LMFDB - Embedded newform 2299.2.a.g.1.1

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 = [2,0,2,6,-1,-10] 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:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 209)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ 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)

$f(q)$	$=$	\(q-2.23607 q^{2} +3.23607 q^{3} +3.00000 q^{4} -1.61803 q^{5} -7.23607 q^{6} -0.381966 q^{7} -2.23607 q^{8} +7.47214 q^{9} +3.61803 q^{10} +9.70820 q^{12} -2.47214 q^{13} +0.854102 q^{14} -5.23607 q^{15} -1.00000 q^{16} -3.09017 q^{17} -16.7082 q^{18} +1.00000 q^{19} -4.85410 q^{20} -1.23607 q^{21} -0.618034 q^{23} -7.23607 q^{24} -2.38197 q^{25} +5.52786 q^{26} +14.4721 q^{27} -1.14590 q^{28} +8.47214 q^{29} +11.7082 q^{30} +8.00000 q^{31} +6.70820 q^{32} +6.90983 q^{34} +0.618034 q^{35} +22.4164 q^{36} -2.00000 q^{37} -2.23607 q^{38} -8.00000 q^{39} +3.61803 q^{40} +11.7082 q^{41} +2.76393 q^{42} -3.09017 q^{43} -12.0902 q^{45} +1.38197 q^{46} +8.61803 q^{47} -3.23607 q^{48} -6.85410 q^{49} +5.32624 q^{50} -10.0000 q^{51} -7.41641 q^{52} +1.52786 q^{53} -32.3607 q^{54} +0.854102 q^{56} +3.23607 q^{57} -18.9443 q^{58} +12.4721 q^{59} -15.7082 q^{60} +8.61803 q^{61} -17.8885 q^{62} -2.85410 q^{63} -13.0000 q^{64} +4.00000 q^{65} -8.47214 q^{67} -9.27051 q^{68} -2.00000 q^{69} -1.38197 q^{70} +11.7082 q^{71} -16.7082 q^{72} +3.52786 q^{73} +4.47214 q^{74} -7.70820 q^{75} +3.00000 q^{76} +17.8885 q^{78} +2.94427 q^{79} +1.61803 q^{80} +24.4164 q^{81} -26.1803 q^{82} +5.32624 q^{83} -3.70820 q^{84} +5.00000 q^{85} +6.90983 q^{86} +27.4164 q^{87} -3.23607 q^{89} +27.0344 q^{90} +0.944272 q^{91} -1.85410 q^{92} +25.8885 q^{93} -19.2705 q^{94} -1.61803 q^{95} +21.7082 q^{96} -12.1803 q^{97} +15.3262 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 6 q^{4} - q^{5} - 10 q^{6} - 3 q^{7} + 6 q^{9} + 5 q^{10} + 6 q^{12} + 4 q^{13} - 5 q^{14} - 6 q^{15} - 2 q^{16} + 5 q^{17} - 20 q^{18} + 2 q^{19} - 3 q^{20} + 2 q^{21} + q^{23} - 10 q^{24}+ \cdots + 15 q^{98}+O(q^{100}) $ 2 * q + 2 * q^3 + 6 * q^4 - q^5 - 10 * q^6 - 3 * q^7 + 6 * q^9 + 5 * q^10 + 6 * q^12 + 4 * q^13 - 5 * q^14 - 6 * q^15 - 2 * q^16 + 5 * q^17 - 20 * q^18 + 2 * q^19 - 3 * q^20 + 2 * q^21 + q^23 - 10 * q^24 - 7 * q^25 + 20 * q^26 + 20 * q^27 - 9 * q^28 + 8 * q^29 + 10 * q^30 + 16 * q^31 + 25 * q^34 - q^35 + 18 * q^36 - 4 * q^37 - 16 * q^39 + 5 * q^40 + 10 * q^41 + 10 * q^42 + 5 * q^43 - 13 * q^45 + 5 * q^46 + 15 * q^47 - 2 * q^48 - 7 * q^49 - 5 * q^50 - 20 * q^51 + 12 * q^52 + 12 * q^53 - 20 * q^54 - 5 * q^56 + 2 * q^57 - 20 * q^58 + 16 * q^59 - 18 * q^60 + 15 * q^61 + q^63 - 26 * q^64 + 8 * q^65 - 8 * q^67 + 15 * q^68 - 4 * q^69 - 5 * q^70 + 10 * q^71 - 20 * q^72 + 16 * q^73 - 2 * q^75 + 6 * q^76 - 12 * q^79 + q^80 + 22 * q^81 - 30 * q^82 - 5 * q^83 + 6 * q^84 + 10 * q^85 + 25 * q^86 + 28 * q^87 - 2 * q^89 + 25 * q^90 - 16 * q^91 + 3 * q^92 + 16 * q^93 - 5 * q^94 - q^95 + 30 * q^96 - 2 * q^97 + 15 * 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 $

$2$	−2.23607	−1.58114	−0.790569	−	0.612372i	$-0.790215\pi$
$2$	−2.23607	−1.58114	−0.790569	+	0.612372i	$0.790215\pi$
$3$	3.23607	1.86834	0.934172	−	0.356822i	$-0.116140\pi$
$3$	3.23607	1.86834	0.934172	+	0.356822i	$0.116140\pi$
$4$	3.00000	1.50000
$5$	−1.61803	−0.723607	−0.361803	−	0.932254i	$-0.617839\pi$
$5$	−1.61803	−0.723607	−0.361803	+	0.932254i	$0.617839\pi$
$6$	−7.23607	−2.95411
$7$	−0.381966	−0.144370	−0.0721848	−	0.997391i	$-0.522997\pi$
$7$	−0.381966	−0.144370	−0.0721848	+	0.997391i	$0.522997\pi$
$8$	−2.23607	−0.790569
$9$	7.47214	2.49071
$10$	3.61803	1.14412
$11$	0	0
$11$	0	0
$12$	9.70820	2.80252
$13$	−2.47214	−0.685647	−0.342824	−	0.939400i	$-0.611383\pi$
$13$	−2.47214	−0.685647	−0.342824	+	0.939400i	$0.611383\pi$
$14$	0.854102	0.228268
$15$	−5.23607	−1.35195
$16$	−1.00000	−0.250000
$17$	−3.09017	−0.749476	−0.374738	−	0.927131i	$-0.622267\pi$
$17$	−3.09017	−0.749476	−0.374738	+	0.927131i	$0.622267\pi$
$18$	−16.7082	−3.93816
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−4.85410	−1.08541
$21$	−1.23607	−0.269732
$22$	0	0
$23$	−0.618034	−0.128869	−0.0644345	−	0.997922i	$-0.520524\pi$
$23$	−0.618034	−0.128869	−0.0644345	+	0.997922i	$0.520524\pi$
$24$	−7.23607	−1.47706
$25$	−2.38197	−0.476393
$26$	5.52786	1.08410
$27$	14.4721	2.78516
$28$	−1.14590	−0.216554
$29$	8.47214	1.57324	0.786618	−	0.617440i	$-0.211830\pi$
$29$	8.47214	1.57324	0.786618	+	0.617440i	$0.211830\pi$
$30$	11.7082	2.13762
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	6.70820	1.18585
$33$	0	0
$34$	6.90983	1.18503
$35$	0.618034	0.104467
$36$	22.4164	3.73607
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	−2.23607	−0.362738
$39$	−8.00000	−1.28103
$40$	3.61803	0.572061
$41$	11.7082	1.82851	0.914257	−	0.405134i	$-0.132775\pi$
$41$	11.7082	1.82851	0.914257	+	0.405134i	$0.132775\pi$
$42$	2.76393	0.426484
$43$	−3.09017	−0.471246	−0.235623	−	0.971844i	$-0.575713\pi$
$43$	−3.09017	−0.471246	−0.235623	+	0.971844i	$0.575713\pi$
$44$	0	0
$45$	−12.0902	−1.80230
$46$	1.38197	0.203760
$47$	8.61803	1.25707	0.628535	−	0.777782i	$-0.283655\pi$
$47$	8.61803	1.25707	0.628535	+	0.777782i	$0.283655\pi$
$48$	−3.23607	−0.467086
$49$	−6.85410	−0.979157
$50$	5.32624	0.753244
$51$	−10.0000	−1.40028
$52$	−7.41641	−1.02847
$53$	1.52786	0.209868	0.104934	−	0.994479i	$-0.466537\pi$
$53$	1.52786	0.209868	0.104934	+	0.994479i	$0.466537\pi$
$54$	−32.3607	−4.40373
$55$	0	0
$56$	0.854102	0.114134
$57$	3.23607	0.428628
$58$	−18.9443	−2.48750
$59$	12.4721	1.62373	0.811867	−	0.583843i	$-0.198451\pi$
