LMFDB - Embedded newform 8001.2.a.w.1.7

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			1.7
Root			$1.78791$ 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)

$f(q)$	$=$	$q-1.78791 q^{2} +1.19664 q^{4} -2.61034 q^{5} +1.00000 q^{7} +1.43634 q^{8} +O(q^{10})$	\(q-1.78791 q^{2} +1.19664 q^{4} -2.61034 q^{5} +1.00000 q^{7} +1.43634 q^{8} +4.66707 q^{10} +3.67427 q^{11} +4.83827 q^{13} -1.78791 q^{14} -4.96133 q^{16} -2.98581 q^{17} +4.34824 q^{19} -3.12363 q^{20} -6.56928 q^{22} -2.59876 q^{23} +1.81388 q^{25} -8.65041 q^{26} +1.19664 q^{28} +1.91937 q^{29} +1.05650 q^{31} +5.99776 q^{32} +5.33837 q^{34} -2.61034 q^{35} -7.32285 q^{37} -7.77428 q^{38} -3.74934 q^{40} +1.13826 q^{41} -7.87546 q^{43} +4.39677 q^{44} +4.64636 q^{46} -2.04802 q^{47} +1.00000 q^{49} -3.24306 q^{50} +5.78966 q^{52} +12.5047 q^{53} -9.59110 q^{55} +1.43634 q^{56} -3.43167 q^{58} -4.81235 q^{59} -10.9771 q^{61} -1.88893 q^{62} -0.800808 q^{64} -12.6295 q^{65} -1.31670 q^{67} -3.57293 q^{68} +4.66707 q^{70} -7.73820 q^{71} -6.77819 q^{73} +13.0926 q^{74} +5.20327 q^{76} +3.67427 q^{77} +0.216655 q^{79} +12.9508 q^{80} -2.03511 q^{82} +3.43302 q^{83} +7.79398 q^{85} +14.0807 q^{86} +5.27751 q^{88} -9.32688 q^{89} +4.83827 q^{91} -3.10977 q^{92} +3.66168 q^{94} -11.3504 q^{95} +0.511430 q^{97} -1.78791 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.78791	−1.26425	−0.632123	−	0.774868i	$-0.717816\pi$
$2$	−1.78791	−1.26425	−0.632123	+	0.774868i	$0.717816\pi$
$3$	0	0
$3$	0	0
$4$	1.19664	0.598319
$5$	−2.61034	−1.16738	−0.583690	−	0.811977i	$-0.698392\pi$
$5$	−2.61034	−1.16738	−0.583690	+	0.811977i	$0.698392\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.43634	0.507824
$9$	0	0
$10$	4.66707	1.47586
$11$	3.67427	1.10783	0.553917	−	0.832572i	$-0.313132\pi$
$11$	3.67427	1.10783	0.553917	+	0.832572i	$0.313132\pi$
$12$	0	0
$13$	4.83827	1.34189	0.670947	−	0.741505i	$-0.265888\pi$
$13$	4.83827	1.34189	0.670947	+	0.741505i	$0.265888\pi$
$14$	−1.78791	−0.477840
$15$	0	0
$16$	−4.96133	−1.24033
$17$	−2.98581	−0.724165	−0.362082	−	0.932146i	$-0.617934\pi$
$17$	−2.98581	−0.724165	−0.362082	+	0.932146i	$0.617934\pi$
$18$	0	0
$19$	4.34824	0.997555	0.498778	−	0.866730i	$-0.333782\pi$
$19$	4.34824	0.997555	0.498778	+	0.866730i	$0.333782\pi$
$20$	−3.12363	−0.698466
$21$	0	0
$22$	−6.56928	−1.40058
$23$	−2.59876	−0.541878	−0.270939	−	0.962596i	$-0.587334\pi$
$23$	−2.59876	−0.541878	−0.270939	+	0.962596i	$0.587334\pi$
$24$	0	0
$25$	1.81388	0.362776
$26$	−8.65041	−1.69648
$27$	0	0
$28$	1.19664	0.226143
$29$	1.91937	0.356418	0.178209	−	0.983993i	$-0.442970\pi$
$29$	1.91937	0.356418	0.178209	+	0.983993i	$0.442970\pi$
$30$	0	0
$31$	1.05650	0.189752	0.0948762	−	0.995489i	$-0.469754\pi$
$31$	1.05650	0.189752	0.0948762	+	0.995489i	$0.469754\pi$
$32$	5.99776	1.06026
$33$	0	0
$34$	5.33837	0.915523
$35$	−2.61034	−0.441228
$36$	0	0
$37$	−7.32285	−1.20387	−0.601934	−	0.798546i	$-0.705603\pi$
$37$	−7.32285	−1.20387	−0.601934	+	0.798546i	$0.705603\pi$
$38$	−7.77428	−1.26116
$39$	0	0
$40$	−3.74934	−0.592823
$41$	1.13826	0.177766	0.0888831	−	0.996042i	$-0.471670\pi$
$41$	1.13826	0.177766	0.0888831	+	0.996042i	$0.471670\pi$
$42$	0	0
$43$	−7.87546	−1.20100	−0.600498	−	0.799626i	$-0.705031\pi$
$43$	−7.87546	−1.20100	−0.600498	+	0.799626i	$0.705031\pi$
$44$	4.39677	0.662839
$45$	0	0
$46$	4.64636	0.685068
$47$	−2.04802	−0.298734	−0.149367	−	0.988782i	$-0.547724\pi$
$47$	−2.04802	−0.298734	−0.149367	+	0.988782i	$0.547724\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−3.24306	−0.458638
$51$	0	0
$52$	5.78966	0.802881
$53$	12.5047	1.71765	0.858823	−	0.512272i	$-0.171196\pi$
$53$	12.5047	1.71765	0.858823	+	0.512272i	$0.171196\pi$
$54$	0	0
$55$	−9.59110	−1.29326
$56$	1.43634	0.191939
$57$	0	0
$58$	−3.43167	−0.450600
$59$	−4.81235	−0.626514	−0.313257	−	0.949668i	$-0.601420\pi$
$59$	−4.81235	−0.626514	−0.313257	+	0.949668i	$0.601420\pi$
$60$	0	0
$61$	−10.9771	−1.40548	−0.702738	−	0.711449i	$-0.748039\pi$
$61$	−10.9771	−1.40548	−0.702738	+	0.711449i	$0.748039\pi$
$62$	−1.88893	−0.239894
$63$	0	0
$64$	−0.800808	−0.100101
$65$	−12.6295	−1.56650
$66$	0	0
$67$	−1.31670	−0.160860	−0.0804302	−	0.996760i	$-0.525629\pi$
$67$	−1.31670	−0.160860	−0.0804302	+	0.996760i	$0.525629\pi$
$68$	−3.57293	−0.433282
$69$	0	0
$70$	4.66707	0.557821
$71$	−7.73820	−0.918356	−0.459178	−	0.888344i	$-0.651856\pi$
$71$	−7.73820	−0.918356	−0.459178	+	0.888344i	$0.651856\pi$
$72$	0	0
$73$	−6.77819	−0.793327	−0.396663	−	0.917964i	$-0.629832\pi$
$73$	−6.77819	−0.793327	−0.396663	+	0.917964i	$0.629832\pi$
$74$	13.0926	1.52199
$75$	0	0
$76$	5.20327	0.596856
$77$	3.67427	0.418722
$78$	0	0
$79$	0.216655	0.0243756	0.0121878	−	0.999926i	$-0.496120\pi$
$79$	0.216655	0.0243756	0.0121878	+	0.999926i	$0.496120\pi$
$80$	12.9508	1.44794
$81$	0	0
