LMFDB - Embedded newform 1287.2.a.p.1.4

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:	$0$
Dimension:	$5$
Coefficient field:	5.5.368464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 6x^{3} + 6x^{2} + 6x - 4 $ x^5 - 2x^4 - 6x^3 + 6x^2 + 6x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.09027$ 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)

$f(q)$	$=$	$q+2.09027 q^{2} +2.36924 q^{4} +3.20929 q^{5} -0.388134 q^{7} +0.771813 q^{8} +O(q^{10})$	\(q+2.09027 q^{2} +2.36924 q^{4} +3.20929 q^{5} -0.388134 q^{7} +0.771813 q^{8} +6.70830 q^{10} -1.00000 q^{11} +1.00000 q^{13} -0.811305 q^{14} -3.12518 q^{16} +5.09027 q^{17} +5.49901 q^{19} +7.60359 q^{20} -2.09027 q^{22} -2.51790 q^{23} +5.29957 q^{25} +2.09027 q^{26} -0.919582 q^{28} +1.17884 q^{29} +2.69755 q^{31} -8.07611 q^{32} +10.6401 q^{34} -1.24564 q^{35} +6.39429 q^{37} +11.4944 q^{38} +2.47697 q^{40} -10.6561 q^{41} -10.7591 q^{43} -2.36924 q^{44} -5.26310 q^{46} +7.61803 q^{47} -6.84935 q^{49} +11.0775 q^{50} +2.36924 q^{52} -3.84935 q^{53} -3.20929 q^{55} -0.299567 q^{56} +2.46410 q^{58} +4.81747 q^{59} +2.68770 q^{61} +5.63863 q^{62} -10.6309 q^{64} +3.20929 q^{65} -12.3202 q^{67} +12.0601 q^{68} -2.60372 q^{70} -9.31332 q^{71} -0.946069 q^{73} +13.3658 q^{74} +13.0285 q^{76} +0.388134 q^{77} -6.41948 q^{79} -10.0296 q^{80} -22.2741 q^{82} -5.18355 q^{83} +16.3362 q^{85} -22.4894 q^{86} -0.771813 q^{88} +10.6090 q^{89} -0.388134 q^{91} -5.96551 q^{92} +15.9238 q^{94} +17.6479 q^{95} +5.95763 q^{97} -14.3170 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{2} + 7 q^{4} + 4 q^{5} - 4 q^{7} + 9 q^{8}+O(q^{10}) $ 5 * q + 3 * q^2 + 7 * q^4 + 4 * q^5 - 4 * q^7 + 9 * q^8	$ 5 q + 3 q^{2} + 7 q^{4} + 4 q^{5} - 4 q^{7} + 9 q^{8} - 6 q^{10} - 5 q^{11} + 5 q^{13} + 6 q^{14} + 19 q^{16} + 18 q^{17} - 2 q^{20} - 3 q^{22} + 8 q^{23} + 7 q^{25} + 3 q^{26} - 8 q^{28} + 20 q^{29} - 8 q^{31} + 31 q^{32} + 26 q^{34} + 6 q^{35} + 4 q^{37} + 18 q^{38} - 20 q^{40} + 8 q^{41} - 22 q^{43} - 7 q^{44} + 6 q^{46} + 6 q^{47} + 5 q^{49} + 11 q^{50} + 7 q^{52} + 20 q^{53} - 4 q^{55} + 18 q^{56} + 12 q^{58} - 16 q^{59} - 4 q^{61} - 26 q^{62} + 11 q^{64} + 4 q^{65} - 20 q^{67} + 36 q^{68} + 14 q^{70} + 6 q^{71} - 12 q^{73} - 20 q^{74} - 16 q^{76} + 4 q^{77} + 6 q^{79} - 52 q^{80} - 18 q^{82} + 4 q^{83} + 6 q^{85} - 46 q^{86} - 9 q^{88} + 10 q^{89} - 4 q^{91} + 68 q^{92} - 2 q^{94} + 2 q^{95} + 4 q^{97} - 45 q^{98}+O(q^{100}) $ 5 * q + 3 * q^2 + 7 * q^4 + 4 * q^5 - 4 * q^7 + 9 * q^8 - 6 * q^10 - 5 * q^11 + 5 * q^13 + 6 * q^14 + 19 * q^16 + 18 * q^17 - 2 * q^20 - 3 * q^22 + 8 * q^23 + 7 * q^25 + 3 * q^26 - 8 * q^28 + 20 * q^29 - 8 * q^31 + 31 * q^32 + 26 * q^34 + 6 * q^35 + 4 * q^37 + 18 * q^38 - 20 * q^40 + 8 * q^41 - 22 * q^43 - 7 * q^44 + 6 * q^46 + 6 * q^47 + 5 * q^49 + 11 * q^50 + 7 * q^52 + 20 * q^53 - 4 * q^55 + 18 * q^56 + 12 * q^58 - 16 * q^59 - 4 * q^61 - 26 * q^62 + 11 * q^64 + 4 * q^65 - 20 * q^67 + 36 * q^68 + 14 * q^70 + 6 * q^71 - 12 * q^73 - 20 * q^74 - 16 * q^76 + 4 * q^77 + 6 * q^79 - 52 * q^80 - 18 * q^82 + 4 * q^83 + 6 * q^85 - 46 * q^86 - 9 * q^88 + 10 * q^89 - 4 * q^91 + 68 * q^92 - 2 * q^94 + 2 * q^95 + 4 * q^97 - 45 * 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 $

$2$	2.09027	1.47805	0.739023	−	0.673680i	$-0.235287\pi$
$2$	2.09027	1.47805	0.739023	+	0.673680i	$0.235287\pi$
$3$	0	0
$3$	0	0
$4$	2.36924	1.18462
$5$	3.20929	1.43524	0.717620	−	0.696435i	$-0.245232\pi$
$5$	3.20929	1.43524	0.717620	+	0.696435i	$0.245232\pi$
$6$	0	0
$7$	−0.388134	−0.146701	−0.0733504	−	0.997306i	$-0.523369\pi$
$7$	−0.388134	−0.146701	−0.0733504	+	0.997306i	$0.523369\pi$
$8$	0.771813	0.272877
$9$	0	0
$10$	6.70830	2.12135
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−0.811305	−0.216830
$15$	0	0
$16$	−3.12518	−0.781295
$17$	5.09027	1.23457	0.617286	−	0.786739i	$-0.288232\pi$
$17$	5.09027	1.23457	0.617286	+	0.786739i	$0.288232\pi$
$18$	0	0
$19$	5.49901	1.26156	0.630779	−	0.775962i	$-0.282735\pi$
$19$	5.49901	1.26156	0.630779	+	0.775962i	$0.282735\pi$
$20$	7.60359	1.70021
$21$	0	0
$22$	−2.09027	−0.445648
$23$	−2.51790	−0.525018	−0.262509	−	0.964930i	$-0.584550\pi$
$23$	−2.51790	−0.525018	−0.262509	+	0.964930i	$0.584550\pi$
$24$	0	0
$25$	5.29957	1.05991
$26$	2.09027	0.409936
$27$	0	0
$28$	−0.919582	−0.173785
$29$	1.17884	0.218905	0.109453	−	0.993992i	$-0.465090\pi$
$29$	1.17884	0.218905	0.109453	+	0.993992i	$0.465090\pi$
$30$	0	0
$31$	2.69755	0.484495	0.242248	−	0.970214i	$-0.422115\pi$
$31$	2.69755	0.484495	0.242248	+	0.970214i	$0.422115\pi$
$32$	−8.07611	−1.42767
$33$	0	0
$34$	10.6401	1.82475
$35$	−1.24564	−0.210551
$36$	0	0
$37$	6.39429	1.05122	0.525608	−	0.850727i	$-0.323838\pi$
$37$	6.39429	1.05122	0.525608	+	0.850727i	$0.323838\pi$
$38$	11.4944	1.86464
$39$	0	0
$40$	2.47697	0.391644
$41$	−10.6561	−1.66420	−0.832100	−	0.554626i	$-0.812861\pi$
$41$	−10.6561	−1.66420	−0.832100	+	0.554626i	$0.812861\pi$
$42$	0	0
$43$	−10.7591	−1.64074	−0.820372	−	0.571830i	$-0.806234\pi$
$43$	−10.7591	−1.64074	−0.820372	+	0.571830i	$0.806234\pi$
$44$	−2.36924	−0.357176
$45$	0	0
$46$	−5.26310	−0.776001
$47$	7.61803	1.11120	0.555602	−	0.831449i	$-0.312488\pi$
$47$	7.61803	1.11120	0.555602	+	0.831449i	$0.312488\pi$
$48$	0	0
$49$	−6.84935	−0.978479
