−42.0124	−2.84544
$219$	0	0
$220$	0.0704847	0.00475208
$221$	−12.3672	−0.831907
$222$	0	0
$223$	−25.1144	−1.68179	−0.840893	−	0.541201i	$-0.817970\pi$
$223$	−25.1144	−1.68179	−0.840893	+	0.541201i	$0.817970\pi$
$224$	6.58247	0.439810
$225$	0	0
$226$	15.5364	1.03346
$227$	23.0330	1.52875	0.764376	−	0.644771i	$-0.223047\pi$
$227$	23.0330	1.52875	0.764376	+	0.644771i	$0.223047\pi$
$228$	0	0
$229$	−13.8748	−0.916874	−0.458437	−	0.888727i	$-0.651591\pi$
$229$	−13.8748	−0.916874	−0.458437	+	0.888727i	$0.651591\pi$
$230$	−0.339565	−0.0223902
$231$	0	0
$232$	17.8328	1.17078
$233$	−1.97593	−0.129448	−0.0647238	−	0.997903i	$-0.520617\pi$
$233$	−1.97593	−0.129448	−0.0647238	+	0.997903i	$0.520617\pi$
$234$	0	0
$235$	−12.2462	−0.798852
$236$	−27.4121	−1.78437
$237$	0	0
$238$	−15.2950	−0.991425
$239$	−20.1115	−1.30090	−0.650452	−	0.759548i	$-0.725420\pi$
$239$	−20.1115	−1.30090	−0.650452	+	0.759548i	$0.725420\pi$
$240$	0	0
$241$	16.5651	1.06705	0.533527	−	0.845783i	$-0.320866\pi$
$241$	16.5651	1.06705	0.533527	+	0.845783i	$0.320866\pi$
$242$	24.6757	1.58621
$243$	0	0
$244$	−5.80433	−0.371584
$245$	1.67114	0.106765
$246$	0	0
$247$	−12.9618	−0.824741
$248$	19.8293	1.25916
$249$	0	0
$250$	27.0190	1.70883
$251$	12.6036	0.795534	0.397767	−	0.917486i	$-0.369785\pi$
$251$	12.6036	0.795534	0.397767	+	0.917486i	$0.369785\pi$
$252$	0	0
$253$	0.00125989	7.92087e−5 0
$254$	2.24329	0.140756
$255$	0	0
$256$	−9.96964	−0.623102
$257$	−21.1699	−1.32054	−0.660272	−	0.751026i	$-0.729559\pi$
$257$	−21.1699	−1.32054	−0.660272	+	0.751026i	$0.729559\pi$
$258$	0	0
$259$	1.24400	0.0772982
$260$	−9.19169	−0.570044
$261$	0	0
$262$	27.1767	1.67898
$263$	15.7598	0.971789	0.485895	−	0.874017i	$-0.338494\pi$
$263$	15.7598	0.971789	0.485895	+	0.874017i	$0.338494\pi$
$264$	0	0
$265$	−21.7550	−1.33640
$266$	−16.0304	−0.982885
$267$	0	0
$268$	−8.88544	−0.542764
$269$	−16.6546	−1.01545	−0.507725	−	0.861519i	$-0.669513\pi$
$269$	−16.6546	−1.01545	−0.507725	+	0.861519i	$0.669513\pi$
$270$	0	0
$271$	−0.191561	−0.0116365	−0.00581824	−	0.999983i	$-0.501852\pi$
$271$	−0.191561	−0.0116365	−0.00581824	+	0.999983i	$0.501852\pi$
$272$	−5.92932	−0.359518
$273$	0	0
$274$	10.5300	0.636142
$275$	−0.0307022	−0.00185141
$276$	0	0
$277$	18.0715	1.08581	0.542906	−	0.839793i	$-0.317324\pi$
$277$	18.0715	1.08581	0.542906	+	0.839793i	$0.317324\pi$
$278$	40.3302	2.41884
$279$	0	0
$280$	−3.87003	−0.231278
$281$	23.4850	1.40100	0.700500	−	0.713652i	$-0.252960\pi$
$281$	23.4850	1.40100	0.700500	+	0.713652i	$0.252960\pi$
$282$	0	0
$283$	−20.9438	−1.24498	−0.622490	−	0.782628i	$-0.713879\pi$
$283$	−20.9438	−1.24498	−0.622490	+	0.782628i	$0.713879\pi$
$284$	−16.8093	−0.997446
$285$	0	0
$286$	0.0565977	0.00334669
$287$	6.73120	0.397330
$288$	0	0
$289$	29.4866	1.73450
$290$	28.8678	1.69517
$291$	0	0
$292$	−9.55889	−0.559392
$293$	17.7504	1.03699	0.518494	−	0.855081i	$-0.326493\pi$
$293$	17.7504	1.03699	0.518494	+	0.855081i	$0.326493\pi$
$294$	0	0
$295$	−15.1070	−0.879562
$296$	−2.88085	−0.167446
$297$	0	0
$298$	−1.23370	−0.0714664
$299$	−0.164298	−0.00950163
$300$	0	0
$301$	−6.87624	−0.396340
$302$	−23.2573	−1.33831
$303$	0	0
$304$	−6.21441	−0.356421
$305$	−3.19880	−0.183163
$306$	0	0
$307$	17.4205	0.994239	0.497119	−	0.867682i	$-0.334391\pi$
$307$	17.4205	0.994239	0.497119	+	0.867682i	$0.334391\pi$
$308$	0.0421777	0.00240330
$309$	0	0
$310$	32.0997	1.82314
$311$	0.0627390	0.00355760	0.00177880	−	0.999998i	$-0.499434\pi$
$311$	0.0627390	0.00355760	0.00177880	+	0.999998i	$0.499434\pi$
$312$	0	0
$313$	−7.66244	−0.433106	−0.216553	−	0.976271i	$-0.569481\pi$
$313$	−7.66244	−0.433106	−0.216553	+	0.976271i	$0.569481\pi$
$314$	−0.371567	−0.0209687
$315$	0	0
$316$	−2.79301	−0.157119
$317$	20.8454	1.17079	0.585396	−	0.810747i	$-0.300939\pi$
