LMFDB - Embedded newform 8001.2.a.l.1.2

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:	$7$
Coefficient field:	7.7.118870813.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 9x^{5} - 3x^{4} + 20x^{3} + 7x^{2} - 13x - 4 $ x^7 - 9x^5 - 3x^4 + 20x^3 + 7x^2 - 13*x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2667)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.69855$ 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)

$f(q)$	$=$	$q-0.840819 q^{2} -1.29302 q^{4} -2.74724 q^{5} +1.00000 q^{7} +2.76884 q^{8} +O(q^{10})$	\(q-0.840819 q^{2} -1.29302 q^{4} -2.74724 q^{5} +1.00000 q^{7} +2.76884 q^{8} +2.30993 q^{10} +5.62069 q^{11} -3.65275 q^{13} -0.840819 q^{14} +0.257957 q^{16} +5.86462 q^{17} -2.05868 q^{19} +3.55225 q^{20} -4.72598 q^{22} -2.78500 q^{23} +2.54733 q^{25} +3.07130 q^{26} -1.29302 q^{28} +4.83208 q^{29} -6.87419 q^{31} -5.75457 q^{32} -4.93108 q^{34} -2.74724 q^{35} -8.71097 q^{37} +1.73097 q^{38} -7.60666 q^{40} +4.09940 q^{41} -2.48024 q^{43} -7.26768 q^{44} +2.34168 q^{46} -12.6576 q^{47} +1.00000 q^{49} -2.14185 q^{50} +4.72310 q^{52} -4.92291 q^{53} -15.4414 q^{55} +2.76884 q^{56} -4.06291 q^{58} +9.41559 q^{59} +4.96498 q^{61} +5.77995 q^{62} +4.32264 q^{64} +10.0350 q^{65} +4.57422 q^{67} -7.58309 q^{68} +2.30993 q^{70} -2.13466 q^{71} +9.59565 q^{73} +7.32435 q^{74} +2.66192 q^{76} +5.62069 q^{77} -10.0298 q^{79} -0.708669 q^{80} -3.44685 q^{82} +5.06691 q^{83} -16.1115 q^{85} +2.08543 q^{86} +15.5628 q^{88} -2.73622 q^{89} -3.65275 q^{91} +3.60107 q^{92} +10.6428 q^{94} +5.65568 q^{95} +14.0813 q^{97} -0.840819 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 2 q^{2} + 4 q^{4} + 8 q^{5} + 7 q^{7} + 9 q^{8}+O(q^{10}) $ 7 * q + 2 * q^2 + 4 * q^4 + 8 * q^5 + 7 * q^7 + 9 * q^8	$ 7 q + 2 q^{2} + 4 q^{4} + 8 q^{5} + 7 q^{7} + 9 q^{8} + 3 q^{11} - 23 q^{13} + 2 q^{14} + 2 q^{16} - 3 q^{17} - 9 q^{19} + 9 q^{20} - 19 q^{22} - 12 q^{23} + 3 q^{25} - 18 q^{26} + 4 q^{28} + 9 q^{29} - 33 q^{31} - 10 q^{32} - 2 q^{34} + 8 q^{35} - 33 q^{37} + 3 q^{38} - 9 q^{40} + 3 q^{41} - 9 q^{43} - 2 q^{44} - 32 q^{46} - 11 q^{47} + 7 q^{49} - 29 q^{50} - 21 q^{52} - q^{53} - 16 q^{55} + 9 q^{56} - 5 q^{58} + 30 q^{59} - 19 q^{61} - 3 q^{62} - 21 q^{64} - 14 q^{65} - 30 q^{67} - 24 q^{68} - 8 q^{71} - 20 q^{73} + 9 q^{74} - 42 q^{76} + 3 q^{77} + 8 q^{79} - 12 q^{80} + 10 q^{82} + 34 q^{83} - 28 q^{85} - 24 q^{86} - q^{88} + 12 q^{89} - 23 q^{91} - 60 q^{92} - 3 q^{94} - 12 q^{95} + 7 q^{97} + 2 q^{98}+O(q^{100}) $ 7 * q + 2 * q^2 + 4 * q^4 + 8 * q^5 + 7 * q^7 + 9 * q^8 + 3 * q^11 - 23 * q^13 + 2 * q^14 + 2 * q^16 - 3 * q^17 - 9 * q^19 + 9 * q^20 - 19 * q^22 - 12 * q^23 + 3 * q^25 - 18 * q^26 + 4 * q^28 + 9 * q^29 - 33 * q^31 - 10 * q^32 - 2 * q^34 + 8 * q^35 - 33 * q^37 + 3 * q^38 - 9 * q^40 + 3 * q^41 - 9 * q^43 - 2 * q^44 - 32 * q^46 - 11 * q^47 + 7 * q^49 - 29 * q^50 - 21 * q^52 - q^53 - 16 * q^55 + 9 * q^56 - 5 * q^58 + 30 * q^59 - 19 * q^61 - 3 * q^62 - 21 * q^64 - 14 * q^65 - 30 * q^67 - 24 * q^68 - 8 * q^71 - 20 * q^73 + 9 * q^74 - 42 * q^76 + 3 * q^77 + 8 * q^79 - 12 * q^80 + 10 * q^82 + 34 * q^83 - 28 * q^85 - 24 * q^86 - q^88 + 12 * q^89 - 23 * q^91 - 60 * q^92 - 3 * q^94 - 12 * q^95 + 7 * q^97 + 2 * 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 $

$2$	−0.840819	−0.594549	−0.297274	−	0.954792i	$-0.596078\pi$
$2$	−0.840819	−0.594549	−0.297274	+	0.954792i	$0.596078\pi$
$3$	0	0
$3$	0	0
$4$	−1.29302	−0.646512
$5$	−2.74724	−1.22860	−0.614302	−	0.789071i	$-0.710562\pi$
$5$	−2.74724	−1.22860	−0.614302	+	0.789071i	$0.710562\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.76884	0.978932
$9$	0	0
$10$	2.30993	0.730465
$11$	5.62069	1.69470	0.847351	−	0.531034i	$-0.178196\pi$
$11$	5.62069	1.69470	0.847351	+	0.531034i	$0.178196\pi$
$12$	0	0
$13$	−3.65275	−1.01309	−0.506546	−	0.862213i	$-0.669078\pi$
$13$	−3.65275	−1.01309	−0.506546	+	0.862213i	$0.669078\pi$
$14$	−0.840819	−0.224718
$15$	0	0
$16$	0.257957	0.0644892
$17$	5.86462	1.42238	0.711189	−	0.703000i	$-0.248157\pi$
$17$	5.86462	1.42238	0.711189	+	0.703000i	$0.248157\pi$
$18$	0	0
$19$	−2.05868	−0.472293	−0.236146	−	0.971718i	$-0.575884\pi$
$19$	−2.05868	−0.472293	−0.236146	+	0.971718i	$0.575884\pi$
$20$	3.55225	0.794307
$21$	0	0
$22$	−4.72598	−1.00758
$23$	−2.78500	−0.580713	−0.290356	−	0.956919i	$-0.593774\pi$
$23$	−2.78500	−0.580713	−0.290356	+	0.956919i	$0.593774\pi$
$24$	0	0
$25$	2.54733	0.509467
$26$	3.07130	0.602332
$27$	0	0
$28$	−1.29302	−0.244358
$29$	4.83208	0.897295	0.448648	−	0.893709i	$-0.351906\pi$
$29$	4.83208	0.897295	0.448648	+	0.893709i	$0.351906\pi$
$30$	0	0
$31$	−6.87419	−1.23464	−0.617321	−	0.786712i	$-0.711782\pi$
$31$	−6.87419	−1.23464	−0.617321	+	0.786712i	$0.711782\pi$
$32$	−5.75457	−1.01727
$33$	0	0
$34$	−4.93108	−0.845674
$35$	−2.74724	−0.464369
$36$	0	0
$37$	−8.71097	−1.43207	−0.716037	−	0.698062i	$-0.754046\pi$
$37$	−8.71097	−1.43207	−0.716037	+	0.698062i	$0.754046\pi$
$38$	1.73097	0.280801
$39$	0	0
$40$	−7.60666	−1.20272
$41$	4.09940	0.640219	0.320109	−	0.947381i	$-0.396280\pi$
$41$	4.09940	0.640219	0.320109	+	0.947381i	$0.396280\pi$
$42$	0	0
$43$	−2.48024	−0.378233	−0.189116	−	0.981955i	$-0.560562\pi$
$43$	−2.48024	−0.378233	−0.189116	+	0.981955i	$0.560562\pi$
