LMFDB - Embedded newform 207.2.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [207,2,Mod(1,207)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(207, 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("207.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 207 = 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	207.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:	$1.65290332184$
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:	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	$\chi$	$=$	207.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23607 q^{2} +3.00000 q^{4} -1.23607 q^{5} +3.23607 q^{7} +2.23607 q^{8} +O(q^{10})$	\(q+2.23607 q^{2} +3.00000 q^{4} -1.23607 q^{5} +3.23607 q^{7} +2.23607 q^{8} -2.76393 q^{10} -4.00000 q^{11} -4.47214 q^{13} +7.23607 q^{14} -1.00000 q^{16} +2.76393 q^{17} +7.23607 q^{19} -3.70820 q^{20} -8.94427 q^{22} -1.00000 q^{23} -3.47214 q^{25} -10.0000 q^{26} +9.70820 q^{28} -4.47214 q^{29} -6.47214 q^{31} -6.70820 q^{32} +6.18034 q^{34} -4.00000 q^{35} +4.47214 q^{37} +16.1803 q^{38} -2.76393 q^{40} +10.9443 q^{41} -5.70820 q^{43} -12.0000 q^{44} -2.23607 q^{46} +4.00000 q^{47} +3.47214 q^{49} -7.76393 q^{50} -13.4164 q^{52} +5.23607 q^{53} +4.94427 q^{55} +7.23607 q^{56} -10.0000 q^{58} +4.94427 q^{59} +4.47214 q^{61} -14.4721 q^{62} -13.0000 q^{64} +5.52786 q^{65} +0.763932 q^{67} +8.29180 q^{68} -8.94427 q^{70} +8.00000 q^{71} +6.94427 q^{73} +10.0000 q^{74} +21.7082 q^{76} -12.9443 q^{77} +9.70820 q^{79} +1.23607 q^{80} +24.4721 q^{82} -4.00000 q^{83} -3.41641 q^{85} -12.7639 q^{86} -8.94427 q^{88} +1.23607 q^{89} -14.4721 q^{91} -3.00000 q^{92} +8.94427 q^{94} -8.94427 q^{95} +8.47214 q^{97} +7.76393 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{4} + 2 q^{5} + 2 q^{7}+O(q^{10}) $ 2 * q + 6 * q^4 + 2 * q^5 + 2 * q^7	$ 2 q + 6 q^{4} + 2 q^{5} + 2 q^{7} - 10 q^{10} - 8 q^{11} + 10 q^{14} - 2 q^{16} + 10 q^{17} + 10 q^{19} + 6 q^{20} - 2 q^{23} + 2 q^{25} - 20 q^{26} + 6 q^{28} - 4 q^{31} - 10 q^{34} - 8 q^{35} + 10 q^{38} - 10 q^{40} + 4 q^{41} + 2 q^{43} - 24 q^{44} + 8 q^{47} - 2 q^{49} - 20 q^{50} + 6 q^{53} - 8 q^{55} + 10 q^{56} - 20 q^{58} - 8 q^{59} - 20 q^{62} - 26 q^{64} + 20 q^{65} + 6 q^{67} + 30 q^{68} + 16 q^{71} - 4 q^{73} + 20 q^{74} + 30 q^{76} - 8 q^{77} + 6 q^{79} - 2 q^{80} + 40 q^{82} - 8 q^{83} + 20 q^{85} - 30 q^{86} - 2 q^{89} - 20 q^{91} - 6 q^{92} + 8 q^{97} + 20 q^{98}+O(q^{100}) $ 2 * q + 6 * q^4 + 2 * q^5 + 2 * q^7 - 10 * q^10 - 8 * q^11 + 10 * q^14 - 2 * q^16 + 10 * q^17 + 10 * q^19 + 6 * q^20 - 2 * q^23 + 2 * q^25 - 20 * q^26 + 6 * q^28 - 4 * q^31 - 10 * q^34 - 8 * q^35 + 10 * q^38 - 10 * q^40 + 4 * q^41 + 2 * q^43 - 24 * q^44 + 8 * q^47 - 2 * q^49 - 20 * q^50 + 6 * q^53 - 8 * q^55 + 10 * q^56 - 20 * q^58 - 8 * q^59 - 20 * q^62 - 26 * q^64 + 20 * q^65 + 6 * q^67 + 30 * q^68 + 16 * q^71 - 4 * q^73 + 20 * q^74 + 30 * q^76 - 8 * q^77 + 6 * q^79 - 2 * q^80 + 40 * q^82 - 8 * q^83 + 20 * q^85 - 30 * q^86 - 2 * q^89 - 20 * q^91 - 6 * q^92 + 8 * q^97 + 20 * q^98

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	$-0.209785\pi$
$2$	2.23607	1.58114	0.790569	+	0.612372i	$0.209785\pi$
$3$	0	0
$3$	0	0
$4$	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
$9$	0	0
$10$	−2.76393	−0.874032
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	−4.47214	−1.24035	−0.620174	−	0.784465i	$-0.712938\pi$
$13$	−4.47214	−1.24035	−0.620174	+	0.784465i	$0.712938\pi$
$14$	7.23607	1.93392
$15$	0	0
$16$	−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	$0.391204\pi$
$18$	0	0
$19$	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
$22$	−8.94427	−1.90693
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−3.47214	−0.694427
$26$	−10.0000	−1.96116
$27$	0	0
$28$	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	$0.636298\pi$
$30$	0	0
$31$	−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
$36$	0	0
$37$	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
$39$	0	0
$40$	−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	$0.173800\pi$
$42$	0	0
$43$	−5.70820	−0.870493	−0.435246	−	0.900311i	$-0.643339\pi$
$43$	−5.70820	−0.870493	−0.435246	+	0.900311i	$0.643339\pi$
$44$	−12.0000	−1.80907
$45$	0	0
$46$	−2.23607	−0.329690
