LMFDB - Embedded newform 2366.2.a.bg.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2366,2,Mod(1,2366)] 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("2366.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2366, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2366 = 2 \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2366.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,6,1,6,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$18.8926051182$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.6052921.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 11x^{4} + 7x^{3} + 33x^{2} - 9x - 27 $ x^6 - x^5 - 11x^4 + 7x^3 + 33x^2 - 9x - 27
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			1.4
Root			$1.21736$ of defining polynomial
Character	$\chi$	$=$	2366.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.21736 q^{3} +1.00000 q^{4} -3.44059 q^{5} +1.21736 q^{6} +1.00000 q^{7} +1.00000 q^{8} -1.51803 q^{9} -3.44059 q^{10} +4.99555 q^{11} +1.21736 q^{12} +1.00000 q^{14} -4.18845 q^{15} +1.00000 q^{16} -7.06822 q^{17} -1.51803 q^{18} +4.99523 q^{19} -3.44059 q^{20} +1.21736 q^{21} +4.99555 q^{22} +5.66813 q^{23} +1.21736 q^{24} +6.83767 q^{25} -5.50008 q^{27} +1.00000 q^{28} +4.73054 q^{29} -4.18845 q^{30} +2.16328 q^{31} +1.00000 q^{32} +6.08140 q^{33} -7.06822 q^{34} -3.44059 q^{35} -1.51803 q^{36} +11.8278 q^{37} +4.99523 q^{38} -3.44059 q^{40} +6.00397 q^{41} +1.21736 q^{42} +7.89705 q^{43} +4.99555 q^{44} +5.22291 q^{45} +5.66813 q^{46} +5.79519 q^{47} +1.21736 q^{48} +1.00000 q^{49} +6.83767 q^{50} -8.60459 q^{51} -3.79017 q^{53} -5.50008 q^{54} -17.1877 q^{55} +1.00000 q^{56} +6.08101 q^{57} +4.73054 q^{58} +0.563865 q^{59} -4.18845 q^{60} -9.60172 q^{61} +2.16328 q^{62} -1.51803 q^{63} +1.00000 q^{64} +6.08140 q^{66} -7.50096 q^{67} -7.06822 q^{68} +6.90017 q^{69} -3.44059 q^{70} -4.20475 q^{71} -1.51803 q^{72} +7.63802 q^{73} +11.8278 q^{74} +8.32393 q^{75} +4.99523 q^{76} +4.99555 q^{77} +0.0666888 q^{79} -3.44059 q^{80} -2.14151 q^{81} +6.00397 q^{82} -9.86513 q^{83} +1.21736 q^{84} +24.3189 q^{85} +7.89705 q^{86} +5.75879 q^{87} +4.99555 q^{88} -6.16227 q^{89} +5.22291 q^{90} +5.66813 q^{92} +2.63350 q^{93} +5.79519 q^{94} -17.1866 q^{95} +1.21736 q^{96} +13.8364 q^{97} +1.00000 q^{98} -7.58338 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{2} + q^{3} + 6 q^{4} + 4 q^{5} + q^{6} + 6 q^{7} + 6 q^{8} + 5 q^{9} + 4 q^{10} + 6 q^{11} + q^{12} + 6 q^{14} - 5 q^{15} + 6 q^{16} - 9 q^{17} + 5 q^{18} + 10 q^{19} + 4 q^{20} + q^{21}+ \cdots - 4 q^{99}+O(q^{100}) $ 6 * q + 6 * q^2 + q^3 + 6 * q^4 + 4 * q^5 + q^6 + 6 * q^7 + 6 * q^8 + 5 * q^9 + 4 * q^10 + 6 * q^11 + q^12 + 6 * q^14 - 5 * q^15 + 6 * q^16 - 9 * q^17 + 5 * q^18 + 10 * q^19 + 4 * q^20 + q^21 + 6 * q^22 + 21 * q^23 + q^24 + 8 * q^25 + 7 * q^27 + 6 * q^28 - q^29 - 5 * q^30 + 20 * q^31 + 6 * q^32 + 9 * q^33 - 9 * q^34 + 4 * q^35 + 5 * q^36 + 16 * q^37 + 10 * q^38 + 4 * q^40 + 2 * q^41 + q^42 + 2 * q^43 + 6 * q^44 + 17 * q^45 + 21 * q^46 - 5 * q^47 + q^48 + 6 * q^49 + 8 * q^50 - 15 * q^51 + 28 * q^53 + 7 * q^54 - 29 * q^55 + 6 * q^56 + 22 * q^57 - q^58 - 12 * q^59 - 5 * q^60 - 27 * q^61 + 20 * q^62 + 5 * q^63 + 6 * q^64 + 9 * q^66 + 16 * q^67 - 9 * q^68 - 15 * q^69 + 4 * q^70 + 5 * q^72 + 38 * q^73 + 16 * q^74 - 7 * q^75 + 10 * q^76 + 6 * q^77 + 6 * q^79 + 4 * q^80 - 26 * q^81 + 2 * q^82 - 6 * q^83 + q^84 + 9 * q^85 + 2 * q^86 - 39 * q^87 + 6 * q^88 - q^89 + 17 * q^90 + 21 * q^92 - 7 * q^93 - 5 * q^94 + 3 * q^95 + q^96 + 16 * q^97 + 6 * q^98 - 4 * q^99

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.21736	0.702845	0.351422	−	0.936217i	$-0.385698\pi$
$3$	1.21736	0.702845	0.351422	+	0.936217i	$0.385698\pi$
$4$	1.00000	0.500000
$5$	−3.44059	−1.53868	−0.769340	−	0.638840i	$-0.779415\pi$
$5$	−3.44059	−1.53868	−0.769340	+	0.638840i	$0.779415\pi$
$6$	1.21736	0.496986
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	−1.51803	−0.506009
$10$	−3.44059	−1.08801
$11$	4.99555	1.50622	0.753108	−	0.657897i	$-0.228554\pi$
$11$	4.99555	1.50622	0.753108	+	0.657897i	$0.228554\pi$
$12$	1.21736	0.351422
$13$	0	0
$13$	0	0
$14$	1.00000	0.267261
$15$	−4.18845	−1.08145
$16$	1.00000	0.250000
$17$	−7.06822	−1.71429	−0.857147	−	0.515071i	$-0.827765\pi$
$17$	−7.06822	−1.71429	−0.857147	+	0.515071i	$0.827765\pi$
$18$	−1.51803	−0.357802
$19$	4.99523	1.14599	0.572993	−	0.819561i	$-0.305782\pi$
$19$	4.99523	1.14599	0.572993	+	0.819561i	$0.305782\pi$
$20$	−3.44059	−0.769340
$21$	1.21736	0.265650
$22$	4.99555	1.06505
$23$	5.66813	1.18189	0.590943	−	0.806713i	$-0.298756\pi$
$23$	5.66813	1.18189	0.590943	+	0.806713i	$0.298756\pi$
$24$	1.21736	0.248493
$25$	6.83767	1.36753
$26$	0	0
$27$	−5.50008	−1.05849
$28$	1.00000	0.188982
$29$	4.73054	0.878440	0.439220	−	0.898380i	$-0.355255\pi$
$29$	4.73054	0.878440	0.439220	+	0.898380i	$0.355255\pi$
$30$	−4.18845	−0.764703
$31$	2.16328	0.388537	0.194269	−	0.980948i	$-0.437767\pi$
$31$	2.16328	0.388537	0.194269	+	0.980948i	$0.437767\pi$
$32$	1.00000	0.176777
$33$	6.08140	1.05864
$34$	−7.06822	−1.21219
$35$	−3.44059	−0.581566
$36$	−1.51803	−0.253004
$37$	11.8278	1.94448	0.972241	−	0.233983i	$-0.0751761\pi$
$37$	11.8278	1.94448	0.972241	+	0.233983i	$0.0751761\pi$
$38$	4.99523	0.810334
$39$	0	0
$40$	−3.44059	−0.544005
$41$	6.00397	0.937662	0.468831	−	0.883288i	$-0.344675\pi$
$41$	6.00397	0.937662	0.468831	+	0.883288i	$0.344675\pi$
$42$	1.21736	0.187843
$43$	7.89705	1.20429	0.602144	−	0.798387i	$-0.294313\pi$
$43$	7.89705	1.20429	0.602144	+	0.798387i	$0.294313\pi$
$44$	4.99555	0.753108
$45$	5.22291	0.778586
$46$	5.66813	0.835720
$47$	5.79519	0.845316	0.422658	−	0.906289i	$-0.361097\pi$
$47$	5.79519	0.845316	0.422658	+	0.906289i	$0.361097\pi$
