LMFDB - Embedded newform 6897.2.a.bc.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6897,2,Mod(1,6897)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6897.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6897, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6897 = 3 \cdot 11^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6897.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,-1,-8,9,-2,1,-6,-6,8,-3,0,-9,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$55.0728222741$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 12x^{6} + 9x^{5} + 40x^{4} - 22x^{3} - 28x^{2} + 12x - 1 $ x^8 - x^7 - 12x^6 + 9x^5 + 40x^4 - 22x^3 - 28x^2 + 12x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.962830$ of defining polynomial
Character	$\chi$	$=$	6897.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.962830 q^{2} -1.00000 q^{3} -1.07296 q^{4} -2.82062 q^{5} -0.962830 q^{6} +3.66641 q^{7} -2.95874 q^{8} +1.00000 q^{9} -2.71578 q^{10} +1.07296 q^{12} +0.962830 q^{13} +3.53013 q^{14} +2.82062 q^{15} -0.702842 q^{16} -5.56155 q^{17} +0.962830 q^{18} +1.00000 q^{19} +3.02641 q^{20} -3.66641 q^{21} -2.40347 q^{23} +2.95874 q^{24} +2.95590 q^{25} +0.927041 q^{26} -1.00000 q^{27} -3.93391 q^{28} +3.20441 q^{29} +2.71578 q^{30} +8.46230 q^{31} +5.24075 q^{32} -5.35483 q^{34} -10.3416 q^{35} -1.07296 q^{36} -4.15265 q^{37} +0.962830 q^{38} -0.962830 q^{39} +8.34547 q^{40} +6.94427 q^{41} -3.53013 q^{42} -1.69751 q^{43} -2.82062 q^{45} -2.31414 q^{46} -0.318191 q^{47} +0.702842 q^{48} +6.44257 q^{49} +2.84603 q^{50} +5.56155 q^{51} -1.03308 q^{52} -3.60355 q^{53} -0.962830 q^{54} -10.8479 q^{56} -1.00000 q^{57} +3.08530 q^{58} -5.94245 q^{59} -3.02641 q^{60} -3.85942 q^{61} +8.14775 q^{62} +3.66641 q^{63} +6.45164 q^{64} -2.71578 q^{65} +7.65808 q^{67} +5.96731 q^{68} +2.40347 q^{69} -9.95716 q^{70} +4.26434 q^{71} -2.95874 q^{72} -2.88048 q^{73} -3.99830 q^{74} -2.95590 q^{75} -1.07296 q^{76} -0.927041 q^{78} +16.1718 q^{79} +1.98245 q^{80} +1.00000 q^{81} +6.68615 q^{82} +12.2304 q^{83} +3.93391 q^{84} +15.6870 q^{85} -1.63441 q^{86} -3.20441 q^{87} -15.1984 q^{89} -2.71578 q^{90} +3.53013 q^{91} +2.57883 q^{92} -8.46230 q^{93} -0.306364 q^{94} -2.82062 q^{95} -5.24075 q^{96} -12.7065 q^{97} +6.20310 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{2} - 8 q^{3} + 9 q^{4} - 2 q^{5} + q^{6} - 6 q^{7} - 6 q^{8} + 8 q^{9} - 3 q^{10} - 9 q^{12} - q^{13} + 7 q^{14} + 2 q^{15} + 23 q^{16} - 15 q^{17} - q^{18} + 8 q^{19} - 4 q^{20} + 6 q^{21}+ \cdots - 55 q^{98}+O(q^{100}) $ 8 * q - q^2 - 8 * q^3 + 9 * q^4 - 2 * q^5 + q^6 - 6 * q^7 - 6 * q^8 + 8 * q^9 - 3 * q^10 - 9 * q^12 - q^13 + 7 * q^14 + 2 * q^15 + 23 * q^16 - 15 * q^17 - q^18 + 8 * q^19 - 4 * q^20 + 6 * q^21 - 5 * q^23 + 6 * q^24 + 4 * q^25 + 25 * q^26 - 8 * q^27 - 8 * q^28 - 17 * q^29 + 3 * q^30 - 4 * q^31 - 38 * q^32 - 18 * q^34 - 2 * q^35 + 9 * q^36 - 4 * q^37 - q^38 + q^39 - 14 * q^40 - 2 * q^41 - 7 * q^42 - 10 * q^43 - 2 * q^45 + 10 * q^46 + 5 * q^47 - 23 * q^48 + 28 * q^49 - 7 * q^50 + 15 * q^51 - 8 * q^52 - 19 * q^53 + q^54 - 49 * q^56 - 8 * q^57 + 35 * q^58 + 12 * q^59 + 4 * q^60 + 5 * q^61 - 22 * q^62 - 6 * q^63 + 24 * q^64 - 3 * q^65 - 12 * q^67 + 3 * q^68 + 5 * q^69 - 22 * q^70 + 40 * q^71 - 6 * q^72 + 12 * q^73 + 7 * q^74 - 4 * q^75 + 9 * q^76 - 25 * q^78 - 14 * q^79 + 37 * q^80 + 8 * q^81 + 31 * q^82 - 17 * q^83 + 8 * q^84 + 35 * q^85 - q^86 + 17 * q^87 - 9 * q^89 - 3 * q^90 + 7 * q^91 + 12 * q^92 + 4 * q^93 - 51 * q^94 - 2 * q^95 + 38 * q^96 + 8 * q^97 - 55 * q^98

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 $

$2$	0.962830	0.680824	0.340412	−	0.940276i	$-0.389434\pi$
$2$	0.962830	0.680824	0.340412	+	0.940276i	$0.389434\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.07296	−0.536479
$5$	−2.82062	−1.26142	−0.630710	−	0.776019i	$-0.717236\pi$
$5$	−2.82062	−1.26142	−0.630710	+	0.776019i	$0.717236\pi$
$6$	−0.962830	−0.393074
$7$	3.66641	1.38577	0.692887	−	0.721047i	$-0.256339\pi$
$7$	3.66641	1.38577	0.692887	+	0.721047i	$0.256339\pi$
$8$	−2.95874	−1.04607
$9$	1.00000	0.333333
$10$	−2.71578	−0.858804
$11$	0	0
$11$	0	0
$12$	1.07296	0.309736
$13$	0.962830	0.267041	0.133520	−	0.991046i	$-0.457372\pi$
$13$	0.962830	0.267041	0.133520	+	0.991046i	$0.457372\pi$
$14$	3.53013	0.943467
$15$	2.82062	0.728281
$16$	−0.702842	−0.175711
$17$	−5.56155	−1.34887	−0.674437	−	0.738333i	$-0.735614\pi$
$17$	−5.56155	−1.34887	−0.674437	+	0.738333i	$0.735614\pi$
$18$	0.962830	0.226941
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	3.02641	0.676726
$21$	−3.66641	−0.800077
$22$	0	0
$23$	−2.40347	−0.501159	−0.250580	−	0.968096i	$-0.580621\pi$
$23$	−2.40347	−0.501159	−0.250580	+	0.968096i	$0.580621\pi$
$24$	2.95874	0.603950
$25$	2.95590	0.591180
$26$	0.927041	0.181808
$27$	−1.00000	−0.192450
$28$	−3.93391	−0.743439
$29$	3.20441	0.595043	0.297522	−	0.954715i	$-0.403840\pi$
$29$	3.20441	0.595043	0.297522	+	0.954715i	$0.403840\pi$
$30$	2.71578	0.495831
$31$	8.46230	1.51987	0.759937	−	0.649997i	$-0.225230\pi$
$31$	8.46230	1.51987	0.759937	+	0.649997i	$0.225230\pi$
$32$	5.24075	0.926443
$33$	0	0
$34$	−5.35483	−0.918345
$35$	−10.3416	−1.74804
$36$	−1.07296	−0.178826
$37$	−4.15265	−0.682692	−0.341346	−	0.939938i	$-0.610883\pi$
$37$	−4.15265	−0.682692	−0.341346	+	0.939938i	$0.610883\pi$
$38$	0.962830	0.156192
$39$	−0.962830	−0.154176
$40$	8.34547	1.31954
$41$	6.94427	1.08451	0.542257	−	0.840213i	$-0.317570\pi$
$41$	6.94427	1.08451	0.542257	+	0.840213i	$0.317570\pi$
$42$	−3.53013	−0.544711
$43$	−1.69751	−0.258868	−0.129434	−	0.991588i	$-0.541316\pi$
$43$	−1.69751	−0.258868	−0.129434	+	0.991588i	$0.541316\pi$
$44$	0	0
$45$	−2.82062	−0.420473
$46$	−2.31414	−0.341201