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	3.47214	0.496019
$50$	−7.76393	−1.09799
$51$	0	0
$52$	−13.4164	−1.86052
$53$	5.23607	0.719229	0.359615	−	0.933101i	$-0.382908\pi$
$53$	5.23607	0.719229	0.359615	+	0.933101i	$0.382908\pi$
$54$	0	0
$55$	4.94427	0.666685
$56$	7.23607	0.966960
$57$	0	0
$58$	−10.0000	−1.31306
$59$	4.94427	0.643689	0.321845	−	0.946792i	$-0.395697\pi$
$59$	4.94427	0.643689	0.321845	+	0.946792i	$0.395697\pi$
$60$	0	0
$61$	4.47214	0.572598	0.286299	−	0.958140i	$-0.407575\pi$
$61$	4.47214	0.572598	0.286299	+	0.958140i	$0.407575\pi$
$62$	−14.4721	−1.83796
$63$	0	0
$64$	−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
$69$	0	0
$70$	−8.94427	−1.06904
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	6.94427	0.812766	0.406383	−	0.913703i	$-0.366790\pi$
$73$	6.94427	0.812766	0.406383	+	0.913703i	$0.366790\pi$
$74$	10.0000	1.16248
$75$	0	0
$76$	21.7082	2.49010
$77$	−12.9443	−1.47514
$78$	0	0
$79$	9.70820	1.09226	0.546129	−	0.837701i	$-0.316101\pi$
$79$	9.70820	1.09226	0.546129	+	0.837701i	$0.316101\pi$
$80$	1.23607	0.138197
$81$	0	0
$82$	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	$0.570452\pi$
$84$	0	0
$85$	−3.41641	−0.370561
$86$	−12.7639	−1.37637
$87$	0	0
$88$	−8.94427	−0.953463
$89$	1.23607	0.131023	0.0655115	−	0.997852i	$-0.479132\pi$
$89$	1.23607	0.131023	0.0655115	+	0.997852i	$0.479132\pi$
$90$	0	0
$91$	−14.4721	−1.51709
$92$	−3.00000	−0.312772
$93$	0	0
$94$	8.94427	0.922531
$95$	−8.94427	−0.917663
$96$	0	0
$97$	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	$0.387713\pi$
$102$	0	0
$103$	−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
$105$	0	0
$106$	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	$0.805489\pi$
$108$	0	0
$109$	−14.9443	−1.43140	−0.715701	−	0.698407i	$-0.753893\pi$
$109$	−14.9443	−1.43140	−0.715701	+	0.698407i	$0.753893\pi$
$110$	11.0557	1.05412
$111$	0	0
$112$	−3.23607	−0.305780
$113$	6.18034	0.581397	0.290699	−	0.956815i	$-0.406112\pi$
$113$	6.18034	0.581397	0.290699	+	0.956815i	$0.406112\pi$
$114$	0	0
$115$	1.23607	0.115264
$116$	−13.4164	−1.24568
$117$	0	0
$118$	11.0557	1.01776
$119$	8.94427	0.819920
$120$	0	0
$121$	5.00000	0.454545
$122$	10.0000	0.905357
$123$	0	0
$124$	−19.4164	−1.74364
$125$	10.4721	0.936656
$126$	0	0
$127$	1.52786	0.135576	0.0677880	−	0.997700i	$-0.478406\pi$
$127$	1.52786	0.135576	0.0677880	+	0.997700i	$0.478406\pi$
$128$	−15.6525	−1.38350
$129$	0	0
$130$	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
$135$	0	0
$136$	6.18034	0.529960
$137$	1.23607	0.105604	0.0528022	−	0.998605i	$-0.483185\pi$
$137$	1.23607	0.105604	0.0528022	+	0.998605i	$0.483185\pi$
$138$	0	0
$139$	−8.94427	−0.758643	−0.379322	−	0.925265i	$-0.623843\pi$
$139$	−8.94427	−0.758643	−0.379322	+	0.925265i	$0.623843\pi$
$140$	−12.0000	−1.01419
$141$	0	0
$142$	17.8885	1.50117
$143$	17.8885	1.49592
$144$	0	0
$145$	5.52786	0.459064
$146$	15.5279	1.28510
$147$	0	0
$148$	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	$0.340787\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	16.1803	1.31240
$153$	0	0
$154$	−28.9443	−2.33240
$155$	8.00000	0.642575
$156$	0	0
$157$	−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
$159$	0	0
$160$	8.29180	0.655524
$161$	−3.23607	−0.255038
$162$	0	0
$163$	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	$0.666971\pi$
$168$	0	0
$169$	7.00000	0.538462
$170$	−7.63932	−0.585909
$171$	0	0
$172$	−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	$0.554378\pi$
$174$	0	0
$175$	−11.2361	−0.849367
$176$	4.00000	0.301511
$177$	0	0
$178$	2.76393	0.207165
$179$	−20.9443	−1.56545	−0.782724	−	0.622369i	$-0.786170\pi$
$179$	−20.9443	−1.56545	−0.782724	+	0.622369i	$0.786170\pi$
$180$	0	0
$181$	−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
$183$	0	0
$184$	−2.23607	−0.164845
$185$	−5.52786	−0.406417
$186$	0	0
$187$	−11.0557	−0.808475
$188$	12.0000	0.875190
$189$	0	0
$190$	−20.0000	−1.45095
$191$	−1.52786	−0.110552	−0.0552762	−	0.998471i	$-0.517604\pi$
$191$	−1.52786	−0.110552	−0.0552762	+	0.998471i	$0.517604\pi$
$192$	0	0
$193$	17.4164	1.25366	0.626830	−	0.779156i	$-0.284352\pi$
$193$	17.4164	1.25366	0.626830	+	0.779156i	$0.284352\pi$
$194$	18.9443	1.36012
$195$	0	0
