LMFDB - Embedded newform 8001.2.a.t.1.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8001,2,Mod(1,8001)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [16,2,0,12,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$63.8883066572$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} - 20 x^{14} + 38 x^{13} + 155 x^{12} - 275 x^{11} - 593 x^{10} + 957 x^{9} + 1177 x^{8} + \cdots + 1 $ x^16 - 2x^15 - 20x^14 + 38x^13 + 155x^12 - 275x^11 - 593x^10 + 957x^9 + 1177x^8 - 1655x^7 - 1150x^6 + 1279x^5 + 474x^4 - 280x^3 - 83x^2 + x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 889)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$1.33134$ 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)

$f(q)$	$=$	\(q+1.33134 q^{2} -0.227522 q^{4} +2.92411 q^{5} -1.00000 q^{7} -2.96560 q^{8} +3.89299 q^{10} -6.16901 q^{11} -0.620036 q^{13} -1.33134 q^{14} -3.49319 q^{16} -0.644462 q^{17} +2.75517 q^{19} -0.665298 q^{20} -8.21308 q^{22} -4.71184 q^{23} +3.55041 q^{25} -0.825482 q^{26} +0.227522 q^{28} +8.85381 q^{29} +2.41753 q^{31} +1.28056 q^{32} -0.858001 q^{34} -2.92411 q^{35} -6.07476 q^{37} +3.66808 q^{38} -8.67173 q^{40} +2.70008 q^{41} +10.3043 q^{43} +1.40358 q^{44} -6.27308 q^{46} +10.7764 q^{47} +1.00000 q^{49} +4.72681 q^{50} +0.141072 q^{52} -6.17224 q^{53} -18.0388 q^{55} +2.96560 q^{56} +11.7875 q^{58} +11.6075 q^{59} -12.3316 q^{61} +3.21857 q^{62} +8.69124 q^{64} -1.81305 q^{65} +12.3886 q^{67} +0.146629 q^{68} -3.89299 q^{70} +10.0499 q^{71} +8.20886 q^{73} -8.08760 q^{74} -0.626862 q^{76} +6.16901 q^{77} +2.30230 q^{79} -10.2145 q^{80} +3.59473 q^{82} +1.22797 q^{83} -1.88448 q^{85} +13.7186 q^{86} +18.2948 q^{88} -11.7296 q^{89} +0.620036 q^{91} +1.07205 q^{92} +14.3471 q^{94} +8.05642 q^{95} +1.19035 q^{97} +1.33134 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8} - 2 q^{10} + 22 q^{11} - 4 q^{13} - 2 q^{14} + 12 q^{16} + 18 q^{17} - 15 q^{19} + 40 q^{20} - 11 q^{22} + 5 q^{23} + 15 q^{25} + 24 q^{26} - 12 q^{28}+ \cdots + 2 q^{98}+O(q^{100}) $ 16 * q + 2 * q^2 + 12 * q^4 + 9 * q^5 - 16 * q^7 + 6 * q^8 - 2 * q^10 + 22 * q^11 - 4 * q^13 - 2 * q^14 + 12 * q^16 + 18 * q^17 - 15 * q^19 + 40 * q^20 - 11 * q^22 + 5 * q^23 + 15 * q^25 + 24 * q^26 - 12 * q^28 + 12 * q^29 - 32 * q^31 + 9 * q^32 - 14 * q^34 - 9 * q^35 - 2 * q^37 - 3 * q^38 - 14 * q^40 + 45 * q^41 - 3 * q^43 + 54 * q^44 + 49 * q^47 + 16 * q^49 + 6 * q^50 + 38 * q^52 - 16 * q^53 + 7 * q^55 - 6 * q^56 + 16 * q^58 + 35 * q^59 - 11 * q^61 - 17 * q^62 - 2 * q^64 - 14 * q^65 + 17 * q^67 + 71 * q^68 + 2 * q^70 + 81 * q^71 - 15 * q^73 - 13 * q^74 + 14 * q^76 - 22 * q^77 - 34 * q^79 + 33 * q^80 - 14 * q^82 + 39 * q^83 - 17 * q^85 - 36 * q^86 + 61 * q^88 + 32 * q^89 + 4 * q^91 - 37 * q^92 + 13 * q^94 + 33 * q^95 - 4 * q^97 + 2 * q^98

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 $

$2$	1.33134	0.941403	0.470701	−	0.882293i	$-0.344001\pi$
$2$	1.33134	0.941403	0.470701	+	0.882293i	$0.344001\pi$
$3$	0	0
$3$	0	0
$4$	−0.227522	−0.113761
$5$	2.92411	1.30770	0.653850	−	0.756624i	$-0.273153\pi$
$5$	2.92411	1.30770	0.653850	+	0.756624i	$0.273153\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−2.96560	−1.04850
$9$	0	0
$10$	3.89299	1.23107
$11$	−6.16901	−1.86003	−0.930013	−	0.367526i	$-0.880205\pi$
$11$	−6.16901	−1.86003	−0.930013	+	0.367526i	$0.880205\pi$
$12$	0	0
$13$	−0.620036	−0.171967	−0.0859836	−	0.996297i	$-0.527403\pi$
$13$	−0.620036	−0.171967	−0.0859836	+	0.996297i	$0.527403\pi$
$14$	−1.33134	−0.355817
$15$	0	0
$16$	−3.49319	−0.873298
$17$	−0.644462	−0.156305	−0.0781525	−	0.996941i	$-0.524902\pi$
$17$	−0.644462	−0.156305	−0.0781525	+	0.996941i	$0.524902\pi$
$18$	0	0
$19$	2.75517	0.632080	0.316040	−	0.948746i	$-0.397647\pi$
$19$	2.75517	0.632080	0.316040	+	0.948746i	$0.397647\pi$
$20$	−0.665298	−0.148765
$21$	0	0
$22$	−8.21308	−1.75103
$23$	−4.71184	−0.982486	−0.491243	−	0.871023i	$-0.663457\pi$
$23$	−4.71184	−0.982486	−0.491243	+	0.871023i	$0.663457\pi$
$24$	0	0
$25$	3.55041	0.710081
$26$	−0.825482	−0.161890
$27$	0	0
$28$	0.227522	0.0429976
$29$	8.85381	1.64411	0.822056	−	0.569407i	$-0.192827\pi$
$29$	8.85381	1.64411	0.822056	+	0.569407i	$0.192827\pi$
$30$	0	0
$31$	2.41753	0.434202	0.217101	−	0.976149i	$-0.430340\pi$
$31$	2.41753	0.434202	0.217101	+	0.976149i	$0.430340\pi$
$32$	1.28056	0.226373
$33$	0	0
$34$	−0.858001	−0.147146
$35$	−2.92411	−0.494264
$36$	0	0
$37$	−6.07476	−0.998685	−0.499343	−	0.866405i	$-0.666425\pi$
$37$	−6.07476	−0.998685	−0.499343	+	0.866405i	$0.666425\pi$
$38$	3.66808	0.595042
$39$	0	0
$40$	−8.67173	−1.37112
$41$	2.70008	0.421681	0.210840	−	0.977520i	$-0.432380\pi$
$41$	2.70008	0.421681	0.210840	+	0.977520i	$0.432380\pi$
$42$	0	0
$43$	10.3043	1.57140	0.785698	−	0.618611i	$-0.212304\pi$
$43$	10.3043	1.57140	0.785698	+	0.618611i	$0.212304\pi$
$44$	1.40358	0.211598
$45$	0	0
$46$	−6.27308	−0.924915
$47$	10.7764	1.57190	0.785950	−	0.618290i	$-0.212174\pi$
$47$	10.7764	1.57190	0.785950	+	0.618290i	$0.212174\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	4.72681	0.668472
$51$	0	0
$52$	0.141072	0.0195631