$317$	20.8454	1.17079	0.585396	+	0.810747i	$0.300939\pi$
$318$	0	0
$319$	−0.107108	−0.00599692
$320$	−21.7700	−1.21698
$321$	0	0
$322$	−0.203194	−0.0113236
$323$	48.7217	2.71095
$324$	0	0
$325$	4.00377	0.222089
$326$	−2.41410	−0.133704
$327$	0	0
$328$	−15.5881	−0.860711
$329$	−7.32805	−0.404008
$330$	0	0
$331$	21.6830	1.19181	0.595903	−	0.803057i	$-0.296794\pi$
$331$	21.6830	1.19181	0.595903	+	0.803057i	$0.296794\pi$
$332$	14.1010	0.773893
$333$	0	0
$334$	−16.5102	−0.903400
$335$	−4.89682	−0.267542
$336$	0	0
$337$	2.65243	0.144487	0.0722435	−	0.997387i	$-0.476984\pi$
$337$	2.65243	0.144487	0.0722435	+	0.997387i	$0.476984\pi$
$338$	21.7820	1.18478
$339$	0	0
$340$	34.5503	1.87375
$341$	−0.119100	−0.00644962
$342$	0	0
$343$	1.00000	0.0539949
$344$	15.9240	0.858566
$345$	0	0
$346$	54.3558	2.92219
$347$	16.9248	0.908570	0.454285	−	0.890856i	$-0.349895\pi$
$347$	16.9248	0.908570	0.454285	+	0.890856i	$0.349895\pi$
$348$	0	0
$349$	24.1725	1.29393	0.646963	−	0.762521i	$-0.276039\pi$
$349$	24.1725	1.29393	0.646963	+	0.762521i	$0.276039\pi$
$350$	4.95161	0.264675
$351$	0	0
$352$	0.0915579	0.00488005
$353$	−23.6903	−1.26091	−0.630455	−	0.776226i	$-0.717132\pi$
$353$	−23.6903	−1.26091	−0.630455	+	0.776226i	$0.717132\pi$
$354$	0	0
$355$	−9.26369	−0.491666
$356$	34.8062	1.84473
$357$	0	0
$358$	31.3755	1.65825
$359$	−12.1671	−0.642153	−0.321077	−	0.947053i	$-0.604045\pi$
$359$	−12.1671	−0.642153	−0.321077	+	0.947053i	$0.604045\pi$
$360$	0	0
$361$	32.0643	1.68760
$362$	−19.4296	−1.02120
$363$	0	0
$364$	−5.50026	−0.288292
$365$	−5.26797	−0.275738
$366$	0	0
$367$	−1.25401	−0.0654590	−0.0327295	−	0.999464i	$-0.510420\pi$
$367$	−1.25401	−0.0654590	−0.0327295	+	0.999464i	$0.510420\pi$
$368$	−0.0787711	−0.00410623
$369$	0	0
$370$	−4.66354	−0.242446
$371$	−13.0181	−0.675866
$372$	0	0
$373$	−19.1384	−0.990947	−0.495473	−	0.868623i	$-0.665005\pi$
$373$	−19.1384	−0.990947	−0.495473	+	0.868623i	$0.665005\pi$
$374$	−0.212743	−0.0110007
$375$	0	0
$376$	16.9703	0.875179
$377$	13.9677	0.719372
$378$	0	0
$379$	1.91817	0.0985298	0.0492649	−	0.998786i	$-0.484312\pi$
$379$	1.91817	0.0985298	0.0492649	+	0.998786i	$0.484312\pi$
$380$	36.2115	1.85761
$381$	0	0
$382$	13.1794	0.674318
$383$	23.7365	1.21288	0.606438	−	0.795131i	$-0.292598\pi$
$383$	23.7365	1.21288	0.606438	+	0.795131i	$0.292598\pi$
$384$	0	0
$385$	0.0232444	0.00118465
$386$	−39.0796	−1.98910
$387$	0	0
$388$	−25.0332	−1.27087
$389$	15.9845	0.810445	0.405223	−	0.914218i	$-0.367194\pi$
$389$	15.9845	0.810445	0.405223	+	0.914218i	$0.367194\pi$
$390$	0	0
$391$	0.617576	0.0312321
$392$	−2.31581	−0.116966
$393$	0	0
$394$	0.308191	0.0155264
$395$	−1.53925	−0.0774479
$396$	0	0
$397$	5.98401	0.300329	0.150164	−	0.988661i	$-0.452020\pi$
$397$	5.98401	0.300329	0.150164	+	0.988661i	$0.452020\pi$
$398$	24.1501	1.21053
$399$	0	0
$400$	1.91957	0.0959783
$401$	17.4822	0.873022	0.436511	−	0.899699i	$-0.356214\pi$
$401$	17.4822	0.873022	0.436511	+	0.899699i	$0.356214\pi$
$402$	0	0
$403$	15.5314	0.773676
$404$	−41.2494	−2.05223
$405$	0	0
$406$	17.2743	0.857311
$407$	0.0173032	0.000857687 0
$408$	0	0
$409$	−22.6180	−1.11839	−0.559194	−	0.829037i	$-0.688889\pi$
$409$	−22.6180	−1.11839	−0.559194	+	0.829037i	$0.688889\pi$
$410$	−25.2341	−1.24623
$411$	0	0
$412$	−33.9686	−1.67351
$413$	−9.03994	−0.444826
$414$	0	0
$415$	7.77116	0.381471
$416$	−11.9398	−0.585395
$417$	0	0
$418$	−0.222972	−0.0109059
$419$	−7.54538	−0.368616	−0.184308	−	0.982869i	$-0.559004\pi$
$419$	−7.54538	−0.368616	−0.184308	+	0.982869i	$0.559004\pi$
$420$	0	0
$421$	−12.6670	−0.617354	−0.308677	−	0.951167i	$-0.599886\pi$
$421$	−12.6670	−0.617354	−0.308677	+	0.951167i	$0.599886\pi$
$422$	−57.4614	−2.79718
$423$	0	0
$424$	30.1474	1.46409
$425$	−15.0496	−0.730014
$426$	0	0
$427$	−1.91415	−0.0926322
$428$	−1.12025	−0.0541494
$429$	0	0
