LMFDB - Embedded newform 8349.2.a.i.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8349,2,Mod(1,8349)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8349, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8349.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8349 = 3 \cdot 11^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8349.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:	$66.6671006476$
Analytic rank:	$1$
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:	no (minimal twist has level 69)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	8349.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.23607 q^{2} -1.00000 q^{3} +3.00000 q^{4} +1.23607 q^{5} -2.23607 q^{6} -3.23607 q^{7} +2.23607 q^{8} +1.00000 q^{9} +2.76393 q^{10} -3.00000 q^{12} +4.47214 q^{13} -7.23607 q^{14} -1.23607 q^{15} -1.00000 q^{16} +2.76393 q^{17} +2.23607 q^{18} -7.23607 q^{19} +3.70820 q^{20} +3.23607 q^{21} +1.00000 q^{23} -2.23607 q^{24} -3.47214 q^{25} +10.0000 q^{26} -1.00000 q^{27} -9.70820 q^{28} -4.47214 q^{29} -2.76393 q^{30} -6.47214 q^{31} -6.70820 q^{32} +6.18034 q^{34} -4.00000 q^{35} +3.00000 q^{36} +4.47214 q^{37} -16.1803 q^{38} -4.47214 q^{39} +2.76393 q^{40} +10.9443 q^{41} +7.23607 q^{42} +5.70820 q^{43} +1.23607 q^{45} +2.23607 q^{46} -4.00000 q^{47} +1.00000 q^{48} +3.47214 q^{49} -7.76393 q^{50} -2.76393 q^{51} +13.4164 q^{52} -5.23607 q^{53} -2.23607 q^{54} -7.23607 q^{56} +7.23607 q^{57} -10.0000 q^{58} -4.94427 q^{59} -3.70820 q^{60} -4.47214 q^{61} -14.4721 q^{62} -3.23607 q^{63} -13.0000 q^{64} +5.52786 q^{65} +0.763932 q^{67} +8.29180 q^{68} -1.00000 q^{69} -8.94427 q^{70} -8.00000 q^{71} +2.23607 q^{72} -6.94427 q^{73} +10.0000 q^{74} +3.47214 q^{75} -21.7082 q^{76} -10.0000 q^{78} -9.70820 q^{79} -1.23607 q^{80} +1.00000 q^{81} +24.4721 q^{82} -4.00000 q^{83} +9.70820 q^{84} +3.41641 q^{85} +12.7639 q^{86} +4.47214 q^{87} -1.23607 q^{89} +2.76393 q^{90} -14.4721 q^{91} +3.00000 q^{92} +6.47214 q^{93} -8.94427 q^{94} -8.94427 q^{95} +6.70820 q^{96} +8.47214 q^{97} +7.76393 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 6 q^{4} - 2 q^{5} - 2 q^{7} + 2 q^{9} + 10 q^{10} - 6 q^{12} - 10 q^{14} + 2 q^{15} - 2 q^{16} + 10 q^{17} - 10 q^{19} - 6 q^{20} + 2 q^{21} + 2 q^{23} + 2 q^{25} + 20 q^{26} - 2 q^{27}+ \cdots + 20 q^{98}+O(q^{100}) $ 2 * q - 2 * q^3 + 6 * q^4 - 2 * q^5 - 2 * q^7 + 2 * q^9 + 10 * q^10 - 6 * q^12 - 10 * q^14 + 2 * q^15 - 2 * q^16 + 10 * q^17 - 10 * q^19 - 6 * q^20 + 2 * q^21 + 2 * q^23 + 2 * q^25 + 20 * q^26 - 2 * q^27 - 6 * q^28 - 10 * q^30 - 4 * q^31 - 10 * q^34 - 8 * q^35 + 6 * q^36 - 10 * q^38 + 10 * q^40 + 4 * q^41 + 10 * q^42 - 2 * q^43 - 2 * q^45 - 8 * q^47 + 2 * q^48 - 2 * q^49 - 20 * q^50 - 10 * q^51 - 6 * q^53 - 10 * q^56 + 10 * q^57 - 20 * q^58 + 8 * q^59 + 6 * q^60 - 20 * q^62 - 2 * q^63 - 26 * q^64 + 20 * q^65 + 6 * q^67 + 30 * q^68 - 2 * q^69 - 16 * q^71 + 4 * q^73 + 20 * q^74 - 2 * q^75 - 30 * q^76 - 20 * q^78 - 6 * q^79 + 2 * q^80 + 2 * q^81 + 40 * q^82 - 8 * q^83 + 6 * q^84 - 20 * q^85 + 30 * q^86 + 2 * q^89 + 10 * q^90 - 20 * q^91 + 6 * q^92 + 4 * q^93 + 8 * q^97 + 20 * 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	$-0.209785\pi$
$2$	2.23607	1.58114	0.790569	+	0.612372i	$0.209785\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	3.00000	1.50000
$5$	1.23607	0.552786	0.276393	−	0.961045i	$-0.410861\pi$
$5$	1.23607	0.552786	0.276393	+	0.961045i	$0.410861\pi$
$6$	−2.23607	−0.912871
$7$	−3.23607	−1.22312	−0.611559	−	0.791199i	$-0.709457\pi$
$7$	−3.23607	−1.22312	−0.611559	+	0.791199i	$0.709457\pi$
$8$	2.23607	0.790569
$9$	1.00000	0.333333
$10$	2.76393	0.874032
$11$	0	0
$11$	0	0
$12$	−3.00000	−0.866025
$13$	4.47214	1.24035	0.620174	−	0.784465i	$-0.287062\pi$
$13$	4.47214	1.24035	0.620174	+	0.784465i	$0.287062\pi$
$14$	−7.23607	−1.93392
$15$	−1.23607	−0.319151
$16$	−1.00000	−0.250000
$17$	2.76393	0.670352	0.335176	−	0.942156i	$-0.391204\pi$
$17$	2.76393	0.670352	0.335176	+	0.942156i	$0.391204\pi$
$18$	2.23607	0.527046
$19$	−7.23607	−1.66007	−0.830034	−	0.557713i	$-0.811679\pi$
$19$	−7.23607	−1.66007	−0.830034	+	0.557713i	$0.811679\pi$
$20$	3.70820	0.829180
$21$	3.23607	0.706168
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−2.23607	−0.456435
$25$	−3.47214	−0.694427
$26$	10.0000	1.96116
$27$	−1.00000	−0.192450
$28$	−9.70820	−1.83468
$29$	−4.47214	−0.830455	−0.415227	−	0.909718i	$-0.636298\pi$
$29$	−4.47214	−0.830455	−0.415227	+	0.909718i	$0.636298\pi$
$30$	−2.76393	−0.504623
$31$	−6.47214	−1.16243	−0.581215	−	0.813750i	$-0.697422\pi$
$31$	−6.47214	−1.16243	−0.581215	+	0.813750i	$0.697422\pi$
$32$	−6.70820	−1.18585
$33$	0	0
$34$	6.18034	1.05992
$35$	−4.00000	−0.676123
$36$	3.00000	0.500000
$37$	4.47214	0.735215	0.367607	−	0.929981i	$-0.380177\pi$
$37$	4.47214	0.735215	0.367607	+	0.929981i	$0.380177\pi$
$38$	−16.1803	−2.62480
$39$	−4.47214	−0.716115
$40$	2.76393	0.437016
$41$	10.9443	1.70921	0.854604	−	0.519280i	$-0.173800\pi$
$41$	10.9443	1.70921	0.854604	+	0.519280i	$0.173800\pi$
$42$	7.23607	1.11655
$43$	5.70820	0.870493	0.435246	−	0.900311i	$-0.356661\pi$
$43$	5.70820	0.870493	0.435246	+	0.900311i	$0.356661\pi$
$44$	0	0
$45$	1.23607	0.184262
$46$	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	$0.594231\pi$
$48$	1.00000	0.144338
$49$	3.47214	0.496019
$50$	−7.76393	−1.09799
$51$	−2.76393	−0.387028
$52$	13.4164	1.86052
$53$	−5.23607	−0.719229	−0.359615	−	0.933101i	$-0.617092\pi$
$53$	−5.23607	−0.719229	−0.359615	+	0.933101i	$0.617092\pi$
$54$	−2.23607	−0.304290
$55$	0	0
$56$	−7.23607	−0.966960
$57$	7.23607	0.958441
$58$	−10.0000	−1.31306
