LMFDB - Embedded newform 299.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [299,2,Mod(1,299)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(299, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("299.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 299 = 13 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	299.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$2.38752702044$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	299.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607 q^{2} +3.00000 q^{4} -1.23607 q^{5} +3.23607 q^{7} -2.23607 q^{8} -3.00000 q^{9} +O(q^{10})$	\(q-2.23607 q^{2} +3.00000 q^{4} -1.23607 q^{5} +3.23607 q^{7} -2.23607 q^{8} -3.00000 q^{9} +2.76393 q^{10} -0.763932 q^{11} +1.00000 q^{13} -7.23607 q^{14} -1.00000 q^{16} +2.00000 q^{17} +6.70820 q^{18} +7.23607 q^{19} -3.70820 q^{20} +1.70820 q^{22} -1.00000 q^{23} -3.47214 q^{25} -2.23607 q^{26} +9.70820 q^{28} +8.47214 q^{29} +10.4721 q^{31} +6.70820 q^{32} -4.47214 q^{34} -4.00000 q^{35} -9.00000 q^{36} -1.23607 q^{37} -16.1803 q^{38} +2.76393 q^{40} +10.0000 q^{41} +10.4721 q^{43} -2.29180 q^{44} +3.70820 q^{45} +2.23607 q^{46} -4.00000 q^{47} +3.47214 q^{49} +7.76393 q^{50} +3.00000 q^{52} -6.94427 q^{53} +0.944272 q^{55} -7.23607 q^{56} -18.9443 q^{58} +6.47214 q^{59} -10.0000 q^{61} -23.4164 q^{62} -9.70820 q^{63} -13.0000 q^{64} -1.23607 q^{65} -7.23607 q^{67} +6.00000 q^{68} +8.94427 q^{70} -5.52786 q^{71} +6.70820 q^{72} +0.472136 q^{73} +2.76393 q^{74} +21.7082 q^{76} -2.47214 q^{77} -4.94427 q^{79} +1.23607 q^{80} +9.00000 q^{81} -22.3607 q^{82} -12.1803 q^{83} -2.47214 q^{85} -23.4164 q^{86} +1.70820 q^{88} +7.70820 q^{89} -8.29180 q^{90} +3.23607 q^{91} -3.00000 q^{92} +8.94427 q^{94} -8.94427 q^{95} +4.29180 q^{97} -7.76393 q^{98} +2.29180 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{4} + 2 q^{5} + 2 q^{7} - 6 q^{9}+O(q^{10}) $ 2 * q + 6 * q^4 + 2 * q^5 + 2 * q^7 - 6 * q^9	$ 2 q + 6 q^{4} + 2 q^{5} + 2 q^{7} - 6 q^{9} + 10 q^{10} - 6 q^{11} + 2 q^{13} - 10 q^{14} - 2 q^{16} + 4 q^{17} + 10 q^{19} + 6 q^{20} - 10 q^{22} - 2 q^{23} + 2 q^{25} + 6 q^{28} + 8 q^{29} + 12 q^{31} - 8 q^{35} - 18 q^{36} + 2 q^{37} - 10 q^{38} + 10 q^{40} + 20 q^{41} + 12 q^{43} - 18 q^{44} - 6 q^{45} - 8 q^{47} - 2 q^{49} + 20 q^{50} + 6 q^{52} + 4 q^{53} - 16 q^{55} - 10 q^{56} - 20 q^{58} + 4 q^{59} - 20 q^{61} - 20 q^{62} - 6 q^{63} - 26 q^{64} + 2 q^{65} - 10 q^{67} + 12 q^{68} - 20 q^{71} - 8 q^{73} + 10 q^{74} + 30 q^{76} + 4 q^{77} + 8 q^{79} - 2 q^{80} + 18 q^{81} - 2 q^{83} + 4 q^{85} - 20 q^{86} - 10 q^{88} + 2 q^{89} - 30 q^{90} + 2 q^{91} - 6 q^{92} + 22 q^{97} - 20 q^{98} + 18 q^{99}+O(q^{100}) $ 2 * q + 6 * q^4 + 2 * q^5 + 2 * q^7 - 6 * q^9 + 10 * q^10 - 6 * q^11 + 2 * q^13 - 10 * q^14 - 2 * q^16 + 4 * q^17 + 10 * q^19 + 6 * q^20 - 10 * q^22 - 2 * q^23 + 2 * q^25 + 6 * q^28 + 8 * q^29 + 12 * q^31 - 8 * q^35 - 18 * q^36 + 2 * q^37 - 10 * q^38 + 10 * q^40 + 20 * q^41 + 12 * q^43 - 18 * q^44 - 6 * q^45 - 8 * q^47 - 2 * q^49 + 20 * q^50 + 6 * q^52 + 4 * q^53 - 16 * q^55 - 10 * q^56 - 20 * q^58 + 4 * q^59 - 20 * q^61 - 20 * q^62 - 6 * q^63 - 26 * q^64 + 2 * q^65 - 10 * q^67 + 12 * q^68 - 20 * q^71 - 8 * q^73 + 10 * q^74 + 30 * q^76 + 4 * q^77 + 8 * q^79 - 2 * q^80 + 18 * q^81 - 2 * q^83 + 4 * q^85 - 20 * q^86 - 10 * q^88 + 2 * q^89 - 30 * q^90 + 2 * q^91 - 6 * q^92 + 22 * q^97 - 20 * q^98 + 18 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.23607	−1.58114	−0.790569	−	0.612372i	$-0.790215\pi$
$2$	−2.23607	−1.58114	−0.790569	+	0.612372i	$0.790215\pi$
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	3.00000	1.50000
$5$	−1.23607	−0.552786	−0.276393	−	0.961045i	$-0.589139\pi$
$5$	−1.23607	−0.552786	−0.276393	+	0.961045i	$0.589139\pi$
$6$	0	0
$7$	3.23607	1.22312	0.611559	−	0.791199i	$-0.290543\pi$
$7$	3.23607	1.22312	0.611559	+	0.791199i	$0.290543\pi$
$8$	−2.23607	−0.790569
$9$	−3.00000	−1.00000
$10$	2.76393	0.874032
$11$	−0.763932	−0.230334	−0.115167	−	0.993346i	$-0.536740\pi$
$11$	−0.763932	−0.230334	−0.115167	+	0.993346i	$0.536740\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−7.23607	−1.93392
$15$	0	0
$16$	−1.00000	−0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	6.70820	1.58114
$19$	7.23607	1.66007	0.830034	−	0.557713i	$-0.188321\pi$
$19$	7.23607	1.66007	0.830034	+	0.557713i	$0.188321\pi$
$20$	−3.70820	−0.829180
$21$	0	0
$22$	1.70820	0.364190
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−3.47214	−0.694427
$26$	−2.23607	−0.438529
$27$	0	0
$28$	9.70820	1.83468
$29$	8.47214	1.57324	0.786618	−	0.617440i	$-0.211830\pi$
$29$	8.47214	1.57324	0.786618	+	0.617440i	$0.211830\pi$
$30$	0	0
$31$	10.4721	1.88085	0.940426	−	0.340000i	$-0.110427\pi$
$31$	10.4721	1.88085	0.940426	+	0.340000i	$0.110427\pi$
$32$	6.70820	1.18585
$33$	0	0
$34$	−4.47214	−0.766965
$35$	−4.00000	−0.676123
$36$	−9.00000	−1.50000
$37$	−1.23607	−0.203208	−0.101604	−	0.994825i	$-0.532398\pi$
$37$	−1.23607	−0.203208	−0.101604	+	0.994825i	$0.532398\pi$
$38$	−16.1803	−2.62480
$39$	0	0
$40$	2.76393	0.437016
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	10.4721	1.59699	0.798493	−	0.602004i	$-0.205631\pi$
$43$	10.4721	1.59699	0.798493	+	0.602004i	$0.205631\pi$
$44$	−2.29180	−0.345501
$45$	3.70820	0.552786
$46$	2.23607	0.329690
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	0	0
$49$	3.47214	0.496019