$59$	12.4721	1.62373	0.811867	+	0.583843i	$0.198451\pi$
$60$	−15.7082	−2.02792
$61$	8.61803	1.10343	0.551713	−	0.834034i	$-0.313974\pi$
$61$	8.61803	1.10343	0.551713	+	0.834034i	$0.313974\pi$
$62$	−17.8885	−2.27185
$63$	−2.85410	−0.359583
$64$	−13.0000	−1.62500
$65$	4.00000	0.496139
$66$	0	0
$67$	−8.47214	−1.03504	−0.517518	−	0.855672i	$-0.673144\pi$
$67$	−8.47214	−1.03504	−0.517518	+	0.855672i	$0.673144\pi$
$68$	−9.27051	−1.12421
$69$	−2.00000	−0.240772
$70$	−1.38197	−0.165177
$71$	11.7082	1.38951	0.694754	−	0.719247i	$-0.255513\pi$
$71$	11.7082	1.38951	0.694754	+	0.719247i	$0.255513\pi$
$72$	−16.7082	−1.96908
$73$	3.52786	0.412905	0.206453	−	0.978457i	$-0.433808\pi$
$73$	3.52786	0.412905	0.206453	+	0.978457i	$0.433808\pi$
$74$	4.47214	0.519875
$75$	−7.70820	−0.890067
$76$	3.00000	0.344124
$77$	0	0
$78$	17.8885	2.02548
$79$	2.94427	0.331256	0.165628	−	0.986188i	$-0.447035\pi$
$79$	2.94427	0.331256	0.165628	+	0.986188i	$0.447035\pi$
$80$	1.61803	0.180902
$81$	24.4164	2.71293
$82$	−26.1803	−2.89113
$83$	5.32624	0.584631	0.292315	−	0.956322i	$-0.405574\pi$
$83$	5.32624	0.584631	0.292315	+	0.956322i	$0.405574\pi$
$84$	−3.70820	−0.404598
$85$	5.00000	0.542326
$86$	6.90983	0.745106
$87$	27.4164	2.93935
$88$	0	0
$89$	−3.23607	−0.343023	−0.171511	−	0.985182i	$-0.554865\pi$
$89$	−3.23607	−0.343023	−0.171511	+	0.985182i	$0.554865\pi$
$90$	27.0344	2.84968
$91$	0.944272	0.0989866
$92$	−1.85410	−0.193303
$93$	25.8885	2.68452
$94$	−19.2705	−1.98760
$95$	−1.61803	−0.166007
$96$	21.7082	2.21558
$97$	−12.1803	−1.23673	−0.618363	−	0.785893i	$-0.712204\pi$
$97$	−12.1803	−1.23673	−0.618363	+	0.785893i	$0.712204\pi$
$98$	15.3262	1.54818
$99$	0	0
$100$	−7.14590	−0.714590
$101$	−6.38197	−0.635029	−0.317515	−	0.948253i	$-0.602848\pi$
$101$	−6.38197	−0.635029	−0.317515	+	0.948253i	$0.602848\pi$
$102$	22.3607	2.21404
$103$	15.7082	1.54778	0.773888	−	0.633323i	$-0.218309\pi$
$103$	15.7082	1.54778	0.773888	+	0.633323i	$0.218309\pi$
$104$	5.52786	0.542052
$105$	2.00000	0.195180
$106$	−3.41641	−0.331831
$107$	−5.23607	−0.506190	−0.253095	−	0.967441i	$-0.581448\pi$
$107$	−5.23607	−0.506190	−0.253095	+	0.967441i	$0.581448\pi$
$108$	43.4164	4.17775
$109$	10.9443	1.04827	0.524136	−	0.851635i	$-0.324389\pi$
$109$	10.9443	1.04827	0.524136	+	0.851635i	$0.324389\pi$
$110$	0	0
$111$	−6.47214	−0.614308
$112$	0.381966	0.0360924
$113$	−11.2361	−1.05700	−0.528500	−	0.848933i	$-0.677245\pi$
$113$	−11.2361	−1.05700	−0.528500	+	0.848933i	$0.677245\pi$
$114$	−7.23607	−0.677720
$115$	1.00000	0.0932505
$116$	25.4164	2.35985
$117$	−18.4721	−1.70775
$118$	−27.8885	−2.56735
$119$	1.18034	0.108202
$120$	11.7082	1.06881
$121$	0	0
$122$	−19.2705	−1.74467
$123$	37.8885	3.41629
$124$	24.0000	2.15526
$125$	11.9443	1.06833
$126$	6.38197	0.568551
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	15.6525	1.38350
$129$	−10.0000	−0.880451
$130$	−8.94427	−0.784465
$131$	11.0344	0.964084	0.482042	−	0.876148i	$-0.339895\pi$
$131$	11.0344	0.964084	0.482042	+	0.876148i	$0.339895\pi$
$132$	0	0
$133$	−0.381966	−0.0331207
$134$	18.9443	1.63654
$135$	−23.4164	−2.01536
$136$	6.90983	0.592513
$137$	−4.85410	−0.414714	−0.207357	−	0.978265i	$-0.566486\pi$
$137$	−4.85410	−0.414714	−0.207357	+	0.978265i	$0.566486\pi$
$138$	4.47214	0.380693
$139$	−8.90983	−0.755722	−0.377861	−	0.925862i	$-0.623340\pi$
$139$	−8.90983	−0.755722	−0.377861	+	0.925862i	$0.623340\pi$
$140$	1.85410	0.156700
$141$	27.8885	2.34864
$142$	−26.1803	−2.19701
$143$	0	0
$144$	−7.47214	−0.622678
$145$	−13.7082	−1.13840
$146$	−7.88854	−0.652861
$147$	−22.1803	−1.82940
$148$	−6.00000	−0.493197
$149$	−9.41641	−0.771422	−0.385711	−	0.922620i	$-0.626044\pi$
$149$	−9.41641	−0.771422	−0.385711	+	0.922620i	$0.626044\pi$
$150$	17.2361	1.40732
$151$	7.23607	0.588863	0.294431	−	0.955673i	$-0.404870\pi$
$151$	7.23607	0.588863	0.294431	+	0.955673i	$0.404870\pi$
$152$	−2.23607	−0.181369
$153$	−23.0902	−1.86673
$154$	0	0
$155$	−12.9443	−1.03971
$156$	−24.0000	−1.92154
$157$	14.5623	1.16220	0.581099	−	0.813833i	$-0.302623\pi$
$157$	14.5623	1.16220	0.581099	+	0.813833i	$0.302623\pi$
$158$	−6.58359	−0.523762
$159$	4.94427	0.392106
$160$	−10.8541	−0.858092
$161$	0.236068	0.0186048
$162$	−54.5967	−4.28953
$163$	−9.32624	−0.730487	−0.365244	−	0.930912i	$-0.619014\pi$
$163$	−9.32624	−0.730487	−0.365244	+	0.930912i	$0.619014\pi$
$164$	35.1246	2.74277
$165$	0	0
$166$	−11.9098	−0.924382
$167$	−24.1803	−1.87113	−0.935565	−	0.353153i	$-0.885109\pi$
$167$	−24.1803	−1.87113	−0.935565	+	0.353153i	$0.885109\pi$
$168$	2.76393	0.213242
$169$	−6.88854	−0.529888
$170$	−11.1803	−0.857493
$171$	7.47214	0.571409
$172$	−9.27051	−0.706870
$173$	9.52786	0.724390	0.362195	−	0.932102i	$-0.382027\pi$
$173$	9.52786	0.724390	0.362195	+	0.932102i	$0.382027\pi$
$174$	−61.3050	−4.64752
$175$	0.909830	0.0687767
$176$	0	0
$177$	40.3607	3.03369
$178$	7.23607	0.542366
$179$	−6.18034	−0.461940	−0.230970	−	0.972961i	$-0.574190\pi$
$179$	−6.18034	−0.461940	−0.230970	+	0.972961i	$0.574190\pi$
$180$	−36.2705	−2.70344
$181$	−6.47214	−0.481070	−0.240535	−	0.970640i	$-0.577323\pi$
$181$	−6.47214	−0.481070	−0.240535	+	0.970640i	$0.577323\pi$
$182$	−2.11146	−0.156512
$183$	27.8885	2.06158
$184$	1.38197	0.101880
$185$	3.23607	0.237920
$186$	−57.8885	−4.24459
$187$	0	0
$188$	25.8541	1.88560
$189$	−5.52786	−0.402093
$190$	3.61803	0.262480
$191$	−1.90983	−0.138190	−0.0690952	−	0.997610i	$-0.522011\pi$
$191$	−1.90983	−0.138190	−0.0690952	+	0.997610i	$0.522011\pi$
$192$	−42.0689	−3.03606
$193$	8.76393	0.630842	0.315421	−	0.948952i	$-0.397854\pi$
$193$	8.76393	0.630842	0.315421	+	0.948952i	$0.397854\pi$