$430$	25.7779	1.24312
$431$	0.821405	0.0395657	0.0197828	−	0.999804i	$-0.493703\pi$
$431$	0.821405	0.0395657	0.0197828	+	0.999804i	$0.493703\pi$
$432$	0	0
$433$	13.4538	0.646549	0.323275	−	0.946305i	$-0.395216\pi$
$433$	13.4538	0.646549	0.323275	+	0.946305i	$0.395216\pi$
$434$	19.2083	0.922027
$435$	0	0
$436$	56.7896	2.71973
$437$	0.647270	0.0309631
$438$	0	0
$439$	−19.1510	−0.914026	−0.457013	−	0.889460i	$-0.651081\pi$
$439$	−19.1510	−0.914026	−0.457013	+	0.889460i	$0.651081\pi$
$440$	−0.0538296	−0.00256622
$441$	0	0
$442$	27.7431	1.31961
$443$	14.7964	0.702999	0.351500	−	0.936188i	$-0.385672\pi$
$443$	14.7964	0.702999	0.351500	+	0.936188i	$0.385672\pi$
$444$	0	0
$445$	19.1819	0.909311
$446$	56.3388	2.66772
$447$	0	0
$448$	−13.0271	−0.615471
$449$	−22.4839	−1.06108	−0.530541	−	0.847659i	$-0.678011\pi$
$449$	−22.4839	−1.06108	−0.530541	+	0.847659i	$0.678011\pi$
$450$	0	0
$451$	0.0936265	0.00440870
$452$	−21.0010	−0.987806
$453$	0	0
$454$	−51.6695	−2.42497
$455$	−3.03123	−0.142106
$456$	0	0
$457$	−6.62597	−0.309950	−0.154975	−	0.987918i	$-0.549530\pi$
$457$	−6.62597	−0.309950	−0.154975	+	0.987918i	$0.549530\pi$
$458$	31.1252	1.45438
$459$	0	0
$460$	0.459002	0.0214011
$461$	17.6433	0.821731	0.410866	−	0.911696i	$-0.365227\pi$
$461$	17.6433	0.821731	0.410866	+	0.911696i	$0.365227\pi$
$462$	0	0
$463$	−3.80804	−0.176975	−0.0884873	−	0.996077i	$-0.528203\pi$
$463$	−3.80804	−0.176975	−0.0884873	+	0.996077i	$0.528203\pi$
$464$	6.69666	0.310884
$465$	0	0
$466$	4.43258	0.205335
$467$	26.0643	1.20611	0.603056	−	0.797699i	$-0.293950\pi$
$467$	26.0643	1.20611	0.603056	+	0.797699i	$0.293950\pi$
$468$	0	0
$469$	−2.93024	−0.135306
$470$	27.4717	1.26717
$471$	0	0
$472$	20.9348	0.963600
$473$	−0.0956440	−0.00439772
$474$	0	0
$475$	−15.7732	−0.723726
$476$	20.6747	0.947625
$477$	0	0
$478$	45.1158	2.06355
$479$	−25.9209	−1.18435	−0.592177	−	0.805808i	$-0.701731\pi$
$479$	−25.9209	−1.18435	−0.592177	+	0.805808i	$0.701731\pi$
$480$	0	0
$481$	−2.25645	−0.102885
$482$	−37.1603	−1.69261
$483$	0	0
$484$	−33.3550	−1.51614
$485$	−13.7959	−0.626441
$486$	0	0
$487$	39.6464	1.79655	0.898276	−	0.439433i	$-0.144820\pi$
$487$	39.6464	1.79655	0.898276	+	0.439433i	$0.144820\pi$
$488$	4.43280	0.200663
$489$	0	0
$490$	−3.74884	−0.169355
$491$	5.88343	0.265515	0.132758	−	0.991149i	$-0.457617\pi$
$491$	5.88343	0.265515	0.132758	+	0.991149i	$0.457617\pi$
$492$	0	0
$493$	−52.5026	−2.36460
$494$	29.0771	1.30824
$495$	0	0
$496$	7.44638	0.334352
$497$	−5.54335	−0.248653
$498$	0	0
$499$	−22.5260	−1.00840	−0.504200	−	0.863587i	$-0.668213\pi$
$499$	−22.5260	−1.00840	−0.504200	+	0.863587i	$0.668213\pi$
$500$	−36.5225	−1.63334
$501$	0	0
$502$	−28.2736	−1.26191
$503$	24.1542	1.07698	0.538492	−	0.842631i	$-0.318994\pi$
$503$	24.1542	1.07698	0.538492	+	0.842631i	$0.318994\pi$
$504$	0	0
$505$	−22.7328	−1.01160
$506$	−0.00282630	−0.000125644 0
$507$	0	0
$508$	−3.03233	−0.134538
$509$	9.10647	0.403637	0.201819	−	0.979423i	$-0.435315\pi$
$509$	9.10647	0.403637	0.201819	+	0.979423i	$0.435315\pi$
$510$	0	0
$511$	−3.15233	−0.139451
$512$	−9.75224	−0.430992
$513$	0	0
$514$	47.4902	2.09470
$515$	−18.7203	−0.824917
$516$	0	0
$517$	−0.101928	−0.00448281
$518$	−2.79064	−0.122614
$519$	0	0
$520$	7.01974	0.307836
$521$	−18.9154	−0.828698	−0.414349	−	0.910118i	$-0.635991\pi$
$521$	−18.9154	−0.828698	−0.414349	+	0.910118i	$0.635991\pi$
$522$	0	0
$523$	43.0010	1.88030	0.940150	−	0.340760i	$-0.110684\pi$
$523$	43.0010	1.88030	0.940150	+	0.340760i	$0.110684\pi$
$524$	−36.7358	−1.60481
$525$	0	0
$526$	−35.3537	−1.54149
$527$	−58.3805	−2.54310
$528$	0	0
$529$	−22.9918	−0.999643
$530$	48.8027	2.11985
$531$	0	0
$532$	21.6688	0.939462
$533$	−12.2095	−0.528854
$534$	0	0
$535$	−0.617378	−0.0266916
$536$	6.78586	0.293104
$537$	0	0
$538$	37.3611	1.61075