$47$	−0.318191	−0.0464130	−0.0232065	−	0.999731i	$-0.507388\pi$
$47$	−0.318191	−0.0464130	−0.0232065	+	0.999731i	$0.507388\pi$
$48$	0.702842	0.101447
$49$	6.44257	0.920368
$50$	2.84603	0.402489
$51$	5.56155	0.778773
$52$	−1.03308	−0.143262
$53$	−3.60355	−0.494986	−0.247493	−	0.968890i	$-0.579607\pi$
$53$	−3.60355	−0.494986	−0.247493	+	0.968890i	$0.579607\pi$
$54$	−0.962830	−0.131025
$55$	0	0
$56$	−10.8479	−1.44962
$57$	−1.00000	−0.132453
$58$	3.08530	0.405120
$59$	−5.94245	−0.773641	−0.386820	−	0.922155i	$-0.626427\pi$
$59$	−5.94245	−0.773641	−0.386820	+	0.922155i	$0.626427\pi$
$60$	−3.02641	−0.390708
$61$	−3.85942	−0.494148	−0.247074	−	0.968997i	$-0.579469\pi$
$61$	−3.85942	−0.494148	−0.247074	+	0.968997i	$0.579469\pi$
$62$	8.14775	1.03477
$63$	3.66641	0.461924
$64$	6.45164	0.806455
$65$	−2.71578	−0.336851
$66$	0	0
$67$	7.65808	0.935583	0.467792	−	0.883839i	$-0.345050\pi$
$67$	7.65808	0.935583	0.467792	+	0.883839i	$0.345050\pi$
$68$	5.96731	0.723643
$69$	2.40347	0.289344
$70$	−9.95716	−1.19011
$71$	4.26434	0.506084	0.253042	−	0.967455i	$-0.418569\pi$
$71$	4.26434	0.506084	0.253042	+	0.967455i	$0.418569\pi$
$72$	−2.95874	−0.348690
$73$	−2.88048	−0.337135	−0.168567	−	0.985690i	$-0.553914\pi$
$73$	−2.88048	−0.337135	−0.168567	+	0.985690i	$0.553914\pi$
$74$	−3.99830	−0.464793
$75$	−2.95590	−0.341318
$76$	−1.07296	−0.123077
$77$	0	0
$78$	−0.927041	−0.104967
$79$	16.1718	1.81946	0.909732	−	0.415196i	$-0.136287\pi$
$79$	16.1718	1.81946	0.909732	+	0.415196i	$0.136287\pi$
$80$	1.98245	0.221645
$81$	1.00000	0.111111
$82$	6.68615	0.738362
$83$	12.2304	1.34246	0.671230	−	0.741249i	$-0.265766\pi$
$83$	12.2304	1.34246	0.671230	+	0.741249i	$0.265766\pi$
$84$	3.93391	0.429225
$85$	15.6870	1.70150
$86$	−1.63441	−0.176243
$87$	−3.20441	−0.343548
$88$	0	0
$89$	−15.1984	−1.61103	−0.805513	−	0.592578i	$-0.798110\pi$
$89$	−15.1984	−1.61103	−0.805513	+	0.592578i	$0.798110\pi$
$90$	−2.71578	−0.286268
$91$	3.53013	0.370058
$92$	2.57883	0.268862
$93$	−8.46230	−0.877499
$94$	−0.306364	−0.0315990
$95$	−2.82062	−0.289390
$96$	−5.24075	−0.534882
$97$	−12.7065	−1.29015	−0.645076	−	0.764119i	$-0.723174\pi$
$97$	−12.7065	−1.29015	−0.645076	+	0.764119i	$0.723174\pi$
$98$	6.20310	0.626608
$99$	0	0
$100$	−3.17156	−0.317156
$101$	−1.93582	−0.192622	−0.0963109	−	0.995351i	$-0.530704\pi$
$101$	−1.93582	−0.192622	−0.0963109	+	0.995351i	$0.530704\pi$
$102$	5.35483	0.530207
$103$	−16.3279	−1.60884	−0.804420	−	0.594061i	$-0.797524\pi$
$103$	−16.3279	−1.60884	−0.804420	+	0.594061i	$0.797524\pi$
$104$	−2.84876	−0.279344
$105$	10.3416	1.00923
$106$	−3.46961	−0.336998
$107$	7.93372	0.766982	0.383491	−	0.923545i	$-0.374722\pi$
$107$	7.93372	0.766982	0.383491	+	0.923545i	$0.374722\pi$
$108$	1.07296	0.103245
$109$	12.2084	1.16935	0.584676	−	0.811267i	$-0.301222\pi$
$109$	12.2084	1.16935	0.584676	+	0.811267i	$0.301222\pi$
$110$	0	0
$111$	4.15265	0.394152
$112$	−2.57691	−0.243495
$113$	7.76336	0.730315	0.365158	−	0.930946i	$-0.381015\pi$
$113$	7.76336	0.730315	0.365158	+	0.930946i	$0.381015\pi$
$114$	−0.962830	−0.0901773
$115$	6.77929	0.632172
$116$	−3.43820	−0.319228
$117$	0.962830	0.0890137
$118$	−5.72157	−0.526713
$119$	−20.3909	−1.86923
$120$	−8.34547	−0.761834
$121$	0	0
$122$	−3.71597	−0.336428
$123$	−6.94427	−0.626144
$124$	−9.07969	−0.815380
$125$	5.76563	0.515694
$126$	3.53013	0.314489
$127$	−18.0045	−1.59764	−0.798821	−	0.601569i	$-0.794543\pi$
$127$	−18.0045	−1.59764	−0.798821	+	0.601569i	$0.794543\pi$
$128$	−4.26968	−0.377390
$129$	1.69751	0.149457
$130$	−2.61483	−0.229336
$131$	−3.09592	−0.270492	−0.135246	−	0.990812i	$-0.543182\pi$
$131$	−3.09592	−0.270492	−0.135246	+	0.990812i	$0.543182\pi$
$132$	0	0
$133$	3.66641	0.317918
$134$	7.37343	0.636967
$135$	2.82062	0.242760
$136$	16.4552	1.41102
$137$	−16.6118	−1.41924	−0.709620	−	0.704585i	$-0.751133\pi$
$137$	−16.6118	−1.41924	−0.709620	+	0.704585i	$0.751133\pi$
$138$	2.31414	0.196992
$139$	16.4703	1.39699	0.698494	−	0.715616i	$-0.253854\pi$
$139$	16.4703	1.39699	0.698494	+	0.715616i	$0.253854\pi$
$140$	11.0961	0.937788
$141$	0.318191	0.0267965
$142$	4.10583	0.344554
$143$	0	0
$144$	−0.702842	−0.0585702
$145$	−9.03842	−0.750600
$146$	−2.77341	−0.229529
$147$	−6.44257	−0.531374
$148$	4.45562	0.366250
$149$	−22.2886	−1.82595	−0.912977	−	0.408010i	$-0.866223\pi$
$149$	−22.2886	−1.82595	−0.912977	+	0.408010i	$0.866223\pi$
$150$	−2.84603	−0.232377
$151$	16.7012	1.35912	0.679561	−	0.733619i	$-0.262170\pi$
$151$	16.7012	1.35912	0.679561	+	0.733619i	$0.262170\pi$
$152$	−2.95874	−0.239985
$153$	−5.56155	−0.449625
$154$	0	0
$155$	−23.8689	−1.91720
$156$	1.03308	0.0827123
$157$	−5.79752	−0.462692	−0.231346	−	0.972872i	$-0.574313\pi$
$157$	−5.79752	−0.462692	−0.231346	+	0.972872i	$0.574313\pi$
$158$	15.5706	1.23873
$159$	3.60355	0.285780
$160$	−14.7822	−1.16863
$161$	−8.81213	−0.694493
$162$	0.962830	0.0756471
$163$	−6.07716	−0.476000	−0.238000	−	0.971265i	$-0.576492\pi$
$163$	−6.07716	−0.476000	−0.238000	+	0.971265i	$0.576492\pi$
$164$	−7.45092	−0.581819
$165$	0	0
$166$	11.7758	0.913978
$167$	−18.4820	−1.43018	−0.715091	−	0.699031i	$-0.753615\pi$
$167$	−18.4820	−1.43018	−0.715091	+	0.699031i	$0.753615\pi$
$168$	10.8479	0.836937
$169$	−12.0730	−0.928689
$170$	15.1039	1.15842
$171$	1.00000	0.0764719
$172$	1.82136	0.138877
$173$	−1.02758	−0.0781256	−0.0390628	−	0.999237i	$-0.512437\pi$
$173$	−1.02758	−0.0781256	−0.0390628	+	0.999237i	$0.512437\pi$
$174$	−3.08530	−0.233896
$175$	10.8375	0.819241
$176$	0	0
$177$	5.94245	0.446662
$178$	−14.6335	−1.09682
$179$	−2.24583	−0.167861	−0.0839305	−	0.996472i	$-0.526747\pi$
$179$	−2.24583	−0.167861	−0.0839305	+	0.996472i	$0.526747\pi$
$180$	3.02641	0.225575
$181$	3.70115	0.275104	0.137552	−	0.990495i	$-0.456077\pi$