$44$	−7.26768	−1.09564
$45$	0	0
$46$	2.34168	0.345262
$47$	−12.6576	−1.84630	−0.923152	−	0.384434i	$-0.874397\pi$
$47$	−12.6576	−1.84630	−0.923152	+	0.384434i	$0.874397\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−2.14185	−0.302903
$51$	0	0
$52$	4.72310	0.654975
$53$	−4.92291	−0.676214	−0.338107	−	0.941108i	$-0.609787\pi$
$53$	−4.92291	−0.676214	−0.338107	+	0.941108i	$0.609787\pi$
$54$	0	0
$55$	−15.4414	−2.08212
$56$	2.76884	0.370001
$57$	0	0
$58$	−4.06291	−0.533486
$59$	9.41559	1.22581	0.612903	−	0.790158i	$-0.290002\pi$
$59$	9.41559	1.22581	0.612903	+	0.790158i	$0.290002\pi$
$60$	0	0
$61$	4.96498	0.635701	0.317850	−	0.948141i	$-0.397039\pi$
$61$	4.96498	0.635701	0.317850	+	0.948141i	$0.397039\pi$
$62$	5.77995	0.734055
$63$	0	0
$64$	4.32264	0.540330
$65$	10.0350	1.24469
$66$	0	0
$67$	4.57422	0.558829	0.279415	−	0.960171i	$-0.409860\pi$
$67$	4.57422	0.558829	0.279415	+	0.960171i	$0.409860\pi$
$68$	−7.58309	−0.919585
$69$	0	0
$70$	2.30993	0.276090
$71$	−2.13466	−0.253338	−0.126669	−	0.991945i	$-0.540429\pi$
$71$	−2.13466	−0.253338	−0.126669	+	0.991945i	$0.540429\pi$
$72$	0	0
$73$	9.59565	1.12309	0.561543	−	0.827448i	$-0.310208\pi$
$73$	9.59565	1.12309	0.561543	+	0.827448i	$0.310208\pi$
$74$	7.32435	0.851438
$75$	0	0
$76$	2.66192	0.305343
$77$	5.62069	0.640537
$78$	0	0
$79$	−10.0298	−1.12844	−0.564220	−	0.825624i	$-0.690823\pi$
$79$	−10.0298	−1.12844	−0.564220	+	0.825624i	$0.690823\pi$
$80$	−0.708669	−0.0792316
$81$	0	0
$82$	−3.44685	−0.380641
$83$	5.06691	0.556166	0.278083	−	0.960557i	$-0.410301\pi$
$83$	5.06691	0.556166	0.278083	+	0.960557i	$0.410301\pi$
$84$	0	0
$85$	−16.1115	−1.74754
$86$	2.08543	0.224878
$87$	0	0
$88$	15.5628	1.65900
$89$	−2.73622	−0.290039	−0.145019	−	0.989429i	$-0.546324\pi$
$89$	−2.73622	−0.290039	−0.145019	+	0.989429i	$0.546324\pi$
$90$	0	0
$91$	−3.65275	−0.382913
$92$	3.60107	0.375437
$93$	0	0
$94$	10.6428	1.09772
$95$	5.65568	0.580260
$96$	0	0
$97$	14.0813	1.42974	0.714872	−	0.699255i	$-0.246485\pi$
$97$	14.0813	1.42974	0.714872	+	0.699255i	$0.246485\pi$
$98$	−0.840819	−0.0849355
$99$	0	0
$100$	−3.29376	−0.329376
$101$	2.91727	0.290279	0.145139	−	0.989411i	$-0.453637\pi$
$101$	2.91727	0.290279	0.145139	+	0.989411i	$0.453637\pi$
$102$	0	0
$103$	17.2096	1.69571	0.847855	−	0.530228i	$-0.177894\pi$
$103$	17.2096	1.69571	0.847855	+	0.530228i	$0.177894\pi$
$104$	−10.1139	−0.991747
$105$	0	0
$106$	4.13928	0.402042
$107$	−3.25655	−0.314823	−0.157411	−	0.987533i	$-0.550315\pi$
$107$	−3.25655	−0.314823	−0.157411	+	0.987533i	$0.550315\pi$
$108$	0	0
$109$	9.69847	0.928945	0.464472	−	0.885588i	$-0.346244\pi$
$109$	9.69847	0.928945	0.464472	+	0.885588i	$0.346244\pi$
$110$	12.9834	1.23792
$111$	0	0
$112$	0.257957	0.0243746
$113$	−1.89566	−0.178329	−0.0891645	−	0.996017i	$-0.528420\pi$
$113$	−1.89566	−0.178329	−0.0891645	+	0.996017i	$0.528420\pi$
$114$	0	0
$115$	7.65107	0.713466
$116$	−6.24800	−0.580112
$117$	0	0
$118$	−7.91681	−0.728801
$119$	5.86462	0.537609
$120$	0	0
$121$	20.5921	1.87201
$122$	−4.17465	−0.377955
$123$	0	0
$124$	8.88849	0.798210
$125$	6.73807	0.602671
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	7.87458	0.696021
$129$	0	0
$130$	−8.43761	−0.740028
$131$	−5.84500	−0.510680	−0.255340	−	0.966851i	$-0.582187\pi$
$131$	−5.84500	−0.510680	−0.255340	+	0.966851i	$0.582187\pi$
$132$	0	0
$133$	−2.05868	−0.178510
$134$	−3.84609	−0.332251
$135$	0	0
$136$	16.2382	1.39241
$137$	10.6774	0.912233	0.456117	−	0.889920i	$-0.349240\pi$
$137$	10.6774	0.912233	0.456117	+	0.889920i	$0.349240\pi$
$138$	0	0
$139$	−21.7313	−1.84323	−0.921613	−	0.388111i	$-0.873128\pi$
$139$	−21.7313	−1.84323	−0.921613	+	0.388111i	$0.873128\pi$
$140$	3.55225	0.300220
$141$	0	0
$142$	1.79486	0.150622
$143$	−20.5310	−1.71689
$144$	0	0
$145$	−13.2749	−1.10242
$146$	−8.06821	−0.667729
$147$	0	0
$148$	11.2635	0.925853
$149$	16.2091	1.32790	0.663952	−	0.747775i	$-0.268878\pi$
$149$	16.2091	1.32790	0.663952	+	0.747775i	$0.268878\pi$
$150$	0	0
$151$	−22.1101	−1.79929	−0.899646	−	0.436621i	$-0.856175\pi$
$151$	−22.1101	−1.79929	−0.899646	+	0.436621i	$0.856175\pi$
$152$	−5.70014	−0.462342
$153$	0	0
$154$	−4.72598	−0.380830
$155$	18.8851	1.51688
$156$	0	0
$157$	23.2986	1.85943	0.929714	−	0.368282i	$-0.120054\pi$
$157$	23.2986	1.85943	0.929714	+	0.368282i	$0.120054\pi$
$158$	8.43325	0.670913
$159$	0	0
$160$	15.8092	1.24983
$161$	−2.78500	−0.219489
$162$	0	0
$163$	−21.5951	−1.69146	−0.845731	−	0.533609i	$-0.820835\pi$
$163$	−21.5951	−1.69146	−0.845731	+	0.533609i	$0.820835\pi$
$164$	−5.30062	−0.413909
$165$	0	0
$166$	−4.26036	−0.330668
$167$	3.26635	0.252758	0.126379	−	0.991982i	$-0.459664\pi$
$167$	3.26635	0.252758	0.126379	+	0.991982i	$0.459664\pi$
$168$	0	0
$169$	0.342604	0.0263542
$170$	13.5469	1.03900
$171$	0	0
$172$	3.20701	0.244532
$173$	−9.97561	−0.758431	−0.379216	−	0.925308i	$-0.623806\pi$
$173$	−9.97561	−0.758431	−0.379216	+	0.925308i	$0.623806\pi$
$174$	0	0
$175$	2.54733	0.192560
$176$	1.44989	0.109290
$177$	0	0
$178$	2.30066	0.172442
$179$	−21.3721	−1.59743	−0.798714	−	0.601711i	$-0.794486\pi$
$179$	−21.3721	−1.59743	−0.798714	+	0.601711i	$0.794486\pi$
$180$	0	0
$181$	−23.7938	−1.76858	−0.884288	−	0.466942i	$-0.845356\pi$
$181$	−23.7938	−1.76858	−0.884288	+	0.466942i	$0.845356\pi$