$539$	0.0139093	0.000599118 0
$540$	0	0
$541$	−11.3893	−0.489665	−0.244832	−	0.969565i	$-0.578733\pi$
$541$	−11.3893	−0.489665	−0.244832	+	0.969565i	$0.578733\pi$
$542$	0.429725	0.0184583
$543$	0	0
$544$	44.8800	1.92421
$545$	31.2971	1.34062
$546$	0	0
$547$	−31.8121	−1.36019	−0.680094	−	0.733125i	$-0.738061\pi$
$547$	−31.8121	−1.36019	−0.680094	+	0.733125i	$0.738061\pi$
$548$	−14.2338	−0.608038
$549$	0	0
$550$	0.0688737	0.00293678
$551$	−55.0270	−2.34423
$552$	0	0
$553$	−0.921078	−0.0391682
$554$	−40.5396	−1.72236
$555$	0	0
$556$	−54.5157	−2.31198
$557$	8.75603	0.371005	0.185502	−	0.982644i	$-0.440609\pi$
$557$	8.75603	0.371005	0.185502	+	0.982644i	$0.440609\pi$
$558$	0	0
$559$	12.4726	0.527536
$560$	−1.45329	−0.0614128
$561$	0	0
$562$	−52.6836	−2.22233
$563$	−17.7767	−0.749199	−0.374599	−	0.927187i	$-0.622220\pi$
$563$	−17.7767	−0.749199	−0.374599	+	0.927187i	$0.622220\pi$
$564$	0	0
$565$	−11.5738	−0.486914
$566$	46.9829	1.97484
$567$	0	0
$568$	12.8373	0.538642
$569$	2.84346	0.119204	0.0596020	−	0.998222i	$-0.481017\pi$
$569$	2.84346	0.119204	0.0596020	+	0.998222i	$0.481017\pi$
$570$	0	0
$571$	−3.84977	−0.161108	−0.0805540	−	0.996750i	$-0.525669\pi$
$571$	−3.84977	−0.161108	−0.0805540	+	0.996750i	$0.525669\pi$
$572$	−0.0765051	−0.00319884
$573$	0	0
$574$	−15.1000	−0.630261
$575$	−0.199935	−0.00833786
$576$	0	0
$577$	12.1129	0.504268	0.252134	−	0.967692i	$-0.418868\pi$
$577$	12.1129	0.504268	0.252134	+	0.967692i	$0.418868\pi$
$578$	−66.1468	−2.75134
$579$	0	0
$580$	−39.0216	−1.62028
$581$	4.65022	0.192924
$582$	0	0
$583$	−0.181073	−0.00749929
$584$	7.30018	0.302084
$585$	0	0
$586$	−39.8191	−1.64491
$587$	−34.0310	−1.40461	−0.702304	−	0.711877i	$-0.747845\pi$
$587$	−34.0310	−1.40461	−0.702304	+	0.711877i	$0.747845\pi$
$588$	0	0
$589$	−61.1876	−2.52119
$590$	33.8893	1.39520
$591$	0	0
$592$	−1.08183	−0.0444630
$593$	−12.3403	−0.506757	−0.253378	−	0.967367i	$-0.581542\pi$
$593$	−12.3403	−0.506757	−0.253378	+	0.967367i	$0.581542\pi$
$594$	0	0
$595$	11.3940	0.467108
$596$	1.66764	0.0683091
$597$	0	0
$598$	0.368568	0.0150719
$599$	13.8192	0.564639	0.282320	−	0.959320i	$-0.408896\pi$
$599$	13.8192	0.564639	0.282320	+	0.959320i	$0.408896\pi$
$600$	0	0
$601$	−24.5683	−1.00216	−0.501081	−	0.865400i	$-0.667064\pi$
$601$	−24.5683	−1.00216	−0.501081	+	0.865400i	$0.667064\pi$
$602$	15.4254	0.628691
$603$	0	0
$604$	31.4378	1.27918
$605$	−18.3822	−0.747342
$606$	0	0
$607$	−5.10336	−0.207139	−0.103570	−	0.994622i	$-0.533026\pi$
$607$	−5.10336	−0.207139	−0.103570	+	0.994622i	$0.533026\pi$
$608$	47.0379	1.90764
$609$	0	0
$610$	7.17583	0.290541
$611$	13.2922	0.537743
$612$	0	0
$613$	24.1198	0.974190	0.487095	−	0.873349i	$-0.338056\pi$
$613$	24.1198	0.974190	0.487095	+	0.873349i	$0.338056\pi$
$614$	−39.0791	−1.57710
$615$	0	0
$616$	−0.0322114	−0.00129783
$617$	37.4334	1.50701	0.753506	−	0.657441i	$-0.228361\pi$
$617$	37.4334	1.50701	0.753506	+	0.657441i	$0.228361\pi$
$618$	0	0
$619$	35.6879	1.43442	0.717209	−	0.696858i	$-0.245419\pi$
$619$	35.6879	1.43442	0.717209	+	0.696858i	$0.245419\pi$
$620$	−43.3903	−1.74259
$621$	0	0
$622$	−0.140742	−0.00564322
$623$	11.4784	0.459872
$624$	0	0
$625$	−9.09130	−0.363652
$626$	17.1890	0.687012
$627$	0	0
$628$	0.502260	0.0200423
$629$	8.48170	0.338188
$630$	0	0
$631$	−6.27915	−0.249969	−0.124985	−	0.992159i	$-0.539888\pi$
$631$	−6.27915	−0.249969	−0.124985	+	0.992159i	$0.539888\pi$
$632$	2.13304	0.0848477
$633$	0	0
$634$	−46.7621	−1.85716
$635$	−1.67114	−0.0663170
$636$	0	0
$637$	−1.81387	−0.0718683
$638$	0.240275	0.00951257
$639$	0	0
$640$	26.8359	1.06078
$641$	6.66828	0.263381	0.131691	−	0.991291i	$-0.457959\pi$
$641$	6.66828	0.263381	0.131691	+	0.991291i	$0.457959\pi$
$642$	0	0
$643$	−21.6864	−0.855229	−0.427615	−	0.903961i	$-0.640646\pi$
$643$	−21.6864	−0.855229	−0.427615	+	0.903961i	$0.640646\pi$