$196$	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	0
$202$	15.5279	1.09254
$203$	−14.4721	−1.01574
$204$	0	0
$205$	−13.5279	−0.944827
$206$	−25.1246	−1.75051
$207$	0	0
$208$	4.47214	0.310087
$209$	−28.9443	−2.00212
$210$	0	0
$211$	13.5279	0.931297	0.465648	−	0.884970i	$-0.345821\pi$
$211$	13.5279	0.931297	0.465648	+	0.884970i	$0.345821\pi$
$212$	15.7082	1.07884
$213$	0	0
$214$	−37.8885	−2.59001
$215$	7.05573	0.481197
$216$	0	0
$217$	−20.9443	−1.42179
$218$	−33.4164	−2.26324
$219$	0	0
$220$	14.8328	1.00003
$221$	−12.3607	−0.831469
$222$	0	0
$223$	−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
$225$	0	0
$226$	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	$0.351802\pi$
$228$	0	0
$229$	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	$0.348356\pi$
$234$	0	0
$235$	−4.94427	−0.322529
$236$	14.8328	0.965534
$237$	0	0
$238$	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	$0.551120\pi$
$240$	0	0
$241$	19.5279	1.25790	0.628950	−	0.777446i	$-0.283485\pi$
$241$	19.5279	1.25790	0.628950	+	0.777446i	$0.283485\pi$
$242$	11.1803	0.718699
$243$	0	0
$244$	13.4164	0.858898
$245$	−4.29180	−0.274193
$246$	0	0
$247$	−32.3607	−2.05906
$248$	−14.4721	−0.918982
$249$	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	$0.392783\pi$
$252$	0	0
$253$	4.00000	0.251478
$254$	3.41641	0.214364
$255$	0	0
$256$	−9.00000	−0.562500
$257$	22.0000	1.37232	0.686161	−	0.727450i	$-0.259294\pi$
$257$	22.0000	1.37232	0.686161	+	0.727450i	$0.259294\pi$
$258$	0	0
$259$	14.4721	0.899255
$260$	16.5836	1.02847
$261$	0	0
$262$	−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
$267$	0	0
$268$	2.29180	0.139994
$269$	0.472136	0.0287866	0.0143933	−	0.999896i	$-0.495418\pi$
$269$	0.472136	0.0287866	0.0143933	+	0.999896i	$0.495418\pi$
$270$	0	0
$271$	−17.5279	−1.06474	−0.532371	−	0.846511i	$-0.678699\pi$
$271$	−17.5279	−1.06474	−0.532371	+	0.846511i	$0.678699\pi$
$272$	−2.76393	−0.167588
$273$	0	0
$274$	2.76393	0.166975
$275$	13.8885	0.837511
$276$	0	0
$277$	15.8885	0.954650	0.477325	−	0.878727i	$-0.341606\pi$
$277$	15.8885	0.954650	0.477325	+	0.878727i	$0.341606\pi$
$278$	−20.0000	−1.19952
$279$	0	0
$280$	−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	$0.762969\pi$
$282$	0	0
$283$	5.70820	0.339318	0.169659	−	0.985503i	$-0.445733\pi$
$283$	5.70820	0.339318	0.169659	+	0.985503i	$0.445733\pi$
$284$	24.0000	1.42414
$285$	0	0
$286$	40.0000	2.36525
$287$	35.4164	2.09056
$288$	0	0
$289$	−9.36068	−0.550628
$290$	12.3607	0.725844
$291$	0	0
$292$	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	$0.572290\pi$
$294$	0	0
$295$	−6.11146	−0.355823
$296$	10.0000	0.581238
$297$	0	0
$298$	26.1803	1.51659
$299$	4.47214	0.258630
$300$	0	0
$301$	−18.4721	−1.06472
$302$	−35.7771	−2.05874
$303$	0	0
$304$	−7.23607	−0.415017
$305$	−5.52786	−0.316525
$306$	0	0
$307$	2.47214	0.141092	0.0705461	−	0.997509i	$-0.477526\pi$
$307$	2.47214	0.141092	0.0705461	+	0.997509i	$0.477526\pi$
$308$	−38.8328	−2.21271
$309$	0	0
$310$	17.8885	1.01600
$311$	−7.05573	−0.400094	−0.200047	−	0.979786i	$-0.564109\pi$
$311$	−7.05573	−0.400094	−0.200047	+	0.979786i	$0.564109\pi$
$312$	0	0
$313$	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
$315$	0	0
$316$	29.1246	1.63839
$317$	−2.58359	−0.145109	−0.0725545	−	0.997364i	$-0.523115\pi$
$317$	−2.58359	−0.145109	−0.0725545	+	0.997364i	$0.523115\pi$
$318$	0	0
$319$	17.8885	1.00157
$320$	16.0689	0.898278
$321$	0	0
$322$	−7.23607	−0.403250
$323$	20.0000	1.11283
$324$	0	0
$325$	15.5279	0.861331
$326$	16.5836	0.918480
$327$	0	0
$328$	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
$333$	0	0
$334$	−28.9443	−1.58376
$335$	−0.944272	−0.0515911
$336$	0	0
$337$	−14.3607	−0.782276	−0.391138	−	0.920332i	$-0.627918\pi$
$337$	−14.3607	−0.782276	−0.391138	+	0.920332i	$0.627918\pi$
$338$	15.6525	0.851382
$339$	0	0
$340$	−10.2492	−0.555842
$341$	25.8885	1.40194
$342$	0	0
$343$	−11.4164	−0.616428
$344$	−12.7639	−0.688185
$345$	0	0
$346$	−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	$0.414489\pi$
$348$	0	0
$349$	−23.5279	−1.25942	−0.629709	−	0.776831i	$-0.716826\pi$
$349$	−23.5279	−1.25942	−0.629709	+	0.776831i	$0.716826\pi$
$350$	−25.1246	−1.34297
$351$	0	0
$352$	26.8328	1.43019
$353$	17.4164	0.926982	0.463491	−	0.886102i	$-0.346597\pi$
