LMFDB - Embedded newform 1287.2.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1287,2,Mod(1,1287)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1287, 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("1287.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1287 = 3^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1287.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:	$10.2767467401$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 429)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	1287.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.53919 q^{2} +0.369102 q^{4} -3.17009 q^{5} +0.630898 q^{7} +2.51026 q^{8} +O(q^{10})$	\(q-1.53919 q^{2} +0.369102 q^{4} -3.17009 q^{5} +0.630898 q^{7} +2.51026 q^{8} +4.87936 q^{10} +1.00000 q^{11} -1.00000 q^{13} -0.971071 q^{14} -4.60197 q^{16} +0.539189 q^{17} -0.630898 q^{19} -1.17009 q^{20} -1.53919 q^{22} -1.55252 q^{23} +5.04945 q^{25} +1.53919 q^{26} +0.232866 q^{28} +8.29791 q^{29} +2.82991 q^{31} +2.06278 q^{32} -0.829914 q^{34} -2.00000 q^{35} -2.15676 q^{37} +0.971071 q^{38} -7.95774 q^{40} +2.63090 q^{41} +10.6381 q^{43} +0.369102 q^{44} +2.38962 q^{46} +3.75872 q^{47} -6.60197 q^{49} -7.77205 q^{50} -0.369102 q^{52} -5.60197 q^{53} -3.17009 q^{55} +1.58372 q^{56} -12.7721 q^{58} -5.65983 q^{59} -4.18342 q^{61} -4.35577 q^{62} +6.02893 q^{64} +3.17009 q^{65} -13.2690 q^{67} +0.199016 q^{68} +3.07838 q^{70} -14.9939 q^{71} -16.2062 q^{73} +3.31965 q^{74} -0.232866 q^{76} +0.630898 q^{77} -15.2195 q^{79} +14.5886 q^{80} -4.04945 q^{82} -1.65983 q^{83} -1.70928 q^{85} -16.3740 q^{86} +2.51026 q^{88} +1.17009 q^{89} -0.630898 q^{91} -0.573039 q^{92} -5.78539 q^{94} +2.00000 q^{95} -1.84324 q^{97} +10.1617 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 5 q^{4} - 4 q^{5} - 2 q^{7} - 9 q^{8}+O(q^{10}) $ 3 * q - 3 * q^2 + 5 * q^4 - 4 * q^5 - 2 * q^7 - 9 * q^8	$ 3 q - 3 q^{2} + 5 q^{4} - 4 q^{5} - 2 q^{7} - 9 q^{8} + 2 q^{10} + 3 q^{11} - 3 q^{13} + 12 q^{14} + 5 q^{16} + 2 q^{19} + 2 q^{20} - 3 q^{22} - 4 q^{23} - 3 q^{25} + 3 q^{26} - 22 q^{28} - 2 q^{29} + 14 q^{31} - 11 q^{32} - 8 q^{34} - 6 q^{35} - 12 q^{38} - 8 q^{40} + 4 q^{41} - 6 q^{43} + 5 q^{44} - 22 q^{46} - 14 q^{47} - q^{49} + q^{50} - 5 q^{52} + 2 q^{53} - 4 q^{55} + 32 q^{56} - 14 q^{58} - 28 q^{59} - 8 q^{61} - 16 q^{62} + 33 q^{64} + 4 q^{65} + 2 q^{67} + 10 q^{68} + 6 q^{70} - 10 q^{71} - 24 q^{73} + 32 q^{74} + 22 q^{76} - 2 q^{77} - 22 q^{79} + 24 q^{80} + 6 q^{82} - 16 q^{83} + 2 q^{85} - 6 q^{86} - 9 q^{88} - 2 q^{89} + 2 q^{91} + 32 q^{92} + 6 q^{94} + 6 q^{95} - 12 q^{97} - 23 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 5 * q^4 - 4 * q^5 - 2 * q^7 - 9 * q^8 + 2 * q^10 + 3 * q^11 - 3 * q^13 + 12 * q^14 + 5 * q^16 + 2 * q^19 + 2 * q^20 - 3 * q^22 - 4 * q^23 - 3 * q^25 + 3 * q^26 - 22 * q^28 - 2 * q^29 + 14 * q^31 - 11 * q^32 - 8 * q^34 - 6 * q^35 - 12 * q^38 - 8 * q^40 + 4 * q^41 - 6 * q^43 + 5 * q^44 - 22 * q^46 - 14 * q^47 - q^49 + q^50 - 5 * q^52 + 2 * q^53 - 4 * q^55 + 32 * q^56 - 14 * q^58 - 28 * q^59 - 8 * q^61 - 16 * q^62 + 33 * q^64 + 4 * q^65 + 2 * q^67 + 10 * q^68 + 6 * q^70 - 10 * q^71 - 24 * q^73 + 32 * q^74 + 22 * q^76 - 2 * q^77 - 22 * q^79 + 24 * q^80 + 6 * q^82 - 16 * q^83 + 2 * q^85 - 6 * q^86 - 9 * q^88 - 2 * q^89 + 2 * q^91 + 32 * q^92 + 6 * q^94 + 6 * q^95 - 12 * q^97 - 23 * 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 $

$2$	−1.53919	−1.08837	−0.544185	−	0.838965i	$-0.683161\pi$
$2$	−1.53919	−1.08837	−0.544185	+	0.838965i	$0.683161\pi$
$3$	0	0
$3$	0	0
$4$	0.369102	0.184551
$5$	−3.17009	−1.41771	−0.708853	−	0.705356i	$-0.750787\pi$
$5$	−3.17009	−1.41771	−0.708853	+	0.705356i	$0.750787\pi$
$6$	0	0
$7$	0.630898	0.238457	0.119228	−	0.992867i	$-0.461958\pi$
$7$	0.630898	0.238457	0.119228	+	0.992867i	$0.461958\pi$
$8$	2.51026	0.887511
$9$	0	0
$10$	4.87936	1.54299
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−0.971071	−0.259530
$15$	0	0
$16$	−4.60197	−1.15049
$17$	0.539189	0.130773	0.0653863	−	0.997860i	$-0.479172\pi$
$17$	0.539189	0.130773	0.0653863	+	0.997860i	$0.479172\pi$
$18$	0	0
$19$	−0.630898	−0.144738	−0.0723689	−	0.997378i	$-0.523056\pi$
$19$	−0.630898	−0.144738	−0.0723689	+	0.997378i	$0.523056\pi$
$20$	−1.17009	−0.261639
$21$	0	0
$22$	−1.53919	−0.328156
$23$	−1.55252	−0.323723	−0.161861	−	0.986814i	$-0.551750\pi$
$23$	−1.55252	−0.323723	−0.161861	+	0.986814i	$0.551750\pi$
$24$	0	0
$25$	5.04945	1.00989
$26$	1.53919	0.301860
$27$	0	0
$28$	0.232866	0.0440075
$29$	8.29791	1.54088	0.770442	−	0.637510i	$-0.220036\pi$
$29$	8.29791	1.54088	0.770442	+	0.637510i	$0.220036\pi$
$30$	0	0
$31$	2.82991	0.508267	0.254134	−	0.967169i	$-0.418210\pi$
$31$	2.82991	0.508267	0.254134	+	0.967169i	$0.418210\pi$
$32$	2.06278	0.364651
$33$	0	0
$34$	−0.829914	−0.142329
$35$	−2.00000	−0.338062
$36$	0	0
$37$	−2.15676	−0.354568	−0.177284	−	0.984160i	$-0.556731\pi$
$37$	−2.15676	−0.354568	−0.177284	+	0.984160i	$0.556731\pi$
$38$	0.971071	0.157528
$39$	0	0
$40$	−7.95774	−1.25823
$41$	2.63090	0.410877	0.205439	−	0.978670i	$-0.434138\pi$
$41$	2.63090	0.410877	0.205439	+	0.978670i	$0.434138\pi$
$42$	0	0
$43$	10.6381	1.62229	0.811146	−	0.584843i	$-0.198844\pi$
$43$	10.6381	1.62229	0.811146	+	0.584843i	$0.198844\pi$
$44$	0.369102	0.0556443
$45$	0	0