$644$	0.274665	0.0108233
$645$	0	0
$646$	−109.297	−4.30022
$647$	33.1186	1.30203	0.651013	−	0.759067i	$-0.274344\pi$
$647$	33.1186	1.30203	0.651013	+	0.759067i	$0.274344\pi$
$648$	0	0
$649$	−0.125740	−0.00493572
$650$	−8.98160	−0.352287
$651$	0	0
$652$	3.26322	0.127797
$653$	23.1772	0.906996	0.453498	−	0.891257i	$-0.350176\pi$
$653$	23.1772	0.906996	0.453498	+	0.891257i	$0.350176\pi$
$654$	0	0
$655$	−20.2453	−0.791050
$656$	−5.85373	−0.228550
$657$	0	0
$658$	16.4389	0.640855
$659$	7.96571	0.310300	0.155150	−	0.987891i	$-0.450414\pi$
$659$	7.96571	0.310300	0.155150	+	0.987891i	$0.450414\pi$
$660$	0	0
$661$	13.7171	0.533532	0.266766	−	0.963761i	$-0.414045\pi$
$661$	13.7171	0.533532	0.266766	+	0.963761i	$0.414045\pi$
$662$	−48.6411	−1.89049
$663$	0	0
$664$	−10.7690	−0.417919
$665$	11.9418	0.463084
$666$	0	0
$667$	−0.697499	−0.0270073
$668$	22.3175	0.863489
$669$	0	0
$670$	10.9850	0.424386
$671$	−0.0266246	−0.00102783
$672$	0	0
$673$	−26.2955	−1.01362	−0.506808	−	0.862059i	$-0.669175\pi$
$673$	−26.2955	−1.01362	−0.506808	+	0.862059i	$0.669175\pi$
$674$	−5.95016	−0.229192
$675$	0	0
$676$	−29.4435	−1.13244
$677$	29.0553	1.11669	0.558343	−	0.829610i	$-0.311437\pi$
$677$	29.0553	1.11669	0.558343	+	0.829610i	$0.311437\pi$
$678$	0	0
$679$	−8.25542	−0.316814
$680$	−26.3863	−1.01187
$681$	0	0
$682$	0.267175	0.0102307
$683$	−37.6783	−1.44172	−0.720861	−	0.693080i	$-0.756253\pi$
$683$	−37.6783	−1.44172	−0.720861	+	0.693080i	$0.756253\pi$
$684$	0	0
$685$	−7.84435	−0.299717
$686$	−2.24329	−0.0856490
$687$	0	0
$688$	5.97987	0.227980
$689$	23.6132	0.899591
$690$	0	0
$691$	38.4706	1.46349	0.731746	−	0.681578i	$-0.238706\pi$
$691$	38.4706	1.46349	0.731746	+	0.681578i	$0.238706\pi$
$692$	−73.4747	−2.79309
$693$	0	0
$694$	−37.9671	−1.44121
$695$	−30.0440	−1.13963
$696$	0	0
$697$	45.8940	1.73836
$698$	−54.2259	−2.05248
$699$	0	0
$700$	−6.69327	−0.252982
$701$	−25.7001	−0.970679	−0.485339	−	0.874326i	$-0.661304\pi$
$701$	−25.7001	−0.970679	−0.485339	+	0.874326i	$0.661304\pi$
$702$	0	0
$703$	8.88951	0.335274
$704$	−0.181198	−0.00682916
$705$	0	0
$706$	53.1442	2.00011
$707$	−13.6032	−0.511601
$708$	0	0
$709$	−48.1204	−1.80720	−0.903599	−	0.428378i	$-0.859085\pi$
$709$	−48.1204	−1.80720	−0.903599	+	0.428378i	$0.859085\pi$
$710$	20.7811	0.779901
$711$	0	0
$712$	−26.5817	−0.996192
$713$	−0.775587	−0.0290460
$714$	0	0
$715$	−0.0421625	−0.00157679
$716$	−42.4114	−1.58499
$717$	0	0
$718$	27.2942	1.01861
$719$	−36.0511	−1.34448	−0.672239	−	0.740334i	$-0.734667\pi$
$719$	−36.0511	−1.34448	−0.672239	+	0.740334i	$0.734667\pi$
$720$	0	0
$721$	−11.2022	−0.417190
$722$	−71.9295	−2.67694
$723$	0	0
$724$	26.2636	0.976080
$725$	16.9973	0.631263
$726$	0	0
$727$	48.2602	1.78987	0.894936	−	0.446195i	$-0.147221\pi$
$727$	48.2602	1.78987	0.894936	+	0.446195i	$0.147221\pi$
$728$	4.20058	0.155684
$729$	0	0
$730$	11.8176	0.437388
$731$	−46.8829	−1.73403
$732$	0	0
$733$	−12.7820	−0.472115	−0.236058	−	0.971739i	$-0.575855\pi$
$733$	−12.7820	−0.472115	−0.236058	+	0.971739i	$0.575855\pi$
$734$	2.81311	0.103834
$735$	0	0
$736$	0.596232	0.0219774
$737$	−0.0407577	−0.00150133
$738$	0	0
$739$	−9.89608	−0.364033	−0.182017	−	0.983295i	$-0.558262\pi$
$739$	−9.89608	−0.364033	−0.182017	+	0.983295i	$0.558262\pi$
$740$	6.30387	0.231735
$741$	0	0
$742$	29.2033	1.07209
$743$	−30.9126	−1.13407	−0.567037	−	0.823692i	$-0.691911\pi$
$743$	−30.9126	−1.13407	−0.567037	+	0.823692i	$0.691911\pi$
$744$	0	0
$745$	0.919047	0.0336713
$746$	42.9328	1.57188
$747$	0	0
$748$	0.287572	0.0105147
$749$	−0.369436	−0.0134989
$750$	0	0
$751$	−40.3924	−1.47394	−0.736969	−	0.675926i	$-0.763744\pi$
$751$	−40.3924	−1.47394	−0.736969	+	0.675926i	$0.763744\pi$
$752$	6.37279	0.232392
$753$	0	0
$754$	−31.3335	−1.14110
$755$	17.3256	0.630542