$50$	7.76393	1.09799
$51$	0	0
$52$	3.00000	0.416025
$53$	−6.94427	−0.953869	−0.476935	−	0.878939i	$-0.658252\pi$
$53$	−6.94427	−0.953869	−0.476935	+	0.878939i	$0.658252\pi$
$54$	0	0
$55$	0.944272	0.127326
$56$	−7.23607	−0.966960
$57$	0	0
$58$	−18.9443	−2.48750
$59$	6.47214	0.842600	0.421300	−	0.906921i	$-0.361574\pi$
$59$	6.47214	0.842600	0.421300	+	0.906921i	$0.361574\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	−23.4164	−2.97389
$63$	−9.70820	−1.22312
$64$	−13.0000	−1.62500
$65$	−1.23607	−0.153315
$66$	0	0
$67$	−7.23607	−0.884026	−0.442013	−	0.897009i	$-0.645736\pi$
$67$	−7.23607	−0.884026	−0.442013	+	0.897009i	$0.645736\pi$
$68$	6.00000	0.727607
$69$	0	0
$70$	8.94427	1.06904
$71$	−5.52786	−0.656037	−0.328018	−	0.944671i	$-0.606381\pi$
$71$	−5.52786	−0.656037	−0.328018	+	0.944671i	$0.606381\pi$
$72$	6.70820	0.790569
$73$	0.472136	0.0552593	0.0276297	−	0.999618i	$-0.491204\pi$
$73$	0.472136	0.0552593	0.0276297	+	0.999618i	$0.491204\pi$
$74$	2.76393	0.321301
$75$	0	0
$76$	21.7082	2.49010
$77$	−2.47214	−0.281726
$78$	0	0
$79$	−4.94427	−0.556274	−0.278137	−	0.960541i	$-0.589717\pi$
$79$	−4.94427	−0.556274	−0.278137	+	0.960541i	$0.589717\pi$
$80$	1.23607	0.138197
$81$	9.00000	1.00000
$82$	−22.3607	−2.46932
$83$	−12.1803	−1.33697	−0.668483	−	0.743727i	$-0.733056\pi$
$83$	−12.1803	−1.33697	−0.668483	+	0.743727i	$0.733056\pi$
$84$	0	0
$85$	−2.47214	−0.268141
$86$	−23.4164	−2.52506
$87$	0	0
$88$	1.70820	0.182095
$89$	7.70820	0.817068	0.408534	−	0.912743i	$-0.366040\pi$
$89$	7.70820	0.817068	0.408534	+	0.912743i	$0.366040\pi$
$90$	−8.29180	−0.874032
$91$	3.23607	0.339232
$92$	−3.00000	−0.312772
$93$	0	0
$94$	8.94427	0.922531
$95$	−8.94427	−0.917663
$96$	0	0
$97$	4.29180	0.435766	0.217883	−	0.975975i	$-0.430085\pi$
$97$	4.29180	0.435766	0.217883	+	0.975975i	$0.430085\pi$
$98$	−7.76393	−0.784276
$99$	2.29180	0.230334
$100$	−10.4164	−1.04164
$101$	8.47214	0.843009	0.421505	−	0.906826i	$-0.361502\pi$
$101$	8.47214	0.843009	0.421505	+	0.906826i	$0.361502\pi$
$102$	0	0
$103$	1.52786	0.150545	0.0752725	−	0.997163i	$-0.476017\pi$
$103$	1.52786	0.150545	0.0752725	+	0.997163i	$0.476017\pi$
$104$	−2.23607	−0.219265
$105$	0	0
$106$	15.5279	1.50820
$107$	2.47214	0.238990	0.119495	−	0.992835i	$-0.461872\pi$
$107$	2.47214	0.238990	0.119495	+	0.992835i	$0.461872\pi$
$108$	0	0
$109$	5.23607	0.501524	0.250762	−	0.968049i	$-0.419319\pi$
$109$	5.23607	0.501524	0.250762	+	0.968049i	$0.419319\pi$
$110$	−2.11146	−0.201319
$111$	0	0
$112$	−3.23607	−0.305780
$113$	−9.41641	−0.885821	−0.442911	−	0.896566i	$-0.646054\pi$
$113$	−9.41641	−0.885821	−0.442911	+	0.896566i	$0.646054\pi$
$114$	0	0
$115$	1.23607	0.115264
$116$	25.4164	2.35985
$117$	−3.00000	−0.277350
$118$	−14.4721	−1.33227
$119$	6.47214	0.593300
$120$	0	0
$121$	−10.4164	−0.946946
$122$	22.3607	2.02444
$123$	0	0
$124$	31.4164	2.82128
$125$	10.4721	0.936656
$126$	21.7082	1.93392
$127$	0.944272	0.0837906	0.0418953	−	0.999122i	$-0.486660\pi$
$127$	0.944272	0.0837906	0.0418953	+	0.999122i	$0.486660\pi$
$128$	15.6525	1.38350
$129$	0	0
$130$	2.76393	0.242413
$131$	−8.94427	−0.781465	−0.390732	−	0.920504i	$-0.627778\pi$
$131$	−8.94427	−0.781465	−0.390732	+	0.920504i	$0.627778\pi$
$132$	0	0
$133$	23.4164	2.03046
$134$	16.1803	1.39777
$135$	0	0
$136$	−4.47214	−0.383482
$137$	−3.70820	−0.316813	−0.158407	−	0.987374i	$-0.550636\pi$
$137$	−3.70820	−0.316813	−0.158407	+	0.987374i	$0.550636\pi$
$138$	0	0
$139$	12.9443	1.09792	0.548959	−	0.835849i	$-0.315024\pi$
$139$	12.9443	1.09792	0.548959	+	0.835849i	$0.315024\pi$
$140$	−12.0000	−1.01419
$141$	0	0
$142$	12.3607	1.03729
$143$	−0.763932	−0.0638832
$144$	3.00000	0.250000
$145$	−10.4721	−0.869664
$146$	−1.05573	−0.0873727
$147$	0	0
$148$	−3.70820	−0.304812
$149$	23.1246	1.89444	0.947221	−	0.320581i	$-0.103878\pi$
$149$	23.1246	1.89444	0.947221	+	0.320581i	$0.103878\pi$
$150$	0	0
$151$	−20.3607	−1.65693	−0.828464	−	0.560042i	$-0.810785\pi$
$151$	−20.3607	−1.65693	−0.828464	+	0.560042i	$0.810785\pi$
$152$	−16.1803	−1.31240
$153$	−6.00000	−0.485071
$154$	5.52786	0.445448
$155$	−12.9443	−1.03971
$156$	0	0
$157$	9.41641	0.751511	0.375756	−	0.926719i	$-0.377383\pi$
$157$	9.41641	0.751511	0.375756	+	0.926719i	$0.377383\pi$
$158$	11.0557	0.879547
$159$	0	0
$160$	−8.29180	−0.655524
$161$	−3.23607	−0.255038
$162$	−20.1246	−1.58114
$163$	−20.9443	−1.64048	−0.820241	−	0.572018i	$-0.806161\pi$
$163$	−20.9443	−1.64048	−0.820241	+	0.572018i	$0.806161\pi$
$164$	30.0000	2.34261
$165$	0	0
$166$	27.2361	2.11393
$167$	−16.9443	−1.31119	−0.655594	−	0.755114i	$-0.727582\pi$
$167$	−16.9443	−1.31119	−0.655594	+	0.755114i	$0.727582\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	5.52786	0.423968
$171$	−21.7082	−1.66007
$172$	31.4164	2.39548
$173$	−5.05573	−0.384380	−0.192190	−	0.981358i	$-0.561559\pi$
$173$	−5.05573	−0.384380	−0.192190	+	0.981358i	$0.561559\pi$
$174$	0	0
$175$	−11.2361	−0.849367
$176$	0.763932	0.0575835
$177$	0	0
$178$	−17.2361	−1.29190
$179$	−3.05573	−0.228396	−0.114198	−	0.993458i	$-0.536430\pi$
$179$	−3.05573	−0.228396	−0.114198	+	0.993458i	$0.536430\pi$
$180$	11.1246	0.829180
$181$	−11.8885	−0.883669	−0.441834	−	0.897097i	$-0.645672\pi$
$181$	−11.8885	−0.883669	−0.441834	+	0.897097i	$0.645672\pi$
$182$	−7.23607	−0.536373
$183$	0	0
$184$	2.23607	0.164845
$185$	1.52786	0.112331
$186$	0	0
$187$	−1.52786	−0.111728
$188$	−12.0000	−0.875190