$194$	27.2361	1.95544
$195$	12.9443	0.926959
$196$	−20.5623	−1.46874
$197$	−3.52786	−0.251350	−0.125675	−	0.992071i	$-0.540110\pi$
$197$	−3.52786	−0.251350	−0.125675	+	0.992071i	$0.540110\pi$
$198$	0	0
$199$	−3.43769	−0.243692	−0.121846	−	0.992549i	$-0.538881\pi$
$199$	−3.43769	−0.243692	−0.121846	+	0.992549i	$0.538881\pi$
$200$	5.32624	0.376622
$201$	−27.4164	−1.93380
$202$	14.2705	1.00407
$203$	−3.23607	−0.227127
$204$	−30.0000	−2.10042
$205$	−18.9443	−1.32313
$206$	−35.1246	−2.44725
$207$	−4.61803	−0.320976
$208$	2.47214	0.171412
$209$	0	0
$210$	−4.47214	−0.308607
$211$	−4.76393	−0.327963	−0.163981	−	0.986463i	$-0.552434\pi$
$211$	−4.76393	−0.327963	−0.163981	+	0.986463i	$0.552434\pi$
$212$	4.58359	0.314802
$213$	37.8885	2.59608
$214$	11.7082	0.800356
$215$	5.00000	0.340997
$216$	−32.3607	−2.20187
$217$	−3.05573	−0.207436
$218$	−24.4721	−1.65746
$219$	11.4164	0.771449
$220$	0	0
$221$	7.63932	0.513876
$222$	14.4721	0.971306
$223$	8.94427	0.598953	0.299476	−	0.954104i	$-0.403188\pi$
$223$	8.94427	0.598953	0.299476	+	0.954104i	$0.403188\pi$
$224$	−2.56231	−0.171201
$225$	−17.7984	−1.18656
$226$	25.1246	1.67126
$227$	23.1246	1.53483	0.767417	−	0.641148i	$-0.221542\pi$
$227$	23.1246	1.53483	0.767417	+	0.641148i	$0.221542\pi$
$228$	9.70820	0.642942
$229$	−11.0902	−0.732859	−0.366430	−	0.930446i	$-0.619420\pi$
$229$	−11.0902	−0.732859	−0.366430	+	0.930446i	$0.619420\pi$
$230$	−2.23607	−0.147442
$231$	0	0
$232$	−18.9443	−1.24375
$233$	−5.61803	−0.368050	−0.184025	−	0.982922i	$-0.558913\pi$
$233$	−5.61803	−0.368050	−0.184025	+	0.982922i	$0.558913\pi$
$234$	41.3050	2.70019
$235$	−13.9443	−0.909624
$236$	37.4164	2.43560
$237$	9.52786	0.618901
$238$	−2.63932	−0.171082
$239$	−19.3820	−1.25372	−0.626858	−	0.779134i	$-0.715659\pi$
$239$	−19.3820	−1.25372	−0.626858	+	0.779134i	$0.715659\pi$
$240$	5.23607	0.337987
$241$	24.4721	1.57639	0.788194	−	0.615426i	$-0.211016\pi$
$241$	24.4721	1.57639	0.788194	+	0.615426i	$0.211016\pi$
$242$	0	0
$243$	35.5967	2.28353
$244$	25.8541	1.65514
$245$	11.0902	0.708525
$246$	−84.7214	−5.40164
$247$	−2.47214	−0.157298
$248$	−17.8885	−1.13592
$249$	17.2361	1.09229
$250$	−26.7082	−1.68918
$251$	19.0902	1.20496	0.602480	−	0.798134i	$-0.294179\pi$
$251$	19.0902	1.20496	0.602480	+	0.798134i	$0.294179\pi$
$252$	−8.56231	−0.539375
$253$	0	0
$254$	−4.47214	−0.280607
$255$	16.1803	1.01325
$256$	−9.00000	−0.562500
$257$	−4.94427	−0.308415	−0.154208	−	0.988038i	$-0.549282\pi$
$257$	−4.94427	−0.308415	−0.154208	+	0.988038i	$0.549282\pi$
$258$	22.3607	1.39212
$259$	0.763932	0.0474684
$260$	12.0000	0.744208
$261$	63.3050	3.91848
$262$	−24.6738	−1.52435
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	−2.47214	−0.151862
$266$	0.854102	0.0523684
$267$	−10.4721	−0.640884
$268$	−25.4164	−1.55255
$269$	1.05573	0.0643689	0.0321844	−	0.999482i	$-0.489754\pi$
$269$	1.05573	0.0643689	0.0321844	+	0.999482i	$0.489754\pi$
$270$	52.3607	3.18657
$271$	16.0344	0.974023	0.487011	−	0.873396i	$-0.338087\pi$
$271$	16.0344	0.974023	0.487011	+	0.873396i	$0.338087\pi$
$272$	3.09017	0.187369
$273$	3.05573	0.184941
$274$	10.8541	0.655720
$275$	0	0
$276$	−6.00000	−0.361158
$277$	6.00000	0.360505	0.180253	−	0.983620i	$-0.442309\pi$
$277$	6.00000	0.360505	0.180253	+	0.983620i	$0.442309\pi$
$278$	19.9230	1.19490
$279$	59.7771	3.57876
$280$	−1.38197	−0.0825883
$281$	4.47214	0.266785	0.133393	−	0.991063i	$-0.457413\pi$
$281$	4.47214	0.266785	0.133393	+	0.991063i	$0.457413\pi$
$282$	−62.3607	−3.71352
$283$	−19.6180	−1.16617	−0.583086	−	0.812411i	$-0.698155\pi$
$283$	−19.6180	−1.16617	−0.583086	+	0.812411i	$0.698155\pi$
$284$	35.1246	2.08426
$285$	−5.23607	−0.310158
$286$	0	0
$287$	−4.47214	−0.263982
$288$	50.1246	2.95362
$289$	−7.45085	−0.438285
$290$	30.6525	1.79998
$291$	−39.4164	−2.31063
$292$	10.5836	0.619358
$293$	−16.4721	−0.962312	−0.481156	−	0.876635i	$-0.659783\pi$
$293$	−16.4721	−0.962312	−0.481156	+	0.876635i	$0.659783\pi$
$294$	49.5967	2.89254
$295$	−20.1803	−1.17494
$296$	4.47214	0.259938
$297$	0	0
$298$	21.0557	1.21973
$299$	1.52786	0.0883587
$300$	−23.1246	−1.33510
$301$	1.18034	0.0680337
$302$	−16.1803	−0.931074
$303$	−20.6525	−1.18645
$304$	−1.00000	−0.0573539
$305$	−13.9443	−0.798447
$306$	51.6312	2.95156
$307$	−25.8885	−1.47754	−0.738769	−	0.673959i	$-0.764592\pi$
$307$	−25.8885	−1.47754	−0.738769	+	0.673959i	$0.764592\pi$
$308$	0	0
$309$	50.8328	2.89178
$310$	28.9443	1.64392
$311$	−1.56231	−0.0885902	−0.0442951	−	0.999018i	$-0.514104\pi$
$311$	−1.56231	−0.0885902	−0.0442951	+	0.999018i	$0.514104\pi$
$312$	17.8885	1.01274
$313$	−7.38197	−0.417253	−0.208627	−	0.977995i	$-0.566899\pi$
$313$	−7.38197	−0.417253	−0.208627	+	0.977995i	$0.566899\pi$
$314$	−32.5623	−1.83760
$315$	4.61803	0.260197
$316$	8.83282	0.496885
$317$	29.5967	1.66232	0.831159	−	0.556034i	$-0.187678\pi$
$317$	29.5967	1.66232	0.831159	+	0.556034i	$0.187678\pi$
$318$	−11.0557	−0.619974
$319$	0	0
$320$	21.0344	1.17586
$321$	−16.9443	−0.945737
$322$	−0.527864	−0.0294167
$323$	−3.09017	−0.171942
$324$	73.2492	4.06940
$325$	5.88854	0.326638
$326$	20.8541	1.15500
$327$	35.4164	1.95853
$328$	−26.1803	−1.44557
$329$	−3.29180	−0.181483
$330$	0	0
$331$	−10.2918	−0.565688	−0.282844	−	0.959166i	$-0.591278\pi$
$331$	−10.2918	−0.565688	−0.282844	+	0.959166i	$0.591278\pi$
$332$	15.9787	0.876946
$333$	−14.9443	−0.818941
$334$	54.0689	2.95852
$335$	13.7082	0.748959
$336$	1.23607	0.0674330
$337$	−7.41641	−0.403997	−0.201999	−	0.979386i	$-0.564744\pi$
$337$	−7.41641	−0.403997	−0.201999	+	0.979386i	$0.564744\pi$
$338$	15.4033	0.837826
$339$	−36.3607	−1.97484