$756$	0	0
$757$	−39.5171	−1.43627	−0.718137	−	0.695902i	$-0.755005\pi$
$757$	−39.5171	−1.43627	−0.718137	+	0.695902i	$0.755005\pi$
$758$	−4.30301	−0.156292
$759$	0	0
$760$	−27.6550	−1.00315
$761$	−19.4721	−0.705864	−0.352932	−	0.935649i	$-0.614815\pi$
$761$	−19.4721	−0.705864	−0.352932	+	0.935649i	$0.614815\pi$
$762$	0	0
$763$	18.7281	0.678001
$764$	−17.8151	−0.644528
$765$	0	0
$766$	−53.2476	−1.92391
$767$	16.3973	0.592073
$768$	0	0
$769$	36.3315	1.31015	0.655074	−	0.755565i	$-0.272638\pi$
$769$	36.3315	1.31015	0.655074	+	0.755565i	$0.272638\pi$
$770$	−0.0521439	−0.00187913
$771$	0	0
$772$	52.8253	1.90122
$773$	3.37486	0.121385	0.0606926	−	0.998157i	$-0.480669\pi$
$773$	3.37486	0.121385	0.0606926	+	0.998157i	$0.480669\pi$
$774$	0	0
$775$	18.9002	0.678915
$776$	19.1180	0.686295
$777$	0	0
$778$	−35.8577	−1.28556
$779$	48.1007	1.72339
$780$	0	0
$781$	−0.0771044	−0.00275901
$782$	−1.38540	−0.0495417
$783$	0	0
$784$	−0.869643	−0.0310587
$785$	0.276799	0.00987937
$786$	0	0
$787$	52.8822	1.88505	0.942523	−	0.334142i	$-0.108446\pi$
$787$	52.8822	1.88505	0.942523	+	0.334142i	$0.108446\pi$
$788$	−0.416593	−0.0148405
$789$	0	0
$790$	3.45297	0.122851
$791$	−6.92572	−0.246250
$792$	0	0
$793$	3.47203	0.123295
$794$	−13.4238	−0.476394
$795$	0	0
$796$	−32.6445	−1.15706
$797$	−37.0800	−1.31344	−0.656721	−	0.754134i	$-0.728057\pi$
$797$	−37.0800	−1.31344	−0.656721	+	0.754134i	$0.728057\pi$
$798$	0	0
$799$	−49.9634	−1.76758
$800$	−14.5295	−0.513696
$801$	0	0
$802$	−39.2177	−1.38482
$803$	−0.0438468	−0.00154732
$804$	0	0
$805$	0.151369	0.00533507
$806$	−34.8414	−1.22724
$807$	0	0
$808$	31.5024	1.10825
$809$	40.1722	1.41238	0.706190	−	0.708023i	$-0.250413\pi$
$809$	40.1722	1.41238	0.706190	+	0.708023i	$0.250413\pi$
$810$	0	0
$811$	1.56656	0.0550093	0.0275047	−	0.999622i	$-0.491244\pi$
$811$	1.56656	0.0550093	0.0275047	+	0.999622i	$0.491244\pi$
$812$	−23.3503	−0.819436
$813$	0	0
$814$	−0.0388160	−0.00136050
$815$	1.79838	0.0629946
$816$	0	0
$817$	−49.1371	−1.71909
$818$	50.7386	1.77403
$819$	0	0
$820$	34.1099	1.19117
$821$	−0.699623	−0.0244170	−0.0122085	−	0.999925i	$-0.503886\pi$
$821$	−0.699623	−0.0244170	−0.0122085	+	0.999925i	$0.503886\pi$
$822$	0	0
$823$	15.3699	0.535760	0.267880	−	0.963452i	$-0.413677\pi$
$823$	15.3699	0.535760	0.267880	+	0.963452i	$0.413677\pi$
$824$	25.9420	0.903734
$825$	0	0
$826$	20.2792	0.705602
$827$	14.6209	0.508420	0.254210	−	0.967149i	$-0.418185\pi$
$827$	14.6209	0.508420	0.254210	+	0.967149i	$0.418185\pi$
$828$	0	0
$829$	10.2497	0.355985	0.177993	−	0.984032i	$-0.443040\pi$
$829$	10.2497	0.355985	0.177993	+	0.984032i	$0.443040\pi$
$830$	−17.4329	−0.605106
$831$	0	0
$832$	23.6295	0.819204
$833$	6.81811	0.236233
$834$	0	0
$835$	12.2993	0.425635
$836$	0.301399	0.0104241
$837$	0	0
$838$	16.9264	0.584714
$839$	18.0184	0.622064	0.311032	−	0.950400i	$-0.399325\pi$
$839$	18.0184	0.622064	0.311032	+	0.950400i	$0.399325\pi$
$840$	0	0
$841$	30.2972	1.04473
$842$	28.4158	0.979273
$843$	0	0
$844$	77.6726	2.67360
$845$	−16.2265	−0.558209
$846$	0	0
$847$	−10.9998	−0.377958
$848$	11.3211	0.388768
$849$	0	0
$850$	33.7606	1.15798
$851$	0.112680	0.00386261
$852$	0	0
$853$	−49.1968	−1.68447	−0.842233	−	0.539113i	$-0.818759\pi$
$853$	−49.1968	−1.68447	−0.842233	+	0.539113i	$0.818759\pi$
$854$	4.29398	0.146937
$855$	0	0
$856$	0.855542	0.0292418
$857$	29.3317	1.00195	0.500976	−	0.865461i	$-0.332975\pi$
$857$	29.3317	1.00195	0.500976	+	0.865461i	$0.332975\pi$
$858$	0	0
$859$	−53.4025	−1.82207	−0.911034	−	0.412330i	$-0.864715\pi$
$859$	−53.4025	−1.82207	−0.911034	+	0.412330i	$0.864715\pi$
$860$	−34.8449	−1.18820
$861$	0	0
$862$	−1.84265	−0.0627607
$863$	−46.7265	−1.59059	−0.795294	−	0.606224i	$-0.792683\pi$
$863$	−46.7265	−1.59059	−0.795294	+	0.606224i	$0.792683\pi$
$864$	0	0
$865$	−40.4924	−1.37678