$182$	3.07130	0.227660
$183$	0	0
$184$	−7.71121	−0.568478
$185$	23.9311	1.75945
$186$	0	0
$187$	32.9632	2.41051
$188$	16.3666	1.19366
$189$	0	0
$190$	−4.75540	−0.344993
$191$	−17.1115	−1.23814	−0.619072	−	0.785335i	$-0.712491\pi$
$191$	−17.1115	−1.23814	−0.619072	+	0.785335i	$0.712491\pi$
$192$	0	0
$193$	−21.1983	−1.52589	−0.762945	−	0.646464i	$-0.776247\pi$
$193$	−21.1983	−1.52589	−0.762945	+	0.646464i	$0.776247\pi$
$194$	−11.8399	−0.850053
$195$	0	0
$196$	−1.29302	−0.0923588
$197$	−9.64407	−0.687111	−0.343556	−	0.939132i	$-0.611631\pi$
$197$	−9.64407	−0.687111	−0.343556	+	0.939132i	$0.611631\pi$
$198$	0	0
$199$	−8.11930	−0.575562	−0.287781	−	0.957696i	$-0.592918\pi$
$199$	−8.11930	−0.575562	−0.287781	+	0.957696i	$0.592918\pi$
$200$	7.05315	0.498733
$201$	0	0
$202$	−2.45289	−0.172585
$203$	4.83208	0.339146
$204$	0	0
$205$	−11.2620	−0.786575
$206$	−14.4701	−1.00818
$207$	0	0
$208$	−0.942252	−0.0653334
$209$	−11.5712	−0.800395
$210$	0	0
$211$	−10.1539	−0.699023	−0.349511	−	0.936932i	$-0.613652\pi$
$211$	−10.1539	−0.699023	−0.349511	+	0.936932i	$0.613652\pi$
$212$	6.36544	0.437180
$213$	0	0
$214$	2.73817	0.187178
$215$	6.81382	0.464698
$216$	0	0
$217$	−6.87419	−0.466651
$218$	−8.15466	−0.552303
$219$	0	0
$220$	19.9661	1.34611
$221$	−21.4220	−1.44100
$222$	0	0
$223$	22.3276	1.49517	0.747583	−	0.664168i	$-0.231214\pi$
$223$	22.3276	1.49517	0.747583	+	0.664168i	$0.231214\pi$
$224$	−5.75457	−0.384493
$225$	0	0
$226$	1.59391	0.106025
$227$	23.3340	1.54873	0.774365	−	0.632739i	$-0.218069\pi$
$227$	23.3340	1.54873	0.774365	+	0.632739i	$0.218069\pi$
$228$	0	0
$229$	−8.57841	−0.566877	−0.283439	−	0.958990i	$-0.591475\pi$
$229$	−8.57841	−0.566877	−0.283439	+	0.958990i	$0.591475\pi$
$230$	−6.43316	−0.424190
$231$	0	0
$232$	13.3793	0.878391
$233$	−14.6183	−0.957677	−0.478839	−	0.877903i	$-0.658942\pi$
$233$	−14.6183	−0.957677	−0.478839	+	0.877903i	$0.658942\pi$
$234$	0	0
$235$	34.7736	2.26838
$236$	−12.1746	−0.792498
$237$	0	0
$238$	−4.93108	−0.319635
$239$	−5.05219	−0.326799	−0.163399	−	0.986560i	$-0.552246\pi$
$239$	−5.05219	−0.326799	−0.163399	+	0.986560i	$0.552246\pi$
$240$	0	0
$241$	−15.9324	−1.02629	−0.513147	−	0.858301i	$-0.671520\pi$
$241$	−15.9324	−1.02629	−0.513147	+	0.858301i	$0.671520\pi$
$242$	−17.3143	−1.11300
$243$	0	0
$244$	−6.41984	−0.410988
$245$	−2.74724	−0.175515
$246$	0	0
$247$	7.51983	0.478475
$248$	−19.0335	−1.20863
$249$	0	0
$250$	−5.66549	−0.358317
$251$	11.8771	0.749678	0.374839	−	0.927090i	$-0.377698\pi$
$251$	11.8771	0.749678	0.374839	+	0.927090i	$0.377698\pi$
$252$	0	0
$253$	−15.6536	−0.984134
$254$	0.840819	0.0527577
$255$	0	0
$256$	−15.2664	−0.954148
$257$	19.3728	1.20844	0.604221	−	0.796817i	$-0.293484\pi$
$257$	19.3728	1.20844	0.604221	+	0.796817i	$0.293484\pi$
$258$	0	0
$259$	−8.71097	−0.541273
$260$	−12.9755	−0.804705
$261$	0	0
$262$	4.91459	0.303624
$263$	25.0402	1.54405	0.772024	−	0.635594i	$-0.219245\pi$
$263$	25.0402	1.54405	0.772024	+	0.635594i	$0.219245\pi$
$264$	0	0
$265$	13.5244	0.830799
$266$	1.73097	0.106133
$267$	0	0
$268$	−5.91457	−0.361290
$269$	20.8253	1.26974	0.634870	−	0.772619i	$-0.281054\pi$
$269$	20.8253	1.26974	0.634870	+	0.772619i	$0.281054\pi$
$270$	0	0
$271$	14.7479	0.895871	0.447936	−	0.894066i	$-0.352159\pi$
$271$	14.7479	0.895871	0.447936	+	0.894066i	$0.352159\pi$
$272$	1.51282	0.0917280
$273$	0	0
$274$	−8.97778	−0.542367
$275$	14.3178	0.863394
$276$	0	0
$277$	−12.6554	−0.760389	−0.380195	−	0.924907i	$-0.624143\pi$
$277$	−12.6554	−0.760389	−0.380195	+	0.924907i	$0.624143\pi$
$278$	18.2721	1.09589
$279$	0	0
$280$	−7.60666	−0.454585
$281$	30.2858	1.80670	0.903350	−	0.428903i	$-0.141100\pi$
$281$	30.2858	1.80670	0.903350	+	0.428903i	$0.141100\pi$
$282$	0	0
$283$	−0.554307	−0.0329501	−0.0164751	−	0.999864i	$-0.505244\pi$
$283$	−0.554307	−0.0329501	−0.0164751	+	0.999864i	$0.505244\pi$
$284$	2.76017	0.163786
$285$	0	0
$286$	17.2628	1.02077
$287$	4.09940	0.241980
$288$	0	0
$289$	17.3937	1.02316
$290$	11.1618	0.655443
$291$	0	0
$292$	−12.4074	−0.726088
$293$	−10.7329	−0.627023	−0.313512	−	0.949584i	$-0.601505\pi$
$293$	−10.7329	−0.627023	−0.313512	+	0.949584i	$0.601505\pi$
$294$	0	0
$295$	−25.8669	−1.50603
$296$	−24.1192	−1.40190
$297$	0	0
$298$	−13.6289	−0.789504
$299$	10.1729	0.588315
$300$	0	0
$301$	−2.48024	−0.142959
$302$	18.5906	1.06977
$303$	0	0
$304$	−0.531049	−0.0304578
$305$	−13.6400	−0.781024
$306$	0	0
$307$	5.96740	0.340577	0.170289	−	0.985394i	$-0.445530\pi$
$307$	5.96740	0.340577	0.170289	+	0.985394i	$0.445530\pi$
$308$	−7.26768	−0.414115
$309$	0	0
$310$	−15.8789	−0.901862
$311$	−26.7130	−1.51475	−0.757377	−	0.652978i	$-0.773519\pi$
$311$	−26.7130	−1.51475	−0.757377	+	0.652978i	$0.773519\pi$
$312$	0	0
$313$	−25.9107	−1.46456	−0.732279	−	0.681005i	$-0.761543\pi$
$313$	−25.9107	−1.46456	−0.732279	+	0.681005i	$0.761543\pi$
$314$	−19.5899	−1.10552
$315$	0	0
$316$	12.9688	0.729550
$317$	6.29184	0.353385	0.176693	−	0.984266i	$-0.443460\pi$
$317$	6.29184	0.353385	0.176693	+	0.984266i	$0.443460\pi$
$318$	0	0
$319$	27.1596	1.52065
$320$	−11.8753	−0.663851
$321$	0	0
$322$	2.34168	0.130497
$323$	−12.0733	−0.671779
$324$	0	0
$325$	−9.30478	−0.516136
$326$	18.1576	1.00566
$327$	0	0
$328$	11.3506	0.626731
$329$	−12.6576	−0.697838
$330$	0	0