$46$	2.38962	0.352330
$47$	3.75872	0.548266	0.274133	−	0.961692i	$-0.411609\pi$
$47$	3.75872	0.548266	0.274133	+	0.961692i	$0.411609\pi$
$48$	0	0
$49$	−6.60197	−0.943138
$50$	−7.77205	−1.09913
$51$	0	0
$52$	−0.369102	−0.0511853
$53$	−5.60197	−0.769490	−0.384745	−	0.923023i	$-0.625711\pi$
$53$	−5.60197	−0.769490	−0.384745	+	0.923023i	$0.625711\pi$
$54$	0	0
$55$	−3.17009	−0.427454
$56$	1.58372	0.211633
$57$	0	0
$58$	−12.7721	−1.67705
$59$	−5.65983	−0.736847	−0.368423	−	0.929658i	$-0.620102\pi$
$59$	−5.65983	−0.736847	−0.368423	+	0.929658i	$0.620102\pi$
$60$	0	0
$61$	−4.18342	−0.535632	−0.267816	−	0.963470i	$-0.586302\pi$
$61$	−4.18342	−0.535632	−0.267816	+	0.963470i	$0.586302\pi$
$62$	−4.35577	−0.553184
$63$	0	0
$64$	6.02893	0.753616
$65$	3.17009	0.393201
$66$	0	0
$67$	−13.2690	−1.62106	−0.810532	−	0.585694i	$-0.800822\pi$
$67$	−13.2690	−1.62106	−0.810532	+	0.585694i	$0.800822\pi$
$68$	0.199016	0.0241342
$69$	0	0
$70$	3.07838	0.367937
$71$	−14.9939	−1.77944	−0.889722	−	0.456503i	$-0.849102\pi$
$71$	−14.9939	−1.77944	−0.889722	+	0.456503i	$0.849102\pi$
$72$	0	0
$73$	−16.2062	−1.89679	−0.948396	−	0.317087i	$-0.897295\pi$
$73$	−16.2062	−1.89679	−0.948396	+	0.317087i	$0.897295\pi$
$74$	3.31965	0.385902
$75$	0	0
$76$	−0.232866	−0.0267115
$77$	0.630898	0.0718975
$78$	0	0
$79$	−15.2195	−1.71233	−0.856166	−	0.516701i	$-0.827160\pi$
$79$	−15.2195	−1.71233	−0.856166	+	0.516701i	$0.827160\pi$
$80$	14.5886	1.63106
$81$	0	0
$82$	−4.04945	−0.447187
$83$	−1.65983	−0.182190	−0.0910948	−	0.995842i	$-0.529037\pi$
$83$	−1.65983	−0.182190	−0.0910948	+	0.995842i	$0.529037\pi$
$84$	0	0
$85$	−1.70928	−0.185397
$86$	−16.3740	−1.76566
$87$	0	0
$88$	2.51026	0.267595
$89$	1.17009	0.124029	0.0620145	−	0.998075i	$-0.480248\pi$
$89$	1.17009	0.124029	0.0620145	+	0.998075i	$0.480248\pi$
$90$	0	0
$91$	−0.630898	−0.0661360
$92$	−0.573039	−0.0597434
$93$	0	0
$94$	−5.78539	−0.596717
$95$	2.00000	0.205196
$96$	0	0
$97$	−1.84324	−0.187153	−0.0935766	−	0.995612i	$-0.529830\pi$
$97$	−1.84324	−0.187153	−0.0935766	+	0.995612i	$0.529830\pi$
$98$	10.1617	1.02648
$99$	0	0
$100$	1.86376	0.186376
$101$	4.04226	0.402220	0.201110	−	0.979569i	$-0.435545\pi$
$101$	4.04226	0.402220	0.201110	+	0.979569i	$0.435545\pi$
$102$	0	0
$103$	10.8371	1.06781	0.533906	−	0.845544i	$-0.320724\pi$
$103$	10.8371	1.06781	0.533906	+	0.845544i	$0.320724\pi$
$104$	−2.51026	−0.246151
$105$	0	0
$106$	8.62249	0.837490
$107$	−9.17727	−0.887201	−0.443600	−	0.896225i	$-0.646299\pi$
$107$	−9.17727	−0.887201	−0.443600	+	0.896225i	$0.646299\pi$
$108$	0	0
$109$	8.57304	0.821148	0.410574	−	0.911827i	$-0.365328\pi$
$109$	8.57304	0.821148	0.410574	+	0.911827i	$0.365328\pi$
$110$	4.87936	0.465229
$111$	0	0
$112$	−2.90337	−0.274343
$113$	0.0266620	0.00250815	0.00125407	−	0.999999i	$-0.499601\pi$
$113$	0.0266620	0.00250815	0.00125407	+	0.999999i	$0.499601\pi$
$114$	0	0
$115$	4.92162	0.458944
$116$	3.06278	0.284372
$117$	0	0
$118$	8.71154	0.801963
$119$	0.340173	0.0311836
$120$	0	0
$121$	1.00000	0.0909091
$122$	6.43907	0.582966
$123$	0	0
$124$	1.04453	0.0938014
$125$	−0.156755	−0.0140206
$126$	0	0
$127$	8.05664	0.714911	0.357455	−	0.933930i	$-0.383644\pi$
$127$	8.05664	0.714911	0.357455	+	0.933930i	$0.383644\pi$
$128$	−13.4052	−1.18487
$129$	0	0
$130$	−4.87936	−0.427948
$131$	−16.0989	−1.40657	−0.703284	−	0.710909i	$-0.748284\pi$
$131$	−16.0989	−1.40657	−0.703284	+	0.710909i	$0.748284\pi$
$132$	0	0
$133$	−0.398032	−0.0345137
$134$	20.4235	1.76432
$135$	0	0
$136$	1.35350	0.116062
$137$	−21.6670	−1.85114	−0.925569	−	0.378579i	$-0.876413\pi$
$137$	−21.6670	−1.85114	−0.925569	+	0.378579i	$0.876413\pi$
$138$	0	0
$139$	−2.51253	−0.213110	−0.106555	−	0.994307i	$-0.533982\pi$
$139$	−2.51253	−0.213110	−0.106555	+	0.994307i	$0.533982\pi$
$140$	−0.738205	−0.0623897
$141$	0	0
$142$	23.0784	1.93669
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−26.3051	−2.18452
$146$	24.9444	2.06441
$147$	0	0
$148$	−0.796064	−0.0654360
$149$	−7.52586	−0.616542	−0.308271	−	0.951299i	$-0.599750\pi$
$149$	−7.52586	−0.616542	−0.308271	+	0.951299i	$0.599750\pi$
$150$	0	0
$151$	−5.31124	−0.432223	−0.216111	−	0.976369i	$-0.569337\pi$
$151$	−5.31124	−0.432223	−0.216111	+	0.976369i	$0.569337\pi$
$152$	−1.58372	−0.128456
$153$	0	0
$154$	−0.971071	−0.0782511
$155$	−8.97107	−0.720574
$156$	0	0
$157$	3.15449	0.251756	0.125878	−	0.992046i	$-0.459825\pi$
$157$	3.15449	0.251756	0.125878	+	0.992046i	$0.459825\pi$
$158$	23.4257	1.86365
$159$	0	0
$160$	−6.53919	−0.516968
$161$	−0.979481	−0.0771939
$162$	0	0
$163$	−11.2690	−0.882655	−0.441327	−	0.897346i	$-0.645492\pi$
$163$	−11.2690	−0.882655	−0.441327	+	0.897346i	$0.645492\pi$
$164$	0.971071	0.0758279
$165$	0	0
$166$	2.55479	0.198290
$167$	−5.94214	−0.459817	−0.229908	−	0.973212i	$-0.573843\pi$
$167$	−5.94214	−0.459817	−0.229908	+	0.973212i	$0.573843\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	2.63090	0.201781
$171$	0	0
$172$	3.92654	0.299396
$173$	18.6647	1.41905	0.709527	−	0.704678i	$-0.248909\pi$
$173$	18.6647	1.41905	0.709527	+	0.704678i	$0.248909\pi$
$174$	0	0
$175$	3.18568	0.240815
$176$	−4.60197	−0.346886
$177$	0	0
$178$	−1.80098	−0.134989
$179$	5.46800	0.408697	0.204349	−	0.978898i	$-0.434492\pi$
$179$	5.46800	0.408697	0.204349	+	0.978898i	$0.434492\pi$
$180$	0	0
$181$	16.0494	1.19295	0.596473	−	0.802633i	$-0.296568\pi$
