LMFDB - Embedded newform 1449.2.a.r.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1449,2,Mod(1,1449)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1449, 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("1449.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1449 = 3^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1449.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:	$11.5703232529$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.2147108.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 9x^{3} + 17x^{2} + 16x - 27 $ x^5 - 2x^4 - 9x^3 + 17x^2 + 16x - 27
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 161)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-2.54577$ of defining polynomial
Character	$\chi$	$=$	1449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.54577 q^{2} +4.48096 q^{4} -2.78847 q^{5} +1.00000 q^{7} +6.31597 q^{8} +O(q^{10})$	\(q+2.54577 q^{2} +4.48096 q^{4} -2.78847 q^{5} +1.00000 q^{7} +6.31597 q^{8} -7.09882 q^{10} +4.70095 q^{11} -2.32579 q^{13} +2.54577 q^{14} +7.11710 q^{16} +1.82655 q^{17} +7.09155 q^{19} -12.4950 q^{20} +11.9675 q^{22} +1.00000 q^{23} +2.77558 q^{25} -5.92093 q^{26} +4.48096 q^{28} +9.98866 q^{29} +3.53732 q^{31} +5.48658 q^{32} +4.64998 q^{34} -2.78847 q^{35} -0.166179 q^{37} +18.0535 q^{38} -17.6119 q^{40} -7.25116 q^{41} -9.57695 q^{43} +21.0648 q^{44} +2.54577 q^{46} -4.66542 q^{47} +1.00000 q^{49} +7.06600 q^{50} -10.4218 q^{52} +0.961924 q^{53} -13.1085 q^{55} +6.31597 q^{56} +25.4289 q^{58} -13.8800 q^{59} +0.954652 q^{61} +9.00521 q^{62} -0.266598 q^{64} +6.48540 q^{65} -11.9221 q^{67} +8.18469 q^{68} -7.09882 q^{70} -4.59958 q^{71} -7.59806 q^{73} -0.423055 q^{74} +31.7770 q^{76} +4.70095 q^{77} +5.73902 q^{79} -19.8458 q^{80} -18.4598 q^{82} +5.57695 q^{83} -5.09328 q^{85} -24.3807 q^{86} +29.6910 q^{88} +11.2284 q^{89} -2.32579 q^{91} +4.48096 q^{92} -11.8771 q^{94} -19.7746 q^{95} -1.68805 q^{97} +2.54577 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{2} + 12 q^{4} + 4 q^{5} + 5 q^{7} - 3 q^{8}+O(q^{10}) $ 5 * q - 2 * q^2 + 12 * q^4 + 4 * q^5 + 5 * q^7 - 3 * q^8	$ 5 q - 2 q^{2} + 12 q^{4} + 4 q^{5} + 5 q^{7} - 3 q^{8} - 8 q^{10} + 4 q^{11} - 6 q^{13} - 2 q^{14} + 10 q^{16} + 12 q^{17} + 6 q^{19} + 14 q^{22} + 5 q^{23} + 19 q^{25} - q^{26} + 12 q^{28} + 4 q^{29} + 30 q^{31} - 8 q^{32} + 6 q^{34} + 4 q^{35} + 4 q^{37} + 40 q^{38} - 50 q^{40} - 6 q^{41} - 12 q^{43} + 26 q^{44} - 2 q^{46} - 10 q^{47} + 5 q^{49} + 2 q^{50} - 21 q^{52} - 16 q^{53} + 18 q^{55} - 3 q^{56} + 13 q^{58} - 22 q^{59} - 18 q^{61} - 15 q^{62} + 25 q^{64} + 26 q^{65} - 2 q^{67} - 12 q^{68} - 8 q^{70} - 4 q^{71} - 2 q^{73} - 38 q^{74} + 10 q^{76} + 4 q^{77} + 30 q^{79} + 10 q^{80} - 7 q^{82} - 8 q^{83} - 12 q^{85} - 8 q^{86} + 4 q^{88} + 20 q^{89} - 6 q^{91} + 12 q^{92} - 25 q^{94} - 8 q^{95} - 12 q^{97} - 2 q^{98}+O(q^{100}) $ 5 * q - 2 * q^2 + 12 * q^4 + 4 * q^5 + 5 * q^7 - 3 * q^8 - 8 * q^10 + 4 * q^11 - 6 * q^13 - 2 * q^14 + 10 * q^16 + 12 * q^17 + 6 * q^19 + 14 * q^22 + 5 * q^23 + 19 * q^25 - q^26 + 12 * q^28 + 4 * q^29 + 30 * q^31 - 8 * q^32 + 6 * q^34 + 4 * q^35 + 4 * q^37 + 40 * q^38 - 50 * q^40 - 6 * q^41 - 12 * q^43 + 26 * q^44 - 2 * q^46 - 10 * q^47 + 5 * q^49 + 2 * q^50 - 21 * q^52 - 16 * q^53 + 18 * q^55 - 3 * q^56 + 13 * q^58 - 22 * q^59 - 18 * q^61 - 15 * q^62 + 25 * q^64 + 26 * q^65 - 2 * q^67 - 12 * q^68 - 8 * q^70 - 4 * q^71 - 2 * q^73 - 38 * q^74 + 10 * q^76 + 4 * q^77 + 30 * q^79 + 10 * q^80 - 7 * q^82 - 8 * q^83 - 12 * q^85 - 8 * q^86 + 4 * q^88 + 20 * q^89 - 6 * q^91 + 12 * q^92 - 25 * q^94 - 8 * q^95 - 12 * q^97 - 2 * 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 $

$2$	2.54577	1.80013	0.900067	−	0.435752i	$-0.143517\pi$
$2$	2.54577	1.80013	0.900067	+	0.435752i	$0.143517\pi$
$3$	0	0
$3$	0	0
$4$	4.48096	2.24048
$5$	−2.78847	−1.24704	−0.623521	−	0.781806i	$-0.714299\pi$
$5$	−2.78847	−1.24704	−0.623521	+	0.781806i	$0.714299\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	6.31597	2.23303
$9$	0	0
$10$	−7.09882	−2.24484
$11$	4.70095	1.41739	0.708694	−	0.705516i	$-0.249285\pi$
$11$	4.70095	1.41739	0.708694	+	0.705516i	$0.249285\pi$
$12$	0	0
$13$	−2.32579	−0.645058	−0.322529	−	0.946560i	$-0.604533\pi$
$13$	−2.32579	−0.645058	−0.322529	+	0.946560i	$0.604533\pi$
$14$	2.54577	0.680387
$15$	0	0
$16$	7.11710	1.77927
$17$	1.82655	0.443003	0.221502	−	0.975160i	$-0.428904\pi$
$17$	1.82655	0.443003	0.221502	+	0.975160i	$0.428904\pi$
$18$	0	0
$19$	7.09155	1.62691	0.813456	−	0.581626i	$-0.197583\pi$
$19$	7.09155	1.62691	0.813456	+	0.581626i	$0.197583\pi$
$20$	−12.4950	−2.79398
$21$	0	0
$22$	11.9675	2.55149
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	2.77558	0.555116
$26$	−5.92093	−1.16119
$27$	0	0
$28$	4.48096	0.846822
$29$	9.98866	1.85485	0.927424	−	0.374012i	$-0.122018\pi$
$29$	9.98866	1.85485	0.927424	+	0.374012i	$0.122018\pi$
$30$	0	0
$31$	3.53732	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$31$	3.53732	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$32$	5.48658	0.969900
$33$	0	0
$34$	4.64998	0.797465
$35$	−2.78847	−0.471338
$36$	0	0
$37$	−0.166179	−0.0273197	−0.0136599	−	0.999907i	$-0.504348\pi$
$37$	−0.166179	−0.0273197	−0.0136599	+	0.999907i	$0.504348\pi$
$38$	18.0535	2.92866
$39$	0	0
$40$	−17.6119	−2.78469