$866$	−30.1808	−1.02558
$867$	0	0
$868$	−25.9645	−0.881293
$869$	−0.0128116	−0.000434603 0
$870$	0	0
$871$	5.31508	0.180095
$872$	−43.3706	−1.46871
$873$	0	0
$874$	−1.45201	−0.0491150
$875$	−12.0444	−0.407175
$876$	0	0
$877$	30.9796	1.04611	0.523053	−	0.852300i	$-0.324793\pi$
$877$	30.9796	1.04611	0.523053	+	0.852300i	$0.324793\pi$
$878$	42.9611	1.44987
$879$	0	0
$880$	−0.0202143	−0.000681425 0
$881$	40.9459	1.37950	0.689752	−	0.724046i	$-0.257720\pi$
$881$	40.9459	1.37950	0.689752	+	0.724046i	$0.257720\pi$
$882$	0	0
$883$	−21.4223	−0.720917	−0.360459	−	0.932775i	$-0.617380\pi$
$883$	−21.4223	−0.720917	−0.360459	+	0.932775i	$0.617380\pi$
$884$	−37.5014	−1.26131
$885$	0	0
$886$	−33.1926	−1.11513
$887$	−26.8839	−0.902674	−0.451337	−	0.892354i	$-0.649053\pi$
$887$	−26.8839	−0.902674	−0.451337	+	0.892354i	$0.649053\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	−43.0306	−1.44239
$891$	0	0
$892$	−76.1552	−2.54986
$893$	−52.3658	−1.75235
$894$	0	0
$895$	−23.3732	−0.781280
$896$	16.0585	0.536477
$897$	0	0
$898$	50.4378	1.68313
$899$	65.9358	2.19908
$900$	0	0
$901$	−88.7588	−2.95698
$902$	−0.210031	−0.00699327
$903$	0	0
$904$	16.0386	0.533436
$905$	14.4741	0.481134
$906$	0	0
$907$	−6.99574	−0.232290	−0.116145	−	0.993232i	$-0.537054\pi$
$907$	−6.99574	−0.232290	−0.116145	+	0.993232i	$0.537054\pi$
$908$	69.8435	2.31784
$909$	0	0
$910$	6.79992	0.225415
$911$	−38.4688	−1.27453	−0.637264	−	0.770645i	$-0.719934\pi$
$911$	−38.4688	−1.27453	−0.637264	+	0.770645i	$0.719934\pi$
$912$	0	0
$913$	0.0646816	0.00214065
$914$	14.8639	0.491655
$915$	0	0
$916$	−42.0730	−1.39013
$917$	−12.1147	−0.400063
$918$	0	0
$919$	−58.9891	−1.94587	−0.972936	−	0.231075i	$-0.925776\pi$
$919$	−58.9891	−1.94587	−0.972936	+	0.231075i	$0.925776\pi$
$920$	−0.350542	−0.0115570
$921$	0	0
$922$	−39.5790	−1.30346
$923$	10.0549	0.330962
$924$	0	0
$925$	−2.74588	−0.0902839
$926$	8.54252	0.280724
$927$	0	0
$928$	−50.6881	−1.66392
$929$	5.77896	0.189601	0.0948007	−	0.995496i	$-0.469779\pi$
$929$	5.77896	0.189601	0.0948007	+	0.995496i	$0.469779\pi$
$930$	0	0
$931$	7.14593	0.234198
$932$	−5.99167	−0.196264
$933$	0	0
$934$	−58.4696	−1.91319
$935$	0.158483	0.00518295
$936$	0	0
$937$	−40.7724	−1.33198	−0.665989	−	0.745962i	$-0.731990\pi$
$937$	−40.7724	−1.33198	−0.665989	+	0.745962i	$0.731990\pi$
$938$	6.57335	0.214628
$939$	0	0
$940$	−37.1344	−1.21119
$941$	14.5328	0.473757	0.236879	−	0.971539i	$-0.423876\pi$
$941$	14.5328	0.473757	0.236879	+	0.971539i	$0.423876\pi$
$942$	0	0
$943$	0.609703	0.0198547
$944$	7.86152	0.255871
$945$	0	0
$946$	0.214557	0.00697584
$947$	20.5170	0.666712	0.333356	−	0.942801i	$-0.391819\pi$
$947$	20.5170	0.666712	0.333356	+	0.942801i	$0.391819\pi$
$948$	0	0
$949$	5.71793	0.185612
$950$	35.3839	1.14800
$951$	0	0
$952$	−15.7894	−0.511738
$953$	60.3450	1.95477	0.977383	−	0.211479i	$-0.0678280\pi$
$953$	60.3450	1.95477	0.977383	+	0.211479i	$0.0678280\pi$
$954$	0	0
$955$	−9.81802	−0.317704
$956$	−60.9846	−1.97238
$957$	0	0
$958$	58.1479	1.87867
$959$	−4.69402	−0.151578
$960$	0	0
$961$	42.3176	1.36508
$962$	5.06187	0.163201
$963$	0	0
$964$	50.2309	1.61783
$965$	29.1124	0.937161
$966$	0	0
$967$	−17.8447	−0.573848	−0.286924	−	0.957953i	$-0.592633\pi$
$967$	−17.8447	−0.573848	−0.286924	+	0.957953i	$0.592633\pi$
$968$	25.4734	0.818747
$969$	0	0
$970$	30.9482	0.993688
$971$	−26.9765	−0.865717	−0.432859	−	0.901462i	$-0.642495\pi$
$971$	−26.9765	−0.865717	−0.432859	+	0.901462i	$0.642495\pi$
$972$	0	0
$973$	−17.9782	−0.576354
$974$	−88.9382	−2.84976
$975$	0	0
$976$	1.66463	0.0532834
$977$	39.6791	1.26945	0.634724	−	0.772739i	$-0.281114\pi$
$977$	39.6791	1.26945	0.634724	+	0.772739i	$0.281114\pi$
$978$	0	0
$979$	0.159657	0.00510265
$980$	5.06743	0.161873
$981$	0	0
$982$	−13.1982	−0.421172
