LMFDB - Embedded newform 1143.2.a.k.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1143,2,Mod(1,1143)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1143, 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("1143.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1143 = 3^{2} \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1143.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:	$9.12690095103$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 26x^{14} + 269x^{12} - 1408x^{10} + 3924x^{8} - 5655x^{6} + 3886x^{4} - 1107x^{2} + 108 $ x^16 - 26x^14 + 269x^12 - 1408x^10 + 3924x^8 - 5655x^6 + 3886x^4 - 1107*x^2 + 108
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$0.487100$ of defining polynomial
Character	$\chi$	$=$	1143.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.487100 q^{2} -1.76273 q^{4} +2.22708 q^{5} -0.937481 q^{7} -1.83283 q^{8} +O(q^{10})$	\(q+0.487100 q^{2} -1.76273 q^{4} +2.22708 q^{5} -0.937481 q^{7} -1.83283 q^{8} +1.08481 q^{10} +2.32765 q^{11} -4.97876 q^{13} -0.456646 q^{14} +2.63270 q^{16} +7.85142 q^{17} +7.13842 q^{19} -3.92575 q^{20} +1.13380 q^{22} -6.70376 q^{23} -0.0401117 q^{25} -2.42515 q^{26} +1.65253 q^{28} +2.92837 q^{29} +3.83622 q^{31} +4.94804 q^{32} +3.82442 q^{34} -2.08785 q^{35} +5.27983 q^{37} +3.47712 q^{38} -4.08185 q^{40} +7.47414 q^{41} +10.3646 q^{43} -4.10303 q^{44} -3.26540 q^{46} -6.09655 q^{47} -6.12113 q^{49} -0.0195384 q^{50} +8.77624 q^{52} +5.27292 q^{53} +5.18387 q^{55} +1.71824 q^{56} +1.42641 q^{58} +0.126428 q^{59} +0.511256 q^{61} +1.86862 q^{62} -2.85521 q^{64} -11.0881 q^{65} +6.08185 q^{67} -13.8400 q^{68} -1.01699 q^{70} -9.50363 q^{71} +15.9981 q^{73} +2.57180 q^{74} -12.5831 q^{76} -2.18213 q^{77} +0.569827 q^{79} +5.86323 q^{80} +3.64065 q^{82} -8.63671 q^{83} +17.4857 q^{85} +5.04859 q^{86} -4.26618 q^{88} -16.6559 q^{89} +4.66750 q^{91} +11.8169 q^{92} -2.96963 q^{94} +15.8978 q^{95} +16.3070 q^{97} -2.98160 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 20 q^{4} + 10 q^{7}+O(q^{10}) $ 16 * q + 20 * q^4 + 10 * q^7	$ 16 q + 20 q^{4} + 10 q^{7} + 14 q^{10} + 20 q^{13} + 28 q^{16} + 12 q^{19} + 18 q^{22} + 52 q^{25} + 42 q^{28} + 18 q^{31} + 10 q^{34} + 16 q^{37} + 6 q^{40} + 26 q^{43} - 24 q^{46} + 54 q^{49} + 52 q^{52} + 20 q^{55} - 14 q^{58} + 36 q^{61} - 4 q^{64} + 26 q^{67} + 36 q^{70} + 60 q^{73} - 20 q^{76} + 12 q^{79} - 20 q^{82} - 12 q^{85} + 8 q^{88} - 24 q^{91} - 26 q^{94} + 108 q^{97}+O(q^{100}) $ 16 * q + 20 * q^4 + 10 * q^7 + 14 * q^10 + 20 * q^13 + 28 * q^16 + 12 * q^19 + 18 * q^22 + 52 * q^25 + 42 * q^28 + 18 * q^31 + 10 * q^34 + 16 * q^37 + 6 * q^40 + 26 * q^43 - 24 * q^46 + 54 * q^49 + 52 * q^52 + 20 * q^55 - 14 * q^58 + 36 * q^61 - 4 * q^64 + 26 * q^67 + 36 * q^70 + 60 * q^73 - 20 * q^76 + 12 * q^79 - 20 * q^82 - 12 * q^85 + 8 * q^88 - 24 * q^91 - 26 * q^94 + 108 * q^97

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 $

$2$	0.487100	0.344431	0.172216	−	0.985059i	$-0.444907\pi$
$2$	0.487100	0.344431	0.172216	+	0.985059i	$0.444907\pi$
$3$	0	0
$3$	0	0
$4$	−1.76273	−0.881367
$5$	2.22708	0.995981	0.497990	−	0.867183i	$-0.334071\pi$
$5$	2.22708	0.995981	0.497990	+	0.867183i	$0.334071\pi$
$6$	0	0
$7$	−0.937481	−0.354334	−0.177167	−	0.984181i	$-0.556693\pi$
$7$	−0.937481	−0.354334	−0.177167	+	0.984181i	$0.556693\pi$
$8$	−1.83283	−0.648002
$9$	0	0
$10$	1.08481	0.343047
$11$	2.32765	0.701814	0.350907	−	0.936410i	$-0.385873\pi$
$11$	2.32765	0.701814	0.350907	+	0.936410i	$0.385873\pi$
$12$	0	0
$13$	−4.97876	−1.38086	−0.690430	−	0.723399i	$-0.742579\pi$
$13$	−4.97876	−1.38086	−0.690430	+	0.723399i	$0.742579\pi$
$14$	−0.456646	−0.122044
$15$	0	0
$16$	2.63270	0.658175
$17$	7.85142	1.90425	0.952124	−	0.305712i	$-0.0988945\pi$
$17$	7.85142	1.90425	0.952124	+	0.305712i	$0.0988945\pi$
$18$	0	0
$19$	7.13842	1.63767	0.818833	−	0.574032i	$-0.194622\pi$
$19$	7.13842	1.63767	0.818833	+	0.574032i	$0.194622\pi$
$20$	−3.92575	−0.877825
$21$	0	0
$22$	1.13380	0.241727
$23$	−6.70376	−1.39783	−0.698915	−	0.715204i	$-0.746334\pi$
$23$	−6.70376	−1.39783	−0.698915	+	0.715204i	$0.746334\pi$
$24$	0	0
$25$	−0.0401117	−0.00802234
$26$	−2.42515	−0.475612
$27$	0	0
$28$	1.65253	0.312299
$29$	2.92837	0.543785	0.271893	−	0.962328i	$-0.412350\pi$
$29$	2.92837	0.543785	0.271893	+	0.962328i	$0.412350\pi$
$30$	0	0
$31$	3.83622	0.689005	0.344502	−	0.938785i	$-0.388048\pi$
$31$	3.83622	0.689005	0.344502	+	0.938785i	$0.388048\pi$
$32$	4.94804	0.874698
$33$	0	0
$34$	3.82442	0.655883
$35$	−2.08785	−0.352910
$36$	0	0
$37$	5.27983	0.867999	0.434000	−	0.900913i	$-0.357102\pi$
$37$	5.27983	0.867999	0.434000	+	0.900913i	$0.357102\pi$
$38$	3.47712	0.564063
$39$	0	0
$40$	−4.08185	−0.645397
$41$	7.47414	1.16726	0.583632	−	0.812018i	$-0.301631\pi$
$41$	7.47414	1.16726	0.583632	+	0.812018i	$0.301631\pi$
$42$	0	0
$43$	10.3646	1.58059	0.790294	−	0.612728i	$-0.209928\pi$
$43$	10.3646	1.58059	0.790294	+	0.612728i	$0.209928\pi$
$44$	−4.10303	−0.618555
$45$	0	0
$46$	−3.26540	−0.481457
$47$	−6.09655	−0.889274	−0.444637	−	0.895711i	$-0.646667\pi$
$47$	−6.09655	−0.889274	−0.444637	+	0.895711i	$0.646667\pi$
$48$	0	0