$340$	15.0000	0.813489
$341$	0	0
$342$	−16.7082	−0.903476
$343$	5.29180	0.285730
$344$	6.90983	0.372553
$345$	3.23607	0.174224
$346$	−21.3050	−1.14536
$347$	−2.14590	−0.115198	−0.0575989	−	0.998340i	$-0.518344\pi$
$347$	−2.14590	−0.115198	−0.0575989	+	0.998340i	$0.518344\pi$
$348$	82.2492	4.40902
$349$	−8.85410	−0.473949	−0.236975	−	0.971516i	$-0.576156\pi$
$349$	−8.85410	−0.473949	−0.236975	+	0.971516i	$0.576156\pi$
$350$	−2.03444	−0.108745
$351$	−35.7771	−1.90964
$352$	0	0
$353$	12.0902	0.643495	0.321747	−	0.946826i	$-0.395730\pi$
$353$	12.0902	0.643495	0.321747	+	0.946826i	$0.395730\pi$
$354$	−90.2492	−4.79669
$355$	−18.9443	−1.00546
$356$	−9.70820	−0.514534
$357$	3.81966	0.202158
$358$	13.8197	0.730392
$359$	26.7984	1.41436	0.707182	−	0.707032i	$-0.249966\pi$
$359$	26.7984	1.41436	0.707182	+	0.707032i	$0.249966\pi$
$360$	27.0344	1.42484
$361$	1.00000	0.0526316
$362$	14.4721	0.760639
$363$	0	0
$364$	2.83282	0.148480
$365$	−5.70820	−0.298781
$366$	−62.3607	−3.25964
$367$	2.67376	0.139569	0.0697846	−	0.997562i	$-0.477769\pi$
$367$	2.67376	0.139569	0.0697846	+	0.997562i	$0.477769\pi$
$368$	0.618034	0.0322172
$369$	87.4853	4.55430
$370$	−7.23607	−0.376185
$371$	−0.583592	−0.0302986
$372$	77.6656	4.02678
$373$	−18.7639	−0.971560	−0.485780	−	0.874081i	$-0.661464\pi$
$373$	−18.7639	−0.971560	−0.485780	+	0.874081i	$0.661464\pi$
$374$	0	0
$375$	38.6525	1.99601
$376$	−19.2705	−0.993801
$377$	−20.9443	−1.07868
$378$	12.3607	0.635765
$379$	20.6525	1.06085	0.530423	−	0.847733i	$-0.322033\pi$
$379$	20.6525	1.06085	0.530423	+	0.847733i	$0.322033\pi$
$380$	−4.85410	−0.249010
$381$	6.47214	0.331578
$382$	4.27051	0.218498
$383$	33.1246	1.69259	0.846294	−	0.532716i	$-0.178828\pi$
$383$	33.1246	1.69259	0.846294	+	0.532716i	$0.178828\pi$
$384$	50.6525	2.58485
$385$	0	0
$386$	−19.5967	−0.997448
$387$	−23.0902	−1.17374
$388$	−36.5410	−1.85509
$389$	−22.7984	−1.15592	−0.577962	−	0.816064i	$-0.696152\pi$
$389$	−22.7984	−1.15592	−0.577962	+	0.816064i	$0.696152\pi$
$390$	−28.9443	−1.46565
$391$	1.90983	0.0965843
$392$	15.3262	0.774092
$393$	35.7082	1.80124
$394$	7.88854	0.397419
$395$	−4.76393	−0.239699
$396$	0	0
$397$	−13.5623	−0.680673	−0.340336	−	0.940304i	$-0.610541\pi$
$397$	−13.5623	−0.680673	−0.340336	+	0.940304i	$0.610541\pi$
$398$	7.68692	0.385310
$399$	−1.23607	−0.0618808
$400$	2.38197	0.119098
$401$	−1.70820	−0.0853036	−0.0426518	−	0.999090i	$-0.513581\pi$
$401$	−1.70820	−0.0853036	−0.0426518	+	0.999090i	$0.513581\pi$
$402$	61.3050	3.05761
$403$	−19.7771	−0.985167
$404$	−19.1459	−0.952544
$405$	−39.5066	−1.96310
$406$	7.23607	0.359120
$407$	0	0
$408$	22.3607	1.10702
$409$	−23.1246	−1.14344	−0.571719	−	0.820449i	$-0.693723\pi$
$409$	−23.1246	−1.14344	−0.571719	+	0.820449i	$0.693723\pi$
$410$	42.3607	2.09204
$411$	−15.7082	−0.774829
$412$	47.1246	2.32166
$413$	−4.76393	−0.234418
$414$	10.3262	0.507507
$415$	−8.61803	−0.423043
$416$	−16.5836	−0.813077
$417$	−28.8328	−1.41195
$418$	0	0
$419$	−13.8541	−0.676817	−0.338409	−	0.940999i	$-0.609889\pi$
$419$	−13.8541	−0.676817	−0.338409	+	0.940999i	$0.609889\pi$
$420$	6.00000	0.292770
$421$	8.76393	0.427128	0.213564	−	0.976929i	$-0.431493\pi$
$421$	8.76393	0.427128	0.213564	+	0.976929i	$0.431493\pi$
$422$	10.6525	0.518554
$423$	64.3951	3.13100
$424$	−3.41641	−0.165915
$425$	7.36068	0.357045
$426$	−84.7214	−4.10476
$427$	−3.29180	−0.159301
$428$	−15.7082	−0.759285
$429$	0	0
$430$	−11.1803	−0.539164
$431$	−34.9443	−1.68321	−0.841603	−	0.540096i	$-0.818388\pi$
$431$	−34.9443	−1.68321	−0.841603	+	0.540096i	$0.818388\pi$
$432$	−14.4721	−0.696291
$433$	−13.1246	−0.630729	−0.315364	−	0.948971i	$-0.602127\pi$
$433$	−13.1246	−0.630729	−0.315364	+	0.948971i	$0.602127\pi$
$434$	6.83282	0.327986
$435$	−44.3607	−2.12693
$436$	32.8328	1.57241
$437$	−0.618034	−0.0295646
$438$	−25.5279	−1.21977
$439$	−13.2361	−0.631723	−0.315862	−	0.948805i	$-0.602294\pi$
$439$	−13.2361	−0.631723	−0.315862	+	0.948805i	$0.602294\pi$
$440$	0	0
$441$	−51.2148	−2.43880
$442$	−17.0820	−0.812510
$443$	27.7984	1.32074	0.660370	−	0.750940i	$-0.270399\pi$
$443$	27.7984	1.32074	0.660370	+	0.750940i	$0.270399\pi$
$444$	−19.4164	−0.921462
$445$	5.23607	0.248213
$446$	−20.0000	−0.947027
$447$	−30.4721	−1.44128
$448$	4.96556	0.234601
$449$	32.4721	1.53245	0.766227	−	0.642569i	$-0.222132\pi$
$449$	32.4721	1.53245	0.766227	+	0.642569i	$0.222132\pi$
$450$	39.7984	1.87611
$451$	0	0
$452$	−33.7082	−1.58550
$453$	23.4164	1.10020
$454$	−51.7082	−2.42679
$455$	−1.52786	−0.0716274
$456$	−7.23607	−0.338860
$457$	−33.6180	−1.57259	−0.786293	−	0.617854i	$-0.788002\pi$
$457$	−33.6180	−1.57259	−0.786293	+	0.617854i	$0.788002\pi$
$458$	24.7984	1.15875
$459$	−44.7214	−2.08741
$460$	3.00000	0.139876
$461$	3.50658	0.163318	0.0816588	−	0.996660i	$-0.473978\pi$
$461$	3.50658	0.163318	0.0816588	+	0.996660i	$0.473978\pi$
$462$	0	0
$463$	7.61803	0.354040	0.177020	−	0.984207i	$-0.443354\pi$
$463$	7.61803	0.354040	0.177020	+	0.984207i	$0.443354\pi$
$464$	−8.47214	−0.393309
$465$	−41.8885	−1.94253
$466$	12.5623	0.581938
$467$	39.0344	1.80630	0.903149	−	0.429327i	$-0.141249\pi$
$467$	39.0344	1.80630	0.903149	+	0.429327i	$0.141249\pi$
$468$	−55.4164	−2.56162
$469$	3.23607	0.149428
$470$	31.1803	1.43824
$471$	47.1246	2.17139
$472$	−27.8885	−1.28367
$473$	0	0
$474$	−21.3050	−0.978569
$475$	−2.38197	−0.109292
$476$	3.54102	0.162302
$477$	11.4164	0.522721
$478$	43.3394	1.98230
$479$	9.85410	0.450245	0.225123	−	0.974330i	$-0.427722\pi$
$479$	9.85410	0.450245	0.225123	+	0.974330i	$0.427722\pi$
$480$	−35.1246	−1.60321
$481$	4.94427	0.225439
$482$	−54.7214	−2.49249