$41$	−7.25116	−1.13244	−0.566220	−	0.824254i	$-0.691595\pi$
$41$	−7.25116	−1.13244	−0.566220	+	0.824254i	$0.691595\pi$
$42$	0	0
$43$	−9.57695	−1.46047	−0.730235	−	0.683196i	$-0.760590\pi$
$43$	−9.57695	−1.46047	−0.730235	+	0.683196i	$0.760590\pi$
$44$	21.0648	3.17563
$45$	0	0
$46$	2.54577	0.375354
$47$	−4.66542	−0.680521	−0.340261	−	0.940331i	$-0.610515\pi$
$47$	−4.66542	−0.680521	−0.340261	+	0.940331i	$0.610515\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	7.06600	0.999283
$51$	0	0
$52$	−10.4218	−1.44524
$53$	0.961924	0.132130	0.0660652	−	0.997815i	$-0.478955\pi$
$53$	0.961924	0.132130	0.0660652	+	0.997815i	$0.478955\pi$
$54$	0	0
$55$	−13.1085	−1.76754
$56$	6.31597	0.844007
$57$	0	0
$58$	25.4289	3.33897
$59$	−13.8800	−1.80702	−0.903512	−	0.428562i	$-0.859020\pi$
$59$	−13.8800	−1.80702	−0.903512	+	0.428562i	$0.859020\pi$
$60$	0	0
$61$	0.954652	0.122231	0.0611153	−	0.998131i	$-0.480534\pi$
$61$	0.954652	0.122231	0.0611153	+	0.998131i	$0.480534\pi$
$62$	9.00521	1.14366
$63$	0	0
$64$	−0.266598	−0.0333248
$65$	6.48540	0.804415
$66$	0	0
$67$	−11.9221	−1.45652	−0.728259	−	0.685302i	$-0.759670\pi$
$67$	−11.9221	−1.45652	−0.728259	+	0.685302i	$0.759670\pi$
$68$	8.18469	0.992540
$69$	0	0
$70$	−7.09882	−0.848471
$71$	−4.59958	−0.545870	−0.272935	−	0.962033i	$-0.587994\pi$
$71$	−4.59958	−0.545870	−0.272935	+	0.962033i	$0.587994\pi$
$72$	0	0
$73$	−7.59806	−0.889286	−0.444643	−	0.895708i	$-0.646669\pi$
$73$	−7.59806	−0.889286	−0.444643	+	0.895708i	$0.646669\pi$
$74$	−0.423055	−0.0491791
$75$	0	0
$76$	31.7770	3.64507
$77$	4.70095	0.535723
$78$	0	0
$79$	5.73902	0.645690	0.322845	−	0.946452i	$-0.395361\pi$
$79$	5.73902	0.645690	0.322845	+	0.946452i	$0.395361\pi$
$80$	−19.8458	−2.21883
$81$	0	0
$82$	−18.4598	−2.03854
$83$	5.57695	0.612149	0.306075	−	0.952008i	$-0.400984\pi$
$83$	5.57695	0.612149	0.306075	+	0.952008i	$0.400984\pi$
$84$	0	0
$85$	−5.09328	−0.552444
$86$	−24.3807	−2.62904
$87$	0	0
$88$	29.6910	3.16507
$89$	11.2284	1.19021	0.595106	−	0.803647i	$-0.297110\pi$
$89$	11.2284	1.19021	0.595106	+	0.803647i	$0.297110\pi$
$90$	0	0
$91$	−2.32579	−0.243809
$92$	4.48096	0.467173
$93$	0	0
$94$	−11.8771	−1.22503
$95$	−19.7746	−2.02883
$96$	0	0
$97$	−1.68805	−0.171396	−0.0856979	−	0.996321i	$-0.527312\pi$
$97$	−1.68805	−0.171396	−0.0856979	+	0.996321i	$0.527312\pi$
$98$	2.54577	0.257162
$99$	0	0
$100$	12.4373	1.24373
$101$	−3.12810	−0.311258	−0.155629	−	0.987816i	$-0.549740\pi$
$101$	−3.12810	−0.311258	−0.155629	+	0.987816i	$0.549740\pi$
$102$	0	0
$103$	1.03808	0.102285	0.0511423	−	0.998691i	$-0.483714\pi$
$103$	1.03808	0.102285	0.0511423	+	0.998691i	$0.483714\pi$
$104$	−14.6896	−1.44043
$105$	0	0
$106$	2.44884	0.237853
$107$	−8.21965	−0.794624	−0.397312	−	0.917684i	$-0.630057\pi$
$107$	−8.21965	−0.794624	−0.397312	+	0.917684i	$0.630057\pi$
$108$	0	0
$109$	6.99848	0.670333	0.335166	−	0.942159i	$-0.391207\pi$
$109$	6.99848	0.670333	0.335166	+	0.942159i	$0.391207\pi$
$110$	−33.3712	−3.18182
$111$	0	0
$112$	7.11710	0.672503
$113$	−2.65158	−0.249439	−0.124720	−	0.992192i	$-0.539803\pi$
$113$	−2.65158	−0.249439	−0.124720	+	0.992192i	$0.539803\pi$
$114$	0	0
$115$	−2.78847	−0.260026
$116$	44.7588	4.15575
$117$	0	0
$118$	−35.3354	−3.25289
$119$	1.82655	0.167439
$120$	0	0
$121$	11.0989	1.00899
$122$	2.43033	0.220031
$123$	0	0
$124$	15.8506	1.42342
$125$	6.20274	0.554790
$126$	0	0
$127$	−0.847744	−0.0752251	−0.0376126	−	0.999292i	$-0.511975\pi$
$127$	−0.847744	−0.0752251	−0.0376126	+	0.999292i	$0.511975\pi$
$128$	−11.6519	−1.02989
$129$	0	0
$130$	16.5104	1.44805
$131$	3.40769	0.297732	0.148866	−	0.988857i	$-0.452438\pi$
$131$	3.40769	0.297732	0.148866	+	0.988857i	$0.452438\pi$
$132$	0	0
$133$	7.09155	0.614915
$134$	−30.3510	−2.62193
$135$	0	0
$136$	11.5364	0.989240
$137$	14.4092	1.23107	0.615533	−	0.788111i	$-0.288941\pi$
$137$	14.4092	1.23107	0.615533	+	0.788111i	$0.288941\pi$
$138$	0	0
$139$	2.60685	0.221110	0.110555	−	0.993870i	$-0.464737\pi$
$139$	2.60685	0.221110	0.110555	+	0.993870i	$0.464737\pi$
$140$	−12.4950	−1.05602
$141$	0	0
$142$	−11.7095	−0.982638
$143$	−10.9334	−0.914298
$144$	0	0
$145$	−27.8531	−2.31307
$146$	−19.3429	−1.60083
$147$	0	0
$148$	−0.744643	−0.0612093
$149$	0.00887233	0.000726849 0	0.000363425	−	1.00000i	$-0.499884\pi$
$149$	0.00887233	0.000726849 0	0.000363425		1.00000i	$0.499884\pi$
$150$	0	0
$151$	13.1070	1.06663	0.533316	−	0.845916i	$-0.320946\pi$
$151$	13.1070	1.06663	0.533316	+	0.845916i	$0.320946\pi$
$152$	44.7900	3.63295
$153$	0	0
$154$	11.9675	0.964372
$155$	−9.86371	−0.792272
$156$	0	0
$157$	−0.249603	−0.0199205	−0.00996025	−	0.999950i	$-0.503170\pi$
$157$	−0.249603	−0.0199205	−0.00996025	+	0.999950i	$0.503170\pi$
$158$	14.6103	1.16233
$159$	0	0
$160$	−15.2992	−1.20951
$161$	1.00000	0.0788110
$162$	0	0
$163$	−0.150737	−0.0118066	−0.00590332	−	0.999983i	$-0.501879\pi$
$163$	−0.150737	−0.0118066	−0.00590332	+	0.999983i	$0.501879\pi$
$164$	−32.4922	−2.53721
$165$	0	0
$166$	14.1976	1.10195
$167$	−11.8235	−0.914931	−0.457465	−	0.889227i	$-0.651243\pi$
$167$	−11.8235	−0.914931	−0.457465	+	0.889227i	$0.651243\pi$
$168$	0	0
$169$	−7.59071	−0.583900
$170$	−12.9663	−0.994473
$171$	0	0
$172$	−42.9139	−3.27216
$173$	−6.78120	−0.515565	−0.257783	−	0.966203i	$-0.582992\pi$
$173$	−6.78120	−0.515565	−0.257783	+	0.966203i	$0.582992\pi$
$174$	0	0