$181$	16.0494	1.19295	0.596473	+	0.802633i	$0.296568\pi$
$182$	0.971071	0.0719805
$183$	0	0
$184$	−3.89723	−0.287307
$185$	6.83710	0.502674
$186$	0	0
$187$	0.539189	0.0394294
$188$	1.38735	0.101183
$189$	0	0
$190$	−3.07838	−0.223329
$191$	−24.0144	−1.73762	−0.868810	−	0.495146i	$-0.835114\pi$
$191$	−24.0144	−1.73762	−0.868810	+	0.495146i	$0.835114\pi$
$192$	0	0
$193$	−6.63090	−0.477302	−0.238651	−	0.971105i	$-0.576705\pi$
$193$	−6.63090	−0.477302	−0.238651	+	0.971105i	$0.576705\pi$
$194$	2.83710	0.203692
$195$	0	0
$196$	−2.43680	−0.174057
$197$	−11.4947	−0.818961	−0.409480	−	0.912319i	$-0.634290\pi$
$197$	−11.4947	−0.818961	−0.409480	+	0.912319i	$0.634290\pi$
$198$	0	0
$199$	−6.83710	−0.484669	−0.242335	−	0.970193i	$-0.577913\pi$
$199$	−6.83710	−0.484669	−0.242335	+	0.970193i	$0.577913\pi$
$200$	12.6754	0.896288
$201$	0	0
$202$	−6.22180	−0.437764
$203$	5.23513	0.367434
$204$	0	0
$205$	−8.34017	−0.582503
$206$	−16.6803	−1.16217
$207$	0	0
$208$	4.60197	0.319089
$209$	−0.630898	−0.0436401
$210$	0	0
$211$	4.19902	0.289072	0.144536	−	0.989500i	$-0.453831\pi$
$211$	4.19902	0.289072	0.144536	+	0.989500i	$0.453831\pi$
$212$	−2.06770	−0.142010
$213$	0	0
$214$	14.1256	0.965603
$215$	−33.7237	−2.29993
$216$	0	0
$217$	1.78539	0.121200
$218$	−13.1955	−0.893714
$219$	0	0
$220$	−1.17009	−0.0788872
$221$	−0.539189	−0.0362698
$222$	0	0
$223$	12.0072	0.804061	0.402030	−	0.915626i	$-0.368305\pi$
$223$	12.0072	0.804061	0.402030	+	0.915626i	$0.368305\pi$
$224$	1.30140	0.0869536
$225$	0	0
$226$	−0.0410378	−0.00272980
$227$	−25.2267	−1.67436	−0.837178	−	0.546930i	$-0.815796\pi$
$227$	−25.2267	−1.67436	−0.837178	+	0.546930i	$0.815796\pi$
$228$	0	0
$229$	13.2762	0.877314	0.438657	−	0.898655i	$-0.355454\pi$
$229$	13.2762	0.877314	0.438657	+	0.898655i	$0.355454\pi$
$230$	−7.57531	−0.499501
$231$	0	0
$232$	20.8299	1.36755
$233$	−8.87936	−0.581706	−0.290853	−	0.956768i	$-0.593939\pi$
$233$	−8.87936	−0.581706	−0.290853	+	0.956768i	$0.593939\pi$
$234$	0	0
$235$	−11.9155	−0.777280
$236$	−2.08906	−0.135986
$237$	0	0
$238$	−0.523590	−0.0339393
$239$	1.34244	0.0868353	0.0434176	−	0.999057i	$-0.486175\pi$
$239$	1.34244	0.0868353	0.0434176	+	0.999057i	$0.486175\pi$
$240$	0	0
$241$	16.4391	1.05893	0.529467	−	0.848331i	$-0.322392\pi$
$241$	16.4391	1.05893	0.529467	+	0.848331i	$0.322392\pi$
$242$	−1.53919	−0.0989428
$243$	0	0
$244$	−1.54411	−0.0988515
$245$	20.9288	1.33709
$246$	0	0
$247$	0.630898	0.0401431
$248$	7.10382	0.451093
$249$	0	0
$250$	0.241276	0.0152597
$251$	24.6947	1.55872	0.779359	−	0.626578i	$-0.215545\pi$
$251$	24.6947	1.55872	0.779359	+	0.626578i	$0.215545\pi$
$252$	0	0
$253$	−1.55252	−0.0976061
$254$	−12.4007	−0.778088
$255$	0	0
$256$	8.57531	0.535957
$257$	3.94214	0.245904	0.122952	−	0.992413i	$-0.460764\pi$
$257$	3.94214	0.245904	0.122952	+	0.992413i	$0.460764\pi$
$258$	0	0
$259$	−1.36069	−0.0845493
$260$	1.17009	0.0725657
$261$	0	0
$262$	24.7792	1.53087
$263$	19.7009	1.21481	0.607404	−	0.794393i	$-0.292211\pi$
$263$	19.7009	1.21481	0.607404	+	0.794393i	$0.292211\pi$
$264$	0	0
$265$	17.7587	1.09091
$266$	0.612646	0.0375637
$267$	0	0
$268$	−4.89761	−0.299169
$269$	22.5958	1.37769	0.688846	−	0.724908i	$-0.258118\pi$
$269$	22.5958	1.37769	0.688846	+	0.724908i	$0.258118\pi$
$270$	0	0
$271$	−6.70313	−0.407186	−0.203593	−	0.979056i	$-0.565262\pi$
$271$	−6.70313	−0.407186	−0.203593	+	0.979056i	$0.565262\pi$
$272$	−2.48133	−0.150453
$273$	0	0
$274$	33.3496	2.01472
$275$	5.04945	0.304493
$276$	0	0
$277$	−13.1050	−0.787406	−0.393703	−	0.919238i	$-0.628806\pi$
$277$	−13.1050	−0.787406	−0.393703	+	0.919238i	$0.628806\pi$
$278$	3.86725	0.231942
$279$	0	0
$280$	−5.02052	−0.300033
$281$	−9.86603	−0.588558	−0.294279	−	0.955720i	$-0.595079\pi$
$281$	−9.86603	−0.588558	−0.294279	+	0.955720i	$0.595079\pi$
$282$	0	0
$283$	−13.2195	−0.785820	−0.392910	−	0.919577i	$-0.628532\pi$
$283$	−13.2195	−0.785820	−0.392910	+	0.919577i	$0.628532\pi$
$284$	−5.53427	−0.328398
$285$	0	0
$286$	1.53919	0.0910141
$287$	1.65983	0.0979765
$288$	0	0
$289$	−16.7093	−0.982899
$290$	40.4885	2.37757
$291$	0	0
$292$	−5.98175	−0.350055
$293$	−30.6453	−1.79032	−0.895158	−	0.445749i	$-0.852937\pi$
$293$	−30.6453	−1.79032	−0.895158	+	0.445749i	$0.852937\pi$
$294$	0	0
$295$	17.9421	1.04463
$296$	−5.41402	−0.314683
$297$	0	0
$298$	11.5837	0.671027
$299$	1.55252	0.0897845
$300$	0	0
$301$	6.71154	0.386847
$302$	8.17501	0.470419
$303$	0	0
$304$	2.90337	0.166520
$305$	13.2618	0.759368
$306$	0	0
$307$	8.26406	0.471655	0.235827	−	0.971795i	$-0.424220\pi$
$307$	8.26406	0.471655	0.235827	+	0.971795i	$0.424220\pi$
$308$	0.232866	0.0132688
$309$	0	0
$310$	13.8082	0.784251
$311$	−11.8082	−0.669580	−0.334790	−	0.942293i	$-0.608665\pi$
$311$	−11.8082	−0.669580	−0.334790	+	0.942293i	$0.608665\pi$
$312$	0	0
$313$	32.7442	1.85081	0.925405	−	0.378980i	$-0.123725\pi$
$313$	32.7442	1.85081	0.925405	+	0.378980i	$0.123725\pi$
$314$	−4.85535	−0.274003
$315$	0	0
$316$	−5.61757	−0.316013
$317$	0.803252	0.0451151	0.0225576	−	0.999746i	$-0.492819\pi$
$317$	0.803252	0.0451151	0.0225576	+	0.999746i	$0.492819\pi$
$318$	0	0
$319$	8.29791	0.464594
$320$	−19.1122	−1.06841
$321$	0	0
$322$	1.50761	0.0840156
$323$	−0.340173	−0.0189277
$324$	0	0
$325$	−5.04945	−0.280093
$326$	17.3451	0.960656
$327$	0	0