$331$	6.26075	0.344122	0.172061	−	0.985086i	$-0.444957\pi$
$331$	6.26075	0.344122	0.172061	+	0.985086i	$0.444957\pi$
$332$	−6.55164	−0.359568
$333$	0	0
$334$	−2.74641	−0.150277
$335$	−12.5665	−0.686580
$336$	0	0
$337$	10.6291	0.579003	0.289502	−	0.957178i	$-0.406510\pi$
$337$	10.6291	0.579003	0.289502	+	0.957178i	$0.406510\pi$
$338$	−0.288068	−0.0156688
$339$	0	0
$340$	20.8326	1.12980
$341$	−38.6377	−2.09235
$342$	0	0
$343$	1.00000	0.0539949
$344$	−6.86738	−0.370264
$345$	0	0
$346$	8.38768	0.450925
$347$	−24.2445	−1.30151	−0.650757	−	0.759286i	$-0.725548\pi$
$347$	−24.2445	−1.30151	−0.650757	+	0.759286i	$0.725548\pi$
$348$	0	0
$349$	19.2656	1.03127	0.515633	−	0.856810i	$-0.327557\pi$
$349$	19.2656	1.03127	0.515633	+	0.856810i	$0.327557\pi$
$350$	−2.14185	−0.114487
$351$	0	0
$352$	−32.3446	−1.72397
$353$	−2.47593	−0.131780	−0.0658902	−	0.997827i	$-0.520989\pi$
$353$	−2.47593	−0.131780	−0.0658902	+	0.997827i	$0.520989\pi$
$354$	0	0
$355$	5.86443	0.311251
$356$	3.53800	0.187513
$357$	0	0
$358$	17.9701	0.949749
$359$	−31.8474	−1.68084	−0.840422	−	0.541933i	$-0.817693\pi$
$359$	−31.8474	−1.68084	−0.840422	+	0.541933i	$0.817693\pi$
$360$	0	0
$361$	−14.7619	−0.776940
$362$	20.0062	1.05150
$363$	0	0
$364$	4.72310	0.247557
$365$	−26.3616	−1.37983
$366$	0	0
$367$	9.45164	0.493372	0.246686	−	0.969095i	$-0.420658\pi$
$367$	9.45164	0.493372	0.246686	+	0.969095i	$0.420658\pi$
$368$	−0.718409	−0.0374497
$369$	0	0
$370$	−20.1217	−1.04608
$371$	−4.92291	−0.255585
$372$	0	0
$373$	−8.74070	−0.452576	−0.226288	−	0.974060i	$-0.572659\pi$
$373$	−8.74070	−0.452576	−0.226288	+	0.974060i	$0.572659\pi$
$374$	−27.7161	−1.43316
$375$	0	0
$376$	−35.0469	−1.80741
$377$	−17.6504	−0.909042
$378$	0	0
$379$	−15.3774	−0.789882	−0.394941	−	0.918707i	$-0.629235\pi$
$379$	−15.3774	−0.789882	−0.394941	+	0.918707i	$0.629235\pi$
$380$	−7.31292	−0.375145
$381$	0	0
$382$	14.3877	0.736137
$383$	−19.4375	−0.993210	−0.496605	−	0.867977i	$-0.665420\pi$
$383$	−19.4375	−0.993210	−0.496605	+	0.867977i	$0.665420\pi$
$384$	0	0
$385$	−15.4414	−0.786966
$386$	17.8240	0.907216
$387$	0	0
$388$	−18.2075	−0.924347
$389$	−20.2992	−1.02921	−0.514604	−	0.857428i	$-0.672061\pi$
$389$	−20.2992	−1.02921	−0.514604	+	0.857428i	$0.672061\pi$
$390$	0	0
$391$	−16.3330	−0.825993
$392$	2.76884	0.139847
$393$	0	0
$394$	8.10891	0.408521
$395$	27.5543	1.38641
$396$	0	0
$397$	−5.04735	−0.253319	−0.126660	−	0.991946i	$-0.540426\pi$
$397$	−5.04735	−0.253319	−0.126660	+	0.991946i	$0.540426\pi$
$398$	6.82686	0.342200
$399$	0	0
$400$	0.657102	0.0328551
$401$	−0.828202	−0.0413584	−0.0206792	−	0.999786i	$-0.506583\pi$
$401$	−0.828202	−0.0413584	−0.0206792	+	0.999786i	$0.506583\pi$
$402$	0	0
$403$	25.1097	1.25080
$404$	−3.77209	−0.187669
$405$	0	0
$406$	−4.06291	−0.201639
$407$	−48.9616	−2.42694
$408$	0	0
$409$	7.63539	0.377546	0.188773	−	0.982021i	$-0.439549\pi$
$409$	7.63539	0.377546	0.188773	+	0.982021i	$0.439549\pi$
$410$	9.46934	0.467657
$411$	0	0
$412$	−22.2524	−1.09630
$413$	9.41559	0.463311
$414$	0	0
$415$	−13.9200	−0.683308
$416$	21.0200	1.03059
$417$	0	0
$418$	9.72926	0.475874
$419$	0.606948	0.0296514	0.0148257	−	0.999890i	$-0.495281\pi$
$419$	0.606948	0.0296514	0.0148257	+	0.999890i	$0.495281\pi$
$420$	0	0
$421$	12.2074	0.594951	0.297476	−	0.954729i	$-0.403855\pi$
$421$	12.2074	0.594951	0.297476	+	0.954729i	$0.403855\pi$
$422$	8.53759	0.415603
$423$	0	0
$424$	−13.6307	−0.661967
$425$	14.9391	0.724655
$426$	0	0
$427$	4.96498	0.240272
$428$	4.21080	0.203537
$429$	0	0
$430$	−5.72919	−0.276286
$431$	−27.6025	−1.32957	−0.664783	−	0.747037i	$-0.731476\pi$
$431$	−27.6025	−1.32957	−0.664783	+	0.747037i	$0.731476\pi$
$432$	0	0
$433$	−21.0747	−1.01278	−0.506392	−	0.862304i	$-0.669021\pi$
$433$	−21.0747	−1.01278	−0.506392	+	0.862304i	$0.669021\pi$
$434$	5.77995	0.277447
$435$	0	0
$436$	−12.5403	−0.600574
$437$	5.73341	0.274266
$438$	0	0
$439$	−11.8630	−0.566189	−0.283094	−	0.959092i	$-0.591361\pi$
$439$	−11.8630	−0.566189	−0.283094	+	0.959092i	$0.591361\pi$
$440$	−42.7547	−2.03825
$441$	0	0
$442$	18.0120	0.856745
$443$	−11.0804	−0.526446	−0.263223	−	0.964735i	$-0.584786\pi$
$443$	−11.0804	−0.526446	−0.263223	+	0.964735i	$0.584786\pi$
$444$	0	0
$445$	7.51705	0.356342
$446$	−18.7735	−0.888949
$447$	0	0
$448$	4.32264	0.204225
$449$	−21.1462	−0.997949	−0.498974	−	0.866617i	$-0.666290\pi$
$449$	−21.1462	−0.997949	−0.498974	+	0.866617i	$0.666290\pi$
$450$	0	0
$451$	23.0415	1.08498
$452$	2.45114	0.115292
$453$	0	0
$454$	−19.6196	−0.920796
$455$	10.0350	0.470448
$456$	0	0
$457$	−0.600191	−0.0280758	−0.0140379	−	0.999901i	$-0.504469\pi$
$457$	−0.600191	−0.0280758	−0.0140379	+	0.999901i	$0.504469\pi$
$458$	7.21289	0.337036
$459$	0	0
$460$	−9.89301	−0.461264
$461$	31.0470	1.44600	0.723001	−	0.690847i	$-0.242762\pi$
$461$	31.0470	1.44600	0.723001	+	0.690847i	$0.242762\pi$
$462$	0	0
$463$	9.93441	0.461691	0.230846	−	0.972990i	$-0.425851\pi$
$463$	9.93441	0.461691	0.230846	+	0.972990i	$0.425851\pi$
$464$	1.24647	0.0578658
$465$	0	0
$466$	12.2913	0.569386
$467$	6.15650	0.284889	0.142444	−	0.989803i	$-0.454504\pi$
$467$	6.15650	0.284889	0.142444	+	0.989803i	$0.454504\pi$
$468$	0	0
$469$	4.57422	0.211218
$470$	−29.2383	−1.34866
$471$	0	0
$472$	26.0702	1.19998
$473$	−13.9407	−0.640992