$175$	2.77558	0.209814
$176$	33.4571	2.52192
$177$	0	0
$178$	28.5851	2.14254
$179$	−20.5624	−1.53690	−0.768451	−	0.639908i	$-0.778972\pi$
$179$	−20.5624	−1.53690	−0.768451	+	0.639908i	$0.778972\pi$
$180$	0	0
$181$	−1.69693	−0.126131	−0.0630657	−	0.998009i	$-0.520088\pi$
$181$	−1.69693	−0.126131	−0.0630657	+	0.998009i	$0.520088\pi$
$182$	−5.92093	−0.438889
$183$	0	0
$184$	6.31597	0.465619
$185$	0.463386	0.0340688
$186$	0	0
$187$	8.58651	0.627907
$188$	−20.9056	−1.52470
$189$	0	0
$190$	−50.3416	−3.65216
$191$	−15.8873	−1.14956	−0.574782	−	0.818307i	$-0.694913\pi$
$191$	−15.8873	−1.14956	−0.574782	+	0.818307i	$0.694913\pi$
$192$	0	0
$193$	−21.1919	−1.52543	−0.762714	−	0.646736i	$-0.776134\pi$
$193$	−21.1919	−1.52543	−0.762714	+	0.646736i	$0.776134\pi$
$194$	−4.29740	−0.308535
$195$	0	0
$196$	4.48096	0.320069
$197$	−18.5146	−1.31911	−0.659554	−	0.751657i	$-0.729255\pi$
$197$	−18.5146	−1.31911	−0.659554	+	0.751657i	$0.729255\pi$
$198$	0	0
$199$	12.3192	0.873286	0.436643	−	0.899635i	$-0.356167\pi$
$199$	12.3192	0.873286	0.436643	+	0.899635i	$0.356167\pi$
$200$	17.5305	1.23959
$201$	0	0
$202$	−7.96344	−0.560306
$203$	9.98866	0.701066
$204$	0	0
$205$	20.2197	1.41220
$206$	2.64271	0.184126
$207$	0	0
$208$	−16.5529	−1.14773
$209$	33.3370	2.30597
$210$	0	0
$211$	−3.49259	−0.240440	−0.120220	−	0.992747i	$-0.538360\pi$
$211$	−3.49259	−0.240440	−0.120220	+	0.992747i	$0.538360\pi$
$212$	4.31035	0.296036
$213$	0	0
$214$	−20.9254	−1.43043
$215$	26.7050	1.82127
$216$	0	0
$217$	3.53732	0.240129
$218$	17.8165	1.20669
$219$	0	0
$220$	−58.7385	−3.96015
$221$	−4.24817	−0.285763
$222$	0	0
$223$	9.77808	0.654789	0.327394	−	0.944888i	$-0.393829\pi$
$223$	9.77808	0.654789	0.327394	+	0.944888i	$0.393829\pi$
$224$	5.48658	0.366588
$225$	0	0
$226$	−6.75032	−0.449024
$227$	−8.25924	−0.548185	−0.274093	−	0.961703i	$-0.588378\pi$
$227$	−8.25924	−0.548185	−0.274093	+	0.961703i	$0.588378\pi$
$228$	0	0
$229$	3.13841	0.207392	0.103696	−	0.994609i	$-0.466933\pi$
$229$	3.13841	0.207392	0.103696	+	0.994609i	$0.466933\pi$
$230$	−7.09882	−0.468082
$231$	0	0
$232$	63.0881	4.14193
$233$	−8.84601	−0.579521	−0.289761	−	0.957099i	$-0.593576\pi$
$233$	−8.84601	−0.579521	−0.289761	+	0.957099i	$0.593576\pi$
$234$	0	0
$235$	13.0094	0.848639
$236$	−62.1958	−4.04860
$237$	0	0
$238$	4.64998	0.301413
$239$	10.3793	0.671379	0.335689	−	0.941973i	$-0.391031\pi$
$239$	10.3793	0.671379	0.335689	+	0.941973i	$0.391031\pi$
$240$	0	0
$241$	−2.47813	−0.159630	−0.0798151	−	0.996810i	$-0.525433\pi$
$241$	−2.47813	−0.159630	−0.0798151	+	0.996810i	$0.525433\pi$
$242$	28.2553	1.81632
$243$	0	0
$244$	4.27776	0.273855
$245$	−2.78847	−0.178149
$246$	0	0
$247$	−16.4934	−1.04945
$248$	22.3416	1.41869
$249$	0	0
$250$	15.7908	0.998695
$251$	9.66697	0.610174	0.305087	−	0.952324i	$-0.401314\pi$
$251$	9.66697	0.610174	0.305087	+	0.952324i	$0.401314\pi$
$252$	0	0
$253$	4.70095	0.295546
$254$	−2.15816	−0.135415
$255$	0	0
$256$	−29.1298	−1.82061
$257$	−16.2773	−1.01535	−0.507676	−	0.861548i	$-0.669495\pi$
$257$	−16.2773	−1.01535	−0.507676	+	0.861548i	$0.669495\pi$
$258$	0	0
$259$	−0.166179	−0.0103259
$260$	29.0608	1.80228
$261$	0	0
$262$	8.67522	0.535957
$263$	9.03331	0.557017	0.278509	−	0.960434i	$-0.410160\pi$
$263$	9.03331	0.557017	0.278509	+	0.960434i	$0.410160\pi$
$264$	0	0
$265$	−2.68230	−0.164772
$266$	18.0535	1.10693
$267$	0	0
$268$	−53.4226	−3.26330
$269$	17.6238	1.07454	0.537272	−	0.843409i	$-0.319455\pi$
$269$	17.6238	1.07454	0.537272	+	0.843409i	$0.319455\pi$
$270$	0	0
$271$	23.9935	1.45750	0.728750	−	0.684780i	$-0.240102\pi$
$271$	23.9935	1.45750	0.728750	+	0.684780i	$0.240102\pi$
$272$	12.9997	0.788224
$273$	0	0
$274$	36.6827	2.21608
$275$	13.0479	0.786815
$276$	0	0
$277$	16.5641	0.995239	0.497620	−	0.867395i	$-0.334208\pi$
$277$	16.5641	0.995239	0.497620	+	0.867395i	$0.334208\pi$
$278$	6.63645	0.398028
$279$	0	0
$280$	−17.6119	−1.05251
$281$	19.5228	1.16463	0.582316	−	0.812962i	$-0.302146\pi$
$281$	19.5228	1.16463	0.582316	+	0.812962i	$0.302146\pi$
$282$	0	0
$283$	−15.7324	−0.935191	−0.467596	−	0.883942i	$-0.654880\pi$
$283$	−15.7324	−0.935191	−0.467596	+	0.883942i	$0.654880\pi$
$284$	−20.6105	−1.22301
$285$	0	0
$286$	−27.8340	−1.64586
$287$	−7.25116	−0.428022
$288$	0	0
$289$	−13.6637	−0.803748
$290$	−70.9077	−4.16384
$291$	0	0
$292$	−34.0466	−1.99243
$293$	8.16268	0.476869	0.238434	−	0.971159i	$-0.423366\pi$
$293$	8.16268	0.476869	0.238434	+	0.971159i	$0.423366\pi$
$294$	0	0
$295$	38.7041	2.25344
$296$	−1.04958	−0.0610058
$297$	0	0
$298$	0.0225869	0.00130843
$299$	−2.32579	−0.134504
$300$	0	0
$301$	−9.57695	−0.552006
$302$	33.3674	1.92008
$303$	0	0
$304$	50.4712	2.89472
$305$	−2.66202	−0.152427
$306$	0	0
$307$	30.5542	1.74382	0.871910	−	0.489667i	$-0.162882\pi$
$307$	30.5542	1.74382	0.871910	+	0.489667i	$0.162882\pi$
$308$	21.0648	1.20028
$309$	0	0
$310$	−25.1108	−1.42620
$311$	−25.3573	−1.43788	−0.718941	−	0.695071i	$-0.755373\pi$
$311$	−25.3573	−1.43788	−0.718941	+	0.695071i	$0.755373\pi$
$312$	0	0
$313$	−6.15654	−0.347988	−0.173994	−	0.984747i	$-0.555667\pi$
$313$	−6.15654	−0.347988	−0.173994	+	0.984747i	$0.555667\pi$
$314$	−0.635433	−0.0358595
$315$	0	0
$316$	25.7163	1.44666
$317$	4.48063	0.251657	0.125829	−	0.992052i	$-0.459841\pi$
$317$	4.48063	0.251657	0.125829	+	0.992052i	$0.459841\pi$
$318$	0	0