$181$	3.70115	0.275104	0.137552	+	0.990495i	$0.456077\pi$
$182$	3.39892	0.251944
$183$	3.85942	0.285297
$184$	7.11125	0.524248
$185$	11.7131	0.861161
$186$	−8.14775	−0.597422
$187$	0	0
$188$	0.341406	0.0248996
$189$	−3.66641	−0.266692
$190$	−2.71578	−0.197023
$191$	−12.2848	−0.888900	−0.444450	−	0.895804i	$-0.646601\pi$
$191$	−12.2848	−0.888900	−0.444450	+	0.895804i	$0.646601\pi$
$192$	−6.45164	−0.465607
$193$	−6.70816	−0.482864	−0.241432	−	0.970418i	$-0.577617\pi$
$193$	−6.70816	−0.482864	−0.241432	+	0.970418i	$0.577617\pi$
$194$	−12.2342	−0.878365
$195$	2.71578	0.194481
$196$	−6.91261	−0.493758
$197$	9.39270	0.669202	0.334601	−	0.942360i	$-0.391398\pi$
$197$	9.39270	0.669202	0.334601	+	0.942360i	$0.391398\pi$
$198$	0	0
$199$	−0.121631	−0.00862222	−0.00431111	−	0.999991i	$-0.501372\pi$
$199$	−0.121631	−0.00862222	−0.00431111	+	0.999991i	$0.501372\pi$
$200$	−8.74573	−0.618416
$201$	−7.65808	−0.540159
$202$	−1.86387	−0.131141
$203$	11.7487	0.824595
$204$	−5.96731	−0.417795
$205$	−19.5872	−1.36803
$206$	−15.7210	−1.09534
$207$	−2.40347	−0.167053
$208$	−0.676718	−0.0469219
$209$	0	0
$210$	9.95716	0.687109
$211$	22.4878	1.54813	0.774063	−	0.633109i	$-0.218221\pi$
$211$	22.4878	1.54813	0.774063	+	0.633109i	$0.218221\pi$
$212$	3.86646	0.265550
$213$	−4.26434	−0.292188
$214$	7.63882	0.522179
$215$	4.78803	0.326541
$216$	2.95874	0.201317
$217$	31.0263	2.10620
$218$	11.7546	0.796123
$219$	2.88048	0.194645
$220$	0	0
$221$	−5.35483	−0.360205
$222$	3.99830	0.268348
$223$	−18.4720	−1.23698	−0.618488	−	0.785794i	$-0.712254\pi$
$223$	−18.4720	−1.23698	−0.618488	+	0.785794i	$0.712254\pi$
$224$	19.2148	1.28384
$225$	2.95590	0.197060
$226$	7.47479	0.497216
$227$	−4.18180	−0.277556	−0.138778	−	0.990324i	$-0.544317\pi$
$227$	−4.18180	−0.277556	−0.138778	+	0.990324i	$0.544317\pi$
$228$	1.07296	0.0710584
$229$	−6.67998	−0.441426	−0.220713	−	0.975339i	$-0.570838\pi$
$229$	−6.67998	−0.441426	−0.220713	+	0.975339i	$0.570838\pi$
$230$	6.52730	0.430398
$231$	0	0
$232$	−9.48100	−0.622458
$233$	−21.3747	−1.40030	−0.700150	−	0.713996i	$-0.746884\pi$
$233$	−21.3747	−1.40030	−0.700150	+	0.713996i	$0.746884\pi$
$234$	0.927041	0.0606026
$235$	0.897497	0.0585462
$236$	6.37600	0.415042
$237$	−16.1718	−1.05047
$238$	−19.6330	−1.27262
$239$	−27.9617	−1.80869	−0.904345	−	0.426801i	$-0.859640\pi$
$239$	−27.9617	−1.80869	−0.904345	+	0.426801i	$0.859640\pi$
$240$	−1.98245	−0.127967
$241$	7.45365	0.480132	0.240066	−	0.970757i	$-0.422831\pi$
$241$	7.45365	0.480132	0.240066	+	0.970757i	$0.422831\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	4.14100	0.265100
$245$	−18.1721	−1.16097
$246$	−6.68615	−0.426294
$247$	0.962830	0.0612634
$248$	−25.0377	−1.58990
$249$	−12.2304	−0.775070
$250$	5.55132	0.351096
$251$	−6.98528	−0.440907	−0.220454	−	0.975397i	$-0.570754\pi$
$251$	−6.98528	−0.440907	−0.220454	+	0.975397i	$0.570754\pi$
$252$	−3.93391	−0.247813
$253$	0	0
$254$	−17.3353	−1.08771
$255$	−15.6870	−0.982359
$256$	−17.0143	−1.06339
$257$	−18.6997	−1.16646	−0.583229	−	0.812308i	$-0.698211\pi$
$257$	−18.6997	−1.16646	−0.583229	+	0.812308i	$0.698211\pi$
$258$	1.63441	0.101754
$259$	−15.2253	−0.946056
$260$	2.91392	0.180713
$261$	3.20441	0.198348
$262$	−2.98085	−0.184157
$263$	−22.6579	−1.39714	−0.698572	−	0.715540i	$-0.746181\pi$
$263$	−22.6579	−1.39714	−0.698572	+	0.715540i	$0.746181\pi$
$264$	0	0
$265$	10.1643	0.624385
$266$	3.53013	0.216446
$267$	15.1984	0.930127
$268$	−8.21680	−0.501921
$269$	5.09792	0.310826	0.155413	−	0.987850i	$-0.450329\pi$
$269$	5.09792	0.310826	0.155413	+	0.987850i	$0.450329\pi$
$270$	2.71578	0.165277
$271$	−2.37684	−0.144383	−0.0721913	−	0.997391i	$-0.522999\pi$
$271$	−2.37684	−0.144383	−0.0721913	+	0.997391i	$0.522999\pi$
$272$	3.90889	0.237011
$273$	−3.53013	−0.213653
$274$	−15.9943	−0.966252
$275$	0	0
$276$	−2.57883	−0.155227
$277$	−17.2921	−1.03898	−0.519490	−	0.854476i	$-0.673878\pi$
$277$	−17.2921	−1.03898	−0.519490	+	0.854476i	$0.673878\pi$
$278$	15.8581	0.951103
$279$	8.46230	0.506624
$280$	30.5979	1.82858
$281$	0.0430852	0.00257025	0.00128512	−	0.999999i	$-0.499591\pi$
$281$	0.0430852	0.00257025	0.00128512	+	0.999999i	$0.499591\pi$
$282$	0.306364	0.0182437
$283$	7.00399	0.416344	0.208172	−	0.978092i	$-0.433249\pi$
$283$	7.00399	0.416344	0.208172	+	0.978092i	$0.433249\pi$
$284$	−4.57546	−0.271503
$285$	2.82062	0.167079
$286$	0	0
$287$	25.4606	1.50289
$288$	5.24075	0.308814
$289$	13.9308	0.819460
$290$	−8.70246	−0.511026
$291$	12.7065	0.744869
$292$	3.09063	0.180866
$293$	−12.0346	−0.703067	−0.351534	−	0.936175i	$-0.614340\pi$
$293$	−12.0346	−0.703067	−0.351534	+	0.936175i	$0.614340\pi$
$294$	−6.20310	−0.361772
$295$	16.7614	0.975886
$296$	12.2866	0.714144
$297$	0	0
$298$	−21.4601	−1.24315
$299$	−2.31414	−0.133830
$300$	3.17156	0.183110
$301$	−6.22377	−0.358732
$302$	16.0804	0.925322
$303$	1.93582	0.111210
$304$	−0.702842	−0.0403108
$305$	10.8860	0.623329
$306$	−5.35483	−0.306115
$307$	−29.7991	−1.70072	−0.850362	−	0.526199i	$-0.823617\pi$
$307$	−29.7991	−1.70072	−0.850362	+	0.526199i	$0.823617\pi$
$308$	0	0
$309$	16.3279	0.928864
$310$	−22.9817	−1.30527
$311$	30.6144	1.73598	0.867991	−	0.496580i	$-0.165411\pi$
$311$	30.6144	1.73598	0.867991	+	0.496580i	$0.165411\pi$
$312$	2.84876	0.161279
$313$	−30.5627	−1.72751	−0.863754	−	0.503914i	$-0.831893\pi$
$313$	−30.5627	−1.72751	−0.863754	+	0.503914i	$0.831893\pi$
$314$	−5.58202	−0.315012
$315$	−10.3416	−0.582681
$316$	−17.3516	−0.976105
$317$	0.811942	0.0456032	0.0228016	−	0.999740i	$-0.492741\pi$
$317$	0.811942	0.0456032	0.0228016	+	0.999740i	$0.492741\pi$
$318$	3.46961	0.194566
$319$	0	0
$320$	−18.1976	−1.01728
$321$	−7.93372	−0.442817
$322$	−8.48458	−0.472827
$323$	−5.56155	−0.309453
$324$	−1.07296	−0.0596088
$325$	2.84603	0.157869