$483$	0.763932	0.0347601
$484$	0	0
$485$	19.7082	0.894903
$486$	−79.5967	−3.61058
$487$	39.3050	1.78108	0.890539	−	0.454908i	$-0.150328\pi$
$487$	39.3050	1.78108	0.890539	+	0.454908i	$0.150328\pi$
$488$	−19.2705	−0.872335
$489$	−30.1803	−1.36480
$490$	−24.7984	−1.12028
$491$	−9.14590	−0.412749	−0.206374	−	0.978473i	$-0.566166\pi$
$491$	−9.14590	−0.412749	−0.206374	+	0.978473i	$0.566166\pi$
$492$	113.666	5.12444
$493$	−26.1803	−1.17910
$494$	5.52786	0.248710
$495$	0	0
$496$	−8.00000	−0.359211
$497$	−4.47214	−0.200603
$498$	−38.5410	−1.72706
$499$	16.5623	0.741431	0.370715	−	0.928747i	$-0.379113\pi$
$499$	16.5623	0.741431	0.370715	+	0.928747i	$0.379113\pi$
$500$	35.8328	1.60249
$501$	−78.2492	−3.49592
$502$	−42.6869	−1.90521
$503$	28.0000	1.24846	0.624229	−	0.781241i	$-0.285413\pi$
$503$	28.0000	1.24846	0.624229	+	0.781241i	$0.285413\pi$
$504$	6.38197	0.284275
$505$	10.3262	0.459512
$506$	0	0
$507$	−22.2918	−0.990013
$508$	6.00000	0.266207
$509$	19.4164	0.860617	0.430309	−	0.902682i	$-0.358405\pi$
$509$	19.4164	0.860617	0.430309	+	0.902682i	$0.358405\pi$
$510$	−36.1803	−1.60209
$511$	−1.34752	−0.0596110
$512$	−11.1803	−0.494106
$513$	14.4721	0.638960
$514$	11.0557	0.487647
$515$	−25.4164	−1.11998
$516$	−30.0000	−1.32068
$517$	0	0
$518$	−1.70820	−0.0750542
$519$	30.8328	1.35341
$520$	−8.94427	−0.392232
$521$	11.8885	0.520847	0.260423	−	0.965495i	$-0.416138\pi$
$521$	11.8885	0.520847	0.260423	+	0.965495i	$0.416138\pi$
$522$	−141.554	−6.19566
$523$	15.0557	0.658341	0.329171	−	0.944270i	$-0.393231\pi$
$523$	15.0557	0.658341	0.329171	+	0.944270i	$0.393231\pi$
$524$	33.1033	1.44613
$525$	2.94427	0.128499
$526$	0	0
$527$	−24.7214	−1.07688
$528$	0	0
$529$	−22.6180	−0.983393
$530$	5.52786	0.240115
$531$	93.1935	4.04425
$532$	−1.14590	−0.0496810
$533$	−28.9443	−1.25372
$534$	23.4164	1.01333
$535$	8.47214	0.366282
$536$	18.9443	0.818268
$537$	−20.0000	−0.863064
$538$	−2.36068	−0.101776
$539$	0	0
$540$	−70.2492	−3.02305
$541$	−2.43769	−0.104805	−0.0524023	−	0.998626i	$-0.516688\pi$
$541$	−2.43769	−0.104805	−0.0524023	+	0.998626i	$0.516688\pi$
$542$	−35.8541	−1.54007
$543$	−20.9443	−0.898805
$544$	−20.7295	−0.888770
$545$	−17.7082	−0.758536
$546$	−6.83282	−0.292418
$547$	−8.58359	−0.367008	−0.183504	−	0.983019i	$-0.558744\pi$
$547$	−8.58359	−0.367008	−0.183504	+	0.983019i	$0.558744\pi$
$548$	−14.5623	−0.622071
$549$	64.3951	2.74832
$550$	0	0
$551$	8.47214	0.360925
$552$	4.47214	0.190347
$553$	−1.12461	−0.0478234
$554$	−13.4164	−0.570009
$555$	10.4721	0.444517
$556$	−26.7295	−1.13358
$557$	−35.3262	−1.49682	−0.748410	−	0.663236i	$-0.769183\pi$
$557$	−35.3262	−1.49682	−0.748410	+	0.663236i	$0.769183\pi$
$558$	−133.666	−5.65852
$559$	7.63932	0.323109
$560$	−0.618034	−0.0261167
$561$	0	0
$562$	−10.0000	−0.421825
$563$	7.81966	0.329559	0.164780	−	0.986330i	$-0.447309\pi$
$563$	7.81966	0.329559	0.164780	+	0.986330i	$0.447309\pi$
$564$	83.6656	3.52296
$565$	18.1803	0.764853
$566$	43.8673	1.84388
$567$	−9.32624	−0.391665
$568$	−26.1803	−1.09850
$569$	−45.7082	−1.91619	−0.958094	−	0.286455i	$-0.907523\pi$
$569$	−45.7082	−1.91619	−0.958094	+	0.286455i	$0.907523\pi$
$570$	11.7082	0.490403
$571$	−21.3262	−0.892475	−0.446238	−	0.894915i	$-0.647236\pi$
$571$	−21.3262	−0.892475	−0.446238	+	0.894915i	$0.647236\pi$
$572$	0	0
$573$	−6.18034	−0.258187
$574$	10.0000	0.417392
$575$	1.47214	0.0613923
$576$	−97.1378	−4.04741
$577$	10.3607	0.431321	0.215660	−	0.976468i	$-0.430810\pi$
$577$	10.3607	0.431321	0.215660	+	0.976468i	$0.430810\pi$
$578$	16.6606	0.692990
$579$	28.3607	1.17863
$580$	−41.1246	−1.70761
$581$	−2.03444	−0.0844029
$582$	88.1378	3.65343
$583$	0	0
$584$	−7.88854	−0.326430
$585$	29.8885	1.23574
$586$	36.8328	1.52155
$587$	7.41641	0.306108	0.153054	−	0.988218i	$-0.451089\pi$
$587$	7.41641	0.306108	0.153054	+	0.988218i	$0.451089\pi$
$588$	−66.5410	−2.74411
$589$	8.00000	0.329634
$590$	45.1246	1.85775
$591$	−11.4164	−0.469608
$592$	2.00000	0.0821995
$593$	−9.27051	−0.380694	−0.190347	−	0.981717i	$-0.560961\pi$
$593$	−9.27051	−0.380694	−0.190347	+	0.981717i	$0.560961\pi$
$594$	0	0
$595$	−1.90983	−0.0782954
$596$	−28.2492	−1.15713
$597$	−11.1246	−0.455300
$598$	−3.41641	−0.139707
$599$	−36.3607	−1.48566	−0.742829	−	0.669482i	$-0.766516\pi$
$599$	−36.3607	−1.48566	−0.742829	+	0.669482i	$0.766516\pi$
$600$	17.2361	0.703660
$601$	−21.0557	−0.858881	−0.429441	−	0.903095i	$-0.641289\pi$
$601$	−21.0557	−0.858881	−0.429441	+	0.903095i	$0.641289\pi$
$602$	−2.63932	−0.107571
$603$	−63.3050	−2.57798
$604$	21.7082	0.883294
$605$	0	0
$606$	46.1803	1.87595
$607$	28.6525	1.16297	0.581484	−	0.813558i	$-0.302472\pi$
$607$	28.6525	1.16297	0.581484	+	0.813558i	$0.302472\pi$
$608$	6.70820	0.272054
$609$	−10.4721	−0.424352
$610$	31.1803	1.26246
$611$	−21.3050	−0.861906
$612$	−69.2705	−2.80009
$613$	−8.97871	−0.362647	−0.181323	−	0.983424i	$-0.558038\pi$
$613$	−8.97871	−0.362647	−0.181323	+	0.983424i	$0.558038\pi$
$614$	57.8885	2.33619
$615$	−61.3050	−2.47205
$616$	0	0
$617$	−6.58359	−0.265045	−0.132523	−	0.991180i	$-0.542308\pi$
$617$	−6.58359	−0.265045	−0.132523	+	0.991180i	$0.542308\pi$
$618$	−113.666	−4.57230
$619$	−25.5623	−1.02744	−0.513718	−	0.857959i	$-0.671732\pi$
$619$	−25.5623	−1.02744	−0.513718	+	0.857959i	$0.671732\pi$
$620$	−38.8328	−1.55956
$621$	−8.94427	−0.358921
$622$	3.49342	0.140073
$623$	1.23607	0.0495220
$624$	8.00000	0.320256
$625$	−7.41641	−0.296656
$626$	16.5066	0.659736
$627$	0	0
$628$	43.6869	1.74330
$629$	6.18034	0.246426
$630$	−10.3262	−0.411407
$631$	−37.3050	−1.48509	−0.742543	−	0.669798i	$-0.766381\pi$
