LMFDB - Embedded newform 2299.2.a.r.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2299,2,Mod(1,2299)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2299.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2299, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2299 = 11^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2299.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,2,-1,6,-1,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

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

Self dual:	yes
Analytic conductor:	$18.3576074247$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 8x^{5} + 14x^{4} + 17x^{3} - 18x^{2} - 16x - 1 $ x^7 - 2x^6 - 8x^5 + 14x^4 + 17x^3 - 18x^2 - 16x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.79190$ of defining polynomial
Character	$\chi$	$=$	2299.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.79190 q^{2} -2.48473 q^{3} +1.21090 q^{4} +0.260416 q^{5} -4.45238 q^{6} +3.76880 q^{7} -1.41399 q^{8} +3.17387 q^{9} +0.466638 q^{10} -3.00874 q^{12} +2.05680 q^{13} +6.75331 q^{14} -0.647062 q^{15} -4.95552 q^{16} +0.278836 q^{17} +5.68725 q^{18} -1.00000 q^{19} +0.315336 q^{20} -9.36445 q^{21} +6.11616 q^{23} +3.51339 q^{24} -4.93218 q^{25} +3.68558 q^{26} -0.432022 q^{27} +4.56363 q^{28} +5.99500 q^{29} -1.15947 q^{30} -6.51669 q^{31} -6.05180 q^{32} +0.499645 q^{34} +0.981456 q^{35} +3.84322 q^{36} +8.75610 q^{37} -1.79190 q^{38} -5.11059 q^{39} -0.368226 q^{40} -3.53945 q^{41} -16.7801 q^{42} -4.65322 q^{43} +0.826526 q^{45} +10.9595 q^{46} +8.70515 q^{47} +12.3131 q^{48} +7.20389 q^{49} -8.83797 q^{50} -0.692830 q^{51} +2.49057 q^{52} -4.68367 q^{53} -0.774138 q^{54} -5.32907 q^{56} +2.48473 q^{57} +10.7424 q^{58} +5.29748 q^{59} -0.783524 q^{60} +11.6024 q^{61} -11.6772 q^{62} +11.9617 q^{63} -0.933152 q^{64} +0.535623 q^{65} +13.0562 q^{67} +0.337641 q^{68} -15.1970 q^{69} +1.75867 q^{70} +7.50689 q^{71} -4.48784 q^{72} +0.611683 q^{73} +15.6900 q^{74} +12.2551 q^{75} -1.21090 q^{76} -9.15765 q^{78} +4.35513 q^{79} -1.29050 q^{80} -8.44816 q^{81} -6.34233 q^{82} -9.53801 q^{83} -11.3394 q^{84} +0.0726132 q^{85} -8.33810 q^{86} -14.8959 q^{87} +7.47422 q^{89} +1.48105 q^{90} +7.75168 q^{91} +7.40603 q^{92} +16.1922 q^{93} +15.5987 q^{94} -0.260416 q^{95} +15.0371 q^{96} +13.4845 q^{97} +12.9086 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 2 q^{2} - q^{3} + 6 q^{4} - q^{5} + 2 q^{7} + 6 q^{8} + 4 q^{9} - 3 q^{10} - 9 q^{12} + 12 q^{13} + 3 q^{14} + 5 q^{15} + 12 q^{17} - 3 q^{18} - 7 q^{19} - 17 q^{20} + 9 q^{21} - 9 q^{23} + 27 q^{24}+ \cdots + 9 q^{98}+O(q^{100}) $ 7 * q + 2 * q^2 - q^3 + 6 * q^4 - q^5 + 2 * q^7 + 6 * q^8 + 4 * q^9 - 3 * q^10 - 9 * q^12 + 12 * q^13 + 3 * q^14 + 5 * q^15 + 12 * q^17 - 3 * q^18 - 7 * q^19 - 17 * q^20 + 9 * q^21 - 9 * q^23 + 27 * q^24 + 4 * q^25 + 18 * q^26 + 23 * q^27 - 10 * q^28 + 18 * q^29 + 4 * q^30 - 5 * q^31 - 16 * q^32 - 6 * q^34 + 20 * q^35 - 2 * q^36 + 4 * q^37 - 2 * q^38 + 21 * q^39 + 17 * q^40 + 25 * q^41 - 19 * q^42 - 10 * q^43 - 16 * q^45 + 29 * q^46 - 14 * q^47 + 30 * q^48 + 25 * q^49 + 13 * q^50 + q^51 - 3 * q^52 - 15 * q^53 - 28 * q^54 - 3 * q^56 + q^57 + 6 * q^58 - 9 * q^59 + 29 * q^60 + 3 * q^61 + 41 * q^62 + 52 * q^63 - 20 * q^64 - 11 * q^65 - 15 * q^67 + 9 * q^68 - 11 * q^69 + 42 * q^70 + 13 * q^71 + 2 * q^72 - 28 * q^73 + 22 * q^74 - 22 * q^75 - 6 * q^76 - 38 * q^78 + 18 * q^79 + 32 * q^80 + 23 * q^81 + 8 * q^82 - 10 * q^83 - 19 * q^84 + 14 * q^85 - 30 * q^86 - 22 * q^87 + 2 * q^89 + 30 * q^90 - 19 * q^91 + 4 * q^92 + 17 * q^93 + 9 * q^94 + q^95 - q^96 + 29 * q^97 + 9 * 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 $

$2$	1.79190	1.26706	0.633531	−	0.773717i	$-0.281605\pi$
$2$	1.79190	1.26706	0.633531	+	0.773717i	$0.281605\pi$
$3$	−2.48473	−1.43456	−0.717279	−	0.696786i	$-0.754613\pi$
$3$	−2.48473	−1.43456	−0.717279	+	0.696786i	$0.754613\pi$
$4$	1.21090	0.605448
$5$	0.260416	0.116461	0.0582307	−	0.998303i	$-0.481454\pi$
$5$	0.260416	0.116461	0.0582307	+	0.998303i	$0.481454\pi$
$6$	−4.45238	−1.81767
$7$	3.76880	1.42447	0.712237	−	0.701939i	$-0.247682\pi$
$7$	3.76880	1.42447	0.712237	+	0.701939i	$0.247682\pi$
$8$	−1.41399	−0.499923
$9$	3.17387	1.05796
$10$	0.466638	0.147564
$11$	0	0
$11$	0	0
$12$	−3.00874	−0.868550
$13$	2.05680	0.570454	0.285227	−	0.958460i	$-0.407931\pi$
$13$	2.05680	0.570454	0.285227	+	0.958460i	$0.407931\pi$
$14$	6.75331	1.80490
$15$	−0.647062	−0.167071
$16$	−4.95552	−1.23888
$17$	0.278836	0.0676276	0.0338138	−	0.999428i	$-0.489235\pi$
$17$	0.278836	0.0676276	0.0338138	+	0.999428i	$0.489235\pi$
$18$	5.68725	1.34050
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0.315336	0.0705113
$21$	−9.36445	−2.04349
$22$	0	0
$23$	6.11616	1.27531	0.637654	−	0.770323i	$-0.279905\pi$
$23$	6.11616	1.27531	0.637654	+	0.770323i	$0.279905\pi$
$24$	3.51339	0.717168
$25$	−4.93218	−0.986437
$26$	3.68558	0.722801
$27$	−0.432022	−0.0831426
$28$	4.56363	0.862444
$29$	5.99500	1.11324	0.556621	−	0.830766i	$-0.312097\pi$
$29$	5.99500	1.11324	0.556621	+	0.830766i	$0.312097\pi$
$30$	−1.15947	−0.211689
$31$	−6.51669	−1.17043	−0.585216	−	0.810878i	$-0.698990\pi$
$31$	−6.51669	−1.17043	−0.585216	+	0.810878i	$0.698990\pi$
$32$	−6.05180	−1.06982
$33$	0	0
$34$	0.499645	0.0856884
$35$	0.981456	0.165896
$36$	3.84322	0.640537
$37$	8.75610	1.43949	0.719747	−	0.694236i	$-0.244258\pi$
$37$	8.75610	1.43949	0.719747	+	0.694236i	$0.244258\pi$
$38$	−1.79190	−0.290684
$39$	−5.11059	−0.818349
$40$	−0.368226	−0.0582217
$41$	−3.53945	−0.552769	−0.276385	−	0.961047i	$-0.589136\pi$
$41$	−3.53945	−0.552769	−0.276385	+	0.961047i	$0.589136\pi$
$42$	−16.7801	−2.58923
$43$	−4.65322	−0.709610	−0.354805	−	0.934940i	$-0.615453\pi$