$319$	46.9562	2.62904
$320$	0.743402	0.0415575
$321$	0	0
$322$	2.54577	0.141870
$323$	12.9531	0.720727
$324$	0	0
$325$	−6.45541	−0.358082
$326$	−0.383743	−0.0212535
$327$	0	0
$328$	−45.7981	−2.52878
$329$	−4.66542	−0.257213
$330$	0	0
$331$	−1.62894	−0.0895349	−0.0447674	−	0.998997i	$-0.514255\pi$
$331$	−1.62894	−0.0895349	−0.0447674	+	0.998997i	$0.514255\pi$
$332$	24.9901	1.37151
$333$	0	0
$334$	−30.1000	−1.64700
$335$	33.2445	1.81634
$336$	0	0
$337$	4.91650	0.267819	0.133909	−	0.990994i	$-0.457247\pi$
$337$	4.91650	0.267819	0.133909	+	0.990994i	$0.457247\pi$
$338$	−19.3242	−1.05110
$339$	0	0
$340$	−22.8228	−1.23774
$341$	16.6287	0.900497
$342$	0	0
$343$	1.00000	0.0539949
$344$	−60.4877	−3.26128
$345$	0	0
$346$	−17.2634	−0.928086
$347$	−33.8004	−1.81450	−0.907249	−	0.420593i	$-0.861822\pi$
$347$	−33.8004	−1.81450	−0.907249	+	0.420593i	$0.861822\pi$
$348$	0	0
$349$	3.26430	0.174734	0.0873669	−	0.996176i	$-0.472155\pi$
$349$	3.26430	0.174734	0.0873669	+	0.996176i	$0.472155\pi$
$350$	7.06600	0.377693
$351$	0	0
$352$	25.7921	1.37473
$353$	−11.1070	−0.591165	−0.295583	−	0.955317i	$-0.595514\pi$
$353$	−11.1070	−0.591165	−0.295583	+	0.955317i	$0.595514\pi$
$354$	0	0
$355$	12.8258	0.680723
$356$	50.3142	2.66665
$357$	0	0
$358$	−52.3471	−2.76663
$359$	19.0760	1.00679	0.503397	−	0.864056i	$-0.332084\pi$
$359$	19.0760	1.00679	0.503397	+	0.864056i	$0.332084\pi$
$360$	0	0
$361$	31.2900	1.64684
$362$	−4.31999	−0.227054
$363$	0	0
$364$	−10.4218	−0.546249
$365$	21.1870	1.10898
$366$	0	0
$367$	−31.9319	−1.66683	−0.833416	−	0.552647i	$-0.813618\pi$
$367$	−31.9319	−1.66683	−0.833416	+	0.552647i	$0.813618\pi$
$368$	7.11710	0.371004
$369$	0	0
$370$	1.17968	0.0613285
$371$	0.961924	0.0499406
$372$	0	0
$373$	20.8038	1.07718	0.538590	−	0.842568i	$-0.318957\pi$
$373$	20.8038	1.07718	0.538590	+	0.842568i	$0.318957\pi$
$374$	21.8593	1.13032
$375$	0	0
$376$	−29.4666	−1.51963
$377$	−23.2315	−1.19648
$378$	0	0
$379$	−26.5314	−1.36282	−0.681412	−	0.731900i	$-0.738634\pi$
$379$	−26.5314	−1.36282	−0.681412	+	0.731900i	$0.738634\pi$
$380$	−88.6092	−4.54555
$381$	0	0
$382$	−40.4454	−2.06937
$383$	20.4224	1.04354	0.521768	−	0.853088i	$-0.325273\pi$
$383$	20.4224	1.04354	0.521768	+	0.853088i	$0.325273\pi$
$384$	0	0
$385$	−13.1085	−0.668069
$386$	−53.9498	−2.74597
$387$	0	0
$388$	−7.56410	−0.384009
$389$	−1.10846	−0.0562012	−0.0281006	−	0.999605i	$-0.508946\pi$
$389$	−1.10846	−0.0562012	−0.0281006	+	0.999605i	$0.508946\pi$
$390$	0	0
$391$	1.82655	0.0923725
$392$	6.31597	0.319005
$393$	0	0
$394$	−47.1339	−2.37457
$395$	−16.0031	−0.805204
$396$	0	0
$397$	−19.3304	−0.970166	−0.485083	−	0.874468i	$-0.661211\pi$
$397$	−19.3304	−0.970166	−0.485083	+	0.874468i	$0.661211\pi$
$398$	31.3619	1.57203
$399$	0	0
$400$	19.7541	0.987703
$401$	−34.3076	−1.71324	−0.856620	−	0.515947i	$-0.827440\pi$
$401$	−34.3076	−1.71324	−0.856620	+	0.515947i	$0.827440\pi$
$402$	0	0
$403$	−8.22705	−0.409819
$404$	−14.0169	−0.697368
$405$	0	0
$406$	25.4289	1.26201
$407$	−0.781200	−0.0387227
$408$	0	0
$409$	−11.1492	−0.551293	−0.275647	−	0.961259i	$-0.588892\pi$
$409$	−11.1492	−0.551293	−0.275647	+	0.961259i	$0.588892\pi$
$410$	51.4746	2.54215
$411$	0	0
$412$	4.65158	0.229167
$413$	−13.8800	−0.682991
$414$	0	0
$415$	−15.5512	−0.763376
$416$	−12.7606	−0.625642
$417$	0	0
$418$	84.8684	4.15105
$419$	36.8881	1.80210	0.901052	−	0.433711i	$-0.142796\pi$
$419$	36.8881	1.80210	0.901052	+	0.433711i	$0.142796\pi$
$420$	0	0
$421$	21.8055	1.06273	0.531367	−	0.847142i	$-0.321679\pi$
$421$	21.8055	1.06273	0.531367	+	0.847142i	$0.321679\pi$
$422$	−8.89134	−0.432824
$423$	0	0
$424$	6.07548	0.295052
$425$	5.06973	0.245918
$426$	0	0
$427$	0.954652	0.0461988
$428$	−36.8319	−1.78034
$429$	0	0
$430$	67.9850	3.27853
$431$	−17.1450	−0.825846	−0.412923	−	0.910766i	$-0.635492\pi$
$431$	−17.1450	−0.825846	−0.412923	+	0.910766i	$0.635492\pi$
$432$	0	0
$433$	2.68890	0.129220	0.0646102	−	0.997911i	$-0.479420\pi$
$433$	2.68890	0.129220	0.0646102	+	0.997911i	$0.479420\pi$
$434$	9.00521	0.432264
$435$	0	0
$436$	31.3599	1.50187
$437$	7.09155	0.339235
$438$	0	0
$439$	23.9686	1.14396	0.571979	−	0.820268i	$-0.306176\pi$
$439$	23.9686	1.14396	0.571979	+	0.820268i	$0.306176\pi$
$440$	−82.7926	−3.94698
$441$	0	0
$442$	−10.8149	−0.514411
$443$	28.5038	1.35426	0.677128	−	0.735865i	$-0.263224\pi$
$443$	28.5038	1.35426	0.677128	+	0.735865i	$0.263224\pi$
$444$	0	0
$445$	−31.3102	−1.48425
$446$	24.8928	1.17871
$447$	0	0
$448$	−0.266598	−0.0125956
$449$	−1.18398	−0.0558753	−0.0279376	−	0.999610i	$-0.508894\pi$
$449$	−1.18398	−0.0558753	−0.0279376	+	0.999610i	$0.508894\pi$
$450$	0	0
$451$	−34.0873	−1.60511
$452$	−11.8816	−0.558864
$453$	0	0
$454$	−21.0262	−0.986807
$455$	6.48540	0.304040
$456$	0	0
$457$	33.3173	1.55852	0.779260	−	0.626701i	$-0.215595\pi$
$457$	33.3173	1.55852	0.779260	+	0.626701i	$0.215595\pi$
$458$	7.98969	0.373334
$459$	0	0
$460$	−12.4950	−0.582584
$461$	1.59002	0.0740545	0.0370273	−	0.999314i	$-0.488211\pi$
$461$	1.59002	0.0740545	0.0370273	+	0.999314i	$0.488211\pi$
$462$	0	0
$463$	30.4569	1.41545	0.707726	−	0.706487i	$-0.249721\pi$
$463$	30.4569	1.41545	0.707726	+	0.706487i	$0.249721\pi$
$464$	71.0903	3.30028
$465$	0	0
$466$	−22.5199	−1.04322
$467$	−27.6450	−1.27926	−0.639628	−	0.768685i	$-0.720912\pi$
$467$	−27.6450	−1.27926	−0.639628	+	0.768685i	$0.720912\pi$