$631$	−37.3050	−1.48509	−0.742543	+	0.669798i	$0.766381\pi$
$632$	−6.58359	−0.261881
$633$	−15.4164	−0.612747
$634$	−66.1803	−2.62836
$635$	−3.23607	−0.128419
$636$	14.8328	0.588159
$637$	16.9443	0.671356
$638$	0	0
$639$	87.4853	3.46086
$640$	−25.3262	−1.00111
$641$	−37.8885	−1.49651	−0.748254	−	0.663413i	$-0.769107\pi$
$641$	−37.8885	−1.49651	−0.748254	+	0.663413i	$0.769107\pi$
$642$	37.8885	1.49534
$643$	15.9656	0.629620	0.314810	−	0.949155i	$-0.398059\pi$
$643$	15.9656	0.629620	0.314810	+	0.949155i	$0.398059\pi$
$644$	0.708204	0.0279071
$645$	16.1803	0.637100
$646$	6.90983	0.271864
$647$	13.8885	0.546015	0.273007	−	0.962012i	$-0.411982\pi$
$647$	13.8885	0.546015	0.273007	+	0.962012i	$0.411982\pi$
$648$	−54.5967	−2.14476
$649$	0	0
$650$	−13.1672	−0.516459
$651$	−9.88854	−0.387563
$652$	−27.9787	−1.09573
$653$	−5.56231	−0.217670	−0.108835	−	0.994060i	$-0.534712\pi$
$653$	−5.56231	−0.217670	−0.108835	+	0.994060i	$0.534712\pi$
$654$	−79.1935	−3.09671
$655$	−17.8541	−0.697617
$656$	−11.7082	−0.457129
$657$	26.3607	1.02843
$658$	7.36068	0.286949
$659$	−19.7082	−0.767723	−0.383861	−	0.923391i	$-0.625406\pi$
$659$	−19.7082	−0.767723	−0.383861	+	0.923391i	$0.625406\pi$
$660$	0	0
$661$	−33.1246	−1.28840	−0.644199	−	0.764858i	$-0.722809\pi$
$661$	−33.1246	−1.28840	−0.644199	+	0.764858i	$0.722809\pi$
$662$	23.0132	0.894432
$663$	24.7214	0.960098
$664$	−11.9098	−0.462191
$665$	0.618034	0.0239663
$666$	33.4164	1.29486
$667$	−5.23607	−0.202741
$668$	−72.5410	−2.80670
$669$	28.9443	1.11905
$670$	−30.6525	−1.18421
$671$	0	0
$672$	−8.29180	−0.319863
$673$	−14.2918	−0.550908	−0.275454	−	0.961314i	$-0.588828\pi$
$673$	−14.2918	−0.550908	−0.275454	+	0.961314i	$0.588828\pi$
$674$	16.5836	0.638776
$675$	−34.4721	−1.32683
$676$	−20.6656	−0.794832
$677$	−27.4164	−1.05370	−0.526849	−	0.849959i	$-0.676627\pi$
$677$	−27.4164	−1.05370	−0.526849	+	0.849959i	$0.676627\pi$
$678$	81.3050	3.12250
$679$	4.65248	0.178546
$680$	−11.1803	−0.428746
$681$	74.8328	2.86760
$682$	0	0
$683$	−14.8328	−0.567562	−0.283781	−	0.958889i	$-0.591589\pi$
$683$	−14.8328	−0.567562	−0.283781	+	0.958889i	$0.591589\pi$
$684$	22.4164	0.857113
$685$	7.85410	0.300090
$686$	−11.8328	−0.451779
$687$	−35.8885	−1.36923
$688$	3.09017	0.117812
$689$	−3.77709	−0.143896
$690$	−7.23607	−0.275472
$691$	19.0344	0.724104	0.362052	−	0.932158i	$-0.382076\pi$
$691$	19.0344	0.724104	0.362052	+	0.932158i	$0.382076\pi$
$692$	28.5836	1.08659
$693$	0	0
$694$	4.79837	0.182144
$695$	14.4164	0.546846
$696$	−61.3050	−2.32376
$697$	−36.1803	−1.37043
$698$	19.7984	0.749380
$699$	−18.1803	−0.687644
$700$	2.72949	0.103165
$701$	13.4934	0.509640	0.254820	−	0.966989i	$-0.417984\pi$
$701$	13.4934	0.509640	0.254820	+	0.966989i	$0.417984\pi$
$702$	80.0000	3.01941
$703$	−2.00000	−0.0754314
$704$	0	0
$705$	−45.1246	−1.69949
$706$	−27.0344	−1.01745
$707$	2.43769	0.0916789
$708$	121.082	4.55054
$709$	8.09017	0.303833	0.151916	−	0.988393i	$-0.451456\pi$
$709$	8.09017	0.303833	0.151916	+	0.988393i	$0.451456\pi$
$710$	42.3607	1.58977
$711$	22.0000	0.825064
$712$	7.23607	0.271183
$713$	−4.94427	−0.185164
$714$	−8.54102	−0.319640
$715$	0	0
$716$	−18.5410	−0.692910
$717$	−62.7214	−2.34237
$718$	−59.9230	−2.23631
$719$	−19.9787	−0.745080	−0.372540	−	0.928016i	$-0.621513\pi$
$719$	−19.9787	−0.745080	−0.372540	+	0.928016i	$0.621513\pi$
$720$	12.0902	0.450574
$721$	−6.00000	−0.223452
$722$	−2.23607	−0.0832178
$723$	79.1935	2.94524
$724$	−19.4164	−0.721605
$725$	−20.1803	−0.749479
$726$	0	0
$727$	−14.1459	−0.524642	−0.262321	−	0.964981i	$-0.584488\pi$
$727$	−14.1459	−0.524642	−0.262321	+	0.964981i	$0.584488\pi$
$728$	−2.11146	−0.0782558
$729$	41.9443	1.55349
$730$	12.7639	0.472414
$731$	9.54915	0.353188
$732$	83.6656	3.09237
$733$	48.8541	1.80447	0.902234	−	0.431247i	$-0.141926\pi$
$733$	48.8541	1.80447	0.902234	+	0.431247i	$0.141926\pi$
$734$	−5.97871	−0.220678
$735$	35.8885	1.32377
$736$	−4.14590	−0.152820
$737$	0	0
$738$	−195.623	−7.20098
$739$	−1.72949	−0.0636203	−0.0318102	−	0.999494i	$-0.510127\pi$
$739$	−1.72949	−0.0636203	−0.0318102	+	0.999494i	$0.510127\pi$
$740$	9.70820	0.356881
$741$	−8.00000	−0.293887
$742$	1.30495	0.0479063
$743$	7.88854	0.289403	0.144701	−	0.989475i	$-0.453778\pi$
$743$	7.88854	0.289403	0.144701	+	0.989475i	$0.453778\pi$
$744$	−57.8885	−2.12230
$745$	15.2361	0.558206
$746$	41.9574	1.53617
$747$	39.7984	1.45615
$748$	0	0
$749$	2.00000	0.0730784
$750$	−86.4296	−3.15596
$751$	6.76393	0.246819	0.123410	−	0.992356i	$-0.460617\pi$
$751$	6.76393	0.246819	0.123410	+	0.992356i	$0.460617\pi$
$752$	−8.61803	−0.314267
$753$	61.7771	2.25128
$754$	46.8328	1.70555
$755$	−11.7082	−0.426105
$756$	−16.5836	−0.603139
$757$	−38.3607	−1.39424	−0.697121	−	0.716953i	$-0.745536\pi$
$757$	−38.3607	−1.39424	−0.697121	+	0.716953i	$0.745536\pi$
$758$	−46.1803	−1.67735
$759$	0	0
$760$	3.61803	0.131240
$761$	−11.3050	−0.409804	−0.204902	−	0.978782i	$-0.565688\pi$
$761$	−11.3050	−0.409804	−0.204902	+	0.978782i	$0.565688\pi$
$762$	−14.4721	−0.524270
$763$	−4.18034	−0.151338
$764$	−5.72949	−0.207286
$765$	37.3607	1.35078
$766$	−74.0689	−2.67622
$767$	−30.8328	−1.11331
$768$	−29.1246	−1.05094
$769$	30.7426	1.10861	0.554304	−	0.832314i	$-0.312984\pi$
$769$	30.7426	1.10861	0.554304	+	0.832314i	$0.312984\pi$
$770$	0	0
$771$	−16.0000	−0.576226
$772$	26.2918	0.946262
$773$	−53.2361	−1.91477	−0.957384	−	0.288818i	$-0.906738\pi$
$773$	−53.2361	−1.91477	−0.957384	+	0.288818i	$0.906738\pi$
$774$	51.6312	1.85584
$775$	−19.0557	−0.684502
$776$	27.2361	0.977718
$777$	2.47214	0.0886874
$778$	50.9787	1.82768
$779$	11.7082	0.419490
$780$	38.8328	1.39044