$43$	−4.65322	−0.709610	−0.354805	+	0.934940i	$0.615453\pi$
$44$	0	0
$45$	0.826526	0.123211
$46$	10.9595	1.61589
$47$	8.70515	1.26978	0.634888	−	0.772604i	$-0.281046\pi$
$47$	8.70515	1.26978	0.634888	+	0.772604i	$0.281046\pi$
$48$	12.3131	1.77725
$49$	7.20389	1.02913
$50$	−8.83797	−1.24988
$51$	−0.692830	−0.0970157
$52$	2.49057	0.345380
$53$	−4.68367	−0.643352	−0.321676	−	0.946850i	$-0.604246\pi$
$53$	−4.68367	−0.643352	−0.321676	+	0.946850i	$0.604246\pi$
$54$	−0.774138	−0.105347
$55$	0	0
$56$	−5.32907	−0.712127
$57$	2.48473	0.329110
$58$	10.7424	1.41055
$59$	5.29748	0.689673	0.344837	−	0.938663i	$-0.387934\pi$
$59$	5.29748	0.689673	0.344837	+	0.938663i	$0.387934\pi$
$60$	−0.783524	−0.101153
$61$	11.6024	1.48553	0.742765	−	0.669553i	$-0.233514\pi$
$61$	11.6024	1.48553	0.742765	+	0.669553i	$0.233514\pi$
$62$	−11.6772	−1.48301
$63$	11.9617	1.50703
$64$	−0.933152	−0.116644
$65$	0.535623	0.0664359
$66$	0	0
$67$	13.0562	1.59507	0.797534	−	0.603274i	$-0.206138\pi$
$67$	13.0562	1.59507	0.797534	+	0.603274i	$0.206138\pi$
$68$	0.337641	0.0409449
$69$	−15.1970	−1.82950
$70$	1.75867	0.210201
$71$	7.50689	0.890904	0.445452	−	0.895306i	$-0.353043\pi$
$71$	7.50689	0.890904	0.445452	+	0.895306i	$0.353043\pi$
$72$	−4.48784	−0.528897
$73$	0.611683	0.0715921	0.0357961	−	0.999359i	$-0.488603\pi$
$73$	0.611683	0.0715921	0.0357961	+	0.999359i	$0.488603\pi$
$74$	15.6900	1.82393
$75$	12.2551	1.41510
$76$	−1.21090	−0.138899
$77$	0	0
$78$	−9.15765	−1.03690
$79$	4.35513	0.489990	0.244995	−	0.969524i	$-0.421214\pi$
$79$	4.35513	0.489990	0.244995	+	0.969524i	$0.421214\pi$
$80$	−1.29050	−0.144282
$81$	−8.44816	−0.938684
$82$	−6.34233	−0.700393
$83$	−9.53801	−1.04693	−0.523467	−	0.852046i	$-0.675361\pi$
$83$	−9.53801	−1.04693	−0.523467	+	0.852046i	$0.675361\pi$
$84$	−11.3394	−1.23723
$85$	0.0726132	0.00787600
$86$	−8.33810	−0.899120
$87$	−14.8959	−1.59701
$88$	0	0
$89$	7.47422	0.792266	0.396133	−	0.918193i	$-0.370352\pi$
$89$	7.47422	0.792266	0.396133	+	0.918193i	$0.370352\pi$
$90$	1.48105	0.156116
$91$	7.75168	0.812597
$92$	7.40603	0.772132
$93$	16.1922	1.67905
$94$	15.5987	1.60889
$95$	−0.260416	−0.0267181
$96$	15.0371	1.53471
$97$	13.4845	1.36914	0.684570	−	0.728947i	$-0.259990\pi$
$97$	13.4845	1.36914	0.684570	+	0.728947i	$0.259990\pi$
$98$	12.9086	1.30397
$99$	0	0
$100$	−5.97236	−0.597236
$101$	−2.48504	−0.247271	−0.123635	−	0.992328i	$-0.539455\pi$
$101$	−2.48504	−0.247271	−0.123635	+	0.992328i	$0.539455\pi$
$102$	−1.24148	−0.122925
$103$	0.449953	0.0443352	0.0221676	−	0.999754i	$-0.492943\pi$
$103$	0.449953	0.0443352	0.0221676	+	0.999754i	$0.492943\pi$
$104$	−2.90831	−0.285183
$105$	−2.43865	−0.237988
$106$	−8.39265	−0.815167
$107$	−1.15345	−0.111508	−0.0557540	−	0.998445i	$-0.517756\pi$
$107$	−1.15345	−0.111508	−0.0557540	+	0.998445i	$0.517756\pi$
$108$	−0.523133	−0.0503385
$109$	13.2279	1.26700	0.633502	−	0.773741i	$-0.281617\pi$
$109$	13.2279	1.26700	0.633502	+	0.773741i	$0.281617\pi$
$110$	0	0
$111$	−21.7565	−2.06504
$112$	−18.6764	−1.76475
$113$	16.7837	1.57888	0.789441	−	0.613827i	$-0.210371\pi$
$113$	16.7837	1.57888	0.789441	+	0.613827i	$0.210371\pi$
$114$	4.45238	0.417003
$115$	1.59274	0.148524
$116$	7.25931	0.674010
$117$	6.52802	0.603516
$118$	9.49254	0.873859
$119$	1.05088	0.0963337
$120$	0.914942	0.0835224
$121$	0	0
$122$	20.7902	1.88226
$123$	8.79457	0.792980
$124$	−7.89103	−0.708635
$125$	−2.58650	−0.231343
$126$	21.4341	1.90950
$127$	6.90332	0.612571	0.306285	−	0.951940i	$-0.400914\pi$
$127$	6.90332	0.612571	0.306285	+	0.951940i	$0.400914\pi$
$128$	10.4315	0.922022
$129$	11.5620	1.01798
$130$	0.959782	0.0841784
$131$	−10.2693	−0.897230	−0.448615	−	0.893725i	$-0.648083\pi$
$131$	−10.2693	−0.897230	−0.448615	+	0.893725i	$0.648083\pi$
$132$	0	0
$133$	−3.76880	−0.326797
$134$	23.3953	2.02105
$135$	−0.112505	−0.00968291
$136$	−0.394272	−0.0338086
$137$	−18.6804	−1.59598	−0.797988	−	0.602673i	$-0.794102\pi$
$137$	−18.6804	−1.59598	−0.797988	+	0.602673i	$0.794102\pi$
$138$	−27.2315	−2.31810
$139$	0.826591	0.0701105	0.0350553	−	0.999385i	$-0.488839\pi$
$139$	0.826591	0.0701105	0.0350553	+	0.999385i	$0.488839\pi$
$140$	1.18844	0.100442
$141$	−21.6299	−1.82157
$142$	13.4516	1.12883
$143$	0	0
$144$	−15.7282	−1.31068
$145$	1.56119	0.129650
$146$	1.09607	0.0907117
$147$	−17.8997	−1.47634
$148$	10.6027	0.871538
$149$	11.5052	0.942544	0.471272	−	0.881988i	$-0.343795\pi$
$149$	11.5052	0.942544	0.471272	+	0.881988i	$0.343795\pi$
$150$	21.9599	1.79302
$151$	−14.2758	−1.16174	−0.580872	−	0.813995i	$-0.697289\pi$
$151$	−14.2758	−1.16174	−0.580872	+	0.813995i	$0.697289\pi$
$152$	1.41399	0.114690
$153$	0.884988	0.0715471
$154$	0	0
$155$	−1.69705	−0.136310
$156$	−6.18839	−0.495468
$157$	−15.9760	−1.27503	−0.637513	−	0.770439i	$-0.720037\pi$
$157$	−15.9760	−1.27503	−0.637513	+	0.770439i	$0.720037\pi$
$158$	7.80394	0.620848
$159$	11.6376	0.922925
$160$	−1.57598	−0.124592
$161$	23.0506	1.81664
$162$	−15.1382	−1.18937
$163$	18.6219	1.45858	0.729290	−	0.684205i	$-0.239851\pi$
$163$	18.6219	1.45858	0.729290	+	0.684205i	$0.239851\pi$
$164$	−4.28590	−0.334673
$165$	0	0
$166$	−17.0911	−1.32653
$167$	18.8050	1.45517	0.727587	−	0.686016i	$-0.240642\pi$
$167$	18.8050	1.45517	0.727587	+	0.686016i	$0.240642\pi$
$168$	13.2413	1.02159
$169$	−8.76957	−0.674582
$170$	0.130115	0.00997939
$171$	−3.17387	−0.242712
$172$	−5.63457	−0.429632
$173$	13.7644	1.04649	0.523244	−	0.852183i	$-0.324722\pi$
$173$	13.7644	1.04649	0.523244	+	0.852183i	$0.324722\pi$
$174$	−26.6920	−2.02351
$175$	−18.5884	−1.40515
$176$	0	0
$177$	−13.1628	−0.989376
$178$	13.3930	1.00385
$179$	−25.7379	−1.92374	−0.961871	−	0.273502i	$-0.911818\pi$
$179$	−25.7379	−1.92374	−0.961871	+	0.273502i	$0.911818\pi$
$180$	1.00084	0.0745979