$49$	−6.12113	−0.874447
$50$	−0.0195384	−0.00276315
$51$	0	0
$52$	8.77624	1.21705
$53$	5.27292	0.724291	0.362145	−	0.932122i	$-0.382044\pi$
$53$	5.27292	0.724291	0.362145	+	0.932122i	$0.382044\pi$
$54$	0	0
$55$	5.18387	0.698993
$56$	1.71824	0.229609
$57$	0	0
$58$	1.42641	0.187297
$59$	0.126428	0.0164595	0.00822974	−	0.999966i	$-0.497380\pi$
$59$	0.126428	0.0164595	0.00822974	+	0.999966i	$0.497380\pi$
$60$	0	0
$61$	0.511256	0.0654596	0.0327298	−	0.999464i	$-0.489580\pi$
$61$	0.511256	0.0654596	0.0327298	+	0.999464i	$0.489580\pi$
$62$	1.86862	0.237315
$63$	0	0
$64$	−2.85521	−0.356901
$65$	−11.0881	−1.37531
$66$	0	0
$67$	6.08185	0.743016	0.371508	−	0.928430i	$-0.378841\pi$
$67$	6.08185	0.743016	0.371508	+	0.928430i	$0.378841\pi$
$68$	−13.8400	−1.67834
$69$	0	0
$70$	−1.01699	−0.121553
$71$	−9.50363	−1.12787	−0.563937	−	0.825818i	$-0.690714\pi$
$71$	−9.50363	−1.12787	−0.563937	+	0.825818i	$0.690714\pi$
$72$	0	0
$73$	15.9981	1.87244	0.936218	−	0.351420i	$-0.114301\pi$
$73$	15.9981	1.87244	0.936218	+	0.351420i	$0.114301\pi$
$74$	2.57180	0.298966
$75$	0	0
$76$	−12.5831	−1.44338
$77$	−2.18213	−0.248677
$78$	0	0
$79$	0.569827	0.0641106	0.0320553	−	0.999486i	$-0.489795\pi$
$79$	0.569827	0.0641106	0.0320553	+	0.999486i	$0.489795\pi$
$80$	5.86323	0.655530
$81$	0	0
$82$	3.64065	0.402043
$83$	−8.63671	−0.948002	−0.474001	−	0.880524i	$-0.657191\pi$
$83$	−8.63671	−0.948002	−0.474001	+	0.880524i	$0.657191\pi$
$84$	0	0
$85$	17.4857	1.89659
$86$	5.04859	0.544404
$87$	0	0
$88$	−4.26618	−0.454776
$89$	−16.6559	−1.76552	−0.882762	−	0.469821i	$-0.844319\pi$
$89$	−16.6559	−1.76552	−0.882762	+	0.469821i	$0.844319\pi$
$90$	0	0
$91$	4.66750	0.489286
$92$	11.8169	1.23200
$93$	0	0
$94$	−2.96963	−0.306294
$95$	15.8978	1.63108
$96$	0	0
$97$	16.3070	1.65573	0.827864	−	0.560929i	$-0.189556\pi$
$97$	16.3070	1.65573	0.827864	+	0.560929i	$0.189556\pi$
$98$	−2.98160	−0.301187
$99$	0	0
$100$	0.0707063	0.00707063
$101$	−1.04891	−0.104371	−0.0521854	−	0.998637i	$-0.516619\pi$
$101$	−1.04891	−0.104371	−0.0521854	+	0.998637i	$0.516619\pi$
$102$	0	0
$103$	−6.77577	−0.667636	−0.333818	−	0.942638i	$-0.608337\pi$
$103$	−6.77577	−0.667636	−0.333818	+	0.942638i	$0.608337\pi$
$104$	9.12521	0.894800
$105$	0	0
$106$	2.56844	0.249468
$107$	19.9785	1.93139	0.965695	−	0.259679i	$-0.0836166\pi$
$107$	19.9785	1.93139	0.965695	+	0.259679i	$0.0836166\pi$
$108$	0	0
$109$	−6.43206	−0.616080	−0.308040	−	0.951373i	$-0.599673\pi$
$109$	−6.43206	−0.616080	−0.308040	+	0.951373i	$0.599673\pi$
$110$	2.52506	0.240755
$111$	0	0
$112$	−2.46810	−0.233214
$113$	10.6703	1.00378	0.501891	−	0.864931i	$-0.332638\pi$
$113$	10.6703	1.00378	0.501891	+	0.864931i	$0.332638\pi$
$114$	0	0
$115$	−14.9298	−1.39221
$116$	−5.16194	−0.479274
$117$	0	0
$118$	0.0615829	0.00566916
$119$	−7.36055	−0.674741
$120$	0	0
$121$	−5.58204	−0.507458
$122$	0.249032	0.0225463
$123$	0	0
$124$	−6.76223	−0.607266
$125$	−11.2247	−1.00397
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−11.2868	−0.997626
$129$	0	0
$130$	−5.40101	−0.473700
$131$	6.04892	0.528497	0.264248	−	0.964455i	$-0.414876\pi$
$131$	6.04892	0.528497	0.264248	+	0.964455i	$0.414876\pi$
$132$	0	0
$133$	−6.69213	−0.580281
$134$	2.96247	0.255918
$135$	0	0
$136$	−14.3903	−1.23396
$137$	−1.35252	−0.115553	−0.0577766	−	0.998330i	$-0.518401\pi$
$137$	−1.35252	−0.115553	−0.0577766	+	0.998330i	$0.518401\pi$
$138$	0	0
$139$	−4.40703	−0.373800	−0.186900	−	0.982379i	$-0.559844\pi$
$139$	−4.40703	−0.373800	−0.186900	+	0.982379i	$0.559844\pi$
$140$	3.68032	0.311043
$141$	0	0
$142$	−4.62921	−0.388475
$143$	−11.5888	−0.969107
$144$	0	0
$145$	6.52172	0.541600
$146$	7.79266	0.644926
$147$	0	0
$148$	−9.30694	−0.765026
$149$	10.9236	0.894893	0.447447	−	0.894311i	$-0.352333\pi$
$149$	10.9236	0.894893	0.447447	+	0.894311i	$0.352333\pi$
$150$	0	0
$151$	−18.2509	−1.48523	−0.742617	−	0.669717i	$-0.766416\pi$
$151$	−18.2509	−1.48523	−0.742617	+	0.669717i	$0.766416\pi$
$152$	−13.0835	−1.06121
$153$	0	0
$154$	−1.06291	−0.0856520
$155$	8.54356	0.686235
$156$	0	0
$157$	9.76043	0.778967	0.389484	−	0.921033i	$-0.372653\pi$
$157$	9.76043	0.778967	0.389484	+	0.921033i	$0.372653\pi$
$158$	0.277563	0.0220817
$159$	0	0
$160$	11.0197	0.871182
$161$	6.28465	0.495300
$162$	0	0
$163$	14.7246	1.15332	0.576658	−	0.816986i	$-0.304356\pi$
$163$	14.7246	1.15332	0.576658	+	0.816986i	$0.304356\pi$
$164$	−13.1749	−1.02879
$165$	0	0
$166$	−4.20694	−0.326522
$167$	−6.39164	−0.494600	−0.247300	−	0.968939i	$-0.579543\pi$
$167$	−6.39164	−0.494600	−0.247300	+	0.968939i	$0.579543\pi$
$168$	0	0
$169$	11.7881	0.906776
$170$	8.51729	0.653247
$171$	0	0
$172$	−18.2700	−1.39308
$173$	−23.3706	−1.77683	−0.888417	−	0.459038i	$-0.848194\pi$
$173$	−23.3706	−1.77683	−0.888417	+	0.459038i	$0.848194\pi$
$174$	0	0
$175$	0.0376040	0.00284259
$176$	6.12801	0.461916
$177$	0	0
$178$	−8.11309	−0.608102
$179$	−20.1683	−1.50745	−0.753726	−	0.657188i	$-0.771746\pi$
$179$	−20.1683	−1.50745	−0.753726	+	0.657188i	$0.771746\pi$
$180$	0	0
$181$	−14.5671	−1.08276	−0.541382	−	0.840777i	$-0.682099\pi$
$181$	−14.5671	−1.08276	−0.541382	+	0.840777i	$0.682099\pi$
$182$	2.27353	0.168526
$183$	0	0