$59$	−4.94427	−0.643689	−0.321845	−	0.946792i	$-0.604303\pi$
$59$	−4.94427	−0.643689	−0.321845	+	0.946792i	$0.604303\pi$
$60$	−3.70820	−0.478727
$61$	−4.47214	−0.572598	−0.286299	−	0.958140i	$-0.592425\pi$
$61$	−4.47214	−0.572598	−0.286299	+	0.958140i	$0.592425\pi$
$62$	−14.4721	−1.83796
$63$	−3.23607	−0.407706
$64$	−13.0000	−1.62500
$65$	5.52786	0.685647
$66$	0	0
$67$	0.763932	0.0933292	0.0466646	−	0.998911i	$-0.485141\pi$
$67$	0.763932	0.0933292	0.0466646	+	0.998911i	$0.485141\pi$
$68$	8.29180	1.00553
$69$	−1.00000	−0.120386
$70$	−8.94427	−1.06904
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	2.23607	0.263523
$73$	−6.94427	−0.812766	−0.406383	−	0.913703i	$-0.633210\pi$
$73$	−6.94427	−0.812766	−0.406383	+	0.913703i	$0.633210\pi$
$74$	10.0000	1.16248
$75$	3.47214	0.400928
$76$	−21.7082	−2.49010
$77$	0	0
$78$	−10.0000	−1.13228
$79$	−9.70820	−1.09226	−0.546129	−	0.837701i	$-0.683899\pi$
$79$	−9.70820	−1.09226	−0.546129	+	0.837701i	$0.683899\pi$
$80$	−1.23607	−0.138197
$81$	1.00000	0.111111
$82$	24.4721	2.70250
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	9.70820	1.05925
$85$	3.41641	0.370561
$86$	12.7639	1.37637
$87$	4.47214	0.479463
$88$	0	0
$89$	−1.23607	−0.131023	−0.0655115	−	0.997852i	$-0.520868\pi$
$89$	−1.23607	−0.131023	−0.0655115	+	0.997852i	$0.520868\pi$
$90$	2.76393	0.291344
$91$	−14.4721	−1.51709
$92$	3.00000	0.312772
$93$	6.47214	0.671129
$94$	−8.94427	−0.922531
$95$	−8.94427	−0.917663
$96$	6.70820	0.684653
$97$	8.47214	0.860215	0.430108	−	0.902778i	$-0.358476\pi$
$97$	8.47214	0.860215	0.430108	+	0.902778i	$0.358476\pi$
$98$	7.76393	0.784276
$99$	0	0
$100$	−10.4164	−1.04164
$101$	6.94427	0.690981	0.345490	−	0.938422i	$-0.387713\pi$
$101$	6.94427	0.690981	0.345490	+	0.938422i	$0.387713\pi$
$102$	−6.18034	−0.611945
$103$	−11.2361	−1.10712	−0.553561	−	0.832808i	$-0.686732\pi$
$103$	−11.2361	−1.10712	−0.553561	+	0.832808i	$0.686732\pi$
$104$	10.0000	0.980581
$105$	4.00000	0.390360
$106$	−11.7082	−1.13720
$107$	−16.9443	−1.63806	−0.819032	−	0.573747i	$-0.805489\pi$
$107$	−16.9443	−1.63806	−0.819032	+	0.573747i	$0.805489\pi$
$108$	−3.00000	−0.288675
$109$	14.9443	1.43140	0.715701	−	0.698407i	$-0.246107\pi$
$109$	14.9443	1.43140	0.715701	+	0.698407i	$0.246107\pi$
$110$	0	0
$111$	−4.47214	−0.424476
$112$	3.23607	0.305780
$113$	−6.18034	−0.581397	−0.290699	−	0.956815i	$-0.593888\pi$
$113$	−6.18034	−0.581397	−0.290699	+	0.956815i	$0.593888\pi$
$114$	16.1803	1.51543
$115$	1.23607	0.115264
$116$	−13.4164	−1.24568
$117$	4.47214	0.413449
$118$	−11.0557	−1.01776
$119$	−8.94427	−0.819920
$120$	−2.76393	−0.252311
$121$	0	0
$122$	−10.0000	−0.905357
$123$	−10.9443	−0.986812
$124$	−19.4164	−1.74364
$125$	−10.4721	−0.936656
$126$	−7.23607	−0.644640
$127$	−1.52786	−0.135576	−0.0677880	−	0.997700i	$-0.521594\pi$
$127$	−1.52786	−0.135576	−0.0677880	+	0.997700i	$0.521594\pi$
$128$	−15.6525	−1.38350
$129$	−5.70820	−0.502579
$130$	12.3607	1.08410
$131$	−16.9443	−1.48043	−0.740214	−	0.672371i	$-0.765276\pi$
$131$	−16.9443	−1.48043	−0.740214	+	0.672371i	$0.765276\pi$
$132$	0	0
$133$	23.4164	2.03046
$134$	1.70820	0.147566
$135$	−1.23607	−0.106384
$136$	6.18034	0.529960
$137$	−1.23607	−0.105604	−0.0528022	−	0.998605i	$-0.516815\pi$
$137$	−1.23607	−0.105604	−0.0528022	+	0.998605i	$0.516815\pi$
$138$	−2.23607	−0.190347
$139$	8.94427	0.758643	0.379322	−	0.925265i	$-0.376157\pi$
$139$	8.94427	0.758643	0.379322	+	0.925265i	$0.376157\pi$
$140$	−12.0000	−1.01419
$141$	4.00000	0.336861
$142$	−17.8885	−1.50117
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	−5.52786	−0.459064
$146$	−15.5279	−1.28510
$147$	−3.47214	−0.286377
$148$	13.4164	1.10282
$149$	11.7082	0.959173	0.479587	−	0.877494i	$-0.340787\pi$
$149$	11.7082	0.959173	0.479587	+	0.877494i	$0.340787\pi$
$150$	7.76393	0.633922
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	−16.1803	−1.31240
$153$	2.76393	0.223451
$154$	0	0
$155$	−8.00000	−0.642575
$156$	−13.4164	−1.07417
$157$	−3.52786	−0.281554	−0.140777	−	0.990041i	$-0.544960\pi$
$157$	−3.52786	−0.281554	−0.140777	+	0.990041i	$0.544960\pi$
$158$	−21.7082	−1.72701
$159$	5.23607	0.415247
$160$	−8.29180	−0.655524
$161$	−3.23607	−0.255038
$162$	2.23607	0.175682
$163$	7.41641	0.580898	0.290449	−	0.956890i	$-0.406195\pi$
$163$	7.41641	0.580898	0.290449	+	0.956890i	$0.406195\pi$
$164$	32.8328	2.56381
$165$	0	0
$166$	−8.94427	−0.694210
$167$	−12.9443	−1.00166	−0.500829	−	0.865546i	$-0.666971\pi$
$167$	−12.9443	−1.00166	−0.500829	+	0.865546i	$0.666971\pi$
$168$	7.23607	0.558275
$169$	7.00000	0.538462
$170$	7.63932	0.585909
$171$	−7.23607	−0.553356
$172$	17.1246	1.30574
$173$	−4.47214	−0.340010	−0.170005	−	0.985443i	$-0.554378\pi$
$173$	−4.47214	−0.340010	−0.170005	+	0.985443i	$0.554378\pi$
$174$	10.0000	0.758098
$175$	11.2361	0.849367
$176$	0	0
$177$	4.94427	0.371634
$178$	−2.76393	−0.207165
$179$	20.9443	1.56545	0.782724	−	0.622369i	$-0.213830\pi$
$179$	20.9443	1.56545	0.782724	+	0.622369i	$0.213830\pi$
$180$	3.70820	0.276393
$181$	−11.8885	−0.883669	−0.441834	−	0.897097i	$-0.645672\pi$
$181$	−11.8885	−0.883669	−0.441834	+	0.897097i	$0.645672\pi$
$182$	−32.3607	−2.39873
$183$	4.47214	0.330590
$184$	2.23607	0.164845
$185$	5.52786	0.406417
$186$	14.4721	1.06115
$187$	0	0
$188$	−12.0000	−0.875190
$189$	3.23607	0.235389
$190$	−20.0000	−1.45095
$191$	1.52786	0.110552	0.0552762	−	0.998471i	$-0.482396\pi$
$191$	1.52786	0.110552	0.0552762	+	0.998471i	$0.482396\pi$
$192$	13.0000	0.938194
$193$	−17.4164	−1.25366	−0.626830	−	0.779156i	$-0.715648\pi$
$193$	−17.4164	−1.25366	−0.626830	+	0.779156i	$0.715648\pi$
$194$	18.9443	1.36012
$195$	−5.52786	−0.395859
$196$	10.4164	0.744029