$181$	14.8940	1.10707	0.553533	−	0.832827i	$-0.313279\pi$
$181$	14.8940	1.10707	0.553533	+	0.832827i	$0.313279\pi$
$182$	13.8902	1.02961
$183$	−28.8287	−2.13108
$184$	−8.64822	−0.637555
$185$	2.28023	0.167646
$186$	29.0147	2.12746
$187$	0	0
$188$	10.5410	0.768783
$189$	−1.62820	−0.118434
$190$	−0.466638	−0.0338535
$191$	−22.4953	−1.62770	−0.813851	−	0.581073i	$-0.802633\pi$
$191$	−22.4953	−1.62770	−0.813851	+	0.581073i	$0.802633\pi$
$192$	2.31863	0.167333
$193$	−5.20478	−0.374648	−0.187324	−	0.982298i	$-0.559982\pi$
$193$	−5.20478	−0.374648	−0.187324	+	0.982298i	$0.559982\pi$
$194$	24.1628	1.73479
$195$	−1.33088	−0.0953061
$196$	8.72315	0.623082
$197$	17.3812	1.23836	0.619179	−	0.785250i	$-0.287465\pi$
$197$	17.3812	1.23836	0.619179	+	0.785250i	$0.287465\pi$
$198$	0	0
$199$	−20.1483	−1.42828	−0.714138	−	0.700005i	$-0.753181\pi$
$199$	−20.1483	−1.42828	−0.714138	+	0.700005i	$0.753181\pi$
$200$	6.97408	0.493142
$201$	−32.4411	−2.28822
$202$	−4.45293	−0.313307
$203$	22.5940	1.58579
$204$	−0.838945	−0.0587379
$205$	−0.921728	−0.0643763
$206$	0.806269	0.0561754
$207$	19.4119	1.34922
$208$	−10.1925	−0.706724
$209$	0	0
$210$	−4.36981	−0.301546
$211$	−1.90770	−0.131332	−0.0656658	−	0.997842i	$-0.520917\pi$
$211$	−1.90770	−0.131332	−0.0656658	+	0.997842i	$0.520917\pi$
$212$	−5.67143	−0.389516
$213$	−18.6526	−1.27805
$214$	−2.06686	−0.141287
$215$	−1.21177	−0.0826422
$216$	0.610876	0.0415649
$217$	−24.5601	−1.66725
$218$	23.7031	1.60537
$219$	−1.51987	−0.102703
$220$	0	0
$221$	0.573509	0.0385784
$222$	−38.9855	−2.61653
$223$	−12.5836	−0.842660	−0.421330	−	0.906907i	$-0.638437\pi$
$223$	−12.5836	−0.842660	−0.421330	+	0.906907i	$0.638437\pi$
$224$	−22.8080	−1.52393
$225$	−15.6541	−1.04361
$226$	30.0747	2.00054
$227$	−19.4974	−1.29408	−0.647042	−	0.762454i	$-0.723994\pi$
$227$	−19.4974	−1.29408	−0.647042	+	0.762454i	$0.723994\pi$
$228$	3.00874	0.199259
$229$	8.71196	0.575702	0.287851	−	0.957675i	$-0.407059\pi$
$229$	8.71196	0.575702	0.287851	+	0.957675i	$0.407059\pi$
$230$	2.85403	0.188189
$231$	0	0
$232$	−8.47689	−0.556535
$233$	18.7602	1.22902	0.614511	−	0.788908i	$-0.289353\pi$
$233$	18.7602	1.22902	0.614511	+	0.788908i	$0.289353\pi$
$234$	11.6975	0.764692
$235$	2.26696	0.147880
$236$	6.41469	0.417561
$237$	−10.8213	−0.702919
$238$	1.88306	0.122061
$239$	−6.13330	−0.396731	−0.198365	−	0.980128i	$-0.563563\pi$
$239$	−6.13330	−0.396731	−0.198365	+	0.980128i	$0.563563\pi$
$240$	3.20653	0.206981
$241$	−27.5582	−1.77518	−0.887591	−	0.460633i	$-0.847622\pi$
$241$	−27.5582	−1.77518	−0.887591	+	0.460633i	$0.847622\pi$
$242$	0	0
$243$	22.2874	1.42974
$244$	14.0492	0.899410
$245$	1.87601	0.119854
$246$	15.7590	1.00475
$247$	−2.05680	−0.130871
$248$	9.21456	0.585125
$249$	23.6994	1.50189
$250$	−4.63474	−0.293126
$251$	−16.6160	−1.04879	−0.524397	−	0.851474i	$-0.675709\pi$
$251$	−16.6160	−1.04879	−0.524397	+	0.851474i	$0.675709\pi$
$252$	14.4844	0.912429
$253$	0	0
$254$	12.3700	0.776166
$255$	−0.180424	−0.0112986
$256$	20.5584	1.28490
$257$	4.84015	0.301920	0.150960	−	0.988540i	$-0.451763\pi$
$257$	4.84015	0.301920	0.150960	+	0.988540i	$0.451763\pi$
$258$	20.7179	1.28984
$259$	33.0000	2.05052
$260$	0.648584	0.0402234
$261$	19.0273	1.17776
$262$	−18.4015	−1.13685
$263$	26.1141	1.61027	0.805133	−	0.593094i	$-0.202094\pi$
$263$	26.1141	1.61027	0.805133	+	0.593094i	$0.202094\pi$
$264$	0	0
$265$	−1.21970	−0.0749256
$266$	−6.75331	−0.414072
$267$	−18.5714	−1.13655
$268$	15.8097	0.965730
$269$	−14.9397	−0.910889	−0.455444	−	0.890264i	$-0.650520\pi$
$269$	−14.9397	−0.910889	−0.455444	+	0.890264i	$0.650520\pi$
$270$	−0.201598	−0.0122688
$271$	−27.6208	−1.67785	−0.838923	−	0.544250i	$-0.816814\pi$
$271$	−27.6208	−1.67785	−0.838923	+	0.544250i	$0.816814\pi$
$272$	−1.38178	−0.0837825
$273$	−19.2608	−1.16572
$274$	−33.4734	−2.02220
$275$	0	0
$276$	−18.4020	−1.10767
$277$	−28.3197	−1.70157	−0.850784	−	0.525515i	$-0.823873\pi$
$277$	−28.3197	−1.70157	−0.850784	+	0.525515i	$0.823873\pi$
$278$	1.48117	0.0888344
$279$	−20.6831	−1.23827
$280$	−1.38777	−0.0829353
$281$	21.8536	1.30368	0.651838	−	0.758359i	$-0.273998\pi$
$281$	21.8536	1.30368	0.651838	+	0.758359i	$0.273998\pi$
$282$	−38.7586	−2.30804
$283$	13.5236	0.803895	0.401947	−	0.915663i	$-0.368333\pi$
$283$	13.5236	0.803895	0.401947	+	0.915663i	$0.368333\pi$
$284$	9.09005	0.539395
$285$	0.647062	0.0383286
$286$	0	0
$287$	−13.3395	−0.787406
$288$	−19.2076	−1.13182
$289$	−16.9223	−0.995427
$290$	2.79749	0.164275
$291$	−33.5052	−1.96411
$292$	0.740684	0.0433453
$293$	−17.0018	−0.993257	−0.496628	−	0.867963i	$-0.665429\pi$
$293$	−17.0018	−0.993257	−0.496628	+	0.867963i	$0.665429\pi$
$294$	−32.0744	−1.87062
$295$	1.37955	0.0803203
$296$	−12.3811	−0.719636
$297$	0	0
$298$	20.6162	1.19426
$299$	12.5797	0.727504
$300$	14.8397	0.856769
$301$	−17.5371	−1.01082
$302$	−25.5807	−1.47200
$303$	6.17464	0.354724
$304$	4.95552	0.284219
$305$	3.02143	0.173007
$306$	1.58581	0.0906546
$307$	16.5331	0.943592	0.471796	−	0.881708i	$-0.343606\pi$
$307$	16.5331	0.943592	0.471796	+	0.881708i	$0.343606\pi$
$308$	0	0
$309$	−1.11801	−0.0636014
$310$	−3.04094	−0.172714
$311$	−3.60078	−0.204182	−0.102091	−	0.994775i	$-0.532553\pi$
$311$	−3.60078	−0.204182	−0.102091	+	0.994775i	$0.532553\pi$
$312$	7.22635	0.409111
$313$	−25.0524	−1.41604	−0.708022	−	0.706191i	$-0.750412\pi$
$313$	−25.0524	−1.41604	−0.708022	+	0.706191i	$0.750412\pi$
$314$	−28.6274	−1.61554
$315$	3.11501	0.175511
$316$	5.27360	0.296663
$317$	−14.9343	−0.838792	−0.419396	−	0.907803i	$-0.637758\pi$
$317$	−14.9343	−0.838792	−0.419396	+	0.907803i	$0.637758\pi$
$318$	20.8535	1.16940
$319$	0	0
$320$	−0.243007	−0.0135845
$321$	2.86600	0.159965
$322$	41.3043	2.30180
$323$	−0.278836	−0.0155148