$184$	12.2868	0.905797
$185$	11.7586	0.864510
$186$	0	0
$187$	18.2754	1.33643
$188$	10.7466	0.783776
$189$	0	0
$190$	7.74383	0.561796
$191$	−3.25859	−0.235783	−0.117892	−	0.993026i	$-0.537614\pi$
$191$	−3.25859	−0.235783	−0.117892	+	0.993026i	$0.537614\pi$
$192$	0	0
$193$	−17.1571	−1.23500	−0.617498	−	0.786573i	$-0.711853\pi$
$193$	−17.1571	−1.23500	−0.617498	+	0.786573i	$0.711853\pi$
$194$	7.94314	0.570284
$195$	0	0
$196$	10.7899	0.770709
$197$	9.22140	0.656997	0.328499	−	0.944504i	$-0.393457\pi$
$197$	9.22140	0.656997	0.328499	+	0.944504i	$0.393457\pi$
$198$	0	0
$199$	−12.3260	−0.873768	−0.436884	−	0.899518i	$-0.643918\pi$
$199$	−12.3260	−0.873768	−0.436884	+	0.899518i	$0.643918\pi$
$200$	0.0735178	0.00519849
$201$	0	0
$202$	−0.510926	−0.0359486
$203$	−2.74529	−0.192682
$204$	0	0
$205$	16.6455	1.16257
$206$	−3.30047	−0.229955
$207$	0	0
$208$	−13.1076	−0.908848
$209$	16.6158	1.14934
$210$	0	0
$211$	−21.3300	−1.46842	−0.734210	−	0.678923i	$-0.762447\pi$
$211$	−21.3300	−1.46842	−0.734210	+	0.678923i	$0.762447\pi$
$212$	−9.29475	−0.638366
$213$	0	0
$214$	9.73150	0.665231
$215$	23.0828	1.57423
$216$	0	0
$217$	−3.59638	−0.244138
$218$	−3.13305	−0.212197
$219$	0	0
$220$	−9.13778	−0.616069
$221$	−39.0903	−2.62950
$222$	0	0
$223$	−7.60732	−0.509424	−0.254712	−	0.967017i	$-0.581981\pi$
$223$	−7.60732	−0.509424	−0.254712	+	0.967017i	$0.581981\pi$
$224$	−4.63869	−0.309936
$225$	0	0
$226$	5.19752	0.345734
$227$	3.96108	0.262906	0.131453	−	0.991322i	$-0.458036\pi$
$227$	3.96108	0.262906	0.131453	+	0.991322i	$0.458036\pi$
$228$	0	0
$229$	2.32590	0.153700	0.0768498	−	0.997043i	$-0.475514\pi$
$229$	2.32590	0.153700	0.0768498	+	0.997043i	$0.475514\pi$
$230$	−7.27231	−0.479522
$231$	0	0
$232$	−5.36720	−0.352374
$233$	9.76241	0.639556	0.319778	−	0.947492i	$-0.396392\pi$
$233$	9.76241	0.639556	0.319778	+	0.947492i	$0.396392\pi$
$234$	0	0
$235$	−13.5775	−0.885699
$236$	−0.222858	−0.0145069
$237$	0	0
$238$	−3.58532	−0.232402
$239$	−30.2266	−1.95520	−0.977598	−	0.210479i	$-0.932498\pi$
$239$	−30.2266	−1.95520	−0.977598	+	0.210479i	$0.932498\pi$
$240$	0	0
$241$	5.61085	0.361426	0.180713	−	0.983536i	$-0.442159\pi$
$241$	5.61085	0.361426	0.180713	+	0.983536i	$0.442159\pi$
$242$	−2.71901	−0.174784
$243$	0	0
$244$	−0.901208	−0.0576939
$245$	−13.6323	−0.870933
$246$	0	0
$247$	−35.5405	−2.26139
$248$	−7.03111	−0.446476
$249$	0	0
$250$	−5.46756	−0.345799
$251$	−17.4505	−1.10147	−0.550734	−	0.834681i	$-0.685652\pi$
$251$	−17.4505	−1.10147	−0.550734	+	0.834681i	$0.685652\pi$
$252$	0	0
$253$	−15.6040	−0.981017
$254$	0.487100	0.0305633
$255$	0	0
$256$	0.212605	0.0132878
$257$	−26.1421	−1.63070	−0.815348	−	0.578971i	$-0.803455\pi$
$257$	−26.1421	−1.63070	−0.815348	+	0.578971i	$0.803455\pi$
$258$	0	0
$259$	−4.94974	−0.307562
$260$	19.5454	1.21215
$261$	0	0
$262$	2.94643	0.182031
$263$	15.4529	0.952868	0.476434	−	0.879210i	$-0.341929\pi$
$263$	15.4529	0.952868	0.476434	+	0.879210i	$0.341929\pi$
$264$	0	0
$265$	11.7432	0.721380
$266$	−3.25973	−0.199867
$267$	0	0
$268$	−10.7207	−0.654870
$269$	4.38996	0.267661	0.133830	−	0.991004i	$-0.457272\pi$
$269$	4.38996	0.267661	0.133830	+	0.991004i	$0.457272\pi$
$270$	0	0
$271$	0.697723	0.0423837	0.0211918	−	0.999775i	$-0.493254\pi$
$271$	0.697723	0.0423837	0.0211918	+	0.999775i	$0.493254\pi$
$272$	20.6704	1.25333
$273$	0	0
$274$	−0.658810	−0.0398002
$275$	−0.0933661	−0.00563019
$276$	0	0
$277$	−7.52784	−0.452304	−0.226152	−	0.974092i	$-0.572615\pi$
$277$	−7.52784	−0.452304	−0.226152	+	0.974092i	$0.572615\pi$
$278$	−2.14666	−0.128748
$279$	0	0
$280$	3.82666	0.228686
$281$	12.6372	0.753871	0.376935	−	0.926240i	$-0.376978\pi$
$281$	12.6372	0.753871	0.376935	+	0.926240i	$0.376978\pi$
$282$	0	0
$283$	9.90031	0.588512	0.294256	−	0.955727i	$-0.404928\pi$
$283$	9.90031	0.588512	0.294256	+	0.955727i	$0.404928\pi$
$284$	16.7524	0.994071
$285$	0	0
$286$	−5.64491	−0.333791
$287$	−7.00686	−0.413602
$288$	0	0
$289$	44.6447	2.62616
$290$	3.17673	0.186544
$291$	0	0
$292$	−28.2004	−1.65030
$293$	33.0008	1.92793	0.963963	−	0.266036i	$-0.0857141\pi$
$293$	33.0008	1.92793	0.963963	+	0.266036i	$0.0857141\pi$
$294$	0	0
$295$	0.281565	0.0163933
$296$	−9.67701	−0.562465
$297$	0	0
$298$	5.32086	0.308229
$299$	33.3764	1.93021
$300$	0	0
$301$	−9.71662	−0.560056
$302$	−8.88998	−0.511561
$303$	0	0
$304$	18.7933	1.07787
$305$	1.13861	0.0651965
$306$	0	0
$307$	−6.09390	−0.347797	−0.173899	−	0.984764i	$-0.555637\pi$
$307$	−6.09390	−0.347797	−0.173899	+	0.984764i	$0.555637\pi$
$308$	3.84651	0.219175
$309$	0	0
$310$	4.16156	0.236361
$311$	−27.2521	−1.54533	−0.772663	−	0.634816i	$-0.781076\pi$
$311$	−27.2521	−1.54533	−0.772663	+	0.634816i	$0.781076\pi$
$312$	0	0
$313$	13.1115	0.741104	0.370552	−	0.928812i	$-0.379169\pi$
$313$	13.1115	0.741104	0.370552	+	0.928812i	$0.379169\pi$
$314$	4.75430	0.268301
$315$	0	0
$316$	−1.00445	−0.0565050
$317$	13.3016	0.747091	0.373546	−	0.927612i	$-0.378142\pi$
$317$	13.3016	0.747091	0.373546	+	0.927612i	$0.378142\pi$
$318$	0	0
$319$	6.81623	0.381636
$320$	−6.35879	−0.355467
$321$	0	0
$322$	3.06125	0.170597
$323$	56.0467	3.11852
$324$	0	0
$325$	0.199707	0.0110777
$326$	7.17232	0.397238