$474$	0	0
$475$	−5.24413	−0.240617
$476$	−7.58309	−0.347570
$477$	0	0
$478$	4.24798	0.194298
$479$	−15.7671	−0.720417	−0.360209	−	0.932872i	$-0.617294\pi$
$479$	−15.7671	−0.720417	−0.360209	+	0.932872i	$0.617294\pi$
$480$	0	0
$481$	31.8190	1.45082
$482$	13.3962	0.610182
$483$	0	0
$484$	−26.6261	−1.21028
$485$	−38.6849	−1.75659
$486$	0	0
$487$	−6.30114	−0.285532	−0.142766	−	0.989756i	$-0.545600\pi$
$487$	−6.30114	−0.285532	−0.142766	+	0.989756i	$0.545600\pi$
$488$	13.7472	0.622308
$489$	0	0
$490$	2.30993	0.104352
$491$	2.75670	0.124408	0.0622040	−	0.998063i	$-0.480187\pi$
$491$	2.75670	0.124408	0.0622040	+	0.998063i	$0.480187\pi$
$492$	0	0
$493$	28.3383	1.27629
$494$	−6.32282	−0.284477
$495$	0	0
$496$	−1.77324	−0.0796210
$497$	−2.13466	−0.0957526
$498$	0	0
$499$	1.20293	0.0538506	0.0269253	−	0.999637i	$-0.491428\pi$
$499$	1.20293	0.0538506	0.0269253	+	0.999637i	$0.491428\pi$
$500$	−8.71248	−0.389634
$501$	0	0
$502$	−9.98652	−0.445720
$503$	−29.0110	−1.29354	−0.646769	−	0.762686i	$-0.723880\pi$
$503$	−29.0110	−1.29354	−0.646769	+	0.762686i	$0.723880\pi$
$504$	0	0
$505$	−8.01443	−0.356637
$506$	13.1619	0.585116
$507$	0	0
$508$	1.29302	0.0573686
$509$	9.23054	0.409136	0.204568	−	0.978852i	$-0.434421\pi$
$509$	9.23054	0.409136	0.204568	+	0.978852i	$0.434421\pi$
$510$	0	0
$511$	9.59565	0.424487
$512$	−2.91291	−0.128734
$513$	0	0
$514$	−16.2890	−0.718478
$515$	−47.2789	−2.08336
$516$	0	0
$517$	−71.1446	−3.12894
$518$	7.32435	0.321813
$519$	0	0
$520$	27.7853	1.21846
$521$	29.3043	1.28385	0.641923	−	0.766769i	$-0.278137\pi$
$521$	29.3043	1.28385	0.641923	+	0.766769i	$0.278137\pi$
$522$	0	0
$523$	5.55943	0.243097	0.121549	−	0.992585i	$-0.461214\pi$
$523$	5.55943	0.243097	0.121549	+	0.992585i	$0.461214\pi$
$524$	7.55772	0.330161
$525$	0	0
$526$	−21.0543	−0.918012
$527$	−40.3145	−1.75613
$528$	0	0
$529$	−15.2438	−0.662773
$530$	−11.3716	−0.493951
$531$	0	0
$532$	2.66192	0.115409
$533$	−14.9741	−0.648600
$534$	0	0
$535$	8.94654	0.386793
$536$	12.6653	0.547056
$537$	0	0
$538$	−17.5103	−0.754922
$539$	5.62069	0.242100
$540$	0	0
$541$	−12.1035	−0.520371	−0.260186	−	0.965559i	$-0.583784\pi$
$541$	−12.1035	−0.520371	−0.260186	+	0.965559i	$0.583784\pi$
$542$	−12.4003	−0.532639
$543$	0	0
$544$	−33.7483	−1.44695
$545$	−26.6440	−1.14130
$546$	0	0
$547$	27.7176	1.18512	0.592561	−	0.805526i	$-0.298117\pi$
$547$	27.7176	1.18512	0.592561	+	0.805526i	$0.298117\pi$
$548$	−13.8062	−0.589770
$549$	0	0
$550$	−12.0387	−0.513330
$551$	−9.94769	−0.423786
$552$	0	0
$553$	−10.0298	−0.426510
$554$	10.6409	0.452088
$555$	0	0
$556$	28.0991	1.19167
$557$	−2.98048	−0.126287	−0.0631435	−	0.998004i	$-0.520113\pi$
$557$	−2.98048	−0.126287	−0.0631435	+	0.998004i	$0.520113\pi$
$558$	0	0
$559$	9.05970	0.383185
$560$	−0.708669	−0.0299467
$561$	0	0
$562$	−25.4649	−1.07417
$563$	36.8092	1.55132	0.775662	−	0.631149i	$-0.217416\pi$
$563$	36.8092	1.55132	0.775662	+	0.631149i	$0.217416\pi$
$564$	0	0
$565$	5.20784	0.219096
$566$	0.466072	0.0195904
$567$	0	0
$568$	−5.91052	−0.248000
$569$	30.9224	1.29633	0.648167	−	0.761499i	$-0.275536\pi$
$569$	30.9224	1.29633	0.648167	+	0.761499i	$0.275536\pi$
$570$	0	0
$571$	13.7621	0.575925	0.287963	−	0.957642i	$-0.407022\pi$
$571$	13.7621	0.575925	0.287963	+	0.957642i	$0.407022\pi$
$572$	26.5470	1.10999
$573$	0	0
$574$	−3.44685	−0.143869
$575$	−7.09432	−0.295854
$576$	0	0
$577$	35.2200	1.46623	0.733114	−	0.680106i	$-0.238066\pi$
$577$	35.2200	1.46623	0.733114	+	0.680106i	$0.238066\pi$
$578$	−14.6250	−0.608320
$579$	0	0
$580$	17.1648	0.712728
$581$	5.06691	0.210211
$582$	0	0
$583$	−27.6702	−1.14598
$584$	26.5688	1.09942
$585$	0	0
$586$	9.02443	0.372796
$587$	−17.1613	−0.708322	−0.354161	−	0.935184i	$-0.615234\pi$
$587$	−17.1613	−0.708322	−0.354161	+	0.935184i	$0.615234\pi$
$588$	0	0
$589$	14.1517	0.583112
$590$	21.7494	0.895408
$591$	0	0
$592$	−2.24705	−0.0923533
$593$	41.1163	1.68845	0.844223	−	0.535992i	$-0.180062\pi$
$593$	41.1163	1.68845	0.844223	+	0.535992i	$0.180062\pi$
$594$	0	0
$595$	−16.1115	−0.660508
$596$	−20.9588	−0.858506
$597$	0	0
$598$	−8.55358	−0.349782
$599$	−15.4682	−0.632013	−0.316007	−	0.948757i	$-0.602342\pi$
$599$	−15.4682	−0.632013	−0.316007	+	0.948757i	$0.602342\pi$
$600$	0	0
$601$	−14.5724	−0.594421	−0.297210	−	0.954812i	$-0.596056\pi$
$601$	−14.5724	−0.594421	−0.297210	+	0.954812i	$0.596056\pi$
$602$	2.08543	0.0849959
$603$	0	0
$604$	28.5888	1.16326
$605$	−56.5716	−2.29996
$606$	0	0
$607$	41.7362	1.69402	0.847009	−	0.531578i	$-0.178401\pi$
$607$	41.7362	1.69402	0.847009	+	0.531578i	$0.178401\pi$
$608$	11.8468	0.480451
$609$	0	0
$610$	11.4688	0.464357
$611$	46.2352	1.87048
$612$	0	0
$613$	−43.5236	−1.75790	−0.878951	−	0.476913i	$-0.841756\pi$
$613$	−43.5236	−1.75790	−0.878951	+	0.476913i	$0.841756\pi$
$614$	−5.01750	−0.202490
$615$	0	0
$616$	15.5628	0.627042
$617$	−18.8239	−0.757823	−0.378912	−	0.925433i	$-0.623702\pi$
$617$	−18.8239	−0.757823	−0.378912	+	0.925433i	$0.623702\pi$
$618$	0	0
$619$	−29.5177	−1.18642	−0.593209	−	0.805049i	$-0.702139\pi$
$619$	−29.5177	−1.18642	−0.593209	+	0.805049i	$0.702139\pi$
$620$	−24.4188	−0.980684
$621$	0	0
$622$	22.4608	0.900595
$623$	−2.73622	−0.109624
$624$	0	0
$625$	−31.2478	−1.24991
$626$	21.7862	0.870751
$627$	0	0