$197$	−17.4164	−1.24087	−0.620434	−	0.784259i	$-0.713043\pi$
$197$	−17.4164	−1.24087	−0.620434	+	0.784259i	$0.713043\pi$
$198$	0	0
$199$	9.70820	0.688196	0.344098	−	0.938934i	$-0.388185\pi$
$199$	9.70820	0.688196	0.344098	+	0.938934i	$0.388185\pi$
$200$	−7.76393	−0.548993
$201$	−0.763932	−0.0538836
$202$	15.5279	1.09254
$203$	14.4721	1.01574
$204$	−8.29180	−0.580542
$205$	13.5279	0.944827
$206$	−25.1246	−1.75051
$207$	1.00000	0.0695048
$208$	−4.47214	−0.310087
$209$	0	0
$210$	8.94427	0.617213
$211$	−13.5279	−0.931297	−0.465648	−	0.884970i	$-0.654179\pi$
$211$	−13.5279	−0.931297	−0.465648	+	0.884970i	$0.654179\pi$
$212$	−15.7082	−1.07884
$213$	8.00000	0.548151
$214$	−37.8885	−2.59001
$215$	7.05573	0.481197
$216$	−2.23607	−0.152145
$217$	20.9443	1.42179
$218$	33.4164	2.26324
$219$	6.94427	0.469250
$220$	0	0
$221$	12.3607	0.831469
$222$	−10.0000	−0.671156
$223$	−25.8885	−1.73363	−0.866813	−	0.498634i	$-0.833835\pi$
$223$	−25.8885	−1.73363	−0.866813	+	0.498634i	$0.833835\pi$
$224$	21.7082	1.45044
$225$	−3.47214	−0.231476
$226$	−13.8197	−0.919270
$227$	13.5279	0.897876	0.448938	−	0.893563i	$-0.351802\pi$
$227$	13.5279	0.897876	0.448938	+	0.893563i	$0.351802\pi$
$228$	21.7082	1.43766
$229$	2.94427	0.194563	0.0972815	−	0.995257i	$-0.468985\pi$
$229$	2.94427	0.194563	0.0972815	+	0.995257i	$0.468985\pi$
$230$	2.76393	0.182248
$231$	0	0
$232$	−10.0000	−0.656532
$233$	14.0000	0.917170	0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000	0.917170	0.458585	+	0.888650i	$0.348356\pi$
$234$	10.0000	0.653720
$235$	−4.94427	−0.322529
$236$	−14.8328	−0.965534
$237$	9.70820	0.630616
$238$	−20.0000	−1.29641
$239$	−4.94427	−0.319818	−0.159909	−	0.987132i	$-0.551120\pi$
$239$	−4.94427	−0.319818	−0.159909	+	0.987132i	$0.551120\pi$
$240$	1.23607	0.0797878
$241$	−19.5279	−1.25790	−0.628950	−	0.777446i	$-0.716515\pi$
$241$	−19.5279	−1.25790	−0.628950	+	0.777446i	$0.716515\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	−13.4164	−0.858898
$245$	4.29180	0.274193
$246$	−24.4721	−1.56029
$247$	−32.3607	−2.05906
$248$	−14.4721	−0.918982
$249$	4.00000	0.253490
$250$	−23.4164	−1.48098
$251$	−10.4721	−0.660995	−0.330498	−	0.943807i	$-0.607217\pi$
$251$	−10.4721	−0.660995	−0.330498	+	0.943807i	$0.607217\pi$
$252$	−9.70820	−0.611559
$253$	0	0
$254$	−3.41641	−0.214364
$255$	−3.41641	−0.213944
$256$	−9.00000	−0.562500
$257$	−22.0000	−1.37232	−0.686161	−	0.727450i	$-0.740706\pi$
$257$	−22.0000	−1.37232	−0.686161	+	0.727450i	$0.740706\pi$
$258$	−12.7639	−0.794648
$259$	−14.4721	−0.899255
$260$	16.5836	1.02847
$261$	−4.47214	−0.276818
$262$	−37.8885	−2.34076
$263$	−6.47214	−0.399089	−0.199544	−	0.979889i	$-0.563946\pi$
$263$	−6.47214	−0.399089	−0.199544	+	0.979889i	$0.563946\pi$
$264$	0	0
$265$	−6.47214	−0.397580
$266$	52.3607	3.21044
$267$	1.23607	0.0756461
$268$	2.29180	0.139994
$269$	−0.472136	−0.0287866	−0.0143933	−	0.999896i	$-0.504582\pi$
$269$	−0.472136	−0.0287866	−0.0143933	+	0.999896i	$0.504582\pi$
$270$	−2.76393	−0.168208
$271$	17.5279	1.06474	0.532371	−	0.846511i	$-0.321301\pi$
$271$	17.5279	1.06474	0.532371	+	0.846511i	$0.321301\pi$
$272$	−2.76393	−0.167588
$273$	14.4721	0.875894
$274$	−2.76393	−0.166975
$275$	0	0
$276$	−3.00000	−0.180579
$277$	−15.8885	−0.954650	−0.477325	−	0.878727i	$-0.658394\pi$
$277$	−15.8885	−0.954650	−0.477325	+	0.878727i	$0.658394\pi$
$278$	20.0000	1.19952
$279$	−6.47214	−0.387477
$280$	−8.94427	−0.534522
$281$	−24.6525	−1.47064	−0.735322	−	0.677718i	$-0.762969\pi$
$281$	−24.6525	−1.47064	−0.735322	+	0.677718i	$0.762969\pi$
$282$	8.94427	0.532624
$283$	−5.70820	−0.339318	−0.169659	−	0.985503i	$-0.554267\pi$
$283$	−5.70820	−0.339318	−0.169659	+	0.985503i	$0.554267\pi$
$284$	−24.0000	−1.42414
$285$	8.94427	0.529813
$286$	0	0
$287$	−35.4164	−2.09056
$288$	−6.70820	−0.395285
$289$	−9.36068	−0.550628
$290$	−12.3607	−0.725844
$291$	−8.47214	−0.496645
$292$	−20.8328	−1.21915
$293$	−7.70820	−0.450318	−0.225159	−	0.974322i	$-0.572290\pi$
$293$	−7.70820	−0.450318	−0.225159	+	0.974322i	$0.572290\pi$
$294$	−7.76393	−0.452802
$295$	−6.11146	−0.355823
$296$	10.0000	0.581238
$297$	0	0
$298$	26.1803	1.51659
$299$	4.47214	0.258630
$300$	10.4164	0.601392
$301$	−18.4721	−1.06472
$302$	35.7771	2.05874
$303$	−6.94427	−0.398938
$304$	7.23607	0.415017
$305$	−5.52786	−0.316525
$306$	6.18034	0.353307
$307$	−2.47214	−0.141092	−0.0705461	−	0.997509i	$-0.522474\pi$
$307$	−2.47214	−0.141092	−0.0705461	+	0.997509i	$0.522474\pi$
$308$	0	0
$309$	11.2361	0.639198
$310$	−17.8885	−1.01600
$311$	7.05573	0.400094	0.200047	−	0.979786i	$-0.435891\pi$
$311$	7.05573	0.400094	0.200047	+	0.979786i	$0.435891\pi$
$312$	−10.0000	−0.566139
$313$	5.05573	0.285767	0.142883	−	0.989740i	$-0.454363\pi$
$313$	5.05573	0.285767	0.142883	+	0.989740i	$0.454363\pi$
$314$	−7.88854	−0.445176
$315$	−4.00000	−0.225374
$316$	−29.1246	−1.63839
$317$	2.58359	0.145109	0.0725545	−	0.997364i	$-0.476885\pi$
$317$	2.58359	0.145109	0.0725545	+	0.997364i	$0.476885\pi$
$318$	11.7082	0.656563
$319$	0	0
$320$	−16.0689	−0.898278
$321$	16.9443	0.945737
$322$	−7.23607	−0.403250
$323$	−20.0000	−1.11283
$324$	3.00000	0.166667
$325$	−15.5279	−0.861331
$326$	16.5836	0.918480
$327$	−14.9443	−0.826420
$328$	24.4721	1.35125
$329$	12.9443	0.713641
$330$	0	0
$331$	23.4164	1.28708	0.643541	−	0.765412i	$-0.277465\pi$
$331$	23.4164	1.28708	0.643541	+	0.765412i	$0.277465\pi$
$332$	−12.0000	−0.658586
$333$	4.47214	0.245072
$334$	−28.9443	−1.58376
$335$	0.944272	0.0515911
$336$	−3.23607	−0.176542
$337$	14.3607	0.782276	0.391138	−	0.920332i	$-0.372082\pi$
$337$	14.3607	0.782276	0.391138	+	0.920332i	$0.372082\pi$
$338$	15.6525	0.851382
$339$	6.18034	0.335670