$327$	0	0
$328$	−13.6988	−0.756390
$329$	5.71540	0.315100
$330$	0	0
$331$	−28.3663	−1.55916	−0.779578	−	0.626305i	$-0.784566\pi$
$331$	−28.3663	−1.55916	−0.779578	+	0.626305i	$0.784566\pi$
$332$	15.2242	0.835537
$333$	0	0
$334$	−3.11337	−0.170356
$335$	13.5448	0.740030
$336$	0	0
$337$	22.0511	1.20120	0.600600	−	0.799550i	$-0.294928\pi$
$337$	22.0511	1.20120	0.600600	+	0.799550i	$0.294928\pi$
$338$	5.74197	0.312322
$339$	0	0
$340$	−30.8227	−1.67160
$341$	8.92937	0.483553
$342$	0	0
$343$	12.3008	0.664181
$344$	−18.9965	−1.02422
$345$	0	0
$346$	−11.3838	−0.611997
$347$	−33.5316	−1.80007	−0.900036	−	0.435816i	$-0.856460\pi$
$347$	−33.5316	−1.80007	−0.900036	+	0.435816i	$0.856460\pi$
$348$	0	0
$349$	30.9899	1.65885	0.829427	−	0.558615i	$-0.188667\pi$
$349$	30.9899	1.65885	0.829427	+	0.558615i	$0.188667\pi$
$350$	0.0183169	0.000979078 0
$351$	0	0
$352$	11.5173	0.613875
$353$	−18.7236	−0.996556	−0.498278	−	0.867017i	$-0.666034\pi$
$353$	−18.7236	−0.996556	−0.498278	+	0.867017i	$0.666034\pi$
$354$	0	0
$355$	−21.1654	−1.12334
$356$	29.3600	1.55607
$357$	0	0
$358$	−9.82399	−0.519214
$359$	−4.48507	−0.236713	−0.118356	−	0.992971i	$-0.537763\pi$
$359$	−4.48507	−0.236713	−0.118356	+	0.992971i	$0.537763\pi$
$360$	0	0
$361$	31.9570	1.68195
$362$	−7.09562	−0.372938
$363$	0	0
$364$	−8.22755	−0.431241
$365$	35.6291	1.86491
$366$	0	0
$367$	28.0326	1.46329	0.731645	−	0.681686i	$-0.238753\pi$
$367$	28.0326	1.46329	0.731645	+	0.681686i	$0.238753\pi$
$368$	−17.6490	−0.920017
$369$	0	0
$370$	5.72761	0.297764
$371$	−4.94326	−0.256641
$372$	0	0
$373$	−4.97074	−0.257375	−0.128688	−	0.991685i	$-0.541076\pi$
$373$	−4.97074	−0.257375	−0.128688	+	0.991685i	$0.541076\pi$
$374$	8.90192	0.460307
$375$	0	0
$376$	11.1739	0.576251
$377$	−14.5797	−0.750892
$378$	0	0
$379$	26.3530	1.35366	0.676831	−	0.736138i	$-0.263353\pi$
$379$	26.3530	1.35366	0.676831	+	0.736138i	$0.263353\pi$
$380$	−28.0237	−1.43758
$381$	0	0
$382$	−1.58726	−0.0812112
$383$	−15.8954	−0.812215	−0.406108	−	0.913825i	$-0.633114\pi$
$383$	−15.8954	−0.812215	−0.406108	+	0.913825i	$0.633114\pi$
$384$	0	0
$385$	−4.85978	−0.247677
$386$	−8.35722	−0.425371
$387$	0	0
$388$	−28.7449	−1.45930
$389$	23.9149	1.21253	0.606267	−	0.795261i	$-0.292666\pi$
$389$	23.9149	1.21253	0.606267	+	0.795261i	$0.292666\pi$
$390$	0	0
$391$	−52.6340	−2.66182
$392$	11.2190	0.566643
$393$	0	0
$394$	4.49174	0.226290
$395$	1.26905	0.0638529
$396$	0	0
$397$	−13.3099	−0.668005	−0.334002	−	0.942572i	$-0.608399\pi$
$397$	−13.3099	−0.668005	−0.334002	+	0.942572i	$0.608399\pi$
$398$	−6.00400	−0.300953
$399$	0	0
$400$	−0.105602	−0.00528010
$401$	5.53456	0.276383	0.138191	−	0.990406i	$-0.455871\pi$
$401$	5.53456	0.276383	0.138191	+	0.990406i	$0.455871\pi$
$402$	0	0
$403$	−19.0996	−0.951419
$404$	1.84896	0.0919890
$405$	0	0
$406$	−1.33723	−0.0663656
$407$	12.2896	0.609174
$408$	0	0
$409$	10.9136	0.539641	0.269820	−	0.962911i	$-0.413036\pi$
$409$	10.9136	0.539641	0.269820	+	0.962911i	$0.413036\pi$
$410$	8.10802	0.400427
$411$	0	0
$412$	11.9439	0.588433
$413$	−0.118524	−0.00583216
$414$	0	0
$415$	−19.2346	−0.944191
$416$	−24.6351	−1.20784
$417$	0	0
$418$	8.09353	0.395867
$419$	−25.2775	−1.23489	−0.617443	−	0.786616i	$-0.711831\pi$
$419$	−25.2775	−1.23489	−0.617443	+	0.786616i	$0.711831\pi$
$420$	0	0
$421$	−16.2978	−0.794307	−0.397154	−	0.917752i	$-0.630002\pi$
$421$	−16.2978	−0.794307	−0.397154	+	0.917752i	$0.630002\pi$
$422$	−10.3898	−0.505770
$423$	0	0
$424$	−9.66434	−0.469342
$425$	−0.314934	−0.0152765
$426$	0	0
$427$	−0.479292	−0.0231946
$428$	−35.2167	−1.70226
$429$	0	0
$430$	11.2436	0.542216
$431$	20.8220	1.00296	0.501480	−	0.865169i	$-0.332789\pi$
$431$	20.8220	1.00296	0.501480	+	0.865169i	$0.332789\pi$
$432$	0	0
$433$	−6.81958	−0.327728	−0.163864	−	0.986483i	$-0.552396\pi$
$433$	−6.81958	−0.327728	−0.163864	+	0.986483i	$0.552396\pi$
$434$	−1.75179	−0.0840888
$435$	0	0
$436$	11.3380	0.542992
$437$	−47.8543	−2.28918
$438$	0	0
$439$	2.81447	0.134328	0.0671638	−	0.997742i	$-0.478605\pi$
$439$	2.81447	0.134328	0.0671638	+	0.997742i	$0.478605\pi$
$440$	−9.50113	−0.452949
$441$	0	0
$442$	−19.0409	−0.905683
$443$	−10.3219	−0.490408	−0.245204	−	0.969471i	$-0.578855\pi$
$443$	−10.3219	−0.490408	−0.245204	+	0.969471i	$0.578855\pi$
$444$	0	0
$445$	−37.0941	−1.75843
$446$	−3.70552	−0.175462
$447$	0	0
$448$	2.67671	0.126462
$449$	−3.70225	−0.174720	−0.0873599	−	0.996177i	$-0.527843\pi$
$449$	−3.70225	−0.174720	−0.0873599	+	0.996177i	$0.527843\pi$
$450$	0	0
$451$	17.3972	0.819202
$452$	−18.8090	−0.884699
$453$	0	0
$454$	1.92944	0.0905531
$455$	10.3949	0.487320
$456$	0	0
$457$	14.6688	0.686177	0.343088	−	0.939303i	$-0.388527\pi$
$457$	14.6688	0.686177	0.343088	+	0.939303i	$0.388527\pi$
$458$	1.13294	0.0529389
$459$	0	0
$460$	26.3173	1.22705
$461$	5.39481	0.251261	0.125631	−	0.992077i	$-0.459905\pi$
$461$	5.39481	0.251261	0.125631	+	0.992077i	$0.459905\pi$
$462$	0	0
$463$	40.9584	1.90350	0.951750	−	0.306876i	$-0.0992836\pi$
$463$	40.9584	1.90350	0.951750	+	0.306876i	$0.0992836\pi$
$464$	7.70953	0.357906
$465$	0	0
$466$	4.75526	0.220283
$467$	1.62697	0.0752871	0.0376435	−	0.999291i	$-0.488015\pi$