$324$	−10.2298	−0.568324
$325$	−10.1445	−0.562717
$326$	33.3685	1.84811
$327$	−32.8678	−1.81759
$328$	5.00476	0.276342
$329$	32.8080	1.80876
$330$	0	0
$331$	−3.63411	−0.199749	−0.0998745	−	0.995000i	$-0.531844\pi$
$331$	−3.63411	−0.199749	−0.0998745	+	0.995000i	$0.531844\pi$
$332$	−11.5495	−0.633863
$333$	27.7907	1.52292
$334$	33.6966	1.84380
$335$	3.40004	0.185764
$336$	46.4058	2.53164
$337$	24.1428	1.31514	0.657570	−	0.753393i	$-0.271584\pi$
$337$	24.1428	1.31514	0.657570	+	0.753393i	$0.271584\pi$
$338$	−15.7142	−0.854738
$339$	−41.7030	−2.26500
$340$	0.0879269	0.00476851
$341$	0	0
$342$	−5.68725	−0.307531
$343$	0.768413	0.0414904
$344$	6.57963	0.354750
$345$	−3.95754	−0.213067
$346$	24.6644	1.32597
$347$	8.99715	0.482992	0.241496	−	0.970402i	$-0.422362\pi$
$347$	8.99715	0.482992	0.241496	+	0.970402i	$0.422362\pi$
$348$	−18.0374	−0.966907
$349$	16.6965	0.893745	0.446872	−	0.894598i	$-0.352538\pi$
$349$	16.6965	0.893745	0.446872	+	0.894598i	$0.352538\pi$
$350$	−33.3086	−1.78042
$351$	−0.888582	−0.0474290
$352$	0	0
$353$	14.5070	0.772132	0.386066	−	0.922471i	$-0.373834\pi$
$353$	14.5070	0.772132	0.386066	+	0.922471i	$0.373834\pi$
$354$	−23.5864	−1.25360
$355$	1.95491	0.103756
$356$	9.05050	0.479675
$357$	−2.61114	−0.138196
$358$	−46.1197	−2.43750
$359$	−20.4038	−1.07687	−0.538436	−	0.842667i	$-0.680984\pi$
$359$	−20.4038	−1.07687	−0.538436	+	0.842667i	$0.680984\pi$
$360$	−1.16870	−0.0615961
$361$	1.00000	0.0526316
$362$	26.6886	1.40272
$363$	0	0
$364$	9.38647	0.491985
$365$	0.159292	0.00833772
$366$	−51.6580	−2.70021
$367$	9.53338	0.497639	0.248819	−	0.968550i	$-0.419957\pi$
$367$	9.53338	0.497639	0.248819	+	0.968550i	$0.419957\pi$
$368$	−30.3088	−1.57995
$369$	−11.2338	−0.584806
$370$	4.08593	0.212417
$371$	−17.6518	−0.916438
$372$	19.6070	1.01658
$373$	−16.7534	−0.867458	−0.433729	−	0.901043i	$-0.642803\pi$
$373$	−16.7534	−0.867458	−0.433729	+	0.901043i	$0.642803\pi$
$374$	0	0
$375$	6.42674	0.331875
$376$	−12.3090	−0.634790
$377$	12.3305	0.635054
$378$	−2.91758	−0.150064
$379$	1.61061	0.0827314	0.0413657	−	0.999144i	$-0.486829\pi$
$379$	1.61061	0.0827314	0.0413657	+	0.999144i	$0.486829\pi$
$380$	−0.315336	−0.0161764
$381$	−17.1529	−0.878768
$382$	−40.3093	−2.06240
$383$	−7.02805	−0.359117	−0.179558	−	0.983747i	$-0.557467\pi$
$383$	−7.02805	−0.359117	−0.179558	+	0.983747i	$0.557467\pi$
$384$	−25.9194	−1.32269
$385$	0	0
$386$	−9.32644	−0.474703
$387$	−14.7687	−0.750737
$388$	16.3283	0.828942
$389$	12.2663	0.621924	0.310962	−	0.950422i	$-0.399349\pi$
$389$	12.2663	0.621924	0.310962	+	0.950422i	$0.399349\pi$
$390$	−2.38480	−0.120759
$391$	1.70540	0.0862460
$392$	−10.1863	−0.514484
$393$	25.5163	1.28713
$394$	31.1453	1.56908
$395$	1.13414	0.0570650
$396$	0	0
$397$	−14.0905	−0.707180	−0.353590	−	0.935400i	$-0.615039\pi$
$397$	−14.0905	−0.707180	−0.353590	+	0.935400i	$0.615039\pi$
$398$	−36.1037	−1.80972
$399$	9.36445	0.468809
$400$	24.4416	1.22208
$401$	17.8578	0.891774	0.445887	−	0.895089i	$-0.352888\pi$
$401$	17.8578	0.891774	0.445887	+	0.895089i	$0.352888\pi$
$402$	−58.1311	−2.89931
$403$	−13.4035	−0.667677
$404$	−3.00912	−0.149709
$405$	−2.20003	−0.109320
$406$	40.4861	2.00929
$407$	0	0
$408$	0.979659	0.0485003
$409$	−35.6237	−1.76148	−0.880739	−	0.473602i	$-0.842953\pi$
$409$	−35.6237	−1.76148	−0.880739	+	0.473602i	$0.842953\pi$
$410$	−1.65164	−0.0815688
$411$	46.4158	2.28952
$412$	0.544845	0.0268426
$413$	19.9652	0.982422
$414$	34.7841	1.70955
$415$	−2.48385	−0.121927
$416$	−12.4473	−0.610281
$417$	−2.05385	−0.100578
$418$	0	0
$419$	9.73712	0.475689	0.237845	−	0.971303i	$-0.423559\pi$
$419$	9.73712	0.475689	0.237845	+	0.971303i	$0.423559\pi$
$420$	−2.95295	−0.144089
$421$	29.5648	1.44090	0.720449	−	0.693508i	$-0.243936\pi$
$421$	29.5648	1.44090	0.720449	+	0.693508i	$0.243936\pi$
$422$	−3.41840	−0.166405
$423$	27.6290	1.34337
$424$	6.62268	0.321626
$425$	−1.37527	−0.0667103
$426$	−33.4235	−1.61937
$427$	43.7270	2.11610
$428$	−1.39670	−0.0675122
$429$	0	0
$430$	−2.17137	−0.104713
$431$	2.84752	0.137160	0.0685800	−	0.997646i	$-0.478153\pi$
$431$	2.84752	0.137160	0.0685800	+	0.997646i	$0.478153\pi$
$432$	2.14089	0.103004
$433$	−2.63979	−0.126860	−0.0634302	−	0.997986i	$-0.520204\pi$
$433$	−2.63979	−0.126860	−0.0634302	+	0.997986i	$0.520204\pi$
$434$	−44.0092	−2.11251
$435$	−3.87914	−0.185990
$436$	16.0176	0.767105
$437$	−6.11616	−0.292576
$438$	−2.72344	−0.130131
$439$	−21.0320	−1.00380	−0.501902	−	0.864925i	$-0.667366\pi$
$439$	−21.0320	−1.00380	−0.501902	+	0.864925i	$0.667366\pi$
$440$	0	0
$441$	22.8642	1.08877
$442$	1.02767	0.0488813
$443$	10.4235	0.495235	0.247618	−	0.968858i	$-0.420352\pi$
$443$	10.4235	0.495235	0.247618	+	0.968858i	$0.420352\pi$
$444$	−26.3449	−1.25027
$445$	1.94640	0.0922684
$446$	−22.5485	−1.06770
$447$	−28.5873	−1.35213
$448$	−3.51687	−0.166156
$449$	26.4268	1.24716	0.623579	−	0.781761i	$-0.285678\pi$
$449$	26.4268	1.24716	0.623579	+	0.781761i	$0.285678\pi$
$450$	−28.0506	−1.32232
$451$	0	0
$452$	20.3233	0.955930
$453$	35.4714	1.66659
$454$	−34.9373	−1.63969
$455$	2.01866	0.0946362
$456$	−3.51339	−0.164530
$457$	13.4092	0.627258	0.313629	−	0.949546i	$-0.398455\pi$
$457$	13.4092	0.627258	0.313629	+	0.949546i	$0.398455\pi$
$458$	15.6109	0.729451
$459$	−0.120463	−0.00562273
$460$	1.92865	0.0899236
$461$	34.0055	1.58379	0.791897	−	0.610655i	$-0.209094\pi$
$461$	34.0055	1.58379	0.791897	+	0.610655i	$0.209094\pi$
$462$	0	0
$463$	−32.1254	−1.49299	−0.746497	−	0.665389i	$-0.768266\pi$
$463$	−32.1254	−1.49299	−0.746497	+	0.665389i	$0.768266\pi$
$464$	−29.7083	−1.37918
$465$	4.21670	0.195545
$466$	33.6164	1.55725
$467$	1.95804	0.0906071	0.0453035	−	0.998973i	$-0.485574\pi$
$467$	1.95804	0.0906071	0.0453035	+	0.998973i	$0.485574\pi$