$53$	−6.17224	−0.847822	−0.423911	−	0.905704i	$-0.639343\pi$
$53$	−6.17224	−0.847822	−0.423911	+	0.905704i	$0.639343\pi$
$54$	0	0
$55$	−18.0388	−2.43236
$56$	2.96560	0.396295
$57$	0	0
$58$	11.7875	1.54777
$59$	11.6075	1.51116	0.755581	−	0.655055i	$-0.227355\pi$
$59$	11.6075	1.51116	0.755581	+	0.655055i	$0.227355\pi$
$60$	0	0
$61$	−12.3316	−1.57890	−0.789448	−	0.613818i	$-0.789633\pi$
$61$	−12.3316	−1.57890	−0.789448	+	0.613818i	$0.789633\pi$
$62$	3.21857	0.408759
$63$	0	0
$64$	8.69124	1.08641
$65$	−1.81305	−0.224882
$66$	0	0
$67$	12.3886	1.51351	0.756753	−	0.653701i	$-0.226785\pi$
$67$	12.3886	1.51351	0.756753	+	0.653701i	$0.226785\pi$
$68$	0.146629	0.0177814
$69$	0	0
$70$	−3.89299	−0.465302
$71$	10.0499	1.19270	0.596351	−	0.802724i	$-0.296617\pi$
$71$	10.0499	1.19270	0.596351	+	0.802724i	$0.296617\pi$
$72$	0	0
$73$	8.20886	0.960775	0.480387	−	0.877056i	$-0.340496\pi$
$73$	8.20886	0.960775	0.480387	+	0.877056i	$0.340496\pi$
$74$	−8.08760	−0.940165
$75$	0	0
$76$	−0.626862	−0.0719060
$77$	6.16901	0.703024
$78$	0	0
$79$	2.30230	0.259029	0.129515	−	0.991578i	$-0.458658\pi$
$79$	2.30230	0.259029	0.129515	+	0.991578i	$0.458658\pi$
$80$	−10.2145	−1.14201
$81$	0	0
$82$	3.59473	0.396972
$83$	1.22797	0.134787	0.0673936	−	0.997726i	$-0.478532\pi$
$83$	1.22797	0.134787	0.0673936	+	0.997726i	$0.478532\pi$
$84$	0	0
$85$	−1.88448	−0.204400
$86$	13.7186	1.47932
$87$	0	0
$88$	18.2948	1.95023
$89$	−11.7296	−1.24334	−0.621668	−	0.783281i	$-0.713544\pi$
$89$	−11.7296	−1.24334	−0.621668	+	0.783281i	$0.713544\pi$
$90$	0	0
$91$	0.620036	0.0649975
$92$	1.07205	0.111768
$93$	0	0
$94$	14.3471	1.47979
$95$	8.05642	0.826571
$96$	0	0
$97$	1.19035	0.120862	0.0604309	−	0.998172i	$-0.480753\pi$
$97$	1.19035	0.120862	0.0604309	+	0.998172i	$0.480753\pi$
$98$	1.33134	0.134486
$99$	0	0
$100$	−0.807795	−0.0807795
$101$	19.3595	1.92635	0.963173	−	0.268882i	$-0.0866542\pi$
$101$	19.3595	1.92635	0.963173	+	0.268882i	$0.0866542\pi$
$102$	0	0
$103$	−10.2002	−1.00506	−0.502528	−	0.864561i	$-0.667597\pi$
$103$	−10.2002	−1.00506	−0.502528	+	0.864561i	$0.667597\pi$
$104$	1.83878	0.180307
$105$	0	0
$106$	−8.21738	−0.798142
$107$	15.3876	1.48758	0.743788	−	0.668415i	$-0.233027\pi$
$107$	15.3876	1.48758	0.743788	+	0.668415i	$0.233027\pi$
$108$	0	0
$109$	−2.40780	−0.230626	−0.115313	−	0.993329i	$-0.536787\pi$
$109$	−2.40780	−0.230626	−0.115313	+	0.993329i	$0.536787\pi$
$110$	−24.0159	−2.28983
$111$	0	0
$112$	3.49319	0.330075
$113$	2.71096	0.255026	0.127513	−	0.991837i	$-0.459301\pi$
$113$	2.71096	0.255026	0.127513	+	0.991837i	$0.459301\pi$
$114$	0	0
$115$	−13.7779	−1.28480
$116$	−2.01443	−0.187036
$117$	0	0
$118$	15.4535	1.42261
$119$	0.644462	0.0590777
$120$	0	0
$121$	27.0567	2.45970
$122$	−16.4176	−1.48638
$123$	0	0
$124$	−0.550041	−0.0493952
$125$	−4.23877	−0.379127
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	9.00992	0.796372
$129$	0	0
$130$	−2.41380	−0.211704
$131$	12.0134	1.04961	0.524807	−	0.851222i	$-0.324138\pi$
$131$	12.0134	1.04961	0.524807	+	0.851222i	$0.324138\pi$
$132$	0	0
$133$	−2.75517	−0.238904
$134$	16.4935	1.42482
$135$	0	0
$136$	1.91122	0.163885
$137$	12.6214	1.07832	0.539161	−	0.842203i	$-0.318742\pi$
$137$	12.6214	1.07832	0.539161	+	0.842203i	$0.318742\pi$
$138$	0	0
$139$	−17.1553	−1.45510	−0.727548	−	0.686057i	$-0.759340\pi$
$139$	−17.1553	−1.45510	−0.727548	+	0.686057i	$0.759340\pi$
$140$	0.665298	0.0562280
$141$	0	0
$142$	13.3799	1.12281
$143$	3.82501	0.319863
$144$	0	0
$145$	25.8895	2.15001
$146$	10.9288	0.904476
$147$	0	0
$148$	1.38214	0.113611
$149$	−5.64770	−0.462677	−0.231339	−	0.972873i	$-0.574311\pi$
$149$	−5.64770	−0.462677	−0.231339	+	0.972873i	$0.574311\pi$
$150$	0	0
$151$	13.0571	1.06257	0.531287	−	0.847192i	$-0.321709\pi$
$151$	13.0571	1.06257	0.531287	+	0.847192i	$0.321709\pi$
$152$	−8.17074	−0.662734
$153$	0	0
$154$	8.21308	0.661829
$155$	7.06913	0.567806
$156$	0	0
$157$	−6.53317	−0.521404	−0.260702	−	0.965419i	$-0.583954\pi$
$157$	−6.53317	−0.521404	−0.260702	+	0.965419i	$0.583954\pi$
$158$	3.06516	0.243851
$159$	0	0
$160$	3.74449	0.296028
$161$	4.71184	0.371345
$162$	0	0
$163$	13.2598	1.03858	0.519292	−	0.854597i	$-0.326196\pi$
$163$	13.2598	1.03858	0.519292	+	0.854597i	$0.326196\pi$
$164$	−0.614326	−0.0479708
$165$	0	0
$166$	1.63485	0.126889
$167$	2.01803	0.156160	0.0780801	−	0.996947i	$-0.475121\pi$
$167$	2.01803	0.156160	0.0780801	+	0.996947i	$0.475121\pi$
$168$	0	0
$169$	−12.6156	−0.970427
$170$	−2.50889	−0.192423
$171$	0	0
$172$	−2.34446	−0.178763
$173$	−1.84088	−0.139960	−0.0699799	−	0.997548i	$-0.522294\pi$
$173$	−1.84088	−0.139960	−0.0699799	+	0.997548i	$0.522294\pi$
$174$	0	0
$175$	−3.55041	−0.268385
$176$	21.5495	1.62436
$177$	0	0
$178$	−15.6161	−1.17048
$179$	−18.7622	−1.40235	−0.701177	−	0.712987i	$-0.747342\pi$
$179$	−18.7622	−1.40235	−0.701177	+	0.712987i	$0.747342\pi$
$180$	0	0
$181$	23.1779	1.72280	0.861399	−	0.507929i	$-0.169589\pi$
$181$	23.1779	1.72280	0.861399	+	0.507929i	$0.169589\pi$
$182$	0.825482	0.0611888
$183$	0	0
$184$	13.9734	1.03013
$185$	−17.7633	−1.30598
$186$	0	0
$187$	3.97569	0.290731
$188$	−2.45187	−0.178821
$189$	0	0