$467$	1.62697	0.0752871	0.0376435	+	0.999291i	$0.488015\pi$
$468$	0	0
$469$	−5.70162	−0.263276
$470$	−6.61360	−0.305063
$471$	0	0
$472$	−0.231720	−0.0106658
$473$	24.1252	1.10928
$474$	0	0
$475$	−0.286334	−0.0131379
$476$	12.9747	0.594694
$477$	0	0
$478$	−14.7234	−0.673431
$479$	−22.9014	−1.04639	−0.523196	−	0.852212i	$-0.675261\pi$
$479$	−22.9014	−1.04639	−0.523196	+	0.852212i	$0.675261\pi$
$480$	0	0
$481$	−26.2870	−1.19859
$482$	2.73304	0.124487
$483$	0	0
$484$	9.83964	0.447257
$485$	36.3171	1.64907
$486$	0	0
$487$	4.58762	0.207885	0.103942	−	0.994583i	$-0.466854\pi$
$487$	4.58762	0.207885	0.103942	+	0.994583i	$0.466854\pi$
$488$	−0.937043	−0.0424179
$489$	0	0
$490$	−6.64026	−0.299976
$491$	−4.77671	−0.215570	−0.107785	−	0.994174i	$-0.534376\pi$
$491$	−4.77671	−0.215570	−0.107785	+	0.994174i	$0.534376\pi$
$492$	0	0
$493$	22.9919	1.03550
$494$	−17.3118	−0.778893
$495$	0	0
$496$	10.0996	0.453486
$497$	8.90947	0.399644
$498$	0	0
$499$	11.6252	0.520417	0.260208	−	0.965552i	$-0.416209\pi$
$499$	11.6252	0.520417	0.260208	+	0.965552i	$0.416209\pi$
$500$	19.7862	0.884867
$501$	0	0
$502$	−8.50015	−0.379380
$503$	2.79419	0.124587	0.0622934	−	0.998058i	$-0.480159\pi$
$503$	2.79419	0.124587	0.0622934	+	0.998058i	$0.480159\pi$
$504$	0	0
$505$	−2.33602	−0.103951
$506$	−7.60071	−0.337893
$507$	0	0
$508$	−1.76273	−0.0782087
$509$	29.4921	1.30721	0.653607	−	0.756835i	$-0.273255\pi$
$509$	29.4921	1.30721	0.653607	+	0.756835i	$0.273255\pi$
$510$	0	0
$511$	−14.9979	−0.663468
$512$	22.6773	1.00220
$513$	0	0
$514$	−12.7338	−0.561663
$515$	−15.0902	−0.664953
$516$	0	0
$517$	−14.1907	−0.624104
$518$	−2.41102	−0.105934
$519$	0	0
$520$	20.3226	0.891204
$521$	0.798756	0.0349941	0.0174971	−	0.999847i	$-0.494430\pi$
$521$	0.798756	0.0349941	0.0174971	+	0.999847i	$0.494430\pi$
$522$	0	0
$523$	19.7926	0.865472	0.432736	−	0.901521i	$-0.357548\pi$
$523$	19.7926	0.865472	0.432736	+	0.901521i	$0.357548\pi$
$524$	−10.6626	−0.465800
$525$	0	0
$526$	7.52711	0.328198
$527$	30.1197	1.31204
$528$	0	0
$529$	21.9404	0.953931
$530$	5.72011	0.248466
$531$	0	0
$532$	11.7964	0.511441
$533$	−37.2120	−1.61183
$534$	0	0
$535$	44.4936	1.92363
$536$	−11.1470	−0.481476
$537$	0	0
$538$	2.13835	0.0921907
$539$	−14.2479	−0.613699
$540$	0	0
$541$	−19.2527	−0.827736	−0.413868	−	0.910337i	$-0.635823\pi$
$541$	−19.2527	−0.827736	−0.413868	+	0.910337i	$0.635823\pi$
$542$	0.339861	0.0145983
$543$	0	0
$544$	38.8491	1.66564
$545$	−14.3247	−0.613603
$546$	0	0
$547$	6.65956	0.284742	0.142371	−	0.989813i	$-0.454527\pi$
$547$	6.65956	0.284742	0.142371	+	0.989813i	$0.454527\pi$
$548$	2.38413	0.101845
$549$	0	0
$550$	−0.0454786	−0.00193921
$551$	20.9039	0.890538
$552$	0	0
$553$	−0.534202	−0.0227166
$554$	−3.66681	−0.155788
$555$	0	0
$556$	7.76842	0.329455
$557$	12.2269	0.518070	0.259035	−	0.965868i	$-0.416596\pi$
$557$	12.2269	0.518070	0.259035	+	0.965868i	$0.416596\pi$
$558$	0	0
$559$	−51.6029	−2.18257
$560$	−5.49667	−0.232277
$561$	0	0
$562$	6.15556	0.259657
$563$	−27.8929	−1.17555	−0.587773	−	0.809026i	$-0.699995\pi$
$563$	−27.8929	−1.17555	−0.587773	+	0.809026i	$0.699995\pi$
$564$	0	0
$565$	23.7637	0.999747
$566$	4.82244	0.202702
$567$	0	0
$568$	17.4185	0.730864
$569$	−43.5239	−1.82461	−0.912307	−	0.409506i	$-0.865701\pi$
$569$	−43.5239	−1.82461	−0.912307	+	0.409506i	$0.865701\pi$
$570$	0	0
$571$	19.5772	0.819281	0.409640	−	0.912247i	$-0.365654\pi$
$571$	19.5772	0.819281	0.409640	+	0.912247i	$0.365654\pi$
$572$	20.4280	0.854139
$573$	0	0
$574$	−3.41304	−0.142458
$575$	0.268899	0.0112139
$576$	0	0
$577$	15.5933	0.649159	0.324579	−	0.945858i	$-0.394777\pi$
$577$	15.5933	0.649159	0.324579	+	0.945858i	$0.394777\pi$
$578$	21.7464	0.904532
$579$	0	0
$580$	−11.4961	−0.477348
$581$	8.09675	0.335910
$582$	0	0
$583$	12.2735	0.508317
$584$	−29.3217	−1.21334
$585$	0	0
$586$	16.0747	0.664038
$587$	−2.30240	−0.0950303	−0.0475152	−	0.998871i	$-0.515130\pi$
$587$	−2.30240	−0.0950303	−0.0475152	+	0.998871i	$0.515130\pi$
$588$	0	0
$589$	27.3845	1.12836
$590$	0.137150	0.00564638
$591$	0	0
$592$	13.9002	0.571295
$593$	−37.9151	−1.55699	−0.778494	−	0.627652i	$-0.784016\pi$
$593$	−37.9151	−1.55699	−0.778494	+	0.627652i	$0.784016\pi$
$594$	0	0
$595$	−16.3925	−0.672029
$596$	−19.2553	−0.788729
$597$	0	0
$598$	16.2577	0.664825
$599$	−4.03478	−0.164857	−0.0824283	−	0.996597i	$-0.526268\pi$
$599$	−4.03478	−0.164857	−0.0824283	+	0.996597i	$0.526268\pi$
$600$	0	0
$601$	9.30991	0.379759	0.189880	−	0.981807i	$-0.439190\pi$
$601$	9.30991	0.379759	0.189880	+	0.981807i	$0.439190\pi$
$602$	−4.73296	−0.192901
$603$	0	0
$604$	32.1714	1.30904
$605$	−12.4316	−0.505418
$606$	0	0
$607$	20.8372	0.845757	0.422878	−	0.906186i	$-0.361020\pi$
$607$	20.8372	0.845757	0.422878	+	0.906186i	$0.361020\pi$
$608$	35.3212	1.43246
$609$	0	0
$610$	0.554615	0.0224557
$611$	30.3533	1.22796
$612$	0	0
$613$	39.6441	1.60121	0.800605	−	0.599192i	$-0.204511\pi$
$613$	39.6441	1.60121	0.800605	+	0.599192i	$0.204511\pi$
$614$	−2.96834	−0.119792
$615$	0	0
$616$	3.99946	0.161143
$617$	−17.0482	−0.686333	−0.343167	−	0.939275i	$-0.611500\pi$
$617$	−17.0482	−0.686333	−0.343167	+	0.939275i	$0.611500\pi$