$468$	7.90475	0.365397
$469$	49.2062	2.27213
$470$	4.06215	0.187373
$471$	39.6961	1.82910
$472$	−7.49061	−0.344783
$473$	0	0
$474$	−19.3907	−0.890643
$475$	4.93218	0.226304
$476$	1.27250	0.0583250
$477$	−14.8654	−0.680638
$478$	−10.9903	−0.502682
$479$	−21.3775	−0.976762	−0.488381	−	0.872631i	$-0.662412\pi$
$479$	−21.3775	−0.976762	−0.488381	+	0.872631i	$0.662412\pi$
$480$	3.91589	0.178735
$481$	18.0096	0.821165
$482$	−49.3815	−2.24927
$483$	−57.2745	−2.60608
$484$	0	0
$485$	3.51156	0.159452
$486$	39.9368	1.81157
$487$	−26.3339	−1.19330	−0.596651	−	0.802501i	$-0.703502\pi$
$487$	−26.3339	−1.19330	−0.596651	+	0.802501i	$0.703502\pi$
$488$	−16.4057	−0.742650
$489$	−46.2703	−2.09242
$490$	3.36161	0.151862
$491$	−12.3994	−0.559578	−0.279789	−	0.960062i	$-0.590265\pi$
$491$	−12.3994	−0.559578	−0.279789	+	0.960062i	$0.590265\pi$
$492$	10.6493	0.480108
$493$	1.67162	0.0752859
$494$	−3.68558	−0.165822
$495$	0	0
$496$	32.2936	1.45003
$497$	28.2920	1.26907
$498$	42.4668	1.90298
$499$	−34.7531	−1.55576	−0.777881	−	0.628411i	$-0.783706\pi$
$499$	−34.7531	−1.55576	−0.777881	+	0.628411i	$0.783706\pi$
$500$	−3.13198	−0.140066
$501$	−46.7253	−2.08753
$502$	−29.7742	−1.32889
$503$	7.99272	0.356378	0.178189	−	0.983996i	$-0.442976\pi$
$503$	7.99272	0.356378	0.178189	+	0.983996i	$0.442976\pi$
$504$	−16.9138	−0.753400
$505$	−0.647143	−0.0287975
$506$	0	0
$507$	21.7900	0.967728
$508$	8.35920	0.370879
$509$	−10.5728	−0.468629	−0.234315	−	0.972161i	$-0.575285\pi$
$509$	−10.5728	−0.468629	−0.234315	+	0.972161i	$0.575285\pi$
$510$	−0.323301	−0.0143160
$511$	2.30532	0.101981
$512$	15.9757	0.706031
$513$	0.432022	0.0190742
$514$	8.67305	0.382552
$515$	0.117175	0.00516334
$516$	14.0004	0.616331
$517$	0	0
$518$	59.1327	2.59814
$519$	−34.2008	−1.50125
$520$	−0.757369	−0.0332128
$521$	−7.50121	−0.328634	−0.164317	−	0.986408i	$-0.552542\pi$
$521$	−7.50121	−0.328634	−0.164317	+	0.986408i	$0.552542\pi$
$522$	34.0950	1.49230
$523$	−19.1032	−0.835325	−0.417663	−	0.908602i	$-0.637151\pi$
$523$	−19.1032	−0.835325	−0.417663	+	0.908602i	$0.637151\pi$
$524$	−12.4350	−0.543226
$525$	46.1872	2.01577
$526$	46.7939	2.04031
$527$	−1.81708	−0.0791535
$528$	0	0
$529$	14.4074	0.626410
$530$	−2.18558	−0.0949355
$531$	16.8135	0.729645
$532$	−4.56363	−0.197858
$533$	−7.27994	−0.315329
$534$	−33.2780	−1.44008
$535$	−0.300376	−0.0129864
$536$	−18.4614	−0.797410
$537$	63.9517	2.75972
$538$	−26.7704	−1.15415
$539$	0	0
$540$	−0.136232	−0.00586249
$541$	3.66193	0.157439	0.0787193	−	0.996897i	$-0.474917\pi$
$541$	3.66193	0.157439	0.0787193	+	0.996897i	$0.474917\pi$
$542$	−49.4937	−2.12594
$543$	−37.0077	−1.58815
$544$	−1.68746	−0.0723491
$545$	3.44476	0.147557
$546$	−34.5134	−1.47704
$547$	3.71881	0.159005	0.0795025	−	0.996835i	$-0.474667\pi$
$547$	3.71881	0.159005	0.0795025	+	0.996835i	$0.474667\pi$
$548$	−22.6200	−0.966280
$549$	36.8244	1.57163
$550$	0	0
$551$	−5.99500	−0.255395
$552$	21.4885	0.914610
$553$	16.4136	0.697978
$554$	−50.7461	−2.15599
$555$	−5.66574	−0.240497
$556$	1.00091	0.0424482
$557$	36.0925	1.52929	0.764643	−	0.644454i	$-0.222915\pi$
$557$	36.0925	1.52929	0.764643	+	0.644454i	$0.222915\pi$
$558$	−37.0620	−1.56896
$559$	−9.57076	−0.404800
$560$	−4.86363	−0.205526
$561$	0	0
$562$	39.1594	1.65184
$563$	−32.9936	−1.39052	−0.695258	−	0.718761i	$-0.744710\pi$
$563$	−32.9936	−1.39052	−0.695258	+	0.718761i	$0.744710\pi$
$564$	−26.1916	−1.10286
$565$	4.37075	0.183879
$566$	24.2329	1.01859
$567$	−31.8395	−1.33713
$568$	−10.6147	−0.445383
$569$	23.6786	0.992658	0.496329	−	0.868134i	$-0.334681\pi$
$569$	23.6786	0.992658	0.496329	+	0.868134i	$0.334681\pi$
$570$	1.15947	0.0485648
$571$	26.7723	1.12039	0.560193	−	0.828362i	$-0.310727\pi$
$571$	26.7723	1.12039	0.560193	+	0.828362i	$0.310727\pi$
$572$	0	0
$573$	55.8947	2.33503
$574$	−23.9030	−0.997692
$575$	−30.1660	−1.25801
$576$	−2.96170	−0.123404
$577$	−21.8266	−0.908655	−0.454327	−	0.890835i	$-0.650120\pi$
$577$	−21.8266	−0.908655	−0.454327	+	0.890835i	$0.650120\pi$
$578$	−30.3229	−1.26127
$579$	12.9325	0.537455
$580$	1.89044	0.0784962
$581$	−35.9469	−1.49133
$582$	−60.0379	−2.48865
$583$	0	0
$584$	−0.864917	−0.0357905
$585$	1.70000	0.0702863
$586$	−30.4655	−1.25852
$587$	−9.12072	−0.376452	−0.188226	−	0.982126i	$-0.560274\pi$
$587$	−9.12072	−0.376452	−0.188226	+	0.982126i	$0.560274\pi$
$588$	−21.6747	−0.893848
$589$	6.51669	0.268515
$590$	2.47201	0.101771
$591$	−43.1875	−1.77650
$592$	−43.3911	−1.78336
$593$	−28.8333	−1.18404	−0.592021	−	0.805923i	$-0.701670\pi$
$593$	−28.8333	−1.18404	−0.592021	+	0.805923i	$0.701670\pi$
$594$	0	0
$595$	0.273665	0.0112192
$596$	13.9316	0.570661
$597$	50.0631	2.04895
$598$	22.5416	0.921794
$599$	22.0967	0.902848	0.451424	−	0.892310i	$-0.350916\pi$
$599$	22.0967	0.902848	0.451424	+	0.892310i	$0.350916\pi$
$600$	−17.3287	−0.707441
$601$	13.9178	0.567720	0.283860	−	0.958866i	$-0.408385\pi$
$601$	13.9178	0.567720	0.283860	+	0.958866i	$0.408385\pi$
$602$	−31.4247	−1.28077
$603$	41.4387	1.68751
$604$	−17.2864	−0.703376
$605$	0	0
$606$	11.0643	0.449457
$607$	−34.5923	−1.40406	−0.702029	−	0.712149i	$-0.747722\pi$
$607$	−34.5923	−1.40406	−0.702029	+	0.712149i	$0.747722\pi$
$608$	6.05180	0.245433
$609$	−56.1399	−2.27490
$610$	5.41410	0.219211
$611$	17.9048	0.724349
$612$	1.07163	0.0433180
$613$	−12.0413	−0.486343	−0.243171	−	0.969983i	$-0.578188\pi$
$613$	−12.0413	−0.486343	−0.243171	+	0.969983i	$0.578188\pi$
$614$	29.6255	1.19559
$615$	2.29024	0.0923516
$616$	0	0
$617$	−38.8403	−1.56365	−0.781825	−	0.623498i	$-0.785711\pi$
$617$	−38.8403	−1.56365	−0.781825	+	0.623498i	$0.785711\pi$
$618$	−2.00336	−0.0805869
$619$	−16.0129	−0.643612	−0.321806	−	0.946806i	$-0.604290\pi$
$619$	−16.0129	−0.643612	−0.321806	+	0.946806i	$0.604290\pi$