$190$	10.7259	0.778137
$191$	0.0673136	0.00487064	0.00243532	−	0.999997i	$-0.499225\pi$
$191$	0.0673136	0.00487064	0.00243532	+	0.999997i	$0.499225\pi$
$192$	0	0
$193$	−22.7803	−1.63976	−0.819879	−	0.572536i	$-0.805960\pi$
$193$	−22.7803	−1.63976	−0.819879	+	0.572536i	$0.805960\pi$
$194$	1.58477	0.113780
$195$	0	0
$196$	−0.227522	−0.0162516
$197$	20.6532	1.47148	0.735741	−	0.677263i	$-0.236834\pi$
$197$	20.6532	1.47148	0.735741	+	0.677263i	$0.236834\pi$
$198$	0	0
$199$	0.546917	0.0387699	0.0193850	−	0.999812i	$-0.493829\pi$
$199$	0.546917	0.0387699	0.0193850	+	0.999812i	$0.493829\pi$
$200$	−10.5291	−0.744518
$201$	0	0
$202$	25.7742	1.81347
$203$	−8.85381	−0.621416
$204$	0	0
$205$	7.89531	0.551432
$206$	−13.5800	−0.946162
$207$	0	0
$208$	2.16590	0.150178
$209$	−16.9967	−1.17569
$210$	0	0
$211$	−7.86912	−0.541733	−0.270866	−	0.962617i	$-0.587310\pi$
$211$	−7.86912	−0.541733	−0.270866	+	0.962617i	$0.587310\pi$
$212$	1.40432	0.0964490
$213$	0	0
$214$	20.4862	1.40041
$215$	30.1310	2.05491
$216$	0	0
$217$	−2.41753	−0.164113
$218$	−3.20561	−0.217112
$219$	0	0
$220$	4.10423	0.276707
$221$	0.399590	0.0268793
$222$	0	0
$223$	0.641278	0.0429432	0.0214716	−	0.999769i	$-0.493165\pi$
$223$	0.641278	0.0429432	0.0214716	+	0.999769i	$0.493165\pi$
$224$	−1.28056	−0.0855609
$225$	0	0
$226$	3.60923	0.240082
$227$	−25.4329	−1.68804	−0.844021	−	0.536310i	$-0.819818\pi$
$227$	−25.4329	−1.68804	−0.844021	+	0.536310i	$0.819818\pi$
$228$	0	0
$229$	−17.2560	−1.14031	−0.570156	−	0.821537i	$-0.693117\pi$
$229$	−17.2560	−1.14031	−0.570156	+	0.821537i	$0.693117\pi$
$230$	−18.3432	−1.20951
$231$	0	0
$232$	−26.2568	−1.72385
$233$	−17.2249	−1.12844	−0.564219	−	0.825625i	$-0.690823\pi$
$233$	−17.2249	−1.12844	−0.564219	+	0.825625i	$0.690823\pi$
$234$	0	0
$235$	31.5114	2.05557
$236$	−2.64095	−0.171911
$237$	0	0
$238$	0.858001	0.0556159
$239$	5.48952	0.355088	0.177544	−	0.984113i	$-0.443185\pi$
$239$	5.48952	0.355088	0.177544	+	0.984113i	$0.443185\pi$
$240$	0	0
$241$	−26.9366	−1.73514	−0.867571	−	0.497314i	$-0.834320\pi$
$241$	−26.9366	−1.73514	−0.867571	+	0.497314i	$0.834320\pi$
$242$	36.0218	2.31557
$243$	0	0
$244$	2.80570	0.179617
$245$	2.92411	0.186814
$246$	0	0
$247$	−1.70831	−0.108697
$248$	−7.16943	−0.455259
$249$	0	0
$250$	−5.64326	−0.356911
$251$	−0.302222	−0.0190761	−0.00953804	−	0.999955i	$-0.503036\pi$
$251$	−0.302222	−0.0190761	−0.00953804	+	0.999955i	$0.503036\pi$
$252$	0	0
$253$	29.0674	1.82745
$254$	−1.33134	−0.0835360
$255$	0	0
$256$	−5.38718	−0.336698
$257$	18.7785	1.17137	0.585685	−	0.810539i	$-0.300826\pi$
$257$	18.7785	1.17137	0.585685	+	0.810539i	$0.300826\pi$
$258$	0	0
$259$	6.07476	0.377467
$260$	0.412509	0.0255827
$261$	0	0
$262$	15.9939	0.988109
$263$	4.41926	0.272503	0.136251	−	0.990674i	$-0.456494\pi$
$263$	4.41926	0.272503	0.136251	+	0.990674i	$0.456494\pi$
$264$	0	0
$265$	−18.0483	−1.10870
$266$	−3.66808	−0.224905
$267$	0	0
$268$	−2.81867	−0.172178
$269$	2.13768	0.130337	0.0651683	−	0.997874i	$-0.479242\pi$
$269$	2.13768	0.130337	0.0651683	+	0.997874i	$0.479242\pi$
$270$	0	0
$271$	−3.17538	−0.192891	−0.0964454	−	0.995338i	$-0.530747\pi$
$271$	−3.17538	−0.192891	−0.0964454	+	0.995338i	$0.530747\pi$
$272$	2.25123	0.136501
$273$	0	0
$274$	16.8035	1.01513
$275$	−21.9025	−1.32077
$276$	0	0
$277$	20.5642	1.23559	0.617793	−	0.786341i	$-0.288027\pi$
$277$	20.5642	1.23559	0.617793	+	0.786341i	$0.288027\pi$
$278$	−22.8396	−1.36983
$279$	0	0
$280$	8.67173	0.518235
$281$	18.9073	1.12791	0.563956	−	0.825805i	$-0.309279\pi$
$281$	18.9073	1.12791	0.563956	+	0.825805i	$0.309279\pi$
$282$	0	0
$283$	−15.0480	−0.894513	−0.447256	−	0.894406i	$-0.647599\pi$
$283$	−15.0480	−0.894513	−0.447256	+	0.894406i	$0.647599\pi$
$284$	−2.28657	−0.135683
$285$	0	0
$286$	5.09241	0.301120
$287$	−2.70008	−0.159380
$288$	0	0
$289$	−16.5847	−0.975569
$290$	34.4678	2.02402
$291$	0	0
$292$	−1.86770	−0.109299
$293$	−12.4842	−0.729334	−0.364667	−	0.931138i	$-0.618817\pi$
$293$	−12.4842	−0.729334	−0.364667	+	0.931138i	$0.618817\pi$
$294$	0	0
$295$	33.9414	1.97615
$296$	18.0153	1.04712
$297$	0	0
$298$	−7.51903	−0.435566
$299$	2.92151	0.168955
$300$	0	0
$301$	−10.3043	−0.593932
$302$	17.3835	1.00031
$303$	0	0
$304$	−9.62434	−0.551994
$305$	−36.0588	−2.06472
$306$	0	0
$307$	27.4111	1.56443	0.782216	−	0.623007i	$-0.214089\pi$
$307$	27.4111	1.56443	0.782216	+	0.623007i	$0.214089\pi$
$308$	−1.40358	−0.0799766
$309$	0	0
$310$	9.41144	0.534534
$311$	25.1878	1.42827	0.714135	−	0.700008i	$-0.246820\pi$
$311$	25.1878	1.42827	0.714135	+	0.700008i	$0.246820\pi$
$312$	0	0
$313$	−11.7164	−0.662251	−0.331125	−	0.943587i	$-0.607428\pi$
$313$	−11.7164	−0.662251	−0.331125	+	0.943587i	$0.607428\pi$
$314$	−8.69790	−0.490851
$315$	0	0
$316$	−0.523824	−0.0294674
$317$	−18.0033	−1.01117	−0.505584	−	0.862777i	$-0.668723\pi$
$317$	−18.0033	−1.01117	−0.505584	+	0.862777i	$0.668723\pi$
$318$	0	0
$319$	−54.6192	−3.05809
$320$	25.4141	1.42069
$321$	0	0
$322$	6.27308	0.349585
$323$	−1.77560	−0.0987973
$324$	0	0
$325$	−2.20138	−0.122111
$326$	17.6533	0.977726
$327$	0	0
$328$	−8.00734	−0.442131
$329$	−10.7764	−0.594122
$330$	0	0