$48$	1.21736	0.175711
$49$	1.00000	0.142857
$50$	6.83767	0.966993
$51$	−8.60459	−1.20488
$52$	0	0
$53$	−3.79017	−0.520620	−0.260310	−	0.965525i	$-0.583825\pi$
$53$	−3.79017	−0.520620	−0.260310	+	0.965525i	$0.583825\pi$
$54$	−5.50008	−0.748466
$55$	−17.1877	−2.31758
$56$	1.00000	0.133631
$57$	6.08101	0.805450
$58$	4.73054	0.621151
$59$	0.563865	0.0734090	0.0367045	−	0.999326i	$-0.488314\pi$
$59$	0.563865	0.0734090	0.0367045	+	0.999326i	$0.488314\pi$
$60$	−4.18845	−0.540727
$61$	−9.60172	−1.22937	−0.614687	−	0.788771i	$-0.710718\pi$
$61$	−9.60172	−1.22937	−0.614687	+	0.788771i	$0.710718\pi$
$62$	2.16328	0.274737
$63$	−1.51803	−0.191253
$64$	1.00000	0.125000
$65$	0	0
$66$	6.08140	0.748568
$67$	−7.50096	−0.916389	−0.458194	−	0.888852i	$-0.651504\pi$
$67$	−7.50096	−0.916389	−0.458194	+	0.888852i	$0.651504\pi$
$68$	−7.06822	−0.857147
$69$	6.90017	0.830683
$70$	−3.44059	−0.411229
$71$	−4.20475	−0.499013	−0.249506	−	0.968373i	$-0.580268\pi$
$71$	−4.20475	−0.499013	−0.249506	+	0.968373i	$0.580268\pi$
$72$	−1.51803	−0.178901
$73$	7.63802	0.893963	0.446981	−	0.894543i	$-0.352499\pi$
$73$	7.63802	0.893963	0.446981	+	0.894543i	$0.352499\pi$
$74$	11.8278	1.37496
$75$	8.32393	0.961165
$76$	4.99523	0.572993
$77$	4.99555	0.569296
$78$	0	0
$79$	0.0666888	0.00750307	0.00375154	−	0.999993i	$-0.498806\pi$
$79$	0.0666888	0.00750307	0.00375154	+	0.999993i	$0.498806\pi$
$80$	−3.44059	−0.384670
$81$	−2.14151	−0.237946
$82$	6.00397	0.663027
$83$	−9.86513	−1.08284	−0.541420	−	0.840753i	$-0.682113\pi$
$83$	−9.86513	−1.08284	−0.541420	+	0.840753i	$0.682113\pi$
$84$	1.21736	0.132825
$85$	24.3189	2.63775
$86$	7.89705	0.851561
$87$	5.75879	0.617407
$88$	4.99555	0.532527
$89$	−6.16227	−0.653199	−0.326600	−	0.945163i	$-0.605903\pi$
$89$	−6.16227	−0.653199	−0.326600	+	0.945163i	$0.605903\pi$
$90$	5.22291	0.550543
$91$	0	0
$92$	5.66813	0.590943
$93$	2.63350	0.273081
$94$	5.79519	0.597729
$95$	−17.1866	−1.76330
$96$	1.21736	0.124247
$97$	13.8364	1.40487	0.702437	−	0.711746i	$-0.252095\pi$
$97$	13.8364	1.40487	0.702437	+	0.711746i	$0.252095\pi$
$98$	1.00000	0.101015
$99$	−7.58338	−0.762158
$100$	6.83767	0.683767
$101$	−6.41713	−0.638528	−0.319264	−	0.947666i	$-0.603436\pi$
$101$	−6.41713	−0.638528	−0.319264	+	0.947666i	$0.603436\pi$
$102$	−8.60459	−0.851981
$103$	5.68463	0.560124	0.280062	−	0.959982i	$-0.409645\pi$
$103$	5.68463	0.560124	0.280062	+	0.959982i	$0.409645\pi$
$104$	0	0
$105$	−4.18845	−0.408751
$106$	−3.79017	−0.368134
$107$	−19.2643	−1.86235	−0.931174	−	0.364574i	$-0.881215\pi$
$107$	−19.2643	−1.86235	−0.931174	+	0.364574i	$0.881215\pi$
$108$	−5.50008	−0.529245
$109$	−10.0195	−0.959690	−0.479845	−	0.877353i	$-0.659307\pi$
$109$	−10.0195	−0.959690	−0.479845	+	0.877353i	$0.659307\pi$
$110$	−17.1877	−1.63878
$111$	14.3987	1.36667
$112$	1.00000	0.0944911
$113$	8.77533	0.825514	0.412757	−	0.910841i	$-0.364566\pi$
$113$	8.77533	0.825514	0.412757	+	0.910841i	$0.364566\pi$
$114$	6.08101	0.569539
$115$	−19.5017	−1.81855
$116$	4.73054	0.439220
$117$	0	0
$118$	0.563865	0.0519080
$119$	−7.06822	−0.647943
$120$	−4.18845	−0.382351
$121$	13.9555	1.26868
$122$	−9.60172	−0.869299
$123$	7.30901	0.659031
$124$	2.16328	0.194269
$125$	−6.32269	−0.565518
$126$	−1.51803	−0.135237
$127$	5.37806	0.477226	0.238613	−	0.971115i	$-0.423307\pi$
$127$	5.37806	0.477226	0.238613	+	0.971115i	$0.423307\pi$
$128$	1.00000	0.0883883
$129$	9.61358	0.846428
$130$	0	0
$131$	16.0652	1.40362	0.701812	−	0.712362i	$-0.252374\pi$
$131$	16.0652	1.40362	0.701812	+	0.712362i	$0.252374\pi$
$132$	6.08140	0.529318
$133$	4.99523	0.433142
$134$	−7.50096	−0.647985
$135$	18.9235	1.62868
$136$	−7.06822	−0.606095
$137$	−9.66486	−0.825725	−0.412862	−	0.910793i	$-0.635471\pi$
$137$	−9.66486	−0.825725	−0.412862	+	0.910793i	$0.635471\pi$
$138$	6.90017	0.587382
$139$	7.87367	0.667836	0.333918	−	0.942602i	$-0.391629\pi$
$139$	7.87367	0.667836	0.333918	+	0.942602i	$0.391629\pi$
$140$	−3.44059	−0.290783
$141$	7.05486	0.594126
$142$	−4.20475	−0.352855
$143$	0	0
$144$	−1.51803	−0.126502
$145$	−16.2759	−1.35164
$146$	7.63802	0.632127
$147$	1.21736	0.100406
$148$	11.8278	0.972241
$149$	21.5447	1.76501	0.882505	−	0.470303i	$-0.155855\pi$
$149$	21.5447	1.76501	0.882505	+	0.470303i	$0.155855\pi$
$150$	8.32393	0.679646
$151$	3.22714	0.262621	0.131311	−	0.991341i	$-0.458081\pi$
$151$	3.22714	0.262621	0.131311	+	0.991341i	$0.458081\pi$
$152$	4.99523	0.405167
$153$	10.7297	0.867449
$154$	4.99555	0.402553
$155$	−7.44298	−0.597834
$156$	0	0
$157$	−1.29645	−0.103468	−0.0517340	−	0.998661i	$-0.516475\pi$
$157$	−1.29645	−0.103468	−0.0517340	+	0.998661i	$0.516475\pi$
$158$	0.0666888	0.00530547
$159$	−4.61402	−0.365915
$160$	−3.44059	−0.272003
$161$	5.66813	0.446711
$162$	−2.14151	−0.168253
$163$	−15.5019	−1.21420	−0.607099	−	0.794626i	$-0.707667\pi$
$163$	−15.5019	−1.21420	−0.607099	+	0.794626i	$0.707667\pi$
$164$	6.00397	0.468831
$165$	−20.9236	−1.62890
$166$	−9.86513	−0.765683
$167$	1.56822	0.121352	0.0606761	−	0.998158i	$-0.480674\pi$
$167$	1.56822	0.121352	0.0606761	+	0.998158i	$0.480674\pi$
$168$	1.21736	0.0939216
$169$	0	0
$170$	24.3189	1.86517
$171$	−7.58290	−0.579879
$172$	7.89705	0.602144
$173$	9.37327	0.712636	0.356318	−	0.934365i	$-0.384032\pi$
$173$	9.37327	0.712636	0.356318	+	0.934365i	$0.384032\pi$
$174$	5.75879	0.436573
$175$	6.83767	0.516880
$176$	4.99555	0.376554
$177$	0.686429	0.0515951
$178$	−6.16227	−0.461882
$179$	3.92185	0.293133	0.146567	−	0.989201i	$-0.453178\pi$
$179$	3.92185	0.293133	0.146567	+	0.989201i	$0.453178\pi$
$180$	5.22291	0.389293
$181$	0.917236	0.0681776	0.0340888	−	0.999419i	$-0.489147\pi$
$181$	0.917236	0.0681776	0.0340888	+	0.999419i	$0.489147\pi$
$182$	0	0
$183$	−11.6888	−0.864060
$184$	5.66813	0.417860
$185$	−40.6947	−2.99193