$82$	−2.03511	−0.224740
$83$	3.43302	0.376823	0.188412	−	0.982090i	$-0.439666\pi$
$83$	3.43302	0.376823	0.188412	+	0.982090i	$0.439666\pi$
$84$	0	0
$85$	7.79398	0.845376
$86$	14.0807	1.51836
$87$	0	0
$88$	5.27751	0.562584
$89$	−9.32688	−0.988648	−0.494324	−	0.869278i	$-0.664584\pi$
$89$	−9.32688	−0.988648	−0.494324	+	0.869278i	$0.664584\pi$
$90$	0	0
$91$	4.83827	0.507188
$92$	−3.10977	−0.324216
$93$	0	0
$94$	3.66168	0.377673
$95$	−11.3504	−1.16453
$96$	0	0
$97$	0.511430	0.0519279	0.0259639	−	0.999663i	$-0.491734\pi$
$97$	0.511430	0.0519279	0.0259639	+	0.999663i	$0.491734\pi$
$98$	−1.78791	−0.180607
$99$	0	0
$100$	2.17056	0.217056
$101$	−6.15736	−0.612680	−0.306340	−	0.951922i	$-0.599104\pi$
$101$	−6.15736	−0.612680	−0.306340	+	0.951922i	$0.599104\pi$
$102$	0	0
$103$	15.9846	1.57501	0.787503	−	0.616311i	$-0.211374\pi$
$103$	15.9846	1.57501	0.787503	+	0.616311i	$0.211374\pi$
$104$	6.94941	0.681445
$105$	0	0
$106$	−22.3573	−2.17153
$107$	−17.6482	−1.70611	−0.853056	−	0.521820i	$-0.825253\pi$
$107$	−17.6482	−1.70611	−0.853056	+	0.521820i	$0.825253\pi$
$108$	0	0
$109$	−19.2144	−1.84040	−0.920202	−	0.391445i	$-0.871975\pi$
$109$	−19.2144	−1.84040	−0.920202	+	0.391445i	$0.871975\pi$
$110$	17.1481	1.63500
$111$	0	0
$112$	−4.96133	−0.468802
$113$	−8.54280	−0.803639	−0.401820	−	0.915719i	$-0.631622\pi$
$113$	−8.54280	−0.803639	−0.401820	+	0.915719i	$0.631622\pi$
$114$	0	0
$115$	6.78364	0.632578
$116$	2.29679	0.213252
$117$	0	0
$118$	8.60407	0.792068
$119$	−2.98581	−0.273709
$120$	0	0
$121$	2.50027	0.227297
$122$	19.6261	1.77687
$123$	0	0
$124$	1.26424	0.113533
$125$	8.31686	0.743883
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−10.5637	−0.933711
$129$	0	0
$130$	22.5805	1.98044
$131$	−8.10827	−0.708422	−0.354211	−	0.935165i	$-0.615251\pi$
$131$	−8.10827	−0.708422	−0.354211	+	0.935165i	$0.615251\pi$
$132$	0	0
$133$	4.34824	0.377040
$134$	2.35415	0.203367
$135$	0	0
$136$	−4.28864	−0.367748
$137$	7.16460	0.612113	0.306057	−	0.952013i	$-0.400990\pi$
$137$	7.16460	0.612113	0.306057	+	0.952013i	$0.400990\pi$
$138$	0	0
$139$	−17.8437	−1.51348	−0.756740	−	0.653716i	$-0.773209\pi$
$139$	−17.8437	−1.51348	−0.756740	+	0.653716i	$0.773209\pi$
$140$	−3.12363	−0.263995
$141$	0	0
$142$	13.8352	1.16103
$143$	17.7771	1.48660
$144$	0	0
$145$	−5.01021	−0.416075
$146$	12.1188	1.00296
$147$	0	0
$148$	−8.76280	−0.720298
$149$	9.67559	0.792655	0.396328	−	0.918109i	$-0.370284\pi$
$149$	9.67559	0.792655	0.396328	+	0.918109i	$0.370284\pi$
$150$	0	0
$151$	7.11519	0.579026	0.289513	−	0.957174i	$-0.406507\pi$
$151$	7.11519	0.579026	0.289513	+	0.957174i	$0.406507\pi$
$152$	6.24556	0.506582
$153$	0	0
$154$	−6.56928	−0.529368
$155$	−2.75782	−0.221513
$156$	0	0
$157$	17.7565	1.41712	0.708562	−	0.705649i	$-0.249344\pi$
$157$	17.7565	1.41712	0.708562	+	0.705649i	$0.249344\pi$
$158$	−0.387361	−0.0308168
$159$	0	0
$160$	−15.6562	−1.23773
$161$	−2.59876	−0.204811
$162$	0	0
$163$	13.3104	1.04255	0.521277	−	0.853388i	$-0.325456\pi$
$163$	13.3104	1.04255	0.521277	+	0.853388i	$0.325456\pi$
$164$	1.36208	0.106361
$165$	0	0
$166$	−6.13795	−0.476397
$167$	−9.83583	−0.761120	−0.380560	−	0.924756i	$-0.624269\pi$
$167$	−9.83583	−0.761120	−0.380560	+	0.924756i	$0.624269\pi$
$168$	0	0
$169$	10.4088	0.800679
$170$	−13.9350	−1.06876
$171$	0	0
$172$	−9.42408	−0.718579
$173$	2.08214	0.158302	0.0791512	−	0.996863i	$-0.474779\pi$
$173$	2.08214	0.158302	0.0791512	+	0.996863i	$0.474779\pi$
$174$	0	0
$175$	1.81388	0.137116
$176$	−18.2293	−1.37408
$177$	0	0
$178$	16.6757	1.24989
$179$	21.6013	1.61456	0.807279	−	0.590170i	$-0.200939\pi$
$179$	21.6013	1.61456	0.807279	+	0.590170i	$0.200939\pi$
$180$	0	0
$181$	1.76440	0.131147	0.0655735	−	0.997848i	$-0.479112\pi$
$181$	1.76440	0.131147	0.0655735	+	0.997848i	$0.479112\pi$
$182$	−8.65041	−0.641211
$183$	0	0
$184$	−3.73270	−0.275179
$185$	19.1151	1.40537
$186$	0	0
$187$	−10.9707	−0.802255
$188$	−2.45073	−0.178738
$189$	0	0
$190$	20.2935	1.47225
$191$	11.2136	0.811384	0.405692	−	0.914010i	$-0.367031\pi$
$191$	11.2136	0.811384	0.405692	+	0.914010i	$0.367031\pi$
$192$	0	0
$193$	14.3214	1.03088	0.515438	−	0.856927i	$-0.327629\pi$
$193$	14.3214	1.03088	0.515438	+	0.856927i	$0.327629\pi$
$194$	−0.914393	−0.0656496
$195$	0	0
$196$	1.19664	0.0854742
$197$	8.77144	0.624939	0.312470	−	0.949928i	$-0.398844\pi$
$197$	8.77144	0.624939	0.312470	+	0.949928i	$0.398844\pi$
$198$	0	0
$199$	−7.29620	−0.517214	−0.258607	−	0.965983i	$-0.583263\pi$
$199$	−7.29620	−0.517214	−0.258607	+	0.965983i	$0.583263\pi$
$200$	2.60535	0.184226
$201$	0	0
$202$	11.0088	0.774578
$203$	1.91937	0.134713
$204$	0	0
$205$	−2.97124	−0.207521
$206$	−28.5790	−1.99120
$207$	0	0
$208$	−24.0043	−1.66440
$209$	15.9766	1.10513
$210$	0	0
$211$	14.4229	0.992913	0.496457	−	0.868062i	$-0.334634\pi$
$211$	14.4229	0.992913	0.496457	+	0.868062i	$0.334634\pi$