$340$	10.2492	0.555842
$341$	0	0
$342$	−16.1803	−0.874933
$343$	11.4164	0.616428
$344$	12.7639	0.688185
$345$	−1.23607	−0.0665477
$346$	−10.0000	−0.537603
$347$	9.88854	0.530845	0.265422	−	0.964132i	$-0.414489\pi$
$347$	9.88854	0.530845	0.265422	+	0.964132i	$0.414489\pi$
$348$	13.4164	0.719195
$349$	23.5279	1.25942	0.629709	−	0.776831i	$-0.283174\pi$
$349$	23.5279	1.25942	0.629709	+	0.776831i	$0.283174\pi$
$350$	25.1246	1.34297
$351$	−4.47214	−0.238705
$352$	0	0
$353$	−17.4164	−0.926982	−0.463491	−	0.886102i	$-0.653403\pi$
$353$	−17.4164	−0.926982	−0.463491	+	0.886102i	$0.653403\pi$
$354$	11.0557	0.587605
$355$	−9.88854	−0.524829
$356$	−3.70820	−0.196534
$357$	8.94427	0.473381
$358$	46.8328	2.47519
$359$	−6.47214	−0.341586	−0.170793	−	0.985307i	$-0.554633\pi$
$359$	−6.47214	−0.341586	−0.170793	+	0.985307i	$0.554633\pi$
$360$	2.76393	0.145672
$361$	33.3607	1.75583
$362$	−26.5836	−1.39720
$363$	0	0
$364$	−43.4164	−2.27564
$365$	−8.58359	−0.449286
$366$	10.0000	0.522708
$367$	25.7082	1.34196	0.670979	−	0.741477i	$-0.265874\pi$
$367$	25.7082	1.34196	0.670979	+	0.741477i	$0.265874\pi$
$368$	−1.00000	−0.0521286
$369$	10.9443	0.569736
$370$	12.3607	0.642601
$371$	16.9443	0.879703
$372$	19.4164	1.00669
$373$	−20.4721	−1.06001	−0.530004	−	0.847995i	$-0.677809\pi$
$373$	−20.4721	−1.06001	−0.530004	+	0.847995i	$0.677809\pi$
$374$	0	0
$375$	10.4721	0.540779
$376$	−8.94427	−0.461266
$377$	−20.0000	−1.03005
$378$	7.23607	0.372183
$379$	30.0689	1.54453	0.772267	−	0.635298i	$-0.219123\pi$
$379$	30.0689	1.54453	0.772267	+	0.635298i	$0.219123\pi$
$380$	−26.8328	−1.37649
$381$	1.52786	0.0782748
$382$	3.41641	0.174799
$383$	25.5279	1.30441	0.652206	−	0.758041i	$-0.273844\pi$
$383$	25.5279	1.30441	0.652206	+	0.758041i	$0.273844\pi$
$384$	15.6525	0.798762
$385$	0	0
$386$	−38.9443	−1.98221
$387$	5.70820	0.290164
$388$	25.4164	1.29032
$389$	20.6525	1.04712	0.523561	−	0.851988i	$-0.324603\pi$
$389$	20.6525	1.04712	0.523561	+	0.851988i	$0.324603\pi$
$390$	−12.3607	−0.625907
$391$	2.76393	0.139778
$392$	7.76393	0.392138
$393$	16.9443	0.854725
$394$	−38.9443	−1.96198
$395$	−12.0000	−0.603786
$396$	0	0
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	21.7082	1.08813
$399$	−23.4164	−1.17229
$400$	3.47214	0.173607
$401$	20.0689	1.00219	0.501096	−	0.865392i	$-0.332930\pi$
$401$	20.0689	1.00219	0.501096	+	0.865392i	$0.332930\pi$
$402$	−1.70820	−0.0851975
$403$	−28.9443	−1.44182
$404$	20.8328	1.03647
$405$	1.23607	0.0614207
$406$	32.3607	1.60603
$407$	0	0
$408$	−6.18034	−0.305972
$409$	−17.4164	−0.861186	−0.430593	−	0.902546i	$-0.641696\pi$
$409$	−17.4164	−0.861186	−0.430593	+	0.902546i	$0.641696\pi$
$410$	30.2492	1.49390
$411$	1.23607	0.0609707
$412$	−33.7082	−1.66068
$413$	16.0000	0.787309
$414$	2.23607	0.109897
$415$	−4.94427	−0.242705
$416$	−30.0000	−1.47087
$417$	−8.94427	−0.438003
$418$	0	0
$419$	13.8885	0.678500	0.339250	−	0.940696i	$-0.389827\pi$
$419$	13.8885	0.678500	0.339250	+	0.940696i	$0.389827\pi$
$420$	12.0000	0.585540
$421$	−30.9443	−1.50813	−0.754066	−	0.656799i	$-0.771910\pi$
$421$	−30.9443	−1.50813	−0.754066	+	0.656799i	$0.771910\pi$
$422$	−30.2492	−1.47251
$423$	−4.00000	−0.194487
$424$	−11.7082	−0.568601
$425$	−9.59675	−0.465511
$426$	17.8885	0.866703
$427$	14.4721	0.700356
$428$	−50.8328	−2.45710
$429$	0	0
$430$	15.7771	0.760839
$431$	−8.00000	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$432$	1.00000	0.0481125
$433$	27.8885	1.34024	0.670119	−	0.742254i	$-0.266243\pi$
$433$	27.8885	1.34024	0.670119	+	0.742254i	$0.266243\pi$
$434$	46.8328	2.24805
$435$	5.52786	0.265041
$436$	44.8328	2.14710
$437$	−7.23607	−0.346148
$438$	15.5279	0.741950
$439$	9.88854	0.471954	0.235977	−	0.971759i	$-0.424171\pi$
$439$	9.88854	0.471954	0.235977	+	0.971759i	$0.424171\pi$
$440$	0	0
$441$	3.47214	0.165340
$442$	27.6393	1.31467
$443$	−34.8328	−1.65496	−0.827479	−	0.561497i	$-0.810225\pi$
$443$	−34.8328	−1.65496	−0.827479	+	0.561497i	$0.810225\pi$
$444$	−13.4164	−0.636715
$445$	−1.52786	−0.0724277
$446$	−57.8885	−2.74110
$447$	−11.7082	−0.553779
$448$	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	$0.385287\pi$
$450$	−7.76393	−0.365995
$451$	0	0
$452$	−18.5410	−0.872096
$453$	−16.0000	−0.751746
$454$	30.2492	1.41967
$455$	−17.8885	−0.838628
$456$	16.1803	0.757714
$457$	−8.47214	−0.396310	−0.198155	−	0.980171i	$-0.563495\pi$
$457$	−8.47214	−0.396310	−0.198155	+	0.980171i	$0.563495\pi$
$458$	6.58359	0.307631
$459$	−2.76393	−0.129009
$460$	3.70820	0.172896
$461$	5.41641	0.252267	0.126134	−	0.992013i	$-0.459743\pi$
$461$	5.41641	0.252267	0.126134	+	0.992013i	$0.459743\pi$
$462$	0	0
$463$	−12.9443	−0.601571	−0.300786	−	0.953692i	$-0.597249\pi$
$463$	−12.9443	−0.601571	−0.300786	+	0.953692i	$0.597249\pi$
$464$	4.47214	0.207614
$465$	8.00000	0.370991
$466$	31.3050	1.45017
$467$	20.3607	0.942180	0.471090	−	0.882085i	$-0.343861\pi$
$467$	20.3607	0.942180	0.471090	+	0.882085i	$0.343861\pi$
$468$	13.4164	0.620174
$469$	−2.47214	−0.114153
$470$	−11.0557	−0.509963
$471$	3.52786	0.162555
$472$	−11.0557	−0.508881
$473$	0	0
$474$	21.7082	0.997091
$475$	25.1246	1.15280
$476$	−26.8328	−1.22988
$477$	−5.23607	−0.239743
$478$	−11.0557	−0.505677
$479$	33.8885	1.54841	0.774204	−	0.632937i	$-0.218151\pi$
$479$	33.8885	1.54841	0.774204	+	0.632937i	$0.218151\pi$
$480$	8.29180	0.378467
$481$	20.0000	0.911922
$482$	−43.6656	−1.98892
$483$	3.23607	0.147246
$484$	0	0
$485$	10.4721	0.475515
$486$	−2.23607	−0.101430
$487$	24.0000	1.08754	0.543772	−	0.839233i	$-0.316996\pi$
$487$	24.0000	1.08754	0.543772	+	0.839233i	$0.316996\pi$
$488$	−10.0000	−0.452679
$489$	−7.41641	−0.335382
$490$	9.59675	0.433537