$983$	47.7047	1.52154	0.760771	−	0.649020i	$-0.224821\pi$
$983$	47.7047	1.52154	0.760771	+	0.649020i	$0.224821\pi$
$984$	0	0
$985$	−0.229587	−0.00731525
$986$	117.778	3.75083
$987$	0	0
$988$	−39.3045	−1.25044
$989$	−0.622841	−0.0198052
$990$	0	0
$991$	−49.4052	−1.56941	−0.784704	−	0.619871i	$-0.787185\pi$
$991$	−49.4052	−1.56941	−0.784704	+	0.619871i	$0.787185\pi$
$992$	−56.3629	−1.78952
$993$	0	0
$994$	12.4353	0.394424
$995$	−17.9906	−0.570341
$996$	0	0
$997$	11.4907	0.363914	0.181957	−	0.983306i	$-0.441757\pi$
$997$	11.4907	0.363914	0.181957	+	0.983306i	$0.441757\pi$
$998$	50.5322	1.59957
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.w.1.4		20
3.2	odd	2		889.2.a.d.1.17	✓	20
21.20	even	2		6223.2.a.l.1.17		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.d.1.17	✓	20	3.2	odd	2
6223.2.a.l.1.17		20	21.20	even	2
8001.2.a.w.1.4		20	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)

\(f(q)\)	\(=\)	\(q-2.24329 q^{2} +3.03233 q^{4} +1.67114 q^{5} +1.00000 q^{7} -2.31581 q^{8} +O(q^{10})\)	\(q-2.24329 q^{2} +3.03233 q^{4} +1.67114 q^{5} +1.00000 q^{7} -2.31581 q^{8} -3.74884 q^{10} +0.0139093 q^{11} -1.81387 q^{13} -2.24329 q^{14} -0.869643 q^{16} +6.81811 q^{17} +7.14593 q^{19} +5.06743 q^{20} -0.0312026 q^{22} +0.0905788 q^{23} -2.20730 q^{25} +4.06904 q^{26} +3.03233 q^{28} -7.70047 q^{29} -8.56257 q^{31} +6.58247 q^{32} -15.2950 q^{34} +1.67114 q^{35} +1.24400 q^{37} -16.0304 q^{38} -3.87003 q^{40} +6.73120 q^{41} -6.87624 q^{43} +0.0421777 q^{44} -0.203194 q^{46} -7.32805 q^{47} +1.00000 q^{49} +4.95161 q^{50} -5.50026 q^{52} -13.0181 q^{53} +0.0232444 q^{55} -2.31581 q^{56} +17.2743 q^{58} -9.03994 q^{59} -1.91415 q^{61} +19.2083 q^{62} -13.0271 q^{64} -3.03123 q^{65} -2.93024 q^{67} +20.6747 q^{68} -3.74884 q^{70} -5.54335 q^{71} -3.15233 q^{73} -2.79064 q^{74} +21.6688 q^{76} +0.0139093 q^{77} -0.921078 q^{79} -1.45329 q^{80} -15.1000 q^{82} +4.65022 q^{83} +11.3940 q^{85} +15.4254 q^{86} -0.0322114 q^{88} +11.4784 q^{89} -1.81387 q^{91} +0.274665 q^{92} +16.4389 q^{94} +11.9418 q^{95} -8.25542 q^{97} -2.24329 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8}+O(q^{10}) \) 20 * q - 8 * q^2 + 24 * q^4 - 3 * q^5 + 20 * q^7 - 24 * q^8	\( 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8} - 8 q^{10} - 26 q^{11} - 4 q^{13} - 8 q^{14} + 24 q^{16} - 4 q^{17} + q^{19} + 2 q^{20} + q^{22} - 31 q^{23} + 27 q^{25} - 4 q^{26} + 24 q^{28} - 16 q^{29} + 6 q^{31} - 41 q^{32} - 10 q^{34} - 3 q^{35} + 2 q^{37} - 3 q^{38} - 38 q^{40} - 25 q^{41} + 13 q^{43} - 66 q^{44} + 20 q^{46} - 19 q^{47} + 20 q^{49} + 4 q^{50} + 20 q^{52} - 24 q^{53} - 3 q^{55} - 24 q^{56} + 12 q^{58} - 23 q^{59} - 27 q^{61} - 7 q^{62} + 2 q^{64} - 26 q^{65} + 9 q^{67} + 25 q^{68} - 8 q^{70} - 63 q^{71} - 21 q^{73} - 21 q^{74} - 10 q^{76} - 26 q^{77} + 18 q^{79} + 23 q^{80} - 42 q^{82} + q^{83} - 41 q^{85} + 12 q^{86} + 57 q^{88} + 16 q^{89} - 4 q^{91} - 17 q^{92} + 7 q^{94} - 75 q^{95} - 32 q^{97} - 8 q^{98}+O(q^{100}) \) 20 * q - 8 * q^2 + 24 * q^4 - 3 * q^5 + 20 * q^7 - 24 * q^8 - 8 * q^10 - 26 * q^11 - 4 * q^13 - 8 * q^14 + 24 * q^16 - 4 * q^17 + q^19 + 2 * q^20 + q^22 - 31 * q^23 + 27 * q^25 - 4 * q^26 + 24 * q^28 - 16 * q^29 + 6 * q^31 - 41 * q^32 - 10 * q^34 - 3 * q^35 + 2 * q^37 - 3 * q^38 - 38 * q^40 - 25 * q^41 + 13 * q^43 - 66 * q^44 + 20 * q^46 - 19 * q^47 + 20 * q^49 + 4 * q^50 + 20 * q^52 - 24 * q^53 - 3 * q^55 - 24 * q^56 + 12 * q^58 - 23 * q^59 - 27 * q^61 - 7 * q^62 + 2 * q^64 - 26 * q^65 + 9 * q^67 + 25 * q^68 - 8 * q^70 - 63 * q^71 - 21 * q^73 - 21 * q^74 - 10 * q^76 - 26 * q^77 + 18 * q^79 + 23 * q^80 - 42 * q^82 + q^83 - 41 * q^85 + 12 * q^86 + 57 * q^88 + 16 * q^89 - 4 * q^91 - 17 * q^92 + 7 * q^94 - 75 * q^95 - 32 * q^97 - 8 * q^98

Introduction

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