$618$	0	0
$619$	17.6663	0.710068	0.355034	−	0.934853i	$-0.384469\pi$
$619$	17.6663	0.710068	0.355034	+	0.934853i	$0.384469\pi$
$620$	−15.0600	−0.604825
$621$	0	0
$622$	−13.2745	−0.532259
$623$	15.6146	0.625586
$624$	0	0
$625$	−24.7978	−0.991913
$626$	6.38659	0.255259
$627$	0	0
$628$	−17.2051	−0.686556
$629$	41.4542	1.65289
$630$	0	0
$631$	−18.8347	−0.749796	−0.374898	−	0.927066i	$-0.622322\pi$
$631$	−18.8347	−0.749796	−0.374898	+	0.927066i	$0.622322\pi$
$632$	−1.04439	−0.0415438
$633$	0	0
$634$	6.47919	0.257322
$635$	2.22708	0.0883790
$636$	0	0
$637$	30.4757	1.20749
$638$	3.32018	0.131447
$639$	0	0
$640$	−25.1367	−0.993616
$641$	27.8887	1.10154	0.550768	−	0.834659i	$-0.314335\pi$
$641$	27.8887	1.10154	0.550768	+	0.834659i	$0.314335\pi$
$642$	0	0
$643$	−23.1067	−0.911241	−0.455621	−	0.890174i	$-0.650583\pi$
$643$	−23.1067	−0.911241	−0.455621	+	0.890174i	$0.650583\pi$
$644$	−11.0782	−0.436541
$645$	0	0
$646$	27.3003	1.07412
$647$	−41.0879	−1.61533	−0.807665	−	0.589641i	$-0.799269\pi$
$647$	−41.0879	−1.61533	−0.807665	+	0.589641i	$0.799269\pi$
$648$	0	0
$649$	0.294280	0.0115515
$650$	0.0972771	0.00381552
$651$	0	0
$652$	−25.9555	−1.01649
$653$	13.7089	0.536472	0.268236	−	0.963353i	$-0.413559\pi$
$653$	13.7089	0.536472	0.268236	+	0.963353i	$0.413559\pi$
$654$	0	0
$655$	13.4714	0.526373
$656$	19.6772	0.768264
$657$	0	0
$658$	2.78397	0.108530
$659$	−3.50221	−0.136427	−0.0682134	−	0.997671i	$-0.521730\pi$
$659$	−3.50221	−0.136427	−0.0682134	+	0.997671i	$0.521730\pi$
$660$	0	0
$661$	−38.1963	−1.48566	−0.742831	−	0.669478i	$-0.766518\pi$
$661$	−38.1963	−1.48566	−0.742831	+	0.669478i	$0.766518\pi$
$662$	−13.8172	−0.537022
$663$	0	0
$664$	15.8296	0.614307
$665$	−14.9039	−0.577949
$666$	0	0
$667$	−19.6311	−0.760120
$668$	11.2668	0.435924
$669$	0	0
$670$	6.59765	0.254890
$671$	1.19003	0.0459404
$672$	0	0
$673$	−12.1185	−0.467136	−0.233568	−	0.972341i	$-0.575040\pi$
$673$	−12.1185	−0.467136	−0.233568	+	0.972341i	$0.575040\pi$
$674$	10.7411	0.413731
$675$	0	0
$676$	−20.7793	−0.799203
$677$	21.7329	0.835262	0.417631	−	0.908617i	$-0.362861\pi$
$677$	21.7329	0.835262	0.417631	+	0.908617i	$0.362861\pi$
$678$	0	0
$679$	−15.2875	−0.586681
$680$	−32.0483	−1.22900
$681$	0	0
$682$	4.34949	0.166551
$683$	−4.83485	−0.185001	−0.0925003	−	0.995713i	$-0.529486\pi$
$683$	−4.83485	−0.185001	−0.0925003	+	0.995713i	$0.529486\pi$
$684$	0	0
$685$	−3.01216	−0.115089
$686$	5.99172	0.228765
$687$	0	0
$688$	27.2869	1.04030
$689$	−26.2526	−1.00014
$690$	0	0
$691$	36.3220	1.38175	0.690877	−	0.722973i	$-0.257225\pi$
$691$	36.3220	1.38175	0.690877	+	0.722973i	$0.257225\pi$
$692$	41.1961	1.56604
$693$	0	0
$694$	−16.3332	−0.620001
$695$	−9.81481	−0.372297
$696$	0	0
$697$	58.6826	2.22276
$698$	15.0952	0.571361
$699$	0	0
$700$	−0.0662858	−0.00250537
$701$	32.3324	1.22118	0.610590	−	0.791947i	$-0.290933\pi$
$701$	32.3324	1.22118	0.610590	+	0.791947i	$0.290933\pi$
$702$	0	0
$703$	37.6897	1.42149
$704$	−6.64594	−0.250478
$705$	0	0
$706$	−9.12025	−0.343245
$707$	0.983337	0.0369822
$708$	0	0
$709$	−15.9096	−0.597496	−0.298748	−	0.954332i	$-0.596569\pi$
$709$	−15.9096	−0.597496	−0.298748	+	0.954332i	$0.596569\pi$
$710$	−10.3096	−0.386914
$711$	0	0
$712$	30.5274	1.14406
$713$	−25.7171	−0.963112
$714$	0	0
$715$	−25.8093	−0.965212
$716$	35.5514	1.32862
$717$	0	0
$718$	−2.18467	−0.0815313
$719$	9.84260	0.367067	0.183533	−	0.983013i	$-0.441246\pi$
$719$	9.84260	0.367067	0.183533	+	0.983013i	$0.441246\pi$
$720$	0	0
$721$	6.35215	0.236566
$722$	15.5662	0.579316
$723$	0	0
$724$	25.6779	0.954312
$725$	−0.117462	−0.00436243
$726$	0	0
$727$	−1.42442	−0.0528290	−0.0264145	−	0.999651i	$-0.508409\pi$
$727$	−1.42442	−0.0528290	−0.0264145	+	0.999651i	$0.508409\pi$
$728$	−8.55471	−0.317058
$729$	0	0
$730$	17.3549	0.642333
$731$	81.3768	3.00983
$732$	0	0
$733$	−21.4810	−0.793421	−0.396710	−	0.917944i	$-0.629848\pi$
$733$	−21.4810	−0.793421	−0.396710	+	0.917944i	$0.629848\pi$
$734$	13.6547	0.504003
$735$	0	0
$736$	−33.1705	−1.22268
$737$	14.1564	0.521459
$738$	0	0
$739$	11.9689	0.440282	0.220141	−	0.975468i	$-0.429348\pi$
$739$	11.9689	0.440282	0.220141	+	0.975468i	$0.429348\pi$
$740$	−20.7273	−0.761951
$741$	0	0
$742$	−2.40786	−0.0883953
$743$	−1.56373	−0.0573676	−0.0286838	−	0.999589i	$-0.509132\pi$
$743$	−1.56373	−0.0573676	−0.0286838	+	0.999589i	$0.509132\pi$
$744$	0	0
$745$	24.3277	0.891296
$746$	−2.42125	−0.0886481
$747$	0	0
$748$	−32.2146	−1.17788
$749$	−18.7294	−0.684358
$750$	0	0
$751$	−2.37258	−0.0865768	−0.0432884	−	0.999063i	$-0.513783\pi$
$751$	−2.37258	−0.0865768	−0.0432884	+	0.999063i	$0.513783\pi$
$752$	−16.0504	−0.585298
$753$	0	0
$754$	−7.10175	−0.258631
$755$	−40.6461	−1.47926
$756$	0	0
$757$	6.93476	0.252048	0.126024	−	0.992027i	$-0.459778\pi$
$757$	6.93476	0.252048	0.126024	+	0.992027i	$0.459778\pi$
$758$	12.8365	0.466244
$759$	0	0
$760$	−29.1380	−1.05694
$761$	10.1807	0.369050	0.184525	−	0.982828i	$-0.440925\pi$
$761$	10.1807	0.369050	0.184525	+	0.982828i	$0.440925\pi$
$762$	0	0
$763$	6.02993	0.218298
$764$	5.74403	0.207812
$765$	0	0
$766$	−7.74262	−0.279752
$767$	−0.629454	−0.0227283
$768$	0	0
$769$	−25.5509	−0.921390	−0.460695	−	0.887558i	$-0.652400\pi$