$781$	0	0
$782$	−4.27051	−0.152713
$783$	122.610	4.38172
$784$	6.85410	0.244789
$785$	−23.5623	−0.840975
$786$	−79.8460	−2.84801
$787$	24.4721	0.872337	0.436169	−	0.899865i	$-0.356335\pi$
$787$	24.4721	0.872337	0.436169	+	0.899865i	$0.356335\pi$
$788$	−10.5836	−0.377025
$789$	0	0
$790$	10.6525	0.378998
$791$	4.29180	0.152599
$792$	0	0
$793$	−21.3050	−0.756561
$794$	30.3262	1.07624
$795$	−8.00000	−0.283731
$796$	−10.3131	−0.365538
$797$	−2.94427	−0.104291	−0.0521457	−	0.998639i	$-0.516606\pi$
$797$	−2.94427	−0.104291	−0.0521457	+	0.998639i	$0.516606\pi$
$798$	2.76393	0.0978421
$799$	−26.6312	−0.942144
$800$	−15.9787	−0.564933
$801$	−24.1803	−0.854370
$802$	3.81966	0.134877
$803$	0	0
$804$	−82.2492	−2.90071
$805$	−0.381966	−0.0134625
$806$	44.2229	1.55769
$807$	3.41641	0.120263
$808$	14.2705	0.502035
$809$	43.6869	1.53595	0.767975	−	0.640480i	$-0.221264\pi$
$809$	43.6869	1.53595	0.767975	+	0.640480i	$0.221264\pi$
$810$	88.3394	3.10393
$811$	9.34752	0.328236	0.164118	−	0.986441i	$-0.447522\pi$
$811$	9.34752	0.328236	0.164118	+	0.986441i	$0.447522\pi$
$812$	−9.70820	−0.340691
$813$	51.8885	1.81981
$814$	0	0
$815$	15.0902	0.528586
$816$	10.0000	0.350070
$817$	−3.09017	−0.108111
$818$	51.7082	1.80793
$819$	7.05573	0.246547
$820$	−56.8328	−1.98469
$821$	−23.6738	−0.826220	−0.413110	−	0.910681i	$-0.635558\pi$
$821$	−23.6738	−0.826220	−0.413110	+	0.910681i	$0.635558\pi$
$822$	35.1246	1.22511
$823$	19.0344	0.663499	0.331749	−	0.943368i	$-0.392361\pi$
$823$	19.0344	0.663499	0.331749	+	0.943368i	$0.392361\pi$
$824$	−35.1246	−1.22362
$825$	0	0
$826$	10.6525	0.370647
$827$	52.7214	1.83330	0.916651	−	0.399689i	$-0.130882\pi$
$827$	52.7214	1.83330	0.916651	+	0.399689i	$0.130882\pi$
$828$	−13.8541	−0.481463
$829$	46.6525	1.62031	0.810154	−	0.586217i	$-0.199384\pi$
$829$	46.6525	1.62031	0.810154	+	0.586217i	$0.199384\pi$
$830$	19.2705	0.668889
$831$	19.4164	0.673548
$832$	32.1378	1.11418
$833$	21.1803	0.733855
$834$	64.4721	2.23249
$835$	39.1246	1.35396
$836$	0	0
$837$	115.777	4.00184
$838$	30.9787	1.07014
$839$	−3.70820	−0.128021	−0.0640107	−	0.997949i	$-0.520389\pi$
$839$	−3.70820	−0.128021	−0.0640107	+	0.997949i	$0.520389\pi$
$840$	−4.47214	−0.154303
$841$	42.7771	1.47507
$842$	−19.5967	−0.675349
$843$	14.4721	0.498447
$844$	−14.2918	−0.491944
$845$	11.1459	0.383431
$846$	−143.992	−4.95054
$847$	0	0
$848$	−1.52786	−0.0524671
$849$	−63.4853	−2.17881
$850$	−16.4590	−0.564538
$851$	1.23607	0.0423719
$852$	113.666	3.89412
$853$	−26.7984	−0.917559	−0.458779	−	0.888550i	$-0.651713\pi$
$853$	−26.7984	−0.917559	−0.458779	+	0.888550i	$0.651713\pi$
$854$	7.36068	0.251877
$855$	−12.0902	−0.413475
$856$	11.7082	0.400178
$857$	12.7639	0.436008	0.218004	−	0.975948i	$-0.430045\pi$
$857$	12.7639	0.436008	0.218004	+	0.975948i	$0.430045\pi$
$858$	0	0
$859$	−23.7426	−0.810089	−0.405044	−	0.914297i	$-0.632744\pi$
$859$	−23.7426	−0.810089	−0.405044	+	0.914297i	$0.632744\pi$
$860$	15.0000	0.511496
$861$	−14.4721	−0.493209
$862$	78.1378	2.66138
$863$	−40.8328	−1.38997	−0.694983	−	0.719027i	$-0.744588\pi$
$863$	−40.8328	−1.38997	−0.694983	+	0.719027i	$0.744588\pi$
$864$	97.0820	3.30280
$865$	−15.4164	−0.524174
$866$	29.3475	0.997269
$867$	−24.1115	−0.818868
$868$	−9.16718	−0.311155
$869$	0	0
$870$	99.1935	3.36297
$871$	20.9443	0.709670
$872$	−24.4721	−0.828731
$873$	−91.0132	−3.08033
$874$	1.38197	0.0467457
$875$	−4.56231	−0.154234
$876$	34.2492	1.15717
$877$	−6.11146	−0.206369	−0.103185	−	0.994662i	$-0.532903\pi$
$877$	−6.11146	−0.206369	−0.103185	+	0.994662i	$0.532903\pi$
$878$	29.5967	0.998842
$879$	−53.3050	−1.79793
$880$	0	0
$881$	−21.7771	−0.733689	−0.366844	−	0.930282i	$-0.619562\pi$
$881$	−21.7771	−0.733689	−0.366844	+	0.930282i	$0.619562\pi$
$882$	114.520	3.85608
$883$	−5.50658	−0.185311	−0.0926555	−	0.995698i	$-0.529536\pi$
$883$	−5.50658	−0.185311	−0.0926555	+	0.995698i	$0.529536\pi$
$884$	22.9180	0.770814
$885$	−65.3050	−2.19520
$886$	−62.1591	−2.08827
$887$	−27.3050	−0.916811	−0.458405	−	0.888743i	$-0.651579\pi$
$887$	−27.3050	−0.916811	−0.458405	+	0.888743i	$0.651579\pi$
$888$	14.4721	0.485653
$889$	−0.763932	−0.0256215
$890$	−11.7082	−0.392460
$891$	0	0
$892$	26.8328	0.898429
$893$	8.61803	0.288392
$894$	68.1378	2.27887
$895$	10.0000	0.334263
$896$	−5.97871	−0.199735
$897$	4.94427	0.165084
$898$	−72.6099	−2.42302
$899$	67.7771	2.26049
$900$	−53.3951	−1.77984
$901$	−4.72136	−0.157291
$902$	0	0
$903$	3.81966	0.127110
$904$	25.1246	0.835632
$905$	10.4721	0.348106
$906$	−52.3607	−1.73957
$907$	−43.3050	−1.43792	−0.718959	−	0.695053i	$-0.755381\pi$
$907$	−43.3050	−1.43792	−0.718959	+	0.695053i	$0.755381\pi$
$908$	69.3738	2.30225
$909$	−47.6869	−1.58168
$910$	3.41641	0.113253
$911$	3.81966	0.126551	0.0632755	−	0.997996i	$-0.479845\pi$
$911$	3.81966	0.126551	0.0632755	+	0.997996i	$0.479845\pi$
$912$	−3.23607	−0.107157
$913$	0	0
$914$	75.1722	2.48648
$915$	−45.1246	−1.49177
$916$	−33.2705	−1.09929
$917$	−4.21478	−0.139184
$918$	100.000	3.30049
$919$	14.4934	0.478094	0.239047	−	0.971008i	$-0.423165\pi$
$919$	14.4934	0.478094	0.239047	+	0.971008i	$0.423165\pi$
$920$	−2.23607	−0.0737210
$921$	−83.7771	−2.76055
$922$	−7.84095	−0.258228
$923$	−28.9443	−0.952712
$924$	0	0
$925$	4.76393	0.156637
$926$	−17.0344	−0.559786
$927$	117.374	3.85506
$928$	56.8328	1.86563
$929$	34.1459	1.12029	0.560145	−	0.828394i	$-0.310745\pi$
$929$	34.1459	1.12029	0.560145	+	0.828394i	$0.310745\pi$
$930$	93.6656	3.07142
$931$	−6.85410	−0.224634
$932$	−16.8541	−0.552074
$933$	−5.05573	−0.165517
$934$	−87.2837	−2.85601
$935$	0	0
$936$	41.3050	1.35009