$189$	0	0
$190$	20.0000	1.45095
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	0	0
$193$	0.472136	0.0339851	0.0169925	−	0.999856i	$-0.494591\pi$
$193$	0.472136	0.0339851	0.0169925	+	0.999856i	$0.494591\pi$
$194$	−9.59675	−0.689006
$195$	0	0
$196$	10.4164	0.744029
$197$	12.4721	0.888603	0.444301	−	0.895877i	$-0.353452\pi$
$197$	12.4721	0.888603	0.444301	+	0.895877i	$0.353452\pi$
$198$	−5.12461	−0.364190
$199$	11.4164	0.809288	0.404644	−	0.914474i	$-0.367395\pi$
$199$	11.4164	0.809288	0.404644	+	0.914474i	$0.367395\pi$
$200$	7.76393	0.548993
$201$	0	0
$202$	−18.9443	−1.33291
$203$	27.4164	1.92425
$204$	0	0
$205$	−12.3607	−0.863307
$206$	−3.41641	−0.238032
$207$	3.00000	0.208514
$208$	−1.00000	−0.0693375
$209$	−5.52786	−0.382370
$210$	0	0
$211$	−13.8885	−0.956127	−0.478063	−	0.878325i	$-0.658661\pi$
$211$	−13.8885	−0.956127	−0.478063	+	0.878325i	$0.658661\pi$
$212$	−20.8328	−1.43080
$213$	0	0
$214$	−5.52786	−0.377877
$215$	−12.9443	−0.882792
$216$	0	0
$217$	33.8885	2.30050
$218$	−11.7082	−0.792980
$219$	0	0
$220$	2.83282	0.190988
$221$	2.00000	0.134535
$222$	0	0
$223$	−28.3607	−1.89917	−0.949586	−	0.313507i	$-0.898496\pi$
$223$	−28.3607	−1.89917	−0.949586	+	0.313507i	$0.898496\pi$
$224$	21.7082	1.45044
$225$	10.4164	0.694427
$226$	21.0557	1.40061
$227$	5.70820	0.378867	0.189433	−	0.981894i	$-0.439335\pi$
$227$	5.70820	0.378867	0.189433	+	0.981894i	$0.439335\pi$
$228$	0	0
$229$	24.6525	1.62908	0.814541	−	0.580106i	$-0.196989\pi$
$229$	24.6525	1.62908	0.814541	+	0.580106i	$0.196989\pi$
$230$	−2.76393	−0.182248
$231$	0	0
$232$	−18.9443	−1.24375
$233$	−0.472136	−0.0309307	−0.0154653	−	0.999880i	$-0.504923\pi$
$233$	−0.472136	−0.0309307	−0.0154653	+	0.999880i	$0.504923\pi$
$234$	6.70820	0.438529
$235$	4.94427	0.322529
$236$	19.4164	1.26390
$237$	0	0
$238$	−14.4721	−0.938089
$239$	8.94427	0.578557	0.289278	−	0.957245i	$-0.406585\pi$
$239$	8.94427	0.578557	0.289278	+	0.957245i	$0.406585\pi$
$240$	0	0
$241$	−26.1803	−1.68642	−0.843212	−	0.537581i	$-0.819338\pi$
$241$	−26.1803	−1.68642	−0.843212	+	0.537581i	$0.819338\pi$
$242$	23.2918	1.49725
$243$	0	0
$244$	−30.0000	−1.92055
$245$	−4.29180	−0.274193
$246$	0	0
$247$	7.23607	0.460420
$248$	−23.4164	−1.48694
$249$	0	0
$250$	−23.4164	−1.48098
$251$	10.4721	0.660995	0.330498	−	0.943807i	$-0.392783\pi$
$251$	10.4721	0.660995	0.330498	+	0.943807i	$0.392783\pi$
$252$	−29.1246	−1.83468
$253$	0.763932	0.0480280
$254$	−2.11146	−0.132485
$255$	0	0
$256$	−9.00000	−0.562500
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	0	0
$259$	−4.00000	−0.248548
$260$	−3.70820	−0.229973
$261$	−25.4164	−1.57324
$262$	20.0000	1.23560
$263$	9.88854	0.609754	0.304877	−	0.952392i	$-0.401385\pi$
$263$	9.88854	0.609754	0.304877	+	0.952392i	$0.401385\pi$
$264$	0	0
$265$	8.58359	0.527286
$266$	−52.3607	−3.21044
$267$	0	0
$268$	−21.7082	−1.32604
$269$	1.05573	0.0643689	0.0321844	−	0.999482i	$-0.489754\pi$
$269$	1.05573	0.0643689	0.0321844	+	0.999482i	$0.489754\pi$
$270$	0	0
$271$	7.05573	0.428605	0.214302	−	0.976767i	$-0.431252\pi$
$271$	7.05573	0.428605	0.214302	+	0.976767i	$0.431252\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	8.29180	0.500926
$275$	2.65248	0.159950
$276$	0	0
$277$	26.3607	1.58386	0.791930	−	0.610612i	$-0.209077\pi$
$277$	26.3607	1.58386	0.791930	+	0.610612i	$0.209077\pi$
$278$	−28.9443	−1.73596
$279$	−31.4164	−1.88085
$280$	8.94427	0.534522
$281$	−6.76393	−0.403502	−0.201751	−	0.979437i	$-0.564663\pi$
$281$	−6.76393	−0.403502	−0.201751	+	0.979437i	$0.564663\pi$
$282$	0	0
$283$	−2.47214	−0.146953	−0.0734766	−	0.997297i	$-0.523409\pi$
$283$	−2.47214	−0.146953	−0.0734766	+	0.997297i	$0.523409\pi$
$284$	−16.5836	−0.984055
$285$	0	0
$286$	1.70820	0.101008
$287$	32.3607	1.91019
$288$	−20.1246	−1.18585
$289$	−13.0000	−0.764706
$290$	23.4164	1.37506
$291$	0	0
$292$	1.41641	0.0828890
$293$	27.7082	1.61873	0.809365	−	0.587306i	$-0.199811\pi$
$293$	27.7082	1.61873	0.809365	+	0.587306i	$0.199811\pi$
$294$	0	0
$295$	−8.00000	−0.465778
$296$	2.76393	0.160650
$297$	0	0
$298$	−51.7082	−2.99538
$299$	−1.00000	−0.0578315
$300$	0	0
$301$	33.8885	1.95330
$302$	45.5279	2.61983
$303$	0	0
$304$	−7.23607	−0.415017
$305$	12.3607	0.707770
$306$	13.4164	0.766965
$307$	−1.52786	−0.0871998	−0.0435999	−	0.999049i	$-0.513883\pi$
$307$	−1.52786	−0.0871998	−0.0435999	+	0.999049i	$0.513883\pi$
$308$	−7.41641	−0.422589
$309$	0	0
$310$	28.9443	1.64392
$311$	−11.0557	−0.626913	−0.313456	−	0.949603i	$-0.601487\pi$
$311$	−11.0557	−0.626913	−0.313456	+	0.949603i	$0.601487\pi$
$312$	0	0
$313$	−20.4721	−1.15715	−0.578577	−	0.815628i	$-0.696392\pi$
$313$	−20.4721	−1.15715	−0.578577	+	0.815628i	$0.696392\pi$
$314$	−21.0557	−1.18824
$315$	12.0000	0.676123
$316$	−14.8328	−0.834411
$317$	−24.4721	−1.37449	−0.687246	−	0.726425i	$-0.741181\pi$
$317$	−24.4721	−1.37449	−0.687246	+	0.726425i	$0.741181\pi$
$318$	0	0
$319$	−6.47214	−0.362370
$320$	16.0689	0.898278
$321$	0	0
$322$	7.23607	0.403250
$323$	14.4721	0.805251
$324$	27.0000	1.50000
$325$	−3.47214	−0.192599
$326$	46.8328	2.59383
$327$	0	0
$328$	−22.3607	−1.23466
$329$	−12.9443	−0.713641
$330$	0	0
$331$	11.0557	0.607678	0.303839	−	0.952723i	$-0.401732\pi$
$331$	11.0557	0.607678	0.303839	+	0.952723i	$0.401732\pi$
$332$	−36.5410	−2.00545
$333$	3.70820	0.203208
$334$	37.8885	2.07317
$335$	8.94427	0.488678
$336$	0	0
$337$	10.0000	0.544735	0.272367	−	0.962193i	$-0.412193\pi$