$212$	14.9636	1.02770
$213$	0	0
$214$	31.5534	2.15695
$215$	20.5576	1.40202
$216$	0	0
$217$	1.05650	0.0717197
$218$	34.3537	2.32672
$219$	0	0
$220$	−11.4771	−0.773784
$221$	−14.4461	−0.971752
$222$	0	0
$223$	12.8364	0.859590	0.429795	−	0.902926i	$-0.358586\pi$
$223$	12.8364	0.859590	0.429795	+	0.902926i	$0.358586\pi$
$224$	5.99776	0.400742
$225$	0	0
$226$	15.2738	1.01600
$227$	−17.7170	−1.17592	−0.587960	−	0.808890i	$-0.700069\pi$
$227$	−17.7170	−1.17592	−0.587960	+	0.808890i	$0.700069\pi$
$228$	0	0
$229$	7.21401	0.476715	0.238358	−	0.971177i	$-0.423391\pi$
$229$	7.21401	0.476715	0.238358	+	0.971177i	$0.423391\pi$
$230$	−12.1286	−0.799734
$231$	0	0
$232$	2.75687	0.180998
$233$	−19.5177	−1.27865	−0.639323	−	0.768938i	$-0.720785\pi$
$233$	−19.5177	−1.27865	−0.639323	+	0.768938i	$0.720785\pi$
$234$	0	0
$235$	5.34602	0.348736
$236$	−5.75864	−0.374855
$237$	0	0
$238$	5.33837	0.346035
$239$	18.9197	1.22381	0.611907	−	0.790930i	$-0.290403\pi$
$239$	18.9197	1.22381	0.611907	+	0.790930i	$0.290403\pi$
$240$	0	0
$241$	22.4343	1.44512	0.722561	−	0.691307i	$-0.242965\pi$
$241$	22.4343	1.44512	0.722561	+	0.691307i	$0.242965\pi$
$242$	−4.47027	−0.287360
$243$	0	0
$244$	−13.1356	−0.840923
$245$	−2.61034	−0.166769
$246$	0	0
$247$	21.0380	1.33861
$248$	1.51749	0.0963608
$249$	0	0
$250$	−14.8698	−0.940451
$251$	−1.97098	−0.124407	−0.0622035	−	0.998063i	$-0.519813\pi$
$251$	−1.97098	−0.124407	−0.0622035	+	0.998063i	$0.519813\pi$
$252$	0	0
$253$	−9.54854	−0.600311
$254$	1.78791	0.112184
$255$	0	0
$256$	20.4887	1.28054
$257$	−19.7692	−1.23317	−0.616585	−	0.787288i	$-0.711484\pi$
$257$	−19.7692	−1.23317	−0.616585	+	0.787288i	$0.711484\pi$
$258$	0	0
$259$	−7.32285	−0.455019
$260$	−15.1130	−0.937267
$261$	0	0
$262$	14.4969	0.895621
$263$	−13.9744	−0.861698	−0.430849	−	0.902424i	$-0.641786\pi$
$263$	−13.9744	−0.861698	−0.430849	+	0.902424i	$0.641786\pi$
$264$	0	0
$265$	−32.6414	−2.00515
$266$	−7.77428	−0.476672
$267$	0	0
$268$	−1.57561	−0.0962459
$269$	−22.2765	−1.35822	−0.679110	−	0.734036i	$-0.737634\pi$
$269$	−22.2765	−1.35822	−0.679110	+	0.734036i	$0.737634\pi$
$270$	0	0
$271$	−22.3273	−1.35628	−0.678142	−	0.734931i	$-0.737215\pi$
$271$	−22.3273	−1.35628	−0.678142	+	0.734931i	$0.737215\pi$
$272$	14.8136	0.898206
$273$	0	0
$274$	−12.8097	−0.773862
$275$	6.66468	0.401895
$276$	0	0
$277$	−30.9057	−1.85694	−0.928472	−	0.371403i	$-0.878877\pi$
$277$	−30.9057	−1.85694	−0.928472	+	0.371403i	$0.878877\pi$
$278$	31.9030	1.91341
$279$	0	0
$280$	−3.74934	−0.224066
$281$	0.0969246	0.00578204	0.00289102	−	0.999996i	$-0.499080\pi$
$281$	0.0969246	0.00578204	0.00289102	+	0.999996i	$0.499080\pi$
$282$	0	0
$283$	18.5673	1.10371	0.551856	−	0.833939i	$-0.313920\pi$
$283$	18.5673	1.10371	0.551856	+	0.833939i	$0.313920\pi$
$284$	−9.25983	−0.549470
$285$	0	0
$286$	−31.7839	−1.87942
$287$	1.13826	0.0671893
$288$	0	0
$289$	−8.08495	−0.475585
$290$	8.95783	0.526022
$291$	0	0
$292$	−8.11104	−0.474663
$293$	12.1959	0.712491	0.356246	−	0.934392i	$-0.384057\pi$
$293$	12.1959	0.712491	0.356246	+	0.934392i	$0.384057\pi$
$294$	0	0
$295$	12.5619	0.731380
$296$	−10.5181	−0.611353
$297$	0	0
$298$	−17.2991	−1.00211
$299$	−12.5735	−0.727143
$300$	0	0
$301$	−7.87546	−0.453934
$302$	−12.7214	−0.732032
$303$	0	0
$304$	−21.5731	−1.23730
$305$	28.6540	1.64072
$306$	0	0
$307$	−9.15964	−0.522768	−0.261384	−	0.965235i	$-0.584179\pi$
$307$	−9.15964	−0.522768	−0.261384	+	0.965235i	$0.584179\pi$
$308$	4.39677	0.250529
$309$	0	0
$310$	4.93074	0.280047
$311$	−19.6147	−1.11225	−0.556125	−	0.831099i	$-0.687712\pi$
$311$	−19.6147	−1.11225	−0.556125	+	0.831099i	$0.687712\pi$
$312$	0	0
$313$	−22.3298	−1.26215	−0.631076	−	0.775721i	$-0.717387\pi$
$313$	−22.3298	−1.26215	−0.631076	+	0.775721i	$0.717387\pi$
$314$	−31.7471	−1.79159
$315$	0	0
$316$	0.259258	0.0145844
$317$	−6.48966	−0.364496	−0.182248	−	0.983253i	$-0.558337\pi$
$317$	−6.48966	−0.364496	−0.182248	+	0.983253i	$0.558337\pi$
$318$	0	0
$319$	7.05229	0.394852
$320$	2.09038	0.116856
$321$	0	0
$322$	4.64636	0.258931
$323$	−12.9830	−0.722394
$324$	0	0
$325$	8.77603	0.486807
$326$	−23.7979	−1.31804
$327$	0	0
$328$	1.63493	0.0902738
$329$	−2.04802	−0.112911
$330$	0	0
$331$	11.6043	0.637830	0.318915	−	0.947783i	$-0.396682\pi$
$331$	11.6043	0.637830	0.318915	+	0.947783i	$0.396682\pi$
$332$	4.10809	0.225461
$333$	0	0
$334$	17.5856	0.962243
$335$	3.43703	0.187785
$336$	0	0
$337$	9.73516	0.530308	0.265154	−	0.964206i	$-0.414577\pi$
$337$	9.73516	0.530308	0.265154	+	0.964206i	$0.414577\pi$
$338$	−18.6101	−1.01226
$339$	0	0
$340$	9.32657	0.505804
$341$	3.88186	0.210214
$342$	0	0
$343$	1.00000	0.0539949
$344$	−11.3119	−0.609894
$345$	0	0
$346$	−3.72269	−0.200133
$347$	−3.20316	−0.171955	−0.0859774	−	0.996297i	$-0.527401\pi$
$347$	−3.20316	−0.171955	−0.0859774	+	0.996297i	$0.527401\pi$