$186$	2.63350	0.193098
$187$	−35.3096	−2.58210
$188$	5.79519	0.422658
$189$	−5.50008	−0.400072
$190$	−17.1866	−1.24684
$191$	7.86239	0.568903	0.284451	−	0.958690i	$-0.408189\pi$
$191$	7.86239	0.568903	0.284451	+	0.958690i	$0.408189\pi$
$192$	1.21736	0.0878556
$193$	12.1952	0.877830	0.438915	−	0.898529i	$-0.355363\pi$
$193$	12.1952	0.877830	0.438915	+	0.898529i	$0.355363\pi$
$194$	13.8364	0.993396
$195$	0	0
$196$	1.00000	0.0714286
$197$	0.499938	0.0356191	0.0178096	−	0.999841i	$-0.494331\pi$
$197$	0.499938	0.0356191	0.0178096	+	0.999841i	$0.494331\pi$
$198$	−7.58338	−0.538927
$199$	−18.7772	−1.33108	−0.665541	−	0.746361i	$-0.731799\pi$
$199$	−18.7772	−1.33108	−0.665541	+	0.746361i	$0.731799\pi$
$200$	6.83767	0.483497
$201$	−9.13140	−0.644079
$202$	−6.41713	−0.451508
$203$	4.73054	0.332019
$204$	−8.60459	−0.602442
$205$	−20.6572	−1.44276
$206$	5.68463	0.396067
$207$	−8.60437	−0.598045
$208$	0	0
$209$	24.9539	1.72610
$210$	−4.18845	−0.289031
$211$	−5.77056	−0.397262	−0.198631	−	0.980074i	$-0.563649\pi$
$211$	−5.77056	−0.397262	−0.198631	+	0.980074i	$0.563649\pi$
$212$	−3.79017	−0.260310
$213$	−5.11871	−0.350728
$214$	−19.2643	−1.31688
$215$	−27.1705	−1.85301
$216$	−5.50008	−0.374233
$217$	2.16328	0.146853
$218$	−10.0195	−0.678603
$219$	9.29824	0.628317
$220$	−17.1877	−1.15879
$221$	0	0
$222$	14.3987	0.966381
$223$	3.21980	0.215614	0.107807	−	0.994172i	$-0.465617\pi$
$223$	3.21980	0.215614	0.107807	+	0.994172i	$0.465617\pi$
$224$	1.00000	0.0668153
$225$	−10.3798	−0.691985
$226$	8.77533	0.583726
$227$	−20.6879	−1.37310	−0.686552	−	0.727081i	$-0.740877\pi$
$227$	−20.6879	−1.37310	−0.686552	+	0.727081i	$0.740877\pi$
$228$	6.08101	0.402725
$229$	−4.86398	−0.321421	−0.160710	−	0.987002i	$-0.551379\pi$
$229$	−4.86398	−0.321421	−0.160710	+	0.987002i	$0.551379\pi$
$230$	−19.5017	−1.28591
$231$	6.08140	0.400127
$232$	4.73054	0.310575
$233$	−25.8858	−1.69583	−0.847917	−	0.530129i	$-0.822143\pi$
$233$	−25.8858	−1.69583	−0.847917	+	0.530129i	$0.822143\pi$
$234$	0	0
$235$	−19.9389	−1.30067
$236$	0.563865	0.0367045
$237$	0.0811845	0.00527350
$238$	−7.06822	−0.458165
$239$	−9.15450	−0.592155	−0.296078	−	0.955164i	$-0.595679\pi$
$239$	−9.15450	−0.592155	−0.296078	+	0.955164i	$0.595679\pi$
$240$	−4.18845	−0.270363
$241$	1.83019	0.117893	0.0589463	−	0.998261i	$-0.481226\pi$
$241$	1.83019	0.117893	0.0589463	+	0.998261i	$0.481226\pi$
$242$	13.9555	0.897095
$243$	13.8932	0.891252
$244$	−9.60172	−0.614687
$245$	−3.44059	−0.219811
$246$	7.30901	0.466005
$247$	0	0
$248$	2.16328	0.137369
$249$	−12.0095	−0.761068
$250$	−6.32269	−0.399882
$251$	10.7479	0.678402	0.339201	−	0.940714i	$-0.389843\pi$
$251$	10.7479	0.678402	0.339201	+	0.940714i	$0.389843\pi$
$252$	−1.51803	−0.0956267
$253$	28.3154	1.78018
$254$	5.37806	0.337450
$255$	29.6049	1.85393
$256$	1.00000	0.0625000
$257$	−28.6797	−1.78899	−0.894496	−	0.447075i	$-0.852466\pi$
$257$	−28.6797	−1.78899	−0.894496	+	0.447075i	$0.852466\pi$
$258$	9.61358	0.598515
$259$	11.8278	0.734945
$260$	0	0
$261$	−7.18109	−0.444499
$262$	16.0652	0.992513
$263$	11.5007	0.709163	0.354581	−	0.935025i	$-0.384623\pi$
$263$	11.5007	0.709163	0.354581	+	0.935025i	$0.384623\pi$
$264$	6.08140	0.374284
$265$	13.0404	0.801068
$266$	4.99523	0.306277
$267$	−7.50172	−0.459098
$268$	−7.50096	−0.458194
$269$	−13.9265	−0.849113	−0.424556	−	0.905402i	$-0.639570\pi$
$269$	−13.9265	−0.849113	−0.424556	+	0.905402i	$0.639570\pi$
$270$	18.9235	1.15165
$271$	7.54822	0.458522	0.229261	−	0.973365i	$-0.426369\pi$
$271$	7.54822	0.458522	0.229261	+	0.973365i	$0.426369\pi$
$272$	−7.06822	−0.428574
$273$	0	0
$274$	−9.66486	−0.583876
$275$	34.1579	2.05980
$276$	6.90017	0.415342
$277$	−26.8940	−1.61591	−0.807953	−	0.589247i	$-0.799424\pi$
$277$	−26.8940	−1.61591	−0.807953	+	0.589247i	$0.799424\pi$
$278$	7.87367	0.472231
$279$	−3.28392	−0.196603
$280$	−3.44059	−0.205615
$281$	−15.1097	−0.901370	−0.450685	−	0.892683i	$-0.648820\pi$
$281$	−15.1097	−0.901370	−0.450685	+	0.892683i	$0.648820\pi$
$282$	7.05486	0.420111
$283$	−23.0460	−1.36994	−0.684971	−	0.728570i	$-0.740185\pi$
$283$	−23.0460	−1.36994	−0.684971	+	0.728570i	$0.740185\pi$
$284$	−4.20475	−0.249506
$285$	−20.9223	−1.23933
$286$	0	0
$287$	6.00397	0.354403
$288$	−1.51803	−0.0894506
$289$	32.9597	1.93881
$290$	−16.2759	−0.955752
$291$	16.8439	0.987409
$292$	7.63802	0.446981
$293$	−6.96795	−0.407072	−0.203536	−	0.979067i	$-0.565243\pi$
$293$	−6.96795	−0.407072	−0.203536	+	0.979067i	$0.565243\pi$
$294$	1.21736	0.0709981
$295$	−1.94003	−0.112953
$296$	11.8278	0.687478
$297$	−27.4759	−1.59431
$298$	21.5447	1.24805
$299$	0	0
$300$	8.32393	0.480582
$301$	7.89705	0.455178
$302$	3.22714	0.185701
$303$	−7.81198	−0.448786
$304$	4.99523	0.286496
$305$	33.0356	1.89161
$306$	10.7297	0.613379
$307$	26.4284	1.50835	0.754174	−	0.656675i	$-0.228038\pi$
$307$	26.4284	1.50835	0.754174	+	0.656675i	$0.228038\pi$
$308$	4.99555	0.284648
$309$	6.92026	0.393680
$310$	−7.44298	−0.422733
$311$	−25.1099	−1.42385	−0.711925	−	0.702256i	$-0.752176\pi$
$311$	−25.1099	−1.42385	−0.711925	+	0.702256i	$0.752176\pi$
$312$	0	0
$313$	−0.185816	−0.0105030	−0.00525148	−	0.999986i	$-0.501672\pi$
$313$	−0.185816	−0.0105030	−0.00525148	+	0.999986i	$0.501672\pi$
$314$	−1.29645	−0.0731629
$315$	5.22291	0.294278
$316$	0.0666888	0.00375154
$317$	−26.0136	−1.46107	−0.730533	−	0.682877i	$-0.760729\pi$
$317$	−26.0136	−1.46107	−0.730533	+	0.682877i	$0.760729\pi$
$318$	−4.61402	−0.258741
$319$	23.6317	1.32312
$320$	−3.44059	−0.192335
$321$	−23.4516	−1.30894
$322$	5.66813	0.315873
$323$	−35.3074	−1.96456
$324$	−2.14151	−0.118973
$325$	0	0
$326$	−15.5019	−0.858568
$327$	−12.1973	−0.674513
$328$	6.00397	0.331514
$329$	5.79519	0.319499
$330$	−20.9236	−1.15181
$331$	−10.2263	−0.562089	−0.281044	−	0.959695i	$-0.590681\pi$