$769$	−25.5509	−0.921390	−0.460695	+	0.887558i	$0.652400\pi$
$770$	−2.36720	−0.0853078
$771$	0	0
$772$	30.2434	1.08848
$773$	−46.2415	−1.66319	−0.831595	−	0.555382i	$-0.812572\pi$
$773$	−46.2415	−1.66319	−0.831595	+	0.555382i	$0.812572\pi$
$774$	0	0
$775$	−0.153877	−0.00552743
$776$	−29.8879	−1.07291
$777$	0	0
$778$	11.6489	0.417635
$779$	53.3535	1.91159
$780$	0	0
$781$	−22.1211	−0.791557
$782$	−25.6380	−0.916813
$783$	0	0
$784$	−16.1151	−0.575539
$785$	21.7373	0.775837
$786$	0	0
$787$	−29.2777	−1.04364	−0.521818	−	0.853057i	$-0.674746\pi$
$787$	−29.2777	−1.04364	−0.521818	+	0.853057i	$0.674746\pi$
$788$	−16.2549	−0.579056
$789$	0	0
$790$	0.618154	0.0219929
$791$	−10.0032	−0.355674
$792$	0	0
$793$	−2.54542	−0.0903906
$794$	−6.48324	−0.230082
$795$	0	0
$796$	21.7275	0.770110
$797$	5.94965	0.210748	0.105374	−	0.994433i	$-0.466396\pi$
$797$	5.94965	0.210748	0.105374	+	0.994433i	$0.466396\pi$
$798$	0	0
$799$	−47.8666	−1.69340
$800$	−0.198474	−0.00701713
$801$	0	0
$802$	2.69588	0.0951949
$803$	37.2380	1.31410
$804$	0	0
$805$	13.9964	0.493309
$806$	−9.30341	−0.327699
$807$	0	0
$808$	1.92248	0.0676325
$809$	−6.26026	−0.220099	−0.110050	−	0.993926i	$-0.535101\pi$
$809$	−6.26026	−0.220099	−0.110050	+	0.993926i	$0.535101\pi$
$810$	0	0
$811$	−22.1026	−0.776128	−0.388064	−	0.921632i	$-0.626856\pi$
$811$	−22.1026	−0.776128	−0.388064	+	0.921632i	$0.626856\pi$
$812$	4.83922	0.169823
$813$	0	0
$814$	5.98626	0.209818
$815$	32.7928	1.14868
$816$	0	0
$817$	73.9869	2.58847
$818$	5.31599	0.185869
$819$	0	0
$820$	−29.3416	−1.02465
$821$	−3.73210	−0.130251	−0.0651256	−	0.997877i	$-0.520745\pi$
$821$	−3.73210	−0.130251	−0.0651256	+	0.997877i	$0.520745\pi$
$822$	0	0
$823$	12.2510	0.427044	0.213522	−	0.976938i	$-0.431507\pi$
$823$	12.2510	0.427044	0.213522	+	0.976938i	$0.431507\pi$
$824$	12.4188	0.432629
$825$	0	0
$826$	−0.0577328	−0.00200878
$827$	−0.214105	−0.00744517	−0.00372259	−	0.999993i	$-0.501185\pi$
$827$	−0.214105	−0.00744517	−0.00372259	+	0.999993i	$0.501185\pi$
$828$	0	0
$829$	34.6831	1.20459	0.602297	−	0.798272i	$-0.294252\pi$
$829$	34.6831	1.20459	0.602297	+	0.798272i	$0.294252\pi$
$830$	−9.36918	−0.325209
$831$	0	0
$832$	14.2154	0.492831
$833$	−48.0595	−1.66516
$834$	0	0
$835$	−14.2347	−0.492612
$836$	−29.2892	−1.01299
$837$	0	0
$838$	−12.3126	−0.425333
$839$	38.6000	1.33262	0.666311	−	0.745674i	$-0.267873\pi$
$839$	38.6000	1.33262	0.666311	+	0.745674i	$0.267873\pi$
$840$	0	0
$841$	−20.4246	−0.704298
$842$	−7.93866	−0.273584
$843$	0	0
$844$	37.5992	1.29422
$845$	26.2530	0.903132
$846$	0	0
$847$	5.23305	0.179810
$848$	13.8820	0.476710
$849$	0	0
$850$	−0.153404	−0.00526172
$851$	−35.3947	−1.21332
$852$	0	0
$853$	−20.1560	−0.690127	−0.345064	−	0.938579i	$-0.612143\pi$
$853$	−20.1560	−0.690127	−0.345064	+	0.938579i	$0.612143\pi$
$854$	−0.233463	−0.00798894
$855$	0	0
$856$	−36.6170	−1.25154
$857$	−21.5609	−0.736505	−0.368252	−	0.929726i	$-0.620044\pi$
$857$	−21.5609	−0.736505	−0.368252	+	0.929726i	$0.620044\pi$
$858$	0	0
$859$	−0.0565266	−0.00192866	−0.000964331	−	1.00000i	$-0.500307\pi$
$859$	−0.0565266	−0.00192866	−0.000964331		1.00000i	$0.500307\pi$
$860$	−40.6889	−1.38748
$861$	0	0
$862$	10.1424	0.345451
$863$	−24.7807	−0.843546	−0.421773	−	0.906701i	$-0.638592\pi$
$863$	−24.7807	−0.843546	−0.421773	+	0.906701i	$0.638592\pi$
$864$	0	0
$865$	−52.0482	−1.76969
$866$	−3.32181	−0.112880
$867$	0	0
$868$	6.33946	0.215175
$869$	1.32636	0.0449937
$870$	0	0
$871$	−30.2801	−1.02600
$872$	11.7888	0.399221
$873$	0	0
$874$	−23.3098	−0.788465
$875$	10.5230	0.355741
$876$	0	0
$877$	15.7648	0.532338	0.266169	−	0.963926i	$-0.414242\pi$
$877$	15.7648	0.532338	0.266169	+	0.963926i	$0.414242\pi$
$878$	1.37093	0.0462666
$879$	0	0
$880$	13.6476	0.460059
$881$	−39.5092	−1.33110	−0.665549	−	0.746354i	$-0.731803\pi$
$881$	−39.5092	−1.33110	−0.665549	+	0.746354i	$0.731803\pi$
$882$	0	0
$883$	−33.8630	−1.13958	−0.569790	−	0.821790i	$-0.692975\pi$
$883$	−33.8630	−1.13958	−0.569790	+	0.821790i	$0.692975\pi$
$884$	68.9059	2.31756
$885$	0	0
$886$	−5.02779	−0.168912
$887$	31.1646	1.04641	0.523203	−	0.852208i	$-0.324737\pi$
$887$	31.1646	1.04641	0.523203	+	0.852208i	$0.324737\pi$
$888$	0	0
$889$	−0.937481	−0.0314421
$890$	−18.0685	−0.605658
$891$	0	0
$892$	13.4097	0.448989
$893$	−43.5197	−1.45633
$894$	0	0
$895$	−44.9165	−1.50139
$896$	10.5812	0.353493
$897$	0	0
$898$	−1.80336	−0.0601790
$899$	11.2339	0.374670
$900$	0	0
$901$	41.3999	1.37923
$902$	8.47417	0.282159
$903$	0	0
$904$	−19.5569	−0.650452
$905$	−32.4421	−1.07841
$906$	0	0
$907$	−46.4005	−1.54070	−0.770352	−	0.637619i	$-0.779920\pi$
$907$	−46.4005	−1.54070	−0.770352	+	0.637619i	$0.779920\pi$
$908$	−6.98233	−0.231717
$909$	0	0
$910$	5.06334	0.167848
$911$	−18.1872	−0.602570	−0.301285	−	0.953534i	$-0.597416\pi$
$911$	−18.1872	−0.602570	−0.301285	+	0.953534i	$0.597416\pi$
$912$	0	0
$913$	−20.1032	−0.665320
$914$	7.14516	0.236341
$915$	0	0
$916$	−4.09994	−0.135466
$917$	−5.67075	−0.187265
$918$	0	0
$919$	16.3273	0.538589	0.269294	−	0.963058i	$-0.413210\pi$
$919$	16.3273	0.538589	0.269294	+	0.963058i	$0.413210\pi$
$920$	27.3638	0.902156