$50$	11.0775	1.56660
$51$	0	0
$52$	2.36924	0.328554
$53$	−3.84935	−0.528749	−0.264375	−	0.964420i	$-0.585166\pi$
$53$	−3.84935	−0.528749	−0.264375	+	0.964420i	$0.585166\pi$
$54$	0	0
$55$	−3.20929	−0.432741
$56$	−0.299567	−0.0400313
$57$	0	0
$58$	2.46410	0.323552
$59$	4.81747	0.627181	0.313590	−	0.949558i	$-0.398468\pi$
$59$	4.81747	0.627181	0.313590	+	0.949558i	$0.398468\pi$
$60$	0	0
$61$	2.68770	0.344125	0.172062	−	0.985086i	$-0.444957\pi$
$61$	2.68770	0.344125	0.172062	+	0.985086i	$0.444957\pi$
$62$	5.63863	0.716106
$63$	0	0
$64$	−10.6309	−1.32886
$65$	3.20929	0.398064
$66$	0	0
$67$	−12.3202	−1.50515	−0.752574	−	0.658508i	$-0.771188\pi$
$67$	−12.3202	−1.50515	−0.752574	+	0.658508i	$0.771188\pi$
$68$	12.0601	1.46250
$69$	0	0
$70$	−2.60372	−0.311204
$71$	−9.31332	−1.10529	−0.552644	−	0.833418i	$-0.686381\pi$
$71$	−9.31332	−1.10529	−0.552644	+	0.833418i	$0.686381\pi$
$72$	0	0
$73$	−0.946069	−0.110729	−0.0553645	−	0.998466i	$-0.517632\pi$
$73$	−0.946069	−0.110729	−0.0553645	+	0.998466i	$0.517632\pi$
$74$	13.3658	1.55374
$75$	0	0
$76$	13.0285	1.49447
$77$	0.388134	0.0442319
$78$	0	0
$79$	−6.41948	−0.722248	−0.361124	−	0.932518i	$-0.617607\pi$
$79$	−6.41948	−0.722248	−0.361124	+	0.932518i	$0.617607\pi$
$80$	−10.0296	−1.12135
$81$	0	0
$82$	−22.2741	−2.45976
$83$	−5.18355	−0.568969	−0.284485	−	0.958681i	$-0.591822\pi$
$83$	−5.18355	−0.568969	−0.284485	+	0.958681i	$0.591822\pi$
$84$	0	0
$85$	16.3362	1.77191
$86$	−22.4894	−2.42510
$87$	0	0
$88$	−0.771813	−0.0822755
$89$	10.6090	1.12455	0.562275	−	0.826950i	$-0.309926\pi$
$89$	10.6090	1.12455	0.562275	+	0.826950i	$0.309926\pi$
$90$	0	0
$91$	−0.388134	−0.0406875
$92$	−5.96551	−0.621947
$93$	0	0
$94$	15.9238	1.64241
$95$	17.6479	1.81064
$96$	0	0
$97$	5.95763	0.604906	0.302453	−	0.953164i	$-0.402195\pi$
$97$	5.95763	0.604906	0.302453	+	0.953164i	$0.402195\pi$
$98$	−14.3170	−1.44624
$99$	0	0
$100$	12.5559	1.25559
$101$	15.8700	1.57912	0.789560	−	0.613674i	$-0.210309\pi$
$101$	15.8700	1.57912	0.789560	+	0.613674i	$0.210309\pi$
$102$	0	0
$103$	−6.75319	−0.665412	−0.332706	−	0.943031i	$-0.607962\pi$
$103$	−6.75319	−0.665412	−0.332706	+	0.943031i	$0.607962\pi$
$104$	0.771813	0.0756825
$105$	0	0
$106$	−8.04620	−0.781516
$107$	10.1517	0.981399	0.490699	−	0.871329i	$-0.336741\pi$
$107$	10.1517	0.981399	0.490699	+	0.871329i	$0.336741\pi$
$108$	0	0
$109$	−14.9421	−1.43119	−0.715596	−	0.698514i	$-0.753845\pi$
$109$	−14.9421	−1.43119	−0.715596	+	0.698514i	$0.753845\pi$
$110$	−6.70830	−0.639611
$111$	0	0
$112$	1.21299	0.114617
$113$	−9.22475	−0.867792	−0.433896	−	0.900963i	$-0.642861\pi$
$113$	−9.22475	−0.867792	−0.433896	+	0.900963i	$0.642861\pi$
$114$	0	0
$115$	−8.08068	−0.753527
$116$	2.79295	0.259319
$117$	0	0
$118$	10.0698	0.927002
$119$	−1.97571	−0.181113
$120$	0	0
$121$	1.00000	0.0909091
$122$	5.61803	0.508632
$123$	0	0
$124$	6.39116	0.573943
$125$	0.961397	0.0859900
$126$	0	0
$127$	−12.1504	−1.07817	−0.539085	−	0.842251i	$-0.681230\pi$
$127$	−12.1504	−1.07817	−0.539085	+	0.842251i	$0.681230\pi$
$128$	−6.06927	−0.536453
$129$	0	0
$130$	6.70830	0.588357
$131$	−8.27924	−0.723361	−0.361680	−	0.932302i	$-0.617797\pi$
$131$	−8.27924	−0.723361	−0.361680	+	0.932302i	$0.617797\pi$
$132$	0	0
$133$	−2.13435	−0.185072
$134$	−25.7525	−2.22468
$135$	0	0
$136$	3.92874	0.336886
$137$	9.74294	0.832395	0.416198	−	0.909274i	$-0.363362\pi$
$137$	9.74294	0.832395	0.416198	+	0.909274i	$0.363362\pi$
$138$	0	0
$139$	−9.82417	−0.833275	−0.416638	−	0.909073i	$-0.636792\pi$
$139$	−9.82417	−0.833275	−0.416638	+	0.909073i	$0.636792\pi$
$140$	−2.95121	−0.249423
$141$	0	0
$142$	−19.4674	−1.63367
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	3.78324	0.314181
$146$	−1.97754	−0.163662
$147$	0	0
$148$	15.1496	1.24529
$149$	11.1586	0.914152	0.457076	−	0.889428i	$-0.348897\pi$
$149$	11.1586	0.914152	0.457076	+	0.889428i	$0.348897\pi$
$150$	0	0
$151$	−14.8744	−1.21046	−0.605231	−	0.796050i	$-0.706919\pi$
$151$	−14.8744	−1.21046	−0.605231	+	0.796050i	$0.706919\pi$
$152$	4.24420	0.344250
$153$	0	0
$154$	0.811305	0.0653768
$155$	8.65725	0.695367
$156$	0	0
$157$	−5.19888	−0.414916	−0.207458	−	0.978244i	$-0.566519\pi$
$157$	−5.19888	−0.414916	−0.207458	+	0.978244i	$0.566519\pi$
$158$	−13.4185	−1.06752
$159$	0	0
$160$	−25.9186	−2.04905
$161$	0.977282	0.0770206
$162$	0	0
$163$	11.2149	0.878419	0.439209	−	0.898385i	$-0.355259\pi$
$163$	11.2149	0.878419	0.439209	+	0.898385i	$0.355259\pi$
$164$	−25.2468	−1.97144
$165$	0	0
$166$	−10.8350	−0.840962
$167$	17.7273	1.37178	0.685891	−	0.727705i	$-0.259413\pi$
$167$	17.7273	1.37178	0.685891	+	0.727705i	$0.259413\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	34.1471	2.61896
$171$	0	0
$172$	−25.4908	−1.94366
$173$	23.2543	1.76799	0.883994	−	0.467498i	$-0.154844\pi$
$173$	23.2543	1.76799	0.883994	+	0.467498i	$0.154844\pi$
$174$	0	0
$175$	−2.05694	−0.155490
$176$	3.12518	0.235569
$177$	0	0
$178$	22.1757	1.66214
$179$	−3.51489	−0.262715	−0.131358	−	0.991335i	$-0.541934\pi$
$179$	−3.51489	−0.262715	−0.131358	+	0.991335i	$0.541934\pi$
$180$	0	0
$181$	12.3211	0.915817	0.457908	−	0.888999i	$-0.348599\pi$
$181$	12.3211	0.915817	0.457908	+	0.888999i	$0.348599\pi$
$182$	−0.811305	−0.0601380
$183$	0	0
$184$	−1.94335	−0.143265
$185$	20.5212	1.50875