$468$	0	0
$469$	−11.9221	−0.550512
$470$	33.1190	1.52766
$471$	0	0
$472$	−87.6658	−4.03514
$473$	−45.0207	−2.07005
$474$	0	0
$475$	19.6832	0.903125
$476$	8.18469	0.375145
$477$	0	0
$478$	26.4232	1.20857
$479$	−36.3245	−1.65971	−0.829855	−	0.557979i	$-0.811577\pi$
$479$	−36.3245	−1.65971	−0.829855	+	0.557979i	$0.811577\pi$
$480$	0	0
$481$	0.386498	0.0176228
$482$	−6.30875	−0.287356
$483$	0	0
$484$	49.7338	2.26063
$485$	4.70709	0.213738
$486$	0	0
$487$	31.6204	1.43286	0.716428	−	0.697661i	$-0.245776\pi$
$487$	31.6204	1.43286	0.716428	+	0.697661i	$0.245776\pi$
$488$	6.02955	0.272945
$489$	0	0
$490$	−7.09882	−0.320692
$491$	41.6773	1.88087	0.940436	−	0.339972i	$-0.110418\pi$
$491$	41.6773	1.88087	0.940436	+	0.339972i	$0.110418\pi$
$492$	0	0
$493$	18.2448	0.821703
$494$	−41.9886	−1.88915
$495$	0	0
$496$	25.1754	1.13041
$497$	−4.59958	−0.206319
$498$	0	0
$499$	−27.7258	−1.24118	−0.620588	−	0.784137i	$-0.713106\pi$
$499$	−27.7258	−1.24118	−0.620588	+	0.784137i	$0.713106\pi$
$500$	27.7942	1.24300
$501$	0	0
$502$	24.6099	1.09839
$503$	29.0492	1.29524	0.647620	−	0.761963i	$-0.275764\pi$
$503$	29.0492	1.29524	0.647620	+	0.761963i	$0.275764\pi$
$504$	0	0
$505$	8.72263	0.388152
$506$	11.9675	0.532022
$507$	0	0
$508$	−3.79871	−0.168540
$509$	8.20421	0.363645	0.181823	−	0.983331i	$-0.441800\pi$
$509$	8.20421	0.363645	0.181823	+	0.983331i	$0.441800\pi$
$510$	0	0
$511$	−7.59806	−0.336118
$512$	−50.8542	−2.24746
$513$	0	0
$514$	−41.4384	−1.82777
$515$	−2.89465	−0.127553
$516$	0	0
$517$	−21.9319	−0.964563
$518$	−0.423055	−0.0185880
$519$	0	0
$520$	40.9616	1.79628
$521$	9.59168	0.420219	0.210110	−	0.977678i	$-0.432618\pi$
$521$	9.59168	0.420219	0.210110	+	0.977678i	$0.432618\pi$
$522$	0	0
$523$	−6.06424	−0.265171	−0.132585	−	0.991172i	$-0.542328\pi$
$523$	−6.06424	−0.265171	−0.132585	+	0.991172i	$0.542328\pi$
$524$	15.2697	0.667062
$525$	0	0
$526$	22.9967	1.00271
$527$	6.46108	0.281449
$528$	0	0
$529$	1.00000	0.0434783
$530$	−6.82853	−0.296612
$531$	0	0
$532$	31.7770	1.37771
$533$	16.8647	0.730489
$534$	0	0
$535$	22.9203	0.990930
$536$	−75.2997	−3.25245
$537$	0	0
$538$	44.8663	1.93432
$539$	4.70095	0.202484
$540$	0	0
$541$	−28.6760	−1.23288	−0.616438	−	0.787403i	$-0.711425\pi$
$541$	−28.6760	−1.23288	−0.616438	+	0.787403i	$0.711425\pi$
$542$	61.0819	2.62369
$543$	0	0
$544$	10.0215	0.429669
$545$	−19.5151	−0.835934
$546$	0	0
$547$	−24.5492	−1.04965	−0.524824	−	0.851211i	$-0.675869\pi$
$547$	−24.5492	−1.04965	−0.524824	+	0.851211i	$0.675869\pi$
$548$	64.5673	2.75818
$549$	0	0
$550$	33.2169	1.41637
$551$	70.8350	3.01767
$552$	0	0
$553$	5.73902	0.244048
$554$	42.1684	1.79156
$555$	0	0
$556$	11.6812	0.495393
$557$	5.82872	0.246971	0.123485	−	0.992346i	$-0.460593\pi$
$557$	5.82872	0.246971	0.123485	+	0.992346i	$0.460593\pi$
$558$	0	0
$559$	22.2740	0.942088
$560$	−19.8458	−0.838639
$561$	0	0
$562$	49.7006	2.09649
$563$	30.9619	1.30489	0.652445	−	0.757836i	$-0.273743\pi$
$563$	30.9619	1.30489	0.652445	+	0.757836i	$0.273743\pi$
$564$	0	0
$565$	7.39385	0.311062
$566$	−40.0510	−1.68347
$567$	0	0
$568$	−29.0508	−1.21894
$569$	−6.77249	−0.283918	−0.141959	−	0.989873i	$-0.545340\pi$
$569$	−6.77249	−0.283918	−0.141959	+	0.989873i	$0.545340\pi$
$570$	0	0
$571$	−26.7220	−1.11828	−0.559140	−	0.829073i	$-0.688868\pi$
$571$	−26.7220	−1.11828	−0.559140	+	0.829073i	$0.688868\pi$
$572$	−48.9922	−2.04847
$573$	0	0
$574$	−18.4598	−0.770497
$575$	2.77558	0.115750
$576$	0	0
$577$	−3.13787	−0.130631	−0.0653157	−	0.997865i	$-0.520805\pi$
$577$	−3.13787	−0.130631	−0.0653157	+	0.997865i	$0.520805\pi$
$578$	−34.7847	−1.44685
$579$	0	0
$580$	−124.809	−5.18240
$581$	5.57695	0.231371
$582$	0	0
$583$	4.52196	0.187280
$584$	−47.9891	−1.98580
$585$	0	0
$586$	20.7803	0.858428
$587$	−31.4784	−1.29925	−0.649626	−	0.760254i	$-0.725075\pi$
$587$	−31.4784	−1.29925	−0.649626	+	0.760254i	$0.725075\pi$
$588$	0	0
$589$	25.0850	1.03361
$590$	98.5317	4.05649
$591$	0	0
$592$	−1.18271	−0.0486093
$593$	−15.4196	−0.633209	−0.316604	−	0.948558i	$-0.602543\pi$
$593$	−15.4196	−0.633209	−0.316604	+	0.948558i	$0.602543\pi$
$594$	0	0
$595$	−5.09328	−0.208804
$596$	0.0397566	0.00162849
$597$	0	0
$598$	−5.92093	−0.242125
$599$	18.7196	0.764862	0.382431	−	0.923984i	$-0.375087\pi$
$599$	18.7196	0.764862	0.382431	+	0.923984i	$0.375087\pi$
$600$	0	0
$601$	12.5523	0.512017	0.256009	−	0.966675i	$-0.417592\pi$
$601$	12.5523	0.512017	0.256009	+	0.966675i	$0.417592\pi$
$602$	−24.3807	−0.993684
$603$	0	0
$604$	58.7319	2.38977
$605$	−30.9490	−1.25825
$606$	0	0
$607$	20.5171	0.832761	0.416381	−	0.909190i	$-0.363298\pi$
$607$	20.5171	0.832761	0.416381	+	0.909190i	$0.363298\pi$
$608$	38.9084	1.57794
$609$	0	0
$610$	−6.77690	−0.274389
$611$	10.8508	0.438976
$612$	0	0
$613$	−24.9165	−1.00637	−0.503184	−	0.864179i	$-0.667838\pi$
$613$	−24.9165	−1.00637	−0.503184	+	0.864179i	$0.667838\pi$
$614$	77.7840	3.13911
$615$	0	0
$616$	29.6910	1.19629
$617$	8.77553	0.353289	0.176645	−	0.984275i	$-0.443476\pi$
$617$	8.77553	0.353289	0.176645	+	0.984275i	$0.443476\pi$
$618$	0	0
$619$	−14.6150	−0.587427	−0.293714	−	0.955893i	$-0.594891\pi$
$619$	−14.6150	−0.587427	−0.293714	+	0.955893i	$0.594891\pi$
$620$	−44.1989	−1.77507
$621$	0	0
$622$	−64.5540	−2.58838
$623$	11.2284	0.449858
$624$	0	0
$625$	−31.1741	−1.24696
$626$	−15.6731	−0.626425