$620$	−2.05495	−0.0825287
$621$	−2.64231	−0.106032
$622$	−6.45223	−0.258711
$623$	28.1689	1.12856
$624$	25.3256	1.01384
$625$	23.9874	0.959494
$626$	−44.8913	−1.79422
$627$	0	0
$628$	−19.3453	−0.771962
$629$	2.44151	0.0973495
$630$	5.58178	0.222384
$631$	30.6829	1.22147	0.610734	−	0.791836i	$-0.290874\pi$
$631$	30.6829	1.22147	0.610734	+	0.791836i	$0.290874\pi$
$632$	−6.15813	−0.244957
$633$	4.74012	0.188403
$634$	−26.7607	−1.06280
$635$	1.79773	0.0713409
$636$	14.0920	0.558783
$637$	14.8170	0.587069
$638$	0	0
$639$	23.8259	0.942538
$640$	2.71652	0.107380
$641$	19.5971	0.774041	0.387020	−	0.922071i	$-0.373504\pi$
$641$	19.5971	0.774041	0.387020	+	0.922071i	$0.373504\pi$
$642$	5.13558	0.202685
$643$	−29.6476	−1.16919	−0.584594	−	0.811326i	$-0.698746\pi$
$643$	−29.6476	−1.16919	−0.584594	+	0.811326i	$0.698746\pi$
$644$	27.9119	1.09988
$645$	3.01092	0.118555
$646$	−0.499645	−0.0196583
$647$	22.9490	0.902219	0.451110	−	0.892469i	$-0.351028\pi$
$647$	22.9490	0.902219	0.451110	+	0.892469i	$0.351028\pi$
$648$	11.9457	0.469269
$649$	0	0
$650$	−18.1779	−0.712997
$651$	61.0252	2.39177
$652$	22.5492	0.883093
$653$	22.6057	0.884631	0.442315	−	0.896860i	$-0.354157\pi$
$653$	22.6057	0.884631	0.442315	+	0.896860i	$0.354157\pi$
$654$	−58.8957	−2.30300
$655$	−2.67428	−0.104493
$656$	17.5398	0.684815
$657$	1.94140	0.0757414
$658$	58.7885	2.29182
$659$	22.8318	0.889399	0.444699	−	0.895680i	$-0.353310\pi$
$659$	22.8318	0.889399	0.444699	+	0.895680i	$0.353310\pi$
$660$	0	0
$661$	−18.5071	−0.719844	−0.359922	−	0.932982i	$-0.617197\pi$
$661$	−18.5071	−0.719844	−0.359922	+	0.932982i	$0.617197\pi$
$662$	−6.51196	−0.253094
$663$	−1.42501	−0.0553430
$664$	13.4867	0.523386
$665$	−0.981456	−0.0380592
$666$	49.7981	1.92964
$667$	36.6664	1.41973
$668$	22.7709	0.881031
$669$	31.2668	1.20884
$670$	6.09252	0.235374
$671$	0	0
$672$	56.6718	2.18616
$673$	−20.2417	−0.780259	−0.390129	−	0.920760i	$-0.627570\pi$
$673$	−20.2417	−0.780259	−0.390129	+	0.920760i	$0.627570\pi$
$674$	43.2613	1.66636
$675$	2.13081	0.0820149
$676$	−10.6190	−0.408424
$677$	32.1530	1.23574	0.617871	−	0.786279i	$-0.287995\pi$
$677$	32.1530	1.23574	0.617871	+	0.786279i	$0.287995\pi$
$678$	−74.7275	−2.86989
$679$	50.8203	1.95030
$680$	−0.102675	−0.00393739
$681$	48.4456	1.85644
$682$	0	0
$683$	14.2346	0.544673	0.272336	−	0.962202i	$-0.412204\pi$
$683$	14.2346	0.544673	0.272336	+	0.962202i	$0.412204\pi$
$684$	−3.84322	−0.146949
$685$	−4.86468	−0.185870
$686$	1.37692	0.0525709
$687$	−21.6468	−0.825878
$688$	23.0592	0.879122
$689$	−9.63338	−0.367002
$690$	−7.09150	−0.269969
$691$	−10.0542	−0.382481	−0.191240	−	0.981543i	$-0.561251\pi$
$691$	−10.0542	−0.382481	−0.191240	+	0.981543i	$0.561251\pi$
$692$	16.6672	0.633594
$693$	0	0
$694$	16.1220	0.611982
$695$	0.215257	0.00816517
$696$	21.0628	0.798382
$697$	−0.986925	−0.0373824
$698$	29.9185	1.13243
$699$	−46.6140	−1.76310
$700$	−22.5086	−0.850747
$701$	43.4165	1.63982	0.819909	−	0.572494i	$-0.194024\pi$
$701$	43.4165	1.63982	0.819909	+	0.572494i	$0.194024\pi$
$702$	−1.59225	−0.0600955
$703$	−8.75610	−0.330243
$704$	0	0
$705$	−5.63277	−0.212142
$706$	25.9951	0.978339
$707$	−9.36562	−0.352230
$708$	−15.9388	−0.599015
$709$	35.7269	1.34175	0.670876	−	0.741570i	$-0.265918\pi$
$709$	35.7269	1.34175	0.670876	+	0.741570i	$0.265918\pi$
$710$	3.50300	0.131465
$711$	13.8226	0.518388
$712$	−10.5685	−0.396072
$713$	−39.8571	−1.49266
$714$	−4.67890	−0.175103
$715$	0	0
$716$	−31.1659	−1.16473
$717$	15.2396	0.569133
$718$	−36.5615	−1.36446
$719$	17.0581	0.636160	0.318080	−	0.948064i	$-0.396962\pi$
$719$	17.0581	0.636160	0.318080	+	0.948064i	$0.396962\pi$
$720$	−4.09587	−0.152644
$721$	1.69578	0.0631543
$722$	1.79190	0.0666875
$723$	68.4747	2.54660
$724$	18.0351	0.670270
$725$	−29.5684	−1.09814
$726$	0	0
$727$	−32.3512	−1.19984	−0.599919	−	0.800061i	$-0.704801\pi$
$727$	−32.3512	−1.19984	−0.599919	+	0.800061i	$0.704801\pi$
$728$	−10.9608	−0.406236
$729$	−30.0337	−1.11236
$730$	0.285435	0.0105644
$731$	−1.29748	−0.0479892
$732$	−34.9085	−1.29026
$733$	−10.1414	−0.374580	−0.187290	−	0.982305i	$-0.559970\pi$
$733$	−10.1414	−0.374580	−0.187290	+	0.982305i	$0.559970\pi$
$734$	17.0828	0.630539
$735$	−4.66136	−0.171937
$736$	−37.0138	−1.36435
$737$	0	0
$738$	−20.1297	−0.740986
$739$	39.3685	1.44819	0.724097	−	0.689698i	$-0.242257\pi$
$739$	39.3685	1.44819	0.724097	+	0.689698i	$0.242257\pi$
$740$	2.76111	0.101501
$741$	5.11059	0.187742
$742$	−31.6303	−1.16118
$743$	−0.416166	−0.0152677	−0.00763383	−	0.999971i	$-0.502430\pi$
$743$	−0.416166	−0.0152677	−0.00763383	+	0.999971i	$0.502430\pi$
$744$	−22.8957	−0.839396
$745$	2.99614	0.109770
$746$	−30.0204	−1.09912
$747$	−30.2724	−1.10761
$748$	0	0
$749$	−4.34712	−0.158840
$750$	11.5161	0.420507
$751$	18.5177	0.675719	0.337860	−	0.941197i	$-0.390297\pi$
$751$	18.5177	0.675719	0.337860	+	0.941197i	$0.390297\pi$
$752$	−43.1386	−1.57310
$753$	41.2863	1.50456
$754$	22.0950	0.804653
$755$	−3.71763	−0.135298
$756$	−1.97159	−0.0717059
$757$	−8.43089	−0.306426	−0.153213	−	0.988193i	$-0.548962\pi$
$757$	−8.43089	−0.306426	−0.153213	+	0.988193i	$0.548962\pi$
$758$	2.88605	0.104826
$759$	0	0
$760$	0.368226	0.0133570
$761$	−26.6054	−0.964444	−0.482222	−	0.876049i	$-0.660170\pi$
$761$	−26.6054	−0.964444	−0.482222	+	0.876049i	$0.660170\pi$
$762$	−30.7362	−1.11345
$763$	49.8534	1.80482
$764$	−27.2394	−0.985489
$765$	0.230465	0.00833247
$766$	−12.5935	−0.455023
$767$	10.8959	0.393427
$768$	−51.0821	−1.84327
$769$	8.62176	0.310908	0.155454	−	0.987843i	$-0.450316\pi$
$769$	8.62176	0.310908	0.155454	+	0.987843i	$0.450316\pi$
$770$	0	0
$771$	−12.0265	−0.433122
$772$	−6.30245	−0.226830
$773$	31.6229	1.13740	0.568698	−	0.822546i	$-0.307447\pi$