$186$	0	0
$187$	−5.09027	−0.372238
$188$	18.0489	1.31635
$189$	0	0
$190$	36.8890	2.67621
$191$	−0.366089	−0.0264893	−0.0132446	−	0.999912i	$-0.504216\pi$
$191$	−0.366089	−0.0264893	−0.0132446	+	0.999912i	$0.504216\pi$
$192$	0	0
$193$	−26.0428	−1.87460	−0.937300	−	0.348524i	$-0.886683\pi$
$193$	−26.0428	−1.87460	−0.937300	+	0.348524i	$0.886683\pi$
$194$	12.4531	0.894078
$195$	0	0
$196$	−16.2278	−1.15913
$197$	−21.9210	−1.56181	−0.780903	−	0.624652i	$-0.785241\pi$
$197$	−21.9210	−1.56181	−0.780903	+	0.624652i	$0.785241\pi$
$198$	0	0
$199$	8.84395	0.626931	0.313466	−	0.949600i	$-0.398510\pi$
$199$	8.84395	0.626931	0.313466	+	0.949600i	$0.398510\pi$
$200$	4.09027	0.289226
$201$	0	0
$202$	33.1725	2.33401
$203$	−0.457548	−0.0321135
$204$	0	0
$205$	−34.1985	−2.38852
$206$	−14.1160	−0.983509
$207$	0	0
$208$	−3.12518	−0.216692
$209$	−5.49901	−0.380374
$210$	0	0
$211$	−12.5164	−0.861664	−0.430832	−	0.902432i	$-0.641780\pi$
$211$	−12.5164	−0.861664	−0.430832	+	0.902432i	$0.641780\pi$
$212$	−9.12004	−0.626367
$213$	0	0
$214$	21.2198	1.45055
$215$	−34.5290	−2.35486
$216$	0	0
$217$	−1.04701	−0.0710758
$218$	−31.2330	−2.11537
$219$	0	0
$220$	−7.60359	−0.512634
$221$	5.09027	0.342409
$222$	0	0
$223$	−13.5788	−0.909303	−0.454652	−	0.890669i	$-0.650236\pi$
$223$	−13.5788	−0.909303	−0.454652	+	0.890669i	$0.650236\pi$
$224$	3.13461	0.209440
$225$	0	0
$226$	−19.2822	−1.28264
$227$	13.5092	0.896637	0.448318	−	0.893874i	$-0.352023\pi$
$227$	13.5092	0.896637	0.448318	+	0.893874i	$0.352023\pi$
$228$	0	0
$229$	8.92676	0.589897	0.294949	−	0.955513i	$-0.404697\pi$
$229$	8.92676	0.589897	0.294949	+	0.955513i	$0.404697\pi$
$230$	−16.8908	−1.11375
$231$	0	0
$232$	0.909844	0.0597342
$233$	25.1912	1.65033	0.825164	−	0.564893i	$-0.191083\pi$
$233$	25.1912	1.65033	0.825164	+	0.564893i	$0.191083\pi$
$234$	0	0
$235$	24.4485	1.59484
$236$	11.4137	0.742971
$237$	0	0
$238$	−4.12977	−0.267693
$239$	−8.20020	−0.530427	−0.265213	−	0.964190i	$-0.585442\pi$
$239$	−8.20020	−0.530427	−0.265213	+	0.964190i	$0.585442\pi$
$240$	0	0
$241$	3.31688	0.213659	0.106830	−	0.994277i	$-0.465930\pi$
$241$	3.31688	0.213659	0.106830	+	0.994277i	$0.465930\pi$
$242$	2.09027	0.134368
$243$	0	0
$244$	6.36781	0.407657
$245$	−21.9816	−1.40435
$246$	0	0
$247$	5.49901	0.349893
$248$	2.08201	0.132208
$249$	0	0
$250$	2.00958	0.127097
$251$	−14.9489	−0.943568	−0.471784	−	0.881714i	$-0.656390\pi$
$251$	−14.9489	−0.943568	−0.471784	+	0.881714i	$0.656390\pi$
$252$	0	0
$253$	2.51790	0.158299
$254$	−25.3976	−1.59359
$255$	0	0
$256$	8.57537	0.535961
$257$	−3.85224	−0.240296	−0.120148	−	0.992756i	$-0.538337\pi$
$257$	−3.85224	−0.240296	−0.120148	+	0.992756i	$0.538337\pi$
$258$	0	0
$259$	−2.48184	−0.154214
$260$	7.60359	0.471554
$261$	0	0
$262$	−17.3059	−1.06916
$263$	−25.5396	−1.57484	−0.787421	−	0.616415i	$-0.788584\pi$
$263$	−25.5396	−1.57484	−0.787421	+	0.616415i	$0.788584\pi$
$264$	0	0
$265$	−12.3537	−0.758882
$266$	−4.46137	−0.273544
$267$	0	0
$268$	−29.1894	−1.78303
$269$	−18.5448	−1.13070	−0.565348	−	0.824853i	$-0.691258\pi$
$269$	−18.5448	−1.13070	−0.565348	+	0.824853i	$0.691258\pi$
$270$	0	0
$271$	24.9043	1.51283	0.756414	−	0.654093i	$-0.226949\pi$
$271$	24.9043	1.51283	0.756414	+	0.654093i	$0.226949\pi$
$272$	−15.9080	−0.964566
$273$	0	0
$274$	20.3654	1.23032
$275$	−5.29957	−0.319576
$276$	0	0
$277$	−1.23290	−0.0740778	−0.0370389	−	0.999314i	$-0.511793\pi$
$277$	−1.23290	−0.0740778	−0.0370389	+	0.999314i	$0.511793\pi$
$278$	−20.5352	−1.23162
$279$	0	0
$280$	−0.961397	−0.0574545
$281$	4.44904	0.265408	0.132704	−	0.991156i	$-0.457634\pi$
$281$	4.44904	0.265408	0.132704	+	0.991156i	$0.457634\pi$
$282$	0	0
$283$	2.76580	0.164409	0.0822047	−	0.996615i	$-0.473804\pi$
$283$	2.76580	0.164409	0.0822047	+	0.996615i	$0.473804\pi$
$284$	−22.0655	−1.30935
$285$	0	0
$286$	−2.09027	−0.123600
$287$	4.13598	0.244139
$288$	0	0
$289$	8.91088	0.524169
$290$	7.90801	0.464374
$291$	0	0
$292$	−2.24146	−0.131172
$293$	−10.7514	−0.628101	−0.314050	−	0.949406i	$-0.601686\pi$
$293$	−10.7514	−0.628101	−0.314050	+	0.949406i	$0.601686\pi$
$294$	0	0
$295$	15.4607	0.900155
$296$	4.93520	0.286853
$297$	0	0
$298$	23.3246	1.35116
$299$	−2.51790	−0.145614
$300$	0	0
$301$	4.17596	0.240698
$302$	−31.0916	−1.78912
$303$	0	0
$304$	−17.1854	−0.985650
$305$	8.62562	0.493902
$306$	0	0
$307$	−33.7562	−1.92657	−0.963284	−	0.268485i	$-0.913477\pi$
$307$	−33.7562	−1.92657	−0.963284	+	0.268485i	$0.913477\pi$
$308$	0.919582	0.0523980
$309$	0	0
$310$	18.0960	1.02778
$311$	−13.0092	−0.737682	−0.368841	−	0.929492i	$-0.620245\pi$
$311$	−13.0092	−0.737682	−0.368841	+	0.929492i	$0.620245\pi$
$312$	0	0
$313$	−5.98808	−0.338466	−0.169233	−	0.985576i	$-0.554129\pi$
$313$	−5.98808	−0.338466	−0.169233	+	0.985576i	$0.554129\pi$
$314$	−10.8671	−0.613265
$315$	0	0
$316$	−15.2093	−0.855589
$317$	20.9352	1.17584	0.587919	−	0.808920i	$-0.299948\pi$
$317$	20.9352	1.17584	0.587919	+	0.808920i	$0.299948\pi$
$318$	0	0
$319$	−1.17884	−0.0660024
$320$	−34.1177	−1.90724
$321$	0	0
$322$	2.04278	0.113840
$323$	27.9914	1.55749
$324$	0	0
$325$	5.29957	0.293967
$326$	23.4422	1.29834
$327$	0	0
$328$	−8.22449	−0.454122
$329$	−2.95681	−0.163014
$330$	0	0
$331$	31.9093	1.75389	0.876947	−	0.480587i	$-0.159576\pi$