$326$	−5.85128	−0.324072
$327$	−12.2084	−0.675126
$328$	−20.5463	−1.13448
$329$	−1.16662	−0.0643179
$330$	0	0
$331$	18.0577	0.992541	0.496271	−	0.868168i	$-0.334702\pi$
$331$	18.0577	0.992541	0.496271	+	0.868168i	$0.334702\pi$
$332$	−13.1227	−0.720202
$333$	−4.15265	−0.227564
$334$	−17.7950	−0.973701
$335$	−21.6005	−1.18016
$336$	2.57691	0.140582
$337$	−2.14441	−0.116813	−0.0584066	−	0.998293i	$-0.518602\pi$
$337$	−2.14441	−0.116813	−0.0584066	+	0.998293i	$0.518602\pi$
$338$	−11.6242	−0.632273
$339$	−7.76336	−0.421648
$340$	−16.8315	−0.912817
$341$	0	0
$342$	0.962830	0.0520639
$343$	−2.04376	−0.110352
$344$	5.02249	0.270794
$345$	−6.77929	−0.364985
$346$	−0.989386	−0.0531897
$347$	−15.2883	−0.820721	−0.410361	−	0.911923i	$-0.634597\pi$
$347$	−15.2883	−0.820721	−0.410361	+	0.911923i	$0.634597\pi$
$348$	3.43820	0.184307
$349$	4.60789	0.246655	0.123327	−	0.992366i	$-0.460643\pi$
$349$	4.60789	0.246655	0.123327	+	0.992366i	$0.460643\pi$
$350$	10.4347	0.557759
$351$	−0.962830	−0.0513921
$352$	0	0
$353$	−16.5059	−0.878520	−0.439260	−	0.898360i	$-0.644759\pi$
$353$	−16.5059	−0.878520	−0.439260	+	0.898360i	$0.644759\pi$
$354$	5.72157	0.304098
$355$	−12.0281	−0.638384
$356$	16.3072	0.864283
$357$	20.3909	1.07920
$358$	−2.16235	−0.114284
$359$	5.10280	0.269316	0.134658	−	0.990892i	$-0.457006\pi$
$359$	5.10280	0.269316	0.134658	+	0.990892i	$0.457006\pi$
$360$	8.34547	0.439845
$361$	1.00000	0.0526316
$362$	3.56358	0.187297
$363$	0	0
$364$	−3.78768	−0.198529
$365$	8.12474	0.425268
$366$	3.71597	0.194237
$367$	25.2179	1.31636	0.658182	−	0.752859i	$-0.271326\pi$
$367$	25.2179	1.31636	0.658182	+	0.752859i	$0.271326\pi$
$368$	1.68926	0.0880590
$369$	6.94427	0.361504
$370$	11.2777	0.586298
$371$	−13.2121	−0.685938
$372$	9.07969	0.470760
$373$	24.9173	1.29017	0.645086	−	0.764110i	$-0.276822\pi$
$373$	24.9173	1.29017	0.645086	+	0.764110i	$0.276822\pi$
$374$	0	0
$375$	−5.76563	−0.297736
$376$	0.941444	0.0485513
$377$	3.08530	0.158901
$378$	−3.53013	−0.181570
$379$	−3.66292	−0.188152	−0.0940758	−	0.995565i	$-0.529990\pi$
$379$	−3.66292	−0.188152	−0.0940758	+	0.995565i	$0.529990\pi$
$380$	3.02641	0.155252
$381$	18.0045	0.922399
$382$	−11.8282	−0.605184
$383$	33.7831	1.72623	0.863117	−	0.505003i	$-0.168509\pi$
$383$	33.7831	1.72623	0.863117	+	0.505003i	$0.168509\pi$
$384$	4.26968	0.217886
$385$	0	0
$386$	−6.45882	−0.328745
$387$	−1.69751	−0.0862893
$388$	13.6336	0.692140
$389$	−34.7163	−1.76018	−0.880092	−	0.474803i	$-0.842519\pi$
$389$	−34.7163	−1.76018	−0.880092	+	0.474803i	$0.842519\pi$
$390$	2.61483	0.132407
$391$	13.3670	0.676000
$392$	−19.0619	−0.962770
$393$	3.09592	0.156169
$394$	9.04357	0.455608
$395$	−45.6144	−2.29511
$396$	0	0
$397$	−3.32667	−0.166961	−0.0834805	−	0.996509i	$-0.526604\pi$
$397$	−3.32667	−0.166961	−0.0834805	+	0.996509i	$0.526604\pi$
$398$	−0.117110	−0.00587021
$399$	−3.66641	−0.183550
$400$	−2.07753	−0.103877
$401$	−10.9214	−0.545389	−0.272695	−	0.962101i	$-0.587915\pi$
$401$	−10.9214	−0.545389	−0.272695	+	0.962101i	$0.587915\pi$
$402$	−7.37343	−0.367753
$403$	8.14775	0.405868
$404$	2.07706	0.103338
$405$	−2.82062	−0.140158
$406$	11.3120	0.561404
$407$	0	0
$408$	−16.4552	−0.814652
$409$	−18.6689	−0.923118	−0.461559	−	0.887109i	$-0.652710\pi$
$409$	−18.6689	−0.923118	−0.461559	+	0.887109i	$0.652710\pi$
$410$	−18.8591	−0.931385
$411$	16.6118	0.819398
$412$	17.5192	0.863109
$413$	−21.7875	−1.07209
$414$	−2.31414	−0.113734
$415$	−34.4973	−1.69341
$416$	5.04596	0.247398
$417$	−16.4703	−0.806552
$418$	0	0
$419$	−5.63200	−0.275141	−0.137571	−	0.990492i	$-0.543929\pi$
$419$	−5.63200	−0.275141	−0.137571	+	0.990492i	$0.543929\pi$
$420$	−11.0961	−0.541432
$421$	−32.5674	−1.58724	−0.793618	−	0.608416i	$-0.791805\pi$
$421$	−32.5674	−1.58724	−0.793618	+	0.608416i	$0.791805\pi$
$422$	21.6519	1.05400
$423$	−0.318191	−0.0154710
$424$	10.6620	0.517790
$425$	−16.4394	−0.797427
$426$	−4.10583	−0.198928
$427$	−14.1502	−0.684778
$428$	−8.51255	−0.411470
$429$	0	0
$430$	4.61006	0.222317
$431$	−1.48197	−0.0713838	−0.0356919	−	0.999363i	$-0.511363\pi$
$431$	−1.48197	−0.0713838	−0.0356919	+	0.999363i	$0.511363\pi$
$432$	0.702842	0.0338155
$433$	−32.4189	−1.55795	−0.778975	−	0.627054i	$-0.784260\pi$
$433$	−32.4189	−1.55795	−0.778975	+	0.627054i	$0.784260\pi$
$434$	29.8730	1.43395
$435$	9.03842	0.433359
$436$	−13.0991	−0.627333
$437$	−2.40347	−0.114974
$438$	2.77341	0.132519
$439$	−8.14799	−0.388883	−0.194441	−	0.980914i	$-0.562289\pi$
$439$	−8.14799	−0.388883	−0.194441	+	0.980914i	$0.562289\pi$
$440$	0	0
$441$	6.44257	0.306789
$442$	−5.15579	−0.245236
$443$	−37.7913	−1.79552	−0.897759	−	0.440487i	$-0.854806\pi$
$443$	−37.7913	−1.79552	−0.897759	+	0.440487i	$0.854806\pi$
$444$	−4.45562	−0.211454
$445$	42.8689	2.03218
$446$	−17.7854	−0.842162
$447$	22.2886	1.05422
$448$	23.6544	1.11756
$449$	26.9471	1.27171	0.635855	−	0.771808i	$-0.280648\pi$
$449$	26.9471	1.27171	0.635855	+	0.771808i	$0.280648\pi$
$450$	2.84603	0.134163
$451$	0	0
$452$	−8.32976	−0.391799
$453$	−16.7012	−0.784689
$454$	−4.02636	−0.188966
$455$	−9.95716	−0.466799
$456$	2.95874	0.138556
$457$	−11.8589	−0.554736	−0.277368	−	0.960764i	$-0.589462\pi$
$457$	−11.8589	−0.554736	−0.277368	+	0.960764i	$0.589462\pi$
$458$	−6.43169	−0.300533
$459$	5.56155	0.259591
$460$	−7.27390	−0.339147
$461$	−23.8895	−1.11264	−0.556322	−	0.830967i	$-0.687788\pi$
$461$	−23.8895	−1.11264	−0.556322	+	0.830967i	$0.687788\pi$
$462$	0	0
$463$	8.83615	0.410651	0.205325	−	0.978694i	$-0.434175\pi$
$463$	8.83615	0.410651	0.205325	+	0.978694i	$0.434175\pi$
$464$	−2.25219	−0.104555
$465$	23.8689	1.10689
$466$	−20.5802	−0.953357
$467$	−8.60334	−0.398115	−0.199057	−	0.979988i	$-0.563788\pi$
$467$	−8.60334	−0.398115	−0.199057	+	0.979988i	$0.563788\pi$