$348$	0	0
$349$	−18.6426	−0.997916	−0.498958	−	0.866626i	$-0.666284\pi$
$349$	−18.6426	−0.997916	−0.498958	+	0.866626i	$0.666284\pi$
$350$	−3.24306	−0.173349
$351$	0	0
$352$	22.0374	1.17460
$353$	23.8815	1.27108	0.635541	−	0.772067i	$-0.280777\pi$
$353$	23.8815	1.27108	0.635541	+	0.772067i	$0.280777\pi$
$354$	0	0
$355$	20.1993	1.07207
$356$	−11.1609	−0.591527
$357$	0	0
$358$	−38.6213	−2.04120
$359$	−4.40641	−0.232561	−0.116281	−	0.993216i	$-0.537097\pi$
$359$	−4.40641	−0.232561	−0.116281	+	0.993216i	$0.537097\pi$
$360$	0	0
$361$	−0.0927956	−0.00488398
$362$	−3.15460	−0.165802
$363$	0	0
$364$	5.78966	0.303460
$365$	17.6934	0.926114
$366$	0	0
$367$	6.38739	0.333419	0.166710	−	0.986006i	$-0.446686\pi$
$367$	6.38739	0.333419	0.166710	+	0.986006i	$0.446686\pi$
$368$	12.8933	0.672110
$369$	0	0
$370$	−34.1762	−1.77674
$371$	12.5047	0.649209
$372$	0	0
$373$	25.2092	1.30528	0.652642	−	0.757666i	$-0.273661\pi$
$373$	25.2092	1.30528	0.652642	+	0.757666i	$0.273661\pi$
$374$	19.6146	1.01425
$375$	0	0
$376$	−2.94165	−0.151704
$377$	9.28643	0.478275
$378$	0	0
$379$	−28.5137	−1.46465	−0.732326	−	0.680955i	$-0.761565\pi$
$379$	−28.5137	−1.46465	−0.732326	+	0.680955i	$0.761565\pi$
$380$	−13.5823	−0.696758
$381$	0	0
$382$	−20.0489	−1.02579
$383$	4.36258	0.222917	0.111459	−	0.993769i	$-0.464448\pi$
$383$	4.36258	0.222917	0.111459	+	0.993769i	$0.464448\pi$
$384$	0	0
$385$	−9.59110	−0.488808
$386$	−25.6054	−1.30328
$387$	0	0
$388$	0.611997	0.0310694
$389$	−18.0882	−0.917109	−0.458554	−	0.888666i	$-0.651633\pi$
$389$	−18.0882	−0.917109	−0.458554	+	0.888666i	$0.651633\pi$
$390$	0	0
$391$	7.75939	0.392409
$392$	1.43634	0.0725462
$393$	0	0
$394$	−15.6826	−0.790077
$395$	−0.565544	−0.0284556
$396$	0	0
$397$	3.34162	0.167711	0.0838556	−	0.996478i	$-0.473277\pi$
$397$	3.34162	0.167711	0.0838556	+	0.996478i	$0.473277\pi$
$398$	13.0450	0.653885
$399$	0	0
$400$	−8.99926	−0.449963
$401$	−9.47634	−0.473226	−0.236613	−	0.971604i	$-0.576037\pi$
$401$	−9.47634	−0.473226	−0.236613	+	0.971604i	$0.576037\pi$
$402$	0	0
$403$	5.11162	0.254628
$404$	−7.36813	−0.366578
$405$	0	0
$406$	−3.43167	−0.170311
$407$	−26.9061	−1.33369
$408$	0	0
$409$	37.4637	1.85246	0.926231	−	0.376957i	$-0.123030\pi$
$409$	37.4637	1.85246	0.926231	+	0.376957i	$0.123030\pi$
$410$	5.31233	0.262357
$411$	0	0
$412$	19.1277	0.942356
$413$	−4.81235	−0.236800
$414$	0	0
$415$	−8.96136	−0.439896
$416$	29.0187	1.42276
$417$	0	0
$418$	−28.5648	−1.39715
$419$	−4.82140	−0.235541	−0.117770	−	0.993041i	$-0.537575\pi$
$419$	−4.82140	−0.235541	−0.117770	+	0.993041i	$0.537575\pi$
$420$	0	0
$421$	−29.3062	−1.42830	−0.714149	−	0.699994i	$-0.753186\pi$
$421$	−29.3062	−1.42830	−0.714149	+	0.699994i	$0.753186\pi$
$422$	−25.7869	−1.25529
$423$	0	0
$424$	17.9610	0.872262
$425$	−5.41589	−0.262709
$426$	0	0
$427$	−10.9771	−0.531220
$428$	−21.1185	−1.02080
$429$	0	0
$430$	−36.7553	−1.77250
$431$	−40.1333	−1.93315	−0.966576	−	0.256380i	$-0.917470\pi$
$431$	−40.1333	−1.93315	−0.966576	+	0.256380i	$0.917470\pi$
$432$	0	0
$433$	−16.8897	−0.811665	−0.405833	−	0.913947i	$-0.633018\pi$
$433$	−16.8897	−0.811665	−0.405833	+	0.913947i	$0.633018\pi$
$434$	−1.88893	−0.0906714
$435$	0	0
$436$	−22.9927	−1.10115
$437$	−11.3000	−0.540553
$438$	0	0
$439$	20.7918	0.992340	0.496170	−	0.868225i	$-0.334739\pi$
$439$	20.7918	0.992340	0.496170	+	0.868225i	$0.334739\pi$
$440$	−13.7761	−0.656750
$441$	0	0
$442$	25.8285	1.22853
$443$	3.18205	0.151184	0.0755918	−	0.997139i	$-0.475915\pi$
$443$	3.18205	0.151184	0.0755918	+	0.997139i	$0.475915\pi$
$444$	0	0
$445$	24.3463	1.15413
$446$	−22.9504	−1.08673
$447$	0	0
$448$	−0.800808	−0.0378346
$449$	21.1543	0.998333	0.499167	−	0.866506i	$-0.333640\pi$
$449$	21.1543	0.998333	0.499167	+	0.866506i	$0.333640\pi$
$450$	0	0
$451$	4.18227	0.196935
$452$	−10.2226	−0.480833
$453$	0	0
$454$	31.6765	1.48665
$455$	−12.6295	−0.592081
$456$	0	0
$457$	−27.3549	−1.27961	−0.639804	−	0.768538i	$-0.720984\pi$
$457$	−27.3549	−1.27961	−0.639804	+	0.768538i	$0.720984\pi$
$458$	−12.8980	−0.602685
$459$	0	0
$460$	8.11757	0.378483
$461$	−17.2554	−0.803663	−0.401831	−	0.915714i	$-0.631626\pi$
$461$	−17.2554	−0.803663	−0.401831	+	0.915714i	$0.631626\pi$
$462$	0	0
$463$	−25.2987	−1.17573	−0.587865	−	0.808959i	$-0.700031\pi$
$463$	−25.2987	−1.17573	−0.587865	+	0.808959i	$0.700031\pi$
$464$	−9.52263	−0.442077
$465$	0	0
$466$	34.8960	1.61652
$467$	−21.6496	−1.00182	−0.500912	−	0.865498i	$-0.667002\pi$
$467$	−21.6496	−1.00182	−0.500912	+	0.865498i	$0.667002\pi$
$468$	0	0
$469$	−1.31670	−0.0607995
$470$	−9.55822	−0.440888
$471$	0	0
$472$	−6.91218	−0.318159
$473$	−28.9366	−1.33051
$474$	0	0
$475$	7.88718	0.361889
$476$	−3.57293	−0.163765
$477$	0	0
$478$	−33.8268	−1.54720
$479$	30.3805	1.38812	0.694060	−	0.719917i	$-0.255820\pi$