$328$	6.60424	0.364658
$329$	2.37137	0.130738
$330$	0	0
$331$	4.34736	0.238953	0.119476	−	0.992837i	$-0.461878\pi$
$331$	4.34736	0.238953	0.119476	+	0.992837i	$0.461878\pi$
$332$	−0.612646	−0.0336233
$333$	0	0
$334$	9.14608	0.500451
$335$	42.0638	2.29819
$336$	0	0
$337$	−11.1773	−0.608865	−0.304432	−	0.952534i	$-0.598467\pi$
$337$	−11.1773	−0.608865	−0.304432	+	0.952534i	$0.598467\pi$
$338$	−1.53919	−0.0837208
$339$	0	0
$340$	−0.630898	−0.0342152
$341$	2.82991	0.153248
$342$	0	0
$343$	−8.58145	−0.463355
$344$	26.7044	1.43980
$345$	0	0
$346$	−28.7286	−1.54446
$347$	6.77924	0.363929	0.181964	−	0.983305i	$-0.441754\pi$
$347$	6.77924	0.363929	0.181964	+	0.983305i	$0.441754\pi$
$348$	0	0
$349$	−4.24128	−0.227030	−0.113515	−	0.993536i	$-0.536211\pi$
$349$	−4.24128	−0.227030	−0.113515	+	0.993536i	$0.536211\pi$
$350$	−4.90337	−0.262096
$351$	0	0
$352$	2.06278	0.109947
$353$	−17.0277	−0.906293	−0.453147	−	0.891436i	$-0.649699\pi$
$353$	−17.0277	−0.906293	−0.453147	+	0.891436i	$0.649699\pi$
$354$	0	0
$355$	47.5318	2.52273
$356$	0.431882	0.0228897
$357$	0	0
$358$	−8.41628	−0.444814
$359$	18.9711	1.00125	0.500627	−	0.865663i	$-0.333103\pi$
$359$	18.9711	1.00125	0.500627	+	0.865663i	$0.333103\pi$
$360$	0	0
$361$	−18.6020	−0.979051
$362$	−24.7031	−1.29837
$363$	0	0
$364$	−0.232866	−0.0122055
$365$	51.3751	2.68909
$366$	0	0
$367$	−4.59583	−0.239900	−0.119950	−	0.992780i	$-0.538273\pi$
$367$	−4.59583	−0.239900	−0.119950	+	0.992780i	$0.538273\pi$
$368$	7.14465	0.372440
$369$	0	0
$370$	−10.5236	−0.547095
$371$	−3.53427	−0.183490
$372$	0	0
$373$	−34.6947	−1.79642	−0.898212	−	0.439562i	$-0.855134\pi$
$373$	−34.6947	−1.79642	−0.898212	+	0.439562i	$0.855134\pi$
$374$	−0.829914	−0.0429138
$375$	0	0
$376$	9.43537	0.486592
$377$	−8.29791	−0.427364
$378$	0	0
$379$	21.7926	1.11941	0.559705	−	0.828692i	$-0.310915\pi$
$379$	21.7926	1.11941	0.559705	+	0.828692i	$0.310915\pi$
$380$	0.738205	0.0378691
$381$	0	0
$382$	36.9627	1.89117
$383$	−9.07838	−0.463883	−0.231942	−	0.972730i	$-0.574508\pi$
$383$	−9.07838	−0.463883	−0.231942	+	0.972730i	$0.574508\pi$
$384$	0	0
$385$	−2.00000	−0.101929
$386$	10.2062	0.519482
$387$	0	0
$388$	−0.680346	−0.0345393
$389$	34.3812	1.74320	0.871598	−	0.490221i	$-0.163084\pi$
$389$	34.3812	1.74320	0.871598	+	0.490221i	$0.163084\pi$
$390$	0	0
$391$	−0.837101	−0.0423340
$392$	−16.5727	−0.837045
$393$	0	0
$394$	17.6925	0.891333
$395$	48.2472	2.42758
$396$	0	0
$397$	−29.3074	−1.47089	−0.735447	−	0.677582i	$-0.763028\pi$
$397$	−29.3074	−1.47089	−0.735447	+	0.677582i	$0.763028\pi$
$398$	10.5236	0.527500
$399$	0	0
$400$	−23.2374	−1.16187
$401$	−2.45854	−0.122774	−0.0613869	−	0.998114i	$-0.519552\pi$
$401$	−2.45854	−0.122774	−0.0613869	+	0.998114i	$0.519552\pi$
$402$	0	0
$403$	−2.82991	−0.140968
$404$	1.49201	0.0742302
$405$	0	0
$406$	−8.05786	−0.399905
$407$	−2.15676	−0.106906
$408$	0	0
$409$	−20.5197	−1.01463	−0.507317	−	0.861759i	$-0.669363\pi$
$409$	−20.5197	−1.01463	−0.507317	+	0.861759i	$0.669363\pi$
$410$	12.8371	0.633979
$411$	0	0
$412$	4.00000	0.197066
$413$	−3.57077	−0.175706
$414$	0	0
$415$	5.26180	0.258291
$416$	−2.06278	−0.101136
$417$	0	0
$418$	0.971071	0.0474966
$419$	−1.68261	−0.0822010	−0.0411005	−	0.999155i	$-0.513086\pi$
$419$	−1.68261	−0.0822010	−0.0411005	+	0.999155i	$0.513086\pi$
$420$	0	0
$421$	2.21461	0.107934	0.0539668	−	0.998543i	$-0.482813\pi$
$421$	2.21461	0.107934	0.0539668	+	0.998543i	$0.482813\pi$
$422$	−6.46308	−0.314618
$423$	0	0
$424$	−14.0624	−0.682930
$425$	2.72261	0.132066
$426$	0	0
$427$	−2.63931	−0.127725
$428$	−3.38735	−0.163734
$429$	0	0
$430$	51.9071	2.50318
$431$	−20.1522	−0.970698	−0.485349	−	0.874320i	$-0.661308\pi$
$431$	−20.1522	−0.970698	−0.485349	+	0.874320i	$0.661308\pi$
$432$	0	0
$433$	−32.5236	−1.56298	−0.781492	−	0.623915i	$-0.785541\pi$
$433$	−32.5236	−1.56298	−0.781492	+	0.623915i	$0.785541\pi$
$434$	−2.74805	−0.131910
$435$	0	0
$436$	3.16433	0.151544
$437$	0.979481	0.0468549
$438$	0	0
$439$	−11.7165	−0.559196	−0.279598	−	0.960117i	$-0.590201\pi$
$439$	−11.7165	−0.559196	−0.279598	+	0.960117i	$0.590201\pi$
$440$	−7.95774	−0.379370
$441$	0	0
$442$	0.829914	0.0394750
$443$	30.3545	1.44219	0.721094	−	0.692837i	$-0.243640\pi$
$443$	30.3545	1.44219	0.721094	+	0.692837i	$0.243640\pi$
$444$	0	0
$445$	−3.70928	−0.175837
$446$	−18.4813	−0.875116
$447$	0	0
$448$	3.80364	0.179705
$449$	−4.79872	−0.226465	−0.113233	−	0.993568i	$-0.536121\pi$
$449$	−4.79872	−0.226465	−0.113233	+	0.993568i	$0.536121\pi$
$450$	0	0
$451$	2.63090	0.123884
$452$	0.00984100	0.000462882 0
$453$	0	0
$454$	38.8287	1.82232
$455$	2.00000	0.0937614
$456$	0	0
$457$	−38.4657	−1.79935	−0.899676	−	0.436559i	$-0.856197\pi$
$457$	−38.4657	−1.79935	−0.899676	+	0.436559i	$0.856197\pi$
$458$	−20.4345	−0.954843
$459$	0	0
$460$	1.81658	0.0846986
$461$	22.6309	1.05403	0.527013	−	0.849857i	$-0.323312\pi$
$461$	22.6309	1.05403	0.527013	+	0.849857i	$0.323312\pi$
$462$	0	0
$463$	24.2940	1.12904	0.564520	−	0.825420i	$-0.309061\pi$
$463$	24.2940	1.12904	0.564520	+	0.825420i	$0.309061\pi$
$464$	−38.1867	−1.77277
$465$	0	0
$466$	13.6670	0.633112
$467$	9.45362	0.437462	0.218731	−	0.975785i	$-0.429808\pi$
$467$	9.45362	0.437462	0.218731	+	0.975785i	$0.429808\pi$
$468$	0	0
$469$	−8.37137	−0.386554
$470$	18.3402	0.845969
$471$	0	0
$472$	−14.2076	−0.653959