$628$	−30.1256	−1.20214
$629$	−51.0865	−2.03695
$630$	0	0
$631$	−24.2239	−0.964337	−0.482169	−	0.876078i	$-0.660151\pi$
$631$	−24.2239	−0.964337	−0.482169	+	0.876078i	$0.660151\pi$
$632$	−27.7709	−1.10467
$633$	0	0
$634$	−5.29030	−0.210105
$635$	2.74724	0.109021
$636$	0	0
$637$	−3.65275	−0.144727
$638$	−22.8363	−0.904099
$639$	0	0
$640$	−21.6334	−0.855134
$641$	−28.0303	−1.10713	−0.553564	−	0.832806i	$-0.686733\pi$
$641$	−28.0303	−1.10713	−0.553564	+	0.832806i	$0.686733\pi$
$642$	0	0
$643$	−42.3142	−1.66871	−0.834354	−	0.551229i	$-0.814159\pi$
$643$	−42.3142	−1.66871	−0.834354	+	0.551229i	$0.814159\pi$
$644$	3.60107	0.141902
$645$	0	0
$646$	10.1515	0.399405
$647$	28.5548	1.12260	0.561302	−	0.827611i	$-0.310301\pi$
$647$	28.5548	1.12260	0.561302	+	0.827611i	$0.310301\pi$
$648$	0	0
$649$	52.9221	2.07737
$650$	7.82364	0.306868
$651$	0	0
$652$	27.9230	1.09355
$653$	0.00473242	0.000185194 0	9.25970e−5	−	1.00000i	$-0.499971\pi$
$653$	0.00473242	0.000185194 0	9.25970e−5		1.00000i	$0.499971\pi$
$654$	0	0
$655$	16.0576	0.627423
$656$	1.05747	0.0412872
$657$	0	0
$658$	10.6428	0.414899
$659$	−1.08891	−0.0424181	−0.0212090	−	0.999775i	$-0.506752\pi$
$659$	−1.08891	−0.0424181	−0.0212090	+	0.999775i	$0.506752\pi$
$660$	0	0
$661$	−34.1649	−1.32886	−0.664431	−	0.747350i	$-0.731326\pi$
$661$	−34.1649	−1.32886	−0.664431	+	0.747350i	$0.731326\pi$
$662$	−5.26415	−0.204597
$663$	0	0
$664$	14.0295	0.544449
$665$	5.65568	0.219318
$666$	0	0
$667$	−13.4574	−0.521071
$668$	−4.22347	−0.163411
$669$	0	0
$670$	10.5661	0.408205
$671$	27.9066	1.07732
$672$	0	0
$673$	−5.87608	−0.226506	−0.113253	−	0.993566i	$-0.536127\pi$
$673$	−5.87608	−0.226506	−0.113253	+	0.993566i	$0.536127\pi$
$674$	−8.93714	−0.344246
$675$	0	0
$676$	−0.442995	−0.0170383
$677$	3.17149	0.121890	0.0609452	−	0.998141i	$-0.480589\pi$
$677$	3.17149	0.121890	0.0609452	+	0.998141i	$0.480589\pi$
$678$	0	0
$679$	14.0813	0.540393
$680$	−44.6102	−1.71072
$681$	0	0
$682$	32.4873	1.24400
$683$	−16.1790	−0.619074	−0.309537	−	0.950887i	$-0.600174\pi$
$683$	−16.1790	−0.619074	−0.309537	+	0.950887i	$0.600174\pi$
$684$	0	0
$685$	−29.3334	−1.12077
$686$	−0.840819	−0.0321026
$687$	0	0
$688$	−0.639794	−0.0243919
$689$	17.9822	0.685067
$690$	0	0
$691$	−20.6251	−0.784617	−0.392309	−	0.919834i	$-0.628323\pi$
$691$	−20.6251	−0.784617	−0.392309	+	0.919834i	$0.628323\pi$
$692$	12.8987	0.490335
$693$	0	0
$694$	20.3852	0.773813
$695$	59.7011	2.26459
$696$	0	0
$697$	24.0414	0.910634
$698$	−16.1989	−0.613138
$699$	0	0
$700$	−3.29376	−0.124493
$701$	4.12137	0.155662	0.0778311	−	0.996967i	$-0.475201\pi$
$701$	4.12137	0.155662	0.0778311	+	0.996967i	$0.475201\pi$
$702$	0	0
$703$	17.9331	0.676358
$704$	24.2962	0.915697
$705$	0	0
$706$	2.08181	0.0783498
$707$	2.91727	0.109715
$708$	0	0
$709$	−0.871563	−0.0327322	−0.0163661	−	0.999866i	$-0.505210\pi$
$709$	−0.871563	−0.0327322	−0.0163661	+	0.999866i	$0.505210\pi$
$710$	−4.93092	−0.185054
$711$	0	0
$712$	−7.57614	−0.283928
$713$	19.1446	0.716972
$714$	0	0
$715$	56.4036	2.10937
$716$	27.6347	1.03276
$717$	0	0
$718$	26.7779	0.999343
$719$	40.4977	1.51031	0.755154	−	0.655547i	$-0.227562\pi$
$719$	40.4977	1.51031	0.755154	+	0.655547i	$0.227562\pi$
$720$	0	0
$721$	17.2096	0.640918
$722$	12.4120	0.461929
$723$	0	0
$724$	30.7659	1.14340
$725$	12.3089	0.457142
$726$	0	0
$727$	20.3846	0.756022	0.378011	−	0.925801i	$-0.376608\pi$
$727$	20.3846	0.756022	0.378011	+	0.925801i	$0.376608\pi$
$728$	−10.1139	−0.374845
$729$	0	0
$730$	22.1653	0.820375
$731$	−14.5457	−0.537991
$732$	0	0
$733$	−45.0929	−1.66554	−0.832772	−	0.553617i	$-0.813247\pi$
$733$	−45.0929	−1.66554	−0.832772	+	0.553617i	$0.813247\pi$
$734$	−7.94712	−0.293334
$735$	0	0
$736$	16.0265	0.590743
$737$	25.7102	0.947049
$738$	0	0
$739$	−18.1927	−0.669230	−0.334615	−	0.942355i	$-0.608606\pi$
$739$	−18.1927	−0.669230	−0.334615	+	0.942355i	$0.608606\pi$
$740$	−30.9435	−1.13751
$741$	0	0
$742$	4.13928	0.151958
$743$	−33.9156	−1.24424	−0.622122	−	0.782921i	$-0.713729\pi$
$743$	−33.9156	−1.24424	−0.622122	+	0.782921i	$0.713729\pi$
$744$	0	0
$745$	−44.5304	−1.63147
$746$	7.34935	0.269079
$747$	0	0
$748$	−42.6222	−1.55842
$749$	−3.25655	−0.118992
$750$	0	0
$751$	26.1611	0.954631	0.477316	−	0.878732i	$-0.341610\pi$
$751$	26.1611	0.954631	0.477316	+	0.878732i	$0.341610\pi$
$752$	−3.26512	−0.119067
$753$	0	0
$754$	14.8408	0.540470
$755$	60.7417	2.21062
$756$	0	0
$757$	−8.54392	−0.310534	−0.155267	−	0.987873i	$-0.549624\pi$
$757$	−8.54392	−0.310534	−0.155267	+	0.987873i	$0.549624\pi$
$758$	12.9296	0.469623
$759$	0	0
$760$	15.6596	0.568035
$761$	−6.45289	−0.233917	−0.116959	−	0.993137i	$-0.537314\pi$
$761$	−6.45289	−0.233917	−0.116959	+	0.993137i	$0.537314\pi$
$762$	0	0
$763$	9.69847	0.351108
$764$	22.1255	0.800474
$765$	0	0
$766$	16.3434	0.590512
$767$	−34.3928	−1.24185
$768$	0	0
$769$	−12.6368	−0.455695	−0.227847	−	0.973697i	$-0.573169\pi$
$769$	−12.6368	−0.455695	−0.227847	+	0.973697i	$0.573169\pi$
$770$	12.9834	0.467890
$771$	0	0
$772$	27.4099	0.986505
$773$	−1.63697	−0.0588776	−0.0294388	−	0.999567i	$-0.509372\pi$
$773$	−1.63697	−0.0588776	−0.0294388	+	0.999567i	$0.509372\pi$
$774$	0	0
$775$	−17.5109	−0.629009
$776$	38.9890	1.39962
$777$	0	0
$778$	17.0679	0.611914
$779$	−8.43934	−0.302371
$780$	0	0