$921$	0	0
$922$	2.62781	0.0865422
$923$	47.3163	1.55744
$924$	0	0
$925$	−0.211783	−0.00696339
$926$	19.9508	0.655625
$927$	0	0
$928$	14.4897	0.475648
$929$	42.1536	1.38302	0.691508	−	0.722369i	$-0.256947\pi$
$929$	42.1536	1.38302	0.691508	+	0.722369i	$0.256947\pi$
$930$	0	0
$931$	−43.6952	−1.43205
$932$	−17.2085	−0.563684
$933$	0	0
$934$	0.792495	0.0259312
$935$	40.7007	1.33106
$936$	0	0
$937$	−50.1135	−1.63714	−0.818569	−	0.574408i	$-0.805232\pi$
$937$	−50.1135	−1.63714	−0.818569	+	0.574408i	$0.805232\pi$
$938$	−2.77726	−0.0906806
$939$	0	0
$940$	23.9335	0.780626
$941$	4.00483	0.130554	0.0652768	−	0.997867i	$-0.479207\pi$
$941$	4.00483	0.130554	0.0652768	+	0.997867i	$0.479207\pi$
$942$	0	0
$943$	−50.1049	−1.63164
$944$	0.332846	0.0108332
$945$	0	0
$946$	11.7514	0.382070
$947$	−33.0686	−1.07458	−0.537292	−	0.843396i	$-0.680553\pi$
$947$	−33.0686	−1.07458	−0.537292	+	0.843396i	$0.680553\pi$
$948$	0	0
$949$	−79.6507	−2.58557
$950$	−0.139473	−0.00452511
$951$	0	0
$952$	13.4906	0.437233
$953$	−55.8818	−1.81019	−0.905094	−	0.425211i	$-0.860200\pi$
$953$	−55.8818	−1.81019	−0.905094	+	0.425211i	$0.860200\pi$
$954$	0	0
$955$	−7.25715	−0.234836
$956$	53.2815	1.72325
$957$	0	0
$958$	−11.1553	−0.360410
$959$	1.26796	0.0409445
$960$	0	0
$961$	−16.2835	−0.525273
$962$	−12.8044	−0.412831
$963$	0	0
$964$	−9.89043	−0.318549
$965$	−38.2102	−1.23003
$966$	0	0
$967$	−8.59071	−0.276259	−0.138129	−	0.990414i	$-0.544109\pi$
$967$	−8.59071	−0.276259	−0.138129	+	0.990414i	$0.544109\pi$
$968$	10.2309	0.328834
$969$	0	0
$970$	17.6900	0.567992
$971$	−21.7780	−0.698890	−0.349445	−	0.936957i	$-0.613630\pi$
$971$	−21.7780	−0.698890	−0.349445	+	0.936957i	$0.613630\pi$
$972$	0	0
$973$	4.13151	0.132450
$974$	2.23463	0.0716021
$975$	0	0
$976$	1.34598	0.0430838
$977$	−4.29100	−0.137281	−0.0686407	−	0.997641i	$-0.521866\pi$
$977$	−4.29100	−0.137281	−0.0686407	+	0.997641i	$0.521866\pi$
$978$	0	0
$979$	−38.7692	−1.23907
$980$	24.0300	0.767611
$981$	0	0
$982$	−2.32673	−0.0742490
$983$	37.7642	1.20449	0.602246	−	0.798311i	$-0.294273\pi$
$983$	37.7642	1.20449	0.602246	+	0.798311i	$0.294273\pi$
$984$	0	0
$985$	20.5368	0.654357
$986$	11.1993	0.356659
$987$	0	0
$988$	62.6485	1.99311
$989$	−69.4818	−2.20939
$990$	0	0
$991$	−11.3791	−0.361468	−0.180734	−	0.983532i	$-0.557847\pi$
$991$	−11.3791	−0.361468	−0.180734	+	0.983532i	$0.557847\pi$
$992$	18.9817	0.602671
$993$	0	0
$994$	4.33980	0.137650
$995$	−27.4510	−0.870256
$996$	0	0
$997$	21.3864	0.677314	0.338657	−	0.940910i	$-0.390027\pi$
$997$	21.3864	0.677314	0.338657	+	0.940910i	$0.390027\pi$
$998$	5.66265	0.179248
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1143.2.a.k.1.9	yes	16
3.2	odd	2	inner	1143.2.a.k.1.8	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1143.2.a.k.1.8	✓	16	3.2	odd	2	inner
1143.2.a.k.1.9	yes	16	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1143 = 3^{2} \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1143.a (trivial)

\(f(q)\)	\(=\)	\(q+0.487100 q^{2} -1.76273 q^{4} +2.22708 q^{5} -0.937481 q^{7} -1.83283 q^{8} +O(q^{10})\)	\(q+0.487100 q^{2} -1.76273 q^{4} +2.22708 q^{5} -0.937481 q^{7} -1.83283 q^{8} +1.08481 q^{10} +2.32765 q^{11} -4.97876 q^{13} -0.456646 q^{14} +2.63270 q^{16} +7.85142 q^{17} +7.13842 q^{19} -3.92575 q^{20} +1.13380 q^{22} -6.70376 q^{23} -0.0401117 q^{25} -2.42515 q^{26} +1.65253 q^{28} +2.92837 q^{29} +3.83622 q^{31} +4.94804 q^{32} +3.82442 q^{34} -2.08785 q^{35} +5.27983 q^{37} +3.47712 q^{38} -4.08185 q^{40} +7.47414 q^{41} +10.3646 q^{43} -4.10303 q^{44} -3.26540 q^{46} -6.09655 q^{47} -6.12113 q^{49} -0.0195384 q^{50} +8.77624 q^{52} +5.27292 q^{53} +5.18387 q^{55} +1.71824 q^{56} +1.42641 q^{58} +0.126428 q^{59} +0.511256 q^{61} +1.86862 q^{62} -2.85521 q^{64} -11.0881 q^{65} +6.08185 q^{67} -13.8400 q^{68} -1.01699 q^{70} -9.50363 q^{71} +15.9981 q^{73} +2.57180 q^{74} -12.5831 q^{76} -2.18213 q^{77} +0.569827 q^{79} +5.86323 q^{80} +3.64065 q^{82} -8.63671 q^{83} +17.4857 q^{85} +5.04859 q^{86} -4.26618 q^{88} -16.6559 q^{89} +4.66750 q^{91} +11.8169 q^{92} -2.96963 q^{94} +15.8978 q^{95} +16.3070 q^{97} -2.98160 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 20 q^{4} + 10 q^{7}+O(q^{10}) \) 16 * q + 20 * q^4 + 10 * q^7	\( 16 q + 20 q^{4} + 10 q^{7} + 14 q^{10} + 20 q^{13} + 28 q^{16} + 12 q^{19} + 18 q^{22} + 52 q^{25} + 42 q^{28} + 18 q^{31} + 10 q^{34} + 16 q^{37} + 6 q^{40} + 26 q^{43} - 24 q^{46} + 54 q^{49} + 52 q^{52} + 20 q^{55} - 14 q^{58} + 36 q^{61} - 4 q^{64} + 26 q^{67} + 36 q^{70} + 60 q^{73} - 20 q^{76} + 12 q^{79} - 20 q^{82} - 12 q^{85} + 8 q^{88} - 24 q^{91} - 26 q^{94} + 108 q^{97}+O(q^{100}) \) 16 * q + 20 * q^4 + 10 * q^7 + 14 * q^10 + 20 * q^13 + 28 * q^16 + 12 * q^19 + 18 * q^22 + 52 * q^25 + 42 * q^28 + 18 * q^31 + 10 * q^34 + 16 * q^37 + 6 * q^40 + 26 * q^43 - 24 * q^46 + 54 * q^49 + 52 * q^52 + 20 * q^55 - 14 * q^58 + 36 * q^61 - 4 * q^64 + 26 * q^67 + 36 * q^70 + 60 * q^73 - 20 * q^76 + 12 * q^79 - 20 * q^82 - 12 * q^85 + 8 * q^88 - 24 * q^91 - 26 * q^94 + 108 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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