$331$	31.9093	1.75389	0.876947	+	0.480587i	$0.159576\pi$
$332$	−12.2811	−0.674012
$333$	0	0
$334$	37.0549	2.02756
$335$	−39.5390	−2.16025
$336$	0	0
$337$	−18.2037	−0.991617	−0.495808	−	0.868432i	$-0.665128\pi$
$337$	−18.2037	−0.991617	−0.495808	+	0.868432i	$0.665128\pi$
$338$	2.09027	0.113696
$339$	0	0
$340$	38.7043	2.09904
$341$	−2.69755	−0.146081
$342$	0	0
$343$	5.37540	0.290244
$344$	−8.30399	−0.447721
$345$	0	0
$346$	48.6077	2.61317
$347$	−1.58263	−0.0849601	−0.0424800	−	0.999097i	$-0.513526\pi$
$347$	−1.58263	−0.0849601	−0.0424800	+	0.999097i	$0.513526\pi$
$348$	0	0
$349$	18.9534	1.01455	0.507277	−	0.861783i	$-0.330652\pi$
$349$	18.9534	1.01455	0.507277	+	0.861783i	$0.330652\pi$
$350$	−4.29957	−0.229821
$351$	0	0
$352$	8.07611	0.430458
$353$	5.03053	0.267748	0.133874	−	0.990998i	$-0.457258\pi$
$353$	5.03053	0.267748	0.133874	+	0.990998i	$0.457258\pi$
$354$	0	0
$355$	−29.8892	−1.58635
$356$	25.1352	1.33217
$357$	0	0
$358$	−7.34708	−0.388305
$359$	14.9862	0.790939	0.395469	−	0.918479i	$-0.370582\pi$
$359$	14.9862	0.790939	0.395469	+	0.918479i	$0.370582\pi$
$360$	0	0
$361$	11.2391	0.591530
$362$	25.7544	1.35362
$363$	0	0
$364$	−0.919582	−0.0481992
$365$	−3.03621	−0.158923
$366$	0	0
$367$	−3.88084	−0.202578	−0.101289	−	0.994857i	$-0.532297\pi$
$367$	−3.88084	−0.202578	−0.101289	+	0.994857i	$0.532297\pi$
$368$	7.86889	0.410194
$369$	0	0
$370$	42.8948	2.23000
$371$	1.49406	0.0775679
$372$	0	0
$373$	34.2149	1.77158	0.885790	−	0.464087i	$-0.153618\pi$
$373$	34.2149	1.77158	0.885790	+	0.464087i	$0.153618\pi$
$374$	−10.6401	−0.550184
$375$	0	0
$376$	5.87969	0.303222
$377$	1.17884	0.0607133
$378$	0	0
$379$	27.2438	1.39942	0.699709	−	0.714428i	$-0.253313\pi$
$379$	27.2438	1.39942	0.699709	+	0.714428i	$0.253313\pi$
$380$	41.8122	2.14492
$381$	0	0
$382$	−0.765226	−0.0391523
$383$	14.3356	0.732516	0.366258	−	0.930513i	$-0.380639\pi$
$383$	14.3356	0.732516	0.366258	+	0.930513i	$0.380639\pi$
$384$	0	0
$385$	1.24564	0.0634834
$386$	−54.4365	−2.77074
$387$	0	0
$388$	14.1151	0.716583
$389$	36.1459	1.83267	0.916335	−	0.400412i	$-0.131133\pi$
$389$	36.1459	1.83267	0.916335	+	0.400412i	$0.131133\pi$
$390$	0	0
$391$	−12.8168	−0.648173
$392$	−5.28642	−0.267004
$393$	0	0
$394$	−45.8209	−2.30842
$395$	−20.6020	−1.03660
$396$	0	0
$397$	18.6360	0.935314	0.467657	−	0.883910i	$-0.345098\pi$
$397$	18.6360	0.935314	0.467657	+	0.883910i	$0.345098\pi$
$398$	18.4863	0.926633
$399$	0	0
$400$	−16.5621	−0.828105
$401$	30.8809	1.54212	0.771059	−	0.636763i	$-0.219727\pi$
$401$	30.8809	1.54212	0.771059	+	0.636763i	$0.219727\pi$
$402$	0	0
$403$	2.69755	0.134375
$404$	37.5997	1.87066
$405$	0	0
$406$	−0.956399	−0.0474653
$407$	−6.39429	−0.316953
$408$	0	0
$409$	8.24426	0.407653	0.203826	−	0.979007i	$-0.434662\pi$
$409$	8.24426	0.407653	0.203826	+	0.979007i	$0.434662\pi$
$410$	−71.4841	−3.53035
$411$	0	0
$412$	−15.9999	−0.788260
$413$	−1.86982	−0.0920079
$414$	0	0
$415$	−16.6355	−0.816607
$416$	−8.07611	−0.395964
$417$	0	0
$418$	−11.4944	−0.562211
$419$	4.06993	0.198829	0.0994146	−	0.995046i	$-0.468303\pi$
$419$	4.06993	0.198829	0.0994146	+	0.995046i	$0.468303\pi$
$420$	0	0
$421$	−8.66291	−0.422204	−0.211102	−	0.977464i	$-0.567705\pi$
$421$	−8.66291	−0.422204	−0.211102	+	0.977464i	$0.567705\pi$
$422$	−26.1627	−1.27358
$423$	0	0
$424$	−2.97098	−0.144283
$425$	26.9762	1.30854
$426$	0	0
$427$	−1.04319	−0.0504834
$428$	24.0517	1.16258
$429$	0	0
$430$	−72.1751	−3.48059
$431$	−13.3316	−0.642160	−0.321080	−	0.947052i	$-0.604046\pi$
$431$	−13.3316	−0.642160	−0.321080	+	0.947052i	$0.604046\pi$
$432$	0	0
$433$	28.6223	1.37550	0.687749	−	0.725949i	$-0.258599\pi$
$433$	28.6223	1.37550	0.687749	+	0.725949i	$0.258599\pi$
$434$	−2.18854	−0.105053
$435$	0	0
$436$	−35.4014	−1.69542
$437$	−13.8459	−0.662341
$438$	0	0
$439$	1.76998	0.0844764	0.0422382	−	0.999108i	$-0.486551\pi$
$439$	1.76998	0.0844764	0.0422382	+	0.999108i	$0.486551\pi$
$440$	−2.47697	−0.118085
$441$	0	0
$442$	10.6401	0.506096
$443$	26.4721	1.25773	0.628864	−	0.777515i	$-0.283520\pi$
$443$	26.4721	1.25773	0.628864	+	0.777515i	$0.283520\pi$
$444$	0	0
$445$	34.0474	1.61400
$446$	−28.3834	−1.34399
$447$	0	0
$448$	4.12621	0.194945
$449$	9.15002	0.431816	0.215908	−	0.976414i	$-0.430729\pi$
$449$	9.15002	0.431816	0.215908	+	0.976414i	$0.430729\pi$
$450$	0	0
$451$	10.6561	0.501775
$452$	−21.8557	−1.02800
$453$	0	0
$454$	28.2379	1.32527
$455$	−1.24564	−0.0583963
$456$	0	0
$457$	−0.148515	−0.00694726	−0.00347363	−	0.999994i	$-0.501106\pi$
$457$	−0.148515	−0.00694726	−0.00347363	+	0.999994i	$0.501106\pi$
$458$	18.6594	0.871895
$459$	0	0
$460$	−19.1451	−0.892643
$461$	−18.7090	−0.871367	−0.435684	−	0.900100i	$-0.643493\pi$
$461$	−18.7090	−0.871367	−0.435684	+	0.900100i	$0.643493\pi$
$462$	0	0
$463$	3.54396	0.164702	0.0823509	−	0.996603i	$-0.473757\pi$
$463$	3.54396	0.164702	0.0823509	+	0.996603i	$0.473757\pi$
$464$	−3.68409	−0.171030
$465$	0	0
$466$	52.6564	2.43926
$467$	21.7978	1.00868	0.504340	−	0.863505i	$-0.331736\pi$
$467$	21.7978	1.00868	0.504340	+	0.863505i	$0.331736\pi$
$468$	0	0
$469$	4.78187	0.220806
$470$	51.1040	2.35725
$471$	0	0
$472$	3.71818	0.171143
$473$	10.7591	0.494703
$474$	0	0
$475$	29.1423	1.33714
$476$	−4.68092	−0.214550
$477$	0	0
$478$	−17.1407	−0.783995