$353$	17.4164	0.926982	0.463491	+	0.886102i	$0.346597\pi$
$354$	0	0
$355$	−9.88854	−0.524829
$356$	3.70820	0.196534
$357$	0	0
$358$	−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	$0.554633\pi$
$360$	0	0
$361$	33.3607	1.75583
$362$	−26.5836	−1.39720
$363$	0	0
$364$	−43.4164	−2.27564
$365$	−8.58359	−0.449286
$366$	0	0
$367$	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
$369$	0	0
$370$	−12.3607	−0.642601
$371$	16.9443	0.879703
$372$	0	0
$373$	20.4721	1.06001	0.530004	−	0.847995i	$-0.322191\pi$
$373$	20.4721	1.06001	0.530004	+	0.847995i	$0.322191\pi$
$374$	−24.7214	−1.27831
$375$	0	0
$376$	8.94427	0.461266
$377$	20.0000	1.03005
$378$	0	0
$379$	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
$381$	0	0
$382$	−3.41641	−0.174799
$383$	−25.5279	−1.30441	−0.652206	−	0.758041i	$-0.726156\pi$
$383$	−25.5279	−1.30441	−0.652206	+	0.758041i	$0.726156\pi$
$384$	0	0
$385$	16.0000	0.815436
$386$	38.9443	1.98221
$387$	0	0
$388$	25.4164	1.29032
$389$	−20.6525	−1.04712	−0.523561	−	0.851988i	$-0.675397\pi$
$389$	−20.6525	−1.04712	−0.523561	+	0.851988i	$0.675397\pi$
$390$	0	0
$391$	−2.76393	−0.139778
$392$	7.76393	0.392138
$393$	0	0
$394$	−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
$399$	0	0
$400$	3.47214	0.173607
$401$	−20.0689	−1.00219	−0.501096	−	0.865392i	$-0.667070\pi$
$401$	−20.0689	−1.00219	−0.501096	+	0.865392i	$0.667070\pi$
$402$	0	0
$403$	28.9443	1.44182
$404$	20.8328	1.03647
$405$	0	0
$406$	−32.3607	−1.60603
$407$	−17.8885	−0.886702
$408$	0	0
$409$	17.4164	0.861186	0.430593	−	0.902546i	$-0.358304\pi$
$409$	17.4164	0.861186	0.430593	+	0.902546i	$0.358304\pi$
$410$	−30.2492	−1.49390
$411$	0	0
$412$	−33.7082	−1.66068
$413$	16.0000	0.787309
$414$	0	0
$415$	4.94427	0.242705
$416$	30.0000	1.47087
$417$	0	0
$418$	−64.7214	−3.16563
$419$	−13.8885	−0.678500	−0.339250	−	0.940696i	$-0.610173\pi$
$419$	−13.8885	−0.678500	−0.339250	+	0.940696i	$0.610173\pi$
$420$	0	0
$421$	−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
$423$	0	0
$424$	11.7082	0.568601
$425$	−9.59675	−0.465511
$426$	0	0
$427$	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	$0.561716\pi$
$432$	0	0
$433$	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
$435$	0	0
$436$	−44.8328	−2.14710
$437$	−7.23607	−0.346148
$438$	0	0
$439$	−9.88854	−0.471954	−0.235977	−	0.971759i	$-0.575829\pi$
$439$	−9.88854	−0.471954	−0.235977	+	0.971759i	$0.575829\pi$
$440$	11.0557	0.527061
$441$	0	0
$442$	−27.6393	−1.31467
$443$	34.8328	1.65496	0.827479	−	0.561497i	$-0.189775\pi$
$443$	34.8328	1.65496	0.827479	+	0.561497i	$0.189775\pi$
$444$	0	0
$445$	−1.52786	−0.0724277
$446$	−57.8885	−2.74110
$447$	0	0
$448$	−42.0689	−1.98757
$449$	−14.9443	−0.705264	−0.352632	−	0.935762i	$-0.614713\pi$
$449$	−14.9443	−0.705264	−0.352632	+	0.935762i	$0.614713\pi$
$450$	0	0
$451$	−43.7771	−2.06138
$452$	18.5410	0.872096
$453$	0	0
$454$	30.2492	1.41967
$455$	17.8885	0.838628
$456$	0	0
$457$	8.47214	0.396310	0.198155	−	0.980171i	$-0.436505\pi$
$457$	8.47214	0.396310	0.198155	+	0.980171i	$0.436505\pi$
$458$	6.58359	0.307631
$459$	0	0
$460$	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
$465$	0	0
$466$	31.3050	1.45017
$467$	−20.3607	−0.942180	−0.471090	−	0.882085i	$-0.656139\pi$
$467$	−20.3607	−0.942180	−0.471090	+	0.882085i	$0.656139\pi$
$468$	0	0
$469$	2.47214	0.114153
$470$	−11.0557	−0.509963
$471$	0	0
$472$	11.0557	0.508881
$473$	22.8328	1.04985
$474$	0	0
$475$	−25.1246	−1.15280
$476$	26.8328	1.22988
$477$	0	0
$478$	−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	$0.218151\pi$
$480$	0	0
$481$	−20.0000	−0.911922
$482$	43.6656	1.98892
$483$	0	0
$484$	15.0000	0.681818
$485$	−10.4721	−0.475515
$486$	0	0
$487$	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
$489$	0	0
$490$	−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	$0.858335\pi$
$492$	0	0
$493$	−12.3607	−0.556697
$494$	−72.3607	−3.25566
$495$	0	0
$496$	6.47214	0.290607
$497$	25.8885	1.16126
$498$	0	0
$499$	−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
$501$	0	0
$502$	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	$0.360335\pi$
$504$	0	0
$505$	−8.58359	−0.381965
$506$	8.94427	0.397621
$507$	0	0
$508$	4.58359	0.203364
$509$	16.4721	0.730115	0.365057	−	0.930985i	$-0.381049\pi$