$331$	−14.0863	−0.774254	−0.387127	−	0.922026i	$-0.626533\pi$
$331$	−14.0863	−0.774254	−0.387127	+	0.922026i	$0.626533\pi$
$332$	−0.279390	−0.0153335
$333$	0	0
$334$	2.68670	0.147010
$335$	36.2255	1.97921
$336$	0	0
$337$	6.85642	0.373493	0.186747	−	0.982408i	$-0.440206\pi$
$337$	6.85642	0.373493	0.186747	+	0.982408i	$0.440206\pi$
$338$	−16.7957	−0.913563
$339$	0	0
$340$	0.428759	0.0232527
$341$	−14.9138	−0.807627
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−30.5585	−1.64760
$345$	0	0
$346$	−2.45085	−0.131759
$347$	−15.2537	−0.818860	−0.409430	−	0.912342i	$-0.634272\pi$
$347$	−15.2537	−0.818860	−0.409430	+	0.912342i	$0.634272\pi$
$348$	0	0
$349$	7.20325	0.385581	0.192791	−	0.981240i	$-0.438246\pi$
$349$	7.20325	0.385581	0.192791	+	0.981240i	$0.438246\pi$
$350$	−4.72681	−0.252659
$351$	0	0
$352$	−7.89977	−0.421059
$353$	4.26996	0.227267	0.113633	−	0.993523i	$-0.463751\pi$
$353$	4.26996	0.227267	0.113633	+	0.993523i	$0.463751\pi$
$354$	0	0
$355$	29.3869	1.55970
$356$	2.66874	0.141443
$357$	0	0
$358$	−24.9790	−1.32018
$359$	31.0817	1.64043	0.820214	−	0.572057i	$-0.193854\pi$
$359$	31.0817	1.64043	0.820214	+	0.572057i	$0.193854\pi$
$360$	0	0
$361$	−11.4090	−0.600475
$362$	30.8577	1.62185
$363$	0	0
$364$	−0.141072	−0.00739417
$365$	24.0036	1.25641
$366$	0	0
$367$	−18.3623	−0.958503	−0.479251	−	0.877678i	$-0.659092\pi$
$367$	−18.3623	−0.958503	−0.479251	+	0.877678i	$0.659092\pi$
$368$	16.4593	0.858002
$369$	0	0
$370$	−23.6490	−1.22945
$371$	6.17224	0.320447
$372$	0	0
$373$	−20.8539	−1.07977	−0.539886	−	0.841738i	$-0.681533\pi$
$373$	−20.8539	−1.07977	−0.539886	+	0.841738i	$0.681533\pi$
$374$	5.29302	0.273695
$375$	0	0
$376$	−31.9585	−1.64813
$377$	−5.48968	−0.282733
$378$	0	0
$379$	6.97802	0.358437	0.179218	−	0.983809i	$-0.442643\pi$
$379$	6.97802	0.358437	0.179218	+	0.983809i	$0.442643\pi$
$380$	−1.83301	−0.0940315
$381$	0	0
$382$	0.0896176	0.00458524
$383$	0.0360913	0.00184418	0.000922089	−	1.00000i	$-0.499706\pi$
$383$	0.0360913	0.00184418	0.000922089		1.00000i	$0.499706\pi$
$384$	0	0
$385$	18.0388	0.919345
$386$	−30.3284	−1.54367
$387$	0	0
$388$	−0.270831	−0.0137494
$389$	19.6007	0.993793	0.496897	−	0.867810i	$-0.334473\pi$
$389$	19.6007	0.993793	0.496897	+	0.867810i	$0.334473\pi$
$390$	0	0
$391$	3.03660	0.153567
$392$	−2.96560	−0.149785
$393$	0	0
$394$	27.4966	1.38526
$395$	6.73218	0.338733
$396$	0	0
$397$	20.6792	1.03786	0.518930	−	0.854817i	$-0.326331\pi$
$397$	20.6792	1.03786	0.518930	+	0.854817i	$0.326331\pi$
$398$	0.728135	0.0364981
$399$	0	0
$400$	−12.4022	−0.620112
$401$	24.7687	1.23689	0.618444	−	0.785829i	$-0.287763\pi$
$401$	24.7687	1.23689	0.618444	+	0.785829i	$0.287763\pi$
$402$	0	0
$403$	−1.49896	−0.0746684
$404$	−4.40472	−0.219143
$405$	0	0
$406$	−11.7875	−0.585002
$407$	37.4753	1.85758
$408$	0	0
$409$	−37.2721	−1.84299	−0.921494	−	0.388392i	$-0.873031\pi$
$409$	−37.2721	−1.84299	−0.921494	+	0.388392i	$0.873031\pi$
$410$	10.5114	0.519120
$411$	0	0
$412$	2.32077	0.114336
$413$	−11.6075	−0.571165
$414$	0	0
$415$	3.59071	0.176261
$416$	−0.793992	−0.0389287
$417$	0	0
$418$	−22.6284	−1.10679
$419$	25.0016	1.22141	0.610705	−	0.791858i	$-0.290886\pi$
$419$	25.0016	1.22141	0.610705	+	0.791858i	$0.290886\pi$
$420$	0	0
$421$	10.6102	0.517108	0.258554	−	0.965997i	$-0.416754\pi$
$421$	10.6102	0.517108	0.258554	+	0.965997i	$0.416754\pi$
$422$	−10.4765	−0.509989
$423$	0	0
$424$	18.3044	0.888940
$425$	−2.28810	−0.110989
$426$	0	0
$427$	12.3316	0.596766
$428$	−3.50102	−0.169228
$429$	0	0
$430$	40.1147	1.93450
$431$	33.4302	1.61028	0.805138	−	0.593088i	$-0.202091\pi$
$431$	33.4302	1.61028	0.805138	+	0.593088i	$0.202091\pi$
$432$	0	0
$433$	4.82890	0.232062	0.116031	−	0.993246i	$-0.462983\pi$
$433$	4.82890	0.232062	0.116031	+	0.993246i	$0.462983\pi$
$434$	−3.21857	−0.154496
$435$	0	0
$436$	0.547827	0.0262362
$437$	−12.9819	−0.621009
$438$	0	0
$439$	5.44232	0.259748	0.129874	−	0.991531i	$-0.458543\pi$
$439$	5.44232	0.259748	0.129874	+	0.991531i	$0.458543\pi$
$440$	53.4960	2.55032
$441$	0	0
$442$	0.531992	0.0253043
$443$	23.6814	1.12514	0.562568	−	0.826751i	$-0.309813\pi$
$443$	23.6814	1.12514	0.562568	+	0.826751i	$0.309813\pi$
$444$	0	0
$445$	−34.2986	−1.62591
$446$	0.853762	0.0404268
$447$	0	0
$448$	−8.69124	−0.410623
$449$	7.30490	0.344740	0.172370	−	0.985032i	$-0.444858\pi$
$449$	7.30490	0.344740	0.172370	+	0.985032i	$0.444858\pi$
$450$	0	0
$451$	−16.6568	−0.784338
$452$	−0.616803	−0.0290120
$453$	0	0
$454$	−33.8600	−1.58913
$455$	1.81305	0.0849972
$456$	0	0
$457$	4.66805	0.218362	0.109181	−	0.994022i	$-0.465177\pi$
$457$	4.66805	0.218362	0.109181	+	0.994022i	$0.465177\pi$
$458$	−22.9737	−1.07349
$459$	0	0
$460$	3.13478	0.146160
$461$	−26.9450	−1.25495	−0.627477	−	0.778635i	$-0.715912\pi$
$461$	−26.9450	−1.25495	−0.627477	+	0.778635i	$0.715912\pi$
$462$	0	0
$463$	9.98472	0.464029	0.232015	−	0.972712i	$-0.425468\pi$
$463$	9.98472	0.464029	0.232015	+	0.972712i	$0.425468\pi$
$464$	−30.9280	−1.43580
$465$	0	0
$466$	−22.9322	−1.06231
$467$	37.0776	1.71575	0.857873	−	0.513861i	$-0.171785\pi$
$467$	37.0776	1.71575	0.857873	+	0.513861i	$0.171785\pi$