$479$	30.3805	1.38812	0.694060	+	0.719917i	$0.255820\pi$
$480$	0	0
$481$	−35.4299	−1.61546
$482$	−40.1107	−1.82699
$483$	0	0
$484$	2.99192	0.135996
$485$	−1.33501	−0.0606195
$486$	0	0
$487$	26.9970	1.22335	0.611675	−	0.791109i	$-0.290496\pi$
$487$	26.9970	1.22335	0.611675	+	0.791109i	$0.290496\pi$
$488$	−15.7669	−0.713734
$489$	0	0
$490$	4.66707	0.210837
$491$	−16.6883	−0.753131	−0.376565	−	0.926390i	$-0.622895\pi$
$491$	−16.6883	−0.753131	−0.376565	+	0.926390i	$0.622895\pi$
$492$	0	0
$493$	−5.73087	−0.258105
$494$	−37.6141	−1.69234
$495$	0	0
$496$	−5.24163	−0.235356
$497$	−7.73820	−0.347106
$498$	0	0
$499$	−16.8527	−0.754429	−0.377214	−	0.926126i	$-0.623118\pi$
$499$	−16.8527	−0.754429	−0.377214	+	0.926126i	$0.623118\pi$
$500$	9.95228	0.445079
$501$	0	0
$502$	3.52394	0.157281
$503$	16.4139	0.731860	0.365930	−	0.930642i	$-0.380751\pi$
$503$	16.4139	0.731860	0.365930	+	0.930642i	$0.380751\pi$
$504$	0	0
$505$	16.0728	0.715230
$506$	17.0720	0.758942
$507$	0	0
$508$	−1.19664	−0.0530922
$509$	−16.1583	−0.716206	−0.358103	−	0.933682i	$-0.616576\pi$
$509$	−16.1583	−0.716206	−0.358103	+	0.933682i	$0.616576\pi$
$510$	0	0
$511$	−6.77819	−0.299849
$512$	−15.5045	−0.685210
$513$	0	0
$514$	35.3457	1.55903
$515$	−41.7251	−1.83863
$516$	0	0
$517$	−7.52496	−0.330948
$518$	13.0926	0.575257
$519$	0	0
$520$	−18.1403	−0.795506
$521$	−13.5649	−0.594290	−0.297145	−	0.954832i	$-0.596035\pi$
$521$	−13.5649	−0.594290	−0.297145	+	0.954832i	$0.596035\pi$
$522$	0	0
$523$	−18.5963	−0.813159	−0.406579	−	0.913615i	$-0.633279\pi$
$523$	−18.5963	−0.813159	−0.406579	+	0.913615i	$0.633279\pi$
$524$	−9.70266	−0.423863
$525$	0	0
$526$	24.9850	1.08940
$527$	−3.15450	−0.137412
$528$	0	0
$529$	−16.2465	−0.706368
$530$	58.3601	2.53500
$531$	0	0
$532$	5.20327	0.225590
$533$	5.50720	0.238543
$534$	0	0
$535$	46.0677	1.99168
$536$	−1.89123	−0.0816887
$537$	0	0
$538$	39.8284	1.71713
$539$	3.67427	0.158262
$540$	0	0
$541$	−18.9711	−0.815631	−0.407815	−	0.913064i	$-0.633709\pi$
$541$	−18.9711	−0.815631	−0.407815	+	0.913064i	$0.633709\pi$
$542$	39.9192	1.71468
$543$	0	0
$544$	−17.9081	−0.767806
$545$	50.1561	2.14845
$546$	0	0
$547$	−32.6463	−1.39585	−0.697927	−	0.716169i	$-0.745894\pi$
$547$	−32.6463	−1.39585	−0.697927	+	0.716169i	$0.745894\pi$
$548$	8.57344	0.366239
$549$	0	0
$550$	−11.9159	−0.508095
$551$	8.34588	0.355547
$552$	0	0
$553$	0.216655	0.00921312
$554$	55.2568	2.34763
$555$	0	0
$556$	−21.3524	−0.905544
$557$	−16.4123	−0.695412	−0.347706	−	0.937604i	$-0.613039\pi$
$557$	−16.4123	−0.695412	−0.347706	+	0.937604i	$0.613039\pi$
$558$	0	0
$559$	−38.1036	−1.61161
$560$	12.9508	0.547270
$561$	0	0
$562$	−0.173293	−0.00730992
$563$	16.9970	0.716340	0.358170	−	0.933656i	$-0.383401\pi$
$563$	16.9970	0.716340	0.358170	+	0.933656i	$0.383401\pi$
$564$	0	0
$565$	22.2996	0.938152
$566$	−33.1968	−1.39536
$567$	0	0
$568$	−11.1147	−0.466363
$569$	3.38733	0.142004	0.0710022	−	0.997476i	$-0.477380\pi$
$569$	3.38733	0.142004	0.0710022	+	0.997476i	$0.477380\pi$
$570$	0	0
$571$	22.6320	0.947118	0.473559	−	0.880762i	$-0.342969\pi$
$571$	22.6320	0.947118	0.473559	+	0.880762i	$0.342969\pi$
$572$	21.2728	0.889459
$573$	0	0
$574$	−2.03511	−0.0849438
$575$	−4.71383	−0.196580
$576$	0	0
$577$	39.6115	1.64905	0.824525	−	0.565826i	$-0.191442\pi$
$577$	39.6115	1.64905	0.824525	+	0.565826i	$0.191442\pi$
$578$	14.4552	0.601257
$579$	0	0
$580$	−5.99541	−0.248946
$581$	3.43302	0.142426
$582$	0	0
$583$	45.9455	1.90287
$584$	−9.73580	−0.402870
$585$	0	0
$586$	−21.8052	−0.900765
$587$	10.6810	0.440850	0.220425	−	0.975404i	$-0.429256\pi$
$587$	10.6810	0.440850	0.220425	+	0.975404i	$0.429256\pi$
$588$	0	0
$589$	4.59390	0.189289
$590$	−22.4595	−0.924645
$591$	0	0
$592$	36.3311	1.49320
$593$	−40.2803	−1.65412	−0.827058	−	0.562117i	$-0.809987\pi$
$593$	−40.2803	−1.65412	−0.827058	+	0.562117i	$0.809987\pi$
$594$	0	0
$595$	7.79398	0.319522
$596$	11.5782	0.474261
$597$	0	0
$598$	22.4803	0.919288
$599$	38.7977	1.58523	0.792616	−	0.609721i	$-0.208719\pi$
$599$	38.7977	1.58523	0.792616	+	0.609721i	$0.208719\pi$
$600$	0	0
$601$	−8.76032	−0.357341	−0.178670	−	0.983909i	$-0.557180\pi$
$601$	−8.76032	−0.357341	−0.178670	+	0.983909i	$0.557180\pi$
$602$	14.0807	0.573884
$603$	0	0
$604$	8.51431	0.346442
$605$	−6.52655	−0.265342
$606$	0	0
$607$	−7.86010	−0.319032	−0.159516	−	0.987195i	$-0.550993\pi$
$607$	−7.86010	−0.319032	−0.159516	+	0.987195i	$0.550993\pi$
$608$	26.0797	1.05767
$609$	0	0
$610$	−51.2309	−2.07428
$611$	−9.90885	−0.400869
$612$	0	0
$613$	17.3298	0.699943	0.349971	−	0.936760i	$-0.386191\pi$
$613$	17.3298	0.699943	0.349971	+	0.936760i	$0.386191\pi$
$614$	16.3766	0.660908
$615$	0	0
$616$	5.27751	0.212637
$617$	13.2526	0.533531	0.266766	−	0.963761i	$-0.414045\pi$
$617$	13.2526	0.533531	0.266766	+	0.963761i	$0.414045\pi$
$618$	0	0