$781$	−11.9983	−0.429331
$782$	13.7331	0.491093
$783$	0	0
$784$	0.257957	0.00921274
$785$	−64.0068	−2.28450
$786$	0	0
$787$	32.4089	1.15525	0.577626	−	0.816301i	$-0.303979\pi$
$787$	32.4089	1.15525	0.577626	+	0.816301i	$0.303979\pi$
$788$	12.4700	0.444225
$789$	0	0
$790$	−23.1682	−0.824286
$791$	−1.89566	−0.0674020
$792$	0	0
$793$	−18.1359	−0.644023
$794$	4.24391	0.150611
$795$	0	0
$796$	10.4985	0.372108
$797$	−28.3883	−1.00557	−0.502783	−	0.864413i	$-0.667690\pi$
$797$	−28.3883	−1.00557	−0.502783	+	0.864413i	$0.667690\pi$
$798$	0	0
$799$	−74.2322	−2.62614
$800$	−14.6588	−0.518267
$801$	0	0
$802$	0.696368	0.0245896
$803$	53.9342	1.90330
$804$	0	0
$805$	7.65107	0.269665
$806$	−21.1127	−0.743664
$807$	0	0
$808$	8.07743	0.284163
$809$	−0.0199237	−0.000700480 0	−0.000350240	−	1.00000i	$-0.500111\pi$
$809$	−0.0199237	−0.000700480 0	−0.000350240		1.00000i	$0.500111\pi$
$810$	0	0
$811$	8.52421	0.299326	0.149663	−	0.988737i	$-0.452181\pi$
$811$	8.52421	0.299326	0.149663	+	0.988737i	$0.452181\pi$
$812$	−6.24800	−0.219262
$813$	0	0
$814$	41.1679	1.44293
$815$	59.3271	2.07814
$816$	0	0
$817$	5.10601	0.178637
$818$	−6.41998	−0.224469
$819$	0	0
$820$	14.5621	0.508530
$821$	21.9384	0.765655	0.382828	−	0.923820i	$-0.374950\pi$
$821$	21.9384	0.765655	0.382828	+	0.923820i	$0.374950\pi$
$822$	0	0
$823$	16.5378	0.576471	0.288236	−	0.957559i	$-0.406931\pi$
$823$	16.5378	0.576471	0.288236	+	0.957559i	$0.406931\pi$
$824$	47.6505	1.65998
$825$	0	0
$826$	−7.91681	−0.275461
$827$	0.791654	0.0275285	0.0137643	−	0.999905i	$-0.495619\pi$
$827$	0.791654	0.0275285	0.0137643	+	0.999905i	$0.495619\pi$
$828$	0	0
$829$	−41.5550	−1.44326	−0.721632	−	0.692277i	$-0.756608\pi$
$829$	−41.5550	−1.44326	−0.721632	+	0.692277i	$0.756608\pi$
$830$	11.7042	0.406260
$831$	0	0
$832$	−15.7895	−0.547403
$833$	5.86462	0.203197
$834$	0	0
$835$	−8.97345	−0.310539
$836$	14.9618	0.517465
$837$	0	0
$838$	−0.510334	−0.0176292
$839$	−44.5008	−1.53634	−0.768169	−	0.640247i	$-0.778832\pi$
$839$	−44.5008	−1.53634	−0.768169	+	0.640247i	$0.778832\pi$
$840$	0	0
$841$	−5.65097	−0.194861
$842$	−10.2642	−0.353728
$843$	0	0
$844$	13.1292	0.451926
$845$	−0.941216	−0.0323788
$846$	0	0
$847$	20.5921	0.707554
$848$	−1.26990	−0.0436085
$849$	0	0
$850$	−12.5611	−0.430843
$851$	24.2600	0.831623
$852$	0	0
$853$	−18.5412	−0.634838	−0.317419	−	0.948285i	$-0.602816\pi$
$853$	−18.5412	−0.634838	−0.317419	+	0.948285i	$0.602816\pi$
$854$	−4.17465	−0.142854
$855$	0	0
$856$	−9.01686	−0.308190
$857$	−37.2675	−1.27304	−0.636518	−	0.771262i	$-0.719626\pi$
$857$	−37.2675	−1.27304	−0.636518	+	0.771262i	$0.719626\pi$
$858$	0	0
$859$	−16.3924	−0.559303	−0.279651	−	0.960102i	$-0.590219\pi$
$859$	−16.3924	−0.559303	−0.279651	+	0.960102i	$0.590219\pi$
$860$	−8.81042	−0.300433
$861$	0	0
$862$	23.2087	0.790491
$863$	47.5599	1.61896	0.809479	−	0.587149i	$-0.199750\pi$
$863$	47.5599	1.61896	0.809479	+	0.587149i	$0.199750\pi$
$864$	0	0
$865$	27.4054	0.931812
$866$	17.7200	0.602149
$867$	0	0
$868$	8.88849	0.301695
$869$	−56.3744	−1.91237
$870$	0	0
$871$	−16.7085	−0.566145
$872$	26.8535	0.909373
$873$	0	0
$874$	−4.82076	−0.163065
$875$	6.73807	0.227788
$876$	0	0
$877$	−22.3076	−0.753275	−0.376637	−	0.926361i	$-0.622920\pi$
$877$	−22.3076	−0.753275	−0.376637	+	0.926361i	$0.622920\pi$
$878$	9.97462	0.336627
$879$	0	0
$880$	−3.98321	−0.134274
$881$	−20.9990	−0.707474	−0.353737	−	0.935345i	$-0.615089\pi$
$881$	−20.9990	−0.707474	−0.353737	+	0.935345i	$0.615089\pi$
$882$	0	0
$883$	24.4512	0.822849	0.411425	−	0.911444i	$-0.365031\pi$
$883$	24.4512	0.822849	0.411425	+	0.911444i	$0.365031\pi$
$884$	27.6992	0.931623
$885$	0	0
$886$	9.31662	0.312998
$887$	−17.1615	−0.576227	−0.288113	−	0.957596i	$-0.593028\pi$
$887$	−17.1615	−0.576227	−0.288113	+	0.957596i	$0.593028\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	−6.32048	−0.211863
$891$	0	0
$892$	−28.8701	−0.966643
$893$	26.0579	0.871996
$894$	0	0
$895$	58.7144	1.96261
$896$	7.87458	0.263071
$897$	0	0
$898$	17.7801	0.593329
$899$	−33.2167	−1.10784
$900$	0	0
$901$	−28.8710	−0.961833
$902$	−19.3737	−0.645073
$903$	0	0
$904$	−5.24878	−0.174572
$905$	65.3672	2.17288
$906$	0	0
$907$	31.9309	1.06025	0.530124	−	0.847920i	$-0.322145\pi$
$907$	31.9309	1.06025	0.530124	+	0.847920i	$0.322145\pi$
$908$	−30.1714	−1.00127
$909$	0	0
$910$	−8.43761	−0.279704
$911$	−37.7485	−1.25067	−0.625333	−	0.780358i	$-0.715037\pi$
$911$	−37.7485	−1.25067	−0.625333	+	0.780358i	$0.715037\pi$
$912$	0	0
$913$	28.4795	0.942535
$914$	0.504652	0.0166924
$915$	0	0
$916$	11.0921	0.366493
$917$	−5.84500	−0.193019
$918$	0	0
$919$	48.2941	1.59308	0.796538	−	0.604588i	$-0.206662\pi$
$919$	48.2941	1.59308	0.796538	+	0.604588i	$0.206662\pi$
$920$	21.1845	0.698434
$921$	0	0
$922$	−26.1049	−0.859718
$923$	7.79739	0.256654
$924$	0	0
$925$	−22.1897	−0.729594
$926$	−8.35304	−0.274498
$927$	0	0
$928$	−27.8066	−0.912795
$929$	−59.7282	−1.95962	−0.979810	−	0.199932i	$-0.935928\pi$
$929$	−59.7282	−1.95962	−0.979810	+	0.199932i	$0.935928\pi$
$930$	0	0
$931$	−2.05868	−0.0674704
$932$	18.9018	0.619149
$933$	0	0
$934$	−5.17650	−0.169380
$935$	−90.5578	−2.96156
$936$	0	0
$937$	−27.0530	−0.883782	−0.441891	−	0.897069i	$-0.645692\pi$
$937$	−27.0530	−0.883782	−0.441891	+	0.897069i	$0.645692\pi$