$468$	0	0
$469$	−12.3886	−0.572052
$470$	41.9525	1.93512
$471$	0	0
$472$	−34.4231	−1.58445
$473$	−63.5675	−2.92284
$474$	0	0
$475$	9.78198	0.448828
$476$	−0.146629	−0.00672074
$477$	0	0
$478$	7.30845	0.334281
$479$	18.8165	0.859748	0.429874	−	0.902889i	$-0.358558\pi$
$479$	18.8165	0.859748	0.429874	+	0.902889i	$0.358558\pi$
$480$	0	0
$481$	3.76657	0.171741
$482$	−35.8619	−1.63347
$483$	0	0
$484$	−6.15598	−0.279817
$485$	3.48072	0.158051
$486$	0	0
$487$	−26.1973	−1.18711	−0.593556	−	0.804793i	$-0.702276\pi$
$487$	−26.1973	−1.18711	−0.593556	+	0.804793i	$0.702276\pi$
$488$	36.5705	1.65547
$489$	0	0
$490$	3.89299	0.175868
$491$	−18.9523	−0.855306	−0.427653	−	0.903943i	$-0.640660\pi$
$491$	−18.9523	−0.855306	−0.427653	+	0.903943i	$0.640660\pi$
$492$	0	0
$493$	−5.70594	−0.256983
$494$	−2.27434	−0.102328
$495$	0	0
$496$	−8.44490	−0.379187
$497$	−10.0499	−0.450799
$498$	0	0
$499$	23.2998	1.04304	0.521520	−	0.853239i	$-0.325365\pi$
$499$	23.2998	1.04304	0.521520	+	0.853239i	$0.325365\pi$
$500$	0.964412	0.0431298
$501$	0	0
$502$	−0.402361	−0.0179583
$503$	−6.11772	−0.272775	−0.136388	−	0.990656i	$-0.543549\pi$
$503$	−6.11772	−0.272775	−0.136388	+	0.990656i	$0.543549\pi$
$504$	0	0
$505$	56.6094	2.51908
$506$	38.6987	1.72037
$507$	0	0
$508$	0.227522	0.0100946
$509$	3.73713	0.165646	0.0828228	−	0.996564i	$-0.473606\pi$
$509$	3.73713	0.165646	0.0828228	+	0.996564i	$0.473606\pi$
$510$	0	0
$511$	−8.20886	−0.363139
$512$	−25.1920	−1.11334
$513$	0	0
$514$	25.0006	1.10273
$515$	−29.8265	−1.31431
$516$	0	0
$517$	−66.4797	−2.92378
$518$	8.08760	0.355349
$519$	0	0
$520$	5.37679	0.235788
$521$	21.0747	0.923299	0.461649	−	0.887062i	$-0.347258\pi$
$521$	21.0747	0.923299	0.461649	+	0.887062i	$0.347258\pi$
$522$	0	0
$523$	−7.26739	−0.317781	−0.158890	−	0.987296i	$-0.550792\pi$
$523$	−7.26739	−0.317781	−0.158890	+	0.987296i	$0.550792\pi$
$524$	−2.73330	−0.119405
$525$	0	0
$526$	5.88355	0.256535
$527$	−1.55801	−0.0678679
$528$	0	0
$529$	−0.798603	−0.0347219
$530$	−24.0285	−1.04373
$531$	0	0
$532$	0.626862	0.0271779
$533$	−1.67414	−0.0725152
$534$	0	0
$535$	44.9950	1.94531
$536$	−36.7396	−1.58691
$537$	0	0
$538$	2.84599	0.122699
$539$	−6.16901	−0.265718
$540$	0	0
$541$	9.77573	0.420291	0.210146	−	0.977670i	$-0.432606\pi$
$541$	9.77573	0.420291	0.210146	+	0.977670i	$0.432606\pi$
$542$	−4.22753	−0.181588
$543$	0	0
$544$	−0.825271	−0.0353832
$545$	−7.04067	−0.301589
$546$	0	0
$547$	19.6478	0.840078	0.420039	−	0.907506i	$-0.362016\pi$
$547$	19.6478	0.840078	0.420039	+	0.907506i	$0.362016\pi$
$548$	−2.87165	−0.122671
$549$	0	0
$550$	−29.1598	−1.24338
$551$	24.3938	1.03921
$552$	0	0
$553$	−2.30230	−0.0979038
$554$	27.3781	1.16318
$555$	0	0
$556$	3.90321	0.165533
$557$	−23.3748	−0.990423	−0.495211	−	0.868773i	$-0.664909\pi$
$557$	−23.3748	−0.990423	−0.495211	+	0.868773i	$0.664909\pi$
$558$	0	0
$559$	−6.38906	−0.270228
$560$	10.2145	0.431640
$561$	0	0
$562$	25.1721	1.06182
$563$	2.16432	0.0912152	0.0456076	−	0.998959i	$-0.485478\pi$
$563$	2.16432	0.0912152	0.0456076	+	0.998959i	$0.485478\pi$
$564$	0	0
$565$	7.92715	0.333498
$566$	−20.0341	−0.842097
$567$	0	0
$568$	−29.8039	−1.25054
$569$	−4.40419	−0.184633	−0.0923165	−	0.995730i	$-0.529427\pi$
$569$	−4.40419	−0.184633	−0.0923165	+	0.995730i	$0.529427\pi$
$570$	0	0
$571$	−2.17029	−0.0908239	−0.0454120	−	0.998968i	$-0.514460\pi$
$571$	−2.17029	−0.0908239	−0.0454120	+	0.998968i	$0.514460\pi$
$572$	−0.870273	−0.0363879
$573$	0	0
$574$	−3.59473	−0.150041
$575$	−16.7289	−0.697645
$576$	0	0
$577$	−18.3420	−0.763587	−0.381794	−	0.924248i	$-0.624694\pi$
$577$	−18.3420	−0.763587	−0.381794	+	0.924248i	$0.624694\pi$
$578$	−22.0799	−0.918403
$579$	0	0
$580$	−5.89042	−0.244586
$581$	−1.22797	−0.0509447
$582$	0	0
$583$	38.0766	1.57697
$584$	−24.3442	−1.00737
$585$	0	0
$586$	−16.6208	−0.686597
$587$	4.64632	0.191774	0.0958871	−	0.995392i	$-0.469431\pi$
$587$	4.64632	0.191774	0.0958871	+	0.995392i	$0.469431\pi$
$588$	0	0
$589$	6.66072	0.274450
$590$	45.1878	1.86035
$591$	0	0
$592$	21.2203	0.872149
$593$	−35.2665	−1.44822	−0.724110	−	0.689684i	$-0.757749\pi$
$593$	−35.2665	−1.44822	−0.724110	+	0.689684i	$0.757749\pi$
$594$	0	0
$595$	1.88448	0.0772560
$596$	1.28497	0.0526346
$597$	0	0
$598$	3.88953	0.159055
$599$	11.1046	0.453720	0.226860	−	0.973927i	$-0.427154\pi$
$599$	11.1046	0.453720	0.226860	+	0.973927i	$0.427154\pi$
$600$	0	0
$601$	27.6727	1.12879	0.564397	−	0.825504i	$-0.309109\pi$
$601$	27.6727	1.12879	0.564397	+	0.825504i	$0.309109\pi$
$602$	−13.7186	−0.559129
$603$	0	0
$604$	−2.97078	−0.120879
$605$	79.1166	3.21655
$606$	0	0
$607$	39.8271	1.61653	0.808266	−	0.588818i	$-0.200406\pi$
$607$	39.8271	1.61653	0.808266	+	0.588818i	$0.200406\pi$
$608$	3.52816	0.143086
$609$	0	0
$610$	−48.0067	−1.94374
$611$	−6.68176	−0.270315
$612$	0	0
$613$	−11.1909	−0.451997	−0.225999	−	0.974128i	$-0.572564\pi$
$613$	−11.1909	−0.451997	−0.225999	+	0.974128i	$0.572564\pi$
$614$	36.4936	1.47276
$615$	0	0
$616$	−18.2948	−0.737119
$617$	21.1289	0.850617	0.425308	−	0.905049i	$-0.360166\pi$
$617$	21.1289	0.850617	0.425308	+	0.905049i	$0.360166\pi$