$337$	10.0000	0.544735	0.272367	+	0.962193i	$0.412193\pi$
$338$	−2.23607	−0.121626
$339$	0	0
$340$	−7.41641	−0.402211
$341$	−8.00000	−0.433224
$342$	48.5410	2.62480
$343$	−11.4164	−0.616428
$344$	−23.4164	−1.26253
$345$	0	0
$346$	11.3050	0.607758
$347$	21.8885	1.17504	0.587519	−	0.809210i	$-0.300105\pi$
$347$	21.8885	1.17504	0.587519	+	0.809210i	$0.300105\pi$
$348$	0	0
$349$	−26.3607	−1.41105	−0.705527	−	0.708683i	$-0.749290\pi$
$349$	−26.3607	−1.41105	−0.705527	+	0.708683i	$0.749290\pi$
$350$	25.1246	1.34297
$351$	0	0
$352$	−5.12461	−0.273143
$353$	−7.52786	−0.400668	−0.200334	−	0.979728i	$-0.564203\pi$
$353$	−7.52786	−0.400668	−0.200334	+	0.979728i	$0.564203\pi$
$354$	0	0
$355$	6.83282	0.362648
$356$	23.1246	1.22560
$357$	0	0
$358$	6.83282	0.361126
$359$	−6.29180	−0.332068	−0.166034	−	0.986120i	$-0.553096\pi$
$359$	−6.29180	−0.332068	−0.166034	+	0.986120i	$0.553096\pi$
$360$	−8.29180	−0.437016
$361$	33.3607	1.75583
$362$	26.5836	1.39720
$363$	0	0
$364$	9.70820	0.508848
$365$	−0.583592	−0.0305466
$366$	0	0
$367$	−27.4164	−1.43112	−0.715562	−	0.698549i	$-0.753830\pi$
$367$	−27.4164	−1.43112	−0.715562	+	0.698549i	$0.753830\pi$
$368$	1.00000	0.0521286
$369$	−30.0000	−1.56174
$370$	−3.41641	−0.177611
$371$	−22.4721	−1.16670
$372$	0	0
$373$	−5.41641	−0.280451	−0.140225	−	0.990120i	$-0.544783\pi$
$373$	−5.41641	−0.280451	−0.140225	+	0.990120i	$0.544783\pi$
$374$	3.41641	0.176658
$375$	0	0
$376$	8.94427	0.461266
$377$	8.47214	0.436337
$378$	0	0
$379$	8.76393	0.450173	0.225086	−	0.974339i	$-0.427734\pi$
$379$	8.76393	0.450173	0.225086	+	0.974339i	$0.427734\pi$
$380$	−26.8328	−1.37649
$381$	0	0
$382$	17.8885	0.915258
$383$	33.7082	1.72241	0.861204	−	0.508259i	$-0.169711\pi$
$383$	33.7082	1.72241	0.861204	+	0.508259i	$0.169711\pi$
$384$	0	0
$385$	3.05573	0.155734
$386$	−1.05573	−0.0537351
$387$	−31.4164	−1.59699
$388$	12.8754	0.653649
$389$	9.05573	0.459144	0.229572	−	0.973292i	$-0.426267\pi$
$389$	9.05573	0.459144	0.229572	+	0.973292i	$0.426267\pi$
$390$	0	0
$391$	−2.00000	−0.101144
$392$	−7.76393	−0.392138
$393$	0	0
$394$	−27.8885	−1.40500
$395$	6.11146	0.307501
$396$	6.87539	0.345501
$397$	9.05573	0.454494	0.227247	−	0.973837i	$-0.427028\pi$
$397$	9.05573	0.454494	0.227247	+	0.973837i	$0.427028\pi$
$398$	−25.5279	−1.27960
$399$	0	0
$400$	3.47214	0.173607
$401$	−34.5410	−1.72490	−0.862448	−	0.506146i	$-0.831070\pi$
$401$	−34.5410	−1.72490	−0.862448	+	0.506146i	$0.831070\pi$
$402$	0	0
$403$	10.4721	0.521654
$404$	25.4164	1.26451
$405$	−11.1246	−0.552786
$406$	−61.3050	−3.04251
$407$	0.944272	0.0468058
$408$	0	0
$409$	5.41641	0.267824	0.133912	−	0.990993i	$-0.457246\pi$
$409$	5.41641	0.267824	0.133912	+	0.990993i	$0.457246\pi$
$410$	27.6393	1.36501
$411$	0	0
$412$	4.58359	0.225817
$413$	20.9443	1.03060
$414$	−6.70820	−0.329690
$415$	15.0557	0.739057
$416$	6.70820	0.328897
$417$	0	0
$418$	12.3607	0.604581
$419$	4.00000	0.195413	0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000	0.195413	0.0977064	+	0.995215i	$0.468849\pi$
$420$	0	0
$421$	1.81966	0.0886848	0.0443424	−	0.999016i	$-0.485881\pi$
$421$	1.81966	0.0886848	0.0443424	+	0.999016i	$0.485881\pi$
$422$	31.0557	1.51177
$423$	12.0000	0.583460
$424$	15.5279	0.754100
$425$	−6.94427	−0.336847
$426$	0	0
$427$	−32.3607	−1.56604
$428$	7.41641	0.358486
$429$	0	0
$430$	28.9443	1.39582
$431$	35.5967	1.71464	0.857318	−	0.514788i	$-0.172129\pi$
$431$	35.5967	1.71464	0.857318	+	0.514788i	$0.172129\pi$
$432$	0	0
$433$	−23.5279	−1.13068	−0.565338	−	0.824859i	$-0.691254\pi$
$433$	−23.5279	−1.13068	−0.565338	+	0.824859i	$0.691254\pi$
$434$	−75.7771	−3.63742
$435$	0	0
$436$	15.7082	0.752287
$437$	−7.23607	−0.346148
$438$	0	0
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	−2.11146	−0.100660
$441$	−10.4164	−0.496019
$442$	−4.47214	−0.212718
$443$	18.8328	0.894774	0.447387	−	0.894340i	$-0.352355\pi$
$443$	18.8328	0.894774	0.447387	+	0.894340i	$0.352355\pi$
$444$	0	0
$445$	−9.52786	−0.451664
$446$	63.4164	3.00285
$447$	0	0
$448$	−42.0689	−1.98757
$449$	14.9443	0.705264	0.352632	−	0.935762i	$-0.385287\pi$
$449$	14.9443	0.705264	0.352632	+	0.935762i	$0.385287\pi$
$450$	−23.2918	−1.09799
$451$	−7.63932	−0.359722
$452$	−28.2492	−1.32873
$453$	0	0
$454$	−12.7639	−0.599041
$455$	−4.00000	−0.187523
$456$	0	0
$457$	38.1803	1.78600	0.893000	−	0.450056i	$-0.148596\pi$
$457$	38.1803	1.78600	0.893000	+	0.450056i	$0.148596\pi$
$458$	−55.1246	−2.57580
$459$	0	0
$460$	3.70820	0.172896
$461$	7.52786	0.350608	0.175304	−	0.984514i	$-0.443909\pi$
$461$	7.52786	0.350608	0.175304	+	0.984514i	$0.443909\pi$
$462$	0	0
$463$	12.0000	0.557687	0.278844	−	0.960337i	$-0.410049\pi$
$463$	12.0000	0.557687	0.278844	+	0.960337i	$0.410049\pi$
$464$	−8.47214	−0.393309
$465$	0	0
$466$	1.05573	0.0489057
$467$	−28.0000	−1.29569	−0.647843	−	0.761774i	$-0.724329\pi$
$467$	−28.0000	−1.29569	−0.647843	+	0.761774i	$0.724329\pi$
$468$	−9.00000	−0.416025
$469$	−23.4164	−1.08127
$470$	−11.0557	−0.509963
$471$	0	0
$472$	−14.4721	−0.666134
$473$	−8.00000	−0.367840
$474$	0	0
$475$	−25.1246	−1.15280
$476$	19.4164	0.889950
$477$	20.8328	0.953869
$478$	−20.0000	−0.914779
$479$	−22.6525	−1.03502	−0.517509	−	0.855678i	$-0.673141\pi$
$479$	−22.6525	−1.03502	−0.517509	+	0.855678i	$0.673141\pi$
$480$	0	0
$481$	−1.23607	−0.0563598
$482$	58.5410	2.66647
$483$	0	0
$484$	−31.2492	−1.42042
$485$	−5.30495	−0.240885