$331$	−10.2263	−0.562089	−0.281044	+	0.959695i	$0.590681\pi$
$332$	−9.86513	−0.541420
$333$	−17.9549	−0.983925
$334$	1.56822	0.0858090
$335$	25.8078	1.41003
$336$	1.21736	0.0664126
$337$	−18.9120	−1.03020	−0.515101	−	0.857130i	$-0.672245\pi$
$337$	−18.9120	−1.03020	−0.515101	+	0.857130i	$0.672245\pi$
$338$	0	0
$339$	10.6828	0.580208
$340$	24.3189	1.31888
$341$	10.8068	0.585221
$342$	−7.58290	−0.410036
$343$	1.00000	0.0539949
$344$	7.89705	0.425780
$345$	−23.7407	−1.27816
$346$	9.37327	0.503910
$347$	−7.34260	−0.394171	−0.197086	−	0.980386i	$-0.563148\pi$
$347$	−7.34260	−0.394171	−0.197086	+	0.980386i	$0.563148\pi$
$348$	5.75879	0.308704
$349$	22.0790	1.18186	0.590930	−	0.806723i	$-0.298761\pi$
$349$	22.0790	1.18186	0.590930	+	0.806723i	$0.298761\pi$
$350$	6.83767	0.365489
$351$	0	0
$352$	4.99555	0.266264
$353$	−6.30884	−0.335785	−0.167893	−	0.985805i	$-0.553696\pi$
$353$	−6.30884	−0.335785	−0.167893	+	0.985805i	$0.553696\pi$
$354$	0.686429	0.0364833
$355$	14.4668	0.767820
$356$	−6.16227	−0.326600
$357$	−8.60459	−0.455403
$358$	3.92185	0.207276
$359$	35.6786	1.88304	0.941521	−	0.336954i	$-0.109397\pi$
$359$	35.6786	1.88304	0.941521	+	0.336954i	$0.109397\pi$
$360$	5.22291	0.275272
$361$	5.95235	0.313282
$362$	0.917236	0.0482089
$363$	16.9889	0.891688
$364$	0	0
$365$	−26.2793	−1.37552
$366$	−11.6888	−0.610983
$367$	5.35362	0.279457	0.139728	−	0.990190i	$-0.455377\pi$
$367$	5.35362	0.279457	0.139728	+	0.990190i	$0.455377\pi$
$368$	5.66813	0.295472
$369$	−9.11419	−0.474466
$370$	−40.6947	−2.11562
$371$	−3.79017	−0.196776
$372$	2.63350	0.136541
$373$	31.7744	1.64522	0.822609	−	0.568607i	$-0.192518\pi$
$373$	31.7744	1.64522	0.822609	+	0.568607i	$0.192518\pi$
$374$	−35.3096	−1.82582
$375$	−7.69701	−0.397472
$376$	5.79519	0.298864
$377$	0	0
$378$	−5.50008	−0.282894
$379$	−15.5529	−0.798900	−0.399450	−	0.916755i	$-0.630799\pi$
$379$	−15.5529	−0.798900	−0.399450	+	0.916755i	$0.630799\pi$
$380$	−17.1866	−0.881652
$381$	6.54706	0.335416
$382$	7.86239	0.402275
$383$	−18.5061	−0.945618	−0.472809	−	0.881165i	$-0.656760\pi$
$383$	−18.5061	−0.945618	−0.472809	+	0.881165i	$0.656760\pi$
$384$	1.21736	0.0621233
$385$	−17.1877	−0.875964
$386$	12.1952	0.620720
$387$	−11.9879	−0.609381
$388$	13.8364	0.702437
$389$	23.3415	1.18346	0.591731	−	0.806136i	$-0.298445\pi$
$389$	23.3415	1.18346	0.591731	+	0.806136i	$0.298445\pi$
$390$	0	0
$391$	−40.0636	−2.02610
$392$	1.00000	0.0505076
$393$	19.5572	0.986531
$394$	0.499938	0.0251865
$395$	−0.229449	−0.0115448
$396$	−7.58338	−0.381079
$397$	3.18220	0.159710	0.0798551	−	0.996806i	$-0.474554\pi$
$397$	3.18220	0.159710	0.0798551	+	0.996806i	$0.474554\pi$
$398$	−18.7772	−0.941217
$399$	6.08101	0.304431
$400$	6.83767	0.341884
$401$	−28.6480	−1.43061	−0.715307	−	0.698811i	$-0.753713\pi$
$401$	−28.6480	−1.43061	−0.715307	+	0.698811i	$0.753713\pi$
$402$	−9.13140	−0.455433
$403$	0	0
$404$	−6.41713	−0.319264
$405$	7.36807	0.366122
$406$	4.73054	0.234773
$407$	59.0865	2.92881
$408$	−8.60459	−0.425991
$409$	2.03868	0.100806	0.0504031	−	0.998729i	$-0.483949\pi$
$409$	2.03868	0.100806	0.0504031	+	0.998729i	$0.483949\pi$
$410$	−20.6572	−1.02019
$411$	−11.7656	−0.580357
$412$	5.68463	0.280062
$413$	0.563865	0.0277460
$414$	−8.60437	−0.422882
$415$	33.9419	1.66614
$416$	0	0
$417$	9.58511	0.469385
$418$	24.9539	1.22054
$419$	−3.58445	−0.175112	−0.0875560	−	0.996160i	$-0.527906\pi$
$419$	−3.58445	−0.175112	−0.0875560	+	0.996160i	$0.527906\pi$
$420$	−4.18845	−0.204375
$421$	6.57788	0.320586	0.160293	−	0.987069i	$-0.448756\pi$
$421$	6.57788	0.320586	0.160293	+	0.987069i	$0.448756\pi$
$422$	−5.77056	−0.280906
$423$	−8.79726	−0.427738
$424$	−3.79017	−0.184067
$425$	−48.3302	−2.34436
$426$	−5.11871	−0.248002
$427$	−9.60172	−0.464660
$428$	−19.2643	−0.931174
$429$	0	0
$430$	−27.1705	−1.31028
$431$	−4.97080	−0.239435	−0.119718	−	0.992808i	$-0.538199\pi$
$431$	−4.97080	−0.239435	−0.119718	+	0.992808i	$0.538199\pi$
$432$	−5.50008	−0.264623
$433$	24.2077	1.16335	0.581674	−	0.813422i	$-0.302398\pi$
$433$	24.2077	1.16335	0.581674	+	0.813422i	$0.302398\pi$
$434$	2.16328	0.103841
$435$	−19.8137	−0.949992
$436$	−10.0195	−0.479845
$437$	28.3136	1.35442
$438$	9.29824	0.444287
$439$	−14.0986	−0.672891	−0.336446	−	0.941703i	$-0.609225\pi$
$439$	−14.0986	−0.672891	−0.336446	+	0.941703i	$0.609225\pi$
$440$	−17.1877	−0.819389
$441$	−1.51803	−0.0722870
$442$	0	0
$443$	−9.14216	−0.434357	−0.217178	−	0.976132i	$-0.569685\pi$
$443$	−9.14216	−0.434357	−0.217178	+	0.976132i	$0.569685\pi$
$444$	14.3987	0.683334
$445$	21.2019	1.00506
$446$	3.21980	0.152462
$447$	26.2277	1.24053
$448$	1.00000	0.0472456
$449$	8.29350	0.391395	0.195697	−	0.980664i	$-0.437303\pi$
$449$	8.29350	0.391395	0.195697	+	0.980664i	$0.437303\pi$
$450$	−10.3798	−0.489307
$451$	29.9931	1.41232
$452$	8.77533	0.412757
$453$	3.92860	0.184582
$454$	−20.6879	−0.970931
$455$	0	0
$456$	6.08101	0.284770
$457$	5.72999	0.268037	0.134019	−	0.990979i	$-0.457212\pi$
$457$	5.72999	0.268037	0.134019	+	0.990979i	$0.457212\pi$
$458$	−4.86398	−0.227279
$459$	38.8758	1.81457
$460$	−19.5017	−0.909273
$461$	−17.7249	−0.825531	−0.412766	−	0.910837i	$-0.635437\pi$
$461$	−17.7249	−0.825531	−0.412766	+	0.910837i	$0.635437\pi$
$462$	6.08140	0.282932
$463$	−15.0103	−0.697589	−0.348794	−	0.937199i	$-0.613409\pi$
$463$	−15.0103	−0.697589	−0.348794	+	0.937199i	$0.613409\pi$
$464$	4.73054	0.219610
$465$	−9.06080	−0.420185
$466$	−25.8858	−1.19914
$467$	−22.1003	−1.02268	−0.511341	−	0.859378i	$-0.670851\pi$
$467$	−22.1003	−1.02268	−0.511341	+	0.859378i	$0.670851\pi$
$468$	0	0
$469$	−7.50096	−0.346362
$470$	−19.9389	−0.919713
$471$	−1.57825	−0.0727220
$472$	0.563865	0.0259540
$473$	39.4501	1.81392
$474$	0.0811845	0.00372893