$619$	−25.3342	−1.01827	−0.509134	−	0.860687i	$-0.670034\pi$
$619$	−25.3342	−1.01827	−0.509134	+	0.860687i	$0.670034\pi$
$620$	−3.30011	−0.132536
$621$	0	0
$622$	35.0695	1.40616
$623$	−9.32688	−0.373674
$624$	0	0
$625$	−30.7792	−1.23117
$626$	39.9237	1.59567
$627$	0	0
$628$	21.2481	0.847892
$629$	21.8646	0.871799
$630$	0	0
$631$	30.7245	1.22312	0.611561	−	0.791197i	$-0.290542\pi$
$631$	30.7245	1.22312	0.611561	+	0.791197i	$0.290542\pi$
$632$	0.311191	0.0123785
$633$	0	0
$634$	11.6030	0.460812
$635$	2.61034	0.103588
$636$	0	0
$637$	4.83827	0.191699
$638$	−12.6089	−0.499191
$639$	0	0
$640$	27.5749	1.09000
$641$	−18.4045	−0.726935	−0.363468	−	0.931607i	$-0.618407\pi$
$641$	−18.4045	−0.726935	−0.363468	+	0.931607i	$0.618407\pi$
$642$	0	0
$643$	35.7664	1.41049	0.705245	−	0.708964i	$-0.250837\pi$
$643$	35.7664	1.41049	0.705245	+	0.708964i	$0.250837\pi$
$644$	−3.10977	−0.122542
$645$	0	0
$646$	23.2125	0.913284
$647$	−20.3216	−0.798923	−0.399461	−	0.916750i	$-0.630803\pi$
$647$	−20.3216	−0.798923	−0.399461	+	0.916750i	$0.630803\pi$
$648$	0	0
$649$	−17.6819	−0.694074
$650$	−15.6908	−0.615444
$651$	0	0
$652$	15.9278	0.623780
$653$	−23.8719	−0.934181	−0.467090	−	0.884210i	$-0.654698\pi$
$653$	−23.8719	−0.934181	−0.467090	+	0.884210i	$0.654698\pi$
$654$	0	0
$655$	21.1653	0.826998
$656$	−5.64728	−0.220489
$657$	0	0
$658$	3.66168	0.142747
$659$	19.6363	0.764923	0.382462	−	0.923971i	$-0.375076\pi$
$659$	19.6363	0.764923	0.382462	+	0.923971i	$0.375076\pi$
$660$	0	0
$661$	−22.5802	−0.878267	−0.439133	−	0.898422i	$-0.644714\pi$
$661$	−22.5802	−0.878267	−0.439133	+	0.898422i	$0.644714\pi$
$662$	−20.7475	−0.806374
$663$	0	0
$664$	4.93100	0.191360
$665$	−11.3504	−0.440149
$666$	0	0
$667$	−4.98798	−0.193135
$668$	−11.7699	−0.455392
$669$	0	0
$670$	−6.14512	−0.237407
$671$	−40.3329	−1.55703
$672$	0	0
$673$	−18.1619	−0.700089	−0.350044	−	0.936733i	$-0.613833\pi$
$673$	−18.1619	−0.700089	−0.350044	+	0.936733i	$0.613833\pi$
$674$	−17.4056	−0.670440
$675$	0	0
$676$	12.4556	0.479062
$677$	−29.6690	−1.14027	−0.570135	−	0.821551i	$-0.693109\pi$
$677$	−29.6690	−1.14027	−0.570135	+	0.821551i	$0.693109\pi$
$678$	0	0
$679$	0.511430	0.0196269
$680$	11.1948	0.429302
$681$	0	0
$682$	−6.94043	−0.265763
$683$	31.8760	1.21970	0.609851	−	0.792516i	$-0.291229\pi$
$683$	31.8760	1.21970	0.609851	+	0.792516i	$0.291229\pi$
$684$	0	0
$685$	−18.7021	−0.714569
$686$	−1.78791	−0.0682629
$687$	0	0
$688$	39.0728	1.48964
$689$	60.5009	2.30490
$690$	0	0
$691$	31.6487	1.20397	0.601987	−	0.798506i	$-0.294376\pi$
$691$	31.6487	1.20397	0.601987	+	0.798506i	$0.294376\pi$
$692$	2.49157	0.0947153
$693$	0	0
$694$	5.72698	0.217393
$695$	46.5781	1.76681
$696$	0	0
$697$	−3.39862	−0.128732
$698$	33.3314	1.26161
$699$	0	0
$700$	2.17056	0.0820393
$701$	−38.3975	−1.45025	−0.725127	−	0.688615i	$-0.758219\pi$
$701$	−38.3975	−1.45025	−0.725127	+	0.688615i	$0.758219\pi$
$702$	0	0
$703$	−31.8415	−1.20092
$704$	−2.94239	−0.110895
$705$	0	0
$706$	−42.6980	−1.60696
$707$	−6.15736	−0.231571
$708$	0	0
$709$	−32.1618	−1.20786	−0.603930	−	0.797037i	$-0.706399\pi$
$709$	−32.1618	−1.20786	−0.603930	+	0.797037i	$0.706399\pi$
$710$	−36.1147	−1.35536
$711$	0	0
$712$	−13.3966	−0.502059
$713$	−2.74558	−0.102823
$714$	0	0
$715$	−46.4043	−1.73542
$716$	25.8490	0.966021
$717$	0	0
$718$	7.87828	0.294015
$719$	−28.0804	−1.04722	−0.523612	−	0.851957i	$-0.675416\pi$
$719$	−28.0804	−1.04722	−0.523612	+	0.851957i	$0.675416\pi$
$720$	0	0
$721$	15.9846	0.595296
$722$	0.165911	0.00617455
$723$	0	0
$724$	2.11135	0.0784678
$725$	3.48150	0.129300
$726$	0	0
$727$	42.9400	1.59256	0.796278	−	0.604930i	$-0.206799\pi$
$727$	42.9400	1.59256	0.796278	+	0.604930i	$0.206799\pi$
$728$	6.94941	0.257562
$729$	0	0
$730$	−31.6342	−1.17084
$731$	23.5146	0.869719
$732$	0	0
$733$	10.9962	0.406153	0.203076	−	0.979163i	$-0.434906\pi$
$733$	10.9962	0.406153	0.203076	+	0.979163i	$0.434906\pi$
$734$	−11.4201	−0.421524
$735$	0	0
$736$	−15.5867	−0.574534
$737$	−4.83791	−0.178207
$738$	0	0
$739$	−3.36023	−0.123608	−0.0618040	−	0.998088i	$-0.519685\pi$
$739$	−3.36023	−0.123608	−0.0618040	+	0.998088i	$0.519685\pi$
$740$	22.8739	0.840861
$741$	0	0
$742$	−22.3573	−0.820761
$743$	−3.20588	−0.117612	−0.0588061	−	0.998269i	$-0.518729\pi$
$743$	−3.20588	−0.117612	−0.0588061	+	0.998269i	$0.518729\pi$
$744$	0	0
$745$	−25.2566	−0.925330
$746$	−45.0719	−1.65020
$747$	0	0
$748$	−13.1279	−0.480004
$749$	−17.6482	−0.644849
$750$	0	0
$751$	−11.9997	−0.437876	−0.218938	−	0.975739i	$-0.570259\pi$
$751$	−11.9997	−0.437876	−0.218938	+	0.975739i	$0.570259\pi$
$752$	10.1609	0.370529
$753$	0	0
$754$	−16.6033	−0.604658
$755$	−18.5731	−0.675943
$756$	0	0
$757$	−45.7149	−1.66154	−0.830768	−	0.556619i	$-0.812098\pi$
$757$	−45.7149	−1.66154	−0.830768	+	0.556619i	$0.812098\pi$
$758$	50.9801	1.85168
$759$	0	0
$760$	−16.3030	−0.591374