$486$	0	0
$487$	−15.4164	−0.698584	−0.349292	−	0.937014i	$-0.613578\pi$
$487$	−15.4164	−0.698584	−0.349292	+	0.937014i	$0.613578\pi$
$488$	22.3607	1.01222
$489$	0	0
$490$	9.59675	0.433537
$491$	−16.9443	−0.764684	−0.382342	−	0.924021i	$-0.624882\pi$
$491$	−16.9443	−0.764684	−0.382342	+	0.924021i	$0.624882\pi$
$492$	0	0
$493$	16.9443	0.763132
$494$	−16.1803	−0.727988
$495$	−2.83282	−0.127326
$496$	−10.4721	−0.470213
$497$	−17.8885	−0.802411
$498$	0	0
$499$	14.4721	0.647862	0.323931	−	0.946081i	$-0.394995\pi$
$499$	14.4721	0.647862	0.323931	+	0.946081i	$0.394995\pi$
$500$	31.4164	1.40498
$501$	0	0
$502$	−23.4164	−1.04513
$503$	−11.0557	−0.492951	−0.246475	−	0.969149i	$-0.579272\pi$
$503$	−11.0557	−0.492951	−0.246475	+	0.969149i	$0.579272\pi$
$504$	21.7082	0.966960
$505$	−10.4721	−0.466004
$506$	−1.70820	−0.0759389
$507$	0	0
$508$	2.83282	0.125686
$509$	28.4721	1.26201	0.631003	−	0.775781i	$-0.282644\pi$
$509$	28.4721	1.26201	0.631003	+	0.775781i	$0.282644\pi$
$510$	0	0
$511$	1.52786	0.0675887
$512$	−11.1803	−0.494106
$513$	0	0
$514$	31.3050	1.38080
$515$	−1.88854	−0.0832192
$516$	0	0
$517$	3.05573	0.134391
$518$	8.94427	0.392989
$519$	0	0
$520$	2.76393	0.121206
$521$	23.3050	1.02101	0.510504	−	0.859875i	$-0.329459\pi$
$521$	23.3050	1.02101	0.510504	+	0.859875i	$0.329459\pi$
$522$	56.8328	2.48750
$523$	−0.583592	−0.0255187	−0.0127594	−	0.999919i	$-0.504062\pi$
$523$	−0.583592	−0.0255187	−0.0127594	+	0.999919i	$0.504062\pi$
$524$	−26.8328	−1.17220
$525$	0	0
$526$	−22.1115	−0.964105
$527$	20.9443	0.912347
$528$	0	0
$529$	1.00000	0.0434783
$530$	−19.1935	−0.833712
$531$	−19.4164	−0.842600
$532$	70.2492	3.04569
$533$	10.0000	0.433148
$534$	0	0
$535$	−3.05573	−0.132111
$536$	16.1803	0.698884
$537$	0	0
$538$	−2.36068	−0.101776
$539$	−2.65248	−0.114250
$540$	0	0
$541$	38.3607	1.64925	0.824627	−	0.565677i	$-0.191385\pi$
$541$	38.3607	1.64925	0.824627	+	0.565677i	$0.191385\pi$
$542$	−15.7771	−0.677684
$543$	0	0
$544$	13.4164	0.575224
$545$	−6.47214	−0.277236
$546$	0	0
$547$	−23.0557	−0.985792	−0.492896	−	0.870088i	$-0.664062\pi$
$547$	−23.0557	−0.985792	−0.492896	+	0.870088i	$0.664062\pi$
$548$	−11.1246	−0.475220
$549$	30.0000	1.28037
$550$	−5.93112	−0.252904
$551$	61.3050	2.61168
$552$	0	0
$553$	−16.0000	−0.680389
$554$	−58.9443	−2.50430
$555$	0	0
$556$	38.8328	1.64688
$557$	3.70820	0.157122	0.0785608	−	0.996909i	$-0.474968\pi$
$557$	3.70820	0.157122	0.0785608	+	0.996909i	$0.474968\pi$
$558$	70.2492	2.97389
$559$	10.4721	0.442924
$560$	4.00000	0.169031
$561$	0	0
$562$	15.1246	0.637993
$563$	7.05573	0.297363	0.148682	−	0.988885i	$-0.452497\pi$
$563$	7.05573	0.297363	0.148682	+	0.988885i	$0.452497\pi$
$564$	0	0
$565$	11.6393	0.489670
$566$	5.52786	0.232353
$567$	29.1246	1.22312
$568$	12.3607	0.518643
$569$	2.36068	0.0989648	0.0494824	−	0.998775i	$-0.484243\pi$
$569$	2.36068	0.0989648	0.0494824	+	0.998775i	$0.484243\pi$
$570$	0	0
$571$	−29.8885	−1.25080	−0.625398	−	0.780306i	$-0.715063\pi$
$571$	−29.8885	−1.25080	−0.625398	+	0.780306i	$0.715063\pi$
$572$	−2.29180	−0.0958248
$573$	0	0
$574$	−72.3607	−3.02028
$575$	3.47214	0.144798
$576$	39.0000	1.62500
$577$	−33.4164	−1.39114	−0.695572	−	0.718457i	$-0.744849\pi$
$577$	−33.4164	−1.39114	−0.695572	+	0.718457i	$0.744849\pi$
$578$	29.0689	1.20911
$579$	0	0
$580$	−31.4164	−1.30450
$581$	−39.4164	−1.63527
$582$	0	0
$583$	5.30495	0.219709
$584$	−1.05573	−0.0436863
$585$	3.70820	0.153315
$586$	−61.9574	−2.55944
$587$	−33.8885	−1.39873	−0.699365	−	0.714765i	$-0.746534\pi$
$587$	−33.8885	−1.39873	−0.699365	+	0.714765i	$0.746534\pi$
$588$	0	0
$589$	75.7771	3.12234
$590$	17.8885	0.736460
$591$	0	0
$592$	1.23607	0.0508021
$593$	−14.0000	−0.574911	−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000	−0.574911	−0.287456	+	0.957794i	$0.592809\pi$
$594$	0	0
$595$	−8.00000	−0.327968
$596$	69.3738	2.84166
$597$	0	0
$598$	2.23607	0.0914396
$599$	−47.7771	−1.95212	−0.976059	−	0.217504i	$-0.930208\pi$
$599$	−47.7771	−1.95212	−0.976059	+	0.217504i	$0.930208\pi$
$600$	0	0
$601$	−28.8328	−1.17612	−0.588058	−	0.808819i	$-0.700107\pi$
$601$	−28.8328	−1.17612	−0.588058	+	0.808819i	$0.700107\pi$
$602$	−75.7771	−3.08844
$603$	21.7082	0.884026
$604$	−61.0820	−2.48539
$605$	12.8754	0.523459
$606$	0	0
$607$	5.88854	0.239009	0.119504	−	0.992834i	$-0.461869\pi$
$607$	5.88854	0.239009	0.119504	+	0.992834i	$0.461869\pi$
$608$	48.5410	1.96860
$609$	0	0
$610$	−27.6393	−1.11908
$611$	−4.00000	−0.161823
$612$	−18.0000	−0.727607
$613$	−17.2361	−0.696158	−0.348079	−	0.937465i	$-0.613166\pi$
$613$	−17.2361	−0.696158	−0.348079	+	0.937465i	$0.613166\pi$
$614$	3.41641	0.137875
$615$	0	0
$616$	5.52786	0.222724
$617$	6.18034	0.248811	0.124406	−	0.992231i	$-0.460298\pi$
$617$	6.18034	0.248811	0.124406	+	0.992231i	$0.460298\pi$
$618$	0	0
$619$	10.6525	0.428159	0.214080	−	0.976816i	$-0.431325\pi$
$619$	10.6525	0.428159	0.214080	+	0.976816i	$0.431325\pi$
$620$	−38.8328	−1.55956
$621$	0	0
$622$	24.7214	0.991236
$623$	24.9443	0.999371
$624$	0	0
$625$	4.41641	0.176656
$626$	45.7771	1.82962
$627$	0	0
$628$	28.2492	1.12727
$629$	−2.47214	−0.0985705
$630$	−26.8328	−1.06904
$631$	32.5410	1.29544	0.647719	−	0.761880i	$-0.275723\pi$
$631$	32.5410	1.29544	0.647719	+	0.761880i	$0.275723\pi$
$632$	11.0557	0.439773
$633$	0	0
$634$	54.7214	2.17326
$635$	−1.16718	−0.0463183
$636$	0	0
$637$	3.47214	0.137571
$638$	14.4721	0.572957