$475$	34.1558	1.56717
$476$	−7.06822	−0.323971
$477$	5.75359	0.263439
$478$	−9.15450	−0.418717
$479$	−8.29945	−0.379212	−0.189606	−	0.981860i	$-0.560721\pi$
$479$	−8.29945	−0.379212	−0.189606	+	0.981860i	$0.560721\pi$
$480$	−4.18845	−0.191176
$481$	0	0
$482$	1.83019	0.0833627
$483$	6.90017	0.313969
$484$	13.9555	0.634342
$485$	−47.6054	−2.16165
$486$	13.8932	0.630210
$487$	4.99054	0.226143	0.113071	−	0.993587i	$-0.463931\pi$
$487$	4.99054	0.226143	0.113071	+	0.993587i	$0.463931\pi$
$488$	−9.60172	−0.434650
$489$	−18.8714	−0.853394
$490$	−3.44059	−0.155430
$491$	2.67530	0.120735	0.0603673	−	0.998176i	$-0.480773\pi$
$491$	2.67530	0.120735	0.0603673	+	0.998176i	$0.480773\pi$
$492$	7.30901	0.329516
$493$	−33.4365	−1.50591
$494$	0	0
$495$	26.0913	1.17272
$496$	2.16328	0.0971343
$497$	−4.20475	−0.188609
$498$	−12.0095	−0.538156
$499$	10.6656	0.477456	0.238728	−	0.971086i	$-0.423270\pi$
$499$	10.6656	0.477456	0.238728	+	0.971086i	$0.423270\pi$
$500$	−6.32269	−0.282759
$501$	1.90909	0.0852918
$502$	10.7479	0.479703
$503$	2.72009	0.121283	0.0606414	−	0.998160i	$-0.480685\pi$
$503$	2.72009	0.121283	0.0606414	+	0.998160i	$0.480685\pi$
$504$	−1.51803	−0.0676183
$505$	22.0787	0.982491
$506$	28.3154	1.25877
$507$	0	0
$508$	5.37806	0.238613
$509$	34.8622	1.54524	0.772620	−	0.634869i	$-0.218946\pi$
$509$	34.8622	1.54524	0.772620	+	0.634869i	$0.218946\pi$
$510$	29.6049	1.31093
$511$	7.63802	0.337886
$512$	1.00000	0.0441942
$513$	−27.4742	−1.21301
$514$	−28.6797	−1.26501
$515$	−19.5585	−0.861851
$516$	9.61358	0.423214
$517$	28.9502	1.27323
$518$	11.8278	0.519684
$519$	11.4107	0.500873
$520$	0	0
$521$	−0.0242185	−0.00106103	−0.000530515	−	1.00000i	$-0.500169\pi$
$521$	−0.0242185	−0.00106103	−0.000530515		1.00000i	$0.500169\pi$
$522$	−7.18109	−0.314308
$523$	−27.6711	−1.20997	−0.604986	−	0.796236i	$-0.706821\pi$
$523$	−27.6711	−1.20997	−0.604986	+	0.796236i	$0.706821\pi$
$524$	16.0652	0.701812
$525$	8.32393	0.363286
$526$	11.5007	0.501454
$527$	−15.2906	−0.666067
$528$	6.08140	0.264659
$529$	9.12769	0.396856
$530$	13.0404	0.566441
$531$	−0.855962	−0.0371456
$532$	4.99523	0.216571
$533$	0	0
$534$	−7.50172	−0.324631
$535$	66.2806	2.86556
$536$	−7.50096	−0.323992
$537$	4.77432	0.206027
$538$	−13.9265	−0.600413
$539$	4.99555	0.215174
$540$	18.9235	0.814339
$541$	42.6311	1.83285	0.916426	−	0.400204i	$-0.131061\pi$
$541$	42.6311	1.83285	0.916426	+	0.400204i	$0.131061\pi$
$542$	7.54822	0.324224
$543$	1.11661	0.0479183
$544$	−7.06822	−0.303047
$545$	34.4729	1.47666
$546$	0	0
$547$	−31.9332	−1.36536	−0.682682	−	0.730716i	$-0.739186\pi$
$547$	−31.9332	−1.36536	−0.682682	+	0.730716i	$0.739186\pi$
$548$	−9.66486	−0.412862
$549$	14.5757	0.622075
$550$	34.1579	1.45650
$551$	23.6302	1.00668
$552$	6.90017	0.293691
$553$	0.0666888	0.00283590
$554$	−26.8940	−1.14262
$555$	−49.5402	−2.10287
$556$	7.87367	0.333918
$557$	36.4922	1.54622	0.773112	−	0.634269i	$-0.218699\pi$
$557$	36.4922	1.54622	0.773112	+	0.634269i	$0.218699\pi$
$558$	−3.28392	−0.139020
$559$	0	0
$560$	−3.44059	−0.145392
$561$	−42.9847	−1.81481
$562$	−15.1097	−0.637365
$563$	21.5178	0.906866	0.453433	−	0.891290i	$-0.350199\pi$
$563$	21.5178	0.906866	0.453433	+	0.891290i	$0.350199\pi$
$564$	7.05486	0.297063
$565$	−30.1923	−1.27020
$566$	−23.0460	−0.968695
$567$	−2.14151	−0.0899351
$568$	−4.20475	−0.176428
$569$	18.9443	0.794185	0.397092	−	0.917779i	$-0.370019\pi$
$569$	18.9443	0.794185	0.397092	+	0.917779i	$0.370019\pi$
$570$	−20.9223	−0.876338
$571$	−30.3868	−1.27165	−0.635824	−	0.771834i	$-0.719340\pi$
$571$	−30.3868	−1.27165	−0.635824	+	0.771834i	$0.719340\pi$
$572$	0	0
$573$	9.57139	0.399850
$574$	6.00397	0.250601
$575$	38.7568	1.61627
$576$	−1.51803	−0.0632511
$577$	10.5080	0.437455	0.218728	−	0.975786i	$-0.429809\pi$
$577$	10.5080	0.437455	0.218728	+	0.975786i	$0.429809\pi$
$578$	32.9597	1.37094
$579$	14.8460	0.616978
$580$	−16.2759	−0.675819
$581$	−9.86513	−0.409275
$582$	16.8439	0.698203
$583$	−18.9340	−0.784166
$584$	7.63802	0.316064
$585$	0	0
$586$	−6.96795	−0.287843
$587$	−42.6299	−1.75952	−0.879762	−	0.475413i	$-0.842298\pi$
$587$	−42.6299	−1.75952	−0.879762	+	0.475413i	$0.842298\pi$
$588$	1.21736	0.0502032
$589$	10.8061	0.445258
$590$	−1.94003	−0.0798698
$591$	0.608606	0.0250347
$592$	11.8278	0.486120
$593$	−47.1372	−1.93569	−0.967846	−	0.251542i	$-0.919062\pi$
$593$	−47.1372	−1.93569	−0.967846	+	0.251542i	$0.919062\pi$
$594$	−27.4759	−1.12735
$595$	24.3189	0.996976
$596$	21.5447	0.882505
$597$	−22.8587	−0.935544
$598$	0	0
$599$	14.4525	0.590513	0.295257	−	0.955418i	$-0.404595\pi$
$599$	14.4525	0.590513	0.295257	+	0.955418i	$0.404595\pi$
$600$	8.32393	0.339823
$601$	29.0649	1.18558	0.592791	−	0.805356i	$-0.298026\pi$
$601$	29.0649	1.18558	0.592791	+	0.805356i	$0.298026\pi$
$602$	7.89705	0.321860
$603$	11.3867	0.463701
$604$	3.22714	0.131311
$605$	−48.0153	−1.95210
$606$	−7.81198	−0.317340
$607$	3.44281	0.139740	0.0698698	−	0.997556i	$-0.477742\pi$
$607$	3.44281	0.139740	0.0698698	+	0.997556i	$0.477742\pi$
$608$	4.99523	0.202583
$609$	5.75879	0.233358
$610$	33.0356	1.33757
$611$	0	0
$612$	10.7297	0.433724
$613$	3.41510	0.137935	0.0689674	−	0.997619i	$-0.478030\pi$
$613$	3.41510	0.137935	0.0689674	+	0.997619i	$0.478030\pi$
$614$	26.4284	1.06656
$615$	−25.1473	−1.01404
$616$	4.99555	0.201276
$617$	−22.8471	−0.919791	−0.459896	−	0.887973i	$-0.652113\pi$
$617$	−22.8471	−0.919791	−0.459896	+	0.887973i	$0.652113\pi$
$618$	6.92026	0.278374
$619$	24.6009	0.988792	0.494396	−	0.869237i	$-0.335389\pi$
$619$	24.6009	0.988792	0.494396	+	0.869237i	$0.335389\pi$
$620$	−7.44298	−0.298917
$621$	−31.1752	−1.25102
$622$	−25.1099	−1.00681
$623$	−6.16227	−0.246886
$624$	0	0
$625$	−12.4346	−0.497383
$626$	−0.185816	−0.00742672
$627$	30.3780	1.21318