$761$	53.7416	1.94813	0.974066	−	0.226264i	$-0.0726513\pi$
$761$	53.7416	1.94813	0.974066	+	0.226264i	$0.0726513\pi$
$762$	0	0
$763$	−19.2144	−0.695607
$764$	13.4186	0.485467
$765$	0	0
$766$	−7.79992	−0.281823
$767$	−23.2834	−0.840716
$768$	0	0
$769$	−40.0369	−1.44377	−0.721884	−	0.692014i	$-0.756724\pi$
$769$	−40.0369	−1.44377	−0.721884	+	0.692014i	$0.756724\pi$
$770$	17.1481	0.617973
$771$	0	0
$772$	17.1375	0.616792
$773$	−26.2445	−0.943948	−0.471974	−	0.881613i	$-0.656458\pi$
$773$	−26.2445	−0.943948	−0.471974	+	0.881613i	$0.656458\pi$
$774$	0	0
$775$	1.91636	0.0688376
$776$	0.734589	0.0263702
$777$	0	0
$778$	32.3402	1.15945
$779$	4.94942	0.177331
$780$	0	0
$781$	−28.4323	−1.01739
$782$	−13.8731	−0.496102
$783$	0	0
$784$	−4.96133	−0.177190
$785$	−46.3505	−1.65432
$786$	0	0
$787$	17.4932	0.623564	0.311782	−	0.950154i	$-0.399074\pi$
$787$	17.4932	0.623564	0.311782	+	0.950154i	$0.399074\pi$
$788$	10.4962	0.373913
$789$	0	0
$790$	1.01114	0.0359749
$791$	−8.54280	−0.303747
$792$	0	0
$793$	−53.1102	−1.88600
$794$	−5.97454	−0.212028
$795$	0	0
$796$	−8.73091	−0.309459
$797$	20.6866	0.732758	0.366379	−	0.930466i	$-0.380597\pi$
$797$	20.6866	0.732758	0.366379	+	0.930466i	$0.380597\pi$
$798$	0	0
$799$	6.11498	0.216333
$800$	10.8792	0.384638
$801$	0	0
$802$	16.9429	0.598274
$803$	−24.9049	−0.878875
$804$	0	0
$805$	6.78364	0.239092
$806$	−9.13913	−0.321912
$807$	0	0
$808$	−8.84407	−0.311133
$809$	24.9709	0.877930	0.438965	−	0.898504i	$-0.355345\pi$
$809$	24.9709	0.877930	0.438965	+	0.898504i	$0.355345\pi$
$810$	0	0
$811$	45.0359	1.58143	0.790713	−	0.612187i	$-0.209710\pi$
$811$	45.0359	1.58143	0.790713	+	0.612187i	$0.209710\pi$
$812$	2.29679	0.0806016
$813$	0	0
$814$	48.1058	1.68611
$815$	−34.7448	−1.21706
$816$	0	0
$817$	−34.2444	−1.19806
$818$	−66.9819	−2.34197
$819$	0	0
$820$	−3.55550	−0.124164
$821$	21.6733	0.756404	0.378202	−	0.925723i	$-0.376542\pi$
$821$	21.6733	0.756404	0.378202	+	0.925723i	$0.376542\pi$
$822$	0	0
$823$	−35.9327	−1.25253	−0.626267	−	0.779609i	$-0.715418\pi$
$823$	−35.9327	−1.25253	−0.626267	+	0.779609i	$0.715418\pi$
$824$	22.9593	0.799825
$825$	0	0
$826$	8.60407	0.299374
$827$	43.0107	1.49563	0.747814	−	0.663908i	$-0.231104\pi$
$827$	43.0107	1.49563	0.747814	+	0.663908i	$0.231104\pi$
$828$	0	0
$829$	5.33521	0.185300	0.0926498	−	0.995699i	$-0.470466\pi$
$829$	5.33521	0.185300	0.0926498	+	0.995699i	$0.470466\pi$
$830$	16.0221	0.556137
$831$	0	0
$832$	−3.87452	−0.134325
$833$	−2.98581	−0.103452
$834$	0	0
$835$	25.6749	0.888516
$836$	19.1182	0.661218
$837$	0	0
$838$	8.62024	0.297781
$839$	−13.1009	−0.452294	−0.226147	−	0.974093i	$-0.572613\pi$
$839$	−13.1009	−0.452294	−0.226147	+	0.974093i	$0.572613\pi$
$840$	0	0
$841$	−25.3160	−0.872966
$842$	52.3970	1.80572
$843$	0	0
$844$	17.2590	0.594079
$845$	−27.1706	−0.934697
$846$	0	0
$847$	2.50027	0.0859103
$848$	−62.0398	−2.13045
$849$	0	0
$850$	9.68316	0.332130
$851$	19.0303	0.652350
$852$	0	0
$853$	−19.2115	−0.657788	−0.328894	−	0.944367i	$-0.606676\pi$
$853$	−19.2115	−0.657788	−0.328894	+	0.944367i	$0.606676\pi$
$854$	19.6261	0.671593
$855$	0	0
$856$	−25.3488	−0.866403
$857$	38.6562	1.32047	0.660236	−	0.751058i	$-0.270456\pi$
$857$	38.6562	1.32047	0.660236	+	0.751058i	$0.270456\pi$
$858$	0	0
$859$	45.0464	1.53696	0.768482	−	0.639872i	$-0.221012\pi$
$859$	45.0464	1.53696	0.768482	+	0.639872i	$0.221012\pi$
$860$	24.6001	0.838855
$861$	0	0
$862$	71.7549	2.44398
$863$	44.3562	1.50990	0.754951	−	0.655782i	$-0.227661\pi$
$863$	44.3562	1.50990	0.754951	+	0.655782i	$0.227661\pi$
$864$	0	0
$865$	−5.43510	−0.184799
$866$	30.1973	1.02615
$867$	0	0
$868$	1.26424	0.0429113
$869$	0.796050	0.0270041
$870$	0	0
$871$	−6.37054	−0.215858
$872$	−27.5984	−0.934600
$873$	0	0
$874$	20.2035	0.683393
$875$	8.31686	0.281161
$876$	0	0
$877$	−50.6435	−1.71011	−0.855055	−	0.518536i	$-0.826477\pi$
$877$	−50.6435	−1.71011	−0.855055	+	0.518536i	$0.826477\pi$
$878$	−37.1740	−1.25456
$879$	0	0
$880$	47.5846	1.60408
$881$	26.3543	0.887900	0.443950	−	0.896052i	$-0.353577\pi$
$881$	26.3543	0.887900	0.443950	+	0.896052i	$0.353577\pi$
$882$	0	0
$883$	−39.0563	−1.31435	−0.657175	−	0.753738i	$-0.728249\pi$
$883$	−39.0563	−1.31435	−0.657175	+	0.753738i	$0.728249\pi$
$884$	−17.2868	−0.581418
$885$	0	0
$886$	−5.68923	−0.191133
$887$	−24.3759	−0.818464	−0.409232	−	0.912430i	$-0.634203\pi$
$887$	−24.3759	−0.818464	−0.409232	+	0.912430i	$0.634203\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	−43.5292	−1.45910
$891$	0	0
$892$	15.3606	0.514309
$893$	−8.90527	−0.298003
$894$	0	0
$895$	−56.3868	−1.88480
$896$	−10.5637	−0.352910
$897$	0	0
$898$	−37.8221	−1.26214
$899$	2.02781	0.0676312
$900$	0	0
$901$	−37.3365	−1.24386
$902$	−7.47754	−0.248975
$903$	0	0
$904$	−12.2704	−0.408107
$905$	−4.60569	−0.153098
$906$	0	0