$468$	−1.03308	−0.0477540
$469$	28.0777	1.29651
$470$	0.864137	0.0398597
$471$	5.79752	0.267136
$472$	17.5821	0.809283
$473$	0	0
$474$	−15.5706	−0.715183
$475$	2.95590	0.135626
$476$	21.8786	1.00280
$477$	−3.60355	−0.164995
$478$	−26.9223	−1.23140
$479$	−1.84184	−0.0841559	−0.0420779	−	0.999114i	$-0.513398\pi$
$479$	−1.84184	−0.0841559	−0.0420779	+	0.999114i	$0.513398\pi$
$480$	14.7822	0.674711
$481$	−3.99830	−0.182307
$482$	7.17659	0.326885
$483$	8.81213	0.400966
$484$	0	0
$485$	35.8403	1.62742
$486$	−0.962830	−0.0436748
$487$	38.7398	1.75547	0.877734	−	0.479149i	$-0.159055\pi$
$487$	38.7398	1.75547	0.877734	+	0.479149i	$0.159055\pi$
$488$	11.4190	0.516914
$489$	6.07716	0.274819
$490$	−17.4966	−0.790416
$491$	23.4360	1.05765	0.528825	−	0.848731i	$-0.322633\pi$
$491$	23.4360	1.05765	0.528825	+	0.848731i	$0.322633\pi$
$492$	7.45092	0.335913
$493$	−17.8215	−0.802638
$494$	0.927041	0.0417096
$495$	0	0
$496$	−5.94766	−0.267058
$497$	15.6348	0.701317
$498$	−11.7758	−0.527686
$499$	−12.6338	−0.565568	−0.282784	−	0.959184i	$-0.591258\pi$
$499$	−12.6338	−0.565568	−0.282784	+	0.959184i	$0.591258\pi$
$500$	−6.18628	−0.276659
$501$	18.4820	0.825716
$502$	−6.72564	−0.300180
$503$	21.7283	0.968819	0.484410	−	0.874841i	$-0.339034\pi$
$503$	21.7283	0.968819	0.484410	+	0.874841i	$0.339034\pi$
$504$	−10.8479	−0.483206
$505$	5.46023	0.242977
$506$	0	0
$507$	12.0730	0.536179
$508$	19.3181	0.857102
$509$	32.4023	1.43621	0.718103	−	0.695937i	$-0.245011\pi$
$509$	32.4023	1.43621	0.718103	+	0.695937i	$0.245011\pi$
$510$	−15.1039	−0.668813
$511$	−10.5610	−0.467192
$512$	−7.84248	−0.346592
$513$	−1.00000	−0.0441511
$514$	−18.0047	−0.794151
$515$	46.0549	2.02942
$516$	−1.82136	−0.0801808
$517$	0	0
$518$	−14.6594	−0.644097
$519$	1.02758	0.0451058
$520$	8.03527	0.352370
$521$	−0.675969	−0.0296147	−0.0148074	−	0.999890i	$-0.504714\pi$
$521$	−0.675969	−0.0296147	−0.0148074	+	0.999890i	$0.504714\pi$
$522$	3.08530	0.135040
$523$	10.9506	0.478837	0.239419	−	0.970916i	$-0.423043\pi$
$523$	10.9506	0.478837	0.239419	+	0.970916i	$0.423043\pi$
$524$	3.32180	0.145113
$525$	−10.8375	−0.472989
$526$	−21.8157	−0.951208
$527$	−47.0635	−2.05012
$528$	0	0
$529$	−17.2233	−0.748840
$530$	9.78644	0.425096
$531$	−5.94245	−0.257880
$532$	−3.93391	−0.170557
$533$	6.68615	0.289609
$534$	14.6335	0.633252
$535$	−22.3780	−0.967486
$536$	−22.6582	−0.978687
$537$	2.24583	0.0969146
$538$	4.90843	0.211618
$539$	0	0
$540$	−3.02641	−0.130236
$541$	21.2992	0.915726	0.457863	−	0.889023i	$-0.348615\pi$
$541$	21.2992	0.915726	0.457863	+	0.889023i	$0.348615\pi$
$542$	−2.28849	−0.0982990
$543$	−3.70115	−0.158831
$544$	−29.1467	−1.24966
$545$	−34.4353	−1.47504
$546$	−3.39892	−0.145460
$547$	−0.992802	−0.0424491	−0.0212246	−	0.999775i	$-0.506756\pi$
$547$	−0.992802	−0.0424491	−0.0212246	+	0.999775i	$0.506756\pi$
$548$	17.8237	0.761393
$549$	−3.85942	−0.164716
$550$	0	0
$551$	3.20441	0.136512
$552$	−7.11125	−0.302675
$553$	59.2923	2.52137
$554$	−16.6493	−0.707362
$555$	−11.7131	−0.497191
$556$	−17.6719	−0.749456
$557$	15.3684	0.651182	0.325591	−	0.945511i	$-0.394437\pi$
$557$	15.3684	0.651182	0.325591	+	0.945511i	$0.394437\pi$
$558$	8.14775	0.344922
$559$	−1.63441	−0.0691283
$560$	7.26848	0.307150
$561$	0	0
$562$	0.0414837	0.00174989
$563$	38.6591	1.62929	0.814644	−	0.579961i	$-0.196932\pi$
$563$	38.6591	1.62929	0.814644	+	0.579961i	$0.196932\pi$
$564$	−0.341406	−0.0143758
$565$	−21.8975	−0.921234
$566$	6.74365	0.283457
$567$	3.66641	0.153975
$568$	−12.6171	−0.529400
$569$	−11.9126	−0.499404	−0.249702	−	0.968323i	$-0.580333\pi$
$569$	−11.9126	−0.499404	−0.249702	+	0.968323i	$0.580333\pi$
$570$	2.71578	0.113751
$571$	39.7008	1.66143	0.830714	−	0.556699i	$-0.187932\pi$
$571$	39.7008	1.66143	0.830714	+	0.556699i	$0.187932\pi$
$572$	0	0
$573$	12.2848	0.513207
$574$	24.5142	1.02320
$575$	−7.10443	−0.296275
$576$	6.45164	0.268818
$577$	−25.6928	−1.06961	−0.534804	−	0.844976i	$-0.679614\pi$
$577$	−25.6928	−1.06961	−0.534804	+	0.844976i	$0.679614\pi$
$578$	13.4130	0.557908
$579$	6.70816	0.278782
$580$	9.69785	0.402681
$581$	44.8416	1.86034
$582$	12.2342	0.507125
$583$	0	0
$584$	8.52258	0.352667
$585$	−2.71578	−0.112284
$586$	−11.5872	−0.478665
$587$	36.2436	1.49593	0.747967	−	0.663736i	$-0.231030\pi$
$587$	36.2436	1.49593	0.747967	+	0.663736i	$0.231030\pi$
$588$	6.91261	0.285071
$589$	8.46230	0.348683
$590$	16.1384	0.664406
$591$	−9.39270	−0.386364
$592$	2.91866	0.119956
$593$	−28.6234	−1.17542	−0.587711	−	0.809071i	$-0.699971\pi$
$593$	−28.6234	−1.17542	−0.587711	+	0.809071i	$0.699971\pi$
$594$	0	0
$595$	57.5151	2.35789
$596$	23.9148	0.979587
$597$	0.121631	0.00497804
$598$	−2.22812	−0.0911146
$599$	0.200807	0.00820477	0.00410239	−	0.999992i	$-0.498694\pi$
$599$	0.200807	0.00820477	0.00410239	+	0.999992i	$0.498694\pi$
$600$	8.74573	0.357043
$601$	−10.1160	−0.412641	−0.206321	−	0.978484i	$-0.566149\pi$
$601$	−10.1160	−0.412641	−0.206321	+	0.978484i	$0.566149\pi$
$602$	−5.99243	−0.244233
$603$	7.65808	0.311861
$604$	−17.9197	−0.729140
$605$	0	0
$606$	1.86387	0.0757145
$607$	−22.4962	−0.913091	−0.456545	−	0.889700i	$-0.650913\pi$
$607$	−22.4962	−0.913091	−0.456545	+	0.889700i	$0.650913\pi$
$608$	5.24075	0.212541
$609$	−11.7487	−0.476080
$610$	10.4813	0.424377
$611$	−0.306364	−0.0123942
$612$	5.96731	0.241214
$613$	−6.59033	−0.266181	−0.133090	−	0.991104i	$-0.542490\pi$
$613$	−6.59033	−0.266181	−0.133090	+	0.991104i	$0.542490\pi$
$614$	−28.6914	−1.15789
$615$	19.5872	0.789831
$616$	0	0
$617$	−44.7996	−1.80357	−0.901783	−	0.432190i	$-0.857741\pi$
$617$	−44.7996	−1.80357	−0.901783	+	0.432190i	$0.857741\pi$
$618$	15.7210	0.632392
$619$	36.9201	1.48395	0.741973	−	0.670430i	$-0.233890\pi$
$619$	36.9201	1.48395	0.741973	+	0.670430i	$0.233890\pi$