$628$	−1.29645	−0.0517340
$629$	−83.6016	−3.33341
$630$	5.22291	0.208086
$631$	−28.6454	−1.14035	−0.570177	−	0.821522i	$-0.693125\pi$
$631$	−28.6454	−1.14035	−0.570177	+	0.821522i	$0.693125\pi$
$632$	0.0666888	0.00265274
$633$	−7.02487	−0.279213
$634$	−26.0136	−1.03313
$635$	−18.5037	−0.734298
$636$	−4.61402	−0.182958
$637$	0	0
$638$	23.6317	0.935587
$639$	6.38293	0.252505
$640$	−3.44059	−0.136001
$641$	−21.0942	−0.833170	−0.416585	−	0.909097i	$-0.636773\pi$
$641$	−21.0942	−0.833170	−0.416585	+	0.909097i	$0.636773\pi$
$642$	−23.4516	−0.925562
$643$	31.1503	1.22845	0.614224	−	0.789132i	$-0.289469\pi$
$643$	31.1503	1.22845	0.614224	+	0.789132i	$0.289469\pi$
$644$	5.66813	0.223356
$645$	−33.0764	−1.30238
$646$	−35.3074	−1.38915
$647$	39.1390	1.53871	0.769356	−	0.638820i	$-0.220577\pi$
$647$	39.1390	1.53871	0.769356	+	0.638820i	$0.220577\pi$
$648$	−2.14151	−0.0841266
$649$	2.81682	0.110570
$650$	0	0
$651$	2.63350	0.103215
$652$	−15.5019	−0.607099
$653$	47.3177	1.85168	0.925842	−	0.377910i	$-0.123357\pi$
$653$	47.3177	1.85168	0.925842	+	0.377910i	$0.123357\pi$
$654$	−12.1973	−0.476953
$655$	−55.2739	−2.15973
$656$	6.00397	0.234416
$657$	−11.5947	−0.452353
$658$	5.79519	0.225920
$659$	−49.1219	−1.91352	−0.956758	−	0.290884i	$-0.906051\pi$
$659$	−49.1219	−1.91352	−0.956758	+	0.290884i	$0.906051\pi$
$660$	−20.9236	−0.814451
$661$	22.7833	0.886168	0.443084	−	0.896480i	$-0.353884\pi$
$661$	22.7833	0.886168	0.443084	+	0.896480i	$0.353884\pi$
$662$	−10.2263	−0.397457
$663$	0	0
$664$	−9.86513	−0.382841
$665$	−17.1866	−0.666466
$666$	−17.9549	−0.695740
$667$	26.8133	1.03822
$668$	1.56822	0.0606761
$669$	3.91967	0.151543
$670$	25.8078	0.997041
$671$	−47.9659	−1.85170
$672$	1.21736	0.0469608
$673$	−11.7325	−0.452255	−0.226128	−	0.974098i	$-0.572607\pi$
$673$	−11.7325	−0.452255	−0.226128	+	0.974098i	$0.572607\pi$
$674$	−18.9120	−0.728462
$675$	−37.6078	−1.44752
$676$	0	0
$677$	−7.16323	−0.275305	−0.137653	−	0.990481i	$-0.543956\pi$
$677$	−7.16323	−0.275305	−0.137653	+	0.990481i	$0.543956\pi$
$678$	10.6828	0.410269
$679$	13.8364	0.530993
$680$	24.3189	0.932586
$681$	−25.1847	−0.965079
$682$	10.8068	0.413813
$683$	4.14299	0.158527	0.0792636	−	0.996854i	$-0.474743\pi$
$683$	4.14299	0.158527	0.0792636	+	0.996854i	$0.474743\pi$
$684$	−7.58290	−0.289939
$685$	33.2528	1.27053
$686$	1.00000	0.0381802
$687$	−5.92123	−0.225909
$688$	7.89705	0.301072
$689$	0	0
$690$	−23.7407	−0.903792
$691$	23.5827	0.897129	0.448564	−	0.893751i	$-0.351936\pi$
$691$	23.5827	0.897129	0.448564	+	0.893751i	$0.351936\pi$
$692$	9.37327	0.356318
$693$	−7.58338	−0.288069
$694$	−7.34260	−0.278721
$695$	−27.0901	−1.02759
$696$	5.75879	0.218286
$697$	−42.4374	−1.60743
$698$	22.0790	0.835702
$699$	−31.5124	−1.19191
$700$	6.83767	0.258440
$701$	36.8468	1.39168	0.695842	−	0.718195i	$-0.255032\pi$
$701$	36.8468	1.39168	0.695842	+	0.718195i	$0.255032\pi$
$702$	0	0
$703$	59.0827	2.22835
$704$	4.99555	0.188277
$705$	−24.2729	−0.914170
$706$	−6.30884	−0.237436
$707$	−6.41713	−0.241341
$708$	0.686429	0.0257976
$709$	25.1620	0.944979	0.472489	−	0.881336i	$-0.343356\pi$
$709$	25.1620	0.944979	0.472489	+	0.881336i	$0.343356\pi$
$710$	14.4668	0.542931
$711$	−0.101235	−0.00379662
$712$	−6.16227	−0.230941
$713$	12.2618	0.459207
$714$	−8.60459	−0.322019
$715$	0	0
$716$	3.92185	0.146567
$717$	−11.1443	−0.416193
$718$	35.6786	1.33151
$719$	−3.37059	−0.125702	−0.0628509	−	0.998023i	$-0.520019\pi$
$719$	−3.37059	−0.125702	−0.0628509	+	0.998023i	$0.520019\pi$
$720$	5.22291	0.194646
$721$	5.68463	0.211707
$722$	5.95235	0.221524
$723$	2.22800	0.0828602
$724$	0.917236	0.0340888
$725$	32.3459	1.20130
$726$	16.9889	0.630519
$727$	27.0316	1.00255	0.501273	−	0.865289i	$-0.332865\pi$
$727$	27.0316	1.00255	0.501273	+	0.865289i	$0.332865\pi$
$728$	0	0
$729$	23.3377	0.864358
$730$	−26.2793	−0.972641
$731$	−55.8181	−2.06451
$732$	−11.6888	−0.432030
$733$	35.2989	1.30380	0.651898	−	0.758307i	$-0.273973\pi$
$733$	35.2989	1.30380	0.651898	+	0.758307i	$0.273973\pi$
$734$	5.35362	0.197606
$735$	−4.18845	−0.154493
$736$	5.66813	0.208930
$737$	−37.4714	−1.38028
$738$	−9.11419	−0.335498
$739$	−21.5248	−0.791801	−0.395901	−	0.918293i	$-0.629568\pi$
$739$	−21.5248	−0.791801	−0.395901	+	0.918293i	$0.629568\pi$
$740$	−40.6947	−1.49597
$741$	0	0
$742$	−3.79017	−0.139142
$743$	35.7822	1.31272	0.656362	−	0.754446i	$-0.272095\pi$
$743$	35.7822	1.31272	0.656362	+	0.754446i	$0.272095\pi$
$744$	2.63350	0.0965488
$745$	−74.1265	−2.71578
$746$	31.7744	1.16335
$747$	14.9755	0.547926
$748$	−35.3096	−1.29105
$749$	−19.2643	−0.703902
$750$	−7.69701	−0.281055
$751$	−47.4173	−1.73028	−0.865141	−	0.501529i	$-0.832771\pi$
$751$	−47.4173	−1.73028	−0.865141	+	0.501529i	$0.832771\pi$
$752$	5.79519	0.211329
$753$	13.0841	0.476812
$754$	0	0
$755$	−11.1033	−0.404090
$756$	−5.50008	−0.200036
$757$	21.0476	0.764989	0.382494	−	0.923958i	$-0.375065\pi$
$757$	21.0476	0.764989	0.382494	+	0.923958i	$0.375065\pi$
$758$	−15.5529	−0.564908
$759$	34.4702	1.25119
$760$	−17.1866	−0.623422
$761$	−51.4526	−1.86516	−0.932578	−	0.360969i	$-0.882446\pi$
$761$	−51.4526	−1.86516	−0.932578	+	0.360969i	$0.882446\pi$
$762$	6.54706	0.237175
$763$	−10.0195	−0.362729
$764$	7.86239	0.284451
$765$	−36.9167	−1.33473
$766$	−18.5061	−0.668653
$767$	0	0
$768$	1.21736	0.0439278
$769$	−46.3035	−1.66975	−0.834873	−	0.550443i	$-0.814459\pi$
$769$	−46.3035	−1.66975	−0.834873	+	0.550443i	$0.814459\pi$
$770$	−17.1877	−0.619400
$771$	−34.9136	−1.25738
$772$	12.1952	0.438915
$773$	−22.2161	−0.799057	−0.399529	−	0.916721i	$-0.630826\pi$
$773$	−22.2161	−0.799057	−0.399529	+	0.916721i	$0.630826\pi$
$774$	−11.9879	−0.430897
$775$	14.7918	0.531338
$776$	13.8364	0.496698
$777$	14.3987	0.516552
$778$	23.3415	0.836833
$779$	29.9912	1.07455