$479$	−0.328858	−0.0150259	−0.00751295	−	0.999972i	$-0.502391\pi$
$479$	−0.328858	−0.0150259	−0.00751295	+	0.999972i	$0.502391\pi$
$480$	0	0
$481$	6.39429	0.291555
$482$	6.93319	0.315798
$483$	0	0
$484$	2.36924	0.107693
$485$	19.1198	0.868185
$486$	0	0
$487$	30.4071	1.37788	0.688940	−	0.724819i	$-0.258076\pi$
$487$	30.4071	1.37788	0.688940	+	0.724819i	$0.258076\pi$
$488$	2.07440	0.0939037
$489$	0	0
$490$	−45.9475	−2.07570
$491$	−24.5303	−1.10704	−0.553518	−	0.832837i	$-0.686715\pi$
$491$	−24.5303	−1.10704	−0.553518	+	0.832837i	$0.686715\pi$
$492$	0	0
$493$	6.00062	0.270254
$494$	11.4944	0.517158
$495$	0	0
$496$	−8.43035	−0.378534
$497$	3.61481	0.162147
$498$	0	0
$499$	−34.1994	−1.53098	−0.765488	−	0.643450i	$-0.777502\pi$
$499$	−34.1994	−1.53098	−0.765488	+	0.643450i	$0.777502\pi$
$500$	2.27778	0.101865
$501$	0	0
$502$	−31.2473	−1.39464
$503$	41.2935	1.84119	0.920594	−	0.390521i	$-0.127705\pi$
$503$	41.2935	1.84119	0.920594	+	0.390521i	$0.127705\pi$
$504$	0	0
$505$	50.9313	2.26641
$506$	5.26310	0.233973
$507$	0	0
$508$	−28.7871	−1.27722
$509$	−10.0539	−0.445633	−0.222817	−	0.974860i	$-0.571525\pi$
$509$	−10.0539	−0.445633	−0.222817	+	0.974860i	$0.571525\pi$
$510$	0	0
$511$	0.367201	0.0162440
$512$	30.0634	1.32863
$513$	0	0
$514$	−8.05224	−0.355169
$515$	−21.6730	−0.955025
$516$	0	0
$517$	−7.61803	−0.335040
$518$	−5.18772	−0.227936
$519$	0	0
$520$	2.47697	0.108622
$521$	20.6692	0.905534	0.452767	−	0.891629i	$-0.350437\pi$
$521$	20.6692	0.905534	0.452767	+	0.891629i	$0.350437\pi$
$522$	0	0
$523$	−20.6953	−0.904943	−0.452471	−	0.891779i	$-0.649457\pi$
$523$	−20.6953	−0.904943	−0.452471	+	0.891779i	$0.649457\pi$
$524$	−19.6155	−0.856907
$525$	0	0
$526$	−53.3848	−2.32769
$527$	13.7313	0.598144
$528$	0	0
$529$	−16.6602	−0.724356
$530$	−25.8226	−1.12166
$531$	0	0
$532$	−5.05679	−0.219239
$533$	−10.6561	−0.461566
$534$	0	0
$535$	32.5797	1.40854
$536$	−9.50886	−0.410720
$537$	0	0
$538$	−38.7637	−1.67122
$539$	6.84935	0.295022
$540$	0	0
$541$	−40.7891	−1.75366	−0.876830	−	0.480800i	$-0.840346\pi$
$541$	−40.7891	−1.75366	−0.876830	+	0.480800i	$0.840346\pi$
$542$	52.0568	2.23603
$543$	0	0
$544$	−41.1096	−1.76256
$545$	−47.9536	−2.05410
$546$	0	0
$547$	31.6431	1.35296	0.676480	−	0.736461i	$-0.263505\pi$
$547$	31.6431	1.35296	0.676480	+	0.736461i	$0.263505\pi$
$548$	23.0834	0.986072
$549$	0	0
$550$	−11.0775	−0.472348
$551$	6.48245	0.276162
$552$	0	0
$553$	2.49162	0.105954
$554$	−2.57710	−0.109490
$555$	0	0
$556$	−23.2758	−0.987114
$557$	17.3017	0.733095	0.366547	−	0.930399i	$-0.380540\pi$
$557$	17.3017	0.733095	0.366547	+	0.930399i	$0.380540\pi$
$558$	0	0
$559$	−10.7591	−0.455061
$560$	3.89284	0.164502
$561$	0	0
$562$	9.29971	0.392285
$563$	10.9980	0.463511	0.231755	−	0.972774i	$-0.425553\pi$
$563$	10.9980	0.463511	0.231755	+	0.972774i	$0.425553\pi$
$564$	0	0
$565$	−29.6049	−1.24549
$566$	5.78127	0.243005
$567$	0	0
$568$	−7.18814	−0.301608
$569$	−5.60756	−0.235081	−0.117540	−	0.993068i	$-0.537501\pi$
$569$	−5.60756	−0.235081	−0.117540	+	0.993068i	$0.537501\pi$
$570$	0	0
$571$	2.14161	0.0896236	0.0448118	−	0.998995i	$-0.485731\pi$
$571$	2.14161	0.0896236	0.0448118	+	0.998995i	$0.485731\pi$
$572$	−2.36924	−0.0990629
$573$	0	0
$574$	8.64533	0.360849
$575$	−13.3438	−0.556474
$576$	0	0
$577$	−26.1236	−1.08754	−0.543770	−	0.839234i	$-0.683004\pi$
$577$	−26.1236	−1.08754	−0.543770	+	0.839234i	$0.683004\pi$
$578$	18.6262	0.774746
$579$	0	0
$580$	8.96341	0.372185
$581$	2.01191	0.0834682
$582$	0	0
$583$	3.84935	0.159424
$584$	−0.730188	−0.0302154
$585$	0	0
$586$	−22.4733	−0.928362
$587$	29.9204	1.23495	0.617475	−	0.786591i	$-0.288156\pi$
$587$	29.9204	1.23495	0.617475	+	0.786591i	$0.288156\pi$
$588$	0	0
$589$	14.8339	0.611219
$590$	32.3170	1.33047
$591$	0	0
$592$	−19.9833	−0.821310
$593$	−6.78201	−0.278504	−0.139252	−	0.990257i	$-0.544470\pi$
$593$	−6.78201	−0.278504	−0.139252	+	0.990257i	$0.544470\pi$
$594$	0	0
$595$	−6.34062	−0.259940
$596$	26.4375	1.08292
$597$	0	0
$598$	−5.26310	−0.215224
$599$	−29.1277	−1.19013	−0.595063	−	0.803679i	$-0.702873\pi$
$599$	−29.1277	−1.19013	−0.595063	+	0.803679i	$0.702873\pi$
$600$	0	0
$601$	−2.28474	−0.0931964	−0.0465982	−	0.998914i	$-0.514838\pi$
$601$	−2.28474	−0.0931964	−0.0465982	+	0.998914i	$0.514838\pi$
$602$	8.72890	0.355763
$603$	0	0
$604$	−35.2410	−1.43394
$605$	3.20929	0.130476
$606$	0	0
$607$	−39.7735	−1.61436	−0.807178	−	0.590308i	$-0.799006\pi$
$607$	−39.7735	−1.61436	−0.807178	+	0.590308i	$0.799006\pi$
$608$	−44.4106	−1.80109
$609$	0	0
$610$	18.0299	0.730009
$611$	7.61803	0.308192
$612$	0	0
$613$	11.4161	0.461093	0.230547	−	0.973061i	$-0.425949\pi$
$613$	11.4161	0.461093	0.230547	+	0.973061i	$0.425949\pi$
$614$	−70.5596	−2.84756
$615$	0	0
$616$	0.299567	0.0120699
$617$	−1.77064	−0.0712833	−0.0356416	−	0.999365i	$-0.511347\pi$
$617$	−1.77064	−0.0712833	−0.0356416	+	0.999365i	$0.511347\pi$
$618$	0	0
$619$	43.5036	1.74856	0.874278	−	0.485426i	$-0.161335\pi$
$619$	43.5036	1.74856	0.874278	+	0.485426i	$0.161335\pi$
$620$	20.5111	0.823745
$621$	0	0
$622$	−27.1927	−1.09033
$623$	−4.11771	−0.164972
$624$	0	0
$625$	−23.4124	−0.936497
$626$	−12.5167	−0.500269
$627$	0	0
$628$	−12.3174	−0.491518
$629$	32.5487	1.29780
$630$	0	0
$631$	11.3311	0.451085	0.225542	−	0.974233i	$-0.427585\pi$