$491$	−40.0000	−1.80517	−0.902587	−	0.430507i	$-0.858335\pi$
$491$	−40.0000	−1.80517	−0.902587	+	0.430507i	$0.858335\pi$
$492$	−32.8328	−1.48022
$493$	−12.3607	−0.556697
$494$	−72.3607	−3.25566
$495$	0	0
$496$	6.47214	0.290607
$497$	25.8885	1.16126
$498$	8.94427	0.400802
$499$	−12.3607	−0.553340	−0.276670	−	0.960965i	$-0.589231\pi$
$499$	−12.3607	−0.553340	−0.276670	+	0.960965i	$0.589231\pi$
$500$	−31.4164	−1.40498
$501$	12.9443	0.578307
$502$	−23.4164	−1.04513
$503$	19.0557	0.849653	0.424826	−	0.905275i	$-0.360335\pi$
$503$	19.0557	0.849653	0.424826	+	0.905275i	$0.360335\pi$
$504$	−7.23607	−0.322320
$505$	8.58359	0.381965
$506$	0	0
$507$	−7.00000	−0.310881
$508$	−4.58359	−0.203364
$509$	−16.4721	−0.730115	−0.365057	−	0.930985i	$-0.618951\pi$
$509$	−16.4721	−0.730115	−0.365057	+	0.930985i	$0.618951\pi$
$510$	−7.63932	−0.338275
$511$	22.4721	0.994109
$512$	11.1803	0.494106
$513$	7.23607	0.319480
$514$	−49.1935	−2.16983
$515$	−13.8885	−0.612002
$516$	−17.1246	−0.753869
$517$	0	0
$518$	−32.3607	−1.42185
$519$	4.47214	0.196305
$520$	12.3607	0.542052
$521$	−33.2361	−1.45610	−0.728049	−	0.685525i	$-0.759573\pi$
$521$	−33.2361	−1.45610	−0.728049	+	0.685525i	$0.759573\pi$
$522$	−10.0000	−0.437688
$523$	−15.5967	−0.681998	−0.340999	−	0.940064i	$-0.610765\pi$
$523$	−15.5967	−0.681998	−0.340999	+	0.940064i	$0.610765\pi$
$524$	−50.8328	−2.22064
$525$	−11.2361	−0.490382
$526$	−14.4721	−0.631015
$527$	−17.8885	−0.779237
$528$	0	0
$529$	1.00000	0.0434783
$530$	−14.4721	−0.628629
$531$	−4.94427	−0.214563
$532$	70.2492	3.04569
$533$	48.9443	2.12001
$534$	2.76393	0.119607
$535$	−20.9443	−0.905500
$536$	1.70820	0.0737832
$537$	−20.9443	−0.903812
$538$	−1.05573	−0.0455157
$539$	0	0
$540$	−3.70820	−0.159576
$541$	15.5279	0.667595	0.333798	−	0.942645i	$-0.391670\pi$
$541$	15.5279	0.667595	0.333798	+	0.942645i	$0.391670\pi$
$542$	39.1935	1.68350
$543$	11.8885	0.510186
$544$	−18.5410	−0.794940
$545$	18.4721	0.791259
$546$	32.3607	1.38491
$547$	−13.5279	−0.578410	−0.289205	−	0.957267i	$-0.593391\pi$
$547$	−13.5279	−0.578410	−0.289205	+	0.957267i	$0.593391\pi$
$548$	−3.70820	−0.158407
$549$	−4.47214	−0.190866
$550$	0	0
$551$	32.3607	1.37861
$552$	−2.23607	−0.0951734
$553$	31.4164	1.33596
$554$	−35.5279	−1.50943
$555$	−5.52786	−0.234645
$556$	26.8328	1.13796
$557$	42.1803	1.78724	0.893619	−	0.448826i	$-0.148158\pi$
$557$	42.1803	1.78724	0.893619	+	0.448826i	$0.148158\pi$
$558$	−14.4721	−0.612654
$559$	25.5279	1.07971
$560$	4.00000	0.169031
$561$	0	0
$562$	−55.1246	−2.32529
$563$	7.41641	0.312564	0.156282	−	0.987712i	$-0.450049\pi$
$563$	7.41641	0.312564	0.156282	+	0.987712i	$0.450049\pi$
$564$	12.0000	0.505291
$565$	−7.63932	−0.321389
$566$	−12.7639	−0.536508
$567$	−3.23607	−0.135902
$568$	−17.8885	−0.750587
$569$	10.7639	0.451248	0.225624	−	0.974215i	$-0.427558\pi$
$569$	10.7639	0.451248	0.225624	+	0.974215i	$0.427558\pi$
$570$	20.0000	0.837708
$571$	−29.7082	−1.24325	−0.621625	−	0.783315i	$-0.713527\pi$
$571$	−29.7082	−1.24325	−0.621625	+	0.783315i	$0.713527\pi$
$572$	0	0
$573$	−1.52786	−0.0638274
$574$	−79.1935	−3.30547
$575$	−3.47214	−0.144798
$576$	−13.0000	−0.541667
$577$	7.52786	0.313389	0.156695	−	0.987647i	$-0.449916\pi$
$577$	7.52786	0.313389	0.156695	+	0.987647i	$0.449916\pi$
$578$	−20.9311	−0.870620
$579$	17.4164	0.723801
$580$	−16.5836	−0.688596
$581$	12.9443	0.537019
$582$	−18.9443	−0.785265
$583$	0	0
$584$	−15.5279	−0.642548
$585$	5.52786	0.228549
$586$	−17.2361	−0.712015
$587$	−0.944272	−0.0389743	−0.0194871	−	0.999810i	$-0.506203\pi$
$587$	−0.944272	−0.0389743	−0.0194871	+	0.999810i	$0.506203\pi$
$588$	−10.4164	−0.429565
$589$	46.8328	1.92971
$590$	−13.6656	−0.562605
$591$	17.4164	0.716415
$592$	−4.47214	−0.183804
$593$	−34.0000	−1.39621	−0.698106	−	0.715994i	$-0.745974\pi$
$593$	−34.0000	−1.39621	−0.698106	+	0.715994i	$0.745974\pi$
$594$	0	0
$595$	−11.0557	−0.453241
$596$	35.1246	1.43876
$597$	−9.70820	−0.397330
$598$	10.0000	0.408930
$599$	−3.05573	−0.124854	−0.0624268	−	0.998050i	$-0.519884\pi$
$599$	−3.05573	−0.124854	−0.0624268	+	0.998050i	$0.519884\pi$
$600$	7.76393	0.316961
$601$	42.3607	1.72793	0.863964	−	0.503553i	$-0.167974\pi$
$601$	42.3607	1.72793	0.863964	+	0.503553i	$0.167974\pi$
$602$	−41.3050	−1.68346
$603$	0.763932	0.0311097
$604$	48.0000	1.95309
$605$	0	0
$606$	−15.5279	−0.630776
$607$	14.8328	0.602045	0.301023	−	0.953617i	$-0.402672\pi$
$607$	14.8328	0.602045	0.301023	+	0.953617i	$0.402672\pi$
$608$	48.5410	1.96860
$609$	−14.4721	−0.586441
$610$	−12.3607	−0.500469
$611$	−17.8885	−0.723693
$612$	8.29180	0.335176
$613$	40.4721	1.63465	0.817327	−	0.576174i	$-0.195455\pi$
$613$	40.4721	1.63465	0.817327	+	0.576174i	$0.195455\pi$
$614$	−5.52786	−0.223086
$615$	−13.5279	−0.545496
$616$	0	0
$617$	−20.2918	−0.816917	−0.408458	−	0.912777i	$-0.633934\pi$
$617$	−20.2918	−0.816917	−0.408458	+	0.912777i	$0.633934\pi$
$618$	25.1246	1.01066
$619$	−9.12461	−0.366749	−0.183375	−	0.983043i	$-0.558702\pi$
$619$	−9.12461	−0.366749	−0.183375	+	0.983043i	$0.558702\pi$
$620$	−24.0000	−0.963863
$621$	−1.00000	−0.0401286
$622$	15.7771	0.632604
$623$	4.00000	0.160257
$624$	4.47214	0.179029
$625$	4.41641	0.176656
$626$	11.3050	0.451837
$627$	0	0
$628$	−10.5836	−0.422331
$629$	12.3607	0.492853
$630$	−8.94427	−0.356348
$631$	−16.1803	−0.644129	−0.322065	−	0.946718i	$-0.604377\pi$
$631$	−16.1803	−0.644129	−0.322065	+	0.946718i	$0.604377\pi$
$632$	−21.7082	−0.863506
$633$	13.5279	0.537684
$634$	5.77709	0.229437
$635$	−1.88854	−0.0749446
$636$	15.7082	0.622871
$637$	15.5279	0.615236
$638$	0	0
$639$	−8.00000	−0.316475
$640$	−19.3475	−0.764778
$641$	49.0132	1.93590	0.967952	−	0.251137i	$-0.0808044\pi$