$473$	10.6381	0.489140
$474$	0	0
$475$	−3.18568	−0.146169
$476$	0.125559	0.00575497
$477$	0	0
$478$	−2.06627	−0.0945090
$479$	39.1155	1.78723	0.893617	−	0.448830i	$-0.148159\pi$
$479$	39.1155	1.78723	0.893617	+	0.448830i	$0.148159\pi$
$480$	0	0
$481$	2.15676	0.0983396
$482$	−25.3028	−1.15251
$483$	0	0
$484$	0.369102	0.0167774
$485$	5.84324	0.265328
$486$	0	0
$487$	−34.6030	−1.56801	−0.784006	−	0.620753i	$-0.786827\pi$
$487$	−34.6030	−1.56801	−0.784006	+	0.620753i	$0.786827\pi$
$488$	−10.5015	−0.475379
$489$	0	0
$490$	−32.2134	−1.45525
$491$	−23.0472	−1.04010	−0.520052	−	0.854134i	$-0.674088\pi$
$491$	−23.0472	−1.04010	−0.520052	+	0.854134i	$0.674088\pi$
$492$	0	0
$493$	4.47414	0.201505
$494$	−0.971071	−0.0436905
$495$	0	0
$496$	−13.0232	−0.584758
$497$	−9.45959	−0.424321
$498$	0	0
$499$	36.5574	1.63654	0.818268	−	0.574837i	$-0.194935\pi$
$499$	36.5574	1.63654	0.818268	+	0.574837i	$0.194935\pi$
$500$	−0.0578588	−0.00258753
$501$	0	0
$502$	−38.0098	−1.69646
$503$	35.0205	1.56149	0.780744	−	0.624851i	$-0.214840\pi$
$503$	35.0205	1.56149	0.780744	+	0.624851i	$0.214840\pi$
$504$	0	0
$505$	−12.8143	−0.570230
$506$	2.38962	0.106232
$507$	0	0
$508$	2.97372	0.131938
$509$	0.190605	0.00844844	0.00422422	−	0.999991i	$-0.498655\pi$
$509$	0.190605	0.00844844	0.00422422	+	0.999991i	$0.498655\pi$
$510$	0	0
$511$	−10.2245	−0.452303
$512$	13.6114	0.601546
$513$	0	0
$514$	−6.06770	−0.267635
$515$	−34.3545	−1.51384
$516$	0	0
$517$	3.75872	0.165308
$518$	2.09436	0.0920210
$519$	0	0
$520$	7.95774	0.348970
$521$	−29.5174	−1.29318	−0.646591	−	0.762837i	$-0.723806\pi$
$521$	−29.5174	−1.29318	−0.646591	+	0.762837i	$0.723806\pi$
$522$	0	0
$523$	22.6693	0.991259	0.495629	−	0.868534i	$-0.334937\pi$
$523$	22.6693	0.991259	0.495629	+	0.868534i	$0.334937\pi$
$524$	−5.94214	−0.259584
$525$	0	0
$526$	−30.3234	−1.32216
$527$	1.52586	0.0664674
$528$	0	0
$529$	−20.5897	−0.895204
$530$	−27.3340	−1.18731
$531$	0	0
$532$	−0.146914	−0.00636955
$533$	−2.63090	−0.113957
$534$	0	0
$535$	29.0928	1.25779
$536$	−33.3086	−1.43871
$537$	0	0
$538$	−34.7792	−1.49944
$539$	−6.60197	−0.284367
$540$	0	0
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	10.3174	0.443170
$543$	0	0
$544$	1.11223	0.0476864
$545$	−27.1773	−1.16415
$546$	0	0
$547$	−32.2713	−1.37982	−0.689910	−	0.723896i	$-0.742350\pi$
$547$	−32.2713	−1.37982	−0.689910	+	0.723896i	$0.742350\pi$
$548$	−7.99735	−0.341630
$549$	0	0
$550$	−7.77205	−0.331402
$551$	−5.23513	−0.223024
$552$	0	0
$553$	−9.60197	−0.408317
$554$	20.1711	0.856989
$555$	0	0
$556$	−0.927380	−0.0393297
$557$	−1.68261	−0.0712946	−0.0356473	−	0.999364i	$-0.511349\pi$
$557$	−1.68261	−0.0712946	−0.0356473	+	0.999364i	$0.511349\pi$
$558$	0	0
$559$	−10.6381	−0.449943
$560$	9.20394	0.388937
$561$	0	0
$562$	15.1857	0.640569
$563$	−43.3028	−1.82500	−0.912498	−	0.409080i	$-0.865850\pi$
$563$	−43.3028	−1.82500	−0.912498	+	0.409080i	$0.865850\pi$
$564$	0	0
$565$	−0.0845208	−0.00355582
$566$	20.3474	0.855263
$567$	0	0
$568$	−37.6385	−1.57928
$569$	16.7682	0.702959	0.351479	−	0.936196i	$-0.385679\pi$
$569$	16.7682	0.702959	0.351479	+	0.936196i	$0.385679\pi$
$570$	0	0
$571$	26.8215	1.12244	0.561222	−	0.827665i	$-0.310331\pi$
$571$	26.8215	1.12244	0.561222	+	0.827665i	$0.310331\pi$
$572$	−0.369102	−0.0154329
$573$	0	0
$574$	−2.55479	−0.106635
$575$	−7.83937	−0.326924
$576$	0	0
$577$	20.7382	0.863343	0.431671	−	0.902031i	$-0.357924\pi$
$577$	20.7382	0.863343	0.431671	+	0.902031i	$0.357924\pi$
$578$	25.7187	1.06976
$579$	0	0
$580$	−9.70928	−0.403156
$581$	−1.04718	−0.0434444
$582$	0	0
$583$	−5.60197	−0.232010
$584$	−40.6818	−1.68342
$585$	0	0
$586$	47.1689	1.94853
$587$	−41.0082	−1.69259	−0.846295	−	0.532714i	$-0.821172\pi$
$587$	−41.0082	−1.69259	−0.846295	+	0.532714i	$0.821172\pi$
$588$	0	0
$589$	−1.78539	−0.0735655
$590$	−27.6163	−1.13695
$591$	0	0
$592$	9.92532	0.407928
$593$	15.5936	0.640351	0.320175	−	0.947358i	$-0.396258\pi$
$593$	15.5936	0.640351	0.320175	+	0.947358i	$0.396258\pi$
$594$	0	0
$595$	−1.07838	−0.0442092
$596$	−2.77781	−0.113784
$597$	0	0
$598$	−2.38962	−0.0977189
$599$	31.0205	1.26746	0.633732	−	0.773553i	$-0.281522\pi$
$599$	31.0205	1.26746	0.633732	+	0.773553i	$0.281522\pi$
$600$	0	0
$601$	20.3545	0.830279	0.415140	−	0.909758i	$-0.363733\pi$
$601$	20.3545	0.830279	0.415140	+	0.909758i	$0.363733\pi$
$602$	−10.3303	−0.421033
$603$	0	0
$604$	−1.96039	−0.0797672
$605$	−3.17009	−0.128882
$606$	0	0
$607$	7.85884	0.318981	0.159490	−	0.987199i	$-0.449015\pi$
$607$	7.85884	0.318981	0.159490	+	0.987199i	$0.449015\pi$
$608$	−1.30140	−0.0527788
$609$	0	0
$610$	−20.4124	−0.826474
$611$	−3.75872	−0.152062
$612$	0	0
$613$	−19.4101	−0.783968	−0.391984	−	0.919972i	$-0.628211\pi$
$613$	−19.4101	−0.783968	−0.391984	+	0.919972i	$0.628211\pi$
$614$	−12.7200	−0.513336
$615$	0	0
$616$	1.58372	0.0638098
$617$	−48.2628	−1.94299	−0.971494	−	0.237063i	$-0.923815\pi$
$617$	−48.2628	−1.94299	−0.971494	+	0.237063i	$0.923815\pi$
$618$	0	0
$619$	30.0072	1.20609	0.603045	−	0.797707i	$-0.293954\pi$
$619$	30.0072	1.20609	0.603045	+	0.797707i	$0.293954\pi$
$620$	−3.31124	−0.132983
$621$	0	0
$622$	18.1750	0.728751
$623$	0.738205	0.0295755
$624$	0	0
$625$	−24.7503	−0.990013
$626$	−50.3995	−2.01437
$627$	0	0
$628$	1.16433	0.0464618
$629$	−1.16290	−0.0463678
$630$	0	0