$620$	25.6104	1.02854
$621$	2.40347	0.0964481
$622$	29.4764	1.18190
$623$	−55.7236	−2.23252
$624$	0.676718	0.0270904
$625$	−31.0422	−1.24169
$626$	−29.4267	−1.17613
$627$	0	0
$628$	6.22050	0.248225
$629$	23.0952	0.920865
$630$	−9.95716	−0.396703
$631$	19.6457	0.782082	0.391041	−	0.920373i	$-0.372115\pi$
$631$	19.6457	0.782082	0.391041	+	0.920373i	$0.372115\pi$
$632$	−47.8480	−1.90329
$633$	−22.4878	−0.893811
$634$	0.781762	0.0310477
$635$	50.7839	2.01530
$636$	−3.86646	−0.153315
$637$	6.20310	0.245776
$638$	0	0
$639$	4.26434	0.168695
$640$	12.0431	0.476047
$641$	−4.70463	−0.185822	−0.0929109	−	0.995674i	$-0.529617\pi$
$641$	−4.70463	−0.185822	−0.0929109	+	0.995674i	$0.529617\pi$
$642$	−7.63882	−0.301480
$643$	−34.5748	−1.36350	−0.681748	−	0.731587i	$-0.738780\pi$
$643$	−34.5748	−1.36350	−0.681748	+	0.731587i	$0.738780\pi$
$644$	9.45505	0.372581
$645$	−4.78803	−0.188529
$646$	−5.35483	−0.210683
$647$	37.1307	1.45976	0.729880	−	0.683576i	$-0.239576\pi$
$647$	37.1307	1.45976	0.729880	+	0.683576i	$0.239576\pi$
$648$	−2.95874	−0.116230
$649$	0	0
$650$	2.74024	0.107481
$651$	−31.0263	−1.21601
$652$	6.52055	0.255364
$653$	3.86208	0.151135	0.0755674	−	0.997141i	$-0.475923\pi$
$653$	3.86208	0.151135	0.0755674	+	0.997141i	$0.475923\pi$
$654$	−11.7546	−0.459642
$655$	8.73242	0.341204
$656$	−4.88073	−0.190561
$657$	−2.88048	−0.112378
$658$	−1.12326	−0.0437891
$659$	36.8683	1.43619	0.718093	−	0.695947i	$-0.245015\pi$
$659$	36.8683	1.43619	0.718093	+	0.695947i	$0.245015\pi$
$660$	0	0
$661$	33.4275	1.30018	0.650089	−	0.759858i	$-0.274732\pi$
$661$	33.4275	1.30018	0.650089	+	0.759858i	$0.274732\pi$
$662$	17.3865	0.675745
$663$	5.35483	0.207964
$664$	−36.1865	−1.40431
$665$	−10.3416	−0.401028
$666$	−3.99830	−0.154931
$667$	−7.70171	−0.298211
$668$	19.8304	0.767263
$669$	18.4720	0.714168
$670$	−20.7976	−0.803483
$671$	0	0
$672$	−19.2148	−0.741226
$673$	−10.7846	−0.415718	−0.207859	−	0.978159i	$-0.566649\pi$
$673$	−10.7846	−0.415718	−0.207859	+	0.978159i	$0.566649\pi$
$674$	−2.06470	−0.0795292
$675$	−2.95590	−0.113773
$676$	12.9538	0.498223
$677$	−10.6469	−0.409193	−0.204597	−	0.978846i	$-0.565588\pi$
$677$	−10.6469	−0.409193	−0.204597	+	0.978846i	$0.565588\pi$
$678$	−7.47479	−0.287068
$679$	−46.5873	−1.78786
$680$	−46.4137	−1.77989
$681$	4.18180	0.160247
$682$	0	0
$683$	−35.1109	−1.34348	−0.671740	−	0.740787i	$-0.734453\pi$
$683$	−35.1109	−1.34348	−0.671740	+	0.740787i	$0.734453\pi$
$684$	−1.07296	−0.0410256
$685$	46.8555	1.79026
$686$	−1.96779	−0.0751306
$687$	6.67998	0.254857
$688$	1.19308	0.0454858
$689$	−3.46961	−0.132181
$690$	−6.52730	−0.248490
$691$	11.1345	0.423576	0.211788	−	0.977316i	$-0.432071\pi$
$691$	11.1345	0.423576	0.211788	+	0.977316i	$0.432071\pi$
$692$	1.10255	0.0419128
$693$	0	0
$694$	−14.7201	−0.558766
$695$	−46.4563	−1.76219
$696$	9.48100	0.359376
$697$	−38.6209	−1.46287
$698$	4.43662	0.167928
$699$	21.3747	0.808464
$700$	−11.6282	−0.439506
$701$	−14.7582	−0.557410	−0.278705	−	0.960377i	$-0.589905\pi$
$701$	−14.7582	−0.557410	−0.278705	+	0.960377i	$0.589905\pi$
$702$	−0.927041	−0.0349889
$703$	−4.15265	−0.156620
$704$	0	0
$705$	−0.897497	−0.0338017
$706$	−15.8924	−0.598117
$707$	−7.09753	−0.266930
$708$	−6.37600	−0.239625
$709$	−48.0337	−1.80394	−0.901972	−	0.431796i	$-0.857880\pi$
$709$	−48.0337	−1.80394	−0.901972	+	0.431796i	$0.857880\pi$
$710$	−11.5810	−0.434627
$711$	16.1718	0.606488
$712$	44.9680	1.68525
$713$	−20.3389	−0.761698
$714$	19.6330	0.734746
$715$	0	0
$716$	2.40968	0.0900540
$717$	27.9617	1.04425
$718$	4.91313	0.183356
$719$	5.42858	0.202452	0.101226	−	0.994863i	$-0.467723\pi$
$719$	5.42858	0.202452	0.101226	+	0.994863i	$0.467723\pi$
$720$	1.98245	0.0738816
$721$	−59.8649	−2.22949
$722$	0.962830	0.0358328
$723$	−7.45365	−0.277204
$724$	−3.97118	−0.147588
$725$	9.47191	0.351778
$726$	0	0
$727$	−26.6951	−0.990068	−0.495034	−	0.868874i	$-0.664844\pi$
$727$	−26.6951	−0.990068	−0.495034	+	0.868874i	$0.664844\pi$
$728$	−10.4447	−0.387107
$729$	1.00000	0.0370370
$730$	7.82274	0.289533
$731$	9.44079	0.349180
$732$	−4.14100	−0.153056
$733$	−41.9342	−1.54888	−0.774438	−	0.632650i	$-0.781967\pi$
$733$	−41.9342	−1.54888	−0.774438	+	0.632650i	$0.781967\pi$
$734$	24.2806	0.896211
$735$	18.1721	0.670286
$736$	−12.5960	−0.464296
$737$	0	0
$738$	6.68615	0.246121
$739$	−4.87247	−0.179237	−0.0896184	−	0.995976i	$-0.528565\pi$
$739$	−4.87247	−0.179237	−0.0896184	+	0.995976i	$0.528565\pi$
$740$	−12.5676	−0.461995
$741$	−0.962830	−0.0353704
$742$	−12.7210	−0.467003
$743$	−4.09643	−0.150283	−0.0751417	−	0.997173i	$-0.523941\pi$
$743$	−4.09643	−0.150283	−0.0751417	+	0.997173i	$0.523941\pi$
$744$	25.0377	0.917927
$745$	62.8677	2.30330
$746$	23.9912	0.878379
$747$	12.2304	0.447487
$748$	0	0
$749$	29.0883	1.06286
$750$	−5.55132	−0.202706
$751$	11.1956	0.408534	0.204267	−	0.978915i	$-0.434519\pi$
$751$	11.1956	0.408534	0.204267	+	0.978915i	$0.434519\pi$
$752$	0.223638	0.00815525
$753$	6.98528	0.254558
$754$	2.97062	0.108184
$755$	−47.1076	−1.71442
$756$	3.93391	0.143075
$757$	48.8585	1.77579	0.887897	−	0.460043i	$-0.152166\pi$
$757$	48.8585	1.77579	0.887897	+	0.460043i	$0.152166\pi$
$758$	−3.52677	−0.128098
$759$	0	0
$760$	8.34547	0.302722
$761$	9.84873	0.357016	0.178508	−	0.983938i	$-0.442873\pi$
$761$	9.84873	0.357016	0.178508	+	0.983938i	$0.442873\pi$
$762$	17.3353	0.627991
$763$	44.7610	1.62046
$764$	13.1811	0.476876
$765$	15.6870	0.567165
$766$	32.5274	1.17526
$767$	−5.72157	−0.206594
$768$	17.0143	0.613949
$769$	38.8348	1.40042	0.700210	−	0.713937i	$-0.253090\pi$
$769$	38.8348	1.40042	0.700210	+	0.713937i	$0.253090\pi$
$770$	0	0
$771$	18.6997	0.673454
$772$	7.19758	0.259047
$773$	−9.72599	−0.349819	−0.174910	−	0.984584i	$-0.555963\pi$