$627$	0	0
$628$	−1.11846	−0.0446315
$629$	−0.303535	−0.0121027
$630$	0	0
$631$	18.5826	0.739760	0.369880	−	0.929079i	$-0.379399\pi$
$631$	18.5826	0.739760	0.369880	+	0.929079i	$0.379399\pi$
$632$	36.2475	1.44185
$633$	0	0
$634$	11.4067	0.453016
$635$	2.36391	0.0938089
$636$	0	0
$637$	−2.32579	−0.0921511
$638$	119.540	4.73262
$639$	0	0
$640$	32.4909	1.28432
$641$	−9.57505	−0.378192	−0.189096	−	0.981959i	$-0.560556\pi$
$641$	−9.57505	−0.378192	−0.189096	+	0.981959i	$0.560556\pi$
$642$	0	0
$643$	23.7204	0.935443	0.467722	−	0.883876i	$-0.345075\pi$
$643$	23.7204	0.935443	0.467722	+	0.883876i	$0.345075\pi$
$644$	4.48096	0.176575
$645$	0	0
$646$	32.9755	1.29741
$647$	20.1385	0.791727	0.395864	−	0.918309i	$-0.370445\pi$
$647$	20.1385	0.791727	0.395864	+	0.918309i	$0.370445\pi$
$648$	0	0
$649$	−65.2492	−2.56126
$650$	−16.4340	−0.644595
$651$	0	0
$652$	−0.675448	−0.0264526
$653$	32.6863	1.27911	0.639557	−	0.768744i	$-0.279118\pi$
$653$	32.6863	1.27911	0.639557	+	0.768744i	$0.279118\pi$
$654$	0	0
$655$	−9.50226	−0.371284
$656$	−51.6072	−2.01492
$657$	0	0
$658$	−11.8771	−0.463018
$659$	−37.3251	−1.45398	−0.726989	−	0.686649i	$-0.759081\pi$
$659$	−37.3251	−1.45398	−0.726989	+	0.686649i	$0.759081\pi$
$660$	0	0
$661$	−9.68312	−0.376630	−0.188315	−	0.982109i	$-0.560303\pi$
$661$	−9.68312	−0.376630	−0.188315	+	0.982109i	$0.560303\pi$
$662$	−4.14692	−0.161175
$663$	0	0
$664$	35.2238	1.36695
$665$	−19.7746	−0.766825
$666$	0	0
$667$	9.98866	0.386762
$668$	−52.9807	−2.04988
$669$	0	0
$670$	84.6329	3.26966
$671$	4.48777	0.173248
$672$	0	0
$673$	38.5622	1.48646	0.743232	−	0.669034i	$-0.233292\pi$
$673$	38.5622	1.48646	0.743232	+	0.669034i	$0.233292\pi$
$674$	12.5163	0.482109
$675$	0	0
$676$	−34.0137	−1.30822
$677$	−2.70428	−0.103934	−0.0519669	−	0.998649i	$-0.516549\pi$
$677$	−2.70428	−0.103934	−0.0519669	+	0.998649i	$0.516549\pi$
$678$	0	0
$679$	−1.68805	−0.0647815
$680$	−32.1690	−1.23362
$681$	0	0
$682$	42.3330	1.62101
$683$	10.6222	0.406446	0.203223	−	0.979132i	$-0.434858\pi$
$683$	10.6222	0.406446	0.203223	+	0.979132i	$0.434858\pi$
$684$	0	0
$685$	−40.1798	−1.53519
$686$	2.54577	0.0971981
$687$	0	0
$688$	−68.1601	−2.59858
$689$	−2.23723	−0.0852318
$690$	0	0
$691$	23.7327	0.902833	0.451416	−	0.892313i	$-0.350919\pi$
$691$	23.7327	0.902833	0.451416	+	0.892313i	$0.350919\pi$
$692$	−30.3863	−1.15511
$693$	0	0
$694$	−86.0481	−3.26634
$695$	−7.26913	−0.275734
$696$	0	0
$697$	−13.2446	−0.501674
$698$	8.31016	0.314544
$699$	0	0
$700$	12.4373	0.470084
$701$	−22.5615	−0.852138	−0.426069	−	0.904691i	$-0.640102\pi$
$701$	−22.5615	−0.852138	−0.426069	+	0.904691i	$0.640102\pi$
$702$	0	0
$703$	−1.17847	−0.0444468
$704$	−1.25327	−0.0472342
$705$	0	0
$706$	−28.2759	−1.06418
$707$	−3.12810	−0.117644
$708$	0	0
$709$	−16.7030	−0.627294	−0.313647	−	0.949540i	$-0.601551\pi$
$709$	−16.7030	−0.627294	−0.313647	+	0.949540i	$0.601551\pi$
$710$	32.6516	1.22539
$711$	0	0
$712$	70.9185	2.65778
$713$	3.53732	0.132474
$714$	0	0
$715$	30.4875	1.14017
$716$	−92.1391	−3.44340
$717$	0	0
$718$	48.5632	1.81236
$719$	−27.7469	−1.03479	−0.517393	−	0.855748i	$-0.673097\pi$
$719$	−27.7469	−1.03479	−0.517393	+	0.855748i	$0.673097\pi$
$720$	0	0
$721$	1.03808	0.0386600
$722$	79.6573	2.96454
$723$	0	0
$724$	−7.60386	−0.282595
$725$	27.7243	1.02966
$726$	0	0
$727$	−41.4981	−1.53908	−0.769539	−	0.638600i	$-0.779514\pi$
$727$	−41.4981	−1.53908	−0.769539	+	0.638600i	$0.779514\pi$
$728$	−14.6896	−0.544433
$729$	0	0
$730$	53.9373	1.99631
$731$	−17.4928	−0.646993
$732$	0	0
$733$	35.7212	1.31939	0.659697	−	0.751532i	$-0.270685\pi$
$733$	35.7212	1.31939	0.659697	+	0.751532i	$0.270685\pi$
$734$	−81.2914	−3.00052
$735$	0	0
$736$	5.48658	0.202238
$737$	−56.0452	−2.06445
$738$	0	0
$739$	−44.4520	−1.63519	−0.817597	−	0.575791i	$-0.804694\pi$
$739$	−44.4520	−1.63519	−0.817597	+	0.575791i	$0.804694\pi$
$740$	2.07642	0.0763306
$741$	0	0
$742$	2.44884	0.0898998
$743$	17.0898	0.626964	0.313482	−	0.949594i	$-0.398504\pi$
$743$	17.0898	0.626964	0.313482	+	0.949594i	$0.398504\pi$
$744$	0	0
$745$	−0.0247402	−0.000906412 0
$746$	52.9617	1.93907
$747$	0	0
$748$	38.4758	1.40681
$749$	−8.21965	−0.300339
$750$	0	0
$751$	−40.9748	−1.49519	−0.747596	−	0.664154i	$-0.768792\pi$
$751$	−40.9748	−1.49519	−0.747596	+	0.664154i	$0.768792\pi$
$752$	−33.2042	−1.21083
$753$	0	0
$754$	−59.1422	−2.15383
$755$	−36.5485	−1.33014
$756$	0	0
$757$	−23.7095	−0.861737	−0.430868	−	0.902415i	$-0.641793\pi$
$757$	−23.7095	−0.861737	−0.430868	+	0.902415i	$0.641793\pi$
$758$	−67.5429	−2.45327
$759$	0	0
$760$	−124.896	−4.53044
$761$	42.7680	1.55034	0.775170	−	0.631753i	$-0.217664\pi$
$761$	42.7680	1.55034	0.775170	+	0.631753i	$0.217664\pi$
$762$	0	0
$763$	6.99848	0.253362
$764$	−71.1904	−2.57558
$765$	0	0
$766$	51.9908	1.87850
$767$	32.2820	1.16564
$768$	0	0
$769$	35.6281	1.28478	0.642392	−	0.766376i	$-0.277942\pi$
$769$	35.6281	1.28478	0.642392	+	0.766376i	$0.277942\pi$
$770$	−33.3712	−1.20261
$771$	0	0
$772$	−94.9602	−3.41769
$773$	−16.9388	−0.609246	−0.304623	−	0.952473i	$-0.598530\pi$
$773$	−16.9388	−0.609246	−0.304623	+	0.952473i	$0.598530\pi$
$774$	0	0
$775$	9.81810	0.352677
$776$	−10.6617	−0.382732
$777$	0	0
$778$	−2.82189	−0.101170
$779$	−51.4219	−1.84238
$780$	0	0
$781$	−21.6224	−0.773709
$782$	4.64998	0.166283
$783$	0	0
$784$	7.11710	0.254182