$773$	31.6229	1.13740	0.568698	+	0.822546i	$0.307447\pi$
$774$	−26.4640	−0.951230
$775$	32.1415	1.15456
$776$	−19.0670	−0.684464
$777$	−81.9961	−2.94159
$778$	21.9799	0.788017
$779$	3.53945	0.126814
$780$	−1.61155	−0.0577029
$781$	0	0
$782$	3.05591	0.109279
$783$	−2.58997	−0.0925579
$784$	−35.6990	−1.27497
$785$	−4.16041	−0.148491
$786$	45.7227	1.63087
$787$	14.2139	0.506670	0.253335	−	0.967379i	$-0.418473\pi$
$787$	14.2139	0.506670	0.253335	+	0.967379i	$0.418473\pi$
$788$	21.0468	0.749761
$789$	−64.8865	−2.31002
$790$	2.03227	0.0723049
$791$	63.2546	2.24908
$792$	0	0
$793$	23.8637	0.847426
$794$	−25.2487	−0.896041
$795$	3.03062	0.107485
$796$	−24.3975	−0.864746
$797$	−14.5762	−0.516317	−0.258158	−	0.966103i	$-0.583116\pi$
$797$	−14.5762	−0.516317	−0.258158	+	0.966103i	$0.583116\pi$
$798$	16.7801	0.594010
$799$	2.42730	0.0858719
$800$	29.8486	1.05531
$801$	23.7222	0.838183
$802$	31.9993	1.12993
$803$	0	0
$804$	−39.2827	−1.38540
$805$	6.00274	0.211569
$806$	−24.0177	−0.845989
$807$	37.1211	1.30672
$808$	3.51383	0.123616
$809$	−17.8912	−0.629021	−0.314511	−	0.949254i	$-0.601840\pi$
$809$	−17.8912	−0.629021	−0.314511	+	0.949254i	$0.601840\pi$
$810$	−3.94223	−0.138516
$811$	41.7192	1.46496	0.732480	−	0.680789i	$-0.238363\pi$
$811$	41.7192	1.46496	0.732480	+	0.680789i	$0.238363\pi$
$812$	27.3589	0.960110
$813$	68.6302	2.40697
$814$	0	0
$815$	4.84943	0.169868
$816$	3.43334	0.120191
$817$	4.65322	0.162796
$818$	−63.8340	−2.23190
$819$	24.6028	0.859693
$820$	−1.11612	−0.0389765
$821$	35.1689	1.22740	0.613701	−	0.789538i	$-0.289680\pi$
$821$	35.1689	1.22740	0.613701	+	0.789538i	$0.289680\pi$
$822$	83.1723	2.90097
$823$	−7.14661	−0.249115	−0.124558	−	0.992212i	$-0.539751\pi$
$823$	−7.14661	−0.249115	−0.124558	+	0.992212i	$0.539751\pi$
$824$	−0.636231	−0.0221642
$825$	0	0
$826$	35.7755	1.24479
$827$	20.5226	0.713641	0.356820	−	0.934173i	$-0.383861\pi$
$827$	20.5226	0.713641	0.356820	+	0.934173i	$0.383861\pi$
$828$	23.5058	0.816882
$829$	3.00661	0.104424	0.0522120	−	0.998636i	$-0.483373\pi$
$829$	3.00661	0.104424	0.0522120	+	0.998636i	$0.483373\pi$
$830$	−4.45080	−0.154490
$831$	70.3669	2.44100
$832$	−1.91931	−0.0665400
$833$	2.00870	0.0695973
$834$	−3.68029	−0.127438
$835$	4.89711	0.169472
$836$	0	0
$837$	2.81535	0.0973127
$838$	17.4479	0.602728
$839$	−33.7640	−1.16566	−0.582831	−	0.812594i	$-0.698055\pi$
$839$	−33.7640	−1.16566	−0.582831	+	0.812594i	$0.698055\pi$
$840$	3.44824	0.118976
$841$	6.93999	0.239310
$842$	52.9770	1.82571
$843$	−54.3002	−1.87020
$844$	−2.31003	−0.0795144
$845$	−2.28373	−0.0785628
$846$	49.5083	1.70213
$847$	0	0
$848$	23.2100	0.797036
$849$	−33.6025	−1.15323
$850$	−2.46434	−0.0845261
$851$	53.5537	1.83580
$852$	−22.5863	−0.773794
$853$	−7.06813	−0.242008	−0.121004	−	0.992652i	$-0.538611\pi$
$853$	−7.06813	−0.242008	−0.121004	+	0.992652i	$0.538611\pi$
$854$	78.3543	2.68123
$855$	−0.826526	−0.0282666
$856$	1.63097	0.0557453
$857$	36.6480	1.25187	0.625935	−	0.779875i	$-0.284717\pi$
$857$	36.6480	1.25187	0.625935	+	0.779875i	$0.284717\pi$
$858$	0	0
$859$	1.24027	0.0423173	0.0211587	−	0.999776i	$-0.493264\pi$
$859$	1.24027	0.0423173	0.0211587	+	0.999776i	$0.493264\pi$
$860$	−1.46733	−0.0500355
$861$	33.1450	1.12958
$862$	5.10246	0.173790
$863$	−30.3554	−1.03331	−0.516654	−	0.856194i	$-0.672823\pi$
$863$	−30.3554	−1.03331	−0.516654	+	0.856194i	$0.672823\pi$
$864$	2.61451	0.0889473
$865$	3.58447	0.121876
$866$	−4.73024	−0.160740
$867$	42.0472	1.42800
$868$	−29.7397	−1.00943
$869$	0	0
$870$	−6.95101	−0.235661
$871$	26.8540	0.909913
$872$	−18.7042	−0.633405
$873$	42.7979	1.44849
$874$	−10.9595	−0.370712
$875$	−9.74800	−0.329543
$876$	−1.84040	−0.0621813
$877$	−35.5881	−1.20173	−0.600863	−	0.799352i	$-0.705176\pi$
$877$	−35.5881	−1.20173	−0.600863	+	0.799352i	$0.705176\pi$
$878$	−37.6872	−1.27188
$879$	42.2449	1.42488
$880$	0	0
$881$	−18.9997	−0.640116	−0.320058	−	0.947398i	$-0.603702\pi$
$881$	−18.9997	−0.640116	−0.320058	+	0.947398i	$0.603702\pi$
$882$	40.9703	1.37954
$883$	−31.2798	−1.05265	−0.526325	−	0.850284i	$-0.676430\pi$
$883$	−31.2798	−1.05265	−0.526325	+	0.850284i	$0.676430\pi$
$884$	0.694460	0.0233572
$885$	−3.42780	−0.115224
$886$	18.6778	0.627494
$887$	−10.6811	−0.358637	−0.179319	−	0.983791i	$-0.557389\pi$
$887$	−10.6811	−0.358637	−0.179319	+	0.983791i	$0.557389\pi$
$888$	30.7636	1.03236
$889$	26.0173	0.872591
$890$	3.48776	0.116910
$891$	0	0
$892$	−15.2374	−0.510186
$893$	−8.70515	−0.291307
$894$	−51.2255	−1.71324
$895$	−6.70256	−0.224042
$896$	39.3142	1.31340
$897$	−31.2572	−1.04365
$898$	47.3541	1.58023
$899$	−39.0675	−1.30297
$900$	−18.9555	−0.631850
$901$	−1.30597	−0.0435083
$902$	0	0
$903$	43.5749	1.45008
$904$	−23.7321	−0.789318
$905$	3.87864	0.128931
$906$	63.5611	2.11167
$907$	27.6869	0.919327	0.459664	−	0.888093i	$-0.347970\pi$
$907$	27.6869	0.919327	0.459664	+	0.888093i	$0.347970\pi$
$908$	−23.6092	−0.783500
$909$	−7.88719	−0.261602
$910$	3.61723	0.119910
$911$	−7.03345	−0.233029	−0.116514	−	0.993189i	$-0.537172\pi$
$911$	−7.03345	−0.233029	−0.116514	+	0.993189i	$0.537172\pi$
$912$	−12.3131	−0.407728
$913$	0	0
$914$	24.0280	0.794775
$915$	−7.50744	−0.248188
$916$	10.5493	0.348558
$917$	−38.7029	−1.27808
$918$	−0.215857	−0.00712435
$919$	−23.3604	−0.770587	−0.385294	−	0.922794i	$-0.625900\pi$
$919$	−23.3604	−0.770587	−0.385294	+	0.922794i	$0.625900\pi$
$920$	−2.25213	−0.0742506
$921$	−41.0802	−1.35364
$922$	60.9343	2.00677
$923$	15.4402	0.508220
$924$	0	0
$925$	−43.1867	−1.41997
$926$	−57.5654	−1.89172
$927$	1.42809	0.0469047
$928$	−36.2805	−1.19097
$929$	20.4047	0.669455	0.334727	−	0.942315i	$-0.391356\pi$
$929$	20.4047	0.669455	0.334727	+	0.942315i	$0.391356\pi$
$930$	7.55590	0.247768
$931$	−7.20389	−0.236098