$780$	0	0
$781$	−21.0051	−0.751620
$782$	−40.0636	−1.43267
$783$	−26.0184	−0.929821
$784$	1.00000	0.0357143
$785$	4.46056	0.159204
$786$	19.5572	0.697582
$787$	24.3469	0.867875	0.433937	−	0.900943i	$-0.357124\pi$
$787$	24.3469	0.867875	0.433937	+	0.900943i	$0.357124\pi$
$788$	0.499938	0.0178096
$789$	14.0005	0.498431
$790$	−0.229449	−0.00816343
$791$	8.77533	0.312015
$792$	−7.58338	−0.269464
$793$	0	0
$794$	3.18220	0.112932
$795$	15.8750	0.563027
$796$	−18.7772	−0.665541
$797$	16.0966	0.570172	0.285086	−	0.958502i	$-0.407978\pi$
$797$	16.0966	0.570172	0.285086	+	0.958502i	$0.407978\pi$
$798$	6.08101	0.215266
$799$	−40.9617	−1.44912
$800$	6.83767	0.241748
$801$	9.35449	0.330525
$802$	−28.6480	−1.01160
$803$	38.1561	1.34650
$804$	−9.13140	−0.322040
$805$	−19.5017	−0.687345
$806$	0	0
$807$	−16.9536	−0.596794
$808$	−6.41713	−0.225754
$809$	−7.31827	−0.257297	−0.128648	−	0.991690i	$-0.541064\pi$
$809$	−7.31827	−0.257297	−0.128648	+	0.991690i	$0.541064\pi$
$810$	7.36807	0.258888
$811$	28.6079	1.00456	0.502279	−	0.864706i	$-0.332495\pi$
$811$	28.6079	1.00456	0.502279	+	0.864706i	$0.332495\pi$
$812$	4.73054	0.166010
$813$	9.18893	0.322270
$814$	59.0865	2.07098
$815$	53.3356	1.86826
$816$	−8.60459	−0.301221
$817$	39.4476	1.38010
$818$	2.03868	0.0712807
$819$	0	0
$820$	−20.6572	−0.721381
$821$	−4.79864	−0.167474	−0.0837368	−	0.996488i	$-0.526686\pi$
$821$	−4.79864	−0.167474	−0.0837368	+	0.996488i	$0.526686\pi$
$822$	−11.7656	−0.410374
$823$	2.13782	0.0745199	0.0372599	−	0.999306i	$-0.488137\pi$
$823$	2.13782	0.0745199	0.0372599	+	0.999306i	$0.488137\pi$
$824$	5.68463	0.198034
$825$	41.5826	1.44772
$826$	0.563865	0.0196194
$827$	−48.8135	−1.69741	−0.848706	−	0.528865i	$-0.822618\pi$
$827$	−48.8135	−1.69741	−0.848706	+	0.528865i	$0.822618\pi$
$828$	−8.60437	−0.299023
$829$	−44.2561	−1.53708	−0.768539	−	0.639803i	$-0.779016\pi$
$829$	−44.2561	−1.53708	−0.768539	+	0.639803i	$0.779016\pi$
$830$	33.9419	1.17814
$831$	−32.7398	−1.13573
$832$	0	0
$833$	−7.06822	−0.244899
$834$	9.58511	0.331905
$835$	−5.39560	−0.186722
$836$	24.9539	0.863050
$837$	−11.8982	−0.411263
$838$	−3.58445	−0.123823
$839$	24.8798	0.858947	0.429474	−	0.903079i	$-0.358699\pi$
$839$	24.8798	0.858947	0.429474	+	0.903079i	$0.358699\pi$
$840$	−4.18845	−0.144515
$841$	−6.62195	−0.228343
$842$	6.57788	0.226689
$843$	−18.3940	−0.633523
$844$	−5.77056	−0.198631
$845$	0	0
$846$	−8.79726	−0.302456
$847$	13.9555	0.479517
$848$	−3.79017	−0.130155
$849$	−28.0553	−0.962857
$850$	−48.3302	−1.65771
$851$	67.0416	2.29816
$852$	−5.11871	−0.175364
$853$	28.2180	0.966166	0.483083	−	0.875574i	$-0.339517\pi$
$853$	28.2180	0.966166	0.483083	+	0.875574i	$0.339517\pi$
$854$	−9.60172	−0.328564
$855$	26.0897	0.892248
$856$	−19.2643	−0.658440
$857$	−21.0991	−0.720733	−0.360366	−	0.932811i	$-0.617348\pi$
$857$	−21.0991	−0.720733	−0.360366	+	0.932811i	$0.617348\pi$
$858$	0	0
$859$	−6.80121	−0.232054	−0.116027	−	0.993246i	$-0.537016\pi$
$859$	−6.80121	−0.232054	−0.116027	+	0.993246i	$0.537016\pi$
$860$	−27.1705	−0.926507
$861$	7.30901	0.249090
$862$	−4.97080	−0.169306
$863$	23.1097	0.786665	0.393332	−	0.919396i	$-0.371322\pi$
$863$	23.1097	0.786665	0.393332	+	0.919396i	$0.371322\pi$
$864$	−5.50008	−0.187117
$865$	−32.2496	−1.09652
$866$	24.2077	0.822611
$867$	40.1239	1.36268
$868$	2.16328	0.0734266
$869$	0.333147	0.0113012
$870$	−19.8137	−0.671746
$871$	0	0
$872$	−10.0195	−0.339302
$873$	−21.0040	−0.710879
$874$	28.3136	0.957723
$875$	−6.32269	−0.213746
$876$	9.29824	0.314159
$877$	−27.3273	−0.922777	−0.461388	−	0.887198i	$-0.652649\pi$
$877$	−27.3273	−0.922777	−0.461388	+	0.887198i	$0.652649\pi$
$878$	−14.0986	−0.475806
$879$	−8.48253	−0.286109
$880$	−17.1877	−0.579396
$881$	−39.2815	−1.32343	−0.661714	−	0.749756i	$-0.730171\pi$
$881$	−39.2815	−1.32343	−0.661714	+	0.749756i	$0.730171\pi$
$882$	−1.51803	−0.0511146
$883$	52.7338	1.77463	0.887316	−	0.461161i	$-0.152567\pi$
$883$	52.7338	1.77463	0.887316	+	0.461161i	$0.152567\pi$
$884$	0	0
$885$	−2.36172	−0.0793884
$886$	−9.14216	−0.307137
$887$	41.3835	1.38952	0.694762	−	0.719240i	$-0.255510\pi$
$887$	41.3835	1.38952	0.694762	+	0.719240i	$0.255510\pi$
$888$	14.3987	0.483190
$889$	5.37806	0.180374
$890$	21.2019	0.710688
$891$	−10.6980	−0.358398
$892$	3.21980	0.107807
$893$	28.9483	0.968720
$894$	26.2277	0.877186
$895$	−13.4935	−0.451038
$896$	1.00000	0.0334077
$897$	0	0
$898$	8.29350	0.276758
$899$	10.2335	0.341307
$900$	−10.3798	−0.345992
$901$	26.7898	0.892497
$902$	29.9931	0.998662
$903$	9.61358	0.319920
$904$	8.77533	0.291863
$905$	−3.15584	−0.104904
$906$	3.92860	0.130519
$907$	39.8937	1.32465	0.662325	−	0.749217i	$-0.269570\pi$
$907$	39.8937	1.32465	0.662325	+	0.749217i	$0.269570\pi$
$908$	−20.6879	−0.686552
$909$	9.74138	0.323101
$910$	0	0
$911$	32.3922	1.07320	0.536601	−	0.843836i	$-0.319708\pi$
$911$	32.3922	1.07320	0.536601	+	0.843836i	$0.319708\pi$
$912$	6.08101	0.201362
$913$	−49.2818	−1.63099
$914$	5.72999	0.189531
$915$	40.2163	1.32951
$916$	−4.86398	−0.160710
$917$	16.0652	0.530520
$918$	38.8758	1.28309
$919$	17.1296	0.565053	0.282527	−	0.959259i	$-0.408827\pi$
$919$	17.1296	0.565053	0.282527	+	0.959259i	$0.408827\pi$
$920$	−19.5017	−0.642953
$921$	32.1729	1.06013
$922$	−17.7249	−0.583739
$923$	0	0
$924$	6.08140	0.200063
$925$	80.8748	2.65915
$926$	−15.0103	−0.493270
$927$	−8.62943	−0.283428
$928$	4.73054	0.155288
$929$	13.8576	0.454652	0.227326	−	0.973819i	$-0.427002\pi$
$929$	13.8576	0.454652	0.227326	+	0.973819i	$0.427002\pi$
$930$	−9.06080	−0.297115
$931$	4.99523	0.163712
$932$	−25.8858	−0.847917
$933$	−30.5678	−1.00075
$934$	−22.1003	−0.723145
$935$	121.486	3.97302
$936$	0	0
$937$	−16.2426	−0.530622	−0.265311	−	0.964163i	$-0.585475\pi$