$509$	16.4721	0.730115	0.365057	+	0.930985i	$0.381049\pi$
$510$	0	0
$511$	22.4721	0.994109
$512$	11.1803	0.494106
$513$	0	0
$514$	49.1935	2.16983
$515$	13.8885	0.612002
$516$	0	0
$517$	−16.0000	−0.703679
$518$	32.3607	1.42185
$519$	0	0
$520$	12.3607	0.542052
$521$	33.2361	1.45610	0.728049	−	0.685525i	$-0.240427\pi$
$521$	33.2361	1.45610	0.728049	+	0.685525i	$0.240427\pi$
$522$	0	0
$523$	15.5967	0.681998	0.340999	−	0.940064i	$-0.389235\pi$
$523$	15.5967	0.681998	0.340999	+	0.940064i	$0.389235\pi$
$524$	−50.8328	−2.22064
$525$	0	0
$526$	−14.4721	−0.631015
$527$	−17.8885	−0.779237
$528$	0	0
$529$	1.00000	0.0434783
$530$	−14.4721	−0.628629
$531$	0	0
$532$	70.2492	3.04569
$533$	−48.9443	−2.12001
$534$	0	0
$535$	20.9443	0.905500
$536$	1.70820	0.0737832
$537$	0	0
$538$	1.05573	0.0455157
$539$	−13.8885	−0.598222
$540$	0	0
$541$	−15.5279	−0.667595	−0.333798	−	0.942645i	$-0.608330\pi$
$541$	−15.5279	−0.667595	−0.333798	+	0.942645i	$0.608330\pi$
$542$	−39.1935	−1.68350
$543$	0	0
$544$	−18.5410	−0.794940
$545$	18.4721	0.791259
$546$	0	0
$547$	13.5279	0.578410	0.289205	−	0.957267i	$-0.406609\pi$
$547$	13.5279	0.578410	0.289205	+	0.957267i	$0.406609\pi$
$548$	3.70820	0.158407
$549$	0	0
$550$	31.0557	1.32422
$551$	−32.3607	−1.37861
$552$	0	0
$553$	31.4164	1.33596
$554$	35.5279	1.50943
$555$	0	0
$556$	−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	$0.148158\pi$
$558$	0	0
$559$	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	$0.450049\pi$
$564$	0	0
$565$	−7.63932	−0.321389
$566$	12.7639	0.536508
$567$	0	0
$568$	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	$0.427558\pi$
$570$	0	0
$571$	29.7082	1.24325	0.621625	−	0.783315i	$-0.286473\pi$
$571$	29.7082	1.24325	0.621625	+	0.783315i	$0.286473\pi$
$572$	53.6656	2.24387
$573$	0	0
$574$	79.1935	3.30547
$575$	3.47214	0.144798
$576$	0	0
$577$	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
$579$	0	0
$580$	16.5836	0.688596
$581$	−12.9443	−0.537019
$582$	0	0
$583$	−20.9443	−0.867423
$584$	15.5279	0.642548
$585$	0	0
$586$	−17.2361	−0.712015
$587$	0.944272	0.0389743	0.0194871	−	0.999810i	$-0.493797\pi$
$587$	0.944272	0.0389743	0.0194871	+	0.999810i	$0.493797\pi$
$588$	0	0
$589$	−46.8328	−1.92971
$590$	−13.6656	−0.562605
$591$	0	0
$592$	−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
$597$	0	0
$598$	10.0000	0.408930
$599$	3.05573	0.124854	0.0624268	−	0.998050i	$-0.480116\pi$
$599$	3.05573	0.124854	0.0624268	+	0.998050i	$0.480116\pi$
$600$	0	0
$601$	−42.3607	−1.72793	−0.863964	−	0.503553i	$-0.832026\pi$
$601$	−42.3607	−1.72793	−0.863964	+	0.503553i	$0.832026\pi$
$602$	−41.3050	−1.68346
$603$	0	0
$604$	−48.0000	−1.95309
$605$	−6.18034	−0.251267
$606$	0	0
$607$	−14.8328	−0.602045	−0.301023	−	0.953617i	$-0.597328\pi$
$607$	−14.8328	−0.602045	−0.301023	+	0.953617i	$0.597328\pi$
$608$	−48.5410	−1.96860
$609$	0	0
$610$	−12.3607	−0.500469
$611$	−17.8885	−0.723693
$612$	0	0
$613$	−40.4721	−1.63465	−0.817327	−	0.576174i	$-0.804545\pi$
$613$	−40.4721	−1.63465	−0.817327	+	0.576174i	$0.804545\pi$
$614$	5.52786	0.223086
$615$	0	0
$616$	−28.9443	−1.16620
$617$	20.2918	0.816917	0.408458	−	0.912777i	$-0.366066\pi$
$617$	20.2918	0.816917	0.408458	+	0.912777i	$0.366066\pi$
$618$	0	0
$619$	−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
$621$	0	0
$622$	−15.7771	−0.632604
$623$	4.00000	0.160257
$624$	0	0
$625$	4.41641	0.176656
$626$	11.3050	0.451837
$627$	0	0
$628$	−10.5836	−0.422331
$629$	12.3607	0.492853
$630$	0	0
$631$	−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
$633$	0	0
$634$	−5.77709	−0.229437
$635$	−1.88854	−0.0749446
$636$	0	0
$637$	−15.5279	−0.615236
$638$	40.0000	1.58362
$639$	0	0
$640$	19.3475	0.764778
$641$	−49.0132	−1.93590	−0.967952	−	0.251137i	$-0.919196\pi$
$641$	−49.0132	−1.93590	−0.967952	+	0.251137i	$0.919196\pi$
$642$	0	0
$643$	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
$645$	0	0
$646$	44.7214	1.75954
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	0	0
$649$	−19.7771	−0.776319
$650$	34.7214	1.36188
$651$	0	0
$652$	22.2492	0.871347
$653$	22.9443	0.897879	0.448939	−	0.893562i	$-0.351802\pi$
$653$	22.9443	0.897879	0.448939	+	0.893562i	$0.351802\pi$
$654$	0	0
$655$	20.9443	0.818360