$641$	49.0132	1.93590	0.967952	+	0.251137i	$0.0808044\pi$
$642$	37.8885	1.49534
$643$	38.0689	1.50129	0.750645	−	0.660706i	$-0.229743\pi$
$643$	38.0689	1.50129	0.750645	+	0.660706i	$0.229743\pi$
$644$	−9.70820	−0.382557
$645$	−7.05573	−0.277819
$646$	−44.7214	−1.75954
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	2.23607	0.0878410
$649$	0	0
$650$	−34.7214	−1.36188
$651$	−20.9443	−0.820871
$652$	22.2492	0.871347
$653$	−22.9443	−0.897879	−0.448939	−	0.893562i	$-0.648198\pi$
$653$	−22.9443	−0.897879	−0.448939	+	0.893562i	$0.648198\pi$
$654$	−33.4164	−1.30668
$655$	−20.9443	−0.818360
$656$	−10.9443	−0.427302
$657$	−6.94427	−0.270922
$658$	28.9443	1.12837
$659$	−36.3607	−1.41641	−0.708205	−	0.706006i	$-0.750495\pi$
$659$	−36.3607	−1.41641	−0.708205	+	0.706006i	$0.750495\pi$
$660$	0	0
$661$	−11.5279	−0.448382	−0.224191	−	0.974545i	$-0.571974\pi$
$661$	−11.5279	−0.448382	−0.224191	+	0.974545i	$0.571974\pi$
$662$	52.3607	2.03506
$663$	−12.3607	−0.480049
$664$	−8.94427	−0.347105
$665$	28.9443	1.12241
$666$	10.0000	0.387492
$667$	−4.47214	−0.173162
$668$	−38.8328	−1.50249
$669$	25.8885	1.00091
$670$	2.11146	0.0815727
$671$	0	0
$672$	−21.7082	−0.837412
$673$	−2.58359	−0.0995902	−0.0497951	−	0.998759i	$-0.515857\pi$
$673$	−2.58359	−0.0995902	−0.0497951	+	0.998759i	$0.515857\pi$
$674$	32.1115	1.23689
$675$	3.47214	0.133643
$676$	21.0000	0.807692
$677$	47.4853	1.82501	0.912504	−	0.409068i	$-0.134146\pi$
$677$	47.4853	1.82501	0.912504	+	0.409068i	$0.134146\pi$
$678$	13.8197	0.530741
$679$	−27.4164	−1.05215
$680$	7.63932	0.292955
$681$	−13.5279	−0.518389
$682$	0	0
$683$	−20.0000	−0.765279	−0.382639	−	0.923898i	$-0.624985\pi$
$683$	−20.0000	−0.765279	−0.382639	+	0.923898i	$0.624985\pi$
$684$	−21.7082	−0.830034
$685$	−1.52786	−0.0583767
$686$	25.5279	0.974658
$687$	−2.94427	−0.112331
$688$	−5.70820	−0.217623
$689$	−23.4164	−0.892094
$690$	−2.76393	−0.105221
$691$	4.36068	0.165888	0.0829440	−	0.996554i	$-0.473568\pi$
$691$	4.36068	0.165888	0.0829440	+	0.996554i	$0.473568\pi$
$692$	−13.4164	−0.510015
$693$	0	0
$694$	22.1115	0.839339
$695$	11.0557	0.419368
$696$	10.0000	0.379049
$697$	30.2492	1.14577
$698$	52.6099	1.99131
$699$	−14.0000	−0.529529
$700$	33.7082	1.27405
$701$	−25.2361	−0.953153	−0.476577	−	0.879133i	$-0.658122\pi$
$701$	−25.2361	−0.953153	−0.476577	+	0.879133i	$0.658122\pi$
$702$	−10.0000	−0.377426
$703$	−32.3607	−1.22051
$704$	0	0
$705$	4.94427	0.186212
$706$	−38.9443	−1.46569
$707$	−22.4721	−0.845152
$708$	14.8328	0.557451
$709$	−32.8328	−1.23306	−0.616531	−	0.787331i	$-0.711463\pi$
$709$	−32.8328	−1.23306	−0.616531	+	0.787331i	$0.711463\pi$
$710$	−22.1115	−0.829828
$711$	−9.70820	−0.364086
$712$	−2.76393	−0.103583
$713$	−6.47214	−0.242383
$714$	20.0000	0.748481
$715$	0	0
$716$	62.8328	2.34817
$717$	4.94427	0.184647
$718$	−14.4721	−0.540095
$719$	4.00000	0.149175	0.0745874	−	0.997214i	$-0.476236\pi$
$719$	4.00000	0.149175	0.0745874	+	0.997214i	$0.476236\pi$
$720$	−1.23607	−0.0460655
$721$	36.3607	1.35414
$722$	74.5967	2.77620
$723$	19.5279	0.726249
$724$	−35.6656	−1.32550
$725$	15.5279	0.576690
$726$	0	0
$727$	1.34752	0.0499769	0.0249885	−	0.999688i	$-0.492045\pi$
$727$	1.34752	0.0499769	0.0249885	+	0.999688i	$0.492045\pi$
$728$	−32.3607	−1.19937
$729$	1.00000	0.0370370
$730$	−19.1935	−0.710383
$731$	15.7771	0.583537
$732$	13.4164	0.495885
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	57.4853	2.12182
$735$	−4.29180	−0.158305
$736$	−6.70820	−0.247268
$737$	0	0
$738$	24.4721	0.900832
$739$	26.8328	0.987061	0.493531	−	0.869728i	$-0.335706\pi$
$739$	26.8328	0.987061	0.493531	+	0.869728i	$0.335706\pi$
$740$	16.5836	0.609625
$741$	32.3607	1.18880
$742$	37.8885	1.39093
$743$	24.3607	0.893707	0.446853	−	0.894607i	$-0.352545\pi$
$743$	24.3607	0.893707	0.446853	+	0.894607i	$0.352545\pi$
$744$	14.4721	0.530574
$745$	14.4721	0.530218
$746$	−45.7771	−1.67602
$747$	−4.00000	−0.146352
$748$	0	0
$749$	54.8328	2.00355
$750$	23.4164	0.855046
$751$	50.0689	1.82704	0.913520	−	0.406794i	$-0.133353\pi$
$751$	50.0689	1.82704	0.913520	+	0.406794i	$0.133353\pi$
$752$	4.00000	0.145865
$753$	10.4721	0.381626
$754$	−44.7214	−1.62866
$755$	19.7771	0.719762
$756$	9.70820	0.353084
$757$	39.8885	1.44977	0.724887	−	0.688868i	$-0.241892\pi$
$757$	39.8885	1.44977	0.724887	+	0.688868i	$0.241892\pi$
$758$	67.2361	2.44212
$759$	0	0
$760$	−20.0000	−0.725476
$761$	−42.3607	−1.53557	−0.767787	−	0.640706i	$-0.778642\pi$
$761$	−42.3607	−1.53557	−0.767787	+	0.640706i	$0.778642\pi$
$762$	3.41641	0.123763
$763$	−48.3607	−1.75077
$764$	4.58359	0.165829
$765$	3.41641	0.123520
$766$	57.0820	2.06246
$767$	−22.1115	−0.798398
$768$	9.00000	0.324760
$769$	−16.8328	−0.607007	−0.303503	−	0.952830i	$-0.598156\pi$
$769$	−16.8328	−0.607007	−0.303503	+	0.952830i	$0.598156\pi$
$770$	0	0
$771$	22.0000	0.792311
$772$	−52.2492	−1.88049
$773$	36.2918	1.30533	0.652663	−	0.757649i	$-0.273652\pi$
$773$	36.2918	1.30533	0.652663	+	0.757649i	$0.273652\pi$
$774$	12.7639	0.458790
$775$	22.4721	0.807223
$776$	18.9443	0.680060
$777$	14.4721	0.519185
$778$	46.1803	1.65565
$779$	−79.1935	−2.83740
$780$	−16.5836	−0.593788
$781$	0	0
$782$	6.18034	0.221009
$783$	4.47214	0.159821
$784$	−3.47214	−0.124005
$785$	−4.36068	−0.155639
$786$	37.8885	1.35144
$787$	13.7082	0.488645	0.244322	−	0.969694i	$-0.421434\pi$
$787$	13.7082	0.488645	0.244322	+	0.969694i	$0.421434\pi$
$788$	−52.2492	−1.86130
$789$	6.47214	0.230414
$790$	−26.8328	−0.954669
$791$	20.0000	0.711118
$792$	0	0
$793$	−20.0000	−0.710221
$794$	−4.47214	−0.158710
$795$	6.47214	0.229543
$796$	29.1246	1.03229
$797$	12.6525	0.448174	0.224087	−	0.974569i	$-0.428060\pi$