$932$	22.7166	0.744108
$933$	8.94696	0.292910
$934$	3.50860	0.114805
$935$	0	0
$936$	−9.23059	−0.301711
$937$	−18.8142	−0.614634	−0.307317	−	0.951607i	$-0.599431\pi$
$937$	−18.8142	−0.614634	−0.307317	+	0.951607i	$0.599431\pi$
$938$	88.1725	2.87893
$939$	62.2483	2.03140
$940$	2.74505	0.0895335
$941$	48.8440	1.59227	0.796135	−	0.605120i	$-0.206875\pi$
$941$	48.8440	1.59227	0.796135	+	0.605120i	$0.206875\pi$
$942$	71.1313	2.31758
$943$	−21.6479	−0.704951
$944$	−26.2518	−0.854423
$945$	−0.424010	−0.0137930
$946$	0	0
$947$	46.9717	1.52638	0.763188	−	0.646177i	$-0.223633\pi$
$947$	46.9717	1.52638	0.763188	+	0.646177i	$0.223633\pi$
$948$	−13.1035	−0.425581
$949$	1.25811	0.0408400
$950$	8.83797	0.286741
$951$	37.1076	1.20330
$952$	−1.48593	−0.0481594
$953$	−10.9130	−0.353506	−0.176753	−	0.984255i	$-0.556559\pi$
$953$	−10.9130	−0.353506	−0.176753	+	0.984255i	$0.556559\pi$
$954$	−26.6372	−0.862411
$955$	−5.85813	−0.189565
$956$	−7.42679	−0.240200
$957$	0	0
$958$	−38.3062	−1.23762
$959$	−70.4029	−2.27343
$960$	0.603807	0.0194878
$961$	11.4672	0.369911
$962$	32.2713	1.04047
$963$	−3.66089	−0.117971
$964$	−33.3701	−1.07478
$965$	−1.35541	−0.0436321
$966$	−102.630	−3.30207
$967$	−41.7908	−1.34390	−0.671950	−	0.740596i	$-0.734543\pi$
$967$	−41.7908	−1.34390	−0.671950	+	0.740596i	$0.734543\pi$
$968$	0	0
$969$	0.692830	0.0222569
$970$	6.29236	0.202036
$971$	−25.0190	−0.802899	−0.401449	−	0.915881i	$-0.631493\pi$
$971$	−25.0190	−0.802899	−0.401449	+	0.915881i	$0.631493\pi$
$972$	26.9877	0.865632
$973$	3.11526	0.0998706
$974$	−47.1876	−1.51199
$975$	25.2064	0.807250
$976$	−57.4957	−1.84039
$977$	−59.1109	−1.89112	−0.945562	−	0.325441i	$-0.894487\pi$
$977$	−59.1109	−1.89112	−0.945562	+	0.325441i	$0.894487\pi$
$978$	−82.9117	−2.65122
$979$	0	0
$980$	2.27165	0.0725651
$981$	41.9837	1.34044
$982$	−22.2185	−0.709020
$983$	−48.0058	−1.53115	−0.765574	−	0.643348i	$-0.777545\pi$
$983$	−48.0058	−1.53115	−0.765574	+	0.643348i	$0.777545\pi$
$984$	−12.4355	−0.396429
$985$	4.52633	0.144221
$986$	2.99537	0.0953920
$987$	−81.5189	−2.59478
$988$	−2.49057	−0.0792356
$989$	−28.4599	−0.904971
$990$	0	0
$991$	−36.6298	−1.16358	−0.581792	−	0.813338i	$-0.697648\pi$
$991$	−36.6298	−1.16358	−0.581792	+	0.813338i	$0.697648\pi$
$992$	39.4377	1.25215
$993$	9.02978	0.286552
$994$	50.6963	1.60799
$995$	−5.24694	−0.166339
$996$	28.6974	0.909313
$997$	52.9863	1.67809	0.839046	−	0.544060i	$-0.183114\pi$
$997$	52.9863	1.67809	0.839046	+	0.544060i	$0.183114\pi$
$998$	−62.2740	−1.97125
$999$	−3.78282	−0.119683

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2299.2.a.r.1.5	yes	7
11.10	odd	2		2299.2.a.p.1.3	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2299.2.a.p.1.3	✓	7	11.10	odd	2
2299.2.a.r.1.5	yes	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2299 = 11^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2299.a (trivial)

\(f(q)\)	\(=\)	\(q+1.79190 q^{2} -2.48473 q^{3} +1.21090 q^{4} +0.260416 q^{5} -4.45238 q^{6} +3.76880 q^{7} -1.41399 q^{8} +3.17387 q^{9} +0.466638 q^{10} -3.00874 q^{12} +2.05680 q^{13} +6.75331 q^{14} -0.647062 q^{15} -4.95552 q^{16} +0.278836 q^{17} +5.68725 q^{18} -1.00000 q^{19} +0.315336 q^{20} -9.36445 q^{21} +6.11616 q^{23} +3.51339 q^{24} -4.93218 q^{25} +3.68558 q^{26} -0.432022 q^{27} +4.56363 q^{28} +5.99500 q^{29} -1.15947 q^{30} -6.51669 q^{31} -6.05180 q^{32} +0.499645 q^{34} +0.981456 q^{35} +3.84322 q^{36} +8.75610 q^{37} -1.79190 q^{38} -5.11059 q^{39} -0.368226 q^{40} -3.53945 q^{41} -16.7801 q^{42} -4.65322 q^{43} +0.826526 q^{45} +10.9595 q^{46} +8.70515 q^{47} +12.3131 q^{48} +7.20389 q^{49} -8.83797 q^{50} -0.692830 q^{51} +2.49057 q^{52} -4.68367 q^{53} -0.774138 q^{54} -5.32907 q^{56} +2.48473 q^{57} +10.7424 q^{58} +5.29748 q^{59} -0.783524 q^{60} +11.6024 q^{61} -11.6772 q^{62} +11.9617 q^{63} -0.933152 q^{64} +0.535623 q^{65} +13.0562 q^{67} +0.337641 q^{68} -15.1970 q^{69} +1.75867 q^{70} +7.50689 q^{71} -4.48784 q^{72} +0.611683 q^{73} +15.6900 q^{74} +12.2551 q^{75} -1.21090 q^{76} -9.15765 q^{78} +4.35513 q^{79} -1.29050 q^{80} -8.44816 q^{81} -6.34233 q^{82} -9.53801 q^{83} -11.3394 q^{84} +0.0726132 q^{85} -8.33810 q^{86} -14.8959 q^{87} +7.47422 q^{89} +1.48105 q^{90} +7.75168 q^{91} +7.40603 q^{92} +16.1922 q^{93} +15.5987 q^{94} -0.260416 q^{95} +15.0371 q^{96} +13.4845 q^{97} +12.9086 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 2 q^{2} - q^{3} + 6 q^{4} - q^{5} + 2 q^{7} + 6 q^{8} + 4 q^{9} - 3 q^{10} - 9 q^{12} + 12 q^{13} + 3 q^{14} + 5 q^{15} + 12 q^{17} - 3 q^{18} - 7 q^{19} - 17 q^{20} + 9 q^{21} - 9 q^{23} + 27 q^{24}+ \cdots + 9 q^{98}+O(q^{100}) \) 7 * q + 2 * q^2 - q^3 + 6 * q^4 - q^5 + 2 * q^7 + 6 * q^8 + 4 * q^9 - 3 * q^10 - 9 * q^12 + 12 * q^13 + 3 * q^14 + 5 * q^15 + 12 * q^17 - 3 * q^18 - 7 * q^19 - 17 * q^20 + 9 * q^21 - 9 * q^23 + 27 * q^24 + 4 * q^25 + 18 * q^26 + 23 * q^27 - 10 * q^28 + 18 * q^29 + 4 * q^30 - 5 * q^31 - 16 * q^32 - 6 * q^34 + 20 * q^35 - 2 * q^36 + 4 * q^37 - 2 * q^38 + 21 * q^39 + 17 * q^40 + 25 * q^41 - 19 * q^42 - 10 * q^43 - 16 * q^45 + 29 * q^46 - 14 * q^47 + 30 * q^48 + 25 * q^49 + 13 * q^50 + q^51 - 3 * q^52 - 15 * q^53 - 28 * q^54 - 3 * q^56 + q^57 + 6 * q^58 - 9 * q^59 + 29 * q^60 + 3 * q^61 + 41 * q^62 + 52 * q^63 - 20 * q^64 - 11 * q^65 - 15 * q^67 + 9 * q^68 - 11 * q^69 + 42 * q^70 + 13 * q^71 + 2 * q^72 - 28 * q^73 + 22 * q^74 - 22 * q^75 - 6 * q^76 - 38 * q^78 + 18 * q^79 + 32 * q^80 + 23 * q^81 + 8 * q^82 - 10 * q^83 - 19 * q^84 + 14 * q^85 - 30 * q^86 - 22 * q^87 + 2 * q^89 + 30 * q^90 - 19 * q^91 + 4 * q^92 + 17 * q^93 + 9 * q^94 + q^95 - q^96 + 29 * q^97 + 9 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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