$631$	11.3311	0.451085	0.225542	+	0.974233i	$0.427585\pi$
$632$	−4.95463	−0.197085
$633$	0	0
$634$	43.7602	1.73794
$635$	−38.9941	−1.54743
$636$	0	0
$637$	−6.84935	−0.271381
$638$	−2.46410	−0.0975545
$639$	0	0
$640$	−19.4781	−0.769939
$641$	7.78345	0.307428	0.153714	−	0.988115i	$-0.450877\pi$
$641$	7.78345	0.307428	0.153714	+	0.988115i	$0.450877\pi$
$642$	0	0
$643$	34.8650	1.37494	0.687470	−	0.726213i	$-0.258721\pi$
$643$	34.8650	1.37494	0.687470	+	0.726213i	$0.258721\pi$
$644$	2.31541	0.0912401
$645$	0	0
$646$	58.5097	2.30203
$647$	13.8454	0.544318	0.272159	−	0.962252i	$-0.412262\pi$
$647$	13.8454	0.544318	0.272159	+	0.962252i	$0.412262\pi$
$648$	0	0
$649$	−4.81747	−0.189102
$650$	11.0775	0.434497
$651$	0	0
$652$	26.5708	1.04059
$653$	13.0259	0.509742	0.254871	−	0.966975i	$-0.417967\pi$
$653$	13.0259	0.509742	0.254871	+	0.966975i	$0.417967\pi$
$654$	0	0
$655$	−26.5705	−1.03820
$656$	33.3022	1.30023
$657$	0	0
$658$	−6.18055	−0.240943
$659$	−3.87817	−0.151072	−0.0755360	−	0.997143i	$-0.524067\pi$
$659$	−3.87817	−0.151072	−0.0755360	+	0.997143i	$0.524067\pi$
$660$	0	0
$661$	−37.1108	−1.44344	−0.721721	−	0.692184i	$-0.756649\pi$
$661$	−37.1108	−1.44344	−0.721721	+	0.692184i	$0.756649\pi$
$662$	66.6992	2.59234
$663$	0	0
$664$	−4.00073	−0.155259
$665$	−6.84975	−0.265622
$666$	0	0
$667$	−2.96820	−0.114929
$668$	42.0003	1.62504
$669$	0	0
$670$	−82.6474	−3.19295
$671$	−2.68770	−0.103758
$672$	0	0
$673$	−31.7752	−1.22485	−0.612423	−	0.790530i	$-0.709805\pi$
$673$	−31.7752	−1.22485	−0.612423	+	0.790530i	$0.709805\pi$
$674$	−38.0506	−1.46566
$675$	0	0
$676$	2.36924	0.0911246
$677$	−17.3554	−0.667023	−0.333511	−	0.942746i	$-0.608234\pi$
$677$	−17.3554	−0.667023	−0.333511	+	0.942746i	$0.608234\pi$
$678$	0	0
$679$	−2.31236	−0.0887401
$680$	12.6085	0.483513
$681$	0	0
$682$	−5.63863	−0.215914
$683$	36.9923	1.41547	0.707736	−	0.706477i	$-0.249717\pi$
$683$	36.9923	1.41547	0.707736	+	0.706477i	$0.249717\pi$
$684$	0	0
$685$	31.2679	1.19469
$686$	11.2361	0.428995
$687$	0	0
$688$	33.6241	1.28191
$689$	−3.84935	−0.146649
$690$	0	0
$691$	−43.3534	−1.64924	−0.824621	−	0.565686i	$-0.808612\pi$
$691$	−43.3534	−1.64924	−0.824621	+	0.565686i	$0.808612\pi$
$692$	55.0949	2.09439
$693$	0	0
$694$	−3.30813	−0.125575
$695$	−31.5286	−1.19595
$696$	0	0
$697$	−54.2423	−2.05457
$698$	39.6179	1.49956
$699$	0	0
$700$	−4.87339	−0.184197
$701$	30.2085	1.14096	0.570479	−	0.821312i	$-0.306758\pi$
$701$	30.2085	1.14096	0.570479	+	0.821312i	$0.306758\pi$
$702$	0	0
$703$	35.1623	1.32617
$704$	10.6309	0.400667
$705$	0	0
$706$	10.5152	0.395744
$707$	−6.15966	−0.231658
$708$	0	0
$709$	46.3434	1.74046	0.870231	−	0.492644i	$-0.163970\pi$
$709$	46.3434	1.74046	0.870231	+	0.492644i	$0.163970\pi$
$710$	−62.4765	−2.34470
$711$	0	0
$712$	8.18815	0.306864
$713$	−6.79217	−0.254369
$714$	0	0
$715$	−3.20929	−0.120021
$716$	−8.32762	−0.311218
$717$	0	0
$718$	31.3252	1.16904
$719$	−8.13537	−0.303398	−0.151699	−	0.988427i	$-0.548474\pi$
$719$	−8.13537	−0.303398	−0.151699	+	0.988427i	$0.548474\pi$
$720$	0	0
$721$	2.62114	0.0976164
$722$	23.4927	0.874308
$723$	0	0
$724$	29.1915	1.08490
$725$	6.24734	0.232020
$726$	0	0
$727$	24.1415	0.895360	0.447680	−	0.894194i	$-0.352250\pi$
$727$	24.1415	0.895360	0.447680	+	0.894194i	$0.352250\pi$
$728$	−0.299567	−0.0111027
$729$	0	0
$730$	−6.34651	−0.234895
$731$	−54.7666	−2.02562
$732$	0	0
$733$	2.16350	0.0799106	0.0399553	−	0.999201i	$-0.487278\pi$
$733$	2.16350	0.0799106	0.0399553	+	0.999201i	$0.487278\pi$
$734$	−8.11201	−0.299420
$735$	0	0
$736$	20.3348	0.749551
$737$	12.3202	0.453819
$738$	0	0
$739$	−42.3965	−1.55958	−0.779790	−	0.626041i	$-0.784674\pi$
$739$	−42.3965	−1.55958	−0.779790	+	0.626041i	$0.784674\pi$
$740$	48.6196	1.78729
$741$	0	0
$742$	3.12300	0.114649
$743$	−32.9534	−1.20894	−0.604472	−	0.796626i	$-0.706616\pi$
$743$	−32.9534	−1.20894	−0.604472	+	0.796626i	$0.706616\pi$
$744$	0	0
$745$	35.8114	1.31203
$746$	71.5184	2.61848
$747$	0	0
$748$	−12.0601	−0.440960
$749$	−3.94020	−0.143972
$750$	0	0
$751$	0.573759	0.0209368	0.0104684	−	0.999945i	$-0.496668\pi$
$751$	0.573759	0.0209368	0.0104684	+	0.999945i	$0.496668\pi$
$752$	−23.8077	−0.868178
$753$	0	0
$754$	2.46410	0.0897371
$755$	−47.7363	−1.73730
$756$	0	0
$757$	−6.83493	−0.248420	−0.124210	−	0.992256i	$-0.539640\pi$
$757$	−6.83493	−0.248420	−0.124210	+	0.992256i	$0.539640\pi$
$758$	56.9469	2.06841
$759$	0	0
$760$	13.6209	0.494082
$761$	−8.62118	−0.312517	−0.156259	−	0.987716i	$-0.549943\pi$
$761$	−8.62118	−0.312517	−0.156259	+	0.987716i	$0.549943\pi$
$762$	0	0
$763$	5.79953	0.209957
$764$	−0.867353	−0.0313797
$765$	0	0
$766$	29.9654	1.08269
$767$	4.81747	0.173949
$768$	0	0
$769$	−20.5566	−0.741289	−0.370644	−	0.928775i	$-0.620863\pi$
$769$	−20.5566	−0.741289	−0.370644	+	0.928775i	$0.620863\pi$
$770$	2.60372	0.0938315
$771$	0	0
$772$	−61.7016	−2.22069
$773$	−37.4711	−1.34774	−0.673870	−	0.738850i	$-0.735369\pi$
$773$	−37.4711	−1.34774	−0.673870	+	0.738850i	$0.735369\pi$
$774$	0	0
$775$	14.2959	0.513523
$776$	4.59817	0.165065
$777$	0	0
$778$	75.5548	2.70877
$779$	−58.5978	−2.09948
$780$	0	0
$781$	9.31332	0.333257
$782$	−26.7906	−0.958030
$783$	0	0
$784$	21.4055	0.764481
$785$	−16.6847	−0.595504
$786$	0	0
$787$	30.0128	1.06984	0.534920	−	0.844903i	$-0.320342\pi$