$618$	0	0
$619$	6.30389	0.253375	0.126687	−	0.991943i	$-0.459566\pi$
$619$	6.30389	0.253375	0.126687	+	0.991943i	$0.459566\pi$
$620$	−1.60838	−0.0645941
$621$	0	0
$622$	33.5337	1.34458
$623$	11.7296	0.469936
$624$	0	0
$625$	−30.1466	−1.20587
$626$	−15.5986	−0.623445
$627$	0	0
$628$	1.48644	0.0593153
$629$	3.91496	0.156099
$630$	0	0
$631$	36.4561	1.45130	0.725648	−	0.688066i	$-0.241540\pi$
$631$	36.4561	1.45130	0.725648	+	0.688066i	$0.241540\pi$
$632$	−6.82770	−0.271591
$633$	0	0
$634$	−23.9687	−0.951917
$635$	−2.92411	−0.116040
$636$	0	0
$637$	−0.620036	−0.0245667
$638$	−72.7170	−2.87889
$639$	0	0
$640$	26.3460	1.04142
$641$	−23.4463	−0.926073	−0.463037	−	0.886339i	$-0.653240\pi$
$641$	−23.4463	−0.926073	−0.463037	+	0.886339i	$0.653240\pi$
$642$	0	0
$643$	−12.6228	−0.497796	−0.248898	−	0.968530i	$-0.580068\pi$
$643$	−12.6228	−0.497796	−0.248898	+	0.968530i	$0.580068\pi$
$644$	−1.07205	−0.0422445
$645$	0	0
$646$	−2.36394	−0.0930080
$647$	−25.2402	−0.992294	−0.496147	−	0.868238i	$-0.665252\pi$
$647$	−25.2402	−0.992294	−0.496147	+	0.868238i	$0.665252\pi$
$648$	0	0
$649$	−71.6065	−2.81080
$650$	−2.93080	−0.114955
$651$	0	0
$652$	−3.01688	−0.118150
$653$	3.66022	0.143236	0.0716178	−	0.997432i	$-0.477184\pi$
$653$	3.66022	0.143236	0.0716178	+	0.997432i	$0.477184\pi$
$654$	0	0
$655$	35.1284	1.37258
$656$	−9.43188	−0.368253
$657$	0	0
$658$	−14.3471	−0.559308
$659$	−9.14308	−0.356164	−0.178082	−	0.984016i	$-0.556989\pi$
$659$	−9.14308	−0.356164	−0.178082	+	0.984016i	$0.556989\pi$
$660$	0	0
$661$	16.9706	0.660078	0.330039	−	0.943967i	$-0.392938\pi$
$661$	16.9706	0.660078	0.330039	+	0.943967i	$0.392938\pi$
$662$	−18.7537	−0.728885
$663$	0	0
$664$	−3.64166	−0.141324
$665$	−8.05642	−0.312415
$666$	0	0
$667$	−41.7177	−1.61532
$668$	−0.459147	−0.0177649
$669$	0	0
$670$	48.2287	1.86324
$671$	76.0736	2.93679
$672$	0	0
$673$	2.01062	0.0775038	0.0387519	−	0.999249i	$-0.487662\pi$
$673$	2.01062	0.0775038	0.0387519	+	0.999249i	$0.487662\pi$
$674$	9.12826	0.351607
$675$	0	0
$676$	2.87031	0.110397
$677$	−7.34704	−0.282370	−0.141185	−	0.989983i	$-0.545091\pi$
$677$	−7.34704	−0.282370	−0.141185	+	0.989983i	$0.545091\pi$
$678$	0	0
$679$	−1.19035	−0.0456815
$680$	5.58860	0.214313
$681$	0	0
$682$	−19.8554	−0.760302
$683$	30.7252	1.17567	0.587833	−	0.808982i	$-0.299981\pi$
$683$	30.7252	1.17567	0.587833	+	0.808982i	$0.299981\pi$
$684$	0	0
$685$	36.9064	1.41012
$686$	−1.33134	−0.0508310
$687$	0	0
$688$	−35.9950	−1.37230
$689$	3.82701	0.145798
$690$	0	0
$691$	41.4079	1.57523	0.787616	−	0.616166i	$-0.211315\pi$
$691$	41.4079	1.57523	0.787616	+	0.616166i	$0.211315\pi$
$692$	0.418841	0.0159220
$693$	0	0
$694$	−20.3079	−0.770877
$695$	−50.1640	−1.90283
$696$	0	0
$697$	−1.74010	−0.0659108
$698$	9.59001	0.362987
$699$	0	0
$700$	0.807795	0.0305318
$701$	−5.45833	−0.206158	−0.103079	−	0.994673i	$-0.532869\pi$
$701$	−5.45833	−0.206158	−0.103079	+	0.994673i	$0.532869\pi$
$702$	0	0
$703$	−16.7370	−0.631249
$704$	−53.6164	−2.02074
$705$	0	0
$706$	5.68479	0.213950
$707$	−19.3595	−0.728090
$708$	0	0
$709$	4.69396	0.176285	0.0881427	−	0.996108i	$-0.471907\pi$
$709$	4.69396	0.176285	0.0881427	+	0.996108i	$0.471907\pi$
$710$	39.1241	1.46830
$711$	0	0
$712$	34.7853	1.30363
$713$	−11.3910	−0.426597
$714$	0	0
$715$	11.1847	0.418286
$716$	4.26882	0.159533
$717$	0	0
$718$	41.3804	1.54430
$719$	32.7760	1.22234	0.611170	−	0.791499i	$-0.290699\pi$
$719$	32.7760	1.22234	0.611170	+	0.791499i	$0.290699\pi$
$720$	0	0
$721$	10.2002	0.379875
$722$	−15.1893	−0.565289
$723$	0	0
$724$	−5.27347	−0.195987
$725$	31.4346	1.16745
$726$	0	0
$727$	−16.3785	−0.607445	−0.303723	−	0.952760i	$-0.598230\pi$
$727$	−16.3785	−0.607445	−0.303723	+	0.952760i	$0.598230\pi$
$728$	−1.83878	−0.0681497
$729$	0	0
$730$	31.9571	1.18278
$731$	−6.64075	−0.245617
$732$	0	0
$733$	10.7267	0.396199	0.198100	−	0.980182i	$-0.436523\pi$
$733$	10.7267	0.396199	0.198100	+	0.980182i	$0.436523\pi$
$734$	−24.4465	−0.902337
$735$	0	0
$736$	−6.03378	−0.222408
$737$	−76.4253	−2.81516
$738$	0	0
$739$	11.9344	0.439014	0.219507	−	0.975611i	$-0.429555\pi$
$739$	11.9344	0.439014	0.219507	+	0.975611i	$0.429555\pi$
$740$	4.04153	0.148570
$741$	0	0
$742$	8.21738	0.301669
$743$	−3.45152	−0.126624	−0.0633119	−	0.997994i	$-0.520166\pi$
$743$	−3.45152	−0.126624	−0.0633119	+	0.997994i	$0.520166\pi$
$744$	0	0
$745$	−16.5145	−0.605043
$746$	−27.7637	−1.01650
$747$	0	0
$748$	−0.904557	−0.0330739
$749$	−15.3876	−0.562251
$750$	0	0
$751$	−0.508332	−0.0185493	−0.00927465	−	0.999957i	$-0.502952\pi$
$751$	−0.508332	−0.0185493	−0.00927465	+	0.999957i	$0.502952\pi$
$752$	−37.6440	−1.37274
$753$	0	0
$754$	−7.30866	−0.266166
$755$	38.1804	1.38953
$756$	0	0
$757$	11.2937	0.410477	0.205238	−	0.978712i	$-0.434203\pi$
$757$	11.2937	0.410477	0.205238	+	0.978712i	$0.434203\pi$
$758$	9.29014	0.337433
$759$	0	0
$760$	−23.8921	−0.866658
$761$	−27.6722	−1.00312	−0.501558	−	0.865124i	$-0.667240\pi$
$761$	−27.6722	−1.00312	−0.501558	+	0.865124i	$0.667240\pi$
$762$	0	0
$763$	2.40780	0.0871683
$764$	−0.0153153	−0.000554089 0
$765$	0	0
$766$	0.0480499	0.00173611
$767$	−7.19704	−0.259870