$631$	8.82991	0.351513	0.175757	−	0.984434i	$-0.443763\pi$
$631$	8.82991	0.351513	0.175757	+	0.984434i	$0.443763\pi$
$632$	−38.2050	−1.51971
$633$	0	0
$634$	−1.23636	−0.0491020
$635$	−25.5402	−1.01353
$636$	0	0
$637$	6.60197	0.261580
$638$	−12.7721	−0.505650
$639$	0	0
$640$	42.4957	1.67979
$641$	15.0928	0.596128	0.298064	−	0.954546i	$-0.403659\pi$
$641$	15.0928	0.596128	0.298064	+	0.954546i	$0.403659\pi$
$642$	0	0
$643$	−41.9637	−1.65489	−0.827443	−	0.561549i	$-0.810206\pi$
$643$	−41.9637	−1.65489	−0.827443	+	0.561549i	$0.810206\pi$
$644$	−0.361529	−0.0142462
$645$	0	0
$646$	0.523590	0.0206004
$647$	−6.02666	−0.236933	−0.118466	−	0.992958i	$-0.537798\pi$
$647$	−6.02666	−0.236933	−0.118466	+	0.992958i	$0.537798\pi$
$648$	0	0
$649$	−5.65983	−0.222168
$650$	7.77205	0.304845
$651$	0	0
$652$	−4.15941	−0.162895
$653$	−35.7731	−1.39991	−0.699955	−	0.714187i	$-0.746797\pi$
$653$	−35.7731	−1.39991	−0.699955	+	0.714187i	$0.746797\pi$
$654$	0	0
$655$	51.0349	1.99410
$656$	−12.1073	−0.472711
$657$	0	0
$658$	−3.64999	−0.142291
$659$	−25.3874	−0.988951	−0.494475	−	0.869192i	$-0.664640\pi$
$659$	−25.3874	−0.988951	−0.494475	+	0.869192i	$0.664640\pi$
$660$	0	0
$661$	18.1133	0.704525	0.352262	−	0.935901i	$-0.385412\pi$
$661$	18.1133	0.704525	0.352262	+	0.935901i	$0.385412\pi$
$662$	−6.69141	−0.260069
$663$	0	0
$664$	−4.16660	−0.161695
$665$	1.26180	0.0489303
$666$	0	0
$667$	−12.8827	−0.498819
$668$	−2.19326	−0.0848597
$669$	0	0
$670$	−64.7442	−2.50129
$671$	−4.18342	−0.161499
$672$	0	0
$673$	−19.4719	−0.750586	−0.375293	−	0.926906i	$-0.622458\pi$
$673$	−19.4719	−0.750586	−0.375293	+	0.926906i	$0.622458\pi$
$674$	17.2039	0.662671
$675$	0	0
$676$	0.369102	0.0141962
$677$	−12.7805	−0.491193	−0.245597	−	0.969372i	$-0.578984\pi$
$677$	−12.7805	−0.491193	−0.245597	+	0.969372i	$0.578984\pi$
$678$	0	0
$679$	−1.16290	−0.0446280
$680$	−4.29072	−0.164542
$681$	0	0
$682$	−4.35577	−0.166791
$683$	44.0410	1.68518	0.842592	−	0.538553i	$-0.181029\pi$
$683$	44.0410	1.68518	0.842592	+	0.538553i	$0.181029\pi$
$684$	0	0
$685$	68.6863	2.62437
$686$	13.2085	0.504302
$687$	0	0
$688$	−48.9561	−1.86644
$689$	5.60197	0.213418
$690$	0	0
$691$	24.1496	0.918693	0.459346	−	0.888257i	$-0.348084\pi$
$691$	24.1496	0.918693	0.459346	+	0.888257i	$0.348084\pi$
$692$	6.88920	0.261888
$693$	0	0
$694$	−10.4345	−0.396090
$695$	7.96493	0.302127
$696$	0	0
$697$	1.41855	0.0537314
$698$	6.52813	0.247093
$699$	0	0
$700$	1.17584	0.0444427
$701$	42.0833	1.58946	0.794732	−	0.606960i	$-0.207611\pi$
$701$	42.0833	1.58946	0.794732	+	0.606960i	$0.207611\pi$
$702$	0	0
$703$	1.36069	0.0513195
$704$	6.02893	0.227224
$705$	0	0
$706$	26.2089	0.986383
$707$	2.55025	0.0959121
$708$	0	0
$709$	21.6598	0.813452	0.406726	−	0.913550i	$-0.366670\pi$
$709$	21.6598	0.813452	0.406726	+	0.913550i	$0.366670\pi$
$710$	−73.1605	−2.74566
$711$	0	0
$712$	2.93722	0.110077
$713$	−4.39350	−0.164538
$714$	0	0
$715$	3.17009	0.118555
$716$	2.01825	0.0754256
$717$	0	0
$718$	−29.2001	−1.08974
$719$	−6.83710	−0.254981	−0.127490	−	0.991840i	$-0.540692\pi$
$719$	−6.83710	−0.254981	−0.127490	+	0.991840i	$0.540692\pi$
$720$	0	0
$721$	6.83710	0.254627
$722$	28.6319	1.06557
$723$	0	0
$724$	5.92389	0.220160
$725$	41.8999	1.55612
$726$	0	0
$727$	−33.4452	−1.24041	−0.620207	−	0.784438i	$-0.712951\pi$
$727$	−33.4452	−1.24041	−0.620207	+	0.784438i	$0.712951\pi$
$728$	−1.58372	−0.0586964
$729$	0	0
$730$	−79.0759	−2.92673
$731$	5.73594	0.212151
$732$	0	0
$733$	52.3051	1.93193	0.965966	−	0.258667i	$-0.0832833\pi$
$733$	52.3051	1.93193	0.965966	+	0.258667i	$0.0832833\pi$
$734$	7.07384	0.261100
$735$	0	0
$736$	−3.20251	−0.118046
$737$	−13.2690	−0.488769
$738$	0	0
$739$	10.2595	0.377403	0.188701	−	0.982035i	$-0.439572\pi$
$739$	10.2595	0.377403	0.188701	+	0.982035i	$0.439572\pi$
$740$	2.52359	0.0927690
$741$	0	0
$742$	5.43991	0.199705
$743$	−6.60811	−0.242428	−0.121214	−	0.992626i	$-0.538679\pi$
$743$	−6.60811	−0.242428	−0.121214	+	0.992626i	$0.538679\pi$
$744$	0	0
$745$	23.8576	0.874076
$746$	53.4017	1.95518
$747$	0	0
$748$	0.199016	0.00727674
$749$	−5.78992	−0.211559
$750$	0	0
$751$	−4.28231	−0.156264	−0.0781319	−	0.996943i	$-0.524896\pi$
$751$	−4.28231	−0.156264	−0.0781319	+	0.996943i	$0.524896\pi$
$752$	−17.2975	−0.630776
$753$	0	0
$754$	12.7721	0.465131
$755$	16.8371	0.612765
$756$	0	0
$757$	−46.8865	−1.70412	−0.852060	−	0.523444i	$-0.824647\pi$
$757$	−46.8865	−1.70412	−0.852060	+	0.523444i	$0.824647\pi$
$758$	−33.5429	−1.21833
$759$	0	0
$760$	5.02052	0.182113
$761$	−30.1795	−1.09401	−0.547004	−	0.837130i	$-0.684232\pi$
$761$	−30.1795	−1.09401	−0.547004	+	0.837130i	$0.684232\pi$
$762$	0	0
$763$	5.40871	0.195808
$764$	−8.86376	−0.320680
$765$	0	0
$766$	13.9733	0.504877
$767$	5.65983	0.204365
$768$	0	0
$769$	7.98771	0.288044	0.144022	−	0.989574i	$-0.453996\pi$
$769$	7.98771	0.288044	0.144022	+	0.989574i	$0.453996\pi$
$770$	3.07838	0.110937
$771$	0	0
$772$	−2.44748	−0.0880867
$773$	16.6342	0.598291	0.299145	−	0.954208i	$-0.403298\pi$
$773$	16.6342	0.598291	0.299145	+	0.954208i	$0.403298\pi$
$774$	0	0
$775$	14.2895	0.513294
$776$	−4.62702	−0.166100
$777$	0	0
$778$	−52.9192	−1.89724
$779$	−1.65983	−0.0594695
$780$	0	0
$781$	−14.9939	−0.536522
$782$	1.28846	0.0460751
$783$	0	0
$784$	30.3820	1.08507
$785$	−10.0000	−0.356915
$786$	0	0