$907$	−35.8315	−1.18977	−0.594883	−	0.803812i	$-0.702802\pi$
$907$	−35.8315	−1.18977	−0.594883	+	0.803812i	$0.702802\pi$
$908$	−21.2009	−0.703575
$909$	0	0
$910$	22.5805	0.748537
$911$	−29.2977	−0.970677	−0.485339	−	0.874326i	$-0.661304\pi$
$911$	−29.2977	−0.970677	−0.485339	+	0.874326i	$0.661304\pi$
$912$	0	0
$913$	12.6139	0.417458
$914$	48.9082	1.61774
$915$	0	0
$916$	8.63256	0.285228
$917$	−8.10827	−0.267759
$918$	0	0
$919$	41.9383	1.38342	0.691709	−	0.722177i	$-0.256858\pi$
$919$	41.9383	1.38342	0.691709	+	0.722177i	$0.256858\pi$
$920$	9.74363	0.321238
$921$	0	0
$922$	30.8511	1.01603
$923$	−37.4395	−1.23234
$924$	0	0
$925$	−13.2828	−0.436734
$926$	45.2319	1.48641
$927$	0	0
$928$	11.5119	0.377897
$929$	32.8282	1.07706	0.538530	−	0.842607i	$-0.318980\pi$
$929$	32.8282	1.07706	0.538530	+	0.842607i	$0.318980\pi$
$930$	0	0
$931$	4.34824	0.142508
$932$	−23.3556	−0.765039
$933$	0	0
$934$	38.7076	1.26655
$935$	28.6372	0.936536
$936$	0	0
$937$	60.4319	1.97422	0.987112	−	0.160030i	$-0.0511593\pi$
$937$	60.4319	1.97422	0.987112	+	0.160030i	$0.0511593\pi$
$938$	2.35415	0.0768656
$939$	0	0
$940$	6.39725	0.208655
$941$	39.9477	1.30226	0.651129	−	0.758967i	$-0.274296\pi$
$941$	39.9477	1.30226	0.651129	+	0.758967i	$0.274296\pi$
$942$	0	0
$943$	−2.95806	−0.0963276
$944$	23.8757	0.777086
$945$	0	0
$946$	51.7361	1.68209
$947$	−48.3521	−1.57123	−0.785616	−	0.618714i	$-0.787654\pi$
$947$	−48.3521	−1.57123	−0.785616	+	0.618714i	$0.787654\pi$
$948$	0	0
$949$	−32.7947	−1.06456
$950$	−14.1016	−0.457517
$951$	0	0
$952$	−4.28864	−0.138996
$953$	−17.5215	−0.567578	−0.283789	−	0.958887i	$-0.591592\pi$
$953$	−17.5215	−0.567578	−0.283789	+	0.958887i	$0.591592\pi$
$954$	0	0
$955$	−29.2712	−0.947194
$956$	22.6400	0.732231
$957$	0	0
$958$	−54.3177	−1.75493
$959$	7.16460	0.231357
$960$	0	0
$961$	−29.8838	−0.963994
$962$	63.3456	2.04234
$963$	0	0
$964$	26.8458	0.864644
$965$	−37.3837	−1.20342
$966$	0	0
$967$	−33.2691	−1.06986	−0.534930	−	0.844896i	$-0.679662\pi$
$967$	−33.2691	−1.06986	−0.534930	+	0.844896i	$0.679662\pi$
$968$	3.59124	0.115427
$969$	0	0
$970$	2.38688	0.0766380
$971$	31.8860	1.02327	0.511635	−	0.859203i	$-0.329040\pi$
$971$	31.8860	1.02327	0.511635	+	0.859203i	$0.329040\pi$
$972$	0	0
$973$	−17.8437	−0.572042
$974$	−48.2683	−1.54662
$975$	0	0
$976$	54.4611	1.74326
$977$	5.55629	0.177762	0.0888808	−	0.996042i	$-0.471671\pi$
$977$	5.55629	0.177762	0.0888808	+	0.996042i	$0.471671\pi$
$978$	0	0
$979$	−34.2695	−1.09526
$980$	−3.12363	−0.0997808
$981$	0	0
$982$	29.8372	0.952143
$983$	−39.4972	−1.25977	−0.629883	−	0.776690i	$-0.716897\pi$
$983$	−39.4972	−1.25977	−0.629883	+	0.776690i	$0.716897\pi$
$984$	0	0
$985$	−22.8964	−0.729541
$986$	10.2463	0.326309
$987$	0	0
$988$	25.1748	0.800918
$989$	20.4664	0.650794
$990$	0	0
$991$	−37.6712	−1.19667	−0.598333	−	0.801248i	$-0.704170\pi$
$991$	−37.6712	−1.19667	−0.598333	+	0.801248i	$0.704170\pi$
$992$	6.33661	0.201188
$993$	0	0
$994$	13.8352	0.438827
$995$	19.0456	0.603785
$996$	0	0
$997$	−34.4421	−1.09079	−0.545396	−	0.838179i	$-0.683621\pi$
$997$	−34.4421	−1.09079	−0.545396	+	0.838179i	$0.683621\pi$
$998$	30.1311	0.953784
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.w.1.7		20
3.2	odd	2		889.2.a.d.1.14	✓	20
21.20	even	2		6223.2.a.l.1.14		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.d.1.14	✓	20	3.2	odd	2
6223.2.a.l.1.14		20	21.20	even	2
8001.2.a.w.1.7		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)

\(f(q)\)	\(=\)	\(q-1.78791 q^{2} +1.19664 q^{4} -2.61034 q^{5} +1.00000 q^{7} +1.43634 q^{8} +O(q^{10})\)	\(q-1.78791 q^{2} +1.19664 q^{4} -2.61034 q^{5} +1.00000 q^{7} +1.43634 q^{8} +4.66707 q^{10} +3.67427 q^{11} +4.83827 q^{13} -1.78791 q^{14} -4.96133 q^{16} -2.98581 q^{17} +4.34824 q^{19} -3.12363 q^{20} -6.56928 q^{22} -2.59876 q^{23} +1.81388 q^{25} -8.65041 q^{26} +1.19664 q^{28} +1.91937 q^{29} +1.05650 q^{31} +5.99776 q^{32} +5.33837 q^{34} -2.61034 q^{35} -7.32285 q^{37} -7.77428 q^{38} -3.74934 q^{40} +1.13826 q^{41} -7.87546 q^{43} +4.39677 q^{44} +4.64636 q^{46} -2.04802 q^{47} +1.00000 q^{49} -3.24306 q^{50} +5.78966 q^{52} +12.5047 q^{53} -9.59110 q^{55} +1.43634 q^{56} -3.43167 q^{58} -4.81235 q^{59} -10.9771 q^{61} -1.88893 q^{62} -0.800808 q^{64} -12.6295 q^{65} -1.31670 q^{67} -3.57293 q^{68} +4.66707 q^{70} -7.73820 q^{71} -6.77819 q^{73} +13.0926 q^{74} +5.20327 q^{76} +3.67427 q^{77} +0.216655 q^{79} +12.9508 q^{80} -2.03511 q^{82} +3.43302 q^{83} +7.79398 q^{85} +14.0807 q^{86} +5.27751 q^{88} -9.32688 q^{89} +4.83827 q^{91} -3.10977 q^{92} +3.66168 q^{94} -11.3504 q^{95} +0.511430 q^{97} -1.78791 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 8 q^{2} + 24 q^{4} - 3 q^{5} + 20 q^{7} - 24 q^{8}+O(q^{10}) \) 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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