$656$	−10.9443	−0.427302
$657$	0	0
$658$	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
$663$	0	0
$664$	−8.94427	−0.347105
$665$	−28.9443	−1.12241
$666$	0	0
$667$	4.47214	0.173162
$668$	−38.8328	−1.50249
$669$	0	0
$670$	−2.11146	−0.0815727
$671$	−17.8885	−0.690580
$672$	0	0
$673$	2.58359	0.0995902	0.0497951	−	0.998759i	$-0.484143\pi$
$673$	2.58359	0.0995902	0.0497951	+	0.998759i	$0.484143\pi$
$674$	−32.1115	−1.23689
$675$	0	0
$676$	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	$0.134146\pi$
$678$	0	0
$679$	27.4164	1.05215
$680$	−7.63932	−0.292955
$681$	0	0
$682$	57.8885	2.21667
$683$	20.0000	0.765279	0.382639	−	0.923898i	$-0.375015\pi$
$683$	20.0000	0.765279	0.382639	+	0.923898i	$0.375015\pi$
$684$	0	0
$685$	−1.52786	−0.0583767
$686$	−25.5279	−0.974658
$687$	0	0
$688$	5.70820	0.217623
$689$	−23.4164	−0.892094
$690$	0	0
$691$	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
$696$	0	0
$697$	30.2492	1.14577
$698$	−52.6099	−1.99131
$699$	0	0
$700$	−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	$0.658122\pi$
$702$	0	0
$703$	32.3607	1.22051
$704$	52.0000	1.95982
$705$	0	0
$706$	38.9443	1.46569
$707$	22.4721	0.845152
$708$	0	0
$709$	−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
$711$	0	0
$712$	2.76393	0.103583
$713$	6.47214	0.242383
$714$	0	0
$715$	−22.1115	−0.826922
$716$	−62.8328	−2.34817
$717$	0	0
$718$	−14.4721	−0.540095
$719$	−4.00000	−0.149175	−0.0745874	−	0.997214i	$-0.523764\pi$
$719$	−4.00000	−0.149175	−0.0745874	+	0.997214i	$0.523764\pi$
$720$	0	0
$721$	−36.3607	−1.35414
$722$	74.5967	2.77620
$723$	0	0
$724$	−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
$729$	0	0
$730$	−19.1935	−0.710383
$731$	−15.7771	−0.583537
$732$	0	0
$733$	22.0000	0.812589	0.406294	−	0.913742i	$-0.366821\pi$
$733$	22.0000	0.812589	0.406294	+	0.913742i	$0.366821\pi$
$734$	57.4853	2.12182
$735$	0	0
$736$	6.70820	0.247268
$737$	−3.05573	−0.112559
$738$	0	0
$739$	−26.8328	−0.987061	−0.493531	−	0.869728i	$-0.664294\pi$
$739$	−26.8328	−0.987061	−0.493531	+	0.869728i	$0.664294\pi$
$740$	−16.5836	−0.609625
$741$	0	0
$742$	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	$0.352545\pi$
$744$	0	0
$745$	−14.4721	−0.530218
$746$	45.7771	1.67602
$747$	0	0
$748$	−33.1672	−1.21271
$749$	−54.8328	−2.00355
$750$	0	0
$751$	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
$753$	0	0
$754$	44.7214	1.62866
$755$	19.7771	0.719762
$756$	0	0
$757$	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	$0.778642\pi$
$762$	0	0
$763$	−48.3607	−1.75077
$764$	−4.58359	−0.165829
$765$	0	0
$766$	−57.0820	−2.06246
$767$	−22.1115	−0.798398
$768$	0	0
$769$	16.8328	0.607007	0.303503	−	0.952830i	$-0.401844\pi$
$769$	16.8328	0.607007	0.303503	+	0.952830i	$0.401844\pi$
$770$	35.7771	1.28932
$771$	0	0
$772$	52.2492	1.88049
$773$	−36.2918	−1.30533	−0.652663	−	0.757649i	$-0.726348\pi$
$773$	−36.2918	−1.30533	−0.652663	+	0.757649i	$0.726348\pi$
$774$	0	0
$775$	22.4721	0.807223
$776$	18.9443	0.680060
$777$	0	0
$778$	−46.1803	−1.65565
$779$	79.1935	2.83740
$780$	0	0
$781$	−32.0000	−1.14505
$782$	−6.18034	−0.221009
$783$	0	0
$784$	−3.47214	−0.124005
$785$	4.36068	0.155639
$786$	0	0
$787$	−13.7082	−0.488645	−0.244322	−	0.969694i	$-0.578566\pi$
$787$	−13.7082	−0.488645	−0.244322	+	0.969694i	$0.578566\pi$
$788$	−52.2492	−1.86130
$789$	0	0
$790$	−26.8328	−0.954669
$791$	20.0000	0.711118
$792$	0	0
$793$	−20.0000	−0.710221
$794$	−4.47214	−0.158710
$795$	0	0
$796$	29.1246	1.03229
$797$	−12.6525	−0.448174	−0.224087	−	0.974569i	$-0.571940\pi$
$797$	−12.6525	−0.448174	−0.224087	+	0.974569i	$0.571940\pi$
$798$	0	0
$799$	11.0557	0.391124
$800$	23.2918	0.823489
$801$	0	0
$802$	−44.8754	−1.58461
$803$	−27.7771	−0.980232
$804$	0	0
$805$	4.00000	0.140981
$806$	64.7214	2.27971
$807$	0	0
$808$	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	$0.224581\pi$
$810$	0	0
$811$	−23.4164	−0.822261	−0.411131	−	0.911576i	$-0.634866\pi$
$811$	−23.4164	−0.822261	−0.411131	+	0.911576i	$0.634866\pi$
$812$	−43.4164	−1.52362
$813$	0	0
$814$	−40.0000	−1.40200
$815$	−9.16718	−0.321112
$816$	0	0
$817$	−41.3050	−1.44508
$818$	38.9443	1.36165
$819$	0	0
$820$	−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	$0.567785\pi$
$822$	0	0