$787$	31.2001	1.11216	0.556081	−	0.831128i	$-0.312304\pi$
$787$	31.2001	1.11216	0.556081	+	0.831128i	$0.312304\pi$
$788$	−4.24271	−0.151140
$789$	0	0
$790$	−74.2616	−2.64211
$791$	0.0168210	0.000598085 0
$792$	0	0
$793$	4.18342	0.148558
$794$	45.1096	1.60088
$795$	0	0
$796$	−2.52359	−0.0894463
$797$	33.1506	1.17425	0.587127	−	0.809494i	$-0.300259\pi$
$797$	33.1506	1.17425	0.587127	+	0.809494i	$0.300259\pi$
$798$	0	0
$799$	2.02666	0.0716981
$800$	10.4159	0.368258
$801$	0	0
$802$	3.78416	0.133623
$803$	−16.2062	−0.571905
$804$	0	0
$805$	3.10504	0.109438
$806$	4.35577	0.153426
$807$	0	0
$808$	10.1471	0.356975
$809$	−7.17850	−0.252383	−0.126191	−	0.992006i	$-0.540275\pi$
$809$	−7.17850	−0.252383	−0.126191	+	0.992006i	$0.540275\pi$
$810$	0	0
$811$	27.1545	0.953523	0.476762	−	0.879033i	$-0.341811\pi$
$811$	27.1545	0.953523	0.476762	+	0.879033i	$0.341811\pi$
$812$	1.93230	0.0678104
$813$	0	0
$814$	3.31965	0.116354
$815$	35.7237	1.25134
$816$	0	0
$817$	−6.71154	−0.234807
$818$	31.5837	1.10430
$819$	0	0
$820$	−3.07838	−0.107502
$821$	−16.1918	−0.565099	−0.282549	−	0.959253i	$-0.591180\pi$
$821$	−16.1918	−0.565099	−0.282549	+	0.959253i	$0.591180\pi$
$822$	0	0
$823$	−6.07223	−0.211665	−0.105832	−	0.994384i	$-0.533751\pi$
$823$	−6.07223	−0.211665	−0.105832	+	0.994384i	$0.533751\pi$
$824$	27.2039	0.947694
$825$	0	0
$826$	5.49609	0.191233
$827$	23.8492	0.829318	0.414659	−	0.909977i	$-0.363901\pi$
$827$	23.8492	0.829318	0.414659	+	0.909977i	$0.363901\pi$
$828$	0	0
$829$	−3.72753	−0.129462	−0.0647312	−	0.997903i	$-0.520619\pi$
$829$	−3.72753	−0.129462	−0.0647312	+	0.997903i	$0.520619\pi$
$830$	−8.09890	−0.281117
$831$	0	0
$832$	−6.02893	−0.209016
$833$	−3.55971	−0.123337
$834$	0	0
$835$	18.8371	0.651885
$836$	−0.232866	−0.00805383
$837$	0	0
$838$	2.58986	0.0894652
$839$	−30.7670	−1.06219	−0.531097	−	0.847311i	$-0.678220\pi$
$839$	−30.7670	−1.06219	−0.531097	+	0.847311i	$0.678220\pi$
$840$	0	0
$841$	39.8554	1.37432
$842$	−3.40871	−0.117472
$843$	0	0
$844$	1.54987	0.0533486
$845$	−3.17009	−0.109054
$846$	0	0
$847$	0.630898	0.0216779
$848$	25.7801	0.885292
$849$	0	0
$850$	−4.19061	−0.143737
$851$	3.34841	0.114782
$852$	0	0
$853$	−37.1338	−1.27144	−0.635718	−	0.771921i	$-0.719296\pi$
$853$	−37.1338	−1.27144	−0.635718	+	0.771921i	$0.719296\pi$
$854$	4.06239	0.139012
$855$	0	0
$856$	−23.0373	−0.787400
$857$	0.611424	0.0208858	0.0104429	−	0.999945i	$-0.496676\pi$
$857$	0.611424	0.0208858	0.0104429	+	0.999945i	$0.496676\pi$
$858$	0	0
$859$	11.1194	0.379390	0.189695	−	0.981843i	$-0.439250\pi$
$859$	11.1194	0.379390	0.189695	+	0.981843i	$0.439250\pi$
$860$	−12.4475	−0.424456
$861$	0	0
$862$	31.0181	1.05648
$863$	36.6491	1.24755	0.623776	−	0.781603i	$-0.285598\pi$
$863$	36.6491	1.24755	0.623776	+	0.781603i	$0.285598\pi$
$864$	0	0
$865$	−59.1689	−2.01180
$866$	50.0599	1.70111
$867$	0	0
$868$	0.658990	0.0223676
$869$	−15.2195	−0.516287
$870$	0	0
$871$	13.2690	0.449602
$872$	21.5206	0.728778
$873$	0	0
$874$	−1.50761	−0.0509955
$875$	−0.0988967	−0.00334332
$876$	0	0
$877$	−41.7998	−1.41148	−0.705739	−	0.708472i	$-0.749385\pi$
$877$	−41.7998	−1.41148	−0.705739	+	0.708472i	$0.749385\pi$
$878$	18.0338	0.608613
$879$	0	0
$880$	14.5886	0.491783
$881$	41.5174	1.39876	0.699379	−	0.714751i	$-0.253460\pi$
$881$	41.5174	1.39876	0.699379	+	0.714751i	$0.253460\pi$
$882$	0	0
$883$	−17.6742	−0.594784	−0.297392	−	0.954755i	$-0.596117\pi$
$883$	−17.6742	−0.594784	−0.297392	+	0.954755i	$0.596117\pi$
$884$	−0.199016	−0.00669363
$885$	0	0
$886$	−46.7214	−1.56964
$887$	9.70086	0.325723	0.162862	−	0.986649i	$-0.447928\pi$
$887$	9.70086	0.325723	0.162862	+	0.986649i	$0.447928\pi$
$888$	0	0
$889$	5.08291	0.170475
$890$	5.70928	0.191375
$891$	0	0
$892$	4.43188	0.148390
$893$	−2.37137	−0.0793549
$894$	0	0
$895$	−17.3340	−0.579413
$896$	−8.45732	−0.282539
$897$	0	0
$898$	7.38613	0.246478
$899$	23.4824	0.783181
$900$	0	0
$901$	−3.02052	−0.100628
$902$	−4.04945	−0.134832
$903$	0	0
$904$	0.0669285	0.00222601
$905$	−50.8781	−1.69125
$906$	0	0
$907$	−21.0784	−0.699896	−0.349948	−	0.936769i	$-0.613801\pi$
$907$	−21.0784	−0.699896	−0.349948	+	0.936769i	$0.613801\pi$
$908$	−9.31124	−0.309004
$909$	0	0
$910$	−3.07838	−0.102047
$911$	−27.2807	−0.903850	−0.451925	−	0.892056i	$-0.649263\pi$
$911$	−27.2807	−0.903850	−0.451925	+	0.892056i	$0.649263\pi$
$912$	0	0
$913$	−1.65983	−0.0549323
$914$	59.2060	1.95836
$915$	0	0
$916$	4.90027	0.161909
$917$	−10.1568	−0.335406
$918$	0	0
$919$	58.4112	1.92681	0.963404	−	0.268055i	$-0.0863809\pi$
$919$	58.4112	1.92681	0.963404	+	0.268055i	$0.0863809\pi$
$920$	12.3545	0.407317
$921$	0	0
$922$	−34.8332	−1.14717
$923$	14.9939	0.493529
$924$	0	0
$925$	−10.8904	−0.358075
$926$	−37.3931	−1.22881
$927$	0	0
$928$	17.1168	0.561885
$929$	−41.5969	−1.36475	−0.682375	−	0.731003i	$-0.739053\pi$
$929$	−41.5969	−1.36475	−0.682375	+	0.731003i	$0.739053\pi$
$930$	0	0
$931$	4.16517	0.136508
$932$	−3.27739	−0.107355
$933$	0	0
$934$	−14.5509	−0.476120
$935$	−1.70928	−0.0558993
$936$	0	0
$937$	−56.7091	−1.85261	−0.926303	−	0.376780i	$-0.877031\pi$
$937$	−56.7091	−1.85261	−0.926303	+	0.376780i	$0.877031\pi$
$938$	12.8851	0.420714
$939$	0	0
$940$	−4.39803	−0.143448
$941$	−29.0556	−0.947185	−0.473593	−	0.880744i	$-0.657043\pi$
$941$	−29.0556	−0.947185	−0.473593	+	0.880744i	$0.657043\pi$
$942$	0	0