$797$	12.6525	0.448174	0.224087	+	0.974569i	$0.428060\pi$
$798$	−52.3607	−1.85355
$799$	−11.0557	−0.391124
$800$	23.2918	0.823489
$801$	−1.23607	−0.0436743
$802$	44.8754	1.58461
$803$	0	0
$804$	−2.29180	−0.0808254
$805$	−4.00000	−0.140981
$806$	−64.7214	−2.27971
$807$	0.472136	0.0166200
$808$	15.5279	0.546268
$809$	43.3050	1.52252	0.761261	−	0.648446i	$-0.224581\pi$
$809$	43.3050	1.52252	0.761261	+	0.648446i	$0.224581\pi$
$810$	2.76393	0.0971147
$811$	23.4164	0.822261	0.411131	−	0.911576i	$-0.365134\pi$
$811$	23.4164	0.822261	0.411131	+	0.911576i	$0.365134\pi$
$812$	43.4164	1.52362
$813$	−17.5279	−0.614729
$814$	0	0
$815$	9.16718	0.321112
$816$	2.76393	0.0967570
$817$	−41.3050	−1.44508
$818$	−38.9443	−1.36165
$819$	−14.4721	−0.505697
$820$	40.5836	1.41724
$821$	−12.1115	−0.422693	−0.211346	−	0.977411i	$-0.567785\pi$
$821$	−12.1115	−0.422693	−0.211346	+	0.977411i	$0.567785\pi$
$822$	2.76393	0.0964032
$823$	−25.5279	−0.889845	−0.444923	−	0.895569i	$-0.646769\pi$
$823$	−25.5279	−0.889845	−0.444923	+	0.895569i	$0.646769\pi$
$824$	−25.1246	−0.875257
$825$	0	0
$826$	35.7771	1.24484
$827$	−8.58359	−0.298481	−0.149240	−	0.988801i	$-0.547683\pi$
$827$	−8.58359	−0.298481	−0.149240	+	0.988801i	$0.547683\pi$
$828$	3.00000	0.104257
$829$	10.3607	0.359841	0.179921	−	0.983681i	$-0.442416\pi$
$829$	10.3607	0.359841	0.179921	+	0.983681i	$0.442416\pi$
$830$	−11.0557	−0.383750
$831$	15.8885	0.551167
$832$	−58.1378	−2.01556
$833$	9.59675	0.332508
$834$	−20.0000	−0.692543
$835$	−16.0000	−0.553703
$836$	0	0
$837$	6.47214	0.223710
$838$	31.0557	1.07280
$839$	−12.5836	−0.434434	−0.217217	−	0.976123i	$-0.569698\pi$
$839$	−12.5836	−0.434434	−0.217217	+	0.976123i	$0.569698\pi$
$840$	8.94427	0.308607
$841$	−9.00000	−0.310345
$842$	−69.1935	−2.38457
$843$	24.6525	0.849076
$844$	−40.5836	−1.39694
$845$	8.65248	0.297654
$846$	−8.94427	−0.307510
$847$	0	0
$848$	5.23607	0.179807
$849$	5.70820	0.195905
$850$	−21.4590	−0.736037
$851$	4.47214	0.153303
$852$	24.0000	0.822226
$853$	−22.0000	−0.753266	−0.376633	−	0.926363i	$-0.622918\pi$
$853$	−22.0000	−0.753266	−0.376633	+	0.926363i	$0.622918\pi$
$854$	32.3607	1.10736
$855$	−8.94427	−0.305888
$856$	−37.8885	−1.29500
$857$	42.9443	1.46695	0.733474	−	0.679717i	$-0.237898\pi$
$857$	42.9443	1.46695	0.733474	+	0.679717i	$0.237898\pi$
$858$	0	0
$859$	−7.05573	−0.240738	−0.120369	−	0.992729i	$-0.538408\pi$
$859$	−7.05573	−0.240738	−0.120369	+	0.992729i	$0.538408\pi$
$860$	21.1672	0.721795
$861$	35.4164	1.20699
$862$	−17.8885	−0.609286
$863$	−4.00000	−0.136162	−0.0680808	−	0.997680i	$-0.521688\pi$
$863$	−4.00000	−0.136162	−0.0680808	+	0.997680i	$0.521688\pi$
$864$	6.70820	0.228218
$865$	−5.52786	−0.187953
$866$	62.3607	2.11910
$867$	9.36068	0.317905
$868$	62.8328	2.13268
$869$	0	0
$870$	12.3607	0.419066
$871$	3.41641	0.115761
$872$	33.4164	1.13162
$873$	8.47214	0.286738
$874$	−16.1803	−0.547308
$875$	33.8885	1.14564
$876$	20.8328	0.703876
$877$	−33.0557	−1.11621	−0.558106	−	0.829769i	$-0.688472\pi$
$877$	−33.0557	−1.11621	−0.558106	+	0.829769i	$0.688472\pi$
$878$	22.1115	0.746226
$879$	7.70820	0.259991
$880$	0	0
$881$	26.5410	0.894190	0.447095	−	0.894487i	$-0.352459\pi$
$881$	26.5410	0.894190	0.447095	+	0.894487i	$0.352459\pi$
$882$	7.76393	0.261425
$883$	−7.41641	−0.249582	−0.124791	−	0.992183i	$-0.539826\pi$
$883$	−7.41641	−0.249582	−0.124791	+	0.992183i	$0.539826\pi$
$884$	37.0820	1.24720
$885$	6.11146	0.205434
$886$	−77.8885	−2.61672
$887$	−31.0557	−1.04275	−0.521375	−	0.853328i	$-0.674581\pi$
$887$	−31.0557	−1.04275	−0.521375	+	0.853328i	$0.674581\pi$
$888$	−10.0000	−0.335578
$889$	4.94427	0.165826
$890$	−3.41641	−0.114518
$891$	0	0
$892$	−77.6656	−2.60044
$893$	28.9443	0.968583
$894$	−26.1803	−0.875602
$895$	25.8885	0.865359
$896$	50.6525	1.69218
$897$	−4.47214	−0.149320
$898$	33.4164	1.11512
$899$	28.9443	0.965346
$900$	−10.4164	−0.347214
$901$	−14.4721	−0.482137
$902$	0	0
$903$	18.4721	0.614714
$904$	−13.8197	−0.459635
$905$	−14.6950	−0.488480
$906$	−35.7771	−1.18861
$907$	−1.12461	−0.0373421	−0.0186711	−	0.999826i	$-0.505944\pi$
$907$	−1.12461	−0.0373421	−0.0186711	+	0.999826i	$0.505944\pi$
$908$	40.5836	1.34681
$909$	6.94427	0.230327
$910$	−40.0000	−1.32599
$911$	−40.7214	−1.34916	−0.674579	−	0.738202i	$-0.735675\pi$
$911$	−40.7214	−1.34916	−0.674579	+	0.738202i	$0.735675\pi$
$912$	−7.23607	−0.239610
$913$	0	0
$914$	−18.9443	−0.626621
$915$	5.52786	0.182746
$916$	8.83282	0.291844
$917$	54.8328	1.81074
$918$	−6.18034	−0.203982
$919$	23.0132	0.759134	0.379567	−	0.925164i	$-0.376073\pi$
$919$	23.0132	0.759134	0.379567	+	0.925164i	$0.376073\pi$
$920$	2.76393	0.0911241
$921$	2.47214	0.0814596
$922$	12.1115	0.398870
$923$	−35.7771	−1.17762
$924$	0	0
$925$	−15.5279	−0.510553
$926$	−28.9443	−0.951168
$927$	−11.2361	−0.369041
$928$	30.0000	0.984798
$929$	−28.8328	−0.945974	−0.472987	−	0.881069i	$-0.656824\pi$
$929$	−28.8328	−0.945974	−0.472987	+	0.881069i	$0.656824\pi$
$930$	17.8885	0.586588
$931$	−25.1246	−0.823426
$932$	42.0000	1.37576
$933$	−7.05573	−0.230994
$934$	45.5279	1.48972
$935$	0	0
$936$	10.0000	0.326860
$937$	12.8328	0.419230	0.209615	−	0.977784i	$-0.432779\pi$
$937$	12.8328	0.419230	0.209615	+	0.977784i	$0.432779\pi$
$938$	−5.52786	−0.180491
$939$	−5.05573	−0.164987
$940$	−14.8328	−0.483793
$941$	42.5410	1.38680	0.693399	−	0.720554i	$-0.256112\pi$
$941$	42.5410	1.38680	0.693399	+	0.720554i	$0.256112\pi$
$942$	7.88854	0.257023
$943$	10.9443	0.356395
$944$	4.94427	0.160922
$945$	4.00000	0.130120
$946$	0	0
$947$	−1.16718	−0.0379284	−0.0189642	−	0.999820i	$-0.506037\pi$
$947$	−1.16718	−0.0379284	−0.0189642	+	0.999820i	$0.506037\pi$