$785$	0.696011	0.0248417
$786$	0	0
$787$	−15.4748	−0.551619	−0.275809	−	0.961212i	$-0.588946\pi$
$787$	−15.4748	−0.551619	−0.275809	+	0.961212i	$0.588946\pi$
$788$	−82.9630	−2.95544
$789$	0	0
$790$	−40.7403	−1.44947
$791$	−2.65158	−0.0942792
$792$	0	0
$793$	−2.22032	−0.0788458
$794$	−49.2109	−1.74643
$795$	0	0
$796$	55.2020	1.95658
$797$	−28.1935	−0.998664	−0.499332	−	0.866411i	$-0.666421\pi$
$797$	−28.1935	−0.998664	−0.499332	+	0.866411i	$0.666421\pi$
$798$	0	0
$799$	−8.52161	−0.301473
$800$	15.2284	0.538407
$801$	0	0
$802$	−87.3394	−3.08406
$803$	−35.7181	−1.26046
$804$	0	0
$805$	−2.78847	−0.0982807
$806$	−20.9442	−0.737728
$807$	0	0
$808$	−19.7570	−0.695049
$809$	30.7565	1.08134	0.540670	−	0.841235i	$-0.318171\pi$
$809$	30.7565	1.08134	0.540670	+	0.841235i	$0.318171\pi$
$810$	0	0
$811$	1.27477	0.0447632	0.0223816	−	0.999750i	$-0.492875\pi$
$811$	1.27477	0.0447632	0.0223816	+	0.999750i	$0.492875\pi$
$812$	44.7588	1.57073
$813$	0	0
$814$	−1.98876	−0.0697059
$815$	0.420327	0.0147234
$816$	0	0
$817$	−67.9154	−2.37606
$818$	−28.3834	−0.992402
$819$	0	0
$820$	90.6035	3.16401
$821$	5.88860	0.205513	0.102757	−	0.994707i	$-0.467234\pi$
$821$	5.88860	0.205513	0.102757	+	0.994707i	$0.467234\pi$
$822$	0	0
$823$	13.3985	0.467043	0.233522	−	0.972352i	$-0.424975\pi$
$823$	13.3985	0.467043	0.233522	+	0.972352i	$0.424975\pi$
$824$	6.55645	0.228405
$825$	0	0
$826$	−35.3354	−1.22948
$827$	−30.5493	−1.06230	−0.531151	−	0.847277i	$-0.678240\pi$
$827$	−30.5493	−1.06230	−0.531151	+	0.847277i	$0.678240\pi$
$828$	0	0
$829$	−33.2608	−1.15520	−0.577598	−	0.816321i	$-0.696010\pi$
$829$	−33.2608	−1.15520	−0.577598	+	0.816321i	$0.696010\pi$
$830$	−39.5897	−1.37418
$831$	0	0
$832$	0.620052	0.0214964
$833$	1.82655	0.0632861
$834$	0	0
$835$	32.9695	1.14096
$836$	149.382	5.16648
$837$	0	0
$838$	93.9089	3.24403
$839$	−19.4516	−0.671543	−0.335772	−	0.941943i	$-0.608997\pi$
$839$	−19.4516	−0.671543	−0.335772	+	0.941943i	$0.608997\pi$
$840$	0	0
$841$	70.7733	2.44046
$842$	55.5118	1.91306
$843$	0	0
$844$	−15.6502	−0.538701
$845$	21.1665	0.728149
$846$	0	0
$847$	11.0989	0.381363
$848$	6.84611	0.235096
$849$	0	0
$850$	12.9064	0.442685
$851$	−0.166179	−0.00569655
$852$	0	0
$853$	−43.9900	−1.50619	−0.753094	−	0.657913i	$-0.771439\pi$
$853$	−43.9900	−1.50619	−0.753094	+	0.657913i	$0.771439\pi$
$854$	2.43033	0.0831641
$855$	0	0
$856$	−51.9150	−1.77442
$857$	50.9265	1.73962	0.869808	−	0.493391i	$-0.164243\pi$
$857$	50.9265	1.73962	0.869808	+	0.493391i	$0.164243\pi$
$858$	0	0
$859$	19.0473	0.649885	0.324943	−	0.945734i	$-0.394655\pi$
$859$	19.0473	0.649885	0.324943	+	0.945734i	$0.394655\pi$
$860$	119.664	4.08052
$861$	0	0
$862$	−43.6473	−1.48663
$863$	30.6481	1.04327	0.521637	−	0.853168i	$-0.325322\pi$
$863$	30.6481	1.04327	0.521637	+	0.853168i	$0.325322\pi$
$864$	0	0
$865$	18.9092	0.642932
$866$	6.84534	0.232614
$867$	0	0
$868$	15.8506	0.538004
$869$	26.9788	0.915194
$870$	0	0
$871$	27.7283	0.939538
$872$	44.2022	1.49687
$873$	0	0
$874$	18.0535	0.610668
$875$	6.20274	0.209691
$876$	0	0
$877$	15.6561	0.528668	0.264334	−	0.964431i	$-0.414848\pi$
$877$	15.6561	0.528668	0.264334	+	0.964431i	$0.414848\pi$
$878$	61.0186	2.05928
$879$	0	0
$880$	−93.2942	−3.14495
$881$	51.9258	1.74943	0.874713	−	0.484642i	$-0.161050\pi$
$881$	51.9258	1.74943	0.874713	+	0.484642i	$0.161050\pi$
$882$	0	0
$883$	18.1603	0.611144	0.305572	−	0.952169i	$-0.401152\pi$
$883$	18.1603	0.611144	0.305572	+	0.952169i	$0.401152\pi$
$884$	−19.0359	−0.640246
$885$	0	0
$886$	72.5642	2.43784
$887$	29.5677	0.992786	0.496393	−	0.868098i	$-0.334658\pi$
$887$	29.5677	0.992786	0.496393	+	0.868098i	$0.334658\pi$
$888$	0	0
$889$	−0.847744	−0.0284324
$890$	−79.7087	−2.67184
$891$	0	0
$892$	43.8152	1.46704
$893$	−33.0850	−1.10715
$894$	0	0
$895$	57.3376	1.91658
$896$	−11.6519	−0.389261
$897$	0	0
$898$	−3.01413	−0.100583
$899$	35.3330	1.17842
$900$	0	0
$901$	1.75700	0.0585342
$902$	−86.7785	−2.88941
$903$	0	0
$904$	−16.7473	−0.557006
$905$	4.73183	0.157291
$906$	0	0
$907$	30.4350	1.01058	0.505289	−	0.862950i	$-0.331386\pi$
$907$	30.4350	1.01058	0.505289	+	0.862950i	$0.331386\pi$
$908$	−37.0094	−1.22820
$909$	0	0
$910$	16.5104	0.547313
$911$	48.2859	1.59978	0.799892	−	0.600144i	$-0.204890\pi$
$911$	48.2859	1.59978	0.799892	+	0.600144i	$0.204890\pi$
$912$	0	0
$913$	26.2169	0.867653
$914$	84.8184	2.80554
$915$	0	0
$916$	14.0631	0.464658
$917$	3.40769	0.112532
$918$	0	0
$919$	−28.4097	−0.937150	−0.468575	−	0.883424i	$-0.655232\pi$
$919$	−28.4097	−0.937150	−0.468575	+	0.883424i	$0.655232\pi$
$920$	−17.6119	−0.580647
$921$	0	0
$922$	4.04783	0.133308
$923$	10.6976	0.352117
$924$	0	0
$925$	−0.461244	−0.0151656
$926$	77.5363	2.54800
$927$	0	0
$928$	54.8036	1.79902
$929$	−27.0080	−0.886104	−0.443052	−	0.896496i	$-0.646104\pi$
$929$	−27.0080	−0.886104	−0.443052	+	0.896496i	$0.646104\pi$
$930$	0	0
$931$	7.09155	0.232416
$932$	−39.6386	−1.29841
$933$	0	0
$934$	−70.3778	−2.30283
$935$	−23.9432	−0.783028
$936$	0	0
$937$	15.3639	0.501917	0.250958	−	0.967998i	$-0.419254\pi$
$937$	15.3639	0.501917	0.250958	+	0.967998i	$0.419254\pi$
$938$	−30.3510	−0.990995
$939$	0	0
$940$	58.2946	1.90136
$941$	46.8015	1.52569	0.762843	−	0.646584i	$-0.223803\pi$
$941$	46.8015	1.52569	0.762843	+	0.646584i	$0.223803\pi$
$942$	0	0
$943$	−7.25116	−0.236130
$944$	−98.7855	−3.21519