$823$	−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	$0.547683\pi$
$828$	0	0
$829$	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
$831$	0	0
$832$	58.1378	2.01556
$833$	9.59675	0.332508
$834$	0	0
$835$	16.0000	0.553703
$836$	−86.8328	−3.00318
$837$	0	0
$838$	−31.0557	−1.07280
$839$	12.5836	0.434434	0.217217	−	0.976123i	$-0.430302\pi$
$839$	12.5836	0.434434	0.217217	+	0.976123i	$0.430302\pi$
$840$	0	0
$841$	−9.00000	−0.310345
$842$	−69.1935	−2.38457
$843$	0	0
$844$	40.5836	1.39694
$845$	−8.65248	−0.297654
$846$	0	0
$847$	16.1803	0.555963
$848$	−5.23607	−0.179807
$849$	0	0
$850$	−21.4590	−0.736037
$851$	−4.47214	−0.153303
$852$	0	0
$853$	22.0000	0.753266	0.376633	−	0.926363i	$-0.377082\pi$
$853$	22.0000	0.753266	0.376633	+	0.926363i	$0.377082\pi$
$854$	32.3607	1.10736
$855$	0	0
$856$	−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
$861$	0	0
$862$	−17.8885	−0.609286
$863$	4.00000	0.136162	0.0680808	−	0.997680i	$-0.478312\pi$
$863$	4.00000	0.136162	0.0680808	+	0.997680i	$0.478312\pi$
$864$	0	0
$865$	5.52786	0.187953
$866$	62.3607	2.11910
$867$	0	0
$868$	−62.8328	−2.13268
$869$	−38.8328	−1.31731
$870$	0	0
$871$	−3.41641	−0.115761
$872$	−33.4164	−1.13162
$873$	0	0
$874$	−16.1803	−0.547308
$875$	33.8885	1.14564
$876$	0	0
$877$	33.0557	1.11621	0.558106	−	0.829769i	$-0.311528\pi$
$877$	33.0557	1.11621	0.558106	+	0.829769i	$0.311528\pi$
$878$	−22.1115	−0.746226
$879$	0	0
$880$	−4.94427	−0.166671
$881$	−26.5410	−0.894190	−0.447095	−	0.894487i	$-0.647541\pi$
$881$	−26.5410	−0.894190	−0.447095	+	0.894487i	$0.647541\pi$
$882$	0	0
$883$	−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
$885$	0	0
$886$	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	$0.674581\pi$
$888$	0	0
$889$	4.94427	0.165826
$890$	−3.41641	−0.114518
$891$	0	0
$892$	−77.6656	−2.60044
$893$	28.9443	0.968583
$894$	0	0
$895$	25.8885	0.865359
$896$	−50.6525	−1.69218
$897$	0	0
$898$	−33.4164	−1.11512
$899$	28.9443	0.965346
$900$	0	0
$901$	14.4721	0.482137
$902$	−97.8885	−3.25933
$903$	0	0
$904$	13.8197	0.459635
$905$	14.6950	0.488480
$906$	0	0
$907$	−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
$909$	0	0
$910$	40.0000	1.32599
$911$	40.7214	1.34916	0.674579	−	0.738202i	$-0.264325\pi$
$911$	40.7214	1.34916	0.674579	+	0.738202i	$0.264325\pi$
$912$	0	0
$913$	16.0000	0.529523
$914$	18.9443	0.626621
$915$	0	0
$916$	8.83282	0.291844
$917$	−54.8328	−1.81074
$918$	0	0
$919$	−23.0132	−0.759134	−0.379567	−	0.925164i	$-0.623927\pi$
$919$	−23.0132	−0.759134	−0.379567	+	0.925164i	$0.623927\pi$
$920$	2.76393	0.0911241
$921$	0	0
$922$	12.1115	0.398870
$923$	−35.7771	−1.17762
$924$	0	0
$925$	−15.5279	−0.510553
$926$	−28.9443	−0.951168
$927$	0	0
$928$	30.0000	0.984798
$929$	28.8328	0.945974	0.472987	−	0.881069i	$-0.343176\pi$
$929$	28.8328	0.945974	0.472987	+	0.881069i	$0.343176\pi$
$930$	0	0
$931$	25.1246	0.823426
$932$	42.0000	1.37576
$933$	0	0
$934$	−45.5279	−1.48972
$935$	13.6656	0.446914
$936$	0	0
$937$	−12.8328	−0.419230	−0.209615	−	0.977784i	$-0.567221\pi$
$937$	−12.8328	−0.419230	−0.209615	+	0.977784i	$0.567221\pi$
$938$	5.52786	0.180491
$939$	0	0
$940$	−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	$0.256112\pi$
$942$	0	0
$943$	−10.9443	−0.356395
$944$	−4.94427	−0.160922
$945$	0	0
$946$	51.0557	1.65996
$947$	1.16718	0.0379284	0.0189642	−	0.999820i	$-0.493963\pi$
$947$	1.16718	0.0379284	0.0189642	+	0.999820i	$0.493963\pi$
$948$	0	0
$949$	−31.0557	−1.00811
$950$	−56.1803	−1.82273
$951$	0	0
$952$	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	$0.562625\pi$
$954$	0	0
$955$	1.88854	0.0611118
$956$	−14.8328	−0.479728
$957$	0	0
$958$	75.7771	2.44825
$959$	4.00000	0.129167
$960$	0	0
$961$	10.8885	0.351243
$962$	−44.7214	−1.44187
$963$	0	0
$964$	58.5836	1.88685
$965$	−21.5279	−0.693006
$966$	0	0
$967$	−7.63932	−0.245664	−0.122832	−	0.992427i	$-0.539198\pi$
$967$	−7.63932	−0.245664	−0.122832	+	0.992427i	$0.539198\pi$
$968$	11.1803	0.359350
$969$	0	0
$970$	−23.4164	−0.751856
$971$	−17.3050	−0.555342	−0.277671	−	0.960676i	$-0.589563\pi$
$971$	−17.3050	−0.555342	−0.277671	+	0.960676i	$0.589563\pi$
$972$	0	0
$973$	−28.9443	−0.927911
$974$	53.6656	1.71956
$975$	0	0
$976$	−4.47214	−0.143150
$977$	32.0689	1.02597	0.512987	−	0.858396i	$-0.328539\pi$