$938$	−3.84609	−0.125579
$939$	0	0
$940$	−44.9630	−1.46653
$941$	−32.2760	−1.05217	−0.526084	−	0.850433i	$-0.676340\pi$
$941$	−32.2760	−1.05217	−0.526084	+	0.850433i	$0.676340\pi$
$942$	0	0
$943$	−11.4168	−0.371783
$944$	2.42882	0.0790512
$945$	0	0
$946$	11.7216	0.381101
$947$	−4.40313	−0.143083	−0.0715413	−	0.997438i	$-0.522792\pi$
$947$	−4.40313	−0.143083	−0.0715413	+	0.997438i	$0.522792\pi$
$948$	0	0
$949$	−35.0505	−1.13779
$950$	4.40937	0.143059
$951$	0	0
$952$	16.2382	0.526282
$953$	−17.0635	−0.552741	−0.276370	−	0.961051i	$-0.589132\pi$
$953$	−17.0635	−0.552741	−0.276370	+	0.961051i	$0.589132\pi$
$954$	0	0
$955$	47.0094	1.52119
$956$	6.53260	0.211279
$957$	0	0
$958$	13.2573	0.428323
$959$	10.6774	0.344792
$960$	0	0
$961$	16.2545	0.524340
$962$	−26.7540	−0.862584
$963$	0	0
$964$	20.6009	0.663511
$965$	58.2369	1.87471
$966$	0	0
$967$	−26.9109	−0.865398	−0.432699	−	0.901539i	$-0.642439\pi$
$967$	−26.9109	−0.865398	−0.432699	+	0.901539i	$0.642439\pi$
$968$	57.0163	1.83257
$969$	0	0
$970$	32.5270	1.04438
$971$	28.9843	0.930151	0.465076	−	0.885271i	$-0.346027\pi$
$971$	28.9843	0.930151	0.465076	+	0.885271i	$0.346027\pi$
$972$	0	0
$973$	−21.7313	−0.696674
$974$	5.29812	0.169763
$975$	0	0
$976$	1.28075	0.0409958
$977$	50.2194	1.60666	0.803330	−	0.595534i	$-0.203059\pi$
$977$	50.2194	1.60666	0.803330	+	0.595534i	$0.203059\pi$
$978$	0	0
$979$	−15.3794	−0.491529
$980$	3.55225	0.113472
$981$	0	0
$982$	−2.31788	−0.0739667
$983$	1.32859	0.0423756	0.0211878	−	0.999776i	$-0.493255\pi$
$983$	1.32859	0.0423756	0.0211878	+	0.999776i	$0.493255\pi$
$984$	0	0
$985$	26.4946	0.844187
$986$	−23.8274	−0.758819
$987$	0	0
$988$	−9.72332	−0.309340
$989$	6.90747	0.219645
$990$	0	0
$991$	−38.1866	−1.21304	−0.606518	−	0.795070i	$-0.707434\pi$
$991$	−38.1866	−1.21304	−0.606518	+	0.795070i	$0.707434\pi$
$992$	39.5580	1.25597
$993$	0	0
$994$	1.79486	0.0569296
$995$	22.3057	0.707138
$996$	0	0
$997$	34.2759	1.08553	0.542764	−	0.839885i	$-0.317378\pi$
$997$	34.2759	1.08553	0.542764	+	0.839885i	$0.317378\pi$
$998$	−1.01145	−0.0320168
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.l.1.2		7
3.2	odd	2		2667.2.a.j.1.6	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2667.2.a.j.1.6	✓	7	3.2	odd	2
8001.2.a.l.1.2		7	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q-0.840819 q^{2} -1.29302 q^{4} -2.74724 q^{5} +1.00000 q^{7} +2.76884 q^{8} +O(q^{10})\)	\(q-0.840819 q^{2} -1.29302 q^{4} -2.74724 q^{5} +1.00000 q^{7} +2.76884 q^{8} +2.30993 q^{10} +5.62069 q^{11} -3.65275 q^{13} -0.840819 q^{14} +0.257957 q^{16} +5.86462 q^{17} -2.05868 q^{19} +3.55225 q^{20} -4.72598 q^{22} -2.78500 q^{23} +2.54733 q^{25} +3.07130 q^{26} -1.29302 q^{28} +4.83208 q^{29} -6.87419 q^{31} -5.75457 q^{32} -4.93108 q^{34} -2.74724 q^{35} -8.71097 q^{37} +1.73097 q^{38} -7.60666 q^{40} +4.09940 q^{41} -2.48024 q^{43} -7.26768 q^{44} +2.34168 q^{46} -12.6576 q^{47} +1.00000 q^{49} -2.14185 q^{50} +4.72310 q^{52} -4.92291 q^{53} -15.4414 q^{55} +2.76884 q^{56} -4.06291 q^{58} +9.41559 q^{59} +4.96498 q^{61} +5.77995 q^{62} +4.32264 q^{64} +10.0350 q^{65} +4.57422 q^{67} -7.58309 q^{68} +2.30993 q^{70} -2.13466 q^{71} +9.59565 q^{73} +7.32435 q^{74} +2.66192 q^{76} +5.62069 q^{77} -10.0298 q^{79} -0.708669 q^{80} -3.44685 q^{82} +5.06691 q^{83} -16.1115 q^{85} +2.08543 q^{86} +15.5628 q^{88} -2.73622 q^{89} -3.65275 q^{91} +3.60107 q^{92} +10.6428 q^{94} +5.65568 q^{95} +14.0813 q^{97} -0.840819 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{2} + 4 q^{4} + 8 q^{5} + 7 q^{7} + 9 q^{8}+O(q^{10}) \) 7 * q + 2 * q^2 + 4 * q^4 + 8 * q^5 + 7 * q^7 + 9 * q^8	\( 7 q + 2 q^{2} + 4 q^{4} + 8 q^{5} + 7 q^{7} + 9 q^{8} + 3 q^{11} - 23 q^{13} + 2 q^{14} + 2 q^{16} - 3 q^{17} - 9 q^{19} + 9 q^{20} - 19 q^{22} - 12 q^{23} + 3 q^{25} - 18 q^{26} + 4 q^{28} + 9 q^{29} - 33 q^{31} - 10 q^{32} - 2 q^{34} + 8 q^{35} - 33 q^{37} + 3 q^{38} - 9 q^{40} + 3 q^{41} - 9 q^{43} - 2 q^{44} - 32 q^{46} - 11 q^{47} + 7 q^{49} - 29 q^{50} - 21 q^{52} - q^{53} - 16 q^{55} + 9 q^{56} - 5 q^{58} + 30 q^{59} - 19 q^{61} - 3 q^{62} - 21 q^{64} - 14 q^{65} - 30 q^{67} - 24 q^{68} - 8 q^{71} - 20 q^{73} + 9 q^{74} - 42 q^{76} + 3 q^{77} + 8 q^{79} - 12 q^{80} + 10 q^{82} + 34 q^{83} - 28 q^{85} - 24 q^{86} - q^{88} + 12 q^{89} - 23 q^{91} - 60 q^{92} - 3 q^{94} - 12 q^{95} + 7 q^{97} + 2 q^{98}+O(q^{100}) \) 7 * q + 2 * q^2 + 4 * q^4 + 8 * q^5 + 7 * q^7 + 9 * q^8 + 3 * q^11 - 23 * q^13 + 2 * q^14 + 2 * q^16 - 3 * q^17 - 9 * q^19 + 9 * q^20 - 19 * q^22 - 12 * q^23 + 3 * q^25 - 18 * q^26 + 4 * q^28 + 9 * q^29 - 33 * q^31 - 10 * q^32 - 2 * q^34 + 8 * q^35 - 33 * q^37 + 3 * q^38 - 9 * q^40 + 3 * q^41 - 9 * q^43 - 2 * q^44 - 32 * q^46 - 11 * q^47 + 7 * q^49 - 29 * q^50 - 21 * q^52 - q^53 - 16 * q^55 + 9 * q^56 - 5 * q^58 + 30 * q^59 - 19 * q^61 - 3 * q^62 - 21 * q^64 - 14 * q^65 - 30 * q^67 - 24 * q^68 - 8 * q^71 - 20 * q^73 + 9 * q^74 - 42 * q^76 + 3 * q^77 + 8 * q^79 - 12 * q^80 + 10 * q^82 + 34 * q^83 - 28 * q^85 - 24 * q^86 - q^88 + 12 * q^89 - 23 * q^91 - 60 * q^92 - 3 * q^94 - 12 * q^95 + 7 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

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Database

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