$639$	16.5836	0.656037
$640$	−19.3475	−0.764778
$641$	5.05573	0.199689	0.0998446	−	0.995003i	$-0.468165\pi$
$641$	5.05573	0.199689	0.0998446	+	0.995003i	$0.468165\pi$
$642$	0	0
$643$	23.2361	0.916341	0.458171	−	0.888864i	$-0.348505\pi$
$643$	23.2361	0.916341	0.458171	+	0.888864i	$0.348505\pi$
$644$	−9.70820	−0.382557
$645$	0	0
$646$	−32.3607	−1.27321
$647$	−42.8328	−1.68393	−0.841966	−	0.539531i	$-0.818602\pi$
$647$	−42.8328	−1.68393	−0.841966	+	0.539531i	$0.818602\pi$
$648$	−20.1246	−0.790569
$649$	−4.94427	−0.194080
$650$	7.76393	0.304526
$651$	0	0
$652$	−62.8328	−2.46072
$653$	−35.3050	−1.38159	−0.690795	−	0.723051i	$-0.742739\pi$
$653$	−35.3050	−1.38159	−0.690795	+	0.723051i	$0.742739\pi$
$654$	0	0
$655$	11.0557	0.431983
$656$	−10.0000	−0.390434
$657$	−1.41641	−0.0552593
$658$	28.9443	1.12837
$659$	−49.3050	−1.92065	−0.960324	−	0.278886i	$-0.910035\pi$
$659$	−49.3050	−1.92065	−0.960324	+	0.278886i	$0.910035\pi$
$660$	0	0
$661$	−30.1803	−1.17388	−0.586940	−	0.809631i	$-0.699667\pi$
$661$	−30.1803	−1.17388	−0.586940	+	0.809631i	$0.699667\pi$
$662$	−24.7214	−0.960823
$663$	0	0
$664$	27.2361	1.05696
$665$	−28.9443	−1.12241
$666$	−8.29180	−0.321301
$667$	−8.47214	−0.328042
$668$	−50.8328	−1.96678
$669$	0	0
$670$	−20.0000	−0.772667
$671$	7.63932	0.294913
$672$	0	0
$673$	−23.8885	−0.920836	−0.460418	−	0.887702i	$-0.652300\pi$
$673$	−23.8885	−0.920836	−0.460418	+	0.887702i	$0.652300\pi$
$674$	−22.3607	−0.861301
$675$	0	0
$676$	3.00000	0.115385
$677$	36.8328	1.41560	0.707800	−	0.706413i	$-0.249688\pi$
$677$	36.8328	1.41560	0.707800	+	0.706413i	$0.249688\pi$
$678$	0	0
$679$	13.8885	0.532993
$680$	5.52786	0.211984
$681$	0	0
$682$	17.8885	0.684988
$683$	−34.2492	−1.31051	−0.655255	−	0.755408i	$-0.727439\pi$
$683$	−34.2492	−1.31051	−0.655255	+	0.755408i	$0.727439\pi$
$684$	−65.1246	−2.49010
$685$	4.58359	0.175130
$686$	25.5279	0.974658
$687$	0	0
$688$	−10.4721	−0.399246
$689$	−6.94427	−0.264556
$690$	0	0
$691$	−19.7771	−0.752356	−0.376178	−	0.926547i	$-0.622762\pi$
$691$	−19.7771	−0.752356	−0.376178	+	0.926547i	$0.622762\pi$
$692$	−15.1672	−0.576570
$693$	7.41641	0.281726
$694$	−48.9443	−1.85790
$695$	−16.0000	−0.606915
$696$	0	0
$697$	20.0000	0.757554
$698$	58.9443	2.23107
$699$	0	0
$700$	−33.7082	−1.27405
$701$	10.9443	0.413359	0.206680	−	0.978409i	$-0.433734\pi$
$701$	10.9443	0.413359	0.206680	+	0.978409i	$0.433734\pi$
$702$	0	0
$703$	−8.94427	−0.337340
$704$	9.93112	0.374293
$705$	0	0
$706$	16.8328	0.633511
$707$	27.4164	1.03110
$708$	0	0
$709$	−12.6525	−0.475174	−0.237587	−	0.971366i	$-0.576356\pi$
$709$	−12.6525	−0.475174	−0.237587	+	0.971366i	$0.576356\pi$
$710$	−15.2786	−0.573397
$711$	14.8328	0.556274
$712$	−17.2361	−0.645949
$713$	−10.4721	−0.392185
$714$	0	0
$715$	0.944272	0.0353138
$716$	−9.16718	−0.342594
$717$	0	0
$718$	14.0689	0.525046
$719$	−8.94427	−0.333565	−0.166783	−	0.985994i	$-0.553338\pi$
$719$	−8.94427	−0.333565	−0.166783	+	0.985994i	$0.553338\pi$
$720$	−3.70820	−0.138197
$721$	4.94427	0.184134
$722$	−74.5967	−2.77620
$723$	0	0
$724$	−35.6656	−1.32550
$725$	−29.4164	−1.09250
$726$	0	0
$727$	38.4721	1.42685	0.713426	−	0.700730i	$-0.247142\pi$
$727$	38.4721	1.42685	0.713426	+	0.700730i	$0.247142\pi$
$728$	−7.23607	−0.268187
$729$	−27.0000	−1.00000
$730$	1.30495	0.0482984
$731$	20.9443	0.774652
$732$	0	0
$733$	2.18034	0.0805327	0.0402663	−	0.999189i	$-0.487179\pi$
$733$	2.18034	0.0805327	0.0402663	+	0.999189i	$0.487179\pi$
$734$	61.3050	2.26281
$735$	0	0
$736$	−6.70820	−0.247268
$737$	5.52786	0.203621
$738$	67.0820	2.46932
$739$	−6.83282	−0.251349	−0.125675	−	0.992072i	$-0.540110\pi$
$739$	−6.83282	−0.251349	−0.125675	+	0.992072i	$0.540110\pi$
$740$	4.58359	0.168496
$741$	0	0
$742$	50.2492	1.84471
$743$	−35.5967	−1.30592	−0.652959	−	0.757393i	$-0.726473\pi$
$743$	−35.5967	−1.30592	−0.652959	+	0.757393i	$0.726473\pi$
$744$	0	0
$745$	−28.5836	−1.04722
$746$	12.1115	0.443432
$747$	36.5410	1.33697
$748$	−4.58359	−0.167593
$749$	8.00000	0.292314
$750$	0	0
$751$	12.5836	0.459182	0.229591	−	0.973287i	$-0.426261\pi$
$751$	12.5836	0.459182	0.229591	+	0.973287i	$0.426261\pi$
$752$	4.00000	0.145865
$753$	0	0
$754$	−18.9443	−0.689910
$755$	25.1672	0.915928
$756$	0	0
$757$	−6.94427	−0.252394	−0.126197	−	0.992005i	$-0.540277\pi$
$757$	−6.94427	−0.252394	−0.126197	+	0.992005i	$0.540277\pi$
$758$	−19.5967	−0.711786
$759$	0	0
$760$	20.0000	0.725476
$761$	−25.4164	−0.921344	−0.460672	−	0.887570i	$-0.652392\pi$
$761$	−25.4164	−0.921344	−0.460672	+	0.887570i	$0.652392\pi$
$762$	0	0
$763$	16.9443	0.613424
$764$	−24.0000	−0.868290
$765$	7.41641	0.268141
$766$	−75.3738	−2.72337
$767$	6.47214	0.233695
$768$	0	0
$769$	−4.87539	−0.175811	−0.0879055	−	0.996129i	$-0.528017\pi$
$769$	−4.87539	−0.175811	−0.0879055	+	0.996129i	$0.528017\pi$
$770$	−6.83282	−0.246238
$771$	0	0
$772$	1.41641	0.0509776
$773$	49.0132	1.76288	0.881440	−	0.472295i	$-0.156574\pi$
$773$	49.0132	1.76288	0.881440	+	0.472295i	$0.156574\pi$
$774$	70.2492	2.52506
$775$	−36.3607	−1.30611
$776$	−9.59675	−0.344503
$777$	0	0
$778$	−20.2492	−0.725970
$779$	72.3607	2.59259
$780$	0	0
$781$	4.22291	0.151108
$782$	4.47214	0.159923
$783$	0	0
$784$	−3.47214	−0.124005
$785$	−11.6393	−0.415425
$786$	0	0
$787$	13.7082	0.488645	0.244322	−	0.969694i	$-0.421434\pi$
$787$	13.7082	0.488645	0.244322	+	0.969694i	$0.421434\pi$
$788$	37.4164	1.33290
$789$	0	0