$773$	−9.72599	−0.349819	−0.174910	+	0.984584i	$0.555963\pi$
$774$	−1.63441	−0.0587478
$775$	25.0137	0.898518
$776$	37.5952	1.34959
$777$	15.2253	0.546206
$778$	−33.4259	−1.19837
$779$	6.94427	0.248804
$780$	−2.91392	−0.104335
$781$	0	0
$782$	12.8702	0.460237
$783$	−3.20441	−0.114516
$784$	−4.52811	−0.161718
$785$	16.3526	0.583649
$786$	2.98085	0.106323
$787$	−52.1724	−1.85975	−0.929873	−	0.367880i	$-0.880084\pi$
$787$	−52.1724	−1.85975	−0.929873	+	0.367880i	$0.880084\pi$
$788$	−10.0780	−0.359013
$789$	22.6579	0.806641
$790$	−43.9189	−1.56256
$791$	28.4637	1.01205
$792$	0	0
$793$	−3.71597	−0.131958
$794$	−3.20302	−0.113671
$795$	−10.1643	−0.360489
$796$	0.130505	0.00462564
$797$	−6.10493	−0.216248	−0.108124	−	0.994137i	$-0.534484\pi$
$797$	−6.10493	−0.216248	−0.108124	+	0.994137i	$0.534484\pi$
$798$	−3.53013	−0.124965
$799$	1.76964	0.0626052
$800$	15.4911	0.547695
$801$	−15.1984	−0.537009
$802$	−10.5155	−0.371314
$803$	0	0
$804$	8.21680	0.289784
$805$	24.8557	0.876047
$806$	7.84490	0.276325
$807$	−5.09792	−0.179455
$808$	5.72759	0.201496
$809$	−29.4137	−1.03413	−0.517065	−	0.855946i	$-0.672975\pi$
$809$	−29.4137	−1.03413	−0.517065	+	0.855946i	$0.672975\pi$
$810$	−2.71578	−0.0954227
$811$	−12.4681	−0.437816	−0.218908	−	0.975746i	$-0.570249\pi$
$811$	−12.4681	−0.437816	−0.218908	+	0.975746i	$0.570249\pi$
$812$	−12.6058	−0.442378
$813$	2.37684	0.0833593
$814$	0	0
$815$	17.1414	0.600436
$816$	−3.90889	−0.136839
$817$	−1.69751	−0.0593884
$818$	−17.9750	−0.628481
$819$	3.53013	0.123353
$820$	21.0162	0.733918
$821$	−2.33364	−0.0814446	−0.0407223	−	0.999171i	$-0.512966\pi$
$821$	−2.33364	−0.0814446	−0.0407223	+	0.999171i	$0.512966\pi$
$822$	15.9943	0.557866
$823$	14.4967	0.505324	0.252662	−	0.967555i	$-0.418694\pi$
$823$	14.4967	0.505324	0.252662	+	0.967555i	$0.418694\pi$
$824$	48.3101	1.68296
$825$	0	0
$826$	−20.9776	−0.729905
$827$	42.8222	1.48907	0.744536	−	0.667582i	$-0.232671\pi$
$827$	42.8222	1.48907	0.744536	+	0.667582i	$0.232671\pi$
$828$	2.57883	0.0896205
$829$	−19.4223	−0.674563	−0.337282	−	0.941404i	$-0.609507\pi$
$829$	−19.4223	−0.674563	−0.337282	+	0.941404i	$0.609507\pi$
$830$	−33.2150	−1.15291
$831$	17.2921	0.599855
$832$	6.21183	0.215357
$833$	−35.8307	−1.24146
$834$	−15.8581	−0.549119
$835$	52.1308	1.80406
$836$	0	0
$837$	−8.46230	−0.292500
$838$	−5.42266	−0.187323
$839$	−30.6144	−1.05693	−0.528463	−	0.848956i	$-0.677231\pi$
$839$	−30.6144	−1.05693	−0.528463	+	0.848956i	$0.677231\pi$
$840$	−30.5979	−1.05573
$841$	−18.7318	−0.645923
$842$	−31.3568	−1.08063
$843$	−0.0430852	−0.00148393
$844$	−24.1285	−0.830537
$845$	34.0532	1.17147
$846$	−0.306364	−0.0105330
$847$	0	0
$848$	2.53273	0.0869743
$849$	−7.00399	−0.240376
$850$	−15.8283	−0.542907
$851$	9.98079	0.342137
$852$	4.57546	0.156753
$853$	34.1278	1.16851	0.584256	−	0.811569i	$-0.301386\pi$
$853$	34.1278	1.16851	0.584256	+	0.811569i	$0.301386\pi$
$854$	−13.6243	−0.466213
$855$	−2.82062	−0.0964632
$856$	−23.4738	−0.802317
$857$	37.8821	1.29403	0.647014	−	0.762478i	$-0.276018\pi$
$857$	37.8821	1.29403	0.647014	+	0.762478i	$0.276018\pi$
$858$	0	0
$859$	−29.2307	−0.997339	−0.498670	−	0.866792i	$-0.666178\pi$
$859$	−29.2307	−0.997339	−0.498670	+	0.866792i	$0.666178\pi$
$860$	−5.13736	−0.175183
$861$	−25.4606	−0.867694
$862$	−1.42688	−0.0485997
$863$	20.7378	0.705923	0.352961	−	0.935638i	$-0.385175\pi$
$863$	20.7378	0.705923	0.352961	+	0.935638i	$0.385175\pi$
$864$	−5.24075	−0.178294
$865$	2.89842	0.0985491
$866$	−31.2138	−1.06069
$867$	−13.9308	−0.473116
$868$	−33.2899	−1.12993
$869$	0	0
$870$	8.70246	0.295041
$871$	7.37343	0.249839
$872$	−36.1214	−1.22323
$873$	−12.7065	−0.430050
$874$	−2.31414	−0.0782769
$875$	21.1392	0.714635
$876$	−3.09063	−0.104423
$877$	−16.4969	−0.557060	−0.278530	−	0.960428i	$-0.589847\pi$
$877$	−16.4969	−0.557060	−0.278530	+	0.960428i	$0.589847\pi$
$878$	−7.84513	−0.264760
$879$	12.0346	0.405916
$880$	0	0
$881$	5.40244	0.182013	0.0910065	−	0.995850i	$-0.470992\pi$
$881$	5.40244	0.182013	0.0910065	+	0.995850i	$0.470992\pi$
$882$	6.20310	0.208869
$883$	43.9467	1.47892	0.739462	−	0.673198i	$-0.235080\pi$
$883$	43.9467	1.47892	0.739462	+	0.673198i	$0.235080\pi$
$884$	5.74551	0.193242
$885$	−16.7614	−0.563428
$886$	−36.3866	−1.22243
$887$	46.7913	1.57110	0.785549	−	0.618799i	$-0.212381\pi$
$887$	46.7913	1.57110	0.785549	+	0.618799i	$0.212381\pi$
$888$	−12.2866	−0.412311
$889$	−66.0120	−2.21397
$890$	41.2755	1.38356
$891$	0	0
$892$	19.8197	0.663612
$893$	−0.318191	−0.0106479
$894$	21.4601	0.717735
$895$	6.33463	0.211743
$896$	−15.6544	−0.522977
$897$	2.31414	0.0772668
$898$	25.9454	0.865810
$899$	27.1166	0.904391
$900$	−3.17156	−0.105719
$901$	20.0413	0.667673
$902$	0	0
$903$	6.22377	0.207114
$904$	−22.9697	−0.763962
$905$	−10.4395	−0.347022
$906$	−16.0804	−0.534235
$907$	47.4163	1.57443	0.787217	−	0.616676i	$-0.211521\pi$
$907$	47.4163	1.57443	0.787217	+	0.616676i	$0.211521\pi$
$908$	4.48689	0.148903
$909$	−1.93582	−0.0642072
$910$	−9.58705	−0.317808
$911$	26.9259	0.892095	0.446047	−	0.895009i	$-0.352831\pi$
$911$	26.9259	0.892095	0.446047	+	0.895009i	$0.352831\pi$
$912$	0.702842	0.0232734
$913$	0	0
$914$	−11.4181	−0.377677
$915$	−10.8860	−0.359879
$916$	7.16735	0.236816
$917$	−11.3509	−0.374841
$918$	5.35483	0.176736
$919$	56.1649	1.85271	0.926354	−	0.376654i	$-0.122925\pi$
$919$	56.1649	1.85271	0.926354	+	0.376654i	$0.122925\pi$
$920$	−20.0581	−0.661297
$921$	29.7991	0.981913
$922$	−23.0015	−0.757514
$923$	4.10583	0.135145
$924$	0	0
$925$	−12.2748	−0.403594
$926$	8.50771	0.279581
$927$	−16.3279	−0.536280
$928$	16.7935	0.551274
$929$	44.1496	1.44850	0.724250	−	0.689537i	$-0.242186\pi$
$929$	44.1496	1.44850	0.724250	+	0.689537i	$0.242186\pi$
$930$	22.9817	0.753600