$787$	30.0128	1.06984	0.534920	+	0.844903i	$0.320342\pi$
$788$	−51.9361	−1.85015
$789$	0	0
$790$	−43.0638	−1.53214
$791$	3.58044	0.127306
$792$	0	0
$793$	2.68770	0.0954430
$794$	38.9543	1.38244
$795$	0	0
$796$	20.9534	0.742675
$797$	−18.7878	−0.665499	−0.332750	−	0.943015i	$-0.607976\pi$
$797$	−18.7878	−0.665499	−0.332750	+	0.943015i	$0.607976\pi$
$798$	0	0
$799$	38.7778	1.37186
$800$	−42.7999	−1.51320
$801$	0	0
$802$	64.5495	2.27932
$803$	0.946069	0.0333860
$804$	0	0
$805$	3.13638	0.110543
$806$	5.63863	0.198612
$807$	0	0
$808$	12.2486	0.430905
$809$	43.1687	1.51773	0.758865	−	0.651248i	$-0.225754\pi$
$809$	43.1687	1.51773	0.758865	+	0.651248i	$0.225754\pi$
$810$	0	0
$811$	−25.6659	−0.901250	−0.450625	−	0.892713i	$-0.648799\pi$
$811$	−25.6659	−0.901250	−0.450625	+	0.892713i	$0.648799\pi$
$812$	−1.08404	−0.0380423
$813$	0	0
$814$	−13.3658	−0.468472
$815$	35.9919	1.26074
$816$	0	0
$817$	−59.1642	−2.06989
$818$	17.2328	0.602529
$819$	0	0
$820$	−81.0244	−2.82949
$821$	7.24599	0.252887	0.126443	−	0.991974i	$-0.459644\pi$
$821$	7.24599	0.252887	0.126443	+	0.991974i	$0.459644\pi$
$822$	0	0
$823$	3.45917	0.120579	0.0602895	−	0.998181i	$-0.480798\pi$
$823$	3.45917	0.120579	0.0602895	+	0.998181i	$0.480798\pi$
$824$	−5.21220	−0.181576
$825$	0	0
$826$	−3.90844	−0.135992
$827$	−38.4164	−1.33587	−0.667935	−	0.744219i	$-0.732822\pi$
$827$	−38.4164	−1.33587	−0.667935	+	0.744219i	$0.732822\pi$
$828$	0	0
$829$	45.6310	1.58483	0.792414	−	0.609983i	$-0.208824\pi$
$829$	45.6310	1.58483	0.792414	+	0.609983i	$0.208824\pi$
$830$	−34.7728	−1.20698
$831$	0	0
$832$	−10.6309	−0.368560
$833$	−34.8651	−1.20800
$834$	0	0
$835$	56.8922	1.96884
$836$	−13.0285	−0.450599
$837$	0	0
$838$	8.50727	0.293879
$839$	15.9565	0.550878	0.275439	−	0.961319i	$-0.411177\pi$
$839$	15.9565	0.550878	0.275439	+	0.961319i	$0.411177\pi$
$840$	0	0
$841$	−27.6103	−0.952081
$842$	−18.1078	−0.624037
$843$	0	0
$844$	−29.6543	−1.02074
$845$	3.20929	0.110403
$846$	0	0
$847$	−0.388134	−0.0133364
$848$	12.0299	0.413109
$849$	0	0
$850$	56.3877	1.93408
$851$	−16.1002	−0.551907
$852$	0	0
$853$	40.9428	1.40186	0.700928	−	0.713232i	$-0.252770\pi$
$853$	40.9428	1.40186	0.700928	+	0.713232i	$0.252770\pi$
$854$	−2.18055	−0.0746167
$855$	0	0
$856$	7.83519	0.267801
$857$	−13.4386	−0.459055	−0.229527	−	0.973302i	$-0.573718\pi$
$857$	−13.4386	−0.459055	−0.229527	+	0.973302i	$0.573718\pi$
$858$	0	0
$859$	19.2704	0.657499	0.328749	−	0.944417i	$-0.393373\pi$
$859$	19.2704	0.657499	0.328749	+	0.944417i	$0.393373\pi$
$860$	−81.8076	−2.78962
$861$	0	0
$862$	−27.8667	−0.949143
$863$	15.2632	0.519564	0.259782	−	0.965667i	$-0.416349\pi$
$863$	15.2632	0.519564	0.259782	+	0.965667i	$0.416349\pi$
$864$	0	0
$865$	74.6298	2.53749
$866$	59.8283	2.03305
$867$	0	0
$868$	−2.48062	−0.0841978
$869$	6.41948	0.217766
$870$	0	0
$871$	−12.3202	−0.417453
$872$	−11.5325	−0.390540
$873$	0	0
$874$	−28.9418	−0.978971
$875$	−0.373151	−0.0126148
$876$	0	0
$877$	−27.5211	−0.929321	−0.464661	−	0.885489i	$-0.653824\pi$
$877$	−27.5211	−0.929321	−0.464661	+	0.885489i	$0.653824\pi$
$878$	3.69974	0.124860
$879$	0	0
$880$	10.0296	0.338099
$881$	54.1530	1.82446	0.912230	−	0.409678i	$-0.134359\pi$
$881$	54.1530	1.82446	0.912230	+	0.409678i	$0.134359\pi$
$882$	0	0
$883$	40.5156	1.36346	0.681729	−	0.731605i	$-0.261229\pi$
$883$	40.5156	1.36346	0.681729	+	0.731605i	$0.261229\pi$
$884$	12.0601	0.405624
$885$	0	0
$886$	55.3339	1.85898
$887$	5.89774	0.198027	0.0990134	−	0.995086i	$-0.468431\pi$
$887$	5.89774	0.198027	0.0990134	+	0.995086i	$0.468431\pi$
$888$	0	0
$889$	4.71597	0.158168
$890$	71.1683	2.38557
$891$	0	0
$892$	−32.1714	−1.07718
$893$	41.8916	1.40185
$894$	0	0
$895$	−11.2803	−0.377059
$896$	2.35569	0.0786981
$897$	0	0
$898$	19.1260	0.638244
$899$	3.17999	0.106058
$900$	0	0
$901$	−19.5943	−0.652779
$902$	22.2741	0.741646
$903$	0	0
$904$	−7.11978	−0.236800
$905$	39.5419	1.31442
$906$	0	0
$907$	−38.6917	−1.28474	−0.642369	−	0.766396i	$-0.722048\pi$
$907$	−38.6917	−1.28474	−0.642369	+	0.766396i	$0.722048\pi$
$908$	32.0065	1.06217
$909$	0	0
$910$	−2.60372	−0.0863124
$911$	−20.1721	−0.668333	−0.334166	−	0.942514i	$-0.608455\pi$
$911$	−20.1721	−0.668333	−0.334166	+	0.942514i	$0.608455\pi$
$912$	0	0
$913$	5.18355	0.171551
$914$	−0.310438	−0.0102684
$915$	0	0
$916$	21.1496	0.698804
$917$	3.21345	0.106118
$918$	0	0
$919$	−0.829308	−0.0273563	−0.0136782	−	0.999906i	$-0.504354\pi$
$919$	−0.829308	−0.0273563	−0.0136782	+	0.999906i	$0.504354\pi$
$920$	−6.23677	−0.205620
$921$	0	0
$922$	−39.1070	−1.28792
$923$	−9.31332	−0.306552
$924$	0	0
$925$	33.8870	1.11420
$926$	7.40784	0.243437
$927$	0	0
$928$	−9.52044	−0.312524
$929$	−42.8173	−1.40479	−0.702395	−	0.711787i	$-0.747886\pi$
$929$	−42.8173	−1.40479	−0.702395	+	0.711787i	$0.747886\pi$
$930$	0	0
$931$	−37.6646	−1.23441
$932$	59.6839	1.95501
$933$	0	0
$934$	45.5633	1.49088
$935$	−16.3362	−0.534250
$936$	0	0
$937$	−39.5976	−1.29360	−0.646798	−	0.762661i	$-0.723892\pi$
$937$	−39.5976	−1.29360	−0.646798	+	0.762661i	$0.723892\pi$
$938$	9.99542	0.326362
$939$	0	0
$940$	57.9243	1.88928
$941$	12.9093	0.420831	0.210416	−	0.977612i	$-0.432518\pi$
$941$	12.9093	0.420831	0.210416	+	0.977612i	$0.432518\pi$
$942$	0	0
$943$	26.8309	0.873735
$944$	−15.0555	−0.490013