$948$	29.1246	0.945923
$949$	−31.0557	−1.00811
$950$	56.1803	1.82273
$951$	−2.58359	−0.0837787
$952$	−20.0000	−0.648204
$953$	−12.0689	−0.390949	−0.195475	−	0.980709i	$-0.562625\pi$
$953$	−12.0689	−0.390949	−0.195475	+	0.980709i	$0.562625\pi$
$954$	−11.7082	−0.379067
$955$	1.88854	0.0611118
$956$	−14.8328	−0.479728
$957$	0	0
$958$	75.7771	2.44825
$959$	4.00000	0.129167
$960$	16.0689	0.518621
$961$	10.8885	0.351243
$962$	44.7214	1.44187
$963$	−16.9443	−0.546022
$964$	−58.5836	−1.88685
$965$	−21.5279	−0.693006
$966$	7.23607	0.232817
$967$	7.63932	0.245664	0.122832	−	0.992427i	$-0.460802\pi$
$967$	7.63932	0.245664	0.122832	+	0.992427i	$0.460802\pi$
$968$	0	0
$969$	20.0000	0.642493
$970$	23.4164	0.751856
$971$	17.3050	0.555342	0.277671	−	0.960676i	$-0.410437\pi$
$971$	17.3050	0.555342	0.277671	+	0.960676i	$0.410437\pi$
$972$	−3.00000	−0.0962250
$973$	−28.9443	−0.927911
$974$	53.6656	1.71956
$975$	15.5279	0.497290
$976$	4.47214	0.143150
$977$	−32.0689	−1.02597	−0.512987	−	0.858396i	$-0.671461\pi$
$977$	−32.0689	−1.02597	−0.512987	+	0.858396i	$0.671461\pi$
$978$	−16.5836	−0.530285
$979$	0	0
$980$	12.8754	0.411289
$981$	14.9443	0.477134
$982$	−89.4427	−2.85423
$983$	22.8328	0.728254	0.364127	−	0.931349i	$-0.381367\pi$
$983$	22.8328	0.728254	0.364127	+	0.931349i	$0.381367\pi$
$984$	−24.4721	−0.780143
$985$	−21.5279	−0.685935
$986$	−27.6393	−0.880215
$987$	−12.9443	−0.412021
$988$	−97.0820	−3.08859
$989$	5.70820	0.181510
$990$	0	0
$991$	−1.52786	−0.0485342	−0.0242671	−	0.999706i	$-0.507725\pi$
$991$	−1.52786	−0.0485342	−0.0242671	+	0.999706i	$0.507725\pi$
$992$	43.4164	1.37847
$993$	−23.4164	−0.743097
$994$	57.8885	1.83611
$995$	12.0000	0.380426
$996$	12.0000	0.380235
$997$	−42.3607	−1.34158	−0.670788	−	0.741649i	$-0.734044\pi$
$997$	−42.3607	−1.34158	−0.670788	+	0.741649i	$0.734044\pi$
$998$	−27.6393	−0.874907
$999$	−4.47214	−0.141492

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8349.2.a.i.1.2		2
11.10	odd	2		69.2.a.b.1.1	✓	2
33.32	even	2		207.2.a.c.1.2		2
44.43	even	2		1104.2.a.m.1.2		2
55.32	even	4		1725.2.b.o.1174.2		4
55.43	even	4		1725.2.b.o.1174.3		4
55.54	odd	2		1725.2.a.ba.1.2		2
77.76	even	2		3381.2.a.t.1.1		2
88.21	odd	2		4416.2.a.bm.1.1		2
88.43	even	2		4416.2.a.bg.1.1		2
132.131	odd	2		3312.2.a.bb.1.1		2
165.164	even	2		5175.2.a.bk.1.1		2
253.252	even	2		1587.2.a.i.1.1		2
759.758	odd	2		4761.2.a.v.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.2.a.b.1.1	✓	2	11.10	odd	2
207.2.a.c.1.2		2	33.32	even	2
1104.2.a.m.1.2		2	44.43	even	2
1587.2.a.i.1.1		2	253.252	even	2
1725.2.a.ba.1.2		2	55.54	odd	2
1725.2.b.o.1174.2		4	55.32	even	4
1725.2.b.o.1174.3		4	55.43	even	4
3312.2.a.bb.1.1		2	132.131	odd	2
3381.2.a.t.1.1		2	77.76	even	2
4416.2.a.bg.1.1		2	88.43	even	2
4416.2.a.bm.1.1		2	88.21	odd	2
4761.2.a.v.1.2		2	759.758	odd	2
5175.2.a.bk.1.1		2	165.164	even	2
8349.2.a.i.1.2		2	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:	\( N \)	\(=\)	\( 8349 = 3 \cdot 11^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8349.a (trivial)

\(f(q)\)	\(=\)	\(q+2.23607 q^{2} -1.00000 q^{3} +3.00000 q^{4} +1.23607 q^{5} -2.23607 q^{6} -3.23607 q^{7} +2.23607 q^{8} +1.00000 q^{9} +2.76393 q^{10} -3.00000 q^{12} +4.47214 q^{13} -7.23607 q^{14} -1.23607 q^{15} -1.00000 q^{16} +2.76393 q^{17} +2.23607 q^{18} -7.23607 q^{19} +3.70820 q^{20} +3.23607 q^{21} +1.00000 q^{23} -2.23607 q^{24} -3.47214 q^{25} +10.0000 q^{26} -1.00000 q^{27} -9.70820 q^{28} -4.47214 q^{29} -2.76393 q^{30} -6.47214 q^{31} -6.70820 q^{32} +6.18034 q^{34} -4.00000 q^{35} +3.00000 q^{36} +4.47214 q^{37} -16.1803 q^{38} -4.47214 q^{39} +2.76393 q^{40} +10.9443 q^{41} +7.23607 q^{42} +5.70820 q^{43} +1.23607 q^{45} +2.23607 q^{46} -4.00000 q^{47} +1.00000 q^{48} +3.47214 q^{49} -7.76393 q^{50} -2.76393 q^{51} +13.4164 q^{52} -5.23607 q^{53} -2.23607 q^{54} -7.23607 q^{56} +7.23607 q^{57} -10.0000 q^{58} -4.94427 q^{59} -3.70820 q^{60} -4.47214 q^{61} -14.4721 q^{62} -3.23607 q^{63} -13.0000 q^{64} +5.52786 q^{65} +0.763932 q^{67} +8.29180 q^{68} -1.00000 q^{69} -8.94427 q^{70} -8.00000 q^{71} +2.23607 q^{72} -6.94427 q^{73} +10.0000 q^{74} +3.47214 q^{75} -21.7082 q^{76} -10.0000 q^{78} -9.70820 q^{79} -1.23607 q^{80} +1.00000 q^{81} +24.4721 q^{82} -4.00000 q^{83} +9.70820 q^{84} +3.41641 q^{85} +12.7639 q^{86} +4.47214 q^{87} -1.23607 q^{89} +2.76393 q^{90} -14.4721 q^{91} +3.00000 q^{92} +6.47214 q^{93} -8.94427 q^{94} -8.94427 q^{95} +6.70820 q^{96} +8.47214 q^{97} +7.76393 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 6 q^{4} - 2 q^{5} - 2 q^{7} + 2 q^{9} + 10 q^{10} - 6 q^{12} - 10 q^{14} + 2 q^{15} - 2 q^{16} + 10 q^{17} - 10 q^{19} - 6 q^{20} + 2 q^{21} + 2 q^{23} + 2 q^{25} + 20 q^{26} - 2 q^{27}+ \cdots + 20 q^{98}+O(q^{100}) \) 2 * q - 2 * q^3 + 6 * q^4 - 2 * q^5 - 2 * q^7 + 2 * q^9 + 10 * q^10 - 6 * q^12 - 10 * q^14 + 2 * q^15 - 2 * q^16 + 10 * q^17 - 10 * q^19 - 6 * q^20 + 2 * q^21 + 2 * q^23 + 2 * q^25 + 20 * q^26 - 2 * q^27 - 6 * q^28 - 10 * q^30 - 4 * q^31 - 10 * q^34 - 8 * q^35 + 6 * q^36 - 10 * q^38 + 10 * q^40 + 4 * q^41 + 10 * q^42 - 2 * q^43 - 2 * q^45 - 8 * q^47 + 2 * q^48 - 2 * q^49 - 20 * q^50 - 10 * q^51 - 6 * q^53 - 10 * q^56 + 10 * q^57 - 20 * q^58 + 8 * q^59 + 6 * q^60 - 20 * q^62 - 2 * q^63 - 26 * q^64 + 20 * q^65 + 6 * q^67 + 30 * q^68 - 2 * q^69 - 16 * q^71 + 4 * q^73 + 20 * q^74 - 2 * q^75 - 30 * q^76 - 20 * q^78 - 6 * q^79 + 2 * q^80 + 2 * q^81 + 40 * q^82 - 8 * q^83 + 6 * q^84 - 20 * q^85 + 30 * q^86 + 2 * q^89 + 10 * q^90 - 20 * q^91 + 6 * q^92 + 4 * q^93 + 8 * q^97 + 20 * q^98

Introduction

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