$768$	0	0
$769$	29.1720	1.05197	0.525984	−	0.850495i	$-0.323697\pi$
$769$	29.1720	1.05197	0.525984	+	0.850495i	$0.323697\pi$
$770$	24.0159	0.865474
$771$	0	0
$772$	5.18300	0.186540
$773$	19.2965	0.694046	0.347023	−	0.937857i	$-0.387193\pi$
$773$	19.2965	0.694046	0.347023	+	0.937857i	$0.387193\pi$
$774$	0	0
$775$	8.58322	0.308318
$776$	−3.53010	−0.126723
$777$	0	0
$778$	26.0952	0.935560
$779$	7.43917	0.266536
$780$	0	0
$781$	−61.9978	−2.21846
$782$	4.04276	0.144569
$783$	0	0
$784$	−3.49319	−0.124757
$785$	−19.1037	−0.681840
$786$	0	0
$787$	−18.6400	−0.664446	−0.332223	−	0.943201i	$-0.607799\pi$
$787$	−18.6400	−0.664446	−0.332223	+	0.943201i	$0.607799\pi$
$788$	−4.69906	−0.167397
$789$	0	0
$790$	8.96285	0.318884
$791$	−2.71096	−0.0963908
$792$	0	0
$793$	7.64602	0.271518
$794$	27.5311	0.977044
$795$	0	0
$796$	−0.124436	−0.00441050
$797$	12.4612	0.441399	0.220699	−	0.975342i	$-0.429166\pi$
$797$	12.4612	0.441399	0.220699	+	0.975342i	$0.429166\pi$
$798$	0	0
$799$	−6.94498	−0.245696
$800$	4.54650	0.160743
$801$	0	0
$802$	32.9756	1.16441
$803$	−50.6405	−1.78707
$804$	0	0
$805$	13.7779	0.485608
$806$	−1.99563	−0.0702930
$807$	0	0
$808$	−57.4126	−2.01977
$809$	27.8109	0.977779	0.488890	−	0.872346i	$-0.337402\pi$
$809$	27.8109	0.977779	0.488890	+	0.872346i	$0.337402\pi$
$810$	0	0
$811$	−27.0417	−0.949561	−0.474781	−	0.880104i	$-0.657473\pi$
$811$	−27.0417	−0.949561	−0.474781	+	0.880104i	$0.657473\pi$
$812$	2.01443	0.0706928
$813$	0	0
$814$	49.8925	1.74873
$815$	38.7730	1.35816
$816$	0	0
$817$	28.3902	0.993247
$818$	−49.6221	−1.73499
$819$	0	0
$820$	−1.79636	−0.0627314
$821$	0.578331	0.0201839	0.0100920	−	0.999949i	$-0.496788\pi$
$821$	0.578331	0.0201839	0.0100920	+	0.999949i	$0.496788\pi$
$822$	0	0
$823$	9.69777	0.338043	0.169022	−	0.985612i	$-0.445939\pi$
$823$	9.69777	0.338043	0.169022	+	0.985612i	$0.445939\pi$
$824$	30.2497	1.05380
$825$	0	0
$826$	−15.4535	−0.537697
$827$	−27.9922	−0.973384	−0.486692	−	0.873574i	$-0.661797\pi$
$827$	−27.9922	−0.973384	−0.486692	+	0.873574i	$0.661797\pi$
$828$	0	0
$829$	26.5981	0.923791	0.461895	−	0.886934i	$-0.347170\pi$
$829$	26.5981	0.923791	0.461895	+	0.886934i	$0.347170\pi$
$830$	4.78048	0.165933
$831$	0	0
$832$	−5.38889	−0.186826
$833$	−0.644462	−0.0223293
$834$	0	0
$835$	5.90095	0.204211
$836$	3.86712	0.133747
$837$	0	0
$838$	33.2858	1.14984
$839$	−4.90573	−0.169365	−0.0846823	−	0.996408i	$-0.526988\pi$
$839$	−4.90573	−0.169365	−0.0846823	+	0.996408i	$0.526988\pi$
$840$	0	0
$841$	49.3899	1.70310
$842$	14.1258	0.486807
$843$	0	0
$844$	1.79040	0.0616280
$845$	−36.8892	−1.26903
$846$	0	0
$847$	−27.0567	−0.929678
$848$	21.5608	0.740401
$849$	0	0
$850$	−3.04625	−0.104486
$851$	28.6233	0.981194
$852$	0	0
$853$	44.0800	1.50927	0.754635	−	0.656145i	$-0.227814\pi$
$853$	44.0800	1.50927	0.754635	+	0.656145i	$0.227814\pi$
$854$	16.4176	0.561798
$855$	0	0
$856$	−45.6335	−1.55972
$857$	−10.7388	−0.366829	−0.183415	−	0.983036i	$-0.558715\pi$
$857$	−10.7388	−0.366829	−0.183415	+	0.983036i	$0.558715\pi$
$858$	0	0
$859$	51.5763	1.75976	0.879880	−	0.475197i	$-0.157623\pi$
$859$	51.5763	1.75976	0.879880	+	0.475197i	$0.157623\pi$
$860$	−6.85545	−0.233769
$861$	0	0
$862$	44.5071	1.51592
$863$	33.0290	1.12432	0.562160	−	0.827029i	$-0.309971\pi$
$863$	33.0290	1.12432	0.562160	+	0.827029i	$0.309971\pi$
$864$	0	0
$865$	−5.38294	−0.183026
$866$	6.42894	0.218464
$867$	0	0
$868$	0.550041	0.0186696
$869$	−14.2029	−0.481801
$870$	0	0
$871$	−7.68137	−0.260273
$872$	7.14058	0.241810
$873$	0	0
$874$	−17.2834	−0.584620
$875$	4.23877	0.143297
$876$	0	0
$877$	30.4104	1.02689	0.513443	−	0.858124i	$-0.328370\pi$
$877$	30.4104	1.02689	0.513443	+	0.858124i	$0.328370\pi$
$878$	7.24560	0.244527
$879$	0	0
$880$	63.0131	2.12417
$881$	2.70216	0.0910381	0.0455190	−	0.998963i	$-0.485506\pi$
$881$	2.70216	0.0910381	0.0455190	+	0.998963i	$0.485506\pi$
$882$	0	0
$883$	57.0235	1.91899	0.959497	−	0.281719i	$-0.0909045\pi$
$883$	57.0235	1.91899	0.959497	+	0.281719i	$0.0909045\pi$
$884$	−0.0909154	−0.00305782
$885$	0	0
$886$	31.5281	1.05921
$887$	−19.5634	−0.656876	−0.328438	−	0.944525i	$-0.606522\pi$
$887$	−19.5634	−0.656876	−0.328438	+	0.944525i	$0.606522\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−45.6633	−1.53064
$891$	0	0
$892$	−0.145905	−0.00488525
$893$	29.6909	0.993566
$894$	0	0
$895$	−54.8628	−1.83386
$896$	−9.00992	−0.301000
$897$	0	0
$898$	9.72534	0.324539
$899$	21.4044	0.713876
$900$	0	0
$901$	3.97777	0.132519
$902$	−22.1759	−0.738378
$903$	0	0
$904$	−8.03963	−0.267394
$905$	67.7746	2.25290
$906$	0	0
$907$	−38.9478	−1.29324	−0.646620	−	0.762812i	$-0.723818\pi$
$907$	−38.9478	−1.29324	−0.646620	+	0.762812i	$0.723818\pi$
$908$	5.78654	0.192033
$909$	0	0
$910$	2.41380	0.0800166
$911$	−49.6820	−1.64604	−0.823020	−	0.568013i	$-0.807712\pi$
$911$	−49.6820	−1.64604	−0.823020	+	0.568013i	$0.807712\pi$
$912$	0	0
$913$	−7.57535	−0.250708
$914$	6.21479	0.205567
$915$	0	0
$916$	3.92613	0.129723
$917$	−12.0134	−0.396716
$918$	0	0
$919$	−17.4080	−0.574238	−0.287119	−	0.957895i	$-0.592698\pi$
$919$	−17.4080	−0.574238	−0.287119	+	0.957895i	$0.592698\pi$