$943$	−4.08452	−0.133010
$944$	26.0463	0.847736
$945$	0	0
$946$	−16.3740	−0.532365
$947$	45.2183	1.46940	0.734699	−	0.678393i	$-0.237323\pi$
$947$	45.2183	1.46940	0.734699	+	0.678393i	$0.237323\pi$
$948$	0	0
$949$	16.2062	0.526076
$950$	4.90337	0.159086
$951$	0	0
$952$	0.853922	0.0276758
$953$	−40.0521	−1.29741	−0.648707	−	0.761038i	$-0.724690\pi$
$953$	−40.0521	−1.29741	−0.648707	+	0.761038i	$0.724690\pi$
$954$	0	0
$955$	76.1276	2.46343
$956$	0.495498	0.0160255
$957$	0	0
$958$	−60.2062	−1.94517
$959$	−13.6697	−0.441417
$960$	0	0
$961$	−22.9916	−0.741664
$962$	−3.31965	−0.107030
$963$	0	0
$964$	6.06770	0.195427
$965$	21.0205	0.676674
$966$	0	0
$967$	−17.8615	−0.574387	−0.287193	−	0.957873i	$-0.592722\pi$
$967$	−17.8615	−0.574387	−0.287193	+	0.957873i	$0.592722\pi$
$968$	2.51026	0.0806828
$969$	0	0
$970$	−8.99386	−0.288775
$971$	−43.1278	−1.38404	−0.692019	−	0.721879i	$-0.743279\pi$
$971$	−43.1278	−1.38404	−0.692019	+	0.721879i	$0.743279\pi$
$972$	0	0
$973$	−1.58515	−0.0508175
$974$	53.2606	1.70658
$975$	0	0
$976$	19.2520	0.616240
$977$	−39.8381	−1.27454	−0.637268	−	0.770643i	$-0.719935\pi$
$977$	−39.8381	−1.27454	−0.637268	+	0.770643i	$0.719935\pi$
$978$	0	0
$979$	1.17009	0.0373961
$980$	7.72487	0.246762
$981$	0	0
$982$	35.4740	1.13202
$983$	38.7670	1.23647	0.618237	−	0.785992i	$-0.287847\pi$
$983$	38.7670	1.23647	0.618237	+	0.785992i	$0.287847\pi$
$984$	0	0
$985$	36.4391	1.16105
$986$	−6.88655	−0.219312
$987$	0	0
$988$	0.232866	0.00740845
$989$	−16.5158	−0.525173
$990$	0	0
$991$	−33.8043	−1.07383	−0.536914	−	0.843637i	$-0.680410\pi$
$991$	−33.8043	−1.07383	−0.536914	+	0.843637i	$0.680410\pi$
$992$	5.83749	0.185340
$993$	0	0
$994$	14.5601	0.461818
$995$	21.6742	0.687118
$996$	0	0
$997$	49.8166	1.57771	0.788853	−	0.614581i	$-0.210675\pi$
$997$	49.8166	1.57771	0.788853	+	0.614581i	$0.210675\pi$
$998$	−56.2688	−1.78116
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1287.2.a.h.1.2		3
3.2	odd	2		429.2.a.g.1.2	✓	3
12.11	even	2		6864.2.a.bs.1.3		3
33.32	even	2		4719.2.a.q.1.2		3
39.38	odd	2		5577.2.a.j.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.a.g.1.2	✓	3	3.2	odd	2
1287.2.a.h.1.2		3	1.1	even	1	trivial
4719.2.a.q.1.2		3	33.32	even	2
5577.2.a.j.1.2		3	39.38	odd	2
6864.2.a.bs.1.3		3	12.11	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:	\( N \)	\(=\)	\( 1287 = 3^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1287.a (trivial)

\(f(q)\)	\(=\)	\(q-1.53919 q^{2} +0.369102 q^{4} -3.17009 q^{5} +0.630898 q^{7} +2.51026 q^{8} +O(q^{10})\)	\(q-1.53919 q^{2} +0.369102 q^{4} -3.17009 q^{5} +0.630898 q^{7} +2.51026 q^{8} +4.87936 q^{10} +1.00000 q^{11} -1.00000 q^{13} -0.971071 q^{14} -4.60197 q^{16} +0.539189 q^{17} -0.630898 q^{19} -1.17009 q^{20} -1.53919 q^{22} -1.55252 q^{23} +5.04945 q^{25} +1.53919 q^{26} +0.232866 q^{28} +8.29791 q^{29} +2.82991 q^{31} +2.06278 q^{32} -0.829914 q^{34} -2.00000 q^{35} -2.15676 q^{37} +0.971071 q^{38} -7.95774 q^{40} +2.63090 q^{41} +10.6381 q^{43} +0.369102 q^{44} +2.38962 q^{46} +3.75872 q^{47} -6.60197 q^{49} -7.77205 q^{50} -0.369102 q^{52} -5.60197 q^{53} -3.17009 q^{55} +1.58372 q^{56} -12.7721 q^{58} -5.65983 q^{59} -4.18342 q^{61} -4.35577 q^{62} +6.02893 q^{64} +3.17009 q^{65} -13.2690 q^{67} +0.199016 q^{68} +3.07838 q^{70} -14.9939 q^{71} -16.2062 q^{73} +3.31965 q^{74} -0.232866 q^{76} +0.630898 q^{77} -15.2195 q^{79} +14.5886 q^{80} -4.04945 q^{82} -1.65983 q^{83} -1.70928 q^{85} -16.3740 q^{86} +2.51026 q^{88} +1.17009 q^{89} -0.630898 q^{91} -0.573039 q^{92} -5.78539 q^{94} +2.00000 q^{95} -1.84324 q^{97} +10.1617 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 5 q^{4} - 4 q^{5} - 2 q^{7} - 9 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 5 * q^4 - 4 * q^5 - 2 * q^7 - 9 * q^8	\( 3 q - 3 q^{2} + 5 q^{4} - 4 q^{5} - 2 q^{7} - 9 q^{8} + 2 q^{10} + 3 q^{11} - 3 q^{13} + 12 q^{14} + 5 q^{16} + 2 q^{19} + 2 q^{20} - 3 q^{22} - 4 q^{23} - 3 q^{25} + 3 q^{26} - 22 q^{28} - 2 q^{29} + 14 q^{31} - 11 q^{32} - 8 q^{34} - 6 q^{35} - 12 q^{38} - 8 q^{40} + 4 q^{41} - 6 q^{43} + 5 q^{44} - 22 q^{46} - 14 q^{47} - q^{49} + q^{50} - 5 q^{52} + 2 q^{53} - 4 q^{55} + 32 q^{56} - 14 q^{58} - 28 q^{59} - 8 q^{61} - 16 q^{62} + 33 q^{64} + 4 q^{65} + 2 q^{67} + 10 q^{68} + 6 q^{70} - 10 q^{71} - 24 q^{73} + 32 q^{74} + 22 q^{76} - 2 q^{77} - 22 q^{79} + 24 q^{80} + 6 q^{82} - 16 q^{83} + 2 q^{85} - 6 q^{86} - 9 q^{88} - 2 q^{89} + 2 q^{91} + 32 q^{92} + 6 q^{94} + 6 q^{95} - 12 q^{97} - 23 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 5 * q^4 - 4 * q^5 - 2 * q^7 - 9 * q^8 + 2 * q^10 + 3 * q^11 - 3 * q^13 + 12 * q^14 + 5 * q^16 + 2 * q^19 + 2 * q^20 - 3 * q^22 - 4 * q^23 - 3 * q^25 + 3 * q^26 - 22 * q^28 - 2 * q^29 + 14 * q^31 - 11 * q^32 - 8 * q^34 - 6 * q^35 - 12 * q^38 - 8 * q^40 + 4 * q^41 - 6 * q^43 + 5 * q^44 - 22 * q^46 - 14 * q^47 - q^49 + q^50 - 5 * q^52 + 2 * q^53 - 4 * q^55 + 32 * q^56 - 14 * q^58 - 28 * q^59 - 8 * q^61 - 16 * q^62 + 33 * q^64 + 4 * q^65 + 2 * q^67 + 10 * q^68 + 6 * q^70 - 10 * q^71 - 24 * q^73 + 32 * q^74 + 22 * q^76 - 2 * q^77 - 22 * q^79 + 24 * q^80 + 6 * q^82 - 16 * q^83 + 2 * q^85 - 6 * q^86 - 9 * q^88 - 2 * q^89 + 2 * q^91 + 32 * q^92 + 6 * q^94 + 6 * q^95 - 12 * q^97 - 23 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more