$931$	6.44257	0.211147
$932$	22.9341	0.751232
$933$	−30.6144	−1.00227
$934$	−8.28355	−0.271046
$935$	0	0
$936$	−2.84876	−0.0931146
$937$	−51.9659	−1.69765	−0.848827	−	0.528671i	$-0.822690\pi$
$937$	−51.9659	−1.69765	−0.848827	+	0.528671i	$0.822690\pi$
$938$	27.0340	0.882692
$939$	30.5627	0.997377
$940$	−0.962977	−0.0314088
$941$	23.8467	0.777380	0.388690	−	0.921369i	$-0.372928\pi$
$941$	23.8467	0.777380	0.388690	+	0.921369i	$0.372928\pi$
$942$	5.58202	0.181872
$943$	−16.6904	−0.543514
$944$	4.17661	0.135937
$945$	10.3416	0.336411
$946$	0	0
$947$	49.8973	1.62145	0.810723	−	0.585430i	$-0.199074\pi$
$947$	49.8973	1.62145	0.810723	+	0.585430i	$0.199074\pi$
$948$	17.3516	0.563554
$949$	−2.77341	−0.0900287
$950$	2.84603	0.0923374
$951$	−0.811942	−0.0263290
$952$	60.3314	1.95535
$953$	−4.68705	−0.151829	−0.0759143	−	0.997114i	$-0.524188\pi$
$953$	−4.68705	−0.151829	−0.0759143	+	0.997114i	$0.524188\pi$
$954$	−3.46961	−0.112333
$955$	34.6509	1.12128
$956$	30.0017	0.970325
$957$	0	0
$958$	−1.77338	−0.0572953
$959$	−60.9056	−1.96674
$960$	18.1976	0.587326
$961$	40.6104	1.31001
$962$	−3.84968	−0.124119
$963$	7.93372	0.255661
$964$	−7.99745	−0.257581
$965$	18.9212	0.609094
$966$	8.48458	0.272987
$967$	−14.2175	−0.457202	−0.228601	−	0.973520i	$-0.573415\pi$
$967$	−14.2175	−0.457202	−0.228601	+	0.973520i	$0.573415\pi$
$968$	0	0
$969$	5.56155	0.178663
$970$	34.5081	1.10799
$971$	7.45842	0.239352	0.119676	−	0.992813i	$-0.461814\pi$
$971$	7.45842	0.239352	0.119676	+	0.992813i	$0.461814\pi$
$972$	1.07296	0.0344152
$973$	60.3867	1.93591
$974$	37.2998	1.19516
$975$	−2.84603	−0.0911459
$976$	2.71257	0.0868271
$977$	36.3774	1.16382	0.581908	−	0.813255i	$-0.302306\pi$
$977$	36.3774	1.16382	0.581908	+	0.813255i	$0.302306\pi$
$978$	5.85128	0.187103
$979$	0	0
$980$	19.4979	0.622836
$981$	12.2084	0.389784
$982$	22.5649	0.720073
$983$	32.5920	1.03952	0.519761	−	0.854312i	$-0.326021\pi$
$983$	32.5920	1.03952	0.519761	+	0.854312i	$0.326021\pi$
$984$	20.5463	0.654991
$985$	−26.4932	−0.844145
$986$	−17.1590	−0.546455
$987$	1.16662	0.0371339
$988$	−1.03308	−0.0328665
$989$	4.07992	0.129734
$990$	0	0
$991$	−20.3347	−0.645954	−0.322977	−	0.946407i	$-0.604684\pi$
$991$	−20.3347	−0.645954	−0.322977	+	0.946407i	$0.604684\pi$
$992$	44.3488	1.40808
$993$	−18.0577	−0.573044
$994$	15.0537	0.477473
$995$	0.343076	0.0108762
$996$	13.1227	0.415809
$997$	46.4927	1.47244	0.736219	−	0.676744i	$-0.236609\pi$
$997$	46.4927	1.47244	0.736219	+	0.676744i	$0.236609\pi$
$998$	−12.1642	−0.385052
$999$	4.15265	0.131384

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6897.2.a.bc.1.6	✓	8
11.10	odd	2		6897.2.a.bd.1.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6897.2.a.bc.1.6	✓	8	1.1	even	1	trivial
6897.2.a.bd.1.3	yes	8	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6897 = 3 \cdot 11^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6897.a (trivial)

\(f(q)\)	\(=\)	\(q+0.962830 q^{2} -1.00000 q^{3} -1.07296 q^{4} -2.82062 q^{5} -0.962830 q^{6} +3.66641 q^{7} -2.95874 q^{8} +1.00000 q^{9} -2.71578 q^{10} +1.07296 q^{12} +0.962830 q^{13} +3.53013 q^{14} +2.82062 q^{15} -0.702842 q^{16} -5.56155 q^{17} +0.962830 q^{18} +1.00000 q^{19} +3.02641 q^{20} -3.66641 q^{21} -2.40347 q^{23} +2.95874 q^{24} +2.95590 q^{25} +0.927041 q^{26} -1.00000 q^{27} -3.93391 q^{28} +3.20441 q^{29} +2.71578 q^{30} +8.46230 q^{31} +5.24075 q^{32} -5.35483 q^{34} -10.3416 q^{35} -1.07296 q^{36} -4.15265 q^{37} +0.962830 q^{38} -0.962830 q^{39} +8.34547 q^{40} +6.94427 q^{41} -3.53013 q^{42} -1.69751 q^{43} -2.82062 q^{45} -2.31414 q^{46} -0.318191 q^{47} +0.702842 q^{48} +6.44257 q^{49} +2.84603 q^{50} +5.56155 q^{51} -1.03308 q^{52} -3.60355 q^{53} -0.962830 q^{54} -10.8479 q^{56} -1.00000 q^{57} +3.08530 q^{58} -5.94245 q^{59} -3.02641 q^{60} -3.85942 q^{61} +8.14775 q^{62} +3.66641 q^{63} +6.45164 q^{64} -2.71578 q^{65} +7.65808 q^{67} +5.96731 q^{68} +2.40347 q^{69} -9.95716 q^{70} +4.26434 q^{71} -2.95874 q^{72} -2.88048 q^{73} -3.99830 q^{74} -2.95590 q^{75} -1.07296 q^{76} -0.927041 q^{78} +16.1718 q^{79} +1.98245 q^{80} +1.00000 q^{81} +6.68615 q^{82} +12.2304 q^{83} +3.93391 q^{84} +15.6870 q^{85} -1.63441 q^{86} -3.20441 q^{87} -15.1984 q^{89} -2.71578 q^{90} +3.53013 q^{91} +2.57883 q^{92} -8.46230 q^{93} -0.306364 q^{94} -2.82062 q^{95} -5.24075 q^{96} -12.7065 q^{97} +6.20310 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} - 8 q^{3} + 9 q^{4} - 2 q^{5} + q^{6} - 6 q^{7} - 6 q^{8} + 8 q^{9} - 3 q^{10} - 9 q^{12} - q^{13} + 7 q^{14} + 2 q^{15} + 23 q^{16} - 15 q^{17} - q^{18} + 8 q^{19} - 4 q^{20} + 6 q^{21}+ \cdots - 55 q^{98}+O(q^{100}) \) 8 * q - q^2 - 8 * q^3 + 9 * q^4 - 2 * q^5 + q^6 - 6 * q^7 - 6 * q^8 + 8 * q^9 - 3 * q^10 - 9 * q^12 - q^13 + 7 * q^14 + 2 * q^15 + 23 * q^16 - 15 * q^17 - q^18 + 8 * q^19 - 4 * q^20 + 6 * q^21 - 5 * q^23 + 6 * q^24 + 4 * q^25 + 25 * q^26 - 8 * q^27 - 8 * q^28 - 17 * q^29 + 3 * q^30 - 4 * q^31 - 38 * q^32 - 18 * q^34 - 2 * q^35 + 9 * q^36 - 4 * q^37 - q^38 + q^39 - 14 * q^40 - 2 * q^41 - 7 * q^42 - 10 * q^43 - 2 * q^45 + 10 * q^46 + 5 * q^47 - 23 * q^48 + 28 * q^49 - 7 * q^50 + 15 * q^51 - 8 * q^52 - 19 * q^53 + q^54 - 49 * q^56 - 8 * q^57 + 35 * q^58 + 12 * q^59 + 4 * q^60 + 5 * q^61 - 22 * q^62 - 6 * q^63 + 24 * q^64 - 3 * q^65 - 12 * q^67 + 3 * q^68 + 5 * q^69 - 22 * q^70 + 40 * q^71 - 6 * q^72 + 12 * q^73 + 7 * q^74 - 4 * q^75 + 9 * q^76 - 25 * q^78 - 14 * q^79 + 37 * q^80 + 8 * q^81 + 31 * q^82 - 17 * q^83 + 8 * q^84 + 35 * q^85 - q^86 + 17 * q^87 - 9 * q^89 - 3 * q^90 + 7 * q^91 + 12 * q^92 + 4 * q^93 - 51 * q^94 - 2 * q^95 + 38 * q^96 + 8 * q^97 - 55 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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