$920$	40.8598	1.34711
$921$	0	0
$922$	−35.8731	−1.18142
$923$	−6.23129	−0.205105
$924$	0	0
$925$	−21.5679	−0.709148
$926$	13.2931	0.436838
$927$	0	0
$928$	11.3378	0.372182
$929$	13.5172	0.443486	0.221743	−	0.975105i	$-0.428825\pi$
$929$	13.5172	0.443486	0.221743	+	0.975105i	$0.428825\pi$
$930$	0	0
$931$	2.75517	0.0902971
$932$	3.91903	0.128372
$933$	0	0
$934$	49.3631	1.61521
$935$	11.6254	0.380190
$936$	0	0
$937$	19.8645	0.648944	0.324472	−	0.945895i	$-0.394813\pi$
$937$	19.8645	0.648944	0.324472	+	0.945895i	$0.394813\pi$
$938$	−16.4935	−0.538531
$939$	0	0
$940$	−7.16952	−0.233844
$941$	−2.52477	−0.0823051	−0.0411525	−	0.999153i	$-0.513103\pi$
$941$	−2.52477	−0.0823051	−0.0411525	+	0.999153i	$0.513103\pi$
$942$	0	0
$943$	−12.7223	−0.414295
$944$	−40.5470	−1.31969
$945$	0	0
$946$	−84.6302	−2.75157
$947$	13.3429	0.433586	0.216793	−	0.976218i	$-0.430440\pi$
$947$	13.3429	0.433586	0.216793	+	0.976218i	$0.430440\pi$
$948$	0	0
$949$	−5.08979	−0.165222
$950$	13.0232	0.422528
$951$	0	0
$952$	−1.91122	−0.0619429
$953$	19.1650	0.620816	0.310408	−	0.950603i	$-0.399534\pi$
$953$	19.1650	0.620816	0.310408	+	0.950603i	$0.399534\pi$
$954$	0	0
$955$	0.196832	0.00636934
$956$	−1.24899	−0.0403951
$957$	0	0
$958$	25.0513	0.809369
$959$	−12.6214	−0.407567
$960$	0	0
$961$	−25.1555	−0.811469
$962$	5.01461	0.161677
$963$	0	0
$964$	6.12867	0.197391
$965$	−66.6119	−2.14431
$966$	0	0
$967$	−41.8133	−1.34463	−0.672313	−	0.740267i	$-0.734699\pi$
$967$	−41.8133	−1.34463	−0.672313	+	0.740267i	$0.734699\pi$
$968$	−80.2392	−2.57899
$969$	0	0
$970$	4.63403	0.148790
$971$	−4.23069	−0.135769	−0.0678846	−	0.997693i	$-0.521625\pi$
$971$	−4.23069	−0.135769	−0.0678846	+	0.997693i	$0.521625\pi$
$972$	0	0
$973$	17.1553	0.549974
$974$	−34.8776	−1.11755
$975$	0	0
$976$	43.0765	1.37885
$977$	2.61078	0.0835263	0.0417631	−	0.999128i	$-0.486703\pi$
$977$	2.61078	0.0835263	0.0417631	+	0.999128i	$0.486703\pi$
$978$	0	0
$979$	72.3600	2.31264
$980$	−0.665298	−0.0212522
$981$	0	0
$982$	−25.2321	−0.805188
$983$	−41.6466	−1.32832	−0.664159	−	0.747591i	$-0.731210\pi$
$983$	−41.6466	−1.32832	−0.664159	+	0.747591i	$0.731210\pi$
$984$	0	0
$985$	60.3923	1.92426
$986$	−7.59658	−0.241924
$987$	0	0
$988$	0.388677	0.0123655
$989$	−48.5523	−1.54387
$990$	0	0
$991$	0.195803	0.00621990	0.00310995	−	0.999995i	$-0.499010\pi$
$991$	0.195803	0.00621990	0.00310995	+	0.999995i	$0.499010\pi$
$992$	3.09579	0.0982915
$993$	0	0
$994$	−13.3799	−0.424383
$995$	1.59925	0.0506995
$996$	0	0
$997$	−0.840968	−0.0266337	−0.0133169	−	0.999911i	$-0.504239\pi$
$997$	−0.840968	−0.0266337	−0.0133169	+	0.999911i	$0.504239\pi$
$998$	31.0200	0.981921
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.t.1.12		16
3.2	odd	2		889.2.a.c.1.5	✓	16
21.20	even	2		6223.2.a.k.1.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.c.1.5	✓	16	3.2	odd	2
6223.2.a.k.1.5		16	21.20	even	2
8001.2.a.t.1.12		16	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+1.33134 q^{2} -0.227522 q^{4} +2.92411 q^{5} -1.00000 q^{7} -2.96560 q^{8} +3.89299 q^{10} -6.16901 q^{11} -0.620036 q^{13} -1.33134 q^{14} -3.49319 q^{16} -0.644462 q^{17} +2.75517 q^{19} -0.665298 q^{20} -8.21308 q^{22} -4.71184 q^{23} +3.55041 q^{25} -0.825482 q^{26} +0.227522 q^{28} +8.85381 q^{29} +2.41753 q^{31} +1.28056 q^{32} -0.858001 q^{34} -2.92411 q^{35} -6.07476 q^{37} +3.66808 q^{38} -8.67173 q^{40} +2.70008 q^{41} +10.3043 q^{43} +1.40358 q^{44} -6.27308 q^{46} +10.7764 q^{47} +1.00000 q^{49} +4.72681 q^{50} +0.141072 q^{52} -6.17224 q^{53} -18.0388 q^{55} +2.96560 q^{56} +11.7875 q^{58} +11.6075 q^{59} -12.3316 q^{61} +3.21857 q^{62} +8.69124 q^{64} -1.81305 q^{65} +12.3886 q^{67} +0.146629 q^{68} -3.89299 q^{70} +10.0499 q^{71} +8.20886 q^{73} -8.08760 q^{74} -0.626862 q^{76} +6.16901 q^{77} +2.30230 q^{79} -10.2145 q^{80} +3.59473 q^{82} +1.22797 q^{83} -1.88448 q^{85} +13.7186 q^{86} +18.2948 q^{88} -11.7296 q^{89} +0.620036 q^{91} +1.07205 q^{92} +14.3471 q^{94} +8.05642 q^{95} +1.19035 q^{97} +1.33134 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8} - 2 q^{10} + 22 q^{11} - 4 q^{13} - 2 q^{14} + 12 q^{16} + 18 q^{17} - 15 q^{19} + 40 q^{20} - 11 q^{22} + 5 q^{23} + 15 q^{25} + 24 q^{26} - 12 q^{28}+ \cdots + 2 q^{98}+O(q^{100}) \) 16 * q + 2 * q^2 + 12 * q^4 + 9 * q^5 - 16 * q^7 + 6 * q^8 - 2 * q^10 + 22 * q^11 - 4 * q^13 - 2 * q^14 + 12 * q^16 + 18 * q^17 - 15 * q^19 + 40 * q^20 - 11 * q^22 + 5 * q^23 + 15 * q^25 + 24 * q^26 - 12 * q^28 + 12 * q^29 - 32 * q^31 + 9 * q^32 - 14 * q^34 - 9 * q^35 - 2 * q^37 - 3 * q^38 - 14 * q^40 + 45 * q^41 - 3 * q^43 + 54 * q^44 + 49 * q^47 + 16 * q^49 + 6 * q^50 + 38 * q^52 - 16 * q^53 + 7 * q^55 - 6 * q^56 + 16 * q^58 + 35 * q^59 - 11 * q^61 - 17 * q^62 - 2 * q^64 - 14 * q^65 + 17 * q^67 + 71 * q^68 + 2 * q^70 + 81 * q^71 - 15 * q^73 - 13 * q^74 + 14 * q^76 - 22 * q^77 - 34 * q^79 + 33 * q^80 - 14 * q^82 + 39 * q^83 - 17 * q^85 - 36 * q^86 + 61 * q^88 + 32 * q^89 + 4 * q^91 - 37 * q^92 + 13 * q^94 + 33 * q^95 - 4 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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