$790$	−13.6656	−0.486201
$791$	−30.4721	−1.08346
$792$	−5.12461	−0.182095
$793$	−10.0000	−0.355110
$794$	−20.2492	−0.718618
$795$	0	0
$796$	34.2492	1.21393
$797$	−8.11146	−0.287323	−0.143661	−	0.989627i	$-0.545888\pi$
$797$	−8.11146	−0.287323	−0.143661	+	0.989627i	$0.545888\pi$
$798$	0	0
$799$	−8.00000	−0.283020
$800$	−23.2918	−0.823489
$801$	−23.1246	−0.817068
$802$	77.2361	2.72730
$803$	−0.360680	−0.0127281
$804$	0	0
$805$	4.00000	0.140981
$806$	−23.4164	−0.824808
$807$	0	0
$808$	−18.9443	−0.666457
$809$	27.3050	0.959991	0.479995	−	0.877271i	$-0.340638\pi$
$809$	27.3050	0.959991	0.479995	+	0.877271i	$0.340638\pi$
$810$	24.8754	0.874032
$811$	24.0000	0.842754	0.421377	−	0.906886i	$-0.361547\pi$
$811$	24.0000	0.842754	0.421377	+	0.906886i	$0.361547\pi$
$812$	82.2492	2.88638
$813$	0	0
$814$	−2.11146	−0.0740065
$815$	25.8885	0.906836
$816$	0	0
$817$	75.7771	2.65110
$818$	−12.1115	−0.423467
$819$	−9.70820	−0.339232
$820$	−37.0820	−1.29496
$821$	−10.0000	−0.349002	−0.174501	−	0.984657i	$-0.555831\pi$
$821$	−10.0000	−0.349002	−0.174501	+	0.984657i	$0.555831\pi$
$822$	0	0
$823$	36.0000	1.25488	0.627441	−	0.778664i	$-0.284103\pi$
$823$	36.0000	1.25488	0.627441	+	0.778664i	$0.284103\pi$
$824$	−3.41641	−0.119016
$825$	0	0
$826$	−46.8328	−1.62952
$827$	−44.5410	−1.54884	−0.774422	−	0.632670i	$-0.781959\pi$
$827$	−44.5410	−1.54884	−0.774422	+	0.632670i	$0.781959\pi$
$828$	9.00000	0.312772
$829$	−14.9443	−0.519036	−0.259518	−	0.965738i	$-0.583564\pi$
$829$	−14.9443	−0.519036	−0.259518	+	0.965738i	$0.583564\pi$
$830$	−33.6656	−1.16855
$831$	0	0
$832$	−13.0000	−0.450694
$833$	6.94427	0.240605
$834$	0	0
$835$	20.9443	0.724806
$836$	−16.5836	−0.573556
$837$	0	0
$838$	−8.94427	−0.308975
$839$	4.76393	0.164469	0.0822346	−	0.996613i	$-0.473794\pi$
$839$	4.76393	0.164469	0.0822346	+	0.996613i	$0.473794\pi$
$840$	0	0
$841$	42.7771	1.47507
$842$	−4.06888	−0.140223
$843$	0	0
$844$	−41.6656	−1.43419
$845$	−1.23607	−0.0425220
$846$	−26.8328	−0.922531
$847$	−33.7082	−1.15823
$848$	6.94427	0.238467
$849$	0	0
$850$	15.5279	0.532601
$851$	1.23607	0.0423719
$852$	0	0
$853$	−18.0000	−0.616308	−0.308154	−	0.951336i	$-0.599711\pi$
$853$	−18.0000	−0.616308	−0.308154	+	0.951336i	$0.599711\pi$
$854$	72.3607	2.47613
$855$	26.8328	0.917663
$856$	−5.52786	−0.188939
$857$	−22.5836	−0.771441	−0.385720	−	0.922616i	$-0.626047\pi$
$857$	−22.5836	−0.771441	−0.385720	+	0.922616i	$0.626047\pi$
$858$	0	0
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	−38.8328	−1.32419
$861$	0	0
$862$	−79.5967	−2.71108
$863$	7.41641	0.252457	0.126229	−	0.992001i	$-0.459713\pi$
$863$	7.41641	0.252457	0.126229	+	0.992001i	$0.459713\pi$
$864$	0	0
$865$	6.24922	0.212480
$866$	52.6099	1.78776
$867$	0	0
$868$	101.666	3.45076
$869$	3.77709	0.128129
$870$	0	0
$871$	−7.23607	−0.245185
$872$	−11.7082	−0.396490
$873$	−12.8754	−0.435766
$874$	16.1803	0.547308
$875$	33.8885	1.14564
$876$	0	0
$877$	12.4721	0.421154	0.210577	−	0.977577i	$-0.432466\pi$
$877$	12.4721	0.421154	0.210577	+	0.977577i	$0.432466\pi$
$878$	−44.7214	−1.50927
$879$	0	0
$880$	−0.944272	−0.0318314
$881$	−1.41641	−0.0477200	−0.0238600	−	0.999715i	$-0.507596\pi$
$881$	−1.41641	−0.0477200	−0.0238600	+	0.999715i	$0.507596\pi$
$882$	23.2918	0.784276
$883$	12.0000	0.403832	0.201916	−	0.979403i	$-0.435283\pi$
$883$	12.0000	0.403832	0.201916	+	0.979403i	$0.435283\pi$
$884$	6.00000	0.201802
$885$	0	0
$886$	−42.1115	−1.41476
$887$	11.0557	0.371215	0.185608	−	0.982624i	$-0.440575\pi$
$887$	11.0557	0.371215	0.185608	+	0.982624i	$0.440575\pi$
$888$	0	0
$889$	3.05573	0.102486
$890$	21.3050	0.714144
$891$	−6.87539	−0.230334
$892$	−85.0820	−2.84876
$893$	−28.9443	−0.968583
$894$	0	0
$895$	3.77709	0.126254
$896$	50.6525	1.69218
$897$	0	0
$898$	−33.4164	−1.11512
$899$	88.7214	2.95902
$900$	31.2492	1.04164
$901$	−13.8885	−0.462694
$902$	17.0820	0.568770
$903$	0	0
$904$	21.0557	0.700303
$905$	14.6950	0.488480
$906$	0	0
$907$	50.8328	1.68788	0.843938	−	0.536441i	$-0.180232\pi$
$907$	50.8328	1.68788	0.843938	+	0.536441i	$0.180232\pi$
$908$	17.1246	0.568300
$909$	−25.4164	−0.843009
$910$	8.94427	0.296500
$911$	−43.4164	−1.43845	−0.719225	−	0.694777i	$-0.755503\pi$
$911$	−43.4164	−1.43845	−0.719225	+	0.694777i	$0.755503\pi$
$912$	0	0
$913$	9.30495	0.307949
$914$	−85.3738	−2.82392
$915$	0	0
$916$	73.9574	2.44362
$917$	−28.9443	−0.955824
$918$	0	0
$919$	37.3050	1.23058	0.615288	−	0.788302i	$-0.289040\pi$
$919$	37.3050	1.23058	0.615288	+	0.788302i	$0.289040\pi$
$920$	−2.76393	−0.0911241
$921$	0	0
$922$	−16.8328	−0.554359
$923$	−5.52786	−0.181952
$924$	0	0
$925$	4.29180	0.141113
$926$	−26.8328	−0.881781
$927$	−4.58359	−0.150545
$928$	56.8328	1.86563
$929$	−12.8328	−0.421031	−0.210516	−	0.977591i	$-0.567514\pi$
$929$	−12.8328	−0.421031	−0.210516	+	0.977591i	$0.567514\pi$
$930$	0	0
$931$	25.1246	0.823426
$932$	−1.41641	−0.0463960
$933$	0	0
$934$	62.6099	2.04866
$935$	1.88854	0.0617620
$936$	6.70820	0.219265
$937$	16.8328	0.549904	0.274952	−	0.961458i	$-0.411338\pi$
$937$	16.8328	0.549904	0.274952	+	0.961458i	$0.411338\pi$
$938$	52.3607	1.70964
$939$	0	0
$940$	14.8328	0.483793
$941$	21.2361	0.692276	0.346138	−	0.938184i	$-0.387493\pi$
$941$	21.2361	0.692276	0.346138	+	0.938184i	$0.387493\pi$
$942$	0	0
$943$	−10.0000	−0.325645
$944$	−6.47214	−0.210650
$945$	0	0
$946$	17.8885	0.581607
$947$	−45.3050	−1.47221	−0.736107	−	0.676866i	$-0.763338\pi$