$945$	0	0
$946$	−114.613	−3.72637
$947$	−13.1326	−0.426753	−0.213376	−	0.976970i	$-0.568446\pi$
$947$	−13.1326	−0.426753	−0.213376	+	0.976970i	$0.568446\pi$
$948$	0	0
$949$	17.6715	0.573641
$950$	50.1088	1.62575
$951$	0	0
$952$	11.5364	0.373897
$953$	−3.29495	−0.106734	−0.0533670	−	0.998575i	$-0.516995\pi$
$953$	−3.29495	−0.106734	−0.0533670	+	0.998575i	$0.516995\pi$
$954$	0	0
$955$	44.3013	1.43356
$956$	46.5091	1.50421
$957$	0	0
$958$	−92.4740	−2.98770
$959$	14.4092	0.465299
$960$	0	0
$961$	−18.4874	−0.596368
$962$	0.983936	0.0317234
$963$	0	0
$964$	−11.1044	−0.357648
$965$	59.0931	1.90227
$966$	0	0
$967$	1.39392	0.0448254	0.0224127	−	0.999749i	$-0.492865\pi$
$967$	1.39392	0.0448254	0.0224127	+	0.999749i	$0.492865\pi$
$968$	70.1003	2.25311
$969$	0	0
$970$	11.9832	0.384757
$971$	10.8178	0.347158	0.173579	−	0.984820i	$-0.444467\pi$
$971$	10.8178	0.347158	0.173579	+	0.984820i	$0.444467\pi$
$972$	0	0
$973$	2.60685	0.0835718
$974$	80.4983	2.57933
$975$	0	0
$976$	6.79435	0.217482
$977$	−23.4628	−0.750642	−0.375321	−	0.926895i	$-0.622468\pi$
$977$	−23.4628	−0.750642	−0.375321	+	0.926895i	$0.622468\pi$
$978$	0	0
$979$	52.7843	1.68699
$980$	−12.4950	−0.399139
$981$	0	0
$982$	106.101	3.38582
$983$	−57.3568	−1.82940	−0.914700	−	0.404134i	$-0.867573\pi$
$983$	−57.3568	−1.82940	−0.914700	+	0.404134i	$0.867573\pi$
$984$	0	0
$985$	51.6273	1.64498
$986$	46.4470	1.47918
$987$	0	0
$988$	−73.9065	−2.35128
$989$	−9.57695	−0.304529
$990$	0	0
$991$	20.1069	0.638718	0.319359	−	0.947634i	$-0.396532\pi$
$991$	20.1069	0.638718	0.319359	+	0.947634i	$0.396532\pi$
$992$	19.4078	0.616198
$993$	0	0
$994$	−11.7095	−0.371402
$995$	−34.3518	−1.08903
$996$	0	0
$997$	−12.2818	−0.388970	−0.194485	−	0.980906i	$-0.562303\pi$
$997$	−12.2818	−0.388970	−0.194485	+	0.980906i	$0.562303\pi$
$998$	−70.5836	−2.23428
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1449.2.a.r.1.5		5
3.2	odd	2		161.2.a.d.1.1	✓	5
12.11	even	2		2576.2.a.bd.1.2		5
15.14	odd	2		4025.2.a.p.1.5		5
21.20	even	2		1127.2.a.h.1.1		5
69.68	even	2		3703.2.a.j.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
161.2.a.d.1.1	✓	5	3.2	odd	2
1127.2.a.h.1.1		5	21.20	even	2
1449.2.a.r.1.5		5	1.1	even	1	trivial
2576.2.a.bd.1.2		5	12.11	even	2
3703.2.a.j.1.1		5	69.68	even	2
4025.2.a.p.1.5		5	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1449 = 3^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1449.a (trivial)

\(f(q)\)	\(=\)	\(q+2.54577 q^{2} +4.48096 q^{4} -2.78847 q^{5} +1.00000 q^{7} +6.31597 q^{8} +O(q^{10})\)	\(q+2.54577 q^{2} +4.48096 q^{4} -2.78847 q^{5} +1.00000 q^{7} +6.31597 q^{8} -7.09882 q^{10} +4.70095 q^{11} -2.32579 q^{13} +2.54577 q^{14} +7.11710 q^{16} +1.82655 q^{17} +7.09155 q^{19} -12.4950 q^{20} +11.9675 q^{22} +1.00000 q^{23} +2.77558 q^{25} -5.92093 q^{26} +4.48096 q^{28} +9.98866 q^{29} +3.53732 q^{31} +5.48658 q^{32} +4.64998 q^{34} -2.78847 q^{35} -0.166179 q^{37} +18.0535 q^{38} -17.6119 q^{40} -7.25116 q^{41} -9.57695 q^{43} +21.0648 q^{44} +2.54577 q^{46} -4.66542 q^{47} +1.00000 q^{49} +7.06600 q^{50} -10.4218 q^{52} +0.961924 q^{53} -13.1085 q^{55} +6.31597 q^{56} +25.4289 q^{58} -13.8800 q^{59} +0.954652 q^{61} +9.00521 q^{62} -0.266598 q^{64} +6.48540 q^{65} -11.9221 q^{67} +8.18469 q^{68} -7.09882 q^{70} -4.59958 q^{71} -7.59806 q^{73} -0.423055 q^{74} +31.7770 q^{76} +4.70095 q^{77} +5.73902 q^{79} -19.8458 q^{80} -18.4598 q^{82} +5.57695 q^{83} -5.09328 q^{85} -24.3807 q^{86} +29.6910 q^{88} +11.2284 q^{89} -2.32579 q^{91} +4.48096 q^{92} -11.8771 q^{94} -19.7746 q^{95} -1.68805 q^{97} +2.54577 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} + 12 q^{4} + 4 q^{5} + 5 q^{7} - 3 q^{8}+O(q^{10}) \) 5 * q - 2 * q^2 + 12 * q^4 + 4 * q^5 + 5 * q^7 - 3 * q^8	\( 5 q - 2 q^{2} + 12 q^{4} + 4 q^{5} + 5 q^{7} - 3 q^{8} - 8 q^{10} + 4 q^{11} - 6 q^{13} - 2 q^{14} + 10 q^{16} + 12 q^{17} + 6 q^{19} + 14 q^{22} + 5 q^{23} + 19 q^{25} - q^{26} + 12 q^{28} + 4 q^{29} + 30 q^{31} - 8 q^{32} + 6 q^{34} + 4 q^{35} + 4 q^{37} + 40 q^{38} - 50 q^{40} - 6 q^{41} - 12 q^{43} + 26 q^{44} - 2 q^{46} - 10 q^{47} + 5 q^{49} + 2 q^{50} - 21 q^{52} - 16 q^{53} + 18 q^{55} - 3 q^{56} + 13 q^{58} - 22 q^{59} - 18 q^{61} - 15 q^{62} + 25 q^{64} + 26 q^{65} - 2 q^{67} - 12 q^{68} - 8 q^{70} - 4 q^{71} - 2 q^{73} - 38 q^{74} + 10 q^{76} + 4 q^{77} + 30 q^{79} + 10 q^{80} - 7 q^{82} - 8 q^{83} - 12 q^{85} - 8 q^{86} + 4 q^{88} + 20 q^{89} - 6 q^{91} + 12 q^{92} - 25 q^{94} - 8 q^{95} - 12 q^{97} - 2 q^{98}+O(q^{100}) \) 5 * q - 2 * q^2 + 12 * q^4 + 4 * q^5 + 5 * q^7 - 3 * q^8 - 8 * q^10 + 4 * q^11 - 6 * q^13 - 2 * q^14 + 10 * q^16 + 12 * q^17 + 6 * q^19 + 14 * q^22 + 5 * q^23 + 19 * q^25 - q^26 + 12 * q^28 + 4 * q^29 + 30 * q^31 - 8 * q^32 + 6 * q^34 + 4 * q^35 + 4 * q^37 + 40 * q^38 - 50 * q^40 - 6 * q^41 - 12 * q^43 + 26 * q^44 - 2 * q^46 - 10 * q^47 + 5 * q^49 + 2 * q^50 - 21 * q^52 - 16 * q^53 + 18 * q^55 - 3 * q^56 + 13 * q^58 - 22 * q^59 - 18 * q^61 - 15 * q^62 + 25 * q^64 + 26 * q^65 - 2 * q^67 - 12 * q^68 - 8 * q^70 - 4 * q^71 - 2 * q^73 - 38 * q^74 + 10 * q^76 + 4 * q^77 + 30 * q^79 + 10 * q^80 - 7 * q^82 - 8 * q^83 - 12 * q^85 - 8 * q^86 + 4 * q^88 + 20 * q^89 - 6 * q^91 + 12 * q^92 - 25 * q^94 - 8 * q^95 - 12 * q^97 - 2 * q^98

Introduction

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