$937$	−16.2426	−0.530622	−0.265311	+	0.964163i	$0.585475\pi$
$938$	−7.50096	−0.244915
$939$	−0.226206	−0.00738196
$940$	−19.9389	−0.650335
$941$	−14.0746	−0.458819	−0.229410	−	0.973330i	$-0.573680\pi$
$941$	−14.0746	−0.458819	−0.229410	+	0.973330i	$0.573680\pi$
$942$	−1.57825	−0.0514222
$943$	34.0313	1.10821
$944$	0.563865	0.0183522
$945$	18.9235	0.615582
$946$	39.4501	1.28263
$947$	3.18365	0.103455	0.0517274	−	0.998661i	$-0.483527\pi$
$947$	3.18365	0.103455	0.0517274	+	0.998661i	$0.483527\pi$
$948$	0.0811845	0.00263675
$949$	0	0
$950$	34.1558	1.10816
$951$	−31.6679	−1.02690
$952$	−7.06822	−0.229082
$953$	−26.5590	−0.860329	−0.430165	−	0.902750i	$-0.641544\pi$
$953$	−26.5590	−0.860329	−0.430165	+	0.902750i	$0.641544\pi$
$954$	5.75359	0.186279
$955$	−27.0513	−0.875359
$956$	−9.15450	−0.296078
$957$	28.7683	0.929948
$958$	−8.29945	−0.268143
$959$	−9.66486	−0.312095
$960$	−4.18845	−0.135182
$961$	−26.3202	−0.849039
$962$	0	0
$963$	29.2437	0.942365
$964$	1.83019	0.0589463
$965$	−41.9587	−1.35070
$966$	6.90017	0.222009
$967$	−24.5993	−0.791061	−0.395531	−	0.918453i	$-0.629439\pi$
$967$	−24.5993	−0.791061	−0.395531	+	0.918453i	$0.629439\pi$
$968$	13.9555	0.448547
$969$	−42.9819	−1.38078
$970$	−47.6054	−1.52852
$971$	15.0671	0.483525	0.241763	−	0.970335i	$-0.422274\pi$
$971$	15.0671	0.483525	0.241763	+	0.970335i	$0.422274\pi$
$972$	13.8932	0.445626
$973$	7.87367	0.252418
$974$	4.99054	0.159907
$975$	0	0
$976$	−9.60172	−0.307344
$977$	14.3564	0.459301	0.229651	−	0.973273i	$-0.426242\pi$
$977$	14.3564	0.459301	0.229651	+	0.973273i	$0.426242\pi$
$978$	−18.8714	−0.603440
$979$	−30.7839	−0.983859
$980$	−3.44059	−0.109906
$981$	15.2098	0.485612
$982$	2.67530	0.0853723
$983$	−31.2374	−0.996319	−0.498160	−	0.867085i	$-0.665991\pi$
$983$	−31.2374	−0.996319	−0.498160	+	0.867085i	$0.665991\pi$
$984$	7.30901	0.233003
$985$	−1.72008	−0.0548064
$986$	−33.4365	−1.06484
$987$	7.05486	0.224559
$988$	0	0
$989$	44.7615	1.42333
$990$	26.0913	0.829237
$991$	−22.7952	−0.724113	−0.362056	−	0.932156i	$-0.617925\pi$
$991$	−22.7952	−0.724113	−0.362056	+	0.932156i	$0.617925\pi$
$992$	2.16328	0.0686843
$993$	−12.4491	−0.395061
$994$	−4.20475	−0.133367
$995$	64.6047	2.04811
$996$	−12.0095	−0.380534
$997$	−2.30401	−0.0729688	−0.0364844	−	0.999334i	$-0.511616\pi$
$997$	−2.30401	−0.0729688	−0.0364844	+	0.999334i	$0.511616\pi$
$998$	10.6656	0.337612
$999$	−65.0539	−2.05822

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2366.2.a.bg.1.4	yes	6
13.5	odd	4		2366.2.d.q.337.4		12
13.8	odd	4		2366.2.d.q.337.10		12
13.12	even	2		2366.2.a.be.1.4	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2366.2.a.be.1.4	✓	6	13.12	even	2
2366.2.a.bg.1.4	yes	6	1.1	even	1	trivial
2366.2.d.q.337.4		12	13.5	odd	4
2366.2.d.q.337.10		12	13.8	odd	4

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:	\( N \)	\(=\)	\( 2366 = 2 \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2366.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.21736 q^{3} +1.00000 q^{4} -3.44059 q^{5} +1.21736 q^{6} +1.00000 q^{7} +1.00000 q^{8} -1.51803 q^{9} -3.44059 q^{10} +4.99555 q^{11} +1.21736 q^{12} +1.00000 q^{14} -4.18845 q^{15} +1.00000 q^{16} -7.06822 q^{17} -1.51803 q^{18} +4.99523 q^{19} -3.44059 q^{20} +1.21736 q^{21} +4.99555 q^{22} +5.66813 q^{23} +1.21736 q^{24} +6.83767 q^{25} -5.50008 q^{27} +1.00000 q^{28} +4.73054 q^{29} -4.18845 q^{30} +2.16328 q^{31} +1.00000 q^{32} +6.08140 q^{33} -7.06822 q^{34} -3.44059 q^{35} -1.51803 q^{36} +11.8278 q^{37} +4.99523 q^{38} -3.44059 q^{40} +6.00397 q^{41} +1.21736 q^{42} +7.89705 q^{43} +4.99555 q^{44} +5.22291 q^{45} +5.66813 q^{46} +5.79519 q^{47} +1.21736 q^{48} +1.00000 q^{49} +6.83767 q^{50} -8.60459 q^{51} -3.79017 q^{53} -5.50008 q^{54} -17.1877 q^{55} +1.00000 q^{56} +6.08101 q^{57} +4.73054 q^{58} +0.563865 q^{59} -4.18845 q^{60} -9.60172 q^{61} +2.16328 q^{62} -1.51803 q^{63} +1.00000 q^{64} +6.08140 q^{66} -7.50096 q^{67} -7.06822 q^{68} +6.90017 q^{69} -3.44059 q^{70} -4.20475 q^{71} -1.51803 q^{72} +7.63802 q^{73} +11.8278 q^{74} +8.32393 q^{75} +4.99523 q^{76} +4.99555 q^{77} +0.0666888 q^{79} -3.44059 q^{80} -2.14151 q^{81} +6.00397 q^{82} -9.86513 q^{83} +1.21736 q^{84} +24.3189 q^{85} +7.89705 q^{86} +5.75879 q^{87} +4.99555 q^{88} -6.16227 q^{89} +5.22291 q^{90} +5.66813 q^{92} +2.63350 q^{93} +5.79519 q^{94} -17.1866 q^{95} +1.21736 q^{96} +13.8364 q^{97} +1.00000 q^{98} -7.58338 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} + q^{3} + 6 q^{4} + 4 q^{5} + q^{6} + 6 q^{7} + 6 q^{8} + 5 q^{9} + 4 q^{10} + 6 q^{11} + q^{12} + 6 q^{14} - 5 q^{15} + 6 q^{16} - 9 q^{17} + 5 q^{18} + 10 q^{19} + 4 q^{20} + q^{21}+ \cdots - 4 q^{99}+O(q^{100}) \) 6 * q + 6 * q^2 + q^3 + 6 * q^4 + 4 * q^5 + q^6 + 6 * q^7 + 6 * q^8 + 5 * q^9 + 4 * q^10 + 6 * q^11 + q^12 + 6 * q^14 - 5 * q^15 + 6 * q^16 - 9 * q^17 + 5 * q^18 + 10 * q^19 + 4 * q^20 + q^21 + 6 * q^22 + 21 * q^23 + q^24 + 8 * q^25 + 7 * q^27 + 6 * q^28 - q^29 - 5 * q^30 + 20 * q^31 + 6 * q^32 + 9 * q^33 - 9 * q^34 + 4 * q^35 + 5 * q^36 + 16 * q^37 + 10 * q^38 + 4 * q^40 + 2 * q^41 + q^42 + 2 * q^43 + 6 * q^44 + 17 * q^45 + 21 * q^46 - 5 * q^47 + q^48 + 6 * q^49 + 8 * q^50 - 15 * q^51 + 28 * q^53 + 7 * q^54 - 29 * q^55 + 6 * q^56 + 22 * q^57 - q^58 - 12 * q^59 - 5 * q^60 - 27 * q^61 + 20 * q^62 + 5 * q^63 + 6 * q^64 + 9 * q^66 + 16 * q^67 - 9 * q^68 - 15 * q^69 + 4 * q^70 + 5 * q^72 + 38 * q^73 + 16 * q^74 - 7 * q^75 + 10 * q^76 + 6 * q^77 + 6 * q^79 + 4 * q^80 - 26 * q^81 + 2 * q^82 - 6 * q^83 + q^84 + 9 * q^85 + 2 * q^86 - 39 * q^87 + 6 * q^88 - q^89 + 17 * q^90 + 21 * q^92 - 7 * q^93 - 5 * q^94 + 3 * q^95 + q^96 + 16 * q^97 + 6 * q^98 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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