$947$	−45.3050	−1.47221	−0.736107	+	0.676866i	$0.763338\pi$
$948$	0	0
$949$	0.472136	0.0153262
$950$	56.1803	1.82273
$951$	0	0
$952$	−14.4721	−0.469045
$953$	−43.3050	−1.40278	−0.701392	−	0.712775i	$-0.747438\pi$
$953$	−43.3050	−1.40278	−0.701392	+	0.712775i	$0.747438\pi$
$954$	−46.5836	−1.50820
$955$	9.88854	0.319986
$956$	26.8328	0.867835
$957$	0	0
$958$	50.6525	1.63651
$959$	−12.0000	−0.387500
$960$	0	0
$961$	78.6656	2.53760
$962$	2.76393	0.0891127
$963$	−7.41641	−0.238990
$964$	−78.5410	−2.52964
$965$	−0.583592	−0.0187865
$966$	0	0
$967$	−3.63932	−0.117033	−0.0585163	−	0.998286i	$-0.518637\pi$
$967$	−3.63932	−0.117033	−0.0585163	+	0.998286i	$0.518637\pi$
$968$	23.2918	0.748627
$969$	0	0
$970$	11.8622	0.380873
$971$	−43.1935	−1.38615	−0.693073	−	0.720868i	$-0.743743\pi$
$971$	−43.1935	−1.38615	−0.693073	+	0.720868i	$0.743743\pi$
$972$	0	0
$973$	41.8885	1.34289
$974$	34.4721	1.10456
$975$	0	0
$976$	10.0000	0.320092
$977$	−45.2361	−1.44723	−0.723615	−	0.690204i	$-0.757521\pi$
$977$	−45.2361	−1.44723	−0.723615	+	0.690204i	$0.757521\pi$
$978$	0	0
$979$	−5.88854	−0.188199
$980$	−12.8754	−0.411289
$981$	−15.7082	−0.501524
$982$	37.8885	1.20907
$983$	10.8754	0.346871	0.173436	−	0.984845i	$-0.444513\pi$
$983$	10.8754	0.346871	0.173436	+	0.984845i	$0.444513\pi$
$984$	0	0
$985$	−15.4164	−0.491208
$986$	−37.8885	−1.20662
$987$	0	0
$988$	21.7082	0.690630
$989$	−10.4721	−0.332995
$990$	6.33437	0.201319
$991$	20.9443	0.665317	0.332658	−	0.943047i	$-0.392054\pi$
$991$	20.9443	0.665317	0.332658	+	0.943047i	$0.392054\pi$
$992$	70.2492	2.23042
$993$	0	0
$994$	40.0000	1.26872
$995$	−14.1115	−0.447363
$996$	0	0
$997$	−33.4164	−1.05831	−0.529154	−	0.848526i	$-0.677491\pi$
$997$	−33.4164	−1.05831	−0.529154	+	0.848526i	$0.677491\pi$
$998$	−32.3607	−1.02436
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	299.2.a.c.1.1	✓	2
3.2	odd	2		2691.2.a.l.1.2		2
4.3	odd	2		4784.2.a.n.1.1		2
5.4	even	2		7475.2.a.l.1.2		2
13.12	even	2		3887.2.a.f.1.2		2
23.22	odd	2		6877.2.a.e.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
299.2.a.c.1.1	✓	2	1.1	even	1	trivial
2691.2.a.l.1.2		2	3.2	odd	2
3887.2.a.f.1.2		2	13.12	even	2
4784.2.a.n.1.1		2	4.3	odd	2
6877.2.a.e.1.1		2	23.22	odd	2
7475.2.a.l.1.2		2	5.4	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 299 = 13 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	299.a (trivial)

\(f(q)\)	\(=\)	\(q-2.23607 q^{2} +3.00000 q^{4} -1.23607 q^{5} +3.23607 q^{7} -2.23607 q^{8} -3.00000 q^{9} +O(q^{10})\)	\(q-2.23607 q^{2} +3.00000 q^{4} -1.23607 q^{5} +3.23607 q^{7} -2.23607 q^{8} -3.00000 q^{9} +2.76393 q^{10} -0.763932 q^{11} +1.00000 q^{13} -7.23607 q^{14} -1.00000 q^{16} +2.00000 q^{17} +6.70820 q^{18} +7.23607 q^{19} -3.70820 q^{20} +1.70820 q^{22} -1.00000 q^{23} -3.47214 q^{25} -2.23607 q^{26} +9.70820 q^{28} +8.47214 q^{29} +10.4721 q^{31} +6.70820 q^{32} -4.47214 q^{34} -4.00000 q^{35} -9.00000 q^{36} -1.23607 q^{37} -16.1803 q^{38} +2.76393 q^{40} +10.0000 q^{41} +10.4721 q^{43} -2.29180 q^{44} +3.70820 q^{45} +2.23607 q^{46} -4.00000 q^{47} +3.47214 q^{49} +7.76393 q^{50} +3.00000 q^{52} -6.94427 q^{53} +0.944272 q^{55} -7.23607 q^{56} -18.9443 q^{58} +6.47214 q^{59} -10.0000 q^{61} -23.4164 q^{62} -9.70820 q^{63} -13.0000 q^{64} -1.23607 q^{65} -7.23607 q^{67} +6.00000 q^{68} +8.94427 q^{70} -5.52786 q^{71} +6.70820 q^{72} +0.472136 q^{73} +2.76393 q^{74} +21.7082 q^{76} -2.47214 q^{77} -4.94427 q^{79} +1.23607 q^{80} +9.00000 q^{81} -22.3607 q^{82} -12.1803 q^{83} -2.47214 q^{85} -23.4164 q^{86} +1.70820 q^{88} +7.70820 q^{89} -8.29180 q^{90} +3.23607 q^{91} -3.00000 q^{92} +8.94427 q^{94} -8.94427 q^{95} +4.29180 q^{97} -7.76393 q^{98} +2.29180 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{4} + 2 q^{5} + 2 q^{7} - 6 q^{9}+O(q^{10}) \) 2 * q + 6 * q^4 + 2 * q^5 + 2 * q^7 - 6 * q^9	\( 2 q + 6 q^{4} + 2 q^{5} + 2 q^{7} - 6 q^{9} + 10 q^{10} - 6 q^{11} + 2 q^{13} - 10 q^{14} - 2 q^{16} + 4 q^{17} + 10 q^{19} + 6 q^{20} - 10 q^{22} - 2 q^{23} + 2 q^{25} + 6 q^{28} + 8 q^{29} + 12 q^{31} - 8 q^{35} - 18 q^{36} + 2 q^{37} - 10 q^{38} + 10 q^{40} + 20 q^{41} + 12 q^{43} - 18 q^{44} - 6 q^{45} - 8 q^{47} - 2 q^{49} + 20 q^{50} + 6 q^{52} + 4 q^{53} - 16 q^{55} - 10 q^{56} - 20 q^{58} + 4 q^{59} - 20 q^{61} - 20 q^{62} - 6 q^{63} - 26 q^{64} + 2 q^{65} - 10 q^{67} + 12 q^{68} - 20 q^{71} - 8 q^{73} + 10 q^{74} + 30 q^{76} + 4 q^{77} + 8 q^{79} - 2 q^{80} + 18 q^{81} - 2 q^{83} + 4 q^{85} - 20 q^{86} - 10 q^{88} + 2 q^{89} - 30 q^{90} + 2 q^{91} - 6 q^{92} + 22 q^{97} - 20 q^{98} + 18 q^{99}+O(q^{100}) \) 2 * q + 6 * q^4 + 2 * q^5 + 2 * q^7 - 6 * q^9 + 10 * q^10 - 6 * q^11 + 2 * q^13 - 10 * q^14 - 2 * q^16 + 4 * q^17 + 10 * q^19 + 6 * q^20 - 10 * q^22 - 2 * q^23 + 2 * q^25 + 6 * q^28 + 8 * q^29 + 12 * q^31 - 8 * q^35 - 18 * q^36 + 2 * q^37 - 10 * q^38 + 10 * q^40 + 20 * q^41 + 12 * q^43 - 18 * q^44 - 6 * q^45 - 8 * q^47 - 2 * q^49 + 20 * q^50 + 6 * q^52 + 4 * q^53 - 16 * q^55 - 10 * q^56 - 20 * q^58 + 4 * q^59 - 20 * q^61 - 20 * q^62 - 6 * q^63 - 26 * q^64 + 2 * q^65 - 10 * q^67 + 12 * q^68 - 20 * q^71 - 8 * q^73 + 10 * q^74 + 30 * q^76 + 4 * q^77 + 8 * q^79 - 2 * q^80 + 18 * q^81 - 2 * q^83 + 4 * q^85 - 20 * q^86 - 10 * q^88 + 2 * q^89 - 30 * q^90 + 2 * q^91 - 6 * q^92 + 22 * q^97 - 20 * q^98 + 18 * q^99

Introduction

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