$937$	−29.5623	−0.965758	−0.482879	−	0.875687i	$-0.660409\pi$
$937$	−29.5623	−0.965758	−0.482879	+	0.875687i	$0.660409\pi$
$938$	−7.23607	−0.236266
$939$	−23.8885	−0.779573
$940$	−41.8328	−1.36444
$941$	−37.4164	−1.21974	−0.609870	−	0.792501i	$-0.708778\pi$
$941$	−37.4164	−1.21974	−0.609870	+	0.792501i	$0.708778\pi$
$942$	−105.374	−3.43327
$943$	−7.23607	−0.235639
$944$	−12.4721	−0.405933
$945$	8.94427	0.290957
$946$	0	0
$947$	49.6869	1.61461	0.807304	−	0.590136i	$-0.200926\pi$
$947$	49.6869	1.61461	0.807304	+	0.590136i	$0.200926\pi$
$948$	28.5836	0.928352
$949$	−8.72136	−0.283107
$950$	5.32624	0.172806
$951$	95.7771	3.10578
$952$	−2.63932	−0.0855409
$953$	−14.2918	−0.462957	−0.231478	−	0.972840i	$-0.574356\pi$
$953$	−14.2918	−0.462957	−0.231478	+	0.972840i	$0.574356\pi$
$954$	−25.5279	−0.826495
$955$	3.09017	0.0999956
$956$	−58.1459	−1.88057
$957$	0	0
$958$	−22.0344	−0.711901
$959$	1.85410	0.0598721
$960$	68.0689	2.19691
$961$	33.0000	1.06452
$962$	−11.0557	−0.356451
$963$	−39.1246	−1.26077
$964$	73.4164	2.36458
$965$	−14.1803	−0.456481
$966$	−1.70820	−0.0549606
$967$	−37.8541	−1.21731	−0.608653	−	0.793437i	$-0.708290\pi$
$967$	−37.8541	−1.21731	−0.608653	+	0.793437i	$0.708290\pi$
$968$	0	0
$969$	−10.0000	−0.321246
$970$	−44.0689	−1.41497
$971$	−6.18034	−0.198337	−0.0991683	−	0.995071i	$-0.531618\pi$
$971$	−6.18034	−0.198337	−0.0991683	+	0.995071i	$0.531618\pi$
$972$	106.790	3.42530
$973$	3.40325	0.109103
$974$	−87.8885	−2.81613
$975$	19.0557	0.610272
$976$	−8.61803	−0.275857
$977$	−39.1246	−1.25171	−0.625854	−	0.779941i	$-0.715249\pi$
$977$	−39.1246	−1.25171	−0.625854	+	0.779941i	$0.715249\pi$
$978$	67.4853	2.15794
$979$	0	0
$980$	33.2705	1.06279
$981$	81.7771	2.61094
$982$	20.4508	0.652613
$983$	−46.1803	−1.47292	−0.736462	−	0.676479i	$-0.763505\pi$
$983$	−46.1803	−1.47292	−0.736462	+	0.676479i	$0.763505\pi$
$984$	−84.7214	−2.70082
$985$	5.70820	0.181879
$986$	58.5410	1.86433
$987$	−10.6525	−0.339072
$988$	−7.41641	−0.235947
$989$	1.90983	0.0607291
$990$	0	0
$991$	8.65248	0.274855	0.137427	−	0.990512i	$-0.456117\pi$
$991$	8.65248	0.274855	0.137427	+	0.990512i	$0.456117\pi$
$992$	53.6656	1.70389
$993$	−33.3050	−1.05690
$994$	10.0000	0.317181
$995$	5.56231	0.176337
$996$	51.7082	1.63844
$997$	−13.2148	−0.418516	−0.209258	−	0.977860i	$-0.567105\pi$
$997$	−13.2148	−0.418516	−0.209258	+	0.977860i	$0.567105\pi$
$998$	−37.0344	−1.17230
$999$	−28.9443	−0.915756

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2299.2.a.g.1.1		2
11.3	even	5		209.2.f.a.20.1	✓	4
11.4	even	5		209.2.f.a.115.1	yes	4
11.10	odd	2		2299.2.a.h.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
209.2.f.a.20.1	✓	4	11.3	even	5
209.2.f.a.115.1	yes	4	11.4	even	5
2299.2.a.g.1.1		2	1.1	even	1	trivial
2299.2.a.h.1.2		2	11.10	odd	2

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)

\(f(q)\)	\(=\)	\(q-2.23607 q^{2} +3.23607 q^{3} +3.00000 q^{4} -1.61803 q^{5} -7.23607 q^{6} -0.381966 q^{7} -2.23607 q^{8} +7.47214 q^{9} +3.61803 q^{10} +9.70820 q^{12} -2.47214 q^{13} +0.854102 q^{14} -5.23607 q^{15} -1.00000 q^{16} -3.09017 q^{17} -16.7082 q^{18} +1.00000 q^{19} -4.85410 q^{20} -1.23607 q^{21} -0.618034 q^{23} -7.23607 q^{24} -2.38197 q^{25} +5.52786 q^{26} +14.4721 q^{27} -1.14590 q^{28} +8.47214 q^{29} +11.7082 q^{30} +8.00000 q^{31} +6.70820 q^{32} +6.90983 q^{34} +0.618034 q^{35} +22.4164 q^{36} -2.00000 q^{37} -2.23607 q^{38} -8.00000 q^{39} +3.61803 q^{40} +11.7082 q^{41} +2.76393 q^{42} -3.09017 q^{43} -12.0902 q^{45} +1.38197 q^{46} +8.61803 q^{47} -3.23607 q^{48} -6.85410 q^{49} +5.32624 q^{50} -10.0000 q^{51} -7.41641 q^{52} +1.52786 q^{53} -32.3607 q^{54} +0.854102 q^{56} +3.23607 q^{57} -18.9443 q^{58} +12.4721 q^{59} -15.7082 q^{60} +8.61803 q^{61} -17.8885 q^{62} -2.85410 q^{63} -13.0000 q^{64} +4.00000 q^{65} -8.47214 q^{67} -9.27051 q^{68} -2.00000 q^{69} -1.38197 q^{70} +11.7082 q^{71} -16.7082 q^{72} +3.52786 q^{73} +4.47214 q^{74} -7.70820 q^{75} +3.00000 q^{76} +17.8885 q^{78} +2.94427 q^{79} +1.61803 q^{80} +24.4164 q^{81} -26.1803 q^{82} +5.32624 q^{83} -3.70820 q^{84} +5.00000 q^{85} +6.90983 q^{86} +27.4164 q^{87} -3.23607 q^{89} +27.0344 q^{90} +0.944272 q^{91} -1.85410 q^{92} +25.8885 q^{93} -19.2705 q^{94} -1.61803 q^{95} +21.7082 q^{96} -12.1803 q^{97} +15.3262 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 6 q^{4} - q^{5} - 10 q^{6} - 3 q^{7} + 6 q^{9} + 5 q^{10} + 6 q^{12} + 4 q^{13} - 5 q^{14} - 6 q^{15} - 2 q^{16} + 5 q^{17} - 20 q^{18} + 2 q^{19} - 3 q^{20} + 2 q^{21} + q^{23} - 10 q^{24}+ \cdots + 15 q^{98}+O(q^{100}) \) 2 * q + 2 * q^3 + 6 * q^4 - q^5 - 10 * q^6 - 3 * q^7 + 6 * q^9 + 5 * q^10 + 6 * q^12 + 4 * q^13 - 5 * q^14 - 6 * q^15 - 2 * q^16 + 5 * q^17 - 20 * q^18 + 2 * q^19 - 3 * q^20 + 2 * q^21 + q^23 - 10 * q^24 - 7 * q^25 + 20 * q^26 + 20 * q^27 - 9 * q^28 + 8 * q^29 + 10 * q^30 + 16 * q^31 + 25 * q^34 - q^35 + 18 * q^36 - 4 * q^37 - 16 * q^39 + 5 * q^40 + 10 * q^41 + 10 * q^42 + 5 * q^43 - 13 * q^45 + 5 * q^46 + 15 * q^47 - 2 * q^48 - 7 * q^49 - 5 * q^50 - 20 * q^51 + 12 * q^52 + 12 * q^53 - 20 * q^54 - 5 * q^56 + 2 * q^57 - 20 * q^58 + 16 * q^59 - 18 * q^60 + 15 * q^61 + q^63 - 26 * q^64 + 8 * q^65 - 8 * q^67 + 15 * q^68 - 4 * q^69 - 5 * q^70 + 10 * q^71 - 20 * q^72 + 16 * q^73 - 2 * q^75 + 6 * q^76 - 12 * q^79 + q^80 + 22 * q^81 - 30 * q^82 - 5 * q^83 + 6 * q^84 + 10 * q^85 + 25 * q^86 + 28 * q^87 - 2 * q^89 + 25 * q^90 - 16 * q^91 + 3 * q^92 + 16 * q^93 - 5 * q^94 - q^95 + 30 * q^96 - 2 * q^97 + 15 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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