$945$	0	0
$946$	22.4894	0.731194
$947$	−31.5894	−1.02652	−0.513259	−	0.858234i	$-0.671562\pi$
$947$	−31.5894	−1.02652	−0.513259	+	0.858234i	$0.671562\pi$
$948$	0	0
$949$	−0.946069	−0.0307107
$950$	60.9155	1.97636
$951$	0	0
$952$	−1.52488	−0.0494215
$953$	59.4223	1.92488	0.962439	−	0.271498i	$-0.0875189\pi$
$953$	59.4223	1.92488	0.962439	+	0.271498i	$0.0875189\pi$
$954$	0	0
$955$	−1.17489	−0.0380184
$956$	−19.4282	−0.628354
$957$	0	0
$958$	−0.687403	−0.0222090
$959$	−3.78156	−0.122113
$960$	0	0
$961$	−23.7232	−0.765264
$962$	13.3658	0.430931
$963$	0	0
$964$	7.85849	0.253105
$965$	−83.5789	−2.69050
$966$	0	0
$967$	−14.6684	−0.471704	−0.235852	−	0.971789i	$-0.575788\pi$
$967$	−14.6684	−0.471704	−0.235852	+	0.971789i	$0.575788\pi$
$968$	0.771813	0.0248070
$969$	0	0
$970$	39.9656	1.28322
$971$	28.8535	0.925954	0.462977	−	0.886370i	$-0.346781\pi$
$971$	28.8535	0.925954	0.462977	+	0.886370i	$0.346781\pi$
$972$	0	0
$973$	3.81309	0.122242
$974$	63.5592	2.03657
$975$	0	0
$976$	−8.39955	−0.268863
$977$	58.5311	1.87257	0.936287	−	0.351235i	$-0.114238\pi$
$977$	58.5311	1.87257	0.936287	+	0.351235i	$0.114238\pi$
$978$	0	0
$979$	−10.6090	−0.339065
$980$	−52.0797	−1.66362
$981$	0	0
$982$	−51.2750	−1.63625
$983$	0.363698	0.0116002	0.00580008	−	0.999983i	$-0.498154\pi$
$983$	0.363698	0.0116002	0.00580008	+	0.999983i	$0.498154\pi$
$984$	0	0
$985$	−70.3509	−2.24157
$986$	12.5429	0.399448
$987$	0	0
$988$	13.0285	0.414491
$989$	27.0903	0.861421
$990$	0	0
$991$	−2.67141	−0.0848601	−0.0424300	−	0.999099i	$-0.513510\pi$
$991$	−2.67141	−0.0848601	−0.0424300	+	0.999099i	$0.513510\pi$
$992$	−21.7857	−0.691698
$993$	0	0
$994$	7.55595	0.239660
$995$	28.3828	0.899797
$996$	0	0
$997$	−40.9706	−1.29755	−0.648776	−	0.760979i	$-0.724719\pi$
$997$	−40.9706	−1.29755	−0.648776	+	0.760979i	$0.724719\pi$
$998$	−71.4861	−2.26285
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1287.2.a.p.1.4	yes	5
3.2	odd	2		1287.2.a.o.1.2	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1287.2.a.o.1.2	✓	5	3.2	odd	2
1287.2.a.p.1.4	yes	5	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:	\( N \)	\(=\)	\( 1287 = 3^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1287.a (trivial)

\(f(q)\)	\(=\)	\(q+2.09027 q^{2} +2.36924 q^{4} +3.20929 q^{5} -0.388134 q^{7} +0.771813 q^{8} +O(q^{10})\)	\(q+2.09027 q^{2} +2.36924 q^{4} +3.20929 q^{5} -0.388134 q^{7} +0.771813 q^{8} +6.70830 q^{10} -1.00000 q^{11} +1.00000 q^{13} -0.811305 q^{14} -3.12518 q^{16} +5.09027 q^{17} +5.49901 q^{19} +7.60359 q^{20} -2.09027 q^{22} -2.51790 q^{23} +5.29957 q^{25} +2.09027 q^{26} -0.919582 q^{28} +1.17884 q^{29} +2.69755 q^{31} -8.07611 q^{32} +10.6401 q^{34} -1.24564 q^{35} +6.39429 q^{37} +11.4944 q^{38} +2.47697 q^{40} -10.6561 q^{41} -10.7591 q^{43} -2.36924 q^{44} -5.26310 q^{46} +7.61803 q^{47} -6.84935 q^{49} +11.0775 q^{50} +2.36924 q^{52} -3.84935 q^{53} -3.20929 q^{55} -0.299567 q^{56} +2.46410 q^{58} +4.81747 q^{59} +2.68770 q^{61} +5.63863 q^{62} -10.6309 q^{64} +3.20929 q^{65} -12.3202 q^{67} +12.0601 q^{68} -2.60372 q^{70} -9.31332 q^{71} -0.946069 q^{73} +13.3658 q^{74} +13.0285 q^{76} +0.388134 q^{77} -6.41948 q^{79} -10.0296 q^{80} -22.2741 q^{82} -5.18355 q^{83} +16.3362 q^{85} -22.4894 q^{86} -0.771813 q^{88} +10.6090 q^{89} -0.388134 q^{91} -5.96551 q^{92} +15.9238 q^{94} +17.6479 q^{95} +5.95763 q^{97} -14.3170 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{2} + 7 q^{4} + 4 q^{5} - 4 q^{7} + 9 q^{8}+O(q^{10}) \) 5 * q + 3 * q^2 + 7 * q^4 + 4 * q^5 - 4 * q^7 + 9 * q^8	\( 5 q + 3 q^{2} + 7 q^{4} + 4 q^{5} - 4 q^{7} + 9 q^{8} - 6 q^{10} - 5 q^{11} + 5 q^{13} + 6 q^{14} + 19 q^{16} + 18 q^{17} - 2 q^{20} - 3 q^{22} + 8 q^{23} + 7 q^{25} + 3 q^{26} - 8 q^{28} + 20 q^{29} - 8 q^{31} + 31 q^{32} + 26 q^{34} + 6 q^{35} + 4 q^{37} + 18 q^{38} - 20 q^{40} + 8 q^{41} - 22 q^{43} - 7 q^{44} + 6 q^{46} + 6 q^{47} + 5 q^{49} + 11 q^{50} + 7 q^{52} + 20 q^{53} - 4 q^{55} + 18 q^{56} + 12 q^{58} - 16 q^{59} - 4 q^{61} - 26 q^{62} + 11 q^{64} + 4 q^{65} - 20 q^{67} + 36 q^{68} + 14 q^{70} + 6 q^{71} - 12 q^{73} - 20 q^{74} - 16 q^{76} + 4 q^{77} + 6 q^{79} - 52 q^{80} - 18 q^{82} + 4 q^{83} + 6 q^{85} - 46 q^{86} - 9 q^{88} + 10 q^{89} - 4 q^{91} + 68 q^{92} - 2 q^{94} + 2 q^{95} + 4 q^{97} - 45 q^{98}+O(q^{100}) \) 5 * q + 3 * q^2 + 7 * q^4 + 4 * q^5 - 4 * q^7 + 9 * q^8 - 6 * q^10 - 5 * q^11 + 5 * q^13 + 6 * q^14 + 19 * q^16 + 18 * q^17 - 2 * q^20 - 3 * q^22 + 8 * q^23 + 7 * q^25 + 3 * q^26 - 8 * q^28 + 20 * q^29 - 8 * q^31 + 31 * q^32 + 26 * q^34 + 6 * q^35 + 4 * q^37 + 18 * q^38 - 20 * q^40 + 8 * q^41 - 22 * q^43 - 7 * q^44 + 6 * q^46 + 6 * q^47 + 5 * q^49 + 11 * q^50 + 7 * q^52 + 20 * q^53 - 4 * q^55 + 18 * q^56 + 12 * q^58 - 16 * q^59 - 4 * q^61 - 26 * q^62 + 11 * q^64 + 4 * q^65 - 20 * q^67 + 36 * q^68 + 14 * q^70 + 6 * q^71 - 12 * q^73 - 20 * q^74 - 16 * q^76 + 4 * q^77 + 6 * q^79 - 52 * q^80 - 18 * q^82 + 4 * q^83 + 6 * q^85 - 46 * q^86 - 9 * q^88 + 10 * q^89 - 4 * q^91 + 68 * q^92 - 2 * q^94 + 2 * q^95 + 4 * q^97 - 45 * q^98

Introduction

L-functions

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Varieties

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Database

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