$977$	32.0689	1.02597	0.512987	+	0.858396i	$0.328539\pi$
$978$	0	0
$979$	−4.94427	−0.158020
$980$	−12.8754	−0.411289
$981$	0	0
$982$	−89.4427	−2.85423
$983$	−22.8328	−0.728254	−0.364127	−	0.931349i	$-0.618633\pi$
$983$	−22.8328	−0.728254	−0.364127	+	0.931349i	$0.618633\pi$
$984$	0	0
$985$	21.5279	0.685935
$986$	−27.6393	−0.880215
$987$	0	0
$988$	−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
$993$	0	0
$994$	57.8885	1.83611
$995$	−12.0000	−0.380426
$996$	0	0
$997$	42.3607	1.34158	0.670788	−	0.741649i	$-0.265956\pi$
$997$	42.3607	1.34158	0.670788	+	0.741649i	$0.265956\pi$
$998$	−27.6393	−0.874907
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.2.a.c.1.2		2
3.2	odd	2		69.2.a.b.1.1	✓	2
4.3	odd	2		3312.2.a.bb.1.1		2
5.4	even	2		5175.2.a.bk.1.1		2
12.11	even	2		1104.2.a.m.1.2		2
15.2	even	4		1725.2.b.o.1174.2		4
15.8	even	4		1725.2.b.o.1174.3		4
15.14	odd	2		1725.2.a.ba.1.2		2
21.20	even	2		3381.2.a.t.1.1		2
23.22	odd	2		4761.2.a.v.1.2		2
24.5	odd	2		4416.2.a.bm.1.1		2
24.11	even	2		4416.2.a.bg.1.1		2
33.32	even	2		8349.2.a.i.1.2		2
69.68	even	2		1587.2.a.i.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.2.a.b.1.1	✓	2	3.2	odd	2
207.2.a.c.1.2		2	1.1	even	1	trivial
1104.2.a.m.1.2		2	12.11	even	2
1587.2.a.i.1.1		2	69.68	even	2
1725.2.a.ba.1.2		2	15.14	odd	2
1725.2.b.o.1174.2		4	15.2	even	4
1725.2.b.o.1174.3		4	15.8	even	4
3312.2.a.bb.1.1		2	4.3	odd	2
3381.2.a.t.1.1		2	21.20	even	2
4416.2.a.bg.1.1		2	24.11	even	2
4416.2.a.bm.1.1		2	24.5	odd	2
4761.2.a.v.1.2		2	23.22	odd	2
5175.2.a.bk.1.1		2	5.4	even	2
8349.2.a.i.1.2		2	33.32	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:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	207.a (trivial)

\(f(q)\)	\(=\)	\(q+2.23607 q^{2} +3.00000 q^{4} -1.23607 q^{5} +3.23607 q^{7} +2.23607 q^{8} +O(q^{10})\)	\(q+2.23607 q^{2} +3.00000 q^{4} -1.23607 q^{5} +3.23607 q^{7} +2.23607 q^{8} -2.76393 q^{10} -4.00000 q^{11} -4.47214 q^{13} +7.23607 q^{14} -1.00000 q^{16} +2.76393 q^{17} +7.23607 q^{19} -3.70820 q^{20} -8.94427 q^{22} -1.00000 q^{23} -3.47214 q^{25} -10.0000 q^{26} +9.70820 q^{28} -4.47214 q^{29} -6.47214 q^{31} -6.70820 q^{32} +6.18034 q^{34} -4.00000 q^{35} +4.47214 q^{37} +16.1803 q^{38} -2.76393 q^{40} +10.9443 q^{41} -5.70820 q^{43} -12.0000 q^{44} -2.23607 q^{46} +4.00000 q^{47} +3.47214 q^{49} -7.76393 q^{50} -13.4164 q^{52} +5.23607 q^{53} +4.94427 q^{55} +7.23607 q^{56} -10.0000 q^{58} +4.94427 q^{59} +4.47214 q^{61} -14.4721 q^{62} -13.0000 q^{64} +5.52786 q^{65} +0.763932 q^{67} +8.29180 q^{68} -8.94427 q^{70} +8.00000 q^{71} +6.94427 q^{73} +10.0000 q^{74} +21.7082 q^{76} -12.9443 q^{77} +9.70820 q^{79} +1.23607 q^{80} +24.4721 q^{82} -4.00000 q^{83} -3.41641 q^{85} -12.7639 q^{86} -8.94427 q^{88} +1.23607 q^{89} -14.4721 q^{91} -3.00000 q^{92} +8.94427 q^{94} -8.94427 q^{95} +8.47214 q^{97} +7.76393 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{4} + 2 q^{5} + 2 q^{7}+O(q^{10}) \) 2 * q + 6 * q^4 + 2 * q^5 + 2 * q^7	\( 2 q + 6 q^{4} + 2 q^{5} + 2 q^{7} - 10 q^{10} - 8 q^{11} + 10 q^{14} - 2 q^{16} + 10 q^{17} + 10 q^{19} + 6 q^{20} - 2 q^{23} + 2 q^{25} - 20 q^{26} + 6 q^{28} - 4 q^{31} - 10 q^{34} - 8 q^{35} + 10 q^{38} - 10 q^{40} + 4 q^{41} + 2 q^{43} - 24 q^{44} + 8 q^{47} - 2 q^{49} - 20 q^{50} + 6 q^{53} - 8 q^{55} + 10 q^{56} - 20 q^{58} - 8 q^{59} - 20 q^{62} - 26 q^{64} + 20 q^{65} + 6 q^{67} + 30 q^{68} + 16 q^{71} - 4 q^{73} + 20 q^{74} + 30 q^{76} - 8 q^{77} + 6 q^{79} - 2 q^{80} + 40 q^{82} - 8 q^{83} + 20 q^{85} - 30 q^{86} - 2 q^{89} - 20 q^{91} - 6 q^{92} + 8 q^{97} + 20 q^{98}+O(q^{100}) \) 2 * q + 6 * q^4 + 2 * q^5 + 2 * q^7 - 10 * q^10 - 8 * q^11 + 10 * q^14 - 2 * q^16 + 10 * q^17 + 10 * q^19 + 6 * q^20 - 2 * q^23 + 2 * q^25 - 20 * q^26 + 6 * q^28 - 4 * q^31 - 10 * q^34 - 8 * q^35 + 10 * q^38 - 10 * q^40 + 4 * q^41 + 2 * q^43 - 24 * q^44 + 8 * q^47 - 2 * q^49 - 20 * q^50 + 6 * q^53 - 8 * q^55 + 10 * q^56 - 20 * q^58 - 8 * q^59 - 20 * q^62 - 26 * q^64 + 20 * q^65 + 6 * q^67 + 30 * q^68 + 16 * q^71 - 4 * q^73 + 20 * q^74 + 30 * q^76 - 8 * q^77 + 6 * q^79 - 2 * q^80 + 40 * q^82 - 8 * q^83 + 20 * q^85 - 30 * q^86 - 2 * q^89 - 20 * q^91 - 6 * q^92 + 8 * q^97 + 20 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more